id	sid	tid	token	lemma	pos
ejpam-4571	1	1	european	european	PROPN
ejpam-4571	1	2	journal	journal	PROPN
ejpam-4571	1	3	of	of	ADP
ejpam-4571	1	4	pure	pure	ADJ
ejpam-4571	1	5	and	and	CCONJ
ejpam-4571	1	6	applied	apply	VERB
ejpam-4571	1	7	mathematics	mathematic	NOUN
ejpam-4571	1	8	vol	vol	NOUN
ejpam-4571	1	9	.	.	PUNCT
ejpam-4571	2	1	16	16	NUM
ejpam-4571	2	2	,	,	PUNCT
ejpam-4571	2	3	no	no	INTJ
ejpam-4571	2	4	.	.	NOUN
ejpam-4571	2	5	1	1	NUM
ejpam-4571	2	6	,	,	PUNCT
ejpam-4571	2	7	2023	2023	NUM
ejpam-4571	2	8	,	,	PUNCT
ejpam-4571	2	9	29	29	NUM
ejpam-4571	2	10	-	-	SYM
ejpam-4571	2	11	43	43	NUM
ejpam-4571	2	12	issn	issn	PROPN
ejpam-4571	2	13	1307	1307	NUM
ejpam-4571	2	14	-	-	SYM
ejpam-4571	2	15	5543	5543	NUM
ejpam-4571	2	16	–	–	PUNCT
ejpam-4571	3	1	ejpam.com	ejpam.com	X
ejpam-4571	3	2	published	publish	VERB
ejpam-4571	3	3	by	by	ADP
ejpam-4571	3	4	new	new	PROPN
ejpam-4571	3	5	york	york	PROPN
ejpam-4571	3	6	business	business	PROPN
ejpam-4571	3	7	global	global	ADJ
ejpam-4571	3	8	weak	weak	ADJ
ejpam-4571	3	9	forms	form	NOUN
ejpam-4571	3	10	of	of	ADP
ejpam-4571	3	11	(	(	PUNCT
ejpam-4571	3	12	λ	λ	PROPN
ejpam-4571	3	13	,	,	PUNCT
ejpam-4571	3	14	b)-open	b)-open	VERB
ejpam-4571	3	15	sets	set	NOUN
ejpam-4571	3	16	and	and	CCONJ
ejpam-4571	3	17	weak	weak	ADJ
ejpam-4571	3	18	(	(	PUNCT
ejpam-4571	3	19	λ	λ	NOUN
ejpam-4571	3	20	,	,	PUNCT
ejpam-4571	3	21	b)-continuity	b)-continuity	NOUN
ejpam-4571	3	22	chawalit	chawalit	VERB
ejpam-4571	3	23	boonpok1	boonpok1	PROPN
ejpam-4571	3	24	,	,	PUNCT
ejpam-4571	3	25	napassanan	napassanan	NOUN
ejpam-4571	3	26	srisarakham1,∗	srisarakham1,∗	NOUN
ejpam-4571	3	27	1	1	NUM
ejpam-4571	3	28	mathematics	mathematic	NOUN
ejpam-4571	3	29	and	and	CCONJ
ejpam-4571	3	30	applied	apply	VERB
ejpam-4571	3	31	mathematics	mathematics	PROPN
ejpam-4571	3	32	research	research	NOUN
ejpam-4571	3	33	unit	unit	NOUN
ejpam-4571	3	34	,	,	PUNCT
ejpam-4571	3	35	department	department	NOUN
ejpam-4571	3	36	of	of	ADP
ejpam-4571	3	37	mathematics	mathematic	NOUN
ejpam-4571	3	38	,	,	PUNCT
ejpam-4571	3	39	faculty	faculty	NOUN
ejpam-4571	3	40	of	of	ADP
ejpam-4571	3	41	science	science	NOUN
ejpam-4571	3	42	,	,	PUNCT
ejpam-4571	3	43	mahasarakham	mahasarakham	PROPN
ejpam-4571	3	44	university	university	PROPN
ejpam-4571	3	45	,	,	PUNCT
ejpam-4571	3	46	maha	maha	PROPN
ejpam-4571	3	47	sarakham	sarakham	PROPN
ejpam-4571	3	48	,	,	PUNCT
ejpam-4571	3	49	44150	44150	NUM
ejpam-4571	3	50	,	,	PUNCT
ejpam-4571	3	51	thailand	thailand	PROPN
ejpam-4571	3	52	abstract	abstract	PROPN
ejpam-4571	3	53	.	.	PUNCT
ejpam-4571	4	1	this	this	DET
ejpam-4571	4	2	paper	paper	NOUN
ejpam-4571	4	3	is	be	AUX
ejpam-4571	4	4	concerned	concern	VERB
ejpam-4571	4	5	with	with	ADP
ejpam-4571	4	6	the	the	DET
ejpam-4571	4	7	concepts	concept	NOUN
ejpam-4571	4	8	of	of	ADP
ejpam-4571	4	9	s(λ	s(λ	PROPN
ejpam-4571	4	10	,	,	PUNCT
ejpam-4571	4	11	b)-open	b)-open	VERB
ejpam-4571	4	12	sets	set	NOUN
ejpam-4571	4	13	,	,	PUNCT
ejpam-4571	4	14	p(λ	p(λ	NOUN
ejpam-4571	4	15	,	,	PUNCT
ejpam-4571	4	16	b)-open	b)-open	PUNCT
ejpam-4571	4	17	sets	set	NOUN
ejpam-4571	4	18	,	,	PUNCT
ejpam-4571	4	19	α(λ	α(λ	PROPN
ejpam-4571	4	20	,	,	PUNCT
ejpam-4571	4	21	b)-open	b)-open	VERB
ejpam-4571	4	22	sets	set	NOUN
ejpam-4571	4	23	,	,	PUNCT
ejpam-4571	4	24	β(λ	β(λ	X
ejpam-4571	4	25	,	,	PUNCT
ejpam-4571	4	26	b)-open	b)-open	VERB
ejpam-4571	4	27	sets	set	NOUN
ejpam-4571	4	28	and	and	CCONJ
ejpam-4571	4	29	b(λ	b(λ	NOUN
ejpam-4571	4	30	,	,	PUNCT
ejpam-4571	4	31	b)-open	b)-open	PUNCT
ejpam-4571	4	32	sets	set	NOUN
ejpam-4571	4	33	.	.	PUNCT
ejpam-4571	5	1	some	some	DET
ejpam-4571	5	2	properties	property	NOUN
ejpam-4571	5	3	of	of	ADP
ejpam-4571	5	4	s(λ	s(λ	PROPN
ejpam-4571	5	5	,	,	PUNCT
ejpam-4571	5	6	b)-open	b)-open	VERB
ejpam-4571	5	7	sets	set	NOUN
ejpam-4571	5	8	,	,	PUNCT
ejpam-4571	5	9	p(λ	p(λ	NOUN
ejpam-4571	5	10	,	,	PUNCT
ejpam-4571	5	11	b)-open	b)-open	PUNCT
ejpam-4571	5	12	sets	set	NOUN
ejpam-4571	5	13	,	,	PUNCT
ejpam-4571	5	14	α(λ	α(λ	PROPN
ejpam-4571	5	15	,	,	PUNCT
ejpam-4571	5	16	b)-open	b)-open	VERB
ejpam-4571	5	17	sets	set	NOUN
ejpam-4571	5	18	,	,	PUNCT
ejpam-4571	5	19	β(λ	β(λ	X
ejpam-4571	5	20	,	,	PUNCT
ejpam-4571	5	21	b)-open	b)-open	VERB
ejpam-4571	5	22	sets	set	NOUN
ejpam-4571	5	23	and	and	CCONJ
ejpam-4571	5	24	b(λ	b(λ	NOUN
ejpam-4571	5	25	,	,	PUNCT
ejpam-4571	5	26	b)-open	b)-open	PUNCT
ejpam-4571	5	27	sets	set	NOUN
ejpam-4571	5	28	are	be	AUX
ejpam-4571	5	29	investigated	investigate	VERB
ejpam-4571	5	30	.	.	PUNCT
ejpam-4571	6	1	moreover	moreover	ADV
ejpam-4571	6	2	,	,	PUNCT
ejpam-4571	6	3	several	several	ADJ
ejpam-4571	6	4	characterizations	characterization	NOUN
ejpam-4571	6	5	of	of	ADP
ejpam-4571	6	6	weakly	weakly	ADJ
ejpam-4571	6	7	(	(	PUNCT
ejpam-4571	6	8	λ	λ	PROPN
ejpam-4571	6	9	,	,	PUNCT
ejpam-4571	6	10	b)-continuous	b)-continuous	ADJ
ejpam-4571	6	11	functions	function	NOUN
ejpam-4571	6	12	are	be	AUX
ejpam-4571	6	13	established	establish	VERB
ejpam-4571	6	14	.	.	PUNCT
ejpam-4571	7	1	2020	2020	NUM
ejpam-4571	7	2	mathematics	mathematics	PROPN
ejpam-4571	7	3	subject	subject	NOUN
ejpam-4571	7	4	classifications	classification	NOUN
ejpam-4571	7	5	:	:	PUNCT
ejpam-4571	7	6	54a05	54a05	NUM
ejpam-4571	7	7	,	,	PUNCT
ejpam-4571	7	8	54c08	54c08	NUM
ejpam-4571	7	9	,	,	PUNCT
ejpam-4571	7	10	54g05	54g05	NUM
ejpam-4571	7	11	key	key	ADJ
ejpam-4571	7	12	words	word	NOUN
ejpam-4571	7	13	and	and	CCONJ
ejpam-4571	7	14	phrases	phrase	NOUN
ejpam-4571	7	15	:	:	PUNCT
ejpam-4571	7	16	(	(	PUNCT
ejpam-4571	7	17	λ	λ	X
ejpam-4571	7	18	,	,	PUNCT
ejpam-4571	7	19	b)-open	b)-open	VERB
ejpam-4571	7	20	set	set	VERB
ejpam-4571	7	21	,	,	PUNCT
ejpam-4571	7	22	(	(	PUNCT
ejpam-4571	7	23	λ	λ	PROPN
ejpam-4571	7	24	,	,	PUNCT
ejpam-4571	7	25	b)-submaximal	b)-submaximal	ADJ
ejpam-4571	7	26	space	space	NOUN
ejpam-4571	7	27	,	,	PUNCT
ejpam-4571	7	28	(	(	PUNCT
ejpam-4571	7	29	λ	λ	NOUN
ejpam-4571	7	30	,	,	PUNCT
ejpam-4571	7	31	b)-extremally	b)-extremally	ADV
ejpam-4571	7	32	disconnected	disconnected	ADJ
ejpam-4571	7	33	space	space	NOUN
ejpam-4571	7	34	,	,	PUNCT
ejpam-4571	7	35	weakly	weakly	ADJ
ejpam-4571	7	36	(	(	PUNCT
ejpam-4571	7	37	λ	λ	PROPN
ejpam-4571	7	38	,	,	PUNCT
ejpam-4571	7	39	b)-continuous	b)-continuous	ADJ
ejpam-4571	7	40	function	function	NOUN
ejpam-4571	7	41	1	1	NUM
ejpam-4571	7	42	.	.	PUNCT
ejpam-4571	8	1	introduction	introduction	NOUN
ejpam-4571	8	2	the	the	DET
ejpam-4571	8	3	concepts	concept	NOUN
ejpam-4571	8	4	of	of	ADP
ejpam-4571	8	5	openness	openness	NOUN
ejpam-4571	8	6	and	and	CCONJ
ejpam-4571	8	7	continuity	continuity	NOUN
ejpam-4571	8	8	are	be	AUX
ejpam-4571	8	9	fundamental	fundamental	ADJ
ejpam-4571	8	10	with	with	ADP
ejpam-4571	8	11	respect	respect	NOUN
ejpam-4571	8	12	to	to	ADP
ejpam-4571	8	13	the	the	DET
ejpam-4571	8	14	investigation	investigation	NOUN
ejpam-4571	8	15	of	of	ADP
ejpam-4571	8	16	topological	topological	ADJ
ejpam-4571	8	17	spaces	space	NOUN
ejpam-4571	8	18	.	.	PUNCT
ejpam-4571	9	1	the	the	DET
ejpam-4571	9	2	notion	notion	NOUN
ejpam-4571	9	3	of	of	ADP
ejpam-4571	9	4	continuity	continuity	NOUN
ejpam-4571	9	5	is	be	AUX
ejpam-4571	9	6	one	one	NUM
ejpam-4571	9	7	of	of	ADP
ejpam-4571	9	8	the	the	DET
ejpam-4571	9	9	most	most	ADV
ejpam-4571	9	10	important	important	ADJ
ejpam-4571	9	11	tools	tool	NOUN
ejpam-4571	9	12	in	in	ADP
ejpam-4571	9	13	mathematics	mathematic	NOUN
ejpam-4571	9	14	and	and	CCONJ
ejpam-4571	9	15	many	many	ADJ
ejpam-4571	9	16	different	different	ADJ
ejpam-4571	9	17	forms	form	NOUN
ejpam-4571	9	18	of	of	ADP
ejpam-4571	9	19	generalizations	generalization	NOUN
ejpam-4571	9	20	of	of	ADP
ejpam-4571	9	21	continuity	continuity	NOUN
ejpam-4571	9	22	have	have	AUX
ejpam-4571	9	23	been	be	AUX
ejpam-4571	9	24	introduced	introduce	VERB
ejpam-4571	9	25	and	and	CCONJ
ejpam-4571	9	26	investigated	investigate	VERB
ejpam-4571	9	27	.	.	PUNCT
ejpam-4571	10	1	the	the	DET
ejpam-4571	10	2	concept	concept	NOUN
ejpam-4571	10	3	of	of	ADP
ejpam-4571	10	4	weakly	weakly	ADJ
ejpam-4571	10	5	continuous	continuous	ADJ
ejpam-4571	10	6	functions	function	NOUN
ejpam-4571	10	7	was	be	AUX
ejpam-4571	10	8	introduced	introduce	VERB
ejpam-4571	10	9	by	by	ADP
ejpam-4571	10	10	levine	levine	PROPN
ejpam-4571	10	11	[	[	X
ejpam-4571	10	12	10	10	NUM
ejpam-4571	10	13	]	]	PUNCT
ejpam-4571	10	14	.	.	PUNCT
ejpam-4571	11	1	rose	rise	VERB
ejpam-4571	12	1	[	[	X
ejpam-4571	12	2	13	13	NUM
ejpam-4571	12	3	]	]	PUNCT
ejpam-4571	12	4	introduced	introduce	VERB
ejpam-4571	12	5	the	the	DET
ejpam-4571	12	6	notion	notion	NOUN
ejpam-4571	12	7	of	of	ADP
ejpam-4571	12	8	subweakly	subweakly	ADJ
ejpam-4571	12	9	continuous	continuous	ADJ
ejpam-4571	12	10	functions	function	NOUN
ejpam-4571	12	11	and	and	CCONJ
ejpam-4571	12	12	investigated	investigate	VERB
ejpam-4571	12	13	the	the	DET
ejpam-4571	12	14	relationships	relationship	NOUN
ejpam-4571	12	15	between	between	ADP
ejpam-4571	12	16	subweak	subweak	NOUN
ejpam-4571	12	17	continuity	continuity	NOUN
ejpam-4571	12	18	and	and	CCONJ
ejpam-4571	12	19	weak	weak	ADJ
ejpam-4571	12	20	continuity	continuity	NOUN
ejpam-4571	12	21	.	.	PUNCT
ejpam-4571	13	1	in	in	ADP
ejpam-4571	13	2	[	[	X
ejpam-4571	13	3	4	4	NUM
ejpam-4571	13	4	]	]	PUNCT
ejpam-4571	13	5	has	have	AUX
ejpam-4571	13	6	obtained	obtain	VERB
ejpam-4571	13	7	further	further	ADJ
ejpam-4571	13	8	properties	property	NOUN
ejpam-4571	13	9	of	of	ADP
ejpam-4571	13	10	subweakly	subweakly	ADJ
ejpam-4571	13	11	continuous	continuous	ADJ
ejpam-4571	13	12	functions	function	NOUN
ejpam-4571	13	13	.	.	PUNCT
ejpam-4571	14	1	in	in	ADP
ejpam-4571	14	2	1970	1970	NUM
ejpam-4571	14	3	,	,	PUNCT
ejpam-4571	14	4	willard	willard	PROPN
ejpam-4571	14	5	[	[	X
ejpam-4571	14	6	16	16	NUM
ejpam-4571	14	7	]	]	PUNCT
ejpam-4571	14	8	introduced	introduce	VERB
ejpam-4571	14	9	the	the	DET
ejpam-4571	14	10	concept	concept	NOUN
ejpam-4571	14	11	of	of	ADP
ejpam-4571	14	12	extremally	extremally	ADV
ejpam-4571	14	13	disconnected	disconnected	ADJ
ejpam-4571	14	14	spaces	space	NOUN
ejpam-4571	14	15	.	.	PUNCT
ejpam-4571	15	1	extremally	extremally	ADV
ejpam-4571	15	2	disconnected	disconnect	VERB
ejpam-4571	15	3	spaces	space	NOUN
ejpam-4571	15	4	play	play	VERB
ejpam-4571	15	5	a	a	DET
ejpam-4571	15	6	prominent	prominent	ADJ
ejpam-4571	15	7	role	role	NOUN
ejpam-4571	15	8	in	in	ADP
ejpam-4571	15	9	set	set	NOUN
ejpam-4571	15	10	-	-	PUNCT
ejpam-4571	15	11	theoretical	theoretical	ADJ
ejpam-4571	15	12	topology	topology	NOUN
ejpam-4571	15	13	.	.	PUNCT
ejpam-4571	16	1	due	due	ADP
ejpam-4571	16	2	to	to	ADP
ejpam-4571	16	3	their	their	PRON
ejpam-4571	16	4	peculiar	peculiar	ADJ
ejpam-4571	16	5	properties	property	NOUN
ejpam-4571	16	6	extremally	extremally	ADV
ejpam-4571	16	7	disconnected	disconnect	VERB
ejpam-4571	16	8	spaces	space	NOUN
ejpam-4571	16	9	provide	provide	VERB
ejpam-4571	16	10	crucial	crucial	ADJ
ejpam-4571	16	11	applications	application	NOUN
ejpam-4571	16	12	in	in	ADP
ejpam-4571	16	13	the	the	DET
ejpam-4571	16	14	theory	theory	NOUN
ejpam-4571	16	15	of	of	ADP
ejpam-4571	16	16	boolean	boolean	ADJ
ejpam-4571	16	17	algebra	algebra	NOUN
ejpam-4571	16	18	,	,	PUNCT
ejpam-4571	16	19	in	in	ADP
ejpam-4571	16	20	axiomatic	axiomatic	ADJ
ejpam-4571	16	21	set	set	NOUN
ejpam-4571	16	22	theory	theory	NOUN
ejpam-4571	16	23	and	and	CCONJ
ejpam-4571	16	24	in	in	ADP
ejpam-4571	16	25	some	some	DET
ejpam-4571	16	26	branches	branch	NOUN
ejpam-4571	16	27	of	of	ADP
ejpam-4571	16	28	functional	functional	ADJ
ejpam-4571	16	29	analysis	analysis	NOUN
ejpam-4571	16	30	.	.	PUNCT
ejpam-4571	17	1	sivaraj	sivaraj	PROPN
ejpam-4571	17	2	[	[	X
ejpam-4571	17	3	14	14	NUM
ejpam-4571	17	4	]	]	PUNCT
ejpam-4571	17	5	investigated	investigate	VERB
ejpam-4571	17	6	some	some	DET
ejpam-4571	17	7	characterizations	characterization	NOUN
ejpam-4571	17	8	of	of	ADP
ejpam-4571	17	9	extremally	extremally	ADV
ejpam-4571	17	10	disconnected	disconnect	VERB
ejpam-4571	17	11	spaces	space	NOUN
ejpam-4571	17	12	by	by	ADP
ejpam-4571	17	13	utilizing	utilize	VERB
ejpam-4571	17	14	semi	semi	ADJ
ejpam-4571	17	15	-	-	ADJ
ejpam-4571	17	16	open	open	ADJ
ejpam-4571	17	17	sets	set	NOUN
ejpam-4571	17	18	due	due	ADP
ejpam-4571	17	19	to	to	ADP
ejpam-4571	17	20	levine	levine	PROPN
ejpam-4571	17	21	[	[	X
ejpam-4571	17	22	11	11	NUM
ejpam-4571	17	23	]	]	PUNCT
ejpam-4571	17	24	.	.	PUNCT
ejpam-4571	18	1	the	the	DET
ejpam-4571	18	2	concept	concept	NOUN
ejpam-4571	18	3	of	of	ADP
ejpam-4571	18	4	submaximality	submaximality	NOUN
ejpam-4571	18	5	of	of	ADP
ejpam-4571	18	6	generalized	generalized	ADJ
ejpam-4571	18	7	topological	topological	ADJ
ejpam-4571	18	8	spaces	space	NOUN
ejpam-4571	18	9	was	be	AUX
ejpam-4571	18	10	introduced	introduce	VERB
ejpam-4571	18	11	by	by	ADP
ejpam-4571	18	12	hewitt	hewitt	PROPN
ejpam-4571	19	1	[	[	X
ejpam-4571	19	2	9	9	NUM
ejpam-4571	19	3	]	]	PUNCT
ejpam-4571	19	4	.	.	PUNCT
ejpam-4571	20	1	he	he	PRON
ejpam-4571	20	2	discovered	discover	VERB
ejpam-4571	20	3	a	a	DET
ejpam-4571	20	4	general	general	ADJ
ejpam-4571	20	5	way	way	NOUN
ejpam-4571	20	6	of	of	ADP
ejpam-4571	20	7	constructing	construct	VERB
ejpam-4571	20	8	maximal	maximal	ADJ
ejpam-4571	20	9	topologies	topology	NOUN
ejpam-4571	20	10	.	.	PUNCT
ejpam-4571	21	1	the	the	DET
ejpam-4571	21	2	first	first	ADJ
ejpam-4571	21	3	systematic	systematic	ADJ
ejpam-4571	21	4	study	study	NOUN
ejpam-4571	21	5	of	of	ADP
ejpam-4571	21	6	submaximal	submaximal	ADJ
ejpam-4571	21	7	spaces	space	NOUN
ejpam-4571	21	8	was	be	AUX
ejpam-4571	21	9	undertaken	undertake	VERB
ejpam-4571	21	10	in	in	ADP
ejpam-4571	21	11	the	the	DET
ejpam-4571	21	12	paper	paper	NOUN
ejpam-4571	21	13	of	of	ADP
ejpam-4571	21	14	arhangel’skĭı	arhangel’skĭı	PROPN
ejpam-4571	21	15	and	and	CCONJ
ejpam-4571	21	16	collins	collin	NOUN
ejpam-4571	21	17	[	[	X
ejpam-4571	21	18	3	3	NUM
ejpam-4571	21	19	]	]	PUNCT
ejpam-4571	21	20	.	.	PUNCT
ejpam-4571	22	1	they	they	PRON
ejpam-4571	22	2	gave	give	VERB
ejpam-4571	22	3	various	various	ADJ
ejpam-4571	22	4	necessary	necessary	ADJ
ejpam-4571	22	5	and	and	CCONJ
ejpam-4571	22	6	suffient	suffient	ADJ
ejpam-4571	22	7	conditions	condition	NOUN
ejpam-4571	22	8	for	for	ADP
ejpam-4571	22	9	a	a	DET
ejpam-4571	22	10	space	space	NOUN
ejpam-4571	22	11	to	to	PART
ejpam-4571	22	12	be	be	AUX
ejpam-4571	22	13	submaximal	submaximal	ADJ
ejpam-4571	22	14	and	and	CCONJ
ejpam-4571	22	15	showed	show	VERB
ejpam-4571	22	16	that	that	SCONJ
ejpam-4571	22	17	every	every	DET
ejpam-4571	22	18	submaximal	submaximal	ADJ
ejpam-4571	22	19	space	space	NOUN
ejpam-4571	22	20	is	be	AUX
ejpam-4571	22	21	left	leave	VERB
ejpam-4571	22	22	-	-	PUNCT
ejpam-4571	22	23	separated	separate	VERB
ejpam-4571	22	24	.	.	PUNCT
ejpam-4571	23	1	in	in	ADP
ejpam-4571	23	2	1996	1996	NUM
ejpam-4571	23	3	,	,	PUNCT
ejpam-4571	23	4	andrijević	andrijević	VERB
ejpam-4571	23	5	[	[	X
ejpam-4571	23	6	2	2	NUM
ejpam-4571	23	7	]	]	PUNCT
ejpam-4571	23	8	introduced	introduce	VERB
ejpam-4571	23	9	a	a	DET
ejpam-4571	23	10	new	new	ADJ
ejpam-4571	23	11	class	class	NOUN
ejpam-4571	23	12	of	of	ADP
ejpam-4571	23	13	generalized	generalized	ADJ
ejpam-4571	23	14	open	open	ADJ
ejpam-4571	23	15	sets	set	NOUN
ejpam-4571	23	16	called	call	VERB
ejpam-4571	23	17	b	b	NOUN
ejpam-4571	23	18	-	-	PUNCT
ejpam-4571	23	19	open	open	ADJ
ejpam-4571	23	20	sets	set	NOUN
ejpam-4571	23	21	into	into	ADP
ejpam-4571	23	22	the	the	DET
ejpam-4571	23	23	∗corresponding	∗corresponde	VERB
ejpam-4571	23	24	author	author	NOUN
ejpam-4571	23	25	.	.	PUNCT
ejpam-4571	24	1	doi	doi	NOUN
ejpam-4571	24	2	:	:	PUNCT
ejpam-4571	24	3	https://doi.org/10.29020/nybg.ejpam.v16i1.4571	https://doi.org/10.29020/nybg.ejpam.v16i1.4571	NUM
ejpam-4571	24	4	email	email	NOUN
ejpam-4571	24	5	addresses	address	NOUN
ejpam-4571	24	6	:	:	PUNCT
ejpam-4571	25	1	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4571	25	2	(	(	PUNCT
ejpam-4571	25	3	c.	c.	PROPN
ejpam-4571	25	4	boonpok	boonpok	PROPN
ejpam-4571	25	5	)	)	PUNCT
ejpam-4571	25	6	,	,	PUNCT
ejpam-4571	25	7	napassanan.sri@msu.ac.th	napassanan.sri@msu.ac.th	PRON
ejpam-4571	25	8	(	(	PUNCT
ejpam-4571	25	9	n.	n.	PROPN
ejpam-4571	25	10	srisarakham	srisarakham	PROPN
ejpam-4571	25	11	)	)	PUNCT
ejpam-4571	25	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4571	26	1	29	29	NUM
ejpam-4571	27	1	©	©	ADP
ejpam-4571	27	2	2023	2023	NUM
ejpam-4571	27	3	ejpam	ejpam	NOUN
ejpam-4571	27	4	all	all	DET
ejpam-4571	27	5	rights	right	NOUN
ejpam-4571	27	6	reserved	reserve	VERB
ejpam-4571	27	7	.	.	PUNCT
ejpam-4571	28	1	c.	c.	PROPN
ejpam-4571	28	2	boonpok	boonpok	PROPN
ejpam-4571	28	3	,	,	PUNCT
ejpam-4571	28	4	n.	n.	PROPN
ejpam-4571	28	5	srisarakham	srisarakham	PROPN
ejpam-4571	28	6	/	/	SYM
ejpam-4571	28	7	eur	eur	PROPN
ejpam-4571	28	8	.	.	PUNCT
ejpam-4571	29	1	j.	j.	PROPN
ejpam-4571	29	2	pure	pure	PROPN
ejpam-4571	29	3	appl	appl	PROPN
ejpam-4571	29	4	.	.	PROPN
ejpam-4571	29	5	math	math	PROPN
ejpam-4571	29	6	,	,	PUNCT
ejpam-4571	29	7	16	16	NUM
ejpam-4571	29	8	(	(	PUNCT
ejpam-4571	29	9	1	1	NUM
ejpam-4571	29	10	)	)	PUNCT
ejpam-4571	29	11	(	(	PUNCT
ejpam-4571	29	12	2023	2023	NUM
ejpam-4571	29	13	)	)	PUNCT
ejpam-4571	29	14	,	,	PUNCT
ejpam-4571	29	15	29	29	NUM
ejpam-4571	29	16	-	-	SYM
ejpam-4571	29	17	43	43	NUM
ejpam-4571	29	18	30	30	NUM
ejpam-4571	29	19	field	field	NOUN
ejpam-4571	29	20	of	of	ADP
ejpam-4571	29	21	topology	topology	NOUN
ejpam-4571	29	22	.	.	PUNCT
ejpam-4571	30	1	this	this	DET
ejpam-4571	30	2	class	class	NOUN
ejpam-4571	30	3	is	be	AUX
ejpam-4571	30	4	a	a	DET
ejpam-4571	30	5	subset	subset	NOUN
ejpam-4571	30	6	of	of	ADP
ejpam-4571	30	7	the	the	DET
ejpam-4571	30	8	class	class	NOUN
ejpam-4571	30	9	of	of	ADP
ejpam-4571	30	10	semi	semi	ADJ
ejpam-4571	30	11	-	-	ADJ
ejpam-4571	30	12	preopen	preopen	ADJ
ejpam-4571	30	13	sets	set	NOUN
ejpam-4571	30	14	[	[	X
ejpam-4571	30	15	1	1	NUM
ejpam-4571	30	16	]	]	PUNCT
ejpam-4571	30	17	,	,	PUNCT
ejpam-4571	30	18	i.e.	i.e.	X
ejpam-4571	30	19	a	a	DET
ejpam-4571	30	20	subset	subset	NOUN
ejpam-4571	30	21	of	of	ADP
ejpam-4571	30	22	a	a	DET
ejpam-4571	30	23	topological	topological	ADJ
ejpam-4571	30	24	space	space	NOUN
ejpam-4571	30	25	which	which	PRON
ejpam-4571	30	26	is	be	AUX
ejpam-4571	30	27	contained	contain	VERB
ejpam-4571	30	28	in	in	ADP
ejpam-4571	30	29	the	the	DET
ejpam-4571	30	30	closure	closure	NOUN
ejpam-4571	30	31	of	of	ADP
ejpam-4571	30	32	the	the	DET
ejpam-4571	30	33	interior	interior	NOUN
ejpam-4571	30	34	of	of	ADP
ejpam-4571	30	35	its	its	PRON
ejpam-4571	30	36	closure	closure	NOUN
ejpam-4571	30	37	.	.	PUNCT
ejpam-4571	31	1	also	also	ADV
ejpam-4571	31	2	the	the	DET
ejpam-4571	31	3	class	class	NOUN
ejpam-4571	31	4	of	of	ADP
ejpam-4571	31	5	b	b	NOUN
ejpam-4571	31	6	-	-	PUNCT
ejpam-4571	31	7	open	open	ADJ
ejpam-4571	31	8	sets	set	NOUN
ejpam-4571	31	9	in	in	ADP
ejpam-4571	31	10	a	a	DET
ejpam-4571	31	11	superset	superset	NOUN
ejpam-4571	31	12	of	of	ADP
ejpam-4571	31	13	the	the	DET
ejpam-4571	31	14	class	class	NOUN
ejpam-4571	31	15	of	of	ADP
ejpam-4571	31	16	semi	semi	ADJ
ejpam-4571	31	17	-	-	ADJ
ejpam-4571	31	18	open	open	ADJ
ejpam-4571	31	19	sets	set	NOUN
ejpam-4571	31	20	[	[	X
ejpam-4571	31	21	11	11	NUM
ejpam-4571	31	22	]	]	PUNCT
ejpam-4571	31	23	,	,	PUNCT
ejpam-4571	31	24	i.e.	i.e.	X
ejpam-4571	31	25	a	a	DET
ejpam-4571	31	26	set	set	NOUN
ejpam-4571	31	27	which	which	PRON
ejpam-4571	31	28	is	be	AUX
ejpam-4571	31	29	contained	contain	VERB
ejpam-4571	31	30	in	in	ADP
ejpam-4571	31	31	the	the	DET
ejpam-4571	31	32	closure	closure	NOUN
ejpam-4571	31	33	of	of	ADP
ejpam-4571	31	34	its	its	PRON
ejpam-4571	31	35	interior	interior	NOUN
ejpam-4571	31	36	and	and	CCONJ
ejpam-4571	31	37	the	the	DET
ejpam-4571	31	38	class	class	NOUN
ejpam-4571	31	39	of	of	ADP
ejpam-4571	31	40	locally	locally	ADV
ejpam-4571	31	41	dense	dense	ADJ
ejpam-4571	31	42	set	set	NOUN
ejpam-4571	31	43	[	[	X
ejpam-4571	31	44	8	8	NUM
ejpam-4571	31	45	]	]	PUNCT
ejpam-4571	31	46	or	or	CCONJ
ejpam-4571	31	47	preopen	preopen	ADJ
ejpam-4571	31	48	sets	set	NOUN
ejpam-4571	31	49	[	[	X
ejpam-4571	31	50	12	12	NUM
ejpam-4571	31	51	]	]	PUNCT
ejpam-4571	31	52	,	,	PUNCT
ejpam-4571	31	53	i.e.	i.e.	X
ejpam-4571	31	54	a	a	DET
ejpam-4571	31	55	set	set	NOUN
ejpam-4571	31	56	which	which	PRON
ejpam-4571	31	57	is	be	AUX
ejpam-4571	31	58	contained	contain	VERB
ejpam-4571	31	59	in	in	ADP
ejpam-4571	31	60	the	the	DET
ejpam-4571	31	61	interior	interior	NOUN
ejpam-4571	31	62	of	of	ADP
ejpam-4571	31	63	its	its	PRON
ejpam-4571	31	64	closure	closure	NOUN
ejpam-4571	31	65	.	.	PUNCT
ejpam-4571	32	1	moreover	moreover	ADV
ejpam-4571	32	2	,	,	PUNCT
ejpam-4571	32	3	andrijević	andrijević	VERB
ejpam-4571	32	4	studied	study	VERB
ejpam-4571	32	5	several	several	ADJ
ejpam-4571	32	6	fundamental	fundamental	ADJ
ejpam-4571	32	7	and	and	CCONJ
ejpam-4571	32	8	interesting	interesting	ADJ
ejpam-4571	32	9	properties	property	NOUN
ejpam-4571	32	10	of	of	ADP
ejpam-4571	32	11	b	b	NOUN
ejpam-4571	32	12	-	-	PUNCT
ejpam-4571	32	13	open	open	ADJ
ejpam-4571	32	14	sets	set	NOUN
ejpam-4571	32	15	.	.	PUNCT
ejpam-4571	33	1	caldas	caldas	PROPN
ejpam-4571	33	2	et	et	PROPN
ejpam-4571	33	3	al	al	PROPN
ejpam-4571	33	4	.	.	PUNCT
ejpam-4571	34	1	[	[	X
ejpam-4571	34	2	7	7	X
ejpam-4571	34	3	]	]	PUNCT
ejpam-4571	34	4	introduced	introduce	VERB
ejpam-4571	34	5	the	the	DET
ejpam-4571	34	6	concept	concept	NOUN
ejpam-4571	34	7	of	of	ADP
ejpam-4571	34	8	λb	λb	NOUN
ejpam-4571	34	9	-	-	PUNCT
ejpam-4571	34	10	sets	set	NOUN
ejpam-4571	34	11	which	which	PRON
ejpam-4571	34	12	is	be	AUX
ejpam-4571	34	13	the	the	DET
ejpam-4571	34	14	intersection	intersection	NOUN
ejpam-4571	34	15	of	of	ADP
ejpam-4571	34	16	b	b	NOUN
ejpam-4571	34	17	-	-	PUNCT
ejpam-4571	34	18	open	open	ADJ
ejpam-4571	34	19	sets	set	NOUN
ejpam-4571	34	20	and	and	CCONJ
ejpam-4571	34	21	investigated	investigate	VERB
ejpam-4571	34	22	the	the	DET
ejpam-4571	34	23	fundamental	fundamental	ADJ
ejpam-4571	34	24	properties	property	NOUN
ejpam-4571	34	25	of	of	ADP
ejpam-4571	34	26	λb	λb	NOUN
ejpam-4571	34	27	-	-	PUNCT
ejpam-4571	34	28	sets	set	NOUN
ejpam-4571	34	29	and	and	CCONJ
ejpam-4571	34	30	vb	vb	NOUN
ejpam-4571	34	31	-	-	PUNCT
ejpam-4571	34	32	sets	set	NOUN
ejpam-4571	34	33	.	.	PUNCT
ejpam-4571	35	1	in	in	ADP
ejpam-4571	35	2	[	[	X
ejpam-4571	35	3	5	5	NUM
ejpam-4571	35	4	]	]	PUNCT
ejpam-4571	35	5	,	,	PUNCT
ejpam-4571	35	6	the	the	DET
ejpam-4571	35	7	present	present	ADJ
ejpam-4571	35	8	authors	author	NOUN
ejpam-4571	35	9	introduced	introduce	VERB
ejpam-4571	35	10	and	and	CCONJ
ejpam-4571	35	11	studied	study	VERB
ejpam-4571	35	12	the	the	DET
ejpam-4571	35	13	notions	notion	NOUN
ejpam-4571	35	14	of	of	ADP
ejpam-4571	35	15	(	(	PUNCT
ejpam-4571	35	16	λ	λ	PROPN
ejpam-4571	35	17	,	,	PUNCT
ejpam-4571	35	18	b)-closed	b)-close	VERB
ejpam-4571	35	19	sets	set	NOUN
ejpam-4571	35	20	and	and	CCONJ
ejpam-4571	35	21	(	(	PUNCT
ejpam-4571	35	22	λ	λ	NOUN
ejpam-4571	35	23	,	,	PUNCT
ejpam-4571	35	24	b)-open	b)-open	VERB
ejpam-4571	35	25	sets	set	NOUN
ejpam-4571	35	26	.	.	PUNCT
ejpam-4571	36	1	the	the	DET
ejpam-4571	36	2	notions	notion	NOUN
ejpam-4571	36	3	of	of	ADP
ejpam-4571	36	4	(	(	PUNCT
ejpam-4571	36	5	λ	λ	PROPN
ejpam-4571	36	6	,	,	PUNCT
ejpam-4571	36	7	b)-extremally	b)-extremally	ADV
ejpam-4571	36	8	disconnected	disconnected	ADJ
ejpam-4571	36	9	spaces	space	NOUN
ejpam-4571	36	10	,	,	PUNCT
ejpam-4571	36	11	(	(	PUNCT
ejpam-4571	36	12	λ	λ	X
ejpam-4571	36	13	,	,	PUNCT
ejpam-4571	36	14	b)-hyperconnected	b)-hyperconnecte	VERB
ejpam-4571	36	15	spaces	space	NOUN
ejpam-4571	36	16	and	and	CCONJ
ejpam-4571	36	17	(	(	PUNCT
ejpam-4571	36	18	λ	λ	NOUN
ejpam-4571	36	19	,	,	PUNCT
ejpam-4571	36	20	b)submaximal	b)submaximal	NOUN
ejpam-4571	36	21	spaces	space	NOUN
ejpam-4571	36	22	were	be	AUX
ejpam-4571	36	23	introduced	introduce	VERB
ejpam-4571	36	24	by	by	ADP
ejpam-4571	36	25	viriyapong	viriyapong	PROPN
ejpam-4571	36	26	and	and	CCONJ
ejpam-4571	36	27	boonpok	boonpok	VERB
ejpam-4571	37	1	[	[	X
ejpam-4571	37	2	15	15	NUM
ejpam-4571	37	3	]	]	PUNCT
ejpam-4571	37	4	.	.	PUNCT
ejpam-4571	38	1	recently	recently	ADV
ejpam-4571	38	2	,	,	PUNCT
ejpam-4571	38	3	boonpok	boonpok	PROPN
ejpam-4571	38	4	and	and	CCONJ
ejpam-4571	38	5	viriyapong	viriyapong	ADJ
ejpam-4571	39	1	[	[	X
ejpam-4571	39	2	6	6	NUM
ejpam-4571	39	3	]	]	PUNCT
ejpam-4571	39	4	introduced	introduce	VERB
ejpam-4571	39	5	and	and	CCONJ
ejpam-4571	39	6	studied	study	VERB
ejpam-4571	39	7	the	the	DET
ejpam-4571	39	8	notion	notion	NOUN
ejpam-4571	39	9	of	of	ADP
ejpam-4571	39	10	weakly	weakly	ADJ
ejpam-4571	39	11	(	(	PUNCT
ejpam-4571	39	12	λ	λ	PROPN
ejpam-4571	39	13	,	,	PUNCT
ejpam-4571	39	14	p)-continuous	p)-continuous	ADJ
ejpam-4571	39	15	functions	function	NOUN
ejpam-4571	39	16	.	.	PUNCT
ejpam-4571	40	1	in	in	ADP
ejpam-4571	40	2	this	this	DET
ejpam-4571	40	3	paper	paper	NOUN
ejpam-4571	40	4	,	,	PUNCT
ejpam-4571	40	5	we	we	PRON
ejpam-4571	40	6	introduce	introduce	VERB
ejpam-4571	40	7	new	new	ADJ
ejpam-4571	40	8	classes	class	NOUN
ejpam-4571	40	9	of	of	ADP
ejpam-4571	40	10	sets	set	NOUN
ejpam-4571	40	11	called	call	VERB
ejpam-4571	40	12	s(λ	s(λ	NOUN
ejpam-4571	40	13	,	,	PUNCT
ejpam-4571	40	14	b)-open	b)-open	VERB
ejpam-4571	40	15	sets	set	NOUN
ejpam-4571	40	16	,	,	PUNCT
ejpam-4571	40	17	p(λ	p(λ	NOUN
ejpam-4571	40	18	,	,	PUNCT
ejpam-4571	40	19	b)-open	b)-open	PUNCT
ejpam-4571	40	20	sets	set	NOUN
ejpam-4571	40	21	,	,	PUNCT
ejpam-4571	40	22	α(λ	α(λ	PROPN
ejpam-4571	40	23	,	,	PUNCT
ejpam-4571	40	24	b)-open	b)-open	VERB
ejpam-4571	40	25	sets	set	NOUN
ejpam-4571	40	26	,	,	PUNCT
ejpam-4571	40	27	β(λ	β(λ	X
ejpam-4571	40	28	,	,	PUNCT
ejpam-4571	40	29	b)-open	b)-open	VERB
ejpam-4571	40	30	sets	set	NOUN
ejpam-4571	40	31	and	and	CCONJ
ejpam-4571	40	32	b(λ	b(λ	NOUN
ejpam-4571	40	33	,	,	PUNCT
ejpam-4571	40	34	b)-open	b)-open	PUNCT
ejpam-4571	40	35	sets	set	NOUN
ejpam-4571	40	36	.	.	PUNCT
ejpam-4571	41	1	we	we	PRON
ejpam-4571	41	2	also	also	ADV
ejpam-4571	41	3	investigate	investigate	VERB
ejpam-4571	41	4	some	some	PRON
ejpam-4571	41	5	of	of	ADP
ejpam-4571	41	6	their	their	PRON
ejpam-4571	41	7	fundamental	fundamental	ADJ
ejpam-4571	41	8	properties	property	NOUN
ejpam-4571	41	9	.	.	PUNCT
ejpam-4571	42	1	furthermore	furthermore	ADV
ejpam-4571	42	2	,	,	PUNCT
ejpam-4571	42	3	we	we	PRON
ejpam-4571	42	4	investigate	investigate	VERB
ejpam-4571	42	5	several	several	ADJ
ejpam-4571	42	6	characterizations	characterization	NOUN
ejpam-4571	42	7	of	of	ADP
ejpam-4571	42	8	(	(	PUNCT
ejpam-4571	42	9	λ	λ	PROPN
ejpam-4571	42	10	,	,	PUNCT
ejpam-4571	42	11	b)-submaximal	b)-submaximal	ADJ
ejpam-4571	42	12	spaces	space	NOUN
ejpam-4571	42	13	and	and	CCONJ
ejpam-4571	42	14	(	(	PUNCT
ejpam-4571	42	15	λ	λ	PROPN
ejpam-4571	42	16	,	,	PUNCT
ejpam-4571	42	17	b)-extremally	b)-extremally	ADV
ejpam-4571	42	18	disconnected	disconnected	ADJ
ejpam-4571	42	19	spaces	space	NOUN
ejpam-4571	42	20	by	by	ADP
ejpam-4571	42	21	utilizing	utilize	VERB
ejpam-4571	42	22	the	the	DET
ejpam-4571	42	23	notion	notion	NOUN
ejpam-4571	42	24	of	of	ADP
ejpam-4571	42	25	(	(	PUNCT
ejpam-4571	42	26	λ	λ	PROPN
ejpam-4571	42	27	,	,	PUNCT
ejpam-4571	42	28	b)-open	b)-open	VERB
ejpam-4571	42	29	sets	set	NOUN
ejpam-4571	42	30	.	.	PUNCT
ejpam-4571	43	1	in	in	ADP
ejpam-4571	43	2	particular	particular	ADJ
ejpam-4571	43	3	,	,	PUNCT
ejpam-4571	43	4	some	some	DET
ejpam-4571	43	5	characterizations	characterization	NOUN
ejpam-4571	43	6	of	of	ADP
ejpam-4571	43	7	weakly	weakly	ADJ
ejpam-4571	43	8	(	(	PUNCT
ejpam-4571	43	9	λ	λ	PROPN
ejpam-4571	43	10	,	,	PUNCT
ejpam-4571	43	11	b)-continuous	b)-continuous	ADJ
ejpam-4571	43	12	functions	function	NOUN
ejpam-4571	43	13	are	be	AUX
ejpam-4571	43	14	discussed	discuss	VERB
ejpam-4571	43	15	.	.	PUNCT
ejpam-4571	44	1	2	2	X
ejpam-4571	44	2	.	.	X
ejpam-4571	44	3	preliminaries	preliminary	NOUN
ejpam-4571	44	4	throughout	throughout	ADP
ejpam-4571	44	5	this	this	DET
ejpam-4571	44	6	paper	paper	NOUN
ejpam-4571	44	7	,	,	PUNCT
ejpam-4571	44	8	the	the	DET
ejpam-4571	44	9	spaces	space	NOUN
ejpam-4571	44	10	(	(	PUNCT
ejpam-4571	44	11	x	x	X
ejpam-4571	44	12	,	,	PUNCT
ejpam-4571	44	13	τ	τ	X
ejpam-4571	44	14	)	)	PUNCT
ejpam-4571	44	15	and	and	CCONJ
ejpam-4571	44	16	(	(	PUNCT
ejpam-4571	44	17	x	x	X
ejpam-4571	44	18	,	,	PUNCT
ejpam-4571	44	19	σ	σ	PROPN
ejpam-4571	44	20	)	)	PUNCT
ejpam-4571	44	21	(	(	PUNCT
ejpam-4571	44	22	or	or	CCONJ
ejpam-4571	44	23	simply	simply	ADV
ejpam-4571	44	24	x	x	X
ejpam-4571	44	25	and	and	CCONJ
ejpam-4571	44	26	y	y	PROPN
ejpam-4571	44	27	)	)	PUNCT
ejpam-4571	44	28	always	always	ADV
ejpam-4571	44	29	means	mean	VERB
ejpam-4571	44	30	topological	topological	ADJ
ejpam-4571	44	31	spaces	space	NOUN
ejpam-4571	44	32	on	on	ADP
ejpam-4571	44	33	which	which	PRON
ejpam-4571	44	34	no	no	DET
ejpam-4571	44	35	separation	separation	NOUN
ejpam-4571	44	36	axioms	axiom	NOUN
ejpam-4571	44	37	are	be	AUX
ejpam-4571	44	38	assumed	assume	VERB
ejpam-4571	44	39	unless	unless	SCONJ
ejpam-4571	44	40	explicitly	explicitly	ADV
ejpam-4571	44	41	stated	state	VERB
ejpam-4571	44	42	.	.	PUNCT
ejpam-4571	45	1	for	for	ADP
ejpam-4571	45	2	a	a	DET
ejpam-4571	45	3	subset	subset	NOUN
ejpam-4571	45	4	a	a	PRON
ejpam-4571	45	5	of	of	ADP
ejpam-4571	45	6	x	x	PRON
ejpam-4571	45	7	,	,	PUNCT
ejpam-4571	45	8	the	the	DET
ejpam-4571	45	9	closure	closure	NOUN
ejpam-4571	45	10	of	of	ADP
ejpam-4571	45	11	a	a	PRON
ejpam-4571	45	12	and	and	CCONJ
ejpam-4571	45	13	the	the	DET
ejpam-4571	45	14	interior	interior	NOUN
ejpam-4571	45	15	of	of	ADP
ejpam-4571	45	16	a	a	PRON
ejpam-4571	45	17	in	in	ADP
ejpam-4571	45	18	(	(	PUNCT
ejpam-4571	45	19	x	x	NOUN
ejpam-4571	45	20	,	,	PUNCT
ejpam-4571	45	21	τ	τ	X
ejpam-4571	45	22	)	)	PUNCT
ejpam-4571	45	23	are	be	AUX
ejpam-4571	45	24	denoted	denote	VERB
ejpam-4571	45	25	by	by	ADP
ejpam-4571	45	26	cl(a	cl(a	NOUN
ejpam-4571	45	27	)	)	PUNCT
ejpam-4571	45	28	and	and	CCONJ
ejpam-4571	45	29	int(a	int(a	PROPN
ejpam-4571	45	30	)	)	PUNCT
ejpam-4571	45	31	,	,	PUNCT
ejpam-4571	45	32	respectively	respectively	ADV
ejpam-4571	45	33	.	.	PUNCT
ejpam-4571	46	1	a	a	DET
ejpam-4571	46	2	subset	subset	NOUN
ejpam-4571	46	3	a	a	PRON
ejpam-4571	46	4	of	of	ADP
ejpam-4571	46	5	a	a	DET
ejpam-4571	46	6	topological	topological	ADJ
ejpam-4571	46	7	space	space	NOUN
ejpam-4571	46	8	(	(	PUNCT
ejpam-4571	46	9	x	x	X
ejpam-4571	46	10	,	,	PUNCT
ejpam-4571	46	11	τ	τ	X
ejpam-4571	46	12	)	)	PUNCT
ejpam-4571	46	13	is	be	AUX
ejpam-4571	46	14	called	call	VERB
ejpam-4571	46	15	b	b	NOUN
ejpam-4571	46	16	-	-	PUNCT
ejpam-4571	46	17	open	open	ADJ
ejpam-4571	46	18	[	[	X
ejpam-4571	46	19	2	2	NUM
ejpam-4571	46	20	]	]	PUNCT
ejpam-4571	46	21	if	if	SCONJ
ejpam-4571	46	22	a	a	DET
ejpam-4571	46	23	⊆	⊆	NUM
ejpam-4571	46	24	cl(int(a	cl(int(a	NOUN
ejpam-4571	46	25	)	)	PUNCT
ejpam-4571	46	26	)	)	PUNCT
ejpam-4571	46	27	∪	∪	ADP
ejpam-4571	46	28	int(cl(a	int(cl(a	PROPN
ejpam-4571	46	29	)	)	PUNCT
ejpam-4571	46	30	)	)	PUNCT
ejpam-4571	46	31	.	.	PUNCT
ejpam-4571	47	1	the	the	DET
ejpam-4571	47	2	complement	complement	NOUN
ejpam-4571	47	3	of	of	ADP
ejpam-4571	47	4	a	a	DET
ejpam-4571	47	5	b	b	NOUN
ejpam-4571	47	6	-	-	PUNCT
ejpam-4571	47	7	open	open	ADJ
ejpam-4571	47	8	set	set	NOUN
ejpam-4571	47	9	is	be	AUX
ejpam-4571	47	10	called	call	VERB
ejpam-4571	47	11	b	b	NOUN
ejpam-4571	47	12	-	-	PUNCT
ejpam-4571	47	13	closed	closed	ADJ
ejpam-4571	47	14	.	.	PUNCT
ejpam-4571	48	1	by	by	ADP
ejpam-4571	48	2	bo(x	bo(x	NUM
ejpam-4571	48	3	,	,	PUNCT
ejpam-4571	48	4	τ	τ	PROPN
ejpam-4571	48	5	)	)	PUNCT
ejpam-4571	48	6	,	,	PUNCT
ejpam-4571	48	7	we	we	PRON
ejpam-4571	48	8	denote	denote	VERB
ejpam-4571	48	9	the	the	DET
ejpam-4571	48	10	collection	collection	NOUN
ejpam-4571	48	11	of	of	ADP
ejpam-4571	48	12	all	all	DET
ejpam-4571	48	13	b	b	NOUN
ejpam-4571	48	14	-	-	PUNCT
ejpam-4571	48	15	open	open	ADJ
ejpam-4571	48	16	sets	set	NOUN
ejpam-4571	48	17	of	of	ADP
ejpam-4571	48	18	a	a	DET
ejpam-4571	48	19	topological	topological	ADJ
ejpam-4571	48	20	space	space	NOUN
ejpam-4571	48	21	(	(	PUNCT
ejpam-4571	48	22	x	x	X
ejpam-4571	48	23	,	,	PUNCT
ejpam-4571	48	24	τ	τ	PROPN
ejpam-4571	48	25	)	)	PUNCT
ejpam-4571	48	26	.	.	PUNCT
ejpam-4571	49	1	let	let	VERB
ejpam-4571	49	2	a	a	DET
ejpam-4571	49	3	be	be	AUX
ejpam-4571	49	4	a	a	DET
ejpam-4571	49	5	subset	subset	NOUN
ejpam-4571	49	6	of	of	ADP
ejpam-4571	49	7	a	a	DET
ejpam-4571	49	8	topological	topological	ADJ
ejpam-4571	49	9	space	space	NOUN
ejpam-4571	49	10	(	(	PUNCT
ejpam-4571	49	11	x	x	X
ejpam-4571	49	12	,	,	PUNCT
ejpam-4571	49	13	τ	τ	PROPN
ejpam-4571	49	14	)	)	PUNCT
ejpam-4571	49	15	.	.	PUNCT
ejpam-4571	50	1	a	a	DET
ejpam-4571	50	2	subset	subset	NOUN
ejpam-4571	50	3	aλb	aλb	NOUN
ejpam-4571	50	4	[	[	X
ejpam-4571	50	5	7	7	NUM
ejpam-4571	50	6	]	]	PUNCT
ejpam-4571	50	7	is	be	AUX
ejpam-4571	50	8	defined	define	VERB
ejpam-4571	50	9	to	to	PART
ejpam-4571	50	10	be	be	AUX
ejpam-4571	50	11	the	the	DET
ejpam-4571	50	12	set	set	NOUN
ejpam-4571	50	13	∩{u	∩{u	PROPN
ejpam-4571	50	14	∈	∈	PROPN
ejpam-4571	50	15	bo(x	bo(x	NUM
ejpam-4571	50	16	,	,	PUNCT
ejpam-4571	50	17	τ)|a	τ)|a	VERB
ejpam-4571	50	18	⊆	⊆	NUM
ejpam-4571	50	19	u	u	NOUN
ejpam-4571	50	20	}	}	PUNCT
ejpam-4571	50	21	.	.	PUNCT
ejpam-4571	51	1	a	a	DET
ejpam-4571	51	2	subset	subset	NOUN
ejpam-4571	51	3	a	a	PRON
ejpam-4571	51	4	of	of	ADP
ejpam-4571	51	5	a	a	DET
ejpam-4571	51	6	topological	topological	ADJ
ejpam-4571	51	7	space	space	NOUN
ejpam-4571	51	8	(	(	PUNCT
ejpam-4571	51	9	x	x	X
ejpam-4571	51	10	,	,	PUNCT
ejpam-4571	51	11	τ	τ	X
ejpam-4571	51	12	)	)	PUNCT
ejpam-4571	51	13	is	be	AUX
ejpam-4571	51	14	called	call	VERB
ejpam-4571	51	15	a	a	DET
ejpam-4571	51	16	λb	λb	ADV
ejpam-4571	51	17	-	-	PUNCT
ejpam-4571	51	18	set	set	NOUN
ejpam-4571	51	19	[	[	X
ejpam-4571	51	20	7	7	X
ejpam-4571	51	21	]	]	X
ejpam-4571	51	22	if	if	SCONJ
ejpam-4571	51	23	a	a	DET
ejpam-4571	51	24	=	=	NOUN
ejpam-4571	51	25	aλb	aλb	NOUN
ejpam-4571	51	26	.	.	PUNCT
ejpam-4571	52	1	a	a	DET
ejpam-4571	52	2	subset	subset	NOUN
ejpam-4571	52	3	a	a	PRON
ejpam-4571	52	4	of	of	ADP
ejpam-4571	52	5	a	a	DET
ejpam-4571	52	6	topological	topological	ADJ
ejpam-4571	52	7	space	space	NOUN
ejpam-4571	52	8	(	(	PUNCT
ejpam-4571	52	9	x	x	X
ejpam-4571	52	10	,	,	PUNCT
ejpam-4571	52	11	τ	τ	X
ejpam-4571	52	12	)	)	PUNCT
ejpam-4571	52	13	is	be	AUX
ejpam-4571	52	14	called	call	VERB
ejpam-4571	52	15	(	(	PUNCT
ejpam-4571	52	16	λ	λ	X
ejpam-4571	52	17	,	,	PUNCT
ejpam-4571	52	18	b)-closed	b)-close	VERB
ejpam-4571	52	19	[	[	X
ejpam-4571	52	20	5	5	NUM
ejpam-4571	52	21	]	]	PUNCT
ejpam-4571	52	22	if	if	SCONJ
ejpam-4571	52	23	a	a	DET
ejpam-4571	52	24	=	=	X
ejpam-4571	52	25	t	t	NOUN
ejpam-4571	52	26	∩c	∩c	NOUN
ejpam-4571	52	27	,	,	PUNCT
ejpam-4571	52	28	where	where	SCONJ
ejpam-4571	52	29	t	t	PROPN
ejpam-4571	52	30	is	be	AUX
ejpam-4571	52	31	a	a	DET
ejpam-4571	52	32	λb	λb	ADV
ejpam-4571	52	33	-	-	PUNCT
ejpam-4571	52	34	set	set	NOUN
ejpam-4571	52	35	and	and	CCONJ
ejpam-4571	52	36	c	c	NOUN
ejpam-4571	52	37	is	be	AUX
ejpam-4571	52	38	a	a	DET
ejpam-4571	52	39	b	b	NOUN
ejpam-4571	52	40	-	-	PUNCT
ejpam-4571	52	41	closed	closed	ADJ
ejpam-4571	52	42	set	set	NOUN
ejpam-4571	52	43	.	.	PUNCT
ejpam-4571	53	1	the	the	DET
ejpam-4571	53	2	complement	complement	NOUN
ejpam-4571	53	3	of	of	ADP
ejpam-4571	53	4	a	a	DET
ejpam-4571	53	5	(	(	PUNCT
ejpam-4571	53	6	λ	λ	PROPN
ejpam-4571	53	7	,	,	PUNCT
ejpam-4571	53	8	b)-closed	b)-close	VERB
ejpam-4571	53	9	set	set	NOUN
ejpam-4571	53	10	is	be	AUX
ejpam-4571	53	11	called	call	VERB
ejpam-4571	53	12	(	(	PUNCT
ejpam-4571	53	13	λ	λ	X
ejpam-4571	53	14	,	,	PUNCT
ejpam-4571	53	15	b)-open	b)-open	VERB
ejpam-4571	53	16	.	.	PUNCT
ejpam-4571	54	1	the	the	DET
ejpam-4571	54	2	collection	collection	NOUN
ejpam-4571	54	3	of	of	ADP
ejpam-4571	54	4	all	all	PRON
ejpam-4571	54	5	(	(	PUNCT
ejpam-4571	54	6	λ	λ	PROPN
ejpam-4571	54	7	,	,	PUNCT
ejpam-4571	54	8	b)-closed	b)-close	VERB
ejpam-4571	54	9	(	(	PUNCT
ejpam-4571	54	10	resp	resp	NOUN
ejpam-4571	54	11	.	.	PUNCT
ejpam-4571	55	1	(	(	PUNCT
ejpam-4571	55	2	λ	λ	INTJ
ejpam-4571	55	3	,	,	PUNCT
ejpam-4571	55	4	b)-open	b)-open	VERB
ejpam-4571	55	5	)	)	PUNCT
ejpam-4571	55	6	sets	set	NOUN
ejpam-4571	55	7	in	in	ADP
ejpam-4571	55	8	a	a	DET
ejpam-4571	55	9	topological	topological	ADJ
ejpam-4571	55	10	space	space	NOUN
ejpam-4571	55	11	(	(	PUNCT
ejpam-4571	55	12	x	x	X
ejpam-4571	55	13	,	,	PUNCT
ejpam-4571	55	14	τ	τ	X
ejpam-4571	55	15	)	)	PUNCT
ejpam-4571	55	16	is	be	AUX
ejpam-4571	55	17	denoted	denote	VERB
ejpam-4571	55	18	by	by	ADP
ejpam-4571	55	19	λbc(x	λbc(x	PROPN
ejpam-4571	55	20	,	,	PUNCT
ejpam-4571	55	21	τ	τ	PROPN
ejpam-4571	55	22	)	)	PUNCT
ejpam-4571	55	23	(	(	PUNCT
ejpam-4571	55	24	resp	resp	NOUN
ejpam-4571	55	25	.	.	PUNCT
ejpam-4571	56	1	λbo(x	λbo(x	PROPN
ejpam-4571	56	2	,	,	PUNCT
ejpam-4571	56	3	τ	τ	PROPN
ejpam-4571	56	4	)	)	PUNCT
ejpam-4571	56	5	)	)	PUNCT
ejpam-4571	56	6	.	.	PUNCT
ejpam-4571	57	1	let	let	VERB
ejpam-4571	57	2	a	a	DET
ejpam-4571	57	3	be	be	AUX
ejpam-4571	57	4	a	a	DET
ejpam-4571	57	5	subset	subset	NOUN
ejpam-4571	57	6	of	of	ADP
ejpam-4571	57	7	a	a	DET
ejpam-4571	57	8	topological	topological	ADJ
ejpam-4571	57	9	space	space	NOUN
ejpam-4571	57	10	(	(	PUNCT
ejpam-4571	57	11	x	x	X
ejpam-4571	57	12	,	,	PUNCT
ejpam-4571	57	13	τ	τ	PROPN
ejpam-4571	57	14	)	)	PUNCT
ejpam-4571	57	15	.	.	PUNCT
ejpam-4571	58	1	a	a	DET
ejpam-4571	58	2	point	point	NOUN
ejpam-4571	58	3	x	x	X
ejpam-4571	58	4	∈	∈	NOUN
ejpam-4571	58	5	x	x	PUNCT
ejpam-4571	58	6	is	be	AUX
ejpam-4571	58	7	called	call	VERB
ejpam-4571	58	8	a	a	DET
ejpam-4571	58	9	(	(	PUNCT
ejpam-4571	58	10	λ	λ	NOUN
ejpam-4571	58	11	,	,	PUNCT
ejpam-4571	58	12	b)-cluster	b)-cluster	NOUN
ejpam-4571	58	13	point	point	NOUN
ejpam-4571	58	14	[	[	X
ejpam-4571	58	15	5	5	NUM
ejpam-4571	58	16	]	]	PUNCT
ejpam-4571	58	17	of	of	ADP
ejpam-4571	58	18	a	a	DET
ejpam-4571	58	19	if	if	NOUN
ejpam-4571	58	20	for	for	SCONJ
ejpam-4571	58	21	every	every	DET
ejpam-4571	58	22	(	(	PUNCT
ejpam-4571	58	23	λ	λ	NOUN
ejpam-4571	58	24	,	,	PUNCT
ejpam-4571	58	25	b)-open	b)-open	AUX
ejpam-4571	58	26	set	set	VERB
ejpam-4571	58	27	u	u	NOUN
ejpam-4571	58	28	of	of	ADP
ejpam-4571	58	29	x	x	PUNCT
ejpam-4571	58	30	containing	contain	VERB
ejpam-4571	58	31	x	x	VERB
ejpam-4571	58	32	we	we	PRON
ejpam-4571	58	33	have	have	AUX
ejpam-4571	58	34	a∩u	a∩u	VERB
ejpam-4571	58	35	̸=	̸=	PROPN
ejpam-4571	58	36	∅.	∅.	ADP
ejpam-4571	58	37	the	the	DET
ejpam-4571	58	38	set	set	NOUN
ejpam-4571	58	39	of	of	ADP
ejpam-4571	58	40	all	all	DET
ejpam-4571	58	41	(	(	PUNCT
ejpam-4571	58	42	λ	λ	NOUN
ejpam-4571	58	43	,	,	PUNCT
ejpam-4571	58	44	b)-cluster	b)-cluster	NOUN
ejpam-4571	58	45	points	point	NOUN
ejpam-4571	58	46	of	of	ADP
ejpam-4571	58	47	a	a	PRON
ejpam-4571	58	48	is	be	AUX
ejpam-4571	58	49	called	call	VERB
ejpam-4571	58	50	the	the	DET
ejpam-4571	58	51	(	(	PUNCT
ejpam-4571	58	52	λ	λ	PROPN
ejpam-4571	58	53	,	,	PUNCT
ejpam-4571	58	54	b)-closure	b)-closure	PUNCT
ejpam-4571	58	55	[	[	X
ejpam-4571	58	56	5	5	NUM
ejpam-4571	58	57	]	]	PUNCT
ejpam-4571	58	58	of	of	ADP
ejpam-4571	58	59	a	a	PRON
ejpam-4571	58	60	and	and	CCONJ
ejpam-4571	58	61	is	be	AUX
ejpam-4571	58	62	denoted	denote	VERB
ejpam-4571	58	63	by	by	ADP
ejpam-4571	58	64	a(λ	a(λ	PROPN
ejpam-4571	58	65	,	,	PUNCT
ejpam-4571	58	66	b	b	NOUN
ejpam-4571	58	67	)	)	PUNCT
ejpam-4571	58	68	.	.	PUNCT
ejpam-4571	59	1	the	the	DET
ejpam-4571	59	2	union	union	NOUN
ejpam-4571	59	3	of	of	ADP
ejpam-4571	59	4	all	all	PRON
ejpam-4571	59	5	(	(	PUNCT
ejpam-4571	59	6	λ	λ	X
ejpam-4571	59	7	,	,	PUNCT
ejpam-4571	59	8	b)-open	b)-open	VERB
ejpam-4571	59	9	sets	set	NOUN
ejpam-4571	59	10	contained	contain	VERB
ejpam-4571	59	11	in	in	ADP
ejpam-4571	59	12	a	a	PRON
ejpam-4571	59	13	is	be	AUX
ejpam-4571	59	14	called	call	VERB
ejpam-4571	59	15	the	the	DET
ejpam-4571	59	16	(	(	PUNCT
ejpam-4571	59	17	λ	λ	PROPN
ejpam-4571	59	18	,	,	PUNCT
ejpam-4571	59	19	b)-interior	b)-interior	PUNCT
ejpam-4571	59	20	[	[	X
ejpam-4571	59	21	5	5	NUM
ejpam-4571	59	22	]	]	PUNCT
ejpam-4571	59	23	of	of	ADP
ejpam-4571	59	24	a	a	PRON
ejpam-4571	59	25	and	and	CCONJ
ejpam-4571	59	26	is	be	AUX
ejpam-4571	59	27	denoted	denote	VERB
ejpam-4571	59	28	by	by	ADP
ejpam-4571	59	29	a(λ	a(λ	PROPN
ejpam-4571	59	30	,	,	PUNCT
ejpam-4571	59	31	b	b	NOUN
ejpam-4571	59	32	)	)	PUNCT
ejpam-4571	59	33	.	.	PUNCT
ejpam-4571	60	1	let	let	VERB
ejpam-4571	60	2	a	a	DET
ejpam-4571	60	3	be	be	AUX
ejpam-4571	60	4	a	a	DET
ejpam-4571	60	5	subset	subset	NOUN
ejpam-4571	60	6	of	of	ADP
ejpam-4571	60	7	a	a	DET
ejpam-4571	60	8	topological	topological	ADJ
ejpam-4571	60	9	space	space	NOUN
ejpam-4571	60	10	(	(	PUNCT
ejpam-4571	60	11	x	x	X
ejpam-4571	60	12	,	,	PUNCT
ejpam-4571	60	13	τ	τ	PROPN
ejpam-4571	60	14	)	)	PUNCT
ejpam-4571	60	15	.	.	PUNCT
ejpam-4571	61	1	a	a	DET
ejpam-4571	61	2	subset	subset	NOUN
ejpam-4571	61	3	λ(λ	λ(λ	ADV
ejpam-4571	61	4	,	,	PUNCT
ejpam-4571	61	5	b)(a	b)(a	ADV
ejpam-4571	61	6	)	)	PUNCT
ejpam-4571	62	1	[	[	X
ejpam-4571	62	2	5	5	NUM
ejpam-4571	62	3	]	]	PUNCT
ejpam-4571	62	4	is	be	AUX
ejpam-4571	62	5	defined	define	VERB
ejpam-4571	62	6	as	as	SCONJ
ejpam-4571	62	7	follows	follow	VERB
ejpam-4571	62	8	:	:	PUNCT
ejpam-4571	62	9	λ(λ	λ(λ	INTJ
ejpam-4571	62	10	,	,	PUNCT
ejpam-4571	62	11	b)(a	b)(a	ADV
ejpam-4571	62	12	)	)	PUNCT
ejpam-4571	63	1	=	=	PUNCT
ejpam-4571	63	2	∩{u	∩{u	PROPN
ejpam-4571	63	3	∈	∈	PROPN
ejpam-4571	64	1	λbo(x	λbo(x	PROPN
ejpam-4571	64	2	,	,	PUNCT
ejpam-4571	64	3	τ	τ	PROPN
ejpam-4571	64	4	)	)	PUNCT
ejpam-4571	65	1	|	|	ADV
ejpam-4571	65	2	a	a	DET
ejpam-4571	65	3	⊆	⊆	NUM
ejpam-4571	65	4	u	u	NOUN
ejpam-4571	65	5	}	}	PUNCT
ejpam-4571	65	6	.	.	PUNCT
ejpam-4571	66	1	lemma	lemma	PROPN
ejpam-4571	66	2	1	1	NUM
ejpam-4571	66	3	.	.	PUNCT
ejpam-4571	67	1	[	[	X
ejpam-4571	67	2	5	5	NUM
ejpam-4571	67	3	]	]	PUNCT
ejpam-4571	67	4	let	let	VERB
ejpam-4571	67	5	a	a	PRON
ejpam-4571	67	6	and	and	CCONJ
ejpam-4571	67	7	b	b	NOUN
ejpam-4571	67	8	be	be	AUX
ejpam-4571	67	9	subsets	subset	NOUN
ejpam-4571	67	10	of	of	ADP
ejpam-4571	67	11	a	a	DET
ejpam-4571	67	12	topological	topological	ADJ
ejpam-4571	67	13	space	space	NOUN
ejpam-4571	67	14	(	(	PUNCT
ejpam-4571	67	15	x	x	X
ejpam-4571	67	16	,	,	PUNCT
ejpam-4571	67	17	τ	τ	PROPN
ejpam-4571	67	18	)	)	PUNCT
ejpam-4571	67	19	.	.	PUNCT
ejpam-4571	68	1	for	for	ADP
ejpam-4571	68	2	the	the	DET
ejpam-4571	68	3	(	(	PUNCT
ejpam-4571	68	4	λ	λ	PROPN
ejpam-4571	68	5	,	,	PUNCT
ejpam-4571	68	6	b)-closure	b)-closure	PROPN
ejpam-4571	68	7	,	,	PUNCT
ejpam-4571	68	8	the	the	DET
ejpam-4571	68	9	following	follow	VERB
ejpam-4571	68	10	properties	property	NOUN
ejpam-4571	68	11	hold	hold	VERB
ejpam-4571	68	12	:	:	PUNCT
ejpam-4571	68	13	(	(	PUNCT
ejpam-4571	68	14	1	1	X
ejpam-4571	68	15	)	)	PUNCT
ejpam-4571	68	16	a	a	DET
ejpam-4571	68	17	⊆	⊆	NUM
ejpam-4571	68	18	a(λ	a(λ	ADJ
ejpam-4571	68	19	,	,	PUNCT
ejpam-4571	68	20	b	b	NOUN
ejpam-4571	68	21	)	)	PUNCT
ejpam-4571	68	22	and	and	CCONJ
ejpam-4571	68	23	[	[	X
ejpam-4571	68	24	a(λ	a(λ	ADV
ejpam-4571	68	25	,	,	PUNCT
ejpam-4571	68	26	b)](λ	b)](λ	NOUN
ejpam-4571	68	27	,	,	PUNCT
ejpam-4571	68	28	p	p	NOUN
ejpam-4571	68	29	)	)	PUNCT
ejpam-4571	68	30	=	=	PUNCT
ejpam-4571	68	31	a(λ	a(λ	PROPN
ejpam-4571	68	32	,	,	PUNCT
ejpam-4571	68	33	b	b	NOUN
ejpam-4571	68	34	)	)	PUNCT
ejpam-4571	68	35	.	.	PUNCT
ejpam-4571	69	1	c.	c.	PROPN
ejpam-4571	69	2	boonpok	boonpok	PROPN
ejpam-4571	69	3	,	,	PUNCT
ejpam-4571	69	4	n.	n.	PROPN
ejpam-4571	69	5	srisarakham	srisarakham	PROPN
ejpam-4571	69	6	/	/	SYM
ejpam-4571	69	7	eur	eur	PROPN
ejpam-4571	69	8	.	.	PUNCT
ejpam-4571	70	1	j.	j.	PROPN
ejpam-4571	70	2	pure	pure	PROPN
ejpam-4571	70	3	appl	appl	PROPN
ejpam-4571	70	4	.	.	PROPN
ejpam-4571	70	5	math	math	PROPN
ejpam-4571	70	6	,	,	PUNCT
ejpam-4571	70	7	16	16	NUM
ejpam-4571	70	8	(	(	PUNCT
ejpam-4571	70	9	1	1	NUM
ejpam-4571	70	10	)	)	PUNCT
ejpam-4571	70	11	(	(	PUNCT
ejpam-4571	70	12	2023	2023	NUM
ejpam-4571	70	13	)	)	PUNCT
ejpam-4571	70	14	,	,	PUNCT
ejpam-4571	70	15	29	29	NUM
ejpam-4571	70	16	-	-	SYM
ejpam-4571	70	17	43	43	NUM
ejpam-4571	70	18	31	31	NUM
ejpam-4571	70	19	(	(	PUNCT
ejpam-4571	70	20	2	2	NUM
ejpam-4571	70	21	)	)	PUNCT
ejpam-4571	70	22	if	if	SCONJ
ejpam-4571	70	23	a	a	DET
ejpam-4571	70	24	⊆	⊆	NUM
ejpam-4571	70	25	b	b	NOUN
ejpam-4571	70	26	,	,	PUNCT
ejpam-4571	70	27	then	then	ADV
ejpam-4571	70	28	a(λ	a(λ	ADV
ejpam-4571	70	29	,	,	PUNCT
ejpam-4571	70	30	b	b	NOUN
ejpam-4571	70	31	)	)	PUNCT
ejpam-4571	70	32	⊆	⊆	NUM
ejpam-4571	70	33	b(λ	b(λ	NOUN
ejpam-4571	70	34	,	,	PUNCT
ejpam-4571	70	35	b	b	NOUN
ejpam-4571	70	36	)	)	PUNCT
ejpam-4571	70	37	.	.	PUNCT
ejpam-4571	71	1	(	(	PUNCT
ejpam-4571	71	2	3	3	X
ejpam-4571	71	3	)	)	PUNCT
ejpam-4571	71	4	a(λ	a(λ	ADV
ejpam-4571	71	5	,	,	PUNCT
ejpam-4571	71	6	b	b	X
ejpam-4571	71	7	)	)	PUNCT
ejpam-4571	71	8	=	=	PUNCT
ejpam-4571	71	9	∩{f	∩{f	NOUN
ejpam-4571	71	10	|a	|a	VERB
ejpam-4571	71	11	⊆	⊆	NUM
ejpam-4571	71	12	f	f	PROPN
ejpam-4571	71	13	and	and	CCONJ
ejpam-4571	71	14	f	f	PROPN
ejpam-4571	71	15	is	be	AUX
ejpam-4571	71	16	(	(	PUNCT
ejpam-4571	71	17	λ	λ	X
ejpam-4571	71	18	,	,	PUNCT
ejpam-4571	71	19	b)-closed	b)-close	VERB
ejpam-4571	71	20	}	}	PUNCT
ejpam-4571	71	21	.	.	PUNCT
ejpam-4571	72	1	(	(	PUNCT
ejpam-4571	72	2	4	4	NUM
ejpam-4571	72	3	)	)	PUNCT
ejpam-4571	72	4	a(λ	a(λ	ADV
ejpam-4571	72	5	,	,	PUNCT
ejpam-4571	72	6	b	b	X
ejpam-4571	72	7	)	)	PUNCT
ejpam-4571	72	8	is	be	AUX
ejpam-4571	72	9	(	(	PUNCT
ejpam-4571	72	10	λ	λ	X
ejpam-4571	72	11	,	,	PUNCT
ejpam-4571	72	12	b)-closed	b)-closed	ADJ
ejpam-4571	72	13	.	.	PUNCT
ejpam-4571	73	1	(	(	PUNCT
ejpam-4571	73	2	5	5	X
ejpam-4571	73	3	)	)	PUNCT
ejpam-4571	73	4	a	a	PRON
ejpam-4571	73	5	is	be	AUX
ejpam-4571	73	6	(	(	PUNCT
ejpam-4571	73	7	λ	λ	X
ejpam-4571	73	8	,	,	PUNCT
ejpam-4571	73	9	b)-closed	b)-close	VERB
ejpam-4571	73	10	if	if	SCONJ
ejpam-4571	73	11	and	and	CCONJ
ejpam-4571	73	12	only	only	ADV
ejpam-4571	73	13	if	if	SCONJ
ejpam-4571	73	14	a	a	PRON
ejpam-4571	73	15	=	=	X
ejpam-4571	73	16	a(λ	a(λ	PROPN
ejpam-4571	73	17	,	,	PUNCT
ejpam-4571	73	18	b	b	NOUN
ejpam-4571	73	19	)	)	PUNCT
ejpam-4571	73	20	.	.	PUNCT
ejpam-4571	74	1	lemma	lemma	PROPN
ejpam-4571	74	2	2	2	NUM
ejpam-4571	74	3	.	.	PUNCT
ejpam-4571	75	1	[	[	X
ejpam-4571	75	2	5	5	NUM
ejpam-4571	75	3	]	]	PUNCT
ejpam-4571	75	4	let	let	VERB
ejpam-4571	75	5	a	a	PRON
ejpam-4571	75	6	and	and	CCONJ
ejpam-4571	75	7	b	b	NOUN
ejpam-4571	75	8	be	be	AUX
ejpam-4571	75	9	subsets	subset	NOUN
ejpam-4571	75	10	of	of	ADP
ejpam-4571	75	11	a	a	DET
ejpam-4571	75	12	topological	topological	ADJ
ejpam-4571	75	13	space	space	NOUN
ejpam-4571	75	14	(	(	PUNCT
ejpam-4571	75	15	x	x	X
ejpam-4571	75	16	,	,	PUNCT
ejpam-4571	75	17	τ	τ	PROPN
ejpam-4571	75	18	)	)	PUNCT
ejpam-4571	75	19	.	.	PUNCT
ejpam-4571	76	1	for	for	ADP
ejpam-4571	76	2	the	the	DET
ejpam-4571	76	3	(	(	PUNCT
ejpam-4571	76	4	λ	λ	PROPN
ejpam-4571	76	5	,	,	PUNCT
ejpam-4571	76	6	b)-interior	b)-interior	PROPN
ejpam-4571	76	7	,	,	PUNCT
ejpam-4571	76	8	the	the	DET
ejpam-4571	76	9	following	follow	VERB
ejpam-4571	76	10	properties	property	NOUN
ejpam-4571	76	11	hold	hold	VERB
ejpam-4571	76	12	:	:	PUNCT
ejpam-4571	76	13	(	(	PUNCT
ejpam-4571	76	14	1	1	X
ejpam-4571	76	15	)	)	PUNCT
ejpam-4571	76	16	a(λ	a(λ	ADV
ejpam-4571	76	17	,	,	PUNCT
ejpam-4571	76	18	b	b	NOUN
ejpam-4571	76	19	)	)	PUNCT
ejpam-4571	76	20	⊆	⊆	NUM
ejpam-4571	76	21	a	a	PRON
ejpam-4571	76	22	and	and	CCONJ
ejpam-4571	76	23	[	[	X
ejpam-4571	76	24	a(λ	a(λ	ADV
ejpam-4571	76	25	,	,	PUNCT
ejpam-4571	76	26	b)](λ	b)](λ	NOUN
ejpam-4571	76	27	,	,	PUNCT
ejpam-4571	76	28	b	b	NOUN
ejpam-4571	76	29	)	)	PUNCT
ejpam-4571	76	30	=	=	PUNCT
ejpam-4571	77	1	a(λ	a(λ	PROPN
ejpam-4571	77	2	,	,	PUNCT
ejpam-4571	77	3	b	b	NOUN
ejpam-4571	77	4	)	)	PUNCT
ejpam-4571	77	5	.	.	PUNCT
ejpam-4571	78	1	(	(	PUNCT
ejpam-4571	78	2	2	2	X
ejpam-4571	78	3	)	)	PUNCT
ejpam-4571	78	4	if	if	SCONJ
ejpam-4571	78	5	a	a	DET
ejpam-4571	78	6	⊆	⊆	NUM
ejpam-4571	78	7	b	b	NOUN
ejpam-4571	78	8	,	,	PUNCT
ejpam-4571	78	9	then	then	ADV
ejpam-4571	78	10	a(λ	a(λ	ADV
ejpam-4571	78	11	,	,	PUNCT
ejpam-4571	78	12	b	b	NOUN
ejpam-4571	78	13	)	)	PUNCT
ejpam-4571	78	14	⊆	⊆	NUM
ejpam-4571	78	15	b(λ	b(λ	NOUN
ejpam-4571	78	16	,	,	PUNCT
ejpam-4571	78	17	b	b	NOUN
ejpam-4571	78	18	)	)	PUNCT
ejpam-4571	78	19	.	.	PUNCT
ejpam-4571	79	1	(	(	PUNCT
ejpam-4571	79	2	3	3	X
ejpam-4571	79	3	)	)	PUNCT
ejpam-4571	79	4	a(λ	a(λ	ADV
ejpam-4571	79	5	,	,	PUNCT
ejpam-4571	79	6	b	b	X
ejpam-4571	79	7	)	)	PUNCT
ejpam-4571	79	8	is	be	AUX
ejpam-4571	79	9	(	(	PUNCT
ejpam-4571	79	10	λ	λ	INTJ
ejpam-4571	79	11	,	,	PUNCT
ejpam-4571	79	12	b)-open	b)-open	VERB
ejpam-4571	79	13	.	.	PUNCT
ejpam-4571	80	1	(	(	PUNCT
ejpam-4571	80	2	4	4	X
ejpam-4571	80	3	)	)	PUNCT
ejpam-4571	80	4	a	a	PRON
ejpam-4571	80	5	is	be	AUX
ejpam-4571	80	6	(	(	PUNCT
ejpam-4571	80	7	λ	λ	X
ejpam-4571	80	8	,	,	PUNCT
ejpam-4571	80	9	b)-open	b)-open	VERB
ejpam-4571	80	10	if	if	SCONJ
ejpam-4571	80	11	and	and	CCONJ
ejpam-4571	80	12	only	only	ADV
ejpam-4571	80	13	if	if	SCONJ
ejpam-4571	80	14	a(λ	a(λ	ADV
ejpam-4571	80	15	,	,	PUNCT
ejpam-4571	80	16	b	b	NOUN
ejpam-4571	80	17	)	)	PUNCT
ejpam-4571	80	18	=	=	SYM
ejpam-4571	80	19	a.	a.	NOUN
ejpam-4571	80	20	lemma	lemma	PROPN
ejpam-4571	80	21	3	3	X
ejpam-4571	80	22	.	.	PUNCT
ejpam-4571	81	1	[	[	X
ejpam-4571	81	2	5	5	NUM
ejpam-4571	81	3	]	]	PUNCT
ejpam-4571	81	4	for	for	ADP
ejpam-4571	81	5	subsets	subset	NOUN
ejpam-4571	81	6	a	a	DET
ejpam-4571	81	7	,	,	PUNCT
ejpam-4571	81	8	b	b	PROPN
ejpam-4571	81	9	of	of	ADP
ejpam-4571	81	10	a	a	DET
ejpam-4571	81	11	topological	topological	ADJ
ejpam-4571	81	12	space	space	NOUN
ejpam-4571	81	13	(	(	PUNCT
ejpam-4571	81	14	x	x	X
ejpam-4571	81	15	,	,	PUNCT
ejpam-4571	81	16	τ	τ	PROPN
ejpam-4571	81	17	)	)	PUNCT
ejpam-4571	81	18	,	,	PUNCT
ejpam-4571	81	19	the	the	DET
ejpam-4571	81	20	following	follow	VERB
ejpam-4571	81	21	properties	property	NOUN
ejpam-4571	81	22	hold	hold	VERB
ejpam-4571	81	23	:	:	PUNCT
ejpam-4571	81	24	(	(	PUNCT
ejpam-4571	81	25	1	1	X
ejpam-4571	81	26	)	)	PUNCT
ejpam-4571	81	27	a	a	DET
ejpam-4571	81	28	⊆	⊆	NUM
ejpam-4571	81	29	λ(λ	λ(λ	NOUN
ejpam-4571	81	30	,	,	PUNCT
ejpam-4571	81	31	b)(a	b)(a	ADV
ejpam-4571	81	32	)	)	PUNCT
ejpam-4571	81	33	.	.	PUNCT
ejpam-4571	82	1	(	(	PUNCT
ejpam-4571	82	2	2	2	X
ejpam-4571	82	3	)	)	PUNCT
ejpam-4571	82	4	if	if	SCONJ
ejpam-4571	82	5	a	a	DET
ejpam-4571	82	6	⊆	⊆	NUM
ejpam-4571	82	7	b	b	NOUN
ejpam-4571	82	8	,	,	PUNCT
ejpam-4571	82	9	then	then	ADV
ejpam-4571	82	10	λ(λ	λ(λ	PROPN
ejpam-4571	82	11	,	,	PUNCT
ejpam-4571	82	12	b)(a	b)(a	ADV
ejpam-4571	82	13	)	)	PUNCT
ejpam-4571	82	14	⊆	⊆	NUM
ejpam-4571	82	15	λ(λ	λ(λ	NOUN
ejpam-4571	82	16	,	,	PUNCT
ejpam-4571	82	17	b)(b	b)(b	ADJ
ejpam-4571	82	18	)	)	PUNCT
ejpam-4571	82	19	.	.	PUNCT
ejpam-4571	83	1	(	(	PUNCT
ejpam-4571	83	2	3	3	X
ejpam-4571	83	3	)	)	PUNCT
ejpam-4571	83	4	λ(λ	λ(λ	ADV
ejpam-4571	83	5	,	,	PUNCT
ejpam-4571	83	6	b)[λ(λ	b)[λ(λ	NOUN
ejpam-4571	83	7	,	,	PUNCT
ejpam-4571	83	8	b)(a	b)(a	ADV
ejpam-4571	83	9	)	)	PUNCT
ejpam-4571	83	10	]	]	PUNCT
ejpam-4571	84	1	=	=	PUNCT
ejpam-4571	84	2	λ(λ	λ(λ	INTJ
ejpam-4571	84	3	,	,	PUNCT
ejpam-4571	84	4	b)(a	b)(a	ADV
ejpam-4571	84	5	)	)	PUNCT
ejpam-4571	84	6	.	.	PUNCT
ejpam-4571	85	1	(	(	PUNCT
ejpam-4571	85	2	4	4	X
ejpam-4571	85	3	)	)	PUNCT
ejpam-4571	85	4	if	if	SCONJ
ejpam-4571	85	5	a	a	PRON
ejpam-4571	85	6	is	be	AUX
ejpam-4571	85	7	(	(	PUNCT
ejpam-4571	85	8	λ	λ	INTJ
ejpam-4571	85	9	,	,	PUNCT
ejpam-4571	85	10	b)-open	b)-open	VERB
ejpam-4571	85	11	,	,	PUNCT
ejpam-4571	85	12	then	then	ADV
ejpam-4571	85	13	λ(λ	λ(λ	INTJ
ejpam-4571	85	14	,	,	PUNCT
ejpam-4571	85	15	b)(a	b)(a	ADV
ejpam-4571	85	16	)	)	PUNCT
ejpam-4571	86	1	=	=	PUNCT
ejpam-4571	86	2	a.	a.	NOUN
ejpam-4571	86	3	3	3	NUM
ejpam-4571	86	4	.	.	PUNCT
ejpam-4571	87	1	some	some	DET
ejpam-4571	87	2	properties	property	NOUN
ejpam-4571	87	3	of	of	ADP
ejpam-4571	87	4	generalized	generalized	ADJ
ejpam-4571	87	5	(	(	PUNCT
ejpam-4571	87	6	λ	λ	NOUN
ejpam-4571	87	7	,	,	PUNCT
ejpam-4571	87	8	b)-open	b)-open	VERB
ejpam-4571	87	9	sets	set	NOUN
ejpam-4571	87	10	in	in	ADP
ejpam-4571	87	11	this	this	DET
ejpam-4571	87	12	section	section	NOUN
ejpam-4571	87	13	,	,	PUNCT
ejpam-4571	87	14	we	we	PRON
ejpam-4571	87	15	introduce	introduce	VERB
ejpam-4571	87	16	new	new	ADJ
ejpam-4571	87	17	classes	class	NOUN
ejpam-4571	87	18	of	of	ADP
ejpam-4571	87	19	sets	set	NOUN
ejpam-4571	87	20	called	call	VERB
ejpam-4571	87	21	s(λ	s(λ	NOUN
ejpam-4571	87	22	,	,	PUNCT
ejpam-4571	87	23	b)-open	b)-open	VERB
ejpam-4571	87	24	sets	set	NOUN
ejpam-4571	87	25	,	,	PUNCT
ejpam-4571	87	26	p(λ	p(λ	NOUN
ejpam-4571	87	27	,	,	PUNCT
ejpam-4571	87	28	b)-open	b)-open	PUNCT
ejpam-4571	87	29	sets	set	NOUN
ejpam-4571	87	30	,	,	PUNCT
ejpam-4571	87	31	α(λ	α(λ	PROPN
ejpam-4571	87	32	,	,	PUNCT
ejpam-4571	87	33	b)-open	b)-open	VERB
ejpam-4571	87	34	sets	set	NOUN
ejpam-4571	87	35	,	,	PUNCT
ejpam-4571	87	36	β(λ	β(λ	X
ejpam-4571	87	37	,	,	PUNCT
ejpam-4571	87	38	b)-open	b)-open	VERB
ejpam-4571	87	39	sets	set	NOUN
ejpam-4571	87	40	and	and	CCONJ
ejpam-4571	87	41	b(λ	b(λ	NOUN
ejpam-4571	87	42	,	,	PUNCT
ejpam-4571	87	43	b)-open	b)-open	PUNCT
ejpam-4571	87	44	sets	set	NOUN
ejpam-4571	87	45	.	.	PUNCT
ejpam-4571	88	1	we	we	PRON
ejpam-4571	88	2	also	also	ADV
ejpam-4571	88	3	investigate	investigate	VERB
ejpam-4571	88	4	some	some	PRON
ejpam-4571	88	5	of	of	ADP
ejpam-4571	88	6	their	their	PRON
ejpam-4571	88	7	fundamental	fundamental	ADJ
ejpam-4571	88	8	properties	property	NOUN
ejpam-4571	88	9	.	.	PUNCT
ejpam-4571	89	1	definition	definition	NOUN
ejpam-4571	89	2	1	1	NUM
ejpam-4571	89	3	.	.	PUNCT
ejpam-4571	90	1	a	a	DET
ejpam-4571	90	2	subset	subset	NOUN
ejpam-4571	90	3	a	a	PRON
ejpam-4571	90	4	of	of	ADP
ejpam-4571	90	5	a	a	DET
ejpam-4571	90	6	topological	topological	ADJ
ejpam-4571	90	7	space	space	NOUN
ejpam-4571	90	8	(	(	PUNCT
ejpam-4571	90	9	x	x	X
ejpam-4571	90	10	,	,	PUNCT
ejpam-4571	90	11	τ	τ	X
ejpam-4571	90	12	)	)	PUNCT
ejpam-4571	90	13	is	be	AUX
ejpam-4571	90	14	said	say	VERB
ejpam-4571	90	15	to	to	PART
ejpam-4571	90	16	be	be	AUX
ejpam-4571	90	17	:	:	PUNCT
ejpam-4571	90	18	(	(	PUNCT
ejpam-4571	90	19	i	i	NOUN
ejpam-4571	90	20	)	)	PUNCT
ejpam-4571	90	21	s(λ	s(λ	PROPN
ejpam-4571	90	22	,	,	PUNCT
ejpam-4571	90	23	b)-open	b)-open	VERB
ejpam-4571	90	24	if	if	SCONJ
ejpam-4571	90	25	a	a	DET
ejpam-4571	90	26	⊆	⊆	NUM
ejpam-4571	90	27	[	[	X
ejpam-4571	90	28	a(λ	a(λ	ADJ
ejpam-4571	90	29	,	,	PUNCT
ejpam-4571	90	30	b	b	NOUN
ejpam-4571	90	31	)	)	PUNCT
ejpam-4571	90	32	]	]	PUNCT
ejpam-4571	91	1	(	(	PUNCT
ejpam-4571	91	2	λ	λ	X
ejpam-4571	91	3	,	,	PUNCT
ejpam-4571	91	4	b	b	NOUN
ejpam-4571	91	5	)	)	PUNCT
ejpam-4571	91	6	;	;	PUNCT
ejpam-4571	91	7	(	(	PUNCT
ejpam-4571	91	8	ii	ii	NOUN
ejpam-4571	91	9	)	)	PUNCT
ejpam-4571	91	10	p(λ	p(λ	NOUN
ejpam-4571	91	11	,	,	PUNCT
ejpam-4571	91	12	b)-open	b)-open	VERB
ejpam-4571	91	13	if	if	SCONJ
ejpam-4571	91	14	a	a	DET
ejpam-4571	91	15	⊆	⊆	NUM
ejpam-4571	91	16	[	[	X
ejpam-4571	91	17	a(λ	a(λ	ADV
ejpam-4571	91	18	,	,	PUNCT
ejpam-4571	91	19	b)](λ	b)](λ	NOUN
ejpam-4571	91	20	,	,	PUNCT
ejpam-4571	91	21	b	b	NOUN
ejpam-4571	91	22	)	)	PUNCT
ejpam-4571	91	23	;	;	PUNCT
ejpam-4571	91	24	(	(	PUNCT
ejpam-4571	91	25	iii	iii	X
ejpam-4571	91	26	)	)	PUNCT
ejpam-4571	91	27	α(λ	α(λ	PROPN
ejpam-4571	91	28	,	,	PUNCT
ejpam-4571	91	29	b)-open	b)-open	VERB
ejpam-4571	91	30	if	if	SCONJ
ejpam-4571	91	31	a	a	DET
ejpam-4571	91	32	⊆	⊆	NUM
ejpam-4571	91	33	[	[	X
ejpam-4571	91	34	[	[	X
ejpam-4571	91	35	a(λ	a(λ	ADJ
ejpam-4571	91	36	,	,	PUNCT
ejpam-4571	91	37	b	b	NOUN
ejpam-4571	91	38	)	)	PUNCT
ejpam-4571	91	39	]	]	PUNCT
ejpam-4571	91	40	(	(	PUNCT
ejpam-4571	91	41	λ	λ	INTJ
ejpam-4571	91	42	,	,	PUNCT
ejpam-4571	91	43	b)](λ	b)](λ	NOUN
ejpam-4571	91	44	,	,	PUNCT
ejpam-4571	91	45	b	b	NOUN
ejpam-4571	91	46	)	)	PUNCT
ejpam-4571	91	47	;	;	PUNCT
ejpam-4571	91	48	(	(	PUNCT
ejpam-4571	91	49	iv	iv	X
ejpam-4571	91	50	)	)	PUNCT
ejpam-4571	91	51	β(λ	β(λ	NOUN
ejpam-4571	91	52	,	,	PUNCT
ejpam-4571	91	53	b)-open	b)-open	VERB
ejpam-4571	91	54	if	if	SCONJ
ejpam-4571	91	55	a	a	DET
ejpam-4571	91	56	⊆	⊆	NUM
ejpam-4571	91	57	[	[	X
ejpam-4571	91	58	[	[	X
ejpam-4571	91	59	a(λ	a(λ	ADV
ejpam-4571	91	60	,	,	PUNCT
ejpam-4571	91	61	b)](λ	b)](λ	NOUN
ejpam-4571	91	62	,	,	PUNCT
ejpam-4571	91	63	b	b	NOUN
ejpam-4571	91	64	)	)	PUNCT
ejpam-4571	91	65	]	]	PUNCT
ejpam-4571	91	66	(	(	PUNCT
ejpam-4571	91	67	λ	λ	X
ejpam-4571	91	68	,	,	PUNCT
ejpam-4571	91	69	b	b	NOUN
ejpam-4571	91	70	)	)	PUNCT
ejpam-4571	91	71	.	.	PUNCT
ejpam-4571	92	1	example	example	NOUN
ejpam-4571	93	1	1	1	X
ejpam-4571	93	2	.	.	PUNCT
ejpam-4571	93	3	let	let	VERB
ejpam-4571	93	4	x	x	PUNCT
ejpam-4571	93	5	=	=	PRON
ejpam-4571	93	6	{	{	PUNCT
ejpam-4571	93	7	−1	−1	NOUN
ejpam-4571	93	8	,	,	PUNCT
ejpam-4571	93	9	0	0	NUM
ejpam-4571	93	10	,	,	PUNCT
ejpam-4571	93	11	1	1	NUM
ejpam-4571	93	12	}	}	PUNCT
ejpam-4571	93	13	with	with	ADP
ejpam-4571	93	14	the	the	DET
ejpam-4571	93	15	topology	topology	NOUN
ejpam-4571	93	16	τ	τ	X
ejpam-4571	93	17	=	=	PUNCT
ejpam-4571	93	18	{	{	PUNCT
ejpam-4571	93	19	∅	∅	NOUN
ejpam-4571	93	20	,	,	PUNCT
ejpam-4571	93	21	{	{	PUNCT
ejpam-4571	93	22	−1	−1	NOUN
ejpam-4571	93	23	}	}	PUNCT
ejpam-4571	93	24	,	,	PUNCT
ejpam-4571	93	25	{	{	PUNCT
ejpam-4571	93	26	0	0	NUM
ejpam-4571	93	27	,	,	PUNCT
ejpam-4571	93	28	1	1	NUM
ejpam-4571	93	29	}	}	PUNCT
ejpam-4571	93	30	,	,	PUNCT
ejpam-4571	93	31	x	x	NOUN
ejpam-4571	93	32	}	}	PUNCT
ejpam-4571	93	33	.	.	PUNCT
ejpam-4571	94	1	let	let	VERB
ejpam-4571	94	2	a	a	PRON
ejpam-4571	94	3	=	=	X
ejpam-4571	94	4	{	{	PUNCT
ejpam-4571	94	5	−1	−1	NOUN
ejpam-4571	94	6	}	}	PUNCT
ejpam-4571	94	7	.	.	PUNCT
ejpam-4571	95	1	then	then	ADV
ejpam-4571	95	2	,	,	PUNCT
ejpam-4571	95	3	a	a	PRON
ejpam-4571	95	4	is	be	AUX
ejpam-4571	95	5	s(λ	s(λ	NOUN
ejpam-4571	95	6	,	,	PUNCT
ejpam-4571	95	7	b)-open	b)-open	PUNCT
ejpam-4571	95	8	and	and	CCONJ
ejpam-4571	95	9	p(λ	p(λ	NOUN
ejpam-4571	95	10	,	,	PUNCT
ejpam-4571	95	11	b)-open	b)-open	VERB
ejpam-4571	95	12	.	.	PUNCT
ejpam-4571	96	1	on	on	ADP
ejpam-4571	96	2	the	the	DET
ejpam-4571	96	3	other	other	ADJ
ejpam-4571	96	4	hand	hand	NOUN
ejpam-4571	96	5	,	,	PUNCT
ejpam-4571	96	6	let	let	VERB
ejpam-4571	96	7	b	b	NOUN
ejpam-4571	96	8	=	=	PRON
ejpam-4571	96	9	{	{	PUNCT
ejpam-4571	96	10	0	0	NUM
ejpam-4571	96	11	,	,	PUNCT
ejpam-4571	96	12	1	1	NUM
ejpam-4571	96	13	}	}	PUNCT
ejpam-4571	96	14	.	.	PUNCT
ejpam-4571	97	1	then	then	ADV
ejpam-4571	97	2	,	,	PUNCT
ejpam-4571	97	3	b	b	PROPN
ejpam-4571	97	4	is	be	AUX
ejpam-4571	97	5	α(λ	α(λ	PROPN
ejpam-4571	97	6	,	,	PUNCT
ejpam-4571	97	7	b)-open	b)-open	PUNCT
ejpam-4571	97	8	and	and	CCONJ
ejpam-4571	97	9	β(λ	β(λ	PROPN
ejpam-4571	97	10	,	,	PUNCT
ejpam-4571	97	11	b)-open	b)-open	PUNCT
ejpam-4571	97	12	.	.	PUNCT
ejpam-4571	98	1	c.	c.	PROPN
ejpam-4571	98	2	boonpok	boonpok	PROPN
ejpam-4571	98	3	,	,	PUNCT
ejpam-4571	98	4	n.	n.	PROPN
ejpam-4571	98	5	srisarakham	srisarakham	PROPN
ejpam-4571	98	6	/	/	SYM
ejpam-4571	98	7	eur	eur	PROPN
ejpam-4571	98	8	.	.	PUNCT
ejpam-4571	99	1	j.	j.	PROPN
ejpam-4571	99	2	pure	pure	PROPN
ejpam-4571	99	3	appl	appl	PROPN
ejpam-4571	99	4	.	.	PROPN
ejpam-4571	99	5	math	math	PROPN
ejpam-4571	99	6	,	,	PUNCT
ejpam-4571	99	7	16	16	NUM
ejpam-4571	99	8	(	(	PUNCT
ejpam-4571	99	9	1	1	NUM
ejpam-4571	99	10	)	)	PUNCT
ejpam-4571	99	11	(	(	PUNCT
ejpam-4571	99	12	2023	2023	NUM
ejpam-4571	99	13	)	)	PUNCT
ejpam-4571	99	14	,	,	PUNCT
ejpam-4571	99	15	29	29	NUM
ejpam-4571	99	16	-	-	SYM
ejpam-4571	99	17	43	43	NUM
ejpam-4571	99	18	32	32	NUM
ejpam-4571	99	19	the	the	DET
ejpam-4571	99	20	family	family	NOUN
ejpam-4571	99	21	of	of	ADP
ejpam-4571	99	22	all	all	DET
ejpam-4571	99	23	s(λ	s(λ	NOUN
ejpam-4571	99	24	,	,	PUNCT
ejpam-4571	99	25	b)-open	b)-open	PUNCT
ejpam-4571	99	26	(	(	PUNCT
ejpam-4571	99	27	resp	resp	NOUN
ejpam-4571	99	28	.	.	PUNCT
ejpam-4571	100	1	p(λ	p(λ	NOUN
ejpam-4571	100	2	,	,	PUNCT
ejpam-4571	100	3	b)-open	b)-open	PUNCT
ejpam-4571	100	4	,	,	PUNCT
ejpam-4571	100	5	α(λ	α(λ	PROPN
ejpam-4571	100	6	,	,	PUNCT
ejpam-4571	100	7	b)-open	b)-open	PUNCT
ejpam-4571	100	8	,	,	PUNCT
ejpam-4571	100	9	β(λ	β(λ	X
ejpam-4571	100	10	,	,	PUNCT
ejpam-4571	100	11	b)-open	b)-open	PUNCT
ejpam-4571	100	12	)	)	PUNCT
ejpam-4571	100	13	sets	set	NOUN
ejpam-4571	100	14	in	in	ADP
ejpam-4571	100	15	a	a	DET
ejpam-4571	100	16	topological	topological	ADJ
ejpam-4571	100	17	space	space	NOUN
ejpam-4571	100	18	(	(	PUNCT
ejpam-4571	100	19	x	x	X
ejpam-4571	100	20	,	,	PUNCT
ejpam-4571	100	21	τ	τ	X
ejpam-4571	100	22	)	)	PUNCT
ejpam-4571	100	23	is	be	AUX
ejpam-4571	100	24	denoted	denote	VERB
ejpam-4571	100	25	by	by	ADP
ejpam-4571	100	26	sλbo(x	sλbo(x	PROPN
ejpam-4571	100	27	,	,	PUNCT
ejpam-4571	100	28	τ	τ	PROPN
ejpam-4571	100	29	)	)	PUNCT
ejpam-4571	100	30	(	(	PUNCT
ejpam-4571	100	31	resp	resp	NOUN
ejpam-4571	100	32	.	.	PUNCT
ejpam-4571	101	1	pλbo(x	pλbo(x	NOUN
ejpam-4571	101	2	,	,	PUNCT
ejpam-4571	101	3	τ	τ	PROPN
ejpam-4571	101	4	)	)	PUNCT
ejpam-4571	101	5	,	,	PUNCT
ejpam-4571	101	6	αλbo(x	αλbo(x	NOUN
ejpam-4571	101	7	,	,	PUNCT
ejpam-4571	101	8	τ	τ	PROPN
ejpam-4571	101	9	)	)	PUNCT
ejpam-4571	101	10	,	,	PUNCT
ejpam-4571	101	11	βλbo(x	βλbo(x	X
ejpam-4571	101	12	,	,	PUNCT
ejpam-4571	101	13	τ	τ	PROPN
ejpam-4571	101	14	)	)	PUNCT
ejpam-4571	101	15	)	)	PUNCT
ejpam-4571	101	16	.	.	PUNCT
ejpam-4571	102	1	the	the	DET
ejpam-4571	102	2	complement	complement	NOUN
ejpam-4571	102	3	of	of	ADP
ejpam-4571	102	4	a	a	DET
ejpam-4571	102	5	s(λ	s(λ	PROPN
ejpam-4571	102	6	,	,	PUNCT
ejpam-4571	102	7	b)-open	b)-open	PUNCT
ejpam-4571	102	8	(	(	PUNCT
ejpam-4571	102	9	resp	resp	NOUN
ejpam-4571	102	10	.	.	PUNCT
ejpam-4571	103	1	p(λ	p(λ	NOUN
ejpam-4571	103	2	,	,	PUNCT
ejpam-4571	103	3	b)-open	b)-open	PUNCT
ejpam-4571	103	4	,	,	PUNCT
ejpam-4571	103	5	α(λ	α(λ	PROPN
ejpam-4571	103	6	,	,	PUNCT
ejpam-4571	103	7	b)-open	b)-open	PUNCT
ejpam-4571	103	8	,	,	PUNCT
ejpam-4571	103	9	β(λ	β(λ	X
ejpam-4571	103	10	,	,	PUNCT
ejpam-4571	103	11	b)open	b)open	NOUN
ejpam-4571	103	12	)	)	PUNCT
ejpam-4571	103	13	set	set	NOUN
ejpam-4571	103	14	is	be	AUX
ejpam-4571	103	15	said	say	VERB
ejpam-4571	103	16	to	to	PART
ejpam-4571	103	17	be	be	AUX
ejpam-4571	103	18	s(λ	s(λ	PROPN
ejpam-4571	103	19	,	,	PUNCT
ejpam-4571	103	20	b)-closed	b)-close	VERB
ejpam-4571	103	21	(	(	PUNCT
ejpam-4571	103	22	resp	resp	NOUN
ejpam-4571	103	23	.	.	PUNCT
ejpam-4571	104	1	p(λ	p(λ	NOUN
ejpam-4571	104	2	,	,	PUNCT
ejpam-4571	104	3	b)-closed	b)-close	VERB
ejpam-4571	104	4	,	,	PUNCT
ejpam-4571	104	5	α(λ	α(λ	PROPN
ejpam-4571	104	6	,	,	PUNCT
ejpam-4571	104	7	b)-closed	b)-close	VERB
ejpam-4571	104	8	,	,	PUNCT
ejpam-4571	104	9	β(λ	β(λ	X
ejpam-4571	104	10	,	,	PUNCT
ejpam-4571	104	11	b)-closed	b)-close	VERB
ejpam-4571	104	12	)	)	PUNCT
ejpam-4571	104	13	.	.	PUNCT
ejpam-4571	105	1	the	the	DET
ejpam-4571	105	2	family	family	NOUN
ejpam-4571	105	3	of	of	ADP
ejpam-4571	105	4	all	all	DET
ejpam-4571	105	5	s(λ	s(λ	NOUN
ejpam-4571	105	6	,	,	PUNCT
ejpam-4571	105	7	b)-closed	b)-close	VERB
ejpam-4571	105	8	(	(	PUNCT
ejpam-4571	105	9	resp	resp	NOUN
ejpam-4571	105	10	.	.	PUNCT
ejpam-4571	106	1	p(λ	p(λ	NOUN
ejpam-4571	106	2	,	,	PUNCT
ejpam-4571	106	3	b)-closed	b)-close	VERB
ejpam-4571	106	4	,	,	PUNCT
ejpam-4571	106	5	α(λ	α(λ	PROPN
ejpam-4571	106	6	,	,	PUNCT
ejpam-4571	106	7	b)-closed	b)-close	VERB
ejpam-4571	106	8	,	,	PUNCT
ejpam-4571	106	9	β(λ	β(λ	X
ejpam-4571	106	10	,	,	PUNCT
ejpam-4571	106	11	b)-closed	b)-closed	ADJ
ejpam-4571	106	12	)	)	PUNCT
ejpam-4571	106	13	sets	set	NOUN
ejpam-4571	106	14	in	in	ADP
ejpam-4571	106	15	a	a	DET
ejpam-4571	106	16	topological	topological	ADJ
ejpam-4571	106	17	space	space	NOUN
ejpam-4571	106	18	(	(	PUNCT
ejpam-4571	106	19	x	x	X
ejpam-4571	106	20	,	,	PUNCT
ejpam-4571	106	21	τ	τ	X
ejpam-4571	106	22	)	)	PUNCT
ejpam-4571	106	23	is	be	AUX
ejpam-4571	106	24	denoted	denote	VERB
ejpam-4571	106	25	by	by	ADP
ejpam-4571	106	26	sλbc(x	sλbc(x	PROPN
ejpam-4571	106	27	,	,	PUNCT
ejpam-4571	106	28	τ	τ	X
ejpam-4571	106	29	)	)	PUNCT
ejpam-4571	106	30	(	(	PUNCT
ejpam-4571	106	31	resp	resp	NOUN
ejpam-4571	106	32	.	.	PUNCT
ejpam-4571	107	1	pλbc(x	pλbc(x	NOUN
ejpam-4571	107	2	,	,	PUNCT
ejpam-4571	107	3	τ	τ	PROPN
ejpam-4571	107	4	)	)	PUNCT
ejpam-4571	107	5	,	,	PUNCT
ejpam-4571	107	6	αλbc(x	αλbc(x	NOUN
ejpam-4571	107	7	,	,	PUNCT
ejpam-4571	107	8	τ	τ	PROPN
ejpam-4571	107	9	)	)	PUNCT
ejpam-4571	107	10	,	,	PUNCT
ejpam-4571	107	11	βλbo(x	βλbo(x	X
ejpam-4571	107	12	,	,	PUNCT
ejpam-4571	107	13	τ	τ	PROPN
ejpam-4571	107	14	)	)	PUNCT
ejpam-4571	107	15	)	)	PUNCT
ejpam-4571	107	16	.	.	PUNCT
ejpam-4571	108	1	proposition	proposition	NOUN
ejpam-4571	108	2	1	1	NUM
ejpam-4571	108	3	.	.	PUNCT
ejpam-4571	109	1	for	for	ADP
ejpam-4571	109	2	a	a	DET
ejpam-4571	109	3	topological	topological	ADJ
ejpam-4571	109	4	space	space	NOUN
ejpam-4571	109	5	(	(	PUNCT
ejpam-4571	109	6	x	x	X
ejpam-4571	109	7	,	,	PUNCT
ejpam-4571	109	8	τ	τ	PROPN
ejpam-4571	109	9	)	)	PUNCT
ejpam-4571	109	10	,	,	PUNCT
ejpam-4571	109	11	the	the	DET
ejpam-4571	109	12	following	follow	VERB
ejpam-4571	109	13	properties	property	NOUN
ejpam-4571	109	14	hold	hold	VERB
ejpam-4571	109	15	:	:	PUNCT
ejpam-4571	109	16	(	(	PUNCT
ejpam-4571	109	17	1	1	X
ejpam-4571	109	18	)	)	PUNCT
ejpam-4571	109	19	λbo(x	λbo(x	PROPN
ejpam-4571	109	20	,	,	PUNCT
ejpam-4571	109	21	τ	τ	PROPN
ejpam-4571	109	22	)	)	PUNCT
ejpam-4571	109	23	⊆	⊆	NUM
ejpam-4571	109	24	αλbo(x	αλbo(x	NOUN
ejpam-4571	109	25	,	,	PUNCT
ejpam-4571	109	26	τ	τ	PROPN
ejpam-4571	109	27	)	)	PUNCT
ejpam-4571	109	28	⊆	⊆	NUM
ejpam-4571	109	29	sλbo(x	sλbo(x	PROPN
ejpam-4571	109	30	,	,	PUNCT
ejpam-4571	109	31	τ	τ	PROPN
ejpam-4571	109	32	)	)	PUNCT
ejpam-4571	109	33	⊆	⊆	NUM
ejpam-4571	109	34	βλbo(x	βλbo(x	NOUN
ejpam-4571	109	35	,	,	PUNCT
ejpam-4571	109	36	τ	τ	PROPN
ejpam-4571	109	37	)	)	PUNCT
ejpam-4571	109	38	.	.	PUNCT
ejpam-4571	110	1	(	(	PUNCT
ejpam-4571	110	2	2	2	X
ejpam-4571	110	3	)	)	PUNCT
ejpam-4571	110	4	αλbo(x	αλbo(x	NOUN
ejpam-4571	110	5	,	,	PUNCT
ejpam-4571	110	6	τ	τ	PROPN
ejpam-4571	110	7	)	)	PUNCT
ejpam-4571	110	8	⊆	⊆	NUM
ejpam-4571	110	9	pλbo(x	pλbo(x	NOUN
ejpam-4571	110	10	,	,	PUNCT
ejpam-4571	110	11	τ	τ	PROPN
ejpam-4571	110	12	)	)	PUNCT
ejpam-4571	110	13	⊆	⊆	NUM
ejpam-4571	110	14	βλbo(x	βλbo(x	NOUN
ejpam-4571	110	15	,	,	PUNCT
ejpam-4571	110	16	τ	τ	PROPN
ejpam-4571	110	17	)	)	PUNCT
ejpam-4571	110	18	.	.	PUNCT
ejpam-4571	111	1	(	(	PUNCT
ejpam-4571	111	2	3	3	X
ejpam-4571	111	3	)	)	PUNCT
ejpam-4571	111	4	αλbo(x	αλbo(x	NOUN
ejpam-4571	111	5	,	,	PUNCT
ejpam-4571	111	6	τ	τ	X
ejpam-4571	111	7	)	)	PUNCT
ejpam-4571	111	8	=	=	SYM
ejpam-4571	111	9	sλbo(x	sλbo(x	PROPN
ejpam-4571	111	10	,	,	PUNCT
ejpam-4571	111	11	τ	τ	NOUN
ejpam-4571	111	12	)	)	PUNCT
ejpam-4571	111	13	∩	∩	ADJ
ejpam-4571	111	14	pλbo(x	pλbo(x	NOUN
ejpam-4571	111	15	,	,	PUNCT
ejpam-4571	111	16	τ	τ	PROPN
ejpam-4571	111	17	)	)	PUNCT
ejpam-4571	111	18	.	.	PUNCT
ejpam-4571	112	1	a	a	DET
ejpam-4571	112	2	subset	subset	NOUN
ejpam-4571	112	3	a	a	PRON
ejpam-4571	112	4	of	of	ADP
ejpam-4571	112	5	a	a	DET
ejpam-4571	112	6	topological	topological	ADJ
ejpam-4571	112	7	space	space	NOUN
ejpam-4571	112	8	(	(	PUNCT
ejpam-4571	112	9	x	x	X
ejpam-4571	112	10	,	,	PUNCT
ejpam-4571	112	11	τ	τ	X
ejpam-4571	112	12	)	)	PUNCT
ejpam-4571	112	13	is	be	AUX
ejpam-4571	112	14	called	call	VERB
ejpam-4571	112	15	r(λ	r(λ	NOUN
ejpam-4571	112	16	,	,	PUNCT
ejpam-4571	112	17	b)-open	b)-open	VERB
ejpam-4571	113	1	[	[	X
ejpam-4571	113	2	15	15	NUM
ejpam-4571	113	3	]	]	X
ejpam-4571	113	4	if	if	SCONJ
ejpam-4571	113	5	a	a	PRON
ejpam-4571	113	6	=	=	X
ejpam-4571	114	1	[	[	X
ejpam-4571	114	2	a(λ	a(λ	ADV
ejpam-4571	114	3	,	,	PUNCT
ejpam-4571	114	4	b)](λ	b)](λ	NOUN
ejpam-4571	114	5	,	,	PUNCT
ejpam-4571	114	6	b	b	NOUN
ejpam-4571	114	7	)	)	PUNCT
ejpam-4571	114	8	.	.	PUNCT
ejpam-4571	115	1	the	the	DET
ejpam-4571	115	2	complement	complement	NOUN
ejpam-4571	115	3	of	of	ADP
ejpam-4571	115	4	a	a	DET
ejpam-4571	115	5	r(λ	r(λ	NOUN
ejpam-4571	115	6	,	,	PUNCT
ejpam-4571	115	7	b)-open	b)-open	VERB
ejpam-4571	115	8	set	set	VERB
ejpam-4571	115	9	is	be	AUX
ejpam-4571	115	10	called	call	VERB
ejpam-4571	115	11	r(λ	r(λ	NOUN
ejpam-4571	115	12	,	,	PUNCT
ejpam-4571	115	13	b)-closed	b)-close	VERB
ejpam-4571	115	14	.	.	PUNCT
ejpam-4571	116	1	the	the	DET
ejpam-4571	116	2	family	family	NOUN
ejpam-4571	116	3	of	of	ADP
ejpam-4571	116	4	all	all	DET
ejpam-4571	116	5	r(λ	r(λ	NOUN
ejpam-4571	116	6	,	,	PUNCT
ejpam-4571	116	7	b)-open	b)-open	PUNCT
ejpam-4571	116	8	(	(	PUNCT
ejpam-4571	116	9	resp	resp	NOUN
ejpam-4571	116	10	.	.	PUNCT
ejpam-4571	117	1	r(λ	r(λ	NOUN
ejpam-4571	117	2	,	,	PUNCT
ejpam-4571	117	3	b)-closed	b)-closed	ADJ
ejpam-4571	117	4	)	)	PUNCT
ejpam-4571	117	5	sets	set	NOUN
ejpam-4571	117	6	in	in	ADP
ejpam-4571	117	7	a	a	DET
ejpam-4571	117	8	topological	topological	ADJ
ejpam-4571	117	9	space	space	NOUN
ejpam-4571	117	10	(	(	PUNCT
ejpam-4571	117	11	x	x	X
ejpam-4571	117	12	,	,	PUNCT
ejpam-4571	117	13	τ	τ	X
ejpam-4571	117	14	)	)	PUNCT
ejpam-4571	117	15	is	be	AUX
ejpam-4571	117	16	denoted	denote	VERB
ejpam-4571	117	17	by	by	ADP
ejpam-4571	117	18	rλbo(x	rλbo(x	PROPN
ejpam-4571	117	19	,	,	PUNCT
ejpam-4571	117	20	τ	τ	X
ejpam-4571	117	21	)	)	PUNCT
ejpam-4571	117	22	(	(	PUNCT
ejpam-4571	117	23	resp	resp	NOUN
ejpam-4571	117	24	.	.	PUNCT
ejpam-4571	118	1	rλbc(x	rλbc(x	PROPN
ejpam-4571	118	2	,	,	PUNCT
ejpam-4571	118	3	τ	τ	PROPN
ejpam-4571	118	4	)	)	PUNCT
ejpam-4571	118	5	)	)	PUNCT
ejpam-4571	118	6	.	.	PUNCT
ejpam-4571	119	1	proposition	proposition	NOUN
ejpam-4571	119	2	2	2	NUM
ejpam-4571	119	3	.	.	X
ejpam-4571	119	4	for	for	ADP
ejpam-4571	119	5	a	a	DET
ejpam-4571	119	6	subset	subset	NOUN
ejpam-4571	119	7	a	a	PRON
ejpam-4571	119	8	of	of	ADP
ejpam-4571	119	9	a	a	DET
ejpam-4571	119	10	topological	topological	ADJ
ejpam-4571	119	11	space	space	NOUN
ejpam-4571	119	12	(	(	PUNCT
ejpam-4571	119	13	x	x	X
ejpam-4571	119	14	,	,	PUNCT
ejpam-4571	119	15	τ	τ	PROPN
ejpam-4571	119	16	)	)	PUNCT
ejpam-4571	119	17	,	,	PUNCT
ejpam-4571	119	18	the	the	DET
ejpam-4571	119	19	following	follow	VERB
ejpam-4571	119	20	properties	property	NOUN
ejpam-4571	119	21	are	be	AUX
ejpam-4571	119	22	equivalent	equivalent	ADJ
ejpam-4571	119	23	:	:	PUNCT
ejpam-4571	119	24	(	(	PUNCT
ejpam-4571	119	25	1	1	X
ejpam-4571	119	26	)	)	PUNCT
ejpam-4571	119	27	a	a	PRON
ejpam-4571	119	28	is	be	AUX
ejpam-4571	119	29	r(λ	r(λ	NOUN
ejpam-4571	119	30	,	,	PUNCT
ejpam-4571	119	31	b)-open	b)-open	VERB
ejpam-4571	119	32	.	.	PUNCT
ejpam-4571	120	1	(	(	PUNCT
ejpam-4571	120	2	2	2	X
ejpam-4571	120	3	)	)	PUNCT
ejpam-4571	120	4	a	a	PRON
ejpam-4571	120	5	is	be	AUX
ejpam-4571	120	6	(	(	PUNCT
ejpam-4571	120	7	λ	λ	X
ejpam-4571	120	8	,	,	PUNCT
ejpam-4571	120	9	b)-open	b)-open	PUNCT
ejpam-4571	120	10	and	and	CCONJ
ejpam-4571	120	11	s(λ	s(λ	NOUN
ejpam-4571	120	12	,	,	PUNCT
ejpam-4571	120	13	b)-closed	b)-close	VERB
ejpam-4571	120	14	.	.	PUNCT
ejpam-4571	121	1	(	(	PUNCT
ejpam-4571	121	2	3	3	X
ejpam-4571	121	3	)	)	PUNCT
ejpam-4571	121	4	a	a	PRON
ejpam-4571	121	5	is	be	AUX
ejpam-4571	121	6	α(λ	α(λ	PROPN
ejpam-4571	121	7	,	,	PUNCT
ejpam-4571	121	8	b)-open	b)-open	PUNCT
ejpam-4571	121	9	and	and	CCONJ
ejpam-4571	121	10	s(λ	s(λ	NOUN
ejpam-4571	121	11	,	,	PUNCT
ejpam-4571	121	12	b)-closed	b)-close	VERB
ejpam-4571	121	13	.	.	PUNCT
ejpam-4571	122	1	(	(	PUNCT
ejpam-4571	122	2	4	4	X
ejpam-4571	122	3	)	)	PUNCT
ejpam-4571	122	4	a	a	PRON
ejpam-4571	122	5	is	be	AUX
ejpam-4571	122	6	p(λ	p(λ	NOUN
ejpam-4571	122	7	,	,	PUNCT
ejpam-4571	122	8	b)-open	b)-open	PUNCT
ejpam-4571	122	9	and	and	CCONJ
ejpam-4571	122	10	s(λ	s(λ	NOUN
ejpam-4571	122	11	,	,	PUNCT
ejpam-4571	122	12	b)-closed	b)-close	VERB
ejpam-4571	122	13	.	.	PUNCT
ejpam-4571	123	1	(	(	PUNCT
ejpam-4571	123	2	5	5	X
ejpam-4571	123	3	)	)	PUNCT
ejpam-4571	123	4	a	a	PRON
ejpam-4571	123	5	is	be	AUX
ejpam-4571	123	6	(	(	PUNCT
ejpam-4571	123	7	λ	λ	X
ejpam-4571	123	8	,	,	PUNCT
ejpam-4571	123	9	b)-open	b)-open	PUNCT
ejpam-4571	123	10	and	and	CCONJ
ejpam-4571	123	11	β(λ	β(λ	NOUN
ejpam-4571	123	12	,	,	PUNCT
ejpam-4571	123	13	b)-closed	b)-close	VERB
ejpam-4571	123	14	.	.	PUNCT
ejpam-4571	124	1	(	(	PUNCT
ejpam-4571	124	2	6	6	NUM
ejpam-4571	124	3	)	)	PUNCT
ejpam-4571	124	4	a	a	PRON
ejpam-4571	124	5	is	be	AUX
ejpam-4571	124	6	α(λ	α(λ	PROPN
ejpam-4571	124	7	,	,	PUNCT
ejpam-4571	124	8	b)-open	b)-open	PUNCT
ejpam-4571	124	9	and	and	CCONJ
ejpam-4571	124	10	β(λ	β(λ	NOUN
ejpam-4571	124	11	,	,	PUNCT
ejpam-4571	124	12	b)-closed	b)-closed	ADJ
ejpam-4571	124	13	.	.	PUNCT
ejpam-4571	125	1	proof	proof	NOUN
ejpam-4571	125	2	.	.	PUNCT
ejpam-4571	126	1	(	(	PUNCT
ejpam-4571	126	2	1	1	X
ejpam-4571	126	3	)	)	PUNCT
ejpam-4571	126	4	⇒	⇒	NOUN
ejpam-4571	126	5	(	(	PUNCT
ejpam-4571	126	6	2	2	NUM
ejpam-4571	126	7	)	)	PUNCT
ejpam-4571	126	8	⇒	⇒	NOUN
ejpam-4571	126	9	(	(	PUNCT
ejpam-4571	126	10	3	3	NUM
ejpam-4571	126	11	)	)	PUNCT
ejpam-4571	126	12	⇒	⇒	NOUN
ejpam-4571	126	13	(	(	PUNCT
ejpam-4571	126	14	4	4	NUM
ejpam-4571	126	15	):	):	PUNCT
ejpam-4571	126	16	obvious	obvious	ADJ
ejpam-4571	126	17	.	.	PUNCT
ejpam-4571	127	1	(	(	PUNCT
ejpam-4571	127	2	4	4	X
ejpam-4571	127	3	)	)	PUNCT
ejpam-4571	127	4	⇒	⇒	NOUN
ejpam-4571	127	5	(	(	PUNCT
ejpam-4571	127	6	5	5	NUM
ejpam-4571	127	7	):	):	PUNCT
ejpam-4571	127	8	let	let	VERB
ejpam-4571	127	9	a	a	DET
ejpam-4571	127	10	be	be	AUX
ejpam-4571	127	11	(	(	PUNCT
ejpam-4571	127	12	λ	λ	X
ejpam-4571	127	13	,	,	PUNCT
ejpam-4571	127	14	b)-open	b)-open	PUNCT
ejpam-4571	127	15	and	and	CCONJ
ejpam-4571	127	16	s(λ	s(λ	NOUN
ejpam-4571	127	17	,	,	PUNCT
ejpam-4571	127	18	b)-closed	b)-close	VERB
ejpam-4571	127	19	.	.	PUNCT
ejpam-4571	128	1	then	then	ADV
ejpam-4571	128	2	,	,	PUNCT
ejpam-4571	128	3	a	a	DET
ejpam-4571	128	4	⊆	⊆	NUM
ejpam-4571	128	5	[	[	X
ejpam-4571	128	6	a(λ	a(λ	ADV
ejpam-4571	128	7	,	,	PUNCT
ejpam-4571	128	8	b)](λ	b)](λ	NOUN
ejpam-4571	128	9	,	,	PUNCT
ejpam-4571	128	10	b	b	NOUN
ejpam-4571	128	11	)	)	PUNCT
ejpam-4571	128	12	and	and	CCONJ
ejpam-4571	128	13	[	[	X
ejpam-4571	128	14	a(λ	a(λ	ADV
ejpam-4571	128	15	,	,	PUNCT
ejpam-4571	128	16	b)](λ	b)](λ	NOUN
ejpam-4571	128	17	,	,	PUNCT
ejpam-4571	128	18	b	b	NOUN
ejpam-4571	128	19	)	)	PUNCT
ejpam-4571	128	20	⊆	⊆	NUM
ejpam-4571	128	21	a.	a.	NOUN
ejpam-4571	128	22	this	this	PRON
ejpam-4571	128	23	implies	imply	VERB
ejpam-4571	128	24	that	that	SCONJ
ejpam-4571	128	25	a	a	PRON
ejpam-4571	128	26	=	=	X
ejpam-4571	129	1	[	[	X
ejpam-4571	129	2	a(λ	a(λ	ADV
ejpam-4571	129	3	,	,	PUNCT
ejpam-4571	129	4	b)](λ	b)](λ	NOUN
ejpam-4571	129	5	,	,	PUNCT
ejpam-4571	129	6	b	b	NOUN
ejpam-4571	129	7	)	)	PUNCT
ejpam-4571	129	8	.	.	PUNCT
ejpam-4571	130	1	thus	thus	ADV
ejpam-4571	130	2	,	,	PUNCT
ejpam-4571	130	3	a	a	PRON
ejpam-4571	130	4	is	be	AUX
ejpam-4571	130	5	r(λ	r(λ	NOUN
ejpam-4571	130	6	,	,	PUNCT
ejpam-4571	130	7	b)-open	b)-open	PUNCT
ejpam-4571	130	8	and	and	CCONJ
ejpam-4571	130	9	hence	hence	ADV
ejpam-4571	130	10	a	a	PRON
ejpam-4571	130	11	is	be	AUX
ejpam-4571	130	12	(	(	PUNCT
ejpam-4571	130	13	λ	λ	INTJ
ejpam-4571	130	14	,	,	PUNCT
ejpam-4571	130	15	b)-open	b)-open	VERB
ejpam-4571	130	16	.	.	PUNCT
ejpam-4571	131	1	since	since	SCONJ
ejpam-4571	131	2	every	every	DET
ejpam-4571	131	3	s(λ	s(λ	NOUN
ejpam-4571	131	4	,	,	PUNCT
ejpam-4571	131	5	b)-closed	b)-close	VERB
ejpam-4571	131	6	set	set	NOUN
ejpam-4571	131	7	is	be	AUX
ejpam-4571	131	8	β(λ	β(λ	NOUN
ejpam-4571	131	9	,	,	PUNCT
ejpam-4571	131	10	b)-closed	b)-close	VERB
ejpam-4571	131	11	.	.	PUNCT
ejpam-4571	132	1	this	this	PRON
ejpam-4571	132	2	shows	show	VERB
ejpam-4571	132	3	that	that	SCONJ
ejpam-4571	132	4	a	a	PRON
ejpam-4571	132	5	is	be	AUX
ejpam-4571	132	6	(	(	PUNCT
ejpam-4571	132	7	λ	λ	X
ejpam-4571	132	8	,	,	PUNCT
ejpam-4571	132	9	b)-open	b)-open	PUNCT
ejpam-4571	132	10	and	and	CCONJ
ejpam-4571	132	11	β(λ	β(λ	NOUN
ejpam-4571	132	12	,	,	PUNCT
ejpam-4571	132	13	b)-closed	b)-close	VERB
ejpam-4571	132	14	.	.	PUNCT
ejpam-4571	133	1	(	(	PUNCT
ejpam-4571	133	2	5	5	X
ejpam-4571	133	3	)	)	PUNCT
ejpam-4571	133	4	⇒	⇒	NOUN
ejpam-4571	133	5	(	(	PUNCT
ejpam-4571	133	6	6	6	NUM
ejpam-4571	133	7	):	):	PUNCT
ejpam-4571	133	8	obvious	obvious	ADJ
ejpam-4571	133	9	.	.	PUNCT
ejpam-4571	134	1	(	(	PUNCT
ejpam-4571	134	2	6	6	NUM
ejpam-4571	134	3	)	)	PUNCT
ejpam-4571	134	4	⇒	⇒	NOUN
ejpam-4571	134	5	(	(	PUNCT
ejpam-4571	134	6	1	1	NUM
ejpam-4571	134	7	):	):	PUNCT
ejpam-4571	134	8	let	let	VERB
ejpam-4571	134	9	a	a	PRON
ejpam-4571	134	10	be	be	AUX
ejpam-4571	134	11	α(λ	α(λ	PROPN
ejpam-4571	134	12	,	,	PUNCT
ejpam-4571	134	13	b)-open	b)-open	PUNCT
ejpam-4571	134	14	and	and	CCONJ
ejpam-4571	134	15	β(λ	β(λ	NOUN
ejpam-4571	134	16	,	,	PUNCT
ejpam-4571	134	17	b)-closed	b)-close	VERB
ejpam-4571	134	18	.	.	PUNCT
ejpam-4571	135	1	then	then	ADV
ejpam-4571	135	2	,	,	PUNCT
ejpam-4571	135	3	a	a	DET
ejpam-4571	135	4	⊆	⊆	NUM
ejpam-4571	135	5	[	[	X
ejpam-4571	135	6	[	[	X
ejpam-4571	135	7	a(λ	a(λ	ADJ
ejpam-4571	135	8	,	,	PUNCT
ejpam-4571	135	9	b	b	NOUN
ejpam-4571	135	10	)	)	PUNCT
ejpam-4571	135	11	]	]	PUNCT
ejpam-4571	136	1	(	(	PUNCT
ejpam-4571	136	2	λ	λ	INTJ
ejpam-4571	136	3	,	,	PUNCT
ejpam-4571	136	4	b)](λ	b)](λ	NOUN
ejpam-4571	136	5	,	,	PUNCT
ejpam-4571	136	6	b	b	NOUN
ejpam-4571	136	7	)	)	PUNCT
ejpam-4571	136	8	and	and	CCONJ
ejpam-4571	137	1	[	[	X
ejpam-4571	137	2	[	[	X
ejpam-4571	137	3	a(λ	a(λ	ADJ
ejpam-4571	137	4	,	,	PUNCT
ejpam-4571	137	5	b	b	NOUN
ejpam-4571	137	6	)	)	PUNCT
ejpam-4571	137	7	]	]	PUNCT
ejpam-4571	138	1	(	(	PUNCT
ejpam-4571	138	2	λ	λ	INTJ
ejpam-4571	138	3	,	,	PUNCT
ejpam-4571	138	4	b)](λ	b)](λ	NOUN
ejpam-4571	138	5	,	,	PUNCT
ejpam-4571	138	6	b	b	NOUN
ejpam-4571	138	7	)	)	PUNCT
ejpam-4571	138	8	⊆	⊆	NUM
ejpam-4571	138	9	a.	a.	NOUN
ejpam-4571	138	10	this	this	PRON
ejpam-4571	138	11	shows	show	VERB
ejpam-4571	138	12	that	that	SCONJ
ejpam-4571	138	13	a	a	PRON
ejpam-4571	138	14	=	=	X
ejpam-4571	139	1	[	[	X
ejpam-4571	139	2	[	[	X
ejpam-4571	139	3	a(λ	a(λ	ADJ
ejpam-4571	139	4	,	,	PUNCT
ejpam-4571	139	5	b	b	NOUN
ejpam-4571	139	6	)	)	PUNCT
ejpam-4571	139	7	]	]	PUNCT
ejpam-4571	140	1	(	(	PUNCT
ejpam-4571	140	2	λ	λ	INTJ
ejpam-4571	140	3	,	,	PUNCT
ejpam-4571	140	4	b)](λ	b)](λ	NOUN
ejpam-4571	140	5	,	,	PUNCT
ejpam-4571	140	6	b	b	NOUN
ejpam-4571	140	7	)	)	PUNCT
ejpam-4571	140	8	.	.	PUNCT
ejpam-4571	141	1	thus	thus	ADV
ejpam-4571	141	2	,	,	PUNCT
ejpam-4571	141	3	a(λ	a(λ	ADV
ejpam-4571	141	4	,	,	PUNCT
ejpam-4571	141	5	b	b	NOUN
ejpam-4571	141	6	)	)	PUNCT
ejpam-4571	141	7	=	=	PUNCT
ejpam-4571	142	1	[	[	X
ejpam-4571	142	2	[	[	X
ejpam-4571	142	3	a(λ	a(λ	ADJ
ejpam-4571	142	4	,	,	PUNCT
ejpam-4571	142	5	b	b	NOUN
ejpam-4571	142	6	)	)	PUNCT
ejpam-4571	142	7	]	]	PUNCT
ejpam-4571	143	1	(	(	PUNCT
ejpam-4571	143	2	λ	λ	INTJ
ejpam-4571	143	3	,	,	PUNCT
ejpam-4571	143	4	b)](λ	b)](λ	NOUN
ejpam-4571	143	5	,	,	PUNCT
ejpam-4571	143	6	b	b	NOUN
ejpam-4571	143	7	)	)	PUNCT
ejpam-4571	143	8	=	=	PUNCT
ejpam-4571	143	9	a	a	PRON
ejpam-4571	143	10	and	and	CCONJ
ejpam-4571	143	11	hence	hence	ADV
ejpam-4571	143	12	[	[	X
ejpam-4571	143	13	a(λ	a(λ	ADV
ejpam-4571	143	14	,	,	PUNCT
ejpam-4571	143	15	b)](λ	b)](λ	NOUN
ejpam-4571	143	16	,	,	PUNCT
ejpam-4571	143	17	b	b	NOUN
ejpam-4571	143	18	)	)	PUNCT
ejpam-4571	143	19	=	=	PUNCT
ejpam-4571	144	1	[	[	X
ejpam-4571	144	2	[	[	X
ejpam-4571	144	3	a(λ	a(λ	ADJ
ejpam-4571	144	4	,	,	PUNCT
ejpam-4571	144	5	b	b	NOUN
ejpam-4571	144	6	)	)	PUNCT
ejpam-4571	144	7	]	]	PUNCT
ejpam-4571	145	1	(	(	PUNCT
ejpam-4571	145	2	λ	λ	INTJ
ejpam-4571	145	3	,	,	PUNCT
ejpam-4571	145	4	b)](λ	b)](λ	NOUN
ejpam-4571	145	5	,	,	PUNCT
ejpam-4571	145	6	b	b	NOUN
ejpam-4571	145	7	)	)	PUNCT
ejpam-4571	145	8	=	=	SYM
ejpam-4571	146	1	a.	a.	NOUN
ejpam-4571	146	2	therefore	therefore	ADV
ejpam-4571	146	3	,	,	PUNCT
ejpam-4571	146	4	a	a	PRON
ejpam-4571	146	5	is	be	AUX
ejpam-4571	146	6	r(λ	r(λ	NOUN
ejpam-4571	146	7	,	,	PUNCT
ejpam-4571	146	8	b)-open	b)-open	PROPN
ejpam-4571	146	9	.	.	PUNCT
ejpam-4571	147	1	c.	c.	PROPN
ejpam-4571	147	2	boonpok	boonpok	PROPN
ejpam-4571	147	3	,	,	PUNCT
ejpam-4571	147	4	n.	n.	PROPN
ejpam-4571	147	5	srisarakham	srisarakham	PROPN
ejpam-4571	147	6	/	/	SYM
ejpam-4571	147	7	eur	eur	PROPN
ejpam-4571	147	8	.	.	PUNCT
ejpam-4571	148	1	j.	j.	PROPN
ejpam-4571	148	2	pure	pure	PROPN
ejpam-4571	148	3	appl	appl	PROPN
ejpam-4571	148	4	.	.	PROPN
ejpam-4571	148	5	math	math	PROPN
ejpam-4571	148	6	,	,	PUNCT
ejpam-4571	148	7	16	16	NUM
ejpam-4571	148	8	(	(	PUNCT
ejpam-4571	148	9	1	1	NUM
ejpam-4571	148	10	)	)	PUNCT
ejpam-4571	148	11	(	(	PUNCT
ejpam-4571	148	12	2023	2023	NUM
ejpam-4571	148	13	)	)	PUNCT
ejpam-4571	148	14	,	,	PUNCT
ejpam-4571	148	15	29	29	NUM
ejpam-4571	148	16	-	-	SYM
ejpam-4571	148	17	43	43	NUM
ejpam-4571	148	18	33	33	NUM
ejpam-4571	148	19	corollary	corollary	ADJ
ejpam-4571	148	20	1	1	NUM
ejpam-4571	148	21	.	.	PUNCT
ejpam-4571	149	1	for	for	ADP
ejpam-4571	149	2	a	a	DET
ejpam-4571	149	3	subset	subset	NOUN
ejpam-4571	149	4	a	a	PRON
ejpam-4571	149	5	of	of	ADP
ejpam-4571	149	6	a	a	DET
ejpam-4571	149	7	topological	topological	ADJ
ejpam-4571	149	8	space	space	NOUN
ejpam-4571	149	9	(	(	PUNCT
ejpam-4571	149	10	x	x	X
ejpam-4571	149	11	,	,	PUNCT
ejpam-4571	149	12	τ	τ	PROPN
ejpam-4571	149	13	)	)	PUNCT
ejpam-4571	149	14	,	,	PUNCT
ejpam-4571	149	15	the	the	DET
ejpam-4571	149	16	following	follow	VERB
ejpam-4571	149	17	properties	property	NOUN
ejpam-4571	149	18	are	be	AUX
ejpam-4571	149	19	equivalent	equivalent	ADJ
ejpam-4571	149	20	:	:	PUNCT
ejpam-4571	149	21	(	(	PUNCT
ejpam-4571	149	22	1	1	X
ejpam-4571	149	23	)	)	PUNCT
ejpam-4571	149	24	a	a	PRON
ejpam-4571	149	25	is	be	AUX
ejpam-4571	149	26	r(λ	r(λ	NOUN
ejpam-4571	149	27	,	,	PUNCT
ejpam-4571	149	28	b)-closed	b)-closed	ADJ
ejpam-4571	149	29	.	.	PUNCT
ejpam-4571	150	1	(	(	PUNCT
ejpam-4571	150	2	2	2	X
ejpam-4571	150	3	)	)	PUNCT
ejpam-4571	150	4	a	a	DET
ejpam-4571	150	5	is	be	AUX
ejpam-4571	150	6	(	(	PUNCT
ejpam-4571	150	7	λ	λ	PROPN
ejpam-4571	150	8	,	,	PUNCT
ejpam-4571	150	9	b)-closed	b)-close	VERB
ejpam-4571	150	10	and	and	CCONJ
ejpam-4571	150	11	s(λ	s(λ	NOUN
ejpam-4571	150	12	,	,	PUNCT
ejpam-4571	150	13	b)-open	b)-open	VERB
ejpam-4571	150	14	.	.	PUNCT
ejpam-4571	151	1	(	(	PUNCT
ejpam-4571	151	2	3	3	X
ejpam-4571	151	3	)	)	PUNCT
ejpam-4571	151	4	a	a	PRON
ejpam-4571	151	5	is	be	AUX
ejpam-4571	151	6	α(λ	α(λ	PROPN
ejpam-4571	151	7	,	,	PUNCT
ejpam-4571	151	8	b)-closed	b)-close	VERB
ejpam-4571	151	9	and	and	CCONJ
ejpam-4571	151	10	s(λ	s(λ	NOUN
ejpam-4571	151	11	,	,	PUNCT
ejpam-4571	151	12	b)-open	b)-open	VERB
ejpam-4571	151	13	.	.	PUNCT
ejpam-4571	152	1	(	(	PUNCT
ejpam-4571	152	2	4	4	X
ejpam-4571	152	3	)	)	PUNCT
ejpam-4571	152	4	a	a	PRON
ejpam-4571	152	5	is	be	AUX
ejpam-4571	152	6	p(λ	p(λ	NOUN
ejpam-4571	152	7	,	,	PUNCT
ejpam-4571	152	8	b)-closed	b)-close	VERB
ejpam-4571	152	9	and	and	CCONJ
ejpam-4571	152	10	s(λ	s(λ	NOUN
ejpam-4571	152	11	,	,	PUNCT
ejpam-4571	152	12	b)-open	b)-open	VERB
ejpam-4571	152	13	.	.	PUNCT
ejpam-4571	153	1	(	(	PUNCT
ejpam-4571	153	2	5	5	X
ejpam-4571	153	3	)	)	PUNCT
ejpam-4571	153	4	a	a	PRON
ejpam-4571	153	5	is	be	AUX
ejpam-4571	153	6	(	(	PUNCT
ejpam-4571	153	7	λ	λ	PROPN
ejpam-4571	153	8	,	,	PUNCT
ejpam-4571	153	9	b)-closed	b)-close	VERB
ejpam-4571	153	10	and	and	CCONJ
ejpam-4571	153	11	β(λ	β(λ	X
ejpam-4571	153	12	,	,	PUNCT
ejpam-4571	153	13	b)-open	b)-open	VERB
ejpam-4571	153	14	.	.	PUNCT
ejpam-4571	154	1	(	(	PUNCT
ejpam-4571	154	2	6	6	NUM
ejpam-4571	154	3	)	)	PUNCT
ejpam-4571	154	4	a	a	PRON
ejpam-4571	154	5	is	be	AUX
ejpam-4571	154	6	α(λ	α(λ	PROPN
ejpam-4571	154	7	,	,	PUNCT
ejpam-4571	154	8	b)-closed	b)-close	VERB
ejpam-4571	154	9	and	and	CCONJ
ejpam-4571	154	10	β(λ	β(λ	PROPN
ejpam-4571	154	11	,	,	PUNCT
ejpam-4571	154	12	b)-open	b)-open	VERB
ejpam-4571	154	13	.	.	PUNCT
ejpam-4571	155	1	definition	definition	NOUN
ejpam-4571	155	2	2	2	NUM
ejpam-4571	155	3	.	.	PUNCT
ejpam-4571	156	1	a	a	DET
ejpam-4571	156	2	subset	subset	NOUN
ejpam-4571	156	3	a	a	PRON
ejpam-4571	156	4	of	of	ADP
ejpam-4571	156	5	a	a	DET
ejpam-4571	156	6	topological	topological	ADJ
ejpam-4571	156	7	space	space	NOUN
ejpam-4571	156	8	(	(	PUNCT
ejpam-4571	156	9	x	x	X
ejpam-4571	156	10	,	,	PUNCT
ejpam-4571	156	11	τ	τ	X
ejpam-4571	156	12	)	)	PUNCT
ejpam-4571	156	13	is	be	AUX
ejpam-4571	156	14	said	say	VERB
ejpam-4571	156	15	to	to	PART
ejpam-4571	156	16	be	be	AUX
ejpam-4571	156	17	b(λ	b(λ	PROPN
ejpam-4571	156	18	,	,	PUNCT
ejpam-4571	156	19	b)-open	b)-open	VERB
ejpam-4571	156	20	if	if	SCONJ
ejpam-4571	156	21	a	a	DET
ejpam-4571	156	22	⊆	⊆	NUM
ejpam-4571	156	23	[	[	X
ejpam-4571	156	24	a(λ	a(λ	ADJ
ejpam-4571	156	25	,	,	PUNCT
ejpam-4571	156	26	b	b	NOUN
ejpam-4571	156	27	)	)	PUNCT
ejpam-4571	156	28	]	]	PUNCT
ejpam-4571	157	1	(	(	PUNCT
ejpam-4571	157	2	λ	λ	X
ejpam-4571	157	3	,	,	PUNCT
ejpam-4571	157	4	b)∪	b)∪	NOUN
ejpam-4571	157	5	[	[	X
ejpam-4571	157	6	a(λ	a(λ	ADV
ejpam-4571	157	7	,	,	PUNCT
ejpam-4571	157	8	b)](λ	b)](λ	NOUN
ejpam-4571	157	9	,	,	PUNCT
ejpam-4571	157	10	b	b	NOUN
ejpam-4571	157	11	)	)	PUNCT
ejpam-4571	157	12	.	.	PUNCT
ejpam-4571	158	1	the	the	DET
ejpam-4571	158	2	complement	complement	NOUN
ejpam-4571	158	3	of	of	ADP
ejpam-4571	158	4	a	a	DET
ejpam-4571	158	5	b(λ	b(λ	NOUN
ejpam-4571	158	6	,	,	PUNCT
ejpam-4571	158	7	b)-open	b)-open	PUNCT
ejpam-4571	158	8	set	set	VERB
ejpam-4571	158	9	is	be	AUX
ejpam-4571	158	10	said	say	VERB
ejpam-4571	158	11	to	to	PART
ejpam-4571	158	12	be	be	AUX
ejpam-4571	158	13	b(λ	b(λ	PROPN
ejpam-4571	158	14	,	,	PUNCT
ejpam-4571	158	15	b)-closed	b)-close	VERB
ejpam-4571	158	16	.	.	PUNCT
ejpam-4571	158	17	example	example	NOUN
ejpam-4571	159	1	2	2	NUM
ejpam-4571	159	2	.	.	PUNCT
ejpam-4571	159	3	let	let	VERB
ejpam-4571	159	4	x	x	PUNCT
ejpam-4571	159	5	=	=	NOUN
ejpam-4571	159	6	{	{	PUNCT
ejpam-4571	159	7	−2,−1	−2,−1	PROPN
ejpam-4571	159	8	,	,	PUNCT
ejpam-4571	159	9	0	0	NUM
ejpam-4571	159	10	,	,	PUNCT
ejpam-4571	159	11	1	1	NUM
ejpam-4571	159	12	,	,	PUNCT
ejpam-4571	159	13	2	2	NUM
ejpam-4571	159	14	}	}	PUNCT
ejpam-4571	159	15	with	with	ADP
ejpam-4571	159	16	the	the	DET
ejpam-4571	159	17	topology	topology	NOUN
ejpam-4571	159	18	τ	τ	X
ejpam-4571	159	19	=	=	PUNCT
ejpam-4571	159	20	{	{	PUNCT
ejpam-4571	159	21	∅	∅	NOUN
ejpam-4571	159	22	,	,	PUNCT
ejpam-4571	159	23	{	{	PUNCT
ejpam-4571	159	24	−2,−1	−2,−1	ADV
ejpam-4571	159	25	}	}	PUNCT
ejpam-4571	159	26	,	,	PUNCT
ejpam-4571	159	27	{	{	PUNCT
ejpam-4571	159	28	0	0	NUM
ejpam-4571	159	29	,	,	PUNCT
ejpam-4571	159	30	1	1	NUM
ejpam-4571	159	31	,	,	PUNCT
ejpam-4571	159	32	2	2	NUM
ejpam-4571	159	33	}	}	PUNCT
ejpam-4571	159	34	,	,	PUNCT
ejpam-4571	159	35	x	x	NOUN
ejpam-4571	159	36	}	}	PUNCT
ejpam-4571	159	37	.	.	PUNCT
ejpam-4571	160	1	let	let	VERB
ejpam-4571	160	2	c	c	NOUN
ejpam-4571	160	3	=	=	PUNCT
ejpam-4571	160	4	{	{	PUNCT
ejpam-4571	160	5	−2,−1	−2,−1	ADV
ejpam-4571	160	6	}	}	PUNCT
ejpam-4571	160	7	.	.	PUNCT
ejpam-4571	161	1	then	then	ADV
ejpam-4571	161	2	,	,	PUNCT
ejpam-4571	161	3	c	c	PROPN
ejpam-4571	161	4	is	be	AUX
ejpam-4571	161	5	b(λ	b(λ	PROPN
ejpam-4571	161	6	,	,	PUNCT
ejpam-4571	161	7	b)-open	b)-open	VERB
ejpam-4571	161	8	.	.	PUNCT
ejpam-4571	162	1	the	the	DET
ejpam-4571	162	2	family	family	NOUN
ejpam-4571	162	3	of	of	ADP
ejpam-4571	162	4	all	all	DET
ejpam-4571	162	5	b(λ	b(λ	NOUN
ejpam-4571	162	6	,	,	PUNCT
ejpam-4571	162	7	b)-open	b)-open	PUNCT
ejpam-4571	162	8	(	(	PUNCT
ejpam-4571	162	9	resp	resp	NOUN
ejpam-4571	162	10	.	.	PUNCT
ejpam-4571	163	1	b(λ	b(λ	NOUN
ejpam-4571	163	2	,	,	PUNCT
ejpam-4571	163	3	b)-closed	b)-closed	ADJ
ejpam-4571	163	4	)	)	PUNCT
ejpam-4571	163	5	sets	set	NOUN
ejpam-4571	163	6	in	in	ADP
ejpam-4571	163	7	a	a	DET
ejpam-4571	163	8	topological	topological	ADJ
ejpam-4571	163	9	space	space	NOUN
ejpam-4571	163	10	(	(	PUNCT
ejpam-4571	163	11	x	x	X
ejpam-4571	163	12	,	,	PUNCT
ejpam-4571	163	13	τ	τ	X
ejpam-4571	163	14	)	)	PUNCT
ejpam-4571	163	15	is	be	AUX
ejpam-4571	163	16	denoted	denote	VERB
ejpam-4571	163	17	by	by	ADP
ejpam-4571	163	18	bλbo(x	bλbo(x	PROPN
ejpam-4571	163	19	,	,	PUNCT
ejpam-4571	163	20	τ	τ	X
ejpam-4571	163	21	)	)	PUNCT
ejpam-4571	163	22	(	(	PUNCT
ejpam-4571	163	23	resp	resp	NOUN
ejpam-4571	163	24	.	.	PUNCT
ejpam-4571	164	1	bλbc(x	bλbc(x	NOUN
ejpam-4571	164	2	,	,	PUNCT
ejpam-4571	164	3	τ	τ	PROPN
ejpam-4571	164	4	)	)	PUNCT
ejpam-4571	164	5	)	)	PUNCT
ejpam-4571	164	6	.	.	PUNCT
ejpam-4571	165	1	proposition	proposition	NOUN
ejpam-4571	165	2	3	3	X
ejpam-4571	165	3	.	.	PUNCT
ejpam-4571	165	4	let	let	VERB
ejpam-4571	165	5	a	a	DET
ejpam-4571	165	6	be	be	AUX
ejpam-4571	165	7	a	a	DET
ejpam-4571	165	8	subset	subset	NOUN
ejpam-4571	165	9	of	of	ADP
ejpam-4571	165	10	a	a	DET
ejpam-4571	165	11	topological	topological	ADJ
ejpam-4571	165	12	space	space	NOUN
ejpam-4571	165	13	(	(	PUNCT
ejpam-4571	165	14	x	x	X
ejpam-4571	165	15	,	,	PUNCT
ejpam-4571	165	16	τ	τ	PROPN
ejpam-4571	165	17	)	)	PUNCT
ejpam-4571	165	18	.	.	PUNCT
ejpam-4571	166	1	if	if	SCONJ
ejpam-4571	166	2	a	a	PRON
ejpam-4571	166	3	is	be	AUX
ejpam-4571	166	4	b(λ	b(λ	NOUN
ejpam-4571	166	5	,	,	PUNCT
ejpam-4571	166	6	b)-open	b)-open	VERB
ejpam-4571	166	7	,	,	PUNCT
ejpam-4571	166	8	then	then	ADV
ejpam-4571	166	9	a(λ	a(λ	ADV
ejpam-4571	166	10	,	,	PUNCT
ejpam-4571	166	11	b	b	NOUN
ejpam-4571	166	12	)	)	PUNCT
ejpam-4571	166	13	is	be	AUX
ejpam-4571	166	14	r(λ	r(λ	NOUN
ejpam-4571	166	15	,	,	PUNCT
ejpam-4571	166	16	b)-closed	b)-closed	ADJ
ejpam-4571	166	17	.	.	PUNCT
ejpam-4571	167	1	proof	proof	NOUN
ejpam-4571	167	2	.	.	PUNCT
ejpam-4571	168	1	suppose	suppose	VERB
ejpam-4571	168	2	that	that	SCONJ
ejpam-4571	168	3	a	a	PRON
ejpam-4571	168	4	is	be	AUX
ejpam-4571	168	5	b(λ	b(λ	NOUN
ejpam-4571	168	6	,	,	PUNCT
ejpam-4571	168	7	b)-open	b)-open	VERB
ejpam-4571	168	8	.	.	PUNCT
ejpam-4571	169	1	then	then	ADV
ejpam-4571	169	2	,	,	PUNCT
ejpam-4571	169	3	a	a	DET
ejpam-4571	169	4	⊆	⊆	NUM
ejpam-4571	169	5	[	[	X
ejpam-4571	169	6	a(λ	a(λ	ADJ
ejpam-4571	169	7	,	,	PUNCT
ejpam-4571	169	8	b	b	NOUN
ejpam-4571	169	9	)	)	PUNCT
ejpam-4571	169	10	]	]	PUNCT
ejpam-4571	170	1	(	(	PUNCT
ejpam-4571	170	2	λ	λ	X
ejpam-4571	170	3	,	,	PUNCT
ejpam-4571	170	4	b	b	NOUN
ejpam-4571	170	5	)	)	PUNCT
ejpam-4571	170	6	∪	∪	ADP
ejpam-4571	170	7	[	[	X
ejpam-4571	170	8	a(λ	a(λ	ADV
ejpam-4571	170	9	,	,	PUNCT
ejpam-4571	170	10	b)](λ	b)](λ	NOUN
ejpam-4571	170	11	,	,	PUNCT
ejpam-4571	170	12	b	b	NOUN
ejpam-4571	170	13	)	)	PUNCT
ejpam-4571	170	14	.	.	PUNCT
ejpam-4571	171	1	thus	thus	ADV
ejpam-4571	171	2	,	,	PUNCT
ejpam-4571	171	3	a(λ	a(λ	ADV
ejpam-4571	171	4	,	,	PUNCT
ejpam-4571	171	5	b	b	NOUN
ejpam-4571	171	6	)	)	PUNCT
ejpam-4571	171	7	⊆	⊆	NUM
ejpam-4571	172	1	[	[	X
ejpam-4571	172	2	[	[	X
ejpam-4571	172	3	a(λ	a(λ	ADJ
ejpam-4571	172	4	,	,	PUNCT
ejpam-4571	172	5	b	b	NOUN
ejpam-4571	172	6	)	)	PUNCT
ejpam-4571	172	7	]	]	PUNCT
ejpam-4571	173	1	(	(	PUNCT
ejpam-4571	173	2	λ	λ	X
ejpam-4571	173	3	,	,	PUNCT
ejpam-4571	173	4	b	b	NOUN
ejpam-4571	173	5	)	)	PUNCT
ejpam-4571	173	6	∪	∪	ADP
ejpam-4571	173	7	[	[	X
ejpam-4571	173	8	a(λ	a(λ	ADV
ejpam-4571	173	9	,	,	PUNCT
ejpam-4571	173	10	b)](λ	b)](λ	NOUN
ejpam-4571	173	11	,	,	PUNCT
ejpam-4571	173	12	b	b	NOUN
ejpam-4571	173	13	)	)	PUNCT
ejpam-4571	173	14	]	]	PUNCT
ejpam-4571	173	15	(	(	PUNCT
ejpam-4571	173	16	λ	λ	X
ejpam-4571	173	17	,	,	PUNCT
ejpam-4571	173	18	b	b	NOUN
ejpam-4571	173	19	)	)	PUNCT
ejpam-4571	173	20	⊆	⊆	NUM
ejpam-4571	174	1	[	[	X
ejpam-4571	174	2	[	[	X
ejpam-4571	174	3	a(λ	a(λ	ADJ
ejpam-4571	174	4	,	,	PUNCT
ejpam-4571	174	5	b	b	NOUN
ejpam-4571	174	6	)	)	PUNCT
ejpam-4571	174	7	]	]	PUNCT
ejpam-4571	175	1	(	(	PUNCT
ejpam-4571	175	2	λ	λ	INTJ
ejpam-4571	175	3	,	,	PUNCT
ejpam-4571	175	4	b)](λ	b)](λ	NOUN
ejpam-4571	175	5	,	,	PUNCT
ejpam-4571	175	6	b	b	NOUN
ejpam-4571	175	7	)	)	PUNCT
ejpam-4571	175	8	∪	∪	ADP
ejpam-4571	175	9	[	[	X
ejpam-4571	175	10	[	[	X
ejpam-4571	175	11	a(λ	a(λ	ADJ
ejpam-4571	175	12	,	,	PUNCT
ejpam-4571	175	13	b)](λ	b)](λ	NOUN
ejpam-4571	175	14	,	,	PUNCT
ejpam-4571	175	15	b	b	NOUN
ejpam-4571	175	16	)	)	PUNCT
ejpam-4571	175	17	]	]	PUNCT
ejpam-4571	175	18	(	(	PUNCT
ejpam-4571	175	19	λ	λ	X
ejpam-4571	175	20	,	,	PUNCT
ejpam-4571	175	21	b	b	NOUN
ejpam-4571	175	22	)	)	PUNCT
ejpam-4571	175	23	=	=	PUNCT
ejpam-4571	176	1	[	[	X
ejpam-4571	176	2	[	[	X
ejpam-4571	176	3	a(λ	a(λ	ADV
ejpam-4571	176	4	,	,	PUNCT
ejpam-4571	176	5	b)](λ	b)](λ	NOUN
ejpam-4571	176	6	,	,	PUNCT
ejpam-4571	176	7	b	b	NOUN
ejpam-4571	176	8	)	)	PUNCT
ejpam-4571	176	9	]	]	PUNCT
ejpam-4571	176	10	(	(	PUNCT
ejpam-4571	176	11	λ	λ	X
ejpam-4571	176	12	,	,	PUNCT
ejpam-4571	176	13	b	b	NOUN
ejpam-4571	176	14	)	)	PUNCT
ejpam-4571	176	15	⊆	⊆	NUM
ejpam-4571	176	16	a(λ	a(λ	ADV
ejpam-4571	176	17	,	,	PUNCT
ejpam-4571	176	18	b	b	NOUN
ejpam-4571	176	19	)	)	PUNCT
ejpam-4571	176	20	and	and	CCONJ
ejpam-4571	176	21	hence	hence	ADV
ejpam-4571	176	22	a(λ	a(λ	ADV
ejpam-4571	176	23	,	,	PUNCT
ejpam-4571	176	24	b	b	NOUN
ejpam-4571	176	25	)	)	PUNCT
ejpam-4571	176	26	=	=	PUNCT
ejpam-4571	177	1	[	[	X
ejpam-4571	177	2	[	[	X
ejpam-4571	177	3	a(λ	a(λ	ADV
ejpam-4571	177	4	,	,	PUNCT
ejpam-4571	177	5	b)](λ	b)](λ	NOUN
ejpam-4571	177	6	,	,	PUNCT
ejpam-4571	177	7	b	b	NOUN
ejpam-4571	177	8	)	)	PUNCT
ejpam-4571	177	9	]	]	PUNCT
ejpam-4571	177	10	(	(	PUNCT
ejpam-4571	177	11	λ	λ	X
ejpam-4571	177	12	,	,	PUNCT
ejpam-4571	177	13	b	b	NOUN
ejpam-4571	177	14	)	)	PUNCT
ejpam-4571	177	15	.	.	PUNCT
ejpam-4571	178	1	therefore	therefore	ADV
ejpam-4571	178	2	,	,	PUNCT
ejpam-4571	178	3	a(λ	a(λ	ADV
ejpam-4571	178	4	,	,	PUNCT
ejpam-4571	178	5	b	b	NOUN
ejpam-4571	178	6	)	)	PUNCT
ejpam-4571	178	7	is	be	AUX
ejpam-4571	178	8	r(λ	r(λ	NOUN
ejpam-4571	178	9	,	,	PUNCT
ejpam-4571	178	10	b)-closed	b)-closed	ADJ
ejpam-4571	178	11	.	.	PUNCT
ejpam-4571	179	1	corollary	corollary	ADJ
ejpam-4571	179	2	2	2	NUM
ejpam-4571	179	3	.	.	PUNCT
ejpam-4571	180	1	for	for	ADP
ejpam-4571	180	2	a	a	DET
ejpam-4571	180	3	subset	subset	NOUN
ejpam-4571	180	4	a	a	PRON
ejpam-4571	180	5	of	of	ADP
ejpam-4571	180	6	a	a	DET
ejpam-4571	180	7	topological	topological	ADJ
ejpam-4571	180	8	space	space	NOUN
ejpam-4571	180	9	(	(	PUNCT
ejpam-4571	180	10	x	x	X
ejpam-4571	180	11	,	,	PUNCT
ejpam-4571	180	12	τ	τ	PROPN
ejpam-4571	180	13	)	)	PUNCT
ejpam-4571	180	14	,	,	PUNCT
ejpam-4571	180	15	the	the	DET
ejpam-4571	180	16	following	follow	VERB
ejpam-4571	180	17	properties	property	NOUN
ejpam-4571	180	18	hold	hold	VERB
ejpam-4571	180	19	:	:	PUNCT
ejpam-4571	180	20	(	(	PUNCT
ejpam-4571	180	21	1	1	X
ejpam-4571	180	22	)	)	PUNCT
ejpam-4571	180	23	if	if	SCONJ
ejpam-4571	180	24	a	a	PRON
ejpam-4571	180	25	is	be	AUX
ejpam-4571	180	26	s(λ	s(λ	NOUN
ejpam-4571	180	27	,	,	PUNCT
ejpam-4571	180	28	b)-open	b)-open	VERB
ejpam-4571	180	29	,	,	PUNCT
ejpam-4571	180	30	then	then	ADV
ejpam-4571	180	31	a(λ	a(λ	ADV
ejpam-4571	180	32	,	,	PUNCT
ejpam-4571	180	33	b	b	NOUN
ejpam-4571	180	34	)	)	PUNCT
ejpam-4571	180	35	is	be	AUX
ejpam-4571	180	36	r(λ	r(λ	NOUN
ejpam-4571	180	37	,	,	PUNCT
ejpam-4571	180	38	b)-closed	b)-closed	ADJ
ejpam-4571	180	39	.	.	PUNCT
ejpam-4571	181	1	(	(	PUNCT
ejpam-4571	181	2	2	2	X
ejpam-4571	181	3	)	)	PUNCT
ejpam-4571	181	4	if	if	SCONJ
ejpam-4571	181	5	a	a	PRON
ejpam-4571	181	6	is	be	AUX
ejpam-4571	181	7	p(λ	p(λ	NOUN
ejpam-4571	181	8	,	,	PUNCT
ejpam-4571	181	9	b)-open	b)-open	VERB
ejpam-4571	181	10	,	,	PUNCT
ejpam-4571	181	11	then	then	ADV
ejpam-4571	181	12	a(λ	a(λ	ADV
ejpam-4571	181	13	,	,	PUNCT
ejpam-4571	181	14	b	b	NOUN
ejpam-4571	181	15	)	)	PUNCT
ejpam-4571	181	16	is	be	AUX
ejpam-4571	181	17	r(λ	r(λ	NOUN
ejpam-4571	181	18	,	,	PUNCT
ejpam-4571	181	19	b)-closed	b)-closed	ADJ
ejpam-4571	181	20	.	.	PUNCT
ejpam-4571	182	1	(	(	PUNCT
ejpam-4571	182	2	3	3	X
ejpam-4571	182	3	)	)	PUNCT
ejpam-4571	182	4	if	if	SCONJ
ejpam-4571	182	5	a	a	PRON
ejpam-4571	182	6	is	be	AUX
ejpam-4571	182	7	α(λ	α(λ	PROPN
ejpam-4571	182	8	,	,	PUNCT
ejpam-4571	182	9	b)-open	b)-open	VERB
ejpam-4571	182	10	,	,	PUNCT
ejpam-4571	182	11	then	then	ADV
ejpam-4571	182	12	a(λ	a(λ	ADV
ejpam-4571	182	13	,	,	PUNCT
ejpam-4571	182	14	b	b	NOUN
ejpam-4571	182	15	)	)	PUNCT
ejpam-4571	182	16	is	be	AUX
ejpam-4571	182	17	r(λ	r(λ	NOUN
ejpam-4571	182	18	,	,	PUNCT
ejpam-4571	182	19	b)-closed	b)-closed	ADJ
ejpam-4571	182	20	.	.	PUNCT
ejpam-4571	183	1	proposition	proposition	NOUN
ejpam-4571	183	2	4	4	NUM
ejpam-4571	183	3	.	.	X
ejpam-4571	183	4	for	for	ADP
ejpam-4571	183	5	a	a	DET
ejpam-4571	183	6	subset	subset	NOUN
ejpam-4571	183	7	a	a	PRON
ejpam-4571	183	8	of	of	ADP
ejpam-4571	183	9	a	a	DET
ejpam-4571	183	10	topological	topological	ADJ
ejpam-4571	183	11	space	space	NOUN
ejpam-4571	183	12	(	(	PUNCT
ejpam-4571	183	13	x	x	X
ejpam-4571	183	14	,	,	PUNCT
ejpam-4571	183	15	τ	τ	PROPN
ejpam-4571	183	16	)	)	PUNCT
ejpam-4571	183	17	,	,	PUNCT
ejpam-4571	183	18	the	the	DET
ejpam-4571	183	19	following	follow	VERB
ejpam-4571	183	20	properties	property	NOUN
ejpam-4571	183	21	are	be	AUX
ejpam-4571	183	22	equivalent	equivalent	ADJ
ejpam-4571	183	23	:	:	PUNCT
ejpam-4571	183	24	(	(	PUNCT
ejpam-4571	183	25	1	1	X
ejpam-4571	183	26	)	)	PUNCT
ejpam-4571	183	27	a	a	DET
ejpam-4571	183	28	∈	∈	PROPN
ejpam-4571	183	29	βλbo(x	βλbo(x	NOUN
ejpam-4571	183	30	,	,	PUNCT
ejpam-4571	183	31	τ	τ	PROPN
ejpam-4571	183	32	)	)	PUNCT
ejpam-4571	183	33	.	.	PUNCT
ejpam-4571	184	1	(	(	PUNCT
ejpam-4571	184	2	2	2	NUM
ejpam-4571	184	3	)	)	PUNCT
ejpam-4571	184	4	a(λ	a(λ	ADV
ejpam-4571	184	5	,	,	PUNCT
ejpam-4571	184	6	b	b	X
ejpam-4571	184	7	)	)	PUNCT
ejpam-4571	184	8	∈	∈	PROPN
ejpam-4571	184	9	rλbc(x	rλbc(x	NOUN
ejpam-4571	184	10	,	,	PUNCT
ejpam-4571	184	11	τ	τ	PROPN
ejpam-4571	184	12	)	)	PUNCT
ejpam-4571	184	13	.	.	PUNCT
ejpam-4571	185	1	(	(	PUNCT
ejpam-4571	185	2	3	3	X
ejpam-4571	185	3	)	)	PUNCT
ejpam-4571	185	4	a(λ	a(λ	ADV
ejpam-4571	185	5	,	,	PUNCT
ejpam-4571	185	6	b	b	X
ejpam-4571	185	7	)	)	PUNCT
ejpam-4571	185	8	∈	∈	NOUN
ejpam-4571	185	9	βλbo(x	βλbo(x	NOUN
ejpam-4571	185	10	,	,	PUNCT
ejpam-4571	185	11	τ	τ	PROPN
ejpam-4571	185	12	)	)	PUNCT
ejpam-4571	185	13	.	.	PUNCT
ejpam-4571	186	1	c.	c.	PROPN
ejpam-4571	186	2	boonpok	boonpok	PROPN
ejpam-4571	186	3	,	,	PUNCT
ejpam-4571	186	4	n.	n.	PROPN
ejpam-4571	186	5	srisarakham	srisarakham	PROPN
ejpam-4571	186	6	/	/	SYM
ejpam-4571	186	7	eur	eur	PROPN
ejpam-4571	186	8	.	.	PUNCT
ejpam-4571	187	1	j.	j.	PROPN
ejpam-4571	187	2	pure	pure	PROPN
ejpam-4571	187	3	appl	appl	PROPN
ejpam-4571	187	4	.	.	PROPN
ejpam-4571	187	5	math	math	PROPN
ejpam-4571	187	6	,	,	PUNCT
ejpam-4571	187	7	16	16	NUM
ejpam-4571	187	8	(	(	PUNCT
ejpam-4571	187	9	1	1	NUM
ejpam-4571	187	10	)	)	PUNCT
ejpam-4571	187	11	(	(	PUNCT
ejpam-4571	187	12	2023	2023	NUM
ejpam-4571	187	13	)	)	PUNCT
ejpam-4571	187	14	,	,	PUNCT
ejpam-4571	187	15	29	29	NUM
ejpam-4571	187	16	-	-	SYM
ejpam-4571	187	17	43	43	NUM
ejpam-4571	187	18	34	34	NUM
ejpam-4571	187	19	(	(	PUNCT
ejpam-4571	187	20	4	4	NUM
ejpam-4571	187	21	)	)	PUNCT
ejpam-4571	187	22	a(λ	a(λ	ADV
ejpam-4571	187	23	,	,	PUNCT
ejpam-4571	187	24	b	b	X
ejpam-4571	187	25	)	)	PUNCT
ejpam-4571	187	26	∈	∈	PROPN
ejpam-4571	187	27	sλbo(x	sλbo(x	PROPN
ejpam-4571	187	28	,	,	PUNCT
ejpam-4571	187	29	τ	τ	PROPN
ejpam-4571	187	30	)	)	PUNCT
ejpam-4571	187	31	.	.	PUNCT
ejpam-4571	188	1	(	(	PUNCT
ejpam-4571	188	2	5	5	NUM
ejpam-4571	188	3	)	)	PUNCT
ejpam-4571	188	4	a(λ	a(λ	ADV
ejpam-4571	188	5	,	,	PUNCT
ejpam-4571	188	6	b	b	X
ejpam-4571	188	7	)	)	PUNCT
ejpam-4571	188	8	∈	∈	PROPN
ejpam-4571	188	9	bλbo(x	bλbo(x	NOUN
ejpam-4571	188	10	,	,	PUNCT
ejpam-4571	188	11	τ	τ	PROPN
ejpam-4571	188	12	)	)	PUNCT
ejpam-4571	188	13	.	.	PUNCT
ejpam-4571	189	1	proof	proof	NOUN
ejpam-4571	189	2	.	.	PUNCT
ejpam-4571	190	1	(	(	PUNCT
ejpam-4571	190	2	1	1	X
ejpam-4571	190	3	)	)	PUNCT
ejpam-4571	190	4	⇒	⇒	NOUN
ejpam-4571	190	5	(	(	PUNCT
ejpam-4571	190	6	2	2	NUM
ejpam-4571	190	7	):	):	PUNCT
ejpam-4571	190	8	let	let	VERB
ejpam-4571	190	9	a	a	DET
ejpam-4571	190	10	∈	∈	ADJ
ejpam-4571	190	11	βλbo(x	βλbo(x	NOUN
ejpam-4571	190	12	,	,	PUNCT
ejpam-4571	190	13	τ	τ	PROPN
ejpam-4571	190	14	)	)	PUNCT
ejpam-4571	190	15	.	.	PUNCT
ejpam-4571	191	1	then	then	ADV
ejpam-4571	191	2	,	,	PUNCT
ejpam-4571	191	3	a	a	DET
ejpam-4571	191	4	⊆	⊆	NUM
ejpam-4571	191	5	[	[	X
ejpam-4571	191	6	[	[	X
ejpam-4571	191	7	a(λ	a(λ	ADV
ejpam-4571	191	8	,	,	PUNCT
ejpam-4571	191	9	b)](λ	b)](λ	NOUN
ejpam-4571	191	10	,	,	PUNCT
ejpam-4571	191	11	b	b	NOUN
ejpam-4571	191	12	)	)	PUNCT
ejpam-4571	191	13	]	]	PUNCT
ejpam-4571	191	14	(	(	PUNCT
ejpam-4571	191	15	λ	λ	X
ejpam-4571	191	16	,	,	PUNCT
ejpam-4571	191	17	b	b	NOUN
ejpam-4571	191	18	)	)	PUNCT
ejpam-4571	191	19	.	.	PUNCT
ejpam-4571	192	1	thus	thus	ADV
ejpam-4571	192	2	,	,	PUNCT
ejpam-4571	192	3	a(λ	a(λ	ADV
ejpam-4571	192	4	,	,	PUNCT
ejpam-4571	192	5	b	b	NOUN
ejpam-4571	192	6	)	)	PUNCT
ejpam-4571	192	7	⊆	⊆	NUM
ejpam-4571	193	1	[	[	X
ejpam-4571	193	2	[	[	X
ejpam-4571	193	3	a(λ	a(λ	ADV
ejpam-4571	193	4	,	,	PUNCT
ejpam-4571	193	5	b)](λ	b)](λ	NOUN
ejpam-4571	193	6	,	,	PUNCT
ejpam-4571	193	7	b	b	NOUN
ejpam-4571	193	8	)	)	PUNCT
ejpam-4571	193	9	]	]	PUNCT
ejpam-4571	193	10	(	(	PUNCT
ejpam-4571	193	11	λ	λ	X
ejpam-4571	193	12	,	,	PUNCT
ejpam-4571	193	13	b	b	NOUN
ejpam-4571	193	14	)	)	PUNCT
ejpam-4571	193	15	⊆	⊆	NUM
ejpam-4571	193	16	a(λ	a(λ	ADV
ejpam-4571	193	17	,	,	PUNCT
ejpam-4571	193	18	b	b	NOUN
ejpam-4571	193	19	)	)	PUNCT
ejpam-4571	193	20	and	and	CCONJ
ejpam-4571	193	21	hence	hence	ADV
ejpam-4571	193	22	a(λ	a(λ	ADV
ejpam-4571	193	23	,	,	PUNCT
ejpam-4571	193	24	b	b	NOUN
ejpam-4571	193	25	)	)	PUNCT
ejpam-4571	193	26	=	=	PUNCT
ejpam-4571	194	1	[	[	X
ejpam-4571	194	2	[	[	X
ejpam-4571	194	3	a(λ	a(λ	ADV
ejpam-4571	194	4	,	,	PUNCT
ejpam-4571	194	5	b)](λ	b)](λ	NOUN
ejpam-4571	194	6	,	,	PUNCT
ejpam-4571	194	7	b	b	NOUN
ejpam-4571	194	8	)	)	PUNCT
ejpam-4571	194	9	]	]	PUNCT
ejpam-4571	194	10	(	(	PUNCT
ejpam-4571	194	11	λ	λ	X
ejpam-4571	194	12	,	,	PUNCT
ejpam-4571	194	13	b	b	NOUN
ejpam-4571	194	14	)	)	PUNCT
ejpam-4571	194	15	.	.	PUNCT
ejpam-4571	195	1	therefore	therefore	ADV
ejpam-4571	195	2	,	,	PUNCT
ejpam-4571	195	3	a(λ	a(λ	ADV
ejpam-4571	195	4	,	,	PUNCT
ejpam-4571	195	5	b	b	X
ejpam-4571	195	6	)	)	PUNCT
ejpam-4571	195	7	∈	∈	PROPN
ejpam-4571	195	8	rλbc(x	rλbc(x	NOUN
ejpam-4571	195	9	,	,	PUNCT
ejpam-4571	195	10	τ	τ	PROPN
ejpam-4571	195	11	)	)	PUNCT
ejpam-4571	195	12	.	.	PUNCT
ejpam-4571	196	1	(	(	PUNCT
ejpam-4571	196	2	2	2	X
ejpam-4571	196	3	)	)	PUNCT
ejpam-4571	196	4	⇒	⇒	NOUN
ejpam-4571	196	5	(	(	PUNCT
ejpam-4571	196	6	3	3	NUM
ejpam-4571	196	7	)	)	PUNCT
ejpam-4571	196	8	⇒	⇒	NOUN
ejpam-4571	196	9	(	(	PUNCT
ejpam-4571	196	10	4	4	NUM
ejpam-4571	196	11	)	)	PUNCT
ejpam-4571	196	12	⇒	⇒	NOUN
ejpam-4571	196	13	(	(	PUNCT
ejpam-4571	196	14	5	5	NUM
ejpam-4571	196	15	):	):	PUNCT
ejpam-4571	196	16	obvious	obvious	ADJ
ejpam-4571	196	17	.	.	PUNCT
ejpam-4571	197	1	(	(	PUNCT
ejpam-4571	197	2	5	5	X
ejpam-4571	197	3	)	)	PUNCT
ejpam-4571	197	4	⇒	⇒	NOUN
ejpam-4571	197	5	(	(	PUNCT
ejpam-4571	197	6	1	1	NUM
ejpam-4571	197	7	):	):	PUNCT
ejpam-4571	197	8	leta(λ	leta(λ	PROPN
ejpam-4571	197	9	,	,	PUNCT
ejpam-4571	197	10	b	b	NOUN
ejpam-4571	197	11	)	)	PUNCT
ejpam-4571	197	12	∈	∈	PROPN
ejpam-4571	197	13	bλbo(x	bλbo(x	NOUN
ejpam-4571	197	14	,	,	PUNCT
ejpam-4571	197	15	τ	τ	PROPN
ejpam-4571	197	16	)	)	PUNCT
ejpam-4571	197	17	.	.	PUNCT
ejpam-4571	198	1	then	then	ADV
ejpam-4571	198	2	,	,	PUNCT
ejpam-4571	198	3	a(λ	a(λ	ADV
ejpam-4571	198	4	,	,	PUNCT
ejpam-4571	198	5	b	b	NOUN
ejpam-4571	198	6	)	)	PUNCT
ejpam-4571	198	7	⊆	⊆	NUM
ejpam-4571	199	1	[	[	X
ejpam-4571	199	2	[	[	X
ejpam-4571	199	3	a(λ	a(λ	ADV
ejpam-4571	199	4	,	,	PUNCT
ejpam-4571	199	5	b)](λ	b)](λ	PROPN
ejpam-4571	199	6	,	,	PUNCT
ejpam-4571	199	7	b)](λ	b)](λ	NOUN
ejpam-4571	199	8	,	,	PUNCT
ejpam-4571	199	9	b)∪[[a(λ	b)∪[[a(λ	NOUN
ejpam-4571	199	10	,	,	PUNCT
ejpam-4571	199	11	b)](λ	b)](λ	NOUN
ejpam-4571	199	12	,	,	PUNCT
ejpam-4571	199	13	b	b	NOUN
ejpam-4571	199	14	)	)	PUNCT
ejpam-4571	199	15	]	]	PUNCT
ejpam-4571	199	16	(	(	PUNCT
ejpam-4571	199	17	λ	λ	X
ejpam-4571	199	18	,	,	PUNCT
ejpam-4571	199	19	b	b	NOUN
ejpam-4571	199	20	)	)	PUNCT
ejpam-4571	199	21	=	=	PUNCT
ejpam-4571	200	1	[	[	X
ejpam-4571	200	2	a(λ	a(λ	ADV
ejpam-4571	200	3	,	,	PUNCT
ejpam-4571	200	4	b)](λ	b)](λ	NOUN
ejpam-4571	200	5	,	,	PUNCT
ejpam-4571	200	6	b	b	NOUN
ejpam-4571	200	7	)	)	PUNCT
ejpam-4571	200	8	∪	∪	ADP
ejpam-4571	200	9	[	[	X
ejpam-4571	200	10	[	[	X
ejpam-4571	200	11	a(λ	a(λ	ADJ
ejpam-4571	200	12	,	,	PUNCT
ejpam-4571	200	13	b)](λ	b)](λ	NOUN
ejpam-4571	200	14	,	,	PUNCT
ejpam-4571	200	15	b	b	NOUN
ejpam-4571	200	16	)	)	PUNCT
ejpam-4571	200	17	]	]	PUNCT
ejpam-4571	200	18	(	(	PUNCT
ejpam-4571	200	19	λ	λ	X
ejpam-4571	200	20	,	,	PUNCT
ejpam-4571	200	21	b	b	NOUN
ejpam-4571	200	22	)	)	PUNCT
ejpam-4571	200	23	=	=	PUNCT
ejpam-4571	201	1	[	[	X
ejpam-4571	201	2	[	[	X
ejpam-4571	201	3	a(λ	a(λ	ADV
ejpam-4571	201	4	,	,	PUNCT
ejpam-4571	201	5	b)](λ	b)](λ	NOUN
ejpam-4571	201	6	,	,	PUNCT
ejpam-4571	201	7	b	b	NOUN
ejpam-4571	201	8	)	)	PUNCT
ejpam-4571	201	9	]	]	PUNCT
ejpam-4571	201	10	(	(	PUNCT
ejpam-4571	201	11	λ	λ	X
ejpam-4571	201	12	,	,	PUNCT
ejpam-4571	201	13	b	b	NOUN
ejpam-4571	201	14	)	)	PUNCT
ejpam-4571	201	15	.	.	PUNCT
ejpam-4571	202	1	thus	thus	ADV
ejpam-4571	202	2	,	,	PUNCT
ejpam-4571	202	3	a	a	DET
ejpam-4571	202	4	⊆	⊆	NUM
ejpam-4571	202	5	[	[	X
ejpam-4571	202	6	[	[	X
ejpam-4571	202	7	a(λ	a(λ	ADV
ejpam-4571	202	8	,	,	PUNCT
ejpam-4571	202	9	b)](λ	b)](λ	NOUN
ejpam-4571	202	10	,	,	PUNCT
ejpam-4571	202	11	b	b	NOUN
ejpam-4571	202	12	)	)	PUNCT
ejpam-4571	202	13	]	]	PUNCT
ejpam-4571	202	14	(	(	PUNCT
ejpam-4571	202	15	λ	λ	X
ejpam-4571	202	16	,	,	PUNCT
ejpam-4571	202	17	b	b	NOUN
ejpam-4571	202	18	)	)	PUNCT
ejpam-4571	202	19	and	and	CCONJ
ejpam-4571	202	20	hence	hence	ADV
ejpam-4571	202	21	a	a	DET
ejpam-4571	202	22	∈	∈	NOUN
ejpam-4571	202	23	βλbo(x	βλbo(x	NOUN
ejpam-4571	202	24	,	,	PUNCT
ejpam-4571	202	25	τ	τ	PROPN
ejpam-4571	202	26	)	)	PUNCT
ejpam-4571	202	27	.	.	PUNCT
ejpam-4571	203	1	definition	definition	NOUN
ejpam-4571	203	2	3	3	X
ejpam-4571	203	3	.	.	PUNCT
ejpam-4571	204	1	let	let	AUX
ejpam-4571	204	2	(	(	PUNCT
ejpam-4571	204	3	x	x	NOUN
ejpam-4571	204	4	,	,	PUNCT
ejpam-4571	204	5	τ	τ	X
ejpam-4571	204	6	)	)	PUNCT
ejpam-4571	204	7	be	be	VERB
ejpam-4571	204	8	a	a	DET
ejpam-4571	204	9	topological	topological	ADJ
ejpam-4571	204	10	space	space	NOUN
ejpam-4571	204	11	and	and	CCONJ
ejpam-4571	204	12	x	x	PUNCT
ejpam-4571	204	13	∈	∈	PROPN
ejpam-4571	204	14	x.	x.	NOUN
ejpam-4571	205	1	a	a	DET
ejpam-4571	205	2	subset	subset	NOUN
ejpam-4571	205	3	⟨x⟩b	⟨x⟩b	ADV
ejpam-4571	205	4	is	be	AUX
ejpam-4571	205	5	defined	define	VERB
ejpam-4571	205	6	as	as	SCONJ
ejpam-4571	205	7	follows	follow	VERB
ejpam-4571	205	8	:	:	PUNCT
ejpam-4571	205	9	⟨x⟩b	⟨x⟩b	ADV
ejpam-4571	205	10	=	=	SYM
ejpam-4571	205	11	λ(λ	λ(λ	PROPN
ejpam-4571	205	12	,	,	PUNCT
ejpam-4571	205	13	b)({x	b)({x	NOUN
ejpam-4571	205	14	}	}	PUNCT
ejpam-4571	205	15	)	)	PUNCT
ejpam-4571	205	16	∩	∩	NOUN
ejpam-4571	205	17	{	{	PUNCT
ejpam-4571	205	18	x}(λ	x}(λ	PROPN
ejpam-4571	205	19	,	,	PUNCT
ejpam-4571	205	20	b	b	NOUN
ejpam-4571	205	21	)	)	PUNCT
ejpam-4571	205	22	.	.	PUNCT
ejpam-4571	206	1	example	example	NOUN
ejpam-4571	207	1	3	3	X
ejpam-4571	207	2	.	.	PUNCT
ejpam-4571	207	3	let	let	VERB
ejpam-4571	207	4	x	x	PUNCT
ejpam-4571	207	5	=	=	PRON
ejpam-4571	207	6	{	{	PUNCT
ejpam-4571	207	7	−1	−1	NOUN
ejpam-4571	207	8	,	,	PUNCT
ejpam-4571	207	9	1	1	NUM
ejpam-4571	207	10	}	}	PUNCT
ejpam-4571	207	11	with	with	ADP
ejpam-4571	207	12	the	the	DET
ejpam-4571	207	13	topology	topology	NOUN
ejpam-4571	207	14	τ	τ	X
ejpam-4571	207	15	=	=	PUNCT
ejpam-4571	207	16	{	{	PUNCT
ejpam-4571	207	17	∅	∅	NOUN
ejpam-4571	207	18	,	,	PUNCT
ejpam-4571	207	19	{	{	PUNCT
ejpam-4571	207	20	−1	−1	NOUN
ejpam-4571	207	21	}	}	PUNCT
ejpam-4571	207	22	,	,	PUNCT
ejpam-4571	207	23	x	x	NOUN
ejpam-4571	207	24	}	}	PUNCT
ejpam-4571	207	25	.	.	PUNCT
ejpam-4571	208	1	then	then	ADV
ejpam-4571	208	2	,	,	PUNCT
ejpam-4571	208	3	⟨1⟩b	⟨1⟩b	NOUN
ejpam-4571	208	4	=	=	PUNCT
ejpam-4571	208	5	{	{	PUNCT
ejpam-4571	208	6	1	1	NUM
ejpam-4571	208	7	}	}	PUNCT
ejpam-4571	208	8	.	.	PUNCT
ejpam-4571	209	1	recall	recall	VERB
ejpam-4571	209	2	that	that	SCONJ
ejpam-4571	209	3	a	a	DET
ejpam-4571	209	4	subset	subset	NOUN
ejpam-4571	209	5	n	n	NOUN
ejpam-4571	209	6	of	of	ADP
ejpam-4571	209	7	a	a	DET
ejpam-4571	209	8	topological	topological	ADJ
ejpam-4571	209	9	space	space	NOUN
ejpam-4571	209	10	(	(	PUNCT
ejpam-4571	209	11	x	x	X
ejpam-4571	209	12	,	,	PUNCT
ejpam-4571	209	13	τ	τ	X
ejpam-4571	209	14	)	)	PUNCT
ejpam-4571	209	15	is	be	AUX
ejpam-4571	209	16	said	say	VERB
ejpam-4571	209	17	to	to	PART
ejpam-4571	209	18	be	be	AUX
ejpam-4571	209	19	a	a	DET
ejpam-4571	209	20	(	(	PUNCT
ejpam-4571	209	21	λ	λ	NOUN
ejpam-4571	209	22	,	,	PUNCT
ejpam-4571	209	23	b)-neighbourhood	b)-neighbourhood	PUNCT
ejpam-4571	209	24	[	[	X
ejpam-4571	209	25	5	5	NUM
ejpam-4571	209	26	]	]	PUNCT
ejpam-4571	209	27	of	of	ADP
ejpam-4571	209	28	a	a	DET
ejpam-4571	209	29	point	point	NOUN
ejpam-4571	209	30	x	x	X
ejpam-4571	209	31	∈	∈	NOUN
ejpam-4571	209	32	x	x	INTJ
ejpam-4571	209	33	if	if	SCONJ
ejpam-4571	209	34	there	there	PRON
ejpam-4571	209	35	exists	exist	VERB
ejpam-4571	209	36	a	a	DET
ejpam-4571	209	37	(	(	PUNCT
ejpam-4571	209	38	λ	λ	NOUN
ejpam-4571	209	39	,	,	PUNCT
ejpam-4571	209	40	b)-open	b)-open	AUX
ejpam-4571	209	41	set	set	VERB
ejpam-4571	209	42	u	u	NOUN
ejpam-4571	209	43	of	of	ADP
ejpam-4571	209	44	x	x	SYM
ejpam-4571	209	45	such	such	ADJ
ejpam-4571	209	46	that	that	SCONJ
ejpam-4571	209	47	x	x	SYM
ejpam-4571	209	48	∈	∈	PROPN
ejpam-4571	209	49	u	u	NOUN
ejpam-4571	209	50	⊆	⊆	NUM
ejpam-4571	209	51	n	n	NOUN
ejpam-4571	209	52	.	.	PUNCT
ejpam-4571	210	1	theorem	theorem	NOUN
ejpam-4571	210	2	1	1	X
ejpam-4571	210	3	.	.	PUNCT
ejpam-4571	211	1	let	let	AUX
ejpam-4571	211	2	(	(	PUNCT
ejpam-4571	211	3	x	x	NOUN
ejpam-4571	211	4	,	,	PUNCT
ejpam-4571	211	5	τ	τ	X
ejpam-4571	211	6	)	)	PUNCT
ejpam-4571	211	7	be	be	VERB
ejpam-4571	211	8	a	a	DET
ejpam-4571	211	9	topological	topological	ADJ
ejpam-4571	211	10	space	space	NOUN
ejpam-4571	211	11	.	.	PUNCT
ejpam-4571	212	1	then	then	ADV
ejpam-4571	212	2	,	,	PUNCT
ejpam-4571	212	3	the	the	DET
ejpam-4571	212	4	following	follow	VERB
ejpam-4571	212	5	properties	property	NOUN
ejpam-4571	212	6	hold	hold	VERB
ejpam-4571	212	7	:	:	PUNCT
ejpam-4571	212	8	(	(	PUNCT
ejpam-4571	212	9	1	1	X
ejpam-4571	212	10	)	)	PUNCT
ejpam-4571	212	11	λ(λ	λ(λ	ADV
ejpam-4571	212	12	,	,	PUNCT
ejpam-4571	212	13	b)(a	b)(a	ADV
ejpam-4571	212	14	)	)	PUNCT
ejpam-4571	212	15	=	=	PRON
ejpam-4571	213	1	{	{	PUNCT
ejpam-4571	213	2	x	x	PUNCT
ejpam-4571	213	3	∈	∈	NOUN
ejpam-4571	213	4	x	x	PUNCT
ejpam-4571	213	5	|	|	ADV
ejpam-4571	213	6	a	a	DET
ejpam-4571	213	7	∩	∩	NOUN
ejpam-4571	213	8	{	{	PUNCT
ejpam-4571	213	9	x}(λ	x}(λ	PROPN
ejpam-4571	213	10	,	,	PUNCT
ejpam-4571	213	11	b	b	NOUN
ejpam-4571	213	12	)	)	PUNCT
ejpam-4571	213	13	̸=	̸=	PROPN
ejpam-4571	213	14	∅	∅	NOUN
ejpam-4571	213	15	}	}	PUNCT
ejpam-4571	213	16	for	for	ADP
ejpam-4571	213	17	each	each	PRON
ejpam-4571	213	18	subset	subset	VERB
ejpam-4571	213	19	a	a	PRON
ejpam-4571	213	20	of	of	ADP
ejpam-4571	213	21	x.	x.	NOUN
ejpam-4571	213	22	(	(	PUNCT
ejpam-4571	213	23	2	2	NUM
ejpam-4571	213	24	)	)	PUNCT
ejpam-4571	213	25	for	for	ADP
ejpam-4571	213	26	each	each	DET
ejpam-4571	213	27	x	x	SYM
ejpam-4571	213	28	∈	∈	PROPN
ejpam-4571	213	29	x	x	X
ejpam-4571	213	30	,	,	PUNCT
ejpam-4571	213	31	λ(λ	λ(λ	PROPN
ejpam-4571	213	32	,	,	PUNCT
ejpam-4571	213	33	b)(⟨x⟩b	b)(⟨x⟩b	PROPN
ejpam-4571	213	34	)	)	PUNCT
ejpam-4571	213	35	=	=	SYM
ejpam-4571	214	1	λ(λ	λ(λ	PROPN
ejpam-4571	214	2	,	,	PUNCT
ejpam-4571	214	3	b)({x	b)({x	NOUN
ejpam-4571	214	4	}	}	PUNCT
ejpam-4571	214	5	)	)	PUNCT
ejpam-4571	214	6	.	.	PUNCT
ejpam-4571	215	1	(	(	PUNCT
ejpam-4571	215	2	3	3	X
ejpam-4571	215	3	)	)	PUNCT
ejpam-4571	215	4	for	for	ADP
ejpam-4571	215	5	each	each	DET
ejpam-4571	215	6	x	x	SYM
ejpam-4571	215	7	∈	∈	PROPN
ejpam-4571	215	8	x	x	X
ejpam-4571	215	9	,	,	PUNCT
ejpam-4571	215	10	(	(	PUNCT
ejpam-4571	215	11	⟨x⟩b)(λ	⟨x⟩b)(λ	NOUN
ejpam-4571	215	12	,	,	PUNCT
ejpam-4571	215	13	b	b	NOUN
ejpam-4571	215	14	)	)	PUNCT
ejpam-4571	215	15	=	=	SYM
ejpam-4571	215	16	{	{	PUNCT
ejpam-4571	215	17	x}(λ	x}(λ	PROPN
ejpam-4571	215	18	,	,	PUNCT
ejpam-4571	215	19	b	b	NOUN
ejpam-4571	215	20	)	)	PUNCT
ejpam-4571	215	21	.	.	PUNCT
ejpam-4571	216	1	(	(	PUNCT
ejpam-4571	216	2	4	4	X
ejpam-4571	216	3	)	)	PUNCT
ejpam-4571	216	4	if	if	SCONJ
ejpam-4571	216	5	u	u	PRON
ejpam-4571	216	6	is	be	AUX
ejpam-4571	216	7	(	(	PUNCT
ejpam-4571	216	8	λ	λ	X
ejpam-4571	216	9	,	,	PUNCT
ejpam-4571	216	10	b)-open	b)-open	VERB
ejpam-4571	216	11	in	in	ADP
ejpam-4571	216	12	(	(	PUNCT
ejpam-4571	216	13	x	x	NOUN
ejpam-4571	216	14	,	,	PUNCT
ejpam-4571	216	15	τ	τ	X
ejpam-4571	216	16	)	)	PUNCT
ejpam-4571	216	17	and	and	CCONJ
ejpam-4571	216	18	x	x	PUNCT
ejpam-4571	216	19	∈	∈	PROPN
ejpam-4571	216	20	u	u	NOUN
ejpam-4571	216	21	,	,	PUNCT
ejpam-4571	216	22	then	then	ADV
ejpam-4571	216	23	⟨x⟩b	⟨x⟩b	ADV
ejpam-4571	216	24	⊆	⊆	NUM
ejpam-4571	216	25	u	u	NOUN
ejpam-4571	216	26	.	.	PUNCT
ejpam-4571	217	1	(	(	PUNCT
ejpam-4571	217	2	5	5	NUM
ejpam-4571	217	3	)	)	PUNCT
ejpam-4571	217	4	if	if	SCONJ
ejpam-4571	217	5	f	f	PROPN
ejpam-4571	217	6	is	be	AUX
ejpam-4571	217	7	(	(	PUNCT
ejpam-4571	217	8	λ	λ	X
ejpam-4571	217	9	,	,	PUNCT
ejpam-4571	217	10	b)-closed	b)-close	VERB
ejpam-4571	217	11	in	in	ADP
ejpam-4571	217	12	(	(	PUNCT
ejpam-4571	217	13	x	x	NOUN
ejpam-4571	217	14	,	,	PUNCT
ejpam-4571	217	15	τ	τ	X
ejpam-4571	217	16	)	)	PUNCT
ejpam-4571	217	17	and	and	CCONJ
ejpam-4571	217	18	x	x	PUNCT
ejpam-4571	217	19	∈	∈	PROPN
ejpam-4571	217	20	f	f	NOUN
ejpam-4571	217	21	,	,	PUNCT
ejpam-4571	217	22	then	then	ADV
ejpam-4571	217	23	⟨x⟩b	⟨x⟩b	ADV
ejpam-4571	217	24	⊆	⊆	NUM
ejpam-4571	217	25	f	f	NOUN
ejpam-4571	217	26	.	.	PUNCT
ejpam-4571	218	1	proof	proof	NOUN
ejpam-4571	218	2	.	.	PUNCT
ejpam-4571	219	1	(	(	PUNCT
ejpam-4571	219	2	1	1	X
ejpam-4571	219	3	)	)	PUNCT
ejpam-4571	219	4	suppose	suppose	VERB
ejpam-4571	219	5	that	that	SCONJ
ejpam-4571	219	6	a	a	DET
ejpam-4571	219	7	∩	∩	NOUN
ejpam-4571	219	8	{	{	PUNCT
ejpam-4571	219	9	x}(λ	x}(λ	PROPN
ejpam-4571	219	10	,	,	PUNCT
ejpam-4571	219	11	b	b	NOUN
ejpam-4571	219	12	)	)	PUNCT
ejpam-4571	219	13	=	=	PUNCT
ejpam-4571	219	14	∅.	∅.	NOUN
ejpam-4571	219	15	then	then	ADV
ejpam-4571	219	16	,	,	PUNCT
ejpam-4571	219	17	we	we	PRON
ejpam-4571	219	18	have	have	VERB
ejpam-4571	219	19	x	x	X
ejpam-4571	219	20	̸∈	̸∈	X
ejpam-4571	219	21	x	x	X
ejpam-4571	219	22	−	−	PROPN
ejpam-4571	219	23	{	{	PUNCT
ejpam-4571	219	24	x}(λ	x}(λ	PROPN
ejpam-4571	219	25	,	,	PUNCT
ejpam-4571	219	26	b	b	NOUN
ejpam-4571	219	27	)	)	PUNCT
ejpam-4571	219	28	which	which	PRON
ejpam-4571	219	29	is	be	AUX
ejpam-4571	219	30	a	a	DET
ejpam-4571	219	31	(	(	PUNCT
ejpam-4571	219	32	λ	λ	NOUN
ejpam-4571	219	33	,	,	PUNCT
ejpam-4571	219	34	b)-open	b)-open	VERB
ejpam-4571	219	35	set	set	VERB
ejpam-4571	219	36	containing	contain	VERB
ejpam-4571	219	37	a.	a.	NOUN
ejpam-4571	219	38	therefore	therefore	ADV
ejpam-4571	219	39	,	,	PUNCT
ejpam-4571	219	40	x	x	PROPN
ejpam-4571	219	41	̸∈	̸∈	PROPN
ejpam-4571	219	42	λ(λ	λ(λ	PROPN
ejpam-4571	219	43	,	,	PUNCT
ejpam-4571	219	44	b)(a	b)(a	ADV
ejpam-4571	219	45	)	)	PUNCT
ejpam-4571	219	46	.	.	PUNCT
ejpam-4571	220	1	consequently	consequently	ADV
ejpam-4571	220	2	,	,	PUNCT
ejpam-4571	220	3	we	we	PRON
ejpam-4571	220	4	obtain	obtain	VERB
ejpam-4571	220	5	λ(λ	λ(λ	ADV
ejpam-4571	220	6	,	,	PUNCT
ejpam-4571	220	7	b)(a	b)(a	ADV
ejpam-4571	220	8	)	)	PUNCT
ejpam-4571	221	1	⊆	⊆	NUM
ejpam-4571	221	2	{	{	PUNCT
ejpam-4571	221	3	x	x	SYM
ejpam-4571	221	4	∈	∈	PROPN
ejpam-4571	221	5	x	x	X
ejpam-4571	221	6	|	|	ADV
ejpam-4571	221	7	a	a	DET
ejpam-4571	221	8	∩	∩	NOUN
ejpam-4571	221	9	{	{	PUNCT
ejpam-4571	221	10	x}(λ	x}(λ	PROPN
ejpam-4571	221	11	,	,	PUNCT
ejpam-4571	221	12	b	b	NOUN
ejpam-4571	221	13	)	)	PUNCT
ejpam-4571	221	14	̸=	̸=	PROPN
ejpam-4571	221	15	∅	∅	NOUN
ejpam-4571	221	16	}	}	PUNCT
ejpam-4571	221	17	.	.	PUNCT
ejpam-4571	222	1	next	next	ADV
ejpam-4571	222	2	,	,	PUNCT
ejpam-4571	222	3	let	let	VERB
ejpam-4571	222	4	x	x	PUNCT
ejpam-4571	222	5	∈	∈	PROPN
ejpam-4571	222	6	x	x	X
ejpam-4571	222	7	such	such	ADJ
ejpam-4571	222	8	that	that	SCONJ
ejpam-4571	222	9	a	a	DET
ejpam-4571	222	10	∩	∩	NOUN
ejpam-4571	222	11	{	{	PUNCT
ejpam-4571	222	12	x}(λ	x}(λ	PROPN
ejpam-4571	222	13	,	,	PUNCT
ejpam-4571	222	14	b	b	NOUN
ejpam-4571	222	15	)	)	PUNCT
ejpam-4571	222	16	̸=	̸=	PROPN
ejpam-4571	222	17	∅	∅	NOUN
ejpam-4571	222	18	and	and	CCONJ
ejpam-4571	222	19	suppose	suppose	VERB
ejpam-4571	222	20	that	that	SCONJ
ejpam-4571	222	21	x	x	PROPN
ejpam-4571	222	22	̸∈	̸∈	PROPN
ejpam-4571	222	23	λ(λ	λ(λ	PROPN
ejpam-4571	222	24	,	,	PUNCT
ejpam-4571	222	25	b)(a	b)(a	ADV
ejpam-4571	222	26	)	)	PUNCT
ejpam-4571	222	27	.	.	PUNCT
ejpam-4571	223	1	then	then	ADV
ejpam-4571	223	2	,	,	PUNCT
ejpam-4571	223	3	there	there	PRON
ejpam-4571	223	4	exists	exist	VERB
ejpam-4571	223	5	a	a	DET
ejpam-4571	223	6	(	(	PUNCT
ejpam-4571	223	7	λ	λ	NOUN
ejpam-4571	223	8	,	,	PUNCT
ejpam-4571	223	9	b)-open	b)-open	AUX
ejpam-4571	223	10	set	set	VERB
ejpam-4571	223	11	u	u	NOUN
ejpam-4571	223	12	containing	contain	VERB
ejpam-4571	223	13	a	a	PRON
ejpam-4571	223	14	and	and	CCONJ
ejpam-4571	223	15	x	x	X
ejpam-4571	223	16	̸∈	̸∈	PROPN
ejpam-4571	223	17	u	u	PROPN
ejpam-4571	223	18	.	.	PUNCT
ejpam-4571	224	1	let	let	VERB
ejpam-4571	224	2	y	y	PRON
ejpam-4571	224	3	∈	∈	PROPN
ejpam-4571	224	4	a	a	DET
ejpam-4571	224	5	∩	∩	NOUN
ejpam-4571	224	6	{	{	PUNCT
ejpam-4571	224	7	x}(λ	x}(λ	PROPN
ejpam-4571	224	8	,	,	PUNCT
ejpam-4571	224	9	b	b	NOUN
ejpam-4571	224	10	)	)	PUNCT
ejpam-4571	224	11	.	.	PUNCT
ejpam-4571	225	1	thus	thus	ADV
ejpam-4571	225	2	,	,	PUNCT
ejpam-4571	225	3	u	u	NOUN
ejpam-4571	225	4	is	be	AUX
ejpam-4571	225	5	a	a	DET
ejpam-4571	225	6	(	(	PUNCT
ejpam-4571	225	7	λ	λ	NOUN
ejpam-4571	225	8	,	,	PUNCT
ejpam-4571	225	9	b)-neighbourhood	b)-neighbourhood	PUNCT
ejpam-4571	225	10	of	of	ADP
ejpam-4571	225	11	y	y	PRON
ejpam-4571	225	12	which	which	PRON
ejpam-4571	225	13	does	do	AUX
ejpam-4571	225	14	not	not	PART
ejpam-4571	225	15	contain	contain	VERB
ejpam-4571	225	16	x.	x.	NOUN
ejpam-4571	225	17	by	by	ADP
ejpam-4571	225	18	this	this	DET
ejpam-4571	225	19	contradiction	contradiction	NOUN
ejpam-4571	225	20	x	x	X
ejpam-4571	225	21	∈	∈	PROPN
ejpam-4571	225	22	λ(λ	λ(λ	PROPN
ejpam-4571	225	23	,	,	PUNCT
ejpam-4571	225	24	b)(a	b)(a	ADV
ejpam-4571	225	25	)	)	PUNCT
ejpam-4571	225	26	.	.	PUNCT
ejpam-4571	226	1	(	(	PUNCT
ejpam-4571	226	2	2	2	X
ejpam-4571	226	3	)	)	PUNCT
ejpam-4571	226	4	let	let	VERB
ejpam-4571	226	5	x	x	PUNCT
ejpam-4571	226	6	∈	∈	PROPN
ejpam-4571	226	7	x	x	NOUN
ejpam-4571	226	8	,	,	PUNCT
ejpam-4571	226	9	then	then	ADV
ejpam-4571	226	10	,	,	PUNCT
ejpam-4571	226	11	we	we	PRON
ejpam-4571	226	12	have	have	VERB
ejpam-4571	226	13	{	{	PUNCT
ejpam-4571	226	14	x	x	NOUN
ejpam-4571	226	15	}	}	PUNCT
ejpam-4571	226	16	⊆	⊆	NUM
ejpam-4571	226	17	{	{	PUNCT
ejpam-4571	226	18	x}(λ	x}(λ	PROPN
ejpam-4571	226	19	,	,	PUNCT
ejpam-4571	226	20	b	b	NOUN
ejpam-4571	226	21	)	)	PUNCT
ejpam-4571	226	22	∩	∩	NOUN
ejpam-4571	226	23	λ(λ	λ(λ	ADP
ejpam-4571	226	24	,	,	PUNCT
ejpam-4571	226	25	b)({x	b)({x	NOUN
ejpam-4571	226	26	}	}	PUNCT
ejpam-4571	226	27	)	)	PUNCT
ejpam-4571	227	1	=	=	PUNCT
ejpam-4571	227	2	⟨x⟩b	⟨x⟩b	ADV
ejpam-4571	227	3	.	.	PUNCT
ejpam-4571	228	1	by	by	ADP
ejpam-4571	228	2	lemma	lemma	PROPN
ejpam-4571	228	3	3	3	NUM
ejpam-4571	228	4	,	,	PUNCT
ejpam-4571	228	5	λ(λ	λ(λ	PROPN
ejpam-4571	228	6	,	,	PUNCT
ejpam-4571	228	7	b)({x	b)({x	NOUN
ejpam-4571	228	8	}	}	PUNCT
ejpam-4571	228	9	)	)	PUNCT
ejpam-4571	229	1	⊆	⊆	NUM
ejpam-4571	229	2	λ(λ	λ(λ	ADP
ejpam-4571	229	3	,	,	PUNCT
ejpam-4571	229	4	b)(⟨x⟩b	b)(⟨x⟩b	PROPN
ejpam-4571	229	5	)	)	PUNCT
ejpam-4571	229	6	.	.	PUNCT
ejpam-4571	230	1	next	next	ADV
ejpam-4571	230	2	,	,	PUNCT
ejpam-4571	230	3	we	we	PRON
ejpam-4571	230	4	show	show	VERB
ejpam-4571	230	5	the	the	DET
ejpam-4571	230	6	opposite	opposite	ADJ
ejpam-4571	230	7	implication	implication	NOUN
ejpam-4571	230	8	.	.	PUNCT
ejpam-4571	230	9	suppose	suppose	VERB
ejpam-4571	230	10	that	that	SCONJ
ejpam-4571	230	11	y	y	PROPN
ejpam-4571	230	12	̸∈	̸∈	PROPN
ejpam-4571	230	13	λ(λ	λ(λ	PROPN
ejpam-4571	230	14	,	,	PUNCT
ejpam-4571	230	15	b)({x	b)({x	NOUN
ejpam-4571	230	16	}	}	PUNCT
ejpam-4571	230	17	)	)	PUNCT
ejpam-4571	230	18	.	.	PUNCT
ejpam-4571	231	1	there	there	PRON
ejpam-4571	231	2	exists	exist	VERB
ejpam-4571	231	3	a	a	DET
ejpam-4571	231	4	(	(	PUNCT
ejpam-4571	231	5	λ	λ	NOUN
ejpam-4571	231	6	,	,	PUNCT
ejpam-4571	231	7	b)-open	b)-open	VERB
ejpam-4571	231	8	set	set	VERB
ejpam-4571	231	9	v	v	ADP
ejpam-4571	231	10	such	such	ADJ
ejpam-4571	231	11	that	that	SCONJ
ejpam-4571	231	12	x	x	SYM
ejpam-4571	231	13	∈	∈	PROPN
ejpam-4571	231	14	v	v	NOUN
ejpam-4571	231	15	and	and	CCONJ
ejpam-4571	231	16	y	y	PROPN
ejpam-4571	231	17	̸∈	̸∈	PROPN
ejpam-4571	231	18	v	v	PROPN
ejpam-4571	231	19	.	.	PUNCT
ejpam-4571	232	1	since	since	SCONJ
ejpam-4571	232	2	⟨x⟩b	⟨x⟩b	ADV
ejpam-4571	232	3	⊆	⊆	NUM
ejpam-4571	232	4	λ(λ	λ(λ	NOUN
ejpam-4571	232	5	,	,	PUNCT
ejpam-4571	232	6	b)({x	b)({x	NOUN
ejpam-4571	232	7	}	}	PUNCT
ejpam-4571	232	8	)	)	PUNCT
ejpam-4571	233	1	⊆	⊆	NUM
ejpam-4571	233	2	λ(λ	λ(λ	ADP
ejpam-4571	233	3	,	,	PUNCT
ejpam-4571	233	4	b)(v	b)(v	X
ejpam-4571	233	5	)	)	PUNCT
ejpam-4571	234	1	=	=	SYM
ejpam-4571	234	2	v	v	X
ejpam-4571	234	3	,	,	PUNCT
ejpam-4571	234	4	we	we	PRON
ejpam-4571	234	5	have	have	VERB
ejpam-4571	234	6	λ(λ	λ(λ	PROPN
ejpam-4571	234	7	,	,	PUNCT
ejpam-4571	234	8	b)(⟨x⟩b	b)(⟨x⟩b	PROPN
ejpam-4571	234	9	)	)	PUNCT
ejpam-4571	234	10	⊆	⊆	NUM
ejpam-4571	234	11	v	v	NOUN
ejpam-4571	234	12	.	.	PUNCT
ejpam-4571	235	1	since	since	SCONJ
ejpam-4571	235	2	y	y	PROPN
ejpam-4571	235	3	̸∈	̸∈	PROPN
ejpam-4571	235	4	v	v	PROPN
ejpam-4571	235	5	,	,	PUNCT
ejpam-4571	235	6	y	y	PROPN
ejpam-4571	235	7	̸∈	̸∈	PROPN
ejpam-4571	235	8	λ(λ	λ(λ	PROPN
ejpam-4571	235	9	,	,	PUNCT
ejpam-4571	235	10	b)(⟨x⟩b	b)(⟨x⟩b	PROPN
ejpam-4571	235	11	)	)	PUNCT
ejpam-4571	235	12	.	.	PUNCT
ejpam-4571	236	1	thus	thus	ADV
ejpam-4571	236	2	,	,	PUNCT
ejpam-4571	236	3	λ(λ	λ(λ	ADV
ejpam-4571	236	4	,	,	PUNCT
ejpam-4571	236	5	b)(⟨x⟩b	b)(⟨x⟩b	PROPN
ejpam-4571	236	6	)	)	PUNCT
ejpam-4571	236	7	⊆	⊆	NUM
ejpam-4571	236	8	λ(λ	λ(λ	PROPN
ejpam-4571	236	9	,	,	PUNCT
ejpam-4571	236	10	b)({x	b)({x	NOUN
ejpam-4571	236	11	}	}	PUNCT
ejpam-4571	236	12	)	)	PUNCT
ejpam-4571	236	13	and	and	CCONJ
ejpam-4571	236	14	hence	hence	ADV
ejpam-4571	236	15	λ(λ	λ(λ	PROPN
ejpam-4571	236	16	,	,	PUNCT
ejpam-4571	236	17	b)({x	b)({x	NOUN
ejpam-4571	236	18	}	}	PUNCT
ejpam-4571	236	19	)	)	PUNCT
ejpam-4571	237	1	=	=	SYM
ejpam-4571	237	2	λ(λ	λ(λ	PROPN
ejpam-4571	237	3	,	,	PUNCT
ejpam-4571	237	4	b)(⟨x⟩b	b)(⟨x⟩b	PROPN
ejpam-4571	237	5	)	)	PUNCT
ejpam-4571	237	6	.	.	PUNCT
ejpam-4571	238	1	(	(	PUNCT
ejpam-4571	238	2	3	3	X
ejpam-4571	238	3	)	)	PUNCT
ejpam-4571	238	4	by	by	ADP
ejpam-4571	238	5	the	the	DET
ejpam-4571	238	6	definition	definition	NOUN
ejpam-4571	238	7	of	of	ADP
ejpam-4571	238	8	⟨x⟩b	⟨x⟩b	ADV
ejpam-4571	238	9	,	,	PUNCT
ejpam-4571	238	10	we	we	PRON
ejpam-4571	238	11	have	have	VERB
ejpam-4571	238	12	{	{	PUNCT
ejpam-4571	238	13	x	x	NOUN
ejpam-4571	238	14	}	}	PUNCT
ejpam-4571	238	15	⊆	⊆	NUM
ejpam-4571	238	16	⟨x⟩b	⟨x⟩b	ADV
ejpam-4571	238	17	and	and	CCONJ
ejpam-4571	238	18	{	{	PUNCT
ejpam-4571	238	19	x}(λ	x}(λ	PROPN
ejpam-4571	238	20	,	,	PUNCT
ejpam-4571	238	21	b	b	NOUN
ejpam-4571	238	22	)	)	PUNCT
ejpam-4571	238	23	⊆	⊆	NUM
ejpam-4571	238	24	(	(	PUNCT
ejpam-4571	238	25	⟨x⟩b)(λ	⟨x⟩b)(λ	NOUN
ejpam-4571	238	26	,	,	PUNCT
ejpam-4571	238	27	b	b	NOUN
ejpam-4571	238	28	)	)	PUNCT
ejpam-4571	238	29	by	by	ADP
ejpam-4571	238	30	lemma	lemma	PROPN
ejpam-4571	238	31	1	1	NUM
ejpam-4571	238	32	.	.	PUNCT
ejpam-4571	239	1	on	on	ADP
ejpam-4571	239	2	the	the	DET
ejpam-4571	239	3	other	other	ADJ
ejpam-4571	239	4	hand	hand	NOUN
ejpam-4571	239	5	,	,	PUNCT
ejpam-4571	239	6	we	we	PRON
ejpam-4571	239	7	have	have	VERB
ejpam-4571	239	8	⟨x⟩b	⟨x⟩b	ADV
ejpam-4571	239	9	⊆	⊆	NUM
ejpam-4571	239	10	{	{	PUNCT
ejpam-4571	239	11	x}(λ	x}(λ	PROPN
ejpam-4571	239	12	,	,	PUNCT
ejpam-4571	239	13	b	b	NOUN
ejpam-4571	239	14	)	)	PUNCT
ejpam-4571	239	15	and	and	CCONJ
ejpam-4571	239	16	(	(	PUNCT
ejpam-4571	239	17	⟨x⟩b)(λ	⟨x⟩b)(λ	NOUN
ejpam-4571	239	18	,	,	PUNCT
ejpam-4571	239	19	b	b	NOUN
ejpam-4571	239	20	)	)	PUNCT
ejpam-4571	239	21	⊆	⊆	NUM
ejpam-4571	239	22	(	(	PUNCT
ejpam-4571	239	23	{	{	PUNCT
ejpam-4571	239	24	x}(λ	x}(λ	PROPN
ejpam-4571	239	25	,	,	PUNCT
ejpam-4571	239	26	b))(λ	b))(λ	PROPN
ejpam-4571	239	27	,	,	PUNCT
ejpam-4571	239	28	b	b	NOUN
ejpam-4571	239	29	)	)	PUNCT
ejpam-4571	239	30	=	=	SYM
ejpam-4571	239	31	{	{	PUNCT
ejpam-4571	239	32	x}(λ	x}(λ	PROPN
ejpam-4571	239	33	,	,	PUNCT
ejpam-4571	239	34	b	b	NOUN
ejpam-4571	239	35	)	)	PUNCT
ejpam-4571	239	36	.	.	PUNCT
ejpam-4571	240	1	thus	thus	ADV
ejpam-4571	240	2	,	,	PUNCT
ejpam-4571	240	3	(	(	PUNCT
ejpam-4571	240	4	⟨x⟩b)(λ	⟨x⟩b)(λ	NOUN
ejpam-4571	240	5	,	,	PUNCT
ejpam-4571	240	6	b	b	NOUN
ejpam-4571	240	7	)	)	PUNCT
ejpam-4571	240	8	⊆	⊆	NUM
ejpam-4571	240	9	{	{	PUNCT
ejpam-4571	240	10	x}(λ	x}(λ	PROPN
ejpam-4571	240	11	,	,	PUNCT
ejpam-4571	240	12	b	b	NOUN
ejpam-4571	240	13	)	)	PUNCT
ejpam-4571	240	14	.	.	PUNCT
ejpam-4571	241	1	c.	c.	PROPN
ejpam-4571	241	2	boonpok	boonpok	PROPN
ejpam-4571	241	3	,	,	PUNCT
ejpam-4571	241	4	n.	n.	PROPN
ejpam-4571	241	5	srisarakham	srisarakham	PROPN
ejpam-4571	241	6	/	/	SYM
ejpam-4571	241	7	eur	eur	PROPN
ejpam-4571	241	8	.	.	PUNCT
ejpam-4571	242	1	j.	j.	PROPN
ejpam-4571	242	2	pure	pure	PROPN
ejpam-4571	242	3	appl	appl	PROPN
ejpam-4571	242	4	.	.	PROPN
ejpam-4571	242	5	math	math	PROPN
ejpam-4571	242	6	,	,	PUNCT
ejpam-4571	242	7	16	16	NUM
ejpam-4571	242	8	(	(	PUNCT
ejpam-4571	242	9	1	1	NUM
ejpam-4571	242	10	)	)	PUNCT
ejpam-4571	242	11	(	(	PUNCT
ejpam-4571	242	12	2023	2023	NUM
ejpam-4571	242	13	)	)	PUNCT
ejpam-4571	242	14	,	,	PUNCT
ejpam-4571	242	15	29	29	NUM
ejpam-4571	242	16	-	-	SYM
ejpam-4571	242	17	43	43	NUM
ejpam-4571	242	18	35	35	NUM
ejpam-4571	242	19	(	(	PUNCT
ejpam-4571	242	20	4	4	NUM
ejpam-4571	242	21	)	)	PUNCT
ejpam-4571	242	22	since	since	SCONJ
ejpam-4571	242	23	x	x	PROPN
ejpam-4571	242	24	∈	∈	PROPN
ejpam-4571	242	25	u	u	NOUN
ejpam-4571	242	26	and	and	CCONJ
ejpam-4571	242	27	u	u	NOUN
ejpam-4571	242	28	is	be	AUX
ejpam-4571	242	29	(	(	PUNCT
ejpam-4571	242	30	λ	λ	INTJ
ejpam-4571	242	31	,	,	PUNCT
ejpam-4571	242	32	b)-open	b)-open	VERB
ejpam-4571	242	33	,	,	PUNCT
ejpam-4571	242	34	we	we	PRON
ejpam-4571	242	35	have	have	VERB
ejpam-4571	242	36	λ(λ	λ(λ	PROPN
ejpam-4571	242	37	,	,	PUNCT
ejpam-4571	242	38	b)({x	b)({x	NOUN
ejpam-4571	242	39	}	}	PUNCT
ejpam-4571	242	40	)	)	PUNCT
ejpam-4571	243	1	⊆	⊆	NUM
ejpam-4571	243	2	u	u	NOUN
ejpam-4571	243	3	.	.	PUNCT
ejpam-4571	244	1	thus	thus	ADV
ejpam-4571	244	2	,	,	PUNCT
ejpam-4571	244	3	⟨x⟩b	⟨x⟩b	ADV
ejpam-4571	244	4	⊆	⊆	NUM
ejpam-4571	244	5	u	u	NOUN
ejpam-4571	244	6	.	.	PUNCT
ejpam-4571	245	1	(	(	PUNCT
ejpam-4571	245	2	5	5	NUM
ejpam-4571	245	3	)	)	PUNCT
ejpam-4571	245	4	since	since	SCONJ
ejpam-4571	245	5	x	x	PROPN
ejpam-4571	245	6	∈	∈	PROPN
ejpam-4571	245	7	f	f	PROPN
ejpam-4571	245	8	and	and	CCONJ
ejpam-4571	245	9	f	f	PROPN
ejpam-4571	245	10	is	be	AUX
ejpam-4571	245	11	(	(	PUNCT
ejpam-4571	245	12	λ	λ	X
ejpam-4571	245	13	,	,	PUNCT
ejpam-4571	245	14	b)-closed	b)-close	VERB
ejpam-4571	245	15	,	,	PUNCT
ejpam-4571	245	16	⟨x⟩b	⟨x⟩b	ADV
ejpam-4571	245	17	=	=	SYM
ejpam-4571	245	18	{	{	PUNCT
ejpam-4571	245	19	x}(λ	x}(λ	PROPN
ejpam-4571	245	20	,	,	PUNCT
ejpam-4571	245	21	b	b	NOUN
ejpam-4571	245	22	)	)	PUNCT
ejpam-4571	245	23	∩	∩	NOUN
ejpam-4571	245	24	λ(λ	λ(λ	ADP
ejpam-4571	245	25	,	,	PUNCT
ejpam-4571	245	26	b)({x	b)({x	NOUN
ejpam-4571	245	27	}	}	PUNCT
ejpam-4571	245	28	)	)	PUNCT
ejpam-4571	246	1	⊆	⊆	X
ejpam-4571	246	2	{	{	PUNCT
ejpam-4571	246	3	x}(λ	x}(λ	PROPN
ejpam-4571	246	4	,	,	PUNCT
ejpam-4571	246	5	b	b	NOUN
ejpam-4571	246	6	)	)	PUNCT
ejpam-4571	246	7	⊆	⊆	NUM
ejpam-4571	246	8	f	f	X
ejpam-4571	246	9	(	(	PUNCT
ejpam-4571	246	10	λ	λ	PROPN
ejpam-4571	246	11	,	,	PUNCT
ejpam-4571	246	12	b	b	NOUN
ejpam-4571	246	13	)	)	PUNCT
ejpam-4571	246	14	=	=	SYM
ejpam-4571	246	15	f	f	PROPN
ejpam-4571	246	16	.	.	PUNCT
ejpam-4571	247	1	definition	definition	NOUN
ejpam-4571	247	2	4	4	NUM
ejpam-4571	247	3	.	.	PUNCT
ejpam-4571	248	1	[	[	X
ejpam-4571	248	2	15	15	NUM
ejpam-4571	248	3	]	]	X
ejpam-4571	248	4	a	a	DET
ejpam-4571	248	5	subset	subset	NOUN
ejpam-4571	248	6	a	a	PRON
ejpam-4571	248	7	of	of	ADP
ejpam-4571	248	8	a	a	DET
ejpam-4571	248	9	topological	topological	ADJ
ejpam-4571	248	10	space	space	NOUN
ejpam-4571	248	11	(	(	PUNCT
ejpam-4571	248	12	x	x	X
ejpam-4571	248	13	,	,	PUNCT
ejpam-4571	248	14	τ	τ	X
ejpam-4571	248	15	)	)	PUNCT
ejpam-4571	248	16	is	be	AUX
ejpam-4571	248	17	said	say	VERB
ejpam-4571	248	18	to	to	PART
ejpam-4571	248	19	be	be	AUX
ejpam-4571	248	20	:	:	PUNCT
ejpam-4571	248	21	(	(	PUNCT
ejpam-4571	248	22	i	i	NOUN
ejpam-4571	248	23	)	)	PUNCT
ejpam-4571	248	24	(	(	PUNCT
ejpam-4571	248	25	λ	λ	NOUN
ejpam-4571	248	26	,	,	PUNCT
ejpam-4571	248	27	b)-dense	b)-dense	NOUN
ejpam-4571	248	28	if	if	SCONJ
ejpam-4571	248	29	a(λ	a(λ	ADV
ejpam-4571	248	30	,	,	PUNCT
ejpam-4571	248	31	b	b	NOUN
ejpam-4571	248	32	)	)	PUNCT
ejpam-4571	248	33	=	=	SYM
ejpam-4571	249	1	x	x	X
ejpam-4571	249	2	;	;	PUNCT
ejpam-4571	249	3	(	(	PUNCT
ejpam-4571	249	4	ii	ii	NOUN
ejpam-4571	249	5	)	)	PUNCT
ejpam-4571	249	6	(	(	PUNCT
ejpam-4571	249	7	λ	λ	INTJ
ejpam-4571	249	8	,	,	PUNCT
ejpam-4571	249	9	b)-codense	b)-codense	PUNCT
ejpam-4571	249	10	if	if	SCONJ
ejpam-4571	249	11	its	its	PRON
ejpam-4571	249	12	complement	complement	NOUN
ejpam-4571	249	13	is	be	AUX
ejpam-4571	249	14	(	(	PUNCT
ejpam-4571	249	15	λ	λ	NOUN
ejpam-4571	249	16	,	,	PUNCT
ejpam-4571	249	17	b)-dense	b)-dense	NOUN
ejpam-4571	249	18	.	.	PUNCT
ejpam-4571	249	19	recall	recall	VERB
ejpam-4571	249	20	that	that	SCONJ
ejpam-4571	249	21	a	a	DET
ejpam-4571	249	22	topological	topological	ADJ
ejpam-4571	249	23	space	space	NOUN
ejpam-4571	249	24	(	(	PUNCT
ejpam-4571	249	25	x	x	X
ejpam-4571	249	26	,	,	PUNCT
ejpam-4571	249	27	τ	τ	X
ejpam-4571	249	28	)	)	PUNCT
ejpam-4571	249	29	is	be	AUX
ejpam-4571	249	30	called	call	VERB
ejpam-4571	249	31	(	(	PUNCT
ejpam-4571	249	32	λ	λ	PROPN
ejpam-4571	249	33	,	,	PUNCT
ejpam-4571	249	34	b)-submaximal	b)-submaximal	PUNCT
ejpam-4571	249	35	[	[	X
ejpam-4571	249	36	15	15	NUM
ejpam-4571	249	37	]	]	X
ejpam-4571	249	38	if	if	SCONJ
ejpam-4571	249	39	each	each	PRON
ejpam-4571	249	40	(	(	PUNCT
ejpam-4571	249	41	λ	λ	PROPN
ejpam-4571	249	42	,	,	PUNCT
ejpam-4571	249	43	b)dense	b)dense	NOUN
ejpam-4571	249	44	subset	subset	NOUN
ejpam-4571	249	45	of	of	ADP
ejpam-4571	249	46	x	x	PUNCT
ejpam-4571	249	47	is	be	AUX
ejpam-4571	249	48	(	(	PUNCT
ejpam-4571	249	49	λ	λ	INTJ
ejpam-4571	249	50	,	,	PUNCT
ejpam-4571	249	51	b)-open	b)-open	VERB
ejpam-4571	249	52	.	.	PUNCT
ejpam-4571	250	1	proposition	proposition	NOUN
ejpam-4571	250	2	5	5	NUM
ejpam-4571	250	3	.	.	PUNCT
ejpam-4571	251	1	let	let	VERB
ejpam-4571	251	2	(	(	PUNCT
ejpam-4571	251	3	x	x	NOUN
ejpam-4571	251	4	,	,	PUNCT
ejpam-4571	251	5	τ	τ	X
ejpam-4571	251	6	)	)	PUNCT
ejpam-4571	251	7	be	be	VERB
ejpam-4571	251	8	a	a	DET
ejpam-4571	251	9	topological	topological	ADJ
ejpam-4571	251	10	space	space	NOUN
ejpam-4571	251	11	.	.	PUNCT
ejpam-4571	252	1	if	if	SCONJ
ejpam-4571	252	2	each	each	DET
ejpam-4571	252	3	p(λ	p(λ	NOUN
ejpam-4571	252	4	,	,	PUNCT
ejpam-4571	252	5	b)-open	b)-open	AUX
ejpam-4571	252	6	set	set	VERB
ejpam-4571	252	7	is	be	AUX
ejpam-4571	252	8	s(λ	s(λ	PROPN
ejpam-4571	252	9	,	,	PUNCT
ejpam-4571	252	10	b)-open	b)-open	PUNCT
ejpam-4571	252	11	and	and	CCONJ
ejpam-4571	252	12	each	each	DET
ejpam-4571	252	13	α(λ	α(λ	PROPN
ejpam-4571	252	14	,	,	PUNCT
ejpam-4571	252	15	b)-open	b)-open	AUX
ejpam-4571	252	16	set	set	VERB
ejpam-4571	252	17	is	be	AUX
ejpam-4571	252	18	(	(	PUNCT
ejpam-4571	252	19	λ	λ	INTJ
ejpam-4571	252	20	,	,	PUNCT
ejpam-4571	252	21	b)-open	b)-open	VERB
ejpam-4571	252	22	,	,	PUNCT
ejpam-4571	252	23	then	then	ADV
ejpam-4571	252	24	(	(	PUNCT
ejpam-4571	252	25	x	x	X
ejpam-4571	252	26	,	,	PUNCT
ejpam-4571	252	27	τ	τ	X
ejpam-4571	252	28	)	)	PUNCT
ejpam-4571	252	29	is	be	AUX
ejpam-4571	252	30	(	(	PUNCT
ejpam-4571	252	31	λ	λ	X
ejpam-4571	252	32	,	,	PUNCT
ejpam-4571	252	33	b)-submaximal	b)-submaximal	ADJ
ejpam-4571	252	34	.	.	PUNCT
ejpam-4571	253	1	proof	proof	NOUN
ejpam-4571	253	2	.	.	PUNCT
ejpam-4571	254	1	let	let	VERB
ejpam-4571	254	2	d	d	PRON
ejpam-4571	254	3	be	be	AUX
ejpam-4571	254	4	(	(	PUNCT
ejpam-4571	254	5	λ	λ	NOUN
ejpam-4571	254	6	,	,	PUNCT
ejpam-4571	254	7	b)-dense	b)-dense	NOUN
ejpam-4571	254	8	subset	subset	NOUN
ejpam-4571	254	9	of	of	ADP
ejpam-4571	254	10	x.	x.	NOUN
ejpam-4571	254	11	since	since	SCONJ
ejpam-4571	254	12	d(λ	d(λ	PROPN
ejpam-4571	254	13	,	,	PUNCT
ejpam-4571	254	14	b	b	NOUN
ejpam-4571	254	15	)	)	PUNCT
ejpam-4571	254	16	=	=	SYM
ejpam-4571	255	1	x	x	X
ejpam-4571	255	2	,	,	PUNCT
ejpam-4571	255	3	we	we	PRON
ejpam-4571	255	4	have	have	VERB
ejpam-4571	255	5	d	d	PROPN
ejpam-4571	255	6	is	be	AUX
ejpam-4571	255	7	p(λ	p(λ	NOUN
ejpam-4571	255	8	,	,	PUNCT
ejpam-4571	255	9	b)open	b)open	VERB
ejpam-4571	255	10	.	.	PUNCT
ejpam-4571	256	1	thus	thus	ADV
ejpam-4571	256	2	,	,	PUNCT
ejpam-4571	256	3	d	d	PROPN
ejpam-4571	256	4	is	be	AUX
ejpam-4571	256	5	s(λ	s(λ	PROPN
ejpam-4571	256	6	,	,	PUNCT
ejpam-4571	256	7	b)-open	b)-open	PUNCT
ejpam-4571	256	8	and	and	CCONJ
ejpam-4571	256	9	hence	hence	ADV
ejpam-4571	256	10	d	d	PROPN
ejpam-4571	256	11	is	be	AUX
ejpam-4571	256	12	α(λ	α(λ	PROPN
ejpam-4571	256	13	,	,	PUNCT
ejpam-4571	256	14	b)-open	b)-open	VERB
ejpam-4571	256	15	.	.	PUNCT
ejpam-4571	257	1	since	since	SCONJ
ejpam-4571	257	2	each	each	DET
ejpam-4571	257	3	α(λ	α(λ	PROPN
ejpam-4571	257	4	,	,	PUNCT
ejpam-4571	257	5	b)-open	b)-open	AUX
ejpam-4571	257	6	set	set	VERB
ejpam-4571	257	7	is	be	AUX
ejpam-4571	257	8	(	(	PUNCT
ejpam-4571	257	9	λ	λ	INTJ
ejpam-4571	257	10	,	,	PUNCT
ejpam-4571	257	11	b)-open	b)-open	VERB
ejpam-4571	257	12	,	,	PUNCT
ejpam-4571	257	13	d	d	X
ejpam-4571	257	14	is	be	AUX
ejpam-4571	257	15	(	(	PUNCT
ejpam-4571	257	16	λ	λ	INTJ
ejpam-4571	257	17	,	,	PUNCT
ejpam-4571	257	18	b)-open	b)-open	VERB
ejpam-4571	257	19	.	.	PUNCT
ejpam-4571	258	1	this	this	PRON
ejpam-4571	258	2	shows	show	VERB
ejpam-4571	258	3	that	that	SCONJ
ejpam-4571	258	4	(	(	PUNCT
ejpam-4571	258	5	x	x	X
ejpam-4571	258	6	,	,	PUNCT
ejpam-4571	258	7	τ	τ	X
ejpam-4571	258	8	)	)	PUNCT
ejpam-4571	258	9	is	be	AUX
ejpam-4571	258	10	(	(	PUNCT
ejpam-4571	258	11	λ	λ	X
ejpam-4571	258	12	,	,	PUNCT
ejpam-4571	258	13	b)-submaximal	b)-submaximal	PROPN
ejpam-4571	258	14	.	.	PUNCT
ejpam-4571	259	1	proposition	proposition	NOUN
ejpam-4571	259	2	6	6	NUM
ejpam-4571	259	3	.	.	PUNCT
ejpam-4571	260	1	let	let	VERB
ejpam-4571	260	2	(	(	PUNCT
ejpam-4571	260	3	x	x	NOUN
ejpam-4571	260	4	,	,	PUNCT
ejpam-4571	260	5	τ	τ	X
ejpam-4571	260	6	)	)	PUNCT
ejpam-4571	260	7	be	be	VERB
ejpam-4571	260	8	a	a	DET
ejpam-4571	260	9	topological	topological	ADJ
ejpam-4571	260	10	space	space	NOUN
ejpam-4571	260	11	.	.	PUNCT
ejpam-4571	261	1	if	if	SCONJ
ejpam-4571	261	2	each	each	DET
ejpam-4571	261	3	p(λ	p(λ	NOUN
ejpam-4571	261	4	,	,	PUNCT
ejpam-4571	261	5	b)-open	b)-open	AUX
ejpam-4571	261	6	set	set	VERB
ejpam-4571	261	7	is	be	AUX
ejpam-4571	261	8	(	(	PUNCT
ejpam-4571	261	9	λ	λ	INTJ
ejpam-4571	261	10	,	,	PUNCT
ejpam-4571	261	11	b)-open	b)-open	VERB
ejpam-4571	261	12	,	,	PUNCT
ejpam-4571	261	13	then	then	ADV
ejpam-4571	261	14	(	(	PUNCT
ejpam-4571	261	15	x	x	X
ejpam-4571	261	16	,	,	PUNCT
ejpam-4571	261	17	τ	τ	X
ejpam-4571	261	18	)	)	PUNCT
ejpam-4571	261	19	is	be	AUX
ejpam-4571	261	20	(	(	PUNCT
ejpam-4571	261	21	λ	λ	X
ejpam-4571	261	22	,	,	PUNCT
ejpam-4571	261	23	b)-submaximal	b)-submaximal	ADJ
ejpam-4571	261	24	.	.	PUNCT
ejpam-4571	262	1	proof	proof	NOUN
ejpam-4571	262	2	.	.	PUNCT
ejpam-4571	263	1	suppose	suppose	VERB
ejpam-4571	263	2	that	that	SCONJ
ejpam-4571	263	3	each	each	DET
ejpam-4571	263	4	p(λ	p(λ	NOUN
ejpam-4571	263	5	,	,	PUNCT
ejpam-4571	263	6	b)-open	b)-open	AUX
ejpam-4571	263	7	set	set	VERB
ejpam-4571	263	8	is	be	AUX
ejpam-4571	263	9	(	(	PUNCT
ejpam-4571	263	10	λ	λ	INTJ
ejpam-4571	263	11	,	,	PUNCT
ejpam-4571	263	12	b)-open	b)-open	AUX
ejpam-4571	263	13	.	.	PUNCT
ejpam-4571	264	1	it	it	PRON
ejpam-4571	264	2	follows	follow	VERB
ejpam-4571	264	3	that	that	SCONJ
ejpam-4571	264	4	every	every	DET
ejpam-4571	264	5	p(λ	p(λ	NOUN
ejpam-4571	264	6	,	,	PUNCT
ejpam-4571	264	7	b)open	b)open	NOUN
ejpam-4571	264	8	set	set	NOUN
ejpam-4571	264	9	is	be	AUX
ejpam-4571	264	10	s(λ	s(λ	PROPN
ejpam-4571	264	11	,	,	PUNCT
ejpam-4571	264	12	b)-open	b)-open	VERB
ejpam-4571	264	13	.	.	PUNCT
ejpam-4571	265	1	since	since	SCONJ
ejpam-4571	265	2	each	each	DET
ejpam-4571	265	3	α(λ	α(λ	PROPN
ejpam-4571	265	4	,	,	PUNCT
ejpam-4571	265	5	b)-open	b)-open	AUX
ejpam-4571	265	6	set	set	VERB
ejpam-4571	265	7	is	be	AUX
ejpam-4571	265	8	p(λ	p(λ	NOUN
ejpam-4571	265	9	,	,	PUNCT
ejpam-4571	265	10	b)-open	b)-open	VERB
ejpam-4571	265	11	,	,	PUNCT
ejpam-4571	265	12	then	then	ADV
ejpam-4571	265	13	each	each	DET
ejpam-4571	265	14	α(λ	α(λ	PROPN
ejpam-4571	265	15	,	,	PUNCT
ejpam-4571	265	16	b)-open	b)-open	AUX
ejpam-4571	265	17	set	set	VERB
ejpam-4571	265	18	is	be	AUX
ejpam-4571	265	19	(	(	PUNCT
ejpam-4571	265	20	λ	λ	INTJ
ejpam-4571	265	21	,	,	PUNCT
ejpam-4571	265	22	b)-open	b)-open	VERB
ejpam-4571	265	23	.	.	PUNCT
ejpam-4571	266	1	thus	thus	ADV
ejpam-4571	266	2	,	,	PUNCT
ejpam-4571	266	3	by	by	ADP
ejpam-4571	266	4	proposition	proposition	NOUN
ejpam-4571	266	5	5	5	NUM
ejpam-4571	266	6	,	,	PUNCT
ejpam-4571	266	7	(	(	PUNCT
ejpam-4571	266	8	x	x	X
ejpam-4571	266	9	,	,	PUNCT
ejpam-4571	266	10	τ	τ	X
ejpam-4571	266	11	)	)	PUNCT
ejpam-4571	266	12	is	be	AUX
ejpam-4571	266	13	(	(	PUNCT
ejpam-4571	266	14	λ	λ	X
ejpam-4571	266	15	,	,	PUNCT
ejpam-4571	266	16	b)-submaximal	b)-submaximal	PROPN
ejpam-4571	266	17	.	.	PUNCT
ejpam-4571	267	1	proposition	proposition	NOUN
ejpam-4571	267	2	7	7	NUM
ejpam-4571	267	3	.	.	X
ejpam-4571	267	4	for	for	ADP
ejpam-4571	267	5	a	a	DET
ejpam-4571	267	6	topological	topological	ADJ
ejpam-4571	267	7	space	space	NOUN
ejpam-4571	267	8	(	(	PUNCT
ejpam-4571	267	9	x	x	X
ejpam-4571	267	10	,	,	PUNCT
ejpam-4571	267	11	τ	τ	PROPN
ejpam-4571	267	12	)	)	PUNCT
ejpam-4571	267	13	,	,	PUNCT
ejpam-4571	267	14	the	the	DET
ejpam-4571	267	15	following	follow	VERB
ejpam-4571	267	16	properties	property	NOUN
ejpam-4571	267	17	are	be	AUX
ejpam-4571	267	18	equivalent	equivalent	ADJ
ejpam-4571	267	19	:	:	PUNCT
ejpam-4571	267	20	(	(	PUNCT
ejpam-4571	267	21	1	1	X
ejpam-4571	267	22	)	)	PUNCT
ejpam-4571	267	23	(	(	PUNCT
ejpam-4571	267	24	x	x	X
ejpam-4571	267	25	,	,	PUNCT
ejpam-4571	267	26	τ	τ	X
ejpam-4571	267	27	)	)	PUNCT
ejpam-4571	267	28	is	be	AUX
ejpam-4571	267	29	(	(	PUNCT
ejpam-4571	267	30	λ	λ	X
ejpam-4571	267	31	,	,	PUNCT
ejpam-4571	267	32	b)-submaximal	b)-submaximal	PROPN
ejpam-4571	267	33	.	.	PUNCT
ejpam-4571	268	1	(	(	PUNCT
ejpam-4571	268	2	2	2	X
ejpam-4571	268	3	)	)	PUNCT
ejpam-4571	268	4	each	each	PRON
ejpam-4571	268	5	(	(	PUNCT
ejpam-4571	268	6	λ	λ	PROPN
ejpam-4571	268	7	,	,	PUNCT
ejpam-4571	268	8	b)-codense	b)-codense	PUNCT
ejpam-4571	268	9	subset	subset	VERB
ejpam-4571	268	10	of	of	ADP
ejpam-4571	268	11	x	x	PUNCT
ejpam-4571	268	12	is	be	AUX
ejpam-4571	268	13	(	(	PUNCT
ejpam-4571	268	14	λ	λ	X
ejpam-4571	268	15	,	,	PUNCT
ejpam-4571	268	16	b)-closed	b)-close	VERB
ejpam-4571	268	17	.	.	PUNCT
ejpam-4571	269	1	recall	recall	VERB
ejpam-4571	269	2	that	that	SCONJ
ejpam-4571	269	3	a	a	DET
ejpam-4571	269	4	topological	topological	ADJ
ejpam-4571	269	5	space	space	NOUN
ejpam-4571	269	6	(	(	PUNCT
ejpam-4571	269	7	x	x	X
ejpam-4571	269	8	,	,	PUNCT
ejpam-4571	269	9	τ	τ	X
ejpam-4571	269	10	)	)	PUNCT
ejpam-4571	269	11	is	be	AUX
ejpam-4571	269	12	called	call	VERB
ejpam-4571	269	13	(	(	PUNCT
ejpam-4571	269	14	λ	λ	PROPN
ejpam-4571	269	15	,	,	PUNCT
ejpam-4571	269	16	b)-extremally	b)-extremally	ADV
ejpam-4571	269	17	disconnected	disconnected	ADJ
ejpam-4571	269	18	[	[	X
ejpam-4571	269	19	15	15	NUM
ejpam-4571	269	20	]	]	X
ejpam-4571	269	21	if	if	SCONJ
ejpam-4571	269	22	u	u	PROPN
ejpam-4571	269	23	(	(	PUNCT
ejpam-4571	269	24	λ	λ	PROPN
ejpam-4571	269	25	,	,	PUNCT
ejpam-4571	269	26	b	b	NOUN
ejpam-4571	269	27	)	)	PUNCT
ejpam-4571	269	28	is	be	AUX
ejpam-4571	269	29	(	(	PUNCT
ejpam-4571	269	30	λ	λ	X
ejpam-4571	269	31	,	,	PUNCT
ejpam-4571	269	32	b)-open	b)-open	VERB
ejpam-4571	269	33	in	in	ADP
ejpam-4571	269	34	x	x	PUNCT
ejpam-4571	269	35	for	for	SCONJ
ejpam-4571	269	36	every	every	DET
ejpam-4571	269	37	(	(	PUNCT
ejpam-4571	269	38	λ	λ	NOUN
ejpam-4571	269	39	,	,	PUNCT
ejpam-4571	269	40	b)-open	b)-open	AUX
ejpam-4571	269	41	set	set	VERB
ejpam-4571	269	42	u	u	NOUN
ejpam-4571	269	43	of	of	ADP
ejpam-4571	269	44	x.	x.	PROPN
ejpam-4571	269	45	lemma	lemma	PROPN
ejpam-4571	269	46	4	4	X
ejpam-4571	269	47	.	.	PUNCT
ejpam-4571	270	1	let	let	VERB
ejpam-4571	270	2	a	a	DET
ejpam-4571	270	3	be	be	AUX
ejpam-4571	270	4	a	a	DET
ejpam-4571	270	5	subset	subset	NOUN
ejpam-4571	270	6	of	of	ADP
ejpam-4571	270	7	a	a	DET
ejpam-4571	270	8	topological	topological	ADJ
ejpam-4571	270	9	space	space	NOUN
ejpam-4571	270	10	(	(	PUNCT
ejpam-4571	270	11	x	x	X
ejpam-4571	270	12	,	,	PUNCT
ejpam-4571	270	13	τ	τ	PROPN
ejpam-4571	270	14	)	)	PUNCT
ejpam-4571	270	15	.	.	PUNCT
ejpam-4571	271	1	if	if	SCONJ
ejpam-4571	271	2	u	u	PROPN
ejpam-4571	271	3	∈	∈	PROPN
ejpam-4571	271	4	λbo(x	λbo(x	PROPN
ejpam-4571	271	5	,	,	PUNCT
ejpam-4571	271	6	τ	τ	X
ejpam-4571	271	7	)	)	PUNCT
ejpam-4571	271	8	and	and	CCONJ
ejpam-4571	271	9	u∩a	u∩a	NOUN
ejpam-4571	271	10	=	=	NOUN
ejpam-4571	271	11	∅	∅	NOUN
ejpam-4571	271	12	,	,	PUNCT
ejpam-4571	271	13	then	then	ADV
ejpam-4571	271	14	u	u	NOUN
ejpam-4571	271	15	∩a(λ	∩a(λ	PROPN
ejpam-4571	271	16	,	,	PUNCT
ejpam-4571	271	17	b	b	NOUN
ejpam-4571	271	18	)	)	PUNCT
ejpam-4571	271	19	=	=	SYM
ejpam-4571	271	20	∅.	∅.	NOUN
ejpam-4571	271	21	theorem	theorem	NOUN
ejpam-4571	271	22	2	2	NUM
ejpam-4571	271	23	.	.	X
ejpam-4571	271	24	for	for	ADP
ejpam-4571	271	25	a	a	DET
ejpam-4571	271	26	topological	topological	ADJ
ejpam-4571	271	27	space	space	NOUN
ejpam-4571	271	28	(	(	PUNCT
ejpam-4571	271	29	x	x	X
ejpam-4571	271	30	,	,	PUNCT
ejpam-4571	271	31	τ	τ	PROPN
ejpam-4571	271	32	)	)	PUNCT
ejpam-4571	271	33	,	,	PUNCT
ejpam-4571	271	34	the	the	DET
ejpam-4571	271	35	following	follow	VERB
ejpam-4571	271	36	properties	property	NOUN
ejpam-4571	271	37	are	be	AUX
ejpam-4571	271	38	equivalent	equivalent	ADJ
ejpam-4571	271	39	:	:	PUNCT
ejpam-4571	271	40	(	(	PUNCT
ejpam-4571	271	41	1	1	X
ejpam-4571	271	42	)	)	PUNCT
ejpam-4571	271	43	(	(	PUNCT
ejpam-4571	271	44	x	x	X
ejpam-4571	271	45	,	,	PUNCT
ejpam-4571	271	46	τ	τ	X
ejpam-4571	271	47	)	)	PUNCT
ejpam-4571	271	48	is	be	AUX
ejpam-4571	271	49	(	(	PUNCT
ejpam-4571	271	50	λ	λ	X
ejpam-4571	271	51	,	,	PUNCT
ejpam-4571	271	52	b)-extremally	b)-extremally	ADV
ejpam-4571	271	53	disconnected	disconnected	ADJ
ejpam-4571	271	54	.	.	PUNCT
ejpam-4571	272	1	(	(	PUNCT
ejpam-4571	272	2	2	2	X
ejpam-4571	272	3	)	)	PUNCT
ejpam-4571	272	4	u	u	NOUN
ejpam-4571	272	5	(	(	PUNCT
ejpam-4571	272	6	λ	λ	PROPN
ejpam-4571	272	7	,	,	PUNCT
ejpam-4571	272	8	b	b	NOUN
ejpam-4571	272	9	)	)	PUNCT
ejpam-4571	272	10	∩	∩	ADJ
ejpam-4571	272	11	v	v	X
ejpam-4571	272	12	(	(	PUNCT
ejpam-4571	272	13	λ	λ	PROPN
ejpam-4571	272	14	,	,	PUNCT
ejpam-4571	272	15	b	b	NOUN
ejpam-4571	272	16	)	)	PUNCT
ejpam-4571	272	17	=	=	NOUN
ejpam-4571	272	18	∅	∅	NOUN
ejpam-4571	272	19	for	for	ADP
ejpam-4571	272	20	all	all	PRON
ejpam-4571	272	21	(	(	PUNCT
ejpam-4571	272	22	λ	λ	NOUN
ejpam-4571	272	23	,	,	PUNCT
ejpam-4571	272	24	b)-open	b)-open	VERB
ejpam-4571	272	25	subsets	subset	NOUN
ejpam-4571	272	26	u	u	NOUN
ejpam-4571	272	27	and	and	CCONJ
ejpam-4571	272	28	v	v	NOUN
ejpam-4571	272	29	of	of	ADP
ejpam-4571	272	30	x	x	PUNCT
ejpam-4571	272	31	with	with	ADP
ejpam-4571	272	32	u	u	NOUN
ejpam-4571	272	33	∩	∩	NOUN
ejpam-4571	272	34	v	v	NOUN
ejpam-4571	272	35	=	=	PUNCT
ejpam-4571	272	36	∅.	∅.	PROPN
ejpam-4571	272	37	c.	c.	PROPN
ejpam-4571	272	38	boonpok	boonpok	PROPN
ejpam-4571	272	39	,	,	PUNCT
ejpam-4571	272	40	n.	n.	PROPN
ejpam-4571	272	41	srisarakham	srisarakham	PROPN
ejpam-4571	272	42	/	/	SYM
ejpam-4571	272	43	eur	eur	PROPN
ejpam-4571	272	44	.	.	PUNCT
ejpam-4571	273	1	j.	j.	PROPN
ejpam-4571	273	2	pure	pure	PROPN
ejpam-4571	273	3	appl	appl	PROPN
ejpam-4571	273	4	.	.	PROPN
ejpam-4571	273	5	math	math	PROPN
ejpam-4571	273	6	,	,	PUNCT
ejpam-4571	273	7	16	16	NUM
ejpam-4571	273	8	(	(	PUNCT
ejpam-4571	273	9	1	1	NUM
ejpam-4571	273	10	)	)	PUNCT
ejpam-4571	273	11	(	(	PUNCT
ejpam-4571	273	12	2023	2023	NUM
ejpam-4571	273	13	)	)	PUNCT
ejpam-4571	273	14	,	,	PUNCT
ejpam-4571	273	15	29	29	NUM
ejpam-4571	273	16	-	-	SYM
ejpam-4571	273	17	43	43	NUM
ejpam-4571	273	18	36	36	NUM
ejpam-4571	273	19	proof	proof	NOUN
ejpam-4571	273	20	.	.	PUNCT
ejpam-4571	274	1	(	(	PUNCT
ejpam-4571	274	2	1	1	X
ejpam-4571	274	3	)	)	PUNCT
ejpam-4571	274	4	⇒	⇒	NOUN
ejpam-4571	274	5	(	(	PUNCT
ejpam-4571	274	6	2	2	NUM
ejpam-4571	274	7	):	):	PUNCT
ejpam-4571	274	8	suppose	suppose	VERB
ejpam-4571	274	9	that	that	SCONJ
ejpam-4571	274	10	u	u	PROPN
ejpam-4571	274	11	and	and	CCONJ
ejpam-4571	274	12	v	v	NOUN
ejpam-4571	274	13	are	be	AUX
ejpam-4571	274	14	(	(	PUNCT
ejpam-4571	274	15	λ	λ	X
ejpam-4571	274	16	,	,	PUNCT
ejpam-4571	274	17	b)-open	b)-open	VERB
ejpam-4571	274	18	sets	set	NOUN
ejpam-4571	274	19	of	of	ADP
ejpam-4571	274	20	x	x	SYM
ejpam-4571	274	21	such	such	ADJ
ejpam-4571	274	22	that	that	DET
ejpam-4571	274	23	u∩v	u∩v	NOUN
ejpam-4571	274	24	=	=	X
ejpam-4571	274	25	∅.	∅.	VERB
ejpam-4571	274	26	thus	thus	ADV
ejpam-4571	274	27	,	,	PUNCT
ejpam-4571	274	28	by	by	ADP
ejpam-4571	274	29	lemma	lemma	PROPN
ejpam-4571	274	30	4	4	NUM
ejpam-4571	274	31	,	,	PUNCT
ejpam-4571	274	32	u	u	NOUN
ejpam-4571	274	33	(	(	PUNCT
ejpam-4571	274	34	λ	λ	PROPN
ejpam-4571	274	35	,	,	PUNCT
ejpam-4571	274	36	b	b	NOUN
ejpam-4571	274	37	)	)	PUNCT
ejpam-4571	274	38	∩	∩	ADJ
ejpam-4571	274	39	v	v	NOUN
ejpam-4571	274	40	=	=	NOUN
ejpam-4571	274	41	∅	∅	NOUN
ejpam-4571	274	42	which	which	PRON
ejpam-4571	274	43	implies	imply	VERB
ejpam-4571	274	44	that	that	SCONJ
ejpam-4571	274	45	[	[	X
ejpam-4571	274	46	u	u	X
ejpam-4571	274	47	(	(	PUNCT
ejpam-4571	274	48	λ	λ	PROPN
ejpam-4571	274	49	,	,	PUNCT
ejpam-4571	274	50	b)](λ	b)](λ	NOUN
ejpam-4571	274	51	,	,	PUNCT
ejpam-4571	274	52	b	b	NOUN
ejpam-4571	274	53	)	)	PUNCT
ejpam-4571	274	54	∩	∩	ADJ
ejpam-4571	274	55	v	v	X
ejpam-4571	274	56	(	(	PUNCT
ejpam-4571	274	57	λ	λ	PROPN
ejpam-4571	274	58	,	,	PUNCT
ejpam-4571	274	59	b	b	NOUN
ejpam-4571	274	60	)	)	PUNCT
ejpam-4571	274	61	=	=	NOUN
ejpam-4571	274	62	∅	∅	NOUN
ejpam-4571	274	63	and	and	CCONJ
ejpam-4571	274	64	hence	hence	ADV
ejpam-4571	274	65	u	u	NOUN
ejpam-4571	274	66	(	(	PUNCT
ejpam-4571	274	67	λ	λ	PROPN
ejpam-4571	274	68	,	,	PUNCT
ejpam-4571	274	69	b	b	NOUN
ejpam-4571	274	70	)	)	PUNCT
ejpam-4571	274	71	∩	∩	ADJ
ejpam-4571	274	72	v	v	X
ejpam-4571	274	73	(	(	PUNCT
ejpam-4571	274	74	λ	λ	PROPN
ejpam-4571	274	75	,	,	PUNCT
ejpam-4571	274	76	b	b	NOUN
ejpam-4571	274	77	)	)	PUNCT
ejpam-4571	274	78	=	=	SYM
ejpam-4571	274	79	∅.	∅.	X
ejpam-4571	274	80	(	(	PUNCT
ejpam-4571	274	81	2	2	NUM
ejpam-4571	274	82	)	)	PUNCT
ejpam-4571	274	83	⇒	⇒	NOUN
ejpam-4571	274	84	(	(	PUNCT
ejpam-4571	274	85	1	1	NUM
ejpam-4571	274	86	):	):	PUNCT
ejpam-4571	274	87	let	let	VERB
ejpam-4571	274	88	u	u	PRON
ejpam-4571	274	89	be	be	AUX
ejpam-4571	274	90	any	any	DET
ejpam-4571	274	91	(	(	PUNCT
ejpam-4571	274	92	λ	λ	NOUN
ejpam-4571	274	93	,	,	PUNCT
ejpam-4571	274	94	b)-open	b)-open	VERB
ejpam-4571	274	95	set	set	VERB
ejpam-4571	274	96	of	of	ADP
ejpam-4571	274	97	x.	x.	NOUN
ejpam-4571	274	98	then	then	ADV
ejpam-4571	274	99	,	,	PUNCT
ejpam-4571	274	100	x	x	X
ejpam-4571	274	101	−	−	NOUN
ejpam-4571	274	102	u	u	NOUN
ejpam-4571	274	103	is	be	AUX
ejpam-4571	274	104	(	(	PUNCT
ejpam-4571	274	105	λ	λ	X
ejpam-4571	274	106	,	,	PUNCT
ejpam-4571	274	107	b)-closed	b)-closed	ADJ
ejpam-4571	274	108	and	and	CCONJ
ejpam-4571	274	109	[	[	X
ejpam-4571	274	110	x−u	x−u	X
ejpam-4571	274	111	]	]	X
ejpam-4571	274	112	(	(	PUNCT
ejpam-4571	274	113	λ	λ	NOUN
ejpam-4571	274	114	,	,	PUNCT
ejpam-4571	274	115	b	b	NOUN
ejpam-4571	274	116	)	)	PUNCT
ejpam-4571	274	117	is	be	AUX
ejpam-4571	274	118	(	(	PUNCT
ejpam-4571	274	119	λ	λ	X
ejpam-4571	274	120	,	,	PUNCT
ejpam-4571	274	121	b)-open	b)-open	VERB
ejpam-4571	274	122	such	such	ADJ
ejpam-4571	274	123	that	that	PRON
ejpam-4571	274	124	u∩[x−u	u∩[x−u	NOUN
ejpam-4571	274	125	]	]	X
ejpam-4571	274	126	(	(	PUNCT
ejpam-4571	274	127	λ	λ	X
ejpam-4571	274	128	,	,	PUNCT
ejpam-4571	274	129	b	b	NOUN
ejpam-4571	274	130	)	)	PUNCT
ejpam-4571	274	131	=	=	PUNCT
ejpam-4571	274	132	∅.	∅.	ADP
ejpam-4571	274	133	thus	thus	ADV
ejpam-4571	274	134	,	,	PUNCT
ejpam-4571	274	135	u	u	PROPN
ejpam-4571	274	136	(	(	PUNCT
ejpam-4571	274	137	λ	λ	PROPN
ejpam-4571	274	138	,	,	PUNCT
ejpam-4571	274	139	b)∩[[x−u	b)∩[[x−u	NOUN
ejpam-4571	274	140	]	]	X
ejpam-4571	274	141	(	(	PUNCT
ejpam-4571	274	142	λ	λ	PROPN
ejpam-4571	274	143	,	,	PUNCT
ejpam-4571	274	144	b	b	NOUN
ejpam-4571	274	145	)	)	PUNCT
ejpam-4571	274	146	]	]	PUNCT
ejpam-4571	274	147	(	(	PUNCT
ejpam-4571	274	148	λ	λ	X
ejpam-4571	274	149	,	,	PUNCT
ejpam-4571	274	150	b	b	NOUN
ejpam-4571	274	151	)	)	PUNCT
ejpam-4571	274	152	=	=	NOUN
ejpam-4571	274	153	∅	∅	NOUN
ejpam-4571	274	154	which	which	PRON
ejpam-4571	274	155	implies	imply	VERB
ejpam-4571	274	156	that	that	SCONJ
ejpam-4571	274	157	u	u	PROPN
ejpam-4571	274	158	(	(	PUNCT
ejpam-4571	274	159	λ	λ	PROPN
ejpam-4571	274	160	,	,	PUNCT
ejpam-4571	274	161	b	b	NOUN
ejpam-4571	274	162	)	)	PUNCT
ejpam-4571	274	163	∩	∩	NOUN
ejpam-4571	274	164	[	[	X
ejpam-4571	274	165	x	x	X
ejpam-4571	274	166	−	−	PROPN
ejpam-4571	274	167	[	[	X
ejpam-4571	274	168	u	u	X
ejpam-4571	274	169	(	(	PUNCT
ejpam-4571	274	170	λ	λ	PROPN
ejpam-4571	274	171	,	,	PUNCT
ejpam-4571	274	172	b)](λ	b)](λ	NOUN
ejpam-4571	274	173	,	,	PUNCT
ejpam-4571	274	174	b	b	NOUN
ejpam-4571	274	175	)	)	PUNCT
ejpam-4571	274	176	=	=	PUNCT
ejpam-4571	274	177	∅.	∅.	VERB
ejpam-4571	274	178	therefore	therefore	ADV
ejpam-4571	274	179	,	,	PUNCT
ejpam-4571	274	180	u	u	PROPN
ejpam-4571	274	181	(	(	PUNCT
ejpam-4571	274	182	λ	λ	PROPN
ejpam-4571	274	183	,	,	PUNCT
ejpam-4571	274	184	b	b	NOUN
ejpam-4571	274	185	)	)	PUNCT
ejpam-4571	274	186	⊆	⊆	NUM
ejpam-4571	274	187	[	[	X
ejpam-4571	274	188	u	u	X
ejpam-4571	274	189	(	(	PUNCT
ejpam-4571	274	190	λ	λ	PROPN
ejpam-4571	274	191	,	,	PUNCT
ejpam-4571	274	192	b)](λ	b)](λ	NOUN
ejpam-4571	274	193	,	,	PUNCT
ejpam-4571	274	194	b	b	NOUN
ejpam-4571	274	195	)	)	PUNCT
ejpam-4571	274	196	and	and	CCONJ
ejpam-4571	274	197	hence	hence	ADV
ejpam-4571	274	198	u	u	NOUN
ejpam-4571	274	199	(	(	PUNCT
ejpam-4571	274	200	λ	λ	PROPN
ejpam-4571	274	201	,	,	PUNCT
ejpam-4571	274	202	b	b	NOUN
ejpam-4571	274	203	)	)	PUNCT
ejpam-4571	274	204	=	=	NOUN
ejpam-4571	275	1	[	[	X
ejpam-4571	275	2	u	u	X
ejpam-4571	275	3	(	(	PUNCT
ejpam-4571	275	4	λ	λ	PROPN
ejpam-4571	275	5	,	,	PUNCT
ejpam-4571	275	6	b)](λ	b)](λ	NOUN
ejpam-4571	275	7	,	,	PUNCT
ejpam-4571	275	8	b	b	NOUN
ejpam-4571	275	9	)	)	PUNCT
ejpam-4571	275	10	.	.	PUNCT
ejpam-4571	276	1	this	this	PRON
ejpam-4571	276	2	shows	show	VERB
ejpam-4571	276	3	that	that	SCONJ
ejpam-4571	276	4	u	u	PROPN
ejpam-4571	276	5	(	(	PUNCT
ejpam-4571	276	6	λ	λ	PROPN
ejpam-4571	276	7	,	,	PUNCT
ejpam-4571	276	8	b	b	NOUN
ejpam-4571	276	9	)	)	PUNCT
ejpam-4571	276	10	is	be	AUX
ejpam-4571	276	11	(	(	PUNCT
ejpam-4571	276	12	λ	λ	INTJ
ejpam-4571	276	13	,	,	PUNCT
ejpam-4571	276	14	b)-open	b)-open	VERB
ejpam-4571	276	15	.	.	PUNCT
ejpam-4571	277	1	thus	thus	ADV
ejpam-4571	277	2	,	,	PUNCT
ejpam-4571	277	3	(	(	PUNCT
ejpam-4571	277	4	x	x	X
ejpam-4571	277	5	,	,	PUNCT
ejpam-4571	277	6	τ	τ	X
ejpam-4571	277	7	)	)	PUNCT
ejpam-4571	277	8	is	be	AUX
ejpam-4571	277	9	(	(	PUNCT
ejpam-4571	277	10	λ	λ	PROPN
ejpam-4571	277	11	,	,	PUNCT
ejpam-4571	277	12	b)extremally	b)extremally	ADV
ejpam-4571	277	13	disconnected	disconnect	VERB
ejpam-4571	277	14	.	.	PUNCT
ejpam-4571	278	1	lemma	lemma	PROPN
ejpam-4571	278	2	5	5	X
ejpam-4571	278	3	.	.	PUNCT
ejpam-4571	278	4	let	let	VERB
ejpam-4571	278	5	a	a	DET
ejpam-4571	278	6	be	be	AUX
ejpam-4571	278	7	a	a	DET
ejpam-4571	278	8	subset	subset	NOUN
ejpam-4571	278	9	of	of	ADP
ejpam-4571	278	10	a	a	DET
ejpam-4571	278	11	topological	topological	ADJ
ejpam-4571	278	12	space	space	NOUN
ejpam-4571	278	13	(	(	PUNCT
ejpam-4571	278	14	x	x	X
ejpam-4571	278	15	,	,	PUNCT
ejpam-4571	278	16	τ	τ	PROPN
ejpam-4571	278	17	)	)	PUNCT
ejpam-4571	278	18	.	.	PUNCT
ejpam-4571	279	1	if	if	SCONJ
ejpam-4571	279	2	u	u	PROPN
ejpam-4571	279	3	∈	∈	PROPN
ejpam-4571	279	4	λbo(x	λbo(x	PROPN
ejpam-4571	279	5	,	,	PUNCT
ejpam-4571	279	6	τ	τ	PROPN
ejpam-4571	279	7	)	)	PUNCT
ejpam-4571	279	8	,	,	PUNCT
ejpam-4571	279	9	then	then	ADV
ejpam-4571	279	10	u	u	X
ejpam-4571	279	11	(	(	PUNCT
ejpam-4571	279	12	λ	λ	PROPN
ejpam-4571	279	13	,	,	PUNCT
ejpam-4571	279	14	b	b	NOUN
ejpam-4571	279	15	)	)	PUNCT
ejpam-4571	279	16	∩a	∩a	NOUN
ejpam-4571	279	17	⊆	⊆	NUM
ejpam-4571	279	18	[	[	X
ejpam-4571	279	19	u	u	NOUN
ejpam-4571	279	20	∩a](λ	∩a](λ	PROPN
ejpam-4571	279	21	,	,	PUNCT
ejpam-4571	279	22	b	b	NOUN
ejpam-4571	279	23	)	)	PUNCT
ejpam-4571	279	24	.	.	PUNCT
ejpam-4571	280	1	lemma	lemma	PROPN
ejpam-4571	280	2	6	6	NUM
ejpam-4571	280	3	.	.	PUNCT
ejpam-4571	281	1	for	for	ADP
ejpam-4571	281	2	a	a	DET
ejpam-4571	281	3	subset	subset	NOUN
ejpam-4571	281	4	a	a	PRON
ejpam-4571	281	5	of	of	ADP
ejpam-4571	281	6	a	a	DET
ejpam-4571	281	7	topological	topological	ADJ
ejpam-4571	281	8	space	space	NOUN
ejpam-4571	281	9	(	(	PUNCT
ejpam-4571	281	10	x	x	X
ejpam-4571	281	11	,	,	PUNCT
ejpam-4571	281	12	τ	τ	PROPN
ejpam-4571	281	13	)	)	PUNCT
ejpam-4571	281	14	,	,	PUNCT
ejpam-4571	281	15	the	the	DET
ejpam-4571	281	16	following	follow	VERB
ejpam-4571	281	17	properties	property	NOUN
ejpam-4571	281	18	hold	hold	VERB
ejpam-4571	281	19	:	:	PUNCT
ejpam-4571	281	20	(	(	PUNCT
ejpam-4571	281	21	1	1	X
ejpam-4571	281	22	)	)	PUNCT
ejpam-4571	282	1	[	[	X
ejpam-4571	282	2	x	x	X
ejpam-4571	282	3	−a](λ	−a](λ	PROPN
ejpam-4571	282	4	,	,	PUNCT
ejpam-4571	282	5	b	b	NOUN
ejpam-4571	282	6	)	)	PUNCT
ejpam-4571	282	7	=	=	SYM
ejpam-4571	282	8	x	x	SYM
ejpam-4571	282	9	−a(λ	−a(λ	NOUN
ejpam-4571	282	10	,	,	PUNCT
ejpam-4571	282	11	b	b	NOUN
ejpam-4571	282	12	)	)	PUNCT
ejpam-4571	282	13	.	.	PUNCT
ejpam-4571	283	1	(	(	PUNCT
ejpam-4571	283	2	2	2	X
ejpam-4571	283	3	)	)	PUNCT
ejpam-4571	283	4	[	[	X
ejpam-4571	283	5	x	x	X
ejpam-4571	283	6	−a](λ	−a](λ	PROPN
ejpam-4571	283	7	,	,	PUNCT
ejpam-4571	283	8	b	b	NOUN
ejpam-4571	283	9	)	)	PUNCT
ejpam-4571	283	10	=	=	SYM
ejpam-4571	283	11	x	x	SYM
ejpam-4571	283	12	−a(λ	−a(λ	NOUN
ejpam-4571	283	13	,	,	PUNCT
ejpam-4571	283	14	b	b	NOUN
ejpam-4571	283	15	)	)	PUNCT
ejpam-4571	283	16	.	.	PUNCT
ejpam-4571	284	1	theorem	theorem	NOUN
ejpam-4571	284	2	3	3	NUM
ejpam-4571	284	3	.	.	X
ejpam-4571	284	4	for	for	ADP
ejpam-4571	284	5	a	a	DET
ejpam-4571	284	6	topological	topological	ADJ
ejpam-4571	284	7	space	space	NOUN
ejpam-4571	284	8	(	(	PUNCT
ejpam-4571	284	9	x	x	X
ejpam-4571	284	10	,	,	PUNCT
ejpam-4571	284	11	τ	τ	PROPN
ejpam-4571	284	12	)	)	PUNCT
ejpam-4571	284	13	,	,	PUNCT
ejpam-4571	284	14	the	the	DET
ejpam-4571	284	15	following	follow	VERB
ejpam-4571	284	16	properties	property	NOUN
ejpam-4571	284	17	are	be	AUX
ejpam-4571	284	18	equivalent	equivalent	ADJ
ejpam-4571	284	19	:	:	PUNCT
ejpam-4571	284	20	(	(	PUNCT
ejpam-4571	284	21	1	1	X
ejpam-4571	284	22	)	)	PUNCT
ejpam-4571	284	23	(	(	PUNCT
ejpam-4571	284	24	x	x	X
ejpam-4571	284	25	,	,	PUNCT
ejpam-4571	284	26	τ	τ	X
ejpam-4571	284	27	)	)	PUNCT
ejpam-4571	284	28	is	be	AUX
ejpam-4571	284	29	(	(	PUNCT
ejpam-4571	284	30	λ	λ	X
ejpam-4571	284	31	,	,	PUNCT
ejpam-4571	284	32	b)-extremally	b)-extremally	ADV
ejpam-4571	284	33	disconnected	disconnected	ADJ
ejpam-4571	284	34	.	.	PUNCT
ejpam-4571	285	1	(	(	PUNCT
ejpam-4571	285	2	2	2	X
ejpam-4571	285	3	)	)	PUNCT
ejpam-4571	285	4	u	u	NOUN
ejpam-4571	285	5	(	(	PUNCT
ejpam-4571	285	6	λ	λ	PROPN
ejpam-4571	285	7	,	,	PUNCT
ejpam-4571	285	8	b	b	NOUN
ejpam-4571	285	9	)	)	PUNCT
ejpam-4571	285	10	∩	∩	ADJ
ejpam-4571	285	11	v	v	X
ejpam-4571	285	12	(	(	PUNCT
ejpam-4571	285	13	λ	λ	PROPN
ejpam-4571	285	14	,	,	PUNCT
ejpam-4571	285	15	b	b	NOUN
ejpam-4571	285	16	)	)	PUNCT
ejpam-4571	285	17	=	=	NOUN
ejpam-4571	286	1	[	[	X
ejpam-4571	286	2	u	u	NOUN
ejpam-4571	286	3	∩	∩	NOUN
ejpam-4571	286	4	v	v	ADP
ejpam-4571	286	5	]	]	PUNCT
ejpam-4571	286	6	(	(	PUNCT
ejpam-4571	286	7	λ	λ	PROPN
ejpam-4571	286	8	,	,	PUNCT
ejpam-4571	286	9	b	b	NOUN
ejpam-4571	286	10	)	)	PUNCT
ejpam-4571	286	11	for	for	ADP
ejpam-4571	286	12	all	all	PRON
ejpam-4571	286	13	(	(	PUNCT
ejpam-4571	286	14	λ	λ	NOUN
ejpam-4571	286	15	,	,	PUNCT
ejpam-4571	286	16	b)-open	b)-open	PUNCT
ejpam-4571	286	17	sets	set	VERB
ejpam-4571	286	18	u	u	NOUN
ejpam-4571	286	19	and	and	CCONJ
ejpam-4571	286	20	v	v	NOUN
ejpam-4571	286	21	of	of	ADP
ejpam-4571	286	22	x.	x.	NOUN
ejpam-4571	286	23	(	(	PUNCT
ejpam-4571	286	24	3	3	X
ejpam-4571	286	25	)	)	PUNCT
ejpam-4571	286	26	e(λ	e(λ	PROPN
ejpam-4571	286	27	,	,	PUNCT
ejpam-4571	286	28	b	b	NOUN
ejpam-4571	286	29	)	)	PUNCT
ejpam-4571	286	30	∪	∪	ADP
ejpam-4571	286	31	f(λ	f(λ	PROPN
ejpam-4571	286	32	,	,	PUNCT
ejpam-4571	286	33	b	b	NOUN
ejpam-4571	286	34	)	)	PUNCT
ejpam-4571	286	35	=	=	PUNCT
ejpam-4571	287	1	[	[	X
ejpam-4571	287	2	e	e	X
ejpam-4571	287	3	∪	∪	VERB
ejpam-4571	287	4	f	f	PROPN
ejpam-4571	287	5	]	]	X
ejpam-4571	287	6	(	(	PUNCT
ejpam-4571	287	7	λ	λ	PROPN
ejpam-4571	287	8	,	,	PUNCT
ejpam-4571	287	9	b	b	NOUN
ejpam-4571	287	10	)	)	PUNCT
ejpam-4571	287	11	for	for	ADP
ejpam-4571	287	12	all	all	DET
ejpam-4571	287	13	(	(	PUNCT
ejpam-4571	287	14	λ	λ	PROPN
ejpam-4571	287	15	,	,	PUNCT
ejpam-4571	287	16	b)-closed	b)-close	VERB
ejpam-4571	287	17	sets	set	NOUN
ejpam-4571	287	18	e	e	NOUN
ejpam-4571	287	19	and	and	CCONJ
ejpam-4571	287	20	f	f	PROPN
ejpam-4571	287	21	of	of	ADP
ejpam-4571	287	22	x.	x.	PROPN
ejpam-4571	287	23	proof	proof	NOUN
ejpam-4571	287	24	.	.	PUNCT
ejpam-4571	288	1	(	(	PUNCT
ejpam-4571	288	2	1	1	X
ejpam-4571	288	3	)	)	PUNCT
ejpam-4571	288	4	⇒	⇒	NOUN
ejpam-4571	288	5	(	(	PUNCT
ejpam-4571	288	6	2	2	NUM
ejpam-4571	288	7	):	):	PUNCT
ejpam-4571	288	8	let	let	VERB
ejpam-4571	288	9	u	u	PRON
ejpam-4571	288	10	and	and	CCONJ
ejpam-4571	288	11	v	v	NOUN
ejpam-4571	288	12	be	be	AUX
ejpam-4571	288	13	(	(	PUNCT
ejpam-4571	288	14	λ	λ	X
ejpam-4571	288	15	,	,	PUNCT
ejpam-4571	288	16	b)-open	b)-open	VERB
ejpam-4571	288	17	sets	set	NOUN
ejpam-4571	288	18	of	of	ADP
ejpam-4571	288	19	x.	x.	NOUN
ejpam-4571	288	20	thus	thus	ADV
ejpam-4571	288	21	,	,	PUNCT
ejpam-4571	288	22	by	by	ADP
ejpam-4571	288	23	lemma	lemma	PROPN
ejpam-4571	288	24	5	5	NUM
ejpam-4571	288	25	,	,	PUNCT
ejpam-4571	288	26	u	u	NOUN
ejpam-4571	288	27	(	(	PUNCT
ejpam-4571	288	28	λ	λ	PROPN
ejpam-4571	288	29	,	,	PUNCT
ejpam-4571	288	30	b	b	NOUN
ejpam-4571	288	31	)	)	PUNCT
ejpam-4571	288	32	∩	∩	ADJ
ejpam-4571	288	33	v	v	X
ejpam-4571	288	34	(	(	PUNCT
ejpam-4571	288	35	λ	λ	PROPN
ejpam-4571	288	36	,	,	PUNCT
ejpam-4571	288	37	b	b	NOUN
ejpam-4571	288	38	)	)	PUNCT
ejpam-4571	288	39	⊆	⊆	NUM
ejpam-4571	289	1	[	[	X
ejpam-4571	289	2	u	u	NOUN
ejpam-4571	289	3	∩	∩	X
ejpam-4571	289	4	v	v	X
ejpam-4571	289	5	(	(	PUNCT
ejpam-4571	289	6	λ	λ	PROPN
ejpam-4571	289	7	,	,	PUNCT
ejpam-4571	289	8	b)](λ	b)](λ	NOUN
ejpam-4571	289	9	,	,	PUNCT
ejpam-4571	289	10	b	b	NOUN
ejpam-4571	289	11	)	)	PUNCT
ejpam-4571	289	12	⊆	⊆	NUM
ejpam-4571	290	1	[	[	X
ejpam-4571	290	2	[	[	X
ejpam-4571	290	3	u	u	NOUN
ejpam-4571	290	4	∩	∩	NOUN
ejpam-4571	290	5	v	v	ADP
ejpam-4571	290	6	]	]	PUNCT
ejpam-4571	290	7	(	(	PUNCT
ejpam-4571	290	8	λ	λ	PROPN
ejpam-4571	290	9	,	,	PUNCT
ejpam-4571	290	10	b)](λ	b)](λ	NOUN
ejpam-4571	290	11	,	,	PUNCT
ejpam-4571	290	12	b	b	NOUN
ejpam-4571	290	13	)	)	PUNCT
ejpam-4571	290	14	=	=	NOUN
ejpam-4571	291	1	[	[	X
ejpam-4571	291	2	u	u	NOUN
ejpam-4571	291	3	∩	∩	NOUN
ejpam-4571	291	4	v	v	ADP
ejpam-4571	291	5	]	]	PUNCT
ejpam-4571	291	6	(	(	PUNCT
ejpam-4571	291	7	λ	λ	PROPN
ejpam-4571	291	8	,	,	PUNCT
ejpam-4571	291	9	b	b	NOUN
ejpam-4571	291	10	)	)	PUNCT
ejpam-4571	291	11	and	and	CCONJ
ejpam-4571	291	12	hence	hence	ADV
ejpam-4571	291	13	u	u	NOUN
ejpam-4571	291	14	(	(	PUNCT
ejpam-4571	291	15	λ	λ	PROPN
ejpam-4571	291	16	,	,	PUNCT
ejpam-4571	291	17	b	b	NOUN
ejpam-4571	291	18	)	)	PUNCT
ejpam-4571	291	19	∩	∩	ADJ
ejpam-4571	291	20	v	v	X
ejpam-4571	291	21	(	(	PUNCT
ejpam-4571	291	22	λ	λ	PROPN
ejpam-4571	291	23	,	,	PUNCT
ejpam-4571	291	24	b	b	NOUN
ejpam-4571	291	25	)	)	PUNCT
ejpam-4571	291	26	=	=	NOUN
ejpam-4571	292	1	[	[	X
ejpam-4571	292	2	u	u	NOUN
ejpam-4571	292	3	∩	∩	NOUN
ejpam-4571	292	4	v	v	ADP
ejpam-4571	292	5	]	]	PUNCT
ejpam-4571	292	6	(	(	PUNCT
ejpam-4571	292	7	λ	λ	PROPN
ejpam-4571	292	8	,	,	PUNCT
ejpam-4571	292	9	b	b	NOUN
ejpam-4571	292	10	)	)	PUNCT
ejpam-4571	292	11	.	.	PUNCT
ejpam-4571	293	1	(	(	PUNCT
ejpam-4571	293	2	2	2	X
ejpam-4571	293	3	)	)	PUNCT
ejpam-4571	293	4	⇒	⇒	NOUN
ejpam-4571	293	5	(	(	PUNCT
ejpam-4571	293	6	3	3	NUM
ejpam-4571	293	7	):	):	PUNCT
ejpam-4571	293	8	let	let	VERB
ejpam-4571	293	9	e	e	NOUN
ejpam-4571	293	10	and	and	CCONJ
ejpam-4571	293	11	f	f	PROPN
ejpam-4571	293	12	be	be	AUX
ejpam-4571	293	13	(	(	PUNCT
ejpam-4571	293	14	λ	λ	PROPN
ejpam-4571	293	15	,	,	PUNCT
ejpam-4571	293	16	b)-closed	b)-close	VERB
ejpam-4571	293	17	sets	set	NOUN
ejpam-4571	293	18	of	of	ADP
ejpam-4571	293	19	x.	x.	NOUN
ejpam-4571	293	20	then	then	ADV
ejpam-4571	293	21	,	,	PUNCT
ejpam-4571	293	22	x	x	PUNCT
ejpam-4571	293	23	−	−	NOUN
ejpam-4571	293	24	e	e	NOUN
ejpam-4571	293	25	and	and	CCONJ
ejpam-4571	293	26	x	x	SYM
ejpam-4571	293	27	−	−	PROPN
ejpam-4571	293	28	f	f	NOUN
ejpam-4571	293	29	are	be	AUX
ejpam-4571	293	30	(	(	PUNCT
ejpam-4571	293	31	λ	λ	X
ejpam-4571	293	32	,	,	PUNCT
ejpam-4571	293	33	b)-open	b)-open	VERB
ejpam-4571	293	34	.	.	PUNCT
ejpam-4571	294	1	by	by	ADP
ejpam-4571	294	2	(	(	PUNCT
ejpam-4571	294	3	2	2	NUM
ejpam-4571	294	4	)	)	PUNCT
ejpam-4571	294	5	and	and	CCONJ
ejpam-4571	294	6	lemma	lemma	PROPN
ejpam-4571	294	7	6	6	NUM
ejpam-4571	294	8	,	,	PUNCT
ejpam-4571	294	9	we	we	PRON
ejpam-4571	294	10	have	have	VERB
ejpam-4571	294	11	e(λ	e(λ	NOUN
ejpam-4571	294	12	,	,	PUNCT
ejpam-4571	294	13	b	b	NOUN
ejpam-4571	294	14	)	)	PUNCT
ejpam-4571	294	15	∪	∪	ADP
ejpam-4571	294	16	f(λ	f(λ	PROPN
ejpam-4571	294	17	,	,	PUNCT
ejpam-4571	294	18	b	b	NOUN
ejpam-4571	294	19	)	)	PUNCT
ejpam-4571	294	20	=	=	PUNCT
ejpam-4571	295	1	x	x	X
ejpam-4571	295	2	−	−	PUNCT
ejpam-4571	296	1	[	[	X
ejpam-4571	296	2	x	x	X
ejpam-4571	296	3	−	−	X
ejpam-4571	297	1	[	[	X
ejpam-4571	297	2	e(λ	e(λ	X
ejpam-4571	297	3	,	,	PUNCT
ejpam-4571	297	4	b	b	NOUN
ejpam-4571	297	5	)	)	PUNCT
ejpam-4571	297	6	∪	∪	ADP
ejpam-4571	297	7	f(λ	f(λ	PROPN
ejpam-4571	297	8	,	,	PUNCT
ejpam-4571	297	9	b	b	NOUN
ejpam-4571	297	10	)	)	PUNCT
ejpam-4571	297	11	]	]	X
ejpam-4571	297	12	]	]	PUNCT
ejpam-4571	298	1	=	=	PUNCT
ejpam-4571	298	2	x	x	X
ejpam-4571	298	3	−	−	PROPN
ejpam-4571	299	1	[	[	X
ejpam-4571	299	2	[	[	X
ejpam-4571	299	3	x	x	X
ejpam-4571	299	4	−	−	ADP
ejpam-4571	299	5	e(λ	e(λ	PROPN
ejpam-4571	299	6	,	,	PUNCT
ejpam-4571	299	7	b	b	NOUN
ejpam-4571	299	8	)	)	PUNCT
ejpam-4571	299	9	]	]	PUNCT
ejpam-4571	299	10	∩	∩	NOUN
ejpam-4571	299	11	[	[	X
ejpam-4571	299	12	x	x	X
ejpam-4571	299	13	−	−	ADP
ejpam-4571	299	14	f(λ	f(λ	NOUN
ejpam-4571	299	15	,	,	PUNCT
ejpam-4571	299	16	b	b	NOUN
ejpam-4571	299	17	)	)	PUNCT
ejpam-4571	299	18	]	]	X
ejpam-4571	299	19	]	]	PUNCT
ejpam-4571	300	1	=	=	PUNCT
ejpam-4571	300	2	x	x	X
ejpam-4571	300	3	−	−	PUNCT
ejpam-4571	301	1	[	[	X
ejpam-4571	301	2	x	x	X
ejpam-4571	301	3	−	−	PROPN
ejpam-4571	302	1	[	[	X
ejpam-4571	302	2	e	e	X
ejpam-4571	302	3	∪	∪	ADP
ejpam-4571	302	4	f	f	PROPN
ejpam-4571	302	5	]	]	X
ejpam-4571	302	6	(	(	PUNCT
ejpam-4571	302	7	λ	λ	PROPN
ejpam-4571	302	8	,	,	PUNCT
ejpam-4571	302	9	b	b	NOUN
ejpam-4571	302	10	)	)	PUNCT
ejpam-4571	302	11	]	]	PUNCT
ejpam-4571	303	1	=	=	PUNCT
ejpam-4571	304	1	[	[	X
ejpam-4571	304	2	e	e	X
ejpam-4571	304	3	∪	∪	VERB
ejpam-4571	304	4	f	f	PROPN
ejpam-4571	304	5	]	]	X
ejpam-4571	304	6	(	(	PUNCT
ejpam-4571	304	7	λ	λ	PROPN
ejpam-4571	304	8	,	,	PUNCT
ejpam-4571	304	9	b	b	NOUN
ejpam-4571	304	10	)	)	PUNCT
ejpam-4571	304	11	.	.	PUNCT
ejpam-4571	305	1	(	(	PUNCT
ejpam-4571	305	2	3	3	X
ejpam-4571	305	3	)	)	PUNCT
ejpam-4571	305	4	⇒	⇒	NOUN
ejpam-4571	305	5	(	(	PUNCT
ejpam-4571	305	6	2	2	NUM
ejpam-4571	305	7	):	):	PUNCT
ejpam-4571	305	8	the	the	DET
ejpam-4571	305	9	proof	proof	NOUN
ejpam-4571	305	10	is	be	AUX
ejpam-4571	305	11	obvious	obvious	ADJ
ejpam-4571	305	12	.	.	PUNCT
ejpam-4571	306	1	(	(	PUNCT
ejpam-4571	306	2	2	2	X
ejpam-4571	306	3	)	)	PUNCT
ejpam-4571	306	4	⇒	⇒	NOUN
ejpam-4571	306	5	(	(	PUNCT
ejpam-4571	306	6	1	1	NUM
ejpam-4571	306	7	):	):	PUNCT
ejpam-4571	306	8	let	let	VERB
ejpam-4571	306	9	u	u	PRON
ejpam-4571	306	10	be	be	AUX
ejpam-4571	306	11	any	any	DET
ejpam-4571	306	12	(	(	PUNCT
ejpam-4571	306	13	λ	λ	NOUN
ejpam-4571	306	14	,	,	PUNCT
ejpam-4571	306	15	b)-open	b)-open	VERB
ejpam-4571	306	16	set	set	VERB
ejpam-4571	306	17	of	of	ADP
ejpam-4571	306	18	x.	x.	NOUN
ejpam-4571	306	19	then	then	ADV
ejpam-4571	306	20	,	,	PUNCT
ejpam-4571	306	21	x	x	X
ejpam-4571	306	22	−	−	NOUN
ejpam-4571	306	23	u	u	NOUN
ejpam-4571	306	24	is	be	AUX
ejpam-4571	306	25	(	(	PUNCT
ejpam-4571	306	26	λ	λ	X
ejpam-4571	306	27	,	,	PUNCT
ejpam-4571	306	28	b)-closed	b)-close	VERB
ejpam-4571	306	29	and	and	CCONJ
ejpam-4571	307	1	[	[	X
ejpam-4571	307	2	x	x	X
ejpam-4571	307	3	−	−	PROPN
ejpam-4571	307	4	u	u	NOUN
ejpam-4571	307	5	]	]	X
ejpam-4571	307	6	(	(	PUNCT
ejpam-4571	307	7	λ	λ	PROPN
ejpam-4571	307	8	,	,	PUNCT
ejpam-4571	307	9	b	b	NOUN
ejpam-4571	307	10	)	)	PUNCT
ejpam-4571	307	11	is	be	AUX
ejpam-4571	307	12	(	(	PUNCT
ejpam-4571	307	13	λ	λ	INTJ
ejpam-4571	307	14	,	,	PUNCT
ejpam-4571	307	15	b)-open	b)-open	VERB
ejpam-4571	307	16	.	.	PUNCT
ejpam-4571	308	1	by	by	ADP
ejpam-4571	308	2	(	(	PUNCT
ejpam-4571	308	3	2	2	NUM
ejpam-4571	308	4	)	)	PUNCT
ejpam-4571	308	5	,	,	PUNCT
ejpam-4571	308	6	u	u	NOUN
ejpam-4571	308	7	(	(	PUNCT
ejpam-4571	308	8	λ	λ	PROPN
ejpam-4571	308	9	,	,	PUNCT
ejpam-4571	308	10	b	b	NOUN
ejpam-4571	308	11	)	)	PUNCT
ejpam-4571	308	12	∩	∩	NOUN
ejpam-4571	308	13	[	[	X
ejpam-4571	308	14	[	[	X
ejpam-4571	308	15	x	x	X
ejpam-4571	308	16	−	−	PROPN
ejpam-4571	308	17	u	u	NOUN
ejpam-4571	308	18	]	]	X
ejpam-4571	308	19	(	(	PUNCT
ejpam-4571	308	20	λ	λ	PROPN
ejpam-4571	308	21	,	,	PUNCT
ejpam-4571	308	22	b	b	NOUN
ejpam-4571	308	23	)	)	PUNCT
ejpam-4571	308	24	]	]	PUNCT
ejpam-4571	308	25	(	(	PUNCT
ejpam-4571	308	26	λ	λ	X
ejpam-4571	308	27	,	,	PUNCT
ejpam-4571	308	28	b	b	NOUN
ejpam-4571	308	29	)	)	PUNCT
ejpam-4571	308	30	=	=	NOUN
ejpam-4571	309	1	[	[	X
ejpam-4571	309	2	u	u	NOUN
ejpam-4571	309	3	∩	∩	NOUN
ejpam-4571	309	4	[	[	X
ejpam-4571	309	5	x	x	X
ejpam-4571	309	6	−	−	PROPN
ejpam-4571	309	7	u	u	NOUN
ejpam-4571	309	8	]	]	X
ejpam-4571	309	9	(	(	PUNCT
ejpam-4571	309	10	λ	λ	PROPN
ejpam-4571	309	11	,	,	PUNCT
ejpam-4571	309	12	b	b	NOUN
ejpam-4571	309	13	)	)	PUNCT
ejpam-4571	309	14	]	]	PUNCT
ejpam-4571	309	15	(	(	PUNCT
ejpam-4571	309	16	λ	λ	X
ejpam-4571	309	17	,	,	PUNCT
ejpam-4571	309	18	b	b	NOUN
ejpam-4571	309	19	)	)	PUNCT
ejpam-4571	309	20	which	which	PRON
ejpam-4571	309	21	implies	imply	VERB
ejpam-4571	309	22	that	that	SCONJ
ejpam-4571	310	1	[	[	X
ejpam-4571	310	2	x	x	X
ejpam-4571	310	3	−	−	X
ejpam-4571	310	4	[	[	X
ejpam-4571	310	5	u	u	X
ejpam-4571	310	6	(	(	PUNCT
ejpam-4571	310	7	λ	λ	PROPN
ejpam-4571	310	8	,	,	PUNCT
ejpam-4571	310	9	b)](λ	b)](λ	NOUN
ejpam-4571	310	10	,	,	PUNCT
ejpam-4571	310	11	b	b	NOUN
ejpam-4571	310	12	)	)	PUNCT
ejpam-4571	310	13	]	]	PUNCT
ejpam-4571	310	14	∩	∩	ADJ
ejpam-4571	310	15	u	u	PROPN
ejpam-4571	310	16	(	(	PUNCT
ejpam-4571	310	17	λ	λ	PROPN
ejpam-4571	310	18	,	,	PUNCT
ejpam-4571	310	19	b	b	NOUN
ejpam-4571	310	20	)	)	PUNCT
ejpam-4571	310	21	=	=	SYM
ejpam-4571	310	22	∅(λ	∅(λ	NOUN
ejpam-4571	310	23	,	,	PUNCT
ejpam-4571	310	24	b	b	NOUN
ejpam-4571	310	25	)	)	PUNCT
ejpam-4571	310	26	=	=	PUNCT
ejpam-4571	310	27	∅.	∅.	ADP
ejpam-4571	310	28	thus	thus	ADV
ejpam-4571	310	29	,	,	PUNCT
ejpam-4571	310	30	u	u	PROPN
ejpam-4571	310	31	(	(	PUNCT
ejpam-4571	310	32	λ	λ	PROPN
ejpam-4571	310	33	,	,	PUNCT
ejpam-4571	310	34	b	b	NOUN
ejpam-4571	310	35	)	)	PUNCT
ejpam-4571	310	36	⊆	⊆	NUM
ejpam-4571	310	37	[	[	X
ejpam-4571	310	38	u	u	X
ejpam-4571	310	39	(	(	PUNCT
ejpam-4571	310	40	λ	λ	PROPN
ejpam-4571	310	41	,	,	PUNCT
ejpam-4571	310	42	b)](λ	b)](λ	NOUN
ejpam-4571	310	43	,	,	PUNCT
ejpam-4571	310	44	b	b	NOUN
ejpam-4571	310	45	)	)	PUNCT
ejpam-4571	310	46	and	and	CCONJ
ejpam-4571	310	47	hence	hence	ADV
ejpam-4571	310	48	u	u	NOUN
ejpam-4571	310	49	(	(	PUNCT
ejpam-4571	310	50	λ	λ	PROPN
ejpam-4571	310	51	,	,	PUNCT
ejpam-4571	310	52	b	b	NOUN
ejpam-4571	310	53	)	)	PUNCT
ejpam-4571	310	54	=	=	NOUN
ejpam-4571	311	1	[	[	X
ejpam-4571	311	2	u	u	X
ejpam-4571	311	3	(	(	PUNCT
ejpam-4571	311	4	λ	λ	PROPN
ejpam-4571	311	5	,	,	PUNCT
ejpam-4571	311	6	b)](λ	b)](λ	NOUN
ejpam-4571	311	7	,	,	PUNCT
ejpam-4571	311	8	b	b	NOUN
ejpam-4571	311	9	)	)	PUNCT
ejpam-4571	311	10	.	.	PUNCT
ejpam-4571	312	1	therefore	therefore	ADV
ejpam-4571	312	2	,	,	PUNCT
ejpam-4571	312	3	u	u	PROPN
ejpam-4571	312	4	(	(	PUNCT
ejpam-4571	312	5	λ	λ	PROPN
ejpam-4571	312	6	,	,	PUNCT
ejpam-4571	312	7	b	b	NOUN
ejpam-4571	312	8	)	)	PUNCT
ejpam-4571	312	9	is	be	AUX
ejpam-4571	312	10	(	(	PUNCT
ejpam-4571	312	11	λ	λ	INTJ
ejpam-4571	312	12	,	,	PUNCT
ejpam-4571	312	13	b)-open	b)-open	VERB
ejpam-4571	312	14	.	.	PUNCT
ejpam-4571	313	1	this	this	PRON
ejpam-4571	313	2	shows	show	VERB
ejpam-4571	313	3	that	that	SCONJ
ejpam-4571	313	4	(	(	PUNCT
ejpam-4571	313	5	x	x	X
ejpam-4571	313	6	,	,	PUNCT
ejpam-4571	313	7	τ	τ	X
ejpam-4571	313	8	)	)	PUNCT
ejpam-4571	313	9	is	be	AUX
ejpam-4571	313	10	(	(	PUNCT
ejpam-4571	313	11	λ	λ	X
ejpam-4571	313	12	,	,	PUNCT
ejpam-4571	313	13	b)-extremally	b)-extremally	ADV
ejpam-4571	313	14	disconnected	disconnected	ADJ
ejpam-4571	313	15	.	.	PUNCT
ejpam-4571	314	1	theorem	theorem	ADJ
ejpam-4571	314	2	4	4	NUM
ejpam-4571	314	3	.	.	X
ejpam-4571	315	1	for	for	ADP
ejpam-4571	315	2	a	a	DET
ejpam-4571	315	3	topological	topological	ADJ
ejpam-4571	315	4	space	space	NOUN
ejpam-4571	315	5	(	(	PUNCT
ejpam-4571	315	6	x	x	X
ejpam-4571	315	7	,	,	PUNCT
ejpam-4571	315	8	τ	τ	PROPN
ejpam-4571	315	9	)	)	PUNCT
ejpam-4571	315	10	,	,	PUNCT
ejpam-4571	315	11	the	the	DET
ejpam-4571	315	12	following	follow	VERB
ejpam-4571	315	13	properties	property	NOUN
ejpam-4571	315	14	are	be	AUX
ejpam-4571	315	15	equivalent	equivalent	ADJ
ejpam-4571	315	16	:	:	PUNCT
ejpam-4571	315	17	(	(	PUNCT
ejpam-4571	315	18	1	1	X
ejpam-4571	315	19	)	)	PUNCT
ejpam-4571	315	20	(	(	PUNCT
ejpam-4571	315	21	x	x	X
ejpam-4571	315	22	,	,	PUNCT
ejpam-4571	315	23	τ	τ	X
ejpam-4571	315	24	)	)	PUNCT
ejpam-4571	315	25	is	be	AUX
ejpam-4571	315	26	(	(	PUNCT
ejpam-4571	315	27	λ	λ	X
ejpam-4571	315	28	,	,	PUNCT
ejpam-4571	315	29	b)-extremally	b)-extremally	ADV
ejpam-4571	315	30	disconnected	disconnected	ADJ
ejpam-4571	315	31	.	.	PUNCT
ejpam-4571	316	1	(	(	PUNCT
ejpam-4571	316	2	2	2	X
ejpam-4571	316	3	)	)	PUNCT
ejpam-4571	316	4	u	u	NOUN
ejpam-4571	316	5	(	(	PUNCT
ejpam-4571	316	6	λ	λ	PROPN
ejpam-4571	316	7	,	,	PUNCT
ejpam-4571	316	8	b	b	NOUN
ejpam-4571	316	9	)	)	PUNCT
ejpam-4571	316	10	∩	∩	ADJ
ejpam-4571	316	11	v	v	X
ejpam-4571	316	12	(	(	PUNCT
ejpam-4571	316	13	λ	λ	PROPN
ejpam-4571	316	14	,	,	PUNCT
ejpam-4571	316	15	b	b	NOUN
ejpam-4571	316	16	)	)	PUNCT
ejpam-4571	316	17	=	=	NOUN
ejpam-4571	317	1	[	[	X
ejpam-4571	317	2	u	u	NOUN
ejpam-4571	317	3	∩	∩	NOUN
ejpam-4571	317	4	v	v	ADP
ejpam-4571	317	5	]	]	PUNCT
ejpam-4571	317	6	(	(	PUNCT
ejpam-4571	317	7	λ	λ	PROPN
ejpam-4571	317	8	,	,	PUNCT
ejpam-4571	317	9	b	b	NOUN
ejpam-4571	317	10	)	)	PUNCT
ejpam-4571	317	11	for	for	ADP
ejpam-4571	317	12	all	all	PRON
ejpam-4571	317	13	(	(	PUNCT
ejpam-4571	317	14	λ	λ	NOUN
ejpam-4571	317	15	,	,	PUNCT
ejpam-4571	317	16	b)-open	b)-open	PUNCT
ejpam-4571	317	17	sets	set	VERB
ejpam-4571	317	18	u	u	NOUN
ejpam-4571	317	19	and	and	CCONJ
ejpam-4571	317	20	v	v	NOUN
ejpam-4571	317	21	of	of	ADP
ejpam-4571	317	22	x.	x.	PROPN
ejpam-4571	317	23	c.	c.	PROPN
ejpam-4571	317	24	boonpok	boonpok	PROPN
ejpam-4571	317	25	,	,	PUNCT
ejpam-4571	317	26	n.	n.	PROPN
ejpam-4571	317	27	srisarakham	srisarakham	PROPN
ejpam-4571	317	28	/	/	SYM
ejpam-4571	317	29	eur	eur	PROPN
ejpam-4571	317	30	.	.	PUNCT
ejpam-4571	318	1	j.	j.	PROPN
ejpam-4571	318	2	pure	pure	PROPN
ejpam-4571	318	3	appl	appl	PROPN
ejpam-4571	318	4	.	.	PROPN
ejpam-4571	318	5	math	math	PROPN
ejpam-4571	318	6	,	,	PUNCT
ejpam-4571	318	7	16	16	NUM
ejpam-4571	318	8	(	(	PUNCT
ejpam-4571	318	9	1	1	NUM
ejpam-4571	318	10	)	)	PUNCT
ejpam-4571	318	11	(	(	PUNCT
ejpam-4571	318	12	2023	2023	NUM
ejpam-4571	318	13	)	)	PUNCT
ejpam-4571	318	14	,	,	PUNCT
ejpam-4571	318	15	29	29	NUM
ejpam-4571	318	16	-	-	SYM
ejpam-4571	318	17	43	43	NUM
ejpam-4571	318	18	37	37	NUM
ejpam-4571	318	19	(	(	PUNCT
ejpam-4571	318	20	3	3	NUM
ejpam-4571	318	21	)	)	PUNCT
ejpam-4571	318	22	u	u	NOUN
ejpam-4571	318	23	(	(	PUNCT
ejpam-4571	318	24	λ	λ	PROPN
ejpam-4571	318	25	,	,	PUNCT
ejpam-4571	318	26	b	b	NOUN
ejpam-4571	318	27	)	)	PUNCT
ejpam-4571	318	28	∩	∩	ADJ
ejpam-4571	318	29	v	v	X
ejpam-4571	318	30	(	(	PUNCT
ejpam-4571	318	31	λ	λ	PROPN
ejpam-4571	318	32	,	,	PUNCT
ejpam-4571	318	33	b	b	NOUN
ejpam-4571	318	34	)	)	PUNCT
ejpam-4571	318	35	=	=	NOUN
ejpam-4571	318	36	∅	∅	NOUN
ejpam-4571	318	37	for	for	ADP
ejpam-4571	318	38	all	all	PRON
ejpam-4571	318	39	(	(	PUNCT
ejpam-4571	318	40	λ	λ	NOUN
ejpam-4571	318	41	,	,	PUNCT
ejpam-4571	318	42	b)-open	b)-open	PUNCT
ejpam-4571	318	43	sets	set	VERB
ejpam-4571	318	44	u	u	NOUN
ejpam-4571	318	45	and	and	CCONJ
ejpam-4571	318	46	v	v	NOUN
ejpam-4571	318	47	of	of	ADP
ejpam-4571	318	48	x	x	PUNCT
ejpam-4571	318	49	with	with	ADP
ejpam-4571	318	50	u	u	NOUN
ejpam-4571	318	51	∩	∩	NOUN
ejpam-4571	318	52	v	v	NOUN
ejpam-4571	318	53	=	=	PUNCT
ejpam-4571	318	54	∅.	∅.	NOUN
ejpam-4571	318	55	proof	proof	NOUN
ejpam-4571	318	56	.	.	PUNCT
ejpam-4571	319	1	this	this	PRON
ejpam-4571	319	2	follows	follow	VERB
ejpam-4571	319	3	from	from	ADP
ejpam-4571	319	4	theorem	theorem	ADJ
ejpam-4571	319	5	2	2	NUM
ejpam-4571	319	6	and	and	CCONJ
ejpam-4571	319	7	theorem	theorem	VERB
ejpam-4571	319	8	3	3	NUM
ejpam-4571	319	9	.	.	PUNCT
ejpam-4571	319	10	theorem	theorem	NOUN
ejpam-4571	319	11	5	5	NUM
ejpam-4571	319	12	.	.	X
ejpam-4571	319	13	for	for	ADP
ejpam-4571	319	14	a	a	DET
ejpam-4571	319	15	topological	topological	ADJ
ejpam-4571	319	16	space	space	NOUN
ejpam-4571	319	17	(	(	PUNCT
ejpam-4571	319	18	x	x	X
ejpam-4571	319	19	,	,	PUNCT
ejpam-4571	319	20	τ	τ	PROPN
ejpam-4571	319	21	)	)	PUNCT
ejpam-4571	319	22	,	,	PUNCT
ejpam-4571	319	23	the	the	DET
ejpam-4571	319	24	following	follow	VERB
ejpam-4571	319	25	properties	property	NOUN
ejpam-4571	319	26	are	be	AUX
ejpam-4571	319	27	equivalent	equivalent	ADJ
ejpam-4571	319	28	:	:	PUNCT
ejpam-4571	319	29	(	(	PUNCT
ejpam-4571	319	30	1	1	X
ejpam-4571	319	31	)	)	PUNCT
ejpam-4571	319	32	(	(	PUNCT
ejpam-4571	319	33	x	x	X
ejpam-4571	319	34	,	,	PUNCT
ejpam-4571	319	35	τ	τ	X
ejpam-4571	319	36	)	)	PUNCT
ejpam-4571	319	37	is	be	AUX
ejpam-4571	319	38	(	(	PUNCT
ejpam-4571	319	39	λ	λ	X
ejpam-4571	319	40	,	,	PUNCT
ejpam-4571	319	41	b)-extremally	b)-extremally	ADV
ejpam-4571	319	42	disconnected	disconnected	ADJ
ejpam-4571	319	43	.	.	PUNCT
ejpam-4571	320	1	(	(	PUNCT
ejpam-4571	320	2	2	2	X
ejpam-4571	320	3	)	)	PUNCT
ejpam-4571	320	4	the	the	DET
ejpam-4571	320	5	(	(	PUNCT
ejpam-4571	320	6	λ	λ	PROPN
ejpam-4571	320	7	,	,	PUNCT
ejpam-4571	320	8	b)-closure	b)-closure	NOUN
ejpam-4571	320	9	of	of	ADP
ejpam-4571	320	10	every	every	DET
ejpam-4571	320	11	β(λ	β(λ	NOUN
ejpam-4571	320	12	,	,	PUNCT
ejpam-4571	320	13	b)-open	b)-open	VERB
ejpam-4571	320	14	set	set	VERB
ejpam-4571	320	15	of	of	ADP
ejpam-4571	320	16	x	x	PUNCT
ejpam-4571	320	17	is	be	AUX
ejpam-4571	320	18	(	(	PUNCT
ejpam-4571	320	19	λ	λ	INTJ
ejpam-4571	320	20	,	,	PUNCT
ejpam-4571	320	21	b)-open	b)-open	VERB
ejpam-4571	320	22	.	.	PUNCT
ejpam-4571	321	1	(	(	PUNCT
ejpam-4571	321	2	3	3	X
ejpam-4571	321	3	)	)	PUNCT
ejpam-4571	321	4	the	the	DET
ejpam-4571	321	5	(	(	PUNCT
ejpam-4571	321	6	λ	λ	PROPN
ejpam-4571	321	7	,	,	PUNCT
ejpam-4571	321	8	b)-closure	b)-closure	NOUN
ejpam-4571	321	9	of	of	ADP
ejpam-4571	321	10	every	every	DET
ejpam-4571	321	11	p(λ	p(λ	NOUN
ejpam-4571	321	12	,	,	PUNCT
ejpam-4571	321	13	b)-open	b)-open	VERB
ejpam-4571	321	14	set	set	VERB
ejpam-4571	321	15	of	of	ADP
ejpam-4571	321	16	x	x	PUNCT
ejpam-4571	321	17	is	be	AUX
ejpam-4571	321	18	(	(	PUNCT
ejpam-4571	321	19	λ	λ	INTJ
ejpam-4571	321	20	,	,	PUNCT
ejpam-4571	321	21	b)-open	b)-open	VERB
ejpam-4571	321	22	.	.	PUNCT
ejpam-4571	322	1	proof	proof	NOUN
ejpam-4571	322	2	.	.	PUNCT
ejpam-4571	323	1	this	this	PRON
ejpam-4571	323	2	follows	follow	VERB
ejpam-4571	323	3	immediately	immediately	ADV
ejpam-4571	323	4	since	since	SCONJ
ejpam-4571	323	5	a(λ	a(λ	PROPN
ejpam-4571	323	6	,	,	PUNCT
ejpam-4571	323	7	b	b	NOUN
ejpam-4571	323	8	)	)	PUNCT
ejpam-4571	323	9	=	=	PUNCT
ejpam-4571	324	1	[	[	X
ejpam-4571	324	2	[	[	X
ejpam-4571	324	3	a(λ	a(λ	ADV
ejpam-4571	324	4	,	,	PUNCT
ejpam-4571	324	5	b)](λ	b)](λ	NOUN
ejpam-4571	324	6	,	,	PUNCT
ejpam-4571	324	7	b	b	NOUN
ejpam-4571	324	8	)	)	PUNCT
ejpam-4571	324	9	]	]	PUNCT
ejpam-4571	324	10	(	(	PUNCT
ejpam-4571	324	11	λ	λ	X
ejpam-4571	324	12	,	,	PUNCT
ejpam-4571	324	13	b	b	NOUN
ejpam-4571	324	14	)	)	PUNCT
ejpam-4571	324	15	.	.	PUNCT
ejpam-4571	325	1	theorem	theorem	VERB
ejpam-4571	325	2	6	6	NUM
ejpam-4571	325	3	.	.	PUNCT
ejpam-4571	326	1	for	for	ADP
ejpam-4571	326	2	a	a	DET
ejpam-4571	326	3	topological	topological	ADJ
ejpam-4571	326	4	space	space	NOUN
ejpam-4571	326	5	(	(	PUNCT
ejpam-4571	326	6	x	x	X
ejpam-4571	326	7	,	,	PUNCT
ejpam-4571	326	8	τ	τ	PROPN
ejpam-4571	326	9	)	)	PUNCT
ejpam-4571	326	10	,	,	PUNCT
ejpam-4571	326	11	the	the	DET
ejpam-4571	326	12	following	follow	VERB
ejpam-4571	326	13	properties	property	NOUN
ejpam-4571	326	14	are	be	AUX
ejpam-4571	326	15	equivalent	equivalent	ADJ
ejpam-4571	326	16	:	:	PUNCT
ejpam-4571	326	17	(	(	PUNCT
ejpam-4571	326	18	1	1	X
ejpam-4571	326	19	)	)	PUNCT
ejpam-4571	326	20	(	(	PUNCT
ejpam-4571	326	21	x	x	X
ejpam-4571	326	22	,	,	PUNCT
ejpam-4571	326	23	τ	τ	X
ejpam-4571	326	24	)	)	PUNCT
ejpam-4571	326	25	is	be	AUX
ejpam-4571	326	26	(	(	PUNCT
ejpam-4571	326	27	λ	λ	X
ejpam-4571	326	28	,	,	PUNCT
ejpam-4571	326	29	b)-extremally	b)-extremally	ADV
ejpam-4571	326	30	disconnected	disconnected	ADJ
ejpam-4571	326	31	.	.	PUNCT
ejpam-4571	327	1	(	(	PUNCT
ejpam-4571	327	2	2	2	X
ejpam-4571	327	3	)	)	PUNCT
ejpam-4571	327	4	if	if	SCONJ
ejpam-4571	327	5	u	u	PROPN
ejpam-4571	327	6	∈	∈	PROPN
ejpam-4571	327	7	βλbo(x	βλbo(x	NOUN
ejpam-4571	327	8	,	,	PUNCT
ejpam-4571	327	9	τ	τ	PROPN
ejpam-4571	327	10	)	)	PUNCT
ejpam-4571	327	11	,	,	PUNCT
ejpam-4571	327	12	v	v	X
ejpam-4571	327	13	∈	∈	PROPN
ejpam-4571	327	14	sλbo(x	sλbo(x	PROPN
ejpam-4571	327	15	,	,	PUNCT
ejpam-4571	327	16	τ	τ	X
ejpam-4571	327	17	)	)	PUNCT
ejpam-4571	327	18	and	and	CCONJ
ejpam-4571	327	19	u	u	NOUN
ejpam-4571	327	20	∩	∩	X
ejpam-4571	327	21	v	v	NOUN
ejpam-4571	327	22	=	=	SYM
ejpam-4571	327	23	∅	∅	NOUN
ejpam-4571	327	24	,	,	PUNCT
ejpam-4571	327	25	then	then	ADV
ejpam-4571	327	26	u	u	X
ejpam-4571	327	27	(	(	PUNCT
ejpam-4571	327	28	λ	λ	PROPN
ejpam-4571	327	29	,	,	PUNCT
ejpam-4571	327	30	b	b	NOUN
ejpam-4571	327	31	)	)	PUNCT
ejpam-4571	327	32	∩	∩	ADJ
ejpam-4571	327	33	v	v	X
ejpam-4571	327	34	(	(	PUNCT
ejpam-4571	327	35	λ	λ	PROPN
ejpam-4571	327	36	,	,	PUNCT
ejpam-4571	327	37	b	b	NOUN
ejpam-4571	327	38	)	)	PUNCT
ejpam-4571	327	39	=	=	SYM
ejpam-4571	327	40	∅.	∅.	X
ejpam-4571	327	41	(	(	PUNCT
ejpam-4571	327	42	3	3	NUM
ejpam-4571	327	43	)	)	PUNCT
ejpam-4571	327	44	if	if	SCONJ
ejpam-4571	327	45	u	u	PROPN
ejpam-4571	327	46	∈	∈	PROPN
ejpam-4571	327	47	bλbo(x	bλbo(x	NOUN
ejpam-4571	327	48	,	,	PUNCT
ejpam-4571	327	49	τ	τ	PROPN
ejpam-4571	327	50	)	)	PUNCT
ejpam-4571	327	51	,	,	PUNCT
ejpam-4571	327	52	v	v	X
ejpam-4571	327	53	∈	∈	PROPN
ejpam-4571	327	54	sλbo(x	sλbo(x	PROPN
ejpam-4571	327	55	,	,	PUNCT
ejpam-4571	327	56	τ	τ	X
ejpam-4571	327	57	)	)	PUNCT
ejpam-4571	327	58	and	and	CCONJ
ejpam-4571	327	59	u	u	NOUN
ejpam-4571	327	60	∩	∩	X
ejpam-4571	327	61	v	v	NOUN
ejpam-4571	327	62	=	=	SYM
ejpam-4571	327	63	∅	∅	NOUN
ejpam-4571	327	64	,	,	PUNCT
ejpam-4571	327	65	then	then	ADV
ejpam-4571	327	66	u	u	X
ejpam-4571	327	67	(	(	PUNCT
ejpam-4571	327	68	λ	λ	PROPN
ejpam-4571	327	69	,	,	PUNCT
ejpam-4571	327	70	b	b	NOUN
ejpam-4571	327	71	)	)	PUNCT
ejpam-4571	327	72	∩	∩	ADJ
ejpam-4571	327	73	v	v	X
ejpam-4571	327	74	(	(	PUNCT
ejpam-4571	327	75	λ	λ	PROPN
ejpam-4571	327	76	,	,	PUNCT
ejpam-4571	327	77	b	b	NOUN
ejpam-4571	327	78	)	)	PUNCT
ejpam-4571	327	79	=	=	SYM
ejpam-4571	327	80	∅.	∅.	X
ejpam-4571	327	81	(	(	PUNCT
ejpam-4571	327	82	4	4	NUM
ejpam-4571	327	83	)	)	PUNCT
ejpam-4571	327	84	if	if	SCONJ
ejpam-4571	327	85	u	u	PROPN
ejpam-4571	327	86	∈	∈	PROPN
ejpam-4571	327	87	pλbo(x	pλbo(x	NOUN
ejpam-4571	327	88	,	,	PUNCT
ejpam-4571	327	89	τ	τ	PROPN
ejpam-4571	327	90	)	)	PUNCT
ejpam-4571	327	91	,	,	PUNCT
ejpam-4571	327	92	v	v	X
ejpam-4571	327	93	∈	∈	PROPN
ejpam-4571	327	94	sλbo(x	sλbo(x	PROPN
ejpam-4571	327	95	,	,	PUNCT
ejpam-4571	327	96	τ	τ	X
ejpam-4571	327	97	)	)	PUNCT
ejpam-4571	327	98	and	and	CCONJ
ejpam-4571	327	99	u	u	NOUN
ejpam-4571	327	100	∩	∩	X
ejpam-4571	327	101	v	v	NOUN
ejpam-4571	327	102	=	=	SYM
ejpam-4571	327	103	∅	∅	NOUN
ejpam-4571	327	104	,	,	PUNCT
ejpam-4571	327	105	then	then	ADV
ejpam-4571	327	106	u	u	X
ejpam-4571	327	107	(	(	PUNCT
ejpam-4571	327	108	λ	λ	PROPN
ejpam-4571	327	109	,	,	PUNCT
ejpam-4571	327	110	b	b	NOUN
ejpam-4571	327	111	)	)	PUNCT
ejpam-4571	327	112	∩	∩	ADJ
ejpam-4571	327	113	v	v	X
ejpam-4571	327	114	(	(	PUNCT
ejpam-4571	327	115	λ	λ	PROPN
ejpam-4571	327	116	,	,	PUNCT
ejpam-4571	327	117	b	b	NOUN
ejpam-4571	327	118	)	)	PUNCT
ejpam-4571	327	119	=	=	SYM
ejpam-4571	327	120	∅.	∅.	X
ejpam-4571	327	121	(	(	PUNCT
ejpam-4571	327	122	5	5	NUM
ejpam-4571	327	123	)	)	PUNCT
ejpam-4571	327	124	if	if	SCONJ
ejpam-4571	327	125	u	u	PROPN
ejpam-4571	327	126	∈	∈	PROPN
ejpam-4571	327	127	rλbo(x	rλbo(x	PROPN
ejpam-4571	327	128	,	,	PUNCT
ejpam-4571	327	129	τ	τ	PROPN
ejpam-4571	327	130	)	)	PUNCT
ejpam-4571	327	131	,	,	PUNCT
ejpam-4571	327	132	v	v	X
ejpam-4571	327	133	∈	∈	PROPN
ejpam-4571	327	134	sλbo(x	sλbo(x	PROPN
ejpam-4571	327	135	,	,	PUNCT
ejpam-4571	327	136	τ	τ	X
ejpam-4571	327	137	)	)	PUNCT
ejpam-4571	327	138	and	and	CCONJ
ejpam-4571	327	139	u	u	NOUN
ejpam-4571	327	140	∩	∩	X
ejpam-4571	327	141	v	v	NOUN
ejpam-4571	327	142	=	=	SYM
ejpam-4571	327	143	∅	∅	NOUN
ejpam-4571	327	144	,	,	PUNCT
ejpam-4571	327	145	then	then	ADV
ejpam-4571	327	146	u	u	X
ejpam-4571	327	147	(	(	PUNCT
ejpam-4571	327	148	λ	λ	PROPN
ejpam-4571	327	149	,	,	PUNCT
ejpam-4571	327	150	b	b	NOUN
ejpam-4571	327	151	)	)	PUNCT
ejpam-4571	327	152	∩	∩	ADJ
ejpam-4571	327	153	v	v	X
ejpam-4571	327	154	(	(	PUNCT
ejpam-4571	327	155	λ	λ	PROPN
ejpam-4571	327	156	,	,	PUNCT
ejpam-4571	327	157	α	α	NOUN
ejpam-4571	327	158	)	)	PUNCT
ejpam-4571	327	159	=	=	PUNCT
ejpam-4571	327	160	∅.	∅.	X
ejpam-4571	327	161	(	(	PUNCT
ejpam-4571	327	162	6	6	NUM
ejpam-4571	327	163	)	)	PUNCT
ejpam-4571	327	164	if	if	SCONJ
ejpam-4571	327	165	u	u	NOUN
ejpam-4571	327	166	,	,	PUNCT
ejpam-4571	327	167	v	v	PROPN
ejpam-4571	327	168	∈	∈	PROPN
ejpam-4571	327	169	rλbo(x	rλbo(x	NOUN
ejpam-4571	327	170	,	,	PUNCT
ejpam-4571	327	171	τ	τ	X
ejpam-4571	327	172	)	)	PUNCT
ejpam-4571	327	173	and	and	CCONJ
ejpam-4571	327	174	u	u	NOUN
ejpam-4571	327	175	∩	∩	X
ejpam-4571	327	176	v	v	NOUN
ejpam-4571	327	177	=	=	SYM
ejpam-4571	327	178	∅	∅	NOUN
ejpam-4571	327	179	,	,	PUNCT
ejpam-4571	327	180	then	then	ADV
ejpam-4571	327	181	u	u	X
ejpam-4571	327	182	(	(	PUNCT
ejpam-4571	327	183	λ	λ	PROPN
ejpam-4571	327	184	,	,	PUNCT
ejpam-4571	327	185	b	b	NOUN
ejpam-4571	327	186	)	)	PUNCT
ejpam-4571	327	187	∩	∩	ADJ
ejpam-4571	327	188	v	v	X
ejpam-4571	327	189	(	(	PUNCT
ejpam-4571	327	190	λ	λ	PROPN
ejpam-4571	327	191	,	,	PUNCT
ejpam-4571	327	192	b	b	NOUN
ejpam-4571	327	193	)	)	PUNCT
ejpam-4571	327	194	=	=	PUNCT
ejpam-4571	327	195	∅.	∅.	NOUN
ejpam-4571	327	196	proof	proof	NOUN
ejpam-4571	327	197	.	.	PUNCT
ejpam-4571	328	1	(	(	PUNCT
ejpam-4571	328	2	1	1	X
ejpam-4571	328	3	)	)	PUNCT
ejpam-4571	328	4	⇒	⇒	NOUN
ejpam-4571	328	5	(	(	PUNCT
ejpam-4571	328	6	2	2	NUM
ejpam-4571	328	7	):	):	PUNCT
ejpam-4571	328	8	suppose	suppose	VERB
ejpam-4571	328	9	that	that	SCONJ
ejpam-4571	328	10	u	u	PROPN
ejpam-4571	328	11	∈	∈	PROPN
ejpam-4571	328	12	βλbo(x	βλbo(x	NOUN
ejpam-4571	328	13	,	,	PUNCT
ejpam-4571	328	14	τ	τ	PROPN
ejpam-4571	328	15	)	)	PUNCT
ejpam-4571	328	16	,	,	PUNCT
ejpam-4571	328	17	v	v	X
ejpam-4571	328	18	∈	∈	PROPN
ejpam-4571	328	19	sλbo(x	sλbo(x	PROPN
ejpam-4571	328	20	,	,	PUNCT
ejpam-4571	328	21	τ	τ	X
ejpam-4571	328	22	)	)	PUNCT
ejpam-4571	328	23	and	and	CCONJ
ejpam-4571	328	24	u	u	NOUN
ejpam-4571	328	25	∩	∩	NOUN
ejpam-4571	328	26	v	v	NOUN
ejpam-4571	328	27	=	=	PUNCT
ejpam-4571	328	28	∅.	∅.	VERB
ejpam-4571	328	29	thus	thus	ADV
ejpam-4571	328	30	,	,	PUNCT
ejpam-4571	328	31	u	u	PROPN
ejpam-4571	328	32	(	(	PUNCT
ejpam-4571	328	33	λ	λ	PROPN
ejpam-4571	328	34	,	,	PUNCT
ejpam-4571	328	35	b	b	NOUN
ejpam-4571	328	36	)	)	PUNCT
ejpam-4571	328	37	∩	∩	PROPN
ejpam-4571	328	38	v(λ	v(λ	PROPN
ejpam-4571	328	39	,	,	PUNCT
ejpam-4571	328	40	b	b	NOUN
ejpam-4571	328	41	)	)	PUNCT
ejpam-4571	328	42	=	=	NOUN
ejpam-4571	328	43	∅	∅	NOUN
ejpam-4571	328	44	,	,	PUNCT
ejpam-4571	328	45	by	by	ADP
ejpam-4571	328	46	theorem	theorem	NOUN
ejpam-4571	328	47	5	5	NUM
ejpam-4571	328	48	,	,	PUNCT
ejpam-4571	328	49	u	u	NOUN
ejpam-4571	328	50	(	(	PUNCT
ejpam-4571	328	51	λ	λ	PROPN
ejpam-4571	328	52	,	,	PUNCT
ejpam-4571	328	53	b	b	NOUN
ejpam-4571	328	54	)	)	PUNCT
ejpam-4571	328	55	is	be	AUX
ejpam-4571	328	56	(	(	PUNCT
ejpam-4571	328	57	λ	λ	X
ejpam-4571	328	58	,	,	PUNCT
ejpam-4571	328	59	b)-open	b)-open	PUNCT
ejpam-4571	328	60	and	and	CCONJ
ejpam-4571	328	61	hence	hence	ADV
ejpam-4571	328	62	u	u	NOUN
ejpam-4571	328	63	(	(	PUNCT
ejpam-4571	328	64	λ	λ	PROPN
ejpam-4571	328	65	,	,	PUNCT
ejpam-4571	328	66	b	b	NOUN
ejpam-4571	328	67	)	)	PUNCT
ejpam-4571	328	68	∩	∩	ADJ
ejpam-4571	328	69	v	v	X
ejpam-4571	328	70	(	(	PUNCT
ejpam-4571	328	71	λ	λ	PROPN
ejpam-4571	328	72	,	,	PUNCT
ejpam-4571	328	73	b	b	NOUN
ejpam-4571	328	74	)	)	PUNCT
ejpam-4571	328	75	=	=	SYM
ejpam-4571	328	76	u	u	NOUN
ejpam-4571	328	77	(	(	PUNCT
ejpam-4571	328	78	λ	λ	PROPN
ejpam-4571	328	79	,	,	PUNCT
ejpam-4571	328	80	p	p	NOUN
ejpam-4571	328	81	)	)	PUNCT
ejpam-4571	328	82	∩	∩	NOUN
ejpam-4571	328	83	[	[	X
ejpam-4571	328	84	v(λ	v(λ	PROPN
ejpam-4571	328	85	,	,	PUNCT
ejpam-4571	328	86	b	b	NOUN
ejpam-4571	328	87	)	)	PUNCT
ejpam-4571	328	88	]	]	PUNCT
ejpam-4571	328	89	(	(	PUNCT
ejpam-4571	328	90	λ	λ	X
ejpam-4571	328	91	,	,	PUNCT
ejpam-4571	328	92	b	b	NOUN
ejpam-4571	328	93	)	)	PUNCT
ejpam-4571	328	94	=	=	SYM
ejpam-4571	328	95	∅.	∅.	X
ejpam-4571	328	96	(	(	PUNCT
ejpam-4571	328	97	2	2	NUM
ejpam-4571	328	98	)	)	PUNCT
ejpam-4571	328	99	⇒	⇒	NOUN
ejpam-4571	328	100	(	(	PUNCT
ejpam-4571	328	101	3	3	NUM
ejpam-4571	328	102	)	)	PUNCT
ejpam-4571	328	103	⇒	⇒	NOUN
ejpam-4571	328	104	(	(	PUNCT
ejpam-4571	328	105	4	4	NUM
ejpam-4571	328	106	)	)	PUNCT
ejpam-4571	328	107	⇒	⇒	NOUN
ejpam-4571	328	108	(	(	PUNCT
ejpam-4571	328	109	5	5	NUM
ejpam-4571	328	110	)	)	PUNCT
ejpam-4571	328	111	⇒	⇒	NOUN
ejpam-4571	328	112	(	(	PUNCT
ejpam-4571	328	113	6	6	NUM
ejpam-4571	328	114	):	):	PUNCT
ejpam-4571	328	115	obvious	obvious	ADJ
ejpam-4571	328	116	.	.	PUNCT
ejpam-4571	329	1	(	(	PUNCT
ejpam-4571	329	2	6	6	NUM
ejpam-4571	329	3	)	)	PUNCT
ejpam-4571	329	4	⇒	⇒	NOUN
ejpam-4571	329	5	(	(	PUNCT
ejpam-4571	329	6	1	1	NUM
ejpam-4571	329	7	):	):	PUNCT
ejpam-4571	329	8	let	let	VERB
ejpam-4571	329	9	u	u	PRON
ejpam-4571	329	10	and	and	CCONJ
ejpam-4571	329	11	v	v	NOUN
ejpam-4571	329	12	be	be	AUX
ejpam-4571	329	13	(	(	PUNCT
ejpam-4571	329	14	λ	λ	X
ejpam-4571	329	15	,	,	PUNCT
ejpam-4571	329	16	b)-open	b)-open	VERB
ejpam-4571	329	17	sets	set	NOUN
ejpam-4571	329	18	of	of	ADP
ejpam-4571	329	19	x.	x.	NOUN
ejpam-4571	329	20	then	then	ADV
ejpam-4571	329	21	,	,	PUNCT
ejpam-4571	329	22	u	u	PROPN
ejpam-4571	329	23	(	(	PUNCT
ejpam-4571	329	24	λ	λ	PROPN
ejpam-4571	329	25	,	,	PUNCT
ejpam-4571	329	26	b	b	NOUN
ejpam-4571	329	27	)	)	PUNCT
ejpam-4571	329	28	,	,	PUNCT
ejpam-4571	329	29	v	v	NOUN
ejpam-4571	329	30	(	(	PUNCT
ejpam-4571	329	31	λ	λ	PROPN
ejpam-4571	329	32	,	,	PUNCT
ejpam-4571	329	33	b	b	NOUN
ejpam-4571	329	34	)	)	PUNCT
ejpam-4571	329	35	∈	∈	PROPN
ejpam-4571	329	36	rλbo(x	rλbo(x	NOUN
ejpam-4571	329	37	,	,	PUNCT
ejpam-4571	329	38	τ	τ	X
ejpam-4571	329	39	)	)	PUNCT
ejpam-4571	329	40	and	and	CCONJ
ejpam-4571	329	41	u	u	PROPN
ejpam-4571	329	42	(	(	PUNCT
ejpam-4571	329	43	λ	λ	PROPN
ejpam-4571	329	44	,	,	PUNCT
ejpam-4571	329	45	b	b	NOUN
ejpam-4571	329	46	)	)	PUNCT
ejpam-4571	329	47	∩	∩	ADJ
ejpam-4571	329	48	v	v	X
ejpam-4571	329	49	(	(	PUNCT
ejpam-4571	329	50	λ	λ	PROPN
ejpam-4571	329	51	,	,	PUNCT
ejpam-4571	329	52	b	b	NOUN
ejpam-4571	329	53	)	)	PUNCT
ejpam-4571	329	54	=	=	PUNCT
ejpam-4571	329	55	∅.	∅.	X
ejpam-4571	329	56	by	by	ADP
ejpam-4571	329	57	(	(	PUNCT
ejpam-4571	329	58	6	6	NUM
ejpam-4571	329	59	)	)	PUNCT
ejpam-4571	329	60	,	,	PUNCT
ejpam-4571	330	1	[	[	X
ejpam-4571	330	2	u	u	X
ejpam-4571	330	3	(	(	PUNCT
ejpam-4571	330	4	λ	λ	PROPN
ejpam-4571	330	5	,	,	PUNCT
ejpam-4571	330	6	b)](λ	b)](λ	NOUN
ejpam-4571	330	7	,	,	PUNCT
ejpam-4571	330	8	b	b	NOUN
ejpam-4571	330	9	)	)	PUNCT
ejpam-4571	330	10	∩	∩	NOUN
ejpam-4571	330	11	[	[	X
ejpam-4571	330	12	v	v	X
ejpam-4571	330	13	(	(	PUNCT
ejpam-4571	330	14	λ	λ	PROPN
ejpam-4571	330	15	,	,	PUNCT
ejpam-4571	330	16	b)](λ	b)](λ	NOUN
ejpam-4571	330	17	,	,	PUNCT
ejpam-4571	330	18	b	b	NOUN
ejpam-4571	330	19	)	)	PUNCT
ejpam-4571	330	20	=	=	SYM
ejpam-4571	330	21	u	u	NOUN
ejpam-4571	330	22	(	(	PUNCT
ejpam-4571	330	23	λ	λ	PROPN
ejpam-4571	330	24	,	,	PUNCT
ejpam-4571	330	25	b	b	NOUN
ejpam-4571	330	26	)	)	PUNCT
ejpam-4571	330	27	∩	∩	ADJ
ejpam-4571	330	28	v	v	X
ejpam-4571	330	29	(	(	PUNCT
ejpam-4571	330	30	λ	λ	PROPN
ejpam-4571	330	31	,	,	PUNCT
ejpam-4571	330	32	b	b	NOUN
ejpam-4571	330	33	)	)	PUNCT
ejpam-4571	330	34	=	=	PUNCT
ejpam-4571	330	35	∅.	∅.	ADP
ejpam-4571	330	36	thus	thus	ADV
ejpam-4571	330	37	,	,	PUNCT
ejpam-4571	330	38	(	(	PUNCT
ejpam-4571	330	39	x	x	X
ejpam-4571	330	40	,	,	PUNCT
ejpam-4571	330	41	τ	τ	X
ejpam-4571	330	42	)	)	PUNCT
ejpam-4571	330	43	is	be	AUX
ejpam-4571	330	44	(	(	PUNCT
ejpam-4571	330	45	λ	λ	X
ejpam-4571	330	46	,	,	PUNCT
ejpam-4571	330	47	b)-extremally	b)-extremally	ADV
ejpam-4571	330	48	disconnected	disconnected	ADJ
ejpam-4571	330	49	.	.	PUNCT
ejpam-4571	331	1	4	4	X
ejpam-4571	331	2	.	.	X
ejpam-4571	331	3	characterizations	characterization	NOUN
ejpam-4571	331	4	of	of	ADP
ejpam-4571	331	5	weakly	weakly	ADJ
ejpam-4571	331	6	(	(	PUNCT
ejpam-4571	331	7	λ	λ	PROPN
ejpam-4571	331	8	,	,	PUNCT
ejpam-4571	331	9	b)-continuous	b)-continuous	ADJ
ejpam-4571	331	10	functions	function	NOUN
ejpam-4571	331	11	in	in	ADP
ejpam-4571	331	12	this	this	DET
ejpam-4571	331	13	section	section	NOUN
ejpam-4571	331	14	,	,	PUNCT
ejpam-4571	331	15	we	we	PRON
ejpam-4571	331	16	introduce	introduce	VERB
ejpam-4571	331	17	the	the	DET
ejpam-4571	331	18	notion	notion	NOUN
ejpam-4571	331	19	of	of	ADP
ejpam-4571	331	20	weakly	weakly	ADJ
ejpam-4571	331	21	(	(	PUNCT
ejpam-4571	331	22	λ	λ	PROPN
ejpam-4571	331	23	,	,	PUNCT
ejpam-4571	331	24	b)-continuous	b)-continuous	ADJ
ejpam-4571	331	25	functions	function	NOUN
ejpam-4571	331	26	.	.	PUNCT
ejpam-4571	332	1	moreover	moreover	ADV
ejpam-4571	332	2	,	,	PUNCT
ejpam-4571	332	3	several	several	ADJ
ejpam-4571	332	4	characterizations	characterization	NOUN
ejpam-4571	332	5	of	of	ADP
ejpam-4571	332	6	weakly	weakly	ADJ
ejpam-4571	332	7	(	(	PUNCT
ejpam-4571	332	8	λ	λ	PROPN
ejpam-4571	332	9	,	,	PUNCT
ejpam-4571	332	10	b)-continuous	b)-continuous	ADJ
ejpam-4571	332	11	functions	function	NOUN
ejpam-4571	332	12	are	be	AUX
ejpam-4571	332	13	discussed	discuss	VERB
ejpam-4571	332	14	.	.	PUNCT
ejpam-4571	333	1	definition	definition	NOUN
ejpam-4571	333	2	5	5	NUM
ejpam-4571	333	3	.	.	PUNCT
ejpam-4571	334	1	a	a	DET
ejpam-4571	334	2	function	function	NOUN
ejpam-4571	334	3	f	f	NOUN
ejpam-4571	334	4	:	:	PUNCT
ejpam-4571	334	5	(	(	PUNCT
ejpam-4571	334	6	x	x	X
ejpam-4571	334	7	,	,	PUNCT
ejpam-4571	334	8	τ	τ	X
ejpam-4571	334	9	)	)	PUNCT
ejpam-4571	334	10	→	→	SYM
ejpam-4571	334	11	(	(	PUNCT
ejpam-4571	334	12	y	y	PROPN
ejpam-4571	334	13	,	,	PUNCT
ejpam-4571	334	14	σ	σ	PROPN
ejpam-4571	334	15	)	)	PUNCT
ejpam-4571	334	16	is	be	AUX
ejpam-4571	334	17	called	call	VERB
ejpam-4571	334	18	weakly	weakly	ADJ
ejpam-4571	334	19	(	(	PUNCT
ejpam-4571	334	20	λ	λ	PROPN
ejpam-4571	334	21	,	,	PUNCT
ejpam-4571	334	22	b)-continuous	b)-continuous	ADJ
ejpam-4571	334	23	at	at	ADP
ejpam-4571	334	24	a	a	DET
ejpam-4571	334	25	point	point	NOUN
ejpam-4571	334	26	x	x	SYM
ejpam-4571	334	27	∈	∈	NOUN
ejpam-4571	334	28	x	x	INTJ
ejpam-4571	334	29	if	if	SCONJ
ejpam-4571	334	30	,	,	PUNCT
ejpam-4571	334	31	for	for	SCONJ
ejpam-4571	334	32	each	each	DET
ejpam-4571	334	33	(	(	PUNCT
ejpam-4571	334	34	λ	λ	PROPN
ejpam-4571	334	35	,	,	PUNCT
ejpam-4571	334	36	b)-open	b)-open	AUX
ejpam-4571	334	37	set	set	VERB
ejpam-4571	334	38	v	v	NOUN
ejpam-4571	334	39	of	of	ADP
ejpam-4571	334	40	y	y	NOUN
ejpam-4571	334	41	containing	contain	VERB
ejpam-4571	334	42	f(x	f(x	PROPN
ejpam-4571	334	43	)	)	PUNCT
ejpam-4571	334	44	,	,	PUNCT
ejpam-4571	334	45	there	there	PRON
ejpam-4571	334	46	exists	exist	VERB
ejpam-4571	334	47	a	a	DET
ejpam-4571	334	48	(	(	PUNCT
ejpam-4571	334	49	λ	λ	NOUN
ejpam-4571	334	50	,	,	PUNCT
ejpam-4571	334	51	b)-open	b)-open	VERB
ejpam-4571	334	52	set	set	VERB
ejpam-4571	334	53	u	u	NOUN
ejpam-4571	334	54	of	of	ADP
ejpam-4571	334	55	x	x	PUNCT
ejpam-4571	334	56	containing	contain	VERB
ejpam-4571	334	57	x	x	PUNCT
ejpam-4571	334	58	such	such	ADJ
ejpam-4571	334	59	that	that	DET
ejpam-4571	334	60	f(u	f(u	PROPN
ejpam-4571	334	61	)	)	PUNCT
ejpam-4571	334	62	⊆	⊆	NUM
ejpam-4571	334	63	v	v	NOUN
ejpam-4571	334	64	(	(	PUNCT
ejpam-4571	334	65	λ	λ	PROPN
ejpam-4571	334	66	,	,	PUNCT
ejpam-4571	334	67	b	b	NOUN
ejpam-4571	334	68	)	)	PUNCT
ejpam-4571	334	69	.	.	PUNCT
ejpam-4571	335	1	a	a	DET
ejpam-4571	335	2	function	function	NOUN
ejpam-4571	335	3	f	f	NOUN
ejpam-4571	335	4	:	:	PUNCT
ejpam-4571	335	5	(	(	PUNCT
ejpam-4571	335	6	x	x	X
ejpam-4571	335	7	,	,	PUNCT
ejpam-4571	335	8	τ	τ	X
ejpam-4571	335	9	)	)	PUNCT
ejpam-4571	335	10	→	→	SYM
ejpam-4571	335	11	(	(	PUNCT
ejpam-4571	335	12	y	y	PROPN
ejpam-4571	335	13	,	,	PUNCT
ejpam-4571	335	14	σ	σ	PROPN
ejpam-4571	335	15	)	)	PUNCT
ejpam-4571	335	16	is	be	AUX
ejpam-4571	335	17	called	call	VERB
ejpam-4571	335	18	(	(	PUNCT
ejpam-4571	335	19	λ	λ	PROPN
ejpam-4571	335	20	,	,	PUNCT
ejpam-4571	335	21	b)-continuous	b)-continuous	ADJ
ejpam-4571	335	22	if	if	SCONJ
ejpam-4571	335	23	f	f	PROPN
ejpam-4571	335	24	has	have	VERB
ejpam-4571	335	25	this	this	DET
ejpam-4571	335	26	property	property	NOUN
ejpam-4571	335	27	at	at	ADP
ejpam-4571	335	28	each	each	DET
ejpam-4571	335	29	point	point	NOUN
ejpam-4571	335	30	x	x	X
ejpam-4571	335	31	∈	∈	PROPN
ejpam-4571	335	32	x.	x.	NOUN
ejpam-4571	335	33	c.	c.	PROPN
ejpam-4571	335	34	boonpok	boonpok	PROPN
ejpam-4571	335	35	,	,	PUNCT
ejpam-4571	335	36	n.	n.	PROPN
ejpam-4571	335	37	srisarakham	srisarakham	PROPN
ejpam-4571	335	38	/	/	SYM
ejpam-4571	335	39	eur	eur	PROPN
ejpam-4571	335	40	.	.	PUNCT
ejpam-4571	336	1	j.	j.	PROPN
ejpam-4571	336	2	pure	pure	PROPN
ejpam-4571	336	3	appl	appl	PROPN
ejpam-4571	336	4	.	.	PROPN
ejpam-4571	336	5	math	math	PROPN
ejpam-4571	336	6	,	,	PUNCT
ejpam-4571	336	7	16	16	NUM
ejpam-4571	336	8	(	(	PUNCT
ejpam-4571	336	9	1	1	NUM
ejpam-4571	336	10	)	)	PUNCT
ejpam-4571	336	11	(	(	PUNCT
ejpam-4571	336	12	2023	2023	NUM
ejpam-4571	336	13	)	)	PUNCT
ejpam-4571	336	14	,	,	PUNCT
ejpam-4571	336	15	29	29	NUM
ejpam-4571	336	16	-	-	SYM
ejpam-4571	336	17	43	43	NUM
ejpam-4571	336	18	38	38	NUM
ejpam-4571	336	19	example	example	NOUN
ejpam-4571	336	20	4	4	NUM
ejpam-4571	336	21	.	.	PUNCT
ejpam-4571	337	1	let	let	VERB
ejpam-4571	337	2	x	x	PUNCT
ejpam-4571	337	3	=	=	PRON
ejpam-4571	337	4	{	{	PUNCT
ejpam-4571	337	5	a	a	PRON
ejpam-4571	337	6	,	,	PUNCT
ejpam-4571	337	7	b	b	NOUN
ejpam-4571	337	8	,	,	PUNCT
ejpam-4571	337	9	c	c	NOUN
ejpam-4571	337	10	}	}	PUNCT
ejpam-4571	337	11	with	with	ADP
ejpam-4571	337	12	the	the	DET
ejpam-4571	337	13	topology	topology	NOUN
ejpam-4571	337	14	τ	τ	X
ejpam-4571	337	15	=	=	PUNCT
ejpam-4571	337	16	{	{	PUNCT
ejpam-4571	337	17	∅	∅	NOUN
ejpam-4571	337	18	,	,	PUNCT
ejpam-4571	337	19	{	{	PUNCT
ejpam-4571	337	20	a	a	X
ejpam-4571	337	21	}	}	PUNCT
ejpam-4571	337	22	,	,	PUNCT
ejpam-4571	337	23	{	{	PUNCT
ejpam-4571	337	24	b	b	NOUN
ejpam-4571	337	25	,	,	PUNCT
ejpam-4571	337	26	c	c	NOUN
ejpam-4571	337	27	}	}	PUNCT
ejpam-4571	337	28	,	,	PUNCT
ejpam-4571	337	29	x	x	NOUN
ejpam-4571	337	30	}	}	PUNCT
ejpam-4571	337	31	.	.	PUNCT
ejpam-4571	338	1	let	let	VERB
ejpam-4571	338	2	y	y	PROPN
ejpam-4571	338	3	=	=	PUNCT
ejpam-4571	338	4	{	{	PUNCT
ejpam-4571	338	5	−1	−1	NOUN
ejpam-4571	338	6	,	,	PUNCT
ejpam-4571	338	7	0	0	NUM
ejpam-4571	338	8	,	,	PUNCT
ejpam-4571	338	9	1	1	NUM
ejpam-4571	338	10	}	}	PUNCT
ejpam-4571	338	11	with	with	ADP
ejpam-4571	338	12	the	the	DET
ejpam-4571	338	13	topology	topology	NOUN
ejpam-4571	338	14	σ	σ	NOUN
ejpam-4571	338	15	=	=	SYM
ejpam-4571	338	16	{	{	PUNCT
ejpam-4571	338	17	∅	∅	NOUN
ejpam-4571	338	18	,	,	PUNCT
ejpam-4571	338	19	{	{	PUNCT
ejpam-4571	338	20	−1	−1	NOUN
ejpam-4571	338	21	}	}	PUNCT
ejpam-4571	338	22	,	,	PUNCT
ejpam-4571	338	23	{	{	PUNCT
ejpam-4571	338	24	0	0	NUM
ejpam-4571	338	25	,	,	PUNCT
ejpam-4571	338	26	1	1	NUM
ejpam-4571	338	27	}	}	PUNCT
ejpam-4571	338	28	,	,	PUNCT
ejpam-4571	338	29	y	y	PROPN
ejpam-4571	338	30	}	}	PUNCT
ejpam-4571	338	31	.	.	PUNCT
ejpam-4571	339	1	let	let	VERB
ejpam-4571	339	2	f	f	NOUN
ejpam-4571	339	3	:	:	PUNCT
ejpam-4571	339	4	(	(	PUNCT
ejpam-4571	339	5	x	x	X
ejpam-4571	339	6	,	,	PUNCT
ejpam-4571	339	7	τ	τ	X
ejpam-4571	339	8	)	)	PUNCT
ejpam-4571	339	9	→	→	SYM
ejpam-4571	339	10	(	(	PUNCT
ejpam-4571	339	11	y	y	PROPN
ejpam-4571	339	12	,	,	PUNCT
ejpam-4571	339	13	σ	σ	PROPN
ejpam-4571	339	14	)	)	PUNCT
ejpam-4571	339	15	be	be	VERB
ejpam-4571	339	16	a	a	DET
ejpam-4571	339	17	function	function	NOUN
ejpam-4571	339	18	defined	define	VERB
ejpam-4571	339	19	as	as	SCONJ
ejpam-4571	339	20	follows	follow	VERB
ejpam-4571	339	21	:	:	PUNCT
ejpam-4571	339	22	f(a	f(a	NOUN
ejpam-4571	339	23	)	)	PUNCT
ejpam-4571	340	1	=	=	SYM
ejpam-4571	340	2	−1	−1	NOUN
ejpam-4571	340	3	,	,	PUNCT
ejpam-4571	340	4	f(b	f(b	PROPN
ejpam-4571	340	5	)	)	PUNCT
ejpam-4571	340	6	=	=	SYM
ejpam-4571	340	7	0	0	NUM
ejpam-4571	340	8	and	and	CCONJ
ejpam-4571	340	9	f(c	f(c	PROPN
ejpam-4571	340	10	)	)	PUNCT
ejpam-4571	340	11	=	=	SYM
ejpam-4571	341	1	1	1	X
ejpam-4571	341	2	.	.	PUNCT
ejpam-4571	341	3	then	then	ADV
ejpam-4571	341	4	,	,	PUNCT
ejpam-4571	341	5	f	f	PROPN
ejpam-4571	341	6	is	be	AUX
ejpam-4571	341	7	weakly	weakly	ADJ
ejpam-4571	341	8	(	(	PUNCT
ejpam-4571	341	9	λ	λ	NOUN
ejpam-4571	341	10	,	,	PUNCT
ejpam-4571	341	11	b)-continuous	b)-continuous	PROPN
ejpam-4571	341	12	.	.	PUNCT
ejpam-4571	342	1	theorem	theorem	VERB
ejpam-4571	342	2	7	7	NUM
ejpam-4571	342	3	.	.	PUNCT
ejpam-4571	343	1	a	a	DET
ejpam-4571	343	2	function	function	NOUN
ejpam-4571	343	3	f	f	NOUN
ejpam-4571	343	4	:	:	PUNCT
ejpam-4571	343	5	(	(	PUNCT
ejpam-4571	343	6	x	x	X
ejpam-4571	343	7	,	,	PUNCT
ejpam-4571	343	8	τ	τ	X
ejpam-4571	343	9	)	)	PUNCT
ejpam-4571	343	10	→	→	SYM
ejpam-4571	343	11	(	(	PUNCT
ejpam-4571	343	12	y	y	PROPN
ejpam-4571	343	13	,	,	PUNCT
ejpam-4571	343	14	σ	σ	PROPN
ejpam-4571	343	15	)	)	PUNCT
ejpam-4571	343	16	is	be	AUX
ejpam-4571	343	17	weakly	weakly	ADJ
ejpam-4571	343	18	(	(	PUNCT
ejpam-4571	343	19	λ	λ	X
ejpam-4571	343	20	,	,	PUNCT
ejpam-4571	343	21	b)-continuous	b)-continuous	ADJ
ejpam-4571	343	22	at	at	ADP
ejpam-4571	343	23	x	x	X
ejpam-4571	343	24	∈	∈	PROPN
ejpam-4571	343	25	x	x	INTJ
ejpam-4571	343	26	if	if	SCONJ
ejpam-4571	343	27	and	and	CCONJ
ejpam-4571	343	28	only	only	ADV
ejpam-4571	343	29	if	if	SCONJ
ejpam-4571	343	30	for	for	SCONJ
ejpam-4571	343	31	each	each	DET
ejpam-4571	343	32	(	(	PUNCT
ejpam-4571	343	33	λ	λ	PROPN
ejpam-4571	343	34	,	,	PUNCT
ejpam-4571	343	35	b)-open	b)-open	AUX
ejpam-4571	343	36	set	set	VERB
ejpam-4571	343	37	v	v	NOUN
ejpam-4571	343	38	of	of	ADP
ejpam-4571	343	39	y	y	NOUN
ejpam-4571	343	40	containing	contain	VERB
ejpam-4571	343	41	f(x	f(x	PROPN
ejpam-4571	343	42	)	)	PUNCT
ejpam-4571	343	43	,	,	PUNCT
ejpam-4571	344	1	x	x	PUNCT
ejpam-4571	344	2	∈	∈	PROPN
ejpam-4571	345	1	[	[	X
ejpam-4571	345	2	f−1(v	f−1(v	NOUN
ejpam-4571	345	3	(	(	PUNCT
ejpam-4571	345	4	λ	λ	PROPN
ejpam-4571	345	5	,	,	PUNCT
ejpam-4571	345	6	b))](λ	b))](λ	PROPN
ejpam-4571	345	7	,	,	PUNCT
ejpam-4571	345	8	b	b	NOUN
ejpam-4571	345	9	)	)	PUNCT
ejpam-4571	345	10	.	.	PUNCT
ejpam-4571	346	1	proof	proof	NOUN
ejpam-4571	346	2	.	.	PUNCT
ejpam-4571	347	1	let	let	VERB
ejpam-4571	347	2	v	v	PART
ejpam-4571	347	3	be	be	AUX
ejpam-4571	347	4	any	any	DET
ejpam-4571	347	5	(	(	PUNCT
ejpam-4571	347	6	λ	λ	NOUN
ejpam-4571	347	7	,	,	PUNCT
ejpam-4571	347	8	b)-open	b)-open	VERB
ejpam-4571	347	9	set	set	VERB
ejpam-4571	347	10	of	of	ADP
ejpam-4571	347	11	y	y	PROPN
ejpam-4571	347	12	containing	contain	VERB
ejpam-4571	347	13	f(x	f(x	PROPN
ejpam-4571	347	14	)	)	PUNCT
ejpam-4571	347	15	.	.	PUNCT
ejpam-4571	348	1	then	then	ADV
ejpam-4571	348	2	,	,	PUNCT
ejpam-4571	348	3	there	there	PRON
ejpam-4571	348	4	exists	exist	VERB
ejpam-4571	348	5	a	a	DET
ejpam-4571	348	6	(	(	PUNCT
ejpam-4571	348	7	λ	λ	NOUN
ejpam-4571	348	8	,	,	PUNCT
ejpam-4571	348	9	b)open	b)open	X
ejpam-4571	348	10	set	set	VERB
ejpam-4571	348	11	u	u	NOUN
ejpam-4571	348	12	of	of	ADP
ejpam-4571	348	13	x	x	PUNCT
ejpam-4571	348	14	containing	contain	VERB
ejpam-4571	348	15	x	x	PUNCT
ejpam-4571	348	16	such	such	ADJ
ejpam-4571	348	17	that	that	DET
ejpam-4571	348	18	f(u	f(u	PROPN
ejpam-4571	348	19	)	)	PUNCT
ejpam-4571	348	20	⊆	⊆	NUM
ejpam-4571	348	21	v	v	NOUN
ejpam-4571	348	22	(	(	PUNCT
ejpam-4571	348	23	λ	λ	PROPN
ejpam-4571	348	24	,	,	PUNCT
ejpam-4571	348	25	b	b	NOUN
ejpam-4571	348	26	)	)	PUNCT
ejpam-4571	348	27	.	.	PUNCT
ejpam-4571	349	1	thus	thus	ADV
ejpam-4571	349	2	,	,	PUNCT
ejpam-4571	349	3	x	x	PUNCT
ejpam-4571	349	4	∈	∈	PROPN
ejpam-4571	349	5	u	u	NOUN
ejpam-4571	349	6	⊆	⊆	NUM
ejpam-4571	349	7	f−1(v	f−1(v	NOUN
ejpam-4571	349	8	(	(	PUNCT
ejpam-4571	349	9	λ	λ	PROPN
ejpam-4571	349	10	,	,	PUNCT
ejpam-4571	349	11	b	b	NOUN
ejpam-4571	349	12	)	)	PUNCT
ejpam-4571	349	13	)	)	PUNCT
ejpam-4571	349	14	and	and	CCONJ
ejpam-4571	349	15	hence	hence	ADV
ejpam-4571	349	16	x	x	X
ejpam-4571	349	17	∈	∈	PROPN
ejpam-4571	350	1	[	[	X
ejpam-4571	350	2	f−1(v	f−1(v	NOUN
ejpam-4571	350	3	(	(	PUNCT
ejpam-4571	350	4	λ	λ	PROPN
ejpam-4571	350	5	,	,	PUNCT
ejpam-4571	350	6	b))](λ	b))](λ	PROPN
ejpam-4571	350	7	,	,	PUNCT
ejpam-4571	350	8	b	b	NOUN
ejpam-4571	350	9	)	)	PUNCT
ejpam-4571	350	10	.	.	PUNCT
ejpam-4571	351	1	conversely	conversely	ADV
ejpam-4571	351	2	,	,	PUNCT
ejpam-4571	351	3	let	let	VERB
ejpam-4571	351	4	v	v	PART
ejpam-4571	351	5	be	be	AUX
ejpam-4571	351	6	any	any	DET
ejpam-4571	351	7	(	(	PUNCT
ejpam-4571	351	8	λ	λ	NOUN
ejpam-4571	351	9	,	,	PUNCT
ejpam-4571	351	10	b)-open	b)-open	VERB
ejpam-4571	351	11	set	set	VERB
ejpam-4571	351	12	of	of	ADP
ejpam-4571	351	13	y	y	PROPN
ejpam-4571	351	14	containing	contain	VERB
ejpam-4571	351	15	f(x	f(x	PROPN
ejpam-4571	351	16	)	)	PUNCT
ejpam-4571	351	17	.	.	PUNCT
ejpam-4571	352	1	then	then	ADV
ejpam-4571	352	2	,	,	PUNCT
ejpam-4571	352	3	x	x	PUNCT
ejpam-4571	352	4	∈	∈	PROPN
ejpam-4571	353	1	[	[	X
ejpam-4571	353	2	f−1(v	f−1(v	NOUN
ejpam-4571	353	3	(	(	PUNCT
ejpam-4571	353	4	λ	λ	PROPN
ejpam-4571	353	5	,	,	PUNCT
ejpam-4571	353	6	b))](λ	b))](λ	PROPN
ejpam-4571	353	7	,	,	PUNCT
ejpam-4571	353	8	b	b	NOUN
ejpam-4571	353	9	)	)	PUNCT
ejpam-4571	353	10	and	and	CCONJ
ejpam-4571	353	11	there	there	PRON
ejpam-4571	353	12	exists	exist	VERB
ejpam-4571	353	13	a	a	DET
ejpam-4571	353	14	(	(	PUNCT
ejpam-4571	353	15	λ	λ	NOUN
ejpam-4571	353	16	,	,	PUNCT
ejpam-4571	353	17	b)-open	b)-open	VERB
ejpam-4571	353	18	set	set	VERB
ejpam-4571	353	19	u	u	NOUN
ejpam-4571	353	20	of	of	ADP
ejpam-4571	353	21	x	x	SYM
ejpam-4571	354	1	such	such	ADJ
ejpam-4571	354	2	that	that	SCONJ
ejpam-4571	354	3	x	x	SYM
ejpam-4571	354	4	∈	∈	PROPN
ejpam-4571	354	5	u	u	NOUN
ejpam-4571	354	6	⊆	⊆	NUM
ejpam-4571	354	7	f−1(v	f−1(v	NOUN
ejpam-4571	354	8	(	(	PUNCT
ejpam-4571	354	9	λ	λ	PROPN
ejpam-4571	354	10	,	,	PUNCT
ejpam-4571	354	11	b	b	NOUN
ejpam-4571	354	12	)	)	PUNCT
ejpam-4571	354	13	)	)	PUNCT
ejpam-4571	354	14	;	;	PUNCT
ejpam-4571	354	15	hence	hence	ADV
ejpam-4571	354	16	f(u	f(u	PROPN
ejpam-4571	354	17	)	)	PUNCT
ejpam-4571	354	18	⊆	⊆	NUM
ejpam-4571	354	19	v	v	NOUN
ejpam-4571	354	20	(	(	PUNCT
ejpam-4571	354	21	λ	λ	PROPN
ejpam-4571	354	22	,	,	PUNCT
ejpam-4571	354	23	b	b	NOUN
ejpam-4571	354	24	)	)	PUNCT
ejpam-4571	354	25	.	.	PUNCT
ejpam-4571	355	1	thus	thus	ADV
ejpam-4571	355	2	,	,	PUNCT
ejpam-4571	355	3	f	f	PROPN
ejpam-4571	355	4	is	be	AUX
ejpam-4571	355	5	weakly	weakly	ADJ
ejpam-4571	355	6	(	(	PUNCT
ejpam-4571	355	7	λ	λ	X
ejpam-4571	355	8	,	,	PUNCT
ejpam-4571	355	9	b)-continuous	b)-continuous	ADJ
ejpam-4571	355	10	at	at	SCONJ
ejpam-4571	355	11	x	x	PART
ejpam-4571	355	12	∈	∈	PROPN
ejpam-4571	355	13	x.	x.	NOUN
ejpam-4571	355	14	theorem	theorem	VERB
ejpam-4571	355	15	8	8	NUM
ejpam-4571	355	16	.	.	PUNCT
ejpam-4571	356	1	a	a	DET
ejpam-4571	356	2	function	function	NOUN
ejpam-4571	356	3	f	f	NOUN
ejpam-4571	356	4	:	:	PUNCT
ejpam-4571	356	5	(	(	PUNCT
ejpam-4571	356	6	x	x	X
ejpam-4571	356	7	,	,	PUNCT
ejpam-4571	356	8	τ	τ	X
ejpam-4571	356	9	)	)	PUNCT
ejpam-4571	356	10	→	→	SYM
ejpam-4571	356	11	(	(	PUNCT
ejpam-4571	356	12	y	y	PROPN
ejpam-4571	356	13	,	,	PUNCT
ejpam-4571	356	14	σ	σ	PROPN
ejpam-4571	356	15	)	)	PUNCT
ejpam-4571	356	16	is	be	AUX
ejpam-4571	356	17	weakly	weakly	ADJ
ejpam-4571	356	18	(	(	PUNCT
ejpam-4571	356	19	λ	λ	NOUN
ejpam-4571	356	20	,	,	PUNCT
ejpam-4571	356	21	b)-continuous	b)-continuous	ADJ
ejpam-4571	356	22	if	if	SCONJ
ejpam-4571	356	23	and	and	CCONJ
ejpam-4571	356	24	only	only	ADV
ejpam-4571	356	25	if	if	SCONJ
ejpam-4571	356	26	f−1(v	f−1(v	PROPN
ejpam-4571	356	27	)	)	PUNCT
ejpam-4571	357	1	⊆	⊆	NUM
ejpam-4571	358	1	[	[	X
ejpam-4571	358	2	f−1(v	f−1(v	NOUN
ejpam-4571	358	3	(	(	PUNCT
ejpam-4571	358	4	λ	λ	PROPN
ejpam-4571	358	5	,	,	PUNCT
ejpam-4571	358	6	b))](λ	b))](λ	PROPN
ejpam-4571	358	7	,	,	PUNCT
ejpam-4571	358	8	b	b	NOUN
ejpam-4571	358	9	)	)	PUNCT
ejpam-4571	358	10	for	for	ADP
ejpam-4571	358	11	every	every	DET
ejpam-4571	358	12	(	(	PUNCT
ejpam-4571	358	13	λ	λ	NOUN
ejpam-4571	358	14	,	,	PUNCT
ejpam-4571	358	15	b)-open	b)-open	AUX
ejpam-4571	358	16	set	set	VERB
ejpam-4571	358	17	v	v	NOUN
ejpam-4571	358	18	of	of	ADP
ejpam-4571	358	19	y	y	PROPN
ejpam-4571	358	20	.	.	PUNCT
ejpam-4571	359	1	proof	proof	NOUN
ejpam-4571	359	2	.	.	PUNCT
ejpam-4571	360	1	let	let	VERB
ejpam-4571	360	2	v	v	PART
ejpam-4571	360	3	be	be	AUX
ejpam-4571	360	4	any	any	DET
ejpam-4571	360	5	(	(	PUNCT
ejpam-4571	360	6	λ	λ	NOUN
ejpam-4571	360	7	,	,	PUNCT
ejpam-4571	360	8	b)-open	b)-open	VERB
ejpam-4571	360	9	set	set	VERB
ejpam-4571	360	10	of	of	ADP
ejpam-4571	360	11	y	y	PROPN
ejpam-4571	360	12	and	and	CCONJ
ejpam-4571	360	13	x	x	PROPN
ejpam-4571	360	14	∈	∈	PROPN
ejpam-4571	360	15	f−1(v	f−1(v	NOUN
ejpam-4571	360	16	)	)	PUNCT
ejpam-4571	360	17	.	.	PUNCT
ejpam-4571	361	1	then	then	ADV
ejpam-4571	361	2	,	,	PUNCT
ejpam-4571	361	3	f(x	f(x	PROPN
ejpam-4571	361	4	)	)	PUNCT
ejpam-4571	361	5	∈	∈	PROPN
ejpam-4571	361	6	v	v	NOUN
ejpam-4571	361	7	.	.	PUNCT
ejpam-4571	362	1	since	since	SCONJ
ejpam-4571	362	2	f	f	PROPN
ejpam-4571	362	3	is	be	AUX
ejpam-4571	362	4	weakly	weakly	ADJ
ejpam-4571	362	5	(	(	PUNCT
ejpam-4571	362	6	λ	λ	X
ejpam-4571	362	7	,	,	PUNCT
ejpam-4571	362	8	b)-continuous	b)-continuous	ADJ
ejpam-4571	362	9	at	at	ADP
ejpam-4571	362	10	x	x	X
ejpam-4571	362	11	and	and	CCONJ
ejpam-4571	362	12	by	by	ADP
ejpam-4571	362	13	theorem	theorem	NOUN
ejpam-4571	362	14	7	7	NUM
ejpam-4571	362	15	,	,	PUNCT
ejpam-4571	362	16	x	x	SYM
ejpam-4571	362	17	∈	∈	PROPN
ejpam-4571	363	1	[	[	X
ejpam-4571	363	2	f−1(v	f−1(v	NOUN
ejpam-4571	363	3	(	(	PUNCT
ejpam-4571	363	4	λ	λ	PROPN
ejpam-4571	363	5	,	,	PUNCT
ejpam-4571	363	6	b))](λ	b))](λ	PROPN
ejpam-4571	363	7	,	,	PUNCT
ejpam-4571	363	8	b	b	NOUN
ejpam-4571	363	9	)	)	PUNCT
ejpam-4571	363	10	.	.	PUNCT
ejpam-4571	364	1	therefore	therefore	ADV
ejpam-4571	364	2	,	,	PUNCT
ejpam-4571	364	3	f−1(v	f−1(v	PROPN
ejpam-4571	364	4	)	)	PUNCT
ejpam-4571	365	1	⊆	⊆	NUM
ejpam-4571	365	2	[	[	X
ejpam-4571	365	3	f−1(v	f−1(v	NOUN
ejpam-4571	365	4	(	(	PUNCT
ejpam-4571	365	5	λ	λ	PROPN
ejpam-4571	365	6	,	,	PUNCT
ejpam-4571	365	7	b))](λ	b))](λ	PROPN
ejpam-4571	365	8	,	,	PUNCT
ejpam-4571	365	9	b	b	NOUN
ejpam-4571	365	10	)	)	PUNCT
ejpam-4571	365	11	.	.	PUNCT
ejpam-4571	366	1	conversely	conversely	ADV
ejpam-4571	366	2	,	,	PUNCT
ejpam-4571	366	3	let	let	VERB
ejpam-4571	366	4	x	x	X
ejpam-4571	366	5	∈	∈	PROPN
ejpam-4571	366	6	x	x	X
ejpam-4571	366	7	and	and	CCONJ
ejpam-4571	366	8	v	v	AUX
ejpam-4571	366	9	be	be	AUX
ejpam-4571	366	10	any	any	DET
ejpam-4571	366	11	(	(	PUNCT
ejpam-4571	366	12	λ	λ	NOUN
ejpam-4571	366	13	,	,	PUNCT
ejpam-4571	366	14	b)-open	b)-open	VERB
ejpam-4571	366	15	set	set	VERB
ejpam-4571	366	16	of	of	ADP
ejpam-4571	366	17	y	y	PROPN
ejpam-4571	366	18	containing	contain	VERB
ejpam-4571	366	19	f(x	f(x	PROPN
ejpam-4571	366	20	)	)	PUNCT
ejpam-4571	366	21	.	.	PUNCT
ejpam-4571	367	1	then	then	ADV
ejpam-4571	367	2	,	,	PUNCT
ejpam-4571	367	3	x	x	PROPN
ejpam-4571	367	4	∈	∈	PROPN
ejpam-4571	367	5	f−1(v	f−1(v	NOUN
ejpam-4571	367	6	)	)	PUNCT
ejpam-4571	368	1	⊆	⊆	NUM
ejpam-4571	368	2	[	[	X
ejpam-4571	368	3	f−1(v	f−1(v	NOUN
ejpam-4571	368	4	(	(	PUNCT
ejpam-4571	368	5	λ	λ	PROPN
ejpam-4571	368	6	,	,	PUNCT
ejpam-4571	368	7	b))](λ	b))](λ	PROPN
ejpam-4571	368	8	,	,	PUNCT
ejpam-4571	368	9	b	b	NOUN
ejpam-4571	368	10	)	)	PUNCT
ejpam-4571	368	11	and	and	CCONJ
ejpam-4571	368	12	hence	hence	ADV
ejpam-4571	368	13	x	x	X
ejpam-4571	368	14	∈	∈	PROPN
ejpam-4571	368	15	[	[	X
ejpam-4571	368	16	f−1(v	f−1(v	NOUN
ejpam-4571	368	17	(	(	PUNCT
ejpam-4571	368	18	λ	λ	PROPN
ejpam-4571	368	19	,	,	PUNCT
ejpam-4571	368	20	b))](λ	b))](λ	PROPN
ejpam-4571	368	21	,	,	PUNCT
ejpam-4571	368	22	b	b	NOUN
ejpam-4571	368	23	)	)	PUNCT
ejpam-4571	368	24	.	.	PUNCT
ejpam-4571	369	1	by	by	ADP
ejpam-4571	369	2	theorem	theorem	NOUN
ejpam-4571	369	3	7	7	NUM
ejpam-4571	369	4	,	,	PUNCT
ejpam-4571	369	5	f	f	PROPN
ejpam-4571	369	6	is	be	AUX
ejpam-4571	369	7	weakly	weakly	ADJ
ejpam-4571	369	8	(	(	PUNCT
ejpam-4571	369	9	λ	λ	NOUN
ejpam-4571	369	10	,	,	PUNCT
ejpam-4571	369	11	b)-continuous	b)-continuous	PROPN
ejpam-4571	369	12	.	.	PUNCT
ejpam-4571	370	1	theorem	theorem	VERB
ejpam-4571	370	2	9	9	NUM
ejpam-4571	370	3	.	.	PUNCT
ejpam-4571	371	1	a	a	DET
ejpam-4571	371	2	function	function	NOUN
ejpam-4571	371	3	f	f	NOUN
ejpam-4571	371	4	:	:	PUNCT
ejpam-4571	371	5	(	(	PUNCT
ejpam-4571	371	6	x	x	X
ejpam-4571	371	7	,	,	PUNCT
ejpam-4571	371	8	τ	τ	X
ejpam-4571	371	9	)	)	PUNCT
ejpam-4571	371	10	→	→	SYM
ejpam-4571	371	11	(	(	PUNCT
ejpam-4571	371	12	y	y	PROPN
ejpam-4571	371	13	,	,	PUNCT
ejpam-4571	371	14	σ	σ	PROPN
ejpam-4571	371	15	)	)	PUNCT
ejpam-4571	371	16	is	be	AUX
ejpam-4571	371	17	weakly	weakly	ADJ
ejpam-4571	371	18	(	(	PUNCT
ejpam-4571	371	19	λ	λ	NOUN
ejpam-4571	371	20	,	,	PUNCT
ejpam-4571	371	21	b)-continuous	b)-continuous	ADJ
ejpam-4571	371	22	if	if	SCONJ
ejpam-4571	371	23	and	and	CCONJ
ejpam-4571	371	24	only	only	ADV
ejpam-4571	371	25	if	if	SCONJ
ejpam-4571	371	26	[	[	X
ejpam-4571	371	27	f−1(v	f−1(v	NOUN
ejpam-4571	371	28	)	)	PUNCT
ejpam-4571	371	29	]	]	PUNCT
ejpam-4571	371	30	(	(	PUNCT
ejpam-4571	371	31	λ	λ	NOUN
ejpam-4571	371	32	,	,	PUNCT
ejpam-4571	371	33	b	b	NOUN
ejpam-4571	371	34	)	)	PUNCT
ejpam-4571	371	35	⊆	⊆	NUM
ejpam-4571	371	36	f−1(v	f−1(v	NOUN
ejpam-4571	371	37	(	(	PUNCT
ejpam-4571	371	38	λ	λ	PROPN
ejpam-4571	371	39	,	,	PUNCT
ejpam-4571	371	40	b	b	NOUN
ejpam-4571	371	41	)	)	PUNCT
ejpam-4571	371	42	)	)	PUNCT
ejpam-4571	371	43	for	for	ADP
ejpam-4571	371	44	every	every	DET
ejpam-4571	371	45	(	(	PUNCT
ejpam-4571	371	46	λ	λ	NOUN
ejpam-4571	371	47	,	,	PUNCT
ejpam-4571	371	48	b)-open	b)-open	AUX
ejpam-4571	371	49	set	set	VERB
ejpam-4571	371	50	v	v	NOUN
ejpam-4571	371	51	of	of	ADP
ejpam-4571	371	52	y	y	PROPN
ejpam-4571	371	53	.	.	PUNCT
ejpam-4571	372	1	proof	proof	NOUN
ejpam-4571	372	2	.	.	PUNCT
ejpam-4571	373	1	this	this	PRON
ejpam-4571	373	2	is	be	AUX
ejpam-4571	373	3	obvious	obvious	ADJ
ejpam-4571	373	4	from	from	ADP
ejpam-4571	373	5	theorem	theorem	ADJ
ejpam-4571	373	6	8	8	NUM
ejpam-4571	373	7	.	.	PUNCT
ejpam-4571	374	1	conversely	conversely	ADV
ejpam-4571	374	2	,	,	PUNCT
ejpam-4571	374	3	let	let	VERB
ejpam-4571	374	4	v	v	PART
ejpam-4571	374	5	be	be	AUX
ejpam-4571	374	6	any	any	DET
ejpam-4571	374	7	(	(	PUNCT
ejpam-4571	374	8	λ	λ	NOUN
ejpam-4571	374	9	,	,	PUNCT
ejpam-4571	374	10	b)-open	b)-open	VERB
ejpam-4571	374	11	set	set	VERB
ejpam-4571	374	12	of	of	ADP
ejpam-4571	374	13	y	y	PROPN
ejpam-4571	374	14	containing	contain	VERB
ejpam-4571	374	15	f(x	f(x	PROPN
ejpam-4571	374	16	)	)	PUNCT
ejpam-4571	374	17	.	.	PUNCT
ejpam-4571	375	1	since	since	SCONJ
ejpam-4571	375	2	v	v	NUM
ejpam-4571	375	3	∩(y−v	∩(y−v	PROPN
ejpam-4571	375	4	(	(	PUNCT
ejpam-4571	375	5	λ	λ	PROPN
ejpam-4571	375	6	,	,	PUNCT
ejpam-4571	375	7	b	b	NOUN
ejpam-4571	375	8	)	)	PUNCT
ejpam-4571	375	9	)	)	PUNCT
ejpam-4571	376	1	=	=	NOUN
ejpam-4571	376	2	∅	∅	NOUN
ejpam-4571	376	3	,	,	PUNCT
ejpam-4571	376	4	f(x	f(x	PROPN
ejpam-4571	376	5	)	)	PUNCT
ejpam-4571	376	6	̸∈	̸∈	PROPN
ejpam-4571	377	1	[	[	X
ejpam-4571	377	2	y	y	PROPN
ejpam-4571	377	3	−	−	PROPN
ejpam-4571	377	4	v	v	PROPN
ejpam-4571	377	5	(	(	PUNCT
ejpam-4571	377	6	λ	λ	PROPN
ejpam-4571	377	7	,	,	PUNCT
ejpam-4571	377	8	b)](λ	b)](λ	NOUN
ejpam-4571	377	9	,	,	PUNCT
ejpam-4571	377	10	b	b	NOUN
ejpam-4571	377	11	)	)	PUNCT
ejpam-4571	377	12	.	.	PUNCT
ejpam-4571	378	1	since	since	SCONJ
ejpam-4571	378	2	y	y	PROPN
ejpam-4571	378	3	−	−	PROPN
ejpam-4571	378	4	v	v	PROPN
ejpam-4571	378	5	(	(	PUNCT
ejpam-4571	378	6	λ	λ	PROPN
ejpam-4571	378	7	,	,	PUNCT
ejpam-4571	378	8	b	b	NOUN
ejpam-4571	378	9	)	)	PUNCT
ejpam-4571	378	10	is	be	AUX
ejpam-4571	378	11	(	(	PUNCT
ejpam-4571	378	12	λ	λ	X
ejpam-4571	378	13	,	,	PUNCT
ejpam-4571	378	14	b)-open	b)-open	PUNCT
ejpam-4571	378	15	and	and	CCONJ
ejpam-4571	378	16	x	x	X
ejpam-4571	378	17	∈	∈	NOUN
ejpam-4571	378	18	f−1([y	f−1([y	NOUN
ejpam-4571	378	19	−	−	NOUN
ejpam-4571	378	20	v	v	NOUN
ejpam-4571	378	21	(	(	PUNCT
ejpam-4571	378	22	λ	λ	PROPN
ejpam-4571	378	23	,	,	PUNCT
ejpam-4571	378	24	b)](λ	b)](λ	NOUN
ejpam-4571	378	25	,	,	PUNCT
ejpam-4571	378	26	b	b	NOUN
ejpam-4571	378	27	)	)	PUNCT
ejpam-4571	378	28	)	)	PUNCT
ejpam-4571	378	29	.	.	PUNCT
ejpam-4571	379	1	by	by	ADP
ejpam-4571	379	2	the	the	DET
ejpam-4571	379	3	hypothesis	hypothesis	NOUN
ejpam-4571	379	4	,	,	PUNCT
ejpam-4571	379	5	x	x	PROPN
ejpam-4571	379	6	̸∈	̸∈	PROPN
ejpam-4571	379	7	[	[	X
ejpam-4571	379	8	f−1([y	f−1([y	NOUN
ejpam-4571	379	9	−	−	NOUN
ejpam-4571	379	10	v	v	NOUN
ejpam-4571	379	11	(	(	PUNCT
ejpam-4571	379	12	λ	λ	PROPN
ejpam-4571	379	13	,	,	PUNCT
ejpam-4571	379	14	b)](λ	b)](λ	PROPN
ejpam-4571	379	15	,	,	PUNCT
ejpam-4571	379	16	b))](λ	b))](λ	PROPN
ejpam-4571	379	17	,	,	PUNCT
ejpam-4571	379	18	b	b	NOUN
ejpam-4571	379	19	)	)	PUNCT
ejpam-4571	379	20	=	=	NOUN
ejpam-4571	380	1	[	[	X
ejpam-4571	380	2	x	x	X
ejpam-4571	380	3	−	−	X
ejpam-4571	380	4	f−1(v	f−1(v	NOUN
ejpam-4571	380	5	(	(	PUNCT
ejpam-4571	380	6	λ	λ	PROPN
ejpam-4571	380	7	,	,	PUNCT
ejpam-4571	380	8	b))](λ	b))](λ	PROPN
ejpam-4571	380	9	,	,	PUNCT
ejpam-4571	380	10	b	b	NOUN
ejpam-4571	380	11	)	)	PUNCT
ejpam-4571	380	12	and	and	CCONJ
ejpam-4571	380	13	there	there	PRON
ejpam-4571	380	14	exists	exist	VERB
ejpam-4571	380	15	a	a	DET
ejpam-4571	380	16	(	(	PUNCT
ejpam-4571	380	17	λ	λ	NOUN
ejpam-4571	380	18	,	,	PUNCT
ejpam-4571	380	19	b)-open	b)-open	VERB
ejpam-4571	380	20	set	set	VERB
ejpam-4571	380	21	u	u	NOUN
ejpam-4571	380	22	of	of	ADP
ejpam-4571	380	23	x	x	PUNCT
ejpam-4571	380	24	containing	contain	VERB
ejpam-4571	380	25	x	x	PUNCT
ejpam-4571	380	26	such	such	ADJ
ejpam-4571	380	27	that	that	SCONJ
ejpam-4571	380	28	u	u	PROPN
ejpam-4571	380	29	∩	∩	NOUN
ejpam-4571	380	30	(	(	PUNCT
ejpam-4571	380	31	x−	x−	PROPN
ejpam-4571	380	32	f−1(v	f−1(v	PROPN
ejpam-4571	380	33	(	(	PUNCT
ejpam-4571	380	34	λ	λ	PROPN
ejpam-4571	380	35	,	,	PUNCT
ejpam-4571	380	36	b	b	NOUN
ejpam-4571	380	37	)	)	PUNCT
ejpam-4571	380	38	)	)	PUNCT
ejpam-4571	380	39	)	)	PUNCT
ejpam-4571	381	1	=	=	PUNCT
ejpam-4571	381	2	∅.	∅.	ADP
ejpam-4571	381	3	this	this	PRON
ejpam-4571	381	4	shows	show	VERB
ejpam-4571	381	5	that	that	SCONJ
ejpam-4571	381	6	f(u	f(u	PROPN
ejpam-4571	381	7	)	)	PUNCT
ejpam-4571	381	8	⊆	⊆	NUM
ejpam-4571	381	9	v	v	NOUN
ejpam-4571	381	10	(	(	PUNCT
ejpam-4571	381	11	λ	λ	PROPN
ejpam-4571	381	12	,	,	PUNCT
ejpam-4571	381	13	b	b	NOUN
ejpam-4571	381	14	)	)	PUNCT
ejpam-4571	381	15	.	.	PUNCT
ejpam-4571	382	1	thus	thus	ADV
ejpam-4571	382	2	,	,	PUNCT
ejpam-4571	382	3	f	f	PROPN
ejpam-4571	382	4	is	be	AUX
ejpam-4571	382	5	is	be	AUX
ejpam-4571	382	6	weakly	weakly	ADJ
ejpam-4571	382	7	(	(	PUNCT
ejpam-4571	382	8	λ	λ	NOUN
ejpam-4571	382	9	,	,	PUNCT
ejpam-4571	382	10	b)-continuous	b)-continuous	PROPN
ejpam-4571	382	11	.	.	PUNCT
ejpam-4571	383	1	theorem	theorem	VERB
ejpam-4571	383	2	10	10	NUM
ejpam-4571	383	3	.	.	PUNCT
ejpam-4571	384	1	for	for	ADP
ejpam-4571	384	2	a	a	DET
ejpam-4571	384	3	function	function	NOUN
ejpam-4571	384	4	f	f	NOUN
ejpam-4571	384	5	:	:	PUNCT
ejpam-4571	384	6	(	(	PUNCT
ejpam-4571	384	7	x	x	X
ejpam-4571	384	8	,	,	PUNCT
ejpam-4571	384	9	τ	τ	X
ejpam-4571	384	10	)	)	PUNCT
ejpam-4571	384	11	→	→	SYM
ejpam-4571	384	12	(	(	PUNCT
ejpam-4571	384	13	y	y	PROPN
ejpam-4571	384	14	,	,	PUNCT
ejpam-4571	384	15	σ	σ	PROPN
ejpam-4571	384	16	)	)	PUNCT
ejpam-4571	384	17	,	,	PUNCT
ejpam-4571	384	18	the	the	DET
ejpam-4571	384	19	following	follow	VERB
ejpam-4571	384	20	properties	property	NOUN
ejpam-4571	384	21	are	be	AUX
ejpam-4571	384	22	equivalent	equivalent	ADJ
ejpam-4571	384	23	:	:	PUNCT
ejpam-4571	384	24	(	(	PUNCT
ejpam-4571	384	25	1	1	X
ejpam-4571	384	26	)	)	PUNCT
ejpam-4571	384	27	f	f	PROPN
ejpam-4571	384	28	is	be	AUX
ejpam-4571	384	29	weakly	weakly	ADJ
ejpam-4571	384	30	(	(	PUNCT
ejpam-4571	384	31	λ	λ	X
ejpam-4571	384	32	,	,	PUNCT
ejpam-4571	384	33	b)-continuous	b)-continuous	ADJ
ejpam-4571	384	34	;	;	PUNCT
ejpam-4571	384	35	(	(	PUNCT
ejpam-4571	384	36	2	2	X
ejpam-4571	384	37	)	)	PUNCT
ejpam-4571	384	38	f−1(u	f−1(u	NOUN
ejpam-4571	384	39	)	)	PUNCT
ejpam-4571	384	40	⊆	⊆	NUM
ejpam-4571	385	1	[	[	X
ejpam-4571	385	2	f−1(u	f−1(u	PROPN
ejpam-4571	385	3	(	(	PUNCT
ejpam-4571	385	4	λ	λ	PROPN
ejpam-4571	385	5	,	,	PUNCT
ejpam-4571	385	6	b))](λ	b))](λ	PROPN
ejpam-4571	385	7	,	,	PUNCT
ejpam-4571	385	8	b	b	NOUN
ejpam-4571	385	9	)	)	PUNCT
ejpam-4571	385	10	for	for	ADP
ejpam-4571	385	11	every	every	DET
ejpam-4571	385	12	(	(	PUNCT
ejpam-4571	385	13	λ	λ	NOUN
ejpam-4571	385	14	,	,	PUNCT
ejpam-4571	385	15	b)-open	b)-open	AUX
ejpam-4571	385	16	set	set	VERB
ejpam-4571	385	17	u	u	NOUN
ejpam-4571	385	18	of	of	ADP
ejpam-4571	385	19	y	y	PROPN
ejpam-4571	385	20	;	;	PUNCT
ejpam-4571	385	21	(	(	PUNCT
ejpam-4571	385	22	3	3	X
ejpam-4571	385	23	)	)	PUNCT
ejpam-4571	385	24	[	[	X
ejpam-4571	385	25	f−1(f(λ	f−1(f(λ	X
ejpam-4571	385	26	,	,	PUNCT
ejpam-4571	385	27	b	b	NOUN
ejpam-4571	385	28	)	)	PUNCT
ejpam-4571	385	29	)	)	PUNCT
ejpam-4571	385	30	]	]	PUNCT
ejpam-4571	386	1	(	(	PUNCT
ejpam-4571	386	2	λ	λ	X
ejpam-4571	386	3	,	,	PUNCT
ejpam-4571	386	4	b	b	NOUN
ejpam-4571	386	5	)	)	PUNCT
ejpam-4571	386	6	⊆	⊆	NUM
ejpam-4571	386	7	f−1(f	f−1(f	PROPN
ejpam-4571	386	8	)	)	PUNCT
ejpam-4571	386	9	for	for	SCONJ
ejpam-4571	386	10	every	every	DET
ejpam-4571	386	11	(	(	PUNCT
ejpam-4571	386	12	λ	λ	PROPN
ejpam-4571	386	13	,	,	PUNCT
ejpam-4571	386	14	b)-closed	b)-close	VERB
ejpam-4571	386	15	set	set	ADJ
ejpam-4571	386	16	f	f	PROPN
ejpam-4571	386	17	of	of	ADP
ejpam-4571	386	18	y	y	PROPN
ejpam-4571	386	19	;	;	PUNCT
ejpam-4571	386	20	(	(	PUNCT
ejpam-4571	386	21	4	4	X
ejpam-4571	386	22	)	)	PUNCT
ejpam-4571	387	1	[	[	X
ejpam-4571	387	2	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4571	387	3	,	,	PUNCT
ejpam-4571	387	4	b)](λ	b)](λ	NOUN
ejpam-4571	387	5	,	,	PUNCT
ejpam-4571	387	6	b	b	NOUN
ejpam-4571	387	7	)	)	PUNCT
ejpam-4571	387	8	)	)	PUNCT
ejpam-4571	387	9	]	]	PUNCT
ejpam-4571	388	1	(	(	PUNCT
ejpam-4571	388	2	λ	λ	X
ejpam-4571	388	3	,	,	PUNCT
ejpam-4571	388	4	b	b	NOUN
ejpam-4571	388	5	)	)	PUNCT
ejpam-4571	388	6	⊆	⊆	NUM
ejpam-4571	388	7	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4571	388	8	,	,	PUNCT
ejpam-4571	388	9	b	b	NOUN
ejpam-4571	388	10	)	)	PUNCT
ejpam-4571	388	11	)	)	PUNCT
ejpam-4571	388	12	for	for	ADP
ejpam-4571	388	13	every	every	DET
ejpam-4571	388	14	subset	subset	NOUN
ejpam-4571	388	15	a	a	PRON
ejpam-4571	388	16	of	of	ADP
ejpam-4571	388	17	y	y	PROPN
ejpam-4571	388	18	;	;	PUNCT
ejpam-4571	388	19	(	(	PUNCT
ejpam-4571	388	20	5	5	X
ejpam-4571	388	21	)	)	PUNCT
ejpam-4571	388	22	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4571	388	23	,	,	PUNCT
ejpam-4571	388	24	b	b	NOUN
ejpam-4571	388	25	)	)	PUNCT
ejpam-4571	388	26	)	)	PUNCT
ejpam-4571	389	1	⊆	⊆	NUM
ejpam-4571	389	2	[	[	X
ejpam-4571	389	3	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4571	389	4	,	,	PUNCT
ejpam-4571	389	5	b	b	NOUN
ejpam-4571	389	6	)	)	PUNCT
ejpam-4571	389	7	]	]	PUNCT
ejpam-4571	389	8	(	(	PUNCT
ejpam-4571	389	9	λ	λ	X
ejpam-4571	389	10	,	,	PUNCT
ejpam-4571	389	11	b))](λ	b))](λ	PROPN
ejpam-4571	389	12	,	,	PUNCT
ejpam-4571	389	13	b	b	NOUN
ejpam-4571	389	14	)	)	PUNCT
ejpam-4571	389	15	for	for	ADP
ejpam-4571	389	16	every	every	DET
ejpam-4571	389	17	subset	subset	NOUN
ejpam-4571	389	18	a	a	PRON
ejpam-4571	389	19	of	of	ADP
ejpam-4571	389	20	y	y	PROPN
ejpam-4571	389	21	;	;	PUNCT
ejpam-4571	389	22	c.	c.	PROPN
ejpam-4571	389	23	boonpok	boonpok	PROPN
ejpam-4571	389	24	,	,	PUNCT
ejpam-4571	389	25	n.	n.	PROPN
ejpam-4571	389	26	srisarakham	srisarakham	PROPN
ejpam-4571	389	27	/	/	SYM
ejpam-4571	389	28	eur	eur	PROPN
ejpam-4571	389	29	.	.	PUNCT
ejpam-4571	390	1	j.	j.	PROPN
ejpam-4571	390	2	pure	pure	PROPN
ejpam-4571	390	3	appl	appl	PROPN
ejpam-4571	390	4	.	.	PROPN
ejpam-4571	390	5	math	math	PROPN
ejpam-4571	390	6	,	,	PUNCT
ejpam-4571	390	7	16	16	NUM
ejpam-4571	390	8	(	(	PUNCT
ejpam-4571	390	9	1	1	NUM
ejpam-4571	390	10	)	)	PUNCT
ejpam-4571	390	11	(	(	PUNCT
ejpam-4571	390	12	2023	2023	NUM
ejpam-4571	390	13	)	)	PUNCT
ejpam-4571	390	14	,	,	PUNCT
ejpam-4571	390	15	29	29	NUM
ejpam-4571	390	16	-	-	SYM
ejpam-4571	390	17	43	43	NUM
ejpam-4571	390	18	39	39	NUM
ejpam-4571	390	19	(	(	PUNCT
ejpam-4571	390	20	6	6	NUM
ejpam-4571	390	21	)	)	PUNCT
ejpam-4571	391	1	[	[	X
ejpam-4571	391	2	f−1(u)](λ	f−1(u)](λ	PROPN
ejpam-4571	391	3	,	,	PUNCT
ejpam-4571	391	4	b	b	NOUN
ejpam-4571	391	5	)	)	PUNCT
ejpam-4571	392	1	⊆	⊆	NUM
ejpam-4571	392	2	f−1(u	f−1(u	NOUN
ejpam-4571	392	3	(	(	PUNCT
ejpam-4571	392	4	λ	λ	PROPN
ejpam-4571	392	5	,	,	PUNCT
ejpam-4571	392	6	b	b	NOUN
ejpam-4571	392	7	)	)	PUNCT
ejpam-4571	392	8	)	)	PUNCT
ejpam-4571	392	9	for	for	ADP
ejpam-4571	392	10	every	every	DET
ejpam-4571	392	11	(	(	PUNCT
ejpam-4571	392	12	λ	λ	NOUN
ejpam-4571	392	13	,	,	PUNCT
ejpam-4571	392	14	b)-open	b)-open	AUX
ejpam-4571	392	15	set	set	VERB
ejpam-4571	392	16	u	u	NOUN
ejpam-4571	392	17	of	of	ADP
ejpam-4571	392	18	y	y	PROPN
ejpam-4571	392	19	.	.	PUNCT
ejpam-4571	393	1	proof	proof	NOUN
ejpam-4571	393	2	.	.	PUNCT
ejpam-4571	394	1	(	(	PUNCT
ejpam-4571	394	2	1	1	X
ejpam-4571	394	3	)	)	PUNCT
ejpam-4571	394	4	⇒	⇒	NOUN
ejpam-4571	394	5	(	(	PUNCT
ejpam-4571	394	6	2	2	NUM
ejpam-4571	394	7	):	):	PUNCT
ejpam-4571	394	8	let	let	VERB
ejpam-4571	394	9	u	u	PRON
ejpam-4571	394	10	be	be	AUX
ejpam-4571	394	11	any	any	DET
ejpam-4571	394	12	(	(	PUNCT
ejpam-4571	394	13	λ	λ	NOUN
ejpam-4571	394	14	,	,	PUNCT
ejpam-4571	394	15	b)-open	b)-open	VERB
ejpam-4571	394	16	set	set	VERB
ejpam-4571	394	17	of	of	ADP
ejpam-4571	394	18	y	y	PROPN
ejpam-4571	394	19	and	and	CCONJ
ejpam-4571	394	20	x	x	PROPN
ejpam-4571	394	21	∈	∈	PROPN
ejpam-4571	394	22	f−1(u	f−1(u	PROPN
ejpam-4571	394	23	)	)	PUNCT
ejpam-4571	394	24	.	.	PUNCT
ejpam-4571	395	1	then	then	ADV
ejpam-4571	395	2	,	,	PUNCT
ejpam-4571	395	3	there	there	PRON
ejpam-4571	395	4	exists	exist	VERB
ejpam-4571	395	5	a	a	DET
ejpam-4571	395	6	(	(	PUNCT
ejpam-4571	395	7	λ	λ	NOUN
ejpam-4571	395	8	,	,	PUNCT
ejpam-4571	395	9	b)-open	b)-open	VERB
ejpam-4571	395	10	set	set	VERB
ejpam-4571	395	11	v	v	NOUN
ejpam-4571	395	12	of	of	ADP
ejpam-4571	395	13	x	x	PUNCT
ejpam-4571	395	14	containing	contain	VERB
ejpam-4571	395	15	x	x	PUNCT
ejpam-4571	395	16	such	such	ADJ
ejpam-4571	395	17	that	that	SCONJ
ejpam-4571	395	18	f(v	f(v	PROPN
ejpam-4571	395	19	)	)	PUNCT
ejpam-4571	396	1	⊆	⊆	NUM
ejpam-4571	396	2	u	u	SYM
ejpam-4571	396	3	(	(	PUNCT
ejpam-4571	396	4	λ	λ	PROPN
ejpam-4571	396	5	,	,	PUNCT
ejpam-4571	396	6	b	b	NOUN
ejpam-4571	396	7	)	)	PUNCT
ejpam-4571	396	8	.	.	PUNCT
ejpam-4571	397	1	since	since	SCONJ
ejpam-4571	397	2	x	x	PROPN
ejpam-4571	397	3	∈	∈	PROPN
ejpam-4571	397	4	v	v	ADP
ejpam-4571	397	5	⊆	⊆	NUM
ejpam-4571	397	6	f−1(u	f−1(u	NOUN
ejpam-4571	397	7	(	(	PUNCT
ejpam-4571	397	8	λ	λ	PROPN
ejpam-4571	397	9	,	,	PUNCT
ejpam-4571	397	10	b	b	NOUN
ejpam-4571	397	11	)	)	PUNCT
ejpam-4571	397	12	)	)	PUNCT
ejpam-4571	397	13	,	,	PUNCT
ejpam-4571	397	14	x	x	PUNCT
ejpam-4571	397	15	∈	∈	PROPN
ejpam-4571	397	16	[	[	X
ejpam-4571	397	17	f−1(u	f−1(u	X
ejpam-4571	397	18	(	(	PUNCT
ejpam-4571	397	19	λ	λ	PROPN
ejpam-4571	397	20	,	,	PUNCT
ejpam-4571	397	21	b))](λ	b))](λ	PROPN
ejpam-4571	397	22	,	,	PUNCT
ejpam-4571	397	23	b	b	NOUN
ejpam-4571	397	24	)	)	PUNCT
ejpam-4571	397	25	.	.	PUNCT
ejpam-4571	398	1	thus	thus	ADV
ejpam-4571	398	2	,	,	PUNCT
ejpam-4571	398	3	f	f	PROPN
ejpam-4571	398	4	−1(u	−1(u	X
ejpam-4571	398	5	)	)	PUNCT
ejpam-4571	398	6	⊆	⊆	NUM
ejpam-4571	398	7	[	[	X
ejpam-4571	398	8	f−1(u	f−1(u	PROPN
ejpam-4571	398	9	(	(	PUNCT
ejpam-4571	398	10	λ	λ	PROPN
ejpam-4571	398	11	,	,	PUNCT
ejpam-4571	398	12	b))](λ	b))](λ	PROPN
ejpam-4571	398	13	,	,	PUNCT
ejpam-4571	398	14	b	b	NOUN
ejpam-4571	398	15	)	)	PUNCT
ejpam-4571	398	16	.	.	PUNCT
ejpam-4571	399	1	(	(	PUNCT
ejpam-4571	399	2	2	2	X
ejpam-4571	399	3	)	)	PUNCT
ejpam-4571	399	4	⇒	⇒	NOUN
ejpam-4571	399	5	(	(	PUNCT
ejpam-4571	399	6	3	3	NUM
ejpam-4571	399	7	):	):	PUNCT
ejpam-4571	399	8	let	let	VERB
ejpam-4571	399	9	f	f	PRON
ejpam-4571	399	10	be	be	AUX
ejpam-4571	399	11	any	any	DET
ejpam-4571	399	12	(	(	PUNCT
ejpam-4571	399	13	λ	λ	PROPN
ejpam-4571	399	14	,	,	PUNCT
ejpam-4571	399	15	b)-closed	b)-close	VERB
ejpam-4571	399	16	set	set	NOUN
ejpam-4571	399	17	of	of	ADP
ejpam-4571	399	18	y	y	PROPN
ejpam-4571	399	19	.	.	PUNCT
ejpam-4571	400	1	then	then	ADV
ejpam-4571	400	2	,	,	PUNCT
ejpam-4571	400	3	y	y	PROPN
ejpam-4571	400	4	−f	−f	PROPN
ejpam-4571	400	5	is	be	AUX
ejpam-4571	400	6	(	(	PUNCT
ejpam-4571	400	7	λ	λ	X
ejpam-4571	400	8	,	,	PUNCT
ejpam-4571	400	9	b)-open	b)-open	PUNCT
ejpam-4571	400	10	and	and	CCONJ
ejpam-4571	400	11	by	by	ADP
ejpam-4571	400	12	(	(	PUNCT
ejpam-4571	400	13	2	2	NUM
ejpam-4571	400	14	)	)	PUNCT
ejpam-4571	400	15	,	,	PUNCT
ejpam-4571	400	16	f−1(x	f−1(x	NOUN
ejpam-4571	400	17	−	−	PROPN
ejpam-4571	400	18	f	f	PROPN
ejpam-4571	400	19	)	)	PUNCT
ejpam-4571	400	20	⊆	⊆	NUM
ejpam-4571	401	1	[	[	X
ejpam-4571	401	2	f−1([x	f−1([x	NOUN
ejpam-4571	401	3	−	−	PROPN
ejpam-4571	401	4	f	f	X
ejpam-4571	401	5	]	]	X
ejpam-4571	401	6	(	(	PUNCT
ejpam-4571	401	7	λ	λ	PROPN
ejpam-4571	401	8	,	,	PUNCT
ejpam-4571	401	9	b))](λ	b))](λ	PROPN
ejpam-4571	401	10	,	,	PUNCT
ejpam-4571	401	11	b	b	NOUN
ejpam-4571	401	12	)	)	PUNCT
ejpam-4571	401	13	=	=	NOUN
ejpam-4571	402	1	[	[	X
ejpam-4571	402	2	f−1(y	f−1(y	NOUN
ejpam-4571	402	3	−	−	PROPN
ejpam-4571	402	4	f(λ	f(λ	PROPN
ejpam-4571	402	5	,	,	PUNCT
ejpam-4571	402	6	b))](λ	b))](λ	PROPN
ejpam-4571	402	7	,	,	PUNCT
ejpam-4571	402	8	b	b	NOUN
ejpam-4571	402	9	)	)	PUNCT
ejpam-4571	402	10	=	=	PUNCT
ejpam-4571	402	11	x	x	X
ejpam-4571	402	12	−	−	PROPN
ejpam-4571	403	1	[	[	X
ejpam-4571	403	2	f−1(f(λ	f−1(f(λ	X
ejpam-4571	403	3	,	,	PUNCT
ejpam-4571	403	4	b	b	NOUN
ejpam-4571	403	5	)	)	PUNCT
ejpam-4571	403	6	)	)	PUNCT
ejpam-4571	403	7	]	]	PUNCT
ejpam-4571	403	8	(	(	PUNCT
ejpam-4571	403	9	λ	λ	X
ejpam-4571	403	10	,	,	PUNCT
ejpam-4571	403	11	b	b	NOUN
ejpam-4571	403	12	)	)	PUNCT
ejpam-4571	403	13	.	.	PUNCT
ejpam-4571	404	1	therefore	therefore	ADV
ejpam-4571	404	2	,	,	PUNCT
ejpam-4571	404	3	[	[	X
ejpam-4571	404	4	f−1(f(λ	f−1(f(λ	X
ejpam-4571	404	5	,	,	PUNCT
ejpam-4571	404	6	b	b	NOUN
ejpam-4571	404	7	)	)	PUNCT
ejpam-4571	404	8	)	)	PUNCT
ejpam-4571	404	9	]	]	PUNCT
ejpam-4571	404	10	(	(	PUNCT
ejpam-4571	404	11	λ	λ	X
ejpam-4571	404	12	,	,	PUNCT
ejpam-4571	404	13	b	b	NOUN
ejpam-4571	404	14	)	)	PUNCT
ejpam-4571	404	15	⊆	⊆	NUM
ejpam-4571	404	16	f−1(f	f−1(f	PROPN
ejpam-4571	404	17	)	)	PUNCT
ejpam-4571	404	18	.	.	PUNCT
ejpam-4571	405	1	(	(	PUNCT
ejpam-4571	405	2	3	3	X
ejpam-4571	405	3	)	)	PUNCT
ejpam-4571	405	4	⇒	⇒	NOUN
ejpam-4571	405	5	(	(	PUNCT
ejpam-4571	405	6	4	4	NUM
ejpam-4571	405	7	):	):	PUNCT
ejpam-4571	405	8	let	let	VERB
ejpam-4571	405	9	a	a	DET
ejpam-4571	405	10	be	be	AUX
ejpam-4571	405	11	any	any	DET
ejpam-4571	405	12	subset	subset	NOUN
ejpam-4571	405	13	of	of	ADP
ejpam-4571	405	14	y	y	PROPN
ejpam-4571	405	15	.	.	PUNCT
ejpam-4571	406	1	since	since	SCONJ
ejpam-4571	406	2	a(λ	a(λ	PROPN
ejpam-4571	406	3	,	,	PUNCT
ejpam-4571	406	4	b	b	NOUN
ejpam-4571	406	5	)	)	PUNCT
ejpam-4571	406	6	is	be	AUX
ejpam-4571	406	7	(	(	PUNCT
ejpam-4571	406	8	λ	λ	X
ejpam-4571	406	9	,	,	PUNCT
ejpam-4571	406	10	b)-closed	b)-close	VERB
ejpam-4571	406	11	in	in	ADP
ejpam-4571	406	12	y	y	PROPN
ejpam-4571	406	13	and	and	CCONJ
ejpam-4571	406	14	by	by	ADP
ejpam-4571	406	15	(	(	PUNCT
ejpam-4571	406	16	3	3	NUM
ejpam-4571	406	17	)	)	PUNCT
ejpam-4571	406	18	,	,	PUNCT
ejpam-4571	406	19	[	[	X
ejpam-4571	406	20	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4571	406	21	,	,	PUNCT
ejpam-4571	406	22	b)](λ	b)](λ	NOUN
ejpam-4571	406	23	,	,	PUNCT
ejpam-4571	406	24	b	b	NOUN
ejpam-4571	406	25	)	)	PUNCT
ejpam-4571	406	26	)	)	PUNCT
ejpam-4571	406	27	]	]	PUNCT
ejpam-4571	407	1	(	(	PUNCT
ejpam-4571	407	2	λ	λ	X
ejpam-4571	407	3	,	,	PUNCT
ejpam-4571	407	4	b	b	NOUN
ejpam-4571	407	5	)	)	PUNCT
ejpam-4571	407	6	⊆	⊆	NUM
ejpam-4571	407	7	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4571	407	8	,	,	PUNCT
ejpam-4571	407	9	b	b	NOUN
ejpam-4571	407	10	)	)	PUNCT
ejpam-4571	407	11	)	)	PUNCT
ejpam-4571	407	12	.	.	PUNCT
ejpam-4571	408	1	(	(	PUNCT
ejpam-4571	408	2	4	4	X
ejpam-4571	408	3	)	)	PUNCT
ejpam-4571	408	4	⇒	⇒	NOUN
ejpam-4571	408	5	(	(	PUNCT
ejpam-4571	408	6	5	5	NUM
ejpam-4571	408	7	):	):	PUNCT
ejpam-4571	408	8	let	let	VERB
ejpam-4571	408	9	a	a	PRON
ejpam-4571	408	10	be	be	AUX
ejpam-4571	408	11	any	any	DET
ejpam-4571	408	12	subset	subset	NOUN
ejpam-4571	408	13	of	of	ADP
ejpam-4571	408	14	y	y	PROPN
ejpam-4571	408	15	.	.	PUNCT
ejpam-4571	409	1	by	by	ADP
ejpam-4571	409	2	(	(	PUNCT
ejpam-4571	409	3	4	4	NUM
ejpam-4571	409	4	)	)	PUNCT
ejpam-4571	409	5	,	,	PUNCT
ejpam-4571	409	6	f−1(a(λ	f−1(a(λ	PROPN
ejpam-4571	409	7	,	,	PUNCT
ejpam-4571	409	8	b	b	NOUN
ejpam-4571	409	9	)	)	PUNCT
ejpam-4571	409	10	)	)	PUNCT
ejpam-4571	409	11	=	=	PUNCT
ejpam-4571	410	1	x	x	X
ejpam-4571	410	2	−	−	NOUN
ejpam-4571	410	3	f−1([y	f−1([y	NOUN
ejpam-4571	410	4	−	−	NOUN
ejpam-4571	410	5	a](λ	a](λ	NOUN
ejpam-4571	410	6	,	,	PUNCT
ejpam-4571	410	7	b	b	NOUN
ejpam-4571	410	8	)	)	PUNCT
ejpam-4571	410	9	)	)	PUNCT
ejpam-4571	411	1	⊆	⊆	NUM
ejpam-4571	411	2	x	x	SYM
ejpam-4571	411	3	−	−	NOUN
ejpam-4571	412	1	[	[	X
ejpam-4571	412	2	f−1([[y	f−1([[y	NOUN
ejpam-4571	412	3	−a](λ	−a](λ	NUM
ejpam-4571	412	4	,	,	PUNCT
ejpam-4571	412	5	b)](λ	b)](λ	NOUN
ejpam-4571	412	6	,	,	PUNCT
ejpam-4571	412	7	b	b	NOUN
ejpam-4571	412	8	)	)	PUNCT
ejpam-4571	412	9	)	)	PUNCT
ejpam-4571	412	10	]	]	PUNCT
ejpam-4571	412	11	(	(	PUNCT
ejpam-4571	412	12	λ	λ	X
ejpam-4571	412	13	,	,	PUNCT
ejpam-4571	412	14	b	b	NOUN
ejpam-4571	412	15	)	)	PUNCT
ejpam-4571	413	1	=	=	NOUN
ejpam-4571	414	1	[	[	X
ejpam-4571	414	2	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4571	414	3	,	,	PUNCT
ejpam-4571	414	4	b	b	NOUN
ejpam-4571	414	5	)	)	PUNCT
ejpam-4571	414	6	]	]	PUNCT
ejpam-4571	415	1	(	(	PUNCT
ejpam-4571	415	2	λ	λ	X
ejpam-4571	415	3	,	,	PUNCT
ejpam-4571	415	4	b))](λ	b))](λ	PROPN
ejpam-4571	415	5	,	,	PUNCT
ejpam-4571	415	6	b	b	NOUN
ejpam-4571	415	7	)	)	PUNCT
ejpam-4571	415	8	.	.	PUNCT
ejpam-4571	416	1	(	(	PUNCT
ejpam-4571	416	2	5	5	X
ejpam-4571	416	3	)	)	PUNCT
ejpam-4571	416	4	⇒	⇒	NOUN
ejpam-4571	416	5	(	(	PUNCT
ejpam-4571	416	6	6	6	NUM
ejpam-4571	416	7	):	):	PUNCT
ejpam-4571	416	8	let	let	VERB
ejpam-4571	416	9	u	u	PRON
ejpam-4571	416	10	be	be	AUX
ejpam-4571	416	11	any	any	DET
ejpam-4571	416	12	(	(	PUNCT
ejpam-4571	416	13	λ	λ	NOUN
ejpam-4571	416	14	,	,	PUNCT
ejpam-4571	416	15	b)-open	b)-open	VERB
ejpam-4571	416	16	set	set	VERB
ejpam-4571	416	17	of	of	ADP
ejpam-4571	416	18	y	y	PROPN
ejpam-4571	416	19	.	.	PUNCT
ejpam-4571	416	20	suppose	suppose	VERB
ejpam-4571	416	21	that	that	SCONJ
ejpam-4571	416	22	x	x	PROPN
ejpam-4571	416	23	̸∈	̸∈	PROPN
ejpam-4571	416	24	f−1(u	f−1(u	PROPN
ejpam-4571	416	25	(	(	PUNCT
ejpam-4571	416	26	λ	λ	PROPN
ejpam-4571	416	27	,	,	PUNCT
ejpam-4571	416	28	b	b	NOUN
ejpam-4571	416	29	)	)	PUNCT
ejpam-4571	416	30	)	)	PUNCT
ejpam-4571	416	31	.	.	PUNCT
ejpam-4571	417	1	then	then	ADV
ejpam-4571	417	2	,	,	PUNCT
ejpam-4571	417	3	f(x	f(x	PROPN
ejpam-4571	417	4	)	)	PUNCT
ejpam-4571	417	5	̸∈	̸∈	PROPN
ejpam-4571	417	6	u	u	PROPN
ejpam-4571	417	7	(	(	PUNCT
ejpam-4571	417	8	λ	λ	PROPN
ejpam-4571	417	9	,	,	PUNCT
ejpam-4571	417	10	b	b	NOUN
ejpam-4571	417	11	)	)	PUNCT
ejpam-4571	417	12	and	and	CCONJ
ejpam-4571	417	13	so	so	ADV
ejpam-4571	417	14	there	there	PRON
ejpam-4571	417	15	exists	exist	VERB
ejpam-4571	417	16	a	a	DET
ejpam-4571	417	17	(	(	PUNCT
ejpam-4571	417	18	λ	λ	NOUN
ejpam-4571	417	19	,	,	PUNCT
ejpam-4571	417	20	b)-open	b)-open	VERB
ejpam-4571	417	21	set	set	VERB
ejpam-4571	417	22	v	v	NOUN
ejpam-4571	417	23	of	of	ADP
ejpam-4571	417	24	y	y	NOUN
ejpam-4571	417	25	containing	contain	VERB
ejpam-4571	417	26	f(x	f(x	PROPN
ejpam-4571	417	27	)	)	PUNCT
ejpam-4571	417	28	such	such	ADJ
ejpam-4571	417	29	that	that	SCONJ
ejpam-4571	417	30	u	u	PROPN
ejpam-4571	417	31	∩	∩	NOUN
ejpam-4571	417	32	v	v	NOUN
ejpam-4571	417	33	=	=	NOUN
ejpam-4571	417	34	∅	∅	NOUN
ejpam-4571	417	35	and	and	CCONJ
ejpam-4571	417	36	hence	hence	ADV
ejpam-4571	417	37	u	u	NOUN
ejpam-4571	417	38	∩	∩	ADJ
ejpam-4571	417	39	v	v	X
ejpam-4571	417	40	(	(	PUNCT
ejpam-4571	417	41	λ	λ	PROPN
ejpam-4571	417	42	,	,	PUNCT
ejpam-4571	417	43	b	b	NOUN
ejpam-4571	417	44	)	)	PUNCT
ejpam-4571	417	45	=	=	PUNCT
ejpam-4571	417	46	∅.	∅.	X
ejpam-4571	417	47	by	by	ADP
ejpam-4571	417	48	(	(	PUNCT
ejpam-4571	417	49	5	5	NUM
ejpam-4571	417	50	)	)	PUNCT
ejpam-4571	417	51	,	,	PUNCT
ejpam-4571	417	52	we	we	PRON
ejpam-4571	417	53	have	have	VERB
ejpam-4571	417	54	x	x	X
ejpam-4571	417	55	∈	∈	PROPN
ejpam-4571	417	56	f−1(v	f−1(v	NOUN
ejpam-4571	417	57	)	)	PUNCT
ejpam-4571	418	1	⊆	⊆	NUM
ejpam-4571	418	2	[	[	X
ejpam-4571	418	3	f−1(v	f−1(v	NOUN
ejpam-4571	418	4	(	(	PUNCT
ejpam-4571	418	5	λ	λ	PROPN
ejpam-4571	418	6	,	,	PUNCT
ejpam-4571	418	7	b))](λ	b))](λ	PROPN
ejpam-4571	418	8	,	,	PUNCT
ejpam-4571	418	9	b	b	NOUN
ejpam-4571	418	10	)	)	PUNCT
ejpam-4571	418	11	.	.	PUNCT
ejpam-4571	419	1	there	there	PRON
ejpam-4571	419	2	exists	exist	VERB
ejpam-4571	419	3	a	a	DET
ejpam-4571	419	4	(	(	PUNCT
ejpam-4571	419	5	λ	λ	NOUN
ejpam-4571	419	6	,	,	PUNCT
ejpam-4571	419	7	b)-open	b)-open	AUX
ejpam-4571	419	8	set	set	VERB
ejpam-4571	419	9	w	w	NOUN
ejpam-4571	419	10	containing	contain	VERB
ejpam-4571	419	11	x	x	PUNCT
ejpam-4571	419	12	such	such	ADJ
ejpam-4571	419	13	that	that	SCONJ
ejpam-4571	419	14	x	x	SYM
ejpam-4571	419	15	∈	∈	PROPN
ejpam-4571	419	16	w	w	NOUN
ejpam-4571	419	17	⊆	⊆	NUM
ejpam-4571	419	18	f−1(v	f−1(v	NOUN
ejpam-4571	419	19	(	(	PUNCT
ejpam-4571	419	20	λ	λ	PROPN
ejpam-4571	419	21	,	,	PUNCT
ejpam-4571	419	22	b	b	NOUN
ejpam-4571	419	23	)	)	PUNCT
ejpam-4571	419	24	)	)	PUNCT
ejpam-4571	419	25	.	.	PUNCT
ejpam-4571	420	1	since	since	SCONJ
ejpam-4571	420	2	u	u	PROPN
ejpam-4571	420	3	∩	∩	PROPN
ejpam-4571	420	4	v	v	X
ejpam-4571	420	5	(	(	PUNCT
ejpam-4571	420	6	λ	λ	PROPN
ejpam-4571	420	7	,	,	PUNCT
ejpam-4571	420	8	b	b	NOUN
ejpam-4571	420	9	)	)	PUNCT
ejpam-4571	420	10	=	=	NOUN
ejpam-4571	420	11	∅	∅	NOUN
ejpam-4571	420	12	and	and	CCONJ
ejpam-4571	420	13	f(w	f(w	PROPN
ejpam-4571	420	14	)	)	PUNCT
ejpam-4571	420	15	⊆	⊆	NUM
ejpam-4571	420	16	v	v	X
ejpam-4571	420	17	(	(	PUNCT
ejpam-4571	420	18	λ	λ	PROPN
ejpam-4571	420	19	,	,	PUNCT
ejpam-4571	420	20	b	b	NOUN
ejpam-4571	420	21	)	)	PUNCT
ejpam-4571	420	22	,	,	PUNCT
ejpam-4571	420	23	we	we	PRON
ejpam-4571	420	24	have	have	VERB
ejpam-4571	420	25	w	w	NOUN
ejpam-4571	420	26	∩	∩	ADJ
ejpam-4571	420	27	f−1(u	f−1(u	NOUN
ejpam-4571	420	28	)	)	PUNCT
ejpam-4571	420	29	=	=	PUNCT
ejpam-4571	420	30	∅.	∅.	ADP
ejpam-4571	420	31	thus	thus	ADV
ejpam-4571	420	32	,	,	PUNCT
ejpam-4571	420	33	x	x	PROPN
ejpam-4571	420	34	̸∈	̸∈	PROPN
ejpam-4571	420	35	[	[	X
ejpam-4571	420	36	f−1(u)](λ	f−1(u)](λ	PROPN
ejpam-4571	420	37	,	,	PUNCT
ejpam-4571	420	38	b	b	NOUN
ejpam-4571	420	39	)	)	PUNCT
ejpam-4571	420	40	and	and	CCONJ
ejpam-4571	420	41	hence	hence	ADV
ejpam-4571	420	42	[	[	X
ejpam-4571	420	43	f−1(u)](λ	f−1(u)](λ	PROPN
ejpam-4571	420	44	,	,	PUNCT
ejpam-4571	420	45	b	b	NOUN
ejpam-4571	420	46	)	)	PUNCT
ejpam-4571	420	47	⊆	⊆	NUM
ejpam-4571	420	48	f−1(u	f−1(u	NOUN
ejpam-4571	420	49	(	(	PUNCT
ejpam-4571	420	50	λ	λ	PROPN
ejpam-4571	420	51	,	,	PUNCT
ejpam-4571	420	52	b	b	NOUN
ejpam-4571	420	53	)	)	PUNCT
ejpam-4571	420	54	)	)	PUNCT
ejpam-4571	420	55	.	.	PUNCT
ejpam-4571	421	1	(	(	PUNCT
ejpam-4571	421	2	6	6	X
ejpam-4571	421	3	)	)	PUNCT
ejpam-4571	421	4	⇒	⇒	NOUN
ejpam-4571	421	5	(	(	PUNCT
ejpam-4571	421	6	1	1	NUM
ejpam-4571	421	7	):	):	PUNCT
ejpam-4571	421	8	let	let	VERB
ejpam-4571	421	9	x	x	PUNCT
ejpam-4571	421	10	∈	∈	PROPN
ejpam-4571	421	11	x	x	X
ejpam-4571	421	12	and	and	CCONJ
ejpam-4571	421	13	u	u	PRON
ejpam-4571	421	14	be	be	VERB
ejpam-4571	421	15	any	any	DET
ejpam-4571	421	16	(	(	PUNCT
ejpam-4571	421	17	λ	λ	NOUN
ejpam-4571	421	18	,	,	PUNCT
ejpam-4571	421	19	b)-open	b)-open	VERB
ejpam-4571	421	20	set	set	VERB
ejpam-4571	421	21	of	of	ADP
ejpam-4571	421	22	y	y	PROPN
ejpam-4571	421	23	containing	contain	VERB
ejpam-4571	421	24	f(x	f(x	PROPN
ejpam-4571	421	25	)	)	PUNCT
ejpam-4571	421	26	.	.	PUNCT
ejpam-4571	422	1	then	then	ADV
ejpam-4571	422	2	,	,	PUNCT
ejpam-4571	422	3	we	we	PRON
ejpam-4571	422	4	have	have	VERB
ejpam-4571	422	5	u	u	NOUN
ejpam-4571	422	6	=	=	PROPN
ejpam-4571	422	7	u(λ	u(λ	PROPN
ejpam-4571	422	8	,	,	PUNCT
ejpam-4571	422	9	b	b	NOUN
ejpam-4571	422	10	)	)	PUNCT
ejpam-4571	422	11	⊆	⊆	NUM
ejpam-4571	423	1	[	[	X
ejpam-4571	423	2	u	u	X
ejpam-4571	423	3	(	(	PUNCT
ejpam-4571	423	4	λ	λ	PROPN
ejpam-4571	423	5	,	,	PUNCT
ejpam-4571	423	6	b)](λ	b)](λ	NOUN
ejpam-4571	423	7	,	,	PUNCT
ejpam-4571	423	8	b	b	NOUN
ejpam-4571	423	9	)	)	PUNCT
ejpam-4571	423	10	.	.	PUNCT
ejpam-4571	424	1	thus	thus	ADV
ejpam-4571	424	2	,	,	PUNCT
ejpam-4571	424	3	by	by	ADP
ejpam-4571	424	4	(	(	PUNCT
ejpam-4571	424	5	6	6	NUM
ejpam-4571	424	6	)	)	PUNCT
ejpam-4571	424	7	,	,	PUNCT
ejpam-4571	424	8	x	x	PUNCT
ejpam-4571	424	9	∈	∈	PROPN
ejpam-4571	424	10	f−1(u	f−1(u	PROPN
ejpam-4571	424	11	)	)	PUNCT
ejpam-4571	424	12	⊆	⊆	NUM
ejpam-4571	424	13	f−1([u	f−1([u	NOUN
ejpam-4571	424	14	(	(	PUNCT
ejpam-4571	424	15	λ	λ	PROPN
ejpam-4571	424	16	,	,	PUNCT
ejpam-4571	424	17	b)](λ	b)](λ	NOUN
ejpam-4571	424	18	,	,	PUNCT
ejpam-4571	424	19	b	b	NOUN
ejpam-4571	424	20	)	)	PUNCT
ejpam-4571	424	21	)	)	PUNCT
ejpam-4571	425	1	=	=	PUNCT
ejpam-4571	425	2	x	x	X
ejpam-4571	426	1	−	−	NOUN
ejpam-4571	426	2	f−1([y	f−1([y	NOUN
ejpam-4571	426	3	−	−	NOUN
ejpam-4571	426	4	u	u	NOUN
ejpam-4571	426	5	(	(	PUNCT
ejpam-4571	426	6	λ	λ	PROPN
ejpam-4571	426	7	,	,	PUNCT
ejpam-4571	426	8	b)](λ	b)](λ	NOUN
ejpam-4571	426	9	,	,	PUNCT
ejpam-4571	426	10	b	b	NOUN
ejpam-4571	426	11	)	)	PUNCT
ejpam-4571	426	12	)	)	PUNCT
ejpam-4571	427	1	⊆	⊆	NUM
ejpam-4571	427	2	x	x	SYM
ejpam-4571	427	3	−	−	PROPN
ejpam-4571	428	1	[	[	X
ejpam-4571	428	2	f−1(y	f−1(y	PROPN
ejpam-4571	428	3	−	−	PROPN
ejpam-4571	428	4	u	u	NOUN
ejpam-4571	428	5	(	(	PUNCT
ejpam-4571	428	6	λ	λ	PROPN
ejpam-4571	428	7	,	,	PUNCT
ejpam-4571	428	8	b))](λ	b))](λ	PROPN
ejpam-4571	428	9	,	,	PUNCT
ejpam-4571	428	10	b	b	NOUN
ejpam-4571	428	11	)	)	PUNCT
ejpam-4571	428	12	=	=	NOUN
ejpam-4571	429	1	[	[	X
ejpam-4571	429	2	f−1(u	f−1(u	X
ejpam-4571	429	3	(	(	PUNCT
ejpam-4571	429	4	λ	λ	PROPN
ejpam-4571	429	5	,	,	PUNCT
ejpam-4571	429	6	b))](λ	b))](λ	PROPN
ejpam-4571	429	7	,	,	PUNCT
ejpam-4571	429	8	b	b	NOUN
ejpam-4571	429	9	)	)	PUNCT
ejpam-4571	429	10	.	.	PUNCT
ejpam-4571	430	1	then	then	ADV
ejpam-4571	430	2	,	,	PUNCT
ejpam-4571	430	3	there	there	PRON
ejpam-4571	430	4	exists	exist	VERB
ejpam-4571	430	5	a	a	DET
ejpam-4571	430	6	(	(	PUNCT
ejpam-4571	430	7	λ	λ	NOUN
ejpam-4571	430	8	,	,	PUNCT
ejpam-4571	430	9	b)-open	b)-open	VERB
ejpam-4571	430	10	set	set	VERB
ejpam-4571	430	11	v	v	NOUN
ejpam-4571	430	12	of	of	ADP
ejpam-4571	430	13	x	x	PUNCT
ejpam-4571	430	14	containing	contain	VERB
ejpam-4571	430	15	x	x	PUNCT
ejpam-4571	430	16	such	such	ADJ
ejpam-4571	430	17	that	that	DET
ejpam-4571	430	18	v	v	ADP
ejpam-4571	430	19	⊆	⊆	NUM
ejpam-4571	430	20	f−1(u	f−1(u	NOUN
ejpam-4571	430	21	(	(	PUNCT
ejpam-4571	430	22	λ	λ	PROPN
ejpam-4571	430	23	,	,	PUNCT
ejpam-4571	430	24	b	b	NOUN
ejpam-4571	430	25	)	)	PUNCT
ejpam-4571	430	26	)	)	PUNCT
ejpam-4571	430	27	.	.	PUNCT
ejpam-4571	431	1	this	this	PRON
ejpam-4571	431	2	shows	show	VERB
ejpam-4571	431	3	that	that	SCONJ
ejpam-4571	431	4	f	f	PROPN
ejpam-4571	431	5	is	be	AUX
ejpam-4571	431	6	weakly	weakly	ADJ
ejpam-4571	431	7	(	(	PUNCT
ejpam-4571	431	8	λ	λ	NOUN
ejpam-4571	431	9	,	,	PUNCT
ejpam-4571	431	10	b)-continuous	b)-continuous	PROPN
ejpam-4571	431	11	.	.	PUNCT
ejpam-4571	432	1	theorem	theorem	VERB
ejpam-4571	432	2	11	11	NUM
ejpam-4571	432	3	.	.	PUNCT
ejpam-4571	433	1	for	for	ADP
ejpam-4571	433	2	a	a	DET
ejpam-4571	433	3	function	function	NOUN
ejpam-4571	433	4	f	f	NOUN
ejpam-4571	433	5	:	:	PUNCT
ejpam-4571	433	6	(	(	PUNCT
ejpam-4571	433	7	x	x	X
ejpam-4571	433	8	,	,	PUNCT
ejpam-4571	433	9	τ	τ	X
ejpam-4571	433	10	)	)	PUNCT
ejpam-4571	433	11	→	→	SYM
ejpam-4571	433	12	(	(	PUNCT
ejpam-4571	433	13	y	y	PROPN
ejpam-4571	433	14	,	,	PUNCT
ejpam-4571	433	15	σ	σ	PROPN
ejpam-4571	433	16	)	)	PUNCT
ejpam-4571	433	17	,	,	PUNCT
ejpam-4571	433	18	the	the	DET
ejpam-4571	433	19	following	follow	VERB
ejpam-4571	433	20	properties	property	NOUN
ejpam-4571	433	21	are	be	AUX
ejpam-4571	433	22	equivalent	equivalent	ADJ
ejpam-4571	433	23	:	:	PUNCT
ejpam-4571	433	24	(	(	PUNCT
ejpam-4571	433	25	1	1	X
ejpam-4571	433	26	)	)	PUNCT
ejpam-4571	433	27	f	f	PROPN
ejpam-4571	433	28	is	be	AUX
ejpam-4571	433	29	weakly	weakly	ADJ
ejpam-4571	433	30	(	(	PUNCT
ejpam-4571	433	31	λ	λ	X
ejpam-4571	433	32	,	,	PUNCT
ejpam-4571	433	33	b)-continuous	b)-continuous	ADJ
ejpam-4571	433	34	;	;	PUNCT
ejpam-4571	433	35	(	(	PUNCT
ejpam-4571	433	36	2	2	X
ejpam-4571	433	37	)	)	PUNCT
ejpam-4571	434	1	[	[	X
ejpam-4571	434	2	f−1(f(λ	f−1(f(λ	X
ejpam-4571	434	3	,	,	PUNCT
ejpam-4571	434	4	b	b	NOUN
ejpam-4571	434	5	)	)	PUNCT
ejpam-4571	434	6	)	)	PUNCT
ejpam-4571	434	7	]	]	PUNCT
ejpam-4571	434	8	(	(	PUNCT
ejpam-4571	434	9	λ	λ	X
ejpam-4571	434	10	,	,	PUNCT
ejpam-4571	434	11	b	b	NOUN
ejpam-4571	434	12	)	)	PUNCT
ejpam-4571	434	13	⊆	⊆	NUM
ejpam-4571	434	14	f−1(f	f−1(f	PROPN
ejpam-4571	434	15	)	)	PUNCT
ejpam-4571	434	16	for	for	ADP
ejpam-4571	434	17	every	every	DET
ejpam-4571	434	18	r(λ	r(λ	NOUN
ejpam-4571	434	19	,	,	PUNCT
ejpam-4571	434	20	b)-closed	b)-close	VERB
ejpam-4571	434	21	set	set	ADJ
ejpam-4571	434	22	f	f	PROPN
ejpam-4571	434	23	of	of	ADP
ejpam-4571	434	24	y	y	PROPN
ejpam-4571	434	25	;	;	PUNCT
ejpam-4571	434	26	(	(	PUNCT
ejpam-4571	434	27	3	3	X
ejpam-4571	434	28	)	)	PUNCT
ejpam-4571	434	29	[	[	X
ejpam-4571	434	30	f−1([u	f−1([u	INTJ
ejpam-4571	434	31	(	(	PUNCT
ejpam-4571	434	32	λ	λ	PROPN
ejpam-4571	434	33	,	,	PUNCT
ejpam-4571	434	34	b)](λ	b)](λ	NOUN
ejpam-4571	434	35	,	,	PUNCT
ejpam-4571	434	36	b	b	NOUN
ejpam-4571	434	37	)	)	PUNCT
ejpam-4571	434	38	)	)	PUNCT
ejpam-4571	434	39	]	]	PUNCT
ejpam-4571	435	1	(	(	PUNCT
ejpam-4571	435	2	λ	λ	X
ejpam-4571	435	3	,	,	PUNCT
ejpam-4571	435	4	b	b	NOUN
ejpam-4571	435	5	)	)	PUNCT
ejpam-4571	435	6	⊆	⊆	NUM
ejpam-4571	435	7	f−1(u	f−1(u	NOUN
ejpam-4571	435	8	(	(	PUNCT
ejpam-4571	435	9	λ	λ	PROPN
ejpam-4571	435	10	,	,	PUNCT
ejpam-4571	435	11	b	b	NOUN
ejpam-4571	435	12	)	)	PUNCT
ejpam-4571	435	13	)	)	PUNCT
ejpam-4571	435	14	for	for	ADP
ejpam-4571	435	15	every	every	DET
ejpam-4571	435	16	β(λ	β(λ	NOUN
ejpam-4571	435	17	,	,	PUNCT
ejpam-4571	435	18	b)-open	b)-open	VERB
ejpam-4571	435	19	set	set	VERB
ejpam-4571	435	20	u	u	NOUN
ejpam-4571	435	21	of	of	ADP
ejpam-4571	435	22	y	y	PROPN
ejpam-4571	435	23	;	;	PUNCT
ejpam-4571	435	24	(	(	PUNCT
ejpam-4571	435	25	4	4	X
ejpam-4571	435	26	)	)	PUNCT
ejpam-4571	435	27	[	[	X
ejpam-4571	435	28	f−1([u	f−1([u	INTJ
ejpam-4571	435	29	(	(	PUNCT
ejpam-4571	435	30	λ	λ	PROPN
ejpam-4571	435	31	,	,	PUNCT
ejpam-4571	435	32	b)](λ	b)](λ	NOUN
ejpam-4571	435	33	,	,	PUNCT
ejpam-4571	435	34	b	b	NOUN
ejpam-4571	435	35	)	)	PUNCT
ejpam-4571	435	36	)	)	PUNCT
ejpam-4571	435	37	]	]	PUNCT
ejpam-4571	435	38	(	(	PUNCT
ejpam-4571	435	39	λ	λ	X
ejpam-4571	435	40	,	,	PUNCT
ejpam-4571	435	41	b	b	NOUN
ejpam-4571	435	42	)	)	PUNCT
ejpam-4571	435	43	⊆	⊆	NUM
ejpam-4571	435	44	f−1(u	f−1(u	NOUN
ejpam-4571	435	45	(	(	PUNCT
ejpam-4571	435	46	λ	λ	PROPN
ejpam-4571	435	47	,	,	PUNCT
ejpam-4571	435	48	b	b	NOUN
ejpam-4571	435	49	)	)	PUNCT
ejpam-4571	435	50	)	)	PUNCT
ejpam-4571	435	51	for	for	ADP
ejpam-4571	435	52	every	every	DET
ejpam-4571	435	53	s(λ	s(λ	PROPN
ejpam-4571	435	54	,	,	PUNCT
ejpam-4571	435	55	b)-open	b)-open	VERB
ejpam-4571	435	56	set	set	VERB
ejpam-4571	435	57	u	u	NOUN
ejpam-4571	435	58	of	of	ADP
ejpam-4571	435	59	y	y	PROPN
ejpam-4571	435	60	.	.	PUNCT
ejpam-4571	436	1	proof	proof	NOUN
ejpam-4571	436	2	.	.	PUNCT
ejpam-4571	437	1	(	(	PUNCT
ejpam-4571	437	2	1	1	X
ejpam-4571	437	3	)	)	PUNCT
ejpam-4571	437	4	⇒	⇒	NOUN
ejpam-4571	437	5	(	(	PUNCT
ejpam-4571	437	6	2	2	NUM
ejpam-4571	437	7	):	):	PUNCT
ejpam-4571	437	8	let	let	VERB
ejpam-4571	437	9	f	f	PRON
ejpam-4571	437	10	be	be	AUX
ejpam-4571	437	11	any	any	DET
ejpam-4571	437	12	r(λ	r(λ	NOUN
ejpam-4571	437	13	,	,	PUNCT
ejpam-4571	437	14	b)-closed	b)-close	VERB
ejpam-4571	437	15	set	set	NOUN
ejpam-4571	437	16	of	of	ADP
ejpam-4571	437	17	y	y	PROPN
ejpam-4571	437	18	.	.	PUNCT
ejpam-4571	438	1	then	then	ADV
ejpam-4571	438	2	,	,	PUNCT
ejpam-4571	438	3	f(λ	f(λ	PROPN
ejpam-4571	438	4	,	,	PUNCT
ejpam-4571	438	5	b	b	NOUN
ejpam-4571	438	6	)	)	PUNCT
ejpam-4571	438	7	is	be	AUX
ejpam-4571	438	8	(	(	PUNCT
ejpam-4571	438	9	λ	λ	X
ejpam-4571	438	10	,	,	PUNCT
ejpam-4571	438	11	b)-open	b)-open	PUNCT
ejpam-4571	438	12	and	and	CCONJ
ejpam-4571	438	13	by	by	ADP
ejpam-4571	438	14	theorem	theorem	NOUN
ejpam-4571	438	15	10	10	NUM
ejpam-4571	438	16	,	,	PUNCT
ejpam-4571	438	17	[	[	X
ejpam-4571	438	18	f−1(f(λ	f−1(f(λ	X
ejpam-4571	438	19	,	,	PUNCT
ejpam-4571	438	20	b	b	NOUN
ejpam-4571	438	21	)	)	PUNCT
ejpam-4571	438	22	)	)	PUNCT
ejpam-4571	438	23	]	]	PUNCT
ejpam-4571	439	1	(	(	PUNCT
ejpam-4571	439	2	λ	λ	X
ejpam-4571	439	3	,	,	PUNCT
ejpam-4571	439	4	b	b	NOUN
ejpam-4571	439	5	)	)	PUNCT
ejpam-4571	439	6	⊆	⊆	NUM
ejpam-4571	439	7	f−1([f(λ	f−1([f(λ	NOUN
ejpam-4571	439	8	,	,	PUNCT
ejpam-4571	439	9	b	b	NOUN
ejpam-4571	439	10	)	)	PUNCT
ejpam-4571	439	11	]	]	PUNCT
ejpam-4571	439	12	(	(	PUNCT
ejpam-4571	439	13	λ	λ	X
ejpam-4571	439	14	,	,	PUNCT
ejpam-4571	439	15	b	b	NOUN
ejpam-4571	439	16	)	)	PUNCT
ejpam-4571	439	17	)	)	PUNCT
ejpam-4571	439	18	.	.	PUNCT
ejpam-4571	440	1	since	since	SCONJ
ejpam-4571	440	2	f	f	PROPN
ejpam-4571	440	3	is	be	AUX
ejpam-4571	440	4	r(λ	r(λ	NOUN
ejpam-4571	440	5	,	,	PUNCT
ejpam-4571	440	6	b)-closed	b)-close	VERB
ejpam-4571	440	7	,	,	PUNCT
ejpam-4571	440	8	we	we	PRON
ejpam-4571	440	9	have	have	VERB
ejpam-4571	440	10	[	[	X
ejpam-4571	440	11	f−1(f(λ	f−1(f(λ	ADJ
ejpam-4571	440	12	,	,	PUNCT
ejpam-4571	440	13	b	b	NOUN
ejpam-4571	440	14	)	)	PUNCT
ejpam-4571	440	15	)	)	PUNCT
ejpam-4571	440	16	]	]	PUNCT
ejpam-4571	441	1	(	(	PUNCT
ejpam-4571	441	2	λ	λ	X
ejpam-4571	441	3	,	,	PUNCT
ejpam-4571	441	4	b	b	NOUN
ejpam-4571	441	5	)	)	PUNCT
ejpam-4571	441	6	⊆	⊆	NUM
ejpam-4571	441	7	f−1([f(λ	f−1([f(λ	NOUN
ejpam-4571	441	8	,	,	PUNCT
ejpam-4571	441	9	b	b	NOUN
ejpam-4571	441	10	)	)	PUNCT
ejpam-4571	441	11	]	]	PUNCT
ejpam-4571	441	12	(	(	PUNCT
ejpam-4571	441	13	λ	λ	X
ejpam-4571	441	14	,	,	PUNCT
ejpam-4571	441	15	b	b	NOUN
ejpam-4571	441	16	)	)	PUNCT
ejpam-4571	441	17	)	)	PUNCT
ejpam-4571	442	1	⊆	⊆	NUM
ejpam-4571	442	2	f−1(f	f−1(f	NOUN
ejpam-4571	442	3	)	)	PUNCT
ejpam-4571	442	4	.	.	PUNCT
ejpam-4571	443	1	(	(	PUNCT
ejpam-4571	443	2	2	2	X
ejpam-4571	443	3	)	)	PUNCT
ejpam-4571	443	4	⇒	⇒	NOUN
ejpam-4571	443	5	(	(	PUNCT
ejpam-4571	443	6	3	3	NUM
ejpam-4571	443	7	):	):	PUNCT
ejpam-4571	443	8	let	let	VERB
ejpam-4571	443	9	u	u	PRON
ejpam-4571	443	10	be	be	AUX
ejpam-4571	443	11	any	any	DET
ejpam-4571	443	12	β(λ	β(λ	NOUN
ejpam-4571	443	13	,	,	PUNCT
ejpam-4571	443	14	b)-open	b)-open	VERB
ejpam-4571	443	15	set	set	VERB
ejpam-4571	443	16	of	of	ADP
ejpam-4571	443	17	y	y	PROPN
ejpam-4571	443	18	.	.	PUNCT
ejpam-4571	444	1	then	then	ADV
ejpam-4571	444	2	,	,	PUNCT
ejpam-4571	444	3	u	u	PROPN
ejpam-4571	444	4	(	(	PUNCT
ejpam-4571	444	5	λ	λ	PROPN
ejpam-4571	444	6	,	,	PUNCT
ejpam-4571	444	7	b	b	NOUN
ejpam-4571	444	8	)	)	PUNCT
ejpam-4571	444	9	⊆	⊆	NUM
ejpam-4571	445	1	[	[	X
ejpam-4571	445	2	[	[	X
ejpam-4571	445	3	u	u	X
ejpam-4571	445	4	(	(	PUNCT
ejpam-4571	445	5	λ	λ	PROPN
ejpam-4571	445	6	,	,	PUNCT
ejpam-4571	445	7	b)](λ	b)](λ	NOUN
ejpam-4571	445	8	,	,	PUNCT
ejpam-4571	445	9	b	b	NOUN
ejpam-4571	445	10	)	)	PUNCT
ejpam-4571	445	11	]	]	PUNCT
ejpam-4571	445	12	(	(	PUNCT
ejpam-4571	445	13	λ	λ	X
ejpam-4571	445	14	,	,	PUNCT
ejpam-4571	445	15	b	b	NOUN
ejpam-4571	445	16	)	)	PUNCT
ejpam-4571	445	17	⊆	⊆	NUM
ejpam-4571	445	18	u	u	NOUN
ejpam-4571	445	19	(	(	PUNCT
ejpam-4571	445	20	λ	λ	PROPN
ejpam-4571	445	21	,	,	PUNCT
ejpam-4571	445	22	b	b	NOUN
ejpam-4571	445	23	)	)	PUNCT
ejpam-4571	445	24	and	and	CCONJ
ejpam-4571	445	25	hence	hence	ADV
ejpam-4571	445	26	u	u	NOUN
ejpam-4571	445	27	(	(	PUNCT
ejpam-4571	445	28	λ	λ	PROPN
ejpam-4571	445	29	,	,	PUNCT
ejpam-4571	445	30	b	b	NOUN
ejpam-4571	445	31	)	)	PUNCT
ejpam-4571	445	32	is	be	AUX
ejpam-4571	445	33	r(λ	r(λ	NOUN
ejpam-4571	445	34	,	,	PUNCT
ejpam-4571	445	35	b)-closed	b)-close	VERB
ejpam-4571	445	36	.	.	PUNCT
ejpam-4571	446	1	by	by	ADP
ejpam-4571	446	2	(	(	PUNCT
ejpam-4571	446	3	2	2	NUM
ejpam-4571	446	4	)	)	PUNCT
ejpam-4571	446	5	,	,	PUNCT
ejpam-4571	446	6	[	[	X
ejpam-4571	446	7	f−1([u	f−1([u	INTJ
ejpam-4571	446	8	(	(	PUNCT
ejpam-4571	446	9	λ	λ	PROPN
ejpam-4571	446	10	,	,	PUNCT
ejpam-4571	446	11	b)](λ	b)](λ	NOUN
ejpam-4571	446	12	,	,	PUNCT
ejpam-4571	446	13	b	b	NOUN
ejpam-4571	446	14	)	)	PUNCT
ejpam-4571	446	15	)	)	PUNCT
ejpam-4571	446	16	]	]	PUNCT
ejpam-4571	446	17	(	(	PUNCT
ejpam-4571	446	18	λ	λ	X
ejpam-4571	446	19	,	,	PUNCT
ejpam-4571	446	20	b	b	NOUN
ejpam-4571	446	21	)	)	PUNCT
ejpam-4571	446	22	⊆	⊆	NUM
ejpam-4571	446	23	f−1(u	f−1(u	NOUN
ejpam-4571	446	24	(	(	PUNCT
ejpam-4571	446	25	λ	λ	PROPN
ejpam-4571	446	26	,	,	PUNCT
ejpam-4571	446	27	b	b	NOUN
ejpam-4571	446	28	)	)	PUNCT
ejpam-4571	446	29	)	)	PUNCT
ejpam-4571	446	30	.	.	PUNCT
ejpam-4571	447	1	(	(	PUNCT
ejpam-4571	447	2	3	3	X
ejpam-4571	447	3	)	)	PUNCT
ejpam-4571	447	4	⇒	⇒	NOUN
ejpam-4571	447	5	(	(	PUNCT
ejpam-4571	447	6	4	4	NUM
ejpam-4571	447	7	):	):	PUNCT
ejpam-4571	447	8	the	the	DET
ejpam-4571	447	9	proof	proof	NOUN
ejpam-4571	447	10	is	be	AUX
ejpam-4571	447	11	obvious	obvious	ADJ
ejpam-4571	447	12	.	.	PUNCT
ejpam-4571	448	1	(	(	PUNCT
ejpam-4571	448	2	4	4	X
ejpam-4571	448	3	)	)	PUNCT
ejpam-4571	448	4	⇒	⇒	NOUN
ejpam-4571	448	5	(	(	PUNCT
ejpam-4571	448	6	1	1	NUM
ejpam-4571	448	7	):	):	PUNCT
ejpam-4571	448	8	let	let	VERB
ejpam-4571	448	9	u	u	PRON
ejpam-4571	448	10	be	be	AUX
ejpam-4571	448	11	any	any	DET
ejpam-4571	448	12	(	(	PUNCT
ejpam-4571	448	13	λ	λ	NOUN
ejpam-4571	448	14	,	,	PUNCT
ejpam-4571	448	15	b)-open	b)-open	VERB
ejpam-4571	448	16	set	set	VERB
ejpam-4571	448	17	of	of	ADP
ejpam-4571	448	18	y	y	PROPN
ejpam-4571	448	19	.	.	PUNCT
ejpam-4571	449	1	thus	thus	ADV
ejpam-4571	449	2	,	,	PUNCT
ejpam-4571	449	3	by	by	ADP
ejpam-4571	449	4	(	(	PUNCT
ejpam-4571	449	5	4	4	NUM
ejpam-4571	449	6	)	)	PUNCT
ejpam-4571	449	7	,	,	PUNCT
ejpam-4571	449	8	[	[	X
ejpam-4571	449	9	f−1(u)](λ	f−1(u)](λ	X
ejpam-4571	449	10	,	,	PUNCT
ejpam-4571	449	11	b	b	NOUN
ejpam-4571	449	12	)	)	PUNCT
ejpam-4571	449	13	⊆	⊆	NUM
ejpam-4571	449	14	[	[	X
ejpam-4571	449	15	f−1([u	f−1([u	INTJ
ejpam-4571	449	16	(	(	PUNCT
ejpam-4571	449	17	λ	λ	PROPN
ejpam-4571	449	18	,	,	PUNCT
ejpam-4571	449	19	b)](λ	b)](λ	NOUN
ejpam-4571	449	20	,	,	PUNCT
ejpam-4571	449	21	b	b	NOUN
ejpam-4571	449	22	)	)	PUNCT
ejpam-4571	449	23	)	)	PUNCT
ejpam-4571	449	24	]	]	PUNCT
ejpam-4571	450	1	(	(	PUNCT
ejpam-4571	450	2	λ	λ	X
ejpam-4571	450	3	,	,	PUNCT
ejpam-4571	450	4	b	b	NOUN
ejpam-4571	450	5	)	)	PUNCT
ejpam-4571	450	6	⊆	⊆	NUM
ejpam-4571	450	7	f−1(u	f−1(u	NOUN
ejpam-4571	450	8	(	(	PUNCT
ejpam-4571	450	9	λ	λ	PROPN
ejpam-4571	450	10	,	,	PUNCT
ejpam-4571	450	11	b	b	NOUN
ejpam-4571	450	12	)	)	PUNCT
ejpam-4571	450	13	)	)	PUNCT
ejpam-4571	450	14	and	and	CCONJ
ejpam-4571	450	15	by	by	ADP
ejpam-4571	450	16	theorem	theorem	NOUN
ejpam-4571	450	17	10	10	NUM
ejpam-4571	450	18	,	,	PUNCT
ejpam-4571	450	19	f	f	PROPN
ejpam-4571	450	20	is	be	AUX
ejpam-4571	450	21	weakly	weakly	ADJ
ejpam-4571	450	22	(	(	PUNCT
ejpam-4571	450	23	λ	λ	NOUN
ejpam-4571	450	24	,	,	PUNCT
ejpam-4571	450	25	b)-continuous	b)-continuous	PROPN
ejpam-4571	450	26	.	.	PUNCT
ejpam-4571	451	1	c.	c.	PROPN
ejpam-4571	451	2	boonpok	boonpok	PROPN
ejpam-4571	451	3	,	,	PUNCT
ejpam-4571	451	4	n.	n.	PROPN
ejpam-4571	451	5	srisarakham	srisarakham	PROPN
ejpam-4571	451	6	/	/	SYM
ejpam-4571	451	7	eur	eur	PROPN
ejpam-4571	451	8	.	.	PUNCT
ejpam-4571	452	1	j.	j.	PROPN
ejpam-4571	452	2	pure	pure	PROPN
ejpam-4571	452	3	appl	appl	PROPN
ejpam-4571	452	4	.	.	PROPN
ejpam-4571	452	5	math	math	PROPN
ejpam-4571	452	6	,	,	PUNCT
ejpam-4571	452	7	16	16	NUM
ejpam-4571	452	8	(	(	PUNCT
ejpam-4571	452	9	1	1	NUM
ejpam-4571	452	10	)	)	PUNCT
ejpam-4571	452	11	(	(	PUNCT
ejpam-4571	452	12	2023	2023	NUM
ejpam-4571	452	13	)	)	PUNCT
ejpam-4571	452	14	,	,	PUNCT
ejpam-4571	452	15	29	29	NUM
ejpam-4571	452	16	-	-	SYM
ejpam-4571	452	17	43	43	NUM
ejpam-4571	452	18	40	40	NUM
ejpam-4571	452	19	theorem	theorem	NOUN
ejpam-4571	452	20	12	12	NUM
ejpam-4571	452	21	.	.	PUNCT
ejpam-4571	453	1	for	for	ADP
ejpam-4571	453	2	a	a	DET
ejpam-4571	453	3	function	function	NOUN
ejpam-4571	453	4	f	f	NOUN
ejpam-4571	453	5	:	:	PUNCT
ejpam-4571	453	6	(	(	PUNCT
ejpam-4571	453	7	x	x	X
ejpam-4571	453	8	,	,	PUNCT
ejpam-4571	453	9	τ	τ	X
ejpam-4571	453	10	)	)	PUNCT
ejpam-4571	453	11	→	→	SYM
ejpam-4571	453	12	(	(	PUNCT
ejpam-4571	453	13	y	y	PROPN
ejpam-4571	453	14	,	,	PUNCT
ejpam-4571	453	15	σ	σ	PROPN
ejpam-4571	453	16	)	)	PUNCT
ejpam-4571	453	17	,	,	PUNCT
ejpam-4571	453	18	the	the	DET
ejpam-4571	453	19	following	follow	VERB
ejpam-4571	453	20	properties	property	NOUN
ejpam-4571	453	21	are	be	AUX
ejpam-4571	453	22	equivalent	equivalent	ADJ
ejpam-4571	453	23	:	:	PUNCT
ejpam-4571	453	24	(	(	PUNCT
ejpam-4571	453	25	1	1	X
ejpam-4571	453	26	)	)	PUNCT
ejpam-4571	453	27	f	f	PROPN
ejpam-4571	453	28	is	be	AUX
ejpam-4571	453	29	weakly	weakly	ADJ
ejpam-4571	453	30	(	(	PUNCT
ejpam-4571	453	31	λ	λ	X
ejpam-4571	453	32	,	,	PUNCT
ejpam-4571	453	33	b)-continuous	b)-continuous	ADJ
ejpam-4571	453	34	;	;	PUNCT
ejpam-4571	453	35	(	(	PUNCT
ejpam-4571	453	36	2	2	X
ejpam-4571	453	37	)	)	PUNCT
ejpam-4571	453	38	[	[	X
ejpam-4571	453	39	f−1([u(λ	f−1([u(λ	X
ejpam-4571	453	40	,	,	PUNCT
ejpam-4571	453	41	b	b	NOUN
ejpam-4571	453	42	)	)	PUNCT
ejpam-4571	453	43	]	]	PUNCT
ejpam-4571	454	1	(	(	PUNCT
ejpam-4571	454	2	λ	λ	X
ejpam-4571	454	3	,	,	PUNCT
ejpam-4571	454	4	b))](λ	b))](λ	PROPN
ejpam-4571	454	5	,	,	PUNCT
ejpam-4571	454	6	b	b	NOUN
ejpam-4571	454	7	)	)	PUNCT
ejpam-4571	454	8	⊆	⊆	NUM
ejpam-4571	454	9	f−1(u	f−1(u	NOUN
ejpam-4571	454	10	(	(	PUNCT
ejpam-4571	454	11	λ	λ	PROPN
ejpam-4571	454	12	,	,	PUNCT
ejpam-4571	454	13	b	b	NOUN
ejpam-4571	454	14	)	)	PUNCT
ejpam-4571	454	15	)	)	PUNCT
ejpam-4571	454	16	for	for	ADP
ejpam-4571	454	17	every	every	DET
ejpam-4571	454	18	p(λ	p(λ	NOUN
ejpam-4571	454	19	,	,	PUNCT
ejpam-4571	454	20	b)-open	b)-open	AUX
ejpam-4571	454	21	set	set	VERB
ejpam-4571	454	22	u	u	NOUN
ejpam-4571	454	23	of	of	ADP
ejpam-4571	454	24	y	y	PROPN
ejpam-4571	454	25	;	;	PUNCT
ejpam-4571	454	26	(	(	PUNCT
ejpam-4571	454	27	3	3	X
ejpam-4571	454	28	)	)	PUNCT
ejpam-4571	455	1	[	[	X
ejpam-4571	455	2	f−1(u)](λ	f−1(u)](λ	PROPN
ejpam-4571	455	3	,	,	PUNCT
ejpam-4571	455	4	b	b	NOUN
ejpam-4571	455	5	)	)	PUNCT
ejpam-4571	456	1	⊆	⊆	NUM
ejpam-4571	456	2	f−1(u	f−1(u	NOUN
ejpam-4571	456	3	(	(	PUNCT
ejpam-4571	456	4	λ	λ	PROPN
ejpam-4571	456	5	,	,	PUNCT
ejpam-4571	456	6	b	b	NOUN
ejpam-4571	456	7	)	)	PUNCT
ejpam-4571	456	8	)	)	PUNCT
ejpam-4571	456	9	for	for	ADP
ejpam-4571	456	10	every	every	DET
ejpam-4571	456	11	p(λ	p(λ	NOUN
ejpam-4571	456	12	,	,	PUNCT
ejpam-4571	456	13	b)-open	b)-open	VERB
ejpam-4571	456	14	set	set	VERB
ejpam-4571	456	15	u	u	NOUN
ejpam-4571	456	16	of	of	ADP
ejpam-4571	456	17	y	y	PROPN
ejpam-4571	456	18	;	;	PUNCT
ejpam-4571	456	19	(	(	PUNCT
ejpam-4571	456	20	4	4	X
ejpam-4571	456	21	)	)	PUNCT
ejpam-4571	456	22	f−1(u	f−1(u	NOUN
ejpam-4571	456	23	)	)	PUNCT
ejpam-4571	456	24	⊆	⊆	NUM
ejpam-4571	457	1	[	[	X
ejpam-4571	457	2	f−1(u	f−1(u	PROPN
ejpam-4571	457	3	(	(	PUNCT
ejpam-4571	457	4	λ	λ	PROPN
ejpam-4571	457	5	,	,	PUNCT
ejpam-4571	457	6	b))](λ	b))](λ	PROPN
ejpam-4571	457	7	,	,	PUNCT
ejpam-4571	457	8	b	b	NOUN
ejpam-4571	457	9	)	)	PUNCT
ejpam-4571	457	10	for	for	ADP
ejpam-4571	457	11	every	every	DET
ejpam-4571	457	12	p(λ	p(λ	NOUN
ejpam-4571	457	13	,	,	PUNCT
ejpam-4571	457	14	b)-open	b)-open	AUX
ejpam-4571	457	15	set	set	VERB
ejpam-4571	457	16	u	u	NOUN
ejpam-4571	457	17	of	of	ADP
ejpam-4571	457	18	y	y	PROPN
ejpam-4571	457	19	.	.	PUNCT
ejpam-4571	458	1	proof	proof	NOUN
ejpam-4571	458	2	.	.	PUNCT
ejpam-4571	459	1	(	(	PUNCT
ejpam-4571	459	2	1	1	X
ejpam-4571	459	3	)	)	PUNCT
ejpam-4571	459	4	⇒	⇒	NOUN
ejpam-4571	459	5	(	(	PUNCT
ejpam-4571	459	6	2	2	NUM
ejpam-4571	459	7	):	):	PUNCT
ejpam-4571	459	8	let	let	VERB
ejpam-4571	459	9	u	u	PRON
ejpam-4571	459	10	be	be	AUX
ejpam-4571	459	11	any	any	DET
ejpam-4571	459	12	p(λ	p(λ	NOUN
ejpam-4571	459	13	,	,	PUNCT
ejpam-4571	459	14	b)-open	b)-open	VERB
ejpam-4571	459	15	set	set	VERB
ejpam-4571	459	16	of	of	ADP
ejpam-4571	459	17	y	y	PROPN
ejpam-4571	459	18	.	.	PUNCT
ejpam-4571	460	1	then	then	ADV
ejpam-4571	460	2	,	,	PUNCT
ejpam-4571	460	3	u	u	PROPN
ejpam-4571	460	4	(	(	PUNCT
ejpam-4571	460	5	λ	λ	PROPN
ejpam-4571	460	6	,	,	PUNCT
ejpam-4571	460	7	b	b	NOUN
ejpam-4571	460	8	)	)	PUNCT
ejpam-4571	460	9	=	=	NOUN
ejpam-4571	461	1	[	[	X
ejpam-4571	461	2	[	[	X
ejpam-4571	461	3	u	u	X
ejpam-4571	461	4	(	(	PUNCT
ejpam-4571	461	5	λ	λ	PROPN
ejpam-4571	461	6	,	,	PUNCT
ejpam-4571	461	7	b)](λ	b)](λ	NOUN
ejpam-4571	461	8	,	,	PUNCT
ejpam-4571	461	9	b	b	NOUN
ejpam-4571	461	10	)	)	PUNCT
ejpam-4571	461	11	]	]	PUNCT
ejpam-4571	461	12	(	(	PUNCT
ejpam-4571	461	13	λ	λ	X
ejpam-4571	461	14	,	,	PUNCT
ejpam-4571	461	15	b	b	NOUN
ejpam-4571	461	16	)	)	PUNCT
ejpam-4571	461	17	and	and	CCONJ
ejpam-4571	461	18	hence	hence	ADV
ejpam-4571	461	19	u	u	NOUN
ejpam-4571	461	20	(	(	PUNCT
ejpam-4571	461	21	λ	λ	PROPN
ejpam-4571	461	22	,	,	PUNCT
ejpam-4571	461	23	b	b	NOUN
ejpam-4571	461	24	)	)	PUNCT
ejpam-4571	461	25	is	be	AUX
ejpam-4571	461	26	r(λ	r(λ	NOUN
ejpam-4571	461	27	,	,	PUNCT
ejpam-4571	461	28	b)-closed	b)-close	VERB
ejpam-4571	461	29	.	.	PUNCT
ejpam-4571	462	1	by	by	ADP
ejpam-4571	462	2	theorem	theorem	NOUN
ejpam-4571	462	3	11	11	NUM
ejpam-4571	462	4	,	,	PUNCT
ejpam-4571	462	5	[	[	X
ejpam-4571	462	6	f−1([u	f−1([u	INTJ
ejpam-4571	462	7	(	(	PUNCT
ejpam-4571	462	8	λ	λ	PROPN
ejpam-4571	462	9	,	,	PUNCT
ejpam-4571	462	10	b)](λ	b)](λ	NOUN
ejpam-4571	462	11	,	,	PUNCT
ejpam-4571	462	12	b	b	NOUN
ejpam-4571	462	13	)	)	PUNCT
ejpam-4571	462	14	)	)	PUNCT
ejpam-4571	462	15	]	]	PUNCT
ejpam-4571	462	16	(	(	PUNCT
ejpam-4571	462	17	λ	λ	X
ejpam-4571	462	18	,	,	PUNCT
ejpam-4571	462	19	b	b	NOUN
ejpam-4571	462	20	)	)	PUNCT
ejpam-4571	462	21	⊆	⊆	NUM
ejpam-4571	462	22	f−1(u	f−1(u	NOUN
ejpam-4571	462	23	(	(	PUNCT
ejpam-4571	462	24	λ	λ	PROPN
ejpam-4571	462	25	,	,	PUNCT
ejpam-4571	462	26	b	b	NOUN
ejpam-4571	462	27	)	)	PUNCT
ejpam-4571	462	28	)	)	PUNCT
ejpam-4571	462	29	.	.	PUNCT
ejpam-4571	463	1	(	(	PUNCT
ejpam-4571	463	2	2	2	X
ejpam-4571	463	3	)	)	PUNCT
ejpam-4571	463	4	⇒	⇒	NOUN
ejpam-4571	463	5	(	(	PUNCT
ejpam-4571	463	6	3	3	NUM
ejpam-4571	463	7	):	):	PUNCT
ejpam-4571	463	8	let	let	VERB
ejpam-4571	463	9	u	u	PRON
ejpam-4571	463	10	be	be	AUX
ejpam-4571	463	11	any	any	DET
ejpam-4571	463	12	p(λ	p(λ	NOUN
ejpam-4571	463	13	,	,	PUNCT
ejpam-4571	463	14	b)-open	b)-open	VERB
ejpam-4571	463	15	set	set	VERB
ejpam-4571	463	16	of	of	ADP
ejpam-4571	463	17	y	y	PROPN
ejpam-4571	463	18	.	.	PUNCT
ejpam-4571	464	1	then	then	ADV
ejpam-4571	464	2	,	,	PUNCT
ejpam-4571	464	3	u	u	NOUN
ejpam-4571	464	4	⊆	⊆	NUM
ejpam-4571	464	5	[	[	X
ejpam-4571	464	6	u	u	X
ejpam-4571	464	7	(	(	PUNCT
ejpam-4571	464	8	λ	λ	PROPN
ejpam-4571	464	9	,	,	PUNCT
ejpam-4571	464	10	b)](λ	b)](λ	NOUN
ejpam-4571	464	11	,	,	PUNCT
ejpam-4571	464	12	b	b	NOUN
ejpam-4571	464	13	)	)	PUNCT
ejpam-4571	464	14	and	and	CCONJ
ejpam-4571	464	15	by	by	ADP
ejpam-4571	464	16	(	(	PUNCT
ejpam-4571	464	17	2	2	NUM
ejpam-4571	464	18	)	)	PUNCT
ejpam-4571	464	19	,	,	PUNCT
ejpam-4571	464	20	we	we	PRON
ejpam-4571	464	21	have	have	VERB
ejpam-4571	464	22	[	[	X
ejpam-4571	464	23	f−1(u)](λ	f−1(u)](λ	PROPN
ejpam-4571	464	24	,	,	PUNCT
ejpam-4571	464	25	b	b	NOUN
ejpam-4571	464	26	)	)	PUNCT
ejpam-4571	464	27	⊆	⊆	NUM
ejpam-4571	464	28	[	[	X
ejpam-4571	464	29	f−1([u	f−1([u	INTJ
ejpam-4571	464	30	(	(	PUNCT
ejpam-4571	464	31	λ	λ	PROPN
ejpam-4571	464	32	,	,	PUNCT
ejpam-4571	464	33	b)](λ	b)](λ	NOUN
ejpam-4571	464	34	,	,	PUNCT
ejpam-4571	464	35	b	b	NOUN
ejpam-4571	464	36	)	)	PUNCT
ejpam-4571	464	37	)	)	PUNCT
ejpam-4571	464	38	]	]	PUNCT
ejpam-4571	464	39	(	(	PUNCT
ejpam-4571	464	40	λ	λ	X
ejpam-4571	464	41	,	,	PUNCT
ejpam-4571	464	42	b	b	NOUN
ejpam-4571	464	43	)	)	PUNCT
ejpam-4571	464	44	⊆	⊆	NUM
ejpam-4571	464	45	f−1(u	f−1(u	NOUN
ejpam-4571	464	46	(	(	PUNCT
ejpam-4571	464	47	λ	λ	PROPN
ejpam-4571	464	48	,	,	PUNCT
ejpam-4571	464	49	b	b	NOUN
ejpam-4571	464	50	)	)	PUNCT
ejpam-4571	464	51	)	)	PUNCT
ejpam-4571	464	52	.	.	PUNCT
ejpam-4571	465	1	(	(	PUNCT
ejpam-4571	465	2	3	3	X
ejpam-4571	465	3	)	)	PUNCT
ejpam-4571	465	4	⇒	⇒	NOUN
ejpam-4571	465	5	(	(	PUNCT
ejpam-4571	465	6	4	4	NUM
ejpam-4571	465	7	):	):	PUNCT
ejpam-4571	465	8	let	let	VERB
ejpam-4571	465	9	u	u	PRON
ejpam-4571	465	10	be	be	AUX
ejpam-4571	465	11	any	any	DET
ejpam-4571	465	12	p(λ	p(λ	NOUN
ejpam-4571	465	13	,	,	PUNCT
ejpam-4571	465	14	b)-open	b)-open	VERB
ejpam-4571	465	15	set	set	VERB
ejpam-4571	465	16	of	of	ADP
ejpam-4571	465	17	y	y	PROPN
ejpam-4571	465	18	.	.	PUNCT
ejpam-4571	466	1	by	by	ADP
ejpam-4571	466	2	(	(	PUNCT
ejpam-4571	466	3	3	3	NUM
ejpam-4571	466	4	)	)	PUNCT
ejpam-4571	466	5	,	,	PUNCT
ejpam-4571	466	6	f−1(u	f−1(u	PROPN
ejpam-4571	466	7	)	)	PUNCT
ejpam-4571	466	8	⊆	⊆	NUM
ejpam-4571	466	9	f−1([u	f−1([u	NOUN
ejpam-4571	466	10	(	(	PUNCT
ejpam-4571	466	11	λ	λ	PROPN
ejpam-4571	466	12	,	,	PUNCT
ejpam-4571	466	13	b)](λ	b)](λ	NOUN
ejpam-4571	466	14	,	,	PUNCT
ejpam-4571	466	15	b	b	NOUN
ejpam-4571	466	16	)	)	PUNCT
ejpam-4571	466	17	)	)	PUNCT
ejpam-4571	466	18	=	=	PUNCT
ejpam-4571	467	1	x	x	X
ejpam-4571	467	2	−	−	NOUN
ejpam-4571	467	3	f−1([y	f−1([y	NOUN
ejpam-4571	467	4	−	−	NOUN
ejpam-4571	467	5	u	u	NOUN
ejpam-4571	467	6	(	(	PUNCT
ejpam-4571	467	7	λ	λ	PROPN
ejpam-4571	467	8	,	,	PUNCT
ejpam-4571	467	9	b)](λ	b)](λ	NOUN
ejpam-4571	467	10	,	,	PUNCT
ejpam-4571	467	11	b	b	NOUN
ejpam-4571	467	12	)	)	PUNCT
ejpam-4571	467	13	)	)	PUNCT
ejpam-4571	468	1	=	=	PUNCT
ejpam-4571	468	2	x	x	X
ejpam-4571	469	1	−	−	PROPN
ejpam-4571	470	1	[	[	X
ejpam-4571	470	2	f−1(y	f−1(y	PROPN
ejpam-4571	470	3	−	−	PROPN
ejpam-4571	470	4	u	u	NOUN
ejpam-4571	470	5	(	(	PUNCT
ejpam-4571	470	6	λ	λ	PROPN
ejpam-4571	470	7	,	,	PUNCT
ejpam-4571	470	8	b))](λ	b))](λ	PROPN
ejpam-4571	470	9	,	,	PUNCT
ejpam-4571	470	10	b	b	NOUN
ejpam-4571	470	11	)	)	PUNCT
ejpam-4571	470	12	=	=	NOUN
ejpam-4571	471	1	[	[	X
ejpam-4571	471	2	f−1(u	f−1(u	X
ejpam-4571	471	3	(	(	PUNCT
ejpam-4571	471	4	λ	λ	PROPN
ejpam-4571	471	5	,	,	PUNCT
ejpam-4571	471	6	b))](λ	b))](λ	PROPN
ejpam-4571	471	7	,	,	PUNCT
ejpam-4571	471	8	b	b	NOUN
ejpam-4571	471	9	)	)	PUNCT
ejpam-4571	471	10	.	.	PUNCT
ejpam-4571	472	1	(	(	PUNCT
ejpam-4571	472	2	4	4	X
ejpam-4571	472	3	)	)	PUNCT
ejpam-4571	472	4	⇒	⇒	NOUN
ejpam-4571	472	5	(	(	PUNCT
ejpam-4571	472	6	1	1	NUM
ejpam-4571	472	7	):	):	PUNCT
ejpam-4571	472	8	since	since	SCONJ
ejpam-4571	472	9	every	every	PRON
ejpam-4571	472	10	(	(	PUNCT
ejpam-4571	472	11	λ	λ	NOUN
ejpam-4571	472	12	,	,	PUNCT
ejpam-4571	472	13	b)-open	b)-open	AUX
ejpam-4571	472	14	set	set	VERB
ejpam-4571	472	15	is	be	AUX
ejpam-4571	472	16	p(λ	p(λ	NOUN
ejpam-4571	472	17	,	,	PUNCT
ejpam-4571	472	18	b)-open	b)-open	VERB
ejpam-4571	472	19	,	,	PUNCT
ejpam-4571	472	20	by	by	ADP
ejpam-4571	472	21	(	(	PUNCT
ejpam-4571	472	22	4	4	NUM
ejpam-4571	472	23	)	)	PUNCT
ejpam-4571	472	24	and	and	CCONJ
ejpam-4571	472	25	theorem	theorem	VERB
ejpam-4571	472	26	10	10	NUM
ejpam-4571	472	27	,	,	PUNCT
ejpam-4571	472	28	f	f	PROPN
ejpam-4571	472	29	is	be	AUX
ejpam-4571	472	30	weakly	weakly	ADJ
ejpam-4571	472	31	(	(	PUNCT
ejpam-4571	472	32	λ	λ	NOUN
ejpam-4571	472	33	,	,	PUNCT
ejpam-4571	472	34	b)-continuous	b)-continuous	PROPN
ejpam-4571	472	35	.	.	PUNCT
ejpam-4571	473	1	theorem	theorem	VERB
ejpam-4571	473	2	13	13	NUM
ejpam-4571	473	3	.	.	PUNCT
ejpam-4571	474	1	for	for	ADP
ejpam-4571	474	2	a	a	DET
ejpam-4571	474	3	function	function	NOUN
ejpam-4571	474	4	f	f	NOUN
ejpam-4571	474	5	:	:	PUNCT
ejpam-4571	474	6	(	(	PUNCT
ejpam-4571	474	7	x	x	X
ejpam-4571	474	8	,	,	PUNCT
ejpam-4571	474	9	τ	τ	X
ejpam-4571	474	10	)	)	PUNCT
ejpam-4571	474	11	→	→	SYM
ejpam-4571	474	12	(	(	PUNCT
ejpam-4571	474	13	y	y	PROPN
ejpam-4571	474	14	,	,	PUNCT
ejpam-4571	474	15	σ	σ	PROPN
ejpam-4571	474	16	)	)	PUNCT
ejpam-4571	474	17	,	,	PUNCT
ejpam-4571	474	18	the	the	DET
ejpam-4571	474	19	following	follow	VERB
ejpam-4571	474	20	properties	property	NOUN
ejpam-4571	474	21	are	be	AUX
ejpam-4571	474	22	equivalent	equivalent	ADJ
ejpam-4571	474	23	:	:	PUNCT
ejpam-4571	474	24	(	(	PUNCT
ejpam-4571	474	25	1	1	X
ejpam-4571	474	26	)	)	PUNCT
ejpam-4571	474	27	f	f	PROPN
ejpam-4571	474	28	is	be	AUX
ejpam-4571	474	29	weakly	weakly	ADJ
ejpam-4571	474	30	(	(	PUNCT
ejpam-4571	474	31	λ	λ	X
ejpam-4571	474	32	,	,	PUNCT
ejpam-4571	474	33	b)-continuous	b)-continuous	ADJ
ejpam-4571	474	34	;	;	PUNCT
ejpam-4571	474	35	(	(	PUNCT
ejpam-4571	474	36	2	2	X
ejpam-4571	474	37	)	)	PUNCT
ejpam-4571	474	38	[	[	X
ejpam-4571	474	39	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4571	474	40	,	,	PUNCT
ejpam-4571	474	41	b)](λ	b)](λ	NOUN
ejpam-4571	474	42	,	,	PUNCT
ejpam-4571	474	43	b	b	NOUN
ejpam-4571	474	44	)	)	PUNCT
ejpam-4571	474	45	)	)	PUNCT
ejpam-4571	474	46	]	]	PUNCT
ejpam-4571	475	1	(	(	PUNCT
ejpam-4571	475	2	λ	λ	X
ejpam-4571	475	3	,	,	PUNCT
ejpam-4571	475	4	b	b	NOUN
ejpam-4571	475	5	)	)	PUNCT
ejpam-4571	475	6	⊆	⊆	NUM
ejpam-4571	475	7	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4571	475	8	,	,	PUNCT
ejpam-4571	475	9	b	b	NOUN
ejpam-4571	475	10	)	)	PUNCT
ejpam-4571	475	11	)	)	PUNCT
ejpam-4571	475	12	for	for	ADP
ejpam-4571	475	13	every	every	DET
ejpam-4571	475	14	subset	subset	NOUN
ejpam-4571	475	15	a	a	PRON
ejpam-4571	475	16	of	of	ADP
ejpam-4571	475	17	y	y	PROPN
ejpam-4571	475	18	;	;	PUNCT
ejpam-4571	475	19	(	(	PUNCT
ejpam-4571	475	20	3	3	X
ejpam-4571	475	21	)	)	PUNCT
ejpam-4571	476	1	[	[	X
ejpam-4571	476	2	f−1(f(λ	f−1(f(λ	X
ejpam-4571	476	3	,	,	PUNCT
ejpam-4571	476	4	b	b	NOUN
ejpam-4571	476	5	)	)	PUNCT
ejpam-4571	476	6	)	)	PUNCT
ejpam-4571	476	7	]	]	PUNCT
ejpam-4571	476	8	(	(	PUNCT
ejpam-4571	476	9	λ	λ	X
ejpam-4571	476	10	,	,	PUNCT
ejpam-4571	476	11	b	b	NOUN
ejpam-4571	476	12	)	)	PUNCT
ejpam-4571	476	13	⊆	⊆	NUM
ejpam-4571	476	14	f−1(f	f−1(f	PROPN
ejpam-4571	476	15	)	)	PUNCT
ejpam-4571	476	16	for	for	ADP
ejpam-4571	476	17	every	every	DET
ejpam-4571	476	18	r(λ	r(λ	NOUN
ejpam-4571	476	19	,	,	PUNCT
ejpam-4571	476	20	b)-closed	b)-close	VERB
ejpam-4571	476	21	set	set	ADJ
ejpam-4571	476	22	f	f	PROPN
ejpam-4571	476	23	of	of	ADP
ejpam-4571	476	24	y	y	PROPN
ejpam-4571	476	25	;	;	PUNCT
ejpam-4571	476	26	(	(	PUNCT
ejpam-4571	476	27	4	4	X
ejpam-4571	476	28	)	)	PUNCT
ejpam-4571	477	1	[	[	X
ejpam-4571	477	2	f−1(u)](λ	f−1(u)](λ	PROPN
ejpam-4571	477	3	,	,	PUNCT
ejpam-4571	477	4	b	b	NOUN
ejpam-4571	477	5	)	)	PUNCT
ejpam-4571	478	1	⊆	⊆	NUM
ejpam-4571	478	2	f−1(u	f−1(u	NOUN
ejpam-4571	478	3	(	(	PUNCT
ejpam-4571	478	4	λ	λ	PROPN
ejpam-4571	478	5	,	,	PUNCT
ejpam-4571	478	6	b	b	NOUN
ejpam-4571	478	7	)	)	PUNCT
ejpam-4571	478	8	)	)	PUNCT
ejpam-4571	478	9	for	for	ADP
ejpam-4571	478	10	every	every	DET
ejpam-4571	478	11	(	(	PUNCT
ejpam-4571	478	12	λ	λ	NOUN
ejpam-4571	478	13	,	,	PUNCT
ejpam-4571	478	14	b)-open	b)-open	AUX
ejpam-4571	478	15	set	set	VERB
ejpam-4571	478	16	u	u	NOUN
ejpam-4571	478	17	of	of	ADP
ejpam-4571	478	18	y	y	PROPN
ejpam-4571	478	19	;	;	PUNCT
ejpam-4571	478	20	(	(	PUNCT
ejpam-4571	478	21	5	5	X
ejpam-4571	478	22	)	)	PUNCT
ejpam-4571	478	23	f−1(u	f−1(u	NOUN
ejpam-4571	478	24	)	)	PUNCT
ejpam-4571	478	25	⊆	⊆	NUM
ejpam-4571	479	1	[	[	X
ejpam-4571	479	2	f−1(u	f−1(u	PROPN
ejpam-4571	479	3	(	(	PUNCT
ejpam-4571	479	4	λ	λ	PROPN
ejpam-4571	479	5	,	,	PUNCT
ejpam-4571	479	6	b))](λ	b))](λ	PROPN
ejpam-4571	479	7	,	,	PUNCT
ejpam-4571	479	8	b	b	NOUN
ejpam-4571	479	9	)	)	PUNCT
ejpam-4571	479	10	for	for	ADP
ejpam-4571	479	11	every	every	DET
ejpam-4571	479	12	(	(	PUNCT
ejpam-4571	479	13	λ	λ	NOUN
ejpam-4571	479	14	,	,	PUNCT
ejpam-4571	479	15	b)-open	b)-open	AUX
ejpam-4571	479	16	set	set	VERB
ejpam-4571	479	17	u	u	NOUN
ejpam-4571	479	18	of	of	ADP
ejpam-4571	479	19	y	y	PROPN
ejpam-4571	479	20	;	;	PUNCT
ejpam-4571	479	21	(	(	PUNCT
ejpam-4571	479	22	6	6	X
ejpam-4571	479	23	)	)	PUNCT
ejpam-4571	480	1	[	[	X
ejpam-4571	480	2	f−1(u)](λ	f−1(u)](λ	PROPN
ejpam-4571	480	3	,	,	PUNCT
ejpam-4571	480	4	b	b	NOUN
ejpam-4571	480	5	)	)	PUNCT
ejpam-4571	481	1	⊆	⊆	NUM
ejpam-4571	481	2	f−1(u	f−1(u	NOUN
ejpam-4571	481	3	(	(	PUNCT
ejpam-4571	481	4	λ	λ	PROPN
ejpam-4571	481	5	,	,	PUNCT
ejpam-4571	481	6	b	b	NOUN
ejpam-4571	481	7	)	)	PUNCT
ejpam-4571	481	8	)	)	PUNCT
ejpam-4571	481	9	for	for	ADP
ejpam-4571	481	10	every	every	DET
ejpam-4571	481	11	p(λ	p(λ	NOUN
ejpam-4571	481	12	,	,	PUNCT
ejpam-4571	481	13	b)-open	b)-open	VERB
ejpam-4571	481	14	set	set	VERB
ejpam-4571	481	15	u	u	NOUN
ejpam-4571	481	16	of	of	ADP
ejpam-4571	481	17	y	y	PROPN
ejpam-4571	481	18	;	;	PUNCT
ejpam-4571	481	19	(	(	PUNCT
ejpam-4571	481	20	7	7	X
ejpam-4571	481	21	)	)	PUNCT
ejpam-4571	481	22	f−1(u	f−1(u	NOUN
ejpam-4571	481	23	)	)	PUNCT
ejpam-4571	481	24	⊆	⊆	NUM
ejpam-4571	482	1	[	[	X
ejpam-4571	482	2	f−1(u	f−1(u	PROPN
ejpam-4571	482	3	(	(	PUNCT
ejpam-4571	482	4	λ	λ	PROPN
ejpam-4571	482	5	,	,	PUNCT
ejpam-4571	482	6	b))](λ	b))](λ	PROPN
ejpam-4571	482	7	,	,	PUNCT
ejpam-4571	482	8	b	b	NOUN
ejpam-4571	482	9	)	)	PUNCT
ejpam-4571	482	10	for	for	ADP
ejpam-4571	482	11	every	every	DET
ejpam-4571	482	12	p(λ	p(λ	NOUN
ejpam-4571	482	13	,	,	PUNCT
ejpam-4571	482	14	b)-open	b)-open	AUX
ejpam-4571	482	15	set	set	VERB
ejpam-4571	482	16	u	u	NOUN
ejpam-4571	482	17	of	of	ADP
ejpam-4571	482	18	y	y	PROPN
ejpam-4571	482	19	.	.	PUNCT
ejpam-4571	483	1	proof	proof	NOUN
ejpam-4571	483	2	.	.	PUNCT
ejpam-4571	484	1	(	(	PUNCT
ejpam-4571	484	2	1	1	X
ejpam-4571	484	3	)	)	PUNCT
ejpam-4571	484	4	⇒	⇒	NOUN
ejpam-4571	484	5	(	(	PUNCT
ejpam-4571	484	6	2	2	NUM
ejpam-4571	484	7	):	):	PUNCT
ejpam-4571	484	8	let	let	VERB
ejpam-4571	484	9	a	a	DET
ejpam-4571	484	10	be	be	AUX
ejpam-4571	484	11	any	any	DET
ejpam-4571	484	12	subset	subset	NOUN
ejpam-4571	484	13	of	of	ADP
ejpam-4571	484	14	y	y	PROPN
ejpam-4571	484	15	and	and	CCONJ
ejpam-4571	484	16	x	x	PUNCT
ejpam-4571	484	17	∈	∈	PROPN
ejpam-4571	484	18	x	x	PUNCT
ejpam-4571	484	19	−	−	PROPN
ejpam-4571	484	20	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4571	484	21	,	,	PUNCT
ejpam-4571	484	22	b	b	NOUN
ejpam-4571	484	23	)	)	PUNCT
ejpam-4571	484	24	)	)	PUNCT
ejpam-4571	484	25	.	.	PUNCT
ejpam-4571	485	1	then	then	ADV
ejpam-4571	485	2	,	,	PUNCT
ejpam-4571	485	3	f(x	f(x	PROPN
ejpam-4571	485	4	)	)	PUNCT
ejpam-4571	485	5	∈	∈	PROPN
ejpam-4571	486	1	y	y	PROPN
ejpam-4571	486	2	−	−	PROPN
ejpam-4571	486	3	a(λ	a(λ	PROPN
ejpam-4571	486	4	,	,	PUNCT
ejpam-4571	486	5	b	b	NOUN
ejpam-4571	486	6	)	)	PUNCT
ejpam-4571	486	7	and	and	CCONJ
ejpam-4571	486	8	there	there	PRON
ejpam-4571	486	9	exists	exist	VERB
ejpam-4571	486	10	a	a	DET
ejpam-4571	486	11	(	(	PUNCT
ejpam-4571	486	12	λ	λ	NOUN
ejpam-4571	486	13	,	,	PUNCT
ejpam-4571	486	14	b)-open	b)-open	AUX
ejpam-4571	486	15	set	set	VERB
ejpam-4571	486	16	u	u	NOUN
ejpam-4571	486	17	of	of	ADP
ejpam-4571	486	18	y	y	PROPN
ejpam-4571	486	19	containing	contain	VERB
ejpam-4571	486	20	f(x	f(x	PROPN
ejpam-4571	486	21	)	)	PUNCT
ejpam-4571	486	22	such	such	ADJ
ejpam-4571	486	23	that	that	SCONJ
ejpam-4571	486	24	u	u	NOUN
ejpam-4571	486	25	∩a	∩a	NOUN
ejpam-4571	486	26	=	=	PUNCT
ejpam-4571	486	27	∅.	∅.	VERB
ejpam-4571	486	28	thus	thus	ADV
ejpam-4571	486	29	,	,	PUNCT
ejpam-4571	486	30	u	u	PROPN
ejpam-4571	486	31	(	(	PUNCT
ejpam-4571	486	32	λ	λ	PROPN
ejpam-4571	486	33	,	,	PUNCT
ejpam-4571	486	34	b)∩	b)∩	PUNCT
ejpam-4571	487	1	[	[	X
ejpam-4571	487	2	a(λ	a(λ	ADV
ejpam-4571	487	3	,	,	PUNCT
ejpam-4571	487	4	b)](λ	b)](λ	NOUN
ejpam-4571	487	5	,	,	PUNCT
ejpam-4571	487	6	b	b	NOUN
ejpam-4571	487	7	)	)	PUNCT
ejpam-4571	487	8	=	=	PUNCT
ejpam-4571	487	9	∅.	∅.	NOUN
ejpam-4571	487	10	since	since	SCONJ
ejpam-4571	487	11	f	f	PROPN
ejpam-4571	487	12	is	be	AUX
ejpam-4571	487	13	weakly	weakly	ADJ
ejpam-4571	487	14	(	(	PUNCT
ejpam-4571	487	15	λ	λ	NOUN
ejpam-4571	487	16	,	,	PUNCT
ejpam-4571	487	17	b)-continuous	b)-continuous	ADJ
ejpam-4571	487	18	,	,	PUNCT
ejpam-4571	487	19	there	there	PRON
ejpam-4571	487	20	exists	exist	VERB
ejpam-4571	487	21	a	a	DET
ejpam-4571	487	22	(	(	PUNCT
ejpam-4571	487	23	λ	λ	NOUN
ejpam-4571	487	24	,	,	PUNCT
ejpam-4571	487	25	b)-open	b)-open	VERB
ejpam-4571	487	26	setw	setw	NOUN
ejpam-4571	487	27	containing	contain	VERB
ejpam-4571	487	28	x	x	PUNCT
ejpam-4571	487	29	such	such	ADJ
ejpam-4571	487	30	that	that	SCONJ
ejpam-4571	487	31	f(w	f(w	PROPN
ejpam-4571	487	32	)	)	PUNCT
ejpam-4571	488	1	⊆	⊆	NUM
ejpam-4571	488	2	u	u	NOUN
ejpam-4571	488	3	(	(	PUNCT
ejpam-4571	488	4	λ	λ	PROPN
ejpam-4571	488	5	,	,	PUNCT
ejpam-4571	488	6	b	b	NOUN
ejpam-4571	488	7	)	)	PUNCT
ejpam-4571	488	8	.	.	PUNCT
ejpam-4571	489	1	then	then	ADV
ejpam-4571	489	2	,	,	PUNCT
ejpam-4571	489	3	w∩f−1([a(λ	w∩f−1([a(λ	PROPN
ejpam-4571	489	4	,	,	PUNCT
ejpam-4571	489	5	b)](λ	b)](λ	NOUN
ejpam-4571	489	6	,	,	PUNCT
ejpam-4571	489	7	b	b	NOUN
ejpam-4571	489	8	)	)	PUNCT
ejpam-4571	489	9	)	)	PUNCT
ejpam-4571	490	1	=	=	NOUN
ejpam-4571	490	2	∅	∅	NOUN
ejpam-4571	490	3	and	and	CCONJ
ejpam-4571	490	4	hence	hence	ADV
ejpam-4571	490	5	x	x	X
ejpam-4571	490	6	∈	∈	PROPN
ejpam-4571	490	7	x−[f−1([a(λ	x−[f−1([a(λ	PROPN
ejpam-4571	490	8	,	,	PUNCT
ejpam-4571	490	9	b)](λ	b)](λ	PROPN
ejpam-4571	490	10	,	,	PUNCT
ejpam-4571	490	11	b	b	NOUN
ejpam-4571	490	12	)	)	PUNCT
ejpam-4571	490	13	)	)	PUNCT
ejpam-4571	490	14	]	]	PUNCT
ejpam-4571	490	15	(	(	PUNCT
ejpam-4571	490	16	λ	λ	X
ejpam-4571	490	17	,	,	PUNCT
ejpam-4571	490	18	b	b	NOUN
ejpam-4571	490	19	)	)	PUNCT
ejpam-4571	490	20	.	.	PUNCT
ejpam-4571	491	1	therefore	therefore	ADV
ejpam-4571	491	2	,	,	PUNCT
ejpam-4571	491	3	[	[	X
ejpam-4571	491	4	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4571	491	5	,	,	PUNCT
ejpam-4571	491	6	b)](λ	b)](λ	NOUN
ejpam-4571	491	7	,	,	PUNCT
ejpam-4571	491	8	b	b	NOUN
ejpam-4571	491	9	)	)	PUNCT
ejpam-4571	491	10	)	)	PUNCT
ejpam-4571	491	11	]	]	PUNCT
ejpam-4571	492	1	(	(	PUNCT
ejpam-4571	492	2	λ	λ	X
ejpam-4571	492	3	,	,	PUNCT
ejpam-4571	492	4	b	b	NOUN
ejpam-4571	492	5	)	)	PUNCT
ejpam-4571	492	6	⊆	⊆	NUM
ejpam-4571	492	7	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4571	492	8	,	,	PUNCT
ejpam-4571	492	9	b	b	NOUN
ejpam-4571	492	10	)	)	PUNCT
ejpam-4571	492	11	)	)	PUNCT
ejpam-4571	492	12	.	.	PUNCT
ejpam-4571	493	1	(	(	PUNCT
ejpam-4571	493	2	2	2	X
ejpam-4571	493	3	)	)	PUNCT
ejpam-4571	493	4	⇒	⇒	NOUN
ejpam-4571	493	5	(	(	PUNCT
ejpam-4571	493	6	3	3	NUM
ejpam-4571	493	7	):	):	PUNCT
ejpam-4571	493	8	let	let	VERB
ejpam-4571	493	9	f	f	PRON
ejpam-4571	493	10	be	be	AUX
ejpam-4571	493	11	any	any	DET
ejpam-4571	493	12	r(λ	r(λ	NOUN
ejpam-4571	493	13	,	,	PUNCT
ejpam-4571	493	14	b)-closed	b)-close	VERB
ejpam-4571	493	15	set	set	NOUN
ejpam-4571	493	16	of	of	ADP
ejpam-4571	493	17	y	y	PROPN
ejpam-4571	493	18	.	.	PUNCT
ejpam-4571	494	1	by	by	ADP
ejpam-4571	494	2	(	(	PUNCT
ejpam-4571	494	3	2	2	NUM
ejpam-4571	494	4	)	)	PUNCT
ejpam-4571	494	5	,	,	PUNCT
ejpam-4571	494	6	we	we	PRON
ejpam-4571	494	7	have	have	VERB
ejpam-4571	494	8	[	[	X
ejpam-4571	494	9	f−1(f(λ	f−1(f(λ	ADJ
ejpam-4571	494	10	,	,	PUNCT
ejpam-4571	494	11	b	b	NOUN
ejpam-4571	494	12	)	)	PUNCT
ejpam-4571	494	13	)	)	PUNCT
ejpam-4571	494	14	]	]	PUNCT
ejpam-4571	495	1	(	(	PUNCT
ejpam-4571	495	2	λ	λ	X
ejpam-4571	495	3	,	,	PUNCT
ejpam-4571	495	4	b	b	NOUN
ejpam-4571	495	5	)	)	PUNCT
ejpam-4571	495	6	=	=	NOUN
ejpam-4571	496	1	[	[	X
ejpam-4571	496	2	f−1([[f(λ	f−1([[f(λ	NOUN
ejpam-4571	496	3	,	,	PUNCT
ejpam-4571	496	4	b	b	NOUN
ejpam-4571	496	5	)	)	PUNCT
ejpam-4571	496	6	]	]	PUNCT
ejpam-4571	496	7	(	(	PUNCT
ejpam-4571	496	8	λ	λ	INTJ
ejpam-4571	496	9	,	,	PUNCT
ejpam-4571	496	10	b)](λ	b)](λ	NOUN
ejpam-4571	496	11	,	,	PUNCT
ejpam-4571	496	12	b	b	NOUN
ejpam-4571	496	13	)	)	PUNCT
ejpam-4571	496	14	)	)	PUNCT
ejpam-4571	496	15	]	]	PUNCT
ejpam-4571	497	1	(	(	PUNCT
ejpam-4571	497	2	λ	λ	X
ejpam-4571	497	3	,	,	PUNCT
ejpam-4571	497	4	b	b	NOUN
ejpam-4571	497	5	)	)	PUNCT
ejpam-4571	497	6	⊆	⊆	NUM
ejpam-4571	497	7	f−1([f(λ	f−1([f(λ	NOUN
ejpam-4571	497	8	,	,	PUNCT
ejpam-4571	497	9	b	b	NOUN
ejpam-4571	497	10	)	)	PUNCT
ejpam-4571	497	11	]	]	PUNCT
ejpam-4571	497	12	(	(	PUNCT
ejpam-4571	497	13	λ	λ	X
ejpam-4571	497	14	,	,	PUNCT
ejpam-4571	497	15	b	b	NOUN
ejpam-4571	497	16	)	)	PUNCT
ejpam-4571	497	17	)	)	PUNCT
ejpam-4571	498	1	=	=	SYM
ejpam-4571	498	2	f−1(f	f−1(f	PROPN
ejpam-4571	498	3	)	)	PUNCT
ejpam-4571	498	4	.	.	PUNCT
ejpam-4571	499	1	(	(	PUNCT
ejpam-4571	499	2	3	3	X
ejpam-4571	499	3	)	)	PUNCT
ejpam-4571	499	4	⇒	⇒	NOUN
ejpam-4571	499	5	(	(	PUNCT
ejpam-4571	499	6	4	4	NUM
ejpam-4571	499	7	):	):	PUNCT
ejpam-4571	499	8	let	let	VERB
ejpam-4571	499	9	u	u	PRON
ejpam-4571	499	10	be	be	AUX
ejpam-4571	499	11	any	any	DET
ejpam-4571	499	12	(	(	PUNCT
ejpam-4571	499	13	λ	λ	NOUN
ejpam-4571	499	14	,	,	PUNCT
ejpam-4571	499	15	b)-open	b)-open	VERB
ejpam-4571	499	16	set	set	VERB
ejpam-4571	499	17	of	of	ADP
ejpam-4571	499	18	y	y	PROPN
ejpam-4571	499	19	.	.	PUNCT
ejpam-4571	500	1	since	since	SCONJ
ejpam-4571	500	2	u	u	PRON
ejpam-4571	500	3	(	(	PUNCT
ejpam-4571	500	4	λ	λ	PROPN
ejpam-4571	500	5	,	,	PUNCT
ejpam-4571	500	6	b	b	NOUN
ejpam-4571	500	7	)	)	PUNCT
ejpam-4571	500	8	is	be	AUX
ejpam-4571	500	9	r(λ	r(λ	NOUN
ejpam-4571	500	10	,	,	PUNCT
ejpam-4571	500	11	b)-closed	b)-close	VERB
ejpam-4571	500	12	,	,	PUNCT
ejpam-4571	500	13	we	we	PRON
ejpam-4571	500	14	have	have	VERB
ejpam-4571	500	15	[	[	NOUN
ejpam-4571	500	16	f−1(u)](λ	f−1(u)](λ	PROPN
ejpam-4571	500	17	,	,	PUNCT
ejpam-4571	500	18	b	b	NOUN
ejpam-4571	500	19	)	)	PUNCT
ejpam-4571	500	20	⊆	⊆	NUM
ejpam-4571	501	1	[	[	X
ejpam-4571	501	2	f−1([u	f−1([u	INTJ
ejpam-4571	501	3	(	(	PUNCT
ejpam-4571	501	4	λ	λ	PROPN
ejpam-4571	501	5	,	,	PUNCT
ejpam-4571	501	6	b)](λ	b)](λ	NOUN
ejpam-4571	501	7	,	,	PUNCT
ejpam-4571	501	8	b	b	NOUN
ejpam-4571	501	9	)	)	PUNCT
ejpam-4571	501	10	)	)	PUNCT
ejpam-4571	501	11	]	]	PUNCT
ejpam-4571	501	12	(	(	PUNCT
ejpam-4571	501	13	λ	λ	X
ejpam-4571	501	14	,	,	PUNCT
ejpam-4571	501	15	b	b	NOUN
ejpam-4571	501	16	)	)	PUNCT
ejpam-4571	501	17	⊆	⊆	NUM
ejpam-4571	501	18	f−1(u	f−1(u	NOUN
ejpam-4571	501	19	(	(	PUNCT
ejpam-4571	501	20	λ	λ	PROPN
ejpam-4571	501	21	,	,	PUNCT
ejpam-4571	501	22	b	b	NOUN
ejpam-4571	501	23	)	)	PUNCT
ejpam-4571	501	24	)	)	PUNCT
ejpam-4571	501	25	.	.	PUNCT
ejpam-4571	502	1	c.	c.	PROPN
ejpam-4571	502	2	boonpok	boonpok	PROPN
ejpam-4571	502	3	,	,	PUNCT
ejpam-4571	502	4	n.	n.	PROPN
ejpam-4571	502	5	srisarakham	srisarakham	PROPN
ejpam-4571	502	6	/	/	SYM
ejpam-4571	502	7	eur	eur	PROPN
ejpam-4571	502	8	.	.	PUNCT
ejpam-4571	503	1	j.	j.	PROPN
ejpam-4571	503	2	pure	pure	PROPN
ejpam-4571	503	3	appl	appl	PROPN
ejpam-4571	503	4	.	.	PROPN
ejpam-4571	503	5	math	math	PROPN
ejpam-4571	503	6	,	,	PUNCT
ejpam-4571	503	7	16	16	NUM
ejpam-4571	503	8	(	(	PUNCT
ejpam-4571	503	9	1	1	NUM
ejpam-4571	503	10	)	)	PUNCT
ejpam-4571	503	11	(	(	PUNCT
ejpam-4571	503	12	2023	2023	NUM
ejpam-4571	503	13	)	)	PUNCT
ejpam-4571	503	14	,	,	PUNCT
ejpam-4571	503	15	29	29	NUM
ejpam-4571	503	16	-	-	SYM
ejpam-4571	503	17	43	43	NUM
ejpam-4571	503	18	41	41	NUM
ejpam-4571	503	19	(	(	PUNCT
ejpam-4571	503	20	4	4	NUM
ejpam-4571	503	21	)	)	PUNCT
ejpam-4571	503	22	⇒	⇒	NOUN
ejpam-4571	503	23	(	(	PUNCT
ejpam-4571	503	24	5	5	NUM
ejpam-4571	503	25	):	):	PUNCT
ejpam-4571	503	26	let	let	VERB
ejpam-4571	503	27	u	u	PRON
ejpam-4571	503	28	be	be	AUX
ejpam-4571	503	29	any	any	DET
ejpam-4571	503	30	(	(	PUNCT
ejpam-4571	503	31	λ	λ	NOUN
ejpam-4571	503	32	,	,	PUNCT
ejpam-4571	503	33	b)-open	b)-open	VERB
ejpam-4571	503	34	set	set	VERB
ejpam-4571	503	35	of	of	ADP
ejpam-4571	503	36	y	y	PROPN
ejpam-4571	503	37	.	.	PUNCT
ejpam-4571	504	1	since	since	SCONJ
ejpam-4571	504	2	y	y	PROPN
ejpam-4571	504	3	−	−	PROPN
ejpam-4571	504	4	u	u	PROPN
ejpam-4571	504	5	(	(	PUNCT
ejpam-4571	504	6	λ	λ	PROPN
ejpam-4571	504	7	,	,	PUNCT
ejpam-4571	504	8	b	b	NOUN
ejpam-4571	504	9	)	)	PUNCT
ejpam-4571	504	10	is	be	AUX
ejpam-4571	504	11	(	(	PUNCT
ejpam-4571	504	12	λ	λ	X
ejpam-4571	504	13	,	,	PUNCT
ejpam-4571	504	14	b)-open	b)-open	PUNCT
ejpam-4571	504	15	and	and	CCONJ
ejpam-4571	504	16	by	by	ADP
ejpam-4571	504	17	(	(	PUNCT
ejpam-4571	504	18	4	4	NUM
ejpam-4571	504	19	)	)	PUNCT
ejpam-4571	504	20	,	,	PUNCT
ejpam-4571	504	21	x	x	X
ejpam-4571	505	1	−	−	NOUN
ejpam-4571	506	1	[	[	X
ejpam-4571	506	2	f−1(u	f−1(u	X
ejpam-4571	506	3	(	(	PUNCT
ejpam-4571	506	4	λ	λ	PROPN
ejpam-4571	506	5	,	,	PUNCT
ejpam-4571	506	6	b))](λ	b))](λ	PROPN
ejpam-4571	506	7	,	,	PUNCT
ejpam-4571	506	8	b	b	NOUN
ejpam-4571	506	9	)	)	PUNCT
ejpam-4571	506	10	=	=	NOUN
ejpam-4571	507	1	[	[	X
ejpam-4571	507	2	f−1(y	f−1(y	X
ejpam-4571	507	3	−u	−u	PROPN
ejpam-4571	507	4	(	(	PUNCT
ejpam-4571	507	5	λ	λ	PROPN
ejpam-4571	507	6	,	,	PUNCT
ejpam-4571	507	7	b))](λ	b))](λ	PROPN
ejpam-4571	507	8	,	,	PUNCT
ejpam-4571	507	9	b	b	NOUN
ejpam-4571	507	10	)	)	PUNCT
ejpam-4571	507	11	⊆	⊆	NUM
ejpam-4571	507	12	f−1([y	f−1([y	NOUN
ejpam-4571	507	13	−u	−u	NOUN
ejpam-4571	507	14	(	(	PUNCT
ejpam-4571	507	15	λ	λ	PROPN
ejpam-4571	507	16	,	,	PUNCT
ejpam-4571	507	17	b)](λ	b)](λ	NOUN
ejpam-4571	507	18	,	,	PUNCT
ejpam-4571	507	19	b	b	NOUN
ejpam-4571	507	20	)	)	PUNCT
ejpam-4571	507	21	)	)	PUNCT
ejpam-4571	508	1	⊆	⊆	NUM
ejpam-4571	508	2	x	x	SYM
ejpam-4571	508	3	−	−	NUM
ejpam-4571	508	4	f−1(u	f−1(u	NOUN
ejpam-4571	508	5	)	)	PUNCT
ejpam-4571	508	6	.	.	PUNCT
ejpam-4571	509	1	thus	thus	ADV
ejpam-4571	509	2	,	,	PUNCT
ejpam-4571	509	3	f−1(u	f−1(u	PROPN
ejpam-4571	509	4	)	)	PUNCT
ejpam-4571	509	5	⊆	⊆	NUM
ejpam-4571	510	1	[	[	X
ejpam-4571	510	2	f−1(u	f−1(u	PROPN
ejpam-4571	510	3	(	(	PUNCT
ejpam-4571	510	4	λ	λ	PROPN
ejpam-4571	510	5	,	,	PUNCT
ejpam-4571	510	6	b))](λ	b))](λ	PROPN
ejpam-4571	510	7	,	,	PUNCT
ejpam-4571	510	8	b	b	NOUN
ejpam-4571	510	9	)	)	PUNCT
ejpam-4571	510	10	.	.	PUNCT
ejpam-4571	511	1	(	(	PUNCT
ejpam-4571	511	2	5	5	X
ejpam-4571	511	3	)	)	PUNCT
ejpam-4571	511	4	⇒	⇒	NOUN
ejpam-4571	511	5	(	(	PUNCT
ejpam-4571	511	6	1	1	NUM
ejpam-4571	511	7	):	):	PUNCT
ejpam-4571	511	8	let	let	VERB
ejpam-4571	511	9	x	x	PUNCT
ejpam-4571	511	10	∈	∈	PROPN
ejpam-4571	511	11	x	x	X
ejpam-4571	511	12	and	and	CCONJ
ejpam-4571	511	13	u	u	PRON
ejpam-4571	511	14	be	be	VERB
ejpam-4571	511	15	any	any	DET
ejpam-4571	511	16	(	(	PUNCT
ejpam-4571	511	17	λ	λ	NOUN
ejpam-4571	511	18	,	,	PUNCT
ejpam-4571	511	19	b)-open	b)-open	VERB
ejpam-4571	511	20	set	set	VERB
ejpam-4571	511	21	of	of	ADP
ejpam-4571	511	22	y	y	PROPN
ejpam-4571	511	23	containing	contain	VERB
ejpam-4571	511	24	f(x	f(x	PROPN
ejpam-4571	511	25	)	)	PUNCT
ejpam-4571	511	26	.	.	PUNCT
ejpam-4571	512	1	then	then	ADV
ejpam-4571	512	2	,	,	PUNCT
ejpam-4571	512	3	x	x	PROPN
ejpam-4571	512	4	∈	∈	PROPN
ejpam-4571	512	5	f−1(u	f−1(u	PROPN
ejpam-4571	512	6	)	)	PUNCT
ejpam-4571	512	7	⊆	⊆	NUM
ejpam-4571	513	1	[	[	X
ejpam-4571	513	2	f−1(u	f−1(u	PROPN
ejpam-4571	513	3	(	(	PUNCT
ejpam-4571	513	4	λ	λ	PROPN
ejpam-4571	513	5	,	,	PUNCT
ejpam-4571	513	6	b))](λ	b))](λ	PROPN
ejpam-4571	513	7	,	,	PUNCT
ejpam-4571	513	8	b	b	NOUN
ejpam-4571	513	9	)	)	PUNCT
ejpam-4571	513	10	.	.	PUNCT
ejpam-4571	514	1	put	put	VERB
ejpam-4571	514	2	w	w	NOUN
ejpam-4571	514	3	=	=	PUNCT
ejpam-4571	515	1	[	[	X
ejpam-4571	515	2	f−1(u	f−1(u	X
ejpam-4571	515	3	(	(	PUNCT
ejpam-4571	515	4	λ	λ	PROPN
ejpam-4571	515	5	,	,	PUNCT
ejpam-4571	515	6	b))](λ	b))](λ	PROPN
ejpam-4571	515	7	,	,	PUNCT
ejpam-4571	515	8	b	b	NOUN
ejpam-4571	515	9	)	)	PUNCT
ejpam-4571	515	10	.	.	PUNCT
ejpam-4571	516	1	then	then	ADV
ejpam-4571	516	2	,	,	PUNCT
ejpam-4571	516	3	w	w	PROPN
ejpam-4571	516	4	is	be	AUX
ejpam-4571	516	5	a	a	DET
ejpam-4571	516	6	(	(	PUNCT
ejpam-4571	516	7	λ	λ	NOUN
ejpam-4571	516	8	,	,	PUNCT
ejpam-4571	516	9	b)-open	b)-open	VERB
ejpam-4571	516	10	set	set	VERB
ejpam-4571	516	11	of	of	ADP
ejpam-4571	516	12	x	x	PUNCT
ejpam-4571	516	13	containing	contain	VERB
ejpam-4571	516	14	x	x	PUNCT
ejpam-4571	516	15	such	such	ADJ
ejpam-4571	516	16	that	that	SCONJ
ejpam-4571	516	17	f(w	f(w	PROPN
ejpam-4571	516	18	)	)	PUNCT
ejpam-4571	517	1	⊆	⊆	NUM
ejpam-4571	517	2	u	u	NOUN
ejpam-4571	517	3	(	(	PUNCT
ejpam-4571	517	4	λ	λ	PROPN
ejpam-4571	517	5	,	,	PUNCT
ejpam-4571	517	6	b	b	NOUN
ejpam-4571	517	7	)	)	PUNCT
ejpam-4571	517	8	.	.	PUNCT
ejpam-4571	518	1	thus	thus	ADV
ejpam-4571	518	2	,	,	PUNCT
ejpam-4571	518	3	f	f	PROPN
ejpam-4571	518	4	is	be	AUX
ejpam-4571	518	5	weakly	weakly	ADJ
ejpam-4571	518	6	(	(	PUNCT
ejpam-4571	518	7	λ	λ	X
ejpam-4571	518	8	,	,	PUNCT
ejpam-4571	518	9	b)-continuous	b)-continuous	ADJ
ejpam-4571	518	10	at	at	ADP
ejpam-4571	518	11	x.	x.	PROPN
ejpam-4571	518	12	this	this	PRON
ejpam-4571	518	13	shows	show	VERB
ejpam-4571	518	14	that	that	SCONJ
ejpam-4571	518	15	f	f	PROPN
ejpam-4571	518	16	is	be	AUX
ejpam-4571	518	17	weakly	weakly	ADJ
ejpam-4571	518	18	(	(	PUNCT
ejpam-4571	518	19	λ	λ	NOUN
ejpam-4571	518	20	,	,	PUNCT
ejpam-4571	518	21	b)-continuous	b)-continuous	ADJ
ejpam-4571	518	22	.	.	PUNCT
ejpam-4571	519	1	(	(	PUNCT
ejpam-4571	519	2	1	1	X
ejpam-4571	519	3	)	)	PUNCT
ejpam-4571	519	4	⇒	⇒	NOUN
ejpam-4571	519	5	(	(	PUNCT
ejpam-4571	519	6	6	6	NUM
ejpam-4571	519	7	):	):	PUNCT
ejpam-4571	519	8	let	let	VERB
ejpam-4571	519	9	u	u	PRON
ejpam-4571	519	10	be	be	AUX
ejpam-4571	519	11	any	any	DET
ejpam-4571	519	12	p(λ	p(λ	NOUN
ejpam-4571	519	13	,	,	PUNCT
ejpam-4571	519	14	b)-open	b)-open	VERB
ejpam-4571	519	15	set	set	VERB
ejpam-4571	519	16	of	of	ADP
ejpam-4571	519	17	y	y	PROPN
ejpam-4571	519	18	and	and	CCONJ
ejpam-4571	519	19	x	x	PUNCT
ejpam-4571	519	20	∈	∈	PROPN
ejpam-4571	519	21	x−f−1(u	x−f−1(u	PUNCT
ejpam-4571	519	22	(	(	PUNCT
ejpam-4571	519	23	λ	λ	X
ejpam-4571	519	24	,	,	PUNCT
ejpam-4571	519	25	b	b	NOUN
ejpam-4571	519	26	)	)	PUNCT
ejpam-4571	519	27	)	)	PUNCT
ejpam-4571	519	28	.	.	PUNCT
ejpam-4571	520	1	there	there	PRON
ejpam-4571	520	2	exists	exist	VERB
ejpam-4571	520	3	a	a	DET
ejpam-4571	520	4	(	(	PUNCT
ejpam-4571	520	5	λ	λ	NOUN
ejpam-4571	520	6	,	,	PUNCT
ejpam-4571	520	7	b)-open	b)-open	VERB
ejpam-4571	520	8	set	set	VERB
ejpam-4571	520	9	v	v	NOUN
ejpam-4571	520	10	of	of	ADP
ejpam-4571	520	11	y	y	NOUN
ejpam-4571	520	12	containing	contain	VERB
ejpam-4571	520	13	f(x	f(x	PROPN
ejpam-4571	520	14	)	)	PUNCT
ejpam-4571	520	15	such	such	ADJ
ejpam-4571	520	16	that	that	PRON
ejpam-4571	520	17	v	v	NOUN
ejpam-4571	520	18	∩u	∩u	NOUN
ejpam-4571	520	19	=	=	PUNCT
ejpam-4571	520	20	∅.	∅.	VERB
ejpam-4571	520	21	thus	thus	ADV
ejpam-4571	520	22	,	,	PUNCT
ejpam-4571	520	23	[	[	X
ejpam-4571	520	24	v	v	X
ejpam-4571	520	25	∩u	∩u	NOUN
ejpam-4571	520	26	]	]	X
ejpam-4571	520	27	(	(	PUNCT
ejpam-4571	520	28	λ	λ	PROPN
ejpam-4571	520	29	,	,	PUNCT
ejpam-4571	520	30	b	b	NOUN
ejpam-4571	520	31	)	)	PUNCT
ejpam-4571	520	32	=	=	PUNCT
ejpam-4571	520	33	∅.	∅.	NOUN
ejpam-4571	520	34	since	since	SCONJ
ejpam-4571	520	35	u	u	NOUN
ejpam-4571	520	36	is	be	AUX
ejpam-4571	520	37	(	(	PUNCT
ejpam-4571	520	38	λ	λ	INTJ
ejpam-4571	520	39	,	,	PUNCT
ejpam-4571	520	40	b)-open	b)-open	VERB
ejpam-4571	520	41	,	,	PUNCT
ejpam-4571	520	42	u	u	PROPN
ejpam-4571	520	43	∩	∩	X
ejpam-4571	520	44	v	v	X
ejpam-4571	520	45	(	(	PUNCT
ejpam-4571	520	46	λ	λ	PROPN
ejpam-4571	520	47	,	,	PUNCT
ejpam-4571	520	48	b	b	NOUN
ejpam-4571	520	49	)	)	PUNCT
ejpam-4571	520	50	⊆	⊆	NUM
ejpam-4571	521	1	[	[	X
ejpam-4571	521	2	u	u	NOUN
ejpam-4571	521	3	∩	∩	NOUN
ejpam-4571	521	4	v	v	ADP
ejpam-4571	521	5	]	]	PUNCT
ejpam-4571	521	6	(	(	PUNCT
ejpam-4571	521	7	λ	λ	PROPN
ejpam-4571	521	8	,	,	PUNCT
ejpam-4571	521	9	b	b	NOUN
ejpam-4571	521	10	)	)	PUNCT
ejpam-4571	521	11	=	=	PUNCT
ejpam-4571	521	12	∅.	∅.	NOUN
ejpam-4571	521	13	since	since	SCONJ
ejpam-4571	521	14	f	f	PROPN
ejpam-4571	521	15	is	be	AUX
ejpam-4571	521	16	weakly	weakly	ADJ
ejpam-4571	521	17	(	(	PUNCT
ejpam-4571	521	18	λ	λ	NOUN
ejpam-4571	521	19	,	,	PUNCT
ejpam-4571	521	20	b)-continuous	b)-continuous	ADJ
ejpam-4571	521	21	and	and	CCONJ
ejpam-4571	521	22	v	v	NOUN
ejpam-4571	521	23	is	be	AUX
ejpam-4571	521	24	a	a	DET
ejpam-4571	521	25	(	(	PUNCT
ejpam-4571	521	26	λ	λ	NOUN
ejpam-4571	521	27	,	,	PUNCT
ejpam-4571	521	28	b)-open	b)-open	VERB
ejpam-4571	521	29	set	set	VERB
ejpam-4571	521	30	of	of	ADP
ejpam-4571	521	31	y	y	PROPN
ejpam-4571	521	32	containing	contain	VERB
ejpam-4571	521	33	f(x	f(x	PROPN
ejpam-4571	521	34	)	)	PUNCT
ejpam-4571	521	35	,	,	PUNCT
ejpam-4571	521	36	there	there	PRON
ejpam-4571	521	37	exists	exist	VERB
ejpam-4571	521	38	a	a	DET
ejpam-4571	521	39	(	(	PUNCT
ejpam-4571	521	40	λ	λ	NOUN
ejpam-4571	521	41	,	,	PUNCT
ejpam-4571	521	42	b)-open	b)-open	VERB
ejpam-4571	521	43	set	set	VERB
ejpam-4571	521	44	w	w	NOUN
ejpam-4571	521	45	of	of	ADP
ejpam-4571	521	46	x	x	PUNCT
ejpam-4571	521	47	containing	contain	VERB
ejpam-4571	521	48	x	x	PUNCT
ejpam-4571	521	49	such	such	ADJ
ejpam-4571	521	50	that	that	SCONJ
ejpam-4571	521	51	f(w	f(w	PROPN
ejpam-4571	521	52	)	)	PUNCT
ejpam-4571	522	1	⊆	⊆	NUM
ejpam-4571	522	2	v	v	X
ejpam-4571	522	3	(	(	PUNCT
ejpam-4571	522	4	λ	λ	PROPN
ejpam-4571	522	5	,	,	PUNCT
ejpam-4571	522	6	b	b	NOUN
ejpam-4571	522	7	)	)	PUNCT
ejpam-4571	522	8	.	.	PUNCT
ejpam-4571	523	1	therefore	therefore	ADV
ejpam-4571	523	2	,	,	PUNCT
ejpam-4571	523	3	f(w	f(w	PROPN
ejpam-4571	523	4	)	)	PUNCT
ejpam-4571	523	5	∩	∩	NOUN
ejpam-4571	523	6	u	u	NOUN
ejpam-4571	523	7	=	=	PUNCT
ejpam-4571	523	8	∅.	∅.	VERB
ejpam-4571	523	9	thus	thus	ADV
ejpam-4571	523	10	,	,	PUNCT
ejpam-4571	523	11	w	w	PROPN
ejpam-4571	523	12	∩	∩	ADJ
ejpam-4571	523	13	f−1(u	f−1(u	NOUN
ejpam-4571	523	14	)	)	PUNCT
ejpam-4571	523	15	=	=	NOUN
ejpam-4571	523	16	∅	∅	NOUN
ejpam-4571	523	17	and	and	CCONJ
ejpam-4571	523	18	hence	hence	ADV
ejpam-4571	523	19	x	x	X
ejpam-4571	523	20	∈	∈	NOUN
ejpam-4571	523	21	x	x	X
ejpam-4571	523	22	−	−	PROPN
ejpam-4571	524	1	[	[	X
ejpam-4571	524	2	f−1(u)](λ	f−1(u)](λ	PROPN
ejpam-4571	524	3	,	,	PUNCT
ejpam-4571	524	4	b	b	NOUN
ejpam-4571	524	5	)	)	PUNCT
ejpam-4571	524	6	.	.	PUNCT
ejpam-4571	525	1	this	this	PRON
ejpam-4571	525	2	shows	show	VERB
ejpam-4571	525	3	that	that	SCONJ
ejpam-4571	526	1	[	[	X
ejpam-4571	526	2	f−1(u)](λ	f−1(u)](λ	PROPN
ejpam-4571	526	3	,	,	PUNCT
ejpam-4571	526	4	b	b	NOUN
ejpam-4571	526	5	)	)	PUNCT
ejpam-4571	526	6	⊆	⊆	NUM
ejpam-4571	526	7	f−1(u	f−1(u	NOUN
ejpam-4571	526	8	(	(	PUNCT
ejpam-4571	526	9	λ	λ	PROPN
ejpam-4571	526	10	,	,	PUNCT
ejpam-4571	526	11	b	b	NOUN
ejpam-4571	526	12	)	)	PUNCT
ejpam-4571	526	13	)	)	PUNCT
ejpam-4571	526	14	.	.	PUNCT
ejpam-4571	527	1	(	(	PUNCT
ejpam-4571	527	2	6	6	X
ejpam-4571	527	3	)	)	PUNCT
ejpam-4571	527	4	⇒	⇒	NOUN
ejpam-4571	527	5	(	(	PUNCT
ejpam-4571	527	6	7	7	NUM
ejpam-4571	527	7	):	):	PUNCT
ejpam-4571	527	8	let	let	VERB
ejpam-4571	527	9	u	u	PRON
ejpam-4571	527	10	be	be	AUX
ejpam-4571	527	11	any	any	DET
ejpam-4571	527	12	p(λ	p(λ	NOUN
ejpam-4571	527	13	,	,	PUNCT
ejpam-4571	527	14	b)-open	b)-open	VERB
ejpam-4571	527	15	set	set	VERB
ejpam-4571	527	16	of	of	ADP
ejpam-4571	527	17	y	y	PROPN
ejpam-4571	527	18	.	.	PUNCT
ejpam-4571	528	1	since	since	SCONJ
ejpam-4571	528	2	y	y	PROPN
ejpam-4571	528	3	−u	−u	PROPN
ejpam-4571	528	4	(	(	PUNCT
ejpam-4571	528	5	λ	λ	PROPN
ejpam-4571	528	6	,	,	PUNCT
ejpam-4571	528	7	b	b	NOUN
ejpam-4571	528	8	)	)	PUNCT
ejpam-4571	528	9	is	be	AUX
ejpam-4571	528	10	(	(	PUNCT
ejpam-4571	528	11	λ	λ	INTJ
ejpam-4571	528	12	,	,	PUNCT
ejpam-4571	528	13	b)-open	b)-open	VERB
ejpam-4571	528	14	,	,	PUNCT
ejpam-4571	528	15	we	we	PRON
ejpam-4571	528	16	have	have	VERB
ejpam-4571	528	17	x	x	X
ejpam-4571	528	18	−	−	PROPN
ejpam-4571	529	1	[	[	X
ejpam-4571	529	2	f−1(u	f−1(u	X
ejpam-4571	529	3	(	(	PUNCT
ejpam-4571	529	4	λ	λ	PROPN
ejpam-4571	529	5	,	,	PUNCT
ejpam-4571	529	6	b))](λ	b))](λ	PROPN
ejpam-4571	529	7	,	,	PUNCT
ejpam-4571	529	8	b	b	NOUN
ejpam-4571	529	9	)	)	PUNCT
ejpam-4571	529	10	=	=	NOUN
ejpam-4571	530	1	[	[	X
ejpam-4571	530	2	f−1(y	f−1(y	X
ejpam-4571	530	3	−u	−u	PROPN
ejpam-4571	530	4	(	(	PUNCT
ejpam-4571	530	5	λ	λ	PROPN
ejpam-4571	530	6	,	,	PUNCT
ejpam-4571	530	7	b))](λ	b))](λ	PROPN
ejpam-4571	530	8	,	,	PUNCT
ejpam-4571	530	9	b	b	NOUN
ejpam-4571	530	10	)	)	PUNCT
ejpam-4571	530	11	⊆	⊆	NUM
ejpam-4571	530	12	f−1([y	f−1([y	NOUN
ejpam-4571	530	13	−u	−u	NOUN
ejpam-4571	530	14	(	(	PUNCT
ejpam-4571	530	15	λ	λ	PROPN
ejpam-4571	530	16	,	,	PUNCT
ejpam-4571	530	17	b)](λ	b)](λ	NOUN
ejpam-4571	530	18	,	,	PUNCT
ejpam-4571	530	19	b	b	NOUN
ejpam-4571	530	20	)	)	PUNCT
ejpam-4571	530	21	)	)	PUNCT
ejpam-4571	531	1	⊆	⊆	NUM
ejpam-4571	531	2	x	x	SYM
ejpam-4571	531	3	−	−	NUM
ejpam-4571	531	4	f−1(u	f−1(u	PROPN
ejpam-4571	531	5	)	)	PUNCT
ejpam-4571	531	6	and	and	CCONJ
ejpam-4571	531	7	hence	hence	ADV
ejpam-4571	531	8	f−1(u	f−1(u	PROPN
ejpam-4571	531	9	)	)	PUNCT
ejpam-4571	532	1	⊆	⊆	NUM
ejpam-4571	532	2	[	[	X
ejpam-4571	532	3	f−1(u	f−1(u	PROPN
ejpam-4571	532	4	(	(	PUNCT
ejpam-4571	532	5	λ	λ	PROPN
ejpam-4571	532	6	,	,	PUNCT
ejpam-4571	532	7	b))](λ	b))](λ	PROPN
ejpam-4571	532	8	,	,	PUNCT
ejpam-4571	532	9	b	b	NOUN
ejpam-4571	532	10	)	)	PUNCT
ejpam-4571	532	11	.	.	PUNCT
ejpam-4571	533	1	(	(	PUNCT
ejpam-4571	533	2	7	7	X
ejpam-4571	533	3	)	)	PUNCT
ejpam-4571	533	4	⇒	⇒	NOUN
ejpam-4571	533	5	(	(	PUNCT
ejpam-4571	533	6	1	1	NUM
ejpam-4571	533	7	):	):	PUNCT
ejpam-4571	533	8	let	let	VERB
ejpam-4571	533	9	x	x	PUNCT
ejpam-4571	533	10	∈	∈	PROPN
ejpam-4571	533	11	x	x	X
ejpam-4571	533	12	and	and	CCONJ
ejpam-4571	533	13	v	v	AUX
ejpam-4571	533	14	be	be	AUX
ejpam-4571	533	15	any	any	DET
ejpam-4571	533	16	(	(	PUNCT
ejpam-4571	533	17	λ	λ	NOUN
ejpam-4571	533	18	,	,	PUNCT
ejpam-4571	533	19	b)-open	b)-open	VERB
ejpam-4571	533	20	set	set	VERB
ejpam-4571	533	21	of	of	ADP
ejpam-4571	533	22	y	y	PROPN
ejpam-4571	533	23	containing	contain	VERB
ejpam-4571	533	24	f(x	f(x	PROPN
ejpam-4571	533	25	)	)	PUNCT
ejpam-4571	533	26	.	.	PUNCT
ejpam-4571	534	1	then	then	ADV
ejpam-4571	534	2	,	,	PUNCT
ejpam-4571	534	3	we	we	PRON
ejpam-4571	534	4	have	have	VERB
ejpam-4571	534	5	x	x	X
ejpam-4571	534	6	∈	∈	PROPN
ejpam-4571	534	7	f−1(v	f−1(v	NOUN
ejpam-4571	534	8	)	)	PUNCT
ejpam-4571	534	9	⊆	⊆	NUM
ejpam-4571	535	1	[	[	X
ejpam-4571	535	2	f−1(v	f−1(v	NOUN
ejpam-4571	535	3	(	(	PUNCT
ejpam-4571	535	4	λ	λ	PROPN
ejpam-4571	535	5	,	,	PUNCT
ejpam-4571	535	6	b))](λ	b))](λ	PROPN
ejpam-4571	535	7	,	,	PUNCT
ejpam-4571	535	8	b	b	NOUN
ejpam-4571	535	9	)	)	PUNCT
ejpam-4571	535	10	.	.	PUNCT
ejpam-4571	536	1	put	put	VERB
ejpam-4571	536	2	u	u	NOUN
ejpam-4571	536	3	=	=	PUNCT
ejpam-4571	537	1	[	[	X
ejpam-4571	537	2	f−1(v	f−1(v	NOUN
ejpam-4571	537	3	(	(	PUNCT
ejpam-4571	537	4	λ	λ	PROPN
ejpam-4571	537	5	,	,	PUNCT
ejpam-4571	537	6	b))](λ	b))](λ	PROPN
ejpam-4571	537	7	,	,	PUNCT
ejpam-4571	537	8	b	b	NOUN
ejpam-4571	537	9	)	)	PUNCT
ejpam-4571	537	10	.	.	PUNCT
ejpam-4571	538	1	then	then	ADV
ejpam-4571	538	2	,	,	PUNCT
ejpam-4571	538	3	u	u	NOUN
ejpam-4571	538	4	is	be	AUX
ejpam-4571	538	5	a	a	DET
ejpam-4571	538	6	(	(	PUNCT
ejpam-4571	538	7	λ	λ	NOUN
ejpam-4571	538	8	,	,	PUNCT
ejpam-4571	538	9	b)-open	b)-open	VERB
ejpam-4571	538	10	set	set	VERB
ejpam-4571	538	11	of	of	ADP
ejpam-4571	538	12	x	x	PUNCT
ejpam-4571	538	13	containing	contain	VERB
ejpam-4571	538	14	x	x	PUNCT
ejpam-4571	538	15	such	such	ADJ
ejpam-4571	538	16	that	that	DET
ejpam-4571	538	17	f(u	f(u	PROPN
ejpam-4571	538	18	)	)	PUNCT
ejpam-4571	538	19	⊆	⊆	NUM
ejpam-4571	538	20	v	v	NOUN
ejpam-4571	538	21	(	(	PUNCT
ejpam-4571	538	22	λ	λ	PROPN
ejpam-4571	538	23	,	,	PUNCT
ejpam-4571	538	24	b	b	NOUN
ejpam-4571	538	25	)	)	PUNCT
ejpam-4571	538	26	.	.	PUNCT
ejpam-4571	539	1	thus	thus	ADV
ejpam-4571	539	2	,	,	PUNCT
ejpam-4571	539	3	f	f	PROPN
ejpam-4571	539	4	is	be	AUX
ejpam-4571	539	5	weakly	weakly	ADJ
ejpam-4571	539	6	(	(	PUNCT
ejpam-4571	539	7	λ	λ	X
ejpam-4571	539	8	,	,	PUNCT
ejpam-4571	539	9	b)-continuous	b)-continuous	ADJ
ejpam-4571	539	10	at	at	ADP
ejpam-4571	539	11	x	x	X
ejpam-4571	539	12	and	and	CCONJ
ejpam-4571	539	13	hence	hence	ADV
ejpam-4571	539	14	f	f	PROPN
ejpam-4571	539	15	is	be	AUX
ejpam-4571	539	16	weakly	weakly	ADJ
ejpam-4571	539	17	(	(	PUNCT
ejpam-4571	539	18	λ	λ	NOUN
ejpam-4571	539	19	,	,	PUNCT
ejpam-4571	539	20	b)-continuous	b)-continuous	PROPN
ejpam-4571	539	21	.	.	PUNCT
ejpam-4571	540	1	definition	definition	NOUN
ejpam-4571	540	2	6	6	NUM
ejpam-4571	540	3	.	.	PUNCT
ejpam-4571	541	1	a	a	DET
ejpam-4571	541	2	topological	topological	ADJ
ejpam-4571	541	3	space	space	NOUN
ejpam-4571	541	4	(	(	PUNCT
ejpam-4571	541	5	x	x	X
ejpam-4571	541	6	,	,	PUNCT
ejpam-4571	541	7	τ	τ	X
ejpam-4571	541	8	)	)	PUNCT
ejpam-4571	541	9	is	be	AUX
ejpam-4571	541	10	called	call	VERB
ejpam-4571	541	11	(	(	PUNCT
ejpam-4571	541	12	λ	λ	PROPN
ejpam-4571	541	13	,	,	PUNCT
ejpam-4571	541	14	b)-connected	b)-connecte	VERB
ejpam-4571	541	15	if	if	SCONJ
ejpam-4571	541	16	x	x	PRON
ejpam-4571	541	17	can	can	AUX
ejpam-4571	541	18	not	not	PART
ejpam-4571	541	19	be	be	AUX
ejpam-4571	541	20	written	write	VERB
ejpam-4571	541	21	as	as	ADP
ejpam-4571	541	22	a	a	DET
ejpam-4571	541	23	disjoint	disjoint	NOUN
ejpam-4571	541	24	union	union	NOUN
ejpam-4571	541	25	of	of	ADP
ejpam-4571	541	26	two	two	NUM
ejpam-4571	541	27	nonempty	nonempty	ADJ
ejpam-4571	541	28	(	(	PUNCT
ejpam-4571	541	29	λ	λ	NOUN
ejpam-4571	541	30	,	,	PUNCT
ejpam-4571	541	31	b)-open	b)-open	VERB
ejpam-4571	541	32	sets	set	NOUN
ejpam-4571	541	33	.	.	PUNCT
ejpam-4571	542	1	example	example	NOUN
ejpam-4571	542	2	5	5	NUM
ejpam-4571	542	3	.	.	PUNCT
ejpam-4571	543	1	let	let	VERB
ejpam-4571	543	2	x	x	PUNCT
ejpam-4571	543	3	=	=	PRON
ejpam-4571	543	4	{	{	PUNCT
ejpam-4571	543	5	a	a	DET
ejpam-4571	543	6	,	,	PUNCT
ejpam-4571	543	7	b	b	NOUN
ejpam-4571	543	8	}	}	PUNCT
ejpam-4571	543	9	with	with	ADP
ejpam-4571	543	10	the	the	DET
ejpam-4571	543	11	topology	topology	NOUN
ejpam-4571	543	12	τ	τ	X
ejpam-4571	543	13	=	=	PUNCT
ejpam-4571	543	14	{	{	PUNCT
ejpam-4571	543	15	∅	∅	NOUN
ejpam-4571	543	16	,	,	PUNCT
ejpam-4571	543	17	{	{	PUNCT
ejpam-4571	543	18	a	a	X
ejpam-4571	543	19	}	}	PUNCT
ejpam-4571	543	20	,	,	PUNCT
ejpam-4571	543	21	x	x	NOUN
ejpam-4571	543	22	}	}	PUNCT
ejpam-4571	543	23	.	.	PUNCT
ejpam-4571	544	1	then	then	ADV
ejpam-4571	544	2	,	,	PUNCT
ejpam-4571	544	3	(	(	PUNCT
ejpam-4571	544	4	x	x	X
ejpam-4571	544	5	,	,	PUNCT
ejpam-4571	544	6	τ	τ	X
ejpam-4571	544	7	)	)	PUNCT
ejpam-4571	544	8	is	be	AUX
ejpam-4571	544	9	(	(	PUNCT
ejpam-4571	544	10	λ	λ	X
ejpam-4571	544	11	,	,	PUNCT
ejpam-4571	544	12	b)connected	b)connecte	VERB
ejpam-4571	544	13	.	.	PUNCT
ejpam-4571	545	1	lemma	lemma	PROPN
ejpam-4571	545	2	7	7	NUM
ejpam-4571	545	3	.	.	X
ejpam-4571	546	1	for	for	ADP
ejpam-4571	546	2	a	a	DET
ejpam-4571	546	3	topological	topological	ADJ
ejpam-4571	546	4	space	space	NOUN
ejpam-4571	546	5	(	(	PUNCT
ejpam-4571	546	6	x	x	X
ejpam-4571	546	7	,	,	PUNCT
ejpam-4571	546	8	τ	τ	PROPN
ejpam-4571	546	9	)	)	PUNCT
ejpam-4571	546	10	,	,	PUNCT
ejpam-4571	546	11	the	the	DET
ejpam-4571	546	12	following	follow	VERB
ejpam-4571	546	13	properties	property	NOUN
ejpam-4571	546	14	are	be	AUX
ejpam-4571	546	15	equivalent	equivalent	ADJ
ejpam-4571	546	16	:	:	PUNCT
ejpam-4571	546	17	(	(	PUNCT
ejpam-4571	546	18	1	1	X
ejpam-4571	546	19	)	)	PUNCT
ejpam-4571	546	20	(	(	PUNCT
ejpam-4571	546	21	x	x	X
ejpam-4571	546	22	,	,	PUNCT
ejpam-4571	546	23	τ	τ	X
ejpam-4571	546	24	)	)	PUNCT
ejpam-4571	546	25	is	be	AUX
ejpam-4571	546	26	(	(	PUNCT
ejpam-4571	546	27	λ	λ	X
ejpam-4571	546	28	,	,	PUNCT
ejpam-4571	546	29	b)-connected	b)-connecte	VERB
ejpam-4571	546	30	.	.	PUNCT
ejpam-4571	547	1	(	(	PUNCT
ejpam-4571	547	2	2	2	X
ejpam-4571	547	3	)	)	PUNCT
ejpam-4571	547	4	the	the	DET
ejpam-4571	547	5	only	only	ADJ
ejpam-4571	547	6	subsets	subset	NOUN
ejpam-4571	547	7	of	of	ADP
ejpam-4571	547	8	x	x	PRON
ejpam-4571	547	9	,	,	PUNCT
ejpam-4571	547	10	which	which	PRON
ejpam-4571	547	11	are	be	AUX
ejpam-4571	547	12	both	both	PRON
ejpam-4571	547	13	(	(	PUNCT
ejpam-4571	547	14	λ	λ	NOUN
ejpam-4571	547	15	,	,	PUNCT
ejpam-4571	547	16	b)-open	b)-open	PUNCT
ejpam-4571	547	17	and	and	CCONJ
ejpam-4571	547	18	(	(	PUNCT
ejpam-4571	547	19	λ	λ	PROPN
ejpam-4571	547	20	,	,	PUNCT
ejpam-4571	547	21	b)-closed	b)-close	VERB
ejpam-4571	547	22	are	be	AUX
ejpam-4571	547	23	∅	∅	NOUN
ejpam-4571	547	24	and	and	CCONJ
ejpam-4571	547	25	x.	x.	NOUN
ejpam-4571	547	26	theorem	theorem	VERB
ejpam-4571	547	27	14	14	NUM
ejpam-4571	547	28	.	.	PUNCT
ejpam-4571	548	1	let	let	VERB
ejpam-4571	548	2	f	f	NOUN
ejpam-4571	548	3	:	:	PUNCT
ejpam-4571	548	4	(	(	PUNCT
ejpam-4571	548	5	x	x	X
ejpam-4571	548	6	,	,	PUNCT
ejpam-4571	548	7	τ	τ	X
ejpam-4571	548	8	)	)	PUNCT
ejpam-4571	548	9	→	→	SYM
ejpam-4571	548	10	(	(	PUNCT
ejpam-4571	548	11	y	y	PROPN
ejpam-4571	548	12	,	,	PUNCT
ejpam-4571	548	13	σ	σ	PROPN
ejpam-4571	548	14	)	)	PUNCT
ejpam-4571	548	15	is	be	AUX
ejpam-4571	548	16	a	a	DET
ejpam-4571	548	17	weakly	weakly	ADJ
ejpam-4571	548	18	(	(	PUNCT
ejpam-4571	548	19	λ	λ	PROPN
ejpam-4571	548	20	,	,	PUNCT
ejpam-4571	548	21	b)-continuous	b)-continuous	PROPN
ejpam-4571	548	22	surjection	surjection	NOUN
ejpam-4571	548	23	.	.	PUNCT
ejpam-4571	549	1	if	if	SCONJ
ejpam-4571	549	2	(	(	PUNCT
ejpam-4571	549	3	x	x	NOUN
ejpam-4571	549	4	,	,	PUNCT
ejpam-4571	549	5	τ	τ	X
ejpam-4571	549	6	)	)	PUNCT
ejpam-4571	549	7	is	be	AUX
ejpam-4571	549	8	(	(	PUNCT
ejpam-4571	549	9	λ	λ	X
ejpam-4571	549	10	,	,	PUNCT
ejpam-4571	549	11	b)-connected	b)-connecte	VERB
ejpam-4571	549	12	,	,	PUNCT
ejpam-4571	549	13	then	then	ADV
ejpam-4571	549	14	(	(	PUNCT
ejpam-4571	549	15	y	y	PROPN
ejpam-4571	549	16	,	,	PUNCT
ejpam-4571	549	17	σ	σ	PROPN
ejpam-4571	549	18	)	)	PUNCT
ejpam-4571	549	19	is	be	AUX
ejpam-4571	549	20	(	(	PUNCT
ejpam-4571	549	21	λ	λ	X
ejpam-4571	549	22	,	,	PUNCT
ejpam-4571	549	23	b)-connected	b)-connecte	VERB
ejpam-4571	549	24	.	.	PUNCT
ejpam-4571	550	1	proof	proof	NOUN
ejpam-4571	550	2	.	.	PUNCT
ejpam-4571	551	1	assume	assume	VERB
ejpam-4571	551	2	that	that	SCONJ
ejpam-4571	551	3	(	(	PUNCT
ejpam-4571	551	4	y	y	PROPN
ejpam-4571	551	5	,	,	PUNCT
ejpam-4571	551	6	σ	σ	PROPN
ejpam-4571	551	7	)	)	PUNCT
ejpam-4571	551	8	is	be	AUX
ejpam-4571	551	9	not	not	PART
ejpam-4571	551	10	(	(	PUNCT
ejpam-4571	551	11	λ	λ	X
ejpam-4571	551	12	,	,	PUNCT
ejpam-4571	551	13	b)-connected	b)-connecte	VERB
ejpam-4571	551	14	.	.	PUNCT
ejpam-4571	552	1	then	then	ADV
ejpam-4571	552	2	,	,	PUNCT
ejpam-4571	552	3	there	there	PRON
ejpam-4571	552	4	exist	exist	VERB
ejpam-4571	552	5	nonempty	nonempty	ADJ
ejpam-4571	552	6	(	(	PUNCT
ejpam-4571	552	7	λ	λ	NOUN
ejpam-4571	552	8	,	,	PUNCT
ejpam-4571	552	9	b)open	b)open	NOUN
ejpam-4571	552	10	sets	set	NOUN
ejpam-4571	552	11	v1	v1	NOUN
ejpam-4571	552	12	,	,	PUNCT
ejpam-4571	552	13	v2	v2	VERB
ejpam-4571	552	14	such	such	ADJ
ejpam-4571	552	15	that	that	DET
ejpam-4571	552	16	v1	v1	NOUN
ejpam-4571	552	17	∩	∩	ADJ
ejpam-4571	552	18	v2	v2	NOUN
ejpam-4571	552	19	=	=	SYM
ejpam-4571	552	20	∅	∅	NOUN
ejpam-4571	552	21	and	and	CCONJ
ejpam-4571	552	22	v1	v1	VERB
ejpam-4571	552	23	∪	∪	ADJ
ejpam-4571	552	24	v2	v2	NOUN
ejpam-4571	552	25	=	=	SYM
ejpam-4571	552	26	y	y	PROPN
ejpam-4571	552	27	.	.	PUNCT
ejpam-4571	553	1	by	by	ADP
ejpam-4571	553	2	theorem	theorem	ADJ
ejpam-4571	553	3	10	10	NUM
ejpam-4571	553	4	,	,	PUNCT
ejpam-4571	553	5	f−1(vi	f−1(vi	PROPN
ejpam-4571	553	6	)	)	PUNCT
ejpam-4571	553	7	⊆	⊆	NUM
ejpam-4571	554	1	[	[	X
ejpam-4571	554	2	f−1[v	f−1[v	NOUN
ejpam-4571	554	3	(	(	PUNCT
ejpam-4571	554	4	λ	λ	PROPN
ejpam-4571	554	5	,	,	PUNCT
ejpam-4571	554	6	b)]](λ	b)]](λ	PRON
ejpam-4571	554	7	,	,	PUNCT
ejpam-4571	554	8	b	b	NOUN
ejpam-4571	554	9	)	)	PUNCT
ejpam-4571	554	10	for	for	ADP
ejpam-4571	554	11	i	i	PROPN
ejpam-4571	554	12	=	=	SYM
ejpam-4571	554	13	1	1	NUM
ejpam-4571	554	14	,	,	PUNCT
ejpam-4571	554	15	2	2	NUM
ejpam-4571	554	16	.	.	PUNCT
ejpam-4571	554	17	since	since	SCONJ
ejpam-4571	554	18	vi	vi	PROPN
ejpam-4571	554	19	is	be	AUX
ejpam-4571	554	20	(	(	PUNCT
ejpam-4571	554	21	λ	λ	X
ejpam-4571	554	22	,	,	PUNCT
ejpam-4571	554	23	b)-closed	b)-close	VERB
ejpam-4571	554	24	in	in	ADP
ejpam-4571	554	25	y	y	PROPN
ejpam-4571	554	26	for	for	ADP
ejpam-4571	554	27	each	each	DET
ejpam-4571	554	28	i	i	NOUN
ejpam-4571	554	29	=	=	NOUN
ejpam-4571	554	30	1	1	NUM
ejpam-4571	554	31	,	,	PUNCT
ejpam-4571	554	32	2	2	NUM
ejpam-4571	554	33	.	.	X
ejpam-4571	554	34	therefore	therefore	ADV
ejpam-4571	554	35	,	,	PUNCT
ejpam-4571	554	36	f−1(vi	f−1(vi	PROPN
ejpam-4571	554	37	)	)	PUNCT
ejpam-4571	554	38	⊆	⊆	NUM
ejpam-4571	555	1	[	[	X
ejpam-4571	555	2	f−1(vi)](λ	f−1(vi)](λ	PROPN
ejpam-4571	555	3	,	,	PUNCT
ejpam-4571	555	4	b	b	NOUN
ejpam-4571	555	5	)	)	PUNCT
ejpam-4571	555	6	and	and	CCONJ
ejpam-4571	555	7	by	by	ADP
ejpam-4571	555	8	lemma	lemma	PROPN
ejpam-4571	555	9	2	2	NUM
ejpam-4571	555	10	,	,	PUNCT
ejpam-4571	555	11	f−1(vi	f−1(vi	X
ejpam-4571	555	12	)	)	PUNCT
ejpam-4571	555	13	is	be	AUX
ejpam-4571	555	14	(	(	PUNCT
ejpam-4571	555	15	λ	λ	X
ejpam-4571	555	16	,	,	PUNCT
ejpam-4571	555	17	b)-open	b)-open	VERB
ejpam-4571	555	18	for	for	ADP
ejpam-4571	555	19	each	each	DET
ejpam-4571	555	20	i	i	NOUN
ejpam-4571	555	21	=	=	NOUN
ejpam-4571	555	22	1	1	NUM
ejpam-4571	555	23	,	,	PUNCT
ejpam-4571	555	24	2	2	NUM
ejpam-4571	555	25	.	.	PUNCT
ejpam-4571	556	1	moreover	moreover	ADV
ejpam-4571	556	2	,	,	PUNCT
ejpam-4571	556	3	x	x	PUNCT
ejpam-4571	556	4	is	be	AUX
ejpam-4571	556	5	union	union	NOUN
ejpam-4571	556	6	of	of	ADP
ejpam-4571	556	7	nonempty	nonempty	ADJ
ejpam-4571	556	8	disjoint	disjoint	NOUN
ejpam-4571	556	9	sets	set	NOUN
ejpam-4571	556	10	f−1(v1	f−1(v1	NOUN
ejpam-4571	556	11	)	)	PUNCT
ejpam-4571	556	12	and	and	CCONJ
ejpam-4571	556	13	f−1(v2	f−1(v2	NOUN
ejpam-4571	556	14	)	)	PUNCT
ejpam-4571	556	15	.	.	PUNCT
ejpam-4571	557	1	this	this	PRON
ejpam-4571	557	2	shows	show	VERB
ejpam-4571	557	3	that	that	SCONJ
ejpam-4571	557	4	(	(	PUNCT
ejpam-4571	557	5	x	x	X
ejpam-4571	557	6	,	,	PUNCT
ejpam-4571	557	7	τ	τ	X
ejpam-4571	557	8	)	)	PUNCT
ejpam-4571	557	9	is	be	AUX
ejpam-4571	557	10	not	not	PART
ejpam-4571	557	11	(	(	PUNCT
ejpam-4571	557	12	λ	λ	X
ejpam-4571	557	13	,	,	PUNCT
ejpam-4571	557	14	b)-connected	b)-connecte	VERB
ejpam-4571	557	15	.	.	PUNCT
ejpam-4571	558	1	this	this	PRON
ejpam-4571	558	2	is	be	AUX
ejpam-4571	558	3	contrary	contrary	ADJ
ejpam-4571	558	4	to	to	ADP
ejpam-4571	558	5	the	the	DET
ejpam-4571	558	6	hypothesis	hypothesis	NOUN
ejpam-4571	558	7	that	that	SCONJ
ejpam-4571	558	8	(	(	PUNCT
ejpam-4571	558	9	x	x	X
ejpam-4571	558	10	,	,	PUNCT
ejpam-4571	558	11	τ	τ	X
ejpam-4571	558	12	)	)	PUNCT
ejpam-4571	558	13	is	be	AUX
ejpam-4571	558	14	(	(	PUNCT
ejpam-4571	558	15	λ	λ	X
ejpam-4571	558	16	,	,	PUNCT
ejpam-4571	558	17	b)-connected	b)-connecte	VERB
ejpam-4571	558	18	.	.	PUNCT
ejpam-4571	559	1	thus	thus	ADV
ejpam-4571	559	2	,	,	PUNCT
ejpam-4571	559	3	(	(	PUNCT
ejpam-4571	559	4	y	y	PROPN
ejpam-4571	559	5	,	,	PUNCT
ejpam-4571	559	6	σ	σ	PROPN
ejpam-4571	559	7	)	)	PUNCT
ejpam-4571	559	8	is	be	AUX
ejpam-4571	559	9	(	(	PUNCT
ejpam-4571	559	10	λ	λ	X
ejpam-4571	559	11	,	,	PUNCT
ejpam-4571	559	12	b)-connected	b)-connecte	VERB
ejpam-4571	559	13	.	.	PUNCT
ejpam-4571	560	1	c.	c.	PROPN
ejpam-4571	560	2	boonpok	boonpok	PROPN
ejpam-4571	560	3	,	,	PUNCT
ejpam-4571	560	4	n.	n.	PROPN
ejpam-4571	560	5	srisarakham	srisarakham	PROPN
ejpam-4571	560	6	/	/	SYM
ejpam-4571	560	7	eur	eur	PROPN
ejpam-4571	560	8	.	.	PUNCT
ejpam-4571	561	1	j.	j.	PROPN
ejpam-4571	561	2	pure	pure	PROPN
ejpam-4571	561	3	appl	appl	PROPN
ejpam-4571	561	4	.	.	PROPN
ejpam-4571	561	5	math	math	PROPN
ejpam-4571	561	6	,	,	PUNCT
ejpam-4571	561	7	16	16	NUM
ejpam-4571	561	8	(	(	PUNCT
ejpam-4571	561	9	1	1	NUM
ejpam-4571	561	10	)	)	PUNCT
ejpam-4571	561	11	(	(	PUNCT
ejpam-4571	561	12	2023	2023	NUM
ejpam-4571	561	13	)	)	PUNCT
ejpam-4571	561	14	,	,	PUNCT
ejpam-4571	561	15	29	29	NUM
ejpam-4571	561	16	-	-	SYM
ejpam-4571	561	17	43	43	NUM
ejpam-4571	561	18	42	42	NUM
ejpam-4571	561	19	definition	definition	NOUN
ejpam-4571	561	20	7	7	NUM
ejpam-4571	561	21	.	.	PUNCT
ejpam-4571	562	1	a	a	DET
ejpam-4571	562	2	subset	subset	NOUN
ejpam-4571	562	3	k	k	PROPN
ejpam-4571	562	4	of	of	ADP
ejpam-4571	562	5	a	a	DET
ejpam-4571	562	6	topological	topological	ADJ
ejpam-4571	562	7	spaces	space	NOUN
ejpam-4571	562	8	(	(	PUNCT
ejpam-4571	562	9	x	x	X
ejpam-4571	562	10	,	,	PUNCT
ejpam-4571	562	11	τ	τ	X
ejpam-4571	562	12	)	)	PUNCT
ejpam-4571	562	13	is	be	AUX
ejpam-4571	562	14	said	say	VERB
ejpam-4571	562	15	to	to	PART
ejpam-4571	562	16	be	be	AUX
ejpam-4571	562	17	λb	λb	ADV
ejpam-4571	562	18	-	-	PUNCT
ejpam-4571	562	19	closed	closed	ADJ
ejpam-4571	562	20	(	(	PUNCT
ejpam-4571	562	21	resp	resp	NOUN
ejpam-4571	562	22	.	.	PUNCT
ejpam-4571	563	1	λbcompact	λbcompact	NOUN
ejpam-4571	563	2	)	)	PUNCT
ejpam-4571	563	3	relative	relative	ADJ
ejpam-4571	563	4	to	to	ADP
ejpam-4571	563	5	(	(	PUNCT
ejpam-4571	563	6	x	x	X
ejpam-4571	563	7	,	,	PUNCT
ejpam-4571	563	8	τ	τ	X
ejpam-4571	563	9	)	)	PUNCT
ejpam-4571	563	10	if	if	SCONJ
ejpam-4571	563	11	for	for	ADP
ejpam-4571	563	12	any	any	DET
ejpam-4571	563	13	cover	cover	NOUN
ejpam-4571	563	14	{	{	PUNCT
ejpam-4571	563	15	vi	vi	NOUN
ejpam-4571	563	16	:	:	PUNCT
ejpam-4571	563	17	i	i	PRON
ejpam-4571	563	18	∈	∈	VERB
ejpam-4571	564	1	i	i	X
ejpam-4571	564	2	}	}	PUNCT
ejpam-4571	564	3	of	of	ADP
ejpam-4571	564	4	k	k	X
ejpam-4571	564	5	by	by	ADP
ejpam-4571	564	6	(	(	PUNCT
ejpam-4571	564	7	λ	λ	INTJ
ejpam-4571	564	8	,	,	PUNCT
ejpam-4571	564	9	b)-open	b)-open	VERB
ejpam-4571	564	10	sets	set	NOUN
ejpam-4571	564	11	of	of	ADP
ejpam-4571	564	12	x	x	NOUN
ejpam-4571	564	13	,	,	PUNCT
ejpam-4571	564	14	there	there	PRON
ejpam-4571	564	15	exist	exist	VERB
ejpam-4571	564	16	a	a	DET
ejpam-4571	564	17	finite	finite	NOUN
ejpam-4571	564	18	subset	subset	NOUN
ejpam-4571	564	19	i0	i0	PROPN
ejpam-4571	564	20	of	of	ADP
ejpam-4571	564	21	i	i	PRON
ejpam-4571	564	22	such	such	ADJ
ejpam-4571	564	23	that	that	SCONJ
ejpam-4571	564	24	k	k	PROPN
ejpam-4571	564	25	⊆	⊆	NUM
ejpam-4571	564	26	∪{v	∪{v	PROPN
ejpam-4571	564	27	(	(	PUNCT
ejpam-4571	564	28	λ	λ	PROPN
ejpam-4571	564	29	,	,	PUNCT
ejpam-4571	564	30	b	b	NOUN
ejpam-4571	564	31	)	)	PUNCT
ejpam-4571	564	32	i	i	PRON
ejpam-4571	564	33	|	|	ADV
ejpam-4571	564	34	i	i	PRON
ejpam-4571	564	35	∈	∈	PROPN
ejpam-4571	564	36	i0	i0	PROPN
ejpam-4571	564	37	}	}	PUNCT
ejpam-4571	564	38	(	(	PUNCT
ejpam-4571	564	39	resp	resp	NOUN
ejpam-4571	564	40	.	.	PUNCT
ejpam-4571	565	1	k	k	PROPN
ejpam-4571	566	1	⊆	⊆	NUM
ejpam-4571	566	2	∪{vi	∪{vi	PROPN
ejpam-4571	566	3	|	|	ADV
ejpam-4571	566	4	i	i	PRON
ejpam-4571	566	5	∈	∈	PROPN
ejpam-4571	566	6	i0	i0	PROPN
ejpam-4571	566	7	}	}	PUNCT
ejpam-4571	566	8	)	)	PUNCT
ejpam-4571	566	9	.	.	PUNCT
ejpam-4571	567	1	if	if	SCONJ
ejpam-4571	567	2	x	x	PRON
ejpam-4571	567	3	is	be	AUX
ejpam-4571	567	4	λb	λb	ADV
ejpam-4571	567	5	-	-	PUNCT
ejpam-4571	567	6	closed	closed	ADJ
ejpam-4571	567	7	(	(	PUNCT
ejpam-4571	567	8	resp	resp	NOUN
ejpam-4571	567	9	.	.	PUNCT
ejpam-4571	568	1	λb	λb	ADP
ejpam-4571	568	2	-	-	PUNCT
ejpam-4571	568	3	compact	compact	ADJ
ejpam-4571	568	4	)	)	PUNCT
ejpam-4571	569	1	relative	relative	ADJ
ejpam-4571	569	2	to	to	ADP
ejpam-4571	569	3	(	(	PUNCT
ejpam-4571	569	4	x	x	X
ejpam-4571	569	5	,	,	PUNCT
ejpam-4571	569	6	τ	τ	PROPN
ejpam-4571	569	7	)	)	PUNCT
ejpam-4571	569	8	,	,	PUNCT
ejpam-4571	569	9	then	then	ADV
ejpam-4571	569	10	(	(	PUNCT
ejpam-4571	569	11	x	x	X
ejpam-4571	569	12	,	,	PUNCT
ejpam-4571	569	13	τ	τ	X
ejpam-4571	569	14	)	)	PUNCT
ejpam-4571	569	15	is	be	AUX
ejpam-4571	569	16	said	say	VERB
ejpam-4571	569	17	to	to	PART
ejpam-4571	569	18	be	be	AUX
ejpam-4571	569	19	λbclosed	λbclose	VERB
ejpam-4571	569	20	(	(	PUNCT
ejpam-4571	569	21	resp	resp	NOUN
ejpam-4571	569	22	.	.	PUNCT
ejpam-4571	570	1	λb	λb	NOUN
ejpam-4571	570	2	-	-	PUNCT
ejpam-4571	570	3	compact	compact	ADJ
ejpam-4571	570	4	)	)	PUNCT
ejpam-4571	570	5	.	.	PUNCT
ejpam-4571	571	1	theorem	theorem	VERB
ejpam-4571	571	2	15	15	NUM
ejpam-4571	571	3	.	.	PUNCT
ejpam-4571	572	1	if	if	SCONJ
ejpam-4571	572	2	f	f	PROPN
ejpam-4571	572	3	:	:	PUNCT
ejpam-4571	572	4	(	(	PUNCT
ejpam-4571	572	5	x	x	X
ejpam-4571	572	6	,	,	PUNCT
ejpam-4571	572	7	τ	τ	X
ejpam-4571	572	8	)	)	PUNCT
ejpam-4571	572	9	→	→	SYM
ejpam-4571	572	10	(	(	PUNCT
ejpam-4571	572	11	y	y	PROPN
ejpam-4571	572	12	,	,	PUNCT
ejpam-4571	572	13	σ	σ	PROPN
ejpam-4571	572	14	)	)	PUNCT
ejpam-4571	572	15	is	be	AUX
ejpam-4571	572	16	weakly	weakly	ADJ
ejpam-4571	572	17	(	(	PUNCT
ejpam-4571	572	18	λ	λ	PROPN
ejpam-4571	572	19	,	,	PUNCT
ejpam-4571	572	20	b)-continuous	b)-continuous	ADJ
ejpam-4571	572	21	and	and	CCONJ
ejpam-4571	572	22	k	k	PROPN
ejpam-4571	572	23	is	be	AUX
ejpam-4571	572	24	λb	λb	ADJ
ejpam-4571	572	25	-	-	ADJ
ejpam-4571	572	26	compact	compact	ADJ
ejpam-4571	572	27	,	,	PUNCT
ejpam-4571	572	28	then	then	ADV
ejpam-4571	572	29	f(k	f(k	VERB
ejpam-4571	572	30	)	)	PUNCT
ejpam-4571	572	31	is	be	AUX
ejpam-4571	572	32	λb	λb	ADV
ejpam-4571	572	33	-	-	PUNCT
ejpam-4571	572	34	closed	closed	ADJ
ejpam-4571	572	35	relative	relative	ADJ
ejpam-4571	572	36	to	to	ADP
ejpam-4571	572	37	(	(	PUNCT
ejpam-4571	572	38	y	y	PROPN
ejpam-4571	572	39	,	,	PUNCT
ejpam-4571	572	40	σ	σ	PROPN
ejpam-4571	572	41	)	)	PUNCT
ejpam-4571	572	42	.	.	PUNCT
ejpam-4571	573	1	proof	proof	NOUN
ejpam-4571	573	2	.	.	PUNCT
ejpam-4571	574	1	let	let	VERB
ejpam-4571	574	2	{	{	PUNCT
ejpam-4571	574	3	vα	vα	X
ejpam-4571	574	4	:	:	PUNCT
ejpam-4571	574	5	α	α	PROPN
ejpam-4571	574	6	∈	∈	PROPN
ejpam-4571	574	7	∇	∇	X
ejpam-4571	574	8	}	}	PUNCT
ejpam-4571	574	9	be	be	AUX
ejpam-4571	574	10	any	any	DET
ejpam-4571	574	11	cover	cover	NOUN
ejpam-4571	574	12	of	of	ADP
ejpam-4571	574	13	f(k	f(k	VERB
ejpam-4571	574	14	)	)	PUNCT
ejpam-4571	574	15	by	by	ADP
ejpam-4571	574	16	(	(	PUNCT
ejpam-4571	574	17	λ	λ	INTJ
ejpam-4571	574	18	,	,	PUNCT
ejpam-4571	574	19	b)-open	b)-open	VERB
ejpam-4571	574	20	sets	set	NOUN
ejpam-4571	574	21	of	of	ADP
ejpam-4571	574	22	y	y	PROPN
ejpam-4571	574	23	.	.	PUNCT
ejpam-4571	575	1	for	for	ADP
ejpam-4571	575	2	each	each	DET
ejpam-4571	575	3	x	x	SYM
ejpam-4571	575	4	∈	∈	PROPN
ejpam-4571	575	5	k	k	NOUN
ejpam-4571	575	6	,	,	PUNCT
ejpam-4571	575	7	there	there	PRON
ejpam-4571	575	8	exists	exist	VERB
ejpam-4571	575	9	i(x	i(x	NOUN
ejpam-4571	575	10	)	)	PUNCT
ejpam-4571	575	11	∈	∈	PROPN
ejpam-4571	576	1	i	i	PRON
ejpam-4571	576	2	such	such	ADJ
ejpam-4571	576	3	that	that	SCONJ
ejpam-4571	576	4	f(x	f(x	PROPN
ejpam-4571	576	5	)	)	PUNCT
ejpam-4571	576	6	∈	∈	PROPN
ejpam-4571	576	7	vi(x	vi(x	NOUN
ejpam-4571	576	8	)	)	PUNCT
ejpam-4571	576	9	.	.	PUNCT
ejpam-4571	577	1	since	since	SCONJ
ejpam-4571	577	2	f	f	PROPN
ejpam-4571	577	3	is	be	AUX
ejpam-4571	577	4	weakly	weakly	ADJ
ejpam-4571	577	5	(	(	PUNCT
ejpam-4571	577	6	λ	λ	NOUN
ejpam-4571	577	7	,	,	PUNCT
ejpam-4571	577	8	b)-continuous	b)-continuous	ADJ
ejpam-4571	577	9	,	,	PUNCT
ejpam-4571	577	10	there	there	PRON
ejpam-4571	577	11	exists	exist	VERB
ejpam-4571	577	12	a	a	DET
ejpam-4571	577	13	(	(	PUNCT
ejpam-4571	577	14	λ	λ	NOUN
ejpam-4571	577	15	,	,	PUNCT
ejpam-4571	577	16	b)-open	b)-open	VERB
ejpam-4571	577	17	set	set	VERB
ejpam-4571	577	18	u(x	u(x	NOUN
ejpam-4571	577	19	)	)	PUNCT
ejpam-4571	577	20	containing	contain	VERB
ejpam-4571	577	21	x	x	PUNCT
ejpam-4571	577	22	such	such	ADJ
ejpam-4571	577	23	that	that	SCONJ
ejpam-4571	577	24	f(u(x	f(u(x	PROPN
ejpam-4571	577	25	)	)	PUNCT
ejpam-4571	577	26	)	)	PUNCT
ejpam-4571	578	1	⊆	⊆	NUM
ejpam-4571	578	2	[	[	X
ejpam-4571	578	3	vi(x	vi(x	NUM
ejpam-4571	578	4	)	)	PUNCT
ejpam-4571	578	5	]	]	PUNCT
ejpam-4571	578	6	(	(	PUNCT
ejpam-4571	578	7	λ	λ	X
ejpam-4571	578	8	,	,	PUNCT
ejpam-4571	578	9	b	b	NOUN
ejpam-4571	578	10	)	)	PUNCT
ejpam-4571	578	11	.	.	PUNCT
ejpam-4571	579	1	the	the	DET
ejpam-4571	579	2	family	family	NOUN
ejpam-4571	579	3	{	{	PUNCT
ejpam-4571	579	4	u(x	u(x	PROPN
ejpam-4571	579	5	)	)	PUNCT
ejpam-4571	579	6	:	:	PUNCT
ejpam-4571	580	1	x	x	X
ejpam-4571	580	2	∈	∈	PROPN
ejpam-4571	580	3	k	k	NOUN
ejpam-4571	580	4	}	}	PUNCT
ejpam-4571	580	5	is	be	AUX
ejpam-4571	580	6	a	a	DET
ejpam-4571	580	7	cover	cover	NOUN
ejpam-4571	580	8	of	of	ADP
ejpam-4571	580	9	k	k	X
ejpam-4571	580	10	by	by	ADP
ejpam-4571	580	11	(	(	PUNCT
ejpam-4571	580	12	λ	λ	INTJ
ejpam-4571	580	13	,	,	PUNCT
ejpam-4571	580	14	b)-open	b)-open	VERB
ejpam-4571	580	15	sets	set	NOUN
ejpam-4571	580	16	of	of	ADP
ejpam-4571	580	17	x.	x.	NOUN
ejpam-4571	580	18	since	since	SCONJ
ejpam-4571	580	19	k	k	PROPN
ejpam-4571	580	20	is	be	AUX
ejpam-4571	580	21	λb	λb	ADJ
ejpam-4571	580	22	-	-	ADJ
ejpam-4571	580	23	compact	compact	ADJ
ejpam-4571	580	24	,	,	PUNCT
ejpam-4571	580	25	there	there	PRON
ejpam-4571	580	26	exist	exist	VERB
ejpam-4571	580	27	a	a	DET
ejpam-4571	580	28	finite	finite	ADJ
ejpam-4571	580	29	number	number	NOUN
ejpam-4571	580	30	of	of	ADP
ejpam-4571	580	31	points	point	NOUN
ejpam-4571	580	32	,	,	PUNCT
ejpam-4571	580	33	say	say	INTJ
ejpam-4571	580	34	,	,	PUNCT
ejpam-4571	580	35	x1	x1	PROPN
ejpam-4571	580	36	,	,	PUNCT
ejpam-4571	580	37	x2	x2	PROPN
ejpam-4571	580	38	,	,	PUNCT
ejpam-4571	580	39	...	...	PUNCT
ejpam-4571	580	40	,	,	PUNCT
ejpam-4571	580	41	xn	xn	PROPN
ejpam-4571	580	42	in	in	ADP
ejpam-4571	580	43	k	k	PROPN
ejpam-4571	580	44	such	such	ADJ
ejpam-4571	580	45	that	that	SCONJ
ejpam-4571	580	46	k	k	PROPN
ejpam-4571	580	47	⊆	⊆	NUM
ejpam-4571	580	48	∪{u(xk	∪{u(xk	NUM
ejpam-4571	580	49	)	)	PUNCT
ejpam-4571	580	50	:	:	PUNCT
ejpam-4571	580	51	xk	xk	PROPN
ejpam-4571	580	52	∈	∈	PROPN
ejpam-4571	580	53	k	k	PROPN
ejpam-4571	580	54	,	,	PUNCT
ejpam-4571	580	55	1	1	NUM
ejpam-4571	580	56	≤	≤	NUM
ejpam-4571	580	57	k	k	X
ejpam-4571	580	58	≤	≤	PROPN
ejpam-4571	580	59	n	n	CCONJ
ejpam-4571	580	60	}	}	PUNCT
ejpam-4571	580	61	.	.	PUNCT
ejpam-4571	581	1	thus	thus	ADV
ejpam-4571	581	2	,	,	PUNCT
ejpam-4571	581	3	f(k	f(k	VERB
ejpam-4571	581	4	)	)	PUNCT
ejpam-4571	581	5	⊆	⊆	NUM
ejpam-4571	581	6	∪{f(u(xk	∪{f(u(xk	NUM
ejpam-4571	581	7	)	)	PUNCT
ejpam-4571	581	8	)	)	PUNCT
ejpam-4571	581	9	:	:	PUNCT
ejpam-4571	582	1	xk	xk	PROPN
ejpam-4571	582	2	∈	∈	PROPN
ejpam-4571	582	3	k	k	PROPN
ejpam-4571	582	4	,	,	PUNCT
ejpam-4571	582	5	1	1	NUM
ejpam-4571	582	6	≤	≤	NUM
ejpam-4571	582	7	k	k	X
ejpam-4571	582	8	≤	≤	PROPN
ejpam-4571	582	9	n	n	CCONJ
ejpam-4571	582	10	}	}	PUNCT
ejpam-4571	582	11	⊆	⊆	NUM
ejpam-4571	582	12	∪{[vi(xk	∪{[vi(xk	NUM
ejpam-4571	582	13	)	)	PUNCT
ejpam-4571	582	14	]	]	PUNCT
ejpam-4571	582	15	(	(	PUNCT
ejpam-4571	582	16	λ	λ	X
ejpam-4571	582	17	,	,	PUNCT
ejpam-4571	582	18	b	b	NOUN
ejpam-4571	582	19	)	)	PUNCT
ejpam-4571	582	20	:	:	PUNCT
ejpam-4571	582	21	xk	xk	PROPN
ejpam-4571	582	22	∈	∈	PROPN
ejpam-4571	582	23	k	k	PROPN
ejpam-4571	582	24	,	,	PUNCT
ejpam-4571	582	25	1	1	NUM
ejpam-4571	582	26	≤	≤	NUM
ejpam-4571	582	27	k	k	X
ejpam-4571	582	28	≤	≤	PROPN
ejpam-4571	582	29	n	n	CCONJ
ejpam-4571	582	30	}	}	PUNCT
ejpam-4571	582	31	.	.	PUNCT
ejpam-4571	583	1	this	this	PRON
ejpam-4571	583	2	shows	show	VERB
ejpam-4571	583	3	that	that	SCONJ
ejpam-4571	583	4	f(k	f(k	VERB
ejpam-4571	583	5	)	)	PUNCT
ejpam-4571	583	6	is	be	AUX
ejpam-4571	583	7	λb	λb	ADV
ejpam-4571	583	8	-	-	PUNCT
ejpam-4571	583	9	closed	closed	ADJ
ejpam-4571	583	10	relative	relative	ADJ
ejpam-4571	583	11	to	to	ADP
ejpam-4571	583	12	(	(	PUNCT
ejpam-4571	583	13	y	y	PROPN
ejpam-4571	583	14	,	,	PUNCT
ejpam-4571	583	15	σ	σ	PROPN
ejpam-4571	583	16	)	)	PUNCT
ejpam-4571	583	17	.	.	PUNCT
ejpam-4571	584	1	corollary	corollary	ADJ
ejpam-4571	584	2	3	3	X
ejpam-4571	584	3	.	.	PUNCT
ejpam-4571	585	1	if	if	SCONJ
ejpam-4571	585	2	f	f	PROPN
ejpam-4571	585	3	:	:	PUNCT
ejpam-4571	585	4	(	(	PUNCT
ejpam-4571	585	5	x	x	X
ejpam-4571	585	6	,	,	PUNCT
ejpam-4571	585	7	τ	τ	X
ejpam-4571	585	8	)	)	PUNCT
ejpam-4571	585	9	→	→	SYM
ejpam-4571	585	10	(	(	PUNCT
ejpam-4571	585	11	y	y	PROPN
ejpam-4571	585	12	,	,	PUNCT
ejpam-4571	585	13	σ	σ	PROPN
ejpam-4571	585	14	)	)	PUNCT
ejpam-4571	585	15	is	be	AUX
ejpam-4571	585	16	a	a	DET
ejpam-4571	585	17	weakly	weakly	ADJ
ejpam-4571	585	18	(	(	PUNCT
ejpam-4571	585	19	λ	λ	PROPN
ejpam-4571	585	20	,	,	PUNCT
ejpam-4571	585	21	b)-continuous	b)-continuous	ADJ
ejpam-4571	585	22	surjection	surjection	NOUN
ejpam-4571	585	23	and	and	CCONJ
ejpam-4571	585	24	(	(	PUNCT
ejpam-4571	585	25	x	x	NOUN
ejpam-4571	585	26	,	,	PUNCT
ejpam-4571	585	27	τ	τ	X
ejpam-4571	585	28	)	)	PUNCT
ejpam-4571	585	29	is	be	AUX
ejpam-4571	585	30	λb	λb	ADJ
ejpam-4571	585	31	-	-	ADJ
ejpam-4571	585	32	compact	compact	ADJ
ejpam-4571	585	33	,	,	PUNCT
ejpam-4571	585	34	then	then	ADV
ejpam-4571	585	35	(	(	PUNCT
ejpam-4571	585	36	y	y	PROPN
ejpam-4571	585	37	,	,	PUNCT
ejpam-4571	585	38	σ	σ	PROPN
ejpam-4571	585	39	)	)	PUNCT
ejpam-4571	585	40	is	be	AUX
ejpam-4571	585	41	λb	λb	ADV
ejpam-4571	585	42	-	-	PUNCT
ejpam-4571	585	43	closed	closed	ADJ
ejpam-4571	585	44	.	.	PUNCT
ejpam-4571	586	1	definition	definition	NOUN
ejpam-4571	586	2	8	8	NUM
ejpam-4571	586	3	.	.	PUNCT
ejpam-4571	587	1	[	[	X
ejpam-4571	587	2	5	5	X
ejpam-4571	587	3	]	]	PUNCT
ejpam-4571	587	4	let	let	VERB
ejpam-4571	587	5	a	a	PRON
ejpam-4571	587	6	be	be	AUX
ejpam-4571	587	7	a	a	DET
ejpam-4571	587	8	subset	subset	NOUN
ejpam-4571	587	9	of	of	ADP
ejpam-4571	587	10	a	a	DET
ejpam-4571	587	11	topological	topological	ADJ
ejpam-4571	587	12	space	space	NOUN
ejpam-4571	587	13	(	(	PUNCT
ejpam-4571	587	14	x	x	X
ejpam-4571	587	15	,	,	PUNCT
ejpam-4571	587	16	τ	τ	PROPN
ejpam-4571	587	17	)	)	PUNCT
ejpam-4571	587	18	.	.	PUNCT
ejpam-4571	588	1	the	the	DET
ejpam-4571	588	2	(	(	PUNCT
ejpam-4571	588	3	λ	λ	PROPN
ejpam-4571	588	4	,	,	PUNCT
ejpam-4571	588	5	b)-frontier	b)-fronti	ADJ
ejpam-4571	588	6	of	of	ADP
ejpam-4571	588	7	a	a	DET
ejpam-4571	588	8	,	,	PUNCT
ejpam-4571	588	9	λbfr(a	λbfr(a	NUM
ejpam-4571	588	10	)	)	PUNCT
ejpam-4571	588	11	,	,	PUNCT
ejpam-4571	588	12	is	be	AUX
ejpam-4571	588	13	defined	define	VERB
ejpam-4571	588	14	as	as	SCONJ
ejpam-4571	588	15	follows	follow	VERB
ejpam-4571	588	16	:	:	PUNCT
ejpam-4571	588	17	λbfr(a	λbfr(a	NUM
ejpam-4571	588	18	)	)	PUNCT
ejpam-4571	588	19	=	=	PUNCT
ejpam-4571	589	1	a(λ	a(λ	PROPN
ejpam-4571	589	2	,	,	PUNCT
ejpam-4571	589	3	b	b	NOUN
ejpam-4571	589	4	)	)	PUNCT
ejpam-4571	589	5	∩	∩	NOUN
ejpam-4571	589	6	[	[	X
ejpam-4571	589	7	x	x	SYM
ejpam-4571	589	8	−a](λ	−a](λ	PROPN
ejpam-4571	589	9	,	,	PUNCT
ejpam-4571	589	10	b	b	NOUN
ejpam-4571	589	11	)	)	PUNCT
ejpam-4571	589	12	.	.	PUNCT
ejpam-4571	590	1	theorem	theorem	VERB
ejpam-4571	590	2	16	16	NUM
ejpam-4571	590	3	.	.	PUNCT
ejpam-4571	591	1	the	the	DET
ejpam-4571	591	2	set	set	NOUN
ejpam-4571	591	3	of	of	ADP
ejpam-4571	591	4	all	all	DET
ejpam-4571	591	5	points	point	NOUN
ejpam-4571	591	6	x	x	X
ejpam-4571	591	7	∈	∈	NOUN
ejpam-4571	591	8	x	x	PUNCT
ejpam-4571	591	9	at	at	ADP
ejpam-4571	591	10	which	which	PRON
ejpam-4571	591	11	a	a	DET
ejpam-4571	591	12	function	function	NOUN
ejpam-4571	591	13	f	f	NOUN
ejpam-4571	591	14	:	:	PUNCT
ejpam-4571	591	15	(	(	PUNCT
ejpam-4571	591	16	x	x	X
ejpam-4571	591	17	,	,	PUNCT
ejpam-4571	591	18	τ	τ	X
ejpam-4571	591	19	)	)	PUNCT
ejpam-4571	591	20	→	→	SYM
ejpam-4571	591	21	(	(	PUNCT
ejpam-4571	591	22	y	y	PROPN
ejpam-4571	591	23	,	,	PUNCT
ejpam-4571	591	24	σ	σ	PROPN
ejpam-4571	591	25	)	)	PUNCT
ejpam-4571	591	26	is	be	AUX
ejpam-4571	591	27	not	not	PART
ejpam-4571	591	28	weakly	weakly	ADJ
ejpam-4571	591	29	(	(	PUNCT
ejpam-4571	591	30	λ	λ	NOUN
ejpam-4571	591	31	,	,	PUNCT
ejpam-4571	591	32	b)-continuous	b)-continuous	ADJ
ejpam-4571	591	33	is	be	AUX
ejpam-4571	591	34	identical	identical	ADJ
ejpam-4571	591	35	with	with	ADP
ejpam-4571	591	36	the	the	DET
ejpam-4571	591	37	union	union	NOUN
ejpam-4571	591	38	of	of	ADP
ejpam-4571	591	39	the	the	DET
ejpam-4571	591	40	(	(	PUNCT
ejpam-4571	591	41	λ	λ	PROPN
ejpam-4571	591	42	,	,	PUNCT
ejpam-4571	591	43	b)-frontiers	b)-frontier	NOUN
ejpam-4571	591	44	of	of	ADP
ejpam-4571	591	45	the	the	DET
ejpam-4571	591	46	inverse	inverse	NOUN
ejpam-4571	591	47	images	image	NOUN
ejpam-4571	591	48	of	of	ADP
ejpam-4571	591	49	the	the	DET
ejpam-4571	591	50	(	(	PUNCT
ejpam-4571	591	51	λ	λ	PROPN
ejpam-4571	591	52	,	,	PUNCT
ejpam-4571	591	53	b)-closure	b)-closure	NOUN
ejpam-4571	591	54	of	of	ADP
ejpam-4571	591	55	the	the	DET
ejpam-4571	591	56	(	(	PUNCT
ejpam-4571	591	57	λ	λ	PROPN
ejpam-4571	591	58	,	,	PUNCT
ejpam-4571	591	59	b)-open	b)-open	VERB
ejpam-4571	591	60	sets	set	NOUN
ejpam-4571	591	61	of	of	ADP
ejpam-4571	591	62	y	y	NOUN
ejpam-4571	591	63	containing	contain	VERB
ejpam-4571	591	64	f(x	f(x	PROPN
ejpam-4571	591	65	)	)	PUNCT
ejpam-4571	591	66	.	.	PUNCT
ejpam-4571	592	1	proof	proof	NOUN
ejpam-4571	592	2	.	.	PUNCT
ejpam-4571	593	1	suppose	suppose	VERB
ejpam-4571	593	2	that	that	SCONJ
ejpam-4571	593	3	f	f	PROPN
ejpam-4571	593	4	is	be	AUX
ejpam-4571	593	5	not	not	PART
ejpam-4571	593	6	weakly	weakly	ADJ
ejpam-4571	593	7	(	(	PUNCT
ejpam-4571	593	8	λ	λ	NOUN
ejpam-4571	593	9	,	,	PUNCT
ejpam-4571	593	10	b)-continuous	b)-continuous	ADJ
ejpam-4571	593	11	at	at	SCONJ
ejpam-4571	593	12	x	x	SYM
ejpam-4571	593	13	∈	∈	PROPN
ejpam-4571	593	14	x.	x.	NOUN
ejpam-4571	593	15	there	there	PRON
ejpam-4571	593	16	exists	exist	VERB
ejpam-4571	593	17	a	a	DET
ejpam-4571	593	18	(	(	PUNCT
ejpam-4571	593	19	λ	λ	NOUN
ejpam-4571	593	20	,	,	PUNCT
ejpam-4571	593	21	b)-open	b)-open	VERB
ejpam-4571	593	22	set	set	VERB
ejpam-4571	593	23	v	v	NOUN
ejpam-4571	593	24	of	of	ADP
ejpam-4571	593	25	y	y	NOUN
ejpam-4571	593	26	containing	contain	VERB
ejpam-4571	593	27	f(x	f(x	PROPN
ejpam-4571	593	28	)	)	PUNCT
ejpam-4571	593	29	such	such	ADJ
ejpam-4571	593	30	that	that	DET
ejpam-4571	593	31	f(u	f(u	PROPN
ejpam-4571	593	32	)	)	PUNCT
ejpam-4571	593	33	is	be	AUX
ejpam-4571	593	34	not	not	PART
ejpam-4571	593	35	contained	contain	VERB
ejpam-4571	593	36	in	in	ADP
ejpam-4571	593	37	v	v	NUM
ejpam-4571	593	38	(	(	PUNCT
ejpam-4571	593	39	λ	λ	PROPN
ejpam-4571	593	40	,	,	PUNCT
ejpam-4571	593	41	b	b	NOUN
ejpam-4571	593	42	)	)	PUNCT
ejpam-4571	593	43	for	for	SCONJ
ejpam-4571	593	44	every	every	DET
ejpam-4571	593	45	(	(	PUNCT
ejpam-4571	593	46	λ	λ	NOUN
ejpam-4571	593	47	,	,	PUNCT
ejpam-4571	593	48	b)-open	b)-open	AUX
ejpam-4571	593	49	set	set	VERB
ejpam-4571	593	50	u	u	NOUN
ejpam-4571	593	51	containing	contain	VERB
ejpam-4571	593	52	x.	x.	NOUN
ejpam-4571	593	53	then	then	ADV
ejpam-4571	593	54	,	,	PUNCT
ejpam-4571	593	55	u	u	NOUN
ejpam-4571	593	56	∩	∩	NOUN
ejpam-4571	593	57	(	(	PUNCT
ejpam-4571	593	58	x	x	SYM
ejpam-4571	593	59	−	−	PROPN
ejpam-4571	593	60	f−1(v	f−1(v	NOUN
ejpam-4571	593	61	(	(	PUNCT
ejpam-4571	593	62	λ	λ	PROPN
ejpam-4571	593	63	,	,	PUNCT
ejpam-4571	593	64	b	b	NOUN
ejpam-4571	593	65	)	)	PUNCT
ejpam-4571	593	66	)	)	PUNCT
ejpam-4571	593	67	)	)	PUNCT
ejpam-4571	594	1	̸=	̸=	NOUN
ejpam-4571	594	2	∅	∅	NOUN
ejpam-4571	594	3	for	for	ADP
ejpam-4571	594	4	every	every	DET
ejpam-4571	594	5	(	(	PUNCT
ejpam-4571	594	6	λ	λ	NOUN
ejpam-4571	594	7	,	,	PUNCT
ejpam-4571	594	8	b)-open	b)-open	VERB
ejpam-4571	594	9	set	set	VERB
ejpam-4571	594	10	u	u	NOUN
ejpam-4571	594	11	of	of	ADP
ejpam-4571	594	12	x	x	PUNCT
ejpam-4571	594	13	containing	contain	VERB
ejpam-4571	594	14	x	x	X
ejpam-4571	594	15	and	and	CCONJ
ejpam-4571	594	16	hence	hence	ADV
ejpam-4571	594	17	x	x	X
ejpam-4571	594	18	∈	∈	PROPN
ejpam-4571	595	1	[	[	X
ejpam-4571	595	2	x	x	X
ejpam-4571	595	3	−	−	PROPN
ejpam-4571	595	4	f−1(v	f−1(v	NOUN
ejpam-4571	595	5	(	(	PUNCT
ejpam-4571	595	6	λ	λ	PROPN
ejpam-4571	595	7	,	,	PUNCT
ejpam-4571	595	8	b))](λ	b))](λ	PROPN
ejpam-4571	595	9	,	,	PUNCT
ejpam-4571	595	10	b	b	NOUN
ejpam-4571	595	11	)	)	PUNCT
ejpam-4571	595	12	.	.	PUNCT
ejpam-4571	596	1	on	on	ADP
ejpam-4571	596	2	the	the	DET
ejpam-4571	596	3	other	other	ADJ
ejpam-4571	596	4	hand	hand	NOUN
ejpam-4571	596	5	,	,	PUNCT
ejpam-4571	596	6	we	we	PRON
ejpam-4571	596	7	have	have	VERB
ejpam-4571	596	8	x	x	X
ejpam-4571	596	9	∈	∈	PROPN
ejpam-4571	596	10	f−1(v	f−1(v	NOUN
ejpam-4571	596	11	)	)	PUNCT
ejpam-4571	596	12	⊆	⊆	NUM
ejpam-4571	597	1	[	[	X
ejpam-4571	597	2	f−1(v	f−1(v	NOUN
ejpam-4571	597	3	(	(	PUNCT
ejpam-4571	597	4	λ	λ	PROPN
ejpam-4571	597	5	,	,	PUNCT
ejpam-4571	597	6	b))](λ	b))](λ	PROPN
ejpam-4571	597	7	,	,	PUNCT
ejpam-4571	597	8	b	b	NOUN
ejpam-4571	597	9	)	)	PUNCT
ejpam-4571	597	10	.	.	PUNCT
ejpam-4571	598	1	thus	thus	ADV
ejpam-4571	598	2	,	,	PUNCT
ejpam-4571	598	3	x	x	PROPN
ejpam-4571	598	4	∈	∈	X
ejpam-4571	598	5	λbfr(f−1(v	λbfr(f−1(v	X
ejpam-4571	598	6	(	(	PUNCT
ejpam-4571	598	7	λ	λ	X
ejpam-4571	598	8	,	,	PUNCT
ejpam-4571	598	9	b	b	NOUN
ejpam-4571	598	10	)	)	PUNCT
ejpam-4571	598	11	)	)	PUNCT
ejpam-4571	598	12	)	)	PUNCT
ejpam-4571	598	13	.	.	PUNCT
ejpam-4571	599	1	conversely	conversely	ADV
ejpam-4571	599	2	,	,	PUNCT
ejpam-4571	599	3	suppose	suppose	VERB
ejpam-4571	599	4	that	that	SCONJ
ejpam-4571	599	5	f	f	PROPN
ejpam-4571	599	6	is	be	AUX
ejpam-4571	599	7	weakly	weakly	ADJ
ejpam-4571	599	8	(	(	PUNCT
ejpam-4571	599	9	λ	λ	X
ejpam-4571	599	10	,	,	PUNCT
ejpam-4571	599	11	b)-continuous	b)-continuous	ADJ
ejpam-4571	599	12	at	at	ADP
ejpam-4571	599	13	x	x	X
ejpam-4571	599	14	∈	∈	PROPN
ejpam-4571	599	15	x	x	PUNCT
ejpam-4571	599	16	and	and	CCONJ
ejpam-4571	599	17	let	let	VERB
ejpam-4571	599	18	v	v	PART
ejpam-4571	599	19	be	be	AUX
ejpam-4571	599	20	any	any	DET
ejpam-4571	599	21	(	(	PUNCT
ejpam-4571	599	22	λ	λ	NOUN
ejpam-4571	599	23	,	,	PUNCT
ejpam-4571	599	24	b)-open	b)-open	VERB
ejpam-4571	599	25	set	set	VERB
ejpam-4571	599	26	of	of	ADP
ejpam-4571	599	27	y	y	PROPN
ejpam-4571	599	28	containing	contain	VERB
ejpam-4571	599	29	f(x	f(x	PROPN
ejpam-4571	599	30	)	)	PUNCT
ejpam-4571	599	31	.	.	PUNCT
ejpam-4571	600	1	thus	thus	ADV
ejpam-4571	600	2	,	,	PUNCT
ejpam-4571	600	3	by	by	ADP
ejpam-4571	600	4	theorem	theorem	NOUN
ejpam-4571	600	5	7	7	NUM
ejpam-4571	600	6	,	,	PUNCT
ejpam-4571	600	7	we	we	PRON
ejpam-4571	600	8	have	have	VERB
ejpam-4571	600	9	x	x	PROPN
ejpam-4571	600	10	∈	∈	PROPN
ejpam-4571	601	1	[	[	X
ejpam-4571	601	2	f−1(v	f−1(v	NOUN
ejpam-4571	601	3	(	(	PUNCT
ejpam-4571	601	4	λ	λ	PROPN
ejpam-4571	601	5	,	,	PUNCT
ejpam-4571	601	6	b))](λ	b))](λ	PROPN
ejpam-4571	601	7	,	,	PUNCT
ejpam-4571	601	8	b	b	NOUN
ejpam-4571	601	9	)	)	PUNCT
ejpam-4571	601	10	.	.	PUNCT
ejpam-4571	602	1	this	this	PRON
ejpam-4571	602	2	shows	show	VERB
ejpam-4571	602	3	that	that	SCONJ
ejpam-4571	602	4	x	x	PROPN
ejpam-4571	602	5	̸∈	̸∈	PROPN
ejpam-4571	602	6	λbfr(f−1(v	λbfr(f−1(v	X
ejpam-4571	602	7	(	(	PUNCT
ejpam-4571	602	8	λ	λ	PROPN
ejpam-4571	602	9	,	,	PUNCT
ejpam-4571	602	10	b	b	NOUN
ejpam-4571	602	11	)	)	PUNCT
ejpam-4571	602	12	)	)	PUNCT
ejpam-4571	602	13	)	)	PUNCT
ejpam-4571	602	14	for	for	ADP
ejpam-4571	602	15	each	each	DET
ejpam-4571	602	16	(	(	PUNCT
ejpam-4571	602	17	λ	λ	PROPN
ejpam-4571	602	18	,	,	PUNCT
ejpam-4571	602	19	b)-open	b)-open	AUX
ejpam-4571	602	20	set	set	VERB
ejpam-4571	602	21	v	v	NOUN
ejpam-4571	602	22	of	of	ADP
ejpam-4571	602	23	y	y	NOUN
ejpam-4571	602	24	containing	contain	VERB
ejpam-4571	602	25	f(x	f(x	PROPN
ejpam-4571	602	26	)	)	PUNCT
ejpam-4571	602	27	.	.	PUNCT
ejpam-4571	603	1	this	this	PRON
ejpam-4571	603	2	completes	complete	VERB
ejpam-4571	603	3	the	the	DET
ejpam-4571	603	4	proof	proof	NOUN
ejpam-4571	603	5	.	.	PUNCT
ejpam-4571	604	1	acknowledgements	acknowledgement	NOUN
ejpam-4571	604	2	this	this	DET
ejpam-4571	604	3	research	research	NOUN
ejpam-4571	604	4	project	project	NOUN
ejpam-4571	604	5	was	be	AUX
ejpam-4571	604	6	financially	financially	ADV
ejpam-4571	604	7	supported	support	VERB
ejpam-4571	604	8	by	by	ADP
ejpam-4571	604	9	mahasarakham	mahasarakham	PROPN
ejpam-4571	604	10	university	university	PROPN
ejpam-4571	604	11	.	.	PUNCT
ejpam-4571	605	1	references	reference	NOUN
ejpam-4571	605	2	43	43	NUM
ejpam-4571	605	3	references	reference	NOUN
ejpam-4571	605	4	[	[	X
ejpam-4571	605	5	1	1	NUM
ejpam-4571	605	6	]	]	PUNCT
ejpam-4571	605	7	d.	d.	PROPN
ejpam-4571	605	8	andrijević.	andrijević.	PROPN
ejpam-4571	605	9	semi	semi	ADJ
ejpam-4571	605	10	-	-	ADJ
ejpam-4571	605	11	preopen	preopen	ADJ
ejpam-4571	605	12	sets	set	NOUN
ejpam-4571	605	13	.	.	PUNCT
ejpam-4571	606	1	matematički	matematički	PROPN
ejpam-4571	606	2	vesnik	vesnik	PROPN
ejpam-4571	606	3	,	,	PUNCT
ejpam-4571	606	4	38:24–32	38:24–32	NUM
ejpam-4571	606	5	,	,	PUNCT
ejpam-4571	606	6	1986	1986	NUM
ejpam-4571	606	7	.	.	PUNCT
ejpam-4571	607	1	[	[	X
ejpam-4571	607	2	2	2	X
ejpam-4571	607	3	]	]	PUNCT
ejpam-4571	607	4	d.	d.	PROPN
ejpam-4571	607	5	andrijević.	andrijević.	PROPN
ejpam-4571	607	6	on	on	ADP
ejpam-4571	607	7	b	b	X
ejpam-4571	607	8	-	-	PUNCT
ejpam-4571	607	9	open	open	ADJ
ejpam-4571	607	10	sets	set	NOUN
ejpam-4571	607	11	.	.	PUNCT
ejpam-4571	608	1	matematički	matematički	PROPN
ejpam-4571	608	2	vesnik	vesnik	PROPN
ejpam-4571	608	3	,	,	PUNCT
ejpam-4571	608	4	48:59–64	48:59–64	PROPN
ejpam-4571	608	5	,	,	PUNCT
ejpam-4571	608	6	1996	1996	NUM
ejpam-4571	608	7	.	.	PUNCT
ejpam-4571	609	1	[	[	X
ejpam-4571	609	2	3	3	NUM
ejpam-4571	609	3	]	]	X
ejpam-4571	609	4	a.	a.	NOUN
ejpam-4571	609	5	v.	v.	PROPN
ejpam-4571	609	6	arhangel’skĭı	arhangel’skĭı	PROPN
ejpam-4571	609	7	and	and	CCONJ
ejpam-4571	609	8	p.	p.	PROPN
ejpam-4571	609	9	j.	j.	PROPN
ejpam-4571	609	10	collins	collins	PROPN
ejpam-4571	609	11	.	.	PUNCT
ejpam-4571	610	1	on	on	ADP
ejpam-4571	610	2	submaximal	submaximal	ADJ
ejpam-4571	610	3	spaces	space	NOUN
ejpam-4571	610	4	.	.	PUNCT
ejpam-4571	611	1	topology	topology	NOUN
ejpam-4571	611	2	and	and	CCONJ
ejpam-4571	611	3	its	its	PRON
ejpam-4571	611	4	applications	application	NOUN
ejpam-4571	611	5	,	,	PUNCT
ejpam-4571	611	6	64:219–241	64:219–241	NUM
ejpam-4571	611	7	,	,	PUNCT
ejpam-4571	611	8	1995	1995	NUM
ejpam-4571	611	9	.	.	PUNCT
ejpam-4571	612	1	[	[	X
ejpam-4571	612	2	4	4	NUM
ejpam-4571	612	3	]	]	PUNCT
ejpam-4571	612	4	c.	c.	PROPN
ejpam-4571	612	5	w.	w.	PROPN
ejpam-4571	612	6	baker	baker	PROPN
ejpam-4571	612	7	.	.	PUNCT
ejpam-4571	613	1	properties	property	NOUN
ejpam-4571	613	2	of	of	ADP
ejpam-4571	613	3	subweakly	subweakly	ADJ
ejpam-4571	613	4	continuous	continuous	ADJ
ejpam-4571	613	5	functions	function	NOUN
ejpam-4571	613	6	.	.	PUNCT
ejpam-4571	614	1	yokohama	yokohama	PROPN
ejpam-4571	614	2	mathematical	mathematical	PROPN
ejpam-4571	614	3	journal	journal	PROPN
ejpam-4571	614	4	,	,	PUNCT
ejpam-4571	614	5	32:39–43	32:39–43	NUM
ejpam-4571	614	6	,	,	PUNCT
ejpam-4571	614	7	1984	1984	NUM
ejpam-4571	614	8	.	.	PUNCT
ejpam-4571	615	1	[	[	X
ejpam-4571	615	2	5	5	X
ejpam-4571	615	3	]	]	PUNCT
ejpam-4571	615	4	c.	c.	PROPN
ejpam-4571	615	5	boonpok	boonpok	PROPN
ejpam-4571	615	6	.	.	PUNCT
ejpam-4571	616	1	generalized	generalize	VERB
ejpam-4571	616	2	(	(	PUNCT
ejpam-4571	616	3	λ	λ	PROPN
ejpam-4571	616	4	,	,	PUNCT
ejpam-4571	616	5	b)-closed	b)-close	VERB
ejpam-4571	616	6	sets	set	NOUN
ejpam-4571	616	7	in	in	ADP
ejpam-4571	616	8	topological	topological	ADJ
ejpam-4571	616	9	spaces	space	NOUN
ejpam-4571	616	10	.	.	PUNCT
ejpam-4571	617	1	korean	korean	ADJ
ejpam-4571	617	2	journal	journal	PROPN
ejpam-4571	617	3	of	of	ADP
ejpam-4571	617	4	mathematics	mathematic	NOUN
ejpam-4571	617	5	,	,	PUNCT
ejpam-4571	617	6	25:437–453	25:437–453	NUM
ejpam-4571	617	7	,	,	PUNCT
ejpam-4571	617	8	2017	2017	NUM
ejpam-4571	617	9	.	.	PUNCT
ejpam-4571	618	1	[	[	X
ejpam-4571	618	2	6	6	NUM
ejpam-4571	618	3	]	]	PUNCT
ejpam-4571	618	4	c.	c.	PROPN
ejpam-4571	618	5	boonpok	boonpok	PROPN
ejpam-4571	618	6	and	and	CCONJ
ejpam-4571	618	7	c.	c.	PROPN
ejpam-4571	618	8	viriyapong	viriyapong	PROPN
ejpam-4571	618	9	.	.	PUNCT
ejpam-4571	619	1	on	on	ADP
ejpam-4571	619	2	(	(	PUNCT
ejpam-4571	619	3	λ	λ	PROPN
ejpam-4571	619	4	,	,	PUNCT
ejpam-4571	619	5	p)-closed	p)-close	VERB
ejpam-4571	619	6	sets	set	NOUN
ejpam-4571	619	7	and	and	CCONJ
ejpam-4571	619	8	the	the	DET
ejpam-4571	619	9	related	related	ADJ
ejpam-4571	619	10	notions	notion	NOUN
ejpam-4571	619	11	in	in	ADP
ejpam-4571	619	12	topological	topological	ADJ
ejpam-4571	619	13	spaces	space	NOUN
ejpam-4571	619	14	.	.	PUNCT
ejpam-4571	620	1	european	european	ADJ
ejpam-4571	620	2	journal	journal	PROPN
ejpam-4571	620	3	of	of	ADP
ejpam-4571	620	4	pure	pure	ADJ
ejpam-4571	620	5	and	and	CCONJ
ejpam-4571	620	6	applied	applied	ADJ
ejpam-4571	620	7	mathematics	mathematic	NOUN
ejpam-4571	620	8	,	,	PUNCT
ejpam-4571	620	9	15(2):415	15(2):415	PROPN
ejpam-4571	620	10	–	–	PUNCT
ejpam-4571	620	11	436	436	NUM
ejpam-4571	620	12	,	,	PUNCT
ejpam-4571	620	13	2022	2022	NUM
ejpam-4571	620	14	.	.	PUNCT
ejpam-4571	621	1	[	[	X
ejpam-4571	621	2	7	7	X
ejpam-4571	621	3	]	]	X
ejpam-4571	621	4	m.	m.	NOUN
ejpam-4571	621	5	caldas	caldas	PROPN
ejpam-4571	621	6	,	,	PUNCT
ejpam-4571	621	7	s.	s.	PROPN
ejpam-4571	621	8	jafari	jafari	PROPN
ejpam-4571	621	9	,	,	PUNCT
ejpam-4571	621	10	and	and	CCONJ
ejpam-4571	621	11	t.	t.	PROPN
ejpam-4571	621	12	noiri	noiri	PROPN
ejpam-4571	621	13	.	.	PUNCT
ejpam-4571	622	1	on	on	ADP
ejpam-4571	622	2	λb	λb	NOUN
ejpam-4571	622	3	-	-	PUNCT
ejpam-4571	622	4	sets	set	NOUN
ejpam-4571	622	5	and	and	CCONJ
ejpam-4571	622	6	the	the	DET
ejpam-4571	622	7	associated	associated	ADJ
ejpam-4571	622	8	topology	topology	NOUN
ejpam-4571	622	9	τλb	τλb	INTJ
ejpam-4571	622	10	.	.	PUNCT
ejpam-4571	623	1	acta	acta	PROPN
ejpam-4571	623	2	mathematica	mathematica	PROPN
ejpam-4571	623	3	hungarica	hungarica	PROPN
ejpam-4571	623	4	,	,	PUNCT
ejpam-4571	623	5	110:337–345	110:337–345	NUM
ejpam-4571	623	6	,	,	PUNCT
ejpam-4571	623	7	2006	2006	NUM
ejpam-4571	623	8	.	.	PUNCT
ejpam-4571	624	1	[	[	X
ejpam-4571	624	2	8	8	X
ejpam-4571	624	3	]	]	X
ejpam-4571	624	4	h.	h.	PROPN
ejpam-4571	624	5	h.	h.	PROPN
ejpam-4571	624	6	corson	corson	PROPN
ejpam-4571	624	7	and	and	CCONJ
ejpam-4571	624	8	e.	e.	PROPN
ejpam-4571	624	9	michael	michael	PROPN
ejpam-4571	624	10	.	.	PUNCT
ejpam-4571	625	1	metrizability	metrizability	NOUN
ejpam-4571	625	2	of	of	ADP
ejpam-4571	625	3	certain	certain	ADJ
ejpam-4571	625	4	countable	countable	ADJ
ejpam-4571	625	5	unions	union	NOUN
ejpam-4571	625	6	.	.	PUNCT
ejpam-4571	626	1	illinois	illinois	PROPN
ejpam-4571	626	2	journal	journal	PROPN
ejpam-4571	626	3	of	of	ADP
ejpam-4571	626	4	mathematics	mathematic	NOUN
ejpam-4571	626	5	,	,	PUNCT
ejpam-4571	626	6	8:351–360	8:351–360	NUM
ejpam-4571	626	7	,	,	PUNCT
ejpam-4571	626	8	1964	1964	NUM
ejpam-4571	626	9	.	.	PUNCT
ejpam-4571	627	1	[	[	X
ejpam-4571	627	2	9	9	X
ejpam-4571	627	3	]	]	PUNCT
ejpam-4571	627	4	e.	e.	PROPN
ejpam-4571	627	5	hewitt	hewitt	PROPN
ejpam-4571	627	6	.	.	PUNCT
ejpam-4571	628	1	a	a	DET
ejpam-4571	628	2	problem	problem	NOUN
ejpam-4571	628	3	of	of	ADP
ejpam-4571	628	4	set	set	NOUN
ejpam-4571	628	5	-	-	PUNCT
ejpam-4571	628	6	theoretic	theoretic	NOUN
ejpam-4571	628	7	topology	topology	NOUN
ejpam-4571	628	8	.	.	PUNCT
ejpam-4571	629	1	duke	duke	PROPN
ejpam-4571	629	2	mathematical	mathematical	PROPN
ejpam-4571	629	3	journal	journal	PROPN
ejpam-4571	629	4	,	,	PUNCT
ejpam-4571	629	5	10:309	10:309	NUM
ejpam-4571	629	6	–	–	PUNCT
ejpam-4571	629	7	333	333	NUM
ejpam-4571	629	8	,	,	PUNCT
ejpam-4571	629	9	1943	1943	NUM
ejpam-4571	629	10	.	.	PUNCT
ejpam-4571	630	1	[	[	X
ejpam-4571	630	2	10	10	NUM
ejpam-4571	630	3	]	]	X
ejpam-4571	630	4	n.	n.	PROPN
ejpam-4571	630	5	levine	levine	PROPN
ejpam-4571	630	6	.	.	PUNCT
ejpam-4571	631	1	a	a	DET
ejpam-4571	631	2	decomposition	decomposition	NOUN
ejpam-4571	631	3	of	of	ADP
ejpam-4571	631	4	continuity	continuity	NOUN
ejpam-4571	631	5	in	in	ADP
ejpam-4571	631	6	topological	topological	ADJ
ejpam-4571	631	7	spaces	space	NOUN
ejpam-4571	631	8	.	.	PUNCT
ejpam-4571	632	1	the	the	DET
ejpam-4571	632	2	american	american	PROPN
ejpam-4571	632	3	mathematical	mathematical	PROPN
ejpam-4571	632	4	monthly	monthly	ADV
ejpam-4571	632	5	,	,	PUNCT
ejpam-4571	632	6	68:44–46	68:44–46	NUM
ejpam-4571	632	7	,	,	PUNCT
ejpam-4571	632	8	1961	1961	NUM
ejpam-4571	632	9	.	.	PUNCT
ejpam-4571	633	1	[	[	X
ejpam-4571	633	2	11	11	NUM
ejpam-4571	633	3	]	]	X
ejpam-4571	633	4	n.	n.	PROPN
ejpam-4571	633	5	levine	levine	PROPN
ejpam-4571	633	6	.	.	PUNCT
ejpam-4571	634	1	semi	semi	ADJ
ejpam-4571	634	2	-	-	ADJ
ejpam-4571	634	3	open	open	ADJ
ejpam-4571	634	4	sets	set	NOUN
ejpam-4571	634	5	and	and	CCONJ
ejpam-4571	634	6	semi	semi	ADJ
ejpam-4571	634	7	-	-	NOUN
ejpam-4571	634	8	continuity	continuity	NOUN
ejpam-4571	634	9	in	in	ADP
ejpam-4571	634	10	topological	topological	ADJ
ejpam-4571	634	11	spaces	space	NOUN
ejpam-4571	634	12	.	.	PUNCT
ejpam-4571	635	1	the	the	DET
ejpam-4571	635	2	american	american	PROPN
ejpam-4571	635	3	mathematical	mathematical	PROPN
ejpam-4571	635	4	monthly	monthly	ADV
ejpam-4571	635	5	,	,	PUNCT
ejpam-4571	635	6	70:36–41	70:36–41	NUM
ejpam-4571	635	7	,	,	PUNCT
ejpam-4571	635	8	1963	1963	NUM
ejpam-4571	635	9	.	.	PUNCT
ejpam-4571	636	1	[	[	X
ejpam-4571	636	2	12	12	NUM
ejpam-4571	636	3	]	]	PUNCT
ejpam-4571	636	4	a.	a.	NOUN
ejpam-4571	636	5	s.	s.	PROPN
ejpam-4571	636	6	mashhour	mashhour	PROPN
ejpam-4571	636	7	,	,	PUNCT
ejpam-4571	636	8	m.	m.	PROPN
ejpam-4571	636	9	e.	e.	PROPN
ejpam-4571	636	10	abd	abd	PROPN
ejpam-4571	636	11	el	el	PROPN
ejpam-4571	636	12	-	-	PROPN
ejpam-4571	636	13	monsef	monsef	ADJ
ejpam-4571	636	14	,	,	PUNCT
ejpam-4571	636	15	and	and	CCONJ
ejpam-4571	636	16	s.	s.	PROPN
ejpam-4571	636	17	n.	n.	PROPN
ejpam-4571	636	18	el	el	PROPN
ejpam-4571	636	19	-	-	PROPN
ejpam-4571	636	20	deeb	deeb	PROPN
ejpam-4571	636	21	.	.	PUNCT
ejpam-4571	637	1	on	on	ADP
ejpam-4571	637	2	precontinuous	precontinuous	ADJ
ejpam-4571	637	3	and	and	CCONJ
ejpam-4571	637	4	weak	weak	ADJ
ejpam-4571	637	5	precontinuous	precontinuous	ADJ
ejpam-4571	637	6	mappings	mapping	NOUN
ejpam-4571	637	7	.	.	PUNCT
ejpam-4571	638	1	proceedings	proceeding	NOUN
ejpam-4571	638	2	of	of	ADP
ejpam-4571	638	3	the	the	DET
ejpam-4571	638	4	mathematical	mathematical	ADJ
ejpam-4571	638	5	and	and	CCONJ
ejpam-4571	638	6	physical	physical	ADJ
ejpam-4571	638	7	society	society	NOUN
ejpam-4571	638	8	of	of	ADP
ejpam-4571	638	9	egypt	egypt	PROPN
ejpam-4571	638	10	,	,	PUNCT
ejpam-4571	638	11	53:47–53	53:47–53	NUM
ejpam-4571	638	12	,	,	PUNCT
ejpam-4571	638	13	1982	1982	NUM
ejpam-4571	638	14	.	.	PUNCT
ejpam-4571	639	1	[	[	X
ejpam-4571	639	2	13	13	NUM
ejpam-4571	639	3	]	]	X
ejpam-4571	639	4	d.	d.	PROPN
ejpam-4571	639	5	a.	a.	PROPN
ejpam-4571	639	6	rose	rise	VERB
ejpam-4571	639	7	.	.	PUNCT
ejpam-4571	640	1	weak	weak	ADJ
ejpam-4571	640	2	continuity	continuity	NOUN
ejpam-4571	640	3	and	and	CCONJ
ejpam-4571	640	4	almost	almost	ADV
ejpam-4571	640	5	continuity	continuity	NOUN
ejpam-4571	640	6	.	.	PUNCT
ejpam-4571	641	1	international	international	ADJ
ejpam-4571	641	2	journal	journal	PROPN
ejpam-4571	641	3	of	of	ADP
ejpam-4571	641	4	mathematics	mathematics	PROPN
ejpam-4571	641	5	and	and	CCONJ
ejpam-4571	641	6	mathematical	mathematical	ADJ
ejpam-4571	641	7	sciences	science	NOUN
ejpam-4571	641	8	,	,	PUNCT
ejpam-4571	641	9	7:311–318	7:311–318	PROPN
ejpam-4571	641	10	,	,	PUNCT
ejpam-4571	641	11	1984	1984	NUM
ejpam-4571	641	12	.	.	PUNCT
ejpam-4571	642	1	[	[	X
ejpam-4571	642	2	14	14	NUM
ejpam-4571	642	3	]	]	X
ejpam-4571	642	4	d.	d.	PROPN
ejpam-4571	642	5	sivaraj	sivaraj	PROPN
ejpam-4571	642	6	.	.	PUNCT
ejpam-4571	643	1	a	a	DET
ejpam-4571	643	2	note	note	NOUN
ejpam-4571	643	3	on	on	ADP
ejpam-4571	643	4	extremally	extremally	ADV
ejpam-4571	643	5	disconnected	disconnected	ADJ
ejpam-4571	643	6	spaces	space	NOUN
ejpam-4571	643	7	.	.	PUNCT
ejpam-4571	644	1	indian	indian	ADJ
ejpam-4571	644	2	journal	journal	PROPN
ejpam-4571	644	3	of	of	ADP
ejpam-4571	644	4	pure	pure	ADJ
ejpam-4571	644	5	and	and	CCONJ
ejpam-4571	644	6	applied	applied	ADJ
ejpam-4571	644	7	mathematics	mathematic	NOUN
ejpam-4571	644	8	,	,	PUNCT
ejpam-4571	644	9	17(12):1373–1375	17(12):1373–1375	NUM
ejpam-4571	644	10	,	,	PUNCT
ejpam-4571	644	11	1986	1986	NUM
ejpam-4571	644	12	.	.	PUNCT
ejpam-4571	645	1	[	[	X
ejpam-4571	645	2	15	15	NUM
ejpam-4571	645	3	]	]	X
ejpam-4571	645	4	c.	c.	PROPN
ejpam-4571	645	5	viriyapong	viriyapong	PROPN
ejpam-4571	645	6	and	and	CCONJ
ejpam-4571	645	7	c.	c.	PROPN
ejpam-4571	645	8	boonpok	boonpok	PROPN
ejpam-4571	645	9	.	.	PUNCT
ejpam-4571	646	1	some	some	DET
ejpam-4571	646	2	open	open	ADJ
ejpam-4571	646	3	sets	set	NOUN
ejpam-4571	646	4	and	and	CCONJ
ejpam-4571	646	5	related	related	ADJ
ejpam-4571	646	6	topics	topic	NOUN
ejpam-4571	646	7	in	in	ADP
ejpam-4571	646	8	topological	topological	ADJ
ejpam-4571	646	9	spaces	space	NOUN
ejpam-4571	646	10	.	.	PUNCT
ejpam-4571	647	1	wseas	wseas	VERB
ejpam-4571	647	2	transactions	transaction	NOUN
ejpam-4571	647	3	on	on	ADP
ejpam-4571	647	4	mathematics	mathematic	NOUN
ejpam-4571	647	5	,	,	PUNCT
ejpam-4571	647	6	21:329–337	21:329–337	PROPN
ejpam-4571	647	7	,	,	PUNCT
ejpam-4571	647	8	2022	2022	NUM
ejpam-4571	647	9	.	.	PUNCT
ejpam-4571	648	1	[	[	X
ejpam-4571	648	2	16	16	NUM
ejpam-4571	648	3	]	]	PUNCT
ejpam-4571	648	4	s.	s.	PROPN
ejpam-4571	648	5	willard	willard	PROPN
ejpam-4571	648	6	.	.	PUNCT
ejpam-4571	648	7	general	general	ADJ
ejpam-4571	648	8	topology	topology	PROPN
ejpam-4571	648	9	.	.	PUNCT
ejpam-4571	649	1	addison	addison	PROPN
ejpam-4571	649	2	-	-	PUNCT
ejpam-4571	649	3	wesley	wesley	PROPN
ejpam-4571	649	4	publishing	publishing	PROPN
ejpam-4571	649	5	company	company	NOUN
ejpam-4571	649	6	,	,	PUNCT
ejpam-4571	649	7	massachusetts	massachusetts	PROPN
ejpam-4571	649	8	,	,	PUNCT
ejpam-4571	649	9	1970	1970	NUM
ejpam-4571	649	10	.	.	PUNCT