id	sid	tid	token	lemma	pos
ejpam-4302	1	1	european	european	PROPN
ejpam-4302	1	2	journal	journal	PROPN
ejpam-4302	1	3	of	of	ADP
ejpam-4302	1	4	pure	pure	ADJ
ejpam-4302	1	5	and	and	CCONJ
ejpam-4302	1	6	applied	apply	VERB
ejpam-4302	1	7	mathematics	mathematic	NOUN
ejpam-4302	1	8	vol	vol	NOUN
ejpam-4302	1	9	.	.	PROPN
ejpam-4302	2	1	15	15	NUM
ejpam-4302	2	2	,	,	PUNCT
ejpam-4302	2	3	no	no	INTJ
ejpam-4302	2	4	.	.	NOUN
ejpam-4302	2	5	4	4	NUM
ejpam-4302	2	6	,	,	PUNCT
ejpam-4302	2	7	2022	2022	NUM
ejpam-4302	2	8	,	,	PUNCT
ejpam-4302	2	9	2127	2127	NUM
ejpam-4302	2	10	-	-	SYM
ejpam-4302	2	11	2140	2140	NUM
ejpam-4302	2	12	issn	issn	PROPN
ejpam-4302	2	13	1307	1307	NUM
ejpam-4302	2	14	-	-	SYM
ejpam-4302	2	15	5543	5543	NUM
ejpam-4302	2	16	–	–	PUNCT
ejpam-4302	2	17	ejpam.com	ejpam.com	X
ejpam-4302	2	18	published	publish	VERB
ejpam-4302	2	19	by	by	ADP
ejpam-4302	2	20	new	new	PROPN
ejpam-4302	2	21	york	york	PROPN
ejpam-4302	2	22	business	business	PROPN
ejpam-4302	2	23	global	global	PROPN
ejpam-4302	2	24	on	on	ADP
ejpam-4302	2	25	generalized	generalized	ADJ
ejpam-4302	2	26	(	(	PUNCT
ejpam-4302	2	27	λ	λ	NOUN
ejpam-4302	2	28	,	,	PUNCT
ejpam-4302	2	29	sp)-closed	sp)-close	VERB
ejpam-4302	2	30	sets	set	NOUN
ejpam-4302	2	31	chawalit	chawalit	VERB
ejpam-4302	2	32	boonpok1	boonpok1	PROPN
ejpam-4302	2	33	,	,	PUNCT
ejpam-4302	2	34	chokchai	chokchai	ADJ
ejpam-4302	2	35	viriyapong1,∗	viriyapong1,∗	NOUN
ejpam-4302	2	36	1	1	NUM
ejpam-4302	2	37	mathematics	mathematic	NOUN
ejpam-4302	2	38	and	and	CCONJ
ejpam-4302	2	39	applied	apply	VERB
ejpam-4302	2	40	mathematics	mathematics	PROPN
ejpam-4302	2	41	research	research	NOUN
ejpam-4302	2	42	unit	unit	NOUN
ejpam-4302	2	43	,	,	PUNCT
ejpam-4302	2	44	department	department	NOUN
ejpam-4302	2	45	of	of	ADP
ejpam-4302	2	46	mathematics	mathematic	NOUN
ejpam-4302	2	47	,	,	PUNCT
ejpam-4302	2	48	faculty	faculty	NOUN
ejpam-4302	2	49	of	of	ADP
ejpam-4302	2	50	science	science	NOUN
ejpam-4302	2	51	,	,	PUNCT
ejpam-4302	2	52	mahasarakham	mahasarakham	PROPN
ejpam-4302	2	53	university	university	PROPN
ejpam-4302	2	54	,	,	PUNCT
ejpam-4302	2	55	maha	maha	PROPN
ejpam-4302	2	56	sarakham	sarakham	PROPN
ejpam-4302	2	57	,	,	PUNCT
ejpam-4302	2	58	44150	44150	NUM
ejpam-4302	2	59	,	,	PUNCT
ejpam-4302	2	60	thailand	thailand	PROPN
ejpam-4302	2	61	abstract	abstract	PROPN
ejpam-4302	2	62	.	.	PUNCT
ejpam-4302	3	1	this	this	DET
ejpam-4302	3	2	paper	paper	NOUN
ejpam-4302	3	3	is	be	AUX
ejpam-4302	3	4	concerned	concern	VERB
ejpam-4302	3	5	with	with	ADP
ejpam-4302	3	6	the	the	DET
ejpam-4302	3	7	concept	concept	NOUN
ejpam-4302	3	8	of	of	ADP
ejpam-4302	3	9	generalized	generalized	ADJ
ejpam-4302	3	10	(	(	PUNCT
ejpam-4302	3	11	λ	λ	PROPN
ejpam-4302	3	12	,	,	PUNCT
ejpam-4302	3	13	sp)-closed	sp)-close	VERB
ejpam-4302	3	14	sets	set	NOUN
ejpam-4302	3	15	.	.	PUNCT
ejpam-4302	4	1	some	some	DET
ejpam-4302	4	2	properties	property	NOUN
ejpam-4302	4	3	of	of	ADP
ejpam-4302	4	4	generalized	generalized	ADJ
ejpam-4302	4	5	(	(	PUNCT
ejpam-4302	4	6	λ	λ	NOUN
ejpam-4302	4	7	,	,	PUNCT
ejpam-4302	4	8	sp)-closed	sp)-close	VERB
ejpam-4302	4	9	sets	set	NOUN
ejpam-4302	4	10	and	and	CCONJ
ejpam-4302	4	11	generalized	generalize	VERB
ejpam-4302	4	12	(	(	PUNCT
ejpam-4302	4	13	λ	λ	NOUN
ejpam-4302	4	14	,	,	PUNCT
ejpam-4302	4	15	sp)-open	sp)-open	ADJ
ejpam-4302	4	16	sets	set	NOUN
ejpam-4302	4	17	are	be	AUX
ejpam-4302	4	18	discussed	discuss	VERB
ejpam-4302	4	19	.	.	PUNCT
ejpam-4302	5	1	moreover	moreover	ADV
ejpam-4302	5	2	,	,	PUNCT
ejpam-4302	5	3	several	several	ADJ
ejpam-4302	5	4	characterizations	characterization	NOUN
ejpam-4302	5	5	of	of	ADP
ejpam-4302	5	6	λsp	λsp	NOUN
ejpam-4302	5	7	-	-	ADJ
ejpam-4302	5	8	normal	normal	ADJ
ejpam-4302	5	9	spaces	space	NOUN
ejpam-4302	5	10	are	be	AUX
ejpam-4302	5	11	investigated	investigate	VERB
ejpam-4302	5	12	.	.	PUNCT
ejpam-4302	6	1	2020	2020	NUM
ejpam-4302	6	2	mathematics	mathematic	NOUN
ejpam-4302	6	3	subject	subject	NOUN
ejpam-4302	6	4	classifications	classification	NOUN
ejpam-4302	6	5	:	:	PUNCT
ejpam-4302	6	6	54a05	54a05	NUM
ejpam-4302	6	7	,	,	PUNCT
ejpam-4302	6	8	54d10	54d10	NUM
ejpam-4302	6	9	key	key	ADJ
ejpam-4302	6	10	words	word	NOUN
ejpam-4302	6	11	and	and	CCONJ
ejpam-4302	6	12	phrases	phrase	NOUN
ejpam-4302	6	13	:	:	PUNCT
ejpam-4302	6	14	(	(	PUNCT
ejpam-4302	6	15	λ	λ	X
ejpam-4302	6	16	,	,	PUNCT
ejpam-4302	6	17	sp)-closed	sp)-close	VERB
ejpam-4302	6	18	set	set	VERB
ejpam-4302	6	19	,	,	PUNCT
ejpam-4302	6	20	(	(	PUNCT
ejpam-4302	6	21	λ	λ	NOUN
ejpam-4302	6	22	,	,	PUNCT
ejpam-4302	6	23	sp)-open	sp)-open	ADJ
ejpam-4302	6	24	set	set	NOUN
ejpam-4302	6	25	,	,	PUNCT
ejpam-4302	6	26	generalized	generalize	VERB
ejpam-4302	6	27	(	(	PUNCT
ejpam-4302	6	28	λ	λ	PROPN
ejpam-4302	6	29	,	,	PUNCT
ejpam-4302	6	30	sp)-closed	sp)-close	VERB
ejpam-4302	6	31	set	set	ADJ
ejpam-4302	6	32	,	,	PUNCT
ejpam-4302	6	33	λsp	λsp	ADJ
ejpam-4302	6	34	-	-	ADJ
ejpam-4302	6	35	normal	normal	ADJ
ejpam-4302	6	36	space	space	NOUN
ejpam-4302	6	37	1	1	NUM
ejpam-4302	6	38	.	.	PUNCT
ejpam-4302	7	1	introduction	introduction	NOUN
ejpam-4302	7	2	general	general	ADJ
ejpam-4302	7	3	topology	topology	NOUN
ejpam-4302	7	4	plays	play	VERB
ejpam-4302	7	5	an	an	DET
ejpam-4302	7	6	important	important	ADJ
ejpam-4302	7	7	role	role	NOUN
ejpam-4302	7	8	in	in	ADP
ejpam-4302	7	9	pure	pure	ADJ
ejpam-4302	7	10	and	and	CCONJ
ejpam-4302	7	11	applied	apply	VERB
ejpam-4302	7	12	sciences	science	NOUN
ejpam-4302	7	13	such	such	ADJ
ejpam-4302	7	14	as	as	ADP
ejpam-4302	7	15	data	datum	NOUN
ejpam-4302	7	16	mining	mining	NOUN
ejpam-4302	7	17	,	,	PUNCT
ejpam-4302	7	18	computational	computational	ADJ
ejpam-4302	7	19	topology	topology	NOUN
ejpam-4302	7	20	for	for	ADP
ejpam-4302	7	21	geometric	geometric	ADJ
ejpam-4302	7	22	design	design	NOUN
ejpam-4302	7	23	and	and	CCONJ
ejpam-4302	7	24	molecular	molecular	ADJ
ejpam-4302	7	25	design	design	NOUN
ejpam-4302	7	26	,	,	PUNCT
ejpam-4302	7	27	computer	computer	NOUN
ejpam-4302	7	28	-	-	PUNCT
ejpam-4302	7	29	aided	aid	VERB
ejpam-4302	7	30	design	design	NOUN
ejpam-4302	7	31	,	,	PUNCT
ejpam-4302	7	32	computer	computer	NOUN
ejpam-4302	7	33	-	-	PUNCT
ejpam-4302	7	34	aided	aid	VERB
ejpam-4302	7	35	geometric	geometric	ADJ
ejpam-4302	7	36	design	design	NOUN
ejpam-4302	7	37	,	,	PUNCT
ejpam-4302	7	38	engineering	engineering	NOUN
ejpam-4302	7	39	design	design	NOUN
ejpam-4302	7	40	,	,	PUNCT
ejpam-4302	7	41	digital	digital	ADJ
ejpam-4302	7	42	topology	topology	NOUN
ejpam-4302	7	43	,	,	PUNCT
ejpam-4302	7	44	information	information	NOUN
ejpam-4302	7	45	systems	system	NOUN
ejpam-4302	7	46	,	,	PUNCT
ejpam-4302	7	47	quantum	quantum	NOUN
ejpam-4302	7	48	physics	physics	NOUN
ejpam-4302	7	49	,	,	PUNCT
ejpam-4302	7	50	high	high	ADJ
ejpam-4302	7	51	energy	energy	NOUN
ejpam-4302	7	52	physics	physics	NOUN
ejpam-4302	7	53	and	and	CCONJ
ejpam-4302	7	54	superstring	superstring	NOUN
ejpam-4302	7	55	theory	theory	NOUN
ejpam-4302	7	56	.	.	PUNCT
ejpam-4302	8	1	the	the	DET
ejpam-4302	8	2	topological	topological	ADJ
ejpam-4302	8	3	structures	structure	NOUN
ejpam-4302	8	4	of	of	ADP
ejpam-4302	8	5	set	set	NOUN
ejpam-4302	8	6	theories	theory	NOUN
ejpam-4302	8	7	dealing	deal	VERB
ejpam-4302	8	8	with	with	ADP
ejpam-4302	8	9	uncertainities	uncertainitie	NOUN
ejpam-4302	8	10	were	be	AUX
ejpam-4302	8	11	first	first	ADV
ejpam-4302	8	12	introduced	introduce	VERB
ejpam-4302	8	13	by	by	ADP
ejpam-4302	8	14	chang	chang	PROPN
ejpam-4302	9	1	[	[	X
ejpam-4302	9	2	3	3	NUM
ejpam-4302	9	3	]	]	PUNCT
ejpam-4302	9	4	.	.	PUNCT
ejpam-4302	10	1	lashin	lashin	PROPN
ejpam-4302	10	2	et	et	PROPN
ejpam-4302	10	3	al	al	PROPN
ejpam-4302	10	4	.	.	PUNCT
ejpam-4302	11	1	[	[	X
ejpam-4302	11	2	8	8	NUM
ejpam-4302	11	3	]	]	PUNCT
ejpam-4302	11	4	investigated	investigate	VERB
ejpam-4302	11	5	topological	topological	ADJ
ejpam-4302	11	6	spaces	space	NOUN
ejpam-4302	11	7	by	by	ADP
ejpam-4302	11	8	generalizing	generalize	VERB
ejpam-4302	11	9	rough	rough	ADJ
ejpam-4302	11	10	set	set	NOUN
ejpam-4302	11	11	theory	theory	NOUN
ejpam-4302	11	12	.	.	PUNCT
ejpam-4302	12	1	the	the	DET
ejpam-4302	12	2	concept	concept	NOUN
ejpam-4302	12	3	of	of	ADP
ejpam-4302	12	4	soft	soft	ADJ
ejpam-4302	12	5	topological	topological	ADJ
ejpam-4302	12	6	spaces	space	NOUN
ejpam-4302	12	7	defined	define	VERB
ejpam-4302	12	8	by	by	ADP
ejpam-4302	12	9	shabir	shabir	PROPN
ejpam-4302	12	10	and	and	CCONJ
ejpam-4302	12	11	naz	naz	PROPN
ejpam-4302	12	12	[	[	X
ejpam-4302	12	13	14	14	NUM
ejpam-4302	12	14	]	]	PUNCT
ejpam-4302	12	15	on	on	ADP
ejpam-4302	12	16	an	an	DET
ejpam-4302	12	17	initial	initial	ADJ
ejpam-4302	12	18	universe	universe	NOUN
ejpam-4302	12	19	with	with	ADP
ejpam-4302	12	20	a	a	DET
ejpam-4302	12	21	fixed	fix	VERB
ejpam-4302	12	22	set	set	NOUN
ejpam-4302	12	23	of	of	ADP
ejpam-4302	12	24	parameters	parameter	NOUN
ejpam-4302	12	25	.	.	PUNCT
ejpam-4302	13	1	şenel	şenel	VERB
ejpam-4302	13	2	and	and	CCONJ
ejpam-4302	13	3	çağman	çağman	NOUN
ejpam-4302	14	1	[	[	X
ejpam-4302	14	2	5	5	NUM
ejpam-4302	14	3	]	]	PUNCT
ejpam-4302	14	4	extended	extend	VERB
ejpam-4302	14	5	the	the	DET
ejpam-4302	14	6	concept	concept	NOUN
ejpam-4302	14	7	of	of	ADP
ejpam-4302	14	8	bitopological	bitopological	ADJ
ejpam-4302	14	9	spaces	space	NOUN
ejpam-4302	14	10	to	to	ADP
ejpam-4302	14	11	soft	soft	ADJ
ejpam-4302	14	12	bitopological	bitopological	ADJ
ejpam-4302	14	13	spaces	space	NOUN
ejpam-4302	14	14	and	and	CCONJ
ejpam-4302	14	15	obtained	obtain	VERB
ejpam-4302	14	16	some	some	DET
ejpam-4302	14	17	relations	relation	NOUN
ejpam-4302	14	18	between	between	ADP
ejpam-4302	14	19	soft	soft	ADJ
ejpam-4302	14	20	topology	topology	NOUN
ejpam-4302	14	21	and	and	CCONJ
ejpam-4302	14	22	soft	soft	ADJ
ejpam-4302	14	23	bitopology	bitopology	NOUN
ejpam-4302	14	24	.	.	PUNCT
ejpam-4302	15	1	in	in	ADP
ejpam-4302	15	2	[	[	X
ejpam-4302	15	3	4	4	NUM
ejpam-4302	15	4	]	]	PUNCT
ejpam-4302	15	5	,	,	PUNCT
ejpam-4302	15	6	the	the	DET
ejpam-4302	15	7	present	present	ADJ
ejpam-4302	15	8	authors	author	NOUN
ejpam-4302	15	9	defined	define	VERB
ejpam-4302	15	10	and	and	CCONJ
ejpam-4302	15	11	studied	study	VERB
ejpam-4302	15	12	the	the	DET
ejpam-4302	15	13	concepts	concept	NOUN
ejpam-4302	15	14	of	of	ADP
ejpam-4302	15	15	soft	soft	ADJ
ejpam-4302	15	16	closed	closed	ADJ
ejpam-4302	15	17	sets	set	NOUN
ejpam-4302	15	18	,	,	PUNCT
ejpam-4302	15	19	soft	soft	ADJ
ejpam-4302	15	20	α	α	NOUN
ejpam-4302	15	21	-	-	PUNCT
ejpam-4302	15	22	closed	closed	ADJ
ejpam-4302	15	23	sets	set	NOUN
ejpam-4302	15	24	,	,	PUNCT
ejpam-4302	15	25	soft	soft	ADJ
ejpam-4302	15	26	semi	semi	ADJ
ejpam-4302	15	27	-	-	ADJ
ejpam-4302	15	28	closed	closed	ADJ
ejpam-4302	15	29	sets	set	NOUN
ejpam-4302	15	30	,	,	PUNCT
ejpam-4302	15	31	soft	soft	ADJ
ejpam-4302	15	32	pre	pre	ADJ
ejpam-4302	15	33	-	-	ADJ
ejpam-4302	15	34	closed	closed	ADJ
ejpam-4302	15	35	sets	set	NOUN
ejpam-4302	15	36	,	,	PUNCT
ejpam-4302	15	37	regular	regular	ADJ
ejpam-4302	15	38	soft	soft	ADJ
ejpam-4302	15	39	closed	closed	ADJ
ejpam-4302	15	40	sets	set	NOUN
ejpam-4302	15	41	,	,	PUNCT
ejpam-4302	15	42	soft	soft	ADJ
ejpam-4302	15	43	g	g	NOUN
ejpam-4302	15	44	-	-	PUNCT
ejpam-4302	15	45	closed	close	VERB
ejpam-4302	15	46	sets	set	NOUN
ejpam-4302	15	47	and	and	CCONJ
ejpam-4302	15	48	soft	soft	ADJ
ejpam-4302	15	49	sg	sg	NOUN
ejpam-4302	15	50	-	-	PUNCT
ejpam-4302	15	51	closed	close	VERB
ejpam-4302	15	52	sets	set	NOUN
ejpam-4302	15	53	in	in	ADP
ejpam-4302	15	54	soft	soft	ADJ
ejpam-4302	15	55	bitopological	bitopological	ADJ
ejpam-4302	15	56	spaces	space	NOUN
ejpam-4302	15	57	.	.	PUNCT
ejpam-4302	16	1	the	the	DET
ejpam-4302	16	2	notions	notion	NOUN
ejpam-4302	16	3	of	of	ADP
ejpam-4302	16	4	closed	closed	ADJ
ejpam-4302	16	5	sets	set	NOUN
ejpam-4302	16	6	and	and	CCONJ
ejpam-4302	16	7	open	open	ADJ
ejpam-4302	16	8	sets	set	NOUN
ejpam-4302	16	9	are	be	AUX
ejpam-4302	16	10	fundamental	fundamental	ADJ
ejpam-4302	16	11	with	with	ADP
ejpam-4302	16	12	respect	respect	NOUN
ejpam-4302	16	13	to	to	ADP
ejpam-4302	16	14	the	the	DET
ejpam-4302	16	15	investigation	investigation	NOUN
ejpam-4302	16	16	of	of	ADP
ejpam-4302	16	17	general	general	ADJ
ejpam-4302	16	18	topology	topology	NOUN
ejpam-4302	16	19	.	.	PUNCT
ejpam-4302	17	1	in	in	ADP
ejpam-4302	17	2	1970	1970	NUM
ejpam-4302	17	3	,	,	PUNCT
ejpam-4302	17	4	levine	levine	PROPN
ejpam-4302	17	5	[	[	X
ejpam-4302	17	6	10	10	NUM
ejpam-4302	17	7	]	]	PUNCT
ejpam-4302	17	8	introduced	introduce	VERB
ejpam-4302	17	9	the	the	DET
ejpam-4302	17	10	concept	concept	NOUN
ejpam-4302	17	11	of	of	ADP
ejpam-4302	17	12	generalized	generalized	ADJ
ejpam-4302	17	13	closed	close	VERB
ejpam-4302	17	14	sets	set	NOUN
ejpam-4302	17	15	in	in	ADP
ejpam-4302	17	16	topological	topological	ADJ
ejpam-4302	17	17	spaces	space	NOUN
ejpam-4302	17	18	and	and	CCONJ
ejpam-4302	17	19	defined	define	VERB
ejpam-4302	17	20	the	the	DET
ejpam-4302	17	21	notion	notion	NOUN
ejpam-4302	17	22	of	of	ADP
ejpam-4302	17	23	a	a	DET
ejpam-4302	17	24	t	t	NOUN
ejpam-4302	17	25	1	1	NUM
ejpam-4302	17	26	2	2	NUM
ejpam-4302	17	27	-space	-space	NOUN
ejpam-4302	17	28	to	to	PART
ejpam-4302	17	29	be	be	AUX
ejpam-4302	17	30	one	one	NUM
ejpam-4302	17	31	in	in	ADP
ejpam-4302	17	32	which	which	PRON
ejpam-4302	17	33	the	the	DET
ejpam-4302	17	34	closed	closed	ADJ
ejpam-4302	17	35	sets	set	NOUN
ejpam-4302	17	36	and	and	CCONJ
ejpam-4302	17	37	the	the	DET
ejpam-4302	17	38	generalized	generalize	VERB
ejpam-4302	17	39	closed	close	VERB
ejpam-4302	17	40	sets	set	NOUN
ejpam-4302	17	41	coincide	coincide	NOUN
ejpam-4302	17	42	.	.	PUNCT
ejpam-4302	18	1	dunham	dunham	PROPN
ejpam-4302	18	2	and	and	CCONJ
ejpam-4302	18	3	levine	levine	PROPN
ejpam-4302	19	1	[	[	X
ejpam-4302	19	2	6	6	NUM
ejpam-4302	19	3	]	]	PUNCT
ejpam-4302	19	4	investigated	investigate	VERB
ejpam-4302	19	5	the	the	DET
ejpam-4302	19	6	further	further	ADJ
ejpam-4302	19	7	properties	property	NOUN
ejpam-4302	19	8	of	of	ADP
ejpam-4302	19	9	generalized	generalized	ADJ
ejpam-4302	19	10	closed	closed	ADJ
ejpam-4302	19	11	sets	set	NOUN
ejpam-4302	19	12	.	.	PUNCT
ejpam-4302	20	1	the	the	DET
ejpam-4302	20	2	concept	concept	NOUN
ejpam-4302	20	3	of	of	ADP
ejpam-4302	20	4	generalized	generalized	ADJ
ejpam-4302	20	5	closed	closed	ADJ
ejpam-4302	20	6	sets	set	NOUN
ejpam-4302	20	7	has	have	AUX
ejpam-4302	20	8	been	be	AUX
ejpam-4302	20	9	modified	modify	VERB
ejpam-4302	20	10	and	and	CCONJ
ejpam-4302	20	11	studied	study	VERB
ejpam-4302	20	12	by	by	ADP
ejpam-4302	20	13	using	use	VERB
ejpam-4302	20	14	weaker	weak	ADJ
ejpam-4302	20	15	forms	form	NOUN
ejpam-4302	20	16	of	of	ADP
ejpam-4302	20	17	open	open	ADJ
ejpam-4302	20	18	sets	set	NOUN
ejpam-4302	20	19	such	such	ADJ
ejpam-4302	20	20	as	as	ADP
ejpam-4302	20	21	α	α	NOUN
ejpam-4302	20	22	-	-	ADJ
ejpam-4302	20	23	open	open	ADJ
ejpam-4302	20	24	sets	set	NOUN
ejpam-4302	20	25	[	[	X
ejpam-4302	20	26	12	12	NUM
ejpam-4302	20	27	]	]	PUNCT
ejpam-4302	20	28	,	,	PUNCT
ejpam-4302	20	29	semi	semi	ADJ
ejpam-4302	20	30	-	-	ADJ
ejpam-4302	20	31	open	open	ADJ
ejpam-4302	20	32	sets	set	NOUN
ejpam-4302	20	33	[	[	X
ejpam-4302	20	34	9	9	NUM
ejpam-4302	20	35	]	]	PUNCT
ejpam-4302	20	36	,	,	PUNCT
ejpam-4302	20	37	preopen	preopen	ADJ
ejpam-4302	20	38	∗corresponding	∗corresponde	VERB
ejpam-4302	20	39	author	author	NOUN
ejpam-4302	20	40	.	.	PUNCT
ejpam-4302	21	1	doi	doi	NOUN
ejpam-4302	21	2	:	:	PUNCT
ejpam-4302	21	3	https://doi.org/10.29020/nybg.ejpam.v15i4.4302	https://doi.org/10.29020/nybg.ejpam.v15i4.4302	ADJ
ejpam-4302	21	4	email	email	NOUN
ejpam-4302	21	5	addresses	address	NOUN
ejpam-4302	21	6	:	:	PUNCT
ejpam-4302	21	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4302	21	8	(	(	PUNCT
ejpam-4302	21	9	c.	c.	PROPN
ejpam-4302	21	10	boonpok	boonpok	PROPN
ejpam-4302	21	11	)	)	PUNCT
ejpam-4302	21	12	,	,	PUNCT
ejpam-4302	21	13	chokchai.v@msu.ac.th	chokchai.v@msu.ac.th	INTJ
ejpam-4302	21	14	(	(	PUNCT
ejpam-4302	21	15	c.	c.	PROPN
ejpam-4302	21	16	viriyapong	viriyapong	PROPN
ejpam-4302	21	17	)	)	PUNCT
ejpam-4302	21	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4302	22	1	2127	2127	NUM
ejpam-4302	22	2	©	©	ADP
ejpam-4302	22	3	2022	2022	NUM
ejpam-4302	22	4	ejpam	ejpam	VERB
ejpam-4302	22	5	all	all	DET
ejpam-4302	22	6	rights	right	NOUN
ejpam-4302	22	7	reserved	reserve	VERB
ejpam-4302	22	8	.	.	PUNCT
ejpam-4302	23	1	c.	c.	PROPN
ejpam-4302	23	2	boonpok	boonpok	PROPN
ejpam-4302	23	3	,	,	PUNCT
ejpam-4302	23	4	c.	c.	PROPN
ejpam-4302	23	5	viriyapong	viriyapong	PROPN
ejpam-4302	23	6	/	/	SYM
ejpam-4302	23	7	eur	eur	PROPN
ejpam-4302	23	8	.	.	PUNCT
ejpam-4302	24	1	j.	j.	PROPN
ejpam-4302	24	2	pure	pure	PROPN
ejpam-4302	24	3	appl	appl	PROPN
ejpam-4302	24	4	.	.	PROPN
ejpam-4302	24	5	math	math	PROPN
ejpam-4302	24	6	,	,	PUNCT
ejpam-4302	24	7	15	15	NUM
ejpam-4302	24	8	(	(	PUNCT
ejpam-4302	24	9	4	4	NUM
ejpam-4302	24	10	)	)	PUNCT
ejpam-4302	24	11	(	(	PUNCT
ejpam-4302	24	12	2022	2022	NUM
ejpam-4302	24	13	)	)	PUNCT
ejpam-4302	24	14	,	,	PUNCT
ejpam-4302	24	15	2127	2127	NUM
ejpam-4302	24	16	-	-	SYM
ejpam-4302	24	17	2140	2140	NUM
ejpam-4302	24	18	2128	2128	NUM
ejpam-4302	24	19	sets	set	NOUN
ejpam-4302	24	20	[	[	X
ejpam-4302	24	21	11	11	NUM
ejpam-4302	24	22	]	]	PUNCT
ejpam-4302	24	23	and	and	CCONJ
ejpam-4302	24	24	semi	semi	ADJ
ejpam-4302	24	25	-	-	ADJ
ejpam-4302	24	26	preopen	preopen	ADJ
ejpam-4302	24	27	sets	set	NOUN
ejpam-4302	24	28	[	[	X
ejpam-4302	24	29	1	1	NUM
ejpam-4302	24	30	]	]	PUNCT
ejpam-4302	24	31	.	.	PUNCT
ejpam-4302	25	1	in	in	ADP
ejpam-4302	25	2	1983	1983	NUM
ejpam-4302	25	3	,	,	PUNCT
ejpam-4302	25	4	abd	abd	PROPN
ejpam-4302	25	5	el	el	PROPN
ejpam-4302	25	6	-	-	PROPN
ejpam-4302	25	7	monsef	monsef	PROPN
ejpam-4302	25	8	et	et	PROPN
ejpam-4302	25	9	al	al	PROPN
ejpam-4302	25	10	.	.	PUNCT
ejpam-4302	26	1	[	[	X
ejpam-4302	26	2	7	7	X
ejpam-4302	26	3	]	]	PUNCT
ejpam-4302	26	4	introduced	introduce	VERB
ejpam-4302	26	5	a	a	DET
ejpam-4302	26	6	weak	weak	ADJ
ejpam-4302	26	7	form	form	NOUN
ejpam-4302	26	8	of	of	ADP
ejpam-4302	26	9	open	open	ADJ
ejpam-4302	26	10	sets	set	NOUN
ejpam-4302	26	11	called	call	VERB
ejpam-4302	26	12	β	β	NOUN
ejpam-4302	26	13	-	-	ADJ
ejpam-4302	26	14	open	open	ADJ
ejpam-4302	26	15	sets	set	NOUN
ejpam-4302	26	16	.	.	PUNCT
ejpam-4302	27	1	the	the	DET
ejpam-4302	27	2	notion	notion	NOUN
ejpam-4302	27	3	of	of	ADP
ejpam-4302	27	4	β	β	ADJ
ejpam-4302	27	5	-	-	ADJ
ejpam-4302	27	6	open	open	ADJ
ejpam-4302	27	7	sets	set	NOUN
ejpam-4302	27	8	is	be	AUX
ejpam-4302	27	9	equivalent	equivalent	ADJ
ejpam-4302	27	10	to	to	ADP
ejpam-4302	27	11	that	that	PRON
ejpam-4302	27	12	of	of	ADP
ejpam-4302	27	13	semi	semi	ADJ
ejpam-4302	27	14	-	-	ADJ
ejpam-4302	27	15	preopen	preopen	ADJ
ejpam-4302	27	16	sets	set	NOUN
ejpam-4302	27	17	[	[	X
ejpam-4302	27	18	1	1	NUM
ejpam-4302	27	19	]	]	PUNCT
ejpam-4302	27	20	.	.	PUNCT
ejpam-4302	28	1	in	in	ADP
ejpam-4302	28	2	2004	2004	NUM
ejpam-4302	28	3	,	,	PUNCT
ejpam-4302	28	4	noiri	noiri	PROPN
ejpam-4302	28	5	and	and	CCONJ
ejpam-4302	28	6	hatir	hatir	NOUN
ejpam-4302	29	1	[	[	X
ejpam-4302	29	2	13	13	NUM
ejpam-4302	29	3	]	]	PUNCT
ejpam-4302	29	4	introduced	introduce	VERB
ejpam-4302	29	5	the	the	DET
ejpam-4302	29	6	notions	notion	NOUN
ejpam-4302	29	7	of	of	ADP
ejpam-4302	29	8	λsp	λsp	NOUN
ejpam-4302	29	9	-	-	PUNCT
ejpam-4302	29	10	sets	set	NOUN
ejpam-4302	29	11	,	,	PUNCT
ejpam-4302	29	12	λsp	λsp	ADV
ejpam-4302	29	13	-	-	PUNCT
ejpam-4302	29	14	closed	close	VERB
ejpam-4302	29	15	sets	set	NOUN
ejpam-4302	29	16	and	and	CCONJ
ejpam-4302	29	17	spg	spg	ADJ
ejpam-4302	29	18	-	-	PUNCT
ejpam-4302	29	19	closed	closed	ADJ
ejpam-4302	29	20	sets	set	NOUN
ejpam-4302	29	21	and	and	CCONJ
ejpam-4302	29	22	investigated	investigate	VERB
ejpam-4302	29	23	properties	property	NOUN
ejpam-4302	29	24	of	of	ADP
ejpam-4302	29	25	these	these	DET
ejpam-4302	29	26	sets	set	NOUN
ejpam-4302	29	27	.	.	PUNCT
ejpam-4302	30	1	in	in	ADP
ejpam-4302	30	2	[	[	X
ejpam-4302	30	3	2	2	NUM
ejpam-4302	30	4	]	]	PUNCT
ejpam-4302	30	5	,	,	PUNCT
ejpam-4302	30	6	the	the	DET
ejpam-4302	30	7	author	author	NOUN
ejpam-4302	30	8	introduced	introduce	VERB
ejpam-4302	30	9	and	and	CCONJ
ejpam-4302	30	10	investigated	investigate	VERB
ejpam-4302	30	11	the	the	DET
ejpam-4302	30	12	concepts	concept	NOUN
ejpam-4302	30	13	of	of	ADP
ejpam-4302	30	14	(	(	PUNCT
ejpam-4302	30	15	λ	λ	PROPN
ejpam-4302	30	16	,	,	PUNCT
ejpam-4302	30	17	sp)-open	sp)-open	ADJ
ejpam-4302	30	18	sets	set	NOUN
ejpam-4302	30	19	and	and	CCONJ
ejpam-4302	30	20	(	(	PUNCT
ejpam-4302	30	21	λ	λ	PROPN
ejpam-4302	30	22	,	,	PUNCT
ejpam-4302	30	23	sp)-closed	sp)-close	VERB
ejpam-4302	30	24	sets	set	NOUN
ejpam-4302	30	25	which	which	PRON
ejpam-4302	30	26	are	be	AUX
ejpam-4302	30	27	defined	define	VERB
ejpam-4302	30	28	by	by	ADP
ejpam-4302	30	29	utilizing	utilize	VERB
ejpam-4302	30	30	the	the	DET
ejpam-4302	30	31	notions	notion	NOUN
ejpam-4302	30	32	of	of	ADP
ejpam-4302	30	33	λsp	λsp	NOUN
ejpam-4302	30	34	-	-	PUNCT
ejpam-4302	30	35	sets	set	NOUN
ejpam-4302	30	36	and	and	CCONJ
ejpam-4302	30	37	β	β	NOUN
ejpam-4302	30	38	-	-	ADJ
ejpam-4302	30	39	closed	closed	ADJ
ejpam-4302	30	40	sets	set	NOUN
ejpam-4302	30	41	.	.	PUNCT
ejpam-4302	31	1	in	in	ADP
ejpam-4302	31	2	the	the	DET
ejpam-4302	31	3	present	present	ADJ
ejpam-4302	31	4	paper	paper	NOUN
ejpam-4302	31	5	,	,	PUNCT
ejpam-4302	31	6	we	we	PRON
ejpam-4302	31	7	introduce	introduce	VERB
ejpam-4302	31	8	the	the	DET
ejpam-4302	31	9	concept	concept	NOUN
ejpam-4302	31	10	of	of	ADP
ejpam-4302	31	11	generalized	generalized	ADJ
ejpam-4302	31	12	(	(	PUNCT
ejpam-4302	31	13	λ	λ	PROPN
ejpam-4302	31	14	,	,	PUNCT
ejpam-4302	31	15	sp)-closed	sp)-close	VERB
ejpam-4302	31	16	sets	set	NOUN
ejpam-4302	31	17	.	.	PUNCT
ejpam-4302	32	1	moreover	moreover	ADV
ejpam-4302	32	2	,	,	PUNCT
ejpam-4302	32	3	some	some	DET
ejpam-4302	32	4	properties	property	NOUN
ejpam-4302	32	5	of	of	ADP
ejpam-4302	32	6	generalized	generalized	ADJ
ejpam-4302	32	7	(	(	PUNCT
ejpam-4302	32	8	λ	λ	NOUN
ejpam-4302	32	9	,	,	PUNCT
ejpam-4302	32	10	sp)-closed	sp)-close	VERB
ejpam-4302	32	11	sets	set	NOUN
ejpam-4302	32	12	and	and	CCONJ
ejpam-4302	32	13	generalized	generalize	VERB
ejpam-4302	32	14	(	(	PUNCT
ejpam-4302	32	15	λ	λ	NOUN
ejpam-4302	32	16	,	,	PUNCT
ejpam-4302	32	17	sp)-open	sp)-open	ADJ
ejpam-4302	32	18	sets	set	NOUN
ejpam-4302	32	19	are	be	AUX
ejpam-4302	32	20	discussed	discuss	VERB
ejpam-4302	32	21	.	.	PUNCT
ejpam-4302	33	1	furthermore	furthermore	ADV
ejpam-4302	33	2	,	,	PUNCT
ejpam-4302	33	3	several	several	ADJ
ejpam-4302	33	4	characterizations	characterization	NOUN
ejpam-4302	33	5	of	of	ADP
ejpam-4302	33	6	λsp	λsp	NOUN
ejpam-4302	33	7	-	-	ADJ
ejpam-4302	33	8	normal	normal	ADJ
ejpam-4302	33	9	spaces	space	NOUN
ejpam-4302	33	10	are	be	AUX
ejpam-4302	33	11	investigated	investigate	VERB
ejpam-4302	33	12	.	.	PUNCT
ejpam-4302	34	1	2	2	X
ejpam-4302	34	2	.	.	X
ejpam-4302	34	3	preliminaries	preliminary	NOUN
ejpam-4302	34	4	we	we	PRON
ejpam-4302	34	5	begin	begin	VERB
ejpam-4302	34	6	with	with	ADP
ejpam-4302	34	7	some	some	DET
ejpam-4302	34	8	definitions	definition	NOUN
ejpam-4302	34	9	and	and	CCONJ
ejpam-4302	34	10	known	know	VERB
ejpam-4302	34	11	results	result	NOUN
ejpam-4302	34	12	which	which	PRON
ejpam-4302	34	13	will	will	AUX
ejpam-4302	34	14	be	be	AUX
ejpam-4302	34	15	used	use	VERB
ejpam-4302	34	16	throughout	throughout	ADP
ejpam-4302	34	17	this	this	DET
ejpam-4302	34	18	paper	paper	NOUN
ejpam-4302	34	19	.	.	PUNCT
ejpam-4302	35	1	in	in	ADP
ejpam-4302	35	2	the	the	DET
ejpam-4302	35	3	present	present	ADJ
ejpam-4302	35	4	paper	paper	NOUN
ejpam-4302	35	5	,	,	PUNCT
ejpam-4302	35	6	spaces	space	NOUN
ejpam-4302	35	7	(	(	PUNCT
ejpam-4302	35	8	x	x	X
ejpam-4302	35	9	,	,	PUNCT
ejpam-4302	35	10	τ	τ	X
ejpam-4302	35	11	)	)	PUNCT
ejpam-4302	35	12	and	and	CCONJ
ejpam-4302	35	13	(	(	PUNCT
ejpam-4302	35	14	y	y	PROPN
ejpam-4302	35	15	,	,	PUNCT
ejpam-4302	35	16	σ	σ	PROPN
ejpam-4302	35	17	)	)	PUNCT
ejpam-4302	35	18	(	(	PUNCT
ejpam-4302	35	19	or	or	CCONJ
ejpam-4302	35	20	simply	simply	ADV
ejpam-4302	35	21	x	x	X
ejpam-4302	35	22	and	and	CCONJ
ejpam-4302	35	23	y	y	PROPN
ejpam-4302	35	24	)	)	PUNCT
ejpam-4302	35	25	always	always	ADV
ejpam-4302	35	26	mean	mean	VERB
ejpam-4302	35	27	topological	topological	ADJ
ejpam-4302	35	28	spaces	space	NOUN
ejpam-4302	35	29	on	on	ADP
ejpam-4302	35	30	which	which	PRON
ejpam-4302	35	31	no	no	DET
ejpam-4302	35	32	separation	separation	NOUN
ejpam-4302	35	33	axioms	axiom	NOUN
ejpam-4302	35	34	are	be	AUX
ejpam-4302	35	35	assumed	assume	VERB
ejpam-4302	35	36	unless	unless	SCONJ
ejpam-4302	35	37	explicitly	explicitly	ADV
ejpam-4302	35	38	stated	state	VERB
ejpam-4302	35	39	.	.	PUNCT
ejpam-4302	36	1	let	let	VERB
ejpam-4302	36	2	a	a	DET
ejpam-4302	36	3	be	be	AUX
ejpam-4302	36	4	a	a	DET
ejpam-4302	36	5	subset	subset	NOUN
ejpam-4302	36	6	of	of	ADP
ejpam-4302	36	7	a	a	DET
ejpam-4302	36	8	topological	topological	ADJ
ejpam-4302	36	9	space	space	NOUN
ejpam-4302	36	10	(	(	PUNCT
ejpam-4302	36	11	x	x	X
ejpam-4302	36	12	,	,	PUNCT
ejpam-4302	36	13	τ	τ	PROPN
ejpam-4302	36	14	)	)	PUNCT
ejpam-4302	36	15	.	.	PUNCT
ejpam-4302	37	1	the	the	DET
ejpam-4302	37	2	closure	closure	NOUN
ejpam-4302	37	3	of	of	ADP
ejpam-4302	37	4	a	a	PRON
ejpam-4302	37	5	and	and	CCONJ
ejpam-4302	37	6	the	the	DET
ejpam-4302	37	7	interior	interior	NOUN
ejpam-4302	37	8	of	of	ADP
ejpam-4302	37	9	a	a	PRON
ejpam-4302	37	10	are	be	AUX
ejpam-4302	37	11	denoted	denote	VERB
ejpam-4302	37	12	by	by	ADP
ejpam-4302	37	13	cl(a	cl(a	NOUN
ejpam-4302	37	14	)	)	PUNCT
ejpam-4302	37	15	and	and	CCONJ
ejpam-4302	37	16	int(a	int(a	PROPN
ejpam-4302	37	17	)	)	PUNCT
ejpam-4302	37	18	,	,	PUNCT
ejpam-4302	37	19	respectively	respectively	ADV
ejpam-4302	37	20	.	.	PUNCT
ejpam-4302	38	1	a	a	DET
ejpam-4302	38	2	subset	subset	NOUN
ejpam-4302	38	3	a	a	PRON
ejpam-4302	38	4	of	of	ADP
ejpam-4302	38	5	a	a	DET
ejpam-4302	38	6	topological	topological	ADJ
ejpam-4302	38	7	space	space	NOUN
ejpam-4302	38	8	(	(	PUNCT
ejpam-4302	38	9	x	x	X
ejpam-4302	38	10	,	,	PUNCT
ejpam-4302	38	11	τ	τ	X
ejpam-4302	38	12	)	)	PUNCT
ejpam-4302	38	13	is	be	AUX
ejpam-4302	38	14	called	call	VERB
ejpam-4302	38	15	β	β	VERB
ejpam-4302	38	16	-	-	ADJ
ejpam-4302	38	17	open	open	ADJ
ejpam-4302	38	18	[	[	X
ejpam-4302	38	19	7	7	NUM
ejpam-4302	38	20	]	]	X
ejpam-4302	38	21	if	if	SCONJ
ejpam-4302	38	22	a	a	DET
ejpam-4302	38	23	⊆	⊆	NUM
ejpam-4302	38	24	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-4302	38	25	)	)	PUNCT
ejpam-4302	38	26	)	)	PUNCT
ejpam-4302	38	27	)	)	PUNCT
ejpam-4302	38	28	.	.	PUNCT
ejpam-4302	39	1	the	the	DET
ejpam-4302	39	2	complement	complement	NOUN
ejpam-4302	39	3	of	of	ADP
ejpam-4302	39	4	a	a	DET
ejpam-4302	39	5	β	β	X
ejpam-4302	39	6	-	-	ADJ
ejpam-4302	39	7	open	open	ADJ
ejpam-4302	39	8	set	set	NOUN
ejpam-4302	39	9	is	be	AUX
ejpam-4302	39	10	called	call	VERB
ejpam-4302	39	11	β	β	NOUN
ejpam-4302	39	12	-	-	VERB
ejpam-4302	39	13	closed	closed	ADJ
ejpam-4302	39	14	.	.	PUNCT
ejpam-4302	40	1	the	the	DET
ejpam-4302	40	2	family	family	NOUN
ejpam-4302	40	3	of	of	ADP
ejpam-4302	40	4	all	all	DET
ejpam-4302	40	5	β	β	ADJ
ejpam-4302	40	6	-	-	ADJ
ejpam-4302	40	7	open	open	ADJ
ejpam-4302	40	8	sets	set	NOUN
ejpam-4302	40	9	in	in	ADP
ejpam-4302	40	10	a	a	DET
ejpam-4302	40	11	topological	topological	ADJ
ejpam-4302	40	12	space	space	NOUN
ejpam-4302	40	13	(	(	PUNCT
ejpam-4302	40	14	x	x	X
ejpam-4302	40	15	,	,	PUNCT
ejpam-4302	40	16	τ	τ	X
ejpam-4302	40	17	)	)	PUNCT
ejpam-4302	40	18	is	be	AUX
ejpam-4302	40	19	denoted	denote	VERB
ejpam-4302	40	20	by	by	ADP
ejpam-4302	40	21	β(x	β(x	PROPN
ejpam-4302	40	22	,	,	PUNCT
ejpam-4302	40	23	τ	τ	PROPN
ejpam-4302	40	24	)	)	PUNCT
ejpam-4302	40	25	.	.	PUNCT
ejpam-4302	41	1	let	let	VERB
ejpam-4302	41	2	a	a	DET
ejpam-4302	41	3	be	be	AUX
ejpam-4302	41	4	a	a	DET
ejpam-4302	41	5	subset	subset	NOUN
ejpam-4302	41	6	of	of	ADP
ejpam-4302	41	7	a	a	DET
ejpam-4302	41	8	topological	topological	ADJ
ejpam-4302	41	9	space	space	NOUN
ejpam-4302	41	10	(	(	PUNCT
ejpam-4302	41	11	x	x	X
ejpam-4302	41	12	,	,	PUNCT
ejpam-4302	41	13	τ	τ	PROPN
ejpam-4302	41	14	)	)	PUNCT
ejpam-4302	41	15	.	.	PUNCT
ejpam-4302	42	1	a	a	DET
ejpam-4302	42	2	subset	subset	NOUN
ejpam-4302	42	3	λsp(a	λsp(a	NOUN
ejpam-4302	42	4	)	)	PUNCT
ejpam-4302	43	1	[	[	X
ejpam-4302	43	2	13	13	NUM
ejpam-4302	43	3	]	]	PUNCT
ejpam-4302	43	4	is	be	AUX
ejpam-4302	43	5	defined	define	VERB
ejpam-4302	43	6	as	as	SCONJ
ejpam-4302	43	7	follows	follow	VERB
ejpam-4302	43	8	:	:	PUNCT
ejpam-4302	43	9	λsp(a	λsp(a	NUM
ejpam-4302	43	10	)	)	PUNCT
ejpam-4302	43	11	=	=	PUNCT
ejpam-4302	44	1	∩{u	∩{u	PROPN
ejpam-4302	44	2	|	|	ADV
ejpam-4302	44	3	a	a	DET
ejpam-4302	44	4	⊆	⊆	NUM
ejpam-4302	44	5	u	u	NOUN
ejpam-4302	44	6	,	,	PUNCT
ejpam-4302	44	7	u	u	NOUN
ejpam-4302	44	8	∈	∈	PROPN
ejpam-4302	44	9	β(x	β(x	PROPN
ejpam-4302	44	10	,	,	PUNCT
ejpam-4302	44	11	τ	τ	X
ejpam-4302	44	12	)	)	PUNCT
ejpam-4302	44	13	}	}	PUNCT
ejpam-4302	44	14	.	.	PUNCT
ejpam-4302	45	1	lemma	lemma	PROPN
ejpam-4302	45	2	1	1	NUM
ejpam-4302	45	3	.	.	PUNCT
ejpam-4302	46	1	[	[	X
ejpam-4302	46	2	13	13	NUM
ejpam-4302	46	3	]	]	PUNCT
ejpam-4302	46	4	for	for	ADP
ejpam-4302	46	5	subsets	subset	NOUN
ejpam-4302	46	6	a	a	PRON
ejpam-4302	46	7	,	,	PUNCT
ejpam-4302	46	8	b	b	PROPN
ejpam-4302	46	9	and	and	CCONJ
ejpam-4302	46	10	aα(α	aα(α	NOUN
ejpam-4302	46	11	∈	∈	PROPN
ejpam-4302	46	12	∇	∇	NOUN
ejpam-4302	46	13	)	)	PUNCT
ejpam-4302	46	14	of	of	ADP
ejpam-4302	46	15	a	a	DET
ejpam-4302	46	16	topological	topological	ADJ
ejpam-4302	46	17	space	space	NOUN
ejpam-4302	46	18	(	(	PUNCT
ejpam-4302	46	19	x	x	X
ejpam-4302	46	20	,	,	PUNCT
ejpam-4302	46	21	τ	τ	PROPN
ejpam-4302	46	22	)	)	PUNCT
ejpam-4302	46	23	,	,	PUNCT
ejpam-4302	46	24	the	the	DET
ejpam-4302	46	25	following	follow	VERB
ejpam-4302	46	26	properties	property	NOUN
ejpam-4302	46	27	hold	hold	VERB
ejpam-4302	46	28	:	:	PUNCT
ejpam-4302	46	29	(	(	PUNCT
ejpam-4302	46	30	1	1	X
ejpam-4302	46	31	)	)	PUNCT
ejpam-4302	46	32	a	a	DET
ejpam-4302	46	33	⊆	⊆	NUM
ejpam-4302	46	34	λsp(a	λsp(a	NOUN
ejpam-4302	46	35	)	)	PUNCT
ejpam-4302	46	36	.	.	PUNCT
ejpam-4302	47	1	(	(	PUNCT
ejpam-4302	47	2	2	2	X
ejpam-4302	47	3	)	)	PUNCT
ejpam-4302	47	4	if	if	SCONJ
ejpam-4302	47	5	a	a	DET
ejpam-4302	47	6	⊆	⊆	NUM
ejpam-4302	47	7	b	b	NOUN
ejpam-4302	47	8	,	,	PUNCT
ejpam-4302	47	9	then	then	ADV
ejpam-4302	47	10	λsp(a	λsp(a	PROPN
ejpam-4302	47	11	)	)	PUNCT
ejpam-4302	47	12	⊆	⊆	NUM
ejpam-4302	47	13	λsp(b	λsp(b	PROPN
ejpam-4302	47	14	)	)	PUNCT
ejpam-4302	47	15	.	.	PUNCT
ejpam-4302	48	1	(	(	PUNCT
ejpam-4302	48	2	3	3	X
ejpam-4302	48	3	)	)	PUNCT
ejpam-4302	48	4	λsp(λsp(a	λsp(λsp(a	NUM
ejpam-4302	48	5	)	)	PUNCT
ejpam-4302	48	6	)	)	PUNCT
ejpam-4302	49	1	=	=	SYM
ejpam-4302	49	2	λsp(a	λsp(a	PROPN
ejpam-4302	49	3	)	)	PUNCT
ejpam-4302	49	4	.	.	PUNCT
ejpam-4302	50	1	(	(	PUNCT
ejpam-4302	50	2	4	4	X
ejpam-4302	50	3	)	)	PUNCT
ejpam-4302	50	4	if	if	SCONJ
ejpam-4302	50	5	u	u	PROPN
ejpam-4302	50	6	∈	∈	PROPN
ejpam-4302	50	7	β(x	β(x	PROPN
ejpam-4302	50	8	,	,	PUNCT
ejpam-4302	50	9	τ	τ	PROPN
ejpam-4302	50	10	)	)	PUNCT
ejpam-4302	50	11	,	,	PUNCT
ejpam-4302	50	12	then	then	ADV
ejpam-4302	50	13	λsp(u	λsp(u	X
ejpam-4302	50	14	)	)	PUNCT
ejpam-4302	50	15	=	=	SYM
ejpam-4302	50	16	u	u	NOUN
ejpam-4302	50	17	.	.	PUNCT
ejpam-4302	51	1	(	(	PUNCT
ejpam-4302	51	2	5	5	NUM
ejpam-4302	51	3	)	)	PUNCT
ejpam-4302	51	4	λsp(∩{aα|α	λsp(∩{aα|α	ADV
ejpam-4302	51	5	∈	∈	NOUN
ejpam-4302	51	6	∇	∇	NOUN
ejpam-4302	51	7	}	}	PUNCT
ejpam-4302	51	8	)	)	PUNCT
ejpam-4302	51	9	⊆	⊆	NUM
ejpam-4302	51	10	∩{λsp(aα)|α	∩{λsp(aα)|α	PROPN
ejpam-4302	51	11	∈	∈	NOUN
ejpam-4302	51	12	∇	∇	X
ejpam-4302	51	13	}	}	PUNCT
ejpam-4302	51	14	.	.	PUNCT
ejpam-4302	52	1	(	(	PUNCT
ejpam-4302	52	2	6	6	NUM
ejpam-4302	52	3	)	)	PUNCT
ejpam-4302	52	4	λsp(∪{aα|α	λsp(∪{aα|α	PROPN
ejpam-4302	52	5	∈	∈	PROPN
ejpam-4302	52	6	∇	∇	X
ejpam-4302	52	7	}	}	PUNCT
ejpam-4302	52	8	)	)	PUNCT
ejpam-4302	52	9	=	=	SYM
ejpam-4302	53	1	∪{λsp(aα)|α	∪{λsp(aα)|α	PROPN
ejpam-4302	53	2	∈	∈	NOUN
ejpam-4302	53	3	∇	∇	NOUN
ejpam-4302	53	4	}	}	PUNCT
ejpam-4302	53	5	.	.	PUNCT
ejpam-4302	54	1	a	a	DET
ejpam-4302	54	2	subset	subset	NOUN
ejpam-4302	54	3	b	b	NOUN
ejpam-4302	54	4	of	of	ADP
ejpam-4302	54	5	a	a	DET
ejpam-4302	54	6	topological	topological	ADJ
ejpam-4302	54	7	space	space	NOUN
ejpam-4302	54	8	(	(	PUNCT
ejpam-4302	54	9	x	x	X
ejpam-4302	54	10	,	,	PUNCT
ejpam-4302	54	11	τ	τ	X
ejpam-4302	54	12	)	)	PUNCT
ejpam-4302	54	13	is	be	AUX
ejpam-4302	54	14	called	call	VERB
ejpam-4302	54	15	a	a	DET
ejpam-4302	54	16	λsp	λsp	NOUN
ejpam-4302	54	17	-	-	PUNCT
ejpam-4302	54	18	set	set	VERB
ejpam-4302	54	19	[	[	X
ejpam-4302	54	20	13	13	NUM
ejpam-4302	54	21	]	]	PUNCT
ejpam-4302	54	22	if	if	SCONJ
ejpam-4302	54	23	b	b	PROPN
ejpam-4302	54	24	=	=	SYM
ejpam-4302	54	25	λsp(b	λsp(b	PROPN
ejpam-4302	54	26	)	)	PUNCT
ejpam-4302	54	27	.	.	PUNCT
ejpam-4302	55	1	the	the	DET
ejpam-4302	55	2	family	family	NOUN
ejpam-4302	55	3	of	of	ADP
ejpam-4302	55	4	all	all	DET
ejpam-4302	55	5	λsp	λsp	NOUN
ejpam-4302	55	6	-	-	PUNCT
ejpam-4302	55	7	sets	set	NOUN
ejpam-4302	55	8	of	of	ADP
ejpam-4302	55	9	a	a	DET
ejpam-4302	55	10	topological	topological	ADJ
ejpam-4302	55	11	space	space	NOUN
ejpam-4302	55	12	(	(	PUNCT
ejpam-4302	55	13	x	x	X
ejpam-4302	55	14	,	,	PUNCT
ejpam-4302	55	15	τ	τ	X
ejpam-4302	55	16	)	)	PUNCT
ejpam-4302	55	17	is	be	AUX
ejpam-4302	55	18	denoted	denote	VERB
ejpam-4302	55	19	by	by	ADP
ejpam-4302	55	20	λsp(x	λsp(x	PROPN
ejpam-4302	55	21	,	,	PUNCT
ejpam-4302	55	22	τ	τ	X
ejpam-4302	55	23	)	)	PUNCT
ejpam-4302	55	24	(	(	PUNCT
ejpam-4302	55	25	or	or	CCONJ
ejpam-4302	55	26	simply	simply	ADV
ejpam-4302	55	27	λsp	λsp	PROPN
ejpam-4302	55	28	)	)	PUNCT
ejpam-4302	55	29	.	.	PUNCT
ejpam-4302	56	1	lemma	lemma	PROPN
ejpam-4302	56	2	2	2	NUM
ejpam-4302	56	3	.	.	PUNCT
ejpam-4302	57	1	[	[	X
ejpam-4302	57	2	13	13	NUM
ejpam-4302	57	3	]	]	PUNCT
ejpam-4302	57	4	for	for	ADP
ejpam-4302	57	5	subsets	subset	NOUN
ejpam-4302	57	6	a	a	DET
ejpam-4302	57	7	and	and	CCONJ
ejpam-4302	57	8	aα(α	aα(α	NOUN
ejpam-4302	57	9	∈	∈	NOUN
ejpam-4302	57	10	∇	∇	NOUN
ejpam-4302	57	11	)	)	PUNCT
ejpam-4302	57	12	of	of	ADP
ejpam-4302	57	13	a	a	DET
ejpam-4302	57	14	topological	topological	ADJ
ejpam-4302	57	15	space	space	NOUN
ejpam-4302	57	16	(	(	PUNCT
ejpam-4302	57	17	x	x	X
ejpam-4302	57	18	,	,	PUNCT
ejpam-4302	57	19	τ	τ	PROPN
ejpam-4302	57	20	)	)	PUNCT
ejpam-4302	57	21	,	,	PUNCT
ejpam-4302	57	22	the	the	DET
ejpam-4302	57	23	following	follow	VERB
ejpam-4302	57	24	properties	property	NOUN
ejpam-4302	57	25	hold	hold	VERB
ejpam-4302	57	26	:	:	PUNCT
ejpam-4302	57	27	(	(	PUNCT
ejpam-4302	57	28	1	1	X
ejpam-4302	57	29	)	)	PUNCT
ejpam-4302	57	30	λsp(a	λsp(a	NOUN
ejpam-4302	57	31	)	)	PUNCT
ejpam-4302	57	32	is	be	AUX
ejpam-4302	57	33	a	a	DET
ejpam-4302	57	34	λsp	λsp	NOUN
ejpam-4302	57	35	-	-	PUNCT
ejpam-4302	57	36	set	set	NOUN
ejpam-4302	57	37	.	.	PUNCT
ejpam-4302	58	1	(	(	PUNCT
ejpam-4302	58	2	2	2	X
ejpam-4302	58	3	)	)	PUNCT
ejpam-4302	58	4	if	if	SCONJ
ejpam-4302	58	5	a	a	PRON
ejpam-4302	58	6	is	be	AUX
ejpam-4302	58	7	β	β	NOUN
ejpam-4302	58	8	-	-	ADJ
ejpam-4302	58	9	open	open	ADJ
ejpam-4302	58	10	,	,	PUNCT
ejpam-4302	58	11	then	then	ADV
ejpam-4302	58	12	a	a	PRON
ejpam-4302	58	13	is	be	AUX
ejpam-4302	58	14	a	a	DET
ejpam-4302	58	15	λsp	λsp	NOUN
ejpam-4302	58	16	-	-	PUNCT
ejpam-4302	58	17	set	set	VERB
ejpam-4302	58	18	.	.	PUNCT
ejpam-4302	59	1	c.	c.	PROPN
ejpam-4302	59	2	boonpok	boonpok	PROPN
ejpam-4302	59	3	,	,	PUNCT
ejpam-4302	59	4	c.	c.	PROPN
ejpam-4302	59	5	viriyapong	viriyapong	PROPN
ejpam-4302	59	6	/	/	SYM
ejpam-4302	59	7	eur	eur	PROPN
ejpam-4302	59	8	.	.	PUNCT
ejpam-4302	60	1	j.	j.	PROPN
ejpam-4302	60	2	pure	pure	PROPN
ejpam-4302	60	3	appl	appl	PROPN
ejpam-4302	60	4	.	.	PROPN
ejpam-4302	60	5	math	math	PROPN
ejpam-4302	60	6	,	,	PUNCT
ejpam-4302	60	7	15	15	NUM
ejpam-4302	60	8	(	(	PUNCT
ejpam-4302	60	9	4	4	NUM
ejpam-4302	60	10	)	)	PUNCT
ejpam-4302	60	11	(	(	PUNCT
ejpam-4302	60	12	2022	2022	NUM
ejpam-4302	60	13	)	)	PUNCT
ejpam-4302	60	14	,	,	PUNCT
ejpam-4302	60	15	2127	2127	NUM
ejpam-4302	60	16	-	-	SYM
ejpam-4302	60	17	2140	2140	NUM
ejpam-4302	60	18	2129	2129	NUM
ejpam-4302	60	19	(	(	PUNCT
ejpam-4302	60	20	3	3	X
ejpam-4302	60	21	)	)	PUNCT
ejpam-4302	60	22	if	if	SCONJ
ejpam-4302	60	23	aα	aα	NOUN
ejpam-4302	60	24	is	be	AUX
ejpam-4302	60	25	a	a	DET
ejpam-4302	60	26	λsp	λsp	NOUN
ejpam-4302	60	27	-	-	PUNCT
ejpam-4302	60	28	set	set	VERB
ejpam-4302	60	29	for	for	ADP
ejpam-4302	60	30	each	each	DET
ejpam-4302	60	31	α	α	PROPN
ejpam-4302	60	32	∈	∈	PROPN
ejpam-4302	60	33	∇	∇	NOUN
ejpam-4302	60	34	,	,	PUNCT
ejpam-4302	60	35	then	then	ADV
ejpam-4302	60	36	∩α∈∇aα	∩α∈∇aα	PROPN
ejpam-4302	60	37	is	be	AUX
ejpam-4302	60	38	a	a	DET
ejpam-4302	60	39	λsp	λsp	NOUN
ejpam-4302	60	40	-	-	PUNCT
ejpam-4302	60	41	set	set	NOUN
ejpam-4302	60	42	.	.	PUNCT
ejpam-4302	61	1	(	(	PUNCT
ejpam-4302	61	2	4	4	X
ejpam-4302	61	3	)	)	PUNCT
ejpam-4302	61	4	if	if	SCONJ
ejpam-4302	61	5	aα	aα	NOUN
ejpam-4302	61	6	is	be	AUX
ejpam-4302	61	7	a	a	DET
ejpam-4302	61	8	λsp	λsp	NOUN
ejpam-4302	61	9	-	-	PUNCT
ejpam-4302	61	10	set	set	VERB
ejpam-4302	61	11	for	for	ADP
ejpam-4302	61	12	each	each	DET
ejpam-4302	61	13	α	α	PROPN
ejpam-4302	61	14	∈	∈	PROPN
ejpam-4302	61	15	∇	∇	NOUN
ejpam-4302	61	16	,	,	PUNCT
ejpam-4302	61	17	then	then	ADV
ejpam-4302	61	18	∪α∈∇aα	∪α∈∇aα	VERB
ejpam-4302	61	19	is	be	AUX
ejpam-4302	61	20	a	a	DET
ejpam-4302	61	21	λsp	λsp	NOUN
ejpam-4302	61	22	-	-	PUNCT
ejpam-4302	61	23	set	set	NOUN
ejpam-4302	61	24	.	.	PUNCT
ejpam-4302	62	1	a	a	DET
ejpam-4302	62	2	subset	subset	NOUN
ejpam-4302	62	3	a	a	PRON
ejpam-4302	62	4	of	of	ADP
ejpam-4302	62	5	a	a	DET
ejpam-4302	62	6	topological	topological	ADJ
ejpam-4302	62	7	space	space	NOUN
ejpam-4302	62	8	(	(	PUNCT
ejpam-4302	62	9	x	x	X
ejpam-4302	62	10	,	,	PUNCT
ejpam-4302	62	11	τ	τ	X
ejpam-4302	62	12	)	)	PUNCT
ejpam-4302	62	13	is	be	AUX
ejpam-4302	62	14	said	say	VERB
ejpam-4302	62	15	to	to	PART
ejpam-4302	62	16	be	be	AUX
ejpam-4302	62	17	(	(	PUNCT
ejpam-4302	62	18	λ	λ	X
ejpam-4302	62	19	,	,	PUNCT
ejpam-4302	62	20	sp)-closed	sp)-close	VERB
ejpam-4302	62	21	[	[	PUNCT
ejpam-4302	62	22	2	2	X
ejpam-4302	62	23	]	]	X
ejpam-4302	62	24	if	if	SCONJ
ejpam-4302	62	25	a	a	DET
ejpam-4302	62	26	=	=	X
ejpam-4302	62	27	t	t	PROPN
ejpam-4302	62	28	∩	∩	ADJ
ejpam-4302	62	29	c	c	NOUN
ejpam-4302	62	30	,	,	PUNCT
ejpam-4302	62	31	where	where	SCONJ
ejpam-4302	62	32	t	t	PROPN
ejpam-4302	62	33	is	be	AUX
ejpam-4302	62	34	a	a	DET
ejpam-4302	62	35	λsp	λsp	NOUN
ejpam-4302	62	36	-	-	PUNCT
ejpam-4302	62	37	set	set	VERB
ejpam-4302	62	38	and	and	CCONJ
ejpam-4302	62	39	c	c	NOUN
ejpam-4302	62	40	is	be	AUX
ejpam-4302	62	41	a	a	DET
ejpam-4302	62	42	β	β	NOUN
ejpam-4302	62	43	-	-	ADJ
ejpam-4302	62	44	closed	closed	ADJ
ejpam-4302	62	45	set	set	NOUN
ejpam-4302	62	46	.	.	PUNCT
ejpam-4302	63	1	the	the	DET
ejpam-4302	63	2	complement	complement	NOUN
ejpam-4302	63	3	of	of	ADP
ejpam-4302	63	4	a	a	DET
ejpam-4302	63	5	(	(	PUNCT
ejpam-4302	63	6	λ	λ	PROPN
ejpam-4302	63	7	,	,	PUNCT
ejpam-4302	63	8	sp)-closed	sp)-close	VERB
ejpam-4302	63	9	set	set	VERB
ejpam-4302	63	10	is	be	AUX
ejpam-4302	63	11	called	call	VERB
ejpam-4302	63	12	(	(	PUNCT
ejpam-4302	63	13	λ	λ	NOUN
ejpam-4302	63	14	,	,	PUNCT
ejpam-4302	63	15	sp)-open	sp)-open	NOUN
ejpam-4302	63	16	.	.	PUNCT
ejpam-4302	64	1	the	the	DET
ejpam-4302	64	2	family	family	NOUN
ejpam-4302	64	3	of	of	ADP
ejpam-4302	64	4	all	all	DET
ejpam-4302	64	5	(	(	PUNCT
ejpam-4302	64	6	λ	λ	NOUN
ejpam-4302	64	7	,	,	PUNCT
ejpam-4302	64	8	sp)-open	sp)-open	ADJ
ejpam-4302	64	9	(	(	PUNCT
ejpam-4302	64	10	resp	resp	NOUN
ejpam-4302	64	11	.	.	PUNCT
ejpam-4302	65	1	(	(	PUNCT
ejpam-4302	65	2	λ	λ	X
ejpam-4302	65	3	,	,	PUNCT
ejpam-4302	65	4	sp)-closed	sp)-closed	ADJ
ejpam-4302	65	5	)	)	PUNCT
ejpam-4302	65	6	sets	set	NOUN
ejpam-4302	65	7	in	in	ADP
ejpam-4302	65	8	a	a	DET
ejpam-4302	65	9	topological	topological	ADJ
ejpam-4302	65	10	space	space	NOUN
ejpam-4302	65	11	(	(	PUNCT
ejpam-4302	65	12	x	x	X
ejpam-4302	65	13	,	,	PUNCT
ejpam-4302	65	14	τ	τ	X
ejpam-4302	65	15	)	)	PUNCT
ejpam-4302	65	16	is	be	AUX
ejpam-4302	65	17	denoted	denote	VERB
ejpam-4302	65	18	by	by	ADP
ejpam-4302	65	19	λspo(x	λspo(x	PROPN
ejpam-4302	65	20	,	,	PUNCT
ejpam-4302	65	21	τ	τ	PROPN
ejpam-4302	65	22	)	)	PUNCT
ejpam-4302	65	23	(	(	PUNCT
ejpam-4302	65	24	resp	resp	NOUN
ejpam-4302	65	25	.	.	PUNCT
ejpam-4302	66	1	λspc(x	λspc(x	NOUN
ejpam-4302	66	2	,	,	PUNCT
ejpam-4302	66	3	τ	τ	PROPN
ejpam-4302	66	4	)	)	PUNCT
ejpam-4302	66	5	)	)	PUNCT
ejpam-4302	66	6	.	.	PUNCT
ejpam-4302	67	1	let	let	VERB
ejpam-4302	67	2	a	a	PRON
ejpam-4302	67	3	be	be	AUX
ejpam-4302	67	4	a	a	DET
ejpam-4302	67	5	subsets	subset	NOUN
ejpam-4302	67	6	of	of	ADP
ejpam-4302	67	7	a	a	DET
ejpam-4302	67	8	topological	topological	ADJ
ejpam-4302	67	9	space	space	NOUN
ejpam-4302	67	10	(	(	PUNCT
ejpam-4302	67	11	x	x	X
ejpam-4302	67	12	,	,	PUNCT
ejpam-4302	67	13	τ	τ	PROPN
ejpam-4302	67	14	)	)	PUNCT
ejpam-4302	67	15	.	.	PUNCT
ejpam-4302	68	1	a	a	DET
ejpam-4302	68	2	point	point	NOUN
ejpam-4302	68	3	x	x	X
ejpam-4302	68	4	∈	∈	NOUN
ejpam-4302	68	5	x	x	PUNCT
ejpam-4302	68	6	is	be	AUX
ejpam-4302	68	7	called	call	VERB
ejpam-4302	68	8	a	a	DET
ejpam-4302	68	9	(	(	PUNCT
ejpam-4302	68	10	λ	λ	NOUN
ejpam-4302	68	11	,	,	PUNCT
ejpam-4302	68	12	sp)-cluster	sp)-cluster	NOUN
ejpam-4302	68	13	point	point	NOUN
ejpam-4302	68	14	[	[	X
ejpam-4302	68	15	2	2	X
ejpam-4302	68	16	]	]	PUNCT
ejpam-4302	68	17	of	of	ADP
ejpam-4302	68	18	a	a	DET
ejpam-4302	68	19	if	if	SCONJ
ejpam-4302	68	20	a∩u	a∩u	VERB
ejpam-4302	68	21	̸=	̸=	NOUN
ejpam-4302	68	22	∅	∅	NOUN
ejpam-4302	68	23	for	for	ADP
ejpam-4302	68	24	every	every	DET
ejpam-4302	68	25	(	(	PUNCT
ejpam-4302	68	26	λ	λ	NOUN
ejpam-4302	68	27	,	,	PUNCT
ejpam-4302	68	28	sp)-open	sp)-open	NOUN
ejpam-4302	68	29	set	set	VERB
ejpam-4302	68	30	u	u	NOUN
ejpam-4302	68	31	of	of	ADP
ejpam-4302	68	32	x	x	SYM
ejpam-4302	68	33	containing	contain	VERB
ejpam-4302	68	34	x.	x.	NOUN
ejpam-4302	68	35	the	the	DET
ejpam-4302	68	36	set	set	NOUN
ejpam-4302	68	37	of	of	ADP
ejpam-4302	68	38	all	all	DET
ejpam-4302	68	39	(	(	PUNCT
ejpam-4302	68	40	λ	λ	PROPN
ejpam-4302	68	41	,	,	PUNCT
ejpam-4302	68	42	sp)-cluster	sp)-cluster	NOUN
ejpam-4302	68	43	points	point	NOUN
ejpam-4302	68	44	of	of	ADP
ejpam-4302	68	45	a	a	PRON
ejpam-4302	68	46	is	be	AUX
ejpam-4302	68	47	called	call	VERB
ejpam-4302	68	48	the	the	DET
ejpam-4302	68	49	(	(	PUNCT
ejpam-4302	68	50	λ	λ	PROPN
ejpam-4302	68	51	,	,	PUNCT
ejpam-4302	68	52	sp)-closure	sp)-closure	NOUN
ejpam-4302	68	53	[	[	X
ejpam-4302	68	54	2	2	NUM
ejpam-4302	68	55	]	]	PUNCT
ejpam-4302	68	56	of	of	ADP
ejpam-4302	68	57	a	a	PRON
ejpam-4302	68	58	and	and	CCONJ
ejpam-4302	68	59	is	be	AUX
ejpam-4302	68	60	denoted	denote	VERB
ejpam-4302	68	61	by	by	ADP
ejpam-4302	68	62	a(λ	a(λ	ADV
ejpam-4302	68	63	,	,	PUNCT
ejpam-4302	68	64	sp	sp	NOUN
ejpam-4302	68	65	)	)	PUNCT
ejpam-4302	68	66	.	.	PUNCT
ejpam-4302	69	1	the	the	DET
ejpam-4302	69	2	union	union	NOUN
ejpam-4302	69	3	of	of	ADP
ejpam-4302	69	4	all	all	DET
ejpam-4302	69	5	(	(	PUNCT
ejpam-4302	69	6	λ	λ	NOUN
ejpam-4302	69	7	,	,	PUNCT
ejpam-4302	69	8	sp)-open	sp)-open	ADJ
ejpam-4302	69	9	sets	set	NOUN
ejpam-4302	69	10	contained	contain	VERB
ejpam-4302	69	11	in	in	ADP
ejpam-4302	69	12	a	a	PRON
ejpam-4302	69	13	is	be	AUX
ejpam-4302	69	14	called	call	VERB
ejpam-4302	69	15	the	the	DET
ejpam-4302	69	16	(	(	PUNCT
ejpam-4302	69	17	λ	λ	PROPN
ejpam-4302	69	18	,	,	PUNCT
ejpam-4302	69	19	sp)-interior	sp)-interior	NOUN
ejpam-4302	69	20	[	[	X
ejpam-4302	69	21	2	2	NUM
ejpam-4302	69	22	]	]	PUNCT
ejpam-4302	69	23	of	of	ADP
ejpam-4302	69	24	a	a	PRON
ejpam-4302	69	25	and	and	CCONJ
ejpam-4302	69	26	is	be	AUX
ejpam-4302	69	27	denoted	denote	VERB
ejpam-4302	69	28	by	by	ADP
ejpam-4302	69	29	a(λ	a(λ	ADV
ejpam-4302	69	30	,	,	PUNCT
ejpam-4302	69	31	sp	sp	NOUN
ejpam-4302	69	32	)	)	PUNCT
ejpam-4302	69	33	.	.	PUNCT
ejpam-4302	70	1	lemma	lemma	PROPN
ejpam-4302	70	2	3	3	X
ejpam-4302	70	3	.	.	PUNCT
ejpam-4302	71	1	[	[	X
ejpam-4302	71	2	2	2	X
ejpam-4302	71	3	]	]	PUNCT
ejpam-4302	71	4	let	let	VERB
ejpam-4302	71	5	a	a	PRON
ejpam-4302	71	6	and	and	CCONJ
ejpam-4302	71	7	b	b	NOUN
ejpam-4302	71	8	be	be	AUX
ejpam-4302	71	9	subsets	subset	NOUN
ejpam-4302	71	10	of	of	ADP
ejpam-4302	71	11	a	a	DET
ejpam-4302	71	12	topological	topological	ADJ
ejpam-4302	71	13	space	space	NOUN
ejpam-4302	71	14	(	(	PUNCT
ejpam-4302	71	15	x	x	X
ejpam-4302	71	16	,	,	PUNCT
ejpam-4302	71	17	τ	τ	PROPN
ejpam-4302	71	18	)	)	PUNCT
ejpam-4302	71	19	.	.	PUNCT
ejpam-4302	72	1	for	for	ADP
ejpam-4302	72	2	the	the	DET
ejpam-4302	72	3	(	(	PUNCT
ejpam-4302	72	4	λ	λ	PROPN
ejpam-4302	72	5	,	,	PUNCT
ejpam-4302	72	6	sp)-closure	sp)-closure	NOUN
ejpam-4302	72	7	,	,	PUNCT
ejpam-4302	72	8	the	the	DET
ejpam-4302	72	9	following	follow	VERB
ejpam-4302	72	10	properties	property	NOUN
ejpam-4302	72	11	hold	hold	VERB
ejpam-4302	72	12	:	:	PUNCT
ejpam-4302	72	13	(	(	PUNCT
ejpam-4302	72	14	1	1	X
ejpam-4302	72	15	)	)	PUNCT
ejpam-4302	72	16	a	a	DET
ejpam-4302	72	17	⊆	⊆	NUM
ejpam-4302	72	18	a(λ	a(λ	ADJ
ejpam-4302	72	19	,	,	PUNCT
ejpam-4302	72	20	sp	sp	NOUN
ejpam-4302	72	21	)	)	PUNCT
ejpam-4302	72	22	and	and	CCONJ
ejpam-4302	72	23	[	[	X
ejpam-4302	72	24	a(λ	a(λ	ADV
ejpam-4302	72	25	,	,	PUNCT
ejpam-4302	72	26	sp)](λ	sp)](λ	PROPN
ejpam-4302	72	27	,	,	PUNCT
ejpam-4302	72	28	sp	sp	NOUN
ejpam-4302	72	29	)	)	PUNCT
ejpam-4302	72	30	=	=	PUNCT
ejpam-4302	72	31	a(λ	a(λ	ADV
ejpam-4302	72	32	,	,	PUNCT
ejpam-4302	72	33	sp	sp	NOUN
ejpam-4302	72	34	)	)	PUNCT
ejpam-4302	72	35	.	.	PUNCT
ejpam-4302	73	1	(	(	PUNCT
ejpam-4302	73	2	2	2	X
ejpam-4302	73	3	)	)	PUNCT
ejpam-4302	73	4	if	if	SCONJ
ejpam-4302	73	5	a	a	DET
ejpam-4302	73	6	⊆	⊆	NUM
ejpam-4302	73	7	b	b	NOUN
ejpam-4302	73	8	,	,	PUNCT
ejpam-4302	73	9	then	then	ADV
ejpam-4302	73	10	a(λ	a(λ	ADV
ejpam-4302	73	11	,	,	PUNCT
ejpam-4302	73	12	sp	sp	NOUN
ejpam-4302	73	13	)	)	PUNCT
ejpam-4302	73	14	⊆	⊆	NUM
ejpam-4302	73	15	b(λ	b(λ	NOUN
ejpam-4302	73	16	,	,	PUNCT
ejpam-4302	73	17	sp	sp	NOUN
ejpam-4302	73	18	)	)	PUNCT
ejpam-4302	73	19	.	.	PUNCT
ejpam-4302	74	1	(	(	PUNCT
ejpam-4302	74	2	3	3	X
ejpam-4302	74	3	)	)	PUNCT
ejpam-4302	74	4	a(λ	a(λ	ADV
ejpam-4302	74	5	,	,	PUNCT
ejpam-4302	74	6	sp	sp	NOUN
ejpam-4302	74	7	)	)	PUNCT
ejpam-4302	74	8	is	be	AUX
ejpam-4302	74	9	(	(	PUNCT
ejpam-4302	74	10	λ	λ	X
ejpam-4302	74	11	,	,	PUNCT
ejpam-4302	74	12	sp)-closed	sp)-close	VERB
ejpam-4302	74	13	.	.	PUNCT
ejpam-4302	75	1	(	(	PUNCT
ejpam-4302	75	2	4	4	X
ejpam-4302	75	3	)	)	PUNCT
ejpam-4302	75	4	a	a	PRON
ejpam-4302	75	5	is	be	AUX
ejpam-4302	75	6	(	(	PUNCT
ejpam-4302	75	7	λ	λ	X
ejpam-4302	75	8	,	,	PUNCT
ejpam-4302	75	9	sp)-closed	sp)-close	VERB
ejpam-4302	75	10	if	if	SCONJ
ejpam-4302	75	11	and	and	CCONJ
ejpam-4302	75	12	only	only	ADV
ejpam-4302	75	13	if	if	SCONJ
ejpam-4302	75	14	a(λ	a(λ	ADV
ejpam-4302	75	15	,	,	PUNCT
ejpam-4302	75	16	sp	sp	NOUN
ejpam-4302	75	17	)	)	PUNCT
ejpam-4302	75	18	=	=	PUNCT
ejpam-4302	75	19	a.	a.	NOUN
ejpam-4302	75	20	lemma	lemma	PROPN
ejpam-4302	75	21	4	4	X
ejpam-4302	75	22	.	.	PUNCT
ejpam-4302	76	1	[	[	X
ejpam-4302	76	2	2	2	NUM
ejpam-4302	76	3	]	]	PUNCT
ejpam-4302	76	4	for	for	ADP
ejpam-4302	76	5	subsets	subset	NOUN
ejpam-4302	76	6	a	a	PRON
ejpam-4302	76	7	and	and	CCONJ
ejpam-4302	76	8	b	b	NOUN
ejpam-4302	76	9	of	of	ADP
ejpam-4302	76	10	a	a	DET
ejpam-4302	76	11	topological	topological	ADJ
ejpam-4302	76	12	space	space	NOUN
ejpam-4302	76	13	(	(	PUNCT
ejpam-4302	76	14	x	x	X
ejpam-4302	76	15	,	,	PUNCT
ejpam-4302	76	16	τ	τ	PROPN
ejpam-4302	76	17	)	)	PUNCT
ejpam-4302	76	18	,	,	PUNCT
ejpam-4302	76	19	the	the	DET
ejpam-4302	76	20	following	follow	VERB
ejpam-4302	76	21	properties	property	NOUN
ejpam-4302	76	22	hold	hold	VERB
ejpam-4302	76	23	:	:	PUNCT
ejpam-4302	76	24	(	(	PUNCT
ejpam-4302	76	25	1	1	X
ejpam-4302	76	26	)	)	PUNCT
ejpam-4302	76	27	a(λ	a(λ	ADV
ejpam-4302	76	28	,	,	PUNCT
ejpam-4302	76	29	sp	sp	NOUN
ejpam-4302	76	30	)	)	PUNCT
ejpam-4302	76	31	⊆	⊆	NUM
ejpam-4302	76	32	a	a	DET
ejpam-4302	76	33	and	and	CCONJ
ejpam-4302	76	34	[	[	X
ejpam-4302	76	35	a(λ	a(λ	ADV
ejpam-4302	76	36	,	,	PUNCT
ejpam-4302	76	37	sp)](λ	sp)](λ	PROPN
ejpam-4302	76	38	,	,	PUNCT
ejpam-4302	76	39	sp	sp	NOUN
ejpam-4302	76	40	)	)	PUNCT
ejpam-4302	76	41	=	=	PUNCT
ejpam-4302	76	42	a(λ	a(λ	ADV
ejpam-4302	76	43	,	,	PUNCT
ejpam-4302	76	44	sp	sp	NOUN
ejpam-4302	76	45	)	)	PUNCT
ejpam-4302	76	46	.	.	PUNCT
ejpam-4302	77	1	(	(	PUNCT
ejpam-4302	77	2	2	2	X
ejpam-4302	77	3	)	)	PUNCT
ejpam-4302	77	4	if	if	SCONJ
ejpam-4302	77	5	a	a	DET
ejpam-4302	77	6	⊆	⊆	NUM
ejpam-4302	77	7	b	b	NOUN
ejpam-4302	77	8	,	,	PUNCT
ejpam-4302	77	9	then	then	ADV
ejpam-4302	77	10	a(λ	a(λ	ADV
ejpam-4302	77	11	,	,	PUNCT
ejpam-4302	77	12	sp	sp	NOUN
ejpam-4302	77	13	)	)	PUNCT
ejpam-4302	77	14	⊆	⊆	NUM
ejpam-4302	77	15	b(λ	b(λ	NOUN
ejpam-4302	77	16	,	,	PUNCT
ejpam-4302	77	17	sp	sp	NOUN
ejpam-4302	77	18	)	)	PUNCT
ejpam-4302	77	19	.	.	PUNCT
ejpam-4302	78	1	(	(	PUNCT
ejpam-4302	78	2	3	3	X
ejpam-4302	78	3	)	)	PUNCT
ejpam-4302	78	4	a(λ	a(λ	ADV
ejpam-4302	78	5	,	,	PUNCT
ejpam-4302	78	6	sp	sp	NOUN
ejpam-4302	78	7	)	)	PUNCT
ejpam-4302	78	8	is	be	AUX
ejpam-4302	78	9	(	(	PUNCT
ejpam-4302	78	10	λ	λ	INTJ
ejpam-4302	78	11	,	,	PUNCT
ejpam-4302	78	12	sp)-open	sp)-open	NOUN
ejpam-4302	78	13	.	.	PUNCT
ejpam-4302	79	1	(	(	PUNCT
ejpam-4302	79	2	4	4	X
ejpam-4302	79	3	)	)	PUNCT
ejpam-4302	79	4	a	a	DET
ejpam-4302	79	5	is	be	AUX
ejpam-4302	79	6	(	(	PUNCT
ejpam-4302	79	7	λ	λ	NOUN
ejpam-4302	79	8	,	,	PUNCT
ejpam-4302	79	9	sp)-open	sp)-open	ADJ
ejpam-4302	79	10	if	if	SCONJ
ejpam-4302	79	11	and	and	CCONJ
ejpam-4302	79	12	only	only	ADV
ejpam-4302	79	13	if	if	SCONJ
ejpam-4302	79	14	a(λ	a(λ	ADV
ejpam-4302	79	15	,	,	PUNCT
ejpam-4302	79	16	sp	sp	NOUN
ejpam-4302	79	17	)	)	PUNCT
ejpam-4302	79	18	=	=	SYM
ejpam-4302	79	19	a.	a.	NOUN
ejpam-4302	79	20	(	(	PUNCT
ejpam-4302	79	21	5	5	NUM
ejpam-4302	79	22	)	)	PUNCT
ejpam-4302	80	1	[	[	X
ejpam-4302	80	2	x	x	X
ejpam-4302	80	3	−a](λ	−a](λ	PROPN
ejpam-4302	80	4	,	,	PUNCT
ejpam-4302	80	5	sp	sp	NOUN
ejpam-4302	80	6	)	)	PUNCT
ejpam-4302	80	7	=	=	SYM
ejpam-4302	80	8	x	x	SYM
ejpam-4302	80	9	−a(λ	−a(λ	NOUN
ejpam-4302	80	10	,	,	PUNCT
ejpam-4302	80	11	sp	sp	NOUN
ejpam-4302	80	12	)	)	PUNCT
ejpam-4302	80	13	.	.	PUNCT
ejpam-4302	81	1	(	(	PUNCT
ejpam-4302	81	2	6	6	NUM
ejpam-4302	81	3	)	)	PUNCT
ejpam-4302	82	1	[	[	X
ejpam-4302	82	2	x	x	X
ejpam-4302	82	3	−a](λ	−a](λ	PROPN
ejpam-4302	82	4	,	,	PUNCT
ejpam-4302	82	5	sp	sp	NOUN
ejpam-4302	82	6	)	)	PUNCT
ejpam-4302	82	7	=	=	SYM
ejpam-4302	82	8	x	x	SYM
ejpam-4302	82	9	−a(λ	−a(λ	NOUN
ejpam-4302	82	10	,	,	PUNCT
ejpam-4302	82	11	sp	sp	NOUN
ejpam-4302	82	12	)	)	PUNCT
ejpam-4302	82	13	.	.	PUNCT
ejpam-4302	83	1	3	3	X
ejpam-4302	83	2	.	.	X
ejpam-4302	83	3	generalized	generalize	VERB
ejpam-4302	83	4	(	(	PUNCT
ejpam-4302	83	5	λ	λ	NOUN
ejpam-4302	83	6	,	,	PUNCT
ejpam-4302	83	7	sp)-closed	sp)-close	VERB
ejpam-4302	83	8	sets	set	NOUN
ejpam-4302	83	9	we	we	PRON
ejpam-4302	83	10	begin	begin	VERB
ejpam-4302	83	11	this	this	DET
ejpam-4302	83	12	section	section	NOUN
ejpam-4302	83	13	by	by	ADP
ejpam-4302	83	14	introducing	introduce	VERB
ejpam-4302	83	15	the	the	DET
ejpam-4302	83	16	concept	concept	NOUN
ejpam-4302	83	17	of	of	ADP
ejpam-4302	83	18	generalized	generalized	ADJ
ejpam-4302	83	19	(	(	PUNCT
ejpam-4302	83	20	λ	λ	PROPN
ejpam-4302	83	21	,	,	PUNCT
ejpam-4302	83	22	sp)-closed	sp)-close	VERB
ejpam-4302	83	23	sets	set	NOUN
ejpam-4302	83	24	.	.	PUNCT
ejpam-4302	84	1	definition	definition	NOUN
ejpam-4302	84	2	1	1	NUM
ejpam-4302	84	3	.	.	PUNCT
ejpam-4302	85	1	a	a	DET
ejpam-4302	85	2	subset	subset	NOUN
ejpam-4302	85	3	a	a	PRON
ejpam-4302	85	4	of	of	ADP
ejpam-4302	85	5	a	a	DET
ejpam-4302	85	6	topological	topological	ADJ
ejpam-4302	85	7	space	space	NOUN
ejpam-4302	85	8	(	(	PUNCT
ejpam-4302	85	9	x	x	X
ejpam-4302	85	10	,	,	PUNCT
ejpam-4302	85	11	τ	τ	X
ejpam-4302	85	12	)	)	PUNCT
ejpam-4302	85	13	is	be	AUX
ejpam-4302	85	14	said	say	VERB
ejpam-4302	85	15	to	to	PART
ejpam-4302	85	16	be	be	AUX
ejpam-4302	85	17	generalized	generalize	VERB
ejpam-4302	85	18	(	(	PUNCT
ejpam-4302	85	19	λ	λ	X
ejpam-4302	85	20	,	,	PUNCT
ejpam-4302	85	21	sp)closed	sp)close	VERB
ejpam-4302	85	22	(	(	PUNCT
ejpam-4302	85	23	briefly	briefly	ADV
ejpam-4302	85	24	g-(λ	g-(λ	PROPN
ejpam-4302	85	25	,	,	PUNCT
ejpam-4302	85	26	sp)-closed	sp)-close	VERB
ejpam-4302	85	27	)	)	PUNCT
ejpam-4302	85	28	if	if	SCONJ
ejpam-4302	85	29	a(λ	a(λ	ADV
ejpam-4302	85	30	,	,	PUNCT
ejpam-4302	85	31	sp	sp	NOUN
ejpam-4302	85	32	)	)	PUNCT
ejpam-4302	85	33	⊆	⊆	NUM
ejpam-4302	85	34	u	u	NOUN
ejpam-4302	85	35	and	and	CCONJ
ejpam-4302	85	36	u	u	NOUN
ejpam-4302	85	37	is	be	AUX
ejpam-4302	85	38	(	(	PUNCT
ejpam-4302	85	39	λ	λ	INTJ
ejpam-4302	85	40	,	,	PUNCT
ejpam-4302	85	41	sp)-open	sp)-open	ADJ
ejpam-4302	85	42	in	in	ADP
ejpam-4302	85	43	x.	x.	NOUN
ejpam-4302	85	44	definition	definition	NOUN
ejpam-4302	85	45	2	2	NUM
ejpam-4302	85	46	.	.	PUNCT
ejpam-4302	86	1	a	a	DET
ejpam-4302	86	2	topological	topological	ADJ
ejpam-4302	86	3	space	space	NOUN
ejpam-4302	86	4	(	(	PUNCT
ejpam-4302	86	5	x	x	X
ejpam-4302	86	6	,	,	PUNCT
ejpam-4302	86	7	τ	τ	X
ejpam-4302	86	8	)	)	PUNCT
ejpam-4302	86	9	is	be	AUX
ejpam-4302	86	10	called	call	VERB
ejpam-4302	86	11	λsp	λsp	INTJ
ejpam-4302	86	12	-	-	NOUN
ejpam-4302	86	13	symmetric	symmetric	ADJ
ejpam-4302	86	14	if	if	SCONJ
ejpam-4302	86	15	,	,	PUNCT
ejpam-4302	86	16	for	for	ADP
ejpam-4302	86	17	each	each	DET
ejpam-4302	86	18	x	x	X
ejpam-4302	86	19	and	and	CCONJ
ejpam-4302	86	20	y	y	PROPN
ejpam-4302	86	21	in	in	ADP
ejpam-4302	86	22	x	x	PRON
ejpam-4302	86	23	,	,	PUNCT
ejpam-4302	86	24	x	x	SYM
ejpam-4302	86	25	∈	∈	PROPN
ejpam-4302	86	26	{	{	PUNCT
ejpam-4302	86	27	y}(λ	y}(λ	PROPN
ejpam-4302	86	28	,	,	PUNCT
ejpam-4302	86	29	sp	sp	NOUN
ejpam-4302	86	30	)	)	PUNCT
ejpam-4302	86	31	implies	imply	VERB
ejpam-4302	86	32	y	y	PROPN
ejpam-4302	86	33	∈	∈	PROPN
ejpam-4302	86	34	{	{	PUNCT
ejpam-4302	86	35	x}(λ	x}(λ	PROPN
ejpam-4302	86	36	,	,	PUNCT
ejpam-4302	86	37	sp	sp	NOUN
ejpam-4302	86	38	)	)	PUNCT
ejpam-4302	86	39	.	.	PUNCT
ejpam-4302	87	1	c.	c.	PROPN
ejpam-4302	87	2	boonpok	boonpok	PROPN
ejpam-4302	87	3	,	,	PUNCT
ejpam-4302	87	4	c.	c.	PROPN
ejpam-4302	87	5	viriyapong	viriyapong	PROPN
ejpam-4302	87	6	/	/	SYM
ejpam-4302	87	7	eur	eur	PROPN
ejpam-4302	87	8	.	.	PUNCT
ejpam-4302	88	1	j.	j.	PROPN
ejpam-4302	88	2	pure	pure	PROPN
ejpam-4302	88	3	appl	appl	PROPN
ejpam-4302	88	4	.	.	PROPN
ejpam-4302	88	5	math	math	PROPN
ejpam-4302	88	6	,	,	PUNCT
ejpam-4302	88	7	15	15	NUM
ejpam-4302	88	8	(	(	PUNCT
ejpam-4302	88	9	4	4	NUM
ejpam-4302	88	10	)	)	PUNCT
ejpam-4302	88	11	(	(	PUNCT
ejpam-4302	88	12	2022	2022	NUM
ejpam-4302	88	13	)	)	PUNCT
ejpam-4302	88	14	,	,	PUNCT
ejpam-4302	88	15	2127	2127	NUM
ejpam-4302	88	16	-	-	SYM
ejpam-4302	88	17	2140	2140	NUM
ejpam-4302	88	18	2130	2130	NUM
ejpam-4302	88	19	theorem	theorem	VERB
ejpam-4302	88	20	1	1	NUM
ejpam-4302	88	21	.	.	PUNCT
ejpam-4302	89	1	a	a	DET
ejpam-4302	89	2	topological	topological	ADJ
ejpam-4302	89	3	space	space	NOUN
ejpam-4302	89	4	(	(	PUNCT
ejpam-4302	89	5	x	x	X
ejpam-4302	89	6	,	,	PUNCT
ejpam-4302	89	7	τ	τ	X
ejpam-4302	89	8	)	)	PUNCT
ejpam-4302	89	9	is	be	AUX
ejpam-4302	89	10	λsp	λsp	NOUN
ejpam-4302	89	11	-	-	PUNCT
ejpam-4302	89	12	symmetric	symmetric	ADJ
ejpam-4302	89	13	if	if	SCONJ
ejpam-4302	89	14	and	and	CCONJ
ejpam-4302	89	15	only	only	ADV
ejpam-4302	89	16	if	if	SCONJ
ejpam-4302	89	17	{	{	PUNCT
ejpam-4302	89	18	x	x	NOUN
ejpam-4302	89	19	}	}	PUNCT
ejpam-4302	89	20	is	be	AUX
ejpam-4302	89	21	g-(λ	g-(λ	PROPN
ejpam-4302	89	22	,	,	PUNCT
ejpam-4302	89	23	sp)closed	sp)close	VERB
ejpam-4302	89	24	for	for	ADP
ejpam-4302	89	25	each	each	DET
ejpam-4302	89	26	x	x	SYM
ejpam-4302	89	27	∈	∈	PROPN
ejpam-4302	89	28	x.	x.	NOUN
ejpam-4302	89	29	proof	proof	NOUN
ejpam-4302	89	30	.	.	PUNCT
ejpam-4302	90	1	assume	assume	VERB
ejpam-4302	90	2	that	that	SCONJ
ejpam-4302	90	3	x	x	SYM
ejpam-4302	90	4	∈	∈	PROPN
ejpam-4302	90	5	{	{	PUNCT
ejpam-4302	90	6	y}(λ	y}(λ	PROPN
ejpam-4302	90	7	,	,	PUNCT
ejpam-4302	90	8	sp	sp	NOUN
ejpam-4302	90	9	)	)	PUNCT
ejpam-4302	90	10	,	,	PUNCT
ejpam-4302	90	11	but	but	CCONJ
ejpam-4302	90	12	y	y	PROPN
ejpam-4302	90	13	̸∈	̸∈	PROPN
ejpam-4302	90	14	{	{	PUNCT
ejpam-4302	90	15	x}(λ	x}(λ	PROPN
ejpam-4302	90	16	,	,	PUNCT
ejpam-4302	90	17	sp	sp	NOUN
ejpam-4302	90	18	)	)	PUNCT
ejpam-4302	90	19	.	.	PUNCT
ejpam-4302	91	1	this	this	PRON
ejpam-4302	91	2	implies	imply	VERB
ejpam-4302	91	3	that	that	SCONJ
ejpam-4302	91	4	the	the	DET
ejpam-4302	91	5	complement	complement	NOUN
ejpam-4302	91	6	of	of	ADP
ejpam-4302	91	7	{	{	PUNCT
ejpam-4302	91	8	x}(λ	x}(λ	PROPN
ejpam-4302	91	9	,	,	PUNCT
ejpam-4302	91	10	sp	sp	NOUN
ejpam-4302	91	11	)	)	PUNCT
ejpam-4302	91	12	contains	contain	VERB
ejpam-4302	91	13	y.	y.	PROPN
ejpam-4302	91	14	therefore	therefore	ADV
ejpam-4302	91	15	,	,	PUNCT
ejpam-4302	91	16	the	the	DET
ejpam-4302	91	17	set	set	NOUN
ejpam-4302	91	18	{	{	PUNCT
ejpam-4302	91	19	y	y	NOUN
ejpam-4302	91	20	}	}	PUNCT
ejpam-4302	91	21	is	be	AUX
ejpam-4302	91	22	a	a	DET
ejpam-4302	91	23	subset	subset	NOUN
ejpam-4302	91	24	of	of	ADP
ejpam-4302	91	25	the	the	DET
ejpam-4302	91	26	complement	complement	NOUN
ejpam-4302	91	27	of	of	ADP
ejpam-4302	91	28	{	{	PUNCT
ejpam-4302	91	29	x}(λ	x}(λ	PROPN
ejpam-4302	91	30	,	,	PUNCT
ejpam-4302	91	31	sp	sp	NOUN
ejpam-4302	91	32	)	)	PUNCT
ejpam-4302	91	33	.	.	PUNCT
ejpam-4302	92	1	this	this	PRON
ejpam-4302	92	2	implies	imply	VERB
ejpam-4302	92	3	that	that	SCONJ
ejpam-4302	92	4	{	{	PUNCT
ejpam-4302	92	5	y}(λ	y}(λ	PROPN
ejpam-4302	92	6	,	,	PUNCT
ejpam-4302	92	7	sp	sp	NOUN
ejpam-4302	92	8	)	)	PUNCT
ejpam-4302	92	9	is	be	AUX
ejpam-4302	92	10	a	a	DET
ejpam-4302	92	11	subset	subset	NOUN
ejpam-4302	92	12	of	of	ADP
ejpam-4302	92	13	the	the	DET
ejpam-4302	92	14	complement	complement	NOUN
ejpam-4302	92	15	of	of	ADP
ejpam-4302	92	16	{	{	PUNCT
ejpam-4302	92	17	x}(λ	x}(λ	PROPN
ejpam-4302	92	18	,	,	PUNCT
ejpam-4302	92	19	sp	sp	NOUN
ejpam-4302	92	20	)	)	PUNCT
ejpam-4302	92	21	.	.	PUNCT
ejpam-4302	93	1	now	now	ADV
ejpam-4302	93	2	,	,	PUNCT
ejpam-4302	93	3	the	the	DET
ejpam-4302	93	4	complement	complement	NOUN
ejpam-4302	93	5	of	of	ADP
ejpam-4302	93	6	{	{	PUNCT
ejpam-4302	93	7	x}(λ	x}(λ	PROPN
ejpam-4302	93	8	,	,	PUNCT
ejpam-4302	93	9	sp	sp	NOUN
ejpam-4302	93	10	)	)	PUNCT
ejpam-4302	93	11	contains	contain	VERB
ejpam-4302	93	12	x	x	PUNCT
ejpam-4302	93	13	which	which	PRON
ejpam-4302	93	14	is	be	AUX
ejpam-4302	93	15	a	a	DET
ejpam-4302	93	16	contradiction	contradiction	NOUN
ejpam-4302	93	17	.	.	PUNCT
ejpam-4302	94	1	conversely	conversely	ADV
ejpam-4302	94	2	,	,	PUNCT
ejpam-4302	94	3	suppose	suppose	VERB
ejpam-4302	94	4	that	that	SCONJ
ejpam-4302	94	5	{	{	PUNCT
ejpam-4302	94	6	x	x	NOUN
ejpam-4302	94	7	}	}	PUNCT
ejpam-4302	94	8	⊆	⊆	NUM
ejpam-4302	94	9	v	v	ADP
ejpam-4302	94	10	∈	∈	PROPN
ejpam-4302	94	11	λspo(x	λspo(x	NOUN
ejpam-4302	94	12	,	,	PUNCT
ejpam-4302	94	13	τ	τ	PROPN
ejpam-4302	94	14	)	)	PUNCT
ejpam-4302	94	15	,	,	PUNCT
ejpam-4302	94	16	but	but	CCONJ
ejpam-4302	94	17	{	{	PUNCT
ejpam-4302	94	18	x}(λ	x}(λ	PROPN
ejpam-4302	94	19	,	,	PUNCT
ejpam-4302	94	20	sp	sp	NOUN
ejpam-4302	94	21	)	)	PUNCT
ejpam-4302	94	22	is	be	AUX
ejpam-4302	94	23	not	not	PART
ejpam-4302	94	24	a	a	DET
ejpam-4302	94	25	subset	subset	NOUN
ejpam-4302	94	26	of	of	ADP
ejpam-4302	94	27	v	v	NOUN
ejpam-4302	94	28	.	.	PUNCT
ejpam-4302	95	1	this	this	PRON
ejpam-4302	95	2	means	mean	VERB
ejpam-4302	95	3	that	that	SCONJ
ejpam-4302	95	4	{	{	PUNCT
ejpam-4302	95	5	x}(λ	x}(λ	PROPN
ejpam-4302	95	6	,	,	PUNCT
ejpam-4302	95	7	sp	sp	NOUN
ejpam-4302	95	8	)	)	PUNCT
ejpam-4302	95	9	and	and	CCONJ
ejpam-4302	95	10	the	the	DET
ejpam-4302	95	11	complement	complement	NOUN
ejpam-4302	95	12	of	of	ADP
ejpam-4302	95	13	v	v	NUM
ejpam-4302	95	14	are	be	AUX
ejpam-4302	95	15	not	not	PART
ejpam-4302	95	16	disjoint	disjoint	ADJ
ejpam-4302	95	17	.	.	PUNCT
ejpam-4302	96	1	let	let	VERB
ejpam-4302	96	2	y	y	PRON
ejpam-4302	96	3	belongs	belong	VERB
ejpam-4302	96	4	to	to	ADP
ejpam-4302	96	5	their	their	PRON
ejpam-4302	96	6	intersection	intersection	NOUN
ejpam-4302	96	7	.	.	PUNCT
ejpam-4302	97	1	now	now	ADV
ejpam-4302	97	2	,	,	PUNCT
ejpam-4302	97	3	we	we	PRON
ejpam-4302	97	4	have	have	VERB
ejpam-4302	97	5	x	x	PART
ejpam-4302	97	6	∈	∈	PROPN
ejpam-4302	97	7	{	{	PUNCT
ejpam-4302	97	8	y}(λ	y}(λ	PROPN
ejpam-4302	97	9	,	,	PUNCT
ejpam-4302	97	10	sp	sp	PROPN
ejpam-4302	97	11	)	)	PUNCT
ejpam-4302	97	12	which	which	PRON
ejpam-4302	97	13	is	be	AUX
ejpam-4302	97	14	a	a	DET
ejpam-4302	97	15	subset	subset	NOUN
ejpam-4302	97	16	of	of	ADP
ejpam-4302	97	17	the	the	DET
ejpam-4302	97	18	complement	complement	NOUN
ejpam-4302	97	19	of	of	ADP
ejpam-4302	97	20	v	v	NOUN
ejpam-4302	97	21	and	and	CCONJ
ejpam-4302	97	22	x	x	PART
ejpam-4302	97	23	̸∈	̸∈	PROPN
ejpam-4302	97	24	v	v	PROPN
ejpam-4302	97	25	.	.	PUNCT
ejpam-4302	98	1	this	this	PRON
ejpam-4302	98	2	is	be	AUX
ejpam-4302	98	3	a	a	DET
ejpam-4302	98	4	contradiction	contradiction	NOUN
ejpam-4302	98	5	.	.	PUNCT
ejpam-4302	99	1	theorem	theorem	NOUN
ejpam-4302	99	2	2	2	NUM
ejpam-4302	99	3	.	.	PUNCT
ejpam-4302	99	4	a	a	DET
ejpam-4302	99	5	subset	subset	NOUN
ejpam-4302	99	6	a	a	PRON
ejpam-4302	99	7	of	of	ADP
ejpam-4302	99	8	a	a	DET
ejpam-4302	99	9	topological	topological	ADJ
ejpam-4302	99	10	space	space	NOUN
ejpam-4302	99	11	(	(	PUNCT
ejpam-4302	99	12	x	x	X
ejpam-4302	99	13	,	,	PUNCT
ejpam-4302	99	14	τ	τ	X
ejpam-4302	99	15	)	)	PUNCT
ejpam-4302	99	16	is	be	AUX
ejpam-4302	99	17	g-(λ	g-(λ	PROPN
ejpam-4302	99	18	,	,	PUNCT
ejpam-4302	99	19	sp)-closed	sp)-close	VERB
ejpam-4302	99	20	if	if	SCONJ
ejpam-4302	99	21	and	and	CCONJ
ejpam-4302	99	22	only	only	ADV
ejpam-4302	99	23	if	if	SCONJ
ejpam-4302	99	24	a(λ	a(λ	ADV
ejpam-4302	99	25	,	,	PUNCT
ejpam-4302	99	26	sp	sp	NOUN
ejpam-4302	99	27	)	)	PUNCT
ejpam-4302	99	28	−a	−a	NOUN
ejpam-4302	99	29	contains	contain	VERB
ejpam-4302	99	30	no	no	DET
ejpam-4302	99	31	nonempty	nonempty	ADJ
ejpam-4302	99	32	(	(	PUNCT
ejpam-4302	99	33	λ	λ	NOUN
ejpam-4302	99	34	,	,	PUNCT
ejpam-4302	99	35	sp)-closed	sp)-close	VERB
ejpam-4302	99	36	set	set	ADJ
ejpam-4302	99	37	.	.	PUNCT
ejpam-4302	100	1	proof	proof	NOUN
ejpam-4302	100	2	.	.	PUNCT
ejpam-4302	101	1	let	let	VERB
ejpam-4302	101	2	f	f	PRON
ejpam-4302	101	3	be	be	AUX
ejpam-4302	101	4	a	a	DET
ejpam-4302	101	5	(	(	PUNCT
ejpam-4302	101	6	λ	λ	PROPN
ejpam-4302	101	7	,	,	PUNCT
ejpam-4302	101	8	sp)-closed	sp)-close	VERB
ejpam-4302	101	9	subset	subset	NOUN
ejpam-4302	101	10	of	of	ADP
ejpam-4302	101	11	a(λ	a(λ	PROPN
ejpam-4302	101	12	,	,	PUNCT
ejpam-4302	101	13	sp	sp	NOUN
ejpam-4302	101	14	)	)	PUNCT
ejpam-4302	101	15	−	−	NOUN
ejpam-4302	101	16	a.	a.	NOUN
ejpam-4302	101	17	since	since	SCONJ
ejpam-4302	101	18	a	a	DET
ejpam-4302	101	19	⊆	⊆	NUM
ejpam-4302	101	20	x	x	SYM
ejpam-4302	101	21	−	−	PROPN
ejpam-4302	101	22	f	f	PROPN
ejpam-4302	101	23	and	and	CCONJ
ejpam-4302	101	24	a	a	PRON
ejpam-4302	101	25	is	be	AUX
ejpam-4302	101	26	g-(λ	g-(λ	PRON
ejpam-4302	101	27	,	,	PUNCT
ejpam-4302	101	28	sp)-closed	sp)-close	VERB
ejpam-4302	101	29	,	,	PUNCT
ejpam-4302	101	30	a(λ	a(λ	ADV
ejpam-4302	101	31	,	,	PUNCT
ejpam-4302	101	32	sp	sp	NOUN
ejpam-4302	101	33	)	)	PUNCT
ejpam-4302	101	34	⊆	⊆	NUM
ejpam-4302	101	35	x	x	SYM
ejpam-4302	101	36	−	−	PROPN
ejpam-4302	101	37	f	f	NOUN
ejpam-4302	101	38	and	and	CCONJ
ejpam-4302	101	39	hence	hence	ADV
ejpam-4302	101	40	f	f	PROPN
ejpam-4302	101	41	⊆	⊆	NUM
ejpam-4302	101	42	x	x	SYM
ejpam-4302	101	43	−a(λ	−a(λ	NOUN
ejpam-4302	101	44	,	,	PUNCT
ejpam-4302	101	45	sp	sp	NOUN
ejpam-4302	101	46	)	)	PUNCT
ejpam-4302	101	47	.	.	PUNCT
ejpam-4302	102	1	thus	thus	ADV
ejpam-4302	102	2	,	,	PUNCT
ejpam-4302	102	3	f	f	PROPN
ejpam-4302	102	4	⊆	⊆	NUM
ejpam-4302	102	5	a(λ	a(λ	ADV
ejpam-4302	102	6	,	,	PUNCT
ejpam-4302	102	7	sp	sp	NOUN
ejpam-4302	102	8	)	)	PUNCT
ejpam-4302	102	9	∩	∩	NOUN
ejpam-4302	102	10	[	[	X
ejpam-4302	102	11	x	x	X
ejpam-4302	102	12	−a(λ	−a(λ	NOUN
ejpam-4302	102	13	,	,	PUNCT
ejpam-4302	102	14	sp	sp	NOUN
ejpam-4302	102	15	)	)	PUNCT
ejpam-4302	102	16	]	]	PUNCT
ejpam-4302	102	17	=	=	PUNCT
ejpam-4302	102	18	∅	∅	NOUN
ejpam-4302	102	19	and	and	CCONJ
ejpam-4302	102	20	f	f	PROPN
ejpam-4302	102	21	is	be	AUX
ejpam-4302	102	22	empty	empty	ADJ
ejpam-4302	102	23	.	.	PUNCT
ejpam-4302	103	1	conversely	conversely	ADV
ejpam-4302	103	2	,	,	PUNCT
ejpam-4302	103	3	suppose	suppose	VERB
ejpam-4302	103	4	that	that	SCONJ
ejpam-4302	103	5	a	a	DET
ejpam-4302	103	6	⊆	⊆	NUM
ejpam-4302	103	7	u	u	NOUN
ejpam-4302	103	8	and	and	CCONJ
ejpam-4302	103	9	u	u	NOUN
ejpam-4302	103	10	is	be	AUX
ejpam-4302	103	11	(	(	PUNCT
ejpam-4302	103	12	λ	λ	NOUN
ejpam-4302	103	13	,	,	PUNCT
ejpam-4302	103	14	sp)-open	sp)-open	NOUN
ejpam-4302	103	15	.	.	PUNCT
ejpam-4302	104	1	if	if	SCONJ
ejpam-4302	104	2	a(λ	a(λ	ADV
ejpam-4302	104	3	,	,	PUNCT
ejpam-4302	104	4	sp	sp	NOUN
ejpam-4302	104	5	)	)	PUNCT
ejpam-4302	104	6	⊈	⊈	PROPN
ejpam-4302	104	7	u	u	NOUN
ejpam-4302	104	8	,	,	PUNCT
ejpam-4302	104	9	then	then	ADV
ejpam-4302	104	10	a(λ	a(λ	ADV
ejpam-4302	104	11	,	,	PUNCT
ejpam-4302	104	12	sp	sp	NOUN
ejpam-4302	104	13	)	)	PUNCT
ejpam-4302	104	14	∩	∩	NOUN
ejpam-4302	104	15	(	(	PUNCT
ejpam-4302	104	16	x	x	SYM
ejpam-4302	104	17	−	−	PROPN
ejpam-4302	104	18	u	u	NOUN
ejpam-4302	104	19	)	)	PUNCT
ejpam-4302	104	20	is	be	AUX
ejpam-4302	104	21	a	a	DET
ejpam-4302	104	22	nonempty	nonempty	ADJ
ejpam-4302	104	23	(	(	PUNCT
ejpam-4302	104	24	λ	λ	NOUN
ejpam-4302	104	25	,	,	PUNCT
ejpam-4302	104	26	sp)-closed	sp)-close	VERB
ejpam-4302	104	27	subset	subset	NOUN
ejpam-4302	104	28	of	of	ADP
ejpam-4302	104	29	a(λ	a(λ	PROPN
ejpam-4302	104	30	,	,	PUNCT
ejpam-4302	104	31	sp	sp	NOUN
ejpam-4302	104	32	)	)	PUNCT
ejpam-4302	104	33	−a	−a	NOUN
ejpam-4302	104	34	.	.	PUNCT
ejpam-4302	105	1	corollary	corollary	ADJ
ejpam-4302	105	2	1	1	NUM
ejpam-4302	105	3	.	.	PUNCT
ejpam-4302	106	1	let	let	VERB
ejpam-4302	106	2	a	a	PRON
ejpam-4302	106	3	be	be	AUX
ejpam-4302	106	4	a	a	DET
ejpam-4302	106	5	g-(λ	g-(λ	NOUN
ejpam-4302	106	6	,	,	PUNCT
ejpam-4302	106	7	sp)-closed	sp)-close	VERB
ejpam-4302	106	8	subset	subset	NOUN
ejpam-4302	106	9	of	of	ADP
ejpam-4302	106	10	a	a	DET
ejpam-4302	106	11	topological	topological	ADJ
ejpam-4302	106	12	space	space	NOUN
ejpam-4302	106	13	(	(	PUNCT
ejpam-4302	106	14	x	x	X
ejpam-4302	106	15	,	,	PUNCT
ejpam-4302	106	16	τ	τ	PROPN
ejpam-4302	106	17	)	)	PUNCT
ejpam-4302	106	18	.	.	PUNCT
ejpam-4302	107	1	then	then	ADV
ejpam-4302	107	2	,	,	PUNCT
ejpam-4302	107	3	a	a	DET
ejpam-4302	107	4	is	be	AUX
ejpam-4302	107	5	(	(	PUNCT
ejpam-4302	107	6	λ	λ	X
ejpam-4302	107	7	,	,	PUNCT
ejpam-4302	107	8	sp)-closed	sp)-close	VERB
ejpam-4302	107	9	if	if	SCONJ
ejpam-4302	107	10	and	and	CCONJ
ejpam-4302	107	11	only	only	ADV
ejpam-4302	107	12	if	if	SCONJ
ejpam-4302	107	13	a(λ	a(λ	ADV
ejpam-4302	107	14	,	,	PUNCT
ejpam-4302	107	15	sp	sp	NOUN
ejpam-4302	107	16	)	)	PUNCT
ejpam-4302	107	17	−a	−a	NOUN
ejpam-4302	107	18	is	be	AUX
ejpam-4302	107	19	(	(	PUNCT
ejpam-4302	107	20	λ	λ	X
ejpam-4302	107	21	,	,	PUNCT
ejpam-4302	107	22	sp)-closed	sp)-closed	ADJ
ejpam-4302	107	23	.	.	PUNCT
ejpam-4302	108	1	proof	proof	NOUN
ejpam-4302	108	2	.	.	PUNCT
ejpam-4302	109	1	if	if	SCONJ
ejpam-4302	109	2	a	a	PRON
ejpam-4302	109	3	is	be	AUX
ejpam-4302	109	4	a	a	DET
ejpam-4302	109	5	(	(	PUNCT
ejpam-4302	109	6	λ	λ	PROPN
ejpam-4302	109	7	,	,	PUNCT
ejpam-4302	109	8	sp)-closed	sp)-close	VERB
ejpam-4302	109	9	set	set	ADJ
ejpam-4302	109	10	,	,	PUNCT
ejpam-4302	109	11	then	then	ADV
ejpam-4302	109	12	a(λ	a(λ	ADV
ejpam-4302	109	13	,	,	PUNCT
ejpam-4302	109	14	sp	sp	NOUN
ejpam-4302	109	15	)	)	PUNCT
ejpam-4302	109	16	−a	−a	NOUN
ejpam-4302	109	17	=	=	PUNCT
ejpam-4302	109	18	∅.	∅.	VERB
ejpam-4302	109	19	conversely	conversely	ADV
ejpam-4302	109	20	,	,	PUNCT
ejpam-4302	109	21	suppose	suppose	VERB
ejpam-4302	109	22	that	that	SCONJ
ejpam-4302	109	23	a(λ	a(λ	ADV
ejpam-4302	109	24	,	,	PUNCT
ejpam-4302	109	25	sp	sp	NOUN
ejpam-4302	109	26	)	)	PUNCT
ejpam-4302	109	27	−	−	NOUN
ejpam-4302	109	28	a	a	PRON
ejpam-4302	109	29	is	be	AUX
ejpam-4302	109	30	(	(	PUNCT
ejpam-4302	109	31	λ	λ	X
ejpam-4302	109	32	,	,	PUNCT
ejpam-4302	109	33	sp)-closed	sp)-close	VERB
ejpam-4302	109	34	.	.	PUNCT
ejpam-4302	110	1	since	since	SCONJ
ejpam-4302	110	2	a	a	PRON
ejpam-4302	110	3	is	be	AUX
ejpam-4302	110	4	g-(λ	g-(λ	PRON
ejpam-4302	110	5	,	,	PUNCT
ejpam-4302	110	6	sp)-closed	sp)-closed	ADJ
ejpam-4302	110	7	and	and	CCONJ
ejpam-4302	110	8	a(λ	a(λ	ADV
ejpam-4302	110	9	,	,	PUNCT
ejpam-4302	110	10	sp	sp	NOUN
ejpam-4302	110	11	)	)	PUNCT
ejpam-4302	110	12	−	−	NOUN
ejpam-4302	110	13	a	a	PRON
ejpam-4302	110	14	is	be	AUX
ejpam-4302	110	15	a	a	DET
ejpam-4302	110	16	(	(	PUNCT
ejpam-4302	110	17	λ	λ	PROPN
ejpam-4302	110	18	,	,	PUNCT
ejpam-4302	110	19	sp)-closed	sp)-close	VERB
ejpam-4302	110	20	subset	subset	NOUN
ejpam-4302	110	21	of	of	ADP
ejpam-4302	110	22	itself	itself	PRON
ejpam-4302	110	23	,	,	PUNCT
ejpam-4302	110	24	by	by	ADP
ejpam-4302	110	25	theorem	theorem	NOUN
ejpam-4302	110	26	2	2	NUM
ejpam-4302	110	27	,	,	PUNCT
ejpam-4302	110	28	a(λ	a(λ	ADV
ejpam-4302	110	29	,	,	PUNCT
ejpam-4302	110	30	sp	sp	NOUN
ejpam-4302	110	31	)	)	PUNCT
ejpam-4302	110	32	−	−	NOUN
ejpam-4302	110	33	a	a	DET
ejpam-4302	110	34	=	=	NOUN
ejpam-4302	110	35	∅	∅	NOUN
ejpam-4302	110	36	and	and	CCONJ
ejpam-4302	110	37	hence	hence	ADV
ejpam-4302	110	38	a(λ	a(λ	ADV
ejpam-4302	110	39	,	,	PUNCT
ejpam-4302	110	40	sp	sp	NOUN
ejpam-4302	110	41	)	)	PUNCT
ejpam-4302	110	42	=	=	NOUN
ejpam-4302	110	43	a.	a.	NOUN
ejpam-4302	110	44	theorem	theorem	NOUN
ejpam-4302	110	45	3	3	NUM
ejpam-4302	110	46	.	.	X
ejpam-4302	110	47	for	for	ADP
ejpam-4302	110	48	a	a	DET
ejpam-4302	110	49	subset	subset	NOUN
ejpam-4302	110	50	a	a	PRON
ejpam-4302	110	51	of	of	ADP
ejpam-4302	110	52	a	a	DET
ejpam-4302	110	53	topological	topological	ADJ
ejpam-4302	110	54	space	space	NOUN
ejpam-4302	110	55	(	(	PUNCT
ejpam-4302	110	56	x	x	X
ejpam-4302	110	57	,	,	PUNCT
ejpam-4302	110	58	τ	τ	PROPN
ejpam-4302	110	59	)	)	PUNCT
ejpam-4302	110	60	,	,	PUNCT
ejpam-4302	110	61	the	the	DET
ejpam-4302	110	62	following	follow	VERB
ejpam-4302	110	63	properties	property	NOUN
ejpam-4302	110	64	hold	hold	VERB
ejpam-4302	110	65	:	:	PUNCT
ejpam-4302	110	66	(	(	PUNCT
ejpam-4302	110	67	1	1	X
ejpam-4302	110	68	)	)	PUNCT
ejpam-4302	110	69	if	if	SCONJ
ejpam-4302	110	70	a	a	PRON
ejpam-4302	110	71	is	be	AUX
ejpam-4302	110	72	(	(	PUNCT
ejpam-4302	110	73	λ	λ	NOUN
ejpam-4302	110	74	,	,	PUNCT
ejpam-4302	110	75	sp)-closed	sp)-close	VERB
ejpam-4302	110	76	,	,	PUNCT
ejpam-4302	110	77	then	then	ADV
ejpam-4302	110	78	a	a	PRON
ejpam-4302	110	79	is	be	AUX
ejpam-4302	110	80	g-(λ	g-(λ	PRON
ejpam-4302	110	81	,	,	PUNCT
ejpam-4302	110	82	sp)-closed	sp)-close	VERB
ejpam-4302	110	83	.	.	PUNCT
ejpam-4302	111	1	(	(	PUNCT
ejpam-4302	111	2	2	2	X
ejpam-4302	111	3	)	)	PUNCT
ejpam-4302	111	4	if	if	SCONJ
ejpam-4302	111	5	a	a	PRON
ejpam-4302	111	6	is	be	AUX
ejpam-4302	111	7	g-(λ	g-(λ	PRON
ejpam-4302	111	8	,	,	PUNCT
ejpam-4302	111	9	sp)-closed	sp)-close	VERB
ejpam-4302	111	10	and	and	CCONJ
ejpam-4302	111	11	(	(	PUNCT
ejpam-4302	111	12	λ	λ	NOUN
ejpam-4302	111	13	,	,	PUNCT
ejpam-4302	111	14	sp)-open	sp)-open	ADJ
ejpam-4302	111	15	,	,	PUNCT
ejpam-4302	111	16	then	then	ADV
ejpam-4302	111	17	a	a	PRON
ejpam-4302	111	18	is	be	AUX
ejpam-4302	111	19	(	(	PUNCT
ejpam-4302	111	20	λ	λ	NOUN
ejpam-4302	111	21	,	,	PUNCT
ejpam-4302	111	22	sp)-closed	sp)-close	VERB
ejpam-4302	111	23	.	.	PUNCT
ejpam-4302	112	1	(	(	PUNCT
ejpam-4302	112	2	3	3	X
ejpam-4302	112	3	)	)	PUNCT
ejpam-4302	112	4	if	if	SCONJ
ejpam-4302	112	5	a	a	PRON
ejpam-4302	112	6	is	be	AUX
ejpam-4302	112	7	g-(λ	g-(λ	PRON
ejpam-4302	112	8	,	,	PUNCT
ejpam-4302	112	9	sp)-closed	sp)-close	VERB
ejpam-4302	112	10	and	and	CCONJ
ejpam-4302	112	11	a	a	DET
ejpam-4302	112	12	⊆	⊆	NUM
ejpam-4302	112	13	b	b	NOUN
ejpam-4302	112	14	⊆	⊆	NUM
ejpam-4302	112	15	a(λ	a(λ	ADJ
ejpam-4302	112	16	,	,	PUNCT
ejpam-4302	112	17	sp	sp	NOUN
ejpam-4302	112	18	)	)	PUNCT
ejpam-4302	112	19	,	,	PUNCT
ejpam-4302	112	20	then	then	ADV
ejpam-4302	112	21	b	b	PROPN
ejpam-4302	112	22	is	be	AUX
ejpam-4302	112	23	g-(λ	g-(λ	PRON
ejpam-4302	112	24	,	,	PUNCT
ejpam-4302	112	25	sp)-closed	sp)-closed	ADJ
ejpam-4302	112	26	.	.	PUNCT
ejpam-4302	113	1	proof	proof	NOUN
ejpam-4302	113	2	.	.	PUNCT
ejpam-4302	114	1	(	(	PUNCT
ejpam-4302	114	2	1	1	X
ejpam-4302	114	3	)	)	PUNCT
ejpam-4302	114	4	let	let	VERB
ejpam-4302	114	5	a	a	DET
ejpam-4302	114	6	be	be	AUX
ejpam-4302	114	7	(	(	PUNCT
ejpam-4302	114	8	λ	λ	X
ejpam-4302	114	9	,	,	PUNCT
ejpam-4302	114	10	sp)-closed	sp)-close	VERB
ejpam-4302	114	11	and	and	CCONJ
ejpam-4302	114	12	a	a	DET
ejpam-4302	114	13	⊆	⊆	NUM
ejpam-4302	114	14	u	u	NOUN
ejpam-4302	114	15	∈	∈	PROPN
ejpam-4302	114	16	λspo(x	λspo(x	PROPN
ejpam-4302	114	17	,	,	PUNCT
ejpam-4302	114	18	τ	τ	PROPN
ejpam-4302	114	19	)	)	PUNCT
ejpam-4302	114	20	.	.	PUNCT
ejpam-4302	115	1	then	then	ADV
ejpam-4302	115	2	,	,	PUNCT
ejpam-4302	115	3	by	by	ADP
ejpam-4302	115	4	lemma	lemma	PROPN
ejpam-4302	115	5	3	3	NUM
ejpam-4302	115	6	,	,	PUNCT
ejpam-4302	115	7	a(λ	a(λ	ADV
ejpam-4302	115	8	,	,	PUNCT
ejpam-4302	115	9	sp	sp	NOUN
ejpam-4302	115	10	)	)	PUNCT
ejpam-4302	115	11	=	=	PUNCT
ejpam-4302	115	12	a	a	DET
ejpam-4302	115	13	⊆	⊆	NUM
ejpam-4302	115	14	u	u	NOUN
ejpam-4302	115	15	and	and	CCONJ
ejpam-4302	115	16	hence	hence	ADV
ejpam-4302	115	17	a	a	PRON
ejpam-4302	115	18	is	be	AUX
ejpam-4302	115	19	g-(λ	g-(λ	PROPN
ejpam-4302	115	20	,	,	PUNCT
ejpam-4302	115	21	sp)-closed	sp)-close	VERB
ejpam-4302	115	22	.	.	PUNCT
ejpam-4302	116	1	(	(	PUNCT
ejpam-4302	116	2	2	2	X
ejpam-4302	116	3	)	)	PUNCT
ejpam-4302	116	4	let	let	VERB
ejpam-4302	116	5	a	a	DET
ejpam-4302	116	6	be	be	AUX
ejpam-4302	116	7	g-(λ	g-(λ	PROPN
ejpam-4302	116	8	,	,	PUNCT
ejpam-4302	116	9	sp)-closed	sp)-close	VERB
ejpam-4302	116	10	and	and	CCONJ
ejpam-4302	116	11	(	(	PUNCT
ejpam-4302	116	12	λ	λ	NOUN
ejpam-4302	116	13	,	,	PUNCT
ejpam-4302	116	14	sp)-open	sp)-open	NOUN
ejpam-4302	116	15	.	.	PUNCT
ejpam-4302	117	1	then	then	ADV
ejpam-4302	117	2	,	,	PUNCT
ejpam-4302	117	3	a(λ	a(λ	ADV
ejpam-4302	117	4	,	,	PUNCT
ejpam-4302	117	5	sp	sp	NOUN
ejpam-4302	117	6	)	)	PUNCT
ejpam-4302	117	7	=	=	PUNCT
ejpam-4302	117	8	a	a	PRON
ejpam-4302	117	9	and	and	CCONJ
ejpam-4302	117	10	by	by	ADP
ejpam-4302	117	11	lemma	lemma	PROPN
ejpam-4302	117	12	3	3	NUM
ejpam-4302	117	13	,	,	PUNCT
ejpam-4302	117	14	a	a	DET
ejpam-4302	117	15	is	be	AUX
ejpam-4302	117	16	(	(	PUNCT
ejpam-4302	117	17	λ	λ	NOUN
ejpam-4302	117	18	,	,	PUNCT
ejpam-4302	117	19	sp)-closed	sp)-close	VERB
ejpam-4302	117	20	.	.	PUNCT
ejpam-4302	118	1	c.	c.	PROPN
ejpam-4302	118	2	boonpok	boonpok	PROPN
ejpam-4302	118	3	,	,	PUNCT
ejpam-4302	118	4	c.	c.	PROPN
ejpam-4302	118	5	viriyapong	viriyapong	PROPN
ejpam-4302	118	6	/	/	SYM
ejpam-4302	118	7	eur	eur	PROPN
ejpam-4302	118	8	.	.	PUNCT
ejpam-4302	119	1	j.	j.	PROPN
ejpam-4302	119	2	pure	pure	PROPN
ejpam-4302	119	3	appl	appl	PROPN
ejpam-4302	119	4	.	.	PROPN
ejpam-4302	119	5	math	math	PROPN
ejpam-4302	119	6	,	,	PUNCT
ejpam-4302	119	7	15	15	NUM
ejpam-4302	119	8	(	(	PUNCT
ejpam-4302	119	9	4	4	NUM
ejpam-4302	119	10	)	)	PUNCT
ejpam-4302	119	11	(	(	PUNCT
ejpam-4302	119	12	2022	2022	NUM
ejpam-4302	119	13	)	)	PUNCT
ejpam-4302	119	14	,	,	PUNCT
ejpam-4302	119	15	2127	2127	NUM
ejpam-4302	119	16	-	-	SYM
ejpam-4302	119	17	2140	2140	NUM
ejpam-4302	119	18	2131	2131	NUM
ejpam-4302	119	19	(	(	PUNCT
ejpam-4302	119	20	3	3	X
ejpam-4302	119	21	)	)	PUNCT
ejpam-4302	119	22	let	let	VERB
ejpam-4302	119	23	b	b	NOUN
ejpam-4302	119	24	⊆	⊆	NUM
ejpam-4302	119	25	u	u	NOUN
ejpam-4302	119	26	and	and	CCONJ
ejpam-4302	119	27	u	u	PROPN
ejpam-4302	119	28	∈	∈	PROPN
ejpam-4302	119	29	λspo(x	λspo(x	PROPN
ejpam-4302	119	30	,	,	PUNCT
ejpam-4302	119	31	τ	τ	PROPN
ejpam-4302	119	32	)	)	PUNCT
ejpam-4302	119	33	.	.	PUNCT
ejpam-4302	120	1	since	since	SCONJ
ejpam-4302	120	2	a	a	DET
ejpam-4302	120	3	⊆	⊆	NUM
ejpam-4302	120	4	u	u	NOUN
ejpam-4302	120	5	and	and	CCONJ
ejpam-4302	120	6	a	a	PRON
ejpam-4302	120	7	is	be	AUX
ejpam-4302	120	8	g-(λ	g-(λ	PROPN
ejpam-4302	120	9	,	,	PUNCT
ejpam-4302	120	10	sp)-closed	sp)-close	VERB
ejpam-4302	120	11	,	,	PUNCT
ejpam-4302	120	12	we	we	PRON
ejpam-4302	120	13	have	have	VERB
ejpam-4302	120	14	a(λ	a(λ	ADV
ejpam-4302	120	15	,	,	PUNCT
ejpam-4302	120	16	sp	sp	NOUN
ejpam-4302	120	17	)	)	PUNCT
ejpam-4302	120	18	⊆	⊆	NUM
ejpam-4302	120	19	u	u	NOUN
ejpam-4302	120	20	.	.	PUNCT
ejpam-4302	121	1	since	since	SCONJ
ejpam-4302	121	2	a	a	DET
ejpam-4302	121	3	⊆	⊆	NUM
ejpam-4302	121	4	b	b	NOUN
ejpam-4302	121	5	⊆	⊆	NUM
ejpam-4302	121	6	a(λ	a(λ	ADJ
ejpam-4302	121	7	,	,	PUNCT
ejpam-4302	121	8	sp	sp	NOUN
ejpam-4302	121	9	)	)	PUNCT
ejpam-4302	121	10	,	,	PUNCT
ejpam-4302	121	11	by	by	ADP
ejpam-4302	121	12	lemma	lemma	PROPN
ejpam-4302	121	13	3	3	NUM
ejpam-4302	121	14	,	,	PUNCT
ejpam-4302	121	15	a(λ	a(λ	ADV
ejpam-4302	121	16	,	,	PUNCT
ejpam-4302	121	17	sp	sp	NOUN
ejpam-4302	121	18	)	)	PUNCT
ejpam-4302	121	19	=	=	SYM
ejpam-4302	121	20	b(λ	b(λ	NOUN
ejpam-4302	121	21	,	,	PUNCT
ejpam-4302	121	22	sp	sp	NOUN
ejpam-4302	121	23	)	)	PUNCT
ejpam-4302	121	24	and	and	CCONJ
ejpam-4302	121	25	hence	hence	ADV
ejpam-4302	121	26	b(λ	b(λ	PROPN
ejpam-4302	121	27	,	,	PUNCT
ejpam-4302	121	28	sp	sp	NOUN
ejpam-4302	121	29	)	)	PUNCT
ejpam-4302	121	30	⊆	⊆	NUM
ejpam-4302	121	31	u	u	NOUN
ejpam-4302	121	32	.	.	PUNCT
ejpam-4302	122	1	thus	thus	ADV
ejpam-4302	122	2	,	,	PUNCT
ejpam-4302	122	3	b	b	PROPN
ejpam-4302	122	4	is	be	AUX
ejpam-4302	122	5	g-(λ	g-(λ	PRON
ejpam-4302	122	6	,	,	PUNCT
ejpam-4302	122	7	sp)-closed	sp)-close	VERB
ejpam-4302	122	8	.	.	PUNCT
ejpam-4302	123	1	corollary	corollary	ADJ
ejpam-4302	123	2	2	2	NUM
ejpam-4302	123	3	.	.	PUNCT
ejpam-4302	124	1	for	for	ADP
ejpam-4302	124	2	a	a	DET
ejpam-4302	124	3	subset	subset	NOUN
ejpam-4302	124	4	a	a	PRON
ejpam-4302	124	5	of	of	ADP
ejpam-4302	124	6	a	a	DET
ejpam-4302	124	7	topological	topological	ADJ
ejpam-4302	124	8	space	space	NOUN
ejpam-4302	124	9	(	(	PUNCT
ejpam-4302	124	10	x	x	X
ejpam-4302	124	11	,	,	PUNCT
ejpam-4302	124	12	τ	τ	PROPN
ejpam-4302	124	13	)	)	PUNCT
ejpam-4302	124	14	,	,	PUNCT
ejpam-4302	124	15	the	the	DET
ejpam-4302	124	16	following	follow	VERB
ejpam-4302	124	17	properties	property	NOUN
ejpam-4302	124	18	hold	hold	VERB
ejpam-4302	124	19	:	:	PUNCT
ejpam-4302	124	20	(	(	PUNCT
ejpam-4302	124	21	1	1	X
ejpam-4302	124	22	)	)	PUNCT
ejpam-4302	124	23	if	if	SCONJ
ejpam-4302	124	24	a	a	PRON
ejpam-4302	124	25	is	be	AUX
ejpam-4302	124	26	(	(	PUNCT
ejpam-4302	124	27	λ	λ	NOUN
ejpam-4302	124	28	,	,	PUNCT
ejpam-4302	124	29	sp)-open	sp)-open	ADJ
ejpam-4302	124	30	,	,	PUNCT
ejpam-4302	124	31	then	then	ADV
ejpam-4302	124	32	a	a	PRON
ejpam-4302	124	33	is	be	AUX
ejpam-4302	124	34	g-(λ	g-(λ	PROPN
ejpam-4302	124	35	,	,	PUNCT
ejpam-4302	124	36	sp)-open	sp)-open	ADJ
ejpam-4302	124	37	.	.	PUNCT
ejpam-4302	125	1	(	(	PUNCT
ejpam-4302	125	2	2	2	X
ejpam-4302	125	3	)	)	PUNCT
ejpam-4302	125	4	if	if	SCONJ
ejpam-4302	125	5	a	a	PRON
ejpam-4302	125	6	is	be	AUX
ejpam-4302	125	7	g-(λ	g-(λ	PROPN
ejpam-4302	125	8	,	,	PUNCT
ejpam-4302	125	9	sp)-open	sp)-open	ADJ
ejpam-4302	125	10	and	and	CCONJ
ejpam-4302	125	11	(	(	PUNCT
ejpam-4302	125	12	λ	λ	PROPN
ejpam-4302	125	13	,	,	PUNCT
ejpam-4302	125	14	sp)-closed	sp)-close	VERB
ejpam-4302	125	15	,	,	PUNCT
ejpam-4302	125	16	then	then	ADV
ejpam-4302	125	17	a	a	PRON
ejpam-4302	125	18	is	be	AUX
ejpam-4302	125	19	(	(	PUNCT
ejpam-4302	125	20	λ	λ	NOUN
ejpam-4302	125	21	,	,	PUNCT
ejpam-4302	125	22	sp)-open	sp)-open	NOUN
ejpam-4302	125	23	.	.	PUNCT
ejpam-4302	126	1	(	(	PUNCT
ejpam-4302	126	2	3	3	X
ejpam-4302	126	3	)	)	PUNCT
ejpam-4302	126	4	if	if	SCONJ
ejpam-4302	126	5	a	a	PRON
ejpam-4302	126	6	is	be	AUX
ejpam-4302	126	7	g-(λ	g-(λ	PROPN
ejpam-4302	126	8	,	,	PUNCT
ejpam-4302	126	9	sp)-open	sp)-open	ADJ
ejpam-4302	126	10	and	and	CCONJ
ejpam-4302	126	11	a(λ	a(λ	ADV
ejpam-4302	126	12	,	,	PUNCT
ejpam-4302	126	13	sp	sp	NOUN
ejpam-4302	126	14	)	)	PUNCT
ejpam-4302	126	15	⊆	⊆	NUM
ejpam-4302	126	16	b	b	NOUN
ejpam-4302	126	17	⊆	⊆	NUM
ejpam-4302	126	18	a	a	PRON
ejpam-4302	126	19	,	,	PUNCT
ejpam-4302	126	20	then	then	ADV
ejpam-4302	126	21	b	b	PROPN
ejpam-4302	126	22	is	be	AUX
ejpam-4302	126	23	g-(λ	g-(λ	PROPN
ejpam-4302	126	24	,	,	PUNCT
ejpam-4302	126	25	sp)-open	sp)-open	ADJ
ejpam-4302	126	26	.	.	PUNCT
ejpam-4302	127	1	proof	proof	NOUN
ejpam-4302	127	2	.	.	PUNCT
ejpam-4302	128	1	this	this	PRON
ejpam-4302	128	2	follows	follow	VERB
ejpam-4302	128	3	from	from	ADP
ejpam-4302	128	4	theorem	theorem	ADJ
ejpam-4302	128	5	3	3	NUM
ejpam-4302	128	6	.	.	PUNCT
ejpam-4302	128	7	definition	definition	NOUN
ejpam-4302	128	8	3	3	X
ejpam-4302	128	9	.	.	PUNCT
ejpam-4302	129	1	let	let	VERB
ejpam-4302	129	2	a	a	DET
ejpam-4302	129	3	be	be	AUX
ejpam-4302	129	4	a	a	DET
ejpam-4302	129	5	subset	subset	NOUN
ejpam-4302	129	6	of	of	ADP
ejpam-4302	129	7	a	a	DET
ejpam-4302	129	8	topological	topological	ADJ
ejpam-4302	129	9	space	space	NOUN
ejpam-4302	129	10	(	(	PUNCT
ejpam-4302	129	11	x	x	X
ejpam-4302	129	12	,	,	PUNCT
ejpam-4302	129	13	τ	τ	PROPN
ejpam-4302	129	14	)	)	PUNCT
ejpam-4302	129	15	.	.	PUNCT
ejpam-4302	130	1	the	the	DET
ejpam-4302	130	2	(	(	PUNCT
ejpam-4302	130	3	λ	λ	NOUN
ejpam-4302	130	4	,	,	PUNCT
ejpam-4302	130	5	sp)-frontier	sp)-fronti	ADJ
ejpam-4302	130	6	of	of	ADP
ejpam-4302	130	7	a	a	DET
ejpam-4302	130	8	,	,	PUNCT
ejpam-4302	130	9	λspfr(a	λspfr(a	NOUN
ejpam-4302	130	10	)	)	PUNCT
ejpam-4302	130	11	,	,	PUNCT
ejpam-4302	130	12	is	be	AUX
ejpam-4302	130	13	defined	define	VERB
ejpam-4302	130	14	as	as	SCONJ
ejpam-4302	130	15	follows	follow	VERB
ejpam-4302	130	16	:	:	PUNCT
ejpam-4302	130	17	λspfr(a	λspfr(a	NUM
ejpam-4302	130	18	)	)	PUNCT
ejpam-4302	130	19	=	=	PUNCT
ejpam-4302	130	20	a(λ	a(λ	ADV
ejpam-4302	130	21	,	,	PUNCT
ejpam-4302	130	22	sp	sp	NOUN
ejpam-4302	130	23	)	)	PUNCT
ejpam-4302	130	24	∩	∩	NOUN
ejpam-4302	131	1	[	[	X
ejpam-4302	131	2	x	x	SYM
ejpam-4302	131	3	−a](λ	−a](λ	PROPN
ejpam-4302	131	4	,	,	PUNCT
ejpam-4302	131	5	sp	sp	NOUN
ejpam-4302	131	6	)	)	PUNCT
ejpam-4302	131	7	.	.	PUNCT
ejpam-4302	132	1	theorem	theorem	ADJ
ejpam-4302	132	2	4	4	NUM
ejpam-4302	132	3	.	.	PUNCT
ejpam-4302	133	1	let	let	VERB
ejpam-4302	133	2	a	a	DET
ejpam-4302	133	3	be	be	AUX
ejpam-4302	133	4	a	a	DET
ejpam-4302	133	5	subset	subset	NOUN
ejpam-4302	133	6	of	of	ADP
ejpam-4302	133	7	a	a	DET
ejpam-4302	133	8	topological	topological	ADJ
ejpam-4302	133	9	space	space	NOUN
ejpam-4302	133	10	(	(	PUNCT
ejpam-4302	133	11	x	x	X
ejpam-4302	133	12	,	,	PUNCT
ejpam-4302	133	13	τ	τ	PROPN
ejpam-4302	133	14	)	)	PUNCT
ejpam-4302	133	15	.	.	PUNCT
ejpam-4302	134	1	if	if	SCONJ
ejpam-4302	134	2	a	a	PRON
ejpam-4302	134	3	is	be	AUX
ejpam-4302	134	4	g-(λ	g-(λ	PRON
ejpam-4302	134	5	,	,	PUNCT
ejpam-4302	134	6	sp)-closed	sp)-close	VERB
ejpam-4302	134	7	and	and	CCONJ
ejpam-4302	134	8	a	a	DET
ejpam-4302	134	9	⊆	⊆	NUM
ejpam-4302	134	10	v	v	NOUN
ejpam-4302	134	11	∈	∈	PROPN
ejpam-4302	134	12	λspo(x	λspo(x	NOUN
ejpam-4302	134	13	,	,	PUNCT
ejpam-4302	134	14	τ	τ	PROPN
ejpam-4302	134	15	)	)	PUNCT
ejpam-4302	134	16	,	,	PUNCT
ejpam-4302	134	17	then	then	ADV
ejpam-4302	134	18	λspfr(v	λspfr(v	NOUN
ejpam-4302	134	19	)	)	PUNCT
ejpam-4302	134	20	⊆	⊆	NUM
ejpam-4302	135	1	[	[	X
ejpam-4302	135	2	x	x	SYM
ejpam-4302	135	3	−a](λ	−a](λ	PROPN
ejpam-4302	135	4	,	,	PUNCT
ejpam-4302	135	5	sp	sp	NOUN
ejpam-4302	135	6	)	)	PUNCT
ejpam-4302	135	7	.	.	PUNCT
ejpam-4302	136	1	proof	proof	NOUN
ejpam-4302	136	2	.	.	PUNCT
ejpam-4302	137	1	let	let	VERB
ejpam-4302	137	2	a	a	DET
ejpam-4302	137	3	be	be	AUX
ejpam-4302	137	4	g-(λ	g-(λ	PROPN
ejpam-4302	137	5	,	,	PUNCT
ejpam-4302	137	6	sp)-closed	sp)-close	VERB
ejpam-4302	137	7	and	and	CCONJ
ejpam-4302	137	8	a	a	DET
ejpam-4302	137	9	⊆	⊆	NUM
ejpam-4302	137	10	v	v	NOUN
ejpam-4302	137	11	∈	∈	PROPN
ejpam-4302	137	12	λspo(x	λspo(x	NOUN
ejpam-4302	137	13	,	,	PUNCT
ejpam-4302	137	14	τ	τ	PROPN
ejpam-4302	137	15	)	)	PUNCT
ejpam-4302	137	16	.	.	PUNCT
ejpam-4302	138	1	then	then	ADV
ejpam-4302	138	2	,	,	PUNCT
ejpam-4302	138	3	a(λ	a(λ	ADV
ejpam-4302	138	4	,	,	PUNCT
ejpam-4302	138	5	sp	sp	NOUN
ejpam-4302	138	6	)	)	PUNCT
ejpam-4302	138	7	⊆	⊆	NUM
ejpam-4302	138	8	v	v	NOUN
ejpam-4302	138	9	.	.	PUNCT
ejpam-4302	139	1	let	let	VERB
ejpam-4302	139	2	x	x	PUNCT
ejpam-4302	139	3	∈	∈	PROPN
ejpam-4302	139	4	λspfr(v	λspfr(v	NOUN
ejpam-4302	139	5	)	)	PUNCT
ejpam-4302	139	6	.	.	PUNCT
ejpam-4302	140	1	since	since	SCONJ
ejpam-4302	140	2	v	v	NUM
ejpam-4302	140	3	∈	∈	X
ejpam-4302	140	4	λspo(x	λspo(x	NOUN
ejpam-4302	140	5	,	,	PUNCT
ejpam-4302	140	6	τ	τ	PROPN
ejpam-4302	140	7	)	)	PUNCT
ejpam-4302	140	8	,	,	PUNCT
ejpam-4302	140	9	we	we	PRON
ejpam-4302	140	10	have	have	VERB
ejpam-4302	140	11	λspfr(v	λspfr(v	NOUN
ejpam-4302	140	12	)	)	PUNCT
ejpam-4302	140	13	=	=	SYM
ejpam-4302	140	14	v	v	X
ejpam-4302	140	15	(	(	PUNCT
ejpam-4302	140	16	λ	λ	PROPN
ejpam-4302	140	17	,	,	PUNCT
ejpam-4302	140	18	sp)−v	sp)−v	NOUN
ejpam-4302	140	19	.	.	PUNCT
ejpam-4302	141	1	thus	thus	ADV
ejpam-4302	141	2	,	,	PUNCT
ejpam-4302	141	3	x	x	PROPN
ejpam-4302	141	4	̸∈	̸∈	PROPN
ejpam-4302	141	5	v	v	NOUN
ejpam-4302	141	6	and	and	CCONJ
ejpam-4302	141	7	hence	hence	ADV
ejpam-4302	141	8	x	x	X
ejpam-4302	141	9	̸∈	̸∈	PROPN
ejpam-4302	141	10	a(λ	a(λ	PROPN
ejpam-4302	141	11	,	,	PUNCT
ejpam-4302	141	12	sp	sp	NOUN
ejpam-4302	141	13	)	)	PUNCT
ejpam-4302	141	14	.	.	PUNCT
ejpam-4302	142	1	therefore	therefore	ADV
ejpam-4302	142	2	,	,	PUNCT
ejpam-4302	142	3	x	x	PUNCT
ejpam-4302	142	4	∈	∈	PROPN
ejpam-4302	142	5	[	[	X
ejpam-4302	142	6	x−a](λ	x−a](λ	PROPN
ejpam-4302	142	7	,	,	PUNCT
ejpam-4302	142	8	sp	sp	NOUN
ejpam-4302	142	9	)	)	PUNCT
ejpam-4302	142	10	.	.	PUNCT
ejpam-4302	143	1	this	this	PRON
ejpam-4302	143	2	shows	show	VERB
ejpam-4302	143	3	that	that	SCONJ
ejpam-4302	143	4	λspfr(v	λspfr(v	NOUN
ejpam-4302	143	5	)	)	PUNCT
ejpam-4302	143	6	⊆	⊆	NUM
ejpam-4302	143	7	[	[	X
ejpam-4302	143	8	x−a](λ	x−a](λ	PROPN
ejpam-4302	143	9	,	,	PUNCT
ejpam-4302	143	10	sp	sp	NOUN
ejpam-4302	143	11	)	)	PUNCT
ejpam-4302	143	12	.	.	PUNCT
ejpam-4302	144	1	theorem	theorem	NOUN
ejpam-4302	144	2	5	5	NUM
ejpam-4302	144	3	.	.	PUNCT
ejpam-4302	145	1	let	let	VERB
ejpam-4302	145	2	(	(	PUNCT
ejpam-4302	145	3	x	x	NOUN
ejpam-4302	145	4	,	,	PUNCT
ejpam-4302	145	5	τ	τ	X
ejpam-4302	145	6	)	)	PUNCT
ejpam-4302	145	7	be	be	VERB
ejpam-4302	145	8	a	a	DET
ejpam-4302	145	9	topological	topological	ADJ
ejpam-4302	145	10	space	space	NOUN
ejpam-4302	145	11	.	.	PUNCT
ejpam-4302	146	1	for	for	SCONJ
ejpam-4302	146	2	each	each	DET
ejpam-4302	146	3	x	x	SYM
ejpam-4302	146	4	∈	∈	PROPN
ejpam-4302	146	5	x	x	NOUN
ejpam-4302	146	6	,	,	PUNCT
ejpam-4302	146	7	either	either	CCONJ
ejpam-4302	146	8	{	{	PUNCT
ejpam-4302	146	9	x	x	X
ejpam-4302	146	10	}	}	PUNCT
ejpam-4302	146	11	is	be	AUX
ejpam-4302	146	12	(	(	PUNCT
ejpam-4302	146	13	λ	λ	X
ejpam-4302	146	14	,	,	PUNCT
ejpam-4302	146	15	sp)-closed	sp)-closed	ADJ
ejpam-4302	146	16	or	or	CCONJ
ejpam-4302	146	17	g-(λ	g-(λ	PROPN
ejpam-4302	146	18	,	,	PUNCT
ejpam-4302	146	19	sp)-open	sp)-open	NOUN
ejpam-4302	146	20	.	.	PUNCT
ejpam-4302	147	1	proof	proof	NOUN
ejpam-4302	147	2	.	.	PUNCT
ejpam-4302	148	1	suppose	suppose	VERB
ejpam-4302	148	2	that	that	SCONJ
ejpam-4302	148	3	{	{	PUNCT
ejpam-4302	148	4	x	x	X
ejpam-4302	148	5	}	}	PUNCT
ejpam-4302	148	6	is	be	AUX
ejpam-4302	148	7	not	not	PART
ejpam-4302	148	8	(	(	PUNCT
ejpam-4302	148	9	λ	λ	X
ejpam-4302	148	10	,	,	PUNCT
ejpam-4302	148	11	sp)-closed	sp)-close	VERB
ejpam-4302	148	12	.	.	PUNCT
ejpam-4302	149	1	then	then	ADV
ejpam-4302	149	2	,	,	PUNCT
ejpam-4302	149	3	x	x	PUNCT
ejpam-4302	149	4	−	−	NOUN
ejpam-4302	149	5	{	{	PUNCT
ejpam-4302	149	6	x	x	NOUN
ejpam-4302	149	7	}	}	PUNCT
ejpam-4302	149	8	is	be	AUX
ejpam-4302	149	9	not	not	PART
ejpam-4302	149	10	(	(	PUNCT
ejpam-4302	149	11	λ	λ	NOUN
ejpam-4302	149	12	,	,	PUNCT
ejpam-4302	149	13	sp)-open	sp)-open	ADJ
ejpam-4302	149	14	and	and	CCONJ
ejpam-4302	149	15	the	the	DET
ejpam-4302	149	16	only	only	ADJ
ejpam-4302	149	17	(	(	PUNCT
ejpam-4302	149	18	λ	λ	NOUN
ejpam-4302	149	19	,	,	PUNCT
ejpam-4302	149	20	sp)-open	sp)-open	ADJ
ejpam-4302	149	21	set	set	NOUN
ejpam-4302	149	22	containing	contain	VERB
ejpam-4302	149	23	x	x	PART
ejpam-4302	149	24	−	−	PROPN
ejpam-4302	149	25	{	{	PUNCT
ejpam-4302	149	26	x	x	NOUN
ejpam-4302	149	27	}	}	PUNCT
ejpam-4302	149	28	is	be	AUX
ejpam-4302	149	29	x	x	X
ejpam-4302	149	30	itself	itself	PRON
ejpam-4302	149	31	.	.	PUNCT
ejpam-4302	150	1	thus	thus	ADV
ejpam-4302	150	2	,	,	PUNCT
ejpam-4302	150	3	[	[	X
ejpam-4302	150	4	x	x	X
ejpam-4302	150	5	−	−	X
ejpam-4302	150	6	{	{	PUNCT
ejpam-4302	150	7	x}](λ	x}](λ	PROPN
ejpam-4302	150	8	,	,	PUNCT
ejpam-4302	150	9	sp	sp	NOUN
ejpam-4302	150	10	)	)	PUNCT
ejpam-4302	150	11	⊆	⊆	NUM
ejpam-4302	150	12	x	x	NOUN
ejpam-4302	150	13	and	and	CCONJ
ejpam-4302	150	14	hence	hence	ADV
ejpam-4302	150	15	x	x	X
ejpam-4302	150	16	−	−	PROPN
ejpam-4302	150	17	{	{	PUNCT
ejpam-4302	150	18	x	x	NOUN
ejpam-4302	150	19	}	}	PUNCT
ejpam-4302	150	20	is	be	AUX
ejpam-4302	150	21	g-(λ	g-(λ	PROPN
ejpam-4302	150	22	,	,	PUNCT
ejpam-4302	150	23	sp)-closed	sp)-close	VERB
ejpam-4302	150	24	.	.	PUNCT
ejpam-4302	151	1	therefore	therefore	ADV
ejpam-4302	151	2	,	,	PUNCT
ejpam-4302	151	3	{	{	PUNCT
ejpam-4302	151	4	x	x	X
ejpam-4302	151	5	}	}	PUNCT
ejpam-4302	151	6	is	be	AUX
ejpam-4302	151	7	g-(λ	g-(λ	PROPN
ejpam-4302	151	8	,	,	PUNCT
ejpam-4302	151	9	sp)-open	sp)-open	ADJ
ejpam-4302	151	10	.	.	PUNCT
ejpam-4302	152	1	theorem	theorem	ADJ
ejpam-4302	152	2	6	6	NUM
ejpam-4302	152	3	.	.	PUNCT
ejpam-4302	153	1	let	let	VERB
ejpam-4302	153	2	a	a	DET
ejpam-4302	153	3	be	be	AUX
ejpam-4302	153	4	a	a	DET
ejpam-4302	153	5	subset	subset	NOUN
ejpam-4302	153	6	of	of	ADP
ejpam-4302	153	7	a	a	DET
ejpam-4302	153	8	topological	topological	ADJ
ejpam-4302	153	9	space	space	NOUN
ejpam-4302	153	10	(	(	PUNCT
ejpam-4302	153	11	x	x	X
ejpam-4302	153	12	,	,	PUNCT
ejpam-4302	153	13	τ	τ	PROPN
ejpam-4302	153	14	)	)	PUNCT
ejpam-4302	153	15	.	.	PUNCT
ejpam-4302	154	1	then	then	ADV
ejpam-4302	154	2	,	,	PUNCT
ejpam-4302	154	3	a	a	PRON
ejpam-4302	154	4	is	be	AUX
ejpam-4302	154	5	g-(λ	g-(λ	PROPN
ejpam-4302	154	6	,	,	PUNCT
ejpam-4302	154	7	sp)-open	sp)-open	ADJ
ejpam-4302	154	8	if	if	SCONJ
ejpam-4302	154	9	and	and	CCONJ
ejpam-4302	154	10	only	only	ADV
ejpam-4302	154	11	if	if	SCONJ
ejpam-4302	154	12	f	f	PROPN
ejpam-4302	154	13	⊆	⊆	NUM
ejpam-4302	154	14	a(λ	a(λ	ADV
ejpam-4302	154	15	,	,	PUNCT
ejpam-4302	154	16	sp	sp	NOUN
ejpam-4302	154	17	)	)	PUNCT
ejpam-4302	154	18	whenever	whenever	SCONJ
ejpam-4302	154	19	f	f	PROPN
ejpam-4302	154	20	⊆	⊆	PROPN
ejpam-4302	154	21	a	a	PRON
ejpam-4302	154	22	and	and	CCONJ
ejpam-4302	154	23	f	f	NOUN
ejpam-4302	154	24	is	be	AUX
ejpam-4302	154	25	(	(	PUNCT
ejpam-4302	154	26	λ	λ	X
ejpam-4302	154	27	,	,	PUNCT
ejpam-4302	154	28	sp)-closed	sp)-closed	ADJ
ejpam-4302	154	29	.	.	PUNCT
ejpam-4302	155	1	proof	proof	NOUN
ejpam-4302	155	2	.	.	PUNCT
ejpam-4302	156	1	suppose	suppose	VERB
ejpam-4302	156	2	that	that	SCONJ
ejpam-4302	156	3	a	a	PRON
ejpam-4302	156	4	is	be	AUX
ejpam-4302	156	5	a	a	DET
ejpam-4302	156	6	g-(λ	g-(λ	PROPN
ejpam-4302	156	7	,	,	PUNCT
ejpam-4302	156	8	sp)-open	sp)-open	ADJ
ejpam-4302	156	9	set	set	NOUN
ejpam-4302	156	10	.	.	PUNCT
ejpam-4302	157	1	let	let	VERB
ejpam-4302	157	2	f	f	PRON
ejpam-4302	157	3	be	be	AUX
ejpam-4302	157	4	a	a	DET
ejpam-4302	157	5	(	(	PUNCT
ejpam-4302	157	6	λ	λ	NOUN
ejpam-4302	157	7	,	,	PUNCT
ejpam-4302	157	8	sp)-closed	sp)-close	VERB
ejpam-4302	157	9	set	set	ADJ
ejpam-4302	157	10	and	and	CCONJ
ejpam-4302	157	11	f	f	PROPN
ejpam-4302	157	12	⊆	⊆	NUM
ejpam-4302	157	13	a.	a.	NOUN
ejpam-4302	157	14	then	then	ADV
ejpam-4302	157	15	,	,	PUNCT
ejpam-4302	157	16	x	x	PUNCT
ejpam-4302	157	17	−a	−a	VERB
ejpam-4302	157	18	⊆	⊆	NUM
ejpam-4302	157	19	x	x	PUNCT
ejpam-4302	157	20	−	−	PROPN
ejpam-4302	157	21	f	f	PROPN
ejpam-4302	157	22	∈	∈	PROPN
ejpam-4302	157	23	λspo(x	λspo(x	PROPN
ejpam-4302	157	24	,	,	PUNCT
ejpam-4302	157	25	τ	τ	PROPN
ejpam-4302	157	26	)	)	PUNCT
ejpam-4302	157	27	and	and	CCONJ
ejpam-4302	157	28	x	x	PART
ejpam-4302	157	29	−a	−a	NOUN
ejpam-4302	157	30	is	be	AUX
ejpam-4302	157	31	g-(λ	g-(λ	PROPN
ejpam-4302	157	32	,	,	PUNCT
ejpam-4302	157	33	sp)-closed	sp)-close	VERB
ejpam-4302	157	34	.	.	PUNCT
ejpam-4302	158	1	thus	thus	ADV
ejpam-4302	158	2	,	,	PUNCT
ejpam-4302	158	3	x	x	SYM
ejpam-4302	158	4	−a(λ	−a(λ	NOUN
ejpam-4302	158	5	,	,	PUNCT
ejpam-4302	158	6	sp	sp	NOUN
ejpam-4302	158	7	)	)	PUNCT
ejpam-4302	158	8	=	=	PUNCT
ejpam-4302	159	1	[	[	X
ejpam-4302	159	2	x	x	X
ejpam-4302	159	3	−a](λ	−a](λ	PROPN
ejpam-4302	159	4	,	,	PUNCT
ejpam-4302	159	5	sp	sp	NOUN
ejpam-4302	159	6	)	)	PUNCT
ejpam-4302	159	7	⊆	⊆	NUM
ejpam-4302	159	8	x	x	SYM
ejpam-4302	159	9	−	−	PROPN
ejpam-4302	159	10	f	f	NOUN
ejpam-4302	159	11	and	and	CCONJ
ejpam-4302	159	12	hence	hence	ADV
ejpam-4302	159	13	f	f	PROPN
ejpam-4302	159	14	⊆	⊆	NUM
ejpam-4302	159	15	a(λ	a(λ	ADV
ejpam-4302	159	16	,	,	PUNCT
ejpam-4302	159	17	sp	sp	NOUN
ejpam-4302	159	18	)	)	PUNCT
ejpam-4302	159	19	.	.	PUNCT
ejpam-4302	160	1	conversely	conversely	ADV
ejpam-4302	160	2	,	,	PUNCT
ejpam-4302	160	3	letx−a	letx−a	PROPN
ejpam-4302	160	4	⊆	⊆	NUM
ejpam-4302	160	5	u	u	NOUN
ejpam-4302	160	6	and	and	CCONJ
ejpam-4302	160	7	u	u	PROPN
ejpam-4302	160	8	∈	∈	PROPN
ejpam-4302	160	9	λspo(x	λspo(x	PROPN
ejpam-4302	160	10	,	,	PUNCT
ejpam-4302	160	11	τ	τ	PROPN
ejpam-4302	160	12	)	)	PUNCT
ejpam-4302	160	13	.	.	PUNCT
ejpam-4302	161	1	then	then	ADV
ejpam-4302	161	2	,	,	PUNCT
ejpam-4302	161	3	x−u	x−u	PROPN
ejpam-4302	161	4	⊆	⊆	NUM
ejpam-4302	161	5	a	a	DET
ejpam-4302	161	6	andx−u	andx−u	PROPN
ejpam-4302	161	7	is	be	AUX
ejpam-4302	161	8	(	(	PUNCT
ejpam-4302	161	9	λ	λ	X
ejpam-4302	161	10	,	,	PUNCT
ejpam-4302	161	11	sp)closed	sp)close	VERB
ejpam-4302	161	12	.	.	PUNCT
ejpam-4302	162	1	by	by	ADP
ejpam-4302	162	2	the	the	DET
ejpam-4302	162	3	hypothesis	hypothesis	NOUN
ejpam-4302	162	4	,	,	PUNCT
ejpam-4302	162	5	x	x	PUNCT
ejpam-4302	162	6	−	−	PUNCT
ejpam-4302	162	7	u	u	NOUN
ejpam-4302	162	8	⊆	⊆	NUM
ejpam-4302	162	9	a(λ	a(λ	ADV
ejpam-4302	162	10	,	,	PUNCT
ejpam-4302	162	11	sp	sp	NOUN
ejpam-4302	162	12	)	)	PUNCT
ejpam-4302	162	13	and	and	CCONJ
ejpam-4302	162	14	hence	hence	ADV
ejpam-4302	162	15	[	[	X
ejpam-4302	162	16	x	x	X
ejpam-4302	162	17	−	−	NOUN
ejpam-4302	162	18	a](λ	a](λ	NOUN
ejpam-4302	162	19	,	,	PUNCT
ejpam-4302	162	20	sp	sp	NOUN
ejpam-4302	162	21	)	)	PUNCT
ejpam-4302	162	22	=	=	PUNCT
ejpam-4302	162	23	x	x	X
ejpam-4302	162	24	−	−	NOUN
ejpam-4302	162	25	a(λ	a(λ	ADV
ejpam-4302	162	26	,	,	PUNCT
ejpam-4302	162	27	sp	sp	NOUN
ejpam-4302	162	28	)	)	PUNCT
ejpam-4302	162	29	⊆	⊆	NUM
ejpam-4302	162	30	u	u	NOUN
ejpam-4302	162	31	.	.	PUNCT
ejpam-4302	163	1	thus	thus	ADV
ejpam-4302	163	2	,	,	PUNCT
ejpam-4302	163	3	x	x	PRON
ejpam-4302	163	4	−a	−a	NOUN
ejpam-4302	163	5	is	be	AUX
ejpam-4302	163	6	g-(λ	g-(λ	PROPN
ejpam-4302	163	7	,	,	PUNCT
ejpam-4302	163	8	sp)-closed	sp)-close	VERB
ejpam-4302	163	9	.	.	PUNCT
ejpam-4302	164	1	this	this	PRON
ejpam-4302	164	2	shows	show	VERB
ejpam-4302	164	3	that	that	SCONJ
ejpam-4302	164	4	a	a	PRON
ejpam-4302	164	5	is	be	AUX
ejpam-4302	164	6	g-(λ	g-(λ	PROPN
ejpam-4302	164	7	,	,	PUNCT
ejpam-4302	164	8	sp)-open	sp)-open	NOUN
ejpam-4302	164	9	.	.	PUNCT
ejpam-4302	165	1	lemma	lemma	PROPN
ejpam-4302	165	2	5	5	X
ejpam-4302	165	3	.	.	PUNCT
ejpam-4302	165	4	let	let	VERB
ejpam-4302	165	5	a	a	DET
ejpam-4302	165	6	be	be	AUX
ejpam-4302	165	7	a	a	DET
ejpam-4302	165	8	subset	subset	NOUN
ejpam-4302	165	9	of	of	ADP
ejpam-4302	165	10	a	a	DET
ejpam-4302	165	11	topological	topological	ADJ
ejpam-4302	165	12	space	space	NOUN
ejpam-4302	165	13	(	(	PUNCT
ejpam-4302	165	14	x	x	X
ejpam-4302	165	15	,	,	PUNCT
ejpam-4302	165	16	τ	τ	PROPN
ejpam-4302	165	17	)	)	PUNCT
ejpam-4302	165	18	.	.	PUNCT
ejpam-4302	166	1	if	if	SCONJ
ejpam-4302	166	2	g	g	PROPN
ejpam-4302	166	3	∈	∈	PROPN
ejpam-4302	166	4	λspo(x	λspo(x	PROPN
ejpam-4302	166	5	,	,	PUNCT
ejpam-4302	166	6	τ	τ	PROPN
ejpam-4302	166	7	)	)	PUNCT
ejpam-4302	166	8	and	and	CCONJ
ejpam-4302	166	9	a	a	DET
ejpam-4302	166	10	∩g	∩g	ADJ
ejpam-4302	166	11	=	=	SYM
ejpam-4302	166	12	∅	∅	NOUN
ejpam-4302	166	13	,	,	PUNCT
ejpam-4302	166	14	then	then	ADV
ejpam-4302	166	15	a(λ	a(λ	ADV
ejpam-4302	166	16	,	,	PUNCT
ejpam-4302	166	17	sp	sp	NOUN
ejpam-4302	166	18	)	)	PUNCT
ejpam-4302	166	19	∩g	∩g	NOUN
ejpam-4302	166	20	=	=	PUNCT
ejpam-4302	166	21	∅.	∅.	PROPN
ejpam-4302	166	22	c.	c.	PROPN
ejpam-4302	166	23	boonpok	boonpok	PROPN
ejpam-4302	166	24	,	,	PUNCT
ejpam-4302	166	25	c.	c.	PROPN
ejpam-4302	166	26	viriyapong	viriyapong	PROPN
ejpam-4302	166	27	/	/	SYM
ejpam-4302	166	28	eur	eur	PROPN
ejpam-4302	166	29	.	.	PUNCT
ejpam-4302	167	1	j.	j.	PROPN
ejpam-4302	167	2	pure	pure	PROPN
ejpam-4302	167	3	appl	appl	PROPN
ejpam-4302	167	4	.	.	PROPN
ejpam-4302	167	5	math	math	PROPN
ejpam-4302	167	6	,	,	PUNCT
ejpam-4302	167	7	15	15	NUM
ejpam-4302	167	8	(	(	PUNCT
ejpam-4302	167	9	4	4	NUM
ejpam-4302	167	10	)	)	PUNCT
ejpam-4302	167	11	(	(	PUNCT
ejpam-4302	167	12	2022	2022	NUM
ejpam-4302	167	13	)	)	PUNCT
ejpam-4302	167	14	,	,	PUNCT
ejpam-4302	167	15	2127	2127	NUM
ejpam-4302	167	16	-	-	SYM
ejpam-4302	167	17	2140	2140	NUM
ejpam-4302	167	18	2132	2132	NUM
ejpam-4302	167	19	theorem	theorem	VERB
ejpam-4302	167	20	7	7	NUM
ejpam-4302	167	21	.	.	X
ejpam-4302	167	22	for	for	ADP
ejpam-4302	167	23	a	a	DET
ejpam-4302	167	24	subset	subset	NOUN
ejpam-4302	167	25	a	a	PRON
ejpam-4302	167	26	of	of	ADP
ejpam-4302	167	27	a	a	DET
ejpam-4302	167	28	topological	topological	ADJ
ejpam-4302	167	29	space	space	NOUN
ejpam-4302	167	30	(	(	PUNCT
ejpam-4302	167	31	x	x	X
ejpam-4302	167	32	,	,	PUNCT
ejpam-4302	167	33	τ	τ	PROPN
ejpam-4302	167	34	)	)	PUNCT
ejpam-4302	167	35	,	,	PUNCT
ejpam-4302	167	36	the	the	DET
ejpam-4302	167	37	following	follow	VERB
ejpam-4302	167	38	properties	property	NOUN
ejpam-4302	167	39	are	be	AUX
ejpam-4302	167	40	equivalent	equivalent	ADJ
ejpam-4302	167	41	:	:	PUNCT
ejpam-4302	167	42	(	(	PUNCT
ejpam-4302	167	43	1	1	X
ejpam-4302	167	44	)	)	PUNCT
ejpam-4302	167	45	a	a	PRON
ejpam-4302	167	46	is	be	AUX
ejpam-4302	167	47	g-(λ	g-(λ	PROPN
ejpam-4302	167	48	,	,	PUNCT
ejpam-4302	167	49	sp)-closed	sp)-close	VERB
ejpam-4302	167	50	.	.	PUNCT
ejpam-4302	168	1	(	(	PUNCT
ejpam-4302	168	2	2	2	NUM
ejpam-4302	168	3	)	)	PUNCT
ejpam-4302	168	4	a(λ	a(λ	ADV
ejpam-4302	168	5	,	,	PUNCT
ejpam-4302	168	6	sp	sp	NOUN
ejpam-4302	168	7	)	)	PUNCT
ejpam-4302	168	8	−a	−a	NOUN
ejpam-4302	168	9	contains	contain	VERB
ejpam-4302	168	10	no	no	DET
ejpam-4302	168	11	nonempty	nonempty	ADJ
ejpam-4302	168	12	(	(	PUNCT
ejpam-4302	168	13	λ	λ	NOUN
ejpam-4302	168	14	,	,	PUNCT
ejpam-4302	168	15	sp)-closed	sp)-close	VERB
ejpam-4302	168	16	set	set	VERB
ejpam-4302	168	17	.	.	PUNCT
ejpam-4302	169	1	(	(	PUNCT
ejpam-4302	169	2	3	3	NUM
ejpam-4302	169	3	)	)	PUNCT
ejpam-4302	169	4	a(λ	a(λ	ADV
ejpam-4302	169	5	,	,	PUNCT
ejpam-4302	169	6	sp	sp	NOUN
ejpam-4302	169	7	)	)	PUNCT
ejpam-4302	169	8	−a	−a	NOUN
ejpam-4302	169	9	is	be	AUX
ejpam-4302	169	10	g-(λ	g-(λ	PROPN
ejpam-4302	169	11	,	,	PUNCT
ejpam-4302	169	12	sp)-open	sp)-open	ADJ
ejpam-4302	169	13	.	.	PUNCT
ejpam-4302	170	1	proof	proof	NOUN
ejpam-4302	170	2	.	.	PUNCT
ejpam-4302	171	1	(	(	PUNCT
ejpam-4302	171	2	1	1	X
ejpam-4302	171	3	)	)	PUNCT
ejpam-4302	171	4	⇒	⇒	NOUN
ejpam-4302	171	5	(	(	PUNCT
ejpam-4302	171	6	2	2	NUM
ejpam-4302	171	7	):	):	PUNCT
ejpam-4302	171	8	this	this	PRON
ejpam-4302	171	9	follows	follow	VERB
ejpam-4302	171	10	from	from	ADP
ejpam-4302	171	11	theorem	theorem	ADJ
ejpam-4302	171	12	2	2	NUM
ejpam-4302	171	13	.	.	PUNCT
ejpam-4302	171	14	(	(	PUNCT
ejpam-4302	171	15	2	2	X
ejpam-4302	171	16	)	)	PUNCT
ejpam-4302	171	17	⇒	⇒	NOUN
ejpam-4302	171	18	(	(	PUNCT
ejpam-4302	171	19	3	3	NUM
ejpam-4302	171	20	):	):	PUNCT
ejpam-4302	171	21	let	let	VERB
ejpam-4302	171	22	f	f	PRON
ejpam-4302	171	23	be	be	AUX
ejpam-4302	171	24	a	a	DET
ejpam-4302	171	25	(	(	PUNCT
ejpam-4302	171	26	λ	λ	NOUN
ejpam-4302	171	27	,	,	PUNCT
ejpam-4302	171	28	sp)-closed	sp)-close	VERB
ejpam-4302	171	29	set	set	ADJ
ejpam-4302	171	30	and	and	CCONJ
ejpam-4302	171	31	f	f	NOUN
ejpam-4302	171	32	⊆	⊆	NUM
ejpam-4302	171	33	a(λ	a(λ	ADV
ejpam-4302	171	34	,	,	PUNCT
ejpam-4302	171	35	sp	sp	NOUN
ejpam-4302	171	36	)	)	PUNCT
ejpam-4302	171	37	−	−	NOUN
ejpam-4302	171	38	a.	a.	NOUN
ejpam-4302	171	39	by	by	ADP
ejpam-4302	171	40	(	(	PUNCT
ejpam-4302	171	41	2	2	NUM
ejpam-4302	171	42	)	)	PUNCT
ejpam-4302	171	43	,	,	PUNCT
ejpam-4302	171	44	we	we	PRON
ejpam-4302	171	45	have	have	VERB
ejpam-4302	171	46	f	f	NOUN
ejpam-4302	171	47	=	=	PUNCT
ejpam-4302	171	48	∅	∅	NOUN
ejpam-4302	171	49	and	and	CCONJ
ejpam-4302	171	50	f	f	NOUN
ejpam-4302	171	51	⊆	⊆	NUM
ejpam-4302	171	52	[	[	X
ejpam-4302	171	53	a(λ	a(λ	ADV
ejpam-4302	171	54	,	,	PUNCT
ejpam-4302	171	55	sp	sp	NOUN
ejpam-4302	171	56	)	)	PUNCT
ejpam-4302	171	57	−a](λ	−a](λ	PROPN
ejpam-4302	171	58	,	,	PUNCT
ejpam-4302	171	59	sp	sp	NOUN
ejpam-4302	171	60	)	)	PUNCT
ejpam-4302	171	61	.	.	PUNCT
ejpam-4302	172	1	it	it	PRON
ejpam-4302	172	2	follows	follow	VERB
ejpam-4302	172	3	from	from	ADP
ejpam-4302	172	4	theorem	theorem	NOUN
ejpam-4302	172	5	6	6	NUM
ejpam-4302	172	6	that	that	PRON
ejpam-4302	172	7	a(λ	a(λ	ADV
ejpam-4302	172	8	,	,	PUNCT
ejpam-4302	172	9	sp	sp	NOUN
ejpam-4302	172	10	)	)	PUNCT
ejpam-4302	172	11	−a	−a	NOUN
ejpam-4302	172	12	is	be	AUX
ejpam-4302	172	13	g-(λ	g-(λ	PROPN
ejpam-4302	172	14	,	,	PUNCT
ejpam-4302	172	15	sp)-open	sp)-open	ADJ
ejpam-4302	172	16	.	.	PUNCT
ejpam-4302	173	1	(	(	PUNCT
ejpam-4302	173	2	3	3	X
ejpam-4302	173	3	)	)	PUNCT
ejpam-4302	173	4	⇒	⇒	NOUN
ejpam-4302	173	5	(	(	PUNCT
ejpam-4302	173	6	1	1	NUM
ejpam-4302	173	7	):	):	PUNCT
ejpam-4302	173	8	suppose	suppose	VERB
ejpam-4302	173	9	that	that	SCONJ
ejpam-4302	173	10	a	a	DET
ejpam-4302	173	11	⊆	⊆	NUM
ejpam-4302	173	12	u	u	NOUN
ejpam-4302	173	13	and	and	CCONJ
ejpam-4302	173	14	u	u	PROPN
ejpam-4302	173	15	∈	∈	PROPN
ejpam-4302	173	16	λspo(x	λspo(x	PROPN
ejpam-4302	173	17	,	,	PUNCT
ejpam-4302	173	18	τ	τ	PROPN
ejpam-4302	173	19	)	)	PUNCT
ejpam-4302	173	20	.	.	PUNCT
ejpam-4302	174	1	then	then	ADV
ejpam-4302	174	2	,	,	PUNCT
ejpam-4302	174	3	a(λ	a(λ	ADV
ejpam-4302	174	4	,	,	PUNCT
ejpam-4302	174	5	sp)−u	sp)−u	NOUN
ejpam-4302	174	6	⊆	⊆	NUM
ejpam-4302	174	7	a(λ	a(λ	ADV
ejpam-4302	174	8	,	,	PUNCT
ejpam-4302	174	9	sp)−a	sp)−a	X
ejpam-4302	174	10	.	.	PUNCT
ejpam-4302	175	1	by	by	ADP
ejpam-4302	175	2	(	(	PUNCT
ejpam-4302	175	3	3	3	NUM
ejpam-4302	175	4	)	)	PUNCT
ejpam-4302	175	5	,	,	PUNCT
ejpam-4302	175	6	we	we	PRON
ejpam-4302	175	7	have	have	VERB
ejpam-4302	175	8	a(λ	a(λ	ADV
ejpam-4302	175	9	,	,	PUNCT
ejpam-4302	175	10	sp	sp	NOUN
ejpam-4302	175	11	)	)	PUNCT
ejpam-4302	175	12	−	−	NOUN
ejpam-4302	175	13	a	a	PRON
ejpam-4302	175	14	is	be	AUX
ejpam-4302	175	15	g-(λ	g-(λ	PROPN
ejpam-4302	175	16	,	,	PUNCT
ejpam-4302	175	17	sp)-open	sp)-open	ADJ
ejpam-4302	175	18	.	.	PUNCT
ejpam-4302	176	1	since	since	SCONJ
ejpam-4302	176	2	a(λ	a(λ	PROPN
ejpam-4302	176	3	,	,	PUNCT
ejpam-4302	176	4	sp	sp	NOUN
ejpam-4302	176	5	)	)	PUNCT
ejpam-4302	176	6	−	−	NOUN
ejpam-4302	176	7	u	u	NOUN
ejpam-4302	176	8	is	be	AUX
ejpam-4302	176	9	(	(	PUNCT
ejpam-4302	176	10	λ	λ	X
ejpam-4302	176	11	,	,	PUNCT
ejpam-4302	176	12	sp)-closed	sp)-close	VERB
ejpam-4302	176	13	,	,	PUNCT
ejpam-4302	176	14	by	by	ADP
ejpam-4302	176	15	theorem	theorem	NOUN
ejpam-4302	176	16	6	6	NUM
ejpam-4302	176	17	,	,	PUNCT
ejpam-4302	176	18	a(λ	a(λ	ADV
ejpam-4302	176	19	,	,	PUNCT
ejpam-4302	176	20	sp	sp	NOUN
ejpam-4302	176	21	)	)	PUNCT
ejpam-4302	176	22	−	−	NOUN
ejpam-4302	176	23	u	u	NOUN
ejpam-4302	176	24	⊆	⊆	NUM
ejpam-4302	176	25	[	[	X
ejpam-4302	176	26	a(λ	a(λ	ADV
ejpam-4302	176	27	,	,	PUNCT
ejpam-4302	176	28	sp	sp	NOUN
ejpam-4302	176	29	)	)	PUNCT
ejpam-4302	176	30	−	−	NOUN
ejpam-4302	176	31	a](λ	a](λ	NOUN
ejpam-4302	176	32	,	,	PUNCT
ejpam-4302	176	33	sp	sp	NOUN
ejpam-4302	176	34	)	)	PUNCT
ejpam-4302	176	35	=	=	NOUN
ejpam-4302	176	36	∅.	∅.	ADP
ejpam-4302	176	37	thus	thus	ADV
ejpam-4302	176	38	,	,	PUNCT
ejpam-4302	176	39	a(λ	a(λ	ADV
ejpam-4302	176	40	,	,	PUNCT
ejpam-4302	176	41	sp	sp	NOUN
ejpam-4302	176	42	)	)	PUNCT
ejpam-4302	176	43	⊆	⊆	NUM
ejpam-4302	176	44	u	u	NOUN
ejpam-4302	176	45	and	and	CCONJ
ejpam-4302	176	46	hence	hence	ADV
ejpam-4302	176	47	a	a	PRON
ejpam-4302	176	48	is	be	AUX
ejpam-4302	176	49	g-(λ	g-(λ	PROPN
ejpam-4302	176	50	,	,	PUNCT
ejpam-4302	176	51	sp)-closed	sp)-close	VERB
ejpam-4302	176	52	.	.	PUNCT
ejpam-4302	177	1	now	now	ADV
ejpam-4302	177	2	,	,	PUNCT
ejpam-4302	177	3	the	the	DET
ejpam-4302	177	4	proof	proof	NOUN
ejpam-4302	177	5	of	of	ADP
ejpam-4302	177	6	[	[	X
ejpam-4302	177	7	a(λ	a(λ	ADV
ejpam-4302	177	8	,	,	PUNCT
ejpam-4302	177	9	sp)−a](λ	sp)−a](λ	PROPN
ejpam-4302	177	10	,	,	PUNCT
ejpam-4302	177	11	sp	sp	NOUN
ejpam-4302	177	12	)	)	PUNCT
ejpam-4302	177	13	=	=	NOUN
ejpam-4302	177	14	∅	∅	NOUN
ejpam-4302	177	15	is	be	AUX
ejpam-4302	177	16	given	give	VERB
ejpam-4302	177	17	as	as	SCONJ
ejpam-4302	177	18	follows	follow	VERB
ejpam-4302	177	19	.	.	PUNCT
ejpam-4302	178	1	suppose	suppose	VERB
ejpam-4302	178	2	that	that	SCONJ
ejpam-4302	178	3	[	[	X
ejpam-4302	178	4	a(λ	a(λ	ADV
ejpam-4302	178	5	,	,	PUNCT
ejpam-4302	178	6	sp	sp	NOUN
ejpam-4302	178	7	)	)	PUNCT
ejpam-4302	178	8	−	−	NOUN
ejpam-4302	178	9	a](λ	a](λ	NOUN
ejpam-4302	178	10	,	,	PUNCT
ejpam-4302	178	11	sp	sp	NOUN
ejpam-4302	178	12	)	)	PUNCT
ejpam-4302	178	13	̸=	̸=	PROPN
ejpam-4302	178	14	∅.	∅.	NOUN
ejpam-4302	178	15	then	then	ADV
ejpam-4302	178	16	,	,	PUNCT
ejpam-4302	178	17	there	there	PRON
ejpam-4302	178	18	exists	exist	VERB
ejpam-4302	178	19	x	x	X
ejpam-4302	178	20	∈	∈	PROPN
ejpam-4302	178	21	[	[	X
ejpam-4302	178	22	a(λ	a(λ	ADV
ejpam-4302	178	23	,	,	PUNCT
ejpam-4302	178	24	sp	sp	NOUN
ejpam-4302	178	25	)	)	PUNCT
ejpam-4302	179	1	−	−	NOUN
ejpam-4302	179	2	a](λ	a](λ	NOUN
ejpam-4302	179	3	,	,	PUNCT
ejpam-4302	179	4	sp	sp	NOUN
ejpam-4302	179	5	)	)	PUNCT
ejpam-4302	179	6	and	and	CCONJ
ejpam-4302	179	7	hence	hence	ADV
ejpam-4302	179	8	there	there	PRON
ejpam-4302	179	9	exists	exist	VERB
ejpam-4302	179	10	g	g	PROPN
ejpam-4302	179	11	∈	∈	PROPN
ejpam-4302	179	12	λspo(x	λspo(x	PROPN
ejpam-4302	179	13	,	,	PUNCT
ejpam-4302	179	14	τ	τ	PROPN
ejpam-4302	179	15	)	)	PUNCT
ejpam-4302	179	16	such	such	ADJ
ejpam-4302	179	17	that	that	SCONJ
ejpam-4302	179	18	x	x	SYM
ejpam-4302	179	19	∈	∈	NOUN
ejpam-4302	179	20	g	g	ADP
ejpam-4302	179	21	⊆	⊆	NUM
ejpam-4302	179	22	a(λ	a(λ	PROPN
ejpam-4302	179	23	,	,	PUNCT
ejpam-4302	179	24	sp	sp	NOUN
ejpam-4302	179	25	)	)	PUNCT
ejpam-4302	179	26	−a	−a	NOUN
ejpam-4302	179	27	.	.	PUNCT
ejpam-4302	180	1	since	since	SCONJ
ejpam-4302	180	2	g	g	PROPN
ejpam-4302	180	3	⊆	⊆	NUM
ejpam-4302	180	4	x	x	SYM
ejpam-4302	180	5	−a	−a	NOUN
ejpam-4302	180	6	,	,	PUNCT
ejpam-4302	180	7	we	we	PRON
ejpam-4302	180	8	have	have	VERB
ejpam-4302	180	9	g∩a	g∩a	NOUN
ejpam-4302	180	10	=	=	SYM
ejpam-4302	180	11	∅	∅	NOUN
ejpam-4302	180	12	,	,	PUNCT
ejpam-4302	180	13	by	by	ADP
ejpam-4302	180	14	lemma	lemma	PROPN
ejpam-4302	180	15	5	5	NUM
ejpam-4302	180	16	,	,	PUNCT
ejpam-4302	180	17	g∩a(λ	g∩a(λ	PROPN
ejpam-4302	180	18	,	,	PUNCT
ejpam-4302	180	19	sp	sp	NOUN
ejpam-4302	180	20	)	)	PUNCT
ejpam-4302	180	21	=	=	NOUN
ejpam-4302	180	22	∅	∅	NOUN
ejpam-4302	180	23	and	and	CCONJ
ejpam-4302	180	24	hence	hence	ADV
ejpam-4302	180	25	g	g	PROPN
ejpam-4302	180	26	⊆	⊆	NUM
ejpam-4302	180	27	x−a(λ	x−a(λ	NOUN
ejpam-4302	180	28	,	,	PUNCT
ejpam-4302	180	29	sp	sp	NOUN
ejpam-4302	180	30	)	)	PUNCT
ejpam-4302	180	31	.	.	PUNCT
ejpam-4302	181	1	thus	thus	ADV
ejpam-4302	181	2	,	,	PUNCT
ejpam-4302	181	3	g	g	PROPN
ejpam-4302	181	4	⊆	⊆	NUM
ejpam-4302	181	5	[	[	X
ejpam-4302	181	6	x−a(λ	x−a(λ	NOUN
ejpam-4302	181	7	,	,	PUNCT
ejpam-4302	181	8	sp)]∩a(λ	sp)]∩a(λ	NOUN
ejpam-4302	181	9	,	,	PUNCT
ejpam-4302	181	10	sp	sp	NOUN
ejpam-4302	181	11	)	)	PUNCT
ejpam-4302	181	12	=	=	PUNCT
ejpam-4302	181	13	∅.	∅.	NOUN
ejpam-4302	181	14	this	this	PRON
ejpam-4302	181	15	is	be	AUX
ejpam-4302	181	16	a	a	DET
ejpam-4302	181	17	contradiction	contradiction	NOUN
ejpam-4302	181	18	.	.	PUNCT
ejpam-4302	182	1	theorem	theorem	ADJ
ejpam-4302	182	2	8	8	NUM
ejpam-4302	182	3	.	.	PUNCT
ejpam-4302	183	1	a	a	DET
ejpam-4302	183	2	subset	subset	NOUN
ejpam-4302	183	3	a	a	PRON
ejpam-4302	183	4	of	of	ADP
ejpam-4302	183	5	a	a	DET
ejpam-4302	183	6	topological	topological	ADJ
ejpam-4302	183	7	space	space	NOUN
ejpam-4302	183	8	(	(	PUNCT
ejpam-4302	183	9	x	x	X
ejpam-4302	183	10	,	,	PUNCT
ejpam-4302	183	11	τ	τ	X
ejpam-4302	183	12	)	)	PUNCT
ejpam-4302	183	13	is	be	AUX
ejpam-4302	183	14	g-(λ	g-(λ	PROPN
ejpam-4302	183	15	,	,	PUNCT
ejpam-4302	183	16	sp)-closed	sp)-close	VERB
ejpam-4302	183	17	if	if	SCONJ
ejpam-4302	183	18	and	and	CCONJ
ejpam-4302	183	19	only	only	ADV
ejpam-4302	183	20	if	if	SCONJ
ejpam-4302	183	21	f	f	PROPN
ejpam-4302	183	22	∩a(λ	∩a(λ	NOUN
ejpam-4302	183	23	,	,	PUNCT
ejpam-4302	183	24	sp	sp	NOUN
ejpam-4302	183	25	)	)	PUNCT
ejpam-4302	183	26	=	=	NOUN
ejpam-4302	183	27	∅	∅	NOUN
ejpam-4302	183	28	whenever	whenever	SCONJ
ejpam-4302	183	29	a	a	DET
ejpam-4302	183	30	∩	∩	ADJ
ejpam-4302	183	31	f	f	NOUN
ejpam-4302	183	32	=	=	NOUN
ejpam-4302	183	33	∅	∅	NOUN
ejpam-4302	183	34	and	and	CCONJ
ejpam-4302	183	35	f	f	PROPN
ejpam-4302	183	36	is	be	AUX
ejpam-4302	183	37	(	(	PUNCT
ejpam-4302	183	38	λ	λ	X
ejpam-4302	183	39	,	,	PUNCT
ejpam-4302	183	40	sp)-closed	sp)-closed	ADJ
ejpam-4302	183	41	.	.	PUNCT
ejpam-4302	184	1	proof	proof	NOUN
ejpam-4302	184	2	.	.	PUNCT
ejpam-4302	185	1	suppose	suppose	VERB
ejpam-4302	185	2	that	that	SCONJ
ejpam-4302	185	3	a	a	PRON
ejpam-4302	185	4	is	be	AUX
ejpam-4302	185	5	a	a	DET
ejpam-4302	185	6	(	(	PUNCT
ejpam-4302	185	7	λ	λ	PROPN
ejpam-4302	185	8	,	,	PUNCT
ejpam-4302	185	9	sp)-closed	sp)-close	VERB
ejpam-4302	185	10	set	set	VERB
ejpam-4302	185	11	.	.	PUNCT
ejpam-4302	186	1	let	let	VERB
ejpam-4302	186	2	f	f	PRON
ejpam-4302	186	3	be	be	AUX
ejpam-4302	186	4	a	a	DET
ejpam-4302	186	5	(	(	PUNCT
ejpam-4302	186	6	λ	λ	NOUN
ejpam-4302	186	7	,	,	PUNCT
ejpam-4302	186	8	sp)-closed	sp)-close	VERB
ejpam-4302	186	9	set	set	ADJ
ejpam-4302	186	10	and	and	CCONJ
ejpam-4302	186	11	a	a	DET
ejpam-4302	186	12	∩	∩	ADJ
ejpam-4302	186	13	f	f	X
ejpam-4302	186	14	=	=	PUNCT
ejpam-4302	186	15	∅.	∅.	NOUN
ejpam-4302	186	16	then	then	ADV
ejpam-4302	186	17	,	,	PUNCT
ejpam-4302	186	18	a	a	DET
ejpam-4302	186	19	⊆	⊆	NUM
ejpam-4302	186	20	x	x	SYM
ejpam-4302	186	21	−	−	PROPN
ejpam-4302	186	22	f	f	PROPN
ejpam-4302	186	23	∈	∈	PROPN
ejpam-4302	186	24	λspo(x	λspo(x	PROPN
ejpam-4302	186	25	,	,	PUNCT
ejpam-4302	186	26	τ	τ	PROPN
ejpam-4302	186	27	)	)	PUNCT
ejpam-4302	186	28	and	and	CCONJ
ejpam-4302	186	29	a(λ	a(λ	ADV
ejpam-4302	186	30	,	,	PUNCT
ejpam-4302	186	31	sp	sp	NOUN
ejpam-4302	186	32	)	)	PUNCT
ejpam-4302	186	33	⊆	⊆	NUM
ejpam-4302	186	34	x	x	SYM
ejpam-4302	186	35	−	−	PROPN
ejpam-4302	186	36	f	f	NOUN
ejpam-4302	186	37	.	.	PUNCT
ejpam-4302	187	1	thus	thus	ADV
ejpam-4302	187	2	,	,	PUNCT
ejpam-4302	187	3	f	f	PROPN
ejpam-4302	187	4	∩a(λ	∩a(λ	PROPN
ejpam-4302	187	5	,	,	PUNCT
ejpam-4302	187	6	sp	sp	NOUN
ejpam-4302	187	7	)	)	PUNCT
ejpam-4302	187	8	=	=	NOUN
ejpam-4302	187	9	∅.	∅.	VERB
ejpam-4302	187	10	conversely	conversely	ADV
ejpam-4302	187	11	,	,	PUNCT
ejpam-4302	187	12	let	let	VERB
ejpam-4302	187	13	a	a	DET
ejpam-4302	187	14	⊆	⊆	NUM
ejpam-4302	187	15	u	u	NOUN
ejpam-4302	187	16	and	and	CCONJ
ejpam-4302	187	17	u	u	PROPN
ejpam-4302	187	18	∈	∈	PROPN
ejpam-4302	187	19	λspo(x	λspo(x	PROPN
ejpam-4302	187	20	,	,	PUNCT
ejpam-4302	187	21	τ	τ	PROPN
ejpam-4302	187	22	)	)	PUNCT
ejpam-4302	187	23	.	.	PUNCT
ejpam-4302	188	1	then	then	ADV
ejpam-4302	188	2	,	,	PUNCT
ejpam-4302	188	3	a	a	DET
ejpam-4302	188	4	∩	∩	NOUN
ejpam-4302	188	5	(	(	PUNCT
ejpam-4302	188	6	x	x	SYM
ejpam-4302	188	7	−	−	PROPN
ejpam-4302	188	8	u	u	NOUN
ejpam-4302	188	9	)	)	PUNCT
ejpam-4302	188	10	=	=	SYM
ejpam-4302	188	11	∅	∅	NOUN
ejpam-4302	188	12	and	and	CCONJ
ejpam-4302	188	13	x	x	X
ejpam-4302	188	14	−	−	NOUN
ejpam-4302	188	15	u	u	NOUN
ejpam-4302	188	16	is	be	AUX
ejpam-4302	188	17	(	(	PUNCT
ejpam-4302	188	18	λ	λ	X
ejpam-4302	188	19	,	,	PUNCT
ejpam-4302	188	20	sp)-closed	sp)-close	VERB
ejpam-4302	188	21	.	.	PUNCT
ejpam-4302	189	1	by	by	ADP
ejpam-4302	189	2	the	the	DET
ejpam-4302	189	3	hypothesis	hypothesis	NOUN
ejpam-4302	189	4	,	,	PUNCT
ejpam-4302	189	5	(	(	PUNCT
ejpam-4302	189	6	x	x	X
ejpam-4302	189	7	−	−	PROPN
ejpam-4302	189	8	u	u	NOUN
ejpam-4302	189	9	)	)	PUNCT
ejpam-4302	189	10	∩a(λ	∩a(λ	NOUN
ejpam-4302	189	11	,	,	PUNCT
ejpam-4302	189	12	sp	sp	NOUN
ejpam-4302	189	13	)	)	PUNCT
ejpam-4302	189	14	=	=	NOUN
ejpam-4302	189	15	∅	∅	NOUN
ejpam-4302	189	16	and	and	CCONJ
ejpam-4302	189	17	hence	hence	ADV
ejpam-4302	189	18	a(λ	a(λ	ADV
ejpam-4302	189	19	,	,	PUNCT
ejpam-4302	189	20	sp	sp	NOUN
ejpam-4302	189	21	)	)	PUNCT
ejpam-4302	189	22	⊆	⊆	NUM
ejpam-4302	189	23	u	u	NOUN
ejpam-4302	189	24	.	.	PUNCT
ejpam-4302	190	1	thus	thus	ADV
ejpam-4302	190	2	,	,	PUNCT
ejpam-4302	190	3	a	a	PRON
ejpam-4302	190	4	is	be	AUX
ejpam-4302	190	5	g-(λ	g-(λ	PRON
ejpam-4302	190	6	,	,	PUNCT
ejpam-4302	190	7	sp)-closed	sp)-close	VERB
ejpam-4302	190	8	.	.	PUNCT
ejpam-4302	191	1	theorem	theorem	VERB
ejpam-4302	191	2	9	9	NUM
ejpam-4302	191	3	.	.	PUNCT
ejpam-4302	192	1	a	a	DET
ejpam-4302	192	2	subset	subset	NOUN
ejpam-4302	192	3	a	a	PRON
ejpam-4302	192	4	of	of	ADP
ejpam-4302	192	5	a	a	DET
ejpam-4302	192	6	topological	topological	ADJ
ejpam-4302	192	7	space	space	NOUN
ejpam-4302	192	8	(	(	PUNCT
ejpam-4302	192	9	x	x	X
ejpam-4302	192	10	,	,	PUNCT
ejpam-4302	192	11	τ	τ	X
ejpam-4302	192	12	)	)	PUNCT
ejpam-4302	192	13	is	be	AUX
ejpam-4302	192	14	g-(λ	g-(λ	PROPN
ejpam-4302	192	15	,	,	PUNCT
ejpam-4302	192	16	sp)-closed	sp)-close	VERB
ejpam-4302	192	17	if	if	SCONJ
ejpam-4302	192	18	and	and	CCONJ
ejpam-4302	192	19	only	only	ADV
ejpam-4302	192	20	if	if	SCONJ
ejpam-4302	192	21	a	a	DET
ejpam-4302	192	22	∩	∩	NOUN
ejpam-4302	192	23	{	{	PUNCT
ejpam-4302	192	24	x}(λ	x}(λ	PROPN
ejpam-4302	192	25	,	,	PUNCT
ejpam-4302	192	26	sp	sp	NOUN
ejpam-4302	192	27	)	)	PUNCT
ejpam-4302	192	28	̸=	̸=	PROPN
ejpam-4302	192	29	∅	∅	NOUN
ejpam-4302	192	30	for	for	ADP
ejpam-4302	192	31	every	every	DET
ejpam-4302	192	32	x	x	PROPN
ejpam-4302	192	33	∈	∈	PROPN
ejpam-4302	192	34	a(λ	a(λ	ADV
ejpam-4302	192	35	,	,	PUNCT
ejpam-4302	192	36	sp	sp	NOUN
ejpam-4302	192	37	)	)	PUNCT
ejpam-4302	192	38	.	.	PUNCT
ejpam-4302	193	1	proof	proof	NOUN
ejpam-4302	193	2	.	.	PUNCT
ejpam-4302	194	1	let	let	VERB
ejpam-4302	194	2	a	a	PRON
ejpam-4302	194	3	be	be	AUX
ejpam-4302	194	4	a	a	DET
ejpam-4302	194	5	g-(λ	g-(λ	NOUN
ejpam-4302	194	6	,	,	PUNCT
ejpam-4302	194	7	sp)-closed	sp)-close	VERB
ejpam-4302	194	8	set	set	ADJ
ejpam-4302	194	9	and	and	CCONJ
ejpam-4302	194	10	suppose	suppose	VERB
ejpam-4302	194	11	that	that	SCONJ
ejpam-4302	194	12	there	there	PRON
ejpam-4302	194	13	exists	exist	VERB
ejpam-4302	194	14	x	x	X
ejpam-4302	194	15	∈	∈	PROPN
ejpam-4302	194	16	a(λ	a(λ	ADV
ejpam-4302	194	17	,	,	PUNCT
ejpam-4302	194	18	sp	sp	NOUN
ejpam-4302	194	19	)	)	PUNCT
ejpam-4302	194	20	such	such	ADJ
ejpam-4302	194	21	that	that	SCONJ
ejpam-4302	194	22	a	a	DET
ejpam-4302	194	23	∩	∩	NOUN
ejpam-4302	194	24	{	{	PUNCT
ejpam-4302	194	25	x}(λ	x}(λ	PROPN
ejpam-4302	194	26	,	,	PUNCT
ejpam-4302	194	27	sp	sp	NOUN
ejpam-4302	194	28	)	)	PUNCT
ejpam-4302	194	29	=	=	NOUN
ejpam-4302	194	30	∅.	∅.	ADP
ejpam-4302	194	31	thus	thus	ADV
ejpam-4302	194	32	,	,	PUNCT
ejpam-4302	194	33	a	a	DET
ejpam-4302	194	34	⊆	⊆	NUM
ejpam-4302	194	35	x	x	SYM
ejpam-4302	194	36	−	−	PROPN
ejpam-4302	194	37	{	{	PUNCT
ejpam-4302	194	38	x}(λ	x}(λ	PROPN
ejpam-4302	194	39	,	,	PUNCT
ejpam-4302	194	40	sp	sp	NOUN
ejpam-4302	194	41	)	)	PUNCT
ejpam-4302	194	42	and	and	CCONJ
ejpam-4302	194	43	hence	hence	ADV
ejpam-4302	194	44	a(λ	a(λ	ADV
ejpam-4302	194	45	,	,	PUNCT
ejpam-4302	194	46	sp	sp	NOUN
ejpam-4302	194	47	)	)	PUNCT
ejpam-4302	194	48	⊆	⊆	NUM
ejpam-4302	194	49	x	x	SYM
ejpam-4302	194	50	−	−	PROPN
ejpam-4302	194	51	{	{	PUNCT
ejpam-4302	194	52	x}(λ	x}(λ	PROPN
ejpam-4302	194	53	,	,	PUNCT
ejpam-4302	194	54	sp	sp	NOUN
ejpam-4302	194	55	)	)	PUNCT
ejpam-4302	194	56	.	.	PUNCT
ejpam-4302	195	1	therefore	therefore	ADV
ejpam-4302	195	2	,	,	PUNCT
ejpam-4302	195	3	x	x	PROPN
ejpam-4302	195	4	̸∈	̸∈	PROPN
ejpam-4302	195	5	a(λ	a(λ	PROPN
ejpam-4302	195	6	,	,	PUNCT
ejpam-4302	195	7	sp	sp	NOUN
ejpam-4302	195	8	)	)	PUNCT
ejpam-4302	195	9	,	,	PUNCT
ejpam-4302	195	10	which	which	PRON
ejpam-4302	195	11	is	be	AUX
ejpam-4302	195	12	a	a	DET
ejpam-4302	195	13	contradiction	contradiction	NOUN
ejpam-4302	195	14	.	.	PUNCT
ejpam-4302	196	1	conversely	conversely	ADV
ejpam-4302	196	2	,	,	PUNCT
ejpam-4302	196	3	suppose	suppose	VERB
ejpam-4302	196	4	that	that	SCONJ
ejpam-4302	196	5	the	the	DET
ejpam-4302	196	6	condition	condition	NOUN
ejpam-4302	196	7	of	of	ADP
ejpam-4302	196	8	the	the	DET
ejpam-4302	196	9	theorem	theorem	NOUN
ejpam-4302	196	10	holds	hold	VERB
ejpam-4302	196	11	and	and	CCONJ
ejpam-4302	196	12	let	let	VERB
ejpam-4302	196	13	u	u	PRON
ejpam-4302	196	14	be	be	AUX
ejpam-4302	196	15	any	any	DET
ejpam-4302	196	16	(	(	PUNCT
ejpam-4302	196	17	λ	λ	NOUN
ejpam-4302	196	18	,	,	PUNCT
ejpam-4302	196	19	sp)open	sp)open	VERB
ejpam-4302	196	20	set	set	VERB
ejpam-4302	196	21	containing	contain	VERB
ejpam-4302	196	22	a.	a.	NOUN
ejpam-4302	196	23	let	let	VERB
ejpam-4302	196	24	x	x	X
ejpam-4302	196	25	∈	∈	PROPN
ejpam-4302	196	26	a(λ	a(λ	ADV
ejpam-4302	196	27	,	,	PUNCT
ejpam-4302	196	28	sp	sp	NOUN
ejpam-4302	196	29	)	)	PUNCT
ejpam-4302	196	30	.	.	PUNCT
ejpam-4302	197	1	by	by	ADP
ejpam-4302	197	2	the	the	DET
ejpam-4302	197	3	hypothesis	hypothesis	NOUN
ejpam-4302	197	4	,	,	PUNCT
ejpam-4302	197	5	a∩a(λ	a∩a(λ	ADV
ejpam-4302	197	6	,	,	PUNCT
ejpam-4302	197	7	sp	sp	NOUN
ejpam-4302	197	8	)	)	PUNCT
ejpam-4302	197	9	̸=	̸=	NOUN
ejpam-4302	197	10	∅	∅	NOUN
ejpam-4302	197	11	,	,	PUNCT
ejpam-4302	197	12	so	so	SCONJ
ejpam-4302	197	13	there	there	PRON
ejpam-4302	197	14	exists	exist	VERB
ejpam-4302	197	15	y	y	PROPN
ejpam-4302	197	16	∈	∈	PROPN
ejpam-4302	197	17	a	a	DET
ejpam-4302	197	18	∩	∩	NOUN
ejpam-4302	197	19	{	{	PUNCT
ejpam-4302	197	20	x}(λ	x}(λ	PROPN
ejpam-4302	197	21	,	,	PUNCT
ejpam-4302	197	22	sp	sp	NOUN
ejpam-4302	197	23	)	)	PUNCT
ejpam-4302	197	24	and	and	CCONJ
ejpam-4302	197	25	hence	hence	ADV
ejpam-4302	197	26	y	y	PROPN
ejpam-4302	197	27	∈	∈	PROPN
ejpam-4302	197	28	a	a	DET
ejpam-4302	197	29	⊆	⊆	NUM
ejpam-4302	197	30	u	u	NOUN
ejpam-4302	197	31	.	.	PUNCT
ejpam-4302	198	1	thus	thus	ADV
ejpam-4302	198	2	,	,	PUNCT
ejpam-4302	198	3	{	{	PUNCT
ejpam-4302	198	4	x	x	NOUN
ejpam-4302	198	5	}	}	PUNCT
ejpam-4302	198	6	∩	∩	NOUN
ejpam-4302	198	7	u	u	NOUN
ejpam-4302	198	8	̸=	̸=	PROPN
ejpam-4302	198	9	∅.	∅.	VERB
ejpam-4302	198	10	therefore	therefore	ADV
ejpam-4302	198	11	,	,	PUNCT
ejpam-4302	198	12	x	x	PUNCT
ejpam-4302	198	13	∈	∈	PROPN
ejpam-4302	198	14	u	u	NOUN
ejpam-4302	198	15	,	,	PUNCT
ejpam-4302	198	16	which	which	PRON
ejpam-4302	198	17	implies	imply	VERB
ejpam-4302	198	18	that	that	SCONJ
ejpam-4302	198	19	a(λ	a(λ	ADV
ejpam-4302	198	20	,	,	PUNCT
ejpam-4302	198	21	sp	sp	NOUN
ejpam-4302	198	22	)	)	PUNCT
ejpam-4302	198	23	⊆	⊆	NUM
ejpam-4302	198	24	u	u	NOUN
ejpam-4302	198	25	.	.	PUNCT
ejpam-4302	199	1	this	this	PRON
ejpam-4302	199	2	shows	show	VERB
ejpam-4302	199	3	that	that	SCONJ
ejpam-4302	199	4	a	a	PRON
ejpam-4302	199	5	is	be	AUX
ejpam-4302	199	6	g-(λ	g-(λ	PRON
ejpam-4302	199	7	,	,	PUNCT
ejpam-4302	199	8	sp)-closed	sp)-close	VERB
ejpam-4302	199	9	.	.	PUNCT
ejpam-4302	200	1	corollary	corollary	ADJ
ejpam-4302	200	2	3	3	NUM
ejpam-4302	200	3	.	.	PUNCT
ejpam-4302	201	1	for	for	ADP
ejpam-4302	201	2	a	a	DET
ejpam-4302	201	3	subset	subset	NOUN
ejpam-4302	201	4	a	a	PRON
ejpam-4302	201	5	of	of	ADP
ejpam-4302	201	6	a	a	DET
ejpam-4302	201	7	topological	topological	ADJ
ejpam-4302	201	8	space	space	NOUN
ejpam-4302	201	9	(	(	PUNCT
ejpam-4302	201	10	x	x	X
ejpam-4302	201	11	,	,	PUNCT
ejpam-4302	201	12	τ	τ	PROPN
ejpam-4302	201	13	)	)	PUNCT
ejpam-4302	201	14	,	,	PUNCT
ejpam-4302	201	15	the	the	DET
ejpam-4302	201	16	following	follow	VERB
ejpam-4302	201	17	properties	property	NOUN
ejpam-4302	201	18	are	be	AUX
ejpam-4302	201	19	equivalent	equivalent	ADJ
ejpam-4302	201	20	:	:	PUNCT
ejpam-4302	201	21	(	(	PUNCT
ejpam-4302	201	22	1	1	X
ejpam-4302	201	23	)	)	PUNCT
ejpam-4302	201	24	a	a	PRON
ejpam-4302	201	25	is	be	AUX
ejpam-4302	201	26	g-(λ	g-(λ	PROPN
ejpam-4302	201	27	,	,	PUNCT
ejpam-4302	201	28	sp)-open	sp)-open	NOUN
ejpam-4302	201	29	.	.	PUNCT
ejpam-4302	202	1	c.	c.	PROPN
ejpam-4302	202	2	boonpok	boonpok	PROPN
ejpam-4302	202	3	,	,	PUNCT
ejpam-4302	202	4	c.	c.	PROPN
ejpam-4302	202	5	viriyapong	viriyapong	PROPN
ejpam-4302	202	6	/	/	SYM
ejpam-4302	202	7	eur	eur	PROPN
ejpam-4302	202	8	.	.	PUNCT
ejpam-4302	203	1	j.	j.	PROPN
ejpam-4302	203	2	pure	pure	PROPN
ejpam-4302	203	3	appl	appl	PROPN
ejpam-4302	203	4	.	.	PROPN
ejpam-4302	203	5	math	math	PROPN
ejpam-4302	203	6	,	,	PUNCT
ejpam-4302	203	7	15	15	NUM
ejpam-4302	203	8	(	(	PUNCT
ejpam-4302	203	9	4	4	NUM
ejpam-4302	203	10	)	)	PUNCT
ejpam-4302	203	11	(	(	PUNCT
ejpam-4302	203	12	2022	2022	NUM
ejpam-4302	203	13	)	)	PUNCT
ejpam-4302	203	14	,	,	PUNCT
ejpam-4302	203	15	2127	2127	NUM
ejpam-4302	203	16	-	-	SYM
ejpam-4302	203	17	2140	2140	NUM
ejpam-4302	203	18	2133	2133	NUM
ejpam-4302	203	19	(	(	PUNCT
ejpam-4302	203	20	2	2	NUM
ejpam-4302	203	21	)	)	PUNCT
ejpam-4302	203	22	a−a(λ	a−a(λ	ADJ
ejpam-4302	203	23	,	,	PUNCT
ejpam-4302	203	24	sp	sp	NOUN
ejpam-4302	203	25	)	)	PUNCT
ejpam-4302	203	26	does	do	AUX
ejpam-4302	203	27	not	not	PART
ejpam-4302	203	28	contain	contain	VERB
ejpam-4302	203	29	any	any	PRON
ejpam-4302	203	30	nonempty	nonempty	ADJ
ejpam-4302	203	31	(	(	PUNCT
ejpam-4302	203	32	λ	λ	NOUN
ejpam-4302	203	33	,	,	PUNCT
ejpam-4302	203	34	sp)-closed	sp)-close	VERB
ejpam-4302	203	35	set	set	VERB
ejpam-4302	203	36	.	.	PUNCT
ejpam-4302	204	1	(	(	PUNCT
ejpam-4302	204	2	3	3	NUM
ejpam-4302	204	3	)	)	PUNCT
ejpam-4302	204	4	(	(	PUNCT
ejpam-4302	204	5	x	x	NOUN
ejpam-4302	204	6	−a	−a	ADJ
ejpam-4302	204	7	)	)	PUNCT
ejpam-4302	204	8	∩	∩	NOUN
ejpam-4302	204	9	{	{	PUNCT
ejpam-4302	204	10	x}(λ	x}(λ	PROPN
ejpam-4302	204	11	,	,	PUNCT
ejpam-4302	204	12	sp	sp	NOUN
ejpam-4302	204	13	)	)	PUNCT
ejpam-4302	204	14	̸=	̸=	PROPN
ejpam-4302	204	15	∅	∅	NOUN
ejpam-4302	204	16	for	for	ADP
ejpam-4302	204	17	every	every	DET
ejpam-4302	204	18	x	x	PROPN
ejpam-4302	204	19	∈	∈	PROPN
ejpam-4302	204	20	a−a(λ	a−a(λ	ADJ
ejpam-4302	204	21	,	,	PUNCT
ejpam-4302	204	22	sp	sp	NOUN
ejpam-4302	204	23	)	)	PUNCT
ejpam-4302	204	24	.	.	PUNCT
ejpam-4302	205	1	theorem	theorem	ADJ
ejpam-4302	205	2	10	10	NUM
ejpam-4302	205	3	.	.	PUNCT
ejpam-4302	206	1	for	for	ADP
ejpam-4302	206	2	a	a	DET
ejpam-4302	206	3	topological	topological	ADJ
ejpam-4302	206	4	space	space	NOUN
ejpam-4302	206	5	(	(	PUNCT
ejpam-4302	206	6	x	x	X
ejpam-4302	206	7	,	,	PUNCT
ejpam-4302	206	8	τ	τ	PROPN
ejpam-4302	206	9	)	)	PUNCT
ejpam-4302	206	10	,	,	PUNCT
ejpam-4302	206	11	the	the	DET
ejpam-4302	206	12	following	follow	VERB
ejpam-4302	206	13	properties	property	NOUN
ejpam-4302	206	14	are	be	AUX
ejpam-4302	206	15	equivalent	equivalent	ADJ
ejpam-4302	206	16	:	:	PUNCT
ejpam-4302	206	17	(	(	PUNCT
ejpam-4302	206	18	1	1	X
ejpam-4302	206	19	)	)	PUNCT
ejpam-4302	206	20	for	for	ADP
ejpam-4302	206	21	every	every	DET
ejpam-4302	206	22	(	(	PUNCT
ejpam-4302	206	23	λ	λ	NOUN
ejpam-4302	206	24	,	,	PUNCT
ejpam-4302	206	25	sp)-open	sp)-open	NOUN
ejpam-4302	206	26	set	set	VERB
ejpam-4302	206	27	u	u	NOUN
ejpam-4302	206	28	of	of	ADP
ejpam-4302	206	29	x	x	PROPN
ejpam-4302	206	30	,	,	PUNCT
ejpam-4302	206	31	u	u	PROPN
ejpam-4302	206	32	(	(	PUNCT
ejpam-4302	206	33	λ	λ	PROPN
ejpam-4302	206	34	,	,	PUNCT
ejpam-4302	206	35	sp	sp	NOUN
ejpam-4302	206	36	)	)	PUNCT
ejpam-4302	206	37	⊆	⊆	NUM
ejpam-4302	206	38	u	u	NOUN
ejpam-4302	206	39	.	.	PUNCT
ejpam-4302	207	1	(	(	PUNCT
ejpam-4302	207	2	2	2	X
ejpam-4302	207	3	)	)	PUNCT
ejpam-4302	207	4	every	every	DET
ejpam-4302	207	5	subset	subset	NOUN
ejpam-4302	207	6	of	of	ADP
ejpam-4302	207	7	x	x	PUNCT
ejpam-4302	207	8	is	be	AUX
ejpam-4302	207	9	g-(λ	g-(λ	PRON
ejpam-4302	207	10	,	,	PUNCT
ejpam-4302	207	11	sp)-closed	sp)-closed	ADJ
ejpam-4302	207	12	.	.	PUNCT
ejpam-4302	208	1	proof	proof	NOUN
ejpam-4302	208	2	.	.	PUNCT
ejpam-4302	209	1	(	(	PUNCT
ejpam-4302	209	2	1	1	X
ejpam-4302	209	3	)	)	PUNCT
ejpam-4302	209	4	⇒	⇒	NOUN
ejpam-4302	209	5	(	(	PUNCT
ejpam-4302	209	6	2	2	NUM
ejpam-4302	209	7	):	):	PUNCT
ejpam-4302	209	8	let	let	VERB
ejpam-4302	209	9	a	a	PRON
ejpam-4302	209	10	be	be	AUX
ejpam-4302	209	11	any	any	DET
ejpam-4302	209	12	subset	subset	NOUN
ejpam-4302	209	13	of	of	ADP
ejpam-4302	209	14	x	x	PUNCT
ejpam-4302	209	15	and	and	CCONJ
ejpam-4302	209	16	a	a	DET
ejpam-4302	209	17	⊆	⊆	NUM
ejpam-4302	209	18	u	u	NOUN
ejpam-4302	209	19	∈	∈	PROPN
ejpam-4302	209	20	λspo(x	λspo(x	PROPN
ejpam-4302	209	21	,	,	PUNCT
ejpam-4302	209	22	τ	τ	PROPN
ejpam-4302	209	23	)	)	PUNCT
ejpam-4302	209	24	.	.	PUNCT
ejpam-4302	210	1	by	by	ADP
ejpam-4302	210	2	(	(	PUNCT
ejpam-4302	210	3	1	1	NUM
ejpam-4302	210	4	)	)	PUNCT
ejpam-4302	210	5	,	,	PUNCT
ejpam-4302	210	6	u	u	NOUN
ejpam-4302	210	7	(	(	PUNCT
ejpam-4302	210	8	λ	λ	PROPN
ejpam-4302	210	9	,	,	PUNCT
ejpam-4302	210	10	sp	sp	NOUN
ejpam-4302	210	11	)	)	PUNCT
ejpam-4302	210	12	⊆	⊆	NUM
ejpam-4302	210	13	u	u	NOUN
ejpam-4302	210	14	and	and	CCONJ
ejpam-4302	210	15	hence	hence	ADV
ejpam-4302	210	16	a(λ	a(λ	ADV
ejpam-4302	210	17	,	,	PUNCT
ejpam-4302	210	18	sp	sp	NOUN
ejpam-4302	210	19	)	)	PUNCT
ejpam-4302	210	20	⊆	⊆	NUM
ejpam-4302	210	21	u	u	NOUN
ejpam-4302	210	22	(	(	PUNCT
ejpam-4302	210	23	λ	λ	PROPN
ejpam-4302	210	24	,	,	PUNCT
ejpam-4302	210	25	sp	sp	NOUN
ejpam-4302	210	26	)	)	PUNCT
ejpam-4302	210	27	⊆	⊆	NUM
ejpam-4302	210	28	u	u	NOUN
ejpam-4302	210	29	.	.	PUNCT
ejpam-4302	211	1	thus	thus	ADV
ejpam-4302	211	2	,	,	PUNCT
ejpam-4302	211	3	a	a	PRON
ejpam-4302	211	4	is	be	AUX
ejpam-4302	211	5	g-(λ	g-(λ	PRON
ejpam-4302	211	6	,	,	PUNCT
ejpam-4302	211	7	sp)-closed	sp)-close	VERB
ejpam-4302	211	8	.	.	PUNCT
ejpam-4302	212	1	(	(	PUNCT
ejpam-4302	212	2	2	2	X
ejpam-4302	212	3	)	)	PUNCT
ejpam-4302	212	4	⇒	⇒	NOUN
ejpam-4302	212	5	(	(	PUNCT
ejpam-4302	212	6	1	1	NUM
ejpam-4302	212	7	):	):	PUNCT
ejpam-4302	212	8	let	let	VERB
ejpam-4302	212	9	u	u	PRON
ejpam-4302	212	10	∈	∈	PROPN
ejpam-4302	212	11	λspo(x	λspo(x	PROPN
ejpam-4302	212	12	,	,	PUNCT
ejpam-4302	212	13	τ	τ	PROPN
ejpam-4302	212	14	)	)	PUNCT
ejpam-4302	212	15	.	.	PUNCT
ejpam-4302	213	1	by	by	ADP
ejpam-4302	213	2	(	(	PUNCT
ejpam-4302	213	3	2	2	NUM
ejpam-4302	213	4	)	)	PUNCT
ejpam-4302	213	5	,	,	PUNCT
ejpam-4302	213	6	u	u	PROPN
ejpam-4302	213	7	is	be	AUX
ejpam-4302	213	8	g-(λ	g-(λ	PROPN
ejpam-4302	213	9	,	,	PUNCT
ejpam-4302	213	10	sp)-closed	sp)-close	VERB
ejpam-4302	213	11	and	and	CCONJ
ejpam-4302	213	12	hence	hence	ADV
ejpam-4302	213	13	u	u	NOUN
ejpam-4302	213	14	(	(	PUNCT
ejpam-4302	213	15	λ	λ	PROPN
ejpam-4302	213	16	,	,	PUNCT
ejpam-4302	213	17	sp	sp	NOUN
ejpam-4302	213	18	)	)	PUNCT
ejpam-4302	213	19	⊆	⊆	NUM
ejpam-4302	213	20	u	u	NOUN
ejpam-4302	213	21	.	.	PUNCT
ejpam-4302	214	1	theorem	theorem	VERB
ejpam-4302	214	2	11	11	NUM
ejpam-4302	214	3	.	.	PUNCT
ejpam-4302	215	1	a	a	DET
ejpam-4302	215	2	subset	subset	NOUN
ejpam-4302	215	3	a	a	PRON
ejpam-4302	215	4	of	of	ADP
ejpam-4302	215	5	a	a	DET
ejpam-4302	215	6	topological	topological	ADJ
ejpam-4302	215	7	space	space	NOUN
ejpam-4302	215	8	(	(	PUNCT
ejpam-4302	215	9	x	x	X
ejpam-4302	215	10	,	,	PUNCT
ejpam-4302	215	11	τ	τ	X
ejpam-4302	215	12	)	)	PUNCT
ejpam-4302	215	13	is	be	AUX
ejpam-4302	215	14	g-(λ	g-(λ	PROPN
ejpam-4302	215	15	,	,	PUNCT
ejpam-4302	215	16	sp)-open	sp)-open	ADJ
ejpam-4302	215	17	if	if	SCONJ
ejpam-4302	215	18	and	and	CCONJ
ejpam-4302	215	19	only	only	ADV
ejpam-4302	215	20	if	if	SCONJ
ejpam-4302	215	21	u	u	NOUN
ejpam-4302	215	22	=	=	NOUN
ejpam-4302	215	23	x	x	INTJ
ejpam-4302	215	24	whenever	whenever	SCONJ
ejpam-4302	215	25	u	u	NOUN
ejpam-4302	215	26	is	be	AUX
ejpam-4302	215	27	(	(	PUNCT
ejpam-4302	215	28	λ	λ	X
ejpam-4302	215	29	,	,	PUNCT
ejpam-4302	215	30	sp)-open	sp)-open	ADJ
ejpam-4302	215	31	and	and	CCONJ
ejpam-4302	215	32	(	(	PUNCT
ejpam-4302	215	33	x	x	NOUN
ejpam-4302	215	34	−a	−a	ADJ
ejpam-4302	215	35	)	)	PUNCT
ejpam-4302	215	36	∩a(λ	∩a(λ	NOUN
ejpam-4302	215	37	,	,	PUNCT
ejpam-4302	215	38	sp	sp	NOUN
ejpam-4302	215	39	)	)	PUNCT
ejpam-4302	215	40	⊆	⊆	NUM
ejpam-4302	215	41	u	u	NOUN
ejpam-4302	215	42	.	.	PUNCT
ejpam-4302	216	1	proof	proof	NOUN
ejpam-4302	216	2	.	.	PUNCT
ejpam-4302	217	1	suppose	suppose	VERB
ejpam-4302	217	2	that	that	SCONJ
ejpam-4302	217	3	a	a	PRON
ejpam-4302	217	4	is	be	AUX
ejpam-4302	217	5	g-(λ	g-(λ	PROPN
ejpam-4302	217	6	,	,	PUNCT
ejpam-4302	217	7	sp)-open	sp)-open	ADJ
ejpam-4302	217	8	and	and	CCONJ
ejpam-4302	217	9	u	u	NOUN
ejpam-4302	217	10	∈	∈	PROPN
ejpam-4302	217	11	λspo(x	λspo(x	PROPN
ejpam-4302	217	12	,	,	PUNCT
ejpam-4302	217	13	τ	τ	PROPN
ejpam-4302	217	14	)	)	PUNCT
ejpam-4302	217	15	such	such	ADJ
ejpam-4302	217	16	that	that	SCONJ
ejpam-4302	217	17	(	(	PUNCT
ejpam-4302	217	18	x	x	NOUN
ejpam-4302	217	19	−a	−a	ADJ
ejpam-4302	217	20	)	)	PUNCT
ejpam-4302	217	21	∩a(λ	∩a(λ	NOUN
ejpam-4302	217	22	,	,	PUNCT
ejpam-4302	217	23	sp	sp	NOUN
ejpam-4302	217	24	)	)	PUNCT
ejpam-4302	217	25	⊆	⊆	NUM
ejpam-4302	217	26	u.	u.	NOUN
ejpam-4302	217	27	thus	thus	ADV
ejpam-4302	217	28	,	,	PUNCT
ejpam-4302	217	29	x	x	PUNCT
ejpam-4302	217	30	−	−	NOUN
ejpam-4302	217	31	u	u	NOUN
ejpam-4302	217	32	⊆	⊆	NUM
ejpam-4302	217	33	[	[	X
ejpam-4302	217	34	x	x	X
ejpam-4302	217	35	−	−	NOUN
ejpam-4302	217	36	a(λ	a(λ	ADV
ejpam-4302	217	37	,	,	PUNCT
ejpam-4302	217	38	sp	sp	NOUN
ejpam-4302	217	39	)	)	PUNCT
ejpam-4302	217	40	]	]	PUNCT
ejpam-4302	217	41	∩	∩	NOUN
ejpam-4302	217	42	a	a	X
ejpam-4302	217	43	and	and	CCONJ
ejpam-4302	217	44	hence	hence	ADV
ejpam-4302	217	45	x	x	ADP
ejpam-4302	217	46	−	−	NOUN
ejpam-4302	217	47	u	u	NOUN
ejpam-4302	217	48	⊆	⊆	NUM
ejpam-4302	217	49	[	[	X
ejpam-4302	217	50	x	x	X
ejpam-4302	217	51	−	−	NOUN
ejpam-4302	217	52	a](λ	a](λ	NOUN
ejpam-4302	217	53	,	,	PUNCT
ejpam-4302	217	54	sp	sp	NOUN
ejpam-4302	217	55	)	)	PUNCT
ejpam-4302	217	56	−	−	PROPN
ejpam-4302	218	1	(	(	PUNCT
ejpam-4302	218	2	x	x	X
ejpam-4302	218	3	−	−	NOUN
ejpam-4302	218	4	a	a	NOUN
ejpam-4302	218	5	)	)	PUNCT
ejpam-4302	218	6	.	.	PUNCT
ejpam-4302	219	1	since	since	SCONJ
ejpam-4302	219	2	x	x	X
ejpam-4302	219	3	−	−	PROPN
ejpam-4302	219	4	a	a	PRON
ejpam-4302	219	5	is	be	AUX
ejpam-4302	219	6	g-(λ	g-(λ	PROPN
ejpam-4302	219	7	,	,	PUNCT
ejpam-4302	219	8	sp)-closed	sp)-close	VERB
ejpam-4302	219	9	and	and	CCONJ
ejpam-4302	220	1	x	x	SYM
ejpam-4302	220	2	−	−	NOUN
ejpam-4302	220	3	u	u	NOUN
ejpam-4302	220	4	is	be	AUX
ejpam-4302	220	5	(	(	PUNCT
ejpam-4302	220	6	λ	λ	X
ejpam-4302	220	7	,	,	PUNCT
ejpam-4302	220	8	sp)-closed	sp)-close	VERB
ejpam-4302	220	9	,	,	PUNCT
ejpam-4302	220	10	by	by	ADP
ejpam-4302	220	11	theorem	theorem	NOUN
ejpam-4302	220	12	2	2	NUM
ejpam-4302	220	13	,	,	PUNCT
ejpam-4302	220	14	x	x	PUNCT
ejpam-4302	220	15	−	−	NOUN
ejpam-4302	220	16	u	u	NOUN
ejpam-4302	220	17	=	=	PUNCT
ejpam-4302	220	18	∅.	∅.	NOUN
ejpam-4302	220	19	this	this	PRON
ejpam-4302	220	20	shows	show	VERB
ejpam-4302	220	21	that	that	SCONJ
ejpam-4302	220	22	x	x	X
ejpam-4302	220	23	=	=	SYM
ejpam-4302	220	24	u	u	NOUN
ejpam-4302	220	25	.	.	PUNCT
ejpam-4302	221	1	conversely	conversely	ADV
ejpam-4302	221	2	,	,	PUNCT
ejpam-4302	221	3	suppose	suppose	VERB
ejpam-4302	221	4	that	that	SCONJ
ejpam-4302	221	5	f	f	PROPN
ejpam-4302	221	6	⊆	⊆	NUM
ejpam-4302	221	7	a	a	PRON
ejpam-4302	221	8	and	and	CCONJ
ejpam-4302	221	9	f	f	NOUN
ejpam-4302	221	10	is	be	AUX
ejpam-4302	221	11	(	(	PUNCT
ejpam-4302	221	12	λ	λ	X
ejpam-4302	221	13	,	,	PUNCT
ejpam-4302	221	14	sp)-closed	sp)-close	VERB
ejpam-4302	221	15	.	.	PUNCT
ejpam-4302	222	1	by	by	ADP
ejpam-4302	222	2	lemma	lemma	PROPN
ejpam-4302	222	3	4	4	NUM
ejpam-4302	222	4	,	,	PUNCT
ejpam-4302	222	5	(	(	PUNCT
ejpam-4302	222	6	x	x	NOUN
ejpam-4302	222	7	−a	−a	NOUN
ejpam-4302	222	8	)	)	PUNCT
ejpam-4302	222	9	∪a(λ	∪a(λ	PROPN
ejpam-4302	222	10	,	,	PUNCT
ejpam-4302	222	11	sp	sp	NOUN
ejpam-4302	222	12	)	)	PUNCT
ejpam-4302	222	13	⊆	⊆	NUM
ejpam-4302	222	14	(	(	PUNCT
ejpam-4302	222	15	x	x	SYM
ejpam-4302	222	16	−	−	PROPN
ejpam-4302	222	17	f	f	PROPN
ejpam-4302	222	18	)	)	PUNCT
ejpam-4302	222	19	∪a(λ	∪a(λ	PROPN
ejpam-4302	222	20	,	,	PUNCT
ejpam-4302	222	21	sp	sp	NOUN
ejpam-4302	222	22	)	)	PUNCT
ejpam-4302	222	23	∈	∈	PROPN
ejpam-4302	222	24	λspo(x	λspo(x	PROPN
ejpam-4302	222	25	,	,	PUNCT
ejpam-4302	222	26	τ	τ	PROPN
ejpam-4302	222	27	)	)	PUNCT
ejpam-4302	222	28	.	.	PUNCT
ejpam-4302	223	1	by	by	ADP
ejpam-4302	223	2	the	the	DET
ejpam-4302	223	3	hypothesis	hypothesis	NOUN
ejpam-4302	223	4	,	,	PUNCT
ejpam-4302	223	5	we	we	PRON
ejpam-4302	223	6	have	have	VERB
ejpam-4302	223	7	x	x	X
ejpam-4302	223	8	=	=	PRON
ejpam-4302	223	9	(	(	PUNCT
ejpam-4302	223	10	x	x	SYM
ejpam-4302	223	11	−	−	PROPN
ejpam-4302	223	12	f	f	PROPN
ejpam-4302	223	13	)	)	PUNCT
ejpam-4302	223	14	∪a(λ	∪a(λ	PROPN
ejpam-4302	223	15	,	,	PUNCT
ejpam-4302	223	16	sp	sp	NOUN
ejpam-4302	223	17	)	)	PUNCT
ejpam-4302	223	18	and	and	CCONJ
ejpam-4302	223	19	hence	hence	ADV
ejpam-4302	223	20	f	f	PROPN
ejpam-4302	223	21	=	=	SYM
ejpam-4302	223	22	f	f	PROPN
ejpam-4302	223	23	∩	∩	X
ejpam-4302	223	24	[	[	X
ejpam-4302	223	25	(	(	PUNCT
ejpam-4302	223	26	x	x	SYM
ejpam-4302	223	27	−	−	PROPN
ejpam-4302	223	28	f	f	PROPN
ejpam-4302	223	29	)	)	PUNCT
ejpam-4302	223	30	∪a(λ	∪a(λ	PROPN
ejpam-4302	223	31	,	,	PUNCT
ejpam-4302	223	32	sp	sp	NOUN
ejpam-4302	223	33	)	)	PUNCT
ejpam-4302	223	34	]	]	PUNCT
ejpam-4302	224	1	=	=	PUNCT
ejpam-4302	224	2	f	f	X
ejpam-4302	224	3	∩a(λ	∩a(λ	NOUN
ejpam-4302	224	4	,	,	PUNCT
ejpam-4302	224	5	sp	sp	NOUN
ejpam-4302	224	6	)	)	PUNCT
ejpam-4302	224	7	⊆	⊆	NUM
ejpam-4302	224	8	a(λ	a(λ	ADV
ejpam-4302	224	9	,	,	PUNCT
ejpam-4302	224	10	sp	sp	NOUN
ejpam-4302	224	11	)	)	PUNCT
ejpam-4302	224	12	.	.	PUNCT
ejpam-4302	225	1	it	it	PRON
ejpam-4302	225	2	follows	follow	VERB
ejpam-4302	225	3	from	from	ADP
ejpam-4302	225	4	theorem	theorem	NOUN
ejpam-4302	225	5	6	6	NUM
ejpam-4302	225	6	that	that	SCONJ
ejpam-4302	225	7	a	a	PRON
ejpam-4302	225	8	is	be	AUX
ejpam-4302	225	9	g-(λ	g-(λ	PROPN
ejpam-4302	225	10	,	,	PUNCT
ejpam-4302	225	11	sp)-open	sp)-open	ADJ
ejpam-4302	225	12	.	.	PUNCT
ejpam-4302	226	1	theorem	theorem	NOUN
ejpam-4302	226	2	12	12	NUM
ejpam-4302	226	3	.	.	PUNCT
ejpam-4302	227	1	let	let	VERB
ejpam-4302	227	2	a	a	DET
ejpam-4302	227	3	be	be	AUX
ejpam-4302	227	4	a	a	DET
ejpam-4302	227	5	subset	subset	NOUN
ejpam-4302	227	6	of	of	ADP
ejpam-4302	227	7	a	a	DET
ejpam-4302	227	8	topological	topological	ADJ
ejpam-4302	227	9	space	space	NOUN
ejpam-4302	227	10	(	(	PUNCT
ejpam-4302	227	11	x	x	X
ejpam-4302	227	12	,	,	PUNCT
ejpam-4302	227	13	τ	τ	PROPN
ejpam-4302	227	14	)	)	PUNCT
ejpam-4302	227	15	.	.	PUNCT
ejpam-4302	228	1	if	if	SCONJ
ejpam-4302	228	2	a	a	PRON
ejpam-4302	228	3	is	be	AUX
ejpam-4302	228	4	g-(λ	g-(λ	PROPN
ejpam-4302	228	5	,	,	PUNCT
ejpam-4302	228	6	sp)-open	sp)-open	ADJ
ejpam-4302	228	7	and	and	CCONJ
ejpam-4302	228	8	a(λ	a(λ	ADV
ejpam-4302	228	9	,	,	PUNCT
ejpam-4302	228	10	sp	sp	NOUN
ejpam-4302	228	11	)	)	PUNCT
ejpam-4302	228	12	⊆	⊆	NUM
ejpam-4302	228	13	b	b	NOUN
ejpam-4302	228	14	⊆	⊆	NUM
ejpam-4302	228	15	a	a	PRON
ejpam-4302	228	16	,	,	PUNCT
ejpam-4302	228	17	then	then	ADV
ejpam-4302	228	18	b	b	PROPN
ejpam-4302	228	19	is	be	AUX
ejpam-4302	228	20	g-(λ	g-(λ	PROPN
ejpam-4302	228	21	,	,	PUNCT
ejpam-4302	228	22	sp)-open	sp)-open	ADJ
ejpam-4302	228	23	.	.	PUNCT
ejpam-4302	229	1	proof	proof	NOUN
ejpam-4302	229	2	.	.	PUNCT
ejpam-4302	230	1	we	we	PRON
ejpam-4302	230	2	have	have	VERB
ejpam-4302	230	3	x	x	ADJ
ejpam-4302	230	4	−	−	VERB
ejpam-4302	230	5	a	a	PRON
ejpam-4302	230	6	⊆	⊆	NUM
ejpam-4302	230	7	x	x	SYM
ejpam-4302	230	8	−	−	PROPN
ejpam-4302	230	9	b	b	NOUN
ejpam-4302	230	10	⊆	⊆	NUM
ejpam-4302	230	11	x	x	SYM
ejpam-4302	230	12	−	−	NOUN
ejpam-4302	230	13	a(λ	a(λ	ADV
ejpam-4302	230	14	,	,	PUNCT
ejpam-4302	230	15	sp	sp	NOUN
ejpam-4302	230	16	)	)	PUNCT
ejpam-4302	230	17	=	=	PUNCT
ejpam-4302	231	1	[	[	X
ejpam-4302	231	2	x	x	X
ejpam-4302	231	3	−	−	NOUN
ejpam-4302	231	4	a](λ	a](λ	NOUN
ejpam-4302	231	5	,	,	PUNCT
ejpam-4302	231	6	sp	sp	NOUN
ejpam-4302	231	7	)	)	PUNCT
ejpam-4302	231	8	.	.	PUNCT
ejpam-4302	232	1	since	since	SCONJ
ejpam-4302	232	2	x	x	X
ejpam-4302	232	3	−	−	PROPN
ejpam-4302	232	4	a	a	PRON
ejpam-4302	232	5	is	be	AUX
ejpam-4302	232	6	g-(λ	g-(λ	PROPN
ejpam-4302	232	7	,	,	PUNCT
ejpam-4302	232	8	sp)-closed	sp)-close	VERB
ejpam-4302	232	9	,	,	PUNCT
ejpam-4302	232	10	it	it	PRON
ejpam-4302	232	11	follows	follow	VERB
ejpam-4302	232	12	from	from	ADP
ejpam-4302	232	13	theorem	theorem	ADJ
ejpam-4302	232	14	3	3	NUM
ejpam-4302	232	15	that	that	PRON
ejpam-4302	232	16	x	x	PRON
ejpam-4302	232	17	−b	−b	NOUN
ejpam-4302	232	18	is	be	AUX
ejpam-4302	232	19	g-(λ	g-(λ	PROPN
ejpam-4302	232	20	,	,	PUNCT
ejpam-4302	232	21	sp)-closed	sp)-close	VERB
ejpam-4302	232	22	and	and	CCONJ
ejpam-4302	232	23	hence	hence	ADV
ejpam-4302	232	24	b	b	PROPN
ejpam-4302	232	25	is	be	AUX
ejpam-4302	232	26	g-(λ	g-(λ	PROPN
ejpam-4302	232	27	,	,	PUNCT
ejpam-4302	232	28	sp)-open	sp)-open	NOUN
ejpam-4302	232	29	.	.	PUNCT
ejpam-4302	233	1	definition	definition	NOUN
ejpam-4302	233	2	4	4	NUM
ejpam-4302	233	3	.	.	PUNCT
ejpam-4302	234	1	[	[	X
ejpam-4302	234	2	2	2	X
ejpam-4302	234	3	]	]	PUNCT
ejpam-4302	234	4	a	a	DET
ejpam-4302	234	5	subset	subset	NOUN
ejpam-4302	234	6	a	a	PRON
ejpam-4302	234	7	of	of	ADP
ejpam-4302	234	8	a	a	DET
ejpam-4302	234	9	topological	topological	ADJ
ejpam-4302	234	10	space	space	NOUN
ejpam-4302	234	11	(	(	PUNCT
ejpam-4302	234	12	x	x	X
ejpam-4302	234	13	,	,	PUNCT
ejpam-4302	234	14	τ	τ	X
ejpam-4302	234	15	)	)	PUNCT
ejpam-4302	234	16	is	be	AUX
ejpam-4302	234	17	said	say	VERB
ejpam-4302	234	18	to	to	PART
ejpam-4302	234	19	be	be	AUX
ejpam-4302	234	20	locally	locally	ADV
ejpam-4302	234	21	(	(	PUNCT
ejpam-4302	234	22	λ	λ	X
ejpam-4302	234	23	,	,	PUNCT
ejpam-4302	234	24	sp)-closed	sp)-close	VERB
ejpam-4302	234	25	if	if	SCONJ
ejpam-4302	234	26	a	a	DET
ejpam-4302	234	27	=	=	X
ejpam-4302	234	28	u	u	NOUN
ejpam-4302	234	29	∩	∩	NOUN
ejpam-4302	234	30	f	f	PROPN
ejpam-4302	234	31	,	,	PUNCT
ejpam-4302	234	32	where	where	SCONJ
ejpam-4302	234	33	u	u	PROPN
ejpam-4302	234	34	∈	∈	PROPN
ejpam-4302	234	35	λspo(x	λspo(x	PROPN
ejpam-4302	234	36	,	,	PUNCT
ejpam-4302	234	37	τ	τ	PROPN
ejpam-4302	234	38	)	)	PUNCT
ejpam-4302	234	39	and	and	CCONJ
ejpam-4302	234	40	f	f	PROPN
ejpam-4302	234	41	is	be	AUX
ejpam-4302	234	42	a	a	DET
ejpam-4302	234	43	(	(	PUNCT
ejpam-4302	234	44	λ	λ	PROPN
ejpam-4302	234	45	,	,	PUNCT
ejpam-4302	234	46	sp)-closed	sp)-close	VERB
ejpam-4302	234	47	set	set	VERB
ejpam-4302	234	48	.	.	PUNCT
ejpam-4302	235	1	lemma	lemma	PROPN
ejpam-4302	235	2	6	6	NUM
ejpam-4302	235	3	.	.	PUNCT
ejpam-4302	236	1	[	[	X
ejpam-4302	236	2	2	2	NUM
ejpam-4302	236	3	]	]	PUNCT
ejpam-4302	236	4	for	for	ADP
ejpam-4302	236	5	a	a	DET
ejpam-4302	236	6	subset	subset	NOUN
ejpam-4302	236	7	a	a	PRON
ejpam-4302	236	8	of	of	ADP
ejpam-4302	236	9	a	a	DET
ejpam-4302	236	10	topological	topological	ADJ
ejpam-4302	236	11	space	space	NOUN
ejpam-4302	236	12	(	(	PUNCT
ejpam-4302	236	13	x	x	X
ejpam-4302	236	14	,	,	PUNCT
ejpam-4302	236	15	τ	τ	PROPN
ejpam-4302	236	16	)	)	PUNCT
ejpam-4302	236	17	,	,	PUNCT
ejpam-4302	236	18	the	the	DET
ejpam-4302	236	19	following	follow	VERB
ejpam-4302	236	20	properties	property	NOUN
ejpam-4302	236	21	are	be	AUX
ejpam-4302	236	22	equivalent	equivalent	ADJ
ejpam-4302	236	23	:	:	PUNCT
ejpam-4302	236	24	c.	c.	PROPN
ejpam-4302	236	25	boonpok	boonpok	PROPN
ejpam-4302	236	26	,	,	PUNCT
ejpam-4302	236	27	c.	c.	PROPN
ejpam-4302	236	28	viriyapong	viriyapong	PROPN
ejpam-4302	236	29	/	/	SYM
ejpam-4302	236	30	eur	eur	PROPN
ejpam-4302	236	31	.	.	PUNCT
ejpam-4302	237	1	j.	j.	PROPN
ejpam-4302	237	2	pure	pure	PROPN
ejpam-4302	237	3	appl	appl	PROPN
ejpam-4302	237	4	.	.	PROPN
ejpam-4302	237	5	math	math	PROPN
ejpam-4302	237	6	,	,	PUNCT
ejpam-4302	237	7	15	15	NUM
ejpam-4302	237	8	(	(	PUNCT
ejpam-4302	237	9	4	4	NUM
ejpam-4302	237	10	)	)	PUNCT
ejpam-4302	237	11	(	(	PUNCT
ejpam-4302	237	12	2022	2022	NUM
ejpam-4302	237	13	)	)	PUNCT
ejpam-4302	237	14	,	,	PUNCT
ejpam-4302	237	15	2127	2127	NUM
ejpam-4302	237	16	-	-	SYM
ejpam-4302	237	17	2140	2140	NUM
ejpam-4302	237	18	2134	2134	NUM
ejpam-4302	237	19	(	(	PUNCT
ejpam-4302	237	20	1	1	X
ejpam-4302	237	21	)	)	PUNCT
ejpam-4302	237	22	a	a	PRON
ejpam-4302	237	23	is	be	AUX
ejpam-4302	237	24	locally	locally	ADV
ejpam-4302	237	25	(	(	PUNCT
ejpam-4302	237	26	λ	λ	NOUN
ejpam-4302	237	27	,	,	PUNCT
ejpam-4302	237	28	sp)-closed	sp)-close	VERB
ejpam-4302	237	29	;	;	PUNCT
ejpam-4302	237	30	(	(	PUNCT
ejpam-4302	237	31	2	2	X
ejpam-4302	237	32	)	)	PUNCT
ejpam-4302	237	33	a	a	DET
ejpam-4302	237	34	=	=	X
ejpam-4302	237	35	u	u	NOUN
ejpam-4302	237	36	∩a(λ	∩a(λ	NOUN
ejpam-4302	237	37	,	,	PUNCT
ejpam-4302	237	38	sp	sp	NOUN
ejpam-4302	237	39	)	)	PUNCT
ejpam-4302	237	40	for	for	ADP
ejpam-4302	237	41	some	some	DET
ejpam-4302	237	42	u	u	PROPN
ejpam-4302	237	43	∈	∈	PROPN
ejpam-4302	237	44	λspo(x	λspo(x	PROPN
ejpam-4302	237	45	,	,	PUNCT
ejpam-4302	237	46	τ	τ	PROPN
ejpam-4302	237	47	)	)	PUNCT
ejpam-4302	237	48	;	;	PUNCT
ejpam-4302	237	49	(	(	PUNCT
ejpam-4302	237	50	3	3	X
ejpam-4302	237	51	)	)	PUNCT
ejpam-4302	237	52	a(λ	a(λ	ADV
ejpam-4302	237	53	,	,	PUNCT
ejpam-4302	237	54	sp	sp	NOUN
ejpam-4302	237	55	)	)	PUNCT
ejpam-4302	237	56	−a	−a	NOUN
ejpam-4302	237	57	is	be	AUX
ejpam-4302	237	58	(	(	PUNCT
ejpam-4302	237	59	λ	λ	X
ejpam-4302	237	60	,	,	PUNCT
ejpam-4302	237	61	sp)-closed	sp)-close	VERB
ejpam-4302	237	62	;	;	PUNCT
ejpam-4302	237	63	(	(	PUNCT
ejpam-4302	237	64	4	4	X
ejpam-4302	237	65	)	)	PUNCT
ejpam-4302	237	66	a	a	DET
ejpam-4302	237	67	∪	∪	ADJ
ejpam-4302	237	68	[	[	X
ejpam-4302	237	69	x	x	X
ejpam-4302	237	70	−a(λ	−a(λ	NOUN
ejpam-4302	237	71	,	,	PUNCT
ejpam-4302	237	72	sp	sp	NOUN
ejpam-4302	237	73	)	)	PUNCT
ejpam-4302	237	74	]	]	PUNCT
ejpam-4302	237	75	∈	∈	PROPN
ejpam-4302	238	1	λspo(x	λspo(x	PROPN
ejpam-4302	238	2	,	,	PUNCT
ejpam-4302	238	3	τ	τ	PROPN
ejpam-4302	238	4	)	)	PUNCT
ejpam-4302	238	5	;	;	PUNCT
ejpam-4302	238	6	(	(	PUNCT
ejpam-4302	238	7	5	5	X
ejpam-4302	238	8	)	)	PUNCT
ejpam-4302	238	9	a	a	DET
ejpam-4302	238	10	⊆	⊆	NUM
ejpam-4302	238	11	[	[	X
ejpam-4302	238	12	a	a	DET
ejpam-4302	238	13	∪	∪	ADJ
ejpam-4302	238	14	[	[	X
ejpam-4302	238	15	x	x	X
ejpam-4302	238	16	−a(λ	−a(λ	NOUN
ejpam-4302	238	17	,	,	PUNCT
ejpam-4302	238	18	sp)]](λ	sp)]](λ	X
ejpam-4302	238	19	,	,	PUNCT
ejpam-4302	238	20	sp	sp	NOUN
ejpam-4302	238	21	)	)	PUNCT
ejpam-4302	238	22	.	.	PUNCT
ejpam-4302	239	1	theorem	theorem	VERB
ejpam-4302	239	2	13	13	NUM
ejpam-4302	239	3	.	.	PUNCT
ejpam-4302	240	1	a	a	DET
ejpam-4302	240	2	subset	subset	NOUN
ejpam-4302	240	3	a	a	PRON
ejpam-4302	240	4	of	of	ADP
ejpam-4302	240	5	a	a	DET
ejpam-4302	240	6	topological	topological	ADJ
ejpam-4302	240	7	space	space	NOUN
ejpam-4302	240	8	(	(	PUNCT
ejpam-4302	240	9	x	x	X
ejpam-4302	240	10	,	,	PUNCT
ejpam-4302	240	11	τ	τ	X
ejpam-4302	240	12	)	)	PUNCT
ejpam-4302	240	13	is	be	AUX
ejpam-4302	240	14	(	(	PUNCT
ejpam-4302	240	15	λ	λ	X
ejpam-4302	240	16	,	,	PUNCT
ejpam-4302	240	17	sp)-closed	sp)-close	VERB
ejpam-4302	240	18	if	if	SCONJ
ejpam-4302	240	19	and	and	CCONJ
ejpam-4302	240	20	only	only	ADV
ejpam-4302	240	21	if	if	SCONJ
ejpam-4302	240	22	a	a	PRON
ejpam-4302	240	23	is	be	AUX
ejpam-4302	240	24	locally	locally	ADV
ejpam-4302	240	25	(	(	PUNCT
ejpam-4302	240	26	λ	λ	NOUN
ejpam-4302	240	27	,	,	PUNCT
ejpam-4302	240	28	sp)-closed	sp)-closed	ADJ
ejpam-4302	240	29	and	and	CCONJ
ejpam-4302	240	30	g-(λ	g-(λ	PROPN
ejpam-4302	240	31	,	,	PUNCT
ejpam-4302	240	32	sp)-closed	sp)-close	VERB
ejpam-4302	240	33	.	.	PUNCT
ejpam-4302	241	1	proof	proof	NOUN
ejpam-4302	241	2	.	.	PUNCT
ejpam-4302	242	1	let	let	VERB
ejpam-4302	242	2	a	a	DET
ejpam-4302	242	3	be	be	AUX
ejpam-4302	242	4	a	a	DET
ejpam-4302	242	5	(	(	PUNCT
ejpam-4302	242	6	λ	λ	PROPN
ejpam-4302	242	7	,	,	PUNCT
ejpam-4302	242	8	sp)-closed	sp)-close	VERB
ejpam-4302	242	9	set	set	VERB
ejpam-4302	242	10	.	.	PUNCT
ejpam-4302	243	1	by	by	ADP
ejpam-4302	243	2	theorem	theorem	NOUN
ejpam-4302	243	3	3	3	NUM
ejpam-4302	243	4	,	,	PUNCT
ejpam-4302	243	5	a	a	PRON
ejpam-4302	243	6	is	be	AUX
ejpam-4302	243	7	g-(λ	g-(λ	PROPN
ejpam-4302	243	8	,	,	PUNCT
ejpam-4302	243	9	sp)-closed	sp)-close	VERB
ejpam-4302	243	10	.	.	PUNCT
ejpam-4302	244	1	since	since	SCONJ
ejpam-4302	244	2	x	x	PRON
ejpam-4302	244	3	is	be	AUX
ejpam-4302	244	4	(	(	PUNCT
ejpam-4302	244	5	λ	λ	X
ejpam-4302	244	6	,	,	PUNCT
ejpam-4302	244	7	sp)-open	sp)-open	NOUN
ejpam-4302	244	8	and	and	CCONJ
ejpam-4302	244	9	a	a	DET
ejpam-4302	244	10	=	=	NOUN
ejpam-4302	244	11	x	x	SYM
ejpam-4302	244	12	∩a	∩a	PROPN
ejpam-4302	244	13	,	,	PUNCT
ejpam-4302	244	14	a	a	PRON
ejpam-4302	244	15	is	be	AUX
ejpam-4302	244	16	locally	locally	ADV
ejpam-4302	244	17	(	(	PUNCT
ejpam-4302	244	18	λ	λ	NOUN
ejpam-4302	244	19	,	,	PUNCT
ejpam-4302	244	20	sp)-closed	sp)-close	VERB
ejpam-4302	244	21	.	.	PUNCT
ejpam-4302	245	1	conversely	conversely	ADV
ejpam-4302	245	2	,	,	PUNCT
ejpam-4302	245	3	suppose	suppose	VERB
ejpam-4302	245	4	that	that	SCONJ
ejpam-4302	245	5	a	a	PRON
ejpam-4302	245	6	is	be	AUX
ejpam-4302	245	7	locally	locally	ADV
ejpam-4302	245	8	(	(	PUNCT
ejpam-4302	245	9	λ	λ	NOUN
ejpam-4302	245	10	,	,	PUNCT
ejpam-4302	245	11	sp)-closed	sp)-closed	ADJ
ejpam-4302	245	12	and	and	CCONJ
ejpam-4302	245	13	g-(λ	g-(λ	PROPN
ejpam-4302	245	14	,	,	PUNCT
ejpam-4302	245	15	sp)-closed	sp)-close	VERB
ejpam-4302	245	16	.	.	PUNCT
ejpam-4302	246	1	since	since	SCONJ
ejpam-4302	246	2	a	a	PRON
ejpam-4302	246	3	is	be	AUX
ejpam-4302	246	4	locally	locally	ADV
ejpam-4302	246	5	(	(	PUNCT
ejpam-4302	246	6	λ	λ	NOUN
ejpam-4302	246	7	,	,	PUNCT
ejpam-4302	246	8	sp)-closed	sp)-close	VERB
ejpam-4302	246	9	,	,	PUNCT
ejpam-4302	246	10	by	by	ADP
ejpam-4302	246	11	lemma	lemma	PROPN
ejpam-4302	246	12	6	6	NUM
ejpam-4302	246	13	,	,	PUNCT
ejpam-4302	246	14	a	a	DET
ejpam-4302	246	15	⊆	⊆	NUM
ejpam-4302	246	16	[	[	X
ejpam-4302	246	17	a	a	DET
ejpam-4302	246	18	∪	∪	ADJ
ejpam-4302	246	19	[	[	X
ejpam-4302	246	20	x	x	X
ejpam-4302	246	21	−a(λ	−a(λ	NOUN
ejpam-4302	246	22	,	,	PUNCT
ejpam-4302	246	23	sp)]](λ	sp)]](λ	X
ejpam-4302	246	24	,	,	PUNCT
ejpam-4302	246	25	sp	sp	NOUN
ejpam-4302	246	26	)	)	PUNCT
ejpam-4302	246	27	.	.	PUNCT
ejpam-4302	247	1	since	since	SCONJ
ejpam-4302	247	2	[	[	X
ejpam-4302	247	3	a	a	DET
ejpam-4302	247	4	∪	∪	ADJ
ejpam-4302	247	5	[	[	X
ejpam-4302	247	6	x	x	X
ejpam-4302	247	7	−a(λ	−a(λ	NOUN
ejpam-4302	247	8	,	,	PUNCT
ejpam-4302	247	9	sp)]](λ	sp)]](λ	X
ejpam-4302	247	10	,	,	PUNCT
ejpam-4302	247	11	sp	sp	NOUN
ejpam-4302	247	12	)	)	PUNCT
ejpam-4302	247	13	∈	∈	PROPN
ejpam-4302	247	14	λspo(x	λspo(x	PROPN
ejpam-4302	247	15	,	,	PUNCT
ejpam-4302	247	16	τ	τ	PROPN
ejpam-4302	247	17	)	)	PUNCT
ejpam-4302	247	18	and	and	CCONJ
ejpam-4302	247	19	a	a	PRON
ejpam-4302	247	20	is	be	AUX
ejpam-4302	247	21	g-(λ	g-(λ	PROPN
ejpam-4302	247	22	,	,	PUNCT
ejpam-4302	247	23	sp)-closed	sp)-close	VERB
ejpam-4302	247	24	,	,	PUNCT
ejpam-4302	247	25	we	we	PRON
ejpam-4302	247	26	have	have	VERB
ejpam-4302	247	27	a(λ	a(λ	ADV
ejpam-4302	247	28	,	,	PUNCT
ejpam-4302	247	29	sp	sp	NOUN
ejpam-4302	247	30	)	)	PUNCT
ejpam-4302	247	31	⊆	⊆	NUM
ejpam-4302	248	1	[	[	X
ejpam-4302	248	2	a	a	DET
ejpam-4302	248	3	∪	∪	ADJ
ejpam-4302	248	4	[	[	X
ejpam-4302	248	5	x	x	X
ejpam-4302	248	6	−	−	NOUN
ejpam-4302	248	7	a(λ	a(λ	ADV
ejpam-4302	248	8	,	,	PUNCT
ejpam-4302	248	9	sp)]](λ	sp)]](λ	X
ejpam-4302	248	10	,	,	PUNCT
ejpam-4302	248	11	sp	sp	NOUN
ejpam-4302	248	12	)	)	PUNCT
ejpam-4302	248	13	⊆	⊆	NOUN
ejpam-4302	248	14	a	a	DET
ejpam-4302	248	15	∪	∪	NOUN
ejpam-4302	248	16	[	[	X
ejpam-4302	248	17	x	x	X
ejpam-4302	248	18	−	−	NOUN
ejpam-4302	248	19	a(λ	a(λ	ADV
ejpam-4302	248	20	,	,	PUNCT
ejpam-4302	248	21	sp	sp	NOUN
ejpam-4302	248	22	)	)	PUNCT
ejpam-4302	248	23	]	]	PUNCT
ejpam-4302	248	24	and	and	CCONJ
ejpam-4302	248	25	hence	hence	ADV
ejpam-4302	248	26	a(λ	a(λ	ADV
ejpam-4302	248	27	,	,	PUNCT
ejpam-4302	248	28	sp	sp	NOUN
ejpam-4302	248	29	)	)	PUNCT
ejpam-4302	248	30	=	=	PUNCT
ejpam-4302	248	31	a.	a.	NOUN
ejpam-4302	248	32	thus	thus	ADV
ejpam-4302	248	33	,	,	PUNCT
ejpam-4302	248	34	by	by	ADP
ejpam-4302	248	35	lemma	lemma	PROPN
ejpam-4302	248	36	3	3	NUM
ejpam-4302	248	37	,	,	PUNCT
ejpam-4302	248	38	a	a	DET
ejpam-4302	248	39	is	be	AUX
ejpam-4302	248	40	(	(	PUNCT
ejpam-4302	248	41	λ	λ	NOUN
ejpam-4302	248	42	,	,	PUNCT
ejpam-4302	248	43	sp)-closed	sp)-close	VERB
ejpam-4302	248	44	.	.	PUNCT
ejpam-4302	249	1	definition	definition	NOUN
ejpam-4302	249	2	5	5	NUM
ejpam-4302	249	3	.	.	PUNCT
ejpam-4302	250	1	let	let	VERB
ejpam-4302	250	2	a	a	DET
ejpam-4302	250	3	be	be	AUX
ejpam-4302	250	4	a	a	DET
ejpam-4302	250	5	subset	subset	NOUN
ejpam-4302	250	6	of	of	ADP
ejpam-4302	250	7	a	a	DET
ejpam-4302	250	8	topological	topological	ADJ
ejpam-4302	250	9	space	space	NOUN
ejpam-4302	250	10	(	(	PUNCT
ejpam-4302	250	11	x	x	X
ejpam-4302	250	12	,	,	PUNCT
ejpam-4302	250	13	τ	τ	PROPN
ejpam-4302	250	14	)	)	PUNCT
ejpam-4302	250	15	.	.	PUNCT
ejpam-4302	251	1	a	a	DET
ejpam-4302	251	2	subset	subset	NOUN
ejpam-4302	251	3	λ(λ	λ(λ	ADP
ejpam-4302	251	4	,	,	PUNCT
ejpam-4302	251	5	sp)(a	sp)(a	PROPN
ejpam-4302	251	6	)	)	PUNCT
ejpam-4302	251	7	is	be	AUX
ejpam-4302	251	8	defined	define	VERB
ejpam-4302	251	9	as	as	SCONJ
ejpam-4302	251	10	follows	follow	VERB
ejpam-4302	251	11	:	:	PUNCT
ejpam-4302	252	1	λ(λ	λ(λ	ADV
ejpam-4302	252	2	,	,	PUNCT
ejpam-4302	252	3	sp)(a	sp)(a	PROPN
ejpam-4302	252	4	)	)	PUNCT
ejpam-4302	253	1	=	=	PUNCT
ejpam-4302	254	1	∩{u	∩{u	PROPN
ejpam-4302	254	2	|	|	ADV
ejpam-4302	254	3	a	a	DET
ejpam-4302	254	4	⊆	⊆	NUM
ejpam-4302	254	5	u	u	NOUN
ejpam-4302	254	6	,	,	PUNCT
ejpam-4302	254	7	u	u	PROPN
ejpam-4302	254	8	∈	∈	PROPN
ejpam-4302	254	9	λspo(x	λspo(x	PROPN
ejpam-4302	254	10	,	,	PUNCT
ejpam-4302	254	11	τ	τ	PROPN
ejpam-4302	254	12	)	)	PUNCT
ejpam-4302	254	13	}	}	PUNCT
ejpam-4302	254	14	.	.	PUNCT
ejpam-4302	255	1	lemma	lemma	PROPN
ejpam-4302	255	2	7	7	NUM
ejpam-4302	255	3	.	.	X
ejpam-4302	255	4	for	for	ADP
ejpam-4302	255	5	subsets	subset	NOUN
ejpam-4302	255	6	a	a	DET
ejpam-4302	255	7	,	,	PUNCT
ejpam-4302	255	8	b	b	PROPN
ejpam-4302	255	9	of	of	ADP
ejpam-4302	255	10	a	a	DET
ejpam-4302	255	11	topological	topological	ADJ
ejpam-4302	255	12	space	space	NOUN
ejpam-4302	255	13	(	(	PUNCT
ejpam-4302	255	14	x	x	X
ejpam-4302	255	15	,	,	PUNCT
ejpam-4302	255	16	τ	τ	PROPN
ejpam-4302	255	17	)	)	PUNCT
ejpam-4302	255	18	,	,	PUNCT
ejpam-4302	255	19	the	the	DET
ejpam-4302	255	20	following	follow	VERB
ejpam-4302	255	21	properties	property	NOUN
ejpam-4302	255	22	hold	hold	VERB
ejpam-4302	255	23	:	:	PUNCT
ejpam-4302	255	24	(	(	PUNCT
ejpam-4302	255	25	1	1	X
ejpam-4302	255	26	)	)	PUNCT
ejpam-4302	255	27	a	a	DET
ejpam-4302	255	28	⊆	⊆	NUM
ejpam-4302	255	29	λ(λ	λ(λ	NOUN
ejpam-4302	255	30	,	,	PUNCT
ejpam-4302	255	31	sp)(a	sp)(a	PROPN
ejpam-4302	255	32	)	)	PUNCT
ejpam-4302	255	33	.	.	PUNCT
ejpam-4302	256	1	(	(	PUNCT
ejpam-4302	256	2	2	2	X
ejpam-4302	256	3	)	)	PUNCT
ejpam-4302	256	4	if	if	SCONJ
ejpam-4302	256	5	a	a	DET
ejpam-4302	256	6	⊆	⊆	NUM
ejpam-4302	256	7	b	b	NOUN
ejpam-4302	256	8	,	,	PUNCT
ejpam-4302	256	9	then	then	ADV
ejpam-4302	256	10	λ(λ	λ(λ	PROPN
ejpam-4302	256	11	,	,	PUNCT
ejpam-4302	256	12	sp)(a	sp)(a	PROPN
ejpam-4302	256	13	)	)	PUNCT
ejpam-4302	256	14	⊆	⊆	NUM
ejpam-4302	256	15	λ(λ	λ(λ	NOUN
ejpam-4302	256	16	,	,	PUNCT
ejpam-4302	256	17	sp)(b	sp)(b	PROPN
ejpam-4302	256	18	)	)	PUNCT
ejpam-4302	256	19	.	.	PUNCT
ejpam-4302	257	1	(	(	PUNCT
ejpam-4302	257	2	3	3	X
ejpam-4302	257	3	)	)	PUNCT
ejpam-4302	257	4	λ(λ	λ(λ	ADV
ejpam-4302	257	5	,	,	PUNCT
ejpam-4302	257	6	sp)[λ(λ	sp)[λ(λ	NOUN
ejpam-4302	257	7	,	,	PUNCT
ejpam-4302	257	8	sp)(a	sp)(a	PROPN
ejpam-4302	257	9	)	)	PUNCT
ejpam-4302	257	10	]	]	PUNCT
ejpam-4302	258	1	=	=	PUNCT
ejpam-4302	258	2	λ(λ	λ(λ	PROPN
ejpam-4302	258	3	,	,	PUNCT
ejpam-4302	258	4	sp)(a	sp)(a	PROPN
ejpam-4302	258	5	)	)	PUNCT
ejpam-4302	258	6	.	.	PUNCT
ejpam-4302	259	1	(	(	PUNCT
ejpam-4302	259	2	4	4	X
ejpam-4302	259	3	)	)	PUNCT
ejpam-4302	259	4	if	if	SCONJ
ejpam-4302	259	5	a	a	PRON
ejpam-4302	259	6	is	be	AUX
ejpam-4302	259	7	(	(	PUNCT
ejpam-4302	259	8	λ	λ	NOUN
ejpam-4302	259	9	,	,	PUNCT
ejpam-4302	259	10	sp)-open	sp)-open	ADJ
ejpam-4302	259	11	,	,	PUNCT
ejpam-4302	259	12	λ(λ	λ(λ	ADV
ejpam-4302	259	13	,	,	PUNCT
ejpam-4302	259	14	sp)(a	sp)(a	PROPN
ejpam-4302	259	15	)	)	PUNCT
ejpam-4302	260	1	=	=	PUNCT
ejpam-4302	260	2	a.	a.	NOUN
ejpam-4302	260	3	a	a	DET
ejpam-4302	260	4	subset	subset	VERB
ejpam-4302	260	5	nx	nx	NOUN
ejpam-4302	260	6	of	of	ADP
ejpam-4302	260	7	a	a	DET
ejpam-4302	260	8	topological	topological	ADJ
ejpam-4302	260	9	space	space	NOUN
ejpam-4302	260	10	(	(	PUNCT
ejpam-4302	260	11	x	x	X
ejpam-4302	260	12	,	,	PUNCT
ejpam-4302	260	13	τ	τ	X
ejpam-4302	260	14	)	)	PUNCT
ejpam-4302	260	15	is	be	AUX
ejpam-4302	260	16	said	say	VERB
ejpam-4302	260	17	to	to	PART
ejpam-4302	260	18	be	be	AUX
ejpam-4302	260	19	(	(	PUNCT
ejpam-4302	260	20	λ	λ	PROPN
ejpam-4302	260	21	,	,	PUNCT
ejpam-4302	260	22	sp)-neighbourhood	sp)-neighbourhood	ADJ
ejpam-4302	260	23	of	of	ADP
ejpam-4302	260	24	a	a	DET
ejpam-4302	260	25	point	point	NOUN
ejpam-4302	260	26	x	x	SYM
ejpam-4302	260	27	∈	∈	NOUN
ejpam-4302	260	28	x	x	INTJ
ejpam-4302	260	29	if	if	SCONJ
ejpam-4302	260	30	there	there	PRON
ejpam-4302	260	31	exists	exist	VERB
ejpam-4302	260	32	a	a	DET
ejpam-4302	260	33	(	(	PUNCT
ejpam-4302	260	34	λ	λ	NOUN
ejpam-4302	260	35	,	,	PUNCT
ejpam-4302	260	36	sp)-open	sp)-open	NOUN
ejpam-4302	260	37	set	set	VERB
ejpam-4302	260	38	u	u	PRON
ejpam-4302	260	39	such	such	ADJ
ejpam-4302	260	40	that	that	SCONJ
ejpam-4302	260	41	x	x	SYM
ejpam-4302	260	42	∈	∈	PROPN
ejpam-4302	260	43	u	u	NOUN
ejpam-4302	260	44	⊆	⊆	NUM
ejpam-4302	260	45	nx	nx	X
ejpam-4302	260	46	.	.	PUNCT
ejpam-4302	261	1	lemma	lemma	PROPN
ejpam-4302	261	2	8	8	NUM
ejpam-4302	261	3	.	.	PUNCT
ejpam-4302	262	1	a	a	DET
ejpam-4302	262	2	subset	subset	NOUN
ejpam-4302	262	3	a	a	PRON
ejpam-4302	262	4	of	of	ADP
ejpam-4302	262	5	a	a	DET
ejpam-4302	262	6	topological	topological	ADJ
ejpam-4302	262	7	space	space	NOUN
ejpam-4302	262	8	(	(	PUNCT
ejpam-4302	262	9	x	x	X
ejpam-4302	262	10	,	,	PUNCT
ejpam-4302	262	11	τ	τ	X
ejpam-4302	262	12	)	)	PUNCT
ejpam-4302	262	13	is	be	AUX
ejpam-4302	262	14	(	(	PUNCT
ejpam-4302	262	15	λ	λ	INTJ
ejpam-4302	262	16	,	,	PUNCT
ejpam-4302	262	17	sp)-open	sp)-open	ADJ
ejpam-4302	262	18	in	in	ADP
ejpam-4302	262	19	x	x	PUNCT
ejpam-4302	262	20	if	if	SCONJ
ejpam-4302	263	1	and	and	CCONJ
ejpam-4302	263	2	only	only	ADV
ejpam-4302	263	3	if	if	SCONJ
ejpam-4302	263	4	a	a	PRON
ejpam-4302	263	5	is	be	AUX
ejpam-4302	263	6	a	a	DET
ejpam-4302	263	7	(	(	PUNCT
ejpam-4302	263	8	λ	λ	PROPN
ejpam-4302	263	9	,	,	PUNCT
ejpam-4302	263	10	sp)-neighbourhood	sp)-neighbourhood	NOUN
ejpam-4302	263	11	of	of	ADP
ejpam-4302	263	12	each	each	DET
ejpam-4302	263	13	point	point	NOUN
ejpam-4302	263	14	of	of	ADP
ejpam-4302	263	15	a.	a.	NOUN
ejpam-4302	263	16	definition	definition	NOUN
ejpam-4302	263	17	6	6	NUM
ejpam-4302	263	18	.	.	PUNCT
ejpam-4302	264	1	let	let	VERB
ejpam-4302	264	2	(	(	PUNCT
ejpam-4302	264	3	x	x	NOUN
ejpam-4302	264	4	,	,	PUNCT
ejpam-4302	264	5	τ	τ	X
ejpam-4302	264	6	)	)	PUNCT
ejpam-4302	264	7	be	be	VERB
ejpam-4302	264	8	a	a	DET
ejpam-4302	264	9	topological	topological	ADJ
ejpam-4302	264	10	space	space	NOUN
ejpam-4302	264	11	and	and	CCONJ
ejpam-4302	265	1	x	x	PUNCT
ejpam-4302	265	2	∈	∈	PROPN
ejpam-4302	265	3	x.	x.	NOUN
ejpam-4302	265	4	a	a	DET
ejpam-4302	265	5	subset	subset	NOUN
ejpam-4302	265	6	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4302	265	7	is	be	AUX
ejpam-4302	265	8	defined	define	VERB
ejpam-4302	265	9	as	as	SCONJ
ejpam-4302	265	10	follows	follow	VERB
ejpam-4302	265	11	:	:	PUNCT
ejpam-4302	265	12	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4302	265	13	=	=	SYM
ejpam-4302	265	14	λ(λ	λ(λ	PROPN
ejpam-4302	265	15	,	,	PUNCT
ejpam-4302	265	16	sp)({x	sp)({x	NOUN
ejpam-4302	265	17	}	}	PUNCT
ejpam-4302	265	18	)	)	PUNCT
ejpam-4302	265	19	∩	∩	NOUN
ejpam-4302	265	20	{	{	PUNCT
ejpam-4302	265	21	x}(λ	x}(λ	PROPN
ejpam-4302	265	22	,	,	PUNCT
ejpam-4302	265	23	sp	sp	NOUN
ejpam-4302	265	24	)	)	PUNCT
ejpam-4302	265	25	.	.	PUNCT
ejpam-4302	266	1	theorem	theorem	VERB
ejpam-4302	266	2	14	14	NUM
ejpam-4302	266	3	.	.	PUNCT
ejpam-4302	267	1	let	let	VERB
ejpam-4302	267	2	(	(	PUNCT
ejpam-4302	267	3	x	x	NOUN
ejpam-4302	267	4	,	,	PUNCT
ejpam-4302	267	5	τ	τ	X
ejpam-4302	267	6	)	)	PUNCT
ejpam-4302	267	7	be	be	VERB
ejpam-4302	267	8	a	a	DET
ejpam-4302	267	9	topological	topological	ADJ
ejpam-4302	267	10	space	space	NOUN
ejpam-4302	267	11	.	.	PUNCT
ejpam-4302	268	1	then	then	ADV
ejpam-4302	268	2	,	,	PUNCT
ejpam-4302	268	3	the	the	DET
ejpam-4302	268	4	following	follow	VERB
ejpam-4302	268	5	properties	property	NOUN
ejpam-4302	268	6	hold	hold	VERB
ejpam-4302	268	7	:	:	PUNCT
ejpam-4302	268	8	(	(	PUNCT
ejpam-4302	268	9	1	1	X
ejpam-4302	268	10	)	)	PUNCT
ejpam-4302	268	11	λ(λ	λ(λ	ADV
ejpam-4302	268	12	,	,	PUNCT
ejpam-4302	268	13	sp)(a	sp)(a	PROPN
ejpam-4302	268	14	)	)	PUNCT
ejpam-4302	269	1	=	=	PRON
ejpam-4302	269	2	{	{	PUNCT
ejpam-4302	269	3	x	x	PUNCT
ejpam-4302	269	4	∈	∈	NOUN
ejpam-4302	269	5	x	x	PUNCT
ejpam-4302	269	6	|	|	ADV
ejpam-4302	269	7	a	a	DET
ejpam-4302	269	8	∩	∩	NOUN
ejpam-4302	269	9	{	{	PUNCT
ejpam-4302	269	10	x}(λ	x}(λ	PROPN
ejpam-4302	269	11	,	,	PUNCT
ejpam-4302	269	12	sp	sp	NOUN
ejpam-4302	269	13	)	)	PUNCT
ejpam-4302	269	14	̸=	̸=	PROPN
ejpam-4302	269	15	∅	∅	NOUN
ejpam-4302	269	16	}	}	PUNCT
ejpam-4302	269	17	for	for	ADP
ejpam-4302	269	18	each	each	PRON
ejpam-4302	269	19	subset	subset	VERB
ejpam-4302	269	20	a	a	PRON
ejpam-4302	269	21	of	of	ADP
ejpam-4302	269	22	x.	x.	PROPN
ejpam-4302	269	23	c.	c.	PROPN
ejpam-4302	269	24	boonpok	boonpok	PROPN
ejpam-4302	269	25	,	,	PUNCT
ejpam-4302	269	26	c.	c.	PROPN
ejpam-4302	269	27	viriyapong	viriyapong	PROPN
ejpam-4302	269	28	/	/	SYM
ejpam-4302	269	29	eur	eur	PROPN
ejpam-4302	269	30	.	.	PUNCT
ejpam-4302	270	1	j.	j.	PROPN
ejpam-4302	270	2	pure	pure	PROPN
ejpam-4302	270	3	appl	appl	PROPN
ejpam-4302	270	4	.	.	PROPN
ejpam-4302	270	5	math	math	PROPN
ejpam-4302	270	6	,	,	PUNCT
ejpam-4302	270	7	15	15	NUM
ejpam-4302	270	8	(	(	PUNCT
ejpam-4302	270	9	4	4	NUM
ejpam-4302	270	10	)	)	PUNCT
ejpam-4302	270	11	(	(	PUNCT
ejpam-4302	270	12	2022	2022	NUM
ejpam-4302	270	13	)	)	PUNCT
ejpam-4302	270	14	,	,	PUNCT
ejpam-4302	270	15	2127	2127	NUM
ejpam-4302	270	16	-	-	SYM
ejpam-4302	270	17	2140	2140	NUM
ejpam-4302	270	18	2135	2135	NUM
ejpam-4302	270	19	(	(	PUNCT
ejpam-4302	270	20	2	2	NUM
ejpam-4302	270	21	)	)	PUNCT
ejpam-4302	270	22	for	for	ADP
ejpam-4302	270	23	each	each	DET
ejpam-4302	270	24	x	x	SYM
ejpam-4302	270	25	∈	∈	PROPN
ejpam-4302	270	26	x	x	SYM
ejpam-4302	270	27	,	,	PUNCT
ejpam-4302	270	28	λ(λ	λ(λ	PROPN
ejpam-4302	270	29	,	,	PUNCT
ejpam-4302	270	30	sp)(⟨x⟩sp	sp)(⟨x⟩sp	PROPN
ejpam-4302	270	31	)	)	PUNCT
ejpam-4302	270	32	=	=	SYM
ejpam-4302	271	1	λ(λ	λ(λ	PROPN
ejpam-4302	271	2	,	,	PUNCT
ejpam-4302	271	3	sp)({x	sp)({x	PROPN
ejpam-4302	271	4	}	}	PUNCT
ejpam-4302	271	5	)	)	PUNCT
ejpam-4302	271	6	.	.	PUNCT
ejpam-4302	272	1	(	(	PUNCT
ejpam-4302	272	2	3	3	X
ejpam-4302	272	3	)	)	PUNCT
ejpam-4302	272	4	for	for	ADP
ejpam-4302	272	5	each	each	DET
ejpam-4302	272	6	x	x	SYM
ejpam-4302	272	7	∈	∈	PROPN
ejpam-4302	272	8	x	x	X
ejpam-4302	272	9	,	,	PUNCT
ejpam-4302	272	10	(	(	PUNCT
ejpam-4302	272	11	⟨x⟩sp)(λ	⟨x⟩sp)(λ	NOUN
ejpam-4302	272	12	,	,	PUNCT
ejpam-4302	272	13	sp	sp	NOUN
ejpam-4302	272	14	)	)	PUNCT
ejpam-4302	272	15	=	=	SYM
ejpam-4302	272	16	{	{	PUNCT
ejpam-4302	272	17	x}(λ	x}(λ	PROPN
ejpam-4302	272	18	,	,	PUNCT
ejpam-4302	272	19	sp	sp	NOUN
ejpam-4302	272	20	)	)	PUNCT
ejpam-4302	272	21	.	.	PUNCT
ejpam-4302	273	1	(	(	PUNCT
ejpam-4302	273	2	4	4	X
ejpam-4302	273	3	)	)	PUNCT
ejpam-4302	273	4	if	if	SCONJ
ejpam-4302	273	5	u	u	PRON
ejpam-4302	273	6	is	be	AUX
ejpam-4302	273	7	(	(	PUNCT
ejpam-4302	273	8	λ	λ	INTJ
ejpam-4302	273	9	,	,	PUNCT
ejpam-4302	273	10	sp)-open	sp)-open	ADJ
ejpam-4302	273	11	in	in	ADP
ejpam-4302	273	12	x	x	PUNCT
ejpam-4302	273	13	and	and	CCONJ
ejpam-4302	273	14	x	x	SYM
ejpam-4302	273	15	∈	∈	PROPN
ejpam-4302	273	16	u	u	NOUN
ejpam-4302	273	17	,	,	PUNCT
ejpam-4302	273	18	then	then	ADV
ejpam-4302	273	19	⟨x⟩sp	⟨x⟩sp	VERB
ejpam-4302	273	20	⊆	⊆	NUM
ejpam-4302	273	21	u	u	NOUN
ejpam-4302	273	22	.	.	PUNCT
ejpam-4302	274	1	(	(	PUNCT
ejpam-4302	274	2	5	5	NUM
ejpam-4302	274	3	)	)	PUNCT
ejpam-4302	274	4	if	if	SCONJ
ejpam-4302	274	5	f	f	PROPN
ejpam-4302	274	6	is	be	AUX
ejpam-4302	274	7	(	(	PUNCT
ejpam-4302	274	8	λ	λ	X
ejpam-4302	274	9	,	,	PUNCT
ejpam-4302	274	10	sp)-closed	sp)-close	VERB
ejpam-4302	274	11	in	in	ADP
ejpam-4302	274	12	x	x	PUNCT
ejpam-4302	274	13	and	and	CCONJ
ejpam-4302	274	14	x	x	SYM
ejpam-4302	274	15	∈	∈	PROPN
ejpam-4302	274	16	f	f	X
ejpam-4302	274	17	,	,	PUNCT
ejpam-4302	274	18	then	then	ADV
ejpam-4302	274	19	⟨x⟩sp	⟨x⟩sp	VERB
ejpam-4302	274	20	⊆	⊆	NUM
ejpam-4302	274	21	f	f	NOUN
ejpam-4302	274	22	.	.	PUNCT
ejpam-4302	275	1	proof	proof	NOUN
ejpam-4302	275	2	.	.	PUNCT
ejpam-4302	276	1	(	(	PUNCT
ejpam-4302	276	2	1	1	X
ejpam-4302	276	3	)	)	PUNCT
ejpam-4302	276	4	suppose	suppose	VERB
ejpam-4302	276	5	that	that	SCONJ
ejpam-4302	276	6	a	a	DET
ejpam-4302	276	7	∩	∩	NOUN
ejpam-4302	276	8	{	{	PUNCT
ejpam-4302	276	9	x}(λ	x}(λ	PROPN
ejpam-4302	276	10	,	,	PUNCT
ejpam-4302	276	11	sp	sp	NOUN
ejpam-4302	276	12	)	)	PUNCT
ejpam-4302	276	13	=	=	PUNCT
ejpam-4302	276	14	∅.	∅.	NOUN
ejpam-4302	276	15	then	then	ADV
ejpam-4302	276	16	,	,	PUNCT
ejpam-4302	276	17	x	x	PROPN
ejpam-4302	276	18	̸∈	̸∈	PROPN
ejpam-4302	276	19	x	x	X
ejpam-4302	276	20	−	−	PROPN
ejpam-4302	276	21	{	{	PUNCT
ejpam-4302	276	22	x}(λ	x}(λ	PROPN
ejpam-4302	276	23	,	,	PUNCT
ejpam-4302	276	24	sp	sp	NOUN
ejpam-4302	276	25	)	)	PUNCT
ejpam-4302	276	26	which	which	PRON
ejpam-4302	276	27	is	be	AUX
ejpam-4302	276	28	a	a	DET
ejpam-4302	276	29	(	(	PUNCT
ejpam-4302	276	30	λ	λ	NOUN
ejpam-4302	276	31	,	,	PUNCT
ejpam-4302	276	32	sp)-open	sp)-open	ADJ
ejpam-4302	276	33	set	set	VERB
ejpam-4302	276	34	containing	contain	VERB
ejpam-4302	276	35	a.	a.	NOUN
ejpam-4302	276	36	thus	thus	ADV
ejpam-4302	276	37	,	,	PUNCT
ejpam-4302	276	38	x	x	PROPN
ejpam-4302	276	39	̸∈	̸∈	PROPN
ejpam-4302	276	40	λ(λ	λ(λ	PROPN
ejpam-4302	276	41	,	,	PUNCT
ejpam-4302	276	42	sp)(a	sp)(a	PROPN
ejpam-4302	276	43	)	)	PUNCT
ejpam-4302	276	44	and	and	CCONJ
ejpam-4302	276	45	hence	hence	ADV
ejpam-4302	276	46	λ(λ	λ(λ	ADV
ejpam-4302	276	47	,	,	PUNCT
ejpam-4302	276	48	sp)(a	sp)(a	PROPN
ejpam-4302	276	49	)	)	PUNCT
ejpam-4302	276	50	⊆	⊆	NUM
ejpam-4302	276	51	{	{	PUNCT
ejpam-4302	276	52	x	x	SYM
ejpam-4302	276	53	∈	∈	PROPN
ejpam-4302	276	54	x	x	X
ejpam-4302	276	55	|	|	ADV
ejpam-4302	276	56	a	a	DET
ejpam-4302	276	57	∩	∩	NOUN
ejpam-4302	276	58	{	{	PUNCT
ejpam-4302	276	59	x}(λ	x}(λ	PROPN
ejpam-4302	276	60	,	,	PUNCT
ejpam-4302	276	61	sp	sp	NOUN
ejpam-4302	276	62	)	)	PUNCT
ejpam-4302	276	63	̸=	̸=	PROPN
ejpam-4302	276	64	∅	∅	NOUN
ejpam-4302	276	65	}	}	PUNCT
ejpam-4302	276	66	.	.	PUNCT
ejpam-4302	277	1	next	next	ADV
ejpam-4302	277	2	,	,	PUNCT
ejpam-4302	277	3	let	let	VERB
ejpam-4302	277	4	x	x	PUNCT
ejpam-4302	277	5	∈	∈	PROPN
ejpam-4302	277	6	x	x	X
ejpam-4302	277	7	such	such	ADJ
ejpam-4302	277	8	that	that	SCONJ
ejpam-4302	277	9	a∩	a∩	PROPN
ejpam-4302	277	10	{	{	PUNCT
ejpam-4302	277	11	x}(λ	x}(λ	PROPN
ejpam-4302	277	12	,	,	PUNCT
ejpam-4302	277	13	sp	sp	NOUN
ejpam-4302	277	14	)	)	PUNCT
ejpam-4302	277	15	̸=	̸=	PROPN
ejpam-4302	277	16	∅	∅	NOUN
ejpam-4302	277	17	and	and	CCONJ
ejpam-4302	277	18	suppose	suppose	VERB
ejpam-4302	277	19	that	that	SCONJ
ejpam-4302	277	20	x	x	PROPN
ejpam-4302	277	21	̸∈	̸∈	PROPN
ejpam-4302	277	22	λ(λ	λ(λ	PROPN
ejpam-4302	277	23	,	,	PUNCT
ejpam-4302	277	24	sp)(a	sp)(a	PROPN
ejpam-4302	277	25	)	)	PUNCT
ejpam-4302	277	26	.	.	PUNCT
ejpam-4302	278	1	then	then	ADV
ejpam-4302	278	2	,	,	PUNCT
ejpam-4302	278	3	there	there	PRON
ejpam-4302	278	4	exists	exist	VERB
ejpam-4302	278	5	a	a	DET
ejpam-4302	278	6	(	(	PUNCT
ejpam-4302	278	7	λ	λ	NOUN
ejpam-4302	278	8	,	,	PUNCT
ejpam-4302	278	9	sp)-open	sp)-open	NOUN
ejpam-4302	278	10	set	set	VERB
ejpam-4302	278	11	u	u	NOUN
ejpam-4302	278	12	containing	contain	VERB
ejpam-4302	278	13	a	a	PRON
ejpam-4302	278	14	and	and	CCONJ
ejpam-4302	278	15	x	x	X
ejpam-4302	278	16	̸∈	̸∈	PROPN
ejpam-4302	278	17	u	u	PROPN
ejpam-4302	278	18	.	.	PUNCT
ejpam-4302	279	1	let	let	VERB
ejpam-4302	279	2	y	y	PROPN
ejpam-4302	279	3	∈	∈	PROPN
ejpam-4302	279	4	a∩{x}(λ	a∩{x}(λ	NUM
ejpam-4302	279	5	,	,	PUNCT
ejpam-4302	279	6	sp	sp	NOUN
ejpam-4302	279	7	)	)	PUNCT
ejpam-4302	279	8	.	.	PUNCT
ejpam-4302	280	1	therefore	therefore	ADV
ejpam-4302	280	2	,	,	PUNCT
ejpam-4302	280	3	u	u	NOUN
ejpam-4302	280	4	is	be	AUX
ejpam-4302	280	5	a	a	DET
ejpam-4302	280	6	(	(	PUNCT
ejpam-4302	280	7	λ	λ	PROPN
ejpam-4302	280	8	,	,	PUNCT
ejpam-4302	280	9	sp)-neighbourhood	sp)-neighbourhood	NOUN
ejpam-4302	280	10	of	of	ADP
ejpam-4302	280	11	y	y	PRON
ejpam-4302	280	12	which	which	PRON
ejpam-4302	280	13	does	do	AUX
ejpam-4302	280	14	not	not	PART
ejpam-4302	280	15	contain	contain	VERB
ejpam-4302	280	16	x.	x.	NOUN
ejpam-4302	280	17	by	by	ADP
ejpam-4302	280	18	this	this	DET
ejpam-4302	280	19	contradiction	contradiction	NOUN
ejpam-4302	280	20	x	x	X
ejpam-4302	280	21	∈	∈	PROPN
ejpam-4302	280	22	λ(λ	λ(λ	PROPN
ejpam-4302	280	23	,	,	PUNCT
ejpam-4302	280	24	sp)(a	sp)(a	PROPN
ejpam-4302	280	25	)	)	PUNCT
ejpam-4302	280	26	.	.	PUNCT
ejpam-4302	281	1	(	(	PUNCT
ejpam-4302	281	2	2	2	X
ejpam-4302	281	3	)	)	PUNCT
ejpam-4302	281	4	let	let	VERB
ejpam-4302	281	5	x	x	SYM
ejpam-4302	281	6	∈	∈	PROPN
ejpam-4302	281	7	x.	x.	NOUN
ejpam-4302	281	8	then	then	ADV
ejpam-4302	281	9	,	,	PUNCT
ejpam-4302	281	10	we	we	PRON
ejpam-4302	281	11	have	have	VERB
ejpam-4302	281	12	{	{	PUNCT
ejpam-4302	281	13	x	x	NOUN
ejpam-4302	281	14	}	}	PUNCT
ejpam-4302	281	15	⊆	⊆	NUM
ejpam-4302	281	16	{	{	PUNCT
ejpam-4302	281	17	x}(λ	x}(λ	PROPN
ejpam-4302	281	18	,	,	PUNCT
ejpam-4302	281	19	sp	sp	NOUN
ejpam-4302	281	20	)	)	PUNCT
ejpam-4302	281	21	∩	∩	NOUN
ejpam-4302	281	22	λ(λ	λ(λ	ADP
ejpam-4302	281	23	,	,	PUNCT
ejpam-4302	281	24	sp)({x	sp)({x	NOUN
ejpam-4302	281	25	}	}	PUNCT
ejpam-4302	281	26	)	)	PUNCT
ejpam-4302	282	1	=	=	SYM
ejpam-4302	282	2	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4302	282	3	.	.	PUNCT
ejpam-4302	283	1	by	by	ADP
ejpam-4302	283	2	lemma	lemma	PROPN
ejpam-4302	283	3	7	7	NUM
ejpam-4302	283	4	,	,	PUNCT
ejpam-4302	283	5	we	we	PRON
ejpam-4302	283	6	obtain	obtain	VERB
ejpam-4302	283	7	λ(λ	λ(λ	ADP
ejpam-4302	283	8	,	,	PUNCT
ejpam-4302	283	9	sp)({x	sp)({x	PROPN
ejpam-4302	283	10	}	}	PUNCT
ejpam-4302	283	11	)	)	PUNCT
ejpam-4302	284	1	⊆	⊆	NUM
ejpam-4302	284	2	λ(λ	λ(λ	ADP
ejpam-4302	284	3	,	,	PUNCT
ejpam-4302	284	4	sp)(⟨x⟩sp	sp)(⟨x⟩sp	PROPN
ejpam-4302	284	5	)	)	PUNCT
ejpam-4302	284	6	.	.	PUNCT
ejpam-4302	285	1	next	next	ADV
ejpam-4302	285	2	,	,	PUNCT
ejpam-4302	285	3	we	we	PRON
ejpam-4302	285	4	show	show	VERB
ejpam-4302	285	5	the	the	DET
ejpam-4302	285	6	opposite	opposite	ADJ
ejpam-4302	285	7	implication	implication	NOUN
ejpam-4302	285	8	.	.	PUNCT
ejpam-4302	285	9	suppose	suppose	VERB
ejpam-4302	285	10	that	that	SCONJ
ejpam-4302	285	11	y	y	PROPN
ejpam-4302	285	12	̸∈	̸∈	PROPN
ejpam-4302	285	13	λ(λ	λ(λ	PROPN
ejpam-4302	285	14	,	,	PUNCT
ejpam-4302	285	15	sp)({x	sp)({x	PROPN
ejpam-4302	285	16	}	}	PUNCT
ejpam-4302	285	17	)	)	PUNCT
ejpam-4302	285	18	.	.	PUNCT
ejpam-4302	286	1	then	then	ADV
ejpam-4302	286	2	,	,	PUNCT
ejpam-4302	286	3	there	there	PRON
ejpam-4302	286	4	exists	exist	VERB
ejpam-4302	286	5	a	a	DET
ejpam-4302	286	6	(	(	PUNCT
ejpam-4302	286	7	λ	λ	NOUN
ejpam-4302	286	8	,	,	PUNCT
ejpam-4302	286	9	sp)-open	sp)-open	NOUN
ejpam-4302	286	10	set	set	VERB
ejpam-4302	286	11	v	v	ADP
ejpam-4302	286	12	such	such	ADJ
ejpam-4302	286	13	that	that	SCONJ
ejpam-4302	286	14	x	x	SYM
ejpam-4302	286	15	∈	∈	PROPN
ejpam-4302	286	16	v	v	NOUN
ejpam-4302	286	17	and	and	CCONJ
ejpam-4302	286	18	y	y	PROPN
ejpam-4302	286	19	̸∈	̸∈	PROPN
ejpam-4302	286	20	v	v	PROPN
ejpam-4302	286	21	.	.	PUNCT
ejpam-4302	287	1	since	since	SCONJ
ejpam-4302	287	2	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4302	287	3	⊆	⊆	NUM
ejpam-4302	287	4	λ(λ	λ(λ	PROPN
ejpam-4302	287	5	,	,	PUNCT
ejpam-4302	287	6	sp)({x	sp)({x	PROPN
ejpam-4302	287	7	}	}	PUNCT
ejpam-4302	287	8	)	)	PUNCT
ejpam-4302	287	9	⊆	⊆	NUM
ejpam-4302	287	10	λ(λ	λ(λ	NOUN
ejpam-4302	287	11	,	,	PUNCT
ejpam-4302	287	12	sp)(v	sp)(v	X
ejpam-4302	287	13	)	)	PUNCT
ejpam-4302	288	1	=	=	SYM
ejpam-4302	288	2	v	v	X
ejpam-4302	288	3	,	,	PUNCT
ejpam-4302	288	4	we	we	PRON
ejpam-4302	288	5	have	have	VERB
ejpam-4302	288	6	λ(λ	λ(λ	PROPN
ejpam-4302	288	7	,	,	PUNCT
ejpam-4302	288	8	sp)(⟨x⟩sp	sp)(⟨x⟩sp	PROPN
ejpam-4302	288	9	)	)	PUNCT
ejpam-4302	288	10	⊆	⊆	NUM
ejpam-4302	288	11	v	v	NOUN
ejpam-4302	288	12	.	.	PUNCT
ejpam-4302	289	1	since	since	SCONJ
ejpam-4302	289	2	y	y	PROPN
ejpam-4302	289	3	̸∈	̸∈	PROPN
ejpam-4302	289	4	v	v	PROPN
ejpam-4302	289	5	,	,	PUNCT
ejpam-4302	289	6	y	y	PROPN
ejpam-4302	289	7	̸∈	̸∈	PROPN
ejpam-4302	289	8	λ(λ	λ(λ	PROPN
ejpam-4302	289	9	,	,	PUNCT
ejpam-4302	289	10	sp)(⟨x⟩sp	sp)(⟨x⟩sp	PROPN
ejpam-4302	289	11	)	)	PUNCT
ejpam-4302	289	12	.	.	PUNCT
ejpam-4302	290	1	consequently	consequently	ADV
ejpam-4302	290	2	,	,	PUNCT
ejpam-4302	290	3	we	we	PRON
ejpam-4302	290	4	have	have	VERB
ejpam-4302	290	5	λ(λ	λ(λ	PROPN
ejpam-4302	290	6	,	,	PUNCT
ejpam-4302	290	7	sp)(⟨x⟩sp	sp)(⟨x⟩sp	PROPN
ejpam-4302	290	8	)	)	PUNCT
ejpam-4302	290	9	⊆	⊆	NUM
ejpam-4302	290	10	λ(λ	λ(λ	PROPN
ejpam-4302	290	11	,	,	PUNCT
ejpam-4302	290	12	sp)({x	sp)({x	PROPN
ejpam-4302	290	13	}	}	PUNCT
ejpam-4302	290	14	)	)	PUNCT
ejpam-4302	290	15	and	and	CCONJ
ejpam-4302	290	16	hence	hence	ADV
ejpam-4302	290	17	λ(λ	λ(λ	PROPN
ejpam-4302	290	18	,	,	PUNCT
ejpam-4302	290	19	sp)({x	sp)({x	PROPN
ejpam-4302	290	20	}	}	PUNCT
ejpam-4302	290	21	)	)	PUNCT
ejpam-4302	290	22	=	=	SYM
ejpam-4302	291	1	λ(λ	λ(λ	PROPN
ejpam-4302	291	2	,	,	PUNCT
ejpam-4302	291	3	sp)(⟨x⟩sp	sp)(⟨x⟩sp	PROPN
ejpam-4302	291	4	)	)	PUNCT
ejpam-4302	291	5	.	.	PUNCT
ejpam-4302	292	1	(	(	PUNCT
ejpam-4302	292	2	3	3	X
ejpam-4302	292	3	)	)	PUNCT
ejpam-4302	292	4	by	by	ADP
ejpam-4302	292	5	the	the	DET
ejpam-4302	292	6	definition	definition	NOUN
ejpam-4302	292	7	of	of	ADP
ejpam-4302	292	8	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4302	292	9	,	,	PUNCT
ejpam-4302	292	10	we	we	PRON
ejpam-4302	292	11	have	have	VERB
ejpam-4302	292	12	{	{	PUNCT
ejpam-4302	292	13	x	x	NOUN
ejpam-4302	292	14	}	}	PUNCT
ejpam-4302	292	15	⊆	⊆	NUM
ejpam-4302	292	16	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4302	292	17	and	and	CCONJ
ejpam-4302	292	18	{	{	PUNCT
ejpam-4302	292	19	x}(λ	x}(λ	PROPN
ejpam-4302	292	20	,	,	PUNCT
ejpam-4302	292	21	sp	sp	NOUN
ejpam-4302	292	22	)	)	PUNCT
ejpam-4302	292	23	⊆	⊆	NUM
ejpam-4302	292	24	(	(	PUNCT
ejpam-4302	292	25	⟨x⟩sp)(λ	⟨x⟩sp)(λ	NOUN
ejpam-4302	292	26	,	,	PUNCT
ejpam-4302	292	27	sp	sp	NOUN
ejpam-4302	292	28	)	)	PUNCT
ejpam-4302	292	29	by	by	ADP
ejpam-4302	292	30	lemma	lemma	PROPN
ejpam-4302	292	31	3	3	NUM
ejpam-4302	292	32	.	.	PUNCT
ejpam-4302	293	1	on	on	ADP
ejpam-4302	293	2	the	the	DET
ejpam-4302	293	3	other	other	ADJ
ejpam-4302	293	4	hand	hand	NOUN
ejpam-4302	293	5	,	,	PUNCT
ejpam-4302	293	6	we	we	PRON
ejpam-4302	293	7	have	have	VERB
ejpam-4302	293	8	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4302	293	9	⊆	⊆	NUM
ejpam-4302	293	10	{	{	SYM
ejpam-4302	293	11	x}(λ	x}(λ	PROPN
ejpam-4302	293	12	,	,	PUNCT
ejpam-4302	293	13	sp	sp	NOUN
ejpam-4302	293	14	)	)	PUNCT
ejpam-4302	293	15	and	and	CCONJ
ejpam-4302	293	16	(	(	PUNCT
ejpam-4302	293	17	⟨x⟩sp)(λ	⟨x⟩sp)(λ	NOUN
ejpam-4302	293	18	,	,	PUNCT
ejpam-4302	293	19	sp	sp	NOUN
ejpam-4302	293	20	)	)	PUNCT
ejpam-4302	293	21	⊆	⊆	NUM
ejpam-4302	293	22	(	(	PUNCT
ejpam-4302	293	23	{	{	PUNCT
ejpam-4302	293	24	x}(λ	x}(λ	PROPN
ejpam-4302	293	25	,	,	PUNCT
ejpam-4302	293	26	sp))(λ	sp))(λ	NOUN
ejpam-4302	293	27	,	,	PUNCT
ejpam-4302	293	28	sp	sp	NOUN
ejpam-4302	293	29	)	)	PUNCT
ejpam-4302	293	30	=	=	SYM
ejpam-4302	293	31	{	{	PUNCT
ejpam-4302	293	32	x}(λ	x}(λ	PROPN
ejpam-4302	293	33	,	,	PUNCT
ejpam-4302	293	34	sp	sp	NOUN
ejpam-4302	293	35	)	)	PUNCT
ejpam-4302	293	36	.	.	PUNCT
ejpam-4302	294	1	thus	thus	ADV
ejpam-4302	294	2	,	,	PUNCT
ejpam-4302	294	3	(	(	PUNCT
ejpam-4302	294	4	⟨x⟩sp)(λ	⟨x⟩sp)(λ	NOUN
ejpam-4302	294	5	,	,	PUNCT
ejpam-4302	294	6	sp	sp	NOUN
ejpam-4302	294	7	)	)	PUNCT
ejpam-4302	294	8	⊆	⊆	NUM
ejpam-4302	294	9	{	{	PUNCT
ejpam-4302	294	10	x}(λ	x}(λ	PROPN
ejpam-4302	294	11	,	,	PUNCT
ejpam-4302	294	12	sp	sp	NOUN
ejpam-4302	294	13	)	)	PUNCT
ejpam-4302	294	14	.	.	PUNCT
ejpam-4302	295	1	(	(	PUNCT
ejpam-4302	295	2	4	4	X
ejpam-4302	295	3	)	)	PUNCT
ejpam-4302	295	4	since	since	SCONJ
ejpam-4302	295	5	x	x	PROPN
ejpam-4302	295	6	∈	∈	PROPN
ejpam-4302	295	7	u	u	NOUN
ejpam-4302	295	8	and	and	CCONJ
ejpam-4302	295	9	u	u	NOUN
ejpam-4302	295	10	is	be	AUX
ejpam-4302	295	11	a	a	DET
ejpam-4302	295	12	(	(	PUNCT
ejpam-4302	295	13	λ	λ	NOUN
ejpam-4302	295	14	,	,	PUNCT
ejpam-4302	295	15	sp)-open	sp)-open	ADJ
ejpam-4302	295	16	set	set	NOUN
ejpam-4302	295	17	,	,	PUNCT
ejpam-4302	295	18	we	we	PRON
ejpam-4302	295	19	have	have	VERB
ejpam-4302	295	20	λ(λ	λ(λ	PROPN
ejpam-4302	295	21	,	,	PUNCT
ejpam-4302	295	22	sp)({x	sp)({x	PROPN
ejpam-4302	295	23	}	}	PUNCT
ejpam-4302	295	24	)	)	PUNCT
ejpam-4302	296	1	⊆	⊆	NUM
ejpam-4302	296	2	u	u	NOUN
ejpam-4302	296	3	.	.	PUNCT
ejpam-4302	297	1	thus	thus	ADV
ejpam-4302	297	2	,	,	PUNCT
ejpam-4302	297	3	⟨x⟩sp	⟨x⟩sp	VERB
ejpam-4302	297	4	⊆	⊆	NUM
ejpam-4302	297	5	u	u	NOUN
ejpam-4302	297	6	.	.	PUNCT
ejpam-4302	298	1	(	(	PUNCT
ejpam-4302	298	2	5	5	NUM
ejpam-4302	298	3	)	)	PUNCT
ejpam-4302	298	4	since	since	SCONJ
ejpam-4302	298	5	x	x	PROPN
ejpam-4302	298	6	∈	∈	PROPN
ejpam-4302	298	7	f	f	PROPN
ejpam-4302	298	8	and	and	CCONJ
ejpam-4302	298	9	f	f	PROPN
ejpam-4302	298	10	is	be	AUX
ejpam-4302	298	11	a	a	DET
ejpam-4302	298	12	(	(	PUNCT
ejpam-4302	298	13	λ	λ	PROPN
ejpam-4302	298	14	,	,	PUNCT
ejpam-4302	298	15	sp)-closed	sp)-close	VERB
ejpam-4302	298	16	set	set	VERB
ejpam-4302	298	17	,	,	PUNCT
ejpam-4302	298	18	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4302	298	19	=	=	SYM
ejpam-4302	298	20	{	{	PUNCT
ejpam-4302	298	21	x}(λ	x}(λ	PROPN
ejpam-4302	298	22	,	,	PUNCT
ejpam-4302	298	23	sp	sp	NOUN
ejpam-4302	298	24	)	)	PUNCT
ejpam-4302	298	25	∩	∩	NOUN
ejpam-4302	298	26	λ(λ	λ(λ	ADP
ejpam-4302	298	27	,	,	PUNCT
ejpam-4302	298	28	sp)({x	sp)({x	NOUN
ejpam-4302	298	29	}	}	PUNCT
ejpam-4302	298	30	)	)	PUNCT
ejpam-4302	299	1	⊆	⊆	X
ejpam-4302	299	2	{	{	PUNCT
ejpam-4302	299	3	x}(λ	x}(λ	PROPN
ejpam-4302	299	4	,	,	PUNCT
ejpam-4302	299	5	sp	sp	NOUN
ejpam-4302	299	6	)	)	PUNCT
ejpam-4302	299	7	⊆	⊆	NUM
ejpam-4302	299	8	f	f	X
ejpam-4302	299	9	(	(	PUNCT
ejpam-4302	299	10	λ	λ	PROPN
ejpam-4302	299	11	,	,	PUNCT
ejpam-4302	299	12	sp	sp	NOUN
ejpam-4302	299	13	)	)	PUNCT
ejpam-4302	299	14	=	=	SYM
ejpam-4302	299	15	f.	f.	PROPN
ejpam-4302	299	16	theorem	theorem	VERB
ejpam-4302	299	17	15	15	NUM
ejpam-4302	299	18	.	.	PUNCT
ejpam-4302	300	1	for	for	ADP
ejpam-4302	300	2	any	any	DET
ejpam-4302	300	3	points	point	NOUN
ejpam-4302	300	4	x	x	PUNCT
ejpam-4302	300	5	and	and	CCONJ
ejpam-4302	300	6	y	y	PROPN
ejpam-4302	300	7	in	in	ADP
ejpam-4302	300	8	a	a	DET
ejpam-4302	300	9	topological	topological	ADJ
ejpam-4302	300	10	space	space	NOUN
ejpam-4302	300	11	(	(	PUNCT
ejpam-4302	300	12	x	x	X
ejpam-4302	300	13	,	,	PUNCT
ejpam-4302	300	14	τ	τ	PROPN
ejpam-4302	300	15	)	)	PUNCT
ejpam-4302	300	16	,	,	PUNCT
ejpam-4302	300	17	the	the	DET
ejpam-4302	300	18	following	follow	VERB
ejpam-4302	300	19	properties	property	NOUN
ejpam-4302	300	20	are	be	AUX
ejpam-4302	300	21	equivalent	equivalent	ADJ
ejpam-4302	300	22	:	:	PUNCT
ejpam-4302	300	23	(	(	PUNCT
ejpam-4302	300	24	1	1	X
ejpam-4302	300	25	)	)	PUNCT
ejpam-4302	300	26	λ(λ	λ(λ	ADV
ejpam-4302	300	27	,	,	PUNCT
ejpam-4302	300	28	sp)({x	sp)({x	PROPN
ejpam-4302	300	29	}	}	PUNCT
ejpam-4302	300	30	)	)	PUNCT
ejpam-4302	301	1	̸=	̸=	PROPN
ejpam-4302	301	2	λ(λ	λ(λ	PROPN
ejpam-4302	301	3	,	,	PUNCT
ejpam-4302	301	4	sp)({y	sp)({y	NOUN
ejpam-4302	301	5	}	}	PUNCT
ejpam-4302	301	6	)	)	PUNCT
ejpam-4302	301	7	.	.	PUNCT
ejpam-4302	302	1	(	(	PUNCT
ejpam-4302	302	2	2	2	X
ejpam-4302	302	3	)	)	PUNCT
ejpam-4302	302	4	{	{	PUNCT
ejpam-4302	302	5	x}(λ	x}(λ	PROPN
ejpam-4302	302	6	,	,	PUNCT
ejpam-4302	302	7	sp	sp	NOUN
ejpam-4302	302	8	)	)	PUNCT
ejpam-4302	302	9	̸=	̸=	PROPN
ejpam-4302	302	10	{	{	PUNCT
ejpam-4302	302	11	y}(λ	y}(λ	PROPN
ejpam-4302	302	12	,	,	PUNCT
ejpam-4302	302	13	sp	sp	NOUN
ejpam-4302	302	14	)	)	PUNCT
ejpam-4302	302	15	.	.	PUNCT
ejpam-4302	303	1	proof	proof	NOUN
ejpam-4302	303	2	.	.	PUNCT
ejpam-4302	304	1	(	(	PUNCT
ejpam-4302	304	2	1	1	X
ejpam-4302	304	3	)	)	PUNCT
ejpam-4302	304	4	⇒	⇒	NOUN
ejpam-4302	304	5	(	(	PUNCT
ejpam-4302	304	6	2	2	NUM
ejpam-4302	304	7	):	):	PUNCT
ejpam-4302	304	8	suppose	suppose	VERB
ejpam-4302	304	9	that	that	SCONJ
ejpam-4302	304	10	λ(λ	λ(λ	PROPN
ejpam-4302	304	11	,	,	PUNCT
ejpam-4302	304	12	sp)({x	sp)({x	PROPN
ejpam-4302	304	13	}	}	PUNCT
ejpam-4302	304	14	)	)	PUNCT
ejpam-4302	304	15	̸=	̸=	PROPN
ejpam-4302	304	16	λ(λ	λ(λ	PROPN
ejpam-4302	304	17	,	,	PUNCT
ejpam-4302	304	18	sp)({y	sp)({y	NOUN
ejpam-4302	304	19	}	}	PUNCT
ejpam-4302	304	20	)	)	PUNCT
ejpam-4302	304	21	.	.	PUNCT
ejpam-4302	305	1	then	then	ADV
ejpam-4302	305	2	,	,	PUNCT
ejpam-4302	305	3	there	there	PRON
ejpam-4302	305	4	exists	exist	VERB
ejpam-4302	305	5	a	a	DET
ejpam-4302	305	6	point	point	NOUN
ejpam-4302	305	7	z	z	NOUN
ejpam-4302	305	8	∈	∈	PROPN
ejpam-4302	305	9	x	x	PUNCT
ejpam-4302	305	10	such	such	ADJ
ejpam-4302	305	11	that	that	SCONJ
ejpam-4302	305	12	z	z	PROPN
ejpam-4302	305	13	∈	∈	PROPN
ejpam-4302	305	14	λ(λ	λ(λ	PROPN
ejpam-4302	305	15	,	,	PUNCT
ejpam-4302	305	16	sp)({x	sp)({x	PROPN
ejpam-4302	305	17	}	}	PUNCT
ejpam-4302	305	18	)	)	PUNCT
ejpam-4302	305	19	and	and	CCONJ
ejpam-4302	305	20	z	z	PROPN
ejpam-4302	305	21	̸∈	̸∈	PROPN
ejpam-4302	305	22	λ(λ	λ(λ	PROPN
ejpam-4302	305	23	,	,	PUNCT
ejpam-4302	305	24	sp)({y	sp)({y	NOUN
ejpam-4302	305	25	}	}	PUNCT
ejpam-4302	305	26	)	)	PUNCT
ejpam-4302	305	27	or	or	CCONJ
ejpam-4302	305	28	z	z	NOUN
ejpam-4302	305	29	∈	∈	PROPN
ejpam-4302	305	30	λ(λ	λ(λ	PROPN
ejpam-4302	305	31	,	,	PUNCT
ejpam-4302	305	32	sp)({y	sp)({y	NOUN
ejpam-4302	305	33	}	}	PUNCT
ejpam-4302	305	34	)	)	PUNCT
ejpam-4302	305	35	and	and	CCONJ
ejpam-4302	305	36	z	z	PROPN
ejpam-4302	305	37	̸∈	̸∈	PROPN
ejpam-4302	305	38	λ(λ	λ(λ	PROPN
ejpam-4302	305	39	,	,	PUNCT
ejpam-4302	305	40	sp)({x	sp)({x	PROPN
ejpam-4302	305	41	}	}	PUNCT
ejpam-4302	305	42	)	)	PUNCT
ejpam-4302	305	43	.	.	PUNCT
ejpam-4302	306	1	we	we	PRON
ejpam-4302	306	2	prove	prove	VERB
ejpam-4302	306	3	only	only	ADV
ejpam-4302	306	4	the	the	DET
ejpam-4302	306	5	first	first	ADJ
ejpam-4302	306	6	case	case	NOUN
ejpam-4302	306	7	being	be	AUX
ejpam-4302	306	8	the	the	DET
ejpam-4302	306	9	second	second	ADJ
ejpam-4302	306	10	analogous	analogous	NOUN
ejpam-4302	306	11	.	.	PUNCT
ejpam-4302	307	1	from	from	ADP
ejpam-4302	307	2	z	z	PROPN
ejpam-4302	307	3	∈	∈	PROPN
ejpam-4302	307	4	λ(λ	λ(λ	PROPN
ejpam-4302	307	5	,	,	PUNCT
ejpam-4302	307	6	sp)({x	sp)({x	PROPN
ejpam-4302	307	7	}	}	PUNCT
ejpam-4302	307	8	)	)	PUNCT
ejpam-4302	308	1	it	it	PRON
ejpam-4302	308	2	follows	follow	VERB
ejpam-4302	308	3	that	that	SCONJ
ejpam-4302	308	4	{	{	PUNCT
ejpam-4302	308	5	x	x	NOUN
ejpam-4302	308	6	}	}	PUNCT
ejpam-4302	308	7	∩	∩	NOUN
ejpam-4302	308	8	{	{	PUNCT
ejpam-4302	308	9	z}(λ	z}(λ	PROPN
ejpam-4302	308	10	,	,	PUNCT
ejpam-4302	308	11	sp	sp	NOUN
ejpam-4302	308	12	)	)	PUNCT
ejpam-4302	308	13	̸=	̸=	PROPN
ejpam-4302	308	14	∅	∅	NOUN
ejpam-4302	308	15	which	which	PRON
ejpam-4302	308	16	implies	imply	VERB
ejpam-4302	308	17	x	x	X
ejpam-4302	308	18	∈	∈	PROPN
ejpam-4302	308	19	{	{	PUNCT
ejpam-4302	308	20	z}(λ	z}(λ	PROPN
ejpam-4302	308	21	,	,	PUNCT
ejpam-4302	308	22	sp	sp	NOUN
ejpam-4302	308	23	)	)	PUNCT
ejpam-4302	308	24	.	.	PUNCT
ejpam-4302	309	1	by	by	ADP
ejpam-4302	309	2	z	z	PROPN
ejpam-4302	309	3	̸∈	̸∈	PROPN
ejpam-4302	309	4	λ(λ	λ(λ	PROPN
ejpam-4302	309	5	,	,	PUNCT
ejpam-4302	309	6	sp)({y	sp)({y	NOUN
ejpam-4302	309	7	}	}	PUNCT
ejpam-4302	309	8	)	)	PUNCT
ejpam-4302	309	9	,	,	PUNCT
ejpam-4302	309	10	we	we	PRON
ejpam-4302	309	11	have	have	AUX
ejpam-4302	309	12	{	{	PUNCT
ejpam-4302	309	13	y}∩{z}(λ	y}∩{z}(λ	NOUN
ejpam-4302	309	14	,	,	PUNCT
ejpam-4302	309	15	sp	sp	NOUN
ejpam-4302	309	16	)	)	PUNCT
ejpam-4302	309	17	=	=	PUNCT
ejpam-4302	309	18	∅.	∅.	NOUN
ejpam-4302	309	19	since	since	SCONJ
ejpam-4302	309	20	x	x	PROPN
ejpam-4302	309	21	∈	∈	PROPN
ejpam-4302	309	22	{	{	PUNCT
ejpam-4302	309	23	z}(λ	z}(λ	PROPN
ejpam-4302	309	24	,	,	PUNCT
ejpam-4302	309	25	sp	sp	NOUN
ejpam-4302	309	26	)	)	PUNCT
ejpam-4302	309	27	,	,	PUNCT
ejpam-4302	309	28	{	{	PUNCT
ejpam-4302	309	29	x}(λ	x}(λ	PROPN
ejpam-4302	309	30	,	,	PUNCT
ejpam-4302	309	31	sp	sp	NOUN
ejpam-4302	309	32	)	)	PUNCT
ejpam-4302	309	33	⊆	⊆	NUM
ejpam-4302	309	34	{	{	PUNCT
ejpam-4302	309	35	z}(λ	z}(λ	PROPN
ejpam-4302	309	36	,	,	PUNCT
ejpam-4302	309	37	sp	sp	NOUN
ejpam-4302	309	38	)	)	PUNCT
ejpam-4302	309	39	and	and	CCONJ
ejpam-4302	309	40	c.	c.	PROPN
ejpam-4302	309	41	boonpok	boonpok	PROPN
ejpam-4302	309	42	,	,	PUNCT
ejpam-4302	309	43	c.	c.	PROPN
ejpam-4302	309	44	viriyapong	viriyapong	PROPN
ejpam-4302	309	45	/	/	SYM
ejpam-4302	309	46	eur	eur	PROPN
ejpam-4302	309	47	.	.	PUNCT
ejpam-4302	310	1	j.	j.	PROPN
ejpam-4302	310	2	pure	pure	PROPN
ejpam-4302	310	3	appl	appl	PROPN
ejpam-4302	310	4	.	.	PROPN
ejpam-4302	310	5	math	math	PROPN
ejpam-4302	310	6	,	,	PUNCT
ejpam-4302	310	7	15	15	NUM
ejpam-4302	310	8	(	(	PUNCT
ejpam-4302	310	9	4	4	NUM
ejpam-4302	310	10	)	)	PUNCT
ejpam-4302	310	11	(	(	PUNCT
ejpam-4302	310	12	2022	2022	NUM
ejpam-4302	310	13	)	)	PUNCT
ejpam-4302	310	14	,	,	PUNCT
ejpam-4302	310	15	2127	2127	NUM
ejpam-4302	310	16	-	-	SYM
ejpam-4302	310	17	2140	2140	NUM
ejpam-4302	310	18	2136	2136	NUM
ejpam-4302	310	19	{	{	PUNCT
ejpam-4302	310	20	y	y	NOUN
ejpam-4302	310	21	}	}	PUNCT
ejpam-4302	310	22	∩	∩	NOUN
ejpam-4302	310	23	{	{	PUNCT
ejpam-4302	310	24	x}(λ	x}(λ	PROPN
ejpam-4302	310	25	,	,	PUNCT
ejpam-4302	310	26	sp	sp	NOUN
ejpam-4302	310	27	)	)	PUNCT
ejpam-4302	310	28	=	=	PUNCT
ejpam-4302	310	29	∅.	∅.	VERB
ejpam-4302	310	30	therefore	therefore	ADV
ejpam-4302	310	31	,	,	PUNCT
ejpam-4302	310	32	{	{	PUNCT
ejpam-4302	310	33	x}(λ	x}(λ	PROPN
ejpam-4302	310	34	,	,	PUNCT
ejpam-4302	310	35	sp	sp	NOUN
ejpam-4302	310	36	)	)	PUNCT
ejpam-4302	310	37	̸=	̸=	PROPN
ejpam-4302	310	38	{	{	PUNCT
ejpam-4302	310	39	y}(λ	y}(λ	PROPN
ejpam-4302	310	40	,	,	PUNCT
ejpam-4302	310	41	sp	sp	NOUN
ejpam-4302	310	42	)	)	PUNCT
ejpam-4302	310	43	.	.	PUNCT
ejpam-4302	311	1	thus	thus	ADV
ejpam-4302	311	2	,	,	PUNCT
ejpam-4302	311	3	λ(λ	λ(λ	ADV
ejpam-4302	311	4	,	,	PUNCT
ejpam-4302	311	5	sp)({x	sp)({x	PROPN
ejpam-4302	311	6	}	}	PUNCT
ejpam-4302	311	7	)	)	PUNCT
ejpam-4302	312	1	̸=	̸=	PROPN
ejpam-4302	312	2	λ(λ	λ(λ	PROPN
ejpam-4302	312	3	,	,	PUNCT
ejpam-4302	312	4	sp)({y	sp)({y	NOUN
ejpam-4302	312	5	}	}	PUNCT
ejpam-4302	312	6	)	)	PUNCT
ejpam-4302	312	7	and	and	CCONJ
ejpam-4302	312	8	hence	hence	ADV
ejpam-4302	312	9	{	{	PUNCT
ejpam-4302	312	10	x}(λ	x}(λ	PROPN
ejpam-4302	312	11	,	,	PUNCT
ejpam-4302	312	12	sp	sp	NOUN
ejpam-4302	312	13	)	)	PUNCT
ejpam-4302	312	14	̸=	̸=	PROPN
ejpam-4302	312	15	{	{	PUNCT
ejpam-4302	312	16	y}(λ	y}(λ	PROPN
ejpam-4302	312	17	,	,	PUNCT
ejpam-4302	312	18	sp	sp	NOUN
ejpam-4302	312	19	)	)	PUNCT
ejpam-4302	312	20	.	.	PUNCT
ejpam-4302	313	1	(	(	PUNCT
ejpam-4302	313	2	2	2	X
ejpam-4302	313	3	)	)	PUNCT
ejpam-4302	313	4	⇒	⇒	NOUN
ejpam-4302	313	5	(	(	PUNCT
ejpam-4302	313	6	1	1	NUM
ejpam-4302	313	7	):	):	PUNCT
ejpam-4302	313	8	suppose	suppose	VERB
ejpam-4302	313	9	that	that	SCONJ
ejpam-4302	313	10	{	{	PUNCT
ejpam-4302	313	11	x}(λ	x}(λ	PROPN
ejpam-4302	313	12	,	,	PUNCT
ejpam-4302	313	13	sp	sp	NOUN
ejpam-4302	313	14	)	)	PUNCT
ejpam-4302	313	15	̸=	̸=	PROPN
ejpam-4302	313	16	{	{	PUNCT
ejpam-4302	313	17	y}(λ	y}(λ	PROPN
ejpam-4302	313	18	,	,	PUNCT
ejpam-4302	313	19	sp	sp	NOUN
ejpam-4302	313	20	)	)	PUNCT
ejpam-4302	313	21	.	.	PUNCT
ejpam-4302	314	1	then	then	ADV
ejpam-4302	314	2	,	,	PUNCT
ejpam-4302	314	3	there	there	PRON
ejpam-4302	314	4	exists	exist	VERB
ejpam-4302	314	5	a	a	DET
ejpam-4302	314	6	point	point	NOUN
ejpam-4302	314	7	z	z	NOUN
ejpam-4302	314	8	∈	∈	PROPN
ejpam-4302	314	9	x	x	PUNCT
ejpam-4302	314	10	such	such	ADJ
ejpam-4302	314	11	that	that	SCONJ
ejpam-4302	314	12	z	z	PROPN
ejpam-4302	314	13	∈	∈	PROPN
ejpam-4302	314	14	{	{	PUNCT
ejpam-4302	314	15	x}(λ	x}(λ	PROPN
ejpam-4302	314	16	,	,	PUNCT
ejpam-4302	314	17	sp	sp	NOUN
ejpam-4302	314	18	)	)	PUNCT
ejpam-4302	314	19	and	and	CCONJ
ejpam-4302	314	20	z	z	PROPN
ejpam-4302	314	21	̸∈	̸∈	PROPN
ejpam-4302	314	22	{	{	PUNCT
ejpam-4302	314	23	y}(λ	y}(λ	PROPN
ejpam-4302	314	24	,	,	PUNCT
ejpam-4302	314	25	sp	sp	NOUN
ejpam-4302	314	26	)	)	PUNCT
ejpam-4302	314	27	or	or	CCONJ
ejpam-4302	314	28	z	z	NOUN
ejpam-4302	314	29	∈	∈	PROPN
ejpam-4302	314	30	{	{	PUNCT
ejpam-4302	314	31	y}(λ	y}(λ	PROPN
ejpam-4302	314	32	,	,	PUNCT
ejpam-4302	314	33	sp	sp	NOUN
ejpam-4302	314	34	)	)	PUNCT
ejpam-4302	314	35	and	and	CCONJ
ejpam-4302	314	36	z	z	PROPN
ejpam-4302	314	37	̸∈	̸∈	PROPN
ejpam-4302	314	38	{	{	PUNCT
ejpam-4302	314	39	x}(λ	x}(λ	PROPN
ejpam-4302	314	40	,	,	PUNCT
ejpam-4302	314	41	sp	sp	NOUN
ejpam-4302	314	42	)	)	PUNCT
ejpam-4302	314	43	.	.	PUNCT
ejpam-4302	315	1	we	we	PRON
ejpam-4302	315	2	prove	prove	VERB
ejpam-4302	315	3	only	only	ADV
ejpam-4302	315	4	the	the	DET
ejpam-4302	315	5	first	first	ADJ
ejpam-4302	315	6	case	case	NOUN
ejpam-4302	315	7	being	be	AUX
ejpam-4302	315	8	the	the	DET
ejpam-4302	315	9	second	second	ADJ
ejpam-4302	315	10	analogous	analogous	NOUN
ejpam-4302	315	11	.	.	PUNCT
ejpam-4302	316	1	it	it	PRON
ejpam-4302	316	2	follows	follow	VERB
ejpam-4302	316	3	that	that	SCONJ
ejpam-4302	316	4	there	there	PRON
ejpam-4302	316	5	exists	exist	VERB
ejpam-4302	316	6	a	a	DET
ejpam-4302	316	7	(	(	PUNCT
ejpam-4302	316	8	λ	λ	NOUN
ejpam-4302	316	9	,	,	PUNCT
ejpam-4302	316	10	sp)open	sp)open	VERB
ejpam-4302	316	11	set	set	VERB
ejpam-4302	316	12	containing	contain	VERB
ejpam-4302	316	13	z	z	NOUN
ejpam-4302	316	14	and	and	CCONJ
ejpam-4302	316	15	therefore	therefore	ADV
ejpam-4302	316	16	x	x	X
ejpam-4302	316	17	but	but	CCONJ
ejpam-4302	316	18	not	not	PART
ejpam-4302	316	19	y	y	NOUN
ejpam-4302	316	20	,	,	PUNCT
ejpam-4302	316	21	namely	namely	ADV
ejpam-4302	316	22	,	,	PUNCT
ejpam-4302	316	23	y	y	PROPN
ejpam-4302	316	24	̸∈	̸∈	PROPN
ejpam-4302	316	25	λ(λ	λ(λ	PROPN
ejpam-4302	316	26	,	,	PUNCT
ejpam-4302	316	27	sp)({x	sp)({x	PROPN
ejpam-4302	316	28	}	}	PUNCT
ejpam-4302	316	29	)	)	PUNCT
ejpam-4302	316	30	and	and	CCONJ
ejpam-4302	316	31	thus	thus	ADV
ejpam-4302	316	32	λ(λ	λ(λ	ADV
ejpam-4302	316	33	,	,	PUNCT
ejpam-4302	316	34	sp)({x	sp)({x	PROPN
ejpam-4302	316	35	}	}	PUNCT
ejpam-4302	316	36	)	)	PUNCT
ejpam-4302	317	1	̸=	̸=	PROPN
ejpam-4302	317	2	λ(λ	λ(λ	PROPN
ejpam-4302	317	3	,	,	PUNCT
ejpam-4302	317	4	sp)({y	sp)({y	NOUN
ejpam-4302	317	5	}	}	PUNCT
ejpam-4302	317	6	)	)	PUNCT
ejpam-4302	317	7	.	.	PUNCT
ejpam-4302	318	1	theorem	theorem	VERB
ejpam-4302	318	2	16	16	NUM
ejpam-4302	318	3	.	.	PUNCT
ejpam-4302	319	1	let	let	VERB
ejpam-4302	319	2	(	(	PUNCT
ejpam-4302	319	3	x	x	NOUN
ejpam-4302	319	4	,	,	PUNCT
ejpam-4302	319	5	τ	τ	X
ejpam-4302	319	6	)	)	PUNCT
ejpam-4302	319	7	be	be	VERB
ejpam-4302	319	8	a	a	DET
ejpam-4302	319	9	topological	topological	ADJ
ejpam-4302	319	10	space	space	NOUN
ejpam-4302	319	11	and	and	CCONJ
ejpam-4302	319	12	x	x	NOUN
ejpam-4302	319	13	,	,	PUNCT
ejpam-4302	319	14	y	y	PROPN
ejpam-4302	319	15	∈	∈	PROPN
ejpam-4302	319	16	x.	x.	NOUN
ejpam-4302	319	17	then	then	ADV
ejpam-4302	319	18	,	,	PUNCT
ejpam-4302	319	19	the	the	DET
ejpam-4302	319	20	following	follow	VERB
ejpam-4302	319	21	properties	property	NOUN
ejpam-4302	319	22	hold	hold	VERB
ejpam-4302	319	23	:	:	PUNCT
ejpam-4302	319	24	(	(	PUNCT
ejpam-4302	319	25	1	1	X
ejpam-4302	319	26	)	)	PUNCT
ejpam-4302	319	27	y	y	PROPN
ejpam-4302	319	28	∈	∈	PROPN
ejpam-4302	320	1	λ(λ	λ(λ	PROPN
ejpam-4302	320	2	,	,	PUNCT
ejpam-4302	320	3	sp)({x	sp)({x	PROPN
ejpam-4302	320	4	}	}	PUNCT
ejpam-4302	320	5	)	)	PUNCT
ejpam-4302	320	6	if	if	SCONJ
ejpam-4302	320	7	and	and	CCONJ
ejpam-4302	320	8	only	only	ADV
ejpam-4302	320	9	if	if	SCONJ
ejpam-4302	320	10	x	x	SYM
ejpam-4302	320	11	∈	∈	PROPN
ejpam-4302	320	12	{	{	PUNCT
ejpam-4302	320	13	y}(λ	y}(λ	PROPN
ejpam-4302	320	14	,	,	PUNCT
ejpam-4302	320	15	sp	sp	NOUN
ejpam-4302	320	16	)	)	PUNCT
ejpam-4302	320	17	.	.	PUNCT
ejpam-4302	321	1	(	(	PUNCT
ejpam-4302	321	2	2	2	X
ejpam-4302	321	3	)	)	PUNCT
ejpam-4302	321	4	λ(λ	λ(λ	ADV
ejpam-4302	321	5	,	,	PUNCT
ejpam-4302	321	6	sp)({x	sp)({x	NOUN
ejpam-4302	321	7	}	}	PUNCT
ejpam-4302	321	8	)	)	PUNCT
ejpam-4302	322	1	=	=	SYM
ejpam-4302	322	2	λ(λ	λ(λ	PROPN
ejpam-4302	322	3	,	,	PUNCT
ejpam-4302	322	4	sp)({y	sp)({y	NOUN
ejpam-4302	322	5	}	}	PUNCT
ejpam-4302	322	6	)	)	PUNCT
ejpam-4302	322	7	if	if	SCONJ
ejpam-4302	322	8	and	and	CCONJ
ejpam-4302	322	9	only	only	ADV
ejpam-4302	322	10	if	if	SCONJ
ejpam-4302	322	11	{	{	PUNCT
ejpam-4302	322	12	x}(λ	x}(λ	PROPN
ejpam-4302	322	13	,	,	PUNCT
ejpam-4302	322	14	sp	sp	NOUN
ejpam-4302	322	15	)	)	PUNCT
ejpam-4302	322	16	=	=	SYM
ejpam-4302	322	17	{	{	PUNCT
ejpam-4302	322	18	y}(λ	y}(λ	PROPN
ejpam-4302	322	19	,	,	PUNCT
ejpam-4302	322	20	sp	sp	NOUN
ejpam-4302	322	21	)	)	PUNCT
ejpam-4302	322	22	.	.	PUNCT
ejpam-4302	323	1	proof	proof	NOUN
ejpam-4302	323	2	.	.	PUNCT
ejpam-4302	324	1	(	(	PUNCT
ejpam-4302	324	2	1	1	X
ejpam-4302	324	3	)	)	PUNCT
ejpam-4302	324	4	let	let	VERB
ejpam-4302	324	5	x	x	SYM
ejpam-4302	324	6	̸∈	̸∈	PROPN
ejpam-4302	324	7	{	{	PUNCT
ejpam-4302	324	8	y}(λ	y}(λ	PROPN
ejpam-4302	324	9	,	,	PUNCT
ejpam-4302	324	10	sp	sp	NOUN
ejpam-4302	324	11	)	)	PUNCT
ejpam-4302	324	12	.	.	PUNCT
ejpam-4302	325	1	then	then	ADV
ejpam-4302	325	2	,	,	PUNCT
ejpam-4302	325	3	there	there	PRON
ejpam-4302	325	4	exists	exist	VERB
ejpam-4302	325	5	u	u	PROPN
ejpam-4302	325	6	∈	∈	PROPN
ejpam-4302	325	7	λspo(x	λspo(x	PROPN
ejpam-4302	325	8	,	,	PUNCT
ejpam-4302	325	9	τ	τ	PROPN
ejpam-4302	325	10	)	)	PUNCT
ejpam-4302	325	11	such	such	ADJ
ejpam-4302	325	12	that	that	SCONJ
ejpam-4302	325	13	x	x	SYM
ejpam-4302	325	14	∈	∈	PROPN
ejpam-4302	325	15	u	u	NOUN
ejpam-4302	325	16	and	and	CCONJ
ejpam-4302	325	17	y	y	PROPN
ejpam-4302	325	18	̸∈	̸∈	PROPN
ejpam-4302	325	19	u	u	PROPN
ejpam-4302	325	20	.	.	PUNCT
ejpam-4302	326	1	thus	thus	ADV
ejpam-4302	326	2	,	,	PUNCT
ejpam-4302	326	3	y	y	PROPN
ejpam-4302	326	4	̸∈	̸∈	PROPN
ejpam-4302	326	5	λ(λ	λ(λ	PROPN
ejpam-4302	326	6	,	,	PUNCT
ejpam-4302	326	7	sp)({x	sp)({x	PROPN
ejpam-4302	326	8	}	}	PUNCT
ejpam-4302	326	9	)	)	PUNCT
ejpam-4302	326	10	.	.	PUNCT
ejpam-4302	327	1	the	the	DET
ejpam-4302	327	2	converse	converse	NOUN
ejpam-4302	327	3	is	be	AUX
ejpam-4302	327	4	similarly	similarly	ADV
ejpam-4302	327	5	shown	show	VERB
ejpam-4302	327	6	.	.	PUNCT
ejpam-4302	328	1	(	(	PUNCT
ejpam-4302	328	2	2	2	X
ejpam-4302	328	3	)	)	PUNCT
ejpam-4302	328	4	suppose	suppose	VERB
ejpam-4302	328	5	that	that	SCONJ
ejpam-4302	328	6	λ(λ	λ(λ	PROPN
ejpam-4302	328	7	,	,	PUNCT
ejpam-4302	328	8	sp)({x	sp)({x	PROPN
ejpam-4302	328	9	}	}	PUNCT
ejpam-4302	328	10	)	)	PUNCT
ejpam-4302	329	1	=	=	SYM
ejpam-4302	329	2	λ(λ	λ(λ	PROPN
ejpam-4302	329	3	,	,	PUNCT
ejpam-4302	329	4	sp)({y	sp)({y	NOUN
ejpam-4302	329	5	}	}	PUNCT
ejpam-4302	329	6	)	)	PUNCT
ejpam-4302	329	7	for	for	ADP
ejpam-4302	329	8	any	any	DET
ejpam-4302	329	9	x	x	NOUN
ejpam-4302	329	10	,	,	PUNCT
ejpam-4302	329	11	y	y	PROPN
ejpam-4302	329	12	∈	∈	PROPN
ejpam-4302	329	13	x.	x.	VERB
ejpam-4302	329	14	since	since	SCONJ
ejpam-4302	329	15	x	x	PROPN
ejpam-4302	329	16	∈	∈	PROPN
ejpam-4302	329	17	λ(λ	λ(λ	PROPN
ejpam-4302	329	18	,	,	PUNCT
ejpam-4302	329	19	sp)({x	sp)({x	PROPN
ejpam-4302	329	20	}	}	PUNCT
ejpam-4302	329	21	)	)	PUNCT
ejpam-4302	329	22	,	,	PUNCT
ejpam-4302	329	23	x	x	PUNCT
ejpam-4302	329	24	∈	∈	PROPN
ejpam-4302	329	25	λ(λ	λ(λ	PROPN
ejpam-4302	329	26	,	,	PUNCT
ejpam-4302	329	27	sp)({y	sp)({y	NOUN
ejpam-4302	329	28	}	}	PUNCT
ejpam-4302	329	29	)	)	PUNCT
ejpam-4302	329	30	,	,	PUNCT
ejpam-4302	329	31	by	by	ADP
ejpam-4302	329	32	(	(	PUNCT
ejpam-4302	329	33	1	1	NUM
ejpam-4302	329	34	)	)	PUNCT
ejpam-4302	329	35	,	,	PUNCT
ejpam-4302	329	36	y	y	PROPN
ejpam-4302	329	37	∈	∈	PROPN
ejpam-4302	329	38	{	{	PUNCT
ejpam-4302	329	39	x}(λ	x}(λ	PROPN
ejpam-4302	329	40	,	,	PUNCT
ejpam-4302	329	41	sp	sp	NOUN
ejpam-4302	329	42	)	)	PUNCT
ejpam-4302	329	43	.	.	PUNCT
ejpam-4302	330	1	by	by	ADP
ejpam-4302	330	2	lemma	lemma	PROPN
ejpam-4302	330	3	3	3	NUM
ejpam-4302	330	4	,	,	PUNCT
ejpam-4302	330	5	{	{	PUNCT
ejpam-4302	330	6	y}(λ	y}(λ	PROPN
ejpam-4302	330	7	,	,	PUNCT
ejpam-4302	330	8	sp	sp	NOUN
ejpam-4302	330	9	)	)	PUNCT
ejpam-4302	330	10	⊆	⊆	NUM
ejpam-4302	330	11	{	{	PUNCT
ejpam-4302	330	12	x}(λ	x}(λ	PROPN
ejpam-4302	330	13	,	,	PUNCT
ejpam-4302	330	14	sp	sp	NOUN
ejpam-4302	330	15	)	)	PUNCT
ejpam-4302	330	16	.	.	PUNCT
ejpam-4302	331	1	similarly	similarly	ADV
ejpam-4302	331	2	,	,	PUNCT
ejpam-4302	331	3	we	we	PRON
ejpam-4302	331	4	have	have	VERB
ejpam-4302	331	5	{	{	PUNCT
ejpam-4302	331	6	x}(λ	x}(λ	PROPN
ejpam-4302	331	7	,	,	PUNCT
ejpam-4302	331	8	sp	sp	NOUN
ejpam-4302	331	9	)	)	PUNCT
ejpam-4302	331	10	⊆	⊆	NUM
ejpam-4302	331	11	{	{	PUNCT
ejpam-4302	331	12	y}(λ	y}(λ	PROPN
ejpam-4302	331	13	,	,	PUNCT
ejpam-4302	331	14	sp	sp	NOUN
ejpam-4302	331	15	)	)	PUNCT
ejpam-4302	331	16	and	and	CCONJ
ejpam-4302	331	17	hence	hence	ADV
ejpam-4302	331	18	{	{	PUNCT
ejpam-4302	331	19	x}(λ	x}(λ	PROPN
ejpam-4302	331	20	,	,	PUNCT
ejpam-4302	331	21	sp	sp	NOUN
ejpam-4302	331	22	)	)	PUNCT
ejpam-4302	331	23	=	=	SYM
ejpam-4302	331	24	{	{	PUNCT
ejpam-4302	331	25	y}(λ	y}(λ	PROPN
ejpam-4302	331	26	,	,	PUNCT
ejpam-4302	331	27	sp	sp	NOUN
ejpam-4302	331	28	)	)	PUNCT
ejpam-4302	331	29	.	.	PUNCT
ejpam-4302	332	1	conversely	conversely	ADV
ejpam-4302	332	2	,	,	PUNCT
ejpam-4302	332	3	suppose	suppose	VERB
ejpam-4302	332	4	that	that	SCONJ
ejpam-4302	332	5	{	{	PUNCT
ejpam-4302	332	6	x}(λ	x}(λ	PROPN
ejpam-4302	332	7	,	,	PUNCT
ejpam-4302	332	8	sp	sp	NOUN
ejpam-4302	332	9	)	)	PUNCT
ejpam-4302	332	10	=	=	SYM
ejpam-4302	332	11	{	{	PUNCT
ejpam-4302	332	12	y}(λ	y}(λ	PROPN
ejpam-4302	332	13	,	,	PUNCT
ejpam-4302	332	14	sp	sp	NOUN
ejpam-4302	332	15	)	)	PUNCT
ejpam-4302	332	16	.	.	PUNCT
ejpam-4302	333	1	since	since	SCONJ
ejpam-4302	333	2	x	x	PROPN
ejpam-4302	333	3	∈	∈	PROPN
ejpam-4302	333	4	{	{	PUNCT
ejpam-4302	333	5	x}(λ	x}(λ	PROPN
ejpam-4302	333	6	,	,	PUNCT
ejpam-4302	333	7	sp	sp	NOUN
ejpam-4302	333	8	)	)	PUNCT
ejpam-4302	333	9	,	,	PUNCT
ejpam-4302	333	10	x	x	PUNCT
ejpam-4302	333	11	∈	∈	PROPN
ejpam-4302	333	12	{	{	PUNCT
ejpam-4302	333	13	y}(λ	y}(λ	PROPN
ejpam-4302	333	14	,	,	PUNCT
ejpam-4302	333	15	sp	sp	NOUN
ejpam-4302	333	16	)	)	PUNCT
ejpam-4302	333	17	,	,	PUNCT
ejpam-4302	333	18	by	by	ADP
ejpam-4302	333	19	(	(	PUNCT
ejpam-4302	333	20	1	1	NUM
ejpam-4302	333	21	)	)	PUNCT
ejpam-4302	333	22	,	,	PUNCT
ejpam-4302	333	23	y	y	PROPN
ejpam-4302	333	24	∈	∈	PROPN
ejpam-4302	333	25	λ(λ	λ(λ	PROPN
ejpam-4302	333	26	,	,	PUNCT
ejpam-4302	333	27	sp)({x	sp)({x	PROPN
ejpam-4302	333	28	}	}	PUNCT
ejpam-4302	333	29	)	)	PUNCT
ejpam-4302	333	30	.	.	PUNCT
ejpam-4302	334	1	by	by	ADP
ejpam-4302	334	2	lemma	lemma	PROPN
ejpam-4302	334	3	7	7	NUM
ejpam-4302	334	4	,	,	PUNCT
ejpam-4302	334	5	λ(λ	λ(λ	ADV
ejpam-4302	334	6	,	,	PUNCT
ejpam-4302	334	7	sp)({y	sp)({y	NOUN
ejpam-4302	334	8	}	}	PUNCT
ejpam-4302	334	9	)	)	PUNCT
ejpam-4302	334	10	⊆	⊆	NUM
ejpam-4302	334	11	λ(λ	λ(λ	ADP
ejpam-4302	334	12	,	,	PUNCT
ejpam-4302	334	13	sp)(λ(λ	sp)(λ(λ	PROPN
ejpam-4302	334	14	,	,	PUNCT
ejpam-4302	334	15	sp)({x	sp)({x	NOUN
ejpam-4302	334	16	}	}	PUNCT
ejpam-4302	334	17	)	)	PUNCT
ejpam-4302	334	18	)	)	PUNCT
ejpam-4302	335	1	=	=	SYM
ejpam-4302	335	2	λ(λ	λ(λ	PROPN
ejpam-4302	335	3	,	,	PUNCT
ejpam-4302	335	4	sp)({x	sp)({x	PROPN
ejpam-4302	335	5	}	}	PUNCT
ejpam-4302	335	6	)	)	PUNCT
ejpam-4302	335	7	.	.	PUNCT
ejpam-4302	336	1	similarly	similarly	ADV
ejpam-4302	336	2	,	,	PUNCT
ejpam-4302	336	3	we	we	PRON
ejpam-4302	336	4	have	have	VERB
ejpam-4302	336	5	λ(λ	λ(λ	PROPN
ejpam-4302	336	6	,	,	PUNCT
ejpam-4302	336	7	sp)({x	sp)({x	PROPN
ejpam-4302	336	8	}	}	PUNCT
ejpam-4302	336	9	)	)	PUNCT
ejpam-4302	337	1	⊆	⊆	NUM
ejpam-4302	337	2	λ(λ	λ(λ	NOUN
ejpam-4302	337	3	,	,	PUNCT
ejpam-4302	337	4	sp)({y	sp)({y	NOUN
ejpam-4302	337	5	}	}	PUNCT
ejpam-4302	337	6	)	)	PUNCT
ejpam-4302	337	7	and	and	CCONJ
ejpam-4302	337	8	hence	hence	ADV
ejpam-4302	337	9	λ(λ	λ(λ	PROPN
ejpam-4302	337	10	,	,	PUNCT
ejpam-4302	337	11	sp)({x	sp)({x	PROPN
ejpam-4302	337	12	}	}	PUNCT
ejpam-4302	337	13	)	)	PUNCT
ejpam-4302	338	1	=	=	SYM
ejpam-4302	338	2	λ(λ	λ(λ	PROPN
ejpam-4302	338	3	,	,	PUNCT
ejpam-4302	338	4	sp)({y	sp)({y	NOUN
ejpam-4302	338	5	}	}	PUNCT
ejpam-4302	338	6	)	)	PUNCT
ejpam-4302	338	7	.	.	PUNCT
ejpam-4302	339	1	definition	definition	NOUN
ejpam-4302	339	2	7	7	NUM
ejpam-4302	339	3	.	.	PUNCT
ejpam-4302	340	1	a	a	DET
ejpam-4302	340	2	subset	subset	NOUN
ejpam-4302	340	3	a	a	PRON
ejpam-4302	340	4	of	of	ADP
ejpam-4302	340	5	a	a	DET
ejpam-4302	340	6	topological	topological	ADJ
ejpam-4302	340	7	space	space	NOUN
ejpam-4302	340	8	(	(	PUNCT
ejpam-4302	340	9	x	x	X
ejpam-4302	340	10	,	,	PUNCT
ejpam-4302	340	11	τ	τ	X
ejpam-4302	340	12	)	)	PUNCT
ejpam-4302	340	13	is	be	AUX
ejpam-4302	340	14	called	call	VERB
ejpam-4302	340	15	a	a	DET
ejpam-4302	340	16	λ(λ	λ(λ	NOUN
ejpam-4302	340	17	,	,	PUNCT
ejpam-4302	340	18	sp)-set	sp)-set	VERB
ejpam-4302	340	19	if	if	SCONJ
ejpam-4302	340	20	a	a	DET
ejpam-4302	340	21	=	=	SYM
ejpam-4302	340	22	λ(λ	λ(λ	PROPN
ejpam-4302	340	23	,	,	PUNCT
ejpam-4302	340	24	sp)(a	sp)(a	PROPN
ejpam-4302	340	25	)	)	PUNCT
ejpam-4302	340	26	.	.	PUNCT
ejpam-4302	341	1	the	the	DET
ejpam-4302	341	2	family	family	NOUN
ejpam-4302	341	3	of	of	ADP
ejpam-4302	341	4	all	all	PRON
ejpam-4302	341	5	λ(λ	λ(λ	ADP
ejpam-4302	341	6	,	,	PUNCT
ejpam-4302	341	7	sp)-sets	sp)-set	NOUN
ejpam-4302	341	8	of	of	ADP
ejpam-4302	341	9	a	a	DET
ejpam-4302	341	10	topological	topological	ADJ
ejpam-4302	341	11	space	space	NOUN
ejpam-4302	341	12	(	(	PUNCT
ejpam-4302	341	13	x	x	X
ejpam-4302	341	14	,	,	PUNCT
ejpam-4302	341	15	τ	τ	X
ejpam-4302	341	16	)	)	PUNCT
ejpam-4302	341	17	is	be	AUX
ejpam-4302	341	18	denoted	denote	VERB
ejpam-4302	341	19	by	by	ADP
ejpam-4302	341	20	λ(λ	λ(λ	PROPN
ejpam-4302	341	21	,	,	PUNCT
ejpam-4302	341	22	sp)(x	sp)(x	PROPN
ejpam-4302	341	23	,	,	PUNCT
ejpam-4302	341	24	τ	τ	PROPN
ejpam-4302	341	25	)	)	PUNCT
ejpam-4302	341	26	(	(	PUNCT
ejpam-4302	341	27	or	or	CCONJ
ejpam-4302	341	28	simply	simply	ADV
ejpam-4302	341	29	λ(λ	λ(λ	ADV
ejpam-4302	341	30	,	,	PUNCT
ejpam-4302	341	31	sp	sp	NOUN
ejpam-4302	341	32	)	)	PUNCT
ejpam-4302	341	33	)	)	PUNCT
ejpam-4302	341	34	.	.	PUNCT
ejpam-4302	342	1	definition	definition	NOUN
ejpam-4302	342	2	8	8	NUM
ejpam-4302	342	3	.	.	PUNCT
ejpam-4302	343	1	a	a	DET
ejpam-4302	343	2	subset	subset	NOUN
ejpam-4302	343	3	a	a	PRON
ejpam-4302	343	4	of	of	ADP
ejpam-4302	343	5	a	a	DET
ejpam-4302	343	6	topological	topological	ADJ
ejpam-4302	343	7	space	space	NOUN
ejpam-4302	343	8	(	(	PUNCT
ejpam-4302	343	9	x	x	X
ejpam-4302	343	10	,	,	PUNCT
ejpam-4302	343	11	τ	τ	X
ejpam-4302	343	12	)	)	PUNCT
ejpam-4302	343	13	is	be	AUX
ejpam-4302	343	14	called	call	VERB
ejpam-4302	343	15	a	a	DET
ejpam-4302	343	16	generalized	generalized	ADJ
ejpam-4302	343	17	λ(λ	λ(λ	ADP
ejpam-4302	343	18	,	,	PUNCT
ejpam-4302	343	19	sp)-set	sp)-set	PROPN
ejpam-4302	343	20	(	(	PUNCT
ejpam-4302	343	21	briefly	briefly	ADV
ejpam-4302	343	22	g	g	NOUN
ejpam-4302	343	23	-	-	PUNCT
ejpam-4302	343	24	λ(λ	λ(λ	NOUN
ejpam-4302	343	25	,	,	PUNCT
ejpam-4302	343	26	sp)-set	sp)-set	PROPN
ejpam-4302	343	27	)	)	PUNCT
ejpam-4302	343	28	if	if	SCONJ
ejpam-4302	343	29	λ(λ	λ(λ	PROPN
ejpam-4302	343	30	,	,	PUNCT
ejpam-4302	343	31	sp)(a	sp)(a	PROPN
ejpam-4302	343	32	)	)	PUNCT
ejpam-4302	344	1	⊆	⊆	NUM
ejpam-4302	344	2	f	f	NOUN
ejpam-4302	344	3	whenever	whenever	SCONJ
ejpam-4302	344	4	a	a	DET
ejpam-4302	344	5	⊆	⊆	NUM
ejpam-4302	344	6	f	f	PROPN
ejpam-4302	344	7	and	and	CCONJ
ejpam-4302	344	8	f	f	PROPN
ejpam-4302	344	9	is	be	AUX
ejpam-4302	344	10	a	a	DET
ejpam-4302	344	11	(	(	PUNCT
ejpam-4302	344	12	λ	λ	PROPN
ejpam-4302	344	13	,	,	PUNCT
ejpam-4302	344	14	sp)-closed	sp)-close	VERB
ejpam-4302	344	15	set	set	NOUN
ejpam-4302	344	16	.	.	PUNCT
ejpam-4302	345	1	definition	definition	NOUN
ejpam-4302	345	2	9	9	NUM
ejpam-4302	345	3	.	.	PUNCT
ejpam-4302	346	1	a	a	DET
ejpam-4302	346	2	topological	topological	ADJ
ejpam-4302	346	3	space	space	NOUN
ejpam-4302	346	4	(	(	PUNCT
ejpam-4302	346	5	x	x	X
ejpam-4302	346	6	,	,	PUNCT
ejpam-4302	346	7	τ	τ	X
ejpam-4302	346	8	)	)	PUNCT
ejpam-4302	346	9	is	be	AUX
ejpam-4302	346	10	called	call	VERB
ejpam-4302	346	11	a	a	DET
ejpam-4302	346	12	λsp	λsp	PROPN
ejpam-4302	346	13	-	-	NOUN
ejpam-4302	346	14	t	t	NOUN
ejpam-4302	346	15	1	1	NUM
ejpam-4302	346	16	2	2	NUM
ejpam-4302	346	17	-space	-space	NOUN
ejpam-4302	346	18	if	if	SCONJ
ejpam-4302	346	19	every	every	DET
ejpam-4302	346	20	g-(λ	g-(λ	PROPN
ejpam-4302	346	21	,	,	PUNCT
ejpam-4302	346	22	sp)-closed	sp)-close	VERB
ejpam-4302	346	23	set	set	NOUN
ejpam-4302	346	24	of	of	ADP
ejpam-4302	346	25	x	x	PUNCT
ejpam-4302	346	26	is	be	AUX
ejpam-4302	346	27	(	(	PUNCT
ejpam-4302	346	28	λ	λ	X
ejpam-4302	346	29	,	,	PUNCT
ejpam-4302	346	30	sp)-closed	sp)-close	VERB
ejpam-4302	346	31	.	.	PUNCT
ejpam-4302	347	1	lemma	lemma	PROPN
ejpam-4302	347	2	9	9	NUM
ejpam-4302	347	3	.	.	PUNCT
ejpam-4302	348	1	for	for	ADP
ejpam-4302	348	2	a	a	DET
ejpam-4302	348	3	topological	topological	ADJ
ejpam-4302	348	4	space	space	NOUN
ejpam-4302	348	5	(	(	PUNCT
ejpam-4302	348	6	x	x	X
ejpam-4302	348	7	,	,	PUNCT
ejpam-4302	348	8	τ	τ	PROPN
ejpam-4302	348	9	)	)	PUNCT
ejpam-4302	348	10	,	,	PUNCT
ejpam-4302	348	11	the	the	DET
ejpam-4302	348	12	following	follow	VERB
ejpam-4302	348	13	properties	property	NOUN
ejpam-4302	348	14	hold	hold	VERB
ejpam-4302	348	15	:	:	PUNCT
ejpam-4302	348	16	(	(	PUNCT
ejpam-4302	348	17	1	1	X
ejpam-4302	348	18	)	)	PUNCT
ejpam-4302	348	19	for	for	ADP
ejpam-4302	348	20	each	each	DET
ejpam-4302	348	21	x	x	SYM
ejpam-4302	348	22	∈	∈	PROPN
ejpam-4302	348	23	x	x	NOUN
ejpam-4302	348	24	,	,	PUNCT
ejpam-4302	348	25	the	the	DET
ejpam-4302	348	26	singleton	singleton	NOUN
ejpam-4302	348	27	{	{	PUNCT
ejpam-4302	348	28	x	x	NOUN
ejpam-4302	348	29	}	}	PUNCT
ejpam-4302	348	30	is	be	AUX
ejpam-4302	348	31	(	(	PUNCT
ejpam-4302	348	32	λ	λ	X
ejpam-4302	348	33	,	,	PUNCT
ejpam-4302	348	34	sp)-closed	sp)-close	VERB
ejpam-4302	348	35	or	or	CCONJ
ejpam-4302	348	36	x	x	PART
ejpam-4302	348	37	−	−	PROPN
ejpam-4302	348	38	{	{	PUNCT
ejpam-4302	348	39	x	x	NOUN
ejpam-4302	348	40	}	}	PUNCT
ejpam-4302	348	41	is	be	AUX
ejpam-4302	348	42	g-(λ	g-(λ	PROPN
ejpam-4302	348	43	,	,	PUNCT
ejpam-4302	348	44	sp)-closed	sp)-close	VERB
ejpam-4302	348	45	.	.	PUNCT
ejpam-4302	349	1	(	(	PUNCT
ejpam-4302	349	2	2	2	X
ejpam-4302	349	3	)	)	PUNCT
ejpam-4302	349	4	for	for	ADP
ejpam-4302	349	5	each	each	DET
ejpam-4302	349	6	x	x	SYM
ejpam-4302	349	7	∈	∈	PROPN
ejpam-4302	349	8	x	x	NOUN
ejpam-4302	349	9	,	,	PUNCT
ejpam-4302	349	10	the	the	DET
ejpam-4302	349	11	singleton	singleton	NOUN
ejpam-4302	349	12	{	{	PUNCT
ejpam-4302	349	13	x	x	NOUN
ejpam-4302	349	14	}	}	PUNCT
ejpam-4302	349	15	is	be	AUX
ejpam-4302	349	16	(	(	PUNCT
ejpam-4302	349	17	λ	λ	X
ejpam-4302	349	18	,	,	PUNCT
ejpam-4302	349	19	sp)-open	sp)-open	ADJ
ejpam-4302	349	20	or	or	CCONJ
ejpam-4302	349	21	x	x	SYM
ejpam-4302	349	22	−	−	PROPN
ejpam-4302	349	23	{	{	PUNCT
ejpam-4302	349	24	x	x	NOUN
ejpam-4302	349	25	}	}	PUNCT
ejpam-4302	349	26	is	be	AUX
ejpam-4302	349	27	a	a	DET
ejpam-4302	349	28	g	g	NOUN
ejpam-4302	349	29	-	-	PUNCT
ejpam-4302	349	30	λ(λ	λ(λ	NOUN
ejpam-4302	349	31	,	,	PUNCT
ejpam-4302	349	32	sp)-set	sp)-set	PROPN
ejpam-4302	349	33	.	.	PUNCT
ejpam-4302	350	1	proof	proof	NOUN
ejpam-4302	350	2	.	.	PUNCT
ejpam-4302	351	1	(	(	PUNCT
ejpam-4302	351	2	1	1	X
ejpam-4302	351	3	)	)	PUNCT
ejpam-4302	351	4	let	let	VERB
ejpam-4302	351	5	x	x	SYM
ejpam-4302	351	6	∈	∈	PROPN
ejpam-4302	351	7	x	x	X
ejpam-4302	351	8	and	and	CCONJ
ejpam-4302	351	9	the	the	DET
ejpam-4302	351	10	singleton	singleton	NOUN
ejpam-4302	351	11	{	{	PUNCT
ejpam-4302	351	12	x	x	NOUN
ejpam-4302	351	13	}	}	PUNCT
ejpam-4302	351	14	be	be	VERB
ejpam-4302	351	15	not	not	PART
ejpam-4302	351	16	(	(	PUNCT
ejpam-4302	351	17	λ	λ	X
ejpam-4302	351	18	,	,	PUNCT
ejpam-4302	351	19	sp)-closed	sp)-close	VERB
ejpam-4302	351	20	.	.	PUNCT
ejpam-4302	352	1	then	then	ADV
ejpam-4302	352	2	,	,	PUNCT
ejpam-4302	352	3	x	x	PUNCT
ejpam-4302	352	4	−	−	NOUN
ejpam-4302	352	5	{	{	PUNCT
ejpam-4302	352	6	x	x	NOUN
ejpam-4302	352	7	}	}	PUNCT
ejpam-4302	352	8	is	be	AUX
ejpam-4302	352	9	not	not	PART
ejpam-4302	352	10	(	(	PUNCT
ejpam-4302	352	11	λ	λ	NOUN
ejpam-4302	352	12	,	,	PUNCT
ejpam-4302	352	13	sp)-open	sp)-open	ADJ
ejpam-4302	352	14	and	and	CCONJ
ejpam-4302	352	15	x	x	X
ejpam-4302	352	16	is	be	AUX
ejpam-4302	352	17	the	the	DET
ejpam-4302	352	18	only	only	ADJ
ejpam-4302	352	19	(	(	PUNCT
ejpam-4302	352	20	λ	λ	NOUN
ejpam-4302	352	21	,	,	PUNCT
ejpam-4302	352	22	sp)-open	sp)-open	ADJ
ejpam-4302	352	23	set	set	NOUN
ejpam-4302	352	24	which	which	PRON
ejpam-4302	352	25	contains	contain	VERB
ejpam-4302	352	26	x	x	X
ejpam-4302	352	27	−	−	PROPN
ejpam-4302	352	28	{	{	PUNCT
ejpam-4302	352	29	x	x	NOUN
ejpam-4302	352	30	}	}	PUNCT
ejpam-4302	352	31	and	and	CCONJ
ejpam-4302	352	32	hence	hence	ADV
ejpam-4302	352	33	x	x	X
ejpam-4302	352	34	−	−	PROPN
ejpam-4302	352	35	{	{	PUNCT
ejpam-4302	352	36	x	x	NOUN
ejpam-4302	352	37	}	}	PUNCT
ejpam-4302	352	38	is	be	AUX
ejpam-4302	352	39	g-(λ	g-(λ	PROPN
ejpam-4302	352	40	,	,	PUNCT
ejpam-4302	352	41	sp)-closed	sp)-close	VERB
ejpam-4302	352	42	.	.	PUNCT
ejpam-4302	353	1	(	(	PUNCT
ejpam-4302	353	2	2	2	X
ejpam-4302	353	3	)	)	PUNCT
ejpam-4302	353	4	let	let	VERB
ejpam-4302	353	5	x	x	SYM
ejpam-4302	353	6	∈	∈	PROPN
ejpam-4302	353	7	x	x	X
ejpam-4302	353	8	and	and	CCONJ
ejpam-4302	353	9	the	the	DET
ejpam-4302	353	10	singleton	singleton	NOUN
ejpam-4302	353	11	{	{	PUNCT
ejpam-4302	353	12	x	x	NOUN
ejpam-4302	353	13	}	}	PUNCT
ejpam-4302	353	14	be	be	VERB
ejpam-4302	353	15	not	not	PART
ejpam-4302	353	16	(	(	PUNCT
ejpam-4302	353	17	λ	λ	NOUN
ejpam-4302	353	18	,	,	PUNCT
ejpam-4302	353	19	sp)-open	sp)-open	NOUN
ejpam-4302	353	20	.	.	PUNCT
ejpam-4302	354	1	then	then	ADV
ejpam-4302	354	2	,	,	PUNCT
ejpam-4302	354	3	x−{x	x−{x	PROPN
ejpam-4302	354	4	}	}	PUNCT
ejpam-4302	354	5	is	be	AUX
ejpam-4302	354	6	not	not	PART
ejpam-4302	354	7	(	(	PUNCT
ejpam-4302	354	8	λ	λ	X
ejpam-4302	354	9	,	,	PUNCT
ejpam-4302	354	10	sp)closed	sp)close	VERB
ejpam-4302	354	11	and	and	CCONJ
ejpam-4302	354	12	x	x	X
ejpam-4302	354	13	is	be	AUX
ejpam-4302	354	14	the	the	DET
ejpam-4302	354	15	only	only	ADJ
ejpam-4302	354	16	(	(	PUNCT
ejpam-4302	354	17	λ	λ	PROPN
ejpam-4302	354	18	,	,	PUNCT
ejpam-4302	354	19	sp)-closed	sp)-close	VERB
ejpam-4302	354	20	set	set	VERB
ejpam-4302	354	21	which	which	PRON
ejpam-4302	354	22	contains	contain	VERB
ejpam-4302	354	23	x	x	X
ejpam-4302	354	24	−	−	PROPN
ejpam-4302	354	25	{	{	PUNCT
ejpam-4302	354	26	x	x	NOUN
ejpam-4302	354	27	}	}	PUNCT
ejpam-4302	354	28	and	and	CCONJ
ejpam-4302	354	29	hence	hence	ADV
ejpam-4302	354	30	x	x	X
ejpam-4302	354	31	−	−	PROPN
ejpam-4302	354	32	{	{	PUNCT
ejpam-4302	354	33	x	x	NOUN
ejpam-4302	354	34	}	}	PUNCT
ejpam-4302	354	35	is	be	AUX
ejpam-4302	354	36	a	a	DET
ejpam-4302	354	37	g	g	NOUN
ejpam-4302	354	38	-	-	PUNCT
ejpam-4302	354	39	λ(λ	λ(λ	NOUN
ejpam-4302	354	40	,	,	PUNCT
ejpam-4302	354	41	sp)-set	sp)-set	PROPN
ejpam-4302	354	42	.	.	PUNCT
ejpam-4302	355	1	c.	c.	PROPN
ejpam-4302	355	2	boonpok	boonpok	PROPN
ejpam-4302	355	3	,	,	PUNCT
ejpam-4302	355	4	c.	c.	PROPN
ejpam-4302	355	5	viriyapong	viriyapong	PROPN
ejpam-4302	355	6	/	/	SYM
ejpam-4302	355	7	eur	eur	PROPN
ejpam-4302	355	8	.	.	PUNCT
ejpam-4302	356	1	j.	j.	PROPN
ejpam-4302	356	2	pure	pure	PROPN
ejpam-4302	356	3	appl	appl	PROPN
ejpam-4302	356	4	.	.	PROPN
ejpam-4302	356	5	math	math	PROPN
ejpam-4302	356	6	,	,	PUNCT
ejpam-4302	356	7	15	15	NUM
ejpam-4302	356	8	(	(	PUNCT
ejpam-4302	356	9	4	4	NUM
ejpam-4302	356	10	)	)	PUNCT
ejpam-4302	356	11	(	(	PUNCT
ejpam-4302	356	12	2022	2022	NUM
ejpam-4302	356	13	)	)	PUNCT
ejpam-4302	356	14	,	,	PUNCT
ejpam-4302	356	15	2127	2127	NUM
ejpam-4302	356	16	-	-	SYM
ejpam-4302	356	17	2140	2140	NUM
ejpam-4302	356	18	2137	2137	NUM
ejpam-4302	356	19	theorem	theorem	VERB
ejpam-4302	356	20	17	17	NUM
ejpam-4302	356	21	.	.	PUNCT
ejpam-4302	357	1	for	for	ADP
ejpam-4302	357	2	a	a	DET
ejpam-4302	357	3	topological	topological	ADJ
ejpam-4302	357	4	space	space	NOUN
ejpam-4302	357	5	(	(	PUNCT
ejpam-4302	357	6	x	x	X
ejpam-4302	357	7	,	,	PUNCT
ejpam-4302	357	8	τ	τ	PROPN
ejpam-4302	357	9	)	)	PUNCT
ejpam-4302	357	10	,	,	PUNCT
ejpam-4302	357	11	the	the	DET
ejpam-4302	357	12	following	follow	VERB
ejpam-4302	357	13	properties	property	NOUN
ejpam-4302	357	14	are	be	AUX
ejpam-4302	357	15	equivalent	equivalent	ADJ
ejpam-4302	357	16	:	:	PUNCT
ejpam-4302	357	17	(	(	PUNCT
ejpam-4302	357	18	1	1	X
ejpam-4302	357	19	)	)	PUNCT
ejpam-4302	357	20	(	(	PUNCT
ejpam-4302	357	21	x	x	X
ejpam-4302	357	22	,	,	PUNCT
ejpam-4302	357	23	τ	τ	X
ejpam-4302	357	24	)	)	PUNCT
ejpam-4302	357	25	is	be	AUX
ejpam-4302	357	26	a	a	DET
ejpam-4302	357	27	λsp	λsp	PROPN
ejpam-4302	357	28	-	-	PUNCT
ejpam-4302	357	29	t	t	NOUN
ejpam-4302	357	30	1	1	NUM
ejpam-4302	357	31	2	2	NUM
ejpam-4302	357	32	-space	-space	NOUN
ejpam-4302	357	33	.	.	PUNCT
ejpam-4302	358	1	(	(	PUNCT
ejpam-4302	358	2	2	2	X
ejpam-4302	358	3	)	)	PUNCT
ejpam-4302	358	4	for	for	ADP
ejpam-4302	358	5	each	each	DET
ejpam-4302	358	6	x	x	SYM
ejpam-4302	358	7	∈	∈	PROPN
ejpam-4302	358	8	x	x	NOUN
ejpam-4302	358	9	,	,	PUNCT
ejpam-4302	358	10	the	the	DET
ejpam-4302	358	11	singleton	singleton	NOUN
ejpam-4302	358	12	{	{	PUNCT
ejpam-4302	358	13	x	x	NOUN
ejpam-4302	358	14	}	}	PUNCT
ejpam-4302	358	15	is	be	AUX
ejpam-4302	358	16	(	(	PUNCT
ejpam-4302	358	17	λ	λ	X
ejpam-4302	358	18	,	,	PUNCT
ejpam-4302	358	19	sp)-open	sp)-open	ADJ
ejpam-4302	358	20	or	or	CCONJ
ejpam-4302	358	21	(	(	PUNCT
ejpam-4302	358	22	λ	λ	PROPN
ejpam-4302	358	23	,	,	PUNCT
ejpam-4302	358	24	sp)-closed	sp)-close	VERB
ejpam-4302	358	25	.	.	PUNCT
ejpam-4302	359	1	(	(	PUNCT
ejpam-4302	359	2	3	3	X
ejpam-4302	359	3	)	)	PUNCT
ejpam-4302	359	4	every	every	DET
ejpam-4302	359	5	g	g	PROPN
ejpam-4302	359	6	-	-	PUNCT
ejpam-4302	359	7	λ(λ	λ(λ	NOUN
ejpam-4302	359	8	,	,	PUNCT
ejpam-4302	359	9	sp)-set	sp)-set	PROPN
ejpam-4302	359	10	is	be	AUX
ejpam-4302	359	11	a	a	DET
ejpam-4302	359	12	λ(λ	λ(λ	NOUN
ejpam-4302	359	13	,	,	PUNCT
ejpam-4302	359	14	sp)-set	sp)-set	PROPN
ejpam-4302	359	15	.	.	PUNCT
ejpam-4302	360	1	proof	proof	NOUN
ejpam-4302	360	2	.	.	PUNCT
ejpam-4302	361	1	(	(	PUNCT
ejpam-4302	361	2	1	1	X
ejpam-4302	361	3	)	)	PUNCT
ejpam-4302	361	4	⇒	⇒	NOUN
ejpam-4302	361	5	(	(	PUNCT
ejpam-4302	361	6	2	2	NUM
ejpam-4302	361	7	):	):	PUNCT
ejpam-4302	361	8	by	by	ADP
ejpam-4302	361	9	lemma	lemma	PROPN
ejpam-4302	361	10	9	9	NUM
ejpam-4302	361	11	,	,	PUNCT
ejpam-4302	361	12	for	for	ADP
ejpam-4302	361	13	each	each	DET
ejpam-4302	361	14	x	x	SYM
ejpam-4302	361	15	∈	∈	PROPN
ejpam-4302	361	16	x	x	NOUN
ejpam-4302	361	17	,	,	PUNCT
ejpam-4302	361	18	the	the	DET
ejpam-4302	361	19	singleton	singleton	NOUN
ejpam-4302	361	20	{	{	PUNCT
ejpam-4302	361	21	x	x	NOUN
ejpam-4302	361	22	}	}	PUNCT
ejpam-4302	361	23	is	be	AUX
ejpam-4302	361	24	(	(	PUNCT
ejpam-4302	361	25	λ	λ	X
ejpam-4302	361	26	,	,	PUNCT
ejpam-4302	361	27	sp)-closed	sp)-close	VERB
ejpam-4302	361	28	or	or	CCONJ
ejpam-4302	361	29	x	x	PART
ejpam-4302	361	30	−	−	PROPN
ejpam-4302	361	31	{	{	PUNCT
ejpam-4302	361	32	x	x	NOUN
ejpam-4302	361	33	}	}	PUNCT
ejpam-4302	361	34	is	be	AUX
ejpam-4302	361	35	g-(λ	g-(λ	PROPN
ejpam-4302	361	36	,	,	PUNCT
ejpam-4302	361	37	sp)-closed	sp)-close	VERB
ejpam-4302	361	38	.	.	PUNCT
ejpam-4302	362	1	since	since	SCONJ
ejpam-4302	362	2	(	(	PUNCT
ejpam-4302	362	3	x	x	X
ejpam-4302	362	4	,	,	PUNCT
ejpam-4302	362	5	τ	τ	X
ejpam-4302	362	6	)	)	PUNCT
ejpam-4302	362	7	is	be	AUX
ejpam-4302	362	8	a	a	DET
ejpam-4302	362	9	λsp	λsp	PROPN
ejpam-4302	362	10	-	-	PUNCT
ejpam-4302	362	11	t	t	NOUN
ejpam-4302	362	12	1	1	NUM
ejpam-4302	362	13	2	2	NUM
ejpam-4302	362	14	-space	-space	NOUN
ejpam-4302	362	15	,	,	PUNCT
ejpam-4302	362	16	x	x	PUNCT
ejpam-4302	362	17	−	−	NOUN
ejpam-4302	362	18	{	{	PUNCT
ejpam-4302	362	19	x	x	NOUN
ejpam-4302	362	20	}	}	PUNCT
ejpam-4302	362	21	is	be	AUX
ejpam-4302	362	22	(	(	PUNCT
ejpam-4302	362	23	λ	λ	X
ejpam-4302	362	24	,	,	PUNCT
ejpam-4302	362	25	sp)-closed	sp)-close	VERB
ejpam-4302	362	26	and	and	CCONJ
ejpam-4302	362	27	hence	hence	ADV
ejpam-4302	362	28	{	{	PUNCT
ejpam-4302	362	29	x	x	X
ejpam-4302	362	30	}	}	PUNCT
ejpam-4302	362	31	is	be	AUX
ejpam-4302	362	32	(	(	PUNCT
ejpam-4302	362	33	λ	λ	X
ejpam-4302	362	34	,	,	PUNCT
ejpam-4302	362	35	sp)-open	sp)-open	ADJ
ejpam-4302	362	36	in	in	ADP
ejpam-4302	362	37	the	the	DET
ejpam-4302	362	38	latter	latter	ADJ
ejpam-4302	362	39	case	case	NOUN
ejpam-4302	362	40	.	.	PUNCT
ejpam-4302	363	1	thus	thus	ADV
ejpam-4302	363	2	,	,	PUNCT
ejpam-4302	363	3	the	the	DET
ejpam-4302	363	4	singleton	singleton	NOUN
ejpam-4302	363	5	{	{	PUNCT
ejpam-4302	363	6	x	x	NOUN
ejpam-4302	363	7	}	}	PUNCT
ejpam-4302	363	8	is	be	AUX
ejpam-4302	363	9	(	(	PUNCT
ejpam-4302	363	10	λ	λ	X
ejpam-4302	363	11	,	,	PUNCT
ejpam-4302	363	12	sp)-open	sp)-open	ADJ
ejpam-4302	363	13	or	or	CCONJ
ejpam-4302	363	14	(	(	PUNCT
ejpam-4302	363	15	λ	λ	PROPN
ejpam-4302	363	16	,	,	PUNCT
ejpam-4302	363	17	sp)-closed	sp)-close	VERB
ejpam-4302	363	18	.	.	PUNCT
ejpam-4302	364	1	(	(	PUNCT
ejpam-4302	364	2	2	2	X
ejpam-4302	364	3	)	)	PUNCT
ejpam-4302	364	4	⇒	⇒	NOUN
ejpam-4302	364	5	(	(	PUNCT
ejpam-4302	364	6	3	3	NUM
ejpam-4302	364	7	):	):	PUNCT
ejpam-4302	364	8	suppose	suppose	VERB
ejpam-4302	364	9	that	that	SCONJ
ejpam-4302	364	10	there	there	PRON
ejpam-4302	364	11	exists	exist	VERB
ejpam-4302	364	12	a	a	DET
ejpam-4302	364	13	g	g	NOUN
ejpam-4302	364	14	-	-	PUNCT
ejpam-4302	364	15	λ(λ	λ(λ	NOUN
ejpam-4302	364	16	,	,	PUNCT
ejpam-4302	364	17	sp)-set	sp)-set	ADP
ejpam-4302	364	18	a	a	DET
ejpam-4302	364	19	which	which	PRON
ejpam-4302	364	20	is	be	AUX
ejpam-4302	364	21	not	not	PART
ejpam-4302	364	22	a	a	DET
ejpam-4302	364	23	λ(λ	λ(λ	NOUN
ejpam-4302	364	24	,	,	PUNCT
ejpam-4302	364	25	sp)-set	sp)-set	PROPN
ejpam-4302	364	26	.	.	PUNCT
ejpam-4302	365	1	there	there	PRON
ejpam-4302	365	2	exists	exist	VERB
ejpam-4302	365	3	x	x	X
ejpam-4302	365	4	∈	∈	PROPN
ejpam-4302	365	5	λ(λ	λ(λ	PROPN
ejpam-4302	365	6	,	,	PUNCT
ejpam-4302	365	7	sp)(a	sp)(a	PROPN
ejpam-4302	365	8	)	)	PUNCT
ejpam-4302	365	9	such	such	ADJ
ejpam-4302	365	10	that	that	SCONJ
ejpam-4302	365	11	x	x	PUNCT
ejpam-4302	365	12	̸∈	̸∈	PROPN
ejpam-4302	365	13	a.	a.	NOUN
ejpam-4302	365	14	in	in	ADP
ejpam-4302	365	15	case	case	NOUN
ejpam-4302	365	16	the	the	DET
ejpam-4302	365	17	singleton	singleton	NOUN
ejpam-4302	365	18	{	{	PUNCT
ejpam-4302	365	19	x	x	NOUN
ejpam-4302	365	20	}	}	PUNCT
ejpam-4302	365	21	is	be	AUX
ejpam-4302	365	22	(	(	PUNCT
ejpam-4302	365	23	λ	λ	NOUN
ejpam-4302	365	24	,	,	PUNCT
ejpam-4302	365	25	sp)-open	sp)-open	ADJ
ejpam-4302	365	26	,	,	PUNCT
ejpam-4302	365	27	a	a	DET
ejpam-4302	365	28	⊆	⊆	NUM
ejpam-4302	365	29	x−{x	x−{x	PROPN
ejpam-4302	365	30	}	}	PUNCT
ejpam-4302	365	31	and	and	CCONJ
ejpam-4302	365	32	x−{x	x−{x	PROPN
ejpam-4302	365	33	}	}	PUNCT
ejpam-4302	365	34	is	be	AUX
ejpam-4302	365	35	(	(	PUNCT
ejpam-4302	365	36	λ	λ	X
ejpam-4302	365	37	,	,	PUNCT
ejpam-4302	365	38	sp)-closed	sp)-close	VERB
ejpam-4302	365	39	.	.	PUNCT
ejpam-4302	366	1	since	since	SCONJ
ejpam-4302	366	2	a	a	PRON
ejpam-4302	366	3	is	be	AUX
ejpam-4302	366	4	a	a	DET
ejpam-4302	366	5	g	g	NOUN
ejpam-4302	366	6	-	-	PUNCT
ejpam-4302	366	7	λ(λ	λ(λ	NOUN
ejpam-4302	366	8	,	,	PUNCT
ejpam-4302	366	9	sp)-set	sp)-set	PROPN
ejpam-4302	366	10	,	,	PUNCT
ejpam-4302	366	11	λ(λ	λ(λ	ADV
ejpam-4302	366	12	,	,	PUNCT
ejpam-4302	366	13	sp)(a	sp)(a	PROPN
ejpam-4302	366	14	)	)	PUNCT
ejpam-4302	367	1	⊆	⊆	NUM
ejpam-4302	367	2	x−{x	x−{x	PROPN
ejpam-4302	367	3	}	}	PUNCT
ejpam-4302	367	4	.	.	PUNCT
ejpam-4302	368	1	this	this	PRON
ejpam-4302	368	2	is	be	AUX
ejpam-4302	368	3	a	a	DET
ejpam-4302	368	4	contradiction	contradiction	NOUN
ejpam-4302	368	5	.	.	PUNCT
ejpam-4302	369	1	in	in	ADP
ejpam-4302	369	2	case	case	NOUN
ejpam-4302	369	3	the	the	DET
ejpam-4302	369	4	singleton	singleton	NOUN
ejpam-4302	369	5	{	{	PUNCT
ejpam-4302	369	6	x	x	NOUN
ejpam-4302	369	7	}	}	PUNCT
ejpam-4302	369	8	is	be	AUX
ejpam-4302	369	9	(	(	PUNCT
ejpam-4302	369	10	λ	λ	X
ejpam-4302	369	11	,	,	PUNCT
ejpam-4302	369	12	sp)-closed	sp)-close	VERB
ejpam-4302	369	13	,	,	PUNCT
ejpam-4302	369	14	a	a	PRON
ejpam-4302	369	15	⊆	⊆	NUM
ejpam-4302	369	16	x	x	SYM
ejpam-4302	369	17	−	−	PROPN
ejpam-4302	369	18	{	{	PUNCT
ejpam-4302	369	19	x	x	NOUN
ejpam-4302	369	20	}	}	PUNCT
ejpam-4302	369	21	and	and	CCONJ
ejpam-4302	369	22	x	x	ADJ
ejpam-4302	369	23	−	−	PROPN
ejpam-4302	369	24	{	{	PUNCT
ejpam-4302	369	25	x	x	NOUN
ejpam-4302	369	26	}	}	PUNCT
ejpam-4302	369	27	is	be	AUX
ejpam-4302	369	28	(	(	PUNCT
ejpam-4302	369	29	λ	λ	NOUN
ejpam-4302	369	30	,	,	PUNCT
ejpam-4302	369	31	sp)-open	sp)-open	NOUN
ejpam-4302	369	32	.	.	PUNCT
ejpam-4302	370	1	by	by	ADP
ejpam-4302	370	2	lemma	lemma	PROPN
ejpam-4302	370	3	7	7	NUM
ejpam-4302	370	4	,	,	PUNCT
ejpam-4302	370	5	λ(λ	λ(λ	ADV
ejpam-4302	370	6	,	,	PUNCT
ejpam-4302	370	7	sp)(a	sp)(a	PROPN
ejpam-4302	370	8	)	)	PUNCT
ejpam-4302	371	1	⊆	⊆	NUM
ejpam-4302	371	2	λ(λ	λ(λ	NOUN
ejpam-4302	371	3	,	,	PUNCT
ejpam-4302	371	4	sp)(x	sp)(x	PROPN
ejpam-4302	371	5	−	−	PROPN
ejpam-4302	371	6	{	{	PUNCT
ejpam-4302	371	7	x	x	NOUN
ejpam-4302	371	8	}	}	PUNCT
ejpam-4302	371	9	)	)	PUNCT
ejpam-4302	371	10	=	=	PUNCT
ejpam-4302	371	11	x	x	X
ejpam-4302	371	12	−	−	PROPN
ejpam-4302	371	13	{	{	PUNCT
ejpam-4302	371	14	x	x	NOUN
ejpam-4302	371	15	}	}	PUNCT
ejpam-4302	371	16	.	.	PUNCT
ejpam-4302	372	1	this	this	PRON
ejpam-4302	372	2	is	be	AUX
ejpam-4302	372	3	a	a	DET
ejpam-4302	372	4	contradiction	contradiction	NOUN
ejpam-4302	372	5	.	.	PUNCT
ejpam-4302	373	1	thus	thus	ADV
ejpam-4302	373	2	,	,	PUNCT
ejpam-4302	373	3	every	every	DET
ejpam-4302	373	4	g	g	PROPN
ejpam-4302	373	5	-	-	PUNCT
ejpam-4302	373	6	λ(λ	λ(λ	NOUN
ejpam-4302	373	7	,	,	PUNCT
ejpam-4302	373	8	sp)-set	sp)-set	PROPN
ejpam-4302	373	9	is	be	AUX
ejpam-4302	373	10	a	a	DET
ejpam-4302	373	11	λ(λ	λ(λ	NOUN
ejpam-4302	373	12	,	,	PUNCT
ejpam-4302	373	13	sp)-set	sp)-set	PROPN
ejpam-4302	373	14	.	.	PUNCT
ejpam-4302	374	1	(	(	PUNCT
ejpam-4302	374	2	3	3	X
ejpam-4302	374	3	)	)	PUNCT
ejpam-4302	374	4	⇒	⇒	NOUN
ejpam-4302	374	5	(	(	PUNCT
ejpam-4302	374	6	1	1	NUM
ejpam-4302	374	7	):	):	PUNCT
ejpam-4302	374	8	suppose	suppose	VERB
ejpam-4302	374	9	that	that	SCONJ
ejpam-4302	374	10	(	(	PUNCT
ejpam-4302	374	11	x	x	X
ejpam-4302	374	12	,	,	PUNCT
ejpam-4302	374	13	τ	τ	X
ejpam-4302	374	14	)	)	PUNCT
ejpam-4302	374	15	is	be	AUX
ejpam-4302	374	16	not	not	PART
ejpam-4302	374	17	a	a	DET
ejpam-4302	374	18	λsp	λsp	NOUN
ejpam-4302	374	19	-	-	NOUN
ejpam-4302	374	20	t	t	NOUN
ejpam-4302	374	21	1	1	NUM
ejpam-4302	374	22	2	2	NUM
ejpam-4302	374	23	-space	-space	NOUN
ejpam-4302	374	24	.	.	PUNCT
ejpam-4302	375	1	then	then	ADV
ejpam-4302	375	2	,	,	PUNCT
ejpam-4302	375	3	there	there	PRON
ejpam-4302	375	4	exists	exist	VERB
ejpam-4302	375	5	a	a	DET
ejpam-4302	375	6	g-(λ	g-(λ	NOUN
ejpam-4302	375	7	,	,	PUNCT
ejpam-4302	375	8	sp)closed	sp)close	VERB
ejpam-4302	375	9	set	set	VERB
ejpam-4302	375	10	a	a	PRON
ejpam-4302	375	11	which	which	PRON
ejpam-4302	375	12	is	be	AUX
ejpam-4302	375	13	not	not	PART
ejpam-4302	375	14	(	(	PUNCT
ejpam-4302	375	15	λ	λ	X
ejpam-4302	375	16	,	,	PUNCT
ejpam-4302	375	17	sp)-closed	sp)-close	VERB
ejpam-4302	375	18	.	.	PUNCT
ejpam-4302	376	1	since	since	SCONJ
ejpam-4302	376	2	a	a	PRON
ejpam-4302	376	3	is	be	AUX
ejpam-4302	376	4	not	not	PART
ejpam-4302	376	5	(	(	PUNCT
ejpam-4302	376	6	λ	λ	X
ejpam-4302	376	7	,	,	PUNCT
ejpam-4302	376	8	sp)-closed	sp)-close	VERB
ejpam-4302	376	9	,	,	PUNCT
ejpam-4302	376	10	there	there	PRON
ejpam-4302	376	11	exists	exist	VERB
ejpam-4302	376	12	a	a	DET
ejpam-4302	376	13	point	point	NOUN
ejpam-4302	376	14	x	x	X
ejpam-4302	376	15	∈	∈	NOUN
ejpam-4302	376	16	a(λ	a(λ	ADV
ejpam-4302	376	17	,	,	PUNCT
ejpam-4302	376	18	sp	sp	NOUN
ejpam-4302	376	19	)	)	PUNCT
ejpam-4302	377	1	such	such	ADJ
ejpam-4302	377	2	that	that	SCONJ
ejpam-4302	377	3	x	x	SYM
ejpam-4302	377	4	̸∈	̸∈	PROPN
ejpam-4302	377	5	a.	a.	NOUN
ejpam-4302	377	6	by	by	ADP
ejpam-4302	377	7	lemma	lemma	PROPN
ejpam-4302	377	8	9	9	NUM
ejpam-4302	377	9	,	,	PUNCT
ejpam-4302	377	10	the	the	DET
ejpam-4302	377	11	singleton	singleton	NOUN
ejpam-4302	377	12	{	{	PUNCT
ejpam-4302	377	13	x	x	NOUN
ejpam-4302	377	14	}	}	PUNCT
ejpam-4302	377	15	is	be	AUX
ejpam-4302	377	16	(	(	PUNCT
ejpam-4302	377	17	λ	λ	X
ejpam-4302	377	18	,	,	PUNCT
ejpam-4302	377	19	sp)-open	sp)-open	ADJ
ejpam-4302	377	20	or	or	CCONJ
ejpam-4302	377	21	x	x	SYM
ejpam-4302	377	22	−	−	PROPN
ejpam-4302	377	23	{	{	PUNCT
ejpam-4302	377	24	x	x	NOUN
ejpam-4302	377	25	}	}	PUNCT
ejpam-4302	377	26	is	be	AUX
ejpam-4302	377	27	a	a	DET
ejpam-4302	377	28	λ(λ	λ(λ	NOUN
ejpam-4302	377	29	,	,	PUNCT
ejpam-4302	377	30	sp)-set	sp)-set	PROPN
ejpam-4302	377	31	.	.	PUNCT
ejpam-4302	378	1	(	(	PUNCT
ejpam-4302	378	2	a	a	X
ejpam-4302	378	3	)	)	PUNCT
ejpam-4302	378	4	in	in	ADP
ejpam-4302	378	5	case	case	NOUN
ejpam-4302	378	6	{	{	PUNCT
ejpam-4302	378	7	x	x	X
ejpam-4302	378	8	}	}	PUNCT
ejpam-4302	378	9	is	be	AUX
ejpam-4302	378	10	(	(	PUNCT
ejpam-4302	378	11	λ	λ	NOUN
ejpam-4302	378	12	,	,	PUNCT
ejpam-4302	378	13	sp)-open	sp)-open	ADJ
ejpam-4302	378	14	,	,	PUNCT
ejpam-4302	378	15	since	since	SCONJ
ejpam-4302	378	16	x	x	PROPN
ejpam-4302	378	17	∈	∈	PROPN
ejpam-4302	378	18	a(λ	a(λ	ADV
ejpam-4302	378	19	,	,	PUNCT
ejpam-4302	378	20	sp	sp	NOUN
ejpam-4302	378	21	)	)	PUNCT
ejpam-4302	378	22	,	,	PUNCT
ejpam-4302	378	23	{	{	PUNCT
ejpam-4302	378	24	x	x	X
ejpam-4302	378	25	}	}	PUNCT
ejpam-4302	378	26	∩a	∩a	PROPN
ejpam-4302	378	27	̸=	̸=	PROPN
ejpam-4302	378	28	∅	∅	NOUN
ejpam-4302	378	29	and	and	CCONJ
ejpam-4302	378	30	x	x	PUNCT
ejpam-4302	378	31	∈	∈	NOUN
ejpam-4302	378	32	a.	a.	NOUN
ejpam-4302	379	1	this	this	PRON
ejpam-4302	379	2	is	be	AUX
ejpam-4302	379	3	a	a	DET
ejpam-4302	379	4	contradiction	contradiction	NOUN
ejpam-4302	379	5	.	.	PUNCT
ejpam-4302	380	1	(	(	PUNCT
ejpam-4302	380	2	b	b	X
ejpam-4302	380	3	)	)	PUNCT
ejpam-4302	380	4	in	in	ADP
ejpam-4302	380	5	case	case	NOUN
ejpam-4302	380	6	x	x	X
ejpam-4302	380	7	−	−	X
ejpam-4302	380	8	{	{	PUNCT
ejpam-4302	380	9	x	x	NOUN
ejpam-4302	380	10	}	}	PUNCT
ejpam-4302	380	11	is	be	AUX
ejpam-4302	380	12	a	a	DET
ejpam-4302	380	13	λ(λ	λ(λ	NOUN
ejpam-4302	380	14	,	,	PUNCT
ejpam-4302	380	15	sp)-set	sp)-set	PROPN
ejpam-4302	380	16	,	,	PUNCT
ejpam-4302	380	17	if	if	SCONJ
ejpam-4302	380	18	{	{	PUNCT
ejpam-4302	380	19	x	x	NOUN
ejpam-4302	380	20	}	}	PUNCT
ejpam-4302	380	21	is	be	AUX
ejpam-4302	380	22	not	not	PART
ejpam-4302	380	23	(	(	PUNCT
ejpam-4302	380	24	λ	λ	X
ejpam-4302	380	25	,	,	PUNCT
ejpam-4302	380	26	sp)-closed	sp)-close	VERB
ejpam-4302	380	27	,	,	PUNCT
ejpam-4302	380	28	x	x	PUNCT
ejpam-4302	380	29	−	−	X
ejpam-4302	380	30	{	{	PUNCT
ejpam-4302	380	31	x	x	NOUN
ejpam-4302	380	32	}	}	PUNCT
ejpam-4302	380	33	is	be	AUX
ejpam-4302	380	34	not	not	PART
ejpam-4302	380	35	(	(	PUNCT
ejpam-4302	380	36	λ	λ	X
ejpam-4302	380	37	,	,	PUNCT
ejpam-4302	380	38	sp)-open	sp)-open	ADJ
ejpam-4302	380	39	and	and	CCONJ
ejpam-4302	380	40	λ(λ	λ(λ	ADV
ejpam-4302	380	41	,	,	PUNCT
ejpam-4302	381	1	sp)(x	sp)(x	PROPN
ejpam-4302	381	2	−	−	PROPN
ejpam-4302	381	3	{	{	PUNCT
ejpam-4302	381	4	x	x	NOUN
ejpam-4302	381	5	}	}	PUNCT
ejpam-4302	381	6	)	)	PUNCT
ejpam-4302	382	1	=	=	PUNCT
ejpam-4302	382	2	x.	x.	PUNCT
ejpam-4302	383	1	thus	thus	ADV
ejpam-4302	383	2	,	,	PUNCT
ejpam-4302	383	3	x	x	PUNCT
ejpam-4302	383	4	−	−	X
ejpam-4302	383	5	{	{	PUNCT
ejpam-4302	383	6	x	x	NOUN
ejpam-4302	383	7	}	}	PUNCT
ejpam-4302	383	8	is	be	AUX
ejpam-4302	383	9	not	not	PART
ejpam-4302	383	10	a	a	DET
ejpam-4302	383	11	λ(λ	λ(λ	NOUN
ejpam-4302	383	12	,	,	PUNCT
ejpam-4302	383	13	sp)set	sp)set	NOUN
ejpam-4302	383	14	.	.	PUNCT
ejpam-4302	384	1	this	this	PRON
ejpam-4302	384	2	contradicts	contradict	VERB
ejpam-4302	384	3	(	(	PUNCT
ejpam-4302	384	4	3	3	NUM
ejpam-4302	384	5	)	)	PUNCT
ejpam-4302	384	6	.	.	PUNCT
ejpam-4302	385	1	if	if	SCONJ
ejpam-4302	385	2	{	{	PUNCT
ejpam-4302	385	3	x	x	NOUN
ejpam-4302	385	4	}	}	PUNCT
ejpam-4302	385	5	is	be	AUX
ejpam-4302	385	6	(	(	PUNCT
ejpam-4302	385	7	λ	λ	X
ejpam-4302	385	8	,	,	PUNCT
ejpam-4302	385	9	sp)-closed	sp)-close	VERB
ejpam-4302	385	10	,	,	PUNCT
ejpam-4302	385	11	a	a	DET
ejpam-4302	385	12	⊆	⊆	NUM
ejpam-4302	385	13	x	x	SYM
ejpam-4302	385	14	−	−	PROPN
ejpam-4302	385	15	{	{	PUNCT
ejpam-4302	385	16	x	x	NOUN
ejpam-4302	385	17	}	}	PUNCT
ejpam-4302	385	18	∈	∈	PROPN
ejpam-4302	385	19	λspo(x	λspo(x	NOUN
ejpam-4302	385	20	,	,	PUNCT
ejpam-4302	385	21	τ	τ	PROPN
ejpam-4302	385	22	)	)	PUNCT
ejpam-4302	385	23	and	and	CCONJ
ejpam-4302	385	24	a	a	PRON
ejpam-4302	385	25	is	be	AUX
ejpam-4302	385	26	g-(λ	g-(λ	PROPN
ejpam-4302	385	27	,	,	PUNCT
ejpam-4302	385	28	sp)-closed	sp)-close	VERB
ejpam-4302	385	29	.	.	PUNCT
ejpam-4302	386	1	hence	hence	ADV
ejpam-4302	386	2	,	,	PUNCT
ejpam-4302	386	3	we	we	PRON
ejpam-4302	386	4	have	have	VERB
ejpam-4302	386	5	a(λ	a(λ	ADV
ejpam-4302	386	6	,	,	PUNCT
ejpam-4302	386	7	sp	sp	NOUN
ejpam-4302	386	8	)	)	PUNCT
ejpam-4302	386	9	⊆	⊆	NUM
ejpam-4302	386	10	x	x	SYM
ejpam-4302	386	11	−	−	PROPN
ejpam-4302	386	12	{	{	PUNCT
ejpam-4302	386	13	x	x	NOUN
ejpam-4302	386	14	}	}	PUNCT
ejpam-4302	386	15	.	.	PUNCT
ejpam-4302	387	1	this	this	PRON
ejpam-4302	387	2	contradicts	contradict	VERB
ejpam-4302	387	3	that	that	SCONJ
ejpam-4302	387	4	x	x	SYM
ejpam-4302	387	5	∈	∈	PROPN
ejpam-4302	387	6	a(λ	a(λ	ADV
ejpam-4302	387	7	,	,	PUNCT
ejpam-4302	387	8	sp	sp	NOUN
ejpam-4302	387	9	)	)	PUNCT
ejpam-4302	387	10	.	.	PUNCT
ejpam-4302	388	1	this	this	PRON
ejpam-4302	388	2	shows	show	VERB
ejpam-4302	388	3	that	that	SCONJ
ejpam-4302	388	4	(	(	PUNCT
ejpam-4302	388	5	x	x	X
ejpam-4302	388	6	,	,	PUNCT
ejpam-4302	388	7	τ	τ	X
ejpam-4302	388	8	)	)	PUNCT
ejpam-4302	388	9	is	be	AUX
ejpam-4302	388	10	a	a	DET
ejpam-4302	388	11	λsp	λsp	PROPN
ejpam-4302	388	12	-	-	PUNCT
ejpam-4302	388	13	t	t	NOUN
ejpam-4302	388	14	1	1	NUM
ejpam-4302	388	15	2	2	NUM
ejpam-4302	388	16	-space	-space	NOUN
ejpam-4302	388	17	.	.	PUNCT
ejpam-4302	389	1	4	4	NUM
ejpam-4302	389	2	.	.	X
ejpam-4302	389	3	an	an	DET
ejpam-4302	389	4	application	application	NOUN
ejpam-4302	389	5	of	of	ADP
ejpam-4302	389	6	generalized	generalized	ADJ
ejpam-4302	389	7	(	(	PUNCT
ejpam-4302	389	8	λ	λ	NOUN
ejpam-4302	389	9	,	,	PUNCT
ejpam-4302	389	10	sp)-closed	sp)-close	VERB
ejpam-4302	389	11	sets	set	NOUN
ejpam-4302	389	12	in	in	ADP
ejpam-4302	389	13	this	this	DET
ejpam-4302	389	14	section	section	NOUN
ejpam-4302	389	15	,	,	PUNCT
ejpam-4302	389	16	we	we	PRON
ejpam-4302	389	17	introduce	introduce	VERB
ejpam-4302	389	18	the	the	DET
ejpam-4302	389	19	notion	notion	NOUN
ejpam-4302	389	20	of	of	ADP
ejpam-4302	389	21	λsp	λsp	NOUN
ejpam-4302	389	22	-	-	ADJ
ejpam-4302	389	23	normal	normal	ADJ
ejpam-4302	389	24	spaces	space	NOUN
ejpam-4302	389	25	and	and	CCONJ
ejpam-4302	389	26	investigate	investigate	VERB
ejpam-4302	389	27	several	several	ADJ
ejpam-4302	389	28	characterizations	characterization	NOUN
ejpam-4302	389	29	of	of	ADP
ejpam-4302	389	30	λsp	λsp	NOUN
ejpam-4302	389	31	-	-	ADJ
ejpam-4302	389	32	normal	normal	ADJ
ejpam-4302	389	33	spaces	space	NOUN
ejpam-4302	389	34	.	.	PUNCT
ejpam-4302	390	1	definition	definition	NOUN
ejpam-4302	390	2	10	10	NUM
ejpam-4302	390	3	.	.	PUNCT
ejpam-4302	391	1	a	a	DET
ejpam-4302	391	2	topological	topological	ADJ
ejpam-4302	391	3	space	space	NOUN
ejpam-4302	391	4	(	(	PUNCT
ejpam-4302	391	5	x	x	X
ejpam-4302	391	6	,	,	PUNCT
ejpam-4302	391	7	τ	τ	X
ejpam-4302	391	8	)	)	PUNCT
ejpam-4302	391	9	is	be	AUX
ejpam-4302	391	10	said	say	VERB
ejpam-4302	391	11	to	to	PART
ejpam-4302	391	12	be	be	AUX
ejpam-4302	391	13	λsp	λsp	NOUN
ejpam-4302	391	14	-	-	ADJ
ejpam-4302	391	15	normal	normal	ADJ
ejpam-4302	391	16	if	if	SCONJ
ejpam-4302	391	17	,	,	PUNCT
ejpam-4302	391	18	for	for	ADP
ejpam-4302	391	19	any	any	DET
ejpam-4302	391	20	pair	pair	NOUN
ejpam-4302	391	21	of	of	ADP
ejpam-4302	391	22	disjoint	disjoint	NOUN
ejpam-4302	391	23	(	(	PUNCT
ejpam-4302	391	24	λ	λ	PROPN
ejpam-4302	391	25	,	,	PUNCT
ejpam-4302	391	26	sp)-closed	sp)-close	VERB
ejpam-4302	391	27	sets	set	VERB
ejpam-4302	391	28	f	f	PROPN
ejpam-4302	391	29	and	and	CCONJ
ejpam-4302	391	30	h	h	NOUN
ejpam-4302	391	31	,	,	PUNCT
ejpam-4302	391	32	there	there	PRON
ejpam-4302	391	33	exist	exist	VERB
ejpam-4302	391	34	disjoint	disjoint	NOUN
ejpam-4302	391	35	(	(	PUNCT
ejpam-4302	391	36	λ	λ	NOUN
ejpam-4302	391	37	,	,	PUNCT
ejpam-4302	391	38	sp)-open	sp)-open	NOUN
ejpam-4302	391	39	sets	set	VERB
ejpam-4302	391	40	u	u	NOUN
ejpam-4302	391	41	and	and	CCONJ
ejpam-4302	391	42	v	v	ADP
ejpam-4302	391	43	such	such	ADJ
ejpam-4302	391	44	that	that	SCONJ
ejpam-4302	391	45	f	f	PROPN
ejpam-4302	391	46	⊆	⊆	NUM
ejpam-4302	391	47	u	u	NOUN
ejpam-4302	391	48	and	and	CCONJ
ejpam-4302	391	49	h	h	NOUN
ejpam-4302	391	50	⊆	⊆	NUM
ejpam-4302	391	51	v	v	NOUN
ejpam-4302	391	52	.	.	PUNCT
ejpam-4302	392	1	lemma	lemma	PROPN
ejpam-4302	392	2	10	10	NUM
ejpam-4302	392	3	.	.	PUNCT
ejpam-4302	393	1	let	let	VERB
ejpam-4302	393	2	(	(	PUNCT
ejpam-4302	393	3	x	x	NOUN
ejpam-4302	393	4	,	,	PUNCT
ejpam-4302	393	5	τ	τ	X
ejpam-4302	393	6	)	)	PUNCT
ejpam-4302	393	7	be	be	VERB
ejpam-4302	393	8	a	a	DET
ejpam-4302	393	9	topological	topological	ADJ
ejpam-4302	393	10	space	space	NOUN
ejpam-4302	393	11	.	.	PUNCT
ejpam-4302	394	1	if	if	SCONJ
ejpam-4302	394	2	u	u	PRON
ejpam-4302	394	3	is	be	AUX
ejpam-4302	394	4	(	(	PUNCT
ejpam-4302	394	5	λ	λ	X
ejpam-4302	394	6	,	,	PUNCT
ejpam-4302	394	7	sp)-open	sp)-open	ADJ
ejpam-4302	394	8	in	in	ADP
ejpam-4302	394	9	x	x	NOUN
ejpam-4302	394	10	,	,	PUNCT
ejpam-4302	394	11	then	then	ADV
ejpam-4302	394	12	u	u	X
ejpam-4302	394	13	(	(	PUNCT
ejpam-4302	394	14	λ	λ	PROPN
ejpam-4302	394	15	,	,	PUNCT
ejpam-4302	394	16	sp	sp	NOUN
ejpam-4302	394	17	)	)	PUNCT
ejpam-4302	394	18	∩a	∩a	NOUN
ejpam-4302	394	19	⊆	⊆	NUM
ejpam-4302	394	20	[	[	X
ejpam-4302	394	21	u	u	NOUN
ejpam-4302	394	22	∩a](λ	∩a](λ	PROPN
ejpam-4302	394	23	,	,	PUNCT
ejpam-4302	394	24	sp	sp	NOUN
ejpam-4302	394	25	)	)	PUNCT
ejpam-4302	394	26	for	for	ADP
ejpam-4302	394	27	every	every	DET
ejpam-4302	394	28	subset	subset	NOUN
ejpam-4302	394	29	a	a	PRON
ejpam-4302	394	30	of	of	ADP
ejpam-4302	394	31	x.	x.	NOUN
ejpam-4302	394	32	theorem	theorem	VERB
ejpam-4302	394	33	18	18	NUM
ejpam-4302	394	34	.	.	PUNCT
ejpam-4302	395	1	for	for	ADP
ejpam-4302	395	2	a	a	DET
ejpam-4302	395	3	topological	topological	ADJ
ejpam-4302	395	4	space	space	NOUN
ejpam-4302	395	5	(	(	PUNCT
ejpam-4302	395	6	x	x	X
ejpam-4302	395	7	,	,	PUNCT
ejpam-4302	395	8	τ	τ	PROPN
ejpam-4302	395	9	)	)	PUNCT
ejpam-4302	395	10	,	,	PUNCT
ejpam-4302	395	11	the	the	DET
ejpam-4302	395	12	following	follow	VERB
ejpam-4302	395	13	properties	property	NOUN
ejpam-4302	395	14	are	be	AUX
ejpam-4302	395	15	equivalent	equivalent	ADJ
ejpam-4302	395	16	:	:	PUNCT
ejpam-4302	395	17	(	(	PUNCT
ejpam-4302	395	18	1	1	X
ejpam-4302	395	19	)	)	PUNCT
ejpam-4302	395	20	(	(	PUNCT
ejpam-4302	395	21	x	x	X
ejpam-4302	395	22	,	,	PUNCT
ejpam-4302	395	23	τ	τ	X
ejpam-4302	395	24	)	)	PUNCT
ejpam-4302	395	25	is	be	AUX
ejpam-4302	395	26	λsp	λsp	ADJ
ejpam-4302	395	27	-	-	ADJ
ejpam-4302	395	28	normal	normal	ADJ
ejpam-4302	395	29	.	.	PUNCT
ejpam-4302	396	1	c.	c.	PROPN
ejpam-4302	396	2	boonpok	boonpok	PROPN
ejpam-4302	396	3	,	,	PUNCT
ejpam-4302	396	4	c.	c.	PROPN
ejpam-4302	396	5	viriyapong	viriyapong	PROPN
ejpam-4302	396	6	/	/	SYM
ejpam-4302	396	7	eur	eur	PROPN
ejpam-4302	396	8	.	.	PUNCT
ejpam-4302	397	1	j.	j.	PROPN
ejpam-4302	397	2	pure	pure	PROPN
ejpam-4302	397	3	appl	appl	PROPN
ejpam-4302	397	4	.	.	PROPN
ejpam-4302	397	5	math	math	PROPN
ejpam-4302	397	6	,	,	PUNCT
ejpam-4302	397	7	15	15	NUM
ejpam-4302	397	8	(	(	PUNCT
ejpam-4302	397	9	4	4	NUM
ejpam-4302	397	10	)	)	PUNCT
ejpam-4302	397	11	(	(	PUNCT
ejpam-4302	397	12	2022	2022	NUM
ejpam-4302	397	13	)	)	PUNCT
ejpam-4302	397	14	,	,	PUNCT
ejpam-4302	397	15	2127	2127	NUM
ejpam-4302	397	16	-	-	SYM
ejpam-4302	397	17	2140	2140	NUM
ejpam-4302	397	18	2138	2138	NUM
ejpam-4302	397	19	(	(	PUNCT
ejpam-4302	397	20	2	2	NUM
ejpam-4302	397	21	)	)	PUNCT
ejpam-4302	397	22	for	for	ADP
ejpam-4302	397	23	every	every	DET
ejpam-4302	397	24	pair	pair	NOUN
ejpam-4302	397	25	of	of	ADP
ejpam-4302	397	26	(	(	PUNCT
ejpam-4302	397	27	λ	λ	PROPN
ejpam-4302	397	28	,	,	PUNCT
ejpam-4302	397	29	sp)-open	sp)-open	NOUN
ejpam-4302	397	30	sets	set	VERB
ejpam-4302	397	31	u	u	NOUN
ejpam-4302	397	32	and	and	CCONJ
ejpam-4302	397	33	v	v	ADP
ejpam-4302	397	34	whose	whose	DET
ejpam-4302	397	35	union	union	NOUN
ejpam-4302	397	36	is	be	AUX
ejpam-4302	397	37	x	x	NOUN
ejpam-4302	397	38	,	,	PUNCT
ejpam-4302	397	39	there	there	PRON
ejpam-4302	397	40	exist	exist	VERB
ejpam-4302	397	41	(	(	PUNCT
ejpam-4302	397	42	λ	λ	NOUN
ejpam-4302	397	43	,	,	PUNCT
ejpam-4302	397	44	sp)closed	sp)close	VERB
ejpam-4302	397	45	sets	set	NOUN
ejpam-4302	397	46	f	f	PROPN
ejpam-4302	398	1	and	and	CCONJ
ejpam-4302	398	2	h	h	NOUN
ejpam-4302	398	3	such	such	ADJ
ejpam-4302	398	4	that	that	SCONJ
ejpam-4302	398	5	f	f	PROPN
ejpam-4302	398	6	⊆	⊆	NUM
ejpam-4302	398	7	u	u	NOUN
ejpam-4302	398	8	,	,	PUNCT
ejpam-4302	398	9	h	h	NOUN
ejpam-4302	398	10	⊆	⊆	NUM
ejpam-4302	398	11	v	v	NOUN
ejpam-4302	398	12	and	and	CCONJ
ejpam-4302	398	13	f	f	PROPN
ejpam-4302	398	14	∪h	∪h	PROPN
ejpam-4302	398	15	=	=	SYM
ejpam-4302	398	16	x.	x.	NOUN
ejpam-4302	398	17	(	(	PUNCT
ejpam-4302	398	18	3	3	NUM
ejpam-4302	398	19	)	)	PUNCT
ejpam-4302	398	20	for	for	ADP
ejpam-4302	398	21	every	every	DET
ejpam-4302	398	22	(	(	PUNCT
ejpam-4302	398	23	λ	λ	PROPN
ejpam-4302	398	24	,	,	PUNCT
ejpam-4302	398	25	sp)-closed	sp)-close	VERB
ejpam-4302	398	26	set	set	VERB
ejpam-4302	398	27	f	f	PROPN
ejpam-4302	398	28	and	and	CCONJ
ejpam-4302	398	29	every	every	DET
ejpam-4302	398	30	(	(	PUNCT
ejpam-4302	398	31	λ	λ	NOUN
ejpam-4302	398	32	,	,	PUNCT
ejpam-4302	398	33	sp)-open	sp)-open	NOUN
ejpam-4302	398	34	set	set	VERB
ejpam-4302	398	35	g	g	NOUN
ejpam-4302	398	36	containing	contain	VERB
ejpam-4302	398	37	f	f	NOUN
ejpam-4302	398	38	,	,	PUNCT
ejpam-4302	398	39	there	there	PRON
ejpam-4302	398	40	exists	exist	VERB
ejpam-4302	398	41	a	a	DET
ejpam-4302	398	42	(	(	PUNCT
ejpam-4302	398	43	λ	λ	NOUN
ejpam-4302	398	44	,	,	PUNCT
ejpam-4302	398	45	sp)-open	sp)-open	NOUN
ejpam-4302	398	46	set	set	VERB
ejpam-4302	398	47	u	u	PRON
ejpam-4302	398	48	such	such	ADJ
ejpam-4302	398	49	that	that	SCONJ
ejpam-4302	398	50	f	f	PROPN
ejpam-4302	398	51	⊆	⊆	NUM
ejpam-4302	398	52	u	u	NOUN
ejpam-4302	398	53	⊆	⊆	NUM
ejpam-4302	398	54	u	u	NOUN
ejpam-4302	398	55	(	(	PUNCT
ejpam-4302	398	56	λ	λ	PROPN
ejpam-4302	398	57	,	,	PUNCT
ejpam-4302	398	58	sp	sp	NOUN
ejpam-4302	398	59	)	)	PUNCT
ejpam-4302	398	60	⊆	⊆	NUM
ejpam-4302	398	61	g.	g.	NOUN
ejpam-4302	398	62	(	(	PUNCT
ejpam-4302	398	63	4	4	NUM
ejpam-4302	398	64	)	)	PUNCT
ejpam-4302	398	65	for	for	ADP
ejpam-4302	398	66	every	every	DET
ejpam-4302	398	67	pair	pair	NOUN
ejpam-4302	398	68	of	of	ADP
ejpam-4302	398	69	disjoint	disjoint	NOUN
ejpam-4302	398	70	(	(	PUNCT
ejpam-4302	398	71	λ	λ	PROPN
ejpam-4302	398	72	,	,	PUNCT
ejpam-4302	398	73	sp)-closed	sp)-close	VERB
ejpam-4302	398	74	sets	set	VERB
ejpam-4302	398	75	f	f	PROPN
ejpam-4302	398	76	and	and	CCONJ
ejpam-4302	398	77	h	h	NOUN
ejpam-4302	398	78	,	,	PUNCT
ejpam-4302	398	79	there	there	PRON
ejpam-4302	398	80	exist	exist	VERB
ejpam-4302	398	81	disjoint	disjoint	NOUN
ejpam-4302	398	82	(	(	PUNCT
ejpam-4302	398	83	λ	λ	NOUN
ejpam-4302	398	84	,	,	PUNCT
ejpam-4302	398	85	sp)open	sp)open	NOUN
ejpam-4302	398	86	sets	set	VERB
ejpam-4302	398	87	u	u	NOUN
ejpam-4302	398	88	and	and	CCONJ
ejpam-4302	398	89	v	v	ADP
ejpam-4302	398	90	such	such	ADJ
ejpam-4302	399	1	that	that	SCONJ
ejpam-4302	399	2	f	f	PROPN
ejpam-4302	399	3	⊆	⊆	NUM
ejpam-4302	399	4	u	u	NOUN
ejpam-4302	399	5	and	and	CCONJ
ejpam-4302	399	6	h	h	NOUN
ejpam-4302	399	7	⊆	⊆	NUM
ejpam-4302	399	8	v	v	NOUN
ejpam-4302	399	9	and	and	CCONJ
ejpam-4302	399	10	u	u	NOUN
ejpam-4302	399	11	(	(	PUNCT
ejpam-4302	399	12	λ	λ	PROPN
ejpam-4302	399	13	,	,	PUNCT
ejpam-4302	399	14	sp	sp	NOUN
ejpam-4302	399	15	)	)	PUNCT
ejpam-4302	399	16	∩	∩	ADJ
ejpam-4302	399	17	v	v	X
ejpam-4302	399	18	(	(	PUNCT
ejpam-4302	399	19	λ	λ	PROPN
ejpam-4302	399	20	,	,	PUNCT
ejpam-4302	399	21	sp	sp	NOUN
ejpam-4302	399	22	)	)	PUNCT
ejpam-4302	399	23	=	=	PUNCT
ejpam-4302	399	24	∅.	∅.	NOUN
ejpam-4302	399	25	proof	proof	NOUN
ejpam-4302	399	26	.	.	PUNCT
ejpam-4302	400	1	(	(	PUNCT
ejpam-4302	400	2	1	1	X
ejpam-4302	400	3	)	)	PUNCT
ejpam-4302	400	4	⇒	⇒	NOUN
ejpam-4302	400	5	(	(	PUNCT
ejpam-4302	400	6	2	2	NUM
ejpam-4302	400	7	):	):	PUNCT
ejpam-4302	400	8	let	let	VERB
ejpam-4302	400	9	u	u	PRON
ejpam-4302	400	10	and	and	CCONJ
ejpam-4302	400	11	v	v	NOUN
ejpam-4302	400	12	be	be	AUX
ejpam-4302	400	13	any	any	DET
ejpam-4302	400	14	pair	pair	NOUN
ejpam-4302	400	15	of	of	ADP
ejpam-4302	400	16	(	(	PUNCT
ejpam-4302	400	17	λ	λ	PROPN
ejpam-4302	400	18	,	,	PUNCT
ejpam-4302	400	19	sp)-open	sp)-open	ADJ
ejpam-4302	400	20	sets	set	NOUN
ejpam-4302	400	21	in	in	ADP
ejpam-4302	400	22	x	x	SYM
ejpam-4302	400	23	such	such	ADJ
ejpam-4302	400	24	that	that	SCONJ
ejpam-4302	400	25	x	x	X
ejpam-4302	401	1	=	=	PUNCT
ejpam-4302	401	2	u	u	NOUN
ejpam-4302	401	3	∪	∪	NOUN
ejpam-4302	401	4	v	v	NOUN
ejpam-4302	401	5	.	.	PUNCT
ejpam-4302	402	1	then	then	ADV
ejpam-4302	402	2	,	,	PUNCT
ejpam-4302	402	3	x	x	PUNCT
ejpam-4302	402	4	−	−	NOUN
ejpam-4302	402	5	u	u	NOUN
ejpam-4302	402	6	and	and	CCONJ
ejpam-4302	402	7	x	x	NOUN
ejpam-4302	402	8	−	−	PROPN
ejpam-4302	402	9	v	v	NOUN
ejpam-4302	402	10	are	be	AUX
ejpam-4302	402	11	disjoint	disjoint	NOUN
ejpam-4302	402	12	(	(	PUNCT
ejpam-4302	402	13	λ	λ	PROPN
ejpam-4302	402	14	,	,	PUNCT
ejpam-4302	402	15	sp)-closed	sp)-close	VERB
ejpam-4302	402	16	sets	set	NOUN
ejpam-4302	402	17	.	.	PUNCT
ejpam-4302	403	1	since	since	SCONJ
ejpam-4302	403	2	(	(	PUNCT
ejpam-4302	403	3	x	x	X
ejpam-4302	403	4	,	,	PUNCT
ejpam-4302	403	5	τ	τ	X
ejpam-4302	403	6	)	)	PUNCT
ejpam-4302	403	7	is	be	AUX
ejpam-4302	403	8	λsp	λsp	ADJ
ejpam-4302	403	9	-	-	ADJ
ejpam-4302	403	10	normal	normal	ADJ
ejpam-4302	403	11	,	,	PUNCT
ejpam-4302	403	12	there	there	PRON
ejpam-4302	403	13	exist	exist	VERB
ejpam-4302	403	14	disjoint	disjoint	NOUN
ejpam-4302	403	15	(	(	PUNCT
ejpam-4302	403	16	λ	λ	NOUN
ejpam-4302	403	17	,	,	PUNCT
ejpam-4302	403	18	sp)-open	sp)-open	NOUN
ejpam-4302	403	19	sets	set	VERB
ejpam-4302	403	20	g	g	NOUN
ejpam-4302	403	21	and	and	CCONJ
ejpam-4302	404	1	w	w	ADP
ejpam-4302	404	2	such	such	ADJ
ejpam-4302	404	3	that	that	SCONJ
ejpam-4302	404	4	x	x	X
ejpam-4302	404	5	−	−	PUNCT
ejpam-4302	404	6	u	u	NOUN
ejpam-4302	404	7	⊆	⊆	NUM
ejpam-4302	404	8	g	g	NOUN
ejpam-4302	404	9	and	and	CCONJ
ejpam-4302	404	10	x	x	NOUN
ejpam-4302	404	11	−	−	NOUN
ejpam-4302	404	12	v	v	NUM
ejpam-4302	404	13	⊆	⊆	NUM
ejpam-4302	404	14	w	w	NOUN
ejpam-4302	404	15	.	.	PUNCT
ejpam-4302	405	1	put	put	VERB
ejpam-4302	405	2	f	f	NOUN
ejpam-4302	406	1	=	=	PUNCT
ejpam-4302	406	2	x	x	PROPN
ejpam-4302	406	3	−	−	NOUN
ejpam-4302	406	4	g	g	NOUN
ejpam-4302	406	5	and	and	CCONJ
ejpam-4302	406	6	h	h	NOUN
ejpam-4302	407	1	=	=	NOUN
ejpam-4302	407	2	x	x	PUNCT
ejpam-4302	407	3	−w	−w	ADV
ejpam-4302	407	4	.	.	PUNCT
ejpam-4302	408	1	then	then	ADV
ejpam-4302	408	2	,	,	PUNCT
ejpam-4302	408	3	f	f	PROPN
ejpam-4302	408	4	and	and	CCONJ
ejpam-4302	408	5	h	h	NOUN
ejpam-4302	408	6	are	be	AUX
ejpam-4302	408	7	(	(	PUNCT
ejpam-4302	408	8	λ	λ	X
ejpam-4302	408	9	,	,	PUNCT
ejpam-4302	408	10	sp)-closed	sp)-close	VERB
ejpam-4302	408	11	sets	set	VERB
ejpam-4302	408	12	such	such	ADJ
ejpam-4302	408	13	that	that	SCONJ
ejpam-4302	408	14	f	f	PROPN
ejpam-4302	408	15	⊆	⊆	NUM
ejpam-4302	408	16	u	u	NOUN
ejpam-4302	408	17	,	,	PUNCT
ejpam-4302	408	18	h	h	NOUN
ejpam-4302	408	19	⊆	⊆	NUM
ejpam-4302	408	20	v	v	NOUN
ejpam-4302	408	21	and	and	CCONJ
ejpam-4302	408	22	f	f	PROPN
ejpam-4302	408	23	∪h	∪h	PROPN
ejpam-4302	408	24	=	=	PUNCT
ejpam-4302	408	25	x.	x.	NOUN
ejpam-4302	408	26	(	(	PUNCT
ejpam-4302	408	27	2	2	NUM
ejpam-4302	408	28	)	)	PUNCT
ejpam-4302	408	29	⇒	⇒	NOUN
ejpam-4302	408	30	(	(	PUNCT
ejpam-4302	408	31	3	3	NUM
ejpam-4302	408	32	):	):	PUNCT
ejpam-4302	408	33	let	let	VERB
ejpam-4302	408	34	f	f	PRON
ejpam-4302	408	35	be	be	AUX
ejpam-4302	408	36	a	a	DET
ejpam-4302	408	37	(	(	PUNCT
ejpam-4302	408	38	λ	λ	NOUN
ejpam-4302	408	39	,	,	PUNCT
ejpam-4302	408	40	sp)-closed	sp)-close	VERB
ejpam-4302	408	41	set	set	ADJ
ejpam-4302	408	42	and	and	CCONJ
ejpam-4302	408	43	let	let	VERB
ejpam-4302	408	44	g	g	PRON
ejpam-4302	408	45	be	be	AUX
ejpam-4302	408	46	a	a	DET
ejpam-4302	408	47	(	(	PUNCT
ejpam-4302	408	48	λ	λ	NOUN
ejpam-4302	408	49	,	,	PUNCT
ejpam-4302	408	50	sp)-open	sp)-open	ADJ
ejpam-4302	408	51	set	set	NOUN
ejpam-4302	408	52	containing	contain	VERB
ejpam-4302	408	53	f	f	PROPN
ejpam-4302	408	54	.	.	PUNCT
ejpam-4302	409	1	then	then	ADV
ejpam-4302	409	2	,	,	PUNCT
ejpam-4302	409	3	x	x	PUNCT
ejpam-4302	409	4	−	−	PROPN
ejpam-4302	409	5	f	f	PROPN
ejpam-4302	409	6	and	and	CCONJ
ejpam-4302	409	7	g	g	PROPN
ejpam-4302	409	8	are	be	AUX
ejpam-4302	409	9	(	(	PUNCT
ejpam-4302	409	10	λ	λ	X
ejpam-4302	409	11	,	,	PUNCT
ejpam-4302	409	12	sp)-open	sp)-open	ADJ
ejpam-4302	409	13	sets	set	NOUN
ejpam-4302	409	14	whose	whose	DET
ejpam-4302	409	15	union	union	NOUN
ejpam-4302	409	16	is	be	AUX
ejpam-4302	409	17	x.	x.	NOUN
ejpam-4302	409	18	then	then	ADV
ejpam-4302	409	19	by	by	ADP
ejpam-4302	409	20	(	(	PUNCT
ejpam-4302	409	21	2	2	NUM
ejpam-4302	409	22	)	)	PUNCT
ejpam-4302	409	23	,	,	PUNCT
ejpam-4302	409	24	there	there	PRON
ejpam-4302	409	25	exist	exist	VERB
ejpam-4302	409	26	(	(	PUNCT
ejpam-4302	409	27	λ	λ	X
ejpam-4302	409	28	,	,	PUNCT
ejpam-4302	409	29	sp)-closed	sp)-close	VERB
ejpam-4302	409	30	sets	set	VERB
ejpam-4302	409	31	m	m	VERB
ejpam-4302	409	32	and	and	CCONJ
ejpam-4302	409	33	n	n	CCONJ
ejpam-4302	409	34	such	such	ADJ
ejpam-4302	409	35	that	that	SCONJ
ejpam-4302	409	36	m	m	PROPN
ejpam-4302	409	37	⊆	⊆	NUM
ejpam-4302	409	38	x	x	SYM
ejpam-4302	409	39	−	−	PROPN
ejpam-4302	409	40	f	f	PROPN
ejpam-4302	409	41	,	,	PUNCT
ejpam-4302	409	42	n	n	PROPN
ejpam-4302	409	43	⊆	⊆	NUM
ejpam-4302	409	44	g	g	NOUN
ejpam-4302	409	45	and	and	CCONJ
ejpam-4302	409	46	m	m	NOUN
ejpam-4302	409	47	∪	∪	ADJ
ejpam-4302	409	48	n	n	NOUN
ejpam-4302	409	49	=	=	SYM
ejpam-4302	409	50	x.	x.	NOUN
ejpam-4302	409	51	then	then	ADV
ejpam-4302	409	52	,	,	PUNCT
ejpam-4302	409	53	f	f	PROPN
ejpam-4302	409	54	⊆	⊆	NUM
ejpam-4302	409	55	x	x	SYM
ejpam-4302	409	56	−	−	PROPN
ejpam-4302	409	57	m	m	PRON
ejpam-4302	409	58	,	,	PUNCT
ejpam-4302	409	59	x	x	X
ejpam-4302	409	60	−	−	NOUN
ejpam-4302	409	61	g	g	PROPN
ejpam-4302	409	62	⊆	⊆	NUM
ejpam-4302	409	63	x	x	SYM
ejpam-4302	409	64	−	−	PROPN
ejpam-4302	409	65	n	n	NOUN
ejpam-4302	409	66	and	and	CCONJ
ejpam-4302	409	67	(	(	PUNCT
ejpam-4302	409	68	x	x	PART
ejpam-4302	409	69	−	−	PROPN
ejpam-4302	409	70	m	m	NOUN
ejpam-4302	409	71	)	)	PUNCT
ejpam-4302	409	72	∩	∩	NOUN
ejpam-4302	409	73	(	(	PUNCT
ejpam-4302	409	74	x	x	SYM
ejpam-4302	409	75	−	−	PROPN
ejpam-4302	409	76	n	n	CCONJ
ejpam-4302	409	77	)	)	PUNCT
ejpam-4302	409	78	=	=	VERB
ejpam-4302	410	1	∅.	∅.	ADV
ejpam-4302	410	2	put	put	VERB
ejpam-4302	410	3	u	u	NOUN
ejpam-4302	410	4	=	=	NOUN
ejpam-4302	410	5	x	x	PROPN
ejpam-4302	410	6	−	−	PROPN
ejpam-4302	410	7	m	m	PROPN
ejpam-4302	410	8	and	and	CCONJ
ejpam-4302	410	9	v	v	NOUN
ejpam-4302	410	10	=	=	SYM
ejpam-4302	410	11	x−n	x−n	PROPN
ejpam-4302	410	12	.	.	PUNCT
ejpam-4302	411	1	then	then	ADV
ejpam-4302	411	2	,	,	PUNCT
ejpam-4302	411	3	u	u	NOUN
ejpam-4302	411	4	and	and	CCONJ
ejpam-4302	411	5	v	v	NOUN
ejpam-4302	411	6	are	be	AUX
ejpam-4302	411	7	disjoint	disjoint	NOUN
ejpam-4302	411	8	(	(	PUNCT
ejpam-4302	411	9	λ	λ	NOUN
ejpam-4302	411	10	,	,	PUNCT
ejpam-4302	411	11	sp)-open	sp)-open	ADJ
ejpam-4302	411	12	sets	set	VERB
ejpam-4302	411	13	such	such	ADJ
ejpam-4302	411	14	that	that	SCONJ
ejpam-4302	411	15	f	f	PROPN
ejpam-4302	411	16	⊆	⊆	NUM
ejpam-4302	411	17	u	u	NOUN
ejpam-4302	411	18	⊆	⊆	NUM
ejpam-4302	411	19	x−v	x−v	PROPN
ejpam-4302	411	20	⊆	⊆	NUM
ejpam-4302	411	21	g.	g.	NOUN
ejpam-4302	411	22	as	as	ADP
ejpam-4302	411	23	x	x	X
ejpam-4302	411	24	−	−	PROPN
ejpam-4302	411	25	v	v	NOUN
ejpam-4302	411	26	is	be	AUX
ejpam-4302	411	27	a	a	DET
ejpam-4302	411	28	(	(	PUNCT
ejpam-4302	411	29	λ	λ	PROPN
ejpam-4302	411	30	,	,	PUNCT
ejpam-4302	411	31	sp)-closed	sp)-close	VERB
ejpam-4302	411	32	set	set	ADJ
ejpam-4302	411	33	,	,	PUNCT
ejpam-4302	411	34	we	we	PRON
ejpam-4302	411	35	have	have	VERB
ejpam-4302	411	36	u	u	NOUN
ejpam-4302	411	37	(	(	PUNCT
ejpam-4302	411	38	λ	λ	PROPN
ejpam-4302	411	39	,	,	PUNCT
ejpam-4302	411	40	sp	sp	NOUN
ejpam-4302	411	41	)	)	PUNCT
ejpam-4302	411	42	⊆	⊆	NUM
ejpam-4302	411	43	x	x	SYM
ejpam-4302	411	44	−	−	NOUN
ejpam-4302	411	45	v	v	NOUN
ejpam-4302	411	46	and	and	CCONJ
ejpam-4302	411	47	f	f	PROPN
ejpam-4302	411	48	⊆	⊆	NUM
ejpam-4302	411	49	u	u	NOUN
ejpam-4302	411	50	⊆	⊆	NUM
ejpam-4302	411	51	u	u	NOUN
ejpam-4302	411	52	(	(	PUNCT
ejpam-4302	411	53	λ	λ	PROPN
ejpam-4302	411	54	,	,	PUNCT
ejpam-4302	411	55	sp	sp	NOUN
ejpam-4302	411	56	)	)	PUNCT
ejpam-4302	411	57	⊆	⊆	NUM
ejpam-4302	411	58	g.	g.	NOUN
ejpam-4302	411	59	(	(	PUNCT
ejpam-4302	411	60	3	3	NUM
ejpam-4302	411	61	)	)	PUNCT
ejpam-4302	411	62	⇒	⇒	NOUN
ejpam-4302	411	63	(	(	PUNCT
ejpam-4302	411	64	4	4	NUM
ejpam-4302	411	65	):	):	PUNCT
ejpam-4302	411	66	let	let	VERB
ejpam-4302	411	67	f	f	PROPN
ejpam-4302	411	68	and	and	CCONJ
ejpam-4302	411	69	h	h	PROPN
ejpam-4302	411	70	be	be	VERB
ejpam-4302	411	71	two	two	NUM
ejpam-4302	411	72	disjoint	disjoint	NOUN
ejpam-4302	411	73	(	(	PUNCT
ejpam-4302	411	74	λ	λ	PROPN
ejpam-4302	411	75	,	,	PUNCT
ejpam-4302	411	76	sp)-closed	sp)-close	VERB
ejpam-4302	411	77	sets	set	NOUN
ejpam-4302	411	78	of	of	ADP
ejpam-4302	411	79	x.	x.	NOUN
ejpam-4302	411	80	then	then	ADV
ejpam-4302	411	81	,	,	PUNCT
ejpam-4302	411	82	f	f	PROPN
ejpam-4302	411	83	⊆	⊆	NUM
ejpam-4302	411	84	x	x	SYM
ejpam-4302	411	85	−h	−h	VERB
ejpam-4302	411	86	and	and	CCONJ
ejpam-4302	411	87	x	x	SYM
ejpam-4302	411	88	−	−	NOUN
ejpam-4302	411	89	h	h	NOUN
ejpam-4302	411	90	is	be	AUX
ejpam-4302	411	91	(	(	PUNCT
ejpam-4302	411	92	λ	λ	INTJ
ejpam-4302	411	93	,	,	PUNCT
ejpam-4302	411	94	sp)-open	sp)-open	ADJ
ejpam-4302	411	95	,	,	PUNCT
ejpam-4302	411	96	by	by	ADP
ejpam-4302	411	97	(	(	PUNCT
ejpam-4302	411	98	3	3	NUM
ejpam-4302	411	99	)	)	PUNCT
ejpam-4302	411	100	,	,	PUNCT
ejpam-4302	411	101	there	there	PRON
ejpam-4302	411	102	exists	exist	VERB
ejpam-4302	411	103	a	a	DET
ejpam-4302	411	104	(	(	PUNCT
ejpam-4302	411	105	λ	λ	NOUN
ejpam-4302	411	106	,	,	PUNCT
ejpam-4302	411	107	sp)-open	sp)-open	NOUN
ejpam-4302	411	108	set	set	VERB
ejpam-4302	411	109	u	u	NOUN
ejpam-4302	411	110	of	of	ADP
ejpam-4302	411	111	x	x	SYM
ejpam-4302	411	112	such	such	ADJ
ejpam-4302	411	113	that	that	SCONJ
ejpam-4302	411	114	f	f	PROPN
ejpam-4302	411	115	⊆	⊆	NUM
ejpam-4302	411	116	u	u	NOUN
ejpam-4302	411	117	⊆	⊆	NUM
ejpam-4302	411	118	u	u	NOUN
ejpam-4302	411	119	(	(	PUNCT
ejpam-4302	411	120	λ	λ	PROPN
ejpam-4302	411	121	,	,	PUNCT
ejpam-4302	411	122	sp	sp	NOUN
ejpam-4302	411	123	)	)	PUNCT
ejpam-4302	411	124	⊆	⊆	NUM
ejpam-4302	411	125	x−h	x−h	NOUN
ejpam-4302	411	126	.	.	PUNCT
ejpam-4302	412	1	put	put	VERB
ejpam-4302	412	2	v	v	NUM
ejpam-4302	412	3	=	=	PRON
ejpam-4302	412	4	x−u	x−u	X
ejpam-4302	412	5	(	(	PUNCT
ejpam-4302	412	6	λ	λ	NOUN
ejpam-4302	412	7	,	,	PUNCT
ejpam-4302	412	8	sp	sp	NOUN
ejpam-4302	412	9	)	)	PUNCT
ejpam-4302	412	10	.	.	PUNCT
ejpam-4302	413	1	then	then	ADV
ejpam-4302	413	2	,	,	PUNCT
ejpam-4302	413	3	u	u	NOUN
ejpam-4302	413	4	and	and	CCONJ
ejpam-4302	413	5	v	v	NOUN
ejpam-4302	413	6	are	be	AUX
ejpam-4302	413	7	disjoint	disjoint	NOUN
ejpam-4302	413	8	(	(	PUNCT
ejpam-4302	413	9	λ	λ	NOUN
ejpam-4302	413	10	,	,	PUNCT
ejpam-4302	413	11	sp)-open	sp)-open	ADJ
ejpam-4302	413	12	sets	set	NOUN
ejpam-4302	413	13	of	of	ADP
ejpam-4302	413	14	x	x	SYM
ejpam-4302	413	15	such	such	ADJ
ejpam-4302	413	16	that	that	SCONJ
ejpam-4302	413	17	f	f	PROPN
ejpam-4302	413	18	⊆	⊆	NUM
ejpam-4302	413	19	u	u	NOUN
ejpam-4302	413	20	,	,	PUNCT
ejpam-4302	413	21	h	h	NOUN
ejpam-4302	413	22	⊆	⊆	NUM
ejpam-4302	413	23	v	v	NOUN
ejpam-4302	413	24	and	and	CCONJ
ejpam-4302	413	25	u	u	NOUN
ejpam-4302	413	26	(	(	PUNCT
ejpam-4302	413	27	λ	λ	PROPN
ejpam-4302	413	28	,	,	PUNCT
ejpam-4302	413	29	sp	sp	NOUN
ejpam-4302	413	30	)	)	PUNCT
ejpam-4302	413	31	∩	∩	ADJ
ejpam-4302	413	32	v	v	X
ejpam-4302	413	33	(	(	PUNCT
ejpam-4302	413	34	λ	λ	PROPN
ejpam-4302	413	35	,	,	PUNCT
ejpam-4302	413	36	sp	sp	NOUN
ejpam-4302	413	37	)	)	PUNCT
ejpam-4302	413	38	=	=	SYM
ejpam-4302	413	39	∅.	∅.	X
ejpam-4302	413	40	(	(	PUNCT
ejpam-4302	413	41	4	4	NUM
ejpam-4302	413	42	)	)	PUNCT
ejpam-4302	413	43	⇒	⇒	NOUN
ejpam-4302	413	44	(	(	PUNCT
ejpam-4302	413	45	1	1	NUM
ejpam-4302	413	46	):	):	PUNCT
ejpam-4302	413	47	the	the	DET
ejpam-4302	413	48	proof	proof	NOUN
ejpam-4302	413	49	is	be	AUX
ejpam-4302	413	50	obvious	obvious	ADJ
ejpam-4302	413	51	.	.	PUNCT
ejpam-4302	414	1	theorem	theorem	NOUN
ejpam-4302	414	2	19	19	NUM
ejpam-4302	414	3	.	.	PUNCT
ejpam-4302	415	1	for	for	ADP
ejpam-4302	415	2	a	a	DET
ejpam-4302	415	3	topological	topological	ADJ
ejpam-4302	415	4	space	space	NOUN
ejpam-4302	415	5	(	(	PUNCT
ejpam-4302	415	6	x	x	X
ejpam-4302	415	7	,	,	PUNCT
ejpam-4302	415	8	τ	τ	PROPN
ejpam-4302	415	9	)	)	PUNCT
ejpam-4302	415	10	,	,	PUNCT
ejpam-4302	415	11	the	the	DET
ejpam-4302	415	12	following	follow	VERB
ejpam-4302	415	13	properties	property	NOUN
ejpam-4302	415	14	are	be	AUX
ejpam-4302	415	15	equivalent	equivalent	ADJ
ejpam-4302	415	16	:	:	PUNCT
ejpam-4302	415	17	(	(	PUNCT
ejpam-4302	415	18	1	1	X
ejpam-4302	415	19	)	)	PUNCT
ejpam-4302	415	20	(	(	PUNCT
ejpam-4302	415	21	x	x	X
ejpam-4302	415	22	,	,	PUNCT
ejpam-4302	415	23	τ	τ	X
ejpam-4302	415	24	)	)	PUNCT
ejpam-4302	415	25	is	be	AUX
ejpam-4302	415	26	λsp	λsp	ADJ
ejpam-4302	415	27	-	-	ADJ
ejpam-4302	415	28	normal	normal	ADJ
ejpam-4302	415	29	.	.	PUNCT
ejpam-4302	416	1	(	(	PUNCT
ejpam-4302	416	2	2	2	X
ejpam-4302	416	3	)	)	PUNCT
ejpam-4302	416	4	for	for	ADP
ejpam-4302	416	5	any	any	DET
ejpam-4302	416	6	pair	pair	NOUN
ejpam-4302	416	7	of	of	ADP
ejpam-4302	416	8	disjoint	disjoint	NOUN
ejpam-4302	416	9	(	(	PUNCT
ejpam-4302	416	10	λ	λ	PROPN
ejpam-4302	416	11	,	,	PUNCT
ejpam-4302	416	12	sp)-closed	sp)-close	VERB
ejpam-4302	416	13	sets	set	VERB
ejpam-4302	416	14	f	f	PROPN
ejpam-4302	416	15	and	and	CCONJ
ejpam-4302	416	16	h	h	NOUN
ejpam-4302	416	17	,	,	PUNCT
ejpam-4302	416	18	there	there	PRON
ejpam-4302	416	19	exist	exist	VERB
ejpam-4302	416	20	disjoint	disjoint	NOUN
ejpam-4302	416	21	g-(λ	g-(λ	NOUN
ejpam-4302	416	22	,	,	PUNCT
ejpam-4302	416	23	sp)open	sp)open	PROPN
ejpam-4302	416	24	sets	set	VERB
ejpam-4302	416	25	u	u	NOUN
ejpam-4302	416	26	and	and	CCONJ
ejpam-4302	416	27	v	v	ADP
ejpam-4302	416	28	such	such	ADJ
ejpam-4302	416	29	that	that	SCONJ
ejpam-4302	416	30	f	f	PROPN
ejpam-4302	416	31	⊆	⊆	NUM
ejpam-4302	416	32	u	u	NOUN
ejpam-4302	416	33	and	and	CCONJ
ejpam-4302	416	34	h	h	NOUN
ejpam-4302	416	35	⊆	⊆	NUM
ejpam-4302	416	36	v	v	NOUN
ejpam-4302	416	37	.	.	PUNCT
ejpam-4302	417	1	(	(	PUNCT
ejpam-4302	417	2	3	3	X
ejpam-4302	417	3	)	)	PUNCT
ejpam-4302	417	4	for	for	ADP
ejpam-4302	417	5	each	each	DET
ejpam-4302	417	6	(	(	PUNCT
ejpam-4302	417	7	λ	λ	PROPN
ejpam-4302	417	8	,	,	PUNCT
ejpam-4302	417	9	sp)-closed	sp)-close	VERB
ejpam-4302	418	1	set	set	VERB
ejpam-4302	418	2	f	f	PROPN
ejpam-4302	418	3	and	and	CCONJ
ejpam-4302	418	4	each	each	DET
ejpam-4302	418	5	(	(	PUNCT
ejpam-4302	418	6	λ	λ	PROPN
ejpam-4302	418	7	,	,	PUNCT
ejpam-4302	418	8	sp)-open	sp)-open	NOUN
ejpam-4302	418	9	set	set	VERB
ejpam-4302	418	10	g	g	NOUN
ejpam-4302	418	11	containing	contain	VERB
ejpam-4302	418	12	f	f	NOUN
ejpam-4302	418	13	,	,	PUNCT
ejpam-4302	418	14	there	there	PRON
ejpam-4302	418	15	exists	exist	VERB
ejpam-4302	418	16	a	a	DET
ejpam-4302	418	17	g-(λ	g-(λ	PROPN
ejpam-4302	418	18	,	,	PUNCT
ejpam-4302	418	19	sp)-open	sp)-open	ADJ
ejpam-4302	418	20	set	set	VERB
ejpam-4302	418	21	u	u	PRON
ejpam-4302	418	22	such	such	ADJ
ejpam-4302	418	23	that	that	SCONJ
ejpam-4302	418	24	f	f	PROPN
ejpam-4302	418	25	⊆	⊆	NUM
ejpam-4302	418	26	u	u	NOUN
ejpam-4302	418	27	⊆	⊆	NUM
ejpam-4302	418	28	u	u	NOUN
ejpam-4302	418	29	(	(	PUNCT
ejpam-4302	418	30	λ	λ	PROPN
ejpam-4302	418	31	,	,	PUNCT
ejpam-4302	418	32	sp	sp	NOUN
ejpam-4302	418	33	)	)	PUNCT
ejpam-4302	418	34	⊆	⊆	NUM
ejpam-4302	418	35	g.	g.	NOUN
ejpam-4302	418	36	(	(	PUNCT
ejpam-4302	418	37	4	4	NUM
ejpam-4302	418	38	)	)	PUNCT
ejpam-4302	418	39	for	for	ADP
ejpam-4302	418	40	each	each	DET
ejpam-4302	418	41	(	(	PUNCT
ejpam-4302	418	42	λ	λ	PROPN
ejpam-4302	418	43	,	,	PUNCT
ejpam-4302	418	44	sp)-closed	sp)-close	VERB
ejpam-4302	418	45	set	set	VERB
ejpam-4302	418	46	f	f	PROPN
ejpam-4302	418	47	and	and	CCONJ
ejpam-4302	418	48	each	each	DET
ejpam-4302	418	49	g-(λ	g-(λ	PROPN
ejpam-4302	418	50	,	,	PUNCT
ejpam-4302	418	51	sp)-open	sp)-open	VERB
ejpam-4302	418	52	set	set	VERB
ejpam-4302	418	53	g	g	NOUN
ejpam-4302	418	54	containing	contain	VERB
ejpam-4302	418	55	f	f	NOUN
ejpam-4302	418	56	,	,	PUNCT
ejpam-4302	418	57	there	there	PRON
ejpam-4302	418	58	exists	exist	VERB
ejpam-4302	418	59	a	a	DET
ejpam-4302	418	60	(	(	PUNCT
ejpam-4302	418	61	λ	λ	NOUN
ejpam-4302	418	62	,	,	PUNCT
ejpam-4302	418	63	sp)-open	sp)-open	NOUN
ejpam-4302	418	64	set	set	VERB
ejpam-4302	418	65	u	u	PRON
ejpam-4302	418	66	such	such	ADJ
ejpam-4302	418	67	that	that	SCONJ
ejpam-4302	418	68	f	f	PROPN
ejpam-4302	418	69	⊆	⊆	NUM
ejpam-4302	418	70	u	u	NOUN
ejpam-4302	418	71	⊆	⊆	NUM
ejpam-4302	418	72	u	u	NOUN
ejpam-4302	418	73	(	(	PUNCT
ejpam-4302	418	74	λ	λ	PROPN
ejpam-4302	418	75	,	,	PUNCT
ejpam-4302	418	76	sp	sp	NOUN
ejpam-4302	418	77	)	)	PUNCT
ejpam-4302	418	78	⊆	⊆	NUM
ejpam-4302	418	79	g(λ	g(λ	PROPN
ejpam-4302	418	80	,	,	PUNCT
ejpam-4302	418	81	sp	sp	NOUN
ejpam-4302	418	82	)	)	PUNCT
ejpam-4302	418	83	.	.	PUNCT
ejpam-4302	419	1	(	(	PUNCT
ejpam-4302	419	2	5	5	NUM
ejpam-4302	419	3	)	)	PUNCT
ejpam-4302	419	4	for	for	SCONJ
ejpam-4302	419	5	each	each	DET
ejpam-4302	419	6	(	(	PUNCT
ejpam-4302	419	7	λ	λ	PROPN
ejpam-4302	419	8	,	,	PUNCT
ejpam-4302	419	9	sp)-closed	sp)-close	VERB
ejpam-4302	420	1	set	set	VERB
ejpam-4302	420	2	f	f	PROPN
ejpam-4302	420	3	and	and	CCONJ
ejpam-4302	420	4	each	each	DET
ejpam-4302	420	5	g-(λ	g-(λ	PROPN
ejpam-4302	420	6	,	,	PUNCT
ejpam-4302	420	7	sp)-open	sp)-open	VERB
ejpam-4302	420	8	set	set	VERB
ejpam-4302	420	9	g	g	NOUN
ejpam-4302	420	10	containing	contain	VERB
ejpam-4302	420	11	f	f	NOUN
ejpam-4302	420	12	,	,	PUNCT
ejpam-4302	420	13	there	there	PRON
ejpam-4302	420	14	exists	exist	VERB
ejpam-4302	420	15	a	a	DET
ejpam-4302	420	16	g-(λ	g-(λ	PROPN
ejpam-4302	420	17	,	,	PUNCT
ejpam-4302	420	18	sp)-open	sp)-open	ADJ
ejpam-4302	420	19	set	set	VERB
ejpam-4302	420	20	u	u	PRON
ejpam-4302	420	21	such	such	ADJ
ejpam-4302	420	22	that	that	SCONJ
ejpam-4302	420	23	f	f	PROPN
ejpam-4302	420	24	⊆	⊆	NUM
ejpam-4302	420	25	u	u	NOUN
ejpam-4302	420	26	⊆	⊆	NUM
ejpam-4302	420	27	u	u	NOUN
ejpam-4302	420	28	(	(	PUNCT
ejpam-4302	420	29	λ	λ	PROPN
ejpam-4302	420	30	,	,	PUNCT
ejpam-4302	420	31	sp	sp	NOUN
ejpam-4302	420	32	)	)	PUNCT
ejpam-4302	420	33	⊆	⊆	NUM
ejpam-4302	420	34	g(λ	g(λ	PROPN
ejpam-4302	420	35	,	,	PUNCT
ejpam-4302	420	36	sp	sp	NOUN
ejpam-4302	420	37	)	)	PUNCT
ejpam-4302	420	38	.	.	PUNCT
ejpam-4302	421	1	(	(	PUNCT
ejpam-4302	421	2	6	6	NUM
ejpam-4302	421	3	)	)	PUNCT
ejpam-4302	421	4	for	for	ADP
ejpam-4302	421	5	each	each	DET
ejpam-4302	421	6	g-(λ	g-(λ	NOUN
ejpam-4302	421	7	,	,	PUNCT
ejpam-4302	421	8	sp)-closed	sp)-close	VERB
ejpam-4302	421	9	set	set	VERB
ejpam-4302	421	10	f	f	PROPN
ejpam-4302	421	11	and	and	CCONJ
ejpam-4302	421	12	each	each	DET
ejpam-4302	421	13	(	(	PUNCT
ejpam-4302	421	14	λ	λ	PROPN
ejpam-4302	421	15	,	,	PUNCT
ejpam-4302	421	16	sp)-open	sp)-open	NOUN
ejpam-4302	421	17	set	set	VERB
ejpam-4302	421	18	g	g	NOUN
ejpam-4302	421	19	containing	contain	VERB
ejpam-4302	421	20	f	f	NOUN
ejpam-4302	421	21	,	,	PUNCT
ejpam-4302	421	22	there	there	PRON
ejpam-4302	421	23	exists	exist	VERB
ejpam-4302	421	24	a	a	DET
ejpam-4302	421	25	(	(	PUNCT
ejpam-4302	421	26	λ	λ	NOUN
ejpam-4302	421	27	,	,	PUNCT
ejpam-4302	421	28	sp)-open	sp)-open	NOUN
ejpam-4302	421	29	set	set	VERB
ejpam-4302	421	30	u	u	PRON
ejpam-4302	421	31	such	such	ADJ
ejpam-4302	421	32	that	that	SCONJ
ejpam-4302	421	33	f	f	PROPN
ejpam-4302	421	34	(	(	PUNCT
ejpam-4302	421	35	λ	λ	PROPN
ejpam-4302	421	36	,	,	PUNCT
ejpam-4302	421	37	sp	sp	NOUN
ejpam-4302	421	38	)	)	PUNCT
ejpam-4302	421	39	⊆	⊆	NUM
ejpam-4302	421	40	u	u	NOUN
ejpam-4302	421	41	⊆	⊆	NUM
ejpam-4302	421	42	u	u	NOUN
ejpam-4302	421	43	(	(	PUNCT
ejpam-4302	421	44	λ	λ	PROPN
ejpam-4302	421	45	,	,	PUNCT
ejpam-4302	421	46	sp	sp	NOUN
ejpam-4302	421	47	)	)	PUNCT
ejpam-4302	421	48	⊆	⊆	NUM
ejpam-4302	421	49	g.	g.	NOUN
ejpam-4302	421	50	(	(	PUNCT
ejpam-4302	421	51	7	7	NUM
ejpam-4302	421	52	)	)	PUNCT
ejpam-4302	421	53	for	for	ADP
ejpam-4302	421	54	each	each	DET
ejpam-4302	421	55	g-(λ	g-(λ	NOUN
ejpam-4302	421	56	,	,	PUNCT
ejpam-4302	421	57	sp)-closed	sp)-close	VERB
ejpam-4302	421	58	set	set	VERB
ejpam-4302	421	59	f	f	PROPN
ejpam-4302	421	60	and	and	CCONJ
ejpam-4302	421	61	each	each	DET
ejpam-4302	421	62	(	(	PUNCT
ejpam-4302	421	63	λ	λ	PROPN
ejpam-4302	421	64	,	,	PUNCT
ejpam-4302	421	65	sp)-open	sp)-open	NOUN
ejpam-4302	421	66	set	set	VERB
ejpam-4302	421	67	g	g	NOUN
ejpam-4302	421	68	containing	contain	VERB
ejpam-4302	421	69	f	f	NOUN
ejpam-4302	421	70	,	,	PUNCT
ejpam-4302	421	71	there	there	PRON
ejpam-4302	421	72	exists	exist	VERB
ejpam-4302	421	73	a	a	DET
ejpam-4302	421	74	g-(λ	g-(λ	PROPN
ejpam-4302	421	75	,	,	PUNCT
ejpam-4302	421	76	sp)-open	sp)-open	ADJ
ejpam-4302	421	77	set	set	VERB
ejpam-4302	421	78	u	u	PRON
ejpam-4302	421	79	such	such	ADJ
ejpam-4302	421	80	that	that	SCONJ
ejpam-4302	421	81	f	f	PROPN
ejpam-4302	421	82	(	(	PUNCT
ejpam-4302	421	83	λ	λ	PROPN
ejpam-4302	421	84	,	,	PUNCT
ejpam-4302	421	85	sp	sp	NOUN
ejpam-4302	421	86	)	)	PUNCT
ejpam-4302	421	87	⊆	⊆	NUM
ejpam-4302	421	88	u	u	NOUN
ejpam-4302	421	89	⊆	⊆	NUM
ejpam-4302	421	90	u	u	NOUN
ejpam-4302	421	91	(	(	PUNCT
ejpam-4302	421	92	λ	λ	PROPN
ejpam-4302	421	93	,	,	PUNCT
ejpam-4302	421	94	sp	sp	NOUN
ejpam-4302	421	95	)	)	PUNCT
ejpam-4302	421	96	⊆	⊆	NUM
ejpam-4302	421	97	g.	g.	NOUN
ejpam-4302	421	98	references	reference	NOUN
ejpam-4302	421	99	2139	2139	NUM
ejpam-4302	421	100	proof	proof	NOUN
ejpam-4302	421	101	.	.	PUNCT
ejpam-4302	422	1	(	(	PUNCT
ejpam-4302	422	2	1	1	X
ejpam-4302	422	3	)	)	PUNCT
ejpam-4302	422	4	⇒	⇒	NOUN
ejpam-4302	422	5	(	(	PUNCT
ejpam-4302	422	6	2	2	NUM
ejpam-4302	422	7	):	):	PUNCT
ejpam-4302	422	8	the	the	DET
ejpam-4302	422	9	proof	proof	NOUN
ejpam-4302	422	10	is	be	AUX
ejpam-4302	422	11	obvious	obvious	ADJ
ejpam-4302	422	12	.	.	PUNCT
ejpam-4302	423	1	(	(	PUNCT
ejpam-4302	423	2	2	2	X
ejpam-4302	423	3	)	)	PUNCT
ejpam-4302	423	4	⇒	⇒	NOUN
ejpam-4302	423	5	(	(	PUNCT
ejpam-4302	423	6	3	3	NUM
ejpam-4302	423	7	):	):	PUNCT
ejpam-4302	423	8	let	let	VERB
ejpam-4302	423	9	f	f	PRON
ejpam-4302	423	10	be	be	AUX
ejpam-4302	423	11	a	a	DET
ejpam-4302	423	12	(	(	PUNCT
ejpam-4302	423	13	λ	λ	NOUN
ejpam-4302	423	14	,	,	PUNCT
ejpam-4302	423	15	sp)-closed	sp)-close	VERB
ejpam-4302	423	16	set	set	ADJ
ejpam-4302	423	17	and	and	CCONJ
ejpam-4302	423	18	let	let	VERB
ejpam-4302	423	19	g	g	PRON
ejpam-4302	423	20	be	be	AUX
ejpam-4302	423	21	a	a	DET
ejpam-4302	423	22	(	(	PUNCT
ejpam-4302	423	23	λ	λ	NOUN
ejpam-4302	423	24	,	,	PUNCT
ejpam-4302	423	25	sp)-open	sp)-open	ADJ
ejpam-4302	423	26	set	set	NOUN
ejpam-4302	423	27	containing	contain	VERB
ejpam-4302	423	28	f	f	PROPN
ejpam-4302	423	29	.	.	PUNCT
ejpam-4302	424	1	then	then	ADV
ejpam-4302	424	2	,	,	PUNCT
ejpam-4302	424	3	f	f	PROPN
ejpam-4302	424	4	and	and	CCONJ
ejpam-4302	424	5	x	x	SYM
ejpam-4302	424	6	−	−	PROPN
ejpam-4302	424	7	g	g	NOUN
ejpam-4302	424	8	are	be	AUX
ejpam-4302	424	9	two	two	NUM
ejpam-4302	424	10	disjoint	disjoint	NOUN
ejpam-4302	424	11	(	(	PUNCT
ejpam-4302	424	12	λ	λ	PROPN
ejpam-4302	424	13	,	,	PUNCT
ejpam-4302	424	14	sp)-closed	sp)-close	VERB
ejpam-4302	424	15	sets	set	NOUN
ejpam-4302	424	16	.	.	PUNCT
ejpam-4302	425	1	hence	hence	ADV
ejpam-4302	425	2	by	by	ADP
ejpam-4302	425	3	(	(	PUNCT
ejpam-4302	425	4	2	2	NUM
ejpam-4302	425	5	)	)	PUNCT
ejpam-4302	425	6	,	,	PUNCT
ejpam-4302	425	7	there	there	PRON
ejpam-4302	425	8	exist	exist	VERB
ejpam-4302	425	9	disjoint	disjoint	NOUN
ejpam-4302	425	10	g-(λ	g-(λ	NOUN
ejpam-4302	425	11	,	,	PUNCT
ejpam-4302	425	12	sp)-open	sp)-open	VERB
ejpam-4302	425	13	sets	set	VERB
ejpam-4302	425	14	u	u	NOUN
ejpam-4302	425	15	and	and	CCONJ
ejpam-4302	425	16	v	v	NOUN
ejpam-4302	425	17	of	of	ADP
ejpam-4302	425	18	x	x	PUNCT
ejpam-4302	425	19	such	such	ADJ
ejpam-4302	425	20	that	that	SCONJ
ejpam-4302	425	21	f	f	PROPN
ejpam-4302	425	22	⊆	⊆	NUM
ejpam-4302	425	23	u	u	NOUN
ejpam-4302	425	24	and	and	CCONJ
ejpam-4302	425	25	x	x	NOUN
ejpam-4302	425	26	−	−	NOUN
ejpam-4302	425	27	g	g	PROPN
ejpam-4302	425	28	⊆	⊆	NUM
ejpam-4302	425	29	v	v	NOUN
ejpam-4302	425	30	.	.	PUNCT
ejpam-4302	426	1	since	since	SCONJ
ejpam-4302	426	2	v	v	NOUN
ejpam-4302	426	3	is	be	AUX
ejpam-4302	426	4	g-(λ	g-(λ	PROPN
ejpam-4302	426	5	,	,	PUNCT
ejpam-4302	426	6	sp)-open	sp)-open	ADJ
ejpam-4302	426	7	and	and	CCONJ
ejpam-4302	426	8	x	x	SYM
ejpam-4302	426	9	−	−	PROPN
ejpam-4302	426	10	g	g	NOUN
ejpam-4302	426	11	is	be	AUX
ejpam-4302	426	12	(	(	PUNCT
ejpam-4302	426	13	λ	λ	X
ejpam-4302	426	14	,	,	PUNCT
ejpam-4302	426	15	sp)-closed	sp)-close	VERB
ejpam-4302	426	16	,	,	PUNCT
ejpam-4302	426	17	by	by	ADP
ejpam-4302	426	18	theorem	theorem	NOUN
ejpam-4302	426	19	6	6	NUM
ejpam-4302	426	20	,	,	PUNCT
ejpam-4302	426	21	x	x	NOUN
ejpam-4302	426	22	−	−	NOUN
ejpam-4302	426	23	g	g	PROPN
ejpam-4302	426	24	⊆	⊆	NUM
ejpam-4302	426	25	v(λ	v(λ	PROPN
ejpam-4302	426	26	,	,	PUNCT
ejpam-4302	426	27	sp	sp	NOUN
ejpam-4302	426	28	)	)	PUNCT
ejpam-4302	426	29	.	.	PUNCT
ejpam-4302	427	1	thus	thus	ADV
ejpam-4302	427	2	,	,	PUNCT
ejpam-4302	427	3	[	[	X
ejpam-4302	427	4	x	x	X
ejpam-4302	427	5	−	−	NOUN
ejpam-4302	427	6	v	v	NOUN
ejpam-4302	427	7	]	]	X
ejpam-4302	427	8	(	(	PUNCT
ejpam-4302	427	9	λ	λ	NOUN
ejpam-4302	427	10	,	,	PUNCT
ejpam-4302	427	11	sp	sp	NOUN
ejpam-4302	427	12	)	)	PUNCT
ejpam-4302	427	13	=	=	PUNCT
ejpam-4302	427	14	x	x	X
ejpam-4302	427	15	−	−	PROPN
ejpam-4302	427	16	v(λ	v(λ	PROPN
ejpam-4302	427	17	,	,	PUNCT
ejpam-4302	427	18	sp	sp	NOUN
ejpam-4302	427	19	)	)	PUNCT
ejpam-4302	427	20	⊆	⊆	NUM
ejpam-4302	427	21	g	g	NOUN
ejpam-4302	427	22	and	and	CCONJ
ejpam-4302	427	23	hence	hence	ADV
ejpam-4302	427	24	f	f	PROPN
ejpam-4302	427	25	⊆	⊆	NUM
ejpam-4302	427	26	u	u	NOUN
ejpam-4302	427	27	⊆	⊆	NUM
ejpam-4302	427	28	u	u	NOUN
ejpam-4302	427	29	(	(	PUNCT
ejpam-4302	427	30	λ	λ	PROPN
ejpam-4302	427	31	,	,	PUNCT
ejpam-4302	427	32	sp	sp	NOUN
ejpam-4302	427	33	)	)	PUNCT
ejpam-4302	427	34	⊆	⊆	NUM
ejpam-4302	427	35	g.	g.	NOUN
ejpam-4302	427	36	(	(	PUNCT
ejpam-4302	427	37	3	3	NUM
ejpam-4302	427	38	)	)	PUNCT
ejpam-4302	427	39	⇒	⇒	NOUN
ejpam-4302	427	40	(	(	PUNCT
ejpam-4302	427	41	1	1	NUM
ejpam-4302	427	42	):	):	PUNCT
ejpam-4302	427	43	let	let	VERB
ejpam-4302	427	44	f	f	PROPN
ejpam-4302	427	45	and	and	CCONJ
ejpam-4302	427	46	h	h	PROPN
ejpam-4302	427	47	be	be	VERB
ejpam-4302	427	48	two	two	NUM
ejpam-4302	427	49	disjoint	disjoint	NOUN
ejpam-4302	427	50	(	(	PUNCT
ejpam-4302	427	51	λ	λ	PROPN
ejpam-4302	427	52	,	,	PUNCT
ejpam-4302	427	53	sp)-closed	sp)-close	VERB
ejpam-4302	427	54	sets	set	NOUN
ejpam-4302	427	55	of	of	ADP
ejpam-4302	427	56	x.	x.	NOUN
ejpam-4302	427	57	then	then	ADV
ejpam-4302	427	58	,	,	PUNCT
ejpam-4302	427	59	f	f	PROPN
ejpam-4302	427	60	is	be	AUX
ejpam-4302	427	61	a	a	DET
ejpam-4302	427	62	(	(	PUNCT
ejpam-4302	427	63	λ	λ	PROPN
ejpam-4302	427	64	,	,	PUNCT
ejpam-4302	427	65	sp)-closed	sp)-close	VERB
ejpam-4302	427	66	set	set	ADJ
ejpam-4302	427	67	and	and	CCONJ
ejpam-4302	427	68	x	x	PRON
ejpam-4302	427	69	−h	−h	ADV
ejpam-4302	427	70	is	be	AUX
ejpam-4302	427	71	a	a	DET
ejpam-4302	427	72	(	(	PUNCT
ejpam-4302	427	73	λ	λ	NOUN
ejpam-4302	427	74	,	,	PUNCT
ejpam-4302	427	75	sp)-open	sp)-open	ADJ
ejpam-4302	427	76	set	set	NOUN
ejpam-4302	427	77	containing	contain	VERB
ejpam-4302	427	78	f	f	X
ejpam-4302	427	79	,	,	PUNCT
ejpam-4302	427	80	by	by	ADP
ejpam-4302	427	81	(	(	PUNCT
ejpam-4302	427	82	3	3	NUM
ejpam-4302	427	83	)	)	PUNCT
ejpam-4302	427	84	,	,	PUNCT
ejpam-4302	427	85	there	there	PRON
ejpam-4302	427	86	exists	exist	VERB
ejpam-4302	427	87	a	a	DET
ejpam-4302	427	88	g(λ	g(λ	PROPN
ejpam-4302	427	89	,	,	PUNCT
ejpam-4302	427	90	sp)-open	sp)-open	ADJ
ejpam-4302	427	91	set	set	VERB
ejpam-4302	427	92	u	u	PRON
ejpam-4302	427	93	such	such	ADJ
ejpam-4302	427	94	that	that	SCONJ
ejpam-4302	427	95	f	f	PROPN
ejpam-4302	427	96	⊆	⊆	NUM
ejpam-4302	427	97	u	u	NOUN
ejpam-4302	427	98	⊆	⊆	NUM
ejpam-4302	427	99	u	u	NOUN
ejpam-4302	427	100	(	(	PUNCT
ejpam-4302	427	101	λ	λ	PROPN
ejpam-4302	427	102	,	,	PUNCT
ejpam-4302	427	103	sp	sp	NOUN
ejpam-4302	427	104	)	)	PUNCT
ejpam-4302	427	105	⊆	⊆	NUM
ejpam-4302	427	106	x−h	x−h	NOUN
ejpam-4302	427	107	.	.	PUNCT
ejpam-4302	428	1	thus	thus	ADV
ejpam-4302	428	2	,	,	PUNCT
ejpam-4302	428	3	by	by	ADP
ejpam-4302	428	4	theorem	theorem	NOUN
ejpam-4302	428	5	6	6	NUM
ejpam-4302	428	6	,	,	PUNCT
ejpam-4302	428	7	f	f	PROPN
ejpam-4302	428	8	⊆	⊆	NUM
ejpam-4302	428	9	u(λ	u(λ	PROPN
ejpam-4302	428	10	,	,	PUNCT
ejpam-4302	428	11	sp	sp	NOUN
ejpam-4302	428	12	)	)	PUNCT
ejpam-4302	428	13	,	,	PUNCT
ejpam-4302	428	14	h	h	NOUN
ejpam-4302	428	15	⊆	⊆	NUM
ejpam-4302	428	16	x	x	SYM
ejpam-4302	428	17	−	−	PROPN
ejpam-4302	428	18	u	u	NOUN
ejpam-4302	428	19	(	(	PUNCT
ejpam-4302	428	20	λ	λ	PROPN
ejpam-4302	428	21	,	,	PUNCT
ejpam-4302	428	22	sp	sp	NOUN
ejpam-4302	428	23	)	)	PUNCT
ejpam-4302	428	24	,	,	PUNCT
ejpam-4302	428	25	where	where	SCONJ
ejpam-4302	428	26	u(λ	u(λ	PROPN
ejpam-4302	428	27	,	,	PUNCT
ejpam-4302	428	28	sp	sp	NOUN
ejpam-4302	428	29	)	)	PUNCT
ejpam-4302	428	30	and	and	CCONJ
ejpam-4302	428	31	x	x	SYM
ejpam-4302	428	32	−	−	PROPN
ejpam-4302	428	33	u	u	NOUN
ejpam-4302	428	34	(	(	PUNCT
ejpam-4302	428	35	λ	λ	PROPN
ejpam-4302	428	36	,	,	PUNCT
ejpam-4302	428	37	sp	sp	NOUN
ejpam-4302	428	38	)	)	PUNCT
ejpam-4302	428	39	are	be	AUX
ejpam-4302	428	40	two	two	NUM
ejpam-4302	428	41	disjoint	disjoint	NOUN
ejpam-4302	428	42	(	(	PUNCT
ejpam-4302	428	43	λ	λ	NOUN
ejpam-4302	428	44	,	,	PUNCT
ejpam-4302	428	45	sp)-open	sp)-open	ADJ
ejpam-4302	428	46	sets	set	NOUN
ejpam-4302	428	47	.	.	PUNCT
ejpam-4302	429	1	(	(	PUNCT
ejpam-4302	429	2	4	4	X
ejpam-4302	429	3	)	)	PUNCT
ejpam-4302	429	4	⇒	⇒	NOUN
ejpam-4302	429	5	(	(	PUNCT
ejpam-4302	429	6	5	5	NUM
ejpam-4302	429	7	)	)	PUNCT
ejpam-4302	429	8	and	and	CCONJ
ejpam-4302	429	9	(	(	PUNCT
ejpam-4302	429	10	5	5	X
ejpam-4302	429	11	)	)	PUNCT
ejpam-4302	429	12	⇒	⇒	NOUN
ejpam-4302	429	13	(	(	PUNCT
ejpam-4302	429	14	2	2	NUM
ejpam-4302	429	15	):	):	PUNCT
ejpam-4302	429	16	the	the	DET
ejpam-4302	429	17	proofs	proof	NOUN
ejpam-4302	429	18	are	be	AUX
ejpam-4302	429	19	obvious	obvious	ADJ
ejpam-4302	429	20	.	.	PUNCT
ejpam-4302	430	1	(	(	PUNCT
ejpam-4302	430	2	6	6	NUM
ejpam-4302	430	3	)	)	PUNCT
ejpam-4302	430	4	⇒	⇒	NOUN
ejpam-4302	430	5	(	(	PUNCT
ejpam-4302	430	6	7	7	NUM
ejpam-4302	430	7	)	)	PUNCT
ejpam-4302	430	8	and	and	CCONJ
ejpam-4302	430	9	(	(	PUNCT
ejpam-4302	430	10	7	7	X
ejpam-4302	430	11	)	)	PUNCT
ejpam-4302	430	12	⇒	⇒	NOUN
ejpam-4302	430	13	(	(	PUNCT
ejpam-4302	430	14	3	3	NUM
ejpam-4302	430	15	):	):	PUNCT
ejpam-4302	430	16	the	the	DET
ejpam-4302	430	17	proofs	proof	NOUN
ejpam-4302	430	18	are	be	AUX
ejpam-4302	430	19	obvious	obvious	ADJ
ejpam-4302	430	20	.	.	PUNCT
ejpam-4302	431	1	(	(	PUNCT
ejpam-4302	431	2	3	3	X
ejpam-4302	431	3	)	)	PUNCT
ejpam-4302	431	4	⇒	⇒	NOUN
ejpam-4302	431	5	(	(	PUNCT
ejpam-4302	431	6	5	5	NUM
ejpam-4302	431	7	):	):	PUNCT
ejpam-4302	431	8	let	let	VERB
ejpam-4302	431	9	f	f	PRON
ejpam-4302	431	10	be	be	AUX
ejpam-4302	431	11	a	a	DET
ejpam-4302	431	12	(	(	PUNCT
ejpam-4302	431	13	λ	λ	NOUN
ejpam-4302	431	14	,	,	PUNCT
ejpam-4302	431	15	sp)-closed	sp)-close	VERB
ejpam-4302	431	16	set	set	ADJ
ejpam-4302	431	17	and	and	CCONJ
ejpam-4302	431	18	let	let	VERB
ejpam-4302	431	19	g	g	PRON
ejpam-4302	431	20	be	be	AUX
ejpam-4302	431	21	a	a	DET
ejpam-4302	431	22	g-(λ	g-(λ	PROPN
ejpam-4302	431	23	,	,	PUNCT
ejpam-4302	431	24	sp)-open	sp)-open	ADJ
ejpam-4302	431	25	set	set	NOUN
ejpam-4302	431	26	containing	contain	VERB
ejpam-4302	431	27	f	f	PROPN
ejpam-4302	431	28	.	.	PUNCT
ejpam-4302	432	1	since	since	SCONJ
ejpam-4302	432	2	g	g	PROPN
ejpam-4302	432	3	is	be	AUX
ejpam-4302	432	4	g-(λ	g-(λ	PROPN
ejpam-4302	432	5	,	,	PUNCT
ejpam-4302	432	6	sp)-open	sp)-open	ADJ
ejpam-4302	432	7	and	and	CCONJ
ejpam-4302	432	8	f	f	PROPN
ejpam-4302	432	9	is	be	AUX
ejpam-4302	432	10	(	(	PUNCT
ejpam-4302	432	11	λ	λ	X
ejpam-4302	432	12	,	,	PUNCT
ejpam-4302	432	13	sp)-closed	sp)-close	VERB
ejpam-4302	432	14	,	,	PUNCT
ejpam-4302	432	15	by	by	ADP
ejpam-4302	432	16	theorem	theorem	NOUN
ejpam-4302	432	17	6	6	NUM
ejpam-4302	432	18	,	,	PUNCT
ejpam-4302	432	19	f	f	PROPN
ejpam-4302	432	20	⊆	⊆	NUM
ejpam-4302	432	21	g(λ	g(λ	PROPN
ejpam-4302	432	22	,	,	PUNCT
ejpam-4302	432	23	sp	sp	NOUN
ejpam-4302	432	24	)	)	PUNCT
ejpam-4302	432	25	.	.	PUNCT
ejpam-4302	433	1	thus	thus	ADV
ejpam-4302	433	2	,	,	PUNCT
ejpam-4302	433	3	by	by	ADP
ejpam-4302	433	4	(	(	PUNCT
ejpam-4302	433	5	3	3	NUM
ejpam-4302	433	6	)	)	PUNCT
ejpam-4302	433	7	,	,	PUNCT
ejpam-4302	433	8	there	there	PRON
ejpam-4302	433	9	exists	exist	VERB
ejpam-4302	433	10	a	a	DET
ejpam-4302	433	11	g-(λ	g-(λ	PROPN
ejpam-4302	433	12	,	,	PUNCT
ejpam-4302	433	13	sp)-open	sp)-open	ADJ
ejpam-4302	433	14	set	set	VERB
ejpam-4302	433	15	u	u	PRON
ejpam-4302	433	16	such	such	ADJ
ejpam-4302	433	17	that	that	SCONJ
ejpam-4302	433	18	f	f	PROPN
ejpam-4302	433	19	⊆	⊆	NUM
ejpam-4302	433	20	u	u	NOUN
ejpam-4302	433	21	⊆	⊆	NUM
ejpam-4302	433	22	u	u	NOUN
ejpam-4302	433	23	(	(	PUNCT
ejpam-4302	433	24	λ	λ	PROPN
ejpam-4302	433	25	,	,	PUNCT
ejpam-4302	433	26	sp	sp	NOUN
ejpam-4302	433	27	)	)	PUNCT
ejpam-4302	433	28	⊆	⊆	NUM
ejpam-4302	433	29	g(λ	g(λ	PROPN
ejpam-4302	433	30	,	,	PUNCT
ejpam-4302	433	31	sp	sp	NOUN
ejpam-4302	433	32	)	)	PUNCT
ejpam-4302	433	33	.	.	PUNCT
ejpam-4302	434	1	(	(	PUNCT
ejpam-4302	434	2	5	5	X
ejpam-4302	434	3	)	)	PUNCT
ejpam-4302	434	4	⇒	⇒	NOUN
ejpam-4302	434	5	(	(	PUNCT
ejpam-4302	434	6	6	6	NUM
ejpam-4302	434	7	):	):	PUNCT
ejpam-4302	434	8	let	let	VERB
ejpam-4302	434	9	f	f	PRON
ejpam-4302	434	10	be	be	AUX
ejpam-4302	434	11	a	a	DET
ejpam-4302	434	12	g-(λ	g-(λ	NOUN
ejpam-4302	434	13	,	,	PUNCT
ejpam-4302	434	14	sp)-closed	sp)-close	VERB
ejpam-4302	434	15	set	set	ADJ
ejpam-4302	434	16	and	and	CCONJ
ejpam-4302	434	17	let	let	VERB
ejpam-4302	434	18	g	g	PRON
ejpam-4302	434	19	be	be	AUX
ejpam-4302	434	20	a	a	DET
ejpam-4302	434	21	(	(	PUNCT
ejpam-4302	434	22	λ	λ	NOUN
ejpam-4302	434	23	,	,	PUNCT
ejpam-4302	434	24	sp)-open	sp)-open	ADJ
ejpam-4302	434	25	set	set	NOUN
ejpam-4302	434	26	containing	contain	VERB
ejpam-4302	434	27	f	f	PROPN
ejpam-4302	434	28	.	.	PUNCT
ejpam-4302	435	1	then	then	ADV
ejpam-4302	435	2	,	,	PUNCT
ejpam-4302	435	3	we	we	PRON
ejpam-4302	435	4	have	have	VERB
ejpam-4302	435	5	f	f	PROPN
ejpam-4302	435	6	(	(	PUNCT
ejpam-4302	435	7	λ	λ	PROPN
ejpam-4302	435	8	,	,	PUNCT
ejpam-4302	435	9	sp	sp	NOUN
ejpam-4302	435	10	)	)	PUNCT
ejpam-4302	435	11	⊆	⊆	NUM
ejpam-4302	435	12	g.	g.	NOUN
ejpam-4302	435	13	since	since	SCONJ
ejpam-4302	435	14	g	g	PROPN
ejpam-4302	435	15	is	be	AUX
ejpam-4302	435	16	g-(λ	g-(λ	PROPN
ejpam-4302	435	17	,	,	PUNCT
ejpam-4302	435	18	sp)-open	sp)-open	ADJ
ejpam-4302	435	19	and	and	CCONJ
ejpam-4302	435	20	f	f	PROPN
ejpam-4302	435	21	(	(	PUNCT
ejpam-4302	435	22	λ	λ	PROPN
ejpam-4302	435	23	,	,	PUNCT
ejpam-4302	435	24	sp	sp	NOUN
ejpam-4302	435	25	)	)	PUNCT
ejpam-4302	435	26	is	be	AUX
ejpam-4302	435	27	(	(	PUNCT
ejpam-4302	435	28	λ	λ	X
ejpam-4302	435	29	,	,	PUNCT
ejpam-4302	435	30	sp)-closed	sp)-close	VERB
ejpam-4302	435	31	,	,	PUNCT
ejpam-4302	435	32	by	by	ADP
ejpam-4302	435	33	(	(	PUNCT
ejpam-4302	435	34	5	5	NUM
ejpam-4302	435	35	)	)	PUNCT
ejpam-4302	435	36	,	,	PUNCT
ejpam-4302	435	37	there	there	PRON
ejpam-4302	435	38	exists	exist	VERB
ejpam-4302	435	39	a	a	DET
ejpam-4302	435	40	g-(λ	g-(λ	PROPN
ejpam-4302	435	41	,	,	PUNCT
ejpam-4302	435	42	sp)-open	sp)-open	ADJ
ejpam-4302	435	43	set	set	VERB
ejpam-4302	435	44	u	u	PRON
ejpam-4302	435	45	such	such	ADJ
ejpam-4302	435	46	that	that	SCONJ
ejpam-4302	435	47	f	f	PROPN
ejpam-4302	435	48	(	(	PUNCT
ejpam-4302	435	49	λ	λ	PROPN
ejpam-4302	435	50	,	,	PUNCT
ejpam-4302	435	51	sp	sp	NOUN
ejpam-4302	435	52	)	)	PUNCT
ejpam-4302	435	53	⊆	⊆	NUM
ejpam-4302	435	54	u	u	NOUN
ejpam-4302	435	55	⊆	⊆	NUM
ejpam-4302	435	56	u	u	NOUN
ejpam-4302	435	57	(	(	PUNCT
ejpam-4302	435	58	λ	λ	PROPN
ejpam-4302	435	59	,	,	PUNCT
ejpam-4302	435	60	sp	sp	NOUN
ejpam-4302	435	61	)	)	PUNCT
ejpam-4302	435	62	⊆	⊆	NUM
ejpam-4302	435	63	g.	g.	NOUN
ejpam-4302	435	64	since	since	SCONJ
ejpam-4302	435	65	u	u	PROPN
ejpam-4302	435	66	is	be	AUX
ejpam-4302	435	67	g-(λ	g-(λ	PROPN
ejpam-4302	435	68	,	,	PUNCT
ejpam-4302	435	69	sp)-open	sp)-open	ADJ
ejpam-4302	435	70	and	and	CCONJ
ejpam-4302	435	71	f	f	PROPN
ejpam-4302	435	72	(	(	PUNCT
ejpam-4302	435	73	λ	λ	PROPN
ejpam-4302	435	74	,	,	PUNCT
ejpam-4302	435	75	sp	sp	NOUN
ejpam-4302	435	76	)	)	PUNCT
ejpam-4302	435	77	⊆	⊆	NUM
ejpam-4302	435	78	u	u	NOUN
ejpam-4302	435	79	,	,	PUNCT
ejpam-4302	435	80	by	by	ADP
ejpam-4302	435	81	theorem	theorem	NOUN
ejpam-4302	435	82	6	6	NUM
ejpam-4302	435	83	,	,	PUNCT
ejpam-4302	435	84	f	f	PROPN
ejpam-4302	435	85	(	(	PUNCT
ejpam-4302	435	86	λ	λ	PROPN
ejpam-4302	435	87	,	,	PUNCT
ejpam-4302	435	88	sp	sp	NOUN
ejpam-4302	435	89	)	)	PUNCT
ejpam-4302	435	90	⊆	⊆	NUM
ejpam-4302	435	91	u(λ	u(λ	PROPN
ejpam-4302	435	92	,	,	PUNCT
ejpam-4302	435	93	sp	sp	NOUN
ejpam-4302	435	94	)	)	PUNCT
ejpam-4302	435	95	.	.	PUNCT
ejpam-4302	436	1	put	put	VERB
ejpam-4302	436	2	v	v	NUM
ejpam-4302	436	3	=	=	SYM
ejpam-4302	436	4	u(λ	u(λ	PROPN
ejpam-4302	436	5	,	,	PUNCT
ejpam-4302	436	6	sp	sp	NOUN
ejpam-4302	436	7	)	)	PUNCT
ejpam-4302	436	8	.	.	PUNCT
ejpam-4302	437	1	then	then	ADV
ejpam-4302	437	2	,	,	PUNCT
ejpam-4302	437	3	v	v	NOUN
ejpam-4302	437	4	is	be	AUX
ejpam-4302	437	5	(	(	PUNCT
ejpam-4302	437	6	λ	λ	X
ejpam-4302	437	7	,	,	PUNCT
ejpam-4302	437	8	sp)-open	sp)-open	NOUN
ejpam-4302	437	9	and	and	CCONJ
ejpam-4302	437	10	f	f	PROPN
ejpam-4302	437	11	(	(	PUNCT
ejpam-4302	437	12	λ	λ	PROPN
ejpam-4302	437	13	,	,	PUNCT
ejpam-4302	437	14	sp	sp	NOUN
ejpam-4302	437	15	)	)	PUNCT
ejpam-4302	437	16	⊆	⊆	NUM
ejpam-4302	437	17	v	v	ADP
ejpam-4302	437	18	⊆	⊆	NUM
ejpam-4302	437	19	v	v	NOUN
ejpam-4302	437	20	(	(	PUNCT
ejpam-4302	437	21	λ	λ	NOUN
ejpam-4302	437	22	,	,	PUNCT
ejpam-4302	437	23	sp	sp	NOUN
ejpam-4302	437	24	)	)	PUNCT
ejpam-4302	437	25	=	=	NOUN
ejpam-4302	438	1	[	[	X
ejpam-4302	438	2	u(λ	u(λ	PROPN
ejpam-4302	438	3	,	,	PUNCT
ejpam-4302	438	4	sp	sp	NOUN
ejpam-4302	438	5	)	)	PUNCT
ejpam-4302	438	6	]	]	PUNCT
ejpam-4302	438	7	(	(	PUNCT
ejpam-4302	438	8	λ	λ	NOUN
ejpam-4302	438	9	,	,	PUNCT
ejpam-4302	438	10	sp	sp	NOUN
ejpam-4302	438	11	)	)	PUNCT
ejpam-4302	438	12	⊆	⊆	NUM
ejpam-4302	438	13	u	u	NOUN
ejpam-4302	438	14	(	(	PUNCT
ejpam-4302	438	15	λ	λ	PROPN
ejpam-4302	438	16	,	,	PUNCT
ejpam-4302	438	17	sp	sp	NOUN
ejpam-4302	438	18	)	)	PUNCT
ejpam-4302	438	19	⊆	⊆	NUM
ejpam-4302	438	20	g.	g.	NOUN
ejpam-4302	438	21	(	(	PUNCT
ejpam-4302	438	22	6	6	NUM
ejpam-4302	438	23	)	)	PUNCT
ejpam-4302	438	24	⇒	⇒	NOUN
ejpam-4302	438	25	(	(	PUNCT
ejpam-4302	438	26	4	4	NUM
ejpam-4302	438	27	):	):	PUNCT
ejpam-4302	438	28	let	let	VERB
ejpam-4302	438	29	f	f	PRON
ejpam-4302	438	30	be	be	AUX
ejpam-4302	438	31	a	a	DET
ejpam-4302	438	32	(	(	PUNCT
ejpam-4302	438	33	λ	λ	NOUN
ejpam-4302	438	34	,	,	PUNCT
ejpam-4302	438	35	sp)-closed	sp)-close	VERB
ejpam-4302	438	36	set	set	ADJ
ejpam-4302	438	37	and	and	CCONJ
ejpam-4302	438	38	let	let	VERB
ejpam-4302	438	39	g	g	PRON
ejpam-4302	438	40	be	be	AUX
ejpam-4302	438	41	a	a	DET
ejpam-4302	438	42	g-(λ	g-(λ	PROPN
ejpam-4302	438	43	,	,	PUNCT
ejpam-4302	438	44	sp)-open	sp)-open	ADJ
ejpam-4302	438	45	set	set	NOUN
ejpam-4302	438	46	containing	contain	VERB
ejpam-4302	438	47	f	f	PROPN
ejpam-4302	438	48	.	.	PUNCT
ejpam-4302	439	1	thus	thus	ADV
ejpam-4302	439	2	,	,	PUNCT
ejpam-4302	439	3	by	by	ADP
ejpam-4302	439	4	theorem	theorem	NOUN
ejpam-4302	439	5	6	6	NUM
ejpam-4302	439	6	,	,	PUNCT
ejpam-4302	439	7	f	f	PROPN
ejpam-4302	439	8	(	(	PUNCT
ejpam-4302	439	9	λ	λ	PROPN
ejpam-4302	439	10	,	,	PUNCT
ejpam-4302	439	11	sp	sp	NOUN
ejpam-4302	439	12	)	)	PUNCT
ejpam-4302	439	13	=	=	SYM
ejpam-4302	439	14	f	f	PROPN
ejpam-4302	439	15	⊆	⊆	NUM
ejpam-4302	439	16	g(λ	g(λ	PROPN
ejpam-4302	439	17	,	,	PUNCT
ejpam-4302	439	18	sp	sp	NOUN
ejpam-4302	439	19	)	)	PUNCT
ejpam-4302	439	20	.	.	PUNCT
ejpam-4302	440	1	since	since	SCONJ
ejpam-4302	440	2	f	f	PROPN
ejpam-4302	440	3	is	be	AUX
ejpam-4302	440	4	g-(λ	g-(λ	PROPN
ejpam-4302	440	5	,	,	PUNCT
ejpam-4302	440	6	sp)-closed	sp)-close	VERB
ejpam-4302	440	7	and	and	CCONJ
ejpam-4302	440	8	g(λ	g(λ	PROPN
ejpam-4302	440	9	,	,	PUNCT
ejpam-4302	440	10	sp	sp	NOUN
ejpam-4302	440	11	)	)	PUNCT
ejpam-4302	440	12	is	be	AUX
ejpam-4302	440	13	(	(	PUNCT
ejpam-4302	440	14	λ	λ	INTJ
ejpam-4302	440	15	,	,	PUNCT
ejpam-4302	440	16	sp)-open	sp)-open	ADJ
ejpam-4302	440	17	,	,	PUNCT
ejpam-4302	440	18	by	by	ADP
ejpam-4302	440	19	(	(	PUNCT
ejpam-4302	440	20	6	6	NUM
ejpam-4302	440	21	)	)	PUNCT
ejpam-4302	440	22	,	,	PUNCT
ejpam-4302	440	23	there	there	PRON
ejpam-4302	440	24	exists	exist	VERB
ejpam-4302	440	25	a	a	DET
ejpam-4302	440	26	(	(	PUNCT
ejpam-4302	440	27	λ	λ	NOUN
ejpam-4302	440	28	,	,	PUNCT
ejpam-4302	440	29	sp)-open	sp)-open	NOUN
ejpam-4302	440	30	set	set	VERB
ejpam-4302	440	31	u	u	PRON
ejpam-4302	440	32	such	such	ADJ
ejpam-4302	440	33	that	that	SCONJ
ejpam-4302	440	34	f	f	PROPN
ejpam-4302	440	35	(	(	PUNCT
ejpam-4302	440	36	λ	λ	PROPN
ejpam-4302	440	37	,	,	PUNCT
ejpam-4302	440	38	sp	sp	NOUN
ejpam-4302	440	39	)	)	PUNCT
ejpam-4302	440	40	⊆	⊆	NUM
ejpam-4302	440	41	u	u	NOUN
ejpam-4302	440	42	⊆	⊆	NUM
ejpam-4302	440	43	u	u	NOUN
ejpam-4302	440	44	(	(	PUNCT
ejpam-4302	440	45	λ	λ	PROPN
ejpam-4302	440	46	,	,	PUNCT
ejpam-4302	440	47	sp	sp	NOUN
ejpam-4302	440	48	)	)	PUNCT
ejpam-4302	440	49	⊆	⊆	NUM
ejpam-4302	440	50	g(λ	g(λ	PROPN
ejpam-4302	440	51	,	,	PUNCT
ejpam-4302	440	52	sp	sp	NOUN
ejpam-4302	440	53	)	)	PUNCT
ejpam-4302	440	54	.	.	PUNCT
ejpam-4302	441	1	acknowledgements	acknowledgement	NOUN
ejpam-4302	441	2	this	this	DET
ejpam-4302	441	3	research	research	NOUN
ejpam-4302	441	4	project	project	NOUN
ejpam-4302	441	5	was	be	AUX
ejpam-4302	441	6	financially	financially	ADV
ejpam-4302	441	7	supported	support	VERB
ejpam-4302	441	8	by	by	ADP
ejpam-4302	441	9	mahasarakham	mahasarakham	PROPN
ejpam-4302	441	10	university	university	PROPN
ejpam-4302	441	11	.	.	PUNCT
ejpam-4302	442	1	references	reference	NOUN
ejpam-4302	442	2	[	[	X
ejpam-4302	442	3	1	1	NUM
ejpam-4302	442	4	]	]	PUNCT
ejpam-4302	442	5	d.	d.	PROPN
ejpam-4302	442	6	andrijević.	andrijević.	PROPN
ejpam-4302	442	7	on	on	ADP
ejpam-4302	442	8	b	b	X
ejpam-4302	442	9	-	-	PUNCT
ejpam-4302	442	10	open	open	ADJ
ejpam-4302	442	11	sets	set	NOUN
ejpam-4302	442	12	.	.	PUNCT
ejpam-4302	443	1	matematički	matematički	PROPN
ejpam-4302	443	2	vesnik	vesnik	PROPN
ejpam-4302	443	3	,	,	PUNCT
ejpam-4302	443	4	48:59–64	48:59–64	PROPN
ejpam-4302	443	5	,	,	PUNCT
ejpam-4302	443	6	1996	1996	NUM
ejpam-4302	443	7	.	.	PUNCT
ejpam-4302	444	1	[	[	X
ejpam-4302	444	2	2	2	NUM
ejpam-4302	444	3	]	]	PUNCT
ejpam-4302	444	4	c.	c.	PROPN
ejpam-4302	444	5	boonpok	boonpok	PROPN
ejpam-4302	444	6	.	.	PUNCT
ejpam-4302	445	1	(	(	PUNCT
ejpam-4302	445	2	λ	λ	NOUN
ejpam-4302	445	3	,	,	PUNCT
ejpam-4302	445	4	sp)-closed	sp)-close	VERB
ejpam-4302	445	5	sets	set	NOUN
ejpam-4302	445	6	and	and	CCONJ
ejpam-4302	445	7	related	related	ADJ
ejpam-4302	445	8	topics	topic	NOUN
ejpam-4302	445	9	in	in	ADP
ejpam-4302	445	10	topological	topological	ADJ
ejpam-4302	445	11	spaces	space	NOUN
ejpam-4302	445	12	.	.	PUNCT
ejpam-4302	446	1	wseas	wseas	VERB
ejpam-4302	446	2	transactions	transaction	NOUN
ejpam-4302	446	3	on	on	ADP
ejpam-4302	446	4	mathematics	mathematic	NOUN
ejpam-4302	446	5	,	,	PUNCT
ejpam-4302	446	6	19:321–322	19:321–322	PROPN
ejpam-4302	446	7	,	,	PUNCT
ejpam-4302	446	8	2020	2020	NUM
ejpam-4302	446	9	.	.	PUNCT
ejpam-4302	447	1	[	[	X
ejpam-4302	447	2	3	3	X
ejpam-4302	447	3	]	]	X
ejpam-4302	447	4	c.	c.	PROPN
ejpam-4302	447	5	l.	l.	PROPN
ejpam-4302	447	6	chang	chang	PROPN
ejpam-4302	447	7	.	.	PUNCT
ejpam-4302	448	1	fuzzy	fuzzy	ADJ
ejpam-4302	448	2	topological	topological	ADJ
ejpam-4302	448	3	spaces	space	NOUN
ejpam-4302	448	4	.	.	PUNCT
ejpam-4302	449	1	journal	journal	PROPN
ejpam-4302	449	2	of	of	ADP
ejpam-4302	449	3	mathematical	mathematical	ADJ
ejpam-4302	449	4	analysis	analysis	NOUN
ejpam-4302	449	5	and	and	CCONJ
ejpam-4302	449	6	applications	application	NOUN
ejpam-4302	449	7	,	,	PUNCT
ejpam-4302	449	8	24:182–190	24:182–190	NUM
ejpam-4302	449	9	,	,	PUNCT
ejpam-4302	449	10	1968	1968	NUM
ejpam-4302	449	11	.	.	PUNCT
ejpam-4302	450	1	[	[	X
ejpam-4302	450	2	4	4	X
ejpam-4302	450	3	]	]	X
ejpam-4302	450	4	g.	g.	NOUN
ejpam-4302	450	5	şenel	şenel	PROPN
ejpam-4302	450	6	and	and	CCONJ
ejpam-4302	450	7	n.	n.	PROPN
ejpam-4302	450	8	çağman	çağman	PROPN
ejpam-4302	450	9	.	.	PUNCT
ejpam-4302	450	10	soft	soft	ADJ
ejpam-4302	450	11	closed	closed	ADJ
ejpam-4302	450	12	sets	set	NOUN
ejpam-4302	450	13	on	on	ADP
ejpam-4302	450	14	soft	soft	ADJ
ejpam-4302	450	15	bitopological	bitopological	ADJ
ejpam-4302	450	16	spaces	space	NOUN
ejpam-4302	450	17	.	.	PUNCT
ejpam-4302	451	1	journal	journal	NOUN
ejpam-4302	451	2	of	of	ADP
ejpam-4302	451	3	new	new	ADJ
ejpam-4302	451	4	results	result	NOUN
ejpam-4302	451	5	in	in	ADP
ejpam-4302	451	6	science	science	NOUN
ejpam-4302	451	7	,	,	PUNCT
ejpam-4302	451	8	5:57–66	5:57–66	NUM
ejpam-4302	451	9	,	,	PUNCT
ejpam-4302	451	10	2014	2014	NUM
ejpam-4302	451	11	.	.	PUNCT
ejpam-4302	452	1	references	reference	NOUN
ejpam-4302	452	2	2140	2140	NUM
ejpam-4302	452	3	[	[	X
ejpam-4302	452	4	5	5	NUM
ejpam-4302	452	5	]	]	PUNCT
ejpam-4302	452	6	g.	g.	NOUN
ejpam-4302	452	7	şenel	şenel	PROPN
ejpam-4302	452	8	and	and	CCONJ
ejpam-4302	452	9	n.	n.	PROPN
ejpam-4302	452	10	çağman	çağman	PROPN
ejpam-4302	452	11	.	.	PUNCT
ejpam-4302	452	12	soft	soft	ADJ
ejpam-4302	452	13	topological	topological	ADJ
ejpam-4302	452	14	subspaces	subspace	NOUN
ejpam-4302	452	15	.	.	PUNCT
ejpam-4302	453	1	annals	annal	NOUN
ejpam-4302	453	2	of	of	ADP
ejpam-4302	453	3	fuzzy	fuzzy	ADJ
ejpam-4302	453	4	mathematics	mathematic	NOUN
ejpam-4302	453	5	and	and	CCONJ
ejpam-4302	453	6	informatics	informatic	NOUN
ejpam-4302	453	7	,	,	PUNCT
ejpam-4302	453	8	10(4):525–535	10(4):525–535	NUM
ejpam-4302	453	9	,	,	PUNCT
ejpam-4302	453	10	2015	2015	NUM
ejpam-4302	453	11	.	.	PUNCT
ejpam-4302	454	1	[	[	X
ejpam-4302	454	2	6	6	NUM
ejpam-4302	454	3	]	]	PUNCT
ejpam-4302	454	4	w.	w.	PROPN
ejpam-4302	454	5	dunham	dunham	PROPN
ejpam-4302	454	6	and	and	CCONJ
ejpam-4302	454	7	n.	n.	PROPN
ejpam-4302	454	8	levine	levine	PROPN
ejpam-4302	454	9	.	.	PUNCT
ejpam-4302	455	1	further	further	ADJ
ejpam-4302	455	2	results	result	NOUN
ejpam-4302	455	3	on	on	ADP
ejpam-4302	455	4	generalized	generalized	ADJ
ejpam-4302	455	5	closed	closed	ADJ
ejpam-4302	455	6	sets	set	NOUN
ejpam-4302	455	7	.	.	PUNCT
ejpam-4302	456	1	kyungpook	kyungpook	PROPN
ejpam-4302	456	2	mathematical	mathematical	PROPN
ejpam-4302	456	3	journal	journal	PROPN
ejpam-4302	456	4	,	,	PUNCT
ejpam-4302	456	5	20:169–175	20:169–175	PROPN
ejpam-4302	456	6	,	,	PUNCT
ejpam-4302	456	7	1980	1980	NUM
ejpam-4302	456	8	.	.	PUNCT
ejpam-4302	457	1	[	[	X
ejpam-4302	457	2	7	7	X
ejpam-4302	457	3	]	]	PUNCT
ejpam-4302	457	4	m.	m.	NOUN
ejpam-4302	457	5	e.	e.	PROPN
ejpam-4302	457	6	abd	abd	PROPN
ejpam-4302	458	1	el	el	PROPN
ejpam-4302	458	2	-	-	PROPN
ejpam-4302	458	3	monsef	monsef	PROPN
ejpam-4302	458	4	,	,	PUNCT
ejpam-4302	458	5	s.	s.	PROPN
ejpam-4302	458	6	n.	n.	PROPN
ejpam-4302	458	7	el	el	PROPN
ejpam-4302	458	8	-	-	PROPN
ejpam-4302	458	9	deeb	deeb	PROPN
ejpam-4302	458	10	,	,	PUNCT
ejpam-4302	458	11	and	and	CCONJ
ejpam-4302	458	12	r.	r.	PROPN
ejpam-4302	458	13	a.	a.	PROPN
ejpam-4302	458	14	mahmoud	mahmoud	PROPN
ejpam-4302	458	15	.	.	PUNCT
ejpam-4302	459	1	β	β	X
ejpam-4302	459	2	-	-	ADJ
ejpam-4302	459	3	open	open	ADJ
ejpam-4302	459	4	sets	set	NOUN
ejpam-4302	459	5	and	and	CCONJ
ejpam-4302	459	6	βcontinuous	βcontinuous	ADJ
ejpam-4302	459	7	mappings	mapping	NOUN
ejpam-4302	459	8	.	.	PUNCT
ejpam-4302	460	1	bulletin	bulletin	NOUN
ejpam-4302	460	2	of	of	ADP
ejpam-4302	460	3	the	the	DET
ejpam-4302	460	4	faculty	faculty	NOUN
ejpam-4302	460	5	of	of	ADP
ejpam-4302	460	6	science	science	NOUN
ejpam-4302	460	7	.	.	PUNCT
ejpam-4302	461	1	assiut	assiut	PROPN
ejpam-4302	461	2	university	university	PROPN
ejpam-4302	461	3	.	.	PUNCT
ejpam-4302	461	4	,	,	PUNCT
ejpam-4302	461	5	12:77–90	12:77–90	NUM
ejpam-4302	461	6	,	,	PUNCT
ejpam-4302	461	7	1983	1983	NUM
ejpam-4302	461	8	.	.	PUNCT
ejpam-4302	462	1	[	[	X
ejpam-4302	462	2	8	8	NUM
ejpam-4302	462	3	]	]	X
ejpam-4302	462	4	e.	e.	PROPN
ejpam-4302	462	5	f.	f.	PROPN
ejpam-4302	462	6	lashin	lashin	PROPN
ejpam-4302	462	7	,	,	PUNCT
ejpam-4302	462	8	a.	a.	NOUN
ejpam-4302	462	9	m.	m.	NOUN
ejpam-4302	462	10	kozae	kozae	PROPN
ejpam-4302	462	11	,	,	PUNCT
ejpam-4302	462	12	a.	a.	NOUN
ejpam-4302	462	13	a.	a.	NOUN
ejpam-4302	462	14	abo	abo	PROPN
ejpam-4302	462	15	khadra	khadra	NOUN
ejpam-4302	462	16	,	,	PUNCT
ejpam-4302	462	17	and	and	CCONJ
ejpam-4302	462	18	t.	t.	PROPN
ejpam-4302	462	19	medhat	medhat	PROPN
ejpam-4302	462	20	.	.	PUNCT
ejpam-4302	463	1	rough	rough	ADJ
ejpam-4302	463	2	set	set	NOUN
ejpam-4302	463	3	for	for	ADP
ejpam-4302	463	4	topological	topological	ADJ
ejpam-4302	463	5	spaces	space	NOUN
ejpam-4302	463	6	.	.	PUNCT
ejpam-4302	464	1	international	international	ADJ
ejpam-4302	464	2	journal	journal	PROPN
ejpam-4302	464	3	of	of	ADP
ejpam-4302	464	4	approximate	approximate	ADJ
ejpam-4302	464	5	reasoning	reasoning	NOUN
ejpam-4302	464	6	,	,	PUNCT
ejpam-4302	464	7	40:35–43	40:35–43	NUM
ejpam-4302	464	8	,	,	PUNCT
ejpam-4302	464	9	2005	2005	NUM
ejpam-4302	464	10	.	.	PUNCT
ejpam-4302	465	1	[	[	X
ejpam-4302	465	2	9	9	NUM
ejpam-4302	465	3	]	]	X
ejpam-4302	465	4	n.	n.	PROPN
ejpam-4302	465	5	levine	levine	PROPN
ejpam-4302	465	6	.	.	PUNCT
ejpam-4302	466	1	semi	semi	ADJ
ejpam-4302	466	2	-	-	ADJ
ejpam-4302	466	3	open	open	ADJ
ejpam-4302	466	4	sets	set	NOUN
ejpam-4302	466	5	and	and	CCONJ
ejpam-4302	466	6	semi	semi	ADJ
ejpam-4302	466	7	-	-	NOUN
ejpam-4302	466	8	continuity	continuity	NOUN
ejpam-4302	466	9	in	in	ADP
ejpam-4302	466	10	topological	topological	ADJ
ejpam-4302	466	11	spaces	space	NOUN
ejpam-4302	466	12	.	.	PUNCT
ejpam-4302	467	1	the	the	DET
ejpam-4302	467	2	american	american	PROPN
ejpam-4302	467	3	mathematical	mathematical	PROPN
ejpam-4302	467	4	monthly	monthly	ADV
ejpam-4302	467	5	,	,	PUNCT
ejpam-4302	467	6	70:36–41	70:36–41	NUM
ejpam-4302	467	7	,	,	PUNCT
ejpam-4302	467	8	1963	1963	NUM
ejpam-4302	467	9	.	.	PUNCT
ejpam-4302	468	1	[	[	X
ejpam-4302	468	2	10	10	NUM
ejpam-4302	468	3	]	]	X
ejpam-4302	468	4	n.	n.	PROPN
ejpam-4302	468	5	levine	levine	PROPN
ejpam-4302	468	6	.	.	PUNCT
ejpam-4302	469	1	generalized	generalize	VERB
ejpam-4302	469	2	closed	closed	ADJ
ejpam-4302	469	3	sets	set	NOUN
ejpam-4302	469	4	in	in	ADP
ejpam-4302	469	5	topology	topology	NOUN
ejpam-4302	469	6	.	.	PUNCT
ejpam-4302	470	1	rendiconti	rendiconti	VERB
ejpam-4302	470	2	del	del	PROPN
ejpam-4302	470	3	circolo	circolo	PROPN
ejpam-4302	470	4	matematico	matematico	X
ejpam-4302	470	5	de	de	X
ejpam-4302	470	6	palermo	palermo	X
ejpam-4302	470	7	(	(	PUNCT
ejpam-4302	470	8	2	2	NUM
ejpam-4302	470	9	)	)	PUNCT
ejpam-4302	470	10	,	,	PUNCT
ejpam-4302	470	11	19:89–96	19:89–96	NUM
ejpam-4302	470	12	,	,	PUNCT
ejpam-4302	470	13	1970	1970	NUM
ejpam-4302	470	14	.	.	PUNCT
ejpam-4302	471	1	[	[	X
ejpam-4302	471	2	11	11	NUM
ejpam-4302	471	3	]	]	PUNCT
ejpam-4302	471	4	a.	a.	NOUN
ejpam-4302	471	5	s.	s.	PROPN
ejpam-4302	471	6	mashhour	mashhour	PROPN
ejpam-4302	471	7	,	,	PUNCT
ejpam-4302	471	8	m.	m.	PROPN
ejpam-4302	471	9	e.	e.	PROPN
ejpam-4302	471	10	el	el	PROPN
ejpam-4302	471	11	-	-	PROPN
ejpam-4302	471	12	monsef	monsef	ADJ
ejpam-4302	471	13	,	,	PUNCT
ejpam-4302	471	14	and	and	CCONJ
ejpam-4302	471	15	s.	s.	PROPN
ejpam-4302	471	16	n.	n.	PROPN
ejpam-4302	471	17	el	el	PROPN
ejpam-4302	471	18	-	-	PROPN
ejpam-4302	471	19	deeb	deeb	PROPN
ejpam-4302	471	20	.	.	PUNCT
ejpam-4302	472	1	on	on	ADP
ejpam-4302	472	2	precontinuous	precontinuous	ADJ
ejpam-4302	472	3	and	and	CCONJ
ejpam-4302	472	4	weak	weak	ADJ
ejpam-4302	472	5	precontinuous	precontinuous	ADJ
ejpam-4302	472	6	mappings	mapping	NOUN
ejpam-4302	472	7	.	.	PUNCT
ejpam-4302	473	1	proceedings	proceeding	NOUN
ejpam-4302	473	2	of	of	ADP
ejpam-4302	473	3	the	the	DET
ejpam-4302	473	4	mathematical	mathematical	ADJ
ejpam-4302	473	5	and	and	CCONJ
ejpam-4302	473	6	physical	physical	ADJ
ejpam-4302	473	7	society	society	NOUN
ejpam-4302	473	8	of	of	ADP
ejpam-4302	473	9	egypt	egypt	PROPN
ejpam-4302	473	10	,	,	PUNCT
ejpam-4302	473	11	53:47–53	53:47–53	NUM
ejpam-4302	473	12	,	,	PUNCT
ejpam-4302	473	13	1982	1982	NUM
ejpam-4302	473	14	.	.	PUNCT
ejpam-4302	474	1	[	[	X
ejpam-4302	474	2	12	12	NUM
ejpam-4302	474	3	]	]	X
ejpam-4302	474	4	o.	o.	NOUN
ejpam-4302	474	5	nj̊astad	nj̊astad	NOUN
ejpam-4302	474	6	.	.	PUNCT
ejpam-4302	475	1	on	on	ADP
ejpam-4302	475	2	some	some	DET
ejpam-4302	475	3	classes	class	NOUN
ejpam-4302	475	4	of	of	ADP
ejpam-4302	475	5	nearly	nearly	ADV
ejpam-4302	475	6	open	open	ADJ
ejpam-4302	475	7	sets	set	NOUN
ejpam-4302	475	8	.	.	PUNCT
ejpam-4302	476	1	pasific	pasific	PROPN
ejpam-4302	476	2	journal	journal	PROPN
ejpam-4302	476	3	of	of	ADP
ejpam-4302	476	4	mathematics	mathematic	NOUN
ejpam-4302	476	5	,	,	PUNCT
ejpam-4302	476	6	15:961–970	15:961–970	PROPN
ejpam-4302	476	7	,	,	PUNCT
ejpam-4302	476	8	1965	1965	NUM
ejpam-4302	476	9	.	.	PUNCT
ejpam-4302	477	1	[	[	X
ejpam-4302	477	2	13	13	NUM
ejpam-4302	477	3	]	]	PUNCT
ejpam-4302	477	4	t.	t.	PROPN
ejpam-4302	477	5	noiri	noiri	PROPN
ejpam-4302	477	6	and	and	CCONJ
ejpam-4302	477	7	e.	e.	PROPN
ejpam-4302	477	8	hatir	hatir	PROPN
ejpam-4302	477	9	.	.	PUNCT
ejpam-4302	478	1	λsp	λsp	NOUN
ejpam-4302	478	2	-	-	PUNCT
ejpam-4302	478	3	sets	set	NOUN
ejpam-4302	478	4	and	and	CCONJ
ejpam-4302	478	5	some	some	DET
ejpam-4302	478	6	weak	weak	ADJ
ejpam-4302	478	7	separation	separation	NOUN
ejpam-4302	478	8	axioms	axiom	NOUN
ejpam-4302	478	9	.	.	PUNCT
ejpam-4302	479	1	acta	acta	PROPN
ejpam-4302	479	2	mathematica	mathematica	PROPN
ejpam-4302	479	3	hungarica	hungarica	PROPN
ejpam-4302	479	4	,	,	PUNCT
ejpam-4302	479	5	103(3):225–232	103(3):225–232	NUM
ejpam-4302	479	6	,	,	PUNCT
ejpam-4302	479	7	2004	2004	NUM
ejpam-4302	479	8	.	.	PUNCT
ejpam-4302	480	1	[	[	X
ejpam-4302	480	2	14	14	NUM
ejpam-4302	480	3	]	]	PUNCT
ejpam-4302	480	4	m.	m.	NOUN
ejpam-4302	480	5	shabir	shabir	PROPN
ejpam-4302	480	6	and	and	CCONJ
ejpam-4302	480	7	m.	m.	PROPN
ejpam-4302	480	8	naz	naz	PROPN
ejpam-4302	480	9	.	.	PUNCT
ejpam-4302	481	1	on	on	ADP
ejpam-4302	481	2	soft	soft	ADJ
ejpam-4302	481	3	topological	topological	ADJ
ejpam-4302	481	4	spaces	space	NOUN
ejpam-4302	481	5	.	.	PUNCT
ejpam-4302	482	1	computers	computer	NOUN
ejpam-4302	482	2	and	and	CCONJ
ejpam-4302	482	3	mathematics	mathematic	NOUN
ejpam-4302	482	4	with	with	ADP
ejpam-4302	482	5	applications	application	NOUN
ejpam-4302	482	6	,	,	PUNCT
ejpam-4302	482	7	61:1786–1799	61:1786–1799	NUM
ejpam-4302	482	8	,	,	PUNCT
ejpam-4302	482	9	2011	2011	NUM
ejpam-4302	482	10	.	.	PUNCT