id	sid	tid	token	lemma	pos
ejpam-4737	1	1	european	european	PROPN
ejpam-4737	1	2	journal	journal	PROPN
ejpam-4737	1	3	of	of	ADP
ejpam-4737	1	4	pure	pure	ADJ
ejpam-4737	1	5	and	and	CCONJ
ejpam-4737	1	6	applied	apply	VERB
ejpam-4737	1	7	mathematics	mathematic	NOUN
ejpam-4737	1	8	vol	vol	NOUN
ejpam-4737	1	9	.	.	PUNCT
ejpam-4737	2	1	16	16	NUM
ejpam-4737	2	2	,	,	PUNCT
ejpam-4737	2	3	no	no	INTJ
ejpam-4737	2	4	.	.	NOUN
ejpam-4737	2	5	3	3	NUM
ejpam-4737	2	6	,	,	PUNCT
ejpam-4737	2	7	2023	2023	NUM
ejpam-4737	2	8	,	,	PUNCT
ejpam-4737	2	9	1434	1434	NUM
ejpam-4737	2	10	-	-	SYM
ejpam-4737	2	11	1447	1447	NUM
ejpam-4737	2	12	issn	issn	PROPN
ejpam-4737	2	13	1307	1307	NUM
ejpam-4737	2	14	-	-	SYM
ejpam-4737	2	15	5543	5543	NUM
ejpam-4737	2	16	–	–	PUNCT
ejpam-4737	3	1	ejpam.com	ejpam.com	X
ejpam-4737	3	2	published	publish	VERB
ejpam-4737	3	3	by	by	ADP
ejpam-4737	3	4	new	new	PROPN
ejpam-4737	3	5	york	york	PROPN
ejpam-4737	3	6	business	business	PROPN
ejpam-4737	3	7	global	global	ADJ
ejpam-4737	3	8	characterizations	characterization	NOUN
ejpam-4737	3	9	of	of	ADP
ejpam-4737	3	10	some	some	DET
ejpam-4737	3	11	topological	topological	ADJ
ejpam-4737	3	12	spaces	space	NOUN
ejpam-4737	3	13	chawalit	chawalit	VERB
ejpam-4737	3	14	boonpok1	boonpok1	PROPN
ejpam-4737	3	15	,	,	PUNCT
ejpam-4737	3	16	montri	montri	PROPN
ejpam-4737	3	17	thongmoon1,∗	thongmoon1,∗	NOUN
ejpam-4737	3	18	1	1	NUM
ejpam-4737	3	19	mathematics	mathematic	NOUN
ejpam-4737	3	20	and	and	CCONJ
ejpam-4737	3	21	applied	apply	VERB
ejpam-4737	3	22	mathematics	mathematics	PROPN
ejpam-4737	3	23	research	research	NOUN
ejpam-4737	3	24	unit	unit	NOUN
ejpam-4737	3	25	,	,	PUNCT
ejpam-4737	3	26	department	department	NOUN
ejpam-4737	3	27	of	of	ADP
ejpam-4737	3	28	mathematics	mathematic	NOUN
ejpam-4737	3	29	,	,	PUNCT
ejpam-4737	3	30	faculty	faculty	NOUN
ejpam-4737	3	31	of	of	ADP
ejpam-4737	3	32	science	science	NOUN
ejpam-4737	3	33	,	,	PUNCT
ejpam-4737	3	34	mahasarakham	mahasarakham	PROPN
ejpam-4737	3	35	university	university	PROPN
ejpam-4737	3	36	,	,	PUNCT
ejpam-4737	3	37	maha	maha	PROPN
ejpam-4737	3	38	sarakham	sarakham	PROPN
ejpam-4737	3	39	,	,	PUNCT
ejpam-4737	3	40	44150	44150	NUM
ejpam-4737	3	41	,	,	PUNCT
ejpam-4737	3	42	thailand	thailand	PROPN
ejpam-4737	3	43	abstract	abstract	PROPN
ejpam-4737	3	44	.	.	PUNCT
ejpam-4737	4	1	this	this	DET
ejpam-4737	4	2	paper	paper	NOUN
ejpam-4737	4	3	is	be	AUX
ejpam-4737	4	4	concerned	concern	VERB
ejpam-4737	4	5	with	with	ADP
ejpam-4737	4	6	the	the	DET
ejpam-4737	4	7	concepts	concept	NOUN
ejpam-4737	4	8	of	of	ADP
ejpam-4737	4	9	some	some	DET
ejpam-4737	4	10	topological	topological	ADJ
ejpam-4737	4	11	spaces	space	NOUN
ejpam-4737	4	12	.	.	PUNCT
ejpam-4737	5	1	firstly	firstly	ADV
ejpam-4737	5	2	,	,	PUNCT
ejpam-4737	5	3	we	we	PRON
ejpam-4737	5	4	introduce	introduce	VERB
ejpam-4737	5	5	the	the	DET
ejpam-4737	5	6	notions	notion	NOUN
ejpam-4737	5	7	of	of	ADP
ejpam-4737	5	8	δs(λ	δs(λ	NOUN
ejpam-4737	5	9	,	,	PUNCT
ejpam-4737	5	10	p)-open	p)-open	VERB
ejpam-4737	5	11	sets	set	NOUN
ejpam-4737	5	12	.	.	PUNCT
ejpam-4737	6	1	some	some	DET
ejpam-4737	6	2	properties	property	NOUN
ejpam-4737	6	3	concerning	concern	VERB
ejpam-4737	6	4	δs(λ	δs(λ	NOUN
ejpam-4737	6	5	,	,	PUNCT
ejpam-4737	6	6	p)-open	p)-open	VERB
ejpam-4737	6	7	sets	set	NOUN
ejpam-4737	6	8	are	be	AUX
ejpam-4737	6	9	discussed	discuss	VERB
ejpam-4737	6	10	.	.	PUNCT
ejpam-4737	7	1	secondly	secondly	ADV
ejpam-4737	7	2	,	,	PUNCT
ejpam-4737	7	3	the	the	DET
ejpam-4737	7	4	concept	concept	NOUN
ejpam-4737	7	5	of	of	ADP
ejpam-4737	7	6	s(λ	s(λ	PROPN
ejpam-4737	7	7	,	,	PUNCT
ejpam-4737	7	8	p)-connected	p)-connecte	VERB
ejpam-4737	7	9	spaces	space	NOUN
ejpam-4737	7	10	is	be	AUX
ejpam-4737	7	11	introduced	introduce	VERB
ejpam-4737	7	12	.	.	PUNCT
ejpam-4737	8	1	moreover	moreover	ADV
ejpam-4737	8	2	,	,	PUNCT
ejpam-4737	8	3	we	we	PRON
ejpam-4737	8	4	give	give	VERB
ejpam-4737	8	5	several	several	ADJ
ejpam-4737	8	6	characterizations	characterization	NOUN
ejpam-4737	8	7	of	of	ADP
ejpam-4737	8	8	s(λ	s(λ	PROPN
ejpam-4737	8	9	,	,	PUNCT
ejpam-4737	8	10	p)-connected	p)-connecte	VERB
ejpam-4737	8	11	spaces	space	NOUN
ejpam-4737	8	12	by	by	ADP
ejpam-4737	8	13	utilizing	utilize	VERB
ejpam-4737	8	14	δs(λ	δs(λ	NOUN
ejpam-4737	8	15	,	,	PUNCT
ejpam-4737	8	16	p)-open	p)-open	VERB
ejpam-4737	8	17	sets	set	NOUN
ejpam-4737	8	18	.	.	PUNCT
ejpam-4737	9	1	thirdly	thirdly	ADV
ejpam-4737	9	2	,	,	PUNCT
ejpam-4737	9	3	we	we	PRON
ejpam-4737	9	4	apply	apply	VERB
ejpam-4737	9	5	the	the	DET
ejpam-4737	9	6	notion	notion	NOUN
ejpam-4737	9	7	of	of	ADP
ejpam-4737	9	8	s(λ	s(λ	PROPN
ejpam-4737	9	9	,	,	PUNCT
ejpam-4737	9	10	p)-open	p)-open	VERB
ejpam-4737	9	11	sets	set	NOUN
ejpam-4737	9	12	to	to	PART
ejpam-4737	9	13	present	present	VERB
ejpam-4737	9	14	and	and	CCONJ
ejpam-4737	9	15	study	study	VERB
ejpam-4737	9	16	new	new	ADJ
ejpam-4737	9	17	classes	class	NOUN
ejpam-4737	9	18	of	of	ADP
ejpam-4737	9	19	spaces	space	NOUN
ejpam-4737	9	20	called	call	VERB
ejpam-4737	9	21	s(λ	s(λ	PROPN
ejpam-4737	9	22	,	,	PUNCT
ejpam-4737	9	23	p)-regular	p)-regular	ADJ
ejpam-4737	9	24	spaces	space	NOUN
ejpam-4737	9	25	and	and	CCONJ
ejpam-4737	9	26	s(λ	s(λ	NOUN
ejpam-4737	9	27	,	,	PUNCT
ejpam-4737	9	28	p)-normal	p)-normal	ADJ
ejpam-4737	9	29	spaces	space	NOUN
ejpam-4737	9	30	.	.	PUNCT
ejpam-4737	10	1	especially	especially	ADV
ejpam-4737	10	2	,	,	PUNCT
ejpam-4737	10	3	some	some	DET
ejpam-4737	10	4	characterizations	characterization	NOUN
ejpam-4737	10	5	of	of	ADP
ejpam-4737	10	6	s(λ	s(λ	PROPN
ejpam-4737	10	7	,	,	PUNCT
ejpam-4737	10	8	p)-regular	p)-regular	ADJ
ejpam-4737	10	9	spaces	space	NOUN
ejpam-4737	10	10	and	and	CCONJ
ejpam-4737	10	11	s(λ	s(λ	NOUN
ejpam-4737	10	12	,	,	PUNCT
ejpam-4737	10	13	p)-normal	p)-normal	PUNCT
ejpam-4737	10	14	spaces	space	NOUN
ejpam-4737	10	15	are	be	AUX
ejpam-4737	10	16	established	establish	VERB
ejpam-4737	10	17	.	.	PUNCT
ejpam-4737	11	1	fourthly	fourthly	ADV
ejpam-4737	11	2	,	,	PUNCT
ejpam-4737	11	3	we	we	PRON
ejpam-4737	11	4	introduce	introduce	VERB
ejpam-4737	11	5	and	and	CCONJ
ejpam-4737	11	6	investigate	investigate	VERB
ejpam-4737	11	7	the	the	DET
ejpam-4737	11	8	concepts	concept	NOUN
ejpam-4737	11	9	of	of	ADP
ejpam-4737	11	10	s(λ	s(λ	PROPN
ejpam-4737	11	11	,	,	PUNCT
ejpam-4737	11	12	p)-t2	p)-t2	ADJ
ejpam-4737	11	13	spaces	space	NOUN
ejpam-4737	11	14	and	and	CCONJ
ejpam-4737	11	15	s(λ	s(λ	PROPN
ejpam-4737	11	16	,	,	PUNCT
ejpam-4737	11	17	p)-urysohn	p)-urysohn	NOUN
ejpam-4737	11	18	spaces	space	NOUN
ejpam-4737	11	19	.	.	PUNCT
ejpam-4737	12	1	finally	finally	ADV
ejpam-4737	12	2	,	,	PUNCT
ejpam-4737	12	3	the	the	DET
ejpam-4737	12	4	notion	notion	NOUN
ejpam-4737	12	5	of	of	ADP
ejpam-4737	12	6	s(λ	s(λ	PROPN
ejpam-4737	12	7	,	,	PUNCT
ejpam-4737	12	8	p)-closed	p)-close	VERB
ejpam-4737	12	9	spaces	space	NOUN
ejpam-4737	12	10	is	be	AUX
ejpam-4737	12	11	studied	study	VERB
ejpam-4737	12	12	.	.	PUNCT
ejpam-4737	13	1	basic	basic	ADJ
ejpam-4737	13	2	properties	property	NOUN
ejpam-4737	13	3	and	and	CCONJ
ejpam-4737	13	4	characterizations	characterization	NOUN
ejpam-4737	13	5	of	of	ADP
ejpam-4737	13	6	s(λ	s(λ	PROPN
ejpam-4737	13	7	,	,	PUNCT
ejpam-4737	13	8	p)-closed	p)-close	VERB
ejpam-4737	13	9	spaces	space	NOUN
ejpam-4737	13	10	are	be	AUX
ejpam-4737	13	11	considered	consider	VERB
ejpam-4737	13	12	.	.	PUNCT
ejpam-4737	14	1	2020	2020	NUM
ejpam-4737	14	2	mathematics	mathematic	NOUN
ejpam-4737	14	3	subject	subject	NOUN
ejpam-4737	14	4	classifications	classification	NOUN
ejpam-4737	14	5	:	:	PUNCT
ejpam-4737	14	6	54a05	54a05	NUM
ejpam-4737	14	7	,	,	PUNCT
ejpam-4737	14	8	54d10	54d10	NUM
ejpam-4737	14	9	key	key	ADJ
ejpam-4737	14	10	words	word	NOUN
ejpam-4737	14	11	and	and	CCONJ
ejpam-4737	14	12	phrases	phrase	NOUN
ejpam-4737	14	13	:	:	PUNCT
ejpam-4737	14	14	δs(λ	δs(λ	NOUN
ejpam-4737	14	15	,	,	PUNCT
ejpam-4737	14	16	p)-open	p)-open	VERB
ejpam-4737	14	17	set	set	VERB
ejpam-4737	14	18	,	,	PUNCT
ejpam-4737	14	19	s(λ	s(λ	PROPN
ejpam-4737	14	20	,	,	PUNCT
ejpam-4737	14	21	p)-connected	p)-connecte	VERB
ejpam-4737	14	22	space	space	NOUN
ejpam-4737	14	23	,	,	PUNCT
ejpam-4737	14	24	s(λ	s(λ	PROPN
ejpam-4737	14	25	,	,	PUNCT
ejpam-4737	14	26	p)-regular	p)-regular	ADJ
ejpam-4737	14	27	space	space	NOUN
ejpam-4737	14	28	,	,	PUNCT
ejpam-4737	14	29	s(λ	s(λ	PROPN
ejpam-4737	14	30	,	,	PUNCT
ejpam-4737	14	31	p)-normal	p)-normal	ADJ
ejpam-4737	14	32	space	space	NOUN
ejpam-4737	14	33	,	,	PUNCT
ejpam-4737	14	34	s(λ	s(λ	PROPN
ejpam-4737	14	35	,	,	PUNCT
ejpam-4737	14	36	p)-t2	p)-t2	ADJ
ejpam-4737	14	37	space	space	NOUN
ejpam-4737	14	38	,	,	PUNCT
ejpam-4737	14	39	s(λ	s(λ	PROPN
ejpam-4737	14	40	,	,	PUNCT
ejpam-4737	14	41	p)-urysohn	p)-urysohn	NOUN
ejpam-4737	14	42	space	space	NOUN
ejpam-4737	14	43	,	,	PUNCT
ejpam-4737	14	44	s(λ	s(λ	PROPN
ejpam-4737	14	45	,	,	PUNCT
ejpam-4737	14	46	p)-closed	p)-close	VERB
ejpam-4737	14	47	space	space	NOUN
ejpam-4737	14	48	1	1	NUM
ejpam-4737	14	49	.	.	PUNCT
ejpam-4737	14	50	introduction	introduction	NOUN
ejpam-4737	14	51	in	in	ADP
ejpam-4737	14	52	1968	1968	NUM
ejpam-4737	14	53	,	,	PUNCT
ejpam-4737	14	54	veličko	veličko	PROPN
ejpam-4737	14	55	[	[	X
ejpam-4737	14	56	14	14	NUM
ejpam-4737	14	57	]	]	PUNCT
ejpam-4737	14	58	introduced	introduce	VERB
ejpam-4737	14	59	δ	δ	PROPN
ejpam-4737	14	60	-	-	ADJ
ejpam-4737	14	61	open	open	ADJ
ejpam-4737	14	62	sets	set	NOUN
ejpam-4737	14	63	,	,	PUNCT
ejpam-4737	14	64	which	which	PRON
ejpam-4737	14	65	are	be	AUX
ejpam-4737	14	66	stronger	strong	ADJ
ejpam-4737	14	67	than	than	ADP
ejpam-4737	14	68	open	open	ADJ
ejpam-4737	14	69	sets	set	NOUN
ejpam-4737	14	70	.	.	PUNCT
ejpam-4737	15	1	in	in	ADP
ejpam-4737	15	2	1982	1982	NUM
ejpam-4737	15	3	,	,	PUNCT
ejpam-4737	15	4	mashhour	mashhour	PROPN
ejpam-4737	15	5	et	et	PROPN
ejpam-4737	15	6	al	al	PROPN
ejpam-4737	15	7	.	.	PUNCT
ejpam-4737	16	1	[	[	X
ejpam-4737	16	2	9	9	NUM
ejpam-4737	16	3	]	]	PUNCT
ejpam-4737	16	4	introduced	introduce	VERB
ejpam-4737	16	5	and	and	CCONJ
ejpam-4737	16	6	investigated	investigate	VERB
ejpam-4737	16	7	the	the	DET
ejpam-4737	16	8	notion	notion	NOUN
ejpam-4737	16	9	of	of	ADP
ejpam-4737	16	10	preopen	preopen	ADJ
ejpam-4737	16	11	sets	set	NOUN
ejpam-4737	16	12	which	which	PRON
ejpam-4737	16	13	is	be	AUX
ejpam-4737	16	14	weaker	weak	ADJ
ejpam-4737	16	15	than	than	ADP
ejpam-4737	16	16	the	the	DET
ejpam-4737	16	17	notion	notion	NOUN
ejpam-4737	16	18	of	of	ADP
ejpam-4737	16	19	open	open	ADJ
ejpam-4737	16	20	sets	set	NOUN
ejpam-4737	16	21	in	in	ADP
ejpam-4737	16	22	topological	topological	ADJ
ejpam-4737	16	23	spaces	space	NOUN
ejpam-4737	16	24	.	.	PUNCT
ejpam-4737	17	1	in	in	ADP
ejpam-4737	17	2	1993	1993	NUM
ejpam-4737	17	3	,	,	PUNCT
ejpam-4737	17	4	raychaudhuri	raychaudhuri	NOUN
ejpam-4737	17	5	and	and	CCONJ
ejpam-4737	17	6	mukherjee	mukherjee	NOUN
ejpam-4737	17	7	[	[	X
ejpam-4737	17	8	11	11	NUM
ejpam-4737	17	9	]	]	PUNCT
ejpam-4737	17	10	introduced	introduce	VERB
ejpam-4737	17	11	and	and	CCONJ
ejpam-4737	17	12	studied	study	VERB
ejpam-4737	17	13	the	the	DET
ejpam-4737	17	14	notions	notion	NOUN
ejpam-4737	17	15	of	of	ADP
ejpam-4737	17	16	δ	δ	PROPN
ejpam-4737	17	17	-	-	PUNCT
ejpam-4737	17	18	preopen	preopen	ADJ
ejpam-4737	17	19	sets	set	NOUN
ejpam-4737	17	20	and	and	CCONJ
ejpam-4737	17	21	δ	δ	NOUN
ejpam-4737	17	22	-	-	NOUN
ejpam-4737	17	23	closures	closure	NOUN
ejpam-4737	17	24	.	.	PUNCT
ejpam-4737	18	1	the	the	DET
ejpam-4737	18	2	class	class	NOUN
ejpam-4737	18	3	of	of	ADP
ejpam-4737	18	4	δ	δ	PROPN
ejpam-4737	18	5	-	-	PUNCT
ejpam-4737	18	6	preopen	preopen	ADJ
ejpam-4737	18	7	sets	set	NOUN
ejpam-4737	18	8	is	be	AUX
ejpam-4737	18	9	larger	large	ADJ
ejpam-4737	18	10	than	than	ADP
ejpam-4737	18	11	that	that	PRON
ejpam-4737	18	12	of	of	ADP
ejpam-4737	18	13	preopen	preopen	ADJ
ejpam-4737	18	14	sets	set	NOUN
ejpam-4737	18	15	.	.	PUNCT
ejpam-4737	19	1	in	in	ADP
ejpam-4737	19	2	1996	1996	NUM
ejpam-4737	19	3	,	,	PUNCT
ejpam-4737	19	4	raychaudhuri	raychaudhuri	NOUN
ejpam-4737	19	5	and	and	CCONJ
ejpam-4737	19	6	mukherjee	mukherjee	NOUN
ejpam-4737	20	1	[	[	X
ejpam-4737	20	2	12	12	NUM
ejpam-4737	20	3	]	]	PUNCT
ejpam-4737	20	4	introduced	introduce	VERB
ejpam-4737	20	5	and	and	CCONJ
ejpam-4737	20	6	investigated	investigate	VERB
ejpam-4737	20	7	the	the	DET
ejpam-4737	20	8	concept	concept	NOUN
ejpam-4737	20	9	of	of	ADP
ejpam-4737	20	10	δp	δp	ADV
ejpam-4737	20	11	-	-	PUNCT
ejpam-4737	20	12	closed	closed	ADJ
ejpam-4737	20	13	spaces	space	NOUN
ejpam-4737	20	14	.	.	PUNCT
ejpam-4737	21	1	in	in	ADP
ejpam-4737	21	2	2005	2005	NUM
ejpam-4737	21	3	,	,	PUNCT
ejpam-4737	21	4	caldas	caldas	PROPN
ejpam-4737	21	5	et	et	PROPN
ejpam-4737	21	6	al	al	PROPN
ejpam-4737	21	7	.	.	PUNCT
ejpam-4737	22	1	[	[	X
ejpam-4737	22	2	4	4	X
ejpam-4737	22	3	]	]	PUNCT
ejpam-4737	22	4	introduced	introduce	VERB
ejpam-4737	22	5	some	some	DET
ejpam-4737	22	6	weak	weak	ADJ
ejpam-4737	22	7	separation	separation	NOUN
ejpam-4737	22	8	axioms	axiom	NOUN
ejpam-4737	22	9	by	by	ADP
ejpam-4737	22	10	utilizing	utilize	VERB
ejpam-4737	22	11	the	the	DET
ejpam-4737	22	12	notions	notion	NOUN
ejpam-4737	22	13	of	of	ADP
ejpam-4737	22	14	δ	δ	PROPN
ejpam-4737	22	15	-	-	PUNCT
ejpam-4737	22	16	preopen	preopen	ADJ
ejpam-4737	22	17	sets	set	NOUN
ejpam-4737	22	18	and	and	CCONJ
ejpam-4737	22	19	the	the	DET
ejpam-4737	22	20	δ	δ	NOUN
ejpam-4737	22	21	-	-	PUNCT
ejpam-4737	22	22	preclosure	preclosure	ADJ
ejpam-4737	22	23	operator	operator	NOUN
ejpam-4737	22	24	.	.	PUNCT
ejpam-4737	23	1	caldas	caldas	PROPN
ejpam-4737	23	2	et	et	PROPN
ejpam-4737	23	3	al	al	PROPN
ejpam-4737	23	4	.	.	PUNCT
ejpam-4737	24	1	[	[	X
ejpam-4737	24	2	4	4	X
ejpam-4737	24	3	]	]	PUNCT
ejpam-4737	24	4	showed	show	VERB
ejpam-4737	24	5	that	that	SCONJ
ejpam-4737	24	6	(	(	PUNCT
ejpam-4737	24	7	δ	δ	PROPN
ejpam-4737	24	8	,	,	PUNCT
ejpam-4737	24	9	p)-t1	p)-t1	VERB
ejpam-4737	24	10	spaces	space	NOUN
ejpam-4737	24	11	,	,	PUNCT
ejpam-4737	24	12	(	(	PUNCT
ejpam-4737	24	13	δ	δ	PROPN
ejpam-4737	24	14	,	,	PUNCT
ejpam-4737	24	15	p)-r0	p)-r0	PROPN
ejpam-4737	24	16	spaces	space	VERB
ejpam-4737	24	17	and	and	CCONJ
ejpam-4737	24	18	(	(	PUNCT
ejpam-4737	24	19	δ	δ	PROPN
ejpam-4737	24	20	,	,	PUNCT
ejpam-4737	24	21	p)-symmetric	p)-symmetric	ADJ
ejpam-4737	24	22	spaces	space	NOUN
ejpam-4737	24	23	are	be	AUX
ejpam-4737	24	24	all	all	ADV
ejpam-4737	24	25	equivalent	equivalent	ADJ
ejpam-4737	24	26	.	.	PUNCT
ejpam-4737	25	1	moreover	moreover	ADV
ejpam-4737	25	2	,	,	PUNCT
ejpam-4737	25	3	caldas	caldas	PROPN
ejpam-4737	25	4	et	et	PROPN
ejpam-4737	25	5	al	al	PROPN
ejpam-4737	25	6	.	.	PUNCT
ejpam-4737	26	1	[	[	X
ejpam-4737	26	2	6	6	NUM
ejpam-4737	26	3	]	]	PUNCT
ejpam-4737	26	4	investigated	investigate	VERB
ejpam-4737	26	5	some	some	DET
ejpam-4737	26	6	weak	weak	ADJ
ejpam-4737	26	7	separation	separation	NOUN
ejpam-4737	26	8	axioms	axiom	NOUN
ejpam-4737	26	9	by	by	ADP
ejpam-4737	26	10	utilizing	utilize	VERB
ejpam-4737	26	11	δ	δ	PROPN
ejpam-4737	26	12	-	-	PUNCT
ejpam-4737	26	13	semiopen	semiopen	ADJ
ejpam-4737	26	14	sets	set	NOUN
ejpam-4737	26	15	and	and	CCONJ
ejpam-4737	26	16	the	the	DET
ejpam-4737	26	17	δ	δ	PROPN
ejpam-4737	26	18	-	-	PUNCT
ejpam-4737	26	19	semiclosure	semiclosure	NOUN
ejpam-4737	26	20	operator	operator	NOUN
ejpam-4737	26	21	.	.	PUNCT
ejpam-4737	27	1	caldas	caldas	PROPN
ejpam-4737	27	2	et	et	PROPN
ejpam-4737	27	3	al	al	PROPN
ejpam-4737	27	4	.	.	PUNCT
ejpam-4737	28	1	[	[	X
ejpam-4737	28	2	5	5	NUM
ejpam-4737	28	3	]	]	PUNCT
ejpam-4737	28	4	investigated	investigate	VERB
ejpam-4737	28	5	the	the	DET
ejpam-4737	28	6	notion	notion	NOUN
ejpam-4737	28	7	of	of	ADP
ejpam-4737	28	8	δ	δ	PROPN
ejpam-4737	28	9	-	-	PUNCT
ejpam-4737	28	10	λs	λs	ADV
ejpam-4737	28	11	-	-	PUNCT
ejpam-4737	28	12	semiclosed	semiclose	VERB
ejpam-4737	28	13	sets	set	NOUN
ejpam-4737	28	14	which	which	PRON
ejpam-4737	28	15	is	be	AUX
ejpam-4737	28	16	defined	define	VERB
ejpam-4737	28	17	as	as	ADP
ejpam-4737	28	18	the	the	DET
ejpam-4737	28	19	intersection	intersection	NOUN
ejpam-4737	28	20	of	of	ADP
ejpam-4737	28	21	a	a	DET
ejpam-4737	28	22	δ	δ	PROPN
ejpam-4737	28	23	-	-	PUNCT
ejpam-4737	28	24	λs	λs	NOUN
ejpam-4737	28	25	-	-	PUNCT
ejpam-4737	28	26	set	set	NOUN
ejpam-4737	28	27	and	and	CCONJ
ejpam-4737	28	28	a	a	DET
ejpam-4737	28	29	δ	δ	NOUN
ejpam-4737	28	30	-	-	PUNCT
ejpam-4737	28	31	semiclosed	semiclose	VERB
ejpam-4737	28	32	set	set	NOUN
ejpam-4737	28	33	.	.	PUNCT
ejpam-4737	29	1	in	in	ADP
ejpam-4737	29	2	2011	2011	NUM
ejpam-4737	29	3	,	,	PUNCT
ejpam-4737	29	4	buadong	buadong	NOUN
ejpam-4737	29	5	et	et	PROPN
ejpam-4737	29	6	al	al	PROPN
ejpam-4737	29	7	.	.	PUNCT
ejpam-4737	30	1	[	[	X
ejpam-4737	30	2	1	1	X
ejpam-4737	30	3	]	]	PUNCT
ejpam-4737	30	4	introduced	introduce	VERB
ejpam-4737	30	5	and	and	CCONJ
ejpam-4737	30	6	investigated	investigate	VERB
ejpam-4737	30	7	some	some	DET
ejpam-4737	30	8	separation	separation	NOUN
ejpam-4737	30	9	axioms	axiom	NOUN
ejpam-4737	30	10	in	in	ADP
ejpam-4737	30	11	generalized	generalized	ADJ
ejpam-4737	30	12	topology	topology	NOUN
ejpam-4737	30	13	and	and	CCONJ
ejpam-4737	30	14	minimal	minimal	ADJ
ejpam-4737	30	15	structure	structure	NOUN
ejpam-4737	30	16	∗corresponding	∗corresponde	VERB
ejpam-4737	30	17	author	author	NOUN
ejpam-4737	30	18	.	.	PUNCT
ejpam-4737	31	1	doi	doi	NOUN
ejpam-4737	31	2	:	:	PUNCT
ejpam-4737	31	3	https://doi.org/10.29020/nybg.ejpam.v16i3.4737	https://doi.org/10.29020/nybg.ejpam.v16i3.4737	ADJ
ejpam-4737	31	4	email	email	NOUN
ejpam-4737	31	5	addresses	address	NOUN
ejpam-4737	31	6	:	:	PUNCT
ejpam-4737	31	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4737	31	8	(	(	PUNCT
ejpam-4737	31	9	c.	c.	PROPN
ejpam-4737	31	10	boonpok	boonpok	PROPN
ejpam-4737	31	11	)	)	PUNCT
ejpam-4737	31	12	,	,	PUNCT
ejpam-4737	31	13	montri.t@msu.ac.th	montri.t@msu.ac.th	PROPN
ejpam-4737	31	14	(	(	PUNCT
ejpam-4737	31	15	m.	m.	NOUN
ejpam-4737	31	16	thongmoon	thongmoon	PROPN
ejpam-4737	31	17	)	)	PUNCT
ejpam-4737	31	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4737	31	19	1434	1434	NUM
ejpam-4737	31	20	©	©	PROPN
ejpam-4737	31	21	2023	2023	NUM
ejpam-4737	31	22	ejpam	ejpam	NOUN
ejpam-4737	31	23	all	all	DET
ejpam-4737	31	24	rights	right	NOUN
ejpam-4737	31	25	reserved	reserve	VERB
ejpam-4737	31	26	.	.	PUNCT
ejpam-4737	32	1	c.	c.	PROPN
ejpam-4737	32	2	boonpok	boonpok	PROPN
ejpam-4737	32	3	,	,	PUNCT
ejpam-4737	32	4	m.	m.	NOUN
ejpam-4737	32	5	thongmoon	thongmoon	PROPN
ejpam-4737	32	6	/	/	SYM
ejpam-4737	32	7	eur	eur	PROPN
ejpam-4737	32	8	.	.	PUNCT
ejpam-4737	33	1	j.	j.	PROPN
ejpam-4737	33	2	pure	pure	PROPN
ejpam-4737	33	3	appl	appl	PROPN
ejpam-4737	33	4	.	.	PROPN
ejpam-4737	33	5	math	math	PROPN
ejpam-4737	33	6	,	,	PUNCT
ejpam-4737	33	7	16	16	NUM
ejpam-4737	33	8	(	(	PUNCT
ejpam-4737	33	9	3	3	NUM
ejpam-4737	33	10	)	)	PUNCT
ejpam-4737	33	11	(	(	PUNCT
ejpam-4737	33	12	2023	2023	NUM
ejpam-4737	33	13	)	)	PUNCT
ejpam-4737	33	14	,	,	PUNCT
ejpam-4737	33	15	1434	1434	NUM
ejpam-4737	33	16	-	-	SYM
ejpam-4737	33	17	1447	1447	NUM
ejpam-4737	33	18	1435	1435	NUM
ejpam-4737	33	19	spaces	space	NOUN
ejpam-4737	33	20	.	.	PUNCT
ejpam-4737	34	1	dungthaisong	dungthaisong	NOUN
ejpam-4737	34	2	et	et	PROPN
ejpam-4737	34	3	al	al	PROPN
ejpam-4737	34	4	.	.	PUNCT
ejpam-4737	35	1	[	[	X
ejpam-4737	35	2	7	7	X
ejpam-4737	35	3	]	]	PUNCT
ejpam-4737	35	4	studied	study	VERB
ejpam-4737	35	5	some	some	DET
ejpam-4737	35	6	properties	property	NOUN
ejpam-4737	35	7	of	of	ADP
ejpam-4737	35	8	pairwise	pairwise	NOUN
ejpam-4737	35	9	µ-t	µ-t	PROPN
ejpam-4737	35	10	1	1	NUM
ejpam-4737	35	11	2	2	NUM
ejpam-4737	35	12	-spaces	-space	NOUN
ejpam-4737	35	13	.	.	PUNCT
ejpam-4737	36	1	torton	torton	PROPN
ejpam-4737	36	2	et	et	PROPN
ejpam-4737	36	3	al	al	PROPN
ejpam-4737	36	4	.	.	PUNCT
ejpam-4737	37	1	[	[	X
ejpam-4737	37	2	13	13	NUM
ejpam-4737	37	3	]	]	PUNCT
ejpam-4737	37	4	introduced	introduce	VERB
ejpam-4737	37	5	and	and	CCONJ
ejpam-4737	37	6	investigated	investigate	VERB
ejpam-4737	37	7	the	the	DET
ejpam-4737	37	8	notions	notion	NOUN
ejpam-4737	37	9	of	of	ADP
ejpam-4737	37	10	µ(m	µ(m	NOUN
ejpam-4737	37	11	,	,	PUNCT
ejpam-4737	37	12	n)-regular	n)-regular	ADJ
ejpam-4737	37	13	spaces	space	NOUN
ejpam-4737	37	14	and	and	CCONJ
ejpam-4737	37	15	µ(m	µ(m	NOUN
ejpam-4737	37	16	,	,	PUNCT
ejpam-4737	37	17	n)normal	n)normal	ADJ
ejpam-4737	37	18	spaces	space	NOUN
ejpam-4737	37	19	.	.	PUNCT
ejpam-4737	38	1	in	in	ADP
ejpam-4737	38	2	[	[	X
ejpam-4737	38	3	3	3	NUM
ejpam-4737	38	4	]	]	PUNCT
ejpam-4737	38	5	,	,	PUNCT
ejpam-4737	38	6	the	the	DET
ejpam-4737	38	7	present	present	ADJ
ejpam-4737	38	8	authors	author	NOUN
ejpam-4737	38	9	introduced	introduce	VERB
ejpam-4737	38	10	the	the	DET
ejpam-4737	38	11	notions	notion	NOUN
ejpam-4737	38	12	of	of	ADP
ejpam-4737	38	13	(	(	PUNCT
ejpam-4737	38	14	λ	λ	INTJ
ejpam-4737	38	15	,	,	PUNCT
ejpam-4737	38	16	p)-open	p)-open	VERB
ejpam-4737	38	17	sets	set	NOUN
ejpam-4737	38	18	and	and	CCONJ
ejpam-4737	38	19	(	(	PUNCT
ejpam-4737	38	20	λ	λ	PROPN
ejpam-4737	38	21	,	,	PUNCT
ejpam-4737	38	22	p)-closed	p)-close	VERB
ejpam-4737	38	23	sets	set	NOUN
ejpam-4737	38	24	which	which	PRON
ejpam-4737	38	25	are	be	AUX
ejpam-4737	38	26	defined	define	VERB
ejpam-4737	38	27	by	by	ADP
ejpam-4737	38	28	utilizing	utilize	VERB
ejpam-4737	38	29	the	the	DET
ejpam-4737	38	30	notions	notion	NOUN
ejpam-4737	38	31	of	of	ADP
ejpam-4737	38	32	λp	λp	NOUN
ejpam-4737	38	33	-	-	PUNCT
ejpam-4737	38	34	sets	set	NOUN
ejpam-4737	38	35	and	and	CCONJ
ejpam-4737	38	36	preclosed	preclose	VERB
ejpam-4737	38	37	sets	set	NOUN
ejpam-4737	38	38	.	.	PUNCT
ejpam-4737	39	1	this	this	DET
ejpam-4737	39	2	paper	paper	NOUN
ejpam-4737	39	3	is	be	AUX
ejpam-4737	39	4	organized	organize	VERB
ejpam-4737	39	5	as	as	SCONJ
ejpam-4737	39	6	follows	follow	VERB
ejpam-4737	39	7	:	:	PUNCT
ejpam-4737	39	8	in	in	ADP
ejpam-4737	39	9	section	section	NOUN
ejpam-4737	39	10	2	2	NUM
ejpam-4737	39	11	is	be	AUX
ejpam-4737	39	12	devoted	devote	VERB
ejpam-4737	39	13	to	to	ADP
ejpam-4737	39	14	basic	basic	ADJ
ejpam-4737	39	15	definitions	definition	NOUN
ejpam-4737	39	16	and	and	CCONJ
ejpam-4737	39	17	preliminaries	preliminary	NOUN
ejpam-4737	39	18	.	.	PUNCT
ejpam-4737	40	1	in	in	ADP
ejpam-4737	40	2	section	section	NOUN
ejpam-4737	40	3	3	3	NUM
ejpam-4737	40	4	,	,	PUNCT
ejpam-4737	40	5	we	we	PRON
ejpam-4737	40	6	introduce	introduce	VERB
ejpam-4737	40	7	the	the	DET
ejpam-4737	40	8	notions	notion	NOUN
ejpam-4737	40	9	of	of	ADP
ejpam-4737	40	10	δs(λ	δs(λ	NOUN
ejpam-4737	40	11	,	,	PUNCT
ejpam-4737	40	12	p)-open	p)-open	VERB
ejpam-4737	40	13	sets	set	NOUN
ejpam-4737	40	14	and	and	CCONJ
ejpam-4737	40	15	δs(λ	δs(λ	NOUN
ejpam-4737	40	16	,	,	PUNCT
ejpam-4737	40	17	p)closed	p)close	VERB
ejpam-4737	40	18	sets	set	NOUN
ejpam-4737	40	19	in	in	ADP
ejpam-4737	40	20	topological	topological	ADJ
ejpam-4737	40	21	spaces	space	NOUN
ejpam-4737	40	22	.	.	PUNCT
ejpam-4737	41	1	moreover	moreover	ADV
ejpam-4737	41	2	,	,	PUNCT
ejpam-4737	41	3	some	some	DET
ejpam-4737	41	4	characterizations	characterization	NOUN
ejpam-4737	41	5	of	of	ADP
ejpam-4737	41	6	δs(λ	δs(λ	NOUN
ejpam-4737	41	7	,	,	PUNCT
ejpam-4737	41	8	p)-t0	p)-t0	NOUN
ejpam-4737	41	9	spaces	space	NOUN
ejpam-4737	41	10	,	,	PUNCT
ejpam-4737	41	11	δs(λ	δs(λ	NOUN
ejpam-4737	41	12	,	,	PUNCT
ejpam-4737	41	13	p)-t1	p)-t1	VERB
ejpam-4737	41	14	spaces	space	NOUN
ejpam-4737	41	15	and	and	CCONJ
ejpam-4737	41	16	δs(λ	δs(λ	NOUN
ejpam-4737	41	17	,	,	PUNCT
ejpam-4737	41	18	p)-symmetric	p)-symmetric	ADJ
ejpam-4737	41	19	spaces	space	NOUN
ejpam-4737	41	20	are	be	AUX
ejpam-4737	41	21	investigated	investigate	VERB
ejpam-4737	41	22	.	.	PUNCT
ejpam-4737	42	1	in	in	ADP
ejpam-4737	42	2	section	section	NOUN
ejpam-4737	42	3	4	4	NUM
ejpam-4737	42	4	,	,	PUNCT
ejpam-4737	42	5	the	the	DET
ejpam-4737	42	6	notion	notion	NOUN
ejpam-4737	42	7	of	of	ADP
ejpam-4737	42	8	s(λ	s(λ	PROPN
ejpam-4737	42	9	,	,	PUNCT
ejpam-4737	42	10	p)-connected	p)-connecte	VERB
ejpam-4737	42	11	spaces	space	NOUN
ejpam-4737	42	12	is	be	AUX
ejpam-4737	42	13	introduced	introduce	VERB
ejpam-4737	42	14	.	.	PUNCT
ejpam-4737	43	1	several	several	ADJ
ejpam-4737	43	2	characterizations	characterization	NOUN
ejpam-4737	43	3	of	of	ADP
ejpam-4737	43	4	s(λ	s(λ	PROPN
ejpam-4737	43	5	,	,	PUNCT
ejpam-4737	43	6	p)-connected	p)-connecte	VERB
ejpam-4737	43	7	spaces	space	NOUN
ejpam-4737	43	8	are	be	AUX
ejpam-4737	43	9	obtained	obtain	VERB
ejpam-4737	43	10	.	.	PUNCT
ejpam-4737	44	1	in	in	ADP
ejpam-4737	44	2	section	section	NOUN
ejpam-4737	44	3	5	5	NUM
ejpam-4737	44	4	,	,	PUNCT
ejpam-4737	44	5	we	we	PRON
ejpam-4737	44	6	introduce	introduce	VERB
ejpam-4737	44	7	the	the	DET
ejpam-4737	44	8	concepts	concept	NOUN
ejpam-4737	44	9	of	of	ADP
ejpam-4737	44	10	s(λ	s(λ	PROPN
ejpam-4737	44	11	,	,	PUNCT
ejpam-4737	44	12	p)-regular	p)-regular	ADJ
ejpam-4737	44	13	spaces	space	NOUN
ejpam-4737	44	14	and	and	CCONJ
ejpam-4737	44	15	s(λ	s(λ	NOUN
ejpam-4737	44	16	,	,	PUNCT
ejpam-4737	44	17	p)-normal	p)-normal	ADJ
ejpam-4737	44	18	spaces	space	NOUN
ejpam-4737	44	19	.	.	PUNCT
ejpam-4737	45	1	furthermore	furthermore	ADV
ejpam-4737	45	2	,	,	PUNCT
ejpam-4737	45	3	we	we	PRON
ejpam-4737	45	4	give	give	VERB
ejpam-4737	45	5	some	some	DET
ejpam-4737	45	6	characterizations	characterization	NOUN
ejpam-4737	45	7	of	of	ADP
ejpam-4737	45	8	s(λ	s(λ	PROPN
ejpam-4737	45	9	,	,	PUNCT
ejpam-4737	45	10	p)-regular	p)-regular	ADJ
ejpam-4737	45	11	spaces	space	NOUN
ejpam-4737	45	12	and	and	CCONJ
ejpam-4737	45	13	s(λ	s(λ	NOUN
ejpam-4737	45	14	,	,	PUNCT
ejpam-4737	45	15	p)-normal	p)-normal	PUNCT
ejpam-4737	45	16	spaces	space	NOUN
ejpam-4737	45	17	by	by	ADP
ejpam-4737	45	18	utilizing	utilize	VERB
ejpam-4737	45	19	δs(λ	δs(λ	NOUN
ejpam-4737	45	20	,	,	PUNCT
ejpam-4737	45	21	p)-open	p)-open	VERB
ejpam-4737	45	22	sets	set	NOUN
ejpam-4737	45	23	.	.	PUNCT
ejpam-4737	46	1	basic	basic	ADJ
ejpam-4737	46	2	properties	property	NOUN
ejpam-4737	46	3	and	and	CCONJ
ejpam-4737	46	4	characterizations	characterization	NOUN
ejpam-4737	46	5	of	of	ADP
ejpam-4737	46	6	s(λ	s(λ	PROPN
ejpam-4737	46	7	,	,	PUNCT
ejpam-4737	46	8	p)-t2	p)-t2	ADJ
ejpam-4737	46	9	spaces	space	NOUN
ejpam-4737	46	10	and	and	CCONJ
ejpam-4737	46	11	s(λ	s(λ	PROPN
ejpam-4737	46	12	,	,	PUNCT
ejpam-4737	46	13	p)-urysohn	p)-urysohn	NOUN
ejpam-4737	46	14	spaces	space	NOUN
ejpam-4737	46	15	are	be	AUX
ejpam-4737	46	16	discussed	discuss	VERB
ejpam-4737	46	17	in	in	ADP
ejpam-4737	46	18	section	section	NOUN
ejpam-4737	46	19	6	6	NUM
ejpam-4737	46	20	.	.	PUNCT
ejpam-4737	47	1	in	in	ADP
ejpam-4737	47	2	the	the	DET
ejpam-4737	47	3	last	last	ADJ
ejpam-4737	47	4	section	section	NOUN
ejpam-4737	47	5	7	7	NUM
ejpam-4737	47	6	,	,	PUNCT
ejpam-4737	47	7	we	we	PRON
ejpam-4737	47	8	define	define	VERB
ejpam-4737	47	9	the	the	DET
ejpam-4737	47	10	notion	notion	NOUN
ejpam-4737	47	11	of	of	ADP
ejpam-4737	47	12	s(λ	s(λ	PROPN
ejpam-4737	47	13	,	,	PUNCT
ejpam-4737	47	14	p)-closed	p)-close	VERB
ejpam-4737	47	15	spaces	space	NOUN
ejpam-4737	47	16	.	.	PUNCT
ejpam-4737	48	1	characterizations	characterization	NOUN
ejpam-4737	48	2	and	and	CCONJ
ejpam-4737	48	3	properties	property	NOUN
ejpam-4737	48	4	concerning	concern	VERB
ejpam-4737	48	5	s(λ	s(λ	PROPN
ejpam-4737	48	6	,	,	PUNCT
ejpam-4737	48	7	p)-closed	p)-close	VERB
ejpam-4737	48	8	spaces	space	NOUN
ejpam-4737	48	9	are	be	AUX
ejpam-4737	48	10	considered	consider	VERB
ejpam-4737	48	11	.	.	PUNCT
ejpam-4737	49	1	2	2	X
ejpam-4737	49	2	.	.	X
ejpam-4737	49	3	preliminaries	preliminary	NOUN
ejpam-4737	49	4	throughout	throughout	ADP
ejpam-4737	49	5	the	the	DET
ejpam-4737	49	6	present	present	ADJ
ejpam-4737	49	7	paper	paper	NOUN
ejpam-4737	49	8	,	,	PUNCT
ejpam-4737	49	9	spaces	space	NOUN
ejpam-4737	49	10	(	(	PUNCT
ejpam-4737	49	11	x	x	X
ejpam-4737	49	12	,	,	PUNCT
ejpam-4737	49	13	τ	τ	X
ejpam-4737	49	14	)	)	PUNCT
ejpam-4737	49	15	and	and	CCONJ
ejpam-4737	49	16	(	(	PUNCT
ejpam-4737	49	17	y	y	PROPN
ejpam-4737	49	18	,	,	PUNCT
ejpam-4737	49	19	σ	σ	PROPN
ejpam-4737	49	20	)	)	PUNCT
ejpam-4737	49	21	(	(	PUNCT
ejpam-4737	49	22	or	or	CCONJ
ejpam-4737	49	23	simply	simply	ADV
ejpam-4737	49	24	x	x	X
ejpam-4737	49	25	and	and	CCONJ
ejpam-4737	49	26	y	y	PROPN
ejpam-4737	49	27	)	)	PUNCT
ejpam-4737	49	28	always	always	ADV
ejpam-4737	49	29	mean	mean	VERB
ejpam-4737	49	30	topological	topological	ADJ
ejpam-4737	49	31	spaces	space	NOUN
ejpam-4737	49	32	on	on	ADP
ejpam-4737	49	33	which	which	PRON
ejpam-4737	49	34	no	no	DET
ejpam-4737	49	35	separation	separation	NOUN
ejpam-4737	49	36	axioms	axiom	NOUN
ejpam-4737	49	37	are	be	AUX
ejpam-4737	49	38	assumed	assume	VERB
ejpam-4737	49	39	unless	unless	SCONJ
ejpam-4737	49	40	explicitly	explicitly	ADV
ejpam-4737	49	41	stated	state	VERB
ejpam-4737	49	42	.	.	PUNCT
ejpam-4737	50	1	let	let	VERB
ejpam-4737	50	2	a	a	DET
ejpam-4737	50	3	be	be	AUX
ejpam-4737	50	4	a	a	DET
ejpam-4737	50	5	subset	subset	NOUN
ejpam-4737	50	6	of	of	ADP
ejpam-4737	50	7	a	a	DET
ejpam-4737	50	8	topological	topological	ADJ
ejpam-4737	50	9	space	space	NOUN
ejpam-4737	50	10	(	(	PUNCT
ejpam-4737	50	11	x	x	X
ejpam-4737	50	12	,	,	PUNCT
ejpam-4737	50	13	τ	τ	PROPN
ejpam-4737	50	14	)	)	PUNCT
ejpam-4737	50	15	.	.	PUNCT
ejpam-4737	51	1	the	the	DET
ejpam-4737	51	2	closure	closure	NOUN
ejpam-4737	51	3	of	of	ADP
ejpam-4737	51	4	a	a	PRON
ejpam-4737	51	5	and	and	CCONJ
ejpam-4737	51	6	the	the	DET
ejpam-4737	51	7	interior	interior	NOUN
ejpam-4737	51	8	of	of	ADP
ejpam-4737	51	9	a	a	PRON
ejpam-4737	51	10	are	be	AUX
ejpam-4737	51	11	denoted	denote	VERB
ejpam-4737	51	12	by	by	ADP
ejpam-4737	51	13	cl(a	cl(a	NOUN
ejpam-4737	51	14	)	)	PUNCT
ejpam-4737	51	15	and	and	CCONJ
ejpam-4737	51	16	int(a	int(a	PROPN
ejpam-4737	51	17	)	)	PUNCT
ejpam-4737	51	18	,	,	PUNCT
ejpam-4737	51	19	respectively	respectively	ADV
ejpam-4737	51	20	.	.	PUNCT
ejpam-4737	52	1	a	a	DET
ejpam-4737	52	2	subset	subset	NOUN
ejpam-4737	52	3	a	a	PRON
ejpam-4737	52	4	of	of	ADP
ejpam-4737	52	5	a	a	DET
ejpam-4737	52	6	topological	topological	ADJ
ejpam-4737	52	7	space	space	NOUN
ejpam-4737	52	8	(	(	PUNCT
ejpam-4737	52	9	x	x	X
ejpam-4737	52	10	,	,	PUNCT
ejpam-4737	52	11	τ	τ	X
ejpam-4737	52	12	)	)	PUNCT
ejpam-4737	52	13	is	be	AUX
ejpam-4737	52	14	said	say	VERB
ejpam-4737	52	15	to	to	PART
ejpam-4737	52	16	be	be	AUX
ejpam-4737	52	17	preopen	preopen	ADJ
ejpam-4737	52	18	[	[	X
ejpam-4737	52	19	9	9	NUM
ejpam-4737	52	20	]	]	X
ejpam-4737	52	21	if	if	SCONJ
ejpam-4737	52	22	a	a	DET
ejpam-4737	52	23	⊆	⊆	NUM
ejpam-4737	52	24	int(cl(a	int(cl(a	PROPN
ejpam-4737	52	25	)	)	PUNCT
ejpam-4737	52	26	)	)	PUNCT
ejpam-4737	52	27	.	.	PUNCT
ejpam-4737	53	1	the	the	DET
ejpam-4737	53	2	complement	complement	NOUN
ejpam-4737	53	3	of	of	ADP
ejpam-4737	53	4	a	a	DET
ejpam-4737	53	5	preopen	preopen	ADJ
ejpam-4737	53	6	set	set	NOUN
ejpam-4737	53	7	is	be	AUX
ejpam-4737	53	8	called	call	VERB
ejpam-4737	53	9	preclosed	preclose	VERB
ejpam-4737	53	10	.	.	PUNCT
ejpam-4737	54	1	the	the	DET
ejpam-4737	54	2	family	family	NOUN
ejpam-4737	54	3	of	of	ADP
ejpam-4737	54	4	all	all	DET
ejpam-4737	54	5	preopen	preopen	ADJ
ejpam-4737	54	6	sets	set	NOUN
ejpam-4737	54	7	of	of	ADP
ejpam-4737	54	8	a	a	DET
ejpam-4737	54	9	topological	topological	ADJ
ejpam-4737	54	10	space	space	NOUN
ejpam-4737	54	11	(	(	PUNCT
ejpam-4737	54	12	x	x	X
ejpam-4737	54	13	,	,	PUNCT
ejpam-4737	54	14	τ	τ	X
ejpam-4737	54	15	)	)	PUNCT
ejpam-4737	54	16	is	be	AUX
ejpam-4737	54	17	denoted	denote	VERB
ejpam-4737	54	18	by	by	ADP
ejpam-4737	54	19	po(x	po(x	NUM
ejpam-4737	54	20	,	,	PUNCT
ejpam-4737	54	21	τ	τ	PROPN
ejpam-4737	54	22	)	)	PUNCT
ejpam-4737	54	23	.	.	PUNCT
ejpam-4737	55	1	a	a	DET
ejpam-4737	55	2	subset	subset	NOUN
ejpam-4737	55	3	λp(a	λp(a	NOUN
ejpam-4737	55	4	)	)	PUNCT
ejpam-4737	56	1	[	[	X
ejpam-4737	56	2	8	8	NUM
ejpam-4737	56	3	]	]	PUNCT
ejpam-4737	56	4	is	be	AUX
ejpam-4737	56	5	defined	define	VERB
ejpam-4737	56	6	as	as	SCONJ
ejpam-4737	56	7	follows	follow	VERB
ejpam-4737	56	8	:	:	PUNCT
ejpam-4737	56	9	λp(a	λp(a	NUM
ejpam-4737	56	10	)	)	PUNCT
ejpam-4737	57	1	=	=	PUNCT
ejpam-4737	58	1	∩{u	∩{u	PROPN
ejpam-4737	58	2	|	|	ADV
ejpam-4737	58	3	a	a	DET
ejpam-4737	58	4	⊆	⊆	NUM
ejpam-4737	58	5	u	u	NOUN
ejpam-4737	58	6	,	,	PUNCT
ejpam-4737	58	7	u	u	PROPN
ejpam-4737	58	8	∈	∈	PROPN
ejpam-4737	58	9	po(x	po(x	NOUN
ejpam-4737	58	10	,	,	PUNCT
ejpam-4737	58	11	τ	τ	NOUN
ejpam-4737	58	12	)	)	PUNCT
ejpam-4737	58	13	}	}	PUNCT
ejpam-4737	58	14	.	.	PUNCT
ejpam-4737	59	1	a	a	DET
ejpam-4737	59	2	subset	subset	NOUN
ejpam-4737	59	3	a	a	PRON
ejpam-4737	59	4	of	of	ADP
ejpam-4737	59	5	a	a	DET
ejpam-4737	59	6	topological	topological	ADJ
ejpam-4737	59	7	space	space	NOUN
ejpam-4737	59	8	(	(	PUNCT
ejpam-4737	59	9	x	x	X
ejpam-4737	59	10	,	,	PUNCT
ejpam-4737	59	11	τ	τ	X
ejpam-4737	59	12	)	)	PUNCT
ejpam-4737	59	13	is	be	AUX
ejpam-4737	59	14	called	call	VERB
ejpam-4737	59	15	a	a	DET
ejpam-4737	59	16	λp	λp	NOUN
ejpam-4737	59	17	-	-	PUNCT
ejpam-4737	59	18	set	set	VERB
ejpam-4737	59	19	[	[	X
ejpam-4737	59	20	3	3	NUM
ejpam-4737	59	21	]	]	PUNCT
ejpam-4737	59	22	(	(	PUNCT
ejpam-4737	59	23	pre	pre	ADJ
ejpam-4737	59	24	-	-	ADJ
ejpam-4737	59	25	λ	λ	NOUN
ejpam-4737	59	26	-	-	NOUN
ejpam-4737	59	27	set	set	NOUN
ejpam-4737	59	28	[	[	NOUN
ejpam-4737	59	29	8	8	NUM
ejpam-4737	59	30	]	]	PUNCT
ejpam-4737	59	31	)	)	PUNCT
ejpam-4737	59	32	if	if	SCONJ
ejpam-4737	59	33	a	a	DET
ejpam-4737	59	34	=	=	NOUN
ejpam-4737	59	35	λp(a	λp(a	NOUN
ejpam-4737	59	36	)	)	PUNCT
ejpam-4737	59	37	.	.	PUNCT
ejpam-4737	60	1	a	a	DET
ejpam-4737	60	2	subset	subset	NOUN
ejpam-4737	60	3	a	a	PRON
ejpam-4737	60	4	of	of	ADP
ejpam-4737	60	5	a	a	DET
ejpam-4737	60	6	topological	topological	ADJ
ejpam-4737	60	7	space	space	NOUN
ejpam-4737	60	8	(	(	PUNCT
ejpam-4737	60	9	x	x	X
ejpam-4737	60	10	,	,	PUNCT
ejpam-4737	60	11	τ	τ	X
ejpam-4737	60	12	)	)	PUNCT
ejpam-4737	60	13	is	be	AUX
ejpam-4737	60	14	called	call	VERB
ejpam-4737	60	15	(	(	PUNCT
ejpam-4737	60	16	λ	λ	X
ejpam-4737	60	17	,	,	PUNCT
ejpam-4737	60	18	p)-closed	p)-close	VERB
ejpam-4737	60	19	[	[	X
ejpam-4737	60	20	3	3	X
ejpam-4737	60	21	]	]	PUNCT
ejpam-4737	60	22	if	if	SCONJ
ejpam-4737	60	23	a	a	DET
ejpam-4737	60	24	=	=	X
ejpam-4737	60	25	t	t	PROPN
ejpam-4737	60	26	∩	∩	ADJ
ejpam-4737	60	27	c	c	NOUN
ejpam-4737	60	28	,	,	PUNCT
ejpam-4737	60	29	where	where	SCONJ
ejpam-4737	60	30	t	t	PROPN
ejpam-4737	60	31	is	be	AUX
ejpam-4737	60	32	a	a	DET
ejpam-4737	60	33	λp	λp	ADV
ejpam-4737	60	34	-	-	PUNCT
ejpam-4737	60	35	set	set	NOUN
ejpam-4737	60	36	and	and	CCONJ
ejpam-4737	60	37	c	c	NOUN
ejpam-4737	60	38	is	be	AUX
ejpam-4737	60	39	a	a	DET
ejpam-4737	60	40	preclosed	preclose	VERB
ejpam-4737	60	41	set	set	NOUN
ejpam-4737	60	42	.	.	PUNCT
ejpam-4737	61	1	the	the	DET
ejpam-4737	61	2	complement	complement	NOUN
ejpam-4737	61	3	of	of	ADP
ejpam-4737	61	4	a	a	DET
ejpam-4737	61	5	(	(	PUNCT
ejpam-4737	61	6	λ	λ	PROPN
ejpam-4737	61	7	,	,	PUNCT
ejpam-4737	61	8	p)-closed	p)-close	VERB
ejpam-4737	61	9	set	set	NOUN
ejpam-4737	61	10	is	be	AUX
ejpam-4737	61	11	called	call	VERB
ejpam-4737	61	12	(	(	PUNCT
ejpam-4737	61	13	λ	λ	X
ejpam-4737	61	14	,	,	PUNCT
ejpam-4737	61	15	p)-open	p)-open	ADJ
ejpam-4737	61	16	.	.	PUNCT
ejpam-4737	62	1	the	the	DET
ejpam-4737	62	2	family	family	NOUN
ejpam-4737	62	3	of	of	ADP
ejpam-4737	62	4	all	all	DET
ejpam-4737	62	5	(	(	PUNCT
ejpam-4737	62	6	λ	λ	X
ejpam-4737	62	7	,	,	PUNCT
ejpam-4737	62	8	p)-open	p)-open	ADJ
ejpam-4737	62	9	(	(	PUNCT
ejpam-4737	62	10	resp	resp	NOUN
ejpam-4737	62	11	.	.	PUNCT
ejpam-4737	63	1	(	(	PUNCT
ejpam-4737	63	2	λ	λ	X
ejpam-4737	63	3	,	,	PUNCT
ejpam-4737	63	4	p)-closed	p)-close	VERB
ejpam-4737	63	5	)	)	PUNCT
ejpam-4737	63	6	sets	set	NOUN
ejpam-4737	63	7	in	in	ADP
ejpam-4737	63	8	a	a	DET
ejpam-4737	63	9	topological	topological	ADJ
ejpam-4737	63	10	space	space	NOUN
ejpam-4737	63	11	(	(	PUNCT
ejpam-4737	63	12	x	x	X
ejpam-4737	63	13	,	,	PUNCT
ejpam-4737	63	14	τ	τ	X
ejpam-4737	63	15	)	)	PUNCT
ejpam-4737	63	16	is	be	AUX
ejpam-4737	63	17	denoted	denote	VERB
ejpam-4737	63	18	by	by	ADP
ejpam-4737	63	19	λpo(x	λpo(x	PROPN
ejpam-4737	63	20	,	,	PUNCT
ejpam-4737	63	21	τ	τ	X
ejpam-4737	63	22	)	)	PUNCT
ejpam-4737	63	23	(	(	PUNCT
ejpam-4737	63	24	resp	resp	NOUN
ejpam-4737	63	25	.	.	PUNCT
ejpam-4737	64	1	λpc(x	λpc(x	PROPN
ejpam-4737	64	2	,	,	PUNCT
ejpam-4737	64	3	τ	τ	PROPN
ejpam-4737	64	4	)	)	PUNCT
ejpam-4737	64	5	)	)	PUNCT
ejpam-4737	64	6	.	.	PUNCT
ejpam-4737	65	1	let	let	VERB
ejpam-4737	65	2	a	a	DET
ejpam-4737	65	3	be	be	AUX
ejpam-4737	65	4	a	a	DET
ejpam-4737	65	5	subset	subset	NOUN
ejpam-4737	65	6	of	of	ADP
ejpam-4737	65	7	a	a	DET
ejpam-4737	65	8	topological	topological	ADJ
ejpam-4737	65	9	space	space	NOUN
ejpam-4737	65	10	(	(	PUNCT
ejpam-4737	65	11	x	x	X
ejpam-4737	65	12	,	,	PUNCT
ejpam-4737	65	13	τ	τ	PROPN
ejpam-4737	65	14	)	)	PUNCT
ejpam-4737	65	15	.	.	PUNCT
ejpam-4737	66	1	a	a	DET
ejpam-4737	66	2	point	point	NOUN
ejpam-4737	66	3	x	x	X
ejpam-4737	66	4	∈	∈	NOUN
ejpam-4737	66	5	x	x	PUNCT
ejpam-4737	66	6	is	be	AUX
ejpam-4737	66	7	called	call	VERB
ejpam-4737	66	8	a	a	DET
ejpam-4737	66	9	(	(	PUNCT
ejpam-4737	66	10	λ	λ	NOUN
ejpam-4737	66	11	,	,	PUNCT
ejpam-4737	66	12	p)-cluster	p)-cluster	NOUN
ejpam-4737	66	13	point	point	NOUN
ejpam-4737	66	14	[	[	X
ejpam-4737	66	15	3	3	X
ejpam-4737	66	16	]	]	PUNCT
ejpam-4737	66	17	of	of	ADP
ejpam-4737	66	18	a	a	DET
ejpam-4737	66	19	if	if	SCONJ
ejpam-4737	66	20	a∩u	a∩u	VERB
ejpam-4737	66	21	̸=	̸=	NOUN
ejpam-4737	66	22	∅	∅	NOUN
ejpam-4737	66	23	for	for	ADP
ejpam-4737	66	24	every	every	DET
ejpam-4737	66	25	(	(	PUNCT
ejpam-4737	66	26	λ	λ	NOUN
ejpam-4737	66	27	,	,	PUNCT
ejpam-4737	66	28	p)-open	p)-open	VERB
ejpam-4737	66	29	set	set	VERB
ejpam-4737	66	30	u	u	NOUN
ejpam-4737	66	31	of	of	ADP
ejpam-4737	66	32	x	x	SYM
ejpam-4737	66	33	containing	contain	VERB
ejpam-4737	66	34	x.	x.	NOUN
ejpam-4737	66	35	the	the	DET
ejpam-4737	66	36	set	set	NOUN
ejpam-4737	66	37	of	of	ADP
ejpam-4737	66	38	all	all	DET
ejpam-4737	66	39	(	(	PUNCT
ejpam-4737	66	40	λ	λ	NOUN
ejpam-4737	66	41	,	,	PUNCT
ejpam-4737	66	42	p)-cluster	p)-cluster	VERB
ejpam-4737	66	43	points	point	NOUN
ejpam-4737	66	44	of	of	ADP
ejpam-4737	66	45	a	a	PRON
ejpam-4737	66	46	is	be	AUX
ejpam-4737	66	47	called	call	VERB
ejpam-4737	66	48	the	the	DET
ejpam-4737	66	49	(	(	PUNCT
ejpam-4737	66	50	λ	λ	PROPN
ejpam-4737	66	51	,	,	PUNCT
ejpam-4737	66	52	p)-closure	p)-closure	PUNCT
ejpam-4737	67	1	[	[	X
ejpam-4737	67	2	3	3	X
ejpam-4737	67	3	]	]	PUNCT
ejpam-4737	67	4	of	of	ADP
ejpam-4737	67	5	a	a	PRON
ejpam-4737	67	6	and	and	CCONJ
ejpam-4737	67	7	is	be	AUX
ejpam-4737	67	8	denoted	denote	VERB
ejpam-4737	67	9	by	by	ADP
ejpam-4737	67	10	a(λ	a(λ	PROPN
ejpam-4737	67	11	,	,	PUNCT
ejpam-4737	67	12	p	p	NOUN
ejpam-4737	67	13	)	)	PUNCT
ejpam-4737	67	14	.	.	PUNCT
ejpam-4737	68	1	the	the	DET
ejpam-4737	68	2	union	union	NOUN
ejpam-4737	68	3	of	of	ADP
ejpam-4737	68	4	all	all	PRON
ejpam-4737	68	5	(	(	PUNCT
ejpam-4737	68	6	λ	λ	NOUN
ejpam-4737	68	7	,	,	PUNCT
ejpam-4737	68	8	p)-open	p)-open	VERB
ejpam-4737	68	9	sets	set	NOUN
ejpam-4737	68	10	of	of	ADP
ejpam-4737	68	11	x	x	PUNCT
ejpam-4737	68	12	contained	contain	VERB
ejpam-4737	68	13	in	in	ADP
ejpam-4737	68	14	a	a	PRON
ejpam-4737	68	15	is	be	AUX
ejpam-4737	68	16	called	call	VERB
ejpam-4737	68	17	the	the	DET
ejpam-4737	68	18	(	(	PUNCT
ejpam-4737	68	19	λ	λ	PROPN
ejpam-4737	68	20	,	,	PUNCT
ejpam-4737	68	21	p)-interior	p)-interior	ADJ
ejpam-4737	68	22	[	[	X
ejpam-4737	68	23	3	3	X
ejpam-4737	68	24	]	]	PUNCT
ejpam-4737	68	25	of	of	ADP
ejpam-4737	68	26	a	a	PRON
ejpam-4737	68	27	and	and	CCONJ
ejpam-4737	68	28	is	be	AUX
ejpam-4737	68	29	denoted	denote	VERB
ejpam-4737	68	30	by	by	ADP
ejpam-4737	68	31	a(λ	a(λ	PROPN
ejpam-4737	68	32	,	,	PUNCT
ejpam-4737	68	33	p	p	NOUN
ejpam-4737	68	34	)	)	PUNCT
ejpam-4737	68	35	.	.	PUNCT
ejpam-4737	69	1	a	a	DET
ejpam-4737	69	2	subset	subset	NOUN
ejpam-4737	69	3	a	a	PRON
ejpam-4737	69	4	of	of	ADP
ejpam-4737	69	5	a	a	DET
ejpam-4737	69	6	topological	topological	ADJ
ejpam-4737	69	7	space	space	NOUN
ejpam-4737	69	8	(	(	PUNCT
ejpam-4737	69	9	x	x	X
ejpam-4737	69	10	,	,	PUNCT
ejpam-4737	69	11	τ	τ	X
ejpam-4737	69	12	)	)	PUNCT
ejpam-4737	69	13	is	be	AUX
ejpam-4737	69	14	said	say	VERB
ejpam-4737	69	15	to	to	PART
ejpam-4737	69	16	be	be	AUX
ejpam-4737	69	17	α(λ	α(λ	PROPN
ejpam-4737	69	18	,	,	PUNCT
ejpam-4737	69	19	p)-open	p)-open	NOUN
ejpam-4737	69	20	(	(	PUNCT
ejpam-4737	69	21	resp	resp	NOUN
ejpam-4737	69	22	.	.	PUNCT
ejpam-4737	70	1	p(λ	p(λ	NOUN
ejpam-4737	70	2	,	,	PUNCT
ejpam-4737	70	3	p)-open	p)-open	NOUN
ejpam-4737	70	4	,	,	PUNCT
ejpam-4737	70	5	s(λ	s(λ	PROPN
ejpam-4737	70	6	,	,	PUNCT
ejpam-4737	70	7	p)open	p)open	ADJ
ejpam-4737	70	8	,	,	PUNCT
ejpam-4737	70	9	β(λ	β(λ	X
ejpam-4737	70	10	,	,	PUNCT
ejpam-4737	70	11	p)-open	p)-open	ADJ
ejpam-4737	70	12	,	,	PUNCT
ejpam-4737	70	13	r(λ	r(λ	NOUN
ejpam-4737	70	14	,	,	PUNCT
ejpam-4737	70	15	p)-open	p)-open	VERB
ejpam-4737	70	16	[	[	X
ejpam-4737	70	17	3	3	NUM
ejpam-4737	70	18	]	]	PUNCT
ejpam-4737	70	19	)	)	PUNCT
ejpam-4737	70	20	if	if	SCONJ
ejpam-4737	70	21	a	a	DET
ejpam-4737	70	22	⊆	⊆	NUM
ejpam-4737	70	23	[	[	X
ejpam-4737	70	24	[	[	X
ejpam-4737	70	25	a(λ	a(λ	ADJ
ejpam-4737	70	26	,	,	PUNCT
ejpam-4737	70	27	p	p	NOUN
ejpam-4737	70	28	)	)	PUNCT
ejpam-4737	70	29	]	]	PUNCT
ejpam-4737	70	30	(	(	PUNCT
ejpam-4737	70	31	λ	λ	X
ejpam-4737	70	32	,	,	PUNCT
ejpam-4737	70	33	p)](λ	p)](λ	ADJ
ejpam-4737	70	34	,	,	PUNCT
ejpam-4737	70	35	p	p	NOUN
ejpam-4737	70	36	)	)	PUNCT
ejpam-4737	70	37	(	(	PUNCT
ejpam-4737	70	38	resp	resp	NOUN
ejpam-4737	70	39	.	.	PUNCT
ejpam-4737	71	1	a	a	DET
ejpam-4737	71	2	⊆	⊆	NUM
ejpam-4737	71	3	[	[	X
ejpam-4737	71	4	a(λ	a(λ	ADV
ejpam-4737	71	5	,	,	PUNCT
ejpam-4737	71	6	p)](λ	p)](λ	X
ejpam-4737	71	7	,	,	PUNCT
ejpam-4737	71	8	p	p	NOUN
ejpam-4737	71	9	)	)	PUNCT
ejpam-4737	71	10	,	,	PUNCT
ejpam-4737	71	11	a	a	DET
ejpam-4737	71	12	⊆	⊆	NUM
ejpam-4737	71	13	[	[	X
ejpam-4737	71	14	a(λ	a(λ	ADV
ejpam-4737	71	15	,	,	PUNCT
ejpam-4737	71	16	p	p	NOUN
ejpam-4737	71	17	)	)	PUNCT
ejpam-4737	71	18	]	]	PUNCT
ejpam-4737	71	19	(	(	PUNCT
ejpam-4737	71	20	λ	λ	X
ejpam-4737	71	21	,	,	PUNCT
ejpam-4737	71	22	p	p	NOUN
ejpam-4737	71	23	)	)	PUNCT
ejpam-4737	71	24	,	,	PUNCT
ejpam-4737	71	25	a	a	DET
ejpam-4737	71	26	⊆	⊆	NUM
ejpam-4737	71	27	[	[	X
ejpam-4737	71	28	[	[	X
ejpam-4737	71	29	a(λ	a(λ	ADJ
ejpam-4737	71	30	,	,	PUNCT
ejpam-4737	71	31	p)](λ	p)](λ	X
ejpam-4737	71	32	,	,	PUNCT
ejpam-4737	71	33	p	p	NOUN
ejpam-4737	71	34	)	)	PUNCT
ejpam-4737	71	35	]	]	PUNCT
ejpam-4737	71	36	(	(	PUNCT
ejpam-4737	71	37	λ	λ	X
ejpam-4737	71	38	,	,	PUNCT
ejpam-4737	71	39	p	p	NOUN
ejpam-4737	71	40	)	)	PUNCT
ejpam-4737	71	41	,	,	PUNCT
ejpam-4737	71	42	a	a	PRON
ejpam-4737	71	43	=	=	X
ejpam-4737	71	44	[	[	X
ejpam-4737	71	45	a(λ	a(λ	ADV
ejpam-4737	71	46	,	,	PUNCT
ejpam-4737	71	47	p)](λ	p)](λ	X
ejpam-4737	71	48	,	,	PUNCT
ejpam-4737	71	49	p	p	NOUN
ejpam-4737	71	50	)	)	PUNCT
ejpam-4737	71	51	)	)	PUNCT
ejpam-4737	71	52	.	.	PUNCT
ejpam-4737	72	1	the	the	DET
ejpam-4737	72	2	family	family	NOUN
ejpam-4737	72	3	of	of	ADP
ejpam-4737	72	4	all	all	DET
ejpam-4737	72	5	α(λ	α(λ	PROPN
ejpam-4737	72	6	,	,	PUNCT
ejpam-4737	72	7	p)-open	p)-open	NOUN
ejpam-4737	72	8	(	(	PUNCT
ejpam-4737	72	9	resp	resp	NOUN
ejpam-4737	72	10	.	.	PUNCT
ejpam-4737	73	1	p(λ	p(λ	NOUN
ejpam-4737	73	2	,	,	PUNCT
ejpam-4737	73	3	p)-open	p)-open	NOUN
ejpam-4737	73	4	,	,	PUNCT
ejpam-4737	73	5	s(λ	s(λ	PROPN
ejpam-4737	73	6	,	,	PUNCT
ejpam-4737	73	7	p)-open	p)-open	ADJ
ejpam-4737	73	8	,	,	PUNCT
ejpam-4737	73	9	β(λ	β(λ	X
ejpam-4737	73	10	,	,	PUNCT
ejpam-4737	73	11	p)-open	p)-open	ADJ
ejpam-4737	73	12	,	,	PUNCT
ejpam-4737	73	13	r(λ	r(λ	NOUN
ejpam-4737	73	14	,	,	PUNCT
ejpam-4737	73	15	p)-open	p)-open	ADJ
ejpam-4737	73	16	)	)	PUNCT
ejpam-4737	73	17	sets	set	NOUN
ejpam-4737	73	18	in	in	ADP
ejpam-4737	73	19	a	a	DET
ejpam-4737	73	20	topological	topological	ADJ
ejpam-4737	73	21	space	space	NOUN
ejpam-4737	73	22	(	(	PUNCT
ejpam-4737	73	23	x	x	X
ejpam-4737	73	24	,	,	PUNCT
ejpam-4737	73	25	τ	τ	X
ejpam-4737	73	26	)	)	PUNCT
ejpam-4737	73	27	is	be	AUX
ejpam-4737	73	28	denoted	denote	VERB
ejpam-4737	73	29	by	by	ADP
ejpam-4737	73	30	α(λ	α(λ	PROPN
ejpam-4737	73	31	,	,	PUNCT
ejpam-4737	73	32	p)o(x	p)o(x	ADJ
ejpam-4737	73	33	,	,	PUNCT
ejpam-4737	73	34	τ	τ	PROPN
ejpam-4737	73	35	)	)	PUNCT
ejpam-4737	73	36	(	(	PUNCT
ejpam-4737	73	37	resp	resp	NOUN
ejpam-4737	73	38	.	.	PUNCT
ejpam-4737	74	1	p(λ	p(λ	PROPN
ejpam-4737	74	2	,	,	PUNCT
ejpam-4737	74	3	p)o(x	p)o(x	ADJ
ejpam-4737	74	4	,	,	PUNCT
ejpam-4737	74	5	τ	τ	PROPN
ejpam-4737	74	6	)	)	PUNCT
ejpam-4737	74	7	,	,	PUNCT
ejpam-4737	74	8	s(λ	s(λ	PROPN
ejpam-4737	74	9	,	,	PUNCT
ejpam-4737	74	10	p)o(x	p)o(x	ADJ
ejpam-4737	74	11	,	,	PUNCT
ejpam-4737	74	12	τ	τ	PROPN
ejpam-4737	74	13	)	)	PUNCT
ejpam-4737	74	14	,	,	PUNCT
ejpam-4737	74	15	β(λ	β(λ	X
ejpam-4737	74	16	,	,	PUNCT
ejpam-4737	74	17	p)o(x	p)o(x	ADJ
ejpam-4737	74	18	,	,	PUNCT
ejpam-4737	74	19	τ	τ	PROPN
ejpam-4737	74	20	)	)	PUNCT
ejpam-4737	74	21	,	,	PUNCT
ejpam-4737	74	22	r(λ	r(λ	PROPN
ejpam-4737	74	23	,	,	PUNCT
ejpam-4737	74	24	p)o(x	p)o(x	ADJ
ejpam-4737	74	25	,	,	PUNCT
ejpam-4737	74	26	τ	τ	PROPN
ejpam-4737	74	27	)	)	PUNCT
ejpam-4737	74	28	)	)	PUNCT
ejpam-4737	74	29	.	.	PUNCT
ejpam-4737	75	1	the	the	DET
ejpam-4737	75	2	complement	complement	NOUN
ejpam-4737	75	3	of	of	ADP
ejpam-4737	75	4	a	a	DET
ejpam-4737	75	5	p(λ	p(λ	NOUN
ejpam-4737	75	6	,	,	PUNCT
ejpam-4737	75	7	p)-open	p)-open	NOUN
ejpam-4737	75	8	(	(	PUNCT
ejpam-4737	75	9	resp	resp	NOUN
ejpam-4737	75	10	.	.	PUNCT
ejpam-4737	76	1	s(λ	s(λ	NOUN
ejpam-4737	76	2	,	,	PUNCT
ejpam-4737	76	3	p)-open	p)-open	NOUN
ejpam-4737	76	4	,	,	PUNCT
ejpam-4737	76	5	α(λ	α(λ	PROPN
ejpam-4737	76	6	,	,	PUNCT
ejpam-4737	76	7	p)-open	p)-open	ADJ
ejpam-4737	76	8	,	,	PUNCT
ejpam-4737	76	9	β(λ	β(λ	X
ejpam-4737	76	10	,	,	PUNCT
ejpam-4737	76	11	p)-open	p)-open	ADJ
ejpam-4737	76	12	,	,	PUNCT
ejpam-4737	76	13	r(λ	r(λ	NOUN
ejpam-4737	76	14	,	,	PUNCT
ejpam-4737	76	15	p)-open	p)-open	NOUN
ejpam-4737	76	16	)	)	PUNCT
ejpam-4737	76	17	set	set	NOUN
ejpam-4737	76	18	is	be	AUX
ejpam-4737	76	19	said	say	VERB
ejpam-4737	76	20	to	to	PART
ejpam-4737	76	21	be	be	AUX
ejpam-4737	76	22	p(λ	p(λ	NOUN
ejpam-4737	76	23	,	,	PUNCT
ejpam-4737	76	24	p)-closed	p)-close	VERB
ejpam-4737	76	25	(	(	PUNCT
ejpam-4737	76	26	resp	resp	NOUN
ejpam-4737	76	27	.	.	PUNCT
ejpam-4737	77	1	s(λ	s(λ	PROPN
ejpam-4737	77	2	,	,	PUNCT
ejpam-4737	77	3	p)-closed	p)-close	VERB
ejpam-4737	77	4	,	,	PUNCT
ejpam-4737	77	5	α(λ	α(λ	PROPN
ejpam-4737	77	6	,	,	PUNCT
ejpam-4737	77	7	p)closed	p)close	VERB
ejpam-4737	77	8	,	,	PUNCT
ejpam-4737	77	9	β(λ	β(λ	PROPN
ejpam-4737	77	10	,	,	PUNCT
ejpam-4737	77	11	p)-closed	p)-close	VERB
ejpam-4737	77	12	,	,	PUNCT
ejpam-4737	77	13	r(λ	r(λ	PROPN
ejpam-4737	77	14	,	,	PUNCT
ejpam-4737	77	15	p)-closed	p)-close	VERB
ejpam-4737	77	16	)	)	PUNCT
ejpam-4737	77	17	.	.	PUNCT
ejpam-4737	78	1	let	let	VERB
ejpam-4737	78	2	a	a	DET
ejpam-4737	78	3	be	be	AUX
ejpam-4737	78	4	a	a	DET
ejpam-4737	78	5	subset	subset	NOUN
ejpam-4737	78	6	of	of	ADP
ejpam-4737	78	7	a	a	DET
ejpam-4737	78	8	topological	topological	ADJ
ejpam-4737	78	9	space	space	NOUN
ejpam-4737	78	10	(	(	PUNCT
ejpam-4737	78	11	x	x	X
ejpam-4737	78	12	,	,	PUNCT
ejpam-4737	78	13	τ	τ	PROPN
ejpam-4737	78	14	)	)	PUNCT
ejpam-4737	78	15	.	.	PUNCT
ejpam-4737	79	1	the	the	DET
ejpam-4737	79	2	intersection	intersection	NOUN
ejpam-4737	79	3	of	of	ADP
ejpam-4737	79	4	all	all	DET
ejpam-4737	79	5	s(λ	s(λ	PROPN
ejpam-4737	79	6	,	,	PUNCT
ejpam-4737	79	7	p)-closed	p)-close	VERB
ejpam-4737	79	8	sets	set	NOUN
ejpam-4737	79	9	of	of	ADP
ejpam-4737	79	10	x	x	PUNCT
ejpam-4737	79	11	containing	contain	VERB
ejpam-4737	79	12	a	a	PRON
ejpam-4737	79	13	is	be	AUX
ejpam-4737	79	14	called	call	VERB
ejpam-4737	79	15	the	the	DET
ejpam-4737	79	16	s(λ	s(λ	PROPN
ejpam-4737	79	17	,	,	PUNCT
ejpam-4737	79	18	p)-closure	p)-closure	PUNCT
ejpam-4737	79	19	c.	c.	PROPN
ejpam-4737	79	20	boonpok	boonpok	PROPN
ejpam-4737	79	21	,	,	PUNCT
ejpam-4737	79	22	m.	m.	NOUN
ejpam-4737	79	23	thongmoon	thongmoon	PROPN
ejpam-4737	79	24	/	/	SYM
ejpam-4737	79	25	eur	eur	PROPN
ejpam-4737	79	26	.	.	PUNCT
ejpam-4737	80	1	j.	j.	PROPN
ejpam-4737	80	2	pure	pure	PROPN
ejpam-4737	80	3	appl	appl	PROPN
ejpam-4737	80	4	.	.	PROPN
ejpam-4737	80	5	math	math	PROPN
ejpam-4737	80	6	,	,	PUNCT
ejpam-4737	80	7	16	16	NUM
ejpam-4737	80	8	(	(	PUNCT
ejpam-4737	80	9	3	3	NUM
ejpam-4737	80	10	)	)	PUNCT
ejpam-4737	80	11	(	(	PUNCT
ejpam-4737	80	12	2023	2023	NUM
ejpam-4737	80	13	)	)	PUNCT
ejpam-4737	80	14	,	,	PUNCT
ejpam-4737	80	15	1434	1434	NUM
ejpam-4737	80	16	-	-	SYM
ejpam-4737	80	17	1447	1447	NUM
ejpam-4737	80	18	1436	1436	NUM
ejpam-4737	80	19	of	of	ADP
ejpam-4737	80	20	a	a	PRON
ejpam-4737	80	21	and	and	CCONJ
ejpam-4737	80	22	is	be	AUX
ejpam-4737	80	23	denoted	denote	VERB
ejpam-4737	80	24	by	by	ADP
ejpam-4737	80	25	as(λ	as(λ	NOUN
ejpam-4737	80	26	,	,	PUNCT
ejpam-4737	80	27	p	p	NOUN
ejpam-4737	80	28	)	)	PUNCT
ejpam-4737	80	29	.	.	PUNCT
ejpam-4737	81	1	a	a	DET
ejpam-4737	81	2	point	point	NOUN
ejpam-4737	81	3	x	x	PUNCT
ejpam-4737	81	4	of	of	ADP
ejpam-4737	81	5	x	x	PROPN
ejpam-4737	81	6	is	be	AUX
ejpam-4737	81	7	called	call	VERB
ejpam-4737	81	8	a	a	DET
ejpam-4737	81	9	δ(λ	δ(λ	PROPN
ejpam-4737	81	10	,	,	PUNCT
ejpam-4737	81	11	p)-cluster	p)-cluster	NOUN
ejpam-4737	81	12	point	point	NOUN
ejpam-4737	81	13	[	[	X
ejpam-4737	81	14	2	2	X
ejpam-4737	81	15	]	]	PUNCT
ejpam-4737	81	16	of	of	ADP
ejpam-4737	81	17	a	a	DET
ejpam-4737	81	18	if	if	SCONJ
ejpam-4737	81	19	a	a	DET
ejpam-4737	81	20	∩	∩	NOUN
ejpam-4737	81	21	[	[	X
ejpam-4737	81	22	v	v	X
ejpam-4737	81	23	(	(	PUNCT
ejpam-4737	81	24	λ	λ	PROPN
ejpam-4737	81	25	,	,	PUNCT
ejpam-4737	81	26	p)](λ	p)](λ	ADJ
ejpam-4737	81	27	,	,	PUNCT
ejpam-4737	81	28	p	p	NOUN
ejpam-4737	81	29	)	)	PUNCT
ejpam-4737	81	30	̸=	̸=	PROPN
ejpam-4737	81	31	∅	∅	NOUN
ejpam-4737	81	32	for	for	ADP
ejpam-4737	81	33	every	every	DET
ejpam-4737	81	34	(	(	PUNCT
ejpam-4737	81	35	λ	λ	NOUN
ejpam-4737	81	36	,	,	PUNCT
ejpam-4737	81	37	p)-open	p)-open	VERB
ejpam-4737	81	38	set	set	VERB
ejpam-4737	81	39	v	v	NOUN
ejpam-4737	81	40	of	of	ADP
ejpam-4737	81	41	x	x	PUNCT
ejpam-4737	81	42	containing	contain	VERB
ejpam-4737	81	43	x.	x.	NOUN
ejpam-4737	81	44	the	the	DET
ejpam-4737	81	45	set	set	NOUN
ejpam-4737	81	46	of	of	ADP
ejpam-4737	81	47	all	all	DET
ejpam-4737	81	48	δ(λ	δ(λ	PROPN
ejpam-4737	81	49	,	,	PUNCT
ejpam-4737	81	50	p)-cluster	p)-cluster	VERB
ejpam-4737	81	51	points	point	NOUN
ejpam-4737	81	52	of	of	ADP
ejpam-4737	81	53	a	a	PRON
ejpam-4737	81	54	is	be	AUX
ejpam-4737	81	55	called	call	VERB
ejpam-4737	81	56	the	the	DET
ejpam-4737	81	57	δ(λ	δ(λ	PROPN
ejpam-4737	81	58	,	,	PUNCT
ejpam-4737	81	59	p)-closure	p)-closure	PUNCT
ejpam-4737	82	1	[	[	X
ejpam-4737	82	2	2	2	X
ejpam-4737	82	3	]	]	PUNCT
ejpam-4737	82	4	of	of	ADP
ejpam-4737	82	5	a	a	PRON
ejpam-4737	82	6	and	and	CCONJ
ejpam-4737	82	7	is	be	AUX
ejpam-4737	82	8	denoted	denote	VERB
ejpam-4737	82	9	by	by	ADP
ejpam-4737	82	10	aδ(λ	aδ(λ	NUM
ejpam-4737	82	11	,	,	PUNCT
ejpam-4737	82	12	p	p	NOUN
ejpam-4737	82	13	)	)	PUNCT
ejpam-4737	82	14	.	.	PUNCT
ejpam-4737	83	1	if	if	SCONJ
ejpam-4737	83	2	a	a	DET
ejpam-4737	83	3	=	=	NOUN
ejpam-4737	83	4	aδ(λ	aδ(λ	NUM
ejpam-4737	83	5	,	,	PUNCT
ejpam-4737	83	6	p	p	NOUN
ejpam-4737	83	7	)	)	PUNCT
ejpam-4737	83	8	,	,	PUNCT
ejpam-4737	83	9	then	then	ADV
ejpam-4737	83	10	a	a	PRON
ejpam-4737	83	11	is	be	AUX
ejpam-4737	83	12	said	say	VERB
ejpam-4737	83	13	to	to	PART
ejpam-4737	83	14	be	be	AUX
ejpam-4737	83	15	δ(λ	δ(λ	PROPN
ejpam-4737	83	16	,	,	PUNCT
ejpam-4737	83	17	p)-closed	p)-close	VERB
ejpam-4737	83	18	[	[	X
ejpam-4737	83	19	2	2	NUM
ejpam-4737	83	20	]	]	PUNCT
ejpam-4737	83	21	.	.	PUNCT
ejpam-4737	84	1	the	the	DET
ejpam-4737	84	2	complement	complement	NOUN
ejpam-4737	84	3	of	of	ADP
ejpam-4737	84	4	a	a	DET
ejpam-4737	84	5	δ(λ	δ(λ	PROPN
ejpam-4737	84	6	,	,	PUNCT
ejpam-4737	84	7	p)-closed	p)-close	VERB
ejpam-4737	84	8	set	set	NOUN
ejpam-4737	84	9	is	be	AUX
ejpam-4737	84	10	said	say	VERB
ejpam-4737	84	11	to	to	PART
ejpam-4737	84	12	be	be	AUX
ejpam-4737	84	13	δ(λ	δ(λ	PROPN
ejpam-4737	84	14	,	,	PUNCT
ejpam-4737	84	15	p)-open	p)-open	NOUN
ejpam-4737	84	16	.	.	PUNCT
ejpam-4737	85	1	the	the	DET
ejpam-4737	85	2	union	union	NOUN
ejpam-4737	85	3	of	of	ADP
ejpam-4737	85	4	all	all	DET
ejpam-4737	85	5	δ(λ	δ(λ	PROPN
ejpam-4737	85	6	,	,	PUNCT
ejpam-4737	85	7	p)-open	p)-open	VERB
ejpam-4737	85	8	sets	set	NOUN
ejpam-4737	85	9	of	of	ADP
ejpam-4737	85	10	x	x	PUNCT
ejpam-4737	85	11	contained	contain	VERB
ejpam-4737	85	12	in	in	ADP
ejpam-4737	85	13	a	a	PRON
ejpam-4737	85	14	is	be	AUX
ejpam-4737	85	15	called	call	VERB
ejpam-4737	85	16	the	the	DET
ejpam-4737	85	17	δ(λ	δ(λ	PROPN
ejpam-4737	85	18	,	,	PUNCT
ejpam-4737	85	19	p)-interior	p)-interior	ADJ
ejpam-4737	86	1	[	[	X
ejpam-4737	86	2	2	2	X
ejpam-4737	86	3	]	]	PUNCT
ejpam-4737	86	4	of	of	ADP
ejpam-4737	86	5	a	a	PRON
ejpam-4737	86	6	and	and	CCONJ
ejpam-4737	86	7	is	be	AUX
ejpam-4737	86	8	denoted	denote	VERB
ejpam-4737	86	9	by	by	ADP
ejpam-4737	86	10	aδ(λ	aδ(λ	NUM
ejpam-4737	86	11	,	,	PUNCT
ejpam-4737	86	12	p	p	NOUN
ejpam-4737	86	13	)	)	PUNCT
ejpam-4737	86	14	.	.	PUNCT
ejpam-4737	87	1	3	3	X
ejpam-4737	87	2	.	.	NUM
ejpam-4737	87	3	δs(λ	δs(λ	NOUN
ejpam-4737	87	4	,	,	PUNCT
ejpam-4737	87	5	p)-open	p)-open	VERB
ejpam-4737	87	6	sets	set	NOUN
ejpam-4737	87	7	in	in	ADP
ejpam-4737	87	8	this	this	DET
ejpam-4737	87	9	section	section	NOUN
ejpam-4737	87	10	,	,	PUNCT
ejpam-4737	87	11	we	we	PRON
ejpam-4737	87	12	introduce	introduce	VERB
ejpam-4737	87	13	the	the	DET
ejpam-4737	87	14	notion	notion	NOUN
ejpam-4737	87	15	of	of	ADP
ejpam-4737	87	16	δs(λ	δs(λ	NOUN
ejpam-4737	87	17	,	,	PUNCT
ejpam-4737	87	18	p)-open	p)-open	VERB
ejpam-4737	87	19	sets	set	NOUN
ejpam-4737	87	20	.	.	PUNCT
ejpam-4737	88	1	moreover	moreover	ADV
ejpam-4737	88	2	,	,	PUNCT
ejpam-4737	88	3	some	some	DET
ejpam-4737	88	4	characterizations	characterization	NOUN
ejpam-4737	88	5	of	of	ADP
ejpam-4737	88	6	δs(λ	δs(λ	NOUN
ejpam-4737	88	7	,	,	PUNCT
ejpam-4737	88	8	p)-t0	p)-t0	NOUN
ejpam-4737	88	9	spaces	space	NOUN
ejpam-4737	88	10	,	,	PUNCT
ejpam-4737	88	11	δs(λ	δs(λ	NOUN
ejpam-4737	88	12	,	,	PUNCT
ejpam-4737	88	13	p)-t1	p)-t1	VERB
ejpam-4737	88	14	spaces	space	NOUN
ejpam-4737	88	15	and	and	CCONJ
ejpam-4737	88	16	δs(λ	δs(λ	NOUN
ejpam-4737	88	17	,	,	PUNCT
ejpam-4737	88	18	p)-symmetric	p)-symmetric	ADJ
ejpam-4737	88	19	spaces	space	NOUN
ejpam-4737	88	20	are	be	AUX
ejpam-4737	88	21	discussed	discuss	VERB
ejpam-4737	88	22	.	.	PUNCT
ejpam-4737	89	1	definition	definition	NOUN
ejpam-4737	89	2	1	1	NUM
ejpam-4737	89	3	.	.	PUNCT
ejpam-4737	90	1	a	a	DET
ejpam-4737	90	2	subset	subset	NOUN
ejpam-4737	90	3	a	a	PRON
ejpam-4737	90	4	of	of	ADP
ejpam-4737	90	5	a	a	DET
ejpam-4737	90	6	topological	topological	ADJ
ejpam-4737	90	7	space	space	NOUN
ejpam-4737	90	8	(	(	PUNCT
ejpam-4737	90	9	x	x	X
ejpam-4737	90	10	,	,	PUNCT
ejpam-4737	90	11	τ	τ	X
ejpam-4737	90	12	)	)	PUNCT
ejpam-4737	90	13	is	be	AUX
ejpam-4737	90	14	said	say	VERB
ejpam-4737	90	15	to	to	PART
ejpam-4737	90	16	be	be	AUX
ejpam-4737	90	17	δs(λ	δs(λ	NOUN
ejpam-4737	90	18	,	,	PUNCT
ejpam-4737	90	19	p)-open	p)-open	VERB
ejpam-4737	90	20	if	if	SCONJ
ejpam-4737	90	21	a	a	DET
ejpam-4737	90	22	⊆	⊆	NUM
ejpam-4737	90	23	[	[	X
ejpam-4737	90	24	a(λ	a(λ	ADV
ejpam-4737	90	25	,	,	PUNCT
ejpam-4737	90	26	p	p	NOUN
ejpam-4737	90	27	)	)	PUNCT
ejpam-4737	90	28	]	]	PUNCT
ejpam-4737	91	1	δ(λ	δ(λ	PROPN
ejpam-4737	91	2	,	,	PUNCT
ejpam-4737	91	3	p	p	NOUN
ejpam-4737	91	4	)	)	PUNCT
ejpam-4737	91	5	.	.	PUNCT
ejpam-4737	92	1	the	the	DET
ejpam-4737	92	2	complement	complement	NOUN
ejpam-4737	92	3	of	of	ADP
ejpam-4737	92	4	a	a	DET
ejpam-4737	92	5	δs(λ	δs(λ	NOUN
ejpam-4737	92	6	,	,	PUNCT
ejpam-4737	92	7	p)-open	p)-open	VERB
ejpam-4737	92	8	set	set	NOUN
ejpam-4737	92	9	is	be	AUX
ejpam-4737	92	10	said	say	VERB
ejpam-4737	92	11	to	to	PART
ejpam-4737	92	12	be	be	AUX
ejpam-4737	92	13	δs(λ	δs(λ	NOUN
ejpam-4737	92	14	,	,	PUNCT
ejpam-4737	92	15	p)-closed	p)-close	VERB
ejpam-4737	92	16	.	.	PUNCT
ejpam-4737	93	1	the	the	DET
ejpam-4737	93	2	family	family	NOUN
ejpam-4737	93	3	of	of	ADP
ejpam-4737	93	4	all	all	DET
ejpam-4737	93	5	δs(λ	δs(λ	NOUN
ejpam-4737	93	6	,	,	PUNCT
ejpam-4737	93	7	p)-open	p)-open	NOUN
ejpam-4737	93	8	(	(	PUNCT
ejpam-4737	93	9	resp	resp	NOUN
ejpam-4737	93	10	.	.	PUNCT
ejpam-4737	93	11	δs(λ	δs(λ	PROPN
ejpam-4737	93	12	,	,	PUNCT
ejpam-4737	93	13	p)-closed	p)-close	VERB
ejpam-4737	93	14	)	)	PUNCT
ejpam-4737	93	15	sets	set	NOUN
ejpam-4737	93	16	in	in	ADP
ejpam-4737	93	17	a	a	DET
ejpam-4737	93	18	topological	topological	ADJ
ejpam-4737	93	19	space	space	NOUN
ejpam-4737	93	20	(	(	PUNCT
ejpam-4737	93	21	x	x	X
ejpam-4737	93	22	,	,	PUNCT
ejpam-4737	93	23	τ	τ	X
ejpam-4737	93	24	)	)	PUNCT
ejpam-4737	93	25	is	be	AUX
ejpam-4737	93	26	denoted	denote	VERB
ejpam-4737	93	27	by	by	ADP
ejpam-4737	93	28	δs(λ	δs(λ	NOUN
ejpam-4737	93	29	,	,	PUNCT
ejpam-4737	93	30	p)o(x	p)o(x	ADJ
ejpam-4737	93	31	,	,	PUNCT
ejpam-4737	93	32	τ	τ	PROPN
ejpam-4737	93	33	)	)	PUNCT
ejpam-4737	93	34	(	(	PUNCT
ejpam-4737	93	35	resp	resp	NOUN
ejpam-4737	93	36	.	.	PUNCT
ejpam-4737	93	37	δs(λ	δs(λ	PROPN
ejpam-4737	93	38	,	,	PUNCT
ejpam-4737	93	39	p)c(x	p)c(x	NOUN
ejpam-4737	93	40	,	,	PUNCT
ejpam-4737	93	41	τ	τ	NOUN
ejpam-4737	93	42	)	)	PUNCT
ejpam-4737	93	43	)	)	PUNCT
ejpam-4737	93	44	.	.	PUNCT
ejpam-4737	94	1	definition	definition	NOUN
ejpam-4737	94	2	2	2	NUM
ejpam-4737	94	3	.	.	PUNCT
ejpam-4737	94	4	let	let	VERB
ejpam-4737	94	5	a	a	DET
ejpam-4737	94	6	be	be	AUX
ejpam-4737	94	7	a	a	DET
ejpam-4737	94	8	subset	subset	NOUN
ejpam-4737	94	9	of	of	ADP
ejpam-4737	94	10	a	a	DET
ejpam-4737	94	11	topological	topological	ADJ
ejpam-4737	94	12	space	space	NOUN
ejpam-4737	94	13	(	(	PUNCT
ejpam-4737	94	14	x	x	X
ejpam-4737	94	15	,	,	PUNCT
ejpam-4737	94	16	τ	τ	PROPN
ejpam-4737	94	17	)	)	PUNCT
ejpam-4737	94	18	.	.	PUNCT
ejpam-4737	95	1	a	a	DET
ejpam-4737	95	2	point	point	NOUN
ejpam-4737	95	3	x	x	PUNCT
ejpam-4737	95	4	of	of	ADP
ejpam-4737	95	5	x	x	PROPN
ejpam-4737	95	6	is	be	AUX
ejpam-4737	95	7	called	call	VERB
ejpam-4737	95	8	a	a	DET
ejpam-4737	95	9	δs(λ	δs(λ	NOUN
ejpam-4737	95	10	,	,	PUNCT
ejpam-4737	95	11	p)-cluster	p)-cluster	NOUN
ejpam-4737	95	12	point	point	NOUN
ejpam-4737	95	13	of	of	ADP
ejpam-4737	95	14	a	a	PRON
ejpam-4737	95	15	if	if	SCONJ
ejpam-4737	95	16	a	a	DET
ejpam-4737	95	17	∩	∩	ADJ
ejpam-4737	95	18	u	u	ADJ
ejpam-4737	95	19	̸=	̸=	PROPN
ejpam-4737	95	20	∅	∅	NOUN
ejpam-4737	95	21	for	for	ADP
ejpam-4737	95	22	every	every	DET
ejpam-4737	95	23	δs(λ	δs(λ	NOUN
ejpam-4737	95	24	,	,	PUNCT
ejpam-4737	95	25	s)-open	s)-open	VERB
ejpam-4737	95	26	set	set	VERB
ejpam-4737	95	27	u	u	NOUN
ejpam-4737	95	28	of	of	ADP
ejpam-4737	95	29	x	x	SYM
ejpam-4737	95	30	containing	contain	VERB
ejpam-4737	95	31	x.	x.	NOUN
ejpam-4737	95	32	the	the	DET
ejpam-4737	95	33	set	set	NOUN
ejpam-4737	95	34	of	of	ADP
ejpam-4737	95	35	all	all	DET
ejpam-4737	95	36	δs(λ	δs(λ	NOUN
ejpam-4737	95	37	,	,	PUNCT
ejpam-4737	95	38	p)-cluster	p)-cluster	VERB
ejpam-4737	95	39	points	point	NOUN
ejpam-4737	95	40	of	of	ADP
ejpam-4737	95	41	a	a	PRON
ejpam-4737	95	42	is	be	AUX
ejpam-4737	95	43	called	call	VERB
ejpam-4737	95	44	the	the	DET
ejpam-4737	95	45	δs(λ	δs(λ	NOUN
ejpam-4737	95	46	,	,	PUNCT
ejpam-4737	95	47	p)-closure	p)-closure	NOUN
ejpam-4737	95	48	of	of	ADP
ejpam-4737	95	49	a	a	PRON
ejpam-4737	95	50	and	and	CCONJ
ejpam-4737	95	51	is	be	AUX
ejpam-4737	95	52	denoted	denote	VERB
ejpam-4737	95	53	by	by	ADP
ejpam-4737	95	54	aδs(λ	aδs(λ	PROPN
ejpam-4737	95	55	,	,	PUNCT
ejpam-4737	95	56	p	p	NOUN
ejpam-4737	95	57	)	)	PUNCT
ejpam-4737	95	58	.	.	PUNCT
ejpam-4737	96	1	lemma	lemma	PROPN
ejpam-4737	96	2	1	1	NUM
ejpam-4737	96	3	.	.	PUNCT
ejpam-4737	97	1	the	the	DET
ejpam-4737	97	2	intersection	intersection	NOUN
ejpam-4737	97	3	of	of	ADP
ejpam-4737	97	4	arbitrary	arbitrary	ADJ
ejpam-4737	97	5	collection	collection	NOUN
ejpam-4737	97	6	of	of	ADP
ejpam-4737	97	7	δs(λ	δs(λ	NOUN
ejpam-4737	97	8	,	,	PUNCT
ejpam-4737	97	9	s)-closed	s)-close	VERB
ejpam-4737	97	10	sets	set	NOUN
ejpam-4737	97	11	in	in	ADP
ejpam-4737	97	12	(	(	PUNCT
ejpam-4737	97	13	x	x	NOUN
ejpam-4737	97	14	,	,	PUNCT
ejpam-4737	97	15	τ	τ	X
ejpam-4737	97	16	)	)	PUNCT
ejpam-4737	97	17	is	be	AUX
ejpam-4737	97	18	δs(λ	δs(λ	NOUN
ejpam-4737	97	19	,	,	PUNCT
ejpam-4737	97	20	p)-closed	p)-close	VERB
ejpam-4737	97	21	.	.	PUNCT
ejpam-4737	98	1	corollary	corollary	ADJ
ejpam-4737	98	2	1	1	NUM
ejpam-4737	98	3	.	.	PUNCT
ejpam-4737	99	1	let	let	VERB
ejpam-4737	99	2	a	a	DET
ejpam-4737	99	3	be	be	AUX
ejpam-4737	99	4	a	a	DET
ejpam-4737	99	5	subset	subset	NOUN
ejpam-4737	99	6	of	of	ADP
ejpam-4737	99	7	a	a	DET
ejpam-4737	99	8	topological	topological	ADJ
ejpam-4737	99	9	space	space	NOUN
ejpam-4737	99	10	(	(	PUNCT
ejpam-4737	99	11	x	x	X
ejpam-4737	99	12	,	,	PUNCT
ejpam-4737	99	13	τ	τ	PROPN
ejpam-4737	99	14	)	)	PUNCT
ejpam-4737	99	15	.	.	PUNCT
ejpam-4737	100	1	then	then	ADV
ejpam-4737	100	2	,	,	PUNCT
ejpam-4737	100	3	aδs(λ	aδs(λ	PROPN
ejpam-4737	100	4	,	,	PUNCT
ejpam-4737	100	5	p	p	NOUN
ejpam-4737	100	6	)	)	PUNCT
ejpam-4737	100	7	=	=	PUNCT
ejpam-4737	100	8	∩{f	∩{f	NOUN
ejpam-4737	100	9	∈	∈	PROPN
ejpam-4737	100	10	δs(λ	δs(λ	NOUN
ejpam-4737	100	11	,	,	PUNCT
ejpam-4737	100	12	p)c(x	p)c(x	NOUN
ejpam-4737	100	13	,	,	PUNCT
ejpam-4737	100	14	τ	τ	X
ejpam-4737	100	15	)	)	PUNCT
ejpam-4737	101	1	|	|	ADV
ejpam-4737	101	2	a	a	DET
ejpam-4737	101	3	⊆	⊆	NUM
ejpam-4737	101	4	f	f	NOUN
ejpam-4737	101	5	}	}	PUNCT
ejpam-4737	101	6	.	.	PUNCT
ejpam-4737	102	1	lemma	lemma	PROPN
ejpam-4737	102	2	2	2	NUM
ejpam-4737	102	3	.	.	X
ejpam-4737	102	4	for	for	ADP
ejpam-4737	102	5	the	the	DET
ejpam-4737	102	6	δs(λ	δs(λ	NOUN
ejpam-4737	102	7	,	,	PUNCT
ejpam-4737	102	8	p)-closure	p)-closure	NOUN
ejpam-4737	102	9	of	of	ADP
ejpam-4737	102	10	subsets	subset	NOUN
ejpam-4737	102	11	a	a	PRON
ejpam-4737	102	12	,	,	PUNCT
ejpam-4737	102	13	b	b	NOUN
ejpam-4737	102	14	in	in	ADP
ejpam-4737	102	15	a	a	DET
ejpam-4737	102	16	topological	topological	ADJ
ejpam-4737	102	17	space	space	NOUN
ejpam-4737	102	18	(	(	PUNCT
ejpam-4737	102	19	x	x	X
ejpam-4737	102	20	,	,	PUNCT
ejpam-4737	102	21	τ	τ	PROPN
ejpam-4737	102	22	)	)	PUNCT
ejpam-4737	102	23	,	,	PUNCT
ejpam-4737	102	24	the	the	DET
ejpam-4737	102	25	following	follow	VERB
ejpam-4737	102	26	properties	property	NOUN
ejpam-4737	102	27	hold	hold	VERB
ejpam-4737	102	28	:	:	PUNCT
ejpam-4737	102	29	(	(	PUNCT
ejpam-4737	102	30	1	1	X
ejpam-4737	102	31	)	)	PUNCT
ejpam-4737	102	32	a	a	PRON
ejpam-4737	102	33	is	be	AUX
ejpam-4737	102	34	δs(λ	δs(λ	NOUN
ejpam-4737	102	35	,	,	PUNCT
ejpam-4737	102	36	p)-closed	p)-close	VERB
ejpam-4737	102	37	in	in	ADP
ejpam-4737	102	38	(	(	PUNCT
ejpam-4737	102	39	x	x	X
ejpam-4737	102	40	,	,	PUNCT
ejpam-4737	102	41	τ	τ	X
ejpam-4737	102	42	)	)	PUNCT
ejpam-4737	103	1	if	if	SCONJ
ejpam-4737	103	2	and	and	CCONJ
ejpam-4737	103	3	only	only	ADV
ejpam-4737	103	4	if	if	SCONJ
ejpam-4737	103	5	a	a	DET
ejpam-4737	103	6	=	=	NOUN
ejpam-4737	103	7	aδs(λ	aδs(λ	PROPN
ejpam-4737	103	8	,	,	PUNCT
ejpam-4737	103	9	p	p	NOUN
ejpam-4737	103	10	)	)	PUNCT
ejpam-4737	103	11	.	.	PUNCT
ejpam-4737	104	1	(	(	PUNCT
ejpam-4737	104	2	2	2	X
ejpam-4737	104	3	)	)	PUNCT
ejpam-4737	104	4	if	if	SCONJ
ejpam-4737	104	5	a	a	DET
ejpam-4737	104	6	⊆	⊆	NUM
ejpam-4737	104	7	b	b	NOUN
ejpam-4737	104	8	,	,	PUNCT
ejpam-4737	104	9	then	then	ADV
ejpam-4737	104	10	aδs(λ	aδs(λ	PROPN
ejpam-4737	104	11	,	,	PUNCT
ejpam-4737	104	12	p	p	NOUN
ejpam-4737	104	13	)	)	PUNCT
ejpam-4737	104	14	⊆	⊆	NUM
ejpam-4737	104	15	bδs(λ	bδs(λ	PROPN
ejpam-4737	104	16	,	,	PUNCT
ejpam-4737	104	17	p	p	NOUN
ejpam-4737	104	18	)	)	PUNCT
ejpam-4737	104	19	.	.	PUNCT
ejpam-4737	105	1	(	(	PUNCT
ejpam-4737	105	2	3	3	X
ejpam-4737	105	3	)	)	PUNCT
ejpam-4737	105	4	aδs(λ	aδs(λ	PROPN
ejpam-4737	105	5	,	,	PUNCT
ejpam-4737	105	6	p	p	NOUN
ejpam-4737	105	7	)	)	PUNCT
ejpam-4737	105	8	is	be	AUX
ejpam-4737	105	9	δs(λ	δs(λ	NOUN
ejpam-4737	105	10	,	,	PUNCT
ejpam-4737	105	11	p)-closed	p)-close	VERB
ejpam-4737	105	12	,	,	PUNCT
ejpam-4737	105	13	that	that	ADV
ejpam-4737	105	14	is	is	ADV
ejpam-4737	105	15	,	,	PUNCT
ejpam-4737	105	16	aδs(λ	aδs(λ	PROPN
ejpam-4737	105	17	,	,	PUNCT
ejpam-4737	105	18	p	p	NOUN
ejpam-4737	105	19	)	)	PUNCT
ejpam-4737	105	20	=	=	PUNCT
ejpam-4737	106	1	[	[	X
ejpam-4737	106	2	aδs(λ	aδs(λ	PROPN
ejpam-4737	106	3	,	,	PUNCT
ejpam-4737	106	4	p)]δs(λ	p)]δs(λ	NOUN
ejpam-4737	106	5	,	,	PUNCT
ejpam-4737	106	6	p	p	NOUN
ejpam-4737	106	7	)	)	PUNCT
ejpam-4737	106	8	.	.	PUNCT
ejpam-4737	107	1	lemma	lemma	PROPN
ejpam-4737	107	2	3	3	X
ejpam-4737	107	3	.	.	PUNCT
ejpam-4737	108	1	for	for	ADP
ejpam-4737	108	2	a	a	DET
ejpam-4737	108	3	family	family	NOUN
ejpam-4737	108	4	{	{	PUNCT
ejpam-4737	108	5	aγ	aγ	INTJ
ejpam-4737	108	6	|	|	ADV
ejpam-4737	108	7	γ	γ	X
ejpam-4737	108	8	∈	∈	PROPN
ejpam-4737	108	9	∇	∇	X
ejpam-4737	108	10	}	}	PUNCT
ejpam-4737	108	11	of	of	ADP
ejpam-4737	108	12	a	a	DET
ejpam-4737	108	13	topological	topological	ADJ
ejpam-4737	108	14	space	space	NOUN
ejpam-4737	108	15	(	(	PUNCT
ejpam-4737	108	16	x	x	X
ejpam-4737	108	17	,	,	PUNCT
ejpam-4737	108	18	τ	τ	PROPN
ejpam-4737	108	19	)	)	PUNCT
ejpam-4737	108	20	,	,	PUNCT
ejpam-4737	108	21	the	the	DET
ejpam-4737	108	22	following	follow	VERB
ejpam-4737	108	23	properties	property	NOUN
ejpam-4737	108	24	hold	hold	VERB
ejpam-4737	108	25	:	:	PUNCT
ejpam-4737	108	26	(	(	PUNCT
ejpam-4737	108	27	1	1	X
ejpam-4737	108	28	)	)	PUNCT
ejpam-4737	109	1	[	[	X
ejpam-4737	109	2	∩{aγ	∩{aγ	VERB
ejpam-4737	109	3	|	|	ADV
ejpam-4737	109	4	γ	γ	X
ejpam-4737	109	5	∈	∈	PROPN
ejpam-4737	109	6	∇}]δs(λ	∇}]δs(λ	PROPN
ejpam-4737	109	7	,	,	PUNCT
ejpam-4737	109	8	p	p	NOUN
ejpam-4737	109	9	)	)	PUNCT
ejpam-4737	109	10	⊆	⊆	NUM
ejpam-4737	109	11	∩{aδs(λ	∩{aδs(λ	NOUN
ejpam-4737	109	12	,	,	PUNCT
ejpam-4737	109	13	p	p	NOUN
ejpam-4737	109	14	)	)	PUNCT
ejpam-4737	109	15	γ	γ	PROPN
ejpam-4737	109	16	|	|	ADV
ejpam-4737	109	17	γ	γ	X
ejpam-4737	109	18	∈	∈	NOUN
ejpam-4737	109	19	∇	∇	X
ejpam-4737	109	20	}	}	PUNCT
ejpam-4737	109	21	.	.	PUNCT
ejpam-4737	110	1	(	(	PUNCT
ejpam-4737	110	2	2	2	X
ejpam-4737	110	3	)	)	PUNCT
ejpam-4737	111	1	[	[	X
ejpam-4737	111	2	∪{aγ	∪{aγ	PROPN
ejpam-4737	111	3	|	|	ADV
ejpam-4737	111	4	γ	γ	PROPN
ejpam-4737	111	5	∈	∈	PROPN
ejpam-4737	111	6	∇}]δs(λ	∇}]δs(λ	PROPN
ejpam-4737	111	7	,	,	PUNCT
ejpam-4737	111	8	p	p	X
ejpam-4737	111	9	)	)	PUNCT
ejpam-4737	111	10	⊇	⊇	PROPN
ejpam-4737	111	11	∪{aδs(λ	∪{aδs(λ	PROPN
ejpam-4737	111	12	,	,	PUNCT
ejpam-4737	111	13	p	p	NOUN
ejpam-4737	111	14	)	)	PUNCT
ejpam-4737	111	15	γ	γ	PROPN
ejpam-4737	111	16	|	|	ADV
ejpam-4737	111	17	γ	γ	X
ejpam-4737	111	18	∈	∈	NOUN
ejpam-4737	111	19	∇	∇	X
ejpam-4737	111	20	}	}	PUNCT
ejpam-4737	111	21	.	.	PUNCT
ejpam-4737	112	1	c.	c.	PROPN
ejpam-4737	112	2	boonpok	boonpok	PROPN
ejpam-4737	112	3	,	,	PUNCT
ejpam-4737	112	4	m.	m.	NOUN
ejpam-4737	112	5	thongmoon	thongmoon	PROPN
ejpam-4737	112	6	/	/	SYM
ejpam-4737	112	7	eur	eur	PROPN
ejpam-4737	112	8	.	.	PUNCT
ejpam-4737	113	1	j.	j.	PROPN
ejpam-4737	113	2	pure	pure	PROPN
ejpam-4737	113	3	appl	appl	PROPN
ejpam-4737	113	4	.	.	PROPN
ejpam-4737	113	5	math	math	PROPN
ejpam-4737	113	6	,	,	PUNCT
ejpam-4737	113	7	16	16	NUM
ejpam-4737	113	8	(	(	PUNCT
ejpam-4737	113	9	3	3	NUM
ejpam-4737	113	10	)	)	PUNCT
ejpam-4737	113	11	(	(	PUNCT
ejpam-4737	113	12	2023	2023	NUM
ejpam-4737	113	13	)	)	PUNCT
ejpam-4737	113	14	,	,	PUNCT
ejpam-4737	113	15	1434	1434	NUM
ejpam-4737	113	16	-	-	SYM
ejpam-4737	113	17	1447	1447	NUM
ejpam-4737	113	18	1437	1437	NUM
ejpam-4737	113	19	definition	definition	NOUN
ejpam-4737	113	20	3	3	NUM
ejpam-4737	113	21	.	.	PUNCT
ejpam-4737	114	1	a	a	DET
ejpam-4737	114	2	subset	subset	NOUN
ejpam-4737	114	3	a	a	PRON
ejpam-4737	114	4	of	of	ADP
ejpam-4737	114	5	a	a	DET
ejpam-4737	114	6	topological	topological	ADJ
ejpam-4737	114	7	space	space	NOUN
ejpam-4737	114	8	(	(	PUNCT
ejpam-4737	114	9	x	x	X
ejpam-4737	114	10	,	,	PUNCT
ejpam-4737	114	11	τ	τ	X
ejpam-4737	114	12	)	)	PUNCT
ejpam-4737	114	13	is	be	AUX
ejpam-4737	114	14	called	call	VERB
ejpam-4737	114	15	s(λ	s(λ	PROPN
ejpam-4737	114	16	,	,	PUNCT
ejpam-4737	114	17	p)-regular	p)-regular	VERB
ejpam-4737	114	18	if	if	SCONJ
ejpam-4737	114	19	a	a	PRON
ejpam-4737	114	20	is	be	AUX
ejpam-4737	114	21	s(λ	s(λ	NOUN
ejpam-4737	114	22	,	,	PUNCT
ejpam-4737	114	23	p)-open	p)-open	PUNCT
ejpam-4737	114	24	and	and	CCONJ
ejpam-4737	114	25	s(λ	s(λ	NOUN
ejpam-4737	114	26	,	,	PUNCT
ejpam-4737	114	27	p)-closed	p)-close	VERB
ejpam-4737	114	28	.	.	PUNCT
ejpam-4737	115	1	the	the	DET
ejpam-4737	115	2	family	family	NOUN
ejpam-4737	115	3	of	of	ADP
ejpam-4737	115	4	all	all	DET
ejpam-4737	115	5	s(λ	s(λ	NOUN
ejpam-4737	115	6	,	,	PUNCT
ejpam-4737	115	7	p)-regular	p)-regular	ADJ
ejpam-4737	115	8	sets	set	NOUN
ejpam-4737	115	9	in	in	ADP
ejpam-4737	115	10	a	a	DET
ejpam-4737	115	11	topological	topological	ADJ
ejpam-4737	115	12	space	space	NOUN
ejpam-4737	115	13	(	(	PUNCT
ejpam-4737	115	14	x	x	X
ejpam-4737	115	15	,	,	PUNCT
ejpam-4737	115	16	τ	τ	X
ejpam-4737	115	17	)	)	PUNCT
ejpam-4737	115	18	is	be	AUX
ejpam-4737	115	19	denoted	denote	VERB
ejpam-4737	115	20	by	by	ADP
ejpam-4737	115	21	s(λ	s(λ	PROPN
ejpam-4737	115	22	,	,	PUNCT
ejpam-4737	115	23	p)r(x	p)r(x	PROPN
ejpam-4737	115	24	,	,	PUNCT
ejpam-4737	115	25	τ	τ	PROPN
ejpam-4737	115	26	)	)	PUNCT
ejpam-4737	115	27	.	.	PUNCT
ejpam-4737	116	1	lemma	lemma	PROPN
ejpam-4737	116	2	4	4	NUM
ejpam-4737	116	3	.	.	X
ejpam-4737	117	1	for	for	ADP
ejpam-4737	117	2	a	a	DET
ejpam-4737	117	3	subset	subset	NOUN
ejpam-4737	117	4	a	a	PRON
ejpam-4737	117	5	of	of	ADP
ejpam-4737	117	6	a	a	DET
ejpam-4737	117	7	topological	topological	ADJ
ejpam-4737	117	8	space	space	NOUN
ejpam-4737	117	9	(	(	PUNCT
ejpam-4737	117	10	x	x	X
ejpam-4737	117	11	,	,	PUNCT
ejpam-4737	117	12	τ	τ	PROPN
ejpam-4737	117	13	)	)	PUNCT
ejpam-4737	117	14	,	,	PUNCT
ejpam-4737	117	15	the	the	DET
ejpam-4737	117	16	following	follow	VERB
ejpam-4737	117	17	properties	property	NOUN
ejpam-4737	117	18	hold	hold	VERB
ejpam-4737	117	19	:	:	PUNCT
ejpam-4737	117	20	(	(	PUNCT
ejpam-4737	117	21	1	1	X
ejpam-4737	117	22	)	)	PUNCT
ejpam-4737	117	23	if	if	SCONJ
ejpam-4737	117	24	a	a	PRON
ejpam-4737	117	25	is	be	AUX
ejpam-4737	117	26	a	a	DET
ejpam-4737	117	27	s(λ	s(λ	PROPN
ejpam-4737	117	28	,	,	PUNCT
ejpam-4737	117	29	p)-regular	p)-regular	ADJ
ejpam-4737	117	30	set	set	NOUN
ejpam-4737	117	31	,	,	PUNCT
ejpam-4737	117	32	then	then	ADV
ejpam-4737	117	33	a	a	PRON
ejpam-4737	117	34	is	be	AUX
ejpam-4737	117	35	δs(λ	δs(λ	NOUN
ejpam-4737	117	36	,	,	PUNCT
ejpam-4737	117	37	p)-open	p)-open	ADJ
ejpam-4737	117	38	.	.	PUNCT
ejpam-4737	118	1	(	(	PUNCT
ejpam-4737	118	2	2	2	X
ejpam-4737	118	3	)	)	PUNCT
ejpam-4737	118	4	if	if	SCONJ
ejpam-4737	118	5	a	a	PRON
ejpam-4737	118	6	is	be	AUX
ejpam-4737	118	7	a	a	DET
ejpam-4737	118	8	δs(λ	δs(λ	NOUN
ejpam-4737	118	9	,	,	PUNCT
ejpam-4737	118	10	p)-open	p)-open	ADJ
ejpam-4737	118	11	set	set	VERB
ejpam-4737	118	12	,	,	PUNCT
ejpam-4737	118	13	then	then	ADV
ejpam-4737	118	14	a	a	PRON
ejpam-4737	118	15	is	be	AUX
ejpam-4737	118	16	s(λ	s(λ	NOUN
ejpam-4737	118	17	,	,	PUNCT
ejpam-4737	118	18	p)-open	p)-open	ADJ
ejpam-4737	118	19	.	.	PUNCT
ejpam-4737	119	1	(	(	PUNCT
ejpam-4737	119	2	3	3	X
ejpam-4737	119	3	)	)	PUNCT
ejpam-4737	119	4	if	if	SCONJ
ejpam-4737	119	5	a	a	PRON
ejpam-4737	119	6	is	be	AUX
ejpam-4737	119	7	a	a	DET
ejpam-4737	119	8	s(λ	s(λ	PROPN
ejpam-4737	119	9	,	,	PUNCT
ejpam-4737	119	10	p)-open	p)-open	ADJ
ejpam-4737	119	11	set	set	VERB
ejpam-4737	119	12	,	,	PUNCT
ejpam-4737	119	13	then	then	ADV
ejpam-4737	119	14	as(λ	as(λ	NUM
ejpam-4737	119	15	,	,	PUNCT
ejpam-4737	119	16	p	p	NOUN
ejpam-4737	119	17	)	)	PUNCT
ejpam-4737	119	18	is	be	AUX
ejpam-4737	119	19	s(λ	s(λ	PROPN
ejpam-4737	119	20	,	,	PUNCT
ejpam-4737	119	21	p)-regular	p)-regular	NOUN
ejpam-4737	119	22	.	.	PUNCT
ejpam-4737	120	1	definition	definition	NOUN
ejpam-4737	120	2	4	4	NUM
ejpam-4737	120	3	.	.	PUNCT
ejpam-4737	120	4	let	let	VERB
ejpam-4737	120	5	a	a	DET
ejpam-4737	120	6	be	be	AUX
ejpam-4737	120	7	a	a	DET
ejpam-4737	120	8	subset	subset	NOUN
ejpam-4737	120	9	of	of	ADP
ejpam-4737	120	10	a	a	DET
ejpam-4737	120	11	topological	topological	ADJ
ejpam-4737	120	12	space	space	NOUN
ejpam-4737	120	13	(	(	PUNCT
ejpam-4737	120	14	x	x	X
ejpam-4737	120	15	,	,	PUNCT
ejpam-4737	120	16	τ	τ	PROPN
ejpam-4737	120	17	)	)	PUNCT
ejpam-4737	120	18	.	.	PUNCT
ejpam-4737	121	1	a	a	DET
ejpam-4737	121	2	point	point	NOUN
ejpam-4737	121	3	x	x	PUNCT
ejpam-4737	121	4	of	of	ADP
ejpam-4737	121	5	x	x	PROPN
ejpam-4737	121	6	is	be	AUX
ejpam-4737	121	7	called	call	VERB
ejpam-4737	121	8	a	a	DET
ejpam-4737	121	9	θs(λ	θs(λ	NOUN
ejpam-4737	121	10	,	,	PUNCT
ejpam-4737	121	11	p)-cluster	p)-cluster	NOUN
ejpam-4737	121	12	point	point	NOUN
ejpam-4737	121	13	of	of	ADP
ejpam-4737	121	14	a	a	DET
ejpam-4737	121	15	if	if	SCONJ
ejpam-4737	121	16	a∩u	a∩u	PROPN
ejpam-4737	121	17	s(λ	s(λ	PROPN
ejpam-4737	121	18	,	,	PUNCT
ejpam-4737	121	19	p	p	NOUN
ejpam-4737	121	20	)	)	PUNCT
ejpam-4737	121	21	̸=	̸=	PROPN
ejpam-4737	121	22	∅	∅	NOUN
ejpam-4737	121	23	for	for	ADP
ejpam-4737	121	24	every	every	DET
ejpam-4737	121	25	s(λ	s(λ	PROPN
ejpam-4737	121	26	,	,	PUNCT
ejpam-4737	121	27	p)-open	p)-open	VERB
ejpam-4737	121	28	set	set	VERB
ejpam-4737	121	29	u	u	NOUN
ejpam-4737	121	30	of	of	ADP
ejpam-4737	121	31	x	x	SYM
ejpam-4737	121	32	containing	contain	VERB
ejpam-4737	121	33	x.	x.	NOUN
ejpam-4737	121	34	the	the	DET
ejpam-4737	121	35	set	set	NOUN
ejpam-4737	121	36	of	of	ADP
ejpam-4737	121	37	all	all	DET
ejpam-4737	121	38	θs(λ	θs(λ	NOUN
ejpam-4737	121	39	,	,	PUNCT
ejpam-4737	121	40	p)-cluster	p)-cluster	VERB
ejpam-4737	121	41	points	point	NOUN
ejpam-4737	121	42	of	of	ADP
ejpam-4737	121	43	a	a	PRON
ejpam-4737	121	44	is	be	AUX
ejpam-4737	121	45	called	call	VERB
ejpam-4737	121	46	the	the	DET
ejpam-4737	121	47	θs(λ	θs(λ	NOUN
ejpam-4737	121	48	,	,	PUNCT
ejpam-4737	121	49	p)-closure	p)-closure	NOUN
ejpam-4737	121	50	of	of	ADP
ejpam-4737	121	51	a	a	PRON
ejpam-4737	121	52	,	,	PUNCT
ejpam-4737	121	53	denoted	denote	VERB
ejpam-4737	121	54	by	by	ADP
ejpam-4737	121	55	aθs(λ	aθs(λ	PROPN
ejpam-4737	121	56	,	,	PUNCT
ejpam-4737	121	57	p	p	NOUN
ejpam-4737	121	58	)	)	PUNCT
ejpam-4737	121	59	.	.	PUNCT
ejpam-4737	122	1	a	a	DET
ejpam-4737	122	2	subset	subset	NOUN
ejpam-4737	122	3	a	a	PRON
ejpam-4737	122	4	of	of	ADP
ejpam-4737	122	5	a	a	DET
ejpam-4737	122	6	topological	topological	ADJ
ejpam-4737	122	7	space	space	NOUN
ejpam-4737	122	8	(	(	PUNCT
ejpam-4737	122	9	x	x	X
ejpam-4737	122	10	,	,	PUNCT
ejpam-4737	122	11	τ	τ	X
ejpam-4737	122	12	)	)	PUNCT
ejpam-4737	122	13	is	be	AUX
ejpam-4737	122	14	said	say	VERB
ejpam-4737	122	15	to	to	PART
ejpam-4737	122	16	be	be	AUX
ejpam-4737	122	17	θs(λ	θs(λ	NOUN
ejpam-4737	122	18	,	,	PUNCT
ejpam-4737	122	19	p)-closed	p)-close	VERB
ejpam-4737	122	20	if	if	SCONJ
ejpam-4737	122	21	a	a	DET
ejpam-4737	122	22	=	=	SYM
ejpam-4737	122	23	aθs(λ	aθs(λ	PROPN
ejpam-4737	122	24	,	,	PUNCT
ejpam-4737	122	25	p	p	NOUN
ejpam-4737	122	26	)	)	PUNCT
ejpam-4737	122	27	.	.	PUNCT
ejpam-4737	123	1	the	the	DET
ejpam-4737	123	2	complement	complement	NOUN
ejpam-4737	123	3	of	of	ADP
ejpam-4737	123	4	a	a	DET
ejpam-4737	123	5	θs(λ	θs(λ	NOUN
ejpam-4737	123	6	,	,	PUNCT
ejpam-4737	123	7	p)-closed	p)-close	VERB
ejpam-4737	123	8	set	set	NOUN
ejpam-4737	123	9	is	be	AUX
ejpam-4737	123	10	said	say	VERB
ejpam-4737	123	11	to	to	PART
ejpam-4737	123	12	be	be	AUX
ejpam-4737	123	13	θs(λ	θs(λ	NOUN
ejpam-4737	123	14	,	,	PUNCT
ejpam-4737	123	15	p)-open	p)-open	ADJ
ejpam-4737	123	16	.	.	PUNCT
ejpam-4737	124	1	lemma	lemma	PROPN
ejpam-4737	124	2	5	5	X
ejpam-4737	124	3	.	.	PUNCT
ejpam-4737	125	1	let	let	VERB
ejpam-4737	125	2	(	(	PUNCT
ejpam-4737	125	3	x	x	NOUN
ejpam-4737	125	4	,	,	PUNCT
ejpam-4737	125	5	τ	τ	X
ejpam-4737	125	6	)	)	PUNCT
ejpam-4737	125	7	be	be	VERB
ejpam-4737	125	8	a	a	DET
ejpam-4737	125	9	topological	topological	ADJ
ejpam-4737	125	10	space	space	NOUN
ejpam-4737	125	11	.	.	PUNCT
ejpam-4737	126	1	then	then	ADV
ejpam-4737	126	2	,	,	PUNCT
ejpam-4737	126	3	v	v	NOUN
ejpam-4737	126	4	θs(λ	θs(λ	NOUN
ejpam-4737	126	5	,	,	PUNCT
ejpam-4737	126	6	p	p	NOUN
ejpam-4737	126	7	)	)	PUNCT
ejpam-4737	126	8	=	=	NOUN
ejpam-4737	126	9	v	v	NOUN
ejpam-4737	126	10	δs(λ	δs(λ	NOUN
ejpam-4737	126	11	,	,	PUNCT
ejpam-4737	126	12	p	p	NOUN
ejpam-4737	126	13	)	)	PUNCT
ejpam-4737	126	14	=	=	SYM
ejpam-4737	126	15	v	v	ADP
ejpam-4737	126	16	s(λ	s(λ	PROPN
ejpam-4737	126	17	,	,	PUNCT
ejpam-4737	126	18	p	p	NOUN
ejpam-4737	126	19	)	)	PUNCT
ejpam-4737	126	20	for	for	ADP
ejpam-4737	126	21	each	each	DET
ejpam-4737	126	22	v	v	X
ejpam-4737	126	23	∈	∈	PROPN
ejpam-4737	126	24	s(λ	s(λ	PROPN
ejpam-4737	126	25	,	,	PUNCT
ejpam-4737	126	26	p)o(x	p)o(x	ADJ
ejpam-4737	126	27	,	,	PUNCT
ejpam-4737	126	28	τ	τ	PROPN
ejpam-4737	126	29	)	)	PUNCT
ejpam-4737	126	30	.	.	PUNCT
ejpam-4737	127	1	definition	definition	NOUN
ejpam-4737	127	2	5	5	NUM
ejpam-4737	127	3	.	.	PUNCT
ejpam-4737	128	1	a	a	DET
ejpam-4737	128	2	topological	topological	ADJ
ejpam-4737	128	3	space	space	NOUN
ejpam-4737	128	4	(	(	PUNCT
ejpam-4737	128	5	x	x	X
ejpam-4737	128	6	,	,	PUNCT
ejpam-4737	128	7	τ	τ	X
ejpam-4737	128	8	)	)	PUNCT
ejpam-4737	128	9	is	be	AUX
ejpam-4737	128	10	called	call	VERB
ejpam-4737	128	11	δs(λ	δs(λ	NOUN
ejpam-4737	128	12	,	,	PUNCT
ejpam-4737	128	13	p)-t0	p)-t0	ADJ
ejpam-4737	128	14	if	if	SCONJ
ejpam-4737	128	15	,	,	PUNCT
ejpam-4737	128	16	for	for	ADP
ejpam-4737	128	17	any	any	DET
ejpam-4737	128	18	distinct	distinct	ADJ
ejpam-4737	128	19	pair	pair	NOUN
ejpam-4737	128	20	of	of	ADP
ejpam-4737	128	21	points	point	NOUN
ejpam-4737	128	22	in	in	ADP
ejpam-4737	128	23	x	x	NOUN
ejpam-4737	128	24	,	,	PUNCT
ejpam-4737	128	25	there	there	PRON
ejpam-4737	128	26	exists	exist	VERB
ejpam-4737	128	27	a	a	DET
ejpam-4737	128	28	δs(λ	δs(λ	NOUN
ejpam-4737	128	29	,	,	PUNCT
ejpam-4737	128	30	p)-open	p)-open	VERB
ejpam-4737	128	31	set	set	VERB
ejpam-4737	128	32	containing	contain	VERB
ejpam-4737	128	33	one	one	NUM
ejpam-4737	128	34	of	of	ADP
ejpam-4737	128	35	the	the	DET
ejpam-4737	128	36	points	point	NOUN
ejpam-4737	128	37	but	but	CCONJ
ejpam-4737	128	38	not	not	PART
ejpam-4737	128	39	the	the	DET
ejpam-4737	128	40	other	other	ADJ
ejpam-4737	128	41	.	.	PUNCT
ejpam-4737	129	1	theorem	theorem	NOUN
ejpam-4737	129	2	1	1	NUM
ejpam-4737	129	3	.	.	PUNCT
ejpam-4737	130	1	a	a	DET
ejpam-4737	130	2	topological	topological	ADJ
ejpam-4737	130	3	space	space	NOUN
ejpam-4737	130	4	(	(	PUNCT
ejpam-4737	130	5	x	x	X
ejpam-4737	130	6	,	,	PUNCT
ejpam-4737	130	7	τ	τ	X
ejpam-4737	130	8	)	)	PUNCT
ejpam-4737	130	9	is	be	AUX
ejpam-4737	130	10	δs(λ	δs(λ	NOUN
ejpam-4737	130	11	,	,	PUNCT
ejpam-4737	130	12	p)-t0	p)-t0	NOUN
ejpam-4737	130	13	if	if	SCONJ
ejpam-4737	130	14	and	and	CCONJ
ejpam-4737	130	15	only	only	ADV
ejpam-4737	130	16	if	if	SCONJ
ejpam-4737	130	17	for	for	ADP
ejpam-4737	130	18	each	each	DET
ejpam-4737	130	19	point	point	NOUN
ejpam-4737	130	20	of	of	ADP
ejpam-4737	130	21	distinct	distinct	ADJ
ejpam-4737	130	22	points	point	NOUN
ejpam-4737	130	23	x	x	X
ejpam-4737	130	24	,	,	PUNCT
ejpam-4737	130	25	y	y	PROPN
ejpam-4737	130	26	of	of	ADP
ejpam-4737	130	27	x	x	PROPN
ejpam-4737	130	28	,	,	PUNCT
ejpam-4737	130	29	{	{	PUNCT
ejpam-4737	130	30	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	130	31	,	,	PUNCT
ejpam-4737	130	32	p	p	NOUN
ejpam-4737	130	33	)	)	PUNCT
ejpam-4737	130	34	̸=	̸=	PROPN
ejpam-4737	130	35	{	{	PUNCT
ejpam-4737	130	36	y}δs(λ	y}δs(λ	PROPN
ejpam-4737	130	37	,	,	PUNCT
ejpam-4737	130	38	p	p	NOUN
ejpam-4737	130	39	)	)	PUNCT
ejpam-4737	130	40	.	.	PUNCT
ejpam-4737	131	1	proof	proof	NOUN
ejpam-4737	131	2	.	.	PUNCT
ejpam-4737	132	1	suppose	suppose	VERB
ejpam-4737	132	2	that	that	SCONJ
ejpam-4737	132	3	x	x	NOUN
ejpam-4737	132	4	,	,	PUNCT
ejpam-4737	132	5	y	y	PROPN
ejpam-4737	132	6	∈	∈	PROPN
ejpam-4737	132	7	x	x	NOUN
ejpam-4737	132	8	,	,	PUNCT
ejpam-4737	132	9	x	x	PROPN
ejpam-4737	132	10	̸=	̸=	PROPN
ejpam-4737	132	11	y	y	PROPN
ejpam-4737	132	12	and	and	CCONJ
ejpam-4737	132	13	{	{	PUNCT
ejpam-4737	132	14	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	132	15	,	,	PUNCT
ejpam-4737	132	16	p	p	NOUN
ejpam-4737	132	17	)	)	PUNCT
ejpam-4737	132	18	̸=	̸=	PROPN
ejpam-4737	132	19	{	{	PUNCT
ejpam-4737	132	20	y}δs(λ	y}δs(λ	PROPN
ejpam-4737	132	21	,	,	PUNCT
ejpam-4737	132	22	p	p	NOUN
ejpam-4737	132	23	)	)	PUNCT
ejpam-4737	132	24	.	.	PUNCT
ejpam-4737	133	1	let	let	VERB
ejpam-4737	133	2	z	z	PRON
ejpam-4737	133	3	be	be	AUX
ejpam-4737	133	4	a	a	DET
ejpam-4737	133	5	point	point	NOUN
ejpam-4737	133	6	of	of	ADP
ejpam-4737	133	7	x	x	PUNCT
ejpam-4737	133	8	such	such	ADJ
ejpam-4737	133	9	that	that	SCONJ
ejpam-4737	133	10	z	z	PROPN
ejpam-4737	133	11	∈	∈	PROPN
ejpam-4737	133	12	{	{	PUNCT
ejpam-4737	133	13	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	133	14	,	,	PUNCT
ejpam-4737	133	15	p	p	NOUN
ejpam-4737	133	16	)	)	PUNCT
ejpam-4737	133	17	but	but	CCONJ
ejpam-4737	133	18	z	z	PROPN
ejpam-4737	133	19	̸∈	̸∈	PROPN
ejpam-4737	133	20	{	{	PUNCT
ejpam-4737	133	21	y}δs(λ	y}δs(λ	PROPN
ejpam-4737	133	22	,	,	PUNCT
ejpam-4737	133	23	p	p	NOUN
ejpam-4737	133	24	)	)	PUNCT
ejpam-4737	133	25	.	.	PUNCT
ejpam-4737	134	1	we	we	PRON
ejpam-4737	134	2	claim	claim	VERB
ejpam-4737	134	3	that	that	SCONJ
ejpam-4737	134	4	x	x	SYM
ejpam-4737	134	5	̸∈	̸∈	PROPN
ejpam-4737	134	6	{	{	PUNCT
ejpam-4737	134	7	y}δs(λ	y}δs(λ	PROPN
ejpam-4737	134	8	,	,	PUNCT
ejpam-4737	134	9	p	p	NOUN
ejpam-4737	134	10	)	)	PUNCT
ejpam-4737	134	11	.	.	PUNCT
ejpam-4737	135	1	for	for	ADP
ejpam-4737	135	2	,	,	PUNCT
ejpam-4737	135	3	if	if	SCONJ
ejpam-4737	135	4	x	x	SYM
ejpam-4737	135	5	∈	∈	PROPN
ejpam-4737	135	6	{	{	PUNCT
ejpam-4737	135	7	y}δs(λ	y}δs(λ	NOUN
ejpam-4737	135	8	,	,	PUNCT
ejpam-4737	135	9	p	p	NOUN
ejpam-4737	135	10	)	)	PUNCT
ejpam-4737	135	11	,	,	PUNCT
ejpam-4737	135	12	then	then	ADV
ejpam-4737	135	13	{	{	PUNCT
ejpam-4737	135	14	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	135	15	,	,	PUNCT
ejpam-4737	135	16	p	p	NOUN
ejpam-4737	135	17	)	)	PUNCT
ejpam-4737	135	18	⊆	⊆	NUM
ejpam-4737	135	19	{	{	PUNCT
ejpam-4737	135	20	y}δs(λ	y}δs(λ	NOUN
ejpam-4737	135	21	,	,	PUNCT
ejpam-4737	135	22	p	p	NOUN
ejpam-4737	135	23	)	)	PUNCT
ejpam-4737	135	24	and	and	CCONJ
ejpam-4737	135	25	this	this	PRON
ejpam-4737	135	26	contradicts	contradict	VERB
ejpam-4737	135	27	the	the	DET
ejpam-4737	135	28	fact	fact	NOUN
ejpam-4737	135	29	that	that	SCONJ
ejpam-4737	135	30	z	z	PROPN
ejpam-4737	135	31	̸∈	̸∈	PROPN
ejpam-4737	135	32	{	{	PUNCT
ejpam-4737	135	33	y}δs(λ	y}δs(λ	PROPN
ejpam-4737	135	34	,	,	PUNCT
ejpam-4737	135	35	p	p	NOUN
ejpam-4737	135	36	)	)	PUNCT
ejpam-4737	135	37	.	.	PUNCT
ejpam-4737	136	1	thus	thus	ADV
ejpam-4737	136	2	,	,	PUNCT
ejpam-4737	136	3	x	x	PRON
ejpam-4737	136	4	belongs	belong	VERB
ejpam-4737	136	5	to	to	ADP
ejpam-4737	136	6	the	the	DET
ejpam-4737	136	7	δs(λ	δs(λ	NOUN
ejpam-4737	136	8	,	,	PUNCT
ejpam-4737	136	9	p)-open	p)-open	VERB
ejpam-4737	136	10	set	set	VERB
ejpam-4737	136	11	x	x	PUNCT
ejpam-4737	136	12	−	−	PROPN
ejpam-4737	136	13	{	{	PUNCT
ejpam-4737	136	14	y}δs(λ	y}δs(λ	NOUN
ejpam-4737	136	15	,	,	PUNCT
ejpam-4737	136	16	p	p	NOUN
ejpam-4737	136	17	)	)	PUNCT
ejpam-4737	136	18	to	to	PART
ejpam-4737	136	19	which	which	PRON
ejpam-4737	136	20	y	y	PROPN
ejpam-4737	136	21	does	do	AUX
ejpam-4737	136	22	not	not	PART
ejpam-4737	136	23	belong	belong	VERB
ejpam-4737	136	24	.	.	PUNCT
ejpam-4737	137	1	conversely	conversely	ADV
ejpam-4737	137	2	,	,	PUNCT
ejpam-4737	137	3	let	let	VERB
ejpam-4737	137	4	(	(	PUNCT
ejpam-4737	137	5	x	x	NOUN
ejpam-4737	137	6	,	,	PUNCT
ejpam-4737	137	7	τ	τ	X
ejpam-4737	137	8	)	)	PUNCT
ejpam-4737	137	9	be	be	VERB
ejpam-4737	137	10	a	a	DET
ejpam-4737	137	11	δs(λ	δs(λ	NOUN
ejpam-4737	137	12	,	,	PUNCT
ejpam-4737	137	13	p)-t0	p)-t0	NOUN
ejpam-4737	137	14	space	space	NOUN
ejpam-4737	137	15	and	and	CCONJ
ejpam-4737	137	16	x	x	NOUN
ejpam-4737	137	17	,	,	PUNCT
ejpam-4737	137	18	y	y	PROPN
ejpam-4737	137	19	be	be	VERB
ejpam-4737	137	20	any	any	DET
ejpam-4737	137	21	two	two	NUM
ejpam-4737	137	22	distinct	distinct	ADJ
ejpam-4737	137	23	points	point	NOUN
ejpam-4737	137	24	of	of	ADP
ejpam-4737	137	25	x.	x.	NOUN
ejpam-4737	137	26	then	then	ADV
ejpam-4737	137	27	,	,	PUNCT
ejpam-4737	137	28	there	there	PRON
ejpam-4737	137	29	exists	exist	VERB
ejpam-4737	137	30	a	a	DET
ejpam-4737	137	31	δs(λ	δs(λ	NOUN
ejpam-4737	137	32	,	,	PUNCT
ejpam-4737	137	33	p)-open	p)-open	VERB
ejpam-4737	137	34	set	set	VERB
ejpam-4737	137	35	u	u	NOUN
ejpam-4737	137	36	containing	contain	VERB
ejpam-4737	137	37	x	x	PUNCT
ejpam-4737	137	38	or	or	CCONJ
ejpam-4737	137	39	y	y	PROPN
ejpam-4737	137	40	,	,	PUNCT
ejpam-4737	137	41	say	say	VERB
ejpam-4737	137	42	x	x	PUNCT
ejpam-4737	137	43	but	but	CCONJ
ejpam-4737	137	44	not	not	PART
ejpam-4737	137	45	y.	y.	NOUN
ejpam-4737	137	46	then	then	ADV
ejpam-4737	137	47	,	,	PUNCT
ejpam-4737	137	48	x	x	PRON
ejpam-4737	137	49	−u	−u	PROPN
ejpam-4737	137	50	is	be	AUX
ejpam-4737	137	51	a	a	DET
ejpam-4737	137	52	δs(λ	δs(λ	NOUN
ejpam-4737	137	53	,	,	PUNCT
ejpam-4737	137	54	p)-closed	p)-close	VERB
ejpam-4737	137	55	set	set	NOUN
ejpam-4737	137	56	which	which	PRON
ejpam-4737	137	57	does	do	AUX
ejpam-4737	137	58	not	not	PART
ejpam-4737	137	59	contain	contain	VERB
ejpam-4737	137	60	x	x	PUNCT
ejpam-4737	137	61	but	but	CCONJ
ejpam-4737	137	62	contains	contain	VERB
ejpam-4737	137	63	y.	y.	PROPN
ejpam-4737	137	64	thus	thus	ADV
ejpam-4737	137	65	,	,	PUNCT
ejpam-4737	137	66	{	{	PUNCT
ejpam-4737	137	67	y}δs(λ	y}δs(λ	NOUN
ejpam-4737	137	68	,	,	PUNCT
ejpam-4737	137	69	p	p	NOUN
ejpam-4737	137	70	)	)	PUNCT
ejpam-4737	137	71	⊆	⊆	NUM
ejpam-4737	137	72	x	x	SYM
ejpam-4737	137	73	−u	−u	NOUN
ejpam-4737	137	74	and	and	CCONJ
ejpam-4737	137	75	hence	hence	ADV
ejpam-4737	137	76	x	x	X
ejpam-4737	137	77	̸∈	̸∈	PROPN
ejpam-4737	137	78	{	{	PUNCT
ejpam-4737	137	79	y}δs(λ	y}δs(λ	PROPN
ejpam-4737	137	80	,	,	PUNCT
ejpam-4737	137	81	p	p	NOUN
ejpam-4737	137	82	)	)	PUNCT
ejpam-4737	137	83	.	.	PUNCT
ejpam-4737	138	1	this	this	PRON
ejpam-4737	138	2	shows	show	VERB
ejpam-4737	138	3	that	that	SCONJ
ejpam-4737	138	4	{	{	PUNCT
ejpam-4737	138	5	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	138	6	,	,	PUNCT
ejpam-4737	138	7	p	p	NOUN
ejpam-4737	138	8	)	)	PUNCT
ejpam-4737	138	9	̸=	̸=	PROPN
ejpam-4737	138	10	{	{	PUNCT
ejpam-4737	138	11	y}δs(λ	y}δs(λ	PROPN
ejpam-4737	138	12	,	,	PUNCT
ejpam-4737	138	13	p	p	NOUN
ejpam-4737	138	14	)	)	PUNCT
ejpam-4737	138	15	.	.	PUNCT
ejpam-4737	139	1	definition	definition	NOUN
ejpam-4737	139	2	6	6	NUM
ejpam-4737	139	3	.	.	PUNCT
ejpam-4737	140	1	a	a	DET
ejpam-4737	140	2	topological	topological	ADJ
ejpam-4737	140	3	space	space	NOUN
ejpam-4737	140	4	(	(	PUNCT
ejpam-4737	140	5	x	x	X
ejpam-4737	140	6	,	,	PUNCT
ejpam-4737	140	7	τ	τ	X
ejpam-4737	140	8	)	)	PUNCT
ejpam-4737	140	9	is	be	AUX
ejpam-4737	140	10	called	call	VERB
ejpam-4737	140	11	δs(λ	δs(λ	NOUN
ejpam-4737	140	12	,	,	PUNCT
ejpam-4737	140	13	p)-t1	p)-t1	VERB
ejpam-4737	140	14	if	if	SCONJ
ejpam-4737	140	15	,	,	PUNCT
ejpam-4737	140	16	for	for	ADP
ejpam-4737	140	17	any	any	DET
ejpam-4737	140	18	distinct	distinct	ADJ
ejpam-4737	140	19	pair	pair	NOUN
ejpam-4737	140	20	of	of	ADP
ejpam-4737	140	21	points	point	NOUN
ejpam-4737	140	22	x	x	PUNCT
ejpam-4737	140	23	and	and	CCONJ
ejpam-4737	140	24	y	y	PROPN
ejpam-4737	140	25	in	in	ADP
ejpam-4737	140	26	x	x	SYM
ejpam-4737	140	27	,	,	PUNCT
ejpam-4737	140	28	there	there	PRON
ejpam-4737	140	29	exist	exist	VERB
ejpam-4737	140	30	a	a	DET
ejpam-4737	140	31	δs(λ	δs(λ	NOUN
ejpam-4737	140	32	,	,	PUNCT
ejpam-4737	140	33	p)-open	p)-open	VERB
ejpam-4737	140	34	set	set	VERB
ejpam-4737	140	35	u	u	NOUN
ejpam-4737	140	36	of	of	ADP
ejpam-4737	140	37	x	x	PUNCT
ejpam-4737	140	38	containing	contain	VERB
ejpam-4737	140	39	x	x	NOUN
ejpam-4737	140	40	but	but	CCONJ
ejpam-4737	140	41	not	not	PART
ejpam-4737	140	42	y	y	PROPN
ejpam-4737	140	43	and	and	CCONJ
ejpam-4737	140	44	a	a	DET
ejpam-4737	140	45	δs(λ	δs(λ	NOUN
ejpam-4737	140	46	,	,	PUNCT
ejpam-4737	140	47	p)-open	p)-open	VERB
ejpam-4737	140	48	set	set	VERB
ejpam-4737	140	49	v	v	NOUN
ejpam-4737	140	50	of	of	ADP
ejpam-4737	140	51	x	x	PUNCT
ejpam-4737	140	52	containing	contain	VERB
ejpam-4737	140	53	y	y	NOUN
ejpam-4737	140	54	but	but	CCONJ
ejpam-4737	140	55	not	not	PART
ejpam-4737	140	56	x.	x.	NOUN
ejpam-4737	140	57	theorem	theorem	VERB
ejpam-4737	140	58	2	2	NUM
ejpam-4737	140	59	.	.	PUNCT
ejpam-4737	141	1	a	a	DET
ejpam-4737	141	2	topological	topological	ADJ
ejpam-4737	141	3	space	space	NOUN
ejpam-4737	141	4	(	(	PUNCT
ejpam-4737	141	5	x	x	X
ejpam-4737	141	6	,	,	PUNCT
ejpam-4737	141	7	τ	τ	X
ejpam-4737	141	8	)	)	PUNCT
ejpam-4737	141	9	is	be	AUX
ejpam-4737	141	10	δs(λ	δs(λ	NOUN
ejpam-4737	141	11	,	,	PUNCT
ejpam-4737	141	12	p)-t1	p)-t1	VERB
ejpam-4737	141	13	if	if	SCONJ
ejpam-4737	141	14	and	and	CCONJ
ejpam-4737	141	15	only	only	ADV
ejpam-4737	141	16	if	if	SCONJ
ejpam-4737	141	17	the	the	DET
ejpam-4737	141	18	singletons	singleton	NOUN
ejpam-4737	141	19	are	be	AUX
ejpam-4737	141	20	δs(λ	δs(λ	NOUN
ejpam-4737	141	21	,	,	PUNCT
ejpam-4737	141	22	p)-closed	p)-close	VERB
ejpam-4737	141	23	sets	set	NOUN
ejpam-4737	141	24	.	.	PUNCT
ejpam-4737	142	1	c.	c.	PROPN
ejpam-4737	142	2	boonpok	boonpok	PROPN
ejpam-4737	142	3	,	,	PUNCT
ejpam-4737	142	4	m.	m.	NOUN
ejpam-4737	142	5	thongmoon	thongmoon	PROPN
ejpam-4737	142	6	/	/	SYM
ejpam-4737	142	7	eur	eur	PROPN
ejpam-4737	142	8	.	.	PUNCT
ejpam-4737	143	1	j.	j.	PROPN
ejpam-4737	143	2	pure	pure	PROPN
ejpam-4737	143	3	appl	appl	PROPN
ejpam-4737	143	4	.	.	PROPN
ejpam-4737	143	5	math	math	PROPN
ejpam-4737	143	6	,	,	PUNCT
ejpam-4737	143	7	16	16	NUM
ejpam-4737	143	8	(	(	PUNCT
ejpam-4737	143	9	3	3	NUM
ejpam-4737	143	10	)	)	PUNCT
ejpam-4737	143	11	(	(	PUNCT
ejpam-4737	143	12	2023	2023	NUM
ejpam-4737	143	13	)	)	PUNCT
ejpam-4737	143	14	,	,	PUNCT
ejpam-4737	143	15	1434	1434	NUM
ejpam-4737	143	16	-	-	SYM
ejpam-4737	143	17	1447	1447	NUM
ejpam-4737	143	18	1438	1438	NUM
ejpam-4737	143	19	proof	proof	NOUN
ejpam-4737	143	20	.	.	PUNCT
ejpam-4737	143	21	suppose	suppose	VERB
ejpam-4737	143	22	that	that	SCONJ
ejpam-4737	143	23	(	(	PUNCT
ejpam-4737	143	24	x	x	X
ejpam-4737	143	25	,	,	PUNCT
ejpam-4737	143	26	τ	τ	X
ejpam-4737	143	27	)	)	PUNCT
ejpam-4737	143	28	is	be	AUX
ejpam-4737	143	29	δs(λ	δs(λ	NOUN
ejpam-4737	143	30	,	,	PUNCT
ejpam-4737	143	31	p)-t1	p)-t1	NOUN
ejpam-4737	143	32	and	and	CCONJ
ejpam-4737	143	33	x	x	AUX
ejpam-4737	143	34	be	be	AUX
ejpam-4737	143	35	any	any	DET
ejpam-4737	143	36	point	point	NOUN
ejpam-4737	143	37	of	of	ADP
ejpam-4737	143	38	x.	x.	NOUN
ejpam-4737	143	39	let	let	VERB
ejpam-4737	143	40	y	y	PROPN
ejpam-4737	143	41	∈	∈	PROPN
ejpam-4737	143	42	x	x	PUNCT
ejpam-4737	143	43	−	−	PROPN
ejpam-4737	143	44	{	{	PUNCT
ejpam-4737	143	45	x	x	NOUN
ejpam-4737	143	46	}	}	PUNCT
ejpam-4737	143	47	.	.	PUNCT
ejpam-4737	144	1	then	then	ADV
ejpam-4737	144	2	,	,	PUNCT
ejpam-4737	144	3	x	x	PROPN
ejpam-4737	144	4	̸=	̸=	PROPN
ejpam-4737	144	5	y	y	PROPN
ejpam-4737	145	1	and	and	CCONJ
ejpam-4737	145	2	so	so	ADV
ejpam-4737	145	3	there	there	PRON
ejpam-4737	145	4	exists	exist	VERB
ejpam-4737	145	5	a	a	DET
ejpam-4737	145	6	δs(λ	δs(λ	NOUN
ejpam-4737	145	7	,	,	PUNCT
ejpam-4737	145	8	p)-open	p)-open	VERB
ejpam-4737	145	9	set	set	VERB
ejpam-4737	145	10	vy	vy	ADP
ejpam-4737	145	11	such	such	ADJ
ejpam-4737	145	12	that	that	SCONJ
ejpam-4737	145	13	y	y	PROPN
ejpam-4737	145	14	∈	∈	PROPN
ejpam-4737	145	15	vy	vy	NOUN
ejpam-4737	145	16	but	but	CCONJ
ejpam-4737	145	17	x	x	PROPN
ejpam-4737	145	18	̸∈	̸∈	PROPN
ejpam-4737	145	19	vy	vy	PROPN
ejpam-4737	145	20	.	.	PUNCT
ejpam-4737	145	21	therefore	therefore	ADV
ejpam-4737	145	22	,	,	PUNCT
ejpam-4737	145	23	y	y	PROPN
ejpam-4737	145	24	∈	∈	PROPN
ejpam-4737	145	25	vy	vy	VERB
ejpam-4737	145	26	⊆	⊆	NUM
ejpam-4737	145	27	x	x	X
ejpam-4737	146	1	−	−	PROPN
ejpam-4737	146	2	{	{	PUNCT
ejpam-4737	146	3	x	x	NOUN
ejpam-4737	146	4	}	}	PUNCT
ejpam-4737	146	5	.	.	PUNCT
ejpam-4737	147	1	thus	thus	ADV
ejpam-4737	147	2	,	,	PUNCT
ejpam-4737	147	3	x	x	PUNCT
ejpam-4737	147	4	−	−	X
ejpam-4737	147	5	{	{	PUNCT
ejpam-4737	147	6	x	x	NOUN
ejpam-4737	147	7	}	}	PUNCT
ejpam-4737	147	8	=	=	SYM
ejpam-4737	147	9	∪{vy	∪{vy	PROPN
ejpam-4737	147	10	|	|	CCONJ
ejpam-4737	147	11	y	y	PROPN
ejpam-4737	147	12	∈	∈	PROPN
ejpam-4737	147	13	(	(	PUNCT
ejpam-4737	147	14	x	x	PART
ejpam-4737	147	15	−	−	PROPN
ejpam-4737	147	16	{	{	PUNCT
ejpam-4737	147	17	x	x	NOUN
ejpam-4737	147	18	}	}	PUNCT
ejpam-4737	147	19	)	)	PUNCT
ejpam-4737	147	20	}	}	PUNCT
ejpam-4737	147	21	which	which	PRON
ejpam-4737	147	22	is	be	AUX
ejpam-4737	147	23	δs(λ	δs(λ	NOUN
ejpam-4737	147	24	,	,	PUNCT
ejpam-4737	147	25	p)-open	p)-open	NOUN
ejpam-4737	147	26	.	.	PUNCT
ejpam-4737	148	1	conversely	conversely	ADV
ejpam-4737	148	2	,	,	PUNCT
ejpam-4737	148	3	suppose	suppose	VERB
ejpam-4737	148	4	that	that	SCONJ
ejpam-4737	148	5	{	{	PUNCT
ejpam-4737	148	6	z	z	NOUN
ejpam-4737	148	7	}	}	PUNCT
ejpam-4737	148	8	is	be	AUX
ejpam-4737	148	9	δs(λ	δs(λ	NOUN
ejpam-4737	148	10	,	,	PUNCT
ejpam-4737	148	11	p)-closed	p)-close	VERB
ejpam-4737	148	12	for	for	ADP
ejpam-4737	148	13	each	each	DET
ejpam-4737	148	14	z	z	NOUN
ejpam-4737	148	15	∈	∈	PROPN
ejpam-4737	148	16	x.	x.	NOUN
ejpam-4737	148	17	let	let	VERB
ejpam-4737	148	18	x	x	PRON
ejpam-4737	148	19	,	,	PUNCT
ejpam-4737	148	20	y	y	PROPN
ejpam-4737	148	21	∈	∈	PROPN
ejpam-4737	148	22	x	x	PUNCT
ejpam-4737	148	23	with	with	ADP
ejpam-4737	148	24	x	x	PUNCT
ejpam-4737	148	25	̸=	̸=	PROPN
ejpam-4737	148	26	y.	y.	NOUN
ejpam-4737	148	27	now	now	ADV
ejpam-4737	148	28	x	x	X
ejpam-4737	148	29	̸=	̸=	PROPN
ejpam-4737	148	30	y	y	PROPN
ejpam-4737	148	31	implies	imply	VERB
ejpam-4737	148	32	y	y	PROPN
ejpam-4737	148	33	∈	∈	PROPN
ejpam-4737	148	34	x	x	PUNCT
ejpam-4737	148	35	−	−	PROPN
ejpam-4737	148	36	{	{	PUNCT
ejpam-4737	148	37	x	x	NOUN
ejpam-4737	148	38	}	}	PUNCT
ejpam-4737	148	39	.	.	PUNCT
ejpam-4737	149	1	thus	thus	ADV
ejpam-4737	149	2	,	,	PUNCT
ejpam-4737	149	3	x	x	PUNCT
ejpam-4737	149	4	−	−	X
ejpam-4737	149	5	{	{	PUNCT
ejpam-4737	149	6	x	x	NOUN
ejpam-4737	149	7	}	}	PUNCT
ejpam-4737	149	8	is	be	AUX
ejpam-4737	149	9	a	a	DET
ejpam-4737	149	10	δs(λ	δs(λ	NOUN
ejpam-4737	149	11	,	,	PUNCT
ejpam-4737	149	12	p)-open	p)-open	VERB
ejpam-4737	149	13	set	set	VERB
ejpam-4737	149	14	containing	contain	VERB
ejpam-4737	149	15	y	y	PROPN
ejpam-4737	149	16	but	but	CCONJ
ejpam-4737	149	17	not	not	PART
ejpam-4737	149	18	containing	contain	VERB
ejpam-4737	149	19	x.	x.	NOUN
ejpam-4737	149	20	similarly	similarly	ADV
ejpam-4737	149	21	,	,	PUNCT
ejpam-4737	149	22	x	x	PUNCT
ejpam-4737	149	23	−	−	PROPN
ejpam-4737	149	24	{	{	PUNCT
ejpam-4737	149	25	y	y	NOUN
ejpam-4737	149	26	}	}	PUNCT
ejpam-4737	149	27	is	be	AUX
ejpam-4737	149	28	a	a	DET
ejpam-4737	149	29	δs(λ	δs(λ	NOUN
ejpam-4737	149	30	,	,	PUNCT
ejpam-4737	149	31	p)-open	p)-open	VERB
ejpam-4737	149	32	set	set	VERB
ejpam-4737	149	33	containing	contain	VERB
ejpam-4737	149	34	x	x	PUNCT
ejpam-4737	149	35	but	but	CCONJ
ejpam-4737	149	36	not	not	PART
ejpam-4737	149	37	containing	contain	VERB
ejpam-4737	149	38	y.	y.	NOUN
ejpam-4737	149	39	this	this	PRON
ejpam-4737	149	40	shows	show	VERB
ejpam-4737	149	41	that	that	SCONJ
ejpam-4737	149	42	(	(	PUNCT
ejpam-4737	149	43	x	x	X
ejpam-4737	149	44	,	,	PUNCT
ejpam-4737	149	45	τ	τ	X
ejpam-4737	149	46	)	)	PUNCT
ejpam-4737	149	47	is	be	AUX
ejpam-4737	149	48	a	a	DET
ejpam-4737	149	49	δs(λ	δs(λ	NOUN
ejpam-4737	149	50	,	,	PUNCT
ejpam-4737	149	51	p)-t1	p)-t1	VERB
ejpam-4737	149	52	space	space	NOUN
ejpam-4737	149	53	.	.	PUNCT
ejpam-4737	150	1	definition	definition	NOUN
ejpam-4737	150	2	7	7	NUM
ejpam-4737	150	3	.	.	PUNCT
ejpam-4737	151	1	a	a	DET
ejpam-4737	151	2	topological	topological	ADJ
ejpam-4737	151	3	space	space	NOUN
ejpam-4737	151	4	(	(	PUNCT
ejpam-4737	151	5	x	x	X
ejpam-4737	151	6	,	,	PUNCT
ejpam-4737	151	7	τ	τ	X
ejpam-4737	151	8	)	)	PUNCT
ejpam-4737	151	9	is	be	AUX
ejpam-4737	151	10	called	call	VERB
ejpam-4737	151	11	δs(λ	δs(λ	NOUN
ejpam-4737	151	12	,	,	PUNCT
ejpam-4737	151	13	p)-symmetric	p)-symmetric	ADJ
ejpam-4737	151	14	if	if	SCONJ
ejpam-4737	151	15	,	,	PUNCT
ejpam-4737	151	16	for	for	ADP
ejpam-4737	151	17	each	each	DET
ejpam-4737	151	18	x	x	X
ejpam-4737	151	19	and	and	CCONJ
ejpam-4737	151	20	y	y	PROPN
ejpam-4737	151	21	in	in	ADP
ejpam-4737	151	22	x	x	PRON
ejpam-4737	151	23	,	,	PUNCT
ejpam-4737	151	24	x	x	SYM
ejpam-4737	151	25	∈	∈	PROPN
ejpam-4737	151	26	{	{	PUNCT
ejpam-4737	151	27	y}δs(λ	y}δs(λ	NOUN
ejpam-4737	151	28	,	,	PUNCT
ejpam-4737	151	29	p	p	NOUN
ejpam-4737	151	30	)	)	PUNCT
ejpam-4737	151	31	implies	imply	VERB
ejpam-4737	151	32	y	y	PROPN
ejpam-4737	151	33	∈	∈	PROPN
ejpam-4737	151	34	{	{	PUNCT
ejpam-4737	151	35	y}δs(λ	y}δs(λ	NOUN
ejpam-4737	151	36	,	,	PUNCT
ejpam-4737	151	37	p	p	NOUN
ejpam-4737	151	38	)	)	PUNCT
ejpam-4737	151	39	.	.	PUNCT
ejpam-4737	152	1	lemma	lemma	PROPN
ejpam-4737	152	2	6	6	NUM
ejpam-4737	152	3	.	.	PUNCT
ejpam-4737	153	1	let	let	VERB
ejpam-4737	153	2	(	(	PUNCT
ejpam-4737	153	3	x	x	NOUN
ejpam-4737	153	4	,	,	PUNCT
ejpam-4737	153	5	τ	τ	X
ejpam-4737	153	6	)	)	PUNCT
ejpam-4737	153	7	be	be	VERB
ejpam-4737	153	8	a	a	DET
ejpam-4737	153	9	topological	topological	ADJ
ejpam-4737	153	10	space	space	NOUN
ejpam-4737	153	11	.	.	PUNCT
ejpam-4737	154	1	for	for	ADP
ejpam-4737	154	2	each	each	DET
ejpam-4737	154	3	point	point	NOUN
ejpam-4737	154	4	x	x	X
ejpam-4737	154	5	∈	∈	NOUN
ejpam-4737	154	6	x	x	X
ejpam-4737	154	7	,	,	PUNCT
ejpam-4737	154	8	{	{	PUNCT
ejpam-4737	154	9	x	x	NOUN
ejpam-4737	154	10	}	}	PUNCT
ejpam-4737	154	11	is	be	AUX
ejpam-4737	154	12	s(λ	s(λ	PROPN
ejpam-4737	154	13	,	,	PUNCT
ejpam-4737	154	14	p)-open	p)-open	ADJ
ejpam-4737	154	15	or	or	CCONJ
ejpam-4737	154	16	s(λ	s(λ	NOUN
ejpam-4737	154	17	,	,	PUNCT
ejpam-4737	154	18	p)-closed	p)-close	VERB
ejpam-4737	154	19	.	.	PUNCT
ejpam-4737	155	1	theorem	theorem	NOUN
ejpam-4737	155	2	3	3	NUM
ejpam-4737	155	3	.	.	X
ejpam-4737	155	4	for	for	ADP
ejpam-4737	155	5	a	a	DET
ejpam-4737	155	6	topological	topological	ADJ
ejpam-4737	155	7	space	space	NOUN
ejpam-4737	155	8	(	(	PUNCT
ejpam-4737	155	9	x	x	X
ejpam-4737	155	10	,	,	PUNCT
ejpam-4737	155	11	τ	τ	PROPN
ejpam-4737	155	12	)	)	PUNCT
ejpam-4737	155	13	,	,	PUNCT
ejpam-4737	155	14	the	the	DET
ejpam-4737	155	15	following	follow	VERB
ejpam-4737	155	16	properties	property	NOUN
ejpam-4737	155	17	are	be	AUX
ejpam-4737	155	18	equivalent	equivalent	ADJ
ejpam-4737	155	19	:	:	PUNCT
ejpam-4737	155	20	(	(	PUNCT
ejpam-4737	155	21	1	1	X
ejpam-4737	155	22	)	)	PUNCT
ejpam-4737	155	23	(	(	PUNCT
ejpam-4737	155	24	x	x	X
ejpam-4737	155	25	,	,	PUNCT
ejpam-4737	155	26	τ	τ	X
ejpam-4737	155	27	)	)	PUNCT
ejpam-4737	155	28	is	be	AUX
ejpam-4737	155	29	δs(λ	δs(λ	NOUN
ejpam-4737	155	30	,	,	PUNCT
ejpam-4737	155	31	p)-symmetric	p)-symmetric	NOUN
ejpam-4737	155	32	.	.	PUNCT
ejpam-4737	156	1	(	(	PUNCT
ejpam-4737	156	2	2	2	X
ejpam-4737	156	3	)	)	PUNCT
ejpam-4737	156	4	for	for	ADP
ejpam-4737	156	5	each	each	DET
ejpam-4737	156	6	x	x	SYM
ejpam-4737	156	7	∈	∈	PROPN
ejpam-4737	156	8	x	x	X
ejpam-4737	156	9	,	,	PUNCT
ejpam-4737	156	10	{	{	PUNCT
ejpam-4737	156	11	x	x	NOUN
ejpam-4737	156	12	}	}	PUNCT
ejpam-4737	156	13	is	be	AUX
ejpam-4737	156	14	δs(λ	δs(λ	NOUN
ejpam-4737	156	15	,	,	PUNCT
ejpam-4737	156	16	p)-closed	p)-close	VERB
ejpam-4737	156	17	.	.	PUNCT
ejpam-4737	157	1	(	(	PUNCT
ejpam-4737	157	2	3	3	NUM
ejpam-4737	157	3	)	)	PUNCT
ejpam-4737	157	4	(	(	PUNCT
ejpam-4737	157	5	x	x	X
ejpam-4737	157	6	,	,	PUNCT
ejpam-4737	157	7	τ	τ	X
ejpam-4737	157	8	)	)	PUNCT
ejpam-4737	157	9	is	be	AUX
ejpam-4737	157	10	δs(λ	δs(λ	NOUN
ejpam-4737	157	11	,	,	PUNCT
ejpam-4737	157	12	p)-t1	p)-t1	NOUN
ejpam-4737	157	13	.	.	PUNCT
ejpam-4737	158	1	proof	proof	NOUN
ejpam-4737	158	2	.	.	PUNCT
ejpam-4737	159	1	(	(	PUNCT
ejpam-4737	159	2	1	1	X
ejpam-4737	159	3	)	)	PUNCT
ejpam-4737	159	4	⇒	⇒	NOUN
ejpam-4737	159	5	(	(	PUNCT
ejpam-4737	159	6	2	2	NUM
ejpam-4737	159	7	):	):	PUNCT
ejpam-4737	159	8	suppose	suppose	VERB
ejpam-4737	159	9	that	that	SCONJ
ejpam-4737	159	10	(	(	PUNCT
ejpam-4737	159	11	x	x	X
ejpam-4737	159	12	,	,	PUNCT
ejpam-4737	159	13	τ	τ	X
ejpam-4737	159	14	)	)	PUNCT
ejpam-4737	159	15	is	be	AUX
ejpam-4737	159	16	δs(λ	δs(λ	NOUN
ejpam-4737	159	17	,	,	PUNCT
ejpam-4737	159	18	p)-symmetric	p)-symmetric	NOUN
ejpam-4737	159	19	.	.	PUNCT
ejpam-4737	160	1	let	let	VERB
ejpam-4737	160	2	x	x	PRON
ejpam-4737	160	3	be	be	AUX
ejpam-4737	160	4	any	any	DET
ejpam-4737	160	5	point	point	NOUN
ejpam-4737	160	6	of	of	ADP
ejpam-4737	160	7	x	x	PUNCT
ejpam-4737	160	8	and	and	CCONJ
ejpam-4737	160	9	y	y	PROPN
ejpam-4737	160	10	be	be	VERB
ejpam-4737	160	11	any	any	DET
ejpam-4737	160	12	distinct	distinct	ADJ
ejpam-4737	160	13	point	point	NOUN
ejpam-4737	160	14	from	from	ADP
ejpam-4737	160	15	x.	x.	NOUN
ejpam-4737	160	16	by	by	ADP
ejpam-4737	160	17	lemma	lemma	PROPN
ejpam-4737	160	18	6	6	NUM
ejpam-4737	160	19	,	,	PUNCT
ejpam-4737	160	20	{	{	PUNCT
ejpam-4737	160	21	y	y	NOUN
ejpam-4737	160	22	}	}	PUNCT
ejpam-4737	160	23	is	be	AUX
ejpam-4737	160	24	s(λ	s(λ	PROPN
ejpam-4737	160	25	,	,	PUNCT
ejpam-4737	160	26	p)-open	p)-open	ADJ
ejpam-4737	160	27	or	or	CCONJ
ejpam-4737	160	28	s(λ	s(λ	NOUN
ejpam-4737	160	29	,	,	PUNCT
ejpam-4737	160	30	p)-closed	p)-close	VERB
ejpam-4737	160	31	in	in	ADP
ejpam-4737	160	32	(	(	PUNCT
ejpam-4737	160	33	x	x	X
ejpam-4737	160	34	,	,	PUNCT
ejpam-4737	160	35	τ	τ	PROPN
ejpam-4737	160	36	)	)	PUNCT
ejpam-4737	160	37	.	.	PUNCT
ejpam-4737	161	1	(	(	PUNCT
ejpam-4737	161	2	i	i	NOUN
ejpam-4737	161	3	)	)	PUNCT
ejpam-4737	161	4	in	in	ADP
ejpam-4737	161	5	case	case	NOUN
ejpam-4737	161	6	{	{	PUNCT
ejpam-4737	161	7	y	y	NOUN
ejpam-4737	161	8	}	}	PUNCT
ejpam-4737	161	9	is	be	AUX
ejpam-4737	161	10	s(λ	s(λ	PROPN
ejpam-4737	161	11	,	,	PUNCT
ejpam-4737	161	12	p)-open	p)-open	ADJ
ejpam-4737	161	13	,	,	PUNCT
ejpam-4737	161	14	put	put	VERB
ejpam-4737	161	15	vy	vy	NOUN
ejpam-4737	161	16	=	=	PUNCT
ejpam-4737	161	17	{	{	PUNCT
ejpam-4737	161	18	y	y	PROPN
ejpam-4737	161	19	}	}	PUNCT
ejpam-4737	161	20	,	,	PUNCT
ejpam-4737	161	21	then	then	ADV
ejpam-4737	161	22	vy	vy	PROPN
ejpam-4737	161	23	∈	∈	PROPN
ejpam-4737	161	24	δs(λ	δs(λ	NOUN
ejpam-4737	161	25	,	,	PUNCT
ejpam-4737	161	26	p)o(x	p)o(x	VERB
ejpam-4737	161	27	,	,	PUNCT
ejpam-4737	161	28	τ	τ	PROPN
ejpam-4737	161	29	)	)	PUNCT
ejpam-4737	161	30	.	.	PUNCT
ejpam-4737	162	1	(	(	PUNCT
ejpam-4737	162	2	ii	ii	NOUN
ejpam-4737	162	3	)	)	PUNCT
ejpam-4737	162	4	in	in	ADP
ejpam-4737	162	5	case	case	NOUN
ejpam-4737	162	6	{	{	PUNCT
ejpam-4737	162	7	y	y	NOUN
ejpam-4737	162	8	}	}	PUNCT
ejpam-4737	162	9	is	be	AUX
ejpam-4737	162	10	s(λ	s(λ	PROPN
ejpam-4737	162	11	,	,	PUNCT
ejpam-4737	162	12	p)-closed	p)-close	VERB
ejpam-4737	162	13	,	,	PUNCT
ejpam-4737	162	14	x	x	PROPN
ejpam-4737	162	15	̸∈	̸∈	PROPN
ejpam-4737	162	16	{	{	PUNCT
ejpam-4737	162	17	y	y	PROPN
ejpam-4737	162	18	}	}	PUNCT
ejpam-4737	162	19	=	=	SYM
ejpam-4737	162	20	{	{	PUNCT
ejpam-4737	162	21	y}s(λ	y}s(λ	NOUN
ejpam-4737	162	22	,	,	PUNCT
ejpam-4737	162	23	p	p	NOUN
ejpam-4737	162	24	)	)	PUNCT
ejpam-4737	162	25	and	and	CCONJ
ejpam-4737	162	26	x	x	PART
ejpam-4737	162	27	̸∈	̸∈	PROPN
ejpam-4737	162	28	{	{	PUNCT
ejpam-4737	162	29	y}δs(λ	y}δs(λ	PROPN
ejpam-4737	162	30	,	,	PUNCT
ejpam-4737	162	31	p	p	NOUN
ejpam-4737	162	32	)	)	PUNCT
ejpam-4737	162	33	.	.	PUNCT
ejpam-4737	163	1	by	by	ADP
ejpam-4737	163	2	(	(	PUNCT
ejpam-4737	163	3	1	1	NUM
ejpam-4737	163	4	)	)	PUNCT
ejpam-4737	163	5	,	,	PUNCT
ejpam-4737	163	6	y	y	PROPN
ejpam-4737	163	7	̸∈	̸∈	PROPN
ejpam-4737	163	8	{	{	PUNCT
ejpam-4737	163	9	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	163	10	,	,	PUNCT
ejpam-4737	163	11	p	p	NOUN
ejpam-4737	163	12	)	)	PUNCT
ejpam-4737	163	13	.	.	PUNCT
ejpam-4737	164	1	now	now	ADV
ejpam-4737	164	2	put	put	VERB
ejpam-4737	164	3	vy	vy	NOUN
ejpam-4737	164	4	=	=	PUNCT
ejpam-4737	164	5	x	x	X
ejpam-4737	164	6	−	−	PROPN
ejpam-4737	164	7	{	{	PUNCT
ejpam-4737	164	8	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	164	9	,	,	PUNCT
ejpam-4737	164	10	p	p	NOUN
ejpam-4737	164	11	)	)	PUNCT
ejpam-4737	164	12	.	.	PUNCT
ejpam-4737	165	1	then	then	ADV
ejpam-4737	165	2	,	,	PUNCT
ejpam-4737	165	3	x	x	PROPN
ejpam-4737	165	4	̸∈	̸∈	PROPN
ejpam-4737	165	5	vy	vy	PROPN
ejpam-4737	165	6	,	,	PUNCT
ejpam-4737	165	7	y	y	PROPN
ejpam-4737	165	8	∈	∈	PROPN
ejpam-4737	165	9	vy	vy	NOUN
ejpam-4737	165	10	and	and	CCONJ
ejpam-4737	165	11	vy	vy	NOUN
ejpam-4737	165	12	∈	∈	PROPN
ejpam-4737	165	13	δs(λ	δs(λ	NOUN
ejpam-4737	165	14	,	,	PUNCT
ejpam-4737	165	15	p)o(x	p)o(x	VERB
ejpam-4737	165	16	,	,	PUNCT
ejpam-4737	165	17	τ	τ	PROPN
ejpam-4737	165	18	)	)	PUNCT
ejpam-4737	165	19	.	.	PUNCT
ejpam-4737	166	1	thus	thus	ADV
ejpam-4737	166	2	,	,	PUNCT
ejpam-4737	166	3	x	x	PUNCT
ejpam-4737	166	4	−	−	X
ejpam-4737	166	5	{	{	PUNCT
ejpam-4737	166	6	x	x	NOUN
ejpam-4737	166	7	}	}	PUNCT
ejpam-4737	166	8	=	=	SYM
ejpam-4737	166	9	∪	∪	X
ejpam-4737	166	10	y∈x−{x	y∈x−{x	PROPN
ejpam-4737	166	11	}	}	PUNCT
ejpam-4737	166	12	vy	vy	NOUN
ejpam-4737	166	13	∈	∈	PROPN
ejpam-4737	166	14	δs(λ	δs(λ	NOUN
ejpam-4737	166	15	,	,	PUNCT
ejpam-4737	166	16	p)o(x	p)o(x	ADJ
ejpam-4737	166	17	,	,	PUNCT
ejpam-4737	166	18	τ	τ	PROPN
ejpam-4737	166	19	)	)	PUNCT
ejpam-4737	166	20	and	and	CCONJ
ejpam-4737	166	21	hence	hence	ADV
ejpam-4737	166	22	{	{	PUNCT
ejpam-4737	166	23	x	x	X
ejpam-4737	166	24	}	}	PUNCT
ejpam-4737	166	25	is	be	AUX
ejpam-4737	166	26	δs(λ	δs(λ	NOUN
ejpam-4737	166	27	,	,	PUNCT
ejpam-4737	166	28	p)-closed	p)-close	VERB
ejpam-4737	166	29	.	.	PUNCT
ejpam-4737	167	1	(	(	PUNCT
ejpam-4737	167	2	2	2	X
ejpam-4737	167	3	)	)	PUNCT
ejpam-4737	167	4	⇒	⇒	NOUN
ejpam-4737	167	5	(	(	PUNCT
ejpam-4737	167	6	3	3	NUM
ejpam-4737	167	7	):	):	PUNCT
ejpam-4737	167	8	suppose	suppose	VERB
ejpam-4737	167	9	that	that	SCONJ
ejpam-4737	167	10	{	{	PUNCT
ejpam-4737	167	11	z	z	NOUN
ejpam-4737	167	12	}	}	PUNCT
ejpam-4737	167	13	is	be	AUX
ejpam-4737	167	14	δs(λ	δs(λ	NOUN
ejpam-4737	167	15	,	,	PUNCT
ejpam-4737	167	16	p)-closed	p)-close	VERB
ejpam-4737	167	17	for	for	ADP
ejpam-4737	167	18	each	each	DET
ejpam-4737	167	19	z	z	NOUN
ejpam-4737	167	20	∈	∈	PROPN
ejpam-4737	167	21	x.	x.	NOUN
ejpam-4737	167	22	let	let	VERB
ejpam-4737	167	23	x	x	PRON
ejpam-4737	167	24	,	,	PUNCT
ejpam-4737	167	25	y	y	PROPN
ejpam-4737	167	26	∈	∈	PROPN
ejpam-4737	167	27	x	x	PUNCT
ejpam-4737	167	28	with	with	ADP
ejpam-4737	167	29	x	x	PUNCT
ejpam-4737	167	30	̸=	̸=	PROPN
ejpam-4737	167	31	y.	y.	NOUN
ejpam-4737	167	32	now	now	ADV
ejpam-4737	167	33	x	x	X
ejpam-4737	167	34	̸=	̸=	PROPN
ejpam-4737	167	35	y	y	PROPN
ejpam-4737	167	36	implies	imply	VERB
ejpam-4737	167	37	y	y	PROPN
ejpam-4737	167	38	∈	∈	PROPN
ejpam-4737	167	39	x	x	SYM
ejpam-4737	167	40	−{x	−{x	NUM
ejpam-4737	167	41	}	}	PUNCT
ejpam-4737	167	42	.	.	PUNCT
ejpam-4737	168	1	thus	thus	ADV
ejpam-4737	168	2	,	,	PUNCT
ejpam-4737	168	3	x	x	PRON
ejpam-4737	168	4	−{x	−{x	PRON
ejpam-4737	168	5	}	}	PUNCT
ejpam-4737	168	6	is	be	AUX
ejpam-4737	168	7	a	a	DET
ejpam-4737	168	8	δs(λ	δs(λ	NOUN
ejpam-4737	168	9	,	,	PUNCT
ejpam-4737	168	10	p)-open	p)-open	VERB
ejpam-4737	168	11	set	set	VERB
ejpam-4737	168	12	containing	contain	VERB
ejpam-4737	168	13	y	y	PROPN
ejpam-4737	168	14	but	but	CCONJ
ejpam-4737	168	15	not	not	PART
ejpam-4737	168	16	containing	contain	VERB
ejpam-4737	168	17	x.	x.	NOUN
ejpam-4737	168	18	similarly	similarly	ADV
ejpam-4737	168	19	,	,	PUNCT
ejpam-4737	168	20	we	we	PRON
ejpam-4737	168	21	have	have	VERB
ejpam-4737	168	22	x	x	PART
ejpam-4737	168	23	−	−	PROPN
ejpam-4737	168	24	{	{	PUNCT
ejpam-4737	168	25	y	y	NOUN
ejpam-4737	168	26	}	}	PUNCT
ejpam-4737	168	27	is	be	AUX
ejpam-4737	168	28	a	a	DET
ejpam-4737	168	29	δs(λ	δs(λ	NOUN
ejpam-4737	168	30	,	,	PUNCT
ejpam-4737	168	31	p)-open	p)-open	VERB
ejpam-4737	168	32	set	set	VERB
ejpam-4737	168	33	containing	contain	VERB
ejpam-4737	168	34	x	x	PUNCT
ejpam-4737	168	35	but	but	CCONJ
ejpam-4737	168	36	not	not	PART
ejpam-4737	168	37	containing	contain	VERB
ejpam-4737	168	38	y.	y.	NOUN
ejpam-4737	168	39	this	this	PRON
ejpam-4737	168	40	shows	show	VERB
ejpam-4737	168	41	that	that	SCONJ
ejpam-4737	168	42	(	(	PUNCT
ejpam-4737	168	43	x	x	X
ejpam-4737	168	44	,	,	PUNCT
ejpam-4737	168	45	τ	τ	X
ejpam-4737	168	46	)	)	PUNCT
ejpam-4737	168	47	is	be	AUX
ejpam-4737	168	48	δs(λ	δs(λ	NOUN
ejpam-4737	168	49	,	,	PUNCT
ejpam-4737	168	50	p)-t1	p)-t1	VERB
ejpam-4737	168	51	.	.	PUNCT
ejpam-4737	169	1	(	(	PUNCT
ejpam-4737	169	2	3	3	X
ejpam-4737	169	3	)	)	PUNCT
ejpam-4737	169	4	⇒	⇒	NOUN
ejpam-4737	169	5	(	(	PUNCT
ejpam-4737	169	6	1	1	NUM
ejpam-4737	169	7	):	):	PUNCT
ejpam-4737	169	8	suppose	suppose	VERB
ejpam-4737	169	9	that	that	SCONJ
ejpam-4737	169	10	y	y	PROPN
ejpam-4737	169	11	̸∈	̸∈	PROPN
ejpam-4737	169	12	{	{	PUNCT
ejpam-4737	169	13	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	169	14	,	,	PUNCT
ejpam-4737	169	15	p	p	NOUN
ejpam-4737	169	16	)	)	PUNCT
ejpam-4737	169	17	.	.	PUNCT
ejpam-4737	170	1	then	then	ADV
ejpam-4737	170	2	,	,	PUNCT
ejpam-4737	170	3	since	since	SCONJ
ejpam-4737	170	4	x	x	PROPN
ejpam-4737	170	5	̸=	̸=	PROPN
ejpam-4737	170	6	y	y	PROPN
ejpam-4737	170	7	,	,	PUNCT
ejpam-4737	170	8	by	by	ADP
ejpam-4737	170	9	(	(	PUNCT
ejpam-4737	170	10	3	3	X
ejpam-4737	170	11	)	)	PUNCT
ejpam-4737	170	12	there	there	PRON
ejpam-4737	170	13	exists	exist	VERB
ejpam-4737	170	14	a	a	DET
ejpam-4737	170	15	δs(λ	δs(λ	NOUN
ejpam-4737	170	16	,	,	PUNCT
ejpam-4737	170	17	p)-open	p)-open	VERB
ejpam-4737	170	18	set	set	VERB
ejpam-4737	170	19	u	u	NOUN
ejpam-4737	170	20	containing	contain	VERB
ejpam-4737	170	21	x	x	PUNCT
ejpam-4737	170	22	such	such	ADJ
ejpam-4737	170	23	that	that	SCONJ
ejpam-4737	170	24	y	y	PROPN
ejpam-4737	170	25	̸∈	̸∈	PROPN
ejpam-4737	170	26	u	u	PROPN
ejpam-4737	170	27	and	and	CCONJ
ejpam-4737	170	28	hence	hence	ADV
ejpam-4737	170	29	x	x	X
ejpam-4737	170	30	̸∈	̸∈	PROPN
ejpam-4737	170	31	{	{	PUNCT
ejpam-4737	170	32	y}δs(λ	y}δs(λ	PROPN
ejpam-4737	170	33	,	,	PUNCT
ejpam-4737	170	34	p	p	NOUN
ejpam-4737	170	35	)	)	PUNCT
ejpam-4737	170	36	.	.	PUNCT
ejpam-4737	171	1	this	this	PRON
ejpam-4737	171	2	shows	show	VERB
ejpam-4737	171	3	that	that	SCONJ
ejpam-4737	171	4	x	x	PUNCT
ejpam-4737	171	5	∈	∈	PROPN
ejpam-4737	171	6	{	{	PUNCT
ejpam-4737	171	7	y}δs(λ	y}δs(λ	NOUN
ejpam-4737	171	8	,	,	PUNCT
ejpam-4737	171	9	p	p	NOUN
ejpam-4737	171	10	)	)	PUNCT
ejpam-4737	171	11	implies	imply	VERB
ejpam-4737	171	12	y	y	PROPN
ejpam-4737	171	13	∈	∈	PROPN
ejpam-4737	171	14	{	{	PUNCT
ejpam-4737	171	15	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	171	16	,	,	PUNCT
ejpam-4737	171	17	p	p	NOUN
ejpam-4737	171	18	)	)	PUNCT
ejpam-4737	171	19	.	.	PUNCT
ejpam-4737	172	1	thus	thus	ADV
ejpam-4737	172	2	,	,	PUNCT
ejpam-4737	172	3	(	(	PUNCT
ejpam-4737	172	4	x	x	X
ejpam-4737	172	5	,	,	PUNCT
ejpam-4737	172	6	τ	τ	X
ejpam-4737	172	7	)	)	PUNCT
ejpam-4737	172	8	is	be	AUX
ejpam-4737	172	9	δs(λ	δs(λ	NOUN
ejpam-4737	172	10	,	,	PUNCT
ejpam-4737	172	11	p)-symmetric	p)-symmetric	ADJ
ejpam-4737	172	12	.	.	PUNCT
ejpam-4737	173	1	definition	definition	NOUN
ejpam-4737	173	2	8	8	NUM
ejpam-4737	173	3	.	.	PUNCT
ejpam-4737	174	1	a	a	DET
ejpam-4737	174	2	subset	subset	NOUN
ejpam-4737	174	3	a	a	PRON
ejpam-4737	174	4	of	of	ADP
ejpam-4737	174	5	a	a	DET
ejpam-4737	174	6	topological	topological	ADJ
ejpam-4737	174	7	space	space	NOUN
ejpam-4737	174	8	(	(	PUNCT
ejpam-4737	174	9	x	x	X
ejpam-4737	174	10	,	,	PUNCT
ejpam-4737	174	11	τ	τ	X
ejpam-4737	174	12	)	)	PUNCT
ejpam-4737	174	13	is	be	AUX
ejpam-4737	174	14	called	call	VERB
ejpam-4737	174	15	generalized	generalized	ADJ
ejpam-4737	174	16	δs(λ	δs(λ	NOUN
ejpam-4737	174	17	,	,	PUNCT
ejpam-4737	174	18	p)-closed	p)-close	VERB
ejpam-4737	174	19	(	(	PUNCT
ejpam-4737	174	20	briefly	briefly	NOUN
ejpam-4737	174	21	g	g	NOUN
ejpam-4737	174	22	-	-	PUNCT
ejpam-4737	174	23	δs(λ	δs(λ	NOUN
ejpam-4737	174	24	,	,	PUNCT
ejpam-4737	174	25	p)-closed	p)-close	VERB
ejpam-4737	174	26	)	)	PUNCT
ejpam-4737	174	27	if	if	SCONJ
ejpam-4737	174	28	aδs(λ	aδs(λ	PROPN
ejpam-4737	174	29	,	,	PUNCT
ejpam-4737	174	30	p	p	NOUN
ejpam-4737	174	31	)	)	PUNCT
ejpam-4737	174	32	⊆	⊆	NUM
ejpam-4737	174	33	u	u	NOUN
ejpam-4737	174	34	whenever	whenever	SCONJ
ejpam-4737	174	35	a	a	DET
ejpam-4737	174	36	⊆	⊆	NUM
ejpam-4737	174	37	u	u	NOUN
ejpam-4737	174	38	and	and	CCONJ
ejpam-4737	174	39	u	u	NOUN
ejpam-4737	174	40	is	be	AUX
ejpam-4737	174	41	δs(λ	δs(λ	NOUN
ejpam-4737	174	42	,	,	PUNCT
ejpam-4737	174	43	p)-open	p)-open	VERB
ejpam-4737	174	44	in	in	ADP
ejpam-4737	174	45	(	(	PUNCT
ejpam-4737	174	46	x	x	NOUN
ejpam-4737	174	47	,	,	PUNCT
ejpam-4737	174	48	τ	τ	PROPN
ejpam-4737	174	49	)	)	PUNCT
ejpam-4737	174	50	.	.	PUNCT
ejpam-4737	175	1	theorem	theorem	VERB
ejpam-4737	175	2	4	4	NUM
ejpam-4737	175	3	.	.	PUNCT
ejpam-4737	176	1	a	a	DET
ejpam-4737	176	2	subset	subset	NOUN
ejpam-4737	176	3	a	a	PRON
ejpam-4737	176	4	of	of	ADP
ejpam-4737	176	5	a	a	DET
ejpam-4737	176	6	topological	topological	ADJ
ejpam-4737	176	7	space	space	NOUN
ejpam-4737	176	8	(	(	PUNCT
ejpam-4737	176	9	x	x	X
ejpam-4737	176	10	,	,	PUNCT
ejpam-4737	176	11	τ	τ	X
ejpam-4737	176	12	)	)	PUNCT
ejpam-4737	176	13	is	be	AUX
ejpam-4737	176	14	g	g	NOUN
ejpam-4737	176	15	-	-	PUNCT
ejpam-4737	176	16	δs(λ	δs(λ	NOUN
ejpam-4737	176	17	,	,	PUNCT
ejpam-4737	176	18	p)-closed	p)-close	VERB
ejpam-4737	176	19	if	if	SCONJ
ejpam-4737	176	20	and	and	CCONJ
ejpam-4737	176	21	only	only	ADV
ejpam-4737	176	22	if	if	SCONJ
ejpam-4737	176	23	aδs(λ	aδs(λ	PROPN
ejpam-4737	176	24	,	,	PUNCT
ejpam-4737	176	25	p	p	NOUN
ejpam-4737	176	26	)	)	PUNCT
ejpam-4737	176	27	−a	−a	NOUN
ejpam-4737	176	28	contains	contain	VERB
ejpam-4737	176	29	no	no	DET
ejpam-4737	176	30	nonempty	nonempty	ADJ
ejpam-4737	176	31	δs(λ	δs(λ	NOUN
ejpam-4737	176	32	,	,	PUNCT
ejpam-4737	176	33	p)-closed	p)-close	VERB
ejpam-4737	176	34	set	set	NOUN
ejpam-4737	176	35	.	.	PUNCT
ejpam-4737	177	1	c.	c.	PROPN
ejpam-4737	177	2	boonpok	boonpok	PROPN
ejpam-4737	177	3	,	,	PUNCT
ejpam-4737	177	4	m.	m.	NOUN
ejpam-4737	177	5	thongmoon	thongmoon	PROPN
ejpam-4737	177	6	/	/	SYM
ejpam-4737	177	7	eur	eur	PROPN
ejpam-4737	177	8	.	.	PUNCT
ejpam-4737	178	1	j.	j.	PROPN
ejpam-4737	178	2	pure	pure	PROPN
ejpam-4737	178	3	appl	appl	PROPN
ejpam-4737	178	4	.	.	PROPN
ejpam-4737	178	5	math	math	PROPN
ejpam-4737	178	6	,	,	PUNCT
ejpam-4737	178	7	16	16	NUM
ejpam-4737	178	8	(	(	PUNCT
ejpam-4737	178	9	3	3	NUM
ejpam-4737	178	10	)	)	PUNCT
ejpam-4737	178	11	(	(	PUNCT
ejpam-4737	178	12	2023	2023	NUM
ejpam-4737	178	13	)	)	PUNCT
ejpam-4737	178	14	,	,	PUNCT
ejpam-4737	178	15	1434	1434	NUM
ejpam-4737	178	16	-	-	SYM
ejpam-4737	178	17	1447	1447	NUM
ejpam-4737	178	18	1439	1439	NUM
ejpam-4737	178	19	proof	proof	NOUN
ejpam-4737	178	20	.	.	PUNCT
ejpam-4737	179	1	let	let	VERB
ejpam-4737	179	2	f	f	PRON
ejpam-4737	179	3	be	be	AUX
ejpam-4737	179	4	a	a	DET
ejpam-4737	179	5	δs(λ	δs(λ	NOUN
ejpam-4737	179	6	,	,	PUNCT
ejpam-4737	179	7	p)-closed	p)-close	VERB
ejpam-4737	179	8	subset	subset	NOUN
ejpam-4737	179	9	of	of	ADP
ejpam-4737	179	10	aδs(λ	aδs(λ	PROPN
ejpam-4737	179	11	,	,	PUNCT
ejpam-4737	179	12	p	p	NOUN
ejpam-4737	179	13	)	)	PUNCT
ejpam-4737	179	14	−	−	NOUN
ejpam-4737	179	15	a.	a.	NOUN
ejpam-4737	179	16	since	since	SCONJ
ejpam-4737	179	17	a	a	DET
ejpam-4737	179	18	⊆	⊆	NUM
ejpam-4737	179	19	x	x	SYM
ejpam-4737	179	20	−	−	PROPN
ejpam-4737	179	21	f	f	PROPN
ejpam-4737	179	22	and	and	CCONJ
ejpam-4737	179	23	a	a	PRON
ejpam-4737	179	24	is	be	AUX
ejpam-4737	179	25	g	g	NOUN
ejpam-4737	179	26	-	-	PUNCT
ejpam-4737	179	27	δs(λ	δs(λ	NOUN
ejpam-4737	179	28	,	,	PUNCT
ejpam-4737	179	29	p)-closed	p)-close	VERB
ejpam-4737	179	30	,	,	PUNCT
ejpam-4737	179	31	aδs(λ	aδs(λ	PROPN
ejpam-4737	179	32	,	,	PUNCT
ejpam-4737	179	33	p	p	NOUN
ejpam-4737	179	34	)	)	PUNCT
ejpam-4737	179	35	⊆	⊆	NUM
ejpam-4737	179	36	x	x	SYM
ejpam-4737	179	37	−	−	PROPN
ejpam-4737	179	38	f	f	NOUN
ejpam-4737	179	39	and	and	CCONJ
ejpam-4737	179	40	hence	hence	ADV
ejpam-4737	179	41	f	f	PROPN
ejpam-4737	179	42	⊆	⊆	NUM
ejpam-4737	179	43	x	x	SYM
ejpam-4737	179	44	−aδs(λ	−aδs(λ	PROPN
ejpam-4737	179	45	,	,	PUNCT
ejpam-4737	179	46	p	p	NOUN
ejpam-4737	179	47	)	)	PUNCT
ejpam-4737	179	48	.	.	PUNCT
ejpam-4737	180	1	thus	thus	ADV
ejpam-4737	180	2	,	,	PUNCT
ejpam-4737	180	3	f	f	PROPN
ejpam-4737	180	4	⊆	⊆	NUM
ejpam-4737	180	5	aδs(λ	aδs(λ	PROPN
ejpam-4737	180	6	,	,	PUNCT
ejpam-4737	180	7	p	p	NOUN
ejpam-4737	180	8	)	)	PUNCT
ejpam-4737	180	9	∩	∩	NOUN
ejpam-4737	180	10	[	[	X
ejpam-4737	180	11	x	x	SYM
ejpam-4737	180	12	−aδs(λ	−aδs(λ	PROPN
ejpam-4737	180	13	,	,	PUNCT
ejpam-4737	180	14	p	p	NOUN
ejpam-4737	180	15	)	)	PUNCT
ejpam-4737	180	16	]	]	PUNCT
ejpam-4737	181	1	=	=	PUNCT
ejpam-4737	181	2	∅	∅	NOUN
ejpam-4737	181	3	and	and	CCONJ
ejpam-4737	181	4	f	f	PROPN
ejpam-4737	181	5	is	be	AUX
ejpam-4737	181	6	empty	empty	ADJ
ejpam-4737	181	7	.	.	PUNCT
ejpam-4737	182	1	conversely	conversely	ADV
ejpam-4737	182	2	,	,	PUNCT
ejpam-4737	182	3	suppose	suppose	VERB
ejpam-4737	182	4	that	that	SCONJ
ejpam-4737	182	5	a	a	DET
ejpam-4737	182	6	⊆	⊆	NUM
ejpam-4737	182	7	u	u	NOUN
ejpam-4737	182	8	and	and	CCONJ
ejpam-4737	182	9	u	u	NOUN
ejpam-4737	182	10	is	be	AUX
ejpam-4737	182	11	δs(λ	δs(λ	NOUN
ejpam-4737	182	12	,	,	PUNCT
ejpam-4737	182	13	p)-open	p)-open	ADJ
ejpam-4737	182	14	.	.	PUNCT
ejpam-4737	183	1	if	if	SCONJ
ejpam-4737	183	2	aδs(λ	aδs(λ	PROPN
ejpam-4737	183	3	,	,	PUNCT
ejpam-4737	183	4	p	p	NOUN
ejpam-4737	183	5	)	)	PUNCT
ejpam-4737	183	6	⊈	⊈	PROPN
ejpam-4737	183	7	u	u	NOUN
ejpam-4737	183	8	,	,	PUNCT
ejpam-4737	183	9	then	then	ADV
ejpam-4737	183	10	aδs(λ	aδs(λ	PROPN
ejpam-4737	183	11	,	,	PUNCT
ejpam-4737	183	12	p	p	NOUN
ejpam-4737	183	13	)	)	PUNCT
ejpam-4737	183	14	∩	∩	NOUN
ejpam-4737	183	15	(	(	PUNCT
ejpam-4737	183	16	x	x	SYM
ejpam-4737	183	17	−	−	PROPN
ejpam-4737	183	18	u	u	NOUN
ejpam-4737	183	19	)	)	PUNCT
ejpam-4737	183	20	is	be	AUX
ejpam-4737	183	21	a	a	DET
ejpam-4737	183	22	nonempty	nonempty	ADJ
ejpam-4737	183	23	δs(λ	δs(λ	NOUN
ejpam-4737	183	24	,	,	PUNCT
ejpam-4737	183	25	p)-closed	p)-close	VERB
ejpam-4737	183	26	subset	subset	NOUN
ejpam-4737	183	27	of	of	ADP
ejpam-4737	183	28	aδs(λ	aδs(λ	PROPN
ejpam-4737	183	29	,	,	PUNCT
ejpam-4737	183	30	p	p	NOUN
ejpam-4737	183	31	)	)	PUNCT
ejpam-4737	183	32	−a	−a	NOUN
ejpam-4737	183	33	.	.	PUNCT
ejpam-4737	184	1	theorem	theorem	VERB
ejpam-4737	184	2	5	5	NUM
ejpam-4737	184	3	.	.	PUNCT
ejpam-4737	185	1	a	a	DET
ejpam-4737	185	2	subset	subset	NOUN
ejpam-4737	185	3	a	a	PRON
ejpam-4737	185	4	of	of	ADP
ejpam-4737	185	5	a	a	DET
ejpam-4737	185	6	topological	topological	ADJ
ejpam-4737	185	7	space	space	NOUN
ejpam-4737	185	8	(	(	PUNCT
ejpam-4737	185	9	x	x	X
ejpam-4737	185	10	,	,	PUNCT
ejpam-4737	185	11	τ	τ	X
ejpam-4737	185	12	)	)	PUNCT
ejpam-4737	185	13	is	be	AUX
ejpam-4737	185	14	g	g	NOUN
ejpam-4737	185	15	-	-	PUNCT
ejpam-4737	185	16	δs(λ	δs(λ	NOUN
ejpam-4737	185	17	,	,	PUNCT
ejpam-4737	185	18	p)-closed	p)-close	VERB
ejpam-4737	185	19	if	if	SCONJ
ejpam-4737	185	20	and	and	CCONJ
ejpam-4737	185	21	only	only	ADV
ejpam-4737	185	22	if	if	SCONJ
ejpam-4737	185	23	f	f	PROPN
ejpam-4737	185	24	∩aδs(λ	∩aδs(λ	X
ejpam-4737	185	25	,	,	PUNCT
ejpam-4737	185	26	p	p	NOUN
ejpam-4737	185	27	)	)	PUNCT
ejpam-4737	185	28	=	=	NOUN
ejpam-4737	185	29	∅	∅	NOUN
ejpam-4737	185	30	whenever	whenever	SCONJ
ejpam-4737	185	31	a	a	DET
ejpam-4737	185	32	∩	∩	ADJ
ejpam-4737	185	33	f	f	NOUN
ejpam-4737	185	34	=	=	NOUN
ejpam-4737	185	35	∅	∅	NOUN
ejpam-4737	185	36	and	and	CCONJ
ejpam-4737	185	37	f	f	PROPN
ejpam-4737	185	38	is	be	AUX
ejpam-4737	185	39	δs(λ	δs(λ	NOUN
ejpam-4737	185	40	,	,	PUNCT
ejpam-4737	185	41	p)-closed	p)-close	VERB
ejpam-4737	185	42	.	.	PUNCT
ejpam-4737	186	1	proof	proof	NOUN
ejpam-4737	186	2	.	.	PUNCT
ejpam-4737	187	1	suppose	suppose	VERB
ejpam-4737	187	2	that	that	SCONJ
ejpam-4737	187	3	a	a	PRON
ejpam-4737	187	4	is	be	AUX
ejpam-4737	187	5	a	a	DET
ejpam-4737	187	6	δs(λ	δs(λ	NOUN
ejpam-4737	187	7	,	,	PUNCT
ejpam-4737	187	8	p)-closed	p)-close	VERB
ejpam-4737	187	9	set	set	NOUN
ejpam-4737	187	10	.	.	PUNCT
ejpam-4737	188	1	let	let	VERB
ejpam-4737	188	2	f	f	PRON
ejpam-4737	188	3	be	be	AUX
ejpam-4737	188	4	a	a	DET
ejpam-4737	188	5	δs(λ	δs(λ	NOUN
ejpam-4737	188	6	,	,	PUNCT
ejpam-4737	188	7	p)-closed	p)-close	VERB
ejpam-4737	188	8	set	set	NOUN
ejpam-4737	188	9	and	and	CCONJ
ejpam-4737	188	10	a	a	DET
ejpam-4737	188	11	∩	∩	ADJ
ejpam-4737	188	12	f	f	X
ejpam-4737	188	13	=	=	PUNCT
ejpam-4737	188	14	∅.	∅.	NOUN
ejpam-4737	188	15	then	then	ADV
ejpam-4737	188	16	,	,	PUNCT
ejpam-4737	188	17	a	a	DET
ejpam-4737	188	18	⊆	⊆	NUM
ejpam-4737	188	19	x	x	SYM
ejpam-4737	188	20	−	−	PROPN
ejpam-4737	188	21	f	f	PROPN
ejpam-4737	188	22	∈	∈	PROPN
ejpam-4737	188	23	δs(λ	δs(λ	NOUN
ejpam-4737	188	24	,	,	PUNCT
ejpam-4737	188	25	p)o(x	p)o(x	ADJ
ejpam-4737	188	26	,	,	PUNCT
ejpam-4737	188	27	τ	τ	PROPN
ejpam-4737	188	28	)	)	PUNCT
ejpam-4737	188	29	and	and	CCONJ
ejpam-4737	188	30	aδs(λ	aδs(λ	PROPN
ejpam-4737	188	31	,	,	PUNCT
ejpam-4737	188	32	p	p	NOUN
ejpam-4737	188	33	)	)	PUNCT
ejpam-4737	188	34	⊆	⊆	NUM
ejpam-4737	188	35	x	x	SYM
ejpam-4737	188	36	−	−	PROPN
ejpam-4737	188	37	f	f	NOUN
ejpam-4737	188	38	.	.	PUNCT
ejpam-4737	189	1	thus	thus	ADV
ejpam-4737	189	2	,	,	PUNCT
ejpam-4737	189	3	f	f	PROPN
ejpam-4737	189	4	∩aδs(λ	∩aδs(λ	X
ejpam-4737	189	5	,	,	PUNCT
ejpam-4737	189	6	p	p	NOUN
ejpam-4737	189	7	)	)	PUNCT
ejpam-4737	189	8	=	=	NOUN
ejpam-4737	189	9	∅.	∅.	VERB
ejpam-4737	189	10	conversely	conversely	ADV
ejpam-4737	189	11	,	,	PUNCT
ejpam-4737	189	12	let	let	VERB
ejpam-4737	189	13	a	a	DET
ejpam-4737	189	14	⊆	⊆	NUM
ejpam-4737	189	15	u	u	NOUN
ejpam-4737	189	16	and	and	CCONJ
ejpam-4737	189	17	u	u	PROPN
ejpam-4737	189	18	∈	∈	PROPN
ejpam-4737	189	19	δs(λ	δs(λ	NOUN
ejpam-4737	189	20	,	,	PUNCT
ejpam-4737	189	21	p)o(x	p)o(x	ADJ
ejpam-4737	189	22	,	,	PUNCT
ejpam-4737	189	23	τ	τ	PROPN
ejpam-4737	189	24	)	)	PUNCT
ejpam-4737	189	25	.	.	PUNCT
ejpam-4737	190	1	then	then	ADV
ejpam-4737	190	2	,	,	PUNCT
ejpam-4737	190	3	a∩	a∩	PROPN
ejpam-4737	190	4	(	(	PUNCT
ejpam-4737	190	5	x	x	NOUN
ejpam-4737	190	6	−u	−u	PROPN
ejpam-4737	190	7	)	)	PUNCT
ejpam-4737	190	8	=	=	NOUN
ejpam-4737	190	9	∅	∅	NOUN
ejpam-4737	190	10	and	and	CCONJ
ejpam-4737	190	11	x	x	SYM
ejpam-4737	190	12	−u	−u	NOUN
ejpam-4737	190	13	is	be	AUX
ejpam-4737	190	14	δs(λ	δs(λ	NOUN
ejpam-4737	190	15	,	,	PUNCT
ejpam-4737	190	16	p)-closed	p)-close	VERB
ejpam-4737	190	17	.	.	PUNCT
ejpam-4737	190	18	by	by	ADP
ejpam-4737	190	19	the	the	DET
ejpam-4737	190	20	hypothesis	hypothesis	NOUN
ejpam-4737	190	21	,	,	PUNCT
ejpam-4737	190	22	(	(	PUNCT
ejpam-4737	190	23	x	x	X
ejpam-4737	190	24	−u	−u	PROPN
ejpam-4737	190	25	)	)	PUNCT
ejpam-4737	190	26	∩aδs(λ	∩aδs(λ	PUNCT
ejpam-4737	190	27	,	,	PUNCT
ejpam-4737	190	28	p	p	X
ejpam-4737	190	29	)	)	PUNCT
ejpam-4737	190	30	=	=	NOUN
ejpam-4737	190	31	∅	∅	NOUN
ejpam-4737	190	32	and	and	CCONJ
ejpam-4737	190	33	hence	hence	ADV
ejpam-4737	190	34	aδs(λ	aδs(λ	PROPN
ejpam-4737	190	35	,	,	PUNCT
ejpam-4737	190	36	p	p	NOUN
ejpam-4737	190	37	)	)	PUNCT
ejpam-4737	190	38	⊆	⊆	NUM
ejpam-4737	190	39	u	u	NOUN
ejpam-4737	190	40	.	.	PUNCT
ejpam-4737	191	1	thus	thus	ADV
ejpam-4737	191	2	,	,	PUNCT
ejpam-4737	191	3	a	a	PRON
ejpam-4737	191	4	is	be	AUX
ejpam-4737	191	5	g	g	NOUN
ejpam-4737	191	6	-	-	PUNCT
ejpam-4737	191	7	δs(λ	δs(λ	NOUN
ejpam-4737	191	8	,	,	PUNCT
ejpam-4737	191	9	p)-closed	p)-close	VERB
ejpam-4737	191	10	.	.	PUNCT
ejpam-4737	192	1	theorem	theorem	VERB
ejpam-4737	192	2	6	6	NUM
ejpam-4737	192	3	.	.	PUNCT
ejpam-4737	193	1	a	a	DET
ejpam-4737	193	2	subset	subset	NOUN
ejpam-4737	193	3	a	a	PRON
ejpam-4737	193	4	of	of	ADP
ejpam-4737	193	5	a	a	DET
ejpam-4737	193	6	topological	topological	ADJ
ejpam-4737	193	7	space	space	NOUN
ejpam-4737	193	8	(	(	PUNCT
ejpam-4737	193	9	x	x	X
ejpam-4737	193	10	,	,	PUNCT
ejpam-4737	193	11	τ	τ	X
ejpam-4737	193	12	)	)	PUNCT
ejpam-4737	193	13	is	be	AUX
ejpam-4737	193	14	g	g	NOUN
ejpam-4737	193	15	-	-	PUNCT
ejpam-4737	193	16	δs(λ	δs(λ	NOUN
ejpam-4737	193	17	,	,	PUNCT
ejpam-4737	193	18	p)-closed	p)-close	VERB
ejpam-4737	193	19	if	if	SCONJ
ejpam-4737	193	20	and	and	CCONJ
ejpam-4737	193	21	only	only	ADV
ejpam-4737	193	22	if	if	SCONJ
ejpam-4737	193	23	a	a	DET
ejpam-4737	193	24	∩	∩	NOUN
ejpam-4737	193	25	{	{	PUNCT
ejpam-4737	193	26	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	193	27	,	,	PUNCT
ejpam-4737	193	28	p	p	NOUN
ejpam-4737	193	29	)	)	PUNCT
ejpam-4737	193	30	̸=	̸=	PROPN
ejpam-4737	193	31	∅	∅	NOUN
ejpam-4737	193	32	for	for	ADP
ejpam-4737	193	33	every	every	DET
ejpam-4737	193	34	x	x	PROPN
ejpam-4737	193	35	∈	∈	PROPN
ejpam-4737	193	36	aδs(λ	aδs(λ	PROPN
ejpam-4737	193	37	,	,	PUNCT
ejpam-4737	193	38	p	p	NOUN
ejpam-4737	193	39	)	)	PUNCT
ejpam-4737	193	40	.	.	PUNCT
ejpam-4737	194	1	proof	proof	NOUN
ejpam-4737	194	2	.	.	PUNCT
ejpam-4737	195	1	let	let	VERB
ejpam-4737	195	2	a	a	PRON
ejpam-4737	195	3	be	be	AUX
ejpam-4737	195	4	a	a	DET
ejpam-4737	195	5	g	g	NOUN
ejpam-4737	195	6	-	-	PUNCT
ejpam-4737	195	7	δs(λ	δs(λ	NOUN
ejpam-4737	195	8	,	,	PUNCT
ejpam-4737	195	9	p)-closed	p)-close	VERB
ejpam-4737	195	10	set	set	NOUN
ejpam-4737	195	11	and	and	CCONJ
ejpam-4737	195	12	suppose	suppose	VERB
ejpam-4737	195	13	that	that	SCONJ
ejpam-4737	195	14	there	there	PRON
ejpam-4737	195	15	exists	exist	VERB
ejpam-4737	195	16	x	x	X
ejpam-4737	195	17	∈	∈	PROPN
ejpam-4737	195	18	aδs(λ	aδs(λ	PROPN
ejpam-4737	195	19	,	,	PUNCT
ejpam-4737	195	20	p	p	NOUN
ejpam-4737	195	21	)	)	PUNCT
ejpam-4737	195	22	such	such	ADJ
ejpam-4737	195	23	that	that	SCONJ
ejpam-4737	195	24	a	a	DET
ejpam-4737	195	25	∩	∩	NOUN
ejpam-4737	195	26	{	{	PUNCT
ejpam-4737	195	27	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	195	28	,	,	PUNCT
ejpam-4737	195	29	p	p	NOUN
ejpam-4737	195	30	)	)	PUNCT
ejpam-4737	195	31	=	=	PUNCT
ejpam-4737	195	32	∅.	∅.	ADP
ejpam-4737	195	33	thus	thus	ADV
ejpam-4737	195	34	,	,	PUNCT
ejpam-4737	195	35	a	a	DET
ejpam-4737	195	36	⊆	⊆	NUM
ejpam-4737	195	37	x	x	SYM
ejpam-4737	195	38	−	−	PROPN
ejpam-4737	195	39	{	{	PUNCT
ejpam-4737	195	40	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	195	41	,	,	PUNCT
ejpam-4737	195	42	p	p	NOUN
ejpam-4737	195	43	)	)	PUNCT
ejpam-4737	195	44	and	and	CCONJ
ejpam-4737	195	45	hence	hence	ADV
ejpam-4737	195	46	aδs(λ	aδs(λ	PROPN
ejpam-4737	195	47	,	,	PUNCT
ejpam-4737	195	48	p	p	NOUN
ejpam-4737	195	49	)	)	PUNCT
ejpam-4737	195	50	⊆	⊆	NUM
ejpam-4737	195	51	x	x	SYM
ejpam-4737	195	52	−	−	PROPN
ejpam-4737	195	53	{	{	PUNCT
ejpam-4737	195	54	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	195	55	,	,	PUNCT
ejpam-4737	195	56	p	p	NOUN
ejpam-4737	195	57	)	)	PUNCT
ejpam-4737	195	58	.	.	PUNCT
ejpam-4737	196	1	therefore	therefore	ADV
ejpam-4737	196	2	,	,	PUNCT
ejpam-4737	196	3	x	x	PROPN
ejpam-4737	196	4	̸∈	̸∈	PROPN
ejpam-4737	196	5	aδs(λ	aδs(λ	PROPN
ejpam-4737	196	6	,	,	PUNCT
ejpam-4737	196	7	p	p	NOUN
ejpam-4737	196	8	)	)	PUNCT
ejpam-4737	196	9	,	,	PUNCT
ejpam-4737	196	10	which	which	PRON
ejpam-4737	196	11	is	be	AUX
ejpam-4737	196	12	a	a	DET
ejpam-4737	196	13	contradiction	contradiction	NOUN
ejpam-4737	196	14	.	.	PUNCT
ejpam-4737	197	1	conversely	conversely	ADV
ejpam-4737	197	2	,	,	PUNCT
ejpam-4737	197	3	suppose	suppose	VERB
ejpam-4737	197	4	that	that	SCONJ
ejpam-4737	197	5	the	the	DET
ejpam-4737	197	6	condition	condition	NOUN
ejpam-4737	197	7	of	of	ADP
ejpam-4737	197	8	the	the	DET
ejpam-4737	197	9	theorem	theorem	NOUN
ejpam-4737	197	10	holds	hold	VERB
ejpam-4737	197	11	and	and	CCONJ
ejpam-4737	197	12	let	let	VERB
ejpam-4737	197	13	u	u	PRON
ejpam-4737	197	14	be	be	AUX
ejpam-4737	197	15	any	any	DET
ejpam-4737	197	16	δs(λ	δs(λ	NOUN
ejpam-4737	197	17	,	,	PUNCT
ejpam-4737	197	18	p)open	p)open	PROPN
ejpam-4737	197	19	set	set	NOUN
ejpam-4737	197	20	containing	contain	VERB
ejpam-4737	197	21	a.	a.	NOUN
ejpam-4737	197	22	let	let	VERB
ejpam-4737	197	23	x	x	X
ejpam-4737	197	24	∈	∈	PROPN
ejpam-4737	197	25	aδs(λ	aδs(λ	PROPN
ejpam-4737	197	26	,	,	PUNCT
ejpam-4737	197	27	p	p	NOUN
ejpam-4737	197	28	)	)	PUNCT
ejpam-4737	197	29	.	.	PUNCT
ejpam-4737	198	1	by	by	ADP
ejpam-4737	198	2	the	the	DET
ejpam-4737	198	3	hypothesis	hypothesis	NOUN
ejpam-4737	198	4	,	,	PUNCT
ejpam-4737	198	5	a	a	DET
ejpam-4737	198	6	∩	∩	ADJ
ejpam-4737	198	7	aδs(λ	aδs(λ	PROPN
ejpam-4737	198	8	,	,	PUNCT
ejpam-4737	198	9	p	p	NOUN
ejpam-4737	198	10	)	)	PUNCT
ejpam-4737	198	11	̸=	̸=	PROPN
ejpam-4737	198	12	∅	∅	NOUN
ejpam-4737	198	13	,	,	PUNCT
ejpam-4737	198	14	so	so	SCONJ
ejpam-4737	198	15	there	there	PRON
ejpam-4737	198	16	exists	exist	VERB
ejpam-4737	198	17	y	y	PROPN
ejpam-4737	198	18	∈	∈	PROPN
ejpam-4737	198	19	a	a	DET
ejpam-4737	198	20	∩	∩	NOUN
ejpam-4737	198	21	{	{	PUNCT
ejpam-4737	198	22	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	198	23	,	,	PUNCT
ejpam-4737	198	24	p	p	NOUN
ejpam-4737	198	25	)	)	PUNCT
ejpam-4737	198	26	and	and	CCONJ
ejpam-4737	198	27	hence	hence	ADV
ejpam-4737	198	28	y	y	PROPN
ejpam-4737	198	29	∈	∈	PROPN
ejpam-4737	198	30	a	a	DET
ejpam-4737	198	31	⊆	⊆	NUM
ejpam-4737	198	32	u	u	NOUN
ejpam-4737	198	33	.	.	PUNCT
ejpam-4737	199	1	thus	thus	ADV
ejpam-4737	199	2	,	,	PUNCT
ejpam-4737	199	3	{	{	PUNCT
ejpam-4737	199	4	x	x	NOUN
ejpam-4737	199	5	}	}	PUNCT
ejpam-4737	199	6	∩	∩	NOUN
ejpam-4737	199	7	u	u	NOUN
ejpam-4737	199	8	̸=	̸=	PROPN
ejpam-4737	199	9	∅.	∅.	VERB
ejpam-4737	199	10	therefore	therefore	ADV
ejpam-4737	199	11	,	,	PUNCT
ejpam-4737	199	12	x	x	PUNCT
ejpam-4737	199	13	∈	∈	PROPN
ejpam-4737	199	14	u	u	NOUN
ejpam-4737	199	15	,	,	PUNCT
ejpam-4737	199	16	which	which	PRON
ejpam-4737	199	17	implies	imply	VERB
ejpam-4737	199	18	that	that	SCONJ
ejpam-4737	199	19	aδs(λ	aδs(λ	PROPN
ejpam-4737	199	20	,	,	PUNCT
ejpam-4737	199	21	p	p	NOUN
ejpam-4737	199	22	)	)	PUNCT
ejpam-4737	199	23	⊆	⊆	NUM
ejpam-4737	199	24	u	u	NOUN
ejpam-4737	199	25	.	.	PUNCT
ejpam-4737	200	1	this	this	PRON
ejpam-4737	200	2	shows	show	VERB
ejpam-4737	200	3	that	that	SCONJ
ejpam-4737	200	4	a	a	PRON
ejpam-4737	200	5	is	be	AUX
ejpam-4737	200	6	g	g	NOUN
ejpam-4737	200	7	-	-	PUNCT
ejpam-4737	200	8	δs(λ	δs(λ	NOUN
ejpam-4737	200	9	,	,	PUNCT
ejpam-4737	200	10	p)-closed	p)-close	VERB
ejpam-4737	200	11	.	.	PUNCT
ejpam-4737	201	1	theorem	theorem	VERB
ejpam-4737	201	2	7	7	NUM
ejpam-4737	201	3	.	.	PUNCT
ejpam-4737	202	1	a	a	DET
ejpam-4737	202	2	topological	topological	ADJ
ejpam-4737	202	3	space	space	NOUN
ejpam-4737	202	4	(	(	PUNCT
ejpam-4737	202	5	x	x	X
ejpam-4737	202	6	,	,	PUNCT
ejpam-4737	202	7	τ	τ	X
ejpam-4737	202	8	)	)	PUNCT
ejpam-4737	202	9	is	be	AUX
ejpam-4737	202	10	δs(λ	δs(λ	NOUN
ejpam-4737	202	11	,	,	PUNCT
ejpam-4737	202	12	p)-symmetric	p)-symmetric	ADJ
ejpam-4737	202	13	if	if	SCONJ
ejpam-4737	202	14	and	and	CCONJ
ejpam-4737	202	15	only	only	ADV
ejpam-4737	202	16	if	if	SCONJ
ejpam-4737	202	17	{	{	PUNCT
ejpam-4737	202	18	x	x	NOUN
ejpam-4737	202	19	}	}	PUNCT
ejpam-4737	202	20	is	be	AUX
ejpam-4737	202	21	gδs(λ	gδs(λ	PROPN
ejpam-4737	202	22	,	,	PUNCT
ejpam-4737	202	23	p)-closed	p)-close	VERB
ejpam-4737	202	24	for	for	ADP
ejpam-4737	202	25	each	each	DET
ejpam-4737	202	26	x	x	SYM
ejpam-4737	202	27	∈	∈	PROPN
ejpam-4737	202	28	x.	x.	NOUN
ejpam-4737	202	29	proof	proof	NOUN
ejpam-4737	202	30	.	.	PUNCT
ejpam-4737	203	1	suppose	suppose	VERB
ejpam-4737	203	2	that	that	SCONJ
ejpam-4737	203	3	x	x	PUNCT
ejpam-4737	203	4	∈	∈	PROPN
ejpam-4737	203	5	{	{	PUNCT
ejpam-4737	203	6	y}δs(λ	y}δs(λ	NOUN
ejpam-4737	203	7	,	,	PUNCT
ejpam-4737	203	8	p	p	NOUN
ejpam-4737	203	9	)	)	PUNCT
ejpam-4737	203	10	but	but	CCONJ
ejpam-4737	203	11	y	y	PROPN
ejpam-4737	203	12	∈	∈	PROPN
ejpam-4737	203	13	{	{	PUNCT
ejpam-4737	203	14	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	203	15	,	,	PUNCT
ejpam-4737	203	16	p	p	NOUN
ejpam-4737	203	17	)	)	PUNCT
ejpam-4737	203	18	.	.	PUNCT
ejpam-4737	204	1	this	this	PRON
ejpam-4737	204	2	means	mean	VERB
ejpam-4737	204	3	that	that	SCONJ
ejpam-4737	204	4	the	the	DET
ejpam-4737	204	5	complement	complement	NOUN
ejpam-4737	204	6	of	of	ADP
ejpam-4737	204	7	{	{	PUNCT
ejpam-4737	204	8	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	204	9	,	,	PUNCT
ejpam-4737	204	10	p	p	NOUN
ejpam-4737	204	11	)	)	PUNCT
ejpam-4737	204	12	contains	contain	VERB
ejpam-4737	204	13	y.	y.	PROPN
ejpam-4737	204	14	thus	thus	ADV
ejpam-4737	204	15	,	,	PUNCT
ejpam-4737	204	16	the	the	DET
ejpam-4737	204	17	set	set	NOUN
ejpam-4737	204	18	{	{	PUNCT
ejpam-4737	204	19	y	y	NOUN
ejpam-4737	204	20	}	}	PUNCT
ejpam-4737	204	21	is	be	AUX
ejpam-4737	204	22	a	a	DET
ejpam-4737	204	23	subset	subset	NOUN
ejpam-4737	204	24	of	of	ADP
ejpam-4737	204	25	the	the	DET
ejpam-4737	204	26	complement	complement	NOUN
ejpam-4737	204	27	of	of	ADP
ejpam-4737	204	28	{	{	PUNCT
ejpam-4737	204	29	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	204	30	,	,	PUNCT
ejpam-4737	204	31	p	p	NOUN
ejpam-4737	204	32	)	)	PUNCT
ejpam-4737	204	33	.	.	PUNCT
ejpam-4737	205	1	this	this	PRON
ejpam-4737	205	2	implies	imply	VERB
ejpam-4737	205	3	that	that	SCONJ
ejpam-4737	205	4	{	{	PUNCT
ejpam-4737	205	5	y}δs(λ	y}δs(λ	NOUN
ejpam-4737	205	6	,	,	PUNCT
ejpam-4737	205	7	p	p	NOUN
ejpam-4737	205	8	)	)	PUNCT
ejpam-4737	205	9	is	be	AUX
ejpam-4737	205	10	a	a	DET
ejpam-4737	205	11	subset	subset	NOUN
ejpam-4737	205	12	of	of	ADP
ejpam-4737	205	13	the	the	DET
ejpam-4737	205	14	complement	complement	NOUN
ejpam-4737	205	15	of	of	ADP
ejpam-4737	205	16	{	{	PUNCT
ejpam-4737	205	17	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	205	18	,	,	PUNCT
ejpam-4737	205	19	p	p	NOUN
ejpam-4737	205	20	)	)	PUNCT
ejpam-4737	205	21	.	.	PUNCT
ejpam-4737	206	1	now	now	ADV
ejpam-4737	206	2	the	the	DET
ejpam-4737	206	3	complement	complement	NOUN
ejpam-4737	206	4	of	of	ADP
ejpam-4737	206	5	{	{	PUNCT
ejpam-4737	206	6	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	206	7	,	,	PUNCT
ejpam-4737	206	8	p	p	NOUN
ejpam-4737	206	9	)	)	PUNCT
ejpam-4737	206	10	contains	contain	VERB
ejpam-4737	206	11	x	x	PUNCT
ejpam-4737	206	12	which	which	PRON
ejpam-4737	206	13	is	be	AUX
ejpam-4737	206	14	a	a	DET
ejpam-4737	206	15	contradiction	contradiction	NOUN
ejpam-4737	206	16	.	.	PUNCT
ejpam-4737	207	1	conversely	conversely	ADV
ejpam-4737	207	2	,	,	PUNCT
ejpam-4737	207	3	suppose	suppose	VERB
ejpam-4737	207	4	that	that	SCONJ
ejpam-4737	207	5	{	{	PUNCT
ejpam-4737	207	6	x	x	NOUN
ejpam-4737	207	7	}	}	PUNCT
ejpam-4737	207	8	⊆	⊆	NUM
ejpam-4737	207	9	u	u	NOUN
ejpam-4737	207	10	∈	∈	NOUN
ejpam-4737	207	11	δs(λ	δs(λ	NOUN
ejpam-4737	207	12	,	,	PUNCT
ejpam-4737	207	13	p)o(x	p)o(x	VERB
ejpam-4737	207	14	,	,	PUNCT
ejpam-4737	207	15	τ	τ	PROPN
ejpam-4737	207	16	)	)	PUNCT
ejpam-4737	207	17	,	,	PUNCT
ejpam-4737	207	18	but	but	CCONJ
ejpam-4737	207	19	{	{	PUNCT
ejpam-4737	207	20	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	207	21	,	,	PUNCT
ejpam-4737	207	22	p	p	NOUN
ejpam-4737	207	23	)	)	PUNCT
ejpam-4737	207	24	is	be	AUX
ejpam-4737	207	25	not	not	PART
ejpam-4737	207	26	a	a	DET
ejpam-4737	207	27	subset	subset	NOUN
ejpam-4737	207	28	of	of	ADP
ejpam-4737	207	29	u	u	PROPN
ejpam-4737	207	30	.	.	PUNCT
ejpam-4737	208	1	this	this	PRON
ejpam-4737	208	2	means	mean	VERB
ejpam-4737	208	3	that	that	SCONJ
ejpam-4737	208	4	{	{	PUNCT
ejpam-4737	208	5	x}δs(λ	x}δs(λ	PROPN
ejpam-4737	208	6	,	,	PUNCT
ejpam-4737	208	7	p	p	NOUN
ejpam-4737	208	8	)	)	PUNCT
ejpam-4737	208	9	and	and	CCONJ
ejpam-4737	208	10	the	the	DET
ejpam-4737	208	11	complement	complement	NOUN
ejpam-4737	208	12	of	of	ADP
ejpam-4737	208	13	u	u	NOUN
ejpam-4737	208	14	are	be	AUX
ejpam-4737	208	15	not	not	PART
ejpam-4737	208	16	disjoint	disjoint	ADJ
ejpam-4737	208	17	.	.	PUNCT
ejpam-4737	209	1	let	let	VERB
ejpam-4737	209	2	y	y	PRON
ejpam-4737	209	3	belongs	belong	VERB
ejpam-4737	209	4	to	to	ADP
ejpam-4737	209	5	their	their	PRON
ejpam-4737	209	6	intersection	intersection	NOUN
ejpam-4737	209	7	.	.	PUNCT
ejpam-4737	210	1	now	now	ADV
ejpam-4737	210	2	we	we	PRON
ejpam-4737	210	3	have	have	VERB
ejpam-4737	210	4	x	x	X
ejpam-4737	210	5	∈	∈	PROPN
ejpam-4737	210	6	{	{	PUNCT
ejpam-4737	210	7	y}δs(λ	y}δs(λ	NOUN
ejpam-4737	210	8	,	,	PUNCT
ejpam-4737	210	9	p	p	NOUN
ejpam-4737	210	10	)	)	PUNCT
ejpam-4737	210	11	which	which	PRON
ejpam-4737	210	12	is	be	AUX
ejpam-4737	210	13	a	a	DET
ejpam-4737	210	14	subset	subset	NOUN
ejpam-4737	210	15	of	of	ADP
ejpam-4737	210	16	the	the	DET
ejpam-4737	210	17	complement	complement	NOUN
ejpam-4737	210	18	of	of	ADP
ejpam-4737	210	19	u	u	PROPN
ejpam-4737	210	20	and	and	CCONJ
ejpam-4737	210	21	x	x	PUNCT
ejpam-4737	210	22	̸∈	̸∈	PROPN
ejpam-4737	210	23	u	u	PROPN
ejpam-4737	210	24	.	.	PUNCT
ejpam-4737	211	1	this	this	PRON
ejpam-4737	211	2	is	be	AUX
ejpam-4737	211	3	a	a	DET
ejpam-4737	211	4	contradiction	contradiction	NOUN
ejpam-4737	211	5	.	.	PUNCT
ejpam-4737	212	1	c.	c.	PROPN
ejpam-4737	212	2	boonpok	boonpok	PROPN
ejpam-4737	212	3	,	,	PUNCT
ejpam-4737	212	4	m.	m.	NOUN
ejpam-4737	212	5	thongmoon	thongmoon	PROPN
ejpam-4737	212	6	/	/	SYM
ejpam-4737	212	7	eur	eur	PROPN
ejpam-4737	212	8	.	.	PUNCT
ejpam-4737	213	1	j.	j.	PROPN
ejpam-4737	213	2	pure	pure	PROPN
ejpam-4737	213	3	appl	appl	PROPN
ejpam-4737	213	4	.	.	PROPN
ejpam-4737	213	5	math	math	PROPN
ejpam-4737	213	6	,	,	PUNCT
ejpam-4737	213	7	16	16	NUM
ejpam-4737	213	8	(	(	PUNCT
ejpam-4737	213	9	3	3	NUM
ejpam-4737	213	10	)	)	PUNCT
ejpam-4737	213	11	(	(	PUNCT
ejpam-4737	213	12	2023	2023	NUM
ejpam-4737	213	13	)	)	PUNCT
ejpam-4737	213	14	,	,	PUNCT
ejpam-4737	213	15	1434	1434	NUM
ejpam-4737	213	16	-	-	SYM
ejpam-4737	213	17	1447	1447	NUM
ejpam-4737	213	18	1440	1440	NUM
ejpam-4737	213	19	4	4	NUM
ejpam-4737	213	20	.	.	PUNCT
ejpam-4737	213	21	characterizations	characterization	NOUN
ejpam-4737	213	22	of	of	ADP
ejpam-4737	213	23	s(λ	s(λ	PROPN
ejpam-4737	213	24	,	,	PUNCT
ejpam-4737	213	25	p)-connected	p)-connecte	VERB
ejpam-4737	213	26	spaces	space	NOUN
ejpam-4737	213	27	we	we	PRON
ejpam-4737	213	28	begin	begin	VERB
ejpam-4737	213	29	this	this	DET
ejpam-4737	213	30	section	section	NOUN
ejpam-4737	213	31	by	by	ADP
ejpam-4737	213	32	introducing	introduce	VERB
ejpam-4737	213	33	the	the	DET
ejpam-4737	213	34	concept	concept	NOUN
ejpam-4737	213	35	of	of	ADP
ejpam-4737	213	36	s(λ	s(λ	PROPN
ejpam-4737	213	37	,	,	PUNCT
ejpam-4737	213	38	p)-connected	p)-connecte	VERB
ejpam-4737	213	39	spaces	space	NOUN
ejpam-4737	213	40	.	.	PUNCT
ejpam-4737	214	1	definition	definition	NOUN
ejpam-4737	214	2	9	9	NUM
ejpam-4737	214	3	.	.	PUNCT
ejpam-4737	215	1	a	a	DET
ejpam-4737	215	2	topological	topological	ADJ
ejpam-4737	215	3	space	space	NOUN
ejpam-4737	215	4	(	(	PUNCT
ejpam-4737	215	5	x	x	X
ejpam-4737	215	6	,	,	PUNCT
ejpam-4737	215	7	τ	τ	X
ejpam-4737	215	8	)	)	PUNCT
ejpam-4737	215	9	is	be	AUX
ejpam-4737	215	10	called	call	VERB
ejpam-4737	215	11	s(λ	s(λ	PROPN
ejpam-4737	215	12	,	,	PUNCT
ejpam-4737	215	13	p)-connected	p)-connecte	VERB
ejpam-4737	215	14	if	if	SCONJ
ejpam-4737	215	15	x	x	PRON
ejpam-4737	215	16	can	can	AUX
ejpam-4737	215	17	not	not	PART
ejpam-4737	215	18	be	be	AUX
ejpam-4737	215	19	expressed	express	VERB
ejpam-4737	215	20	by	by	ADP
ejpam-4737	215	21	the	the	DET
ejpam-4737	215	22	disjoint	disjoint	PROPN
ejpam-4737	215	23	union	union	NOUN
ejpam-4737	215	24	of	of	ADP
ejpam-4737	215	25	two	two	NUM
ejpam-4737	215	26	nonempty	nonempty	ADJ
ejpam-4737	215	27	s(λ	s(λ	NOUN
ejpam-4737	215	28	,	,	PUNCT
ejpam-4737	215	29	p)-open	p)-open	VERB
ejpam-4737	215	30	sets	set	NOUN
ejpam-4737	215	31	.	.	PUNCT
ejpam-4737	216	1	theorem	theorem	ADJ
ejpam-4737	216	2	8	8	NUM
ejpam-4737	216	3	.	.	PUNCT
ejpam-4737	217	1	for	for	ADP
ejpam-4737	217	2	a	a	DET
ejpam-4737	217	3	topological	topological	ADJ
ejpam-4737	217	4	space	space	NOUN
ejpam-4737	217	5	(	(	PUNCT
ejpam-4737	217	6	x	x	X
ejpam-4737	217	7	,	,	PUNCT
ejpam-4737	217	8	τ	τ	PROPN
ejpam-4737	217	9	)	)	PUNCT
ejpam-4737	217	10	,	,	PUNCT
ejpam-4737	217	11	the	the	DET
ejpam-4737	217	12	following	follow	VERB
ejpam-4737	217	13	properties	property	NOUN
ejpam-4737	217	14	are	be	AUX
ejpam-4737	217	15	equivalent	equivalent	ADJ
ejpam-4737	217	16	:	:	PUNCT
ejpam-4737	217	17	(	(	PUNCT
ejpam-4737	217	18	1	1	X
ejpam-4737	217	19	)	)	PUNCT
ejpam-4737	217	20	v	v	NOUN
ejpam-4737	217	21	(	(	PUNCT
ejpam-4737	217	22	λ	λ	PROPN
ejpam-4737	217	23	,	,	PUNCT
ejpam-4737	217	24	p	p	NOUN
ejpam-4737	217	25	)	)	PUNCT
ejpam-4737	217	26	=	=	PUNCT
ejpam-4737	218	1	x	x	PUNCT
ejpam-4737	218	2	for	for	ADP
ejpam-4737	218	3	every	every	DET
ejpam-4737	218	4	nonempty	nonempty	ADJ
ejpam-4737	218	5	(	(	PUNCT
ejpam-4737	218	6	λ	λ	NOUN
ejpam-4737	218	7	,	,	PUNCT
ejpam-4737	218	8	p)-open	p)-open	VERB
ejpam-4737	218	9	set	set	VERB
ejpam-4737	218	10	v	v	NOUN
ejpam-4737	218	11	of	of	ADP
ejpam-4737	218	12	x	x	PRON
ejpam-4737	218	13	;	;	PUNCT
ejpam-4737	218	14	(	(	PUNCT
ejpam-4737	218	15	2	2	X
ejpam-4737	218	16	)	)	PUNCT
ejpam-4737	218	17	(	(	PUNCT
ejpam-4737	218	18	x	x	X
ejpam-4737	218	19	,	,	PUNCT
ejpam-4737	218	20	τ	τ	X
ejpam-4737	218	21	)	)	PUNCT
ejpam-4737	218	22	is	be	AUX
ejpam-4737	218	23	s(λ	s(λ	PROPN
ejpam-4737	218	24	,	,	PUNCT
ejpam-4737	218	25	p)-connected	p)-connecte	VERB
ejpam-4737	218	26	;	;	PUNCT
ejpam-4737	218	27	(	(	PUNCT
ejpam-4737	218	28	3	3	X
ejpam-4737	218	29	)	)	PUNCT
ejpam-4737	218	30	x	x	PRON
ejpam-4737	218	31	can	can	AUX
ejpam-4737	218	32	not	not	PART
ejpam-4737	218	33	be	be	AUX
ejpam-4737	218	34	expressed	express	VERB
ejpam-4737	218	35	by	by	ADP
ejpam-4737	218	36	the	the	DET
ejpam-4737	218	37	disjoint	disjoint	PROPN
ejpam-4737	218	38	union	union	NOUN
ejpam-4737	218	39	of	of	ADP
ejpam-4737	218	40	two	two	NUM
ejpam-4737	218	41	nonempty	nonempty	ADJ
ejpam-4737	218	42	δs(λ	δs(λ	NOUN
ejpam-4737	218	43	,	,	PUNCT
ejpam-4737	218	44	p)-open	p)-open	VERB
ejpam-4737	218	45	sets	set	NOUN
ejpam-4737	218	46	;	;	PUNCT
ejpam-4737	218	47	(	(	PUNCT
ejpam-4737	218	48	4	4	X
ejpam-4737	218	49	)	)	PUNCT
ejpam-4737	218	50	v	v	NOUN
ejpam-4737	218	51	δs(λ	δs(λ	NOUN
ejpam-4737	218	52	,	,	PUNCT
ejpam-4737	218	53	p	p	NOUN
ejpam-4737	218	54	)	)	PUNCT
ejpam-4737	218	55	=	=	PUNCT
ejpam-4737	219	1	x	x	PUNCT
ejpam-4737	219	2	for	for	ADP
ejpam-4737	219	3	every	every	DET
ejpam-4737	219	4	nonempty	nonempty	ADJ
ejpam-4737	219	5	δs(λ	δs(λ	NOUN
ejpam-4737	219	6	,	,	PUNCT
ejpam-4737	219	7	p)-open	p)-open	VERB
ejpam-4737	219	8	set	set	VERB
ejpam-4737	219	9	v	v	NOUN
ejpam-4737	219	10	of	of	ADP
ejpam-4737	219	11	x.	x.	NOUN
ejpam-4737	219	12	proof	proof	NOUN
ejpam-4737	219	13	.	.	PUNCT
ejpam-4737	220	1	(	(	PUNCT
ejpam-4737	220	2	1	1	X
ejpam-4737	220	3	)	)	PUNCT
ejpam-4737	220	4	⇔	⇔	NOUN
ejpam-4737	220	5	(	(	PUNCT
ejpam-4737	220	6	2	2	NUM
ejpam-4737	220	7	):	):	PUNCT
ejpam-4737	220	8	the	the	DET
ejpam-4737	220	9	proof	proof	NOUN
ejpam-4737	220	10	follows	follow	VERB
ejpam-4737	220	11	from	from	ADP
ejpam-4737	220	12	theorem	theorem	ADJ
ejpam-4737	220	13	4.3	4.3	NUM
ejpam-4737	220	14	of	of	ADP
ejpam-4737	220	15	[	[	X
ejpam-4737	220	16	10	10	NUM
ejpam-4737	220	17	]	]	PUNCT
ejpam-4737	220	18	.	.	PUNCT
ejpam-4737	221	1	(	(	PUNCT
ejpam-4737	221	2	2	2	X
ejpam-4737	221	3	)	)	PUNCT
ejpam-4737	221	4	⇒	⇒	NOUN
ejpam-4737	221	5	(	(	PUNCT
ejpam-4737	221	6	3	3	NUM
ejpam-4737	221	7	):	):	PUNCT
ejpam-4737	221	8	suppose	suppose	VERB
ejpam-4737	221	9	that	that	SCONJ
ejpam-4737	221	10	there	there	PRON
ejpam-4737	221	11	exist	exist	VERB
ejpam-4737	221	12	two	two	NUM
ejpam-4737	221	13	nonempty	nonempty	ADJ
ejpam-4737	221	14	δs(λ	δs(λ	NOUN
ejpam-4737	221	15	,	,	PUNCT
ejpam-4737	221	16	p)-open	p)-open	VERB
ejpam-4737	221	17	sets	set	VERB
ejpam-4737	221	18	v1	v1	NOUN
ejpam-4737	221	19	,	,	PUNCT
ejpam-4737	221	20	v2	v2	VERB
ejpam-4737	221	21	such	such	ADJ
ejpam-4737	221	22	that	that	DET
ejpam-4737	221	23	v1	v1	NOUN
ejpam-4737	221	24	∩	∩	ADJ
ejpam-4737	221	25	v2	v2	NOUN
ejpam-4737	221	26	=	=	SYM
ejpam-4737	221	27	∅	∅	NOUN
ejpam-4737	221	28	and	and	CCONJ
ejpam-4737	221	29	v1	v1	VERB
ejpam-4737	221	30	∪	∪	ADJ
ejpam-4737	221	31	v2	v2	NOUN
ejpam-4737	221	32	=	=	SYM
ejpam-4737	221	33	x.	x.	NOUN
ejpam-4737	221	34	since	since	SCONJ
ejpam-4737	221	35	δs(λ	δs(λ	NOUN
ejpam-4737	221	36	,	,	PUNCT
ejpam-4737	221	37	p)o(x	p)o(x	ADJ
ejpam-4737	221	38	,	,	PUNCT
ejpam-4737	221	39	τ	τ	PROPN
ejpam-4737	221	40	)	)	PUNCT
ejpam-4737	221	41	⊆	⊆	NUM
ejpam-4737	221	42	s(λ	s(λ	PROPN
ejpam-4737	221	43	,	,	PUNCT
ejpam-4737	221	44	p)o(x	p)o(x	ADJ
ejpam-4737	221	45	,	,	PUNCT
ejpam-4737	221	46	τ	τ	PROPN
ejpam-4737	221	47	)	)	PUNCT
ejpam-4737	221	48	,	,	PUNCT
ejpam-4737	221	49	this	this	PRON
ejpam-4737	221	50	shows	show	VERB
ejpam-4737	221	51	that	that	SCONJ
ejpam-4737	221	52	(	(	PUNCT
ejpam-4737	221	53	x	x	X
ejpam-4737	221	54	,	,	PUNCT
ejpam-4737	221	55	τ	τ	X
ejpam-4737	221	56	)	)	PUNCT
ejpam-4737	221	57	is	be	AUX
ejpam-4737	221	58	not	not	PART
ejpam-4737	221	59	s(λ	s(λ	NOUN
ejpam-4737	221	60	,	,	PUNCT
ejpam-4737	221	61	p)-connected	p)-connecte	VERB
ejpam-4737	221	62	.	.	PUNCT
ejpam-4737	222	1	(	(	PUNCT
ejpam-4737	222	2	3	3	X
ejpam-4737	222	3	)	)	PUNCT
ejpam-4737	222	4	⇒	⇒	NOUN
ejpam-4737	222	5	(	(	PUNCT
ejpam-4737	222	6	4	4	NUM
ejpam-4737	222	7	):	):	PUNCT
ejpam-4737	222	8	suppose	suppose	VERB
ejpam-4737	222	9	that	that	SCONJ
ejpam-4737	222	10	v	v	NOUN
ejpam-4737	222	11	δs(λ	δs(λ	NOUN
ejpam-4737	222	12	,	,	PUNCT
ejpam-4737	222	13	p	p	NOUN
ejpam-4737	222	14	)	)	PUNCT
ejpam-4737	222	15	̸=	̸=	PROPN
ejpam-4737	222	16	x	x	PUNCT
ejpam-4737	222	17	for	for	ADP
ejpam-4737	222	18	some	some	DET
ejpam-4737	222	19	nonempty	nonempty	ADJ
ejpam-4737	222	20	δs(λ	δs(λ	NOUN
ejpam-4737	222	21	,	,	PUNCT
ejpam-4737	222	22	p)-open	p)-open	VERB
ejpam-4737	222	23	set	set	VERB
ejpam-4737	222	24	v	v	NUM
ejpam-4737	222	25	of	of	ADP
ejpam-4737	222	26	x.	x.	NOUN
ejpam-4737	222	27	then	then	ADV
ejpam-4737	222	28	,	,	PUNCT
ejpam-4737	222	29	x	x	PUNCT
ejpam-4737	222	30	−	−	NOUN
ejpam-4737	222	31	v	v	NUM
ejpam-4737	222	32	δs(λ	δs(λ	NOUN
ejpam-4737	222	33	,	,	PUNCT
ejpam-4737	222	34	p	p	NOUN
ejpam-4737	222	35	)	)	PUNCT
ejpam-4737	222	36	̸=	̸=	PROPN
ejpam-4737	222	37	∅	∅	NOUN
ejpam-4737	222	38	and	and	CCONJ
ejpam-4737	222	39	x	x	X
ejpam-4737	222	40	=	=	SYM
ejpam-4737	222	41	(	(	PUNCT
ejpam-4737	222	42	x	x	X
ejpam-4737	222	43	−	−	NOUN
ejpam-4737	222	44	v	v	NUM
ejpam-4737	222	45	δs(λ	δs(λ	NOUN
ejpam-4737	222	46	,	,	PUNCT
ejpam-4737	222	47	p	p	NOUN
ejpam-4737	222	48	)	)	PUNCT
ejpam-4737	222	49	)	)	PUNCT
ejpam-4737	222	50	∪	∪	ADP
ejpam-4737	222	51	v	v	NUM
ejpam-4737	222	52	δs(λ	δs(λ	NOUN
ejpam-4737	222	53	,	,	PUNCT
ejpam-4737	222	54	p	p	NOUN
ejpam-4737	222	55	)	)	PUNCT
ejpam-4737	222	56	.	.	PUNCT
ejpam-4737	223	1	since	since	SCONJ
ejpam-4737	223	2	δs(λ	δs(λ	NOUN
ejpam-4737	223	3	,	,	PUNCT
ejpam-4737	223	4	p)o(x	p)o(x	ADJ
ejpam-4737	223	5	,	,	PUNCT
ejpam-4737	223	6	τ	τ	PROPN
ejpam-4737	223	7	)	)	PUNCT
ejpam-4737	223	8	⊆	⊆	NUM
ejpam-4737	223	9	s(λ	s(λ	NOUN
ejpam-4737	223	10	,	,	PUNCT
ejpam-4737	223	11	p)r(x	p)r(x	PROPN
ejpam-4737	223	12	,	,	PUNCT
ejpam-4737	223	13	τ	τ	PROPN
ejpam-4737	223	14	)	)	PUNCT
ejpam-4737	223	15	,	,	PUNCT
ejpam-4737	223	16	by	by	ADP
ejpam-4737	223	17	lemma	lemma	PROPN
ejpam-4737	223	18	4	4	NUM
ejpam-4737	223	19	and	and	CCONJ
ejpam-4737	223	20	5	5	NUM
ejpam-4737	223	21	,	,	PUNCT
ejpam-4737	223	22	v	v	NOUN
ejpam-4737	223	23	δs(λ	δs(λ	NOUN
ejpam-4737	223	24	,	,	PUNCT
ejpam-4737	223	25	p	p	NOUN
ejpam-4737	223	26	)	)	PUNCT
ejpam-4737	223	27	=	=	SYM
ejpam-4737	223	28	v	v	ADP
ejpam-4737	223	29	s(λ	s(λ	PROPN
ejpam-4737	223	30	,	,	PUNCT
ejpam-4737	223	31	p	p	NOUN
ejpam-4737	223	32	)	)	PUNCT
ejpam-4737	223	33	∈	∈	PROPN
ejpam-4737	223	34	s(λ	s(λ	PROPN
ejpam-4737	223	35	,	,	PUNCT
ejpam-4737	223	36	p)r(x	p)r(x	PROPN
ejpam-4737	223	37	,	,	PUNCT
ejpam-4737	223	38	τ	τ	PROPN
ejpam-4737	223	39	)	)	PUNCT
ejpam-4737	223	40	.	.	PUNCT
ejpam-4737	224	1	moreover	moreover	ADV
ejpam-4737	224	2	,	,	PUNCT
ejpam-4737	224	3	since	since	SCONJ
ejpam-4737	224	4	s(λ	s(λ	PROPN
ejpam-4737	224	5	,	,	PUNCT
ejpam-4737	224	6	p)r(x	p)r(x	PROPN
ejpam-4737	224	7	,	,	PUNCT
ejpam-4737	224	8	τ	τ	PROPN
ejpam-4737	224	9	)	)	PUNCT
ejpam-4737	224	10	⊆	⊆	NUM
ejpam-4737	224	11	δs(λ	δs(λ	NOUN
ejpam-4737	224	12	,	,	PUNCT
ejpam-4737	224	13	p)o(x	p)o(x	ADJ
ejpam-4737	224	14	,	,	PUNCT
ejpam-4737	224	15	τ	τ	PROPN
ejpam-4737	224	16	)	)	PUNCT
ejpam-4737	224	17	,	,	PUNCT
ejpam-4737	224	18	(	(	PUNCT
ejpam-4737	224	19	x	x	X
ejpam-4737	224	20	−	−	NOUN
ejpam-4737	224	21	v	v	NUM
ejpam-4737	224	22	δs(λ	δs(λ	NOUN
ejpam-4737	224	23	,	,	PUNCT
ejpam-4737	224	24	p	p	NOUN
ejpam-4737	224	25	)	)	PUNCT
ejpam-4737	224	26	)	)	PUNCT
ejpam-4737	224	27	and	and	CCONJ
ejpam-4737	224	28	v	v	X
ejpam-4737	224	29	δs(λ	δs(λ	NOUN
ejpam-4737	224	30	,	,	PUNCT
ejpam-4737	224	31	p	p	NOUN
ejpam-4737	224	32	)	)	PUNCT
ejpam-4737	224	33	are	be	AUX
ejpam-4737	224	34	δs(λ	δs(λ	NOUN
ejpam-4737	224	35	,	,	PUNCT
ejpam-4737	224	36	p)-open	p)-open	ADJ
ejpam-4737	224	37	.	.	PUNCT
ejpam-4737	225	1	(	(	PUNCT
ejpam-4737	225	2	4	4	X
ejpam-4737	225	3	)	)	PUNCT
ejpam-4737	225	4	⇒	⇒	NOUN
ejpam-4737	225	5	(	(	PUNCT
ejpam-4737	225	6	1	1	NUM
ejpam-4737	225	7	):	):	PUNCT
ejpam-4737	225	8	let	let	VERB
ejpam-4737	225	9	v	v	PART
ejpam-4737	225	10	be	be	AUX
ejpam-4737	225	11	any	any	PRON
ejpam-4737	225	12	nonempty	nonempty	ADJ
ejpam-4737	225	13	(	(	PUNCT
ejpam-4737	225	14	λ	λ	NOUN
ejpam-4737	225	15	,	,	PUNCT
ejpam-4737	225	16	p)-open	p)-open	VERB
ejpam-4737	225	17	set	set	VERB
ejpam-4737	225	18	of	of	ADP
ejpam-4737	225	19	x.	x.	NOUN
ejpam-4737	225	20	then	then	ADV
ejpam-4737	225	21	,	,	PUNCT
ejpam-4737	225	22	v	v	INTJ
ejpam-4737	225	23	(	(	PUNCT
ejpam-4737	225	24	λ	λ	PROPN
ejpam-4737	225	25	,	,	PUNCT
ejpam-4737	225	26	p	p	NOUN
ejpam-4737	225	27	)	)	PUNCT
ejpam-4737	225	28	is	be	AUX
ejpam-4737	225	29	r(λ	r(λ	NOUN
ejpam-4737	225	30	,	,	PUNCT
ejpam-4737	225	31	p)-closed	p)-close	VERB
ejpam-4737	225	32	and	and	CCONJ
ejpam-4737	225	33	hence	hence	ADV
ejpam-4737	225	34	s(λ	s(λ	PROPN
ejpam-4737	225	35	,	,	PUNCT
ejpam-4737	225	36	p)-regular	p)-regular	NOUN
ejpam-4737	225	37	.	.	PUNCT
ejpam-4737	226	1	thus	thus	ADV
ejpam-4737	226	2	,	,	PUNCT
ejpam-4737	226	3	v	v	INTJ
ejpam-4737	226	4	(	(	PUNCT
ejpam-4737	226	5	λ	λ	PROPN
ejpam-4737	226	6	,	,	PUNCT
ejpam-4737	226	7	p	p	NOUN
ejpam-4737	226	8	)	)	PUNCT
ejpam-4737	226	9	is	be	AUX
ejpam-4737	226	10	δs(λ	δs(λ	NOUN
ejpam-4737	226	11	,	,	PUNCT
ejpam-4737	226	12	p)-open	p)-open	VERB
ejpam-4737	226	13	and	and	CCONJ
ejpam-4737	226	14	x	x	X
ejpam-4737	226	15	=	=	PUNCT
ejpam-4737	227	1	[	[	X
ejpam-4737	227	2	v	v	X
ejpam-4737	227	3	(	(	PUNCT
ejpam-4737	227	4	λ	λ	PROPN
ejpam-4737	227	5	,	,	PUNCT
ejpam-4737	227	6	p)]δs(λ	p)]δs(λ	NOUN
ejpam-4737	227	7	,	,	PUNCT
ejpam-4737	227	8	p	p	NOUN
ejpam-4737	227	9	)	)	PUNCT
ejpam-4737	227	10	=	=	PUNCT
ejpam-4737	228	1	[	[	X
ejpam-4737	228	2	v	v	X
ejpam-4737	228	3	(	(	PUNCT
ejpam-4737	228	4	λ	λ	PROPN
ejpam-4737	228	5	,	,	PUNCT
ejpam-4737	228	6	p)]s(λ	p)]s(λ	NOUN
ejpam-4737	228	7	,	,	PUNCT
ejpam-4737	228	8	p	p	NOUN
ejpam-4737	228	9	)	)	PUNCT
ejpam-4737	228	10	=	=	SYM
ejpam-4737	228	11	v	v	X
ejpam-4737	228	12	(	(	PUNCT
ejpam-4737	228	13	λ	λ	PROPN
ejpam-4737	228	14	,	,	PUNCT
ejpam-4737	228	15	p	p	NOUN
ejpam-4737	228	16	)	)	PUNCT
ejpam-4737	228	17	.	.	PUNCT
ejpam-4737	229	1	theorem	theorem	VERB
ejpam-4737	229	2	9	9	NUM
ejpam-4737	229	3	.	.	X
ejpam-4737	230	1	for	for	ADP
ejpam-4737	230	2	a	a	DET
ejpam-4737	230	3	topological	topological	ADJ
ejpam-4737	230	4	space	space	NOUN
ejpam-4737	230	5	(	(	PUNCT
ejpam-4737	230	6	x	x	X
ejpam-4737	230	7	,	,	PUNCT
ejpam-4737	230	8	τ	τ	PROPN
ejpam-4737	230	9	)	)	PUNCT
ejpam-4737	230	10	,	,	PUNCT
ejpam-4737	230	11	the	the	DET
ejpam-4737	230	12	following	follow	VERB
ejpam-4737	230	13	properties	property	NOUN
ejpam-4737	230	14	are	be	AUX
ejpam-4737	230	15	equivalent	equivalent	ADJ
ejpam-4737	230	16	:	:	PUNCT
ejpam-4737	230	17	(	(	PUNCT
ejpam-4737	230	18	1	1	X
ejpam-4737	230	19	)	)	PUNCT
ejpam-4737	230	20	(	(	PUNCT
ejpam-4737	230	21	x	x	X
ejpam-4737	230	22	,	,	PUNCT
ejpam-4737	230	23	τ	τ	X
ejpam-4737	230	24	)	)	PUNCT
ejpam-4737	230	25	is	be	AUX
ejpam-4737	230	26	s(λ	s(λ	PROPN
ejpam-4737	230	27	,	,	PUNCT
ejpam-4737	230	28	p)-connected	p)-connecte	VERB
ejpam-4737	230	29	;	;	PUNCT
ejpam-4737	230	30	(	(	PUNCT
ejpam-4737	230	31	2	2	X
ejpam-4737	230	32	)	)	PUNCT
ejpam-4737	230	33	v	v	NOUN
ejpam-4737	230	34	δs(λ	δs(λ	NOUN
ejpam-4737	230	35	,	,	PUNCT
ejpam-4737	230	36	p	p	NOUN
ejpam-4737	230	37	)	)	PUNCT
ejpam-4737	230	38	=	=	PUNCT
ejpam-4737	231	1	x	x	PUNCT
ejpam-4737	231	2	for	for	ADP
ejpam-4737	231	3	every	every	DET
ejpam-4737	231	4	nonempty	nonempty	NOUN
ejpam-4737	231	5	v	v	ADP
ejpam-4737	231	6	∈	∈	PROPN
ejpam-4737	231	7	β(λ	β(λ	X
ejpam-4737	231	8	,	,	PUNCT
ejpam-4737	231	9	p)o(x	p)o(x	ADJ
ejpam-4737	231	10	,	,	PUNCT
ejpam-4737	231	11	τ	τ	PROPN
ejpam-4737	231	12	)	)	PUNCT
ejpam-4737	231	13	;	;	PUNCT
ejpam-4737	231	14	(	(	PUNCT
ejpam-4737	231	15	3	3	X
ejpam-4737	231	16	)	)	PUNCT
ejpam-4737	231	17	v	v	NOUN
ejpam-4737	231	18	δs(λ	δs(λ	NOUN
ejpam-4737	231	19	,	,	PUNCT
ejpam-4737	231	20	p	p	NOUN
ejpam-4737	231	21	)	)	PUNCT
ejpam-4737	231	22	=	=	PUNCT
ejpam-4737	232	1	x	x	PUNCT
ejpam-4737	232	2	for	for	ADP
ejpam-4737	232	3	every	every	DET
ejpam-4737	232	4	nonempty	nonempty	NOUN
ejpam-4737	232	5	v	v	ADP
ejpam-4737	232	6	∈	∈	PROPN
ejpam-4737	232	7	s(λ	s(λ	PROPN
ejpam-4737	232	8	,	,	PUNCT
ejpam-4737	232	9	p)o(x	p)o(x	ADJ
ejpam-4737	232	10	,	,	PUNCT
ejpam-4737	232	11	τ	τ	PROPN
ejpam-4737	232	12	)	)	PUNCT
ejpam-4737	232	13	;	;	PUNCT
ejpam-4737	232	14	(	(	PUNCT
ejpam-4737	232	15	4	4	X
ejpam-4737	232	16	)	)	PUNCT
ejpam-4737	232	17	v	v	NOUN
ejpam-4737	232	18	δs(λ	δs(λ	NOUN
ejpam-4737	232	19	,	,	PUNCT
ejpam-4737	232	20	p	p	NOUN
ejpam-4737	232	21	)	)	PUNCT
ejpam-4737	232	22	=	=	PUNCT
ejpam-4737	233	1	x	x	PUNCT
ejpam-4737	233	2	for	for	ADP
ejpam-4737	233	3	every	every	DET
ejpam-4737	233	4	nonempty	nonempty	ADJ
ejpam-4737	233	5	v	v	ADP
ejpam-4737	233	6	∈	∈	PROPN
ejpam-4737	233	7	p(λ	p(λ	NOUN
ejpam-4737	233	8	,	,	PUNCT
ejpam-4737	233	9	p)o(x	p)o(x	ADJ
ejpam-4737	233	10	,	,	PUNCT
ejpam-4737	233	11	τ	τ	PROPN
ejpam-4737	233	12	)	)	PUNCT
ejpam-4737	233	13	;	;	PUNCT
ejpam-4737	233	14	(	(	PUNCT
ejpam-4737	233	15	5	5	X
ejpam-4737	233	16	)	)	PUNCT
ejpam-4737	233	17	v	v	NOUN
ejpam-4737	233	18	δs(λ	δs(λ	NOUN
ejpam-4737	233	19	,	,	PUNCT
ejpam-4737	233	20	p	p	NOUN
ejpam-4737	233	21	)	)	PUNCT
ejpam-4737	233	22	=	=	PUNCT
ejpam-4737	234	1	x	x	PUNCT
ejpam-4737	234	2	for	for	ADP
ejpam-4737	234	3	every	every	DET
ejpam-4737	234	4	nonempty	nonempty	ADJ
ejpam-4737	234	5	v	v	ADP
ejpam-4737	234	6	∈	∈	PROPN
ejpam-4737	234	7	α(λ	α(λ	PROPN
ejpam-4737	234	8	,	,	PUNCT
ejpam-4737	234	9	p)o(x	p)o(x	ADJ
ejpam-4737	234	10	,	,	PUNCT
ejpam-4737	234	11	τ	τ	PROPN
ejpam-4737	234	12	)	)	PUNCT
ejpam-4737	234	13	;	;	PUNCT
ejpam-4737	234	14	(	(	PUNCT
ejpam-4737	234	15	6	6	X
ejpam-4737	234	16	)	)	PUNCT
ejpam-4737	234	17	v	v	NOUN
ejpam-4737	234	18	δs(λ	δs(λ	NOUN
ejpam-4737	234	19	,	,	PUNCT
ejpam-4737	234	20	p	p	NOUN
ejpam-4737	234	21	)	)	PUNCT
ejpam-4737	234	22	=	=	PUNCT
ejpam-4737	235	1	x	x	PUNCT
ejpam-4737	235	2	for	for	ADP
ejpam-4737	235	3	every	every	DET
ejpam-4737	235	4	nonempty	nonempty	NOUN
ejpam-4737	235	5	v	v	ADP
ejpam-4737	235	6	∈	∈	PROPN
ejpam-4737	235	7	λpo(x	λpo(x	PROPN
ejpam-4737	235	8	,	,	PUNCT
ejpam-4737	235	9	τ	τ	PROPN
ejpam-4737	235	10	)	)	PUNCT
ejpam-4737	235	11	.	.	PUNCT
ejpam-4737	236	1	c.	c.	PROPN
ejpam-4737	236	2	boonpok	boonpok	PROPN
ejpam-4737	236	3	,	,	PUNCT
ejpam-4737	236	4	m.	m.	NOUN
ejpam-4737	236	5	thongmoon	thongmoon	PROPN
ejpam-4737	236	6	/	/	SYM
ejpam-4737	236	7	eur	eur	PROPN
ejpam-4737	236	8	.	.	PUNCT
ejpam-4737	237	1	j.	j.	PROPN
ejpam-4737	237	2	pure	pure	PROPN
ejpam-4737	237	3	appl	appl	PROPN
ejpam-4737	237	4	.	.	PROPN
ejpam-4737	237	5	math	math	PROPN
ejpam-4737	237	6	,	,	PUNCT
ejpam-4737	237	7	16	16	NUM
ejpam-4737	237	8	(	(	PUNCT
ejpam-4737	237	9	3	3	NUM
ejpam-4737	237	10	)	)	PUNCT
ejpam-4737	237	11	(	(	PUNCT
ejpam-4737	237	12	2023	2023	NUM
ejpam-4737	237	13	)	)	PUNCT
ejpam-4737	237	14	,	,	PUNCT
ejpam-4737	237	15	1434	1434	NUM
ejpam-4737	237	16	-	-	SYM
ejpam-4737	237	17	1447	1447	NUM
ejpam-4737	237	18	1441	1441	NUM
ejpam-4737	237	19	proof	proof	NOUN
ejpam-4737	237	20	.	.	PUNCT
ejpam-4737	238	1	(	(	PUNCT
ejpam-4737	238	2	1	1	X
ejpam-4737	238	3	)	)	PUNCT
ejpam-4737	238	4	⇒	⇒	NOUN
ejpam-4737	238	5	(	(	PUNCT
ejpam-4737	238	6	2	2	NUM
ejpam-4737	238	7	):	):	PUNCT
ejpam-4737	238	8	let	let	VERB
ejpam-4737	238	9	v	v	PART
ejpam-4737	238	10	be	be	AUX
ejpam-4737	238	11	any	any	PRON
ejpam-4737	238	12	nonempty	nonempty	ADJ
ejpam-4737	238	13	β(λ	β(λ	NOUN
ejpam-4737	238	14	,	,	PUNCT
ejpam-4737	238	15	p)-open	p)-open	VERB
ejpam-4737	238	16	set	set	VERB
ejpam-4737	238	17	and	and	CCONJ
ejpam-4737	238	18	u	u	NOUN
ejpam-4737	238	19	be	be	VERB
ejpam-4737	238	20	any	any	DET
ejpam-4737	238	21	nonempty	nonempty	ADJ
ejpam-4737	238	22	δs(λ	δs(λ	NOUN
ejpam-4737	238	23	,	,	PUNCT
ejpam-4737	238	24	p)-open	p)-open	VERB
ejpam-4737	238	25	set	set	VERB
ejpam-4737	238	26	.	.	PUNCT
ejpam-4737	239	1	then	then	ADV
ejpam-4737	239	2	,	,	PUNCT
ejpam-4737	239	3	[	[	X
ejpam-4737	239	4	v	v	X
ejpam-4737	239	5	(	(	PUNCT
ejpam-4737	239	6	λ	λ	PROPN
ejpam-4737	239	7	,	,	PUNCT
ejpam-4737	239	8	p)](λ	p)](λ	ADJ
ejpam-4737	239	9	,	,	PUNCT
ejpam-4737	239	10	p	p	NOUN
ejpam-4737	239	11	)	)	PUNCT
ejpam-4737	239	12	̸=	̸=	PROPN
ejpam-4737	239	13	∅	∅	NOUN
ejpam-4737	239	14	and	and	CCONJ
ejpam-4737	239	15	u(λ	u(λ	PROPN
ejpam-4737	239	16	,	,	PUNCT
ejpam-4737	239	17	p	p	NOUN
ejpam-4737	239	18	)	)	PUNCT
ejpam-4737	239	19	̸=	̸=	PROPN
ejpam-4737	239	20	∅.	∅.	ADV
ejpam-4737	239	21	thus	thus	ADV
ejpam-4737	239	22	,	,	PUNCT
ejpam-4737	239	23	by	by	ADP
ejpam-4737	239	24	theorem	theorem	NOUN
ejpam-4737	239	25	8	8	NUM
ejpam-4737	239	26	,	,	PUNCT
ejpam-4737	239	27	∅	∅	NOUN
ejpam-4737	239	28	=	=	NOUN
ejpam-4737	239	29	̸	̸	X
ejpam-4737	239	30	u(λ	u(λ	PROPN
ejpam-4737	239	31	,	,	PUNCT
ejpam-4737	239	32	p	p	NOUN
ejpam-4737	239	33	)	)	PUNCT
ejpam-4737	239	34	∩	∩	NOUN
ejpam-4737	239	35	[	[	X
ejpam-4737	239	36	v	v	X
ejpam-4737	239	37	(	(	PUNCT
ejpam-4737	239	38	λ	λ	PROPN
ejpam-4737	239	39	,	,	PUNCT
ejpam-4737	239	40	p)](λ	p)](λ	ADJ
ejpam-4737	239	41	,	,	PUNCT
ejpam-4737	239	42	p	p	NOUN
ejpam-4737	239	43	)	)	PUNCT
ejpam-4737	239	44	⊆	⊆	NUM
ejpam-4737	239	45	u	u	NOUN
ejpam-4737	239	46	∩	∩	NOUN
ejpam-4737	239	47	[	[	X
ejpam-4737	239	48	v	v	X
ejpam-4737	239	49	(	(	PUNCT
ejpam-4737	239	50	λ	λ	PROPN
ejpam-4737	239	51	,	,	PUNCT
ejpam-4737	239	52	p)](λ	p)](λ	ADJ
ejpam-4737	239	53	,	,	PUNCT
ejpam-4737	239	54	p	p	NOUN
ejpam-4737	239	55	)	)	PUNCT
ejpam-4737	239	56	⊆	⊆	NUM
ejpam-4737	239	57	u	u	NOUN
ejpam-4737	239	58	∩	∩	NOUN
ejpam-4737	239	59	(	(	PUNCT
ejpam-4737	239	60	v	v	NOUN
ejpam-4737	239	61	∪	∪	ADP
ejpam-4737	239	62	[	[	X
ejpam-4737	239	63	v	v	X
ejpam-4737	239	64	(	(	PUNCT
ejpam-4737	239	65	λ	λ	PROPN
ejpam-4737	239	66	,	,	PUNCT
ejpam-4737	239	67	p)](λ	p)](λ	ADJ
ejpam-4737	239	68	,	,	PUNCT
ejpam-4737	239	69	p	p	NOUN
ejpam-4737	239	70	)	)	PUNCT
ejpam-4737	239	71	)	)	PUNCT
ejpam-4737	240	1	=	=	SYM
ejpam-4737	240	2	u	u	PROPN
ejpam-4737	240	3	∩	∩	X
ejpam-4737	240	4	v	v	ADP
ejpam-4737	240	5	s(λ	s(λ	PROPN
ejpam-4737	240	6	,	,	PUNCT
ejpam-4737	240	7	p	p	NOUN
ejpam-4737	240	8	)	)	PUNCT
ejpam-4737	240	9	⊆	⊆	NUM
ejpam-4737	240	10	u	u	NOUN
ejpam-4737	240	11	∩	∩	NOUN
ejpam-4737	240	12	v	v	NOUN
ejpam-4737	240	13	δs(λ	δs(λ	NOUN
ejpam-4737	240	14	,	,	PUNCT
ejpam-4737	240	15	p	p	NOUN
ejpam-4737	240	16	)	)	PUNCT
ejpam-4737	240	17	.	.	PUNCT
ejpam-4737	241	1	since	since	SCONJ
ejpam-4737	241	2	u	u	PROPN
ejpam-4737	241	3	∈	∈	PROPN
ejpam-4737	241	4	δs(λ	δs(λ	NOUN
ejpam-4737	241	5	,	,	PUNCT
ejpam-4737	241	6	p)o(x	p)o(x	VERB
ejpam-4737	241	7	,	,	PUNCT
ejpam-4737	241	8	τ	τ	PROPN
ejpam-4737	241	9	)	)	PUNCT
ejpam-4737	241	10	,	,	PUNCT
ejpam-4737	241	11	u	u	PROPN
ejpam-4737	241	12	∩	∩	NOUN
ejpam-4737	241	13	v	v	ADP
ejpam-4737	241	14	̸=	̸=	PROPN
ejpam-4737	241	15	∅.	∅.	ADP
ejpam-4737	241	16	this	this	DET
ejpam-4737	241	17	shows	show	VERB
ejpam-4737	241	18	that	that	SCONJ
ejpam-4737	241	19	v	v	X
ejpam-4737	241	20	δs(λ	δs(λ	NOUN
ejpam-4737	241	21	,	,	PUNCT
ejpam-4737	241	22	p	p	NOUN
ejpam-4737	241	23	)	)	PUNCT
ejpam-4737	241	24	=	=	PUNCT
ejpam-4737	241	25	x.	x.	NOUN
ejpam-4737	241	26	(	(	PUNCT
ejpam-4737	241	27	6	6	NUM
ejpam-4737	241	28	)	)	PUNCT
ejpam-4737	241	29	⇒	⇒	NOUN
ejpam-4737	241	30	(	(	PUNCT
ejpam-4737	241	31	1	1	NUM
ejpam-4737	241	32	):	):	PUNCT
ejpam-4737	241	33	let	let	VERB
ejpam-4737	241	34	u	u	NOUN
ejpam-4737	241	35	,	,	PUNCT
ejpam-4737	241	36	v	v	AUX
ejpam-4737	241	37	be	be	AUX
ejpam-4737	241	38	any	any	DET
ejpam-4737	241	39	nonempty	nonempty	ADJ
ejpam-4737	241	40	δs(λ	δs(λ	NOUN
ejpam-4737	241	41	,	,	PUNCT
ejpam-4737	241	42	p)-open	p)-open	VERB
ejpam-4737	241	43	sets	set	NOUN
ejpam-4737	241	44	.	.	PUNCT
ejpam-4737	242	1	since	since	SCONJ
ejpam-4737	242	2	δs(λ	δs(λ	NOUN
ejpam-4737	242	3	,	,	PUNCT
ejpam-4737	242	4	p)o(x	p)o(x	ADJ
ejpam-4737	242	5	,	,	PUNCT
ejpam-4737	242	6	τ	τ	PROPN
ejpam-4737	242	7	)	)	PUNCT
ejpam-4737	242	8	⊆	⊆	NUM
ejpam-4737	242	9	s(λ	s(λ	PROPN
ejpam-4737	242	10	,	,	PUNCT
ejpam-4737	242	11	p)o(x	p)o(x	ADJ
ejpam-4737	242	12	,	,	PUNCT
ejpam-4737	242	13	τ	τ	PROPN
ejpam-4737	242	14	)	)	PUNCT
ejpam-4737	242	15	and	and	CCONJ
ejpam-4737	242	16	v(λ	v(λ	PROPN
ejpam-4737	242	17	,	,	PUNCT
ejpam-4737	242	18	p	p	NOUN
ejpam-4737	242	19	)	)	PUNCT
ejpam-4737	242	20	̸=	̸=	NOUN
ejpam-4737	242	21	∅	∅	NOUN
ejpam-4737	242	22	,	,	PUNCT
ejpam-4737	242	23	we	we	PRON
ejpam-4737	242	24	have	have	VERB
ejpam-4737	242	25	∅	∅	NOUN
ejpam-4737	242	26	=	=	NOUN
ejpam-4737	242	27	̸	̸	NUM
ejpam-4737	242	28	u	u	NOUN
ejpam-4737	242	29	∩	∩	NOUN
ejpam-4737	242	30	v(λ	v(λ	PROPN
ejpam-4737	242	31	,	,	PUNCT
ejpam-4737	242	32	p	p	NOUN
ejpam-4737	242	33	)	)	PUNCT
ejpam-4737	243	1	⊆	⊆	NUM
ejpam-4737	243	2	u	u	NOUN
ejpam-4737	243	3	∩	∩	NOUN
ejpam-4737	243	4	v	v	NOUN
ejpam-4737	243	5	.	.	PUNCT
ejpam-4737	244	1	this	this	PRON
ejpam-4737	244	2	shows	show	VERB
ejpam-4737	244	3	that	that	SCONJ
ejpam-4737	244	4	v	v	NOUN
ejpam-4737	244	5	δs(λ	δs(λ	NOUN
ejpam-4737	244	6	,	,	PUNCT
ejpam-4737	244	7	p	p	NOUN
ejpam-4737	244	8	)	)	PUNCT
ejpam-4737	244	9	=	=	PUNCT
ejpam-4737	245	1	x	x	PUNCT
ejpam-4737	245	2	for	for	ADP
ejpam-4737	245	3	every	every	DET
ejpam-4737	245	4	nonempty	nonempty	NOUN
ejpam-4737	245	5	v	v	ADP
ejpam-4737	245	6	∈	∈	PROPN
ejpam-4737	245	7	δs(λ	δs(λ	NOUN
ejpam-4737	245	8	,	,	PUNCT
ejpam-4737	245	9	p)o(x	p)o(x	VERB
ejpam-4737	245	10	,	,	PUNCT
ejpam-4737	245	11	τ	τ	PROPN
ejpam-4737	245	12	)	)	PUNCT
ejpam-4737	245	13	.	.	PUNCT
ejpam-4737	246	1	thus	thus	ADV
ejpam-4737	246	2	,	,	PUNCT
ejpam-4737	246	3	by	by	ADP
ejpam-4737	246	4	theorem	theorem	NOUN
ejpam-4737	246	5	8	8	NUM
ejpam-4737	246	6	,	,	PUNCT
ejpam-4737	246	7	(	(	PUNCT
ejpam-4737	246	8	x	x	X
ejpam-4737	246	9	,	,	PUNCT
ejpam-4737	246	10	τ	τ	X
ejpam-4737	246	11	)	)	PUNCT
ejpam-4737	246	12	is	be	AUX
ejpam-4737	246	13	s(λ	s(λ	PROPN
ejpam-4737	246	14	,	,	PUNCT
ejpam-4737	246	15	p)-connected	p)-connecte	VERB
ejpam-4737	246	16	.	.	PUNCT
ejpam-4737	247	1	other	other	ADJ
ejpam-4737	247	2	implications	implication	NOUN
ejpam-4737	247	3	are	be	AUX
ejpam-4737	247	4	obvious	obvious	ADJ
ejpam-4737	247	5	since	since	SCONJ
ejpam-4737	247	6	λpo(x	λpo(x	PROPN
ejpam-4737	247	7	,	,	PUNCT
ejpam-4737	247	8	τ	τ	X
ejpam-4737	247	9	)	)	PUNCT
ejpam-4737	247	10	⊆	⊆	NUM
ejpam-4737	247	11	α(λ	α(λ	PROPN
ejpam-4737	247	12	,	,	PUNCT
ejpam-4737	247	13	p)o(x	p)o(x	ADJ
ejpam-4737	247	14	,	,	PUNCT
ejpam-4737	247	15	τ	τ	PROPN
ejpam-4737	247	16	)	)	PUNCT
ejpam-4737	247	17	⊆	⊆	NUM
ejpam-4737	247	18	s(λ	s(λ	PROPN
ejpam-4737	247	19	,	,	PUNCT
ejpam-4737	247	20	p)o(x	p)o(x	ADJ
ejpam-4737	247	21	,	,	PUNCT
ejpam-4737	247	22	τ	τ	NOUN
ejpam-4737	247	23	)	)	PUNCT
ejpam-4737	247	24	∩	∩	ADJ
ejpam-4737	247	25	p(λ	p(λ	NOUN
ejpam-4737	247	26	,	,	PUNCT
ejpam-4737	247	27	p)o(x	p)o(x	ADJ
ejpam-4737	247	28	,	,	PUNCT
ejpam-4737	247	29	τ	τ	PROPN
ejpam-4737	247	30	)	)	PUNCT
ejpam-4737	247	31	and	and	CCONJ
ejpam-4737	247	32	s(λ	s(λ	PROPN
ejpam-4737	247	33	,	,	PUNCT
ejpam-4737	247	34	p)o(x	p)o(x	ADJ
ejpam-4737	247	35	,	,	PUNCT
ejpam-4737	247	36	τ	τ	PROPN
ejpam-4737	247	37	)	)	PUNCT
ejpam-4737	247	38	∪	∪	ADJ
ejpam-4737	247	39	p(λ	p(λ	NOUN
ejpam-4737	247	40	,	,	PUNCT
ejpam-4737	247	41	p)o(x	p)o(x	ADJ
ejpam-4737	247	42	,	,	PUNCT
ejpam-4737	247	43	τ	τ	PROPN
ejpam-4737	247	44	)	)	PUNCT
ejpam-4737	247	45	⊆	⊆	NUM
ejpam-4737	247	46	β(λ	β(λ	NOUN
ejpam-4737	247	47	,	,	PUNCT
ejpam-4737	247	48	p)o(x	p)o(x	ADJ
ejpam-4737	247	49	,	,	PUNCT
ejpam-4737	247	50	τ	τ	PROPN
ejpam-4737	247	51	)	)	PUNCT
ejpam-4737	247	52	.	.	PUNCT
ejpam-4737	248	1	corollary	corollary	ADJ
ejpam-4737	248	2	2	2	NUM
ejpam-4737	248	3	.	.	PUNCT
ejpam-4737	249	1	for	for	ADP
ejpam-4737	249	2	a	a	DET
ejpam-4737	249	3	topological	topological	ADJ
ejpam-4737	249	4	space	space	NOUN
ejpam-4737	249	5	(	(	PUNCT
ejpam-4737	249	6	x	x	X
ejpam-4737	249	7	,	,	PUNCT
ejpam-4737	249	8	τ	τ	PROPN
ejpam-4737	249	9	)	)	PUNCT
ejpam-4737	249	10	,	,	PUNCT
ejpam-4737	249	11	the	the	DET
ejpam-4737	249	12	following	follow	VERB
ejpam-4737	249	13	properties	property	NOUN
ejpam-4737	249	14	are	be	AUX
ejpam-4737	249	15	equivalent	equivalent	ADJ
ejpam-4737	249	16	:	:	PUNCT
ejpam-4737	249	17	(	(	PUNCT
ejpam-4737	249	18	1	1	X
ejpam-4737	249	19	)	)	PUNCT
ejpam-4737	249	20	(	(	PUNCT
ejpam-4737	249	21	x	x	X
ejpam-4737	249	22	,	,	PUNCT
ejpam-4737	249	23	τ	τ	X
ejpam-4737	249	24	)	)	PUNCT
ejpam-4737	249	25	is	be	AUX
ejpam-4737	249	26	s(λ	s(λ	PROPN
ejpam-4737	249	27	,	,	PUNCT
ejpam-4737	249	28	p)-connected	p)-connecte	VERB
ejpam-4737	249	29	;	;	PUNCT
ejpam-4737	249	30	(	(	PUNCT
ejpam-4737	249	31	2	2	X
ejpam-4737	249	32	)	)	PUNCT
ejpam-4737	249	33	u	u	NOUN
ejpam-4737	249	34	∩	∩	NOUN
ejpam-4737	249	35	v	v	ADP
ejpam-4737	249	36	̸=	̸=	PROPN
ejpam-4737	249	37	∅	∅	NOUN
ejpam-4737	249	38	for	for	ADP
ejpam-4737	249	39	every	every	DET
ejpam-4737	249	40	nonempty	nonempty	NOUN
ejpam-4737	249	41	sets	set	VERB
ejpam-4737	249	42	u	u	PRON
ejpam-4737	249	43	∈	∈	PROPN
ejpam-4737	249	44	β(λ	β(λ	X
ejpam-4737	249	45	,	,	PUNCT
ejpam-4737	249	46	p)o(x	p)o(x	ADJ
ejpam-4737	249	47	,	,	PUNCT
ejpam-4737	249	48	τ	τ	PROPN
ejpam-4737	249	49	)	)	PUNCT
ejpam-4737	249	50	and	and	CCONJ
ejpam-4737	249	51	v	v	ADP
ejpam-4737	249	52	∈	∈	PROPN
ejpam-4737	249	53	δs(λ	δs(λ	NOUN
ejpam-4737	249	54	,	,	PUNCT
ejpam-4737	249	55	p)o(x	p)o(x	VERB
ejpam-4737	249	56	,	,	PUNCT
ejpam-4737	249	57	τ	τ	PROPN
ejpam-4737	249	58	)	)	PUNCT
ejpam-4737	249	59	;	;	PUNCT
ejpam-4737	249	60	(	(	PUNCT
ejpam-4737	249	61	3	3	X
ejpam-4737	249	62	)	)	PUNCT
ejpam-4737	249	63	u	u	NOUN
ejpam-4737	249	64	∩	∩	NOUN
ejpam-4737	249	65	v	v	ADP
ejpam-4737	249	66	̸=	̸=	PROPN
ejpam-4737	249	67	∅	∅	NOUN
ejpam-4737	249	68	for	for	ADP
ejpam-4737	249	69	every	every	DET
ejpam-4737	249	70	nonempty	nonempty	NOUN
ejpam-4737	249	71	sets	set	VERB
ejpam-4737	249	72	u	u	PRON
ejpam-4737	249	73	∈	∈	NOUN
ejpam-4737	249	74	p(λ	p(λ	NOUN
ejpam-4737	249	75	,	,	PUNCT
ejpam-4737	249	76	p)o(x	p)o(x	ADJ
ejpam-4737	249	77	,	,	PUNCT
ejpam-4737	249	78	τ	τ	PROPN
ejpam-4737	249	79	)	)	PUNCT
ejpam-4737	249	80	and	and	CCONJ
ejpam-4737	249	81	v	v	ADP
ejpam-4737	249	82	∈	∈	PROPN
ejpam-4737	249	83	δs(λ	δs(λ	NOUN
ejpam-4737	249	84	,	,	PUNCT
ejpam-4737	249	85	p)o(x	p)o(x	VERB
ejpam-4737	249	86	,	,	PUNCT
ejpam-4737	249	87	τ	τ	PROPN
ejpam-4737	249	88	)	)	PUNCT
ejpam-4737	249	89	;	;	PUNCT
ejpam-4737	249	90	(	(	PUNCT
ejpam-4737	249	91	4	4	X
ejpam-4737	249	92	)	)	PUNCT
ejpam-4737	249	93	u	u	NOUN
ejpam-4737	249	94	∩	∩	NOUN
ejpam-4737	249	95	v	v	ADP
ejpam-4737	249	96	̸=	̸=	PROPN
ejpam-4737	249	97	∅	∅	NOUN
ejpam-4737	249	98	for	for	ADP
ejpam-4737	249	99	every	every	DET
ejpam-4737	249	100	nonempty	nonempty	NOUN
ejpam-4737	249	101	sets	set	VERB
ejpam-4737	249	102	u	u	PROPN
ejpam-4737	249	103	∈	∈	PROPN
ejpam-4737	249	104	s(λ	s(λ	PROPN
ejpam-4737	249	105	,	,	PUNCT
ejpam-4737	249	106	p)o(x	p)o(x	ADJ
ejpam-4737	249	107	,	,	PUNCT
ejpam-4737	249	108	τ	τ	PROPN
ejpam-4737	249	109	)	)	PUNCT
ejpam-4737	249	110	and	and	CCONJ
ejpam-4737	249	111	v	v	ADP
ejpam-4737	249	112	∈	∈	PROPN
ejpam-4737	249	113	δs(λ	δs(λ	NOUN
ejpam-4737	249	114	,	,	PUNCT
ejpam-4737	249	115	p)o(x	p)o(x	VERB
ejpam-4737	249	116	,	,	PUNCT
ejpam-4737	249	117	τ	τ	PROPN
ejpam-4737	249	118	)	)	PUNCT
ejpam-4737	249	119	;	;	PUNCT
ejpam-4737	249	120	(	(	PUNCT
ejpam-4737	249	121	5	5	X
ejpam-4737	249	122	)	)	PUNCT
ejpam-4737	249	123	u	u	NOUN
ejpam-4737	249	124	∩	∩	NOUN
ejpam-4737	249	125	v	v	ADP
ejpam-4737	249	126	̸=	̸=	PROPN
ejpam-4737	249	127	∅	∅	NOUN
ejpam-4737	249	128	for	for	ADP
ejpam-4737	249	129	every	every	DET
ejpam-4737	249	130	nonempty	nonempty	NOUN
ejpam-4737	249	131	sets	set	VERB
ejpam-4737	249	132	u	u	PROPN
ejpam-4737	249	133	∈	∈	PROPN
ejpam-4737	249	134	α(λ	α(λ	PROPN
ejpam-4737	249	135	,	,	PUNCT
ejpam-4737	249	136	p)o(x	p)o(x	ADJ
ejpam-4737	249	137	,	,	PUNCT
ejpam-4737	249	138	τ	τ	PROPN
ejpam-4737	249	139	)	)	PUNCT
ejpam-4737	249	140	and	and	CCONJ
ejpam-4737	249	141	v	v	ADP
ejpam-4737	249	142	∈	∈	PROPN
ejpam-4737	249	143	δs(λ	δs(λ	NOUN
ejpam-4737	249	144	,	,	PUNCT
ejpam-4737	249	145	p)o(x	p)o(x	VERB
ejpam-4737	249	146	,	,	PUNCT
ejpam-4737	249	147	τ	τ	PROPN
ejpam-4737	249	148	)	)	PUNCT
ejpam-4737	249	149	;	;	PUNCT
ejpam-4737	249	150	(	(	PUNCT
ejpam-4737	249	151	6	6	X
ejpam-4737	249	152	)	)	PUNCT
ejpam-4737	249	153	u	u	NOUN
ejpam-4737	249	154	∩	∩	NOUN
ejpam-4737	249	155	v	v	ADP
ejpam-4737	249	156	̸=	̸=	PROPN
ejpam-4737	249	157	∅	∅	NOUN
ejpam-4737	249	158	for	for	ADP
ejpam-4737	249	159	every	every	DET
ejpam-4737	249	160	nonempty	nonempty	NOUN
ejpam-4737	249	161	sets	set	VERB
ejpam-4737	249	162	u	u	PRON
ejpam-4737	249	163	∈	∈	PROPN
ejpam-4737	249	164	λpo(x	λpo(x	PROPN
ejpam-4737	249	165	,	,	PUNCT
ejpam-4737	249	166	τ	τ	X
ejpam-4737	249	167	)	)	PUNCT
ejpam-4737	249	168	and	and	CCONJ
ejpam-4737	249	169	v	v	ADP
ejpam-4737	249	170	∈	∈	PROPN
ejpam-4737	249	171	δs(λ	δs(λ	NOUN
ejpam-4737	249	172	,	,	PUNCT
ejpam-4737	249	173	p)o(x	p)o(x	VERB
ejpam-4737	249	174	,	,	PUNCT
ejpam-4737	249	175	τ	τ	PROPN
ejpam-4737	249	176	)	)	PUNCT
ejpam-4737	249	177	;	;	PUNCT
ejpam-4737	249	178	(	(	PUNCT
ejpam-4737	249	179	7	7	X
ejpam-4737	249	180	)	)	PUNCT
ejpam-4737	249	181	u	u	NOUN
ejpam-4737	249	182	∩	∩	NOUN
ejpam-4737	249	183	v	v	ADP
ejpam-4737	249	184	̸=	̸=	PROPN
ejpam-4737	249	185	∅	∅	NOUN
ejpam-4737	249	186	for	for	ADP
ejpam-4737	249	187	every	every	DET
ejpam-4737	249	188	nonempty	nonempty	NOUN
ejpam-4737	249	189	sets	set	VERB
ejpam-4737	249	190	u	u	PRON
ejpam-4737	249	191	∈	∈	PROPN
ejpam-4737	249	192	δs(λ	δs(λ	NOUN
ejpam-4737	249	193	,	,	PUNCT
ejpam-4737	249	194	p)o(x	p)o(x	ADJ
ejpam-4737	249	195	,	,	PUNCT
ejpam-4737	249	196	τ	τ	PROPN
ejpam-4737	249	197	)	)	PUNCT
ejpam-4737	249	198	and	and	CCONJ
ejpam-4737	249	199	v	v	ADP
ejpam-4737	249	200	∈	∈	PROPN
ejpam-4737	249	201	δs(λ	δs(λ	NOUN
ejpam-4737	249	202	,	,	PUNCT
ejpam-4737	249	203	p)o(x	p)o(x	VERB
ejpam-4737	249	204	,	,	PUNCT
ejpam-4737	249	205	τ	τ	PROPN
ejpam-4737	249	206	)	)	PUNCT
ejpam-4737	249	207	.	.	PUNCT
ejpam-4737	250	1	proof	proof	NOUN
ejpam-4737	250	2	.	.	PUNCT
ejpam-4737	251	1	this	this	PRON
ejpam-4737	251	2	is	be	AUX
ejpam-4737	251	3	immediate	immediate	ADJ
ejpam-4737	251	4	consequence	consequence	NOUN
ejpam-4737	251	5	of	of	ADP
ejpam-4737	251	6	theorem	theorem	ADJ
ejpam-4737	251	7	8	8	NUM
ejpam-4737	251	8	and	and	CCONJ
ejpam-4737	251	9	9	9	NUM
ejpam-4737	251	10	.	.	PUNCT
ejpam-4737	251	11	c.	c.	PROPN
ejpam-4737	251	12	boonpok	boonpok	PROPN
ejpam-4737	251	13	,	,	PUNCT
ejpam-4737	251	14	m.	m.	NOUN
ejpam-4737	251	15	thongmoon	thongmoon	PROPN
ejpam-4737	251	16	/	/	SYM
ejpam-4737	251	17	eur	eur	PROPN
ejpam-4737	251	18	.	.	PUNCT
ejpam-4737	252	1	j.	j.	PROPN
ejpam-4737	252	2	pure	pure	PROPN
ejpam-4737	252	3	appl	appl	PROPN
ejpam-4737	252	4	.	.	PROPN
ejpam-4737	252	5	math	math	PROPN
ejpam-4737	252	6	,	,	PUNCT
ejpam-4737	252	7	16	16	NUM
ejpam-4737	252	8	(	(	PUNCT
ejpam-4737	252	9	3	3	NUM
ejpam-4737	252	10	)	)	PUNCT
ejpam-4737	252	11	(	(	PUNCT
ejpam-4737	252	12	2023	2023	NUM
ejpam-4737	252	13	)	)	PUNCT
ejpam-4737	252	14	,	,	PUNCT
ejpam-4737	252	15	1434	1434	NUM
ejpam-4737	252	16	-	-	SYM
ejpam-4737	252	17	1447	1447	NUM
ejpam-4737	252	18	1442	1442	NUM
ejpam-4737	252	19	5	5	NUM
ejpam-4737	252	20	.	.	PUNCT
ejpam-4737	253	1	characterizations	characterization	NOUN
ejpam-4737	253	2	of	of	ADP
ejpam-4737	253	3	s(λ	s(λ	PROPN
ejpam-4737	253	4	,	,	PUNCT
ejpam-4737	253	5	p)-regular	p)-regular	ADJ
ejpam-4737	253	6	spaces	space	NOUN
ejpam-4737	253	7	and	and	CCONJ
ejpam-4737	253	8	s(λ	s(λ	NOUN
ejpam-4737	253	9	,	,	PUNCT
ejpam-4737	253	10	p)-normal	p)-normal	PUNCT
ejpam-4737	253	11	spaces	space	NOUN
ejpam-4737	253	12	in	in	ADP
ejpam-4737	253	13	this	this	DET
ejpam-4737	253	14	section	section	NOUN
ejpam-4737	253	15	,	,	PUNCT
ejpam-4737	253	16	we	we	PRON
ejpam-4737	253	17	introduce	introduce	VERB
ejpam-4737	253	18	the	the	DET
ejpam-4737	253	19	notions	notion	NOUN
ejpam-4737	253	20	of	of	ADP
ejpam-4737	253	21	s(λ	s(λ	PROPN
ejpam-4737	253	22	,	,	PUNCT
ejpam-4737	253	23	p)-regular	p)-regular	ADJ
ejpam-4737	253	24	spaces	space	NOUN
ejpam-4737	253	25	and	and	CCONJ
ejpam-4737	253	26	s(λ	s(λ	NOUN
ejpam-4737	253	27	,	,	PUNCT
ejpam-4737	253	28	p)-normal	p)-normal	ADJ
ejpam-4737	253	29	spaces	space	NOUN
ejpam-4737	253	30	.	.	PUNCT
ejpam-4737	254	1	moreover	moreover	ADV
ejpam-4737	254	2	,	,	PUNCT
ejpam-4737	254	3	several	several	ADJ
ejpam-4737	254	4	characterizations	characterization	NOUN
ejpam-4737	254	5	of	of	ADP
ejpam-4737	254	6	s(λ	s(λ	PROPN
ejpam-4737	254	7	,	,	PUNCT
ejpam-4737	254	8	p)-regular	p)-regular	ADJ
ejpam-4737	254	9	spaces	space	NOUN
ejpam-4737	254	10	and	and	CCONJ
ejpam-4737	254	11	s(λ	s(λ	NOUN
ejpam-4737	254	12	,	,	PUNCT
ejpam-4737	254	13	p)-normal	p)-normal	PUNCT
ejpam-4737	254	14	spaces	space	NOUN
ejpam-4737	254	15	are	be	AUX
ejpam-4737	254	16	discussed	discuss	VERB
ejpam-4737	254	17	.	.	PUNCT
ejpam-4737	255	1	definition	definition	NOUN
ejpam-4737	255	2	10	10	NUM
ejpam-4737	255	3	.	.	PUNCT
ejpam-4737	256	1	a	a	DET
ejpam-4737	256	2	topological	topological	ADJ
ejpam-4737	256	3	space	space	NOUN
ejpam-4737	256	4	(	(	PUNCT
ejpam-4737	256	5	x	x	X
ejpam-4737	256	6	,	,	PUNCT
ejpam-4737	256	7	τ	τ	X
ejpam-4737	256	8	)	)	PUNCT
ejpam-4737	256	9	is	be	AUX
ejpam-4737	256	10	said	say	VERB
ejpam-4737	256	11	to	to	PART
ejpam-4737	256	12	be	be	AUX
ejpam-4737	256	13	s(λ	s(λ	NOUN
ejpam-4737	256	14	,	,	PUNCT
ejpam-4737	256	15	p)-regular	p)-regular	ADJ
ejpam-4737	256	16	if	if	SCONJ
ejpam-4737	256	17	,	,	PUNCT
ejpam-4737	256	18	for	for	ADP
ejpam-4737	256	19	each	each	DET
ejpam-4737	256	20	s(λ	s(λ	PROPN
ejpam-4737	256	21	,	,	PUNCT
ejpam-4737	256	22	p)closed	p)close	VERB
ejpam-4737	256	23	set	set	VERB
ejpam-4737	256	24	f	f	PROPN
ejpam-4737	256	25	of	of	ADP
ejpam-4737	256	26	x	x	PUNCT
ejpam-4737	256	27	and	and	CCONJ
ejpam-4737	256	28	each	each	DET
ejpam-4737	256	29	point	point	NOUN
ejpam-4737	256	30	x	x	PUNCT
ejpam-4737	256	31	̸∈	̸∈	PROPN
ejpam-4737	256	32	f	f	PROPN
ejpam-4737	256	33	,	,	PUNCT
ejpam-4737	256	34	there	there	PRON
ejpam-4737	256	35	exist	exist	VERB
ejpam-4737	256	36	u	u	NOUN
ejpam-4737	256	37	,	,	PUNCT
ejpam-4737	256	38	v	v	PROPN
ejpam-4737	256	39	∈	∈	PROPN
ejpam-4737	256	40	s(λ	s(λ	PROPN
ejpam-4737	256	41	,	,	PUNCT
ejpam-4737	256	42	p)o(x	p)o(x	ADJ
ejpam-4737	256	43	,	,	PUNCT
ejpam-4737	256	44	τ	τ	PROPN
ejpam-4737	256	45	)	)	PUNCT
ejpam-4737	256	46	such	such	ADJ
ejpam-4737	256	47	that	that	SCONJ
ejpam-4737	256	48	x	x	SYM
ejpam-4737	256	49	∈	∈	PROPN
ejpam-4737	256	50	u	u	PROPN
ejpam-4737	256	51	,	,	PUNCT
ejpam-4737	256	52	f	f	PROPN
ejpam-4737	256	53	⊆	⊆	PROPN
ejpam-4737	256	54	v	v	NOUN
ejpam-4737	256	55	and	and	CCONJ
ejpam-4737	256	56	u	u	NOUN
ejpam-4737	256	57	∩	∩	NOUN
ejpam-4737	256	58	v	v	NOUN
ejpam-4737	256	59	=	=	PUNCT
ejpam-4737	256	60	∅.	∅.	NOUN
ejpam-4737	256	61	theorem	theorem	VERB
ejpam-4737	256	62	10	10	NUM
ejpam-4737	256	63	.	.	PUNCT
ejpam-4737	257	1	for	for	ADP
ejpam-4737	257	2	a	a	DET
ejpam-4737	257	3	topological	topological	ADJ
ejpam-4737	257	4	space	space	NOUN
ejpam-4737	257	5	(	(	PUNCT
ejpam-4737	257	6	x	x	X
ejpam-4737	257	7	,	,	PUNCT
ejpam-4737	257	8	τ	τ	PROPN
ejpam-4737	257	9	)	)	PUNCT
ejpam-4737	257	10	,	,	PUNCT
ejpam-4737	257	11	the	the	DET
ejpam-4737	257	12	following	follow	VERB
ejpam-4737	257	13	properties	property	NOUN
ejpam-4737	257	14	are	be	AUX
ejpam-4737	257	15	equivalent	equivalent	ADJ
ejpam-4737	257	16	:	:	PUNCT
ejpam-4737	257	17	(	(	PUNCT
ejpam-4737	257	18	1	1	X
ejpam-4737	257	19	)	)	PUNCT
ejpam-4737	257	20	(	(	PUNCT
ejpam-4737	257	21	x	x	X
ejpam-4737	257	22	,	,	PUNCT
ejpam-4737	257	23	τ	τ	X
ejpam-4737	257	24	)	)	PUNCT
ejpam-4737	257	25	is	be	AUX
ejpam-4737	257	26	s(λ	s(λ	PROPN
ejpam-4737	257	27	,	,	PUNCT
ejpam-4737	257	28	p)-regular	p)-regular	NOUN
ejpam-4737	257	29	.	.	PUNCT
ejpam-4737	258	1	(	(	PUNCT
ejpam-4737	258	2	2	2	X
ejpam-4737	258	3	)	)	PUNCT
ejpam-4737	258	4	for	for	ADP
ejpam-4737	258	5	each	each	DET
ejpam-4737	258	6	s(λ	s(λ	PROPN
ejpam-4737	258	7	,	,	PUNCT
ejpam-4737	258	8	p)-closed	p)-close	VERB
ejpam-4737	258	9	set	set	VERB
ejpam-4737	258	10	f	f	NOUN
ejpam-4737	258	11	and	and	CCONJ
ejpam-4737	258	12	each	each	DET
ejpam-4737	258	13	point	point	NOUN
ejpam-4737	258	14	x	x	PUNCT
ejpam-4737	258	15	̸∈	̸∈	PROPN
ejpam-4737	258	16	f	f	PROPN
ejpam-4737	258	17	,	,	PUNCT
ejpam-4737	258	18	there	there	PRON
ejpam-4737	258	19	exist	exist	VERB
ejpam-4737	258	20	u	u	NOUN
ejpam-4737	258	21	,	,	PUNCT
ejpam-4737	258	22	v	v	PROPN
ejpam-4737	258	23	∈	∈	PROPN
ejpam-4737	258	24	δs(λ	δs(λ	NOUN
ejpam-4737	258	25	,	,	PUNCT
ejpam-4737	258	26	p)o(x	p)o(x	ADJ
ejpam-4737	258	27	,	,	PUNCT
ejpam-4737	258	28	τ	τ	PROPN
ejpam-4737	258	29	)	)	PUNCT
ejpam-4737	258	30	such	such	ADJ
ejpam-4737	258	31	that	that	SCONJ
ejpam-4737	258	32	x	x	SYM
ejpam-4737	258	33	∈	∈	PROPN
ejpam-4737	258	34	u	u	PROPN
ejpam-4737	258	35	,	,	PUNCT
ejpam-4737	258	36	f	f	PROPN
ejpam-4737	258	37	⊆	⊆	PROPN
ejpam-4737	258	38	v	v	NOUN
ejpam-4737	258	39	and	and	CCONJ
ejpam-4737	258	40	u	u	NOUN
ejpam-4737	258	41	∩	∩	NOUN
ejpam-4737	258	42	v	v	NOUN
ejpam-4737	258	43	=	=	PUNCT
ejpam-4737	258	44	∅.	∅.	X
ejpam-4737	258	45	(	(	PUNCT
ejpam-4737	258	46	3	3	NUM
ejpam-4737	258	47	)	)	PUNCT
ejpam-4737	258	48	for	for	ADP
ejpam-4737	258	49	each	each	DET
ejpam-4737	258	50	point	point	NOUN
ejpam-4737	258	51	x	x	X
ejpam-4737	258	52	∈	∈	NOUN
ejpam-4737	258	53	x	x	X
ejpam-4737	258	54	and	and	CCONJ
ejpam-4737	258	55	each	each	DET
ejpam-4737	258	56	s(λ	s(λ	PROPN
ejpam-4737	258	57	,	,	PUNCT
ejpam-4737	258	58	p)-open	p)-open	VERB
ejpam-4737	258	59	set	set	VERB
ejpam-4737	258	60	v	v	NOUN
ejpam-4737	258	61	containing	contain	VERB
ejpam-4737	258	62	x	x	X
ejpam-4737	258	63	,	,	PUNCT
ejpam-4737	258	64	there	there	PRON
ejpam-4737	258	65	exists	exist	VERB
ejpam-4737	258	66	u	u	PROPN
ejpam-4737	258	67	∈	∈	PROPN
ejpam-4737	258	68	δs(λ	δs(λ	NOUN
ejpam-4737	258	69	,	,	PUNCT
ejpam-4737	258	70	p)o(x	p)o(x	ADJ
ejpam-4737	258	71	,	,	PUNCT
ejpam-4737	258	72	τ	τ	PROPN
ejpam-4737	258	73	)	)	PUNCT
ejpam-4737	258	74	such	such	ADJ
ejpam-4737	258	75	that	that	SCONJ
ejpam-4737	258	76	x	x	SYM
ejpam-4737	258	77	∈	∈	NOUN
ejpam-4737	258	78	u	u	NOUN
ejpam-4737	258	79	⊆	⊆	NUM
ejpam-4737	258	80	u	u	NOUN
ejpam-4737	258	81	δs(λ	δs(λ	NOUN
ejpam-4737	258	82	,	,	PUNCT
ejpam-4737	258	83	p	p	NOUN
ejpam-4737	258	84	)	)	PUNCT
ejpam-4737	258	85	⊆	⊆	NUM
ejpam-4737	258	86	v	v	NOUN
ejpam-4737	258	87	.	.	PUNCT
ejpam-4737	259	1	proof	proof	NOUN
ejpam-4737	259	2	.	.	PUNCT
ejpam-4737	260	1	(	(	PUNCT
ejpam-4737	260	2	1	1	X
ejpam-4737	260	3	)	)	PUNCT
ejpam-4737	260	4	⇒	⇒	NOUN
ejpam-4737	260	5	(	(	PUNCT
ejpam-4737	260	6	2	2	NUM
ejpam-4737	260	7	):	):	PUNCT
ejpam-4737	260	8	let	let	VERB
ejpam-4737	260	9	f	f	PRON
ejpam-4737	260	10	be	be	AUX
ejpam-4737	260	11	a	a	DET
ejpam-4737	260	12	s(λ	s(λ	PROPN
ejpam-4737	260	13	,	,	PUNCT
ejpam-4737	260	14	p)-closed	p)-close	VERB
ejpam-4737	260	15	set	set	NOUN
ejpam-4737	260	16	and	and	CCONJ
ejpam-4737	260	17	x	x	PART
ejpam-4737	260	18	̸∈	̸∈	PROPN
ejpam-4737	260	19	f	f	PROPN
ejpam-4737	260	20	.	.	PUNCT
ejpam-4737	261	1	then	then	ADV
ejpam-4737	261	2	,	,	PUNCT
ejpam-4737	261	3	there	there	PRON
ejpam-4737	261	4	exist	exist	VERB
ejpam-4737	261	5	g	g	NOUN
ejpam-4737	261	6	,	,	PUNCT
ejpam-4737	261	7	h	h	PROPN
ejpam-4737	261	8	∈	∈	PROPN
ejpam-4737	261	9	s(λ	s(λ	PROPN
ejpam-4737	261	10	,	,	PUNCT
ejpam-4737	261	11	p)o(x	p)o(x	ADJ
ejpam-4737	261	12	,	,	PUNCT
ejpam-4737	261	13	τ	τ	PROPN
ejpam-4737	261	14	)	)	PUNCT
ejpam-4737	261	15	such	such	ADJ
ejpam-4737	261	16	that	that	SCONJ
ejpam-4737	261	17	x	x	SYM
ejpam-4737	261	18	∈	∈	PROPN
ejpam-4737	261	19	g	g	PROPN
ejpam-4737	261	20	,	,	PUNCT
ejpam-4737	261	21	f	f	PROPN
ejpam-4737	261	22	⊆	⊆	NUM
ejpam-4737	261	23	h	h	NOUN
ejpam-4737	261	24	and	and	CCONJ
ejpam-4737	261	25	g	g	NOUN
ejpam-4737	261	26	∩h	∩h	NOUN
ejpam-4737	261	27	=	=	PUNCT
ejpam-4737	261	28	∅.	∅.	X
ejpam-4737	261	29	by	by	ADP
ejpam-4737	261	30	lemma	lemma	PROPN
ejpam-4737	261	31	4	4	NUM
ejpam-4737	261	32	,	,	PUNCT
ejpam-4737	261	33	gs(λ	gs(λ	VERB
ejpam-4737	261	34	,	,	PUNCT
ejpam-4737	261	35	p	p	NOUN
ejpam-4737	261	36	)	)	PUNCT
ejpam-4737	262	1	is	be	AUX
ejpam-4737	262	2	s(λ	s(λ	PROPN
ejpam-4737	262	3	,	,	PUNCT
ejpam-4737	262	4	p)-regular	p)-regular	PUNCT
ejpam-4737	262	5	and	and	CCONJ
ejpam-4737	262	6	gs(λ	gs(λ	VERB
ejpam-4737	262	7	,	,	PUNCT
ejpam-4737	262	8	p)∩h	p)∩h	NOUN
ejpam-4737	262	9	=	=	PUNCT
ejpam-4737	262	10	∅.	∅.	VERB
ejpam-4737	262	11	thus	thus	ADV
ejpam-4737	262	12	,	,	PUNCT
ejpam-4737	262	13	gs(λ	gs(λ	VERB
ejpam-4737	262	14	,	,	PUNCT
ejpam-4737	262	15	p)∩hs(λ	p)∩hs(λ	PROPN
ejpam-4737	262	16	,	,	PUNCT
ejpam-4737	262	17	p	p	NOUN
ejpam-4737	262	18	)	)	PUNCT
ejpam-4737	262	19	=	=	PUNCT
ejpam-4737	262	20	∅.	∅.	ADP
ejpam-4737	262	21	now	now	ADV
ejpam-4737	262	22	,	,	PUNCT
ejpam-4737	262	23	we	we	PRON
ejpam-4737	262	24	put	put	VERB
ejpam-4737	262	25	u	u	PRON
ejpam-4737	262	26	=	=	X
ejpam-4737	262	27	gs(λ	gs(λ	X
ejpam-4737	262	28	,	,	PUNCT
ejpam-4737	262	29	p	p	NOUN
ejpam-4737	262	30	)	)	PUNCT
ejpam-4737	262	31	and	and	CCONJ
ejpam-4737	262	32	v	v	NOUN
ejpam-4737	262	33	=	=	SYM
ejpam-4737	262	34	hs(λ	hs(λ	NOUN
ejpam-4737	262	35	,	,	PUNCT
ejpam-4737	262	36	p	p	NOUN
ejpam-4737	262	37	)	)	PUNCT
ejpam-4737	262	38	,	,	PUNCT
ejpam-4737	262	39	then	then	ADV
ejpam-4737	262	40	u	u	NOUN
ejpam-4737	262	41	and	and	CCONJ
ejpam-4737	262	42	v	v	NOUN
ejpam-4737	262	43	are	be	AUX
ejpam-4737	262	44	δs(λ	δs(λ	NOUN
ejpam-4737	262	45	,	,	PUNCT
ejpam-4737	262	46	p)-open	p)-open	VERB
ejpam-4737	262	47	sets	set	VERB
ejpam-4737	262	48	such	such	ADJ
ejpam-4737	262	49	that	that	SCONJ
ejpam-4737	262	50	x	x	SYM
ejpam-4737	262	51	∈	∈	PROPN
ejpam-4737	262	52	u	u	PROPN
ejpam-4737	262	53	,	,	PUNCT
ejpam-4737	262	54	f	f	PROPN
ejpam-4737	262	55	⊆	⊆	PROPN
ejpam-4737	262	56	v	v	NOUN
ejpam-4737	262	57	and	and	CCONJ
ejpam-4737	262	58	u	u	NOUN
ejpam-4737	262	59	∩	∩	NOUN
ejpam-4737	262	60	v	v	NOUN
ejpam-4737	262	61	=	=	PUNCT
ejpam-4737	262	62	∅.	∅.	X
ejpam-4737	262	63	(	(	PUNCT
ejpam-4737	262	64	2	2	NUM
ejpam-4737	262	65	)	)	PUNCT
ejpam-4737	262	66	⇒	⇒	NOUN
ejpam-4737	262	67	(	(	PUNCT
ejpam-4737	262	68	3	3	NUM
ejpam-4737	262	69	):	):	PUNCT
ejpam-4737	262	70	let	let	VERB
ejpam-4737	262	71	x	x	PUNCT
ejpam-4737	262	72	∈	∈	PROPN
ejpam-4737	262	73	x	x	X
ejpam-4737	262	74	and	and	CCONJ
ejpam-4737	262	75	v	v	X
ejpam-4737	262	76	be	be	AUX
ejpam-4737	262	77	any	any	DET
ejpam-4737	262	78	s(λ	s(λ	NOUN
ejpam-4737	262	79	,	,	PUNCT
ejpam-4737	262	80	p)-open	p)-open	VERB
ejpam-4737	262	81	set	set	VERB
ejpam-4737	262	82	containing	contain	VERB
ejpam-4737	262	83	x.	x.	NOUN
ejpam-4737	262	84	since	since	SCONJ
ejpam-4737	262	85	x	x	PROPN
ejpam-4737	262	86	̸∈	̸∈	PROPN
ejpam-4737	262	87	x	x	X
ejpam-4737	262	88	−	−	PROPN
ejpam-4737	262	89	v	v	INTJ
ejpam-4737	262	90	,	,	PUNCT
ejpam-4737	262	91	there	there	PRON
ejpam-4737	262	92	exist	exist	VERB
ejpam-4737	262	93	u	u	NOUN
ejpam-4737	262	94	,	,	PUNCT
ejpam-4737	262	95	g	g	PROPN
ejpam-4737	262	96	∈	∈	PROPN
ejpam-4737	262	97	δs(λ	δs(λ	NOUN
ejpam-4737	262	98	,	,	PUNCT
ejpam-4737	262	99	p)o(x	p)o(x	ADJ
ejpam-4737	262	100	,	,	PUNCT
ejpam-4737	262	101	τ	τ	PROPN
ejpam-4737	262	102	)	)	PUNCT
ejpam-4737	262	103	such	such	ADJ
ejpam-4737	262	104	that	that	SCONJ
ejpam-4737	262	105	x	x	SYM
ejpam-4737	262	106	∈	∈	PROPN
ejpam-4737	262	107	u	u	NOUN
ejpam-4737	262	108	,	,	PUNCT
ejpam-4737	262	109	x	x	PROPN
ejpam-4737	262	110	−	−	NOUN
ejpam-4737	262	111	v	v	VERB
ejpam-4737	262	112	⊆	⊆	NUM
ejpam-4737	262	113	g	g	NOUN
ejpam-4737	262	114	and	and	CCONJ
ejpam-4737	262	115	u	u	PROPN
ejpam-4737	262	116	∩	∩	NOUN
ejpam-4737	262	117	g	g	NOUN
ejpam-4737	262	118	=	=	PUNCT
ejpam-4737	262	119	∅.	∅.	NOUN
ejpam-4737	262	120	since	since	SCONJ
ejpam-4737	262	121	x	x	SYM
ejpam-4737	262	122	−g	−g	NOUN
ejpam-4737	262	123	is	be	AUX
ejpam-4737	262	124	δs(λ	δs(λ	NOUN
ejpam-4737	262	125	,	,	PUNCT
ejpam-4737	262	126	p)-closed	p)-close	VERB
ejpam-4737	262	127	and	and	CCONJ
ejpam-4737	262	128	u	u	NOUN
ejpam-4737	262	129	⊆	⊆	NUM
ejpam-4737	262	130	x	x	SYM
ejpam-4737	262	131	−g	−g	NOUN
ejpam-4737	262	132	,	,	PUNCT
ejpam-4737	262	133	x	x	SYM
ejpam-4737	262	134	∈	∈	NOUN
ejpam-4737	262	135	u	u	NOUN
ejpam-4737	262	136	⊆	⊆	NUM
ejpam-4737	262	137	u	u	NOUN
ejpam-4737	262	138	δs(λ	δs(λ	NOUN
ejpam-4737	262	139	,	,	PUNCT
ejpam-4737	262	140	p	p	NOUN
ejpam-4737	262	141	)	)	PUNCT
ejpam-4737	262	142	⊆	⊆	NUM
ejpam-4737	262	143	x	x	SYM
ejpam-4737	262	144	−g	−g	VERB
ejpam-4737	262	145	⊆	⊆	NUM
ejpam-4737	262	146	v	v	NOUN
ejpam-4737	262	147	.	.	PUNCT
ejpam-4737	263	1	(	(	PUNCT
ejpam-4737	263	2	3	3	X
ejpam-4737	263	3	)	)	PUNCT
ejpam-4737	263	4	⇒	⇒	NOUN
ejpam-4737	263	5	(	(	PUNCT
ejpam-4737	263	6	1	1	NUM
ejpam-4737	263	7	):	):	PUNCT
ejpam-4737	263	8	let	let	VERB
ejpam-4737	263	9	f	f	PRON
ejpam-4737	263	10	be	be	AUX
ejpam-4737	263	11	a	a	DET
ejpam-4737	263	12	s(λ	s(λ	PROPN
ejpam-4737	263	13	,	,	PUNCT
ejpam-4737	263	14	p)-closed	p)-close	VERB
ejpam-4737	263	15	set	set	NOUN
ejpam-4737	263	16	and	and	CCONJ
ejpam-4737	263	17	x	x	PART
ejpam-4737	263	18	̸∈	̸∈	PROPN
ejpam-4737	263	19	f	f	PROPN
ejpam-4737	263	20	.	.	PUNCT
ejpam-4737	264	1	then	then	ADV
ejpam-4737	264	2	,	,	PUNCT
ejpam-4737	264	3	x	x	PUNCT
ejpam-4737	264	4	−	−	PROPN
ejpam-4737	264	5	f	f	PROPN
ejpam-4737	264	6	is	be	AUX
ejpam-4737	264	7	s(λ	s(λ	PROPN
ejpam-4737	264	8	,	,	PUNCT
ejpam-4737	264	9	p)-open	p)-open	VERB
ejpam-4737	264	10	set	set	VERB
ejpam-4737	264	11	containing	contain	VERB
ejpam-4737	264	12	x.	x.	NOUN
ejpam-4737	264	13	by	by	ADP
ejpam-4737	264	14	(	(	PUNCT
ejpam-4737	264	15	3	3	NUM
ejpam-4737	264	16	)	)	PUNCT
ejpam-4737	264	17	,	,	PUNCT
ejpam-4737	264	18	there	there	PRON
ejpam-4737	264	19	exists	exist	VERB
ejpam-4737	264	20	u	u	PROPN
ejpam-4737	264	21	∈	∈	PROPN
ejpam-4737	264	22	δs(λ	δs(λ	NOUN
ejpam-4737	264	23	,	,	PUNCT
ejpam-4737	264	24	p)o(x	p)o(x	ADJ
ejpam-4737	264	25	,	,	PUNCT
ejpam-4737	264	26	τ	τ	PROPN
ejpam-4737	264	27	)	)	PUNCT
ejpam-4737	264	28	such	such	ADJ
ejpam-4737	264	29	that	that	SCONJ
ejpam-4737	264	30	x	x	SYM
ejpam-4737	264	31	∈	∈	NOUN
ejpam-4737	264	32	u	u	NOUN
ejpam-4737	264	33	⊆	⊆	NUM
ejpam-4737	264	34	u	u	NOUN
ejpam-4737	264	35	δs(λ	δs(λ	NOUN
ejpam-4737	264	36	,	,	PUNCT
ejpam-4737	264	37	p	p	NOUN
ejpam-4737	264	38	)	)	PUNCT
ejpam-4737	264	39	⊆	⊆	NUM
ejpam-4737	264	40	x−f	x−f	PROPN
ejpam-4737	264	41	.	.	PUNCT
ejpam-4737	265	1	thus	thus	ADV
ejpam-4737	265	2	,	,	PUNCT
ejpam-4737	265	3	x	x	PUNCT
ejpam-4737	265	4	∈	∈	PROPN
ejpam-4737	265	5	u	u	PROPN
ejpam-4737	265	6	,	,	PUNCT
ejpam-4737	265	7	f	f	PROPN
ejpam-4737	265	8	⊆	⊆	NUM
ejpam-4737	265	9	x	x	SYM
ejpam-4737	265	10	−	−	PROPN
ejpam-4737	265	11	u	u	NOUN
ejpam-4737	265	12	δs(λ	δs(λ	NOUN
ejpam-4737	265	13	,	,	PUNCT
ejpam-4737	265	14	p	p	NOUN
ejpam-4737	265	15	)	)	PUNCT
ejpam-4737	265	16	and	and	CCONJ
ejpam-4737	265	17	u	u	NOUN
ejpam-4737	265	18	∩	∩	NOUN
ejpam-4737	265	19	(	(	PUNCT
ejpam-4737	265	20	x	x	SYM
ejpam-4737	265	21	−	−	PROPN
ejpam-4737	265	22	u	u	NOUN
ejpam-4737	265	23	δs(λ	δs(λ	NOUN
ejpam-4737	265	24	,	,	PUNCT
ejpam-4737	265	25	p	p	NOUN
ejpam-4737	265	26	)	)	PUNCT
ejpam-4737	265	27	)	)	PUNCT
ejpam-4737	266	1	=	=	PUNCT
ejpam-4737	266	2	∅.	∅.	NOUN
ejpam-4737	266	3	since	since	SCONJ
ejpam-4737	266	4	δs(λ	δs(λ	NOUN
ejpam-4737	266	5	,	,	PUNCT
ejpam-4737	266	6	p)o(x	p)o(x	ADJ
ejpam-4737	266	7	,	,	PUNCT
ejpam-4737	266	8	τ	τ	PROPN
ejpam-4737	266	9	)	)	PUNCT
ejpam-4737	266	10	⊆	⊆	NUM
ejpam-4737	266	11	s(λ	s(λ	PROPN
ejpam-4737	266	12	,	,	PUNCT
ejpam-4737	266	13	p)o(x	p)o(x	ADJ
ejpam-4737	266	14	,	,	PUNCT
ejpam-4737	266	15	τ	τ	PROPN
ejpam-4737	266	16	)	)	PUNCT
ejpam-4737	266	17	,	,	PUNCT
ejpam-4737	266	18	(	(	PUNCT
ejpam-4737	266	19	x	x	X
ejpam-4737	266	20	,	,	PUNCT
ejpam-4737	266	21	τ	τ	X
ejpam-4737	266	22	)	)	PUNCT
ejpam-4737	266	23	is	be	AUX
ejpam-4737	266	24	s(λ	s(λ	PROPN
ejpam-4737	266	25	,	,	PUNCT
ejpam-4737	266	26	p)-regular	p)-regular	NOUN
ejpam-4737	266	27	.	.	PUNCT
ejpam-4737	267	1	definition	definition	NOUN
ejpam-4737	267	2	11	11	NUM
ejpam-4737	267	3	.	.	PUNCT
ejpam-4737	268	1	a	a	DET
ejpam-4737	268	2	topological	topological	ADJ
ejpam-4737	268	3	space	space	NOUN
ejpam-4737	268	4	(	(	PUNCT
ejpam-4737	268	5	x	x	X
ejpam-4737	268	6	,	,	PUNCT
ejpam-4737	268	7	τ	τ	X
ejpam-4737	268	8	)	)	PUNCT
ejpam-4737	268	9	is	be	AUX
ejpam-4737	268	10	said	say	VERB
ejpam-4737	268	11	to	to	PART
ejpam-4737	268	12	be	be	AUX
ejpam-4737	268	13	s(λ	s(λ	PROPN
ejpam-4737	268	14	,	,	PUNCT
ejpam-4737	268	15	p)-normal	p)-normal	PUNCT
ejpam-4737	268	16	if	if	SCONJ
ejpam-4737	268	17	,	,	PUNCT
ejpam-4737	268	18	for	for	ADP
ejpam-4737	268	19	each	each	DET
ejpam-4737	268	20	disjoint	disjoint	NOUN
ejpam-4737	268	21	s(λ	s(λ	PROPN
ejpam-4737	268	22	,	,	PUNCT
ejpam-4737	268	23	p)-closed	p)-close	VERB
ejpam-4737	268	24	sets	set	NOUN
ejpam-4737	268	25	f	f	PROPN
ejpam-4737	268	26	and	and	CCONJ
ejpam-4737	268	27	k	k	PROPN
ejpam-4737	268	28	of	of	ADP
ejpam-4737	268	29	x	x	PRON
ejpam-4737	268	30	,	,	PUNCT
ejpam-4737	268	31	there	there	PRON
ejpam-4737	268	32	exist	exist	VERB
ejpam-4737	268	33	u	u	NOUN
ejpam-4737	268	34	,	,	PUNCT
ejpam-4737	268	35	v	v	PROPN
ejpam-4737	268	36	∈	∈	PROPN
ejpam-4737	268	37	s(λ	s(λ	PROPN
ejpam-4737	268	38	,	,	PUNCT
ejpam-4737	268	39	p)o(x	p)o(x	ADJ
ejpam-4737	268	40	,	,	PUNCT
ejpam-4737	268	41	τ	τ	PROPN
ejpam-4737	268	42	)	)	PUNCT
ejpam-4737	268	43	such	such	ADJ
ejpam-4737	268	44	that	that	SCONJ
ejpam-4737	268	45	f	f	PROPN
ejpam-4737	268	46	⊆	⊆	NUM
ejpam-4737	268	47	u	u	NOUN
ejpam-4737	268	48	,	,	PUNCT
ejpam-4737	268	49	k	k	PROPN
ejpam-4737	268	50	⊆	⊆	NUM
ejpam-4737	268	51	v	v	NOUN
ejpam-4737	268	52	and	and	CCONJ
ejpam-4737	268	53	u	u	NOUN
ejpam-4737	268	54	∩	∩	NOUN
ejpam-4737	268	55	v	v	NOUN
ejpam-4737	268	56	=	=	PUNCT
ejpam-4737	268	57	∅.	∅.	NOUN
ejpam-4737	268	58	theorem	theorem	VERB
ejpam-4737	268	59	11	11	NUM
ejpam-4737	268	60	.	.	PUNCT
ejpam-4737	269	1	for	for	ADP
ejpam-4737	269	2	a	a	DET
ejpam-4737	269	3	topological	topological	ADJ
ejpam-4737	269	4	space	space	NOUN
ejpam-4737	269	5	(	(	PUNCT
ejpam-4737	269	6	x	x	X
ejpam-4737	269	7	,	,	PUNCT
ejpam-4737	269	8	τ	τ	PROPN
ejpam-4737	269	9	)	)	PUNCT
ejpam-4737	269	10	,	,	PUNCT
ejpam-4737	269	11	the	the	DET
ejpam-4737	269	12	following	follow	VERB
ejpam-4737	269	13	properties	property	NOUN
ejpam-4737	269	14	are	be	AUX
ejpam-4737	269	15	equivalent	equivalent	ADJ
ejpam-4737	269	16	:	:	PUNCT
ejpam-4737	269	17	(	(	PUNCT
ejpam-4737	269	18	1	1	X
ejpam-4737	269	19	)	)	PUNCT
ejpam-4737	269	20	(	(	PUNCT
ejpam-4737	269	21	x	x	X
ejpam-4737	269	22	,	,	PUNCT
ejpam-4737	269	23	τ	τ	X
ejpam-4737	269	24	)	)	PUNCT
ejpam-4737	269	25	is	be	AUX
ejpam-4737	269	26	s(λ	s(λ	PROPN
ejpam-4737	269	27	,	,	PUNCT
ejpam-4737	269	28	p)-normal	p)-normal	NOUN
ejpam-4737	269	29	.	.	PUNCT
ejpam-4737	270	1	c.	c.	PROPN
ejpam-4737	270	2	boonpok	boonpok	PROPN
ejpam-4737	270	3	,	,	PUNCT
ejpam-4737	270	4	m.	m.	NOUN
ejpam-4737	270	5	thongmoon	thongmoon	PROPN
ejpam-4737	270	6	/	/	SYM
ejpam-4737	270	7	eur	eur	PROPN
ejpam-4737	270	8	.	.	PUNCT
ejpam-4737	271	1	j.	j.	PROPN
ejpam-4737	271	2	pure	pure	PROPN
ejpam-4737	271	3	appl	appl	PROPN
ejpam-4737	271	4	.	.	PROPN
ejpam-4737	271	5	math	math	PROPN
ejpam-4737	271	6	,	,	PUNCT
ejpam-4737	271	7	16	16	NUM
ejpam-4737	271	8	(	(	PUNCT
ejpam-4737	271	9	3	3	NUM
ejpam-4737	271	10	)	)	PUNCT
ejpam-4737	271	11	(	(	PUNCT
ejpam-4737	271	12	2023	2023	NUM
ejpam-4737	271	13	)	)	PUNCT
ejpam-4737	271	14	,	,	PUNCT
ejpam-4737	271	15	1434	1434	NUM
ejpam-4737	271	16	-	-	SYM
ejpam-4737	271	17	1447	1447	NUM
ejpam-4737	271	18	1443	1443	NUM
ejpam-4737	271	19	(	(	PUNCT
ejpam-4737	271	20	2	2	NUM
ejpam-4737	271	21	)	)	PUNCT
ejpam-4737	271	22	for	for	ADP
ejpam-4737	271	23	each	each	DET
ejpam-4737	271	24	disjoint	disjoint	NOUN
ejpam-4737	271	25	s(λ	s(λ	PROPN
ejpam-4737	271	26	,	,	PUNCT
ejpam-4737	271	27	p)-closed	p)-close	VERB
ejpam-4737	271	28	sets	set	NOUN
ejpam-4737	271	29	f	f	PROPN
ejpam-4737	271	30	and	and	CCONJ
ejpam-4737	271	31	k	k	PROPN
ejpam-4737	271	32	of	of	ADP
ejpam-4737	271	33	x	x	PRON
ejpam-4737	271	34	,	,	PUNCT
ejpam-4737	271	35	there	there	PRON
ejpam-4737	271	36	exist	exist	VERB
ejpam-4737	271	37	u	u	NOUN
ejpam-4737	271	38	,	,	PUNCT
ejpam-4737	271	39	v	v	PROPN
ejpam-4737	271	40	∈	∈	PROPN
ejpam-4737	271	41	δs(λ	δs(λ	NOUN
ejpam-4737	271	42	,	,	PUNCT
ejpam-4737	271	43	p)o(x	p)o(x	ADJ
ejpam-4737	271	44	,	,	PUNCT
ejpam-4737	271	45	τ	τ	PROPN
ejpam-4737	271	46	)	)	PUNCT
ejpam-4737	271	47	such	such	ADJ
ejpam-4737	271	48	that	that	SCONJ
ejpam-4737	271	49	f	f	PROPN
ejpam-4737	271	50	⊆	⊆	NUM
ejpam-4737	271	51	u	u	NOUN
ejpam-4737	271	52	,	,	PUNCT
ejpam-4737	271	53	k	k	PROPN
ejpam-4737	271	54	⊆	⊆	NUM
ejpam-4737	271	55	v	v	NOUN
ejpam-4737	271	56	and	and	CCONJ
ejpam-4737	271	57	u	u	NOUN
ejpam-4737	271	58	∩	∩	NOUN
ejpam-4737	271	59	v	v	NOUN
ejpam-4737	271	60	=	=	PUNCT
ejpam-4737	271	61	∅.	∅.	X
ejpam-4737	271	62	(	(	PUNCT
ejpam-4737	271	63	3	3	NUM
ejpam-4737	271	64	)	)	PUNCT
ejpam-4737	271	65	for	for	ADP
ejpam-4737	271	66	each	each	DET
ejpam-4737	271	67	s(λ	s(λ	PROPN
ejpam-4737	271	68	,	,	PUNCT
ejpam-4737	271	69	p)-closed	p)-close	VERB
ejpam-4737	271	70	set	set	VERB
ejpam-4737	271	71	f	f	PROPN
ejpam-4737	271	72	and	and	CCONJ
ejpam-4737	271	73	each	each	DET
ejpam-4737	271	74	s(λ	s(λ	PROPN
ejpam-4737	271	75	,	,	PUNCT
ejpam-4737	271	76	p)-open	p)-open	VERB
ejpam-4737	271	77	set	set	VERB
ejpam-4737	271	78	v	v	NOUN
ejpam-4737	271	79	containing	contain	VERB
ejpam-4737	271	80	f	f	NOUN
ejpam-4737	271	81	,	,	PUNCT
ejpam-4737	271	82	there	there	PRON
ejpam-4737	271	83	exists	exist	VERB
ejpam-4737	271	84	u	u	PROPN
ejpam-4737	271	85	∈	∈	PROPN
ejpam-4737	271	86	δs(λ	δs(λ	NOUN
ejpam-4737	271	87	,	,	PUNCT
ejpam-4737	271	88	p)o(x	p)o(x	ADJ
ejpam-4737	271	89	,	,	PUNCT
ejpam-4737	271	90	τ	τ	PROPN
ejpam-4737	271	91	)	)	PUNCT
ejpam-4737	271	92	such	such	ADJ
ejpam-4737	271	93	that	that	SCONJ
ejpam-4737	271	94	f	f	PROPN
ejpam-4737	271	95	⊆	⊆	NUM
ejpam-4737	271	96	u	u	NOUN
ejpam-4737	271	97	⊆	⊆	NUM
ejpam-4737	271	98	u	u	NOUN
ejpam-4737	271	99	δs(λ	δs(λ	NOUN
ejpam-4737	271	100	,	,	PUNCT
ejpam-4737	271	101	p	p	NOUN
ejpam-4737	271	102	)	)	PUNCT
ejpam-4737	271	103	⊆	⊆	NUM
ejpam-4737	271	104	v	v	NOUN
ejpam-4737	271	105	.	.	PUNCT
ejpam-4737	272	1	proof	proof	NOUN
ejpam-4737	272	2	.	.	PUNCT
ejpam-4737	273	1	the	the	DET
ejpam-4737	273	2	proof	proof	NOUN
ejpam-4737	273	3	is	be	AUX
ejpam-4737	273	4	analogous	analogous	ADJ
ejpam-4737	273	5	to	to	ADP
ejpam-4737	273	6	that	that	PRON
ejpam-4737	273	7	of	of	ADP
ejpam-4737	273	8	theorem	theorem	ADJ
ejpam-4737	273	9	10	10	NUM
ejpam-4737	273	10	and	and	CCONJ
ejpam-4737	273	11	is	be	AUX
ejpam-4737	273	12	omitted	omit	VERB
ejpam-4737	273	13	.	.	PUNCT
ejpam-4737	274	1	6	6	X
ejpam-4737	274	2	.	.	PUNCT
ejpam-4737	274	3	characterizations	characterization	NOUN
ejpam-4737	274	4	of	of	ADP
ejpam-4737	274	5	s(λ	s(λ	PROPN
ejpam-4737	274	6	,	,	PUNCT
ejpam-4737	274	7	p)-t2	p)-t2	ADJ
ejpam-4737	274	8	spaces	space	NOUN
ejpam-4737	274	9	and	and	CCONJ
ejpam-4737	274	10	s(λ	s(λ	PROPN
ejpam-4737	274	11	,	,	PUNCT
ejpam-4737	274	12	p)-urysohn	p)-urysohn	NOUN
ejpam-4737	274	13	spaces	space	NOUN
ejpam-4737	274	14	in	in	ADP
ejpam-4737	274	15	this	this	DET
ejpam-4737	274	16	section	section	NOUN
ejpam-4737	274	17	,	,	PUNCT
ejpam-4737	274	18	we	we	PRON
ejpam-4737	274	19	introduce	introduce	VERB
ejpam-4737	274	20	the	the	DET
ejpam-4737	274	21	notions	notion	NOUN
ejpam-4737	274	22	of	of	ADP
ejpam-4737	274	23	s(λ	s(λ	PROPN
ejpam-4737	274	24	,	,	PUNCT
ejpam-4737	274	25	p)-t2	p)-t2	ADJ
ejpam-4737	274	26	spaces	space	NOUN
ejpam-4737	274	27	and	and	CCONJ
ejpam-4737	274	28	s(λ	s(λ	PROPN
ejpam-4737	274	29	,	,	PUNCT
ejpam-4737	274	30	p)-urysohn	p)-urysohn	NOUN
ejpam-4737	274	31	spaces	space	NOUN
ejpam-4737	274	32	.	.	PUNCT
ejpam-4737	275	1	furthermore	furthermore	ADV
ejpam-4737	275	2	,	,	PUNCT
ejpam-4737	275	3	some	some	DET
ejpam-4737	275	4	characterizations	characterization	NOUN
ejpam-4737	275	5	of	of	ADP
ejpam-4737	275	6	s(λ	s(λ	PROPN
ejpam-4737	275	7	,	,	PUNCT
ejpam-4737	275	8	p)-t2	p)-t2	ADJ
ejpam-4737	275	9	spaces	space	NOUN
ejpam-4737	275	10	and	and	CCONJ
ejpam-4737	275	11	s(λ	s(λ	PROPN
ejpam-4737	275	12	,	,	PUNCT
ejpam-4737	275	13	p)-urysohn	p)-urysohn	NOUN
ejpam-4737	275	14	spaces	space	NOUN
ejpam-4737	275	15	are	be	AUX
ejpam-4737	275	16	investigated	investigate	VERB
ejpam-4737	275	17	.	.	PUNCT
ejpam-4737	276	1	definition	definition	NOUN
ejpam-4737	276	2	12	12	NUM
ejpam-4737	276	3	.	.	PUNCT
ejpam-4737	277	1	a	a	DET
ejpam-4737	277	2	topological	topological	ADJ
ejpam-4737	277	3	space	space	NOUN
ejpam-4737	277	4	(	(	PUNCT
ejpam-4737	277	5	x	x	X
ejpam-4737	277	6	,	,	PUNCT
ejpam-4737	277	7	τ	τ	X
ejpam-4737	277	8	)	)	PUNCT
ejpam-4737	277	9	is	be	AUX
ejpam-4737	277	10	said	say	VERB
ejpam-4737	277	11	to	to	PART
ejpam-4737	277	12	be	be	AUX
ejpam-4737	277	13	s(λ	s(λ	PROPN
ejpam-4737	277	14	,	,	PUNCT
ejpam-4737	277	15	p)-t2	p)-t2	ADJ
ejpam-4737	277	16	if	if	SCONJ
ejpam-4737	277	17	,	,	PUNCT
ejpam-4737	277	18	for	for	ADP
ejpam-4737	277	19	each	each	DET
ejpam-4737	277	20	pair	pair	NOUN
ejpam-4737	277	21	of	of	ADP
ejpam-4737	277	22	distinct	distinct	ADJ
ejpam-4737	277	23	points	point	NOUN
ejpam-4737	277	24	x	x	X
ejpam-4737	277	25	,	,	PUNCT
ejpam-4737	277	26	y	y	PROPN
ejpam-4737	277	27	∈	∈	PROPN
ejpam-4737	277	28	x	x	PRON
ejpam-4737	277	29	,	,	PUNCT
ejpam-4737	277	30	there	there	PRON
ejpam-4737	277	31	exist	exist	VERB
ejpam-4737	277	32	u	u	NOUN
ejpam-4737	277	33	,	,	PUNCT
ejpam-4737	277	34	v	v	PROPN
ejpam-4737	277	35	∈	∈	PROPN
ejpam-4737	277	36	s(λ	s(λ	PROPN
ejpam-4737	277	37	,	,	PUNCT
ejpam-4737	277	38	p)o(x	p)o(x	ADJ
ejpam-4737	277	39	,	,	PUNCT
ejpam-4737	277	40	τ	τ	PROPN
ejpam-4737	277	41	)	)	PUNCT
ejpam-4737	277	42	such	such	ADJ
ejpam-4737	277	43	that	that	SCONJ
ejpam-4737	277	44	x	x	SYM
ejpam-4737	277	45	∈	∈	PROPN
ejpam-4737	277	46	u	u	NOUN
ejpam-4737	277	47	,	,	PUNCT
ejpam-4737	277	48	y	y	PROPN
ejpam-4737	277	49	∈	∈	PROPN
ejpam-4737	277	50	v	v	NOUN
ejpam-4737	277	51	and	and	CCONJ
ejpam-4737	277	52	u	u	NOUN
ejpam-4737	277	53	∩	∩	NOUN
ejpam-4737	277	54	v	v	NOUN
ejpam-4737	277	55	=	=	PUNCT
ejpam-4737	277	56	∅.	∅.	NOUN
ejpam-4737	277	57	theorem	theorem	VERB
ejpam-4737	277	58	12	12	NUM
ejpam-4737	277	59	.	.	PUNCT
ejpam-4737	278	1	for	for	ADP
ejpam-4737	278	2	a	a	DET
ejpam-4737	278	3	topological	topological	ADJ
ejpam-4737	278	4	space	space	NOUN
ejpam-4737	278	5	(	(	PUNCT
ejpam-4737	278	6	x	x	X
ejpam-4737	278	7	,	,	PUNCT
ejpam-4737	278	8	τ	τ	PROPN
ejpam-4737	278	9	)	)	PUNCT
ejpam-4737	278	10	,	,	PUNCT
ejpam-4737	278	11	the	the	DET
ejpam-4737	278	12	following	follow	VERB
ejpam-4737	278	13	properties	property	NOUN
ejpam-4737	278	14	are	be	AUX
ejpam-4737	278	15	equivalent	equivalent	ADJ
ejpam-4737	278	16	:	:	PUNCT
ejpam-4737	278	17	(	(	PUNCT
ejpam-4737	278	18	1	1	X
ejpam-4737	278	19	)	)	PUNCT
ejpam-4737	278	20	(	(	PUNCT
ejpam-4737	278	21	x	x	X
ejpam-4737	278	22	,	,	PUNCT
ejpam-4737	278	23	τ	τ	X
ejpam-4737	278	24	)	)	PUNCT
ejpam-4737	278	25	is	be	AUX
ejpam-4737	278	26	s(λ	s(λ	PROPN
ejpam-4737	278	27	,	,	PUNCT
ejpam-4737	278	28	p)-t2	p)-t2	NOUN
ejpam-4737	278	29	.	.	PUNCT
ejpam-4737	279	1	(	(	PUNCT
ejpam-4737	279	2	2	2	X
ejpam-4737	279	3	)	)	PUNCT
ejpam-4737	279	4	for	for	ADP
ejpam-4737	279	5	each	each	DET
ejpam-4737	279	6	pair	pair	NOUN
ejpam-4737	279	7	of	of	ADP
ejpam-4737	279	8	distinct	distinct	ADJ
ejpam-4737	279	9	points	point	NOUN
ejpam-4737	279	10	x	x	X
ejpam-4737	279	11	,	,	PUNCT
ejpam-4737	279	12	y	y	PROPN
ejpam-4737	279	13	∈	∈	PROPN
ejpam-4737	279	14	x	x	PRON
ejpam-4737	279	15	,	,	PUNCT
ejpam-4737	279	16	there	there	PRON
ejpam-4737	279	17	exist	exist	VERB
ejpam-4737	279	18	u	u	NOUN
ejpam-4737	279	19	,	,	PUNCT
ejpam-4737	279	20	v	v	PROPN
ejpam-4737	279	21	∈	∈	PROPN
ejpam-4737	279	22	s(λ	s(λ	PROPN
ejpam-4737	279	23	,	,	PUNCT
ejpam-4737	279	24	p)r(x	p)r(x	PROPN
ejpam-4737	279	25	,	,	PUNCT
ejpam-4737	279	26	τ	τ	PROPN
ejpam-4737	279	27	)	)	PUNCT
ejpam-4737	279	28	such	such	ADJ
ejpam-4737	279	29	that	that	SCONJ
ejpam-4737	279	30	x	x	SYM
ejpam-4737	279	31	∈	∈	PROPN
ejpam-4737	279	32	u	u	NOUN
ejpam-4737	279	33	,	,	PUNCT
ejpam-4737	279	34	y	y	PROPN
ejpam-4737	279	35	∈	∈	PROPN
ejpam-4737	279	36	v	v	NOUN
ejpam-4737	279	37	and	and	CCONJ
ejpam-4737	279	38	u	u	NOUN
ejpam-4737	279	39	∩	∩	NOUN
ejpam-4737	279	40	v	v	NOUN
ejpam-4737	279	41	=	=	PUNCT
ejpam-4737	279	42	∅.	∅.	X
ejpam-4737	279	43	(	(	PUNCT
ejpam-4737	279	44	3	3	NUM
ejpam-4737	279	45	)	)	PUNCT
ejpam-4737	279	46	for	for	ADP
ejpam-4737	279	47	each	each	DET
ejpam-4737	279	48	pair	pair	NOUN
ejpam-4737	279	49	of	of	ADP
ejpam-4737	279	50	distinct	distinct	ADJ
ejpam-4737	279	51	points	point	NOUN
ejpam-4737	279	52	x	x	X
ejpam-4737	279	53	,	,	PUNCT
ejpam-4737	279	54	y	y	PROPN
ejpam-4737	279	55	∈	∈	PROPN
ejpam-4737	280	1	x	x	PRON
ejpam-4737	280	2	,	,	PUNCT
ejpam-4737	280	3	there	there	PRON
ejpam-4737	280	4	exist	exist	VERB
ejpam-4737	280	5	u	u	NOUN
ejpam-4737	280	6	,	,	PUNCT
ejpam-4737	280	7	v	v	PROPN
ejpam-4737	280	8	∈	∈	PROPN
ejpam-4737	280	9	δs(λ	δs(λ	NOUN
ejpam-4737	280	10	,	,	PUNCT
ejpam-4737	280	11	p)o(x	p)o(x	ADJ
ejpam-4737	280	12	,	,	PUNCT
ejpam-4737	280	13	τ	τ	PROPN
ejpam-4737	280	14	)	)	PUNCT
ejpam-4737	280	15	such	such	ADJ
ejpam-4737	280	16	that	that	SCONJ
ejpam-4737	280	17	x	x	SYM
ejpam-4737	280	18	∈	∈	PROPN
ejpam-4737	280	19	u	u	NOUN
ejpam-4737	280	20	,	,	PUNCT
ejpam-4737	280	21	y	y	PROPN
ejpam-4737	280	22	∈	∈	PROPN
ejpam-4737	280	23	v	v	NOUN
ejpam-4737	280	24	and	and	CCONJ
ejpam-4737	280	25	u	u	NOUN
ejpam-4737	280	26	δs(λ	δs(λ	NOUN
ejpam-4737	280	27	,	,	PUNCT
ejpam-4737	280	28	p	p	NOUN
ejpam-4737	280	29	)	)	PUNCT
ejpam-4737	280	30	∩	∩	PROPN
ejpam-4737	280	31	v	v	NOUN
ejpam-4737	280	32	δs(λ	δs(λ	NOUN
ejpam-4737	280	33	,	,	PUNCT
ejpam-4737	280	34	p	p	NOUN
ejpam-4737	280	35	)	)	PUNCT
ejpam-4737	280	36	=	=	SYM
ejpam-4737	280	37	∅.	∅.	X
ejpam-4737	280	38	(	(	PUNCT
ejpam-4737	280	39	4	4	NUM
ejpam-4737	280	40	)	)	PUNCT
ejpam-4737	280	41	for	for	ADP
ejpam-4737	280	42	each	each	DET
ejpam-4737	280	43	pair	pair	NOUN
ejpam-4737	280	44	of	of	ADP
ejpam-4737	280	45	distinct	distinct	ADJ
ejpam-4737	280	46	points	point	NOUN
ejpam-4737	280	47	x	x	X
ejpam-4737	280	48	,	,	PUNCT
ejpam-4737	280	49	y	y	PROPN
ejpam-4737	280	50	∈	∈	PROPN
ejpam-4737	281	1	x	x	PRON
ejpam-4737	281	2	,	,	PUNCT
ejpam-4737	281	3	there	there	PRON
ejpam-4737	281	4	exist	exist	VERB
ejpam-4737	281	5	u	u	NOUN
ejpam-4737	281	6	,	,	PUNCT
ejpam-4737	281	7	v	v	PROPN
ejpam-4737	281	8	∈	∈	PROPN
ejpam-4737	281	9	δs(λ	δs(λ	NOUN
ejpam-4737	281	10	,	,	PUNCT
ejpam-4737	281	11	p)o(x	p)o(x	ADJ
ejpam-4737	281	12	,	,	PUNCT
ejpam-4737	281	13	τ	τ	PROPN
ejpam-4737	281	14	)	)	PUNCT
ejpam-4737	281	15	such	such	ADJ
ejpam-4737	281	16	that	that	SCONJ
ejpam-4737	281	17	x	x	SYM
ejpam-4737	281	18	∈	∈	PROPN
ejpam-4737	281	19	u	u	NOUN
ejpam-4737	281	20	,	,	PUNCT
ejpam-4737	281	21	y	y	PROPN
ejpam-4737	281	22	∈	∈	PROPN
ejpam-4737	281	23	v	v	NOUN
ejpam-4737	281	24	and	and	CCONJ
ejpam-4737	281	25	u	u	NOUN
ejpam-4737	281	26	s(λ	s(λ	PROPN
ejpam-4737	281	27	,	,	PUNCT
ejpam-4737	281	28	p	p	NOUN
ejpam-4737	281	29	)	)	PUNCT
ejpam-4737	281	30	∩	∩	PROPN
ejpam-4737	281	31	v	v	ADP
ejpam-4737	281	32	s(λ	s(λ	PROPN
ejpam-4737	281	33	,	,	PUNCT
ejpam-4737	281	34	p	p	NOUN
ejpam-4737	281	35	)	)	PUNCT
ejpam-4737	281	36	=	=	SYM
ejpam-4737	281	37	∅.	∅.	X
ejpam-4737	281	38	(	(	PUNCT
ejpam-4737	281	39	5	5	NUM
ejpam-4737	281	40	)	)	PUNCT
ejpam-4737	281	41	for	for	ADP
ejpam-4737	281	42	each	each	DET
ejpam-4737	281	43	pair	pair	NOUN
ejpam-4737	281	44	of	of	ADP
ejpam-4737	281	45	distinct	distinct	ADJ
ejpam-4737	281	46	points	point	NOUN
ejpam-4737	281	47	x	x	X
ejpam-4737	281	48	,	,	PUNCT
ejpam-4737	281	49	y	y	PROPN
ejpam-4737	281	50	∈	∈	PROPN
ejpam-4737	282	1	x	x	PRON
ejpam-4737	282	2	,	,	PUNCT
ejpam-4737	282	3	there	there	PRON
ejpam-4737	282	4	exist	exist	VERB
ejpam-4737	282	5	u	u	NOUN
ejpam-4737	282	6	,	,	PUNCT
ejpam-4737	282	7	v	v	PROPN
ejpam-4737	282	8	∈	∈	PROPN
ejpam-4737	282	9	δs(λ	δs(λ	NOUN
ejpam-4737	282	10	,	,	PUNCT
ejpam-4737	282	11	p)o(x	p)o(x	ADJ
ejpam-4737	282	12	,	,	PUNCT
ejpam-4737	282	13	τ	τ	PROPN
ejpam-4737	282	14	)	)	PUNCT
ejpam-4737	282	15	such	such	ADJ
ejpam-4737	282	16	that	that	SCONJ
ejpam-4737	282	17	x	x	SYM
ejpam-4737	282	18	∈	∈	PROPN
ejpam-4737	282	19	u	u	NOUN
ejpam-4737	282	20	,	,	PUNCT
ejpam-4737	282	21	y	y	PROPN
ejpam-4737	282	22	∈	∈	PROPN
ejpam-4737	282	23	v	v	NOUN
ejpam-4737	282	24	and	and	CCONJ
ejpam-4737	282	25	u	u	NOUN
ejpam-4737	282	26	∩	∩	NOUN
ejpam-4737	282	27	v	v	NOUN
ejpam-4737	282	28	=	=	PUNCT
ejpam-4737	282	29	∅.	∅.	NOUN
ejpam-4737	282	30	proof	proof	NOUN
ejpam-4737	282	31	.	.	PUNCT
ejpam-4737	283	1	(	(	PUNCT
ejpam-4737	283	2	1	1	X
ejpam-4737	283	3	)	)	PUNCT
ejpam-4737	283	4	⇒	⇒	NOUN
ejpam-4737	283	5	(	(	PUNCT
ejpam-4737	283	6	2	2	NUM
ejpam-4737	283	7	):	):	PUNCT
ejpam-4737	283	8	suppose	suppose	VERB
ejpam-4737	283	9	that	that	SCONJ
ejpam-4737	283	10	(	(	PUNCT
ejpam-4737	283	11	x	x	X
ejpam-4737	283	12	,	,	PUNCT
ejpam-4737	283	13	τ	τ	X
ejpam-4737	283	14	)	)	PUNCT
ejpam-4737	283	15	is	be	AUX
ejpam-4737	283	16	s(λ	s(λ	PROPN
ejpam-4737	283	17	,	,	PUNCT
ejpam-4737	283	18	p)-t2	p)-t2	NOUN
ejpam-4737	283	19	.	.	PUNCT
ejpam-4737	284	1	then	then	ADV
ejpam-4737	284	2	,	,	PUNCT
ejpam-4737	284	3	for	for	SCONJ
ejpam-4737	284	4	each	each	DET
ejpam-4737	284	5	pair	pair	NOUN
ejpam-4737	284	6	of	of	ADP
ejpam-4737	284	7	distinct	distinct	ADJ
ejpam-4737	284	8	points	point	NOUN
ejpam-4737	284	9	x	x	X
ejpam-4737	284	10	,	,	PUNCT
ejpam-4737	284	11	y	y	PROPN
ejpam-4737	284	12	∈	∈	PROPN
ejpam-4737	284	13	x	x	PRON
ejpam-4737	284	14	,	,	PUNCT
ejpam-4737	284	15	there	there	PRON
ejpam-4737	284	16	exist	exist	VERB
ejpam-4737	284	17	g	g	NOUN
ejpam-4737	284	18	,	,	PUNCT
ejpam-4737	284	19	h	h	PROPN
ejpam-4737	284	20	∈	∈	PROPN
ejpam-4737	284	21	s(λ	s(λ	PROPN
ejpam-4737	284	22	,	,	PUNCT
ejpam-4737	284	23	p)o(x	p)o(x	ADJ
ejpam-4737	284	24	,	,	PUNCT
ejpam-4737	284	25	τ	τ	PROPN
ejpam-4737	284	26	)	)	PUNCT
ejpam-4737	284	27	such	such	ADJ
ejpam-4737	284	28	that	that	SCONJ
ejpam-4737	284	29	x	x	SYM
ejpam-4737	284	30	∈	∈	PROPN
ejpam-4737	284	31	g	g	PROPN
ejpam-4737	284	32	,	,	PUNCT
ejpam-4737	284	33	y	y	PROPN
ejpam-4737	284	34	∈	∈	PROPN
ejpam-4737	284	35	h	h	NOUN
ejpam-4737	284	36	and	and	CCONJ
ejpam-4737	284	37	g∩h	g∩h	PROPN
ejpam-4737	284	38	=	=	PUNCT
ejpam-4737	284	39	∅.	∅.	VERB
ejpam-4737	284	40	thus	thus	ADV
ejpam-4737	284	41	,	,	PUNCT
ejpam-4737	284	42	gs(λ	gs(λ	VERB
ejpam-4737	284	43	,	,	PUNCT
ejpam-4737	284	44	p	p	NOUN
ejpam-4737	284	45	)	)	PUNCT
ejpam-4737	284	46	∩h	∩h	NOUN
ejpam-4737	284	47	=	=	PUNCT
ejpam-4737	285	1	∅.	∅.	NOUN
ejpam-4737	285	2	by	by	ADP
ejpam-4737	285	3	lemma	lemma	PROPN
ejpam-4737	285	4	4	4	NUM
ejpam-4737	285	5	,	,	PUNCT
ejpam-4737	285	6	we	we	PRON
ejpam-4737	285	7	have	have	AUX
ejpam-4737	285	8	gs(λ	gs(λ	NOUN
ejpam-4737	285	9	,	,	PUNCT
ejpam-4737	285	10	p	p	X
ejpam-4737	285	11	)	)	PUNCT
ejpam-4737	285	12	∈	∈	PROPN
ejpam-4737	285	13	s(λ	s(λ	PROPN
ejpam-4737	285	14	,	,	PUNCT
ejpam-4737	285	15	p)r(x	p)r(x	PROPN
ejpam-4737	285	16	,	,	PUNCT
ejpam-4737	285	17	τ	τ	PROPN
ejpam-4737	285	18	)	)	PUNCT
ejpam-4737	285	19	and	and	CCONJ
ejpam-4737	285	20	gs(λ	gs(λ	VERB
ejpam-4737	285	21	,	,	PUNCT
ejpam-4737	285	22	p	p	X
ejpam-4737	285	23	)	)	PUNCT
ejpam-4737	285	24	∩hs(λ	∩hs(λ	PROPN
ejpam-4737	285	25	,	,	PUNCT
ejpam-4737	285	26	p	p	NOUN
ejpam-4737	285	27	)	)	PUNCT
ejpam-4737	285	28	=	=	PUNCT
ejpam-4737	285	29	∅.	∅.	NOUN
ejpam-4737	285	30	now	now	ADV
ejpam-4737	285	31	set	set	VERB
ejpam-4737	285	32	u	u	PRON
ejpam-4737	285	33	=	=	X
ejpam-4737	285	34	gs(λ	gs(λ	X
ejpam-4737	285	35	,	,	PUNCT
ejpam-4737	285	36	p	p	NOUN
ejpam-4737	285	37	)	)	PUNCT
ejpam-4737	285	38	and	and	CCONJ
ejpam-4737	285	39	v	v	NOUN
ejpam-4737	285	40	=	=	SYM
ejpam-4737	285	41	hs(λ	hs(λ	NOUN
ejpam-4737	285	42	,	,	PUNCT
ejpam-4737	285	43	p	p	NOUN
ejpam-4737	285	44	)	)	PUNCT
ejpam-4737	285	45	.	.	PUNCT
ejpam-4737	286	1	then	then	ADV
ejpam-4737	286	2	,	,	PUNCT
ejpam-4737	286	3	u	u	NOUN
ejpam-4737	286	4	and	and	CCONJ
ejpam-4737	286	5	v	v	NOUN
ejpam-4737	286	6	are	be	AUX
ejpam-4737	286	7	s(λ	s(λ	PROPN
ejpam-4737	286	8	,	,	PUNCT
ejpam-4737	286	9	p)-regular	p)-regular	NOUN
ejpam-4737	286	10	sets	set	VERB
ejpam-4737	286	11	such	such	ADJ
ejpam-4737	286	12	that	that	SCONJ
ejpam-4737	286	13	x	x	SYM
ejpam-4737	286	14	∈	∈	PROPN
ejpam-4737	286	15	u	u	NOUN
ejpam-4737	286	16	,	,	PUNCT
ejpam-4737	286	17	y	y	PROPN
ejpam-4737	286	18	∈	∈	PROPN
ejpam-4737	286	19	v	v	NOUN
ejpam-4737	286	20	and	and	CCONJ
ejpam-4737	286	21	u	u	NOUN
ejpam-4737	286	22	∩	∩	NOUN
ejpam-4737	286	23	v	v	NOUN
ejpam-4737	286	24	=	=	PUNCT
ejpam-4737	286	25	∅.	∅.	X
ejpam-4737	286	26	(	(	PUNCT
ejpam-4737	286	27	2	2	NUM
ejpam-4737	286	28	)	)	PUNCT
ejpam-4737	286	29	⇒	⇒	NOUN
ejpam-4737	286	30	(	(	PUNCT
ejpam-4737	286	31	3	3	NUM
ejpam-4737	286	32	):	):	PUNCT
ejpam-4737	286	33	this	this	PRON
ejpam-4737	286	34	is	be	AUX
ejpam-4737	286	35	follows	follow	VERB
ejpam-4737	286	36	from	from	ADP
ejpam-4737	286	37	the	the	DET
ejpam-4737	286	38	facts	fact	NOUN
ejpam-4737	286	39	that	that	PRON
ejpam-4737	286	40	s(λ	s(λ	PROPN
ejpam-4737	286	41	,	,	PUNCT
ejpam-4737	286	42	p)r(x	p)r(x	PROPN
ejpam-4737	286	43	,	,	PUNCT
ejpam-4737	286	44	τ	τ	PROPN
ejpam-4737	286	45	)	)	PUNCT
ejpam-4737	286	46	⊆	⊆	NUM
ejpam-4737	286	47	δs(λ	δs(λ	NOUN
ejpam-4737	286	48	,	,	PUNCT
ejpam-4737	286	49	p)o(x	p)o(x	ADJ
ejpam-4737	286	50	,	,	PUNCT
ejpam-4737	286	51	τ	τ	X
ejpam-4737	286	52	)	)	PUNCT
ejpam-4737	286	53	and	and	CCONJ
ejpam-4737	286	54	u	u	NOUN
ejpam-4737	286	55	δs(λ	δs(λ	NOUN
ejpam-4737	286	56	,	,	PUNCT
ejpam-4737	286	57	p	p	NOUN
ejpam-4737	286	58	)	)	PUNCT
ejpam-4737	286	59	=	=	SYM
ejpam-4737	286	60	u	u	PROPN
ejpam-4737	286	61	s(λ	s(λ	PROPN
ejpam-4737	286	62	,	,	PUNCT
ejpam-4737	286	63	p	p	NOUN
ejpam-4737	286	64	)	)	PUNCT
ejpam-4737	286	65	=	=	SYM
ejpam-4737	286	66	u	u	NOUN
ejpam-4737	286	67	for	for	ADP
ejpam-4737	286	68	every	every	DET
ejpam-4737	286	69	u	u	PROPN
ejpam-4737	286	70	∈	∈	PROPN
ejpam-4737	286	71	s(λ	s(λ	PROPN
ejpam-4737	286	72	,	,	PUNCT
ejpam-4737	286	73	p)r(x	p)r(x	PROPN
ejpam-4737	286	74	,	,	PUNCT
ejpam-4737	286	75	τ	τ	PROPN
ejpam-4737	286	76	)	)	PUNCT
ejpam-4737	286	77	.	.	PUNCT
ejpam-4737	287	1	(	(	PUNCT
ejpam-4737	287	2	3	3	X
ejpam-4737	287	3	)	)	PUNCT
ejpam-4737	287	4	⇒	⇒	NOUN
ejpam-4737	287	5	(	(	PUNCT
ejpam-4737	287	6	4	4	NUM
ejpam-4737	287	7	):	):	PUNCT
ejpam-4737	287	8	this	this	PRON
ejpam-4737	287	9	follows	follow	VERB
ejpam-4737	287	10	from	from	ADP
ejpam-4737	287	11	the	the	DET
ejpam-4737	287	12	fact	fact	NOUN
ejpam-4737	287	13	that	that	SCONJ
ejpam-4737	287	14	u	u	PROPN
ejpam-4737	287	15	δs(λ	δs(λ	NOUN
ejpam-4737	287	16	,	,	PUNCT
ejpam-4737	287	17	p	p	NOUN
ejpam-4737	287	18	)	)	PUNCT
ejpam-4737	287	19	=	=	SYM
ejpam-4737	287	20	u	u	PROPN
ejpam-4737	287	21	s(λ	s(λ	PROPN
ejpam-4737	287	22	,	,	PUNCT
ejpam-4737	287	23	p	p	NOUN
ejpam-4737	287	24	)	)	PUNCT
ejpam-4737	287	25	for	for	ADP
ejpam-4737	287	26	every	every	DET
ejpam-4737	287	27	u	u	PROPN
ejpam-4737	287	28	∈	∈	PROPN
ejpam-4737	287	29	δs(λ	δs(λ	NOUN
ejpam-4737	287	30	,	,	PUNCT
ejpam-4737	287	31	p)o(x	p)o(x	ADJ
ejpam-4737	287	32	,	,	PUNCT
ejpam-4737	287	33	τ	τ	PROPN
ejpam-4737	287	34	)	)	PUNCT
ejpam-4737	287	35	.	.	PUNCT
ejpam-4737	288	1	c.	c.	PROPN
ejpam-4737	288	2	boonpok	boonpok	PROPN
ejpam-4737	288	3	,	,	PUNCT
ejpam-4737	288	4	m.	m.	NOUN
ejpam-4737	288	5	thongmoon	thongmoon	PROPN
ejpam-4737	288	6	/	/	SYM
ejpam-4737	288	7	eur	eur	PROPN
ejpam-4737	288	8	.	.	PUNCT
ejpam-4737	289	1	j.	j.	PROPN
ejpam-4737	289	2	pure	pure	PROPN
ejpam-4737	289	3	appl	appl	PROPN
ejpam-4737	289	4	.	.	PROPN
ejpam-4737	289	5	math	math	PROPN
ejpam-4737	289	6	,	,	PUNCT
ejpam-4737	289	7	16	16	NUM
ejpam-4737	289	8	(	(	PUNCT
ejpam-4737	289	9	3	3	NUM
ejpam-4737	289	10	)	)	PUNCT
ejpam-4737	289	11	(	(	PUNCT
ejpam-4737	289	12	2023	2023	NUM
ejpam-4737	289	13	)	)	PUNCT
ejpam-4737	289	14	,	,	PUNCT
ejpam-4737	289	15	1434	1434	NUM
ejpam-4737	289	16	-	-	SYM
ejpam-4737	289	17	1447	1447	NUM
ejpam-4737	289	18	1444	1444	NUM
ejpam-4737	289	19	(	(	PUNCT
ejpam-4737	289	20	4	4	NUM
ejpam-4737	289	21	)	)	PUNCT
ejpam-4737	289	22	⇒	⇒	NOUN
ejpam-4737	289	23	(	(	PUNCT
ejpam-4737	289	24	5	5	NUM
ejpam-4737	289	25	):	):	PUNCT
ejpam-4737	289	26	this	this	PRON
ejpam-4737	289	27	is	be	AUX
ejpam-4737	289	28	obvious	obvious	ADJ
ejpam-4737	289	29	.	.	PUNCT
ejpam-4737	290	1	(	(	PUNCT
ejpam-4737	290	2	5	5	X
ejpam-4737	290	3	)	)	PUNCT
ejpam-4737	290	4	⇒	⇒	NOUN
ejpam-4737	290	5	(	(	PUNCT
ejpam-4737	290	6	1	1	NUM
ejpam-4737	290	7	):	):	PUNCT
ejpam-4737	290	8	this	this	PRON
ejpam-4737	290	9	is	be	AUX
ejpam-4737	290	10	obvious	obvious	ADJ
ejpam-4737	290	11	since	since	SCONJ
ejpam-4737	290	12	δs(λ	δs(λ	NOUN
ejpam-4737	290	13	,	,	PUNCT
ejpam-4737	290	14	p)o(x	p)o(x	ADJ
ejpam-4737	290	15	,	,	PUNCT
ejpam-4737	290	16	τ	τ	PROPN
ejpam-4737	290	17	)	)	PUNCT
ejpam-4737	290	18	⊆	⊆	NUM
ejpam-4737	290	19	s(λ	s(λ	PROPN
ejpam-4737	290	20	,	,	PUNCT
ejpam-4737	290	21	p)o(x	p)o(x	ADJ
ejpam-4737	290	22	,	,	PUNCT
ejpam-4737	290	23	τ	τ	PROPN
ejpam-4737	290	24	)	)	PUNCT
ejpam-4737	290	25	.	.	PUNCT
ejpam-4737	291	1	definition	definition	NOUN
ejpam-4737	291	2	13	13	NUM
ejpam-4737	291	3	.	.	PUNCT
ejpam-4737	292	1	a	a	DET
ejpam-4737	292	2	topological	topological	ADJ
ejpam-4737	292	3	space	space	NOUN
ejpam-4737	292	4	(	(	PUNCT
ejpam-4737	292	5	x	x	X
ejpam-4737	292	6	,	,	PUNCT
ejpam-4737	292	7	τ	τ	X
ejpam-4737	292	8	)	)	PUNCT
ejpam-4737	292	9	is	be	AUX
ejpam-4737	292	10	said	say	VERB
ejpam-4737	292	11	to	to	PART
ejpam-4737	292	12	be	be	AUX
ejpam-4737	292	13	s(λ	s(λ	PROPN
ejpam-4737	292	14	,	,	PUNCT
ejpam-4737	292	15	p)-urysohn	p)-urysohn	PUNCT
ejpam-4737	292	16	if	if	SCONJ
ejpam-4737	292	17	,	,	PUNCT
ejpam-4737	292	18	for	for	ADP
ejpam-4737	292	19	each	each	DET
ejpam-4737	292	20	pair	pair	NOUN
ejpam-4737	292	21	of	of	ADP
ejpam-4737	292	22	distinct	distinct	ADJ
ejpam-4737	292	23	points	point	NOUN
ejpam-4737	292	24	x	x	X
ejpam-4737	292	25	,	,	PUNCT
ejpam-4737	292	26	y	y	PROPN
ejpam-4737	292	27	∈	∈	PROPN
ejpam-4737	292	28	x	x	PRON
ejpam-4737	292	29	,	,	PUNCT
ejpam-4737	292	30	there	there	PRON
ejpam-4737	292	31	exist	exist	VERB
ejpam-4737	292	32	u	u	NOUN
ejpam-4737	292	33	,	,	PUNCT
ejpam-4737	292	34	v	v	PROPN
ejpam-4737	292	35	∈	∈	PROPN
ejpam-4737	292	36	s(λ	s(λ	PROPN
ejpam-4737	292	37	,	,	PUNCT
ejpam-4737	292	38	p)o(x	p)o(x	ADJ
ejpam-4737	292	39	,	,	PUNCT
ejpam-4737	292	40	τ	τ	PROPN
ejpam-4737	292	41	)	)	PUNCT
ejpam-4737	292	42	such	such	ADJ
ejpam-4737	292	43	that	that	SCONJ
ejpam-4737	292	44	x	x	SYM
ejpam-4737	292	45	∈	∈	PROPN
ejpam-4737	292	46	u	u	NOUN
ejpam-4737	292	47	,	,	PUNCT
ejpam-4737	292	48	y	y	PROPN
ejpam-4737	292	49	∈	∈	PROPN
ejpam-4737	292	50	v	v	NOUN
ejpam-4737	292	51	and	and	CCONJ
ejpam-4737	292	52	u	u	NOUN
ejpam-4737	292	53	(	(	PUNCT
ejpam-4737	292	54	λ	λ	PROPN
ejpam-4737	292	55	,	,	PUNCT
ejpam-4737	292	56	p	p	NOUN
ejpam-4737	292	57	)	)	PUNCT
ejpam-4737	292	58	∩	∩	ADJ
ejpam-4737	292	59	v	v	X
ejpam-4737	292	60	(	(	PUNCT
ejpam-4737	292	61	λ	λ	PROPN
ejpam-4737	292	62	,	,	PUNCT
ejpam-4737	292	63	p	p	NOUN
ejpam-4737	292	64	)	)	PUNCT
ejpam-4737	292	65	=	=	PUNCT
ejpam-4737	292	66	∅.	∅.	NOUN
ejpam-4737	292	67	theorem	theorem	VERB
ejpam-4737	292	68	13	13	NUM
ejpam-4737	292	69	.	.	PUNCT
ejpam-4737	293	1	a	a	DET
ejpam-4737	293	2	topological	topological	ADJ
ejpam-4737	293	3	space	space	NOUN
ejpam-4737	293	4	(	(	PUNCT
ejpam-4737	293	5	x	x	X
ejpam-4737	293	6	,	,	PUNCT
ejpam-4737	293	7	τ	τ	X
ejpam-4737	293	8	)	)	PUNCT
ejpam-4737	293	9	is	be	AUX
ejpam-4737	293	10	s(λ	s(λ	PROPN
ejpam-4737	293	11	,	,	PUNCT
ejpam-4737	293	12	p)-urysohn	p)-urysohn	PUNCT
ejpam-4737	293	13	if	if	SCONJ
ejpam-4737	293	14	and	and	CCONJ
ejpam-4737	293	15	only	only	ADV
ejpam-4737	293	16	if	if	SCONJ
ejpam-4737	293	17	for	for	ADP
ejpam-4737	293	18	each	each	DET
ejpam-4737	293	19	pair	pair	NOUN
ejpam-4737	293	20	of	of	ADP
ejpam-4737	293	21	distinct	distinct	ADJ
ejpam-4737	293	22	points	point	NOUN
ejpam-4737	293	23	x	x	X
ejpam-4737	293	24	,	,	PUNCT
ejpam-4737	293	25	y	y	PROPN
ejpam-4737	293	26	of	of	ADP
ejpam-4737	293	27	x	x	PRON
ejpam-4737	293	28	,	,	PUNCT
ejpam-4737	293	29	there	there	PRON
ejpam-4737	293	30	exist	exist	VERB
ejpam-4737	293	31	u	u	NOUN
ejpam-4737	293	32	,	,	PUNCT
ejpam-4737	293	33	v	v	PROPN
ejpam-4737	293	34	∈	∈	PROPN
ejpam-4737	293	35	δs(λ	δs(λ	NOUN
ejpam-4737	293	36	,	,	PUNCT
ejpam-4737	293	37	p)o(x	p)o(x	ADJ
ejpam-4737	293	38	,	,	PUNCT
ejpam-4737	293	39	τ	τ	PROPN
ejpam-4737	293	40	)	)	PUNCT
ejpam-4737	293	41	such	such	ADJ
ejpam-4737	293	42	that	that	SCONJ
ejpam-4737	293	43	x	x	SYM
ejpam-4737	293	44	∈	∈	PROPN
ejpam-4737	293	45	u	u	NOUN
ejpam-4737	293	46	,	,	PUNCT
ejpam-4737	293	47	y	y	PROPN
ejpam-4737	293	48	∈	∈	PROPN
ejpam-4737	293	49	v	v	NOUN
ejpam-4737	293	50	and	and	CCONJ
ejpam-4737	293	51	u	u	NOUN
ejpam-4737	293	52	(	(	PUNCT
ejpam-4737	293	53	λ	λ	PROPN
ejpam-4737	293	54	,	,	PUNCT
ejpam-4737	293	55	p	p	NOUN
ejpam-4737	293	56	)	)	PUNCT
ejpam-4737	293	57	∩	∩	ADJ
ejpam-4737	293	58	v	v	X
ejpam-4737	293	59	(	(	PUNCT
ejpam-4737	293	60	λ	λ	PROPN
ejpam-4737	293	61	,	,	PUNCT
ejpam-4737	293	62	p	p	NOUN
ejpam-4737	293	63	)	)	PUNCT
ejpam-4737	293	64	=	=	PUNCT
ejpam-4737	293	65	∅.	∅.	NOUN
ejpam-4737	293	66	proof	proof	NOUN
ejpam-4737	293	67	.	.	PUNCT
ejpam-4737	294	1	suppose	suppose	VERB
ejpam-4737	294	2	that	that	SCONJ
ejpam-4737	294	3	(	(	PUNCT
ejpam-4737	294	4	x	x	X
ejpam-4737	294	5	,	,	PUNCT
ejpam-4737	294	6	τ	τ	X
ejpam-4737	294	7	)	)	PUNCT
ejpam-4737	294	8	is	be	AUX
ejpam-4737	294	9	s(λ	s(λ	PROPN
ejpam-4737	294	10	,	,	PUNCT
ejpam-4737	294	11	p)-urysohn	p)-urysohn	NOUN
ejpam-4737	294	12	.	.	PUNCT
ejpam-4737	295	1	then	then	ADV
ejpam-4737	295	2	,	,	PUNCT
ejpam-4737	295	3	for	for	SCONJ
ejpam-4737	295	4	each	each	DET
ejpam-4737	295	5	pair	pair	NOUN
ejpam-4737	295	6	of	of	ADP
ejpam-4737	295	7	distinct	distinct	ADJ
ejpam-4737	295	8	points	point	NOUN
ejpam-4737	295	9	x	x	X
ejpam-4737	295	10	,	,	PUNCT
ejpam-4737	295	11	y	y	PROPN
ejpam-4737	295	12	of	of	ADP
ejpam-4737	295	13	x	x	PRON
ejpam-4737	295	14	,	,	PUNCT
ejpam-4737	295	15	there	there	PRON
ejpam-4737	295	16	exist	exist	VERB
ejpam-4737	295	17	u	u	NOUN
ejpam-4737	295	18	,	,	PUNCT
ejpam-4737	295	19	v	v	PROPN
ejpam-4737	295	20	∈	∈	PROPN
ejpam-4737	295	21	s(λ	s(λ	PROPN
ejpam-4737	295	22	,	,	PUNCT
ejpam-4737	295	23	p)o(x	p)o(x	ADJ
ejpam-4737	295	24	,	,	PUNCT
ejpam-4737	295	25	τ	τ	PROPN
ejpam-4737	295	26	)	)	PUNCT
ejpam-4737	295	27	such	such	ADJ
ejpam-4737	295	28	that	that	SCONJ
ejpam-4737	295	29	x	x	SYM
ejpam-4737	295	30	∈	∈	PROPN
ejpam-4737	295	31	u	u	NOUN
ejpam-4737	295	32	,	,	PUNCT
ejpam-4737	295	33	y	y	PROPN
ejpam-4737	295	34	∈	∈	PROPN
ejpam-4737	295	35	v	v	NOUN
ejpam-4737	295	36	and	and	CCONJ
ejpam-4737	295	37	u	u	NOUN
ejpam-4737	295	38	(	(	PUNCT
ejpam-4737	295	39	λ	λ	PROPN
ejpam-4737	295	40	,	,	PUNCT
ejpam-4737	295	41	p	p	NOUN
ejpam-4737	295	42	)	)	PUNCT
ejpam-4737	295	43	∩	∩	ADJ
ejpam-4737	295	44	v	v	X
ejpam-4737	295	45	(	(	PUNCT
ejpam-4737	295	46	λ	λ	PROPN
ejpam-4737	295	47	,	,	PUNCT
ejpam-4737	295	48	p	p	NOUN
ejpam-4737	295	49	)	)	PUNCT
ejpam-4737	295	50	=	=	PUNCT
ejpam-4737	295	51	∅.	∅.	NOUN
ejpam-4737	295	52	since	since	SCONJ
ejpam-4737	295	53	u	u	PROPN
ejpam-4737	295	54	∈	∈	PROPN
ejpam-4737	295	55	s(λ	s(λ	PROPN
ejpam-4737	295	56	,	,	PUNCT
ejpam-4737	295	57	p)o(x	p)o(x	ADJ
ejpam-4737	295	58	,	,	PUNCT
ejpam-4737	295	59	τ	τ	PROPN
ejpam-4737	295	60	)	)	PUNCT
ejpam-4737	295	61	,	,	PUNCT
ejpam-4737	295	62	u	u	NOUN
ejpam-4737	295	63	(	(	PUNCT
ejpam-4737	295	64	λ	λ	PROPN
ejpam-4737	295	65	,	,	PUNCT
ejpam-4737	295	66	p	p	NOUN
ejpam-4737	295	67	)	)	PUNCT
ejpam-4737	295	68	=	=	NOUN
ejpam-4737	296	1	[	[	X
ejpam-4737	296	2	u(λ	u(λ	PROPN
ejpam-4737	296	3	,	,	PUNCT
ejpam-4737	296	4	p	p	NOUN
ejpam-4737	296	5	)	)	PUNCT
ejpam-4737	296	6	]	]	PUNCT
ejpam-4737	296	7	(	(	PUNCT
ejpam-4737	296	8	λ	λ	X
ejpam-4737	296	9	,	,	PUNCT
ejpam-4737	296	10	p	p	NOUN
ejpam-4737	296	11	)	)	PUNCT
ejpam-4737	296	12	and	and	CCONJ
ejpam-4737	296	13	u	u	PROPN
ejpam-4737	296	14	(	(	PUNCT
ejpam-4737	296	15	λ	λ	PROPN
ejpam-4737	296	16	,	,	PUNCT
ejpam-4737	296	17	p	p	NOUN
ejpam-4737	296	18	)	)	PUNCT
ejpam-4737	296	19	is	be	AUX
ejpam-4737	296	20	r(λ	r(λ	PROPN
ejpam-4737	296	21	,	,	PUNCT
ejpam-4737	296	22	p)-closed	p)-close	VERB
ejpam-4737	296	23	.	.	PUNCT
ejpam-4737	297	1	thus	thus	ADV
ejpam-4737	297	2	,	,	PUNCT
ejpam-4737	297	3	u	u	PROPN
ejpam-4737	297	4	(	(	PUNCT
ejpam-4737	297	5	λ	λ	PROPN
ejpam-4737	297	6	,	,	PUNCT
ejpam-4737	297	7	p	p	NOUN
ejpam-4737	297	8	)	)	PUNCT
ejpam-4737	297	9	,	,	PUNCT
ejpam-4737	297	10	v	v	X
ejpam-4737	297	11	(	(	PUNCT
ejpam-4737	297	12	λ	λ	PROPN
ejpam-4737	297	13	,	,	PUNCT
ejpam-4737	297	14	p	p	NOUN
ejpam-4737	297	15	)	)	PUNCT
ejpam-4737	297	16	∈	∈	PROPN
ejpam-4737	297	17	s(λ	s(λ	PROPN
ejpam-4737	297	18	,	,	PUNCT
ejpam-4737	297	19	p)r(x	p)r(x	PROPN
ejpam-4737	297	20	,	,	PUNCT
ejpam-4737	297	21	τ	τ	PROPN
ejpam-4737	297	22	)	)	PUNCT
ejpam-4737	297	23	⊆	⊆	NUM
ejpam-4737	297	24	δs(λ	δs(λ	NOUN
ejpam-4737	297	25	,	,	PUNCT
ejpam-4737	297	26	p)o(x	p)o(x	ADJ
ejpam-4737	297	27	,	,	PUNCT
ejpam-4737	297	28	τ	τ	PROPN
ejpam-4737	297	29	)	)	PUNCT
ejpam-4737	297	30	.	.	PUNCT
ejpam-4737	298	1	it	it	PRON
ejpam-4737	298	2	is	be	AUX
ejpam-4737	298	3	obvious	obvious	ADJ
ejpam-4737	298	4	that	that	SCONJ
ejpam-4737	298	5	x	x	PUNCT
ejpam-4737	298	6	∈	∈	PROPN
ejpam-4737	298	7	u	u	NOUN
ejpam-4737	298	8	(	(	PUNCT
ejpam-4737	298	9	λ	λ	PROPN
ejpam-4737	298	10	,	,	PUNCT
ejpam-4737	298	11	p	p	NOUN
ejpam-4737	298	12	)	)	PUNCT
ejpam-4737	298	13	,	,	PUNCT
ejpam-4737	298	14	y	y	PROPN
ejpam-4737	298	15	∈	∈	PROPN
ejpam-4737	298	16	v	v	ADP
ejpam-4737	298	17	(	(	PUNCT
ejpam-4737	298	18	λ	λ	PROPN
ejpam-4737	298	19	,	,	PUNCT
ejpam-4737	298	20	p	p	NOUN
ejpam-4737	298	21	)	)	PUNCT
ejpam-4737	298	22	and	and	CCONJ
ejpam-4737	299	1	[	[	X
ejpam-4737	299	2	u	u	X
ejpam-4737	299	3	(	(	PUNCT
ejpam-4737	299	4	λ	λ	PROPN
ejpam-4737	299	5	,	,	PUNCT
ejpam-4737	299	6	p)](λ	p)](λ	ADJ
ejpam-4737	299	7	,	,	PUNCT
ejpam-4737	299	8	p)∩[v	p)∩[v	NUM
ejpam-4737	299	9	(	(	PUNCT
ejpam-4737	299	10	λ	λ	NOUN
ejpam-4737	299	11	,	,	PUNCT
ejpam-4737	299	12	p)](λ	p)](λ	ADJ
ejpam-4737	299	13	,	,	PUNCT
ejpam-4737	299	14	p	p	NOUN
ejpam-4737	299	15	)	)	PUNCT
ejpam-4737	299	16	=	=	SYM
ejpam-4737	299	17	u	u	NOUN
ejpam-4737	299	18	(	(	PUNCT
ejpam-4737	299	19	λ	λ	PROPN
ejpam-4737	299	20	,	,	PUNCT
ejpam-4737	299	21	p)∩v	p)∩v	PROPN
ejpam-4737	299	22	(	(	PUNCT
ejpam-4737	299	23	λ	λ	PROPN
ejpam-4737	299	24	,	,	PUNCT
ejpam-4737	299	25	p	p	NOUN
ejpam-4737	299	26	)	)	PUNCT
ejpam-4737	299	27	=	=	NOUN
ejpam-4737	299	28	∅.	∅.	VERB
ejpam-4737	299	29	conversely	conversely	ADV
ejpam-4737	299	30	,	,	PUNCT
ejpam-4737	299	31	the	the	DET
ejpam-4737	299	32	proof	proof	NOUN
ejpam-4737	299	33	is	be	AUX
ejpam-4737	299	34	obvious	obvious	ADJ
ejpam-4737	299	35	since	since	SCONJ
ejpam-4737	299	36	δs(λ	δs(λ	NOUN
ejpam-4737	299	37	,	,	PUNCT
ejpam-4737	299	38	p)o(x	p)o(x	ADJ
ejpam-4737	299	39	,	,	PUNCT
ejpam-4737	299	40	τ	τ	PROPN
ejpam-4737	299	41	)	)	PUNCT
ejpam-4737	299	42	⊆	⊆	NUM
ejpam-4737	299	43	s(λ	s(λ	PROPN
ejpam-4737	299	44	,	,	PUNCT
ejpam-4737	299	45	p)o(x	p)o(x	ADJ
ejpam-4737	299	46	,	,	PUNCT
ejpam-4737	299	47	τ	τ	PROPN
ejpam-4737	299	48	)	)	PUNCT
ejpam-4737	299	49	.	.	PUNCT
ejpam-4737	300	1	7	7	X
ejpam-4737	300	2	.	.	X
ejpam-4737	300	3	characterizations	characterization	NOUN
ejpam-4737	300	4	of	of	ADP
ejpam-4737	300	5	s(λ	s(λ	PROPN
ejpam-4737	300	6	,	,	PUNCT
ejpam-4737	300	7	p)-closed	p)-close	VERB
ejpam-4737	300	8	spaces	space	NOUN
ejpam-4737	300	9	in	in	ADP
ejpam-4737	300	10	this	this	DET
ejpam-4737	300	11	section	section	NOUN
ejpam-4737	300	12	,	,	PUNCT
ejpam-4737	300	13	we	we	PRON
ejpam-4737	300	14	introduce	introduce	VERB
ejpam-4737	300	15	the	the	DET
ejpam-4737	300	16	notion	notion	NOUN
ejpam-4737	300	17	of	of	ADP
ejpam-4737	300	18	s(λ	s(λ	PROPN
ejpam-4737	300	19	,	,	PUNCT
ejpam-4737	300	20	p)-closed	p)-close	VERB
ejpam-4737	300	21	spaces	space	NOUN
ejpam-4737	300	22	.	.	PUNCT
ejpam-4737	301	1	in	in	ADP
ejpam-4737	301	2	particular	particular	ADJ
ejpam-4737	301	3	,	,	PUNCT
ejpam-4737	301	4	several	several	ADJ
ejpam-4737	301	5	characterizations	characterization	NOUN
ejpam-4737	301	6	of	of	ADP
ejpam-4737	301	7	s(λ	s(λ	PROPN
ejpam-4737	301	8	,	,	PUNCT
ejpam-4737	301	9	p)-closed	p)-close	VERB
ejpam-4737	301	10	spaces	space	NOUN
ejpam-4737	301	11	are	be	AUX
ejpam-4737	301	12	discussed	discuss	VERB
ejpam-4737	301	13	.	.	PUNCT
ejpam-4737	302	1	definition	definition	NOUN
ejpam-4737	302	2	14	14	NUM
ejpam-4737	302	3	.	.	PUNCT
ejpam-4737	303	1	a	a	DET
ejpam-4737	303	2	topological	topological	ADJ
ejpam-4737	303	3	space	space	NOUN
ejpam-4737	303	4	(	(	PUNCT
ejpam-4737	303	5	x	x	X
ejpam-4737	303	6	,	,	PUNCT
ejpam-4737	303	7	τ	τ	X
ejpam-4737	303	8	)	)	PUNCT
ejpam-4737	303	9	is	be	AUX
ejpam-4737	303	10	said	say	VERB
ejpam-4737	303	11	to	to	PART
ejpam-4737	303	12	be	be	AUX
ejpam-4737	303	13	s(λ	s(λ	PROPN
ejpam-4737	303	14	,	,	PUNCT
ejpam-4737	303	15	p)-closed	p)-close	VERB
ejpam-4737	303	16	if	if	SCONJ
ejpam-4737	303	17	,	,	PUNCT
ejpam-4737	303	18	for	for	ADP
ejpam-4737	303	19	every	every	DET
ejpam-4737	303	20	cover	cover	NOUN
ejpam-4737	303	21	{	{	PUNCT
ejpam-4737	303	22	vγ	vγ	NOUN
ejpam-4737	303	23	|	|	ADV
ejpam-4737	303	24	γ	γ	X
ejpam-4737	303	25	∈	∈	PROPN
ejpam-4737	303	26	∇	∇	X
ejpam-4737	303	27	}	}	PUNCT
ejpam-4737	303	28	of	of	ADP
ejpam-4737	303	29	x	x	PUNCT
ejpam-4737	303	30	by	by	ADP
ejpam-4737	303	31	s(λ	s(λ	NOUN
ejpam-4737	303	32	,	,	PUNCT
ejpam-4737	303	33	p)-open	p)-open	VERB
ejpam-4737	303	34	sets	set	NOUN
ejpam-4737	303	35	of	of	ADP
ejpam-4737	303	36	x	x	PRON
ejpam-4737	303	37	,	,	PUNCT
ejpam-4737	303	38	there	there	PRON
ejpam-4737	303	39	exists	exist	VERB
ejpam-4737	303	40	a	a	DET
ejpam-4737	303	41	finite	finite	NOUN
ejpam-4737	303	42	subset	subset	NOUN
ejpam-4737	303	43	∇0	∇0	NUM
ejpam-4737	303	44	of	of	ADP
ejpam-4737	303	45	∇	∇	NOUN
ejpam-4737	304	1	such	such	ADJ
ejpam-4737	304	2	that	that	SCONJ
ejpam-4737	304	3	x	x	X
ejpam-4737	304	4	=	=	SYM
ejpam-4737	304	5	∪	∪	X
ejpam-4737	304	6	γ∈∇0	γ∈∇0	X
ejpam-4737	304	7	v	v	ADP
ejpam-4737	304	8	s(λ	s(λ	PROPN
ejpam-4737	304	9	,	,	PUNCT
ejpam-4737	304	10	p	p	NOUN
ejpam-4737	304	11	)	)	PUNCT
ejpam-4737	304	12	γ	γ	NOUN
ejpam-4737	304	13	.	.	PUNCT
ejpam-4737	304	14	theorem	theorem	PROPN
ejpam-4737	304	15	14	14	NUM
ejpam-4737	304	16	.	.	PUNCT
ejpam-4737	305	1	for	for	ADP
ejpam-4737	305	2	a	a	DET
ejpam-4737	305	3	topological	topological	ADJ
ejpam-4737	305	4	space	space	NOUN
ejpam-4737	305	5	(	(	PUNCT
ejpam-4737	305	6	x	x	X
ejpam-4737	305	7	,	,	PUNCT
ejpam-4737	305	8	τ	τ	PROPN
ejpam-4737	305	9	)	)	PUNCT
ejpam-4737	305	10	,	,	PUNCT
ejpam-4737	305	11	the	the	DET
ejpam-4737	305	12	following	follow	VERB
ejpam-4737	305	13	properties	property	NOUN
ejpam-4737	305	14	are	be	AUX
ejpam-4737	305	15	equivalent	equivalent	ADJ
ejpam-4737	305	16	:	:	PUNCT
ejpam-4737	305	17	(	(	PUNCT
ejpam-4737	305	18	1	1	X
ejpam-4737	305	19	)	)	PUNCT
ejpam-4737	305	20	(	(	PUNCT
ejpam-4737	305	21	x	x	X
ejpam-4737	305	22	,	,	PUNCT
ejpam-4737	305	23	τ	τ	X
ejpam-4737	305	24	)	)	PUNCT
ejpam-4737	305	25	is	be	AUX
ejpam-4737	305	26	s(λ	s(λ	PROPN
ejpam-4737	305	27	,	,	PUNCT
ejpam-4737	305	28	p)-closed	p)-close	VERB
ejpam-4737	305	29	.	.	PUNCT
ejpam-4737	306	1	(	(	PUNCT
ejpam-4737	306	2	2	2	X
ejpam-4737	306	3	)	)	PUNCT
ejpam-4737	306	4	for	for	ADP
ejpam-4737	306	5	every	every	DET
ejpam-4737	306	6	δs(λ	δs(λ	NOUN
ejpam-4737	306	7	,	,	PUNCT
ejpam-4737	306	8	p)-open	p)-open	VERB
ejpam-4737	306	9	cover	cover	NOUN
ejpam-4737	306	10	{	{	PUNCT
ejpam-4737	306	11	vγ	vγ	NOUN
ejpam-4737	306	12	|	|	ADV
ejpam-4737	306	13	γ	γ	X
ejpam-4737	306	14	∈	∈	PROPN
ejpam-4737	306	15	∇	∇	X
ejpam-4737	306	16	}	}	PUNCT
ejpam-4737	306	17	of	of	ADP
ejpam-4737	306	18	x	x	NOUN
ejpam-4737	306	19	,	,	PUNCT
ejpam-4737	306	20	there	there	PRON
ejpam-4737	306	21	exists	exist	VERB
ejpam-4737	306	22	a	a	DET
ejpam-4737	306	23	finite	finite	NOUN
ejpam-4737	306	24	subset	subset	NOUN
ejpam-4737	306	25	∇0	∇0	NUM
ejpam-4737	306	26	of	of	ADP
ejpam-4737	306	27	∇	∇	NOUN
ejpam-4737	306	28	such	such	ADJ
ejpam-4737	306	29	that	that	SCONJ
ejpam-4737	306	30	x	x	X
ejpam-4737	306	31	=	=	SYM
ejpam-4737	306	32	∪	∪	X
ejpam-4737	306	33	γ∈∇0	γ∈∇0	X
ejpam-4737	306	34	v	v	ADP
ejpam-4737	306	35	s(λ	s(λ	PROPN
ejpam-4737	306	36	,	,	PUNCT
ejpam-4737	306	37	p	p	NOUN
ejpam-4737	306	38	)	)	PUNCT
ejpam-4737	306	39	γ	γ	NOUN
ejpam-4737	306	40	.	.	PUNCT
ejpam-4737	307	1	(	(	PUNCT
ejpam-4737	307	2	3	3	X
ejpam-4737	307	3	)	)	PUNCT
ejpam-4737	307	4	for	for	ADP
ejpam-4737	307	5	every	every	DET
ejpam-4737	307	6	δs(λ	δs(λ	NOUN
ejpam-4737	307	7	,	,	PUNCT
ejpam-4737	307	8	p)-open	p)-open	VERB
ejpam-4737	307	9	cover	cover	NOUN
ejpam-4737	307	10	{	{	PUNCT
ejpam-4737	307	11	vγ	vγ	NOUN
ejpam-4737	307	12	|	|	ADV
ejpam-4737	307	13	γ	γ	X
ejpam-4737	307	14	∈	∈	PROPN
ejpam-4737	307	15	∇	∇	X
ejpam-4737	307	16	}	}	PUNCT
ejpam-4737	307	17	of	of	ADP
ejpam-4737	307	18	x	x	NOUN
ejpam-4737	307	19	,	,	PUNCT
ejpam-4737	307	20	there	there	PRON
ejpam-4737	307	21	exists	exist	VERB
ejpam-4737	307	22	a	a	DET
ejpam-4737	307	23	finite	finite	NOUN
ejpam-4737	307	24	subset	subset	NOUN
ejpam-4737	307	25	∇0	∇0	NUM
ejpam-4737	307	26	of	of	ADP
ejpam-4737	307	27	∇	∇	NOUN
ejpam-4737	307	28	such	such	ADJ
ejpam-4737	307	29	that	that	SCONJ
ejpam-4737	308	1	x	x	X
ejpam-4737	308	2	=	=	PRON
ejpam-4737	308	3	∪	∪	ADJ
ejpam-4737	308	4	γ∈∇0	γ∈∇0	X
ejpam-4737	308	5	v	v	NOUN
ejpam-4737	308	6	δs(λ	δs(λ	NOUN
ejpam-4737	308	7	,	,	PUNCT
ejpam-4737	308	8	p	p	NOUN
ejpam-4737	308	9	)	)	PUNCT
ejpam-4737	308	10	γ	γ	NOUN
ejpam-4737	308	11	.	.	PUNCT
ejpam-4737	309	1	c.	c.	PROPN
ejpam-4737	309	2	boonpok	boonpok	PROPN
ejpam-4737	309	3	,	,	PUNCT
ejpam-4737	309	4	m.	m.	NOUN
ejpam-4737	309	5	thongmoon	thongmoon	PROPN
ejpam-4737	309	6	/	/	SYM
ejpam-4737	309	7	eur	eur	PROPN
ejpam-4737	309	8	.	.	PUNCT
ejpam-4737	310	1	j.	j.	PROPN
ejpam-4737	310	2	pure	pure	PROPN
ejpam-4737	310	3	appl	appl	PROPN
ejpam-4737	310	4	.	.	PROPN
ejpam-4737	310	5	math	math	PROPN
ejpam-4737	310	6	,	,	PUNCT
ejpam-4737	310	7	16	16	NUM
ejpam-4737	310	8	(	(	PUNCT
ejpam-4737	310	9	3	3	NUM
ejpam-4737	310	10	)	)	PUNCT
ejpam-4737	310	11	(	(	PUNCT
ejpam-4737	310	12	2023	2023	NUM
ejpam-4737	310	13	)	)	PUNCT
ejpam-4737	310	14	,	,	PUNCT
ejpam-4737	310	15	1434	1434	NUM
ejpam-4737	310	16	-	-	SYM
ejpam-4737	310	17	1447	1447	NUM
ejpam-4737	310	18	1445	1445	NUM
ejpam-4737	310	19	proof	proof	NOUN
ejpam-4737	310	20	.	.	PUNCT
ejpam-4737	311	1	(	(	PUNCT
ejpam-4737	311	2	1	1	X
ejpam-4737	311	3	)	)	PUNCT
ejpam-4737	311	4	⇒	⇒	NOUN
ejpam-4737	311	5	(	(	PUNCT
ejpam-4737	311	6	2	2	NUM
ejpam-4737	311	7	):	):	PUNCT
ejpam-4737	311	8	suppose	suppose	VERB
ejpam-4737	311	9	that	that	SCONJ
ejpam-4737	311	10	(	(	PUNCT
ejpam-4737	311	11	x	x	X
ejpam-4737	311	12	,	,	PUNCT
ejpam-4737	311	13	τ	τ	X
ejpam-4737	311	14	)	)	PUNCT
ejpam-4737	311	15	is	be	AUX
ejpam-4737	311	16	s(λ	s(λ	PROPN
ejpam-4737	311	17	,	,	PUNCT
ejpam-4737	311	18	p)-closed	p)-close	VERB
ejpam-4737	311	19	.	.	PUNCT
ejpam-4737	312	1	let	let	AUX
ejpam-4737	312	2	{	{	PUNCT
ejpam-4737	312	3	vγ	vγ	VERB
ejpam-4737	312	4	|	|	ADV
ejpam-4737	312	5	γ	γ	X
ejpam-4737	312	6	∈	∈	PROPN
ejpam-4737	312	7	∇	∇	X
ejpam-4737	312	8	}	}	PUNCT
ejpam-4737	312	9	be	be	AUX
ejpam-4737	312	10	a	a	DET
ejpam-4737	312	11	δs(λ	δs(λ	NOUN
ejpam-4737	312	12	,	,	PUNCT
ejpam-4737	312	13	p)-open	p)-open	VERB
ejpam-4737	312	14	cover	cover	NOUN
ejpam-4737	312	15	of	of	ADP
ejpam-4737	312	16	x.	x.	NOUN
ejpam-4737	312	17	by	by	ADP
ejpam-4737	312	18	lemma	lemma	PROPN
ejpam-4737	312	19	4	4	NUM
ejpam-4737	312	20	,	,	PUNCT
ejpam-4737	312	21	δs(λ	δs(λ	NOUN
ejpam-4737	312	22	,	,	PUNCT
ejpam-4737	312	23	p)o(x	p)o(x	ADJ
ejpam-4737	312	24	,	,	PUNCT
ejpam-4737	312	25	τ	τ	PROPN
ejpam-4737	312	26	)	)	PUNCT
ejpam-4737	312	27	⊆	⊆	NUM
ejpam-4737	312	28	s(λ	s(λ	PROPN
ejpam-4737	312	29	,	,	PUNCT
ejpam-4737	312	30	p)o(x	p)o(x	ADJ
ejpam-4737	312	31	,	,	PUNCT
ejpam-4737	312	32	τ	τ	PROPN
ejpam-4737	312	33	)	)	PUNCT
ejpam-4737	312	34	and	and	CCONJ
ejpam-4737	312	35	there	there	PRON
ejpam-4737	312	36	exists	exist	VERB
ejpam-4737	312	37	a	a	DET
ejpam-4737	312	38	finite	finite	NOUN
ejpam-4737	312	39	subset	subset	NOUN
ejpam-4737	312	40	∇0	∇0	NUM
ejpam-4737	312	41	of	of	ADP
ejpam-4737	312	42	∇	∇	NOUN
ejpam-4737	312	43	such	such	ADJ
ejpam-4737	312	44	that	that	SCONJ
ejpam-4737	312	45	x	x	X
ejpam-4737	312	46	=	=	SYM
ejpam-4737	312	47	∪	∪	X
ejpam-4737	312	48	γ∈∇0	γ∈∇0	X
ejpam-4737	312	49	v	v	ADP
ejpam-4737	312	50	s(λ	s(λ	PROPN
ejpam-4737	312	51	,	,	PUNCT
ejpam-4737	312	52	p	p	NOUN
ejpam-4737	312	53	)	)	PUNCT
ejpam-4737	312	54	γ	γ	NOUN
ejpam-4737	312	55	.	.	PUNCT
ejpam-4737	313	1	(	(	PUNCT
ejpam-4737	313	2	2	2	X
ejpam-4737	313	3	)	)	PUNCT
ejpam-4737	313	4	⇒	⇒	NOUN
ejpam-4737	313	5	(	(	PUNCT
ejpam-4737	313	6	3	3	NUM
ejpam-4737	313	7	):	):	PUNCT
ejpam-4737	313	8	let	let	VERB
ejpam-4737	313	9	{	{	PUNCT
ejpam-4737	313	10	vγ	vγ	VERB
ejpam-4737	313	11	|	|	ADV
ejpam-4737	313	12	γ	γ	X
ejpam-4737	313	13	∈	∈	PROPN
ejpam-4737	313	14	∇	∇	X
ejpam-4737	313	15	}	}	PUNCT
ejpam-4737	313	16	be	be	AUX
ejpam-4737	313	17	a	a	DET
ejpam-4737	313	18	δs(λ	δs(λ	NOUN
ejpam-4737	313	19	,	,	PUNCT
ejpam-4737	313	20	p)-open	p)-open	VERB
ejpam-4737	313	21	cover	cover	NOUN
ejpam-4737	313	22	of	of	ADP
ejpam-4737	313	23	x.	x.	NOUN
ejpam-4737	313	24	by	by	ADP
ejpam-4737	313	25	lemma	lemma	PROPN
ejpam-4737	313	26	4	4	NUM
ejpam-4737	313	27	,	,	PUNCT
ejpam-4737	313	28	δs(λ	δs(λ	NOUN
ejpam-4737	313	29	,	,	PUNCT
ejpam-4737	313	30	p)o(x	p)o(x	ADJ
ejpam-4737	313	31	,	,	PUNCT
ejpam-4737	313	32	τ	τ	PROPN
ejpam-4737	313	33	)	)	PUNCT
ejpam-4737	313	34	⊆	⊆	NUM
ejpam-4737	313	35	s(λ	s(λ	PROPN
ejpam-4737	313	36	,	,	PUNCT
ejpam-4737	313	37	p)o(x	p)o(x	ADJ
ejpam-4737	313	38	,	,	PUNCT
ejpam-4737	313	39	τ	τ	X
ejpam-4737	313	40	)	)	PUNCT
ejpam-4737	313	41	and	and	CCONJ
ejpam-4737	313	42	it	it	PRON
ejpam-4737	313	43	follows	follow	VERB
ejpam-4737	313	44	from	from	ADP
ejpam-4737	313	45	lemma	lemma	PROPN
ejpam-4737	313	46	5	5	NUM
ejpam-4737	313	47	that	that	PRON
ejpam-4737	313	48	v	v	ADP
ejpam-4737	313	49	δs(λ	δs(λ	NOUN
ejpam-4737	313	50	,	,	PUNCT
ejpam-4737	313	51	p	p	NOUN
ejpam-4737	313	52	)	)	PUNCT
ejpam-4737	313	53	γ	γ	PROPN
ejpam-4737	313	54	=	=	PROPN
ejpam-4737	313	55	v	v	ADP
ejpam-4737	313	56	s(λ	s(λ	PROPN
ejpam-4737	313	57	,	,	PUNCT
ejpam-4737	313	58	p	p	NOUN
ejpam-4737	313	59	)	)	PUNCT
ejpam-4737	313	60	γ	γ	NOUN
ejpam-4737	313	61	for	for	ADP
ejpam-4737	313	62	each	each	DET
ejpam-4737	313	63	γ	γ	X
ejpam-4737	313	64	∈	∈	NOUN
ejpam-4737	313	65	∇.	∇.	X
ejpam-4737	313	66	(	(	PUNCT
ejpam-4737	313	67	3	3	X
ejpam-4737	313	68	)	)	PUNCT
ejpam-4737	313	69	⇒	⇒	NOUN
ejpam-4737	313	70	(	(	PUNCT
ejpam-4737	313	71	1	1	NUM
ejpam-4737	313	72	):	):	PUNCT
ejpam-4737	313	73	let	let	VERB
ejpam-4737	313	74	{	{	PUNCT
ejpam-4737	313	75	vγ	vγ	VERB
ejpam-4737	313	76	|	|	ADV
ejpam-4737	313	77	γ	γ	X
ejpam-4737	313	78	∈	∈	PROPN
ejpam-4737	313	79	∇	∇	X
ejpam-4737	313	80	}	}	PUNCT
ejpam-4737	313	81	be	be	VERB
ejpam-4737	313	82	a	a	DET
ejpam-4737	313	83	s(λ	s(λ	PROPN
ejpam-4737	313	84	,	,	PUNCT
ejpam-4737	313	85	p)-open	p)-open	VERB
ejpam-4737	313	86	cover	cover	NOUN
ejpam-4737	313	87	of	of	ADP
ejpam-4737	313	88	x.	x.	NOUN
ejpam-4737	313	89	then	then	ADV
ejpam-4737	313	90	,	,	PUNCT
ejpam-4737	313	91	x	x	SYM
ejpam-4737	313	92	=	=	SYM
ejpam-4737	313	93	∪	∪	X
ejpam-4737	313	94	γ∈∇0	γ∈∇0	X
ejpam-4737	313	95	v	v	ADP
ejpam-4737	313	96	s(λ	s(λ	PROPN
ejpam-4737	313	97	,	,	PUNCT
ejpam-4737	313	98	p	p	NOUN
ejpam-4737	313	99	)	)	PUNCT
ejpam-4737	313	100	γ	γ	NOUN
ejpam-4737	313	101	.	.	PUNCT
ejpam-4737	314	1	by	by	ADP
ejpam-4737	314	2	lemma	lemma	PROPN
ejpam-4737	314	3	4	4	NUM
ejpam-4737	314	4	,	,	PUNCT
ejpam-4737	314	5	v	v	X
ejpam-4737	314	6	s(λ	s(λ	PROPN
ejpam-4737	314	7	,	,	PUNCT
ejpam-4737	314	8	p	p	NOUN
ejpam-4737	314	9	)	)	PUNCT
ejpam-4737	314	10	γ	γ	PROPN
ejpam-4737	314	11	∈	∈	PROPN
ejpam-4737	314	12	s(λ	s(λ	PROPN
ejpam-4737	314	13	,	,	PUNCT
ejpam-4737	314	14	p)r(x	p)r(x	PROPN
ejpam-4737	314	15	,	,	PUNCT
ejpam-4737	314	16	τ	τ	PROPN
ejpam-4737	314	17	)	)	PUNCT
ejpam-4737	314	18	⊆	⊆	NUM
ejpam-4737	314	19	δs(λ	δs(λ	NOUN
ejpam-4737	314	20	,	,	PUNCT
ejpam-4737	314	21	p)o(x	p)o(x	ADJ
ejpam-4737	314	22	,	,	PUNCT
ejpam-4737	314	23	τ	τ	PROPN
ejpam-4737	314	24	)	)	PUNCT
ejpam-4737	314	25	and	and	CCONJ
ejpam-4737	314	26	there	there	PRON
ejpam-4737	314	27	exists	exist	VERB
ejpam-4737	314	28	a	a	DET
ejpam-4737	314	29	finite	finite	NOUN
ejpam-4737	314	30	subset	subset	NOUN
ejpam-4737	314	31	∇0	∇0	NUM
ejpam-4737	314	32	of	of	ADP
ejpam-4737	314	33	∇	∇	NOUN
ejpam-4737	314	34	such	such	ADJ
ejpam-4737	314	35	that	that	SCONJ
ejpam-4737	314	36	x	x	X
ejpam-4737	314	37	=	=	PRON
ejpam-4737	314	38	∪	∪	ADJ
ejpam-4737	314	39	γ∈∇0	γ∈∇0	NOUN
ejpam-4737	315	1	[	[	X
ejpam-4737	315	2	v	v	X
ejpam-4737	315	3	s(λ	s(λ	PROPN
ejpam-4737	315	4	,	,	PUNCT
ejpam-4737	315	5	p	p	NOUN
ejpam-4737	315	6	)	)	PUNCT
ejpam-4737	315	7	γ	γ	NOUN
ejpam-4737	315	8	]	]	X
ejpam-4737	315	9	δs(λ	δs(λ	NOUN
ejpam-4737	315	10	,	,	PUNCT
ejpam-4737	315	11	p	p	NOUN
ejpam-4737	315	12	)	)	PUNCT
ejpam-4737	315	13	.	.	PUNCT
ejpam-4737	316	1	by	by	ADP
ejpam-4737	316	2	lemma	lemma	PROPN
ejpam-4737	316	3	5	5	NUM
ejpam-4737	316	4	,	,	PUNCT
ejpam-4737	316	5	[	[	X
ejpam-4737	316	6	v	v	ADP
ejpam-4737	316	7	s(λ	s(λ	PROPN
ejpam-4737	316	8	,	,	PUNCT
ejpam-4737	316	9	p	p	NOUN
ejpam-4737	316	10	)	)	PUNCT
ejpam-4737	316	11	γ	γ	NOUN
ejpam-4737	316	12	]	]	X
ejpam-4737	316	13	δs(λ	δs(λ	NOUN
ejpam-4737	316	14	,	,	PUNCT
ejpam-4737	316	15	p	p	NOUN
ejpam-4737	316	16	)	)	PUNCT
ejpam-4737	316	17	=	=	PUNCT
ejpam-4737	317	1	[	[	X
ejpam-4737	317	2	v	v	ADP
ejpam-4737	317	3	s(λ	s(λ	PROPN
ejpam-4737	317	4	,	,	PUNCT
ejpam-4737	317	5	p	p	NOUN
ejpam-4737	317	6	)	)	PUNCT
ejpam-4737	317	7	γ	γ	X
ejpam-4737	317	8	]	]	PUNCT
ejpam-4737	317	9	s(λ	s(λ	PROPN
ejpam-4737	317	10	,	,	PUNCT
ejpam-4737	317	11	p	p	NOUN
ejpam-4737	317	12	)	)	PUNCT
ejpam-4737	317	13	=	=	SYM
ejpam-4737	317	14	v	v	ADP
ejpam-4737	317	15	s(λ	s(λ	PROPN
ejpam-4737	317	16	,	,	PUNCT
ejpam-4737	317	17	p	p	NOUN
ejpam-4737	317	18	)	)	PUNCT
ejpam-4737	317	19	γ	γ	NOUN
ejpam-4737	317	20	and	and	CCONJ
ejpam-4737	317	21	hence	hence	ADV
ejpam-4737	317	22	x	x	PUNCT
ejpam-4737	317	23	=	=	SYM
ejpam-4737	317	24	∪	∪	X
ejpam-4737	317	25	γ∈∇0	γ∈∇0	X
ejpam-4737	317	26	v	v	ADP
ejpam-4737	317	27	s(λ	s(λ	PROPN
ejpam-4737	317	28	,	,	PUNCT
ejpam-4737	317	29	p	p	NOUN
ejpam-4737	317	30	)	)	PUNCT
ejpam-4737	317	31	γ	γ	NOUN
ejpam-4737	317	32	.	.	PUNCT
ejpam-4737	318	1	thus	thus	ADV
ejpam-4737	318	2	,	,	PUNCT
ejpam-4737	318	3	(	(	PUNCT
ejpam-4737	318	4	x	x	X
ejpam-4737	318	5	,	,	PUNCT
ejpam-4737	318	6	τ	τ	X
ejpam-4737	318	7	)	)	PUNCT
ejpam-4737	318	8	is	be	AUX
ejpam-4737	318	9	s(λ	s(λ	PROPN
ejpam-4737	318	10	,	,	PUNCT
ejpam-4737	318	11	p)-closed	p)-close	VERB
ejpam-4737	318	12	.	.	PUNCT
ejpam-4737	319	1	theorem	theorem	VERB
ejpam-4737	319	2	15	15	NUM
ejpam-4737	319	3	.	.	PUNCT
ejpam-4737	320	1	a	a	DET
ejpam-4737	320	2	topological	topological	ADJ
ejpam-4737	320	3	space	space	NOUN
ejpam-4737	320	4	(	(	PUNCT
ejpam-4737	320	5	x	x	X
ejpam-4737	320	6	,	,	PUNCT
ejpam-4737	320	7	τ	τ	X
ejpam-4737	320	8	)	)	PUNCT
ejpam-4737	320	9	is	be	AUX
ejpam-4737	320	10	s(λ	s(λ	PROPN
ejpam-4737	320	11	,	,	PUNCT
ejpam-4737	320	12	p)-closed	p)-close	VERB
ejpam-4737	320	13	if	if	SCONJ
ejpam-4737	320	14	and	and	CCONJ
ejpam-4737	320	15	only	only	ADV
ejpam-4737	320	16	if	if	SCONJ
ejpam-4737	320	17	for	for	ADP
ejpam-4737	320	18	every	every	DET
ejpam-4737	320	19	θs(λ	θs(λ	NOUN
ejpam-4737	320	20	,	,	PUNCT
ejpam-4737	320	21	p)open	p)open	ADJ
ejpam-4737	320	22	cover	cover	NOUN
ejpam-4737	320	23	{	{	PUNCT
ejpam-4737	320	24	vγ	vγ	NOUN
ejpam-4737	320	25	|	|	ADV
ejpam-4737	320	26	γ	γ	X
ejpam-4737	320	27	∈	∈	PROPN
ejpam-4737	320	28	∇	∇	X
ejpam-4737	320	29	}	}	PUNCT
ejpam-4737	320	30	of	of	ADP
ejpam-4737	320	31	x	x	NOUN
ejpam-4737	320	32	,	,	PUNCT
ejpam-4737	320	33	there	there	PRON
ejpam-4737	320	34	exists	exist	VERB
ejpam-4737	320	35	a	a	DET
ejpam-4737	320	36	finite	finite	NOUN
ejpam-4737	320	37	subset	subset	NOUN
ejpam-4737	320	38	∇0	∇0	NUM
ejpam-4737	320	39	of	of	ADP
ejpam-4737	320	40	∇	∇	NOUN
ejpam-4737	321	1	such	such	ADJ
ejpam-4737	321	2	that	that	SCONJ
ejpam-4737	321	3	x	x	X
ejpam-4737	321	4	=	=	SYM
ejpam-4737	321	5	∪	∪	ADJ
ejpam-4737	321	6	γ∈∇0	γ∈∇0	X
ejpam-4737	321	7	vγ	vγ	NOUN
ejpam-4737	321	8	.	.	PUNCT
ejpam-4737	322	1	proof	proof	NOUN
ejpam-4737	322	2	.	.	PUNCT
ejpam-4737	323	1	let	let	VERB
ejpam-4737	323	2	{	{	PUNCT
ejpam-4737	323	3	vγ	vγ	VERB
ejpam-4737	323	4	|	|	ADV
ejpam-4737	323	5	γ	γ	X
ejpam-4737	323	6	∈	∈	PROPN
ejpam-4737	323	7	∇	∇	X
ejpam-4737	323	8	}	}	PUNCT
ejpam-4737	323	9	be	be	AUX
ejpam-4737	323	10	a	a	DET
ejpam-4737	323	11	θs(λ	θs(λ	NOUN
ejpam-4737	323	12	,	,	PUNCT
ejpam-4737	323	13	p)-open	p)-open	VERB
ejpam-4737	323	14	cover	cover	NOUN
ejpam-4737	323	15	of	of	ADP
ejpam-4737	323	16	x.	x.	NOUN
ejpam-4737	323	17	for	for	ADP
ejpam-4737	323	18	each	each	DET
ejpam-4737	323	19	x	x	SYM
ejpam-4737	323	20	∈	∈	PROPN
ejpam-4737	323	21	x	x	NOUN
ejpam-4737	323	22	,	,	PUNCT
ejpam-4737	323	23	there	there	PRON
ejpam-4737	323	24	exists	exist	VERB
ejpam-4737	323	25	γ(x	γ(x	NOUN
ejpam-4737	323	26	)	)	PUNCT
ejpam-4737	323	27	∈	∈	PROPN
ejpam-4737	323	28	∇	∇	X
ejpam-4737	323	29	such	such	ADJ
ejpam-4737	323	30	that	that	SCONJ
ejpam-4737	323	31	x	x	SYM
ejpam-4737	323	32	∈	∈	PROPN
ejpam-4737	323	33	vγ(x	vγ(x	NOUN
ejpam-4737	323	34	)	)	PUNCT
ejpam-4737	323	35	.	.	PUNCT
ejpam-4737	324	1	since	since	SCONJ
ejpam-4737	324	2	vγ(x	vγ(x	NOUN
ejpam-4737	324	3	)	)	PUNCT
ejpam-4737	324	4	is	be	AUX
ejpam-4737	324	5	θs(λ	θs(λ	NOUN
ejpam-4737	324	6	,	,	PUNCT
ejpam-4737	324	7	p)-open	p)-open	VERB
ejpam-4737	324	8	,	,	PUNCT
ejpam-4737	324	9	there	there	PRON
ejpam-4737	324	10	exists	exist	VERB
ejpam-4737	324	11	gγ(x	gγ(x	PUNCT
ejpam-4737	324	12	)	)	PUNCT
ejpam-4737	324	13	∈	∈	PROPN
ejpam-4737	324	14	s(λ	s(λ	PROPN
ejpam-4737	324	15	,	,	PUNCT
ejpam-4737	324	16	p)o(x	p)o(x	ADJ
ejpam-4737	324	17	,	,	PUNCT
ejpam-4737	324	18	τ	τ	PROPN
ejpam-4737	324	19	)	)	PUNCT
ejpam-4737	325	1	such	such	ADJ
ejpam-4737	325	2	that	that	SCONJ
ejpam-4737	325	3	x	x	SYM
ejpam-4737	325	4	∈	∈	NOUN
ejpam-4737	325	5	gγ(x	gγ(x	NOUN
ejpam-4737	325	6	)	)	PUNCT
ejpam-4737	325	7	⊆	⊆	NUM
ejpam-4737	325	8	g	g	PROPN
ejpam-4737	325	9	s(λ	s(λ	PROPN
ejpam-4737	325	10	,	,	PUNCT
ejpam-4737	325	11	p	p	NOUN
ejpam-4737	325	12	)	)	PUNCT
ejpam-4737	325	13	γ(x	γ(x	NOUN
ejpam-4737	325	14	)	)	PUNCT
ejpam-4737	325	15	⊆	⊆	NUM
ejpam-4737	325	16	vγ(x	vγ(x	NOUN
ejpam-4737	325	17	)	)	PUNCT
ejpam-4737	326	1	.	.	PUNCT
ejpam-4737	327	1	since	since	SCONJ
ejpam-4737	327	2	{	{	PUNCT
ejpam-4737	327	3	gγ(x	gγ(x	X
ejpam-4737	327	4	)	)	PUNCT
ejpam-4737	327	5	|	|	ADV
ejpam-4737	327	6	x	x	SYM
ejpam-4737	327	7	∈	∈	NOUN
ejpam-4737	327	8	x	x	X
ejpam-4737	327	9	}	}	PUNCT
ejpam-4737	327	10	is	be	AUX
ejpam-4737	327	11	a	a	DET
ejpam-4737	327	12	s(λ	s(λ	PROPN
ejpam-4737	327	13	,	,	PUNCT
ejpam-4737	327	14	p)-open	p)-open	VERB
ejpam-4737	327	15	cover	cover	NOUN
ejpam-4737	327	16	of	of	ADP
ejpam-4737	327	17	x	x	NOUN
ejpam-4737	327	18	,	,	PUNCT
ejpam-4737	327	19	there	there	PRON
ejpam-4737	327	20	exist	exist	VERB
ejpam-4737	327	21	finite	finite	ADJ
ejpam-4737	327	22	points	point	NOUN
ejpam-4737	327	23	,	,	PUNCT
ejpam-4737	327	24	say	say	INTJ
ejpam-4737	327	25	,	,	PUNCT
ejpam-4737	327	26	x1	x1	PROPN
ejpam-4737	327	27	,	,	PUNCT
ejpam-4737	327	28	x2	x2	PROPN
ejpam-4737	327	29	,	,	PUNCT
ejpam-4737	327	30	...	...	PUNCT
ejpam-4737	327	31	,	,	PUNCT
ejpam-4737	327	32	xn	xn	PROPN
ejpam-4737	327	33	such	such	ADJ
ejpam-4737	327	34	that	that	SCONJ
ejpam-4737	327	35	x	x	X
ejpam-4737	327	36	=	=	NOUN
ejpam-4737	327	37	n	n	PRON
ejpam-4737	327	38	∪	∪	VERB
ejpam-4737	327	39	i=1	i=1	PROPN
ejpam-4737	327	40	g	g	PROPN
ejpam-4737	327	41	s(λ	s(λ	PROPN
ejpam-4737	327	42	,	,	PUNCT
ejpam-4737	327	43	p	p	NOUN
ejpam-4737	327	44	)	)	PUNCT
ejpam-4737	327	45	γ(xi	γ(xi	PROPN
ejpam-4737	327	46	)	)	PUNCT
ejpam-4737	327	47	.	.	PUNCT
ejpam-4737	328	1	thus	thus	ADV
ejpam-4737	328	2	,	,	PUNCT
ejpam-4737	328	3	x	x	SYM
ejpam-4737	328	4	=	=	NOUN
ejpam-4737	328	5	n	n	PRON
ejpam-4737	328	6	∪	∪	VERB
ejpam-4737	328	7	i=1	i=1	PROPN
ejpam-4737	328	8	vγ(xi	vγ(xi	PROPN
ejpam-4737	328	9	)	)	PUNCT
ejpam-4737	328	10	.	.	PUNCT
ejpam-4737	329	1	conversely	conversely	ADV
ejpam-4737	329	2	,	,	PUNCT
ejpam-4737	329	3	let	let	VERB
ejpam-4737	329	4	{	{	PUNCT
ejpam-4737	329	5	vγ	vγ	VERB
ejpam-4737	329	6	|	|	ADV
ejpam-4737	329	7	γ	γ	X
ejpam-4737	329	8	∈	∈	PROPN
ejpam-4737	329	9	∇	∇	X
ejpam-4737	329	10	}	}	PUNCT
ejpam-4737	329	11	be	be	VERB
ejpam-4737	329	12	a	a	DET
ejpam-4737	329	13	s(λ	s(λ	PROPN
ejpam-4737	329	14	,	,	PUNCT
ejpam-4737	329	15	p)-open	p)-open	VERB
ejpam-4737	329	16	cover	cover	NOUN
ejpam-4737	329	17	of	of	ADP
ejpam-4737	329	18	x.	x.	NOUN
ejpam-4737	329	19	by	by	ADP
ejpam-4737	329	20	lemma	lemma	PROPN
ejpam-4737	329	21	4	4	NUM
ejpam-4737	329	22	,	,	PUNCT
ejpam-4737	329	23	{	{	PUNCT
ejpam-4737	329	24	v	v	X
ejpam-4737	329	25	s(λ	s(λ	PROPN
ejpam-4737	329	26	,	,	PUNCT
ejpam-4737	329	27	p	p	NOUN
ejpam-4737	329	28	)	)	PUNCT
ejpam-4737	329	29	γ	γ	PROPN
ejpam-4737	329	30	|	|	ADV
ejpam-4737	329	31	γ	γ	PROPN
ejpam-4737	329	32	∈	∈	PROPN
ejpam-4737	329	33	∇	∇	X
ejpam-4737	329	34	}	}	PUNCT
ejpam-4737	329	35	is	be	AUX
ejpam-4737	329	36	a	a	DET
ejpam-4737	329	37	s(λ	s(λ	PROPN
ejpam-4737	329	38	,	,	PUNCT
ejpam-4737	329	39	p)-regular	p)-regular	ADJ
ejpam-4737	329	40	cover	cover	NOUN
ejpam-4737	329	41	of	of	ADP
ejpam-4737	329	42	x	x	X
ejpam-4737	329	43	and	and	CCONJ
ejpam-4737	329	44	hence	hence	ADV
ejpam-4737	329	45	a	a	DET
ejpam-4737	329	46	θs(λ	θs(λ	NOUN
ejpam-4737	329	47	,	,	PUNCT
ejpam-4737	329	48	p)-open	p)-open	VERB
ejpam-4737	329	49	cover	cover	NOUN
ejpam-4737	329	50	of	of	ADP
ejpam-4737	329	51	x.	x.	NOUN
ejpam-4737	329	52	thus	thus	ADV
ejpam-4737	329	53	,	,	PUNCT
ejpam-4737	329	54	there	there	PRON
ejpam-4737	329	55	exists	exist	VERB
ejpam-4737	329	56	a	a	DET
ejpam-4737	329	57	finite	finite	NOUN
ejpam-4737	329	58	subset	subset	NOUN
ejpam-4737	329	59	∇0	∇0	NUM
ejpam-4737	329	60	of	of	ADP
ejpam-4737	329	61	∇	∇	NOUN
ejpam-4737	329	62	such	such	ADJ
ejpam-4737	329	63	that	that	SCONJ
ejpam-4737	329	64	x	x	X
ejpam-4737	329	65	=	=	SYM
ejpam-4737	329	66	∪	∪	X
ejpam-4737	329	67	γ∈∇0	γ∈∇0	X
ejpam-4737	329	68	v	v	ADP
ejpam-4737	329	69	s(λ	s(λ	PROPN
ejpam-4737	329	70	,	,	PUNCT
ejpam-4737	329	71	p	p	NOUN
ejpam-4737	329	72	)	)	PUNCT
ejpam-4737	329	73	γ	γ	NOUN
ejpam-4737	329	74	.	.	PUNCT
ejpam-4737	330	1	this	this	PRON
ejpam-4737	330	2	shows	show	VERB
ejpam-4737	330	3	that	that	SCONJ
ejpam-4737	330	4	(	(	PUNCT
ejpam-4737	330	5	x	x	X
ejpam-4737	330	6	,	,	PUNCT
ejpam-4737	330	7	τ	τ	X
ejpam-4737	330	8	)	)	PUNCT
ejpam-4737	330	9	is	be	AUX
ejpam-4737	330	10	s(λ	s(λ	PROPN
ejpam-4737	330	11	,	,	PUNCT
ejpam-4737	330	12	p)-closed	p)-close	VERB
ejpam-4737	330	13	.	.	PUNCT
ejpam-4737	331	1	acknowledgements	acknowledgement	NOUN
ejpam-4737	331	2	this	this	DET
ejpam-4737	331	3	research	research	NOUN
ejpam-4737	331	4	project	project	NOUN
ejpam-4737	331	5	was	be	AUX
ejpam-4737	331	6	financially	financially	ADV
ejpam-4737	331	7	supported	support	VERB
ejpam-4737	331	8	by	by	ADP
ejpam-4737	331	9	mahasarakham	mahasarakham	PROPN
ejpam-4737	331	10	university	university	PROPN
ejpam-4737	331	11	.	.	PUNCT
ejpam-4737	332	1	references	reference	NOUN
ejpam-4737	332	2	1446	1446	NUM
ejpam-4737	332	3	references	reference	NOUN
ejpam-4737	332	4	[	[	X
ejpam-4737	332	5	1	1	NUM
ejpam-4737	332	6	]	]	PUNCT
ejpam-4737	332	7	s.	s.	PROPN
ejpam-4737	332	8	baudong	baudong	PROPN
ejpam-4737	332	9	,	,	PUNCT
ejpam-4737	332	10	c.	c.	PROPN
ejpam-4737	332	11	viriyapong	viriyapong	PROPN
ejpam-4737	332	12	,	,	PUNCT
ejpam-4737	332	13	and	and	CCONJ
ejpam-4737	332	14	c.	c.	PROPN
ejpam-4737	332	15	boonpok	boonpok	PROPN
ejpam-4737	332	16	.	.	PUNCT
ejpam-4737	333	1	on	on	ADP
ejpam-4737	333	2	generalized	generalized	ADJ
ejpam-4737	333	3	topology	topology	NOUN
ejpam-4737	333	4	and	and	CCONJ
ejpam-4737	333	5	minimal	minimal	ADJ
ejpam-4737	333	6	structure	structure	NOUN
ejpam-4737	333	7	spaces	space	NOUN
ejpam-4737	333	8	.	.	PUNCT
ejpam-4737	334	1	international	international	ADJ
ejpam-4737	334	2	journal	journal	PROPN
ejpam-4737	334	3	of	of	ADP
ejpam-4737	334	4	mathematical	mathematical	ADJ
ejpam-4737	334	5	analysis	analysis	NOUN
ejpam-4737	334	6	,	,	PUNCT
ejpam-4737	334	7	5(31):1507–1516	5(31):1507–1516	NUM
ejpam-4737	334	8	,	,	PUNCT
ejpam-4737	334	9	2011	2011	NUM
ejpam-4737	334	10	.	.	PUNCT
ejpam-4737	335	1	[	[	X
ejpam-4737	335	2	2	2	NUM
ejpam-4737	335	3	]	]	PUNCT
ejpam-4737	335	4	c.	c.	PROPN
ejpam-4737	335	5	boonpok	boonpok	PROPN
ejpam-4737	335	6	and	and	CCONJ
ejpam-4737	335	7	m.	m.	NOUN
ejpam-4737	335	8	thongmoon	thongmoon	NOUN
ejpam-4737	335	9	.	.	PUNCT
ejpam-4737	336	1	δp(λ	δp(λ	NOUN
ejpam-4737	336	2	,	,	PUNCT
ejpam-4737	336	3	p)-open	p)-open	VERB
ejpam-4737	336	4	sets	set	NOUN
ejpam-4737	336	5	in	in	ADP
ejpam-4737	336	6	topological	topological	ADJ
ejpam-4737	336	7	spaces	space	NOUN
ejpam-4737	336	8	.	.	PUNCT
ejpam-4737	337	1	european	european	ADJ
ejpam-4737	337	2	journal	journal	PROPN
ejpam-4737	337	3	of	of	ADP
ejpam-4737	337	4	pure	pure	ADJ
ejpam-4737	337	5	and	and	CCONJ
ejpam-4737	337	6	applied	applied	ADJ
ejpam-4737	337	7	mathematics	mathematic	NOUN
ejpam-4737	337	8	,	,	PUNCT
ejpam-4737	337	9	in	in	ADP
ejpam-4737	337	10	press	press	NOUN
ejpam-4737	337	11	,	,	PUNCT
ejpam-4737	337	12	2023	2023	NUM
ejpam-4737	337	13	.	.	PUNCT
ejpam-4737	338	1	https://doi.org/10.29020/nybg.ejpam.v16i3.4737	https://doi.org/10.29020/nybg.ejpam.v16i3.4737	X
ejpam-4737	338	2	.	.	PUNCT
ejpam-4737	339	1	[	[	X
ejpam-4737	339	2	3	3	X
ejpam-4737	339	3	]	]	PUNCT
ejpam-4737	339	4	c.	c.	PROPN
ejpam-4737	339	5	boonpok	boonpok	PROPN
ejpam-4737	339	6	and	and	CCONJ
ejpam-4737	339	7	c.	c.	PROPN
ejpam-4737	339	8	viriyapong	viriyapong	PROPN
ejpam-4737	339	9	.	.	PUNCT
ejpam-4737	340	1	on	on	ADP
ejpam-4737	340	2	(	(	PUNCT
ejpam-4737	340	3	λ	λ	PROPN
ejpam-4737	340	4	,	,	PUNCT
ejpam-4737	340	5	p)-closed	p)-close	VERB
ejpam-4737	340	6	sets	set	NOUN
ejpam-4737	340	7	and	and	CCONJ
ejpam-4737	340	8	the	the	DET
ejpam-4737	340	9	related	related	ADJ
ejpam-4737	340	10	notions	notion	NOUN
ejpam-4737	340	11	in	in	ADP
ejpam-4737	340	12	topological	topological	ADJ
ejpam-4737	340	13	spaces	space	NOUN
ejpam-4737	340	14	.	.	PUNCT
ejpam-4737	341	1	european	european	ADJ
ejpam-4737	341	2	journal	journal	PROPN
ejpam-4737	341	3	of	of	ADP
ejpam-4737	341	4	pure	pure	ADJ
ejpam-4737	341	5	and	and	CCONJ
ejpam-4737	341	6	applied	applied	ADJ
ejpam-4737	341	7	mathematics	mathematic	NOUN
ejpam-4737	341	8	,	,	PUNCT
ejpam-4737	341	9	15(2):415	15(2):415	PROPN
ejpam-4737	341	10	–	–	PUNCT
ejpam-4737	341	11	436	436	NUM
ejpam-4737	341	12	,	,	PUNCT
ejpam-4737	341	13	2022	2022	NUM
ejpam-4737	341	14	.	.	PUNCT
ejpam-4737	342	1	[	[	X
ejpam-4737	342	2	4	4	NUM
ejpam-4737	342	3	]	]	PUNCT
ejpam-4737	342	4	m.	m.	NOUN
ejpam-4737	342	5	caldas	caldas	PROPN
ejpam-4737	342	6	,	,	PUNCT
ejpam-4737	342	7	t.	t.	NOUN
ejpam-4737	342	8	fukutake	fukutake	NOUN
ejpam-4737	342	9	,	,	PUNCT
ejpam-4737	342	10	s.	s.	PROPN
ejpam-4737	342	11	jafari	jafari	PROPN
ejpam-4737	342	12	,	,	PUNCT
ejpam-4737	342	13	and	and	CCONJ
ejpam-4737	342	14	t.	t.	PROPN
ejpam-4737	342	15	noiri	noiri	PROPN
ejpam-4737	342	16	.	.	PUNCT
ejpam-4737	343	1	some	some	DET
ejpam-4737	343	2	applications	application	NOUN
ejpam-4737	343	3	of	of	ADP
ejpam-4737	343	4	δ	δ	NOUN
ejpam-4737	343	5	-	-	PUNCT
ejpam-4737	343	6	preopen	preopen	ADJ
ejpam-4737	343	7	sets	set	NOUN
ejpam-4737	343	8	in	in	ADP
ejpam-4737	343	9	topological	topological	ADJ
ejpam-4737	343	10	spaces	space	NOUN
ejpam-4737	343	11	.	.	PUNCT
ejpam-4737	344	1	bulletin	bulletin	NOUN
ejpam-4737	344	2	of	of	ADP
ejpam-4737	344	3	the	the	DET
ejpam-4737	344	4	institute	institute	NOUN
ejpam-4737	344	5	of	of	ADP
ejpam-4737	344	6	mathematics	mathematics	PROPN
ejpam-4737	344	7	,	,	PUNCT
ejpam-4737	344	8	academia	academia	PROPN
ejpam-4737	344	9	sinica	sinica	PROPN
ejpam-4737	344	10	,	,	PUNCT
ejpam-4737	344	11	33(3):261–276	33(3):261–276	NOUN
ejpam-4737	344	12	,	,	PUNCT
ejpam-4737	344	13	2005	2005	NUM
ejpam-4737	344	14	.	.	PUNCT
ejpam-4737	345	1	[	[	X
ejpam-4737	345	2	5	5	NUM
ejpam-4737	345	3	]	]	PUNCT
ejpam-4737	345	4	m.	m.	NOUN
ejpam-4737	345	5	caldas	caldas	PROPN
ejpam-4737	345	6	,	,	PUNCT
ejpam-4737	345	7	m.	m.	NOUN
ejpam-4737	345	8	ganster	ganster	NOUN
ejpam-4737	345	9	,	,	PUNCT
ejpam-4737	345	10	d.	d.	PROPN
ejpam-4737	345	11	n.	n.	PROPN
ejpam-4737	345	12	georgiou	georgiou	PROPN
ejpam-4737	345	13	,	,	PUNCT
ejpam-4737	345	14	s.	s.	PROPN
ejpam-4737	345	15	jafari	jafari	PROPN
ejpam-4737	345	16	,	,	PUNCT
ejpam-4737	345	17	and	and	CCONJ
ejpam-4737	345	18	t.	t.	PROPN
ejpam-4737	345	19	noiri	noiri	PROPN
ejpam-4737	345	20	.	.	PUNCT
ejpam-4737	346	1	δ	δ	PROPN
ejpam-4737	346	2	-	-	PUNCT
ejpam-4737	346	3	semiopen	semiopen	ADJ
ejpam-4737	346	4	sets	set	NOUN
ejpam-4737	346	5	in	in	ADP
ejpam-4737	346	6	topological	topological	ADJ
ejpam-4737	346	7	spaces	space	NOUN
ejpam-4737	346	8	.	.	PUNCT
ejpam-4737	347	1	topology	topology	NOUN
ejpam-4737	347	2	proceedings	proceeding	NOUN
ejpam-4737	347	3	,	,	PUNCT
ejpam-4737	347	4	29(2):369–383	29(2):369–383	NUM
ejpam-4737	347	5	,	,	PUNCT
ejpam-4737	347	6	2005	2005	NUM
ejpam-4737	347	7	.	.	PUNCT
ejpam-4737	348	1	[	[	X
ejpam-4737	348	2	6	6	NUM
ejpam-4737	348	3	]	]	PUNCT
ejpam-4737	348	4	m.	m.	NOUN
ejpam-4737	348	5	caldas	caldas	PROPN
ejpam-4737	348	6	,	,	PUNCT
ejpam-4737	348	7	d.	d.	PROPN
ejpam-4737	348	8	n.	n.	PROPN
ejpam-4737	348	9	georgiou	georgiou	PROPN
ejpam-4737	348	10	,	,	PUNCT
ejpam-4737	348	11	s.	s.	PROPN
ejpam-4737	348	12	jafari	jafari	PROPN
ejpam-4737	348	13	,	,	PUNCT
ejpam-4737	348	14	and	and	CCONJ
ejpam-4737	348	15	t.	t.	PROPN
ejpam-4737	348	16	noiri	noiri	PROPN
ejpam-4737	348	17	.	.	PUNCT
ejpam-4737	349	1	more	more	ADV
ejpam-4737	349	2	on	on	ADP
ejpam-4737	349	3	δ	δ	PROPN
ejpam-4737	349	4	-	-	PUNCT
ejpam-4737	349	5	semiopen	semiopen	ADJ
ejpam-4737	349	6	sets	set	NOUN
ejpam-4737	349	7	.	.	PUNCT
ejpam-4737	350	1	note	note	VERB
ejpam-4737	350	2	di	di	PROPN
ejpam-4737	350	3	matematica	matematica	PROPN
ejpam-4737	350	4	,	,	PUNCT
ejpam-4737	350	5	22(2):1–14	22(2):1–14	PROPN
ejpam-4737	350	6	,	,	PUNCT
ejpam-4737	350	7	2003	2003	NUM
ejpam-4737	350	8	.	.	PUNCT
ejpam-4737	351	1	[	[	X
ejpam-4737	351	2	7	7	X
ejpam-4737	351	3	]	]	X
ejpam-4737	351	4	w.	w.	PROPN
ejpam-4737	351	5	dungthaisong	dungthaisong	PROPN
ejpam-4737	351	6	,	,	PUNCT
ejpam-4737	351	7	c.	c.	PROPN
ejpam-4737	351	8	boonpok	boonpok	PROPN
ejpam-4737	351	9	,	,	PUNCT
ejpam-4737	351	10	and	and	CCONJ
ejpam-4737	351	11	c.	c.	PROPN
ejpam-4737	351	12	viriyapong	viriyapong	PROPN
ejpam-4737	351	13	.	.	PUNCT
ejpam-4737	352	1	generalized	generalize	VERB
ejpam-4737	352	2	closed	close	VERB
ejpam-4737	352	3	sets	set	NOUN
ejpam-4737	352	4	in	in	ADP
ejpam-4737	352	5	bigeneralized	bigeneralize	VERB
ejpam-4737	352	6	topological	topological	ADJ
ejpam-4737	352	7	spaces	space	NOUN
ejpam-4737	352	8	.	.	PUNCT
ejpam-4737	353	1	international	international	ADJ
ejpam-4737	353	2	journal	journal	PROPN
ejpam-4737	353	3	of	of	ADP
ejpam-4737	353	4	mathematical	mathematical	ADJ
ejpam-4737	353	5	analysis	analysis	NOUN
ejpam-4737	353	6	,	,	PUNCT
ejpam-4737	353	7	5(24):1175–1184	5(24):1175–1184	NUM
ejpam-4737	353	8	,	,	PUNCT
ejpam-4737	353	9	2011	2011	NUM
ejpam-4737	353	10	.	.	PUNCT
ejpam-4737	354	1	[	[	X
ejpam-4737	354	2	8	8	NUM
ejpam-4737	354	3	]	]	PUNCT
ejpam-4737	354	4	m.	m.	NOUN
ejpam-4737	354	5	ganster	ganster	NOUN
ejpam-4737	354	6	,	,	PUNCT
ejpam-4737	354	7	s.	s.	PROPN
ejpam-4737	354	8	jafari	jafari	PROPN
ejpam-4737	354	9	,	,	PUNCT
ejpam-4737	354	10	and	and	CCONJ
ejpam-4737	354	11	t.	t.	PROPN
ejpam-4737	354	12	noiri	noiri	PROPN
ejpam-4737	354	13	.	.	PUNCT
ejpam-4737	355	1	on	on	ADP
ejpam-4737	355	2	pre	pre	ADJ
ejpam-4737	355	3	-	-	ADJ
ejpam-4737	355	4	λ	λ	NOUN
ejpam-4737	355	5	-	-	NOUN
ejpam-4737	355	6	sets	set	NOUN
ejpam-4737	355	7	and	and	CCONJ
ejpam-4737	355	8	pre	pre	ADJ
ejpam-4737	355	9	-	-	ADJ
ejpam-4737	355	10	v	v	ADJ
ejpam-4737	355	11	-sets	-set	NOUN
ejpam-4737	355	12	.	.	PUNCT
ejpam-4737	356	1	acta	acta	PROPN
ejpam-4737	356	2	mathematica	mathematica	PROPN
ejpam-4737	356	3	hungarica	hungarica	PROPN
ejpam-4737	356	4	,	,	PUNCT
ejpam-4737	356	5	95:337–343	95:337–343	PROPN
ejpam-4737	356	6	,	,	PUNCT
ejpam-4737	356	7	2002	2002	NUM
ejpam-4737	356	8	.	.	PUNCT
ejpam-4737	357	1	[	[	X
ejpam-4737	357	2	9	9	NUM
ejpam-4737	357	3	]	]	PUNCT
ejpam-4737	357	4	a.	a.	NOUN
ejpam-4737	357	5	s.	s.	PROPN
ejpam-4737	357	6	mashhour	mashhour	PROPN
ejpam-4737	357	7	,	,	PUNCT
ejpam-4737	357	8	m.	m.	PROPN
ejpam-4737	357	9	e.	e.	PROPN
ejpam-4737	357	10	abd	abd	PROPN
ejpam-4737	357	11	el	el	PROPN
ejpam-4737	357	12	-	-	PROPN
ejpam-4737	357	13	monsef	monsef	ADJ
ejpam-4737	357	14	,	,	PUNCT
ejpam-4737	357	15	and	and	CCONJ
ejpam-4737	357	16	s.	s.	PROPN
ejpam-4737	357	17	n.	n.	PROPN
ejpam-4737	357	18	el	el	PROPN
ejpam-4737	357	19	-	-	PROPN
ejpam-4737	357	20	deeb	deeb	PROPN
ejpam-4737	357	21	.	.	PUNCT
ejpam-4737	358	1	on	on	ADP
ejpam-4737	358	2	precontinuous	precontinuous	ADJ
ejpam-4737	358	3	and	and	CCONJ
ejpam-4737	358	4	weak	weak	ADJ
ejpam-4737	358	5	precontinuous	precontinuous	ADJ
ejpam-4737	358	6	mappings	mapping	NOUN
ejpam-4737	358	7	.	.	PUNCT
ejpam-4737	359	1	proceedings	proceeding	NOUN
ejpam-4737	359	2	of	of	ADP
ejpam-4737	359	3	the	the	DET
ejpam-4737	359	4	mathematical	mathematical	ADJ
ejpam-4737	359	5	and	and	CCONJ
ejpam-4737	359	6	physical	physical	ADJ
ejpam-4737	359	7	society	society	NOUN
ejpam-4737	359	8	of	of	ADP
ejpam-4737	359	9	egypt	egypt	PROPN
ejpam-4737	359	10	,	,	PUNCT
ejpam-4737	359	11	53:47–53	53:47–53	NUM
ejpam-4737	359	12	,	,	PUNCT
ejpam-4737	359	13	1982	1982	NUM
ejpam-4737	359	14	.	.	PUNCT
ejpam-4737	360	1	[	[	X
ejpam-4737	360	2	10	10	NUM
ejpam-4737	360	3	]	]	X
ejpam-4737	360	4	v.	v.	ADP
ejpam-4737	360	5	pipitone	pipitone	NOUN
ejpam-4737	360	6	and	and	CCONJ
ejpam-4737	360	7	g.	g.	PROPN
ejpam-4737	360	8	russo	russo	PROPN
ejpam-4737	360	9	.	.	PUNCT
ejpam-4737	361	1	spazi	spazi	PROPN
ejpam-4737	361	2	semiconnessi	semiconnessi	PROPN
ejpam-4737	361	3	e	e	PROPN
ejpam-4737	361	4	spazi	spazi	X
ejpam-4737	361	5	semiaperti	semiaperti	PROPN
ejpam-4737	361	6	.	.	PUNCT
ejpam-4737	362	1	rendiconti	rendiconti	PROPN
ejpam-4737	362	2	del	del	PROPN
ejpam-4737	362	3	circolo	circolo	PROPN
ejpam-4737	362	4	matematico	matematico	NOUN
ejpam-4737	362	5	di	di	PROPN
ejpam-4737	362	6	palermo	palermo	PROPN
ejpam-4737	362	7	series	series	PROPN
ejpam-4737	362	8	2	2	NUM
ejpam-4737	362	9	,	,	PUNCT
ejpam-4737	362	10	24:273–285	24:273–285	NUM
ejpam-4737	362	11	,	,	PUNCT
ejpam-4737	362	12	1975	1975	NUM
ejpam-4737	362	13	.	.	PUNCT
ejpam-4737	363	1	[	[	X
ejpam-4737	363	2	11	11	NUM
ejpam-4737	363	3	]	]	PUNCT
ejpam-4737	363	4	s.	s.	PROPN
ejpam-4737	363	5	raychaudhuri	raychaudhuri	PROPN
ejpam-4737	363	6	and	and	CCONJ
ejpam-4737	363	7	m.	m.	PROPN
ejpam-4737	363	8	n.	n.	PROPN
ejpam-4737	363	9	mukherjee	mukherjee	PROPN
ejpam-4737	363	10	.	.	PUNCT
ejpam-4737	364	1	on	on	ADP
ejpam-4737	364	2	δ	δ	PROPN
ejpam-4737	364	3	-	-	PUNCT
ejpam-4737	364	4	almost	almost	ADV
ejpam-4737	364	5	continuity	continuity	NOUN
ejpam-4737	364	6	and	and	CCONJ
ejpam-4737	364	7	δ	δ	NOUN
ejpam-4737	364	8	-	-	PUNCT
ejpam-4737	364	9	preopen	preopen	ADJ
ejpam-4737	364	10	sets	set	NOUN
ejpam-4737	364	11	.	.	PUNCT
ejpam-4737	365	1	bulletin	bulletin	NOUN
ejpam-4737	365	2	of	of	ADP
ejpam-4737	365	3	the	the	DET
ejpam-4737	365	4	institute	institute	NOUN
ejpam-4737	365	5	of	of	ADP
ejpam-4737	365	6	mathematics	mathematics	PROPN
ejpam-4737	365	7	,	,	PUNCT
ejpam-4737	365	8	academia	academia	PROPN
ejpam-4737	365	9	sinica	sinica	PROPN
ejpam-4737	365	10	,	,	PUNCT
ejpam-4737	365	11	21:357–366	21:357–366	PROPN
ejpam-4737	365	12	,	,	PUNCT
ejpam-4737	365	13	1993	1993	NUM
ejpam-4737	365	14	.	.	PUNCT
ejpam-4737	366	1	[	[	X
ejpam-4737	366	2	12	12	NUM
ejpam-4737	366	3	]	]	PUNCT
ejpam-4737	366	4	s.	s.	PROPN
ejpam-4737	366	5	raychaudhuri	raychaudhuri	PROPN
ejpam-4737	366	6	and	and	CCONJ
ejpam-4737	366	7	m.	m.	PROPN
ejpam-4737	366	8	n.	n.	PROPN
ejpam-4737	366	9	mukherjee	mukherjee	PROPN
ejpam-4737	366	10	.	.	PUNCT
ejpam-4737	367	1	δp	δp	PRON
ejpam-4737	367	2	-	-	PUNCT
ejpam-4737	367	3	closedness	closedness	NOUN
ejpam-4737	367	4	for	for	ADP
ejpam-4737	367	5	topological	topological	ADJ
ejpam-4737	367	6	spaces	space	NOUN
ejpam-4737	367	7	.	.	PUNCT
ejpam-4737	368	1	the	the	DET
ejpam-4737	368	2	journal	journal	NOUN
ejpam-4737	368	3	of	of	ADP
ejpam-4737	368	4	the	the	DET
ejpam-4737	368	5	indian	indian	PROPN
ejpam-4737	368	6	academy	academy	PROPN
ejpam-4737	368	7	of	of	ADP
ejpam-4737	368	8	mathematics	mathematic	NOUN
ejpam-4737	368	9	,	,	PUNCT
ejpam-4737	368	10	18:89–99	18:89–99	NUM
ejpam-4737	368	11	,	,	PUNCT
ejpam-4737	368	12	1996	1996	NUM
ejpam-4737	368	13	.	.	PUNCT
ejpam-4737	369	1	[	[	X
ejpam-4737	369	2	13	13	NUM
ejpam-4737	369	3	]	]	PUNCT
ejpam-4737	369	4	p.	p.	NOUN
ejpam-4737	369	5	torton	torton	PROPN
ejpam-4737	369	6	,	,	PUNCT
ejpam-4737	369	7	c.	c.	PROPN
ejpam-4737	369	8	viriyapong	viriyapong	PROPN
ejpam-4737	369	9	,	,	PUNCT
ejpam-4737	369	10	and	and	CCONJ
ejpam-4737	369	11	c.	c.	PROPN
ejpam-4737	369	12	boonpok	boonpok	PROPN
ejpam-4737	369	13	.	.	PUNCT
ejpam-4737	370	1	some	some	DET
ejpam-4737	370	2	separation	separation	NOUN
ejpam-4737	370	3	axioms	axiom	VERB
ejpam-4737	370	4	in	in	ADP
ejpam-4737	370	5	bigeneralized	bigeneralize	VERB
ejpam-4737	370	6	topological	topological	ADJ
ejpam-4737	370	7	spaces	space	NOUN
ejpam-4737	370	8	.	.	PUNCT
ejpam-4737	371	1	international	international	ADJ
ejpam-4737	371	2	journal	journal	PROPN
ejpam-4737	371	3	of	of	ADP
ejpam-4737	371	4	mathematical	mathematical	ADJ
ejpam-4737	371	5	analysis	analysis	NOUN
ejpam-4737	371	6	,	,	PUNCT
ejpam-4737	371	7	6(56):2789–2796	6(56):2789–2796	NOUN
ejpam-4737	371	8	,	,	PUNCT
ejpam-4737	371	9	2012	2012	NUM
ejpam-4737	371	10	.	.	PUNCT
ejpam-4737	372	1	references	reference	NOUN
ejpam-4737	372	2	1447	1447	NUM
ejpam-4737	372	3	[	[	X
ejpam-4737	372	4	14	14	NUM
ejpam-4737	372	5	]	]	X
ejpam-4737	372	6	n.	n.	NOUN
ejpam-4737	372	7	v.	v.	ADP
ejpam-4737	372	8	veličko	veličko	PROPN
ejpam-4737	372	9	.	.	PUNCT
ejpam-4737	373	1	h	h	NOUN
ejpam-4737	373	2	-	-	PUNCT
ejpam-4737	373	3	closed	close	VERB
ejpam-4737	373	4	topological	topological	ADJ
ejpam-4737	373	5	spaces	space	NOUN
ejpam-4737	373	6	.	.	PUNCT
ejpam-4737	374	1	american	american	PROPN
ejpam-4737	374	2	mathematical	mathematical	ADJ
ejpam-4737	374	3	society	society	NOUN
ejpam-4737	374	4	translations	translation	NOUN
ejpam-4737	374	5	,	,	PUNCT
ejpam-4737	374	6	78(2):102–118	78(2):102–118	NUM
ejpam-4737	374	7	,	,	PUNCT
ejpam-4737	374	8	1968	1968	NUM
ejpam-4737	374	9	.	.	PUNCT