id	sid	tid	token	lemma	pos
ejpam-4582	1	1	european	european	PROPN
ejpam-4582	1	2	journal	journal	PROPN
ejpam-4582	1	3	of	of	ADP
ejpam-4582	1	4	pure	pure	ADJ
ejpam-4582	1	5	and	and	CCONJ
ejpam-4582	1	6	applied	apply	VERB
ejpam-4582	1	7	mathematics	mathematic	NOUN
ejpam-4582	1	8	vol	vol	NOUN
ejpam-4582	1	9	.	.	PUNCT
ejpam-4582	2	1	16	16	NUM
ejpam-4582	2	2	,	,	PUNCT
ejpam-4582	2	3	no	no	INTJ
ejpam-4582	2	4	.	.	NOUN
ejpam-4582	2	5	1	1	NUM
ejpam-4582	2	6	,	,	PUNCT
ejpam-4582	2	7	2023	2023	NUM
ejpam-4582	2	8	,	,	PUNCT
ejpam-4582	2	9	336	336	NUM
ejpam-4582	2	10	-	-	SYM
ejpam-4582	2	11	362	362	NUM
ejpam-4582	3	1	issn	issn	PROPN
ejpam-4582	3	2	1307	1307	NUM
ejpam-4582	3	3	-	-	SYM
ejpam-4582	3	4	5543	5543	NUM
ejpam-4582	3	5	–	–	PUNCT
ejpam-4582	3	6	ejpam.com	ejpam.com	X
ejpam-4582	3	7	published	publish	VERB
ejpam-4582	3	8	by	by	ADP
ejpam-4582	3	9	new	new	PROPN
ejpam-4582	3	10	york	york	PROPN
ejpam-4582	3	11	business	business	PROPN
ejpam-4582	3	12	global	global	PROPN
ejpam-4582	3	13	on	on	ADP
ejpam-4582	3	14	some	some	DET
ejpam-4582	3	15	forms	form	NOUN
ejpam-4582	3	16	of	of	ADP
ejpam-4582	3	17	closed	closed	ADJ
ejpam-4582	3	18	sets	set	NOUN
ejpam-4582	3	19	and	and	CCONJ
ejpam-4582	3	20	related	related	ADJ
ejpam-4582	3	21	topics	topic	NOUN
ejpam-4582	3	22	chawalit	chawalit	VERB
ejpam-4582	3	23	boonpok1	boonpok1	PROPN
ejpam-4582	3	24	,	,	PUNCT
ejpam-4582	3	25	chokchai	chokchai	ADJ
ejpam-4582	3	26	viriyapong1,∗	viriyapong1,∗	NOUN
ejpam-4582	3	27	1	1	NUM
ejpam-4582	3	28	mathematics	mathematic	NOUN
ejpam-4582	3	29	and	and	CCONJ
ejpam-4582	3	30	applied	apply	VERB
ejpam-4582	3	31	mathematics	mathematics	PROPN
ejpam-4582	3	32	research	research	NOUN
ejpam-4582	3	33	unit	unit	NOUN
ejpam-4582	3	34	,	,	PUNCT
ejpam-4582	3	35	department	department	NOUN
ejpam-4582	3	36	of	of	ADP
ejpam-4582	3	37	mathematics	mathematic	NOUN
ejpam-4582	3	38	,	,	PUNCT
ejpam-4582	3	39	faculty	faculty	NOUN
ejpam-4582	3	40	of	of	ADP
ejpam-4582	3	41	science	science	NOUN
ejpam-4582	3	42	,	,	PUNCT
ejpam-4582	3	43	mahasarakham	mahasarakham	PROPN
ejpam-4582	3	44	university	university	PROPN
ejpam-4582	3	45	,	,	PUNCT
ejpam-4582	3	46	maha	maha	PROPN
ejpam-4582	3	47	sarakham	sarakham	PROPN
ejpam-4582	3	48	,	,	PUNCT
ejpam-4582	3	49	44150	44150	NUM
ejpam-4582	3	50	,	,	PUNCT
ejpam-4582	3	51	thailand	thailand	PROPN
ejpam-4582	3	52	abstract	abstract	PROPN
ejpam-4582	3	53	.	.	PUNCT
ejpam-4582	4	1	the	the	DET
ejpam-4582	4	2	purpose	purpose	NOUN
ejpam-4582	4	3	of	of	ADP
ejpam-4582	4	4	the	the	DET
ejpam-4582	4	5	present	present	ADJ
ejpam-4582	4	6	article	article	NOUN
ejpam-4582	4	7	is	be	AUX
ejpam-4582	4	8	to	to	PART
ejpam-4582	4	9	introduce	introduce	VERB
ejpam-4582	4	10	the	the	DET
ejpam-4582	4	11	notion	notion	NOUN
ejpam-4582	4	12	of	of	ADP
ejpam-4582	4	13	(	(	PUNCT
ejpam-4582	4	14	λ	λ	PROPN
ejpam-4582	4	15	,	,	PUNCT
ejpam-4582	4	16	s)-closed	s)-close	VERB
ejpam-4582	4	17	sets	set	NOUN
ejpam-4582	4	18	.	.	PUNCT
ejpam-4582	5	1	especially	especially	ADV
ejpam-4582	5	2	,	,	PUNCT
ejpam-4582	5	3	some	some	DET
ejpam-4582	5	4	properties	property	NOUN
ejpam-4582	5	5	of	of	ADP
ejpam-4582	5	6	generalized	generalized	ADJ
ejpam-4582	5	7	(	(	PUNCT
ejpam-4582	5	8	λ	λ	X
ejpam-4582	5	9	,	,	PUNCT
ejpam-4582	5	10	s)-closed	s)-close	VERB
ejpam-4582	5	11	sets	set	NOUN
ejpam-4582	5	12	are	be	AUX
ejpam-4582	5	13	obtained	obtain	VERB
ejpam-4582	5	14	.	.	PUNCT
ejpam-4582	6	1	several	several	ADJ
ejpam-4582	6	2	characterizations	characterization	NOUN
ejpam-4582	6	3	of	of	ADP
ejpam-4582	6	4	some	some	DET
ejpam-4582	6	5	low	low	ADJ
ejpam-4582	6	6	separation	separation	NOUN
ejpam-4582	6	7	axioms	axiom	NOUN
ejpam-4582	6	8	are	be	AUX
ejpam-4582	6	9	given	give	VERB
ejpam-4582	6	10	.	.	PUNCT
ejpam-4582	7	1	characterizations	characterization	NOUN
ejpam-4582	7	2	of	of	ADP
ejpam-4582	7	3	(	(	PUNCT
ejpam-4582	7	4	λ	λ	PROPN
ejpam-4582	7	5	,	,	PUNCT
ejpam-4582	7	6	s)-extremally	s)-extremally	ADV
ejpam-4582	7	7	disconnected	disconnected	ADJ
ejpam-4582	7	8	spaces	space	NOUN
ejpam-4582	7	9	are	be	AUX
ejpam-4582	7	10	investigated	investigate	VERB
ejpam-4582	7	11	.	.	PUNCT
ejpam-4582	8	1	furthermore	furthermore	ADV
ejpam-4582	8	2	,	,	PUNCT
ejpam-4582	8	3	some	some	DET
ejpam-4582	8	4	characterizations	characterization	NOUN
ejpam-4582	8	5	of	of	ADP
ejpam-4582	8	6	almost	almost	ADV
ejpam-4582	8	7	(	(	PUNCT
ejpam-4582	8	8	λ	λ	PROPN
ejpam-4582	8	9	,	,	PUNCT
ejpam-4582	8	10	s)-continuous	s)-continuous	ADJ
ejpam-4582	8	11	functions	function	NOUN
ejpam-4582	8	12	are	be	AUX
ejpam-4582	8	13	discussed	discuss	VERB
ejpam-4582	8	14	.	.	PUNCT
ejpam-4582	9	1	2020	2020	NUM
ejpam-4582	9	2	mathematics	mathematic	NOUN
ejpam-4582	9	3	subject	subject	NOUN
ejpam-4582	9	4	classifications	classification	NOUN
ejpam-4582	9	5	:	:	PUNCT
ejpam-4582	9	6	54a05	54a05	NUM
ejpam-4582	9	7	,	,	PUNCT
ejpam-4582	9	8	54d10	54d10	NUM
ejpam-4582	9	9	,	,	PUNCT
ejpam-4582	9	10	54g05	54g05	NUM
ejpam-4582	9	11	key	key	ADJ
ejpam-4582	9	12	words	word	NOUN
ejpam-4582	9	13	and	and	CCONJ
ejpam-4582	9	14	phrases	phrase	NOUN
ejpam-4582	9	15	:	:	PUNCT
ejpam-4582	9	16	(	(	PUNCT
ejpam-4582	9	17	λ	λ	X
ejpam-4582	9	18	,	,	PUNCT
ejpam-4582	9	19	s)-closed	s)-close	VERB
ejpam-4582	9	20	set	set	NOUN
ejpam-4582	9	21	,	,	PUNCT
ejpam-4582	9	22	generalized	generalize	VERB
ejpam-4582	9	23	(	(	PUNCT
ejpam-4582	9	24	λ	λ	X
ejpam-4582	9	25	,	,	PUNCT
ejpam-4582	9	26	s)-closed	s)-close	VERB
ejpam-4582	9	27	set	set	NOUN
ejpam-4582	9	28	,	,	PUNCT
ejpam-4582	9	29	(	(	PUNCT
ejpam-4582	9	30	λ	λ	INTJ
ejpam-4582	9	31	,	,	PUNCT
ejpam-4582	9	32	s)-r0	s)-r0	NOUN
ejpam-4582	9	33	space	space	NOUN
ejpam-4582	9	34	,	,	PUNCT
ejpam-4582	9	35	(	(	PUNCT
ejpam-4582	9	36	λ	λ	NOUN
ejpam-4582	9	37	,	,	PUNCT
ejpam-4582	9	38	s)extremally	s)extremally	ADV
ejpam-4582	9	39	disconnected	disconnected	ADJ
ejpam-4582	9	40	space	space	NOUN
ejpam-4582	9	41	,	,	PUNCT
ejpam-4582	9	42	almost	almost	ADV
ejpam-4582	9	43	(	(	PUNCT
ejpam-4582	9	44	λ	λ	NOUN
ejpam-4582	9	45	,	,	PUNCT
ejpam-4582	9	46	s)-continuous	s)-continuous	ADJ
ejpam-4582	9	47	function	function	NOUN
ejpam-4582	9	48	1	1	NUM
ejpam-4582	9	49	.	.	PUNCT
ejpam-4582	10	1	introduction	introduction	NOUN
ejpam-4582	10	2	general	general	ADJ
ejpam-4582	10	3	topology	topology	NOUN
ejpam-4582	10	4	plays	play	VERB
ejpam-4582	10	5	an	an	DET
ejpam-4582	10	6	important	important	ADJ
ejpam-4582	10	7	role	role	NOUN
ejpam-4582	10	8	in	in	ADP
ejpam-4582	10	9	many	many	ADJ
ejpam-4582	10	10	fields	field	NOUN
ejpam-4582	10	11	of	of	ADP
ejpam-4582	10	12	applied	apply	VERB
ejpam-4582	10	13	sciences	science	NOUN
ejpam-4582	10	14	as	as	ADV
ejpam-4582	10	15	well	well	ADV
ejpam-4582	10	16	as	as	ADP
ejpam-4582	10	17	branches	branch	NOUN
ejpam-4582	10	18	of	of	ADP
ejpam-4582	10	19	mathematics	mathematic	NOUN
ejpam-4582	10	20	.	.	PUNCT
ejpam-4582	11	1	the	the	DET
ejpam-4582	11	2	concepts	concept	NOUN
ejpam-4582	11	3	of	of	ADP
ejpam-4582	11	4	maximality	maximality	NOUN
ejpam-4582	11	5	and	and	CCONJ
ejpam-4582	11	6	submaximality	submaximality	NOUN
ejpam-4582	11	7	of	of	ADP
ejpam-4582	11	8	general	general	ADJ
ejpam-4582	11	9	topological	topological	ADJ
ejpam-4582	11	10	spaces	space	NOUN
ejpam-4582	11	11	were	be	AUX
ejpam-4582	11	12	introduced	introduce	VERB
ejpam-4582	11	13	by	by	ADP
ejpam-4582	11	14	hewitt	hewitt	PROPN
ejpam-4582	12	1	[	[	X
ejpam-4582	12	2	16	16	NUM
ejpam-4582	12	3	]	]	PUNCT
ejpam-4582	12	4	.	.	PUNCT
ejpam-4582	13	1	he	he	PRON
ejpam-4582	13	2	discovered	discover	VERB
ejpam-4582	13	3	a	a	DET
ejpam-4582	13	4	general	general	ADJ
ejpam-4582	13	5	way	way	NOUN
ejpam-4582	13	6	of	of	ADP
ejpam-4582	13	7	constructing	construct	VERB
ejpam-4582	13	8	maximal	maximal	ADJ
ejpam-4582	13	9	topologies	topology	NOUN
ejpam-4582	13	10	.	.	PUNCT
ejpam-4582	14	1	the	the	DET
ejpam-4582	14	2	existence	existence	NOUN
ejpam-4582	14	3	of	of	ADP
ejpam-4582	14	4	a	a	DET
ejpam-4582	14	5	maximal	maximal	ADJ
ejpam-4582	14	6	space	space	NOUN
ejpam-4582	14	7	that	that	PRON
ejpam-4582	14	8	is	be	AUX
ejpam-4582	14	9	tychonoff	tychonoff	NOUN
ejpam-4582	14	10	is	be	AUX
ejpam-4582	14	11	nontrivial	nontrivial	ADJ
ejpam-4582	14	12	and	and	CCONJ
ejpam-4582	14	13	due	due	ADP
ejpam-4582	14	14	to	to	ADP
ejpam-4582	14	15	van	van	PROPN
ejpam-4582	14	16	douwen	douwen	NOUN
ejpam-4582	14	17	[	[	X
ejpam-4582	14	18	28	28	NUM
ejpam-4582	14	19	]	]	PUNCT
ejpam-4582	14	20	.	.	PUNCT
ejpam-4582	15	1	the	the	DET
ejpam-4582	15	2	first	first	ADJ
ejpam-4582	15	3	systematic	systematic	ADJ
ejpam-4582	15	4	study	study	NOUN
ejpam-4582	15	5	of	of	ADP
ejpam-4582	15	6	submaximal	submaximal	ADJ
ejpam-4582	15	7	spaces	space	NOUN
ejpam-4582	15	8	was	be	AUX
ejpam-4582	15	9	undertaken	undertake	VERB
ejpam-4582	15	10	in	in	ADP
ejpam-4582	15	11	the	the	DET
ejpam-4582	15	12	paper	paper	NOUN
ejpam-4582	15	13	of	of	ADP
ejpam-4582	15	14	arhangel’skĭi	arhangel’skĭi	PROPN
ejpam-4582	15	15	and	and	CCONJ
ejpam-4582	15	16	collins	collin	VERB
ejpam-4582	15	17	[	[	X
ejpam-4582	15	18	2	2	NUM
ejpam-4582	15	19	]	]	PUNCT
ejpam-4582	15	20	.	.	PUNCT
ejpam-4582	16	1	they	they	PRON
ejpam-4582	16	2	gave	give	VERB
ejpam-4582	16	3	various	various	ADJ
ejpam-4582	16	4	necessary	necessary	ADJ
ejpam-4582	16	5	and	and	CCONJ
ejpam-4582	16	6	sufficient	sufficient	ADJ
ejpam-4582	16	7	conditions	condition	NOUN
ejpam-4582	16	8	for	for	ADP
ejpam-4582	16	9	a	a	DET
ejpam-4582	16	10	space	space	NOUN
ejpam-4582	16	11	to	to	PART
ejpam-4582	16	12	be	be	AUX
ejpam-4582	16	13	submaximal	submaximal	ADJ
ejpam-4582	16	14	and	and	CCONJ
ejpam-4582	16	15	showed	show	VERB
ejpam-4582	16	16	that	that	SCONJ
ejpam-4582	16	17	every	every	DET
ejpam-4582	16	18	submaximal	submaximal	ADJ
ejpam-4582	16	19	space	space	NOUN
ejpam-4582	16	20	is	be	AUX
ejpam-4582	16	21	left	leave	VERB
ejpam-4582	16	22	-	-	PUNCT
ejpam-4582	16	23	separated	separate	VERB
ejpam-4582	16	24	.	.	PUNCT
ejpam-4582	17	1	this	this	PRON
ejpam-4582	17	2	led	lead	VERB
ejpam-4582	17	3	to	to	ADP
ejpam-4582	17	4	the	the	DET
ejpam-4582	17	5	question	question	NOUN
ejpam-4582	17	6	whether	whether	SCONJ
ejpam-4582	17	7	every	every	DET
ejpam-4582	17	8	submaximal	submaximal	ADJ
ejpam-4582	17	9	space	space	NOUN
ejpam-4582	17	10	is	be	AUX
ejpam-4582	17	11	σ	σ	NOUN
ejpam-4582	17	12	-	-	NOUN
ejpam-4582	17	13	discrete	discrete	NOUN
ejpam-4582	17	14	[	[	X
ejpam-4582	17	15	2	2	NUM
ejpam-4582	17	16	]	]	PUNCT
ejpam-4582	17	17	.	.	PUNCT
ejpam-4582	18	1	gillman	gillman	PROPN
ejpam-4582	18	2	and	and	CCONJ
ejpam-4582	18	3	jerison	jerison	PROPN
ejpam-4582	19	1	[	[	X
ejpam-4582	19	2	14	14	NUM
ejpam-4582	19	3	]	]	PUNCT
ejpam-4582	19	4	introduced	introduce	VERB
ejpam-4582	19	5	the	the	DET
ejpam-4582	19	6	notion	notion	NOUN
ejpam-4582	19	7	of	of	ADP
ejpam-4582	19	8	extremally	extremally	ADV
ejpam-4582	19	9	disconnected	disconnect	VERB
ejpam-4582	19	10	topological	topological	ADJ
ejpam-4582	19	11	spaces	space	NOUN
ejpam-4582	19	12	.	.	PUNCT
ejpam-4582	20	1	thompson	thompson	NOUN
ejpam-4582	21	1	[	[	X
ejpam-4582	21	2	26	26	NUM
ejpam-4582	21	3	]	]	PUNCT
ejpam-4582	21	4	introduced	introduce	VERB
ejpam-4582	21	5	the	the	DET
ejpam-4582	21	6	notion	notion	NOUN
ejpam-4582	21	7	of	of	ADP
ejpam-4582	21	8	s	s	NOUN
ejpam-4582	21	9	-	-	PUNCT
ejpam-4582	21	10	closed	closed	ADJ
ejpam-4582	21	11	spaces	space	NOUN
ejpam-4582	21	12	.	.	PUNCT
ejpam-4582	22	1	herrman	herrman	PROPN
ejpam-4582	23	1	[	[	X
ejpam-4582	23	2	15	15	NUM
ejpam-4582	23	3	,	,	PUNCT
ejpam-4582	23	4	16	16	NUM
ejpam-4582	23	5	]	]	PUNCT
ejpam-4582	23	6	showed	show	VERB
ejpam-4582	23	7	that	that	SCONJ
ejpam-4582	23	8	every	every	DET
ejpam-4582	23	9	s	s	NOUN
ejpam-4582	23	10	-	-	PUNCT
ejpam-4582	23	11	closed	closed	ADJ
ejpam-4582	23	12	weakly	weakly	ADJ
ejpam-4582	23	13	hausdorff	hausdorff	NOUN
ejpam-4582	23	14	(	(	PUNCT
ejpam-4582	23	15	or	or	CCONJ
ejpam-4582	23	16	almost	almost	ADV
ejpam-4582	23	17	regular	regular	ADJ
ejpam-4582	23	18	)	)	PUNCT
ejpam-4582	23	19	space	space	NOUN
ejpam-4582	23	20	is	be	AUX
ejpam-4582	23	21	extremally	extremally	ADV
ejpam-4582	23	22	disconnected	disconnect	VERB
ejpam-4582	23	23	.	.	PUNCT
ejpam-4582	24	1	cameron	cameron	PROPN
ejpam-4582	25	1	[	[	X
ejpam-4582	25	2	6	6	NUM
ejpam-4582	25	3	]	]	PUNCT
ejpam-4582	25	4	proved	prove	VERB
ejpam-4582	25	5	that	that	SCONJ
ejpam-4582	25	6	every	every	DET
ejpam-4582	25	7	maximally	maximally	ADV
ejpam-4582	25	8	s	s	ADJ
ejpam-4582	25	9	-	-	PUNCT
ejpam-4582	25	10	closed	closed	ADJ
ejpam-4582	25	11	space	space	NOUN
ejpam-4582	25	12	is	be	AUX
ejpam-4582	25	13	extremally	extremally	ADV
ejpam-4582	25	14	disconnected	disconnect	VERB
ejpam-4582	25	15	.	.	PUNCT
ejpam-4582	26	1	in	in	ADP
ejpam-4582	26	2	[	[	X
ejpam-4582	26	3	21	21	NUM
ejpam-4582	26	4	]	]	PUNCT
ejpam-4582	26	5	,	,	PUNCT
ejpam-4582	26	6	the	the	DET
ejpam-4582	26	7	present	present	ADJ
ejpam-4582	26	8	author	author	NOUN
ejpam-4582	26	9	introduced	introduce	VERB
ejpam-4582	26	10	the	the	DET
ejpam-4582	26	11	concept	concept	NOUN
ejpam-4582	26	12	of	of	ADP
ejpam-4582	26	13	locally	locally	ADV
ejpam-4582	26	14	s	s	NOUN
ejpam-4582	26	15	-	-	PUNCT
ejpam-4582	26	16	closed	closed	ADJ
ejpam-4582	26	17	spaces	space	NOUN
ejpam-4582	26	18	which	which	PRON
ejpam-4582	26	19	is	be	AUX
ejpam-4582	26	20	strictly	strictly	ADV
ejpam-4582	26	21	weaker	weak	ADJ
ejpam-4582	26	22	than	than	ADP
ejpam-4582	26	23	that	that	PRON
ejpam-4582	26	24	of	of	ADP
ejpam-4582	26	25	s	s	NOUN
ejpam-4582	26	26	-	-	PUNCT
ejpam-4582	26	27	closed	closed	ADJ
ejpam-4582	26	28	spaces	space	NOUN
ejpam-4582	26	29	.	.	PUNCT
ejpam-4582	27	1	noiri	noiri	PROPN
ejpam-4582	28	1	[	[	X
ejpam-4582	28	2	22	22	NUM
ejpam-4582	28	3	]	]	PUNCT
ejpam-4582	28	4	showed	show	VERB
ejpam-4582	28	5	that	that	SCONJ
ejpam-4582	28	6	every	every	DET
ejpam-4582	28	7	locally	locally	ADV
ejpam-4582	28	8	s	s	NOUN
ejpam-4582	28	9	-	-	PUNCT
ejpam-4582	28	10	closed	closed	ADJ
ejpam-4582	28	11	weakly	weakly	ADJ
ejpam-4582	28	12	hausdorff	hausdorff	NOUN
ejpam-4582	28	13	(	(	PUNCT
ejpam-4582	28	14	or	or	CCONJ
ejpam-4582	28	15	almost	almost	ADV
ejpam-4582	28	16	regular	regular	ADJ
ejpam-4582	28	17	)	)	PUNCT
ejpam-4582	28	18	space	space	NOUN
ejpam-4582	28	19	is	be	AUX
ejpam-4582	28	20	extremally	extremally	ADV
ejpam-4582	28	21	disconnected	disconnect	VERB
ejpam-4582	28	22	.	.	PUNCT
ejpam-4582	29	1	sivaraj	sivaraj	PROPN
ejpam-4582	30	1	[	[	X
ejpam-4582	30	2	25	25	NUM
ejpam-4582	30	3	]	]	PUNCT
ejpam-4582	30	4	has	have	AUX
ejpam-4582	30	5	obtained	obtain	VERB
ejpam-4582	30	6	some	some	DET
ejpam-4582	30	7	characterizations	characterization	NOUN
ejpam-4582	30	8	of	of	ADP
ejpam-4582	30	9	extremally	extremally	ADV
ejpam-4582	30	10	disconnected	disconnect	VERB
ejpam-4582	30	11	spaces	space	NOUN
ejpam-4582	30	12	by	by	ADP
ejpam-4582	30	13	utilizing	utilize	VERB
ejpam-4582	30	14	semi	semi	ADJ
ejpam-4582	30	15	-	-	ADJ
ejpam-4582	30	16	open	open	ADJ
ejpam-4582	30	17	sets	set	NOUN
ejpam-4582	30	18	due	due	ADJ
ejpam-4582	30	19	∗corresponding	∗corresponde	VERB
ejpam-4582	30	20	author	author	NOUN
ejpam-4582	30	21	.	.	PUNCT
ejpam-4582	31	1	doi	doi	NOUN
ejpam-4582	31	2	:	:	PUNCT
ejpam-4582	31	3	https://doi.org/10.29020/nybg.ejpam.v16i1.4582	https://doi.org/10.29020/nybg.ejpam.v16i1.4582	ADP
ejpam-4582	31	4	email	email	NOUN
ejpam-4582	31	5	addresses	address	NOUN
ejpam-4582	31	6	:	:	PUNCT
ejpam-4582	31	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4582	31	8	(	(	PUNCT
ejpam-4582	31	9	c.	c.	PROPN
ejpam-4582	31	10	boonpok	boonpok	PROPN
ejpam-4582	31	11	)	)	PUNCT
ejpam-4582	31	12	,	,	PUNCT
ejpam-4582	31	13	chokchai.v@msu.ac.th	chokchai.v@msu.ac.th	INTJ
ejpam-4582	31	14	(	(	PUNCT
ejpam-4582	31	15	c.	c.	PROPN
ejpam-4582	31	16	viriyapong	viriyapong	PROPN
ejpam-4582	31	17	)	)	PUNCT
ejpam-4582	31	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4582	32	1	336	336	NUM
ejpam-4582	33	1	©	©	ADP
ejpam-4582	33	2	2023	2023	NUM
ejpam-4582	33	3	ejpam	ejpam	NOUN
ejpam-4582	33	4	all	all	DET
ejpam-4582	33	5	rights	right	NOUN
ejpam-4582	33	6	reserved	reserve	VERB
ejpam-4582	33	7	.	.	PUNCT
ejpam-4582	34	1	c.	c.	PROPN
ejpam-4582	34	2	boonpok	boonpok	PROPN
ejpam-4582	34	3	,	,	PUNCT
ejpam-4582	34	4	c.	c.	PROPN
ejpam-4582	34	5	viriyapong	viriyapong	PROPN
ejpam-4582	34	6	/	/	SYM
ejpam-4582	34	7	eur	eur	PROPN
ejpam-4582	34	8	.	.	PUNCT
ejpam-4582	35	1	j.	j.	PROPN
ejpam-4582	35	2	pure	pure	PROPN
ejpam-4582	35	3	appl	appl	PROPN
ejpam-4582	35	4	.	.	PROPN
ejpam-4582	35	5	math	math	PROPN
ejpam-4582	35	6	,	,	PUNCT
ejpam-4582	35	7	16	16	NUM
ejpam-4582	35	8	(	(	PUNCT
ejpam-4582	35	9	1	1	NUM
ejpam-4582	35	10	)	)	PUNCT
ejpam-4582	35	11	(	(	PUNCT
ejpam-4582	35	12	2023	2023	NUM
ejpam-4582	35	13	)	)	PUNCT
ejpam-4582	35	14	,	,	PUNCT
ejpam-4582	35	15	336	336	NUM
ejpam-4582	35	16	-	-	SYM
ejpam-4582	35	17	362	362	NUM
ejpam-4582	35	18	337	337	NUM
ejpam-4582	35	19	to	to	ADP
ejpam-4582	35	20	levine	levine	PROPN
ejpam-4582	35	21	[	[	X
ejpam-4582	35	22	17	17	NUM
ejpam-4582	35	23	]	]	PUNCT
ejpam-4582	35	24	.	.	PUNCT
ejpam-4582	36	1	in	in	ADP
ejpam-4582	36	2	[	[	X
ejpam-4582	36	3	23	23	NUM
ejpam-4582	36	4	]	]	PUNCT
ejpam-4582	36	5	,	,	PUNCT
ejpam-4582	36	6	the	the	DET
ejpam-4582	36	7	present	present	ADJ
ejpam-4582	36	8	author	author	NOUN
ejpam-4582	36	9	obtained	obtain	VERB
ejpam-4582	36	10	several	several	ADJ
ejpam-4582	36	11	characterizations	characterization	NOUN
ejpam-4582	36	12	of	of	ADP
ejpam-4582	36	13	extremally	extremally	ADV
ejpam-4582	36	14	disconnected	disconnect	VERB
ejpam-4582	36	15	spaces	space	NOUN
ejpam-4582	36	16	by	by	ADP
ejpam-4582	36	17	utilizing	utilize	VERB
ejpam-4582	36	18	preopen	preopen	ADJ
ejpam-4582	36	19	sets	set	NOUN
ejpam-4582	36	20	and	and	CCONJ
ejpam-4582	36	21	semi	semi	ADJ
ejpam-4582	36	22	-	-	ADJ
ejpam-4582	36	23	preopen	preopen	ADJ
ejpam-4582	36	24	sets	set	NOUN
ejpam-4582	36	25	.	.	PUNCT
ejpam-4582	37	1	the	the	DET
ejpam-4582	37	2	notion	notion	NOUN
ejpam-4582	37	3	of	of	ADP
ejpam-4582	37	4	r0	r0	NOUN
ejpam-4582	37	5	topological	topological	ADJ
ejpam-4582	37	6	spaces	space	NOUN
ejpam-4582	37	7	was	be	AUX
ejpam-4582	37	8	first	first	ADV
ejpam-4582	37	9	introduced	introduce	VERB
ejpam-4582	37	10	by	by	ADP
ejpam-4582	37	11	shanin	shanin	PROPN
ejpam-4582	37	12	[	[	X
ejpam-4582	37	13	24	24	NUM
ejpam-4582	37	14	]	]	PUNCT
ejpam-4582	37	15	.	.	PUNCT
ejpam-4582	38	1	davis	davis	PROPN
ejpam-4582	39	1	[	[	X
ejpam-4582	39	2	8	8	NUM
ejpam-4582	39	3	]	]	PUNCT
ejpam-4582	39	4	introduced	introduce	VERB
ejpam-4582	39	5	the	the	DET
ejpam-4582	39	6	notion	notion	NOUN
ejpam-4582	39	7	of	of	ADP
ejpam-4582	39	8	a	a	DET
ejpam-4582	39	9	separation	separation	NOUN
ejpam-4582	39	10	axiom	axiom	NOUN
ejpam-4582	39	11	called	call	VERB
ejpam-4582	39	12	r1	r1	PROPN
ejpam-4582	39	13	.	.	PUNCT
ejpam-4582	40	1	this	this	DET
ejpam-4582	40	2	notions	notion	NOUN
ejpam-4582	40	3	were	be	AUX
ejpam-4582	40	4	further	far	ADV
ejpam-4582	40	5	investigated	investigate	VERB
ejpam-4582	40	6	by	by	ADP
ejpam-4582	40	7	naimpally	naimpally	ADV
ejpam-4582	40	8	[	[	X
ejpam-4582	40	9	20	20	NUM
ejpam-4582	40	10	]	]	PUNCT
ejpam-4582	40	11	,	,	PUNCT
ejpam-4582	40	12	dube	dube	PROPN
ejpam-4582	41	1	[	[	X
ejpam-4582	41	2	10	10	NUM
ejpam-4582	41	3	]	]	PUNCT
ejpam-4582	41	4	and	and	CCONJ
ejpam-4582	41	5	dorsett	dorsett	PROPN
ejpam-4582	42	1	[	[	X
ejpam-4582	42	2	9	9	NUM
ejpam-4582	42	3	]	]	PUNCT
ejpam-4582	42	4	.	.	PUNCT
ejpam-4582	43	1	cammaroto	cammaroto	NOUN
ejpam-4582	43	2	and	and	CCONJ
ejpam-4582	43	3	noiri	noiri	ADV
ejpam-4582	44	1	[	[	X
ejpam-4582	44	2	7	7	X
ejpam-4582	44	3	]	]	PUNCT
ejpam-4582	44	4	have	have	AUX
ejpam-4582	44	5	defined	define	VERB
ejpam-4582	44	6	a	a	DET
ejpam-4582	44	7	weak	weak	ADJ
ejpam-4582	44	8	separation	separation	NOUN
ejpam-4582	44	9	axioms	axiom	VERB
ejpam-4582	44	10	m	m	NOUN
ejpam-4582	44	11	-	-	PUNCT
ejpam-4582	44	12	r0	r0	NOUN
ejpam-4582	44	13	in	in	ADP
ejpam-4582	44	14	m	m	NOUN
ejpam-4582	44	15	-	-	NOUN
ejpam-4582	44	16	spaces	space	NOUN
ejpam-4582	44	17	which	which	PRON
ejpam-4582	44	18	are	be	AUX
ejpam-4582	44	19	equivalent	equivalent	ADJ
ejpam-4582	44	20	to	to	ADP
ejpam-4582	44	21	generalized	generalize	VERB
ejpam-4582	44	22	topological	topological	ADJ
ejpam-4582	44	23	spaces	space	NOUN
ejpam-4582	44	24	due	due	ADP
ejpam-4582	44	25	to	to	ADP
ejpam-4582	44	26	lugojan	lugojan	NOUN
ejpam-4582	44	27	[	[	X
ejpam-4582	44	28	19	19	NUM
ejpam-4582	44	29	]	]	PUNCT
ejpam-4582	44	30	.	.	PUNCT
ejpam-4582	45	1	levine	levine	PROPN
ejpam-4582	46	1	[	[	X
ejpam-4582	46	2	18	18	NUM
ejpam-4582	46	3	]	]	PUNCT
ejpam-4582	46	4	introduced	introduce	VERB
ejpam-4582	46	5	the	the	DET
ejpam-4582	46	6	concept	concept	NOUN
ejpam-4582	46	7	of	of	ADP
ejpam-4582	46	8	generalized	generalized	ADJ
ejpam-4582	46	9	closed	closed	ADJ
ejpam-4582	46	10	sets	set	NOUN
ejpam-4582	46	11	of	of	ADP
ejpam-4582	46	12	a	a	DET
ejpam-4582	46	13	topological	topological	ADJ
ejpam-4582	46	14	space	space	NOUN
ejpam-4582	46	15	and	and	CCONJ
ejpam-4582	46	16	a	a	DET
ejpam-4582	46	17	class	class	NOUN
ejpam-4582	46	18	of	of	ADP
ejpam-4582	46	19	topological	topological	ADJ
ejpam-4582	46	20	spaces	space	NOUN
ejpam-4582	46	21	called	call	VERB
ejpam-4582	46	22	t	t	PROPN
ejpam-4582	46	23	1	1	NUM
ejpam-4582	46	24	2	2	NUM
ejpam-4582	46	25	-spaces	-space	NOUN
ejpam-4582	46	26	.	.	PUNCT
ejpam-4582	47	1	dunham	dunham	PROPN
ejpam-4582	48	1	[	[	X
ejpam-4582	48	2	12	12	NUM
ejpam-4582	48	3	]	]	PUNCT
ejpam-4582	48	4	and	and	CCONJ
ejpam-4582	48	5	dunham	dunham	PROPN
ejpam-4582	48	6	and	and	CCONJ
ejpam-4582	48	7	levine	levine	PROPN
ejpam-4582	48	8	[	[	X
ejpam-4582	48	9	13	13	NUM
ejpam-4582	48	10	]	]	PUNCT
ejpam-4582	48	11	further	far	ADV
ejpam-4582	48	12	studied	study	VERB
ejpam-4582	48	13	some	some	DET
ejpam-4582	48	14	properties	property	NOUN
ejpam-4582	48	15	of	of	ADP
ejpam-4582	48	16	generalized	generalized	ADJ
ejpam-4582	48	17	closed	closed	ADJ
ejpam-4582	48	18	sets	set	NOUN
ejpam-4582	48	19	and	and	CCONJ
ejpam-4582	48	20	t	t	NOUN
ejpam-4582	48	21	1	1	NUM
ejpam-4582	48	22	2	2	NUM
ejpam-4582	48	23	-spaces	-space	NOUN
ejpam-4582	48	24	.	.	PUNCT
ejpam-4582	49	1	in	in	ADP
ejpam-4582	49	2	[	[	X
ejpam-4582	49	3	17	17	NUM
ejpam-4582	49	4	]	]	PUNCT
ejpam-4582	49	5	,	,	PUNCT
ejpam-4582	49	6	the	the	DET
ejpam-4582	49	7	present	present	ADJ
ejpam-4582	49	8	author	author	NOUN
ejpam-4582	49	9	offered	offer	VERB
ejpam-4582	49	10	a	a	DET
ejpam-4582	49	11	new	new	ADJ
ejpam-4582	49	12	concept	concept	NOUN
ejpam-4582	49	13	to	to	ADP
ejpam-4582	49	14	the	the	DET
ejpam-4582	49	15	field	field	NOUN
ejpam-4582	49	16	of	of	ADP
ejpam-4582	49	17	generalized	generalized	ADJ
ejpam-4582	49	18	topology	topology	NOUN
ejpam-4582	49	19	by	by	ADP
ejpam-4582	49	20	introducing	introduce	VERB
ejpam-4582	49	21	semi	semi	ADJ
ejpam-4582	49	22	-	-	ADJ
ejpam-4582	49	23	open	open	ADJ
ejpam-4582	49	24	sets	set	NOUN
ejpam-4582	49	25	,	,	PUNCT
ejpam-4582	49	26	i.e.	i.e.	X
ejpam-4582	49	27	,	,	PUNCT
ejpam-4582	49	28	a	a	DET
ejpam-4582	49	29	subset	subset	NOUN
ejpam-4582	49	30	of	of	ADP
ejpam-4582	49	31	a	a	DET
ejpam-4582	49	32	topological	topological	ADJ
ejpam-4582	49	33	space	space	NOUN
ejpam-4582	49	34	which	which	PRON
ejpam-4582	49	35	is	be	AUX
ejpam-4582	49	36	contained	contain	VERB
ejpam-4582	49	37	in	in	ADP
ejpam-4582	49	38	the	the	DET
ejpam-4582	49	39	closure	closure	NOUN
ejpam-4582	49	40	of	of	ADP
ejpam-4582	49	41	the	the	DET
ejpam-4582	49	42	interior	interior	NOUN
ejpam-4582	49	43	of	of	ADP
ejpam-4582	49	44	its	its	PRON
ejpam-4582	49	45	closure	closure	NOUN
ejpam-4582	49	46	.	.	PUNCT
ejpam-4582	50	1	caldas	caldas	PROPN
ejpam-4582	50	2	and	and	CCONJ
ejpam-4582	50	3	dontchev	dontchev	ADJ
ejpam-4582	50	4	[	[	X
ejpam-4582	50	5	5	5	NUM
ejpam-4582	50	6	]	]	PUNCT
ejpam-4582	50	7	introduced	introduce	VERB
ejpam-4582	50	8	the	the	DET
ejpam-4582	50	9	notions	notion	NOUN
ejpam-4582	50	10	of	of	ADP
ejpam-4582	50	11	λs	λs	NOUN
ejpam-4582	50	12	-	-	PUNCT
ejpam-4582	50	13	sets	set	NOUN
ejpam-4582	50	14	and	and	CCONJ
ejpam-4582	50	15	generalized	generalize	VERB
ejpam-4582	50	16	λs	λs	NOUN
ejpam-4582	50	17	-	-	PUNCT
ejpam-4582	50	18	sets	set	NOUN
ejpam-4582	50	19	and	and	CCONJ
ejpam-4582	50	20	studied	study	VERB
ejpam-4582	50	21	some	some	DET
ejpam-4582	50	22	characterizations	characterization	NOUN
ejpam-4582	50	23	of	of	ADP
ejpam-4582	50	24	semi	semi	ADJ
ejpam-4582	50	25	-	-	NOUN
ejpam-4582	50	26	t	t	ADJ
ejpam-4582	50	27	1	1	NUM
ejpam-4582	50	28	2	2	NUM
ejpam-4582	50	29	-spaces	-space	NOUN
ejpam-4582	50	30	.	.	PUNCT
ejpam-4582	51	1	buadong	buadong	NOUN
ejpam-4582	51	2	et	et	PROPN
ejpam-4582	51	3	al	al	PROPN
ejpam-4582	51	4	.	.	PUNCT
ejpam-4582	52	1	[	[	X
ejpam-4582	52	2	4	4	X
ejpam-4582	52	3	]	]	PUNCT
ejpam-4582	52	4	introduced	introduce	VERB
ejpam-4582	52	5	and	and	CCONJ
ejpam-4582	52	6	investigated	investigate	VERB
ejpam-4582	52	7	some	some	DET
ejpam-4582	52	8	separation	separation	NOUN
ejpam-4582	52	9	axioms	axiom	NOUN
ejpam-4582	52	10	in	in	ADP
ejpam-4582	52	11	generalized	generalized	ADJ
ejpam-4582	52	12	topology	topology	NOUN
ejpam-4582	52	13	and	and	CCONJ
ejpam-4582	52	14	minimal	minimal	ADJ
ejpam-4582	52	15	structure	structure	NOUN
ejpam-4582	52	16	spaces	space	NOUN
ejpam-4582	52	17	.	.	PUNCT
ejpam-4582	53	1	dungthaisong	dungthaisong	NOUN
ejpam-4582	53	2	et	et	PROPN
ejpam-4582	53	3	al	al	PROPN
ejpam-4582	53	4	.	.	PUNCT
ejpam-4582	54	1	[	[	X
ejpam-4582	54	2	11	11	NUM
ejpam-4582	54	3	]	]	PUNCT
ejpam-4582	54	4	investigated	investigate	VERB
ejpam-4582	54	5	several	several	ADJ
ejpam-4582	54	6	characterizations	characterization	NOUN
ejpam-4582	54	7	of	of	ADP
ejpam-4582	54	8	pairwise	pairwise	NOUN
ejpam-4582	54	9	µ-t	µ-t	PROPN
ejpam-4582	54	10	1	1	NUM
ejpam-4582	54	11	2	2	NUM
ejpam-4582	54	12	-spaces	-space	NOUN
ejpam-4582	54	13	.	.	PUNCT
ejpam-4582	55	1	torton	torton	PROPN
ejpam-4582	55	2	et	et	PROPN
ejpam-4582	55	3	al	al	PROPN
ejpam-4582	55	4	.	.	PUNCT
ejpam-4582	56	1	[	[	X
ejpam-4582	56	2	27	27	NUM
ejpam-4582	56	3	]	]	PUNCT
ejpam-4582	56	4	introduced	introduce	VERB
ejpam-4582	56	5	and	and	CCONJ
ejpam-4582	56	6	studied	study	VERB
ejpam-4582	56	7	the	the	DET
ejpam-4582	56	8	concepts	concept	NOUN
ejpam-4582	56	9	of	of	ADP
ejpam-4582	56	10	µ(m	µ(m	NOUN
ejpam-4582	56	11	,	,	PUNCT
ejpam-4582	56	12	n)-regular	n)-regular	ADJ
ejpam-4582	56	13	spaces	space	NOUN
ejpam-4582	56	14	and	and	CCONJ
ejpam-4582	56	15	µ(m	µ(m	NOUN
ejpam-4582	56	16	,	,	PUNCT
ejpam-4582	56	17	n)-normal	n)-normal	ADJ
ejpam-4582	56	18	spaces	space	NOUN
ejpam-4582	56	19	.	.	PUNCT
ejpam-4582	57	1	the	the	DET
ejpam-4582	57	2	article	article	NOUN
ejpam-4582	57	3	is	be	AUX
ejpam-4582	57	4	organized	organize	VERB
ejpam-4582	57	5	as	as	SCONJ
ejpam-4582	57	6	follows	follow	VERB
ejpam-4582	57	7	.	.	PUNCT
ejpam-4582	58	1	in	in	ADP
ejpam-4582	58	2	section	section	NOUN
ejpam-4582	58	3	3	3	NUM
ejpam-4582	58	4	,	,	PUNCT
ejpam-4582	58	5	we	we	PRON
ejpam-4582	58	6	introduce	introduce	VERB
ejpam-4582	58	7	the	the	DET
ejpam-4582	58	8	notion	notion	NOUN
ejpam-4582	58	9	of	of	ADP
ejpam-4582	58	10	(	(	PUNCT
ejpam-4582	58	11	λ	λ	PROPN
ejpam-4582	58	12	,	,	PUNCT
ejpam-4582	58	13	s)closed	s)close	VERB
ejpam-4582	58	14	sets	set	NOUN
ejpam-4582	58	15	.	.	PUNCT
ejpam-4582	59	1	moreover	moreover	ADV
ejpam-4582	59	2	,	,	PUNCT
ejpam-4582	59	3	some	some	DET
ejpam-4582	59	4	properties	property	NOUN
ejpam-4582	59	5	of	of	ADP
ejpam-4582	59	6	(	(	PUNCT
ejpam-4582	59	7	λ	λ	PROPN
ejpam-4582	59	8	,	,	PUNCT
ejpam-4582	59	9	s)-closed	s)-close	VERB
ejpam-4582	59	10	sets	set	NOUN
ejpam-4582	59	11	are	be	AUX
ejpam-4582	59	12	discussed	discuss	VERB
ejpam-4582	59	13	.	.	PUNCT
ejpam-4582	60	1	in	in	ADP
ejpam-4582	60	2	section	section	NOUN
ejpam-4582	60	3	4	4	NUM
ejpam-4582	60	4	,	,	PUNCT
ejpam-4582	60	5	we	we	PRON
ejpam-4582	60	6	introduce	introduce	VERB
ejpam-4582	60	7	the	the	DET
ejpam-4582	60	8	notion	notion	NOUN
ejpam-4582	60	9	of	of	ADP
ejpam-4582	60	10	(	(	PUNCT
ejpam-4582	60	11	λ	λ	PROPN
ejpam-4582	60	12	,	,	PUNCT
ejpam-4582	60	13	s)-r0	s)-r0	PRON
ejpam-4582	60	14	spaces	space	VERB
ejpam-4582	60	15	and	and	CCONJ
ejpam-4582	60	16	investigate	investigate	VERB
ejpam-4582	60	17	some	some	DET
ejpam-4582	60	18	characterizations	characterization	NOUN
ejpam-4582	60	19	of	of	ADP
ejpam-4582	60	20	(	(	PUNCT
ejpam-4582	60	21	λ	λ	PROPN
ejpam-4582	60	22	,	,	PUNCT
ejpam-4582	60	23	s)r0	s)r0	PROPN
ejpam-4582	60	24	spaces	space	NOUN
ejpam-4582	60	25	.	.	PUNCT
ejpam-4582	61	1	in	in	ADP
ejpam-4582	61	2	section	section	NOUN
ejpam-4582	61	3	5	5	NUM
ejpam-4582	61	4	,	,	PUNCT
ejpam-4582	61	5	we	we	PRON
ejpam-4582	61	6	introduce	introduce	VERB
ejpam-4582	61	7	the	the	DET
ejpam-4582	61	8	notion	notion	NOUN
ejpam-4582	61	9	of	of	ADP
ejpam-4582	61	10	generalized	generalized	ADJ
ejpam-4582	61	11	(	(	PUNCT
ejpam-4582	61	12	λ	λ	X
ejpam-4582	61	13	,	,	PUNCT
ejpam-4582	61	14	s)-closed	s)-close	VERB
ejpam-4582	61	15	sets	set	NOUN
ejpam-4582	61	16	and	and	CCONJ
ejpam-4582	61	17	investigate	investigate	VERB
ejpam-4582	61	18	several	several	ADJ
ejpam-4582	61	19	fundamental	fundamental	ADJ
ejpam-4582	61	20	properties	property	NOUN
ejpam-4582	61	21	of	of	ADP
ejpam-4582	61	22	generalized	generalized	ADJ
ejpam-4582	61	23	(	(	PUNCT
ejpam-4582	61	24	λ	λ	X
ejpam-4582	61	25	,	,	PUNCT
ejpam-4582	61	26	s)-closed	s)-close	VERB
ejpam-4582	61	27	sets	set	NOUN
ejpam-4582	61	28	.	.	PUNCT
ejpam-4582	62	1	furthermore	furthermore	ADV
ejpam-4582	62	2	,	,	PUNCT
ejpam-4582	62	3	some	some	DET
ejpam-4582	62	4	characterizations	characterization	NOUN
ejpam-4582	62	5	of	of	ADP
ejpam-4582	62	6	(	(	PUNCT
ejpam-4582	62	7	λ	λ	PROPN
ejpam-4582	62	8	,	,	PUNCT
ejpam-4582	62	9	s)-normal	s)-normal	ADJ
ejpam-4582	62	10	spaces	space	NOUN
ejpam-4582	62	11	are	be	AUX
ejpam-4582	62	12	explored	explore	VERB
ejpam-4582	62	13	.	.	PUNCT
ejpam-4582	63	1	in	in	ADP
ejpam-4582	63	2	section	section	NOUN
ejpam-4582	63	3	6	6	NUM
ejpam-4582	63	4	,	,	PUNCT
ejpam-4582	63	5	we	we	PRON
ejpam-4582	63	6	introduce	introduce	VERB
ejpam-4582	63	7	the	the	DET
ejpam-4582	63	8	notion	notion	NOUN
ejpam-4582	63	9	of	of	ADP
ejpam-4582	63	10	(	(	PUNCT
ejpam-4582	63	11	λ	λ	PROPN
ejpam-4582	63	12	,	,	PUNCT
ejpam-4582	63	13	s)-extremally	s)-extremally	ADV
ejpam-4582	63	14	disconnected	disconnected	ADJ
ejpam-4582	63	15	spaces	space	NOUN
ejpam-4582	63	16	and	and	CCONJ
ejpam-4582	63	17	investigate	investigate	VERB
ejpam-4582	63	18	several	several	ADJ
ejpam-4582	63	19	characterizations	characterization	NOUN
ejpam-4582	63	20	of	of	ADP
ejpam-4582	63	21	such	such	ADJ
ejpam-4582	63	22	spaces	space	NOUN
ejpam-4582	63	23	.	.	PUNCT
ejpam-4582	64	1	in	in	ADP
ejpam-4582	64	2	section	section	NOUN
ejpam-4582	64	3	7	7	NUM
ejpam-4582	64	4	,	,	PUNCT
ejpam-4582	64	5	we	we	PRON
ejpam-4582	64	6	introduce	introduce	VERB
ejpam-4582	64	7	the	the	DET
ejpam-4582	64	8	notion	notion	NOUN
ejpam-4582	64	9	of	of	ADP
ejpam-4582	64	10	almost	almost	ADV
ejpam-4582	64	11	(	(	PUNCT
ejpam-4582	64	12	λ	λ	PROPN
ejpam-4582	64	13	,	,	PUNCT
ejpam-4582	64	14	s)-continuous	s)-continuous	ADJ
ejpam-4582	64	15	functions	function	NOUN
ejpam-4582	64	16	and	and	CCONJ
ejpam-4582	64	17	investigate	investigate	VERB
ejpam-4582	64	18	several	several	ADJ
ejpam-4582	64	19	characterizations	characterization	NOUN
ejpam-4582	64	20	of	of	ADP
ejpam-4582	64	21	such	such	ADJ
ejpam-4582	64	22	functions	function	NOUN
ejpam-4582	64	23	.	.	PUNCT
ejpam-4582	65	1	2	2	X
ejpam-4582	65	2	.	.	X
ejpam-4582	65	3	preliminaries	preliminary	NOUN
ejpam-4582	65	4	throughout	throughout	ADP
ejpam-4582	65	5	the	the	DET
ejpam-4582	65	6	present	present	ADJ
ejpam-4582	65	7	paper	paper	NOUN
ejpam-4582	65	8	,	,	PUNCT
ejpam-4582	65	9	spaces	space	NOUN
ejpam-4582	65	10	(	(	PUNCT
ejpam-4582	65	11	x	x	X
ejpam-4582	65	12	,	,	PUNCT
ejpam-4582	65	13	τ	τ	X
ejpam-4582	65	14	)	)	PUNCT
ejpam-4582	65	15	and	and	CCONJ
ejpam-4582	65	16	(	(	PUNCT
ejpam-4582	65	17	y	y	PROPN
ejpam-4582	65	18	,	,	PUNCT
ejpam-4582	65	19	σ	σ	PROPN
ejpam-4582	65	20	)	)	PUNCT
ejpam-4582	65	21	(	(	PUNCT
ejpam-4582	65	22	or	or	CCONJ
ejpam-4582	65	23	simply	simply	ADV
ejpam-4582	65	24	x	x	X
ejpam-4582	65	25	and	and	CCONJ
ejpam-4582	65	26	y	y	PROPN
ejpam-4582	65	27	)	)	PUNCT
ejpam-4582	65	28	always	always	ADV
ejpam-4582	65	29	mean	mean	VERB
ejpam-4582	65	30	topological	topological	ADJ
ejpam-4582	65	31	spaces	space	NOUN
ejpam-4582	65	32	on	on	ADP
ejpam-4582	65	33	which	which	PRON
ejpam-4582	65	34	no	no	DET
ejpam-4582	65	35	separation	separation	NOUN
ejpam-4582	65	36	axioms	axiom	NOUN
ejpam-4582	65	37	are	be	AUX
ejpam-4582	65	38	assumed	assume	VERB
ejpam-4582	65	39	unless	unless	SCONJ
ejpam-4582	65	40	explicitly	explicitly	ADV
ejpam-4582	65	41	stated	state	VERB
ejpam-4582	65	42	.	.	PUNCT
ejpam-4582	66	1	for	for	ADP
ejpam-4582	66	2	a	a	DET
ejpam-4582	66	3	subset	subset	NOUN
ejpam-4582	66	4	a	a	PRON
ejpam-4582	66	5	of	of	ADP
ejpam-4582	66	6	a	a	DET
ejpam-4582	66	7	topological	topological	ADJ
ejpam-4582	66	8	space	space	NOUN
ejpam-4582	66	9	(	(	PUNCT
ejpam-4582	66	10	x	x	X
ejpam-4582	66	11	,	,	PUNCT
ejpam-4582	66	12	τ	τ	PROPN
ejpam-4582	66	13	)	)	PUNCT
ejpam-4582	66	14	,	,	PUNCT
ejpam-4582	66	15	cl(a	cl(a	NUM
ejpam-4582	66	16	)	)	PUNCT
ejpam-4582	66	17	and	and	CCONJ
ejpam-4582	66	18	int(a	int(a	PROPN
ejpam-4582	66	19	)	)	PUNCT
ejpam-4582	66	20	represent	represent	VERB
ejpam-4582	66	21	the	the	DET
ejpam-4582	66	22	closure	closure	NOUN
ejpam-4582	66	23	and	and	CCONJ
ejpam-4582	66	24	the	the	DET
ejpam-4582	66	25	interior	interior	NOUN
ejpam-4582	66	26	of	of	ADP
ejpam-4582	66	27	a	a	PRON
ejpam-4582	66	28	,	,	PUNCT
ejpam-4582	66	29	respectively	respectively	ADV
ejpam-4582	66	30	.	.	PUNCT
ejpam-4582	67	1	a	a	DET
ejpam-4582	67	2	subset	subset	NOUN
ejpam-4582	67	3	a	a	PRON
ejpam-4582	67	4	of	of	ADP
ejpam-4582	67	5	a	a	DET
ejpam-4582	67	6	topological	topological	ADJ
ejpam-4582	67	7	space	space	NOUN
ejpam-4582	67	8	(	(	PUNCT
ejpam-4582	67	9	x	x	X
ejpam-4582	67	10	,	,	PUNCT
ejpam-4582	67	11	τ	τ	X
ejpam-4582	67	12	)	)	PUNCT
ejpam-4582	67	13	is	be	AUX
ejpam-4582	67	14	called	call	VERB
ejpam-4582	67	15	semi	semi	ADJ
ejpam-4582	67	16	-	-	ADJ
ejpam-4582	67	17	open	open	ADJ
ejpam-4582	67	18	[	[	X
ejpam-4582	67	19	17	17	NUM
ejpam-4582	67	20	]	]	PUNCT
ejpam-4582	67	21	if	if	SCONJ
ejpam-4582	67	22	a	a	DET
ejpam-4582	67	23	⊆	⊆	NUM
ejpam-4582	67	24	cl(int(a	cl(int(a	NOUN
ejpam-4582	67	25	)	)	PUNCT
ejpam-4582	67	26	)	)	PUNCT
ejpam-4582	67	27	.	.	PUNCT
ejpam-4582	68	1	the	the	DET
ejpam-4582	68	2	complement	complement	NOUN
ejpam-4582	68	3	of	of	ADP
ejpam-4582	68	4	a	a	DET
ejpam-4582	68	5	semi	semi	ADJ
ejpam-4582	68	6	-	-	ADJ
ejpam-4582	68	7	open	open	ADJ
ejpam-4582	68	8	set	set	NOUN
ejpam-4582	68	9	is	be	AUX
ejpam-4582	68	10	called	call	VERB
ejpam-4582	68	11	semi	semi	ADV
ejpam-4582	68	12	-	-	ADJ
ejpam-4582	68	13	closed	closed	ADJ
ejpam-4582	68	14	.	.	PUNCT
ejpam-4582	69	1	by	by	ADP
ejpam-4582	69	2	so(x	so(x	PROPN
ejpam-4582	69	3	,	,	PUNCT
ejpam-4582	69	4	τ	τ	X
ejpam-4582	69	5	)	)	PUNCT
ejpam-4582	69	6	and	and	CCONJ
ejpam-4582	69	7	sc(x	sc(x	PROPN
ejpam-4582	69	8	,	,	PUNCT
ejpam-4582	69	9	τ	τ	X
ejpam-4582	69	10	)	)	PUNCT
ejpam-4582	69	11	we	we	PRON
ejpam-4582	69	12	denote	denote	VERB
ejpam-4582	69	13	the	the	DET
ejpam-4582	69	14	family	family	NOUN
ejpam-4582	69	15	of	of	ADP
ejpam-4582	69	16	all	all	DET
ejpam-4582	69	17	semi	semi	ADJ
ejpam-4582	69	18	-	-	ADJ
ejpam-4582	69	19	open	open	ADJ
ejpam-4582	69	20	sets	set	NOUN
ejpam-4582	69	21	and	and	CCONJ
ejpam-4582	69	22	the	the	DET
ejpam-4582	69	23	family	family	NOUN
ejpam-4582	69	24	of	of	ADP
ejpam-4582	69	25	all	all	DET
ejpam-4582	69	26	semi	semi	ADJ
ejpam-4582	69	27	-	-	ADJ
ejpam-4582	69	28	closed	closed	ADJ
ejpam-4582	69	29	sets	set	NOUN
ejpam-4582	69	30	in	in	ADP
ejpam-4582	69	31	a	a	DET
ejpam-4582	69	32	topological	topological	ADJ
ejpam-4582	69	33	space	space	NOUN
ejpam-4582	69	34	(	(	PUNCT
ejpam-4582	69	35	x	x	X
ejpam-4582	69	36	,	,	PUNCT
ejpam-4582	69	37	τ	τ	PROPN
ejpam-4582	69	38	)	)	PUNCT
ejpam-4582	69	39	,	,	PUNCT
ejpam-4582	69	40	respectively	respectively	ADV
ejpam-4582	69	41	.	.	PUNCT
ejpam-4582	70	1	the	the	DET
ejpam-4582	70	2	semi	semi	NOUN
ejpam-4582	70	3	-	-	NOUN
ejpam-4582	70	4	closure	closure	NOUN
ejpam-4582	70	5	of	of	ADP
ejpam-4582	70	6	a	a	DET
ejpam-4582	70	7	set	set	NOUN
ejpam-4582	70	8	a	a	PRON
ejpam-4582	70	9	,	,	PUNCT
ejpam-4582	70	10	denoted	denote	VERB
ejpam-4582	70	11	by	by	ADP
ejpam-4582	70	12	scl(a	scl(a	PROPN
ejpam-4582	70	13	)	)	PUNCT
ejpam-4582	70	14	,	,	PUNCT
ejpam-4582	70	15	is	be	AUX
ejpam-4582	70	16	the	the	DET
ejpam-4582	70	17	intersection	intersection	NOUN
ejpam-4582	70	18	of	of	ADP
ejpam-4582	70	19	all	all	DET
ejpam-4582	70	20	semi	semi	ADJ
ejpam-4582	70	21	-	-	ADJ
ejpam-4582	70	22	closed	closed	ADJ
ejpam-4582	70	23	sets	set	NOUN
ejpam-4582	70	24	containing	contain	VERB
ejpam-4582	70	25	a.	a.	NOUN
ejpam-4582	70	26	the	the	DET
ejpam-4582	70	27	semi	semi	ADJ
ejpam-4582	70	28	-	-	ADJ
ejpam-4582	70	29	interior	interior	ADJ
ejpam-4582	70	30	of	of	ADP
ejpam-4582	70	31	a	a	DET
ejpam-4582	70	32	set	set	NOUN
ejpam-4582	70	33	a	a	PRON
ejpam-4582	70	34	,	,	PUNCT
ejpam-4582	70	35	denoted	denote	VERB
ejpam-4582	70	36	by	by	ADP
ejpam-4582	70	37	sint(a	sint(a	PROPN
ejpam-4582	70	38	)	)	PUNCT
ejpam-4582	70	39	,	,	PUNCT
ejpam-4582	70	40	is	be	AUX
ejpam-4582	70	41	the	the	DET
ejpam-4582	70	42	union	union	NOUN
ejpam-4582	70	43	of	of	ADP
ejpam-4582	70	44	all	all	DET
ejpam-4582	70	45	semi	semi	ADJ
ejpam-4582	70	46	-	-	ADJ
ejpam-4582	70	47	open	open	ADJ
ejpam-4582	70	48	sets	set	NOUN
ejpam-4582	70	49	contained	contain	VERB
ejpam-4582	70	50	in	in	ADP
ejpam-4582	70	51	a.	a.	NOUN
ejpam-4582	71	1	a	a	DET
ejpam-4582	71	2	subset	subset	NOUN
ejpam-4582	71	3	aλs	aλs	NOUN
ejpam-4582	72	1	[	[	X
ejpam-4582	72	2	5	5	NUM
ejpam-4582	72	3	]	]	PUNCT
ejpam-4582	72	4	(	(	PUNCT
ejpam-4582	72	5	resp	resp	NOUN
ejpam-4582	72	6	.	.	PUNCT
ejpam-4582	73	1	aλs	aλs	PROPN
ejpam-4582	73	2	)	)	PUNCT
ejpam-4582	73	3	is	be	AUX
ejpam-4582	73	4	defined	define	VERB
ejpam-4582	73	5	as	as	SCONJ
ejpam-4582	73	6	follows	follow	VERB
ejpam-4582	73	7	:	:	PUNCT
ejpam-4582	73	8	aλs	aλs	PROPN
ejpam-4582	74	1	=	=	SYM
ejpam-4582	74	2	∩{u	∩{u	PROPN
ejpam-4582	75	1	|	|	ADV
ejpam-4582	75	2	u	u	PROPN
ejpam-4582	75	3	⊇	⊇	PROPN
ejpam-4582	75	4	a	a	PROPN
ejpam-4582	75	5	,	,	PUNCT
ejpam-4582	75	6	a	a	DET
ejpam-4582	75	7	∈	∈	PROPN
ejpam-4582	75	8	so(x	so(x	NOUN
ejpam-4582	75	9	,	,	PUNCT
ejpam-4582	75	10	τ	τ	PROPN
ejpam-4582	75	11	)	)	PUNCT
ejpam-4582	75	12	}	}	PUNCT
ejpam-4582	75	13	(	(	PUNCT
ejpam-4582	75	14	resp	resp	NOUN
ejpam-4582	75	15	.	.	PUNCT
ejpam-4582	76	1	avs	avs	PROPN
ejpam-4582	77	1	=	=	PROPN
ejpam-4582	77	2	∪{f	∪{f	PROPN
ejpam-4582	77	3	|	|	ADV
ejpam-4582	77	4	f	f	PROPN
ejpam-4582	77	5	⊆	⊆	NUM
ejpam-4582	77	6	a	a	PRON
ejpam-4582	77	7	,	,	PUNCT
ejpam-4582	77	8	x	x	PUNCT
ejpam-4582	77	9	−	−	PROPN
ejpam-4582	77	10	f	f	PROPN
ejpam-4582	77	11	∈	∈	PROPN
ejpam-4582	77	12	so(x	so(x	NOUN
ejpam-4582	77	13	,	,	PUNCT
ejpam-4582	77	14	τ	τ	PROPN
ejpam-4582	77	15	)	)	PUNCT
ejpam-4582	77	16	}	}	PUNCT
ejpam-4582	77	17	)	)	PUNCT
ejpam-4582	77	18	.	.	PUNCT
ejpam-4582	78	1	lemma	lemma	PROPN
ejpam-4582	78	2	1	1	NUM
ejpam-4582	78	3	.	.	PUNCT
ejpam-4582	79	1	[	[	X
ejpam-4582	79	2	5	5	NUM
ejpam-4582	79	3	]	]	PUNCT
ejpam-4582	79	4	for	for	ADP
ejpam-4582	79	5	subsets	subset	NOUN
ejpam-4582	79	6	a	a	PRON
ejpam-4582	79	7	,	,	PUNCT
ejpam-4582	79	8	b	b	NOUN
ejpam-4582	79	9	and	and	CCONJ
ejpam-4582	79	10	aγ(γ	aγ(γ	NOUN
ejpam-4582	79	11	∈	∈	PROPN
ejpam-4582	79	12	∇	∇	NOUN
ejpam-4582	79	13	)	)	PUNCT
ejpam-4582	79	14	of	of	ADP
ejpam-4582	79	15	a	a	DET
ejpam-4582	79	16	topological	topological	ADJ
ejpam-4582	79	17	space	space	NOUN
ejpam-4582	79	18	(	(	PUNCT
ejpam-4582	79	19	x	x	X
ejpam-4582	79	20	,	,	PUNCT
ejpam-4582	79	21	τ	τ	PROPN
ejpam-4582	79	22	)	)	PUNCT
ejpam-4582	79	23	,	,	PUNCT
ejpam-4582	79	24	the	the	DET
ejpam-4582	79	25	following	follow	VERB
ejpam-4582	79	26	properties	property	NOUN
ejpam-4582	79	27	hold	hold	VERB
ejpam-4582	79	28	:	:	PUNCT
ejpam-4582	79	29	c.	c.	PROPN
ejpam-4582	79	30	boonpok	boonpok	PROPN
ejpam-4582	79	31	,	,	PUNCT
ejpam-4582	79	32	c.	c.	PROPN
ejpam-4582	79	33	viriyapong	viriyapong	PROPN
ejpam-4582	79	34	/	/	SYM
ejpam-4582	79	35	eur	eur	PROPN
ejpam-4582	79	36	.	.	PUNCT
ejpam-4582	80	1	j.	j.	PROPN
ejpam-4582	80	2	pure	pure	PROPN
ejpam-4582	80	3	appl	appl	PROPN
ejpam-4582	80	4	.	.	PROPN
ejpam-4582	80	5	math	math	PROPN
ejpam-4582	80	6	,	,	PUNCT
ejpam-4582	80	7	16	16	NUM
ejpam-4582	80	8	(	(	PUNCT
ejpam-4582	80	9	1	1	NUM
ejpam-4582	80	10	)	)	PUNCT
ejpam-4582	80	11	(	(	PUNCT
ejpam-4582	80	12	2023	2023	NUM
ejpam-4582	80	13	)	)	PUNCT
ejpam-4582	80	14	,	,	PUNCT
ejpam-4582	80	15	336	336	NUM
ejpam-4582	80	16	-	-	SYM
ejpam-4582	80	17	362	362	NUM
ejpam-4582	80	18	338	338	NUM
ejpam-4582	80	19	(	(	PUNCT
ejpam-4582	80	20	1	1	NUM
ejpam-4582	80	21	)	)	PUNCT
ejpam-4582	80	22	a	a	DET
ejpam-4582	80	23	⊆	⊆	NUM
ejpam-4582	80	24	aλs	aλs	NOUN
ejpam-4582	80	25	.	.	PUNCT
ejpam-4582	81	1	(	(	PUNCT
ejpam-4582	81	2	2	2	X
ejpam-4582	81	3	)	)	PUNCT
ejpam-4582	81	4	if	if	SCONJ
ejpam-4582	81	5	a	a	DET
ejpam-4582	81	6	⊆	⊆	NUM
ejpam-4582	81	7	b	b	NOUN
ejpam-4582	81	8	,	,	PUNCT
ejpam-4582	81	9	then	then	ADV
ejpam-4582	81	10	aλs	aλs	NOUN
ejpam-4582	81	11	⊆	⊆	NUM
ejpam-4582	81	12	bλs	bλ	NOUN
ejpam-4582	81	13	.	.	PUNCT
ejpam-4582	82	1	(	(	PUNCT
ejpam-4582	82	2	3	3	X
ejpam-4582	82	3	)	)	PUNCT
ejpam-4582	82	4	(	(	PUNCT
ejpam-4582	82	5	aλs)λs	aλs)λs	ADV
ejpam-4582	82	6	=	=	SYM
ejpam-4582	82	7	aλs	aλs	PROPN
ejpam-4582	82	8	.	.	PUNCT
ejpam-4582	83	1	(	(	PUNCT
ejpam-4582	83	2	4	4	NUM
ejpam-4582	83	3	)	)	PUNCT
ejpam-4582	83	4	[	[	PUNCT
ejpam-4582	83	5	∪	∪	X
ejpam-4582	83	6	γ∈∇	γ∈∇	PUNCT
ejpam-4582	83	7	aγ	aγ	NOUN
ejpam-4582	83	8	]	]	PUNCT
ejpam-4582	83	9	λs	λs	NOUN
ejpam-4582	83	10	=	=	SYM
ejpam-4582	83	11	∪	∪	NOUN
ejpam-4582	83	12	γ∈∇	γ∈∇	PUNCT
ejpam-4582	83	13	aλs	aλs	PROPN
ejpam-4582	83	14	γ	γ	X
ejpam-4582	83	15	.	.	PUNCT
ejpam-4582	84	1	(	(	PUNCT
ejpam-4582	84	2	5	5	X
ejpam-4582	84	3	)	)	PUNCT
ejpam-4582	84	4	if	if	SCONJ
ejpam-4582	84	5	a	a	DET
ejpam-4582	84	6	∈	∈	PROPN
ejpam-4582	84	7	so(x	so(x	NOUN
ejpam-4582	84	8	,	,	PUNCT
ejpam-4582	84	9	τ	τ	PROPN
ejpam-4582	84	10	)	)	PUNCT
ejpam-4582	84	11	,	,	PUNCT
ejpam-4582	84	12	then	then	ADV
ejpam-4582	84	13	a	a	DET
ejpam-4582	84	14	=	=	ADJ
ejpam-4582	84	15	aλs	aλs	PROPN
ejpam-4582	84	16	.	.	PUNCT
ejpam-4582	85	1	(	(	PUNCT
ejpam-4582	85	2	6	6	NUM
ejpam-4582	85	3	)	)	PUNCT
ejpam-4582	85	4	(	(	PUNCT
ejpam-4582	85	5	x	x	X
ejpam-4582	85	6	−a)λs	−a)λs	ADV
ejpam-4582	85	7	=	=	X
ejpam-4582	85	8	x	x	X
ejpam-4582	85	9	−avs	−avs	ADJ
ejpam-4582	85	10	.	.	PUNCT
ejpam-4582	86	1	(	(	PUNCT
ejpam-4582	86	2	7	7	X
ejpam-4582	86	3	)	)	PUNCT
ejpam-4582	86	4	avs	avs	NOUN
ejpam-4582	86	5	⊆	⊆	NUM
ejpam-4582	86	6	a.	a.	NOUN
ejpam-4582	86	7	(	(	PUNCT
ejpam-4582	86	8	8)	8)	NUM
ejpam-4582	86	9	if	if	SCONJ
ejpam-4582	86	10	a	a	DET
ejpam-4582	86	11	∈	∈	PROPN
ejpam-4582	86	12	sc(x	sc(x	NOUN
ejpam-4582	86	13	,	,	PUNCT
ejpam-4582	86	14	τ	τ	PROPN
ejpam-4582	86	15	)	)	PUNCT
ejpam-4582	86	16	,	,	PUNCT
ejpam-4582	86	17	then	then	ADV
ejpam-4582	86	18	a	a	DET
ejpam-4582	86	19	=	=	X
ejpam-4582	86	20	avs	avs	PROPN
ejpam-4582	86	21	.	.	PUNCT
ejpam-4582	87	1	(	(	PUNCT
ejpam-4582	87	2	9	9	NUM
ejpam-4582	87	3	)	)	PUNCT
ejpam-4582	87	4	[	[	PUNCT
ejpam-4582	87	5	∩	∩	NOUN
ejpam-4582	87	6	γ∈∇	γ∈∇	PUNCT
ejpam-4582	87	7	aγ	aγ	NOUN
ejpam-4582	87	8	]	]	PUNCT
ejpam-4582	87	9	λs	λs	ADP
ejpam-4582	87	10	⊆	⊆	NUM
ejpam-4582	87	11	∩	∩	X
ejpam-4582	87	12	γ∈∇	γ∈∇	PUNCT
ejpam-4582	87	13	aλs	aλs	PROPN
ejpam-4582	87	14	γ	γ	X
ejpam-4582	87	15	.	.	PUNCT
ejpam-4582	88	1	(	(	PUNCT
ejpam-4582	88	2	10	10	NUM
ejpam-4582	88	3	)	)	PUNCT
ejpam-4582	88	4	[	[	PUNCT
ejpam-4582	88	5	∪	∪	X
ejpam-4582	88	6	γ∈∇	γ∈∇	PUNCT
ejpam-4582	88	7	aγ	aγ	NOUN
ejpam-4582	88	8	]	]	PUNCT
ejpam-4582	88	9	vs	vs	ADP
ejpam-4582	88	10	⊇	⊇	PROPN
ejpam-4582	88	11	∪	∪	X
ejpam-4582	88	12	γ∈∇	γ∈∇	PUNCT
ejpam-4582	88	13	avs	avs	PROPN
ejpam-4582	88	14	γ	γ	PROPN
ejpam-4582	88	15	.	.	PUNCT
ejpam-4582	89	1	definition	definition	NOUN
ejpam-4582	89	2	1	1	NUM
ejpam-4582	89	3	.	.	PUNCT
ejpam-4582	90	1	[	[	X
ejpam-4582	90	2	5	5	NUM
ejpam-4582	90	3	]	]	PUNCT
ejpam-4582	90	4	a	a	DET
ejpam-4582	90	5	subset	subset	NOUN
ejpam-4582	90	6	a	a	PRON
ejpam-4582	90	7	of	of	ADP
ejpam-4582	90	8	a	a	DET
ejpam-4582	90	9	topological	topological	ADJ
ejpam-4582	90	10	space	space	NOUN
ejpam-4582	90	11	(	(	PUNCT
ejpam-4582	90	12	x	x	X
ejpam-4582	90	13	,	,	PUNCT
ejpam-4582	90	14	τ	τ	X
ejpam-4582	90	15	)	)	PUNCT
ejpam-4582	90	16	is	be	AUX
ejpam-4582	90	17	called	call	VERB
ejpam-4582	90	18	a	a	DET
ejpam-4582	90	19	λs	λs	ADV
ejpam-4582	90	20	-	-	PUNCT
ejpam-4582	90	21	set	set	VERB
ejpam-4582	90	22	(	(	PUNCT
ejpam-4582	90	23	resp	resp	NOUN
ejpam-4582	90	24	.	.	PUNCT
ejpam-4582	91	1	vs	vs	ADP
ejpam-4582	91	2	-	-	PUNCT
ejpam-4582	91	3	set	set	NOUN
ejpam-4582	91	4	)	)	PUNCT
ejpam-4582	91	5	if	if	SCONJ
ejpam-4582	91	6	a	a	DET
ejpam-4582	91	7	=	=	X
ejpam-4582	91	8	aλs	aλs	NOUN
ejpam-4582	91	9	(	(	PUNCT
ejpam-4582	91	10	resp	resp	NOUN
ejpam-4582	91	11	.	.	PUNCT
ejpam-4582	92	1	a	a	DET
ejpam-4582	92	2	=	=	SYM
ejpam-4582	92	3	avs	avs	PROPN
ejpam-4582	92	4	)	)	PUNCT
ejpam-4582	92	5	.	.	PUNCT
ejpam-4582	93	1	lemma	lemma	PROPN
ejpam-4582	93	2	2	2	NUM
ejpam-4582	93	3	.	.	PUNCT
ejpam-4582	94	1	[	[	X
ejpam-4582	94	2	5	5	NUM
ejpam-4582	94	3	]	]	PUNCT
ejpam-4582	94	4	for	for	ADP
ejpam-4582	94	5	a	a	DET
ejpam-4582	94	6	topological	topological	ADJ
ejpam-4582	94	7	space	space	NOUN
ejpam-4582	94	8	(	(	PUNCT
ejpam-4582	94	9	x	x	X
ejpam-4582	94	10	,	,	PUNCT
ejpam-4582	94	11	τ	τ	PROPN
ejpam-4582	94	12	)	)	PUNCT
ejpam-4582	94	13	,	,	PUNCT
ejpam-4582	94	14	the	the	DET
ejpam-4582	94	15	following	follow	VERB
ejpam-4582	94	16	properties	property	NOUN
ejpam-4582	94	17	hold	hold	VERB
ejpam-4582	94	18	:	:	PUNCT
ejpam-4582	94	19	(	(	PUNCT
ejpam-4582	94	20	1	1	X
ejpam-4582	94	21	)	)	PUNCT
ejpam-4582	94	22	the	the	DET
ejpam-4582	94	23	subsets	subset	NOUN
ejpam-4582	94	24	∅	∅	NOUN
ejpam-4582	94	25	and	and	CCONJ
ejpam-4582	94	26	x	x	NOUN
ejpam-4582	94	27	are	be	AUX
ejpam-4582	94	28	λs	λs	NOUN
ejpam-4582	94	29	-	-	PUNCT
ejpam-4582	94	30	sets	set	NOUN
ejpam-4582	94	31	and	and	CCONJ
ejpam-4582	94	32	vs	vs	NOUN
ejpam-4582	94	33	-	-	PUNCT
ejpam-4582	94	34	sets	set	NOUN
ejpam-4582	94	35	.	.	PUNCT
ejpam-4582	95	1	(	(	PUNCT
ejpam-4582	95	2	2	2	X
ejpam-4582	95	3	)	)	PUNCT
ejpam-4582	95	4	every	every	DET
ejpam-4582	95	5	union	union	NOUN
ejpam-4582	95	6	of	of	ADP
ejpam-4582	95	7	λs	λs	NOUN
ejpam-4582	95	8	-	-	PUNCT
ejpam-4582	95	9	sets	set	NOUN
ejpam-4582	95	10	(	(	PUNCT
ejpam-4582	95	11	resp	resp	NOUN
ejpam-4582	95	12	.	.	PUNCT
ejpam-4582	96	1	vs	vs	ADP
ejpam-4582	96	2	-	-	PUNCT
ejpam-4582	96	3	sets	set	NOUN
ejpam-4582	96	4	)	)	PUNCT
ejpam-4582	96	5	is	be	AUX
ejpam-4582	96	6	a	a	DET
ejpam-4582	96	7	λs	λs	ADV
ejpam-4582	96	8	-	-	PUNCT
ejpam-4582	96	9	set	set	VERB
ejpam-4582	96	10	(	(	PUNCT
ejpam-4582	96	11	resp	resp	NOUN
ejpam-4582	96	12	.	.	PUNCT
ejpam-4582	97	1	vs	vs	ADP
ejpam-4582	97	2	-	-	PUNCT
ejpam-4582	97	3	set	set	NOUN
ejpam-4582	97	4	)	)	PUNCT
ejpam-4582	97	5	.	.	PUNCT
ejpam-4582	98	1	(	(	PUNCT
ejpam-4582	98	2	3	3	X
ejpam-4582	98	3	)	)	PUNCT
ejpam-4582	98	4	every	every	DET
ejpam-4582	98	5	intersection	intersection	NOUN
ejpam-4582	98	6	of	of	ADP
ejpam-4582	98	7	λs	λs	NOUN
ejpam-4582	98	8	-	-	PUNCT
ejpam-4582	98	9	sets	set	NOUN
ejpam-4582	98	10	(	(	PUNCT
ejpam-4582	98	11	resp	resp	NOUN
ejpam-4582	98	12	.	.	PUNCT
ejpam-4582	99	1	vs	vs	ADP
ejpam-4582	99	2	-	-	PUNCT
ejpam-4582	99	3	sets	set	NOUN
ejpam-4582	99	4	)	)	PUNCT
ejpam-4582	99	5	is	be	AUX
ejpam-4582	99	6	a	a	DET
ejpam-4582	99	7	λs	λs	ADV
ejpam-4582	99	8	-	-	PUNCT
ejpam-4582	99	9	set	set	VERB
ejpam-4582	99	10	(	(	PUNCT
ejpam-4582	99	11	resp	resp	NOUN
ejpam-4582	99	12	.	.	PUNCT
ejpam-4582	100	1	vs	vs	ADP
ejpam-4582	100	2	-	-	PUNCT
ejpam-4582	100	3	set	set	NOUN
ejpam-4582	100	4	)	)	PUNCT
ejpam-4582	100	5	.	.	PUNCT
ejpam-4582	101	1	(	(	PUNCT
ejpam-4582	101	2	4	4	X
ejpam-4582	101	3	)	)	PUNCT
ejpam-4582	101	4	a	a	DET
ejpam-4582	101	5	subset	subset	NOUN
ejpam-4582	101	6	a	a	PRON
ejpam-4582	101	7	is	be	AUX
ejpam-4582	101	8	a	a	DET
ejpam-4582	101	9	λs	λs	ADV
ejpam-4582	101	10	-	-	PUNCT
ejpam-4582	101	11	set	set	VERB
ejpam-4582	101	12	if	if	SCONJ
ejpam-4582	101	13	and	and	CCONJ
ejpam-4582	101	14	only	only	ADV
ejpam-4582	101	15	x	x	PRON
ejpam-4582	101	16	−a	−a	NOUN
ejpam-4582	101	17	is	be	AUX
ejpam-4582	101	18	a	a	DET
ejpam-4582	101	19	vs	vs	NOUN
ejpam-4582	101	20	-	-	PUNCT
ejpam-4582	101	21	set	set	NOUN
ejpam-4582	101	22	.	.	PUNCT
ejpam-4582	102	1	3	3	X
ejpam-4582	102	2	.	.	X
ejpam-4582	102	3	on	on	ADP
ejpam-4582	102	4	(	(	PUNCT
ejpam-4582	102	5	λ	λ	X
ejpam-4582	102	6	,	,	PUNCT
ejpam-4582	102	7	s)-closed	s)-close	VERB
ejpam-4582	102	8	sets	set	NOUN
ejpam-4582	102	9	in	in	ADP
ejpam-4582	102	10	this	this	DET
ejpam-4582	102	11	section	section	NOUN
ejpam-4582	102	12	,	,	PUNCT
ejpam-4582	102	13	we	we	PRON
ejpam-4582	102	14	introduce	introduce	VERB
ejpam-4582	102	15	the	the	DET
ejpam-4582	102	16	notion	notion	NOUN
ejpam-4582	102	17	of	of	ADP
ejpam-4582	102	18	(	(	PUNCT
ejpam-4582	102	19	λ	λ	PROPN
ejpam-4582	102	20	,	,	PUNCT
ejpam-4582	102	21	s)-closed	s)-close	VERB
ejpam-4582	102	22	sets	set	NOUN
ejpam-4582	102	23	.	.	PUNCT
ejpam-4582	103	1	moreover	moreover	ADV
ejpam-4582	103	2	,	,	PUNCT
ejpam-4582	103	3	some	some	DET
ejpam-4582	103	4	properties	property	NOUN
ejpam-4582	103	5	of	of	ADP
ejpam-4582	103	6	(	(	PUNCT
ejpam-4582	103	7	λ	λ	PROPN
ejpam-4582	103	8	,	,	PUNCT
ejpam-4582	103	9	s)-closed	s)-close	VERB
ejpam-4582	103	10	sets	set	NOUN
ejpam-4582	103	11	are	be	AUX
ejpam-4582	103	12	discussed	discuss	VERB
ejpam-4582	103	13	.	.	PUNCT
ejpam-4582	104	1	definition	definition	NOUN
ejpam-4582	104	2	2	2	NUM
ejpam-4582	104	3	.	.	PUNCT
ejpam-4582	105	1	a	a	DET
ejpam-4582	105	2	subset	subset	NOUN
ejpam-4582	105	3	a	a	PRON
ejpam-4582	105	4	of	of	ADP
ejpam-4582	105	5	a	a	DET
ejpam-4582	105	6	topological	topological	ADJ
ejpam-4582	105	7	space	space	NOUN
ejpam-4582	105	8	(	(	PUNCT
ejpam-4582	105	9	x	x	X
ejpam-4582	105	10	,	,	PUNCT
ejpam-4582	105	11	τ	τ	X
ejpam-4582	105	12	)	)	PUNCT
ejpam-4582	105	13	is	be	AUX
ejpam-4582	105	14	called	call	VERB
ejpam-4582	105	15	(	(	PUNCT
ejpam-4582	105	16	λ	λ	X
ejpam-4582	105	17	,	,	PUNCT
ejpam-4582	105	18	s)-closed	s)-close	VERB
ejpam-4582	105	19	if	if	SCONJ
ejpam-4582	105	20	a	a	DET
ejpam-4582	105	21	=	=	X
ejpam-4582	105	22	t	t	NOUN
ejpam-4582	105	23	∩c	∩c	NOUN
ejpam-4582	105	24	,	,	PUNCT
ejpam-4582	105	25	where	where	SCONJ
ejpam-4582	105	26	t	t	PROPN
ejpam-4582	105	27	is	be	AUX
ejpam-4582	105	28	a	a	DET
ejpam-4582	105	29	λs	λs	ADV
ejpam-4582	105	30	-	-	PUNCT
ejpam-4582	105	31	set	set	VERB
ejpam-4582	105	32	and	and	CCONJ
ejpam-4582	105	33	c	c	NOUN
ejpam-4582	105	34	is	be	AUX
ejpam-4582	105	35	a	a	DET
ejpam-4582	105	36	semi	semi	ADJ
ejpam-4582	105	37	-	-	ADJ
ejpam-4582	105	38	closed	closed	ADJ
ejpam-4582	105	39	set	set	NOUN
ejpam-4582	105	40	.	.	PUNCT
ejpam-4582	106	1	the	the	DET
ejpam-4582	106	2	family	family	NOUN
ejpam-4582	106	3	of	of	ADP
ejpam-4582	106	4	all	all	PRON
ejpam-4582	106	5	(	(	PUNCT
ejpam-4582	106	6	λ	λ	X
ejpam-4582	106	7	,	,	PUNCT
ejpam-4582	106	8	s)-closed	s)-close	VERB
ejpam-4582	106	9	sets	set	NOUN
ejpam-4582	106	10	in	in	ADP
ejpam-4582	106	11	a	a	DET
ejpam-4582	106	12	topological	topological	ADJ
ejpam-4582	106	13	space	space	NOUN
ejpam-4582	106	14	(	(	PUNCT
ejpam-4582	106	15	x	x	X
ejpam-4582	106	16	,	,	PUNCT
ejpam-4582	106	17	τ	τ	X
ejpam-4582	106	18	)	)	PUNCT
ejpam-4582	106	19	is	be	AUX
ejpam-4582	106	20	denoted	denote	VERB
ejpam-4582	106	21	by	by	ADP
ejpam-4582	106	22	(	(	PUNCT
ejpam-4582	106	23	λ	λ	PROPN
ejpam-4582	106	24	,	,	PUNCT
ejpam-4582	106	25	s)c(x	s)c(x	NOUN
ejpam-4582	106	26	)	)	PUNCT
ejpam-4582	106	27	.	.	PUNCT
ejpam-4582	107	1	lemma	lemma	PROPN
ejpam-4582	107	2	3	3	X
ejpam-4582	107	3	.	.	PUNCT
ejpam-4582	108	1	[	[	X
ejpam-4582	108	2	1	1	X
ejpam-4582	108	3	]	]	PUNCT
ejpam-4582	108	4	for	for	ADP
ejpam-4582	108	5	a	a	DET
ejpam-4582	108	6	subset	subset	NOUN
ejpam-4582	108	7	a	a	PRON
ejpam-4582	108	8	of	of	ADP
ejpam-4582	108	9	a	a	DET
ejpam-4582	108	10	topological	topological	ADJ
ejpam-4582	108	11	space	space	NOUN
ejpam-4582	108	12	(	(	PUNCT
ejpam-4582	108	13	x	x	X
ejpam-4582	108	14	,	,	PUNCT
ejpam-4582	108	15	τ	τ	PROPN
ejpam-4582	108	16	)	)	PUNCT
ejpam-4582	108	17	,	,	PUNCT
ejpam-4582	108	18	the	the	DET
ejpam-4582	108	19	following	follow	VERB
ejpam-4582	108	20	properties	property	NOUN
ejpam-4582	108	21	hold	hold	VERB
ejpam-4582	108	22	:	:	PUNCT
ejpam-4582	108	23	(	(	PUNCT
ejpam-4582	108	24	1	1	X
ejpam-4582	108	25	)	)	PUNCT
ejpam-4582	108	26	scl(a	scl(a	PROPN
ejpam-4582	108	27	)	)	PUNCT
ejpam-4582	108	28	=	=	NOUN
ejpam-4582	108	29	a	a	PRON
ejpam-4582	108	30	∪	∪	X
ejpam-4582	108	31	int(cl(a	int(cl(a	PROPN
ejpam-4582	108	32	)	)	PUNCT
ejpam-4582	108	33	)	)	PUNCT
ejpam-4582	108	34	;	;	PUNCT
ejpam-4582	108	35	(	(	PUNCT
ejpam-4582	108	36	2	2	X
ejpam-4582	108	37	)	)	PUNCT
ejpam-4582	108	38	sint(a	sint(a	NOUN
ejpam-4582	108	39	)	)	PUNCT
ejpam-4582	108	40	=	=	PUNCT
ejpam-4582	109	1	a	a	DET
ejpam-4582	109	2	∩	∩	ADJ
ejpam-4582	109	3	cl(int(a	cl(int(a	NOUN
ejpam-4582	109	4	)	)	PUNCT
ejpam-4582	109	5	)	)	PUNCT
ejpam-4582	109	6	.	.	PUNCT
ejpam-4582	110	1	c.	c.	PROPN
ejpam-4582	110	2	boonpok	boonpok	PROPN
ejpam-4582	110	3	,	,	PUNCT
ejpam-4582	110	4	c.	c.	PROPN
ejpam-4582	110	5	viriyapong	viriyapong	PROPN
ejpam-4582	110	6	/	/	SYM
ejpam-4582	110	7	eur	eur	PROPN
ejpam-4582	110	8	.	.	PUNCT
ejpam-4582	111	1	j.	j.	PROPN
ejpam-4582	111	2	pure	pure	PROPN
ejpam-4582	111	3	appl	appl	PROPN
ejpam-4582	111	4	.	.	PROPN
ejpam-4582	111	5	math	math	PROPN
ejpam-4582	111	6	,	,	PUNCT
ejpam-4582	111	7	16	16	NUM
ejpam-4582	111	8	(	(	PUNCT
ejpam-4582	111	9	1	1	NUM
ejpam-4582	111	10	)	)	PUNCT
ejpam-4582	111	11	(	(	PUNCT
ejpam-4582	111	12	2023	2023	NUM
ejpam-4582	111	13	)	)	PUNCT
ejpam-4582	111	14	,	,	PUNCT
ejpam-4582	111	15	336	336	NUM
ejpam-4582	111	16	-	-	SYM
ejpam-4582	111	17	362	362	NUM
ejpam-4582	111	18	339	339	NUM
ejpam-4582	111	19	theorem	theorem	NOUN
ejpam-4582	111	20	1	1	NUM
ejpam-4582	111	21	.	.	X
ejpam-4582	112	1	for	for	ADP
ejpam-4582	112	2	a	a	DET
ejpam-4582	112	3	subset	subset	NOUN
ejpam-4582	112	4	a	a	PRON
ejpam-4582	112	5	of	of	ADP
ejpam-4582	112	6	a	a	DET
ejpam-4582	112	7	topological	topological	ADJ
ejpam-4582	112	8	space	space	NOUN
ejpam-4582	112	9	(	(	PUNCT
ejpam-4582	112	10	x	x	X
ejpam-4582	112	11	,	,	PUNCT
ejpam-4582	112	12	τ	τ	PROPN
ejpam-4582	112	13	)	)	PUNCT
ejpam-4582	112	14	,	,	PUNCT
ejpam-4582	112	15	the	the	DET
ejpam-4582	112	16	following	follow	VERB
ejpam-4582	112	17	properties	property	NOUN
ejpam-4582	112	18	are	be	AUX
ejpam-4582	112	19	equivalent	equivalent	ADJ
ejpam-4582	112	20	:	:	PUNCT
ejpam-4582	112	21	(	(	PUNCT
ejpam-4582	112	22	1	1	X
ejpam-4582	112	23	)	)	PUNCT
ejpam-4582	112	24	a	a	PRON
ejpam-4582	112	25	is	be	AUX
ejpam-4582	112	26	(	(	PUNCT
ejpam-4582	112	27	λ	λ	X
ejpam-4582	112	28	,	,	PUNCT
ejpam-4582	112	29	s)-closed	s)-close	VERB
ejpam-4582	112	30	;	;	PUNCT
ejpam-4582	112	31	(	(	PUNCT
ejpam-4582	112	32	2	2	X
ejpam-4582	112	33	)	)	PUNCT
ejpam-4582	112	34	a	a	DET
ejpam-4582	112	35	=	=	SYM
ejpam-4582	112	36	t	t	PROPN
ejpam-4582	112	37	∩	∩	X
ejpam-4582	112	38	scl(a	scl(a	PROPN
ejpam-4582	112	39	)	)	PUNCT
ejpam-4582	112	40	,	,	PUNCT
ejpam-4582	112	41	where	where	SCONJ
ejpam-4582	112	42	t	t	PROPN
ejpam-4582	112	43	is	be	AUX
ejpam-4582	112	44	a	a	DET
ejpam-4582	112	45	λs	λs	NOUN
ejpam-4582	112	46	-	-	PUNCT
ejpam-4582	112	47	set	set	NOUN
ejpam-4582	112	48	;	;	PUNCT
ejpam-4582	112	49	(	(	PUNCT
ejpam-4582	112	50	3	3	X
ejpam-4582	112	51	)	)	PUNCT
ejpam-4582	112	52	a	a	DET
ejpam-4582	112	53	=	=	PUNCT
ejpam-4582	112	54	aλs	aλs	NOUN
ejpam-4582	112	55	∩	∩	NOUN
ejpam-4582	112	56	scl(a	scl(a	PROPN
ejpam-4582	112	57	)	)	PUNCT
ejpam-4582	112	58	;	;	PUNCT
ejpam-4582	112	59	(	(	PUNCT
ejpam-4582	112	60	4	4	X
ejpam-4582	112	61	)	)	PUNCT
ejpam-4582	112	62	int(cl(a	int(cl(a	PROPN
ejpam-4582	112	63	)	)	PUNCT
ejpam-4582	112	64	)	)	PUNCT
ejpam-4582	113	1	∩aλs	∩aλ	VERB
ejpam-4582	113	2	⊆	⊆	NUM
ejpam-4582	113	3	a.	a.	NOUN
ejpam-4582	113	4	proof	proof	NOUN
ejpam-4582	113	5	.	.	PUNCT
ejpam-4582	114	1	(	(	PUNCT
ejpam-4582	114	2	1	1	X
ejpam-4582	114	3	)	)	PUNCT
ejpam-4582	114	4	⇒	⇒	NOUN
ejpam-4582	114	5	(	(	PUNCT
ejpam-4582	114	6	2	2	NUM
ejpam-4582	114	7	):	):	PUNCT
ejpam-4582	114	8	suppose	suppose	VERB
ejpam-4582	114	9	that	that	SCONJ
ejpam-4582	114	10	a	a	DET
ejpam-4582	114	11	=	=	X
ejpam-4582	114	12	t	t	NOUN
ejpam-4582	114	13	∩c	∩c	NOUN
ejpam-4582	114	14	,	,	PUNCT
ejpam-4582	114	15	where	where	SCONJ
ejpam-4582	114	16	t	t	PROPN
ejpam-4582	114	17	is	be	AUX
ejpam-4582	114	18	a	a	DET
ejpam-4582	114	19	λs	λs	ADV
ejpam-4582	114	20	-	-	PUNCT
ejpam-4582	114	21	set	set	VERB
ejpam-4582	114	22	and	and	CCONJ
ejpam-4582	114	23	c	c	NOUN
ejpam-4582	114	24	is	be	AUX
ejpam-4582	114	25	a	a	DET
ejpam-4582	114	26	semi	semi	ADJ
ejpam-4582	114	27	-	-	ADJ
ejpam-4582	114	28	closed	closed	ADJ
ejpam-4582	114	29	set	set	NOUN
ejpam-4582	114	30	.	.	PUNCT
ejpam-4582	115	1	since	since	SCONJ
ejpam-4582	115	2	a	a	DET
ejpam-4582	115	3	⊆	⊆	NUM
ejpam-4582	115	4	c	c	NOUN
ejpam-4582	115	5	,	,	PUNCT
ejpam-4582	115	6	we	we	PRON
ejpam-4582	115	7	have	have	VERB
ejpam-4582	115	8	scl(a	scl(a	PROPN
ejpam-4582	115	9	)	)	PUNCT
ejpam-4582	115	10	⊆	⊆	NUM
ejpam-4582	115	11	c	c	NOUN
ejpam-4582	115	12	and	and	CCONJ
ejpam-4582	115	13	hence	hence	ADV
ejpam-4582	115	14	a	a	DET
ejpam-4582	115	15	=	=	SYM
ejpam-4582	115	16	t	t	PROPN
ejpam-4582	115	17	∩	∩	PROPN
ejpam-4582	115	18	c	c	PROPN
ejpam-4582	115	19	⊇	⊇	PROPN
ejpam-4582	115	20	t	t	PROPN
ejpam-4582	115	21	∩	∩	ADJ
ejpam-4582	115	22	scl(a	scl(a	PROPN
ejpam-4582	115	23	)	)	PUNCT
ejpam-4582	115	24	⊇	⊇	PROPN
ejpam-4582	115	25	a.	a.	NOUN
ejpam-4582	115	26	consequently	consequently	ADV
ejpam-4582	115	27	,	,	PUNCT
ejpam-4582	115	28	we	we	PRON
ejpam-4582	115	29	obtain	obtain	VERB
ejpam-4582	115	30	a	a	DET
ejpam-4582	115	31	=	=	SYM
ejpam-4582	115	32	t	t	NOUN
ejpam-4582	115	33	∩	∩	X
ejpam-4582	115	34	scl(a	scl(a	PROPN
ejpam-4582	115	35	)	)	PUNCT
ejpam-4582	115	36	.	.	PUNCT
ejpam-4582	116	1	(	(	PUNCT
ejpam-4582	116	2	2	2	X
ejpam-4582	116	3	)	)	PUNCT
ejpam-4582	116	4	⇒	⇒	NOUN
ejpam-4582	116	5	(	(	PUNCT
ejpam-4582	116	6	3	3	NUM
ejpam-4582	116	7	):	):	PUNCT
ejpam-4582	116	8	suppose	suppose	VERB
ejpam-4582	116	9	that	that	SCONJ
ejpam-4582	116	10	a	a	DET
ejpam-4582	116	11	=	=	SYM
ejpam-4582	116	12	t	t	NOUN
ejpam-4582	116	13	∩	∩	X
ejpam-4582	116	14	scl(a	scl(a	PROPN
ejpam-4582	116	15	)	)	PUNCT
ejpam-4582	116	16	,	,	PUNCT
ejpam-4582	116	17	where	where	SCONJ
ejpam-4582	116	18	t	t	PROPN
ejpam-4582	116	19	is	be	AUX
ejpam-4582	116	20	a	a	DET
ejpam-4582	116	21	λs	λs	NOUN
ejpam-4582	116	22	-	-	PUNCT
ejpam-4582	116	23	set	set	NOUN
ejpam-4582	116	24	.	.	PUNCT
ejpam-4582	117	1	since	since	SCONJ
ejpam-4582	117	2	a	a	DET
ejpam-4582	117	3	⊆	⊆	NUM
ejpam-4582	117	4	t	t	NOUN
ejpam-4582	117	5	,	,	PUNCT
ejpam-4582	117	6	we	we	PRON
ejpam-4582	117	7	have	have	VERB
ejpam-4582	117	8	aλs	aλs	NOUN
ejpam-4582	117	9	⊆	⊆	NUM
ejpam-4582	117	10	tλs	tλs	NOUN
ejpam-4582	117	11	=	=	SYM
ejpam-4582	117	12	t	t	PROPN
ejpam-4582	117	13	and	and	CCONJ
ejpam-4582	117	14	hence	hence	ADV
ejpam-4582	117	15	a	a	DET
ejpam-4582	117	16	⊆	⊆	NUM
ejpam-4582	117	17	aλs	aλs	NOUN
ejpam-4582	117	18	∩scl(a	∩scl(a	NUM
ejpam-4582	117	19	)	)	PUNCT
ejpam-4582	117	20	⊆	⊆	NUM
ejpam-4582	117	21	t	t	NOUN
ejpam-4582	117	22	∩scl(a	∩scl(a	NUM
ejpam-4582	117	23	)	)	PUNCT
ejpam-4582	117	24	=	=	SYM
ejpam-4582	117	25	a.	a.	NOUN
ejpam-4582	117	26	thus	thus	ADV
ejpam-4582	117	27	,	,	PUNCT
ejpam-4582	117	28	a	a	DET
ejpam-4582	117	29	=	=	PUNCT
ejpam-4582	117	30	aλs	aλs	NOUN
ejpam-4582	117	31	∩scl(a	∩scl(a	NUM
ejpam-4582	117	32	)	)	PUNCT
ejpam-4582	117	33	.	.	PUNCT
ejpam-4582	118	1	(	(	PUNCT
ejpam-4582	118	2	3	3	X
ejpam-4582	118	3	)	)	PUNCT
ejpam-4582	118	4	⇒	⇒	NOUN
ejpam-4582	118	5	(	(	PUNCT
ejpam-4582	118	6	4	4	NUM
ejpam-4582	118	7	):	):	PUNCT
ejpam-4582	118	8	let	let	VERB
ejpam-4582	118	9	a	a	DET
ejpam-4582	118	10	=	=	PUNCT
ejpam-4582	118	11	aλs	aλs	NOUN
ejpam-4582	118	12	∩	∩	NOUN
ejpam-4582	118	13	scl(a	scl(a	PROPN
ejpam-4582	118	14	)	)	PUNCT
ejpam-4582	118	15	.	.	PUNCT
ejpam-4582	119	1	thus	thus	ADV
ejpam-4582	119	2	,	,	PUNCT
ejpam-4582	119	3	by	by	ADP
ejpam-4582	119	4	lemma	lemma	PROPN
ejpam-4582	119	5	3	3	NUM
ejpam-4582	119	6	,	,	PUNCT
ejpam-4582	119	7	a	a	DET
ejpam-4582	119	8	=	=	PUNCT
ejpam-4582	119	9	aλs	aλs	NOUN
ejpam-4582	119	10	∩	∩	NOUN
ejpam-4582	119	11	[	[	X
ejpam-4582	119	12	a	a	DET
ejpam-4582	119	13	∪	∪	X
ejpam-4582	119	14	int(cl(a	int(cl(a	PROPN
ejpam-4582	119	15	)	)	PUNCT
ejpam-4582	119	16	)	)	PUNCT
ejpam-4582	119	17	]	]	PUNCT
ejpam-4582	120	1	=	=	PUNCT
ejpam-4582	120	2	(	(	PUNCT
ejpam-4582	120	3	aλs	aλs	PROPN
ejpam-4582	120	4	∩a	∩a	PROPN
ejpam-4582	120	5	)	)	PUNCT
ejpam-4582	120	6	∪	∪	ADP
ejpam-4582	120	7	[	[	X
ejpam-4582	120	8	aλs	aλs	NOUN
ejpam-4582	120	9	∩	∩	ADJ
ejpam-4582	120	10	int(cl(a	int(cl(a	PROPN
ejpam-4582	120	11	)	)	PUNCT
ejpam-4582	120	12	)	)	PUNCT
ejpam-4582	120	13	]	]	PUNCT
ejpam-4582	121	1	=	=	PUNCT
ejpam-4582	121	2	a	a	DET
ejpam-4582	121	3	∪	∪	ADJ
ejpam-4582	121	4	[	[	X
ejpam-4582	121	5	aλs	aλs	NOUN
ejpam-4582	121	6	∩	∩	ADJ
ejpam-4582	121	7	int(cl(a	int(cl(a	PROPN
ejpam-4582	121	8	)	)	PUNCT
ejpam-4582	121	9	)	)	PUNCT
ejpam-4582	121	10	]	]	PUNCT
ejpam-4582	121	11	and	and	CCONJ
ejpam-4582	121	12	hence	hence	ADV
ejpam-4582	121	13	int(cl(a	int(cl(a	PROPN
ejpam-4582	121	14	)	)	PUNCT
ejpam-4582	121	15	)	)	PUNCT
ejpam-4582	121	16	∩aλs	∩aλ	VERB
ejpam-4582	121	17	⊆	⊆	NUM
ejpam-4582	121	18	a.	a.	NOUN
ejpam-4582	121	19	(	(	PUNCT
ejpam-4582	121	20	4	4	NUM
ejpam-4582	121	21	)	)	PUNCT
ejpam-4582	121	22	⇒	⇒	NOUN
ejpam-4582	121	23	(	(	PUNCT
ejpam-4582	121	24	1	1	NUM
ejpam-4582	121	25	):	):	PUNCT
ejpam-4582	121	26	let	let	VERB
ejpam-4582	121	27	int(cl(a	int(cl(a	PROPN
ejpam-4582	121	28	)	)	PUNCT
ejpam-4582	121	29	)	)	PUNCT
ejpam-4582	121	30	∩aλs	∩aλ	VERB
ejpam-4582	121	31	⊆	⊆	NUM
ejpam-4582	121	32	a.	a.	NOUN
ejpam-4582	121	33	then	then	ADV
ejpam-4582	121	34	,	,	PUNCT
ejpam-4582	121	35	we	we	PRON
ejpam-4582	121	36	have	have	VERB
ejpam-4582	121	37	a	a	DET
ejpam-4582	121	38	∪	∪	ADJ
ejpam-4582	121	39	[	[	X
ejpam-4582	121	40	aλs	aλs	NOUN
ejpam-4582	121	41	∩	∩	ADJ
ejpam-4582	121	42	int(cl(a	int(cl(a	PROPN
ejpam-4582	121	43	)	)	PUNCT
ejpam-4582	121	44	)	)	PUNCT
ejpam-4582	121	45	]	]	PUNCT
ejpam-4582	122	1	=	=	PUNCT
ejpam-4582	122	2	a	a	PRON
ejpam-4582	122	3	and	and	CCONJ
ejpam-4582	122	4	by	by	ADP
ejpam-4582	122	5	lemma	lemma	PROPN
ejpam-4582	122	6	3	3	NUM
ejpam-4582	122	7	,	,	PUNCT
ejpam-4582	122	8	a	a	PRON
ejpam-4582	122	9	=	=	X
ejpam-4582	122	10	(	(	PUNCT
ejpam-4582	122	11	a	a	DET
ejpam-4582	122	12	∪	∪	ADJ
ejpam-4582	122	13	aλs	aλs	NOUN
ejpam-4582	122	14	)	)	PUNCT
ejpam-4582	122	15	∩	∩	NOUN
ejpam-4582	122	16	[	[	X
ejpam-4582	122	17	a	a	DET
ejpam-4582	122	18	∪	∪	X
ejpam-4582	122	19	int(cl(a	int(cl(a	PROPN
ejpam-4582	122	20	)	)	PUNCT
ejpam-4582	122	21	)	)	PUNCT
ejpam-4582	122	22	]	]	PUNCT
ejpam-4582	123	1	=	=	PUNCT
ejpam-4582	123	2	aλs	aλs	PROPN
ejpam-4582	123	3	∩	∩	NOUN
ejpam-4582	123	4	scl(a	scl(a	PROPN
ejpam-4582	123	5	)	)	PUNCT
ejpam-4582	123	6	.	.	PUNCT
ejpam-4582	124	1	this	this	PRON
ejpam-4582	124	2	shows	show	VERB
ejpam-4582	124	3	that	that	SCONJ
ejpam-4582	124	4	a	a	PRON
ejpam-4582	124	5	is	be	AUX
ejpam-4582	124	6	(	(	PUNCT
ejpam-4582	124	7	λ	λ	X
ejpam-4582	124	8	,	,	PUNCT
ejpam-4582	124	9	s)-closed	s)-close	VERB
ejpam-4582	124	10	.	.	PUNCT
ejpam-4582	125	1	definition	definition	NOUN
ejpam-4582	125	2	3	3	NUM
ejpam-4582	125	3	.	.	PUNCT
ejpam-4582	126	1	a	a	DET
ejpam-4582	126	2	subset	subset	NOUN
ejpam-4582	126	3	a	a	PRON
ejpam-4582	126	4	of	of	ADP
ejpam-4582	126	5	a	a	DET
ejpam-4582	126	6	topological	topological	ADJ
ejpam-4582	126	7	space	space	NOUN
ejpam-4582	126	8	(	(	PUNCT
ejpam-4582	126	9	x	x	X
ejpam-4582	126	10	,	,	PUNCT
ejpam-4582	126	11	τ	τ	X
ejpam-4582	126	12	)	)	PUNCT
ejpam-4582	126	13	is	be	AUX
ejpam-4582	126	14	said	say	VERB
ejpam-4582	126	15	to	to	PART
ejpam-4582	126	16	be	be	AUX
ejpam-4582	126	17	(	(	PUNCT
ejpam-4582	126	18	λ	λ	X
ejpam-4582	126	19	,	,	PUNCT
ejpam-4582	126	20	s)-open	s)-open	VERB
ejpam-4582	126	21	if	if	SCONJ
ejpam-4582	126	22	the	the	DET
ejpam-4582	126	23	complement	complement	NOUN
ejpam-4582	126	24	of	of	ADP
ejpam-4582	126	25	a	a	DET
ejpam-4582	126	26	is	is	NOUN
ejpam-4582	126	27	(	(	PUNCT
ejpam-4582	126	28	λ	λ	X
ejpam-4582	126	29	,	,	PUNCT
ejpam-4582	126	30	s)-closed	s)-close	VERB
ejpam-4582	126	31	.	.	PUNCT
ejpam-4582	127	1	the	the	DET
ejpam-4582	127	2	family	family	NOUN
ejpam-4582	127	3	of	of	ADP
ejpam-4582	127	4	all	all	DET
ejpam-4582	127	5	(	(	PUNCT
ejpam-4582	127	6	λ	λ	X
ejpam-4582	127	7	,	,	PUNCT
ejpam-4582	127	8	s)-open	s)-open	PUNCT
ejpam-4582	127	9	sets	set	NOUN
ejpam-4582	127	10	in	in	ADP
ejpam-4582	127	11	a	a	DET
ejpam-4582	127	12	topological	topological	ADJ
ejpam-4582	127	13	space	space	NOUN
ejpam-4582	127	14	(	(	PUNCT
ejpam-4582	127	15	x	x	X
ejpam-4582	127	16	,	,	PUNCT
ejpam-4582	127	17	τ	τ	X
ejpam-4582	127	18	)	)	PUNCT
ejpam-4582	127	19	is	be	AUX
ejpam-4582	127	20	denoted	denote	VERB
ejpam-4582	127	21	by	by	ADP
ejpam-4582	127	22	(	(	PUNCT
ejpam-4582	127	23	λ	λ	NOUN
ejpam-4582	127	24	,	,	PUNCT
ejpam-4582	127	25	s)o(x	s)o(x	NOUN
ejpam-4582	127	26	)	)	PUNCT
ejpam-4582	127	27	.	.	PUNCT
ejpam-4582	128	1	proposition	proposition	NOUN
ejpam-4582	128	2	1	1	NUM
ejpam-4582	128	3	.	.	PUNCT
ejpam-4582	129	1	let	let	VERB
ejpam-4582	129	2	aγ(γ	aγ(γ	NUM
ejpam-4582	129	3	∈	∈	PROPN
ejpam-4582	129	4	∇	∇	X
ejpam-4582	129	5	)	)	PUNCT
ejpam-4582	129	6	be	be	AUX
ejpam-4582	129	7	a	a	DET
ejpam-4582	129	8	subset	subset	NOUN
ejpam-4582	129	9	of	of	ADP
ejpam-4582	129	10	a	a	DET
ejpam-4582	129	11	topological	topological	ADJ
ejpam-4582	129	12	space	space	NOUN
ejpam-4582	129	13	(	(	PUNCT
ejpam-4582	129	14	x	x	X
ejpam-4582	129	15	,	,	PUNCT
ejpam-4582	129	16	τ	τ	PROPN
ejpam-4582	129	17	)	)	PUNCT
ejpam-4582	129	18	.	.	PUNCT
ejpam-4582	130	1	then	then	ADV
ejpam-4582	130	2	,	,	PUNCT
ejpam-4582	130	3	the	the	DET
ejpam-4582	130	4	following	follow	VERB
ejpam-4582	130	5	properties	property	NOUN
ejpam-4582	130	6	hold	hold	VERB
ejpam-4582	130	7	:	:	PUNCT
ejpam-4582	130	8	(	(	PUNCT
ejpam-4582	130	9	1	1	X
ejpam-4582	130	10	)	)	PUNCT
ejpam-4582	130	11	if	if	SCONJ
ejpam-4582	130	12	aγ	aγ	PRON
ejpam-4582	130	13	is	be	AUX
ejpam-4582	130	14	(	(	PUNCT
ejpam-4582	130	15	λ	λ	X
ejpam-4582	130	16	,	,	PUNCT
ejpam-4582	130	17	s)-closed	s)-close	VERB
ejpam-4582	130	18	for	for	ADP
ejpam-4582	130	19	each	each	DET
ejpam-4582	130	20	γ	γ	PROPN
ejpam-4582	130	21	∈	∈	PROPN
ejpam-4582	130	22	∇	∇	X
ejpam-4582	130	23	,	,	PUNCT
ejpam-4582	130	24	then	then	ADV
ejpam-4582	130	25	∩{aγ	∩{aγ	VERB
ejpam-4582	130	26	|	|	ADV
ejpam-4582	130	27	γ	γ	X
ejpam-4582	130	28	∈	∈	PROPN
ejpam-4582	130	29	∇	∇	X
ejpam-4582	130	30	}	}	PUNCT
ejpam-4582	130	31	is	be	AUX
ejpam-4582	130	32	(	(	PUNCT
ejpam-4582	130	33	λ	λ	X
ejpam-4582	130	34	,	,	PUNCT
ejpam-4582	130	35	s)-closed	s)-close	VERB
ejpam-4582	130	36	.	.	PUNCT
ejpam-4582	131	1	(	(	PUNCT
ejpam-4582	131	2	2	2	X
ejpam-4582	131	3	)	)	PUNCT
ejpam-4582	131	4	if	if	SCONJ
ejpam-4582	131	5	aγ	aγ	PRON
ejpam-4582	131	6	is	be	AUX
ejpam-4582	131	7	(	(	PUNCT
ejpam-4582	131	8	λ	λ	X
ejpam-4582	131	9	,	,	PUNCT
ejpam-4582	131	10	s)-open	s)-open	VERB
ejpam-4582	131	11	for	for	ADP
ejpam-4582	131	12	each	each	DET
ejpam-4582	131	13	γ	γ	PROPN
ejpam-4582	131	14	∈	∈	PROPN
ejpam-4582	131	15	∇	∇	X
ejpam-4582	131	16	,	,	PUNCT
ejpam-4582	131	17	then	then	ADV
ejpam-4582	131	18	∪{aγ	∪{aγ	PROPN
ejpam-4582	131	19	|	|	ADV
ejpam-4582	131	20	γ	γ	PROPN
ejpam-4582	131	21	∈	∈	PROPN
ejpam-4582	131	22	∇	∇	X
ejpam-4582	131	23	}	}	PUNCT
ejpam-4582	131	24	is	be	AUX
ejpam-4582	131	25	(	(	PUNCT
ejpam-4582	131	26	λ	λ	X
ejpam-4582	131	27	,	,	PUNCT
ejpam-4582	131	28	s)-open	s)-open	PUNCT
ejpam-4582	131	29	.	.	PUNCT
ejpam-4582	132	1	proof	proof	NOUN
ejpam-4582	132	2	.	.	PUNCT
ejpam-4582	133	1	(	(	PUNCT
ejpam-4582	133	2	1	1	X
ejpam-4582	133	3	)	)	PUNCT
ejpam-4582	133	4	suppose	suppose	VERB
ejpam-4582	133	5	that	that	SCONJ
ejpam-4582	133	6	aγ	aγ	PRON
ejpam-4582	133	7	is	be	AUX
ejpam-4582	133	8	(	(	PUNCT
ejpam-4582	133	9	λ	λ	X
ejpam-4582	133	10	,	,	PUNCT
ejpam-4582	133	11	s)-closed	s)-close	VERB
ejpam-4582	133	12	for	for	ADP
ejpam-4582	133	13	each	each	DET
ejpam-4582	133	14	γ	γ	X
ejpam-4582	133	15	∈	∈	PROPN
ejpam-4582	133	16	∇.	∇.	NOUN
ejpam-4582	133	17	then	then	ADV
ejpam-4582	133	18	,	,	PUNCT
ejpam-4582	133	19	for	for	ADP
ejpam-4582	133	20	each	each	DET
ejpam-4582	133	21	γ	γ	NOUN
ejpam-4582	133	22	,	,	PUNCT
ejpam-4582	133	23	there	there	PRON
ejpam-4582	133	24	exist	exist	VERB
ejpam-4582	133	25	a	a	DET
ejpam-4582	133	26	λs	λs	ADV
ejpam-4582	133	27	-	-	PUNCT
ejpam-4582	133	28	set	set	VERB
ejpam-4582	133	29	tγ	tγ	NOUN
ejpam-4582	133	30	and	and	CCONJ
ejpam-4582	133	31	a	a	DET
ejpam-4582	133	32	semi	semi	ADJ
ejpam-4582	133	33	-	-	ADJ
ejpam-4582	133	34	closed	closed	ADJ
ejpam-4582	133	35	set	set	NOUN
ejpam-4582	133	36	cγ	cγ	NOUN
ejpam-4582	133	37	such	such	ADJ
ejpam-4582	133	38	that	that	SCONJ
ejpam-4582	133	39	aγ	aγ	PRON
ejpam-4582	133	40	=	=	SYM
ejpam-4582	133	41	tγ	tγ	PROPN
ejpam-4582	133	42	∩	∩	PROPN
ejpam-4582	133	43	cγ	cγ	NOUN
ejpam-4582	133	44	.	.	PUNCT
ejpam-4582	134	1	we	we	PRON
ejpam-4582	134	2	have	have	VERB
ejpam-4582	134	3	∩γ∈∇aγ	∩γ∈∇aγ	PROPN
ejpam-4582	134	4	=	=	SYM
ejpam-4582	134	5	∩γ∈∇(tγ	∩γ∈∇(tγ	NOUN
ejpam-4582	134	6	∩	∩	NOUN
ejpam-4582	134	7	cγ	cγ	NOUN
ejpam-4582	134	8	)	)	PUNCT
ejpam-4582	134	9	=	=	SYM
ejpam-4582	134	10	(	(	PUNCT
ejpam-4582	134	11	∩γ∈∇tγ	∩γ∈∇tγ	NOUN
ejpam-4582	134	12	)	)	PUNCT
ejpam-4582	134	13	∩	∩	NOUN
ejpam-4582	134	14	(	(	PUNCT
ejpam-4582	134	15	∩γ∈∇cγ	∩γ∈∇cγ	PROPN
ejpam-4582	134	16	)	)	PUNCT
ejpam-4582	134	17	.	.	PUNCT
ejpam-4582	135	1	by	by	ADP
ejpam-4582	135	2	lemma	lemma	PROPN
ejpam-4582	135	3	2	2	NUM
ejpam-4582	135	4	,	,	PUNCT
ejpam-4582	135	5	∩γ∈∇tγ	∩γ∈∇tγ	PROPN
ejpam-4582	135	6	is	be	AUX
ejpam-4582	135	7	a	a	DET
ejpam-4582	135	8	λs	λs	ADV
ejpam-4582	135	9	-	-	PUNCT
ejpam-4582	135	10	set	set	VERB
ejpam-4582	135	11	and	and	CCONJ
ejpam-4582	135	12	∩γ∈∇cγ	∩γ∈∇cγ	PROPN
ejpam-4582	135	13	is	be	AUX
ejpam-4582	135	14	a	a	DET
ejpam-4582	135	15	semi	semi	ADJ
ejpam-4582	135	16	-	-	ADJ
ejpam-4582	135	17	closed	closed	ADJ
ejpam-4582	135	18	set	set	NOUN
ejpam-4582	135	19	.	.	PUNCT
ejpam-4582	136	1	thus	thus	ADV
ejpam-4582	136	2	,	,	PUNCT
ejpam-4582	136	3	∩γ∈∇aγ	∩γ∈∇aγ	PROPN
ejpam-4582	136	4	is	be	AUX
ejpam-4582	136	5	(	(	PUNCT
ejpam-4582	136	6	λ	λ	X
ejpam-4582	136	7	,	,	PUNCT
ejpam-4582	136	8	s)-closed	s)-close	VERB
ejpam-4582	136	9	.	.	PUNCT
ejpam-4582	137	1	(	(	PUNCT
ejpam-4582	137	2	2	2	X
ejpam-4582	137	3	)	)	PUNCT
ejpam-4582	137	4	let	let	VERB
ejpam-4582	137	5	aγ	aγ	PRON
ejpam-4582	137	6	is	be	AUX
ejpam-4582	137	7	(	(	PUNCT
ejpam-4582	137	8	λ	λ	X
ejpam-4582	137	9	,	,	PUNCT
ejpam-4582	137	10	s)-open	s)-open	VERB
ejpam-4582	137	11	for	for	ADP
ejpam-4582	137	12	each	each	DET
ejpam-4582	137	13	γ	γ	X
ejpam-4582	137	14	∈	∈	PROPN
ejpam-4582	137	15	∇.	∇.	ADV
ejpam-4582	137	16	then	then	ADV
ejpam-4582	137	17	,	,	PUNCT
ejpam-4582	137	18	x−aγ	x−aγ	PROPN
ejpam-4582	137	19	is	be	AUX
ejpam-4582	137	20	(	(	PUNCT
ejpam-4582	137	21	λ	λ	X
ejpam-4582	137	22	,	,	PUNCT
ejpam-4582	137	23	s)-closed	s)-close	VERB
ejpam-4582	137	24	for	for	ADP
ejpam-4582	137	25	each	each	DET
ejpam-4582	137	26	γ	γ	X
ejpam-4582	137	27	∈	∈	NOUN
ejpam-4582	137	28	∇.	∇.	ADV
ejpam-4582	137	29	thus	thus	ADV
ejpam-4582	137	30	,	,	PUNCT
ejpam-4582	137	31	by	by	ADP
ejpam-4582	137	32	(	(	PUNCT
ejpam-4582	137	33	1	1	NUM
ejpam-4582	137	34	)	)	PUNCT
ejpam-4582	137	35	,	,	PUNCT
ejpam-4582	137	36	we	we	PRON
ejpam-4582	137	37	have	have	AUX
ejpam-4582	137	38	x	x	X
ejpam-4582	137	39	−	−	VERB
ejpam-4582	137	40	∪γ∈∇aγ	∪γ∈∇aγ	PROPN
ejpam-4582	137	41	=	=	PUNCT
ejpam-4582	137	42	∩γ∈∇(x	∩γ∈∇(x	PUNCT
ejpam-4582	137	43	−	−	X
ejpam-4582	137	44	aγ	aγ	NOUN
ejpam-4582	137	45	)	)	PUNCT
ejpam-4582	137	46	is	be	AUX
ejpam-4582	137	47	(	(	PUNCT
ejpam-4582	137	48	λ	λ	X
ejpam-4582	137	49	,	,	PUNCT
ejpam-4582	137	50	s)-closed	s)-close	VERB
ejpam-4582	137	51	and	and	CCONJ
ejpam-4582	137	52	hence	hence	ADV
ejpam-4582	137	53	∪γ∈∇aγ	∪γ∈∇aγ	PROPN
ejpam-4582	137	54	is	be	AUX
ejpam-4582	137	55	(	(	PUNCT
ejpam-4582	137	56	λ	λ	X
ejpam-4582	137	57	,	,	PUNCT
ejpam-4582	137	58	s)-open	s)-open	PUNCT
ejpam-4582	137	59	.	.	PUNCT
ejpam-4582	138	1	c.	c.	PROPN
ejpam-4582	138	2	boonpok	boonpok	PROPN
ejpam-4582	138	3	,	,	PUNCT
ejpam-4582	138	4	c.	c.	PROPN
ejpam-4582	138	5	viriyapong	viriyapong	PROPN
ejpam-4582	138	6	/	/	SYM
ejpam-4582	138	7	eur	eur	PROPN
ejpam-4582	138	8	.	.	PUNCT
ejpam-4582	139	1	j.	j.	PROPN
ejpam-4582	139	2	pure	pure	PROPN
ejpam-4582	139	3	appl	appl	PROPN
ejpam-4582	139	4	.	.	PROPN
ejpam-4582	139	5	math	math	PROPN
ejpam-4582	139	6	,	,	PUNCT
ejpam-4582	139	7	16	16	NUM
ejpam-4582	139	8	(	(	PUNCT
ejpam-4582	139	9	1	1	NUM
ejpam-4582	139	10	)	)	PUNCT
ejpam-4582	139	11	(	(	PUNCT
ejpam-4582	139	12	2023	2023	NUM
ejpam-4582	139	13	)	)	PUNCT
ejpam-4582	139	14	,	,	PUNCT
ejpam-4582	139	15	336	336	NUM
ejpam-4582	139	16	-	-	SYM
ejpam-4582	139	17	362	362	NUM
ejpam-4582	139	18	340	340	NUM
ejpam-4582	139	19	theorem	theorem	NOUN
ejpam-4582	139	20	2	2	NUM
ejpam-4582	139	21	.	.	X
ejpam-4582	140	1	for	for	ADP
ejpam-4582	140	2	a	a	DET
ejpam-4582	140	3	subset	subset	NOUN
ejpam-4582	140	4	a	a	PRON
ejpam-4582	140	5	of	of	ADP
ejpam-4582	140	6	a	a	DET
ejpam-4582	140	7	topological	topological	ADJ
ejpam-4582	140	8	space	space	NOUN
ejpam-4582	140	9	(	(	PUNCT
ejpam-4582	140	10	x	x	X
ejpam-4582	140	11	,	,	PUNCT
ejpam-4582	140	12	τ	τ	PROPN
ejpam-4582	140	13	)	)	PUNCT
ejpam-4582	140	14	,	,	PUNCT
ejpam-4582	140	15	the	the	DET
ejpam-4582	140	16	following	follow	VERB
ejpam-4582	140	17	properties	property	NOUN
ejpam-4582	140	18	are	be	AUX
ejpam-4582	140	19	equivalent	equivalent	ADJ
ejpam-4582	140	20	:	:	PUNCT
ejpam-4582	140	21	(	(	PUNCT
ejpam-4582	140	22	1	1	X
ejpam-4582	140	23	)	)	PUNCT
ejpam-4582	140	24	a	a	PRON
ejpam-4582	140	25	is	be	AUX
ejpam-4582	140	26	(	(	PUNCT
ejpam-4582	140	27	λ	λ	X
ejpam-4582	140	28	,	,	PUNCT
ejpam-4582	140	29	s)-open	s)-open	PUNCT
ejpam-4582	140	30	;	;	PUNCT
ejpam-4582	140	31	(	(	PUNCT
ejpam-4582	140	32	2	2	X
ejpam-4582	140	33	)	)	PUNCT
ejpam-4582	140	34	a	a	DET
ejpam-4582	140	35	=	=	SYM
ejpam-4582	140	36	t	t	PROPN
ejpam-4582	140	37	∪g	∪g	NUM
ejpam-4582	140	38	,	,	PUNCT
ejpam-4582	140	39	where	where	SCONJ
ejpam-4582	140	40	t	t	PROPN
ejpam-4582	140	41	is	be	AUX
ejpam-4582	140	42	a	a	DET
ejpam-4582	140	43	v	v	NOUN
ejpam-4582	140	44	s	s	NOUN
ejpam-4582	140	45	-	-	PUNCT
ejpam-4582	140	46	set	set	VERB
ejpam-4582	140	47	and	and	CCONJ
ejpam-4582	140	48	u	u	NOUN
ejpam-4582	140	49	is	be	AUX
ejpam-4582	140	50	a	a	DET
ejpam-4582	140	51	semi	semi	ADJ
ejpam-4582	140	52	-	-	ADJ
ejpam-4582	140	53	open	open	ADJ
ejpam-4582	140	54	set	set	NOUN
ejpam-4582	140	55	;	;	PUNCT
ejpam-4582	140	56	(	(	PUNCT
ejpam-4582	140	57	3	3	X
ejpam-4582	140	58	)	)	PUNCT
ejpam-4582	140	59	a	a	DET
ejpam-4582	140	60	=	=	X
ejpam-4582	140	61	t	t	PROPN
ejpam-4582	140	62	∪	∪	ADJ
ejpam-4582	140	63	sint(a	sint(a	PROPN
ejpam-4582	140	64	)	)	PUNCT
ejpam-4582	140	65	,	,	PUNCT
ejpam-4582	140	66	where	where	SCONJ
ejpam-4582	140	67	t	t	PROPN
ejpam-4582	140	68	is	be	AUX
ejpam-4582	140	69	a	a	DET
ejpam-4582	140	70	v	v	NOUN
ejpam-4582	140	71	s	s	NOUN
ejpam-4582	140	72	-	-	PUNCT
ejpam-4582	140	73	set	set	ADJ
ejpam-4582	140	74	;	;	PUNCT
ejpam-4582	140	75	(	(	PUNCT
ejpam-4582	140	76	4	4	X
ejpam-4582	140	77	)	)	PUNCT
ejpam-4582	140	78	a	a	DET
ejpam-4582	140	79	=	=	X
ejpam-4582	140	80	av	av	PROPN
ejpam-4582	140	81	s	s	NOUN
ejpam-4582	140	82	∪	∪	ADJ
ejpam-4582	140	83	sint(a	sint(a	NOUN
ejpam-4582	140	84	)	)	PUNCT
ejpam-4582	140	85	;	;	PUNCT
ejpam-4582	140	86	(	(	PUNCT
ejpam-4582	140	87	5	5	X
ejpam-4582	140	88	)	)	PUNCT
ejpam-4582	140	89	a	a	DET
ejpam-4582	140	90	⊆	⊆	NUM
ejpam-4582	140	91	cl(int(a	cl(int(a	NOUN
ejpam-4582	140	92	)	)	PUNCT
ejpam-4582	140	93	)	)	PUNCT
ejpam-4582	141	1	∪avs	∪avs	ADV
ejpam-4582	141	2	.	.	PUNCT
ejpam-4582	142	1	proof	proof	NOUN
ejpam-4582	142	2	.	.	PUNCT
ejpam-4582	143	1	(	(	PUNCT
ejpam-4582	143	2	1	1	X
ejpam-4582	143	3	)	)	PUNCT
ejpam-4582	143	4	⇒	⇒	NOUN
ejpam-4582	143	5	(	(	PUNCT
ejpam-4582	143	6	2	2	NUM
ejpam-4582	143	7	):	):	PUNCT
ejpam-4582	143	8	suppose	suppose	VERB
ejpam-4582	143	9	that	that	SCONJ
ejpam-4582	143	10	a	a	PRON
ejpam-4582	143	11	is	be	AUX
ejpam-4582	143	12	(	(	PUNCT
ejpam-4582	143	13	λ	λ	X
ejpam-4582	143	14	,	,	PUNCT
ejpam-4582	143	15	s)-open	s)-open	VERB
ejpam-4582	143	16	.	.	PUNCT
ejpam-4582	144	1	then	then	ADV
ejpam-4582	144	2	,	,	PUNCT
ejpam-4582	144	3	x	x	PUNCT
ejpam-4582	144	4	−	−	NOUN
ejpam-4582	144	5	a	a	PRON
ejpam-4582	144	6	is	be	AUX
ejpam-4582	144	7	(	(	PUNCT
ejpam-4582	144	8	λ	λ	X
ejpam-4582	144	9	,	,	PUNCT
ejpam-4582	144	10	s)-closed	s)-close	VERB
ejpam-4582	144	11	and	and	CCONJ
ejpam-4582	144	12	hence	hence	ADV
ejpam-4582	144	13	x	x	NOUN
ejpam-4582	144	14	−	−	NOUN
ejpam-4582	144	15	a	a	DET
ejpam-4582	144	16	=	=	SYM
ejpam-4582	144	17	t	t	PROPN
ejpam-4582	144	18	∩	∩	PROPN
ejpam-4582	144	19	f	f	PROPN
ejpam-4582	144	20	,	,	PUNCT
ejpam-4582	144	21	where	where	SCONJ
ejpam-4582	144	22	t	t	PROPN
ejpam-4582	144	23	is	be	AUX
ejpam-4582	144	24	a	a	DET
ejpam-4582	144	25	λs	λs	ADV
ejpam-4582	144	26	-	-	PUNCT
ejpam-4582	144	27	set	set	VERB
ejpam-4582	144	28	and	and	CCONJ
ejpam-4582	144	29	f	f	PROPN
ejpam-4582	144	30	is	be	AUX
ejpam-4582	144	31	a	a	DET
ejpam-4582	144	32	semi	semi	ADJ
ejpam-4582	144	33	-	-	ADJ
ejpam-4582	144	34	closed	closed	ADJ
ejpam-4582	144	35	set	set	NOUN
ejpam-4582	144	36	.	.	PUNCT
ejpam-4582	145	1	thus	thus	ADV
ejpam-4582	145	2	,	,	PUNCT
ejpam-4582	145	3	we	we	PRON
ejpam-4582	145	4	have	have	VERB
ejpam-4582	145	5	a	a	DET
ejpam-4582	145	6	=	=	SYM
ejpam-4582	145	7	(	(	PUNCT
ejpam-4582	145	8	x	x	NOUN
ejpam-4582	145	9	−a	−a	ADJ
ejpam-4582	145	10	)	)	PUNCT
ejpam-4582	145	11	∪	∪	NOUN
ejpam-4582	145	12	(	(	PUNCT
ejpam-4582	145	13	x	x	SYM
ejpam-4582	145	14	−	−	PROPN
ejpam-4582	145	15	f	f	PROPN
ejpam-4582	145	16	)	)	PUNCT
ejpam-4582	145	17	,	,	PUNCT
ejpam-4582	145	18	where	where	SCONJ
ejpam-4582	145	19	x	x	PUNCT
ejpam-4582	145	20	−	−	PROPN
ejpam-4582	145	21	t	t	PROPN
ejpam-4582	145	22	is	be	AUX
ejpam-4582	145	23	a	a	DET
ejpam-4582	145	24	vs	vs	NOUN
ejpam-4582	145	25	-	-	PUNCT
ejpam-4582	145	26	set	set	VERB
ejpam-4582	145	27	and	and	CCONJ
ejpam-4582	145	28	x	x	SYM
ejpam-4582	145	29	−	−	PROPN
ejpam-4582	145	30	f	f	PROPN
ejpam-4582	145	31	is	be	AUX
ejpam-4582	145	32	a	a	DET
ejpam-4582	145	33	semi	semi	ADJ
ejpam-4582	145	34	-	-	ADJ
ejpam-4582	145	35	open	open	ADJ
ejpam-4582	145	36	set	set	NOUN
ejpam-4582	145	37	.	.	PUNCT
ejpam-4582	146	1	(	(	PUNCT
ejpam-4582	146	2	2	2	X
ejpam-4582	146	3	)	)	PUNCT
ejpam-4582	146	4	⇒	⇒	NOUN
ejpam-4582	146	5	(	(	PUNCT
ejpam-4582	146	6	3	3	NUM
ejpam-4582	146	7	):	):	PUNCT
ejpam-4582	146	8	suppose	suppose	VERB
ejpam-4582	146	9	that	that	SCONJ
ejpam-4582	146	10	a	a	DET
ejpam-4582	146	11	=	=	NOUN
ejpam-4582	146	12	t∪u	t∪u	NOUN
ejpam-4582	146	13	,	,	PUNCT
ejpam-4582	146	14	where	where	SCONJ
ejpam-4582	146	15	t	t	PROPN
ejpam-4582	146	16	is	be	AUX
ejpam-4582	146	17	a	a	DET
ejpam-4582	146	18	v	v	NOUN
ejpam-4582	146	19	s	s	NOUN
ejpam-4582	146	20	-	-	PUNCT
ejpam-4582	146	21	set	set	VERB
ejpam-4582	146	22	and	and	CCONJ
ejpam-4582	146	23	u	u	NOUN
ejpam-4582	146	24	is	be	AUX
ejpam-4582	146	25	a	a	DET
ejpam-4582	146	26	semi	semi	ADJ
ejpam-4582	146	27	-	-	ADJ
ejpam-4582	146	28	open	open	ADJ
ejpam-4582	146	29	set	set	NOUN
ejpam-4582	146	30	.	.	PUNCT
ejpam-4582	147	1	since	since	SCONJ
ejpam-4582	147	2	u	u	PRON
ejpam-4582	147	3	⊆	⊆	NUM
ejpam-4582	147	4	a	a	PRON
ejpam-4582	147	5	and	and	CCONJ
ejpam-4582	147	6	u	u	NOUN
ejpam-4582	147	7	is	be	AUX
ejpam-4582	147	8	semi	semi	ADJ
ejpam-4582	147	9	-	-	ADJ
ejpam-4582	147	10	open	open	ADJ
ejpam-4582	147	11	,	,	PUNCT
ejpam-4582	147	12	we	we	PRON
ejpam-4582	147	13	have	have	VERB
ejpam-4582	147	14	u	u	NOUN
ejpam-4582	147	15	⊆	⊆	NUM
ejpam-4582	147	16	sint(a	sint(a	NOUN
ejpam-4582	147	17	)	)	PUNCT
ejpam-4582	147	18	and	and	CCONJ
ejpam-4582	147	19	hence	hence	ADV
ejpam-4582	147	20	a	a	DET
ejpam-4582	147	21	=	=	X
ejpam-4582	147	22	t	t	PROPN
ejpam-4582	147	23	∪u	∪u	NUM
ejpam-4582	147	24	⊆	⊆	NUM
ejpam-4582	147	25	t	t	NOUN
ejpam-4582	147	26	∪sint(a	∪sint(a	NUM
ejpam-4582	147	27	)	)	PUNCT
ejpam-4582	147	28	⊆	⊆	NUM
ejpam-4582	147	29	a.	a.	NOUN
ejpam-4582	147	30	thus	thus	ADV
ejpam-4582	147	31	,	,	PUNCT
ejpam-4582	147	32	a	a	DET
ejpam-4582	147	33	=	=	X
ejpam-4582	147	34	t	t	PROPN
ejpam-4582	147	35	∪	∪	ADJ
ejpam-4582	147	36	sint(a	sint(a	PROPN
ejpam-4582	147	37	)	)	PUNCT
ejpam-4582	147	38	.	.	PUNCT
ejpam-4582	148	1	(	(	PUNCT
ejpam-4582	148	2	3	3	X
ejpam-4582	148	3	)	)	PUNCT
ejpam-4582	148	4	⇒	⇒	NOUN
ejpam-4582	148	5	(	(	PUNCT
ejpam-4582	148	6	4	4	NUM
ejpam-4582	148	7	):	):	PUNCT
ejpam-4582	148	8	suppose	suppose	VERB
ejpam-4582	148	9	that	that	SCONJ
ejpam-4582	148	10	a	a	DET
ejpam-4582	148	11	=	=	X
ejpam-4582	148	12	t	t	X
ejpam-4582	148	13	∪	∪	ADJ
ejpam-4582	148	14	sint(a	sint(a	PROPN
ejpam-4582	148	15	)	)	PUNCT
ejpam-4582	148	16	,	,	PUNCT
ejpam-4582	148	17	where	where	SCONJ
ejpam-4582	148	18	t	t	PROPN
ejpam-4582	148	19	is	be	AUX
ejpam-4582	148	20	a	a	DET
ejpam-4582	148	21	v	v	NOUN
ejpam-4582	148	22	s	s	NOUN
ejpam-4582	148	23	-	-	PUNCT
ejpam-4582	148	24	set	set	NOUN
ejpam-4582	148	25	.	.	PUNCT
ejpam-4582	149	1	since	since	SCONJ
ejpam-4582	149	2	t	t	PROPN
ejpam-4582	149	3	⊆	⊆	NUM
ejpam-4582	149	4	a	a	PRON
ejpam-4582	149	5	,	,	PUNCT
ejpam-4582	149	6	we	we	PRON
ejpam-4582	149	7	have	have	VERB
ejpam-4582	149	8	av	av	PROPN
ejpam-4582	149	9	s	s	PROPN
ejpam-4582	149	10	⊇	⊇	PROPN
ejpam-4582	149	11	t	t	PROPN
ejpam-4582	149	12	v	v	NOUN
ejpam-4582	149	13	s	s	X
ejpam-4582	149	14	and	and	CCONJ
ejpam-4582	149	15	hence	hence	ADV
ejpam-4582	149	16	a	a	DET
ejpam-4582	149	17	⊇	⊇	NOUN
ejpam-4582	149	18	av	av	PROPN
ejpam-4582	149	19	s	s	PART
ejpam-4582	149	20	∪sint(a	∪sint(a	NUM
ejpam-4582	149	21	)	)	PUNCT
ejpam-4582	150	1	⊇	⊇	PROPN
ejpam-4582	150	2	t	t	PROPN
ejpam-4582	150	3	v	v	NOUN
ejpam-4582	150	4	s	s	NOUN
ejpam-4582	150	5	∪sint(a	∪sint(a	NUM
ejpam-4582	150	6	)	)	PUNCT
ejpam-4582	151	1	=	=	SYM
ejpam-4582	151	2	t	t	PROPN
ejpam-4582	151	3	∪sint(a	∪sint(a	NUM
ejpam-4582	151	4	)	)	PUNCT
ejpam-4582	152	1	=	=	SYM
ejpam-4582	152	2	a.	a.	NOUN
ejpam-4582	152	3	this	this	PRON
ejpam-4582	152	4	shows	show	VERB
ejpam-4582	152	5	that	that	SCONJ
ejpam-4582	152	6	a	a	DET
ejpam-4582	152	7	=	=	X
ejpam-4582	152	8	av	av	X
ejpam-4582	152	9	s	s	NOUN
ejpam-4582	152	10	∪	∪	ADJ
ejpam-4582	152	11	sint(a	sint(a	NOUN
ejpam-4582	152	12	)	)	PUNCT
ejpam-4582	152	13	.	.	PUNCT
ejpam-4582	153	1	(	(	PUNCT
ejpam-4582	153	2	4	4	X
ejpam-4582	153	3	)	)	PUNCT
ejpam-4582	153	4	⇒	⇒	NOUN
ejpam-4582	153	5	(	(	PUNCT
ejpam-4582	153	6	5	5	NUM
ejpam-4582	153	7	):	):	PUNCT
ejpam-4582	153	8	let	let	VERB
ejpam-4582	153	9	a	a	DET
ejpam-4582	153	10	=	=	X
ejpam-4582	153	11	av	av	PROPN
ejpam-4582	153	12	s	s	NOUN
ejpam-4582	153	13	∪	∪	ADJ
ejpam-4582	153	14	sint(a	sint(a	NOUN
ejpam-4582	153	15	)	)	PUNCT
ejpam-4582	153	16	.	.	PUNCT
ejpam-4582	154	1	thus	thus	ADV
ejpam-4582	154	2	,	,	PUNCT
ejpam-4582	154	3	by	by	ADP
ejpam-4582	154	4	lemma	lemma	PROPN
ejpam-4582	154	5	3	3	NUM
ejpam-4582	154	6	,	,	PUNCT
ejpam-4582	154	7	a	a	DET
ejpam-4582	154	8	=	=	X
ejpam-4582	154	9	av	av	X
ejpam-4582	154	10	s	s	NOUN
ejpam-4582	154	11	∪	∪	ADJ
ejpam-4582	154	12	sint(a	sint(a	NOUN
ejpam-4582	154	13	)	)	PUNCT
ejpam-4582	154	14	=	=	SYM
ejpam-4582	154	15	av	av	PROPN
ejpam-4582	154	16	s	s	VERB
ejpam-4582	154	17	∪	∪	NOUN
ejpam-4582	154	18	[	[	PUNCT
ejpam-4582	154	19	a	a	DET
ejpam-4582	154	20	∩	∩	ADJ
ejpam-4582	154	21	cl(int(a	cl(int(a	NOUN
ejpam-4582	154	22	)	)	PUNCT
ejpam-4582	154	23	)	)	PUNCT
ejpam-4582	154	24	]	]	PUNCT
ejpam-4582	155	1	=	=	PUNCT
ejpam-4582	156	1	[	[	X
ejpam-4582	156	2	avs	avs	X
ejpam-4582	156	3	∪a	∪a	NUM
ejpam-4582	156	4	]	]	X
ejpam-4582	156	5	∩	∩	NOUN
ejpam-4582	156	6	[	[	X
ejpam-4582	156	7	cl(int(a	cl(int(a	NOUN
ejpam-4582	156	8	)	)	PUNCT
ejpam-4582	156	9	)	)	PUNCT
ejpam-4582	156	10	∪avs	∪avs	ADP
ejpam-4582	156	11	]	]	PUNCT
ejpam-4582	157	1	=	=	PUNCT
ejpam-4582	157	2	a	a	DET
ejpam-4582	157	3	∩	∩	NOUN
ejpam-4582	157	4	[	[	X
ejpam-4582	157	5	cl(int(a	cl(int(a	NOUN
ejpam-4582	157	6	)	)	PUNCT
ejpam-4582	157	7	)	)	PUNCT
ejpam-4582	157	8	∪avs	∪avs	ADP
ejpam-4582	157	9	]	]	PUNCT
ejpam-4582	157	10	and	and	CCONJ
ejpam-4582	157	11	hence	hence	ADV
ejpam-4582	157	12	a	a	DET
ejpam-4582	157	13	⊆	⊆	NUM
ejpam-4582	157	14	cl(int(a	cl(int(a	NOUN
ejpam-4582	157	15	)	)	PUNCT
ejpam-4582	157	16	)	)	PUNCT
ejpam-4582	157	17	∪avs	∪avs	ADV
ejpam-4582	157	18	.	.	PUNCT
ejpam-4582	158	1	(	(	PUNCT
ejpam-4582	158	2	5	5	X
ejpam-4582	158	3	)	)	PUNCT
ejpam-4582	158	4	⇒	⇒	NOUN
ejpam-4582	158	5	(	(	PUNCT
ejpam-4582	158	6	1	1	NUM
ejpam-4582	158	7	):	):	PUNCT
ejpam-4582	158	8	let	let	VERB
ejpam-4582	158	9	a	a	DET
ejpam-4582	158	10	⊆	⊆	NUM
ejpam-4582	158	11	cl(int(a	cl(int(a	NOUN
ejpam-4582	158	12	)	)	PUNCT
ejpam-4582	158	13	)	)	PUNCT
ejpam-4582	158	14	∪avs	∪avs	ADV
ejpam-4582	158	15	.	.	PUNCT
ejpam-4582	159	1	then	then	ADV
ejpam-4582	159	2	,	,	PUNCT
ejpam-4582	159	3	we	we	PRON
ejpam-4582	159	4	have	have	VERB
ejpam-4582	159	5	int(cl(x	int(cl(x	PROPN
ejpam-4582	159	6	−a	−a	NOUN
ejpam-4582	159	7	)	)	PUNCT
ejpam-4582	159	8	)	)	PUNCT
ejpam-4582	160	1	∩	∩	NOUN
ejpam-4582	161	1	[	[	X
ejpam-4582	161	2	x	x	X
ejpam-4582	161	3	−a]λs	−a]λ	NOUN
ejpam-4582	161	4	=	=	PUNCT
ejpam-4582	162	1	[	[	X
ejpam-4582	162	2	x	x	X
ejpam-4582	162	3	−	−	PROPN
ejpam-4582	162	4	cl(int(a	cl(int(a	PROPN
ejpam-4582	162	5	)	)	PUNCT
ejpam-4582	162	6	)	)	PUNCT
ejpam-4582	162	7	]	]	PUNCT
ejpam-4582	163	1	∩	∩	NOUN
ejpam-4582	164	1	[	[	X
ejpam-4582	164	2	x	x	X
ejpam-4582	164	3	−avs	−avs	X
ejpam-4582	164	4	]	]	PUNCT
ejpam-4582	165	1	=	=	PUNCT
ejpam-4582	165	2	x	x	X
ejpam-4582	165	3	−	−	PROPN
ejpam-4582	166	1	[	[	X
ejpam-4582	166	2	cl(int(a	cl(int(a	PROPN
ejpam-4582	166	3	)	)	PUNCT
ejpam-4582	166	4	)	)	PUNCT
ejpam-4582	167	1	∪avs	∪avs	ADP
ejpam-4582	167	2	]	]	PUNCT
ejpam-4582	168	1	⊆	⊆	NUM
ejpam-4582	168	2	x	x	SYM
ejpam-4582	168	3	−a	−a	NOUN
ejpam-4582	168	4	and	and	CCONJ
ejpam-4582	168	5	by	by	ADP
ejpam-4582	168	6	theorem	theorem	NOUN
ejpam-4582	168	7	1	1	NUM
ejpam-4582	168	8	,	,	PUNCT
ejpam-4582	168	9	x	x	PRON
ejpam-4582	168	10	−a	−a	NOUN
ejpam-4582	168	11	is	be	AUX
ejpam-4582	168	12	(	(	PUNCT
ejpam-4582	168	13	λ	λ	X
ejpam-4582	168	14	,	,	PUNCT
ejpam-4582	168	15	s)-closed	s)-close	VERB
ejpam-4582	168	16	.	.	PUNCT
ejpam-4582	169	1	thus	thus	ADV
ejpam-4582	169	2	,	,	PUNCT
ejpam-4582	169	3	a	a	PRON
ejpam-4582	169	4	is	be	AUX
ejpam-4582	169	5	(	(	PUNCT
ejpam-4582	169	6	λ	λ	X
ejpam-4582	169	7	,	,	PUNCT
ejpam-4582	169	8	s)-open	s)-open	PUNCT
ejpam-4582	169	9	.	.	PUNCT
ejpam-4582	170	1	definition	definition	NOUN
ejpam-4582	170	2	4	4	NUM
ejpam-4582	170	3	.	.	PUNCT
ejpam-4582	170	4	let	let	VERB
ejpam-4582	170	5	a	a	PRON
ejpam-4582	170	6	be	be	AUX
ejpam-4582	170	7	a	a	DET
ejpam-4582	170	8	subsets	subset	NOUN
ejpam-4582	170	9	of	of	ADP
ejpam-4582	170	10	a	a	DET
ejpam-4582	170	11	topological	topological	ADJ
ejpam-4582	170	12	space	space	NOUN
ejpam-4582	170	13	(	(	PUNCT
ejpam-4582	170	14	x	x	X
ejpam-4582	170	15	,	,	PUNCT
ejpam-4582	170	16	τ	τ	PROPN
ejpam-4582	170	17	)	)	PUNCT
ejpam-4582	170	18	.	.	PUNCT
ejpam-4582	171	1	a	a	DET
ejpam-4582	171	2	point	point	NOUN
ejpam-4582	171	3	x	x	X
ejpam-4582	171	4	∈	∈	NOUN
ejpam-4582	171	5	x	x	PUNCT
ejpam-4582	171	6	is	be	AUX
ejpam-4582	171	7	called	call	VERB
ejpam-4582	171	8	a	a	DET
ejpam-4582	171	9	(	(	PUNCT
ejpam-4582	171	10	λ	λ	NOUN
ejpam-4582	171	11	,	,	PUNCT
ejpam-4582	171	12	s)-cluster	s)-cluster	PUNCT
ejpam-4582	171	13	point	point	NOUN
ejpam-4582	171	14	of	of	ADP
ejpam-4582	171	15	a	a	DET
ejpam-4582	171	16	if	if	NOUN
ejpam-4582	171	17	for	for	ADP
ejpam-4582	171	18	every	every	DET
ejpam-4582	171	19	(	(	PUNCT
ejpam-4582	171	20	λ	λ	NOUN
ejpam-4582	171	21	,	,	PUNCT
ejpam-4582	171	22	s)-open	s)-open	VERB
ejpam-4582	171	23	set	set	VERB
ejpam-4582	171	24	u	u	NOUN
ejpam-4582	171	25	of	of	ADP
ejpam-4582	171	26	x	x	PUNCT
ejpam-4582	171	27	containing	contain	VERB
ejpam-4582	171	28	x	x	VERB
ejpam-4582	171	29	we	we	PRON
ejpam-4582	171	30	have	have	AUX
ejpam-4582	171	31	a∩u	a∩u	VERB
ejpam-4582	171	32	̸=	̸=	PROPN
ejpam-4582	171	33	∅.	∅.	ADP
ejpam-4582	171	34	the	the	DET
ejpam-4582	171	35	set	set	NOUN
ejpam-4582	171	36	of	of	ADP
ejpam-4582	171	37	all	all	DET
ejpam-4582	171	38	(	(	PUNCT
ejpam-4582	171	39	λ	λ	NOUN
ejpam-4582	171	40	,	,	PUNCT
ejpam-4582	172	1	s)-cluster	s)-cluster	PUNCT
ejpam-4582	172	2	points	point	NOUN
ejpam-4582	172	3	of	of	ADP
ejpam-4582	172	4	a	a	PRON
ejpam-4582	172	5	is	be	AUX
ejpam-4582	172	6	called	call	VERB
ejpam-4582	172	7	the	the	DET
ejpam-4582	172	8	(	(	PUNCT
ejpam-4582	172	9	λ	λ	PROPN
ejpam-4582	172	10	,	,	PUNCT
ejpam-4582	172	11	s)-closure	s)-closure	NOUN
ejpam-4582	172	12	of	of	ADP
ejpam-4582	172	13	a	a	PRON
ejpam-4582	172	14	and	and	CCONJ
ejpam-4582	172	15	is	be	AUX
ejpam-4582	172	16	denoted	denote	VERB
ejpam-4582	172	17	by	by	ADP
ejpam-4582	172	18	a(λ	a(λ	PROPN
ejpam-4582	172	19	,	,	PUNCT
ejpam-4582	172	20	s	s	PART
ejpam-4582	172	21	)	)	PUNCT
ejpam-4582	172	22	.	.	PUNCT
ejpam-4582	173	1	lemma	lemma	PROPN
ejpam-4582	173	2	4	4	NUM
ejpam-4582	173	3	.	.	X
ejpam-4582	174	1	for	for	ADP
ejpam-4582	174	2	subsets	subset	NOUN
ejpam-4582	174	3	a	a	PRON
ejpam-4582	174	4	and	and	CCONJ
ejpam-4582	174	5	b	b	NOUN
ejpam-4582	174	6	of	of	ADP
ejpam-4582	174	7	a	a	DET
ejpam-4582	174	8	topological	topological	ADJ
ejpam-4582	174	9	space	space	NOUN
ejpam-4582	174	10	(	(	PUNCT
ejpam-4582	174	11	x	x	X
ejpam-4582	174	12	,	,	PUNCT
ejpam-4582	174	13	τ	τ	PROPN
ejpam-4582	174	14	)	)	PUNCT
ejpam-4582	174	15	,	,	PUNCT
ejpam-4582	174	16	the	the	DET
ejpam-4582	174	17	following	follow	VERB
ejpam-4582	174	18	properties	property	NOUN
ejpam-4582	174	19	hold	hold	VERB
ejpam-4582	174	20	:	:	PUNCT
ejpam-4582	174	21	c.	c.	PROPN
ejpam-4582	174	22	boonpok	boonpok	PROPN
ejpam-4582	174	23	,	,	PUNCT
ejpam-4582	174	24	c.	c.	PROPN
ejpam-4582	174	25	viriyapong	viriyapong	PROPN
ejpam-4582	174	26	/	/	SYM
ejpam-4582	174	27	eur	eur	PROPN
ejpam-4582	174	28	.	.	PUNCT
ejpam-4582	175	1	j.	j.	PROPN
ejpam-4582	175	2	pure	pure	PROPN
ejpam-4582	175	3	appl	appl	PROPN
ejpam-4582	175	4	.	.	PROPN
ejpam-4582	175	5	math	math	PROPN
ejpam-4582	175	6	,	,	PUNCT
ejpam-4582	175	7	16	16	NUM
ejpam-4582	175	8	(	(	PUNCT
ejpam-4582	175	9	1	1	NUM
ejpam-4582	175	10	)	)	PUNCT
ejpam-4582	175	11	(	(	PUNCT
ejpam-4582	175	12	2023	2023	NUM
ejpam-4582	175	13	)	)	PUNCT
ejpam-4582	175	14	,	,	PUNCT
ejpam-4582	175	15	336	336	NUM
ejpam-4582	175	16	-	-	SYM
ejpam-4582	175	17	362	362	NUM
ejpam-4582	175	18	341	341	NUM
ejpam-4582	175	19	(	(	PUNCT
ejpam-4582	175	20	1	1	NUM
ejpam-4582	175	21	)	)	PUNCT
ejpam-4582	175	22	a	a	DET
ejpam-4582	175	23	⊆	⊆	NUM
ejpam-4582	175	24	a(λ	a(λ	ADJ
ejpam-4582	175	25	,	,	PUNCT
ejpam-4582	175	26	s	s	PART
ejpam-4582	175	27	)	)	PUNCT
ejpam-4582	175	28	and	and	CCONJ
ejpam-4582	175	29	[	[	X
ejpam-4582	175	30	a(λ	a(λ	ADV
ejpam-4582	175	31	,	,	PUNCT
ejpam-4582	175	32	s)](λ	s)](λ	PROPN
ejpam-4582	175	33	,	,	PUNCT
ejpam-4582	175	34	s	s	NOUN
ejpam-4582	175	35	)	)	PUNCT
ejpam-4582	175	36	=	=	PUNCT
ejpam-4582	175	37	a(λ	a(λ	PROPN
ejpam-4582	175	38	,	,	PUNCT
ejpam-4582	175	39	s	s	NOUN
ejpam-4582	175	40	)	)	PUNCT
ejpam-4582	175	41	.	.	PUNCT
ejpam-4582	176	1	(	(	PUNCT
ejpam-4582	176	2	2	2	X
ejpam-4582	176	3	)	)	PUNCT
ejpam-4582	176	4	if	if	SCONJ
ejpam-4582	176	5	a	a	DET
ejpam-4582	176	6	⊆	⊆	NUM
ejpam-4582	176	7	b	b	NOUN
ejpam-4582	176	8	,	,	PUNCT
ejpam-4582	176	9	then	then	ADV
ejpam-4582	176	10	a(λ	a(λ	ADV
ejpam-4582	176	11	,	,	PUNCT
ejpam-4582	176	12	s	s	X
ejpam-4582	176	13	)	)	PUNCT
ejpam-4582	176	14	⊆	⊆	NUM
ejpam-4582	176	15	b(λ	b(λ	NOUN
ejpam-4582	176	16	,	,	PUNCT
ejpam-4582	176	17	s	s	NOUN
ejpam-4582	176	18	)	)	PUNCT
ejpam-4582	176	19	.	.	PUNCT
ejpam-4582	177	1	(	(	PUNCT
ejpam-4582	177	2	3	3	X
ejpam-4582	177	3	)	)	PUNCT
ejpam-4582	177	4	a(λ	a(λ	ADV
ejpam-4582	177	5	,	,	PUNCT
ejpam-4582	177	6	s	s	PART
ejpam-4582	177	7	)	)	PUNCT
ejpam-4582	177	8	=	=	SYM
ejpam-4582	177	9	∩{f	∩{f	NOUN
ejpam-4582	177	10	|a	|a	VERB
ejpam-4582	177	11	⊆	⊆	NUM
ejpam-4582	177	12	f	f	PROPN
ejpam-4582	177	13	and	and	CCONJ
ejpam-4582	177	14	f	f	PROPN
ejpam-4582	177	15	is	be	AUX
ejpam-4582	177	16	(	(	PUNCT
ejpam-4582	177	17	λ	λ	X
ejpam-4582	177	18	,	,	PUNCT
ejpam-4582	177	19	s)-closed	s)-close	VERB
ejpam-4582	177	20	}	}	PUNCT
ejpam-4582	177	21	.	.	PUNCT
ejpam-4582	178	1	(	(	PUNCT
ejpam-4582	178	2	4	4	NUM
ejpam-4582	178	3	)	)	PUNCT
ejpam-4582	178	4	a(λ	a(λ	ADV
ejpam-4582	178	5	,	,	PUNCT
ejpam-4582	178	6	s	s	PART
ejpam-4582	178	7	)	)	PUNCT
ejpam-4582	178	8	is	be	AUX
ejpam-4582	178	9	(	(	PUNCT
ejpam-4582	178	10	λ	λ	X
ejpam-4582	178	11	,	,	PUNCT
ejpam-4582	178	12	s)-closed	s)-close	VERB
ejpam-4582	178	13	.	.	PUNCT
ejpam-4582	179	1	(	(	PUNCT
ejpam-4582	179	2	5	5	X
ejpam-4582	179	3	)	)	PUNCT
ejpam-4582	179	4	a	a	PRON
ejpam-4582	179	5	is	be	AUX
ejpam-4582	179	6	(	(	PUNCT
ejpam-4582	179	7	λ	λ	X
ejpam-4582	179	8	,	,	PUNCT
ejpam-4582	179	9	s)-closed	s)-close	VERB
ejpam-4582	179	10	if	if	SCONJ
ejpam-4582	179	11	and	and	CCONJ
ejpam-4582	179	12	only	only	ADV
ejpam-4582	179	13	if	if	SCONJ
ejpam-4582	179	14	a	a	PRON
ejpam-4582	179	15	=	=	X
ejpam-4582	179	16	a(λ	a(λ	PROPN
ejpam-4582	179	17	,	,	PUNCT
ejpam-4582	179	18	s	s	NOUN
ejpam-4582	179	19	)	)	PUNCT
ejpam-4582	179	20	.	.	PUNCT
ejpam-4582	180	1	proposition	proposition	NOUN
ejpam-4582	180	2	2	2	NUM
ejpam-4582	180	3	.	.	X
ejpam-4582	180	4	for	for	ADP
ejpam-4582	180	5	a	a	DET
ejpam-4582	180	6	subset	subset	NOUN
ejpam-4582	180	7	a	a	PRON
ejpam-4582	180	8	of	of	ADP
ejpam-4582	180	9	a	a	DET
ejpam-4582	180	10	topological	topological	ADJ
ejpam-4582	180	11	space	space	NOUN
ejpam-4582	180	12	(	(	PUNCT
ejpam-4582	180	13	x	x	X
ejpam-4582	180	14	,	,	PUNCT
ejpam-4582	180	15	τ	τ	PROPN
ejpam-4582	180	16	)	)	PUNCT
ejpam-4582	180	17	,	,	PUNCT
ejpam-4582	180	18	the	the	DET
ejpam-4582	180	19	following	follow	VERB
ejpam-4582	180	20	properties	property	NOUN
ejpam-4582	180	21	hold	hold	VERB
ejpam-4582	180	22	:	:	PUNCT
ejpam-4582	180	23	(	(	PUNCT
ejpam-4582	180	24	1	1	X
ejpam-4582	180	25	)	)	PUNCT
ejpam-4582	180	26	if	if	SCONJ
ejpam-4582	180	27	a	a	PRON
ejpam-4582	180	28	is	be	AUX
ejpam-4582	180	29	(	(	PUNCT
ejpam-4582	180	30	λ	λ	X
ejpam-4582	180	31	,	,	PUNCT
ejpam-4582	180	32	s)-closed	s)-close	VERB
ejpam-4582	180	33	,	,	PUNCT
ejpam-4582	180	34	then	then	ADV
ejpam-4582	180	35	a	a	DET
ejpam-4582	180	36	=	=	PUNCT
ejpam-4582	180	37	aλs	aλs	NOUN
ejpam-4582	180	38	∩a(λ	∩a(λ	NOUN
ejpam-4582	180	39	,	,	PUNCT
ejpam-4582	180	40	s	s	NOUN
ejpam-4582	180	41	)	)	PUNCT
ejpam-4582	180	42	.	.	PUNCT
ejpam-4582	181	1	(	(	PUNCT
ejpam-4582	181	2	2	2	X
ejpam-4582	181	3	)	)	PUNCT
ejpam-4582	181	4	if	if	SCONJ
ejpam-4582	181	5	a	a	PRON
ejpam-4582	181	6	is	be	AUX
ejpam-4582	181	7	semi	semi	ADJ
ejpam-4582	181	8	-	-	ADJ
ejpam-4582	181	9	closed	closed	ADJ
ejpam-4582	181	10	,	,	PUNCT
ejpam-4582	181	11	then	then	ADV
ejpam-4582	181	12	a	a	DET
ejpam-4582	181	13	is	be	AUX
ejpam-4582	181	14	(	(	PUNCT
ejpam-4582	181	15	λ	λ	X
ejpam-4582	181	16	,	,	PUNCT
ejpam-4582	181	17	s)-closed	s)-close	VERB
ejpam-4582	181	18	.	.	PUNCT
ejpam-4582	182	1	proof	proof	NOUN
ejpam-4582	182	2	.	.	PUNCT
ejpam-4582	183	1	(	(	PUNCT
ejpam-4582	183	2	1	1	X
ejpam-4582	183	3	)	)	PUNCT
ejpam-4582	183	4	let	let	VERB
ejpam-4582	183	5	a	a	DET
ejpam-4582	183	6	be	be	AUX
ejpam-4582	183	7	a	a	DET
ejpam-4582	183	8	(	(	PUNCT
ejpam-4582	183	9	λ	λ	X
ejpam-4582	183	10	,	,	PUNCT
ejpam-4582	183	11	s)-closed	s)-close	VERB
ejpam-4582	183	12	set	set	NOUN
ejpam-4582	183	13	.	.	PUNCT
ejpam-4582	184	1	then	then	ADV
ejpam-4582	184	2	,	,	PUNCT
ejpam-4582	184	3	there	there	PRON
ejpam-4582	184	4	exist	exist	VERB
ejpam-4582	184	5	a	a	DET
ejpam-4582	184	6	λs	λs	ADV
ejpam-4582	184	7	-	-	PUNCT
ejpam-4582	184	8	set	set	VERB
ejpam-4582	184	9	t	t	NOUN
ejpam-4582	184	10	and	and	CCONJ
ejpam-4582	184	11	a	a	DET
ejpam-4582	184	12	semi	semi	ADJ
ejpam-4582	184	13	-	-	ADJ
ejpam-4582	184	14	closed	closed	ADJ
ejpam-4582	184	15	set	set	VERB
ejpam-4582	184	16	c	c	NOUN
ejpam-4582	184	17	such	such	ADJ
ejpam-4582	184	18	that	that	SCONJ
ejpam-4582	184	19	a	a	DET
ejpam-4582	184	20	=	=	X
ejpam-4582	184	21	t	t	NOUN
ejpam-4582	184	22	∩c	∩c	NOUN
ejpam-4582	184	23	.	.	PUNCT
ejpam-4582	185	1	by	by	ADP
ejpam-4582	185	2	a	a	DET
ejpam-4582	185	3	⊆	⊆	NUM
ejpam-4582	185	4	t	t	NOUN
ejpam-4582	185	5	,	,	PUNCT
ejpam-4582	185	6	we	we	PRON
ejpam-4582	185	7	have	have	VERB
ejpam-4582	185	8	a	a	DET
ejpam-4582	185	9	⊆	⊆	NUM
ejpam-4582	185	10	aλs	aλs	NOUN
ejpam-4582	185	11	⊆	⊆	NUM
ejpam-4582	185	12	tλs	tλs	NOUN
ejpam-4582	185	13	=	=	SYM
ejpam-4582	185	14	t	t	PROPN
ejpam-4582	185	15	,	,	PUNCT
ejpam-4582	185	16	and	and	CCONJ
ejpam-4582	185	17	also	also	ADV
ejpam-4582	185	18	by	by	ADP
ejpam-4582	185	19	a	a	DET
ejpam-4582	185	20	⊆	⊆	NUM
ejpam-4582	185	21	c	c	NOUN
ejpam-4582	185	22	,	,	PUNCT
ejpam-4582	185	23	a	a	DET
ejpam-4582	185	24	⊂	⊂	PROPN
ejpam-4582	185	25	a(λ	a(λ	PROPN
ejpam-4582	185	26	,	,	PUNCT
ejpam-4582	185	27	s	s	X
ejpam-4582	185	28	)	)	PUNCT
ejpam-4582	185	29	⊆	⊆	NUM
ejpam-4582	185	30	c(λ	c(λ	PROPN
ejpam-4582	185	31	,	,	PUNCT
ejpam-4582	185	32	s	s	NOUN
ejpam-4582	185	33	)	)	PUNCT
ejpam-4582	185	34	=	=	SYM
ejpam-4582	185	35	c.	c.	NOUN
ejpam-4582	185	36	now	now	ADV
ejpam-4582	185	37	a	a	DET
ejpam-4582	185	38	⊆	⊆	NUM
ejpam-4582	185	39	aλs	aλs	NOUN
ejpam-4582	185	40	∩a(λ	∩a(λ	NOUN
ejpam-4582	185	41	,	,	PUNCT
ejpam-4582	185	42	s	s	PART
ejpam-4582	185	43	)	)	PUNCT
ejpam-4582	185	44	⊆	⊆	NUM
ejpam-4582	185	45	t	t	NOUN
ejpam-4582	185	46	∩	∩	NOUN
ejpam-4582	185	47	c	c	NOUN
ejpam-4582	185	48	=	=	PUNCT
ejpam-4582	185	49	a.	a.	NOUN
ejpam-4582	185	50	thus	thus	ADV
ejpam-4582	185	51	,	,	PUNCT
ejpam-4582	185	52	a	a	DET
ejpam-4582	185	53	=	=	PUNCT
ejpam-4582	185	54	aλs	aλs	NOUN
ejpam-4582	185	55	∩a(λ	∩a(λ	NOUN
ejpam-4582	185	56	,	,	PUNCT
ejpam-4582	185	57	s	s	NOUN
ejpam-4582	185	58	)	)	PUNCT
ejpam-4582	185	59	.	.	PUNCT
ejpam-4582	186	1	(	(	PUNCT
ejpam-4582	186	2	2	2	X
ejpam-4582	186	3	)	)	PUNCT
ejpam-4582	186	4	it	it	PRON
ejpam-4582	186	5	is	be	AUX
ejpam-4582	186	6	sufficient	sufficient	ADJ
ejpam-4582	186	7	to	to	PART
ejpam-4582	186	8	observe	observe	VERB
ejpam-4582	186	9	that	that	SCONJ
ejpam-4582	186	10	a	a	DET
ejpam-4582	186	11	=	=	X
ejpam-4582	186	12	x	x	SYM
ejpam-4582	186	13	∩a	∩a	PROPN
ejpam-4582	186	14	,	,	PUNCT
ejpam-4582	186	15	where	where	SCONJ
ejpam-4582	186	16	the	the	DET
ejpam-4582	186	17	whole	whole	ADJ
ejpam-4582	186	18	set	set	NOUN
ejpam-4582	186	19	x	x	PUNCT
ejpam-4582	186	20	is	be	AUX
ejpam-4582	186	21	a	a	DET
ejpam-4582	186	22	λs	λs	NOUN
ejpam-4582	186	23	-	-	PUNCT
ejpam-4582	186	24	set	set	NOUN
ejpam-4582	186	25	.	.	PUNCT
ejpam-4582	187	1	definition	definition	NOUN
ejpam-4582	187	2	5	5	NUM
ejpam-4582	187	3	.	.	PUNCT
ejpam-4582	188	1	let	let	VERB
ejpam-4582	188	2	a	a	DET
ejpam-4582	188	3	be	be	AUX
ejpam-4582	188	4	a	a	DET
ejpam-4582	188	5	subset	subset	NOUN
ejpam-4582	188	6	of	of	ADP
ejpam-4582	188	7	a	a	DET
ejpam-4582	188	8	topological	topological	ADJ
ejpam-4582	188	9	space	space	NOUN
ejpam-4582	188	10	(	(	PUNCT
ejpam-4582	188	11	x	x	X
ejpam-4582	188	12	,	,	PUNCT
ejpam-4582	188	13	τ	τ	PROPN
ejpam-4582	188	14	)	)	PUNCT
ejpam-4582	188	15	.	.	PUNCT
ejpam-4582	189	1	then	then	ADV
ejpam-4582	189	2	,	,	PUNCT
ejpam-4582	189	3	γ(λ	γ(λ	ADV
ejpam-4582	189	4	,	,	PUNCT
ejpam-4582	189	5	s)(a	s)(a	NUM
ejpam-4582	189	6	)	)	PUNCT
ejpam-4582	189	7	is	be	AUX
ejpam-4582	189	8	defined	define	VERB
ejpam-4582	189	9	as	as	SCONJ
ejpam-4582	189	10	follows	follow	VERB
ejpam-4582	189	11	:	:	PUNCT
ejpam-4582	190	1	γ(λ	γ(λ	ADV
ejpam-4582	190	2	,	,	PUNCT
ejpam-4582	190	3	s)(a	s)(a	NUM
ejpam-4582	190	4	)	)	PUNCT
ejpam-4582	191	1	=	=	PUNCT
ejpam-4582	191	2	∩{u	∩{u	NUM
ejpam-4582	191	3	∈	∈	PROPN
ejpam-4582	191	4	(	(	PUNCT
ejpam-4582	191	5	λ	λ	NOUN
ejpam-4582	191	6	,	,	PUNCT
ejpam-4582	191	7	s)o(x	s)o(x	ADJ
ejpam-4582	191	8	)	)	PUNCT
ejpam-4582	191	9	|	|	ADV
ejpam-4582	191	10	a	a	DET
ejpam-4582	191	11	⊆	⊆	NUM
ejpam-4582	191	12	u	u	NOUN
ejpam-4582	191	13	}	}	PUNCT
ejpam-4582	191	14	.	.	PUNCT
ejpam-4582	192	1	proposition	proposition	NOUN
ejpam-4582	192	2	3	3	NUM
ejpam-4582	192	3	.	.	X
ejpam-4582	193	1	for	for	ADP
ejpam-4582	193	2	subsets	subset	NOUN
ejpam-4582	193	3	a	a	PRON
ejpam-4582	193	4	and	and	CCONJ
ejpam-4582	193	5	b	b	NOUN
ejpam-4582	193	6	of	of	ADP
ejpam-4582	193	7	a	a	DET
ejpam-4582	193	8	topological	topological	ADJ
ejpam-4582	193	9	space	space	NOUN
ejpam-4582	193	10	(	(	PUNCT
ejpam-4582	193	11	x	x	X
ejpam-4582	193	12	,	,	PUNCT
ejpam-4582	193	13	τ	τ	PROPN
ejpam-4582	193	14	)	)	PUNCT
ejpam-4582	193	15	,	,	PUNCT
ejpam-4582	193	16	the	the	DET
ejpam-4582	193	17	following	follow	VERB
ejpam-4582	193	18	properties	property	NOUN
ejpam-4582	193	19	hold	hold	VERB
ejpam-4582	193	20	:	:	PUNCT
ejpam-4582	193	21	(	(	PUNCT
ejpam-4582	193	22	1	1	X
ejpam-4582	193	23	)	)	PUNCT
ejpam-4582	193	24	if	if	SCONJ
ejpam-4582	193	25	a	a	DET
ejpam-4582	193	26	⊆	⊆	NUM
ejpam-4582	193	27	b	b	NOUN
ejpam-4582	193	28	,	,	PUNCT
ejpam-4582	193	29	then	then	ADV
ejpam-4582	193	30	γ(λ	γ(λ	PROPN
ejpam-4582	193	31	,	,	PUNCT
ejpam-4582	193	32	s)(a	s)(a	NUM
ejpam-4582	193	33	)	)	PUNCT
ejpam-4582	193	34	⊆	⊆	NUM
ejpam-4582	193	35	γ(λ	γ(λ	PROPN
ejpam-4582	193	36	,	,	PUNCT
ejpam-4582	193	37	s)(b	s)(b	PROPN
ejpam-4582	193	38	)	)	PUNCT
ejpam-4582	193	39	.	.	PUNCT
ejpam-4582	194	1	(	(	PUNCT
ejpam-4582	194	2	2	2	X
ejpam-4582	194	3	)	)	PUNCT
ejpam-4582	194	4	if	if	SCONJ
ejpam-4582	194	5	a	a	DET
ejpam-4582	194	6	∈	∈	PROPN
ejpam-4582	194	7	(	(	PUNCT
ejpam-4582	194	8	λ	λ	NOUN
ejpam-4582	194	9	,	,	PUNCT
ejpam-4582	194	10	s)o(x	s)o(x	NOUN
ejpam-4582	194	11	)	)	PUNCT
ejpam-4582	194	12	,	,	PUNCT
ejpam-4582	194	13	then	then	ADV
ejpam-4582	194	14	γ(λ	γ(λ	PROPN
ejpam-4582	194	15	,	,	PUNCT
ejpam-4582	194	16	s)(a	s)(a	NUM
ejpam-4582	194	17	)	)	PUNCT
ejpam-4582	195	1	=	=	SYM
ejpam-4582	195	2	a.	a.	NOUN
ejpam-4582	195	3	(	(	PUNCT
ejpam-4582	195	4	3	3	NUM
ejpam-4582	195	5	)	)	PUNCT
ejpam-4582	195	6	γ(λ	γ(λ	PROPN
ejpam-4582	195	7	,	,	PUNCT
ejpam-4582	195	8	s)[γ(λ	s)[γ(λ	NOUN
ejpam-4582	195	9	,	,	PUNCT
ejpam-4582	195	10	s)(a	s)(a	NUM
ejpam-4582	195	11	)	)	PUNCT
ejpam-4582	195	12	]	]	PUNCT
ejpam-4582	196	1	=	=	PUNCT
ejpam-4582	196	2	γ(λ	γ(λ	PROPN
ejpam-4582	196	3	,	,	PUNCT
ejpam-4582	196	4	s)(a	s)(a	NUM
ejpam-4582	196	5	)	)	PUNCT
ejpam-4582	196	6	.	.	PUNCT
ejpam-4582	197	1	proposition	proposition	NOUN
ejpam-4582	197	2	4	4	NUM
ejpam-4582	197	3	.	.	PUNCT
ejpam-4582	198	1	let	let	AUX
ejpam-4582	198	2	(	(	PUNCT
ejpam-4582	198	3	x	x	NOUN
ejpam-4582	198	4	,	,	PUNCT
ejpam-4582	198	5	τ	τ	X
ejpam-4582	198	6	)	)	PUNCT
ejpam-4582	198	7	be	be	VERB
ejpam-4582	198	8	a	a	DET
ejpam-4582	198	9	topological	topological	ADJ
ejpam-4582	198	10	space	space	NOUN
ejpam-4582	198	11	and	and	CCONJ
ejpam-4582	198	12	x	x	NOUN
ejpam-4582	198	13	,	,	PUNCT
ejpam-4582	198	14	y	y	PROPN
ejpam-4582	198	15	∈	∈	PROPN
ejpam-4582	198	16	x.	x.	NOUN
ejpam-4582	199	1	then	then	ADV
ejpam-4582	199	2	,	,	PUNCT
ejpam-4582	199	3	y	y	PROPN
ejpam-4582	199	4	∈	∈	PROPN
ejpam-4582	199	5	γ(λ	γ(λ	PROPN
ejpam-4582	199	6	,	,	PUNCT
ejpam-4582	199	7	s)({x	s)({x	PROPN
ejpam-4582	199	8	}	}	PUNCT
ejpam-4582	199	9	)	)	PUNCT
ejpam-4582	200	1	if	if	SCONJ
ejpam-4582	200	2	and	and	CCONJ
ejpam-4582	200	3	only	only	ADV
ejpam-4582	200	4	if	if	SCONJ
ejpam-4582	200	5	x	x	SYM
ejpam-4582	200	6	∈	∈	PROPN
ejpam-4582	200	7	{	{	PUNCT
ejpam-4582	200	8	y}(λ	y}(λ	PROPN
ejpam-4582	200	9	,	,	PUNCT
ejpam-4582	200	10	s	s	PART
ejpam-4582	200	11	)	)	PUNCT
ejpam-4582	200	12	.	.	PUNCT
ejpam-4582	201	1	proof	proof	NOUN
ejpam-4582	201	2	.	.	PUNCT
ejpam-4582	202	1	let	let	VERB
ejpam-4582	202	2	y	y	PROPN
ejpam-4582	202	3	̸∈	̸∈	PROPN
ejpam-4582	202	4	γ(λ	γ(λ	PROPN
ejpam-4582	202	5	,	,	PUNCT
ejpam-4582	202	6	s)({x	s)({x	PROPN
ejpam-4582	202	7	}	}	PUNCT
ejpam-4582	202	8	)	)	PUNCT
ejpam-4582	202	9	.	.	PUNCT
ejpam-4582	203	1	then	then	ADV
ejpam-4582	203	2	,	,	PUNCT
ejpam-4582	203	3	there	there	PRON
ejpam-4582	203	4	exists	exist	VERB
ejpam-4582	203	5	a	a	DET
ejpam-4582	203	6	(	(	PUNCT
ejpam-4582	203	7	λ	λ	X
ejpam-4582	203	8	,	,	PUNCT
ejpam-4582	203	9	s)-open	s)-open	PUNCT
ejpam-4582	203	10	set	set	VERB
ejpam-4582	203	11	v	v	NOUN
ejpam-4582	203	12	containing	contain	VERB
ejpam-4582	203	13	x	x	PUNCT
ejpam-4582	203	14	such	such	ADJ
ejpam-4582	203	15	that	that	SCONJ
ejpam-4582	203	16	y	y	PROPN
ejpam-4582	203	17	̸∈	̸∈	PROPN
ejpam-4582	203	18	v	v	PROPN
ejpam-4582	203	19	.	.	PUNCT
ejpam-4582	204	1	thus	thus	ADV
ejpam-4582	204	2	,	,	PUNCT
ejpam-4582	204	3	x	x	PROPN
ejpam-4582	204	4	̸∈	̸∈	PROPN
ejpam-4582	204	5	{	{	PUNCT
ejpam-4582	204	6	y}(λ	y}(λ	PROPN
ejpam-4582	204	7	,	,	PUNCT
ejpam-4582	204	8	s	s	PROPN
ejpam-4582	204	9	)	)	PUNCT
ejpam-4582	204	10	.	.	PUNCT
ejpam-4582	205	1	the	the	DET
ejpam-4582	205	2	converse	converse	NOUN
ejpam-4582	205	3	is	be	AUX
ejpam-4582	205	4	similarly	similarly	ADV
ejpam-4582	205	5	shown	show	VERB
ejpam-4582	205	6	.	.	PUNCT
ejpam-4582	206	1	definition	definition	NOUN
ejpam-4582	206	2	6	6	NUM
ejpam-4582	206	3	.	.	PUNCT
ejpam-4582	207	1	let	let	VERB
ejpam-4582	207	2	(	(	PUNCT
ejpam-4582	207	3	x	x	NOUN
ejpam-4582	207	4	,	,	PUNCT
ejpam-4582	207	5	τ	τ	X
ejpam-4582	207	6	)	)	PUNCT
ejpam-4582	207	7	be	be	VERB
ejpam-4582	207	8	a	a	DET
ejpam-4582	207	9	topological	topological	ADJ
ejpam-4582	207	10	space	space	NOUN
ejpam-4582	207	11	and	and	CCONJ
ejpam-4582	207	12	x	x	AUX
ejpam-4582	207	13	∈	∈	PROPN
ejpam-4582	207	14	x.	x.	NOUN
ejpam-4582	207	15	then	then	ADV
ejpam-4582	207	16	,	,	PUNCT
ejpam-4582	207	17	⟨x⟩(λ	⟨x⟩(λ	PROPN
ejpam-4582	207	18	,	,	PUNCT
ejpam-4582	207	19	s	s	PART
ejpam-4582	207	20	)	)	PUNCT
ejpam-4582	207	21	is	be	AUX
ejpam-4582	207	22	defined	define	VERB
ejpam-4582	207	23	as	as	ADP
ejpam-4582	207	24	follows	follow	VERB
ejpam-4582	207	25	:	:	PUNCT
ejpam-4582	207	26	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	207	27	,	,	PUNCT
ejpam-4582	207	28	s	s	NOUN
ejpam-4582	207	29	)	)	PUNCT
ejpam-4582	207	30	=	=	SYM
ejpam-4582	207	31	γ(λ	γ(λ	PROPN
ejpam-4582	207	32	,	,	PUNCT
ejpam-4582	207	33	s)({x	s)({x	NOUN
ejpam-4582	207	34	}	}	PUNCT
ejpam-4582	207	35	)	)	PUNCT
ejpam-4582	207	36	∩	∩	NOUN
ejpam-4582	207	37	{	{	PUNCT
ejpam-4582	207	38	x}(λ	x}(λ	PROPN
ejpam-4582	207	39	,	,	PUNCT
ejpam-4582	207	40	s	s	PART
ejpam-4582	207	41	)	)	PUNCT
ejpam-4582	207	42	.	.	PUNCT
ejpam-4582	208	1	proposition	proposition	NOUN
ejpam-4582	208	2	5	5	NUM
ejpam-4582	208	3	.	.	PUNCT
ejpam-4582	208	4	for	for	ADP
ejpam-4582	208	5	a	a	DET
ejpam-4582	208	6	topological	topological	ADJ
ejpam-4582	208	7	space	space	NOUN
ejpam-4582	208	8	(	(	PUNCT
ejpam-4582	208	9	x	x	X
ejpam-4582	208	10	,	,	PUNCT
ejpam-4582	208	11	τ	τ	PROPN
ejpam-4582	208	12	)	)	PUNCT
ejpam-4582	208	13	,	,	PUNCT
ejpam-4582	208	14	the	the	DET
ejpam-4582	208	15	following	follow	VERB
ejpam-4582	208	16	properties	property	NOUN
ejpam-4582	208	17	hold	hold	VERB
ejpam-4582	208	18	:	:	PUNCT
ejpam-4582	208	19	(	(	PUNCT
ejpam-4582	208	20	1	1	X
ejpam-4582	208	21	)	)	PUNCT
ejpam-4582	208	22	for	for	ADP
ejpam-4582	208	23	each	each	DET
ejpam-4582	208	24	x	x	SYM
ejpam-4582	208	25	∈	∈	PROPN
ejpam-4582	208	26	x	x	X
ejpam-4582	208	27	,	,	PUNCT
ejpam-4582	208	28	γ(λ	γ(λ	ADV
ejpam-4582	208	29	,	,	PUNCT
ejpam-4582	208	30	s)[⟨x⟩(λ	s)[⟨x⟩(λ	NOUN
ejpam-4582	208	31	,	,	PUNCT
ejpam-4582	208	32	s	s	NOUN
ejpam-4582	208	33	)	)	PUNCT
ejpam-4582	208	34	]	]	PUNCT
ejpam-4582	209	1	=	=	SYM
ejpam-4582	209	2	γ(λ	γ(λ	PROPN
ejpam-4582	209	3	,	,	PUNCT
ejpam-4582	209	4	s)({x	s)({x	PROPN
ejpam-4582	209	5	}	}	PUNCT
ejpam-4582	209	6	)	)	PUNCT
ejpam-4582	209	7	;	;	PUNCT
ejpam-4582	209	8	(	(	PUNCT
ejpam-4582	209	9	2	2	X
ejpam-4582	209	10	)	)	PUNCT
ejpam-4582	209	11	for	for	ADP
ejpam-4582	209	12	each	each	DET
ejpam-4582	209	13	x	x	SYM
ejpam-4582	209	14	∈	∈	PROPN
ejpam-4582	209	15	x	x	X
ejpam-4582	209	16	,	,	PUNCT
ejpam-4582	209	17	[	[	X
ejpam-4582	209	18	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	209	19	,	,	PUNCT
ejpam-4582	209	20	s)](λ	s)](λ	NOUN
ejpam-4582	209	21	,	,	PUNCT
ejpam-4582	209	22	s	s	PART
ejpam-4582	209	23	)	)	PUNCT
ejpam-4582	209	24	=	=	SYM
ejpam-4582	209	25	{	{	PUNCT
ejpam-4582	209	26	x}(λ	x}(λ	PROPN
ejpam-4582	209	27	,	,	PUNCT
ejpam-4582	209	28	s	s	PART
ejpam-4582	209	29	)	)	PUNCT
ejpam-4582	209	30	.	.	PUNCT
ejpam-4582	210	1	c.	c.	PROPN
ejpam-4582	210	2	boonpok	boonpok	PROPN
ejpam-4582	210	3	,	,	PUNCT
ejpam-4582	210	4	c.	c.	PROPN
ejpam-4582	210	5	viriyapong	viriyapong	PROPN
ejpam-4582	210	6	/	/	SYM
ejpam-4582	210	7	eur	eur	PROPN
ejpam-4582	210	8	.	.	PUNCT
ejpam-4582	211	1	j.	j.	PROPN
ejpam-4582	211	2	pure	pure	PROPN
ejpam-4582	211	3	appl	appl	PROPN
ejpam-4582	211	4	.	.	PROPN
ejpam-4582	211	5	math	math	PROPN
ejpam-4582	211	6	,	,	PUNCT
ejpam-4582	211	7	16	16	NUM
ejpam-4582	211	8	(	(	PUNCT
ejpam-4582	211	9	1	1	NUM
ejpam-4582	211	10	)	)	PUNCT
ejpam-4582	211	11	(	(	PUNCT
ejpam-4582	211	12	2023	2023	NUM
ejpam-4582	211	13	)	)	PUNCT
ejpam-4582	211	14	,	,	PUNCT
ejpam-4582	211	15	336	336	NUM
ejpam-4582	211	16	-	-	SYM
ejpam-4582	211	17	362	362	NUM
ejpam-4582	211	18	342	342	NUM
ejpam-4582	211	19	proof	proof	NOUN
ejpam-4582	211	20	.	.	PUNCT
ejpam-4582	212	1	(	(	PUNCT
ejpam-4582	212	2	1	1	X
ejpam-4582	212	3	)	)	PUNCT
ejpam-4582	212	4	let	let	VERB
ejpam-4582	212	5	x	x	PUNCT
ejpam-4582	212	6	∈	∈	PROPN
ejpam-4582	212	7	x.	x.	NOUN
ejpam-4582	212	8	then	then	ADV
ejpam-4582	212	9	,	,	PUNCT
ejpam-4582	212	10	we	we	PRON
ejpam-4582	212	11	have	have	VERB
ejpam-4582	212	12	{	{	PUNCT
ejpam-4582	212	13	x	x	NOUN
ejpam-4582	212	14	}	}	PUNCT
ejpam-4582	212	15	⊆	⊆	NUM
ejpam-4582	212	16	{	{	PUNCT
ejpam-4582	212	17	x}(λ	x}(λ	PROPN
ejpam-4582	212	18	,	,	PUNCT
ejpam-4582	212	19	s	s	PART
ejpam-4582	212	20	)	)	PUNCT
ejpam-4582	212	21	∩	∩	ADJ
ejpam-4582	212	22	γ(λ	γ(λ	PROPN
ejpam-4582	212	23	,	,	PUNCT
ejpam-4582	212	24	s)({x	s)({x	NOUN
ejpam-4582	212	25	}	}	PUNCT
ejpam-4582	212	26	)	)	PUNCT
ejpam-4582	213	1	=	=	SYM
ejpam-4582	213	2	⟨x⟩s	⟨x⟩s	PROPN
ejpam-4582	213	3	.	.	PUNCT
ejpam-4582	214	1	by	by	ADP
ejpam-4582	214	2	proposition	proposition	NOUN
ejpam-4582	214	3	3	3	NUM
ejpam-4582	214	4	,	,	PUNCT
ejpam-4582	214	5	γ(λ	γ(λ	PROPN
ejpam-4582	214	6	,	,	PUNCT
ejpam-4582	214	7	s)({x	s)({x	PROPN
ejpam-4582	214	8	}	}	PUNCT
ejpam-4582	214	9	)	)	PUNCT
ejpam-4582	214	10	⊆	⊆	NUM
ejpam-4582	214	11	γ(λ	γ(λ	NOUN
ejpam-4582	214	12	,	,	PUNCT
ejpam-4582	214	13	s)[⟨x⟩(λ	s)[⟨x⟩(λ	NOUN
ejpam-4582	214	14	,	,	PUNCT
ejpam-4582	214	15	s	s	NOUN
ejpam-4582	214	16	)	)	PUNCT
ejpam-4582	214	17	]	]	PUNCT
ejpam-4582	214	18	.	.	PUNCT
ejpam-4582	215	1	next	next	ADV
ejpam-4582	215	2	,	,	PUNCT
ejpam-4582	215	3	we	we	PRON
ejpam-4582	215	4	show	show	VERB
ejpam-4582	215	5	the	the	DET
ejpam-4582	215	6	opposite	opposite	ADJ
ejpam-4582	215	7	implication	implication	NOUN
ejpam-4582	215	8	.	.	PUNCT
ejpam-4582	215	9	suppose	suppose	VERB
ejpam-4582	215	10	that	that	SCONJ
ejpam-4582	215	11	y	y	PROPN
ejpam-4582	215	12	̸∈	̸∈	PROPN
ejpam-4582	215	13	γ(λ	γ(λ	PROPN
ejpam-4582	215	14	,	,	PUNCT
ejpam-4582	215	15	s)({x	s)({x	PROPN
ejpam-4582	215	16	}	}	PUNCT
ejpam-4582	215	17	)	)	PUNCT
ejpam-4582	215	18	.	.	PUNCT
ejpam-4582	216	1	there	there	PRON
ejpam-4582	216	2	exists	exist	VERB
ejpam-4582	216	3	a	a	DET
ejpam-4582	216	4	(	(	PUNCT
ejpam-4582	216	5	λ	λ	X
ejpam-4582	216	6	,	,	PUNCT
ejpam-4582	216	7	s)-open	s)-open	PUNCT
ejpam-4582	216	8	set	set	VERB
ejpam-4582	216	9	v	v	ADP
ejpam-4582	216	10	such	such	ADJ
ejpam-4582	216	11	that	that	SCONJ
ejpam-4582	216	12	x	x	SYM
ejpam-4582	216	13	∈	∈	PROPN
ejpam-4582	216	14	v	v	NOUN
ejpam-4582	216	15	and	and	CCONJ
ejpam-4582	216	16	y	y	PROPN
ejpam-4582	216	17	̸∈	̸∈	PROPN
ejpam-4582	216	18	v	v	PROPN
ejpam-4582	216	19	.	.	PUNCT
ejpam-4582	217	1	since	since	SCONJ
ejpam-4582	217	2	⟨x⟩s	⟨x⟩s	PROPN
ejpam-4582	217	3	⊆	⊆	NUM
ejpam-4582	217	4	γ(λ	γ(λ	PROPN
ejpam-4582	217	5	,	,	PUNCT
ejpam-4582	217	6	s)({x	s)({x	PROPN
ejpam-4582	217	7	}	}	PUNCT
ejpam-4582	217	8	)	)	PUNCT
ejpam-4582	218	1	⊆	⊆	NUM
ejpam-4582	218	2	γ(λ	γ(λ	PROPN
ejpam-4582	218	3	,	,	PUNCT
ejpam-4582	218	4	s)(v	s)(v	PUNCT
ejpam-4582	218	5	)	)	PUNCT
ejpam-4582	218	6	=	=	SYM
ejpam-4582	218	7	v	v	X
ejpam-4582	218	8	,	,	PUNCT
ejpam-4582	218	9	we	we	PRON
ejpam-4582	218	10	have	have	VERB
ejpam-4582	218	11	γ(λ	γ(λ	NOUN
ejpam-4582	218	12	,	,	PUNCT
ejpam-4582	218	13	s)[⟨x⟩(λ	s)[⟨x⟩(λ	NOUN
ejpam-4582	218	14	,	,	PUNCT
ejpam-4582	218	15	s	s	NOUN
ejpam-4582	218	16	)	)	PUNCT
ejpam-4582	218	17	]	]	PUNCT
ejpam-4582	219	1	⊆	⊆	NUM
ejpam-4582	219	2	v	v	NOUN
ejpam-4582	219	3	.	.	PUNCT
ejpam-4582	220	1	since	since	SCONJ
ejpam-4582	220	2	y	y	PROPN
ejpam-4582	220	3	̸∈	̸∈	PROPN
ejpam-4582	220	4	v	v	PROPN
ejpam-4582	220	5	,	,	PUNCT
ejpam-4582	220	6	y	y	PROPN
ejpam-4582	220	7	̸∈	̸∈	PROPN
ejpam-4582	220	8	γ(λ	γ(λ	PROPN
ejpam-4582	220	9	,	,	PUNCT
ejpam-4582	220	10	s)[⟨x⟩(λ	s)[⟨x⟩(λ	NOUN
ejpam-4582	220	11	,	,	PUNCT
ejpam-4582	220	12	s	s	NOUN
ejpam-4582	220	13	)	)	PUNCT
ejpam-4582	220	14	]	]	PUNCT
ejpam-4582	220	15	.	.	PUNCT
ejpam-4582	221	1	thus	thus	ADV
ejpam-4582	221	2	,	,	PUNCT
ejpam-4582	221	3	γ(λ	γ(λ	ADV
ejpam-4582	221	4	,	,	PUNCT
ejpam-4582	221	5	s)[⟨x⟩(λ	s)[⟨x⟩(λ	NOUN
ejpam-4582	221	6	,	,	PUNCT
ejpam-4582	221	7	s	s	NOUN
ejpam-4582	221	8	)	)	PUNCT
ejpam-4582	221	9	]	]	PUNCT
ejpam-4582	221	10	⊆	⊆	NUM
ejpam-4582	221	11	γ(λ	γ(λ	PROPN
ejpam-4582	221	12	,	,	PUNCT
ejpam-4582	221	13	s)({x	s)({x	NOUN
ejpam-4582	221	14	}	}	PUNCT
ejpam-4582	221	15	)	)	PUNCT
ejpam-4582	221	16	and	and	CCONJ
ejpam-4582	221	17	hence	hence	ADV
ejpam-4582	221	18	γ(λ	γ(λ	PROPN
ejpam-4582	221	19	,	,	PUNCT
ejpam-4582	221	20	s)({x	s)({x	NOUN
ejpam-4582	221	21	}	}	PUNCT
ejpam-4582	221	22	)	)	PUNCT
ejpam-4582	221	23	=	=	SYM
ejpam-4582	221	24	γ(λ	γ(λ	NOUN
ejpam-4582	221	25	,	,	PUNCT
ejpam-4582	221	26	s)[⟨x⟩(λ	s)[⟨x⟩(λ	NOUN
ejpam-4582	221	27	,	,	PUNCT
ejpam-4582	221	28	s	s	NOUN
ejpam-4582	221	29	)	)	PUNCT
ejpam-4582	221	30	]	]	PUNCT
ejpam-4582	221	31	.	.	PUNCT
ejpam-4582	222	1	(	(	PUNCT
ejpam-4582	222	2	2	2	X
ejpam-4582	222	3	)	)	PUNCT
ejpam-4582	222	4	by	by	ADP
ejpam-4582	222	5	the	the	DET
ejpam-4582	222	6	definition	definition	NOUN
ejpam-4582	222	7	of	of	ADP
ejpam-4582	222	8	⟨x⟩(λ	⟨x⟩(λ	PROPN
ejpam-4582	222	9	,	,	PUNCT
ejpam-4582	222	10	s	s	NOUN
ejpam-4582	222	11	)	)	PUNCT
ejpam-4582	222	12	,	,	PUNCT
ejpam-4582	222	13	we	we	PRON
ejpam-4582	222	14	have	have	VERB
ejpam-4582	222	15	{	{	PUNCT
ejpam-4582	222	16	x	x	NOUN
ejpam-4582	222	17	}	}	PUNCT
ejpam-4582	222	18	⊆	⊆	NUM
ejpam-4582	222	19	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	222	20	,	,	PUNCT
ejpam-4582	222	21	s	s	NOUN
ejpam-4582	222	22	)	)	PUNCT
ejpam-4582	222	23	and	and	CCONJ
ejpam-4582	222	24	{	{	PUNCT
ejpam-4582	222	25	x}(λ	x}(λ	PROPN
ejpam-4582	222	26	,	,	PUNCT
ejpam-4582	222	27	s	s	PART
ejpam-4582	222	28	)	)	PUNCT
ejpam-4582	223	1	⊆	⊆	NUM
ejpam-4582	224	1	[	[	X
ejpam-4582	224	2	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	224	3	,	,	PUNCT
ejpam-4582	224	4	s)](λ	s)](λ	NOUN
ejpam-4582	224	5	,	,	PUNCT
ejpam-4582	224	6	s	s	PART
ejpam-4582	224	7	)	)	PUNCT
ejpam-4582	224	8	by	by	ADP
ejpam-4582	224	9	lemma	lemma	PROPN
ejpam-4582	224	10	4	4	NUM
ejpam-4582	224	11	.	.	PUNCT
ejpam-4582	225	1	on	on	ADP
ejpam-4582	225	2	the	the	DET
ejpam-4582	225	3	other	other	ADJ
ejpam-4582	225	4	hand	hand	NOUN
ejpam-4582	225	5	,	,	PUNCT
ejpam-4582	225	6	we	we	PRON
ejpam-4582	225	7	have	have	VERB
ejpam-4582	225	8	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	225	9	,	,	PUNCT
ejpam-4582	225	10	s	s	NOUN
ejpam-4582	225	11	)	)	PUNCT
ejpam-4582	225	12	⊆	⊆	NUM
ejpam-4582	225	13	{	{	PUNCT
ejpam-4582	225	14	x}(λ	x}(λ	PROPN
ejpam-4582	225	15	,	,	PUNCT
ejpam-4582	225	16	s	s	PART
ejpam-4582	225	17	)	)	PUNCT
ejpam-4582	225	18	and	and	CCONJ
ejpam-4582	226	1	[	[	X
ejpam-4582	226	2	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	226	3	,	,	PUNCT
ejpam-4582	226	4	s)](λ	s)](λ	NOUN
ejpam-4582	226	5	,	,	PUNCT
ejpam-4582	226	6	s	s	PART
ejpam-4582	226	7	)	)	PUNCT
ejpam-4582	226	8	⊆	⊆	NUM
ejpam-4582	226	9	[	[	X
ejpam-4582	226	10	{	{	PUNCT
ejpam-4582	226	11	x}(λ	x}(λ	PROPN
ejpam-4582	226	12	,	,	PUNCT
ejpam-4582	226	13	s)](λ	s)](λ	PROPN
ejpam-4582	226	14	,	,	PUNCT
ejpam-4582	226	15	s	s	PART
ejpam-4582	226	16	)	)	PUNCT
ejpam-4582	226	17	=	=	SYM
ejpam-4582	226	18	{	{	PUNCT
ejpam-4582	226	19	x}(λ	x}(λ	PROPN
ejpam-4582	226	20	,	,	PUNCT
ejpam-4582	226	21	s	s	PART
ejpam-4582	226	22	)	)	PUNCT
ejpam-4582	226	23	.	.	PUNCT
ejpam-4582	227	1	this	this	PRON
ejpam-4582	227	2	shows	show	VERB
ejpam-4582	227	3	that	that	SCONJ
ejpam-4582	227	4	[	[	X
ejpam-4582	227	5	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	227	6	,	,	PUNCT
ejpam-4582	227	7	s)](λ	s)](λ	NOUN
ejpam-4582	227	8	,	,	PUNCT
ejpam-4582	227	9	s	s	PART
ejpam-4582	227	10	)	)	PUNCT
ejpam-4582	227	11	⊆	⊆	NUM
ejpam-4582	227	12	{	{	PUNCT
ejpam-4582	227	13	x}(λ	x}(λ	PROPN
ejpam-4582	227	14	,	,	PUNCT
ejpam-4582	227	15	s	s	PART
ejpam-4582	227	16	)	)	PUNCT
ejpam-4582	227	17	.	.	PUNCT
ejpam-4582	228	1	theorem	theorem	NOUN
ejpam-4582	228	2	3	3	NUM
ejpam-4582	228	3	.	.	X
ejpam-4582	228	4	for	for	ADP
ejpam-4582	228	5	any	any	DET
ejpam-4582	228	6	points	point	NOUN
ejpam-4582	228	7	x	x	PUNCT
ejpam-4582	228	8	and	and	CCONJ
ejpam-4582	228	9	y	y	PROPN
ejpam-4582	228	10	in	in	ADP
ejpam-4582	228	11	a	a	DET
ejpam-4582	228	12	topological	topological	ADJ
ejpam-4582	228	13	space	space	NOUN
ejpam-4582	228	14	(	(	PUNCT
ejpam-4582	228	15	x	x	X
ejpam-4582	228	16	,	,	PUNCT
ejpam-4582	228	17	τ	τ	PROPN
ejpam-4582	228	18	)	)	PUNCT
ejpam-4582	228	19	,	,	PUNCT
ejpam-4582	228	20	the	the	DET
ejpam-4582	228	21	following	follow	VERB
ejpam-4582	228	22	properties	property	NOUN
ejpam-4582	228	23	are	be	AUX
ejpam-4582	228	24	equivalent	equivalent	ADJ
ejpam-4582	228	25	:	:	PUNCT
ejpam-4582	228	26	(	(	PUNCT
ejpam-4582	228	27	1	1	X
ejpam-4582	228	28	)	)	PUNCT
ejpam-4582	228	29	γ(λ	γ(λ	PROPN
ejpam-4582	228	30	,	,	PUNCT
ejpam-4582	228	31	s)({x	s)({x	PROPN
ejpam-4582	228	32	}	}	PUNCT
ejpam-4582	228	33	)	)	PUNCT
ejpam-4582	229	1	̸=	̸=	PROPN
ejpam-4582	229	2	γ(λ	γ(λ	PROPN
ejpam-4582	229	3	,	,	PUNCT
ejpam-4582	229	4	s)({y	s)({y	NOUN
ejpam-4582	229	5	}	}	PUNCT
ejpam-4582	229	6	)	)	PUNCT
ejpam-4582	229	7	.	.	PUNCT
ejpam-4582	230	1	(	(	PUNCT
ejpam-4582	230	2	2	2	X
ejpam-4582	230	3	)	)	PUNCT
ejpam-4582	230	4	{	{	PUNCT
ejpam-4582	230	5	x}(λ	x}(λ	PROPN
ejpam-4582	230	6	,	,	PUNCT
ejpam-4582	230	7	s	s	PART
ejpam-4582	230	8	)	)	PUNCT
ejpam-4582	230	9	̸=	̸=	PROPN
ejpam-4582	230	10	{	{	PUNCT
ejpam-4582	230	11	y}(λ	y}(λ	PROPN
ejpam-4582	230	12	,	,	PUNCT
ejpam-4582	230	13	s	s	PART
ejpam-4582	230	14	)	)	PUNCT
ejpam-4582	230	15	.	.	PUNCT
ejpam-4582	231	1	proof	proof	NOUN
ejpam-4582	231	2	.	.	PUNCT
ejpam-4582	232	1	(	(	PUNCT
ejpam-4582	232	2	1	1	X
ejpam-4582	232	3	)	)	PUNCT
ejpam-4582	232	4	⇒	⇒	NOUN
ejpam-4582	232	5	(	(	PUNCT
ejpam-4582	232	6	2	2	NUM
ejpam-4582	232	7	):	):	PUNCT
ejpam-4582	232	8	suppose	suppose	VERB
ejpam-4582	232	9	that	that	SCONJ
ejpam-4582	232	10	γ(λ	γ(λ	PROPN
ejpam-4582	232	11	,	,	PUNCT
ejpam-4582	232	12	s)({x	s)({x	PROPN
ejpam-4582	232	13	}	}	PUNCT
ejpam-4582	232	14	)	)	PUNCT
ejpam-4582	232	15	̸=	̸=	PROPN
ejpam-4582	232	16	γ(λ	γ(λ	PROPN
ejpam-4582	232	17	,	,	PUNCT
ejpam-4582	232	18	s)({y	s)({y	NOUN
ejpam-4582	232	19	}	}	PUNCT
ejpam-4582	232	20	)	)	PUNCT
ejpam-4582	232	21	.	.	PUNCT
ejpam-4582	233	1	there	there	PRON
ejpam-4582	233	2	exists	exist	VERB
ejpam-4582	233	3	a	a	DET
ejpam-4582	233	4	point	point	NOUN
ejpam-4582	233	5	z	z	NOUN
ejpam-4582	233	6	∈	∈	PROPN
ejpam-4582	233	7	x	x	PUNCT
ejpam-4582	233	8	such	such	ADJ
ejpam-4582	233	9	that	that	SCONJ
ejpam-4582	233	10	z	z	PROPN
ejpam-4582	233	11	∈	∈	PROPN
ejpam-4582	233	12	γ(λ	γ(λ	PROPN
ejpam-4582	233	13	,	,	PUNCT
ejpam-4582	233	14	s)({x	s)({x	NOUN
ejpam-4582	233	15	}	}	PUNCT
ejpam-4582	233	16	)	)	PUNCT
ejpam-4582	233	17	and	and	CCONJ
ejpam-4582	233	18	z	z	PROPN
ejpam-4582	233	19	̸∈	̸∈	PROPN
ejpam-4582	233	20	γ(λ	γ(λ	PROPN
ejpam-4582	233	21	,	,	PUNCT
ejpam-4582	233	22	s)({y	s)({y	NOUN
ejpam-4582	233	23	}	}	PUNCT
ejpam-4582	233	24	)	)	PUNCT
ejpam-4582	233	25	or	or	CCONJ
ejpam-4582	233	26	z	z	NOUN
ejpam-4582	233	27	∈	∈	PROPN
ejpam-4582	233	28	γ(λ	γ(λ	PROPN
ejpam-4582	233	29	,	,	PUNCT
ejpam-4582	233	30	s)({y	s)({y	NOUN
ejpam-4582	233	31	}	}	PUNCT
ejpam-4582	233	32	)	)	PUNCT
ejpam-4582	233	33	and	and	CCONJ
ejpam-4582	233	34	z	z	PROPN
ejpam-4582	233	35	̸∈	̸∈	PROPN
ejpam-4582	233	36	γ(λ	γ(λ	PROPN
ejpam-4582	233	37	,	,	PUNCT
ejpam-4582	233	38	s)({x	s)({x	PROPN
ejpam-4582	233	39	}	}	PUNCT
ejpam-4582	233	40	)	)	PUNCT
ejpam-4582	233	41	.	.	PUNCT
ejpam-4582	234	1	we	we	PRON
ejpam-4582	234	2	prove	prove	VERB
ejpam-4582	234	3	only	only	ADV
ejpam-4582	234	4	the	the	DET
ejpam-4582	234	5	first	first	ADJ
ejpam-4582	234	6	case	case	NOUN
ejpam-4582	234	7	being	be	AUX
ejpam-4582	234	8	the	the	DET
ejpam-4582	234	9	second	second	ADJ
ejpam-4582	234	10	analogous	analogous	NOUN
ejpam-4582	234	11	.	.	PUNCT
ejpam-4582	235	1	from	from	ADP
ejpam-4582	235	2	z	z	PROPN
ejpam-4582	235	3	∈	∈	PROPN
ejpam-4582	235	4	γ(λ	γ(λ	PROPN
ejpam-4582	235	5	,	,	PUNCT
ejpam-4582	235	6	s)({x	s)({x	PROPN
ejpam-4582	235	7	}	}	PUNCT
ejpam-4582	235	8	)	)	PUNCT
ejpam-4582	236	1	it	it	PRON
ejpam-4582	236	2	follows	follow	VERB
ejpam-4582	236	3	that	that	SCONJ
ejpam-4582	236	4	{	{	PUNCT
ejpam-4582	236	5	x}∩{z}(λ	x}∩{z}(λ	PROPN
ejpam-4582	236	6	,	,	PUNCT
ejpam-4582	236	7	s	s	PART
ejpam-4582	236	8	)	)	PUNCT
ejpam-4582	236	9	̸=	̸=	PROPN
ejpam-4582	236	10	∅	∅	NOUN
ejpam-4582	236	11	which	which	PRON
ejpam-4582	236	12	implies	imply	VERB
ejpam-4582	236	13	x	x	X
ejpam-4582	236	14	∈	∈	PROPN
ejpam-4582	236	15	{	{	PUNCT
ejpam-4582	236	16	z}(λ	z}(λ	PROPN
ejpam-4582	236	17	,	,	PUNCT
ejpam-4582	236	18	s	s	NOUN
ejpam-4582	236	19	)	)	PUNCT
ejpam-4582	236	20	.	.	PUNCT
ejpam-4582	237	1	by	by	ADP
ejpam-4582	237	2	z	z	PROPN
ejpam-4582	237	3	̸∈	̸∈	PROPN
ejpam-4582	237	4	γ(λ	γ(λ	PROPN
ejpam-4582	237	5	,	,	PUNCT
ejpam-4582	237	6	s)({y	s)({y	NOUN
ejpam-4582	237	7	}	}	PUNCT
ejpam-4582	237	8	)	)	PUNCT
ejpam-4582	237	9	,	,	PUNCT
ejpam-4582	237	10	we	we	PRON
ejpam-4582	237	11	have	have	AUX
ejpam-4582	237	12	{	{	PUNCT
ejpam-4582	237	13	y}∩{z}(λ	y}∩{z}(λ	VERB
ejpam-4582	237	14	,	,	PUNCT
ejpam-4582	237	15	s	s	PART
ejpam-4582	237	16	)	)	PUNCT
ejpam-4582	237	17	=	=	PUNCT
ejpam-4582	237	18	∅.	∅.	NOUN
ejpam-4582	237	19	since	since	SCONJ
ejpam-4582	237	20	x	x	PROPN
ejpam-4582	237	21	∈	∈	PROPN
ejpam-4582	237	22	{	{	PUNCT
ejpam-4582	237	23	z}(λ	z}(λ	PROPN
ejpam-4582	237	24	,	,	PUNCT
ejpam-4582	237	25	s	s	PART
ejpam-4582	237	26	)	)	PUNCT
ejpam-4582	237	27	,	,	PUNCT
ejpam-4582	237	28	{	{	PUNCT
ejpam-4582	237	29	x}(λ	x}(λ	PROPN
ejpam-4582	237	30	,	,	PUNCT
ejpam-4582	237	31	s	s	PART
ejpam-4582	237	32	)	)	PUNCT
ejpam-4582	237	33	⊆	⊆	NUM
ejpam-4582	237	34	{	{	PUNCT
ejpam-4582	237	35	z}(λ	z}(λ	PROPN
ejpam-4582	237	36	,	,	PUNCT
ejpam-4582	237	37	s	s	PART
ejpam-4582	237	38	)	)	PUNCT
ejpam-4582	237	39	and	and	CCONJ
ejpam-4582	237	40	{	{	PUNCT
ejpam-4582	237	41	y	y	NOUN
ejpam-4582	237	42	}	}	PUNCT
ejpam-4582	237	43	∩	∩	NOUN
ejpam-4582	237	44	{	{	PUNCT
ejpam-4582	237	45	x}(λ	x}(λ	PROPN
ejpam-4582	237	46	,	,	PUNCT
ejpam-4582	237	47	s	s	PART
ejpam-4582	237	48	)	)	PUNCT
ejpam-4582	237	49	=	=	PUNCT
ejpam-4582	237	50	∅.	∅.	PRON
ejpam-4582	237	51	therefore	therefore	ADV
ejpam-4582	237	52	,	,	PUNCT
ejpam-4582	237	53	it	it	PRON
ejpam-4582	237	54	follows	follow	VERB
ejpam-4582	237	55	that	that	SCONJ
ejpam-4582	237	56	{	{	PUNCT
ejpam-4582	237	57	x}(λ	x}(λ	PROPN
ejpam-4582	237	58	,	,	PUNCT
ejpam-4582	237	59	s	s	PART
ejpam-4582	237	60	)	)	PUNCT
ejpam-4582	237	61	̸=	̸=	PROPN
ejpam-4582	237	62	{	{	PUNCT
ejpam-4582	237	63	y}(λ	y}(λ	PROPN
ejpam-4582	237	64	,	,	PUNCT
ejpam-4582	237	65	s	s	PROPN
ejpam-4582	237	66	)	)	PUNCT
ejpam-4582	237	67	.	.	PUNCT
ejpam-4582	238	1	thus	thus	ADV
ejpam-4582	238	2	,	,	PUNCT
ejpam-4582	238	3	γ(λ	γ(λ	PROPN
ejpam-4582	238	4	,	,	PUNCT
ejpam-4582	238	5	s)({x	s)({x	PROPN
ejpam-4582	238	6	}	}	PUNCT
ejpam-4582	238	7	)	)	PUNCT
ejpam-4582	238	8	̸=	̸=	PROPN
ejpam-4582	238	9	γ(λ	γ(λ	PROPN
ejpam-4582	238	10	,	,	PUNCT
ejpam-4582	238	11	s)({y	s)({y	NOUN
ejpam-4582	238	12	}	}	PUNCT
ejpam-4582	238	13	)	)	PUNCT
ejpam-4582	238	14	implies	imply	VERB
ejpam-4582	238	15	that	that	SCONJ
ejpam-4582	238	16	{	{	PUNCT
ejpam-4582	238	17	x}(λ	x}(λ	PROPN
ejpam-4582	238	18	,	,	PUNCT
ejpam-4582	238	19	s	s	PART
ejpam-4582	238	20	)	)	PUNCT
ejpam-4582	238	21	̸=	̸=	PROPN
ejpam-4582	238	22	{	{	PUNCT
ejpam-4582	238	23	y}(λ	y}(λ	PROPN
ejpam-4582	238	24	,	,	PUNCT
ejpam-4582	238	25	s	s	PROPN
ejpam-4582	238	26	)	)	PUNCT
ejpam-4582	238	27	.	.	PUNCT
ejpam-4582	239	1	(	(	PUNCT
ejpam-4582	239	2	2	2	X
ejpam-4582	239	3	)	)	PUNCT
ejpam-4582	239	4	⇒	⇒	NOUN
ejpam-4582	239	5	(	(	PUNCT
ejpam-4582	239	6	1	1	NUM
ejpam-4582	239	7	):	):	PUNCT
ejpam-4582	239	8	suppose	suppose	VERB
ejpam-4582	239	9	that	that	SCONJ
ejpam-4582	239	10	{	{	PUNCT
ejpam-4582	239	11	x}(λ	x}(λ	PROPN
ejpam-4582	239	12	,	,	PUNCT
ejpam-4582	239	13	s	s	PART
ejpam-4582	239	14	)	)	PUNCT
ejpam-4582	239	15	̸=	̸=	PROPN
ejpam-4582	239	16	{	{	PUNCT
ejpam-4582	239	17	y}(λ	y}(λ	PROPN
ejpam-4582	239	18	,	,	PUNCT
ejpam-4582	239	19	s	s	PROPN
ejpam-4582	239	20	)	)	PUNCT
ejpam-4582	239	21	.	.	PUNCT
ejpam-4582	240	1	then	then	ADV
ejpam-4582	240	2	,	,	PUNCT
ejpam-4582	240	3	there	there	PRON
ejpam-4582	240	4	exists	exist	VERB
ejpam-4582	240	5	a	a	DET
ejpam-4582	240	6	point	point	NOUN
ejpam-4582	240	7	z	z	NOUN
ejpam-4582	240	8	∈	∈	PROPN
ejpam-4582	240	9	x	x	PUNCT
ejpam-4582	240	10	such	such	ADJ
ejpam-4582	240	11	that	that	SCONJ
ejpam-4582	240	12	z	z	PROPN
ejpam-4582	240	13	∈	∈	PROPN
ejpam-4582	240	14	{	{	PUNCT
ejpam-4582	240	15	x}(λ	x}(λ	PROPN
ejpam-4582	240	16	,	,	PUNCT
ejpam-4582	240	17	s	s	PART
ejpam-4582	240	18	)	)	PUNCT
ejpam-4582	240	19	and	and	CCONJ
ejpam-4582	240	20	z	z	PROPN
ejpam-4582	240	21	̸∈	̸∈	PROPN
ejpam-4582	240	22	{	{	PUNCT
ejpam-4582	240	23	y}(λ	y}(λ	PROPN
ejpam-4582	240	24	,	,	PUNCT
ejpam-4582	240	25	s	s	PART
ejpam-4582	240	26	)	)	PUNCT
ejpam-4582	240	27	or	or	CCONJ
ejpam-4582	240	28	z	z	NOUN
ejpam-4582	240	29	∈	∈	PROPN
ejpam-4582	240	30	{	{	PUNCT
ejpam-4582	240	31	y}(λ	y}(λ	PROPN
ejpam-4582	240	32	,	,	PUNCT
ejpam-4582	240	33	s	s	PART
ejpam-4582	240	34	)	)	PUNCT
ejpam-4582	240	35	and	and	CCONJ
ejpam-4582	240	36	z	z	PROPN
ejpam-4582	240	37	̸∈	̸∈	PROPN
ejpam-4582	240	38	{	{	PUNCT
ejpam-4582	240	39	x}(λ	x}(λ	PROPN
ejpam-4582	240	40	,	,	PUNCT
ejpam-4582	240	41	s	s	PART
ejpam-4582	240	42	)	)	PUNCT
ejpam-4582	240	43	.	.	PUNCT
ejpam-4582	241	1	we	we	PRON
ejpam-4582	241	2	prove	prove	VERB
ejpam-4582	241	3	only	only	ADV
ejpam-4582	241	4	the	the	DET
ejpam-4582	241	5	first	first	ADJ
ejpam-4582	241	6	case	case	NOUN
ejpam-4582	241	7	being	be	AUX
ejpam-4582	241	8	the	the	DET
ejpam-4582	241	9	second	second	ADJ
ejpam-4582	241	10	analogous	analogous	NOUN
ejpam-4582	241	11	.	.	PUNCT
ejpam-4582	242	1	it	it	PRON
ejpam-4582	242	2	follows	follow	VERB
ejpam-4582	242	3	that	that	SCONJ
ejpam-4582	242	4	there	there	PRON
ejpam-4582	242	5	exists	exist	VERB
ejpam-4582	242	6	a	a	DET
ejpam-4582	242	7	(	(	PUNCT
ejpam-4582	242	8	λ	λ	NOUN
ejpam-4582	242	9	,	,	PUNCT
ejpam-4582	242	10	s)-open	s)-open	VERB
ejpam-4582	242	11	set	set	VERB
ejpam-4582	242	12	containing	contain	VERB
ejpam-4582	242	13	z	z	NOUN
ejpam-4582	242	14	and	and	CCONJ
ejpam-4582	242	15	therefore	therefore	ADV
ejpam-4582	242	16	x	x	X
ejpam-4582	242	17	but	but	CCONJ
ejpam-4582	242	18	not	not	PART
ejpam-4582	242	19	y	y	NOUN
ejpam-4582	242	20	,	,	PUNCT
ejpam-4582	242	21	namely	namely	ADV
ejpam-4582	242	22	,	,	PUNCT
ejpam-4582	242	23	y	y	PROPN
ejpam-4582	242	24	̸∈	̸∈	PROPN
ejpam-4582	242	25	γ(λ	γ(λ	PROPN
ejpam-4582	242	26	,	,	PUNCT
ejpam-4582	242	27	s)({x	s)({x	PROPN
ejpam-4582	242	28	}	}	PUNCT
ejpam-4582	242	29	)	)	PUNCT
ejpam-4582	242	30	and	and	CCONJ
ejpam-4582	242	31	thus	thus	ADV
ejpam-4582	242	32	γ(λ	γ(λ	PROPN
ejpam-4582	242	33	,	,	PUNCT
ejpam-4582	242	34	s)({x	s)({x	PROPN
ejpam-4582	242	35	}	}	PUNCT
ejpam-4582	242	36	)	)	PUNCT
ejpam-4582	243	1	̸=	̸=	PROPN
ejpam-4582	243	2	γ(λ	γ(λ	PROPN
ejpam-4582	243	3	,	,	PUNCT
ejpam-4582	243	4	s)({y	s)({y	NOUN
ejpam-4582	243	5	}	}	PUNCT
ejpam-4582	243	6	)	)	PUNCT
ejpam-4582	243	7	.	.	PUNCT
ejpam-4582	244	1	theorem	theorem	ADJ
ejpam-4582	244	2	4	4	NUM
ejpam-4582	244	3	.	.	X
ejpam-4582	245	1	for	for	ADP
ejpam-4582	245	2	any	any	DET
ejpam-4582	245	3	points	point	NOUN
ejpam-4582	245	4	x	x	PUNCT
ejpam-4582	245	5	and	and	CCONJ
ejpam-4582	245	6	y	y	PROPN
ejpam-4582	245	7	in	in	ADP
ejpam-4582	245	8	a	a	DET
ejpam-4582	245	9	topological	topological	ADJ
ejpam-4582	245	10	space	space	NOUN
ejpam-4582	245	11	(	(	PUNCT
ejpam-4582	245	12	x	x	X
ejpam-4582	245	13	,	,	PUNCT
ejpam-4582	245	14	τ	τ	PROPN
ejpam-4582	245	15	)	)	PUNCT
ejpam-4582	245	16	,	,	PUNCT
ejpam-4582	245	17	the	the	DET
ejpam-4582	245	18	following	follow	VERB
ejpam-4582	245	19	properties	property	NOUN
ejpam-4582	245	20	hold	hold	VERB
ejpam-4582	245	21	:	:	PUNCT
ejpam-4582	245	22	(	(	PUNCT
ejpam-4582	245	23	1	1	X
ejpam-4582	245	24	)	)	PUNCT
ejpam-4582	245	25	y	y	PROPN
ejpam-4582	245	26	∈	∈	PROPN
ejpam-4582	245	27	γ(λ	γ(λ	PROPN
ejpam-4582	245	28	,	,	PUNCT
ejpam-4582	245	29	s)({x	s)({x	PROPN
ejpam-4582	245	30	}	}	PUNCT
ejpam-4582	245	31	)	)	PUNCT
ejpam-4582	246	1	if	if	SCONJ
ejpam-4582	246	2	and	and	CCONJ
ejpam-4582	246	3	only	only	ADV
ejpam-4582	246	4	if	if	SCONJ
ejpam-4582	246	5	x	x	SYM
ejpam-4582	246	6	∈	∈	PROPN
ejpam-4582	246	7	{	{	PUNCT
ejpam-4582	246	8	y}(λ	y}(λ	PROPN
ejpam-4582	246	9	,	,	PUNCT
ejpam-4582	246	10	s	s	PROPN
ejpam-4582	246	11	)	)	PUNCT
ejpam-4582	246	12	.	.	PUNCT
ejpam-4582	247	1	(	(	PUNCT
ejpam-4582	247	2	2	2	X
ejpam-4582	247	3	)	)	PUNCT
ejpam-4582	247	4	γ(λ	γ(λ	PROPN
ejpam-4582	247	5	,	,	PUNCT
ejpam-4582	247	6	s)({x	s)({x	NOUN
ejpam-4582	247	7	}	}	PUNCT
ejpam-4582	247	8	)	)	PUNCT
ejpam-4582	248	1	=	=	SYM
ejpam-4582	248	2	γ(λ	γ(λ	PROPN
ejpam-4582	248	3	,	,	PUNCT
ejpam-4582	248	4	s)({y	s)({y	NOUN
ejpam-4582	248	5	}	}	PUNCT
ejpam-4582	248	6	)	)	PUNCT
ejpam-4582	248	7	if	if	SCONJ
ejpam-4582	248	8	and	and	CCONJ
ejpam-4582	248	9	only	only	ADV
ejpam-4582	248	10	if	if	SCONJ
ejpam-4582	248	11	{	{	PUNCT
ejpam-4582	248	12	x}(λ	x}(λ	PROPN
ejpam-4582	248	13	,	,	PUNCT
ejpam-4582	248	14	s	s	PART
ejpam-4582	248	15	)	)	PUNCT
ejpam-4582	248	16	=	=	SYM
ejpam-4582	248	17	{	{	PUNCT
ejpam-4582	248	18	y}(λ	y}(λ	PROPN
ejpam-4582	248	19	,	,	PUNCT
ejpam-4582	248	20	s	s	PART
ejpam-4582	248	21	)	)	PUNCT
ejpam-4582	248	22	.	.	PUNCT
ejpam-4582	249	1	proof	proof	NOUN
ejpam-4582	249	2	.	.	PUNCT
ejpam-4582	250	1	(	(	PUNCT
ejpam-4582	250	2	1	1	X
ejpam-4582	250	3	)	)	PUNCT
ejpam-4582	250	4	let	let	VERB
ejpam-4582	250	5	x	x	SYM
ejpam-4582	250	6	̸∈	̸∈	PROPN
ejpam-4582	250	7	{	{	PUNCT
ejpam-4582	250	8	y}(λ	y}(λ	PROPN
ejpam-4582	250	9	,	,	PUNCT
ejpam-4582	250	10	s	s	PROPN
ejpam-4582	250	11	)	)	PUNCT
ejpam-4582	250	12	.	.	PUNCT
ejpam-4582	251	1	then	then	ADV
ejpam-4582	251	2	,	,	PUNCT
ejpam-4582	251	3	there	there	PRON
ejpam-4582	251	4	exists	exist	VERB
ejpam-4582	251	5	u	u	PROPN
ejpam-4582	251	6	∈	∈	PROPN
ejpam-4582	251	7	(	(	PUNCT
ejpam-4582	251	8	λ	λ	NOUN
ejpam-4582	251	9	,	,	PUNCT
ejpam-4582	251	10	s)o(x	s)o(x	ADJ
ejpam-4582	251	11	)	)	PUNCT
ejpam-4582	251	12	such	such	ADJ
ejpam-4582	251	13	that	that	SCONJ
ejpam-4582	251	14	x	x	SYM
ejpam-4582	251	15	∈	∈	PROPN
ejpam-4582	251	16	u	u	NOUN
ejpam-4582	251	17	and	and	CCONJ
ejpam-4582	251	18	y	y	PROPN
ejpam-4582	251	19	̸∈	̸∈	PROPN
ejpam-4582	251	20	u	u	PROPN
ejpam-4582	251	21	.	.	PUNCT
ejpam-4582	252	1	thus	thus	ADV
ejpam-4582	252	2	,	,	PUNCT
ejpam-4582	252	3	y	y	PROPN
ejpam-4582	252	4	̸∈	̸∈	PROPN
ejpam-4582	252	5	γ(λ	γ(λ	PROPN
ejpam-4582	252	6	,	,	PUNCT
ejpam-4582	252	7	s)({x	s)({x	PROPN
ejpam-4582	252	8	}	}	PUNCT
ejpam-4582	252	9	)	)	PUNCT
ejpam-4582	252	10	.	.	PUNCT
ejpam-4582	253	1	the	the	DET
ejpam-4582	253	2	converse	converse	NOUN
ejpam-4582	253	3	is	be	AUX
ejpam-4582	253	4	similarly	similarly	ADV
ejpam-4582	253	5	shown	show	VERB
ejpam-4582	253	6	.	.	PUNCT
ejpam-4582	254	1	(	(	PUNCT
ejpam-4582	254	2	2	2	X
ejpam-4582	254	3	)	)	PUNCT
ejpam-4582	254	4	suppose	suppose	VERB
ejpam-4582	254	5	that	that	SCONJ
ejpam-4582	254	6	γ(λ	γ(λ	PROPN
ejpam-4582	254	7	,	,	PUNCT
ejpam-4582	254	8	s)({x	s)({x	NOUN
ejpam-4582	254	9	}	}	PUNCT
ejpam-4582	254	10	)	)	PUNCT
ejpam-4582	254	11	=	=	SYM
ejpam-4582	254	12	γ(λ	γ(λ	PROPN
ejpam-4582	254	13	,	,	PUNCT
ejpam-4582	254	14	s)({y	s)({y	NOUN
ejpam-4582	254	15	}	}	PUNCT
ejpam-4582	254	16	)	)	PUNCT
ejpam-4582	254	17	for	for	ADP
ejpam-4582	254	18	any	any	DET
ejpam-4582	254	19	x	x	NOUN
ejpam-4582	254	20	,	,	PUNCT
ejpam-4582	254	21	y	y	PROPN
ejpam-4582	254	22	∈	∈	PROPN
ejpam-4582	254	23	x.	x.	VERB
ejpam-4582	254	24	since	since	SCONJ
ejpam-4582	254	25	x	x	PROPN
ejpam-4582	254	26	∈	∈	PROPN
ejpam-4582	254	27	γ(λ	γ(λ	PROPN
ejpam-4582	254	28	,	,	PUNCT
ejpam-4582	254	29	s)({x	s)({x	PROPN
ejpam-4582	254	30	}	}	PUNCT
ejpam-4582	254	31	)	)	PUNCT
ejpam-4582	254	32	,	,	PUNCT
ejpam-4582	254	33	x	x	PUNCT
ejpam-4582	254	34	∈	∈	PROPN
ejpam-4582	254	35	γ(λ	γ(λ	PROPN
ejpam-4582	254	36	,	,	PUNCT
ejpam-4582	254	37	s)({y	s)({y	NOUN
ejpam-4582	254	38	}	}	PUNCT
ejpam-4582	254	39	)	)	PUNCT
ejpam-4582	254	40	and	and	CCONJ
ejpam-4582	254	41	by	by	ADP
ejpam-4582	254	42	(	(	PUNCT
ejpam-4582	254	43	1	1	NUM
ejpam-4582	254	44	)	)	PUNCT
ejpam-4582	254	45	,	,	PUNCT
ejpam-4582	254	46	y	y	PROPN
ejpam-4582	254	47	∈	∈	PROPN
ejpam-4582	254	48	{	{	PUNCT
ejpam-4582	254	49	x}(λ	x}(λ	PROPN
ejpam-4582	254	50	,	,	PUNCT
ejpam-4582	254	51	s	s	PART
ejpam-4582	254	52	)	)	PUNCT
ejpam-4582	254	53	.	.	PUNCT
ejpam-4582	255	1	by	by	ADP
ejpam-4582	255	2	lemma	lemma	PROPN
ejpam-4582	255	3	4	4	NUM
ejpam-4582	255	4	,	,	PUNCT
ejpam-4582	255	5	we	we	PRON
ejpam-4582	255	6	have	have	VERB
ejpam-4582	255	7	{	{	PUNCT
ejpam-4582	255	8	y}(λ	y}(λ	PROPN
ejpam-4582	255	9	,	,	PUNCT
ejpam-4582	255	10	s	s	PART
ejpam-4582	255	11	)	)	PUNCT
ejpam-4582	255	12	⊆	⊆	NUM
ejpam-4582	255	13	{	{	PUNCT
ejpam-4582	255	14	x}(λ	x}(λ	PROPN
ejpam-4582	255	15	,	,	PUNCT
ejpam-4582	255	16	s	s	PART
ejpam-4582	255	17	)	)	PUNCT
ejpam-4582	255	18	.	.	PUNCT
ejpam-4582	256	1	similarly	similarly	ADV
ejpam-4582	256	2	,	,	PUNCT
ejpam-4582	256	3	we	we	PRON
ejpam-4582	256	4	have	have	VERB
ejpam-4582	256	5	{	{	PUNCT
ejpam-4582	256	6	x}(λ	x}(λ	PROPN
ejpam-4582	256	7	,	,	PUNCT
ejpam-4582	256	8	s	s	PART
ejpam-4582	256	9	)	)	PUNCT
ejpam-4582	256	10	⊆	⊆	NUM
ejpam-4582	256	11	{	{	PUNCT
ejpam-4582	256	12	y}(λ	y}(λ	PROPN
ejpam-4582	256	13	,	,	PUNCT
ejpam-4582	256	14	s	s	PART
ejpam-4582	256	15	)	)	PUNCT
ejpam-4582	256	16	and	and	CCONJ
ejpam-4582	256	17	hence	hence	ADV
ejpam-4582	256	18	{	{	PUNCT
ejpam-4582	256	19	x}(λ	x}(λ	PROPN
ejpam-4582	256	20	,	,	PUNCT
ejpam-4582	256	21	s	s	PART
ejpam-4582	256	22	)	)	PUNCT
ejpam-4582	256	23	=	=	SYM
ejpam-4582	256	24	{	{	PUNCT
ejpam-4582	256	25	y}(λ	y}(λ	PROPN
ejpam-4582	256	26	,	,	PUNCT
ejpam-4582	256	27	s	s	PROPN
ejpam-4582	256	28	)	)	PUNCT
ejpam-4582	256	29	.	.	PUNCT
ejpam-4582	257	1	c.	c.	PROPN
ejpam-4582	257	2	boonpok	boonpok	PROPN
ejpam-4582	257	3	,	,	PUNCT
ejpam-4582	257	4	c.	c.	PROPN
ejpam-4582	257	5	viriyapong	viriyapong	PROPN
ejpam-4582	257	6	/	/	SYM
ejpam-4582	257	7	eur	eur	PROPN
ejpam-4582	257	8	.	.	PUNCT
ejpam-4582	258	1	j.	j.	PROPN
ejpam-4582	258	2	pure	pure	PROPN
ejpam-4582	258	3	appl	appl	PROPN
ejpam-4582	258	4	.	.	PROPN
ejpam-4582	258	5	math	math	PROPN
ejpam-4582	258	6	,	,	PUNCT
ejpam-4582	258	7	16	16	NUM
ejpam-4582	258	8	(	(	PUNCT
ejpam-4582	258	9	1	1	NUM
ejpam-4582	258	10	)	)	PUNCT
ejpam-4582	258	11	(	(	PUNCT
ejpam-4582	258	12	2023	2023	NUM
ejpam-4582	258	13	)	)	PUNCT
ejpam-4582	258	14	,	,	PUNCT
ejpam-4582	258	15	336	336	NUM
ejpam-4582	258	16	-	-	SYM
ejpam-4582	258	17	362	362	NUM
ejpam-4582	258	18	343	343	NUM
ejpam-4582	258	19	conversely	conversely	ADV
ejpam-4582	258	20	,	,	PUNCT
ejpam-4582	258	21	suppose	suppose	VERB
ejpam-4582	258	22	that	that	SCONJ
ejpam-4582	258	23	{	{	PUNCT
ejpam-4582	258	24	x}(λ	x}(λ	PROPN
ejpam-4582	258	25	,	,	PUNCT
ejpam-4582	258	26	s	s	PART
ejpam-4582	258	27	)	)	PUNCT
ejpam-4582	258	28	=	=	SYM
ejpam-4582	258	29	{	{	PUNCT
ejpam-4582	258	30	y}(λ	y}(λ	PROPN
ejpam-4582	258	31	,	,	PUNCT
ejpam-4582	258	32	s	s	PROPN
ejpam-4582	258	33	)	)	PUNCT
ejpam-4582	258	34	.	.	PUNCT
ejpam-4582	259	1	since	since	SCONJ
ejpam-4582	259	2	x	x	PROPN
ejpam-4582	259	3	∈	∈	PROPN
ejpam-4582	259	4	{	{	PUNCT
ejpam-4582	259	5	x}(λ	x}(λ	PROPN
ejpam-4582	259	6	,	,	PUNCT
ejpam-4582	259	7	s	s	PART
ejpam-4582	259	8	)	)	PUNCT
ejpam-4582	259	9	,	,	PUNCT
ejpam-4582	259	10	we	we	PRON
ejpam-4582	259	11	have	have	VERB
ejpam-4582	259	12	x	x	PART
ejpam-4582	259	13	∈	∈	PROPN
ejpam-4582	259	14	{	{	PUNCT
ejpam-4582	259	15	y}(λ	y}(λ	PROPN
ejpam-4582	259	16	,	,	PUNCT
ejpam-4582	259	17	s	s	PART
ejpam-4582	259	18	)	)	PUNCT
ejpam-4582	259	19	and	and	CCONJ
ejpam-4582	259	20	by	by	ADP
ejpam-4582	259	21	(	(	PUNCT
ejpam-4582	259	22	1	1	NUM
ejpam-4582	259	23	)	)	PUNCT
ejpam-4582	259	24	,	,	PUNCT
ejpam-4582	259	25	y	y	PROPN
ejpam-4582	259	26	∈	∈	PROPN
ejpam-4582	259	27	γ(λ	γ(λ	PROPN
ejpam-4582	259	28	,	,	PUNCT
ejpam-4582	259	29	s)({x	s)({x	PROPN
ejpam-4582	259	30	}	}	PUNCT
ejpam-4582	259	31	)	)	PUNCT
ejpam-4582	259	32	.	.	PUNCT
ejpam-4582	260	1	by	by	ADP
ejpam-4582	260	2	proposition	proposition	NOUN
ejpam-4582	260	3	3	3	NUM
ejpam-4582	260	4	,	,	PUNCT
ejpam-4582	260	5	γ(λ	γ(λ	PROPN
ejpam-4582	260	6	,	,	PUNCT
ejpam-4582	260	7	s)({y	s)({y	NOUN
ejpam-4582	260	8	}	}	PUNCT
ejpam-4582	260	9	)	)	PUNCT
ejpam-4582	260	10	⊆	⊆	NUM
ejpam-4582	260	11	γ(λ	γ(λ	NOUN
ejpam-4582	260	12	,	,	PUNCT
ejpam-4582	260	13	s)(γ(λ	s)(γ(λ	NOUN
ejpam-4582	260	14	,	,	PUNCT
ejpam-4582	260	15	s)({x	s)({x	PROPN
ejpam-4582	260	16	}	}	PUNCT
ejpam-4582	260	17	)	)	PUNCT
ejpam-4582	260	18	)	)	PUNCT
ejpam-4582	261	1	=	=	SYM
ejpam-4582	261	2	γ(λ	γ(λ	PROPN
ejpam-4582	261	3	,	,	PUNCT
ejpam-4582	261	4	s)({x	s)({x	PROPN
ejpam-4582	261	5	}	}	PUNCT
ejpam-4582	261	6	)	)	PUNCT
ejpam-4582	261	7	.	.	PUNCT
ejpam-4582	262	1	similarly	similarly	ADV
ejpam-4582	262	2	,	,	PUNCT
ejpam-4582	262	3	we	we	PRON
ejpam-4582	262	4	have	have	VERB
ejpam-4582	262	5	γ(λ	γ(λ	PROPN
ejpam-4582	262	6	,	,	PUNCT
ejpam-4582	262	7	s)({x	s)({x	PROPN
ejpam-4582	262	8	}	}	PUNCT
ejpam-4582	262	9	)	)	PUNCT
ejpam-4582	263	1	⊆	⊆	NUM
ejpam-4582	263	2	γ(λ	γ(λ	PROPN
ejpam-4582	263	3	,	,	PUNCT
ejpam-4582	263	4	s)({y	s)({y	NOUN
ejpam-4582	263	5	}	}	PUNCT
ejpam-4582	263	6	)	)	PUNCT
ejpam-4582	263	7	.	.	PUNCT
ejpam-4582	264	1	thus	thus	ADV
ejpam-4582	264	2	,	,	PUNCT
ejpam-4582	264	3	γ(λ	γ(λ	PROPN
ejpam-4582	264	4	,	,	PUNCT
ejpam-4582	264	5	s)({x	s)({x	NOUN
ejpam-4582	264	6	}	}	PUNCT
ejpam-4582	264	7	)	)	PUNCT
ejpam-4582	264	8	=	=	SYM
ejpam-4582	264	9	γ(λ	γ(λ	PROPN
ejpam-4582	264	10	,	,	PUNCT
ejpam-4582	264	11	s)({y	s)({y	NOUN
ejpam-4582	264	12	}	}	PUNCT
ejpam-4582	264	13	)	)	PUNCT
ejpam-4582	264	14	.	.	PUNCT
ejpam-4582	265	1	4	4	X
ejpam-4582	265	2	.	.	X
ejpam-4582	265	3	on	on	ADP
ejpam-4582	265	4	some	some	DET
ejpam-4582	265	5	low	low	ADJ
ejpam-4582	265	6	separation	separation	NOUN
ejpam-4582	265	7	axioms	axiom	NOUN
ejpam-4582	265	8	in	in	ADP
ejpam-4582	265	9	this	this	DET
ejpam-4582	265	10	section	section	NOUN
ejpam-4582	265	11	,	,	PUNCT
ejpam-4582	265	12	we	we	PRON
ejpam-4582	265	13	introduce	introduce	VERB
ejpam-4582	265	14	the	the	DET
ejpam-4582	265	15	notion	notion	NOUN
ejpam-4582	265	16	of	of	ADP
ejpam-4582	265	17	(	(	PUNCT
ejpam-4582	265	18	λ	λ	PROPN
ejpam-4582	265	19	,	,	PUNCT
ejpam-4582	265	20	s)-r0	s)-r0	PRON
ejpam-4582	265	21	spaces	space	VERB
ejpam-4582	265	22	and	and	CCONJ
ejpam-4582	265	23	investigate	investigate	VERB
ejpam-4582	265	24	some	some	DET
ejpam-4582	265	25	characterizations	characterization	NOUN
ejpam-4582	265	26	of	of	ADP
ejpam-4582	265	27	(	(	PUNCT
ejpam-4582	265	28	λ	λ	PROPN
ejpam-4582	265	29	,	,	PUNCT
ejpam-4582	265	30	s)-r0	s)-r0	PRON
ejpam-4582	265	31	spaces	space	VERB
ejpam-4582	265	32	.	.	PUNCT
ejpam-4582	266	1	definition	definition	NOUN
ejpam-4582	266	2	7	7	NUM
ejpam-4582	266	3	.	.	PUNCT
ejpam-4582	267	1	a	a	DET
ejpam-4582	267	2	topological	topological	ADJ
ejpam-4582	267	3	space	space	NOUN
ejpam-4582	267	4	(	(	PUNCT
ejpam-4582	267	5	x	x	X
ejpam-4582	267	6	,	,	PUNCT
ejpam-4582	267	7	τ	τ	X
ejpam-4582	267	8	)	)	PUNCT
ejpam-4582	267	9	is	be	AUX
ejpam-4582	267	10	called	call	VERB
ejpam-4582	267	11	(	(	PUNCT
ejpam-4582	267	12	λ	λ	PROPN
ejpam-4582	267	13	,	,	PUNCT
ejpam-4582	267	14	s)-r0	s)-r0	PRON
ejpam-4582	267	15	if	if	SCONJ
ejpam-4582	267	16	,	,	PUNCT
ejpam-4582	267	17	for	for	SCONJ
ejpam-4582	267	18	each	each	DET
ejpam-4582	267	19	(	(	PUNCT
ejpam-4582	267	20	λ	λ	PROPN
ejpam-4582	267	21	,	,	PUNCT
ejpam-4582	267	22	s)-open	s)-open	VERB
ejpam-4582	267	23	set	set	VERB
ejpam-4582	267	24	u	u	NOUN
ejpam-4582	267	25	and	and	CCONJ
ejpam-4582	267	26	each	each	DET
ejpam-4582	267	27	x	x	SYM
ejpam-4582	267	28	∈	∈	PROPN
ejpam-4582	267	29	u	u	NOUN
ejpam-4582	267	30	,	,	PUNCT
ejpam-4582	267	31	{	{	PUNCT
ejpam-4582	267	32	x}(λ	x}(λ	PROPN
ejpam-4582	267	33	,	,	PUNCT
ejpam-4582	267	34	s	s	PART
ejpam-4582	267	35	)	)	PUNCT
ejpam-4582	267	36	⊆	⊆	NUM
ejpam-4582	267	37	u	u	NOUN
ejpam-4582	267	38	.	.	PUNCT
ejpam-4582	268	1	theorem	theorem	ADJ
ejpam-4582	268	2	5	5	NUM
ejpam-4582	268	3	.	.	X
ejpam-4582	269	1	for	for	ADP
ejpam-4582	269	2	a	a	DET
ejpam-4582	269	3	topological	topological	ADJ
ejpam-4582	269	4	space	space	NOUN
ejpam-4582	269	5	(	(	PUNCT
ejpam-4582	269	6	x	x	X
ejpam-4582	269	7	,	,	PUNCT
ejpam-4582	269	8	τ	τ	PROPN
ejpam-4582	269	9	)	)	PUNCT
ejpam-4582	269	10	,	,	PUNCT
ejpam-4582	269	11	the	the	DET
ejpam-4582	269	12	following	follow	VERB
ejpam-4582	269	13	properties	property	NOUN
ejpam-4582	269	14	are	be	AUX
ejpam-4582	269	15	equivalent	equivalent	ADJ
ejpam-4582	269	16	:	:	PUNCT
ejpam-4582	269	17	(	(	PUNCT
ejpam-4582	269	18	1	1	X
ejpam-4582	269	19	)	)	PUNCT
ejpam-4582	269	20	(	(	PUNCT
ejpam-4582	269	21	x	x	X
ejpam-4582	269	22	,	,	PUNCT
ejpam-4582	269	23	τ	τ	X
ejpam-4582	269	24	)	)	PUNCT
ejpam-4582	269	25	is	be	AUX
ejpam-4582	269	26	(	(	PUNCT
ejpam-4582	269	27	λ	λ	INTJ
ejpam-4582	269	28	,	,	PUNCT
ejpam-4582	269	29	s)-r0	s)-r0	X
ejpam-4582	269	30	;	;	PUNCT
ejpam-4582	269	31	(	(	PUNCT
ejpam-4582	269	32	2	2	X
ejpam-4582	269	33	)	)	PUNCT
ejpam-4582	269	34	for	for	ADP
ejpam-4582	269	35	any	any	DET
ejpam-4582	269	36	f	f	PROPN
ejpam-4582	269	37	∈	∈	PROPN
ejpam-4582	269	38	(	(	PUNCT
ejpam-4582	269	39	λ	λ	NOUN
ejpam-4582	269	40	,	,	PUNCT
ejpam-4582	269	41	s)c(x	s)c(x	NOUN
ejpam-4582	269	42	)	)	PUNCT
ejpam-4582	269	43	,	,	PUNCT
ejpam-4582	269	44	x	x	PROPN
ejpam-4582	269	45	̸∈	̸∈	PROPN
ejpam-4582	269	46	f	f	PROPN
ejpam-4582	269	47	implies	imply	VERB
ejpam-4582	269	48	f	f	PROPN
ejpam-4582	269	49	⊆	⊆	NUM
ejpam-4582	269	50	u	u	NOUN
ejpam-4582	269	51	and	and	CCONJ
ejpam-4582	269	52	x	x	PUNCT
ejpam-4582	269	53	̸∈	̸∈	PROPN
ejpam-4582	269	54	u	u	PROPN
ejpam-4582	269	55	for	for	ADP
ejpam-4582	269	56	some	some	DET
ejpam-4582	269	57	u	u	NOUN
ejpam-4582	269	58	∈	∈	PROPN
ejpam-4582	269	59	(	(	PUNCT
ejpam-4582	269	60	λ	λ	NOUN
ejpam-4582	269	61	,	,	PUNCT
ejpam-4582	269	62	s)o(x	s)o(x	PROPN
ejpam-4582	269	63	)	)	PUNCT
ejpam-4582	269	64	;	;	PUNCT
ejpam-4582	269	65	(	(	PUNCT
ejpam-4582	269	66	3	3	X
ejpam-4582	269	67	)	)	PUNCT
ejpam-4582	269	68	for	for	ADP
ejpam-4582	269	69	any	any	DET
ejpam-4582	269	70	f	f	PROPN
ejpam-4582	269	71	∈	∈	PROPN
ejpam-4582	269	72	(	(	PUNCT
ejpam-4582	269	73	λ	λ	NOUN
ejpam-4582	269	74	,	,	PUNCT
ejpam-4582	269	75	s)c(x	s)c(x	NOUN
ejpam-4582	269	76	)	)	PUNCT
ejpam-4582	269	77	,	,	PUNCT
ejpam-4582	269	78	x	x	PROPN
ejpam-4582	269	79	̸∈	̸∈	PROPN
ejpam-4582	269	80	f	f	PROPN
ejpam-4582	269	81	implies	imply	VERB
ejpam-4582	269	82	f	f	PROPN
ejpam-4582	269	83	∩	∩	PROPN
ejpam-4582	269	84	{	{	PUNCT
ejpam-4582	269	85	x}(λ	x}(λ	PROPN
ejpam-4582	269	86	,	,	PUNCT
ejpam-4582	269	87	s	s	PART
ejpam-4582	269	88	)	)	PUNCT
ejpam-4582	269	89	=	=	SYM
ejpam-4582	269	90	∅	∅	NOUN
ejpam-4582	269	91	;	;	PUNCT
ejpam-4582	269	92	(	(	PUNCT
ejpam-4582	269	93	4	4	X
ejpam-4582	269	94	)	)	PUNCT
ejpam-4582	269	95	for	for	ADP
ejpam-4582	269	96	any	any	DET
ejpam-4582	269	97	distinct	distinct	ADJ
ejpam-4582	269	98	points	point	NOUN
ejpam-4582	269	99	x	x	PUNCT
ejpam-4582	269	100	and	and	CCONJ
ejpam-4582	269	101	y	y	PROPN
ejpam-4582	269	102	of	of	ADP
ejpam-4582	269	103	x	x	PRON
ejpam-4582	269	104	,	,	PUNCT
ejpam-4582	269	105	{	{	PUNCT
ejpam-4582	269	106	x}(λ	x}(λ	PROPN
ejpam-4582	269	107	,	,	PUNCT
ejpam-4582	269	108	s	s	PART
ejpam-4582	269	109	)	)	PUNCT
ejpam-4582	269	110	=	=	SYM
ejpam-4582	269	111	{	{	PUNCT
ejpam-4582	269	112	y}(λ	y}(λ	PROPN
ejpam-4582	269	113	,	,	PUNCT
ejpam-4582	269	114	s	s	PART
ejpam-4582	269	115	)	)	PUNCT
ejpam-4582	269	116	or	or	CCONJ
ejpam-4582	269	117	{	{	PUNCT
ejpam-4582	269	118	x}(λ	x}(λ	PROPN
ejpam-4582	269	119	,	,	PUNCT
ejpam-4582	269	120	s	s	PART
ejpam-4582	269	121	)	)	PUNCT
ejpam-4582	269	122	∩	∩	NOUN
ejpam-4582	269	123	{	{	PUNCT
ejpam-4582	269	124	y}(λ	y}(λ	PROPN
ejpam-4582	269	125	,	,	PUNCT
ejpam-4582	269	126	s	s	PART
ejpam-4582	269	127	)	)	PUNCT
ejpam-4582	269	128	=	=	PUNCT
ejpam-4582	269	129	∅.	∅.	NOUN
ejpam-4582	269	130	proof	proof	NOUN
ejpam-4582	269	131	.	.	PUNCT
ejpam-4582	270	1	(	(	PUNCT
ejpam-4582	270	2	1	1	X
ejpam-4582	270	3	)	)	PUNCT
ejpam-4582	270	4	⇒	⇒	NOUN
ejpam-4582	270	5	(	(	PUNCT
ejpam-4582	270	6	2	2	NUM
ejpam-4582	270	7	):	):	PUNCT
ejpam-4582	270	8	let	let	VERB
ejpam-4582	270	9	f	f	PROPN
ejpam-4582	270	10	∈	∈	PROPN
ejpam-4582	270	11	(	(	PUNCT
ejpam-4582	270	12	λ	λ	PROPN
ejpam-4582	270	13	,	,	PUNCT
ejpam-4582	270	14	s)c(x	s)c(x	NOUN
ejpam-4582	270	15	)	)	PUNCT
ejpam-4582	270	16	and	and	CCONJ
ejpam-4582	270	17	x	x	PUNCT
ejpam-4582	270	18	̸∈	̸∈	PROPN
ejpam-4582	270	19	f	f	PROPN
ejpam-4582	270	20	.	.	PUNCT
ejpam-4582	271	1	then	then	ADV
ejpam-4582	271	2	by	by	ADP
ejpam-4582	271	3	(	(	PUNCT
ejpam-4582	271	4	1	1	NUM
ejpam-4582	271	5	)	)	PUNCT
ejpam-4582	271	6	,	,	PUNCT
ejpam-4582	271	7	{	{	PUNCT
ejpam-4582	271	8	x}(λ	x}(λ	PROPN
ejpam-4582	271	9	,	,	PUNCT
ejpam-4582	271	10	s	s	PART
ejpam-4582	271	11	)	)	PUNCT
ejpam-4582	271	12	⊆	⊆	NUM
ejpam-4582	271	13	x	x	SYM
ejpam-4582	271	14	−	−	PROPN
ejpam-4582	271	15	f	f	X
ejpam-4582	271	16	.	.	PUNCT
ejpam-4582	272	1	put	put	VERB
ejpam-4582	272	2	u	u	NOUN
ejpam-4582	272	3	=	=	NOUN
ejpam-4582	272	4	x	x	SYM
ejpam-4582	272	5	−	−	PROPN
ejpam-4582	272	6	{	{	PUNCT
ejpam-4582	272	7	x}(λ	x}(λ	PROPN
ejpam-4582	272	8	,	,	PUNCT
ejpam-4582	272	9	s	s	PART
ejpam-4582	272	10	)	)	PUNCT
ejpam-4582	272	11	,	,	PUNCT
ejpam-4582	272	12	then	then	ADV
ejpam-4582	272	13	u	u	PROPN
ejpam-4582	272	14	∈	∈	PROPN
ejpam-4582	272	15	(	(	PUNCT
ejpam-4582	272	16	λ	λ	NOUN
ejpam-4582	272	17	,	,	PUNCT
ejpam-4582	272	18	s)o(x	s)o(x	NOUN
ejpam-4582	272	19	)	)	PUNCT
ejpam-4582	272	20	,	,	PUNCT
ejpam-4582	272	21	f	f	PROPN
ejpam-4582	272	22	⊆	⊆	NUM
ejpam-4582	272	23	u	u	NOUN
ejpam-4582	272	24	and	and	CCONJ
ejpam-4582	272	25	x	x	PUNCT
ejpam-4582	272	26	̸∈	̸∈	PROPN
ejpam-4582	272	27	u	u	PROPN
ejpam-4582	272	28	.	.	PUNCT
ejpam-4582	273	1	(	(	PUNCT
ejpam-4582	273	2	2	2	X
ejpam-4582	273	3	)	)	PUNCT
ejpam-4582	273	4	⇒	⇒	NOUN
ejpam-4582	273	5	(	(	PUNCT
ejpam-4582	273	6	3	3	NUM
ejpam-4582	273	7	):	):	PUNCT
ejpam-4582	273	8	let	let	VERB
ejpam-4582	273	9	f	f	PROPN
ejpam-4582	273	10	∈	∈	PROPN
ejpam-4582	273	11	(	(	PUNCT
ejpam-4582	273	12	λ	λ	PROPN
ejpam-4582	273	13	,	,	PUNCT
ejpam-4582	273	14	s)c(x	s)c(x	NOUN
ejpam-4582	273	15	)	)	PUNCT
ejpam-4582	273	16	and	and	CCONJ
ejpam-4582	273	17	x	x	PUNCT
ejpam-4582	273	18	̸∈	̸∈	PROPN
ejpam-4582	273	19	f	f	PROPN
ejpam-4582	273	20	.	.	PUNCT
ejpam-4582	274	1	there	there	PRON
ejpam-4582	274	2	exists	exist	VERB
ejpam-4582	274	3	u	u	PROPN
ejpam-4582	274	4	∈	∈	PROPN
ejpam-4582	274	5	(	(	PUNCT
ejpam-4582	274	6	λ	λ	NOUN
ejpam-4582	274	7	,	,	PUNCT
ejpam-4582	274	8	s)o(x	s)o(x	ADJ
ejpam-4582	274	9	)	)	PUNCT
ejpam-4582	274	10	such	such	ADJ
ejpam-4582	274	11	that	that	SCONJ
ejpam-4582	274	12	f	f	PROPN
ejpam-4582	274	13	⊆	⊆	NUM
ejpam-4582	274	14	u	u	NOUN
ejpam-4582	274	15	and	and	CCONJ
ejpam-4582	274	16	x	x	PUNCT
ejpam-4582	274	17	̸∈	̸∈	PROPN
ejpam-4582	274	18	u	u	PROPN
ejpam-4582	274	19	.	.	PUNCT
ejpam-4582	275	1	since	since	SCONJ
ejpam-4582	275	2	u	u	PROPN
ejpam-4582	275	3	∈	∈	PROPN
ejpam-4582	275	4	(	(	PUNCT
ejpam-4582	275	5	λ	λ	NOUN
ejpam-4582	275	6	,	,	PUNCT
ejpam-4582	275	7	s)o(x	s)o(x	ADJ
ejpam-4582	275	8	)	)	PUNCT
ejpam-4582	275	9	,	,	PUNCT
ejpam-4582	275	10	u	u	PROPN
ejpam-4582	275	11	∩	∩	NOUN
ejpam-4582	275	12	{	{	PUNCT
ejpam-4582	275	13	x}(λ	x}(λ	PROPN
ejpam-4582	275	14	,	,	PUNCT
ejpam-4582	275	15	s	s	PART
ejpam-4582	275	16	)	)	PUNCT
ejpam-4582	275	17	=	=	SYM
ejpam-4582	275	18	∅	∅	NOUN
ejpam-4582	275	19	and	and	CCONJ
ejpam-4582	275	20	f	f	PROPN
ejpam-4582	275	21	∩	∩	NOUN
ejpam-4582	275	22	{	{	PUNCT
ejpam-4582	275	23	x}(λ	x}(λ	PROPN
ejpam-4582	275	24	,	,	PUNCT
ejpam-4582	275	25	s	s	PART
ejpam-4582	275	26	)	)	PUNCT
ejpam-4582	275	27	=	=	SYM
ejpam-4582	275	28	∅.	∅.	X
ejpam-4582	275	29	(	(	PUNCT
ejpam-4582	275	30	3	3	NUM
ejpam-4582	275	31	)	)	PUNCT
ejpam-4582	275	32	⇒	⇒	NOUN
ejpam-4582	275	33	(	(	PUNCT
ejpam-4582	275	34	4	4	NUM
ejpam-4582	275	35	):	):	PUNCT
ejpam-4582	275	36	let	let	VERB
ejpam-4582	275	37	x	x	PRON
ejpam-4582	275	38	,	,	PUNCT
ejpam-4582	275	39	y	y	PROPN
ejpam-4582	275	40	be	be	VERB
ejpam-4582	275	41	distinct	distinct	ADJ
ejpam-4582	275	42	points	point	NOUN
ejpam-4582	275	43	of	of	ADP
ejpam-4582	275	44	x.	x.	NOUN
ejpam-4582	275	45	suppose	suppose	VERB
ejpam-4582	275	46	that	that	SCONJ
ejpam-4582	275	47	{	{	PUNCT
ejpam-4582	275	48	x}(λ	x}(λ	PROPN
ejpam-4582	275	49	,	,	PUNCT
ejpam-4582	275	50	s	s	PART
ejpam-4582	275	51	)	)	PUNCT
ejpam-4582	275	52	̸=	̸=	PROPN
ejpam-4582	275	53	{	{	PUNCT
ejpam-4582	275	54	y}(λ	y}(λ	PROPN
ejpam-4582	275	55	,	,	PUNCT
ejpam-4582	275	56	s	s	PROPN
ejpam-4582	275	57	)	)	PUNCT
ejpam-4582	275	58	.	.	PUNCT
ejpam-4582	276	1	by	by	ADP
ejpam-4582	276	2	(	(	PUNCT
ejpam-4582	276	3	3	3	NUM
ejpam-4582	276	4	)	)	PUNCT
ejpam-4582	276	5	,	,	PUNCT
ejpam-4582	276	6	x	x	PUNCT
ejpam-4582	276	7	∈	∈	PROPN
ejpam-4582	276	8	{	{	PUNCT
ejpam-4582	276	9	y}(λ	y}(λ	PROPN
ejpam-4582	276	10	,	,	PUNCT
ejpam-4582	276	11	s	s	PART
ejpam-4582	276	12	)	)	PUNCT
ejpam-4582	276	13	and	and	CCONJ
ejpam-4582	276	14	y	y	PROPN
ejpam-4582	276	15	∈	∈	PROPN
ejpam-4582	276	16	{	{	PUNCT
ejpam-4582	276	17	x}(λ	x}(λ	PROPN
ejpam-4582	276	18	,	,	PUNCT
ejpam-4582	276	19	s	s	PART
ejpam-4582	276	20	)	)	PUNCT
ejpam-4582	276	21	.	.	PUNCT
ejpam-4582	277	1	thus	thus	ADV
ejpam-4582	277	2	,	,	PUNCT
ejpam-4582	277	3	{	{	PUNCT
ejpam-4582	277	4	x}(λ	x}(λ	PROPN
ejpam-4582	277	5	,	,	PUNCT
ejpam-4582	277	6	s	s	PART
ejpam-4582	277	7	)	)	PUNCT
ejpam-4582	277	8	⊆	⊆	NUM
ejpam-4582	277	9	{	{	PUNCT
ejpam-4582	277	10	y}(λ	y}(λ	PROPN
ejpam-4582	277	11	,	,	PUNCT
ejpam-4582	277	12	s	s	PART
ejpam-4582	277	13	)	)	PUNCT
ejpam-4582	277	14	⊆	⊆	NUM
ejpam-4582	277	15	{	{	PUNCT
ejpam-4582	277	16	x}(λ	x}(λ	PROPN
ejpam-4582	277	17	,	,	PUNCT
ejpam-4582	277	18	s	s	PART
ejpam-4582	277	19	)	)	PUNCT
ejpam-4582	277	20	.	.	PUNCT
ejpam-4582	278	1	consequently	consequently	ADV
ejpam-4582	278	2	,	,	PUNCT
ejpam-4582	278	3	we	we	PRON
ejpam-4582	278	4	obtain	obtain	VERB
ejpam-4582	278	5	{	{	PUNCT
ejpam-4582	278	6	x}(λ	x}(λ	PROPN
ejpam-4582	278	7	,	,	PUNCT
ejpam-4582	278	8	s	s	PART
ejpam-4582	278	9	)	)	PUNCT
ejpam-4582	278	10	=	=	SYM
ejpam-4582	278	11	{	{	PUNCT
ejpam-4582	278	12	y}(λ	y}(λ	PROPN
ejpam-4582	278	13	,	,	PUNCT
ejpam-4582	278	14	s	s	PROPN
ejpam-4582	278	15	)	)	PUNCT
ejpam-4582	278	16	.	.	PUNCT
ejpam-4582	279	1	(	(	PUNCT
ejpam-4582	279	2	4	4	X
ejpam-4582	279	3	)	)	PUNCT
ejpam-4582	279	4	⇒	⇒	NOUN
ejpam-4582	279	5	(	(	PUNCT
ejpam-4582	279	6	1	1	NUM
ejpam-4582	279	7	):	):	PUNCT
ejpam-4582	279	8	let	let	VERB
ejpam-4582	279	9	v	v	X
ejpam-4582	279	10	∈	∈	PROPN
ejpam-4582	279	11	(	(	PUNCT
ejpam-4582	279	12	λ	λ	NOUN
ejpam-4582	279	13	,	,	PUNCT
ejpam-4582	279	14	s)o(x	s)o(x	ADJ
ejpam-4582	279	15	)	)	PUNCT
ejpam-4582	279	16	and	and	CCONJ
ejpam-4582	279	17	x	x	PUNCT
ejpam-4582	279	18	∈	∈	NOUN
ejpam-4582	279	19	v	v	NOUN
ejpam-4582	279	20	.	.	PUNCT
ejpam-4582	280	1	for	for	ADP
ejpam-4582	280	2	each	each	DET
ejpam-4582	280	3	y	y	PROPN
ejpam-4582	280	4	̸∈	̸∈	PROPN
ejpam-4582	280	5	v	v	PROPN
ejpam-4582	280	6	,	,	PUNCT
ejpam-4582	280	7	v	v	X
ejpam-4582	280	8	∩{y}(λ	∩{y}(λ	NOUN
ejpam-4582	280	9	,	,	PUNCT
ejpam-4582	280	10	s	s	PART
ejpam-4582	280	11	)	)	PUNCT
ejpam-4582	280	12	=	=	NOUN
ejpam-4582	280	13	∅	∅	NOUN
ejpam-4582	280	14	and	and	CCONJ
ejpam-4582	280	15	hence	hence	ADV
ejpam-4582	280	16	x	x	X
ejpam-4582	280	17	̸∈	̸∈	PROPN
ejpam-4582	280	18	{	{	PUNCT
ejpam-4582	280	19	y}(λ	y}(λ	PROPN
ejpam-4582	280	20	,	,	PUNCT
ejpam-4582	280	21	s	s	PROPN
ejpam-4582	280	22	)	)	PUNCT
ejpam-4582	280	23	.	.	PUNCT
ejpam-4582	281	1	thus	thus	ADV
ejpam-4582	281	2	,	,	PUNCT
ejpam-4582	281	3	{	{	PUNCT
ejpam-4582	281	4	x}(λ	x}(λ	PROPN
ejpam-4582	281	5	,	,	PUNCT
ejpam-4582	281	6	s	s	PART
ejpam-4582	281	7	)	)	PUNCT
ejpam-4582	281	8	̸=	̸=	PROPN
ejpam-4582	281	9	{	{	PUNCT
ejpam-4582	281	10	y}(λ	y}(λ	PROPN
ejpam-4582	281	11	,	,	PUNCT
ejpam-4582	281	12	s	s	PROPN
ejpam-4582	281	13	)	)	PUNCT
ejpam-4582	281	14	.	.	PUNCT
ejpam-4582	282	1	by	by	ADP
ejpam-4582	282	2	(	(	PUNCT
ejpam-4582	282	3	4	4	NUM
ejpam-4582	282	4	)	)	PUNCT
ejpam-4582	282	5	,	,	PUNCT
ejpam-4582	282	6	for	for	ADP
ejpam-4582	282	7	each	each	DET
ejpam-4582	282	8	y	y	PROPN
ejpam-4582	282	9	̸∈	̸∈	PROPN
ejpam-4582	282	10	v	v	PROPN
ejpam-4582	282	11	,	,	PUNCT
ejpam-4582	282	12	{	{	PUNCT
ejpam-4582	282	13	x}(λ	x}(λ	PROPN
ejpam-4582	282	14	,	,	PUNCT
ejpam-4582	282	15	s	s	PART
ejpam-4582	282	16	)	)	PUNCT
ejpam-4582	282	17	∩	∩	NOUN
ejpam-4582	282	18	{	{	PUNCT
ejpam-4582	282	19	y}(λ	y}(λ	PROPN
ejpam-4582	282	20	,	,	PUNCT
ejpam-4582	282	21	s	s	PART
ejpam-4582	282	22	)	)	PUNCT
ejpam-4582	282	23	=	=	PUNCT
ejpam-4582	282	24	∅.	∅.	ADV
ejpam-4582	282	25	since	since	SCONJ
ejpam-4582	282	26	x	x	PROPN
ejpam-4582	282	27	−	−	PROPN
ejpam-4582	282	28	v	v	NOUN
ejpam-4582	282	29	is	be	AUX
ejpam-4582	282	30	(	(	PUNCT
ejpam-4582	282	31	λ	λ	X
ejpam-4582	282	32	,	,	PUNCT
ejpam-4582	282	33	s)-closed	s)-close	VERB
ejpam-4582	282	34	,	,	PUNCT
ejpam-4582	282	35	{	{	PUNCT
ejpam-4582	282	36	y}(λ	y}(λ	PROPN
ejpam-4582	282	37	,	,	PUNCT
ejpam-4582	282	38	s	s	PART
ejpam-4582	282	39	)	)	PUNCT
ejpam-4582	282	40	⊆	⊆	NUM
ejpam-4582	282	41	x	x	SYM
ejpam-4582	282	42	−	−	NOUN
ejpam-4582	282	43	v	v	NOUN
ejpam-4582	282	44	and	and	CCONJ
ejpam-4582	282	45	x	x	NOUN
ejpam-4582	283	1	−	−	NOUN
ejpam-4582	283	2	v	v	NOUN
ejpam-4582	283	3	=	=	SYM
ejpam-4582	283	4	∪y∈x−v	∪y∈x−v	PROPN
ejpam-4582	283	5	{	{	PUNCT
ejpam-4582	283	6	y}(λ	y}(λ	PROPN
ejpam-4582	283	7	,	,	PUNCT
ejpam-4582	283	8	s	s	PROPN
ejpam-4582	283	9	)	)	PUNCT
ejpam-4582	283	10	.	.	PUNCT
ejpam-4582	284	1	thus	thus	ADV
ejpam-4582	284	2	,	,	PUNCT
ejpam-4582	284	3	(	(	PUNCT
ejpam-4582	284	4	x	x	X
ejpam-4582	284	5	−	−	PROPN
ejpam-4582	284	6	v	v	NOUN
ejpam-4582	284	7	)	)	PUNCT
ejpam-4582	284	8	∩	∩	NOUN
ejpam-4582	284	9	{	{	PUNCT
ejpam-4582	284	10	x}(λ	x}(λ	PROPN
ejpam-4582	284	11	,	,	PUNCT
ejpam-4582	284	12	s	s	PART
ejpam-4582	284	13	)	)	PUNCT
ejpam-4582	284	14	=	=	PUNCT
ejpam-4582	285	1	[	[	X
ejpam-4582	285	2	∪y∈x−v	∪y∈x−v	PROPN
ejpam-4582	285	3	{	{	PUNCT
ejpam-4582	285	4	y}(λ	y}(λ	PROPN
ejpam-4582	285	5	,	,	PUNCT
ejpam-4582	285	6	s	s	PART
ejpam-4582	285	7	)	)	PUNCT
ejpam-4582	285	8	]	]	PUNCT
ejpam-4582	285	9	∩	∩	NOUN
ejpam-4582	285	10	{	{	PUNCT
ejpam-4582	285	11	x}(λ	x}(λ	PROPN
ejpam-4582	285	12	,	,	PUNCT
ejpam-4582	285	13	s	s	PART
ejpam-4582	285	14	)	)	PUNCT
ejpam-4582	285	15	=	=	SYM
ejpam-4582	285	16	∪y∈x−v	∪y∈x−v	PROPN
ejpam-4582	286	1	[	[	X
ejpam-4582	286	2	{	{	PUNCT
ejpam-4582	286	3	y}(λ	y}(λ	PROPN
ejpam-4582	286	4	,	,	PUNCT
ejpam-4582	286	5	s	s	PART
ejpam-4582	286	6	)	)	PUNCT
ejpam-4582	286	7	∩	∩	NOUN
ejpam-4582	286	8	{	{	PUNCT
ejpam-4582	286	9	x}(λ	x}(λ	PROPN
ejpam-4582	286	10	,	,	PUNCT
ejpam-4582	286	11	s	s	PART
ejpam-4582	286	12	)	)	PUNCT
ejpam-4582	286	13	]	]	PUNCT
ejpam-4582	287	1	=	=	PUNCT
ejpam-4582	287	2	∅	∅	NOUN
ejpam-4582	287	3	and	and	CCONJ
ejpam-4582	287	4	hence	hence	ADV
ejpam-4582	287	5	{	{	PUNCT
ejpam-4582	287	6	x}(λ	x}(λ	PROPN
ejpam-4582	287	7	,	,	PUNCT
ejpam-4582	287	8	s	s	PART
ejpam-4582	287	9	)	)	PUNCT
ejpam-4582	287	10	⊆	⊆	NUM
ejpam-4582	287	11	v	v	NOUN
ejpam-4582	287	12	.	.	PUNCT
ejpam-4582	288	1	this	this	PRON
ejpam-4582	288	2	shows	show	VERB
ejpam-4582	288	3	that	that	SCONJ
ejpam-4582	288	4	(	(	PUNCT
ejpam-4582	288	5	x	x	X
ejpam-4582	288	6	,	,	PUNCT
ejpam-4582	288	7	τ	τ	X
ejpam-4582	288	8	)	)	PUNCT
ejpam-4582	288	9	is	be	AUX
ejpam-4582	288	10	a	a	DET
ejpam-4582	288	11	(	(	PUNCT
ejpam-4582	288	12	λ	λ	PROPN
ejpam-4582	288	13	,	,	PUNCT
ejpam-4582	288	14	s)-r0	s)-r0	NOUN
ejpam-4582	288	15	space	space	NOUN
ejpam-4582	288	16	.	.	PUNCT
ejpam-4582	289	1	corollary	corollary	ADJ
ejpam-4582	289	2	1	1	NUM
ejpam-4582	289	3	.	.	PUNCT
ejpam-4582	290	1	a	a	DET
ejpam-4582	290	2	topological	topological	ADJ
ejpam-4582	290	3	space	space	NOUN
ejpam-4582	290	4	(	(	PUNCT
ejpam-4582	290	5	x	x	X
ejpam-4582	290	6	,	,	PUNCT
ejpam-4582	290	7	τ	τ	X
ejpam-4582	290	8	)	)	PUNCT
ejpam-4582	290	9	is	be	AUX
ejpam-4582	290	10	(	(	PUNCT
ejpam-4582	290	11	λ	λ	X
ejpam-4582	290	12	,	,	PUNCT
ejpam-4582	290	13	s)-r0	s)-r0	PRON
ejpam-4582	290	14	if	if	SCONJ
ejpam-4582	290	15	and	and	CCONJ
ejpam-4582	290	16	only	only	ADV
ejpam-4582	290	17	if	if	SCONJ
ejpam-4582	290	18	for	for	ADP
ejpam-4582	290	19	any	any	DET
ejpam-4582	290	20	points	point	NOUN
ejpam-4582	290	21	x	x	X
ejpam-4582	290	22	,	,	PUNCT
ejpam-4582	290	23	y	y	PROPN
ejpam-4582	290	24	in	in	ADP
ejpam-4582	290	25	x	x	X
ejpam-4582	290	26	,	,	PUNCT
ejpam-4582	290	27	{	{	PUNCT
ejpam-4582	290	28	x}(λ	x}(λ	PROPN
ejpam-4582	290	29	,	,	PUNCT
ejpam-4582	290	30	s	s	PART
ejpam-4582	290	31	)	)	PUNCT
ejpam-4582	290	32	̸=	̸=	PROPN
ejpam-4582	290	33	{	{	PUNCT
ejpam-4582	290	34	y}(λ	y}(λ	PROPN
ejpam-4582	290	35	,	,	PUNCT
ejpam-4582	290	36	s	s	PART
ejpam-4582	290	37	)	)	PUNCT
ejpam-4582	290	38	implies	imply	VERB
ejpam-4582	290	39	{	{	PUNCT
ejpam-4582	290	40	x}(λ	x}(λ	PROPN
ejpam-4582	290	41	,	,	PUNCT
ejpam-4582	290	42	s	s	PART
ejpam-4582	290	43	)	)	PUNCT
ejpam-4582	290	44	∩	∩	NOUN
ejpam-4582	290	45	{	{	PUNCT
ejpam-4582	290	46	y}(λ	y}(λ	PROPN
ejpam-4582	290	47	,	,	PUNCT
ejpam-4582	290	48	s	s	PART
ejpam-4582	290	49	)	)	PUNCT
ejpam-4582	290	50	=	=	PUNCT
ejpam-4582	290	51	∅.	∅.	NOUN
ejpam-4582	290	52	proof	proof	NOUN
ejpam-4582	290	53	.	.	PUNCT
ejpam-4582	291	1	this	this	PRON
ejpam-4582	291	2	is	be	AUX
ejpam-4582	291	3	obvious	obvious	ADJ
ejpam-4582	291	4	by	by	ADP
ejpam-4582	291	5	theorem	theorem	NOUN
ejpam-4582	291	6	5	5	NUM
ejpam-4582	291	7	.	.	PUNCT
ejpam-4582	291	8	conversely	conversely	ADV
ejpam-4582	291	9	,	,	PUNCT
ejpam-4582	291	10	let	let	VERB
ejpam-4582	291	11	u	u	PRON
ejpam-4582	291	12	∈	∈	PROPN
ejpam-4582	291	13	(	(	PUNCT
ejpam-4582	291	14	λ	λ	NOUN
ejpam-4582	291	15	,	,	PUNCT
ejpam-4582	291	16	s)o(x	s)o(x	ADJ
ejpam-4582	291	17	)	)	PUNCT
ejpam-4582	291	18	and	and	CCONJ
ejpam-4582	291	19	x	x	PUNCT
ejpam-4582	291	20	∈	∈	PROPN
ejpam-4582	291	21	u	u	NOUN
ejpam-4582	291	22	.	.	PUNCT
ejpam-4582	292	1	if	if	SCONJ
ejpam-4582	292	2	y	y	PROPN
ejpam-4582	292	3	̸∈	̸∈	PROPN
ejpam-4582	292	4	u	u	PROPN
ejpam-4582	292	5	,	,	PUNCT
ejpam-4582	292	6	then	then	ADV
ejpam-4582	292	7	u	u	NOUN
ejpam-4582	292	8	∩	∩	NOUN
ejpam-4582	292	9	{	{	PUNCT
ejpam-4582	292	10	y}(λ	y}(λ	PROPN
ejpam-4582	292	11	,	,	PUNCT
ejpam-4582	292	12	s	s	PART
ejpam-4582	292	13	)	)	PUNCT
ejpam-4582	292	14	=	=	PUNCT
ejpam-4582	292	15	∅.	∅.	ADP
ejpam-4582	292	16	thus	thus	ADV
ejpam-4582	292	17	,	,	PUNCT
ejpam-4582	292	18	x	x	PROPN
ejpam-4582	292	19	̸∈	̸∈	PROPN
ejpam-4582	292	20	{	{	PUNCT
ejpam-4582	292	21	y}(λ	y}(λ	PROPN
ejpam-4582	292	22	,	,	PUNCT
ejpam-4582	292	23	s	s	PART
ejpam-4582	292	24	)	)	PUNCT
ejpam-4582	292	25	and	and	CCONJ
ejpam-4582	292	26	{	{	PUNCT
ejpam-4582	292	27	x}(λ	x}(λ	PROPN
ejpam-4582	292	28	,	,	PUNCT
ejpam-4582	292	29	s	s	PART
ejpam-4582	292	30	)	)	PUNCT
ejpam-4582	292	31	̸=	̸=	PROPN
ejpam-4582	292	32	{	{	PUNCT
ejpam-4582	292	33	y}(λ	y}(λ	PROPN
ejpam-4582	292	34	,	,	PUNCT
ejpam-4582	292	35	s	s	PROPN
ejpam-4582	292	36	)	)	PUNCT
ejpam-4582	292	37	.	.	PUNCT
ejpam-4582	293	1	by	by	ADP
ejpam-4582	293	2	the	the	DET
ejpam-4582	293	3	hypothesis	hypothesis	NOUN
ejpam-4582	293	4	,	,	PUNCT
ejpam-4582	293	5	{	{	PUNCT
ejpam-4582	293	6	x}(λ	x}(λ	PROPN
ejpam-4582	293	7	,	,	PUNCT
ejpam-4582	293	8	s	s	PART
ejpam-4582	293	9	)	)	PUNCT
ejpam-4582	293	10	∩	∩	NOUN
ejpam-4582	293	11	{	{	PUNCT
ejpam-4582	293	12	y}(λ	y}(λ	PROPN
ejpam-4582	293	13	,	,	PUNCT
ejpam-4582	293	14	s	s	PART
ejpam-4582	293	15	)	)	PUNCT
ejpam-4582	293	16	=	=	NOUN
ejpam-4582	293	17	∅	∅	NOUN
ejpam-4582	293	18	and	and	CCONJ
ejpam-4582	293	19	hence	hence	ADV
ejpam-4582	293	20	y	y	PROPN
ejpam-4582	293	21	̸∈	̸∈	PROPN
ejpam-4582	293	22	{	{	PUNCT
ejpam-4582	293	23	x}(λ	x}(λ	PROPN
ejpam-4582	293	24	,	,	PUNCT
ejpam-4582	293	25	s	s	PART
ejpam-4582	293	26	)	)	PUNCT
ejpam-4582	293	27	.	.	PUNCT
ejpam-4582	294	1	this	this	PRON
ejpam-4582	294	2	shows	show	VERB
ejpam-4582	294	3	that	that	SCONJ
ejpam-4582	294	4	{	{	PUNCT
ejpam-4582	294	5	x}(λ	x}(λ	PROPN
ejpam-4582	294	6	,	,	PUNCT
ejpam-4582	294	7	s	s	PART
ejpam-4582	294	8	)	)	PUNCT
ejpam-4582	294	9	⊆	⊆	NUM
ejpam-4582	294	10	u	u	NOUN
ejpam-4582	294	11	.	.	PUNCT
ejpam-4582	295	1	consequently	consequently	ADV
ejpam-4582	295	2	,	,	PUNCT
ejpam-4582	295	3	we	we	PRON
ejpam-4582	295	4	obtain	obtain	VERB
ejpam-4582	295	5	(	(	PUNCT
ejpam-4582	295	6	x	x	NOUN
ejpam-4582	295	7	,	,	PUNCT
ejpam-4582	295	8	τ	τ	X
ejpam-4582	295	9	)	)	PUNCT
ejpam-4582	295	10	is	be	AUX
ejpam-4582	295	11	(	(	PUNCT
ejpam-4582	295	12	λ	λ	X
ejpam-4582	295	13	,	,	PUNCT
ejpam-4582	295	14	s)-r0	s)-r0	X
ejpam-4582	295	15	.	.	PUNCT
ejpam-4582	296	1	c.	c.	PROPN
ejpam-4582	296	2	boonpok	boonpok	PROPN
ejpam-4582	296	3	,	,	PUNCT
ejpam-4582	296	4	c.	c.	PROPN
ejpam-4582	296	5	viriyapong	viriyapong	PROPN
ejpam-4582	296	6	/	/	SYM
ejpam-4582	296	7	eur	eur	PROPN
ejpam-4582	296	8	.	.	PUNCT
ejpam-4582	297	1	j.	j.	PROPN
ejpam-4582	297	2	pure	pure	PROPN
ejpam-4582	297	3	appl	appl	PROPN
ejpam-4582	297	4	.	.	PROPN
ejpam-4582	297	5	math	math	PROPN
ejpam-4582	297	6	,	,	PUNCT
ejpam-4582	297	7	16	16	NUM
ejpam-4582	297	8	(	(	PUNCT
ejpam-4582	297	9	1	1	NUM
ejpam-4582	297	10	)	)	PUNCT
ejpam-4582	297	11	(	(	PUNCT
ejpam-4582	297	12	2023	2023	NUM
ejpam-4582	297	13	)	)	PUNCT
ejpam-4582	297	14	,	,	PUNCT
ejpam-4582	297	15	336	336	NUM
ejpam-4582	297	16	-	-	SYM
ejpam-4582	297	17	362	362	NUM
ejpam-4582	297	18	344	344	NUM
ejpam-4582	297	19	theorem	theorem	VERB
ejpam-4582	297	20	6	6	NUM
ejpam-4582	297	21	.	.	PUNCT
ejpam-4582	298	1	a	a	DET
ejpam-4582	298	2	topological	topological	ADJ
ejpam-4582	298	3	space	space	NOUN
ejpam-4582	298	4	(	(	PUNCT
ejpam-4582	298	5	x	x	X
ejpam-4582	298	6	,	,	PUNCT
ejpam-4582	298	7	τ	τ	X
ejpam-4582	298	8	)	)	PUNCT
ejpam-4582	298	9	is	be	AUX
ejpam-4582	298	10	(	(	PUNCT
ejpam-4582	298	11	λ	λ	X
ejpam-4582	298	12	,	,	PUNCT
ejpam-4582	298	13	s)-r0	s)-r0	PRON
ejpam-4582	298	14	if	if	SCONJ
ejpam-4582	298	15	and	and	CCONJ
ejpam-4582	298	16	only	only	ADV
ejpam-4582	298	17	if	if	SCONJ
ejpam-4582	298	18	for	for	ADP
ejpam-4582	298	19	any	any	DET
ejpam-4582	298	20	x	x	NOUN
ejpam-4582	298	21	,	,	PUNCT
ejpam-4582	298	22	y	y	PROPN
ejpam-4582	298	23	∈	∈	PROPN
ejpam-4582	298	24	x	x	X
ejpam-4582	298	25	,	,	PUNCT
ejpam-4582	298	26	γ(λ	γ(λ	PROPN
ejpam-4582	298	27	,	,	PUNCT
ejpam-4582	298	28	s)({x	s)({x	PROPN
ejpam-4582	298	29	}	}	PUNCT
ejpam-4582	298	30	)	)	PUNCT
ejpam-4582	299	1	̸=	̸=	PROPN
ejpam-4582	299	2	γ(λ	γ(λ	PROPN
ejpam-4582	299	3	,	,	PUNCT
ejpam-4582	299	4	s)({y	s)({y	NOUN
ejpam-4582	299	5	}	}	PUNCT
ejpam-4582	299	6	)	)	PUNCT
ejpam-4582	299	7	implies	imply	VERB
ejpam-4582	299	8	γ(λ	γ(λ	PROPN
ejpam-4582	299	9	,	,	PUNCT
ejpam-4582	299	10	s)({x	s)({x	NOUN
ejpam-4582	299	11	}	}	PUNCT
ejpam-4582	299	12	)	)	PUNCT
ejpam-4582	299	13	∩	∩	NOUN
ejpam-4582	299	14	γ(λ	γ(λ	PROPN
ejpam-4582	299	15	,	,	PUNCT
ejpam-4582	299	16	s)({y	s)({y	NOUN
ejpam-4582	299	17	}	}	PUNCT
ejpam-4582	299	18	)	)	PUNCT
ejpam-4582	300	1	=	=	PUNCT
ejpam-4582	300	2	∅.	∅.	NOUN
ejpam-4582	300	3	proof	proof	NOUN
ejpam-4582	300	4	.	.	PUNCT
ejpam-4582	301	1	let	let	VERB
ejpam-4582	301	2	(	(	PUNCT
ejpam-4582	301	3	x	x	NOUN
ejpam-4582	301	4	,	,	PUNCT
ejpam-4582	301	5	τ	τ	X
ejpam-4582	301	6	)	)	PUNCT
ejpam-4582	301	7	be	be	VERB
ejpam-4582	301	8	a	a	DET
ejpam-4582	301	9	(	(	PUNCT
ejpam-4582	301	10	λ	λ	PROPN
ejpam-4582	301	11	,	,	PUNCT
ejpam-4582	301	12	s)-r0	s)-r0	NOUN
ejpam-4582	301	13	space	space	NOUN
ejpam-4582	301	14	.	.	PUNCT
ejpam-4582	302	1	suppose	suppose	VERB
ejpam-4582	302	2	that	that	SCONJ
ejpam-4582	302	3	γ(λ	γ(λ	PROPN
ejpam-4582	302	4	,	,	PUNCT
ejpam-4582	302	5	s)({x})∩γ(λ	s)({x})∩γ(λ	NOUN
ejpam-4582	302	6	,	,	PUNCT
ejpam-4582	302	7	s)({y	s)({y	PROPN
ejpam-4582	302	8	}	}	PUNCT
ejpam-4582	302	9	)	)	PUNCT
ejpam-4582	302	10	̸=	̸=	PROPN
ejpam-4582	302	11	∅.	∅.	ADV
ejpam-4582	302	12	let	let	VERB
ejpam-4582	302	13	z	z	NOUN
ejpam-4582	302	14	∈	∈	PROPN
ejpam-4582	302	15	γ(λ	γ(λ	PROPN
ejpam-4582	302	16	,	,	PUNCT
ejpam-4582	302	17	s)({x	s)({x	NOUN
ejpam-4582	302	18	}	}	PUNCT
ejpam-4582	302	19	)	)	PUNCT
ejpam-4582	302	20	∩	∩	NOUN
ejpam-4582	302	21	γ(λ	γ(λ	PROPN
ejpam-4582	302	22	,	,	PUNCT
ejpam-4582	302	23	s)({y	s)({y	NOUN
ejpam-4582	302	24	}	}	PUNCT
ejpam-4582	302	25	)	)	PUNCT
ejpam-4582	302	26	.	.	PUNCT
ejpam-4582	303	1	then	then	ADV
ejpam-4582	303	2	,	,	PUNCT
ejpam-4582	303	3	we	we	PRON
ejpam-4582	303	4	have	have	VERB
ejpam-4582	303	5	z	z	NOUN
ejpam-4582	303	6	∈	∈	PROPN
ejpam-4582	303	7	γ(λ	γ(λ	PROPN
ejpam-4582	303	8	,	,	PUNCT
ejpam-4582	303	9	s)({x	s)({x	NOUN
ejpam-4582	303	10	}	}	PUNCT
ejpam-4582	303	11	)	)	PUNCT
ejpam-4582	303	12	and	and	CCONJ
ejpam-4582	303	13	theorem	theorem	VERB
ejpam-4582	303	14	4	4	NUM
ejpam-4582	303	15	,	,	PUNCT
ejpam-4582	303	16	x	x	SYM
ejpam-4582	303	17	∈	∈	NOUN
ejpam-4582	303	18	{	{	PUNCT
ejpam-4582	303	19	z}(λ	z}(λ	PROPN
ejpam-4582	303	20	,	,	PUNCT
ejpam-4582	303	21	s	s	PART
ejpam-4582	303	22	)	)	PUNCT
ejpam-4582	303	23	.	.	PUNCT
ejpam-4582	304	1	thus	thus	ADV
ejpam-4582	304	2	,	,	PUNCT
ejpam-4582	304	3	x	x	SYM
ejpam-4582	304	4	∈	∈	PROPN
ejpam-4582	304	5	{	{	PUNCT
ejpam-4582	304	6	z}(λ	z}(λ	PROPN
ejpam-4582	304	7	,	,	PUNCT
ejpam-4582	304	8	s	s	PART
ejpam-4582	304	9	)	)	PUNCT
ejpam-4582	304	10	∩	∩	NOUN
ejpam-4582	304	11	{	{	PUNCT
ejpam-4582	304	12	x}(λ	x}(λ	PROPN
ejpam-4582	304	13	,	,	PUNCT
ejpam-4582	304	14	s	s	PART
ejpam-4582	304	15	)	)	PUNCT
ejpam-4582	304	16	and	and	CCONJ
ejpam-4582	304	17	by	by	ADP
ejpam-4582	304	18	corollary	corollary	ADJ
ejpam-4582	304	19	1	1	NUM
ejpam-4582	304	20	,	,	PUNCT
ejpam-4582	304	21	{	{	PUNCT
ejpam-4582	304	22	z}(λ	z}(λ	PROPN
ejpam-4582	304	23	,	,	PUNCT
ejpam-4582	304	24	s	s	PART
ejpam-4582	304	25	)	)	PUNCT
ejpam-4582	304	26	=	=	SYM
ejpam-4582	304	27	{	{	PUNCT
ejpam-4582	304	28	x}(λ	x}(λ	PROPN
ejpam-4582	304	29	,	,	PUNCT
ejpam-4582	304	30	s	s	PART
ejpam-4582	304	31	)	)	PUNCT
ejpam-4582	304	32	.	.	PUNCT
ejpam-4582	305	1	similarly	similarly	ADV
ejpam-4582	305	2	,	,	PUNCT
ejpam-4582	305	3	we	we	PRON
ejpam-4582	305	4	have	have	AUX
ejpam-4582	305	5	{	{	PUNCT
ejpam-4582	305	6	z}(λ	z}(λ	PROPN
ejpam-4582	305	7	,	,	PUNCT
ejpam-4582	305	8	s	s	PART
ejpam-4582	305	9	)	)	PUNCT
ejpam-4582	305	10	=	=	SYM
ejpam-4582	305	11	{	{	PUNCT
ejpam-4582	305	12	y}(λ	y}(λ	PROPN
ejpam-4582	305	13	,	,	PUNCT
ejpam-4582	305	14	s	s	PART
ejpam-4582	305	15	)	)	PUNCT
ejpam-4582	305	16	and	and	CCONJ
ejpam-4582	305	17	hence	hence	ADV
ejpam-4582	305	18	{	{	PUNCT
ejpam-4582	305	19	x}(λ	x}(λ	PROPN
ejpam-4582	305	20	,	,	PUNCT
ejpam-4582	305	21	s	s	PART
ejpam-4582	305	22	)	)	PUNCT
ejpam-4582	305	23	=	=	SYM
ejpam-4582	305	24	{	{	PUNCT
ejpam-4582	305	25	y}(λ	y}(λ	PROPN
ejpam-4582	305	26	,	,	PUNCT
ejpam-4582	305	27	s	s	PROPN
ejpam-4582	305	28	)	)	PUNCT
ejpam-4582	305	29	.	.	PUNCT
ejpam-4582	306	1	by	by	ADP
ejpam-4582	306	2	theorem	theorem	NOUN
ejpam-4582	306	3	4	4	NUM
ejpam-4582	306	4	,	,	PUNCT
ejpam-4582	306	5	γ(λ	γ(λ	PROPN
ejpam-4582	306	6	,	,	PUNCT
ejpam-4582	306	7	s)({x	s)({x	NOUN
ejpam-4582	306	8	}	}	PUNCT
ejpam-4582	306	9	)	)	PUNCT
ejpam-4582	306	10	=	=	SYM
ejpam-4582	306	11	γ(λ	γ(λ	PROPN
ejpam-4582	306	12	,	,	PUNCT
ejpam-4582	306	13	s)({y	s)({y	NOUN
ejpam-4582	306	14	}	}	PUNCT
ejpam-4582	306	15	)	)	PUNCT
ejpam-4582	306	16	.	.	PUNCT
ejpam-4582	307	1	conversely	conversely	ADV
ejpam-4582	307	2	,	,	PUNCT
ejpam-4582	307	3	suppose	suppose	VERB
ejpam-4582	307	4	that	that	SCONJ
ejpam-4582	307	5	{	{	PUNCT
ejpam-4582	307	6	x}(λ	x}(λ	PROPN
ejpam-4582	307	7	,	,	PUNCT
ejpam-4582	307	8	s	s	PART
ejpam-4582	307	9	)	)	PUNCT
ejpam-4582	307	10	̸=	̸=	PROPN
ejpam-4582	307	11	{	{	PUNCT
ejpam-4582	307	12	y}(λ	y}(λ	PROPN
ejpam-4582	307	13	,	,	PUNCT
ejpam-4582	307	14	s	s	PROPN
ejpam-4582	307	15	)	)	PUNCT
ejpam-4582	307	16	.	.	PUNCT
ejpam-4582	308	1	by	by	ADP
ejpam-4582	308	2	theorem	theorem	NOUN
ejpam-4582	308	3	4	4	NUM
ejpam-4582	308	4	,	,	PUNCT
ejpam-4582	308	5	γ(λ	γ(λ	PROPN
ejpam-4582	308	6	,	,	PUNCT
ejpam-4582	308	7	s)({x	s)({x	PROPN
ejpam-4582	308	8	}	}	PUNCT
ejpam-4582	308	9	)	)	PUNCT
ejpam-4582	308	10	̸=	̸=	PROPN
ejpam-4582	308	11	γ(λ	γ(λ	PROPN
ejpam-4582	308	12	,	,	PUNCT
ejpam-4582	308	13	s)({y	s)({y	NOUN
ejpam-4582	308	14	}	}	PUNCT
ejpam-4582	308	15	)	)	PUNCT
ejpam-4582	308	16	and	and	CCONJ
ejpam-4582	308	17	hence	hence	ADV
ejpam-4582	308	18	γ(λ	γ(λ	PROPN
ejpam-4582	308	19	,	,	PUNCT
ejpam-4582	308	20	s)({x	s)({x	NOUN
ejpam-4582	308	21	}	}	PUNCT
ejpam-4582	308	22	)	)	PUNCT
ejpam-4582	308	23	∩	∩	NOUN
ejpam-4582	308	24	γ(λ	γ(λ	PROPN
ejpam-4582	308	25	,	,	PUNCT
ejpam-4582	308	26	s)({y	s)({y	NOUN
ejpam-4582	308	27	}	}	PUNCT
ejpam-4582	308	28	)	)	PUNCT
ejpam-4582	308	29	=	=	PUNCT
ejpam-4582	308	30	∅.	∅.	ADP
ejpam-4582	308	31	thus	thus	ADV
ejpam-4582	308	32	,	,	PUNCT
ejpam-4582	308	33	{	{	PUNCT
ejpam-4582	308	34	x}(λ	x}(λ	PROPN
ejpam-4582	308	35	,	,	PUNCT
ejpam-4582	308	36	s	s	PART
ejpam-4582	308	37	)	)	PUNCT
ejpam-4582	308	38	∩	∩	NOUN
ejpam-4582	308	39	{	{	PUNCT
ejpam-4582	308	40	y}(λ	y}(λ	PROPN
ejpam-4582	308	41	,	,	PUNCT
ejpam-4582	308	42	s	s	PART
ejpam-4582	308	43	)	)	PUNCT
ejpam-4582	308	44	=	=	PUNCT
ejpam-4582	308	45	∅.	∅.	NOUN
ejpam-4582	308	46	in	in	ADP
ejpam-4582	308	47	fact	fact	NOUN
ejpam-4582	308	48	,	,	PUNCT
ejpam-4582	308	49	assume	assume	VERB
ejpam-4582	308	50	z	z	PROPN
ejpam-4582	308	51	∈	∈	PROPN
ejpam-4582	308	52	{	{	PUNCT
ejpam-4582	308	53	x}(λ	x}(λ	PROPN
ejpam-4582	308	54	,	,	PUNCT
ejpam-4582	308	55	s	s	PART
ejpam-4582	308	56	)	)	PUNCT
ejpam-4582	308	57	∩	∩	NOUN
ejpam-4582	308	58	{	{	PUNCT
ejpam-4582	308	59	y}(λ	y}(λ	PROPN
ejpam-4582	308	60	,	,	PUNCT
ejpam-4582	308	61	s	s	PROPN
ejpam-4582	308	62	)	)	PUNCT
ejpam-4582	308	63	.	.	PUNCT
ejpam-4582	309	1	then	then	ADV
ejpam-4582	309	2	,	,	PUNCT
ejpam-4582	309	3	we	we	PRON
ejpam-4582	309	4	have	have	VERB
ejpam-4582	309	5	z	z	PROPN
ejpam-4582	309	6	∈	∈	PROPN
ejpam-4582	309	7	{	{	PUNCT
ejpam-4582	309	8	x}(λ	x}(λ	PROPN
ejpam-4582	309	9	,	,	PUNCT
ejpam-4582	309	10	s	s	PART
ejpam-4582	309	11	)	)	PUNCT
ejpam-4582	309	12	implies	imply	VERB
ejpam-4582	309	13	x	x	X
ejpam-4582	309	14	∈	∈	PROPN
ejpam-4582	309	15	γ(λ	γ(λ	PROPN
ejpam-4582	309	16	,	,	PUNCT
ejpam-4582	309	17	s)({z	s)({z	ADJ
ejpam-4582	309	18	}	}	PUNCT
ejpam-4582	309	19	)	)	PUNCT
ejpam-4582	309	20	and	and	CCONJ
ejpam-4582	309	21	hence	hence	ADV
ejpam-4582	309	22	x	x	X
ejpam-4582	309	23	∈	∈	PROPN
ejpam-4582	309	24	γ(λ	γ(λ	PROPN
ejpam-4582	309	25	,	,	PUNCT
ejpam-4582	309	26	s)({z})∩γ(λ	s)({z})∩γ(λ	NOUN
ejpam-4582	309	27	,	,	PUNCT
ejpam-4582	309	28	s)({x	s)({x	PROPN
ejpam-4582	309	29	}	}	PUNCT
ejpam-4582	309	30	)	)	PUNCT
ejpam-4582	309	31	.	.	PUNCT
ejpam-4582	310	1	by	by	ADP
ejpam-4582	310	2	the	the	DET
ejpam-4582	310	3	hypothesis	hypothesis	NOUN
ejpam-4582	310	4	,	,	PUNCT
ejpam-4582	310	5	γ(λ	γ(λ	PROPN
ejpam-4582	310	6	,	,	PUNCT
ejpam-4582	310	7	s)({z	s)({z	ADJ
ejpam-4582	310	8	}	}	PUNCT
ejpam-4582	310	9	)	)	PUNCT
ejpam-4582	310	10	=	=	SYM
ejpam-4582	310	11	γ(λ	γ(λ	PROPN
ejpam-4582	310	12	,	,	PUNCT
ejpam-4582	310	13	s)({x	s)({x	NOUN
ejpam-4582	310	14	}	}	PUNCT
ejpam-4582	310	15	)	)	PUNCT
ejpam-4582	310	16	and	and	CCONJ
ejpam-4582	310	17	by	by	ADP
ejpam-4582	310	18	theorem	theorem	NOUN
ejpam-4582	310	19	4	4	NUM
ejpam-4582	310	20	,	,	PUNCT
ejpam-4582	310	21	{	{	PUNCT
ejpam-4582	310	22	z}(λ	z}(λ	PROPN
ejpam-4582	310	23	,	,	PUNCT
ejpam-4582	310	24	s	s	PART
ejpam-4582	310	25	)	)	PUNCT
ejpam-4582	310	26	=	=	SYM
ejpam-4582	310	27	{	{	PUNCT
ejpam-4582	310	28	x}(λ	x}(λ	PROPN
ejpam-4582	310	29	,	,	PUNCT
ejpam-4582	310	30	s	s	PART
ejpam-4582	310	31	)	)	PUNCT
ejpam-4582	310	32	.	.	PUNCT
ejpam-4582	311	1	similarly	similarly	ADV
ejpam-4582	311	2	,	,	PUNCT
ejpam-4582	311	3	we	we	PRON
ejpam-4582	311	4	have	have	AUX
ejpam-4582	311	5	{	{	PUNCT
ejpam-4582	311	6	z}(λ	z}(λ	PROPN
ejpam-4582	311	7	,	,	PUNCT
ejpam-4582	311	8	s	s	PART
ejpam-4582	311	9	)	)	PUNCT
ejpam-4582	311	10	=	=	SYM
ejpam-4582	311	11	{	{	PUNCT
ejpam-4582	311	12	y}(λ	y}(λ	PROPN
ejpam-4582	311	13	,	,	PUNCT
ejpam-4582	311	14	s	s	PART
ejpam-4582	311	15	)	)	PUNCT
ejpam-4582	311	16	and	and	CCONJ
ejpam-4582	311	17	hence	hence	ADV
ejpam-4582	311	18	{	{	PUNCT
ejpam-4582	311	19	x}(λ	x}(λ	PROPN
ejpam-4582	311	20	,	,	PUNCT
ejpam-4582	311	21	s	s	PART
ejpam-4582	311	22	)	)	PUNCT
ejpam-4582	311	23	=	=	SYM
ejpam-4582	311	24	{	{	PUNCT
ejpam-4582	311	25	y}(λ	y}(λ	PROPN
ejpam-4582	311	26	,	,	PUNCT
ejpam-4582	311	27	s	s	PROPN
ejpam-4582	311	28	)	)	PUNCT
ejpam-4582	311	29	.	.	PUNCT
ejpam-4582	312	1	this	this	PRON
ejpam-4582	312	2	contradicts	contradict	VERB
ejpam-4582	312	3	that	that	SCONJ
ejpam-4582	312	4	{	{	PUNCT
ejpam-4582	312	5	x}(λ	x}(λ	PROPN
ejpam-4582	312	6	,	,	PUNCT
ejpam-4582	312	7	s	s	PART
ejpam-4582	312	8	)	)	PUNCT
ejpam-4582	312	9	̸=	̸=	PROPN
ejpam-4582	312	10	{	{	PUNCT
ejpam-4582	312	11	y}(λ	y}(λ	PROPN
ejpam-4582	312	12	,	,	PUNCT
ejpam-4582	312	13	s	s	PROPN
ejpam-4582	312	14	)	)	PUNCT
ejpam-4582	312	15	.	.	PUNCT
ejpam-4582	313	1	thus	thus	ADV
ejpam-4582	313	2	,	,	PUNCT
ejpam-4582	313	3	{	{	PUNCT
ejpam-4582	313	4	x}(λ	x}(λ	PROPN
ejpam-4582	313	5	,	,	PUNCT
ejpam-4582	313	6	s	s	PART
ejpam-4582	313	7	)	)	PUNCT
ejpam-4582	313	8	∩{y}(λ	∩{y}(λ	PROPN
ejpam-4582	313	9	,	,	PUNCT
ejpam-4582	313	10	s	s	PART
ejpam-4582	313	11	)	)	PUNCT
ejpam-4582	313	12	=	=	SYM
ejpam-4582	313	13	∅	∅	NOUN
ejpam-4582	313	14	and	and	CCONJ
ejpam-4582	313	15	by	by	ADP
ejpam-4582	313	16	theorem	theorem	NOUN
ejpam-4582	313	17	4	4	NUM
ejpam-4582	313	18	,	,	PUNCT
ejpam-4582	313	19	(	(	PUNCT
ejpam-4582	313	20	x	x	X
ejpam-4582	313	21	,	,	PUNCT
ejpam-4582	313	22	τ	τ	X
ejpam-4582	313	23	)	)	PUNCT
ejpam-4582	313	24	is	be	AUX
ejpam-4582	313	25	a	a	DET
ejpam-4582	313	26	(	(	PUNCT
ejpam-4582	313	27	λ	λ	PROPN
ejpam-4582	313	28	,	,	PUNCT
ejpam-4582	313	29	s)-r0	s)-r0	NOUN
ejpam-4582	313	30	space	space	NOUN
ejpam-4582	313	31	.	.	PUNCT
ejpam-4582	314	1	theorem	theorem	VERB
ejpam-4582	314	2	7	7	NUM
ejpam-4582	314	3	.	.	X
ejpam-4582	314	4	for	for	ADP
ejpam-4582	314	5	a	a	DET
ejpam-4582	314	6	topological	topological	ADJ
ejpam-4582	314	7	space	space	NOUN
ejpam-4582	314	8	(	(	PUNCT
ejpam-4582	314	9	x	x	X
ejpam-4582	314	10	,	,	PUNCT
ejpam-4582	314	11	τ	τ	PROPN
ejpam-4582	314	12	)	)	PUNCT
ejpam-4582	314	13	,	,	PUNCT
ejpam-4582	314	14	the	the	DET
ejpam-4582	314	15	following	follow	VERB
ejpam-4582	314	16	properties	property	NOUN
ejpam-4582	314	17	are	be	AUX
ejpam-4582	314	18	equivalent	equivalent	ADJ
ejpam-4582	314	19	:	:	PUNCT
ejpam-4582	314	20	(	(	PUNCT
ejpam-4582	314	21	1	1	X
ejpam-4582	314	22	)	)	PUNCT
ejpam-4582	314	23	(	(	PUNCT
ejpam-4582	314	24	x	x	X
ejpam-4582	314	25	,	,	PUNCT
ejpam-4582	314	26	τ	τ	X
ejpam-4582	314	27	)	)	PUNCT
ejpam-4582	314	28	is	be	AUX
ejpam-4582	314	29	(	(	PUNCT
ejpam-4582	314	30	λ	λ	INTJ
ejpam-4582	314	31	,	,	PUNCT
ejpam-4582	314	32	s)-r0	s)-r0	X
ejpam-4582	314	33	;	;	PUNCT
ejpam-4582	314	34	(	(	PUNCT
ejpam-4582	314	35	2	2	X
ejpam-4582	314	36	)	)	PUNCT
ejpam-4582	314	37	{	{	PUNCT
ejpam-4582	314	38	x}(λ	x}(λ	PROPN
ejpam-4582	314	39	,	,	PUNCT
ejpam-4582	314	40	s	s	PART
ejpam-4582	314	41	)	)	PUNCT
ejpam-4582	314	42	=	=	SYM
ejpam-4582	315	1	γ(λ	γ(λ	PROPN
ejpam-4582	315	2	,	,	PUNCT
ejpam-4582	315	3	s)({x	s)({x	PROPN
ejpam-4582	315	4	}	}	PUNCT
ejpam-4582	315	5	)	)	PUNCT
ejpam-4582	315	6	for	for	ADP
ejpam-4582	315	7	each	each	DET
ejpam-4582	315	8	x	x	SYM
ejpam-4582	315	9	∈	∈	PROPN
ejpam-4582	315	10	x.	x.	NOUN
ejpam-4582	315	11	proof	proof	NOUN
ejpam-4582	315	12	.	.	PUNCT
ejpam-4582	316	1	(	(	PUNCT
ejpam-4582	316	2	1	1	X
ejpam-4582	316	3	)	)	PUNCT
ejpam-4582	316	4	⇒	⇒	NOUN
ejpam-4582	316	5	(	(	PUNCT
ejpam-4582	316	6	2	2	NUM
ejpam-4582	316	7	):	):	PUNCT
ejpam-4582	316	8	let	let	VERB
ejpam-4582	316	9	x	x	PUNCT
ejpam-4582	316	10	∈	∈	PROPN
ejpam-4582	316	11	x	x	X
ejpam-4582	316	12	and	and	CCONJ
ejpam-4582	316	13	x	x	X
ejpam-4582	316	14	̸∈	̸∈	PROPN
ejpam-4582	316	15	γ(λ	γ(λ	PROPN
ejpam-4582	316	16	,	,	PUNCT
ejpam-4582	316	17	s)({x	s)({x	PROPN
ejpam-4582	316	18	}	}	PUNCT
ejpam-4582	316	19	)	)	PUNCT
ejpam-4582	316	20	.	.	PUNCT
ejpam-4582	317	1	there	there	PRON
ejpam-4582	317	2	exists	exist	VERB
ejpam-4582	317	3	v	v	ADP
ejpam-4582	317	4	∈	∈	PROPN
ejpam-4582	317	5	(	(	PUNCT
ejpam-4582	317	6	λ	λ	NOUN
ejpam-4582	317	7	,	,	PUNCT
ejpam-4582	317	8	s)o(x	s)o(x	ADJ
ejpam-4582	317	9	)	)	PUNCT
ejpam-4582	317	10	such	such	ADJ
ejpam-4582	317	11	that	that	SCONJ
ejpam-4582	317	12	x	x	SYM
ejpam-4582	317	13	∈	∈	PROPN
ejpam-4582	317	14	v	v	NOUN
ejpam-4582	317	15	and	and	CCONJ
ejpam-4582	317	16	y	y	PROPN
ejpam-4582	317	17	̸∈	̸∈	PROPN
ejpam-4582	317	18	v	v	PROPN
ejpam-4582	317	19	;	;	PUNCT
ejpam-4582	317	20	hence	hence	ADV
ejpam-4582	317	21	{	{	PUNCT
ejpam-4582	317	22	y}(λ	y}(λ	PROPN
ejpam-4582	317	23	,	,	PUNCT
ejpam-4582	317	24	s	s	PART
ejpam-4582	317	25	)	)	PUNCT
ejpam-4582	317	26	∩	∩	ADJ
ejpam-4582	317	27	v	v	NOUN
ejpam-4582	317	28	=	=	PUNCT
ejpam-4582	317	29	∅.	∅.	NOUN
ejpam-4582	317	30	since	since	SCONJ
ejpam-4582	317	31	(	(	PUNCT
ejpam-4582	317	32	x	x	X
ejpam-4582	317	33	,	,	PUNCT
ejpam-4582	317	34	τ	τ	X
ejpam-4582	317	35	)	)	PUNCT
ejpam-4582	317	36	is	be	AUX
ejpam-4582	317	37	(	(	PUNCT
ejpam-4582	317	38	λ	λ	INTJ
ejpam-4582	317	39	,	,	PUNCT
ejpam-4582	317	40	s)-r0	s)-r0	PRON
ejpam-4582	317	41	,	,	PUNCT
ejpam-4582	317	42	we	we	PRON
ejpam-4582	317	43	have	have	VERB
ejpam-4582	317	44	{	{	PUNCT
ejpam-4582	317	45	x}(λ	x}(λ	PROPN
ejpam-4582	317	46	,	,	PUNCT
ejpam-4582	317	47	s	s	PART
ejpam-4582	317	48	)	)	PUNCT
ejpam-4582	317	49	⊆	⊆	NUM
ejpam-4582	317	50	v	v	NOUN
ejpam-4582	317	51	.	.	PUNCT
ejpam-4582	318	1	therefore	therefore	ADV
ejpam-4582	318	2	,	,	PUNCT
ejpam-4582	318	3	{	{	PUNCT
ejpam-4582	318	4	y}(λ	y}(λ	PROPN
ejpam-4582	318	5	,	,	PUNCT
ejpam-4582	318	6	s	s	PART
ejpam-4582	318	7	)	)	PUNCT
ejpam-4582	318	8	∩	∩	NOUN
ejpam-4582	318	9	{	{	PUNCT
ejpam-4582	318	10	x}(λ	x}(λ	PROPN
ejpam-4582	318	11	,	,	PUNCT
ejpam-4582	318	12	s	s	PART
ejpam-4582	318	13	)	)	PUNCT
ejpam-4582	318	14	=	=	PUNCT
ejpam-4582	318	15	∅.	∅.	ADP
ejpam-4582	318	16	thus	thus	ADV
ejpam-4582	318	17	,	,	PUNCT
ejpam-4582	318	18	y	y	PROPN
ejpam-4582	318	19	̸∈	̸∈	PROPN
ejpam-4582	318	20	{	{	PUNCT
ejpam-4582	318	21	x}(λ	x}(λ	PROPN
ejpam-4582	318	22	,	,	PUNCT
ejpam-4582	318	23	s	s	PART
ejpam-4582	318	24	)	)	PUNCT
ejpam-4582	318	25	and	and	CCONJ
ejpam-4582	318	26	hence	hence	ADV
ejpam-4582	318	27	{	{	PUNCT
ejpam-4582	318	28	x}(λ	x}(λ	PROPN
ejpam-4582	318	29	,	,	PUNCT
ejpam-4582	318	30	s	s	PART
ejpam-4582	318	31	)	)	PUNCT
ejpam-4582	318	32	⊆	⊆	NUM
ejpam-4582	318	33	γ(λ	γ(λ	PROPN
ejpam-4582	318	34	,	,	PUNCT
ejpam-4582	318	35	s)({x	s)({x	PROPN
ejpam-4582	318	36	}	}	PUNCT
ejpam-4582	318	37	)	)	PUNCT
ejpam-4582	318	38	.	.	PUNCT
ejpam-4582	319	1	by	by	ADP
ejpam-4582	319	2	corollary	corollary	ADJ
ejpam-4582	319	3	1	1	NUM
ejpam-4582	319	4	,	,	PUNCT
ejpam-4582	319	5	{	{	PUNCT
ejpam-4582	319	6	x}(λ	x}(λ	PROPN
ejpam-4582	319	7	,	,	PUNCT
ejpam-4582	319	8	s	s	PART
ejpam-4582	319	9	)	)	PUNCT
ejpam-4582	319	10	=	=	SYM
ejpam-4582	319	11	{	{	PUNCT
ejpam-4582	319	12	y}(λ	y}(λ	PROPN
ejpam-4582	319	13	,	,	PUNCT
ejpam-4582	319	14	s	s	PROPN
ejpam-4582	319	15	)	)	PUNCT
ejpam-4582	319	16	.	.	PUNCT
ejpam-4582	320	1	thus	thus	ADV
ejpam-4582	320	2	,	,	PUNCT
ejpam-4582	320	3	y	y	PROPN
ejpam-4582	320	4	∈	∈	PROPN
ejpam-4582	320	5	{	{	PUNCT
ejpam-4582	320	6	x}(λ	x}(λ	PROPN
ejpam-4582	320	7	,	,	PUNCT
ejpam-4582	320	8	s	s	PART
ejpam-4582	320	9	)	)	PUNCT
ejpam-4582	320	10	and	and	CCONJ
ejpam-4582	320	11	so	so	ADV
ejpam-4582	320	12	γ(λ	γ(λ	PROPN
ejpam-4582	320	13	,	,	PUNCT
ejpam-4582	320	14	s)({x	s)({x	PROPN
ejpam-4582	320	15	}	}	PUNCT
ejpam-4582	320	16	)	)	PUNCT
ejpam-4582	321	1	⊆	⊆	X
ejpam-4582	321	2	{	{	PUNCT
ejpam-4582	321	3	x}(λ	x}(λ	PROPN
ejpam-4582	321	4	,	,	PUNCT
ejpam-4582	321	5	s	s	PART
ejpam-4582	321	6	)	)	PUNCT
ejpam-4582	321	7	.	.	PUNCT
ejpam-4582	322	1	this	this	PRON
ejpam-4582	322	2	shows	show	VERB
ejpam-4582	322	3	that	that	SCONJ
ejpam-4582	322	4	{	{	PUNCT
ejpam-4582	322	5	x}(λ	x}(λ	PROPN
ejpam-4582	322	6	,	,	PUNCT
ejpam-4582	322	7	s	s	PART
ejpam-4582	322	8	)	)	PUNCT
ejpam-4582	322	9	=	=	SYM
ejpam-4582	322	10	γ(λ	γ(λ	PROPN
ejpam-4582	322	11	,	,	PUNCT
ejpam-4582	322	12	s)({x	s)({x	PROPN
ejpam-4582	322	13	}	}	PUNCT
ejpam-4582	322	14	)	)	PUNCT
ejpam-4582	322	15	.	.	PUNCT
ejpam-4582	323	1	(	(	PUNCT
ejpam-4582	323	2	2	2	X
ejpam-4582	323	3	)	)	PUNCT
ejpam-4582	323	4	⇒	⇒	NOUN
ejpam-4582	323	5	(	(	PUNCT
ejpam-4582	323	6	1	1	NUM
ejpam-4582	323	7	):	):	PUNCT
ejpam-4582	323	8	let	let	VERB
ejpam-4582	323	9	u	u	PRON
ejpam-4582	323	10	∈	∈	PROPN
ejpam-4582	323	11	(	(	PUNCT
ejpam-4582	323	12	λ	λ	NOUN
ejpam-4582	323	13	,	,	PUNCT
ejpam-4582	323	14	s)o(x	s)o(x	ADJ
ejpam-4582	323	15	)	)	PUNCT
ejpam-4582	323	16	and	and	CCONJ
ejpam-4582	323	17	x	x	PUNCT
ejpam-4582	323	18	∈	∈	PROPN
ejpam-4582	323	19	u	u	NOUN
ejpam-4582	323	20	.	.	PUNCT
ejpam-4582	324	1	by	by	ADP
ejpam-4582	324	2	(	(	PUNCT
ejpam-4582	324	3	2	2	NUM
ejpam-4582	324	4	)	)	PUNCT
ejpam-4582	324	5	and	and	CCONJ
ejpam-4582	324	6	proposition	proposition	NOUN
ejpam-4582	324	7	3	3	NUM
ejpam-4582	324	8	,	,	PUNCT
ejpam-4582	324	9	{	{	PUNCT
ejpam-4582	324	10	x}(λ	x}(λ	PROPN
ejpam-4582	324	11	,	,	PUNCT
ejpam-4582	324	12	s	s	PART
ejpam-4582	324	13	)	)	PUNCT
ejpam-4582	324	14	=	=	SYM
ejpam-4582	324	15	γ(λ	γ(λ	PROPN
ejpam-4582	324	16	,	,	PUNCT
ejpam-4582	324	17	s)({x	s)({x	NOUN
ejpam-4582	324	18	}	}	PUNCT
ejpam-4582	324	19	)	)	PUNCT
ejpam-4582	324	20	⊆	⊆	NUM
ejpam-4582	324	21	γ(λ	γ(λ	NOUN
ejpam-4582	324	22	,	,	PUNCT
ejpam-4582	324	23	s)(u	s)(u	NOUN
ejpam-4582	324	24	)	)	PUNCT
ejpam-4582	324	25	=	=	VERB
ejpam-4582	324	26	u.	u.	PROPN
ejpam-4582	324	27	consequently	consequently	ADV
ejpam-4582	324	28	,	,	PUNCT
ejpam-4582	324	29	we	we	PRON
ejpam-4582	324	30	obtain	obtain	VERB
ejpam-4582	324	31	(	(	PUNCT
ejpam-4582	324	32	x	x	NOUN
ejpam-4582	324	33	,	,	PUNCT
ejpam-4582	324	34	τ	τ	X
ejpam-4582	324	35	)	)	PUNCT
ejpam-4582	324	36	is	be	AUX
ejpam-4582	324	37	a	a	DET
ejpam-4582	324	38	(	(	PUNCT
ejpam-4582	324	39	λ	λ	PROPN
ejpam-4582	324	40	,	,	PUNCT
ejpam-4582	324	41	s)-r0	s)-r0	NOUN
ejpam-4582	324	42	space	space	NOUN
ejpam-4582	324	43	.	.	PUNCT
ejpam-4582	325	1	corollary	corollary	ADJ
ejpam-4582	325	2	2	2	NUM
ejpam-4582	325	3	.	.	PUNCT
ejpam-4582	326	1	let	let	VERB
ejpam-4582	326	2	(	(	PUNCT
ejpam-4582	326	3	x	x	NOUN
ejpam-4582	326	4	,	,	PUNCT
ejpam-4582	326	5	τ	τ	X
ejpam-4582	326	6	)	)	PUNCT
ejpam-4582	326	7	be	be	VERB
ejpam-4582	326	8	a	a	DET
ejpam-4582	326	9	(	(	PUNCT
ejpam-4582	326	10	λ	λ	NOUN
ejpam-4582	326	11	,	,	PUNCT
ejpam-4582	326	12	s)-r0	s)-r0	X
ejpam-4582	326	13	topological	topological	ADJ
ejpam-4582	326	14	space	space	NOUN
ejpam-4582	326	15	and	and	CCONJ
ejpam-4582	326	16	x	x	PUNCT
ejpam-4582	326	17	∈	∈	PROPN
ejpam-4582	326	18	x.	x.	NOUN
ejpam-4582	327	1	if	if	SCONJ
ejpam-4582	327	2	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	327	3	,	,	PUNCT
ejpam-4582	327	4	s	s	NOUN
ejpam-4582	327	5	)	)	PUNCT
ejpam-4582	327	6	=	=	SYM
ejpam-4582	327	7	{	{	PUNCT
ejpam-4582	327	8	x	x	NOUN
ejpam-4582	327	9	}	}	PUNCT
ejpam-4582	327	10	,	,	PUNCT
ejpam-4582	327	11	then	then	ADV
ejpam-4582	327	12	{	{	PUNCT
ejpam-4582	327	13	x}(λ	x}(λ	PROPN
ejpam-4582	327	14	,	,	PUNCT
ejpam-4582	327	15	s	s	PART
ejpam-4582	327	16	)	)	PUNCT
ejpam-4582	327	17	=	=	SYM
ejpam-4582	327	18	{	{	PUNCT
ejpam-4582	327	19	x	x	NOUN
ejpam-4582	327	20	}	}	PUNCT
ejpam-4582	327	21	.	.	PUNCT
ejpam-4582	328	1	proof	proof	NOUN
ejpam-4582	328	2	.	.	PUNCT
ejpam-4582	329	1	this	this	PRON
ejpam-4582	329	2	is	be	AUX
ejpam-4582	329	3	a	a	DET
ejpam-4582	329	4	consequence	consequence	NOUN
ejpam-4582	329	5	of	of	ADP
ejpam-4582	329	6	theorem	theorem	ADJ
ejpam-4582	329	7	7	7	NUM
ejpam-4582	329	8	.	.	PUNCT
ejpam-4582	329	9	theorem	theorem	NOUN
ejpam-4582	329	10	8	8	NUM
ejpam-4582	329	11	.	.	PUNCT
ejpam-4582	330	1	for	for	ADP
ejpam-4582	330	2	a	a	DET
ejpam-4582	330	3	topological	topological	ADJ
ejpam-4582	330	4	space	space	NOUN
ejpam-4582	330	5	(	(	PUNCT
ejpam-4582	330	6	x	x	X
ejpam-4582	330	7	,	,	PUNCT
ejpam-4582	330	8	τ	τ	PROPN
ejpam-4582	330	9	)	)	PUNCT
ejpam-4582	330	10	,	,	PUNCT
ejpam-4582	330	11	the	the	DET
ejpam-4582	330	12	following	follow	VERB
ejpam-4582	330	13	properties	property	NOUN
ejpam-4582	330	14	are	be	AUX
ejpam-4582	330	15	equivalent	equivalent	ADJ
ejpam-4582	330	16	:	:	PUNCT
ejpam-4582	330	17	(	(	PUNCT
ejpam-4582	330	18	1	1	X
ejpam-4582	330	19	)	)	PUNCT
ejpam-4582	330	20	(	(	PUNCT
ejpam-4582	330	21	x	x	X
ejpam-4582	330	22	,	,	PUNCT
ejpam-4582	330	23	τ	τ	X
ejpam-4582	330	24	)	)	PUNCT
ejpam-4582	330	25	is	be	AUX
ejpam-4582	330	26	(	(	PUNCT
ejpam-4582	330	27	λ	λ	INTJ
ejpam-4582	330	28	,	,	PUNCT
ejpam-4582	330	29	s)-r0	s)-r0	X
ejpam-4582	330	30	;	;	PUNCT
ejpam-4582	330	31	(	(	PUNCT
ejpam-4582	330	32	2	2	X
ejpam-4582	330	33	)	)	PUNCT
ejpam-4582	330	34	x	x	SYM
ejpam-4582	330	35	∈	∈	PROPN
ejpam-4582	330	36	{	{	PUNCT
ejpam-4582	330	37	y}(λ	y}(λ	PROPN
ejpam-4582	330	38	,	,	PUNCT
ejpam-4582	330	39	s	s	PART
ejpam-4582	330	40	)	)	PUNCT
ejpam-4582	330	41	if	if	SCONJ
ejpam-4582	330	42	and	and	CCONJ
ejpam-4582	330	43	only	only	ADV
ejpam-4582	330	44	if	if	SCONJ
ejpam-4582	330	45	y	y	PROPN
ejpam-4582	330	46	∈	∈	PROPN
ejpam-4582	330	47	{	{	PUNCT
ejpam-4582	330	48	x}(λ	x}(λ	PROPN
ejpam-4582	330	49	,	,	PUNCT
ejpam-4582	330	50	s	s	PART
ejpam-4582	330	51	)	)	PUNCT
ejpam-4582	330	52	.	.	PUNCT
ejpam-4582	331	1	c.	c.	PROPN
ejpam-4582	331	2	boonpok	boonpok	PROPN
ejpam-4582	331	3	,	,	PUNCT
ejpam-4582	331	4	c.	c.	PROPN
ejpam-4582	331	5	viriyapong	viriyapong	PROPN
ejpam-4582	331	6	/	/	SYM
ejpam-4582	331	7	eur	eur	PROPN
ejpam-4582	331	8	.	.	PUNCT
ejpam-4582	332	1	j.	j.	PROPN
ejpam-4582	332	2	pure	pure	PROPN
ejpam-4582	332	3	appl	appl	PROPN
ejpam-4582	332	4	.	.	PROPN
ejpam-4582	332	5	math	math	PROPN
ejpam-4582	332	6	,	,	PUNCT
ejpam-4582	332	7	16	16	NUM
ejpam-4582	332	8	(	(	PUNCT
ejpam-4582	332	9	1	1	NUM
ejpam-4582	332	10	)	)	PUNCT
ejpam-4582	332	11	(	(	PUNCT
ejpam-4582	332	12	2023	2023	NUM
ejpam-4582	332	13	)	)	PUNCT
ejpam-4582	332	14	,	,	PUNCT
ejpam-4582	332	15	336	336	NUM
ejpam-4582	332	16	-	-	SYM
ejpam-4582	332	17	362	362	NUM
ejpam-4582	332	18	345	345	NUM
ejpam-4582	332	19	proof	proof	NOUN
ejpam-4582	332	20	.	.	PUNCT
ejpam-4582	333	1	(	(	PUNCT
ejpam-4582	333	2	1	1	X
ejpam-4582	333	3	)	)	PUNCT
ejpam-4582	333	4	⇒	⇒	NOUN
ejpam-4582	333	5	(	(	PUNCT
ejpam-4582	333	6	2	2	NUM
ejpam-4582	333	7	):	):	PUNCT
ejpam-4582	333	8	suppose	suppose	VERB
ejpam-4582	333	9	that	that	SCONJ
ejpam-4582	333	10	x	x	SYM
ejpam-4582	333	11	∈	∈	PROPN
ejpam-4582	333	12	{	{	PUNCT
ejpam-4582	333	13	y}(λ	y}(λ	PROPN
ejpam-4582	333	14	,	,	PUNCT
ejpam-4582	333	15	s	s	PROPN
ejpam-4582	333	16	)	)	PUNCT
ejpam-4582	333	17	.	.	PUNCT
ejpam-4582	334	1	by	by	ADP
ejpam-4582	334	2	theorem	theorem	NOUN
ejpam-4582	334	3	4	4	NUM
ejpam-4582	334	4	,	,	PUNCT
ejpam-4582	334	5	y	y	PROPN
ejpam-4582	334	6	∈	∈	PROPN
ejpam-4582	334	7	γ(λ	γ(λ	PROPN
ejpam-4582	334	8	,	,	PUNCT
ejpam-4582	334	9	s)({x	s)({x	NOUN
ejpam-4582	334	10	}	}	PUNCT
ejpam-4582	334	11	)	)	PUNCT
ejpam-4582	334	12	and	and	CCONJ
ejpam-4582	334	13	hence	hence	ADV
ejpam-4582	334	14	γ(λ	γ(λ	PROPN
ejpam-4582	334	15	,	,	PUNCT
ejpam-4582	334	16	s)({x	s)({x	NOUN
ejpam-4582	334	17	}	}	PUNCT
ejpam-4582	334	18	)	)	PUNCT
ejpam-4582	334	19	∩	∩	NOUN
ejpam-4582	334	20	γ(λ	γ(λ	PROPN
ejpam-4582	334	21	,	,	PUNCT
ejpam-4582	334	22	s)({y	s)({y	NOUN
ejpam-4582	334	23	}	}	PUNCT
ejpam-4582	334	24	)	)	PUNCT
ejpam-4582	334	25	̸=	̸=	PROPN
ejpam-4582	334	26	∅.	∅.	VERB
ejpam-4582	334	27	by	by	ADP
ejpam-4582	334	28	theorem	theorem	ADJ
ejpam-4582	334	29	6	6	NUM
ejpam-4582	334	30	,	,	PUNCT
ejpam-4582	334	31	γ(λ	γ(λ	PROPN
ejpam-4582	334	32	,	,	PUNCT
ejpam-4582	334	33	s)({x	s)({x	NOUN
ejpam-4582	334	34	}	}	PUNCT
ejpam-4582	334	35	)	)	PUNCT
ejpam-4582	334	36	=	=	SYM
ejpam-4582	334	37	γ(λ	γ(λ	PROPN
ejpam-4582	334	38	,	,	PUNCT
ejpam-4582	334	39	s)({y	s)({y	NOUN
ejpam-4582	334	40	}	}	PUNCT
ejpam-4582	334	41	)	)	PUNCT
ejpam-4582	334	42	and	and	CCONJ
ejpam-4582	334	43	so	so	ADV
ejpam-4582	334	44	x	x	SYM
ejpam-4582	334	45	∈	∈	PROPN
ejpam-4582	334	46	γ(λ	γ(λ	PROPN
ejpam-4582	334	47	,	,	PUNCT
ejpam-4582	334	48	s)({y	s)({y	NOUN
ejpam-4582	334	49	}	}	PUNCT
ejpam-4582	334	50	)	)	PUNCT
ejpam-4582	334	51	.	.	PUNCT
ejpam-4582	335	1	thus	thus	ADV
ejpam-4582	335	2	,	,	PUNCT
ejpam-4582	335	3	by	by	ADP
ejpam-4582	335	4	theorem	theorem	NOUN
ejpam-4582	335	5	4	4	NUM
ejpam-4582	335	6	,	,	PUNCT
ejpam-4582	335	7	y	y	PROPN
ejpam-4582	335	8	∈	∈	PROPN
ejpam-4582	335	9	{	{	PUNCT
ejpam-4582	335	10	x}(λ	x}(λ	PROPN
ejpam-4582	335	11	,	,	PUNCT
ejpam-4582	335	12	s	s	PART
ejpam-4582	335	13	)	)	PUNCT
ejpam-4582	335	14	.	.	PUNCT
ejpam-4582	336	1	the	the	DET
ejpam-4582	336	2	converse	converse	NOUN
ejpam-4582	336	3	is	be	AUX
ejpam-4582	336	4	similarly	similarly	ADV
ejpam-4582	336	5	shown	show	VERB
ejpam-4582	336	6	.	.	PUNCT
ejpam-4582	337	1	(	(	PUNCT
ejpam-4582	337	2	2	2	X
ejpam-4582	337	3	)	)	PUNCT
ejpam-4582	337	4	⇒	⇒	NOUN
ejpam-4582	337	5	(	(	PUNCT
ejpam-4582	337	6	1	1	NUM
ejpam-4582	337	7	):	):	PUNCT
ejpam-4582	337	8	let	let	VERB
ejpam-4582	337	9	u	u	PRON
ejpam-4582	337	10	∈	∈	PROPN
ejpam-4582	337	11	(	(	PUNCT
ejpam-4582	337	12	λ	λ	NOUN
ejpam-4582	337	13	,	,	PUNCT
ejpam-4582	337	14	s)o(x	s)o(x	ADJ
ejpam-4582	337	15	)	)	PUNCT
ejpam-4582	337	16	and	and	CCONJ
ejpam-4582	337	17	x	x	PUNCT
ejpam-4582	337	18	∈	∈	PROPN
ejpam-4582	337	19	u	u	NOUN
ejpam-4582	337	20	.	.	PUNCT
ejpam-4582	338	1	if	if	SCONJ
ejpam-4582	338	2	y	y	PROPN
ejpam-4582	338	3	̸∈	̸∈	PROPN
ejpam-4582	338	4	u	u	PROPN
ejpam-4582	338	5	,	,	PUNCT
ejpam-4582	338	6	then	then	ADV
ejpam-4582	338	7	x	x	PROPN
ejpam-4582	338	8	̸∈	̸∈	PROPN
ejpam-4582	338	9	{	{	PUNCT
ejpam-4582	338	10	y}(λ	y}(λ	PROPN
ejpam-4582	338	11	,	,	PUNCT
ejpam-4582	338	12	s	s	PART
ejpam-4582	338	13	)	)	PUNCT
ejpam-4582	338	14	and	and	CCONJ
ejpam-4582	338	15	hence	hence	ADV
ejpam-4582	338	16	y	y	PROPN
ejpam-4582	338	17	̸∈	̸∈	PROPN
ejpam-4582	338	18	{	{	PUNCT
ejpam-4582	338	19	x}(λ	x}(λ	PROPN
ejpam-4582	338	20	,	,	PUNCT
ejpam-4582	338	21	s	s	PART
ejpam-4582	338	22	)	)	PUNCT
ejpam-4582	338	23	.	.	PUNCT
ejpam-4582	339	1	this	this	PRON
ejpam-4582	339	2	implies	imply	VERB
ejpam-4582	339	3	that	that	SCONJ
ejpam-4582	339	4	{	{	PUNCT
ejpam-4582	339	5	x}(λ	x}(λ	PROPN
ejpam-4582	339	6	,	,	PUNCT
ejpam-4582	339	7	s	s	PART
ejpam-4582	339	8	)	)	PUNCT
ejpam-4582	339	9	⊆	⊆	NUM
ejpam-4582	339	10	u	u	NOUN
ejpam-4582	339	11	.	.	PUNCT
ejpam-4582	340	1	hence	hence	ADV
ejpam-4582	340	2	,	,	PUNCT
ejpam-4582	340	3	(	(	PUNCT
ejpam-4582	340	4	x	x	X
ejpam-4582	340	5	,	,	PUNCT
ejpam-4582	340	6	τ	τ	X
ejpam-4582	340	7	)	)	PUNCT
ejpam-4582	340	8	is	be	AUX
ejpam-4582	340	9	a	a	DET
ejpam-4582	340	10	(	(	PUNCT
ejpam-4582	340	11	λ	λ	PROPN
ejpam-4582	340	12	,	,	PUNCT
ejpam-4582	340	13	s)-r0	s)-r0	NOUN
ejpam-4582	340	14	space	space	NOUN
ejpam-4582	340	15	.	.	PUNCT
ejpam-4582	341	1	lemma	lemma	PROPN
ejpam-4582	341	2	5	5	X
ejpam-4582	341	3	.	.	PUNCT
ejpam-4582	342	1	let	let	VERB
ejpam-4582	342	2	(	(	PUNCT
ejpam-4582	342	3	x	x	NOUN
ejpam-4582	342	4	,	,	PUNCT
ejpam-4582	342	5	τ	τ	X
ejpam-4582	342	6	)	)	PUNCT
ejpam-4582	342	7	be	be	VERB
ejpam-4582	342	8	a	a	DET
ejpam-4582	342	9	topological	topological	ADJ
ejpam-4582	342	10	space	space	NOUN
ejpam-4582	342	11	.	.	PUNCT
ejpam-4582	343	1	then	then	ADV
ejpam-4582	343	2	,	,	PUNCT
ejpam-4582	343	3	[	[	X
ejpam-4582	343	4	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	343	5	,	,	PUNCT
ejpam-4582	343	6	s)](λ	s)](λ	NOUN
ejpam-4582	343	7	,	,	PUNCT
ejpam-4582	343	8	s	s	PART
ejpam-4582	343	9	)	)	PUNCT
ejpam-4582	343	10	=	=	SYM
ejpam-4582	343	11	{	{	PUNCT
ejpam-4582	343	12	x}(λ	x}(λ	PROPN
ejpam-4582	343	13	,	,	PUNCT
ejpam-4582	343	14	s	s	PART
ejpam-4582	343	15	)	)	PUNCT
ejpam-4582	343	16	for	for	ADP
ejpam-4582	343	17	each	each	DET
ejpam-4582	343	18	x	x	SYM
ejpam-4582	343	19	∈	∈	PROPN
ejpam-4582	343	20	x.	x.	NOUN
ejpam-4582	343	21	proof	proof	NOUN
ejpam-4582	343	22	.	.	PUNCT
ejpam-4582	344	1	since	since	SCONJ
ejpam-4582	344	2	{	{	PUNCT
ejpam-4582	344	3	x	x	NOUN
ejpam-4582	344	4	}	}	PUNCT
ejpam-4582	344	5	⊆	⊆	NUM
ejpam-4582	344	6	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	344	7	,	,	PUNCT
ejpam-4582	344	8	s	s	NOUN
ejpam-4582	344	9	)	)	PUNCT
ejpam-4582	344	10	,	,	PUNCT
ejpam-4582	344	11	{	{	PUNCT
ejpam-4582	344	12	x}(λ	x}(λ	PROPN
ejpam-4582	344	13	,	,	PUNCT
ejpam-4582	344	14	s	s	PART
ejpam-4582	344	15	)	)	PUNCT
ejpam-4582	344	16	⊆	⊆	NUM
ejpam-4582	344	17	[	[	X
ejpam-4582	344	18	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	344	19	,	,	PUNCT
ejpam-4582	344	20	s)](λ	s)](λ	NOUN
ejpam-4582	344	21	,	,	PUNCT
ejpam-4582	344	22	s	s	PART
ejpam-4582	344	23	)	)	PUNCT
ejpam-4582	344	24	by	by	ADP
ejpam-4582	344	25	lemma	lemma	PROPN
ejpam-4582	344	26	4	4	NUM
ejpam-4582	344	27	.	.	PUNCT
ejpam-4582	344	28	on	on	ADP
ejpam-4582	344	29	the	the	DET
ejpam-4582	344	30	other	other	ADJ
ejpam-4582	344	31	hand	hand	NOUN
ejpam-4582	344	32	,	,	PUNCT
ejpam-4582	344	33	we	we	PRON
ejpam-4582	344	34	have	have	VERB
ejpam-4582	344	35	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	344	36	,	,	PUNCT
ejpam-4582	344	37	s	s	NOUN
ejpam-4582	344	38	)	)	PUNCT
ejpam-4582	344	39	⊆	⊆	NUM
ejpam-4582	344	40	{	{	PUNCT
ejpam-4582	344	41	x}(λ	x}(λ	PROPN
ejpam-4582	344	42	,	,	PUNCT
ejpam-4582	344	43	s	s	PART
ejpam-4582	344	44	)	)	PUNCT
ejpam-4582	344	45	and	and	CCONJ
ejpam-4582	344	46	[	[	X
ejpam-4582	344	47	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	344	48	,	,	PUNCT
ejpam-4582	344	49	s)](λ	s)](λ	NOUN
ejpam-4582	344	50	,	,	PUNCT
ejpam-4582	344	51	s	s	PART
ejpam-4582	344	52	)	)	PUNCT
ejpam-4582	344	53	⊆	⊆	NUM
ejpam-4582	345	1	[	[	X
ejpam-4582	345	2	{	{	PUNCT
ejpam-4582	345	3	x}(λ	x}(λ	PROPN
ejpam-4582	345	4	,	,	PUNCT
ejpam-4582	345	5	s)](λ	s)](λ	PROPN
ejpam-4582	345	6	,	,	PUNCT
ejpam-4582	345	7	s	s	PART
ejpam-4582	345	8	)	)	PUNCT
ejpam-4582	345	9	=	=	SYM
ejpam-4582	345	10	{	{	PUNCT
ejpam-4582	345	11	x}(λ	x}(λ	PROPN
ejpam-4582	345	12	,	,	PUNCT
ejpam-4582	345	13	s	s	PART
ejpam-4582	345	14	)	)	PUNCT
ejpam-4582	345	15	.	.	PUNCT
ejpam-4582	346	1	thus	thus	ADV
ejpam-4582	346	2	,	,	PUNCT
ejpam-4582	346	3	[	[	X
ejpam-4582	346	4	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	346	5	,	,	PUNCT
ejpam-4582	346	6	s)](λ	s)](λ	NOUN
ejpam-4582	346	7	,	,	PUNCT
ejpam-4582	346	8	s	s	PART
ejpam-4582	346	9	)	)	PUNCT
ejpam-4582	346	10	=	=	SYM
ejpam-4582	346	11	{	{	PUNCT
ejpam-4582	346	12	x}(λ	x}(λ	PROPN
ejpam-4582	346	13	,	,	PUNCT
ejpam-4582	346	14	s	s	PART
ejpam-4582	346	15	)	)	PUNCT
ejpam-4582	346	16	.	.	PUNCT
ejpam-4582	347	1	theorem	theorem	VERB
ejpam-4582	347	2	9	9	NUM
ejpam-4582	347	3	.	.	X
ejpam-4582	348	1	for	for	ADP
ejpam-4582	348	2	a	a	DET
ejpam-4582	348	3	topological	topological	ADJ
ejpam-4582	348	4	space	space	NOUN
ejpam-4582	348	5	(	(	PUNCT
ejpam-4582	348	6	x	x	X
ejpam-4582	348	7	,	,	PUNCT
ejpam-4582	348	8	τ	τ	PROPN
ejpam-4582	348	9	)	)	PUNCT
ejpam-4582	348	10	,	,	PUNCT
ejpam-4582	348	11	the	the	DET
ejpam-4582	348	12	following	follow	VERB
ejpam-4582	348	13	properties	property	NOUN
ejpam-4582	348	14	are	be	AUX
ejpam-4582	348	15	equivalent	equivalent	ADJ
ejpam-4582	348	16	:	:	PUNCT
ejpam-4582	348	17	(	(	PUNCT
ejpam-4582	348	18	1	1	X
ejpam-4582	348	19	)	)	PUNCT
ejpam-4582	348	20	(	(	PUNCT
ejpam-4582	348	21	x	x	X
ejpam-4582	348	22	,	,	PUNCT
ejpam-4582	348	23	τ	τ	X
ejpam-4582	348	24	)	)	PUNCT
ejpam-4582	348	25	is	be	AUX
ejpam-4582	348	26	(	(	PUNCT
ejpam-4582	348	27	λ	λ	INTJ
ejpam-4582	348	28	,	,	PUNCT
ejpam-4582	348	29	s)-r0	s)-r0	X
ejpam-4582	348	30	;	;	PUNCT
ejpam-4582	348	31	(	(	PUNCT
ejpam-4582	348	32	2	2	X
ejpam-4582	348	33	)	)	PUNCT
ejpam-4582	348	34	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	348	35	,	,	PUNCT
ejpam-4582	348	36	s	s	NOUN
ejpam-4582	348	37	)	)	PUNCT
ejpam-4582	348	38	=	=	SYM
ejpam-4582	348	39	{	{	PUNCT
ejpam-4582	348	40	x}(λ	x}(λ	PROPN
ejpam-4582	348	41	,	,	PUNCT
ejpam-4582	348	42	s	s	PART
ejpam-4582	348	43	)	)	PUNCT
ejpam-4582	348	44	for	for	ADP
ejpam-4582	348	45	each	each	DET
ejpam-4582	348	46	x	x	SYM
ejpam-4582	348	47	∈	∈	PROPN
ejpam-4582	348	48	x	x	X
ejpam-4582	348	49	;	;	PUNCT
ejpam-4582	348	50	(	(	PUNCT
ejpam-4582	348	51	3	3	X
ejpam-4582	348	52	)	)	PUNCT
ejpam-4582	348	53	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	348	54	,	,	PUNCT
ejpam-4582	348	55	s	s	PART
ejpam-4582	348	56	)	)	PUNCT
ejpam-4582	348	57	is	be	AUX
ejpam-4582	348	58	(	(	PUNCT
ejpam-4582	348	59	λ	λ	X
ejpam-4582	348	60	,	,	PUNCT
ejpam-4582	348	61	s)-closed	s)-close	VERB
ejpam-4582	348	62	for	for	ADP
ejpam-4582	348	63	each	each	DET
ejpam-4582	348	64	x	x	SYM
ejpam-4582	348	65	∈	∈	PROPN
ejpam-4582	348	66	x.	x.	NOUN
ejpam-4582	348	67	proof	proof	NOUN
ejpam-4582	348	68	.	.	PUNCT
ejpam-4582	349	1	(	(	PUNCT
ejpam-4582	349	2	1	1	X
ejpam-4582	349	3	)	)	PUNCT
ejpam-4582	349	4	⇒	⇒	NOUN
ejpam-4582	349	5	(	(	PUNCT
ejpam-4582	349	6	2	2	NUM
ejpam-4582	349	7	):	):	PUNCT
ejpam-4582	349	8	by	by	ADP
ejpam-4582	349	9	theorem	theorem	NOUN
ejpam-4582	349	10	7	7	NUM
ejpam-4582	349	11	,	,	PUNCT
ejpam-4582	349	12	{	{	PUNCT
ejpam-4582	349	13	x}(λ	x}(λ	PROPN
ejpam-4582	349	14	,	,	PUNCT
ejpam-4582	349	15	s	s	PART
ejpam-4582	349	16	)	)	PUNCT
ejpam-4582	349	17	=	=	SYM
ejpam-4582	349	18	γ(λ	γ(λ	PROPN
ejpam-4582	349	19	,	,	PUNCT
ejpam-4582	349	20	s)({x	s)({x	PROPN
ejpam-4582	349	21	}	}	PUNCT
ejpam-4582	349	22	)	)	PUNCT
ejpam-4582	349	23	for	for	ADP
ejpam-4582	349	24	each	each	DET
ejpam-4582	349	25	x	x	SYM
ejpam-4582	349	26	∈	∈	PROPN
ejpam-4582	349	27	x.	x.	NOUN
ejpam-4582	349	28	thus	thus	ADV
ejpam-4582	349	29	,	,	PUNCT
ejpam-4582	349	30	{	{	PUNCT
ejpam-4582	349	31	x}(λ	x}(λ	PROPN
ejpam-4582	349	32	,	,	PUNCT
ejpam-4582	349	33	s	s	PART
ejpam-4582	349	34	)	)	PUNCT
ejpam-4582	349	35	=	=	SYM
ejpam-4582	349	36	{	{	PUNCT
ejpam-4582	349	37	x}(λ	x}(λ	PROPN
ejpam-4582	349	38	,	,	PUNCT
ejpam-4582	349	39	s	s	PART
ejpam-4582	349	40	)	)	PUNCT
ejpam-4582	349	41	∩	∩	ADJ
ejpam-4582	349	42	γ(λ	γ(λ	PROPN
ejpam-4582	349	43	,	,	PUNCT
ejpam-4582	349	44	s)({x	s)({x	NOUN
ejpam-4582	349	45	}	}	PUNCT
ejpam-4582	349	46	)	)	PUNCT
ejpam-4582	349	47	=	=	SYM
ejpam-4582	349	48	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	349	49	,	,	PUNCT
ejpam-4582	349	50	s	s	NOUN
ejpam-4582	349	51	)	)	PUNCT
ejpam-4582	349	52	.	.	PUNCT
ejpam-4582	350	1	(	(	PUNCT
ejpam-4582	350	2	2	2	X
ejpam-4582	350	3	)	)	PUNCT
ejpam-4582	350	4	⇒	⇒	NOUN
ejpam-4582	350	5	(	(	PUNCT
ejpam-4582	350	6	1	1	NUM
ejpam-4582	350	7	):	):	PUNCT
ejpam-4582	350	8	let	let	VERB
ejpam-4582	350	9	u	u	PRON
ejpam-4582	350	10	∈	∈	PROPN
ejpam-4582	350	11	(	(	PUNCT
ejpam-4582	350	12	λ	λ	NOUN
ejpam-4582	350	13	,	,	PUNCT
ejpam-4582	350	14	s)o(x	s)o(x	ADJ
ejpam-4582	350	15	)	)	PUNCT
ejpam-4582	350	16	and	and	CCONJ
ejpam-4582	350	17	x	x	PUNCT
ejpam-4582	350	18	∈	∈	PROPN
ejpam-4582	350	19	u	u	NOUN
ejpam-4582	350	20	.	.	PUNCT
ejpam-4582	351	1	by	by	ADP
ejpam-4582	351	2	(	(	PUNCT
ejpam-4582	351	3	2	2	NUM
ejpam-4582	351	4	)	)	PUNCT
ejpam-4582	351	5	,	,	PUNCT
ejpam-4582	351	6	we	we	PRON
ejpam-4582	351	7	have	have	VERB
ejpam-4582	351	8	{	{	PUNCT
ejpam-4582	351	9	x}(λ	x}(λ	PROPN
ejpam-4582	351	10	,	,	PUNCT
ejpam-4582	351	11	s	s	PART
ejpam-4582	351	12	)	)	PUNCT
ejpam-4582	351	13	=	=	SYM
ejpam-4582	351	14	⟨x⟩(λ	⟨x⟩(λ	NOUN
ejpam-4582	351	15	,	,	PUNCT
ejpam-4582	351	16	s	s	NOUN
ejpam-4582	351	17	)	)	PUNCT
ejpam-4582	351	18	=	=	SYM
ejpam-4582	351	19	{	{	PUNCT
ejpam-4582	351	20	x}(λ	x}(λ	PROPN
ejpam-4582	351	21	,	,	PUNCT
ejpam-4582	351	22	s	s	PART
ejpam-4582	351	23	)	)	PUNCT
ejpam-4582	351	24	∩	∩	ADJ
ejpam-4582	351	25	γ(λ	γ(λ	PROPN
ejpam-4582	351	26	,	,	PUNCT
ejpam-4582	351	27	s)({x	s)({x	PROPN
ejpam-4582	351	28	}	}	PUNCT
ejpam-4582	351	29	)	)	PUNCT
ejpam-4582	351	30	⊆	⊆	NUM
ejpam-4582	351	31	γ(λ	γ(λ	PROPN
ejpam-4582	351	32	,	,	PUNCT
ejpam-4582	351	33	s)({x	s)({x	PROPN
ejpam-4582	351	34	}	}	PUNCT
ejpam-4582	351	35	)	)	PUNCT
ejpam-4582	352	1	⊆	⊆	NUM
ejpam-4582	352	2	γ(λ	γ(λ	NOUN
ejpam-4582	352	3	,	,	PUNCT
ejpam-4582	352	4	s)(u	s)(u	ADJ
ejpam-4582	352	5	)	)	PUNCT
ejpam-4582	352	6	=	=	SYM
ejpam-4582	352	7	u	u	NOUN
ejpam-4582	352	8	and	and	CCONJ
ejpam-4582	352	9	hence	hence	ADV
ejpam-4582	352	10	(	(	PUNCT
ejpam-4582	352	11	x	x	X
ejpam-4582	352	12	,	,	PUNCT
ejpam-4582	352	13	τ	τ	X
ejpam-4582	352	14	)	)	PUNCT
ejpam-4582	352	15	is	be	AUX
ejpam-4582	352	16	(	(	PUNCT
ejpam-4582	352	17	λ	λ	X
ejpam-4582	352	18	,	,	PUNCT
ejpam-4582	352	19	s)-r0	s)-r0	X
ejpam-4582	352	20	.	.	PUNCT
ejpam-4582	353	1	(	(	PUNCT
ejpam-4582	353	2	2	2	X
ejpam-4582	353	3	)	)	PUNCT
ejpam-4582	353	4	⇔	⇔	X
ejpam-4582	353	5	(	(	PUNCT
ejpam-4582	353	6	3	3	NUM
ejpam-4582	353	7	):	):	PUNCT
ejpam-4582	353	8	this	this	PRON
ejpam-4582	353	9	is	be	AUX
ejpam-4582	353	10	a	a	DET
ejpam-4582	353	11	consequence	consequence	NOUN
ejpam-4582	353	12	of	of	ADP
ejpam-4582	353	13	lemma	lemma	PROPN
ejpam-4582	353	14	5	5	NUM
ejpam-4582	353	15	.	.	PUNCT
ejpam-4582	353	16	theorem	theorem	VERB
ejpam-4582	353	17	10	10	NUM
ejpam-4582	353	18	.	.	PUNCT
ejpam-4582	354	1	for	for	ADP
ejpam-4582	354	2	a	a	DET
ejpam-4582	354	3	topological	topological	ADJ
ejpam-4582	354	4	space	space	NOUN
ejpam-4582	354	5	(	(	PUNCT
ejpam-4582	354	6	x	x	X
ejpam-4582	354	7	,	,	PUNCT
ejpam-4582	354	8	τ	τ	PROPN
ejpam-4582	354	9	)	)	PUNCT
ejpam-4582	354	10	,	,	PUNCT
ejpam-4582	354	11	the	the	DET
ejpam-4582	354	12	following	follow	VERB
ejpam-4582	354	13	properties	property	NOUN
ejpam-4582	354	14	are	be	AUX
ejpam-4582	354	15	equivalent	equivalent	ADJ
ejpam-4582	354	16	:	:	PUNCT
ejpam-4582	354	17	(	(	PUNCT
ejpam-4582	354	18	1	1	X
ejpam-4582	354	19	)	)	PUNCT
ejpam-4582	354	20	(	(	PUNCT
ejpam-4582	354	21	x	x	X
ejpam-4582	354	22	,	,	PUNCT
ejpam-4582	354	23	τ	τ	X
ejpam-4582	354	24	)	)	PUNCT
ejpam-4582	354	25	is	be	AUX
ejpam-4582	354	26	(	(	PUNCT
ejpam-4582	354	27	λ	λ	INTJ
ejpam-4582	354	28	,	,	PUNCT
ejpam-4582	354	29	s)-r0	s)-r0	X
ejpam-4582	354	30	;	;	PUNCT
ejpam-4582	354	31	(	(	PUNCT
ejpam-4582	354	32	2	2	X
ejpam-4582	354	33	)	)	PUNCT
ejpam-4582	354	34	for	for	ADP
ejpam-4582	354	35	each	each	DET
ejpam-4582	354	36	nonempty	nonempty	ADV
ejpam-4582	354	37	set	set	VERB
ejpam-4582	354	38	a	a	PRON
ejpam-4582	354	39	of	of	ADP
ejpam-4582	354	40	x	x	PUNCT
ejpam-4582	354	41	and	and	CCONJ
ejpam-4582	354	42	each	each	DET
ejpam-4582	354	43	u	u	NOUN
ejpam-4582	354	44	∈	∈	PROPN
ejpam-4582	354	45	(	(	PUNCT
ejpam-4582	354	46	λ	λ	NOUN
ejpam-4582	354	47	,	,	PUNCT
ejpam-4582	354	48	s)o(x	s)o(x	ADJ
ejpam-4582	354	49	)	)	PUNCT
ejpam-4582	354	50	such	such	ADJ
ejpam-4582	354	51	that	that	SCONJ
ejpam-4582	354	52	a	a	DET
ejpam-4582	354	53	∩	∩	ADJ
ejpam-4582	354	54	u	u	NOUN
ejpam-4582	354	55	̸=	̸=	PROPN
ejpam-4582	354	56	∅	∅	NOUN
ejpam-4582	354	57	,	,	PUNCT
ejpam-4582	354	58	there	there	PRON
ejpam-4582	354	59	exists	exist	VERB
ejpam-4582	354	60	a	a	DET
ejpam-4582	354	61	(	(	PUNCT
ejpam-4582	354	62	λ	λ	X
ejpam-4582	354	63	,	,	PUNCT
ejpam-4582	354	64	s)-closed	s)-close	VERB
ejpam-4582	354	65	set	set	NOUN
ejpam-4582	354	66	f	f	PRON
ejpam-4582	355	1	such	such	ADJ
ejpam-4582	355	2	that	that	SCONJ
ejpam-4582	355	3	a	a	DET
ejpam-4582	355	4	∩	∩	ADJ
ejpam-4582	355	5	f	f	PROPN
ejpam-4582	355	6	̸=	̸=	PROPN
ejpam-4582	355	7	∅	∅	NOUN
ejpam-4582	355	8	and	and	CCONJ
ejpam-4582	355	9	f	f	PROPN
ejpam-4582	355	10	⊆	⊆	NUM
ejpam-4582	355	11	u	u	NOUN
ejpam-4582	355	12	;	;	PUNCT
ejpam-4582	355	13	(	(	PUNCT
ejpam-4582	355	14	3	3	X
ejpam-4582	355	15	)	)	PUNCT
ejpam-4582	355	16	f	f	NOUN
ejpam-4582	355	17	=	=	PUNCT
ejpam-4582	355	18	γ(λ	γ(λ	PROPN
ejpam-4582	355	19	,	,	PUNCT
ejpam-4582	355	20	s)(f	s)(f	PROPN
ejpam-4582	355	21	)	)	PUNCT
ejpam-4582	356	1	for	for	ADP
ejpam-4582	356	2	every	every	DET
ejpam-4582	356	3	(	(	PUNCT
ejpam-4582	356	4	λ	λ	PROPN
ejpam-4582	356	5	,	,	PUNCT
ejpam-4582	356	6	s)-closed	s)-close	VERB
ejpam-4582	356	7	set	set	NOUN
ejpam-4582	356	8	f	f	X
ejpam-4582	356	9	;	;	PUNCT
ejpam-4582	356	10	(	(	PUNCT
ejpam-4582	356	11	4	4	X
ejpam-4582	356	12	)	)	PUNCT
ejpam-4582	356	13	{	{	PUNCT
ejpam-4582	356	14	x}(λ	x}(λ	PROPN
ejpam-4582	356	15	,	,	PUNCT
ejpam-4582	356	16	s	s	PART
ejpam-4582	356	17	)	)	PUNCT
ejpam-4582	356	18	=	=	SYM
ejpam-4582	356	19	γ(λ	γ(λ	PROPN
ejpam-4582	356	20	,	,	PUNCT
ejpam-4582	356	21	s)({x	s)({x	PROPN
ejpam-4582	356	22	}	}	PUNCT
ejpam-4582	356	23	)	)	PUNCT
ejpam-4582	356	24	for	for	ADP
ejpam-4582	356	25	each	each	DET
ejpam-4582	356	26	x	x	SYM
ejpam-4582	356	27	∈	∈	PROPN
ejpam-4582	356	28	x.	x.	NOUN
ejpam-4582	356	29	proof	proof	NOUN
ejpam-4582	356	30	.	.	PUNCT
ejpam-4582	357	1	(	(	PUNCT
ejpam-4582	357	2	1	1	X
ejpam-4582	357	3	)	)	PUNCT
ejpam-4582	357	4	⇒	⇒	NOUN
ejpam-4582	357	5	(	(	PUNCT
ejpam-4582	357	6	2	2	NUM
ejpam-4582	357	7	):	):	PUNCT
ejpam-4582	357	8	let	let	VERB
ejpam-4582	357	9	a	a	DET
ejpam-4582	357	10	be	be	AUX
ejpam-4582	357	11	any	any	DET
ejpam-4582	357	12	nonempty	nonempty	ADJ
ejpam-4582	357	13	set	set	NOUN
ejpam-4582	357	14	of	of	ADP
ejpam-4582	357	15	x	x	PUNCT
ejpam-4582	357	16	and	and	CCONJ
ejpam-4582	357	17	u	u	PROPN
ejpam-4582	357	18	∈	∈	PROPN
ejpam-4582	357	19	(	(	PUNCT
ejpam-4582	357	20	λ	λ	NOUN
ejpam-4582	357	21	,	,	PUNCT
ejpam-4582	357	22	s)o(x	s)o(x	ADJ
ejpam-4582	357	23	)	)	PUNCT
ejpam-4582	357	24	such	such	ADJ
ejpam-4582	357	25	that	that	SCONJ
ejpam-4582	357	26	a	a	DET
ejpam-4582	357	27	∩	∩	ADJ
ejpam-4582	357	28	u	u	NOUN
ejpam-4582	357	29	̸=	̸=	PROPN
ejpam-4582	357	30	∅.	∅.	VERB
ejpam-4582	357	31	then	then	ADV
ejpam-4582	357	32	,	,	PUNCT
ejpam-4582	357	33	there	there	PRON
ejpam-4582	357	34	exists	exist	VERB
ejpam-4582	357	35	x	x	X
ejpam-4582	357	36	∈	∈	PROPN
ejpam-4582	357	37	u	u	NOUN
ejpam-4582	357	38	∩a	∩a	PROPN
ejpam-4582	357	39	and	and	CCONJ
ejpam-4582	357	40	hence	hence	ADV
ejpam-4582	357	41	{	{	PUNCT
ejpam-4582	357	42	x}(λ	x}(λ	PROPN
ejpam-4582	357	43	,	,	PUNCT
ejpam-4582	357	44	s	s	PART
ejpam-4582	357	45	)	)	PUNCT
ejpam-4582	357	46	⊆	⊆	NUM
ejpam-4582	357	47	u	u	NOUN
ejpam-4582	357	48	.	.	PUNCT
ejpam-4582	358	1	put	put	VERB
ejpam-4582	358	2	f	f	X
ejpam-4582	358	3	=	=	SYM
ejpam-4582	358	4	{	{	PUNCT
ejpam-4582	358	5	x}(λ	x}(λ	PROPN
ejpam-4582	358	6	,	,	PUNCT
ejpam-4582	358	7	s	s	PART
ejpam-4582	358	8	)	)	PUNCT
ejpam-4582	358	9	,	,	PUNCT
ejpam-4582	358	10	then	then	ADV
ejpam-4582	358	11	f	f	PROPN
ejpam-4582	358	12	is	be	AUX
ejpam-4582	358	13	(	(	PUNCT
ejpam-4582	358	14	λ	λ	X
ejpam-4582	358	15	,	,	PUNCT
ejpam-4582	358	16	s)-closed	s)-close	VERB
ejpam-4582	358	17	,	,	PUNCT
ejpam-4582	358	18	a	a	DET
ejpam-4582	358	19	∩	∩	ADJ
ejpam-4582	358	20	f	f	PROPN
ejpam-4582	358	21	̸=	̸=	PROPN
ejpam-4582	358	22	∅	∅	NOUN
ejpam-4582	358	23	and	and	CCONJ
ejpam-4582	358	24	f	f	PROPN
ejpam-4582	358	25	⊆	⊆	NUM
ejpam-4582	358	26	u	u	NOUN
ejpam-4582	358	27	.	.	PUNCT
ejpam-4582	359	1	(	(	PUNCT
ejpam-4582	359	2	2	2	X
ejpam-4582	359	3	)	)	PUNCT
ejpam-4582	359	4	⇒	⇒	NOUN
ejpam-4582	359	5	(	(	PUNCT
ejpam-4582	359	6	3	3	NUM
ejpam-4582	359	7	):	):	PUNCT
ejpam-4582	359	8	let	let	VERB
ejpam-4582	359	9	f	f	PRON
ejpam-4582	359	10	be	be	AUX
ejpam-4582	359	11	any	any	DET
ejpam-4582	359	12	(	(	PUNCT
ejpam-4582	359	13	λ	λ	X
ejpam-4582	359	14	,	,	PUNCT
ejpam-4582	359	15	s)-closed	s)-close	VERB
ejpam-4582	359	16	set	set	NOUN
ejpam-4582	359	17	of	of	ADP
ejpam-4582	359	18	x.	x.	NOUN
ejpam-4582	359	19	on	on	ADP
ejpam-4582	359	20	the	the	DET
ejpam-4582	359	21	other	other	ADJ
ejpam-4582	359	22	hand	hand	NOUN
ejpam-4582	359	23	,	,	PUNCT
ejpam-4582	359	24	we	we	PRON
ejpam-4582	359	25	have	have	VERB
ejpam-4582	359	26	f	f	PROPN
ejpam-4582	359	27	⊆	⊆	NUM
ejpam-4582	359	28	γ(λ	γ(λ	PROPN
ejpam-4582	359	29	,	,	PUNCT
ejpam-4582	359	30	s)(f	s)(f	PROPN
ejpam-4582	359	31	)	)	PUNCT
ejpam-4582	359	32	.	.	PUNCT
ejpam-4582	360	1	next	next	ADV
ejpam-4582	360	2	,	,	PUNCT
ejpam-4582	360	3	we	we	PRON
ejpam-4582	360	4	show	show	VERB
ejpam-4582	360	5	that	that	SCONJ
ejpam-4582	360	6	f	f	PROPN
ejpam-4582	360	7	⊇	⊇	PROPN
ejpam-4582	360	8	γ(λ	γ(λ	PROPN
ejpam-4582	360	9	,	,	PUNCT
ejpam-4582	360	10	s)(f	s)(f	PROPN
ejpam-4582	360	11	)	)	PUNCT
ejpam-4582	360	12	.	.	PUNCT
ejpam-4582	361	1	suppose	suppose	VERB
ejpam-4582	361	2	that	that	SCONJ
ejpam-4582	361	3	x	x	PROPN
ejpam-4582	361	4	̸∈	̸∈	PROPN
ejpam-4582	361	5	f	f	PROPN
ejpam-4582	361	6	.	.	PUNCT
ejpam-4582	362	1	then	then	ADV
ejpam-4582	362	2	,	,	PUNCT
ejpam-4582	362	3	x	x	PUNCT
ejpam-4582	362	4	∈	∈	NOUN
ejpam-4582	362	5	x	x	X
ejpam-4582	362	6	−	−	PROPN
ejpam-4582	362	7	f	f	PROPN
ejpam-4582	362	8	c.	c.	PROPN
ejpam-4582	362	9	boonpok	boonpok	PROPN
ejpam-4582	362	10	,	,	PUNCT
ejpam-4582	362	11	c.	c.	PROPN
ejpam-4582	362	12	viriyapong	viriyapong	PROPN
ejpam-4582	362	13	/	/	SYM
ejpam-4582	362	14	eur	eur	PROPN
ejpam-4582	362	15	.	.	PUNCT
ejpam-4582	363	1	j.	j.	PROPN
ejpam-4582	363	2	pure	pure	PROPN
ejpam-4582	363	3	appl	appl	PROPN
ejpam-4582	363	4	.	.	PROPN
ejpam-4582	363	5	math	math	PROPN
ejpam-4582	363	6	,	,	PUNCT
ejpam-4582	363	7	16	16	NUM
ejpam-4582	363	8	(	(	PUNCT
ejpam-4582	363	9	1	1	NUM
ejpam-4582	363	10	)	)	PUNCT
ejpam-4582	363	11	(	(	PUNCT
ejpam-4582	363	12	2023	2023	NUM
ejpam-4582	363	13	)	)	PUNCT
ejpam-4582	363	14	,	,	PUNCT
ejpam-4582	363	15	336	336	NUM
ejpam-4582	363	16	-	-	SYM
ejpam-4582	363	17	362	362	NUM
ejpam-4582	363	18	346	346	NUM
ejpam-4582	363	19	and	and	CCONJ
ejpam-4582	363	20	x	x	SYM
ejpam-4582	363	21	−	−	PROPN
ejpam-4582	363	22	f	f	PROPN
ejpam-4582	363	23	∈	∈	PROPN
ejpam-4582	363	24	(	(	PUNCT
ejpam-4582	363	25	λ	λ	NOUN
ejpam-4582	363	26	,	,	PUNCT
ejpam-4582	363	27	s)o(x	s)o(x	NOUN
ejpam-4582	363	28	)	)	PUNCT
ejpam-4582	363	29	.	.	PUNCT
ejpam-4582	364	1	by	by	ADP
ejpam-4582	364	2	(	(	PUNCT
ejpam-4582	364	3	2	2	NUM
ejpam-4582	364	4	)	)	PUNCT
ejpam-4582	364	5	,	,	PUNCT
ejpam-4582	364	6	there	there	PRON
ejpam-4582	364	7	exists	exist	VERB
ejpam-4582	364	8	a	a	DET
ejpam-4582	364	9	(	(	PUNCT
ejpam-4582	364	10	λ	λ	X
ejpam-4582	364	11	,	,	PUNCT
ejpam-4582	364	12	s)-closed	s)-close	VERB
ejpam-4582	364	13	set	set	NOUN
ejpam-4582	364	14	k	k	ADP
ejpam-4582	364	15	such	such	ADJ
ejpam-4582	364	16	that	that	SCONJ
ejpam-4582	364	17	x	x	SYM
ejpam-4582	364	18	∈	∈	PROPN
ejpam-4582	364	19	k	k	PROPN
ejpam-4582	364	20	and	and	CCONJ
ejpam-4582	364	21	k	k	PROPN
ejpam-4582	364	22	⊆	⊆	NUM
ejpam-4582	364	23	x	x	SYM
ejpam-4582	364	24	−	−	PROPN
ejpam-4582	364	25	f	f	NOUN
ejpam-4582	364	26	.	.	PUNCT
ejpam-4582	365	1	now	now	ADV
ejpam-4582	365	2	,	,	PUNCT
ejpam-4582	365	3	put	put	VERB
ejpam-4582	365	4	u	u	NOUN
ejpam-4582	365	5	=	=	NOUN
ejpam-4582	365	6	x	x	PROPN
ejpam-4582	365	7	−	−	PROPN
ejpam-4582	366	1	k.	k.	NOUN
ejpam-4582	366	2	then	then	ADV
ejpam-4582	366	3	,	,	PUNCT
ejpam-4582	366	4	we	we	PRON
ejpam-4582	366	5	have	have	VERB
ejpam-4582	366	6	f	f	PROPN
ejpam-4582	366	7	⊆	⊆	NUM
ejpam-4582	366	8	u	u	X
ejpam-4582	366	9	∈	∈	PROPN
ejpam-4582	366	10	(	(	PUNCT
ejpam-4582	366	11	λ	λ	NOUN
ejpam-4582	366	12	,	,	PUNCT
ejpam-4582	366	13	s)o(x	s)o(x	ADJ
ejpam-4582	366	14	)	)	PUNCT
ejpam-4582	366	15	and	and	CCONJ
ejpam-4582	366	16	x	x	PUNCT
ejpam-4582	366	17	̸∈	̸∈	PROPN
ejpam-4582	366	18	u	u	PROPN
ejpam-4582	366	19	.	.	PUNCT
ejpam-4582	367	1	thus	thus	ADV
ejpam-4582	367	2	,	,	PUNCT
ejpam-4582	367	3	x	x	PROPN
ejpam-4582	367	4	̸∈	̸∈	PROPN
ejpam-4582	367	5	γ(λ	γ(λ	PROPN
ejpam-4582	367	6	,	,	PUNCT
ejpam-4582	367	7	s)(f	s)(f	PROPN
ejpam-4582	367	8	)	)	PUNCT
ejpam-4582	367	9	and	and	CCONJ
ejpam-4582	367	10	hence	hence	ADV
ejpam-4582	367	11	f	f	PROPN
ejpam-4582	367	12	⊆	⊆	NUM
ejpam-4582	367	13	γ(λ	γ(λ	PROPN
ejpam-4582	367	14	,	,	PUNCT
ejpam-4582	367	15	s)(f	s)(f	PROPN
ejpam-4582	367	16	)	)	PUNCT
ejpam-4582	367	17	.	.	PUNCT
ejpam-4582	368	1	this	this	PRON
ejpam-4582	368	2	shows	show	VERB
ejpam-4582	368	3	that	that	SCONJ
ejpam-4582	368	4	f	f	PROPN
ejpam-4582	368	5	=	=	PUNCT
ejpam-4582	368	6	γ(λ	γ(λ	PROPN
ejpam-4582	368	7	,	,	PUNCT
ejpam-4582	368	8	s)(f	s)(f	PROPN
ejpam-4582	368	9	)	)	PUNCT
ejpam-4582	368	10	.	.	PUNCT
ejpam-4582	369	1	(	(	PUNCT
ejpam-4582	369	2	3	3	X
ejpam-4582	369	3	)	)	PUNCT
ejpam-4582	369	4	⇒	⇒	NOUN
ejpam-4582	369	5	(	(	PUNCT
ejpam-4582	369	6	4	4	NUM
ejpam-4582	369	7	):	):	PUNCT
ejpam-4582	369	8	let	let	VERB
ejpam-4582	369	9	x	x	PUNCT
ejpam-4582	369	10	∈	∈	PROPN
ejpam-4582	369	11	x	x	X
ejpam-4582	369	12	and	and	CCONJ
ejpam-4582	369	13	y	y	PROPN
ejpam-4582	369	14	̸∈	̸∈	PROPN
ejpam-4582	369	15	γ(λ	γ(λ	PROPN
ejpam-4582	369	16	,	,	PUNCT
ejpam-4582	369	17	s)({x	s)({x	PROPN
ejpam-4582	369	18	}	}	PUNCT
ejpam-4582	369	19	)	)	PUNCT
ejpam-4582	369	20	.	.	PUNCT
ejpam-4582	370	1	there	there	PRON
ejpam-4582	370	2	exists	exist	VERB
ejpam-4582	370	3	a	a	DET
ejpam-4582	370	4	(	(	PUNCT
ejpam-4582	370	5	λ	λ	X
ejpam-4582	370	6	,	,	PUNCT
ejpam-4582	370	7	s)-open	s)-open	VERB
ejpam-4582	370	8	set	set	VERB
ejpam-4582	370	9	u	u	PRON
ejpam-4582	370	10	such	such	ADJ
ejpam-4582	370	11	that	that	SCONJ
ejpam-4582	370	12	x	x	SYM
ejpam-4582	370	13	∈	∈	PROPN
ejpam-4582	370	14	u	u	NOUN
ejpam-4582	370	15	and	and	CCONJ
ejpam-4582	370	16	y	y	PROPN
ejpam-4582	370	17	̸∈	̸∈	PROPN
ejpam-4582	370	18	u	u	PROPN
ejpam-4582	370	19	.	.	PUNCT
ejpam-4582	371	1	thus	thus	ADV
ejpam-4582	371	2	,	,	PUNCT
ejpam-4582	371	3	{	{	PUNCT
ejpam-4582	371	4	y}(λ	y}(λ	PROPN
ejpam-4582	371	5	,	,	PUNCT
ejpam-4582	371	6	s	s	PART
ejpam-4582	371	7	)	)	PUNCT
ejpam-4582	371	8	∩	∩	ADJ
ejpam-4582	371	9	u	u	NOUN
ejpam-4582	371	10	=	=	NOUN
ejpam-4582	371	11	∅	∅	NOUN
ejpam-4582	371	12	and	and	CCONJ
ejpam-4582	371	13	by	by	ADP
ejpam-4582	371	14	(	(	PUNCT
ejpam-4582	371	15	3	3	NUM
ejpam-4582	371	16	)	)	PUNCT
ejpam-4582	371	17	,	,	PUNCT
ejpam-4582	371	18	γ(λ	γ(λ	PROPN
ejpam-4582	371	19	,	,	PUNCT
ejpam-4582	371	20	s)({y}(λ	s)({y}(λ	VERB
ejpam-4582	371	21	,	,	PUNCT
ejpam-4582	371	22	s	s	NOUN
ejpam-4582	371	23	)	)	PUNCT
ejpam-4582	371	24	)	)	PUNCT
ejpam-4582	371	25	∩	∩	NOUN
ejpam-4582	371	26	u	u	NOUN
ejpam-4582	371	27	=	=	PUNCT
ejpam-4582	371	28	∅.	∅.	NOUN
ejpam-4582	371	29	since	since	SCONJ
ejpam-4582	371	30	x	x	PROPN
ejpam-4582	371	31	̸∈	̸∈	PROPN
ejpam-4582	371	32	γ(λ	γ(λ	PROPN
ejpam-4582	371	33	,	,	PUNCT
ejpam-4582	371	34	s)({y}(λ	s)({y}(λ	VERB
ejpam-4582	371	35	,	,	PUNCT
ejpam-4582	371	36	s	s	NOUN
ejpam-4582	371	37	)	)	PUNCT
ejpam-4582	371	38	)	)	PUNCT
ejpam-4582	371	39	,	,	PUNCT
ejpam-4582	371	40	there	there	PRON
ejpam-4582	371	41	exists	exist	VERB
ejpam-4582	371	42	a	a	DET
ejpam-4582	371	43	(	(	PUNCT
ejpam-4582	371	44	λ	λ	X
ejpam-4582	371	45	,	,	PUNCT
ejpam-4582	371	46	s)-open	s)-open	PUNCT
ejpam-4582	371	47	set	set	VERB
ejpam-4582	371	48	g	g	PRON
ejpam-4582	371	49	such	such	ADJ
ejpam-4582	371	50	that	that	SCONJ
ejpam-4582	371	51	{	{	PUNCT
ejpam-4582	371	52	y}(λ	y}(λ	PROPN
ejpam-4582	371	53	,	,	PUNCT
ejpam-4582	371	54	s	s	PART
ejpam-4582	371	55	)	)	PUNCT
ejpam-4582	371	56	⊆	⊆	NUM
ejpam-4582	371	57	g	g	NOUN
ejpam-4582	371	58	and	and	CCONJ
ejpam-4582	371	59	x	x	PUNCT
ejpam-4582	371	60	̸∈	̸∈	PROPN
ejpam-4582	371	61	g.	g.	PROPN
ejpam-4582	371	62	hence	hence	ADV
ejpam-4582	371	63	,	,	PUNCT
ejpam-4582	371	64	{	{	PUNCT
ejpam-4582	371	65	x}(λ	x}(λ	PROPN
ejpam-4582	371	66	,	,	PUNCT
ejpam-4582	371	67	s	s	PART
ejpam-4582	371	68	)	)	PUNCT
ejpam-4582	371	69	∩	∩	NOUN
ejpam-4582	371	70	g	g	NOUN
ejpam-4582	371	71	=	=	PUNCT
ejpam-4582	371	72	∅.	∅.	NOUN
ejpam-4582	371	73	since	since	SCONJ
ejpam-4582	371	74	y	y	PROPN
ejpam-4582	371	75	∈	∈	PROPN
ejpam-4582	371	76	g	g	PROPN
ejpam-4582	371	77	,	,	PUNCT
ejpam-4582	371	78	y	y	PROPN
ejpam-4582	371	79	̸∈	̸∈	PROPN
ejpam-4582	371	80	{	{	PUNCT
ejpam-4582	371	81	x}(λ	x}(λ	PROPN
ejpam-4582	371	82	,	,	PUNCT
ejpam-4582	371	83	s	s	PART
ejpam-4582	371	84	)	)	PUNCT
ejpam-4582	371	85	.	.	PUNCT
ejpam-4582	372	1	therefore	therefore	ADV
ejpam-4582	372	2	,	,	PUNCT
ejpam-4582	372	3	{	{	PUNCT
ejpam-4582	372	4	x}(λ	x}(λ	PROPN
ejpam-4582	372	5	,	,	PUNCT
ejpam-4582	372	6	s	s	PART
ejpam-4582	372	7	)	)	PUNCT
ejpam-4582	372	8	⊆	⊆	NUM
ejpam-4582	372	9	γ(λ	γ(λ	PROPN
ejpam-4582	372	10	,	,	PUNCT
ejpam-4582	372	11	s)({x	s)({x	PROPN
ejpam-4582	372	12	}	}	PUNCT
ejpam-4582	372	13	)	)	PUNCT
ejpam-4582	372	14	.	.	PUNCT
ejpam-4582	373	1	moreover	moreover	ADV
ejpam-4582	373	2	,	,	PUNCT
ejpam-4582	373	3	{	{	PUNCT
ejpam-4582	373	4	x}(λ	x}(λ	PROPN
ejpam-4582	373	5	,	,	PUNCT
ejpam-4582	373	6	s	s	PART
ejpam-4582	373	7	)	)	PUNCT
ejpam-4582	373	8	⊆	⊆	NUM
ejpam-4582	373	9	γ(λ	γ(λ	PROPN
ejpam-4582	373	10	,	,	PUNCT
ejpam-4582	373	11	s)({x	s)({x	PROPN
ejpam-4582	373	12	}	}	PUNCT
ejpam-4582	373	13	)	)	PUNCT
ejpam-4582	374	1	⊆	⊆	NUM
ejpam-4582	374	2	γ(λ	γ(λ	NOUN
ejpam-4582	374	3	,	,	PUNCT
ejpam-4582	374	4	s)[{x}(λ	s)[{x}(λ	ADJ
ejpam-4582	374	5	,	,	PUNCT
ejpam-4582	374	6	s	s	NOUN
ejpam-4582	374	7	)	)	PUNCT
ejpam-4582	374	8	]	]	PUNCT
ejpam-4582	375	1	=	=	PRON
ejpam-4582	375	2	{	{	PUNCT
ejpam-4582	375	3	x}(λ	x}(λ	PROPN
ejpam-4582	375	4	,	,	PUNCT
ejpam-4582	375	5	s	s	PART
ejpam-4582	375	6	)	)	PUNCT
ejpam-4582	375	7	.	.	PUNCT
ejpam-4582	376	1	this	this	PRON
ejpam-4582	376	2	shows	show	VERB
ejpam-4582	376	3	that	that	SCONJ
ejpam-4582	376	4	{	{	PUNCT
ejpam-4582	376	5	x}(λ	x}(λ	PROPN
ejpam-4582	376	6	,	,	PUNCT
ejpam-4582	376	7	s	s	PART
ejpam-4582	376	8	)	)	PUNCT
ejpam-4582	376	9	=	=	SYM
ejpam-4582	376	10	γ(λ	γ(λ	PROPN
ejpam-4582	376	11	,	,	PUNCT
ejpam-4582	376	12	s)({x	s)({x	PROPN
ejpam-4582	376	13	}	}	PUNCT
ejpam-4582	376	14	)	)	PUNCT
ejpam-4582	376	15	.	.	PUNCT
ejpam-4582	377	1	(	(	PUNCT
ejpam-4582	377	2	4	4	X
ejpam-4582	377	3	)	)	PUNCT
ejpam-4582	377	4	⇒	⇒	NOUN
ejpam-4582	377	5	(	(	PUNCT
ejpam-4582	377	6	1	1	NUM
ejpam-4582	377	7	):	):	PUNCT
ejpam-4582	377	8	this	this	PRON
ejpam-4582	377	9	is	be	AUX
ejpam-4582	377	10	obvious	obvious	ADJ
ejpam-4582	377	11	by	by	ADP
ejpam-4582	377	12	theorem	theorem	NOUN
ejpam-4582	377	13	7	7	NUM
ejpam-4582	377	14	.	.	PUNCT
ejpam-4582	377	15	definition	definition	NOUN
ejpam-4582	377	16	8	8	NUM
ejpam-4582	377	17	.	.	PUNCT
ejpam-4582	378	1	a	a	DET
ejpam-4582	378	2	topological	topological	ADJ
ejpam-4582	378	3	space	space	NOUN
ejpam-4582	378	4	(	(	PUNCT
ejpam-4582	378	5	x	x	X
ejpam-4582	378	6	,	,	PUNCT
ejpam-4582	378	7	τ	τ	X
ejpam-4582	378	8	)	)	PUNCT
ejpam-4582	378	9	is	be	AUX
ejpam-4582	378	10	called	call	VERB
ejpam-4582	378	11	(	(	PUNCT
ejpam-4582	378	12	λ	λ	PROPN
ejpam-4582	378	13	,	,	PUNCT
ejpam-4582	378	14	s)-symmetric	s)-symmetric	ADJ
ejpam-4582	378	15	if	if	SCONJ
ejpam-4582	378	16	,	,	PUNCT
ejpam-4582	378	17	for	for	ADP
ejpam-4582	378	18	each	each	DET
ejpam-4582	378	19	x	x	NOUN
ejpam-4582	378	20	,	,	PUNCT
ejpam-4582	378	21	y	y	PROPN
ejpam-4582	378	22	∈	∈	PROPN
ejpam-4582	378	23	x	x	X
ejpam-4582	378	24	,	,	PUNCT
ejpam-4582	378	25	x	x	SYM
ejpam-4582	378	26	∈	∈	PROPN
ejpam-4582	378	27	{	{	PUNCT
ejpam-4582	378	28	y}(λ	y}(λ	PROPN
ejpam-4582	378	29	,	,	PUNCT
ejpam-4582	378	30	s	s	PART
ejpam-4582	378	31	)	)	PUNCT
ejpam-4582	378	32	implies	imply	VERB
ejpam-4582	378	33	y	y	PROPN
ejpam-4582	378	34	∈	∈	PROPN
ejpam-4582	378	35	{	{	PUNCT
ejpam-4582	378	36	x}(λ	x}(λ	PROPN
ejpam-4582	378	37	,	,	PUNCT
ejpam-4582	378	38	s	s	PART
ejpam-4582	378	39	)	)	PUNCT
ejpam-4582	378	40	.	.	PUNCT
ejpam-4582	379	1	theorem	theorem	VERB
ejpam-4582	379	2	11	11	NUM
ejpam-4582	379	3	.	.	PUNCT
ejpam-4582	380	1	a	a	DET
ejpam-4582	380	2	topological	topological	ADJ
ejpam-4582	380	3	space	space	NOUN
ejpam-4582	380	4	(	(	PUNCT
ejpam-4582	380	5	x	x	X
ejpam-4582	380	6	,	,	PUNCT
ejpam-4582	380	7	τ	τ	X
ejpam-4582	380	8	)	)	PUNCT
ejpam-4582	380	9	is	be	AUX
ejpam-4582	380	10	(	(	PUNCT
ejpam-4582	380	11	λ	λ	X
ejpam-4582	380	12	,	,	PUNCT
ejpam-4582	380	13	s)-r0	s)-r0	PRON
ejpam-4582	380	14	if	if	SCONJ
ejpam-4582	380	15	and	and	CCONJ
ejpam-4582	380	16	only	only	ADV
ejpam-4582	380	17	if	if	SCONJ
ejpam-4582	380	18	(	(	PUNCT
ejpam-4582	380	19	x	x	NOUN
ejpam-4582	380	20	,	,	PUNCT
ejpam-4582	380	21	τ	τ	X
ejpam-4582	380	22	)	)	PUNCT
ejpam-4582	380	23	is	be	AUX
ejpam-4582	380	24	(	(	PUNCT
ejpam-4582	380	25	λ	λ	INTJ
ejpam-4582	380	26	,	,	PUNCT
ejpam-4582	380	27	s)symmetric	s)symmetric	ADJ
ejpam-4582	380	28	.	.	PUNCT
ejpam-4582	381	1	proof	proof	NOUN
ejpam-4582	381	2	.	.	PUNCT
ejpam-4582	382	1	let	let	VERB
ejpam-4582	382	2	x	x	PRON
ejpam-4582	382	3	∈	∈	PROPN
ejpam-4582	382	4	{	{	PUNCT
ejpam-4582	382	5	y}(λ	y}(λ	PROPN
ejpam-4582	382	6	,	,	PUNCT
ejpam-4582	382	7	s	s	PART
ejpam-4582	382	8	)	)	PUNCT
ejpam-4582	382	9	and	and	CCONJ
ejpam-4582	382	10	u	u	PRON
ejpam-4582	382	11	be	be	VERB
ejpam-4582	382	12	a	a	DET
ejpam-4582	382	13	(	(	PUNCT
ejpam-4582	382	14	λ	λ	X
ejpam-4582	382	15	,	,	PUNCT
ejpam-4582	382	16	s)-open	s)-open	PUNCT
ejpam-4582	382	17	set	set	VERB
ejpam-4582	382	18	such	such	ADJ
ejpam-4582	382	19	that	that	SCONJ
ejpam-4582	382	20	y	y	PROPN
ejpam-4582	382	21	∈	∈	PROPN
ejpam-4582	382	22	u	u	PROPN
ejpam-4582	382	23	.	.	PUNCT
ejpam-4582	383	1	since	since	SCONJ
ejpam-4582	383	2	(	(	PUNCT
ejpam-4582	383	3	x	x	X
ejpam-4582	383	4	,	,	PUNCT
ejpam-4582	383	5	τ	τ	X
ejpam-4582	383	6	)	)	PUNCT
ejpam-4582	383	7	is	be	AUX
ejpam-4582	383	8	(	(	PUNCT
ejpam-4582	383	9	λ	λ	INTJ
ejpam-4582	383	10	,	,	PUNCT
ejpam-4582	383	11	s)-r0	s)-r0	PRON
ejpam-4582	383	12	,	,	PUNCT
ejpam-4582	383	13	we	we	PRON
ejpam-4582	383	14	have	have	VERB
ejpam-4582	383	15	x	x	PART
ejpam-4582	383	16	∈	∈	PROPN
ejpam-4582	383	17	{	{	PUNCT
ejpam-4582	383	18	x}(λ	x}(λ	PROPN
ejpam-4582	383	19	,	,	PUNCT
ejpam-4582	383	20	s	s	PART
ejpam-4582	383	21	)	)	PUNCT
ejpam-4582	383	22	⊆	⊆	NUM
ejpam-4582	383	23	u	u	NOUN
ejpam-4582	383	24	.	.	PUNCT
ejpam-4582	384	1	thus	thus	ADV
ejpam-4582	384	2	,	,	PUNCT
ejpam-4582	384	3	every	every	DET
ejpam-4582	384	4	(	(	PUNCT
ejpam-4582	384	5	λ	λ	NOUN
ejpam-4582	384	6	,	,	PUNCT
ejpam-4582	384	7	s)-open	s)-open	PUNCT
ejpam-4582	384	8	set	set	VERB
ejpam-4582	384	9	which	which	PRON
ejpam-4582	384	10	contains	contain	VERB
ejpam-4582	384	11	y	y	PROPN
ejpam-4582	384	12	contains	contain	VERB
ejpam-4582	384	13	x.	x.	NOUN
ejpam-4582	384	14	conversely	conversely	ADV
ejpam-4582	384	15	,	,	PUNCT
ejpam-4582	384	16	let	let	VERB
ejpam-4582	384	17	u	u	PRON
ejpam-4582	384	18	∈	∈	PROPN
ejpam-4582	384	19	(	(	PUNCT
ejpam-4582	384	20	λ	λ	NOUN
ejpam-4582	384	21	,	,	PUNCT
ejpam-4582	384	22	s)o(x	s)o(x	ADJ
ejpam-4582	384	23	)	)	PUNCT
ejpam-4582	384	24	and	and	CCONJ
ejpam-4582	384	25	x	x	PUNCT
ejpam-4582	384	26	∈	∈	PROPN
ejpam-4582	384	27	u	u	NOUN
ejpam-4582	384	28	.	.	PUNCT
ejpam-4582	385	1	if	if	SCONJ
ejpam-4582	385	2	y	y	PROPN
ejpam-4582	385	3	̸∈	̸∈	PROPN
ejpam-4582	385	4	u	u	PROPN
ejpam-4582	385	5	,	,	PUNCT
ejpam-4582	385	6	then	then	ADV
ejpam-4582	385	7	x	x	PROPN
ejpam-4582	385	8	̸∈	̸∈	PROPN
ejpam-4582	385	9	{	{	PUNCT
ejpam-4582	385	10	y}(λ	y}(λ	PROPN
ejpam-4582	385	11	,	,	PUNCT
ejpam-4582	385	12	s	s	PART
ejpam-4582	385	13	)	)	PUNCT
ejpam-4582	385	14	and	and	CCONJ
ejpam-4582	385	15	hence	hence	ADV
ejpam-4582	385	16	y	y	PROPN
ejpam-4582	385	17	̸∈	̸∈	PROPN
ejpam-4582	385	18	{	{	PUNCT
ejpam-4582	385	19	x}(λ	x}(λ	PROPN
ejpam-4582	385	20	,	,	PUNCT
ejpam-4582	385	21	s	s	PART
ejpam-4582	385	22	)	)	PUNCT
ejpam-4582	385	23	.	.	PUNCT
ejpam-4582	386	1	this	this	PRON
ejpam-4582	386	2	implies	imply	VERB
ejpam-4582	386	3	that	that	SCONJ
ejpam-4582	386	4	{	{	PUNCT
ejpam-4582	386	5	y}(λ	y}(λ	PROPN
ejpam-4582	386	6	,	,	PUNCT
ejpam-4582	386	7	s	s	PART
ejpam-4582	386	8	)	)	PUNCT
ejpam-4582	386	9	⊆	⊆	NUM
ejpam-4582	386	10	u	u	NOUN
ejpam-4582	386	11	.	.	PUNCT
ejpam-4582	387	1	thus	thus	ADV
ejpam-4582	387	2	,	,	PUNCT
ejpam-4582	387	3	(	(	PUNCT
ejpam-4582	387	4	x	x	X
ejpam-4582	387	5	,	,	PUNCT
ejpam-4582	387	6	τ	τ	X
ejpam-4582	387	7	)	)	PUNCT
ejpam-4582	387	8	is	be	AUX
ejpam-4582	387	9	a	a	DET
ejpam-4582	387	10	(	(	PUNCT
ejpam-4582	387	11	λ	λ	PROPN
ejpam-4582	387	12	,	,	PUNCT
ejpam-4582	387	13	s)-r0	s)-r0	NOUN
ejpam-4582	387	14	space	space	NOUN
ejpam-4582	387	15	.	.	PUNCT
ejpam-4582	388	1	5	5	X
ejpam-4582	388	2	.	.	X
ejpam-4582	388	3	on	on	ADP
ejpam-4582	388	4	generalized	generalized	ADJ
ejpam-4582	388	5	(	(	PUNCT
ejpam-4582	388	6	λ	λ	X
ejpam-4582	388	7	,	,	PUNCT
ejpam-4582	388	8	s)-closed	s)-close	VERB
ejpam-4582	388	9	sets	set	NOUN
ejpam-4582	388	10	in	in	ADP
ejpam-4582	388	11	this	this	DET
ejpam-4582	388	12	section	section	NOUN
ejpam-4582	388	13	,	,	PUNCT
ejpam-4582	388	14	we	we	PRON
ejpam-4582	388	15	introduce	introduce	VERB
ejpam-4582	388	16	the	the	DET
ejpam-4582	388	17	notion	notion	NOUN
ejpam-4582	388	18	of	of	ADP
ejpam-4582	388	19	generalized	generalized	ADJ
ejpam-4582	388	20	(	(	PUNCT
ejpam-4582	388	21	λ	λ	X
ejpam-4582	388	22	,	,	PUNCT
ejpam-4582	388	23	s)-closed	s)-close	VERB
ejpam-4582	388	24	sets	set	NOUN
ejpam-4582	388	25	and	and	CCONJ
ejpam-4582	388	26	investigate	investigate	VERB
ejpam-4582	388	27	several	several	ADJ
ejpam-4582	388	28	fundamental	fundamental	ADJ
ejpam-4582	388	29	properties	property	NOUN
ejpam-4582	388	30	of	of	ADP
ejpam-4582	388	31	generalized	generalized	ADJ
ejpam-4582	388	32	(	(	PUNCT
ejpam-4582	388	33	λ	λ	X
ejpam-4582	388	34	,	,	PUNCT
ejpam-4582	388	35	s)-closed	s)-close	VERB
ejpam-4582	388	36	sets	set	NOUN
ejpam-4582	388	37	.	.	PUNCT
ejpam-4582	389	1	furthermore	furthermore	ADV
ejpam-4582	389	2	,	,	PUNCT
ejpam-4582	389	3	some	some	DET
ejpam-4582	389	4	characterizations	characterization	NOUN
ejpam-4582	389	5	of	of	ADP
ejpam-4582	389	6	(	(	PUNCT
ejpam-4582	389	7	λ	λ	PROPN
ejpam-4582	389	8	,	,	PUNCT
ejpam-4582	389	9	s)-normal	s)-normal	ADJ
ejpam-4582	389	10	spaces	space	NOUN
ejpam-4582	389	11	are	be	AUX
ejpam-4582	389	12	discussed	discuss	VERB
ejpam-4582	389	13	.	.	PUNCT
ejpam-4582	390	1	definition	definition	NOUN
ejpam-4582	390	2	9	9	NUM
ejpam-4582	390	3	.	.	PUNCT
ejpam-4582	391	1	a	a	DET
ejpam-4582	391	2	subset	subset	NOUN
ejpam-4582	391	3	a	a	PRON
ejpam-4582	391	4	of	of	ADP
ejpam-4582	391	5	a	a	DET
ejpam-4582	391	6	topological	topological	ADJ
ejpam-4582	391	7	space	space	NOUN
ejpam-4582	391	8	(	(	PUNCT
ejpam-4582	391	9	x	x	X
ejpam-4582	391	10	,	,	PUNCT
ejpam-4582	391	11	τ	τ	X
ejpam-4582	391	12	)	)	PUNCT
ejpam-4582	391	13	is	be	AUX
ejpam-4582	391	14	said	say	VERB
ejpam-4582	391	15	to	to	PART
ejpam-4582	391	16	be	be	AUX
ejpam-4582	391	17	generalized	generalize	VERB
ejpam-4582	391	18	(	(	PUNCT
ejpam-4582	391	19	λ	λ	X
ejpam-4582	391	20	,	,	PUNCT
ejpam-4582	391	21	s)-closed	s)-close	VERB
ejpam-4582	391	22	(	(	PUNCT
ejpam-4582	391	23	briefly	briefly	ADV
ejpam-4582	391	24	g-(λ	g-(λ	PROPN
ejpam-4582	391	25	,	,	PUNCT
ejpam-4582	391	26	s)-closed	s)-close	VERB
ejpam-4582	391	27	)	)	PUNCT
ejpam-4582	391	28	if	if	SCONJ
ejpam-4582	391	29	a(λ	a(λ	ADV
ejpam-4582	391	30	,	,	PUNCT
ejpam-4582	391	31	s	s	PART
ejpam-4582	391	32	)	)	PUNCT
ejpam-4582	391	33	⊆	⊆	NUM
ejpam-4582	391	34	u	u	NOUN
ejpam-4582	391	35	whenever	whenever	SCONJ
ejpam-4582	391	36	a	a	DET
ejpam-4582	391	37	⊆	⊆	NUM
ejpam-4582	391	38	u	u	NOUN
ejpam-4582	391	39	and	and	CCONJ
ejpam-4582	391	40	u	u	PROPN
ejpam-4582	391	41	∈	∈	PROPN
ejpam-4582	391	42	(	(	PUNCT
ejpam-4582	391	43	λ	λ	NOUN
ejpam-4582	391	44	,	,	PUNCT
ejpam-4582	391	45	s)o(x	s)o(x	NOUN
ejpam-4582	391	46	)	)	PUNCT
ejpam-4582	391	47	.	.	PUNCT
ejpam-4582	392	1	remark	remark	PROPN
ejpam-4582	392	2	1	1	NUM
ejpam-4582	392	3	.	.	PUNCT
ejpam-4582	393	1	every	every	DET
ejpam-4582	393	2	(	(	PUNCT
ejpam-4582	393	3	λ	λ	NOUN
ejpam-4582	393	4	,	,	PUNCT
ejpam-4582	393	5	s)-closed	s)-close	VERB
ejpam-4582	393	6	set	set	NOUN
ejpam-4582	393	7	is	be	AUX
ejpam-4582	393	8	g-(λ	g-(λ	PROPN
ejpam-4582	393	9	,	,	PUNCT
ejpam-4582	393	10	s)-closed	s)-close	VERB
ejpam-4582	393	11	.	.	PUNCT
ejpam-4582	394	1	the	the	DET
ejpam-4582	394	2	converse	converse	NOUN
ejpam-4582	394	3	of	of	ADP
ejpam-4582	394	4	remark	remark	NOUN
ejpam-4582	394	5	1	1	NUM
ejpam-4582	394	6	need	need	AUX
ejpam-4582	394	7	not	not	PART
ejpam-4582	394	8	be	be	AUX
ejpam-4582	394	9	true	true	ADJ
ejpam-4582	394	10	as	as	SCONJ
ejpam-4582	394	11	shown	show	VERB
ejpam-4582	394	12	in	in	ADP
ejpam-4582	394	13	the	the	DET
ejpam-4582	394	14	following	follow	VERB
ejpam-4582	394	15	example	example	NOUN
ejpam-4582	394	16	.	.	PUNCT
ejpam-4582	395	1	example	example	NOUN
ejpam-4582	396	1	1	1	NUM
ejpam-4582	396	2	.	.	PUNCT
ejpam-4582	396	3	let	let	VERB
ejpam-4582	396	4	x	x	PUNCT
ejpam-4582	396	5	=	=	PRON
ejpam-4582	396	6	{	{	PUNCT
ejpam-4582	396	7	1	1	NUM
ejpam-4582	396	8	,	,	PUNCT
ejpam-4582	396	9	2	2	NUM
ejpam-4582	396	10	,	,	PUNCT
ejpam-4582	396	11	3	3	NUM
ejpam-4582	396	12	}	}	PUNCT
ejpam-4582	396	13	and	and	CCONJ
ejpam-4582	396	14	τ	τ	PROPN
ejpam-4582	396	15	=	=	SYM
ejpam-4582	396	16	{	{	PUNCT
ejpam-4582	396	17	∅	∅	NOUN
ejpam-4582	396	18	,	,	PUNCT
ejpam-4582	396	19	{	{	PUNCT
ejpam-4582	396	20	1	1	NUM
ejpam-4582	396	21	,	,	PUNCT
ejpam-4582	396	22	2	2	NUM
ejpam-4582	396	23	}	}	PUNCT
ejpam-4582	396	24	,	,	PUNCT
ejpam-4582	396	25	x	x	NOUN
ejpam-4582	396	26	}	}	PUNCT
ejpam-4582	396	27	.	.	PUNCT
ejpam-4582	397	1	then	then	ADV
ejpam-4582	397	2	,	,	PUNCT
ejpam-4582	397	3	a	a	PRON
ejpam-4582	397	4	=	=	X
ejpam-4582	397	5	{	{	PUNCT
ejpam-4582	397	6	1	1	NUM
ejpam-4582	397	7	}	}	PUNCT
ejpam-4582	397	8	is	be	AUX
ejpam-4582	397	9	a	a	DET
ejpam-4582	397	10	g-(λ	g-(λ	NOUN
ejpam-4582	397	11	,	,	PUNCT
ejpam-4582	397	12	s)-closed	s)-close	VERB
ejpam-4582	397	13	set	set	NOUN
ejpam-4582	397	14	,	,	PUNCT
ejpam-4582	397	15	which	which	PRON
ejpam-4582	397	16	is	be	AUX
ejpam-4582	397	17	not	not	PART
ejpam-4582	397	18	(	(	PUNCT
ejpam-4582	397	19	λ	λ	X
ejpam-4582	397	20	,	,	PUNCT
ejpam-4582	397	21	s)-closed	s)-close	VERB
ejpam-4582	397	22	.	.	PUNCT
ejpam-4582	398	1	theorem	theorem	NOUN
ejpam-4582	398	2	12	12	NUM
ejpam-4582	398	3	.	.	PUNCT
ejpam-4582	399	1	a	a	DET
ejpam-4582	399	2	topological	topological	ADJ
ejpam-4582	399	3	space	space	NOUN
ejpam-4582	399	4	(	(	PUNCT
ejpam-4582	399	5	x	x	X
ejpam-4582	399	6	,	,	PUNCT
ejpam-4582	399	7	τ	τ	X
ejpam-4582	399	8	)	)	PUNCT
ejpam-4582	399	9	is	be	AUX
ejpam-4582	399	10	(	(	PUNCT
ejpam-4582	399	11	λ	λ	X
ejpam-4582	399	12	,	,	PUNCT
ejpam-4582	399	13	s)-symmetric	s)-symmetric	ADJ
ejpam-4582	399	14	if	if	SCONJ
ejpam-4582	399	15	and	and	CCONJ
ejpam-4582	399	16	only	only	ADV
ejpam-4582	399	17	if	if	SCONJ
ejpam-4582	399	18	{	{	PUNCT
ejpam-4582	399	19	x	x	NOUN
ejpam-4582	399	20	}	}	PUNCT
ejpam-4582	399	21	is	be	AUX
ejpam-4582	399	22	g-(λ	g-(λ	PROPN
ejpam-4582	399	23	,	,	PUNCT
ejpam-4582	399	24	s)closed	s)close	VERB
ejpam-4582	399	25	for	for	ADP
ejpam-4582	399	26	each	each	DET
ejpam-4582	399	27	x	x	SYM
ejpam-4582	399	28	∈	∈	PROPN
ejpam-4582	399	29	x.	x.	NOUN
ejpam-4582	399	30	proof	proof	NOUN
ejpam-4582	399	31	.	.	PUNCT
ejpam-4582	400	1	let	let	VERB
ejpam-4582	400	2	v	v	X
ejpam-4582	400	3	∈	∈	PROPN
ejpam-4582	400	4	(	(	PUNCT
ejpam-4582	400	5	λ	λ	NOUN
ejpam-4582	400	6	,	,	PUNCT
ejpam-4582	400	7	s)o(x	s)o(x	ADJ
ejpam-4582	400	8	)	)	PUNCT
ejpam-4582	400	9	and	and	CCONJ
ejpam-4582	400	10	x	x	PUNCT
ejpam-4582	400	11	∈	∈	NOUN
ejpam-4582	400	12	v	v	NOUN
ejpam-4582	400	13	.	.	PUNCT
ejpam-4582	401	1	suppose	suppose	VERB
ejpam-4582	401	2	that	that	SCONJ
ejpam-4582	401	3	{	{	PUNCT
ejpam-4582	401	4	x}(λ	x}(λ	PROPN
ejpam-4582	401	5	,	,	PUNCT
ejpam-4582	401	6	s	s	PART
ejpam-4582	401	7	)	)	PUNCT
ejpam-4582	401	8	⊈	⊈	PROPN
ejpam-4582	401	9	v	v	NOUN
ejpam-4582	401	10	.	.	PUNCT
ejpam-4582	402	1	therefore	therefore	ADV
ejpam-4582	402	2	,	,	PUNCT
ejpam-4582	402	3	(	(	PUNCT
ejpam-4582	402	4	x	x	X
ejpam-4582	402	5	−	−	PROPN
ejpam-4582	402	6	v	v	NOUN
ejpam-4582	402	7	)	)	PUNCT
ejpam-4582	402	8	∩	∩	NOUN
ejpam-4582	402	9	{	{	PUNCT
ejpam-4582	402	10	x}(λ	x}(λ	PROPN
ejpam-4582	402	11	,	,	PUNCT
ejpam-4582	402	12	s	s	PART
ejpam-4582	402	13	)	)	PUNCT
ejpam-4582	402	14	̸=	̸=	PROPN
ejpam-4582	402	15	∅.	∅.	ADV
ejpam-4582	402	16	let	let	VERB
ejpam-4582	402	17	y	y	PROPN
ejpam-4582	402	18	∈	∈	PROPN
ejpam-4582	402	19	(	(	PUNCT
ejpam-4582	402	20	x	x	PROPN
ejpam-4582	402	21	−	−	PROPN
ejpam-4582	402	22	v	v	NOUN
ejpam-4582	402	23	)	)	PUNCT
ejpam-4582	402	24	∩	∩	NOUN
ejpam-4582	402	25	{	{	PUNCT
ejpam-4582	402	26	x}(λ	x}(λ	PROPN
ejpam-4582	402	27	,	,	PUNCT
ejpam-4582	402	28	s	s	PART
ejpam-4582	402	29	)	)	PUNCT
ejpam-4582	402	30	.	.	PUNCT
ejpam-4582	403	1	thus	thus	ADV
ejpam-4582	403	2	,	,	PUNCT
ejpam-4582	403	3	y	y	PROPN
ejpam-4582	403	4	∈	∈	PROPN
ejpam-4582	403	5	{	{	PUNCT
ejpam-4582	403	6	x}(λ	x}(λ	PROPN
ejpam-4582	403	7	,	,	PUNCT
ejpam-4582	403	8	s	s	PART
ejpam-4582	403	9	)	)	PUNCT
ejpam-4582	403	10	and	and	CCONJ
ejpam-4582	403	11	hence	hence	ADV
ejpam-4582	403	12	x	x	X
ejpam-4582	403	13	∈	∈	PROPN
ejpam-4582	403	14	{	{	PUNCT
ejpam-4582	403	15	y}(λ	y}(λ	PROPN
ejpam-4582	403	16	,	,	PUNCT
ejpam-4582	403	17	s	s	PROPN
ejpam-4582	403	18	)	)	PUNCT
ejpam-4582	403	19	.	.	PUNCT
ejpam-4582	404	1	now	now	ADV
ejpam-4582	404	2	,	,	PUNCT
ejpam-4582	404	3	we	we	PRON
ejpam-4582	404	4	have	have	VERB
ejpam-4582	404	5	x	x	PART
ejpam-4582	404	6	∈	∈	PROPN
ejpam-4582	404	7	{	{	PUNCT
ejpam-4582	404	8	y}(λ	y}(λ	PROPN
ejpam-4582	404	9	,	,	PUNCT
ejpam-4582	404	10	s	s	PROPN
ejpam-4582	404	11	)	)	PUNCT
ejpam-4582	404	12	which	which	PRON
ejpam-4582	404	13	is	be	AUX
ejpam-4582	404	14	a	a	DET
ejpam-4582	404	15	subset	subset	NOUN
ejpam-4582	404	16	of	of	ADP
ejpam-4582	404	17	the	the	DET
ejpam-4582	404	18	complement	complement	NOUN
ejpam-4582	404	19	of	of	ADP
ejpam-4582	404	20	v	v	NOUN
ejpam-4582	404	21	and	and	CCONJ
ejpam-4582	404	22	x	x	PART
ejpam-4582	404	23	̸∈	̸∈	PROPN
ejpam-4582	404	24	v	v	PROPN
ejpam-4582	404	25	.	.	PUNCT
ejpam-4582	405	1	this	this	PRON
ejpam-4582	405	2	is	be	AUX
ejpam-4582	405	3	a	a	DET
ejpam-4582	405	4	contradiction	contradiction	NOUN
ejpam-4582	405	5	.	.	PUNCT
ejpam-4582	406	1	c.	c.	PROPN
ejpam-4582	406	2	boonpok	boonpok	PROPN
ejpam-4582	406	3	,	,	PUNCT
ejpam-4582	406	4	c.	c.	PROPN
ejpam-4582	406	5	viriyapong	viriyapong	PROPN
ejpam-4582	406	6	/	/	SYM
ejpam-4582	406	7	eur	eur	PROPN
ejpam-4582	406	8	.	.	PUNCT
ejpam-4582	407	1	j.	j.	PROPN
ejpam-4582	407	2	pure	pure	PROPN
ejpam-4582	407	3	appl	appl	PROPN
ejpam-4582	407	4	.	.	PROPN
ejpam-4582	407	5	math	math	PROPN
ejpam-4582	407	6	,	,	PUNCT
ejpam-4582	407	7	16	16	NUM
ejpam-4582	407	8	(	(	PUNCT
ejpam-4582	407	9	1	1	NUM
ejpam-4582	407	10	)	)	PUNCT
ejpam-4582	407	11	(	(	PUNCT
ejpam-4582	407	12	2023	2023	NUM
ejpam-4582	407	13	)	)	PUNCT
ejpam-4582	407	14	,	,	PUNCT
ejpam-4582	407	15	336	336	NUM
ejpam-4582	407	16	-	-	SYM
ejpam-4582	407	17	362	362	NUM
ejpam-4582	407	18	347	347	NUM
ejpam-4582	407	19	conversely	conversely	ADV
ejpam-4582	407	20	,	,	PUNCT
ejpam-4582	407	21	let	let	VERB
ejpam-4582	407	22	x	x	X
ejpam-4582	407	23	∈	∈	PROPN
ejpam-4582	407	24	{	{	PUNCT
ejpam-4582	407	25	y}(λ	y}(λ	PROPN
ejpam-4582	407	26	,	,	PUNCT
ejpam-4582	407	27	s	s	PROPN
ejpam-4582	407	28	)	)	PUNCT
ejpam-4582	407	29	.	.	PUNCT
ejpam-4582	408	1	suppose	suppose	VERB
ejpam-4582	408	2	that	that	SCONJ
ejpam-4582	408	3	y	y	PROPN
ejpam-4582	408	4	̸∈	̸∈	PROPN
ejpam-4582	408	5	{	{	PUNCT
ejpam-4582	408	6	x}(λ	x}(λ	PROPN
ejpam-4582	408	7	,	,	PUNCT
ejpam-4582	408	8	s	s	PART
ejpam-4582	408	9	)	)	PUNCT
ejpam-4582	408	10	.	.	PUNCT
ejpam-4582	409	1	thus	thus	ADV
ejpam-4582	409	2	,	,	PUNCT
ejpam-4582	409	3	y	y	PROPN
ejpam-4582	409	4	∈	∈	PROPN
ejpam-4582	409	5	x	x	PUNCT
ejpam-4582	409	6	−	−	PROPN
ejpam-4582	409	7	{	{	PUNCT
ejpam-4582	409	8	x}(λ	x}(λ	PROPN
ejpam-4582	409	9	,	,	PUNCT
ejpam-4582	409	10	s	s	PART
ejpam-4582	409	11	)	)	PUNCT
ejpam-4582	409	12	and	and	CCONJ
ejpam-4582	409	13	hence	hence	ADV
ejpam-4582	409	14	{	{	PUNCT
ejpam-4582	409	15	y}(λ	y}(λ	PROPN
ejpam-4582	409	16	,	,	PUNCT
ejpam-4582	409	17	s	s	PART
ejpam-4582	409	18	)	)	PUNCT
ejpam-4582	409	19	⊆	⊆	NUM
ejpam-4582	409	20	x	x	SYM
ejpam-4582	409	21	−	−	PROPN
ejpam-4582	409	22	{	{	PUNCT
ejpam-4582	409	23	x}(λ	x}(λ	PROPN
ejpam-4582	409	24	,	,	PUNCT
ejpam-4582	409	25	s	s	PART
ejpam-4582	409	26	)	)	PUNCT
ejpam-4582	409	27	.	.	PUNCT
ejpam-4582	410	1	now	now	ADV
ejpam-4582	410	2	,	,	PUNCT
ejpam-4582	410	3	the	the	DET
ejpam-4582	410	4	complement	complement	NOUN
ejpam-4582	410	5	of	of	ADP
ejpam-4582	410	6	{	{	PUNCT
ejpam-4582	410	7	x}(λ	x}(λ	PROPN
ejpam-4582	410	8	,	,	PUNCT
ejpam-4582	410	9	s	s	PART
ejpam-4582	410	10	)	)	PUNCT
ejpam-4582	410	11	contains	contain	VERB
ejpam-4582	410	12	x	x	PUNCT
ejpam-4582	410	13	which	which	PRON
ejpam-4582	410	14	is	be	AUX
ejpam-4582	410	15	a	a	DET
ejpam-4582	410	16	contradiction	contradiction	NOUN
ejpam-4582	410	17	.	.	PUNCT
ejpam-4582	411	1	theorem	theorem	VERB
ejpam-4582	411	2	13	13	NUM
ejpam-4582	411	3	.	.	PUNCT
ejpam-4582	412	1	a	a	DET
ejpam-4582	412	2	subset	subset	NOUN
ejpam-4582	412	3	a	a	PRON
ejpam-4582	412	4	of	of	ADP
ejpam-4582	412	5	a	a	DET
ejpam-4582	412	6	topological	topological	ADJ
ejpam-4582	412	7	space	space	NOUN
ejpam-4582	412	8	(	(	PUNCT
ejpam-4582	412	9	x	x	X
ejpam-4582	412	10	,	,	PUNCT
ejpam-4582	412	11	τ	τ	X
ejpam-4582	412	12	)	)	PUNCT
ejpam-4582	412	13	is	be	AUX
ejpam-4582	412	14	g-(λ	g-(λ	PROPN
ejpam-4582	412	15	,	,	PUNCT
ejpam-4582	412	16	s)-closed	s)-close	VERB
ejpam-4582	412	17	if	if	SCONJ
ejpam-4582	412	18	and	and	CCONJ
ejpam-4582	412	19	only	only	ADV
ejpam-4582	412	20	if	if	SCONJ
ejpam-4582	412	21	a(λ	a(λ	PROPN
ejpam-4582	412	22	,	,	PUNCT
ejpam-4582	412	23	s	s	X
ejpam-4582	412	24	)	)	PUNCT
ejpam-4582	412	25	−a	−a	NOUN
ejpam-4582	412	26	contains	contain	VERB
ejpam-4582	412	27	no	no	DET
ejpam-4582	412	28	nonempty	nonempty	ADJ
ejpam-4582	412	29	(	(	PUNCT
ejpam-4582	412	30	λ	λ	X
ejpam-4582	412	31	,	,	PUNCT
ejpam-4582	412	32	s)-closed	s)-close	VERB
ejpam-4582	412	33	set	set	NOUN
ejpam-4582	412	34	.	.	PUNCT
ejpam-4582	413	1	proof	proof	NOUN
ejpam-4582	413	2	.	.	PUNCT
ejpam-4582	414	1	let	let	VERB
ejpam-4582	414	2	f	f	PRON
ejpam-4582	414	3	be	be	AUX
ejpam-4582	414	4	a	a	DET
ejpam-4582	414	5	(	(	PUNCT
ejpam-4582	414	6	λ	λ	X
ejpam-4582	414	7	,	,	PUNCT
ejpam-4582	414	8	s)-closed	s)-close	VERB
ejpam-4582	414	9	subset	subset	NOUN
ejpam-4582	414	10	of	of	ADP
ejpam-4582	414	11	a(λ	a(λ	PROPN
ejpam-4582	414	12	,	,	PUNCT
ejpam-4582	414	13	s	s	NOUN
ejpam-4582	414	14	)	)	PUNCT
ejpam-4582	414	15	−	−	NOUN
ejpam-4582	414	16	a.	a.	NOUN
ejpam-4582	414	17	now	now	ADV
ejpam-4582	414	18	,	,	PUNCT
ejpam-4582	414	19	a	a	DET
ejpam-4582	414	20	⊆	⊆	NUM
ejpam-4582	414	21	x	x	SYM
ejpam-4582	414	22	−	−	PROPN
ejpam-4582	414	23	f	f	PROPN
ejpam-4582	414	24	and	and	CCONJ
ejpam-4582	414	25	since	since	SCONJ
ejpam-4582	414	26	a	a	PRON
ejpam-4582	414	27	is	be	AUX
ejpam-4582	414	28	g-(λ	g-(λ	PRON
ejpam-4582	414	29	,	,	PUNCT
ejpam-4582	414	30	s)-closed	s)-close	VERB
ejpam-4582	414	31	,	,	PUNCT
ejpam-4582	414	32	we	we	PRON
ejpam-4582	414	33	have	have	VERB
ejpam-4582	414	34	a(λ	a(λ	ADV
ejpam-4582	414	35	,	,	PUNCT
ejpam-4582	414	36	s	s	PART
ejpam-4582	414	37	)	)	PUNCT
ejpam-4582	414	38	⊆	⊆	NUM
ejpam-4582	414	39	x	x	SYM
ejpam-4582	414	40	−	−	PROPN
ejpam-4582	414	41	f	f	PROPN
ejpam-4582	414	42	and	and	CCONJ
ejpam-4582	414	43	f	f	PROPN
ejpam-4582	414	44	⊆	⊆	NUM
ejpam-4582	414	45	x	x	SYM
ejpam-4582	414	46	−a(λ	−a(λ	NOUN
ejpam-4582	414	47	,	,	PUNCT
ejpam-4582	414	48	s	s	NOUN
ejpam-4582	414	49	)	)	PUNCT
ejpam-4582	414	50	.	.	PUNCT
ejpam-4582	415	1	thus	thus	ADV
ejpam-4582	415	2	,	,	PUNCT
ejpam-4582	415	3	f	f	PROPN
ejpam-4582	415	4	⊆	⊆	NUM
ejpam-4582	415	5	a(λ	a(λ	PROPN
ejpam-4582	415	6	,	,	PUNCT
ejpam-4582	415	7	s	s	NOUN
ejpam-4582	415	8	)	)	PUNCT
ejpam-4582	415	9	∩	∩	NOUN
ejpam-4582	415	10	(	(	PUNCT
ejpam-4582	415	11	x	x	SYM
ejpam-4582	415	12	−a(λ	−a(λ	NOUN
ejpam-4582	415	13	,	,	PUNCT
ejpam-4582	415	14	s	s	NOUN
ejpam-4582	415	15	)	)	PUNCT
ejpam-4582	415	16	)	)	PUNCT
ejpam-4582	416	1	=	=	NOUN
ejpam-4582	416	2	∅	∅	NOUN
ejpam-4582	416	3	and	and	CCONJ
ejpam-4582	416	4	f	f	PROPN
ejpam-4582	416	5	is	be	AUX
ejpam-4582	416	6	empty	empty	ADJ
ejpam-4582	416	7	.	.	PUNCT
ejpam-4582	417	1	conversely	conversely	ADV
ejpam-4582	417	2	,	,	PUNCT
ejpam-4582	417	3	let	let	VERB
ejpam-4582	417	4	a	a	DET
ejpam-4582	417	5	⊆	⊆	NUM
ejpam-4582	417	6	u	u	NOUN
ejpam-4582	417	7	and	and	CCONJ
ejpam-4582	417	8	u	u	PRON
ejpam-4582	417	9	be	be	VERB
ejpam-4582	417	10	(	(	PUNCT
ejpam-4582	417	11	λ	λ	X
ejpam-4582	417	12	,	,	PUNCT
ejpam-4582	417	13	s)-open	s)-open	ADJ
ejpam-4582	417	14	.	.	PUNCT
ejpam-4582	418	1	if	if	SCONJ
ejpam-4582	418	2	a(λ	a(λ	ADV
ejpam-4582	418	3	,	,	PUNCT
ejpam-4582	418	4	s	s	PART
ejpam-4582	418	5	)	)	PUNCT
ejpam-4582	418	6	⊈	⊈	PROPN
ejpam-4582	418	7	u	u	NOUN
ejpam-4582	418	8	,	,	PUNCT
ejpam-4582	418	9	then	then	ADV
ejpam-4582	418	10	a(λ	a(λ	ADV
ejpam-4582	418	11	,	,	PUNCT
ejpam-4582	418	12	s	s	NOUN
ejpam-4582	418	13	)	)	PUNCT
ejpam-4582	418	14	∩	∩	NOUN
ejpam-4582	418	15	(	(	PUNCT
ejpam-4582	418	16	x	x	SYM
ejpam-4582	418	17	−	−	PROPN
ejpam-4582	418	18	u	u	NOUN
ejpam-4582	418	19	)	)	PUNCT
ejpam-4582	418	20	is	be	AUX
ejpam-4582	418	21	a	a	DET
ejpam-4582	418	22	nonempty	nonempty	ADJ
ejpam-4582	418	23	(	(	PUNCT
ejpam-4582	418	24	λ	λ	X
ejpam-4582	418	25	,	,	PUNCT
ejpam-4582	418	26	s)-closed	s)-close	VERB
ejpam-4582	418	27	subset	subset	NOUN
ejpam-4582	418	28	of	of	ADP
ejpam-4582	418	29	a(λ	a(λ	PROPN
ejpam-4582	418	30	,	,	PUNCT
ejpam-4582	418	31	s	s	X
ejpam-4582	418	32	)	)	PUNCT
ejpam-4582	418	33	−a	−a	NOUN
ejpam-4582	418	34	.	.	PUNCT
ejpam-4582	419	1	proposition	proposition	NOUN
ejpam-4582	419	2	6	6	NUM
ejpam-4582	419	3	.	.	PUNCT
ejpam-4582	420	1	let	let	VERB
ejpam-4582	420	2	a	a	DET
ejpam-4582	420	3	,	,	PUNCT
ejpam-4582	420	4	b	b	NOUN
ejpam-4582	420	5	subsets	subset	NOUN
ejpam-4582	420	6	of	of	ADP
ejpam-4582	420	7	a	a	DET
ejpam-4582	420	8	topological	topological	ADJ
ejpam-4582	420	9	space	space	NOUN
ejpam-4582	420	10	(	(	PUNCT
ejpam-4582	420	11	x	x	X
ejpam-4582	420	12	,	,	PUNCT
ejpam-4582	420	13	τ	τ	PROPN
ejpam-4582	420	14	)	)	PUNCT
ejpam-4582	420	15	.	.	PUNCT
ejpam-4582	421	1	if	if	SCONJ
ejpam-4582	421	2	a	a	PRON
ejpam-4582	421	3	is	be	AUX
ejpam-4582	421	4	g-(λ	g-(λ	PRON
ejpam-4582	421	5	,	,	PUNCT
ejpam-4582	421	6	s)-closed	s)-close	VERB
ejpam-4582	421	7	and	and	CCONJ
ejpam-4582	421	8	a	a	DET
ejpam-4582	421	9	⊆	⊆	NUM
ejpam-4582	421	10	b	b	NOUN
ejpam-4582	421	11	⊆	⊆	NUM
ejpam-4582	421	12	a(λ	a(λ	ADJ
ejpam-4582	421	13	,	,	PUNCT
ejpam-4582	421	14	s	s	NOUN
ejpam-4582	421	15	)	)	PUNCT
ejpam-4582	421	16	,	,	PUNCT
ejpam-4582	421	17	then	then	ADV
ejpam-4582	421	18	b	b	PROPN
ejpam-4582	421	19	is	be	AUX
ejpam-4582	421	20	g-(λ	g-(λ	PRON
ejpam-4582	421	21	,	,	PUNCT
ejpam-4582	421	22	s)-closed	s)-close	VERB
ejpam-4582	421	23	.	.	PUNCT
ejpam-4582	422	1	proof	proof	NOUN
ejpam-4582	422	2	.	.	PUNCT
ejpam-4582	423	1	let	let	VERB
ejpam-4582	423	2	b	b	NOUN
ejpam-4582	423	3	⊆	⊆	NUM
ejpam-4582	423	4	u	u	NOUN
ejpam-4582	423	5	and	and	CCONJ
ejpam-4582	423	6	u	u	PROPN
ejpam-4582	423	7	∈	∈	PROPN
ejpam-4582	423	8	(	(	PUNCT
ejpam-4582	423	9	λ	λ	NOUN
ejpam-4582	423	10	,	,	PUNCT
ejpam-4582	423	11	s)o(x	s)o(x	NOUN
ejpam-4582	423	12	)	)	PUNCT
ejpam-4582	423	13	.	.	PUNCT
ejpam-4582	424	1	then	then	ADV
ejpam-4582	424	2	,	,	PUNCT
ejpam-4582	424	3	we	we	PRON
ejpam-4582	424	4	have	have	VERB
ejpam-4582	424	5	a	a	DET
ejpam-4582	424	6	⊆	⊆	NUM
ejpam-4582	424	7	u	u	NOUN
ejpam-4582	424	8	.	.	PUNCT
ejpam-4582	425	1	since	since	SCONJ
ejpam-4582	425	2	a	a	PRON
ejpam-4582	425	3	is	be	AUX
ejpam-4582	425	4	g-(λ	g-(λ	PRON
ejpam-4582	425	5	,	,	PUNCT
ejpam-4582	425	6	s)closed	s)close	VERB
ejpam-4582	425	7	,	,	PUNCT
ejpam-4582	425	8	a(λ	a(λ	ADV
ejpam-4582	425	9	,	,	PUNCT
ejpam-4582	425	10	s	s	PART
ejpam-4582	425	11	)	)	PUNCT
ejpam-4582	425	12	⊆	⊆	NUM
ejpam-4582	425	13	u	u	NOUN
ejpam-4582	425	14	.	.	PUNCT
ejpam-4582	426	1	since	since	SCONJ
ejpam-4582	426	2	a	a	DET
ejpam-4582	426	3	⊆	⊆	NUM
ejpam-4582	426	4	b	b	NOUN
ejpam-4582	426	5	⊆	⊆	NUM
ejpam-4582	426	6	a(λ	a(λ	ADJ
ejpam-4582	426	7	,	,	PUNCT
ejpam-4582	426	8	s	s	NOUN
ejpam-4582	426	9	)	)	PUNCT
ejpam-4582	426	10	,	,	PUNCT
ejpam-4582	426	11	b(λ	b(λ	PROPN
ejpam-4582	426	12	,	,	PUNCT
ejpam-4582	426	13	s	s	NOUN
ejpam-4582	426	14	)	)	PUNCT
ejpam-4582	426	15	=	=	PUNCT
ejpam-4582	426	16	a(λ	a(λ	PROPN
ejpam-4582	426	17	,	,	PUNCT
ejpam-4582	426	18	s	s	PART
ejpam-4582	426	19	)	)	PUNCT
ejpam-4582	426	20	and	and	CCONJ
ejpam-4582	426	21	hence	hence	ADV
ejpam-4582	426	22	b(λ	b(λ	PROPN
ejpam-4582	426	23	,	,	PUNCT
ejpam-4582	426	24	s	s	PART
ejpam-4582	426	25	)	)	PUNCT
ejpam-4582	426	26	⊆	⊆	NUM
ejpam-4582	426	27	u	u	NOUN
ejpam-4582	426	28	.	.	PUNCT
ejpam-4582	427	1	thus	thus	ADV
ejpam-4582	427	2	,	,	PUNCT
ejpam-4582	427	3	b	b	PROPN
ejpam-4582	427	4	is	be	AUX
ejpam-4582	427	5	g-(λ	g-(λ	PRON
ejpam-4582	427	6	,	,	PUNCT
ejpam-4582	427	7	s)-closed	s)-close	VERB
ejpam-4582	427	8	.	.	PUNCT
ejpam-4582	428	1	definition	definition	NOUN
ejpam-4582	428	2	10	10	NUM
ejpam-4582	428	3	.	.	PUNCT
ejpam-4582	429	1	let	let	VERB
ejpam-4582	429	2	a	a	DET
ejpam-4582	429	3	be	be	AUX
ejpam-4582	429	4	a	a	DET
ejpam-4582	429	5	subset	subset	NOUN
ejpam-4582	429	6	of	of	ADP
ejpam-4582	429	7	a	a	DET
ejpam-4582	429	8	topological	topological	ADJ
ejpam-4582	429	9	space	space	NOUN
ejpam-4582	429	10	(	(	PUNCT
ejpam-4582	429	11	x	x	X
ejpam-4582	429	12	,	,	PUNCT
ejpam-4582	429	13	τ	τ	PROPN
ejpam-4582	429	14	)	)	PUNCT
ejpam-4582	429	15	.	.	PUNCT
ejpam-4582	430	1	the	the	DET
ejpam-4582	430	2	union	union	NOUN
ejpam-4582	430	3	of	of	ADP
ejpam-4582	430	4	all	all	DET
ejpam-4582	430	5	(	(	PUNCT
ejpam-4582	430	6	λ	λ	X
ejpam-4582	430	7	,	,	PUNCT
ejpam-4582	430	8	s)-open	s)-open	PUNCT
ejpam-4582	430	9	sets	set	NOUN
ejpam-4582	430	10	contained	contain	VERB
ejpam-4582	430	11	in	in	ADP
ejpam-4582	430	12	a	a	PRON
ejpam-4582	430	13	is	be	AUX
ejpam-4582	430	14	called	call	VERB
ejpam-4582	430	15	the	the	DET
ejpam-4582	430	16	(	(	PUNCT
ejpam-4582	430	17	λ	λ	PROPN
ejpam-4582	430	18	,	,	PUNCT
ejpam-4582	430	19	s)-interior	s)-interior	ADJ
ejpam-4582	430	20	of	of	ADP
ejpam-4582	430	21	a	a	PRON
ejpam-4582	430	22	and	and	CCONJ
ejpam-4582	430	23	is	be	AUX
ejpam-4582	430	24	denoted	denote	VERB
ejpam-4582	430	25	by	by	ADP
ejpam-4582	430	26	a(λ	a(λ	PROPN
ejpam-4582	430	27	,	,	PUNCT
ejpam-4582	430	28	s	s	PART
ejpam-4582	430	29	)	)	PUNCT
ejpam-4582	430	30	.	.	PUNCT
ejpam-4582	431	1	lemma	lemma	PROPN
ejpam-4582	431	2	6	6	NUM
ejpam-4582	431	3	.	.	PUNCT
ejpam-4582	432	1	let	let	VERB
ejpam-4582	432	2	a	a	PRON
ejpam-4582	432	3	and	and	CCONJ
ejpam-4582	432	4	b	b	NOUN
ejpam-4582	432	5	be	be	AUX
ejpam-4582	432	6	subsets	subset	NOUN
ejpam-4582	432	7	of	of	ADP
ejpam-4582	432	8	a	a	DET
ejpam-4582	432	9	topological	topological	ADJ
ejpam-4582	432	10	space	space	NOUN
ejpam-4582	432	11	(	(	PUNCT
ejpam-4582	432	12	x	x	X
ejpam-4582	432	13	,	,	PUNCT
ejpam-4582	432	14	τ	τ	PROPN
ejpam-4582	432	15	)	)	PUNCT
ejpam-4582	432	16	.	.	PUNCT
ejpam-4582	433	1	for	for	ADP
ejpam-4582	433	2	the	the	DET
ejpam-4582	433	3	(	(	PUNCT
ejpam-4582	433	4	λ	λ	PROPN
ejpam-4582	433	5	,	,	PUNCT
ejpam-4582	433	6	s)-interior	s)-interior	ADJ
ejpam-4582	433	7	,	,	PUNCT
ejpam-4582	433	8	the	the	DET
ejpam-4582	433	9	following	follow	VERB
ejpam-4582	433	10	properties	property	NOUN
ejpam-4582	433	11	hold	hold	VERB
ejpam-4582	433	12	:	:	PUNCT
ejpam-4582	433	13	(	(	PUNCT
ejpam-4582	433	14	1	1	X
ejpam-4582	433	15	)	)	PUNCT
ejpam-4582	433	16	a(λ	a(λ	ADV
ejpam-4582	433	17	,	,	PUNCT
ejpam-4582	433	18	s	s	X
ejpam-4582	433	19	)	)	PUNCT
ejpam-4582	433	20	⊆	⊆	NUM
ejpam-4582	433	21	a	a	PRON
ejpam-4582	433	22	and	and	CCONJ
ejpam-4582	433	23	[	[	X
ejpam-4582	433	24	a(λ	a(λ	ADV
ejpam-4582	433	25	,	,	PUNCT
ejpam-4582	433	26	s)](λ	s)](λ	PROPN
ejpam-4582	433	27	,	,	PUNCT
ejpam-4582	433	28	s	s	NOUN
ejpam-4582	433	29	)	)	PUNCT
ejpam-4582	433	30	=	=	PUNCT
ejpam-4582	433	31	a(λ	a(λ	PROPN
ejpam-4582	433	32	,	,	PUNCT
ejpam-4582	433	33	s	s	NOUN
ejpam-4582	433	34	)	)	PUNCT
ejpam-4582	433	35	.	.	PUNCT
ejpam-4582	434	1	(	(	PUNCT
ejpam-4582	434	2	2	2	X
ejpam-4582	434	3	)	)	PUNCT
ejpam-4582	434	4	if	if	SCONJ
ejpam-4582	434	5	a	a	DET
ejpam-4582	434	6	⊆	⊆	NUM
ejpam-4582	434	7	b	b	NOUN
ejpam-4582	434	8	,	,	PUNCT
ejpam-4582	434	9	then	then	ADV
ejpam-4582	434	10	a(λ	a(λ	ADV
ejpam-4582	434	11	,	,	PUNCT
ejpam-4582	434	12	s	s	X
ejpam-4582	434	13	)	)	PUNCT
ejpam-4582	434	14	⊆	⊆	NUM
ejpam-4582	434	15	b(λ	b(λ	NOUN
ejpam-4582	434	16	,	,	PUNCT
ejpam-4582	434	17	s	s	NOUN
ejpam-4582	434	18	)	)	PUNCT
ejpam-4582	434	19	.	.	PUNCT
ejpam-4582	435	1	(	(	PUNCT
ejpam-4582	435	2	3	3	X
ejpam-4582	435	3	)	)	PUNCT
ejpam-4582	435	4	a(λ	a(λ	ADV
ejpam-4582	435	5	,	,	PUNCT
ejpam-4582	435	6	s	s	PART
ejpam-4582	435	7	)	)	PUNCT
ejpam-4582	435	8	is	be	AUX
ejpam-4582	435	9	(	(	PUNCT
ejpam-4582	435	10	λ	λ	X
ejpam-4582	435	11	,	,	PUNCT
ejpam-4582	435	12	s)-open	s)-open	ADJ
ejpam-4582	435	13	.	.	PUNCT
ejpam-4582	436	1	(	(	PUNCT
ejpam-4582	436	2	4	4	X
ejpam-4582	436	3	)	)	PUNCT
ejpam-4582	436	4	a	a	PRON
ejpam-4582	436	5	is	be	AUX
ejpam-4582	436	6	(	(	PUNCT
ejpam-4582	436	7	λ	λ	X
ejpam-4582	436	8	,	,	PUNCT
ejpam-4582	436	9	s)-open	s)-open	VERB
ejpam-4582	436	10	if	if	SCONJ
ejpam-4582	436	11	and	and	CCONJ
ejpam-4582	436	12	only	only	ADV
ejpam-4582	436	13	if	if	SCONJ
ejpam-4582	436	14	a(λ	a(λ	NOUN
ejpam-4582	436	15	,	,	PUNCT
ejpam-4582	436	16	s	s	NOUN
ejpam-4582	436	17	)	)	PUNCT
ejpam-4582	436	18	=	=	SYM
ejpam-4582	436	19	a.	a.	NOUN
ejpam-4582	436	20	theorem	theorem	VERB
ejpam-4582	436	21	14	14	NUM
ejpam-4582	436	22	.	.	PUNCT
ejpam-4582	437	1	a	a	DET
ejpam-4582	437	2	subset	subset	NOUN
ejpam-4582	437	3	a	a	PRON
ejpam-4582	437	4	of	of	ADP
ejpam-4582	437	5	a	a	DET
ejpam-4582	437	6	topological	topological	ADJ
ejpam-4582	437	7	space	space	NOUN
ejpam-4582	437	8	(	(	PUNCT
ejpam-4582	437	9	x	x	X
ejpam-4582	437	10	,	,	PUNCT
ejpam-4582	437	11	τ	τ	X
ejpam-4582	437	12	)	)	PUNCT
ejpam-4582	437	13	is	be	AUX
ejpam-4582	437	14	g-(λ	g-(λ	PROPN
ejpam-4582	437	15	,	,	PUNCT
ejpam-4582	437	16	s)-open	s)-open	VERB
ejpam-4582	437	17	if	if	SCONJ
ejpam-4582	437	18	and	and	CCONJ
ejpam-4582	437	19	only	only	ADV
ejpam-4582	437	20	if	if	SCONJ
ejpam-4582	437	21	f	f	PROPN
ejpam-4582	437	22	⊆	⊆	NUM
ejpam-4582	437	23	a(λ	a(λ	PROPN
ejpam-4582	437	24	,	,	PUNCT
ejpam-4582	437	25	s	s	PART
ejpam-4582	437	26	)	)	PUNCT
ejpam-4582	437	27	whenever	whenever	SCONJ
ejpam-4582	437	28	f	f	PROPN
ejpam-4582	437	29	⊆	⊆	PROPN
ejpam-4582	437	30	a	a	PRON
ejpam-4582	437	31	and	and	CCONJ
ejpam-4582	437	32	f	f	NOUN
ejpam-4582	437	33	is	be	AUX
ejpam-4582	437	34	(	(	PUNCT
ejpam-4582	437	35	λ	λ	X
ejpam-4582	437	36	,	,	PUNCT
ejpam-4582	437	37	s)-closed	s)-close	VERB
ejpam-4582	437	38	.	.	PUNCT
ejpam-4582	438	1	proof	proof	NOUN
ejpam-4582	438	2	.	.	PUNCT
ejpam-4582	439	1	suppose	suppose	VERB
ejpam-4582	439	2	that	that	SCONJ
ejpam-4582	439	3	a	a	PRON
ejpam-4582	439	4	is	be	AUX
ejpam-4582	439	5	g-(λ	g-(λ	PRON
ejpam-4582	439	6	,	,	PUNCT
ejpam-4582	439	7	s)-open	s)-open	VERB
ejpam-4582	439	8	.	.	PUNCT
ejpam-4582	440	1	let	let	VERB
ejpam-4582	440	2	f	f	PROPN
ejpam-4582	440	3	⊆	⊆	NUM
ejpam-4582	440	4	a	a	PROPN
ejpam-4582	440	5	and	and	CCONJ
ejpam-4582	440	6	f	f	NOUN
ejpam-4582	440	7	be	be	AUX
ejpam-4582	440	8	(	(	PUNCT
ejpam-4582	440	9	λ	λ	X
ejpam-4582	440	10	,	,	PUNCT
ejpam-4582	440	11	s)-closed	s)-close	VERB
ejpam-4582	440	12	.	.	PUNCT
ejpam-4582	441	1	then	then	ADV
ejpam-4582	441	2	,	,	PUNCT
ejpam-4582	441	3	we	we	PRON
ejpam-4582	441	4	have	have	VERB
ejpam-4582	441	5	x	x	ADJ
ejpam-4582	441	6	−	−	VERB
ejpam-4582	441	7	a	a	DET
ejpam-4582	441	8	⊆	⊆	NUM
ejpam-4582	441	9	x	x	SYM
ejpam-4582	441	10	−	−	PROPN
ejpam-4582	441	11	f	f	NOUN
ejpam-4582	441	12	.	.	PUNCT
ejpam-4582	442	1	since	since	SCONJ
ejpam-4582	442	2	x	x	PRON
ejpam-4582	442	3	−	−	PROPN
ejpam-4582	442	4	f	f	PROPN
ejpam-4582	442	5	is	be	AUX
ejpam-4582	442	6	(	(	PUNCT
ejpam-4582	442	7	λ	λ	X
ejpam-4582	442	8	,	,	PUNCT
ejpam-4582	442	9	s)-open	s)-open	PUNCT
ejpam-4582	442	10	and	and	CCONJ
ejpam-4582	442	11	x	x	X
ejpam-4582	442	12	−	−	NOUN
ejpam-4582	442	13	a	a	PRON
ejpam-4582	442	14	is	be	AUX
ejpam-4582	442	15	g-(λ	g-(λ	PRON
ejpam-4582	442	16	,	,	PUNCT
ejpam-4582	442	17	s)-closed	s)-close	VERB
ejpam-4582	442	18	,	,	PUNCT
ejpam-4582	442	19	x	x	SYM
ejpam-4582	442	20	−a(λ	−a(λ	NOUN
ejpam-4582	442	21	,	,	PUNCT
ejpam-4582	442	22	s	s	PART
ejpam-4582	442	23	)	)	PUNCT
ejpam-4582	442	24	=	=	PUNCT
ejpam-4582	443	1	[	[	X
ejpam-4582	443	2	x	x	X
ejpam-4582	443	3	−a](λ	−a](λ	PROPN
ejpam-4582	443	4	,	,	PUNCT
ejpam-4582	443	5	s	s	PART
ejpam-4582	443	6	)	)	PUNCT
ejpam-4582	443	7	⊆	⊆	NUM
ejpam-4582	443	8	x	x	SYM
ejpam-4582	443	9	−	−	PROPN
ejpam-4582	443	10	f	f	NOUN
ejpam-4582	443	11	and	and	CCONJ
ejpam-4582	443	12	hence	hence	ADV
ejpam-4582	443	13	f	f	PROPN
ejpam-4582	443	14	⊆	⊆	NUM
ejpam-4582	443	15	a(λ	a(λ	PROPN
ejpam-4582	443	16	,	,	PUNCT
ejpam-4582	443	17	s	s	NOUN
ejpam-4582	443	18	)	)	PUNCT
ejpam-4582	443	19	.	.	PUNCT
ejpam-4582	444	1	conversely	conversely	ADV
ejpam-4582	444	2	,	,	PUNCT
ejpam-4582	444	3	letx−a	letx−a	PROPN
ejpam-4582	444	4	⊆	⊆	NUM
ejpam-4582	444	5	u	u	NOUN
ejpam-4582	444	6	and	and	CCONJ
ejpam-4582	444	7	u	u	PROPN
ejpam-4582	444	8	∈	∈	PROPN
ejpam-4582	444	9	(	(	PUNCT
ejpam-4582	444	10	λ	λ	NOUN
ejpam-4582	444	11	,	,	PUNCT
ejpam-4582	444	12	s)o(x	s)o(x	NOUN
ejpam-4582	444	13	)	)	PUNCT
ejpam-4582	444	14	.	.	PUNCT
ejpam-4582	445	1	then	then	ADV
ejpam-4582	445	2	,	,	PUNCT
ejpam-4582	445	3	we	we	PRON
ejpam-4582	445	4	havex−u	havex−u	NOUN
ejpam-4582	445	5	⊆	⊆	NOUN
ejpam-4582	445	6	a	a	DET
ejpam-4582	445	7	andx−u	andx−u	PROPN
ejpam-4582	445	8	is	be	AUX
ejpam-4582	445	9	(	(	PUNCT
ejpam-4582	445	10	λ	λ	X
ejpam-4582	445	11	,	,	PUNCT
ejpam-4582	445	12	s)-closed	s)-close	VERB
ejpam-4582	445	13	.	.	PUNCT
ejpam-4582	446	1	by	by	ADP
ejpam-4582	446	2	the	the	DET
ejpam-4582	446	3	hypothesis	hypothesis	NOUN
ejpam-4582	446	4	,	,	PUNCT
ejpam-4582	446	5	x−u	x−u	X
ejpam-4582	446	6	⊆	⊆	NUM
ejpam-4582	446	7	a(λ	a(λ	ADV
ejpam-4582	446	8	,	,	PUNCT
ejpam-4582	446	9	s	s	NOUN
ejpam-4582	446	10	)	)	PUNCT
ejpam-4582	446	11	and	and	CCONJ
ejpam-4582	446	12	hence	hence	ADV
ejpam-4582	446	13	[	[	X
ejpam-4582	446	14	x−a](λ	x−a](λ	PROPN
ejpam-4582	446	15	,	,	PUNCT
ejpam-4582	446	16	s	s	PART
ejpam-4582	446	17	)	)	PUNCT
ejpam-4582	446	18	=	=	SYM
ejpam-4582	446	19	x−a(λ	x−a(λ	NOUN
ejpam-4582	446	20	,	,	PUNCT
ejpam-4582	446	21	s	s	PART
ejpam-4582	446	22	)	)	PUNCT
ejpam-4582	446	23	⊆	⊆	NUM
ejpam-4582	446	24	u	u	NOUN
ejpam-4582	446	25	.	.	PUNCT
ejpam-4582	447	1	thus	thus	ADV
ejpam-4582	447	2	,	,	PUNCT
ejpam-4582	447	3	x	x	PRON
ejpam-4582	447	4	−a	−a	NOUN
ejpam-4582	447	5	is	be	AUX
ejpam-4582	447	6	g-(λ	g-(λ	PROPN
ejpam-4582	447	7	,	,	PUNCT
ejpam-4582	447	8	s)-closed	s)-close	VERB
ejpam-4582	447	9	and	and	CCONJ
ejpam-4582	447	10	so	so	ADV
ejpam-4582	447	11	a	a	PRON
ejpam-4582	447	12	is	be	AUX
ejpam-4582	447	13	g-(λ	g-(λ	PROPN
ejpam-4582	447	14	,	,	PUNCT
ejpam-4582	447	15	s)-open	s)-open	PUNCT
ejpam-4582	447	16	.	.	PUNCT
ejpam-4582	448	1	c.	c.	PROPN
ejpam-4582	448	2	boonpok	boonpok	PROPN
ejpam-4582	448	3	,	,	PUNCT
ejpam-4582	448	4	c.	c.	PROPN
ejpam-4582	448	5	viriyapong	viriyapong	PROPN
ejpam-4582	448	6	/	/	SYM
ejpam-4582	448	7	eur	eur	PROPN
ejpam-4582	448	8	.	.	PUNCT
ejpam-4582	449	1	j.	j.	PROPN
ejpam-4582	449	2	pure	pure	PROPN
ejpam-4582	449	3	appl	appl	PROPN
ejpam-4582	449	4	.	.	PROPN
ejpam-4582	449	5	math	math	PROPN
ejpam-4582	449	6	,	,	PUNCT
ejpam-4582	449	7	16	16	NUM
ejpam-4582	449	8	(	(	PUNCT
ejpam-4582	449	9	1	1	NUM
ejpam-4582	449	10	)	)	PUNCT
ejpam-4582	449	11	(	(	PUNCT
ejpam-4582	449	12	2023	2023	NUM
ejpam-4582	449	13	)	)	PUNCT
ejpam-4582	449	14	,	,	PUNCT
ejpam-4582	449	15	336	336	NUM
ejpam-4582	449	16	-	-	SYM
ejpam-4582	449	17	362	362	NUM
ejpam-4582	449	18	348	348	NUM
ejpam-4582	449	19	corollary	corollary	NOUN
ejpam-4582	449	20	3	3	NUM
ejpam-4582	449	21	.	.	PUNCT
ejpam-4582	450	1	let	let	VERB
ejpam-4582	450	2	a	a	DET
ejpam-4582	450	3	,	,	PUNCT
ejpam-4582	450	4	b	b	NOUN
ejpam-4582	450	5	subsets	subset	NOUN
ejpam-4582	450	6	of	of	ADP
ejpam-4582	450	7	a	a	DET
ejpam-4582	450	8	topological	topological	ADJ
ejpam-4582	450	9	space	space	NOUN
ejpam-4582	450	10	(	(	PUNCT
ejpam-4582	450	11	x	x	X
ejpam-4582	450	12	,	,	PUNCT
ejpam-4582	450	13	τ	τ	PROPN
ejpam-4582	450	14	)	)	PUNCT
ejpam-4582	450	15	.	.	PUNCT
ejpam-4582	451	1	if	if	SCONJ
ejpam-4582	451	2	a	a	PRON
ejpam-4582	451	3	is	be	AUX
ejpam-4582	451	4	g-(λ	g-(λ	PRON
ejpam-4582	451	5	,	,	PUNCT
ejpam-4582	451	6	s)-open	s)-open	PUNCT
ejpam-4582	451	7	and	and	CCONJ
ejpam-4582	451	8	a(λ	a(λ	ADV
ejpam-4582	451	9	,	,	PUNCT
ejpam-4582	451	10	s	s	X
ejpam-4582	451	11	)	)	PUNCT
ejpam-4582	451	12	⊆	⊆	NUM
ejpam-4582	451	13	b	b	NOUN
ejpam-4582	451	14	⊆	⊆	NUM
ejpam-4582	451	15	a	a	PRON
ejpam-4582	451	16	,	,	PUNCT
ejpam-4582	451	17	then	then	ADV
ejpam-4582	451	18	b	b	PROPN
ejpam-4582	451	19	is	be	AUX
ejpam-4582	451	20	g-(λ	g-(λ	PROPN
ejpam-4582	451	21	,	,	PUNCT
ejpam-4582	451	22	s)-open	s)-open	PUNCT
ejpam-4582	451	23	.	.	PUNCT
ejpam-4582	452	1	proof	proof	NOUN
ejpam-4582	452	2	.	.	PUNCT
ejpam-4582	453	1	this	this	PRON
ejpam-4582	453	2	follows	follow	VERB
ejpam-4582	453	3	from	from	ADP
ejpam-4582	453	4	proposition	proposition	NOUN
ejpam-4582	453	5	6	6	NUM
ejpam-4582	453	6	.	.	PUNCT
ejpam-4582	454	1	lemma	lemma	PROPN
ejpam-4582	454	2	7	7	X
ejpam-4582	454	3	.	.	PUNCT
ejpam-4582	454	4	let	let	VERB
ejpam-4582	454	5	a	a	DET
ejpam-4582	454	6	be	be	AUX
ejpam-4582	454	7	a	a	DET
ejpam-4582	454	8	subset	subset	NOUN
ejpam-4582	454	9	of	of	ADP
ejpam-4582	454	10	a	a	DET
ejpam-4582	454	11	topological	topological	ADJ
ejpam-4582	454	12	space	space	NOUN
ejpam-4582	454	13	(	(	PUNCT
ejpam-4582	454	14	x	x	X
ejpam-4582	454	15	,	,	PUNCT
ejpam-4582	454	16	τ	τ	X
ejpam-4582	454	17	)	)	PUNCT
ejpam-4582	454	18	and	and	CCONJ
ejpam-4582	454	19	g	g	PROPN
ejpam-4582	454	20	∈	∈	PROPN
ejpam-4582	454	21	(	(	PUNCT
ejpam-4582	454	22	λ	λ	NOUN
ejpam-4582	454	23	,	,	PUNCT
ejpam-4582	454	24	s)o(x	s)o(x	NOUN
ejpam-4582	454	25	)	)	PUNCT
ejpam-4582	454	26	.	.	PUNCT
ejpam-4582	455	1	if	if	SCONJ
ejpam-4582	455	2	a	a	DET
ejpam-4582	455	3	∩g	∩g	ADJ
ejpam-4582	455	4	=	=	SYM
ejpam-4582	455	5	∅	∅	NOUN
ejpam-4582	455	6	,	,	PUNCT
ejpam-4582	455	7	then	then	ADV
ejpam-4582	455	8	a(λ	a(λ	ADV
ejpam-4582	455	9	,	,	PUNCT
ejpam-4582	455	10	s	s	NOUN
ejpam-4582	455	11	)	)	PUNCT
ejpam-4582	455	12	∩g	∩g	NOUN
ejpam-4582	455	13	=	=	SYM
ejpam-4582	455	14	∅.	∅.	AUX
ejpam-4582	455	15	theorem	theorem	VERB
ejpam-4582	455	16	15	15	NUM
ejpam-4582	455	17	.	.	PUNCT
ejpam-4582	456	1	for	for	ADP
ejpam-4582	456	2	a	a	DET
ejpam-4582	456	3	subset	subset	NOUN
ejpam-4582	456	4	a	a	PRON
ejpam-4582	456	5	of	of	ADP
ejpam-4582	456	6	a	a	DET
ejpam-4582	456	7	topological	topological	ADJ
ejpam-4582	456	8	space	space	NOUN
ejpam-4582	456	9	(	(	PUNCT
ejpam-4582	456	10	x	x	X
ejpam-4582	456	11	,	,	PUNCT
ejpam-4582	456	12	τ	τ	PROPN
ejpam-4582	456	13	)	)	PUNCT
ejpam-4582	456	14	,	,	PUNCT
ejpam-4582	456	15	the	the	DET
ejpam-4582	456	16	following	follow	VERB
ejpam-4582	456	17	properties	property	NOUN
ejpam-4582	456	18	are	be	AUX
ejpam-4582	456	19	equivalent	equivalent	ADJ
ejpam-4582	456	20	:	:	PUNCT
ejpam-4582	456	21	(	(	PUNCT
ejpam-4582	456	22	1	1	X
ejpam-4582	456	23	)	)	PUNCT
ejpam-4582	456	24	a	a	PRON
ejpam-4582	456	25	is	be	AUX
ejpam-4582	456	26	g-(λ	g-(λ	PROPN
ejpam-4582	456	27	,	,	PUNCT
ejpam-4582	456	28	s)-closed	s)-close	VERB
ejpam-4582	456	29	;	;	PUNCT
ejpam-4582	456	30	(	(	PUNCT
ejpam-4582	456	31	2	2	X
ejpam-4582	456	32	)	)	PUNCT
ejpam-4582	456	33	a(λ	a(λ	ADV
ejpam-4582	456	34	,	,	PUNCT
ejpam-4582	456	35	s	s	X
ejpam-4582	456	36	)	)	PUNCT
ejpam-4582	456	37	−a	−a	NOUN
ejpam-4582	456	38	contains	contain	VERB
ejpam-4582	456	39	no	no	DET
ejpam-4582	456	40	nonempty	nonempty	ADJ
ejpam-4582	456	41	(	(	PUNCT
ejpam-4582	456	42	λ	λ	X
ejpam-4582	456	43	,	,	PUNCT
ejpam-4582	456	44	s)-closed	s)-close	VERB
ejpam-4582	456	45	set	set	VERB
ejpam-4582	456	46	;	;	PUNCT
ejpam-4582	456	47	(	(	PUNCT
ejpam-4582	456	48	3	3	X
ejpam-4582	456	49	)	)	PUNCT
ejpam-4582	456	50	a(λ	a(λ	ADV
ejpam-4582	456	51	,	,	PUNCT
ejpam-4582	456	52	s	s	X
ejpam-4582	456	53	)	)	PUNCT
ejpam-4582	456	54	−a	−a	NOUN
ejpam-4582	456	55	is	be	AUX
ejpam-4582	456	56	g-(λ	g-(λ	PROPN
ejpam-4582	456	57	,	,	PUNCT
ejpam-4582	456	58	s)-open	s)-open	PUNCT
ejpam-4582	456	59	.	.	PUNCT
ejpam-4582	457	1	proof	proof	NOUN
ejpam-4582	457	2	.	.	PUNCT
ejpam-4582	458	1	(	(	PUNCT
ejpam-4582	458	2	1	1	X
ejpam-4582	458	3	)	)	PUNCT
ejpam-4582	458	4	⇒	⇒	NOUN
ejpam-4582	458	5	(	(	PUNCT
ejpam-4582	458	6	2	2	NUM
ejpam-4582	458	7	):	):	PUNCT
ejpam-4582	458	8	this	this	PRON
ejpam-4582	458	9	follows	follow	VERB
ejpam-4582	458	10	from	from	ADP
ejpam-4582	458	11	theorem	theorem	ADJ
ejpam-4582	458	12	13	13	NUM
ejpam-4582	458	13	.	.	PUNCT
ejpam-4582	459	1	(	(	PUNCT
ejpam-4582	459	2	2	2	X
ejpam-4582	459	3	)	)	PUNCT
ejpam-4582	459	4	⇒	⇒	NOUN
ejpam-4582	459	5	(	(	PUNCT
ejpam-4582	459	6	3	3	NUM
ejpam-4582	459	7	):	):	PUNCT
ejpam-4582	459	8	let	let	VERB
ejpam-4582	459	9	f	f	PROPN
ejpam-4582	459	10	⊆	⊆	NUM
ejpam-4582	459	11	a(λ	a(λ	PROPN
ejpam-4582	459	12	,	,	PUNCT
ejpam-4582	459	13	s	s	NOUN
ejpam-4582	459	14	)	)	PUNCT
ejpam-4582	459	15	−	−	PROPN
ejpam-4582	459	16	a	a	PRON
ejpam-4582	459	17	and	and	CCONJ
ejpam-4582	459	18	f	f	AUX
ejpam-4582	459	19	be	be	AUX
ejpam-4582	459	20	(	(	PUNCT
ejpam-4582	459	21	λ	λ	X
ejpam-4582	459	22	,	,	PUNCT
ejpam-4582	459	23	s)-closed	s)-close	VERB
ejpam-4582	459	24	.	.	PUNCT
ejpam-4582	460	1	by	by	ADP
ejpam-4582	460	2	(	(	PUNCT
ejpam-4582	460	3	2	2	NUM
ejpam-4582	460	4	)	)	PUNCT
ejpam-4582	460	5	,	,	PUNCT
ejpam-4582	460	6	we	we	PRON
ejpam-4582	460	7	have	have	VERB
ejpam-4582	460	8	f	f	NOUN
ejpam-4582	460	9	=	=	PUNCT
ejpam-4582	460	10	∅	∅	NOUN
ejpam-4582	460	11	and	and	CCONJ
ejpam-4582	460	12	f	f	NOUN
ejpam-4582	460	13	⊆	⊆	NUM
ejpam-4582	460	14	[	[	X
ejpam-4582	460	15	a(λ	a(λ	ADJ
ejpam-4582	460	16	,	,	PUNCT
ejpam-4582	460	17	s	s	NOUN
ejpam-4582	460	18	)	)	PUNCT
ejpam-4582	460	19	−a](λ	−a](λ	PROPN
ejpam-4582	460	20	,	,	PUNCT
ejpam-4582	460	21	s	s	PART
ejpam-4582	460	22	)	)	PUNCT
ejpam-4582	460	23	.	.	PUNCT
ejpam-4582	461	1	it	it	PRON
ejpam-4582	461	2	follows	follow	VERB
ejpam-4582	461	3	from	from	ADP
ejpam-4582	461	4	theorem	theorem	ADJ
ejpam-4582	461	5	14	14	NUM
ejpam-4582	461	6	that	that	SCONJ
ejpam-4582	461	7	a(λ	a(λ	ADV
ejpam-4582	461	8	,	,	PUNCT
ejpam-4582	461	9	s	s	X
ejpam-4582	461	10	)	)	PUNCT
ejpam-4582	461	11	−a	−a	NOUN
ejpam-4582	461	12	is	be	AUX
ejpam-4582	461	13	g-(λ	g-(λ	PROPN
ejpam-4582	461	14	,	,	PUNCT
ejpam-4582	461	15	s)-open	s)-open	PUNCT
ejpam-4582	461	16	.	.	PUNCT
ejpam-4582	462	1	(	(	PUNCT
ejpam-4582	462	2	3	3	X
ejpam-4582	462	3	)	)	PUNCT
ejpam-4582	462	4	⇒	⇒	NOUN
ejpam-4582	462	5	(	(	PUNCT
ejpam-4582	462	6	1	1	NUM
ejpam-4582	462	7	):	):	PUNCT
ejpam-4582	462	8	let	let	VERB
ejpam-4582	462	9	a	a	DET
ejpam-4582	462	10	⊆	⊆	NUM
ejpam-4582	462	11	u	u	NOUN
ejpam-4582	462	12	and	and	CCONJ
ejpam-4582	462	13	u	u	PROPN
ejpam-4582	462	14	∈	∈	PROPN
ejpam-4582	462	15	(	(	PUNCT
ejpam-4582	462	16	λ	λ	NOUN
ejpam-4582	462	17	,	,	PUNCT
ejpam-4582	462	18	s)o(x	s)o(x	NOUN
ejpam-4582	462	19	)	)	PUNCT
ejpam-4582	462	20	.	.	PUNCT
ejpam-4582	463	1	thus	thus	ADV
ejpam-4582	463	2	,	,	PUNCT
ejpam-4582	463	3	a(λ	a(λ	ADV
ejpam-4582	463	4	,	,	PUNCT
ejpam-4582	463	5	s	s	NOUN
ejpam-4582	463	6	)	)	PUNCT
ejpam-4582	463	7	−	−	PROPN
ejpam-4582	463	8	u	u	NOUN
ejpam-4582	463	9	⊆	⊆	NUM
ejpam-4582	463	10	a(λ	a(λ	PROPN
ejpam-4582	463	11	,	,	PUNCT
ejpam-4582	463	12	s	s	NOUN
ejpam-4582	463	13	)	)	PUNCT
ejpam-4582	463	14	−	−	NOUN
ejpam-4582	463	15	a.	a.	NOUN
ejpam-4582	463	16	since	since	SCONJ
ejpam-4582	463	17	a(λ	a(λ	PROPN
ejpam-4582	463	18	,	,	PUNCT
ejpam-4582	463	19	s	s	PART
ejpam-4582	463	20	)	)	PUNCT
ejpam-4582	463	21	−a	−a	NOUN
ejpam-4582	463	22	is	be	AUX
ejpam-4582	463	23	g-(λ	g-(λ	PROPN
ejpam-4582	463	24	,	,	PUNCT
ejpam-4582	463	25	s)-open	s)-open	PUNCT
ejpam-4582	463	26	and	and	CCONJ
ejpam-4582	463	27	a(λ	a(λ	ADV
ejpam-4582	463	28	,	,	PUNCT
ejpam-4582	463	29	s	s	NOUN
ejpam-4582	463	30	)	)	PUNCT
ejpam-4582	464	1	−	−	PROPN
ejpam-4582	464	2	u	u	NOUN
ejpam-4582	464	3	is	be	AUX
ejpam-4582	464	4	(	(	PUNCT
ejpam-4582	464	5	λ	λ	X
ejpam-4582	464	6	,	,	PUNCT
ejpam-4582	464	7	s)-closed	s)-close	VERB
ejpam-4582	464	8	.	.	PUNCT
ejpam-4582	465	1	by	by	ADP
ejpam-4582	465	2	theorem	theorem	ADJ
ejpam-4582	465	3	14	14	NUM
ejpam-4582	465	4	,	,	PUNCT
ejpam-4582	465	5	a(λ	a(λ	ADV
ejpam-4582	465	6	,	,	PUNCT
ejpam-4582	465	7	s	s	NOUN
ejpam-4582	465	8	)	)	PUNCT
ejpam-4582	465	9	−	−	PROPN
ejpam-4582	465	10	u	u	NOUN
ejpam-4582	465	11	⊆	⊆	NUM
ejpam-4582	465	12	[	[	X
ejpam-4582	465	13	a(λ	a(λ	ADJ
ejpam-4582	465	14	,	,	PUNCT
ejpam-4582	465	15	s	s	NOUN
ejpam-4582	465	16	)	)	PUNCT
ejpam-4582	465	17	−a](λ	−a](λ	NUM
ejpam-4582	465	18	,	,	PUNCT
ejpam-4582	465	19	s	s	PART
ejpam-4582	465	20	)	)	PUNCT
ejpam-4582	465	21	=	=	PUNCT
ejpam-4582	465	22	∅.	∅.	ADP
ejpam-4582	465	23	thus	thus	ADV
ejpam-4582	465	24	,	,	PUNCT
ejpam-4582	465	25	a(λ	a(λ	ADV
ejpam-4582	465	26	,	,	PUNCT
ejpam-4582	465	27	s	s	PART
ejpam-4582	465	28	)	)	PUNCT
ejpam-4582	465	29	⊆	⊆	NUM
ejpam-4582	465	30	u	u	NOUN
ejpam-4582	465	31	and	and	CCONJ
ejpam-4582	465	32	hence	hence	ADV
ejpam-4582	465	33	a	a	PRON
ejpam-4582	465	34	is	be	AUX
ejpam-4582	465	35	g-(λ	g-(λ	PRON
ejpam-4582	465	36	,	,	PUNCT
ejpam-4582	465	37	s)-closed	s)-close	VERB
ejpam-4582	465	38	.	.	PUNCT
ejpam-4582	466	1	now	now	ADV
ejpam-4582	466	2	,	,	PUNCT
ejpam-4582	466	3	the	the	DET
ejpam-4582	466	4	proof	proof	NOUN
ejpam-4582	466	5	of	of	ADP
ejpam-4582	466	6	[	[	X
ejpam-4582	466	7	a(λ	a(λ	PROPN
ejpam-4582	466	8	,	,	PUNCT
ejpam-4582	466	9	s	s	NOUN
ejpam-4582	466	10	)	)	PUNCT
ejpam-4582	466	11	−	−	NOUN
ejpam-4582	466	12	a](λ	a](λ	NOUN
ejpam-4582	466	13	,	,	PUNCT
ejpam-4582	466	14	s	s	NOUN
ejpam-4582	466	15	)	)	PUNCT
ejpam-4582	466	16	=	=	NOUN
ejpam-4582	466	17	∅	∅	NOUN
ejpam-4582	466	18	is	be	AUX
ejpam-4582	466	19	given	give	VERB
ejpam-4582	466	20	as	as	SCONJ
ejpam-4582	466	21	follows	follow	VERB
ejpam-4582	466	22	.	.	PUNCT
ejpam-4582	467	1	suppose	suppose	VERB
ejpam-4582	467	2	that	that	SCONJ
ejpam-4582	467	3	[	[	X
ejpam-4582	467	4	a(λ	a(λ	ADV
ejpam-4582	467	5	,	,	PUNCT
ejpam-4582	467	6	s	s	NOUN
ejpam-4582	467	7	)	)	PUNCT
ejpam-4582	467	8	−	−	NOUN
ejpam-4582	467	9	a](λ	a](λ	NOUN
ejpam-4582	467	10	,	,	PUNCT
ejpam-4582	467	11	s	s	NOUN
ejpam-4582	467	12	)	)	PUNCT
ejpam-4582	467	13	̸=	̸=	PROPN
ejpam-4582	467	14	∅.	∅.	ADV
ejpam-4582	467	15	let	let	VERB
ejpam-4582	467	16	x	x	PUNCT
ejpam-4582	467	17	∈	∈	PROPN
ejpam-4582	467	18	[	[	X
ejpam-4582	467	19	a(λ	a(λ	ADJ
ejpam-4582	467	20	,	,	PUNCT
ejpam-4582	467	21	s	s	NOUN
ejpam-4582	467	22	)	)	PUNCT
ejpam-4582	467	23	−	−	NOUN
ejpam-4582	467	24	a](λ	a](λ	NOUN
ejpam-4582	467	25	,	,	PUNCT
ejpam-4582	467	26	s	s	NOUN
ejpam-4582	467	27	)	)	PUNCT
ejpam-4582	467	28	.	.	PUNCT
ejpam-4582	468	1	then	then	ADV
ejpam-4582	468	2	,	,	PUNCT
ejpam-4582	468	3	there	there	PRON
ejpam-4582	468	4	exists	exist	VERB
ejpam-4582	468	5	g	g	PROPN
ejpam-4582	468	6	∈	∈	PROPN
ejpam-4582	468	7	(	(	PUNCT
ejpam-4582	468	8	λ	λ	NOUN
ejpam-4582	468	9	,	,	PUNCT
ejpam-4582	468	10	s)o(x	s)o(x	ADJ
ejpam-4582	468	11	)	)	PUNCT
ejpam-4582	468	12	such	such	ADJ
ejpam-4582	468	13	that	that	SCONJ
ejpam-4582	468	14	x	x	SYM
ejpam-4582	468	15	∈	∈	NOUN
ejpam-4582	468	16	g	g	ADP
ejpam-4582	468	17	⊆	⊆	NUM
ejpam-4582	468	18	a(λ	a(λ	PROPN
ejpam-4582	468	19	,	,	PUNCT
ejpam-4582	468	20	s	s	NOUN
ejpam-4582	468	21	)	)	PUNCT
ejpam-4582	468	22	−	−	NOUN
ejpam-4582	468	23	a.	a.	NOUN
ejpam-4582	468	24	since	since	SCONJ
ejpam-4582	468	25	g	g	PROPN
ejpam-4582	468	26	⊆	⊆	NUM
ejpam-4582	468	27	x	x	SYM
ejpam-4582	468	28	−	−	NOUN
ejpam-4582	468	29	a	a	X
ejpam-4582	468	30	,	,	PUNCT
ejpam-4582	468	31	we	we	PRON
ejpam-4582	468	32	have	have	VERB
ejpam-4582	468	33	g	g	NOUN
ejpam-4582	468	34	∩	∩	NOUN
ejpam-4582	468	35	a	a	DET
ejpam-4582	468	36	=	=	NOUN
ejpam-4582	468	37	∅	∅	NOUN
ejpam-4582	468	38	and	and	CCONJ
ejpam-4582	468	39	by	by	ADP
ejpam-4582	468	40	lemma	lemma	PROPN
ejpam-4582	468	41	7	7	NUM
ejpam-4582	468	42	,	,	PUNCT
ejpam-4582	468	43	g	g	NOUN
ejpam-4582	468	44	∩	∩	NOUN
ejpam-4582	468	45	a(λ	a(λ	SYM
ejpam-4582	468	46	,	,	PUNCT
ejpam-4582	468	47	s	s	PART
ejpam-4582	468	48	)	)	PUNCT
ejpam-4582	468	49	=	=	PUNCT
ejpam-4582	468	50	∅.	∅.	ADP
ejpam-4582	468	51	this	this	PRON
ejpam-4582	468	52	implies	imply	VERB
ejpam-4582	468	53	that	that	SCONJ
ejpam-4582	468	54	g	g	PROPN
ejpam-4582	468	55	⊆	⊆	NUM
ejpam-4582	468	56	x	x	SYM
ejpam-4582	468	57	−	−	PROPN
ejpam-4582	468	58	a(λ	a(λ	ADV
ejpam-4582	468	59	,	,	PUNCT
ejpam-4582	468	60	s	s	NOUN
ejpam-4582	468	61	)	)	PUNCT
ejpam-4582	468	62	.	.	PUNCT
ejpam-4582	469	1	thus	thus	ADV
ejpam-4582	469	2	,	,	PUNCT
ejpam-4582	469	3	g	g	PROPN
ejpam-4582	469	4	⊆	⊆	NUM
ejpam-4582	469	5	[	[	X
ejpam-4582	469	6	x	x	X
ejpam-4582	469	7	−a(λ	−a(λ	NOUN
ejpam-4582	469	8	,	,	PUNCT
ejpam-4582	469	9	s	s	NOUN
ejpam-4582	469	10	)	)	PUNCT
ejpam-4582	469	11	]	]	PUNCT
ejpam-4582	470	1	∩a(λ	∩a(λ	NOUN
ejpam-4582	470	2	,	,	PUNCT
ejpam-4582	470	3	s	s	PART
ejpam-4582	470	4	)	)	PUNCT
ejpam-4582	470	5	=	=	PUNCT
ejpam-4582	470	6	∅.	∅.	ADP
ejpam-4582	470	7	this	this	PRON
ejpam-4582	470	8	is	be	AUX
ejpam-4582	470	9	a	a	DET
ejpam-4582	470	10	contradiction	contradiction	NOUN
ejpam-4582	470	11	.	.	PUNCT
ejpam-4582	471	1	theorem	theorem	VERB
ejpam-4582	471	2	16	16	NUM
ejpam-4582	471	3	.	.	PUNCT
ejpam-4582	472	1	a	a	DET
ejpam-4582	472	2	subset	subset	NOUN
ejpam-4582	472	3	a	a	PRON
ejpam-4582	472	4	of	of	ADP
ejpam-4582	472	5	a	a	DET
ejpam-4582	472	6	topological	topological	ADJ
ejpam-4582	472	7	space	space	NOUN
ejpam-4582	472	8	(	(	PUNCT
ejpam-4582	472	9	x	x	X
ejpam-4582	472	10	,	,	PUNCT
ejpam-4582	472	11	τ	τ	X
ejpam-4582	472	12	)	)	PUNCT
ejpam-4582	472	13	is	be	AUX
ejpam-4582	472	14	g-(λ	g-(λ	PROPN
ejpam-4582	472	15	,	,	PUNCT
ejpam-4582	472	16	s)-closed	s)-close	VERB
ejpam-4582	472	17	if	if	SCONJ
ejpam-4582	472	18	and	and	CCONJ
ejpam-4582	472	19	only	only	ADV
ejpam-4582	472	20	if	if	SCONJ
ejpam-4582	472	21	f	f	PROPN
ejpam-4582	472	22	∩a(λ	∩a(λ	PROPN
ejpam-4582	472	23	,	,	PUNCT
ejpam-4582	472	24	s	s	PART
ejpam-4582	472	25	)	)	PUNCT
ejpam-4582	472	26	=	=	PUNCT
ejpam-4582	472	27	∅	∅	NOUN
ejpam-4582	472	28	whenever	whenever	SCONJ
ejpam-4582	472	29	a	a	DET
ejpam-4582	472	30	∩	∩	ADJ
ejpam-4582	472	31	f	f	NOUN
ejpam-4582	472	32	=	=	NOUN
ejpam-4582	472	33	∅	∅	NOUN
ejpam-4582	472	34	and	and	CCONJ
ejpam-4582	472	35	f	f	PROPN
ejpam-4582	472	36	is	be	AUX
ejpam-4582	472	37	(	(	PUNCT
ejpam-4582	472	38	λ	λ	X
ejpam-4582	472	39	,	,	PUNCT
ejpam-4582	472	40	s)-closed	s)-close	VERB
ejpam-4582	472	41	.	.	PUNCT
ejpam-4582	473	1	proof	proof	NOUN
ejpam-4582	473	2	.	.	PUNCT
ejpam-4582	474	1	let	let	VERB
ejpam-4582	474	2	f	f	PRON
ejpam-4582	474	3	be	be	AUX
ejpam-4582	474	4	a	a	DET
ejpam-4582	474	5	(	(	PUNCT
ejpam-4582	474	6	λ	λ	X
ejpam-4582	474	7	,	,	PUNCT
ejpam-4582	474	8	s)-closed	s)-close	VERB
ejpam-4582	474	9	set	set	VERB
ejpam-4582	474	10	such	such	ADJ
ejpam-4582	474	11	that	that	SCONJ
ejpam-4582	474	12	a	a	DET
ejpam-4582	474	13	∩	∩	ADJ
ejpam-4582	474	14	f	f	X
ejpam-4582	474	15	=	=	PUNCT
ejpam-4582	474	16	∅.	∅.	NOUN
ejpam-4582	474	17	then	then	ADV
ejpam-4582	474	18	,	,	PUNCT
ejpam-4582	474	19	we	we	PRON
ejpam-4582	474	20	have	have	VERB
ejpam-4582	474	21	a	a	DET
ejpam-4582	474	22	⊆	⊆	NUM
ejpam-4582	474	23	x	x	SYM
ejpam-4582	474	24	−	−	PROPN
ejpam-4582	474	25	f	f	PROPN
ejpam-4582	474	26	.	.	PUNCT
ejpam-4582	475	1	since	since	SCONJ
ejpam-4582	475	2	a	a	PRON
ejpam-4582	475	3	is	be	AUX
ejpam-4582	475	4	g-(λ	g-(λ	PRON
ejpam-4582	475	5	,	,	PUNCT
ejpam-4582	475	6	s)-closed	s)-close	VERB
ejpam-4582	475	7	and	and	CCONJ
ejpam-4582	475	8	x	x	SYM
ejpam-4582	475	9	−f	−f	NOUN
ejpam-4582	475	10	is	be	AUX
ejpam-4582	475	11	(	(	PUNCT
ejpam-4582	475	12	λ	λ	INTJ
ejpam-4582	475	13	,	,	PUNCT
ejpam-4582	475	14	s)-open	s)-open	ADJ
ejpam-4582	475	15	,	,	PUNCT
ejpam-4582	475	16	a(λ	a(λ	ADV
ejpam-4582	475	17	,	,	PUNCT
ejpam-4582	475	18	s	s	NOUN
ejpam-4582	475	19	)	)	PUNCT
ejpam-4582	475	20	⊆	⊆	NUM
ejpam-4582	475	21	x	x	SYM
ejpam-4582	475	22	−f	−f	NOUN
ejpam-4582	475	23	.	.	PUNCT
ejpam-4582	476	1	thus	thus	ADV
ejpam-4582	476	2	,	,	PUNCT
ejpam-4582	476	3	f	f	PROPN
ejpam-4582	476	4	∩a(λ	∩a(λ	PROPN
ejpam-4582	476	5	,	,	PUNCT
ejpam-4582	476	6	s	s	PART
ejpam-4582	476	7	)	)	PUNCT
ejpam-4582	476	8	=	=	PUNCT
ejpam-4582	476	9	∅.	∅.	VERB
ejpam-4582	476	10	conversely	conversely	ADV
ejpam-4582	476	11	,	,	PUNCT
ejpam-4582	476	12	let	let	VERB
ejpam-4582	476	13	a	a	DET
ejpam-4582	476	14	⊆	⊆	NUM
ejpam-4582	476	15	u	u	NOUN
ejpam-4582	476	16	and	and	CCONJ
ejpam-4582	476	17	u	u	PROPN
ejpam-4582	476	18	∈	∈	PROPN
ejpam-4582	476	19	(	(	PUNCT
ejpam-4582	476	20	λ	λ	NOUN
ejpam-4582	476	21	,	,	PUNCT
ejpam-4582	476	22	s)o(x	s)o(x	NOUN
ejpam-4582	476	23	)	)	PUNCT
ejpam-4582	476	24	.	.	PUNCT
ejpam-4582	477	1	then	then	ADV
ejpam-4582	477	2	,	,	PUNCT
ejpam-4582	477	3	we	we	PRON
ejpam-4582	477	4	have	have	VERB
ejpam-4582	477	5	a	a	DET
ejpam-4582	477	6	∩	∩	NOUN
ejpam-4582	477	7	(	(	PUNCT
ejpam-4582	477	8	x	x	SYM
ejpam-4582	477	9	−	−	PROPN
ejpam-4582	477	10	u	u	NOUN
ejpam-4582	477	11	)	)	PUNCT
ejpam-4582	477	12	=	=	SYM
ejpam-4582	477	13	∅	∅	NOUN
ejpam-4582	477	14	and	and	CCONJ
ejpam-4582	477	15	x	x	X
ejpam-4582	477	16	−	−	NOUN
ejpam-4582	477	17	u	u	NOUN
ejpam-4582	477	18	is	be	AUX
ejpam-4582	477	19	(	(	PUNCT
ejpam-4582	477	20	λ	λ	X
ejpam-4582	477	21	,	,	PUNCT
ejpam-4582	477	22	s)-closed	s)-close	VERB
ejpam-4582	477	23	.	.	PUNCT
ejpam-4582	478	1	by	by	ADP
ejpam-4582	478	2	the	the	DET
ejpam-4582	478	3	hypothesis	hypothesis	NOUN
ejpam-4582	478	4	,	,	PUNCT
ejpam-4582	478	5	(	(	PUNCT
ejpam-4582	478	6	x	x	X
ejpam-4582	478	7	−	−	PROPN
ejpam-4582	478	8	u	u	NOUN
ejpam-4582	478	9	)	)	PUNCT
ejpam-4582	478	10	∩	∩	NOUN
ejpam-4582	478	11	a(λ	a(λ	PROPN
ejpam-4582	478	12	,	,	PUNCT
ejpam-4582	478	13	s	s	PART
ejpam-4582	478	14	)	)	PUNCT
ejpam-4582	478	15	=	=	NOUN
ejpam-4582	478	16	∅	∅	NOUN
ejpam-4582	478	17	and	and	CCONJ
ejpam-4582	478	18	hence	hence	ADV
ejpam-4582	478	19	a(λ	a(λ	ADV
ejpam-4582	478	20	,	,	PUNCT
ejpam-4582	478	21	s	s	PART
ejpam-4582	478	22	)	)	PUNCT
ejpam-4582	478	23	⊆	⊆	NUM
ejpam-4582	478	24	u	u	NOUN
ejpam-4582	478	25	.	.	PUNCT
ejpam-4582	479	1	this	this	PRON
ejpam-4582	479	2	shows	show	VERB
ejpam-4582	479	3	that	that	SCONJ
ejpam-4582	479	4	a	a	PRON
ejpam-4582	479	5	is	be	AUX
ejpam-4582	479	6	(	(	PUNCT
ejpam-4582	479	7	λ	λ	X
ejpam-4582	479	8	,	,	PUNCT
ejpam-4582	479	9	s)-closed	s)-close	VERB
ejpam-4582	479	10	.	.	PUNCT
ejpam-4582	480	1	theorem	theorem	VERB
ejpam-4582	480	2	17	17	NUM
ejpam-4582	480	3	.	.	PUNCT
ejpam-4582	481	1	a	a	DET
ejpam-4582	481	2	subset	subset	NOUN
ejpam-4582	481	3	a	a	PRON
ejpam-4582	481	4	of	of	ADP
ejpam-4582	481	5	a	a	DET
ejpam-4582	481	6	topological	topological	ADJ
ejpam-4582	481	7	space	space	NOUN
ejpam-4582	481	8	(	(	PUNCT
ejpam-4582	481	9	x	x	X
ejpam-4582	481	10	,	,	PUNCT
ejpam-4582	481	11	τ	τ	X
ejpam-4582	481	12	)	)	PUNCT
ejpam-4582	481	13	is	be	AUX
ejpam-4582	481	14	g-(λ	g-(λ	PROPN
ejpam-4582	481	15	,	,	PUNCT
ejpam-4582	481	16	s)-closed	s)-close	VERB
ejpam-4582	481	17	if	if	SCONJ
ejpam-4582	481	18	and	and	CCONJ
ejpam-4582	481	19	only	only	ADV
ejpam-4582	481	20	if	if	SCONJ
ejpam-4582	481	21	a	a	DET
ejpam-4582	481	22	∩	∩	NOUN
ejpam-4582	481	23	{	{	PUNCT
ejpam-4582	481	24	x}(λ	x}(λ	PROPN
ejpam-4582	481	25	,	,	PUNCT
ejpam-4582	481	26	s	s	PART
ejpam-4582	481	27	)	)	PUNCT
ejpam-4582	481	28	̸=	̸=	NOUN
ejpam-4582	481	29	∅	∅	NOUN
ejpam-4582	481	30	for	for	ADP
ejpam-4582	481	31	every	every	DET
ejpam-4582	481	32	x	x	PROPN
ejpam-4582	481	33	∈	∈	PROPN
ejpam-4582	481	34	a(λ	a(λ	PROPN
ejpam-4582	481	35	,	,	PUNCT
ejpam-4582	481	36	s	s	PART
ejpam-4582	481	37	)	)	PUNCT
ejpam-4582	481	38	.	.	PUNCT
ejpam-4582	482	1	proof	proof	NOUN
ejpam-4582	482	2	.	.	PUNCT
ejpam-4582	483	1	suppose	suppose	VERB
ejpam-4582	483	2	that	that	SCONJ
ejpam-4582	483	3	a	a	DET
ejpam-4582	483	4	∩	∩	NOUN
ejpam-4582	483	5	{	{	PUNCT
ejpam-4582	483	6	x}(λ	x}(λ	PROPN
ejpam-4582	483	7	,	,	PUNCT
ejpam-4582	483	8	s	s	PART
ejpam-4582	483	9	)	)	PUNCT
ejpam-4582	483	10	=	=	NOUN
ejpam-4582	483	11	∅	∅	NOUN
ejpam-4582	483	12	for	for	ADP
ejpam-4582	483	13	some	some	DET
ejpam-4582	483	14	x	x	SYM
ejpam-4582	483	15	∈	∈	PROPN
ejpam-4582	483	16	a(λ	a(λ	PROPN
ejpam-4582	483	17	,	,	PUNCT
ejpam-4582	483	18	s	s	PART
ejpam-4582	483	19	)	)	PUNCT
ejpam-4582	483	20	.	.	PUNCT
ejpam-4582	484	1	then	then	ADV
ejpam-4582	484	2	,	,	PUNCT
ejpam-4582	484	3	a	a	DET
ejpam-4582	484	4	⊆	⊆	NUM
ejpam-4582	484	5	x	x	SYM
ejpam-4582	484	6	−	−	PROPN
ejpam-4582	484	7	{	{	PUNCT
ejpam-4582	484	8	x}(λ	x}(λ	PROPN
ejpam-4582	484	9	,	,	PUNCT
ejpam-4582	484	10	s	s	PART
ejpam-4582	484	11	)	)	PUNCT
ejpam-4582	484	12	.	.	PUNCT
ejpam-4582	485	1	since	since	SCONJ
ejpam-4582	485	2	a	a	PRON
ejpam-4582	485	3	is	be	AUX
ejpam-4582	485	4	g-(λ	g-(λ	PRON
ejpam-4582	485	5	,	,	PUNCT
ejpam-4582	485	6	s)-closed	s)-close	VERB
ejpam-4582	485	7	and	and	CCONJ
ejpam-4582	485	8	x−{x}(λ	x−{x}(λ	PROPN
ejpam-4582	485	9	,	,	PUNCT
ejpam-4582	485	10	s	s	PART
ejpam-4582	485	11	)	)	PUNCT
ejpam-4582	485	12	is	be	AUX
ejpam-4582	485	13	(	(	PUNCT
ejpam-4582	485	14	λ	λ	INTJ
ejpam-4582	485	15	,	,	PUNCT
ejpam-4582	485	16	s)-open	s)-open	ADJ
ejpam-4582	485	17	,	,	PUNCT
ejpam-4582	485	18	a(λ	a(λ	ADV
ejpam-4582	485	19	,	,	PUNCT
ejpam-4582	485	20	s	s	PART
ejpam-4582	485	21	)	)	PUNCT
ejpam-4582	485	22	⊆	⊆	NUM
ejpam-4582	485	23	x−{x}(λ	x−{x}(λ	NOUN
ejpam-4582	485	24	,	,	PUNCT
ejpam-4582	485	25	s	s	PART
ejpam-4582	485	26	)	)	PUNCT
ejpam-4582	485	27	⊆	⊆	NUM
ejpam-4582	485	28	x−{x	x−{x	PROPN
ejpam-4582	485	29	}	}	PUNCT
ejpam-4582	485	30	.	.	PUNCT
ejpam-4582	486	1	this	this	PRON
ejpam-4582	486	2	contradicts	contradict	VERB
ejpam-4582	486	3	that	that	SCONJ
ejpam-4582	486	4	x	x	SYM
ejpam-4582	486	5	∈	∈	PROPN
ejpam-4582	486	6	a(λ	a(λ	PROPN
ejpam-4582	486	7	,	,	PUNCT
ejpam-4582	486	8	s	s	NOUN
ejpam-4582	486	9	)	)	PUNCT
ejpam-4582	486	10	.	.	PUNCT
ejpam-4582	487	1	c.	c.	PROPN
ejpam-4582	487	2	boonpok	boonpok	PROPN
ejpam-4582	487	3	,	,	PUNCT
ejpam-4582	487	4	c.	c.	PROPN
ejpam-4582	487	5	viriyapong	viriyapong	PROPN
ejpam-4582	487	6	/	/	SYM
ejpam-4582	487	7	eur	eur	PROPN
ejpam-4582	487	8	.	.	PUNCT
ejpam-4582	488	1	j.	j.	PROPN
ejpam-4582	488	2	pure	pure	PROPN
ejpam-4582	488	3	appl	appl	PROPN
ejpam-4582	488	4	.	.	PROPN
ejpam-4582	488	5	math	math	PROPN
ejpam-4582	488	6	,	,	PUNCT
ejpam-4582	488	7	16	16	NUM
ejpam-4582	488	8	(	(	PUNCT
ejpam-4582	488	9	1	1	NUM
ejpam-4582	488	10	)	)	PUNCT
ejpam-4582	488	11	(	(	PUNCT
ejpam-4582	488	12	2023	2023	NUM
ejpam-4582	488	13	)	)	PUNCT
ejpam-4582	488	14	,	,	PUNCT
ejpam-4582	488	15	336	336	NUM
ejpam-4582	488	16	-	-	SYM
ejpam-4582	488	17	362	362	NUM
ejpam-4582	488	18	349	349	NUM
ejpam-4582	488	19	conversely	conversely	ADV
ejpam-4582	488	20	,	,	PUNCT
ejpam-4582	488	21	suppose	suppose	VERB
ejpam-4582	488	22	that	that	SCONJ
ejpam-4582	488	23	a	a	PRON
ejpam-4582	488	24	is	be	AUX
ejpam-4582	488	25	not	not	PART
ejpam-4582	488	26	g-(λ	g-(λ	PRON
ejpam-4582	488	27	,	,	PUNCT
ejpam-4582	488	28	s)-closed	s)-close	VERB
ejpam-4582	488	29	.	.	PUNCT
ejpam-4582	489	1	thus	thus	ADV
ejpam-4582	489	2	,	,	PUNCT
ejpam-4582	489	3	∅	∅	NOUN
ejpam-4582	489	4	̸=	̸=	PROPN
ejpam-4582	489	5	a(λ	a(λ	ADV
ejpam-4582	489	6	,	,	PUNCT
ejpam-4582	489	7	s	s	NOUN
ejpam-4582	489	8	)	)	PUNCT
ejpam-4582	489	9	−	−	PROPN
ejpam-4582	489	10	u	u	NOUN
ejpam-4582	489	11	for	for	ADP
ejpam-4582	489	12	some	some	DET
ejpam-4582	489	13	u	u	NOUN
ejpam-4582	489	14	∈	∈	PROPN
ejpam-4582	489	15	(	(	PUNCT
ejpam-4582	489	16	λ	λ	NOUN
ejpam-4582	489	17	,	,	PUNCT
ejpam-4582	489	18	s)o(x	s)o(x	NOUN
ejpam-4582	489	19	)	)	PUNCT
ejpam-4582	489	20	containing	contain	VERB
ejpam-4582	489	21	a.	a.	NOUN
ejpam-4582	489	22	there	there	PRON
ejpam-4582	489	23	exists	exist	VERB
ejpam-4582	489	24	x	x	X
ejpam-4582	489	25	∈	∈	PROPN
ejpam-4582	489	26	a(λ	a(λ	PROPN
ejpam-4582	489	27	,	,	PUNCT
ejpam-4582	489	28	s	s	NOUN
ejpam-4582	489	29	)	)	PUNCT
ejpam-4582	489	30	−	−	PROPN
ejpam-4582	489	31	u	u	NOUN
ejpam-4582	489	32	.	.	PUNCT
ejpam-4582	490	1	since	since	SCONJ
ejpam-4582	490	2	x	x	PROPN
ejpam-4582	490	3	̸∈	̸∈	PROPN
ejpam-4582	490	4	u	u	PROPN
ejpam-4582	490	5	,	,	PUNCT
ejpam-4582	490	6	by	by	ADP
ejpam-4582	490	7	lemma	lemma	PROPN
ejpam-4582	490	8	7	7	NUM
ejpam-4582	490	9	,	,	PUNCT
ejpam-4582	490	10	u∩{x}(λ	u∩{x}(λ	ADJ
ejpam-4582	490	11	,	,	PUNCT
ejpam-4582	490	12	s	s	PART
ejpam-4582	490	13	)	)	PUNCT
ejpam-4582	490	14	=	=	NOUN
ejpam-4582	490	15	∅	∅	NOUN
ejpam-4582	490	16	and	and	CCONJ
ejpam-4582	490	17	hence	hence	ADV
ejpam-4582	490	18	a∩{x}(λ	a∩{x}(λ	NUM
ejpam-4582	490	19	,	,	PUNCT
ejpam-4582	490	20	s	s	PART
ejpam-4582	490	21	)	)	PUNCT
ejpam-4582	490	22	⊆	⊆	NUM
ejpam-4582	490	23	u∩{x}(λ	u∩{x}(λ	NUM
ejpam-4582	490	24	,	,	PUNCT
ejpam-4582	490	25	s	s	PART
ejpam-4582	490	26	)	)	PUNCT
ejpam-4582	490	27	=	=	PUNCT
ejpam-4582	490	28	∅.	∅.	ADP
ejpam-4582	490	29	this	this	PRON
ejpam-4582	490	30	shows	show	VERB
ejpam-4582	490	31	that	that	SCONJ
ejpam-4582	490	32	a∩{x}(λ	a∩{x}(λ	NOUN
ejpam-4582	490	33	,	,	PUNCT
ejpam-4582	490	34	s	s	PART
ejpam-4582	490	35	)	)	PUNCT
ejpam-4582	490	36	=	=	NOUN
ejpam-4582	490	37	∅	∅	NOUN
ejpam-4582	490	38	for	for	ADP
ejpam-4582	490	39	some	some	DET
ejpam-4582	490	40	x	x	SYM
ejpam-4582	490	41	∈	∈	PROPN
ejpam-4582	490	42	a(λ	a(λ	PROPN
ejpam-4582	490	43	,	,	PUNCT
ejpam-4582	490	44	s	s	PART
ejpam-4582	490	45	)	)	PUNCT
ejpam-4582	490	46	.	.	PUNCT
ejpam-4582	491	1	corollary	corollary	ADJ
ejpam-4582	491	2	4	4	NUM
ejpam-4582	491	3	.	.	PUNCT
ejpam-4582	492	1	for	for	ADP
ejpam-4582	492	2	a	a	DET
ejpam-4582	492	3	subset	subset	NOUN
ejpam-4582	492	4	a	a	PRON
ejpam-4582	492	5	of	of	ADP
ejpam-4582	492	6	a	a	DET
ejpam-4582	492	7	topological	topological	ADJ
ejpam-4582	492	8	space	space	NOUN
ejpam-4582	492	9	(	(	PUNCT
ejpam-4582	492	10	x	x	X
ejpam-4582	492	11	,	,	PUNCT
ejpam-4582	492	12	τ	τ	PROPN
ejpam-4582	492	13	)	)	PUNCT
ejpam-4582	492	14	,	,	PUNCT
ejpam-4582	492	15	the	the	DET
ejpam-4582	492	16	following	follow	VERB
ejpam-4582	492	17	properties	property	NOUN
ejpam-4582	492	18	are	be	AUX
ejpam-4582	492	19	equivalent	equivalent	ADJ
ejpam-4582	492	20	:	:	PUNCT
ejpam-4582	492	21	(	(	PUNCT
ejpam-4582	492	22	1	1	X
ejpam-4582	492	23	)	)	PUNCT
ejpam-4582	492	24	a	a	PRON
ejpam-4582	492	25	is	be	AUX
ejpam-4582	492	26	g-(λ	g-(λ	PROPN
ejpam-4582	492	27	,	,	PUNCT
ejpam-4582	492	28	s)-open	s)-open	PUNCT
ejpam-4582	492	29	;	;	PUNCT
ejpam-4582	492	30	(	(	PUNCT
ejpam-4582	492	31	2	2	X
ejpam-4582	492	32	)	)	PUNCT
ejpam-4582	492	33	a−a(λ	a−a(λ	ADJ
ejpam-4582	492	34	,	,	PUNCT
ejpam-4582	492	35	s	s	PART
ejpam-4582	492	36	)	)	PUNCT
ejpam-4582	492	37	contains	contain	VERB
ejpam-4582	492	38	no	no	DET
ejpam-4582	492	39	nonempty	nonempty	ADJ
ejpam-4582	492	40	(	(	PUNCT
ejpam-4582	492	41	λ	λ	X
ejpam-4582	492	42	,	,	PUNCT
ejpam-4582	492	43	s)-closed	s)-close	VERB
ejpam-4582	492	44	set	set	VERB
ejpam-4582	492	45	;	;	PUNCT
ejpam-4582	492	46	(	(	PUNCT
ejpam-4582	492	47	3	3	X
ejpam-4582	492	48	)	)	PUNCT
ejpam-4582	492	49	a−a(λ	a−a(λ	ADJ
ejpam-4582	492	50	,	,	PUNCT
ejpam-4582	492	51	s	s	PART
ejpam-4582	492	52	)	)	PUNCT
ejpam-4582	492	53	is	be	AUX
ejpam-4582	492	54	g-(λ	g-(λ	PROPN
ejpam-4582	492	55	,	,	PUNCT
ejpam-4582	492	56	s)-open	s)-open	PUNCT
ejpam-4582	492	57	;	;	PUNCT
ejpam-4582	492	58	(	(	PUNCT
ejpam-4582	492	59	4	4	X
ejpam-4582	492	60	)	)	PUNCT
ejpam-4582	492	61	(	(	PUNCT
ejpam-4582	492	62	x	x	NOUN
ejpam-4582	492	63	−a	−a	ADJ
ejpam-4582	492	64	)	)	PUNCT
ejpam-4582	492	65	∩	∩	NOUN
ejpam-4582	492	66	{	{	PUNCT
ejpam-4582	492	67	x}(λ	x}(λ	PROPN
ejpam-4582	492	68	,	,	PUNCT
ejpam-4582	492	69	s	s	PART
ejpam-4582	492	70	)	)	PUNCT
ejpam-4582	492	71	̸=	̸=	NOUN
ejpam-4582	492	72	∅	∅	NOUN
ejpam-4582	492	73	for	for	ADP
ejpam-4582	492	74	every	every	DET
ejpam-4582	492	75	x	x	SYM
ejpam-4582	492	76	∈	∈	PROPN
ejpam-4582	492	77	x	x	SYM
ejpam-4582	492	78	−a(λ	−a(λ	NOUN
ejpam-4582	492	79	,	,	PUNCT
ejpam-4582	492	80	s	s	NOUN
ejpam-4582	492	81	)	)	PUNCT
ejpam-4582	492	82	.	.	PUNCT
ejpam-4582	493	1	proof	proof	NOUN
ejpam-4582	493	2	.	.	PUNCT
ejpam-4582	494	1	this	this	PRON
ejpam-4582	494	2	follows	follow	VERB
ejpam-4582	494	3	from	from	ADP
ejpam-4582	494	4	theorem	theorem	ADJ
ejpam-4582	494	5	15	15	NUM
ejpam-4582	494	6	and	and	CCONJ
ejpam-4582	494	7	theorem	theorem	VERB
ejpam-4582	494	8	16	16	NUM
ejpam-4582	494	9	.	.	PUNCT
ejpam-4582	495	1	theorem	theorem	VERB
ejpam-4582	495	2	18	18	NUM
ejpam-4582	495	3	.	.	PUNCT
ejpam-4582	496	1	a	a	DET
ejpam-4582	496	2	subset	subset	NOUN
ejpam-4582	496	3	a	a	PRON
ejpam-4582	496	4	of	of	ADP
ejpam-4582	496	5	a	a	DET
ejpam-4582	496	6	topological	topological	ADJ
ejpam-4582	496	7	space	space	NOUN
ejpam-4582	496	8	(	(	PUNCT
ejpam-4582	496	9	x	x	X
ejpam-4582	496	10	,	,	PUNCT
ejpam-4582	496	11	τ	τ	X
ejpam-4582	496	12	)	)	PUNCT
ejpam-4582	496	13	is	be	AUX
ejpam-4582	496	14	g-(λ	g-(λ	PROPN
ejpam-4582	496	15	,	,	PUNCT
ejpam-4582	496	16	s)-open	s)-open	VERB
ejpam-4582	496	17	if	if	SCONJ
ejpam-4582	496	18	and	and	CCONJ
ejpam-4582	496	19	only	only	ADV
ejpam-4582	496	20	if	if	SCONJ
ejpam-4582	496	21	u	u	NOUN
ejpam-4582	496	22	=	=	NOUN
ejpam-4582	496	23	x	x	INTJ
ejpam-4582	496	24	whenever	whenever	SCONJ
ejpam-4582	496	25	u	u	NOUN
ejpam-4582	496	26	is	be	AUX
ejpam-4582	496	27	(	(	PUNCT
ejpam-4582	496	28	λ	λ	X
ejpam-4582	496	29	,	,	PUNCT
ejpam-4582	496	30	s)-open	s)-open	PUNCT
ejpam-4582	496	31	and	and	CCONJ
ejpam-4582	496	32	(	(	PUNCT
ejpam-4582	496	33	x	x	NOUN
ejpam-4582	496	34	−a	−a	NOUN
ejpam-4582	496	35	)	)	PUNCT
ejpam-4582	496	36	∪a(λ	∪a(λ	PROPN
ejpam-4582	496	37	,	,	PUNCT
ejpam-4582	496	38	s	s	PART
ejpam-4582	496	39	)	)	PUNCT
ejpam-4582	496	40	⊆	⊆	NUM
ejpam-4582	496	41	u	u	NOUN
ejpam-4582	496	42	.	.	PUNCT
ejpam-4582	497	1	proof	proof	NOUN
ejpam-4582	497	2	.	.	PUNCT
ejpam-4582	498	1	let	let	VERB
ejpam-4582	498	2	u	u	PRON
ejpam-4582	498	3	be	be	AUX
ejpam-4582	498	4	a	a	DET
ejpam-4582	498	5	(	(	PUNCT
ejpam-4582	498	6	λ	λ	X
ejpam-4582	498	7	,	,	PUNCT
ejpam-4582	498	8	s)-open	s)-open	VERB
ejpam-4582	498	9	set	set	VERB
ejpam-4582	498	10	and	and	CCONJ
ejpam-4582	498	11	(	(	PUNCT
ejpam-4582	498	12	x	x	NOUN
ejpam-4582	498	13	−a	−a	NOUN
ejpam-4582	498	14	)	)	PUNCT
ejpam-4582	498	15	∪a(λ	∪a(λ	PROPN
ejpam-4582	498	16	,	,	PUNCT
ejpam-4582	498	17	s	s	PART
ejpam-4582	498	18	)	)	PUNCT
ejpam-4582	498	19	⊆	⊆	NUM
ejpam-4582	498	20	u	u	NOUN
ejpam-4582	498	21	.	.	PUNCT
ejpam-4582	499	1	then	then	ADV
ejpam-4582	499	2	,	,	PUNCT
ejpam-4582	499	3	we	we	PRON
ejpam-4582	499	4	have	have	VERB
ejpam-4582	499	5	x	x	ADJ
ejpam-4582	499	6	−	−	VERB
ejpam-4582	499	7	u	u	NOUN
ejpam-4582	499	8	⊆	⊆	NUM
ejpam-4582	499	9	(	(	PUNCT
ejpam-4582	499	10	x	x	SYM
ejpam-4582	499	11	−a)(λ	−a)(λ	PROPN
ejpam-4582	499	12	,	,	PUNCT
ejpam-4582	499	13	s	s	NOUN
ejpam-4582	499	14	)	)	PUNCT
ejpam-4582	499	15	−	−	PROPN
ejpam-4582	500	1	(	(	PUNCT
ejpam-4582	500	2	x	x	NOUN
ejpam-4582	500	3	−a	−a	NOUN
ejpam-4582	500	4	)	)	PUNCT
ejpam-4582	500	5	.	.	PUNCT
ejpam-4582	501	1	since	since	SCONJ
ejpam-4582	501	2	x	x	PRON
ejpam-4582	501	3	−a	−a	NOUN
ejpam-4582	501	4	is	be	AUX
ejpam-4582	501	5	g-(λ	g-(λ	PROPN
ejpam-4582	501	6	,	,	PUNCT
ejpam-4582	501	7	s)-closed	s)-close	VERB
ejpam-4582	501	8	and	and	CCONJ
ejpam-4582	501	9	x	x	X
ejpam-4582	501	10	−u	−u	NOUN
ejpam-4582	501	11	is	be	AUX
ejpam-4582	501	12	(	(	PUNCT
ejpam-4582	501	13	λ	λ	X
ejpam-4582	501	14	,	,	PUNCT
ejpam-4582	501	15	s)-closed	s)-close	VERB
ejpam-4582	501	16	.	.	PUNCT
ejpam-4582	502	1	by	by	ADP
ejpam-4582	502	2	theorem	theorem	NOUN
ejpam-4582	502	3	13	13	NUM
ejpam-4582	502	4	,	,	PUNCT
ejpam-4582	502	5	x	x	X
ejpam-4582	502	6	−u	−u	NOUN
ejpam-4582	502	7	=	=	PUNCT
ejpam-4582	502	8	∅	∅	NOUN
ejpam-4582	502	9	and	and	CCONJ
ejpam-4582	502	10	hence	hence	ADV
ejpam-4582	502	11	x	x	X
ejpam-4582	503	1	=	=	PUNCT
ejpam-4582	503	2	u	u	NOUN
ejpam-4582	503	3	.	.	PUNCT
ejpam-4582	504	1	conversely	conversely	ADV
ejpam-4582	504	2	,	,	PUNCT
ejpam-4582	504	3	suppose	suppose	VERB
ejpam-4582	504	4	that	that	SCONJ
ejpam-4582	504	5	f	f	PROPN
ejpam-4582	504	6	⊆	⊆	NUM
ejpam-4582	504	7	a	a	PRON
ejpam-4582	504	8	and	and	CCONJ
ejpam-4582	504	9	f	f	NOUN
ejpam-4582	504	10	is	be	AUX
ejpam-4582	504	11	(	(	PUNCT
ejpam-4582	504	12	λ	λ	X
ejpam-4582	504	13	,	,	PUNCT
ejpam-4582	504	14	s)-closed	s)-close	VERB
ejpam-4582	504	15	.	.	PUNCT
ejpam-4582	505	1	then	then	ADV
ejpam-4582	505	2	,	,	PUNCT
ejpam-4582	505	3	(	(	PUNCT
ejpam-4582	505	4	x	x	NOUN
ejpam-4582	505	5	−a	−a	NOUN
ejpam-4582	505	6	)	)	PUNCT
ejpam-4582	505	7	∪a(λ	∪a(λ	PROPN
ejpam-4582	505	8	,	,	PUNCT
ejpam-4582	505	9	s	s	PART
ejpam-4582	505	10	)	)	PUNCT
ejpam-4582	505	11	⊆	⊆	NUM
ejpam-4582	505	12	(	(	PUNCT
ejpam-4582	505	13	x	x	SYM
ejpam-4582	505	14	−	−	PROPN
ejpam-4582	505	15	f	f	PROPN
ejpam-4582	505	16	)	)	PUNCT
ejpam-4582	505	17	∪a(λ	∪a(λ	PROPN
ejpam-4582	505	18	,	,	PUNCT
ejpam-4582	505	19	s	s	PART
ejpam-4582	505	20	)	)	PUNCT
ejpam-4582	505	21	∈	∈	PROPN
ejpam-4582	505	22	(	(	PUNCT
ejpam-4582	505	23	λ	λ	NOUN
ejpam-4582	505	24	,	,	PUNCT
ejpam-4582	505	25	s)o(x	s)o(x	NOUN
ejpam-4582	505	26	)	)	PUNCT
ejpam-4582	505	27	.	.	PUNCT
ejpam-4582	506	1	by	by	ADP
ejpam-4582	506	2	the	the	DET
ejpam-4582	506	3	hypothesis	hypothesis	NOUN
ejpam-4582	506	4	,	,	PUNCT
ejpam-4582	506	5	we	we	PRON
ejpam-4582	506	6	have	have	VERB
ejpam-4582	506	7	x	x	X
ejpam-4582	506	8	=	=	PRON
ejpam-4582	506	9	(	(	PUNCT
ejpam-4582	506	10	x	x	SYM
ejpam-4582	506	11	−	−	PROPN
ejpam-4582	506	12	f	f	PROPN
ejpam-4582	506	13	)	)	PUNCT
ejpam-4582	506	14	∪a(λ	∪a(λ	PROPN
ejpam-4582	506	15	,	,	PUNCT
ejpam-4582	506	16	s	s	PART
ejpam-4582	506	17	)	)	PUNCT
ejpam-4582	506	18	and	and	CCONJ
ejpam-4582	506	19	hence	hence	ADV
ejpam-4582	506	20	f	f	PROPN
ejpam-4582	506	21	=	=	SYM
ejpam-4582	506	22	f	f	PROPN
ejpam-4582	506	23	∩	∩	X
ejpam-4582	506	24	[	[	X
ejpam-4582	506	25	(	(	PUNCT
ejpam-4582	506	26	x	x	SYM
ejpam-4582	506	27	−	−	PROPN
ejpam-4582	506	28	f	f	PROPN
ejpam-4582	506	29	)	)	PUNCT
ejpam-4582	506	30	∪a(λ	∪a(λ	PROPN
ejpam-4582	506	31	,	,	PUNCT
ejpam-4582	506	32	s	s	NOUN
ejpam-4582	506	33	)	)	PUNCT
ejpam-4582	506	34	]	]	PUNCT
ejpam-4582	507	1	=	=	SYM
ejpam-4582	507	2	f	f	X
ejpam-4582	507	3	∩a(λ	∩a(λ	PROPN
ejpam-4582	507	4	,	,	PUNCT
ejpam-4582	507	5	s	s	PART
ejpam-4582	507	6	)	)	PUNCT
ejpam-4582	507	7	⊆	⊆	NUM
ejpam-4582	507	8	a(λ	a(λ	PROPN
ejpam-4582	507	9	,	,	PUNCT
ejpam-4582	507	10	s	s	NOUN
ejpam-4582	507	11	)	)	PUNCT
ejpam-4582	507	12	.	.	PUNCT
ejpam-4582	508	1	it	it	PRON
ejpam-4582	508	2	follows	follow	VERB
ejpam-4582	508	3	from	from	ADP
ejpam-4582	508	4	theorem	theorem	ADJ
ejpam-4582	508	5	14	14	NUM
ejpam-4582	508	6	that	that	SCONJ
ejpam-4582	508	7	a	a	PRON
ejpam-4582	508	8	is	be	AUX
ejpam-4582	508	9	g-(λ	g-(λ	PRON
ejpam-4582	508	10	,	,	PUNCT
ejpam-4582	508	11	s)-open	s)-open	VERB
ejpam-4582	508	12	.	.	PUNCT
ejpam-4582	509	1	definition	definition	NOUN
ejpam-4582	509	2	11	11	NUM
ejpam-4582	509	3	.	.	PUNCT
ejpam-4582	510	1	a	a	DET
ejpam-4582	510	2	subset	subset	NOUN
ejpam-4582	510	3	a	a	PRON
ejpam-4582	510	4	of	of	ADP
ejpam-4582	510	5	a	a	DET
ejpam-4582	510	6	topological	topological	ADJ
ejpam-4582	510	7	space	space	NOUN
ejpam-4582	510	8	(	(	PUNCT
ejpam-4582	510	9	x	x	X
ejpam-4582	510	10	,	,	PUNCT
ejpam-4582	510	11	τ	τ	X
ejpam-4582	510	12	)	)	PUNCT
ejpam-4582	510	13	is	be	AUX
ejpam-4582	510	14	said	say	VERB
ejpam-4582	510	15	to	to	PART
ejpam-4582	510	16	be	be	AUX
ejpam-4582	510	17	locally	locally	ADV
ejpam-4582	510	18	(	(	PUNCT
ejpam-4582	510	19	λ	λ	X
ejpam-4582	510	20	,	,	PUNCT
ejpam-4582	510	21	s)-closed	s)-close	VERB
ejpam-4582	510	22	if	if	SCONJ
ejpam-4582	510	23	a	a	DET
ejpam-4582	510	24	=	=	X
ejpam-4582	510	25	u	u	NOUN
ejpam-4582	510	26	∩	∩	NOUN
ejpam-4582	510	27	f	f	PROPN
ejpam-4582	510	28	,	,	PUNCT
ejpam-4582	510	29	where	where	SCONJ
ejpam-4582	510	30	u	u	PROPN
ejpam-4582	510	31	∈	∈	PROPN
ejpam-4582	510	32	(	(	PUNCT
ejpam-4582	510	33	λ	λ	NOUN
ejpam-4582	510	34	,	,	PUNCT
ejpam-4582	510	35	s)o(x	s)o(x	ADJ
ejpam-4582	510	36	)	)	PUNCT
ejpam-4582	510	37	and	and	CCONJ
ejpam-4582	510	38	f	f	PROPN
ejpam-4582	510	39	is	be	AUX
ejpam-4582	510	40	a	a	DET
ejpam-4582	510	41	(	(	PUNCT
ejpam-4582	510	42	λ	λ	X
ejpam-4582	510	43	,	,	PUNCT
ejpam-4582	510	44	s)-closed	s)-close	VERB
ejpam-4582	510	45	set	set	NOUN
ejpam-4582	510	46	.	.	PUNCT
ejpam-4582	511	1	theorem	theorem	PROPN
ejpam-4582	511	2	19	19	NUM
ejpam-4582	511	3	.	.	PUNCT
ejpam-4582	512	1	for	for	ADP
ejpam-4582	512	2	a	a	DET
ejpam-4582	512	3	subset	subset	NOUN
ejpam-4582	512	4	a	a	PRON
ejpam-4582	512	5	of	of	ADP
ejpam-4582	512	6	a	a	DET
ejpam-4582	512	7	topological	topological	ADJ
ejpam-4582	512	8	space	space	NOUN
ejpam-4582	512	9	(	(	PUNCT
ejpam-4582	512	10	x	x	X
ejpam-4582	512	11	,	,	PUNCT
ejpam-4582	512	12	τ	τ	PROPN
ejpam-4582	512	13	)	)	PUNCT
ejpam-4582	512	14	,	,	PUNCT
ejpam-4582	512	15	the	the	DET
ejpam-4582	512	16	following	follow	VERB
ejpam-4582	512	17	properties	property	NOUN
ejpam-4582	512	18	are	be	AUX
ejpam-4582	512	19	equivalent	equivalent	ADJ
ejpam-4582	512	20	:	:	PUNCT
ejpam-4582	512	21	(	(	PUNCT
ejpam-4582	512	22	1	1	X
ejpam-4582	512	23	)	)	PUNCT
ejpam-4582	512	24	a	a	PRON
ejpam-4582	512	25	is	be	AUX
ejpam-4582	512	26	locally	locally	ADV
ejpam-4582	512	27	(	(	PUNCT
ejpam-4582	512	28	λ	λ	X
ejpam-4582	512	29	,	,	PUNCT
ejpam-4582	512	30	s)-closed	s)-close	VERB
ejpam-4582	512	31	;	;	PUNCT
ejpam-4582	512	32	(	(	PUNCT
ejpam-4582	512	33	2	2	X
ejpam-4582	512	34	)	)	PUNCT
ejpam-4582	512	35	a	a	DET
ejpam-4582	512	36	=	=	X
ejpam-4582	512	37	u	u	NOUN
ejpam-4582	512	38	∩a(λ	∩a(λ	NOUN
ejpam-4582	512	39	,	,	PUNCT
ejpam-4582	512	40	s	s	NOUN
ejpam-4582	512	41	)	)	PUNCT
ejpam-4582	512	42	for	for	ADP
ejpam-4582	512	43	some	some	DET
ejpam-4582	512	44	u	u	NOUN
ejpam-4582	512	45	∈	∈	PROPN
ejpam-4582	512	46	(	(	PUNCT
ejpam-4582	512	47	λ	λ	NOUN
ejpam-4582	512	48	,	,	PUNCT
ejpam-4582	512	49	s)o(x	s)o(x	PROPN
ejpam-4582	512	50	)	)	PUNCT
ejpam-4582	512	51	;	;	PUNCT
ejpam-4582	512	52	(	(	PUNCT
ejpam-4582	512	53	3	3	X
ejpam-4582	512	54	)	)	PUNCT
ejpam-4582	512	55	a(λ	a(λ	ADV
ejpam-4582	512	56	,	,	PUNCT
ejpam-4582	512	57	s	s	X
ejpam-4582	512	58	)	)	PUNCT
ejpam-4582	512	59	−a	−a	NOUN
ejpam-4582	512	60	is	be	AUX
ejpam-4582	512	61	(	(	PUNCT
ejpam-4582	512	62	λ	λ	X
ejpam-4582	512	63	,	,	PUNCT
ejpam-4582	512	64	s)-closed	s)-close	VERB
ejpam-4582	512	65	;	;	PUNCT
ejpam-4582	512	66	c.	c.	PROPN
ejpam-4582	512	67	boonpok	boonpok	PROPN
ejpam-4582	512	68	,	,	PUNCT
ejpam-4582	512	69	c.	c.	PROPN
ejpam-4582	512	70	viriyapong	viriyapong	PROPN
ejpam-4582	512	71	/	/	SYM
ejpam-4582	512	72	eur	eur	PROPN
ejpam-4582	512	73	.	.	PUNCT
ejpam-4582	513	1	j.	j.	PROPN
ejpam-4582	513	2	pure	pure	PROPN
ejpam-4582	513	3	appl	appl	PROPN
ejpam-4582	513	4	.	.	PROPN
ejpam-4582	513	5	math	math	PROPN
ejpam-4582	513	6	,	,	PUNCT
ejpam-4582	513	7	16	16	NUM
ejpam-4582	513	8	(	(	PUNCT
ejpam-4582	513	9	1	1	NUM
ejpam-4582	513	10	)	)	PUNCT
ejpam-4582	513	11	(	(	PUNCT
ejpam-4582	513	12	2023	2023	NUM
ejpam-4582	513	13	)	)	PUNCT
ejpam-4582	513	14	,	,	PUNCT
ejpam-4582	513	15	336	336	NUM
ejpam-4582	513	16	-	-	SYM
ejpam-4582	513	17	362	362	NUM
ejpam-4582	513	18	350	350	NUM
ejpam-4582	513	19	(	(	PUNCT
ejpam-4582	513	20	4	4	NUM
ejpam-4582	513	21	)	)	PUNCT
ejpam-4582	513	22	a	a	DET
ejpam-4582	513	23	∪	∪	ADJ
ejpam-4582	513	24	[	[	X
ejpam-4582	513	25	x	x	X
ejpam-4582	513	26	−a(λ	−a(λ	NOUN
ejpam-4582	513	27	,	,	PUNCT
ejpam-4582	513	28	s	s	NOUN
ejpam-4582	513	29	)	)	PUNCT
ejpam-4582	513	30	]	]	PUNCT
ejpam-4582	513	31	is	be	AUX
ejpam-4582	513	32	(	(	PUNCT
ejpam-4582	513	33	λ	λ	X
ejpam-4582	513	34	,	,	PUNCT
ejpam-4582	513	35	s)-open	s)-open	PUNCT
ejpam-4582	513	36	;	;	PUNCT
ejpam-4582	513	37	(	(	PUNCT
ejpam-4582	513	38	5	5	X
ejpam-4582	513	39	)	)	PUNCT
ejpam-4582	513	40	a	a	DET
ejpam-4582	513	41	⊆	⊆	NUM
ejpam-4582	513	42	[	[	X
ejpam-4582	513	43	a	a	DET
ejpam-4582	513	44	∪	∪	ADJ
ejpam-4582	513	45	[	[	X
ejpam-4582	513	46	x	x	X
ejpam-4582	513	47	−a(λ	−a(λ	NOUN
ejpam-4582	513	48	,	,	PUNCT
ejpam-4582	513	49	s)]](λ	s)]](λ	PROPN
ejpam-4582	513	50	,	,	PUNCT
ejpam-4582	513	51	s	s	PART
ejpam-4582	513	52	)	)	PUNCT
ejpam-4582	513	53	.	.	PUNCT
ejpam-4582	514	1	proof	proof	NOUN
ejpam-4582	514	2	.	.	PUNCT
ejpam-4582	515	1	(	(	PUNCT
ejpam-4582	515	2	1	1	X
ejpam-4582	515	3	)	)	PUNCT
ejpam-4582	515	4	⇒	⇒	NOUN
ejpam-4582	515	5	(	(	PUNCT
ejpam-4582	515	6	2	2	NUM
ejpam-4582	515	7	):	):	PUNCT
ejpam-4582	515	8	suppose	suppose	VERB
ejpam-4582	515	9	that	that	SCONJ
ejpam-4582	515	10	a	a	DET
ejpam-4582	515	11	=	=	SYM
ejpam-4582	515	12	u	u	NOUN
ejpam-4582	515	13	∩	∩	NOUN
ejpam-4582	515	14	f	f	PROPN
ejpam-4582	515	15	,	,	PUNCT
ejpam-4582	515	16	where	where	SCONJ
ejpam-4582	515	17	u	u	PROPN
ejpam-4582	515	18	∈	∈	PROPN
ejpam-4582	515	19	(	(	PUNCT
ejpam-4582	515	20	λ	λ	NOUN
ejpam-4582	515	21	,	,	PUNCT
ejpam-4582	515	22	s)o(x	s)o(x	ADJ
ejpam-4582	515	23	)	)	PUNCT
ejpam-4582	515	24	and	and	CCONJ
ejpam-4582	515	25	f	f	PROPN
ejpam-4582	515	26	is	be	AUX
ejpam-4582	515	27	(	(	PUNCT
ejpam-4582	515	28	λ	λ	X
ejpam-4582	515	29	,	,	PUNCT
ejpam-4582	515	30	s)closed	s)closed	ADJ
ejpam-4582	515	31	.	.	PUNCT
ejpam-4582	516	1	since	since	SCONJ
ejpam-4582	516	2	a	a	DET
ejpam-4582	516	3	⊆	⊆	NUM
ejpam-4582	516	4	f	f	NOUN
ejpam-4582	516	5	,	,	PUNCT
ejpam-4582	516	6	we	we	PRON
ejpam-4582	516	7	have	have	VERB
ejpam-4582	516	8	a(λ	a(λ	ADV
ejpam-4582	516	9	,	,	PUNCT
ejpam-4582	516	10	s	s	PART
ejpam-4582	516	11	)	)	PUNCT
ejpam-4582	516	12	⊆	⊆	NUM
ejpam-4582	516	13	f	f	X
ejpam-4582	516	14	(	(	PUNCT
ejpam-4582	516	15	λ	λ	PROPN
ejpam-4582	516	16	,	,	PUNCT
ejpam-4582	516	17	s	s	PART
ejpam-4582	516	18	)	)	PUNCT
ejpam-4582	516	19	=	=	SYM
ejpam-4582	516	20	f	f	PROPN
ejpam-4582	516	21	.	.	PUNCT
ejpam-4582	517	1	since	since	SCONJ
ejpam-4582	517	2	a	a	DET
ejpam-4582	517	3	⊆	⊆	NUM
ejpam-4582	517	4	u	u	NOUN
ejpam-4582	517	5	,	,	PUNCT
ejpam-4582	517	6	a	a	DET
ejpam-4582	517	7	⊆	⊆	NUM
ejpam-4582	517	8	u	u	NOUN
ejpam-4582	517	9	∩a(λ	∩a(λ	NOUN
ejpam-4582	517	10	,	,	PUNCT
ejpam-4582	517	11	s	s	PART
ejpam-4582	517	12	)	)	PUNCT
ejpam-4582	517	13	⊆	⊆	NUM
ejpam-4582	517	14	u	u	NOUN
ejpam-4582	517	15	∩	∩	NOUN
ejpam-4582	517	16	f	f	PROPN
ejpam-4582	517	17	=	=	PUNCT
ejpam-4582	517	18	a.	a.	NOUN
ejpam-4582	517	19	thus	thus	ADV
ejpam-4582	517	20	,	,	PUNCT
ejpam-4582	517	21	a	a	DET
ejpam-4582	517	22	=	=	X
ejpam-4582	517	23	u	u	NOUN
ejpam-4582	517	24	∩a(λ	∩a(λ	NOUN
ejpam-4582	517	25	,	,	PUNCT
ejpam-4582	517	26	s	s	NOUN
ejpam-4582	517	27	)	)	PUNCT
ejpam-4582	517	28	.	.	PUNCT
ejpam-4582	518	1	(	(	PUNCT
ejpam-4582	518	2	2	2	X
ejpam-4582	518	3	)	)	PUNCT
ejpam-4582	518	4	⇒	⇒	NOUN
ejpam-4582	518	5	(	(	PUNCT
ejpam-4582	518	6	3	3	NUM
ejpam-4582	518	7	):	):	PUNCT
ejpam-4582	518	8	suppose	suppose	VERB
ejpam-4582	518	9	that	that	SCONJ
ejpam-4582	518	10	a	a	DET
ejpam-4582	518	11	=	=	SYM
ejpam-4582	518	12	u	u	NOUN
ejpam-4582	518	13	∩	∩	NOUN
ejpam-4582	518	14	a(λ	a(λ	SYM
ejpam-4582	518	15	,	,	PUNCT
ejpam-4582	518	16	s	s	NOUN
ejpam-4582	518	17	)	)	PUNCT
ejpam-4582	518	18	for	for	ADP
ejpam-4582	518	19	some	some	DET
ejpam-4582	518	20	u	u	NOUN
ejpam-4582	518	21	∈	∈	PROPN
ejpam-4582	518	22	(	(	PUNCT
ejpam-4582	518	23	λ	λ	NOUN
ejpam-4582	518	24	,	,	PUNCT
ejpam-4582	518	25	s)o(x	s)o(x	NOUN
ejpam-4582	518	26	)	)	PUNCT
ejpam-4582	518	27	.	.	PUNCT
ejpam-4582	519	1	then	then	ADV
ejpam-4582	519	2	,	,	PUNCT
ejpam-4582	519	3	we	we	PRON
ejpam-4582	519	4	have	have	VERB
ejpam-4582	519	5	a(λ	a(λ	ADV
ejpam-4582	519	6	,	,	PUNCT
ejpam-4582	519	7	s	s	X
ejpam-4582	519	8	)	)	PUNCT
ejpam-4582	519	9	−a	−a	NOUN
ejpam-4582	519	10	=	=	SYM
ejpam-4582	519	11	(	(	PUNCT
ejpam-4582	519	12	x	x	X
ejpam-4582	519	13	−	−	PUNCT
ejpam-4582	520	1	[	[	X
ejpam-4582	520	2	u	u	X
ejpam-4582	520	3	∩a(λ	∩a(λ	NOUN
ejpam-4582	520	4	,	,	PUNCT
ejpam-4582	520	5	s)])∩a(λ	s)])∩a(λ	NOUN
ejpam-4582	520	6	,	,	PUNCT
ejpam-4582	520	7	s	s	PART
ejpam-4582	520	8	)	)	PUNCT
ejpam-4582	520	9	=	=	SYM
ejpam-4582	521	1	(	(	PUNCT
ejpam-4582	521	2	x	x	SYM
ejpam-4582	521	3	−u)∩a(λ	−u)∩a(λ	NOUN
ejpam-4582	521	4	,	,	PUNCT
ejpam-4582	521	5	s	s	NOUN
ejpam-4582	521	6	)	)	PUNCT
ejpam-4582	521	7	.	.	PUNCT
ejpam-4582	522	1	this	this	PRON
ejpam-4582	522	2	shows	show	VERB
ejpam-4582	522	3	that	that	SCONJ
ejpam-4582	522	4	a(λ	a(λ	ADV
ejpam-4582	522	5	,	,	PUNCT
ejpam-4582	522	6	s	s	X
ejpam-4582	522	7	)	)	PUNCT
ejpam-4582	522	8	−a	−a	NOUN
ejpam-4582	522	9	is	be	AUX
ejpam-4582	522	10	(	(	PUNCT
ejpam-4582	522	11	λ	λ	X
ejpam-4582	522	12	,	,	PUNCT
ejpam-4582	522	13	s)-closed	s)-close	VERB
ejpam-4582	522	14	.	.	PUNCT
ejpam-4582	523	1	(	(	PUNCT
ejpam-4582	523	2	3	3	X
ejpam-4582	523	3	)	)	PUNCT
ejpam-4582	523	4	⇒	⇒	NOUN
ejpam-4582	523	5	(	(	PUNCT
ejpam-4582	523	6	4	4	NUM
ejpam-4582	523	7	):	):	PUNCT
ejpam-4582	523	8	sincex−[a(λ	sincex−[a(λ	NOUN
ejpam-4582	523	9	,	,	PUNCT
ejpam-4582	523	10	s)−a	s)−a	NOUN
ejpam-4582	523	11	]	]	X
ejpam-4582	523	12	=	=	PUNCT
ejpam-4582	524	1	[	[	X
ejpam-4582	524	2	x−a(λ	x−a(λ	NOUN
ejpam-4582	524	3	,	,	PUNCT
ejpam-4582	524	4	s)]∪a	s)]∪a	NOUN
ejpam-4582	524	5	and	and	CCONJ
ejpam-4582	524	6	by	by	ADP
ejpam-4582	524	7	(	(	PUNCT
ejpam-4582	524	8	3	3	NUM
ejpam-4582	524	9	)	)	PUNCT
ejpam-4582	524	10	,	,	PUNCT
ejpam-4582	524	11	we	we	PRON
ejpam-4582	524	12	obtain	obtain	VERB
ejpam-4582	524	13	a∪[x−a(λ	a∪[x−a(λ	NOUN
ejpam-4582	524	14	,	,	PUNCT
ejpam-4582	524	15	s	s	PART
ejpam-4582	524	16	)	)	PUNCT
ejpam-4582	524	17	]	]	PUNCT
ejpam-4582	524	18	is	be	AUX
ejpam-4582	524	19	(	(	PUNCT
ejpam-4582	524	20	λ	λ	X
ejpam-4582	524	21	,	,	PUNCT
ejpam-4582	524	22	s)-open	s)-open	ADJ
ejpam-4582	524	23	.	.	PUNCT
ejpam-4582	525	1	(	(	PUNCT
ejpam-4582	525	2	4	4	X
ejpam-4582	525	3	)	)	PUNCT
ejpam-4582	525	4	⇒	⇒	NOUN
ejpam-4582	525	5	(	(	PUNCT
ejpam-4582	525	6	5	5	NUM
ejpam-4582	525	7	):	):	PUNCT
ejpam-4582	525	8	by	by	ADP
ejpam-4582	525	9	(	(	PUNCT
ejpam-4582	525	10	4	4	NUM
ejpam-4582	525	11	)	)	PUNCT
ejpam-4582	525	12	,	,	PUNCT
ejpam-4582	525	13	a	a	DET
ejpam-4582	525	14	⊆	⊆	NUM
ejpam-4582	525	15	a	a	DET
ejpam-4582	525	16	∪	∪	ADJ
ejpam-4582	525	17	[	[	X
ejpam-4582	525	18	x	x	X
ejpam-4582	525	19	−a(λ	−a(λ	NOUN
ejpam-4582	525	20	,	,	PUNCT
ejpam-4582	525	21	s	s	NOUN
ejpam-4582	525	22	)	)	PUNCT
ejpam-4582	525	23	]	]	PUNCT
ejpam-4582	526	1	=	=	PUNCT
ejpam-4582	527	1	[	[	X
ejpam-4582	527	2	a	a	DET
ejpam-4582	527	3	∪	∪	X
ejpam-4582	527	4	(	(	PUNCT
ejpam-4582	527	5	x	x	SYM
ejpam-4582	527	6	−a(λ	−a(λ	NOUN
ejpam-4582	527	7	,	,	PUNCT
ejpam-4582	527	8	s))](λ	s))](λ	PROPN
ejpam-4582	527	9	,	,	PUNCT
ejpam-4582	527	10	s	s	PART
ejpam-4582	527	11	)	)	PUNCT
ejpam-4582	527	12	.	.	PUNCT
ejpam-4582	528	1	(	(	PUNCT
ejpam-4582	528	2	5	5	X
ejpam-4582	528	3	)	)	PUNCT
ejpam-4582	528	4	⇒	⇒	NOUN
ejpam-4582	528	5	(	(	PUNCT
ejpam-4582	528	6	1	1	NUM
ejpam-4582	528	7	):	):	PUNCT
ejpam-4582	528	8	we	we	PRON
ejpam-4582	528	9	put	put	VERB
ejpam-4582	528	10	u	u	NOUN
ejpam-4582	528	11	=	=	PUNCT
ejpam-4582	529	1	[	[	X
ejpam-4582	529	2	a	a	DET
ejpam-4582	529	3	∪	∪	ADJ
ejpam-4582	529	4	[	[	X
ejpam-4582	529	5	x	x	X
ejpam-4582	529	6	−	−	NOUN
ejpam-4582	529	7	a(λ	a(λ	ADV
ejpam-4582	529	8	,	,	PUNCT
ejpam-4582	529	9	s)]](λ	s)]](λ	PROPN
ejpam-4582	529	10	,	,	PUNCT
ejpam-4582	529	11	s	s	PART
ejpam-4582	529	12	)	)	PUNCT
ejpam-4582	529	13	.	.	PUNCT
ejpam-4582	530	1	then	then	ADV
ejpam-4582	530	2	,	,	PUNCT
ejpam-4582	530	3	we	we	PRON
ejpam-4582	530	4	have	have	VERB
ejpam-4582	530	5	u	u	NOUN
ejpam-4582	530	6	is	be	AUX
ejpam-4582	530	7	(	(	PUNCT
ejpam-4582	530	8	λ	λ	X
ejpam-4582	530	9	,	,	PUNCT
ejpam-4582	530	10	s)-open	s)-open	PUNCT
ejpam-4582	530	11	and	and	CCONJ
ejpam-4582	530	12	a	a	DET
ejpam-4582	530	13	=	=	X
ejpam-4582	530	14	a	a	DET
ejpam-4582	530	15	∩	∩	ADJ
ejpam-4582	530	16	u	u	ADJ
ejpam-4582	530	17	⊆	⊆	NUM
ejpam-4582	530	18	u	u	NOUN
ejpam-4582	530	19	∩	∩	NOUN
ejpam-4582	530	20	a(λ	a(λ	SYM
ejpam-4582	530	21	,	,	PUNCT
ejpam-4582	530	22	s	s	NOUN
ejpam-4582	530	23	)	)	PUNCT
ejpam-4582	530	24	⊆	⊆	NUM
ejpam-4582	531	1	[	[	X
ejpam-4582	531	2	a	a	DET
ejpam-4582	531	3	∪	∪	ADJ
ejpam-4582	531	4	[	[	X
ejpam-4582	531	5	x	x	X
ejpam-4582	531	6	−	−	NOUN
ejpam-4582	531	7	a(λ	a(λ	PROPN
ejpam-4582	531	8	,	,	PUNCT
ejpam-4582	531	9	s	s	NOUN
ejpam-4582	531	10	)	)	PUNCT
ejpam-4582	531	11	]	]	PUNCT
ejpam-4582	531	12	]	]	X
ejpam-4582	531	13	∩	∩	X
ejpam-4582	531	14	a(λ	a(λ	SYM
ejpam-4582	531	15	,	,	PUNCT
ejpam-4582	531	16	s	s	PART
ejpam-4582	531	17	)	)	PUNCT
ejpam-4582	531	18	=	=	SYM
ejpam-4582	531	19	a	a	DET
ejpam-4582	531	20	∩	∩	NOUN
ejpam-4582	531	21	a(λ	a(λ	SYM
ejpam-4582	531	22	,	,	PUNCT
ejpam-4582	531	23	s	s	PART
ejpam-4582	531	24	)	)	PUNCT
ejpam-4582	531	25	=	=	SYM
ejpam-4582	531	26	a.	a.	NOUN
ejpam-4582	531	27	therefore	therefore	ADV
ejpam-4582	531	28	,	,	PUNCT
ejpam-4582	531	29	we	we	PRON
ejpam-4582	531	30	obtain	obtain	VERB
ejpam-4582	531	31	a	a	DET
ejpam-4582	531	32	=	=	PUNCT
ejpam-4582	531	33	u∩a(λ	u∩a(λ	NOUN
ejpam-4582	531	34	,	,	PUNCT
ejpam-4582	531	35	s	s	PART
ejpam-4582	531	36	)	)	PUNCT
ejpam-4582	531	37	,	,	PUNCT
ejpam-4582	531	38	where	where	SCONJ
ejpam-4582	531	39	u	u	PROPN
ejpam-4582	531	40	∈	∈	PROPN
ejpam-4582	531	41	(	(	PUNCT
ejpam-4582	531	42	λ	λ	NOUN
ejpam-4582	531	43	,	,	PUNCT
ejpam-4582	531	44	s)o(x	s)o(x	ADJ
ejpam-4582	531	45	)	)	PUNCT
ejpam-4582	531	46	and	and	CCONJ
ejpam-4582	531	47	a(λ	a(λ	ADV
ejpam-4582	531	48	,	,	PUNCT
ejpam-4582	531	49	s	s	PART
ejpam-4582	531	50	)	)	PUNCT
ejpam-4582	531	51	is	be	AUX
ejpam-4582	531	52	(	(	PUNCT
ejpam-4582	531	53	λ	λ	X
ejpam-4582	531	54	,	,	PUNCT
ejpam-4582	531	55	s)-closed	s)-close	VERB
ejpam-4582	531	56	.	.	PUNCT
ejpam-4582	532	1	thus	thus	ADV
ejpam-4582	532	2	,	,	PUNCT
ejpam-4582	532	3	a	a	PRON
ejpam-4582	532	4	is	be	AUX
ejpam-4582	532	5	locally	locally	ADV
ejpam-4582	532	6	(	(	PUNCT
ejpam-4582	532	7	λ	λ	X
ejpam-4582	532	8	,	,	PUNCT
ejpam-4582	532	9	s)-closed	s)-close	VERB
ejpam-4582	532	10	.	.	PUNCT
ejpam-4582	533	1	theorem	theorem	ADJ
ejpam-4582	533	2	20	20	NUM
ejpam-4582	533	3	.	.	PUNCT
ejpam-4582	534	1	a	a	DET
ejpam-4582	534	2	subset	subset	NOUN
ejpam-4582	534	3	a	a	PRON
ejpam-4582	534	4	of	of	ADP
ejpam-4582	534	5	a	a	DET
ejpam-4582	534	6	topological	topological	ADJ
ejpam-4582	534	7	space	space	NOUN
ejpam-4582	534	8	(	(	PUNCT
ejpam-4582	534	9	x	x	X
ejpam-4582	534	10	,	,	PUNCT
ejpam-4582	534	11	τ	τ	X
ejpam-4582	534	12	)	)	PUNCT
ejpam-4582	534	13	is	be	AUX
ejpam-4582	534	14	(	(	PUNCT
ejpam-4582	534	15	λ	λ	X
ejpam-4582	534	16	,	,	PUNCT
ejpam-4582	534	17	s)-closed	s)-close	VERB
ejpam-4582	534	18	if	if	SCONJ
ejpam-4582	534	19	and	and	CCONJ
ejpam-4582	534	20	only	only	ADV
ejpam-4582	534	21	if	if	SCONJ
ejpam-4582	534	22	a	a	PRON
ejpam-4582	534	23	is	be	AUX
ejpam-4582	534	24	locally	locally	ADV
ejpam-4582	534	25	(	(	PUNCT
ejpam-4582	534	26	λ	λ	X
ejpam-4582	534	27	,	,	PUNCT
ejpam-4582	534	28	s)-closed	s)-close	VERB
ejpam-4582	534	29	and	and	CCONJ
ejpam-4582	534	30	g-(λ	g-(λ	PROPN
ejpam-4582	534	31	,	,	PUNCT
ejpam-4582	534	32	s)-closed	s)-close	VERB
ejpam-4582	534	33	.	.	PUNCT
ejpam-4582	535	1	proof	proof	NOUN
ejpam-4582	535	2	.	.	PUNCT
ejpam-4582	536	1	let	let	VERB
ejpam-4582	536	2	a	a	DET
ejpam-4582	536	3	be	be	AUX
ejpam-4582	536	4	(	(	PUNCT
ejpam-4582	536	5	λ	λ	X
ejpam-4582	536	6	,	,	PUNCT
ejpam-4582	536	7	s)-closed	s)-close	VERB
ejpam-4582	536	8	.	.	PUNCT
ejpam-4582	537	1	by	by	ADP
ejpam-4582	537	2	remark	remark	NOUN
ejpam-4582	537	3	1	1	NUM
ejpam-4582	537	4	,	,	PUNCT
ejpam-4582	537	5	a	a	PRON
ejpam-4582	537	6	is	be	AUX
ejpam-4582	537	7	g-(λ	g-(λ	PRON
ejpam-4582	537	8	,	,	PUNCT
ejpam-4582	537	9	s)-closed	s)-close	VERB
ejpam-4582	537	10	.	.	PUNCT
ejpam-4582	538	1	sincex	sincex	PROPN
ejpam-4582	538	2	is	be	AUX
ejpam-4582	538	3	(	(	PUNCT
ejpam-4582	538	4	λ	λ	X
ejpam-4582	538	5	,	,	PUNCT
ejpam-4582	538	6	s)-open	s)-open	PUNCT
ejpam-4582	538	7	and	and	CCONJ
ejpam-4582	538	8	a	a	DET
ejpam-4582	538	9	=	=	NOUN
ejpam-4582	538	10	x	x	SYM
ejpam-4582	538	11	∩a	∩a	PROPN
ejpam-4582	538	12	,	,	PUNCT
ejpam-4582	538	13	we	we	PRON
ejpam-4582	538	14	have	have	VERB
ejpam-4582	538	15	a	a	PRON
ejpam-4582	538	16	is	be	AUX
ejpam-4582	538	17	locally	locally	ADV
ejpam-4582	538	18	(	(	PUNCT
ejpam-4582	538	19	λ	λ	X
ejpam-4582	538	20	,	,	PUNCT
ejpam-4582	538	21	s)-closed	s)-close	VERB
ejpam-4582	538	22	.	.	PUNCT
ejpam-4582	539	1	conversely	conversely	ADV
ejpam-4582	539	2	,	,	PUNCT
ejpam-4582	539	3	suppose	suppose	VERB
ejpam-4582	539	4	that	that	SCONJ
ejpam-4582	539	5	a	a	PRON
ejpam-4582	539	6	is	be	AUX
ejpam-4582	539	7	locally	locally	ADV
ejpam-4582	539	8	(	(	PUNCT
ejpam-4582	539	9	λ	λ	X
ejpam-4582	539	10	,	,	PUNCT
ejpam-4582	539	11	s)-closed	s)-close	VERB
ejpam-4582	539	12	and	and	CCONJ
ejpam-4582	539	13	g-(λ	g-(λ	PROPN
ejpam-4582	539	14	,	,	PUNCT
ejpam-4582	539	15	s)-closed	s)-close	VERB
ejpam-4582	539	16	.	.	PUNCT
ejpam-4582	540	1	since	since	SCONJ
ejpam-4582	540	2	a	a	PRON
ejpam-4582	540	3	is	be	AUX
ejpam-4582	540	4	locally	locally	ADV
ejpam-4582	540	5	(	(	PUNCT
ejpam-4582	540	6	λ	λ	X
ejpam-4582	540	7	,	,	PUNCT
ejpam-4582	540	8	s)-closed	s)-close	VERB
ejpam-4582	540	9	and	and	CCONJ
ejpam-4582	540	10	by	by	ADP
ejpam-4582	540	11	theorem	theorem	NOUN
ejpam-4582	540	12	19	19	NUM
ejpam-4582	540	13	,	,	PUNCT
ejpam-4582	540	14	a	a	DET
ejpam-4582	540	15	⊆	⊆	NUM
ejpam-4582	540	16	[	[	X
ejpam-4582	540	17	a∪	a∪	INTJ
ejpam-4582	540	18	(	(	PUNCT
ejpam-4582	540	19	x	x	SYM
ejpam-4582	540	20	−a(λ	−a(λ	NOUN
ejpam-4582	540	21	,	,	PUNCT
ejpam-4582	540	22	s))](λ	s))](λ	PROPN
ejpam-4582	540	23	,	,	PUNCT
ejpam-4582	540	24	s	s	PART
ejpam-4582	540	25	)	)	PUNCT
ejpam-4582	540	26	.	.	PUNCT
ejpam-4582	541	1	since	since	SCONJ
ejpam-4582	541	2	[	[	X
ejpam-4582	541	3	a∪	a∪	INTJ
ejpam-4582	541	4	(	(	PUNCT
ejpam-4582	541	5	x	x	SYM
ejpam-4582	541	6	−a(λ	−a(λ	NOUN
ejpam-4582	541	7	,	,	PUNCT
ejpam-4582	541	8	s))](λ	s))](λ	PROPN
ejpam-4582	541	9	,	,	PUNCT
ejpam-4582	541	10	s	s	PART
ejpam-4582	541	11	)	)	PUNCT
ejpam-4582	541	12	is	be	AUX
ejpam-4582	541	13	(	(	PUNCT
ejpam-4582	541	14	λ	λ	X
ejpam-4582	541	15	,	,	PUNCT
ejpam-4582	541	16	s)-open	s)-open	PUNCT
ejpam-4582	541	17	and	and	CCONJ
ejpam-4582	541	18	a	a	PRON
ejpam-4582	541	19	is	be	AUX
ejpam-4582	541	20	g-(λ	g-(λ	PRON
ejpam-4582	541	21	,	,	PUNCT
ejpam-4582	541	22	s)-closed	s)-close	VERB
ejpam-4582	541	23	,	,	PUNCT
ejpam-4582	541	24	a(λ	a(λ	ADV
ejpam-4582	541	25	,	,	PUNCT
ejpam-4582	541	26	s	s	NOUN
ejpam-4582	541	27	)	)	PUNCT
ejpam-4582	541	28	⊆	⊆	NUM
ejpam-4582	542	1	[	[	X
ejpam-4582	542	2	a	a	DET
ejpam-4582	542	3	∪	∪	ADJ
ejpam-4582	542	4	[	[	X
ejpam-4582	542	5	x	x	X
ejpam-4582	542	6	−a(λ	−a(λ	NOUN
ejpam-4582	542	7	,	,	PUNCT
ejpam-4582	542	8	s)]](λ	s)]](λ	PROPN
ejpam-4582	542	9	,	,	PUNCT
ejpam-4582	542	10	s	s	PART
ejpam-4582	542	11	)	)	PUNCT
ejpam-4582	542	12	⊆	⊆	NOUN
ejpam-4582	542	13	a	a	DET
ejpam-4582	542	14	∪	∪	ADJ
ejpam-4582	542	15	[	[	X
ejpam-4582	542	16	x	x	X
ejpam-4582	542	17	−a(λ	−a(λ	NOUN
ejpam-4582	542	18	,	,	PUNCT
ejpam-4582	542	19	s	s	NOUN
ejpam-4582	542	20	)	)	PUNCT
ejpam-4582	542	21	]	]	PUNCT
ejpam-4582	542	22	and	and	CCONJ
ejpam-4582	542	23	hence	hence	ADV
ejpam-4582	542	24	a(λ	a(λ	ADV
ejpam-4582	542	25	,	,	PUNCT
ejpam-4582	542	26	s	s	X
ejpam-4582	542	27	)	)	PUNCT
ejpam-4582	542	28	⊆	⊆	NUM
ejpam-4582	542	29	a.	a.	NOUN
ejpam-4582	542	30	thus	thus	ADV
ejpam-4582	542	31	,	,	PUNCT
ejpam-4582	542	32	a(λ	a(λ	ADV
ejpam-4582	542	33	,	,	PUNCT
ejpam-4582	542	34	s	s	PART
ejpam-4582	542	35	)	)	PUNCT
ejpam-4582	542	36	=	=	SYM
ejpam-4582	542	37	a	a	PRON
ejpam-4582	542	38	and	and	CCONJ
ejpam-4582	542	39	by	by	ADP
ejpam-4582	542	40	lemma	lemma	PROPN
ejpam-4582	542	41	4	4	NUM
ejpam-4582	542	42	,	,	PUNCT
ejpam-4582	542	43	a	a	DET
ejpam-4582	542	44	is	be	AUX
ejpam-4582	542	45	(	(	PUNCT
ejpam-4582	542	46	λ	λ	X
ejpam-4582	542	47	,	,	PUNCT
ejpam-4582	542	48	s)-closed	s)-close	VERB
ejpam-4582	542	49	.	.	PUNCT
ejpam-4582	543	1	definition	definition	NOUN
ejpam-4582	543	2	12	12	NUM
ejpam-4582	543	3	.	.	PUNCT
ejpam-4582	544	1	a	a	DET
ejpam-4582	544	2	subset	subset	NOUN
ejpam-4582	544	3	a	a	PRON
ejpam-4582	544	4	of	of	ADP
ejpam-4582	544	5	a	a	DET
ejpam-4582	544	6	topological	topological	ADJ
ejpam-4582	544	7	space	space	NOUN
ejpam-4582	544	8	(	(	PUNCT
ejpam-4582	544	9	x	x	X
ejpam-4582	544	10	,	,	PUNCT
ejpam-4582	544	11	τ	τ	X
ejpam-4582	544	12	)	)	PUNCT
ejpam-4582	544	13	is	be	AUX
ejpam-4582	544	14	said	say	VERB
ejpam-4582	544	15	to	to	PART
ejpam-4582	544	16	be	be	AUX
ejpam-4582	544	17	:	:	PUNCT
ejpam-4582	544	18	(	(	PUNCT
ejpam-4582	544	19	i	i	NOUN
ejpam-4582	544	20	)	)	PUNCT
ejpam-4582	544	21	(	(	PUNCT
ejpam-4582	544	22	λ	λ	NOUN
ejpam-4582	544	23	,	,	PUNCT
ejpam-4582	544	24	s)-dense	s)-dense	VERB
ejpam-4582	544	25	if	if	SCONJ
ejpam-4582	544	26	a(λ	a(λ	ADV
ejpam-4582	544	27	,	,	PUNCT
ejpam-4582	544	28	s	s	NOUN
ejpam-4582	544	29	)	)	PUNCT
ejpam-4582	544	30	=	=	SYM
ejpam-4582	545	1	x	x	X
ejpam-4582	545	2	;	;	PUNCT
ejpam-4582	545	3	(	(	PUNCT
ejpam-4582	545	4	ii	ii	NOUN
ejpam-4582	545	5	)	)	PUNCT
ejpam-4582	545	6	(	(	PUNCT
ejpam-4582	545	7	λ	λ	PROPN
ejpam-4582	545	8	,	,	PUNCT
ejpam-4582	545	9	s)-codense	s)-codense	VERB
ejpam-4582	545	10	if	if	SCONJ
ejpam-4582	545	11	its	its	PRON
ejpam-4582	545	12	complement	complement	NOUN
ejpam-4582	545	13	is	be	AUX
ejpam-4582	545	14	(	(	PUNCT
ejpam-4582	545	15	λ	λ	INTJ
ejpam-4582	545	16	,	,	PUNCT
ejpam-4582	545	17	s)-dense	s)-dense	NOUN
ejpam-4582	545	18	.	.	PUNCT
ejpam-4582	546	1	definition	definition	NOUN
ejpam-4582	546	2	13	13	NUM
ejpam-4582	546	3	.	.	PUNCT
ejpam-4582	547	1	a	a	DET
ejpam-4582	547	2	topological	topological	ADJ
ejpam-4582	547	3	space	space	NOUN
ejpam-4582	547	4	(	(	PUNCT
ejpam-4582	547	5	x	x	X
ejpam-4582	547	6	,	,	PUNCT
ejpam-4582	547	7	τ	τ	X
ejpam-4582	547	8	)	)	PUNCT
ejpam-4582	547	9	is	be	AUX
ejpam-4582	547	10	said	say	VERB
ejpam-4582	547	11	to	to	PART
ejpam-4582	547	12	be	be	AUX
ejpam-4582	547	13	(	(	PUNCT
ejpam-4582	547	14	λ	λ	X
ejpam-4582	547	15	,	,	PUNCT
ejpam-4582	547	16	s)-submaximal	s)-submaximal	ADJ
ejpam-4582	547	17	if	if	SCONJ
ejpam-4582	547	18	,	,	PUNCT
ejpam-4582	547	19	for	for	SCONJ
ejpam-4582	547	20	each	each	DET
ejpam-4582	547	21	(	(	PUNCT
ejpam-4582	547	22	λ	λ	PROPN
ejpam-4582	547	23	,	,	PUNCT
ejpam-4582	547	24	s)-dense	s)-dense	NOUN
ejpam-4582	547	25	subset	subset	NOUN
ejpam-4582	547	26	of	of	ADP
ejpam-4582	547	27	x	x	PUNCT
ejpam-4582	547	28	is	be	AUX
ejpam-4582	547	29	(	(	PUNCT
ejpam-4582	547	30	λ	λ	X
ejpam-4582	547	31	,	,	PUNCT
ejpam-4582	547	32	s)-open	s)-open	PUNCT
ejpam-4582	547	33	.	.	PUNCT
ejpam-4582	548	1	theorem	theorem	VERB
ejpam-4582	548	2	21	21	NUM
ejpam-4582	548	3	.	.	PUNCT
ejpam-4582	549	1	for	for	ADP
ejpam-4582	549	2	a	a	DET
ejpam-4582	549	3	topological	topological	ADJ
ejpam-4582	549	4	space	space	NOUN
ejpam-4582	549	5	(	(	PUNCT
ejpam-4582	549	6	x	x	X
ejpam-4582	549	7	,	,	PUNCT
ejpam-4582	549	8	τ	τ	PROPN
ejpam-4582	549	9	)	)	PUNCT
ejpam-4582	549	10	,	,	PUNCT
ejpam-4582	549	11	the	the	DET
ejpam-4582	549	12	following	follow	VERB
ejpam-4582	549	13	properties	property	NOUN
ejpam-4582	549	14	are	be	AUX
ejpam-4582	549	15	equivalent	equivalent	ADJ
ejpam-4582	549	16	:	:	PUNCT
ejpam-4582	549	17	(	(	PUNCT
ejpam-4582	549	18	1	1	X
ejpam-4582	549	19	)	)	PUNCT
ejpam-4582	549	20	(	(	PUNCT
ejpam-4582	549	21	x	x	X
ejpam-4582	549	22	,	,	PUNCT
ejpam-4582	549	23	τ	τ	X
ejpam-4582	549	24	)	)	PUNCT
ejpam-4582	549	25	is	be	AUX
ejpam-4582	549	26	(	(	PUNCT
ejpam-4582	549	27	λ	λ	X
ejpam-4582	549	28	,	,	PUNCT
ejpam-4582	549	29	s)-submaximal	s)-submaximal	ADJ
ejpam-4582	549	30	;	;	PUNCT
ejpam-4582	549	31	c.	c.	PROPN
ejpam-4582	549	32	boonpok	boonpok	PROPN
ejpam-4582	549	33	,	,	PUNCT
ejpam-4582	549	34	c.	c.	PROPN
ejpam-4582	549	35	viriyapong	viriyapong	PROPN
ejpam-4582	549	36	/	/	SYM
ejpam-4582	549	37	eur	eur	PROPN
ejpam-4582	549	38	.	.	PUNCT
ejpam-4582	550	1	j.	j.	PROPN
ejpam-4582	550	2	pure	pure	PROPN
ejpam-4582	550	3	appl	appl	PROPN
ejpam-4582	550	4	.	.	PROPN
ejpam-4582	550	5	math	math	PROPN
ejpam-4582	550	6	,	,	PUNCT
ejpam-4582	550	7	16	16	NUM
ejpam-4582	550	8	(	(	PUNCT
ejpam-4582	550	9	1	1	NUM
ejpam-4582	550	10	)	)	PUNCT
ejpam-4582	550	11	(	(	PUNCT
ejpam-4582	550	12	2023	2023	NUM
ejpam-4582	550	13	)	)	PUNCT
ejpam-4582	550	14	,	,	PUNCT
ejpam-4582	550	15	336	336	NUM
ejpam-4582	550	16	-	-	SYM
ejpam-4582	550	17	362	362	NUM
ejpam-4582	550	18	351	351	NUM
ejpam-4582	550	19	(	(	PUNCT
ejpam-4582	550	20	2	2	NUM
ejpam-4582	550	21	)	)	PUNCT
ejpam-4582	550	22	every	every	DET
ejpam-4582	550	23	subset	subset	NOUN
ejpam-4582	550	24	of	of	ADP
ejpam-4582	550	25	x	x	PUNCT
ejpam-4582	550	26	is	be	AUX
ejpam-4582	550	27	a	a	DET
ejpam-4582	550	28	locally	locally	ADV
ejpam-4582	550	29	(	(	PUNCT
ejpam-4582	550	30	λ	λ	NOUN
ejpam-4582	550	31	,	,	PUNCT
ejpam-4582	550	32	s)-closed	s)-close	VERB
ejpam-4582	550	33	set	set	VERB
ejpam-4582	550	34	;	;	PUNCT
ejpam-4582	550	35	(	(	PUNCT
ejpam-4582	550	36	3	3	X
ejpam-4582	550	37	)	)	PUNCT
ejpam-4582	550	38	every	every	DET
ejpam-4582	550	39	subset	subset	NOUN
ejpam-4582	550	40	of	of	ADP
ejpam-4582	550	41	x	x	SYM
ejpam-4582	550	42	is	be	AUX
ejpam-4582	550	43	the	the	DET
ejpam-4582	550	44	union	union	NOUN
ejpam-4582	550	45	of	of	ADP
ejpam-4582	550	46	a	a	DET
ejpam-4582	550	47	(	(	PUNCT
ejpam-4582	550	48	λ	λ	NOUN
ejpam-4582	550	49	,	,	PUNCT
ejpam-4582	550	50	s)-open	s)-open	PUNCT
ejpam-4582	550	51	set	set	VERB
ejpam-4582	550	52	and	and	CCONJ
ejpam-4582	550	53	a	a	DET
ejpam-4582	550	54	(	(	PUNCT
ejpam-4582	550	55	λ	λ	PROPN
ejpam-4582	550	56	,	,	PUNCT
ejpam-4582	550	57	s)-closed	s)-close	VERB
ejpam-4582	550	58	set	set	VERB
ejpam-4582	550	59	;	;	PUNCT
ejpam-4582	550	60	(	(	PUNCT
ejpam-4582	550	61	4	4	X
ejpam-4582	550	62	)	)	PUNCT
ejpam-4582	550	63	every	every	PRON
ejpam-4582	550	64	(	(	PUNCT
ejpam-4582	550	65	λ	λ	NOUN
ejpam-4582	550	66	,	,	PUNCT
ejpam-4582	550	67	s)-dense	s)-dense	NOUN
ejpam-4582	550	68	set	set	VERB
ejpam-4582	550	69	of	of	ADP
ejpam-4582	550	70	x	x	SYM
ejpam-4582	550	71	is	be	AUX
ejpam-4582	550	72	the	the	DET
ejpam-4582	550	73	intersection	intersection	NOUN
ejpam-4582	550	74	of	of	ADP
ejpam-4582	550	75	a	a	DET
ejpam-4582	550	76	(	(	PUNCT
ejpam-4582	550	77	λ	λ	PROPN
ejpam-4582	550	78	,	,	PUNCT
ejpam-4582	550	79	s)-closed	s)-close	VERB
ejpam-4582	550	80	set	set	NOUN
ejpam-4582	550	81	and	and	CCONJ
ejpam-4582	550	82	a	a	DET
ejpam-4582	550	83	(	(	PUNCT
ejpam-4582	550	84	λ	λ	NOUN
ejpam-4582	550	85	,	,	PUNCT
ejpam-4582	550	86	s)-open	s)-open	PUNCT
ejpam-4582	550	87	set	set	VERB
ejpam-4582	550	88	;	;	PUNCT
ejpam-4582	550	89	(	(	PUNCT
ejpam-4582	550	90	5	5	X
ejpam-4582	550	91	)	)	PUNCT
ejpam-4582	550	92	every	every	DET
ejpam-4582	550	93	(	(	PUNCT
ejpam-4582	550	94	λ	λ	NOUN
ejpam-4582	550	95	,	,	PUNCT
ejpam-4582	550	96	s)-codense	s)-codense	NOUN
ejpam-4582	550	97	set	set	VERB
ejpam-4582	550	98	of	of	ADP
ejpam-4582	550	99	x	x	SYM
ejpam-4582	550	100	is	be	AUX
ejpam-4582	550	101	the	the	DET
ejpam-4582	550	102	union	union	NOUN
ejpam-4582	550	103	of	of	ADP
ejpam-4582	550	104	a	a	DET
ejpam-4582	550	105	(	(	PUNCT
ejpam-4582	550	106	λ	λ	NOUN
ejpam-4582	550	107	,	,	PUNCT
ejpam-4582	550	108	s)-open	s)-open	PUNCT
ejpam-4582	550	109	set	set	VERB
ejpam-4582	550	110	and	and	CCONJ
ejpam-4582	550	111	a	a	DET
ejpam-4582	550	112	(	(	PUNCT
ejpam-4582	550	113	λ	λ	PROPN
ejpam-4582	550	114	,	,	PUNCT
ejpam-4582	550	115	s)-closed	s)-close	VERB
ejpam-4582	550	116	set	set	NOUN
ejpam-4582	550	117	.	.	PUNCT
ejpam-4582	551	1	proof	proof	NOUN
ejpam-4582	551	2	.	.	PUNCT
ejpam-4582	552	1	(	(	PUNCT
ejpam-4582	552	2	1	1	X
ejpam-4582	552	3	)	)	PUNCT
ejpam-4582	552	4	⇒	⇒	NOUN
ejpam-4582	552	5	(	(	PUNCT
ejpam-4582	552	6	2	2	NUM
ejpam-4582	552	7	):	):	PUNCT
ejpam-4582	552	8	suppose	suppose	VERB
ejpam-4582	552	9	that	that	SCONJ
ejpam-4582	552	10	(	(	PUNCT
ejpam-4582	552	11	x	x	X
ejpam-4582	552	12	,	,	PUNCT
ejpam-4582	552	13	τ	τ	X
ejpam-4582	552	14	)	)	PUNCT
ejpam-4582	552	15	is	be	AUX
ejpam-4582	552	16	(	(	PUNCT
ejpam-4582	552	17	λ	λ	X
ejpam-4582	552	18	,	,	PUNCT
ejpam-4582	552	19	s)-submaximal	s)-submaximal	ADJ
ejpam-4582	552	20	.	.	PUNCT
ejpam-4582	553	1	let	let	VERB
ejpam-4582	553	2	a	a	DET
ejpam-4582	553	3	be	be	AUX
ejpam-4582	553	4	any	any	DET
ejpam-4582	553	5	subset	subset	NOUN
ejpam-4582	553	6	of	of	ADP
ejpam-4582	553	7	x.	x.	NOUN
ejpam-4582	553	8	then	then	ADV
ejpam-4582	553	9	,	,	PUNCT
ejpam-4582	553	10	[	[	X
ejpam-4582	553	11	x−	x−	NOUN
ejpam-4582	554	1	[	[	X
ejpam-4582	554	2	a(λ	a(λ	PROPN
ejpam-4582	554	3	,	,	PUNCT
ejpam-4582	554	4	s)−a]](λ	s)−a]](λ	X
ejpam-4582	554	5	,	,	PUNCT
ejpam-4582	554	6	s	s	PART
ejpam-4582	554	7	)	)	PUNCT
ejpam-4582	554	8	=	=	PUNCT
ejpam-4582	555	1	[	[	X
ejpam-4582	555	2	a∪	a∪	NOUN
ejpam-4582	555	3	[	[	X
ejpam-4582	555	4	x−a(λ	x−a(λ	NOUN
ejpam-4582	555	5	,	,	PUNCT
ejpam-4582	555	6	s)]](λ	s)]](λ	PROPN
ejpam-4582	555	7	,	,	PUNCT
ejpam-4582	555	8	s	s	PART
ejpam-4582	555	9	)	)	PUNCT
ejpam-4582	555	10	=	=	SYM
ejpam-4582	555	11	x.	x.	NOUN
ejpam-4582	555	12	therefore	therefore	ADV
ejpam-4582	555	13	,	,	PUNCT
ejpam-4582	555	14	x−	x−	PROPN
ejpam-4582	556	1	[	[	X
ejpam-4582	556	2	a(λ	a(λ	ADV
ejpam-4582	556	3	,	,	PUNCT
ejpam-4582	556	4	s)−a	s)−a	PROPN
ejpam-4582	556	5	]	]	X
ejpam-4582	556	6	is	be	AUX
ejpam-4582	556	7	(	(	PUNCT
ejpam-4582	556	8	λ	λ	X
ejpam-4582	556	9	,	,	PUNCT
ejpam-4582	556	10	s)-dense	s)-dense	NOUN
ejpam-4582	556	11	and	and	CCONJ
ejpam-4582	556	12	so	so	ADV
ejpam-4582	556	13	x−	x−	PROPN
ejpam-4582	557	1	[	[	X
ejpam-4582	557	2	a(λ	a(λ	ADV
ejpam-4582	557	3	,	,	PUNCT
ejpam-4582	557	4	s)−a	s)−a	PROPN
ejpam-4582	557	5	]	]	X
ejpam-4582	557	6	is	be	AUX
ejpam-4582	557	7	(	(	PUNCT
ejpam-4582	557	8	λ	λ	X
ejpam-4582	557	9	,	,	PUNCT
ejpam-4582	557	10	s)-open	s)-open	ADJ
ejpam-4582	557	11	.	.	PUNCT
ejpam-4582	558	1	thus	thus	ADV
ejpam-4582	558	2	,	,	PUNCT
ejpam-4582	558	3	x−	x−	PROPN
ejpam-4582	559	1	[	[	X
ejpam-4582	559	2	a(λ	a(λ	ADV
ejpam-4582	559	3	,	,	PUNCT
ejpam-4582	559	4	s)−a	s)−a	NOUN
ejpam-4582	559	5	]	]	X
ejpam-4582	559	6	=	=	PUNCT
ejpam-4582	559	7	a∪	a∪	PROPN
ejpam-4582	560	1	[	[	X
ejpam-4582	560	2	x−a(λ	x−a(λ	NOUN
ejpam-4582	560	3	,	,	PUNCT
ejpam-4582	560	4	s	s	PART
ejpam-4582	560	5	)	)	PUNCT
ejpam-4582	560	6	]	]	PUNCT
ejpam-4582	560	7	is	be	AUX
ejpam-4582	560	8	(	(	PUNCT
ejpam-4582	560	9	λ	λ	X
ejpam-4582	560	10	,	,	PUNCT
ejpam-4582	560	11	s)-open	s)-open	ADJ
ejpam-4582	560	12	.	.	PUNCT
ejpam-4582	561	1	this	this	PRON
ejpam-4582	561	2	shows	show	VERB
ejpam-4582	561	3	that	that	SCONJ
ejpam-4582	561	4	a	a	PRON
ejpam-4582	561	5	=	=	X
ejpam-4582	562	1	[	[	X
ejpam-4582	562	2	a	a	DET
ejpam-4582	562	3	∪	∪	ADJ
ejpam-4582	562	4	[	[	X
ejpam-4582	562	5	x	x	X
ejpam-4582	562	6	−a(λ	−a(λ	NOUN
ejpam-4582	562	7	,	,	PUNCT
ejpam-4582	562	8	s	s	NOUN
ejpam-4582	562	9	)	)	PUNCT
ejpam-4582	562	10	]	]	X
ejpam-4582	562	11	]	]	X
ejpam-4582	562	12	∩a(λ	∩a(λ	NOUN
ejpam-4582	562	13	,	,	PUNCT
ejpam-4582	562	14	s	s	PART
ejpam-4582	562	15	)	)	PUNCT
ejpam-4582	562	16	is	be	AUX
ejpam-4582	562	17	locally	locally	ADV
ejpam-4582	562	18	(	(	PUNCT
ejpam-4582	562	19	λ	λ	X
ejpam-4582	562	20	,	,	PUNCT
ejpam-4582	562	21	s)-closed	s)-close	VERB
ejpam-4582	562	22	.	.	PUNCT
ejpam-4582	563	1	(	(	PUNCT
ejpam-4582	563	2	2	2	X
ejpam-4582	563	3	)	)	PUNCT
ejpam-4582	563	4	⇔	⇔	X
ejpam-4582	563	5	(	(	PUNCT
ejpam-4582	563	6	3	3	NUM
ejpam-4582	563	7	):	):	PUNCT
ejpam-4582	563	8	suppose	suppose	VERB
ejpam-4582	563	9	that	that	SCONJ
ejpam-4582	563	10	every	every	DET
ejpam-4582	563	11	subset	subset	NOUN
ejpam-4582	563	12	of	of	ADP
ejpam-4582	563	13	x	x	PUNCT
ejpam-4582	563	14	is	be	AUX
ejpam-4582	563	15	a	a	DET
ejpam-4582	563	16	locally	locally	ADV
ejpam-4582	563	17	(	(	PUNCT
ejpam-4582	563	18	λ	λ	NOUN
ejpam-4582	563	19	,	,	PUNCT
ejpam-4582	563	20	s)-closed	s)-close	VERB
ejpam-4582	563	21	set	set	NOUN
ejpam-4582	563	22	.	.	PUNCT
ejpam-4582	564	1	let	let	VERB
ejpam-4582	564	2	a	a	DET
ejpam-4582	564	3	be	be	AUX
ejpam-4582	564	4	any	any	DET
ejpam-4582	564	5	subset	subset	NOUN
ejpam-4582	564	6	of	of	ADP
ejpam-4582	564	7	x.	x.	NOUN
ejpam-4582	564	8	by	by	ADP
ejpam-4582	564	9	(	(	PUNCT
ejpam-4582	564	10	2	2	NUM
ejpam-4582	564	11	)	)	PUNCT
ejpam-4582	564	12	,	,	PUNCT
ejpam-4582	564	13	we	we	PRON
ejpam-4582	564	14	have	have	VERB
ejpam-4582	564	15	x	x	ADJ
ejpam-4582	564	16	−	−	PROPN
ejpam-4582	564	17	a	a	DET
ejpam-4582	564	18	=	=	X
ejpam-4582	564	19	u	u	NOUN
ejpam-4582	564	20	∩	∩	NOUN
ejpam-4582	564	21	f	f	PROPN
ejpam-4582	564	22	,	,	PUNCT
ejpam-4582	564	23	where	where	SCONJ
ejpam-4582	564	24	u	u	NOUN
ejpam-4582	564	25	is	be	AUX
ejpam-4582	564	26	a	a	DET
ejpam-4582	564	27	(	(	PUNCT
ejpam-4582	564	28	λ	λ	X
ejpam-4582	564	29	,	,	PUNCT
ejpam-4582	564	30	s)-open	s)-open	PUNCT
ejpam-4582	564	31	set	set	VERB
ejpam-4582	564	32	and	and	CCONJ
ejpam-4582	564	33	f	f	PROPN
ejpam-4582	564	34	is	be	AUX
ejpam-4582	564	35	a	a	DET
ejpam-4582	564	36	(	(	PUNCT
ejpam-4582	564	37	λ	λ	X
ejpam-4582	564	38	,	,	PUNCT
ejpam-4582	564	39	s)-closed	s)-close	VERB
ejpam-4582	564	40	set	set	NOUN
ejpam-4582	564	41	.	.	PUNCT
ejpam-4582	565	1	this	this	PRON
ejpam-4582	565	2	implies	imply	VERB
ejpam-4582	565	3	that	that	SCONJ
ejpam-4582	565	4	a	a	PRON
ejpam-4582	565	5	=	=	X
ejpam-4582	565	6	(	(	PUNCT
ejpam-4582	565	7	x−u)∪	x−u)∪	PROPN
ejpam-4582	565	8	(	(	PUNCT
ejpam-4582	565	9	x−k	x−k	PROPN
ejpam-4582	565	10	)	)	PUNCT
ejpam-4582	565	11	,	,	PUNCT
ejpam-4582	565	12	where	where	SCONJ
ejpam-4582	565	13	x−u	x−u	PROPN
ejpam-4582	565	14	is	be	AUX
ejpam-4582	565	15	a	a	DET
ejpam-4582	565	16	(	(	PUNCT
ejpam-4582	565	17	λ	λ	X
ejpam-4582	565	18	,	,	PUNCT
ejpam-4582	565	19	s)-closed	s)-close	VERB
ejpam-4582	565	20	set	set	NOUN
ejpam-4582	565	21	and	and	CCONJ
ejpam-4582	565	22	x	x	SYM
ejpam-4582	565	23	−	−	PROPN
ejpam-4582	565	24	f	f	PROPN
ejpam-4582	565	25	is	be	AUX
ejpam-4582	565	26	a	a	DET
ejpam-4582	565	27	(	(	PUNCT
ejpam-4582	565	28	λ	λ	X
ejpam-4582	565	29	,	,	PUNCT
ejpam-4582	565	30	s)-open	s)-open	PUNCT
ejpam-4582	565	31	set	set	VERB
ejpam-4582	565	32	.	.	PUNCT
ejpam-4582	566	1	the	the	DET
ejpam-4582	566	2	converse	converse	NOUN
ejpam-4582	566	3	is	be	AUX
ejpam-4582	566	4	similar	similar	ADJ
ejpam-4582	566	5	.	.	PUNCT
ejpam-4582	567	1	(	(	PUNCT
ejpam-4582	567	2	2	2	X
ejpam-4582	567	3	)	)	PUNCT
ejpam-4582	567	4	⇒	⇒	NOUN
ejpam-4582	567	5	(	(	PUNCT
ejpam-4582	567	6	4	4	NUM
ejpam-4582	567	7	)	)	PUNCT
ejpam-4582	567	8	and	and	CCONJ
ejpam-4582	567	9	(	(	PUNCT
ejpam-4582	567	10	4	4	X
ejpam-4582	567	11	)	)	PUNCT
ejpam-4582	567	12	⇔	⇔	X
ejpam-4582	567	13	(	(	PUNCT
ejpam-4582	567	14	5	5	NUM
ejpam-4582	567	15	)	)	PUNCT
ejpam-4582	567	16	are	be	AUX
ejpam-4582	567	17	obvious	obvious	ADJ
ejpam-4582	567	18	.	.	PUNCT
ejpam-4582	568	1	(	(	PUNCT
ejpam-4582	568	2	4	4	X
ejpam-4582	568	3	)	)	PUNCT
ejpam-4582	568	4	⇒	⇒	NOUN
ejpam-4582	568	5	(	(	PUNCT
ejpam-4582	568	6	1	1	NUM
ejpam-4582	568	7	):	):	PUNCT
ejpam-4582	568	8	let	let	VERB
ejpam-4582	568	9	a	a	PRON
ejpam-4582	568	10	be	be	AUX
ejpam-4582	568	11	a	a	DET
ejpam-4582	568	12	(	(	PUNCT
ejpam-4582	568	13	λ	λ	NOUN
ejpam-4582	568	14	,	,	PUNCT
ejpam-4582	568	15	s)-dense	s)-dense	NOUN
ejpam-4582	568	16	set	set	NOUN
ejpam-4582	568	17	.	.	PUNCT
ejpam-4582	569	1	by	by	ADP
ejpam-4582	569	2	(	(	PUNCT
ejpam-4582	569	3	4	4	NUM
ejpam-4582	569	4	)	)	PUNCT
ejpam-4582	569	5	,	,	PUNCT
ejpam-4582	569	6	there	there	PRON
ejpam-4582	569	7	exist	exist	VERB
ejpam-4582	569	8	a	a	DET
ejpam-4582	569	9	(	(	PUNCT
ejpam-4582	569	10	λ	λ	NOUN
ejpam-4582	569	11	,	,	PUNCT
ejpam-4582	569	12	s)-open	s)-open	VERB
ejpam-4582	569	13	set	set	VERB
ejpam-4582	569	14	u	u	NOUN
ejpam-4582	569	15	and	and	CCONJ
ejpam-4582	569	16	a	a	DET
ejpam-4582	569	17	(	(	PUNCT
ejpam-4582	569	18	λ	λ	NOUN
ejpam-4582	569	19	,	,	PUNCT
ejpam-4582	569	20	s)-closed	s)-close	VERB
ejpam-4582	569	21	set	set	NOUN
ejpam-4582	569	22	f	f	PRON
ejpam-4582	569	23	such	such	ADJ
ejpam-4582	569	24	that	that	SCONJ
ejpam-4582	569	25	a	a	DET
ejpam-4582	569	26	=	=	X
ejpam-4582	569	27	u	u	NOUN
ejpam-4582	569	28	∩	∩	NOUN
ejpam-4582	569	29	f	f	PROPN
ejpam-4582	569	30	.	.	PUNCT
ejpam-4582	570	1	since	since	SCONJ
ejpam-4582	570	2	a	a	DET
ejpam-4582	570	3	⊆	⊆	NUM
ejpam-4582	570	4	f	f	NOUN
ejpam-4582	570	5	and	and	CCONJ
ejpam-4582	570	6	a	a	PRON
ejpam-4582	570	7	is	be	AUX
ejpam-4582	570	8	a	a	DET
ejpam-4582	570	9	(	(	PUNCT
ejpam-4582	570	10	λ	λ	NOUN
ejpam-4582	570	11	,	,	PUNCT
ejpam-4582	570	12	s)-dense	s)-dense	NOUN
ejpam-4582	570	13	set	set	VERB
ejpam-4582	570	14	,	,	PUNCT
ejpam-4582	570	15	x	x	X
ejpam-4582	570	16	⊆	⊆	NUM
ejpam-4582	570	17	f	f	NOUN
ejpam-4582	570	18	.	.	PUNCT
ejpam-4582	571	1	thus	thus	ADV
ejpam-4582	571	2	,	,	PUNCT
ejpam-4582	571	3	f	f	PROPN
ejpam-4582	571	4	=	=	PUNCT
ejpam-4582	571	5	x	x	X
ejpam-4582	571	6	and	and	CCONJ
ejpam-4582	571	7	hence	hence	ADV
ejpam-4582	571	8	a	a	DET
ejpam-4582	571	9	=	=	X
ejpam-4582	571	10	u	u	NOUN
ejpam-4582	571	11	is	be	AUX
ejpam-4582	571	12	(	(	PUNCT
ejpam-4582	571	13	λ	λ	X
ejpam-4582	571	14	,	,	PUNCT
ejpam-4582	571	15	s)-open	s)-open	ADJ
ejpam-4582	571	16	.	.	PUNCT
ejpam-4582	572	1	this	this	PRON
ejpam-4582	572	2	shows	show	VERB
ejpam-4582	572	3	that	that	SCONJ
ejpam-4582	572	4	(	(	PUNCT
ejpam-4582	572	5	x	x	X
ejpam-4582	572	6	,	,	PUNCT
ejpam-4582	572	7	τ	τ	X
ejpam-4582	572	8	)	)	PUNCT
ejpam-4582	572	9	is	be	AUX
ejpam-4582	572	10	(	(	PUNCT
ejpam-4582	572	11	λ	λ	X
ejpam-4582	572	12	,	,	PUNCT
ejpam-4582	572	13	s)-submaximal	s)-submaximal	ADJ
ejpam-4582	572	14	.	.	PUNCT
ejpam-4582	573	1	definition	definition	NOUN
ejpam-4582	573	2	14	14	NUM
ejpam-4582	573	3	.	.	PUNCT
ejpam-4582	574	1	a	a	DET
ejpam-4582	574	2	subset	subset	NOUN
ejpam-4582	574	3	a	a	PRON
ejpam-4582	574	4	of	of	ADP
ejpam-4582	574	5	a	a	DET
ejpam-4582	574	6	topological	topological	ADJ
ejpam-4582	574	7	space	space	NOUN
ejpam-4582	574	8	(	(	PUNCT
ejpam-4582	574	9	x	x	X
ejpam-4582	574	10	,	,	PUNCT
ejpam-4582	574	11	τ	τ	X
ejpam-4582	574	12	)	)	PUNCT
ejpam-4582	574	13	is	be	AUX
ejpam-4582	574	14	said	say	VERB
ejpam-4582	574	15	to	to	PART
ejpam-4582	574	16	be	be	AUX
ejpam-4582	574	17	:	:	PUNCT
ejpam-4582	574	18	(	(	PUNCT
ejpam-4582	574	19	i	i	NOUN
ejpam-4582	574	20	)	)	PUNCT
ejpam-4582	574	21	a	a	DET
ejpam-4582	574	22	γ(λ	γ(λ	NOUN
ejpam-4582	574	23	,	,	PUNCT
ejpam-4582	574	24	s)-set	s)-set	VERB
ejpam-4582	574	25	if	if	SCONJ
ejpam-4582	574	26	a	a	DET
ejpam-4582	574	27	=	=	X
ejpam-4582	574	28	γ(λ	γ(λ	NOUN
ejpam-4582	574	29	,	,	PUNCT
ejpam-4582	574	30	s)(a	s)(a	NUM
ejpam-4582	574	31	)	)	PUNCT
ejpam-4582	574	32	;	;	PUNCT
ejpam-4582	574	33	(	(	PUNCT
ejpam-4582	574	34	ii	ii	NOUN
ejpam-4582	574	35	)	)	PUNCT
ejpam-4582	574	36	a	a	DET
ejpam-4582	574	37	⊛γ(λ	⊛γ(λ	PROPN
ejpam-4582	574	38	,	,	PUNCT
ejpam-4582	574	39	s)-set	s)-set	VERB
ejpam-4582	574	40	if	if	SCONJ
ejpam-4582	574	41	γ(λ	γ(λ	PROPN
ejpam-4582	574	42	,	,	PUNCT
ejpam-4582	574	43	s)(a	s)(a	NUM
ejpam-4582	574	44	)	)	PUNCT
ejpam-4582	575	1	⊆	⊆	NUM
ejpam-4582	575	2	f	f	NOUN
ejpam-4582	575	3	whenever	whenever	SCONJ
ejpam-4582	575	4	a	a	DET
ejpam-4582	575	5	⊆	⊆	NUM
ejpam-4582	575	6	f	f	PROPN
ejpam-4582	575	7	and	and	CCONJ
ejpam-4582	575	8	f	f	PROPN
ejpam-4582	575	9	is	be	AUX
ejpam-4582	575	10	a	a	DET
ejpam-4582	575	11	(	(	PUNCT
ejpam-4582	575	12	λ	λ	X
ejpam-4582	575	13	,	,	PUNCT
ejpam-4582	575	14	s)-closed	s)-close	VERB
ejpam-4582	575	15	set	set	NOUN
ejpam-4582	575	16	.	.	PUNCT
ejpam-4582	576	1	definition	definition	NOUN
ejpam-4582	576	2	15	15	NUM
ejpam-4582	576	3	.	.	PUNCT
ejpam-4582	577	1	a	a	DET
ejpam-4582	577	2	topological	topological	ADJ
ejpam-4582	577	3	space	space	NOUN
ejpam-4582	577	4	(	(	PUNCT
ejpam-4582	577	5	x	x	X
ejpam-4582	577	6	,	,	PUNCT
ejpam-4582	577	7	τ	τ	X
ejpam-4582	577	8	)	)	PUNCT
ejpam-4582	577	9	is	be	AUX
ejpam-4582	577	10	called	call	VERB
ejpam-4582	577	11	(	(	PUNCT
ejpam-4582	577	12	λ	λ	PROPN
ejpam-4582	577	13	,	,	PUNCT
ejpam-4582	577	14	s)-t	s)-t	VERB
ejpam-4582	577	15	1	1	NUM
ejpam-4582	577	16	2	2	NUM
ejpam-4582	577	17	if	if	SCONJ
ejpam-4582	577	18	every	every	DET
ejpam-4582	577	19	g-(λ	g-(λ	PROPN
ejpam-4582	577	20	,	,	PUNCT
ejpam-4582	577	21	s)-closed	s)-close	VERB
ejpam-4582	577	22	set	set	NOUN
ejpam-4582	577	23	of	of	ADP
ejpam-4582	577	24	x	x	PUNCT
ejpam-4582	577	25	is	be	AUX
ejpam-4582	577	26	(	(	PUNCT
ejpam-4582	577	27	λ	λ	X
ejpam-4582	577	28	,	,	PUNCT
ejpam-4582	577	29	s)-closed	s)-close	VERB
ejpam-4582	577	30	.	.	PUNCT
ejpam-4582	578	1	lemma	lemma	PROPN
ejpam-4582	578	2	8	8	NUM
ejpam-4582	578	3	.	.	PUNCT
ejpam-4582	579	1	for	for	ADP
ejpam-4582	579	2	a	a	DET
ejpam-4582	579	3	topological	topological	ADJ
ejpam-4582	579	4	space	space	NOUN
ejpam-4582	579	5	(	(	PUNCT
ejpam-4582	579	6	x	x	X
ejpam-4582	579	7	,	,	PUNCT
ejpam-4582	579	8	τ	τ	PROPN
ejpam-4582	579	9	)	)	PUNCT
ejpam-4582	579	10	,	,	PUNCT
ejpam-4582	579	11	the	the	DET
ejpam-4582	579	12	following	follow	VERB
ejpam-4582	579	13	properties	property	NOUN
ejpam-4582	579	14	hold	hold	VERB
ejpam-4582	579	15	:	:	PUNCT
ejpam-4582	579	16	(	(	PUNCT
ejpam-4582	579	17	1	1	X
ejpam-4582	579	18	)	)	PUNCT
ejpam-4582	579	19	for	for	ADP
ejpam-4582	579	20	each	each	DET
ejpam-4582	579	21	x	x	SYM
ejpam-4582	579	22	∈	∈	PROPN
ejpam-4582	579	23	x	x	NOUN
ejpam-4582	579	24	,	,	PUNCT
ejpam-4582	579	25	the	the	DET
ejpam-4582	579	26	singleton	singleton	NOUN
ejpam-4582	579	27	{	{	PUNCT
ejpam-4582	579	28	x	x	NOUN
ejpam-4582	579	29	}	}	PUNCT
ejpam-4582	579	30	is	be	AUX
ejpam-4582	579	31	(	(	PUNCT
ejpam-4582	579	32	λ	λ	X
ejpam-4582	579	33	,	,	PUNCT
ejpam-4582	579	34	s)-closed	s)-close	VERB
ejpam-4582	579	35	or	or	CCONJ
ejpam-4582	579	36	x	x	SYM
ejpam-4582	579	37	−	−	PROPN
ejpam-4582	579	38	{	{	PUNCT
ejpam-4582	579	39	x	x	NOUN
ejpam-4582	579	40	}	}	PUNCT
ejpam-4582	579	41	is	be	AUX
ejpam-4582	579	42	g-(λ	g-(λ	PROPN
ejpam-4582	579	43	,	,	PUNCT
ejpam-4582	579	44	s)-closed	s)-close	VERB
ejpam-4582	579	45	;	;	PUNCT
ejpam-4582	579	46	(	(	PUNCT
ejpam-4582	579	47	2	2	X
ejpam-4582	579	48	)	)	PUNCT
ejpam-4582	579	49	for	for	ADP
ejpam-4582	579	50	each	each	DET
ejpam-4582	579	51	x	x	SYM
ejpam-4582	579	52	∈	∈	PROPN
ejpam-4582	579	53	x	x	NOUN
ejpam-4582	579	54	,	,	PUNCT
ejpam-4582	579	55	the	the	DET
ejpam-4582	579	56	singleton	singleton	NOUN
ejpam-4582	579	57	{	{	PUNCT
ejpam-4582	579	58	x	x	NOUN
ejpam-4582	579	59	}	}	PUNCT
ejpam-4582	579	60	is	be	AUX
ejpam-4582	579	61	(	(	PUNCT
ejpam-4582	579	62	λ	λ	X
ejpam-4582	579	63	,	,	PUNCT
ejpam-4582	579	64	s)-open	s)-open	PUNCT
ejpam-4582	579	65	or	or	CCONJ
ejpam-4582	579	66	x	x	X
ejpam-4582	579	67	−	−	PROPN
ejpam-4582	579	68	{	{	PUNCT
ejpam-4582	579	69	x	x	NOUN
ejpam-4582	579	70	}	}	PUNCT
ejpam-4582	579	71	is	be	AUX
ejpam-4582	579	72	a	a	DET
ejpam-4582	579	73	⊛γ(λ	⊛γ(λ	PROPN
ejpam-4582	579	74	,	,	PUNCT
ejpam-4582	579	75	s)-set	s)-set	NOUN
ejpam-4582	579	76	.	.	PUNCT
ejpam-4582	580	1	proof	proof	NOUN
ejpam-4582	580	2	.	.	PUNCT
ejpam-4582	581	1	(	(	PUNCT
ejpam-4582	581	2	1	1	X
ejpam-4582	581	3	)	)	PUNCT
ejpam-4582	581	4	let	let	VERB
ejpam-4582	581	5	x	x	SYM
ejpam-4582	581	6	∈	∈	PROPN
ejpam-4582	581	7	x	x	X
ejpam-4582	581	8	and	and	CCONJ
ejpam-4582	581	9	the	the	DET
ejpam-4582	581	10	singleton	singleton	NOUN
ejpam-4582	581	11	{	{	PUNCT
ejpam-4582	581	12	x	x	NOUN
ejpam-4582	581	13	}	}	PUNCT
ejpam-4582	581	14	be	be	VERB
ejpam-4582	581	15	not	not	PART
ejpam-4582	581	16	(	(	PUNCT
ejpam-4582	581	17	λ	λ	X
ejpam-4582	581	18	,	,	PUNCT
ejpam-4582	581	19	s)-closed	s)-close	VERB
ejpam-4582	581	20	.	.	PUNCT
ejpam-4582	582	1	then	then	ADV
ejpam-4582	582	2	,	,	PUNCT
ejpam-4582	582	3	we	we	PRON
ejpam-4582	582	4	have	have	VERB
ejpam-4582	582	5	x	x	PART
ejpam-4582	582	6	−	−	X
ejpam-4582	582	7	{	{	PUNCT
ejpam-4582	582	8	x	x	NOUN
ejpam-4582	582	9	}	}	PUNCT
ejpam-4582	582	10	is	be	AUX
ejpam-4582	582	11	not	not	PART
ejpam-4582	582	12	(	(	PUNCT
ejpam-4582	582	13	λ	λ	X
ejpam-4582	582	14	,	,	PUNCT
ejpam-4582	582	15	s)-open	s)-open	PUNCT
ejpam-4582	582	16	and	and	CCONJ
ejpam-4582	582	17	x	x	X
ejpam-4582	582	18	is	be	AUX
ejpam-4582	582	19	the	the	DET
ejpam-4582	582	20	only	only	ADJ
ejpam-4582	582	21	(	(	PUNCT
ejpam-4582	582	22	λ	λ	NOUN
ejpam-4582	582	23	,	,	PUNCT
ejpam-4582	582	24	s)-open	s)-open	PUNCT
ejpam-4582	582	25	set	set	VERB
ejpam-4582	582	26	which	which	PRON
ejpam-4582	582	27	contains	contain	VERB
ejpam-4582	582	28	x	x	X
ejpam-4582	582	29	−	−	PROPN
ejpam-4582	582	30	{	{	PUNCT
ejpam-4582	582	31	x	x	NOUN
ejpam-4582	582	32	}	}	PUNCT
ejpam-4582	582	33	and	and	CCONJ
ejpam-4582	582	34	hence	hence	ADV
ejpam-4582	582	35	x	x	X
ejpam-4582	582	36	−	−	PROPN
ejpam-4582	582	37	{	{	PUNCT
ejpam-4582	582	38	x	x	NOUN
ejpam-4582	582	39	}	}	PUNCT
ejpam-4582	582	40	is	be	AUX
ejpam-4582	582	41	g-(λ	g-(λ	PROPN
ejpam-4582	582	42	,	,	PUNCT
ejpam-4582	582	43	s)-closed	s)-close	VERB
ejpam-4582	582	44	.	.	PUNCT
ejpam-4582	583	1	(	(	PUNCT
ejpam-4582	583	2	2	2	X
ejpam-4582	583	3	)	)	PUNCT
ejpam-4582	583	4	let	let	VERB
ejpam-4582	583	5	x	x	SYM
ejpam-4582	583	6	∈	∈	PROPN
ejpam-4582	583	7	x	x	X
ejpam-4582	583	8	and	and	CCONJ
ejpam-4582	583	9	the	the	DET
ejpam-4582	583	10	singleton	singleton	NOUN
ejpam-4582	583	11	{	{	PUNCT
ejpam-4582	583	12	x	x	NOUN
ejpam-4582	583	13	}	}	PUNCT
ejpam-4582	583	14	be	be	VERB
ejpam-4582	583	15	not	not	PART
ejpam-4582	583	16	(	(	PUNCT
ejpam-4582	583	17	λ	λ	X
ejpam-4582	583	18	,	,	PUNCT
ejpam-4582	583	19	s)-open	s)-open	VERB
ejpam-4582	583	20	.	.	PUNCT
ejpam-4582	584	1	then	then	ADV
ejpam-4582	584	2	,	,	PUNCT
ejpam-4582	584	3	we	we	PRON
ejpam-4582	584	4	have	have	VERB
ejpam-4582	584	5	x	x	PART
ejpam-4582	584	6	−	−	X
ejpam-4582	584	7	{	{	PUNCT
ejpam-4582	584	8	x	x	NOUN
ejpam-4582	584	9	}	}	PUNCT
ejpam-4582	584	10	is	be	AUX
ejpam-4582	584	11	not	not	PART
ejpam-4582	584	12	(	(	PUNCT
ejpam-4582	584	13	λ	λ	X
ejpam-4582	584	14	,	,	PUNCT
ejpam-4582	584	15	s)-closed	s)-close	VERB
ejpam-4582	584	16	and	and	CCONJ
ejpam-4582	584	17	the	the	DET
ejpam-4582	584	18	only	only	ADJ
ejpam-4582	584	19	(	(	PUNCT
ejpam-4582	584	20	λ	λ	X
ejpam-4582	584	21	,	,	PUNCT
ejpam-4582	584	22	s)-closed	s)-close	VERB
ejpam-4582	584	23	set	set	NOUN
ejpam-4582	584	24	which	which	PRON
ejpam-4582	584	25	contains	contain	VERB
ejpam-4582	584	26	x	x	X
ejpam-4582	584	27	−	−	PROPN
ejpam-4582	584	28	{	{	PUNCT
ejpam-4582	584	29	x	x	NOUN
ejpam-4582	584	30	}	}	PUNCT
ejpam-4582	584	31	is	be	AUX
ejpam-4582	584	32	x	x	PUNCT
ejpam-4582	584	33	and	and	CCONJ
ejpam-4582	584	34	hence	hence	ADV
ejpam-4582	584	35	x	x	X
ejpam-4582	584	36	−	−	PROPN
ejpam-4582	584	37	{	{	PUNCT
ejpam-4582	584	38	x	x	NOUN
ejpam-4582	584	39	}	}	PUNCT
ejpam-4582	584	40	is	be	AUX
ejpam-4582	584	41	a	a	DET
ejpam-4582	584	42	⊛γ(λ	⊛γ(λ	PROPN
ejpam-4582	584	43	,	,	PUNCT
ejpam-4582	584	44	s)-set	s)-set	NOUN
ejpam-4582	584	45	.	.	PUNCT
ejpam-4582	585	1	theorem	theorem	VERB
ejpam-4582	585	2	22	22	NUM
ejpam-4582	585	3	.	.	PUNCT
ejpam-4582	586	1	for	for	ADP
ejpam-4582	586	2	a	a	DET
ejpam-4582	586	3	topological	topological	ADJ
ejpam-4582	586	4	space	space	NOUN
ejpam-4582	586	5	(	(	PUNCT
ejpam-4582	586	6	x	x	X
ejpam-4582	586	7	,	,	PUNCT
ejpam-4582	586	8	τ	τ	PROPN
ejpam-4582	586	9	)	)	PUNCT
ejpam-4582	586	10	,	,	PUNCT
ejpam-4582	586	11	the	the	DET
ejpam-4582	586	12	following	follow	VERB
ejpam-4582	586	13	properties	property	NOUN
ejpam-4582	586	14	are	be	AUX
ejpam-4582	586	15	equivalent	equivalent	ADJ
ejpam-4582	586	16	:	:	PUNCT
ejpam-4582	586	17	c.	c.	PROPN
ejpam-4582	586	18	boonpok	boonpok	PROPN
ejpam-4582	586	19	,	,	PUNCT
ejpam-4582	586	20	c.	c.	PROPN
ejpam-4582	586	21	viriyapong	viriyapong	PROPN
ejpam-4582	586	22	/	/	SYM
ejpam-4582	586	23	eur	eur	PROPN
ejpam-4582	586	24	.	.	PUNCT
ejpam-4582	587	1	j.	j.	PROPN
ejpam-4582	587	2	pure	pure	PROPN
ejpam-4582	587	3	appl	appl	PROPN
ejpam-4582	587	4	.	.	PROPN
ejpam-4582	587	5	math	math	PROPN
ejpam-4582	587	6	,	,	PUNCT
ejpam-4582	587	7	16	16	NUM
ejpam-4582	587	8	(	(	PUNCT
ejpam-4582	587	9	1	1	NUM
ejpam-4582	587	10	)	)	PUNCT
ejpam-4582	587	11	(	(	PUNCT
ejpam-4582	587	12	2023	2023	NUM
ejpam-4582	587	13	)	)	PUNCT
ejpam-4582	587	14	,	,	PUNCT
ejpam-4582	587	15	336	336	NUM
ejpam-4582	587	16	-	-	SYM
ejpam-4582	587	17	362	362	NUM
ejpam-4582	587	18	352	352	NUM
ejpam-4582	587	19	(	(	PUNCT
ejpam-4582	587	20	1	1	NUM
ejpam-4582	587	21	)	)	PUNCT
ejpam-4582	587	22	(	(	PUNCT
ejpam-4582	587	23	x	x	X
ejpam-4582	587	24	,	,	PUNCT
ejpam-4582	587	25	τ	τ	X
ejpam-4582	587	26	)	)	PUNCT
ejpam-4582	587	27	is	be	AUX
ejpam-4582	587	28	(	(	PUNCT
ejpam-4582	587	29	λ	λ	X
ejpam-4582	587	30	,	,	PUNCT
ejpam-4582	587	31	s)-t	s)-t	VERB
ejpam-4582	587	32	1	1	NUM
ejpam-4582	587	33	2	2	NUM
ejpam-4582	587	34	;	;	PUNCT
ejpam-4582	587	35	(	(	PUNCT
ejpam-4582	587	36	2	2	X
ejpam-4582	587	37	)	)	PUNCT
ejpam-4582	587	38	for	for	ADP
ejpam-4582	587	39	each	each	DET
ejpam-4582	587	40	x	x	SYM
ejpam-4582	587	41	∈	∈	PROPN
ejpam-4582	587	42	x	x	NOUN
ejpam-4582	587	43	,	,	PUNCT
ejpam-4582	587	44	the	the	DET
ejpam-4582	587	45	singleton	singleton	NOUN
ejpam-4582	587	46	{	{	PUNCT
ejpam-4582	587	47	x	x	NOUN
ejpam-4582	587	48	}	}	PUNCT
ejpam-4582	587	49	is	be	AUX
ejpam-4582	587	50	(	(	PUNCT
ejpam-4582	587	51	λ	λ	X
ejpam-4582	587	52	,	,	PUNCT
ejpam-4582	587	53	s)-open	s)-open	PUNCT
ejpam-4582	587	54	or	or	CCONJ
ejpam-4582	587	55	(	(	PUNCT
ejpam-4582	587	56	λ	λ	X
ejpam-4582	587	57	,	,	PUNCT
ejpam-4582	587	58	s)-closed	s)-close	VERB
ejpam-4582	587	59	;	;	PUNCT
ejpam-4582	587	60	(	(	PUNCT
ejpam-4582	587	61	3	3	X
ejpam-4582	587	62	)	)	PUNCT
ejpam-4582	587	63	every	every	DET
ejpam-4582	587	64	⊛γ(λ	⊛γ(λ	PROPN
ejpam-4582	587	65	,	,	PUNCT
ejpam-4582	587	66	s)-set	s)-set	VERB
ejpam-4582	587	67	is	be	AUX
ejpam-4582	587	68	a	a	DET
ejpam-4582	587	69	γ(λ	γ(λ	NOUN
ejpam-4582	587	70	,	,	PUNCT
ejpam-4582	587	71	s)-set	s)-set	NOUN
ejpam-4582	587	72	.	.	PUNCT
ejpam-4582	588	1	proof	proof	NOUN
ejpam-4582	588	2	.	.	PUNCT
ejpam-4582	589	1	(	(	PUNCT
ejpam-4582	589	2	1	1	X
ejpam-4582	589	3	)	)	PUNCT
ejpam-4582	589	4	⇒	⇒	NOUN
ejpam-4582	589	5	(	(	PUNCT
ejpam-4582	589	6	2	2	NUM
ejpam-4582	589	7	):	):	PUNCT
ejpam-4582	589	8	by	by	ADP
ejpam-4582	589	9	lemma	lemma	PROPN
ejpam-4582	589	10	8	8	NUM
ejpam-4582	589	11	,	,	PUNCT
ejpam-4582	589	12	for	for	ADP
ejpam-4582	589	13	each	each	DET
ejpam-4582	589	14	x	x	SYM
ejpam-4582	589	15	∈	∈	PROPN
ejpam-4582	589	16	x	x	NOUN
ejpam-4582	589	17	,	,	PUNCT
ejpam-4582	589	18	the	the	DET
ejpam-4582	589	19	singleton	singleton	NOUN
ejpam-4582	589	20	{	{	PUNCT
ejpam-4582	589	21	x	x	NOUN
ejpam-4582	589	22	}	}	PUNCT
ejpam-4582	589	23	is	be	AUX
ejpam-4582	589	24	(	(	PUNCT
ejpam-4582	589	25	λ	λ	X
ejpam-4582	589	26	,	,	PUNCT
ejpam-4582	589	27	s)-closed	s)-close	VERB
ejpam-4582	589	28	or	or	CCONJ
ejpam-4582	589	29	x	x	SYM
ejpam-4582	589	30	−	−	PROPN
ejpam-4582	589	31	{	{	PUNCT
ejpam-4582	589	32	x	x	NOUN
ejpam-4582	589	33	}	}	PUNCT
ejpam-4582	589	34	is	be	AUX
ejpam-4582	589	35	g-(λ	g-(λ	PROPN
ejpam-4582	589	36	,	,	PUNCT
ejpam-4582	589	37	s)-closed	s)-close	VERB
ejpam-4582	589	38	.	.	PUNCT
ejpam-4582	590	1	since	since	SCONJ
ejpam-4582	590	2	(	(	PUNCT
ejpam-4582	590	3	x	x	X
ejpam-4582	590	4	,	,	PUNCT
ejpam-4582	590	5	τ	τ	X
ejpam-4582	590	6	)	)	PUNCT
ejpam-4582	590	7	is	be	AUX
ejpam-4582	590	8	a	a	DET
ejpam-4582	590	9	(	(	PUNCT
ejpam-4582	590	10	λ	λ	NOUN
ejpam-4582	590	11	,	,	PUNCT
ejpam-4582	590	12	s)-t	s)-t	VERB
ejpam-4582	590	13	1	1	NUM
ejpam-4582	590	14	2	2	NUM
ejpam-4582	590	15	-space	-space	NOUN
ejpam-4582	590	16	,	,	PUNCT
ejpam-4582	590	17	x	x	PUNCT
ejpam-4582	590	18	−	−	NOUN
ejpam-4582	590	19	{	{	PUNCT
ejpam-4582	590	20	x	x	NOUN
ejpam-4582	590	21	}	}	PUNCT
ejpam-4582	590	22	is	be	AUX
ejpam-4582	590	23	(	(	PUNCT
ejpam-4582	590	24	λ	λ	X
ejpam-4582	590	25	,	,	PUNCT
ejpam-4582	590	26	s)-closed	s)-close	VERB
ejpam-4582	590	27	and	and	CCONJ
ejpam-4582	590	28	hence	hence	ADV
ejpam-4582	590	29	{	{	PUNCT
ejpam-4582	590	30	x	x	X
ejpam-4582	590	31	}	}	PUNCT
ejpam-4582	590	32	is	be	AUX
ejpam-4582	590	33	(	(	PUNCT
ejpam-4582	590	34	λ	λ	X
ejpam-4582	590	35	,	,	PUNCT
ejpam-4582	590	36	s)-open	s)-open	VERB
ejpam-4582	590	37	in	in	ADP
ejpam-4582	590	38	the	the	DET
ejpam-4582	590	39	latter	latter	ADJ
ejpam-4582	590	40	case	case	NOUN
ejpam-4582	590	41	.	.	PUNCT
ejpam-4582	591	1	therefore	therefore	ADV
ejpam-4582	591	2	,	,	PUNCT
ejpam-4582	591	3	the	the	DET
ejpam-4582	591	4	singleton	singleton	NOUN
ejpam-4582	591	5	{	{	PUNCT
ejpam-4582	591	6	x	x	NOUN
ejpam-4582	591	7	}	}	PUNCT
ejpam-4582	591	8	is	be	AUX
ejpam-4582	591	9	(	(	PUNCT
ejpam-4582	591	10	λ	λ	X
ejpam-4582	591	11	,	,	PUNCT
ejpam-4582	591	12	s)-open	s)-open	PUNCT
ejpam-4582	591	13	or	or	CCONJ
ejpam-4582	591	14	(	(	PUNCT
ejpam-4582	591	15	λ	λ	X
ejpam-4582	591	16	,	,	PUNCT
ejpam-4582	591	17	s)-closed	s)-close	VERB
ejpam-4582	591	18	.	.	PUNCT
ejpam-4582	592	1	(	(	PUNCT
ejpam-4582	592	2	2	2	X
ejpam-4582	592	3	)	)	PUNCT
ejpam-4582	592	4	⇒	⇒	NOUN
ejpam-4582	592	5	(	(	PUNCT
ejpam-4582	592	6	3	3	NUM
ejpam-4582	592	7	):	):	PUNCT
ejpam-4582	592	8	suppose	suppose	VERB
ejpam-4582	592	9	that	that	SCONJ
ejpam-4582	592	10	there	there	PRON
ejpam-4582	592	11	exists	exist	VERB
ejpam-4582	592	12	a	a	DET
ejpam-4582	592	13	⊛γ(λ	⊛γ(λ	PROPN
ejpam-4582	592	14	,	,	PUNCT
ejpam-4582	592	15	s)-set	s)-set	VERB
ejpam-4582	592	16	a	a	PRON
ejpam-4582	592	17	which	which	PRON
ejpam-4582	592	18	is	be	AUX
ejpam-4582	592	19	not	not	PART
ejpam-4582	592	20	a	a	DET
ejpam-4582	592	21	γ(λ	γ(λ	NOUN
ejpam-4582	592	22	,	,	PUNCT
ejpam-4582	592	23	s)-set	s)-set	NOUN
ejpam-4582	592	24	.	.	PUNCT
ejpam-4582	593	1	there	there	PRON
ejpam-4582	593	2	exists	exist	VERB
ejpam-4582	593	3	x	x	X
ejpam-4582	593	4	∈	∈	PROPN
ejpam-4582	593	5	γ(λ	γ(λ	PROPN
ejpam-4582	593	6	,	,	PUNCT
ejpam-4582	593	7	s)(a	s)(a	NUM
ejpam-4582	593	8	)	)	PUNCT
ejpam-4582	593	9	such	such	ADJ
ejpam-4582	593	10	that	that	SCONJ
ejpam-4582	593	11	x	x	PUNCT
ejpam-4582	593	12	̸∈	̸∈	PROPN
ejpam-4582	593	13	a.	a.	NOUN
ejpam-4582	593	14	in	in	ADP
ejpam-4582	593	15	case	case	NOUN
ejpam-4582	593	16	the	the	DET
ejpam-4582	593	17	singleton	singleton	NOUN
ejpam-4582	593	18	{	{	PUNCT
ejpam-4582	593	19	x	x	NOUN
ejpam-4582	593	20	}	}	PUNCT
ejpam-4582	593	21	is	be	AUX
ejpam-4582	593	22	(	(	PUNCT
ejpam-4582	593	23	λ	λ	X
ejpam-4582	593	24	,	,	PUNCT
ejpam-4582	593	25	s)-open	s)-open	PUNCT
ejpam-4582	593	26	,	,	PUNCT
ejpam-4582	593	27	a	a	DET
ejpam-4582	593	28	⊆	⊆	NUM
ejpam-4582	593	29	x−{x	x−{x	PROPN
ejpam-4582	593	30	}	}	PUNCT
ejpam-4582	593	31	and	and	CCONJ
ejpam-4582	593	32	x	x	ADJ
ejpam-4582	593	33	−	−	PROPN
ejpam-4582	593	34	{	{	PUNCT
ejpam-4582	593	35	x	x	NOUN
ejpam-4582	593	36	}	}	PUNCT
ejpam-4582	593	37	is	be	AUX
ejpam-4582	593	38	(	(	PUNCT
ejpam-4582	593	39	λ	λ	X
ejpam-4582	593	40	,	,	PUNCT
ejpam-4582	593	41	s)-closed	s)-close	VERB
ejpam-4582	593	42	.	.	PUNCT
ejpam-4582	594	1	since	since	SCONJ
ejpam-4582	594	2	a	a	PRON
ejpam-4582	594	3	is	be	AUX
ejpam-4582	594	4	a	a	DET
ejpam-4582	594	5	⊛γ(λ	⊛γ(λ	PROPN
ejpam-4582	594	6	,	,	PUNCT
ejpam-4582	594	7	s)-set	s)-set	NOUN
ejpam-4582	594	8	,	,	PUNCT
ejpam-4582	594	9	γ(λ	γ(λ	PROPN
ejpam-4582	594	10	,	,	PUNCT
ejpam-4582	594	11	s)(a	s)(a	NUM
ejpam-4582	594	12	)	)	PUNCT
ejpam-4582	595	1	⊆	⊆	NUM
ejpam-4582	595	2	x	x	SYM
ejpam-4582	595	3	−	−	PROPN
ejpam-4582	595	4	{	{	PUNCT
ejpam-4582	595	5	x	x	NOUN
ejpam-4582	595	6	}	}	PUNCT
ejpam-4582	595	7	.	.	PUNCT
ejpam-4582	596	1	this	this	PRON
ejpam-4582	596	2	is	be	AUX
ejpam-4582	596	3	a	a	DET
ejpam-4582	596	4	contradiction	contradiction	NOUN
ejpam-4582	596	5	.	.	PUNCT
ejpam-4582	597	1	in	in	ADP
ejpam-4582	597	2	case	case	NOUN
ejpam-4582	597	3	the	the	DET
ejpam-4582	597	4	singleton	singleton	NOUN
ejpam-4582	597	5	{	{	PUNCT
ejpam-4582	597	6	x	x	NOUN
ejpam-4582	597	7	}	}	PUNCT
ejpam-4582	597	8	is	be	AUX
ejpam-4582	597	9	(	(	PUNCT
ejpam-4582	597	10	λ	λ	X
ejpam-4582	597	11	,	,	PUNCT
ejpam-4582	597	12	s)-closed	s)-close	VERB
ejpam-4582	597	13	,	,	PUNCT
ejpam-4582	597	14	a	a	DET
ejpam-4582	597	15	⊆	⊆	NUM
ejpam-4582	597	16	x−{x	x−{x	PROPN
ejpam-4582	597	17	}	}	PUNCT
ejpam-4582	597	18	and	and	CCONJ
ejpam-4582	597	19	x−{x	x−{x	PROPN
ejpam-4582	597	20	}	}	PUNCT
ejpam-4582	597	21	is	be	AUX
ejpam-4582	597	22	(	(	PUNCT
ejpam-4582	597	23	λ	λ	PROPN
ejpam-4582	597	24	,	,	PUNCT
ejpam-4582	597	25	s)open	s)open	NOUN
ejpam-4582	597	26	.	.	PUNCT
ejpam-4582	598	1	by	by	ADP
ejpam-4582	598	2	proposition	proposition	NOUN
ejpam-4582	598	3	3	3	NUM
ejpam-4582	598	4	,	,	PUNCT
ejpam-4582	598	5	γ(λ	γ(λ	ADV
ejpam-4582	598	6	,	,	PUNCT
ejpam-4582	598	7	s)(a	s)(a	NUM
ejpam-4582	598	8	)	)	PUNCT
ejpam-4582	598	9	⊆	⊆	NUM
ejpam-4582	598	10	γ(λ	γ(λ	NOUN
ejpam-4582	598	11	,	,	PUNCT
ejpam-4582	598	12	s)(x	s)(x	PROPN
ejpam-4582	598	13	−	−	PROPN
ejpam-4582	598	14	{	{	PUNCT
ejpam-4582	598	15	x	x	NOUN
ejpam-4582	598	16	}	}	PUNCT
ejpam-4582	598	17	)	)	PUNCT
ejpam-4582	598	18	=	=	PUNCT
ejpam-4582	599	1	x	x	X
ejpam-4582	599	2	−	−	PROPN
ejpam-4582	599	3	{	{	PUNCT
ejpam-4582	599	4	x	x	NOUN
ejpam-4582	599	5	}	}	PUNCT
ejpam-4582	599	6	.	.	PUNCT
ejpam-4582	600	1	this	this	PRON
ejpam-4582	600	2	is	be	AUX
ejpam-4582	600	3	a	a	DET
ejpam-4582	600	4	contradiction	contradiction	NOUN
ejpam-4582	600	5	.	.	PUNCT
ejpam-4582	601	1	thus	thus	ADV
ejpam-4582	601	2	,	,	PUNCT
ejpam-4582	601	3	every	every	DET
ejpam-4582	601	4	⊛γ(λ	⊛γ(λ	PROPN
ejpam-4582	601	5	,	,	PUNCT
ejpam-4582	601	6	s)-set	s)-set	VERB
ejpam-4582	601	7	is	be	AUX
ejpam-4582	601	8	a	a	DET
ejpam-4582	601	9	γ(λ	γ(λ	NOUN
ejpam-4582	601	10	,	,	PUNCT
ejpam-4582	601	11	s)-set	s)-set	VERB
ejpam-4582	601	12	.	.	PUNCT
ejpam-4582	602	1	(	(	PUNCT
ejpam-4582	602	2	3	3	X
ejpam-4582	602	3	)	)	PUNCT
ejpam-4582	602	4	⇒	⇒	NOUN
ejpam-4582	602	5	(	(	PUNCT
ejpam-4582	602	6	1	1	NUM
ejpam-4582	602	7	):	):	PUNCT
ejpam-4582	602	8	suppose	suppose	VERB
ejpam-4582	602	9	that	that	SCONJ
ejpam-4582	602	10	(	(	PUNCT
ejpam-4582	602	11	x	x	X
ejpam-4582	602	12	,	,	PUNCT
ejpam-4582	602	13	τ	τ	X
ejpam-4582	602	14	)	)	PUNCT
ejpam-4582	602	15	is	be	AUX
ejpam-4582	602	16	not	not	PART
ejpam-4582	602	17	a	a	DET
ejpam-4582	602	18	(	(	PUNCT
ejpam-4582	602	19	λ	λ	NOUN
ejpam-4582	602	20	,	,	PUNCT
ejpam-4582	602	21	s)-t	s)-t	VERB
ejpam-4582	602	22	1	1	NUM
ejpam-4582	602	23	2	2	NUM
ejpam-4582	602	24	-space	-space	NOUN
ejpam-4582	602	25	.	.	PUNCT
ejpam-4582	603	1	then	then	ADV
ejpam-4582	603	2	,	,	PUNCT
ejpam-4582	603	3	there	there	PRON
ejpam-4582	603	4	exists	exist	VERB
ejpam-4582	603	5	a	a	PRON
ejpam-4582	603	6	g-(λ	g-(λ	PROPN
ejpam-4582	603	7	,	,	PUNCT
ejpam-4582	603	8	s)closed	s)close	VERB
ejpam-4582	603	9	set	set	VERB
ejpam-4582	603	10	a	a	PRON
ejpam-4582	603	11	which	which	PRON
ejpam-4582	603	12	is	be	AUX
ejpam-4582	603	13	not	not	PART
ejpam-4582	603	14	(	(	PUNCT
ejpam-4582	603	15	λ	λ	X
ejpam-4582	603	16	,	,	PUNCT
ejpam-4582	603	17	s)-closed	s)-close	VERB
ejpam-4582	603	18	.	.	PUNCT
ejpam-4582	604	1	since	since	SCONJ
ejpam-4582	604	2	a	a	PRON
ejpam-4582	604	3	is	be	AUX
ejpam-4582	604	4	not	not	PART
ejpam-4582	604	5	(	(	PUNCT
ejpam-4582	604	6	λ	λ	X
ejpam-4582	604	7	,	,	PUNCT
ejpam-4582	604	8	s)-closed	s)-close	VERB
ejpam-4582	604	9	,	,	PUNCT
ejpam-4582	604	10	there	there	PRON
ejpam-4582	604	11	exists	exist	VERB
ejpam-4582	604	12	x	x	X
ejpam-4582	604	13	∈	∈	PROPN
ejpam-4582	604	14	a(λ	a(λ	PROPN
ejpam-4582	604	15	,	,	PUNCT
ejpam-4582	604	16	s	s	NOUN
ejpam-4582	604	17	)	)	PUNCT
ejpam-4582	604	18	such	such	ADJ
ejpam-4582	604	19	that	that	SCONJ
ejpam-4582	604	20	x	x	SYM
ejpam-4582	604	21	̸∈	̸∈	PROPN
ejpam-4582	604	22	a.	a.	NOUN
ejpam-4582	604	23	by	by	ADP
ejpam-4582	604	24	lemma	lemma	PROPN
ejpam-4582	604	25	8	8	NUM
ejpam-4582	604	26	,	,	PUNCT
ejpam-4582	604	27	the	the	DET
ejpam-4582	604	28	singleton	singleton	NOUN
ejpam-4582	604	29	{	{	PUNCT
ejpam-4582	604	30	x	x	NOUN
ejpam-4582	604	31	}	}	PUNCT
ejpam-4582	604	32	is	be	AUX
ejpam-4582	604	33	(	(	PUNCT
ejpam-4582	604	34	λ	λ	X
ejpam-4582	604	35	,	,	PUNCT
ejpam-4582	604	36	s)-open	s)-open	PUNCT
ejpam-4582	604	37	or	or	CCONJ
ejpam-4582	604	38	x	x	X
ejpam-4582	604	39	−	−	PROPN
ejpam-4582	604	40	{	{	PUNCT
ejpam-4582	604	41	x	x	NOUN
ejpam-4582	604	42	}	}	PUNCT
ejpam-4582	604	43	is	be	AUX
ejpam-4582	604	44	a	a	DET
ejpam-4582	604	45	γ(λ	γ(λ	NOUN
ejpam-4582	604	46	,	,	PUNCT
ejpam-4582	604	47	s)set	s)set	NOUN
ejpam-4582	604	48	.	.	PUNCT
ejpam-4582	605	1	(	(	PUNCT
ejpam-4582	605	2	a	a	X
ejpam-4582	605	3	)	)	PUNCT
ejpam-4582	605	4	in	in	ADP
ejpam-4582	605	5	case	case	NOUN
ejpam-4582	605	6	{	{	PUNCT
ejpam-4582	605	7	x	x	X
ejpam-4582	605	8	}	}	PUNCT
ejpam-4582	605	9	is	be	AUX
ejpam-4582	605	10	(	(	PUNCT
ejpam-4582	605	11	λ	λ	X
ejpam-4582	605	12	,	,	PUNCT
ejpam-4582	605	13	s)-open	s)-open	VERB
ejpam-4582	605	14	,	,	PUNCT
ejpam-4582	605	15	since	since	SCONJ
ejpam-4582	605	16	x	x	PROPN
ejpam-4582	605	17	∈	∈	PROPN
ejpam-4582	605	18	a(λ	a(λ	PROPN
ejpam-4582	605	19	,	,	PUNCT
ejpam-4582	605	20	s	s	PART
ejpam-4582	605	21	)	)	PUNCT
ejpam-4582	605	22	,	,	PUNCT
ejpam-4582	605	23	{	{	PUNCT
ejpam-4582	605	24	x	x	X
ejpam-4582	605	25	}	}	PUNCT
ejpam-4582	605	26	∩	∩	NOUN
ejpam-4582	605	27	a	a	DET
ejpam-4582	605	28	̸=	̸=	PROPN
ejpam-4582	605	29	∅	∅	NOUN
ejpam-4582	605	30	and	and	CCONJ
ejpam-4582	605	31	x	x	PUNCT
ejpam-4582	605	32	∈	∈	NOUN
ejpam-4582	605	33	a.	a.	NOUN
ejpam-4582	605	34	this	this	PRON
ejpam-4582	605	35	is	be	AUX
ejpam-4582	605	36	a	a	DET
ejpam-4582	605	37	contradiction	contradiction	NOUN
ejpam-4582	605	38	.	.	PUNCT
ejpam-4582	606	1	(	(	PUNCT
ejpam-4582	606	2	b	b	X
ejpam-4582	606	3	)	)	PUNCT
ejpam-4582	606	4	in	in	ADP
ejpam-4582	606	5	case	case	NOUN
ejpam-4582	606	6	x−{x	x−{x	PROPN
ejpam-4582	606	7	}	}	PUNCT
ejpam-4582	606	8	is	be	AUX
ejpam-4582	606	9	a	a	DET
ejpam-4582	606	10	γ(λ	γ(λ	NOUN
ejpam-4582	606	11	,	,	PUNCT
ejpam-4582	606	12	s)-set	s)-set	VERB
ejpam-4582	606	13	,	,	PUNCT
ejpam-4582	606	14	if	if	SCONJ
ejpam-4582	606	15	{	{	PUNCT
ejpam-4582	606	16	x	x	NOUN
ejpam-4582	606	17	}	}	PUNCT
ejpam-4582	606	18	is	be	AUX
ejpam-4582	606	19	not	not	PART
ejpam-4582	606	20	(	(	PUNCT
ejpam-4582	606	21	λ	λ	X
ejpam-4582	606	22	,	,	PUNCT
ejpam-4582	606	23	s)-closed	s)-close	VERB
ejpam-4582	606	24	,	,	PUNCT
ejpam-4582	606	25	x−{x	x−{x	PROPN
ejpam-4582	606	26	}	}	PUNCT
ejpam-4582	606	27	is	be	AUX
ejpam-4582	606	28	not	not	PART
ejpam-4582	606	29	(	(	PUNCT
ejpam-4582	606	30	λ	λ	X
ejpam-4582	606	31	,	,	PUNCT
ejpam-4582	606	32	s)-open	s)-open	PUNCT
ejpam-4582	606	33	and	and	CCONJ
ejpam-4582	606	34	γ(λ	γ(λ	PROPN
ejpam-4582	606	35	,	,	PUNCT
ejpam-4582	606	36	s)(x−{x	s)(x−{x	NOUN
ejpam-4582	606	37	}	}	PUNCT
ejpam-4582	606	38	)	)	PUNCT
ejpam-4582	607	1	=	=	PUNCT
ejpam-4582	607	2	x.	x.	NOUN
ejpam-4582	607	3	hence	hence	ADV
ejpam-4582	607	4	,	,	PUNCT
ejpam-4582	607	5	x−{x	x−{x	PROPN
ejpam-4582	607	6	}	}	PUNCT
ejpam-4582	607	7	is	be	AUX
ejpam-4582	607	8	not	not	PART
ejpam-4582	607	9	a	a	DET
ejpam-4582	607	10	γ(λ	γ(λ	NOUN
ejpam-4582	607	11	,	,	PUNCT
ejpam-4582	607	12	s)-set	s)-set	NOUN
ejpam-4582	607	13	.	.	PUNCT
ejpam-4582	608	1	this	this	PRON
ejpam-4582	608	2	contradicts	contradict	VERB
ejpam-4582	608	3	(	(	PUNCT
ejpam-4582	608	4	3	3	NUM
ejpam-4582	608	5	)	)	PUNCT
ejpam-4582	608	6	.	.	PUNCT
ejpam-4582	609	1	if	if	SCONJ
ejpam-4582	609	2	{	{	PUNCT
ejpam-4582	609	3	x	x	NOUN
ejpam-4582	609	4	}	}	PUNCT
ejpam-4582	609	5	is	be	AUX
ejpam-4582	609	6	(	(	PUNCT
ejpam-4582	609	7	λ	λ	X
ejpam-4582	609	8	,	,	PUNCT
ejpam-4582	609	9	s)-closed	s)-close	VERB
ejpam-4582	609	10	,	,	PUNCT
ejpam-4582	609	11	a	a	DET
ejpam-4582	609	12	⊆	⊆	NUM
ejpam-4582	609	13	x	x	SYM
ejpam-4582	609	14	−	−	PROPN
ejpam-4582	609	15	{	{	PUNCT
ejpam-4582	609	16	x	x	NOUN
ejpam-4582	609	17	}	}	PUNCT
ejpam-4582	609	18	∈	∈	PROPN
ejpam-4582	609	19	(	(	PUNCT
ejpam-4582	609	20	λ	λ	NOUN
ejpam-4582	609	21	,	,	PUNCT
ejpam-4582	609	22	s)o(x	s)o(x	ADJ
ejpam-4582	609	23	)	)	PUNCT
ejpam-4582	609	24	and	and	CCONJ
ejpam-4582	609	25	a	a	PRON
ejpam-4582	609	26	is	be	AUX
ejpam-4582	609	27	g-(λ	g-(λ	PRON
ejpam-4582	609	28	,	,	PUNCT
ejpam-4582	609	29	s)-closed	s)-close	VERB
ejpam-4582	609	30	.	.	PUNCT
ejpam-4582	610	1	thus	thus	ADV
ejpam-4582	610	2	,	,	PUNCT
ejpam-4582	610	3	we	we	PRON
ejpam-4582	610	4	have	have	VERB
ejpam-4582	610	5	a(λ	a(λ	ADV
ejpam-4582	610	6	,	,	PUNCT
ejpam-4582	610	7	s	s	PART
ejpam-4582	610	8	)	)	PUNCT
ejpam-4582	610	9	⊆	⊆	NUM
ejpam-4582	610	10	x	x	SYM
ejpam-4582	610	11	−	−	PROPN
ejpam-4582	610	12	{	{	PUNCT
ejpam-4582	610	13	x	x	NOUN
ejpam-4582	610	14	}	}	PUNCT
ejpam-4582	610	15	.	.	PUNCT
ejpam-4582	611	1	this	this	PRON
ejpam-4582	611	2	contradicts	contradict	VERB
ejpam-4582	611	3	that	that	SCONJ
ejpam-4582	611	4	x	x	SYM
ejpam-4582	611	5	∈	∈	PROPN
ejpam-4582	611	6	a(λ	a(λ	PROPN
ejpam-4582	611	7	,	,	PUNCT
ejpam-4582	611	8	s	s	PART
ejpam-4582	611	9	)	)	PUNCT
ejpam-4582	611	10	.	.	PUNCT
ejpam-4582	612	1	therefore	therefore	ADV
ejpam-4582	612	2	,	,	PUNCT
ejpam-4582	612	3	(	(	PUNCT
ejpam-4582	612	4	x	x	X
ejpam-4582	612	5	,	,	PUNCT
ejpam-4582	612	6	τ	τ	X
ejpam-4582	612	7	)	)	PUNCT
ejpam-4582	612	8	is	be	AUX
ejpam-4582	612	9	(	(	PUNCT
ejpam-4582	612	10	λ	λ	X
ejpam-4582	612	11	,	,	PUNCT
ejpam-4582	612	12	s)-t	s)-t	VERB
ejpam-4582	612	13	1	1	NUM
ejpam-4582	612	14	2	2	NUM
ejpam-4582	612	15	.	.	PUNCT
ejpam-4582	613	1	now	now	ADV
ejpam-4582	613	2	,	,	PUNCT
ejpam-4582	613	3	as	as	ADP
ejpam-4582	613	4	an	an	DET
ejpam-4582	613	5	application	application	NOUN
ejpam-4582	613	6	of	of	ADP
ejpam-4582	613	7	g-(λ	g-(λ	PROPN
ejpam-4582	613	8	,	,	PUNCT
ejpam-4582	613	9	s)-closed	s)-close	VERB
ejpam-4582	613	10	sets	set	NOUN
ejpam-4582	613	11	,	,	PUNCT
ejpam-4582	613	12	we	we	PRON
ejpam-4582	613	13	introduce	introduce	VERB
ejpam-4582	613	14	the	the	DET
ejpam-4582	613	15	concept	concept	NOUN
ejpam-4582	613	16	of	of	ADP
ejpam-4582	613	17	(	(	PUNCT
ejpam-4582	613	18	λ	λ	PROPN
ejpam-4582	613	19	,	,	PUNCT
ejpam-4582	613	20	s)normality	s)normality	NOUN
ejpam-4582	613	21	in	in	ADP
ejpam-4582	613	22	a	a	DET
ejpam-4582	613	23	topological	topological	ADJ
ejpam-4582	613	24	space	space	NOUN
ejpam-4582	613	25	(	(	PUNCT
ejpam-4582	613	26	x	x	X
ejpam-4582	613	27	,	,	PUNCT
ejpam-4582	613	28	τ	τ	PROPN
ejpam-4582	613	29	)	)	PUNCT
ejpam-4582	613	30	.	.	PUNCT
ejpam-4582	614	1	this	this	DET
ejpam-4582	614	2	concept	concept	NOUN
ejpam-4582	614	3	enables	enable	VERB
ejpam-4582	614	4	us	we	PRON
ejpam-4582	614	5	to	to	PART
ejpam-4582	614	6	unify	unify	VERB
ejpam-4582	614	7	several	several	ADJ
ejpam-4582	614	8	modifications	modification	NOUN
ejpam-4582	614	9	of	of	ADP
ejpam-4582	614	10	normal	normal	ADJ
ejpam-4582	614	11	spaces	space	NOUN
ejpam-4582	614	12	.	.	PUNCT
ejpam-4582	615	1	definition	definition	NOUN
ejpam-4582	615	2	16	16	NUM
ejpam-4582	615	3	.	.	PUNCT
ejpam-4582	616	1	a	a	DET
ejpam-4582	616	2	topological	topological	ADJ
ejpam-4582	616	3	space	space	NOUN
ejpam-4582	616	4	(	(	PUNCT
ejpam-4582	616	5	x	x	X
ejpam-4582	616	6	,	,	PUNCT
ejpam-4582	616	7	τ	τ	X
ejpam-4582	616	8	)	)	PUNCT
ejpam-4582	616	9	is	be	AUX
ejpam-4582	616	10	said	say	VERB
ejpam-4582	616	11	to	to	PART
ejpam-4582	616	12	be	be	AUX
ejpam-4582	616	13	(	(	PUNCT
ejpam-4582	616	14	λ	λ	PROPN
ejpam-4582	616	15	,	,	PUNCT
ejpam-4582	616	16	s)-normal	s)-normal	ADJ
ejpam-4582	616	17	if	if	SCONJ
ejpam-4582	616	18	,	,	PUNCT
ejpam-4582	616	19	for	for	ADP
ejpam-4582	616	20	any	any	DET
ejpam-4582	616	21	disjoint	disjoint	NOUN
ejpam-4582	616	22	(	(	PUNCT
ejpam-4582	616	23	λ	λ	PROPN
ejpam-4582	616	24	,	,	PUNCT
ejpam-4582	616	25	s)-closed	s)-close	VERB
ejpam-4582	616	26	sets	set	NOUN
ejpam-4582	616	27	f1	f1	NOUN
ejpam-4582	616	28	and	and	CCONJ
ejpam-4582	616	29	f2	f2	PROPN
ejpam-4582	616	30	,	,	PUNCT
ejpam-4582	616	31	there	there	PRON
ejpam-4582	616	32	exist	exist	VERB
ejpam-4582	616	33	disjoint	disjoint	NOUN
ejpam-4582	616	34	(	(	PUNCT
ejpam-4582	616	35	λ	λ	X
ejpam-4582	616	36	,	,	PUNCT
ejpam-4582	616	37	s)-open	s)-open	PUNCT
ejpam-4582	616	38	sets	set	VERB
ejpam-4582	616	39	u1	u1	NOUN
ejpam-4582	616	40	and	and	CCONJ
ejpam-4582	616	41	u2	u2	NOUN
ejpam-4582	616	42	such	such	ADJ
ejpam-4582	616	43	that	that	DET
ejpam-4582	616	44	f1	f1	PROPN
ejpam-4582	616	45	⊆	⊆	NUM
ejpam-4582	616	46	u1	u1	NOUN
ejpam-4582	616	47	and	and	CCONJ
ejpam-4582	616	48	f2	f2	PRON
ejpam-4582	616	49	⊆	⊆	NUM
ejpam-4582	616	50	u2	u2	PROPN
ejpam-4582	616	51	.	.	PUNCT
ejpam-4582	616	52	theorem	theorem	PROPN
ejpam-4582	616	53	23	23	NUM
ejpam-4582	616	54	.	.	PUNCT
ejpam-4582	617	1	for	for	ADP
ejpam-4582	617	2	a	a	DET
ejpam-4582	617	3	topological	topological	ADJ
ejpam-4582	617	4	space	space	NOUN
ejpam-4582	617	5	(	(	PUNCT
ejpam-4582	617	6	x	x	X
ejpam-4582	617	7	,	,	PUNCT
ejpam-4582	617	8	τ	τ	PROPN
ejpam-4582	617	9	)	)	PUNCT
ejpam-4582	617	10	,	,	PUNCT
ejpam-4582	617	11	the	the	DET
ejpam-4582	617	12	following	follow	VERB
ejpam-4582	617	13	properties	property	NOUN
ejpam-4582	617	14	are	be	AUX
ejpam-4582	617	15	equivalent	equivalent	ADJ
ejpam-4582	617	16	:	:	PUNCT
ejpam-4582	617	17	(	(	PUNCT
ejpam-4582	617	18	1	1	X
ejpam-4582	617	19	)	)	PUNCT
ejpam-4582	617	20	(	(	PUNCT
ejpam-4582	617	21	x	x	X
ejpam-4582	617	22	,	,	PUNCT
ejpam-4582	617	23	τ	τ	X
ejpam-4582	617	24	)	)	PUNCT
ejpam-4582	617	25	is	be	AUX
ejpam-4582	617	26	(	(	PUNCT
ejpam-4582	617	27	λ	λ	X
ejpam-4582	617	28	,	,	PUNCT
ejpam-4582	617	29	s)-normal	s)-normal	ADJ
ejpam-4582	617	30	;	;	PUNCT
ejpam-4582	617	31	(	(	PUNCT
ejpam-4582	617	32	2	2	X
ejpam-4582	617	33	)	)	PUNCT
ejpam-4582	617	34	for	for	ADP
ejpam-4582	617	35	every	every	DET
ejpam-4582	617	36	disjoint	disjoint	NOUN
ejpam-4582	617	37	(	(	PUNCT
ejpam-4582	617	38	λ	λ	PROPN
ejpam-4582	617	39	,	,	PUNCT
ejpam-4582	617	40	s)-closed	s)-close	VERB
ejpam-4582	617	41	sets	set	NOUN
ejpam-4582	617	42	f1	f1	NOUN
ejpam-4582	617	43	and	and	CCONJ
ejpam-4582	617	44	f2	f2	PROPN
ejpam-4582	617	45	,	,	PUNCT
ejpam-4582	617	46	there	there	PRON
ejpam-4582	617	47	exist	exist	VERB
ejpam-4582	617	48	disjoint	disjoint	NOUN
ejpam-4582	617	49	g-(λ	g-(λ	PROPN
ejpam-4582	617	50	,	,	PUNCT
ejpam-4582	617	51	s)-open	s)-open	PUNCT
ejpam-4582	617	52	sets	set	VERB
ejpam-4582	617	53	u1	u1	NOUN
ejpam-4582	617	54	and	and	CCONJ
ejpam-4582	617	55	u2	u2	NOUN
ejpam-4582	617	56	such	such	ADJ
ejpam-4582	617	57	that	that	DET
ejpam-4582	617	58	f1	f1	PROPN
ejpam-4582	617	59	⊆	⊆	NUM
ejpam-4582	617	60	u1	u1	NOUN
ejpam-4582	617	61	and	and	CCONJ
ejpam-4582	617	62	f2	f2	PRON
ejpam-4582	617	63	⊆	⊆	NUM
ejpam-4582	617	64	u2	u2	NOUN
ejpam-4582	617	65	;	;	PUNCT
ejpam-4582	617	66	(	(	PUNCT
ejpam-4582	617	67	3	3	X
ejpam-4582	617	68	)	)	PUNCT
ejpam-4582	617	69	for	for	ADP
ejpam-4582	617	70	each	each	DET
ejpam-4582	617	71	(	(	PUNCT
ejpam-4582	617	72	λ	λ	PROPN
ejpam-4582	617	73	,	,	PUNCT
ejpam-4582	617	74	s)-closed	s)-close	VERB
ejpam-4582	617	75	set	set	ADJ
ejpam-4582	617	76	f	f	PROPN
ejpam-4582	617	77	and	and	CCONJ
ejpam-4582	617	78	each	each	DET
ejpam-4582	617	79	(	(	PUNCT
ejpam-4582	617	80	λ	λ	PROPN
ejpam-4582	617	81	,	,	PUNCT
ejpam-4582	617	82	s)-open	s)-open	VERB
ejpam-4582	617	83	set	set	VERB
ejpam-4582	617	84	g	g	NOUN
ejpam-4582	617	85	containing	contain	VERB
ejpam-4582	617	86	f	f	NOUN
ejpam-4582	617	87	,	,	PUNCT
ejpam-4582	617	88	there	there	PRON
ejpam-4582	617	89	exists	exist	VERB
ejpam-4582	617	90	a	a	DET
ejpam-4582	617	91	g-(λ	g-(λ	NOUN
ejpam-4582	617	92	,	,	PUNCT
ejpam-4582	617	93	s)-open	s)-open	VERB
ejpam-4582	617	94	set	set	VERB
ejpam-4582	617	95	u	u	PRON
ejpam-4582	617	96	such	such	ADJ
ejpam-4582	617	97	that	that	SCONJ
ejpam-4582	617	98	f	f	PROPN
ejpam-4582	617	99	⊆	⊆	NUM
ejpam-4582	617	100	u	u	NOUN
ejpam-4582	617	101	⊆	⊆	NUM
ejpam-4582	617	102	u	u	NOUN
ejpam-4582	617	103	(	(	PUNCT
ejpam-4582	617	104	λ	λ	PROPN
ejpam-4582	617	105	,	,	PUNCT
ejpam-4582	617	106	s	s	PART
ejpam-4582	617	107	)	)	PUNCT
ejpam-4582	617	108	⊆	⊆	NUM
ejpam-4582	617	109	g	g	NOUN
ejpam-4582	617	110	;	;	PUNCT
ejpam-4582	617	111	(	(	PUNCT
ejpam-4582	617	112	4	4	X
ejpam-4582	617	113	)	)	PUNCT
ejpam-4582	617	114	for	for	ADP
ejpam-4582	617	115	each	each	DET
ejpam-4582	617	116	(	(	PUNCT
ejpam-4582	617	117	λ	λ	PROPN
ejpam-4582	617	118	,	,	PUNCT
ejpam-4582	617	119	s)-closed	s)-close	VERB
ejpam-4582	617	120	set	set	ADJ
ejpam-4582	617	121	f	f	PROPN
ejpam-4582	617	122	and	and	CCONJ
ejpam-4582	617	123	each	each	DET
ejpam-4582	617	124	g-(λ	g-(λ	PROPN
ejpam-4582	617	125	,	,	PUNCT
ejpam-4582	617	126	s)-open	s)-open	VERB
ejpam-4582	617	127	set	set	VERB
ejpam-4582	617	128	g	g	NOUN
ejpam-4582	617	129	containing	contain	VERB
ejpam-4582	617	130	f	f	NOUN
ejpam-4582	617	131	,	,	PUNCT
ejpam-4582	617	132	there	there	PRON
ejpam-4582	617	133	exists	exist	VERB
ejpam-4582	617	134	a	a	DET
ejpam-4582	617	135	(	(	PUNCT
ejpam-4582	617	136	λ	λ	X
ejpam-4582	617	137	,	,	PUNCT
ejpam-4582	617	138	s)-open	s)-open	VERB
ejpam-4582	617	139	set	set	VERB
ejpam-4582	617	140	u	u	PRON
ejpam-4582	617	141	such	such	ADJ
ejpam-4582	617	142	that	that	SCONJ
ejpam-4582	617	143	f	f	PROPN
ejpam-4582	617	144	⊆	⊆	NUM
ejpam-4582	617	145	u	u	NOUN
ejpam-4582	617	146	⊆	⊆	NUM
ejpam-4582	617	147	u	u	NOUN
ejpam-4582	617	148	(	(	PUNCT
ejpam-4582	617	149	λ	λ	PROPN
ejpam-4582	617	150	,	,	PUNCT
ejpam-4582	617	151	s	s	PART
ejpam-4582	617	152	)	)	PUNCT
ejpam-4582	617	153	⊆	⊆	NUM
ejpam-4582	617	154	g(λ	g(λ	PROPN
ejpam-4582	617	155	,	,	PUNCT
ejpam-4582	617	156	s	s	PART
ejpam-4582	617	157	)	)	PUNCT
ejpam-4582	617	158	;	;	PUNCT
ejpam-4582	617	159	c.	c.	PROPN
ejpam-4582	617	160	boonpok	boonpok	PROPN
ejpam-4582	617	161	,	,	PUNCT
ejpam-4582	617	162	c.	c.	PROPN
ejpam-4582	617	163	viriyapong	viriyapong	PROPN
ejpam-4582	617	164	/	/	SYM
ejpam-4582	617	165	eur	eur	PROPN
ejpam-4582	617	166	.	.	PUNCT
ejpam-4582	618	1	j.	j.	PROPN
ejpam-4582	618	2	pure	pure	PROPN
ejpam-4582	618	3	appl	appl	PROPN
ejpam-4582	618	4	.	.	PROPN
ejpam-4582	618	5	math	math	PROPN
ejpam-4582	618	6	,	,	PUNCT
ejpam-4582	618	7	16	16	NUM
ejpam-4582	618	8	(	(	PUNCT
ejpam-4582	618	9	1	1	NUM
ejpam-4582	618	10	)	)	PUNCT
ejpam-4582	618	11	(	(	PUNCT
ejpam-4582	618	12	2023	2023	NUM
ejpam-4582	618	13	)	)	PUNCT
ejpam-4582	618	14	,	,	PUNCT
ejpam-4582	618	15	336	336	NUM
ejpam-4582	618	16	-	-	SYM
ejpam-4582	618	17	362	362	NUM
ejpam-4582	618	18	353	353	NUM
ejpam-4582	618	19	(	(	PUNCT
ejpam-4582	618	20	5	5	NUM
ejpam-4582	618	21	)	)	PUNCT
ejpam-4582	618	22	for	for	ADP
ejpam-4582	618	23	each	each	DET
ejpam-4582	618	24	(	(	PUNCT
ejpam-4582	618	25	λ	λ	PROPN
ejpam-4582	618	26	,	,	PUNCT
ejpam-4582	618	27	s)-closed	s)-close	VERB
ejpam-4582	618	28	set	set	ADJ
ejpam-4582	618	29	f	f	PROPN
ejpam-4582	618	30	and	and	CCONJ
ejpam-4582	618	31	each	each	DET
ejpam-4582	618	32	g-(λ	g-(λ	PROPN
ejpam-4582	618	33	,	,	PUNCT
ejpam-4582	618	34	s)-open	s)-open	VERB
ejpam-4582	618	35	set	set	VERB
ejpam-4582	618	36	g	g	NOUN
ejpam-4582	618	37	containing	contain	VERB
ejpam-4582	618	38	f	f	NOUN
ejpam-4582	618	39	,	,	PUNCT
ejpam-4582	618	40	there	there	PRON
ejpam-4582	618	41	exists	exist	VERB
ejpam-4582	618	42	a	a	DET
ejpam-4582	618	43	g-(λ	g-(λ	NOUN
ejpam-4582	618	44	,	,	PUNCT
ejpam-4582	618	45	s)-open	s)-open	VERB
ejpam-4582	618	46	set	set	VERB
ejpam-4582	618	47	u	u	PRON
ejpam-4582	618	48	such	such	ADJ
ejpam-4582	618	49	that	that	SCONJ
ejpam-4582	618	50	f	f	PROPN
ejpam-4582	618	51	⊆	⊆	NUM
ejpam-4582	618	52	u	u	NOUN
ejpam-4582	618	53	⊆	⊆	NUM
ejpam-4582	618	54	u	u	NOUN
ejpam-4582	618	55	(	(	PUNCT
ejpam-4582	618	56	λ	λ	PROPN
ejpam-4582	618	57	,	,	PUNCT
ejpam-4582	618	58	s	s	PART
ejpam-4582	618	59	)	)	PUNCT
ejpam-4582	618	60	⊆	⊆	NUM
ejpam-4582	618	61	g(λ	g(λ	PROPN
ejpam-4582	618	62	,	,	PUNCT
ejpam-4582	618	63	s	s	PART
ejpam-4582	618	64	)	)	PUNCT
ejpam-4582	618	65	;	;	PUNCT
ejpam-4582	618	66	(	(	PUNCT
ejpam-4582	618	67	6	6	NUM
ejpam-4582	618	68	)	)	PUNCT
ejpam-4582	618	69	for	for	ADP
ejpam-4582	618	70	each	each	DET
ejpam-4582	618	71	g-(λ	g-(λ	NOUN
ejpam-4582	618	72	,	,	PUNCT
ejpam-4582	618	73	s)-closed	s)-close	VERB
ejpam-4582	618	74	set	set	ADJ
ejpam-4582	618	75	f	f	PROPN
ejpam-4582	618	76	and	and	CCONJ
ejpam-4582	618	77	each	each	DET
ejpam-4582	618	78	(	(	PUNCT
ejpam-4582	618	79	λ	λ	PROPN
ejpam-4582	618	80	,	,	PUNCT
ejpam-4582	618	81	s)-open	s)-open	VERB
ejpam-4582	618	82	set	set	VERB
ejpam-4582	618	83	g	g	NOUN
ejpam-4582	618	84	containing	contain	VERB
ejpam-4582	618	85	f	f	NOUN
ejpam-4582	618	86	,	,	PUNCT
ejpam-4582	618	87	there	there	PRON
ejpam-4582	618	88	exists	exist	VERB
ejpam-4582	618	89	a	a	DET
ejpam-4582	618	90	(	(	PUNCT
ejpam-4582	618	91	λ	λ	X
ejpam-4582	618	92	,	,	PUNCT
ejpam-4582	618	93	s)-open	s)-open	VERB
ejpam-4582	618	94	set	set	VERB
ejpam-4582	618	95	u	u	PRON
ejpam-4582	618	96	such	such	ADJ
ejpam-4582	618	97	that	that	SCONJ
ejpam-4582	618	98	f	f	PROPN
ejpam-4582	618	99	(	(	PUNCT
ejpam-4582	618	100	λ	λ	PROPN
ejpam-4582	618	101	,	,	PUNCT
ejpam-4582	618	102	s	s	PART
ejpam-4582	618	103	)	)	PUNCT
ejpam-4582	618	104	⊆	⊆	NUM
ejpam-4582	618	105	u	u	NOUN
ejpam-4582	618	106	⊆	⊆	NUM
ejpam-4582	618	107	u	u	NOUN
ejpam-4582	618	108	(	(	PUNCT
ejpam-4582	618	109	λ	λ	PROPN
ejpam-4582	618	110	,	,	PUNCT
ejpam-4582	618	111	s	s	PART
ejpam-4582	618	112	)	)	PUNCT
ejpam-4582	618	113	⊆	⊆	NUM
ejpam-4582	618	114	g	g	NOUN
ejpam-4582	618	115	;	;	PUNCT
ejpam-4582	618	116	(	(	PUNCT
ejpam-4582	618	117	7	7	X
ejpam-4582	618	118	)	)	PUNCT
ejpam-4582	618	119	for	for	ADP
ejpam-4582	618	120	each	each	DET
ejpam-4582	618	121	g-(λ	g-(λ	NOUN
ejpam-4582	618	122	,	,	PUNCT
ejpam-4582	618	123	s)-closed	s)-close	VERB
ejpam-4582	618	124	set	set	ADJ
ejpam-4582	618	125	f	f	PROPN
ejpam-4582	618	126	and	and	CCONJ
ejpam-4582	618	127	each	each	DET
ejpam-4582	618	128	(	(	PUNCT
ejpam-4582	618	129	λ	λ	PROPN
ejpam-4582	618	130	,	,	PUNCT
ejpam-4582	618	131	s)-open	s)-open	VERB
ejpam-4582	618	132	set	set	VERB
ejpam-4582	618	133	g	g	NOUN
ejpam-4582	618	134	containing	contain	VERB
ejpam-4582	618	135	f	f	NOUN
ejpam-4582	618	136	,	,	PUNCT
ejpam-4582	618	137	there	there	PRON
ejpam-4582	618	138	exists	exist	VERB
ejpam-4582	618	139	a	a	DET
ejpam-4582	618	140	g-(λ	g-(λ	NOUN
ejpam-4582	618	141	,	,	PUNCT
ejpam-4582	618	142	s)-open	s)-open	VERB
ejpam-4582	618	143	set	set	VERB
ejpam-4582	618	144	u	u	PRON
ejpam-4582	618	145	such	such	ADJ
ejpam-4582	618	146	that	that	SCONJ
ejpam-4582	618	147	f	f	PROPN
ejpam-4582	618	148	(	(	PUNCT
ejpam-4582	618	149	λ	λ	PROPN
ejpam-4582	618	150	,	,	PUNCT
ejpam-4582	618	151	s	s	PART
ejpam-4582	618	152	)	)	PUNCT
ejpam-4582	618	153	⊆	⊆	NUM
ejpam-4582	618	154	u	u	NOUN
ejpam-4582	618	155	⊆	⊆	NUM
ejpam-4582	618	156	u	u	NOUN
ejpam-4582	618	157	(	(	PUNCT
ejpam-4582	618	158	λ	λ	PROPN
ejpam-4582	618	159	,	,	PUNCT
ejpam-4582	618	160	s	s	PART
ejpam-4582	618	161	)	)	PUNCT
ejpam-4582	618	162	⊆	⊆	NUM
ejpam-4582	618	163	g.	g.	NOUN
ejpam-4582	618	164	proof	proof	NOUN
ejpam-4582	618	165	.	.	PUNCT
ejpam-4582	619	1	(	(	PUNCT
ejpam-4582	619	2	1	1	X
ejpam-4582	619	3	)	)	PUNCT
ejpam-4582	619	4	⇒	⇒	NOUN
ejpam-4582	619	5	(	(	PUNCT
ejpam-4582	619	6	2	2	NUM
ejpam-4582	619	7	):	):	PUNCT
ejpam-4582	619	8	the	the	DET
ejpam-4582	619	9	proof	proof	NOUN
ejpam-4582	619	10	is	be	AUX
ejpam-4582	619	11	obvious	obvious	ADJ
ejpam-4582	619	12	.	.	PUNCT
ejpam-4582	620	1	(	(	PUNCT
ejpam-4582	620	2	2	2	X
ejpam-4582	620	3	)	)	PUNCT
ejpam-4582	620	4	⇒	⇒	NOUN
ejpam-4582	620	5	(	(	PUNCT
ejpam-4582	620	6	3	3	NUM
ejpam-4582	620	7	):	):	PUNCT
ejpam-4582	620	8	let	let	VERB
ejpam-4582	620	9	f	f	PRON
ejpam-4582	620	10	be	be	AUX
ejpam-4582	620	11	a	a	DET
ejpam-4582	620	12	(	(	PUNCT
ejpam-4582	620	13	λ	λ	X
ejpam-4582	620	14	,	,	PUNCT
ejpam-4582	620	15	s)-closed	s)-close	VERB
ejpam-4582	620	16	set	set	NOUN
ejpam-4582	620	17	and	and	CCONJ
ejpam-4582	620	18	g	g	NOUN
ejpam-4582	620	19	be	be	AUX
ejpam-4582	620	20	a	a	DET
ejpam-4582	620	21	(	(	PUNCT
ejpam-4582	620	22	λ	λ	X
ejpam-4582	620	23	,	,	PUNCT
ejpam-4582	620	24	s)-open	s)-open	VERB
ejpam-4582	620	25	set	set	VERB
ejpam-4582	620	26	containing	contain	VERB
ejpam-4582	620	27	f	f	PROPN
ejpam-4582	620	28	.	.	PUNCT
ejpam-4582	621	1	then	then	ADV
ejpam-4582	621	2	,	,	PUNCT
ejpam-4582	621	3	f	f	PROPN
ejpam-4582	621	4	and	and	CCONJ
ejpam-4582	621	5	x	x	PROPN
ejpam-4582	621	6	−g	−g	NOUN
ejpam-4582	621	7	are	be	AUX
ejpam-4582	621	8	two	two	NUM
ejpam-4582	621	9	disjoint	disjoint	NOUN
ejpam-4582	621	10	(	(	PUNCT
ejpam-4582	621	11	λ	λ	PROPN
ejpam-4582	621	12	,	,	PUNCT
ejpam-4582	621	13	s)-closed	s)-close	VERB
ejpam-4582	621	14	sets	set	NOUN
ejpam-4582	621	15	.	.	PUNCT
ejpam-4582	622	1	hence	hence	ADV
ejpam-4582	622	2	by	by	ADP
ejpam-4582	622	3	(	(	PUNCT
ejpam-4582	622	4	2	2	NUM
ejpam-4582	622	5	)	)	PUNCT
ejpam-4582	622	6	,	,	PUNCT
ejpam-4582	622	7	there	there	PRON
ejpam-4582	622	8	exist	exist	VERB
ejpam-4582	622	9	disjoint	disjoint	NOUN
ejpam-4582	622	10	g-(λ	g-(λ	PROPN
ejpam-4582	622	11	,	,	PUNCT
ejpam-4582	622	12	s)-open	s)-open	PUNCT
ejpam-4582	622	13	sets	set	VERB
ejpam-4582	622	14	u	u	NOUN
ejpam-4582	622	15	and	and	CCONJ
ejpam-4582	622	16	v	v	ADP
ejpam-4582	622	17	such	such	ADJ
ejpam-4582	622	18	that	that	SCONJ
ejpam-4582	622	19	f	f	PROPN
ejpam-4582	622	20	⊆	⊆	NUM
ejpam-4582	622	21	u	u	NOUN
ejpam-4582	622	22	and	and	CCONJ
ejpam-4582	622	23	x	x	NOUN
ejpam-4582	622	24	−	−	NOUN
ejpam-4582	622	25	g	g	PROPN
ejpam-4582	622	26	⊆	⊆	NUM
ejpam-4582	622	27	v	v	NOUN
ejpam-4582	622	28	.	.	PUNCT
ejpam-4582	623	1	since	since	SCONJ
ejpam-4582	623	2	v	v	NOUN
ejpam-4582	623	3	is	be	AUX
ejpam-4582	623	4	g-(λ	g-(λ	PROPN
ejpam-4582	623	5	,	,	PUNCT
ejpam-4582	623	6	s)-open	s)-open	PUNCT
ejpam-4582	623	7	and	and	CCONJ
ejpam-4582	623	8	x	x	X
ejpam-4582	623	9	−	−	NOUN
ejpam-4582	623	10	g	g	NOUN
ejpam-4582	623	11	is	be	AUX
ejpam-4582	623	12	(	(	PUNCT
ejpam-4582	623	13	λ	λ	X
ejpam-4582	623	14	,	,	PUNCT
ejpam-4582	623	15	s)-closed	s)-close	VERB
ejpam-4582	623	16	,	,	PUNCT
ejpam-4582	623	17	by	by	ADP
ejpam-4582	623	18	theorem	theorem	NOUN
ejpam-4582	623	19	14	14	NUM
ejpam-4582	623	20	,	,	PUNCT
ejpam-4582	623	21	x	x	NOUN
ejpam-4582	623	22	−	−	NOUN
ejpam-4582	623	23	g	g	PROPN
ejpam-4582	623	24	⊆	⊆	NUM
ejpam-4582	623	25	v(λ	v(λ	PROPN
ejpam-4582	623	26	,	,	PUNCT
ejpam-4582	623	27	s	s	PART
ejpam-4582	623	28	)	)	PUNCT
ejpam-4582	623	29	.	.	PUNCT
ejpam-4582	624	1	since	since	SCONJ
ejpam-4582	624	2	u	u	PROPN
ejpam-4582	624	3	∩	∩	NOUN
ejpam-4582	624	4	v	v	NOUN
ejpam-4582	624	5	=	=	SYM
ejpam-4582	624	6	∅	∅	NOUN
ejpam-4582	624	7	,	,	PUNCT
ejpam-4582	624	8	we	we	PRON
ejpam-4582	624	9	have	have	VERB
ejpam-4582	624	10	u	u	NOUN
ejpam-4582	624	11	(	(	PUNCT
ejpam-4582	624	12	λ	λ	PROPN
ejpam-4582	624	13	,	,	PUNCT
ejpam-4582	624	14	s	s	PART
ejpam-4582	624	15	)	)	PUNCT
ejpam-4582	624	16	⊆	⊆	NUM
ejpam-4582	624	17	(	(	PUNCT
ejpam-4582	624	18	x	x	NOUN
ejpam-4582	624	19	−	−	NOUN
ejpam-4582	624	20	v	v	NOUN
ejpam-4582	624	21	)	)	PUNCT
ejpam-4582	624	22	(	(	PUNCT
ejpam-4582	624	23	λ	λ	PROPN
ejpam-4582	624	24	,	,	PUNCT
ejpam-4582	624	25	s	s	PART
ejpam-4582	624	26	)	)	PUNCT
ejpam-4582	624	27	=	=	SYM
ejpam-4582	624	28	x	x	X
ejpam-4582	625	1	−	−	PROPN
ejpam-4582	625	2	v(λ	v(λ	PROPN
ejpam-4582	625	3	,	,	PUNCT
ejpam-4582	625	4	s	s	PART
ejpam-4582	625	5	)	)	PUNCT
ejpam-4582	626	1	⊆	⊆	NUM
ejpam-4582	626	2	g.	g.	NOUN
ejpam-4582	626	3	thus	thus	ADV
ejpam-4582	626	4	,	,	PUNCT
ejpam-4582	626	5	f	f	PROPN
ejpam-4582	626	6	⊆	⊆	NUM
ejpam-4582	626	7	u	u	NOUN
ejpam-4582	626	8	⊆	⊆	NUM
ejpam-4582	626	9	u	u	NOUN
ejpam-4582	626	10	(	(	PUNCT
ejpam-4582	626	11	λ	λ	PROPN
ejpam-4582	626	12	,	,	PUNCT
ejpam-4582	626	13	s	s	PART
ejpam-4582	626	14	)	)	PUNCT
ejpam-4582	626	15	⊆	⊆	NUM
ejpam-4582	626	16	g.	g.	NOUN
ejpam-4582	626	17	(	(	PUNCT
ejpam-4582	626	18	3	3	NUM
ejpam-4582	626	19	)	)	PUNCT
ejpam-4582	626	20	⇒	⇒	NOUN
ejpam-4582	626	21	(	(	PUNCT
ejpam-4582	626	22	1	1	NUM
ejpam-4582	626	23	):	):	PUNCT
ejpam-4582	626	24	let	let	VERB
ejpam-4582	626	25	f1	f1	NOUN
ejpam-4582	626	26	and	and	CCONJ
ejpam-4582	626	27	f2	f2	PRON
ejpam-4582	626	28	be	be	AUX
ejpam-4582	626	29	any	any	DET
ejpam-4582	626	30	disjoint	disjoint	NOUN
ejpam-4582	626	31	(	(	PUNCT
ejpam-4582	626	32	λ	λ	PROPN
ejpam-4582	626	33	,	,	PUNCT
ejpam-4582	626	34	s)-closed	s)-close	VERB
ejpam-4582	626	35	sets	set	NOUN
ejpam-4582	626	36	.	.	PUNCT
ejpam-4582	627	1	then	then	ADV
ejpam-4582	627	2	,	,	PUNCT
ejpam-4582	627	3	we	we	PRON
ejpam-4582	627	4	have	have	VERB
ejpam-4582	627	5	x	x	NOUN
ejpam-4582	627	6	−	−	PROPN
ejpam-4582	627	7	f2	f2	PROPN
ejpam-4582	627	8	is	be	AUX
ejpam-4582	627	9	a	a	DET
ejpam-4582	627	10	(	(	PUNCT
ejpam-4582	627	11	λ	λ	X
ejpam-4582	627	12	,	,	PUNCT
ejpam-4582	627	13	s)-open	s)-open	VERB
ejpam-4582	627	14	set	set	VERB
ejpam-4582	627	15	containing	contain	VERB
ejpam-4582	627	16	f1	f1	NOUN
ejpam-4582	627	17	.	.	PUNCT
ejpam-4582	628	1	thus	thus	ADV
ejpam-4582	628	2	by	by	ADP
ejpam-4582	628	3	(	(	PUNCT
ejpam-4582	628	4	3	3	NUM
ejpam-4582	628	5	)	)	PUNCT
ejpam-4582	628	6	,	,	PUNCT
ejpam-4582	628	7	there	there	PRON
ejpam-4582	628	8	exists	exist	VERB
ejpam-4582	628	9	a	a	DET
ejpam-4582	628	10	g-(λ	g-(λ	NOUN
ejpam-4582	628	11	,	,	PUNCT
ejpam-4582	628	12	s)-open	s)-open	VERB
ejpam-4582	628	13	set	set	VERB
ejpam-4582	628	14	u	u	PRON
ejpam-4582	628	15	such	such	ADJ
ejpam-4582	628	16	that	that	DET
ejpam-4582	628	17	f1	f1	NOUN
ejpam-4582	628	18	⊆	⊆	NUM
ejpam-4582	628	19	u	u	NOUN
ejpam-4582	628	20	⊆	⊆	NUM
ejpam-4582	628	21	u	u	NOUN
ejpam-4582	628	22	(	(	PUNCT
ejpam-4582	628	23	λ	λ	PROPN
ejpam-4582	628	24	,	,	PUNCT
ejpam-4582	628	25	s	s	PART
ejpam-4582	628	26	)	)	PUNCT
ejpam-4582	628	27	⊆	⊆	NUM
ejpam-4582	628	28	x	x	SYM
ejpam-4582	628	29	−	−	NOUN
ejpam-4582	628	30	f2	f2	ADJ
ejpam-4582	628	31	and	and	CCONJ
ejpam-4582	628	32	hence	hence	ADV
ejpam-4582	628	33	f2	f2	VERB
ejpam-4582	628	34	⊆	⊆	NUM
ejpam-4582	628	35	x	x	SYM
ejpam-4582	628	36	−	−	PROPN
ejpam-4582	628	37	u	u	NOUN
ejpam-4582	628	38	(	(	PUNCT
ejpam-4582	628	39	λ	λ	PROPN
ejpam-4582	628	40	,	,	PUNCT
ejpam-4582	628	41	s	s	PART
ejpam-4582	628	42	)	)	PUNCT
ejpam-4582	628	43	.	.	PUNCT
ejpam-4582	629	1	since	since	SCONJ
ejpam-4582	629	2	f1	f1	PROPN
ejpam-4582	629	3	is	be	AUX
ejpam-4582	629	4	(	(	PUNCT
ejpam-4582	629	5	λ	λ	X
ejpam-4582	629	6	,	,	PUNCT
ejpam-4582	629	7	s)-closed	s)-close	VERB
ejpam-4582	629	8	and	and	CCONJ
ejpam-4582	629	9	u	u	NOUN
ejpam-4582	629	10	is	be	AUX
ejpam-4582	629	11	g-(λ	g-(λ	PROPN
ejpam-4582	629	12	,	,	PUNCT
ejpam-4582	629	13	s)-open	s)-open	PUNCT
ejpam-4582	629	14	,	,	PUNCT
ejpam-4582	629	15	by	by	ADP
ejpam-4582	629	16	theorem	theorem	NOUN
ejpam-4582	629	17	14	14	NUM
ejpam-4582	629	18	,	,	PUNCT
ejpam-4582	629	19	we	we	PRON
ejpam-4582	629	20	have	have	VERB
ejpam-4582	629	21	f1	f1	PROPN
ejpam-4582	629	22	⊆	⊆	NUM
ejpam-4582	629	23	u(λ	u(λ	PROPN
ejpam-4582	629	24	,	,	PUNCT
ejpam-4582	629	25	s	s	PART
ejpam-4582	629	26	)	)	PUNCT
ejpam-4582	629	27	.	.	PUNCT
ejpam-4582	630	1	this	this	PRON
ejpam-4582	630	2	shows	show	VERB
ejpam-4582	630	3	that	that	SCONJ
ejpam-4582	630	4	(	(	PUNCT
ejpam-4582	630	5	x	x	X
ejpam-4582	630	6	,	,	PUNCT
ejpam-4582	630	7	τ	τ	X
ejpam-4582	630	8	)	)	PUNCT
ejpam-4582	630	9	is	be	AUX
ejpam-4582	630	10	(	(	PUNCT
ejpam-4582	630	11	λ	λ	X
ejpam-4582	630	12	,	,	PUNCT
ejpam-4582	630	13	s)-normal	s)-normal	ADJ
ejpam-4582	630	14	.	.	PUNCT
ejpam-4582	631	1	(	(	PUNCT
ejpam-4582	631	2	6	6	NUM
ejpam-4582	631	3	)	)	PUNCT
ejpam-4582	631	4	⇒	⇒	NOUN
ejpam-4582	631	5	(	(	PUNCT
ejpam-4582	631	6	7	7	NUM
ejpam-4582	631	7	)	)	PUNCT
ejpam-4582	631	8	and	and	CCONJ
ejpam-4582	631	9	(	(	PUNCT
ejpam-4582	631	10	7	7	X
ejpam-4582	631	11	)	)	PUNCT
ejpam-4582	631	12	⇒	⇒	NOUN
ejpam-4582	631	13	(	(	PUNCT
ejpam-4582	631	14	3	3	NUM
ejpam-4582	631	15	):	):	PUNCT
ejpam-4582	631	16	the	the	DET
ejpam-4582	631	17	proofs	proof	NOUN
ejpam-4582	631	18	are	be	AUX
ejpam-4582	631	19	obvious	obvious	ADJ
ejpam-4582	631	20	.	.	PUNCT
ejpam-4582	632	1	(	(	PUNCT
ejpam-4582	632	2	3	3	X
ejpam-4582	632	3	)	)	PUNCT
ejpam-4582	632	4	⇒	⇒	NOUN
ejpam-4582	632	5	(	(	PUNCT
ejpam-4582	632	6	5	5	NUM
ejpam-4582	632	7	):	):	PUNCT
ejpam-4582	632	8	let	let	VERB
ejpam-4582	632	9	f	f	PRON
ejpam-4582	632	10	be	be	AUX
ejpam-4582	632	11	a	a	DET
ejpam-4582	632	12	(	(	PUNCT
ejpam-4582	632	13	λ	λ	X
ejpam-4582	632	14	,	,	PUNCT
ejpam-4582	632	15	s)-closed	s)-close	VERB
ejpam-4582	632	16	set	set	NOUN
ejpam-4582	632	17	and	and	CCONJ
ejpam-4582	632	18	g	g	NOUN
ejpam-4582	632	19	be	be	AUX
ejpam-4582	632	20	a	a	DET
ejpam-4582	632	21	g-(λ	g-(λ	NOUN
ejpam-4582	632	22	,	,	PUNCT
ejpam-4582	632	23	s)-open	s)-open	VERB
ejpam-4582	632	24	set	set	VERB
ejpam-4582	632	25	containing	contain	VERB
ejpam-4582	632	26	f	f	PROPN
ejpam-4582	632	27	.	.	PUNCT
ejpam-4582	633	1	since	since	SCONJ
ejpam-4582	633	2	g	g	PROPN
ejpam-4582	633	3	is	be	AUX
ejpam-4582	633	4	g-(λ	g-(λ	PROPN
ejpam-4582	633	5	,	,	PUNCT
ejpam-4582	633	6	s)-open	s)-open	PUNCT
ejpam-4582	633	7	and	and	CCONJ
ejpam-4582	633	8	f	f	PROPN
ejpam-4582	633	9	is	be	AUX
ejpam-4582	633	10	(	(	PUNCT
ejpam-4582	633	11	λ	λ	X
ejpam-4582	633	12	,	,	PUNCT
ejpam-4582	633	13	s)-closed	s)-close	VERB
ejpam-4582	633	14	,	,	PUNCT
ejpam-4582	633	15	by	by	ADP
ejpam-4582	633	16	theorem	theorem	NOUN
ejpam-4582	633	17	14	14	NUM
ejpam-4582	633	18	,	,	PUNCT
ejpam-4582	633	19	f	f	PROPN
ejpam-4582	633	20	⊆	⊆	NUM
ejpam-4582	633	21	g(λ	g(λ	PROPN
ejpam-4582	633	22	,	,	PUNCT
ejpam-4582	633	23	s	s	NOUN
ejpam-4582	633	24	)	)	PUNCT
ejpam-4582	633	25	.	.	PUNCT
ejpam-4582	634	1	thus	thus	ADV
ejpam-4582	634	2	by	by	ADP
ejpam-4582	634	3	(	(	PUNCT
ejpam-4582	634	4	3	3	NUM
ejpam-4582	634	5	)	)	PUNCT
ejpam-4582	634	6	,	,	PUNCT
ejpam-4582	634	7	there	there	PRON
ejpam-4582	634	8	exists	exist	VERB
ejpam-4582	634	9	a	a	DET
ejpam-4582	634	10	g-(λ	g-(λ	NOUN
ejpam-4582	634	11	,	,	PUNCT
ejpam-4582	634	12	s)-open	s)-open	VERB
ejpam-4582	634	13	set	set	VERB
ejpam-4582	634	14	u	u	PRON
ejpam-4582	634	15	such	such	ADJ
ejpam-4582	634	16	that	that	SCONJ
ejpam-4582	634	17	f	f	PROPN
ejpam-4582	634	18	⊆	⊆	NUM
ejpam-4582	634	19	u	u	NOUN
ejpam-4582	634	20	⊆	⊆	NUM
ejpam-4582	634	21	u	u	NOUN
ejpam-4582	634	22	(	(	PUNCT
ejpam-4582	634	23	λ	λ	PROPN
ejpam-4582	634	24	,	,	PUNCT
ejpam-4582	634	25	s	s	PART
ejpam-4582	634	26	)	)	PUNCT
ejpam-4582	634	27	⊆	⊆	NUM
ejpam-4582	634	28	g(λ	g(λ	PROPN
ejpam-4582	634	29	,	,	PUNCT
ejpam-4582	634	30	s	s	NOUN
ejpam-4582	634	31	)	)	PUNCT
ejpam-4582	634	32	.	.	PUNCT
ejpam-4582	635	1	(	(	PUNCT
ejpam-4582	635	2	5	5	X
ejpam-4582	635	3	)	)	PUNCT
ejpam-4582	635	4	⇒	⇒	NOUN
ejpam-4582	635	5	(	(	PUNCT
ejpam-4582	635	6	6	6	NUM
ejpam-4582	635	7	):	):	PUNCT
ejpam-4582	635	8	let	let	VERB
ejpam-4582	635	9	f	f	PRON
ejpam-4582	635	10	be	be	AUX
ejpam-4582	635	11	a	a	DET
ejpam-4582	635	12	g-(λ	g-(λ	NOUN
ejpam-4582	635	13	,	,	PUNCT
ejpam-4582	635	14	s)-closed	s)-close	VERB
ejpam-4582	635	15	set	set	NOUN
ejpam-4582	635	16	and	and	CCONJ
ejpam-4582	635	17	g	g	NOUN
ejpam-4582	635	18	be	be	AUX
ejpam-4582	635	19	a	a	DET
ejpam-4582	635	20	(	(	PUNCT
ejpam-4582	635	21	λ	λ	X
ejpam-4582	635	22	,	,	PUNCT
ejpam-4582	635	23	s)-open	s)-open	VERB
ejpam-4582	635	24	set	set	VERB
ejpam-4582	635	25	containing	contain	VERB
ejpam-4582	635	26	f	f	PROPN
ejpam-4582	635	27	.	.	PUNCT
ejpam-4582	636	1	then	then	ADV
ejpam-4582	636	2	,	,	PUNCT
ejpam-4582	636	3	we	we	PRON
ejpam-4582	636	4	have	have	VERB
ejpam-4582	636	5	f	f	PROPN
ejpam-4582	636	6	(	(	PUNCT
ejpam-4582	636	7	λ	λ	PROPN
ejpam-4582	636	8	,	,	PUNCT
ejpam-4582	636	9	s	s	PART
ejpam-4582	636	10	)	)	PUNCT
ejpam-4582	636	11	⊆	⊆	NUM
ejpam-4582	636	12	g.	g.	NOUN
ejpam-4582	636	13	since	since	SCONJ
ejpam-4582	636	14	g	g	PROPN
ejpam-4582	636	15	is	be	AUX
ejpam-4582	636	16	g-(λ	g-(λ	PROPN
ejpam-4582	636	17	,	,	PUNCT
ejpam-4582	636	18	s)-open	s)-open	PUNCT
ejpam-4582	636	19	and	and	CCONJ
ejpam-4582	636	20	by	by	ADP
ejpam-4582	636	21	(	(	PUNCT
ejpam-4582	636	22	5	5	NUM
ejpam-4582	636	23	)	)	PUNCT
ejpam-4582	636	24	,	,	PUNCT
ejpam-4582	636	25	there	there	PRON
ejpam-4582	636	26	exists	exist	VERB
ejpam-4582	636	27	a	a	DET
ejpam-4582	636	28	g-(λ	g-(λ	PROPN
ejpam-4582	636	29	,	,	PUNCT
ejpam-4582	636	30	s)open	s)open	PROPN
ejpam-4582	636	31	set	set	VERB
ejpam-4582	636	32	u	u	PRON
ejpam-4582	636	33	such	such	ADJ
ejpam-4582	636	34	that	that	SCONJ
ejpam-4582	636	35	f	f	PROPN
ejpam-4582	636	36	(	(	PUNCT
ejpam-4582	636	37	λ	λ	PROPN
ejpam-4582	636	38	,	,	PUNCT
ejpam-4582	636	39	s	s	PART
ejpam-4582	636	40	)	)	PUNCT
ejpam-4582	636	41	⊆	⊆	NUM
ejpam-4582	636	42	u	u	NOUN
ejpam-4582	636	43	⊆	⊆	NUM
ejpam-4582	636	44	u	u	NOUN
ejpam-4582	636	45	(	(	PUNCT
ejpam-4582	636	46	λ	λ	PROPN
ejpam-4582	636	47	,	,	PUNCT
ejpam-4582	636	48	s	s	PART
ejpam-4582	636	49	)	)	PUNCT
ejpam-4582	636	50	⊆	⊆	NUM
ejpam-4582	636	51	g.	g.	NOUN
ejpam-4582	636	52	since	since	SCONJ
ejpam-4582	636	53	u	u	PROPN
ejpam-4582	636	54	is	be	AUX
ejpam-4582	636	55	g-(λ	g-(λ	PRON
ejpam-4582	636	56	,	,	PUNCT
ejpam-4582	636	57	s)-open	s)-open	PUNCT
ejpam-4582	636	58	and	and	CCONJ
ejpam-4582	636	59	f	f	PROPN
ejpam-4582	636	60	(	(	PUNCT
ejpam-4582	636	61	λ	λ	PROPN
ejpam-4582	636	62	,	,	PUNCT
ejpam-4582	636	63	s	s	PART
ejpam-4582	636	64	)	)	PUNCT
ejpam-4582	636	65	is	be	AUX
ejpam-4582	636	66	(	(	PUNCT
ejpam-4582	636	67	λ	λ	X
ejpam-4582	636	68	,	,	PUNCT
ejpam-4582	636	69	s)-closed	s)-close	VERB
ejpam-4582	636	70	,	,	PUNCT
ejpam-4582	636	71	by	by	ADP
ejpam-4582	636	72	theorem	theorem	NOUN
ejpam-4582	636	73	14	14	NUM
ejpam-4582	636	74	,	,	PUNCT
ejpam-4582	636	75	f	f	PROPN
ejpam-4582	636	76	(	(	PUNCT
ejpam-4582	636	77	λ	λ	PROPN
ejpam-4582	636	78	,	,	PUNCT
ejpam-4582	636	79	s	s	PART
ejpam-4582	636	80	)	)	PUNCT
ejpam-4582	637	1	⊆	⊆	PROPN
ejpam-4582	637	2	u(λ	u(λ	PROPN
ejpam-4582	637	3	,	,	PUNCT
ejpam-4582	637	4	s	s	PART
ejpam-4582	637	5	)	)	PUNCT
ejpam-4582	637	6	.	.	PUNCT
ejpam-4582	638	1	put	put	VERB
ejpam-4582	638	2	v	v	NUM
ejpam-4582	638	3	=	=	SYM
ejpam-4582	638	4	u(λ	u(λ	PROPN
ejpam-4582	638	5	,	,	PUNCT
ejpam-4582	638	6	s	s	PART
ejpam-4582	638	7	)	)	PUNCT
ejpam-4582	638	8	.	.	PUNCT
ejpam-4582	639	1	then	then	ADV
ejpam-4582	639	2	,	,	PUNCT
ejpam-4582	639	3	v	v	NOUN
ejpam-4582	639	4	is	be	AUX
ejpam-4582	639	5	(	(	PUNCT
ejpam-4582	639	6	λ	λ	X
ejpam-4582	639	7	,	,	PUNCT
ejpam-4582	639	8	s)-open	s)-open	PUNCT
ejpam-4582	639	9	and	and	CCONJ
ejpam-4582	639	10	f	f	PROPN
ejpam-4582	639	11	(	(	PUNCT
ejpam-4582	639	12	λ	λ	PROPN
ejpam-4582	639	13	,	,	PUNCT
ejpam-4582	639	14	s	s	PART
ejpam-4582	639	15	)	)	PUNCT
ejpam-4582	639	16	⊆	⊆	NUM
ejpam-4582	639	17	v	v	ADP
ejpam-4582	639	18	⊆	⊆	NUM
ejpam-4582	639	19	v	v	NOUN
ejpam-4582	639	20	(	(	PUNCT
ejpam-4582	639	21	λ	λ	PROPN
ejpam-4582	639	22	,	,	PUNCT
ejpam-4582	639	23	s	s	PART
ejpam-4582	639	24	)	)	PUNCT
ejpam-4582	639	25	=	=	NOUN
ejpam-4582	640	1	[	[	X
ejpam-4582	640	2	u(λ	u(λ	PROPN
ejpam-4582	640	3	,	,	PUNCT
ejpam-4582	640	4	s	s	PART
ejpam-4582	640	5	)	)	PUNCT
ejpam-4582	640	6	]	]	PUNCT
ejpam-4582	640	7	(	(	PUNCT
ejpam-4582	640	8	λ	λ	X
ejpam-4582	640	9	,	,	PUNCT
ejpam-4582	640	10	s	s	PART
ejpam-4582	640	11	)	)	PUNCT
ejpam-4582	640	12	⊆	⊆	NUM
ejpam-4582	640	13	u	u	NOUN
ejpam-4582	640	14	(	(	PUNCT
ejpam-4582	640	15	λ	λ	PROPN
ejpam-4582	640	16	,	,	PUNCT
ejpam-4582	640	17	s	s	PART
ejpam-4582	640	18	)	)	PUNCT
ejpam-4582	640	19	⊆	⊆	NUM
ejpam-4582	640	20	g.	g.	NOUN
ejpam-4582	640	21	(	(	PUNCT
ejpam-4582	640	22	4	4	NUM
ejpam-4582	640	23	)	)	PUNCT
ejpam-4582	640	24	⇒	⇒	NOUN
ejpam-4582	640	25	(	(	PUNCT
ejpam-4582	640	26	5	5	NUM
ejpam-4582	640	27	)	)	PUNCT
ejpam-4582	640	28	and	and	CCONJ
ejpam-4582	640	29	(	(	PUNCT
ejpam-4582	640	30	5	5	X
ejpam-4582	640	31	)	)	PUNCT
ejpam-4582	640	32	⇒	⇒	NOUN
ejpam-4582	640	33	(	(	PUNCT
ejpam-4582	640	34	2	2	NUM
ejpam-4582	640	35	):	):	PUNCT
ejpam-4582	640	36	the	the	DET
ejpam-4582	640	37	proofs	proof	NOUN
ejpam-4582	640	38	are	be	AUX
ejpam-4582	640	39	obvious	obvious	ADJ
ejpam-4582	640	40	.	.	PUNCT
ejpam-4582	641	1	(	(	PUNCT
ejpam-4582	641	2	6	6	NUM
ejpam-4582	641	3	)	)	PUNCT
ejpam-4582	641	4	⇒	⇒	NOUN
ejpam-4582	641	5	(	(	PUNCT
ejpam-4582	641	6	4	4	NUM
ejpam-4582	641	7	):	):	PUNCT
ejpam-4582	641	8	let	let	VERB
ejpam-4582	641	9	f	f	PRON
ejpam-4582	641	10	be	be	AUX
ejpam-4582	641	11	a	a	DET
ejpam-4582	641	12	(	(	PUNCT
ejpam-4582	641	13	λ	λ	X
ejpam-4582	641	14	,	,	PUNCT
ejpam-4582	641	15	s)-closed	s)-close	VERB
ejpam-4582	641	16	set	set	NOUN
ejpam-4582	641	17	and	and	CCONJ
ejpam-4582	641	18	g	g	NOUN
ejpam-4582	641	19	be	be	AUX
ejpam-4582	641	20	a	a	DET
ejpam-4582	641	21	g-(λ	g-(λ	NOUN
ejpam-4582	641	22	,	,	PUNCT
ejpam-4582	641	23	s)-open	s)-open	VERB
ejpam-4582	641	24	set	set	VERB
ejpam-4582	641	25	containing	contain	VERB
ejpam-4582	641	26	f	f	PROPN
ejpam-4582	641	27	.	.	PUNCT
ejpam-4582	642	1	by	by	ADP
ejpam-4582	642	2	theorem	theorem	NOUN
ejpam-4582	642	3	14	14	NUM
ejpam-4582	642	4	,	,	PUNCT
ejpam-4582	642	5	f	f	PROPN
ejpam-4582	642	6	⊆	⊆	NUM
ejpam-4582	642	7	g(λ	g(λ	PROPN
ejpam-4582	642	8	,	,	PUNCT
ejpam-4582	642	9	s	s	NOUN
ejpam-4582	642	10	)	)	PUNCT
ejpam-4582	642	11	.	.	PUNCT
ejpam-4582	643	1	since	since	SCONJ
ejpam-4582	643	2	f	f	PROPN
ejpam-4582	643	3	is	be	AUX
ejpam-4582	643	4	g-(λ	g-(λ	PROPN
ejpam-4582	643	5	,	,	PUNCT
ejpam-4582	643	6	s)-closed	s)-close	VERB
ejpam-4582	643	7	and	and	CCONJ
ejpam-4582	643	8	g(λ	g(λ	PROPN
ejpam-4582	643	9	,	,	PUNCT
ejpam-4582	643	10	s	s	PART
ejpam-4582	643	11	)	)	PUNCT
ejpam-4582	643	12	is	be	AUX
ejpam-4582	643	13	(	(	PUNCT
ejpam-4582	643	14	λ	λ	X
ejpam-4582	643	15	,	,	PUNCT
ejpam-4582	643	16	s)-open	s)-open	VERB
ejpam-4582	643	17	,	,	PUNCT
ejpam-4582	643	18	by	by	ADP
ejpam-4582	643	19	(	(	PUNCT
ejpam-4582	643	20	6	6	NUM
ejpam-4582	643	21	)	)	PUNCT
ejpam-4582	643	22	,	,	PUNCT
ejpam-4582	643	23	there	there	PRON
ejpam-4582	643	24	exists	exist	VERB
ejpam-4582	643	25	a	a	DET
ejpam-4582	643	26	(	(	PUNCT
ejpam-4582	643	27	λ	λ	X
ejpam-4582	643	28	,	,	PUNCT
ejpam-4582	643	29	s)-open	s)-open	VERB
ejpam-4582	643	30	set	set	VERB
ejpam-4582	643	31	u	u	PRON
ejpam-4582	643	32	such	such	ADJ
ejpam-4582	643	33	that	that	SCONJ
ejpam-4582	643	34	f	f	PROPN
ejpam-4582	643	35	=	=	SYM
ejpam-4582	643	36	f	f	PROPN
ejpam-4582	643	37	(	(	PUNCT
ejpam-4582	643	38	λ	λ	PROPN
ejpam-4582	643	39	,	,	PUNCT
ejpam-4582	643	40	s	s	PART
ejpam-4582	643	41	)	)	PUNCT
ejpam-4582	643	42	⊆	⊆	NUM
ejpam-4582	643	43	u	u	NOUN
ejpam-4582	643	44	⊆	⊆	NUM
ejpam-4582	643	45	u	u	NOUN
ejpam-4582	643	46	(	(	PUNCT
ejpam-4582	643	47	λ	λ	PROPN
ejpam-4582	643	48	,	,	PUNCT
ejpam-4582	643	49	s	s	PART
ejpam-4582	643	50	)	)	PUNCT
ejpam-4582	643	51	⊆	⊆	NUM
ejpam-4582	643	52	g(λ	g(λ	PROPN
ejpam-4582	643	53	,	,	PUNCT
ejpam-4582	643	54	s	s	NOUN
ejpam-4582	643	55	)	)	PUNCT
ejpam-4582	643	56	.	.	PUNCT
ejpam-4582	644	1	6	6	X
ejpam-4582	644	2	.	.	X
ejpam-4582	644	3	on	on	ADP
ejpam-4582	644	4	(	(	PUNCT
ejpam-4582	644	5	λ	λ	NOUN
ejpam-4582	644	6	,	,	PUNCT
ejpam-4582	644	7	s)-extremally	s)-extremally	ADV
ejpam-4582	644	8	disconnected	disconnected	ADJ
ejpam-4582	644	9	spaces	space	NOUN
ejpam-4582	644	10	in	in	ADP
ejpam-4582	644	11	this	this	DET
ejpam-4582	644	12	section	section	NOUN
ejpam-4582	644	13	,	,	PUNCT
ejpam-4582	644	14	we	we	PRON
ejpam-4582	644	15	introduce	introduce	VERB
ejpam-4582	644	16	the	the	DET
ejpam-4582	644	17	notion	notion	NOUN
ejpam-4582	644	18	of	of	ADP
ejpam-4582	644	19	(	(	PUNCT
ejpam-4582	644	20	λ	λ	PROPN
ejpam-4582	644	21	,	,	PUNCT
ejpam-4582	644	22	s)-extremally	s)-extremally	ADV
ejpam-4582	644	23	disconnected	disconnected	ADJ
ejpam-4582	644	24	spaces	space	NOUN
ejpam-4582	644	25	and	and	CCONJ
ejpam-4582	644	26	investigate	investigate	VERB
ejpam-4582	644	27	several	several	ADJ
ejpam-4582	644	28	characterizations	characterization	NOUN
ejpam-4582	644	29	of	of	ADP
ejpam-4582	644	30	such	such	ADJ
ejpam-4582	644	31	spaces	space	NOUN
ejpam-4582	644	32	.	.	PUNCT
ejpam-4582	645	1	definition	definition	NOUN
ejpam-4582	645	2	17	17	NUM
ejpam-4582	645	3	.	.	PUNCT
ejpam-4582	646	1	a	a	DET
ejpam-4582	646	2	subset	subset	NOUN
ejpam-4582	646	3	a	a	PRON
ejpam-4582	646	4	of	of	ADP
ejpam-4582	646	5	a	a	DET
ejpam-4582	646	6	topological	topological	ADJ
ejpam-4582	646	7	space	space	NOUN
ejpam-4582	646	8	(	(	PUNCT
ejpam-4582	646	9	x	x	X
ejpam-4582	646	10	,	,	PUNCT
ejpam-4582	646	11	τ	τ	X
ejpam-4582	646	12	)	)	PUNCT
ejpam-4582	646	13	is	be	AUX
ejpam-4582	646	14	said	say	VERB
ejpam-4582	646	15	to	to	PART
ejpam-4582	646	16	be	be	AUX
ejpam-4582	646	17	:	:	PUNCT
ejpam-4582	646	18	(	(	PUNCT
ejpam-4582	646	19	i	i	NOUN
ejpam-4582	646	20	)	)	PUNCT
ejpam-4582	646	21	s(λ	s(λ	PROPN
ejpam-4582	646	22	,	,	PUNCT
ejpam-4582	646	23	s)-open	s)-open	VERB
ejpam-4582	646	24	if	if	SCONJ
ejpam-4582	646	25	a	a	DET
ejpam-4582	646	26	⊆	⊆	NUM
ejpam-4582	646	27	[	[	X
ejpam-4582	646	28	a(λ	a(λ	ADJ
ejpam-4582	646	29	,	,	PUNCT
ejpam-4582	646	30	s	s	NOUN
ejpam-4582	646	31	)	)	PUNCT
ejpam-4582	646	32	]	]	PUNCT
ejpam-4582	647	1	(	(	PUNCT
ejpam-4582	647	2	λ	λ	X
ejpam-4582	647	3	,	,	PUNCT
ejpam-4582	647	4	s	s	PART
ejpam-4582	647	5	)	)	PUNCT
ejpam-4582	647	6	;	;	PUNCT
ejpam-4582	647	7	(	(	PUNCT
ejpam-4582	647	8	ii	ii	NOUN
ejpam-4582	647	9	)	)	PUNCT
ejpam-4582	647	10	p(λ	p(λ	NOUN
ejpam-4582	647	11	,	,	PUNCT
ejpam-4582	647	12	s)-open	s)-open	VERB
ejpam-4582	647	13	if	if	SCONJ
ejpam-4582	647	14	a	a	DET
ejpam-4582	647	15	⊆	⊆	NUM
ejpam-4582	647	16	[	[	X
ejpam-4582	647	17	a(λ	a(λ	ADV
ejpam-4582	647	18	,	,	PUNCT
ejpam-4582	647	19	s)](λ	s)](λ	PROPN
ejpam-4582	647	20	,	,	PUNCT
ejpam-4582	647	21	s	s	PART
ejpam-4582	647	22	)	)	PUNCT
ejpam-4582	647	23	;	;	PUNCT
ejpam-4582	647	24	c.	c.	PROPN
ejpam-4582	647	25	boonpok	boonpok	PROPN
ejpam-4582	647	26	,	,	PUNCT
ejpam-4582	647	27	c.	c.	PROPN
ejpam-4582	647	28	viriyapong	viriyapong	PROPN
ejpam-4582	647	29	/	/	SYM
ejpam-4582	647	30	eur	eur	PROPN
ejpam-4582	647	31	.	.	PUNCT
ejpam-4582	648	1	j.	j.	PROPN
ejpam-4582	648	2	pure	pure	PROPN
ejpam-4582	648	3	appl	appl	PROPN
ejpam-4582	648	4	.	.	PROPN
ejpam-4582	648	5	math	math	PROPN
ejpam-4582	648	6	,	,	PUNCT
ejpam-4582	648	7	16	16	NUM
ejpam-4582	648	8	(	(	PUNCT
ejpam-4582	648	9	1	1	NUM
ejpam-4582	648	10	)	)	PUNCT
ejpam-4582	648	11	(	(	PUNCT
ejpam-4582	648	12	2023	2023	NUM
ejpam-4582	648	13	)	)	PUNCT
ejpam-4582	648	14	,	,	PUNCT
ejpam-4582	648	15	336	336	NUM
ejpam-4582	648	16	-	-	SYM
ejpam-4582	648	17	362	362	NUM
ejpam-4582	648	18	354	354	NUM
ejpam-4582	648	19	(	(	PUNCT
ejpam-4582	648	20	iii	iii	NOUN
ejpam-4582	648	21	)	)	PUNCT
ejpam-4582	648	22	α(λ	α(λ	PROPN
ejpam-4582	648	23	,	,	PUNCT
ejpam-4582	648	24	s)-open	s)-open	VERB
ejpam-4582	648	25	if	if	SCONJ
ejpam-4582	648	26	a	a	DET
ejpam-4582	648	27	⊆	⊆	NUM
ejpam-4582	648	28	[	[	X
ejpam-4582	648	29	[	[	X
ejpam-4582	648	30	a(λ	a(λ	ADJ
ejpam-4582	648	31	,	,	PUNCT
ejpam-4582	648	32	s	s	NOUN
ejpam-4582	648	33	)	)	PUNCT
ejpam-4582	648	34	]	]	PUNCT
ejpam-4582	648	35	(	(	PUNCT
ejpam-4582	648	36	λ	λ	X
ejpam-4582	648	37	,	,	PUNCT
ejpam-4582	648	38	s)](λ	s)](λ	PROPN
ejpam-4582	648	39	,	,	PUNCT
ejpam-4582	648	40	s	s	PART
ejpam-4582	648	41	)	)	PUNCT
ejpam-4582	648	42	;	;	PUNCT
ejpam-4582	648	43	(	(	PUNCT
ejpam-4582	648	44	iv	iv	X
ejpam-4582	648	45	)	)	PUNCT
ejpam-4582	648	46	β(λ	β(λ	NOUN
ejpam-4582	648	47	,	,	PUNCT
ejpam-4582	648	48	s)-open	s)-open	VERB
ejpam-4582	648	49	if	if	SCONJ
ejpam-4582	648	50	a	a	DET
ejpam-4582	648	51	⊆	⊆	NUM
ejpam-4582	648	52	[	[	X
ejpam-4582	648	53	[	[	X
ejpam-4582	648	54	a(λ	a(λ	ADV
ejpam-4582	648	55	,	,	PUNCT
ejpam-4582	648	56	s)](λ	s)](λ	PROPN
ejpam-4582	648	57	,	,	PUNCT
ejpam-4582	648	58	s	s	NOUN
ejpam-4582	648	59	)	)	PUNCT
ejpam-4582	648	60	]	]	PUNCT
ejpam-4582	648	61	(	(	PUNCT
ejpam-4582	648	62	λ	λ	X
ejpam-4582	648	63	,	,	PUNCT
ejpam-4582	648	64	s	s	PART
ejpam-4582	648	65	)	)	PUNCT
ejpam-4582	648	66	;	;	PUNCT
ejpam-4582	648	67	(	(	PUNCT
ejpam-4582	648	68	v	v	NOUN
ejpam-4582	648	69	)	)	PUNCT
ejpam-4582	648	70	b(λ	b(λ	NOUN
ejpam-4582	648	71	,	,	PUNCT
ejpam-4582	648	72	s)-open	s)-open	VERB
ejpam-4582	648	73	set	set	VERB
ejpam-4582	648	74	if	if	SCONJ
ejpam-4582	648	75	a	a	DET
ejpam-4582	648	76	⊆	⊆	NUM
ejpam-4582	648	77	[	[	X
ejpam-4582	648	78	a(λ	a(λ	ADJ
ejpam-4582	648	79	,	,	PUNCT
ejpam-4582	648	80	s	s	NOUN
ejpam-4582	648	81	)	)	PUNCT
ejpam-4582	648	82	]	]	PUNCT
ejpam-4582	648	83	(	(	PUNCT
ejpam-4582	648	84	λ	λ	X
ejpam-4582	648	85	,	,	PUNCT
ejpam-4582	648	86	s	s	PART
ejpam-4582	648	87	)	)	PUNCT
ejpam-4582	648	88	∪	∪	ADP
ejpam-4582	648	89	[	[	X
ejpam-4582	648	90	a(λ	a(λ	ADV
ejpam-4582	648	91	,	,	PUNCT
ejpam-4582	648	92	s)](λ	s)](λ	PROPN
ejpam-4582	648	93	,	,	PUNCT
ejpam-4582	648	94	s	s	NOUN
ejpam-4582	648	95	)	)	PUNCT
ejpam-4582	648	96	.	.	PUNCT
ejpam-4582	649	1	the	the	DET
ejpam-4582	649	2	family	family	NOUN
ejpam-4582	649	3	of	of	ADP
ejpam-4582	649	4	all	all	DET
ejpam-4582	649	5	s(λ	s(λ	NOUN
ejpam-4582	649	6	,	,	PUNCT
ejpam-4582	649	7	s)-open	s)-open	PUNCT
ejpam-4582	649	8	(	(	PUNCT
ejpam-4582	649	9	resp	resp	NOUN
ejpam-4582	649	10	.	.	PUNCT
ejpam-4582	650	1	p(λ	p(λ	NOUN
ejpam-4582	650	2	,	,	PUNCT
ejpam-4582	650	3	s)-open	s)-open	PUNCT
ejpam-4582	650	4	,	,	PUNCT
ejpam-4582	650	5	α(λ	α(λ	PROPN
ejpam-4582	650	6	,	,	PUNCT
ejpam-4582	650	7	s)-open	s)-open	PUNCT
ejpam-4582	650	8	,	,	PUNCT
ejpam-4582	650	9	β(λ	β(λ	X
ejpam-4582	650	10	,	,	PUNCT
ejpam-4582	650	11	s)-open	s)-open	PUNCT
ejpam-4582	650	12	,	,	PUNCT
ejpam-4582	650	13	b(λ	b(λ	PROPN
ejpam-4582	650	14	,	,	PUNCT
ejpam-4582	650	15	s)open	s)open	NOUN
ejpam-4582	650	16	)	)	PUNCT
ejpam-4582	650	17	sets	set	NOUN
ejpam-4582	650	18	in	in	ADP
ejpam-4582	650	19	a	a	DET
ejpam-4582	650	20	topological	topological	ADJ
ejpam-4582	650	21	space	space	NOUN
ejpam-4582	650	22	(	(	PUNCT
ejpam-4582	650	23	x	x	X
ejpam-4582	650	24	,	,	PUNCT
ejpam-4582	650	25	τ	τ	X
ejpam-4582	650	26	)	)	PUNCT
ejpam-4582	650	27	is	be	AUX
ejpam-4582	650	28	denoted	denote	VERB
ejpam-4582	650	29	by	by	ADP
ejpam-4582	650	30	s(λ	s(λ	NOUN
ejpam-4582	650	31	,	,	PUNCT
ejpam-4582	650	32	s)o(x	s)o(x	ADJ
ejpam-4582	650	33	)	)	PUNCT
ejpam-4582	650	34	(	(	PUNCT
ejpam-4582	650	35	resp	resp	NOUN
ejpam-4582	650	36	.	.	PUNCT
ejpam-4582	651	1	p(λ	p(λ	NOUN
ejpam-4582	651	2	,	,	PUNCT
ejpam-4582	651	3	s)o(x	s)o(x	NOUN
ejpam-4582	651	4	)	)	PUNCT
ejpam-4582	651	5	,	,	PUNCT
ejpam-4582	651	6	α(λ	α(λ	PROPN
ejpam-4582	651	7	,	,	PUNCT
ejpam-4582	651	8	s)o(x	s)o(x	NOUN
ejpam-4582	651	9	)	)	PUNCT
ejpam-4582	651	10	,	,	PUNCT
ejpam-4582	651	11	β(λ	β(λ	X
ejpam-4582	651	12	,	,	PUNCT
ejpam-4582	651	13	s)o(x	s)o(x	NOUN
ejpam-4582	651	14	)	)	PUNCT
ejpam-4582	651	15	,	,	PUNCT
ejpam-4582	651	16	b(λ	b(λ	PROPN
ejpam-4582	651	17	,	,	PUNCT
ejpam-4582	651	18	s)o(x	s)o(x	NOUN
ejpam-4582	651	19	)	)	PUNCT
ejpam-4582	651	20	)	)	PUNCT
ejpam-4582	651	21	.	.	PUNCT
ejpam-4582	652	1	the	the	DET
ejpam-4582	652	2	complement	complement	NOUN
ejpam-4582	652	3	of	of	ADP
ejpam-4582	652	4	a	a	DET
ejpam-4582	652	5	s(λ	s(λ	PROPN
ejpam-4582	652	6	,	,	PUNCT
ejpam-4582	652	7	s)-open	s)-open	PUNCT
ejpam-4582	652	8	(	(	PUNCT
ejpam-4582	652	9	resp	resp	NOUN
ejpam-4582	652	10	.	.	PUNCT
ejpam-4582	653	1	p(λ	p(λ	NOUN
ejpam-4582	653	2	,	,	PUNCT
ejpam-4582	653	3	s)-open	s)-open	PUNCT
ejpam-4582	653	4	,	,	PUNCT
ejpam-4582	653	5	α(λ	α(λ	PROPN
ejpam-4582	653	6	,	,	PUNCT
ejpam-4582	653	7	s)-open	s)-open	PUNCT
ejpam-4582	653	8	,	,	PUNCT
ejpam-4582	653	9	β(λ	β(λ	X
ejpam-4582	653	10	,	,	PUNCT
ejpam-4582	653	11	s)-open	s)-open	PUNCT
ejpam-4582	653	12	,	,	PUNCT
ejpam-4582	653	13	b(λ	b(λ	PROPN
ejpam-4582	653	14	,	,	PUNCT
ejpam-4582	653	15	s)-open	s)-open	PUNCT
ejpam-4582	653	16	)	)	PUNCT
ejpam-4582	653	17	set	set	NOUN
ejpam-4582	653	18	is	be	AUX
ejpam-4582	653	19	called	call	VERB
ejpam-4582	653	20	s(λ	s(λ	PROPN
ejpam-4582	653	21	,	,	PUNCT
ejpam-4582	653	22	s)-closed	s)-close	VERB
ejpam-4582	653	23	(	(	PUNCT
ejpam-4582	653	24	resp	resp	NOUN
ejpam-4582	653	25	.	.	PUNCT
ejpam-4582	654	1	p(λ	p(λ	NOUN
ejpam-4582	654	2	,	,	PUNCT
ejpam-4582	654	3	s)-closed	s)-close	VERB
ejpam-4582	654	4	,	,	PUNCT
ejpam-4582	654	5	α(λ	α(λ	PROPN
ejpam-4582	654	6	,	,	PUNCT
ejpam-4582	654	7	s)-closed	s)-close	VERB
ejpam-4582	654	8	,	,	PUNCT
ejpam-4582	654	9	β(λ	β(λ	X
ejpam-4582	654	10	,	,	PUNCT
ejpam-4582	654	11	s)closed	s)close	VERB
ejpam-4582	654	12	,	,	PUNCT
ejpam-4582	654	13	b(λ	b(λ	PROPN
ejpam-4582	654	14	,	,	PUNCT
ejpam-4582	654	15	s)-closed	s)-close	VERB
ejpam-4582	654	16	)	)	PUNCT
ejpam-4582	654	17	.	.	PUNCT
ejpam-4582	655	1	the	the	DET
ejpam-4582	655	2	family	family	NOUN
ejpam-4582	655	3	of	of	ADP
ejpam-4582	655	4	all	all	DET
ejpam-4582	655	5	s(λ	s(λ	NOUN
ejpam-4582	655	6	,	,	PUNCT
ejpam-4582	655	7	s)-closed	s)-close	VERB
ejpam-4582	655	8	(	(	PUNCT
ejpam-4582	655	9	resp	resp	NOUN
ejpam-4582	655	10	.	.	PUNCT
ejpam-4582	656	1	p(λ	p(λ	NOUN
ejpam-4582	656	2	,	,	PUNCT
ejpam-4582	656	3	s)-closed	s)-close	VERB
ejpam-4582	656	4	,	,	PUNCT
ejpam-4582	656	5	α(λ	α(λ	PROPN
ejpam-4582	656	6	,	,	PUNCT
ejpam-4582	656	7	s)-closed	s)-close	VERB
ejpam-4582	656	8	,	,	PUNCT
ejpam-4582	656	9	β(λ	β(λ	X
ejpam-4582	656	10	,	,	PUNCT
ejpam-4582	656	11	s)-closed	s)-close	VERB
ejpam-4582	656	12	,	,	PUNCT
ejpam-4582	656	13	b(λ	b(λ	PROPN
ejpam-4582	656	14	,	,	PUNCT
ejpam-4582	656	15	s)-closed	s)-close	VERB
ejpam-4582	656	16	)	)	PUNCT
ejpam-4582	656	17	sets	set	NOUN
ejpam-4582	656	18	in	in	ADP
ejpam-4582	656	19	a	a	DET
ejpam-4582	656	20	topological	topological	ADJ
ejpam-4582	656	21	space	space	NOUN
ejpam-4582	656	22	(	(	PUNCT
ejpam-4582	656	23	x	x	X
ejpam-4582	656	24	,	,	PUNCT
ejpam-4582	656	25	τ	τ	X
ejpam-4582	656	26	)	)	PUNCT
ejpam-4582	656	27	is	be	AUX
ejpam-4582	656	28	denoted	denote	VERB
ejpam-4582	656	29	by	by	ADP
ejpam-4582	656	30	s(λ	s(λ	PROPN
ejpam-4582	656	31	,	,	PUNCT
ejpam-4582	656	32	s)c(x	s)c(x	NOUN
ejpam-4582	656	33	)	)	PUNCT
ejpam-4582	656	34	(	(	PUNCT
ejpam-4582	656	35	resp	resp	NOUN
ejpam-4582	656	36	.	.	PUNCT
ejpam-4582	657	1	p(λ	p(λ	NOUN
ejpam-4582	657	2	,	,	PUNCT
ejpam-4582	657	3	s)c(x	s)c(x	NOUN
ejpam-4582	657	4	)	)	PUNCT
ejpam-4582	657	5	,	,	PUNCT
ejpam-4582	657	6	α(λ	α(λ	PROPN
ejpam-4582	657	7	,	,	PUNCT
ejpam-4582	657	8	s)c(x	s)c(x	NOUN
ejpam-4582	657	9	)	)	PUNCT
ejpam-4582	657	10	,	,	PUNCT
ejpam-4582	657	11	β(λ	β(λ	X
ejpam-4582	657	12	,	,	PUNCT
ejpam-4582	657	13	s)c(x	s)c(x	NOUN
ejpam-4582	657	14	)	)	PUNCT
ejpam-4582	657	15	,	,	PUNCT
ejpam-4582	657	16	b(λ	b(λ	PROPN
ejpam-4582	657	17	,	,	PUNCT
ejpam-4582	657	18	s)c(x	s)c(x	NOUN
ejpam-4582	657	19	)	)	PUNCT
ejpam-4582	657	20	)	)	PUNCT
ejpam-4582	657	21	.	.	PUNCT
ejpam-4582	658	1	definition	definition	NOUN
ejpam-4582	658	2	18	18	NUM
ejpam-4582	658	3	.	.	PUNCT
ejpam-4582	659	1	a	a	DET
ejpam-4582	659	2	subset	subset	NOUN
ejpam-4582	659	3	a	a	PRON
ejpam-4582	659	4	of	of	ADP
ejpam-4582	659	5	a	a	DET
ejpam-4582	659	6	topological	topological	ADJ
ejpam-4582	659	7	space	space	NOUN
ejpam-4582	659	8	(	(	PUNCT
ejpam-4582	659	9	x	x	X
ejpam-4582	659	10	,	,	PUNCT
ejpam-4582	659	11	τ	τ	X
ejpam-4582	659	12	)	)	PUNCT
ejpam-4582	659	13	is	be	AUX
ejpam-4582	659	14	said	say	VERB
ejpam-4582	659	15	to	to	PART
ejpam-4582	659	16	be	be	AUX
ejpam-4582	659	17	r(λ	r(λ	NOUN
ejpam-4582	659	18	,	,	PUNCT
ejpam-4582	659	19	s)-open	s)-open	PUNCT
ejpam-4582	659	20	(	(	PUNCT
ejpam-4582	659	21	resp	resp	NOUN
ejpam-4582	659	22	.	.	PUNCT
ejpam-4582	660	1	r(λ	r(λ	NOUN
ejpam-4582	660	2	,	,	PUNCT
ejpam-4582	660	3	s)-closed	s)-close	VERB
ejpam-4582	660	4	)	)	PUNCT
ejpam-4582	660	5	if	if	SCONJ
ejpam-4582	660	6	a	a	PRON
ejpam-4582	660	7	=	=	X
ejpam-4582	661	1	[	[	X
ejpam-4582	661	2	a(λ	a(λ	ADV
ejpam-4582	661	3	,	,	PUNCT
ejpam-4582	661	4	s)](λ	s)](λ	PROPN
ejpam-4582	661	5	,	,	PUNCT
ejpam-4582	661	6	s	s	PART
ejpam-4582	661	7	)	)	PUNCT
ejpam-4582	661	8	(	(	PUNCT
ejpam-4582	661	9	resp	resp	NOUN
ejpam-4582	661	10	.	.	PUNCT
ejpam-4582	662	1	a	a	PRON
ejpam-4582	662	2	=	=	X
ejpam-4582	663	1	[	[	X
ejpam-4582	663	2	a(λ	a(λ	ADJ
ejpam-4582	663	3	,	,	PUNCT
ejpam-4582	663	4	s	s	NOUN
ejpam-4582	663	5	)	)	PUNCT
ejpam-4582	663	6	]	]	PUNCT
ejpam-4582	664	1	(	(	PUNCT
ejpam-4582	664	2	λ	λ	X
ejpam-4582	664	3	,	,	PUNCT
ejpam-4582	664	4	s	s	NOUN
ejpam-4582	664	5	)	)	PUNCT
ejpam-4582	664	6	)	)	PUNCT
ejpam-4582	664	7	.	.	PUNCT
ejpam-4582	665	1	the	the	DET
ejpam-4582	665	2	family	family	NOUN
ejpam-4582	665	3	of	of	ADP
ejpam-4582	665	4	all	all	DET
ejpam-4582	665	5	r(λ	r(λ	NOUN
ejpam-4582	665	6	,	,	PUNCT
ejpam-4582	665	7	s)-open	s)-open	PUNCT
ejpam-4582	665	8	(	(	PUNCT
ejpam-4582	665	9	resp	resp	NOUN
ejpam-4582	665	10	.	.	PUNCT
ejpam-4582	666	1	r(λ	r(λ	NOUN
ejpam-4582	666	2	,	,	PUNCT
ejpam-4582	666	3	s)-closed	s)-close	VERB
ejpam-4582	666	4	)	)	PUNCT
ejpam-4582	666	5	sets	set	NOUN
ejpam-4582	666	6	in	in	ADP
ejpam-4582	666	7	a	a	DET
ejpam-4582	666	8	topological	topological	ADJ
ejpam-4582	666	9	space	space	NOUN
ejpam-4582	666	10	(	(	PUNCT
ejpam-4582	666	11	x	x	X
ejpam-4582	666	12	,	,	PUNCT
ejpam-4582	666	13	τ	τ	X
ejpam-4582	666	14	)	)	PUNCT
ejpam-4582	666	15	is	be	AUX
ejpam-4582	666	16	denoted	denote	VERB
ejpam-4582	666	17	by	by	ADP
ejpam-4582	666	18	r(λ	r(λ	PROPN
ejpam-4582	666	19	,	,	PUNCT
ejpam-4582	666	20	s)o(x	s)o(x	PROPN
ejpam-4582	666	21	)	)	PUNCT
ejpam-4582	666	22	(	(	PUNCT
ejpam-4582	666	23	resp	resp	NOUN
ejpam-4582	666	24	.	.	PUNCT
ejpam-4582	667	1	r(λ	r(λ	NOUN
ejpam-4582	667	2	,	,	PUNCT
ejpam-4582	667	3	s)c(x	s)c(x	NOUN
ejpam-4582	667	4	)	)	PUNCT
ejpam-4582	667	5	)	)	PUNCT
ejpam-4582	667	6	.	.	PUNCT
ejpam-4582	668	1	proposition	proposition	NOUN
ejpam-4582	668	2	7	7	NUM
ejpam-4582	668	3	.	.	X
ejpam-4582	668	4	for	for	ADP
ejpam-4582	668	5	a	a	DET
ejpam-4582	668	6	subset	subset	NOUN
ejpam-4582	668	7	a	a	PRON
ejpam-4582	668	8	of	of	ADP
ejpam-4582	668	9	a	a	DET
ejpam-4582	668	10	topological	topological	ADJ
ejpam-4582	668	11	space	space	NOUN
ejpam-4582	668	12	(	(	PUNCT
ejpam-4582	668	13	x	x	X
ejpam-4582	668	14	,	,	PUNCT
ejpam-4582	668	15	τ	τ	PROPN
ejpam-4582	668	16	)	)	PUNCT
ejpam-4582	668	17	,	,	PUNCT
ejpam-4582	668	18	the	the	DET
ejpam-4582	668	19	following	follow	VERB
ejpam-4582	668	20	properties	property	NOUN
ejpam-4582	668	21	are	be	AUX
ejpam-4582	668	22	equivalent	equivalent	ADJ
ejpam-4582	668	23	:	:	PUNCT
ejpam-4582	668	24	(	(	PUNCT
ejpam-4582	668	25	1	1	X
ejpam-4582	668	26	)	)	PUNCT
ejpam-4582	668	27	a	a	DET
ejpam-4582	668	28	∈	∈	NOUN
ejpam-4582	668	29	β(λ	β(λ	X
ejpam-4582	668	30	,	,	PUNCT
ejpam-4582	668	31	s)o(x	s)o(x	NOUN
ejpam-4582	668	32	)	)	PUNCT
ejpam-4582	668	33	;	;	PUNCT
ejpam-4582	668	34	(	(	PUNCT
ejpam-4582	668	35	2	2	X
ejpam-4582	668	36	)	)	PUNCT
ejpam-4582	668	37	a(λ	a(λ	ADV
ejpam-4582	668	38	,	,	PUNCT
ejpam-4582	668	39	s	s	X
ejpam-4582	668	40	)	)	PUNCT
ejpam-4582	668	41	∈	∈	PROPN
ejpam-4582	668	42	r(λ	r(λ	NOUN
ejpam-4582	668	43	,	,	PUNCT
ejpam-4582	668	44	s)c(x	s)c(x	NOUN
ejpam-4582	668	45	)	)	PUNCT
ejpam-4582	668	46	;	;	PUNCT
ejpam-4582	668	47	(	(	PUNCT
ejpam-4582	668	48	3	3	X
ejpam-4582	668	49	)	)	PUNCT
ejpam-4582	668	50	a(λ	a(λ	ADV
ejpam-4582	668	51	,	,	PUNCT
ejpam-4582	668	52	s	s	X
ejpam-4582	668	53	)	)	PUNCT
ejpam-4582	668	54	∈	∈	PROPN
ejpam-4582	668	55	β(λ	β(λ	X
ejpam-4582	668	56	,	,	PUNCT
ejpam-4582	668	57	s)o(x	s)o(x	NOUN
ejpam-4582	668	58	)	)	PUNCT
ejpam-4582	668	59	;	;	PUNCT
ejpam-4582	668	60	(	(	PUNCT
ejpam-4582	668	61	4	4	X
ejpam-4582	668	62	)	)	PUNCT
ejpam-4582	668	63	a(λ	a(λ	ADV
ejpam-4582	668	64	,	,	PUNCT
ejpam-4582	668	65	s	s	X
ejpam-4582	668	66	)	)	PUNCT
ejpam-4582	668	67	∈	∈	PROPN
ejpam-4582	668	68	s(λ	s(λ	PROPN
ejpam-4582	668	69	,	,	PUNCT
ejpam-4582	668	70	s)o(x	s)o(x	NOUN
ejpam-4582	668	71	)	)	PUNCT
ejpam-4582	668	72	;	;	PUNCT
ejpam-4582	668	73	(	(	PUNCT
ejpam-4582	668	74	5	5	NUM
ejpam-4582	668	75	)	)	PUNCT
ejpam-4582	668	76	a(λ	a(λ	ADV
ejpam-4582	668	77	,	,	PUNCT
ejpam-4582	668	78	s	s	X
ejpam-4582	668	79	)	)	PUNCT
ejpam-4582	668	80	∈	∈	PROPN
ejpam-4582	668	81	b(λ	b(λ	NOUN
ejpam-4582	668	82	,	,	PUNCT
ejpam-4582	668	83	s)o(x	s)o(x	NOUN
ejpam-4582	668	84	)	)	PUNCT
ejpam-4582	668	85	.	.	PUNCT
ejpam-4582	669	1	proof	proof	NOUN
ejpam-4582	669	2	.	.	PUNCT
ejpam-4582	670	1	(	(	PUNCT
ejpam-4582	670	2	1	1	X
ejpam-4582	670	3	)	)	PUNCT
ejpam-4582	670	4	⇒	⇒	NOUN
ejpam-4582	670	5	(	(	PUNCT
ejpam-4582	670	6	2	2	NUM
ejpam-4582	670	7	):	):	PUNCT
ejpam-4582	670	8	let	let	VERB
ejpam-4582	670	9	a	a	DET
ejpam-4582	670	10	∈	∈	NOUN
ejpam-4582	670	11	β(λ	β(λ	X
ejpam-4582	670	12	,	,	PUNCT
ejpam-4582	670	13	s)o(x	s)o(x	NOUN
ejpam-4582	670	14	)	)	PUNCT
ejpam-4582	670	15	.	.	PUNCT
ejpam-4582	671	1	then	then	ADV
ejpam-4582	671	2	,	,	PUNCT
ejpam-4582	671	3	we	we	PRON
ejpam-4582	671	4	have	have	VERB
ejpam-4582	671	5	a	a	DET
ejpam-4582	671	6	⊆	⊆	NUM
ejpam-4582	671	7	[	[	X
ejpam-4582	671	8	[	[	X
ejpam-4582	671	9	a(λ	a(λ	ADV
ejpam-4582	671	10	,	,	PUNCT
ejpam-4582	671	11	s)](λ	s)](λ	PROPN
ejpam-4582	671	12	,	,	PUNCT
ejpam-4582	671	13	s	s	NOUN
ejpam-4582	671	14	)	)	PUNCT
ejpam-4582	671	15	]	]	PUNCT
ejpam-4582	672	1	(	(	PUNCT
ejpam-4582	672	2	λ	λ	X
ejpam-4582	672	3	,	,	PUNCT
ejpam-4582	672	4	s	s	PART
ejpam-4582	672	5	)	)	PUNCT
ejpam-4582	672	6	and	and	CCONJ
ejpam-4582	672	7	hence	hence	ADV
ejpam-4582	672	8	a(λ	a(λ	ADV
ejpam-4582	672	9	,	,	PUNCT
ejpam-4582	672	10	s	s	NOUN
ejpam-4582	672	11	)	)	PUNCT
ejpam-4582	672	12	⊆	⊆	NUM
ejpam-4582	673	1	[	[	X
ejpam-4582	673	2	[	[	X
ejpam-4582	673	3	a(λ	a(λ	ADV
ejpam-4582	673	4	,	,	PUNCT
ejpam-4582	673	5	s)](λ	s)](λ	PROPN
ejpam-4582	673	6	,	,	PUNCT
ejpam-4582	673	7	s	s	NOUN
ejpam-4582	673	8	)	)	PUNCT
ejpam-4582	673	9	]	]	PUNCT
ejpam-4582	673	10	(	(	PUNCT
ejpam-4582	673	11	λ	λ	X
ejpam-4582	673	12	,	,	PUNCT
ejpam-4582	673	13	s	s	PART
ejpam-4582	673	14	)	)	PUNCT
ejpam-4582	673	15	⊆	⊆	NUM
ejpam-4582	673	16	a(λ	a(λ	PROPN
ejpam-4582	673	17	,	,	PUNCT
ejpam-4582	673	18	s	s	NOUN
ejpam-4582	673	19	)	)	PUNCT
ejpam-4582	673	20	.	.	PUNCT
ejpam-4582	674	1	thus	thus	ADV
ejpam-4582	674	2	,	,	PUNCT
ejpam-4582	674	3	a(λ	a(λ	ADV
ejpam-4582	674	4	,	,	PUNCT
ejpam-4582	674	5	s	s	PART
ejpam-4582	674	6	)	)	PUNCT
ejpam-4582	674	7	=	=	PUNCT
ejpam-4582	675	1	[	[	X
ejpam-4582	675	2	[	[	X
ejpam-4582	675	3	a(λ	a(λ	ADV
ejpam-4582	675	4	,	,	PUNCT
ejpam-4582	675	5	s)](λ	s)](λ	PROPN
ejpam-4582	675	6	,	,	PUNCT
ejpam-4582	675	7	s	s	NOUN
ejpam-4582	675	8	)	)	PUNCT
ejpam-4582	675	9	]	]	PUNCT
ejpam-4582	675	10	(	(	PUNCT
ejpam-4582	675	11	λ	λ	X
ejpam-4582	675	12	,	,	PUNCT
ejpam-4582	675	13	s	s	PART
ejpam-4582	675	14	)	)	PUNCT
ejpam-4582	675	15	.	.	PUNCT
ejpam-4582	676	1	therefore	therefore	ADV
ejpam-4582	676	2	,	,	PUNCT
ejpam-4582	676	3	a(λ	a(λ	ADV
ejpam-4582	676	4	,	,	PUNCT
ejpam-4582	676	5	s	s	X
ejpam-4582	676	6	)	)	PUNCT
ejpam-4582	676	7	∈	∈	PROPN
ejpam-4582	676	8	r(λ	r(λ	NOUN
ejpam-4582	676	9	,	,	PUNCT
ejpam-4582	676	10	s)c(x	s)c(x	NOUN
ejpam-4582	676	11	)	)	PUNCT
ejpam-4582	676	12	.	.	PUNCT
ejpam-4582	677	1	(	(	PUNCT
ejpam-4582	677	2	2	2	X
ejpam-4582	677	3	)	)	PUNCT
ejpam-4582	677	4	⇒	⇒	NOUN
ejpam-4582	677	5	(	(	PUNCT
ejpam-4582	677	6	3	3	NUM
ejpam-4582	677	7	)	)	PUNCT
ejpam-4582	677	8	⇒	⇒	NOUN
ejpam-4582	677	9	(	(	PUNCT
ejpam-4582	677	10	4	4	NUM
ejpam-4582	677	11	)	)	PUNCT
ejpam-4582	677	12	⇒	⇒	NOUN
ejpam-4582	677	13	(	(	PUNCT
ejpam-4582	677	14	5	5	NUM
ejpam-4582	677	15	):	):	PUNCT
ejpam-4582	677	16	obvious	obvious	ADJ
ejpam-4582	677	17	.	.	PUNCT
ejpam-4582	678	1	(	(	PUNCT
ejpam-4582	678	2	5	5	X
ejpam-4582	678	3	)	)	PUNCT
ejpam-4582	678	4	⇒	⇒	NOUN
ejpam-4582	678	5	(	(	PUNCT
ejpam-4582	678	6	1	1	NUM
ejpam-4582	678	7	):	):	PUNCT
ejpam-4582	678	8	let	let	VERB
ejpam-4582	678	9	a(λ	a(λ	ADV
ejpam-4582	678	10	,	,	PUNCT
ejpam-4582	678	11	s	s	X
ejpam-4582	678	12	)	)	PUNCT
ejpam-4582	678	13	∈	∈	PROPN
ejpam-4582	678	14	b(λ	b(λ	NOUN
ejpam-4582	678	15	,	,	PUNCT
ejpam-4582	678	16	s)o(x	s)o(x	NOUN
ejpam-4582	678	17	)	)	PUNCT
ejpam-4582	678	18	.	.	PUNCT
ejpam-4582	679	1	then	then	ADV
ejpam-4582	679	2	,	,	PUNCT
ejpam-4582	679	3	we	we	PRON
ejpam-4582	679	4	have	have	VERB
ejpam-4582	679	5	a(λ	a(λ	ADV
ejpam-4582	679	6	,	,	PUNCT
ejpam-4582	679	7	s	s	NOUN
ejpam-4582	679	8	)	)	PUNCT
ejpam-4582	679	9	⊆	⊆	NUM
ejpam-4582	680	1	[	[	X
ejpam-4582	680	2	[	[	X
ejpam-4582	680	3	a(λ	a(λ	ADV
ejpam-4582	680	4	,	,	PUNCT
ejpam-4582	680	5	s)](λ	s)](λ	PROPN
ejpam-4582	680	6	,	,	PUNCT
ejpam-4582	680	7	s)](λ	s)](λ	PROPN
ejpam-4582	680	8	,	,	PUNCT
ejpam-4582	680	9	s	s	PART
ejpam-4582	680	10	)	)	PUNCT
ejpam-4582	680	11	∪	∪	ADP
ejpam-4582	680	12	[	[	X
ejpam-4582	680	13	[	[	X
ejpam-4582	680	14	a(λ	a(λ	ADJ
ejpam-4582	680	15	,	,	PUNCT
ejpam-4582	680	16	s)](λ	s)](λ	PROPN
ejpam-4582	680	17	,	,	PUNCT
ejpam-4582	680	18	s	s	NOUN
ejpam-4582	680	19	)	)	PUNCT
ejpam-4582	680	20	]	]	PUNCT
ejpam-4582	680	21	(	(	PUNCT
ejpam-4582	680	22	λ	λ	X
ejpam-4582	680	23	,	,	PUNCT
ejpam-4582	680	24	s	s	PART
ejpam-4582	680	25	)	)	PUNCT
ejpam-4582	680	26	=	=	PUNCT
ejpam-4582	681	1	[	[	X
ejpam-4582	681	2	a(λ	a(λ	ADV
ejpam-4582	681	3	,	,	PUNCT
ejpam-4582	681	4	s)](λ	s)](λ	PROPN
ejpam-4582	681	5	,	,	PUNCT
ejpam-4582	681	6	s	s	PART
ejpam-4582	681	7	)	)	PUNCT
ejpam-4582	681	8	∪	∪	ADP
ejpam-4582	681	9	[	[	X
ejpam-4582	681	10	[	[	X
ejpam-4582	681	11	a(λ	a(λ	ADJ
ejpam-4582	681	12	,	,	PUNCT
ejpam-4582	681	13	s)](λ	s)](λ	PROPN
ejpam-4582	681	14	,	,	PUNCT
ejpam-4582	681	15	s	s	NOUN
ejpam-4582	681	16	)	)	PUNCT
ejpam-4582	681	17	]	]	PUNCT
ejpam-4582	681	18	(	(	PUNCT
ejpam-4582	681	19	λ	λ	X
ejpam-4582	681	20	,	,	PUNCT
ejpam-4582	681	21	s	s	PART
ejpam-4582	681	22	)	)	PUNCT
ejpam-4582	681	23	=	=	PUNCT
ejpam-4582	682	1	[	[	X
ejpam-4582	682	2	[	[	X
ejpam-4582	682	3	a(λ	a(λ	ADV
ejpam-4582	682	4	,	,	PUNCT
ejpam-4582	682	5	s)](λ	s)](λ	PROPN
ejpam-4582	682	6	,	,	PUNCT
ejpam-4582	682	7	s	s	NOUN
ejpam-4582	682	8	)	)	PUNCT
ejpam-4582	682	9	]	]	PUNCT
ejpam-4582	682	10	(	(	PUNCT
ejpam-4582	682	11	λ	λ	X
ejpam-4582	682	12	,	,	PUNCT
ejpam-4582	682	13	s	s	PART
ejpam-4582	682	14	)	)	PUNCT
ejpam-4582	682	15	and	and	CCONJ
ejpam-4582	682	16	hence	hence	ADV
ejpam-4582	682	17	a	a	DET
ejpam-4582	682	18	⊆	⊆	NUM
ejpam-4582	682	19	[	[	X
ejpam-4582	682	20	[	[	X
ejpam-4582	682	21	a(λ	a(λ	ADV
ejpam-4582	682	22	,	,	PUNCT
ejpam-4582	682	23	s)](λ	s)](λ	PROPN
ejpam-4582	682	24	,	,	PUNCT
ejpam-4582	682	25	s	s	NOUN
ejpam-4582	682	26	)	)	PUNCT
ejpam-4582	682	27	]	]	PUNCT
ejpam-4582	682	28	(	(	PUNCT
ejpam-4582	682	29	λ	λ	X
ejpam-4582	682	30	,	,	PUNCT
ejpam-4582	682	31	s	s	PART
ejpam-4582	682	32	)	)	PUNCT
ejpam-4582	682	33	.	.	PUNCT
ejpam-4582	683	1	thus	thus	ADV
ejpam-4582	683	2	,	,	PUNCT
ejpam-4582	683	3	a	a	DET
ejpam-4582	683	4	∈	∈	NOUN
ejpam-4582	683	5	β(λ	β(λ	X
ejpam-4582	683	6	,	,	PUNCT
ejpam-4582	683	7	s)o(x	s)o(x	NOUN
ejpam-4582	683	8	)	)	PUNCT
ejpam-4582	683	9	.	.	PUNCT
ejpam-4582	684	1	c.	c.	PROPN
ejpam-4582	684	2	boonpok	boonpok	PROPN
ejpam-4582	684	3	,	,	PUNCT
ejpam-4582	684	4	c.	c.	PROPN
ejpam-4582	684	5	viriyapong	viriyapong	PROPN
ejpam-4582	684	6	/	/	SYM
ejpam-4582	684	7	eur	eur	PROPN
ejpam-4582	684	8	.	.	PUNCT
ejpam-4582	685	1	j.	j.	PROPN
ejpam-4582	685	2	pure	pure	PROPN
ejpam-4582	685	3	appl	appl	PROPN
ejpam-4582	685	4	.	.	PROPN
ejpam-4582	685	5	math	math	PROPN
ejpam-4582	685	6	,	,	PUNCT
ejpam-4582	685	7	16	16	NUM
ejpam-4582	685	8	(	(	PUNCT
ejpam-4582	685	9	1	1	NUM
ejpam-4582	685	10	)	)	PUNCT
ejpam-4582	685	11	(	(	PUNCT
ejpam-4582	685	12	2023	2023	NUM
ejpam-4582	685	13	)	)	PUNCT
ejpam-4582	685	14	,	,	PUNCT
ejpam-4582	685	15	336	336	NUM
ejpam-4582	685	16	-	-	SYM
ejpam-4582	685	17	362	362	NUM
ejpam-4582	685	18	355	355	NUM
ejpam-4582	685	19	corollary	corollary	NOUN
ejpam-4582	685	20	5	5	NUM
ejpam-4582	685	21	.	.	PUNCT
ejpam-4582	686	1	for	for	ADP
ejpam-4582	686	2	a	a	DET
ejpam-4582	686	3	subset	subset	NOUN
ejpam-4582	686	4	a	a	PRON
ejpam-4582	686	5	of	of	ADP
ejpam-4582	686	6	a	a	DET
ejpam-4582	686	7	topological	topological	ADJ
ejpam-4582	686	8	space	space	NOUN
ejpam-4582	686	9	(	(	PUNCT
ejpam-4582	686	10	x	x	X
ejpam-4582	686	11	,	,	PUNCT
ejpam-4582	686	12	τ	τ	PROPN
ejpam-4582	686	13	)	)	PUNCT
ejpam-4582	686	14	,	,	PUNCT
ejpam-4582	686	15	the	the	DET
ejpam-4582	686	16	following	follow	VERB
ejpam-4582	686	17	properties	property	NOUN
ejpam-4582	686	18	are	be	AUX
ejpam-4582	686	19	equivalent	equivalent	ADJ
ejpam-4582	686	20	:	:	PUNCT
ejpam-4582	686	21	(	(	PUNCT
ejpam-4582	686	22	1	1	X
ejpam-4582	686	23	)	)	PUNCT
ejpam-4582	686	24	a	a	DET
ejpam-4582	686	25	∈	∈	NOUN
ejpam-4582	686	26	β(λ	β(λ	X
ejpam-4582	686	27	,	,	PUNCT
ejpam-4582	686	28	s)c(x	s)c(x	NOUN
ejpam-4582	686	29	)	)	PUNCT
ejpam-4582	686	30	;	;	PUNCT
ejpam-4582	686	31	(	(	PUNCT
ejpam-4582	686	32	2	2	X
ejpam-4582	686	33	)	)	PUNCT
ejpam-4582	686	34	a(λ	a(λ	ADV
ejpam-4582	686	35	,	,	PUNCT
ejpam-4582	686	36	s	s	X
ejpam-4582	686	37	)	)	PUNCT
ejpam-4582	686	38	∈	∈	PROPN
ejpam-4582	686	39	r(λ	r(λ	NOUN
ejpam-4582	686	40	,	,	PUNCT
ejpam-4582	686	41	s)o(x	s)o(x	PROPN
ejpam-4582	686	42	)	)	PUNCT
ejpam-4582	686	43	;	;	PUNCT
ejpam-4582	686	44	(	(	PUNCT
ejpam-4582	686	45	3	3	X
ejpam-4582	686	46	)	)	PUNCT
ejpam-4582	686	47	a(λ	a(λ	ADV
ejpam-4582	686	48	,	,	PUNCT
ejpam-4582	686	49	s	s	X
ejpam-4582	686	50	)	)	PUNCT
ejpam-4582	686	51	∈	∈	PROPN
ejpam-4582	686	52	β(λ	β(λ	NOUN
ejpam-4582	686	53	,	,	PUNCT
ejpam-4582	686	54	s)c(x	s)c(x	NOUN
ejpam-4582	686	55	)	)	PUNCT
ejpam-4582	686	56	;	;	PUNCT
ejpam-4582	686	57	(	(	PUNCT
ejpam-4582	686	58	4	4	X
ejpam-4582	686	59	)	)	PUNCT
ejpam-4582	686	60	a(λ	a(λ	ADV
ejpam-4582	686	61	,	,	PUNCT
ejpam-4582	686	62	s	s	X
ejpam-4582	686	63	)	)	PUNCT
ejpam-4582	686	64	∈	∈	PROPN
ejpam-4582	686	65	s(λ	s(λ	PROPN
ejpam-4582	686	66	,	,	PUNCT
ejpam-4582	686	67	s)c(x	s)c(x	NOUN
ejpam-4582	686	68	)	)	PUNCT
ejpam-4582	686	69	;	;	PUNCT
ejpam-4582	686	70	(	(	PUNCT
ejpam-4582	686	71	5	5	NUM
ejpam-4582	686	72	)	)	PUNCT
ejpam-4582	686	73	a(λ	a(λ	ADV
ejpam-4582	686	74	,	,	PUNCT
ejpam-4582	686	75	s	s	X
ejpam-4582	686	76	)	)	PUNCT
ejpam-4582	686	77	∈	∈	PROPN
ejpam-4582	686	78	b(λ	b(λ	NOUN
ejpam-4582	686	79	,	,	PUNCT
ejpam-4582	686	80	s)c(x	s)c(x	NOUN
ejpam-4582	686	81	)	)	PUNCT
ejpam-4582	686	82	.	.	PUNCT
ejpam-4582	687	1	definition	definition	NOUN
ejpam-4582	687	2	19	19	NUM
ejpam-4582	687	3	.	.	PUNCT
ejpam-4582	688	1	a	a	DET
ejpam-4582	688	2	topological	topological	ADJ
ejpam-4582	688	3	space	space	NOUN
ejpam-4582	688	4	(	(	PUNCT
ejpam-4582	688	5	x	x	X
ejpam-4582	688	6	,	,	PUNCT
ejpam-4582	688	7	τ	τ	X
ejpam-4582	688	8	)	)	PUNCT
ejpam-4582	688	9	is	be	AUX
ejpam-4582	688	10	called	call	VERB
ejpam-4582	688	11	(	(	PUNCT
ejpam-4582	688	12	λ	λ	PROPN
ejpam-4582	688	13	,	,	PUNCT
ejpam-4582	688	14	s)-extremally	s)-extremally	ADV
ejpam-4582	688	15	disconnected	disconnected	ADJ
ejpam-4582	688	16	if	if	SCONJ
ejpam-4582	688	17	u	u	PROPN
ejpam-4582	688	18	(	(	PUNCT
ejpam-4582	688	19	λ	λ	PROPN
ejpam-4582	688	20	,	,	PUNCT
ejpam-4582	688	21	s	s	PART
ejpam-4582	688	22	)	)	PUNCT
ejpam-4582	688	23	is	be	AUX
ejpam-4582	688	24	(	(	PUNCT
ejpam-4582	688	25	λ	λ	X
ejpam-4582	688	26	,	,	PUNCT
ejpam-4582	688	27	s)-open	s)-open	PUNCT
ejpam-4582	688	28	in	in	ADP
ejpam-4582	688	29	x	x	PUNCT
ejpam-4582	688	30	for	for	ADP
ejpam-4582	688	31	every	every	DET
ejpam-4582	688	32	(	(	PUNCT
ejpam-4582	688	33	λ	λ	NOUN
ejpam-4582	688	34	,	,	PUNCT
ejpam-4582	688	35	s)-open	s)-open	VERB
ejpam-4582	688	36	set	set	VERB
ejpam-4582	688	37	u	u	NOUN
ejpam-4582	688	38	of	of	ADP
ejpam-4582	688	39	x.	x.	PROPN
ejpam-4582	688	40	theorem	theorem	VERB
ejpam-4582	688	41	24	24	NUM
ejpam-4582	688	42	.	.	PUNCT
ejpam-4582	689	1	for	for	ADP
ejpam-4582	689	2	a	a	DET
ejpam-4582	689	3	topological	topological	ADJ
ejpam-4582	689	4	space	space	NOUN
ejpam-4582	689	5	(	(	PUNCT
ejpam-4582	689	6	x	x	X
ejpam-4582	689	7	,	,	PUNCT
ejpam-4582	689	8	τ	τ	PROPN
ejpam-4582	689	9	)	)	PUNCT
ejpam-4582	689	10	,	,	PUNCT
ejpam-4582	689	11	the	the	DET
ejpam-4582	689	12	following	follow	VERB
ejpam-4582	689	13	properties	property	NOUN
ejpam-4582	689	14	are	be	AUX
ejpam-4582	689	15	equivalent	equivalent	ADJ
ejpam-4582	689	16	:	:	PUNCT
ejpam-4582	689	17	(	(	PUNCT
ejpam-4582	689	18	1	1	X
ejpam-4582	689	19	)	)	PUNCT
ejpam-4582	689	20	(	(	PUNCT
ejpam-4582	689	21	x	x	X
ejpam-4582	689	22	,	,	PUNCT
ejpam-4582	689	23	τ	τ	X
ejpam-4582	689	24	)	)	PUNCT
ejpam-4582	689	25	is	be	AUX
ejpam-4582	689	26	(	(	PUNCT
ejpam-4582	689	27	λ	λ	X
ejpam-4582	689	28	,	,	PUNCT
ejpam-4582	689	29	s)-extremally	s)-extremally	ADV
ejpam-4582	689	30	disconnected	disconnect	VERB
ejpam-4582	689	31	;	;	PUNCT
ejpam-4582	689	32	(	(	PUNCT
ejpam-4582	689	33	2	2	X
ejpam-4582	689	34	)	)	PUNCT
ejpam-4582	689	35	for	for	ADP
ejpam-4582	689	36	each	each	DET
ejpam-4582	689	37	v	v	X
ejpam-4582	689	38	∈	∈	PROPN
ejpam-4582	689	39	β(λ	β(λ	X
ejpam-4582	689	40	,	,	PUNCT
ejpam-4582	689	41	s)o(x	s)o(x	NOUN
ejpam-4582	689	42	)	)	PUNCT
ejpam-4582	689	43	,	,	PUNCT
ejpam-4582	689	44	v	v	NOUN
ejpam-4582	689	45	(	(	PUNCT
ejpam-4582	689	46	λ	λ	PROPN
ejpam-4582	689	47	,	,	PUNCT
ejpam-4582	689	48	s	s	PART
ejpam-4582	689	49	)	)	PUNCT
ejpam-4582	689	50	∈	∈	PROPN
ejpam-4582	689	51	r(λ	r(λ	NOUN
ejpam-4582	689	52	,	,	PUNCT
ejpam-4582	689	53	s)o(x	s)o(x	PROPN
ejpam-4582	689	54	)	)	PUNCT
ejpam-4582	689	55	;	;	PUNCT
ejpam-4582	689	56	(	(	PUNCT
ejpam-4582	689	57	3	3	X
ejpam-4582	689	58	)	)	PUNCT
ejpam-4582	689	59	for	for	ADP
ejpam-4582	689	60	each	each	DET
ejpam-4582	689	61	v	v	PROPN
ejpam-4582	689	62	∈	∈	PROPN
ejpam-4582	689	63	b(λ	b(λ	NOUN
ejpam-4582	689	64	,	,	PUNCT
ejpam-4582	689	65	s)o(x	s)o(x	NOUN
ejpam-4582	689	66	)	)	PUNCT
ejpam-4582	689	67	,	,	PUNCT
ejpam-4582	689	68	v	v	NOUN
ejpam-4582	689	69	(	(	PUNCT
ejpam-4582	689	70	λ	λ	PROPN
ejpam-4582	689	71	,	,	PUNCT
ejpam-4582	689	72	s	s	PART
ejpam-4582	689	73	)	)	PUNCT
ejpam-4582	689	74	∈	∈	PROPN
ejpam-4582	689	75	r(λ	r(λ	NOUN
ejpam-4582	689	76	,	,	PUNCT
ejpam-4582	689	77	s)o(x	s)o(x	PROPN
ejpam-4582	689	78	)	)	PUNCT
ejpam-4582	689	79	;	;	PUNCT
ejpam-4582	689	80	(	(	PUNCT
ejpam-4582	689	81	4	4	X
ejpam-4582	689	82	)	)	PUNCT
ejpam-4582	689	83	for	for	ADP
ejpam-4582	689	84	each	each	DET
ejpam-4582	689	85	v	v	X
ejpam-4582	689	86	∈	∈	PROPN
ejpam-4582	689	87	s(λ	s(λ	PROPN
ejpam-4582	689	88	,	,	PUNCT
ejpam-4582	689	89	s)o(x	s)o(x	NOUN
ejpam-4582	689	90	)	)	PUNCT
ejpam-4582	689	91	,	,	PUNCT
ejpam-4582	689	92	v	v	NOUN
ejpam-4582	689	93	(	(	PUNCT
ejpam-4582	689	94	λ	λ	PROPN
ejpam-4582	689	95	,	,	PUNCT
ejpam-4582	689	96	s	s	PART
ejpam-4582	689	97	)	)	PUNCT
ejpam-4582	689	98	∈	∈	PROPN
ejpam-4582	689	99	r(λ	r(λ	NOUN
ejpam-4582	689	100	,	,	PUNCT
ejpam-4582	689	101	s)o(x	s)o(x	PROPN
ejpam-4582	689	102	)	)	PUNCT
ejpam-4582	689	103	;	;	PUNCT
ejpam-4582	689	104	(	(	PUNCT
ejpam-4582	689	105	5	5	X
ejpam-4582	689	106	)	)	PUNCT
ejpam-4582	689	107	for	for	ADP
ejpam-4582	689	108	each	each	DET
ejpam-4582	689	109	v	v	X
ejpam-4582	689	110	∈	∈	PROPN
ejpam-4582	689	111	α(λ	α(λ	PROPN
ejpam-4582	689	112	,	,	PUNCT
ejpam-4582	689	113	s)o(x	s)o(x	NOUN
ejpam-4582	689	114	)	)	PUNCT
ejpam-4582	689	115	,	,	PUNCT
ejpam-4582	689	116	v	v	NOUN
ejpam-4582	689	117	(	(	PUNCT
ejpam-4582	689	118	λ	λ	PROPN
ejpam-4582	689	119	,	,	PUNCT
ejpam-4582	689	120	s	s	PART
ejpam-4582	689	121	)	)	PUNCT
ejpam-4582	689	122	∈	∈	PROPN
ejpam-4582	689	123	r(λ	r(λ	NOUN
ejpam-4582	689	124	,	,	PUNCT
ejpam-4582	689	125	s)o(x	s)o(x	PROPN
ejpam-4582	689	126	)	)	PUNCT
ejpam-4582	689	127	;	;	PUNCT
ejpam-4582	689	128	(	(	PUNCT
ejpam-4582	689	129	6	6	X
ejpam-4582	689	130	)	)	PUNCT
ejpam-4582	689	131	for	for	ADP
ejpam-4582	689	132	each	each	PRON
ejpam-4582	689	133	v	v	X
ejpam-4582	689	134	∈	∈	NOUN
ejpam-4582	689	135	(	(	PUNCT
ejpam-4582	689	136	λ	λ	NOUN
ejpam-4582	689	137	,	,	PUNCT
ejpam-4582	689	138	s)o(x	s)o(x	ADJ
ejpam-4582	689	139	)	)	PUNCT
ejpam-4582	689	140	,	,	PUNCT
ejpam-4582	689	141	v	v	NOUN
ejpam-4582	689	142	(	(	PUNCT
ejpam-4582	689	143	λ	λ	PROPN
ejpam-4582	689	144	,	,	PUNCT
ejpam-4582	689	145	s	s	PART
ejpam-4582	689	146	)	)	PUNCT
ejpam-4582	689	147	∈	∈	PROPN
ejpam-4582	689	148	r(λ	r(λ	NOUN
ejpam-4582	689	149	,	,	PUNCT
ejpam-4582	689	150	s)o(x	s)o(x	PROPN
ejpam-4582	689	151	)	)	PUNCT
ejpam-4582	689	152	;	;	PUNCT
ejpam-4582	689	153	(	(	PUNCT
ejpam-4582	689	154	7	7	X
ejpam-4582	689	155	)	)	PUNCT
ejpam-4582	689	156	for	for	ADP
ejpam-4582	689	157	each	each	DET
ejpam-4582	689	158	v	v	NUM
ejpam-4582	689	159	∈	∈	PROPN
ejpam-4582	689	160	r(λ	r(λ	NOUN
ejpam-4582	689	161	,	,	PUNCT
ejpam-4582	689	162	s)o(x	s)o(x	NOUN
ejpam-4582	689	163	)	)	PUNCT
ejpam-4582	689	164	,	,	PUNCT
ejpam-4582	689	165	v	v	NOUN
ejpam-4582	689	166	(	(	PUNCT
ejpam-4582	689	167	λ	λ	PROPN
ejpam-4582	689	168	,	,	PUNCT
ejpam-4582	689	169	s	s	PART
ejpam-4582	689	170	)	)	PUNCT
ejpam-4582	689	171	∈	∈	PROPN
ejpam-4582	689	172	r(λ	r(λ	NOUN
ejpam-4582	689	173	,	,	PUNCT
ejpam-4582	689	174	s)o(x	s)o(x	PROPN
ejpam-4582	689	175	)	)	PUNCT
ejpam-4582	689	176	;	;	PUNCT
ejpam-4582	689	177	(	(	PUNCT
ejpam-4582	689	178	8)	8)	NUM
ejpam-4582	689	179	for	for	ADP
ejpam-4582	689	180	each	each	PRON
ejpam-4582	689	181	v	v	ADP
ejpam-4582	689	182	∈	∈	PROPN
ejpam-4582	689	183	p(λ	p(λ	NOUN
ejpam-4582	689	184	,	,	PUNCT
ejpam-4582	689	185	s)o(x	s)o(x	NOUN
ejpam-4582	689	186	)	)	PUNCT
ejpam-4582	689	187	,	,	PUNCT
ejpam-4582	689	188	v	v	NOUN
ejpam-4582	689	189	(	(	PUNCT
ejpam-4582	689	190	λ	λ	PROPN
ejpam-4582	689	191	,	,	PUNCT
ejpam-4582	689	192	s	s	PART
ejpam-4582	689	193	)	)	PUNCT
ejpam-4582	689	194	∈	∈	PROPN
ejpam-4582	689	195	r(λ	r(λ	NOUN
ejpam-4582	689	196	,	,	PUNCT
ejpam-4582	689	197	s)o(x	s)o(x	NOUN
ejpam-4582	689	198	)	)	PUNCT
ejpam-4582	689	199	.	.	PUNCT
ejpam-4582	690	1	proof	proof	NOUN
ejpam-4582	690	2	.	.	PUNCT
ejpam-4582	691	1	the	the	DET
ejpam-4582	691	2	proof	proof	NOUN
ejpam-4582	691	3	follows	follow	VERB
ejpam-4582	691	4	from	from	ADP
ejpam-4582	691	5	theorem	theorem	ADJ
ejpam-4582	691	6	2	2	NUM
ejpam-4582	691	7	of	of	ADP
ejpam-4582	691	8	[	[	X
ejpam-4582	691	9	3	3	NUM
ejpam-4582	691	10	]	]	PUNCT
ejpam-4582	691	11	.	.	PUNCT
ejpam-4582	692	1	theorem	theorem	PROPN
ejpam-4582	692	2	25	25	NUM
ejpam-4582	692	3	.	.	PUNCT
ejpam-4582	693	1	for	for	ADP
ejpam-4582	693	2	a	a	DET
ejpam-4582	693	3	topological	topological	ADJ
ejpam-4582	693	4	space	space	NOUN
ejpam-4582	693	5	(	(	PUNCT
ejpam-4582	693	6	x	x	X
ejpam-4582	693	7	,	,	PUNCT
ejpam-4582	693	8	τ	τ	PROPN
ejpam-4582	693	9	)	)	PUNCT
ejpam-4582	693	10	,	,	PUNCT
ejpam-4582	693	11	the	the	DET
ejpam-4582	693	12	following	follow	VERB
ejpam-4582	693	13	properties	property	NOUN
ejpam-4582	693	14	are	be	AUX
ejpam-4582	693	15	equivalent	equivalent	ADJ
ejpam-4582	693	16	:	:	PUNCT
ejpam-4582	693	17	(	(	PUNCT
ejpam-4582	693	18	1	1	X
ejpam-4582	693	19	)	)	PUNCT
ejpam-4582	693	20	(	(	PUNCT
ejpam-4582	693	21	x	x	X
ejpam-4582	693	22	,	,	PUNCT
ejpam-4582	693	23	τ	τ	X
ejpam-4582	693	24	)	)	PUNCT
ejpam-4582	693	25	is	be	AUX
ejpam-4582	693	26	(	(	PUNCT
ejpam-4582	693	27	λ	λ	X
ejpam-4582	693	28	,	,	PUNCT
ejpam-4582	693	29	s)-extremally	s)-extremally	ADV
ejpam-4582	693	30	disconnected	disconnect	VERB
ejpam-4582	693	31	;	;	PUNCT
ejpam-4582	693	32	(	(	PUNCT
ejpam-4582	693	33	2	2	X
ejpam-4582	693	34	)	)	PUNCT
ejpam-4582	693	35	r(λ	r(λ	NOUN
ejpam-4582	693	36	,	,	PUNCT
ejpam-4582	693	37	s)c(x	s)c(x	NOUN
ejpam-4582	693	38	)	)	PUNCT
ejpam-4582	693	39	⊆	⊆	NUM
ejpam-4582	693	40	(	(	PUNCT
ejpam-4582	693	41	λ	λ	NOUN
ejpam-4582	693	42	,	,	PUNCT
ejpam-4582	693	43	s)o(x	s)o(x	PROPN
ejpam-4582	693	44	)	)	PUNCT
ejpam-4582	693	45	;	;	PUNCT
ejpam-4582	693	46	(	(	PUNCT
ejpam-4582	693	47	3	3	X
ejpam-4582	693	48	)	)	PUNCT
ejpam-4582	693	49	r(λ	r(λ	NOUN
ejpam-4582	693	50	,	,	PUNCT
ejpam-4582	693	51	s)c(x	s)c(x	NOUN
ejpam-4582	693	52	)	)	PUNCT
ejpam-4582	693	53	⊆	⊆	NUM
ejpam-4582	693	54	α(λ	α(λ	PROPN
ejpam-4582	693	55	,	,	PUNCT
ejpam-4582	693	56	s)o(x	s)o(x	PROPN
ejpam-4582	693	57	)	)	PUNCT
ejpam-4582	693	58	;	;	PUNCT
ejpam-4582	693	59	(	(	PUNCT
ejpam-4582	693	60	4	4	X
ejpam-4582	693	61	)	)	PUNCT
ejpam-4582	693	62	r(λ	r(λ	NOUN
ejpam-4582	693	63	,	,	PUNCT
ejpam-4582	693	64	s)c(x	s)c(x	NOUN
ejpam-4582	693	65	)	)	PUNCT
ejpam-4582	693	66	⊆	⊆	NUM
ejpam-4582	693	67	p(λ	p(λ	NOUN
ejpam-4582	693	68	,	,	PUNCT
ejpam-4582	693	69	s)o(x	s)o(x	NOUN
ejpam-4582	693	70	)	)	PUNCT
ejpam-4582	693	71	;	;	PUNCT
ejpam-4582	693	72	(	(	PUNCT
ejpam-4582	693	73	5	5	X
ejpam-4582	693	74	)	)	PUNCT
ejpam-4582	693	75	s(λ	s(λ	NOUN
ejpam-4582	693	76	,	,	PUNCT
ejpam-4582	693	77	s)o(x	s)o(x	NOUN
ejpam-4582	693	78	)	)	PUNCT
ejpam-4582	693	79	⊆	⊆	NUM
ejpam-4582	693	80	α(λ	α(λ	PROPN
ejpam-4582	693	81	,	,	PUNCT
ejpam-4582	693	82	s)o(x	s)o(x	PROPN
ejpam-4582	693	83	)	)	PUNCT
ejpam-4582	693	84	;	;	PUNCT
ejpam-4582	693	85	(	(	PUNCT
ejpam-4582	693	86	6	6	X
ejpam-4582	693	87	)	)	PUNCT
ejpam-4582	693	88	s(λ	s(λ	NOUN
ejpam-4582	693	89	,	,	PUNCT
ejpam-4582	693	90	s)c(x	s)c(x	NOUN
ejpam-4582	693	91	)	)	PUNCT
ejpam-4582	693	92	⊆	⊆	NUM
ejpam-4582	693	93	α(λ	α(λ	PROPN
ejpam-4582	693	94	,	,	PUNCT
ejpam-4582	693	95	s)c(x	s)c(x	NOUN
ejpam-4582	693	96	)	)	PUNCT
ejpam-4582	693	97	;	;	PUNCT
ejpam-4582	693	98	(	(	PUNCT
ejpam-4582	693	99	7	7	X
ejpam-4582	693	100	)	)	PUNCT
ejpam-4582	693	101	s(λ	s(λ	NOUN
ejpam-4582	693	102	,	,	PUNCT
ejpam-4582	693	103	s)c(x	s)c(x	NOUN
ejpam-4582	693	104	)	)	PUNCT
ejpam-4582	693	105	⊆	⊆	NUM
ejpam-4582	693	106	p(λ	p(λ	NOUN
ejpam-4582	693	107	,	,	PUNCT
ejpam-4582	693	108	s)c(x	s)c(x	NOUN
ejpam-4582	693	109	)	)	PUNCT
ejpam-4582	693	110	;	;	PUNCT
ejpam-4582	693	111	c.	c.	PROPN
ejpam-4582	693	112	boonpok	boonpok	PROPN
ejpam-4582	693	113	,	,	PUNCT
ejpam-4582	693	114	c.	c.	PROPN
ejpam-4582	693	115	viriyapong	viriyapong	PROPN
ejpam-4582	693	116	/	/	SYM
ejpam-4582	693	117	eur	eur	PROPN
ejpam-4582	693	118	.	.	PUNCT
ejpam-4582	694	1	j.	j.	PROPN
ejpam-4582	694	2	pure	pure	PROPN
ejpam-4582	694	3	appl	appl	PROPN
ejpam-4582	694	4	.	.	PROPN
ejpam-4582	694	5	math	math	PROPN
ejpam-4582	694	6	,	,	PUNCT
ejpam-4582	694	7	16	16	NUM
ejpam-4582	694	8	(	(	PUNCT
ejpam-4582	694	9	1	1	NUM
ejpam-4582	694	10	)	)	PUNCT
ejpam-4582	694	11	(	(	PUNCT
ejpam-4582	694	12	2023	2023	NUM
ejpam-4582	694	13	)	)	PUNCT
ejpam-4582	694	14	,	,	PUNCT
ejpam-4582	694	15	336	336	NUM
ejpam-4582	694	16	-	-	SYM
ejpam-4582	694	17	362	362	NUM
ejpam-4582	694	18	356	356	NUM
ejpam-4582	694	19	(	(	PUNCT
ejpam-4582	694	20	8)	8)	NUM
ejpam-4582	694	21	s(λ	s(λ	NOUN
ejpam-4582	694	22	,	,	PUNCT
ejpam-4582	694	23	s)o(x	s)o(x	NOUN
ejpam-4582	694	24	)	)	PUNCT
ejpam-4582	694	25	⊆	⊆	NUM
ejpam-4582	694	26	p(λ	p(λ	NOUN
ejpam-4582	694	27	,	,	PUNCT
ejpam-4582	694	28	s)o(x	s)o(x	NOUN
ejpam-4582	694	29	)	)	PUNCT
ejpam-4582	694	30	;	;	PUNCT
ejpam-4582	694	31	(	(	PUNCT
ejpam-4582	694	32	9	9	X
ejpam-4582	694	33	)	)	PUNCT
ejpam-4582	694	34	β(λ	β(λ	NOUN
ejpam-4582	694	35	,	,	PUNCT
ejpam-4582	694	36	s)o(x	s)o(x	ADJ
ejpam-4582	694	37	)	)	PUNCT
ejpam-4582	694	38	⊆	⊆	NUM
ejpam-4582	694	39	p(λ	p(λ	NOUN
ejpam-4582	694	40	,	,	PUNCT
ejpam-4582	694	41	s)o(x	s)o(x	NOUN
ejpam-4582	694	42	)	)	PUNCT
ejpam-4582	694	43	;	;	PUNCT
ejpam-4582	694	44	(	(	PUNCT
ejpam-4582	694	45	10	10	NUM
ejpam-4582	694	46	)	)	PUNCT
ejpam-4582	694	47	β(λ	β(λ	NOUN
ejpam-4582	694	48	,	,	PUNCT
ejpam-4582	694	49	s)c(x	s)c(x	NOUN
ejpam-4582	694	50	)	)	PUNCT
ejpam-4582	694	51	⊆	⊆	NUM
ejpam-4582	694	52	p(λ	p(λ	NOUN
ejpam-4582	694	53	,	,	PUNCT
ejpam-4582	694	54	s)c(x	s)c(x	NOUN
ejpam-4582	694	55	)	)	PUNCT
ejpam-4582	694	56	;	;	PUNCT
ejpam-4582	694	57	(	(	PUNCT
ejpam-4582	694	58	11	11	NUM
ejpam-4582	694	59	)	)	PUNCT
ejpam-4582	694	60	b(λ	b(λ	NOUN
ejpam-4582	694	61	,	,	PUNCT
ejpam-4582	694	62	s)c(x	s)c(x	NOUN
ejpam-4582	694	63	)	)	PUNCT
ejpam-4582	694	64	⊆	⊆	NUM
ejpam-4582	694	65	p(λ	p(λ	NOUN
ejpam-4582	694	66	,	,	PUNCT
ejpam-4582	694	67	s)c(x	s)c(x	NOUN
ejpam-4582	694	68	)	)	PUNCT
ejpam-4582	694	69	;	;	PUNCT
ejpam-4582	694	70	(	(	PUNCT
ejpam-4582	694	71	12	12	NUM
ejpam-4582	694	72	)	)	PUNCT
ejpam-4582	694	73	b(λ	b(λ	NOUN
ejpam-4582	694	74	,	,	PUNCT
ejpam-4582	694	75	s)o(x	s)o(x	NOUN
ejpam-4582	694	76	)	)	PUNCT
ejpam-4582	694	77	⊆	⊆	NUM
ejpam-4582	694	78	p(λ	p(λ	NOUN
ejpam-4582	694	79	,	,	PUNCT
ejpam-4582	694	80	s)o(x	s)o(x	NOUN
ejpam-4582	694	81	)	)	PUNCT
ejpam-4582	694	82	;	;	PUNCT
ejpam-4582	694	83	(	(	PUNCT
ejpam-4582	694	84	13	13	X
ejpam-4582	694	85	)	)	PUNCT
ejpam-4582	694	86	r(λ	r(λ	NOUN
ejpam-4582	694	87	,	,	PUNCT
ejpam-4582	694	88	s)o(x	s)o(x	NOUN
ejpam-4582	694	89	)	)	PUNCT
ejpam-4582	694	90	⊆	⊆	NUM
ejpam-4582	694	91	p(λ	p(λ	NOUN
ejpam-4582	694	92	,	,	PUNCT
ejpam-4582	694	93	s)c(x	s)c(x	NOUN
ejpam-4582	694	94	)	)	PUNCT
ejpam-4582	694	95	;	;	PUNCT
ejpam-4582	694	96	(	(	PUNCT
ejpam-4582	694	97	14	14	NUM
ejpam-4582	694	98	)	)	PUNCT
ejpam-4582	694	99	r(λ	r(λ	NOUN
ejpam-4582	694	100	,	,	PUNCT
ejpam-4582	694	101	s)o(x	s)o(x	NOUN
ejpam-4582	694	102	)	)	PUNCT
ejpam-4582	694	103	⊆	⊆	NUM
ejpam-4582	694	104	(	(	PUNCT
ejpam-4582	694	105	λ	λ	PROPN
ejpam-4582	694	106	,	,	PUNCT
ejpam-4582	694	107	s)c(x	s)c(x	NOUN
ejpam-4582	694	108	)	)	PUNCT
ejpam-4582	694	109	;	;	PUNCT
ejpam-4582	694	110	(	(	PUNCT
ejpam-4582	694	111	15	15	X
ejpam-4582	694	112	)	)	PUNCT
ejpam-4582	694	113	r(λ	r(λ	NOUN
ejpam-4582	694	114	,	,	PUNCT
ejpam-4582	694	115	s)o(x	s)o(x	NOUN
ejpam-4582	694	116	)	)	PUNCT
ejpam-4582	694	117	⊆	⊆	NUM
ejpam-4582	694	118	α(λ	α(λ	PROPN
ejpam-4582	694	119	,	,	PUNCT
ejpam-4582	694	120	s)c(x	s)c(x	NOUN
ejpam-4582	694	121	)	)	PUNCT
ejpam-4582	694	122	.	.	PUNCT
ejpam-4582	695	1	proof	proof	NOUN
ejpam-4582	695	2	.	.	PUNCT
ejpam-4582	696	1	the	the	DET
ejpam-4582	696	2	proof	proof	NOUN
ejpam-4582	696	3	follows	follow	VERB
ejpam-4582	696	4	from	from	ADP
ejpam-4582	696	5	theorem	theorem	ADJ
ejpam-4582	696	6	3	3	NUM
ejpam-4582	696	7	of	of	ADP
ejpam-4582	696	8	[	[	X
ejpam-4582	696	9	3	3	NUM
ejpam-4582	696	10	]	]	PUNCT
ejpam-4582	696	11	.	.	PUNCT
ejpam-4582	697	1	theorem	theorem	PROPN
ejpam-4582	697	2	26	26	NUM
ejpam-4582	697	3	.	.	PUNCT
ejpam-4582	698	1	for	for	ADP
ejpam-4582	698	2	a	a	DET
ejpam-4582	698	3	topological	topological	ADJ
ejpam-4582	698	4	space	space	NOUN
ejpam-4582	698	5	(	(	PUNCT
ejpam-4582	698	6	x	x	X
ejpam-4582	698	7	,	,	PUNCT
ejpam-4582	698	8	τ	τ	PROPN
ejpam-4582	698	9	)	)	PUNCT
ejpam-4582	698	10	,	,	PUNCT
ejpam-4582	698	11	the	the	DET
ejpam-4582	698	12	following	follow	VERB
ejpam-4582	698	13	properties	property	NOUN
ejpam-4582	698	14	are	be	AUX
ejpam-4582	698	15	equivalent	equivalent	ADJ
ejpam-4582	698	16	:	:	PUNCT
ejpam-4582	698	17	(	(	PUNCT
ejpam-4582	698	18	1	1	X
ejpam-4582	698	19	)	)	PUNCT
ejpam-4582	698	20	(	(	PUNCT
ejpam-4582	698	21	x	x	X
ejpam-4582	698	22	,	,	PUNCT
ejpam-4582	698	23	τ	τ	X
ejpam-4582	698	24	)	)	PUNCT
ejpam-4582	698	25	is	be	AUX
ejpam-4582	698	26	(	(	PUNCT
ejpam-4582	698	27	λ	λ	X
ejpam-4582	698	28	,	,	PUNCT
ejpam-4582	698	29	s)-extremally	s)-extremally	ADV
ejpam-4582	698	30	disconnected	disconnect	VERB
ejpam-4582	698	31	;	;	PUNCT
ejpam-4582	698	32	(	(	PUNCT
ejpam-4582	698	33	2	2	X
ejpam-4582	698	34	)	)	PUNCT
ejpam-4582	698	35	f(λ	f(λ	NOUN
ejpam-4582	698	36	,	,	PUNCT
ejpam-4582	698	37	s	s	PART
ejpam-4582	698	38	)	)	PUNCT
ejpam-4582	698	39	is	be	AUX
ejpam-4582	698	40	(	(	PUNCT
ejpam-4582	698	41	λ	λ	X
ejpam-4582	698	42	,	,	PUNCT
ejpam-4582	698	43	s)-closed	s)-close	VERB
ejpam-4582	698	44	for	for	SCONJ
ejpam-4582	698	45	every	every	DET
ejpam-4582	698	46	(	(	PUNCT
ejpam-4582	698	47	λ	λ	PROPN
ejpam-4582	698	48	,	,	PUNCT
ejpam-4582	698	49	s)-closed	s)-close	VERB
ejpam-4582	698	50	set	set	ADJ
ejpam-4582	698	51	f	f	PROPN
ejpam-4582	698	52	of	of	ADP
ejpam-4582	698	53	x	x	PRON
ejpam-4582	698	54	;	;	PUNCT
ejpam-4582	698	55	(	(	PUNCT
ejpam-4582	698	56	3	3	X
ejpam-4582	698	57	)	)	PUNCT
ejpam-4582	699	1	[	[	X
ejpam-4582	699	2	a(λ	a(λ	ADV
ejpam-4582	699	3	,	,	PUNCT
ejpam-4582	699	4	s	s	NOUN
ejpam-4582	699	5	)	)	PUNCT
ejpam-4582	699	6	]	]	PUNCT
ejpam-4582	700	1	(	(	PUNCT
ejpam-4582	700	2	λ	λ	X
ejpam-4582	700	3	,	,	PUNCT
ejpam-4582	700	4	s	s	PART
ejpam-4582	700	5	)	)	PUNCT
ejpam-4582	700	6	⊆	⊆	NUM
ejpam-4582	701	1	[	[	X
ejpam-4582	701	2	a(λ	a(λ	ADV
ejpam-4582	701	3	,	,	PUNCT
ejpam-4582	701	4	s)](λ	s)](λ	PROPN
ejpam-4582	701	5	,	,	PUNCT
ejpam-4582	701	6	s	s	PART
ejpam-4582	701	7	)	)	PUNCT
ejpam-4582	701	8	for	for	ADP
ejpam-4582	701	9	every	every	DET
ejpam-4582	701	10	subset	subset	NOUN
ejpam-4582	701	11	a	a	PRON
ejpam-4582	701	12	of	of	ADP
ejpam-4582	701	13	x.	x.	NOUN
ejpam-4582	701	14	proof	proof	NOUN
ejpam-4582	701	15	.	.	PUNCT
ejpam-4582	702	1	(	(	PUNCT
ejpam-4582	702	2	1	1	X
ejpam-4582	702	3	)	)	PUNCT
ejpam-4582	702	4	⇒	⇒	NOUN
ejpam-4582	702	5	(	(	PUNCT
ejpam-4582	702	6	2	2	NUM
ejpam-4582	702	7	):	):	PUNCT
ejpam-4582	702	8	let	let	VERB
ejpam-4582	702	9	f	f	PRON
ejpam-4582	702	10	be	be	AUX
ejpam-4582	702	11	any	any	DET
ejpam-4582	702	12	(	(	PUNCT
ejpam-4582	702	13	λ	λ	X
ejpam-4582	702	14	,	,	PUNCT
ejpam-4582	702	15	s)-closed	s)-close	VERB
ejpam-4582	702	16	set	set	NOUN
ejpam-4582	702	17	.	.	PUNCT
ejpam-4582	703	1	then	then	ADV
ejpam-4582	703	2	,	,	PUNCT
ejpam-4582	703	3	we	we	PRON
ejpam-4582	703	4	have	have	VERB
ejpam-4582	703	5	x	x	PROPN
ejpam-4582	703	6	−f	−f	PROPN
ejpam-4582	703	7	is	be	AUX
ejpam-4582	703	8	(	(	PUNCT
ejpam-4582	703	9	λ	λ	X
ejpam-4582	703	10	,	,	PUNCT
ejpam-4582	703	11	s)-open	s)-open	ADJ
ejpam-4582	703	12	.	.	PUNCT
ejpam-4582	704	1	since	since	SCONJ
ejpam-4582	704	2	(	(	PUNCT
ejpam-4582	704	3	x	x	X
ejpam-4582	704	4	,	,	PUNCT
ejpam-4582	704	5	τ	τ	X
ejpam-4582	704	6	)	)	PUNCT
ejpam-4582	704	7	is	be	AUX
ejpam-4582	704	8	(	(	PUNCT
ejpam-4582	704	9	λ	λ	X
ejpam-4582	704	10	,	,	PUNCT
ejpam-4582	704	11	s)-extremally	s)-extremally	ADV
ejpam-4582	704	12	disconnected	disconnect	VERB
ejpam-4582	704	13	,	,	PUNCT
ejpam-4582	704	14	[	[	X
ejpam-4582	704	15	x−f	x−f	X
ejpam-4582	704	16	]	]	X
ejpam-4582	704	17	(	(	PUNCT
ejpam-4582	704	18	λ	λ	X
ejpam-4582	704	19	,	,	PUNCT
ejpam-4582	704	20	s	s	PART
ejpam-4582	704	21	)	)	PUNCT
ejpam-4582	704	22	=	=	SYM
ejpam-4582	704	23	x−f(λ	x−f(λ	PROPN
ejpam-4582	704	24	,	,	PUNCT
ejpam-4582	704	25	s	s	PART
ejpam-4582	704	26	)	)	PUNCT
ejpam-4582	704	27	is	be	AUX
ejpam-4582	704	28	(	(	PUNCT
ejpam-4582	704	29	λ	λ	X
ejpam-4582	704	30	,	,	PUNCT
ejpam-4582	704	31	s)-open	s)-open	PUNCT
ejpam-4582	704	32	and	and	CCONJ
ejpam-4582	704	33	hence	hence	ADV
ejpam-4582	704	34	f(λ	f(λ	NOUN
ejpam-4582	704	35	,	,	PUNCT
ejpam-4582	704	36	s	s	PART
ejpam-4582	704	37	)	)	PUNCT
ejpam-4582	704	38	is	be	AUX
ejpam-4582	704	39	(	(	PUNCT
ejpam-4582	704	40	λ	λ	X
ejpam-4582	704	41	,	,	PUNCT
ejpam-4582	704	42	s)-closed	s)-close	VERB
ejpam-4582	704	43	.	.	PUNCT
ejpam-4582	705	1	(	(	PUNCT
ejpam-4582	705	2	2	2	X
ejpam-4582	705	3	)	)	PUNCT
ejpam-4582	705	4	⇒	⇒	NOUN
ejpam-4582	705	5	(	(	PUNCT
ejpam-4582	705	6	3	3	NUM
ejpam-4582	705	7	):	):	PUNCT
ejpam-4582	705	8	let	let	VERB
ejpam-4582	705	9	a	a	DET
ejpam-4582	705	10	be	be	AUX
ejpam-4582	705	11	any	any	DET
ejpam-4582	705	12	subset	subset	NOUN
ejpam-4582	705	13	of	of	ADP
ejpam-4582	705	14	x.	x.	NOUN
ejpam-4582	705	15	then	then	ADV
ejpam-4582	705	16	,	,	PUNCT
ejpam-4582	705	17	x	x	NOUN
ejpam-4582	705	18	−	−	NOUN
ejpam-4582	705	19	a(λ	a(λ	ADV
ejpam-4582	705	20	,	,	PUNCT
ejpam-4582	705	21	s	s	PART
ejpam-4582	705	22	)	)	PUNCT
ejpam-4582	705	23	is	be	AUX
ejpam-4582	705	24	(	(	PUNCT
ejpam-4582	705	25	λ	λ	X
ejpam-4582	705	26	,	,	PUNCT
ejpam-4582	705	27	s)-closed	s)-close	VERB
ejpam-4582	705	28	and	and	CCONJ
ejpam-4582	705	29	by	by	ADP
ejpam-4582	705	30	(	(	PUNCT
ejpam-4582	705	31	2	2	NUM
ejpam-4582	705	32	)	)	PUNCT
ejpam-4582	705	33	,	,	PUNCT
ejpam-4582	706	1	[	[	X
ejpam-4582	706	2	x	x	X
ejpam-4582	706	3	−a(λ	−a(λ	NOUN
ejpam-4582	706	4	,	,	PUNCT
ejpam-4582	706	5	s)](λ	s)](λ	PROPN
ejpam-4582	706	6	,	,	PUNCT
ejpam-4582	706	7	s	s	PART
ejpam-4582	706	8	)	)	PUNCT
ejpam-4582	706	9	is	be	AUX
ejpam-4582	706	10	(	(	PUNCT
ejpam-4582	706	11	λ	λ	X
ejpam-4582	706	12	,	,	PUNCT
ejpam-4582	706	13	s)-closed	s)-close	VERB
ejpam-4582	706	14	.	.	PUNCT
ejpam-4582	707	1	thus	thus	ADV
ejpam-4582	707	2	,	,	PUNCT
ejpam-4582	707	3	[	[	X
ejpam-4582	707	4	a(λ	a(λ	ADV
ejpam-4582	707	5	,	,	PUNCT
ejpam-4582	707	6	s	s	NOUN
ejpam-4582	707	7	)	)	PUNCT
ejpam-4582	707	8	]	]	PUNCT
ejpam-4582	708	1	(	(	PUNCT
ejpam-4582	708	2	λ	λ	X
ejpam-4582	708	3	,	,	PUNCT
ejpam-4582	708	4	s	s	PART
ejpam-4582	708	5	)	)	PUNCT
ejpam-4582	708	6	is	be	AUX
ejpam-4582	708	7	(	(	PUNCT
ejpam-4582	708	8	λ	λ	X
ejpam-4582	708	9	,	,	PUNCT
ejpam-4582	708	10	s)-open	s)-open	PUNCT
ejpam-4582	708	11	and	and	CCONJ
ejpam-4582	708	12	hence	hence	ADV
ejpam-4582	708	13	[	[	X
ejpam-4582	708	14	a(λ	a(λ	PROPN
ejpam-4582	708	15	,	,	PUNCT
ejpam-4582	708	16	s	s	NOUN
ejpam-4582	708	17	)	)	PUNCT
ejpam-4582	708	18	]	]	PUNCT
ejpam-4582	708	19	(	(	PUNCT
ejpam-4582	708	20	λ	λ	X
ejpam-4582	708	21	,	,	PUNCT
ejpam-4582	708	22	s	s	PART
ejpam-4582	708	23	)	)	PUNCT
ejpam-4582	708	24	⊆	⊆	NUM
ejpam-4582	709	1	[	[	X
ejpam-4582	709	2	a(λ	a(λ	ADV
ejpam-4582	709	3	,	,	PUNCT
ejpam-4582	709	4	s)](λ	s)](λ	PROPN
ejpam-4582	709	5	,	,	PUNCT
ejpam-4582	709	6	s	s	NOUN
ejpam-4582	709	7	)	)	PUNCT
ejpam-4582	709	8	.	.	PUNCT
ejpam-4582	710	1	(	(	PUNCT
ejpam-4582	710	2	3	3	X
ejpam-4582	710	3	)	)	PUNCT
ejpam-4582	710	4	⇒	⇒	NOUN
ejpam-4582	710	5	(	(	PUNCT
ejpam-4582	710	6	1	1	NUM
ejpam-4582	710	7	):	):	PUNCT
ejpam-4582	710	8	let	let	VERB
ejpam-4582	710	9	u	u	PRON
ejpam-4582	710	10	be	be	AUX
ejpam-4582	710	11	any	any	DET
ejpam-4582	710	12	s(λ	s(λ	NOUN
ejpam-4582	710	13	,	,	PUNCT
ejpam-4582	710	14	s)-open	s)-open	PUNCT
ejpam-4582	710	15	set	set	VERB
ejpam-4582	710	16	.	.	PUNCT
ejpam-4582	711	1	thus	thus	ADV
ejpam-4582	711	2	,	,	PUNCT
ejpam-4582	711	3	by	by	ADP
ejpam-4582	711	4	(	(	PUNCT
ejpam-4582	711	5	3	3	NUM
ejpam-4582	711	6	)	)	PUNCT
ejpam-4582	711	7	,	,	PUNCT
ejpam-4582	711	8	u	u	NOUN
ejpam-4582	711	9	(	(	PUNCT
ejpam-4582	711	10	λ	λ	PROPN
ejpam-4582	711	11	,	,	PUNCT
ejpam-4582	711	12	s	s	PART
ejpam-4582	711	13	)	)	PUNCT
ejpam-4582	711	14	=	=	NOUN
ejpam-4582	712	1	[	[	X
ejpam-4582	712	2	u(λ	u(λ	PROPN
ejpam-4582	712	3	,	,	PUNCT
ejpam-4582	712	4	s	s	PART
ejpam-4582	712	5	)	)	PUNCT
ejpam-4582	712	6	]	]	PUNCT
ejpam-4582	712	7	(	(	PUNCT
ejpam-4582	712	8	λ	λ	X
ejpam-4582	712	9	,	,	PUNCT
ejpam-4582	712	10	s	s	PART
ejpam-4582	712	11	)	)	PUNCT
ejpam-4582	712	12	⊆	⊆	NUM
ejpam-4582	712	13	[	[	X
ejpam-4582	712	14	u	u	X
ejpam-4582	712	15	(	(	PUNCT
ejpam-4582	712	16	λ	λ	PROPN
ejpam-4582	712	17	,	,	PUNCT
ejpam-4582	712	18	s)](λ	s)](λ	PROPN
ejpam-4582	712	19	,	,	PUNCT
ejpam-4582	712	20	s	s	PART
ejpam-4582	712	21	)	)	PUNCT
ejpam-4582	712	22	and	and	CCONJ
ejpam-4582	712	23	so	so	ADV
ejpam-4582	712	24	u	u	PROPN
ejpam-4582	712	25	(	(	PUNCT
ejpam-4582	712	26	λ	λ	PROPN
ejpam-4582	712	27	,	,	PUNCT
ejpam-4582	712	28	s	s	PART
ejpam-4582	712	29	)	)	PUNCT
ejpam-4582	712	30	is	be	AUX
ejpam-4582	712	31	(	(	PUNCT
ejpam-4582	712	32	λ	λ	X
ejpam-4582	712	33	,	,	PUNCT
ejpam-4582	712	34	s)-open	s)-open	ADJ
ejpam-4582	712	35	.	.	PUNCT
ejpam-4582	713	1	this	this	PRON
ejpam-4582	713	2	shows	show	VERB
ejpam-4582	713	3	that	that	SCONJ
ejpam-4582	713	4	(	(	PUNCT
ejpam-4582	713	5	x	x	X
ejpam-4582	713	6	,	,	PUNCT
ejpam-4582	713	7	τ	τ	X
ejpam-4582	713	8	)	)	PUNCT
ejpam-4582	713	9	is	be	AUX
ejpam-4582	713	10	(	(	PUNCT
ejpam-4582	713	11	λ	λ	X
ejpam-4582	713	12	,	,	PUNCT
ejpam-4582	713	13	s)-extremally	s)-extremally	ADV
ejpam-4582	713	14	disconnected	disconnected	ADJ
ejpam-4582	713	15	.	.	PUNCT
ejpam-4582	714	1	theorem	theorem	VERB
ejpam-4582	714	2	27	27	NUM
ejpam-4582	714	3	.	.	PUNCT
ejpam-4582	715	1	for	for	ADP
ejpam-4582	715	2	a	a	DET
ejpam-4582	715	3	topological	topological	ADJ
ejpam-4582	715	4	space	space	NOUN
ejpam-4582	715	5	(	(	PUNCT
ejpam-4582	715	6	x	x	X
ejpam-4582	715	7	,	,	PUNCT
ejpam-4582	715	8	τ	τ	PROPN
ejpam-4582	715	9	)	)	PUNCT
ejpam-4582	715	10	,	,	PUNCT
ejpam-4582	715	11	the	the	DET
ejpam-4582	715	12	following	follow	VERB
ejpam-4582	715	13	properties	property	NOUN
ejpam-4582	715	14	are	be	AUX
ejpam-4582	715	15	equivalent	equivalent	ADJ
ejpam-4582	715	16	:	:	PUNCT
ejpam-4582	715	17	(	(	PUNCT
ejpam-4582	715	18	1	1	X
ejpam-4582	715	19	)	)	PUNCT
ejpam-4582	715	20	(	(	PUNCT
ejpam-4582	715	21	x	x	X
ejpam-4582	715	22	,	,	PUNCT
ejpam-4582	715	23	τ	τ	X
ejpam-4582	715	24	)	)	PUNCT
ejpam-4582	715	25	is	be	AUX
ejpam-4582	715	26	(	(	PUNCT
ejpam-4582	715	27	λ	λ	X
ejpam-4582	715	28	,	,	PUNCT
ejpam-4582	715	29	s)-extremally	s)-extremally	ADV
ejpam-4582	715	30	disconnected	disconnect	VERB
ejpam-4582	715	31	;	;	PUNCT
ejpam-4582	715	32	(	(	PUNCT
ejpam-4582	715	33	2	2	X
ejpam-4582	715	34	)	)	PUNCT
ejpam-4582	715	35	for	for	ADP
ejpam-4582	715	36	every	every	DET
ejpam-4582	715	37	(	(	PUNCT
ejpam-4582	715	38	λ	λ	PROPN
ejpam-4582	715	39	,	,	PUNCT
ejpam-4582	715	40	s)-open	s)-open	PUNCT
ejpam-4582	715	41	sets	set	VERB
ejpam-4582	715	42	u1	u1	NOUN
ejpam-4582	715	43	and	and	CCONJ
ejpam-4582	715	44	u2	u2	NOUN
ejpam-4582	715	45	such	such	ADJ
ejpam-4582	715	46	that	that	DET
ejpam-4582	715	47	u1	u1	NOUN
ejpam-4582	715	48	∩	∩	ADJ
ejpam-4582	715	49	u2	u2	NOUN
ejpam-4582	715	50	=	=	PUNCT
ejpam-4582	715	51	∅	∅	NOUN
ejpam-4582	715	52	,	,	PUNCT
ejpam-4582	715	53	there	there	PRON
ejpam-4582	715	54	exist	exist	VERB
ejpam-4582	715	55	disjoint	disjoint	NOUN
ejpam-4582	715	56	(	(	PUNCT
ejpam-4582	715	57	λ	λ	PROPN
ejpam-4582	715	58	,	,	PUNCT
ejpam-4582	715	59	s)-closed	s)-close	VERB
ejpam-4582	715	60	sets	set	NOUN
ejpam-4582	715	61	f1	f1	NOUN
ejpam-4582	715	62	and	and	CCONJ
ejpam-4582	715	63	f2	f2	NOUN
ejpam-4582	715	64	such	such	ADJ
ejpam-4582	715	65	that	that	DET
ejpam-4582	715	66	u1	u1	NOUN
ejpam-4582	715	67	⊆	⊆	NUM
ejpam-4582	715	68	f1	f1	NOUN
ejpam-4582	715	69	and	and	CCONJ
ejpam-4582	715	70	u2	u2	PROPN
ejpam-4582	715	71	⊆	⊆	NUM
ejpam-4582	715	72	f2	f2	NOUN
ejpam-4582	715	73	;	;	PUNCT
ejpam-4582	715	74	(	(	PUNCT
ejpam-4582	715	75	3	3	X
ejpam-4582	715	76	)	)	PUNCT
ejpam-4582	715	77	u	u	NOUN
ejpam-4582	715	78	(	(	PUNCT
ejpam-4582	715	79	λ	λ	PROPN
ejpam-4582	715	80	,	,	PUNCT
ejpam-4582	715	81	s	s	PART
ejpam-4582	715	82	)	)	PUNCT
ejpam-4582	715	83	1	1	NUM
ejpam-4582	715	84	∩	∩	X
ejpam-4582	715	85	u	u	NOUN
ejpam-4582	715	86	(	(	PUNCT
ejpam-4582	715	87	λ	λ	PROPN
ejpam-4582	715	88	,	,	PUNCT
ejpam-4582	715	89	s	s	PART
ejpam-4582	715	90	)	)	PUNCT
ejpam-4582	715	91	2	2	NUM
ejpam-4582	715	92	=	=	NOUN
ejpam-4582	715	93	∅	∅	NOUN
ejpam-4582	715	94	for	for	ADP
ejpam-4582	715	95	every	every	DET
ejpam-4582	715	96	(	(	PUNCT
ejpam-4582	715	97	λ	λ	PROPN
ejpam-4582	715	98	,	,	PUNCT
ejpam-4582	715	99	s)-open	s)-open	PUNCT
ejpam-4582	715	100	sets	set	VERB
ejpam-4582	715	101	u1	u1	NOUN
ejpam-4582	715	102	and	and	CCONJ
ejpam-4582	715	103	u2	u2	NOUN
ejpam-4582	715	104	such	such	ADJ
ejpam-4582	715	105	that	that	DET
ejpam-4582	715	106	u1	u1	NOUN
ejpam-4582	715	107	∩	∩	ADJ
ejpam-4582	715	108	u2	u2	NOUN
ejpam-4582	715	109	=	=	PUNCT
ejpam-4582	715	110	∅	∅	NOUN
ejpam-4582	715	111	;	;	PUNCT
ejpam-4582	715	112	c.	c.	PROPN
ejpam-4582	715	113	boonpok	boonpok	PROPN
ejpam-4582	715	114	,	,	PUNCT
ejpam-4582	715	115	c.	c.	PROPN
ejpam-4582	715	116	viriyapong	viriyapong	PROPN
ejpam-4582	715	117	/	/	SYM
ejpam-4582	715	118	eur	eur	PROPN
ejpam-4582	715	119	.	.	PUNCT
ejpam-4582	716	1	j.	j.	PROPN
ejpam-4582	716	2	pure	pure	PROPN
ejpam-4582	716	3	appl	appl	PROPN
ejpam-4582	716	4	.	.	PROPN
ejpam-4582	716	5	math	math	PROPN
ejpam-4582	716	6	,	,	PUNCT
ejpam-4582	716	7	16	16	NUM
ejpam-4582	716	8	(	(	PUNCT
ejpam-4582	716	9	1	1	NUM
ejpam-4582	716	10	)	)	PUNCT
ejpam-4582	716	11	(	(	PUNCT
ejpam-4582	716	12	2023	2023	NUM
ejpam-4582	716	13	)	)	PUNCT
ejpam-4582	716	14	,	,	PUNCT
ejpam-4582	716	15	336	336	NUM
ejpam-4582	716	16	-	-	SYM
ejpam-4582	716	17	362	362	NUM
ejpam-4582	716	18	357	357	NUM
ejpam-4582	716	19	(	(	PUNCT
ejpam-4582	716	20	4	4	NUM
ejpam-4582	716	21	)	)	PUNCT
ejpam-4582	717	1	[	[	X
ejpam-4582	717	2	[	[	X
ejpam-4582	717	3	a(λ	a(λ	ADV
ejpam-4582	717	4	,	,	PUNCT
ejpam-4582	717	5	s)](λ	s)](λ	PROPN
ejpam-4582	717	6	,	,	PUNCT
ejpam-4582	717	7	s	s	NOUN
ejpam-4582	717	8	)	)	PUNCT
ejpam-4582	717	9	]	]	PUNCT
ejpam-4582	717	10	(	(	PUNCT
ejpam-4582	717	11	λ	λ	X
ejpam-4582	717	12	,	,	PUNCT
ejpam-4582	717	13	s	s	PART
ejpam-4582	717	14	)	)	PUNCT
ejpam-4582	717	15	∩	∩	ADJ
ejpam-4582	717	16	u	u	NOUN
ejpam-4582	717	17	(	(	PUNCT
ejpam-4582	717	18	λ	λ	PROPN
ejpam-4582	717	19	,	,	PUNCT
ejpam-4582	717	20	s	s	PART
ejpam-4582	717	21	)	)	PUNCT
ejpam-4582	717	22	=	=	NOUN
ejpam-4582	717	23	∅	∅	NOUN
ejpam-4582	717	24	for	for	ADP
ejpam-4582	717	25	every	every	DET
ejpam-4582	717	26	subset	subset	NOUN
ejpam-4582	717	27	a	a	PRON
ejpam-4582	717	28	of	of	ADP
ejpam-4582	717	29	x	x	PUNCT
ejpam-4582	717	30	and	and	CCONJ
ejpam-4582	717	31	every	every	DET
ejpam-4582	717	32	(	(	PUNCT
ejpam-4582	717	33	λ	λ	NOUN
ejpam-4582	717	34	,	,	PUNCT
ejpam-4582	717	35	s)-open	s)-open	VERB
ejpam-4582	717	36	set	set	VERB
ejpam-4582	717	37	u	u	PRON
ejpam-4582	717	38	such	such	ADJ
ejpam-4582	717	39	that	that	SCONJ
ejpam-4582	717	40	a	a	DET
ejpam-4582	717	41	∩	∩	ADJ
ejpam-4582	717	42	u	u	NOUN
ejpam-4582	717	43	=	=	NOUN
ejpam-4582	717	44	∅.	∅.	NOUN
ejpam-4582	717	45	proof	proof	NOUN
ejpam-4582	717	46	.	.	PUNCT
ejpam-4582	718	1	(	(	PUNCT
ejpam-4582	718	2	1	1	X
ejpam-4582	718	3	)	)	PUNCT
ejpam-4582	718	4	⇒	⇒	NOUN
ejpam-4582	718	5	(	(	PUNCT
ejpam-4582	718	6	2	2	NUM
ejpam-4582	718	7	):	):	PUNCT
ejpam-4582	718	8	suppose	suppose	VERB
ejpam-4582	718	9	that	that	SCONJ
ejpam-4582	718	10	(	(	PUNCT
ejpam-4582	718	11	x	x	X
ejpam-4582	718	12	,	,	PUNCT
ejpam-4582	718	13	τ	τ	X
ejpam-4582	718	14	)	)	PUNCT
ejpam-4582	718	15	is	be	AUX
ejpam-4582	718	16	(	(	PUNCT
ejpam-4582	718	17	λ	λ	X
ejpam-4582	718	18	,	,	PUNCT
ejpam-4582	718	19	s)-extremally	s)-extremally	ADV
ejpam-4582	718	20	disconnected	disconnected	ADJ
ejpam-4582	718	21	.	.	PUNCT
ejpam-4582	719	1	let	let	VERB
ejpam-4582	719	2	u1	u1	NOUN
ejpam-4582	719	3	and	and	CCONJ
ejpam-4582	719	4	u2	u2	PROPN
ejpam-4582	719	5	be	be	AUX
ejpam-4582	719	6	(	(	PUNCT
ejpam-4582	719	7	λ	λ	X
ejpam-4582	719	8	,	,	PUNCT
ejpam-4582	719	9	s)-open	s)-open	PUNCT
ejpam-4582	719	10	sets	set	VERB
ejpam-4582	719	11	such	such	ADJ
ejpam-4582	719	12	that	that	DET
ejpam-4582	719	13	u1	u1	NOUN
ejpam-4582	719	14	∩	∩	ADJ
ejpam-4582	719	15	u2	u2	NOUN
ejpam-4582	719	16	=	=	PUNCT
ejpam-4582	719	17	∅.	∅.	NOUN
ejpam-4582	719	18	then	then	ADV
ejpam-4582	719	19	,	,	PUNCT
ejpam-4582	719	20	we	we	PRON
ejpam-4582	719	21	have	have	VERB
ejpam-4582	719	22	u	u	NOUN
ejpam-4582	719	23	(	(	PUNCT
ejpam-4582	719	24	λ	λ	PROPN
ejpam-4582	719	25	,	,	PUNCT
ejpam-4582	719	26	s	s	PART
ejpam-4582	719	27	)	)	PUNCT
ejpam-4582	719	28	1	1	NUM
ejpam-4582	719	29	and	and	CCONJ
ejpam-4582	719	30	x	x	SYM
ejpam-4582	719	31	−	−	PROPN
ejpam-4582	719	32	u	u	NOUN
ejpam-4582	719	33	(	(	PUNCT
ejpam-4582	719	34	λ	λ	PROPN
ejpam-4582	719	35	,	,	PUNCT
ejpam-4582	719	36	s	s	PART
ejpam-4582	719	37	)	)	PUNCT
ejpam-4582	719	38	1	1	NUM
ejpam-4582	719	39	are	be	AUX
ejpam-4582	719	40	disjoint	disjoint	NOUN
ejpam-4582	719	41	(	(	PUNCT
ejpam-4582	719	42	λ	λ	X
ejpam-4582	719	43	,	,	PUNCT
ejpam-4582	719	44	s)-closed	s)-close	VERB
ejpam-4582	719	45	sets	set	NOUN
ejpam-4582	719	46	containing	contain	VERB
ejpam-4582	719	47	u1	u1	NOUN
ejpam-4582	719	48	and	and	CCONJ
ejpam-4582	719	49	u2	u2	NOUN
ejpam-4582	719	50	,	,	PUNCT
ejpam-4582	719	51	respectively	respectively	ADV
ejpam-4582	719	52	.	.	PUNCT
ejpam-4582	720	1	(	(	PUNCT
ejpam-4582	720	2	2	2	X
ejpam-4582	720	3	)	)	PUNCT
ejpam-4582	720	4	⇒	⇒	NOUN
ejpam-4582	720	5	(	(	PUNCT
ejpam-4582	720	6	3	3	NUM
ejpam-4582	720	7	):	):	PUNCT
ejpam-4582	720	8	let	let	VERB
ejpam-4582	720	9	u1	u1	NOUN
ejpam-4582	720	10	and	and	CCONJ
ejpam-4582	720	11	u2	u2	PROPN
ejpam-4582	720	12	be	be	AUX
ejpam-4582	720	13	(	(	PUNCT
ejpam-4582	720	14	λ	λ	X
ejpam-4582	720	15	,	,	PUNCT
ejpam-4582	720	16	s)-open	s)-open	PUNCT
ejpam-4582	720	17	sets	set	VERB
ejpam-4582	720	18	such	such	ADJ
ejpam-4582	720	19	that	that	DET
ejpam-4582	720	20	u1	u1	NOUN
ejpam-4582	720	21	∩	∩	ADJ
ejpam-4582	720	22	u2	u2	NOUN
ejpam-4582	720	23	=	=	PUNCT
ejpam-4582	720	24	∅.	∅.	NOUN
ejpam-4582	720	25	by	by	ADP
ejpam-4582	720	26	(	(	PUNCT
ejpam-4582	720	27	2	2	NUM
ejpam-4582	720	28	)	)	PUNCT
ejpam-4582	720	29	,	,	PUNCT
ejpam-4582	720	30	there	there	PRON
ejpam-4582	720	31	exist	exist	VERB
ejpam-4582	720	32	disjoint	disjoint	NOUN
ejpam-4582	720	33	(	(	PUNCT
ejpam-4582	720	34	λ	λ	PROPN
ejpam-4582	720	35	,	,	PUNCT
ejpam-4582	720	36	s)-closed	s)-close	VERB
ejpam-4582	720	37	sets	set	NOUN
ejpam-4582	720	38	f1	f1	NOUN
ejpam-4582	720	39	and	and	CCONJ
ejpam-4582	720	40	f2	f2	NOUN
ejpam-4582	720	41	such	such	ADJ
ejpam-4582	720	42	that	that	DET
ejpam-4582	720	43	u1	u1	NOUN
ejpam-4582	720	44	⊆	⊆	NUM
ejpam-4582	720	45	f1	f1	NOUN
ejpam-4582	720	46	and	and	CCONJ
ejpam-4582	720	47	u2	u2	PROPN
ejpam-4582	720	48	⊆	⊆	NUM
ejpam-4582	720	49	f2	f2	PROPN
ejpam-4582	720	50	.	.	PUNCT
ejpam-4582	721	1	thus	thus	ADV
ejpam-4582	721	2	,	,	PUNCT
ejpam-4582	721	3	u	u	PROPN
ejpam-4582	721	4	(	(	PUNCT
ejpam-4582	721	5	λ	λ	PROPN
ejpam-4582	721	6	,	,	PUNCT
ejpam-4582	721	7	s	s	PART
ejpam-4582	721	8	)	)	PUNCT
ejpam-4582	721	9	1	1	NUM
ejpam-4582	721	10	∩	∩	X
ejpam-4582	721	11	u	u	NOUN
ejpam-4582	721	12	(	(	PUNCT
ejpam-4582	721	13	λ	λ	PROPN
ejpam-4582	721	14	,	,	PUNCT
ejpam-4582	721	15	s	s	PART
ejpam-4582	721	16	)	)	PUNCT
ejpam-4582	721	17	2	2	NUM
ejpam-4582	721	18	⊆	⊆	NUM
ejpam-4582	721	19	f1	f1	NOUN
ejpam-4582	721	20	∩	∩	NOUN
ejpam-4582	721	21	f2	f2	NOUN
ejpam-4582	721	22	=	=	NOUN
ejpam-4582	721	23	∅	∅	NOUN
ejpam-4582	721	24	and	and	CCONJ
ejpam-4582	721	25	hence	hence	ADV
ejpam-4582	721	26	u	u	NOUN
ejpam-4582	721	27	(	(	PUNCT
ejpam-4582	721	28	λ	λ	PROPN
ejpam-4582	721	29	,	,	PUNCT
ejpam-4582	721	30	s	s	PART
ejpam-4582	721	31	)	)	PUNCT
ejpam-4582	721	32	1	1	NUM
ejpam-4582	721	33	∩	∩	X
ejpam-4582	721	34	u	u	NOUN
ejpam-4582	721	35	(	(	PUNCT
ejpam-4582	721	36	λ	λ	PROPN
ejpam-4582	721	37	,	,	PUNCT
ejpam-4582	721	38	s	s	PART
ejpam-4582	721	39	)	)	PUNCT
ejpam-4582	721	40	2	2	NUM
ejpam-4582	721	41	=	=	SYM
ejpam-4582	721	42	∅.	∅.	X
ejpam-4582	721	43	(	(	PUNCT
ejpam-4582	721	44	3	3	NUM
ejpam-4582	721	45	)	)	PUNCT
ejpam-4582	721	46	⇒	⇒	NOUN
ejpam-4582	721	47	(	(	PUNCT
ejpam-4582	721	48	4	4	NUM
ejpam-4582	721	49	):	):	PUNCT
ejpam-4582	721	50	let	let	VERB
ejpam-4582	721	51	a	a	PRON
ejpam-4582	721	52	be	be	AUX
ejpam-4582	721	53	any	any	DET
ejpam-4582	721	54	subset	subset	NOUN
ejpam-4582	721	55	of	of	ADP
ejpam-4582	721	56	x	x	PUNCT
ejpam-4582	721	57	and	and	CCONJ
ejpam-4582	721	58	u	u	NOUN
ejpam-4582	721	59	be	be	VERB
ejpam-4582	721	60	any	any	DET
ejpam-4582	721	61	(	(	PUNCT
ejpam-4582	721	62	λ	λ	X
ejpam-4582	721	63	,	,	PUNCT
ejpam-4582	721	64	s)-open	s)-open	PUNCT
ejpam-4582	721	65	set	set	VERB
ejpam-4582	721	66	such	such	ADJ
ejpam-4582	721	67	that	that	DET
ejpam-4582	721	68	a∩u	a∩u	PROPN
ejpam-4582	721	69	=	=	PUNCT
ejpam-4582	721	70	∅.	∅.	NOUN
ejpam-4582	721	71	since	since	SCONJ
ejpam-4582	721	72	[	[	X
ejpam-4582	721	73	a(λ	a(λ	ADV
ejpam-4582	721	74	,	,	PUNCT
ejpam-4582	721	75	s)](λ	s)](λ	PROPN
ejpam-4582	721	76	,	,	PUNCT
ejpam-4582	721	77	s	s	PART
ejpam-4582	721	78	)	)	PUNCT
ejpam-4582	721	79	is	be	AUX
ejpam-4582	721	80	(	(	PUNCT
ejpam-4582	721	81	λ	λ	X
ejpam-4582	721	82	,	,	PUNCT
ejpam-4582	721	83	s)-open	s)-open	PUNCT
ejpam-4582	721	84	and	and	CCONJ
ejpam-4582	721	85	[	[	X
ejpam-4582	721	86	a(λ	a(λ	ADV
ejpam-4582	721	87	,	,	PUNCT
ejpam-4582	721	88	s)](λ	s)](λ	PROPN
ejpam-4582	721	89	,	,	PUNCT
ejpam-4582	721	90	s	s	NOUN
ejpam-4582	721	91	)	)	PUNCT
ejpam-4582	721	92	∩	∩	ADJ
ejpam-4582	721	93	u	u	NOUN
ejpam-4582	721	94	=	=	PUNCT
ejpam-4582	721	95	∅.	∅.	X
ejpam-4582	721	96	by	by	ADP
ejpam-4582	721	97	(	(	PUNCT
ejpam-4582	721	98	3	3	NUM
ejpam-4582	721	99	)	)	PUNCT
ejpam-4582	721	100	,	,	PUNCT
ejpam-4582	722	1	[	[	X
ejpam-4582	722	2	[	[	X
ejpam-4582	722	3	a(λ	a(λ	ADV
ejpam-4582	722	4	,	,	PUNCT
ejpam-4582	722	5	s)](λ	s)](λ	PROPN
ejpam-4582	722	6	,	,	PUNCT
ejpam-4582	722	7	s	s	NOUN
ejpam-4582	722	8	)	)	PUNCT
ejpam-4582	722	9	]	]	PUNCT
ejpam-4582	722	10	(	(	PUNCT
ejpam-4582	722	11	λ	λ	X
ejpam-4582	722	12	,	,	PUNCT
ejpam-4582	722	13	s	s	PART
ejpam-4582	722	14	)	)	PUNCT
ejpam-4582	722	15	∩	∩	ADJ
ejpam-4582	722	16	u	u	NOUN
ejpam-4582	722	17	(	(	PUNCT
ejpam-4582	722	18	λ	λ	PROPN
ejpam-4582	722	19	,	,	PUNCT
ejpam-4582	722	20	s	s	PART
ejpam-4582	722	21	)	)	PUNCT
ejpam-4582	722	22	=	=	SYM
ejpam-4582	722	23	∅.	∅.	X
ejpam-4582	722	24	(	(	PUNCT
ejpam-4582	722	25	4	4	NUM
ejpam-4582	722	26	)	)	PUNCT
ejpam-4582	722	27	⇒	⇒	NOUN
ejpam-4582	722	28	(	(	PUNCT
ejpam-4582	722	29	1	1	NUM
ejpam-4582	722	30	):	):	PUNCT
ejpam-4582	722	31	let	let	VERB
ejpam-4582	722	32	u	u	PRON
ejpam-4582	722	33	be	be	AUX
ejpam-4582	722	34	any	any	DET
ejpam-4582	722	35	(	(	PUNCT
ejpam-4582	722	36	λ	λ	NOUN
ejpam-4582	722	37	,	,	PUNCT
ejpam-4582	722	38	s)-open	s)-open	PUNCT
ejpam-4582	722	39	set	set	VERB
ejpam-4582	722	40	.	.	PUNCT
ejpam-4582	723	1	then	then	ADV
ejpam-4582	723	2	,	,	PUNCT
ejpam-4582	723	3	we	we	PRON
ejpam-4582	723	4	have	have	VERB
ejpam-4582	723	5	[	[	X
ejpam-4582	723	6	x	x	X
ejpam-4582	723	7	−	−	PROPN
ejpam-4582	723	8	u	u	NOUN
ejpam-4582	723	9	(	(	PUNCT
ejpam-4582	723	10	λ	λ	PROPN
ejpam-4582	723	11	,	,	PUNCT
ejpam-4582	723	12	s	s	PART
ejpam-4582	723	13	)	)	PUNCT
ejpam-4582	723	14	]	]	PUNCT
ejpam-4582	723	15	∩	∩	ADJ
ejpam-4582	723	16	u	u	NOUN
ejpam-4582	723	17	=	=	PUNCT
ejpam-4582	723	18	∅.	∅.	NOUN
ejpam-4582	723	19	since	since	SCONJ
ejpam-4582	723	20	x	x	INTJ
ejpam-4582	723	21	−	−	PROPN
ejpam-4582	723	22	u	u	NOUN
ejpam-4582	723	23	(	(	PUNCT
ejpam-4582	723	24	λ	λ	PROPN
ejpam-4582	723	25	,	,	PUNCT
ejpam-4582	723	26	s	s	PART
ejpam-4582	723	27	)	)	PUNCT
ejpam-4582	723	28	is	be	AUX
ejpam-4582	723	29	(	(	PUNCT
ejpam-4582	723	30	λ	λ	X
ejpam-4582	723	31	,	,	PUNCT
ejpam-4582	723	32	s)-open	s)-open	PUNCT
ejpam-4582	723	33	and	and	CCONJ
ejpam-4582	723	34	by	by	ADP
ejpam-4582	723	35	(	(	PUNCT
ejpam-4582	723	36	4	4	NUM
ejpam-4582	723	37	)	)	PUNCT
ejpam-4582	723	38	,	,	PUNCT
ejpam-4582	724	1	[	[	X
ejpam-4582	724	2	[	[	X
ejpam-4582	724	3	u	u	X
ejpam-4582	724	4	(	(	PUNCT
ejpam-4582	724	5	λ	λ	PROPN
ejpam-4582	724	6	,	,	PUNCT
ejpam-4582	724	7	s)](λ	s)](λ	PROPN
ejpam-4582	724	8	,	,	PUNCT
ejpam-4582	724	9	s	s	NOUN
ejpam-4582	724	10	)	)	PUNCT
ejpam-4582	724	11	]	]	PUNCT
ejpam-4582	724	12	(	(	PUNCT
ejpam-4582	724	13	λ	λ	X
ejpam-4582	724	14	,	,	PUNCT
ejpam-4582	724	15	s	s	PART
ejpam-4582	724	16	)	)	PUNCT
ejpam-4582	724	17	∩	∩	NOUN
ejpam-4582	724	18	[	[	X
ejpam-4582	724	19	x	x	X
ejpam-4582	724	20	−	−	PROPN
ejpam-4582	724	21	u	u	NOUN
ejpam-4582	724	22	(	(	PUNCT
ejpam-4582	724	23	λ	λ	PROPN
ejpam-4582	724	24	,	,	PUNCT
ejpam-4582	724	25	s)](λ	s)](λ	PROPN
ejpam-4582	724	26	,	,	PUNCT
ejpam-4582	724	27	s	s	PART
ejpam-4582	724	28	)	)	PUNCT
ejpam-4582	724	29	=	=	PUNCT
ejpam-4582	724	30	∅.	∅.	NOUN
ejpam-4582	724	31	since	since	SCONJ
ejpam-4582	724	32	u	u	NOUN
ejpam-4582	724	33	is	be	AUX
ejpam-4582	724	34	(	(	PUNCT
ejpam-4582	724	35	λ	λ	X
ejpam-4582	724	36	,	,	PUNCT
ejpam-4582	724	37	s)-open	s)-open	PUNCT
ejpam-4582	724	38	,	,	PUNCT
ejpam-4582	724	39	we	we	PRON
ejpam-4582	724	40	have	have	VERB
ejpam-4582	724	41	u	u	NOUN
ejpam-4582	724	42	(	(	PUNCT
ejpam-4582	724	43	λ	λ	PROPN
ejpam-4582	724	44	,	,	PUNCT
ejpam-4582	724	45	s	s	PART
ejpam-4582	724	46	)	)	PUNCT
ejpam-4582	724	47	∩	∩	NOUN
ejpam-4582	725	1	[	[	X
ejpam-4582	725	2	x	x	X
ejpam-4582	725	3	−	−	PROPN
ejpam-4582	725	4	[	[	X
ejpam-4582	725	5	u	u	X
ejpam-4582	725	6	(	(	PUNCT
ejpam-4582	725	7	λ	λ	PROPN
ejpam-4582	725	8	,	,	PUNCT
ejpam-4582	725	9	s)](λ	s)](λ	PROPN
ejpam-4582	725	10	,	,	PUNCT
ejpam-4582	725	11	s	s	NOUN
ejpam-4582	725	12	)	)	PUNCT
ejpam-4582	725	13	]	]	PUNCT
ejpam-4582	725	14	=	=	PUNCT
ejpam-4582	725	15	∅	∅	NOUN
ejpam-4582	725	16	and	and	CCONJ
ejpam-4582	725	17	hence	hence	ADV
ejpam-4582	725	18	u	u	NOUN
ejpam-4582	725	19	(	(	PUNCT
ejpam-4582	725	20	λ	λ	PROPN
ejpam-4582	725	21	,	,	PUNCT
ejpam-4582	725	22	s	s	PART
ejpam-4582	725	23	)	)	PUNCT
ejpam-4582	725	24	⊆	⊆	NUM
ejpam-4582	726	1	[	[	X
ejpam-4582	726	2	u	u	X
ejpam-4582	726	3	(	(	PUNCT
ejpam-4582	726	4	λ	λ	PROPN
ejpam-4582	726	5	,	,	PUNCT
ejpam-4582	726	6	s)](λ	s)](λ	PROPN
ejpam-4582	726	7	,	,	PUNCT
ejpam-4582	726	8	s	s	NOUN
ejpam-4582	726	9	)	)	PUNCT
ejpam-4582	726	10	.	.	PUNCT
ejpam-4582	727	1	this	this	PRON
ejpam-4582	727	2	implies	imply	VERB
ejpam-4582	727	3	that	that	SCONJ
ejpam-4582	727	4	u	u	PROPN
ejpam-4582	727	5	(	(	PUNCT
ejpam-4582	727	6	λ	λ	PROPN
ejpam-4582	727	7	,	,	PUNCT
ejpam-4582	727	8	s	s	PART
ejpam-4582	727	9	)	)	PUNCT
ejpam-4582	727	10	is	be	AUX
ejpam-4582	727	11	(	(	PUNCT
ejpam-4582	727	12	λ	λ	X
ejpam-4582	727	13	,	,	PUNCT
ejpam-4582	727	14	s)-open	s)-open	ADJ
ejpam-4582	727	15	.	.	PUNCT
ejpam-4582	728	1	thus	thus	ADV
ejpam-4582	728	2	,	,	PUNCT
ejpam-4582	728	3	(	(	PUNCT
ejpam-4582	728	4	x	x	X
ejpam-4582	728	5	,	,	PUNCT
ejpam-4582	728	6	τ	τ	X
ejpam-4582	728	7	)	)	PUNCT
ejpam-4582	728	8	is	be	AUX
ejpam-4582	728	9	(	(	PUNCT
ejpam-4582	728	10	λ	λ	X
ejpam-4582	728	11	,	,	PUNCT
ejpam-4582	728	12	s)-extremally	s)-extremally	ADV
ejpam-4582	728	13	disconnected	disconnected	ADJ
ejpam-4582	728	14	.	.	PUNCT
ejpam-4582	729	1	theorem	theorem	PROPN
ejpam-4582	729	2	28	28	NUM
ejpam-4582	729	3	.	.	PUNCT
ejpam-4582	730	1	for	for	ADP
ejpam-4582	730	2	a	a	DET
ejpam-4582	730	3	topological	topological	ADJ
ejpam-4582	730	4	space	space	NOUN
ejpam-4582	730	5	(	(	PUNCT
ejpam-4582	730	6	x	x	X
ejpam-4582	730	7	,	,	PUNCT
ejpam-4582	730	8	τ	τ	PROPN
ejpam-4582	730	9	)	)	PUNCT
ejpam-4582	730	10	,	,	PUNCT
ejpam-4582	730	11	the	the	DET
ejpam-4582	730	12	following	follow	VERB
ejpam-4582	730	13	properties	property	NOUN
ejpam-4582	730	14	are	be	AUX
ejpam-4582	730	15	equivalent	equivalent	ADJ
ejpam-4582	730	16	:	:	PUNCT
ejpam-4582	730	17	(	(	PUNCT
ejpam-4582	730	18	1	1	X
ejpam-4582	730	19	)	)	PUNCT
ejpam-4582	730	20	(	(	PUNCT
ejpam-4582	730	21	x	x	X
ejpam-4582	730	22	,	,	PUNCT
ejpam-4582	730	23	τ	τ	X
ejpam-4582	730	24	)	)	PUNCT
ejpam-4582	730	25	is	be	AUX
ejpam-4582	730	26	(	(	PUNCT
ejpam-4582	730	27	λ	λ	X
ejpam-4582	730	28	,	,	PUNCT
ejpam-4582	730	29	s)-extremally	s)-extremally	ADV
ejpam-4582	730	30	disconnected	disconnect	VERB
ejpam-4582	730	31	;	;	PUNCT
ejpam-4582	730	32	(	(	PUNCT
ejpam-4582	730	33	2	2	X
ejpam-4582	730	34	)	)	PUNCT
ejpam-4582	730	35	for	for	ADP
ejpam-4582	730	36	every	every	DET
ejpam-4582	730	37	r(λ	r(λ	NOUN
ejpam-4582	730	38	,	,	PUNCT
ejpam-4582	730	39	s)-open	s)-open	VERB
ejpam-4582	730	40	set	set	VERB
ejpam-4582	730	41	of	of	ADP
ejpam-4582	730	42	x	x	PUNCT
ejpam-4582	730	43	is	be	AUX
ejpam-4582	730	44	(	(	PUNCT
ejpam-4582	730	45	λ	λ	X
ejpam-4582	730	46	,	,	PUNCT
ejpam-4582	730	47	s)-closed	s)-close	VERB
ejpam-4582	730	48	;	;	PUNCT
ejpam-4582	730	49	(	(	PUNCT
ejpam-4582	730	50	3	3	X
ejpam-4582	730	51	)	)	PUNCT
ejpam-4582	730	52	for	for	ADP
ejpam-4582	730	53	every	every	DET
ejpam-4582	730	54	r(λ	r(λ	NOUN
ejpam-4582	730	55	,	,	PUNCT
ejpam-4582	730	56	s)-closed	s)-close	VERB
ejpam-4582	730	57	set	set	NOUN
ejpam-4582	730	58	of	of	ADP
ejpam-4582	730	59	x	x	PUNCT
ejpam-4582	730	60	is	be	AUX
ejpam-4582	730	61	(	(	PUNCT
ejpam-4582	730	62	λ	λ	X
ejpam-4582	730	63	,	,	PUNCT
ejpam-4582	730	64	s)-open	s)-open	PUNCT
ejpam-4582	730	65	.	.	PUNCT
ejpam-4582	731	1	proof	proof	NOUN
ejpam-4582	731	2	.	.	PUNCT
ejpam-4582	732	1	(	(	PUNCT
ejpam-4582	732	2	1	1	X
ejpam-4582	732	3	)	)	PUNCT
ejpam-4582	732	4	⇒	⇒	NOUN
ejpam-4582	732	5	(	(	PUNCT
ejpam-4582	732	6	2	2	NUM
ejpam-4582	732	7	):	):	PUNCT
ejpam-4582	732	8	suppose	suppose	VERB
ejpam-4582	732	9	that	that	SCONJ
ejpam-4582	732	10	(	(	PUNCT
ejpam-4582	732	11	x	x	X
ejpam-4582	732	12	,	,	PUNCT
ejpam-4582	732	13	τ	τ	X
ejpam-4582	732	14	)	)	PUNCT
ejpam-4582	732	15	is	be	AUX
ejpam-4582	732	16	(	(	PUNCT
ejpam-4582	732	17	λ	λ	X
ejpam-4582	732	18	,	,	PUNCT
ejpam-4582	732	19	s)-extremally	s)-extremally	ADV
ejpam-4582	732	20	disconnected	disconnected	ADJ
ejpam-4582	732	21	.	.	PUNCT
ejpam-4582	733	1	let	let	VERB
ejpam-4582	733	2	u	u	PRON
ejpam-4582	733	3	be	be	AUX
ejpam-4582	733	4	any	any	DET
ejpam-4582	733	5	r(λ	r(λ	NOUN
ejpam-4582	733	6	,	,	PUNCT
ejpam-4582	733	7	s)-open	s)-open	PUNCT
ejpam-4582	733	8	set	set	VERB
ejpam-4582	733	9	of	of	ADP
ejpam-4582	733	10	x.	x.	NOUN
ejpam-4582	733	11	then	then	ADV
ejpam-4582	733	12	,	,	PUNCT
ejpam-4582	733	13	we	we	PRON
ejpam-4582	733	14	have	have	VERB
ejpam-4582	733	15	u	u	NOUN
ejpam-4582	733	16	=	=	PUNCT
ejpam-4582	734	1	[	[	X
ejpam-4582	734	2	u	u	X
ejpam-4582	734	3	(	(	PUNCT
ejpam-4582	734	4	λ	λ	PROPN
ejpam-4582	734	5	,	,	PUNCT
ejpam-4582	734	6	s)](λ	s)](λ	PROPN
ejpam-4582	734	7	,	,	PUNCT
ejpam-4582	734	8	s	s	NOUN
ejpam-4582	734	9	)	)	PUNCT
ejpam-4582	734	10	.	.	PUNCT
ejpam-4582	735	1	since	since	SCONJ
ejpam-4582	735	2	u	u	NOUN
ejpam-4582	735	3	is	be	AUX
ejpam-4582	735	4	(	(	PUNCT
ejpam-4582	735	5	λ	λ	INTJ
ejpam-4582	735	6	,	,	PUNCT
ejpam-4582	735	7	s)-open	s)-open	PUNCT
ejpam-4582	735	8	,	,	PUNCT
ejpam-4582	735	9	u	u	NOUN
ejpam-4582	735	10	(	(	PUNCT
ejpam-4582	735	11	λ	λ	PROPN
ejpam-4582	735	12	,	,	PUNCT
ejpam-4582	735	13	s	s	PART
ejpam-4582	735	14	)	)	PUNCT
ejpam-4582	735	15	is	be	AUX
ejpam-4582	735	16	(	(	PUNCT
ejpam-4582	735	17	λ	λ	X
ejpam-4582	735	18	,	,	PUNCT
ejpam-4582	735	19	s)-open	s)-open	VERB
ejpam-4582	735	20	.	.	PUNCT
ejpam-4582	736	1	thus	thus	ADV
ejpam-4582	736	2	,	,	PUNCT
ejpam-4582	736	3	u	u	NOUN
ejpam-4582	736	4	=	=	PUNCT
ejpam-4582	737	1	[	[	X
ejpam-4582	737	2	u	u	X
ejpam-4582	737	3	(	(	PUNCT
ejpam-4582	737	4	λ	λ	PROPN
ejpam-4582	737	5	,	,	PUNCT
ejpam-4582	737	6	s)](λ	s)](λ	PROPN
ejpam-4582	737	7	,	,	PUNCT
ejpam-4582	737	8	s	s	PART
ejpam-4582	737	9	)	)	PUNCT
ejpam-4582	737	10	=	=	SYM
ejpam-4582	737	11	u	u	NOUN
ejpam-4582	737	12	(	(	PUNCT
ejpam-4582	737	13	λ	λ	PROPN
ejpam-4582	737	14	,	,	PUNCT
ejpam-4582	737	15	s	s	PART
ejpam-4582	737	16	)	)	PUNCT
ejpam-4582	737	17	and	and	CCONJ
ejpam-4582	737	18	hence	hence	ADV
ejpam-4582	737	19	u	u	NOUN
ejpam-4582	737	20	is	be	AUX
ejpam-4582	737	21	(	(	PUNCT
ejpam-4582	737	22	λ	λ	X
ejpam-4582	737	23	,	,	PUNCT
ejpam-4582	737	24	s)-closed	s)-close	VERB
ejpam-4582	737	25	.	.	PUNCT
ejpam-4582	738	1	(	(	PUNCT
ejpam-4582	738	2	2	2	X
ejpam-4582	738	3	)	)	PUNCT
ejpam-4582	738	4	⇒	⇒	NOUN
ejpam-4582	738	5	(	(	PUNCT
ejpam-4582	738	6	1	1	NUM
ejpam-4582	738	7	):	):	PUNCT
ejpam-4582	738	8	suppose	suppose	VERB
ejpam-4582	738	9	that	that	SCONJ
ejpam-4582	738	10	for	for	ADP
ejpam-4582	738	11	every	every	DET
ejpam-4582	738	12	r(λ	r(λ	NOUN
ejpam-4582	738	13	,	,	PUNCT
ejpam-4582	738	14	s)-open	s)-open	VERB
ejpam-4582	738	15	set	set	VERB
ejpam-4582	738	16	of	of	ADP
ejpam-4582	738	17	x	x	PUNCT
ejpam-4582	738	18	is	be	AUX
ejpam-4582	738	19	(	(	PUNCT
ejpam-4582	738	20	λ	λ	X
ejpam-4582	738	21	,	,	PUNCT
ejpam-4582	738	22	s)-closed	s)-close	VERB
ejpam-4582	738	23	.	.	PUNCT
ejpam-4582	739	1	let	let	VERB
ejpam-4582	739	2	u	u	PRON
ejpam-4582	739	3	be	be	AUX
ejpam-4582	739	4	any	any	DET
ejpam-4582	739	5	(	(	PUNCT
ejpam-4582	739	6	λ	λ	NOUN
ejpam-4582	739	7	,	,	PUNCT
ejpam-4582	739	8	s)-open	s)-open	PUNCT
ejpam-4582	739	9	set	set	VERB
ejpam-4582	739	10	.	.	PUNCT
ejpam-4582	740	1	since	since	SCONJ
ejpam-4582	740	2	[	[	X
ejpam-4582	740	3	u	u	X
ejpam-4582	740	4	(	(	PUNCT
ejpam-4582	740	5	λ	λ	PROPN
ejpam-4582	740	6	,	,	PUNCT
ejpam-4582	740	7	s)](λ	s)](λ	PROPN
ejpam-4582	740	8	,	,	PUNCT
ejpam-4582	740	9	s	s	PART
ejpam-4582	740	10	)	)	PUNCT
ejpam-4582	740	11	is	be	AUX
ejpam-4582	740	12	r(λ	r(λ	NOUN
ejpam-4582	740	13	,	,	PUNCT
ejpam-4582	740	14	s)-open	s)-open	PUNCT
ejpam-4582	740	15	,	,	PUNCT
ejpam-4582	740	16	we	we	PRON
ejpam-4582	740	17	have	have	VERB
ejpam-4582	740	18	[	[	X
ejpam-4582	740	19	u	u	X
ejpam-4582	740	20	(	(	PUNCT
ejpam-4582	740	21	λ	λ	PROPN
ejpam-4582	740	22	,	,	PUNCT
ejpam-4582	740	23	s)](λ	s)](λ	PROPN
ejpam-4582	740	24	,	,	PUNCT
ejpam-4582	740	25	s	s	PART
ejpam-4582	740	26	)	)	PUNCT
ejpam-4582	740	27	is	be	AUX
ejpam-4582	740	28	(	(	PUNCT
ejpam-4582	740	29	λ	λ	X
ejpam-4582	740	30	,	,	PUNCT
ejpam-4582	740	31	s)-closed	s)-close	VERB
ejpam-4582	740	32	and	and	CCONJ
ejpam-4582	740	33	hence	hence	ADV
ejpam-4582	740	34	u	u	NOUN
ejpam-4582	740	35	(	(	PUNCT
ejpam-4582	740	36	λ	λ	PROPN
ejpam-4582	740	37	,	,	PUNCT
ejpam-4582	740	38	s	s	PART
ejpam-4582	740	39	)	)	PUNCT
ejpam-4582	740	40	⊆	⊆	NUM
ejpam-4582	741	1	[	[	X
ejpam-4582	741	2	[	[	X
ejpam-4582	741	3	u	u	X
ejpam-4582	741	4	(	(	PUNCT
ejpam-4582	741	5	λ	λ	PROPN
ejpam-4582	741	6	,	,	PUNCT
ejpam-4582	741	7	s)](λ	s)](λ	PROPN
ejpam-4582	741	8	,	,	PUNCT
ejpam-4582	741	9	s	s	NOUN
ejpam-4582	741	10	)	)	PUNCT
ejpam-4582	741	11	]	]	PUNCT
ejpam-4582	741	12	(	(	PUNCT
ejpam-4582	741	13	λ	λ	X
ejpam-4582	741	14	,	,	PUNCT
ejpam-4582	741	15	s	s	PART
ejpam-4582	741	16	)	)	PUNCT
ejpam-4582	741	17	=	=	PUNCT
ejpam-4582	742	1	[	[	X
ejpam-4582	742	2	u	u	X
ejpam-4582	742	3	(	(	PUNCT
ejpam-4582	742	4	λ	λ	PROPN
ejpam-4582	742	5	,	,	PUNCT
ejpam-4582	742	6	s)](λ	s)](λ	PROPN
ejpam-4582	742	7	,	,	PUNCT
ejpam-4582	742	8	s	s	NOUN
ejpam-4582	742	9	)	)	PUNCT
ejpam-4582	742	10	.	.	PUNCT
ejpam-4582	743	1	thus	thus	ADV
ejpam-4582	743	2	,	,	PUNCT
ejpam-4582	743	3	u	u	PROPN
ejpam-4582	743	4	(	(	PUNCT
ejpam-4582	743	5	λ	λ	PROPN
ejpam-4582	743	6	,	,	PUNCT
ejpam-4582	743	7	s	s	PART
ejpam-4582	743	8	)	)	PUNCT
ejpam-4582	743	9	is	be	AUX
ejpam-4582	743	10	(	(	PUNCT
ejpam-4582	743	11	λ	λ	X
ejpam-4582	743	12	,	,	PUNCT
ejpam-4582	743	13	s)-open	s)-open	ADJ
ejpam-4582	743	14	.	.	PUNCT
ejpam-4582	744	1	this	this	PRON
ejpam-4582	744	2	shows	show	VERB
ejpam-4582	744	3	that	that	SCONJ
ejpam-4582	744	4	(	(	PUNCT
ejpam-4582	744	5	x	x	X
ejpam-4582	744	6	,	,	PUNCT
ejpam-4582	744	7	τ	τ	X
ejpam-4582	744	8	)	)	PUNCT
ejpam-4582	744	9	is	be	AUX
ejpam-4582	744	10	(	(	PUNCT
ejpam-4582	744	11	λ	λ	X
ejpam-4582	744	12	,	,	PUNCT
ejpam-4582	744	13	s)-extremally	s)-extremally	ADV
ejpam-4582	744	14	disconnected	disconnected	ADJ
ejpam-4582	744	15	.	.	PUNCT
ejpam-4582	745	1	(	(	PUNCT
ejpam-4582	745	2	2	2	X
ejpam-4582	745	3	)	)	PUNCT
ejpam-4582	745	4	⇔	⇔	X
ejpam-4582	745	5	(	(	PUNCT
ejpam-4582	745	6	3	3	NUM
ejpam-4582	745	7	):	):	PUNCT
ejpam-4582	745	8	the	the	DET
ejpam-4582	745	9	proof	proof	NOUN
ejpam-4582	745	10	is	be	AUX
ejpam-4582	745	11	obvious	obvious	ADJ
ejpam-4582	745	12	.	.	PUNCT
ejpam-4582	746	1	7	7	X
ejpam-4582	746	2	.	.	X
ejpam-4582	746	3	characterizations	characterization	NOUN
ejpam-4582	746	4	of	of	ADP
ejpam-4582	746	5	almost	almost	ADV
ejpam-4582	746	6	(	(	PUNCT
ejpam-4582	746	7	λ	λ	PROPN
ejpam-4582	746	8	,	,	PUNCT
ejpam-4582	746	9	s)-continuous	s)-continuous	ADJ
ejpam-4582	746	10	functions	function	NOUN
ejpam-4582	746	11	in	in	ADP
ejpam-4582	746	12	this	this	DET
ejpam-4582	746	13	section	section	NOUN
ejpam-4582	746	14	,	,	PUNCT
ejpam-4582	746	15	we	we	PRON
ejpam-4582	746	16	introduce	introduce	VERB
ejpam-4582	746	17	the	the	DET
ejpam-4582	746	18	notion	notion	NOUN
ejpam-4582	746	19	of	of	ADP
ejpam-4582	746	20	almost	almost	ADV
ejpam-4582	746	21	(	(	PUNCT
ejpam-4582	746	22	λ	λ	PROPN
ejpam-4582	746	23	,	,	PUNCT
ejpam-4582	746	24	s)-continuous	s)-continuous	ADJ
ejpam-4582	746	25	functions	function	NOUN
ejpam-4582	746	26	.	.	PUNCT
ejpam-4582	747	1	moreover	moreover	ADV
ejpam-4582	747	2	,	,	PUNCT
ejpam-4582	747	3	some	some	DET
ejpam-4582	747	4	characterizations	characterization	NOUN
ejpam-4582	747	5	of	of	ADP
ejpam-4582	747	6	almost	almost	ADV
ejpam-4582	747	7	(	(	PUNCT
ejpam-4582	747	8	λ	λ	PROPN
ejpam-4582	747	9	,	,	PUNCT
ejpam-4582	747	10	s)-continuous	s)-continuous	ADJ
ejpam-4582	747	11	functions	function	NOUN
ejpam-4582	747	12	are	be	AUX
ejpam-4582	747	13	discussed	discuss	VERB
ejpam-4582	747	14	.	.	PUNCT
ejpam-4582	748	1	c.	c.	PROPN
ejpam-4582	748	2	boonpok	boonpok	PROPN
ejpam-4582	748	3	,	,	PUNCT
ejpam-4582	748	4	c.	c.	PROPN
ejpam-4582	748	5	viriyapong	viriyapong	PROPN
ejpam-4582	748	6	/	/	SYM
ejpam-4582	748	7	eur	eur	PROPN
ejpam-4582	748	8	.	.	PUNCT
ejpam-4582	749	1	j.	j.	PROPN
ejpam-4582	749	2	pure	pure	PROPN
ejpam-4582	749	3	appl	appl	PROPN
ejpam-4582	749	4	.	.	PROPN
ejpam-4582	749	5	math	math	PROPN
ejpam-4582	749	6	,	,	PUNCT
ejpam-4582	749	7	16	16	NUM
ejpam-4582	749	8	(	(	PUNCT
ejpam-4582	749	9	1	1	NUM
ejpam-4582	749	10	)	)	PUNCT
ejpam-4582	749	11	(	(	PUNCT
ejpam-4582	749	12	2023	2023	NUM
ejpam-4582	749	13	)	)	PUNCT
ejpam-4582	749	14	,	,	PUNCT
ejpam-4582	749	15	336	336	NUM
ejpam-4582	749	16	-	-	SYM
ejpam-4582	749	17	362	362	NUM
ejpam-4582	749	18	358	358	NUM
ejpam-4582	749	19	definition	definition	NOUN
ejpam-4582	749	20	20	20	NUM
ejpam-4582	749	21	.	.	PUNCT
ejpam-4582	750	1	a	a	DET
ejpam-4582	750	2	function	function	NOUN
ejpam-4582	750	3	f	f	NOUN
ejpam-4582	750	4	:	:	PUNCT
ejpam-4582	750	5	(	(	PUNCT
ejpam-4582	750	6	x	x	X
ejpam-4582	750	7	,	,	PUNCT
ejpam-4582	750	8	τ	τ	X
ejpam-4582	750	9	)	)	PUNCT
ejpam-4582	750	10	→	→	SYM
ejpam-4582	750	11	(	(	PUNCT
ejpam-4582	750	12	y	y	PROPN
ejpam-4582	750	13	,	,	PUNCT
ejpam-4582	750	14	σ	σ	PROPN
ejpam-4582	750	15	)	)	PUNCT
ejpam-4582	750	16	is	be	AUX
ejpam-4582	750	17	said	say	VERB
ejpam-4582	750	18	to	to	PART
ejpam-4582	750	19	be	be	AUX
ejpam-4582	750	20	almost	almost	ADV
ejpam-4582	750	21	(	(	PUNCT
ejpam-4582	750	22	λ	λ	X
ejpam-4582	750	23	,	,	PUNCT
ejpam-4582	750	24	s)-continuous	s)-continuous	ADJ
ejpam-4582	750	25	at	at	ADP
ejpam-4582	750	26	a	a	DET
ejpam-4582	750	27	point	point	NOUN
ejpam-4582	750	28	x	x	SYM
ejpam-4582	750	29	∈	∈	NOUN
ejpam-4582	750	30	x	x	INTJ
ejpam-4582	750	31	if	if	SCONJ
ejpam-4582	750	32	,	,	PUNCT
ejpam-4582	750	33	for	for	ADP
ejpam-4582	750	34	each	each	DET
ejpam-4582	750	35	(	(	PUNCT
ejpam-4582	750	36	λ	λ	PROPN
ejpam-4582	750	37	,	,	PUNCT
ejpam-4582	750	38	s)-open	s)-open	PUNCT
ejpam-4582	750	39	set	set	VERB
ejpam-4582	750	40	v	v	NUM
ejpam-4582	750	41	of	of	ADP
ejpam-4582	750	42	y	y	NOUN
ejpam-4582	750	43	containing	contain	VERB
ejpam-4582	750	44	f(x	f(x	PROPN
ejpam-4582	750	45	)	)	PUNCT
ejpam-4582	750	46	,	,	PUNCT
ejpam-4582	750	47	there	there	PRON
ejpam-4582	750	48	exists	exist	VERB
ejpam-4582	750	49	a	a	DET
ejpam-4582	750	50	(	(	PUNCT
ejpam-4582	750	51	λ	λ	X
ejpam-4582	750	52	,	,	PUNCT
ejpam-4582	750	53	s)-open	s)-open	VERB
ejpam-4582	750	54	set	set	VERB
ejpam-4582	750	55	u	u	NOUN
ejpam-4582	750	56	of	of	ADP
ejpam-4582	750	57	x	x	PUNCT
ejpam-4582	750	58	containing	contain	VERB
ejpam-4582	750	59	x	x	PUNCT
ejpam-4582	750	60	such	such	ADJ
ejpam-4582	750	61	that	that	DET
ejpam-4582	750	62	f(u	f(u	PROPN
ejpam-4582	750	63	)	)	PUNCT
ejpam-4582	751	1	⊆	⊆	NUM
ejpam-4582	751	2	[	[	X
ejpam-4582	751	3	v	v	X
ejpam-4582	751	4	(	(	PUNCT
ejpam-4582	751	5	λ	λ	PROPN
ejpam-4582	751	6	,	,	PUNCT
ejpam-4582	751	7	s)](λ	s)](λ	PROPN
ejpam-4582	751	8	,	,	PUNCT
ejpam-4582	751	9	s	s	NOUN
ejpam-4582	751	10	)	)	PUNCT
ejpam-4582	751	11	.	.	PUNCT
ejpam-4582	752	1	a	a	DET
ejpam-4582	752	2	function	function	NOUN
ejpam-4582	752	3	f	f	NOUN
ejpam-4582	752	4	:	:	PUNCT
ejpam-4582	752	5	(	(	PUNCT
ejpam-4582	752	6	x	x	X
ejpam-4582	752	7	,	,	PUNCT
ejpam-4582	752	8	τ	τ	X
ejpam-4582	752	9	)	)	PUNCT
ejpam-4582	752	10	→	→	SYM
ejpam-4582	752	11	(	(	PUNCT
ejpam-4582	752	12	y	y	PROPN
ejpam-4582	752	13	,	,	PUNCT
ejpam-4582	752	14	σ	σ	PROPN
ejpam-4582	752	15	)	)	PUNCT
ejpam-4582	752	16	is	be	AUX
ejpam-4582	752	17	said	say	VERB
ejpam-4582	752	18	to	to	PART
ejpam-4582	752	19	be	be	AUX
ejpam-4582	752	20	almost	almost	ADV
ejpam-4582	752	21	(	(	PUNCT
ejpam-4582	752	22	λ	λ	X
ejpam-4582	752	23	,	,	PUNCT
ejpam-4582	752	24	s)-continuous	s)-continuous	ADJ
ejpam-4582	752	25	if	if	SCONJ
ejpam-4582	752	26	f	f	PROPN
ejpam-4582	752	27	has	have	VERB
ejpam-4582	752	28	this	this	DET
ejpam-4582	752	29	property	property	NOUN
ejpam-4582	752	30	at	at	ADP
ejpam-4582	752	31	each	each	DET
ejpam-4582	752	32	point	point	NOUN
ejpam-4582	752	33	x	x	X
ejpam-4582	752	34	∈	∈	PROPN
ejpam-4582	752	35	x.	x.	NOUN
ejpam-4582	752	36	theorem	theorem	VERB
ejpam-4582	752	37	29	29	NUM
ejpam-4582	752	38	.	.	PUNCT
ejpam-4582	753	1	for	for	ADP
ejpam-4582	753	2	a	a	DET
ejpam-4582	753	3	function	function	NOUN
ejpam-4582	753	4	f	f	NOUN
ejpam-4582	753	5	:	:	PUNCT
ejpam-4582	753	6	(	(	PUNCT
ejpam-4582	753	7	x	x	X
ejpam-4582	753	8	,	,	PUNCT
ejpam-4582	753	9	τ	τ	X
ejpam-4582	753	10	)	)	PUNCT
ejpam-4582	753	11	→	→	SYM
ejpam-4582	753	12	(	(	PUNCT
ejpam-4582	753	13	y	y	PROPN
ejpam-4582	753	14	,	,	PUNCT
ejpam-4582	753	15	σ	σ	PROPN
ejpam-4582	753	16	)	)	PUNCT
ejpam-4582	753	17	,	,	PUNCT
ejpam-4582	753	18	the	the	DET
ejpam-4582	753	19	following	follow	VERB
ejpam-4582	753	20	properties	property	NOUN
ejpam-4582	753	21	are	be	AUX
ejpam-4582	753	22	equivalent	equivalent	ADJ
ejpam-4582	753	23	:	:	PUNCT
ejpam-4582	753	24	(	(	PUNCT
ejpam-4582	753	25	1	1	X
ejpam-4582	753	26	)	)	PUNCT
ejpam-4582	753	27	f	f	NOUN
ejpam-4582	753	28	is	be	AUX
ejpam-4582	753	29	almost	almost	ADV
ejpam-4582	753	30	(	(	PUNCT
ejpam-4582	753	31	λ	λ	X
ejpam-4582	753	32	,	,	PUNCT
ejpam-4582	753	33	s)-continuous	s)-continuous	ADJ
ejpam-4582	753	34	at	at	ADP
ejpam-4582	753	35	x	x	X
ejpam-4582	753	36	∈	∈	PROPN
ejpam-4582	753	37	x	x	X
ejpam-4582	753	38	;	;	PUNCT
ejpam-4582	753	39	(	(	PUNCT
ejpam-4582	753	40	2	2	X
ejpam-4582	753	41	)	)	PUNCT
ejpam-4582	753	42	x	x	SYM
ejpam-4582	753	43	∈	∈	PROPN
ejpam-4582	754	1	[	[	X
ejpam-4582	754	2	f−1([v	f−1([v	ADJ
ejpam-4582	754	3	(	(	PUNCT
ejpam-4582	754	4	λ	λ	NOUN
ejpam-4582	754	5	,	,	PUNCT
ejpam-4582	754	6	s)](λ	s)](λ	PROPN
ejpam-4582	754	7	,	,	PUNCT
ejpam-4582	754	8	s))](λ	s))](λ	PROPN
ejpam-4582	754	9	,	,	PUNCT
ejpam-4582	754	10	s	s	PART
ejpam-4582	754	11	)	)	PUNCT
ejpam-4582	754	12	for	for	ADP
ejpam-4582	754	13	every	every	DET
ejpam-4582	754	14	(	(	PUNCT
ejpam-4582	754	15	λ	λ	NOUN
ejpam-4582	754	16	,	,	PUNCT
ejpam-4582	754	17	s)-open	s)-open	PUNCT
ejpam-4582	754	18	set	set	VERB
ejpam-4582	754	19	v	v	NUM
ejpam-4582	754	20	of	of	ADP
ejpam-4582	754	21	y	y	NOUN
ejpam-4582	754	22	containing	contain	VERB
ejpam-4582	754	23	f(x	f(x	PROPN
ejpam-4582	754	24	)	)	PUNCT
ejpam-4582	754	25	;	;	PUNCT
ejpam-4582	754	26	(	(	PUNCT
ejpam-4582	754	27	3	3	X
ejpam-4582	754	28	)	)	PUNCT
ejpam-4582	754	29	x	x	SYM
ejpam-4582	754	30	∈	∈	PROPN
ejpam-4582	754	31	[	[	X
ejpam-4582	754	32	f−1(v	f−1(v	NOUN
ejpam-4582	754	33	)	)	PUNCT
ejpam-4582	754	34	]	]	PUNCT
ejpam-4582	754	35	(	(	PUNCT
ejpam-4582	754	36	λ	λ	X
ejpam-4582	754	37	,	,	PUNCT
ejpam-4582	754	38	s	s	PART
ejpam-4582	754	39	)	)	PUNCT
ejpam-4582	754	40	for	for	ADP
ejpam-4582	754	41	every	every	DET
ejpam-4582	754	42	r(λ	r(λ	NOUN
ejpam-4582	754	43	,	,	PUNCT
ejpam-4582	754	44	s)-open	s)-open	PUNCT
ejpam-4582	754	45	set	set	VERB
ejpam-4582	754	46	v	v	NUM
ejpam-4582	754	47	of	of	ADP
ejpam-4582	754	48	y	y	NOUN
ejpam-4582	754	49	containing	contain	VERB
ejpam-4582	754	50	f(x	f(x	PROPN
ejpam-4582	754	51	)	)	PUNCT
ejpam-4582	754	52	;	;	PUNCT
ejpam-4582	754	53	(	(	PUNCT
ejpam-4582	754	54	4	4	X
ejpam-4582	754	55	)	)	PUNCT
ejpam-4582	754	56	for	for	ADP
ejpam-4582	754	57	every	every	DET
ejpam-4582	754	58	r(λ	r(λ	NOUN
ejpam-4582	754	59	,	,	PUNCT
ejpam-4582	754	60	s)-open	s)-open	PUNCT
ejpam-4582	754	61	set	set	VERB
ejpam-4582	754	62	v	v	NUM
ejpam-4582	754	63	of	of	ADP
ejpam-4582	754	64	y	y	NOUN
ejpam-4582	754	65	containing	contain	VERB
ejpam-4582	754	66	f(x	f(x	PROPN
ejpam-4582	754	67	)	)	PUNCT
ejpam-4582	754	68	,	,	PUNCT
ejpam-4582	754	69	there	there	PRON
ejpam-4582	754	70	exists	exist	VERB
ejpam-4582	754	71	a	a	DET
ejpam-4582	754	72	(	(	PUNCT
ejpam-4582	754	73	λ	λ	X
ejpam-4582	754	74	,	,	PUNCT
ejpam-4582	754	75	s)-open	s)-open	VERB
ejpam-4582	754	76	set	set	VERB
ejpam-4582	754	77	u	u	NOUN
ejpam-4582	754	78	of	of	ADP
ejpam-4582	754	79	x	x	PUNCT
ejpam-4582	754	80	containing	contain	VERB
ejpam-4582	754	81	x	x	PUNCT
ejpam-4582	754	82	such	such	ADJ
ejpam-4582	754	83	that	that	DET
ejpam-4582	754	84	f(u	f(u	PROPN
ejpam-4582	754	85	)	)	PUNCT
ejpam-4582	754	86	⊆	⊆	NUM
ejpam-4582	754	87	v	v	NOUN
ejpam-4582	754	88	.	.	PUNCT
ejpam-4582	755	1	proof	proof	NOUN
ejpam-4582	755	2	.	.	PUNCT
ejpam-4582	756	1	(	(	PUNCT
ejpam-4582	756	2	1	1	X
ejpam-4582	756	3	)	)	PUNCT
ejpam-4582	756	4	⇒	⇒	NOUN
ejpam-4582	756	5	(	(	PUNCT
ejpam-4582	756	6	2	2	NUM
ejpam-4582	756	7	):	):	PUNCT
ejpam-4582	756	8	let	let	VERB
ejpam-4582	756	9	v	v	PART
ejpam-4582	756	10	be	be	AUX
ejpam-4582	756	11	any	any	DET
ejpam-4582	756	12	(	(	PUNCT
ejpam-4582	756	13	λ	λ	X
ejpam-4582	756	14	,	,	PUNCT
ejpam-4582	756	15	s)-open	s)-open	PUNCT
ejpam-4582	756	16	set	set	VERB
ejpam-4582	756	17	of	of	ADP
ejpam-4582	756	18	y	y	PROPN
ejpam-4582	756	19	containing	contain	VERB
ejpam-4582	756	20	f(x	f(x	PROPN
ejpam-4582	756	21	)	)	PUNCT
ejpam-4582	756	22	.	.	PUNCT
ejpam-4582	757	1	then	then	ADV
ejpam-4582	757	2	,	,	PUNCT
ejpam-4582	757	3	there	there	PRON
ejpam-4582	757	4	exists	exist	VERB
ejpam-4582	757	5	a	a	DET
ejpam-4582	757	6	(	(	PUNCT
ejpam-4582	757	7	λ	λ	X
ejpam-4582	757	8	,	,	PUNCT
ejpam-4582	757	9	s)-open	s)-open	VERB
ejpam-4582	757	10	set	set	VERB
ejpam-4582	757	11	u	u	NOUN
ejpam-4582	757	12	of	of	ADP
ejpam-4582	757	13	x	x	PUNCT
ejpam-4582	757	14	containing	contain	VERB
ejpam-4582	757	15	x	x	PUNCT
ejpam-4582	757	16	such	such	ADJ
ejpam-4582	757	17	that	that	DET
ejpam-4582	757	18	f(u	f(u	PROPN
ejpam-4582	757	19	)	)	PUNCT
ejpam-4582	757	20	⊆	⊆	NUM
ejpam-4582	758	1	[	[	X
ejpam-4582	758	2	v	v	X
ejpam-4582	758	3	(	(	PUNCT
ejpam-4582	758	4	λ	λ	PROPN
ejpam-4582	758	5	,	,	PUNCT
ejpam-4582	758	6	s)](λ	s)](λ	PROPN
ejpam-4582	758	7	,	,	PUNCT
ejpam-4582	758	8	s	s	NOUN
ejpam-4582	758	9	)	)	PUNCT
ejpam-4582	758	10	.	.	PUNCT
ejpam-4582	759	1	thus	thus	ADV
ejpam-4582	759	2	,	,	PUNCT
ejpam-4582	759	3	x	x	PUNCT
ejpam-4582	759	4	∈	∈	PROPN
ejpam-4582	759	5	u	u	NOUN
ejpam-4582	759	6	⊆	⊆	NUM
ejpam-4582	759	7	f−1([v	f−1([v	ADJ
ejpam-4582	759	8	(	(	PUNCT
ejpam-4582	759	9	λ	λ	NOUN
ejpam-4582	759	10	,	,	PUNCT
ejpam-4582	759	11	s)](λ	s)](λ	PROPN
ejpam-4582	759	12	,	,	PUNCT
ejpam-4582	759	13	s	s	NOUN
ejpam-4582	759	14	)	)	PUNCT
ejpam-4582	759	15	)	)	PUNCT
ejpam-4582	759	16	.	.	PUNCT
ejpam-4582	760	1	since	since	SCONJ
ejpam-4582	760	2	u	u	PROPN
ejpam-4582	760	3	∈	∈	PROPN
ejpam-4582	760	4	(	(	PUNCT
ejpam-4582	760	5	λ	λ	NOUN
ejpam-4582	760	6	,	,	PUNCT
ejpam-4582	760	7	s)o(x	s)o(x	NOUN
ejpam-4582	760	8	)	)	PUNCT
ejpam-4582	760	9	,	,	PUNCT
ejpam-4582	760	10	we	we	PRON
ejpam-4582	760	11	have	have	VERB
ejpam-4582	760	12	x	x	X
ejpam-4582	760	13	∈	∈	PROPN
ejpam-4582	760	14	[	[	X
ejpam-4582	760	15	f−1([v	f−1([v	ADJ
ejpam-4582	760	16	(	(	PUNCT
ejpam-4582	760	17	λ	λ	NOUN
ejpam-4582	760	18	,	,	PUNCT
ejpam-4582	760	19	s)](λ	s)](λ	PROPN
ejpam-4582	760	20	,	,	PUNCT
ejpam-4582	760	21	s))](λ	s))](λ	PROPN
ejpam-4582	760	22	,	,	PUNCT
ejpam-4582	760	23	s	s	PART
ejpam-4582	760	24	)	)	PUNCT
ejpam-4582	760	25	.	.	PUNCT
ejpam-4582	761	1	(	(	PUNCT
ejpam-4582	761	2	2	2	X
ejpam-4582	761	3	)	)	PUNCT
ejpam-4582	761	4	⇒	⇒	NOUN
ejpam-4582	761	5	(	(	PUNCT
ejpam-4582	761	6	3	3	NUM
ejpam-4582	761	7	):	):	PUNCT
ejpam-4582	761	8	let	let	VERB
ejpam-4582	761	9	v	v	PART
ejpam-4582	761	10	be	be	AUX
ejpam-4582	761	11	any	any	DET
ejpam-4582	761	12	r(λ	r(λ	NOUN
ejpam-4582	761	13	,	,	PUNCT
ejpam-4582	761	14	s)-open	s)-open	PUNCT
ejpam-4582	761	15	set	set	VERB
ejpam-4582	761	16	of	of	ADP
ejpam-4582	761	17	y	y	PROPN
ejpam-4582	761	18	containing	contain	VERB
ejpam-4582	761	19	f(x	f(x	PROPN
ejpam-4582	761	20	)	)	PUNCT
ejpam-4582	761	21	.	.	PUNCT
ejpam-4582	762	1	since	since	SCONJ
ejpam-4582	762	2	v	v	NOUN
ejpam-4582	762	3	=	=	SYM
ejpam-4582	763	1	[	[	X
ejpam-4582	763	2	v	v	X
ejpam-4582	763	3	(	(	PUNCT
ejpam-4582	763	4	λ	λ	PROPN
ejpam-4582	763	5	,	,	PUNCT
ejpam-4582	763	6	s)](λ	s)](λ	PROPN
ejpam-4582	763	7	,	,	PUNCT
ejpam-4582	763	8	s	s	PART
ejpam-4582	763	9	)	)	PUNCT
ejpam-4582	763	10	and	and	CCONJ
ejpam-4582	763	11	by	by	ADP
ejpam-4582	763	12	(	(	PUNCT
ejpam-4582	763	13	2	2	NUM
ejpam-4582	763	14	)	)	PUNCT
ejpam-4582	763	15	,	,	PUNCT
ejpam-4582	763	16	we	we	PRON
ejpam-4582	763	17	have	have	VERB
ejpam-4582	763	18	x	x	X
ejpam-4582	763	19	∈	∈	PROPN
ejpam-4582	763	20	[	[	X
ejpam-4582	763	21	f−1(v	f−1(v	NOUN
ejpam-4582	763	22	)	)	PUNCT
ejpam-4582	763	23	]	]	PUNCT
ejpam-4582	763	24	(	(	PUNCT
ejpam-4582	763	25	λ	λ	X
ejpam-4582	763	26	,	,	PUNCT
ejpam-4582	763	27	s	s	PART
ejpam-4582	763	28	)	)	PUNCT
ejpam-4582	763	29	.	.	PUNCT
ejpam-4582	764	1	(	(	PUNCT
ejpam-4582	764	2	3	3	X
ejpam-4582	764	3	)	)	PUNCT
ejpam-4582	764	4	⇒	⇒	NOUN
ejpam-4582	764	5	(	(	PUNCT
ejpam-4582	764	6	4	4	NUM
ejpam-4582	764	7	):	):	PUNCT
ejpam-4582	764	8	let	let	VERB
ejpam-4582	764	9	v	v	PART
ejpam-4582	764	10	be	be	AUX
ejpam-4582	764	11	any	any	DET
ejpam-4582	764	12	r(λ	r(λ	NOUN
ejpam-4582	764	13	,	,	PUNCT
ejpam-4582	764	14	s)-open	s)-open	PUNCT
ejpam-4582	764	15	set	set	VERB
ejpam-4582	764	16	of	of	ADP
ejpam-4582	764	17	y	y	PROPN
ejpam-4582	764	18	containing	contain	VERB
ejpam-4582	764	19	f(x	f(x	PROPN
ejpam-4582	764	20	)	)	PUNCT
ejpam-4582	764	21	.	.	PUNCT
ejpam-4582	765	1	thus	thus	ADV
ejpam-4582	765	2	,	,	PUNCT
ejpam-4582	765	3	by	by	ADP
ejpam-4582	765	4	(	(	PUNCT
ejpam-4582	765	5	3	3	NUM
ejpam-4582	765	6	)	)	PUNCT
ejpam-4582	765	7	,	,	PUNCT
ejpam-4582	765	8	we	we	PRON
ejpam-4582	765	9	have	have	VERB
ejpam-4582	765	10	x	x	X
ejpam-4582	765	11	∈	∈	PROPN
ejpam-4582	765	12	[	[	X
ejpam-4582	765	13	f−1(v	f−1(v	NOUN
ejpam-4582	765	14	)	)	PUNCT
ejpam-4582	765	15	]	]	PUNCT
ejpam-4582	765	16	(	(	PUNCT
ejpam-4582	765	17	λ	λ	X
ejpam-4582	765	18	,	,	PUNCT
ejpam-4582	765	19	s	s	PART
ejpam-4582	765	20	)	)	PUNCT
ejpam-4582	765	21	.	.	PUNCT
ejpam-4582	766	1	then	then	ADV
ejpam-4582	766	2	,	,	PUNCT
ejpam-4582	766	3	there	there	PRON
ejpam-4582	766	4	exists	exist	VERB
ejpam-4582	766	5	a	a	DET
ejpam-4582	766	6	(	(	PUNCT
ejpam-4582	766	7	λ	λ	X
ejpam-4582	766	8	,	,	PUNCT
ejpam-4582	766	9	s)-open	s)-open	VERB
ejpam-4582	766	10	set	set	VERB
ejpam-4582	766	11	u	u	NOUN
ejpam-4582	766	12	of	of	ADP
ejpam-4582	766	13	x	x	PUNCT
ejpam-4582	766	14	containing	contain	VERB
ejpam-4582	766	15	x	x	PUNCT
ejpam-4582	766	16	such	such	ADJ
ejpam-4582	766	17	that	that	SCONJ
ejpam-4582	766	18	u	u	PROPN
ejpam-4582	766	19	⊆	⊆	NUM
ejpam-4582	766	20	f−1(v	f−1(v	NOUN
ejpam-4582	766	21	)	)	PUNCT
ejpam-4582	766	22	and	and	CCONJ
ejpam-4582	766	23	hence	hence	ADV
ejpam-4582	766	24	f(u	f(u	PROPN
ejpam-4582	766	25	)	)	PUNCT
ejpam-4582	766	26	⊆	⊆	NUM
ejpam-4582	766	27	v	v	NOUN
ejpam-4582	766	28	.	.	PUNCT
ejpam-4582	767	1	(	(	PUNCT
ejpam-4582	767	2	4	4	X
ejpam-4582	767	3	)	)	PUNCT
ejpam-4582	767	4	⇒	⇒	NOUN
ejpam-4582	767	5	(	(	PUNCT
ejpam-4582	767	6	1	1	NUM
ejpam-4582	767	7	):	):	PUNCT
ejpam-4582	767	8	let	let	VERB
ejpam-4582	767	9	v	v	PART
ejpam-4582	767	10	be	be	AUX
ejpam-4582	767	11	any	any	DET
ejpam-4582	767	12	(	(	PUNCT
ejpam-4582	767	13	λ	λ	X
ejpam-4582	767	14	,	,	PUNCT
ejpam-4582	767	15	s)-open	s)-open	PUNCT
ejpam-4582	767	16	set	set	VERB
ejpam-4582	767	17	of	of	ADP
ejpam-4582	767	18	y	y	PROPN
ejpam-4582	767	19	containing	contain	VERB
ejpam-4582	767	20	f(x	f(x	PROPN
ejpam-4582	767	21	)	)	PUNCT
ejpam-4582	767	22	.	.	PUNCT
ejpam-4582	768	1	then	then	ADV
ejpam-4582	768	2	,	,	PUNCT
ejpam-4582	768	3	f(x	f(x	PROPN
ejpam-4582	768	4	)	)	PUNCT
ejpam-4582	768	5	∈	∈	PROPN
ejpam-4582	768	6	v	v	ADP
ejpam-4582	768	7	⊆	⊆	NUM
ejpam-4582	768	8	[	[	X
ejpam-4582	768	9	v	v	X
ejpam-4582	768	10	(	(	PUNCT
ejpam-4582	768	11	λ	λ	PROPN
ejpam-4582	768	12	,	,	PUNCT
ejpam-4582	768	13	s)](λ	s)](λ	PROPN
ejpam-4582	768	14	,	,	PUNCT
ejpam-4582	768	15	s	s	NOUN
ejpam-4582	768	16	)	)	PUNCT
ejpam-4582	768	17	.	.	PUNCT
ejpam-4582	769	1	since	since	SCONJ
ejpam-4582	769	2	[	[	X
ejpam-4582	769	3	v	v	X
ejpam-4582	769	4	(	(	PUNCT
ejpam-4582	769	5	λ	λ	PROPN
ejpam-4582	769	6	,	,	PUNCT
ejpam-4582	769	7	s)](λ	s)](λ	PROPN
ejpam-4582	769	8	,	,	PUNCT
ejpam-4582	769	9	s	s	PART
ejpam-4582	769	10	)	)	PUNCT
ejpam-4582	769	11	is	be	AUX
ejpam-4582	769	12	r(λ	r(λ	NOUN
ejpam-4582	769	13	,	,	PUNCT
ejpam-4582	769	14	s)-open	s)-open	PUNCT
ejpam-4582	769	15	and	and	CCONJ
ejpam-4582	769	16	by	by	ADP
ejpam-4582	769	17	(	(	PUNCT
ejpam-4582	769	18	4	4	NUM
ejpam-4582	769	19	)	)	PUNCT
ejpam-4582	769	20	,	,	PUNCT
ejpam-4582	769	21	there	there	PRON
ejpam-4582	769	22	exists	exist	VERB
ejpam-4582	769	23	a	a	DET
ejpam-4582	769	24	(	(	PUNCT
ejpam-4582	769	25	λ	λ	X
ejpam-4582	769	26	,	,	PUNCT
ejpam-4582	769	27	s)-open	s)-open	VERB
ejpam-4582	769	28	set	set	VERB
ejpam-4582	769	29	u	u	PRON
ejpam-4582	769	30	ofx	ofx	NOUN
ejpam-4582	769	31	containing	contain	VERB
ejpam-4582	769	32	x	x	PUNCT
ejpam-4582	769	33	such	such	ADJ
ejpam-4582	769	34	that	that	DET
ejpam-4582	769	35	f(u	f(u	PROPN
ejpam-4582	769	36	)	)	PUNCT
ejpam-4582	770	1	⊆	⊆	NUM
ejpam-4582	771	1	[	[	X
ejpam-4582	771	2	v	v	X
ejpam-4582	771	3	(	(	PUNCT
ejpam-4582	771	4	λ	λ	PROPN
ejpam-4582	771	5	,	,	PUNCT
ejpam-4582	771	6	s)](λ	s)](λ	PROPN
ejpam-4582	771	7	,	,	PUNCT
ejpam-4582	771	8	s	s	NOUN
ejpam-4582	771	9	)	)	PUNCT
ejpam-4582	771	10	.	.	PUNCT
ejpam-4582	772	1	this	this	PRON
ejpam-4582	772	2	shows	show	VERB
ejpam-4582	772	3	that	that	SCONJ
ejpam-4582	772	4	f	f	PROPN
ejpam-4582	772	5	is	be	AUX
ejpam-4582	772	6	almost	almost	ADV
ejpam-4582	772	7	(	(	PUNCT
ejpam-4582	772	8	λ	λ	X
ejpam-4582	772	9	,	,	PUNCT
ejpam-4582	772	10	s)-continuous	s)-continuous	ADJ
ejpam-4582	772	11	.	.	PUNCT
ejpam-4582	772	12	theorem	theorem	VERB
ejpam-4582	772	13	30	30	NUM
ejpam-4582	772	14	.	.	PUNCT
ejpam-4582	773	1	for	for	ADP
ejpam-4582	773	2	a	a	DET
ejpam-4582	773	3	function	function	NOUN
ejpam-4582	773	4	f	f	NOUN
ejpam-4582	773	5	:	:	PUNCT
ejpam-4582	773	6	(	(	PUNCT
ejpam-4582	773	7	x	x	X
ejpam-4582	773	8	,	,	PUNCT
ejpam-4582	773	9	τ	τ	X
ejpam-4582	773	10	)	)	PUNCT
ejpam-4582	773	11	→	→	SYM
ejpam-4582	773	12	(	(	PUNCT
ejpam-4582	773	13	y	y	PROPN
ejpam-4582	773	14	,	,	PUNCT
ejpam-4582	773	15	σ	σ	PROPN
ejpam-4582	773	16	)	)	PUNCT
ejpam-4582	773	17	,	,	PUNCT
ejpam-4582	773	18	the	the	DET
ejpam-4582	773	19	following	follow	VERB
ejpam-4582	773	20	properties	property	NOUN
ejpam-4582	773	21	are	be	AUX
ejpam-4582	773	22	equivalent	equivalent	ADJ
ejpam-4582	773	23	:	:	PUNCT
ejpam-4582	773	24	(	(	PUNCT
ejpam-4582	773	25	1	1	X
ejpam-4582	773	26	)	)	PUNCT
ejpam-4582	773	27	f	f	NOUN
ejpam-4582	773	28	is	be	AUX
ejpam-4582	773	29	almost	almost	ADV
ejpam-4582	773	30	(	(	PUNCT
ejpam-4582	773	31	λ	λ	X
ejpam-4582	773	32	,	,	PUNCT
ejpam-4582	773	33	s)-continuous	s)-continuous	ADJ
ejpam-4582	773	34	;	;	PUNCT
ejpam-4582	773	35	(	(	PUNCT
ejpam-4582	773	36	2	2	X
ejpam-4582	773	37	)	)	PUNCT
ejpam-4582	773	38	f−1(v	f−1(v	NOUN
ejpam-4582	773	39	)	)	PUNCT
ejpam-4582	774	1	⊆	⊆	NUM
ejpam-4582	775	1	[	[	X
ejpam-4582	775	2	f−1([v	f−1([v	ADJ
ejpam-4582	775	3	(	(	PUNCT
ejpam-4582	775	4	λ	λ	NOUN
ejpam-4582	775	5	,	,	PUNCT
ejpam-4582	775	6	s)](λ	s)](λ	PROPN
ejpam-4582	775	7	,	,	PUNCT
ejpam-4582	775	8	s))](λ	s))](λ	PROPN
ejpam-4582	775	9	,	,	PUNCT
ejpam-4582	775	10	s	s	PART
ejpam-4582	775	11	)	)	PUNCT
ejpam-4582	775	12	for	for	ADP
ejpam-4582	775	13	every	every	DET
ejpam-4582	775	14	(	(	PUNCT
ejpam-4582	775	15	λ	λ	NOUN
ejpam-4582	775	16	,	,	PUNCT
ejpam-4582	775	17	s)-open	s)-open	PUNCT
ejpam-4582	775	18	set	set	VERB
ejpam-4582	775	19	v	v	NUM
ejpam-4582	775	20	of	of	ADP
ejpam-4582	775	21	y	y	PROPN
ejpam-4582	775	22	;	;	PUNCT
ejpam-4582	775	23	(	(	PUNCT
ejpam-4582	775	24	3	3	X
ejpam-4582	775	25	)	)	PUNCT
ejpam-4582	775	26	[	[	X
ejpam-4582	775	27	f−1([f(λ	f−1([f(λ	X
ejpam-4582	775	28	,	,	PUNCT
ejpam-4582	775	29	s	s	NOUN
ejpam-4582	775	30	)	)	PUNCT
ejpam-4582	775	31	]	]	PUNCT
ejpam-4582	775	32	(	(	PUNCT
ejpam-4582	775	33	λ	λ	X
ejpam-4582	775	34	,	,	PUNCT
ejpam-4582	775	35	s))](λ	s))](λ	X
ejpam-4582	775	36	,	,	PUNCT
ejpam-4582	775	37	s	s	PART
ejpam-4582	775	38	)	)	PUNCT
ejpam-4582	775	39	⊆	⊆	NUM
ejpam-4582	775	40	f−1(f	f−1(f	PROPN
ejpam-4582	775	41	)	)	PUNCT
ejpam-4582	775	42	for	for	SCONJ
ejpam-4582	775	43	every	every	DET
ejpam-4582	775	44	(	(	PUNCT
ejpam-4582	775	45	λ	λ	PROPN
ejpam-4582	775	46	,	,	PUNCT
ejpam-4582	775	47	s)-closed	s)-close	VERB
ejpam-4582	775	48	set	set	ADJ
ejpam-4582	775	49	f	f	PROPN
ejpam-4582	775	50	of	of	ADP
ejpam-4582	775	51	y	y	PROPN
ejpam-4582	775	52	;	;	PUNCT
ejpam-4582	775	53	(	(	PUNCT
ejpam-4582	775	54	4	4	X
ejpam-4582	775	55	)	)	PUNCT
ejpam-4582	776	1	[	[	X
ejpam-4582	776	2	f−1([[b(λ	f−1([[b(λ	X
ejpam-4582	776	3	,	,	PUNCT
ejpam-4582	776	4	s)](λ	s)](λ	PROPN
ejpam-4582	776	5	,	,	PUNCT
ejpam-4582	776	6	s	s	PART
ejpam-4582	776	7	)	)	PUNCT
ejpam-4582	776	8	]	]	PUNCT
ejpam-4582	776	9	(	(	PUNCT
ejpam-4582	776	10	λ	λ	X
ejpam-4582	776	11	,	,	PUNCT
ejpam-4582	776	12	s))](λ	s))](λ	X
ejpam-4582	776	13	,	,	PUNCT
ejpam-4582	776	14	s	s	PART
ejpam-4582	776	15	)	)	PUNCT
ejpam-4582	776	16	⊆	⊆	NUM
ejpam-4582	776	17	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4582	776	18	,	,	PUNCT
ejpam-4582	776	19	s	s	NOUN
ejpam-4582	776	20	)	)	PUNCT
ejpam-4582	776	21	)	)	PUNCT
ejpam-4582	776	22	for	for	ADP
ejpam-4582	776	23	every	every	DET
ejpam-4582	776	24	subset	subset	NOUN
ejpam-4582	776	25	b	b	PROPN
ejpam-4582	776	26	of	of	ADP
ejpam-4582	776	27	y	y	PROPN
ejpam-4582	776	28	;	;	PUNCT
ejpam-4582	776	29	(	(	PUNCT
ejpam-4582	776	30	5	5	X
ejpam-4582	776	31	)	)	PUNCT
ejpam-4582	776	32	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4582	776	33	,	,	PUNCT
ejpam-4582	776	34	s	s	NOUN
ejpam-4582	776	35	)	)	PUNCT
ejpam-4582	776	36	)	)	PUNCT
ejpam-4582	777	1	⊆	⊆	NUM
ejpam-4582	777	2	[	[	X
ejpam-4582	777	3	f−1([[b(λ	f−1([[b(λ	PROPN
ejpam-4582	777	4	,	,	PUNCT
ejpam-4582	777	5	s	s	PART
ejpam-4582	777	6	)	)	PUNCT
ejpam-4582	777	7	]	]	PUNCT
ejpam-4582	777	8	(	(	PUNCT
ejpam-4582	777	9	λ	λ	X
ejpam-4582	777	10	,	,	PUNCT
ejpam-4582	777	11	s)](λ	s)](λ	PROPN
ejpam-4582	777	12	,	,	PUNCT
ejpam-4582	777	13	s))](λ	s))](λ	PROPN
ejpam-4582	777	14	,	,	PUNCT
ejpam-4582	777	15	s	s	PART
ejpam-4582	777	16	)	)	PUNCT
ejpam-4582	777	17	for	for	ADP
ejpam-4582	777	18	every	every	DET
ejpam-4582	777	19	subset	subset	NOUN
ejpam-4582	777	20	b	b	PROPN
ejpam-4582	777	21	of	of	ADP
ejpam-4582	777	22	y	y	PROPN
ejpam-4582	777	23	;	;	PUNCT
ejpam-4582	777	24	(	(	PUNCT
ejpam-4582	777	25	6	6	X
ejpam-4582	777	26	)	)	PUNCT
ejpam-4582	777	27	f−1(v	f−1(v	NOUN
ejpam-4582	777	28	)	)	PUNCT
ejpam-4582	777	29	is	be	AUX
ejpam-4582	777	30	(	(	PUNCT
ejpam-4582	777	31	λ	λ	X
ejpam-4582	777	32	,	,	PUNCT
ejpam-4582	777	33	s)-open	s)-open	PUNCT
ejpam-4582	777	34	in	in	ADP
ejpam-4582	777	35	x	x	PUNCT
ejpam-4582	777	36	for	for	ADP
ejpam-4582	777	37	every	every	DET
ejpam-4582	777	38	r(λ	r(λ	NOUN
ejpam-4582	777	39	,	,	PUNCT
ejpam-4582	777	40	s)-open	s)-open	PUNCT
ejpam-4582	777	41	set	set	VERB
ejpam-4582	777	42	v	v	NOUN
ejpam-4582	777	43	of	of	ADP
ejpam-4582	777	44	y	y	PROPN
ejpam-4582	777	45	;	;	PUNCT
ejpam-4582	777	46	c.	c.	PROPN
ejpam-4582	777	47	boonpok	boonpok	PROPN
ejpam-4582	777	48	,	,	PUNCT
ejpam-4582	777	49	c.	c.	PROPN
ejpam-4582	777	50	viriyapong	viriyapong	PROPN
ejpam-4582	777	51	/	/	SYM
ejpam-4582	777	52	eur	eur	PROPN
ejpam-4582	777	53	.	.	PUNCT
ejpam-4582	778	1	j.	j.	PROPN
ejpam-4582	778	2	pure	pure	PROPN
ejpam-4582	778	3	appl	appl	PROPN
ejpam-4582	778	4	.	.	PROPN
ejpam-4582	778	5	math	math	PROPN
ejpam-4582	778	6	,	,	PUNCT
ejpam-4582	778	7	16	16	NUM
ejpam-4582	778	8	(	(	PUNCT
ejpam-4582	778	9	1	1	NUM
ejpam-4582	778	10	)	)	PUNCT
ejpam-4582	778	11	(	(	PUNCT
ejpam-4582	778	12	2023	2023	NUM
ejpam-4582	778	13	)	)	PUNCT
ejpam-4582	778	14	,	,	PUNCT
ejpam-4582	778	15	336	336	NUM
ejpam-4582	778	16	-	-	SYM
ejpam-4582	778	17	362	362	NUM
ejpam-4582	778	18	359	359	NUM
ejpam-4582	778	19	(	(	PUNCT
ejpam-4582	778	20	7	7	X
ejpam-4582	778	21	)	)	PUNCT
ejpam-4582	778	22	f−1(f	f−1(f	NOUN
ejpam-4582	778	23	)	)	PUNCT
ejpam-4582	778	24	is	be	AUX
ejpam-4582	778	25	(	(	PUNCT
ejpam-4582	778	26	λ	λ	X
ejpam-4582	778	27	,	,	PUNCT
ejpam-4582	778	28	s)-closed	s)-close	VERB
ejpam-4582	778	29	in	in	ADP
ejpam-4582	778	30	x	x	PUNCT
ejpam-4582	778	31	for	for	ADP
ejpam-4582	778	32	every	every	DET
ejpam-4582	778	33	r(λ	r(λ	NOUN
ejpam-4582	778	34	,	,	PUNCT
ejpam-4582	778	35	s)-closed	s)-close	VERB
ejpam-4582	778	36	set	set	ADJ
ejpam-4582	778	37	f	f	PROPN
ejpam-4582	778	38	of	of	ADP
ejpam-4582	778	39	y	y	PROPN
ejpam-4582	778	40	.	.	PUNCT
ejpam-4582	779	1	proof	proof	NOUN
ejpam-4582	779	2	.	.	PUNCT
ejpam-4582	780	1	(	(	PUNCT
ejpam-4582	780	2	1	1	X
ejpam-4582	780	3	)	)	PUNCT
ejpam-4582	780	4	⇒	⇒	NOUN
ejpam-4582	780	5	(	(	PUNCT
ejpam-4582	780	6	2	2	NUM
ejpam-4582	780	7	):	):	PUNCT
ejpam-4582	780	8	let	let	VERB
ejpam-4582	780	9	v	v	PART
ejpam-4582	780	10	be	be	AUX
ejpam-4582	780	11	any	any	DET
ejpam-4582	780	12	(	(	PUNCT
ejpam-4582	780	13	λ	λ	X
ejpam-4582	780	14	,	,	PUNCT
ejpam-4582	780	15	s)-open	s)-open	PUNCT
ejpam-4582	780	16	set	set	VERB
ejpam-4582	780	17	of	of	ADP
ejpam-4582	780	18	y	y	PROPN
ejpam-4582	780	19	and	and	CCONJ
ejpam-4582	780	20	x	x	PROPN
ejpam-4582	780	21	∈	∈	PROPN
ejpam-4582	780	22	f−1(v	f−1(v	NOUN
ejpam-4582	780	23	)	)	PUNCT
ejpam-4582	780	24	.	.	PUNCT
ejpam-4582	781	1	by	by	ADP
ejpam-4582	781	2	(	(	PUNCT
ejpam-4582	781	3	1	1	NUM
ejpam-4582	781	4	)	)	PUNCT
ejpam-4582	781	5	,	,	PUNCT
ejpam-4582	781	6	there	there	PRON
ejpam-4582	781	7	exists	exist	VERB
ejpam-4582	781	8	a	a	DET
ejpam-4582	781	9	(	(	PUNCT
ejpam-4582	781	10	λ	λ	X
ejpam-4582	781	11	,	,	PUNCT
ejpam-4582	781	12	s)-open	s)-open	VERB
ejpam-4582	781	13	set	set	VERB
ejpam-4582	781	14	u	u	NOUN
ejpam-4582	781	15	of	of	ADP
ejpam-4582	781	16	x	x	PUNCT
ejpam-4582	781	17	containing	contain	VERB
ejpam-4582	781	18	x	x	PUNCT
ejpam-4582	781	19	such	such	ADJ
ejpam-4582	781	20	that	that	DET
ejpam-4582	781	21	f(u	f(u	PROPN
ejpam-4582	781	22	)	)	PUNCT
ejpam-4582	781	23	⊆	⊆	NUM
ejpam-4582	782	1	[	[	X
ejpam-4582	782	2	v	v	X
ejpam-4582	782	3	(	(	PUNCT
ejpam-4582	782	4	λ	λ	PROPN
ejpam-4582	782	5	,	,	PUNCT
ejpam-4582	782	6	s)](λ	s)](λ	PROPN
ejpam-4582	782	7	,	,	PUNCT
ejpam-4582	782	8	s	s	NOUN
ejpam-4582	782	9	)	)	PUNCT
ejpam-4582	782	10	.	.	PUNCT
ejpam-4582	783	1	this	this	PRON
ejpam-4582	783	2	implies	imply	VERB
ejpam-4582	783	3	that	that	SCONJ
ejpam-4582	783	4	x	x	PUNCT
ejpam-4582	783	5	∈	∈	PROPN
ejpam-4582	783	6	[	[	X
ejpam-4582	783	7	f−1([v	f−1([v	ADJ
ejpam-4582	783	8	(	(	PUNCT
ejpam-4582	783	9	λ	λ	NOUN
ejpam-4582	783	10	,	,	PUNCT
ejpam-4582	783	11	s)](λ	s)](λ	PROPN
ejpam-4582	783	12	,	,	PUNCT
ejpam-4582	783	13	s))](λ	s))](λ	PROPN
ejpam-4582	783	14	,	,	PUNCT
ejpam-4582	783	15	s	s	PART
ejpam-4582	783	16	)	)	PUNCT
ejpam-4582	783	17	.	.	PUNCT
ejpam-4582	784	1	thus	thus	ADV
ejpam-4582	784	2	,	,	PUNCT
ejpam-4582	784	3	f	f	PROPN
ejpam-4582	784	4	−1(v	−1(v	PROPN
ejpam-4582	784	5	)	)	PUNCT
ejpam-4582	784	6	⊆	⊆	NUM
ejpam-4582	784	7	[	[	X
ejpam-4582	784	8	f−1([v	f−1([v	ADJ
ejpam-4582	784	9	(	(	PUNCT
ejpam-4582	784	10	λ	λ	NOUN
ejpam-4582	784	11	,	,	PUNCT
ejpam-4582	784	12	s)](λ	s)](λ	PROPN
ejpam-4582	784	13	,	,	PUNCT
ejpam-4582	784	14	s))](λ	s))](λ	PROPN
ejpam-4582	784	15	,	,	PUNCT
ejpam-4582	784	16	s	s	PART
ejpam-4582	784	17	)	)	PUNCT
ejpam-4582	784	18	.	.	PUNCT
ejpam-4582	785	1	(	(	PUNCT
ejpam-4582	785	2	2	2	X
ejpam-4582	785	3	)	)	PUNCT
ejpam-4582	785	4	⇒	⇒	NOUN
ejpam-4582	785	5	(	(	PUNCT
ejpam-4582	785	6	3	3	NUM
ejpam-4582	785	7	):	):	PUNCT
ejpam-4582	785	8	let	let	VERB
ejpam-4582	785	9	f	f	PRON
ejpam-4582	785	10	be	be	AUX
ejpam-4582	785	11	any	any	DET
ejpam-4582	785	12	(	(	PUNCT
ejpam-4582	785	13	λ	λ	X
ejpam-4582	785	14	,	,	PUNCT
ejpam-4582	785	15	s)-closed	s)-close	VERB
ejpam-4582	785	16	set	set	NOUN
ejpam-4582	785	17	of	of	ADP
ejpam-4582	785	18	y	y	PROPN
ejpam-4582	785	19	.	.	PUNCT
ejpam-4582	786	1	thus	thus	ADV
ejpam-4582	786	2	,	,	PUNCT
ejpam-4582	786	3	by	by	ADP
ejpam-4582	786	4	(	(	PUNCT
ejpam-4582	786	5	2	2	NUM
ejpam-4582	786	6	)	)	PUNCT
ejpam-4582	786	7	,	,	PUNCT
ejpam-4582	786	8	we	we	PRON
ejpam-4582	786	9	have	have	VERB
ejpam-4582	786	10	x	x	X
ejpam-4582	786	11	−	−	PROPN
ejpam-4582	786	12	f−1(v	f−1(v	NOUN
ejpam-4582	786	13	)	)	PUNCT
ejpam-4582	787	1	=	=	PUNCT
ejpam-4582	787	2	f−1(y	f−1(y	PROPN
ejpam-4582	787	3	−	−	PROPN
ejpam-4582	787	4	f	f	PROPN
ejpam-4582	787	5	)	)	PUNCT
ejpam-4582	788	1	⊆	⊆	NUM
ejpam-4582	789	1	[	[	X
ejpam-4582	789	2	f−1([[y	f−1([[y	NOUN
ejpam-4582	789	3	−	−	NOUN
ejpam-4582	789	4	f	f	X
ejpam-4582	789	5	]	]	X
ejpam-4582	789	6	(	(	PUNCT
ejpam-4582	789	7	λ	λ	NOUN
ejpam-4582	789	8	,	,	PUNCT
ejpam-4582	789	9	s)](λ	s)](λ	PROPN
ejpam-4582	789	10	,	,	PUNCT
ejpam-4582	789	11	s))](λ	s))](λ	PROPN
ejpam-4582	789	12	,	,	PUNCT
ejpam-4582	789	13	s	s	PART
ejpam-4582	789	14	)	)	PUNCT
ejpam-4582	789	15	=	=	PUNCT
ejpam-4582	790	1	[	[	X
ejpam-4582	790	2	f−1(y	f−1(y	NOUN
ejpam-4582	790	3	−	−	PROPN
ejpam-4582	791	1	[	[	X
ejpam-4582	791	2	f(λ	f(λ	X
ejpam-4582	791	3	,	,	PUNCT
ejpam-4582	791	4	s	s	NOUN
ejpam-4582	791	5	)	)	PUNCT
ejpam-4582	791	6	]	]	PUNCT
ejpam-4582	791	7	(	(	PUNCT
ejpam-4582	791	8	λ	λ	X
ejpam-4582	791	9	,	,	PUNCT
ejpam-4582	791	10	s))](λ	s))](λ	X
ejpam-4582	791	11	,	,	PUNCT
ejpam-4582	791	12	s	s	PART
ejpam-4582	791	13	)	)	PUNCT
ejpam-4582	791	14	=	=	PUNCT
ejpam-4582	791	15	x	x	X
ejpam-4582	791	16	−	−	PROPN
ejpam-4582	792	1	[	[	X
ejpam-4582	792	2	f−1([f(λ	f−1([f(λ	X
ejpam-4582	792	3	,	,	PUNCT
ejpam-4582	792	4	s	s	NOUN
ejpam-4582	792	5	)	)	PUNCT
ejpam-4582	792	6	]	]	PUNCT
ejpam-4582	792	7	(	(	PUNCT
ejpam-4582	792	8	λ	λ	X
ejpam-4582	792	9	,	,	PUNCT
ejpam-4582	792	10	s))](λ	s))](λ	X
ejpam-4582	792	11	,	,	PUNCT
ejpam-4582	792	12	s	s	PART
ejpam-4582	792	13	)	)	PUNCT
ejpam-4582	792	14	and	and	CCONJ
ejpam-4582	792	15	hence	hence	ADV
ejpam-4582	792	16	[	[	X
ejpam-4582	792	17	f−1([f(λ	f−1([f(λ	ADJ
ejpam-4582	792	18	,	,	PUNCT
ejpam-4582	792	19	s	s	NOUN
ejpam-4582	792	20	)	)	PUNCT
ejpam-4582	792	21	]	]	PUNCT
ejpam-4582	792	22	(	(	PUNCT
ejpam-4582	792	23	λ	λ	X
ejpam-4582	792	24	,	,	PUNCT
ejpam-4582	792	25	s))](λ	s))](λ	X
ejpam-4582	792	26	,	,	PUNCT
ejpam-4582	792	27	s	s	PART
ejpam-4582	792	28	)	)	PUNCT
ejpam-4582	792	29	⊆	⊆	NUM
ejpam-4582	792	30	f−1(f	f−1(f	PROPN
ejpam-4582	792	31	)	)	PUNCT
ejpam-4582	792	32	.	.	PUNCT
ejpam-4582	793	1	(	(	PUNCT
ejpam-4582	793	2	3	3	X
ejpam-4582	793	3	)	)	PUNCT
ejpam-4582	793	4	⇒	⇒	NOUN
ejpam-4582	793	5	(	(	PUNCT
ejpam-4582	793	6	4	4	NUM
ejpam-4582	793	7	):	):	PUNCT
ejpam-4582	793	8	let	let	VERB
ejpam-4582	793	9	b	b	X
ejpam-4582	793	10	be	be	AUX
ejpam-4582	793	11	any	any	DET
ejpam-4582	793	12	subset	subset	NOUN
ejpam-4582	793	13	of	of	ADP
ejpam-4582	793	14	y	y	PROPN
ejpam-4582	793	15	.	.	PUNCT
ejpam-4582	794	1	since	since	SCONJ
ejpam-4582	794	2	b(λ	b(λ	PROPN
ejpam-4582	794	3	,	,	PUNCT
ejpam-4582	794	4	s	s	PART
ejpam-4582	794	5	)	)	PUNCT
ejpam-4582	794	6	is	be	AUX
ejpam-4582	794	7	(	(	PUNCT
ejpam-4582	794	8	λ	λ	X
ejpam-4582	794	9	,	,	PUNCT
ejpam-4582	794	10	s)-closed	s)-close	VERB
ejpam-4582	794	11	and	and	CCONJ
ejpam-4582	794	12	by	by	ADP
ejpam-4582	794	13	(	(	PUNCT
ejpam-4582	794	14	3	3	NUM
ejpam-4582	794	15	)	)	PUNCT
ejpam-4582	794	16	,	,	PUNCT
ejpam-4582	794	17	we	we	PRON
ejpam-4582	794	18	have	have	VERB
ejpam-4582	794	19	[	[	X
ejpam-4582	794	20	f−1([b(λ	f−1([b(λ	X
ejpam-4582	794	21	,	,	PUNCT
ejpam-4582	794	22	s)](λ	s)](λ	PROPN
ejpam-4582	794	23	,	,	PUNCT
ejpam-4582	794	24	s	s	NOUN
ejpam-4582	794	25	)	)	PUNCT
ejpam-4582	794	26	]	]	PUNCT
ejpam-4582	795	1	(	(	PUNCT
ejpam-4582	795	2	λ	λ	X
ejpam-4582	795	3	,	,	PUNCT
ejpam-4582	795	4	s))](λ	s))](λ	X
ejpam-4582	795	5	,	,	PUNCT
ejpam-4582	795	6	s	s	PART
ejpam-4582	795	7	)	)	PUNCT
ejpam-4582	795	8	⊆	⊆	NUM
ejpam-4582	795	9	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4582	795	10	,	,	PUNCT
ejpam-4582	795	11	s	s	NOUN
ejpam-4582	795	12	)	)	PUNCT
ejpam-4582	795	13	)	)	PUNCT
ejpam-4582	795	14	.	.	PUNCT
ejpam-4582	796	1	(	(	PUNCT
ejpam-4582	796	2	4	4	X
ejpam-4582	796	3	)	)	PUNCT
ejpam-4582	796	4	⇒	⇒	NOUN
ejpam-4582	796	5	(	(	PUNCT
ejpam-4582	796	6	5	5	NUM
ejpam-4582	796	7	):	):	PUNCT
ejpam-4582	796	8	let	let	VERB
ejpam-4582	796	9	b	b	X
ejpam-4582	796	10	be	be	AUX
ejpam-4582	796	11	any	any	DET
ejpam-4582	796	12	subset	subset	NOUN
ejpam-4582	796	13	of	of	ADP
ejpam-4582	796	14	y	y	PROPN
ejpam-4582	796	15	.	.	PUNCT
ejpam-4582	797	1	by	by	ADP
ejpam-4582	797	2	(	(	PUNCT
ejpam-4582	797	3	4	4	NUM
ejpam-4582	797	4	)	)	PUNCT
ejpam-4582	797	5	,	,	PUNCT
ejpam-4582	797	6	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4582	797	7	,	,	PUNCT
ejpam-4582	797	8	s	s	NOUN
ejpam-4582	797	9	)	)	PUNCT
ejpam-4582	797	10	)	)	PUNCT
ejpam-4582	797	11	=	=	PUNCT
ejpam-4582	798	1	x	x	X
ejpam-4582	798	2	−	−	NOUN
ejpam-4582	798	3	f−1([y	f−1([y	NOUN
ejpam-4582	798	4	−b](λ	−b](λ	NOUN
ejpam-4582	798	5	,	,	PUNCT
ejpam-4582	798	6	s	s	NOUN
ejpam-4582	798	7	)	)	PUNCT
ejpam-4582	798	8	)	)	PUNCT
ejpam-4582	799	1	⊆	⊆	NUM
ejpam-4582	799	2	x	x	SYM
ejpam-4582	799	3	−	−	PROPN
ejpam-4582	800	1	[	[	X
ejpam-4582	800	2	f−1([[[y	f−1([[[y	NOUN
ejpam-4582	800	3	−b](λ	−b](λ	PROPN
ejpam-4582	800	4	,	,	PUNCT
ejpam-4582	800	5	s)](λ	s)](λ	PROPN
ejpam-4582	800	6	,	,	PUNCT
ejpam-4582	800	7	s	s	NOUN
ejpam-4582	800	8	)	)	PUNCT
ejpam-4582	800	9	]	]	PUNCT
ejpam-4582	800	10	(	(	PUNCT
ejpam-4582	800	11	λ	λ	X
ejpam-4582	800	12	,	,	PUNCT
ejpam-4582	800	13	s))](λ	s))](λ	X
ejpam-4582	800	14	,	,	PUNCT
ejpam-4582	800	15	s	s	PART
ejpam-4582	800	16	)	)	PUNCT
ejpam-4582	800	17	=	=	PUNCT
ejpam-4582	801	1	[	[	X
ejpam-4582	801	2	f−1([[b(λ	f−1([[b(λ	PROPN
ejpam-4582	801	3	,	,	PUNCT
ejpam-4582	801	4	s	s	PART
ejpam-4582	801	5	)	)	PUNCT
ejpam-4582	801	6	]	]	PUNCT
ejpam-4582	802	1	(	(	PUNCT
ejpam-4582	802	2	λ	λ	X
ejpam-4582	802	3	,	,	PUNCT
ejpam-4582	802	4	s)](λ	s)](λ	PROPN
ejpam-4582	802	5	,	,	PUNCT
ejpam-4582	802	6	s))](λ	s))](λ	PROPN
ejpam-4582	802	7	,	,	PUNCT
ejpam-4582	802	8	s	s	PART
ejpam-4582	802	9	)	)	PUNCT
ejpam-4582	802	10	.	.	PUNCT
ejpam-4582	803	1	(	(	PUNCT
ejpam-4582	803	2	5	5	X
ejpam-4582	803	3	)	)	PUNCT
ejpam-4582	803	4	⇒	⇒	NOUN
ejpam-4582	803	5	(	(	PUNCT
ejpam-4582	803	6	6	6	NUM
ejpam-4582	803	7	):	):	PUNCT
ejpam-4582	803	8	let	let	VERB
ejpam-4582	803	9	v	v	PART
ejpam-4582	803	10	be	be	AUX
ejpam-4582	803	11	any	any	DET
ejpam-4582	803	12	r(λ	r(λ	NOUN
ejpam-4582	803	13	,	,	PUNCT
ejpam-4582	803	14	s)-open	s)-open	PUNCT
ejpam-4582	803	15	set	set	VERB
ejpam-4582	803	16	of	of	ADP
ejpam-4582	803	17	y	y	PROPN
ejpam-4582	803	18	.	.	PUNCT
ejpam-4582	804	1	since	since	SCONJ
ejpam-4582	804	2	[	[	X
ejpam-4582	804	3	[	[	X
ejpam-4582	804	4	v(λ	v(λ	PROPN
ejpam-4582	804	5	,	,	PUNCT
ejpam-4582	804	6	s	s	PART
ejpam-4582	804	7	)	)	PUNCT
ejpam-4582	804	8	]	]	PUNCT
ejpam-4582	804	9	(	(	PUNCT
ejpam-4582	804	10	λ	λ	X
ejpam-4582	804	11	,	,	PUNCT
ejpam-4582	804	12	s)](λ	s)](λ	PROPN
ejpam-4582	804	13	,	,	PUNCT
ejpam-4582	804	14	s	s	PART
ejpam-4582	804	15	)	)	PUNCT
ejpam-4582	804	16	=	=	SYM
ejpam-4582	804	17	v	v	NOUN
ejpam-4582	804	18	and	and	CCONJ
ejpam-4582	804	19	by	by	ADP
ejpam-4582	804	20	(	(	PUNCT
ejpam-4582	804	21	5	5	NUM
ejpam-4582	804	22	)	)	PUNCT
ejpam-4582	804	23	,	,	PUNCT
ejpam-4582	804	24	f−1(v	f−1(v	NOUN
ejpam-4582	804	25	)	)	PUNCT
ejpam-4582	804	26	⊆	⊆	NUM
ejpam-4582	804	27	[	[	X
ejpam-4582	804	28	f−1(v	f−1(v	NOUN
ejpam-4582	804	29	)	)	PUNCT
ejpam-4582	804	30	]	]	PUNCT
ejpam-4582	804	31	(	(	PUNCT
ejpam-4582	804	32	λ	λ	X
ejpam-4582	804	33	,	,	PUNCT
ejpam-4582	804	34	s	s	PART
ejpam-4582	804	35	)	)	PUNCT
ejpam-4582	804	36	.	.	PUNCT
ejpam-4582	805	1	thus	thus	ADV
ejpam-4582	805	2	,	,	PUNCT
ejpam-4582	805	3	f	f	PROPN
ejpam-4582	805	4	−1(v	−1(v	PROPN
ejpam-4582	805	5	)	)	PUNCT
ejpam-4582	806	1	=	=	PUNCT
ejpam-4582	807	1	[	[	X
ejpam-4582	807	2	f−1(v	f−1(v	NOUN
ejpam-4582	807	3	)	)	PUNCT
ejpam-4582	807	4	]	]	PUNCT
ejpam-4582	807	5	(	(	PUNCT
ejpam-4582	807	6	λ	λ	X
ejpam-4582	807	7	,	,	PUNCT
ejpam-4582	807	8	s	s	PART
ejpam-4582	807	9	)	)	PUNCT
ejpam-4582	807	10	and	and	CCONJ
ejpam-4582	807	11	hence	hence	ADV
ejpam-4582	807	12	f−1(v	f−1(v	PROPN
ejpam-4582	807	13	)	)	PUNCT
ejpam-4582	807	14	is	be	AUX
ejpam-4582	807	15	(	(	PUNCT
ejpam-4582	807	16	λ	λ	X
ejpam-4582	807	17	,	,	PUNCT
ejpam-4582	807	18	s)-open	s)-open	ADJ
ejpam-4582	807	19	.	.	PUNCT
ejpam-4582	808	1	(	(	PUNCT
ejpam-4582	808	2	6	6	NUM
ejpam-4582	808	3	)	)	PUNCT
ejpam-4582	808	4	⇒	⇒	NOUN
ejpam-4582	808	5	(	(	PUNCT
ejpam-4582	808	6	7	7	NUM
ejpam-4582	808	7	):	):	PUNCT
ejpam-4582	808	8	the	the	DET
ejpam-4582	808	9	proof	proof	NOUN
ejpam-4582	808	10	is	be	AUX
ejpam-4582	808	11	obvious	obvious	ADJ
ejpam-4582	808	12	.	.	PUNCT
ejpam-4582	809	1	(	(	PUNCT
ejpam-4582	809	2	7	7	X
ejpam-4582	809	3	)	)	PUNCT
ejpam-4582	809	4	⇒	⇒	NOUN
ejpam-4582	809	5	(	(	PUNCT
ejpam-4582	809	6	1	1	NUM
ejpam-4582	809	7	):	):	PUNCT
ejpam-4582	809	8	let	let	VERB
ejpam-4582	809	9	v	v	PART
ejpam-4582	809	10	be	be	AUX
ejpam-4582	809	11	any	any	DET
ejpam-4582	809	12	r(λ	r(λ	NOUN
ejpam-4582	809	13	,	,	PUNCT
ejpam-4582	809	14	s)-open	s)-open	PUNCT
ejpam-4582	809	15	set	set	VERB
ejpam-4582	809	16	of	of	ADP
ejpam-4582	809	17	y	y	PROPN
ejpam-4582	809	18	containing	contain	VERB
ejpam-4582	809	19	f(x	f(x	PROPN
ejpam-4582	809	20	)	)	PUNCT
ejpam-4582	809	21	.	.	PUNCT
ejpam-4582	810	1	thus	thus	ADV
ejpam-4582	810	2	,	,	PUNCT
ejpam-4582	810	3	by	by	ADP
ejpam-4582	810	4	(	(	PUNCT
ejpam-4582	810	5	7	7	NUM
ejpam-4582	810	6	)	)	PUNCT
ejpam-4582	810	7	,	,	PUNCT
ejpam-4582	810	8	we	we	PRON
ejpam-4582	810	9	have	have	VERB
ejpam-4582	810	10	x	x	X
ejpam-4582	810	11	−	−	PROPN
ejpam-4582	810	12	f−1(v	f−1(v	NOUN
ejpam-4582	810	13	)	)	PUNCT
ejpam-4582	811	1	=	=	PUNCT
ejpam-4582	811	2	f−1(y	f−1(y	PROPN
ejpam-4582	811	3	−	−	PROPN
ejpam-4582	811	4	v	v	NOUN
ejpam-4582	811	5	)	)	PUNCT
ejpam-4582	811	6	=	=	PUNCT
ejpam-4582	812	1	[	[	X
ejpam-4582	812	2	f−1(y	f−1(y	NOUN
ejpam-4582	812	3	−	−	PROPN
ejpam-4582	812	4	v	v	NOUN
ejpam-4582	812	5	)	)	PUNCT
ejpam-4582	812	6	]	]	PUNCT
ejpam-4582	812	7	(	(	PUNCT
ejpam-4582	812	8	λ	λ	X
ejpam-4582	812	9	,	,	PUNCT
ejpam-4582	812	10	s	s	PART
ejpam-4582	812	11	)	)	PUNCT
ejpam-4582	812	12	=	=	PUNCT
ejpam-4582	812	13	x	x	X
ejpam-4582	812	14	−	−	PROPN
ejpam-4582	813	1	[	[	X
ejpam-4582	813	2	f−1(v	f−1(v	NOUN
ejpam-4582	813	3	)	)	PUNCT
ejpam-4582	813	4	]	]	PUNCT
ejpam-4582	813	5	(	(	PUNCT
ejpam-4582	813	6	λ	λ	X
ejpam-4582	813	7	,	,	PUNCT
ejpam-4582	813	8	s	s	PART
ejpam-4582	813	9	)	)	PUNCT
ejpam-4582	813	10	and	and	CCONJ
ejpam-4582	813	11	hence	hence	ADV
ejpam-4582	813	12	f−1(v	f−1(v	NOUN
ejpam-4582	813	13	)	)	PUNCT
ejpam-4582	813	14	=	=	PUNCT
ejpam-4582	814	1	[	[	X
ejpam-4582	814	2	f−1(v	f−1(v	NOUN
ejpam-4582	814	3	)	)	PUNCT
ejpam-4582	814	4	]	]	PUNCT
ejpam-4582	814	5	(	(	PUNCT
ejpam-4582	814	6	λ	λ	X
ejpam-4582	814	7	,	,	PUNCT
ejpam-4582	814	8	s	s	PART
ejpam-4582	814	9	)	)	PUNCT
ejpam-4582	814	10	.	.	PUNCT
ejpam-4582	815	1	since	since	SCONJ
ejpam-4582	815	2	x	x	PROPN
ejpam-4582	815	3	∈	∈	PROPN
ejpam-4582	815	4	[	[	X
ejpam-4582	815	5	f−1(v	f−1(v	NOUN
ejpam-4582	815	6	)	)	PUNCT
ejpam-4582	815	7	]	]	PUNCT
ejpam-4582	815	8	(	(	PUNCT
ejpam-4582	815	9	λ	λ	X
ejpam-4582	815	10	,	,	PUNCT
ejpam-4582	815	11	s	s	PART
ejpam-4582	815	12	)	)	PUNCT
ejpam-4582	815	13	,	,	PUNCT
ejpam-4582	815	14	there	there	PRON
ejpam-4582	815	15	exists	exist	VERB
ejpam-4582	815	16	a	a	DET
ejpam-4582	815	17	(	(	PUNCT
ejpam-4582	815	18	λ	λ	X
ejpam-4582	815	19	,	,	PUNCT
ejpam-4582	815	20	s)-open	s)-open	VERB
ejpam-4582	815	21	set	set	VERB
ejpam-4582	815	22	u	u	NOUN
ejpam-4582	815	23	of	of	ADP
ejpam-4582	815	24	x	x	PUNCT
ejpam-4582	815	25	containing	contain	VERB
ejpam-4582	815	26	x	x	PUNCT
ejpam-4582	815	27	such	such	ADJ
ejpam-4582	815	28	that	that	SCONJ
ejpam-4582	815	29	u	u	PROPN
ejpam-4582	815	30	⊆	⊆	NUM
ejpam-4582	815	31	f−1(v	f−1(v	NOUN
ejpam-4582	815	32	)	)	PUNCT
ejpam-4582	815	33	.	.	PUNCT
ejpam-4582	816	1	thus	thus	ADV
ejpam-4582	816	2	,	,	PUNCT
ejpam-4582	816	3	f(u	f(u	PROPN
ejpam-4582	816	4	)	)	PUNCT
ejpam-4582	816	5	⊆	⊆	NUM
ejpam-4582	816	6	v	v	NOUN
ejpam-4582	816	7	and	and	CCONJ
ejpam-4582	816	8	by	by	ADP
ejpam-4582	816	9	theorem	theorem	NOUN
ejpam-4582	816	10	29	29	NUM
ejpam-4582	816	11	,	,	PUNCT
ejpam-4582	816	12	f	f	PROPN
ejpam-4582	816	13	is	be	AUX
ejpam-4582	816	14	almost	almost	ADV
ejpam-4582	816	15	(	(	PUNCT
ejpam-4582	816	16	λ	λ	X
ejpam-4582	816	17	,	,	PUNCT
ejpam-4582	816	18	s)-continuous	s)-continuous	ADJ
ejpam-4582	816	19	.	.	PUNCT
ejpam-4582	817	1	theorem	theorem	VERB
ejpam-4582	817	2	31	31	NUM
ejpam-4582	817	3	.	.	PUNCT
ejpam-4582	818	1	for	for	ADP
ejpam-4582	818	2	a	a	DET
ejpam-4582	818	3	function	function	NOUN
ejpam-4582	818	4	f	f	NOUN
ejpam-4582	818	5	:	:	PUNCT
ejpam-4582	818	6	(	(	PUNCT
ejpam-4582	818	7	x	x	X
ejpam-4582	818	8	,	,	PUNCT
ejpam-4582	818	9	τ	τ	X
ejpam-4582	818	10	)	)	PUNCT
ejpam-4582	818	11	→	→	SYM
ejpam-4582	818	12	(	(	PUNCT
ejpam-4582	818	13	y	y	PROPN
ejpam-4582	818	14	,	,	PUNCT
ejpam-4582	818	15	σ	σ	PROPN
ejpam-4582	818	16	)	)	PUNCT
ejpam-4582	818	17	,	,	PUNCT
ejpam-4582	818	18	the	the	DET
ejpam-4582	818	19	following	follow	VERB
ejpam-4582	818	20	properties	property	NOUN
ejpam-4582	818	21	are	be	AUX
ejpam-4582	818	22	equivalent	equivalent	ADJ
ejpam-4582	818	23	:	:	PUNCT
ejpam-4582	818	24	(	(	PUNCT
ejpam-4582	818	25	1	1	X
ejpam-4582	818	26	)	)	PUNCT
ejpam-4582	818	27	f	f	NOUN
ejpam-4582	818	28	is	be	AUX
ejpam-4582	818	29	almost	almost	ADV
ejpam-4582	818	30	(	(	PUNCT
ejpam-4582	818	31	λ	λ	X
ejpam-4582	818	32	,	,	PUNCT
ejpam-4582	818	33	s)-continuous	s)-continuous	ADJ
ejpam-4582	818	34	;	;	PUNCT
ejpam-4582	818	35	(	(	PUNCT
ejpam-4582	818	36	2	2	X
ejpam-4582	818	37	)	)	PUNCT
ejpam-4582	818	38	[	[	X
ejpam-4582	818	39	f−1(u)](λ	f−1(u)](λ	X
ejpam-4582	818	40	,	,	PUNCT
ejpam-4582	818	41	s	s	PART
ejpam-4582	818	42	)	)	PUNCT
ejpam-4582	818	43	⊆	⊆	NUM
ejpam-4582	818	44	f−1(u	f−1(u	NOUN
ejpam-4582	818	45	(	(	PUNCT
ejpam-4582	818	46	λ	λ	PROPN
ejpam-4582	818	47	,	,	PUNCT
ejpam-4582	818	48	s	s	NOUN
ejpam-4582	818	49	)	)	PUNCT
ejpam-4582	818	50	)	)	PUNCT
ejpam-4582	818	51	for	for	ADP
ejpam-4582	818	52	every	every	DET
ejpam-4582	818	53	β(λ	β(λ	NOUN
ejpam-4582	818	54	,	,	PUNCT
ejpam-4582	818	55	s)-open	s)-open	PUNCT
ejpam-4582	818	56	set	set	VERB
ejpam-4582	818	57	u	u	NOUN
ejpam-4582	818	58	of	of	ADP
ejpam-4582	818	59	y	y	PROPN
ejpam-4582	818	60	;	;	PUNCT
ejpam-4582	818	61	(	(	PUNCT
ejpam-4582	818	62	3	3	X
ejpam-4582	818	63	)	)	PUNCT
ejpam-4582	818	64	[	[	X
ejpam-4582	818	65	f−1(u)](λ	f−1(u)](λ	X
ejpam-4582	818	66	,	,	PUNCT
ejpam-4582	818	67	s	s	PART
ejpam-4582	818	68	)	)	PUNCT
ejpam-4582	818	69	⊆	⊆	NUM
ejpam-4582	818	70	f−1(u	f−1(u	NOUN
ejpam-4582	818	71	(	(	PUNCT
ejpam-4582	818	72	λ	λ	PROPN
ejpam-4582	818	73	,	,	PUNCT
ejpam-4582	818	74	s	s	NOUN
ejpam-4582	818	75	)	)	PUNCT
ejpam-4582	818	76	)	)	PUNCT
ejpam-4582	818	77	for	for	ADP
ejpam-4582	818	78	every	every	DET
ejpam-4582	818	79	s(λ	s(λ	PROPN
ejpam-4582	818	80	,	,	PUNCT
ejpam-4582	818	81	s)-open	s)-open	VERB
ejpam-4582	818	82	set	set	VERB
ejpam-4582	818	83	u	u	NOUN
ejpam-4582	818	84	of	of	ADP
ejpam-4582	818	85	y	y	PROPN
ejpam-4582	818	86	;	;	PUNCT
ejpam-4582	818	87	(	(	PUNCT
ejpam-4582	818	88	4	4	X
ejpam-4582	818	89	)	)	PUNCT
ejpam-4582	818	90	f−1(u	f−1(u	NOUN
ejpam-4582	818	91	)	)	PUNCT
ejpam-4582	819	1	⊆	⊆	NUM
ejpam-4582	820	1	[	[	X
ejpam-4582	820	2	f−1([u	f−1([u	INTJ
ejpam-4582	820	3	(	(	PUNCT
ejpam-4582	820	4	λ	λ	PROPN
ejpam-4582	820	5	,	,	PUNCT
ejpam-4582	820	6	s)](λ	s)](λ	PROPN
ejpam-4582	820	7	,	,	PUNCT
ejpam-4582	820	8	s))](λ	s))](λ	PROPN
ejpam-4582	820	9	,	,	PUNCT
ejpam-4582	820	10	s	s	PART
ejpam-4582	820	11	)	)	PUNCT
ejpam-4582	820	12	for	for	ADP
ejpam-4582	820	13	every	every	DET
ejpam-4582	820	14	p(λ	p(λ	NOUN
ejpam-4582	820	15	,	,	PUNCT
ejpam-4582	820	16	s)-open	s)-open	VERB
ejpam-4582	820	17	set	set	VERB
ejpam-4582	820	18	u	u	NOUN
ejpam-4582	820	19	of	of	ADP
ejpam-4582	820	20	y	y	PROPN
ejpam-4582	820	21	.	.	PUNCT
ejpam-4582	821	1	proof	proof	NOUN
ejpam-4582	821	2	.	.	PUNCT
ejpam-4582	822	1	(	(	PUNCT
ejpam-4582	822	2	1	1	X
ejpam-4582	822	3	)	)	PUNCT
ejpam-4582	822	4	⇒	⇒	NOUN
ejpam-4582	822	5	(	(	PUNCT
ejpam-4582	822	6	2	2	NUM
ejpam-4582	822	7	):	):	PUNCT
ejpam-4582	822	8	let	let	VERB
ejpam-4582	822	9	u	u	PRON
ejpam-4582	822	10	be	be	AUX
ejpam-4582	822	11	any	any	DET
ejpam-4582	822	12	β(λ	β(λ	NOUN
ejpam-4582	822	13	,	,	PUNCT
ejpam-4582	822	14	s)-open	s)-open	PUNCT
ejpam-4582	822	15	set	set	VERB
ejpam-4582	822	16	of	of	ADP
ejpam-4582	822	17	y	y	PROPN
ejpam-4582	822	18	.	.	PUNCT
ejpam-4582	823	1	since	since	SCONJ
ejpam-4582	823	2	u	u	PRON
ejpam-4582	823	3	(	(	PUNCT
ejpam-4582	823	4	λ	λ	PROPN
ejpam-4582	823	5	,	,	PUNCT
ejpam-4582	823	6	s	s	PART
ejpam-4582	823	7	)	)	PUNCT
ejpam-4582	823	8	is	be	AUX
ejpam-4582	823	9	r(λ	r(λ	NOUN
ejpam-4582	823	10	,	,	PUNCT
ejpam-4582	823	11	s)-closed	s)-close	VERB
ejpam-4582	823	12	,	,	PUNCT
ejpam-4582	823	13	by	by	ADP
ejpam-4582	823	14	theorem	theorem	NOUN
ejpam-4582	823	15	30	30	NUM
ejpam-4582	823	16	,	,	PUNCT
ejpam-4582	823	17	[	[	X
ejpam-4582	823	18	f−1(u)](λ	f−1(u)](λ	X
ejpam-4582	823	19	,	,	PUNCT
ejpam-4582	823	20	s	s	PART
ejpam-4582	823	21	)	)	PUNCT
ejpam-4582	823	22	⊆	⊆	NUM
ejpam-4582	823	23	[	[	X
ejpam-4582	823	24	f−1(u	f−1(u	PROPN
ejpam-4582	823	25	(	(	PUNCT
ejpam-4582	823	26	λ	λ	PROPN
ejpam-4582	823	27	,	,	PUNCT
ejpam-4582	823	28	s))](λ	s))](λ	X
ejpam-4582	823	29	,	,	PUNCT
ejpam-4582	823	30	s	s	PART
ejpam-4582	823	31	)	)	PUNCT
ejpam-4582	824	1	=	=	SYM
ejpam-4582	824	2	f−1(u	f−1(u	PROPN
ejpam-4582	824	3	(	(	PUNCT
ejpam-4582	824	4	λ	λ	PROPN
ejpam-4582	824	5	,	,	PUNCT
ejpam-4582	824	6	s	s	NOUN
ejpam-4582	824	7	)	)	PUNCT
ejpam-4582	824	8	)	)	PUNCT
ejpam-4582	824	9	.	.	PUNCT
ejpam-4582	825	1	(	(	PUNCT
ejpam-4582	825	2	2	2	X
ejpam-4582	825	3	)	)	PUNCT
ejpam-4582	825	4	⇒	⇒	NOUN
ejpam-4582	825	5	(	(	PUNCT
ejpam-4582	825	6	3	3	NUM
ejpam-4582	825	7	):	):	PUNCT
ejpam-4582	825	8	the	the	DET
ejpam-4582	825	9	proof	proof	NOUN
ejpam-4582	825	10	is	be	AUX
ejpam-4582	825	11	obvious	obvious	ADJ
ejpam-4582	825	12	.	.	PUNCT
ejpam-4582	826	1	references	reference	NOUN
ejpam-4582	826	2	360	360	NUM
ejpam-4582	826	3	(	(	PUNCT
ejpam-4582	826	4	3	3	NUM
ejpam-4582	826	5	)	)	PUNCT
ejpam-4582	826	6	⇒	⇒	NOUN
ejpam-4582	826	7	(	(	PUNCT
ejpam-4582	826	8	1	1	NUM
ejpam-4582	826	9	):	):	PUNCT
ejpam-4582	826	10	let	let	VERB
ejpam-4582	826	11	f	f	PRON
ejpam-4582	826	12	be	be	AUX
ejpam-4582	826	13	any	any	DET
ejpam-4582	826	14	r(λ	r(λ	NOUN
ejpam-4582	826	15	,	,	PUNCT
ejpam-4582	826	16	s)-closed	s)-close	VERB
ejpam-4582	826	17	set	set	NOUN
ejpam-4582	826	18	of	of	ADP
ejpam-4582	826	19	y	y	PROPN
ejpam-4582	826	20	.	.	PUNCT
ejpam-4582	827	1	then	then	ADV
ejpam-4582	827	2	,	,	PUNCT
ejpam-4582	827	3	we	we	PRON
ejpam-4582	827	4	have	have	VERB
ejpam-4582	827	5	f	f	PROPN
ejpam-4582	827	6	is	be	AUX
ejpam-4582	827	7	s(λ	s(λ	PROPN
ejpam-4582	827	8	,	,	PUNCT
ejpam-4582	827	9	s)-open	s)-open	PUNCT
ejpam-4582	827	10	.	.	PUNCT
ejpam-4582	828	1	by	by	ADP
ejpam-4582	828	2	(	(	PUNCT
ejpam-4582	828	3	3	3	NUM
ejpam-4582	828	4	)	)	PUNCT
ejpam-4582	828	5	,	,	PUNCT
ejpam-4582	828	6	[	[	X
ejpam-4582	828	7	f−1(f	f−1(f	NOUN
ejpam-4582	828	8	)	)	PUNCT
ejpam-4582	828	9	]	]	PUNCT
ejpam-4582	828	10	(	(	PUNCT
ejpam-4582	828	11	λ	λ	X
ejpam-4582	828	12	,	,	PUNCT
ejpam-4582	828	13	s	s	PART
ejpam-4582	828	14	)	)	PUNCT
ejpam-4582	828	15	⊆	⊆	NUM
ejpam-4582	828	16	f−1[f	f−1[f	NOUN
ejpam-4582	828	17	(	(	PUNCT
ejpam-4582	828	18	λ	λ	X
ejpam-4582	828	19	,	,	PUNCT
ejpam-4582	828	20	s	s	PART
ejpam-4582	828	21	)	)	PUNCT
ejpam-4582	828	22	]	]	PUNCT
ejpam-4582	828	23	=	=	SYM
ejpam-4582	828	24	f−1(f	f−1(f	PROPN
ejpam-4582	828	25	)	)	PUNCT
ejpam-4582	828	26	.	.	PUNCT
ejpam-4582	829	1	therefore	therefore	ADV
ejpam-4582	829	2	,	,	PUNCT
ejpam-4582	829	3	f−1(f	f−1(f	PROPN
ejpam-4582	829	4	)	)	PUNCT
ejpam-4582	829	5	is	be	AUX
ejpam-4582	829	6	(	(	PUNCT
ejpam-4582	829	7	λ	λ	X
ejpam-4582	829	8	,	,	PUNCT
ejpam-4582	829	9	s)-closed	s)-close	VERB
ejpam-4582	829	10	and	and	CCONJ
ejpam-4582	829	11	by	by	ADP
ejpam-4582	829	12	theorem	theorem	NOUN
ejpam-4582	829	13	30	30	NUM
ejpam-4582	829	14	,	,	PUNCT
ejpam-4582	829	15	f	f	PROPN
ejpam-4582	829	16	is	be	AUX
ejpam-4582	829	17	almost	almost	ADV
ejpam-4582	829	18	(	(	PUNCT
ejpam-4582	829	19	λ	λ	X
ejpam-4582	829	20	,	,	PUNCT
ejpam-4582	829	21	s)-continuous	s)-continuous	ADJ
ejpam-4582	829	22	.	.	PUNCT
ejpam-4582	830	1	(	(	PUNCT
ejpam-4582	830	2	1	1	X
ejpam-4582	830	3	)	)	PUNCT
ejpam-4582	830	4	⇒	⇒	NOUN
ejpam-4582	830	5	(	(	PUNCT
ejpam-4582	830	6	4	4	NUM
ejpam-4582	830	7	):	):	PUNCT
ejpam-4582	830	8	let	let	VERB
ejpam-4582	830	9	u	u	PRON
ejpam-4582	830	10	be	be	AUX
ejpam-4582	830	11	any	any	DET
ejpam-4582	830	12	p(λ	p(λ	NOUN
ejpam-4582	830	13	,	,	PUNCT
ejpam-4582	830	14	s)-open	s)-open	PUNCT
ejpam-4582	830	15	set	set	VERB
ejpam-4582	830	16	of	of	ADP
ejpam-4582	830	17	y	y	PROPN
ejpam-4582	830	18	.	.	PUNCT
ejpam-4582	831	1	then	then	ADV
ejpam-4582	831	2	,	,	PUNCT
ejpam-4582	831	3	u	u	NOUN
ejpam-4582	831	4	⊆	⊆	NUM
ejpam-4582	831	5	[	[	X
ejpam-4582	831	6	u	u	X
ejpam-4582	831	7	(	(	PUNCT
ejpam-4582	831	8	λ	λ	PROPN
ejpam-4582	831	9	,	,	PUNCT
ejpam-4582	831	10	s)](λ	s)](λ	PROPN
ejpam-4582	831	11	,	,	PUNCT
ejpam-4582	831	12	s	s	PART
ejpam-4582	831	13	)	)	PUNCT
ejpam-4582	831	14	and	and	CCONJ
ejpam-4582	831	15	[	[	X
ejpam-4582	831	16	u	u	X
ejpam-4582	831	17	(	(	PUNCT
ejpam-4582	831	18	λ	λ	PROPN
ejpam-4582	831	19	,	,	PUNCT
ejpam-4582	831	20	s)](λ	s)](λ	PROPN
ejpam-4582	831	21	,	,	PUNCT
ejpam-4582	831	22	s	s	PART
ejpam-4582	831	23	)	)	PUNCT
ejpam-4582	831	24	is	be	AUX
ejpam-4582	831	25	r(λ	r(λ	NOUN
ejpam-4582	831	26	,	,	PUNCT
ejpam-4582	831	27	s)-open	s)-open	ADJ
ejpam-4582	831	28	.	.	PUNCT
ejpam-4582	832	1	by	by	ADP
ejpam-4582	832	2	theorem	theorem	ADJ
ejpam-4582	832	3	30	30	NUM
ejpam-4582	832	4	,	,	PUNCT
ejpam-4582	832	5	f−1(u	f−1(u	PROPN
ejpam-4582	832	6	)	)	PUNCT
ejpam-4582	832	7	⊆	⊆	NUM
ejpam-4582	832	8	f−1([u	f−1([u	NOUN
ejpam-4582	832	9	(	(	PUNCT
ejpam-4582	832	10	λ	λ	NOUN
ejpam-4582	832	11	,	,	PUNCT
ejpam-4582	832	12	s)](λ	s)](λ	PROPN
ejpam-4582	832	13	,	,	PUNCT
ejpam-4582	832	14	s	s	NOUN
ejpam-4582	832	15	)	)	PUNCT
ejpam-4582	832	16	)	)	PUNCT
ejpam-4582	833	1	=	=	PUNCT
ejpam-4582	834	1	[	[	X
ejpam-4582	834	2	f−1([u	f−1([u	INTJ
ejpam-4582	834	3	(	(	PUNCT
ejpam-4582	834	4	λ	λ	PROPN
ejpam-4582	834	5	,	,	PUNCT
ejpam-4582	834	6	s)](λ	s)](λ	PROPN
ejpam-4582	834	7	,	,	PUNCT
ejpam-4582	834	8	s))](λ	s))](λ	PROPN
ejpam-4582	834	9	,	,	PUNCT
ejpam-4582	834	10	s	s	PART
ejpam-4582	834	11	)	)	PUNCT
ejpam-4582	834	12	.	.	PUNCT
ejpam-4582	835	1	(	(	PUNCT
ejpam-4582	835	2	4	4	X
ejpam-4582	835	3	)	)	PUNCT
ejpam-4582	835	4	⇒	⇒	NOUN
ejpam-4582	835	5	(	(	PUNCT
ejpam-4582	835	6	1	1	NUM
ejpam-4582	835	7	):	):	PUNCT
ejpam-4582	835	8	let	let	VERB
ejpam-4582	835	9	u	u	PRON
ejpam-4582	835	10	be	be	AUX
ejpam-4582	835	11	any	any	DET
ejpam-4582	835	12	r(λ	r(λ	NOUN
ejpam-4582	835	13	,	,	PUNCT
ejpam-4582	835	14	s)-open	s)-open	PUNCT
ejpam-4582	835	15	set	set	VERB
ejpam-4582	835	16	of	of	ADP
ejpam-4582	835	17	y	y	PROPN
ejpam-4582	835	18	.	.	PUNCT
ejpam-4582	836	1	then	then	ADV
ejpam-4582	836	2	,	,	PUNCT
ejpam-4582	836	3	we	we	PRON
ejpam-4582	836	4	have	have	VERB
ejpam-4582	836	5	u	u	NOUN
ejpam-4582	836	6	is	be	AUX
ejpam-4582	836	7	p(λ	p(λ	NOUN
ejpam-4582	836	8	,	,	PUNCT
ejpam-4582	836	9	s)-open	s)-open	PUNCT
ejpam-4582	836	10	and	and	CCONJ
ejpam-4582	836	11	by	by	ADP
ejpam-4582	836	12	(	(	PUNCT
ejpam-4582	836	13	4	4	NUM
ejpam-4582	836	14	)	)	PUNCT
ejpam-4582	836	15	,	,	PUNCT
ejpam-4582	836	16	f−1(u	f−1(u	PROPN
ejpam-4582	836	17	)	)	PUNCT
ejpam-4582	836	18	⊆	⊆	NUM
ejpam-4582	836	19	[	[	X
ejpam-4582	836	20	f−1([u	f−1([u	INTJ
ejpam-4582	836	21	(	(	PUNCT
ejpam-4582	836	22	λ	λ	PROPN
ejpam-4582	836	23	,	,	PUNCT
ejpam-4582	836	24	s)](λ	s)](λ	PROPN
ejpam-4582	836	25	,	,	PUNCT
ejpam-4582	836	26	s))](λ	s))](λ	PROPN
ejpam-4582	836	27	,	,	PUNCT
ejpam-4582	836	28	s	s	PART
ejpam-4582	836	29	)	)	PUNCT
ejpam-4582	836	30	=	=	PUNCT
ejpam-4582	837	1	[	[	X
ejpam-4582	837	2	f−1(u)](λ	f−1(u)](λ	X
ejpam-4582	837	3	,	,	PUNCT
ejpam-4582	837	4	s	s	NOUN
ejpam-4582	837	5	)	)	PUNCT
ejpam-4582	837	6	.	.	PUNCT
ejpam-4582	838	1	thus	thus	ADV
ejpam-4582	838	2	,	,	PUNCT
ejpam-4582	838	3	f	f	PROPN
ejpam-4582	838	4	−1(u	−1(u	X
ejpam-4582	838	5	)	)	PUNCT
ejpam-4582	838	6	is	be	AUX
ejpam-4582	838	7	(	(	PUNCT
ejpam-4582	838	8	λ	λ	X
ejpam-4582	838	9	,	,	PUNCT
ejpam-4582	838	10	s)-open	s)-open	PUNCT
ejpam-4582	838	11	and	and	CCONJ
ejpam-4582	838	12	by	by	ADP
ejpam-4582	838	13	theorem	theorem	NOUN
ejpam-4582	838	14	30	30	NUM
ejpam-4582	838	15	,	,	PUNCT
ejpam-4582	838	16	f	f	PROPN
ejpam-4582	838	17	is	be	AUX
ejpam-4582	838	18	almost	almost	ADV
ejpam-4582	838	19	(	(	PUNCT
ejpam-4582	838	20	λ	λ	X
ejpam-4582	838	21	,	,	PUNCT
ejpam-4582	838	22	s)-continuous	s)-continuous	ADJ
ejpam-4582	838	23	.	.	NOUN
ejpam-4582	839	1	8	8	NUM
ejpam-4582	839	2	.	.	X
ejpam-4582	839	3	conclusion	conclusion	VERB
ejpam-4582	839	4	the	the	DET
ejpam-4582	839	5	notions	notion	NOUN
ejpam-4582	839	6	of	of	ADP
ejpam-4582	839	7	closed	closed	ADJ
ejpam-4582	839	8	sets	set	NOUN
ejpam-4582	839	9	and	and	CCONJ
ejpam-4582	839	10	low	low	ADJ
ejpam-4582	839	11	separation	separation	NOUN
ejpam-4582	839	12	axioms	axiom	NOUN
ejpam-4582	839	13	are	be	AUX
ejpam-4582	839	14	fundamental	fundamental	ADJ
ejpam-4582	839	15	with	with	ADP
ejpam-4582	839	16	respect	respect	NOUN
ejpam-4582	839	17	to	to	ADP
ejpam-4582	839	18	the	the	DET
ejpam-4582	839	19	investigation	investigation	NOUN
ejpam-4582	839	20	of	of	ADP
ejpam-4582	839	21	topological	topological	ADJ
ejpam-4582	839	22	spaces	space	NOUN
ejpam-4582	839	23	.	.	PUNCT
ejpam-4582	840	1	various	various	ADJ
ejpam-4582	840	2	types	type	NOUN
ejpam-4582	840	3	of	of	ADP
ejpam-4582	840	4	generalizations	generalization	NOUN
ejpam-4582	840	5	of	of	ADP
ejpam-4582	840	6	closed	closed	ADJ
ejpam-4582	840	7	sets	set	NOUN
ejpam-4582	840	8	and	and	CCONJ
ejpam-4582	840	9	some	some	DET
ejpam-4582	840	10	new	new	ADJ
ejpam-4582	840	11	separation	separation	NOUN
ejpam-4582	840	12	axioms	axiom	NOUN
ejpam-4582	840	13	have	have	AUX
ejpam-4582	840	14	been	be	AUX
ejpam-4582	840	15	researched	research	VERB
ejpam-4582	840	16	by	by	ADP
ejpam-4582	840	17	many	many	ADJ
ejpam-4582	840	18	mathematicians	mathematician	NOUN
ejpam-4582	840	19	.	.	PUNCT
ejpam-4582	841	1	semi	semi	ADJ
ejpam-4582	841	2	-	-	ADJ
ejpam-4582	841	3	open	open	ADJ
ejpam-4582	841	4	sets	set	NOUN
ejpam-4582	841	5	,	,	PUNCT
ejpam-4582	841	6	preopen	preopen	ADJ
ejpam-4582	841	7	sets	set	NOUN
ejpam-4582	841	8	,	,	PUNCT
ejpam-4582	841	9	α	α	NOUN
ejpam-4582	841	10	-	-	ADJ
ejpam-4582	841	11	open	open	ADJ
ejpam-4582	841	12	sets	set	NOUN
ejpam-4582	841	13	and	and	CCONJ
ejpam-4582	841	14	β	β	NOUN
ejpam-4582	841	15	-	-	ADJ
ejpam-4582	841	16	open	open	ADJ
ejpam-4582	841	17	sets	set	NOUN
ejpam-4582	841	18	play	play	VERB
ejpam-4582	841	19	an	an	DET
ejpam-4582	841	20	important	important	ADJ
ejpam-4582	841	21	role	role	NOUN
ejpam-4582	841	22	in	in	ADP
ejpam-4582	841	23	the	the	DET
ejpam-4582	841	24	researching	researching	NOUN
ejpam-4582	841	25	of	of	ADP
ejpam-4582	841	26	generalizations	generalization	NOUN
ejpam-4582	841	27	of	of	ADP
ejpam-4582	841	28	continuity	continuity	NOUN
ejpam-4582	841	29	in	in	ADP
ejpam-4582	841	30	topological	topological	ADJ
ejpam-4582	841	31	spaces	space	NOUN
ejpam-4582	841	32	.	.	PUNCT
ejpam-4582	842	1	using	use	VERB
ejpam-4582	842	2	different	different	ADJ
ejpam-4582	842	3	forms	form	NOUN
ejpam-4582	842	4	of	of	ADP
ejpam-4582	842	5	open	open	ADJ
ejpam-4582	842	6	sets	set	NOUN
ejpam-4582	842	7	,	,	PUNCT
ejpam-4582	842	8	several	several	ADJ
ejpam-4582	842	9	authors	author	NOUN
ejpam-4582	842	10	have	have	AUX
ejpam-4582	842	11	introduced	introduce	VERB
ejpam-4582	842	12	and	and	CCONJ
ejpam-4582	842	13	studied	study	VERB
ejpam-4582	842	14	various	various	ADJ
ejpam-4582	842	15	types	type	NOUN
ejpam-4582	842	16	of	of	ADP
ejpam-4582	842	17	weak	weak	ADJ
ejpam-4582	842	18	forms	form	NOUN
ejpam-4582	842	19	of	of	ADP
ejpam-4582	842	20	continuity	continuity	NOUN
ejpam-4582	842	21	.	.	PUNCT
ejpam-4582	843	1	this	this	DET
ejpam-4582	843	2	work	work	NOUN
ejpam-4582	843	3	is	be	AUX
ejpam-4582	843	4	concerned	concern	VERB
ejpam-4582	843	5	with	with	ADP
ejpam-4582	843	6	the	the	DET
ejpam-4582	843	7	concepts	concept	NOUN
ejpam-4582	843	8	of	of	ADP
ejpam-4582	843	9	(	(	PUNCT
ejpam-4582	843	10	λ	λ	PROPN
ejpam-4582	843	11	,	,	PUNCT
ejpam-4582	843	12	s)-closed	s)-close	VERB
ejpam-4582	843	13	sets	set	NOUN
ejpam-4582	843	14	.	.	PUNCT
ejpam-4582	844	1	moreover	moreover	ADV
ejpam-4582	844	2	,	,	PUNCT
ejpam-4582	844	3	some	some	DET
ejpam-4582	844	4	properties	property	NOUN
ejpam-4582	844	5	of	of	ADP
ejpam-4582	844	6	generalized	generalized	ADJ
ejpam-4582	844	7	(	(	PUNCT
ejpam-4582	844	8	λ	λ	X
ejpam-4582	844	9	,	,	PUNCT
ejpam-4582	844	10	s)-closed	s)-close	VERB
ejpam-4582	844	11	sets	set	NOUN
ejpam-4582	844	12	are	be	AUX
ejpam-4582	844	13	obtained	obtain	VERB
ejpam-4582	844	14	.	.	PUNCT
ejpam-4582	845	1	several	several	ADJ
ejpam-4582	845	2	characterizations	characterization	NOUN
ejpam-4582	845	3	of	of	ADP
ejpam-4582	845	4	some	some	DET
ejpam-4582	845	5	low	low	ADJ
ejpam-4582	845	6	separation	separation	NOUN
ejpam-4582	845	7	axioms	axiom	NOUN
ejpam-4582	845	8	are	be	AUX
ejpam-4582	845	9	established	establish	VERB
ejpam-4582	845	10	.	.	PUNCT
ejpam-4582	846	1	characterizations	characterization	NOUN
ejpam-4582	846	2	of	of	ADP
ejpam-4582	846	3	(	(	PUNCT
ejpam-4582	846	4	λ	λ	PROPN
ejpam-4582	846	5	,	,	PUNCT
ejpam-4582	846	6	s)-extremally	s)-extremally	ADV
ejpam-4582	846	7	disconnected	disconnected	ADJ
ejpam-4582	846	8	are	be	AUX
ejpam-4582	846	9	obtained	obtain	VERB
ejpam-4582	846	10	.	.	PUNCT
ejpam-4582	847	1	furthermore	furthermore	ADV
ejpam-4582	847	2	,	,	PUNCT
ejpam-4582	847	3	some	some	DET
ejpam-4582	847	4	characterizations	characterization	NOUN
ejpam-4582	847	5	of	of	ADP
ejpam-4582	847	6	almost	almost	ADV
ejpam-4582	847	7	(	(	PUNCT
ejpam-4582	847	8	λ	λ	PROPN
ejpam-4582	847	9	,	,	PUNCT
ejpam-4582	847	10	s)-continuous	s)-continuous	ADJ
ejpam-4582	847	11	functions	function	NOUN
ejpam-4582	847	12	are	be	AUX
ejpam-4582	847	13	explored	explore	VERB
ejpam-4582	847	14	.	.	PUNCT
ejpam-4582	848	1	the	the	DET
ejpam-4582	848	2	ideas	idea	NOUN
ejpam-4582	848	3	and	and	CCONJ
ejpam-4582	848	4	results	result	NOUN
ejpam-4582	848	5	of	of	ADP
ejpam-4582	848	6	this	this	DET
ejpam-4582	848	7	work	work	NOUN
ejpam-4582	848	8	may	may	AUX
ejpam-4582	848	9	motivate	motivate	VERB
ejpam-4582	848	10	further	further	ADJ
ejpam-4582	848	11	research	research	NOUN
ejpam-4582	848	12	.	.	PUNCT
ejpam-4582	849	1	acknowledgements	acknowledgement	NOUN
ejpam-4582	849	2	this	this	DET
ejpam-4582	849	3	research	research	NOUN
ejpam-4582	849	4	project	project	NOUN
ejpam-4582	849	5	was	be	AUX
ejpam-4582	849	6	financially	financially	ADV
ejpam-4582	849	7	supported	support	VERB
ejpam-4582	849	8	by	by	ADP
ejpam-4582	849	9	mahasarakham	mahasarakham	PROPN
ejpam-4582	849	10	university	university	PROPN
ejpam-4582	849	11	.	.	PUNCT
ejpam-4582	850	1	references	reference	NOUN
ejpam-4582	850	2	[	[	X
ejpam-4582	850	3	1	1	NUM
ejpam-4582	850	4	]	]	PUNCT
ejpam-4582	850	5	d.	d.	PROPN
ejpam-4582	850	6	andrijević.	andrijević.	PROPN
ejpam-4582	850	7	semi	semi	ADJ
ejpam-4582	850	8	-	-	ADJ
ejpam-4582	850	9	preopen	preopen	ADJ
ejpam-4582	850	10	sets	set	NOUN
ejpam-4582	850	11	.	.	PUNCT
ejpam-4582	851	1	matematički	matematički	PROPN
ejpam-4582	851	2	vesnik	vesnik	PROPN
ejpam-4582	851	3	,	,	PUNCT
ejpam-4582	851	4	38:24–32	38:24–32	NUM
ejpam-4582	851	5	,	,	PUNCT
ejpam-4582	851	6	1986	1986	NUM
ejpam-4582	851	7	.	.	PUNCT
ejpam-4582	852	1	[	[	X
ejpam-4582	852	2	2	2	NUM
ejpam-4582	852	3	]	]	PUNCT
ejpam-4582	852	4	a.	a.	NOUN
ejpam-4582	852	5	v.	v.	PROPN
ejpam-4582	852	6	arhangel’skĭı	arhangel’skĭı	PROPN
ejpam-4582	852	7	and	and	CCONJ
ejpam-4582	852	8	p.	p.	PROPN
ejpam-4582	852	9	j.	j.	PROPN
ejpam-4582	852	10	collins	collins	PROPN
ejpam-4582	852	11	.	.	PUNCT
ejpam-4582	853	1	on	on	ADP
ejpam-4582	853	2	submaximal	submaximal	ADJ
ejpam-4582	853	3	spaces	space	NOUN
ejpam-4582	853	4	.	.	PUNCT
ejpam-4582	854	1	topology	topology	NOUN
ejpam-4582	854	2	and	and	CCONJ
ejpam-4582	854	3	its	its	PRON
ejpam-4582	854	4	applicaions	applicaion	NOUN
ejpam-4582	854	5	,	,	PUNCT
ejpam-4582	854	6	64:219–241	64:219–241	PROPN
ejpam-4582	854	7	,	,	PUNCT
ejpam-4582	854	8	1995	1995	NUM
ejpam-4582	854	9	.	.	PUNCT
ejpam-4582	855	1	[	[	X
ejpam-4582	855	2	3	3	X
ejpam-4582	855	3	]	]	PUNCT
ejpam-4582	855	4	c.	c.	PROPN
ejpam-4582	855	5	boonpok	boonpok	PROPN
ejpam-4582	855	6	and	and	CCONJ
ejpam-4582	855	7	j.	j.	PROPN
ejpam-4582	855	8	khampakdee	khampakdee	PROPN
ejpam-4582	855	9	.	.	PUNCT
ejpam-4582	856	1	(	(	PUNCT
ejpam-4582	856	2	λ	λ	NOUN
ejpam-4582	856	3	,	,	PUNCT
ejpam-4582	856	4	sp)-open	sp)-open	ADJ
ejpam-4582	856	5	sets	set	NOUN
ejpam-4582	856	6	in	in	ADP
ejpam-4582	856	7	topological	topological	ADJ
ejpam-4582	856	8	spaces	space	NOUN
ejpam-4582	856	9	.	.	PUNCT
ejpam-4582	857	1	european	european	ADJ
ejpam-4582	857	2	journal	journal	PROPN
ejpam-4582	857	3	of	of	ADP
ejpam-4582	857	4	pure	pure	ADJ
ejpam-4582	857	5	and	and	CCONJ
ejpam-4582	857	6	applied	applied	ADJ
ejpam-4582	857	7	mathematics	mathematic	NOUN
ejpam-4582	857	8	,	,	PUNCT
ejpam-4582	857	9	15(2):572–588	15(2):572–588	NUM
ejpam-4582	857	10	,	,	PUNCT
ejpam-4582	857	11	2022	2022	NUM
ejpam-4582	857	12	.	.	PUNCT
ejpam-4582	858	1	[	[	X
ejpam-4582	858	2	4	4	X
ejpam-4582	858	3	]	]	X
ejpam-4582	858	4	s.	s.	PROPN
ejpam-4582	858	5	buadong	buadong	PROPN
ejpam-4582	858	6	,	,	PUNCT
ejpam-4582	858	7	c.	c.	PROPN
ejpam-4582	858	8	viriyapong	viriyapong	PROPN
ejpam-4582	858	9	,	,	PUNCT
ejpam-4582	858	10	and	and	CCONJ
ejpam-4582	858	11	c.	c.	PROPN
ejpam-4582	858	12	boonpok	boonpok	PROPN
ejpam-4582	858	13	.	.	PUNCT
ejpam-4582	859	1	on	on	ADP
ejpam-4582	859	2	generalized	generalized	ADJ
ejpam-4582	859	3	topology	topology	NOUN
ejpam-4582	859	4	and	and	CCONJ
ejpam-4582	859	5	minimal	minimal	ADJ
ejpam-4582	859	6	structure	structure	NOUN
ejpam-4582	859	7	spaces	space	NOUN
ejpam-4582	859	8	.	.	PUNCT
ejpam-4582	860	1	international	international	ADJ
ejpam-4582	860	2	journal	journal	PROPN
ejpam-4582	860	3	of	of	ADP
ejpam-4582	860	4	mathematical	mathematical	ADJ
ejpam-4582	860	5	analysis	analysis	NOUN
ejpam-4582	860	6	,	,	PUNCT
ejpam-4582	860	7	5(31):1507–1516	5(31):1507–1516	NUM
ejpam-4582	860	8	,	,	PUNCT
ejpam-4582	860	9	2011	2011	NUM
ejpam-4582	860	10	.	.	PUNCT
ejpam-4582	861	1	[	[	X
ejpam-4582	861	2	5	5	NUM
ejpam-4582	861	3	]	]	PUNCT
ejpam-4582	861	4	m.	m.	NOUN
ejpam-4582	861	5	caldas	caldas	PROPN
ejpam-4582	861	6	and	and	CCONJ
ejpam-4582	861	7	j.	j.	PROPN
ejpam-4582	861	8	dontchev	dontchev	PROPN
ejpam-4582	861	9	.	.	PUNCT
ejpam-4582	862	1	g.λs	g.λs	ADJ
ejpam-4582	862	2	-	-	PUNCT
ejpam-4582	862	3	sets	set	NOUN
ejpam-4582	862	4	and	and	CCONJ
ejpam-4582	862	5	g.vs	g.vs	NOUN
ejpam-4582	862	6	-	-	PUNCT
ejpam-4582	862	7	sets	set	NOUN
ejpam-4582	862	8	.	.	PUNCT
ejpam-4582	863	1	arxiv	arxiv	NOUN
ejpam-4582	863	2	:	:	PUNCT
ejpam-4582	863	3	math/9810080v1	math/9810080v1	PROPN
ejpam-4582	864	1	[	[	X
ejpam-4582	864	2	math	math	NOUN
ejpam-4582	864	3	:	:	PUNCT
ejpam-4582	864	4	gn	gn	PROPN
ejpam-4582	864	5	]	]	X
ejpam-4582	864	6	,	,	PUNCT
ejpam-4582	864	7	1998	1998	NUM
ejpam-4582	864	8	.	.	PUNCT
ejpam-4582	865	1	references	reference	NOUN
ejpam-4582	865	2	361	361	NUM
ejpam-4582	865	3	[	[	X
ejpam-4582	865	4	6	6	NUM
ejpam-4582	865	5	]	]	PUNCT
ejpam-4582	865	6	d.	d.	PROPN
ejpam-4582	865	7	cameron	cameron	PROPN
ejpam-4582	865	8	.	.	PUNCT
ejpam-4582	866	1	properties	property	NOUN
ejpam-4582	866	2	of	of	ADP
ejpam-4582	866	3	s	s	NOUN
ejpam-4582	866	4	-	-	PUNCT
ejpam-4582	866	5	closed	closed	ADJ
ejpam-4582	866	6	spaces	space	NOUN
ejpam-4582	866	7	.	.	PUNCT
ejpam-4582	867	1	proceedings	proceeding	NOUN
ejpam-4582	867	2	of	of	ADP
ejpam-4582	867	3	the	the	DET
ejpam-4582	867	4	university	university	PROPN
ejpam-4582	867	5	of	of	ADP
ejpam-4582	867	6	oklahoma	oklahoma	PROPN
ejpam-4582	867	7	topology	topology	PROPN
ejpam-4582	867	8	conference	conference	PROPN
ejpam-4582	867	9	,	,	PUNCT
ejpam-4582	867	10	oklahoma	oklahoma	PROPN
ejpam-4582	867	11	,	,	PUNCT
ejpam-4582	867	12	1978	1978	NUM
ejpam-4582	867	13	.	.	PUNCT
ejpam-4582	868	1	[	[	X
ejpam-4582	868	2	7	7	X
ejpam-4582	868	3	]	]	X
ejpam-4582	868	4	f.	f.	PROPN
ejpam-4582	868	5	cammaroto	cammaroto	NOUN
ejpam-4582	868	6	and	and	CCONJ
ejpam-4582	868	7	t.	t.	PROPN
ejpam-4582	868	8	noiri	noiri	PROPN
ejpam-4582	868	9	.	.	PUNCT
ejpam-4582	869	1	on	on	ADP
ejpam-4582	869	2	λm	λm	NOUN
ejpam-4582	869	3	-	-	PUNCT
ejpam-4582	869	4	sets	set	NOUN
ejpam-4582	869	5	and	and	CCONJ
ejpam-4582	869	6	related	relate	VERB
ejpam-4582	869	7	topological	topological	ADJ
ejpam-4582	869	8	spaces	space	NOUN
ejpam-4582	869	9	.	.	PUNCT
ejpam-4582	870	1	acta	acta	PROPN
ejpam-4582	870	2	mathematica	mathematica	PROPN
ejpam-4582	870	3	hungarica	hungarica	PROPN
ejpam-4582	870	4	,	,	PUNCT
ejpam-4582	870	5	109:261–279	109:261–279	NUM
ejpam-4582	870	6	,	,	PUNCT
ejpam-4582	870	7	2005	2005	NUM
ejpam-4582	870	8	.	.	PUNCT
ejpam-4582	871	1	[	[	X
ejpam-4582	871	2	8	8	NUM
ejpam-4582	871	3	]	]	PUNCT
ejpam-4582	871	4	a.	a.	PROPN
ejpam-4582	871	5	s.	s.	PROPN
ejpam-4582	871	6	davis	davis	PROPN
ejpam-4582	871	7	.	.	PUNCT
ejpam-4582	872	1	indexed	index	VERB
ejpam-4582	872	2	systems	system	NOUN
ejpam-4582	872	3	of	of	ADP
ejpam-4582	872	4	neighborhoods	neighborhood	NOUN
ejpam-4582	872	5	for	for	ADP
ejpam-4582	872	6	general	general	ADJ
ejpam-4582	872	7	topological	topological	ADJ
ejpam-4582	872	8	spaces	space	NOUN
ejpam-4582	872	9	.	.	PUNCT
ejpam-4582	873	1	the	the	DET
ejpam-4582	873	2	american	american	PROPN
ejpam-4582	873	3	mathematical	mathematical	PROPN
ejpam-4582	873	4	monthly	monthly	ADV
ejpam-4582	873	5	,	,	PUNCT
ejpam-4582	873	6	68:886–893	68:886–893	PROPN
ejpam-4582	873	7	,	,	PUNCT
ejpam-4582	873	8	1961	1961	NUM
ejpam-4582	873	9	.	.	PUNCT
ejpam-4582	874	1	[	[	X
ejpam-4582	874	2	9	9	NUM
ejpam-4582	874	3	]	]	PUNCT
ejpam-4582	874	4	c.	c.	PROPN
ejpam-4582	874	5	dorsett	dorsett	PROPN
ejpam-4582	874	6	.	.	PUNCT
ejpam-4582	875	1	r0	r0	NOUN
ejpam-4582	875	2	and	and	CCONJ
ejpam-4582	875	3	r1	r1	PROPN
ejpam-4582	875	4	topological	topological	ADJ
ejpam-4582	875	5	spaces	space	NOUN
ejpam-4582	875	6	.	.	PUNCT
ejpam-4582	876	1	matematički	matematički	PROPN
ejpam-4582	876	2	vesnik	vesnik	PROPN
ejpam-4582	876	3	,	,	PUNCT
ejpam-4582	876	4	2(15)(30):117–122	2(15)(30):117–122	NUM
ejpam-4582	876	5	,	,	PUNCT
ejpam-4582	876	6	1978	1978	NUM
ejpam-4582	876	7	.	.	PUNCT
ejpam-4582	877	1	[	[	X
ejpam-4582	877	2	10	10	NUM
ejpam-4582	877	3	]	]	PUNCT
ejpam-4582	877	4	k.	k.	PROPN
ejpam-4582	877	5	k.	k.	PROPN
ejpam-4582	877	6	dube	dube	PROPN
ejpam-4582	877	7	.	.	PUNCT
ejpam-4582	878	1	a	a	DET
ejpam-4582	878	2	note	note	NOUN
ejpam-4582	878	3	on	on	ADP
ejpam-4582	878	4	r0	r0	PROPN
ejpam-4582	878	5	topological	topological	ADJ
ejpam-4582	878	6	spaces	space	NOUN
ejpam-4582	878	7	.	.	PUNCT
ejpam-4582	879	1	matematički	matematički	PROPN
ejpam-4582	879	2	vesnik	vesnik	PROPN
ejpam-4582	879	3	,	,	PUNCT
ejpam-4582	879	4	11:203–208	11:203–208	NUM
ejpam-4582	879	5	,	,	PUNCT
ejpam-4582	879	6	1974	1974	NUM
ejpam-4582	879	7	.	.	PUNCT
ejpam-4582	880	1	[	[	X
ejpam-4582	880	2	11	11	NUM
ejpam-4582	880	3	]	]	X
ejpam-4582	880	4	w.	w.	PROPN
ejpam-4582	880	5	dungthaisong	dungthaisong	PROPN
ejpam-4582	880	6	,	,	PUNCT
ejpam-4582	880	7	c.	c.	PROPN
ejpam-4582	880	8	boonpok	boonpok	PROPN
ejpam-4582	880	9	,	,	PUNCT
ejpam-4582	880	10	and	and	CCONJ
ejpam-4582	880	11	c.	c.	PROPN
ejpam-4582	880	12	viriyapong	viriyapong	PROPN
ejpam-4582	880	13	.	.	PUNCT
ejpam-4582	881	1	generalized	generalize	VERB
ejpam-4582	881	2	closed	close	VERB
ejpam-4582	881	3	sets	set	NOUN
ejpam-4582	881	4	in	in	ADP
ejpam-4582	881	5	bigeneralized	bigeneralize	VERB
ejpam-4582	881	6	topological	topological	ADJ
ejpam-4582	881	7	spaces	space	NOUN
ejpam-4582	881	8	.	.	PUNCT
ejpam-4582	882	1	international	international	ADJ
ejpam-4582	882	2	journal	journal	PROPN
ejpam-4582	882	3	of	of	ADP
ejpam-4582	882	4	mathematical	mathematical	ADJ
ejpam-4582	882	5	analysis	analysis	NOUN
ejpam-4582	882	6	,	,	PUNCT
ejpam-4582	882	7	5(24):1175–1184	5(24):1175–1184	NUM
ejpam-4582	882	8	,	,	PUNCT
ejpam-4582	882	9	2011	2011	NUM
ejpam-4582	882	10	.	.	PUNCT
ejpam-4582	883	1	[	[	X
ejpam-4582	883	2	12	12	NUM
ejpam-4582	883	3	]	]	X
ejpam-4582	883	4	w.	w.	PROPN
ejpam-4582	883	5	dunham	dunham	PROPN
ejpam-4582	883	6	.	.	PUNCT
ejpam-4582	884	1	t	t	PROPN
ejpam-4582	884	2	1	1	NUM
ejpam-4582	884	3	2	2	NUM
ejpam-4582	884	4	-spaces	-space	NOUN
ejpam-4582	884	5	.	.	PUNCT
ejpam-4582	885	1	kyungpook	kyungpook	PROPN
ejpam-4582	885	2	mathematical	mathematical	PROPN
ejpam-4582	885	3	journal	journal	PROPN
ejpam-4582	885	4	,	,	PUNCT
ejpam-4582	885	5	17:161–169	17:161–169	PROPN
ejpam-4582	885	6	,	,	PUNCT
ejpam-4582	885	7	1977	1977	NUM
ejpam-4582	885	8	.	.	PUNCT
ejpam-4582	886	1	[	[	X
ejpam-4582	886	2	13	13	NUM
ejpam-4582	886	3	]	]	X
ejpam-4582	886	4	w.	w.	PROPN
ejpam-4582	886	5	dunham	dunham	PROPN
ejpam-4582	886	6	and	and	CCONJ
ejpam-4582	886	7	n.	n.	PROPN
ejpam-4582	886	8	levine	levine	PROPN
ejpam-4582	886	9	.	.	PUNCT
ejpam-4582	887	1	further	further	ADJ
ejpam-4582	887	2	results	result	NOUN
ejpam-4582	887	3	on	on	ADP
ejpam-4582	887	4	generalized	generalized	ADJ
ejpam-4582	887	5	closed	closed	ADJ
ejpam-4582	887	6	sets	set	NOUN
ejpam-4582	887	7	in	in	ADP
ejpam-4582	887	8	topology	topology	NOUN
ejpam-4582	887	9	.	.	PUNCT
ejpam-4582	888	1	kyungpook	kyungpook	PROPN
ejpam-4582	888	2	mathematical	mathematical	PROPN
ejpam-4582	888	3	journal	journal	PROPN
ejpam-4582	888	4	,	,	PUNCT
ejpam-4582	888	5	20:169–175	20:169–175	PROPN
ejpam-4582	888	6	,	,	PUNCT
ejpam-4582	888	7	1980	1980	NUM
ejpam-4582	888	8	.	.	PUNCT
ejpam-4582	889	1	[	[	X
ejpam-4582	889	2	14	14	NUM
ejpam-4582	889	3	]	]	X
ejpam-4582	889	4	l.	l.	PROPN
ejpam-4582	889	5	gillman	gillman	PROPN
ejpam-4582	889	6	and	and	CCONJ
ejpam-4582	889	7	m.	m.	PROPN
ejpam-4582	889	8	jerison	jerison	PROPN
ejpam-4582	889	9	.	.	PUNCT
ejpam-4582	890	1	rings	ring	NOUN
ejpam-4582	890	2	of	of	ADP
ejpam-4582	890	3	continuous	continuous	ADJ
ejpam-4582	890	4	functions	function	NOUN
ejpam-4582	890	5	,	,	PUNCT
ejpam-4582	890	6	the	the	DET
ejpam-4582	890	7	university	university	NOUN
ejpam-4582	890	8	series	series	NOUN
ejpam-4582	890	9	in	in	ADP
ejpam-4582	890	10	higher	high	ADJ
ejpam-4582	890	11	mathematics	mathematic	NOUN
ejpam-4582	890	12	.	.	PUNCT
ejpam-4582	891	1	van	van	PROPN
ejpam-4582	891	2	nostrand	nostrand	PROPN
ejpam-4582	891	3	,	,	PUNCT
ejpam-4582	891	4	princeton	princeton	PROPN
ejpam-4582	891	5	,	,	PUNCT
ejpam-4582	891	6	new	new	PROPN
ejpam-4582	891	7	york	york	PROPN
ejpam-4582	891	8	,	,	PUNCT
ejpam-4582	891	9	1960	1960	NUM
ejpam-4582	891	10	.	.	PUNCT
ejpam-4582	892	1	[	[	X
ejpam-4582	892	2	15	15	NUM
ejpam-4582	892	3	]	]	X
ejpam-4582	892	4	r.	r.	PROPN
ejpam-4582	892	5	a.	a.	PROPN
ejpam-4582	892	6	herrman	herrman	PROPN
ejpam-4582	892	7	.	.	PUNCT
ejpam-4582	893	1	rc	rc	NOUN
ejpam-4582	893	2	-	-	NOUN
ejpam-4582	893	3	convergence	convergence	NOUN
ejpam-4582	893	4	.	.	PUNCT
ejpam-4582	894	1	proceedings	proceeding	NOUN
ejpam-4582	894	2	of	of	ADP
ejpam-4582	894	3	the	the	DET
ejpam-4582	894	4	american	american	PROPN
ejpam-4582	894	5	mathematical	mathematical	ADJ
ejpam-4582	894	6	soceity	soceity	NOUN
ejpam-4582	894	7	,	,	PUNCT
ejpam-4582	894	8	75:311–317	75:311–317	PROPN
ejpam-4582	894	9	,	,	PUNCT
ejpam-4582	894	10	1979	1979	NUM
ejpam-4582	894	11	.	.	PUNCT
ejpam-4582	895	1	[	[	X
ejpam-4582	895	2	16	16	NUM
ejpam-4582	895	3	]	]	X
ejpam-4582	895	4	e.	e.	PROPN
ejpam-4582	895	5	hewitt	hewitt	PROPN
ejpam-4582	895	6	.	.	PUNCT
ejpam-4582	896	1	a	a	DET
ejpam-4582	896	2	problem	problem	NOUN
ejpam-4582	896	3	of	of	ADP
ejpam-4582	896	4	set	set	NOUN
ejpam-4582	896	5	-	-	PUNCT
ejpam-4582	896	6	theoretic	theoretic	NOUN
ejpam-4582	896	7	topology	topology	NOUN
ejpam-4582	896	8	.	.	PUNCT
ejpam-4582	897	1	duke	duke	PROPN
ejpam-4582	897	2	mathematical	mathematical	PROPN
ejpam-4582	897	3	journal	journal	PROPN
ejpam-4582	897	4	,	,	PUNCT
ejpam-4582	897	5	10:309	10:309	NUM
ejpam-4582	897	6	–	–	PUNCT
ejpam-4582	897	7	333	333	NUM
ejpam-4582	897	8	,	,	PUNCT
ejpam-4582	897	9	1943	1943	NUM
ejpam-4582	897	10	.	.	PUNCT
ejpam-4582	898	1	[	[	X
ejpam-4582	898	2	17	17	NUM
ejpam-4582	898	3	]	]	X
ejpam-4582	898	4	n.	n.	PROPN
ejpam-4582	898	5	levine	levine	PROPN
ejpam-4582	898	6	.	.	PUNCT
ejpam-4582	899	1	semi	semi	ADJ
ejpam-4582	899	2	-	-	ADJ
ejpam-4582	899	3	open	open	ADJ
ejpam-4582	899	4	sets	set	NOUN
ejpam-4582	899	5	and	and	CCONJ
ejpam-4582	899	6	semi	semi	ADJ
ejpam-4582	899	7	-	-	NOUN
ejpam-4582	899	8	continuity	continuity	NOUN
ejpam-4582	899	9	in	in	ADP
ejpam-4582	899	10	topological	topological	ADJ
ejpam-4582	899	11	spaces	space	NOUN
ejpam-4582	899	12	.	.	PUNCT
ejpam-4582	900	1	the	the	DET
ejpam-4582	900	2	american	american	PROPN
ejpam-4582	900	3	mathematical	mathematical	PROPN
ejpam-4582	900	4	monthly	monthly	ADV
ejpam-4582	900	5	,	,	PUNCT
ejpam-4582	900	6	70:36–41	70:36–41	NUM
ejpam-4582	900	7	,	,	PUNCT
ejpam-4582	900	8	1963	1963	NUM
ejpam-4582	900	9	.	.	PUNCT
ejpam-4582	901	1	[	[	X
ejpam-4582	901	2	18	18	NUM
ejpam-4582	901	3	]	]	X
ejpam-4582	901	4	n.	n.	PROPN
ejpam-4582	901	5	levine	levine	PROPN
ejpam-4582	901	6	.	.	PUNCT
ejpam-4582	902	1	generalized	generalize	VERB
ejpam-4582	902	2	closed	closed	ADJ
ejpam-4582	902	3	sets	set	NOUN
ejpam-4582	902	4	in	in	ADP
ejpam-4582	902	5	topology	topology	NOUN
ejpam-4582	902	6	.	.	PUNCT
ejpam-4582	903	1	rendiconti	rendiconti	VERB
ejpam-4582	903	2	del	del	PROPN
ejpam-4582	903	3	circolo	circolo	PROPN
ejpam-4582	903	4	matematico	matematico	X
ejpam-4582	903	5	de	de	X
ejpam-4582	903	6	palermo	palermo	X
ejpam-4582	903	7	(	(	PUNCT
ejpam-4582	903	8	2	2	NUM
ejpam-4582	903	9	)	)	PUNCT
ejpam-4582	903	10	,	,	PUNCT
ejpam-4582	903	11	19:89–96	19:89–96	NUM
ejpam-4582	903	12	,	,	PUNCT
ejpam-4582	903	13	1970	1970	NUM
ejpam-4582	903	14	.	.	PUNCT
ejpam-4582	904	1	[	[	X
ejpam-4582	904	2	19	19	NUM
ejpam-4582	904	3	]	]	X
ejpam-4582	904	4	s.	s.	PROPN
ejpam-4582	904	5	lugojan	lugojan	PROPN
ejpam-4582	904	6	.	.	PUNCT
ejpam-4582	905	1	generalized	generalized	ADJ
ejpam-4582	905	2	topology	topology	NOUN
ejpam-4582	905	3	.	.	PUNCT
ejpam-4582	906	1	studii	studii	PROPN
ejpam-4582	906	2	şi	şi	PROPN
ejpam-4582	906	3	cercetǎri	cercetǎri	NOUN
ejpam-4582	906	4	de	de	X
ejpam-4582	906	5	matematicǎ	matematicǎ	NOUN
ejpam-4582	906	6	,	,	PUNCT
ejpam-4582	906	7	34:348–360	34:348–360	PROPN
ejpam-4582	906	8	,	,	PUNCT
ejpam-4582	906	9	1982	1982	NUM
ejpam-4582	906	10	.	.	PUNCT
ejpam-4582	907	1	[	[	X
ejpam-4582	907	2	20	20	NUM
ejpam-4582	907	3	]	]	PUNCT
ejpam-4582	907	4	s.	s.	PROPN
ejpam-4582	907	5	a.	a.	PROPN
ejpam-4582	907	6	naimpally	naimpally	ADV
ejpam-4582	907	7	.	.	PUNCT
ejpam-4582	908	1	on	on	ADP
ejpam-4582	908	2	r0	r0	NOUN
ejpam-4582	908	3	-	-	PUNCT
ejpam-4582	908	4	topological	topological	ADJ
ejpam-4582	908	5	spaces	space	NOUN
ejpam-4582	908	6	.	.	PUNCT
ejpam-4582	909	1	annales	annales	PROPN
ejpam-4582	909	2	universitatis	universitatis	PROPN
ejpam-4582	909	3	scientiarum	scientiarum	PROPN
ejpam-4582	909	4	budapestinensis	budapestinensis	PROPN
ejpam-4582	909	5	de	de	PROPN
ejpam-4582	909	6	rolando	rolando	PROPN
ejpam-4582	909	7	eötvös	eötvös	PROPN
ejpam-4582	909	8	nominatae	nominatae	NOUN
ejpam-4582	909	9	sectio	sectio	NOUN
ejpam-4582	909	10	mathematica	mathematica	PROPN
ejpam-4582	909	11	,	,	PUNCT
ejpam-4582	909	12	10:53–54	10:53–54	NUM
ejpam-4582	909	13	,	,	PUNCT
ejpam-4582	909	14	1967	1967	NUM
ejpam-4582	909	15	.	.	PUNCT
ejpam-4582	910	1	[	[	X
ejpam-4582	910	2	21	21	NUM
ejpam-4582	910	3	]	]	PUNCT
ejpam-4582	910	4	t.	t.	PROPN
ejpam-4582	910	5	noiri	noiri	PROPN
ejpam-4582	910	6	.	.	PUNCT
ejpam-4582	911	1	on	on	ADP
ejpam-4582	911	2	s	s	NOUN
ejpam-4582	911	3	-	-	PUNCT
ejpam-4582	911	4	closed	closed	ADJ
ejpam-4582	911	5	subspaces	subspace	NOUN
ejpam-4582	911	6	.	.	PUNCT
ejpam-4582	912	1	atti	atti	PROPN
ejpam-4582	912	2	della	della	PROPN
ejpam-4582	912	3	accademia	accademia	PROPN
ejpam-4582	912	4	nazionale	nazionale	PROPN
ejpam-4582	912	5	dei	dei	PROPN
ejpam-4582	912	6	lincei	lincei	NOUN
ejpam-4582	912	7	.	.	PUNCT
ejpam-4582	913	1	rendiconti	rendiconti	PROPN
ejpam-4582	913	2	.	.	PUNCT
ejpam-4582	913	3	classe	classe	PROPN
ejpam-4582	913	4	di	di	PROPN
ejpam-4582	913	5	scienze	scienze	PROPN
ejpam-4582	913	6	fisiche	fisiche	PROPN
ejpam-4582	913	7	,	,	PUNCT
ejpam-4582	913	8	matematiche	matematiche	PROPN
ejpam-4582	913	9	e	e	X
ejpam-4582	913	10	naturali	naturali	PROPN
ejpam-4582	913	11	,	,	PUNCT
ejpam-4582	913	12	64:273–285	64:273–285	PROPN
ejpam-4582	913	13	,	,	PUNCT
ejpam-4582	913	14	1978	1978	NUM
ejpam-4582	913	15	.	.	PUNCT
ejpam-4582	914	1	[	[	X
ejpam-4582	914	2	22	22	NUM
ejpam-4582	914	3	]	]	PUNCT
ejpam-4582	914	4	t.	t.	PROPN
ejpam-4582	914	5	noiri	noiri	PROPN
ejpam-4582	914	6	.	.	PUNCT
ejpam-4582	915	1	a	a	DET
ejpam-4582	915	2	note	note	NOUN
ejpam-4582	915	3	on	on	ADP
ejpam-4582	915	4	extremally	extremally	ADV
ejpam-4582	915	5	disconnected	disconnected	ADJ
ejpam-4582	915	6	spaces	space	NOUN
ejpam-4582	915	7	.	.	PUNCT
ejpam-4582	916	1	proceedings	proceeding	NOUN
ejpam-4582	916	2	of	of	ADP
ejpam-4582	916	3	the	the	DET
ejpam-4582	916	4	american	american	PROPN
ejpam-4582	916	5	mathematical	mathematical	PROPN
ejpam-4582	916	6	society	society	NOUN
ejpam-4582	916	7	,	,	PUNCT
ejpam-4582	916	8	79(2):327–330	79(2):327–330	NUM
ejpam-4582	916	9	,	,	PUNCT
ejpam-4582	916	10	1980	1980	NUM
ejpam-4582	916	11	.	.	PUNCT
ejpam-4582	917	1	references	reference	NOUN
ejpam-4582	917	2	362	362	NUM
ejpam-4582	918	1	[	[	X
ejpam-4582	918	2	23	23	NUM
ejpam-4582	918	3	]	]	PUNCT
ejpam-4582	918	4	t.	t.	PROPN
ejpam-4582	918	5	noiri	noiri	PROPN
ejpam-4582	918	6	.	.	PUNCT
ejpam-4582	919	1	characterizations	characterization	NOUN
ejpam-4582	919	2	of	of	ADP
ejpam-4582	919	3	extremally	extremally	ADV
ejpam-4582	919	4	disconnected	disconnected	ADJ
ejpam-4582	919	5	spaces	space	NOUN
ejpam-4582	919	6	.	.	PUNCT
ejpam-4582	920	1	indian	indian	ADJ
ejpam-4582	920	2	journal	journal	PROPN
ejpam-4582	920	3	of	of	ADP
ejpam-4582	920	4	pure	pure	ADJ
ejpam-4582	920	5	and	and	CCONJ
ejpam-4582	920	6	applied	applied	ADJ
ejpam-4582	920	7	mathematics	mathematic	NOUN
ejpam-4582	920	8	,	,	PUNCT
ejpam-4582	920	9	19:325–329	19:325–329	NUM
ejpam-4582	920	10	,	,	PUNCT
ejpam-4582	920	11	1988	1988	NUM
ejpam-4582	920	12	.	.	PUNCT
ejpam-4582	921	1	[	[	X
ejpam-4582	921	2	24	24	NUM
ejpam-4582	921	3	]	]	X
ejpam-4582	921	4	n.	n.	NOUN
ejpam-4582	921	5	a.	a.	NOUN
ejpam-4582	921	6	shanin	shanin	PROPN
ejpam-4582	921	7	.	.	PUNCT
ejpam-4582	922	1	on	on	ADP
ejpam-4582	922	2	separation	separation	NOUN
ejpam-4582	922	3	in	in	ADP
ejpam-4582	922	4	topological	topological	ADJ
ejpam-4582	922	5	spaces	space	NOUN
ejpam-4582	922	6	.	.	PUNCT
ejpam-4582	923	1	doklady	doklady	PROPN
ejpam-4582	923	2	akademii	akademii	NOUN
ejpam-4582	923	3	nauk	nauk	NOUN
ejpam-4582	923	4	sssr	sssr	NOUN
ejpam-4582	923	5	,	,	PUNCT
ejpam-4582	923	6	38:110–113	38:110–113	NUM
ejpam-4582	923	7	,	,	PUNCT
ejpam-4582	923	8	1943	1943	NUM
ejpam-4582	923	9	.	.	PUNCT
ejpam-4582	924	1	[	[	X
ejpam-4582	924	2	25	25	NUM
ejpam-4582	924	3	]	]	PUNCT
ejpam-4582	924	4	d.	d.	PROPN
ejpam-4582	924	5	sivaraj	sivaraj	PROPN
ejpam-4582	924	6	.	.	PUNCT
ejpam-4582	925	1	a	a	DET
ejpam-4582	925	2	note	note	NOUN
ejpam-4582	925	3	on	on	ADP
ejpam-4582	925	4	extremally	extremally	ADV
ejpam-4582	925	5	disconnected	disconnected	ADJ
ejpam-4582	925	6	spaces	space	NOUN
ejpam-4582	925	7	.	.	PUNCT
ejpam-4582	926	1	indian	indian	ADJ
ejpam-4582	926	2	journal	journal	PROPN
ejpam-4582	926	3	of	of	ADP
ejpam-4582	926	4	pure	pure	ADJ
ejpam-4582	926	5	and	and	CCONJ
ejpam-4582	926	6	applied	applied	ADJ
ejpam-4582	926	7	mathematics	mathematic	NOUN
ejpam-4582	926	8	,	,	PUNCT
ejpam-4582	926	9	17(12):1373–1375	17(12):1373–1375	NUM
ejpam-4582	926	10	,	,	PUNCT
ejpam-4582	926	11	1986	1986	NUM
ejpam-4582	926	12	.	.	PUNCT
ejpam-4582	927	1	[	[	X
ejpam-4582	927	2	26	26	NUM
ejpam-4582	927	3	]	]	PUNCT
ejpam-4582	927	4	t.	t.	PROPN
ejpam-4582	927	5	thompson	thompson	PROPN
ejpam-4582	927	6	.	.	PUNCT
ejpam-4582	928	1	s	s	X
ejpam-4582	928	2	-	-	PUNCT
ejpam-4582	928	3	closed	closed	ADJ
ejpam-4582	928	4	spaces	space	NOUN
ejpam-4582	928	5	.	.	PUNCT
ejpam-4582	929	1	proceedings	proceeding	NOUN
ejpam-4582	929	2	of	of	ADP
ejpam-4582	929	3	the	the	DET
ejpam-4582	929	4	american	american	PROPN
ejpam-4582	929	5	mathematical	mathematical	PROPN
ejpam-4582	929	6	society	society	NOUN
ejpam-4582	929	7	,	,	PUNCT
ejpam-4582	929	8	60:335–338	60:335–338	PROPN
ejpam-4582	929	9	,	,	PUNCT
ejpam-4582	929	10	1976	1976	NUM
ejpam-4582	929	11	.	.	PUNCT
ejpam-4582	930	1	[	[	X
ejpam-4582	930	2	27	27	NUM
ejpam-4582	930	3	]	]	X
ejpam-4582	930	4	p.	p.	NOUN
ejpam-4582	930	5	torton	torton	PROPN
ejpam-4582	930	6	,	,	PUNCT
ejpam-4582	930	7	c.	c.	PROPN
ejpam-4582	930	8	viriyapong	viriyapong	PROPN
ejpam-4582	930	9	,	,	PUNCT
ejpam-4582	930	10	and	and	CCONJ
ejpam-4582	930	11	c.	c.	PROPN
ejpam-4582	930	12	boonpok	boonpok	PROPN
ejpam-4582	930	13	.	.	PUNCT
ejpam-4582	931	1	some	some	DET
ejpam-4582	931	2	separation	separation	NOUN
ejpam-4582	931	3	axioms	axiom	VERB
ejpam-4582	931	4	in	in	ADP
ejpam-4582	931	5	bigeneralized	bigeneralize	VERB
ejpam-4582	931	6	topological	topological	ADJ
ejpam-4582	931	7	spaces	space	NOUN
ejpam-4582	931	8	.	.	PUNCT
ejpam-4582	932	1	international	international	ADJ
ejpam-4582	932	2	journal	journal	PROPN
ejpam-4582	932	3	of	of	ADP
ejpam-4582	932	4	mathematical	mathematical	ADJ
ejpam-4582	932	5	analysis	analysis	NOUN
ejpam-4582	932	6	,	,	PUNCT
ejpam-4582	932	7	6(56):2789–2796	6(56):2789–2796	NOUN
ejpam-4582	932	8	,	,	PUNCT
ejpam-4582	932	9	2012	2012	NUM
ejpam-4582	932	10	.	.	PUNCT
ejpam-4582	933	1	[	[	X
ejpam-4582	933	2	28	28	NUM
ejpam-4582	933	3	]	]	X
ejpam-4582	933	4	e.	e.	PROPN
ejpam-4582	933	5	k.	k.	PROPN
ejpam-4582	933	6	van	van	PROPN
ejpam-4582	933	7	douwen	douwen	PROPN
ejpam-4582	933	8	.	.	PUNCT
ejpam-4582	934	1	applications	application	NOUN
ejpam-4582	934	2	of	of	ADP
ejpam-4582	934	3	maximal	maximal	ADJ
ejpam-4582	934	4	topologies	topology	NOUN
ejpam-4582	934	5	.	.	PUNCT
ejpam-4582	935	1	topology	topology	NOUN
ejpam-4582	935	2	and	and	CCONJ
ejpam-4582	935	3	its	its	PRON
ejpam-4582	935	4	applications	application	NOUN
ejpam-4582	935	5	,	,	PUNCT
ejpam-4582	935	6	51:125–139	51:125–139	PROPN
ejpam-4582	935	7	,	,	PUNCT
ejpam-4582	935	8	1993	1993	NUM
ejpam-4582	935	9	.	.	PUNCT