id	sid	tid	token	lemma	pos
ejpam-4736	1	1	european	european	PROPN
ejpam-4736	1	2	journal	journal	PROPN
ejpam-4736	1	3	of	of	ADP
ejpam-4736	1	4	pure	pure	ADJ
ejpam-4736	1	5	and	and	CCONJ
ejpam-4736	1	6	applied	apply	VERB
ejpam-4736	1	7	mathematics	mathematic	NOUN
ejpam-4736	1	8	vol	vol	NOUN
ejpam-4736	1	9	.	.	PUNCT
ejpam-4736	2	1	16	16	NUM
ejpam-4736	2	2	,	,	PUNCT
ejpam-4736	2	3	no	no	INTJ
ejpam-4736	2	4	.	.	NOUN
ejpam-4736	2	5	4	4	NUM
ejpam-4736	2	6	,	,	PUNCT
ejpam-4736	2	7	2023	2023	NUM
ejpam-4736	2	8	,	,	PUNCT
ejpam-4736	2	9	2581	2581	NUM
ejpam-4736	2	10	-	-	SYM
ejpam-4736	2	11	2596	2596	NUM
ejpam-4736	2	12	issn	issn	PROPN
ejpam-4736	2	13	1307	1307	NUM
ejpam-4736	2	14	-	-	SYM
ejpam-4736	2	15	5543	5543	NUM
ejpam-4736	2	16	–	–	PUNCT
ejpam-4736	2	17	ejpam.com	ejpam.com	X
ejpam-4736	2	18	published	publish	VERB
ejpam-4736	2	19	by	by	ADP
ejpam-4736	2	20	new	new	PROPN
ejpam-4736	2	21	york	york	PROPN
ejpam-4736	2	22	business	business	PROPN
ejpam-4736	2	23	global	global	ADJ
ejpam-4736	2	24	properties	property	NOUN
ejpam-4736	2	25	of	of	ADP
ejpam-4736	2	26	generalized	generalized	ADJ
ejpam-4736	2	27	δp(λ	δp(λ	NOUN
ejpam-4736	2	28	,	,	PUNCT
ejpam-4736	2	29	s)-closed	s)-close	VERB
ejpam-4736	2	30	sets	set	NOUN
ejpam-4736	2	31	chawalit	chawalit	VERB
ejpam-4736	2	32	boonpok1	boonpok1	PROPN
ejpam-4736	2	33	,	,	PUNCT
ejpam-4736	2	34	napassanan	napassanan	NOUN
ejpam-4736	2	35	srisarakham1,∗	srisarakham1,∗	NOUN
ejpam-4736	2	36	1	1	NUM
ejpam-4736	2	37	mathematics	mathematic	NOUN
ejpam-4736	2	38	and	and	CCONJ
ejpam-4736	2	39	applied	apply	VERB
ejpam-4736	2	40	mathematics	mathematics	PROPN
ejpam-4736	2	41	research	research	NOUN
ejpam-4736	2	42	unit	unit	NOUN
ejpam-4736	2	43	,	,	PUNCT
ejpam-4736	2	44	department	department	NOUN
ejpam-4736	2	45	of	of	ADP
ejpam-4736	2	46	mathematics	mathematic	NOUN
ejpam-4736	2	47	,	,	PUNCT
ejpam-4736	2	48	faculty	faculty	NOUN
ejpam-4736	2	49	of	of	ADP
ejpam-4736	2	50	science	science	NOUN
ejpam-4736	2	51	,	,	PUNCT
ejpam-4736	2	52	mahasarakham	mahasarakham	PROPN
ejpam-4736	2	53	university	university	PROPN
ejpam-4736	2	54	,	,	PUNCT
ejpam-4736	2	55	maha	maha	PROPN
ejpam-4736	2	56	sarakham	sarakham	PROPN
ejpam-4736	2	57	,	,	PUNCT
ejpam-4736	2	58	44150	44150	NUM
ejpam-4736	2	59	,	,	PUNCT
ejpam-4736	2	60	thailand	thailand	PROPN
ejpam-4736	2	61	abstract	abstract	PROPN
ejpam-4736	2	62	.	.	PUNCT
ejpam-4736	3	1	this	this	DET
ejpam-4736	3	2	paper	paper	NOUN
ejpam-4736	3	3	deals	deal	NOUN
ejpam-4736	3	4	with	with	ADP
ejpam-4736	3	5	the	the	DET
ejpam-4736	3	6	concept	concept	NOUN
ejpam-4736	3	7	of	of	ADP
ejpam-4736	3	8	generalized	generalized	ADJ
ejpam-4736	3	9	δp(λ	δp(λ	NOUN
ejpam-4736	3	10	,	,	PUNCT
ejpam-4736	3	11	s)-closed	s)-close	VERB
ejpam-4736	3	12	sets	set	NOUN
ejpam-4736	3	13	.	.	PUNCT
ejpam-4736	4	1	especially	especially	ADV
ejpam-4736	4	2	,	,	PUNCT
ejpam-4736	4	3	some	some	DET
ejpam-4736	4	4	properties	property	NOUN
ejpam-4736	4	5	of	of	ADP
ejpam-4736	4	6	generalized	generalized	ADJ
ejpam-4736	4	7	δp(λ	δp(λ	NOUN
ejpam-4736	4	8	,	,	PUNCT
ejpam-4736	4	9	s)-closed	s)-close	VERB
ejpam-4736	4	10	sets	set	NOUN
ejpam-4736	4	11	are	be	AUX
ejpam-4736	4	12	discussed	discuss	VERB
ejpam-4736	4	13	.	.	PUNCT
ejpam-4736	5	1	moreover	moreover	ADV
ejpam-4736	5	2	,	,	PUNCT
ejpam-4736	5	3	we	we	PRON
ejpam-4736	5	4	apply	apply	VERB
ejpam-4736	5	5	the	the	DET
ejpam-4736	5	6	notion	notion	NOUN
ejpam-4736	5	7	of	of	ADP
ejpam-4736	5	8	generalized	generalized	ADJ
ejpam-4736	5	9	δp(λ	δp(λ	NOUN
ejpam-4736	5	10	,	,	PUNCT
ejpam-4736	5	11	s)-closed	s)-close	VERB
ejpam-4736	5	12	sets	set	NOUN
ejpam-4736	5	13	to	to	PART
ejpam-4736	5	14	present	present	VERB
ejpam-4736	5	15	and	and	CCONJ
ejpam-4736	5	16	study	study	VERB
ejpam-4736	5	17	new	new	ADJ
ejpam-4736	5	18	classes	class	NOUN
ejpam-4736	5	19	of	of	ADP
ejpam-4736	5	20	spaces	space	NOUN
ejpam-4736	5	21	called	call	VERB
ejpam-4736	5	22	δp(λ	δp(λ	NOUN
ejpam-4736	5	23	,	,	PUNCT
ejpam-4736	5	24	s)-t	s)-t	VERB
ejpam-4736	5	25	1	1	NUM
ejpam-4736	5	26	2	2	NUM
ejpam-4736	5	27	-spaces	-space	NOUN
ejpam-4736	5	28	and	and	CCONJ
ejpam-4736	5	29	δp(λ	δp(λ	NOUN
ejpam-4736	5	30	,	,	PUNCT
ejpam-4736	5	31	s)-normal	s)-normal	ADJ
ejpam-4736	5	32	spaces	space	NOUN
ejpam-4736	5	33	.	.	PUNCT
ejpam-4736	6	1	several	several	ADJ
ejpam-4736	6	2	properties	property	NOUN
ejpam-4736	6	3	and	and	CCONJ
ejpam-4736	6	4	characterizations	characterization	NOUN
ejpam-4736	6	5	concerning	concern	VERB
ejpam-4736	6	6	δp(λ	δp(λ	NOUN
ejpam-4736	6	7	,	,	PUNCT
ejpam-4736	6	8	s)-t	s)-t	VERB
ejpam-4736	6	9	1	1	NUM
ejpam-4736	6	10	2	2	NUM
ejpam-4736	6	11	-spaces	-space	NOUN
ejpam-4736	6	12	and	and	CCONJ
ejpam-4736	6	13	δp(λ	δp(λ	NOUN
ejpam-4736	6	14	,	,	PUNCT
ejpam-4736	6	15	s)-normal	s)-normal	ADJ
ejpam-4736	6	16	spaces	space	NOUN
ejpam-4736	6	17	are	be	AUX
ejpam-4736	6	18	established	establish	VERB
ejpam-4736	6	19	.	.	PUNCT
ejpam-4736	7	1	2020	2020	NUM
ejpam-4736	7	2	mathematics	mathematics	PROPN
ejpam-4736	7	3	subject	subject	NOUN
ejpam-4736	7	4	classifications	classification	NOUN
ejpam-4736	7	5	:	:	PUNCT
ejpam-4736	7	6	54a05	54a05	NUM
ejpam-4736	7	7	,	,	PUNCT
ejpam-4736	7	8	54d10	54d10	NUM
ejpam-4736	7	9	key	key	ADJ
ejpam-4736	7	10	words	word	NOUN
ejpam-4736	7	11	and	and	CCONJ
ejpam-4736	7	12	phrases	phrase	NOUN
ejpam-4736	7	13	:	:	PUNCT
ejpam-4736	7	14	δp(λ	δp(λ	NOUN
ejpam-4736	7	15	,	,	PUNCT
ejpam-4736	7	16	s)-open	s)-open	PUNCT
ejpam-4736	7	17	set	set	VERB
ejpam-4736	7	18	,	,	PUNCT
ejpam-4736	7	19	generalized	generalized	ADJ
ejpam-4736	7	20	δp(λ	δp(λ	NOUN
ejpam-4736	7	21	,	,	PUNCT
ejpam-4736	7	22	s)-closed	s)-close	VERB
ejpam-4736	7	23	set	set	VERB
ejpam-4736	7	24	1	1	NUM
ejpam-4736	7	25	.	.	PUNCT
ejpam-4736	7	26	introduction	introduction	NOUN
ejpam-4736	7	27	in	in	ADP
ejpam-4736	7	28	1970	1970	NUM
ejpam-4736	7	29	,	,	PUNCT
ejpam-4736	7	30	levine	levine	PROPN
ejpam-4736	8	1	[	[	X
ejpam-4736	8	2	11	11	NUM
ejpam-4736	8	3	]	]	PUNCT
ejpam-4736	8	4	introduced	introduce	VERB
ejpam-4736	8	5	the	the	DET
ejpam-4736	8	6	concept	concept	NOUN
ejpam-4736	8	7	of	of	ADP
ejpam-4736	8	8	generalized	generalized	ADJ
ejpam-4736	8	9	closed	close	VERB
ejpam-4736	8	10	sets	set	NOUN
ejpam-4736	8	11	in	in	ADP
ejpam-4736	8	12	topological	topological	ADJ
ejpam-4736	8	13	spaces	space	NOUN
ejpam-4736	8	14	and	and	CCONJ
ejpam-4736	8	15	defined	define	VERB
ejpam-4736	8	16	the	the	DET
ejpam-4736	8	17	notion	notion	NOUN
ejpam-4736	8	18	of	of	ADP
ejpam-4736	8	19	a	a	DET
ejpam-4736	8	20	t	t	NOUN
ejpam-4736	8	21	1	1	NUM
ejpam-4736	8	22	2	2	NUM
ejpam-4736	8	23	-space	-space	NOUN
ejpam-4736	8	24	to	to	PART
ejpam-4736	8	25	be	be	AUX
ejpam-4736	8	26	one	one	NUM
ejpam-4736	8	27	in	in	ADP
ejpam-4736	8	28	which	which	PRON
ejpam-4736	8	29	the	the	DET
ejpam-4736	8	30	closed	closed	ADJ
ejpam-4736	8	31	sets	set	NOUN
ejpam-4736	8	32	and	and	CCONJ
ejpam-4736	8	33	the	the	DET
ejpam-4736	8	34	generalized	generalize	VERB
ejpam-4736	8	35	closed	close	VERB
ejpam-4736	8	36	sets	set	NOUN
ejpam-4736	8	37	coincide	coincide	NOUN
ejpam-4736	8	38	.	.	PUNCT
ejpam-4736	9	1	dunham	dunham	PROPN
ejpam-4736	9	2	and	and	CCONJ
ejpam-4736	9	3	levine	levine	PROPN
ejpam-4736	10	1	[	[	X
ejpam-4736	10	2	9	9	NUM
ejpam-4736	10	3	]	]	PUNCT
ejpam-4736	10	4	investigated	investigate	VERB
ejpam-4736	10	5	the	the	DET
ejpam-4736	10	6	further	further	ADJ
ejpam-4736	10	7	properties	property	NOUN
ejpam-4736	10	8	of	of	ADP
ejpam-4736	10	9	generalized	generalized	ADJ
ejpam-4736	10	10	closed	closed	ADJ
ejpam-4736	10	11	sets	set	NOUN
ejpam-4736	10	12	.	.	PUNCT
ejpam-4736	11	1	the	the	DET
ejpam-4736	11	2	concept	concept	NOUN
ejpam-4736	11	3	of	of	ADP
ejpam-4736	11	4	generalized	generalized	ADJ
ejpam-4736	11	5	closed	closed	ADJ
ejpam-4736	11	6	sets	set	NOUN
ejpam-4736	11	7	has	have	AUX
ejpam-4736	11	8	been	be	AUX
ejpam-4736	11	9	modified	modify	VERB
ejpam-4736	11	10	and	and	CCONJ
ejpam-4736	11	11	studied	study	VERB
ejpam-4736	11	12	by	by	ADP
ejpam-4736	11	13	using	use	VERB
ejpam-4736	11	14	weaker	weak	ADJ
ejpam-4736	11	15	forms	form	NOUN
ejpam-4736	11	16	of	of	ADP
ejpam-4736	11	17	open	open	ADJ
ejpam-4736	11	18	sets	set	NOUN
ejpam-4736	11	19	such	such	ADJ
ejpam-4736	11	20	as	as	ADP
ejpam-4736	11	21	α	α	NOUN
ejpam-4736	11	22	-	-	ADJ
ejpam-4736	11	23	open	open	ADJ
ejpam-4736	11	24	sets	set	NOUN
ejpam-4736	11	25	[	[	X
ejpam-4736	11	26	13	13	NUM
ejpam-4736	11	27	]	]	PUNCT
ejpam-4736	11	28	,	,	PUNCT
ejpam-4736	11	29	semi	semi	ADJ
ejpam-4736	11	30	-	-	ADJ
ejpam-4736	11	31	open	open	ADJ
ejpam-4736	11	32	sets	set	NOUN
ejpam-4736	11	33	[	[	X
ejpam-4736	11	34	10	10	NUM
ejpam-4736	11	35	]	]	PUNCT
ejpam-4736	11	36	,	,	PUNCT
ejpam-4736	11	37	preopen	preopen	ADJ
ejpam-4736	11	38	sets	set	NOUN
ejpam-4736	11	39	[	[	X
ejpam-4736	11	40	12	12	NUM
ejpam-4736	11	41	]	]	PUNCT
ejpam-4736	11	42	and	and	CCONJ
ejpam-4736	11	43	semi	semi	ADJ
ejpam-4736	11	44	-	-	ADJ
ejpam-4736	11	45	preopen	preopen	ADJ
ejpam-4736	11	46	sets	set	NOUN
ejpam-4736	11	47	[	[	X
ejpam-4736	11	48	1	1	NUM
ejpam-4736	11	49	]	]	PUNCT
ejpam-4736	11	50	.	.	PUNCT
ejpam-4736	12	1	levine	levine	PROPN
ejpam-4736	13	1	[	[	X
ejpam-4736	13	2	10	10	NUM
ejpam-4736	13	3	]	]	PUNCT
ejpam-4736	13	4	introduced	introduce	VERB
ejpam-4736	13	5	the	the	DET
ejpam-4736	13	6	concept	concept	NOUN
ejpam-4736	13	7	of	of	ADP
ejpam-4736	13	8	semiopen	semiopen	ADJ
ejpam-4736	13	9	sets	set	NOUN
ejpam-4736	13	10	which	which	PRON
ejpam-4736	13	11	is	be	AUX
ejpam-4736	13	12	weaker	weak	ADJ
ejpam-4736	13	13	than	than	ADP
ejpam-4736	13	14	the	the	DET
ejpam-4736	13	15	concept	concept	NOUN
ejpam-4736	13	16	of	of	ADP
ejpam-4736	13	17	open	open	ADJ
ejpam-4736	13	18	sets	set	NOUN
ejpam-4736	13	19	in	in	ADP
ejpam-4736	13	20	topological	topological	ADJ
ejpam-4736	13	21	spaces	space	NOUN
ejpam-4736	13	22	.	.	PUNCT
ejpam-4736	14	1	veličko	veličko	PROPN
ejpam-4736	15	1	[	[	X
ejpam-4736	15	2	19	19	NUM
ejpam-4736	15	3	]	]	PUNCT
ejpam-4736	15	4	introduced	introduce	VERB
ejpam-4736	15	5	δ	δ	PROPN
ejpam-4736	15	6	-	-	ADJ
ejpam-4736	15	7	open	open	ADJ
ejpam-4736	15	8	sets	set	NOUN
ejpam-4736	15	9	,	,	PUNCT
ejpam-4736	15	10	which	which	PRON
ejpam-4736	15	11	are	be	AUX
ejpam-4736	15	12	stronger	strong	ADJ
ejpam-4736	15	13	than	than	ADP
ejpam-4736	15	14	open	open	ADJ
ejpam-4736	15	15	sets	set	NOUN
ejpam-4736	15	16	.	.	PUNCT
ejpam-4736	16	1	park	park	NOUN
ejpam-4736	16	2	et	et	PROPN
ejpam-4736	16	3	al	al	PROPN
ejpam-4736	16	4	.	.	PUNCT
ejpam-4736	17	1	[	[	X
ejpam-4736	17	2	14	14	NUM
ejpam-4736	17	3	]	]	PUNCT
ejpam-4736	17	4	have	have	AUX
ejpam-4736	17	5	offered	offer	VERB
ejpam-4736	17	6	new	new	ADJ
ejpam-4736	17	7	notion	notion	NOUN
ejpam-4736	17	8	called	call	VERB
ejpam-4736	17	9	δ	δ	PROPN
ejpam-4736	17	10	-	-	PUNCT
ejpam-4736	17	11	semiopen	semiopen	VERB
ejpam-4736	17	12	sets	set	NOUN
ejpam-4736	17	13	which	which	PRON
ejpam-4736	17	14	are	be	AUX
ejpam-4736	17	15	stronger	strong	ADJ
ejpam-4736	17	16	than	than	ADP
ejpam-4736	17	17	semi	semi	ADJ
ejpam-4736	17	18	-	-	ADJ
ejpam-4736	17	19	open	open	ADJ
ejpam-4736	17	20	sets	set	NOUN
ejpam-4736	17	21	but	but	CCONJ
ejpam-4736	17	22	weaker	weak	ADJ
ejpam-4736	17	23	than	than	ADP
ejpam-4736	17	24	δ	δ	NOUN
ejpam-4736	17	25	-	-	ADJ
ejpam-4736	17	26	open	open	ADJ
ejpam-4736	17	27	sets	set	NOUN
ejpam-4736	17	28	and	and	CCONJ
ejpam-4736	17	29	investigated	investigate	VERB
ejpam-4736	17	30	the	the	DET
ejpam-4736	17	31	relationships	relationship	NOUN
ejpam-4736	17	32	between	between	ADP
ejpam-4736	17	33	several	several	ADJ
ejpam-4736	17	34	types	type	NOUN
ejpam-4736	17	35	of	of	ADP
ejpam-4736	17	36	these	these	DET
ejpam-4736	17	37	open	open	ADJ
ejpam-4736	17	38	sets	set	NOUN
ejpam-4736	17	39	.	.	PUNCT
ejpam-4736	18	1	caldas	caldas	PROPN
ejpam-4736	18	2	and	and	CCONJ
ejpam-4736	18	3	dontchev	dontchev	ADJ
ejpam-4736	18	4	[	[	X
ejpam-4736	18	5	4	4	NUM
ejpam-4736	18	6	]	]	PUNCT
ejpam-4736	18	7	introduced	introduce	VERB
ejpam-4736	18	8	and	and	CCONJ
ejpam-4736	18	9	investigated	investigate	VERB
ejpam-4736	18	10	the	the	DET
ejpam-4736	18	11	notions	notion	NOUN
ejpam-4736	18	12	of	of	ADP
ejpam-4736	18	13	λs	λs	NOUN
ejpam-4736	18	14	-	-	PUNCT
ejpam-4736	18	15	sets	set	NOUN
ejpam-4736	18	16	and	and	CCONJ
ejpam-4736	18	17	vs	vs	NOUN
ejpam-4736	18	18	-	-	PUNCT
ejpam-4736	18	19	sets	set	NOUN
ejpam-4736	18	20	in	in	ADP
ejpam-4736	18	21	topological	topological	ADJ
ejpam-4736	18	22	spaces	space	NOUN
ejpam-4736	18	23	.	.	PUNCT
ejpam-4736	19	1	moreover	moreover	ADV
ejpam-4736	19	2	,	,	PUNCT
ejpam-4736	19	3	caldas	caldas	PROPN
ejpam-4736	19	4	et	et	PROPN
ejpam-4736	19	5	al	al	PROPN
ejpam-4736	19	6	.	.	PUNCT
ejpam-4736	20	1	[	[	X
ejpam-4736	20	2	7	7	NUM
ejpam-4736	20	3	]	]	PUNCT
ejpam-4736	20	4	investigated	investigate	VERB
ejpam-4736	20	5	some	some	DET
ejpam-4736	20	6	weak	weak	ADJ
ejpam-4736	20	7	separation	separation	NOUN
ejpam-4736	20	8	axioms	axiom	NOUN
ejpam-4736	20	9	by	by	ADP
ejpam-4736	20	10	utilizing	utilize	VERB
ejpam-4736	20	11	δ	δ	PROPN
ejpam-4736	20	12	-	-	PUNCT
ejpam-4736	20	13	semiopen	semiopen	ADJ
ejpam-4736	20	14	sets	set	NOUN
ejpam-4736	20	15	and	and	CCONJ
ejpam-4736	20	16	the	the	DET
ejpam-4736	20	17	δ	δ	PROPN
ejpam-4736	20	18	-	-	PUNCT
ejpam-4736	20	19	semiclosure	semiclosure	NOUN
ejpam-4736	20	20	operator	operator	NOUN
ejpam-4736	20	21	.	.	PUNCT
ejpam-4736	21	1	caldas	caldas	PROPN
ejpam-4736	21	2	et	et	PROPN
ejpam-4736	21	3	al	al	PROPN
ejpam-4736	21	4	.	.	PUNCT
ejpam-4736	22	1	[	[	X
ejpam-4736	22	2	6	6	NUM
ejpam-4736	22	3	]	]	PUNCT
ejpam-4736	22	4	investigated	investigate	VERB
ejpam-4736	22	5	the	the	DET
ejpam-4736	22	6	notion	notion	NOUN
ejpam-4736	22	7	of	of	ADP
ejpam-4736	22	8	δ	δ	PROPN
ejpam-4736	22	9	-	-	PUNCT
ejpam-4736	22	10	λs	λs	ADV
ejpam-4736	22	11	-	-	PUNCT
ejpam-4736	22	12	semiclosed	semiclose	VERB
ejpam-4736	22	13	sets	set	NOUN
ejpam-4736	22	14	which	which	PRON
ejpam-4736	22	15	is	be	AUX
ejpam-4736	22	16	defined	define	VERB
ejpam-4736	22	17	as	as	ADP
ejpam-4736	22	18	the	the	DET
ejpam-4736	22	19	intersection	intersection	NOUN
ejpam-4736	22	20	of	of	ADP
ejpam-4736	22	21	a	a	DET
ejpam-4736	22	22	δ	δ	PROPN
ejpam-4736	22	23	-	-	PUNCT
ejpam-4736	22	24	λs	λs	NOUN
ejpam-4736	22	25	-	-	PUNCT
ejpam-4736	22	26	set	set	NOUN
ejpam-4736	22	27	and	and	CCONJ
ejpam-4736	22	28	a	a	DET
ejpam-4736	22	29	δ	δ	NOUN
ejpam-4736	22	30	-	-	PUNCT
ejpam-4736	22	31	semiclosed	semiclose	VERB
ejpam-4736	22	32	set	set	NOUN
ejpam-4736	22	33	.	.	PUNCT
ejpam-4736	23	1	mashhour	mashhour	PROPN
ejpam-4736	23	2	et	et	PROPN
ejpam-4736	23	3	al	al	PROPN
ejpam-4736	23	4	.	.	PUNCT
ejpam-4736	24	1	[	[	X
ejpam-4736	24	2	12	12	NUM
ejpam-4736	24	3	]	]	PUNCT
ejpam-4736	24	4	introduced	introduce	VERB
ejpam-4736	24	5	and	and	CCONJ
ejpam-4736	24	6	studied	study	VERB
ejpam-4736	24	7	the	the	DET
ejpam-4736	24	8	concept	concept	NOUN
ejpam-4736	24	9	of	of	ADP
ejpam-4736	24	10	preopen	preopen	ADJ
ejpam-4736	24	11	sets	set	NOUN
ejpam-4736	24	12	.	.	PUNCT
ejpam-4736	25	1	raychaudhuri	raychaudhuri	PROPN
ejpam-4736	25	2	and	and	CCONJ
ejpam-4736	25	3	mukherjee	mukherjee	NOUN
ejpam-4736	26	1	[	[	X
ejpam-4736	26	2	15	15	NUM
ejpam-4736	26	3	]	]	PUNCT
ejpam-4736	26	4	introduced	introduce	VERB
ejpam-4736	26	5	the	the	DET
ejpam-4736	26	6	notions	notion	NOUN
ejpam-4736	26	7	of	of	ADP
ejpam-4736	26	8	δ	δ	PROPN
ejpam-4736	26	9	-	-	PUNCT
ejpam-4736	26	10	preopen	preopen	ADJ
ejpam-4736	26	11	sets	set	NOUN
ejpam-4736	26	12	and	and	CCONJ
ejpam-4736	26	13	δpreclosure	δpreclosure	NOUN
ejpam-4736	26	14	.	.	PUNCT
ejpam-4736	27	1	the	the	DET
ejpam-4736	27	2	class	class	NOUN
ejpam-4736	27	3	of	of	ADP
ejpam-4736	27	4	δ	δ	PROPN
ejpam-4736	27	5	-	-	PUNCT
ejpam-4736	27	6	preopen	preopen	ADJ
ejpam-4736	27	7	sets	set	NOUN
ejpam-4736	27	8	is	be	AUX
ejpam-4736	27	9	larger	large	ADJ
ejpam-4736	27	10	than	than	ADP
ejpam-4736	27	11	that	that	PRON
ejpam-4736	27	12	of	of	ADP
ejpam-4736	27	13	preopen	preopen	ADJ
ejpam-4736	27	14	sets	set	NOUN
ejpam-4736	27	15	.	.	PUNCT
ejpam-4736	28	1	caldas	caldas	PROPN
ejpam-4736	28	2	et	et	PROPN
ejpam-4736	28	3	al	al	PROPN
ejpam-4736	28	4	.	.	PUNCT
ejpam-4736	29	1	[	[	X
ejpam-4736	29	2	5	5	NUM
ejpam-4736	29	3	]	]	PUNCT
ejpam-4736	29	4	∗corresponding	∗corresponde	VERB
ejpam-4736	29	5	author	author	NOUN
ejpam-4736	29	6	.	.	PUNCT
ejpam-4736	30	1	doi	doi	NOUN
ejpam-4736	30	2	:	:	PUNCT
ejpam-4736	30	3	https://doi.org/10.29020/nybg.ejpam.v16i4.4736	https://doi.org/10.29020/nybg.ejpam.v16i4.4736	NOUN
ejpam-4736	30	4	email	email	NOUN
ejpam-4736	30	5	addresses	address	NOUN
ejpam-4736	30	6	:	:	PUNCT
ejpam-4736	30	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4736	30	8	(	(	PUNCT
ejpam-4736	30	9	c.	c.	PROPN
ejpam-4736	30	10	boonpok	boonpok	PROPN
ejpam-4736	30	11	)	)	PUNCT
ejpam-4736	30	12	,	,	PUNCT
ejpam-4736	30	13	napassanan.sri@msu.ac.th	napassanan.sri@msu.ac.th	PRON
ejpam-4736	30	14	(	(	PUNCT
ejpam-4736	30	15	n.	n.	PROPN
ejpam-4736	30	16	srisarakham	srisarakham	PROPN
ejpam-4736	30	17	)	)	PUNCT
ejpam-4736	30	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4736	30	19	2581	2581	NUM
ejpam-4736	31	1	©	©	ADP
ejpam-4736	31	2	2023	2023	NUM
ejpam-4736	31	3	ejpam	ejpam	NOUN
ejpam-4736	31	4	all	all	DET
ejpam-4736	31	5	rights	right	NOUN
ejpam-4736	31	6	reserved	reserve	VERB
ejpam-4736	31	7	.	.	PUNCT
ejpam-4736	32	1	c.	c.	PROPN
ejpam-4736	32	2	boonpok	boonpok	PROPN
ejpam-4736	32	3	,	,	PUNCT
ejpam-4736	32	4	n.	n.	PROPN
ejpam-4736	32	5	srisarakham	srisarakham	PROPN
ejpam-4736	32	6	/	/	SYM
ejpam-4736	32	7	eur	eur	PROPN
ejpam-4736	32	8	.	.	PUNCT
ejpam-4736	33	1	j.	j.	PROPN
ejpam-4736	33	2	pure	pure	PROPN
ejpam-4736	33	3	appl	appl	PROPN
ejpam-4736	33	4	.	.	PROPN
ejpam-4736	33	5	math	math	PROPN
ejpam-4736	33	6	,	,	PUNCT
ejpam-4736	33	7	16	16	NUM
ejpam-4736	33	8	(	(	PUNCT
ejpam-4736	33	9	4	4	NUM
ejpam-4736	33	10	)	)	PUNCT
ejpam-4736	33	11	(	(	PUNCT
ejpam-4736	33	12	2023	2023	NUM
ejpam-4736	33	13	)	)	PUNCT
ejpam-4736	33	14	,	,	PUNCT
ejpam-4736	33	15	2581	2581	NUM
ejpam-4736	33	16	-	-	SYM
ejpam-4736	33	17	2596	2596	NUM
ejpam-4736	33	18	2582	2582	NUM
ejpam-4736	33	19	introduced	introduce	VERB
ejpam-4736	33	20	some	some	DET
ejpam-4736	33	21	weak	weak	ADJ
ejpam-4736	33	22	separation	separation	NOUN
ejpam-4736	33	23	axioms	axiom	NOUN
ejpam-4736	33	24	by	by	ADP
ejpam-4736	33	25	utilizing	utilize	VERB
ejpam-4736	33	26	the	the	DET
ejpam-4736	33	27	notions	notion	NOUN
ejpam-4736	33	28	of	of	ADP
ejpam-4736	33	29	δ	δ	PROPN
ejpam-4736	33	30	-	-	PUNCT
ejpam-4736	33	31	preopen	preopen	ADJ
ejpam-4736	33	32	sets	set	NOUN
ejpam-4736	33	33	and	and	CCONJ
ejpam-4736	33	34	the	the	DET
ejpam-4736	33	35	δ	δ	NOUN
ejpam-4736	33	36	-	-	PUNCT
ejpam-4736	33	37	preclosure	preclosure	ADJ
ejpam-4736	33	38	operator	operator	NOUN
ejpam-4736	33	39	.	.	PUNCT
ejpam-4736	34	1	buadong	buadong	PROPN
ejpam-4736	34	2	et	et	PROPN
ejpam-4736	34	3	al	al	PROPN
ejpam-4736	34	4	.	.	PUNCT
ejpam-4736	35	1	[	[	X
ejpam-4736	35	2	2	2	X
ejpam-4736	35	3	]	]	PUNCT
ejpam-4736	35	4	introduced	introduce	VERB
ejpam-4736	35	5	and	and	CCONJ
ejpam-4736	35	6	studied	study	VERB
ejpam-4736	35	7	some	some	DET
ejpam-4736	35	8	separation	separation	NOUN
ejpam-4736	35	9	axioms	axiom	NOUN
ejpam-4736	35	10	in	in	ADP
ejpam-4736	35	11	generalized	generalized	ADJ
ejpam-4736	35	12	topology	topology	NOUN
ejpam-4736	35	13	and	and	CCONJ
ejpam-4736	35	14	minimal	minimal	ADJ
ejpam-4736	35	15	structure	structure	NOUN
ejpam-4736	35	16	spaces	space	NOUN
ejpam-4736	35	17	.	.	PUNCT
ejpam-4736	36	1	dungthaisong	dungthaisong	NOUN
ejpam-4736	36	2	et	et	PROPN
ejpam-4736	36	3	al	al	PROPN
ejpam-4736	36	4	.	.	PUNCT
ejpam-4736	37	1	[	[	X
ejpam-4736	37	2	8	8	NUM
ejpam-4736	37	3	]	]	PUNCT
ejpam-4736	37	4	investigated	investigate	VERB
ejpam-4736	37	5	some	some	DET
ejpam-4736	37	6	properties	property	NOUN
ejpam-4736	37	7	of	of	ADP
ejpam-4736	37	8	pairwise	pairwise	NOUN
ejpam-4736	37	9	µ-t	µ-t	PROPN
ejpam-4736	37	10	1	1	NUM
ejpam-4736	37	11	2	2	NUM
ejpam-4736	37	12	-spaces	-space	NOUN
ejpam-4736	37	13	.	.	PUNCT
ejpam-4736	38	1	torton	torton	PROPN
ejpam-4736	38	2	et	et	PROPN
ejpam-4736	38	3	al	al	PROPN
ejpam-4736	38	4	.	.	PUNCT
ejpam-4736	39	1	[	[	X
ejpam-4736	39	2	18	18	NUM
ejpam-4736	39	3	]	]	PUNCT
ejpam-4736	39	4	introduced	introduce	VERB
ejpam-4736	39	5	and	and	CCONJ
ejpam-4736	39	6	studied	study	VERB
ejpam-4736	39	7	the	the	DET
ejpam-4736	39	8	notions	notion	NOUN
ejpam-4736	39	9	of	of	ADP
ejpam-4736	39	10	µ(m	µ(m	NOUN
ejpam-4736	39	11	,	,	PUNCT
ejpam-4736	39	12	n)-regular	n)-regular	ADJ
ejpam-4736	39	13	spaces	space	NOUN
ejpam-4736	39	14	and	and	CCONJ
ejpam-4736	39	15	µ(m	µ(m	NOUN
ejpam-4736	39	16	,	,	PUNCT
ejpam-4736	39	17	n)-normal	n)-normal	ADJ
ejpam-4736	39	18	spaces	space	NOUN
ejpam-4736	39	19	.	.	PUNCT
ejpam-4736	40	1	viriyapong	viriyapong	PROPN
ejpam-4736	40	2	and	and	CCONJ
ejpam-4736	40	3	boonpok	boonpok	VERB
ejpam-4736	40	4	[	[	X
ejpam-4736	40	5	20	20	NUM
ejpam-4736	40	6	]	]	PUNCT
ejpam-4736	40	7	defined	define	VERB
ejpam-4736	40	8	and	and	CCONJ
ejpam-4736	40	9	investigated	investigate	VERB
ejpam-4736	40	10	the	the	DET
ejpam-4736	40	11	notion	notion	NOUN
ejpam-4736	40	12	of	of	ADP
ejpam-4736	40	13	generalized	generalized	ADJ
ejpam-4736	40	14	(	(	PUNCT
ejpam-4736	40	15	λ	λ	PROPN
ejpam-4736	40	16	,	,	PUNCT
ejpam-4736	40	17	p)-closed	p)-close	VERB
ejpam-4736	40	18	sets	set	NOUN
ejpam-4736	40	19	in	in	ADP
ejpam-4736	40	20	topological	topological	ADJ
ejpam-4736	40	21	spaces	space	NOUN
ejpam-4736	40	22	.	.	PUNCT
ejpam-4736	41	1	in	in	ADP
ejpam-4736	41	2	[	[	X
ejpam-4736	41	3	3	3	NUM
ejpam-4736	41	4	]	]	PUNCT
ejpam-4736	41	5	,	,	PUNCT
ejpam-4736	41	6	the	the	DET
ejpam-4736	41	7	present	present	ADJ
ejpam-4736	41	8	authors	author	NOUN
ejpam-4736	41	9	introduced	introduce	VERB
ejpam-4736	41	10	and	and	CCONJ
ejpam-4736	41	11	investigated	investigate	VERB
ejpam-4736	41	12	the	the	DET
ejpam-4736	41	13	concept	concept	NOUN
ejpam-4736	41	14	of	of	ADP
ejpam-4736	41	15	(	(	PUNCT
ejpam-4736	41	16	λ	λ	PROPN
ejpam-4736	41	17	,	,	PUNCT
ejpam-4736	41	18	s)-closed	s)-close	VERB
ejpam-4736	41	19	sets	set	NOUN
ejpam-4736	41	20	by	by	ADP
ejpam-4736	41	21	utilizing	utilize	VERB
ejpam-4736	41	22	the	the	DET
ejpam-4736	41	23	notions	notion	NOUN
ejpam-4736	41	24	of	of	ADP
ejpam-4736	41	25	λs	λs	NOUN
ejpam-4736	41	26	-	-	PUNCT
ejpam-4736	41	27	sets	set	NOUN
ejpam-4736	41	28	and	and	CCONJ
ejpam-4736	41	29	semi	semi	ADJ
ejpam-4736	41	30	-	-	ADJ
ejpam-4736	41	31	closed	closed	ADJ
ejpam-4736	41	32	sets	set	NOUN
ejpam-4736	41	33	.	.	PUNCT
ejpam-4736	42	1	in	in	ADP
ejpam-4736	42	2	this	this	DET
ejpam-4736	42	3	paper	paper	NOUN
ejpam-4736	42	4	,	,	PUNCT
ejpam-4736	42	5	we	we	PRON
ejpam-4736	42	6	introduce	introduce	VERB
ejpam-4736	42	7	the	the	DET
ejpam-4736	42	8	concept	concept	NOUN
ejpam-4736	42	9	of	of	ADP
ejpam-4736	42	10	generalized	generalized	ADJ
ejpam-4736	42	11	δp(λ	δp(λ	NOUN
ejpam-4736	42	12	,	,	PUNCT
ejpam-4736	42	13	s)-closed	s)-close	VERB
ejpam-4736	42	14	sets	set	NOUN
ejpam-4736	42	15	.	.	PUNCT
ejpam-4736	43	1	moreover	moreover	ADV
ejpam-4736	43	2	,	,	PUNCT
ejpam-4736	43	3	some	some	DET
ejpam-4736	43	4	properties	property	NOUN
ejpam-4736	43	5	of	of	ADP
ejpam-4736	43	6	generalized	generalized	ADJ
ejpam-4736	43	7	δp(λ	δp(λ	NOUN
ejpam-4736	43	8	,	,	PUNCT
ejpam-4736	43	9	s)-closed	s)-close	VERB
ejpam-4736	43	10	sets	set	NOUN
ejpam-4736	43	11	are	be	AUX
ejpam-4736	43	12	discussed	discuss	VERB
ejpam-4736	43	13	.	.	PUNCT
ejpam-4736	44	1	in	in	ADP
ejpam-4736	44	2	particular	particular	ADJ
ejpam-4736	44	3	,	,	PUNCT
ejpam-4736	44	4	we	we	PRON
ejpam-4736	44	5	give	give	VERB
ejpam-4736	44	6	several	several	ADJ
ejpam-4736	44	7	characterizations	characterization	NOUN
ejpam-4736	44	8	of	of	ADP
ejpam-4736	44	9	δp(λ	δp(λ	NOUN
ejpam-4736	44	10	,	,	PUNCT
ejpam-4736	44	11	s)-t	s)-t	VERB
ejpam-4736	44	12	1	1	NUM
ejpam-4736	44	13	2	2	NUM
ejpam-4736	44	14	-spaces	-space	NOUN
ejpam-4736	44	15	and	and	CCONJ
ejpam-4736	44	16	δp(λ	δp(λ	NOUN
ejpam-4736	44	17	,	,	PUNCT
ejpam-4736	44	18	s)-normal	s)-normal	ADJ
ejpam-4736	44	19	spaces	space	NOUN
ejpam-4736	44	20	by	by	ADP
ejpam-4736	44	21	utilizing	utilize	VERB
ejpam-4736	44	22	the	the	DET
ejpam-4736	44	23	concept	concept	NOUN
ejpam-4736	44	24	of	of	ADP
ejpam-4736	44	25	generalized	generalized	ADJ
ejpam-4736	44	26	δp(λ	δp(λ	NOUN
ejpam-4736	44	27	,	,	PUNCT
ejpam-4736	44	28	s)-closed	s)-close	VERB
ejpam-4736	44	29	sets	set	NOUN
ejpam-4736	44	30	.	.	PUNCT
ejpam-4736	45	1	2	2	X
ejpam-4736	45	2	.	.	NUM
ejpam-4736	45	3	preliminaries	preliminary	NOUN
ejpam-4736	45	4	throughout	throughout	ADP
ejpam-4736	45	5	the	the	DET
ejpam-4736	45	6	present	present	ADJ
ejpam-4736	45	7	paper	paper	NOUN
ejpam-4736	45	8	,	,	PUNCT
ejpam-4736	45	9	spaces	space	NOUN
ejpam-4736	45	10	(	(	PUNCT
ejpam-4736	45	11	x	x	X
ejpam-4736	45	12	,	,	PUNCT
ejpam-4736	45	13	τ	τ	X
ejpam-4736	45	14	)	)	PUNCT
ejpam-4736	45	15	and	and	CCONJ
ejpam-4736	45	16	(	(	PUNCT
ejpam-4736	45	17	y	y	PROPN
ejpam-4736	45	18	,	,	PUNCT
ejpam-4736	45	19	σ	σ	PROPN
ejpam-4736	45	20	)	)	PUNCT
ejpam-4736	45	21	(	(	PUNCT
ejpam-4736	45	22	or	or	CCONJ
ejpam-4736	45	23	simply	simply	ADV
ejpam-4736	45	24	x	x	X
ejpam-4736	45	25	and	and	CCONJ
ejpam-4736	45	26	y	y	PROPN
ejpam-4736	45	27	)	)	PUNCT
ejpam-4736	45	28	always	always	ADV
ejpam-4736	45	29	mean	mean	VERB
ejpam-4736	45	30	topological	topological	ADJ
ejpam-4736	45	31	spaces	space	NOUN
ejpam-4736	45	32	on	on	ADP
ejpam-4736	45	33	which	which	PRON
ejpam-4736	45	34	no	no	DET
ejpam-4736	45	35	separation	separation	NOUN
ejpam-4736	45	36	axioms	axiom	NOUN
ejpam-4736	45	37	are	be	AUX
ejpam-4736	45	38	assumed	assume	VERB
ejpam-4736	45	39	unless	unless	SCONJ
ejpam-4736	45	40	explicitly	explicitly	ADV
ejpam-4736	45	41	stated	state	VERB
ejpam-4736	45	42	.	.	PUNCT
ejpam-4736	46	1	let	let	VERB
ejpam-4736	46	2	a	a	DET
ejpam-4736	46	3	be	be	AUX
ejpam-4736	46	4	a	a	DET
ejpam-4736	46	5	subset	subset	NOUN
ejpam-4736	46	6	of	of	ADP
ejpam-4736	46	7	a	a	DET
ejpam-4736	46	8	topological	topological	ADJ
ejpam-4736	46	9	space	space	NOUN
ejpam-4736	46	10	(	(	PUNCT
ejpam-4736	46	11	x	x	X
ejpam-4736	46	12	,	,	PUNCT
ejpam-4736	46	13	τ	τ	PROPN
ejpam-4736	46	14	)	)	PUNCT
ejpam-4736	46	15	.	.	PUNCT
ejpam-4736	47	1	the	the	DET
ejpam-4736	47	2	closure	closure	NOUN
ejpam-4736	47	3	of	of	ADP
ejpam-4736	47	4	a	a	PRON
ejpam-4736	47	5	and	and	CCONJ
ejpam-4736	47	6	the	the	DET
ejpam-4736	47	7	interior	interior	NOUN
ejpam-4736	47	8	of	of	ADP
ejpam-4736	47	9	a	a	PRON
ejpam-4736	47	10	are	be	AUX
ejpam-4736	47	11	denoted	denote	VERB
ejpam-4736	47	12	by	by	ADP
ejpam-4736	47	13	cl(a	cl(a	NOUN
ejpam-4736	47	14	)	)	PUNCT
ejpam-4736	47	15	and	and	CCONJ
ejpam-4736	47	16	int(a	int(a	PROPN
ejpam-4736	47	17	)	)	PUNCT
ejpam-4736	47	18	,	,	PUNCT
ejpam-4736	47	19	respectively	respectively	ADV
ejpam-4736	47	20	.	.	PUNCT
ejpam-4736	48	1	a	a	DET
ejpam-4736	48	2	subset	subset	NOUN
ejpam-4736	48	3	a	a	PRON
ejpam-4736	48	4	of	of	ADP
ejpam-4736	48	5	a	a	DET
ejpam-4736	48	6	topological	topological	ADJ
ejpam-4736	48	7	space	space	NOUN
ejpam-4736	48	8	(	(	PUNCT
ejpam-4736	48	9	x	x	X
ejpam-4736	48	10	,	,	PUNCT
ejpam-4736	48	11	τ	τ	X
ejpam-4736	48	12	)	)	PUNCT
ejpam-4736	48	13	is	be	AUX
ejpam-4736	48	14	called	call	VERB
ejpam-4736	48	15	semi	semi	ADJ
ejpam-4736	48	16	-	-	ADJ
ejpam-4736	48	17	open	open	ADJ
ejpam-4736	48	18	[	[	X
ejpam-4736	48	19	10	10	NUM
ejpam-4736	48	20	]	]	X
ejpam-4736	48	21	if	if	SCONJ
ejpam-4736	48	22	a	a	DET
ejpam-4736	48	23	⊆	⊆	NUM
ejpam-4736	48	24	cl(int(a	cl(int(a	NOUN
ejpam-4736	48	25	)	)	PUNCT
ejpam-4736	48	26	)	)	PUNCT
ejpam-4736	48	27	.	.	PUNCT
ejpam-4736	49	1	the	the	DET
ejpam-4736	49	2	complement	complement	NOUN
ejpam-4736	49	3	of	of	ADP
ejpam-4736	49	4	a	a	DET
ejpam-4736	49	5	semi	semi	ADJ
ejpam-4736	49	6	-	-	ADJ
ejpam-4736	49	7	open	open	ADJ
ejpam-4736	49	8	set	set	NOUN
ejpam-4736	49	9	is	be	AUX
ejpam-4736	49	10	called	call	VERB
ejpam-4736	49	11	semiclosed	semiclose	VERB
ejpam-4736	49	12	.	.	PUNCT
ejpam-4736	50	1	the	the	DET
ejpam-4736	50	2	family	family	NOUN
ejpam-4736	50	3	of	of	ADP
ejpam-4736	50	4	all	all	PRON
ejpam-4736	50	5	semi	semi	ADJ
ejpam-4736	50	6	-	-	ADJ
ejpam-4736	50	7	open	open	ADJ
ejpam-4736	50	8	(	(	PUNCT
ejpam-4736	50	9	resp	resp	NOUN
ejpam-4736	50	10	.	.	PUNCT
ejpam-4736	51	1	semi	semi	ADJ
ejpam-4736	51	2	-	-	ADJ
ejpam-4736	51	3	closed	closed	ADJ
ejpam-4736	51	4	)	)	PUNCT
ejpam-4736	51	5	sets	set	NOUN
ejpam-4736	51	6	in	in	ADP
ejpam-4736	51	7	a	a	DET
ejpam-4736	51	8	topological	topological	ADJ
ejpam-4736	51	9	space	space	NOUN
ejpam-4736	51	10	(	(	PUNCT
ejpam-4736	51	11	x	x	X
ejpam-4736	51	12	,	,	PUNCT
ejpam-4736	51	13	τ	τ	X
ejpam-4736	51	14	)	)	PUNCT
ejpam-4736	51	15	is	be	AUX
ejpam-4736	51	16	denoted	denote	VERB
ejpam-4736	51	17	by	by	ADP
ejpam-4736	51	18	so(x	so(x	NOUN
ejpam-4736	51	19	,	,	PUNCT
ejpam-4736	51	20	τ	τ	X
ejpam-4736	51	21	)	)	PUNCT
ejpam-4736	51	22	(	(	PUNCT
ejpam-4736	51	23	resp	resp	NOUN
ejpam-4736	51	24	.	.	PUNCT
ejpam-4736	52	1	sc(x	sc(x	PROPN
ejpam-4736	52	2	,	,	PUNCT
ejpam-4736	52	3	τ	τ	PROPN
ejpam-4736	52	4	)	)	PUNCT
ejpam-4736	52	5	)	)	PUNCT
ejpam-4736	52	6	.	.	PUNCT
ejpam-4736	53	1	a	a	DET
ejpam-4736	53	2	subset	subset	NOUN
ejpam-4736	53	3	aλs	aλs	NOUN
ejpam-4736	54	1	[	[	X
ejpam-4736	54	2	4	4	NUM
ejpam-4736	54	3	]	]	X
ejpam-4736	54	4	(	(	PUNCT
ejpam-4736	54	5	resp	resp	NOUN
ejpam-4736	54	6	.	.	PUNCT
ejpam-4736	54	7	avs	avs	PROPN
ejpam-4736	54	8	)	)	PUNCT
ejpam-4736	54	9	is	be	AUX
ejpam-4736	54	10	defined	define	VERB
ejpam-4736	54	11	as	as	SCONJ
ejpam-4736	54	12	follows	follow	VERB
ejpam-4736	54	13	:	:	PUNCT
ejpam-4736	54	14	aλs	aλs	PROPN
ejpam-4736	55	1	=	=	SYM
ejpam-4736	55	2	∩{u	∩{u	PROPN
ejpam-4736	56	1	|	|	ADV
ejpam-4736	56	2	u	u	PROPN
ejpam-4736	56	3	⊇	⊇	PROPN
ejpam-4736	56	4	a	a	PROPN
ejpam-4736	56	5	,	,	PUNCT
ejpam-4736	56	6	u	u	PROPN
ejpam-4736	56	7	∈	∈	PROPN
ejpam-4736	56	8	so(x	so(x	NOUN
ejpam-4736	56	9	,	,	PUNCT
ejpam-4736	56	10	τ	τ	PROPN
ejpam-4736	56	11	)	)	PUNCT
ejpam-4736	56	12	}	}	PUNCT
ejpam-4736	56	13	(	(	PUNCT
ejpam-4736	56	14	resp	resp	NOUN
ejpam-4736	56	15	.	.	PUNCT
ejpam-4736	57	1	avs	avs	PROPN
ejpam-4736	58	1	=	=	PROPN
ejpam-4736	58	2	∪{f	∪{f	PROPN
ejpam-4736	58	3	|	|	ADV
ejpam-4736	58	4	f	f	PROPN
ejpam-4736	59	1	⊆	⊆	NUM
ejpam-4736	59	2	a	a	PRON
ejpam-4736	59	3	,	,	PUNCT
ejpam-4736	59	4	f	f	PROPN
ejpam-4736	59	5	∈	∈	PROPN
ejpam-4736	59	6	sc(x	sc(x	PROPN
ejpam-4736	59	7	,	,	PUNCT
ejpam-4736	59	8	τ	τ	NOUN
ejpam-4736	59	9	)	)	PUNCT
ejpam-4736	59	10	}	}	PUNCT
ejpam-4736	59	11	)	)	PUNCT
ejpam-4736	59	12	.	.	PUNCT
ejpam-4736	60	1	a	a	DET
ejpam-4736	60	2	subset	subset	NOUN
ejpam-4736	60	3	a	a	PRON
ejpam-4736	60	4	of	of	ADP
ejpam-4736	60	5	a	a	DET
ejpam-4736	60	6	topological	topological	ADJ
ejpam-4736	60	7	space	space	NOUN
ejpam-4736	60	8	(	(	PUNCT
ejpam-4736	60	9	x	x	X
ejpam-4736	60	10	,	,	PUNCT
ejpam-4736	60	11	τ	τ	X
ejpam-4736	60	12	)	)	PUNCT
ejpam-4736	60	13	is	be	AUX
ejpam-4736	60	14	called	call	VERB
ejpam-4736	60	15	a	a	DET
ejpam-4736	60	16	λs	λs	ADV
ejpam-4736	60	17	-	-	PUNCT
ejpam-4736	60	18	set	set	VERB
ejpam-4736	60	19	(	(	PUNCT
ejpam-4736	60	20	resp	resp	NOUN
ejpam-4736	60	21	.	.	PUNCT
ejpam-4736	61	1	vs	vs	ADP
ejpam-4736	61	2	-	-	PUNCT
ejpam-4736	61	3	set	set	NOUN
ejpam-4736	61	4	)	)	PUNCT
ejpam-4736	62	1	[	[	X
ejpam-4736	62	2	4	4	X
ejpam-4736	62	3	]	]	X
ejpam-4736	62	4	if	if	SCONJ
ejpam-4736	62	5	a	a	PRON
ejpam-4736	62	6	=	=	X
ejpam-4736	62	7	aλs	aλs	NOUN
ejpam-4736	62	8	(	(	PUNCT
ejpam-4736	62	9	resp	resp	NOUN
ejpam-4736	62	10	.	.	PUNCT
ejpam-4736	63	1	a	a	DET
ejpam-4736	63	2	=	=	SYM
ejpam-4736	63	3	avs	avs	PROPN
ejpam-4736	63	4	)	)	PUNCT
ejpam-4736	63	5	.	.	PUNCT
ejpam-4736	64	1	a	a	DET
ejpam-4736	64	2	subset	subset	NOUN
ejpam-4736	64	3	a	a	PRON
ejpam-4736	64	4	of	of	ADP
ejpam-4736	64	5	a	a	DET
ejpam-4736	64	6	topological	topological	ADJ
ejpam-4736	64	7	space	space	NOUN
ejpam-4736	64	8	(	(	PUNCT
ejpam-4736	64	9	x	x	X
ejpam-4736	64	10	,	,	PUNCT
ejpam-4736	64	11	τ	τ	X
ejpam-4736	64	12	)	)	PUNCT
ejpam-4736	64	13	is	be	AUX
ejpam-4736	64	14	called	call	VERB
ejpam-4736	64	15	(	(	PUNCT
ejpam-4736	64	16	λ	λ	X
ejpam-4736	64	17	,	,	PUNCT
ejpam-4736	64	18	s)-closed	s)-close	VERB
ejpam-4736	64	19	[	[	X
ejpam-4736	64	20	3	3	X
ejpam-4736	64	21	]	]	PUNCT
ejpam-4736	64	22	if	if	SCONJ
ejpam-4736	64	23	a	a	DET
ejpam-4736	64	24	=	=	X
ejpam-4736	64	25	t	t	NOUN
ejpam-4736	64	26	∩c	∩c	NOUN
ejpam-4736	64	27	,	,	PUNCT
ejpam-4736	64	28	where	where	SCONJ
ejpam-4736	64	29	t	t	PROPN
ejpam-4736	64	30	is	be	AUX
ejpam-4736	64	31	a	a	DET
ejpam-4736	64	32	λs	λs	ADV
ejpam-4736	64	33	-	-	PUNCT
ejpam-4736	64	34	set	set	VERB
ejpam-4736	64	35	and	and	CCONJ
ejpam-4736	64	36	c	c	NOUN
ejpam-4736	64	37	is	be	AUX
ejpam-4736	64	38	a	a	DET
ejpam-4736	64	39	semi	semi	ADJ
ejpam-4736	64	40	-	-	ADJ
ejpam-4736	64	41	closed	closed	ADJ
ejpam-4736	64	42	set	set	NOUN
ejpam-4736	64	43	.	.	PUNCT
ejpam-4736	65	1	the	the	DET
ejpam-4736	65	2	complement	complement	NOUN
ejpam-4736	65	3	of	of	ADP
ejpam-4736	65	4	a	a	DET
ejpam-4736	65	5	(	(	PUNCT
ejpam-4736	65	6	λ	λ	PROPN
ejpam-4736	65	7	,	,	PUNCT
ejpam-4736	65	8	s)-closed	s)-close	VERB
ejpam-4736	65	9	set	set	NOUN
ejpam-4736	65	10	is	be	AUX
ejpam-4736	65	11	called	call	VERB
ejpam-4736	65	12	(	(	PUNCT
ejpam-4736	65	13	λ	λ	X
ejpam-4736	65	14	,	,	PUNCT
ejpam-4736	65	15	s)-open	s)-open	VERB
ejpam-4736	65	16	.	.	PUNCT
ejpam-4736	66	1	the	the	DET
ejpam-4736	66	2	family	family	NOUN
ejpam-4736	66	3	of	of	ADP
ejpam-4736	66	4	all	all	PRON
ejpam-4736	66	5	(	(	PUNCT
ejpam-4736	66	6	λ	λ	X
ejpam-4736	66	7	,	,	PUNCT
ejpam-4736	66	8	s)-closed	s)-close	VERB
ejpam-4736	66	9	(	(	PUNCT
ejpam-4736	66	10	resp	resp	NOUN
ejpam-4736	66	11	.	.	PUNCT
ejpam-4736	67	1	(	(	PUNCT
ejpam-4736	67	2	λ	λ	X
ejpam-4736	67	3	,	,	PUNCT
ejpam-4736	67	4	s)-open	s)-open	PUNCT
ejpam-4736	67	5	)	)	PUNCT
ejpam-4736	67	6	sets	set	NOUN
ejpam-4736	67	7	in	in	ADP
ejpam-4736	67	8	a	a	DET
ejpam-4736	67	9	topological	topological	ADJ
ejpam-4736	67	10	space	space	NOUN
ejpam-4736	67	11	(	(	PUNCT
ejpam-4736	67	12	x	x	X
ejpam-4736	67	13	,	,	PUNCT
ejpam-4736	67	14	τ	τ	X
ejpam-4736	67	15	)	)	PUNCT
ejpam-4736	67	16	is	be	AUX
ejpam-4736	67	17	denoted	denote	VERB
ejpam-4736	67	18	by	by	ADP
ejpam-4736	67	19	λsc(x	λsc(x	PROPN
ejpam-4736	67	20	,	,	PUNCT
ejpam-4736	67	21	τ	τ	PROPN
ejpam-4736	67	22	)	)	PUNCT
ejpam-4736	67	23	(	(	PUNCT
ejpam-4736	67	24	resp	resp	NOUN
ejpam-4736	67	25	.	.	PUNCT
ejpam-4736	68	1	λso(x	λso(x	NUM
ejpam-4736	68	2	,	,	PUNCT
ejpam-4736	68	3	τ	τ	NOUN
ejpam-4736	68	4	)	)	PUNCT
ejpam-4736	68	5	)	)	PUNCT
ejpam-4736	68	6	.	.	PUNCT
ejpam-4736	69	1	let	let	VERB
ejpam-4736	69	2	a	a	DET
ejpam-4736	69	3	be	be	AUX
ejpam-4736	69	4	a	a	DET
ejpam-4736	69	5	subset	subset	NOUN
ejpam-4736	69	6	of	of	ADP
ejpam-4736	69	7	a	a	DET
ejpam-4736	69	8	topological	topological	ADJ
ejpam-4736	69	9	space	space	NOUN
ejpam-4736	69	10	(	(	PUNCT
ejpam-4736	69	11	x	x	X
ejpam-4736	69	12	,	,	PUNCT
ejpam-4736	69	13	τ	τ	PROPN
ejpam-4736	69	14	)	)	PUNCT
ejpam-4736	69	15	.	.	PUNCT
ejpam-4736	70	1	a	a	DET
ejpam-4736	70	2	point	point	NOUN
ejpam-4736	70	3	x	x	X
ejpam-4736	70	4	∈	∈	NOUN
ejpam-4736	70	5	x	x	PUNCT
ejpam-4736	70	6	is	be	AUX
ejpam-4736	70	7	called	call	VERB
ejpam-4736	70	8	a	a	DET
ejpam-4736	70	9	(	(	PUNCT
ejpam-4736	70	10	λ	λ	NOUN
ejpam-4736	70	11	,	,	PUNCT
ejpam-4736	70	12	s)-cluster	s)-cluster	PUNCT
ejpam-4736	70	13	point	point	VERB
ejpam-4736	70	14	[	[	X
ejpam-4736	70	15	3	3	X
ejpam-4736	70	16	]	]	PUNCT
ejpam-4736	70	17	of	of	ADP
ejpam-4736	70	18	a	a	DET
ejpam-4736	70	19	if	if	NOUN
ejpam-4736	70	20	for	for	ADP
ejpam-4736	70	21	every	every	DET
ejpam-4736	70	22	(	(	PUNCT
ejpam-4736	70	23	λ	λ	NOUN
ejpam-4736	70	24	,	,	PUNCT
ejpam-4736	70	25	s)-open	s)-open	VERB
ejpam-4736	70	26	set	set	VERB
ejpam-4736	70	27	u	u	NOUN
ejpam-4736	70	28	of	of	ADP
ejpam-4736	70	29	x	x	PUNCT
ejpam-4736	70	30	containing	contain	VERB
ejpam-4736	70	31	x	x	VERB
ejpam-4736	70	32	we	we	PRON
ejpam-4736	70	33	have	have	VERB
ejpam-4736	70	34	a	a	DET
ejpam-4736	70	35	∩	∩	ADJ
ejpam-4736	70	36	u	u	ADJ
ejpam-4736	70	37	̸=	̸=	PROPN
ejpam-4736	70	38	∅.	∅.	ADP
ejpam-4736	70	39	the	the	DET
ejpam-4736	70	40	set	set	NOUN
ejpam-4736	70	41	of	of	ADP
ejpam-4736	70	42	all	all	DET
ejpam-4736	70	43	(	(	PUNCT
ejpam-4736	70	44	λ	λ	NOUN
ejpam-4736	70	45	,	,	PUNCT
ejpam-4736	71	1	s)-cluster	s)-cluster	PUNCT
ejpam-4736	71	2	points	point	NOUN
ejpam-4736	71	3	of	of	ADP
ejpam-4736	71	4	a	a	PRON
ejpam-4736	71	5	is	be	AUX
ejpam-4736	71	6	called	call	VERB
ejpam-4736	71	7	the	the	DET
ejpam-4736	71	8	(	(	PUNCT
ejpam-4736	71	9	λ	λ	PROPN
ejpam-4736	71	10	,	,	PUNCT
ejpam-4736	71	11	s)-closure	s)-closure	PUNCT
ejpam-4736	71	12	[	[	X
ejpam-4736	71	13	3	3	X
ejpam-4736	71	14	]	]	PUNCT
ejpam-4736	71	15	of	of	ADP
ejpam-4736	71	16	a	a	PRON
ejpam-4736	71	17	and	and	CCONJ
ejpam-4736	71	18	is	be	AUX
ejpam-4736	71	19	denoted	denote	VERB
ejpam-4736	71	20	by	by	ADP
ejpam-4736	71	21	a(λ	a(λ	PROPN
ejpam-4736	71	22	,	,	PUNCT
ejpam-4736	71	23	s	s	PART
ejpam-4736	71	24	)	)	PUNCT
ejpam-4736	71	25	.	.	PUNCT
ejpam-4736	72	1	the	the	DET
ejpam-4736	72	2	union	union	NOUN
ejpam-4736	72	3	of	of	ADP
ejpam-4736	72	4	all	all	DET
ejpam-4736	72	5	(	(	PUNCT
ejpam-4736	72	6	λ	λ	X
ejpam-4736	72	7	,	,	PUNCT
ejpam-4736	72	8	s)-open	s)-open	PUNCT
ejpam-4736	72	9	sets	set	NOUN
ejpam-4736	72	10	contained	contain	VERB
ejpam-4736	72	11	in	in	ADP
ejpam-4736	72	12	a	a	PRON
ejpam-4736	72	13	is	be	AUX
ejpam-4736	72	14	called	call	VERB
ejpam-4736	72	15	the	the	DET
ejpam-4736	72	16	(	(	PUNCT
ejpam-4736	72	17	λ	λ	PROPN
ejpam-4736	72	18	,	,	PUNCT
ejpam-4736	72	19	s)-interior	s)-interior	X
ejpam-4736	72	20	[	[	X
ejpam-4736	72	21	3	3	X
ejpam-4736	72	22	]	]	PUNCT
ejpam-4736	72	23	of	of	ADP
ejpam-4736	72	24	a	a	PRON
ejpam-4736	72	25	and	and	CCONJ
ejpam-4736	72	26	is	be	AUX
ejpam-4736	72	27	denoted	denote	VERB
ejpam-4736	72	28	by	by	ADP
ejpam-4736	72	29	a(λ	a(λ	PROPN
ejpam-4736	72	30	,	,	PUNCT
ejpam-4736	72	31	s	s	PART
ejpam-4736	72	32	)	)	PUNCT
ejpam-4736	72	33	.	.	PUNCT
ejpam-4736	73	1	let	let	VERB
ejpam-4736	73	2	a	a	DET
ejpam-4736	73	3	be	be	AUX
ejpam-4736	73	4	a	a	DET
ejpam-4736	73	5	subset	subset	NOUN
ejpam-4736	73	6	of	of	ADP
ejpam-4736	73	7	a	a	DET
ejpam-4736	73	8	topological	topological	ADJ
ejpam-4736	73	9	space	space	NOUN
ejpam-4736	73	10	(	(	PUNCT
ejpam-4736	73	11	x	x	X
ejpam-4736	73	12	,	,	PUNCT
ejpam-4736	73	13	τ	τ	PROPN
ejpam-4736	73	14	)	)	PUNCT
ejpam-4736	73	15	.	.	PUNCT
ejpam-4736	74	1	a	a	DET
ejpam-4736	74	2	point	point	NOUN
ejpam-4736	74	3	x	x	PUNCT
ejpam-4736	74	4	of	of	ADP
ejpam-4736	74	5	x	x	PROPN
ejpam-4736	74	6	is	be	AUX
ejpam-4736	74	7	called	call	VERB
ejpam-4736	74	8	a	a	DET
ejpam-4736	74	9	δ(λ	δ(λ	PROPN
ejpam-4736	74	10	,	,	PUNCT
ejpam-4736	74	11	s)-cluster	s)-cluster	PUNCT
ejpam-4736	74	12	point	point	VERB
ejpam-4736	74	13	[	[	X
ejpam-4736	74	14	16	16	NUM
ejpam-4736	74	15	]	]	PUNCT
ejpam-4736	74	16	of	of	ADP
ejpam-4736	74	17	a	a	DET
ejpam-4736	74	18	if	if	SCONJ
ejpam-4736	74	19	a	a	DET
ejpam-4736	74	20	∩	∩	NOUN
ejpam-4736	74	21	[	[	X
ejpam-4736	74	22	v	v	X
ejpam-4736	74	23	(	(	PUNCT
ejpam-4736	74	24	λ	λ	PROPN
ejpam-4736	74	25	,	,	PUNCT
ejpam-4736	74	26	s)](λ	s)](λ	PROPN
ejpam-4736	74	27	,	,	PUNCT
ejpam-4736	74	28	s	s	PART
ejpam-4736	74	29	)	)	PUNCT
ejpam-4736	74	30	̸=	̸=	NOUN
ejpam-4736	74	31	∅	∅	NOUN
ejpam-4736	74	32	for	for	ADP
ejpam-4736	74	33	every	every	DET
ejpam-4736	74	34	(	(	PUNCT
ejpam-4736	74	35	λ	λ	NOUN
ejpam-4736	74	36	,	,	PUNCT
ejpam-4736	74	37	s)-open	s)-open	PUNCT
ejpam-4736	74	38	set	set	VERB
ejpam-4736	74	39	v	v	NUM
ejpam-4736	74	40	of	of	ADP
ejpam-4736	74	41	x	x	PUNCT
ejpam-4736	74	42	containing	contain	VERB
ejpam-4736	74	43	x.	x.	NOUN
ejpam-4736	74	44	the	the	DET
ejpam-4736	74	45	set	set	NOUN
ejpam-4736	74	46	of	of	ADP
ejpam-4736	74	47	all	all	DET
ejpam-4736	74	48	δ(λ	δ(λ	PROPN
ejpam-4736	74	49	,	,	PUNCT
ejpam-4736	74	50	s)-cluster	s)-cluster	PUNCT
ejpam-4736	74	51	points	point	NOUN
ejpam-4736	74	52	of	of	ADP
ejpam-4736	74	53	a	a	PRON
ejpam-4736	74	54	is	be	AUX
ejpam-4736	74	55	called	call	VERB
ejpam-4736	74	56	the	the	DET
ejpam-4736	74	57	δ(λ	δ(λ	PROPN
ejpam-4736	74	58	,	,	PUNCT
ejpam-4736	74	59	s)-closure	s)-closure	PUNCT
ejpam-4736	75	1	[	[	X
ejpam-4736	75	2	16	16	NUM
ejpam-4736	75	3	]	]	PUNCT
ejpam-4736	75	4	of	of	ADP
ejpam-4736	75	5	a	a	PRON
ejpam-4736	75	6	and	and	CCONJ
ejpam-4736	75	7	is	be	AUX
ejpam-4736	75	8	denoted	denote	VERB
ejpam-4736	75	9	by	by	ADP
ejpam-4736	75	10	aδ(λ	aδ(λ	NUM
ejpam-4736	75	11	,	,	PUNCT
ejpam-4736	75	12	s	s	NOUN
ejpam-4736	75	13	)	)	PUNCT
ejpam-4736	75	14	.	.	PUNCT
ejpam-4736	76	1	if	if	SCONJ
ejpam-4736	76	2	a	a	DET
ejpam-4736	76	3	=	=	NOUN
ejpam-4736	76	4	aδ(λ	aδ(λ	NUM
ejpam-4736	76	5	,	,	PUNCT
ejpam-4736	76	6	s	s	PART
ejpam-4736	76	7	)	)	PUNCT
ejpam-4736	76	8	,	,	PUNCT
ejpam-4736	76	9	then	then	ADV
ejpam-4736	76	10	a	a	PRON
ejpam-4736	76	11	is	be	AUX
ejpam-4736	76	12	said	say	VERB
ejpam-4736	76	13	to	to	PART
ejpam-4736	76	14	be	be	AUX
ejpam-4736	76	15	δ(λ	δ(λ	PROPN
ejpam-4736	76	16	,	,	PUNCT
ejpam-4736	76	17	s)-closed	s)-close	VERB
ejpam-4736	76	18	[	[	X
ejpam-4736	76	19	16	16	NUM
ejpam-4736	76	20	]	]	PUNCT
ejpam-4736	76	21	.	.	PUNCT
ejpam-4736	77	1	the	the	DET
ejpam-4736	77	2	complement	complement	NOUN
ejpam-4736	77	3	of	of	ADP
ejpam-4736	77	4	a	a	DET
ejpam-4736	77	5	δ(λ	δ(λ	PROPN
ejpam-4736	77	6	,	,	PUNCT
ejpam-4736	77	7	s)-closed	s)-close	VERB
ejpam-4736	77	8	set	set	NOUN
ejpam-4736	77	9	is	be	AUX
ejpam-4736	77	10	said	say	VERB
ejpam-4736	77	11	to	to	PART
ejpam-4736	77	12	be	be	AUX
ejpam-4736	77	13	δ(λ	δ(λ	PROPN
ejpam-4736	77	14	,	,	PUNCT
ejpam-4736	77	15	s)-open	s)-open	VERB
ejpam-4736	77	16	.	.	PUNCT
ejpam-4736	78	1	the	the	DET
ejpam-4736	78	2	union	union	NOUN
ejpam-4736	78	3	of	of	ADP
ejpam-4736	78	4	all	all	DET
ejpam-4736	78	5	δ(λ	δ(λ	PROPN
ejpam-4736	78	6	,	,	PUNCT
ejpam-4736	78	7	s)-open	s)-open	PUNCT
ejpam-4736	78	8	sets	set	NOUN
ejpam-4736	78	9	contained	contain	VERB
ejpam-4736	78	10	in	in	ADP
ejpam-4736	78	11	a	a	PRON
ejpam-4736	78	12	is	be	AUX
ejpam-4736	78	13	called	call	VERB
ejpam-4736	78	14	the	the	DET
ejpam-4736	78	15	δ(λ	δ(λ	PROPN
ejpam-4736	78	16	,	,	PUNCT
ejpam-4736	78	17	s)-interior	s)-interior	VERB
ejpam-4736	79	1	[	[	X
ejpam-4736	79	2	16	16	NUM
ejpam-4736	79	3	]	]	PUNCT
ejpam-4736	79	4	of	of	ADP
ejpam-4736	79	5	a	a	PRON
ejpam-4736	79	6	and	and	CCONJ
ejpam-4736	79	7	is	be	AUX
ejpam-4736	79	8	denoted	denote	VERB
ejpam-4736	79	9	by	by	ADP
ejpam-4736	79	10	aδ(λ	aδ(λ	NUM
ejpam-4736	79	11	,	,	PUNCT
ejpam-4736	79	12	s	s	NOUN
ejpam-4736	79	13	)	)	PUNCT
ejpam-4736	79	14	.	.	PUNCT
ejpam-4736	80	1	definition	definition	NOUN
ejpam-4736	80	2	1	1	NUM
ejpam-4736	80	3	.	.	PUNCT
ejpam-4736	81	1	[	[	X
ejpam-4736	81	2	16	16	NUM
ejpam-4736	81	3	]	]	PUNCT
ejpam-4736	81	4	a	a	DET
ejpam-4736	81	5	subset	subset	NOUN
ejpam-4736	81	6	a	a	PRON
ejpam-4736	81	7	of	of	ADP
ejpam-4736	81	8	a	a	DET
ejpam-4736	81	9	topological	topological	ADJ
ejpam-4736	81	10	space	space	NOUN
ejpam-4736	81	11	(	(	PUNCT
ejpam-4736	81	12	x	x	X
ejpam-4736	81	13	,	,	PUNCT
ejpam-4736	81	14	τ	τ	X
ejpam-4736	81	15	)	)	PUNCT
ejpam-4736	81	16	is	be	AUX
ejpam-4736	81	17	said	say	VERB
ejpam-4736	81	18	to	to	PART
ejpam-4736	81	19	be	be	AUX
ejpam-4736	81	20	δp(λ	δp(λ	NOUN
ejpam-4736	81	21	,	,	PUNCT
ejpam-4736	81	22	s)-open	s)-open	VERB
ejpam-4736	81	23	if	if	SCONJ
ejpam-4736	81	24	a	a	DET
ejpam-4736	81	25	⊆	⊆	NUM
ejpam-4736	81	26	[	[	X
ejpam-4736	81	27	a(λ	a(λ	ADJ
ejpam-4736	81	28	,	,	PUNCT
ejpam-4736	81	29	s)]δ(λ	s)]δ(λ	ADJ
ejpam-4736	81	30	,	,	PUNCT
ejpam-4736	81	31	s	s	PART
ejpam-4736	81	32	)	)	PUNCT
ejpam-4736	81	33	.	.	PUNCT
ejpam-4736	82	1	the	the	DET
ejpam-4736	82	2	complement	complement	NOUN
ejpam-4736	82	3	of	of	ADP
ejpam-4736	82	4	a	a	DET
ejpam-4736	82	5	δp(λ	δp(λ	NOUN
ejpam-4736	82	6	,	,	PUNCT
ejpam-4736	82	7	s)-open	s)-open	PUNCT
ejpam-4736	82	8	set	set	VERB
ejpam-4736	82	9	is	be	AUX
ejpam-4736	82	10	said	say	VERB
ejpam-4736	82	11	to	to	PART
ejpam-4736	82	12	be	be	AUX
ejpam-4736	82	13	δp(λ	δp(λ	NOUN
ejpam-4736	82	14	,	,	PUNCT
ejpam-4736	82	15	s)-closed	s)-close	VERB
ejpam-4736	82	16	.	.	PUNCT
ejpam-4736	83	1	the	the	DET
ejpam-4736	83	2	family	family	NOUN
ejpam-4736	83	3	of	of	ADP
ejpam-4736	83	4	all	all	DET
ejpam-4736	83	5	δp(λ	δp(λ	NOUN
ejpam-4736	83	6	,	,	PUNCT
ejpam-4736	83	7	s)-open	s)-open	PUNCT
ejpam-4736	83	8	(	(	PUNCT
ejpam-4736	83	9	resp	resp	NOUN
ejpam-4736	83	10	.	.	PUNCT
ejpam-4736	84	1	δp(λ	δp(λ	NOUN
ejpam-4736	84	2	,	,	PUNCT
ejpam-4736	84	3	s)-closed	s)-close	VERB
ejpam-4736	84	4	)	)	PUNCT
ejpam-4736	84	5	sets	set	NOUN
ejpam-4736	84	6	in	in	ADP
ejpam-4736	84	7	a	a	DET
ejpam-4736	84	8	topological	topological	ADJ
ejpam-4736	84	9	space	space	NOUN
ejpam-4736	84	10	(	(	PUNCT
ejpam-4736	84	11	x	x	X
ejpam-4736	84	12	,	,	PUNCT
ejpam-4736	84	13	τ	τ	X
ejpam-4736	84	14	)	)	PUNCT
ejpam-4736	84	15	is	be	AUX
ejpam-4736	84	16	denoted	denote	VERB
ejpam-4736	84	17	by	by	ADP
ejpam-4736	84	18	δp(λ	δp(λ	NOUN
ejpam-4736	84	19	,	,	PUNCT
ejpam-4736	84	20	s)o(x	s)o(x	PROPN
ejpam-4736	84	21	,	,	PUNCT
ejpam-4736	84	22	τ	τ	X
ejpam-4736	84	23	)	)	PUNCT
ejpam-4736	84	24	(	(	PUNCT
ejpam-4736	84	25	resp	resp	NOUN
ejpam-4736	84	26	.	.	PUNCT
ejpam-4736	85	1	δp(λ	δp(λ	PROPN
ejpam-4736	85	2	,	,	PUNCT
ejpam-4736	85	3	s)c(x	s)c(x	NOUN
ejpam-4736	85	4	,	,	PUNCT
ejpam-4736	85	5	τ	τ	PROPN
ejpam-4736	85	6	)	)	PUNCT
ejpam-4736	85	7	)	)	PUNCT
ejpam-4736	85	8	.	.	PUNCT
ejpam-4736	86	1	let	let	VERB
ejpam-4736	86	2	a	a	DET
ejpam-4736	86	3	be	be	AUX
ejpam-4736	86	4	a	a	DET
ejpam-4736	86	5	subset	subset	NOUN
ejpam-4736	86	6	of	of	ADP
ejpam-4736	86	7	a	a	DET
ejpam-4736	86	8	topological	topological	ADJ
ejpam-4736	86	9	c.	c.	PROPN
ejpam-4736	86	10	boonpok	boonpok	PROPN
ejpam-4736	86	11	,	,	PUNCT
ejpam-4736	86	12	n.	n.	PROPN
ejpam-4736	86	13	srisarakham	srisarakham	PROPN
ejpam-4736	86	14	/	/	SYM
ejpam-4736	86	15	eur	eur	PROPN
ejpam-4736	86	16	.	.	PUNCT
ejpam-4736	87	1	j.	j.	PROPN
ejpam-4736	87	2	pure	pure	PROPN
ejpam-4736	87	3	appl	appl	PROPN
ejpam-4736	87	4	.	.	PROPN
ejpam-4736	87	5	math	math	PROPN
ejpam-4736	87	6	,	,	PUNCT
ejpam-4736	87	7	16	16	NUM
ejpam-4736	87	8	(	(	PUNCT
ejpam-4736	87	9	4	4	NUM
ejpam-4736	87	10	)	)	PUNCT
ejpam-4736	87	11	(	(	PUNCT
ejpam-4736	87	12	2023	2023	NUM
ejpam-4736	87	13	)	)	PUNCT
ejpam-4736	87	14	,	,	PUNCT
ejpam-4736	87	15	2581	2581	NUM
ejpam-4736	87	16	-	-	SYM
ejpam-4736	87	17	2596	2596	NUM
ejpam-4736	87	18	2583	2583	NUM
ejpam-4736	87	19	space	space	NOUN
ejpam-4736	87	20	(	(	PUNCT
ejpam-4736	87	21	x	x	X
ejpam-4736	87	22	,	,	PUNCT
ejpam-4736	87	23	τ	τ	PROPN
ejpam-4736	87	24	)	)	PUNCT
ejpam-4736	87	25	.	.	PUNCT
ejpam-4736	88	1	the	the	DET
ejpam-4736	88	2	intersection	intersection	NOUN
ejpam-4736	88	3	of	of	ADP
ejpam-4736	88	4	all	all	DET
ejpam-4736	88	5	δp(λ	δp(λ	NOUN
ejpam-4736	88	6	,	,	PUNCT
ejpam-4736	88	7	s)-closed	s)-close	VERB
ejpam-4736	88	8	sets	set	NOUN
ejpam-4736	88	9	containing	contain	VERB
ejpam-4736	88	10	a	a	PRON
ejpam-4736	88	11	is	be	AUX
ejpam-4736	88	12	called	call	VERB
ejpam-4736	88	13	the	the	DET
ejpam-4736	88	14	δp(λ	δp(λ	NOUN
ejpam-4736	88	15	,	,	PUNCT
ejpam-4736	88	16	s)closure	s)closure	NOUN
ejpam-4736	88	17	of	of	ADP
ejpam-4736	88	18	a	a	PRON
ejpam-4736	88	19	and	and	CCONJ
ejpam-4736	88	20	is	be	AUX
ejpam-4736	88	21	denoted	denote	VERB
ejpam-4736	88	22	by	by	ADP
ejpam-4736	88	23	aδp(λ	aδp(λ	PROPN
ejpam-4736	88	24	,	,	PUNCT
ejpam-4736	88	25	s	s	NOUN
ejpam-4736	88	26	)	)	PUNCT
ejpam-4736	88	27	.	.	PUNCT
ejpam-4736	89	1	lemma	lemma	PROPN
ejpam-4736	89	2	1	1	NUM
ejpam-4736	89	3	.	.	PUNCT
ejpam-4736	90	1	[	[	X
ejpam-4736	90	2	16	16	NUM
ejpam-4736	90	3	]	]	PUNCT
ejpam-4736	90	4	for	for	ADP
ejpam-4736	90	5	the	the	DET
ejpam-4736	90	6	δp(λ	δp(λ	NOUN
ejpam-4736	90	7	,	,	PUNCT
ejpam-4736	90	8	s)-closure	s)-closure	NOUN
ejpam-4736	90	9	of	of	ADP
ejpam-4736	90	10	subsets	subset	NOUN
ejpam-4736	90	11	a	a	PRON
ejpam-4736	90	12	,	,	PUNCT
ejpam-4736	90	13	b	b	NOUN
ejpam-4736	90	14	in	in	ADP
ejpam-4736	90	15	a	a	DET
ejpam-4736	90	16	topological	topological	ADJ
ejpam-4736	90	17	space	space	NOUN
ejpam-4736	90	18	(	(	PUNCT
ejpam-4736	90	19	x	x	X
ejpam-4736	90	20	,	,	PUNCT
ejpam-4736	90	21	τ	τ	PROPN
ejpam-4736	90	22	)	)	PUNCT
ejpam-4736	90	23	,	,	PUNCT
ejpam-4736	90	24	the	the	DET
ejpam-4736	90	25	following	follow	VERB
ejpam-4736	90	26	properties	property	NOUN
ejpam-4736	90	27	hold	hold	VERB
ejpam-4736	90	28	:	:	PUNCT
ejpam-4736	90	29	(	(	PUNCT
ejpam-4736	90	30	1	1	X
ejpam-4736	90	31	)	)	PUNCT
ejpam-4736	90	32	if	if	SCONJ
ejpam-4736	90	33	a	a	DET
ejpam-4736	90	34	⊆	⊆	NUM
ejpam-4736	90	35	b	b	NOUN
ejpam-4736	90	36	,	,	PUNCT
ejpam-4736	90	37	then	then	ADV
ejpam-4736	90	38	aδp(λ	aδp(λ	PROPN
ejpam-4736	90	39	,	,	PUNCT
ejpam-4736	90	40	s	s	PART
ejpam-4736	90	41	)	)	PUNCT
ejpam-4736	90	42	⊆	⊆	NUM
ejpam-4736	90	43	bδp(λ	bδp(λ	PROPN
ejpam-4736	90	44	,	,	PUNCT
ejpam-4736	90	45	s	s	PART
ejpam-4736	90	46	)	)	PUNCT
ejpam-4736	90	47	.	.	PUNCT
ejpam-4736	91	1	(	(	PUNCT
ejpam-4736	91	2	2	2	X
ejpam-4736	91	3	)	)	PUNCT
ejpam-4736	91	4	a	a	PRON
ejpam-4736	91	5	is	is	NOUN
ejpam-4736	91	6	δp(λ	δp(λ	NOUN
ejpam-4736	91	7	,	,	PUNCT
ejpam-4736	91	8	s)-closed	s)-close	VERB
ejpam-4736	91	9	in	in	ADP
ejpam-4736	91	10	(	(	PUNCT
ejpam-4736	91	11	x	x	X
ejpam-4736	91	12	,	,	PUNCT
ejpam-4736	91	13	τ	τ	X
ejpam-4736	91	14	)	)	PUNCT
ejpam-4736	91	15	if	if	SCONJ
ejpam-4736	91	16	and	and	CCONJ
ejpam-4736	91	17	only	only	ADV
ejpam-4736	91	18	if	if	SCONJ
ejpam-4736	91	19	a	a	DET
ejpam-4736	91	20	=	=	X
ejpam-4736	91	21	aδp(λ	aδp(λ	PROPN
ejpam-4736	91	22	,	,	PUNCT
ejpam-4736	91	23	s	s	NOUN
ejpam-4736	91	24	)	)	PUNCT
ejpam-4736	91	25	.	.	PUNCT
ejpam-4736	92	1	(	(	PUNCT
ejpam-4736	92	2	3	3	X
ejpam-4736	92	3	)	)	PUNCT
ejpam-4736	92	4	aδp(λ	aδp(λ	PROPN
ejpam-4736	92	5	,	,	PUNCT
ejpam-4736	92	6	s	s	PART
ejpam-4736	92	7	)	)	PUNCT
ejpam-4736	92	8	is	be	AUX
ejpam-4736	92	9	δp(λ	δp(λ	NOUN
ejpam-4736	92	10	,	,	PUNCT
ejpam-4736	92	11	s)-closed	s)-close	VERB
ejpam-4736	92	12	,	,	PUNCT
ejpam-4736	92	13	that	that	ADV
ejpam-4736	92	14	is	is	ADV
ejpam-4736	92	15	,	,	PUNCT
ejpam-4736	92	16	aδp(λ	aδp(λ	PROPN
ejpam-4736	92	17	,	,	PUNCT
ejpam-4736	92	18	s	s	PART
ejpam-4736	92	19	)	)	PUNCT
ejpam-4736	92	20	=	=	PUNCT
ejpam-4736	93	1	[	[	X
ejpam-4736	93	2	aδp(λ	aδp(λ	PROPN
ejpam-4736	93	3	,	,	PUNCT
ejpam-4736	93	4	s)]δp(λ	s)]δp(λ	NOUN
ejpam-4736	93	5	,	,	PUNCT
ejpam-4736	93	6	s	s	NOUN
ejpam-4736	93	7	)	)	PUNCT
ejpam-4736	93	8	.	.	PUNCT
ejpam-4736	94	1	(	(	PUNCT
ejpam-4736	94	2	4	4	X
ejpam-4736	94	3	)	)	PUNCT
ejpam-4736	94	4	x	x	SYM
ejpam-4736	94	5	∈	∈	PROPN
ejpam-4736	94	6	aδp(λ	aδp(λ	PROPN
ejpam-4736	94	7	,	,	PUNCT
ejpam-4736	94	8	s	s	PART
ejpam-4736	94	9	)	)	PUNCT
ejpam-4736	94	10	if	if	SCONJ
ejpam-4736	94	11	and	and	CCONJ
ejpam-4736	94	12	only	only	ADV
ejpam-4736	94	13	if	if	SCONJ
ejpam-4736	94	14	a	a	DET
ejpam-4736	94	15	∩	∩	NOUN
ejpam-4736	94	16	v	v	ADP
ejpam-4736	94	17	̸=	̸=	PROPN
ejpam-4736	94	18	∅	∅	NOUN
ejpam-4736	94	19	for	for	ADP
ejpam-4736	94	20	every	every	DET
ejpam-4736	94	21	v	v	NOUN
ejpam-4736	94	22	∈	∈	PROPN
ejpam-4736	94	23	δp(λ	δp(λ	NOUN
ejpam-4736	94	24	,	,	PUNCT
ejpam-4736	94	25	s)o(x	s)o(x	PROPN
ejpam-4736	94	26	,	,	PUNCT
ejpam-4736	94	27	τ	τ	X
ejpam-4736	94	28	)	)	PUNCT
ejpam-4736	94	29	containing	contain	VERB
ejpam-4736	94	30	x.	x.	PROPN
ejpam-4736	94	31	lemma	lemma	PROPN
ejpam-4736	94	32	2	2	X
ejpam-4736	94	33	.	.	PUNCT
ejpam-4736	95	1	[	[	X
ejpam-4736	95	2	16	16	NUM
ejpam-4736	95	3	]	]	PUNCT
ejpam-4736	95	4	for	for	ADP
ejpam-4736	95	5	a	a	DET
ejpam-4736	95	6	family	family	NOUN
ejpam-4736	95	7	{	{	PUNCT
ejpam-4736	95	8	aγ	aγ	INTJ
ejpam-4736	95	9	|	|	ADV
ejpam-4736	95	10	γ	γ	X
ejpam-4736	95	11	∈	∈	PROPN
ejpam-4736	95	12	∇	∇	X
ejpam-4736	95	13	}	}	PUNCT
ejpam-4736	95	14	of	of	ADP
ejpam-4736	95	15	a	a	DET
ejpam-4736	95	16	topological	topological	ADJ
ejpam-4736	95	17	space	space	NOUN
ejpam-4736	95	18	(	(	PUNCT
ejpam-4736	95	19	x	x	X
ejpam-4736	95	20	,	,	PUNCT
ejpam-4736	95	21	τ	τ	PROPN
ejpam-4736	95	22	)	)	PUNCT
ejpam-4736	95	23	,	,	PUNCT
ejpam-4736	95	24	the	the	DET
ejpam-4736	95	25	following	follow	VERB
ejpam-4736	95	26	properties	property	NOUN
ejpam-4736	95	27	hold	hold	VERB
ejpam-4736	95	28	:	:	PUNCT
ejpam-4736	95	29	(	(	PUNCT
ejpam-4736	95	30	1	1	X
ejpam-4736	95	31	)	)	PUNCT
ejpam-4736	96	1	[	[	X
ejpam-4736	96	2	∩{aγ	∩{aγ	VERB
ejpam-4736	96	3	|	|	ADV
ejpam-4736	96	4	γ	γ	X
ejpam-4736	96	5	∈	∈	PROPN
ejpam-4736	96	6	∇}]δp(λ	∇}]δp(λ	X
ejpam-4736	96	7	,	,	PUNCT
ejpam-4736	96	8	s	s	PART
ejpam-4736	96	9	)	)	PUNCT
ejpam-4736	96	10	⊆	⊆	NUM
ejpam-4736	96	11	∩{aδp(λ	∩{aδp(λ	NOUN
ejpam-4736	96	12	,	,	PUNCT
ejpam-4736	96	13	s	s	PART
ejpam-4736	96	14	)	)	PUNCT
ejpam-4736	96	15	γ	γ	PROPN
ejpam-4736	96	16	|	|	ADV
ejpam-4736	96	17	γ	γ	X
ejpam-4736	96	18	∈	∈	NOUN
ejpam-4736	96	19	∇	∇	X
ejpam-4736	96	20	}	}	PUNCT
ejpam-4736	96	21	.	.	PUNCT
ejpam-4736	97	1	(	(	PUNCT
ejpam-4736	97	2	2	2	X
ejpam-4736	97	3	)	)	PUNCT
ejpam-4736	97	4	[	[	X
ejpam-4736	97	5	∪{aγ	∪{aγ	PROPN
ejpam-4736	97	6	|	|	ADV
ejpam-4736	97	7	γ	γ	X
ejpam-4736	97	8	∈	∈	PROPN
ejpam-4736	97	9	∇}]δp(λ	∇}]δp(λ	X
ejpam-4736	97	10	,	,	PUNCT
ejpam-4736	97	11	s	s	NOUN
ejpam-4736	97	12	)	)	PUNCT
ejpam-4736	97	13	⊇	⊇	NOUN
ejpam-4736	97	14	∪{aδp(λ	∪{aδp(λ	NOUN
ejpam-4736	97	15	,	,	PUNCT
ejpam-4736	97	16	s	s	PART
ejpam-4736	97	17	)	)	PUNCT
ejpam-4736	97	18	γ	γ	PROPN
ejpam-4736	97	19	|	|	ADV
ejpam-4736	97	20	γ	γ	X
ejpam-4736	97	21	∈	∈	NOUN
ejpam-4736	97	22	∇	∇	X
ejpam-4736	97	23	}	}	PUNCT
ejpam-4736	97	24	.	.	PUNCT
ejpam-4736	98	1	definition	definition	NOUN
ejpam-4736	98	2	2	2	NUM
ejpam-4736	98	3	.	.	PUNCT
ejpam-4736	98	4	let	let	VERB
ejpam-4736	98	5	a	a	DET
ejpam-4736	98	6	be	be	AUX
ejpam-4736	98	7	a	a	DET
ejpam-4736	98	8	subset	subset	NOUN
ejpam-4736	98	9	of	of	ADP
ejpam-4736	98	10	a	a	DET
ejpam-4736	98	11	topological	topological	ADJ
ejpam-4736	98	12	space	space	NOUN
ejpam-4736	98	13	(	(	PUNCT
ejpam-4736	98	14	x	x	X
ejpam-4736	98	15	,	,	PUNCT
ejpam-4736	98	16	τ	τ	PROPN
ejpam-4736	98	17	)	)	PUNCT
ejpam-4736	98	18	.	.	PUNCT
ejpam-4736	99	1	the	the	DET
ejpam-4736	99	2	union	union	NOUN
ejpam-4736	99	3	of	of	ADP
ejpam-4736	99	4	all	all	DET
ejpam-4736	99	5	(	(	PUNCT
ejpam-4736	99	6	λ	λ	NOUN
ejpam-4736	99	7	,	,	PUNCT
ejpam-4736	99	8	sp)-open	sp)-open	ADJ
ejpam-4736	99	9	sets	set	NOUN
ejpam-4736	99	10	contained	contain	VERB
ejpam-4736	99	11	in	in	ADP
ejpam-4736	99	12	a	a	PRON
ejpam-4736	99	13	is	be	AUX
ejpam-4736	99	14	called	call	VERB
ejpam-4736	99	15	the	the	DET
ejpam-4736	99	16	δp(λ	δp(λ	NOUN
ejpam-4736	99	17	,	,	PUNCT
ejpam-4736	99	18	s)-interior	s)-interior	ADJ
ejpam-4736	99	19	of	of	ADP
ejpam-4736	99	20	a	a	PRON
ejpam-4736	99	21	and	and	CCONJ
ejpam-4736	99	22	is	be	AUX
ejpam-4736	99	23	denoted	denote	VERB
ejpam-4736	99	24	by	by	ADP
ejpam-4736	99	25	aδp(λ	aδp(λ	PROPN
ejpam-4736	99	26	,	,	PUNCT
ejpam-4736	99	27	s	s	NOUN
ejpam-4736	99	28	)	)	PUNCT
ejpam-4736	99	29	.	.	PUNCT
ejpam-4736	100	1	lemma	lemma	PROPN
ejpam-4736	100	2	3	3	X
ejpam-4736	100	3	.	.	X
ejpam-4736	101	1	for	for	ADP
ejpam-4736	101	2	subsets	subset	NOUN
ejpam-4736	101	3	a	a	PRON
ejpam-4736	101	4	and	and	CCONJ
ejpam-4736	101	5	b	b	NOUN
ejpam-4736	101	6	of	of	ADP
ejpam-4736	101	7	a	a	DET
ejpam-4736	101	8	topological	topological	ADJ
ejpam-4736	101	9	space	space	NOUN
ejpam-4736	101	10	(	(	PUNCT
ejpam-4736	101	11	x	x	X
ejpam-4736	101	12	,	,	PUNCT
ejpam-4736	101	13	τ	τ	PROPN
ejpam-4736	101	14	)	)	PUNCT
ejpam-4736	101	15	,	,	PUNCT
ejpam-4736	101	16	the	the	DET
ejpam-4736	101	17	following	follow	VERB
ejpam-4736	101	18	properties	property	NOUN
ejpam-4736	101	19	hold	hold	VERB
ejpam-4736	101	20	:	:	PUNCT
ejpam-4736	101	21	(	(	PUNCT
ejpam-4736	101	22	1	1	X
ejpam-4736	101	23	)	)	PUNCT
ejpam-4736	101	24	aδp(λ	aδp(λ	PROPN
ejpam-4736	101	25	,	,	PUNCT
ejpam-4736	101	26	s	s	PART
ejpam-4736	101	27	)	)	PUNCT
ejpam-4736	101	28	⊆	⊆	NUM
ejpam-4736	101	29	a	a	PRON
ejpam-4736	101	30	and	and	CCONJ
ejpam-4736	101	31	[	[	X
ejpam-4736	101	32	aδp(λ	aδp(λ	PROPN
ejpam-4736	101	33	,	,	PUNCT
ejpam-4736	101	34	s)]δp(λ	s)]δp(λ	NOUN
ejpam-4736	101	35	,	,	PUNCT
ejpam-4736	101	36	s	s	NOUN
ejpam-4736	101	37	)	)	PUNCT
ejpam-4736	101	38	=	=	SYM
ejpam-4736	101	39	aδp(λ	aδp(λ	PROPN
ejpam-4736	101	40	,	,	PUNCT
ejpam-4736	101	41	s	s	NOUN
ejpam-4736	101	42	)	)	PUNCT
ejpam-4736	101	43	.	.	PUNCT
ejpam-4736	102	1	(	(	PUNCT
ejpam-4736	102	2	2	2	X
ejpam-4736	102	3	)	)	PUNCT
ejpam-4736	102	4	if	if	SCONJ
ejpam-4736	102	5	a	a	DET
ejpam-4736	102	6	⊆	⊆	NUM
ejpam-4736	102	7	b	b	NOUN
ejpam-4736	102	8	,	,	PUNCT
ejpam-4736	102	9	then	then	ADV
ejpam-4736	102	10	aδp(λ	aδp(λ	PROPN
ejpam-4736	102	11	,	,	PUNCT
ejpam-4736	102	12	s	s	PART
ejpam-4736	102	13	)	)	PUNCT
ejpam-4736	102	14	⊆	⊆	NUM
ejpam-4736	102	15	bδp(λ	bδp(λ	PROPN
ejpam-4736	102	16	,	,	PUNCT
ejpam-4736	102	17	s	s	PART
ejpam-4736	102	18	)	)	PUNCT
ejpam-4736	102	19	.	.	PUNCT
ejpam-4736	103	1	(	(	PUNCT
ejpam-4736	103	2	3	3	X
ejpam-4736	103	3	)	)	PUNCT
ejpam-4736	103	4	aδp(λ	aδp(λ	PROPN
ejpam-4736	103	5	,	,	PUNCT
ejpam-4736	103	6	s	s	PART
ejpam-4736	103	7	)	)	PUNCT
ejpam-4736	103	8	is	be	AUX
ejpam-4736	103	9	δp(λ	δp(λ	NOUN
ejpam-4736	103	10	,	,	PUNCT
ejpam-4736	103	11	s)-open	s)-open	ADJ
ejpam-4736	103	12	.	.	PUNCT
ejpam-4736	104	1	(	(	PUNCT
ejpam-4736	104	2	4	4	X
ejpam-4736	104	3	)	)	PUNCT
ejpam-4736	104	4	a	a	PRON
ejpam-4736	104	5	is	is	NOUN
ejpam-4736	104	6	δp(λ	δp(λ	NOUN
ejpam-4736	104	7	,	,	PUNCT
ejpam-4736	104	8	s)-open	s)-open	VERB
ejpam-4736	104	9	if	if	SCONJ
ejpam-4736	104	10	and	and	CCONJ
ejpam-4736	104	11	only	only	ADV
ejpam-4736	104	12	if	if	SCONJ
ejpam-4736	104	13	aδp(λ	aδp(λ	PROPN
ejpam-4736	104	14	,	,	PUNCT
ejpam-4736	104	15	s	s	PART
ejpam-4736	104	16	)	)	PUNCT
ejpam-4736	104	17	=	=	SYM
ejpam-4736	104	18	a.	a.	NOUN
ejpam-4736	104	19	(	(	PUNCT
ejpam-4736	104	20	5	5	NUM
ejpam-4736	104	21	)	)	PUNCT
ejpam-4736	104	22	[	[	X
ejpam-4736	104	23	x	x	X
ejpam-4736	104	24	−a]δp(λ	−a]δp(λ	PROPN
ejpam-4736	104	25	,	,	PUNCT
ejpam-4736	104	26	s	s	NOUN
ejpam-4736	104	27	)	)	PUNCT
ejpam-4736	104	28	=	=	SYM
ejpam-4736	104	29	x	x	SYM
ejpam-4736	104	30	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	104	31	,	,	PUNCT
ejpam-4736	104	32	s	s	NOUN
ejpam-4736	104	33	)	)	PUNCT
ejpam-4736	104	34	.	.	PUNCT
ejpam-4736	105	1	(	(	PUNCT
ejpam-4736	105	2	6	6	X
ejpam-4736	105	3	)	)	PUNCT
ejpam-4736	106	1	[	[	X
ejpam-4736	106	2	x	x	X
ejpam-4736	106	3	−a]δp(λ	−a]δp(λ	PROPN
ejpam-4736	106	4	,	,	PUNCT
ejpam-4736	106	5	s	s	NOUN
ejpam-4736	106	6	)	)	PUNCT
ejpam-4736	106	7	=	=	SYM
ejpam-4736	106	8	x	x	SYM
ejpam-4736	106	9	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	106	10	,	,	PUNCT
ejpam-4736	106	11	s	s	PART
ejpam-4736	106	12	)	)	PUNCT
ejpam-4736	106	13	.	.	PUNCT
ejpam-4736	107	1	3	3	X
ejpam-4736	107	2	.	.	NUM
ejpam-4736	107	3	generalized	generalize	VERB
ejpam-4736	107	4	δp(λ	δp(λ	NOUN
ejpam-4736	107	5	,	,	PUNCT
ejpam-4736	107	6	s)-closed	s)-close	VERB
ejpam-4736	107	7	sets	set	NOUN
ejpam-4736	107	8	we	we	PRON
ejpam-4736	107	9	begin	begin	VERB
ejpam-4736	107	10	this	this	DET
ejpam-4736	107	11	section	section	NOUN
ejpam-4736	107	12	by	by	ADP
ejpam-4736	107	13	introducing	introduce	VERB
ejpam-4736	107	14	the	the	DET
ejpam-4736	107	15	concept	concept	NOUN
ejpam-4736	107	16	of	of	ADP
ejpam-4736	107	17	generalized	generalized	ADJ
ejpam-4736	107	18	δp(λ	δp(λ	NOUN
ejpam-4736	107	19	,	,	PUNCT
ejpam-4736	107	20	s)-closed	s)-close	VERB
ejpam-4736	107	21	sets	set	NOUN
ejpam-4736	107	22	.	.	PUNCT
ejpam-4736	108	1	definition	definition	NOUN
ejpam-4736	108	2	3	3	NUM
ejpam-4736	108	3	.	.	PUNCT
ejpam-4736	109	1	a	a	DET
ejpam-4736	109	2	subset	subset	NOUN
ejpam-4736	109	3	a	a	PRON
ejpam-4736	109	4	of	of	ADP
ejpam-4736	109	5	a	a	DET
ejpam-4736	109	6	topological	topological	ADJ
ejpam-4736	109	7	space	space	NOUN
ejpam-4736	109	8	(	(	PUNCT
ejpam-4736	109	9	x	x	X
ejpam-4736	109	10	,	,	PUNCT
ejpam-4736	109	11	τ	τ	X
ejpam-4736	109	12	)	)	PUNCT
ejpam-4736	109	13	is	be	AUX
ejpam-4736	109	14	said	say	VERB
ejpam-4736	109	15	to	to	PART
ejpam-4736	109	16	be	be	AUX
ejpam-4736	109	17	generalized	generalize	VERB
ejpam-4736	109	18	δp(λ	δp(λ	NOUN
ejpam-4736	109	19	,	,	PUNCT
ejpam-4736	109	20	s)closed	s)close	VERB
ejpam-4736	109	21	(	(	PUNCT
ejpam-4736	109	22	briefly	briefly	ADV
ejpam-4736	109	23	,	,	PUNCT
ejpam-4736	109	24	g	g	NOUN
ejpam-4736	109	25	-	-	PUNCT
ejpam-4736	109	26	δp(λ	δp(λ	NOUN
ejpam-4736	109	27	,	,	PUNCT
ejpam-4736	109	28	s)-closed	s)-close	VERB
ejpam-4736	109	29	)	)	PUNCT
ejpam-4736	109	30	if	if	SCONJ
ejpam-4736	109	31	aδp(λ	aδp(λ	PROPN
ejpam-4736	109	32	,	,	PUNCT
ejpam-4736	109	33	s	s	PART
ejpam-4736	109	34	)	)	PUNCT
ejpam-4736	110	1	⊆	⊆	NUM
ejpam-4736	110	2	u	u	NOUN
ejpam-4736	110	3	whenever	whenever	SCONJ
ejpam-4736	110	4	a	a	DET
ejpam-4736	110	5	⊆	⊆	NUM
ejpam-4736	110	6	u	u	NOUN
ejpam-4736	110	7	and	and	CCONJ
ejpam-4736	110	8	u	u	NOUN
ejpam-4736	110	9	is	be	AUX
ejpam-4736	110	10	δp(λ	δp(λ	NOUN
ejpam-4736	110	11	,	,	PUNCT
ejpam-4736	110	12	s)-open	s)-open	PUNCT
ejpam-4736	110	13	in	in	ADP
ejpam-4736	110	14	(	(	PUNCT
ejpam-4736	110	15	x	x	NOUN
ejpam-4736	110	16	,	,	PUNCT
ejpam-4736	110	17	τ	τ	PROPN
ejpam-4736	110	18	)	)	PUNCT
ejpam-4736	110	19	.	.	PUNCT
ejpam-4736	111	1	the	the	DET
ejpam-4736	111	2	complement	complement	NOUN
ejpam-4736	111	3	of	of	ADP
ejpam-4736	111	4	a	a	DET
ejpam-4736	111	5	generalized	generalized	ADJ
ejpam-4736	111	6	δp(λ	δp(λ	NOUN
ejpam-4736	111	7	,	,	PUNCT
ejpam-4736	111	8	s)-closed	s)-close	VERB
ejpam-4736	111	9	set	set	NOUN
ejpam-4736	111	10	is	be	AUX
ejpam-4736	111	11	said	say	VERB
ejpam-4736	111	12	to	to	PART
ejpam-4736	111	13	be	be	AUX
ejpam-4736	111	14	generalized	generalize	VERB
ejpam-4736	111	15	δp(λ	δp(λ	NOUN
ejpam-4736	111	16	,	,	PUNCT
ejpam-4736	111	17	s)-open	s)-open	PUNCT
ejpam-4736	111	18	(	(	PUNCT
ejpam-4736	111	19	briefly	briefly	ADV
ejpam-4736	111	20	,	,	PUNCT
ejpam-4736	111	21	g	g	NOUN
ejpam-4736	111	22	-	-	PUNCT
ejpam-4736	111	23	δp(λ	δp(λ	NOUN
ejpam-4736	111	24	,	,	PUNCT
ejpam-4736	111	25	s)-open	s)-open	PUNCT
ejpam-4736	111	26	)	)	PUNCT
ejpam-4736	111	27	.	.	PUNCT
ejpam-4736	112	1	theorem	theorem	NOUN
ejpam-4736	112	2	1	1	NUM
ejpam-4736	112	3	.	.	PUNCT
ejpam-4736	113	1	a	a	DET
ejpam-4736	113	2	subset	subset	NOUN
ejpam-4736	113	3	a	a	PRON
ejpam-4736	113	4	of	of	ADP
ejpam-4736	113	5	a	a	DET
ejpam-4736	113	6	topological	topological	ADJ
ejpam-4736	113	7	space	space	NOUN
ejpam-4736	113	8	(	(	PUNCT
ejpam-4736	113	9	x	x	X
ejpam-4736	113	10	,	,	PUNCT
ejpam-4736	113	11	τ	τ	X
ejpam-4736	113	12	)	)	PUNCT
ejpam-4736	113	13	is	be	AUX
ejpam-4736	113	14	g	g	NOUN
ejpam-4736	113	15	-	-	PUNCT
ejpam-4736	113	16	δp(λ	δp(λ	NOUN
ejpam-4736	113	17	,	,	PUNCT
ejpam-4736	113	18	s)-closed	s)-close	VERB
ejpam-4736	113	19	if	if	SCONJ
ejpam-4736	113	20	and	and	CCONJ
ejpam-4736	113	21	only	only	ADV
ejpam-4736	113	22	if	if	SCONJ
ejpam-4736	113	23	aδp(λ	aδp(λ	PROPN
ejpam-4736	113	24	,	,	PUNCT
ejpam-4736	113	25	s	s	PART
ejpam-4736	113	26	)	)	PUNCT
ejpam-4736	113	27	−a	−a	NOUN
ejpam-4736	113	28	contains	contain	VERB
ejpam-4736	113	29	no	no	DET
ejpam-4736	113	30	nonempty	nonempty	ADJ
ejpam-4736	113	31	δp(λ	δp(λ	NOUN
ejpam-4736	113	32	,	,	PUNCT
ejpam-4736	113	33	s)-closed	s)-close	VERB
ejpam-4736	113	34	set	set	NOUN
ejpam-4736	113	35	.	.	PUNCT
ejpam-4736	114	1	c.	c.	PROPN
ejpam-4736	114	2	boonpok	boonpok	PROPN
ejpam-4736	114	3	,	,	PUNCT
ejpam-4736	114	4	n.	n.	PROPN
ejpam-4736	114	5	srisarakham	srisarakham	PROPN
ejpam-4736	114	6	/	/	SYM
ejpam-4736	114	7	eur	eur	PROPN
ejpam-4736	114	8	.	.	PUNCT
ejpam-4736	115	1	j.	j.	PROPN
ejpam-4736	115	2	pure	pure	PROPN
ejpam-4736	115	3	appl	appl	PROPN
ejpam-4736	115	4	.	.	PROPN
ejpam-4736	115	5	math	math	PROPN
ejpam-4736	115	6	,	,	PUNCT
ejpam-4736	115	7	16	16	NUM
ejpam-4736	115	8	(	(	PUNCT
ejpam-4736	115	9	4	4	NUM
ejpam-4736	115	10	)	)	PUNCT
ejpam-4736	115	11	(	(	PUNCT
ejpam-4736	115	12	2023	2023	NUM
ejpam-4736	115	13	)	)	PUNCT
ejpam-4736	115	14	,	,	PUNCT
ejpam-4736	115	15	2581	2581	NUM
ejpam-4736	115	16	-	-	SYM
ejpam-4736	115	17	2596	2596	NUM
ejpam-4736	115	18	2584	2584	NUM
ejpam-4736	115	19	proof	proof	NOUN
ejpam-4736	115	20	.	.	PUNCT
ejpam-4736	116	1	let	let	VERB
ejpam-4736	116	2	f	f	PRON
ejpam-4736	116	3	be	be	AUX
ejpam-4736	116	4	a	a	DET
ejpam-4736	116	5	δp(λ	δp(λ	NOUN
ejpam-4736	116	6	,	,	PUNCT
ejpam-4736	116	7	s)-closed	s)-close	VERB
ejpam-4736	116	8	subset	subset	NOUN
ejpam-4736	116	9	of	of	ADP
ejpam-4736	116	10	aδp(λ	aδp(λ	PROPN
ejpam-4736	116	11	,	,	PUNCT
ejpam-4736	116	12	s	s	PART
ejpam-4736	116	13	)	)	PUNCT
ejpam-4736	116	14	−	−	NOUN
ejpam-4736	116	15	a.	a.	NOUN
ejpam-4736	116	16	since	since	SCONJ
ejpam-4736	116	17	a	a	DET
ejpam-4736	116	18	⊆	⊆	NUM
ejpam-4736	116	19	x	x	SYM
ejpam-4736	116	20	−	−	PROPN
ejpam-4736	116	21	f	f	PROPN
ejpam-4736	116	22	and	and	CCONJ
ejpam-4736	116	23	a	a	PRON
ejpam-4736	116	24	is	be	AUX
ejpam-4736	116	25	g	g	NOUN
ejpam-4736	116	26	-	-	PUNCT
ejpam-4736	116	27	δp(λ	δp(λ	NOUN
ejpam-4736	116	28	,	,	PUNCT
ejpam-4736	116	29	s)-closed	s)-close	VERB
ejpam-4736	116	30	,	,	PUNCT
ejpam-4736	116	31	aδp(λ	aδp(λ	PROPN
ejpam-4736	116	32	,	,	PUNCT
ejpam-4736	116	33	s	s	PART
ejpam-4736	116	34	)	)	PUNCT
ejpam-4736	116	35	⊆	⊆	NUM
ejpam-4736	116	36	x	x	SYM
ejpam-4736	116	37	−	−	PROPN
ejpam-4736	116	38	f	f	NOUN
ejpam-4736	116	39	and	and	CCONJ
ejpam-4736	116	40	hence	hence	ADV
ejpam-4736	116	41	f	f	PROPN
ejpam-4736	116	42	⊆	⊆	NUM
ejpam-4736	116	43	x	x	SYM
ejpam-4736	116	44	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	116	45	,	,	PUNCT
ejpam-4736	116	46	s	s	NOUN
ejpam-4736	116	47	)	)	PUNCT
ejpam-4736	116	48	.	.	PUNCT
ejpam-4736	117	1	thus	thus	ADV
ejpam-4736	117	2	,	,	PUNCT
ejpam-4736	117	3	f	f	PROPN
ejpam-4736	117	4	⊆	⊆	NUM
ejpam-4736	117	5	aδp(λ	aδp(λ	PROPN
ejpam-4736	117	6	,	,	PUNCT
ejpam-4736	117	7	s	s	NOUN
ejpam-4736	117	8	)	)	PUNCT
ejpam-4736	117	9	∩	∩	NOUN
ejpam-4736	117	10	[	[	X
ejpam-4736	117	11	x	x	SYM
ejpam-4736	117	12	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	117	13	,	,	PUNCT
ejpam-4736	117	14	s	s	NOUN
ejpam-4736	117	15	)	)	PUNCT
ejpam-4736	117	16	]	]	PUNCT
ejpam-4736	117	17	=	=	PUNCT
ejpam-4736	117	18	∅	∅	NOUN
ejpam-4736	117	19	and	and	CCONJ
ejpam-4736	117	20	f	f	PROPN
ejpam-4736	117	21	is	be	AUX
ejpam-4736	117	22	empty	empty	ADJ
ejpam-4736	117	23	.	.	PUNCT
ejpam-4736	118	1	conversely	conversely	ADV
ejpam-4736	118	2	,	,	PUNCT
ejpam-4736	118	3	suppose	suppose	VERB
ejpam-4736	118	4	that	that	SCONJ
ejpam-4736	118	5	a	a	DET
ejpam-4736	118	6	⊆	⊆	NUM
ejpam-4736	118	7	u	u	NOUN
ejpam-4736	118	8	and	and	CCONJ
ejpam-4736	118	9	u	u	NOUN
ejpam-4736	118	10	is	be	AUX
ejpam-4736	118	11	δp(λ	δp(λ	NOUN
ejpam-4736	118	12	,	,	PUNCT
ejpam-4736	118	13	s)-open	s)-open	VERB
ejpam-4736	118	14	.	.	PUNCT
ejpam-4736	119	1	if	if	SCONJ
ejpam-4736	119	2	aδp(λ	aδp(λ	PROPN
ejpam-4736	119	3	,	,	PUNCT
ejpam-4736	119	4	s	s	PART
ejpam-4736	119	5	)	)	PUNCT
ejpam-4736	119	6	⊈	⊈	PROPN
ejpam-4736	119	7	u	u	NOUN
ejpam-4736	119	8	,	,	PUNCT
ejpam-4736	119	9	then	then	ADV
ejpam-4736	119	10	aδp(λ	aδp(λ	PROPN
ejpam-4736	119	11	,	,	PUNCT
ejpam-4736	119	12	s	s	NOUN
ejpam-4736	119	13	)	)	PUNCT
ejpam-4736	119	14	∩	∩	NOUN
ejpam-4736	119	15	(	(	PUNCT
ejpam-4736	119	16	x	x	SYM
ejpam-4736	119	17	−	−	PROPN
ejpam-4736	119	18	u	u	NOUN
ejpam-4736	119	19	)	)	PUNCT
ejpam-4736	119	20	is	be	AUX
ejpam-4736	119	21	a	a	DET
ejpam-4736	119	22	nonempty	nonempty	ADJ
ejpam-4736	119	23	δp(λ	δp(λ	NOUN
ejpam-4736	119	24	,	,	PUNCT
ejpam-4736	119	25	s)-closed	s)-close	VERB
ejpam-4736	119	26	subset	subset	NOUN
ejpam-4736	119	27	of	of	ADP
ejpam-4736	119	28	aδp(λ	aδp(λ	PROPN
ejpam-4736	119	29	,	,	PUNCT
ejpam-4736	119	30	s	s	PART
ejpam-4736	119	31	)	)	PUNCT
ejpam-4736	119	32	−a	−a	NOUN
ejpam-4736	119	33	.	.	PUNCT
ejpam-4736	120	1	corollary	corollary	ADJ
ejpam-4736	120	2	1	1	NUM
ejpam-4736	120	3	.	.	PUNCT
ejpam-4736	121	1	let	let	VERB
ejpam-4736	121	2	a	a	PRON
ejpam-4736	121	3	be	be	AUX
ejpam-4736	121	4	a	a	DET
ejpam-4736	121	5	g	g	NOUN
ejpam-4736	121	6	-	-	PUNCT
ejpam-4736	121	7	δp(λ	δp(λ	NOUN
ejpam-4736	121	8	,	,	PUNCT
ejpam-4736	121	9	s)-closed	s)-close	VERB
ejpam-4736	121	10	subset	subset	NOUN
ejpam-4736	121	11	of	of	ADP
ejpam-4736	121	12	a	a	DET
ejpam-4736	121	13	topological	topological	ADJ
ejpam-4736	121	14	space	space	NOUN
ejpam-4736	121	15	(	(	PUNCT
ejpam-4736	121	16	x	x	X
ejpam-4736	121	17	,	,	PUNCT
ejpam-4736	121	18	τ	τ	PROPN
ejpam-4736	121	19	)	)	PUNCT
ejpam-4736	121	20	.	.	PUNCT
ejpam-4736	122	1	then	then	ADV
ejpam-4736	122	2	,	,	PUNCT
ejpam-4736	122	3	a	a	PRON
ejpam-4736	122	4	is	is	NOUN
ejpam-4736	122	5	δp(λ	δp(λ	NOUN
ejpam-4736	122	6	,	,	PUNCT
ejpam-4736	122	7	s)-closed	s)-close	VERB
ejpam-4736	122	8	if	if	SCONJ
ejpam-4736	122	9	and	and	CCONJ
ejpam-4736	122	10	only	only	ADV
ejpam-4736	122	11	if	if	SCONJ
ejpam-4736	122	12	aδp(λ	aδp(λ	PROPN
ejpam-4736	122	13	,	,	PUNCT
ejpam-4736	122	14	s	s	PART
ejpam-4736	122	15	)	)	PUNCT
ejpam-4736	122	16	−a	−a	NOUN
ejpam-4736	122	17	is	be	AUX
ejpam-4736	122	18	δp(λ	δp(λ	NOUN
ejpam-4736	122	19	,	,	PUNCT
ejpam-4736	122	20	s)-closed	s)-close	VERB
ejpam-4736	122	21	.	.	PUNCT
ejpam-4736	123	1	proof	proof	NOUN
ejpam-4736	123	2	.	.	PUNCT
ejpam-4736	124	1	if	if	SCONJ
ejpam-4736	124	2	a	a	PRON
ejpam-4736	124	3	is	be	AUX
ejpam-4736	124	4	a	a	DET
ejpam-4736	124	5	δp(λ	δp(λ	NOUN
ejpam-4736	124	6	,	,	PUNCT
ejpam-4736	124	7	s)-closed	s)-close	VERB
ejpam-4736	124	8	set	set	NOUN
ejpam-4736	124	9	,	,	PUNCT
ejpam-4736	124	10	then	then	ADV
ejpam-4736	124	11	aδp(λ	aδp(λ	PROPN
ejpam-4736	124	12	,	,	PUNCT
ejpam-4736	124	13	s	s	PART
ejpam-4736	124	14	)	)	PUNCT
ejpam-4736	124	15	−a	−a	NOUN
ejpam-4736	124	16	=	=	PUNCT
ejpam-4736	124	17	∅.	∅.	VERB
ejpam-4736	124	18	conversely	conversely	ADV
ejpam-4736	124	19	,	,	PUNCT
ejpam-4736	124	20	suppose	suppose	VERB
ejpam-4736	124	21	that	that	SCONJ
ejpam-4736	124	22	aδp(λ	aδp(λ	PROPN
ejpam-4736	124	23	,	,	PUNCT
ejpam-4736	124	24	s	s	PART
ejpam-4736	124	25	)	)	PUNCT
ejpam-4736	124	26	−	−	NOUN
ejpam-4736	124	27	a	a	PRON
ejpam-4736	124	28	is	be	AUX
ejpam-4736	124	29	δp(λ	δp(λ	NOUN
ejpam-4736	124	30	,	,	PUNCT
ejpam-4736	124	31	s)-closed	s)-close	VERB
ejpam-4736	124	32	.	.	PUNCT
ejpam-4736	125	1	since	since	SCONJ
ejpam-4736	125	2	a	a	PRON
ejpam-4736	125	3	is	be	AUX
ejpam-4736	125	4	g	g	NOUN
ejpam-4736	125	5	-	-	PUNCT
ejpam-4736	125	6	δp(λ	δp(λ	NOUN
ejpam-4736	125	7	,	,	PUNCT
ejpam-4736	125	8	s)-closed	s)-close	VERB
ejpam-4736	125	9	and	and	CCONJ
ejpam-4736	125	10	aδp(λ	aδp(λ	PROPN
ejpam-4736	125	11	,	,	PUNCT
ejpam-4736	125	12	s	s	PART
ejpam-4736	125	13	)	)	PUNCT
ejpam-4736	125	14	−	−	NOUN
ejpam-4736	125	15	a	a	PRON
ejpam-4736	125	16	is	be	AUX
ejpam-4736	125	17	a	a	DET
ejpam-4736	125	18	δp(λ	δp(λ	NOUN
ejpam-4736	125	19	,	,	PUNCT
ejpam-4736	125	20	s)-closed	s)-close	VERB
ejpam-4736	125	21	subset	subset	NOUN
ejpam-4736	125	22	of	of	ADP
ejpam-4736	125	23	itself	itself	PRON
ejpam-4736	125	24	,	,	PUNCT
ejpam-4736	125	25	by	by	ADP
ejpam-4736	125	26	theorem	theorem	NOUN
ejpam-4736	125	27	1	1	NUM
ejpam-4736	125	28	,	,	PUNCT
ejpam-4736	125	29	aδp(λ	aδp(λ	PROPN
ejpam-4736	125	30	,	,	PUNCT
ejpam-4736	125	31	s	s	PART
ejpam-4736	125	32	)	)	PUNCT
ejpam-4736	125	33	−	−	NOUN
ejpam-4736	125	34	a	a	DET
ejpam-4736	125	35	=	=	NOUN
ejpam-4736	125	36	∅	∅	NOUN
ejpam-4736	125	37	and	and	CCONJ
ejpam-4736	125	38	hence	hence	ADV
ejpam-4736	125	39	aδp(λ	aδp(λ	PROPN
ejpam-4736	125	40	,	,	PUNCT
ejpam-4736	125	41	s	s	PART
ejpam-4736	125	42	)	)	PUNCT
ejpam-4736	125	43	=	=	SYM
ejpam-4736	125	44	a.	a.	NOUN
ejpam-4736	125	45	theorem	theorem	NOUN
ejpam-4736	125	46	2	2	NUM
ejpam-4736	125	47	.	.	X
ejpam-4736	125	48	for	for	ADP
ejpam-4736	125	49	a	a	DET
ejpam-4736	125	50	subset	subset	NOUN
ejpam-4736	125	51	a	a	PRON
ejpam-4736	125	52	of	of	ADP
ejpam-4736	125	53	a	a	DET
ejpam-4736	125	54	topological	topological	ADJ
ejpam-4736	125	55	space	space	NOUN
ejpam-4736	125	56	(	(	PUNCT
ejpam-4736	125	57	x	x	X
ejpam-4736	125	58	,	,	PUNCT
ejpam-4736	125	59	τ	τ	PROPN
ejpam-4736	125	60	)	)	PUNCT
ejpam-4736	125	61	,	,	PUNCT
ejpam-4736	125	62	the	the	DET
ejpam-4736	125	63	following	follow	VERB
ejpam-4736	125	64	properties	property	NOUN
ejpam-4736	125	65	hold	hold	VERB
ejpam-4736	125	66	:	:	PUNCT
ejpam-4736	125	67	(	(	PUNCT
ejpam-4736	125	68	1	1	X
ejpam-4736	125	69	)	)	PUNCT
ejpam-4736	125	70	if	if	SCONJ
ejpam-4736	125	71	a	a	PRON
ejpam-4736	125	72	is	be	AUX
ejpam-4736	125	73	δp(λ	δp(λ	NOUN
ejpam-4736	125	74	,	,	PUNCT
ejpam-4736	125	75	s)-closed	s)-close	VERB
ejpam-4736	125	76	,	,	PUNCT
ejpam-4736	125	77	then	then	ADV
ejpam-4736	125	78	a	a	PRON
ejpam-4736	125	79	is	be	AUX
ejpam-4736	125	80	g	g	NOUN
ejpam-4736	125	81	-	-	PUNCT
ejpam-4736	125	82	δp(λ	δp(λ	NOUN
ejpam-4736	125	83	,	,	PUNCT
ejpam-4736	125	84	s)-closed	s)-close	VERB
ejpam-4736	125	85	.	.	PUNCT
ejpam-4736	126	1	(	(	PUNCT
ejpam-4736	126	2	2	2	X
ejpam-4736	126	3	)	)	PUNCT
ejpam-4736	126	4	if	if	SCONJ
ejpam-4736	126	5	a	a	PRON
ejpam-4736	126	6	is	be	AUX
ejpam-4736	126	7	g	g	NOUN
ejpam-4736	126	8	-	-	PUNCT
ejpam-4736	126	9	δp(λ	δp(λ	NOUN
ejpam-4736	126	10	,	,	PUNCT
ejpam-4736	126	11	s)-closed	s)-close	VERB
ejpam-4736	126	12	and	and	CCONJ
ejpam-4736	126	13	δp(λ	δp(λ	NOUN
ejpam-4736	126	14	,	,	PUNCT
ejpam-4736	126	15	s)-open	s)-open	PUNCT
ejpam-4736	126	16	,	,	PUNCT
ejpam-4736	126	17	then	then	ADV
ejpam-4736	126	18	a	a	PRON
ejpam-4736	126	19	is	is	NOUN
ejpam-4736	126	20	δp(λ	δp(λ	NOUN
ejpam-4736	126	21	,	,	PUNCT
ejpam-4736	126	22	s)-closed	s)-close	VERB
ejpam-4736	126	23	.	.	PUNCT
ejpam-4736	127	1	(	(	PUNCT
ejpam-4736	127	2	3	3	X
ejpam-4736	127	3	)	)	PUNCT
ejpam-4736	127	4	if	if	SCONJ
ejpam-4736	127	5	a	a	PRON
ejpam-4736	127	6	is	be	AUX
ejpam-4736	127	7	g	g	NOUN
ejpam-4736	127	8	-	-	PUNCT
ejpam-4736	127	9	δp(λ	δp(λ	NOUN
ejpam-4736	127	10	,	,	PUNCT
ejpam-4736	127	11	s)-closed	s)-close	VERB
ejpam-4736	127	12	and	and	CCONJ
ejpam-4736	127	13	a	a	DET
ejpam-4736	127	14	⊆	⊆	NUM
ejpam-4736	127	15	b	b	NOUN
ejpam-4736	127	16	⊆	⊆	NUM
ejpam-4736	127	17	aδp(λ	aδp(λ	PROPN
ejpam-4736	127	18	,	,	PUNCT
ejpam-4736	127	19	s	s	PART
ejpam-4736	127	20	)	)	PUNCT
ejpam-4736	127	21	,	,	PUNCT
ejpam-4736	127	22	then	then	ADV
ejpam-4736	127	23	b	b	PROPN
ejpam-4736	127	24	is	be	AUX
ejpam-4736	127	25	g	g	NOUN
ejpam-4736	127	26	-	-	PUNCT
ejpam-4736	127	27	δp(λ	δp(λ	NOUN
ejpam-4736	127	28	,	,	PUNCT
ejpam-4736	127	29	s)-closed	s)-close	VERB
ejpam-4736	127	30	.	.	PUNCT
ejpam-4736	128	1	proof	proof	NOUN
ejpam-4736	128	2	.	.	PUNCT
ejpam-4736	129	1	(	(	PUNCT
ejpam-4736	129	2	1	1	X
ejpam-4736	129	3	)	)	PUNCT
ejpam-4736	129	4	let	let	VERB
ejpam-4736	129	5	a	a	DET
ejpam-4736	129	6	be	be	AUX
ejpam-4736	129	7	δp(λ	δp(λ	NOUN
ejpam-4736	129	8	,	,	PUNCT
ejpam-4736	129	9	s)-closed	s)-close	VERB
ejpam-4736	129	10	and	and	CCONJ
ejpam-4736	129	11	a	a	DET
ejpam-4736	129	12	⊆	⊆	NUM
ejpam-4736	129	13	u	u	NOUN
ejpam-4736	129	14	∈	∈	PROPN
ejpam-4736	129	15	δp(λ	δp(λ	NOUN
ejpam-4736	129	16	,	,	PUNCT
ejpam-4736	129	17	s)o(x	s)o(x	PROPN
ejpam-4736	129	18	,	,	PUNCT
ejpam-4736	129	19	τ	τ	PROPN
ejpam-4736	129	20	)	)	PUNCT
ejpam-4736	129	21	.	.	PUNCT
ejpam-4736	130	1	then	then	ADV
ejpam-4736	130	2	,	,	PUNCT
ejpam-4736	130	3	by	by	ADP
ejpam-4736	130	4	lemma	lemma	PROPN
ejpam-4736	130	5	1	1	NUM
ejpam-4736	130	6	,	,	PUNCT
ejpam-4736	130	7	aδp(λ	aδp(λ	PROPN
ejpam-4736	130	8	,	,	PUNCT
ejpam-4736	130	9	s	s	PART
ejpam-4736	130	10	)	)	PUNCT
ejpam-4736	130	11	=	=	PUNCT
ejpam-4736	130	12	a	a	DET
ejpam-4736	130	13	⊆	⊆	NUM
ejpam-4736	130	14	u	u	NOUN
ejpam-4736	130	15	and	and	CCONJ
ejpam-4736	130	16	hence	hence	ADV
ejpam-4736	130	17	a	a	PRON
ejpam-4736	130	18	is	be	AUX
ejpam-4736	130	19	g	g	NOUN
ejpam-4736	130	20	-	-	PUNCT
ejpam-4736	130	21	δp(λ	δp(λ	NOUN
ejpam-4736	130	22	,	,	PUNCT
ejpam-4736	130	23	s)-closed	s)-close	VERB
ejpam-4736	130	24	.	.	PUNCT
ejpam-4736	131	1	(	(	PUNCT
ejpam-4736	131	2	2	2	X
ejpam-4736	131	3	)	)	PUNCT
ejpam-4736	131	4	let	let	VERB
ejpam-4736	131	5	a	a	PRON
ejpam-4736	131	6	be	be	AUX
ejpam-4736	131	7	g	g	NOUN
ejpam-4736	131	8	-	-	PUNCT
ejpam-4736	131	9	δp(λ	δp(λ	NOUN
ejpam-4736	131	10	,	,	PUNCT
ejpam-4736	131	11	s)-closed	s)-close	VERB
ejpam-4736	131	12	and	and	CCONJ
ejpam-4736	131	13	δp(λ	δp(λ	NOUN
ejpam-4736	131	14	,	,	PUNCT
ejpam-4736	131	15	s)-open	s)-open	VERB
ejpam-4736	131	16	.	.	PUNCT
ejpam-4736	132	1	then	then	ADV
ejpam-4736	132	2	,	,	PUNCT
ejpam-4736	132	3	aδp(λ	aδp(λ	PROPN
ejpam-4736	132	4	,	,	PUNCT
ejpam-4736	132	5	s	s	PART
ejpam-4736	132	6	)	)	PUNCT
ejpam-4736	132	7	=	=	SYM
ejpam-4736	132	8	a	a	PRON
ejpam-4736	132	9	and	and	CCONJ
ejpam-4736	132	10	by	by	ADP
ejpam-4736	132	11	lemma	lemma	PROPN
ejpam-4736	132	12	1	1	NUM
ejpam-4736	132	13	,	,	PUNCT
ejpam-4736	132	14	a	a	DET
ejpam-4736	132	15	is	is	NOUN
ejpam-4736	132	16	δp(λ	δp(λ	NOUN
ejpam-4736	132	17	,	,	PUNCT
ejpam-4736	132	18	s)-closed	s)-close	VERB
ejpam-4736	132	19	.	.	PUNCT
ejpam-4736	133	1	(	(	PUNCT
ejpam-4736	133	2	3	3	X
ejpam-4736	133	3	)	)	PUNCT
ejpam-4736	133	4	let	let	VERB
ejpam-4736	133	5	b	b	NOUN
ejpam-4736	133	6	⊆	⊆	NUM
ejpam-4736	133	7	u	u	NOUN
ejpam-4736	133	8	and	and	CCONJ
ejpam-4736	133	9	u	u	PROPN
ejpam-4736	133	10	∈	∈	PROPN
ejpam-4736	133	11	δp(λ	δp(λ	NOUN
ejpam-4736	133	12	,	,	PUNCT
ejpam-4736	133	13	s)o(x	s)o(x	PROPN
ejpam-4736	133	14	,	,	PUNCT
ejpam-4736	133	15	τ	τ	PROPN
ejpam-4736	133	16	)	)	PUNCT
ejpam-4736	133	17	.	.	PUNCT
ejpam-4736	134	1	since	since	SCONJ
ejpam-4736	134	2	a	a	DET
ejpam-4736	134	3	⊆	⊆	NUM
ejpam-4736	134	4	u	u	NOUN
ejpam-4736	134	5	and	and	CCONJ
ejpam-4736	134	6	a	a	PRON
ejpam-4736	134	7	is	be	AUX
ejpam-4736	134	8	g	g	NOUN
ejpam-4736	134	9	-	-	PUNCT
ejpam-4736	134	10	δp(λ	δp(λ	NOUN
ejpam-4736	134	11	,	,	PUNCT
ejpam-4736	134	12	s)-closed	s)-close	VERB
ejpam-4736	134	13	,	,	PUNCT
ejpam-4736	134	14	we	we	PRON
ejpam-4736	134	15	have	have	VERB
ejpam-4736	134	16	aδp(λ	aδp(λ	PROPN
ejpam-4736	134	17	,	,	PUNCT
ejpam-4736	134	18	s	s	PART
ejpam-4736	134	19	)	)	PUNCT
ejpam-4736	134	20	⊆	⊆	NUM
ejpam-4736	134	21	u	u	NOUN
ejpam-4736	134	22	.	.	PUNCT
ejpam-4736	135	1	since	since	SCONJ
ejpam-4736	135	2	a	a	DET
ejpam-4736	135	3	⊆	⊆	NUM
ejpam-4736	135	4	b	b	NOUN
ejpam-4736	135	5	⊆	⊆	NUM
ejpam-4736	135	6	aδp(λ	aδp(λ	PROPN
ejpam-4736	135	7	,	,	PUNCT
ejpam-4736	135	8	s	s	PART
ejpam-4736	135	9	)	)	PUNCT
ejpam-4736	135	10	,	,	PUNCT
ejpam-4736	135	11	by	by	ADP
ejpam-4736	135	12	lemma	lemma	PROPN
ejpam-4736	135	13	1	1	NUM
ejpam-4736	135	14	,	,	PUNCT
ejpam-4736	135	15	aδp(λ	aδp(λ	PROPN
ejpam-4736	135	16	,	,	PUNCT
ejpam-4736	135	17	s	s	PART
ejpam-4736	135	18	)	)	PUNCT
ejpam-4736	135	19	=	=	SYM
ejpam-4736	135	20	bδp(λ	bδp(λ	PROPN
ejpam-4736	135	21	,	,	PUNCT
ejpam-4736	135	22	s	s	PART
ejpam-4736	135	23	)	)	PUNCT
ejpam-4736	135	24	and	and	CCONJ
ejpam-4736	135	25	hence	hence	ADV
ejpam-4736	135	26	bδp(λ	bδp(λ	PROPN
ejpam-4736	135	27	,	,	PUNCT
ejpam-4736	135	28	s	s	PART
ejpam-4736	135	29	)	)	PUNCT
ejpam-4736	135	30	⊆	⊆	NUM
ejpam-4736	135	31	u	u	NOUN
ejpam-4736	135	32	.	.	PUNCT
ejpam-4736	136	1	thus	thus	ADV
ejpam-4736	136	2	,	,	PUNCT
ejpam-4736	136	3	b	b	PROPN
ejpam-4736	136	4	is	be	AUX
ejpam-4736	136	5	g	g	NOUN
ejpam-4736	136	6	-	-	PUNCT
ejpam-4736	136	7	δp(λ	δp(λ	NOUN
ejpam-4736	136	8	,	,	PUNCT
ejpam-4736	136	9	s)-closed	s)-close	VERB
ejpam-4736	136	10	.	.	PUNCT
ejpam-4736	137	1	corollary	corollary	ADJ
ejpam-4736	137	2	2	2	NUM
ejpam-4736	137	3	.	.	PUNCT
ejpam-4736	138	1	for	for	ADP
ejpam-4736	138	2	a	a	DET
ejpam-4736	138	3	subset	subset	NOUN
ejpam-4736	138	4	a	a	PRON
ejpam-4736	138	5	of	of	ADP
ejpam-4736	138	6	a	a	DET
ejpam-4736	138	7	topological	topological	ADJ
ejpam-4736	138	8	space	space	NOUN
ejpam-4736	138	9	(	(	PUNCT
ejpam-4736	138	10	x	x	X
ejpam-4736	138	11	,	,	PUNCT
ejpam-4736	138	12	τ	τ	PROPN
ejpam-4736	138	13	)	)	PUNCT
ejpam-4736	138	14	,	,	PUNCT
ejpam-4736	138	15	the	the	DET
ejpam-4736	138	16	following	follow	VERB
ejpam-4736	138	17	properties	property	NOUN
ejpam-4736	138	18	hold	hold	VERB
ejpam-4736	138	19	:	:	PUNCT
ejpam-4736	138	20	(	(	PUNCT
ejpam-4736	138	21	1	1	X
ejpam-4736	138	22	)	)	PUNCT
ejpam-4736	138	23	if	if	SCONJ
ejpam-4736	138	24	a	a	PRON
ejpam-4736	138	25	is	be	AUX
ejpam-4736	138	26	δp(λ	δp(λ	NOUN
ejpam-4736	138	27	,	,	PUNCT
ejpam-4736	138	28	s)-open	s)-open	PUNCT
ejpam-4736	138	29	,	,	PUNCT
ejpam-4736	138	30	then	then	ADV
ejpam-4736	138	31	a	a	PRON
ejpam-4736	138	32	is	be	AUX
ejpam-4736	138	33	g	g	NOUN
ejpam-4736	138	34	-	-	PUNCT
ejpam-4736	138	35	δp(λ	δp(λ	NOUN
ejpam-4736	138	36	,	,	PUNCT
ejpam-4736	138	37	s)-open	s)-open	ADJ
ejpam-4736	138	38	.	.	PUNCT
ejpam-4736	139	1	(	(	PUNCT
ejpam-4736	139	2	2	2	X
ejpam-4736	139	3	)	)	PUNCT
ejpam-4736	139	4	if	if	SCONJ
ejpam-4736	139	5	a	a	PRON
ejpam-4736	139	6	is	be	AUX
ejpam-4736	139	7	g	g	NOUN
ejpam-4736	139	8	-	-	PUNCT
ejpam-4736	139	9	δp(λ	δp(λ	NOUN
ejpam-4736	139	10	,	,	PUNCT
ejpam-4736	139	11	s)-open	s)-open	PUNCT
ejpam-4736	139	12	and	and	CCONJ
ejpam-4736	139	13	δp(λ	δp(λ	NOUN
ejpam-4736	139	14	,	,	PUNCT
ejpam-4736	139	15	s)-closed	s)-close	VERB
ejpam-4736	139	16	,	,	PUNCT
ejpam-4736	139	17	then	then	ADV
ejpam-4736	139	18	a	a	PRON
ejpam-4736	139	19	is	be	AUX
ejpam-4736	139	20	δp(λ	δp(λ	NOUN
ejpam-4736	139	21	,	,	PUNCT
ejpam-4736	139	22	s)-open	s)-open	ADJ
ejpam-4736	139	23	.	.	PUNCT
ejpam-4736	140	1	(	(	PUNCT
ejpam-4736	140	2	3	3	X
ejpam-4736	140	3	)	)	PUNCT
ejpam-4736	140	4	if	if	SCONJ
ejpam-4736	140	5	a	a	PRON
ejpam-4736	140	6	is	be	AUX
ejpam-4736	140	7	g	g	NOUN
ejpam-4736	140	8	-	-	PUNCT
ejpam-4736	140	9	δp(λ	δp(λ	NOUN
ejpam-4736	140	10	,	,	PUNCT
ejpam-4736	140	11	s)-open	s)-open	PUNCT
ejpam-4736	140	12	and	and	CCONJ
ejpam-4736	140	13	aδp(λ	aδp(λ	PROPN
ejpam-4736	140	14	,	,	PUNCT
ejpam-4736	140	15	s	s	PART
ejpam-4736	140	16	)	)	PUNCT
ejpam-4736	140	17	⊆	⊆	NUM
ejpam-4736	140	18	b	b	NOUN
ejpam-4736	140	19	⊆	⊆	NUM
ejpam-4736	140	20	a	a	PRON
ejpam-4736	140	21	,	,	PUNCT
ejpam-4736	140	22	then	then	ADV
ejpam-4736	140	23	b	b	PROPN
ejpam-4736	140	24	is	be	AUX
ejpam-4736	140	25	g	g	NOUN
ejpam-4736	140	26	-	-	PUNCT
ejpam-4736	140	27	δp(λ	δp(λ	NOUN
ejpam-4736	140	28	,	,	PUNCT
ejpam-4736	140	29	s)-open	s)-open	PUNCT
ejpam-4736	140	30	.	.	PUNCT
ejpam-4736	141	1	proof	proof	NOUN
ejpam-4736	141	2	.	.	PUNCT
ejpam-4736	142	1	this	this	PRON
ejpam-4736	142	2	follows	follow	VERB
ejpam-4736	142	3	from	from	ADP
ejpam-4736	142	4	theorem	theorem	ADJ
ejpam-4736	142	5	2	2	NUM
ejpam-4736	142	6	.	.	PUNCT
ejpam-4736	142	7	c.	c.	PROPN
ejpam-4736	142	8	boonpok	boonpok	PROPN
ejpam-4736	142	9	,	,	PUNCT
ejpam-4736	142	10	n.	n.	PROPN
ejpam-4736	142	11	srisarakham	srisarakham	PROPN
ejpam-4736	142	12	/	/	SYM
ejpam-4736	142	13	eur	eur	PROPN
ejpam-4736	142	14	.	.	PUNCT
ejpam-4736	143	1	j.	j.	PROPN
ejpam-4736	143	2	pure	pure	PROPN
ejpam-4736	143	3	appl	appl	PROPN
ejpam-4736	143	4	.	.	PROPN
ejpam-4736	143	5	math	math	PROPN
ejpam-4736	143	6	,	,	PUNCT
ejpam-4736	143	7	16	16	NUM
ejpam-4736	143	8	(	(	PUNCT
ejpam-4736	143	9	4	4	NUM
ejpam-4736	143	10	)	)	PUNCT
ejpam-4736	143	11	(	(	PUNCT
ejpam-4736	143	12	2023	2023	NUM
ejpam-4736	143	13	)	)	PUNCT
ejpam-4736	143	14	,	,	PUNCT
ejpam-4736	143	15	2581	2581	NUM
ejpam-4736	143	16	-	-	SYM
ejpam-4736	143	17	2596	2596	NUM
ejpam-4736	143	18	2585	2585	NUM
ejpam-4736	143	19	definition	definition	NOUN
ejpam-4736	143	20	4	4	NUM
ejpam-4736	143	21	.	.	PUNCT
ejpam-4736	144	1	let	let	VERB
ejpam-4736	144	2	a	a	DET
ejpam-4736	144	3	be	be	AUX
ejpam-4736	144	4	a	a	DET
ejpam-4736	144	5	subset	subset	NOUN
ejpam-4736	144	6	of	of	ADP
ejpam-4736	144	7	a	a	DET
ejpam-4736	144	8	topological	topological	ADJ
ejpam-4736	144	9	space	space	NOUN
ejpam-4736	144	10	(	(	PUNCT
ejpam-4736	144	11	x	x	X
ejpam-4736	144	12	,	,	PUNCT
ejpam-4736	144	13	τ	τ	PROPN
ejpam-4736	144	14	)	)	PUNCT
ejpam-4736	144	15	.	.	PUNCT
ejpam-4736	145	1	the	the	DET
ejpam-4736	145	2	δp(λ	δp(λ	NOUN
ejpam-4736	145	3	,	,	PUNCT
ejpam-4736	145	4	s)-frontier	s)-fronti	ADJ
ejpam-4736	145	5	of	of	ADP
ejpam-4736	145	6	a	a	DET
ejpam-4736	145	7	,	,	PUNCT
ejpam-4736	145	8	δp(λ	δp(λ	NOUN
ejpam-4736	145	9	,	,	PUNCT
ejpam-4736	145	10	s)fr(a	s)fr(a	NOUN
ejpam-4736	145	11	)	)	PUNCT
ejpam-4736	145	12	,	,	PUNCT
ejpam-4736	145	13	is	be	AUX
ejpam-4736	145	14	defined	define	VERB
ejpam-4736	145	15	as	as	SCONJ
ejpam-4736	145	16	follows	follow	VERB
ejpam-4736	145	17	:	:	PUNCT
ejpam-4736	146	1	δp(λ	δp(λ	NOUN
ejpam-4736	146	2	,	,	PUNCT
ejpam-4736	146	3	s)fr(a	s)fr(a	NOUN
ejpam-4736	146	4	)	)	PUNCT
ejpam-4736	146	5	=	=	SYM
ejpam-4736	146	6	aδp(λ	aδp(λ	PROPN
ejpam-4736	146	7	,	,	PUNCT
ejpam-4736	146	8	s	s	NOUN
ejpam-4736	146	9	)	)	PUNCT
ejpam-4736	146	10	∩	∩	NOUN
ejpam-4736	147	1	[	[	X
ejpam-4736	147	2	x	x	SYM
ejpam-4736	147	3	−a]δp(λ	−a]δp(λ	PROPN
ejpam-4736	147	4	,	,	PUNCT
ejpam-4736	147	5	s	s	NOUN
ejpam-4736	147	6	)	)	PUNCT
ejpam-4736	147	7	.	.	PUNCT
ejpam-4736	148	1	theorem	theorem	NOUN
ejpam-4736	148	2	3	3	X
ejpam-4736	148	3	.	.	PUNCT
ejpam-4736	149	1	let	let	VERB
ejpam-4736	149	2	a	a	DET
ejpam-4736	149	3	be	be	AUX
ejpam-4736	149	4	a	a	DET
ejpam-4736	149	5	subset	subset	NOUN
ejpam-4736	149	6	of	of	ADP
ejpam-4736	149	7	a	a	DET
ejpam-4736	149	8	topological	topological	ADJ
ejpam-4736	149	9	space	space	NOUN
ejpam-4736	149	10	(	(	PUNCT
ejpam-4736	149	11	x	x	X
ejpam-4736	149	12	,	,	PUNCT
ejpam-4736	149	13	τ	τ	PROPN
ejpam-4736	149	14	)	)	PUNCT
ejpam-4736	149	15	.	.	PUNCT
ejpam-4736	150	1	if	if	SCONJ
ejpam-4736	150	2	a	a	PRON
ejpam-4736	150	3	is	be	AUX
ejpam-4736	150	4	g	g	NOUN
ejpam-4736	150	5	-	-	PUNCT
ejpam-4736	150	6	δp(λ	δp(λ	NOUN
ejpam-4736	150	7	,	,	PUNCT
ejpam-4736	150	8	s)-closed	s)-close	VERB
ejpam-4736	150	9	and	and	CCONJ
ejpam-4736	150	10	a	a	DET
ejpam-4736	150	11	⊆	⊆	NUM
ejpam-4736	150	12	v	v	NOUN
ejpam-4736	150	13	∈	∈	PROPN
ejpam-4736	150	14	δp(λ	δp(λ	NOUN
ejpam-4736	150	15	,	,	PUNCT
ejpam-4736	150	16	s)o(x	s)o(x	PROPN
ejpam-4736	150	17	,	,	PUNCT
ejpam-4736	150	18	τ	τ	PROPN
ejpam-4736	150	19	)	)	PUNCT
ejpam-4736	150	20	,	,	PUNCT
ejpam-4736	150	21	then	then	ADV
ejpam-4736	150	22	δp(λ	δp(λ	NUM
ejpam-4736	150	23	,	,	PUNCT
ejpam-4736	150	24	s)fr(v	s)fr(v	ADJ
ejpam-4736	150	25	)	)	PUNCT
ejpam-4736	150	26	⊆	⊆	NUM
ejpam-4736	150	27	[	[	X
ejpam-4736	150	28	x	x	SYM
ejpam-4736	150	29	−a]δp(λ	−a]δp(λ	PROPN
ejpam-4736	150	30	,	,	PUNCT
ejpam-4736	150	31	s	s	NOUN
ejpam-4736	150	32	)	)	PUNCT
ejpam-4736	150	33	.	.	PUNCT
ejpam-4736	151	1	proof	proof	NOUN
ejpam-4736	151	2	.	.	PUNCT
ejpam-4736	152	1	let	let	VERB
ejpam-4736	152	2	a	a	PRON
ejpam-4736	152	3	be	be	AUX
ejpam-4736	152	4	g	g	NOUN
ejpam-4736	152	5	-	-	PUNCT
ejpam-4736	152	6	δp(λ	δp(λ	NOUN
ejpam-4736	152	7	,	,	PUNCT
ejpam-4736	152	8	s)-closed	s)-close	VERB
ejpam-4736	152	9	and	and	CCONJ
ejpam-4736	152	10	a	a	DET
ejpam-4736	152	11	⊆	⊆	NUM
ejpam-4736	152	12	v	v	NOUN
ejpam-4736	152	13	∈	∈	PROPN
ejpam-4736	152	14	δp(λ	δp(λ	NOUN
ejpam-4736	152	15	,	,	PUNCT
ejpam-4736	152	16	s)o(x	s)o(x	PROPN
ejpam-4736	152	17	,	,	PUNCT
ejpam-4736	152	18	τ	τ	PROPN
ejpam-4736	152	19	)	)	PUNCT
ejpam-4736	152	20	.	.	PUNCT
ejpam-4736	153	1	then	then	ADV
ejpam-4736	153	2	,	,	PUNCT
ejpam-4736	153	3	aδp(λ	aδp(λ	PROPN
ejpam-4736	153	4	,	,	PUNCT
ejpam-4736	153	5	s	s	PART
ejpam-4736	153	6	)	)	PUNCT
ejpam-4736	153	7	⊆	⊆	NUM
ejpam-4736	153	8	v	v	NOUN
ejpam-4736	153	9	.	.	PUNCT
ejpam-4736	154	1	let	let	VERB
ejpam-4736	154	2	x	x	PUNCT
ejpam-4736	154	3	∈	∈	PROPN
ejpam-4736	154	4	δp(λ	δp(λ	NOUN
ejpam-4736	154	5	,	,	PUNCT
ejpam-4736	154	6	s)fr(v	s)fr(v	ADJ
ejpam-4736	154	7	)	)	PUNCT
ejpam-4736	154	8	.	.	PUNCT
ejpam-4736	155	1	since	since	SCONJ
ejpam-4736	155	2	v	v	NUM
ejpam-4736	155	3	∈	∈	PROPN
ejpam-4736	155	4	δp(λ	δp(λ	NOUN
ejpam-4736	155	5	,	,	PUNCT
ejpam-4736	155	6	s)o(x	s)o(x	PROPN
ejpam-4736	155	7	,	,	PUNCT
ejpam-4736	155	8	τ	τ	PROPN
ejpam-4736	155	9	)	)	PUNCT
ejpam-4736	155	10	,	,	PUNCT
ejpam-4736	155	11	we	we	PRON
ejpam-4736	155	12	have	have	VERB
ejpam-4736	155	13	δp(λ	δp(λ	NOUN
ejpam-4736	155	14	,	,	PUNCT
ejpam-4736	155	15	s)fr(v	s)fr(v	ADJ
ejpam-4736	155	16	)	)	PUNCT
ejpam-4736	156	1	=	=	SYM
ejpam-4736	156	2	v	v	NOUN
ejpam-4736	156	3	δp(λ	δp(λ	NOUN
ejpam-4736	156	4	,	,	PUNCT
ejpam-4736	156	5	s)−v	s)−v	X
ejpam-4736	156	6	.	.	PUNCT
ejpam-4736	157	1	thus	thus	ADV
ejpam-4736	157	2	,	,	PUNCT
ejpam-4736	157	3	x	x	PROPN
ejpam-4736	157	4	̸∈	̸∈	PROPN
ejpam-4736	157	5	v	v	NOUN
ejpam-4736	157	6	and	and	CCONJ
ejpam-4736	157	7	hence	hence	ADV
ejpam-4736	157	8	x	x	X
ejpam-4736	157	9	̸∈	̸∈	PROPN
ejpam-4736	157	10	aδp(λ	aδp(λ	PROPN
ejpam-4736	157	11	,	,	PUNCT
ejpam-4736	157	12	s	s	PROPN
ejpam-4736	157	13	)	)	PUNCT
ejpam-4736	157	14	.	.	PUNCT
ejpam-4736	158	1	therefore	therefore	ADV
ejpam-4736	158	2	,	,	PUNCT
ejpam-4736	158	3	x	x	PUNCT
ejpam-4736	158	4	∈	∈	PROPN
ejpam-4736	159	1	[	[	X
ejpam-4736	159	2	x	x	X
ejpam-4736	159	3	−	−	NOUN
ejpam-4736	159	4	a]δp(λ	a]δp(λ	NOUN
ejpam-4736	159	5	,	,	PUNCT
ejpam-4736	159	6	s	s	NOUN
ejpam-4736	159	7	)	)	PUNCT
ejpam-4736	159	8	.	.	PUNCT
ejpam-4736	160	1	this	this	PRON
ejpam-4736	160	2	shows	show	VERB
ejpam-4736	160	3	that	that	SCONJ
ejpam-4736	160	4	δp(λ	δp(λ	NOUN
ejpam-4736	160	5	,	,	PUNCT
ejpam-4736	160	6	s)fr(v	s)fr(v	ADJ
ejpam-4736	160	7	)	)	PUNCT
ejpam-4736	160	8	⊆	⊆	NUM
ejpam-4736	161	1	[	[	X
ejpam-4736	161	2	x	x	SYM
ejpam-4736	161	3	−a]δp(λ	−a]δp(λ	PROPN
ejpam-4736	161	4	,	,	PUNCT
ejpam-4736	161	5	s	s	NOUN
ejpam-4736	161	6	)	)	PUNCT
ejpam-4736	161	7	.	.	PUNCT
ejpam-4736	162	1	theorem	theorem	ADJ
ejpam-4736	162	2	4	4	NUM
ejpam-4736	162	3	.	.	PUNCT
ejpam-4736	163	1	let	let	AUX
ejpam-4736	163	2	(	(	PUNCT
ejpam-4736	163	3	x	x	NOUN
ejpam-4736	163	4	,	,	PUNCT
ejpam-4736	163	5	τ	τ	X
ejpam-4736	163	6	)	)	PUNCT
ejpam-4736	163	7	be	be	VERB
ejpam-4736	163	8	a	a	DET
ejpam-4736	163	9	topological	topological	ADJ
ejpam-4736	163	10	space	space	NOUN
ejpam-4736	163	11	.	.	PUNCT
ejpam-4736	164	1	for	for	SCONJ
ejpam-4736	164	2	each	each	DET
ejpam-4736	164	3	x	x	SYM
ejpam-4736	164	4	∈	∈	PROPN
ejpam-4736	164	5	x	x	NOUN
ejpam-4736	164	6	,	,	PUNCT
ejpam-4736	164	7	either	either	CCONJ
ejpam-4736	164	8	{	{	PUNCT
ejpam-4736	164	9	x	x	X
ejpam-4736	164	10	}	}	PUNCT
ejpam-4736	164	11	is	be	AUX
ejpam-4736	164	12	δp(λ	δp(λ	NOUN
ejpam-4736	164	13	,	,	PUNCT
ejpam-4736	164	14	s)closed	s)closed	ADJ
ejpam-4736	164	15	or	or	CCONJ
ejpam-4736	164	16	g	g	NOUN
ejpam-4736	164	17	-	-	PUNCT
ejpam-4736	164	18	δp(λ	δp(λ	NOUN
ejpam-4736	164	19	,	,	PUNCT
ejpam-4736	164	20	s)-open	s)-open	PUNCT
ejpam-4736	164	21	.	.	PUNCT
ejpam-4736	165	1	proof	proof	NOUN
ejpam-4736	165	2	.	.	PUNCT
ejpam-4736	166	1	suppose	suppose	VERB
ejpam-4736	166	2	that	that	SCONJ
ejpam-4736	166	3	{	{	PUNCT
ejpam-4736	166	4	x	x	X
ejpam-4736	166	5	}	}	PUNCT
ejpam-4736	166	6	is	be	AUX
ejpam-4736	166	7	not	not	PART
ejpam-4736	166	8	δp(λ	δp(λ	NOUN
ejpam-4736	166	9	,	,	PUNCT
ejpam-4736	166	10	s)-closed	s)-close	VERB
ejpam-4736	166	11	.	.	PUNCT
ejpam-4736	167	1	then	then	ADV
ejpam-4736	167	2	,	,	PUNCT
ejpam-4736	167	3	x−{x	x−{x	PROPN
ejpam-4736	167	4	}	}	PUNCT
ejpam-4736	167	5	is	be	AUX
ejpam-4736	167	6	not	not	PART
ejpam-4736	167	7	δp(λ	δp(λ	NOUN
ejpam-4736	167	8	,	,	PUNCT
ejpam-4736	167	9	s)-open	s)-open	PUNCT
ejpam-4736	167	10	and	and	CCONJ
ejpam-4736	167	11	the	the	DET
ejpam-4736	167	12	only	only	ADJ
ejpam-4736	167	13	δp(λ	δp(λ	NOUN
ejpam-4736	167	14	,	,	PUNCT
ejpam-4736	167	15	s)-open	s)-open	VERB
ejpam-4736	167	16	set	set	VERB
ejpam-4736	167	17	containing	contain	VERB
ejpam-4736	167	18	x−{x	x−{x	PROPN
ejpam-4736	167	19	}	}	PUNCT
ejpam-4736	167	20	is	be	AUX
ejpam-4736	167	21	x	x	X
ejpam-4736	167	22	itself	itself	PRON
ejpam-4736	167	23	.	.	PUNCT
ejpam-4736	168	1	therefore	therefore	ADV
ejpam-4736	168	2	,	,	PUNCT
ejpam-4736	168	3	[	[	X
ejpam-4736	168	4	x−{x}]δp(λ	x−{x}]δp(λ	PROPN
ejpam-4736	168	5	,	,	PUNCT
ejpam-4736	168	6	s	s	PART
ejpam-4736	168	7	)	)	PUNCT
ejpam-4736	168	8	⊆	⊆	NUM
ejpam-4736	168	9	x.	x.	NOUN
ejpam-4736	168	10	thus	thus	ADV
ejpam-4736	168	11	,	,	PUNCT
ejpam-4736	168	12	x	x	PUNCT
ejpam-4736	168	13	−	−	X
ejpam-4736	168	14	{	{	PUNCT
ejpam-4736	168	15	x	x	NOUN
ejpam-4736	168	16	}	}	PUNCT
ejpam-4736	168	17	is	be	AUX
ejpam-4736	168	18	g	g	NOUN
ejpam-4736	168	19	-	-	PUNCT
ejpam-4736	168	20	δp(λ	δp(λ	NOUN
ejpam-4736	168	21	,	,	PUNCT
ejpam-4736	168	22	s)-closed	s)-close	VERB
ejpam-4736	168	23	and	and	CCONJ
ejpam-4736	168	24	hence	hence	ADV
ejpam-4736	168	25	{	{	PUNCT
ejpam-4736	168	26	x	x	X
ejpam-4736	168	27	}	}	PUNCT
ejpam-4736	168	28	is	be	AUX
ejpam-4736	168	29	g	g	NOUN
ejpam-4736	168	30	-	-	PUNCT
ejpam-4736	168	31	δp(λ	δp(λ	NOUN
ejpam-4736	168	32	,	,	PUNCT
ejpam-4736	168	33	s)-open	s)-open	PUNCT
ejpam-4736	168	34	.	.	PUNCT
ejpam-4736	169	1	theorem	theorem	NOUN
ejpam-4736	169	2	5	5	NUM
ejpam-4736	169	3	.	.	PUNCT
ejpam-4736	170	1	let	let	VERB
ejpam-4736	170	2	a	a	DET
ejpam-4736	170	3	be	be	AUX
ejpam-4736	170	4	a	a	DET
ejpam-4736	170	5	subset	subset	NOUN
ejpam-4736	170	6	of	of	ADP
ejpam-4736	170	7	a	a	DET
ejpam-4736	170	8	topological	topological	ADJ
ejpam-4736	170	9	space	space	NOUN
ejpam-4736	170	10	(	(	PUNCT
ejpam-4736	170	11	x	x	X
ejpam-4736	170	12	,	,	PUNCT
ejpam-4736	170	13	τ	τ	PROPN
ejpam-4736	170	14	)	)	PUNCT
ejpam-4736	170	15	.	.	PUNCT
ejpam-4736	171	1	then	then	ADV
ejpam-4736	171	2	,	,	PUNCT
ejpam-4736	171	3	a	a	PRON
ejpam-4736	171	4	is	be	AUX
ejpam-4736	171	5	g	g	NOUN
ejpam-4736	171	6	-	-	PUNCT
ejpam-4736	171	7	δp(λ	δp(λ	NOUN
ejpam-4736	171	8	,	,	PUNCT
ejpam-4736	171	9	s)-open	s)-open	VERB
ejpam-4736	171	10	if	if	SCONJ
ejpam-4736	171	11	and	and	CCONJ
ejpam-4736	171	12	only	only	ADV
ejpam-4736	171	13	if	if	SCONJ
ejpam-4736	171	14	f	f	PROPN
ejpam-4736	171	15	⊆	⊆	NUM
ejpam-4736	171	16	aδp(λ	aδp(λ	PROPN
ejpam-4736	171	17	,	,	PUNCT
ejpam-4736	171	18	s	s	PART
ejpam-4736	171	19	)	)	PUNCT
ejpam-4736	171	20	whenever	whenever	SCONJ
ejpam-4736	171	21	f	f	PROPN
ejpam-4736	171	22	⊆	⊆	PROPN
ejpam-4736	171	23	a	a	PRON
ejpam-4736	171	24	and	and	CCONJ
ejpam-4736	171	25	f	f	PROPN
ejpam-4736	171	26	is	be	AUX
ejpam-4736	171	27	δp(λ	δp(λ	NOUN
ejpam-4736	171	28	,	,	PUNCT
ejpam-4736	171	29	s)-closed	s)-close	VERB
ejpam-4736	171	30	.	.	PUNCT
ejpam-4736	172	1	proof	proof	NOUN
ejpam-4736	172	2	.	.	PUNCT
ejpam-4736	173	1	suppose	suppose	VERB
ejpam-4736	173	2	that	that	SCONJ
ejpam-4736	173	3	a	a	PRON
ejpam-4736	173	4	is	be	AUX
ejpam-4736	173	5	a	a	DET
ejpam-4736	173	6	g	g	NOUN
ejpam-4736	173	7	-	-	PUNCT
ejpam-4736	173	8	δp(λ	δp(λ	NOUN
ejpam-4736	173	9	,	,	PUNCT
ejpam-4736	173	10	s)-open	s)-open	PUNCT
ejpam-4736	173	11	set	set	VERB
ejpam-4736	173	12	.	.	PUNCT
ejpam-4736	174	1	let	let	VERB
ejpam-4736	174	2	f	f	PRON
ejpam-4736	174	3	be	be	AUX
ejpam-4736	174	4	a	a	DET
ejpam-4736	174	5	δp(λ	δp(λ	NOUN
ejpam-4736	174	6	,	,	PUNCT
ejpam-4736	174	7	s)-closed	s)-close	VERB
ejpam-4736	174	8	set	set	NOUN
ejpam-4736	174	9	and	and	CCONJ
ejpam-4736	174	10	f	f	PROPN
ejpam-4736	174	11	⊆	⊆	NUM
ejpam-4736	174	12	a.	a.	NOUN
ejpam-4736	174	13	then	then	ADV
ejpam-4736	174	14	,	,	PUNCT
ejpam-4736	174	15	x	x	PUNCT
ejpam-4736	174	16	−	−	NOUN
ejpam-4736	174	17	a	a	DET
ejpam-4736	174	18	⊆	⊆	NUM
ejpam-4736	174	19	x	x	SYM
ejpam-4736	174	20	−	−	PROPN
ejpam-4736	174	21	f	f	PROPN
ejpam-4736	174	22	∈	∈	PROPN
ejpam-4736	174	23	δp(λ	δp(λ	NOUN
ejpam-4736	174	24	,	,	PUNCT
ejpam-4736	174	25	s)o(x	s)o(x	PROPN
ejpam-4736	174	26	,	,	PUNCT
ejpam-4736	174	27	τ	τ	X
ejpam-4736	174	28	)	)	PUNCT
ejpam-4736	174	29	and	and	CCONJ
ejpam-4736	174	30	x	x	X
ejpam-4736	174	31	−	−	NOUN
ejpam-4736	174	32	a	a	PRON
ejpam-4736	174	33	is	be	AUX
ejpam-4736	174	34	g	g	NOUN
ejpam-4736	174	35	-	-	PUNCT
ejpam-4736	174	36	δp(λ	δp(λ	NOUN
ejpam-4736	174	37	,	,	PUNCT
ejpam-4736	174	38	s)-closed	s)-close	VERB
ejpam-4736	174	39	.	.	PUNCT
ejpam-4736	175	1	thus	thus	ADV
ejpam-4736	175	2	,	,	PUNCT
ejpam-4736	175	3	x	x	PROPN
ejpam-4736	175	4	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	175	5	,	,	PUNCT
ejpam-4736	175	6	s	s	PART
ejpam-4736	175	7	)	)	PUNCT
ejpam-4736	175	8	=	=	PUNCT
ejpam-4736	176	1	[	[	X
ejpam-4736	176	2	x	x	X
ejpam-4736	176	3	−a]δp(λ	−a]δp(λ	PROPN
ejpam-4736	176	4	,	,	PUNCT
ejpam-4736	176	5	s	s	PART
ejpam-4736	176	6	)	)	PUNCT
ejpam-4736	176	7	⊆	⊆	NUM
ejpam-4736	176	8	x	x	SYM
ejpam-4736	176	9	−	−	PROPN
ejpam-4736	176	10	f	f	NOUN
ejpam-4736	176	11	and	and	CCONJ
ejpam-4736	176	12	hence	hence	ADV
ejpam-4736	176	13	f	f	PROPN
ejpam-4736	176	14	⊆	⊆	NUM
ejpam-4736	176	15	aδp(λ	aδp(λ	PROPN
ejpam-4736	176	16	,	,	PUNCT
ejpam-4736	176	17	s	s	NOUN
ejpam-4736	176	18	)	)	PUNCT
ejpam-4736	176	19	.	.	PUNCT
ejpam-4736	177	1	conversely	conversely	ADV
ejpam-4736	177	2	,	,	PUNCT
ejpam-4736	177	3	let	let	VERB
ejpam-4736	177	4	x	x	PART
ejpam-4736	177	5	−a	−a	VERB
ejpam-4736	177	6	⊆	⊆	NUM
ejpam-4736	177	7	u	u	NOUN
ejpam-4736	177	8	and	and	CCONJ
ejpam-4736	177	9	u	u	PROPN
ejpam-4736	177	10	∈	∈	PROPN
ejpam-4736	177	11	δp(λ	δp(λ	NOUN
ejpam-4736	177	12	,	,	PUNCT
ejpam-4736	177	13	s)o(x	s)o(x	PROPN
ejpam-4736	177	14	,	,	PUNCT
ejpam-4736	177	15	τ	τ	PROPN
ejpam-4736	177	16	)	)	PUNCT
ejpam-4736	177	17	.	.	PUNCT
ejpam-4736	178	1	then	then	ADV
ejpam-4736	178	2	,	,	PUNCT
ejpam-4736	178	3	x	x	PROPN
ejpam-4736	178	4	−u	−u	NOUN
ejpam-4736	178	5	⊆	⊆	NUM
ejpam-4736	178	6	a	a	PRON
ejpam-4736	178	7	and	and	CCONJ
ejpam-4736	178	8	x	x	SYM
ejpam-4736	178	9	−u	−u	NOUN
ejpam-4736	178	10	is	be	AUX
ejpam-4736	178	11	δp(λ	δp(λ	NOUN
ejpam-4736	178	12	,	,	PUNCT
ejpam-4736	178	13	s)-closed	s)-close	VERB
ejpam-4736	178	14	.	.	PUNCT
ejpam-4736	179	1	by	by	ADP
ejpam-4736	179	2	the	the	DET
ejpam-4736	179	3	hypothesis	hypothesis	NOUN
ejpam-4736	179	4	,	,	PUNCT
ejpam-4736	179	5	x	x	PUNCT
ejpam-4736	179	6	−	−	PROPN
ejpam-4736	179	7	u	u	NOUN
ejpam-4736	179	8	⊆	⊆	NUM
ejpam-4736	179	9	aδp(λ	aδp(λ	PROPN
ejpam-4736	179	10	,	,	PUNCT
ejpam-4736	179	11	s	s	PART
ejpam-4736	179	12	)	)	PUNCT
ejpam-4736	179	13	and	and	CCONJ
ejpam-4736	179	14	hence	hence	ADV
ejpam-4736	179	15	[	[	X
ejpam-4736	179	16	x	x	X
ejpam-4736	179	17	−a]δp(λ	−a]δp(λ	PROPN
ejpam-4736	179	18	,	,	PUNCT
ejpam-4736	179	19	s	s	NOUN
ejpam-4736	179	20	)	)	PUNCT
ejpam-4736	179	21	=	=	SYM
ejpam-4736	179	22	x	x	SYM
ejpam-4736	179	23	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	179	24	,	,	PUNCT
ejpam-4736	179	25	s	s	PART
ejpam-4736	179	26	)	)	PUNCT
ejpam-4736	179	27	⊆	⊆	NUM
ejpam-4736	179	28	u.	u.	NOUN
ejpam-4736	179	29	thus	thus	ADV
ejpam-4736	179	30	,	,	PUNCT
ejpam-4736	179	31	x	x	PRON
ejpam-4736	179	32	−a	−a	NOUN
ejpam-4736	179	33	is	be	AUX
ejpam-4736	179	34	g	g	NOUN
ejpam-4736	179	35	-	-	PUNCT
ejpam-4736	179	36	δp(λ	δp(λ	NOUN
ejpam-4736	179	37	,	,	PUNCT
ejpam-4736	179	38	s)-closed	s)-close	VERB
ejpam-4736	179	39	.	.	PUNCT
ejpam-4736	180	1	this	this	PRON
ejpam-4736	180	2	shows	show	VERB
ejpam-4736	180	3	that	that	SCONJ
ejpam-4736	180	4	a	a	PRON
ejpam-4736	180	5	is	be	AUX
ejpam-4736	180	6	g	g	NOUN
ejpam-4736	180	7	-	-	PUNCT
ejpam-4736	180	8	δp(λ	δp(λ	NOUN
ejpam-4736	180	9	,	,	PUNCT
ejpam-4736	180	10	s)-open	s)-open	VERB
ejpam-4736	180	11	.	.	PUNCT
ejpam-4736	181	1	lemma	lemma	PROPN
ejpam-4736	181	2	4	4	X
ejpam-4736	181	3	.	.	PUNCT
ejpam-4736	182	1	let	let	VERB
ejpam-4736	182	2	a	a	DET
ejpam-4736	182	3	be	be	AUX
ejpam-4736	182	4	a	a	DET
ejpam-4736	182	5	subset	subset	NOUN
ejpam-4736	182	6	of	of	ADP
ejpam-4736	182	7	a	a	DET
ejpam-4736	182	8	topological	topological	ADJ
ejpam-4736	182	9	space	space	NOUN
ejpam-4736	182	10	(	(	PUNCT
ejpam-4736	182	11	x	x	X
ejpam-4736	182	12	,	,	PUNCT
ejpam-4736	182	13	τ	τ	PROPN
ejpam-4736	182	14	)	)	PUNCT
ejpam-4736	182	15	.	.	PUNCT
ejpam-4736	183	1	if	if	SCONJ
ejpam-4736	183	2	g	g	PROPN
ejpam-4736	183	3	∈	∈	PROPN
ejpam-4736	183	4	δp(λ	δp(λ	NOUN
ejpam-4736	183	5	,	,	PUNCT
ejpam-4736	183	6	s)o(x	s)o(x	PROPN
ejpam-4736	183	7	,	,	PUNCT
ejpam-4736	183	8	τ	τ	X
ejpam-4736	183	9	)	)	PUNCT
ejpam-4736	183	10	and	and	CCONJ
ejpam-4736	183	11	a	a	DET
ejpam-4736	183	12	∩g	∩g	ADJ
ejpam-4736	183	13	=	=	SYM
ejpam-4736	183	14	∅	∅	NOUN
ejpam-4736	183	15	,	,	PUNCT
ejpam-4736	183	16	then	then	ADV
ejpam-4736	183	17	aδp(λ	aδp(λ	PROPN
ejpam-4736	183	18	,	,	PUNCT
ejpam-4736	183	19	s	s	NOUN
ejpam-4736	183	20	)	)	PUNCT
ejpam-4736	183	21	∩g	∩g	NOUN
ejpam-4736	183	22	=	=	SYM
ejpam-4736	183	23	∅.	∅.	NOUN
ejpam-4736	183	24	theorem	theorem	VERB
ejpam-4736	183	25	6	6	NUM
ejpam-4736	183	26	.	.	PUNCT
ejpam-4736	183	27	for	for	ADP
ejpam-4736	183	28	a	a	DET
ejpam-4736	183	29	subset	subset	NOUN
ejpam-4736	183	30	a	a	PRON
ejpam-4736	183	31	of	of	ADP
ejpam-4736	183	32	a	a	DET
ejpam-4736	183	33	topological	topological	ADJ
ejpam-4736	183	34	space	space	NOUN
ejpam-4736	183	35	(	(	PUNCT
ejpam-4736	183	36	x	x	X
ejpam-4736	183	37	,	,	PUNCT
ejpam-4736	183	38	τ	τ	PROPN
ejpam-4736	183	39	)	)	PUNCT
ejpam-4736	183	40	,	,	PUNCT
ejpam-4736	183	41	the	the	DET
ejpam-4736	183	42	following	follow	VERB
ejpam-4736	183	43	properties	property	NOUN
ejpam-4736	183	44	are	be	AUX
ejpam-4736	183	45	equivalent	equivalent	ADJ
ejpam-4736	183	46	:	:	PUNCT
ejpam-4736	183	47	(	(	PUNCT
ejpam-4736	183	48	1	1	X
ejpam-4736	183	49	)	)	PUNCT
ejpam-4736	183	50	a	a	PRON
ejpam-4736	183	51	is	be	AUX
ejpam-4736	183	52	g	g	NOUN
ejpam-4736	183	53	-	-	PUNCT
ejpam-4736	183	54	δp(λ	δp(λ	NOUN
ejpam-4736	183	55	,	,	PUNCT
ejpam-4736	183	56	s)-closed	s)-close	VERB
ejpam-4736	183	57	.	.	PUNCT
ejpam-4736	184	1	(	(	PUNCT
ejpam-4736	184	2	2	2	X
ejpam-4736	184	3	)	)	PUNCT
ejpam-4736	184	4	aδp(λ	aδp(λ	PROPN
ejpam-4736	184	5	,	,	PUNCT
ejpam-4736	184	6	s	s	PART
ejpam-4736	184	7	)	)	PUNCT
ejpam-4736	184	8	−a	−a	NOUN
ejpam-4736	184	9	contains	contain	VERB
ejpam-4736	184	10	no	no	DET
ejpam-4736	184	11	nonempty	nonempty	ADJ
ejpam-4736	184	12	δp(λ	δp(λ	NOUN
ejpam-4736	184	13	,	,	PUNCT
ejpam-4736	184	14	s)-closed	s)-close	VERB
ejpam-4736	184	15	set	set	NOUN
ejpam-4736	184	16	.	.	PUNCT
ejpam-4736	185	1	(	(	PUNCT
ejpam-4736	185	2	3	3	X
ejpam-4736	185	3	)	)	PUNCT
ejpam-4736	185	4	aδp(λ	aδp(λ	PROPN
ejpam-4736	185	5	,	,	PUNCT
ejpam-4736	185	6	s	s	PART
ejpam-4736	185	7	)	)	PUNCT
ejpam-4736	185	8	−a	−a	NOUN
ejpam-4736	185	9	is	be	AUX
ejpam-4736	185	10	g	g	NOUN
ejpam-4736	185	11	-	-	PUNCT
ejpam-4736	185	12	δp(λ	δp(λ	NOUN
ejpam-4736	185	13	,	,	PUNCT
ejpam-4736	185	14	s)-open	s)-open	PUNCT
ejpam-4736	185	15	.	.	PUNCT
ejpam-4736	186	1	proof	proof	NOUN
ejpam-4736	186	2	.	.	PUNCT
ejpam-4736	187	1	(	(	PUNCT
ejpam-4736	187	2	1	1	X
ejpam-4736	187	3	)	)	PUNCT
ejpam-4736	187	4	⇒	⇒	NOUN
ejpam-4736	187	5	(	(	PUNCT
ejpam-4736	187	6	2	2	NUM
ejpam-4736	187	7	):	):	PUNCT
ejpam-4736	187	8	this	this	PRON
ejpam-4736	187	9	follows	follow	VERB
ejpam-4736	187	10	from	from	ADP
ejpam-4736	187	11	theorem	theorem	ADJ
ejpam-4736	187	12	1	1	NUM
ejpam-4736	187	13	.	.	PUNCT
ejpam-4736	187	14	(	(	PUNCT
ejpam-4736	187	15	2	2	X
ejpam-4736	187	16	)	)	PUNCT
ejpam-4736	187	17	⇒	⇒	NOUN
ejpam-4736	187	18	(	(	PUNCT
ejpam-4736	187	19	3	3	NUM
ejpam-4736	187	20	):	):	PUNCT
ejpam-4736	187	21	let	let	VERB
ejpam-4736	187	22	f	f	PRON
ejpam-4736	187	23	be	be	AUX
ejpam-4736	187	24	a	a	DET
ejpam-4736	187	25	δp(λ	δp(λ	NOUN
ejpam-4736	187	26	,	,	PUNCT
ejpam-4736	187	27	s)-closed	s)-close	VERB
ejpam-4736	187	28	set	set	NOUN
ejpam-4736	187	29	and	and	CCONJ
ejpam-4736	187	30	f	f	PROPN
ejpam-4736	187	31	⊆	⊆	NUM
ejpam-4736	187	32	aδp(λ	aδp(λ	PROPN
ejpam-4736	187	33	,	,	PUNCT
ejpam-4736	187	34	s	s	PART
ejpam-4736	187	35	)	)	PUNCT
ejpam-4736	188	1	−	−	NOUN
ejpam-4736	188	2	a.	a.	NOUN
ejpam-4736	188	3	by	by	ADP
ejpam-4736	188	4	(	(	PUNCT
ejpam-4736	188	5	2	2	NUM
ejpam-4736	188	6	)	)	PUNCT
ejpam-4736	188	7	,	,	PUNCT
ejpam-4736	188	8	we	we	PRON
ejpam-4736	188	9	have	have	VERB
ejpam-4736	188	10	f	f	NOUN
ejpam-4736	188	11	=	=	PUNCT
ejpam-4736	188	12	∅	∅	NOUN
ejpam-4736	188	13	and	and	CCONJ
ejpam-4736	188	14	f	f	PROPN
ejpam-4736	188	15	⊆	⊆	NUM
ejpam-4736	188	16	[	[	X
ejpam-4736	188	17	aδp(λ	aδp(λ	PROPN
ejpam-4736	188	18	,	,	PUNCT
ejpam-4736	188	19	s	s	NOUN
ejpam-4736	188	20	)	)	PUNCT
ejpam-4736	188	21	−	−	PROPN
ejpam-4736	188	22	a]δp(λ	a]δp(λ	NOUN
ejpam-4736	188	23	,	,	PUNCT
ejpam-4736	188	24	s	s	NOUN
ejpam-4736	188	25	)	)	PUNCT
ejpam-4736	188	26	.	.	PUNCT
ejpam-4736	189	1	it	it	PRON
ejpam-4736	189	2	follows	follow	VERB
ejpam-4736	189	3	from	from	ADP
ejpam-4736	189	4	theorem	theorem	NOUN
ejpam-4736	189	5	5	5	NUM
ejpam-4736	189	6	that	that	SCONJ
ejpam-4736	189	7	aδp(λ	aδp(λ	PROPN
ejpam-4736	189	8	,	,	PUNCT
ejpam-4736	189	9	s	s	PART
ejpam-4736	189	10	)	)	PUNCT
ejpam-4736	189	11	−	−	NOUN
ejpam-4736	189	12	a	a	PRON
ejpam-4736	189	13	is	be	AUX
ejpam-4736	189	14	g	g	NOUN
ejpam-4736	189	15	-	-	PUNCT
ejpam-4736	189	16	δp(λ	δp(λ	NOUN
ejpam-4736	189	17	,	,	PUNCT
ejpam-4736	189	18	s)-open	s)-open	PUNCT
ejpam-4736	189	19	.	.	PUNCT
ejpam-4736	190	1	c.	c.	PROPN
ejpam-4736	190	2	boonpok	boonpok	PROPN
ejpam-4736	190	3	,	,	PUNCT
ejpam-4736	190	4	n.	n.	PROPN
ejpam-4736	190	5	srisarakham	srisarakham	PROPN
ejpam-4736	190	6	/	/	SYM
ejpam-4736	190	7	eur	eur	PROPN
ejpam-4736	190	8	.	.	PUNCT
ejpam-4736	191	1	j.	j.	PROPN
ejpam-4736	191	2	pure	pure	PROPN
ejpam-4736	191	3	appl	appl	PROPN
ejpam-4736	191	4	.	.	PROPN
ejpam-4736	191	5	math	math	PROPN
ejpam-4736	191	6	,	,	PUNCT
ejpam-4736	191	7	16	16	NUM
ejpam-4736	191	8	(	(	PUNCT
ejpam-4736	191	9	4	4	NUM
ejpam-4736	191	10	)	)	PUNCT
ejpam-4736	191	11	(	(	PUNCT
ejpam-4736	191	12	2023	2023	NUM
ejpam-4736	191	13	)	)	PUNCT
ejpam-4736	191	14	,	,	PUNCT
ejpam-4736	191	15	2581	2581	NUM
ejpam-4736	191	16	-	-	SYM
ejpam-4736	191	17	2596	2596	NUM
ejpam-4736	191	18	2586	2586	NUM
ejpam-4736	191	19	(	(	PUNCT
ejpam-4736	191	20	3	3	X
ejpam-4736	191	21	)	)	PUNCT
ejpam-4736	191	22	⇒	⇒	NOUN
ejpam-4736	191	23	(	(	PUNCT
ejpam-4736	191	24	1	1	NUM
ejpam-4736	191	25	):	):	PUNCT
ejpam-4736	191	26	suppose	suppose	VERB
ejpam-4736	191	27	that	that	SCONJ
ejpam-4736	191	28	a	a	DET
ejpam-4736	191	29	⊆	⊆	NUM
ejpam-4736	191	30	u	u	NOUN
ejpam-4736	191	31	and	and	CCONJ
ejpam-4736	191	32	u	u	PROPN
ejpam-4736	191	33	∈	∈	PROPN
ejpam-4736	191	34	δp(λ	δp(λ	NOUN
ejpam-4736	191	35	,	,	PUNCT
ejpam-4736	191	36	s)o(x	s)o(x	PROPN
ejpam-4736	191	37	,	,	PUNCT
ejpam-4736	191	38	τ	τ	PROPN
ejpam-4736	191	39	)	)	PUNCT
ejpam-4736	191	40	.	.	PUNCT
ejpam-4736	192	1	then	then	ADV
ejpam-4736	192	2	,	,	PUNCT
ejpam-4736	192	3	aδp(λ	aδp(λ	PROPN
ejpam-4736	192	4	,	,	PUNCT
ejpam-4736	192	5	s	s	PART
ejpam-4736	192	6	)	)	PUNCT
ejpam-4736	192	7	−	−	PROPN
ejpam-4736	192	8	u	u	NOUN
ejpam-4736	192	9	⊆	⊆	NUM
ejpam-4736	192	10	aδp(λ	aδp(λ	PROPN
ejpam-4736	192	11	,	,	PUNCT
ejpam-4736	192	12	s	s	PART
ejpam-4736	192	13	)	)	PUNCT
ejpam-4736	192	14	−a	−a	NOUN
ejpam-4736	192	15	.	.	PUNCT
ejpam-4736	193	1	by	by	ADP
ejpam-4736	193	2	(	(	PUNCT
ejpam-4736	193	3	3	3	NUM
ejpam-4736	193	4	)	)	PUNCT
ejpam-4736	193	5	,	,	PUNCT
ejpam-4736	193	6	we	we	PRON
ejpam-4736	193	7	have	have	VERB
ejpam-4736	193	8	aδp(λ	aδp(λ	PROPN
ejpam-4736	193	9	,	,	PUNCT
ejpam-4736	193	10	s	s	PART
ejpam-4736	193	11	)	)	PUNCT
ejpam-4736	193	12	−	−	NOUN
ejpam-4736	193	13	a	a	PRON
ejpam-4736	193	14	is	be	AUX
ejpam-4736	193	15	g	g	NOUN
ejpam-4736	193	16	-	-	PUNCT
ejpam-4736	193	17	δp(λ	δp(λ	NOUN
ejpam-4736	193	18	,	,	PUNCT
ejpam-4736	193	19	s)-open	s)-open	VERB
ejpam-4736	193	20	.	.	PUNCT
ejpam-4736	194	1	since	since	SCONJ
ejpam-4736	194	2	aδp(λ	aδp(λ	PROPN
ejpam-4736	194	3	,	,	PUNCT
ejpam-4736	194	4	s	s	PART
ejpam-4736	194	5	)	)	PUNCT
ejpam-4736	194	6	−	−	PROPN
ejpam-4736	194	7	u	u	NOUN
ejpam-4736	194	8	is	be	AUX
ejpam-4736	194	9	δp(λ	δp(λ	NOUN
ejpam-4736	194	10	,	,	PUNCT
ejpam-4736	194	11	s)-closed	s)-close	VERB
ejpam-4736	194	12	,	,	PUNCT
ejpam-4736	194	13	by	by	ADP
ejpam-4736	194	14	theorem	theorem	NOUN
ejpam-4736	194	15	5	5	NUM
ejpam-4736	194	16	,	,	PUNCT
ejpam-4736	194	17	aδp(λ	aδp(λ	PROPN
ejpam-4736	194	18	,	,	PUNCT
ejpam-4736	194	19	s	s	PART
ejpam-4736	194	20	)	)	PUNCT
ejpam-4736	194	21	−	−	PROPN
ejpam-4736	194	22	u	u	NOUN
ejpam-4736	194	23	⊆	⊆	NUM
ejpam-4736	194	24	[	[	X
ejpam-4736	194	25	aδp(λ	aδp(λ	PROPN
ejpam-4736	194	26	,	,	PUNCT
ejpam-4736	194	27	s	s	NOUN
ejpam-4736	194	28	)	)	PUNCT
ejpam-4736	194	29	−	−	PROPN
ejpam-4736	194	30	a]δp(λ	a]δp(λ	NOUN
ejpam-4736	194	31	,	,	PUNCT
ejpam-4736	194	32	s	s	PART
ejpam-4736	194	33	)	)	PUNCT
ejpam-4736	194	34	=	=	PUNCT
ejpam-4736	194	35	∅.	∅.	ADP
ejpam-4736	194	36	thus	thus	ADV
ejpam-4736	194	37	,	,	PUNCT
ejpam-4736	194	38	aδp(λ	aδp(λ	PROPN
ejpam-4736	194	39	,	,	PUNCT
ejpam-4736	194	40	s	s	PART
ejpam-4736	194	41	)	)	PUNCT
ejpam-4736	194	42	⊆	⊆	NUM
ejpam-4736	194	43	u	u	NOUN
ejpam-4736	194	44	and	and	CCONJ
ejpam-4736	194	45	hence	hence	ADV
ejpam-4736	194	46	a	a	PRON
ejpam-4736	194	47	is	be	AUX
ejpam-4736	194	48	g	g	NOUN
ejpam-4736	194	49	-	-	PUNCT
ejpam-4736	194	50	δp(λ	δp(λ	NOUN
ejpam-4736	194	51	,	,	PUNCT
ejpam-4736	194	52	s)-closed	s)-close	VERB
ejpam-4736	194	53	.	.	PUNCT
ejpam-4736	195	1	now	now	ADV
ejpam-4736	195	2	,	,	PUNCT
ejpam-4736	195	3	the	the	DET
ejpam-4736	195	4	proof	proof	NOUN
ejpam-4736	195	5	of	of	ADP
ejpam-4736	195	6	[	[	X
ejpam-4736	195	7	aδp(λ	aδp(λ	PROPN
ejpam-4736	195	8	,	,	PUNCT
ejpam-4736	195	9	s	s	NOUN
ejpam-4736	195	10	)	)	PUNCT
ejpam-4736	195	11	−	−	PROPN
ejpam-4736	195	12	a]δp(λ	a]δp(λ	NOUN
ejpam-4736	195	13	,	,	PUNCT
ejpam-4736	195	14	s	s	PART
ejpam-4736	195	15	)	)	PUNCT
ejpam-4736	195	16	=	=	NOUN
ejpam-4736	195	17	∅	∅	NOUN
ejpam-4736	195	18	is	be	AUX
ejpam-4736	195	19	given	give	VERB
ejpam-4736	195	20	as	as	SCONJ
ejpam-4736	195	21	follows	follow	VERB
ejpam-4736	195	22	.	.	PUNCT
ejpam-4736	196	1	suppose	suppose	VERB
ejpam-4736	196	2	that	that	SCONJ
ejpam-4736	196	3	[	[	X
ejpam-4736	196	4	aδp(λ	aδp(λ	PROPN
ejpam-4736	196	5	,	,	PUNCT
ejpam-4736	196	6	s	s	NOUN
ejpam-4736	196	7	)	)	PUNCT
ejpam-4736	196	8	−	−	PROPN
ejpam-4736	196	9	a]δp(λ	a]δp(λ	PROPN
ejpam-4736	196	10	,	,	PUNCT
ejpam-4736	196	11	s	s	PART
ejpam-4736	196	12	)	)	PUNCT
ejpam-4736	196	13	̸=	̸=	PROPN
ejpam-4736	196	14	∅.	∅.	NOUN
ejpam-4736	196	15	then	then	ADV
ejpam-4736	196	16	,	,	PUNCT
ejpam-4736	196	17	there	there	PRON
ejpam-4736	196	18	exists	exist	VERB
ejpam-4736	196	19	x	x	X
ejpam-4736	196	20	∈	∈	PROPN
ejpam-4736	196	21	[	[	X
ejpam-4736	196	22	aδp(λ	aδp(λ	PROPN
ejpam-4736	196	23	,	,	PUNCT
ejpam-4736	196	24	s	s	NOUN
ejpam-4736	196	25	)	)	PUNCT
ejpam-4736	196	26	−	−	PROPN
ejpam-4736	196	27	a]δp(λ	a]δp(λ	NOUN
ejpam-4736	196	28	,	,	PUNCT
ejpam-4736	196	29	s	s	PART
ejpam-4736	196	30	)	)	PUNCT
ejpam-4736	196	31	and	and	CCONJ
ejpam-4736	196	32	hence	hence	ADV
ejpam-4736	196	33	there	there	PRON
ejpam-4736	196	34	exists	exist	VERB
ejpam-4736	196	35	g	g	PROPN
ejpam-4736	196	36	∈	∈	PROPN
ejpam-4736	196	37	δp(λ	δp(λ	NOUN
ejpam-4736	196	38	,	,	PUNCT
ejpam-4736	196	39	s)o(x	s)o(x	PROPN
ejpam-4736	196	40	,	,	PUNCT
ejpam-4736	196	41	τ	τ	X
ejpam-4736	196	42	)	)	PUNCT
ejpam-4736	196	43	such	such	ADJ
ejpam-4736	196	44	that	that	SCONJ
ejpam-4736	196	45	x	x	SYM
ejpam-4736	196	46	∈	∈	NOUN
ejpam-4736	196	47	g	g	NOUN
ejpam-4736	196	48	⊆	⊆	NUM
ejpam-4736	196	49	aδp(λ	aδp(λ	PROPN
ejpam-4736	196	50	,	,	PUNCT
ejpam-4736	196	51	s	s	PART
ejpam-4736	196	52	)	)	PUNCT
ejpam-4736	196	53	−	−	NOUN
ejpam-4736	196	54	a.	a.	NOUN
ejpam-4736	196	55	since	since	SCONJ
ejpam-4736	196	56	g	g	PROPN
ejpam-4736	196	57	⊆	⊆	NUM
ejpam-4736	196	58	x	x	SYM
ejpam-4736	196	59	−	−	NOUN
ejpam-4736	196	60	a	a	X
ejpam-4736	196	61	,	,	PUNCT
ejpam-4736	196	62	we	we	PRON
ejpam-4736	196	63	have	have	VERB
ejpam-4736	196	64	g	g	NOUN
ejpam-4736	196	65	∩	∩	NOUN
ejpam-4736	196	66	a	a	DET
ejpam-4736	196	67	=	=	NOUN
ejpam-4736	196	68	∅	∅	NOUN
ejpam-4736	196	69	,	,	PUNCT
ejpam-4736	196	70	by	by	ADP
ejpam-4736	196	71	lemma	lemma	PROPN
ejpam-4736	196	72	4	4	NUM
ejpam-4736	196	73	,	,	PUNCT
ejpam-4736	196	74	g	g	PROPN
ejpam-4736	196	75	∩	∩	ADJ
ejpam-4736	196	76	aδp(λ	aδp(λ	PROPN
ejpam-4736	196	77	,	,	PUNCT
ejpam-4736	196	78	s	s	PART
ejpam-4736	196	79	)	)	PUNCT
ejpam-4736	196	80	=	=	NOUN
ejpam-4736	196	81	∅	∅	NOUN
ejpam-4736	196	82	and	and	CCONJ
ejpam-4736	196	83	hence	hence	ADV
ejpam-4736	196	84	g	g	PROPN
ejpam-4736	196	85	⊆	⊆	NUM
ejpam-4736	197	1	x	x	SYM
ejpam-4736	197	2	−	−	PROPN
ejpam-4736	197	3	aδp(λ	aδp(λ	PROPN
ejpam-4736	197	4	,	,	PUNCT
ejpam-4736	197	5	s	s	NOUN
ejpam-4736	197	6	)	)	PUNCT
ejpam-4736	197	7	.	.	PUNCT
ejpam-4736	198	1	thus	thus	ADV
ejpam-4736	198	2	,	,	PUNCT
ejpam-4736	198	3	g	g	PROPN
ejpam-4736	198	4	⊆	⊆	NUM
ejpam-4736	198	5	[	[	X
ejpam-4736	198	6	x	x	SYM
ejpam-4736	198	7	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	198	8	,	,	PUNCT
ejpam-4736	198	9	s	s	NOUN
ejpam-4736	198	10	)	)	PUNCT
ejpam-4736	198	11	]	]	PUNCT
ejpam-4736	198	12	∩aδp(λ	∩aδp(λ	NUM
ejpam-4736	198	13	,	,	PUNCT
ejpam-4736	198	14	s	s	NOUN
ejpam-4736	198	15	)	)	PUNCT
ejpam-4736	198	16	=	=	PUNCT
ejpam-4736	198	17	∅.	∅.	ADP
ejpam-4736	198	18	this	this	PRON
ejpam-4736	198	19	is	be	AUX
ejpam-4736	198	20	a	a	DET
ejpam-4736	198	21	contradiction	contradiction	NOUN
ejpam-4736	198	22	.	.	PUNCT
ejpam-4736	199	1	theorem	theorem	VERB
ejpam-4736	199	2	7	7	NUM
ejpam-4736	199	3	.	.	PUNCT
ejpam-4736	200	1	a	a	DET
ejpam-4736	200	2	subset	subset	NOUN
ejpam-4736	200	3	a	a	PRON
ejpam-4736	200	4	of	of	ADP
ejpam-4736	200	5	a	a	DET
ejpam-4736	200	6	topological	topological	ADJ
ejpam-4736	200	7	space	space	NOUN
ejpam-4736	200	8	(	(	PUNCT
ejpam-4736	200	9	x	x	X
ejpam-4736	200	10	,	,	PUNCT
ejpam-4736	200	11	τ	τ	X
ejpam-4736	200	12	)	)	PUNCT
ejpam-4736	200	13	is	be	AUX
ejpam-4736	200	14	g	g	NOUN
ejpam-4736	200	15	-	-	PUNCT
ejpam-4736	200	16	δp(λ	δp(λ	NOUN
ejpam-4736	200	17	,	,	PUNCT
ejpam-4736	200	18	s)-closed	s)-close	VERB
ejpam-4736	200	19	if	if	SCONJ
ejpam-4736	200	20	and	and	CCONJ
ejpam-4736	200	21	only	only	ADV
ejpam-4736	201	1	if	if	SCONJ
ejpam-4736	201	2	f	f	PROPN
ejpam-4736	201	3	∩aδp(λ	∩aδp(λ	X
ejpam-4736	201	4	,	,	PUNCT
ejpam-4736	201	5	s	s	NOUN
ejpam-4736	201	6	)	)	PUNCT
ejpam-4736	201	7	=	=	PUNCT
ejpam-4736	201	8	∅	∅	NOUN
ejpam-4736	201	9	whenever	whenever	SCONJ
ejpam-4736	201	10	a	a	DET
ejpam-4736	201	11	∩	∩	ADJ
ejpam-4736	201	12	f	f	NOUN
ejpam-4736	201	13	=	=	NOUN
ejpam-4736	201	14	∅	∅	NOUN
ejpam-4736	201	15	and	and	CCONJ
ejpam-4736	201	16	f	f	PROPN
ejpam-4736	201	17	is	be	AUX
ejpam-4736	201	18	δp(λ	δp(λ	NOUN
ejpam-4736	201	19	,	,	PUNCT
ejpam-4736	201	20	s)-closed	s)-close	VERB
ejpam-4736	201	21	.	.	PUNCT
ejpam-4736	202	1	proof	proof	NOUN
ejpam-4736	202	2	.	.	PUNCT
ejpam-4736	203	1	suppose	suppose	VERB
ejpam-4736	203	2	that	that	SCONJ
ejpam-4736	203	3	a	a	PRON
ejpam-4736	203	4	is	be	AUX
ejpam-4736	203	5	a	a	DET
ejpam-4736	203	6	δp(λ	δp(λ	NOUN
ejpam-4736	203	7	,	,	PUNCT
ejpam-4736	203	8	s)-closed	s)-close	VERB
ejpam-4736	203	9	set	set	NOUN
ejpam-4736	203	10	.	.	PUNCT
ejpam-4736	204	1	let	let	VERB
ejpam-4736	204	2	f	f	PRON
ejpam-4736	204	3	be	be	AUX
ejpam-4736	204	4	a	a	DET
ejpam-4736	204	5	δp(λ	δp(λ	NOUN
ejpam-4736	204	6	,	,	PUNCT
ejpam-4736	204	7	s)-closed	s)-close	VERB
ejpam-4736	204	8	set	set	NOUN
ejpam-4736	204	9	and	and	CCONJ
ejpam-4736	204	10	a	a	DET
ejpam-4736	204	11	∩	∩	ADJ
ejpam-4736	204	12	f	f	X
ejpam-4736	204	13	=	=	PUNCT
ejpam-4736	204	14	∅.	∅.	NOUN
ejpam-4736	204	15	then	then	ADV
ejpam-4736	204	16	,	,	PUNCT
ejpam-4736	204	17	a	a	DET
ejpam-4736	204	18	⊆	⊆	NUM
ejpam-4736	204	19	x	x	SYM
ejpam-4736	204	20	−	−	PROPN
ejpam-4736	204	21	f	f	PROPN
ejpam-4736	204	22	∈	∈	PROPN
ejpam-4736	204	23	δp(λ	δp(λ	NOUN
ejpam-4736	204	24	,	,	PUNCT
ejpam-4736	204	25	s)o(x	s)o(x	PROPN
ejpam-4736	204	26	,	,	PUNCT
ejpam-4736	204	27	τ	τ	X
ejpam-4736	204	28	)	)	PUNCT
ejpam-4736	204	29	and	and	CCONJ
ejpam-4736	204	30	aδp(λ	aδp(λ	PROPN
ejpam-4736	204	31	,	,	PUNCT
ejpam-4736	204	32	s	s	PART
ejpam-4736	204	33	)	)	PUNCT
ejpam-4736	204	34	⊆	⊆	NUM
ejpam-4736	204	35	x	x	SYM
ejpam-4736	204	36	−	−	PROPN
ejpam-4736	204	37	f	f	NOUN
ejpam-4736	204	38	.	.	PUNCT
ejpam-4736	205	1	thus	thus	ADV
ejpam-4736	205	2	,	,	PUNCT
ejpam-4736	205	3	f	f	PROPN
ejpam-4736	205	4	∩aδp(λ	∩aδp(λ	PROPN
ejpam-4736	205	5	,	,	PUNCT
ejpam-4736	205	6	s	s	NOUN
ejpam-4736	205	7	)	)	PUNCT
ejpam-4736	205	8	=	=	PUNCT
ejpam-4736	205	9	∅.	∅.	VERB
ejpam-4736	205	10	conversely	conversely	ADV
ejpam-4736	205	11	,	,	PUNCT
ejpam-4736	205	12	let	let	VERB
ejpam-4736	205	13	a	a	DET
ejpam-4736	205	14	⊆	⊆	NUM
ejpam-4736	205	15	u	u	NOUN
ejpam-4736	205	16	and	and	CCONJ
ejpam-4736	205	17	u	u	PROPN
ejpam-4736	205	18	∈	∈	PROPN
ejpam-4736	205	19	δp(λ	δp(λ	NOUN
ejpam-4736	205	20	,	,	PUNCT
ejpam-4736	205	21	s)o(x	s)o(x	PROPN
ejpam-4736	205	22	,	,	PUNCT
ejpam-4736	205	23	τ	τ	PROPN
ejpam-4736	205	24	)	)	PUNCT
ejpam-4736	205	25	.	.	PUNCT
ejpam-4736	206	1	then	then	ADV
ejpam-4736	206	2	,	,	PUNCT
ejpam-4736	206	3	a∩	a∩	PROPN
ejpam-4736	206	4	(	(	PUNCT
ejpam-4736	206	5	x	x	NOUN
ejpam-4736	206	6	−u	−u	PROPN
ejpam-4736	206	7	)	)	PUNCT
ejpam-4736	206	8	=	=	NOUN
ejpam-4736	206	9	∅	∅	NOUN
ejpam-4736	206	10	and	and	CCONJ
ejpam-4736	206	11	x	x	SYM
ejpam-4736	206	12	−u	−u	NOUN
ejpam-4736	206	13	is	be	AUX
ejpam-4736	206	14	δp(λ	δp(λ	NOUN
ejpam-4736	206	15	,	,	PUNCT
ejpam-4736	206	16	s)-closed	s)-close	VERB
ejpam-4736	206	17	.	.	PUNCT
ejpam-4736	207	1	by	by	ADP
ejpam-4736	207	2	the	the	DET
ejpam-4736	207	3	hypothesis	hypothesis	NOUN
ejpam-4736	207	4	,	,	PUNCT
ejpam-4736	207	5	(	(	PUNCT
ejpam-4736	207	6	x	x	X
ejpam-4736	207	7	−u	−u	PROPN
ejpam-4736	207	8	)	)	PUNCT
ejpam-4736	207	9	∩aδp(λ	∩aδp(λ	VERB
ejpam-4736	207	10	,	,	PUNCT
ejpam-4736	207	11	s	s	NOUN
ejpam-4736	207	12	)	)	PUNCT
ejpam-4736	207	13	=	=	NOUN
ejpam-4736	207	14	∅	∅	NOUN
ejpam-4736	207	15	and	and	CCONJ
ejpam-4736	207	16	hence	hence	ADV
ejpam-4736	207	17	aδp(λ	aδp(λ	PROPN
ejpam-4736	207	18	,	,	PUNCT
ejpam-4736	207	19	s	s	PART
ejpam-4736	207	20	)	)	PUNCT
ejpam-4736	207	21	⊆	⊆	NUM
ejpam-4736	207	22	u	u	NOUN
ejpam-4736	207	23	.	.	PUNCT
ejpam-4736	208	1	thus	thus	ADV
ejpam-4736	208	2	,	,	PUNCT
ejpam-4736	208	3	a	a	PRON
ejpam-4736	208	4	is	be	AUX
ejpam-4736	208	5	g	g	NOUN
ejpam-4736	208	6	-	-	PUNCT
ejpam-4736	208	7	δp(λ	δp(λ	NOUN
ejpam-4736	208	8	,	,	PUNCT
ejpam-4736	208	9	s)-closed	s)-close	VERB
ejpam-4736	208	10	.	.	PUNCT
ejpam-4736	209	1	theorem	theorem	VERB
ejpam-4736	209	2	8	8	NUM
ejpam-4736	209	3	.	.	PUNCT
ejpam-4736	210	1	a	a	DET
ejpam-4736	210	2	subset	subset	NOUN
ejpam-4736	210	3	a	a	PRON
ejpam-4736	210	4	of	of	ADP
ejpam-4736	210	5	a	a	DET
ejpam-4736	210	6	topological	topological	ADJ
ejpam-4736	210	7	space	space	NOUN
ejpam-4736	210	8	(	(	PUNCT
ejpam-4736	210	9	x	x	X
ejpam-4736	210	10	,	,	PUNCT
ejpam-4736	210	11	τ	τ	X
ejpam-4736	210	12	)	)	PUNCT
ejpam-4736	210	13	is	be	AUX
ejpam-4736	210	14	g	g	NOUN
ejpam-4736	210	15	-	-	PUNCT
ejpam-4736	210	16	δp(λ	δp(λ	NOUN
ejpam-4736	210	17	,	,	PUNCT
ejpam-4736	210	18	s)-closed	s)-close	VERB
ejpam-4736	210	19	if	if	SCONJ
ejpam-4736	210	20	and	and	CCONJ
ejpam-4736	210	21	only	only	ADV
ejpam-4736	210	22	if	if	SCONJ
ejpam-4736	210	23	a	a	DET
ejpam-4736	210	24	∩	∩	NOUN
ejpam-4736	210	25	{	{	PUNCT
ejpam-4736	210	26	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	210	27	,	,	PUNCT
ejpam-4736	210	28	s	s	PART
ejpam-4736	210	29	)	)	PUNCT
ejpam-4736	210	30	̸=	̸=	NOUN
ejpam-4736	210	31	∅	∅	NOUN
ejpam-4736	210	32	for	for	ADP
ejpam-4736	210	33	every	every	DET
ejpam-4736	210	34	x	x	PROPN
ejpam-4736	210	35	∈	∈	PROPN
ejpam-4736	210	36	aδp(λ	aδp(λ	PROPN
ejpam-4736	210	37	,	,	PUNCT
ejpam-4736	210	38	s	s	NOUN
ejpam-4736	210	39	)	)	PUNCT
ejpam-4736	210	40	.	.	PUNCT
ejpam-4736	211	1	proof	proof	NOUN
ejpam-4736	211	2	.	.	PUNCT
ejpam-4736	212	1	let	let	VERB
ejpam-4736	212	2	a	a	PRON
ejpam-4736	212	3	be	be	AUX
ejpam-4736	212	4	a	a	DET
ejpam-4736	212	5	g	g	NOUN
ejpam-4736	212	6	-	-	PUNCT
ejpam-4736	212	7	δp(λ	δp(λ	NOUN
ejpam-4736	212	8	,	,	PUNCT
ejpam-4736	212	9	s)-closed	s)-close	VERB
ejpam-4736	212	10	set	set	VERB
ejpam-4736	212	11	and	and	CCONJ
ejpam-4736	212	12	suppose	suppose	VERB
ejpam-4736	212	13	that	that	SCONJ
ejpam-4736	212	14	there	there	PRON
ejpam-4736	212	15	exists	exist	VERB
ejpam-4736	212	16	x	x	X
ejpam-4736	212	17	∈	∈	PROPN
ejpam-4736	212	18	aδp(λ	aδp(λ	PROPN
ejpam-4736	212	19	,	,	PUNCT
ejpam-4736	212	20	s	s	PART
ejpam-4736	212	21	)	)	PUNCT
ejpam-4736	212	22	such	such	ADJ
ejpam-4736	212	23	that	that	SCONJ
ejpam-4736	212	24	a	a	DET
ejpam-4736	212	25	∩	∩	NOUN
ejpam-4736	212	26	{	{	PUNCT
ejpam-4736	212	27	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	212	28	,	,	PUNCT
ejpam-4736	212	29	s	s	PART
ejpam-4736	212	30	)	)	PUNCT
ejpam-4736	212	31	=	=	PUNCT
ejpam-4736	212	32	∅.	∅.	ADP
ejpam-4736	212	33	thus	thus	ADV
ejpam-4736	212	34	,	,	PUNCT
ejpam-4736	212	35	a	a	DET
ejpam-4736	212	36	⊆	⊆	NUM
ejpam-4736	212	37	x	x	SYM
ejpam-4736	212	38	−	−	PROPN
ejpam-4736	212	39	{	{	PUNCT
ejpam-4736	212	40	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	212	41	,	,	PUNCT
ejpam-4736	212	42	s	s	PART
ejpam-4736	212	43	)	)	PUNCT
ejpam-4736	212	44	and	and	CCONJ
ejpam-4736	212	45	hence	hence	ADV
ejpam-4736	212	46	aδp(λ	aδp(λ	PROPN
ejpam-4736	212	47	,	,	PUNCT
ejpam-4736	212	48	s	s	PART
ejpam-4736	212	49	)	)	PUNCT
ejpam-4736	212	50	⊆	⊆	NUM
ejpam-4736	212	51	x	x	SYM
ejpam-4736	212	52	−	−	PROPN
ejpam-4736	212	53	{	{	PUNCT
ejpam-4736	212	54	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	212	55	,	,	PUNCT
ejpam-4736	212	56	s	s	NOUN
ejpam-4736	212	57	)	)	PUNCT
ejpam-4736	212	58	.	.	PUNCT
ejpam-4736	213	1	therefore	therefore	ADV
ejpam-4736	213	2	,	,	PUNCT
ejpam-4736	213	3	x	x	PROPN
ejpam-4736	213	4	̸∈	̸∈	PROPN
ejpam-4736	213	5	aδp(λ	aδp(λ	PROPN
ejpam-4736	213	6	,	,	PUNCT
ejpam-4736	213	7	s	s	PART
ejpam-4736	213	8	)	)	PUNCT
ejpam-4736	213	9	,	,	PUNCT
ejpam-4736	213	10	which	which	PRON
ejpam-4736	213	11	is	be	AUX
ejpam-4736	213	12	a	a	DET
ejpam-4736	213	13	contradiction	contradiction	NOUN
ejpam-4736	213	14	.	.	PUNCT
ejpam-4736	214	1	conversely	conversely	ADV
ejpam-4736	214	2	,	,	PUNCT
ejpam-4736	214	3	suppose	suppose	VERB
ejpam-4736	214	4	that	that	SCONJ
ejpam-4736	214	5	the	the	DET
ejpam-4736	214	6	condition	condition	NOUN
ejpam-4736	214	7	of	of	ADP
ejpam-4736	214	8	the	the	DET
ejpam-4736	214	9	theorem	theorem	NOUN
ejpam-4736	214	10	holds	hold	VERB
ejpam-4736	214	11	and	and	CCONJ
ejpam-4736	214	12	let	let	VERB
ejpam-4736	214	13	u	u	PRON
ejpam-4736	214	14	be	be	AUX
ejpam-4736	214	15	any	any	DET
ejpam-4736	214	16	δp(λ	δp(λ	NOUN
ejpam-4736	214	17	,	,	PUNCT
ejpam-4736	214	18	s)open	s)open	NOUN
ejpam-4736	214	19	set	set	NOUN
ejpam-4736	214	20	containing	contain	VERB
ejpam-4736	214	21	a.	a.	NOUN
ejpam-4736	214	22	let	let	VERB
ejpam-4736	214	23	x	x	X
ejpam-4736	214	24	∈	∈	PROPN
ejpam-4736	214	25	aδp(λ	aδp(λ	PROPN
ejpam-4736	214	26	,	,	PUNCT
ejpam-4736	214	27	s	s	NOUN
ejpam-4736	214	28	)	)	PUNCT
ejpam-4736	214	29	.	.	PUNCT
ejpam-4736	215	1	by	by	ADP
ejpam-4736	215	2	the	the	DET
ejpam-4736	215	3	hypothesis	hypothesis	NOUN
ejpam-4736	215	4	,	,	PUNCT
ejpam-4736	215	5	a	a	DET
ejpam-4736	215	6	∩	∩	ADJ
ejpam-4736	215	7	aδp(λ	aδp(λ	PROPN
ejpam-4736	215	8	,	,	PUNCT
ejpam-4736	215	9	s	s	PART
ejpam-4736	215	10	)	)	PUNCT
ejpam-4736	215	11	̸=	̸=	NOUN
ejpam-4736	215	12	∅	∅	NOUN
ejpam-4736	215	13	,	,	PUNCT
ejpam-4736	215	14	so	so	SCONJ
ejpam-4736	215	15	there	there	PRON
ejpam-4736	215	16	exists	exist	VERB
ejpam-4736	215	17	y	y	PROPN
ejpam-4736	215	18	∈	∈	PROPN
ejpam-4736	215	19	a	a	DET
ejpam-4736	215	20	∩	∩	NOUN
ejpam-4736	215	21	{	{	PUNCT
ejpam-4736	215	22	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	215	23	,	,	PUNCT
ejpam-4736	215	24	s	s	PART
ejpam-4736	215	25	)	)	PUNCT
ejpam-4736	215	26	and	and	CCONJ
ejpam-4736	215	27	hence	hence	ADV
ejpam-4736	215	28	y	y	PROPN
ejpam-4736	215	29	∈	∈	PROPN
ejpam-4736	215	30	a	a	DET
ejpam-4736	215	31	⊆	⊆	NUM
ejpam-4736	215	32	u	u	NOUN
ejpam-4736	215	33	.	.	PUNCT
ejpam-4736	216	1	thus	thus	ADV
ejpam-4736	216	2	,	,	PUNCT
ejpam-4736	216	3	{	{	PUNCT
ejpam-4736	216	4	x	x	NOUN
ejpam-4736	216	5	}	}	PUNCT
ejpam-4736	216	6	∩	∩	NOUN
ejpam-4736	216	7	u	u	NOUN
ejpam-4736	216	8	̸=	̸=	PROPN
ejpam-4736	216	9	∅.	∅.	VERB
ejpam-4736	216	10	therefore	therefore	ADV
ejpam-4736	216	11	,	,	PUNCT
ejpam-4736	216	12	x	x	PUNCT
ejpam-4736	216	13	∈	∈	PROPN
ejpam-4736	216	14	u	u	NOUN
ejpam-4736	216	15	,	,	PUNCT
ejpam-4736	216	16	which	which	PRON
ejpam-4736	216	17	implies	imply	VERB
ejpam-4736	216	18	that	that	SCONJ
ejpam-4736	216	19	aδp(λ	aδp(λ	PROPN
ejpam-4736	216	20	,	,	PUNCT
ejpam-4736	216	21	s	s	PART
ejpam-4736	216	22	)	)	PUNCT
ejpam-4736	216	23	⊆	⊆	NUM
ejpam-4736	216	24	u	u	NOUN
ejpam-4736	216	25	.	.	PUNCT
ejpam-4736	217	1	this	this	PRON
ejpam-4736	217	2	shows	show	VERB
ejpam-4736	217	3	that	that	SCONJ
ejpam-4736	217	4	a	a	PRON
ejpam-4736	217	5	is	be	AUX
ejpam-4736	217	6	g	g	NOUN
ejpam-4736	217	7	-	-	PUNCT
ejpam-4736	217	8	δp(λ	δp(λ	NOUN
ejpam-4736	217	9	,	,	PUNCT
ejpam-4736	217	10	s)-closed	s)-close	VERB
ejpam-4736	217	11	.	.	PUNCT
ejpam-4736	218	1	corollary	corollary	ADJ
ejpam-4736	218	2	3	3	NUM
ejpam-4736	218	3	.	.	PUNCT
ejpam-4736	219	1	for	for	ADP
ejpam-4736	219	2	a	a	DET
ejpam-4736	219	3	subset	subset	NOUN
ejpam-4736	219	4	a	a	PRON
ejpam-4736	219	5	of	of	ADP
ejpam-4736	219	6	a	a	DET
ejpam-4736	219	7	topological	topological	ADJ
ejpam-4736	219	8	space	space	NOUN
ejpam-4736	219	9	(	(	PUNCT
ejpam-4736	219	10	x	x	X
ejpam-4736	219	11	,	,	PUNCT
ejpam-4736	219	12	τ	τ	PROPN
ejpam-4736	219	13	)	)	PUNCT
ejpam-4736	219	14	,	,	PUNCT
ejpam-4736	219	15	the	the	DET
ejpam-4736	219	16	following	follow	VERB
ejpam-4736	219	17	properties	property	NOUN
ejpam-4736	219	18	are	be	AUX
ejpam-4736	219	19	equivalent	equivalent	ADJ
ejpam-4736	219	20	:	:	PUNCT
ejpam-4736	219	21	(	(	PUNCT
ejpam-4736	219	22	1	1	X
ejpam-4736	219	23	)	)	PUNCT
ejpam-4736	219	24	a	a	PRON
ejpam-4736	219	25	is	be	AUX
ejpam-4736	219	26	g	g	NOUN
ejpam-4736	219	27	-	-	PUNCT
ejpam-4736	219	28	δp(λ	δp(λ	NOUN
ejpam-4736	219	29	,	,	PUNCT
ejpam-4736	219	30	s)-open	s)-open	ADJ
ejpam-4736	219	31	.	.	PUNCT
ejpam-4736	220	1	(	(	PUNCT
ejpam-4736	220	2	2	2	X
ejpam-4736	220	3	)	)	PUNCT
ejpam-4736	220	4	a−aδp(λ	a−aδp(λ	NUM
ejpam-4736	220	5	,	,	PUNCT
ejpam-4736	220	6	s	s	PART
ejpam-4736	220	7	)	)	PUNCT
ejpam-4736	220	8	does	do	AUX
ejpam-4736	220	9	not	not	PART
ejpam-4736	220	10	contain	contain	VERB
ejpam-4736	220	11	any	any	DET
ejpam-4736	220	12	nonempty	nonempty	ADJ
ejpam-4736	220	13	δp(λ	δp(λ	NOUN
ejpam-4736	220	14	,	,	PUNCT
ejpam-4736	220	15	s)-closed	s)-close	VERB
ejpam-4736	220	16	set	set	NOUN
ejpam-4736	220	17	.	.	PUNCT
ejpam-4736	221	1	(	(	PUNCT
ejpam-4736	221	2	3	3	NUM
ejpam-4736	221	3	)	)	PUNCT
ejpam-4736	221	4	(	(	PUNCT
ejpam-4736	221	5	x	x	NOUN
ejpam-4736	221	6	−a	−a	ADJ
ejpam-4736	221	7	)	)	PUNCT
ejpam-4736	221	8	∩	∩	NOUN
ejpam-4736	221	9	{	{	PUNCT
ejpam-4736	221	10	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	221	11	,	,	PUNCT
ejpam-4736	221	12	s	s	PART
ejpam-4736	221	13	)	)	PUNCT
ejpam-4736	221	14	̸=	̸=	NOUN
ejpam-4736	221	15	∅	∅	NOUN
ejpam-4736	221	16	for	for	ADP
ejpam-4736	221	17	every	every	DET
ejpam-4736	221	18	x	x	SYM
ejpam-4736	221	19	∈	∈	PROPN
ejpam-4736	221	20	a−aδp(λ	a−aδp(λ	NUM
ejpam-4736	221	21	,	,	PUNCT
ejpam-4736	221	22	s	s	NOUN
ejpam-4736	221	23	)	)	PUNCT
ejpam-4736	221	24	.	.	PUNCT
ejpam-4736	222	1	theorem	theorem	VERB
ejpam-4736	222	2	9	9	NUM
ejpam-4736	222	3	.	.	X
ejpam-4736	223	1	for	for	ADP
ejpam-4736	223	2	a	a	DET
ejpam-4736	223	3	topological	topological	ADJ
ejpam-4736	223	4	space	space	NOUN
ejpam-4736	223	5	(	(	PUNCT
ejpam-4736	223	6	x	x	X
ejpam-4736	223	7	,	,	PUNCT
ejpam-4736	223	8	τ	τ	PROPN
ejpam-4736	223	9	)	)	PUNCT
ejpam-4736	223	10	,	,	PUNCT
ejpam-4736	223	11	the	the	DET
ejpam-4736	223	12	following	follow	VERB
ejpam-4736	223	13	properties	property	NOUN
ejpam-4736	223	14	are	be	AUX
ejpam-4736	223	15	equivalent	equivalent	ADJ
ejpam-4736	223	16	:	:	PUNCT
ejpam-4736	223	17	c.	c.	PROPN
ejpam-4736	223	18	boonpok	boonpok	PROPN
ejpam-4736	223	19	,	,	PUNCT
ejpam-4736	223	20	n.	n.	PROPN
ejpam-4736	223	21	srisarakham	srisarakham	PROPN
ejpam-4736	223	22	/	/	SYM
ejpam-4736	223	23	eur	eur	PROPN
ejpam-4736	223	24	.	.	PUNCT
ejpam-4736	224	1	j.	j.	PROPN
ejpam-4736	224	2	pure	pure	PROPN
ejpam-4736	224	3	appl	appl	PROPN
ejpam-4736	224	4	.	.	PROPN
ejpam-4736	224	5	math	math	PROPN
ejpam-4736	224	6	,	,	PUNCT
ejpam-4736	224	7	16	16	NUM
ejpam-4736	224	8	(	(	PUNCT
ejpam-4736	224	9	4	4	NUM
ejpam-4736	224	10	)	)	PUNCT
ejpam-4736	224	11	(	(	PUNCT
ejpam-4736	224	12	2023	2023	NUM
ejpam-4736	224	13	)	)	PUNCT
ejpam-4736	224	14	,	,	PUNCT
ejpam-4736	224	15	2581	2581	NUM
ejpam-4736	224	16	-	-	SYM
ejpam-4736	224	17	2596	2596	NUM
ejpam-4736	224	18	2587	2587	NUM
ejpam-4736	224	19	(	(	PUNCT
ejpam-4736	224	20	1	1	NUM
ejpam-4736	224	21	)	)	PUNCT
ejpam-4736	224	22	for	for	ADP
ejpam-4736	224	23	every	every	DET
ejpam-4736	224	24	δp(λ	δp(λ	NOUN
ejpam-4736	224	25	,	,	PUNCT
ejpam-4736	224	26	s)-open	s)-open	VERB
ejpam-4736	224	27	set	set	VERB
ejpam-4736	224	28	u	u	NOUN
ejpam-4736	224	29	of	of	ADP
ejpam-4736	224	30	x	x	PROPN
ejpam-4736	224	31	,	,	PUNCT
ejpam-4736	224	32	u	u	NOUN
ejpam-4736	224	33	δp(λ	δp(λ	NOUN
ejpam-4736	224	34	,	,	PUNCT
ejpam-4736	224	35	s	s	PART
ejpam-4736	224	36	)	)	PUNCT
ejpam-4736	224	37	⊆	⊆	NUM
ejpam-4736	224	38	u	u	NOUN
ejpam-4736	224	39	.	.	PUNCT
ejpam-4736	225	1	(	(	PUNCT
ejpam-4736	225	2	2	2	X
ejpam-4736	225	3	)	)	PUNCT
ejpam-4736	225	4	every	every	DET
ejpam-4736	225	5	subset	subset	NOUN
ejpam-4736	225	6	of	of	ADP
ejpam-4736	225	7	x	x	PUNCT
ejpam-4736	225	8	is	be	AUX
ejpam-4736	225	9	g	g	NOUN
ejpam-4736	225	10	-	-	PUNCT
ejpam-4736	225	11	δp(λ	δp(λ	NOUN
ejpam-4736	225	12	,	,	PUNCT
ejpam-4736	225	13	s)-closed	s)-close	VERB
ejpam-4736	225	14	.	.	PUNCT
ejpam-4736	226	1	proof	proof	NOUN
ejpam-4736	226	2	.	.	PUNCT
ejpam-4736	227	1	(	(	PUNCT
ejpam-4736	227	2	1	1	X
ejpam-4736	227	3	)	)	PUNCT
ejpam-4736	227	4	⇒	⇒	NOUN
ejpam-4736	227	5	(	(	PUNCT
ejpam-4736	227	6	2	2	NUM
ejpam-4736	227	7	):	):	PUNCT
ejpam-4736	227	8	let	let	VERB
ejpam-4736	227	9	a	a	PRON
ejpam-4736	227	10	be	be	AUX
ejpam-4736	227	11	any	any	DET
ejpam-4736	227	12	subset	subset	NOUN
ejpam-4736	227	13	of	of	ADP
ejpam-4736	227	14	x	x	PUNCT
ejpam-4736	227	15	and	and	CCONJ
ejpam-4736	227	16	a	a	DET
ejpam-4736	227	17	⊆	⊆	NUM
ejpam-4736	227	18	u	u	NOUN
ejpam-4736	227	19	∈	∈	PROPN
ejpam-4736	227	20	δ(λ	δ(λ	PROPN
ejpam-4736	227	21	,	,	PUNCT
ejpam-4736	227	22	s)o(x	s)o(x	PROPN
ejpam-4736	227	23	,	,	PUNCT
ejpam-4736	227	24	τ	τ	X
ejpam-4736	227	25	)	)	PUNCT
ejpam-4736	227	26	.	.	PUNCT
ejpam-4736	228	1	by	by	ADP
ejpam-4736	228	2	(	(	PUNCT
ejpam-4736	228	3	1	1	NUM
ejpam-4736	228	4	)	)	PUNCT
ejpam-4736	228	5	,	,	PUNCT
ejpam-4736	228	6	u	u	NOUN
ejpam-4736	228	7	δp(λ	δp(λ	NOUN
ejpam-4736	228	8	,	,	PUNCT
ejpam-4736	228	9	s	s	PART
ejpam-4736	228	10	)	)	PUNCT
ejpam-4736	228	11	⊆	⊆	NUM
ejpam-4736	228	12	u	u	NOUN
ejpam-4736	228	13	and	and	CCONJ
ejpam-4736	228	14	hence	hence	ADV
ejpam-4736	228	15	aδp(λ	aδp(λ	PROPN
ejpam-4736	228	16	,	,	PUNCT
ejpam-4736	228	17	s	s	PART
ejpam-4736	228	18	)	)	PUNCT
ejpam-4736	228	19	⊆	⊆	NUM
ejpam-4736	228	20	u	u	NOUN
ejpam-4736	228	21	δp(λ	δp(λ	NOUN
ejpam-4736	228	22	,	,	PUNCT
ejpam-4736	228	23	s	s	PART
ejpam-4736	228	24	)	)	PUNCT
ejpam-4736	228	25	⊆	⊆	NUM
ejpam-4736	228	26	u	u	NOUN
ejpam-4736	228	27	.	.	PUNCT
ejpam-4736	229	1	thus	thus	ADV
ejpam-4736	229	2	,	,	PUNCT
ejpam-4736	229	3	a	a	PRON
ejpam-4736	229	4	is	be	AUX
ejpam-4736	229	5	g	g	NOUN
ejpam-4736	229	6	-	-	PUNCT
ejpam-4736	229	7	δp(λ	δp(λ	NOUN
ejpam-4736	229	8	,	,	PUNCT
ejpam-4736	229	9	s)-closed	s)-close	VERB
ejpam-4736	229	10	.	.	PUNCT
ejpam-4736	230	1	(	(	PUNCT
ejpam-4736	230	2	2	2	X
ejpam-4736	230	3	)	)	PUNCT
ejpam-4736	230	4	⇒	⇒	NOUN
ejpam-4736	230	5	(	(	PUNCT
ejpam-4736	230	6	1	1	NUM
ejpam-4736	230	7	):	):	PUNCT
ejpam-4736	230	8	let	let	VERB
ejpam-4736	230	9	u	u	PRON
ejpam-4736	230	10	∈	∈	PROPN
ejpam-4736	230	11	δp(λ	δp(λ	NOUN
ejpam-4736	230	12	,	,	PUNCT
ejpam-4736	230	13	s)o(x	s)o(x	PROPN
ejpam-4736	230	14	,	,	PUNCT
ejpam-4736	230	15	τ	τ	X
ejpam-4736	230	16	)	)	PUNCT
ejpam-4736	230	17	.	.	PUNCT
ejpam-4736	231	1	by	by	ADP
ejpam-4736	231	2	(	(	PUNCT
ejpam-4736	231	3	2	2	NUM
ejpam-4736	231	4	)	)	PUNCT
ejpam-4736	231	5	,	,	PUNCT
ejpam-4736	231	6	u	u	NOUN
ejpam-4736	231	7	is	be	AUX
ejpam-4736	231	8	g	g	NOUN
ejpam-4736	231	9	-	-	PUNCT
ejpam-4736	231	10	δp(λ	δp(λ	NOUN
ejpam-4736	231	11	,	,	PUNCT
ejpam-4736	231	12	s)-closed	s)-close	VERB
ejpam-4736	231	13	and	and	CCONJ
ejpam-4736	231	14	hence	hence	ADV
ejpam-4736	231	15	u	u	PRON
ejpam-4736	231	16	δp(λ	δp(λ	NOUN
ejpam-4736	231	17	,	,	PUNCT
ejpam-4736	231	18	s	s	PART
ejpam-4736	231	19	)	)	PUNCT
ejpam-4736	232	1	⊆	⊆	NUM
ejpam-4736	232	2	u.	u.	NOUN
ejpam-4736	232	3	theorem	theorem	VERB
ejpam-4736	232	4	10	10	NUM
ejpam-4736	232	5	.	.	PUNCT
ejpam-4736	233	1	a	a	DET
ejpam-4736	233	2	subset	subset	NOUN
ejpam-4736	233	3	a	a	PRON
ejpam-4736	233	4	of	of	ADP
ejpam-4736	233	5	a	a	DET
ejpam-4736	233	6	topological	topological	ADJ
ejpam-4736	233	7	space	space	NOUN
ejpam-4736	233	8	(	(	PUNCT
ejpam-4736	233	9	x	x	X
ejpam-4736	233	10	,	,	PUNCT
ejpam-4736	233	11	τ	τ	X
ejpam-4736	233	12	)	)	PUNCT
ejpam-4736	233	13	is	be	AUX
ejpam-4736	233	14	g	g	NOUN
ejpam-4736	233	15	-	-	PUNCT
ejpam-4736	233	16	δp(λ	δp(λ	NOUN
ejpam-4736	233	17	,	,	PUNCT
ejpam-4736	233	18	s)-open	s)-open	VERB
ejpam-4736	233	19	if	if	SCONJ
ejpam-4736	233	20	and	and	CCONJ
ejpam-4736	233	21	only	only	ADV
ejpam-4736	233	22	if	if	SCONJ
ejpam-4736	233	23	u	u	NOUN
ejpam-4736	233	24	=	=	NOUN
ejpam-4736	233	25	x	x	INTJ
ejpam-4736	233	26	whenever	whenever	SCONJ
ejpam-4736	233	27	u	u	NOUN
ejpam-4736	233	28	is	be	AUX
ejpam-4736	233	29	δp(λ	δp(λ	NOUN
ejpam-4736	233	30	,	,	PUNCT
ejpam-4736	233	31	s)-open	s)-open	PUNCT
ejpam-4736	233	32	and	and	CCONJ
ejpam-4736	233	33	(	(	PUNCT
ejpam-4736	233	34	x	x	NOUN
ejpam-4736	233	35	−a	−a	NOUN
ejpam-4736	233	36	)	)	PUNCT
ejpam-4736	233	37	∩aδp(λ	∩aδp(λ	NOUN
ejpam-4736	233	38	,	,	PUNCT
ejpam-4736	233	39	s	s	PART
ejpam-4736	233	40	)	)	PUNCT
ejpam-4736	233	41	⊆	⊆	NUM
ejpam-4736	233	42	u	u	NOUN
ejpam-4736	233	43	.	.	PUNCT
ejpam-4736	234	1	proof	proof	NOUN
ejpam-4736	234	2	.	.	PUNCT
ejpam-4736	235	1	suppose	suppose	VERB
ejpam-4736	235	2	that	that	SCONJ
ejpam-4736	235	3	a	a	PRON
ejpam-4736	235	4	is	be	AUX
ejpam-4736	235	5	g	g	NOUN
ejpam-4736	235	6	-	-	PUNCT
ejpam-4736	235	7	δp(λ	δp(λ	NOUN
ejpam-4736	235	8	,	,	PUNCT
ejpam-4736	235	9	s)-open	s)-open	PUNCT
ejpam-4736	235	10	and	and	CCONJ
ejpam-4736	235	11	u	u	PROPN
ejpam-4736	235	12	∈	∈	PROPN
ejpam-4736	235	13	δp(λ	δp(λ	NOUN
ejpam-4736	235	14	,	,	PUNCT
ejpam-4736	235	15	s)o(x	s)o(x	PROPN
ejpam-4736	235	16	,	,	PUNCT
ejpam-4736	235	17	τ	τ	X
ejpam-4736	235	18	)	)	PUNCT
ejpam-4736	235	19	such	such	ADJ
ejpam-4736	235	20	that	that	SCONJ
ejpam-4736	235	21	(	(	PUNCT
ejpam-4736	235	22	x	x	NOUN
ejpam-4736	235	23	−a	−a	NOUN
ejpam-4736	235	24	)	)	PUNCT
ejpam-4736	235	25	∩aδp(λ	∩aδp(λ	NOUN
ejpam-4736	235	26	,	,	PUNCT
ejpam-4736	235	27	s	s	PART
ejpam-4736	235	28	)	)	PUNCT
ejpam-4736	235	29	⊆	⊆	NUM
ejpam-4736	235	30	u.	u.	NOUN
ejpam-4736	235	31	thus	thus	ADV
ejpam-4736	235	32	,	,	PUNCT
ejpam-4736	235	33	x	x	PUNCT
ejpam-4736	235	34	−	−	NOUN
ejpam-4736	235	35	u	u	NOUN
ejpam-4736	235	36	⊆	⊆	NUM
ejpam-4736	235	37	[	[	X
ejpam-4736	235	38	x	x	X
ejpam-4736	235	39	−	−	PROPN
ejpam-4736	235	40	aδp(λ	aδp(λ	PROPN
ejpam-4736	235	41	,	,	PUNCT
ejpam-4736	235	42	s	s	NOUN
ejpam-4736	235	43	)	)	PUNCT
ejpam-4736	235	44	]	]	PUNCT
ejpam-4736	235	45	∩	∩	NOUN
ejpam-4736	235	46	a	a	X
ejpam-4736	235	47	and	and	CCONJ
ejpam-4736	235	48	hence	hence	ADV
ejpam-4736	235	49	x	x	ADP
ejpam-4736	235	50	−	−	NOUN
ejpam-4736	235	51	u	u	NOUN
ejpam-4736	235	52	⊆	⊆	NUM
ejpam-4736	235	53	[	[	X
ejpam-4736	235	54	x	x	X
ejpam-4736	235	55	−	−	NOUN
ejpam-4736	235	56	a]δp(λ	a]δp(λ	NOUN
ejpam-4736	235	57	,	,	PUNCT
ejpam-4736	235	58	s	s	NOUN
ejpam-4736	235	59	)	)	PUNCT
ejpam-4736	235	60	−	−	PROPN
ejpam-4736	236	1	(	(	PUNCT
ejpam-4736	236	2	x	x	X
ejpam-4736	236	3	−	−	NOUN
ejpam-4736	236	4	a	a	NOUN
ejpam-4736	236	5	)	)	PUNCT
ejpam-4736	236	6	.	.	PUNCT
ejpam-4736	237	1	since	since	SCONJ
ejpam-4736	237	2	x	x	PRON
ejpam-4736	237	3	−a	−a	NOUN
ejpam-4736	237	4	is	be	AUX
ejpam-4736	237	5	g	g	NOUN
ejpam-4736	237	6	-	-	PUNCT
ejpam-4736	237	7	δp(λ	δp(λ	NOUN
ejpam-4736	237	8	,	,	PUNCT
ejpam-4736	237	9	s)-closed	s)-close	VERB
ejpam-4736	237	10	and	and	CCONJ
ejpam-4736	237	11	x	x	X
ejpam-4736	238	1	−	−	NOUN
ejpam-4736	238	2	u	u	NOUN
ejpam-4736	238	3	is	be	AUX
ejpam-4736	238	4	δp(λ	δp(λ	NOUN
ejpam-4736	238	5	,	,	PUNCT
ejpam-4736	238	6	s)-closed	s)-close	VERB
ejpam-4736	238	7	,	,	PUNCT
ejpam-4736	238	8	by	by	ADP
ejpam-4736	238	9	theorem	theorem	NOUN
ejpam-4736	238	10	1	1	NUM
ejpam-4736	238	11	,	,	PUNCT
ejpam-4736	238	12	x	x	PUNCT
ejpam-4736	238	13	−	−	NOUN
ejpam-4736	238	14	u	u	NOUN
ejpam-4736	238	15	=	=	PUNCT
ejpam-4736	238	16	∅.	∅.	NOUN
ejpam-4736	238	17	this	this	PRON
ejpam-4736	238	18	shows	show	VERB
ejpam-4736	238	19	that	that	SCONJ
ejpam-4736	238	20	x	x	X
ejpam-4736	238	21	=	=	SYM
ejpam-4736	238	22	u	u	NOUN
ejpam-4736	238	23	.	.	PUNCT
ejpam-4736	239	1	conversely	conversely	ADV
ejpam-4736	239	2	,	,	PUNCT
ejpam-4736	239	3	suppose	suppose	VERB
ejpam-4736	239	4	that	that	SCONJ
ejpam-4736	239	5	f	f	PROPN
ejpam-4736	239	6	⊆	⊆	NUM
ejpam-4736	239	7	a	a	PRON
ejpam-4736	239	8	and	and	CCONJ
ejpam-4736	239	9	f	f	PROPN
ejpam-4736	239	10	is	be	AUX
ejpam-4736	239	11	δp(λ	δp(λ	NOUN
ejpam-4736	239	12	,	,	PUNCT
ejpam-4736	239	13	s)-closed	s)-close	VERB
ejpam-4736	239	14	.	.	PUNCT
ejpam-4736	240	1	by	by	ADP
ejpam-4736	240	2	lemma	lemma	PROPN
ejpam-4736	240	3	3	3	NUM
ejpam-4736	240	4	,	,	PUNCT
ejpam-4736	240	5	(	(	PUNCT
ejpam-4736	240	6	x	x	NOUN
ejpam-4736	240	7	−a	−a	NOUN
ejpam-4736	240	8	)	)	PUNCT
ejpam-4736	240	9	∪aδp(λ	∪aδp(λ	PROPN
ejpam-4736	240	10	,	,	PUNCT
ejpam-4736	240	11	s	s	PART
ejpam-4736	240	12	)	)	PUNCT
ejpam-4736	240	13	⊆	⊆	NUM
ejpam-4736	240	14	(	(	PUNCT
ejpam-4736	240	15	x	x	SYM
ejpam-4736	240	16	−	−	PROPN
ejpam-4736	240	17	f	f	PROPN
ejpam-4736	240	18	)	)	PUNCT
ejpam-4736	240	19	∪aδp(λ	∪aδp(λ	PROPN
ejpam-4736	240	20	,	,	PUNCT
ejpam-4736	240	21	s	s	PART
ejpam-4736	240	22	)	)	PUNCT
ejpam-4736	240	23	∈	∈	PROPN
ejpam-4736	240	24	δp(λ	δp(λ	NOUN
ejpam-4736	240	25	,	,	PUNCT
ejpam-4736	240	26	s)o(x	s)o(x	PROPN
ejpam-4736	240	27	,	,	PUNCT
ejpam-4736	240	28	τ	τ	X
ejpam-4736	240	29	)	)	PUNCT
ejpam-4736	240	30	.	.	PUNCT
ejpam-4736	241	1	by	by	ADP
ejpam-4736	241	2	the	the	DET
ejpam-4736	241	3	hypothesis	hypothesis	NOUN
ejpam-4736	241	4	,	,	PUNCT
ejpam-4736	241	5	we	we	PRON
ejpam-4736	241	6	have	have	VERB
ejpam-4736	241	7	x	x	X
ejpam-4736	241	8	=	=	PRON
ejpam-4736	241	9	(	(	PUNCT
ejpam-4736	241	10	x	x	SYM
ejpam-4736	241	11	−	−	PROPN
ejpam-4736	241	12	f	f	PROPN
ejpam-4736	241	13	)	)	PUNCT
ejpam-4736	241	14	∪aδp(λ	∪aδp(λ	PROPN
ejpam-4736	241	15	,	,	PUNCT
ejpam-4736	241	16	s	s	PART
ejpam-4736	241	17	)	)	PUNCT
ejpam-4736	241	18	and	and	CCONJ
ejpam-4736	241	19	hence	hence	ADV
ejpam-4736	241	20	f	f	PROPN
ejpam-4736	241	21	=	=	SYM
ejpam-4736	241	22	f	f	PROPN
ejpam-4736	241	23	∩	∩	X
ejpam-4736	241	24	[	[	X
ejpam-4736	241	25	(	(	PUNCT
ejpam-4736	241	26	x	x	SYM
ejpam-4736	241	27	−	−	PROPN
ejpam-4736	241	28	f	f	PROPN
ejpam-4736	241	29	)	)	PUNCT
ejpam-4736	241	30	∪aδp(λ	∪aδp(λ	PROPN
ejpam-4736	241	31	,	,	PUNCT
ejpam-4736	241	32	s	s	NOUN
ejpam-4736	241	33	)	)	PUNCT
ejpam-4736	241	34	]	]	PUNCT
ejpam-4736	242	1	=	=	SYM
ejpam-4736	242	2	f	f	PROPN
ejpam-4736	242	3	∩aδp(λ	∩aδp(λ	X
ejpam-4736	242	4	,	,	PUNCT
ejpam-4736	242	5	s	s	PART
ejpam-4736	242	6	)	)	PUNCT
ejpam-4736	242	7	⊆	⊆	NUM
ejpam-4736	242	8	aδp(λ	aδp(λ	PROPN
ejpam-4736	242	9	,	,	PUNCT
ejpam-4736	242	10	s	s	NOUN
ejpam-4736	242	11	)	)	PUNCT
ejpam-4736	242	12	.	.	PUNCT
ejpam-4736	243	1	it	it	PRON
ejpam-4736	243	2	follows	follow	VERB
ejpam-4736	243	3	from	from	ADP
ejpam-4736	243	4	theorem	theorem	ADJ
ejpam-4736	243	5	5	5	NUM
ejpam-4736	243	6	that	that	SCONJ
ejpam-4736	243	7	a	a	PRON
ejpam-4736	243	8	is	be	AUX
ejpam-4736	243	9	g	g	NOUN
ejpam-4736	243	10	-	-	PUNCT
ejpam-4736	243	11	δp(λ	δp(λ	NOUN
ejpam-4736	243	12	,	,	PUNCT
ejpam-4736	243	13	s)-open	s)-open	PUNCT
ejpam-4736	243	14	.	.	PUNCT
ejpam-4736	244	1	theorem	theorem	VERB
ejpam-4736	244	2	11	11	NUM
ejpam-4736	244	3	.	.	PUNCT
ejpam-4736	245	1	let	let	VERB
ejpam-4736	245	2	a	a	DET
ejpam-4736	245	3	be	be	AUX
ejpam-4736	245	4	a	a	DET
ejpam-4736	245	5	subset	subset	NOUN
ejpam-4736	245	6	of	of	ADP
ejpam-4736	245	7	a	a	DET
ejpam-4736	245	8	topological	topological	ADJ
ejpam-4736	245	9	space	space	NOUN
ejpam-4736	245	10	(	(	PUNCT
ejpam-4736	245	11	x	x	X
ejpam-4736	245	12	,	,	PUNCT
ejpam-4736	245	13	τ	τ	PROPN
ejpam-4736	245	14	)	)	PUNCT
ejpam-4736	245	15	.	.	PUNCT
ejpam-4736	246	1	if	if	SCONJ
ejpam-4736	246	2	a	a	PRON
ejpam-4736	246	3	is	be	AUX
ejpam-4736	246	4	g	g	NOUN
ejpam-4736	246	5	-	-	PUNCT
ejpam-4736	246	6	δp(λ	δp(λ	NOUN
ejpam-4736	246	7	,	,	PUNCT
ejpam-4736	246	8	s)-open	s)-open	PUNCT
ejpam-4736	246	9	and	and	CCONJ
ejpam-4736	246	10	aδp(λ	aδp(λ	PROPN
ejpam-4736	246	11	,	,	PUNCT
ejpam-4736	246	12	s	s	PART
ejpam-4736	246	13	)	)	PUNCT
ejpam-4736	246	14	⊆	⊆	NUM
ejpam-4736	246	15	b	b	NOUN
ejpam-4736	246	16	⊆	⊆	NUM
ejpam-4736	246	17	a	a	PRON
ejpam-4736	246	18	,	,	PUNCT
ejpam-4736	246	19	then	then	ADV
ejpam-4736	246	20	b	b	PROPN
ejpam-4736	246	21	is	be	AUX
ejpam-4736	246	22	g	g	NOUN
ejpam-4736	246	23	-	-	PUNCT
ejpam-4736	246	24	δp(λ	δp(λ	NOUN
ejpam-4736	246	25	,	,	PUNCT
ejpam-4736	246	26	s)-open	s)-open	PUNCT
ejpam-4736	246	27	.	.	PUNCT
ejpam-4736	247	1	proof	proof	NOUN
ejpam-4736	247	2	.	.	PUNCT
ejpam-4736	248	1	we	we	PRON
ejpam-4736	248	2	have	have	VERB
ejpam-4736	248	3	x	x	ADJ
ejpam-4736	248	4	−	−	VERB
ejpam-4736	248	5	a	a	DET
ejpam-4736	248	6	⊆	⊆	NUM
ejpam-4736	248	7	x	x	SYM
ejpam-4736	248	8	−	−	PROPN
ejpam-4736	248	9	b	b	NOUN
ejpam-4736	248	10	⊆	⊆	NUM
ejpam-4736	248	11	x	x	SYM
ejpam-4736	248	12	−	−	PROPN
ejpam-4736	248	13	aδp(λ	aδp(λ	PROPN
ejpam-4736	248	14	,	,	PUNCT
ejpam-4736	248	15	s	s	PART
ejpam-4736	248	16	)	)	PUNCT
ejpam-4736	248	17	=	=	PUNCT
ejpam-4736	249	1	[	[	X
ejpam-4736	249	2	x	x	X
ejpam-4736	249	3	−	−	NOUN
ejpam-4736	249	4	a]δp(λ	a]δp(λ	NOUN
ejpam-4736	249	5	,	,	PUNCT
ejpam-4736	249	6	s	s	NOUN
ejpam-4736	249	7	)	)	PUNCT
ejpam-4736	249	8	.	.	PUNCT
ejpam-4736	250	1	since	since	SCONJ
ejpam-4736	250	2	x	x	X
ejpam-4736	250	3	−	−	PROPN
ejpam-4736	250	4	a	a	PRON
ejpam-4736	250	5	is	be	AUX
ejpam-4736	250	6	g	g	NOUN
ejpam-4736	250	7	-	-	PUNCT
ejpam-4736	250	8	δp(λ	δp(λ	NOUN
ejpam-4736	250	9	,	,	PUNCT
ejpam-4736	250	10	s)-closed	s)-close	VERB
ejpam-4736	250	11	,	,	PUNCT
ejpam-4736	250	12	it	it	PRON
ejpam-4736	250	13	follows	follow	VERB
ejpam-4736	250	14	from	from	ADP
ejpam-4736	250	15	theorem	theorem	ADJ
ejpam-4736	250	16	2	2	NUM
ejpam-4736	250	17	that	that	PRON
ejpam-4736	250	18	x	x	PRON
ejpam-4736	250	19	−b	−b	NOUN
ejpam-4736	250	20	is	be	AUX
ejpam-4736	250	21	g	g	NOUN
ejpam-4736	250	22	-	-	PUNCT
ejpam-4736	250	23	δp(λ	δp(λ	NOUN
ejpam-4736	250	24	,	,	PUNCT
ejpam-4736	250	25	s)-closed	s)-close	VERB
ejpam-4736	250	26	and	and	CCONJ
ejpam-4736	250	27	hence	hence	ADV
ejpam-4736	250	28	b	b	PROPN
ejpam-4736	250	29	is	be	AUX
ejpam-4736	250	30	g	g	NOUN
ejpam-4736	250	31	-	-	PUNCT
ejpam-4736	250	32	δp(λ	δp(λ	NOUN
ejpam-4736	250	33	,	,	PUNCT
ejpam-4736	250	34	s)-open	s)-open	PUNCT
ejpam-4736	250	35	.	.	PUNCT
ejpam-4736	251	1	definition	definition	NOUN
ejpam-4736	251	2	5	5	NUM
ejpam-4736	251	3	.	.	PUNCT
ejpam-4736	252	1	a	a	DET
ejpam-4736	252	2	subset	subset	NOUN
ejpam-4736	252	3	a	a	PRON
ejpam-4736	252	4	of	of	ADP
ejpam-4736	252	5	a	a	DET
ejpam-4736	252	6	topological	topological	ADJ
ejpam-4736	252	7	space	space	NOUN
ejpam-4736	252	8	(	(	PUNCT
ejpam-4736	252	9	x	x	X
ejpam-4736	252	10	,	,	PUNCT
ejpam-4736	252	11	τ	τ	X
ejpam-4736	252	12	)	)	PUNCT
ejpam-4736	252	13	is	be	AUX
ejpam-4736	252	14	said	say	VERB
ejpam-4736	252	15	to	to	PART
ejpam-4736	252	16	be	be	AUX
ejpam-4736	252	17	locally	locally	ADV
ejpam-4736	252	18	δp(λ	δp(λ	NOUN
ejpam-4736	252	19	,	,	PUNCT
ejpam-4736	252	20	s)-closed	s)-close	VERB
ejpam-4736	252	21	if	if	SCONJ
ejpam-4736	252	22	a	a	DET
ejpam-4736	252	23	=	=	X
ejpam-4736	252	24	u	u	NOUN
ejpam-4736	252	25	∩	∩	NOUN
ejpam-4736	252	26	f	f	PROPN
ejpam-4736	252	27	,	,	PUNCT
ejpam-4736	252	28	where	where	SCONJ
ejpam-4736	252	29	u	u	PROPN
ejpam-4736	252	30	∈	∈	PROPN
ejpam-4736	252	31	δp(λ	δp(λ	NOUN
ejpam-4736	252	32	,	,	PUNCT
ejpam-4736	252	33	s)o(x	s)o(x	PROPN
ejpam-4736	252	34	,	,	PUNCT
ejpam-4736	252	35	τ	τ	X
ejpam-4736	252	36	)	)	PUNCT
ejpam-4736	252	37	and	and	CCONJ
ejpam-4736	252	38	f	f	PROPN
ejpam-4736	252	39	is	be	AUX
ejpam-4736	252	40	a	a	DET
ejpam-4736	252	41	δp(λ	δp(λ	NOUN
ejpam-4736	252	42	,	,	PUNCT
ejpam-4736	252	43	s)-closed	s)-close	VERB
ejpam-4736	252	44	set	set	NOUN
ejpam-4736	252	45	.	.	PUNCT
ejpam-4736	253	1	lemma	lemma	PROPN
ejpam-4736	253	2	5	5	NUM
ejpam-4736	253	3	.	.	PUNCT
ejpam-4736	254	1	for	for	ADP
ejpam-4736	254	2	a	a	DET
ejpam-4736	254	3	subset	subset	NOUN
ejpam-4736	254	4	a	a	PRON
ejpam-4736	254	5	of	of	ADP
ejpam-4736	254	6	a	a	DET
ejpam-4736	254	7	topological	topological	ADJ
ejpam-4736	254	8	space	space	NOUN
ejpam-4736	254	9	(	(	PUNCT
ejpam-4736	254	10	x	x	X
ejpam-4736	254	11	,	,	PUNCT
ejpam-4736	254	12	τ	τ	PROPN
ejpam-4736	254	13	)	)	PUNCT
ejpam-4736	254	14	,	,	PUNCT
ejpam-4736	254	15	the	the	DET
ejpam-4736	254	16	following	follow	VERB
ejpam-4736	254	17	properties	property	NOUN
ejpam-4736	254	18	are	be	AUX
ejpam-4736	254	19	equivalent	equivalent	ADJ
ejpam-4736	254	20	:	:	PUNCT
ejpam-4736	254	21	(	(	PUNCT
ejpam-4736	254	22	1	1	X
ejpam-4736	254	23	)	)	PUNCT
ejpam-4736	254	24	a	a	PRON
ejpam-4736	254	25	is	be	AUX
ejpam-4736	254	26	locally	locally	ADV
ejpam-4736	254	27	δp(λ	δp(λ	NOUN
ejpam-4736	254	28	,	,	PUNCT
ejpam-4736	254	29	s)-closed	s)-close	VERB
ejpam-4736	254	30	;	;	PUNCT
ejpam-4736	254	31	(	(	PUNCT
ejpam-4736	254	32	2	2	X
ejpam-4736	254	33	)	)	PUNCT
ejpam-4736	254	34	a	a	DET
ejpam-4736	254	35	=	=	X
ejpam-4736	254	36	u	u	NOUN
ejpam-4736	254	37	∩aδp(λ	∩aδp(λ	NOUN
ejpam-4736	254	38	,	,	PUNCT
ejpam-4736	254	39	s	s	NOUN
ejpam-4736	254	40	)	)	PUNCT
ejpam-4736	254	41	for	for	ADP
ejpam-4736	254	42	some	some	DET
ejpam-4736	254	43	u	u	PROPN
ejpam-4736	254	44	∈	∈	PROPN
ejpam-4736	254	45	δp(λ	δp(λ	NOUN
ejpam-4736	254	46	,	,	PUNCT
ejpam-4736	254	47	s)o(x	s)o(x	PROPN
ejpam-4736	254	48	,	,	PUNCT
ejpam-4736	254	49	τ	τ	X
ejpam-4736	254	50	)	)	PUNCT
ejpam-4736	254	51	;	;	PUNCT
ejpam-4736	254	52	c.	c.	PROPN
ejpam-4736	254	53	boonpok	boonpok	PROPN
ejpam-4736	254	54	,	,	PUNCT
ejpam-4736	254	55	n.	n.	PROPN
ejpam-4736	254	56	srisarakham	srisarakham	PROPN
ejpam-4736	254	57	/	/	SYM
ejpam-4736	254	58	eur	eur	PROPN
ejpam-4736	254	59	.	.	PUNCT
ejpam-4736	255	1	j.	j.	PROPN
ejpam-4736	255	2	pure	pure	PROPN
ejpam-4736	255	3	appl	appl	PROPN
ejpam-4736	255	4	.	.	PROPN
ejpam-4736	255	5	math	math	PROPN
ejpam-4736	255	6	,	,	PUNCT
ejpam-4736	255	7	16	16	NUM
ejpam-4736	255	8	(	(	PUNCT
ejpam-4736	255	9	4	4	NUM
ejpam-4736	255	10	)	)	PUNCT
ejpam-4736	255	11	(	(	PUNCT
ejpam-4736	255	12	2023	2023	NUM
ejpam-4736	255	13	)	)	PUNCT
ejpam-4736	255	14	,	,	PUNCT
ejpam-4736	255	15	2581	2581	NUM
ejpam-4736	255	16	-	-	SYM
ejpam-4736	255	17	2596	2596	NUM
ejpam-4736	255	18	2588	2588	NUM
ejpam-4736	255	19	(	(	PUNCT
ejpam-4736	255	20	3	3	X
ejpam-4736	255	21	)	)	PUNCT
ejpam-4736	255	22	aδp(λ	aδp(λ	PROPN
ejpam-4736	255	23	,	,	PUNCT
ejpam-4736	255	24	s	s	PART
ejpam-4736	255	25	)	)	PUNCT
ejpam-4736	255	26	−a	−a	NOUN
ejpam-4736	255	27	is	be	AUX
ejpam-4736	255	28	δp(λ	δp(λ	NOUN
ejpam-4736	255	29	,	,	PUNCT
ejpam-4736	255	30	s)-closed	s)-close	VERB
ejpam-4736	255	31	;	;	PUNCT
ejpam-4736	255	32	(	(	PUNCT
ejpam-4736	255	33	4	4	X
ejpam-4736	255	34	)	)	PUNCT
ejpam-4736	256	1	[	[	X
ejpam-4736	256	2	a	a	DET
ejpam-4736	256	3	∪	∪	X
ejpam-4736	256	4	(	(	PUNCT
ejpam-4736	256	5	x	x	PROPN
ejpam-4736	256	6	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	256	7	,	,	PUNCT
ejpam-4736	256	8	s	s	NOUN
ejpam-4736	256	9	)	)	PUNCT
ejpam-4736	256	10	)	)	PUNCT
ejpam-4736	256	11	]	]	PUNCT
ejpam-4736	256	12	∈	∈	PROPN
ejpam-4736	256	13	δp(λ	δp(λ	NOUN
ejpam-4736	256	14	,	,	PUNCT
ejpam-4736	256	15	s)o(x	s)o(x	PROPN
ejpam-4736	256	16	,	,	PUNCT
ejpam-4736	256	17	τ	τ	X
ejpam-4736	256	18	)	)	PUNCT
ejpam-4736	256	19	;	;	PUNCT
ejpam-4736	256	20	(	(	PUNCT
ejpam-4736	256	21	5	5	X
ejpam-4736	256	22	)	)	PUNCT
ejpam-4736	256	23	a	a	DET
ejpam-4736	256	24	⊆	⊆	NUM
ejpam-4736	256	25	[	[	X
ejpam-4736	256	26	a	a	DET
ejpam-4736	256	27	∪	∪	ADJ
ejpam-4736	256	28	[	[	X
ejpam-4736	256	29	x	x	SYM
ejpam-4736	256	30	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	256	31	,	,	PUNCT
ejpam-4736	256	32	s)]]δp(λ	s)]]δp(λ	PROPN
ejpam-4736	256	33	,	,	PUNCT
ejpam-4736	256	34	s	s	NOUN
ejpam-4736	256	35	)	)	PUNCT
ejpam-4736	256	36	.	.	PUNCT
ejpam-4736	257	1	proof	proof	NOUN
ejpam-4736	257	2	.	.	PUNCT
ejpam-4736	258	1	(	(	PUNCT
ejpam-4736	258	2	1	1	X
ejpam-4736	258	3	)	)	PUNCT
ejpam-4736	258	4	⇒	⇒	NOUN
ejpam-4736	258	5	(	(	PUNCT
ejpam-4736	258	6	2	2	NUM
ejpam-4736	258	7	):	):	PUNCT
ejpam-4736	258	8	let	let	VERB
ejpam-4736	258	9	a	a	DET
ejpam-4736	258	10	=	=	X
ejpam-4736	258	11	u	u	NOUN
ejpam-4736	258	12	∩f	∩f	NOUN
ejpam-4736	258	13	,	,	PUNCT
ejpam-4736	258	14	where	where	SCONJ
ejpam-4736	258	15	u	u	PROPN
ejpam-4736	258	16	∈	∈	PROPN
ejpam-4736	258	17	δp(λ	δp(λ	NOUN
ejpam-4736	258	18	,	,	PUNCT
ejpam-4736	258	19	s)o(x	s)o(x	PROPN
ejpam-4736	258	20	,	,	PUNCT
ejpam-4736	258	21	τ	τ	X
ejpam-4736	258	22	)	)	PUNCT
ejpam-4736	258	23	and	and	CCONJ
ejpam-4736	258	24	f	f	PROPN
ejpam-4736	258	25	is	be	AUX
ejpam-4736	258	26	δp(λ	δp(λ	NOUN
ejpam-4736	258	27	,	,	PUNCT
ejpam-4736	258	28	s)-closed	s)-close	VERB
ejpam-4736	258	29	.	.	PUNCT
ejpam-4736	259	1	since	since	SCONJ
ejpam-4736	259	2	a	a	DET
ejpam-4736	259	3	⊆	⊆	NUM
ejpam-4736	259	4	f	f	NOUN
ejpam-4736	259	5	,	,	PUNCT
ejpam-4736	259	6	we	we	PRON
ejpam-4736	259	7	have	have	VERB
ejpam-4736	259	8	aδp(λ	aδp(λ	PROPN
ejpam-4736	259	9	,	,	PUNCT
ejpam-4736	259	10	s	s	PART
ejpam-4736	259	11	)	)	PUNCT
ejpam-4736	259	12	⊆	⊆	NUM
ejpam-4736	259	13	f	f	PROPN
ejpam-4736	259	14	δp(λ	δp(λ	NOUN
ejpam-4736	259	15	,	,	PUNCT
ejpam-4736	259	16	s	s	PART
ejpam-4736	259	17	)	)	PUNCT
ejpam-4736	259	18	=	=	SYM
ejpam-4736	259	19	f	f	PROPN
ejpam-4736	259	20	.	.	PUNCT
ejpam-4736	260	1	since	since	SCONJ
ejpam-4736	260	2	a	a	DET
ejpam-4736	260	3	⊆	⊆	NUM
ejpam-4736	260	4	u	u	NOUN
ejpam-4736	260	5	,	,	PUNCT
ejpam-4736	260	6	a	a	DET
ejpam-4736	260	7	⊆	⊆	NUM
ejpam-4736	260	8	u∩aδp(λ	u∩aδp(λ	PROPN
ejpam-4736	260	9	,	,	PUNCT
ejpam-4736	260	10	s	s	PART
ejpam-4736	260	11	)	)	PUNCT
ejpam-4736	260	12	⊆	⊆	NUM
ejpam-4736	260	13	u∩f	u∩f	ADJ
ejpam-4736	260	14	=	=	NOUN
ejpam-4736	260	15	a.	a.	NOUN
ejpam-4736	260	16	thus	thus	ADV
ejpam-4736	260	17	,	,	PUNCT
ejpam-4736	260	18	a	a	DET
ejpam-4736	260	19	=	=	X
ejpam-4736	260	20	u	u	NOUN
ejpam-4736	260	21	∩aδp(λ	∩aδp(λ	NOUN
ejpam-4736	260	22	,	,	PUNCT
ejpam-4736	260	23	s	s	NOUN
ejpam-4736	260	24	)	)	PUNCT
ejpam-4736	260	25	.	.	PUNCT
ejpam-4736	261	1	(	(	PUNCT
ejpam-4736	261	2	2	2	X
ejpam-4736	261	3	)	)	PUNCT
ejpam-4736	261	4	⇒	⇒	NOUN
ejpam-4736	261	5	(	(	PUNCT
ejpam-4736	261	6	3	3	NUM
ejpam-4736	261	7	):	):	PUNCT
ejpam-4736	261	8	suppose	suppose	VERB
ejpam-4736	261	9	that	that	SCONJ
ejpam-4736	261	10	a	a	DET
ejpam-4736	261	11	=	=	SYM
ejpam-4736	261	12	u	u	NOUN
ejpam-4736	261	13	∩	∩	X
ejpam-4736	261	14	aδp(λ	aδp(λ	PROPN
ejpam-4736	261	15	,	,	PUNCT
ejpam-4736	261	16	s	s	PART
ejpam-4736	261	17	)	)	PUNCT
ejpam-4736	261	18	for	for	ADP
ejpam-4736	261	19	some	some	DET
ejpam-4736	261	20	u	u	PROPN
ejpam-4736	261	21	∈	∈	PROPN
ejpam-4736	261	22	δp(λ	δp(λ	NOUN
ejpam-4736	261	23	,	,	PUNCT
ejpam-4736	261	24	s)o(x	s)o(x	PROPN
ejpam-4736	261	25	,	,	PUNCT
ejpam-4736	261	26	τ	τ	PROPN
ejpam-4736	261	27	)	)	PUNCT
ejpam-4736	261	28	.	.	PUNCT
ejpam-4736	262	1	then	then	ADV
ejpam-4736	262	2	,	,	PUNCT
ejpam-4736	262	3	we	we	PRON
ejpam-4736	262	4	have	have	VERB
ejpam-4736	262	5	aδp(λ	aδp(λ	PROPN
ejpam-4736	262	6	,	,	PUNCT
ejpam-4736	262	7	s)−a	s)−a	PROPN
ejpam-4736	262	8	=	=	SYM
ejpam-4736	262	9	(	(	PUNCT
ejpam-4736	262	10	x−	x−	PROPN
ejpam-4736	263	1	[	[	X
ejpam-4736	263	2	u	u	NOUN
ejpam-4736	263	3	∩aδp(λ	∩aδp(λ	NOUN
ejpam-4736	263	4	,	,	PUNCT
ejpam-4736	263	5	s)])∩aδp(λ	s)])∩aδp(λ	NOUN
ejpam-4736	263	6	,	,	PUNCT
ejpam-4736	263	7	s	s	PART
ejpam-4736	263	8	)	)	PUNCT
ejpam-4736	263	9	=	=	SYM
ejpam-4736	263	10	(	(	PUNCT
ejpam-4736	263	11	x−u)∩aδp(λ	x−u)∩aδp(λ	NUM
ejpam-4736	263	12	,	,	PUNCT
ejpam-4736	263	13	s	s	NOUN
ejpam-4736	263	14	)	)	PUNCT
ejpam-4736	263	15	.	.	PUNCT
ejpam-4736	264	1	thus	thus	ADV
ejpam-4736	264	2	,	,	PUNCT
ejpam-4736	264	3	aδp(λ	aδp(λ	PROPN
ejpam-4736	264	4	,	,	PUNCT
ejpam-4736	264	5	s)−a	s)−a	PROPN
ejpam-4736	264	6	is	be	AUX
ejpam-4736	264	7	δp(λ	δp(λ	NOUN
ejpam-4736	264	8	,	,	PUNCT
ejpam-4736	264	9	s)-closed	s)-close	VERB
ejpam-4736	264	10	.	.	PUNCT
ejpam-4736	265	1	(	(	PUNCT
ejpam-4736	265	2	3	3	X
ejpam-4736	265	3	)	)	PUNCT
ejpam-4736	265	4	⇒	⇒	NOUN
ejpam-4736	265	5	(	(	PUNCT
ejpam-4736	265	6	4	4	NUM
ejpam-4736	265	7	):	):	PUNCT
ejpam-4736	265	8	since	since	SCONJ
ejpam-4736	265	9	x−	x−	PROPN
ejpam-4736	265	10	(	(	PUNCT
ejpam-4736	265	11	aδp(λ	aδp(λ	PROPN
ejpam-4736	265	12	,	,	PUNCT
ejpam-4736	265	13	s)−a	s)−a	NOUN
ejpam-4736	265	14	)	)	PUNCT
ejpam-4736	265	15	=	=	SYM
ejpam-4736	265	16	(	(	PUNCT
ejpam-4736	265	17	x−aδp(λ	x−aδp(λ	ADP
ejpam-4736	265	18	,	,	PUNCT
ejpam-4736	265	19	s))∪a	s))∪a	NOUN
ejpam-4736	265	20	and	and	CCONJ
ejpam-4736	265	21	by	by	ADP
ejpam-4736	265	22	(	(	PUNCT
ejpam-4736	265	23	3	3	NUM
ejpam-4736	265	24	)	)	PUNCT
ejpam-4736	265	25	,	,	PUNCT
ejpam-4736	265	26	a∪	a∪	X
ejpam-4736	265	27	(	(	PUNCT
ejpam-4736	265	28	x−aδp(λ	x−aδp(λ	PROPN
ejpam-4736	265	29	,	,	PUNCT
ejpam-4736	265	30	s	s	PART
ejpam-4736	265	31	)	)	PUNCT
ejpam-4736	265	32	)	)	PUNCT
ejpam-4736	265	33	is	be	AUX
ejpam-4736	265	34	δp(λ	δp(λ	NOUN
ejpam-4736	265	35	,	,	PUNCT
ejpam-4736	265	36	s)-open	s)-open	ADJ
ejpam-4736	265	37	.	.	PUNCT
ejpam-4736	266	1	(	(	PUNCT
ejpam-4736	266	2	4	4	X
ejpam-4736	266	3	)	)	PUNCT
ejpam-4736	266	4	⇒	⇒	NOUN
ejpam-4736	266	5	(	(	PUNCT
ejpam-4736	266	6	5	5	NUM
ejpam-4736	266	7	):	):	PUNCT
ejpam-4736	266	8	by	by	ADP
ejpam-4736	266	9	(	(	PUNCT
ejpam-4736	266	10	4	4	NUM
ejpam-4736	266	11	)	)	PUNCT
ejpam-4736	266	12	,	,	PUNCT
ejpam-4736	266	13	a	a	DET
ejpam-4736	266	14	⊆	⊆	NUM
ejpam-4736	266	15	a	a	DET
ejpam-4736	266	16	∪	∪	NOUN
ejpam-4736	266	17	(	(	PUNCT
ejpam-4736	266	18	x	x	PROPN
ejpam-4736	266	19	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	266	20	,	,	PUNCT
ejpam-4736	266	21	s	s	NOUN
ejpam-4736	266	22	)	)	PUNCT
ejpam-4736	266	23	)	)	PUNCT
ejpam-4736	267	1	=	=	PUNCT
ejpam-4736	268	1	[	[	X
ejpam-4736	268	2	a	a	DET
ejpam-4736	268	3	∪	∪	X
ejpam-4736	268	4	(	(	PUNCT
ejpam-4736	268	5	x	x	PROPN
ejpam-4736	268	6	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	268	7	,	,	PUNCT
ejpam-4736	268	8	s))]δp(λ	s))]δp(λ	PROPN
ejpam-4736	268	9	,	,	PUNCT
ejpam-4736	268	10	s	s	NOUN
ejpam-4736	268	11	)	)	PUNCT
ejpam-4736	268	12	.	.	PUNCT
ejpam-4736	269	1	(	(	PUNCT
ejpam-4736	269	2	5	5	X
ejpam-4736	269	3	)	)	PUNCT
ejpam-4736	269	4	⇒	⇒	NOUN
ejpam-4736	269	5	(	(	PUNCT
ejpam-4736	269	6	1	1	NUM
ejpam-4736	269	7	):	):	PUNCT
ejpam-4736	269	8	we	we	PRON
ejpam-4736	269	9	put	put	VERB
ejpam-4736	269	10	u	u	NOUN
ejpam-4736	269	11	=	=	PUNCT
ejpam-4736	270	1	[	[	X
ejpam-4736	270	2	a	a	DET
ejpam-4736	270	3	∪	∪	X
ejpam-4736	270	4	(	(	PUNCT
ejpam-4736	270	5	x	x	SYM
ejpam-4736	270	6	−	−	PROPN
ejpam-4736	270	7	aδp(λ	aδp(λ	PROPN
ejpam-4736	270	8	,	,	PUNCT
ejpam-4736	270	9	s))]δp(λ	s))]δp(λ	PROPN
ejpam-4736	270	10	,	,	PUNCT
ejpam-4736	270	11	s	s	NOUN
ejpam-4736	270	12	)	)	PUNCT
ejpam-4736	270	13	.	.	PUNCT
ejpam-4736	271	1	then	then	ADV
ejpam-4736	271	2	,	,	PUNCT
ejpam-4736	271	3	u	u	NOUN
ejpam-4736	271	4	is	be	AUX
ejpam-4736	271	5	δp(λ	δp(λ	NOUN
ejpam-4736	271	6	,	,	PUNCT
ejpam-4736	271	7	s)-open	s)-open	PUNCT
ejpam-4736	271	8	and	and	CCONJ
ejpam-4736	271	9	a	a	DET
ejpam-4736	271	10	=	=	X
ejpam-4736	271	11	a	a	DET
ejpam-4736	271	12	∩	∩	ADJ
ejpam-4736	271	13	u	u	ADJ
ejpam-4736	271	14	⊆	⊆	NUM
ejpam-4736	271	15	u	u	NOUN
ejpam-4736	271	16	∩	∩	X
ejpam-4736	271	17	aδp(λ	aδp(λ	PROPN
ejpam-4736	271	18	,	,	PUNCT
ejpam-4736	271	19	s	s	PART
ejpam-4736	271	20	)	)	PUNCT
ejpam-4736	271	21	⊆	⊆	NUM
ejpam-4736	272	1	[	[	X
ejpam-4736	272	2	a	a	DET
ejpam-4736	272	3	∪	∪	X
ejpam-4736	272	4	(	(	PUNCT
ejpam-4736	272	5	x	x	SYM
ejpam-4736	272	6	−	−	PROPN
ejpam-4736	272	7	aδp(λ	aδp(λ	PROPN
ejpam-4736	272	8	,	,	PUNCT
ejpam-4736	272	9	s	s	NOUN
ejpam-4736	272	10	)	)	PUNCT
ejpam-4736	272	11	)	)	PUNCT
ejpam-4736	272	12	]	]	PUNCT
ejpam-4736	272	13	∩	∩	PROPN
ejpam-4736	272	14	aδp(λ	aδp(λ	PROPN
ejpam-4736	272	15	,	,	PUNCT
ejpam-4736	272	16	s	s	PART
ejpam-4736	272	17	)	)	PUNCT
ejpam-4736	272	18	=	=	PUNCT
ejpam-4736	272	19	a	a	DET
ejpam-4736	272	20	∩	∩	ADJ
ejpam-4736	272	21	aδp(λ	aδp(λ	PROPN
ejpam-4736	272	22	,	,	PUNCT
ejpam-4736	272	23	s	s	PART
ejpam-4736	272	24	)	)	PUNCT
ejpam-4736	272	25	=	=	SYM
ejpam-4736	272	26	a.	a.	NOUN
ejpam-4736	272	27	thus	thus	ADV
ejpam-4736	272	28	,	,	PUNCT
ejpam-4736	272	29	a	a	PRON
ejpam-4736	272	30	=	=	X
ejpam-4736	272	31	u	u	NOUN
ejpam-4736	272	32	∩	∩	X
ejpam-4736	272	33	aδp(λ	aδp(λ	PROPN
ejpam-4736	272	34	,	,	PUNCT
ejpam-4736	272	35	s	s	PART
ejpam-4736	272	36	)	)	PUNCT
ejpam-4736	272	37	,	,	PUNCT
ejpam-4736	272	38	where	where	SCONJ
ejpam-4736	272	39	u	u	PROPN
ejpam-4736	272	40	∈	∈	PROPN
ejpam-4736	272	41	δp(λ	δp(λ	NOUN
ejpam-4736	272	42	,	,	PUNCT
ejpam-4736	272	43	s)o(x	s)o(x	PROPN
ejpam-4736	272	44	,	,	PUNCT
ejpam-4736	272	45	τ	τ	X
ejpam-4736	272	46	)	)	PUNCT
ejpam-4736	272	47	and	and	CCONJ
ejpam-4736	272	48	aδp(λ	aδp(λ	PROPN
ejpam-4736	272	49	,	,	PUNCT
ejpam-4736	272	50	s	s	PART
ejpam-4736	272	51	)	)	PUNCT
ejpam-4736	272	52	is	be	AUX
ejpam-4736	272	53	δp(λ	δp(λ	NOUN
ejpam-4736	272	54	,	,	PUNCT
ejpam-4736	272	55	s)-closed	s)-close	VERB
ejpam-4736	272	56	.	.	PUNCT
ejpam-4736	273	1	this	this	PRON
ejpam-4736	273	2	shows	show	VERB
ejpam-4736	273	3	that	that	SCONJ
ejpam-4736	273	4	a	a	PRON
ejpam-4736	273	5	is	be	AUX
ejpam-4736	273	6	locally	locally	ADV
ejpam-4736	273	7	δp(λ	δp(λ	NOUN
ejpam-4736	273	8	,	,	PUNCT
ejpam-4736	273	9	s)-closed	s)-close	VERB
ejpam-4736	273	10	.	.	PUNCT
ejpam-4736	274	1	theorem	theorem	NOUN
ejpam-4736	274	2	12	12	NUM
ejpam-4736	274	3	.	.	PUNCT
ejpam-4736	275	1	a	a	DET
ejpam-4736	275	2	subset	subset	NOUN
ejpam-4736	275	3	a	a	PRON
ejpam-4736	275	4	of	of	ADP
ejpam-4736	275	5	a	a	DET
ejpam-4736	275	6	topological	topological	ADJ
ejpam-4736	275	7	space	space	NOUN
ejpam-4736	275	8	(	(	PUNCT
ejpam-4736	275	9	x	x	X
ejpam-4736	275	10	,	,	PUNCT
ejpam-4736	275	11	τ	τ	X
ejpam-4736	275	12	)	)	PUNCT
ejpam-4736	275	13	is	be	AUX
ejpam-4736	275	14	δp(λ	δp(λ	NOUN
ejpam-4736	275	15	,	,	PUNCT
ejpam-4736	275	16	s)-closed	s)-close	VERB
ejpam-4736	275	17	if	if	SCONJ
ejpam-4736	275	18	and	and	CCONJ
ejpam-4736	275	19	only	only	ADV
ejpam-4736	275	20	if	if	SCONJ
ejpam-4736	275	21	a	a	PRON
ejpam-4736	275	22	is	be	AUX
ejpam-4736	275	23	locally	locally	ADV
ejpam-4736	275	24	δp(λ	δp(λ	NOUN
ejpam-4736	275	25	,	,	PUNCT
ejpam-4736	275	26	s)-closed	s)-close	VERB
ejpam-4736	275	27	and	and	CCONJ
ejpam-4736	275	28	g	g	NOUN
ejpam-4736	275	29	-	-	PUNCT
ejpam-4736	275	30	δp(λ	δp(λ	NOUN
ejpam-4736	275	31	,	,	PUNCT
ejpam-4736	275	32	s)-closed	s)-close	VERB
ejpam-4736	275	33	.	.	PUNCT
ejpam-4736	276	1	proof	proof	NOUN
ejpam-4736	276	2	.	.	PUNCT
ejpam-4736	277	1	let	let	VERB
ejpam-4736	277	2	a	a	PRON
ejpam-4736	277	3	be	be	AUX
ejpam-4736	277	4	a	a	DET
ejpam-4736	277	5	δp(λ	δp(λ	NOUN
ejpam-4736	277	6	,	,	PUNCT
ejpam-4736	277	7	s)-closed	s)-close	VERB
ejpam-4736	277	8	set	set	NOUN
ejpam-4736	277	9	.	.	PUNCT
ejpam-4736	278	1	by	by	ADP
ejpam-4736	278	2	theorem	theorem	NOUN
ejpam-4736	278	3	2	2	NUM
ejpam-4736	278	4	,	,	PUNCT
ejpam-4736	278	5	a	a	PRON
ejpam-4736	278	6	is	be	AUX
ejpam-4736	278	7	g	g	NOUN
ejpam-4736	278	8	-	-	PUNCT
ejpam-4736	278	9	δp(λ	δp(λ	NOUN
ejpam-4736	278	10	,	,	PUNCT
ejpam-4736	278	11	s)-closed	s)-close	VERB
ejpam-4736	278	12	.	.	PUNCT
ejpam-4736	279	1	since	since	SCONJ
ejpam-4736	279	2	x	x	PRON
ejpam-4736	279	3	is	be	AUX
ejpam-4736	279	4	δp(λ	δp(λ	NOUN
ejpam-4736	279	5	,	,	PUNCT
ejpam-4736	279	6	s)-open	s)-open	PUNCT
ejpam-4736	279	7	and	and	CCONJ
ejpam-4736	279	8	a	a	PRON
ejpam-4736	279	9	=	=	NOUN
ejpam-4736	279	10	x	x	SYM
ejpam-4736	279	11	∩a	∩a	PROPN
ejpam-4736	279	12	,	,	PUNCT
ejpam-4736	279	13	a	a	PRON
ejpam-4736	279	14	is	be	AUX
ejpam-4736	279	15	locally	locally	ADV
ejpam-4736	279	16	δp(λ	δp(λ	NOUN
ejpam-4736	279	17	,	,	PUNCT
ejpam-4736	279	18	s)-closed	s)-close	VERB
ejpam-4736	279	19	.	.	PUNCT
ejpam-4736	280	1	conversely	conversely	ADV
ejpam-4736	280	2	,	,	PUNCT
ejpam-4736	280	3	suppose	suppose	VERB
ejpam-4736	280	4	that	that	SCONJ
ejpam-4736	280	5	a	a	PRON
ejpam-4736	280	6	is	be	AUX
ejpam-4736	280	7	locally	locally	ADV
ejpam-4736	280	8	δp(λ	δp(λ	NOUN
ejpam-4736	280	9	,	,	PUNCT
ejpam-4736	280	10	s)-closed	s)-close	VERB
ejpam-4736	280	11	and	and	CCONJ
ejpam-4736	280	12	g	g	NOUN
ejpam-4736	280	13	-	-	PUNCT
ejpam-4736	280	14	δp(λ	δp(λ	NOUN
ejpam-4736	280	15	,	,	PUNCT
ejpam-4736	280	16	s)-closed	s)-close	VERB
ejpam-4736	280	17	.	.	PUNCT
ejpam-4736	281	1	since	since	SCONJ
ejpam-4736	281	2	a	a	PRON
ejpam-4736	281	3	is	be	AUX
ejpam-4736	281	4	locally	locally	ADV
ejpam-4736	281	5	δp(λ	δp(λ	NOUN
ejpam-4736	281	6	,	,	PUNCT
ejpam-4736	281	7	s)-closed	s)-close	VERB
ejpam-4736	281	8	,	,	PUNCT
ejpam-4736	281	9	by	by	ADP
ejpam-4736	281	10	lemma	lemma	PROPN
ejpam-4736	281	11	5	5	NUM
ejpam-4736	281	12	,	,	PUNCT
ejpam-4736	281	13	a	a	DET
ejpam-4736	281	14	⊆	⊆	NUM
ejpam-4736	281	15	[	[	X
ejpam-4736	281	16	a	a	DET
ejpam-4736	281	17	∪	∪	ADJ
ejpam-4736	281	18	[	[	X
ejpam-4736	281	19	x	x	SYM
ejpam-4736	281	20	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	281	21	,	,	PUNCT
ejpam-4736	281	22	s)]]δp(λ	s)]]δp(λ	PROPN
ejpam-4736	281	23	,	,	PUNCT
ejpam-4736	281	24	s	s	NOUN
ejpam-4736	281	25	)	)	PUNCT
ejpam-4736	281	26	.	.	PUNCT
ejpam-4736	282	1	since	since	SCONJ
ejpam-4736	282	2	[	[	X
ejpam-4736	282	3	a	a	DET
ejpam-4736	282	4	∪	∪	ADJ
ejpam-4736	282	5	[	[	X
ejpam-4736	282	6	x	x	SYM
ejpam-4736	282	7	−aδp(λ	−aδp(λ	PROPN
ejpam-4736	282	8	,	,	PUNCT
ejpam-4736	282	9	s)]]δp(λ	s)]]δp(λ	PROPN
ejpam-4736	282	10	,	,	PUNCT
ejpam-4736	282	11	s	s	PART
ejpam-4736	282	12	)	)	PUNCT
ejpam-4736	282	13	∈	∈	PROPN
ejpam-4736	282	14	δp(λ	δp(λ	NOUN
ejpam-4736	282	15	,	,	PUNCT
ejpam-4736	282	16	s)o(x	s)o(x	PROPN
ejpam-4736	282	17	,	,	PUNCT
ejpam-4736	282	18	τ	τ	X
ejpam-4736	282	19	)	)	PUNCT
ejpam-4736	282	20	and	and	CCONJ
ejpam-4736	282	21	a	a	PRON
ejpam-4736	282	22	is	be	AUX
ejpam-4736	282	23	g	g	NOUN
ejpam-4736	282	24	-	-	PUNCT
ejpam-4736	282	25	δp(λ	δp(λ	NOUN
ejpam-4736	282	26	,	,	PUNCT
ejpam-4736	282	27	s)-closed	s)-close	VERB
ejpam-4736	282	28	,	,	PUNCT
ejpam-4736	282	29	we	we	PRON
ejpam-4736	282	30	have	have	VERB
ejpam-4736	282	31	aδp(λ	aδp(λ	PROPN
ejpam-4736	282	32	,	,	PUNCT
ejpam-4736	282	33	s	s	PART
ejpam-4736	282	34	)	)	PUNCT
ejpam-4736	282	35	⊆	⊆	NUM
ejpam-4736	283	1	[	[	X
ejpam-4736	283	2	a∪	a∪	NOUN
ejpam-4736	283	3	[	[	X
ejpam-4736	283	4	x−aδp(λ	x−aδp(λ	PROPN
ejpam-4736	283	5	,	,	PUNCT
ejpam-4736	283	6	s)]]δp(λ	s)]]δp(λ	PROPN
ejpam-4736	283	7	,	,	PUNCT
ejpam-4736	283	8	s	s	PART
ejpam-4736	283	9	)	)	PUNCT
ejpam-4736	283	10	⊆	⊆	NUM
ejpam-4736	283	11	a∪	a∪	PUNCT
ejpam-4736	284	1	[	[	X
ejpam-4736	284	2	x−aδp(λ	x−aδp(λ	PROPN
ejpam-4736	284	3	,	,	PUNCT
ejpam-4736	284	4	s	s	NOUN
ejpam-4736	284	5	)	)	PUNCT
ejpam-4736	284	6	]	]	PUNCT
ejpam-4736	284	7	and	and	CCONJ
ejpam-4736	284	8	hence	hence	ADV
ejpam-4736	284	9	aδp(λ	aδp(λ	PROPN
ejpam-4736	284	10	,	,	PUNCT
ejpam-4736	284	11	s	s	PART
ejpam-4736	284	12	)	)	PUNCT
ejpam-4736	284	13	=	=	SYM
ejpam-4736	284	14	a.	a.	NOUN
ejpam-4736	284	15	thus	thus	ADV
ejpam-4736	284	16	,	,	PUNCT
ejpam-4736	284	17	by	by	ADP
ejpam-4736	284	18	lemma	lemma	PROPN
ejpam-4736	284	19	1	1	NUM
ejpam-4736	284	20	,	,	PUNCT
ejpam-4736	284	21	a	a	DET
ejpam-4736	284	22	is	is	NOUN
ejpam-4736	284	23	δp(λ	δp(λ	NOUN
ejpam-4736	284	24	,	,	PUNCT
ejpam-4736	284	25	s)-closed	s)-close	VERB
ejpam-4736	284	26	.	.	PUNCT
ejpam-4736	285	1	definition	definition	NOUN
ejpam-4736	285	2	6	6	NUM
ejpam-4736	285	3	.	.	PUNCT
ejpam-4736	286	1	[	[	X
ejpam-4736	286	2	17	17	NUM
ejpam-4736	286	3	]	]	PUNCT
ejpam-4736	286	4	let	let	VERB
ejpam-4736	286	5	a	a	PRON
ejpam-4736	286	6	be	be	AUX
ejpam-4736	286	7	a	a	DET
ejpam-4736	286	8	subset	subset	NOUN
ejpam-4736	286	9	of	of	ADP
ejpam-4736	286	10	a	a	DET
ejpam-4736	286	11	topological	topological	ADJ
ejpam-4736	286	12	space	space	NOUN
ejpam-4736	286	13	(	(	PUNCT
ejpam-4736	286	14	x	x	X
ejpam-4736	286	15	,	,	PUNCT
ejpam-4736	286	16	τ	τ	PROPN
ejpam-4736	286	17	)	)	PUNCT
ejpam-4736	286	18	.	.	PUNCT
ejpam-4736	287	1	a	a	DET
ejpam-4736	287	2	subset	subset	NOUN
ejpam-4736	287	3	δp(λ	δp(λ	NOUN
ejpam-4736	287	4	,	,	PUNCT
ejpam-4736	287	5	s)ker(a	s)ker(a	X
ejpam-4736	287	6	)	)	PUNCT
ejpam-4736	287	7	is	be	AUX
ejpam-4736	287	8	defined	define	VERB
ejpam-4736	287	9	as	as	ADP
ejpam-4736	287	10	follows	follow	VERB
ejpam-4736	287	11	:	:	PUNCT
ejpam-4736	287	12	δp(λ	δp(λ	NOUN
ejpam-4736	287	13	,	,	PUNCT
ejpam-4736	287	14	s)ker(a	s)ker(a	NOUN
ejpam-4736	287	15	)	)	PUNCT
ejpam-4736	287	16	=	=	PUNCT
ejpam-4736	288	1	∩{u	∩{u	PROPN
ejpam-4736	288	2	|	|	ADV
ejpam-4736	288	3	a	a	DET
ejpam-4736	288	4	⊆	⊆	NUM
ejpam-4736	288	5	u	u	NOUN
ejpam-4736	288	6	,	,	PUNCT
ejpam-4736	288	7	u	u	PROPN
ejpam-4736	288	8	∈	∈	PROPN
ejpam-4736	288	9	δp(λ	δp(λ	NOUN
ejpam-4736	288	10	,	,	PUNCT
ejpam-4736	288	11	s)o(x	s)o(x	PROPN
ejpam-4736	288	12	,	,	PUNCT
ejpam-4736	288	13	τ	τ	NOUN
ejpam-4736	288	14	)	)	PUNCT
ejpam-4736	288	15	}	}	PUNCT
ejpam-4736	288	16	.	.	PUNCT
ejpam-4736	289	1	lemma	lemma	PROPN
ejpam-4736	289	2	6	6	NUM
ejpam-4736	289	3	.	.	PUNCT
ejpam-4736	290	1	for	for	ADP
ejpam-4736	290	2	subsets	subset	NOUN
ejpam-4736	290	3	a	a	DET
ejpam-4736	290	4	,	,	PUNCT
ejpam-4736	290	5	b	b	PROPN
ejpam-4736	290	6	of	of	ADP
ejpam-4736	290	7	a	a	DET
ejpam-4736	290	8	topological	topological	ADJ
ejpam-4736	290	9	space	space	NOUN
ejpam-4736	290	10	(	(	PUNCT
ejpam-4736	290	11	x	x	X
ejpam-4736	290	12	,	,	PUNCT
ejpam-4736	290	13	τ	τ	PROPN
ejpam-4736	290	14	)	)	PUNCT
ejpam-4736	290	15	,	,	PUNCT
ejpam-4736	290	16	the	the	DET
ejpam-4736	290	17	following	follow	VERB
ejpam-4736	290	18	properties	property	NOUN
ejpam-4736	290	19	hold	hold	VERB
ejpam-4736	290	20	:	:	PUNCT
ejpam-4736	290	21	(	(	PUNCT
ejpam-4736	290	22	1	1	X
ejpam-4736	290	23	)	)	PUNCT
ejpam-4736	290	24	a	a	DET
ejpam-4736	290	25	⊆	⊆	NUM
ejpam-4736	290	26	δp(λ	δp(λ	NOUN
ejpam-4736	290	27	,	,	PUNCT
ejpam-4736	290	28	s)ker(a	s)ker(a	NOUN
ejpam-4736	290	29	)	)	PUNCT
ejpam-4736	290	30	.	.	PUNCT
ejpam-4736	291	1	(	(	PUNCT
ejpam-4736	291	2	2	2	X
ejpam-4736	291	3	)	)	PUNCT
ejpam-4736	291	4	if	if	SCONJ
ejpam-4736	291	5	a	a	DET
ejpam-4736	291	6	⊆	⊆	NUM
ejpam-4736	291	7	b	b	NOUN
ejpam-4736	291	8	,	,	PUNCT
ejpam-4736	291	9	then	then	ADV
ejpam-4736	291	10	δp(λ	δp(λ	NOUN
ejpam-4736	291	11	,	,	PUNCT
ejpam-4736	291	12	s)ker(a	s)ker(a	NOUN
ejpam-4736	291	13	)	)	PUNCT
ejpam-4736	291	14	⊆	⊆	NUM
ejpam-4736	291	15	δp(λ	δp(λ	NOUN
ejpam-4736	291	16	,	,	PUNCT
ejpam-4736	291	17	s)ker(b	s)ker(b	NUM
ejpam-4736	291	18	)	)	PUNCT
ejpam-4736	291	19	.	.	PUNCT
ejpam-4736	292	1	(	(	PUNCT
ejpam-4736	292	2	3	3	NUM
ejpam-4736	292	3	)	)	PUNCT
ejpam-4736	292	4	δp(λ	δp(λ	NOUN
ejpam-4736	292	5	,	,	PUNCT
ejpam-4736	292	6	s)ker[δp(λ	s)ker[δp(λ	NOUN
ejpam-4736	292	7	,	,	PUNCT
ejpam-4736	292	8	s)ker(a	s)ker(a	NOUN
ejpam-4736	292	9	)	)	PUNCT
ejpam-4736	292	10	]	]	PUNCT
ejpam-4736	293	1	=	=	SYM
ejpam-4736	293	2	δp(λ	δp(λ	NOUN
ejpam-4736	293	3	,	,	PUNCT
ejpam-4736	293	4	s)ker(a	s)ker(a	NOUN
ejpam-4736	293	5	)	)	PUNCT
ejpam-4736	293	6	.	.	PUNCT
ejpam-4736	294	1	(	(	PUNCT
ejpam-4736	294	2	4	4	X
ejpam-4736	294	3	)	)	PUNCT
ejpam-4736	294	4	if	if	SCONJ
ejpam-4736	294	5	a	a	PRON
ejpam-4736	294	6	is	be	AUX
ejpam-4736	294	7	δp(λ	δp(λ	NOUN
ejpam-4736	294	8	,	,	PUNCT
ejpam-4736	294	9	s)-open	s)-open	PUNCT
ejpam-4736	294	10	,	,	PUNCT
ejpam-4736	294	11	δp(λ	δp(λ	NOUN
ejpam-4736	294	12	,	,	PUNCT
ejpam-4736	294	13	s)ker(a	s)ker(a	NOUN
ejpam-4736	294	14	)	)	PUNCT
ejpam-4736	294	15	=	=	SYM
ejpam-4736	294	16	a.	a.	NOUN
ejpam-4736	294	17	c.	c.	PROPN
ejpam-4736	294	18	boonpok	boonpok	PROPN
ejpam-4736	294	19	,	,	PUNCT
ejpam-4736	294	20	n.	n.	PROPN
ejpam-4736	294	21	srisarakham	srisarakham	PROPN
ejpam-4736	294	22	/	/	SYM
ejpam-4736	294	23	eur	eur	PROPN
ejpam-4736	294	24	.	.	PUNCT
ejpam-4736	295	1	j.	j.	PROPN
ejpam-4736	295	2	pure	pure	PROPN
ejpam-4736	295	3	appl	appl	PROPN
ejpam-4736	295	4	.	.	PROPN
ejpam-4736	295	5	math	math	PROPN
ejpam-4736	295	6	,	,	PUNCT
ejpam-4736	295	7	16	16	NUM
ejpam-4736	295	8	(	(	PUNCT
ejpam-4736	295	9	4	4	NUM
ejpam-4736	295	10	)	)	PUNCT
ejpam-4736	295	11	(	(	PUNCT
ejpam-4736	295	12	2023	2023	NUM
ejpam-4736	295	13	)	)	PUNCT
ejpam-4736	295	14	,	,	PUNCT
ejpam-4736	295	15	2581	2581	NUM
ejpam-4736	295	16	-	-	SYM
ejpam-4736	295	17	2596	2596	NUM
ejpam-4736	295	18	2589	2589	NUM
ejpam-4736	295	19	a	a	DET
ejpam-4736	295	20	subset	subset	NOUN
ejpam-4736	295	21	nx	nx	NOUN
ejpam-4736	295	22	of	of	ADP
ejpam-4736	295	23	a	a	DET
ejpam-4736	295	24	topological	topological	ADJ
ejpam-4736	295	25	space	space	NOUN
ejpam-4736	295	26	(	(	PUNCT
ejpam-4736	295	27	x	x	X
ejpam-4736	295	28	,	,	PUNCT
ejpam-4736	295	29	τ	τ	X
ejpam-4736	295	30	)	)	PUNCT
ejpam-4736	295	31	is	be	AUX
ejpam-4736	295	32	said	say	VERB
ejpam-4736	295	33	to	to	PART
ejpam-4736	295	34	be	be	AUX
ejpam-4736	295	35	a	a	DET
ejpam-4736	295	36	δp(λ	δp(λ	NOUN
ejpam-4736	295	37	,	,	PUNCT
ejpam-4736	295	38	s)-neighbourhood	s)-neighbourhood	PUNCT
ejpam-4736	296	1	[	[	X
ejpam-4736	296	2	16	16	NUM
ejpam-4736	296	3	]	]	PUNCT
ejpam-4736	296	4	of	of	ADP
ejpam-4736	296	5	a	a	DET
ejpam-4736	296	6	point	point	NOUN
ejpam-4736	296	7	x	x	X
ejpam-4736	296	8	∈	∈	NOUN
ejpam-4736	296	9	x	x	INTJ
ejpam-4736	296	10	if	if	SCONJ
ejpam-4736	296	11	there	there	PRON
ejpam-4736	296	12	exists	exist	VERB
ejpam-4736	296	13	a	a	DET
ejpam-4736	296	14	δp(λ	δp(λ	NOUN
ejpam-4736	296	15	,	,	PUNCT
ejpam-4736	296	16	s)-open	s)-open	PUNCT
ejpam-4736	296	17	set	set	VERB
ejpam-4736	296	18	u	u	PRON
ejpam-4736	296	19	such	such	ADJ
ejpam-4736	296	20	that	that	SCONJ
ejpam-4736	296	21	x	x	SYM
ejpam-4736	296	22	∈	∈	PROPN
ejpam-4736	296	23	u	u	NOUN
ejpam-4736	296	24	⊆	⊆	NUM
ejpam-4736	296	25	nx	nx	X
ejpam-4736	296	26	.	.	PUNCT
ejpam-4736	297	1	lemma	lemma	PROPN
ejpam-4736	297	2	7	7	NUM
ejpam-4736	297	3	.	.	PUNCT
ejpam-4736	298	1	a	a	DET
ejpam-4736	298	2	subset	subset	NOUN
ejpam-4736	298	3	a	a	PRON
ejpam-4736	298	4	of	of	ADP
ejpam-4736	298	5	a	a	DET
ejpam-4736	298	6	topological	topological	ADJ
ejpam-4736	298	7	space	space	NOUN
ejpam-4736	298	8	(	(	PUNCT
ejpam-4736	298	9	x	x	X
ejpam-4736	298	10	,	,	PUNCT
ejpam-4736	298	11	τ	τ	X
ejpam-4736	298	12	)	)	PUNCT
ejpam-4736	298	13	is	be	AUX
ejpam-4736	298	14	δp(λ	δp(λ	NOUN
ejpam-4736	298	15	,	,	PUNCT
ejpam-4736	298	16	s)-open	s)-open	VERB
ejpam-4736	298	17	in	in	ADP
ejpam-4736	298	18	x	x	PART
ejpam-4736	298	19	if	if	SCONJ
ejpam-4736	298	20	and	and	CCONJ
ejpam-4736	298	21	only	only	ADV
ejpam-4736	298	22	if	if	SCONJ
ejpam-4736	298	23	a	a	PRON
ejpam-4736	298	24	is	be	AUX
ejpam-4736	298	25	a	a	DET
ejpam-4736	298	26	δp(λ	δp(λ	NOUN
ejpam-4736	298	27	,	,	PUNCT
ejpam-4736	298	28	s)-neighbourhood	s)-neighbourhood	PUNCT
ejpam-4736	298	29	of	of	ADP
ejpam-4736	298	30	each	each	DET
ejpam-4736	298	31	point	point	NOUN
ejpam-4736	298	32	of	of	ADP
ejpam-4736	298	33	a.	a.	NOUN
ejpam-4736	298	34	definition	definition	NOUN
ejpam-4736	298	35	7	7	NUM
ejpam-4736	298	36	.	.	PUNCT
ejpam-4736	299	1	let	let	VERB
ejpam-4736	299	2	(	(	PUNCT
ejpam-4736	299	3	x	x	NOUN
ejpam-4736	299	4	,	,	PUNCT
ejpam-4736	299	5	τ	τ	X
ejpam-4736	299	6	)	)	PUNCT
ejpam-4736	299	7	be	be	VERB
ejpam-4736	299	8	a	a	DET
ejpam-4736	299	9	topological	topological	ADJ
ejpam-4736	299	10	space	space	NOUN
ejpam-4736	299	11	and	and	CCONJ
ejpam-4736	299	12	x	x	PUNCT
ejpam-4736	299	13	∈	∈	PROPN
ejpam-4736	299	14	x.	x.	NOUN
ejpam-4736	299	15	a	a	DET
ejpam-4736	299	16	subset	subset	NOUN
ejpam-4736	299	17	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	299	18	,	,	PUNCT
ejpam-4736	299	19	s	s	PART
ejpam-4736	299	20	)	)	PUNCT
ejpam-4736	299	21	is	be	AUX
ejpam-4736	299	22	defined	define	VERB
ejpam-4736	299	23	as	as	SCONJ
ejpam-4736	299	24	follows	follow	VERB
ejpam-4736	299	25	:	:	PUNCT
ejpam-4736	299	26	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	299	27	,	,	PUNCT
ejpam-4736	299	28	s	s	NOUN
ejpam-4736	299	29	)	)	PUNCT
ejpam-4736	299	30	=	=	SYM
ejpam-4736	299	31	δp(λ	δp(λ	NOUN
ejpam-4736	299	32	,	,	PUNCT
ejpam-4736	299	33	s)ker({x	s)ker({x	NOUN
ejpam-4736	299	34	}	}	PUNCT
ejpam-4736	299	35	)	)	PUNCT
ejpam-4736	299	36	∩	∩	NOUN
ejpam-4736	299	37	{	{	PUNCT
ejpam-4736	299	38	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	299	39	,	,	PUNCT
ejpam-4736	299	40	s	s	PART
ejpam-4736	299	41	)	)	PUNCT
ejpam-4736	299	42	.	.	PUNCT
ejpam-4736	300	1	theorem	theorem	VERB
ejpam-4736	300	2	13	13	NUM
ejpam-4736	300	3	.	.	PUNCT
ejpam-4736	301	1	for	for	ADP
ejpam-4736	301	2	a	a	DET
ejpam-4736	301	3	topological	topological	ADJ
ejpam-4736	301	4	space	space	NOUN
ejpam-4736	301	5	(	(	PUNCT
ejpam-4736	301	6	x	x	X
ejpam-4736	301	7	,	,	PUNCT
ejpam-4736	301	8	τ	τ	PROPN
ejpam-4736	301	9	)	)	PUNCT
ejpam-4736	301	10	,	,	PUNCT
ejpam-4736	301	11	the	the	DET
ejpam-4736	301	12	following	follow	VERB
ejpam-4736	301	13	properties	property	NOUN
ejpam-4736	301	14	hold	hold	VERB
ejpam-4736	301	15	:	:	PUNCT
ejpam-4736	301	16	(	(	PUNCT
ejpam-4736	301	17	1	1	X
ejpam-4736	301	18	)	)	PUNCT
ejpam-4736	301	19	λδp(λ	λδp(λ	PROPN
ejpam-4736	301	20	,	,	PUNCT
ejpam-4736	301	21	s)(a	s)(a	NUM
ejpam-4736	301	22	)	)	PUNCT
ejpam-4736	302	1	=	=	PRON
ejpam-4736	302	2	{	{	PUNCT
ejpam-4736	302	3	x	x	PUNCT
ejpam-4736	302	4	∈	∈	NOUN
ejpam-4736	302	5	x	x	PUNCT
ejpam-4736	302	6	|	|	ADV
ejpam-4736	302	7	a	a	DET
ejpam-4736	302	8	∩	∩	NOUN
ejpam-4736	302	9	{	{	PUNCT
ejpam-4736	302	10	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	302	11	,	,	PUNCT
ejpam-4736	302	12	s	s	PART
ejpam-4736	302	13	)	)	PUNCT
ejpam-4736	302	14	̸=	̸=	PROPN
ejpam-4736	302	15	∅	∅	NOUN
ejpam-4736	302	16	}	}	PUNCT
ejpam-4736	302	17	for	for	ADP
ejpam-4736	302	18	each	each	PRON
ejpam-4736	302	19	subset	subset	VERB
ejpam-4736	302	20	a	a	PRON
ejpam-4736	302	21	of	of	ADP
ejpam-4736	302	22	x.	x.	NOUN
ejpam-4736	302	23	(	(	PUNCT
ejpam-4736	302	24	2	2	NUM
ejpam-4736	302	25	)	)	PUNCT
ejpam-4736	302	26	for	for	ADP
ejpam-4736	302	27	each	each	DET
ejpam-4736	302	28	x	x	SYM
ejpam-4736	302	29	∈	∈	PROPN
ejpam-4736	302	30	x	x	X
ejpam-4736	302	31	,	,	PUNCT
ejpam-4736	302	32	δp(λ	δp(λ	NOUN
ejpam-4736	302	33	,	,	PUNCT
ejpam-4736	302	34	s)ker(⟨x⟩δp(λ	s)ker(⟨x⟩δp(λ	PROPN
ejpam-4736	302	35	,	,	PUNCT
ejpam-4736	302	36	s	s	NOUN
ejpam-4736	302	37	)	)	PUNCT
ejpam-4736	302	38	)	)	PUNCT
ejpam-4736	302	39	=	=	SYM
ejpam-4736	303	1	δp(λ	δp(λ	NOUN
ejpam-4736	303	2	,	,	PUNCT
ejpam-4736	303	3	s)ker({x	s)ker({x	NOUN
ejpam-4736	303	4	}	}	PUNCT
ejpam-4736	303	5	)	)	PUNCT
ejpam-4736	303	6	.	.	PUNCT
ejpam-4736	304	1	(	(	PUNCT
ejpam-4736	304	2	3	3	X
ejpam-4736	304	3	)	)	PUNCT
ejpam-4736	304	4	for	for	ADP
ejpam-4736	304	5	each	each	DET
ejpam-4736	304	6	x	x	SYM
ejpam-4736	304	7	∈	∈	PROPN
ejpam-4736	304	8	x	x	X
ejpam-4736	304	9	,	,	PUNCT
ejpam-4736	304	10	(	(	PUNCT
ejpam-4736	304	11	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	304	12	,	,	PUNCT
ejpam-4736	304	13	s))δp(λ	s))δp(λ	NOUN
ejpam-4736	304	14	,	,	PUNCT
ejpam-4736	304	15	s	s	PART
ejpam-4736	304	16	)	)	PUNCT
ejpam-4736	304	17	=	=	SYM
ejpam-4736	304	18	{	{	PUNCT
ejpam-4736	304	19	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	304	20	,	,	PUNCT
ejpam-4736	304	21	s	s	NOUN
ejpam-4736	304	22	)	)	PUNCT
ejpam-4736	304	23	.	.	PUNCT
ejpam-4736	305	1	(	(	PUNCT
ejpam-4736	305	2	4	4	X
ejpam-4736	305	3	)	)	PUNCT
ejpam-4736	305	4	if	if	SCONJ
ejpam-4736	305	5	u	u	NOUN
ejpam-4736	305	6	is	be	AUX
ejpam-4736	305	7	δp(λ	δp(λ	NOUN
ejpam-4736	305	8	,	,	PUNCT
ejpam-4736	305	9	s)-open	s)-open	VERB
ejpam-4736	305	10	in	in	ADP
ejpam-4736	305	11	x	x	PUNCT
ejpam-4736	305	12	and	and	CCONJ
ejpam-4736	305	13	x	x	SYM
ejpam-4736	305	14	∈	∈	PROPN
ejpam-4736	305	15	u	u	NOUN
ejpam-4736	305	16	,	,	PUNCT
ejpam-4736	305	17	then	then	ADV
ejpam-4736	305	18	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	305	19	,	,	PUNCT
ejpam-4736	305	20	s	s	PART
ejpam-4736	305	21	)	)	PUNCT
ejpam-4736	305	22	⊆	⊆	NUM
ejpam-4736	305	23	u	u	NOUN
ejpam-4736	305	24	.	.	PUNCT
ejpam-4736	306	1	(	(	PUNCT
ejpam-4736	306	2	5	5	NUM
ejpam-4736	306	3	)	)	PUNCT
ejpam-4736	306	4	if	if	SCONJ
ejpam-4736	306	5	f	f	PROPN
ejpam-4736	306	6	is	be	AUX
ejpam-4736	306	7	δp(λ	δp(λ	NOUN
ejpam-4736	306	8	,	,	PUNCT
ejpam-4736	306	9	s)-closed	s)-close	VERB
ejpam-4736	306	10	in	in	ADP
ejpam-4736	306	11	x	x	PUNCT
ejpam-4736	306	12	and	and	CCONJ
ejpam-4736	306	13	x	x	SYM
ejpam-4736	306	14	∈	∈	PROPN
ejpam-4736	306	15	f	f	X
ejpam-4736	306	16	,	,	PUNCT
ejpam-4736	306	17	then	then	ADV
ejpam-4736	306	18	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	306	19	,	,	PUNCT
ejpam-4736	306	20	s	s	PART
ejpam-4736	306	21	)	)	PUNCT
ejpam-4736	306	22	⊆	⊆	NUM
ejpam-4736	306	23	f	f	NOUN
ejpam-4736	306	24	.	.	PUNCT
ejpam-4736	307	1	proof	proof	NOUN
ejpam-4736	307	2	.	.	PUNCT
ejpam-4736	308	1	(	(	PUNCT
ejpam-4736	308	2	1	1	X
ejpam-4736	308	3	)	)	PUNCT
ejpam-4736	308	4	suppose	suppose	VERB
ejpam-4736	308	5	that	that	SCONJ
ejpam-4736	308	6	a	a	DET
ejpam-4736	308	7	∩	∩	NOUN
ejpam-4736	308	8	{	{	PUNCT
ejpam-4736	308	9	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	308	10	,	,	PUNCT
ejpam-4736	308	11	s	s	PART
ejpam-4736	308	12	)	)	PUNCT
ejpam-4736	308	13	=	=	PUNCT
ejpam-4736	308	14	∅.	∅.	ADP
ejpam-4736	308	15	then	then	ADV
ejpam-4736	308	16	,	,	PUNCT
ejpam-4736	308	17	x	x	PROPN
ejpam-4736	308	18	̸∈	̸∈	PROPN
ejpam-4736	308	19	x	x	X
ejpam-4736	308	20	−	−	PROPN
ejpam-4736	308	21	{	{	PUNCT
ejpam-4736	308	22	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	308	23	,	,	PUNCT
ejpam-4736	308	24	s	s	PART
ejpam-4736	308	25	)	)	PUNCT
ejpam-4736	308	26	which	which	PRON
ejpam-4736	308	27	is	be	AUX
ejpam-4736	308	28	a	a	DET
ejpam-4736	308	29	δp(λ	δp(λ	NOUN
ejpam-4736	308	30	,	,	PUNCT
ejpam-4736	308	31	s)-open	s)-open	PUNCT
ejpam-4736	308	32	set	set	VERB
ejpam-4736	308	33	containing	contain	VERB
ejpam-4736	308	34	a.	a.	NOUN
ejpam-4736	308	35	thus	thus	ADV
ejpam-4736	308	36	,	,	PUNCT
ejpam-4736	308	37	x	x	PROPN
ejpam-4736	308	38	̸∈	̸∈	PROPN
ejpam-4736	308	39	δp(λ	δp(λ	NOUN
ejpam-4736	308	40	,	,	PUNCT
ejpam-4736	308	41	s)ker(a	s)ker(a	NOUN
ejpam-4736	308	42	)	)	PUNCT
ejpam-4736	308	43	and	and	CCONJ
ejpam-4736	308	44	hence	hence	ADV
ejpam-4736	308	45	δp(λ	δp(λ	NOUN
ejpam-4736	308	46	,	,	PUNCT
ejpam-4736	308	47	s)ker(a	s)ker(a	NOUN
ejpam-4736	308	48	)	)	PUNCT
ejpam-4736	308	49	⊆	⊆	NUM
ejpam-4736	308	50	{	{	PUNCT
ejpam-4736	308	51	x	x	SYM
ejpam-4736	308	52	∈	∈	PROPN
ejpam-4736	308	53	x	x	X
ejpam-4736	308	54	|	|	ADV
ejpam-4736	308	55	a	a	DET
ejpam-4736	308	56	∩	∩	NOUN
ejpam-4736	308	57	{	{	PUNCT
ejpam-4736	308	58	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	308	59	,	,	PUNCT
ejpam-4736	308	60	s	s	PART
ejpam-4736	308	61	)	)	PUNCT
ejpam-4736	308	62	̸=	̸=	PROPN
ejpam-4736	308	63	∅	∅	NOUN
ejpam-4736	308	64	}	}	PUNCT
ejpam-4736	308	65	.	.	PUNCT
ejpam-4736	309	1	next	next	ADV
ejpam-4736	309	2	,	,	PUNCT
ejpam-4736	309	3	let	let	VERB
ejpam-4736	309	4	x	x	PUNCT
ejpam-4736	309	5	∈	∈	PROPN
ejpam-4736	309	6	x	x	X
ejpam-4736	309	7	such	such	ADJ
ejpam-4736	309	8	that	that	DET
ejpam-4736	309	9	a∩{x}δp(λ	a∩{x}δp(λ	NOUN
ejpam-4736	309	10	,	,	PUNCT
ejpam-4736	309	11	s	s	PART
ejpam-4736	309	12	)	)	PUNCT
ejpam-4736	309	13	̸=	̸=	PROPN
ejpam-4736	309	14	∅	∅	NOUN
ejpam-4736	309	15	and	and	CCONJ
ejpam-4736	309	16	suppose	suppose	VERB
ejpam-4736	309	17	that	that	SCONJ
ejpam-4736	309	18	x	x	PROPN
ejpam-4736	309	19	̸∈	̸∈	PROPN
ejpam-4736	309	20	δp(λ	δp(λ	NOUN
ejpam-4736	309	21	,	,	PUNCT
ejpam-4736	309	22	s)ker(a	s)ker(a	NOUN
ejpam-4736	309	23	)	)	PUNCT
ejpam-4736	309	24	.	.	PUNCT
ejpam-4736	310	1	then	then	ADV
ejpam-4736	310	2	,	,	PUNCT
ejpam-4736	310	3	there	there	PRON
ejpam-4736	310	4	exists	exist	VERB
ejpam-4736	310	5	a	a	DET
ejpam-4736	310	6	δp(λ	δp(λ	NOUN
ejpam-4736	310	7	,	,	PUNCT
ejpam-4736	310	8	s)-open	s)-open	VERB
ejpam-4736	310	9	set	set	VERB
ejpam-4736	310	10	u	u	NOUN
ejpam-4736	310	11	containing	contain	VERB
ejpam-4736	310	12	a	a	PRON
ejpam-4736	310	13	and	and	CCONJ
ejpam-4736	310	14	x	x	X
ejpam-4736	310	15	̸∈	̸∈	PROPN
ejpam-4736	310	16	u	u	PROPN
ejpam-4736	310	17	.	.	PUNCT
ejpam-4736	311	1	let	let	VERB
ejpam-4736	311	2	y	y	PROPN
ejpam-4736	311	3	∈	∈	PROPN
ejpam-4736	311	4	a∩	a∩	PROPN
ejpam-4736	311	5	{	{	PUNCT
ejpam-4736	311	6	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	311	7	,	,	PUNCT
ejpam-4736	311	8	s	s	NOUN
ejpam-4736	311	9	)	)	PUNCT
ejpam-4736	311	10	.	.	PUNCT
ejpam-4736	312	1	therefore	therefore	ADV
ejpam-4736	312	2	,	,	PUNCT
ejpam-4736	312	3	u	u	NOUN
ejpam-4736	312	4	is	be	AUX
ejpam-4736	312	5	a	a	DET
ejpam-4736	312	6	δp(λ	δp(λ	NOUN
ejpam-4736	312	7	,	,	PUNCT
ejpam-4736	312	8	s)-neighbourhood	s)-neighbourhood	PUNCT
ejpam-4736	312	9	of	of	ADP
ejpam-4736	312	10	y	y	PRON
ejpam-4736	312	11	which	which	PRON
ejpam-4736	312	12	does	do	AUX
ejpam-4736	312	13	not	not	PART
ejpam-4736	312	14	contain	contain	VERB
ejpam-4736	312	15	x.	x.	NOUN
ejpam-4736	312	16	by	by	ADP
ejpam-4736	312	17	this	this	DET
ejpam-4736	312	18	contradiction	contradiction	NOUN
ejpam-4736	312	19	x	x	PUNCT
ejpam-4736	312	20	∈	∈	PROPN
ejpam-4736	312	21	δp(λ	δp(λ	NOUN
ejpam-4736	312	22	,	,	PUNCT
ejpam-4736	312	23	s)ker(a	s)ker(a	NOUN
ejpam-4736	312	24	)	)	PUNCT
ejpam-4736	312	25	.	.	PUNCT
ejpam-4736	313	1	(	(	PUNCT
ejpam-4736	313	2	2	2	X
ejpam-4736	313	3	)	)	PUNCT
ejpam-4736	313	4	let	let	VERB
ejpam-4736	313	5	x	x	SYM
ejpam-4736	313	6	∈	∈	PROPN
ejpam-4736	313	7	x.	x.	NOUN
ejpam-4736	313	8	then	then	ADV
ejpam-4736	313	9	,	,	PUNCT
ejpam-4736	313	10	we	we	PRON
ejpam-4736	313	11	have	have	VERB
ejpam-4736	313	12	{	{	PUNCT
ejpam-4736	313	13	x	x	NOUN
ejpam-4736	313	14	}	}	PUNCT
ejpam-4736	313	15	⊆	⊆	NUM
ejpam-4736	313	16	{	{	PUNCT
ejpam-4736	313	17	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	313	18	,	,	PUNCT
ejpam-4736	313	19	s	s	NOUN
ejpam-4736	313	20	)	)	PUNCT
ejpam-4736	313	21	∩	∩	NOUN
ejpam-4736	313	22	δp(λ	δp(λ	NOUN
ejpam-4736	313	23	,	,	PUNCT
ejpam-4736	313	24	s)ker({x	s)ker({x	NOUN
ejpam-4736	313	25	}	}	PUNCT
ejpam-4736	313	26	)	)	PUNCT
ejpam-4736	314	1	=	=	SYM
ejpam-4736	314	2	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	314	3	,	,	PUNCT
ejpam-4736	314	4	s	s	NOUN
ejpam-4736	314	5	)	)	PUNCT
ejpam-4736	314	6	.	.	PUNCT
ejpam-4736	315	1	by	by	ADP
ejpam-4736	315	2	lemma	lemma	PROPN
ejpam-4736	315	3	6	6	NUM
ejpam-4736	315	4	,	,	PUNCT
ejpam-4736	315	5	we	we	PRON
ejpam-4736	315	6	obtain	obtain	VERB
ejpam-4736	315	7	δp(λ	δp(λ	NOUN
ejpam-4736	315	8	,	,	PUNCT
ejpam-4736	315	9	s)ker({x	s)ker({x	NUM
ejpam-4736	315	10	}	}	PUNCT
ejpam-4736	315	11	)	)	PUNCT
ejpam-4736	315	12	⊆	⊆	NUM
ejpam-4736	315	13	δp(λ	δp(λ	NOUN
ejpam-4736	315	14	,	,	PUNCT
ejpam-4736	315	15	s)ker(⟨x⟩δp(λ	s)ker(⟨x⟩δp(λ	PROPN
ejpam-4736	315	16	,	,	PUNCT
ejpam-4736	315	17	s	s	NOUN
ejpam-4736	315	18	)	)	PUNCT
ejpam-4736	315	19	)	)	PUNCT
ejpam-4736	315	20	.	.	PUNCT
ejpam-4736	316	1	next	next	ADV
ejpam-4736	316	2	,	,	PUNCT
ejpam-4736	316	3	we	we	PRON
ejpam-4736	316	4	show	show	VERB
ejpam-4736	316	5	the	the	DET
ejpam-4736	316	6	opposite	opposite	ADJ
ejpam-4736	316	7	implication	implication	NOUN
ejpam-4736	316	8	.	.	PUNCT
ejpam-4736	316	9	suppose	suppose	VERB
ejpam-4736	316	10	that	that	SCONJ
ejpam-4736	316	11	y	y	PROPN
ejpam-4736	316	12	̸∈	̸∈	PROPN
ejpam-4736	316	13	δp(λ	δp(λ	NOUN
ejpam-4736	316	14	,	,	PUNCT
ejpam-4736	316	15	s)ker({x	s)ker({x	NUM
ejpam-4736	316	16	}	}	PUNCT
ejpam-4736	316	17	)	)	PUNCT
ejpam-4736	316	18	.	.	PUNCT
ejpam-4736	317	1	then	then	ADV
ejpam-4736	317	2	,	,	PUNCT
ejpam-4736	317	3	there	there	PRON
ejpam-4736	317	4	exists	exist	VERB
ejpam-4736	317	5	a	a	DET
ejpam-4736	317	6	δp(λ	δp(λ	NOUN
ejpam-4736	317	7	,	,	PUNCT
ejpam-4736	317	8	s)open	s)open	NOUN
ejpam-4736	317	9	set	set	VERB
ejpam-4736	317	10	v	v	ADP
ejpam-4736	317	11	such	such	ADJ
ejpam-4736	317	12	that	that	SCONJ
ejpam-4736	317	13	x	x	SYM
ejpam-4736	317	14	∈	∈	PROPN
ejpam-4736	317	15	v	v	NOUN
ejpam-4736	317	16	and	and	CCONJ
ejpam-4736	317	17	y	y	PROPN
ejpam-4736	317	18	̸∈	̸∈	PROPN
ejpam-4736	317	19	v	v	PROPN
ejpam-4736	317	20	.	.	PUNCT
ejpam-4736	318	1	since	since	SCONJ
ejpam-4736	318	2	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	318	3	,	,	PUNCT
ejpam-4736	318	4	s	s	PART
ejpam-4736	318	5	)	)	PUNCT
ejpam-4736	318	6	⊆	⊆	NUM
ejpam-4736	318	7	δp(λ	δp(λ	NOUN
ejpam-4736	318	8	,	,	PUNCT
ejpam-4736	318	9	s)ker({x	s)ker({x	NUM
ejpam-4736	318	10	}	}	PUNCT
ejpam-4736	318	11	)	)	PUNCT
ejpam-4736	318	12	⊆	⊆	NUM
ejpam-4736	318	13	δp(λ	δp(λ	NOUN
ejpam-4736	318	14	,	,	PUNCT
ejpam-4736	318	15	s)ker(v	s)ker(v	X
ejpam-4736	318	16	)	)	PUNCT
ejpam-4736	318	17	=	=	SYM
ejpam-4736	318	18	v	v	NOUN
ejpam-4736	318	19	,	,	PUNCT
ejpam-4736	318	20	we	we	PRON
ejpam-4736	318	21	have	have	VERB
ejpam-4736	318	22	δp(λ	δp(λ	NOUN
ejpam-4736	318	23	,	,	PUNCT
ejpam-4736	318	24	s)ker(⟨x⟩δp(λ	s)ker(⟨x⟩δp(λ	PROPN
ejpam-4736	318	25	,	,	PUNCT
ejpam-4736	318	26	s	s	NOUN
ejpam-4736	318	27	)	)	PUNCT
ejpam-4736	318	28	)	)	PUNCT
ejpam-4736	319	1	⊆	⊆	NUM
ejpam-4736	319	2	v	v	NOUN
ejpam-4736	319	3	.	.	PUNCT
ejpam-4736	320	1	since	since	SCONJ
ejpam-4736	320	2	y	y	PROPN
ejpam-4736	320	3	̸∈	̸∈	PROPN
ejpam-4736	320	4	v	v	PROPN
ejpam-4736	320	5	,	,	PUNCT
ejpam-4736	320	6	y	y	PROPN
ejpam-4736	320	7	̸∈	̸∈	PROPN
ejpam-4736	320	8	δp(λ	δp(λ	PROPN
ejpam-4736	320	9	,	,	PUNCT
ejpam-4736	320	10	s)ker(⟨x⟩δp(λ	s)ker(⟨x⟩δp(λ	PROPN
ejpam-4736	320	11	,	,	PUNCT
ejpam-4736	320	12	s	s	NOUN
ejpam-4736	320	13	)	)	PUNCT
ejpam-4736	320	14	)	)	PUNCT
ejpam-4736	320	15	.	.	PUNCT
ejpam-4736	321	1	thus	thus	ADV
ejpam-4736	321	2	,	,	PUNCT
ejpam-4736	321	3	δp(λ	δp(λ	NOUN
ejpam-4736	321	4	,	,	PUNCT
ejpam-4736	321	5	s)ker(⟨x⟩δp(λ	s)ker(⟨x⟩δp(λ	PROPN
ejpam-4736	321	6	,	,	PUNCT
ejpam-4736	321	7	s	s	NOUN
ejpam-4736	321	8	)	)	PUNCT
ejpam-4736	321	9	)	)	PUNCT
ejpam-4736	321	10	⊆	⊆	NUM
ejpam-4736	321	11	δp(λ	δp(λ	NOUN
ejpam-4736	321	12	,	,	PUNCT
ejpam-4736	321	13	s)ker({x	s)ker({x	NUM
ejpam-4736	321	14	}	}	PUNCT
ejpam-4736	321	15	)	)	PUNCT
ejpam-4736	321	16	and	and	CCONJ
ejpam-4736	321	17	hence	hence	ADV
ejpam-4736	321	18	δp(λ	δp(λ	NOUN
ejpam-4736	321	19	,	,	PUNCT
ejpam-4736	321	20	s)ker({x	s)ker({x	NUM
ejpam-4736	321	21	}	}	PUNCT
ejpam-4736	321	22	)	)	PUNCT
ejpam-4736	322	1	=	=	SYM
ejpam-4736	322	2	δp(λ	δp(λ	NOUN
ejpam-4736	322	3	,	,	PUNCT
ejpam-4736	322	4	s)ker(⟨x⟩δp(λ	s)ker(⟨x⟩δp(λ	PROPN
ejpam-4736	322	5	,	,	PUNCT
ejpam-4736	322	6	s	s	NOUN
ejpam-4736	322	7	)	)	PUNCT
ejpam-4736	322	8	)	)	PUNCT
ejpam-4736	322	9	.	.	PUNCT
ejpam-4736	323	1	(	(	PUNCT
ejpam-4736	323	2	3	3	X
ejpam-4736	323	3	)	)	PUNCT
ejpam-4736	323	4	by	by	ADP
ejpam-4736	323	5	the	the	DET
ejpam-4736	323	6	definition	definition	NOUN
ejpam-4736	323	7	of	of	ADP
ejpam-4736	323	8	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	323	9	,	,	PUNCT
ejpam-4736	323	10	s	s	PART
ejpam-4736	323	11	)	)	PUNCT
ejpam-4736	323	12	,	,	PUNCT
ejpam-4736	323	13	we	we	PRON
ejpam-4736	323	14	have	have	VERB
ejpam-4736	323	15	{	{	PUNCT
ejpam-4736	323	16	x	x	NOUN
ejpam-4736	323	17	}	}	PUNCT
ejpam-4736	323	18	⊆	⊆	NUM
ejpam-4736	323	19	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	323	20	,	,	PUNCT
ejpam-4736	323	21	s	s	PART
ejpam-4736	323	22	)	)	PUNCT
ejpam-4736	323	23	and	and	CCONJ
ejpam-4736	323	24	{	{	PUNCT
ejpam-4736	323	25	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	323	26	,	,	PUNCT
ejpam-4736	323	27	s	s	PART
ejpam-4736	323	28	)	)	PUNCT
ejpam-4736	323	29	⊆	⊆	NUM
ejpam-4736	323	30	(	(	PUNCT
ejpam-4736	323	31	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	323	32	,	,	PUNCT
ejpam-4736	323	33	s))δp(λ	s))δp(λ	NOUN
ejpam-4736	323	34	,	,	PUNCT
ejpam-4736	323	35	s	s	PART
ejpam-4736	323	36	)	)	PUNCT
ejpam-4736	323	37	by	by	ADP
ejpam-4736	323	38	lemma	lemma	PROPN
ejpam-4736	323	39	1	1	NUM
ejpam-4736	323	40	.	.	PUNCT
ejpam-4736	324	1	on	on	ADP
ejpam-4736	324	2	the	the	DET
ejpam-4736	324	3	other	other	ADJ
ejpam-4736	324	4	hand	hand	NOUN
ejpam-4736	324	5	,	,	PUNCT
ejpam-4736	324	6	we	we	PRON
ejpam-4736	324	7	have	have	VERB
ejpam-4736	324	8	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	324	9	,	,	PUNCT
ejpam-4736	324	10	s	s	PART
ejpam-4736	324	11	)	)	PUNCT
ejpam-4736	324	12	⊆	⊆	NUM
ejpam-4736	324	13	{	{	PUNCT
ejpam-4736	324	14	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	324	15	,	,	PUNCT
ejpam-4736	324	16	s	s	PART
ejpam-4736	324	17	)	)	PUNCT
ejpam-4736	324	18	and	and	CCONJ
ejpam-4736	324	19	(	(	PUNCT
ejpam-4736	324	20	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	324	21	,	,	PUNCT
ejpam-4736	324	22	s))δp(λ	s))δp(λ	NOUN
ejpam-4736	324	23	,	,	PUNCT
ejpam-4736	324	24	s	s	PART
ejpam-4736	324	25	)	)	PUNCT
ejpam-4736	324	26	⊆	⊆	NUM
ejpam-4736	324	27	(	(	PUNCT
ejpam-4736	324	28	{	{	PUNCT
ejpam-4736	324	29	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	324	30	,	,	PUNCT
ejpam-4736	324	31	s))δp(λ	s))δp(λ	NOUN
ejpam-4736	324	32	,	,	PUNCT
ejpam-4736	324	33	s	s	PART
ejpam-4736	324	34	)	)	PUNCT
ejpam-4736	324	35	=	=	SYM
ejpam-4736	324	36	{	{	PUNCT
ejpam-4736	324	37	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	324	38	,	,	PUNCT
ejpam-4736	324	39	s	s	NOUN
ejpam-4736	324	40	)	)	PUNCT
ejpam-4736	324	41	.	.	PUNCT
ejpam-4736	325	1	c.	c.	PROPN
ejpam-4736	325	2	boonpok	boonpok	PROPN
ejpam-4736	325	3	,	,	PUNCT
ejpam-4736	325	4	n.	n.	PROPN
ejpam-4736	325	5	srisarakham	srisarakham	PROPN
ejpam-4736	325	6	/	/	SYM
ejpam-4736	325	7	eur	eur	PROPN
ejpam-4736	325	8	.	.	PUNCT
ejpam-4736	326	1	j.	j.	PROPN
ejpam-4736	326	2	pure	pure	PROPN
ejpam-4736	326	3	appl	appl	PROPN
ejpam-4736	326	4	.	.	PROPN
ejpam-4736	326	5	math	math	PROPN
ejpam-4736	326	6	,	,	PUNCT
ejpam-4736	326	7	16	16	NUM
ejpam-4736	326	8	(	(	PUNCT
ejpam-4736	326	9	4	4	NUM
ejpam-4736	326	10	)	)	PUNCT
ejpam-4736	326	11	(	(	PUNCT
ejpam-4736	326	12	2023	2023	NUM
ejpam-4736	326	13	)	)	PUNCT
ejpam-4736	326	14	,	,	PUNCT
ejpam-4736	326	15	2581	2581	NUM
ejpam-4736	326	16	-	-	SYM
ejpam-4736	326	17	2596	2596	NUM
ejpam-4736	326	18	2590	2590	NUM
ejpam-4736	326	19	thus	thus	ADV
ejpam-4736	326	20	,	,	PUNCT
ejpam-4736	326	21	(	(	PUNCT
ejpam-4736	326	22	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	326	23	,	,	PUNCT
ejpam-4736	326	24	s))δp(λ	s))δp(λ	NOUN
ejpam-4736	326	25	,	,	PUNCT
ejpam-4736	326	26	s	s	PART
ejpam-4736	326	27	)	)	PUNCT
ejpam-4736	326	28	⊆	⊆	NUM
ejpam-4736	326	29	{	{	PUNCT
ejpam-4736	326	30	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	326	31	,	,	PUNCT
ejpam-4736	326	32	s	s	NOUN
ejpam-4736	326	33	)	)	PUNCT
ejpam-4736	326	34	.	.	PUNCT
ejpam-4736	327	1	(	(	PUNCT
ejpam-4736	327	2	4	4	X
ejpam-4736	327	3	)	)	PUNCT
ejpam-4736	327	4	since	since	SCONJ
ejpam-4736	327	5	x	x	PROPN
ejpam-4736	327	6	∈	∈	PROPN
ejpam-4736	327	7	u	u	NOUN
ejpam-4736	327	8	and	and	CCONJ
ejpam-4736	327	9	u	u	NOUN
ejpam-4736	327	10	is	be	AUX
ejpam-4736	327	11	a	a	DET
ejpam-4736	327	12	δp(λ	δp(λ	NOUN
ejpam-4736	327	13	,	,	PUNCT
ejpam-4736	327	14	s)-open	s)-open	PUNCT
ejpam-4736	327	15	set	set	VERB
ejpam-4736	327	16	,	,	PUNCT
ejpam-4736	327	17	we	we	PRON
ejpam-4736	327	18	have	have	VERB
ejpam-4736	327	19	δp(λ	δp(λ	NOUN
ejpam-4736	327	20	,	,	PUNCT
ejpam-4736	327	21	s)ker({x	s)ker({x	NUM
ejpam-4736	327	22	}	}	PUNCT
ejpam-4736	327	23	)	)	PUNCT
ejpam-4736	328	1	⊆	⊆	NUM
ejpam-4736	328	2	u	u	NOUN
ejpam-4736	328	3	.	.	PUNCT
ejpam-4736	329	1	thus	thus	ADV
ejpam-4736	329	2	,	,	PUNCT
ejpam-4736	329	3	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	329	4	,	,	PUNCT
ejpam-4736	329	5	s	s	PART
ejpam-4736	329	6	)	)	PUNCT
ejpam-4736	329	7	⊆	⊆	NUM
ejpam-4736	329	8	u	u	NOUN
ejpam-4736	329	9	.	.	PUNCT
ejpam-4736	330	1	(	(	PUNCT
ejpam-4736	330	2	5	5	NUM
ejpam-4736	330	3	)	)	PUNCT
ejpam-4736	330	4	since	since	SCONJ
ejpam-4736	330	5	x	x	PROPN
ejpam-4736	330	6	∈	∈	PROPN
ejpam-4736	330	7	f	f	PROPN
ejpam-4736	330	8	and	and	CCONJ
ejpam-4736	330	9	f	f	PROPN
ejpam-4736	330	10	is	be	AUX
ejpam-4736	330	11	a	a	DET
ejpam-4736	330	12	δp(λ	δp(λ	NOUN
ejpam-4736	330	13	,	,	PUNCT
ejpam-4736	330	14	s)-closed	s)-close	VERB
ejpam-4736	330	15	set	set	NOUN
ejpam-4736	330	16	,	,	PUNCT
ejpam-4736	330	17	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4736	330	18	,	,	PUNCT
ejpam-4736	330	19	s	s	PART
ejpam-4736	330	20	)	)	PUNCT
ejpam-4736	330	21	=	=	SYM
ejpam-4736	330	22	{	{	PUNCT
ejpam-4736	330	23	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	330	24	,	,	PUNCT
ejpam-4736	330	25	s	s	NOUN
ejpam-4736	330	26	)	)	PUNCT
ejpam-4736	330	27	∩	∩	NOUN
ejpam-4736	330	28	δp(λ	δp(λ	NOUN
ejpam-4736	330	29	,	,	PUNCT
ejpam-4736	330	30	s)ker({x	s)ker({x	NOUN
ejpam-4736	330	31	}	}	PUNCT
ejpam-4736	330	32	)	)	PUNCT
ejpam-4736	331	1	⊆	⊆	NUM
ejpam-4736	331	2	{	{	PUNCT
ejpam-4736	331	3	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	331	4	,	,	PUNCT
ejpam-4736	331	5	s	s	PART
ejpam-4736	331	6	)	)	PUNCT
ejpam-4736	331	7	⊆	⊆	NUM
ejpam-4736	331	8	f	f	PROPN
ejpam-4736	331	9	δp(λ	δp(λ	NOUN
ejpam-4736	331	10	,	,	PUNCT
ejpam-4736	331	11	s	s	PART
ejpam-4736	331	12	)	)	PUNCT
ejpam-4736	331	13	=	=	SYM
ejpam-4736	331	14	f.	f.	PROPN
ejpam-4736	331	15	theorem	theorem	VERB
ejpam-4736	331	16	14	14	NUM
ejpam-4736	331	17	.	.	PUNCT
ejpam-4736	332	1	for	for	ADP
ejpam-4736	332	2	any	any	DET
ejpam-4736	332	3	points	point	NOUN
ejpam-4736	332	4	x	x	PUNCT
ejpam-4736	332	5	and	and	CCONJ
ejpam-4736	332	6	y	y	PROPN
ejpam-4736	332	7	in	in	ADP
ejpam-4736	332	8	a	a	DET
ejpam-4736	332	9	topological	topological	ADJ
ejpam-4736	332	10	space	space	NOUN
ejpam-4736	332	11	(	(	PUNCT
ejpam-4736	332	12	x	x	X
ejpam-4736	332	13	,	,	PUNCT
ejpam-4736	332	14	τ	τ	PROPN
ejpam-4736	332	15	)	)	PUNCT
ejpam-4736	332	16	,	,	PUNCT
ejpam-4736	332	17	the	the	DET
ejpam-4736	332	18	following	follow	VERB
ejpam-4736	332	19	properties	property	NOUN
ejpam-4736	332	20	are	be	AUX
ejpam-4736	332	21	equivalent	equivalent	ADJ
ejpam-4736	332	22	:	:	PUNCT
ejpam-4736	332	23	(	(	PUNCT
ejpam-4736	332	24	1	1	NUM
ejpam-4736	332	25	)	)	PUNCT
ejpam-4736	332	26	δp(λ	δp(λ	NOUN
ejpam-4736	332	27	,	,	PUNCT
ejpam-4736	332	28	s)ker({x	s)ker({x	NOUN
ejpam-4736	332	29	}	}	PUNCT
ejpam-4736	332	30	)	)	PUNCT
ejpam-4736	332	31	̸=	̸=	PROPN
ejpam-4736	332	32	δp(λ	δp(λ	NOUN
ejpam-4736	332	33	,	,	PUNCT
ejpam-4736	332	34	s)ker({y	s)ker({y	NOUN
ejpam-4736	332	35	}	}	PUNCT
ejpam-4736	332	36	)	)	PUNCT
ejpam-4736	332	37	.	.	PUNCT
ejpam-4736	333	1	(	(	PUNCT
ejpam-4736	333	2	2	2	X
ejpam-4736	333	3	)	)	PUNCT
ejpam-4736	333	4	{	{	PUNCT
ejpam-4736	333	5	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	333	6	,	,	PUNCT
ejpam-4736	333	7	s	s	PART
ejpam-4736	333	8	)	)	PUNCT
ejpam-4736	333	9	̸=	̸=	PROPN
ejpam-4736	333	10	{	{	PUNCT
ejpam-4736	333	11	y}δp(λ	y}δp(λ	PROPN
ejpam-4736	333	12	,	,	PUNCT
ejpam-4736	333	13	s	s	NOUN
ejpam-4736	333	14	)	)	PUNCT
ejpam-4736	333	15	.	.	PUNCT
ejpam-4736	334	1	proof	proof	NOUN
ejpam-4736	334	2	.	.	PUNCT
ejpam-4736	335	1	(	(	PUNCT
ejpam-4736	335	2	1	1	X
ejpam-4736	335	3	)	)	PUNCT
ejpam-4736	335	4	⇒	⇒	NOUN
ejpam-4736	335	5	(	(	PUNCT
ejpam-4736	335	6	2	2	NUM
ejpam-4736	335	7	):	):	PUNCT
ejpam-4736	335	8	suppose	suppose	VERB
ejpam-4736	335	9	that	that	SCONJ
ejpam-4736	335	10	δp(λ	δp(λ	NOUN
ejpam-4736	335	11	,	,	PUNCT
ejpam-4736	335	12	s)ker({x	s)ker({x	NOUN
ejpam-4736	335	13	}	}	PUNCT
ejpam-4736	335	14	)	)	PUNCT
ejpam-4736	335	15	̸=	̸=	PROPN
ejpam-4736	335	16	δp(λ	δp(λ	NOUN
ejpam-4736	335	17	,	,	PUNCT
ejpam-4736	335	18	s)ker({y	s)ker({y	NOUN
ejpam-4736	335	19	}	}	PUNCT
ejpam-4736	335	20	)	)	PUNCT
ejpam-4736	335	21	.	.	PUNCT
ejpam-4736	336	1	then	then	ADV
ejpam-4736	336	2	,	,	PUNCT
ejpam-4736	336	3	there	there	PRON
ejpam-4736	336	4	exists	exist	VERB
ejpam-4736	336	5	a	a	DET
ejpam-4736	336	6	point	point	NOUN
ejpam-4736	336	7	z	z	NOUN
ejpam-4736	336	8	∈	∈	PROPN
ejpam-4736	336	9	x	x	PUNCT
ejpam-4736	336	10	such	such	ADJ
ejpam-4736	336	11	that	that	SCONJ
ejpam-4736	336	12	z	z	PROPN
ejpam-4736	336	13	∈	∈	PROPN
ejpam-4736	336	14	δp(λ	δp(λ	NOUN
ejpam-4736	336	15	,	,	PUNCT
ejpam-4736	336	16	s)ker({x	s)ker({x	NOUN
ejpam-4736	336	17	}	}	PUNCT
ejpam-4736	336	18	)	)	PUNCT
ejpam-4736	336	19	and	and	CCONJ
ejpam-4736	336	20	z	z	PROPN
ejpam-4736	336	21	̸∈	̸∈	PROPN
ejpam-4736	336	22	δp(λ	δp(λ	NOUN
ejpam-4736	336	23	,	,	PUNCT
ejpam-4736	336	24	s)ker({y	s)ker({y	NOUN
ejpam-4736	336	25	}	}	PUNCT
ejpam-4736	336	26	)	)	PUNCT
ejpam-4736	336	27	or	or	CCONJ
ejpam-4736	336	28	z	z	NOUN
ejpam-4736	336	29	∈	∈	PROPN
ejpam-4736	336	30	δp(λ	δp(λ	NOUN
ejpam-4736	336	31	,	,	PUNCT
ejpam-4736	336	32	s)ker({y	s)ker({y	NOUN
ejpam-4736	336	33	}	}	PUNCT
ejpam-4736	336	34	)	)	PUNCT
ejpam-4736	336	35	and	and	CCONJ
ejpam-4736	336	36	z	z	PROPN
ejpam-4736	336	37	̸∈	̸∈	PROPN
ejpam-4736	336	38	δp(λ	δp(λ	NOUN
ejpam-4736	336	39	,	,	PUNCT
ejpam-4736	336	40	s)ker({x	s)ker({x	NUM
ejpam-4736	336	41	}	}	PUNCT
ejpam-4736	336	42	)	)	PUNCT
ejpam-4736	336	43	.	.	PUNCT
ejpam-4736	337	1	we	we	PRON
ejpam-4736	337	2	prove	prove	VERB
ejpam-4736	337	3	only	only	ADV
ejpam-4736	337	4	the	the	DET
ejpam-4736	337	5	first	first	ADJ
ejpam-4736	337	6	case	case	NOUN
ejpam-4736	337	7	being	be	AUX
ejpam-4736	337	8	the	the	DET
ejpam-4736	337	9	second	second	ADJ
ejpam-4736	337	10	analogous	analogous	NOUN
ejpam-4736	337	11	.	.	PUNCT
ejpam-4736	338	1	from	from	ADP
ejpam-4736	338	2	z	z	PROPN
ejpam-4736	338	3	∈	∈	PROPN
ejpam-4736	338	4	δp(λ	δp(λ	NOUN
ejpam-4736	338	5	,	,	PUNCT
ejpam-4736	338	6	s)ker({x	s)ker({x	NUM
ejpam-4736	338	7	}	}	PUNCT
ejpam-4736	338	8	)	)	PUNCT
ejpam-4736	339	1	it	it	PRON
ejpam-4736	339	2	follows	follow	VERB
ejpam-4736	339	3	that	that	SCONJ
ejpam-4736	339	4	{	{	PUNCT
ejpam-4736	339	5	x	x	NOUN
ejpam-4736	339	6	}	}	PUNCT
ejpam-4736	339	7	∩	∩	ADJ
ejpam-4736	339	8	{	{	PUNCT
ejpam-4736	339	9	z}δp(λ	z}δp(λ	PROPN
ejpam-4736	339	10	,	,	PUNCT
ejpam-4736	339	11	s	s	PART
ejpam-4736	339	12	)	)	PUNCT
ejpam-4736	339	13	̸=	̸=	PROPN
ejpam-4736	339	14	∅	∅	NOUN
ejpam-4736	339	15	which	which	PRON
ejpam-4736	339	16	implies	imply	VERB
ejpam-4736	339	17	x	x	X
ejpam-4736	339	18	∈	∈	PROPN
ejpam-4736	339	19	{	{	PUNCT
ejpam-4736	339	20	z}δp(λ	z}δp(λ	PROPN
ejpam-4736	339	21	,	,	PUNCT
ejpam-4736	339	22	s	s	NOUN
ejpam-4736	339	23	)	)	PUNCT
ejpam-4736	339	24	.	.	PUNCT
ejpam-4736	340	1	by	by	ADP
ejpam-4736	340	2	z	z	PROPN
ejpam-4736	340	3	̸∈	̸∈	PROPN
ejpam-4736	340	4	δp(λ	δp(λ	NOUN
ejpam-4736	340	5	,	,	PUNCT
ejpam-4736	340	6	s)ker({y	s)ker({y	NOUN
ejpam-4736	340	7	}	}	PUNCT
ejpam-4736	340	8	)	)	PUNCT
ejpam-4736	340	9	,	,	PUNCT
ejpam-4736	340	10	we	we	PRON
ejpam-4736	340	11	have	have	VERB
ejpam-4736	340	12	{	{	PUNCT
ejpam-4736	340	13	y	y	NOUN
ejpam-4736	340	14	}	}	PUNCT
ejpam-4736	340	15	∩	∩	NOUN
ejpam-4736	340	16	{	{	PUNCT
ejpam-4736	340	17	z}δp(λ	z}δp(λ	PROPN
ejpam-4736	340	18	,	,	PUNCT
ejpam-4736	340	19	s	s	PART
ejpam-4736	340	20	)	)	PUNCT
ejpam-4736	340	21	=	=	PUNCT
ejpam-4736	340	22	∅.	∅.	NOUN
ejpam-4736	340	23	since	since	SCONJ
ejpam-4736	340	24	x	x	PROPN
ejpam-4736	340	25	∈	∈	PROPN
ejpam-4736	340	26	{	{	PUNCT
ejpam-4736	340	27	z}δp(λ	z}δp(λ	PROPN
ejpam-4736	340	28	,	,	PUNCT
ejpam-4736	340	29	s	s	PART
ejpam-4736	340	30	)	)	PUNCT
ejpam-4736	340	31	,	,	PUNCT
ejpam-4736	340	32	{	{	PUNCT
ejpam-4736	340	33	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	340	34	,	,	PUNCT
ejpam-4736	340	35	s	s	PART
ejpam-4736	340	36	)	)	PUNCT
ejpam-4736	340	37	⊆	⊆	NUM
ejpam-4736	340	38	{	{	PUNCT
ejpam-4736	340	39	z}δp(λ	z}δp(λ	PROPN
ejpam-4736	340	40	,	,	PUNCT
ejpam-4736	340	41	s	s	PART
ejpam-4736	340	42	)	)	PUNCT
ejpam-4736	340	43	and	and	CCONJ
ejpam-4736	340	44	{	{	PUNCT
ejpam-4736	340	45	y	y	NOUN
ejpam-4736	340	46	}	}	PUNCT
ejpam-4736	340	47	∩	∩	NOUN
ejpam-4736	340	48	{	{	PUNCT
ejpam-4736	340	49	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	340	50	,	,	PUNCT
ejpam-4736	340	51	s	s	PART
ejpam-4736	340	52	)	)	PUNCT
ejpam-4736	340	53	=	=	PUNCT
ejpam-4736	340	54	∅.	∅.	VERB
ejpam-4736	340	55	therefore	therefore	ADV
ejpam-4736	340	56	,	,	PUNCT
ejpam-4736	340	57	{	{	PUNCT
ejpam-4736	340	58	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	340	59	,	,	PUNCT
ejpam-4736	340	60	s	s	PART
ejpam-4736	340	61	)	)	PUNCT
ejpam-4736	340	62	̸=	̸=	PROPN
ejpam-4736	340	63	{	{	PUNCT
ejpam-4736	340	64	y}δp(λ	y}δp(λ	PROPN
ejpam-4736	340	65	,	,	PUNCT
ejpam-4736	340	66	s	s	NOUN
ejpam-4736	340	67	)	)	PUNCT
ejpam-4736	340	68	.	.	PUNCT
ejpam-4736	341	1	thus	thus	ADV
ejpam-4736	341	2	,	,	PUNCT
ejpam-4736	341	3	δp(λ	δp(λ	NOUN
ejpam-4736	341	4	,	,	PUNCT
ejpam-4736	341	5	s)ker({x	s)ker({x	NOUN
ejpam-4736	341	6	}	}	PUNCT
ejpam-4736	341	7	)	)	PUNCT
ejpam-4736	342	1	̸=	̸=	PROPN
ejpam-4736	342	2	δp(λ	δp(λ	NOUN
ejpam-4736	342	3	,	,	PUNCT
ejpam-4736	342	4	s)ker({y	s)ker({y	NOUN
ejpam-4736	342	5	}	}	PUNCT
ejpam-4736	342	6	)	)	PUNCT
ejpam-4736	342	7	implies	imply	VERB
ejpam-4736	342	8	that	that	SCONJ
ejpam-4736	342	9	{	{	PUNCT
ejpam-4736	342	10	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	342	11	,	,	PUNCT
ejpam-4736	342	12	s	s	PART
ejpam-4736	342	13	)	)	PUNCT
ejpam-4736	342	14	̸=	̸=	PROPN
ejpam-4736	342	15	{	{	PUNCT
ejpam-4736	342	16	y}δp(λ	y}δp(λ	PROPN
ejpam-4736	342	17	,	,	PUNCT
ejpam-4736	342	18	s	s	NOUN
ejpam-4736	342	19	)	)	PUNCT
ejpam-4736	342	20	.	.	PUNCT
ejpam-4736	343	1	(	(	PUNCT
ejpam-4736	343	2	2	2	X
ejpam-4736	343	3	)	)	PUNCT
ejpam-4736	343	4	⇒	⇒	NOUN
ejpam-4736	343	5	(	(	PUNCT
ejpam-4736	343	6	1	1	NUM
ejpam-4736	343	7	):	):	PUNCT
ejpam-4736	343	8	suppose	suppose	VERB
ejpam-4736	343	9	that	that	SCONJ
ejpam-4736	343	10	{	{	PUNCT
ejpam-4736	343	11	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	343	12	,	,	PUNCT
ejpam-4736	343	13	s	s	PART
ejpam-4736	343	14	)	)	PUNCT
ejpam-4736	343	15	̸=	̸=	PROPN
ejpam-4736	343	16	{	{	PUNCT
ejpam-4736	343	17	y}δp(λ	y}δp(λ	PROPN
ejpam-4736	343	18	,	,	PUNCT
ejpam-4736	343	19	s	s	NOUN
ejpam-4736	343	20	)	)	PUNCT
ejpam-4736	343	21	.	.	PUNCT
ejpam-4736	344	1	there	there	PRON
ejpam-4736	344	2	exists	exist	VERB
ejpam-4736	344	3	a	a	DET
ejpam-4736	344	4	point	point	NOUN
ejpam-4736	344	5	z	z	NOUN
ejpam-4736	344	6	∈	∈	PROPN
ejpam-4736	344	7	x	x	PUNCT
ejpam-4736	344	8	such	such	ADJ
ejpam-4736	344	9	that	that	SCONJ
ejpam-4736	344	10	z	z	PROPN
ejpam-4736	344	11	∈	∈	PROPN
ejpam-4736	344	12	{	{	PUNCT
ejpam-4736	344	13	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	344	14	,	,	PUNCT
ejpam-4736	344	15	s	s	PART
ejpam-4736	344	16	)	)	PUNCT
ejpam-4736	344	17	and	and	CCONJ
ejpam-4736	344	18	z	z	PROPN
ejpam-4736	344	19	̸∈	̸∈	PROPN
ejpam-4736	344	20	{	{	PUNCT
ejpam-4736	344	21	y}δp(λ	y}δp(λ	PROPN
ejpam-4736	344	22	,	,	PUNCT
ejpam-4736	344	23	s	s	NOUN
ejpam-4736	344	24	)	)	PUNCT
ejpam-4736	344	25	or	or	CCONJ
ejpam-4736	344	26	z	z	NOUN
ejpam-4736	344	27	∈	∈	PROPN
ejpam-4736	344	28	{	{	PUNCT
ejpam-4736	344	29	y}δp(λ	y}δp(λ	NOUN
ejpam-4736	344	30	,	,	PUNCT
ejpam-4736	344	31	s	s	PART
ejpam-4736	344	32	)	)	PUNCT
ejpam-4736	344	33	and	and	CCONJ
ejpam-4736	344	34	z	z	PROPN
ejpam-4736	344	35	̸∈	̸∈	PROPN
ejpam-4736	344	36	{	{	PUNCT
ejpam-4736	344	37	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	344	38	,	,	PUNCT
ejpam-4736	344	39	s	s	NOUN
ejpam-4736	344	40	)	)	PUNCT
ejpam-4736	344	41	.	.	PUNCT
ejpam-4736	345	1	we	we	PRON
ejpam-4736	345	2	prove	prove	VERB
ejpam-4736	345	3	only	only	ADV
ejpam-4736	345	4	the	the	DET
ejpam-4736	345	5	first	first	ADJ
ejpam-4736	345	6	case	case	NOUN
ejpam-4736	345	7	being	be	AUX
ejpam-4736	345	8	the	the	DET
ejpam-4736	345	9	second	second	ADJ
ejpam-4736	345	10	analogous	analogous	NOUN
ejpam-4736	345	11	.	.	PUNCT
ejpam-4736	346	1	it	it	PRON
ejpam-4736	346	2	follows	follow	VERB
ejpam-4736	346	3	that	that	SCONJ
ejpam-4736	346	4	there	there	PRON
ejpam-4736	346	5	exists	exist	VERB
ejpam-4736	346	6	a	a	DET
ejpam-4736	346	7	δp(λ	δp(λ	NOUN
ejpam-4736	346	8	,	,	PUNCT
ejpam-4736	346	9	s)open	s)open	NOUN
ejpam-4736	346	10	set	set	NOUN
ejpam-4736	346	11	containing	contain	VERB
ejpam-4736	346	12	z	z	NOUN
ejpam-4736	346	13	and	and	CCONJ
ejpam-4736	346	14	therefore	therefore	ADV
ejpam-4736	346	15	x	x	X
ejpam-4736	346	16	but	but	CCONJ
ejpam-4736	346	17	not	not	PART
ejpam-4736	346	18	y	y	NOUN
ejpam-4736	346	19	,	,	PUNCT
ejpam-4736	346	20	namely	namely	ADV
ejpam-4736	346	21	,	,	PUNCT
ejpam-4736	346	22	y	y	PROPN
ejpam-4736	346	23	̸∈	̸∈	PROPN
ejpam-4736	346	24	δp(λ	δp(λ	NOUN
ejpam-4736	346	25	,	,	PUNCT
ejpam-4736	346	26	s)ker({x	s)ker({x	NUM
ejpam-4736	346	27	}	}	PUNCT
ejpam-4736	346	28	)	)	PUNCT
ejpam-4736	346	29	and	and	CCONJ
ejpam-4736	346	30	thus	thus	ADV
ejpam-4736	346	31	δp(λ	δp(λ	NOUN
ejpam-4736	346	32	,	,	PUNCT
ejpam-4736	346	33	s)ker({x	s)ker({x	NOUN
ejpam-4736	346	34	}	}	PUNCT
ejpam-4736	346	35	)	)	PUNCT
ejpam-4736	346	36	̸=	̸=	PROPN
ejpam-4736	346	37	δp(λ	δp(λ	NOUN
ejpam-4736	346	38	,	,	PUNCT
ejpam-4736	346	39	s)ker({y	s)ker({y	NOUN
ejpam-4736	346	40	}	}	PUNCT
ejpam-4736	346	41	)	)	PUNCT
ejpam-4736	346	42	.	.	PUNCT
ejpam-4736	347	1	theorem	theorem	NOUN
ejpam-4736	347	2	15	15	NUM
ejpam-4736	347	3	.	.	PUNCT
ejpam-4736	348	1	let	let	AUX
ejpam-4736	348	2	(	(	PUNCT
ejpam-4736	348	3	x	x	NOUN
ejpam-4736	348	4	,	,	PUNCT
ejpam-4736	348	5	τ	τ	X
ejpam-4736	348	6	)	)	PUNCT
ejpam-4736	348	7	be	be	VERB
ejpam-4736	348	8	a	a	DET
ejpam-4736	348	9	topological	topological	ADJ
ejpam-4736	348	10	space	space	NOUN
ejpam-4736	348	11	and	and	CCONJ
ejpam-4736	348	12	x	x	NOUN
ejpam-4736	348	13	,	,	PUNCT
ejpam-4736	348	14	y	y	PROPN
ejpam-4736	348	15	∈	∈	PROPN
ejpam-4736	348	16	x.	x.	NOUN
ejpam-4736	348	17	then	then	ADV
ejpam-4736	348	18	,	,	PUNCT
ejpam-4736	348	19	the	the	DET
ejpam-4736	348	20	following	follow	VERB
ejpam-4736	348	21	properties	property	NOUN
ejpam-4736	348	22	hold	hold	VERB
ejpam-4736	348	23	:	:	PUNCT
ejpam-4736	348	24	(	(	PUNCT
ejpam-4736	348	25	1	1	X
ejpam-4736	348	26	)	)	PUNCT
ejpam-4736	348	27	y	y	PROPN
ejpam-4736	348	28	∈	∈	PROPN
ejpam-4736	348	29	δp(λ	δp(λ	NOUN
ejpam-4736	348	30	,	,	PUNCT
ejpam-4736	348	31	s)ker({x	s)ker({x	NUM
ejpam-4736	348	32	}	}	PUNCT
ejpam-4736	348	33	)	)	PUNCT
ejpam-4736	349	1	if	if	SCONJ
ejpam-4736	349	2	and	and	CCONJ
ejpam-4736	349	3	only	only	ADV
ejpam-4736	349	4	if	if	SCONJ
ejpam-4736	349	5	x	x	SYM
ejpam-4736	349	6	∈	∈	NOUN
ejpam-4736	349	7	{	{	PUNCT
ejpam-4736	349	8	y}δp(λ	y}δp(λ	NOUN
ejpam-4736	349	9	,	,	PUNCT
ejpam-4736	349	10	s	s	NOUN
ejpam-4736	349	11	)	)	PUNCT
ejpam-4736	349	12	.	.	PUNCT
ejpam-4736	350	1	(	(	PUNCT
ejpam-4736	350	2	2	2	NUM
ejpam-4736	350	3	)	)	PUNCT
ejpam-4736	350	4	δp(λ	δp(λ	NOUN
ejpam-4736	350	5	,	,	PUNCT
ejpam-4736	350	6	s)ker({x	s)ker({x	NUM
ejpam-4736	350	7	}	}	PUNCT
ejpam-4736	350	8	)	)	PUNCT
ejpam-4736	351	1	=	=	SYM
ejpam-4736	351	2	δp(λ	δp(λ	NOUN
ejpam-4736	351	3	,	,	PUNCT
ejpam-4736	351	4	s)ker({y	s)ker({y	NOUN
ejpam-4736	351	5	}	}	PUNCT
ejpam-4736	351	6	)	)	PUNCT
ejpam-4736	351	7	if	if	SCONJ
ejpam-4736	351	8	and	and	CCONJ
ejpam-4736	351	9	only	only	ADV
ejpam-4736	351	10	if	if	SCONJ
ejpam-4736	351	11	{	{	PUNCT
ejpam-4736	351	12	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	351	13	,	,	PUNCT
ejpam-4736	351	14	s	s	PART
ejpam-4736	351	15	)	)	PUNCT
ejpam-4736	351	16	=	=	PRON
ejpam-4736	351	17	{	{	PUNCT
ejpam-4736	351	18	y}δp(λ	y}δp(λ	PROPN
ejpam-4736	351	19	,	,	PUNCT
ejpam-4736	351	20	s	s	NOUN
ejpam-4736	351	21	)	)	PUNCT
ejpam-4736	351	22	.	.	PUNCT
ejpam-4736	352	1	c.	c.	PROPN
ejpam-4736	352	2	boonpok	boonpok	PROPN
ejpam-4736	352	3	,	,	PUNCT
ejpam-4736	352	4	n.	n.	PROPN
ejpam-4736	352	5	srisarakham	srisarakham	PROPN
ejpam-4736	352	6	/	/	SYM
ejpam-4736	352	7	eur	eur	PROPN
ejpam-4736	352	8	.	.	PUNCT
ejpam-4736	353	1	j.	j.	PROPN
ejpam-4736	353	2	pure	pure	PROPN
ejpam-4736	353	3	appl	appl	PROPN
ejpam-4736	353	4	.	.	PROPN
ejpam-4736	353	5	math	math	PROPN
ejpam-4736	353	6	,	,	PUNCT
ejpam-4736	353	7	16	16	NUM
ejpam-4736	353	8	(	(	PUNCT
ejpam-4736	353	9	4	4	NUM
ejpam-4736	353	10	)	)	PUNCT
ejpam-4736	353	11	(	(	PUNCT
ejpam-4736	353	12	2023	2023	NUM
ejpam-4736	353	13	)	)	PUNCT
ejpam-4736	353	14	,	,	PUNCT
ejpam-4736	353	15	2581	2581	NUM
ejpam-4736	353	16	-	-	SYM
ejpam-4736	353	17	2596	2596	NUM
ejpam-4736	353	18	2591	2591	NUM
ejpam-4736	353	19	proof	proof	NOUN
ejpam-4736	353	20	.	.	PUNCT
ejpam-4736	354	1	(	(	PUNCT
ejpam-4736	354	2	1	1	X
ejpam-4736	354	3	)	)	PUNCT
ejpam-4736	354	4	let	let	VERB
ejpam-4736	354	5	x	x	SYM
ejpam-4736	354	6	̸∈	̸∈	PROPN
ejpam-4736	354	7	{	{	PUNCT
ejpam-4736	354	8	y}δp(λ	y}δp(λ	PROPN
ejpam-4736	354	9	,	,	PUNCT
ejpam-4736	354	10	s	s	NOUN
ejpam-4736	354	11	)	)	PUNCT
ejpam-4736	354	12	.	.	PUNCT
ejpam-4736	355	1	then	then	ADV
ejpam-4736	355	2	,	,	PUNCT
ejpam-4736	355	3	there	there	PRON
ejpam-4736	355	4	exists	exist	VERB
ejpam-4736	355	5	u	u	PROPN
ejpam-4736	355	6	∈	∈	PROPN
ejpam-4736	355	7	δp(λ	δp(λ	NOUN
ejpam-4736	355	8	,	,	PUNCT
ejpam-4736	355	9	s)o(x	s)o(x	PROPN
ejpam-4736	355	10	,	,	PUNCT
ejpam-4736	355	11	τ	τ	X
ejpam-4736	355	12	)	)	PUNCT
ejpam-4736	355	13	such	such	ADJ
ejpam-4736	355	14	that	that	SCONJ
ejpam-4736	355	15	x	x	SYM
ejpam-4736	355	16	∈	∈	PROPN
ejpam-4736	355	17	u	u	NOUN
ejpam-4736	355	18	and	and	CCONJ
ejpam-4736	355	19	y	y	PROPN
ejpam-4736	355	20	̸∈	̸∈	PROPN
ejpam-4736	355	21	u	u	PROPN
ejpam-4736	355	22	.	.	PUNCT
ejpam-4736	356	1	thus	thus	ADV
ejpam-4736	356	2	,	,	PUNCT
ejpam-4736	356	3	y	y	PROPN
ejpam-4736	356	4	̸∈	̸∈	PROPN
ejpam-4736	356	5	δp(λ	δp(λ	NOUN
ejpam-4736	356	6	,	,	PUNCT
ejpam-4736	356	7	s)ker({x	s)ker({x	NUM
ejpam-4736	356	8	}	}	PUNCT
ejpam-4736	356	9	)	)	PUNCT
ejpam-4736	356	10	.	.	PUNCT
ejpam-4736	357	1	the	the	DET
ejpam-4736	357	2	converse	converse	NOUN
ejpam-4736	357	3	is	be	AUX
ejpam-4736	357	4	similarly	similarly	ADV
ejpam-4736	357	5	shown	show	VERB
ejpam-4736	357	6	.	.	PUNCT
ejpam-4736	358	1	(	(	PUNCT
ejpam-4736	358	2	2	2	X
ejpam-4736	358	3	)	)	PUNCT
ejpam-4736	358	4	suppose	suppose	VERB
ejpam-4736	358	5	that	that	SCONJ
ejpam-4736	358	6	δp(λ	δp(λ	NOUN
ejpam-4736	358	7	,	,	PUNCT
ejpam-4736	358	8	s)ker({x	s)ker({x	NUM
ejpam-4736	358	9	}	}	PUNCT
ejpam-4736	358	10	)	)	PUNCT
ejpam-4736	358	11	=	=	SYM
ejpam-4736	359	1	δp(λ	δp(λ	NOUN
ejpam-4736	359	2	,	,	PUNCT
ejpam-4736	359	3	s)ker({y	s)ker({y	NOUN
ejpam-4736	359	4	}	}	PUNCT
ejpam-4736	359	5	)	)	PUNCT
ejpam-4736	359	6	for	for	ADP
ejpam-4736	359	7	any	any	DET
ejpam-4736	359	8	x	x	NOUN
ejpam-4736	359	9	,	,	PUNCT
ejpam-4736	359	10	y	y	PROPN
ejpam-4736	359	11	∈	∈	PROPN
ejpam-4736	359	12	x.	x.	VERB
ejpam-4736	360	1	since	since	SCONJ
ejpam-4736	360	2	x	x	PROPN
ejpam-4736	360	3	∈	∈	PROPN
ejpam-4736	360	4	δp(λ	δp(λ	NOUN
ejpam-4736	360	5	,	,	PUNCT
ejpam-4736	360	6	s)ker({x	s)ker({x	NOUN
ejpam-4736	360	7	}	}	PUNCT
ejpam-4736	360	8	)	)	PUNCT
ejpam-4736	360	9	,	,	PUNCT
ejpam-4736	360	10	x	x	PUNCT
ejpam-4736	360	11	∈	∈	PROPN
ejpam-4736	360	12	δp(λ	δp(λ	NOUN
ejpam-4736	360	13	,	,	PUNCT
ejpam-4736	360	14	s)ker({y	s)ker({y	NOUN
ejpam-4736	360	15	}	}	PUNCT
ejpam-4736	360	16	)	)	PUNCT
ejpam-4736	360	17	,	,	PUNCT
ejpam-4736	360	18	by	by	ADP
ejpam-4736	360	19	(	(	PUNCT
ejpam-4736	360	20	1	1	NUM
ejpam-4736	360	21	)	)	PUNCT
ejpam-4736	360	22	,	,	PUNCT
ejpam-4736	360	23	y	y	PROPN
ejpam-4736	360	24	∈	∈	PROPN
ejpam-4736	360	25	{	{	PUNCT
ejpam-4736	360	26	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	360	27	,	,	PUNCT
ejpam-4736	360	28	s	s	NOUN
ejpam-4736	360	29	)	)	PUNCT
ejpam-4736	360	30	.	.	PUNCT
ejpam-4736	361	1	by	by	ADP
ejpam-4736	361	2	lemma	lemma	PROPN
ejpam-4736	361	3	1	1	NUM
ejpam-4736	361	4	,	,	PUNCT
ejpam-4736	361	5	{	{	PUNCT
ejpam-4736	361	6	y}δp(λ	y}δp(λ	NOUN
ejpam-4736	361	7	,	,	PUNCT
ejpam-4736	361	8	s	s	NOUN
ejpam-4736	361	9	)	)	PUNCT
ejpam-4736	361	10	⊆	⊆	NUM
ejpam-4736	361	11	{	{	PUNCT
ejpam-4736	361	12	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	361	13	,	,	PUNCT
ejpam-4736	361	14	s	s	NOUN
ejpam-4736	361	15	)	)	PUNCT
ejpam-4736	361	16	.	.	PUNCT
ejpam-4736	362	1	similarly	similarly	ADV
ejpam-4736	362	2	,	,	PUNCT
ejpam-4736	362	3	we	we	PRON
ejpam-4736	362	4	have	have	VERB
ejpam-4736	362	5	{	{	PUNCT
ejpam-4736	362	6	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	362	7	,	,	PUNCT
ejpam-4736	362	8	s	s	PART
ejpam-4736	362	9	)	)	PUNCT
ejpam-4736	362	10	⊆	⊆	NUM
ejpam-4736	362	11	{	{	PUNCT
ejpam-4736	362	12	y}δp(λ	y}δp(λ	PROPN
ejpam-4736	362	13	,	,	PUNCT
ejpam-4736	362	14	s	s	NOUN
ejpam-4736	362	15	)	)	PUNCT
ejpam-4736	362	16	and	and	CCONJ
ejpam-4736	362	17	hence	hence	ADV
ejpam-4736	362	18	{	{	PUNCT
ejpam-4736	362	19	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	362	20	,	,	PUNCT
ejpam-4736	362	21	s	s	PART
ejpam-4736	362	22	)	)	PUNCT
ejpam-4736	362	23	=	=	PRON
ejpam-4736	362	24	{	{	PUNCT
ejpam-4736	362	25	y}δp(λ	y}δp(λ	PROPN
ejpam-4736	362	26	,	,	PUNCT
ejpam-4736	362	27	s	s	NOUN
ejpam-4736	362	28	)	)	PUNCT
ejpam-4736	362	29	.	.	PUNCT
ejpam-4736	363	1	conversely	conversely	ADV
ejpam-4736	363	2	,	,	PUNCT
ejpam-4736	363	3	suppose	suppose	VERB
ejpam-4736	363	4	that	that	SCONJ
ejpam-4736	363	5	{	{	PUNCT
ejpam-4736	363	6	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	363	7	,	,	PUNCT
ejpam-4736	363	8	s	s	PART
ejpam-4736	363	9	)	)	PUNCT
ejpam-4736	363	10	=	=	PRON
ejpam-4736	363	11	{	{	PUNCT
ejpam-4736	363	12	y}δp(λ	y}δp(λ	PROPN
ejpam-4736	363	13	,	,	PUNCT
ejpam-4736	363	14	s	s	NOUN
ejpam-4736	363	15	)	)	PUNCT
ejpam-4736	363	16	.	.	PUNCT
ejpam-4736	364	1	since	since	SCONJ
ejpam-4736	364	2	x	x	PROPN
ejpam-4736	364	3	∈	∈	PROPN
ejpam-4736	364	4	{	{	PUNCT
ejpam-4736	364	5	x}δp(λ	x}δp(λ	PROPN
ejpam-4736	364	6	,	,	PUNCT
ejpam-4736	364	7	s	s	PART
ejpam-4736	364	8	)	)	PUNCT
ejpam-4736	364	9	,	,	PUNCT
ejpam-4736	364	10	x	x	PUNCT
ejpam-4736	364	11	∈	∈	NOUN
ejpam-4736	364	12	{	{	PUNCT
ejpam-4736	364	13	y}δp(λ	y}δp(λ	NOUN
ejpam-4736	364	14	,	,	PUNCT
ejpam-4736	364	15	s	s	NOUN
ejpam-4736	364	16	)	)	PUNCT
ejpam-4736	364	17	and	and	CCONJ
ejpam-4736	364	18	by	by	ADP
ejpam-4736	364	19	(	(	PUNCT
ejpam-4736	364	20	1	1	NUM
ejpam-4736	364	21	)	)	PUNCT
ejpam-4736	364	22	,	,	PUNCT
ejpam-4736	364	23	y	y	PROPN
ejpam-4736	364	24	∈	∈	PROPN
ejpam-4736	364	25	δp(λ	δp(λ	NOUN
ejpam-4736	364	26	,	,	PUNCT
ejpam-4736	364	27	s)ker({x	s)ker({x	NOUN
ejpam-4736	364	28	}	}	PUNCT
ejpam-4736	364	29	)	)	PUNCT
ejpam-4736	364	30	.	.	PUNCT
ejpam-4736	365	1	by	by	ADP
ejpam-4736	365	2	lemma	lemma	PROPN
ejpam-4736	365	3	6	6	NUM
ejpam-4736	365	4	,	,	PUNCT
ejpam-4736	365	5	δp(λ	δp(λ	NOUN
ejpam-4736	365	6	,	,	PUNCT
ejpam-4736	365	7	s)ker({y	s)ker({y	NOUN
ejpam-4736	365	8	}	}	PUNCT
ejpam-4736	365	9	)	)	PUNCT
ejpam-4736	365	10	⊆	⊆	NUM
ejpam-4736	365	11	δp(λ	δp(λ	NOUN
ejpam-4736	365	12	,	,	PUNCT
ejpam-4736	365	13	s)ker(δp(λ	s)ker(δp(λ	NOUN
ejpam-4736	365	14	,	,	PUNCT
ejpam-4736	365	15	s)ker({x	s)ker({x	NOUN
ejpam-4736	365	16	}	}	PUNCT
ejpam-4736	365	17	)	)	PUNCT
ejpam-4736	365	18	)	)	PUNCT
ejpam-4736	366	1	=	=	SYM
ejpam-4736	366	2	δp(λ	δp(λ	NOUN
ejpam-4736	366	3	,	,	PUNCT
ejpam-4736	366	4	s)ker({x	s)ker({x	NOUN
ejpam-4736	366	5	}	}	PUNCT
ejpam-4736	366	6	)	)	PUNCT
ejpam-4736	366	7	.	.	PUNCT
ejpam-4736	367	1	similarly	similarly	ADV
ejpam-4736	367	2	,	,	PUNCT
ejpam-4736	367	3	we	we	PRON
ejpam-4736	367	4	have	have	VERB
ejpam-4736	367	5	δp(λ	δp(λ	NOUN
ejpam-4736	367	6	,	,	PUNCT
ejpam-4736	367	7	s)ker({x	s)ker({x	NUM
ejpam-4736	367	8	}	}	PUNCT
ejpam-4736	367	9	)	)	PUNCT
ejpam-4736	368	1	⊆	⊆	NUM
ejpam-4736	368	2	δp(λ	δp(λ	NOUN
ejpam-4736	368	3	,	,	PUNCT
ejpam-4736	368	4	s)ker({y	s)ker({y	NOUN
ejpam-4736	368	5	}	}	PUNCT
ejpam-4736	368	6	)	)	PUNCT
ejpam-4736	368	7	and	and	CCONJ
ejpam-4736	368	8	hence	hence	ADV
ejpam-4736	368	9	δp(λ	δp(λ	NOUN
ejpam-4736	368	10	,	,	PUNCT
ejpam-4736	368	11	s)ker({x	s)ker({x	NUM
ejpam-4736	368	12	}	}	PUNCT
ejpam-4736	368	13	)	)	PUNCT
ejpam-4736	369	1	=	=	SYM
ejpam-4736	369	2	δp(λ	δp(λ	NOUN
ejpam-4736	369	3	,	,	PUNCT
ejpam-4736	369	4	s)ker({y	s)ker({y	NOUN
ejpam-4736	369	5	}	}	PUNCT
ejpam-4736	369	6	)	)	PUNCT
ejpam-4736	369	7	.	.	PUNCT
ejpam-4736	370	1	definition	definition	NOUN
ejpam-4736	370	2	8	8	NUM
ejpam-4736	370	3	.	.	PUNCT
ejpam-4736	371	1	a	a	DET
ejpam-4736	371	2	subset	subset	NOUN
ejpam-4736	371	3	a	a	PRON
ejpam-4736	371	4	of	of	ADP
ejpam-4736	371	5	a	a	DET
ejpam-4736	371	6	topological	topological	ADJ
ejpam-4736	371	7	space	space	NOUN
ejpam-4736	371	8	(	(	PUNCT
ejpam-4736	371	9	x	x	X
ejpam-4736	371	10	,	,	PUNCT
ejpam-4736	371	11	τ	τ	X
ejpam-4736	371	12	)	)	PUNCT
ejpam-4736	371	13	is	be	AUX
ejpam-4736	371	14	called	call	VERB
ejpam-4736	371	15	a	a	DET
ejpam-4736	371	16	λδp(λ	λδp(λ	PROPN
ejpam-4736	371	17	,	,	PUNCT
ejpam-4736	371	18	s)-set	s)-set	VERB
ejpam-4736	371	19	if	if	SCONJ
ejpam-4736	371	20	a	a	DET
ejpam-4736	371	21	=	=	NOUN
ejpam-4736	371	22	δp(λ	δp(λ	NOUN
ejpam-4736	371	23	,	,	PUNCT
ejpam-4736	371	24	s)ker(a	s)ker(a	NOUN
ejpam-4736	371	25	)	)	PUNCT
ejpam-4736	371	26	.	.	PUNCT
ejpam-4736	372	1	the	the	DET
ejpam-4736	372	2	family	family	NOUN
ejpam-4736	372	3	of	of	ADP
ejpam-4736	372	4	all	all	DET
ejpam-4736	372	5	λδp(λ	λδp(λ	PROPN
ejpam-4736	372	6	,	,	PUNCT
ejpam-4736	372	7	s)-sets	s)-set	NOUN
ejpam-4736	372	8	of	of	ADP
ejpam-4736	372	9	a	a	DET
ejpam-4736	372	10	topological	topological	ADJ
ejpam-4736	372	11	space	space	NOUN
ejpam-4736	372	12	(	(	PUNCT
ejpam-4736	372	13	x	x	X
ejpam-4736	372	14	,	,	PUNCT
ejpam-4736	372	15	τ	τ	X
ejpam-4736	372	16	)	)	PUNCT
ejpam-4736	372	17	is	be	AUX
ejpam-4736	372	18	denoted	denote	VERB
ejpam-4736	372	19	by	by	ADP
ejpam-4736	372	20	λδp(λ	λδp(λ	PROPN
ejpam-4736	372	21	,	,	PUNCT
ejpam-4736	372	22	s)(x	s)(x	PROPN
ejpam-4736	372	23	,	,	PUNCT
ejpam-4736	372	24	τ	τ	X
ejpam-4736	372	25	)	)	PUNCT
ejpam-4736	372	26	(	(	PUNCT
ejpam-4736	372	27	or	or	CCONJ
ejpam-4736	372	28	simply	simply	ADV
ejpam-4736	372	29	λδp(λ	λδp(λ	PROPN
ejpam-4736	372	30	,	,	PUNCT
ejpam-4736	372	31	s	s	NOUN
ejpam-4736	372	32	)	)	PUNCT
ejpam-4736	372	33	)	)	PUNCT
ejpam-4736	372	34	.	.	PUNCT
ejpam-4736	373	1	definition	definition	NOUN
ejpam-4736	373	2	9	9	NUM
ejpam-4736	373	3	.	.	PUNCT
ejpam-4736	374	1	a	a	DET
ejpam-4736	374	2	subset	subset	NOUN
ejpam-4736	374	3	a	a	PRON
ejpam-4736	374	4	of	of	ADP
ejpam-4736	374	5	a	a	DET
ejpam-4736	374	6	topological	topological	ADJ
ejpam-4736	374	7	space	space	NOUN
ejpam-4736	374	8	(	(	PUNCT
ejpam-4736	374	9	x	x	X
ejpam-4736	374	10	,	,	PUNCT
ejpam-4736	374	11	τ	τ	X
ejpam-4736	374	12	)	)	PUNCT
ejpam-4736	374	13	is	be	AUX
ejpam-4736	374	14	called	call	VERB
ejpam-4736	374	15	a	a	DET
ejpam-4736	374	16	generalized	generalized	ADJ
ejpam-4736	374	17	λδp(λ	λδp(λ	PROPN
ejpam-4736	374	18	,	,	PUNCT
ejpam-4736	374	19	s)-set	s)-set	VERB
ejpam-4736	374	20	(	(	PUNCT
ejpam-4736	374	21	briefly	briefly	ADV
ejpam-4736	374	22	g	g	PROPN
ejpam-4736	374	23	-	-	PUNCT
ejpam-4736	374	24	λδp(λ	λδp(λ	PROPN
ejpam-4736	374	25	,	,	PUNCT
ejpam-4736	374	26	s)-set	s)-set	NOUN
ejpam-4736	374	27	)	)	PUNCT
ejpam-4736	374	28	if	if	SCONJ
ejpam-4736	374	29	δp(λ	δp(λ	NOUN
ejpam-4736	374	30	,	,	PUNCT
ejpam-4736	374	31	s)ker(a	s)ker(a	NOUN
ejpam-4736	374	32	)	)	PUNCT
ejpam-4736	374	33	⊆	⊆	NUM
ejpam-4736	374	34	f	f	NOUN
ejpam-4736	374	35	whenever	whenever	SCONJ
ejpam-4736	374	36	a	a	DET
ejpam-4736	374	37	⊆	⊆	NUM
ejpam-4736	374	38	f	f	PROPN
ejpam-4736	374	39	and	and	CCONJ
ejpam-4736	374	40	f	f	PROPN
ejpam-4736	374	41	is	be	AUX
ejpam-4736	374	42	a	a	DET
ejpam-4736	374	43	δp(λ	δp(λ	NOUN
ejpam-4736	374	44	,	,	PUNCT
ejpam-4736	374	45	s)-closed	s)-close	VERB
ejpam-4736	374	46	set	set	NOUN
ejpam-4736	374	47	.	.	PUNCT
ejpam-4736	375	1	definition	definition	NOUN
ejpam-4736	375	2	10	10	NUM
ejpam-4736	375	3	.	.	PUNCT
ejpam-4736	376	1	a	a	DET
ejpam-4736	376	2	topological	topological	ADJ
ejpam-4736	376	3	space	space	NOUN
ejpam-4736	376	4	(	(	PUNCT
ejpam-4736	376	5	x	x	X
ejpam-4736	376	6	,	,	PUNCT
ejpam-4736	376	7	τ	τ	X
ejpam-4736	376	8	)	)	PUNCT
ejpam-4736	376	9	is	be	AUX
ejpam-4736	376	10	called	call	VERB
ejpam-4736	376	11	a	a	DET
ejpam-4736	376	12	δp(λ	δp(λ	NOUN
ejpam-4736	376	13	,	,	PUNCT
ejpam-4736	376	14	s)-t	s)-t	VERB
ejpam-4736	376	15	1	1	NUM
ejpam-4736	376	16	2	2	NUM
ejpam-4736	376	17	-space	-space	NOUN
ejpam-4736	376	18	if	if	SCONJ
ejpam-4736	376	19	every	every	DET
ejpam-4736	376	20	g	g	NOUN
ejpam-4736	376	21	-	-	PUNCT
ejpam-4736	376	22	δp(λ	δp(λ	NOUN
ejpam-4736	376	23	,	,	PUNCT
ejpam-4736	376	24	s)closed	s)close	VERB
ejpam-4736	376	25	set	set	NOUN
ejpam-4736	376	26	of	of	ADP
ejpam-4736	376	27	x	x	PUNCT
ejpam-4736	376	28	is	be	AUX
ejpam-4736	376	29	δp(λ	δp(λ	NOUN
ejpam-4736	376	30	,	,	PUNCT
ejpam-4736	376	31	s)-closed	s)-close	VERB
ejpam-4736	376	32	.	.	PUNCT
ejpam-4736	377	1	lemma	lemma	PROPN
ejpam-4736	377	2	8	8	NUM
ejpam-4736	377	3	.	.	PUNCT
ejpam-4736	378	1	for	for	ADP
ejpam-4736	378	2	a	a	DET
ejpam-4736	378	3	topological	topological	ADJ
ejpam-4736	378	4	space	space	NOUN
ejpam-4736	378	5	(	(	PUNCT
ejpam-4736	378	6	x	x	X
ejpam-4736	378	7	,	,	PUNCT
ejpam-4736	378	8	τ	τ	PROPN
ejpam-4736	378	9	)	)	PUNCT
ejpam-4736	378	10	,	,	PUNCT
ejpam-4736	378	11	the	the	DET
ejpam-4736	378	12	following	follow	VERB
ejpam-4736	378	13	properties	property	NOUN
ejpam-4736	378	14	hold	hold	VERB
ejpam-4736	378	15	:	:	PUNCT
ejpam-4736	378	16	(	(	PUNCT
ejpam-4736	378	17	1	1	X
ejpam-4736	378	18	)	)	PUNCT
ejpam-4736	378	19	for	for	SCONJ
ejpam-4736	378	20	each	each	DET
ejpam-4736	378	21	x	x	SYM
ejpam-4736	378	22	∈	∈	PROPN
ejpam-4736	378	23	x	x	NOUN
ejpam-4736	378	24	,	,	PUNCT
ejpam-4736	378	25	the	the	DET
ejpam-4736	378	26	singleton	singleton	NOUN
ejpam-4736	378	27	{	{	PUNCT
ejpam-4736	378	28	x	x	NOUN
ejpam-4736	378	29	}	}	PUNCT
ejpam-4736	378	30	is	be	AUX
ejpam-4736	378	31	δp(λ	δp(λ	NOUN
ejpam-4736	378	32	,	,	PUNCT
ejpam-4736	378	33	s)-closed	s)-close	VERB
ejpam-4736	378	34	or	or	CCONJ
ejpam-4736	378	35	x	x	SYM
ejpam-4736	378	36	−{x	−{x	NUM
ejpam-4736	378	37	}	}	PUNCT
ejpam-4736	378	38	is	be	AUX
ejpam-4736	378	39	g	g	NOUN
ejpam-4736	378	40	-	-	PUNCT
ejpam-4736	378	41	δp(λ	δp(λ	NOUN
ejpam-4736	378	42	,	,	PUNCT
ejpam-4736	378	43	s)-closed	s)-close	VERB
ejpam-4736	378	44	.	.	PUNCT
ejpam-4736	379	1	(	(	PUNCT
ejpam-4736	379	2	2	2	X
ejpam-4736	379	3	)	)	PUNCT
ejpam-4736	379	4	for	for	SCONJ
ejpam-4736	379	5	each	each	DET
ejpam-4736	379	6	x	x	SYM
ejpam-4736	379	7	∈	∈	PROPN
ejpam-4736	379	8	x	x	NOUN
ejpam-4736	379	9	,	,	PUNCT
ejpam-4736	379	10	the	the	DET
ejpam-4736	379	11	singleton	singleton	NOUN
ejpam-4736	379	12	{	{	PUNCT
ejpam-4736	379	13	x	x	NOUN
ejpam-4736	379	14	}	}	PUNCT
ejpam-4736	379	15	is	be	AUX
ejpam-4736	379	16	δp(λ	δp(λ	NOUN
ejpam-4736	379	17	,	,	PUNCT
ejpam-4736	379	18	s)-open	s)-open	PUNCT
ejpam-4736	379	19	or	or	CCONJ
ejpam-4736	379	20	x	x	X
ejpam-4736	379	21	−	−	PROPN
ejpam-4736	379	22	{	{	PUNCT
ejpam-4736	379	23	x	x	NOUN
ejpam-4736	379	24	}	}	PUNCT
ejpam-4736	379	25	is	be	AUX
ejpam-4736	379	26	a	a	DET
ejpam-4736	379	27	g	g	NOUN
ejpam-4736	379	28	-	-	PUNCT
ejpam-4736	379	29	λδp(λ	λδp(λ	PROPN
ejpam-4736	379	30	,	,	PUNCT
ejpam-4736	379	31	s)-set	s)-set	NOUN
ejpam-4736	379	32	.	.	PUNCT
ejpam-4736	380	1	proof	proof	NOUN
ejpam-4736	380	2	.	.	PUNCT
ejpam-4736	381	1	(	(	PUNCT
ejpam-4736	381	2	1	1	X
ejpam-4736	381	3	)	)	PUNCT
ejpam-4736	381	4	let	let	VERB
ejpam-4736	381	5	x	x	SYM
ejpam-4736	381	6	∈	∈	PROPN
ejpam-4736	381	7	x	x	X
ejpam-4736	381	8	and	and	CCONJ
ejpam-4736	381	9	the	the	DET
ejpam-4736	381	10	singleton	singleton	NOUN
ejpam-4736	381	11	{	{	PUNCT
ejpam-4736	381	12	x	x	NOUN
ejpam-4736	381	13	}	}	PUNCT
ejpam-4736	381	14	be	be	AUX
ejpam-4736	381	15	not	not	PART
ejpam-4736	381	16	δp(λ	δp(λ	NOUN
ejpam-4736	381	17	,	,	PUNCT
ejpam-4736	381	18	s)-closed	s)-close	VERB
ejpam-4736	381	19	.	.	PUNCT
ejpam-4736	382	1	then	then	ADV
ejpam-4736	382	2	,	,	PUNCT
ejpam-4736	382	3	x	x	PUNCT
ejpam-4736	382	4	−	−	NOUN
ejpam-4736	382	5	{	{	PUNCT
ejpam-4736	382	6	x	x	NOUN
ejpam-4736	382	7	}	}	PUNCT
ejpam-4736	382	8	is	be	AUX
ejpam-4736	382	9	not	not	PART
ejpam-4736	382	10	δp(λ	δp(λ	NOUN
ejpam-4736	382	11	,	,	PUNCT
ejpam-4736	382	12	s)-open	s)-open	PUNCT
ejpam-4736	382	13	and	and	CCONJ
ejpam-4736	382	14	x	x	X
ejpam-4736	382	15	is	be	AUX
ejpam-4736	382	16	the	the	DET
ejpam-4736	382	17	only	only	ADJ
ejpam-4736	382	18	δp(λ	δp(λ	NOUN
ejpam-4736	382	19	,	,	PUNCT
ejpam-4736	382	20	s)-open	s)-open	PUNCT
ejpam-4736	382	21	set	set	VERB
ejpam-4736	382	22	which	which	PRON
ejpam-4736	382	23	contains	contain	VERB
ejpam-4736	382	24	x	x	X
ejpam-4736	382	25	−	−	PROPN
ejpam-4736	382	26	{	{	PUNCT
ejpam-4736	382	27	x	x	NOUN
ejpam-4736	382	28	}	}	PUNCT
ejpam-4736	382	29	and	and	CCONJ
ejpam-4736	382	30	hence	hence	ADV
ejpam-4736	382	31	x	x	X
ejpam-4736	382	32	−	−	PROPN
ejpam-4736	382	33	{	{	PUNCT
ejpam-4736	382	34	x	x	NOUN
ejpam-4736	382	35	}	}	PUNCT
ejpam-4736	382	36	is	be	AUX
ejpam-4736	382	37	g	g	NOUN
ejpam-4736	382	38	-	-	PUNCT
ejpam-4736	382	39	δp(λ	δp(λ	NOUN
ejpam-4736	382	40	,	,	PUNCT
ejpam-4736	382	41	s)-closed	s)-close	VERB
ejpam-4736	382	42	.	.	PUNCT
ejpam-4736	383	1	(	(	PUNCT
ejpam-4736	383	2	2	2	X
ejpam-4736	383	3	)	)	PUNCT
ejpam-4736	383	4	let	let	VERB
ejpam-4736	383	5	x	x	SYM
ejpam-4736	383	6	∈	∈	PROPN
ejpam-4736	383	7	x	x	X
ejpam-4736	383	8	and	and	CCONJ
ejpam-4736	383	9	the	the	DET
ejpam-4736	383	10	singleton	singleton	NOUN
ejpam-4736	383	11	{	{	PUNCT
ejpam-4736	383	12	x	x	NOUN
ejpam-4736	383	13	}	}	PUNCT
ejpam-4736	383	14	be	be	AUX
ejpam-4736	383	15	not	not	PART
ejpam-4736	383	16	δp(λ	δp(λ	NOUN
ejpam-4736	383	17	,	,	PUNCT
ejpam-4736	383	18	s)-open	s)-open	VERB
ejpam-4736	383	19	.	.	PUNCT
ejpam-4736	384	1	then	then	ADV
ejpam-4736	384	2	,	,	PUNCT
ejpam-4736	384	3	x	x	PUNCT
ejpam-4736	384	4	−	−	NOUN
ejpam-4736	384	5	{	{	PUNCT
ejpam-4736	384	6	x	x	NOUN
ejpam-4736	384	7	}	}	PUNCT
ejpam-4736	384	8	is	be	AUX
ejpam-4736	384	9	not	not	PART
ejpam-4736	384	10	δp(λ	δp(λ	NOUN
ejpam-4736	384	11	,	,	PUNCT
ejpam-4736	384	12	s)-closed	s)-close	VERB
ejpam-4736	384	13	and	and	CCONJ
ejpam-4736	384	14	x	x	X
ejpam-4736	384	15	is	be	AUX
ejpam-4736	384	16	the	the	DET
ejpam-4736	384	17	only	only	ADJ
ejpam-4736	384	18	δp(λ	δp(λ	NOUN
ejpam-4736	384	19	,	,	PUNCT
ejpam-4736	384	20	s)-closed	s)-close	VERB
ejpam-4736	384	21	set	set	NOUN
ejpam-4736	384	22	which	which	PRON
ejpam-4736	384	23	contains	contain	VERB
ejpam-4736	384	24	x	x	X
ejpam-4736	384	25	−	−	PROPN
ejpam-4736	384	26	{	{	PUNCT
ejpam-4736	384	27	x	x	NOUN
ejpam-4736	384	28	}	}	PUNCT
ejpam-4736	384	29	and	and	CCONJ
ejpam-4736	384	30	hence	hence	ADV
ejpam-4736	384	31	x	x	X
ejpam-4736	384	32	−	−	PROPN
ejpam-4736	384	33	{	{	PUNCT
ejpam-4736	384	34	x	x	NOUN
ejpam-4736	384	35	}	}	PUNCT
ejpam-4736	384	36	is	be	AUX
ejpam-4736	384	37	a	a	DET
ejpam-4736	384	38	g	g	NOUN
ejpam-4736	384	39	-	-	PUNCT
ejpam-4736	384	40	λδp(λ	λδp(λ	PROPN
ejpam-4736	384	41	,	,	PUNCT
ejpam-4736	384	42	s)-set	s)-set	NOUN
ejpam-4736	384	43	.	.	PUNCT
ejpam-4736	385	1	theorem	theorem	VERB
ejpam-4736	385	2	16	16	NUM
ejpam-4736	385	3	.	.	PUNCT
ejpam-4736	386	1	for	for	ADP
ejpam-4736	386	2	a	a	DET
ejpam-4736	386	3	topological	topological	ADJ
ejpam-4736	386	4	space	space	NOUN
ejpam-4736	386	5	(	(	PUNCT
ejpam-4736	386	6	x	x	X
ejpam-4736	386	7	,	,	PUNCT
ejpam-4736	386	8	τ	τ	PROPN
ejpam-4736	386	9	)	)	PUNCT
ejpam-4736	386	10	,	,	PUNCT
ejpam-4736	386	11	the	the	DET
ejpam-4736	386	12	following	follow	VERB
ejpam-4736	386	13	properties	property	NOUN
ejpam-4736	386	14	are	be	AUX
ejpam-4736	386	15	equivalent	equivalent	ADJ
ejpam-4736	386	16	:	:	PUNCT
ejpam-4736	386	17	(	(	PUNCT
ejpam-4736	386	18	1	1	X
ejpam-4736	386	19	)	)	PUNCT
ejpam-4736	386	20	(	(	PUNCT
ejpam-4736	386	21	x	x	X
ejpam-4736	386	22	,	,	PUNCT
ejpam-4736	386	23	τ	τ	X
ejpam-4736	386	24	)	)	PUNCT
ejpam-4736	386	25	is	be	AUX
ejpam-4736	386	26	a	a	DET
ejpam-4736	386	27	δp(λ	δp(λ	NOUN
ejpam-4736	386	28	,	,	PUNCT
ejpam-4736	386	29	s)-t	s)-t	VERB
ejpam-4736	386	30	1	1	NUM
ejpam-4736	386	31	2	2	NUM
ejpam-4736	386	32	-space	-space	NOUN
ejpam-4736	386	33	.	.	PUNCT
ejpam-4736	387	1	c.	c.	PROPN
ejpam-4736	387	2	boonpok	boonpok	PROPN
ejpam-4736	387	3	,	,	PUNCT
ejpam-4736	387	4	n.	n.	PROPN
ejpam-4736	387	5	srisarakham	srisarakham	PROPN
ejpam-4736	387	6	/	/	SYM
ejpam-4736	387	7	eur	eur	PROPN
ejpam-4736	387	8	.	.	PUNCT
ejpam-4736	388	1	j.	j.	PROPN
ejpam-4736	388	2	pure	pure	PROPN
ejpam-4736	388	3	appl	appl	PROPN
ejpam-4736	388	4	.	.	PROPN
ejpam-4736	388	5	math	math	PROPN
ejpam-4736	388	6	,	,	PUNCT
ejpam-4736	388	7	16	16	NUM
ejpam-4736	388	8	(	(	PUNCT
ejpam-4736	388	9	4	4	NUM
ejpam-4736	388	10	)	)	PUNCT
ejpam-4736	388	11	(	(	PUNCT
ejpam-4736	388	12	2023	2023	NUM
ejpam-4736	388	13	)	)	PUNCT
ejpam-4736	388	14	,	,	PUNCT
ejpam-4736	388	15	2581	2581	NUM
ejpam-4736	388	16	-	-	SYM
ejpam-4736	388	17	2596	2596	NUM
ejpam-4736	388	18	2592	2592	NUM
ejpam-4736	388	19	(	(	PUNCT
ejpam-4736	388	20	2	2	NUM
ejpam-4736	388	21	)	)	PUNCT
ejpam-4736	388	22	for	for	ADP
ejpam-4736	388	23	each	each	DET
ejpam-4736	388	24	x	x	SYM
ejpam-4736	388	25	∈	∈	PROPN
ejpam-4736	388	26	x	x	NOUN
ejpam-4736	388	27	,	,	PUNCT
ejpam-4736	388	28	the	the	DET
ejpam-4736	388	29	singleton	singleton	NOUN
ejpam-4736	388	30	{	{	PUNCT
ejpam-4736	388	31	x	x	NOUN
ejpam-4736	388	32	}	}	PUNCT
ejpam-4736	388	33	is	be	AUX
ejpam-4736	388	34	δp(λ	δp(λ	NOUN
ejpam-4736	388	35	,	,	PUNCT
ejpam-4736	388	36	s)-open	s)-open	PUNCT
ejpam-4736	388	37	or	or	CCONJ
ejpam-4736	388	38	δp(λ	δp(λ	NOUN
ejpam-4736	388	39	,	,	PUNCT
ejpam-4736	388	40	s)-closed	s)-close	VERB
ejpam-4736	388	41	.	.	PUNCT
ejpam-4736	389	1	(	(	PUNCT
ejpam-4736	389	2	3	3	X
ejpam-4736	389	3	)	)	PUNCT
ejpam-4736	389	4	every	every	DET
ejpam-4736	389	5	g	g	PROPN
ejpam-4736	389	6	-	-	PUNCT
ejpam-4736	389	7	λδp(λ	λδp(λ	PROPN
ejpam-4736	389	8	,	,	PUNCT
ejpam-4736	389	9	s)-set	s)-set	VERB
ejpam-4736	389	10	is	be	AUX
ejpam-4736	389	11	a	a	DET
ejpam-4736	389	12	λδp(λ	λδp(λ	PROPN
ejpam-4736	389	13	,	,	PUNCT
ejpam-4736	389	14	s)-set	s)-set	NOUN
ejpam-4736	389	15	.	.	PUNCT
ejpam-4736	390	1	proof	proof	NOUN
ejpam-4736	390	2	.	.	PUNCT
ejpam-4736	391	1	(	(	PUNCT
ejpam-4736	391	2	1	1	X
ejpam-4736	391	3	)	)	PUNCT
ejpam-4736	391	4	⇒	⇒	NOUN
ejpam-4736	391	5	(	(	PUNCT
ejpam-4736	391	6	2	2	NUM
ejpam-4736	391	7	):	):	PUNCT
ejpam-4736	391	8	by	by	ADP
ejpam-4736	391	9	lemma	lemma	PROPN
ejpam-4736	391	10	8	8	NUM
ejpam-4736	391	11	,	,	PUNCT
ejpam-4736	391	12	for	for	ADP
ejpam-4736	391	13	each	each	DET
ejpam-4736	391	14	x	x	SYM
ejpam-4736	391	15	∈	∈	PROPN
ejpam-4736	391	16	x	x	NOUN
ejpam-4736	391	17	,	,	PUNCT
ejpam-4736	391	18	the	the	DET
ejpam-4736	391	19	singleton	singleton	NOUN
ejpam-4736	391	20	{	{	PUNCT
ejpam-4736	391	21	x	x	NOUN
ejpam-4736	391	22	}	}	PUNCT
ejpam-4736	391	23	is	be	AUX
ejpam-4736	391	24	δp(λ	δp(λ	NOUN
ejpam-4736	391	25	,	,	PUNCT
ejpam-4736	391	26	s)-closed	s)-close	VERB
ejpam-4736	391	27	or	or	CCONJ
ejpam-4736	391	28	x−{x	x−{x	PROPN
ejpam-4736	391	29	}	}	PUNCT
ejpam-4736	391	30	is	be	AUX
ejpam-4736	391	31	g	g	NOUN
ejpam-4736	391	32	-	-	PUNCT
ejpam-4736	391	33	δp(λ	δp(λ	NOUN
ejpam-4736	391	34	,	,	PUNCT
ejpam-4736	391	35	s)-closed	s)-close	VERB
ejpam-4736	391	36	.	.	PUNCT
ejpam-4736	392	1	since	since	SCONJ
ejpam-4736	392	2	(	(	PUNCT
ejpam-4736	392	3	x	x	X
ejpam-4736	392	4	,	,	PUNCT
ejpam-4736	392	5	τ	τ	X
ejpam-4736	392	6	)	)	PUNCT
ejpam-4736	392	7	is	be	AUX
ejpam-4736	392	8	a	a	DET
ejpam-4736	392	9	δp(λ	δp(λ	NOUN
ejpam-4736	392	10	,	,	PUNCT
ejpam-4736	392	11	s)-t	s)-t	VERB
ejpam-4736	392	12	1	1	NUM
ejpam-4736	392	13	2	2	NUM
ejpam-4736	392	14	-space	-space	NOUN
ejpam-4736	392	15	,	,	PUNCT
ejpam-4736	392	16	x−{x	x−{x	PROPN
ejpam-4736	392	17	}	}	PUNCT
ejpam-4736	392	18	is	be	AUX
ejpam-4736	392	19	δp(λ	δp(λ	NOUN
ejpam-4736	392	20	,	,	PUNCT
ejpam-4736	392	21	s)-closed	s)-close	VERB
ejpam-4736	392	22	and	and	CCONJ
ejpam-4736	392	23	hence	hence	ADV
ejpam-4736	392	24	{	{	PUNCT
ejpam-4736	392	25	x	x	X
ejpam-4736	392	26	}	}	PUNCT
ejpam-4736	392	27	is	be	AUX
ejpam-4736	392	28	δp(λ	δp(λ	NOUN
ejpam-4736	392	29	,	,	PUNCT
ejpam-4736	392	30	s)-open	s)-open	VERB
ejpam-4736	392	31	in	in	ADP
ejpam-4736	392	32	the	the	DET
ejpam-4736	392	33	latter	latter	ADJ
ejpam-4736	392	34	case	case	NOUN
ejpam-4736	392	35	.	.	PUNCT
ejpam-4736	393	1	thus	thus	ADV
ejpam-4736	393	2	,	,	PUNCT
ejpam-4736	393	3	the	the	DET
ejpam-4736	393	4	singleton	singleton	NOUN
ejpam-4736	393	5	{	{	PUNCT
ejpam-4736	393	6	x	x	NOUN
ejpam-4736	393	7	}	}	PUNCT
ejpam-4736	393	8	is	be	AUX
ejpam-4736	393	9	δp(λ	δp(λ	NOUN
ejpam-4736	393	10	,	,	PUNCT
ejpam-4736	393	11	s)-open	s)-open	PUNCT
ejpam-4736	393	12	or	or	CCONJ
ejpam-4736	393	13	δp(λ	δp(λ	NOUN
ejpam-4736	393	14	,	,	PUNCT
ejpam-4736	393	15	s)-closed	s)-close	VERB
ejpam-4736	393	16	.	.	PUNCT
ejpam-4736	394	1	(	(	PUNCT
ejpam-4736	394	2	2	2	X
ejpam-4736	394	3	)	)	PUNCT
ejpam-4736	394	4	⇒	⇒	NOUN
ejpam-4736	394	5	(	(	PUNCT
ejpam-4736	394	6	3	3	NUM
ejpam-4736	394	7	):	):	PUNCT
ejpam-4736	394	8	suppose	suppose	VERB
ejpam-4736	394	9	that	that	SCONJ
ejpam-4736	394	10	there	there	PRON
ejpam-4736	394	11	exists	exist	VERB
ejpam-4736	394	12	a	a	DET
ejpam-4736	394	13	g	g	NOUN
ejpam-4736	394	14	-	-	PUNCT
ejpam-4736	394	15	λδp(λ	λδp(λ	PROPN
ejpam-4736	394	16	,	,	PUNCT
ejpam-4736	394	17	s)-set	s)-set	VERB
ejpam-4736	394	18	a	a	PRON
ejpam-4736	394	19	which	which	PRON
ejpam-4736	394	20	is	be	AUX
ejpam-4736	394	21	not	not	PART
ejpam-4736	394	22	a	a	DET
ejpam-4736	394	23	λδp(λ	λδp(λ	PROPN
ejpam-4736	394	24	,	,	PUNCT
ejpam-4736	394	25	s)-set	s)-set	NOUN
ejpam-4736	394	26	.	.	PUNCT
ejpam-4736	395	1	there	there	PRON
ejpam-4736	395	2	exists	exist	VERB
ejpam-4736	395	3	x	x	X
ejpam-4736	395	4	∈	∈	PROPN
ejpam-4736	395	5	δp(λ	δp(λ	NOUN
ejpam-4736	395	6	,	,	PUNCT
ejpam-4736	395	7	s)ker(a	s)ker(a	NOUN
ejpam-4736	395	8	)	)	PUNCT
ejpam-4736	395	9	such	such	ADJ
ejpam-4736	395	10	that	that	SCONJ
ejpam-4736	395	11	x	x	PUNCT
ejpam-4736	395	12	̸∈	̸∈	PROPN
ejpam-4736	395	13	a.	a.	NOUN
ejpam-4736	395	14	in	in	ADP
ejpam-4736	395	15	case	case	NOUN
ejpam-4736	395	16	the	the	DET
ejpam-4736	395	17	singleton	singleton	NOUN
ejpam-4736	395	18	{	{	PUNCT
ejpam-4736	395	19	x	x	NOUN
ejpam-4736	395	20	}	}	PUNCT
ejpam-4736	395	21	is	be	AUX
ejpam-4736	395	22	δp(λ	δp(λ	NOUN
ejpam-4736	395	23	,	,	PUNCT
ejpam-4736	395	24	s)-open	s)-open	PUNCT
ejpam-4736	395	25	,	,	PUNCT
ejpam-4736	395	26	a	a	DET
ejpam-4736	395	27	⊆	⊆	NUM
ejpam-4736	395	28	x	x	SYM
ejpam-4736	395	29	−	−	PROPN
ejpam-4736	395	30	{	{	PUNCT
ejpam-4736	395	31	x	x	NOUN
ejpam-4736	395	32	}	}	PUNCT
ejpam-4736	395	33	and	and	CCONJ
ejpam-4736	395	34	x	x	ADJ
ejpam-4736	395	35	−	−	PROPN
ejpam-4736	395	36	{	{	PUNCT
ejpam-4736	395	37	x	x	NOUN
ejpam-4736	395	38	}	}	PUNCT
ejpam-4736	395	39	is	be	AUX
ejpam-4736	395	40	δp(λ	δp(λ	NOUN
ejpam-4736	395	41	,	,	PUNCT
ejpam-4736	395	42	s)-closed	s)-close	VERB
ejpam-4736	395	43	.	.	PUNCT
ejpam-4736	396	1	since	since	SCONJ
ejpam-4736	396	2	a	a	PRON
ejpam-4736	396	3	is	be	AUX
ejpam-4736	396	4	a	a	DET
ejpam-4736	396	5	g	g	NOUN
ejpam-4736	396	6	-	-	PUNCT
ejpam-4736	396	7	λδp(λ	λδp(λ	PROPN
ejpam-4736	396	8	,	,	PUNCT
ejpam-4736	396	9	s)-set	s)-set	NOUN
ejpam-4736	396	10	,	,	PUNCT
ejpam-4736	396	11	δp(λ	δp(λ	NOUN
ejpam-4736	396	12	,	,	PUNCT
ejpam-4736	396	13	s)ker(a	s)ker(a	NOUN
ejpam-4736	396	14	)	)	PUNCT
ejpam-4736	396	15	⊆	⊆	NUM
ejpam-4736	396	16	x	x	SYM
ejpam-4736	396	17	−	−	PROPN
ejpam-4736	396	18	{	{	PUNCT
ejpam-4736	396	19	x	x	NOUN
ejpam-4736	396	20	}	}	PUNCT
ejpam-4736	396	21	.	.	PUNCT
ejpam-4736	397	1	this	this	PRON
ejpam-4736	397	2	is	be	AUX
ejpam-4736	397	3	a	a	DET
ejpam-4736	397	4	contradiction	contradiction	NOUN
ejpam-4736	397	5	.	.	PUNCT
ejpam-4736	398	1	in	in	ADP
ejpam-4736	398	2	case	case	NOUN
ejpam-4736	398	3	the	the	DET
ejpam-4736	398	4	singleton	singleton	NOUN
ejpam-4736	398	5	{	{	PUNCT
ejpam-4736	398	6	x	x	NOUN
ejpam-4736	398	7	}	}	PUNCT
ejpam-4736	398	8	is	be	AUX
ejpam-4736	398	9	δp(λ	δp(λ	NOUN
ejpam-4736	398	10	,	,	PUNCT
ejpam-4736	398	11	s)-closed	s)-close	VERB
ejpam-4736	398	12	,	,	PUNCT
ejpam-4736	398	13	a	a	DET
ejpam-4736	398	14	⊆	⊆	NUM
ejpam-4736	398	15	x	x	SYM
ejpam-4736	398	16	−	−	PROPN
ejpam-4736	398	17	{	{	PUNCT
ejpam-4736	398	18	x	x	NOUN
ejpam-4736	398	19	}	}	PUNCT
ejpam-4736	398	20	and	and	CCONJ
ejpam-4736	398	21	x	x	ADJ
ejpam-4736	398	22	−	−	PROPN
ejpam-4736	398	23	{	{	PUNCT
ejpam-4736	398	24	x	x	NOUN
ejpam-4736	398	25	}	}	PUNCT
ejpam-4736	398	26	is	be	AUX
ejpam-4736	398	27	δp(λ	δp(λ	NOUN
ejpam-4736	398	28	,	,	PUNCT
ejpam-4736	398	29	s)-open	s)-open	PUNCT
ejpam-4736	398	30	.	.	PUNCT
ejpam-4736	399	1	by	by	ADP
ejpam-4736	399	2	lemma	lemma	PROPN
ejpam-4736	399	3	6	6	NUM
ejpam-4736	399	4	,	,	PUNCT
ejpam-4736	399	5	δp(λ	δp(λ	NOUN
ejpam-4736	399	6	,	,	PUNCT
ejpam-4736	399	7	s)ker(a	s)ker(a	NOUN
ejpam-4736	399	8	)	)	PUNCT
ejpam-4736	399	9	⊆	⊆	NUM
ejpam-4736	399	10	δp(λ	δp(λ	NOUN
ejpam-4736	399	11	,	,	PUNCT
ejpam-4736	399	12	s)ker(x	s)ker(x	PROPN
ejpam-4736	399	13	−	−	PROPN
ejpam-4736	399	14	{	{	PUNCT
ejpam-4736	399	15	x	x	NOUN
ejpam-4736	399	16	}	}	PUNCT
ejpam-4736	399	17	)	)	PUNCT
ejpam-4736	400	1	=	=	PUNCT
ejpam-4736	400	2	x	x	X
ejpam-4736	400	3	−	−	PROPN
ejpam-4736	400	4	{	{	PUNCT
ejpam-4736	400	5	x	x	NOUN
ejpam-4736	400	6	}	}	PUNCT
ejpam-4736	400	7	.	.	PUNCT
ejpam-4736	401	1	this	this	PRON
ejpam-4736	401	2	is	be	AUX
ejpam-4736	401	3	a	a	DET
ejpam-4736	401	4	contradiction	contradiction	NOUN
ejpam-4736	401	5	.	.	PUNCT
ejpam-4736	402	1	thus	thus	ADV
ejpam-4736	402	2	,	,	PUNCT
ejpam-4736	402	3	every	every	DET
ejpam-4736	402	4	g	g	PROPN
ejpam-4736	402	5	-	-	PUNCT
ejpam-4736	402	6	λδp(λ	λδp(λ	PROPN
ejpam-4736	402	7	,	,	PUNCT
ejpam-4736	402	8	s)-set	s)-set	VERB
ejpam-4736	402	9	is	be	AUX
ejpam-4736	402	10	a	a	DET
ejpam-4736	402	11	λδp(λ	λδp(λ	PROPN
ejpam-4736	402	12	,	,	PUNCT
ejpam-4736	402	13	s)-set	s)-set	NOUN
ejpam-4736	402	14	.	.	PUNCT
ejpam-4736	403	1	(	(	PUNCT
ejpam-4736	403	2	3	3	X
ejpam-4736	403	3	)	)	PUNCT
ejpam-4736	403	4	⇒	⇒	NOUN
ejpam-4736	403	5	(	(	PUNCT
ejpam-4736	403	6	1	1	NUM
ejpam-4736	403	7	):	):	PUNCT
ejpam-4736	403	8	suppose	suppose	VERB
ejpam-4736	403	9	that	that	SCONJ
ejpam-4736	403	10	(	(	PUNCT
ejpam-4736	403	11	x	x	X
ejpam-4736	403	12	,	,	PUNCT
ejpam-4736	403	13	τ	τ	X
ejpam-4736	403	14	)	)	PUNCT
ejpam-4736	403	15	is	be	AUX
ejpam-4736	403	16	not	not	PART
ejpam-4736	403	17	a	a	DET
ejpam-4736	403	18	δp(λ	δp(λ	NOUN
ejpam-4736	403	19	,	,	PUNCT
ejpam-4736	403	20	s)-t	s)-t	VERB
ejpam-4736	403	21	1	1	NUM
ejpam-4736	403	22	2	2	NUM
ejpam-4736	403	23	-space	-space	NOUN
ejpam-4736	403	24	.	.	PUNCT
ejpam-4736	404	1	then	then	ADV
ejpam-4736	404	2	,	,	PUNCT
ejpam-4736	404	3	there	there	PRON
ejpam-4736	404	4	exists	exist	VERB
ejpam-4736	404	5	a	a	DET
ejpam-4736	404	6	gδp(λ	gδp(λ	PROPN
ejpam-4736	404	7	,	,	PUNCT
ejpam-4736	404	8	s)-closed	s)-close	VERB
ejpam-4736	404	9	set	set	VERB
ejpam-4736	404	10	a	a	PRON
ejpam-4736	404	11	which	which	PRON
ejpam-4736	404	12	is	be	AUX
ejpam-4736	404	13	not	not	PART
ejpam-4736	404	14	δp(λ	δp(λ	NOUN
ejpam-4736	404	15	,	,	PUNCT
ejpam-4736	404	16	s)-closed	s)-close	VERB
ejpam-4736	404	17	.	.	PUNCT
ejpam-4736	405	1	since	since	SCONJ
ejpam-4736	405	2	a	a	PRON
ejpam-4736	405	3	is	be	AUX
ejpam-4736	405	4	not	not	PART
ejpam-4736	405	5	δp(λ	δp(λ	NOUN
ejpam-4736	405	6	,	,	PUNCT
ejpam-4736	405	7	s)-closed	s)-close	VERB
ejpam-4736	405	8	,	,	PUNCT
ejpam-4736	405	9	there	there	PRON
ejpam-4736	405	10	exists	exist	VERB
ejpam-4736	405	11	a	a	DET
ejpam-4736	405	12	point	point	NOUN
ejpam-4736	405	13	x	x	X
ejpam-4736	405	14	∈	∈	PROPN
ejpam-4736	405	15	aδp(λ	aδp(λ	PROPN
ejpam-4736	405	16	,	,	PUNCT
ejpam-4736	405	17	s	s	PART
ejpam-4736	405	18	)	)	PUNCT
ejpam-4736	406	1	such	such	ADJ
ejpam-4736	406	2	that	that	SCONJ
ejpam-4736	406	3	x	x	SYM
ejpam-4736	406	4	̸∈	̸∈	PROPN
ejpam-4736	406	5	a.	a.	NOUN
ejpam-4736	406	6	by	by	ADP
ejpam-4736	406	7	lemma	lemma	PROPN
ejpam-4736	406	8	8	8	NUM
ejpam-4736	406	9	,	,	PUNCT
ejpam-4736	406	10	the	the	DET
ejpam-4736	406	11	singleton	singleton	NOUN
ejpam-4736	406	12	{	{	PUNCT
ejpam-4736	406	13	x	x	NOUN
ejpam-4736	406	14	}	}	PUNCT
ejpam-4736	406	15	is	be	AUX
ejpam-4736	406	16	δp(λ	δp(λ	NOUN
ejpam-4736	406	17	,	,	PUNCT
ejpam-4736	406	18	s)open	s)open	ADJ
ejpam-4736	406	19	or	or	CCONJ
ejpam-4736	406	20	x	x	SYM
ejpam-4736	406	21	−	−	PROPN
ejpam-4736	406	22	{	{	PUNCT
ejpam-4736	406	23	x	x	NOUN
ejpam-4736	406	24	}	}	PUNCT
ejpam-4736	406	25	is	be	AUX
ejpam-4736	406	26	a	a	DET
ejpam-4736	406	27	λδp(λ	λδp(λ	PROPN
ejpam-4736	406	28	,	,	PUNCT
ejpam-4736	406	29	s)-set	s)-set	NOUN
ejpam-4736	406	30	.	.	PUNCT
ejpam-4736	407	1	(	(	PUNCT
ejpam-4736	407	2	a	a	X
ejpam-4736	407	3	)	)	PUNCT
ejpam-4736	407	4	in	in	ADP
ejpam-4736	407	5	case	case	NOUN
ejpam-4736	407	6	{	{	PUNCT
ejpam-4736	407	7	x	x	NOUN
ejpam-4736	407	8	}	}	PUNCT
ejpam-4736	407	9	is	be	AUX
ejpam-4736	407	10	δp(λ	δp(λ	NOUN
ejpam-4736	407	11	,	,	PUNCT
ejpam-4736	407	12	s)-open	s)-open	VERB
ejpam-4736	407	13	,	,	PUNCT
ejpam-4736	407	14	since	since	SCONJ
ejpam-4736	407	15	x	x	PROPN
ejpam-4736	407	16	∈	∈	PROPN
ejpam-4736	407	17	aδp(λ	aδp(λ	PROPN
ejpam-4736	407	18	,	,	PUNCT
ejpam-4736	407	19	s	s	PART
ejpam-4736	407	20	)	)	PUNCT
ejpam-4736	407	21	,	,	PUNCT
ejpam-4736	407	22	{	{	PUNCT
ejpam-4736	407	23	x	x	X
ejpam-4736	407	24	}	}	PUNCT
ejpam-4736	407	25	∩	∩	NOUN
ejpam-4736	407	26	a	a	DET
ejpam-4736	407	27	̸=	̸=	PROPN
ejpam-4736	407	28	∅	∅	NOUN
ejpam-4736	407	29	and	and	CCONJ
ejpam-4736	407	30	x	x	PUNCT
ejpam-4736	407	31	∈	∈	NOUN
ejpam-4736	407	32	a.	a.	NOUN
ejpam-4736	407	33	this	this	PRON
ejpam-4736	407	34	is	be	AUX
ejpam-4736	407	35	a	a	DET
ejpam-4736	407	36	contradiction	contradiction	NOUN
ejpam-4736	407	37	.	.	PUNCT
ejpam-4736	408	1	(	(	PUNCT
ejpam-4736	408	2	b	b	X
ejpam-4736	408	3	)	)	PUNCT
ejpam-4736	408	4	in	in	ADP
ejpam-4736	408	5	case	case	NOUN
ejpam-4736	408	6	x	x	X
ejpam-4736	408	7	−	−	X
ejpam-4736	408	8	{	{	PUNCT
ejpam-4736	408	9	x	x	NOUN
ejpam-4736	408	10	}	}	PUNCT
ejpam-4736	408	11	is	be	AUX
ejpam-4736	408	12	a	a	DET
ejpam-4736	408	13	λδp(λ	λδp(λ	PROPN
ejpam-4736	408	14	,	,	PUNCT
ejpam-4736	408	15	s)-set	s)-set	VERB
ejpam-4736	408	16	,	,	PUNCT
ejpam-4736	408	17	if	if	SCONJ
ejpam-4736	408	18	{	{	PUNCT
ejpam-4736	408	19	x	x	NOUN
ejpam-4736	408	20	}	}	PUNCT
ejpam-4736	408	21	is	be	AUX
ejpam-4736	408	22	not	not	PART
ejpam-4736	408	23	δp(λ	δp(λ	NOUN
ejpam-4736	408	24	,	,	PUNCT
ejpam-4736	408	25	s)-closed	s)-close	VERB
ejpam-4736	408	26	,	,	PUNCT
ejpam-4736	408	27	x	x	PUNCT
ejpam-4736	408	28	−	−	NOUN
ejpam-4736	408	29	{	{	PUNCT
ejpam-4736	408	30	x	x	NOUN
ejpam-4736	408	31	}	}	PUNCT
ejpam-4736	408	32	is	be	AUX
ejpam-4736	408	33	not	not	PART
ejpam-4736	408	34	δp(λ	δp(λ	NOUN
ejpam-4736	408	35	,	,	PUNCT
ejpam-4736	408	36	s)-open	s)-open	PUNCT
ejpam-4736	408	37	and	and	CCONJ
ejpam-4736	408	38	δp(λ	δp(λ	NOUN
ejpam-4736	408	39	,	,	PUNCT
ejpam-4736	408	40	s)ker(x	s)ker(x	PROPN
ejpam-4736	408	41	−	−	PROPN
ejpam-4736	408	42	{	{	PUNCT
ejpam-4736	408	43	x	x	NOUN
ejpam-4736	408	44	}	}	PUNCT
ejpam-4736	408	45	)	)	PUNCT
ejpam-4736	408	46	=	=	PUNCT
ejpam-4736	409	1	x.	x.	PUNCT
ejpam-4736	409	2	thus	thus	ADV
ejpam-4736	409	3	,	,	PUNCT
ejpam-4736	409	4	x	x	PUNCT
ejpam-4736	409	5	−	−	X
ejpam-4736	409	6	{	{	PUNCT
ejpam-4736	409	7	x	x	NOUN
ejpam-4736	409	8	}	}	PUNCT
ejpam-4736	409	9	is	be	AUX
ejpam-4736	409	10	not	not	PART
ejpam-4736	409	11	a	a	DET
ejpam-4736	409	12	λδp(λ	λδp(λ	PROPN
ejpam-4736	409	13	,	,	PUNCT
ejpam-4736	409	14	s)-set	s)-set	NOUN
ejpam-4736	409	15	.	.	PUNCT
ejpam-4736	410	1	this	this	PRON
ejpam-4736	410	2	contradicts	contradict	VERB
ejpam-4736	410	3	(	(	PUNCT
ejpam-4736	410	4	3	3	NUM
ejpam-4736	410	5	)	)	PUNCT
ejpam-4736	410	6	.	.	PUNCT
ejpam-4736	411	1	if	if	SCONJ
ejpam-4736	411	2	{	{	PUNCT
ejpam-4736	411	3	x	x	NOUN
ejpam-4736	411	4	}	}	PUNCT
ejpam-4736	411	5	is	be	AUX
ejpam-4736	411	6	δp(λ	δp(λ	NOUN
ejpam-4736	411	7	,	,	PUNCT
ejpam-4736	411	8	s)-closed	s)-close	VERB
ejpam-4736	411	9	,	,	PUNCT
ejpam-4736	411	10	a	a	DET
ejpam-4736	411	11	⊆	⊆	NUM
ejpam-4736	411	12	x	x	SYM
ejpam-4736	411	13	−	−	PROPN
ejpam-4736	411	14	{	{	PUNCT
ejpam-4736	411	15	x	x	NOUN
ejpam-4736	411	16	}	}	PUNCT
ejpam-4736	411	17	∈	∈	PROPN
ejpam-4736	411	18	δp(λ	δp(λ	NOUN
ejpam-4736	411	19	,	,	PUNCT
ejpam-4736	411	20	s)o(x	s)o(x	PROPN
ejpam-4736	411	21	,	,	PUNCT
ejpam-4736	411	22	τ	τ	X
ejpam-4736	411	23	)	)	PUNCT
ejpam-4736	411	24	and	and	CCONJ
ejpam-4736	411	25	a	a	PRON
ejpam-4736	411	26	is	be	AUX
ejpam-4736	411	27	g	g	NOUN
ejpam-4736	411	28	-	-	PUNCT
ejpam-4736	411	29	δp(λ	δp(λ	NOUN
ejpam-4736	411	30	,	,	PUNCT
ejpam-4736	411	31	s)-closed	s)-close	VERB
ejpam-4736	411	32	.	.	PUNCT
ejpam-4736	412	1	hence	hence	ADV
ejpam-4736	412	2	,	,	PUNCT
ejpam-4736	412	3	we	we	PRON
ejpam-4736	412	4	have	have	VERB
ejpam-4736	412	5	aδp(λ	aδp(λ	PROPN
ejpam-4736	412	6	,	,	PUNCT
ejpam-4736	412	7	s	s	PART
ejpam-4736	412	8	)	)	PUNCT
ejpam-4736	412	9	⊆	⊆	NUM
ejpam-4736	412	10	x	x	SYM
ejpam-4736	412	11	−	−	PROPN
ejpam-4736	412	12	{	{	PUNCT
ejpam-4736	412	13	x	x	NOUN
ejpam-4736	412	14	}	}	PUNCT
ejpam-4736	412	15	.	.	PUNCT
ejpam-4736	413	1	this	this	PRON
ejpam-4736	413	2	contradicts	contradict	VERB
ejpam-4736	413	3	that	that	SCONJ
ejpam-4736	413	4	x	x	SYM
ejpam-4736	413	5	∈	∈	PROPN
ejpam-4736	413	6	aδp(λ	aδp(λ	PROPN
ejpam-4736	413	7	,	,	PUNCT
ejpam-4736	413	8	s	s	PROPN
ejpam-4736	413	9	)	)	PUNCT
ejpam-4736	413	10	.	.	PUNCT
ejpam-4736	414	1	this	this	PRON
ejpam-4736	414	2	shows	show	VERB
ejpam-4736	414	3	that	that	SCONJ
ejpam-4736	414	4	(	(	PUNCT
ejpam-4736	414	5	x	x	X
ejpam-4736	414	6	,	,	PUNCT
ejpam-4736	414	7	τ	τ	X
ejpam-4736	414	8	)	)	PUNCT
ejpam-4736	414	9	is	be	AUX
ejpam-4736	414	10	a	a	DET
ejpam-4736	414	11	δp(λ	δp(λ	NOUN
ejpam-4736	414	12	,	,	PUNCT
ejpam-4736	414	13	s)-t	s)-t	VERB
ejpam-4736	414	14	1	1	NUM
ejpam-4736	414	15	2	2	NUM
ejpam-4736	414	16	-space	-space	NOUN
ejpam-4736	414	17	.	.	PUNCT
ejpam-4736	415	1	definition	definition	NOUN
ejpam-4736	415	2	11	11	NUM
ejpam-4736	415	3	.	.	PUNCT
ejpam-4736	416	1	a	a	DET
ejpam-4736	416	2	topological	topological	ADJ
ejpam-4736	416	3	space	space	NOUN
ejpam-4736	416	4	(	(	PUNCT
ejpam-4736	416	5	x	x	X
ejpam-4736	416	6	,	,	PUNCT
ejpam-4736	416	7	τ	τ	X
ejpam-4736	416	8	)	)	PUNCT
ejpam-4736	416	9	is	be	AUX
ejpam-4736	416	10	said	say	VERB
ejpam-4736	416	11	to	to	PART
ejpam-4736	416	12	be	be	AUX
ejpam-4736	416	13	δp(λ	δp(λ	NOUN
ejpam-4736	416	14	,	,	PUNCT
ejpam-4736	416	15	s)-normal	s)-normal	ADJ
ejpam-4736	416	16	if	if	SCONJ
ejpam-4736	416	17	for	for	ADP
ejpam-4736	416	18	any	any	DET
ejpam-4736	416	19	pair	pair	NOUN
ejpam-4736	416	20	of	of	ADP
ejpam-4736	416	21	disjoint	disjoint	NOUN
ejpam-4736	416	22	δp(λ	δp(λ	NOUN
ejpam-4736	416	23	,	,	PUNCT
ejpam-4736	416	24	s)-closed	s)-close	VERB
ejpam-4736	416	25	sets	set	NOUN
ejpam-4736	416	26	f	f	PROPN
ejpam-4736	416	27	and	and	CCONJ
ejpam-4736	416	28	h	h	NOUN
ejpam-4736	416	29	,	,	PUNCT
ejpam-4736	416	30	there	there	PRON
ejpam-4736	416	31	exist	exist	VERB
ejpam-4736	416	32	disjoint	disjoint	NOUN
ejpam-4736	416	33	δp(λ	δp(λ	NOUN
ejpam-4736	416	34	,	,	PUNCT
ejpam-4736	416	35	s)-open	s)-open	PUNCT
ejpam-4736	416	36	sets	set	VERB
ejpam-4736	416	37	u	u	NOUN
ejpam-4736	416	38	and	and	CCONJ
ejpam-4736	416	39	v	v	ADP
ejpam-4736	416	40	such	such	ADJ
ejpam-4736	416	41	that	that	SCONJ
ejpam-4736	416	42	f	f	PROPN
ejpam-4736	416	43	⊆	⊆	NUM
ejpam-4736	416	44	u	u	NOUN
ejpam-4736	416	45	and	and	CCONJ
ejpam-4736	416	46	h	h	NOUN
ejpam-4736	416	47	⊆	⊆	NUM
ejpam-4736	416	48	v	v	NOUN
ejpam-4736	416	49	.	.	PUNCT
ejpam-4736	417	1	lemma	lemma	PROPN
ejpam-4736	417	2	9	9	NUM
ejpam-4736	417	3	.	.	PUNCT
ejpam-4736	418	1	let	let	VERB
ejpam-4736	418	2	(	(	PUNCT
ejpam-4736	418	3	x	x	NOUN
ejpam-4736	418	4	,	,	PUNCT
ejpam-4736	418	5	τ	τ	X
ejpam-4736	418	6	)	)	PUNCT
ejpam-4736	418	7	be	be	VERB
ejpam-4736	418	8	a	a	DET
ejpam-4736	418	9	topological	topological	ADJ
ejpam-4736	418	10	space	space	NOUN
ejpam-4736	418	11	.	.	PUNCT
ejpam-4736	419	1	if	if	SCONJ
ejpam-4736	419	2	u	u	NOUN
ejpam-4736	419	3	is	be	AUX
ejpam-4736	419	4	δp(λ	δp(λ	NOUN
ejpam-4736	419	5	,	,	PUNCT
ejpam-4736	419	6	s)-open	s)-open	VERB
ejpam-4736	419	7	in	in	ADP
ejpam-4736	419	8	x	x	NOUN
ejpam-4736	419	9	,	,	PUNCT
ejpam-4736	419	10	then	then	ADV
ejpam-4736	419	11	u	u	NOUN
ejpam-4736	419	12	δp(λ	δp(λ	NOUN
ejpam-4736	419	13	,	,	PUNCT
ejpam-4736	419	14	s	s	PART
ejpam-4736	419	15	)	)	PUNCT
ejpam-4736	419	16	∩a	∩a	NOUN
ejpam-4736	419	17	⊆	⊆	NUM
ejpam-4736	420	1	[	[	X
ejpam-4736	420	2	u	u	NOUN
ejpam-4736	420	3	∩a]δp(λ	∩a]δp(λ	SYM
ejpam-4736	420	4	,	,	PUNCT
ejpam-4736	420	5	s	s	PART
ejpam-4736	420	6	)	)	PUNCT
ejpam-4736	420	7	for	for	ADP
ejpam-4736	420	8	every	every	DET
ejpam-4736	420	9	subset	subset	NOUN
ejpam-4736	420	10	a	a	PRON
ejpam-4736	420	11	of	of	ADP
ejpam-4736	420	12	x.	x.	NOUN
ejpam-4736	420	13	theorem	theorem	VERB
ejpam-4736	420	14	17	17	NUM
ejpam-4736	420	15	.	.	PUNCT
ejpam-4736	421	1	for	for	ADP
ejpam-4736	421	2	a	a	DET
ejpam-4736	421	3	topological	topological	ADJ
ejpam-4736	421	4	space	space	NOUN
ejpam-4736	421	5	(	(	PUNCT
ejpam-4736	421	6	x	x	X
ejpam-4736	421	7	,	,	PUNCT
ejpam-4736	421	8	τ	τ	PROPN
ejpam-4736	421	9	)	)	PUNCT
ejpam-4736	421	10	,	,	PUNCT
ejpam-4736	421	11	the	the	DET
ejpam-4736	421	12	following	follow	VERB
ejpam-4736	421	13	properties	property	NOUN
ejpam-4736	421	14	are	be	AUX
ejpam-4736	421	15	equivalent	equivalent	ADJ
ejpam-4736	421	16	:	:	PUNCT
ejpam-4736	421	17	(	(	PUNCT
ejpam-4736	421	18	1	1	X
ejpam-4736	421	19	)	)	PUNCT
ejpam-4736	421	20	(	(	PUNCT
ejpam-4736	421	21	x	x	X
ejpam-4736	421	22	,	,	PUNCT
ejpam-4736	421	23	τ	τ	X
ejpam-4736	421	24	)	)	PUNCT
ejpam-4736	421	25	is	be	AUX
ejpam-4736	421	26	δp(λ	δp(λ	NOUN
ejpam-4736	421	27	,	,	PUNCT
ejpam-4736	421	28	s)-normal	s)-normal	ADJ
ejpam-4736	421	29	.	.	PUNCT
ejpam-4736	422	1	c.	c.	PROPN
ejpam-4736	422	2	boonpok	boonpok	PROPN
ejpam-4736	422	3	,	,	PUNCT
ejpam-4736	422	4	n.	n.	PROPN
ejpam-4736	422	5	srisarakham	srisarakham	PROPN
ejpam-4736	422	6	/	/	SYM
ejpam-4736	422	7	eur	eur	PROPN
ejpam-4736	422	8	.	.	PUNCT
ejpam-4736	423	1	j.	j.	PROPN
ejpam-4736	423	2	pure	pure	PROPN
ejpam-4736	423	3	appl	appl	PROPN
ejpam-4736	423	4	.	.	PROPN
ejpam-4736	423	5	math	math	PROPN
ejpam-4736	423	6	,	,	PUNCT
ejpam-4736	423	7	16	16	NUM
ejpam-4736	423	8	(	(	PUNCT
ejpam-4736	423	9	4	4	NUM
ejpam-4736	423	10	)	)	PUNCT
ejpam-4736	423	11	(	(	PUNCT
ejpam-4736	423	12	2023	2023	NUM
ejpam-4736	423	13	)	)	PUNCT
ejpam-4736	423	14	,	,	PUNCT
ejpam-4736	423	15	2581	2581	NUM
ejpam-4736	423	16	-	-	SYM
ejpam-4736	423	17	2596	2596	NUM
ejpam-4736	423	18	2593	2593	NUM
ejpam-4736	423	19	(	(	PUNCT
ejpam-4736	423	20	2	2	NUM
ejpam-4736	423	21	)	)	PUNCT
ejpam-4736	423	22	for	for	ADP
ejpam-4736	423	23	every	every	DET
ejpam-4736	423	24	pair	pair	NOUN
ejpam-4736	423	25	of	of	ADP
ejpam-4736	423	26	δp(λ	δp(λ	NOUN
ejpam-4736	423	27	,	,	PUNCT
ejpam-4736	423	28	s)-open	s)-open	PUNCT
ejpam-4736	423	29	sets	set	VERB
ejpam-4736	423	30	u	u	NOUN
ejpam-4736	423	31	and	and	CCONJ
ejpam-4736	423	32	v	v	ADP
ejpam-4736	423	33	whose	whose	DET
ejpam-4736	423	34	union	union	NOUN
ejpam-4736	423	35	is	be	AUX
ejpam-4736	423	36	x	x	NOUN
ejpam-4736	423	37	,	,	PUNCT
ejpam-4736	423	38	there	there	PRON
ejpam-4736	423	39	exist	exist	VERB
ejpam-4736	423	40	δp(λ	δp(λ	NOUN
ejpam-4736	423	41	,	,	PUNCT
ejpam-4736	423	42	s)closed	s)close	VERB
ejpam-4736	423	43	sets	set	NOUN
ejpam-4736	423	44	f	f	PROPN
ejpam-4736	423	45	and	and	CCONJ
ejpam-4736	423	46	h	h	NOUN
ejpam-4736	423	47	such	such	ADJ
ejpam-4736	423	48	that	that	SCONJ
ejpam-4736	423	49	f	f	PROPN
ejpam-4736	423	50	⊆	⊆	NUM
ejpam-4736	423	51	u	u	NOUN
ejpam-4736	423	52	,	,	PUNCT
ejpam-4736	423	53	h	h	NOUN
ejpam-4736	423	54	⊆	⊆	NUM
ejpam-4736	423	55	v	v	NOUN
ejpam-4736	423	56	and	and	CCONJ
ejpam-4736	423	57	f	f	PROPN
ejpam-4736	423	58	∪h	∪h	PROPN
ejpam-4736	423	59	=	=	SYM
ejpam-4736	423	60	x.	x.	NOUN
ejpam-4736	423	61	(	(	PUNCT
ejpam-4736	423	62	3	3	NUM
ejpam-4736	423	63	)	)	PUNCT
ejpam-4736	423	64	for	for	ADP
ejpam-4736	423	65	every	every	DET
ejpam-4736	423	66	δp(λ	δp(λ	NOUN
ejpam-4736	423	67	,	,	PUNCT
ejpam-4736	423	68	s)-closed	s)-close	VERB
ejpam-4736	423	69	set	set	ADJ
ejpam-4736	423	70	f	f	PROPN
ejpam-4736	423	71	and	and	CCONJ
ejpam-4736	423	72	every	every	DET
ejpam-4736	423	73	δp(λ	δp(λ	NOUN
ejpam-4736	423	74	,	,	PUNCT
ejpam-4736	423	75	s)-open	s)-open	PUNCT
ejpam-4736	423	76	set	set	VERB
ejpam-4736	423	77	g	g	NOUN
ejpam-4736	423	78	containing	contain	VERB
ejpam-4736	423	79	f	f	NOUN
ejpam-4736	423	80	,	,	PUNCT
ejpam-4736	423	81	there	there	PRON
ejpam-4736	423	82	exists	exist	VERB
ejpam-4736	423	83	a	a	DET
ejpam-4736	423	84	δp(λ	δp(λ	NOUN
ejpam-4736	423	85	,	,	PUNCT
ejpam-4736	423	86	s)-open	s)-open	PUNCT
ejpam-4736	423	87	set	set	VERB
ejpam-4736	423	88	u	u	PRON
ejpam-4736	423	89	such	such	ADJ
ejpam-4736	423	90	that	that	SCONJ
ejpam-4736	423	91	f	f	PROPN
ejpam-4736	423	92	⊆	⊆	NUM
ejpam-4736	423	93	u	u	NOUN
ejpam-4736	423	94	⊆	⊆	NUM
ejpam-4736	423	95	u	u	NOUN
ejpam-4736	423	96	δp(λ	δp(λ	NOUN
ejpam-4736	423	97	,	,	PUNCT
ejpam-4736	423	98	s	s	PART
ejpam-4736	423	99	)	)	PUNCT
ejpam-4736	423	100	⊆	⊆	NUM
ejpam-4736	423	101	g.	g.	NOUN
ejpam-4736	423	102	(	(	PUNCT
ejpam-4736	423	103	4	4	NUM
ejpam-4736	423	104	)	)	PUNCT
ejpam-4736	423	105	for	for	ADP
ejpam-4736	423	106	every	every	DET
ejpam-4736	423	107	pair	pair	NOUN
ejpam-4736	423	108	of	of	ADP
ejpam-4736	423	109	disjoint	disjoint	NOUN
ejpam-4736	423	110	δp(λ	δp(λ	NOUN
ejpam-4736	423	111	,	,	PUNCT
ejpam-4736	423	112	s)-closed	s)-close	VERB
ejpam-4736	423	113	sets	set	NOUN
ejpam-4736	423	114	f	f	PROPN
ejpam-4736	423	115	and	and	CCONJ
ejpam-4736	423	116	h	h	NOUN
ejpam-4736	423	117	,	,	PUNCT
ejpam-4736	423	118	there	there	PRON
ejpam-4736	423	119	exist	exist	VERB
ejpam-4736	423	120	disjoint	disjoint	NOUN
ejpam-4736	423	121	δp(λ	δp(λ	NOUN
ejpam-4736	423	122	,	,	PUNCT
ejpam-4736	423	123	s)open	s)open	NOUN
ejpam-4736	423	124	sets	set	VERB
ejpam-4736	423	125	u	u	NOUN
ejpam-4736	423	126	and	and	CCONJ
ejpam-4736	423	127	v	v	ADP
ejpam-4736	423	128	such	such	ADJ
ejpam-4736	423	129	that	that	SCONJ
ejpam-4736	423	130	f	f	PROPN
ejpam-4736	423	131	⊆	⊆	NUM
ejpam-4736	423	132	u	u	NOUN
ejpam-4736	423	133	and	and	CCONJ
ejpam-4736	423	134	h	h	NOUN
ejpam-4736	423	135	⊆	⊆	NUM
ejpam-4736	423	136	v	v	NOUN
ejpam-4736	423	137	and	and	CCONJ
ejpam-4736	423	138	u	u	PRON
ejpam-4736	423	139	δp(λ	δp(λ	NOUN
ejpam-4736	423	140	,	,	PUNCT
ejpam-4736	423	141	s	s	NOUN
ejpam-4736	423	142	)	)	PUNCT
ejpam-4736	423	143	∩	∩	NOUN
ejpam-4736	423	144	v	v	ADP
ejpam-4736	423	145	δp(λ	δp(λ	NOUN
ejpam-4736	423	146	,	,	PUNCT
ejpam-4736	423	147	s	s	PART
ejpam-4736	423	148	)	)	PUNCT
ejpam-4736	423	149	=	=	PUNCT
ejpam-4736	423	150	∅.	∅.	NOUN
ejpam-4736	423	151	proof	proof	NOUN
ejpam-4736	423	152	.	.	PUNCT
ejpam-4736	424	1	(	(	PUNCT
ejpam-4736	424	2	1	1	X
ejpam-4736	424	3	)	)	PUNCT
ejpam-4736	424	4	⇒	⇒	NOUN
ejpam-4736	424	5	(	(	PUNCT
ejpam-4736	424	6	2	2	NUM
ejpam-4736	424	7	):	):	PUNCT
ejpam-4736	424	8	let	let	VERB
ejpam-4736	424	9	u	u	PRON
ejpam-4736	424	10	and	and	CCONJ
ejpam-4736	424	11	v	v	NOUN
ejpam-4736	424	12	be	be	AUX
ejpam-4736	424	13	any	any	DET
ejpam-4736	424	14	pair	pair	NOUN
ejpam-4736	424	15	of	of	ADP
ejpam-4736	424	16	δp(λ	δp(λ	NOUN
ejpam-4736	424	17	,	,	PUNCT
ejpam-4736	424	18	s)-open	s)-open	PUNCT
ejpam-4736	424	19	sets	set	NOUN
ejpam-4736	424	20	in	in	ADP
ejpam-4736	424	21	x	x	SYM
ejpam-4736	424	22	such	such	ADJ
ejpam-4736	424	23	that	that	SCONJ
ejpam-4736	424	24	x	x	X
ejpam-4736	424	25	=	=	PUNCT
ejpam-4736	424	26	u	u	NOUN
ejpam-4736	424	27	∪	∪	NOUN
ejpam-4736	424	28	v	v	NOUN
ejpam-4736	424	29	.	.	PUNCT
ejpam-4736	425	1	then	then	ADV
ejpam-4736	425	2	,	,	PUNCT
ejpam-4736	425	3	x	x	PUNCT
ejpam-4736	425	4	−	−	NOUN
ejpam-4736	425	5	u	u	NOUN
ejpam-4736	425	6	and	and	CCONJ
ejpam-4736	425	7	x	x	NOUN
ejpam-4736	425	8	−	−	PROPN
ejpam-4736	425	9	v	v	NOUN
ejpam-4736	425	10	are	be	AUX
ejpam-4736	425	11	disjoint	disjoint	NOUN
ejpam-4736	425	12	δp(λ	δp(λ	NOUN
ejpam-4736	425	13	,	,	PUNCT
ejpam-4736	425	14	s)-closed	s)-close	VERB
ejpam-4736	425	15	sets	set	NOUN
ejpam-4736	425	16	.	.	PUNCT
ejpam-4736	426	1	since	since	SCONJ
ejpam-4736	426	2	(	(	PUNCT
ejpam-4736	426	3	x	x	X
ejpam-4736	426	4	,	,	PUNCT
ejpam-4736	426	5	τ	τ	X
ejpam-4736	426	6	)	)	PUNCT
ejpam-4736	426	7	is	be	AUX
ejpam-4736	426	8	δp(λ	δp(λ	NOUN
ejpam-4736	426	9	,	,	PUNCT
ejpam-4736	426	10	s)-normal	s)-normal	ADJ
ejpam-4736	426	11	,	,	PUNCT
ejpam-4736	426	12	there	there	PRON
ejpam-4736	426	13	exist	exist	VERB
ejpam-4736	426	14	disjoint	disjoint	NOUN
ejpam-4736	426	15	δp(λ	δp(λ	NOUN
ejpam-4736	426	16	,	,	PUNCT
ejpam-4736	426	17	s)-open	s)-open	PUNCT
ejpam-4736	426	18	sets	set	VERB
ejpam-4736	426	19	g	g	NOUN
ejpam-4736	426	20	and	and	CCONJ
ejpam-4736	427	1	w	w	ADP
ejpam-4736	427	2	such	such	ADJ
ejpam-4736	427	3	that	that	SCONJ
ejpam-4736	427	4	x	x	X
ejpam-4736	427	5	−	−	PUNCT
ejpam-4736	427	6	u	u	NOUN
ejpam-4736	427	7	⊆	⊆	NUM
ejpam-4736	427	8	g	g	NOUN
ejpam-4736	427	9	and	and	CCONJ
ejpam-4736	427	10	x	x	NOUN
ejpam-4736	427	11	−	−	NOUN
ejpam-4736	427	12	v	v	NUM
ejpam-4736	427	13	⊆	⊆	NUM
ejpam-4736	427	14	w	w	NOUN
ejpam-4736	427	15	.	.	PUNCT
ejpam-4736	428	1	put	put	VERB
ejpam-4736	428	2	f	f	NOUN
ejpam-4736	429	1	=	=	NOUN
ejpam-4736	429	2	x	x	NOUN
ejpam-4736	429	3	−g	−g	NOUN
ejpam-4736	429	4	and	and	CCONJ
ejpam-4736	429	5	h	h	NOUN
ejpam-4736	430	1	=	=	SYM
ejpam-4736	430	2	x	x	PUNCT
ejpam-4736	430	3	−w	−w	ADV
ejpam-4736	430	4	.	.	PUNCT
ejpam-4736	431	1	then	then	ADV
ejpam-4736	431	2	,	,	PUNCT
ejpam-4736	431	3	f	f	PROPN
ejpam-4736	431	4	and	and	CCONJ
ejpam-4736	431	5	h	h	PROPN
ejpam-4736	431	6	are	be	AUX
ejpam-4736	431	7	δp(λ	δp(λ	NOUN
ejpam-4736	431	8	,	,	PUNCT
ejpam-4736	431	9	s)-closed	s)-close	VERB
ejpam-4736	431	10	sets	set	NOUN
ejpam-4736	431	11	such	such	ADJ
ejpam-4736	431	12	that	that	SCONJ
ejpam-4736	431	13	f	f	PROPN
ejpam-4736	431	14	⊆	⊆	NUM
ejpam-4736	431	15	u	u	NOUN
ejpam-4736	431	16	,	,	PUNCT
ejpam-4736	431	17	h	h	NOUN
ejpam-4736	431	18	⊆	⊆	NUM
ejpam-4736	431	19	v	v	NOUN
ejpam-4736	431	20	and	and	CCONJ
ejpam-4736	431	21	f	f	PROPN
ejpam-4736	431	22	∪h	∪h	PROPN
ejpam-4736	431	23	=	=	PUNCT
ejpam-4736	431	24	x.	x.	NOUN
ejpam-4736	431	25	(	(	PUNCT
ejpam-4736	431	26	2	2	NUM
ejpam-4736	431	27	)	)	PUNCT
ejpam-4736	431	28	⇒	⇒	NOUN
ejpam-4736	431	29	(	(	PUNCT
ejpam-4736	431	30	3	3	NUM
ejpam-4736	431	31	):	):	PUNCT
ejpam-4736	431	32	let	let	VERB
ejpam-4736	431	33	f	f	PRON
ejpam-4736	431	34	be	be	AUX
ejpam-4736	431	35	a	a	DET
ejpam-4736	431	36	δp(λ	δp(λ	NOUN
ejpam-4736	431	37	,	,	PUNCT
ejpam-4736	431	38	s)-closed	s)-close	VERB
ejpam-4736	431	39	set	set	NOUN
ejpam-4736	431	40	and	and	CCONJ
ejpam-4736	431	41	g	g	NOUN
ejpam-4736	431	42	be	be	AUX
ejpam-4736	431	43	a	a	DET
ejpam-4736	431	44	δp(λ	δp(λ	NOUN
ejpam-4736	431	45	,	,	PUNCT
ejpam-4736	431	46	s)-open	s)-open	PUNCT
ejpam-4736	431	47	set	set	VERB
ejpam-4736	431	48	containing	contain	VERB
ejpam-4736	431	49	f	f	PROPN
ejpam-4736	431	50	.	.	PUNCT
ejpam-4736	432	1	then	then	ADV
ejpam-4736	432	2	,	,	PUNCT
ejpam-4736	432	3	x	x	PUNCT
ejpam-4736	432	4	−	−	PROPN
ejpam-4736	432	5	f	f	PROPN
ejpam-4736	432	6	and	and	CCONJ
ejpam-4736	432	7	g	g	PROPN
ejpam-4736	432	8	are	be	AUX
ejpam-4736	432	9	δp(λ	δp(λ	NOUN
ejpam-4736	432	10	,	,	PUNCT
ejpam-4736	432	11	s)-open	s)-open	PUNCT
ejpam-4736	432	12	sets	set	VERB
ejpam-4736	432	13	whose	whose	DET
ejpam-4736	432	14	union	union	NOUN
ejpam-4736	432	15	is	be	AUX
ejpam-4736	432	16	x.	x.	NOUN
ejpam-4736	432	17	then	then	ADV
ejpam-4736	432	18	by	by	ADP
ejpam-4736	432	19	(	(	PUNCT
ejpam-4736	432	20	2	2	NUM
ejpam-4736	432	21	)	)	PUNCT
ejpam-4736	432	22	,	,	PUNCT
ejpam-4736	432	23	there	there	PRON
ejpam-4736	432	24	exist	exist	VERB
ejpam-4736	432	25	δp(λ	δp(λ	NOUN
ejpam-4736	432	26	,	,	PUNCT
ejpam-4736	432	27	s)-closed	s)-close	VERB
ejpam-4736	432	28	sets	set	NOUN
ejpam-4736	432	29	m	m	VERB
ejpam-4736	432	30	and	and	CCONJ
ejpam-4736	432	31	n	n	CCONJ
ejpam-4736	432	32	such	such	ADJ
ejpam-4736	432	33	that	that	SCONJ
ejpam-4736	432	34	m	m	PROPN
ejpam-4736	432	35	⊆	⊆	NUM
ejpam-4736	432	36	x	x	SYM
ejpam-4736	432	37	−	−	PROPN
ejpam-4736	432	38	f	f	PROPN
ejpam-4736	432	39	,	,	PUNCT
ejpam-4736	432	40	n	n	PROPN
ejpam-4736	432	41	⊆	⊆	NUM
ejpam-4736	432	42	g	g	NOUN
ejpam-4736	432	43	and	and	CCONJ
ejpam-4736	432	44	m	m	NOUN
ejpam-4736	432	45	∪	∪	ADJ
ejpam-4736	432	46	n	n	NOUN
ejpam-4736	432	47	=	=	SYM
ejpam-4736	432	48	x.	x.	NOUN
ejpam-4736	432	49	then	then	ADV
ejpam-4736	432	50	,	,	PUNCT
ejpam-4736	432	51	f	f	PROPN
ejpam-4736	432	52	⊆	⊆	NUM
ejpam-4736	432	53	x−m	x−m	PROPN
ejpam-4736	432	54	,	,	PUNCT
ejpam-4736	432	55	x−g	x−g	PROPN
ejpam-4736	432	56	⊆	⊆	NUM
ejpam-4736	432	57	x−n	x−n	PROPN
ejpam-4736	432	58	and	and	CCONJ
ejpam-4736	432	59	(	(	PUNCT
ejpam-4736	432	60	x−m)∩(x−n	x−m)∩(x−n	X
ejpam-4736	432	61	)	)	PUNCT
ejpam-4736	432	62	=	=	VERB
ejpam-4736	432	63	∅.	∅.	AUX
ejpam-4736	432	64	put	put	VERB
ejpam-4736	432	65	u	u	NOUN
ejpam-4736	432	66	=	=	PROPN
ejpam-4736	432	67	x−m	x−m	PROPN
ejpam-4736	432	68	and	and	CCONJ
ejpam-4736	432	69	v	v	NOUN
ejpam-4736	432	70	=	=	SYM
ejpam-4736	432	71	x−n	x−n	PROPN
ejpam-4736	432	72	.	.	PUNCT
ejpam-4736	433	1	then	then	ADV
ejpam-4736	433	2	,	,	PUNCT
ejpam-4736	433	3	u	u	NOUN
ejpam-4736	433	4	and	and	CCONJ
ejpam-4736	433	5	v	v	NOUN
ejpam-4736	433	6	are	be	AUX
ejpam-4736	433	7	disjoint	disjoint	NOUN
ejpam-4736	433	8	δp(λ	δp(λ	NOUN
ejpam-4736	433	9	,	,	PUNCT
ejpam-4736	433	10	s)-open	s)-open	PUNCT
ejpam-4736	433	11	sets	set	VERB
ejpam-4736	433	12	such	such	ADJ
ejpam-4736	433	13	that	that	SCONJ
ejpam-4736	433	14	f	f	PROPN
ejpam-4736	433	15	⊆	⊆	NUM
ejpam-4736	433	16	u	u	NOUN
ejpam-4736	433	17	⊆	⊆	NUM
ejpam-4736	433	18	x	x	SYM
ejpam-4736	433	19	−	−	NOUN
ejpam-4736	433	20	v	v	NUM
ejpam-4736	433	21	⊆	⊆	NUM
ejpam-4736	433	22	g.	g.	NOUN
ejpam-4736	433	23	as	as	SCONJ
ejpam-4736	433	24	x	x	X
ejpam-4736	433	25	−	−	PROPN
ejpam-4736	433	26	v	v	NOUN
ejpam-4736	433	27	is	be	AUX
ejpam-4736	433	28	a	a	DET
ejpam-4736	433	29	δp(λ	δp(λ	NOUN
ejpam-4736	433	30	,	,	PUNCT
ejpam-4736	433	31	s)-closed	s)-close	VERB
ejpam-4736	433	32	set	set	NOUN
ejpam-4736	433	33	,	,	PUNCT
ejpam-4736	433	34	we	we	PRON
ejpam-4736	433	35	have	have	VERB
ejpam-4736	433	36	u	u	PRON
ejpam-4736	433	37	δp(λ	δp(λ	NOUN
ejpam-4736	433	38	,	,	PUNCT
ejpam-4736	433	39	s	s	PART
ejpam-4736	433	40	)	)	PUNCT
ejpam-4736	433	41	⊆	⊆	NUM
ejpam-4736	433	42	x	x	SYM
ejpam-4736	433	43	−	−	NUM
ejpam-4736	433	44	v	v	NOUN
ejpam-4736	433	45	and	and	CCONJ
ejpam-4736	433	46	f	f	PROPN
ejpam-4736	434	1	⊆	⊆	NUM
ejpam-4736	434	2	u	u	NOUN
ejpam-4736	434	3	⊆	⊆	NUM
ejpam-4736	434	4	u	u	NOUN
ejpam-4736	434	5	δp(λ	δp(λ	NOUN
ejpam-4736	434	6	,	,	PUNCT
ejpam-4736	434	7	s	s	PART
ejpam-4736	434	8	)	)	PUNCT
ejpam-4736	434	9	⊆	⊆	NUM
ejpam-4736	434	10	g.	g.	NOUN
ejpam-4736	434	11	(	(	PUNCT
ejpam-4736	434	12	3	3	NUM
ejpam-4736	434	13	)	)	PUNCT
ejpam-4736	434	14	⇒	⇒	NOUN
ejpam-4736	434	15	(	(	PUNCT
ejpam-4736	434	16	4	4	NUM
ejpam-4736	434	17	):	):	PUNCT
ejpam-4736	434	18	let	let	VERB
ejpam-4736	434	19	f	f	PROPN
ejpam-4736	434	20	and	and	CCONJ
ejpam-4736	434	21	h	h	PROPN
ejpam-4736	434	22	be	be	VERB
ejpam-4736	434	23	two	two	NUM
ejpam-4736	434	24	disjoint	disjoint	NOUN
ejpam-4736	434	25	δp(λ	δp(λ	NOUN
ejpam-4736	434	26	,	,	PUNCT
ejpam-4736	434	27	s)-closed	s)-close	VERB
ejpam-4736	434	28	sets	set	NOUN
ejpam-4736	434	29	of	of	ADP
ejpam-4736	434	30	x.	x.	NOUN
ejpam-4736	434	31	then	then	ADV
ejpam-4736	434	32	,	,	PUNCT
ejpam-4736	434	33	f	f	PROPN
ejpam-4736	434	34	⊆	⊆	NUM
ejpam-4736	434	35	x	x	SYM
ejpam-4736	434	36	−h	−h	VERB
ejpam-4736	434	37	and	and	CCONJ
ejpam-4736	434	38	x	x	SYM
ejpam-4736	434	39	−	−	NOUN
ejpam-4736	434	40	h	h	NOUN
ejpam-4736	434	41	is	be	AUX
ejpam-4736	434	42	δp(λ	δp(λ	NOUN
ejpam-4736	434	43	,	,	PUNCT
ejpam-4736	434	44	s)-open	s)-open	VERB
ejpam-4736	434	45	,	,	PUNCT
ejpam-4736	434	46	by	by	ADP
ejpam-4736	434	47	(	(	PUNCT
ejpam-4736	434	48	3	3	NUM
ejpam-4736	434	49	)	)	PUNCT
ejpam-4736	434	50	,	,	PUNCT
ejpam-4736	434	51	there	there	PRON
ejpam-4736	434	52	exists	exist	VERB
ejpam-4736	434	53	a	a	DET
ejpam-4736	434	54	δp(λ	δp(λ	NOUN
ejpam-4736	434	55	,	,	PUNCT
ejpam-4736	434	56	s)-open	s)-open	VERB
ejpam-4736	434	57	set	set	VERB
ejpam-4736	434	58	u	u	NOUN
ejpam-4736	434	59	of	of	ADP
ejpam-4736	434	60	x	x	SYM
ejpam-4736	434	61	such	such	ADJ
ejpam-4736	434	62	that	that	SCONJ
ejpam-4736	434	63	f	f	PROPN
ejpam-4736	434	64	⊆	⊆	NUM
ejpam-4736	434	65	u	u	NOUN
ejpam-4736	434	66	⊆	⊆	NUM
ejpam-4736	434	67	u	u	NOUN
ejpam-4736	434	68	δp(λ	δp(λ	NOUN
ejpam-4736	434	69	,	,	PUNCT
ejpam-4736	434	70	s	s	PART
ejpam-4736	434	71	)	)	PUNCT
ejpam-4736	434	72	⊆	⊆	NUM
ejpam-4736	434	73	x	x	SYM
ejpam-4736	434	74	−	−	PROPN
ejpam-4736	434	75	h.	h.	NOUN
ejpam-4736	434	76	put	put	VERB
ejpam-4736	434	77	v	v	NOUN
ejpam-4736	434	78	=	=	NOUN
ejpam-4736	434	79	x	x	SYM
ejpam-4736	434	80	−	−	PROPN
ejpam-4736	434	81	u	u	PROPN
ejpam-4736	434	82	δp(λ	δp(λ	NOUN
ejpam-4736	434	83	,	,	PUNCT
ejpam-4736	434	84	s	s	PART
ejpam-4736	434	85	)	)	PUNCT
ejpam-4736	434	86	.	.	PUNCT
ejpam-4736	435	1	then	then	ADV
ejpam-4736	435	2	,	,	PUNCT
ejpam-4736	435	3	u	u	NOUN
ejpam-4736	435	4	and	and	CCONJ
ejpam-4736	435	5	v	v	NOUN
ejpam-4736	435	6	are	be	AUX
ejpam-4736	435	7	disjoint	disjoint	NOUN
ejpam-4736	435	8	δp(λ	δp(λ	NOUN
ejpam-4736	435	9	,	,	PUNCT
ejpam-4736	435	10	s)-open	s)-open	PUNCT
ejpam-4736	435	11	sets	set	NOUN
ejpam-4736	435	12	of	of	ADP
ejpam-4736	435	13	x	x	SYM
ejpam-4736	435	14	such	such	ADJ
ejpam-4736	435	15	that	that	SCONJ
ejpam-4736	435	16	f	f	PROPN
ejpam-4736	435	17	⊆	⊆	NUM
ejpam-4736	435	18	u	u	NOUN
ejpam-4736	435	19	,	,	PUNCT
ejpam-4736	435	20	h	h	NOUN
ejpam-4736	435	21	⊆	⊆	NUM
ejpam-4736	435	22	v	v	NOUN
ejpam-4736	435	23	and	and	CCONJ
ejpam-4736	435	24	u	u	PRON
ejpam-4736	435	25	δp(λ	δp(λ	NOUN
ejpam-4736	435	26	,	,	PUNCT
ejpam-4736	435	27	s	s	NOUN
ejpam-4736	435	28	)	)	PUNCT
ejpam-4736	435	29	∩	∩	NOUN
ejpam-4736	435	30	v	v	ADP
ejpam-4736	435	31	δp(λ	δp(λ	NOUN
ejpam-4736	435	32	,	,	PUNCT
ejpam-4736	435	33	s	s	PART
ejpam-4736	435	34	)	)	PUNCT
ejpam-4736	435	35	=	=	SYM
ejpam-4736	435	36	∅.	∅.	X
ejpam-4736	435	37	(	(	PUNCT
ejpam-4736	435	38	4	4	NUM
ejpam-4736	435	39	)	)	PUNCT
ejpam-4736	435	40	⇒	⇒	NOUN
ejpam-4736	435	41	(	(	PUNCT
ejpam-4736	435	42	1	1	NUM
ejpam-4736	435	43	):	):	PUNCT
ejpam-4736	435	44	the	the	DET
ejpam-4736	435	45	proof	proof	NOUN
ejpam-4736	435	46	is	be	AUX
ejpam-4736	435	47	obvious	obvious	ADJ
ejpam-4736	435	48	.	.	PUNCT
ejpam-4736	436	1	theorem	theorem	NOUN
ejpam-4736	436	2	18	18	NUM
ejpam-4736	436	3	.	.	PUNCT
ejpam-4736	437	1	for	for	ADP
ejpam-4736	437	2	a	a	DET
ejpam-4736	437	3	topological	topological	ADJ
ejpam-4736	437	4	space	space	NOUN
ejpam-4736	437	5	(	(	PUNCT
ejpam-4736	437	6	x	x	X
ejpam-4736	437	7	,	,	PUNCT
ejpam-4736	437	8	τ	τ	PROPN
ejpam-4736	437	9	)	)	PUNCT
ejpam-4736	437	10	,	,	PUNCT
ejpam-4736	437	11	the	the	DET
ejpam-4736	437	12	following	follow	VERB
ejpam-4736	437	13	properties	property	NOUN
ejpam-4736	437	14	are	be	AUX
ejpam-4736	437	15	equivalent	equivalent	ADJ
ejpam-4736	437	16	:	:	PUNCT
ejpam-4736	437	17	(	(	PUNCT
ejpam-4736	437	18	1	1	X
ejpam-4736	437	19	)	)	PUNCT
ejpam-4736	437	20	(	(	PUNCT
ejpam-4736	437	21	x	x	X
ejpam-4736	437	22	,	,	PUNCT
ejpam-4736	437	23	τ	τ	X
ejpam-4736	437	24	)	)	PUNCT
ejpam-4736	437	25	is	be	AUX
ejpam-4736	437	26	δp(λ	δp(λ	NOUN
ejpam-4736	437	27	,	,	PUNCT
ejpam-4736	437	28	s)-normal	s)-normal	ADJ
ejpam-4736	437	29	.	.	PUNCT
ejpam-4736	438	1	(	(	PUNCT
ejpam-4736	438	2	2	2	X
ejpam-4736	438	3	)	)	PUNCT
ejpam-4736	438	4	for	for	ADP
ejpam-4736	438	5	any	any	DET
ejpam-4736	438	6	pair	pair	NOUN
ejpam-4736	438	7	of	of	ADP
ejpam-4736	438	8	disjoint	disjoint	NOUN
ejpam-4736	438	9	δp(λ	δp(λ	NOUN
ejpam-4736	438	10	,	,	PUNCT
ejpam-4736	438	11	s)-closed	s)-close	VERB
ejpam-4736	438	12	sets	set	NOUN
ejpam-4736	438	13	f	f	PROPN
ejpam-4736	438	14	and	and	CCONJ
ejpam-4736	438	15	h	h	NOUN
ejpam-4736	438	16	,	,	PUNCT
ejpam-4736	438	17	there	there	PRON
ejpam-4736	438	18	exist	exist	VERB
ejpam-4736	438	19	disjoint	disjoint	NOUN
ejpam-4736	438	20	g	g	NOUN
ejpam-4736	438	21	-	-	PUNCT
ejpam-4736	438	22	δp(λ	δp(λ	NOUN
ejpam-4736	438	23	,	,	PUNCT
ejpam-4736	438	24	s)open	s)open	NOUN
ejpam-4736	438	25	sets	set	VERB
ejpam-4736	438	26	u	u	NOUN
ejpam-4736	438	27	and	and	CCONJ
ejpam-4736	438	28	v	v	ADP
ejpam-4736	438	29	such	such	ADJ
ejpam-4736	438	30	that	that	SCONJ
ejpam-4736	438	31	f	f	PROPN
ejpam-4736	438	32	⊆	⊆	NUM
ejpam-4736	438	33	u	u	NOUN
ejpam-4736	438	34	and	and	CCONJ
ejpam-4736	438	35	h	h	NOUN
ejpam-4736	438	36	⊆	⊆	NUM
ejpam-4736	438	37	v	v	NOUN
ejpam-4736	438	38	.	.	PUNCT
ejpam-4736	439	1	(	(	PUNCT
ejpam-4736	439	2	3	3	X
ejpam-4736	439	3	)	)	PUNCT
ejpam-4736	439	4	for	for	ADP
ejpam-4736	439	5	each	each	DET
ejpam-4736	439	6	δp(λ	δp(λ	NOUN
ejpam-4736	439	7	,	,	PUNCT
ejpam-4736	439	8	s)-closed	s)-close	VERB
ejpam-4736	439	9	set	set	ADJ
ejpam-4736	439	10	f	f	PROPN
ejpam-4736	439	11	and	and	CCONJ
ejpam-4736	439	12	each	each	DET
ejpam-4736	439	13	δp(λ	δp(λ	NOUN
ejpam-4736	439	14	,	,	PUNCT
ejpam-4736	439	15	s)-open	s)-open	PUNCT
ejpam-4736	439	16	set	set	VERB
ejpam-4736	439	17	g	g	NOUN
ejpam-4736	439	18	containing	contain	VERB
ejpam-4736	439	19	f	f	NOUN
ejpam-4736	439	20	,	,	PUNCT
ejpam-4736	439	21	there	there	PRON
ejpam-4736	439	22	exists	exist	VERB
ejpam-4736	439	23	a	a	DET
ejpam-4736	439	24	g	g	NOUN
ejpam-4736	439	25	-	-	PUNCT
ejpam-4736	439	26	δp(λ	δp(λ	NOUN
ejpam-4736	439	27	,	,	PUNCT
ejpam-4736	439	28	s)-open	s)-open	PUNCT
ejpam-4736	439	29	set	set	VERB
ejpam-4736	439	30	u	u	PRON
ejpam-4736	439	31	such	such	ADJ
ejpam-4736	439	32	that	that	SCONJ
ejpam-4736	439	33	f	f	PROPN
ejpam-4736	439	34	⊆	⊆	NUM
ejpam-4736	439	35	u	u	NOUN
ejpam-4736	439	36	⊆	⊆	NUM
ejpam-4736	439	37	u	u	NOUN
ejpam-4736	439	38	δp(λ	δp(λ	NOUN
ejpam-4736	439	39	,	,	PUNCT
ejpam-4736	439	40	s	s	PART
ejpam-4736	439	41	)	)	PUNCT
ejpam-4736	439	42	⊆	⊆	NUM
ejpam-4736	439	43	g.	g.	NOUN
ejpam-4736	439	44	(	(	PUNCT
ejpam-4736	439	45	4	4	NUM
ejpam-4736	439	46	)	)	PUNCT
ejpam-4736	439	47	for	for	ADP
ejpam-4736	439	48	each	each	DET
ejpam-4736	439	49	δp(λ	δp(λ	NOUN
ejpam-4736	439	50	,	,	PUNCT
ejpam-4736	439	51	s)-closed	s)-close	VERB
ejpam-4736	439	52	set	set	ADJ
ejpam-4736	439	53	f	f	NOUN
ejpam-4736	439	54	and	and	CCONJ
ejpam-4736	439	55	each	each	DET
ejpam-4736	439	56	g	g	NOUN
ejpam-4736	439	57	-	-	PUNCT
ejpam-4736	439	58	δp(λ	δp(λ	NOUN
ejpam-4736	439	59	,	,	PUNCT
ejpam-4736	439	60	s)-open	s)-open	PUNCT
ejpam-4736	439	61	set	set	VERB
ejpam-4736	439	62	g	g	NOUN
ejpam-4736	439	63	containing	contain	VERB
ejpam-4736	439	64	f	f	NOUN
ejpam-4736	439	65	,	,	PUNCT
ejpam-4736	439	66	there	there	PRON
ejpam-4736	439	67	exists	exist	VERB
ejpam-4736	439	68	a	a	DET
ejpam-4736	439	69	δp(λ	δp(λ	NOUN
ejpam-4736	439	70	,	,	PUNCT
ejpam-4736	439	71	s)-open	s)-open	PUNCT
ejpam-4736	439	72	set	set	VERB
ejpam-4736	439	73	u	u	PRON
ejpam-4736	439	74	such	such	ADJ
ejpam-4736	439	75	that	that	SCONJ
ejpam-4736	439	76	f	f	PROPN
ejpam-4736	439	77	⊆	⊆	NUM
ejpam-4736	439	78	u	u	NOUN
ejpam-4736	439	79	⊆	⊆	NUM
ejpam-4736	439	80	u	u	NOUN
ejpam-4736	439	81	δp(λ	δp(λ	NOUN
ejpam-4736	439	82	,	,	PUNCT
ejpam-4736	439	83	s	s	PART
ejpam-4736	439	84	)	)	PUNCT
ejpam-4736	439	85	⊆	⊆	NUM
ejpam-4736	439	86	gδp(λ	gδp(λ	PROPN
ejpam-4736	439	87	,	,	PUNCT
ejpam-4736	439	88	s	s	NOUN
ejpam-4736	439	89	)	)	PUNCT
ejpam-4736	439	90	.	.	PUNCT
ejpam-4736	440	1	(	(	PUNCT
ejpam-4736	440	2	5	5	NUM
ejpam-4736	440	3	)	)	PUNCT
ejpam-4736	440	4	for	for	ADP
ejpam-4736	440	5	each	each	DET
ejpam-4736	440	6	δp(λ	δp(λ	NOUN
ejpam-4736	440	7	,	,	PUNCT
ejpam-4736	440	8	s)-closed	s)-close	VERB
ejpam-4736	440	9	set	set	ADJ
ejpam-4736	440	10	f	f	NOUN
ejpam-4736	440	11	and	and	CCONJ
ejpam-4736	440	12	each	each	DET
ejpam-4736	440	13	g	g	NOUN
ejpam-4736	440	14	-	-	PUNCT
ejpam-4736	440	15	δp(λ	δp(λ	NOUN
ejpam-4736	440	16	,	,	PUNCT
ejpam-4736	440	17	s)-open	s)-open	PUNCT
ejpam-4736	440	18	set	set	VERB
ejpam-4736	440	19	g	g	NOUN
ejpam-4736	440	20	containing	contain	VERB
ejpam-4736	440	21	f	f	NOUN
ejpam-4736	440	22	,	,	PUNCT
ejpam-4736	440	23	there	there	PRON
ejpam-4736	440	24	exists	exist	VERB
ejpam-4736	440	25	a	a	DET
ejpam-4736	440	26	g	g	NOUN
ejpam-4736	440	27	-	-	PUNCT
ejpam-4736	440	28	δp(λ	δp(λ	NOUN
ejpam-4736	440	29	,	,	PUNCT
ejpam-4736	440	30	s)-open	s)-open	PUNCT
ejpam-4736	440	31	set	set	VERB
ejpam-4736	440	32	u	u	PRON
ejpam-4736	440	33	such	such	ADJ
ejpam-4736	440	34	that	that	SCONJ
ejpam-4736	440	35	f	f	PROPN
ejpam-4736	440	36	⊆	⊆	NUM
ejpam-4736	440	37	u	u	NOUN
ejpam-4736	440	38	⊆	⊆	NUM
ejpam-4736	440	39	u	u	NOUN
ejpam-4736	440	40	δp(λ	δp(λ	NOUN
ejpam-4736	440	41	,	,	PUNCT
ejpam-4736	440	42	s	s	PART
ejpam-4736	440	43	)	)	PUNCT
ejpam-4736	440	44	⊆	⊆	NUM
ejpam-4736	440	45	gδp(λ	gδp(λ	PROPN
ejpam-4736	440	46	,	,	PUNCT
ejpam-4736	440	47	s	s	NOUN
ejpam-4736	440	48	)	)	PUNCT
ejpam-4736	440	49	.	.	PUNCT
ejpam-4736	441	1	(	(	PUNCT
ejpam-4736	441	2	6	6	NUM
ejpam-4736	441	3	)	)	PUNCT
ejpam-4736	441	4	for	for	ADP
ejpam-4736	441	5	each	each	DET
ejpam-4736	441	6	g	g	NOUN
ejpam-4736	441	7	-	-	PUNCT
ejpam-4736	441	8	δp(λ	δp(λ	NOUN
ejpam-4736	441	9	,	,	PUNCT
ejpam-4736	441	10	s)-closed	s)-close	VERB
ejpam-4736	441	11	set	set	ADJ
ejpam-4736	441	12	f	f	PROPN
ejpam-4736	441	13	and	and	CCONJ
ejpam-4736	441	14	each	each	DET
ejpam-4736	441	15	δp(λ	δp(λ	NOUN
ejpam-4736	441	16	,	,	PUNCT
ejpam-4736	441	17	s)-open	s)-open	PUNCT
ejpam-4736	441	18	set	set	VERB
ejpam-4736	441	19	g	g	NOUN
ejpam-4736	441	20	containing	contain	VERB
ejpam-4736	441	21	f	f	NOUN
ejpam-4736	441	22	,	,	PUNCT
ejpam-4736	441	23	there	there	PRON
ejpam-4736	441	24	exists	exist	VERB
ejpam-4736	441	25	a	a	DET
ejpam-4736	441	26	δp(λ	δp(λ	NOUN
ejpam-4736	441	27	,	,	PUNCT
ejpam-4736	441	28	s)-open	s)-open	PUNCT
ejpam-4736	441	29	set	set	VERB
ejpam-4736	441	30	u	u	PRON
ejpam-4736	441	31	such	such	ADJ
ejpam-4736	441	32	that	that	SCONJ
ejpam-4736	441	33	f	f	PROPN
ejpam-4736	441	34	δp(λ	δp(λ	PROPN
ejpam-4736	441	35	,	,	PUNCT
ejpam-4736	441	36	s	s	PART
ejpam-4736	441	37	)	)	PUNCT
ejpam-4736	441	38	⊆	⊆	NUM
ejpam-4736	441	39	u	u	NOUN
ejpam-4736	441	40	⊆	⊆	NUM
ejpam-4736	441	41	u	u	NOUN
ejpam-4736	441	42	δp(λ	δp(λ	NOUN
ejpam-4736	441	43	,	,	PUNCT
ejpam-4736	441	44	s	s	PART
ejpam-4736	441	45	)	)	PUNCT
ejpam-4736	441	46	⊆	⊆	NUM
ejpam-4736	441	47	g.	g.	NOUN
ejpam-4736	441	48	(	(	PUNCT
ejpam-4736	441	49	7	7	NUM
ejpam-4736	441	50	)	)	PUNCT
ejpam-4736	441	51	for	for	ADP
ejpam-4736	441	52	each	each	DET
ejpam-4736	441	53	g	g	NOUN
ejpam-4736	441	54	-	-	PUNCT
ejpam-4736	441	55	δp(λ	δp(λ	NOUN
ejpam-4736	441	56	,	,	PUNCT
ejpam-4736	441	57	s)-closed	s)-close	VERB
ejpam-4736	441	58	set	set	ADJ
ejpam-4736	441	59	f	f	PROPN
ejpam-4736	441	60	and	and	CCONJ
ejpam-4736	441	61	each	each	DET
ejpam-4736	441	62	δp(λ	δp(λ	NOUN
ejpam-4736	441	63	,	,	PUNCT
ejpam-4736	441	64	s)-open	s)-open	PUNCT
ejpam-4736	441	65	set	set	VERB
ejpam-4736	441	66	g	g	NOUN
ejpam-4736	441	67	containing	contain	VERB
ejpam-4736	441	68	f	f	NOUN
ejpam-4736	441	69	,	,	PUNCT
ejpam-4736	441	70	there	there	PRON
ejpam-4736	441	71	exists	exist	VERB
ejpam-4736	441	72	a	a	DET
ejpam-4736	441	73	g	g	NOUN
ejpam-4736	441	74	-	-	PUNCT
ejpam-4736	441	75	δp(λ	δp(λ	NOUN
ejpam-4736	441	76	,	,	PUNCT
ejpam-4736	441	77	s)-open	s)-open	PUNCT
ejpam-4736	441	78	set	set	VERB
ejpam-4736	441	79	u	u	PRON
ejpam-4736	441	80	such	such	ADJ
ejpam-4736	441	81	that	that	SCONJ
ejpam-4736	441	82	f	f	PROPN
ejpam-4736	441	83	δp(λ	δp(λ	PROPN
ejpam-4736	441	84	,	,	PUNCT
ejpam-4736	441	85	s	s	PART
ejpam-4736	441	86	)	)	PUNCT
ejpam-4736	441	87	⊆	⊆	NUM
ejpam-4736	441	88	u	u	NOUN
ejpam-4736	441	89	⊆	⊆	NUM
ejpam-4736	441	90	u	u	NOUN
ejpam-4736	441	91	δp(λ	δp(λ	NOUN
ejpam-4736	441	92	,	,	PUNCT
ejpam-4736	441	93	s	s	PART
ejpam-4736	441	94	)	)	PUNCT
ejpam-4736	441	95	⊆	⊆	NUM
ejpam-4736	441	96	g.	g.	NOUN
ejpam-4736	441	97	references	reference	NOUN
ejpam-4736	441	98	2594	2594	NUM
ejpam-4736	441	99	proof	proof	NOUN
ejpam-4736	441	100	.	.	PUNCT
ejpam-4736	442	1	(	(	PUNCT
ejpam-4736	442	2	1	1	X
ejpam-4736	442	3	)	)	PUNCT
ejpam-4736	442	4	⇒	⇒	NOUN
ejpam-4736	442	5	(	(	PUNCT
ejpam-4736	442	6	2	2	NUM
ejpam-4736	442	7	):	):	PUNCT
ejpam-4736	442	8	the	the	DET
ejpam-4736	442	9	proof	proof	NOUN
ejpam-4736	442	10	is	be	AUX
ejpam-4736	442	11	obvious	obvious	ADJ
ejpam-4736	442	12	.	.	PUNCT
ejpam-4736	443	1	(	(	PUNCT
ejpam-4736	443	2	2	2	X
ejpam-4736	443	3	)	)	PUNCT
ejpam-4736	443	4	⇒	⇒	NOUN
ejpam-4736	443	5	(	(	PUNCT
ejpam-4736	443	6	3	3	NUM
ejpam-4736	443	7	):	):	PUNCT
ejpam-4736	443	8	let	let	VERB
ejpam-4736	443	9	f	f	PRON
ejpam-4736	443	10	be	be	AUX
ejpam-4736	443	11	a	a	DET
ejpam-4736	443	12	δp(λ	δp(λ	NOUN
ejpam-4736	443	13	,	,	PUNCT
ejpam-4736	443	14	s)-closed	s)-close	VERB
ejpam-4736	443	15	set	set	NOUN
ejpam-4736	443	16	and	and	CCONJ
ejpam-4736	443	17	g	g	NOUN
ejpam-4736	443	18	be	be	AUX
ejpam-4736	443	19	a	a	DET
ejpam-4736	443	20	δp(λ	δp(λ	NOUN
ejpam-4736	443	21	,	,	PUNCT
ejpam-4736	443	22	s)-open	s)-open	PUNCT
ejpam-4736	443	23	set	set	VERB
ejpam-4736	443	24	containing	contain	VERB
ejpam-4736	443	25	f	f	PROPN
ejpam-4736	443	26	.	.	PUNCT
ejpam-4736	444	1	then	then	ADV
ejpam-4736	444	2	,	,	PUNCT
ejpam-4736	444	3	f	f	PROPN
ejpam-4736	444	4	and	and	CCONJ
ejpam-4736	444	5	x	x	SYM
ejpam-4736	444	6	−	−	PROPN
ejpam-4736	444	7	g	g	NOUN
ejpam-4736	444	8	are	be	AUX
ejpam-4736	444	9	two	two	NUM
ejpam-4736	444	10	disjoint	disjoint	NOUN
ejpam-4736	444	11	δp(λ	δp(λ	NOUN
ejpam-4736	444	12	,	,	PUNCT
ejpam-4736	444	13	s)-closed	s)-close	VERB
ejpam-4736	444	14	sets	set	NOUN
ejpam-4736	444	15	.	.	PUNCT
ejpam-4736	445	1	hence	hence	ADV
ejpam-4736	445	2	by	by	ADP
ejpam-4736	445	3	(	(	PUNCT
ejpam-4736	445	4	2	2	NUM
ejpam-4736	445	5	)	)	PUNCT
ejpam-4736	445	6	,	,	PUNCT
ejpam-4736	445	7	there	there	PRON
ejpam-4736	445	8	exist	exist	VERB
ejpam-4736	445	9	disjoint	disjoint	NOUN
ejpam-4736	445	10	g	g	NOUN
ejpam-4736	445	11	-	-	PUNCT
ejpam-4736	445	12	δp(λ	δp(λ	NOUN
ejpam-4736	445	13	,	,	PUNCT
ejpam-4736	445	14	s)-open	s)-open	PUNCT
ejpam-4736	445	15	sets	set	VERB
ejpam-4736	445	16	u	u	NOUN
ejpam-4736	445	17	and	and	CCONJ
ejpam-4736	445	18	v	v	NOUN
ejpam-4736	445	19	of	of	ADP
ejpam-4736	445	20	x	x	PUNCT
ejpam-4736	445	21	such	such	ADJ
ejpam-4736	445	22	that	that	SCONJ
ejpam-4736	445	23	f	f	PROPN
ejpam-4736	445	24	⊆	⊆	NUM
ejpam-4736	445	25	u	u	NOUN
ejpam-4736	445	26	and	and	CCONJ
ejpam-4736	445	27	x	x	NOUN
ejpam-4736	445	28	−	−	NOUN
ejpam-4736	445	29	g	g	PROPN
ejpam-4736	445	30	⊆	⊆	NUM
ejpam-4736	445	31	v	v	NOUN
ejpam-4736	445	32	.	.	PUNCT
ejpam-4736	446	1	since	since	SCONJ
ejpam-4736	446	2	v	v	NOUN
ejpam-4736	446	3	is	be	AUX
ejpam-4736	446	4	g	g	NOUN
ejpam-4736	446	5	-	-	PUNCT
ejpam-4736	446	6	δp(λ	δp(λ	NOUN
ejpam-4736	446	7	,	,	PUNCT
ejpam-4736	446	8	s)-open	s)-open	PUNCT
ejpam-4736	446	9	and	and	CCONJ
ejpam-4736	446	10	x	x	X
ejpam-4736	446	11	−	−	NOUN
ejpam-4736	446	12	g	g	NOUN
ejpam-4736	446	13	is	be	AUX
ejpam-4736	446	14	δp(λ	δp(λ	NOUN
ejpam-4736	446	15	,	,	PUNCT
ejpam-4736	446	16	s)-closed	s)-close	VERB
ejpam-4736	446	17	,	,	PUNCT
ejpam-4736	446	18	by	by	ADP
ejpam-4736	446	19	theorem	theorem	NOUN
ejpam-4736	446	20	5	5	NUM
ejpam-4736	446	21	,	,	PUNCT
ejpam-4736	446	22	x	x	NOUN
ejpam-4736	446	23	−	−	NOUN
ejpam-4736	446	24	g	g	NOUN
ejpam-4736	446	25	⊆	⊆	NUM
ejpam-4736	446	26	vδp(λ	vδp(λ	NOUN
ejpam-4736	446	27	,	,	PUNCT
ejpam-4736	446	28	s	s	NOUN
ejpam-4736	446	29	)	)	PUNCT
ejpam-4736	446	30	.	.	PUNCT
ejpam-4736	447	1	thus	thus	ADV
ejpam-4736	447	2	,	,	PUNCT
ejpam-4736	447	3	[	[	X
ejpam-4736	447	4	x	x	X
ejpam-4736	447	5	−	−	NOUN
ejpam-4736	447	6	v	v	NOUN
ejpam-4736	447	7	]	]	X
ejpam-4736	447	8	δp(λ	δp(λ	NUM
ejpam-4736	447	9	,	,	PUNCT
ejpam-4736	447	10	s	s	PART
ejpam-4736	447	11	)	)	PUNCT
ejpam-4736	447	12	=	=	PUNCT
ejpam-4736	447	13	x	x	X
ejpam-4736	447	14	−	−	NOUN
ejpam-4736	447	15	vδp(λ	vδp(λ	NOUN
ejpam-4736	447	16	,	,	PUNCT
ejpam-4736	447	17	s	s	NOUN
ejpam-4736	447	18	)	)	PUNCT
ejpam-4736	447	19	⊆	⊆	NUM
ejpam-4736	447	20	g	g	NOUN
ejpam-4736	447	21	and	and	CCONJ
ejpam-4736	447	22	hence	hence	ADV
ejpam-4736	447	23	f	f	PROPN
ejpam-4736	447	24	⊆	⊆	NUM
ejpam-4736	447	25	u	u	NOUN
ejpam-4736	447	26	⊆	⊆	NUM
ejpam-4736	447	27	u	u	NOUN
ejpam-4736	447	28	δp(λ	δp(λ	NOUN
ejpam-4736	447	29	,	,	PUNCT
ejpam-4736	447	30	s	s	PART
ejpam-4736	447	31	)	)	PUNCT
ejpam-4736	447	32	⊆	⊆	NUM
ejpam-4736	447	33	g.	g.	NOUN
ejpam-4736	447	34	(	(	PUNCT
ejpam-4736	447	35	3	3	NUM
ejpam-4736	447	36	)	)	PUNCT
ejpam-4736	447	37	⇒	⇒	NOUN
ejpam-4736	447	38	(	(	PUNCT
ejpam-4736	447	39	5	5	NUM
ejpam-4736	447	40	):	):	PUNCT
ejpam-4736	447	41	let	let	VERB
ejpam-4736	447	42	f	f	PRON
ejpam-4736	447	43	be	be	AUX
ejpam-4736	447	44	a	a	DET
ejpam-4736	447	45	δp(λ	δp(λ	NOUN
ejpam-4736	447	46	,	,	PUNCT
ejpam-4736	447	47	s)-closed	s)-close	VERB
ejpam-4736	447	48	set	set	NOUN
ejpam-4736	447	49	and	and	CCONJ
ejpam-4736	447	50	g	g	NOUN
ejpam-4736	447	51	be	be	AUX
ejpam-4736	447	52	a	a	DET
ejpam-4736	447	53	g	g	NOUN
ejpam-4736	447	54	-	-	PUNCT
ejpam-4736	447	55	δp(λ	δp(λ	NOUN
ejpam-4736	447	56	,	,	PUNCT
ejpam-4736	447	57	s)-open	s)-open	VERB
ejpam-4736	447	58	set	set	VERB
ejpam-4736	447	59	containing	contain	VERB
ejpam-4736	447	60	f	f	PROPN
ejpam-4736	447	61	.	.	PUNCT
ejpam-4736	448	1	since	since	SCONJ
ejpam-4736	448	2	g	g	PROPN
ejpam-4736	448	3	is	be	AUX
ejpam-4736	448	4	g	g	NOUN
ejpam-4736	448	5	-	-	PUNCT
ejpam-4736	448	6	δp(λ	δp(λ	NOUN
ejpam-4736	448	7	,	,	PUNCT
ejpam-4736	448	8	s)-open	s)-open	PUNCT
ejpam-4736	448	9	and	and	CCONJ
ejpam-4736	448	10	f	f	PROPN
ejpam-4736	448	11	is	be	AUX
ejpam-4736	448	12	δp(λ	δp(λ	NOUN
ejpam-4736	448	13	,	,	PUNCT
ejpam-4736	448	14	s)-closed	s)-close	VERB
ejpam-4736	448	15	,	,	PUNCT
ejpam-4736	448	16	by	by	ADP
ejpam-4736	448	17	theorem	theorem	NOUN
ejpam-4736	448	18	5	5	NUM
ejpam-4736	448	19	,	,	PUNCT
ejpam-4736	448	20	f	f	PROPN
ejpam-4736	448	21	⊆	⊆	NUM
ejpam-4736	448	22	gδp(λ	gδp(λ	PROPN
ejpam-4736	448	23	,	,	PUNCT
ejpam-4736	448	24	s	s	NOUN
ejpam-4736	448	25	)	)	PUNCT
ejpam-4736	448	26	.	.	PUNCT
ejpam-4736	449	1	thus	thus	ADV
ejpam-4736	449	2	,	,	PUNCT
ejpam-4736	449	3	by	by	ADP
ejpam-4736	449	4	(	(	PUNCT
ejpam-4736	449	5	3	3	NUM
ejpam-4736	449	6	)	)	PUNCT
ejpam-4736	449	7	,	,	PUNCT
ejpam-4736	449	8	there	there	PRON
ejpam-4736	449	9	exists	exist	VERB
ejpam-4736	449	10	a	a	DET
ejpam-4736	449	11	g	g	NOUN
ejpam-4736	449	12	-	-	PUNCT
ejpam-4736	449	13	δp(λ	δp(λ	NOUN
ejpam-4736	449	14	,	,	PUNCT
ejpam-4736	449	15	s)-open	s)-open	PUNCT
ejpam-4736	449	16	set	set	VERB
ejpam-4736	449	17	u	u	PRON
ejpam-4736	449	18	such	such	ADJ
ejpam-4736	449	19	that	that	SCONJ
ejpam-4736	449	20	f	f	PROPN
ejpam-4736	449	21	⊆	⊆	NUM
ejpam-4736	449	22	u	u	NOUN
ejpam-4736	449	23	⊆	⊆	NUM
ejpam-4736	449	24	u	u	NOUN
ejpam-4736	449	25	δp(λ	δp(λ	NOUN
ejpam-4736	449	26	,	,	PUNCT
ejpam-4736	449	27	s	s	PART
ejpam-4736	449	28	)	)	PUNCT
ejpam-4736	449	29	⊆	⊆	NUM
ejpam-4736	449	30	gδp(λ	gδp(λ	PROPN
ejpam-4736	449	31	,	,	PUNCT
ejpam-4736	449	32	s	s	NOUN
ejpam-4736	449	33	)	)	PUNCT
ejpam-4736	449	34	.	.	PUNCT
ejpam-4736	450	1	(	(	PUNCT
ejpam-4736	450	2	5	5	X
ejpam-4736	450	3	)	)	PUNCT
ejpam-4736	450	4	⇒	⇒	NOUN
ejpam-4736	450	5	(	(	PUNCT
ejpam-4736	450	6	6	6	NUM
ejpam-4736	450	7	):	):	PUNCT
ejpam-4736	450	8	let	let	VERB
ejpam-4736	450	9	f	f	PRON
ejpam-4736	450	10	be	be	AUX
ejpam-4736	450	11	a	a	DET
ejpam-4736	450	12	g	g	NOUN
ejpam-4736	450	13	-	-	PUNCT
ejpam-4736	450	14	δp(λ	δp(λ	NOUN
ejpam-4736	450	15	,	,	PUNCT
ejpam-4736	450	16	s)-closed	s)-close	VERB
ejpam-4736	450	17	set	set	NOUN
ejpam-4736	450	18	and	and	CCONJ
ejpam-4736	450	19	g	g	NOUN
ejpam-4736	450	20	be	be	AUX
ejpam-4736	450	21	a	a	DET
ejpam-4736	450	22	δp(λ	δp(λ	NOUN
ejpam-4736	450	23	,	,	PUNCT
ejpam-4736	450	24	s)-open	s)-open	PUNCT
ejpam-4736	450	25	set	set	VERB
ejpam-4736	450	26	containing	contain	VERB
ejpam-4736	450	27	f	f	PROPN
ejpam-4736	450	28	.	.	PUNCT
ejpam-4736	451	1	then	then	ADV
ejpam-4736	451	2	,	,	PUNCT
ejpam-4736	451	3	we	we	PRON
ejpam-4736	451	4	have	have	VERB
ejpam-4736	451	5	f	f	PROPN
ejpam-4736	451	6	δp(λ	δp(λ	NOUN
ejpam-4736	451	7	,	,	PUNCT
ejpam-4736	451	8	s	s	PART
ejpam-4736	451	9	)	)	PUNCT
ejpam-4736	451	10	⊆	⊆	NUM
ejpam-4736	451	11	g.	g.	NOUN
ejpam-4736	451	12	since	since	SCONJ
ejpam-4736	451	13	g	g	PROPN
ejpam-4736	451	14	is	be	AUX
ejpam-4736	451	15	g	g	NOUN
ejpam-4736	451	16	-	-	PUNCT
ejpam-4736	451	17	δp(λ	δp(λ	NOUN
ejpam-4736	451	18	,	,	PUNCT
ejpam-4736	451	19	s)-open	s)-open	PUNCT
ejpam-4736	451	20	and	and	CCONJ
ejpam-4736	451	21	f	f	PROPN
ejpam-4736	451	22	δp(λ	δp(λ	PROPN
ejpam-4736	451	23	,	,	PUNCT
ejpam-4736	451	24	s	s	PART
ejpam-4736	451	25	)	)	PUNCT
ejpam-4736	451	26	is	be	AUX
ejpam-4736	451	27	δp(λ	δp(λ	NOUN
ejpam-4736	451	28	,	,	PUNCT
ejpam-4736	451	29	s)-closed	s)-close	VERB
ejpam-4736	451	30	,	,	PUNCT
ejpam-4736	451	31	by	by	ADP
ejpam-4736	451	32	(	(	PUNCT
ejpam-4736	451	33	5	5	NUM
ejpam-4736	451	34	)	)	PUNCT
ejpam-4736	451	35	,	,	PUNCT
ejpam-4736	451	36	there	there	PRON
ejpam-4736	451	37	exists	exist	VERB
ejpam-4736	451	38	a	a	DET
ejpam-4736	451	39	g	g	NOUN
ejpam-4736	451	40	-	-	PUNCT
ejpam-4736	451	41	δp(λ	δp(λ	NOUN
ejpam-4736	451	42	,	,	PUNCT
ejpam-4736	451	43	s)-open	s)-open	PUNCT
ejpam-4736	451	44	set	set	VERB
ejpam-4736	451	45	u	u	PRON
ejpam-4736	451	46	such	such	ADJ
ejpam-4736	451	47	that	that	SCONJ
ejpam-4736	451	48	f	f	PROPN
ejpam-4736	451	49	δp(λ	δp(λ	PROPN
ejpam-4736	451	50	,	,	PUNCT
ejpam-4736	451	51	s	s	PART
ejpam-4736	451	52	)	)	PUNCT
ejpam-4736	451	53	⊆	⊆	NUM
ejpam-4736	451	54	u	u	NOUN
ejpam-4736	451	55	⊆	⊆	NUM
ejpam-4736	451	56	u	u	NOUN
ejpam-4736	451	57	δp(λ	δp(λ	NOUN
ejpam-4736	451	58	,	,	PUNCT
ejpam-4736	451	59	s	s	PART
ejpam-4736	451	60	)	)	PUNCT
ejpam-4736	451	61	⊆	⊆	NUM
ejpam-4736	451	62	g.	g.	NOUN
ejpam-4736	451	63	since	since	SCONJ
ejpam-4736	451	64	u	u	NOUN
ejpam-4736	451	65	is	be	AUX
ejpam-4736	451	66	g	g	NOUN
ejpam-4736	451	67	-	-	PUNCT
ejpam-4736	451	68	δp(λ	δp(λ	NOUN
ejpam-4736	451	69	,	,	PUNCT
ejpam-4736	451	70	s)-open	s)-open	PUNCT
ejpam-4736	451	71	and	and	CCONJ
ejpam-4736	451	72	f	f	PROPN
ejpam-4736	451	73	δp(λ	δp(λ	PROPN
ejpam-4736	451	74	,	,	PUNCT
ejpam-4736	451	75	s	s	PART
ejpam-4736	451	76	)	)	PUNCT
ejpam-4736	451	77	⊆	⊆	NUM
ejpam-4736	451	78	u	u	NOUN
ejpam-4736	451	79	,	,	PUNCT
ejpam-4736	451	80	by	by	ADP
ejpam-4736	451	81	theorem	theorem	NOUN
ejpam-4736	451	82	5	5	NUM
ejpam-4736	451	83	,	,	PUNCT
ejpam-4736	451	84	f	f	PROPN
ejpam-4736	451	85	δp(λ	δp(λ	PROPN
ejpam-4736	451	86	,	,	PUNCT
ejpam-4736	451	87	s	s	PART
ejpam-4736	451	88	)	)	PUNCT
ejpam-4736	451	89	⊆	⊆	NUM
ejpam-4736	451	90	uδp(λ	uδp(λ	PROPN
ejpam-4736	451	91	,	,	PUNCT
ejpam-4736	451	92	s	s	PART
ejpam-4736	451	93	)	)	PUNCT
ejpam-4736	451	94	.	.	PUNCT
ejpam-4736	452	1	put	put	VERB
ejpam-4736	452	2	v	v	NUM
ejpam-4736	452	3	=	=	SYM
ejpam-4736	452	4	uδp(λ	uδp(λ	PROPN
ejpam-4736	452	5	,	,	PUNCT
ejpam-4736	452	6	s	s	NOUN
ejpam-4736	452	7	)	)	PUNCT
ejpam-4736	452	8	.	.	PUNCT
ejpam-4736	453	1	then	then	ADV
ejpam-4736	453	2	,	,	PUNCT
ejpam-4736	453	3	v	v	NOUN
ejpam-4736	453	4	is	be	AUX
ejpam-4736	453	5	δp(λ	δp(λ	NOUN
ejpam-4736	453	6	,	,	PUNCT
ejpam-4736	453	7	s)-open	s)-open	PUNCT
ejpam-4736	453	8	and	and	CCONJ
ejpam-4736	453	9	f	f	PROPN
ejpam-4736	453	10	δp(λ	δp(λ	PROPN
ejpam-4736	453	11	,	,	PUNCT
ejpam-4736	453	12	s	s	PART
ejpam-4736	453	13	)	)	PUNCT
ejpam-4736	453	14	⊆	⊆	NUM
ejpam-4736	453	15	v	v	ADP
ejpam-4736	453	16	⊆	⊆	NUM
ejpam-4736	453	17	v	v	NOUN
ejpam-4736	453	18	δp(λ	δp(λ	NOUN
ejpam-4736	453	19	,	,	PUNCT
ejpam-4736	453	20	s	s	NOUN
ejpam-4736	453	21	)	)	PUNCT
ejpam-4736	453	22	=	=	PUNCT
ejpam-4736	454	1	[	[	X
ejpam-4736	454	2	uδp(λ	uδp(λ	PROPN
ejpam-4736	454	3	,	,	PUNCT
ejpam-4736	454	4	s	s	NOUN
ejpam-4736	454	5	)	)	PUNCT
ejpam-4736	454	6	]	]	PUNCT
ejpam-4736	454	7	δp(λ	δp(λ	NOUN
ejpam-4736	454	8	,	,	PUNCT
ejpam-4736	454	9	s	s	PART
ejpam-4736	454	10	)	)	PUNCT
ejpam-4736	454	11	⊆	⊆	NUM
ejpam-4736	454	12	u	u	NOUN
ejpam-4736	454	13	δp(λ	δp(λ	NOUN
ejpam-4736	454	14	,	,	PUNCT
ejpam-4736	454	15	s	s	PART
ejpam-4736	454	16	)	)	PUNCT
ejpam-4736	454	17	⊆	⊆	NUM
ejpam-4736	454	18	g.	g.	NOUN
ejpam-4736	454	19	(	(	PUNCT
ejpam-4736	454	20	6	6	NUM
ejpam-4736	454	21	)	)	PUNCT
ejpam-4736	454	22	⇒	⇒	NOUN
ejpam-4736	454	23	(	(	PUNCT
ejpam-4736	454	24	4	4	NUM
ejpam-4736	454	25	):	):	PUNCT
ejpam-4736	454	26	let	let	VERB
ejpam-4736	454	27	f	f	PRON
ejpam-4736	454	28	be	be	AUX
ejpam-4736	454	29	a	a	DET
ejpam-4736	454	30	δp(λ	δp(λ	NOUN
ejpam-4736	454	31	,	,	PUNCT
ejpam-4736	454	32	s)-closed	s)-close	VERB
ejpam-4736	454	33	set	set	NOUN
ejpam-4736	454	34	and	and	CCONJ
ejpam-4736	454	35	g	g	NOUN
ejpam-4736	454	36	be	be	AUX
ejpam-4736	454	37	a	a	DET
ejpam-4736	454	38	g	g	NOUN
ejpam-4736	454	39	-	-	PUNCT
ejpam-4736	454	40	δp(λ	δp(λ	NOUN
ejpam-4736	454	41	,	,	PUNCT
ejpam-4736	454	42	s)-open	s)-open	VERB
ejpam-4736	454	43	set	set	VERB
ejpam-4736	454	44	containing	contain	VERB
ejpam-4736	454	45	f	f	PROPN
ejpam-4736	454	46	.	.	PUNCT
ejpam-4736	455	1	thus	thus	ADV
ejpam-4736	455	2	,	,	PUNCT
ejpam-4736	455	3	by	by	ADP
ejpam-4736	455	4	theorem	theorem	NOUN
ejpam-4736	455	5	5	5	NUM
ejpam-4736	455	6	,	,	PUNCT
ejpam-4736	455	7	f	f	PROPN
ejpam-4736	455	8	δp(λ	δp(λ	PROPN
ejpam-4736	455	9	,	,	PUNCT
ejpam-4736	455	10	s	s	PART
ejpam-4736	455	11	)	)	PUNCT
ejpam-4736	455	12	=	=	SYM
ejpam-4736	455	13	f	f	PROPN
ejpam-4736	455	14	⊆	⊆	NUM
ejpam-4736	455	15	gδp(λ	gδp(λ	PROPN
ejpam-4736	455	16	,	,	PUNCT
ejpam-4736	455	17	s	s	NOUN
ejpam-4736	455	18	)	)	PUNCT
ejpam-4736	455	19	.	.	PUNCT
ejpam-4736	456	1	since	since	SCONJ
ejpam-4736	456	2	f	f	PROPN
ejpam-4736	456	3	is	be	AUX
ejpam-4736	456	4	g	g	NOUN
ejpam-4736	456	5	-	-	PUNCT
ejpam-4736	456	6	δp(λ	δp(λ	NOUN
ejpam-4736	456	7	,	,	PUNCT
ejpam-4736	456	8	s)-closed	s)-close	VERB
ejpam-4736	456	9	and	and	CCONJ
ejpam-4736	456	10	gδp(λ	gδp(λ	PROPN
ejpam-4736	456	11	,	,	PUNCT
ejpam-4736	456	12	s	s	PART
ejpam-4736	456	13	)	)	PUNCT
ejpam-4736	456	14	is	be	AUX
ejpam-4736	456	15	δp(λ	δp(λ	NOUN
ejpam-4736	456	16	,	,	PUNCT
ejpam-4736	456	17	s)-open	s)-open	VERB
ejpam-4736	456	18	,	,	PUNCT
ejpam-4736	456	19	by	by	ADP
ejpam-4736	456	20	(	(	PUNCT
ejpam-4736	456	21	6	6	NUM
ejpam-4736	456	22	)	)	PUNCT
ejpam-4736	456	23	,	,	PUNCT
ejpam-4736	456	24	there	there	PRON
ejpam-4736	456	25	exists	exist	VERB
ejpam-4736	456	26	a	a	DET
ejpam-4736	456	27	δp(λ	δp(λ	NOUN
ejpam-4736	456	28	,	,	PUNCT
ejpam-4736	456	29	s)-open	s)-open	PUNCT
ejpam-4736	456	30	set	set	VERB
ejpam-4736	456	31	u	u	PRON
ejpam-4736	456	32	such	such	ADJ
ejpam-4736	456	33	that	that	SCONJ
ejpam-4736	456	34	f	f	PROPN
ejpam-4736	456	35	δp(λ	δp(λ	PROPN
ejpam-4736	456	36	,	,	PUNCT
ejpam-4736	456	37	s	s	PART
ejpam-4736	456	38	)	)	PUNCT
ejpam-4736	456	39	⊆	⊆	NUM
ejpam-4736	456	40	u	u	NOUN
ejpam-4736	456	41	⊆	⊆	NUM
ejpam-4736	456	42	u	u	NOUN
ejpam-4736	456	43	δp(λ	δp(λ	NOUN
ejpam-4736	456	44	,	,	PUNCT
ejpam-4736	456	45	s	s	PART
ejpam-4736	456	46	)	)	PUNCT
ejpam-4736	456	47	⊆	⊆	NUM
ejpam-4736	456	48	gδp(λ	gδp(λ	PROPN
ejpam-4736	456	49	,	,	PUNCT
ejpam-4736	456	50	s	s	NOUN
ejpam-4736	456	51	)	)	PUNCT
ejpam-4736	456	52	.	.	PUNCT
ejpam-4736	457	1	(	(	PUNCT
ejpam-4736	457	2	4	4	X
ejpam-4736	457	3	)	)	PUNCT
ejpam-4736	457	4	⇒	⇒	NOUN
ejpam-4736	457	5	(	(	PUNCT
ejpam-4736	457	6	5	5	NUM
ejpam-4736	457	7	):	):	PUNCT
ejpam-4736	457	8	the	the	DET
ejpam-4736	457	9	proof	proof	NOUN
ejpam-4736	457	10	is	be	AUX
ejpam-4736	457	11	obvious	obvious	ADJ
ejpam-4736	457	12	.	.	PUNCT
ejpam-4736	458	1	(	(	PUNCT
ejpam-4736	458	2	6	6	NUM
ejpam-4736	458	3	)	)	PUNCT
ejpam-4736	458	4	⇒	⇒	NOUN
ejpam-4736	458	5	(	(	PUNCT
ejpam-4736	458	6	7	7	NUM
ejpam-4736	458	7	)	)	PUNCT
ejpam-4736	458	8	and	and	CCONJ
ejpam-4736	458	9	(	(	PUNCT
ejpam-4736	458	10	7	7	X
ejpam-4736	458	11	)	)	PUNCT
ejpam-4736	458	12	⇒	⇒	NOUN
ejpam-4736	458	13	(	(	PUNCT
ejpam-4736	458	14	3	3	NUM
ejpam-4736	458	15	):	):	PUNCT
ejpam-4736	458	16	the	the	DET
ejpam-4736	458	17	proofs	proof	NOUN
ejpam-4736	458	18	are	be	AUX
ejpam-4736	458	19	obvious	obvious	ADJ
ejpam-4736	458	20	.	.	PUNCT
ejpam-4736	459	1	(	(	PUNCT
ejpam-4736	459	2	3	3	X
ejpam-4736	459	3	)	)	PUNCT
ejpam-4736	459	4	⇒	⇒	NOUN
ejpam-4736	459	5	(	(	PUNCT
ejpam-4736	459	6	1	1	NUM
ejpam-4736	459	7	):	):	PUNCT
ejpam-4736	459	8	let	let	VERB
ejpam-4736	459	9	f	f	PROPN
ejpam-4736	459	10	and	and	CCONJ
ejpam-4736	459	11	h	h	PROPN
ejpam-4736	459	12	be	be	VERB
ejpam-4736	459	13	two	two	NUM
ejpam-4736	459	14	disjoint	disjoint	NOUN
ejpam-4736	459	15	δp(λ	δp(λ	NOUN
ejpam-4736	459	16	,	,	PUNCT
ejpam-4736	459	17	s)-closed	s)-close	VERB
ejpam-4736	459	18	sets	set	NOUN
ejpam-4736	459	19	of	of	ADP
ejpam-4736	459	20	x.	x.	NOUN
ejpam-4736	459	21	then	then	ADV
ejpam-4736	459	22	,	,	PUNCT
ejpam-4736	459	23	f	f	PROPN
ejpam-4736	459	24	is	be	AUX
ejpam-4736	459	25	a	a	DET
ejpam-4736	459	26	δp(λ	δp(λ	NOUN
ejpam-4736	459	27	,	,	PUNCT
ejpam-4736	459	28	s)-closed	s)-close	VERB
ejpam-4736	459	29	set	set	NOUN
ejpam-4736	459	30	and	and	CCONJ
ejpam-4736	459	31	x	x	AUX
ejpam-4736	459	32	−	−	NOUN
ejpam-4736	459	33	h	h	NOUN
ejpam-4736	459	34	is	be	AUX
ejpam-4736	459	35	a	a	DET
ejpam-4736	459	36	δp(λ	δp(λ	NOUN
ejpam-4736	459	37	,	,	PUNCT
ejpam-4736	459	38	s)-open	s)-open	PUNCT
ejpam-4736	459	39	set	set	VERB
ejpam-4736	459	40	containing	contain	VERB
ejpam-4736	459	41	f	f	X
ejpam-4736	459	42	,	,	PUNCT
ejpam-4736	459	43	by	by	ADP
ejpam-4736	459	44	(	(	PUNCT
ejpam-4736	459	45	3	3	NUM
ejpam-4736	459	46	)	)	PUNCT
ejpam-4736	459	47	,	,	PUNCT
ejpam-4736	459	48	there	there	PRON
ejpam-4736	459	49	exists	exist	VERB
ejpam-4736	459	50	a	a	DET
ejpam-4736	459	51	g	g	NOUN
ejpam-4736	459	52	-	-	PUNCT
ejpam-4736	459	53	δp(λ	δp(λ	NOUN
ejpam-4736	459	54	,	,	PUNCT
ejpam-4736	459	55	s)-open	s)-open	PUNCT
ejpam-4736	459	56	set	set	VERB
ejpam-4736	459	57	u	u	PRON
ejpam-4736	459	58	such	such	ADJ
ejpam-4736	459	59	that	that	SCONJ
ejpam-4736	459	60	f	f	PROPN
ejpam-4736	459	61	⊆	⊆	NUM
ejpam-4736	459	62	u	u	NOUN
ejpam-4736	459	63	⊆	⊆	NUM
ejpam-4736	459	64	u	u	NOUN
ejpam-4736	459	65	δp(λ	δp(λ	NOUN
ejpam-4736	459	66	,	,	PUNCT
ejpam-4736	459	67	s	s	PART
ejpam-4736	459	68	)	)	PUNCT
ejpam-4736	459	69	⊆	⊆	NUM
ejpam-4736	459	70	x	x	SYM
ejpam-4736	459	71	−	−	PROPN
ejpam-4736	459	72	h.	h.	PROPN
ejpam-4736	459	73	thus	thus	ADV
ejpam-4736	459	74	,	,	PUNCT
ejpam-4736	459	75	by	by	ADP
ejpam-4736	459	76	theorem	theorem	NOUN
ejpam-4736	459	77	5	5	NUM
ejpam-4736	459	78	,	,	PUNCT
ejpam-4736	459	79	f	f	PROPN
ejpam-4736	459	80	⊆	⊆	NUM
ejpam-4736	459	81	uδp(λ	uδp(λ	PROPN
ejpam-4736	459	82	,	,	PUNCT
ejpam-4736	459	83	s	s	PART
ejpam-4736	459	84	)	)	PUNCT
ejpam-4736	459	85	,	,	PUNCT
ejpam-4736	459	86	h	h	NOUN
ejpam-4736	459	87	⊆	⊆	NUM
ejpam-4736	459	88	x	x	X
ejpam-4736	459	89	−u	−u	PROPN
ejpam-4736	459	90	δp(λ	δp(λ	NOUN
ejpam-4736	459	91	,	,	PUNCT
ejpam-4736	459	92	s	s	PART
ejpam-4736	459	93	)	)	PUNCT
ejpam-4736	459	94	,	,	PUNCT
ejpam-4736	460	1	where	where	SCONJ
ejpam-4736	460	2	uδp(λ	uδp(λ	PROPN
ejpam-4736	460	3	,	,	PUNCT
ejpam-4736	460	4	s	s	PART
ejpam-4736	460	5	)	)	PUNCT
ejpam-4736	460	6	and	and	CCONJ
ejpam-4736	460	7	x	x	X
ejpam-4736	460	8	−u	−u	PROPN
ejpam-4736	460	9	δp(λ	δp(λ	NOUN
ejpam-4736	460	10	,	,	PUNCT
ejpam-4736	460	11	s	s	PART
ejpam-4736	460	12	)	)	PUNCT
ejpam-4736	460	13	are	be	AUX
ejpam-4736	460	14	two	two	NUM
ejpam-4736	460	15	disjoint	disjoint	ADJ
ejpam-4736	460	16	δp(λ	δp(λ	NOUN
ejpam-4736	460	17	,	,	PUNCT
ejpam-4736	460	18	s)open	s)open	ADJ
ejpam-4736	460	19	sets	set	NOUN
ejpam-4736	460	20	.	.	PUNCT
ejpam-4736	461	1	this	this	PRON
ejpam-4736	461	2	shows	show	VERB
ejpam-4736	461	3	that	that	SCONJ
ejpam-4736	461	4	(	(	PUNCT
ejpam-4736	461	5	x	x	X
ejpam-4736	461	6	,	,	PUNCT
ejpam-4736	461	7	τ	τ	X
ejpam-4736	461	8	)	)	PUNCT
ejpam-4736	461	9	is	be	AUX
ejpam-4736	461	10	δp(λ	δp(λ	NOUN
ejpam-4736	461	11	,	,	PUNCT
ejpam-4736	461	12	s)-normal	s)-normal	ADJ
ejpam-4736	461	13	.	.	PUNCT
ejpam-4736	462	1	acknowledgements	acknowledgement	NOUN
ejpam-4736	462	2	this	this	DET
ejpam-4736	462	3	research	research	NOUN
ejpam-4736	462	4	project	project	NOUN
ejpam-4736	462	5	was	be	AUX
ejpam-4736	462	6	financially	financially	ADV
ejpam-4736	462	7	supported	support	VERB
ejpam-4736	462	8	by	by	ADP
ejpam-4736	462	9	mahasarakham	mahasarakham	PROPN
ejpam-4736	462	10	university	university	PROPN
ejpam-4736	462	11	.	.	PUNCT
ejpam-4736	463	1	references	reference	NOUN
ejpam-4736	463	2	[	[	X
ejpam-4736	463	3	1	1	NUM
ejpam-4736	463	4	]	]	PUNCT
ejpam-4736	463	5	d.	d.	PROPN
ejpam-4736	463	6	andrijević.	andrijević.	PROPN
ejpam-4736	463	7	on	on	ADP
ejpam-4736	463	8	b	b	X
ejpam-4736	463	9	-	-	PUNCT
ejpam-4736	463	10	open	open	ADJ
ejpam-4736	463	11	sets	set	NOUN
ejpam-4736	463	12	.	.	PUNCT
ejpam-4736	464	1	matematički	matematički	PROPN
ejpam-4736	464	2	vesnik	vesnik	PROPN
ejpam-4736	464	3	,	,	PUNCT
ejpam-4736	464	4	48:59–64	48:59–64	PROPN
ejpam-4736	464	5	,	,	PUNCT
ejpam-4736	464	6	1996	1996	NUM
ejpam-4736	464	7	.	.	PUNCT
ejpam-4736	465	1	[	[	X
ejpam-4736	465	2	2	2	NUM
ejpam-4736	465	3	]	]	PUNCT
ejpam-4736	465	4	s.	s.	PROPN
ejpam-4736	465	5	baudong	baudong	PROPN
ejpam-4736	465	6	,	,	PUNCT
ejpam-4736	465	7	c.	c.	PROPN
ejpam-4736	465	8	viriyapong	viriyapong	PROPN
ejpam-4736	465	9	,	,	PUNCT
ejpam-4736	465	10	and	and	CCONJ
ejpam-4736	465	11	c.	c.	PROPN
ejpam-4736	465	12	boonpok	boonpok	PROPN
ejpam-4736	465	13	.	.	PUNCT
ejpam-4736	466	1	on	on	ADP
ejpam-4736	466	2	generalized	generalized	ADJ
ejpam-4736	466	3	topology	topology	NOUN
ejpam-4736	466	4	and	and	CCONJ
ejpam-4736	466	5	minimal	minimal	ADJ
ejpam-4736	466	6	structure	structure	NOUN
ejpam-4736	466	7	spaces	space	NOUN
ejpam-4736	466	8	.	.	PUNCT
ejpam-4736	467	1	international	international	ADJ
ejpam-4736	467	2	journal	journal	PROPN
ejpam-4736	467	3	of	of	ADP
ejpam-4736	467	4	mathematical	mathematical	ADJ
ejpam-4736	467	5	analysis	analysis	NOUN
ejpam-4736	467	6	,	,	PUNCT
ejpam-4736	467	7	5(31):1507–1516	5(31):1507–1516	NUM
ejpam-4736	467	8	,	,	PUNCT
ejpam-4736	467	9	2011	2011	NUM
ejpam-4736	467	10	.	.	PUNCT
ejpam-4736	468	1	[	[	X
ejpam-4736	468	2	3	3	X
ejpam-4736	468	3	]	]	PUNCT
ejpam-4736	468	4	c.	c.	PROPN
ejpam-4736	468	5	boonpok	boonpok	PROPN
ejpam-4736	468	6	and	and	CCONJ
ejpam-4736	468	7	c.	c.	PROPN
ejpam-4736	468	8	viriyapong	viriyapong	PROPN
ejpam-4736	468	9	.	.	PUNCT
ejpam-4736	469	1	on	on	ADP
ejpam-4736	469	2	some	some	DET
ejpam-4736	469	3	forms	form	NOUN
ejpam-4736	469	4	of	of	ADP
ejpam-4736	469	5	closed	closed	ADJ
ejpam-4736	469	6	sets	set	NOUN
ejpam-4736	469	7	and	and	CCONJ
ejpam-4736	469	8	related	related	ADJ
ejpam-4736	469	9	topics	topic	NOUN
ejpam-4736	469	10	.	.	PUNCT
ejpam-4736	470	1	european	european	ADJ
ejpam-4736	470	2	journal	journal	PROPN
ejpam-4736	470	3	of	of	ADP
ejpam-4736	470	4	pure	pure	ADJ
ejpam-4736	470	5	and	and	CCONJ
ejpam-4736	470	6	applied	applied	ADJ
ejpam-4736	470	7	mathematics	mathematic	NOUN
ejpam-4736	470	8	,	,	PUNCT
ejpam-4736	470	9	16(1):336–362	16(1):336–362	NUM
ejpam-4736	470	10	,	,	PUNCT
ejpam-4736	470	11	2023	2023	NUM
ejpam-4736	470	12	.	.	PUNCT
ejpam-4736	471	1	references	reference	NOUN
ejpam-4736	471	2	2595	2595	NUM
ejpam-4736	471	3	[	[	X
ejpam-4736	471	4	4	4	NUM
ejpam-4736	471	5	]	]	PUNCT
ejpam-4736	471	6	m.	m.	NOUN
ejpam-4736	471	7	caldas	caldas	PROPN
ejpam-4736	471	8	and	and	CCONJ
ejpam-4736	471	9	j.	j.	PROPN
ejpam-4736	471	10	dontchev	dontchev	PROPN
ejpam-4736	471	11	.	.	PUNCT
ejpam-4736	472	1	g.λs	g.λs	ADJ
ejpam-4736	472	2	-	-	PUNCT
ejpam-4736	472	3	sets	set	NOUN
ejpam-4736	472	4	and	and	CCONJ
ejpam-4736	472	5	g.vs	g.vs	NOUN
ejpam-4736	472	6	-	-	PUNCT
ejpam-4736	472	7	sets	set	NOUN
ejpam-4736	472	8	.	.	PUNCT
ejpam-4736	473	1	arxiv	arxiv	NOUN
ejpam-4736	473	2	:	:	PUNCT
ejpam-4736	473	3	math/9810080v1	math/9810080v1	PROPN
ejpam-4736	474	1	[	[	X
ejpam-4736	474	2	math.gn	math.gn	X
ejpam-4736	474	3	]	]	X
ejpam-4736	474	4	,	,	PUNCT
ejpam-4736	474	5	1998	1998	NUM
ejpam-4736	474	6	.	.	PUNCT
ejpam-4736	475	1	[	[	X
ejpam-4736	475	2	5	5	NUM
ejpam-4736	475	3	]	]	PUNCT
ejpam-4736	475	4	m.	m.	NOUN
ejpam-4736	475	5	caldas	caldas	PROPN
ejpam-4736	475	6	,	,	PUNCT
ejpam-4736	475	7	t.	t.	NOUN
ejpam-4736	475	8	fukutake	fukutake	NOUN
ejpam-4736	475	9	,	,	PUNCT
ejpam-4736	475	10	s.	s.	PROPN
ejpam-4736	475	11	jafari	jafari	PROPN
ejpam-4736	475	12	,	,	PUNCT
ejpam-4736	475	13	and	and	CCONJ
ejpam-4736	475	14	t.	t.	PROPN
ejpam-4736	475	15	noiri	noiri	PROPN
ejpam-4736	475	16	.	.	PUNCT
ejpam-4736	476	1	some	some	DET
ejpam-4736	476	2	applications	application	NOUN
ejpam-4736	476	3	of	of	ADP
ejpam-4736	476	4	δ	δ	NOUN
ejpam-4736	476	5	-	-	PUNCT
ejpam-4736	476	6	preopen	preopen	ADJ
ejpam-4736	476	7	sets	set	NOUN
ejpam-4736	476	8	in	in	ADP
ejpam-4736	476	9	topological	topological	ADJ
ejpam-4736	476	10	spaces	space	NOUN
ejpam-4736	476	11	.	.	PUNCT
ejpam-4736	477	1	bulletin	bulletin	NOUN
ejpam-4736	477	2	of	of	ADP
ejpam-4736	477	3	the	the	DET
ejpam-4736	477	4	institute	institute	NOUN
ejpam-4736	477	5	of	of	ADP
ejpam-4736	477	6	mathematics	mathematics	PROPN
ejpam-4736	477	7	,	,	PUNCT
ejpam-4736	477	8	academia	academia	PROPN
ejpam-4736	477	9	sinica	sinica	PROPN
ejpam-4736	477	10	,	,	PUNCT
ejpam-4736	477	11	33(3):261–276	33(3):261–276	NOUN
ejpam-4736	477	12	,	,	PUNCT
ejpam-4736	477	13	2005	2005	NUM
ejpam-4736	477	14	.	.	PUNCT
ejpam-4736	478	1	[	[	X
ejpam-4736	478	2	6	6	NUM
ejpam-4736	478	3	]	]	PUNCT
ejpam-4736	478	4	m.	m.	NOUN
ejpam-4736	478	5	caldas	caldas	PROPN
ejpam-4736	478	6	,	,	PUNCT
ejpam-4736	478	7	m.	m.	NOUN
ejpam-4736	478	8	ganster	ganster	NOUN
ejpam-4736	478	9	,	,	PUNCT
ejpam-4736	478	10	d.	d.	PROPN
ejpam-4736	478	11	n.	n.	PROPN
ejpam-4736	478	12	georgiou	georgiou	PROPN
ejpam-4736	478	13	,	,	PUNCT
ejpam-4736	478	14	s.	s.	PROPN
ejpam-4736	478	15	jafari	jafari	PROPN
ejpam-4736	478	16	,	,	PUNCT
ejpam-4736	478	17	and	and	CCONJ
ejpam-4736	478	18	t.	t.	PROPN
ejpam-4736	478	19	noiri	noiri	PROPN
ejpam-4736	478	20	.	.	PUNCT
ejpam-4736	479	1	δ	δ	PROPN
ejpam-4736	479	2	-	-	PUNCT
ejpam-4736	479	3	semiopen	semiopen	ADJ
ejpam-4736	479	4	sets	set	NOUN
ejpam-4736	479	5	in	in	ADP
ejpam-4736	479	6	topological	topological	ADJ
ejpam-4736	479	7	spaces	space	NOUN
ejpam-4736	479	8	.	.	PUNCT
ejpam-4736	480	1	topology	topology	NOUN
ejpam-4736	480	2	proceedings	proceeding	NOUN
ejpam-4736	480	3	,	,	PUNCT
ejpam-4736	480	4	29(2):369–383	29(2):369–383	NUM
ejpam-4736	480	5	,	,	PUNCT
ejpam-4736	480	6	2005	2005	NUM
ejpam-4736	480	7	.	.	PUNCT
ejpam-4736	481	1	[	[	X
ejpam-4736	481	2	7	7	X
ejpam-4736	481	3	]	]	X
ejpam-4736	481	4	m.	m.	NOUN
ejpam-4736	481	5	caldas	caldas	PROPN
ejpam-4736	481	6	,	,	PUNCT
ejpam-4736	481	7	d.	d.	PROPN
ejpam-4736	481	8	n.	n.	PROPN
ejpam-4736	481	9	georgiou	georgiou	PROPN
ejpam-4736	481	10	,	,	PUNCT
ejpam-4736	481	11	s.	s.	PROPN
ejpam-4736	481	12	jafari	jafari	PROPN
ejpam-4736	481	13	,	,	PUNCT
ejpam-4736	481	14	and	and	CCONJ
ejpam-4736	481	15	t.	t.	PROPN
ejpam-4736	481	16	noiri	noiri	PROPN
ejpam-4736	481	17	.	.	PUNCT
ejpam-4736	482	1	more	more	ADV
ejpam-4736	482	2	on	on	ADP
ejpam-4736	482	3	δ	δ	PROPN
ejpam-4736	482	4	-	-	PUNCT
ejpam-4736	482	5	semiopen	semiopen	ADJ
ejpam-4736	482	6	sets	set	NOUN
ejpam-4736	482	7	.	.	PUNCT
ejpam-4736	483	1	note	note	VERB
ejpam-4736	483	2	di	di	PROPN
ejpam-4736	483	3	matematica	matematica	PROPN
ejpam-4736	483	4	,	,	PUNCT
ejpam-4736	483	5	22(2):1–14	22(2):1–14	PROPN
ejpam-4736	483	6	,	,	PUNCT
ejpam-4736	483	7	2003	2003	NUM
ejpam-4736	483	8	.	.	PUNCT
ejpam-4736	484	1	[	[	X
ejpam-4736	484	2	8	8	NUM
ejpam-4736	484	3	]	]	X
ejpam-4736	484	4	w.	w.	PROPN
ejpam-4736	484	5	dungthaisong	dungthaisong	PROPN
ejpam-4736	484	6	,	,	PUNCT
ejpam-4736	484	7	c.	c.	PROPN
ejpam-4736	484	8	boonpok	boonpok	PROPN
ejpam-4736	484	9	,	,	PUNCT
ejpam-4736	484	10	and	and	CCONJ
ejpam-4736	484	11	c.	c.	PROPN
ejpam-4736	484	12	viriyapong	viriyapong	PROPN
ejpam-4736	484	13	.	.	PUNCT
ejpam-4736	485	1	generalized	generalize	VERB
ejpam-4736	485	2	closed	close	VERB
ejpam-4736	485	3	sets	set	NOUN
ejpam-4736	485	4	in	in	ADP
ejpam-4736	485	5	bigeneralized	bigeneralize	VERB
ejpam-4736	485	6	topological	topological	ADJ
ejpam-4736	485	7	spaces	space	NOUN
ejpam-4736	485	8	.	.	PUNCT
ejpam-4736	486	1	international	international	ADJ
ejpam-4736	486	2	journal	journal	PROPN
ejpam-4736	486	3	of	of	ADP
ejpam-4736	486	4	mathematical	mathematical	ADJ
ejpam-4736	486	5	analysis	analysis	NOUN
ejpam-4736	486	6	,	,	PUNCT
ejpam-4736	486	7	5(24):1175–1184	5(24):1175–1184	NUM
ejpam-4736	486	8	,	,	PUNCT
ejpam-4736	486	9	2011	2011	NUM
ejpam-4736	486	10	.	.	PUNCT
ejpam-4736	487	1	[	[	X
ejpam-4736	487	2	9	9	NUM
ejpam-4736	487	3	]	]	X
ejpam-4736	487	4	w.	w.	PROPN
ejpam-4736	487	5	dunham	dunham	PROPN
ejpam-4736	487	6	and	and	CCONJ
ejpam-4736	487	7	n.	n.	PROPN
ejpam-4736	487	8	levine	levine	PROPN
ejpam-4736	487	9	.	.	PUNCT
ejpam-4736	488	1	further	further	ADJ
ejpam-4736	488	2	results	result	NOUN
ejpam-4736	488	3	on	on	ADP
ejpam-4736	488	4	generalized	generalized	ADJ
ejpam-4736	488	5	closed	closed	ADJ
ejpam-4736	488	6	sets	set	NOUN
ejpam-4736	488	7	.	.	PUNCT
ejpam-4736	489	1	kyungpook	kyungpook	PROPN
ejpam-4736	489	2	mathematical	mathematical	PROPN
ejpam-4736	489	3	journal	journal	PROPN
ejpam-4736	489	4	,	,	PUNCT
ejpam-4736	489	5	20:169–175	20:169–175	PROPN
ejpam-4736	489	6	,	,	PUNCT
ejpam-4736	489	7	1980	1980	NUM
ejpam-4736	489	8	.	.	PUNCT
ejpam-4736	490	1	[	[	X
ejpam-4736	490	2	10	10	NUM
ejpam-4736	490	3	]	]	X
ejpam-4736	490	4	n.	n.	PROPN
ejpam-4736	490	5	levine	levine	PROPN
ejpam-4736	490	6	.	.	PUNCT
ejpam-4736	491	1	semi	semi	ADJ
ejpam-4736	491	2	-	-	ADJ
ejpam-4736	491	3	open	open	ADJ
ejpam-4736	491	4	sets	set	NOUN
ejpam-4736	491	5	and	and	CCONJ
ejpam-4736	491	6	semi	semi	ADJ
ejpam-4736	491	7	-	-	NOUN
ejpam-4736	491	8	continuity	continuity	NOUN
ejpam-4736	491	9	in	in	ADP
ejpam-4736	491	10	topological	topological	ADJ
ejpam-4736	491	11	spaces	space	NOUN
ejpam-4736	491	12	.	.	PUNCT
ejpam-4736	492	1	the	the	DET
ejpam-4736	492	2	american	american	PROPN
ejpam-4736	492	3	mathematical	mathematical	PROPN
ejpam-4736	492	4	monthly	monthly	ADV
ejpam-4736	492	5	,	,	PUNCT
ejpam-4736	492	6	70:36–41	70:36–41	NUM
ejpam-4736	492	7	,	,	PUNCT
ejpam-4736	492	8	1963	1963	NUM
ejpam-4736	492	9	.	.	PUNCT
ejpam-4736	493	1	[	[	X
ejpam-4736	493	2	11	11	NUM
ejpam-4736	493	3	]	]	X
ejpam-4736	493	4	n.	n.	PROPN
ejpam-4736	493	5	levine	levine	PROPN
ejpam-4736	493	6	.	.	PUNCT
ejpam-4736	494	1	generalized	generalize	VERB
ejpam-4736	494	2	closed	closed	ADJ
ejpam-4736	494	3	sets	set	NOUN
ejpam-4736	494	4	in	in	ADP
ejpam-4736	494	5	topology	topology	NOUN
ejpam-4736	494	6	.	.	PUNCT
ejpam-4736	495	1	rendiconti	rendiconti	VERB
ejpam-4736	495	2	del	del	PROPN
ejpam-4736	495	3	circolo	circolo	PROPN
ejpam-4736	495	4	matematico	matematico	NOUN
ejpam-4736	495	5	di	di	PROPN
ejpam-4736	495	6	palermo	palermo	PROPN
ejpam-4736	495	7	series	series	PROPN
ejpam-4736	495	8	2	2	NUM
ejpam-4736	495	9	,	,	PUNCT
ejpam-4736	495	10	19:89–96	19:89–96	NUM
ejpam-4736	495	11	,	,	PUNCT
ejpam-4736	495	12	1970	1970	NUM
ejpam-4736	495	13	.	.	PUNCT
ejpam-4736	496	1	[	[	X
ejpam-4736	496	2	12	12	NUM
ejpam-4736	496	3	]	]	PUNCT
ejpam-4736	496	4	a.	a.	NOUN
ejpam-4736	496	5	s.	s.	PROPN
ejpam-4736	496	6	mashhour	mashhour	PROPN
ejpam-4736	496	7	,	,	PUNCT
ejpam-4736	496	8	m.	m.	PROPN
ejpam-4736	496	9	e.	e.	PROPN
ejpam-4736	496	10	abd	abd	PROPN
ejpam-4736	496	11	el	el	PROPN
ejpam-4736	496	12	-	-	PROPN
ejpam-4736	496	13	monsef	monsef	ADJ
ejpam-4736	496	14	,	,	PUNCT
ejpam-4736	496	15	and	and	CCONJ
ejpam-4736	496	16	s.	s.	PROPN
ejpam-4736	496	17	n.	n.	PROPN
ejpam-4736	496	18	el	el	PROPN
ejpam-4736	496	19	-	-	PROPN
ejpam-4736	496	20	deeb	deeb	PROPN
ejpam-4736	496	21	.	.	PUNCT
ejpam-4736	497	1	on	on	ADP
ejpam-4736	497	2	precontinuous	precontinuous	ADJ
ejpam-4736	497	3	and	and	CCONJ
ejpam-4736	497	4	weak	weak	ADJ
ejpam-4736	497	5	precontinuous	precontinuous	ADJ
ejpam-4736	497	6	functions	function	NOUN
ejpam-4736	497	7	.	.	PUNCT
ejpam-4736	498	1	proceedings	proceeding	NOUN
ejpam-4736	498	2	of	of	ADP
ejpam-4736	498	3	the	the	DET
ejpam-4736	498	4	mathematical	mathematical	ADJ
ejpam-4736	498	5	and	and	CCONJ
ejpam-4736	498	6	physical	physical	ADJ
ejpam-4736	498	7	society	society	NOUN
ejpam-4736	498	8	of	of	ADP
ejpam-4736	498	9	egypt	egypt	PROPN
ejpam-4736	498	10	,	,	PUNCT
ejpam-4736	498	11	53:47–53	53:47–53	NUM
ejpam-4736	498	12	,	,	PUNCT
ejpam-4736	498	13	1982	1982	NUM
ejpam-4736	498	14	.	.	PUNCT
ejpam-4736	499	1	[	[	X
ejpam-4736	499	2	13	13	NUM
ejpam-4736	499	3	]	]	X
ejpam-4736	499	4	o.	o.	NOUN
ejpam-4736	499	5	nj̊astad	nj̊astad	NOUN
ejpam-4736	499	6	.	.	PUNCT
ejpam-4736	500	1	on	on	ADP
ejpam-4736	500	2	some	some	DET
ejpam-4736	500	3	classes	class	NOUN
ejpam-4736	500	4	of	of	ADP
ejpam-4736	500	5	nearly	nearly	ADV
ejpam-4736	500	6	open	open	ADJ
ejpam-4736	500	7	sets	set	NOUN
ejpam-4736	500	8	.	.	PUNCT
ejpam-4736	501	1	pasific	pasific	PROPN
ejpam-4736	501	2	journal	journal	PROPN
ejpam-4736	501	3	of	of	ADP
ejpam-4736	501	4	mathematics	mathematic	NOUN
ejpam-4736	501	5	,	,	PUNCT
ejpam-4736	501	6	15:961–970	15:961–970	PROPN
ejpam-4736	501	7	,	,	PUNCT
ejpam-4736	501	8	1965	1965	NUM
ejpam-4736	501	9	.	.	PUNCT
ejpam-4736	502	1	[	[	X
ejpam-4736	502	2	14	14	NUM
ejpam-4736	502	3	]	]	PUNCT
ejpam-4736	502	4	j.	j.	PROPN
ejpam-4736	502	5	h.	h.	PROPN
ejpam-4736	502	6	park	park	PROPN
ejpam-4736	502	7	,	,	PUNCT
ejpam-4736	502	8	b.	b.	PROPN
ejpam-4736	502	9	y.	y.	PROPN
ejpam-4736	502	10	lee	lee	PROPN
ejpam-4736	502	11	,	,	PUNCT
ejpam-4736	502	12	and	and	CCONJ
ejpam-4736	502	13	m.	m.	PROPN
ejpam-4736	502	14	j.	j.	PROPN
ejpam-4736	502	15	son	son	PROPN
ejpam-4736	502	16	.	.	PUNCT
ejpam-4736	503	1	on	on	ADP
ejpam-4736	503	2	δ	δ	PROPN
ejpam-4736	503	3	-	-	PUNCT
ejpam-4736	503	4	semiopen	semiopen	ADJ
ejpam-4736	503	5	sets	set	NOUN
ejpam-4736	503	6	in	in	ADP
ejpam-4736	503	7	topological	topological	ADJ
ejpam-4736	503	8	spaces	space	NOUN
ejpam-4736	503	9	.	.	PUNCT
ejpam-4736	504	1	the	the	DET
ejpam-4736	504	2	journal	journal	NOUN
ejpam-4736	504	3	of	of	ADP
ejpam-4736	504	4	the	the	DET
ejpam-4736	504	5	indian	indian	PROPN
ejpam-4736	504	6	academy	academy	PROPN
ejpam-4736	504	7	of	of	ADP
ejpam-4736	504	8	mathematics	mathematic	NOUN
ejpam-4736	504	9	,	,	PUNCT
ejpam-4736	504	10	19:59–67	19:59–67	NUM
ejpam-4736	504	11	,	,	PUNCT
ejpam-4736	504	12	1997	1997	NUM
ejpam-4736	504	13	.	.	PUNCT
ejpam-4736	505	1	[	[	X
ejpam-4736	505	2	15	15	NUM
ejpam-4736	505	3	]	]	X
ejpam-4736	505	4	s.	s.	PROPN
ejpam-4736	505	5	raychaudhuri	raychaudhuri	PROPN
ejpam-4736	505	6	and	and	CCONJ
ejpam-4736	505	7	m.	m.	PROPN
ejpam-4736	505	8	n.	n.	PROPN
ejpam-4736	505	9	mukherjee	mukherjee	PROPN
ejpam-4736	505	10	.	.	PUNCT
ejpam-4736	506	1	on	on	ADP
ejpam-4736	506	2	δ	δ	PROPN
ejpam-4736	506	3	-	-	PUNCT
ejpam-4736	506	4	almost	almost	ADV
ejpam-4736	506	5	continuity	continuity	NOUN
ejpam-4736	506	6	and	and	CCONJ
ejpam-4736	506	7	δ	δ	NOUN
ejpam-4736	506	8	-	-	PUNCT
ejpam-4736	506	9	preopen	preopen	ADJ
ejpam-4736	506	10	sets	set	NOUN
ejpam-4736	506	11	.	.	PUNCT
ejpam-4736	507	1	bulletin	bulletin	NOUN
ejpam-4736	507	2	of	of	ADP
ejpam-4736	507	3	the	the	DET
ejpam-4736	507	4	institute	institute	NOUN
ejpam-4736	507	5	of	of	ADP
ejpam-4736	507	6	mathematics	mathematics	PROPN
ejpam-4736	507	7	,	,	PUNCT
ejpam-4736	507	8	academia	academia	PROPN
ejpam-4736	507	9	sinica	sinica	PROPN
ejpam-4736	507	10	,	,	PUNCT
ejpam-4736	507	11	21:357–366	21:357–366	PROPN
ejpam-4736	507	12	,	,	PUNCT
ejpam-4736	507	13	1993	1993	NUM
ejpam-4736	507	14	.	.	PUNCT
ejpam-4736	508	1	[	[	X
ejpam-4736	508	2	16	16	NUM
ejpam-4736	508	3	]	]	X
ejpam-4736	508	4	n.	n.	PROPN
ejpam-4736	508	5	srisarakham	srisarakham	PROPN
ejpam-4736	508	6	and	and	CCONJ
ejpam-4736	508	7	c.	c.	PROPN
ejpam-4736	508	8	boonpok	boonpok	PROPN
ejpam-4736	508	9	.	.	PUNCT
ejpam-4736	509	1	on	on	ADP
ejpam-4736	509	2	characterizations	characterization	NOUN
ejpam-4736	509	3	of	of	ADP
ejpam-4736	509	4	δp(λ	δp(λ	NOUN
ejpam-4736	509	5	,	,	PUNCT
ejpam-4736	509	6	s)-d1	s)-d1	NOUN
ejpam-4736	509	7	spaces	space	NOUN
ejpam-4736	509	8	.	.	PUNCT
ejpam-4736	510	1	international	international	ADJ
ejpam-4736	510	2	journal	journal	PROPN
ejpam-4736	510	3	of	of	ADP
ejpam-4736	510	4	mathematics	mathematic	NOUN
ejpam-4736	510	5	and	and	CCONJ
ejpam-4736	510	6	computer	computer	NOUN
ejpam-4736	510	7	science	science	NOUN
ejpam-4736	510	8	,	,	PUNCT
ejpam-4736	510	9	18(4):743–747	18(4):743–747	PROPN
ejpam-4736	510	10	,	,	PUNCT
ejpam-4736	510	11	2023	2023	NUM
ejpam-4736	510	12	.	.	PUNCT
ejpam-4736	511	1	[	[	X
ejpam-4736	511	2	17	17	NUM
ejpam-4736	511	3	]	]	PUNCT
ejpam-4736	511	4	m.	m.	NOUN
ejpam-4736	511	5	thongmoon	thongmoon	NOUN
ejpam-4736	511	6	and	and	CCONJ
ejpam-4736	511	7	c.	c.	PROPN
ejpam-4736	511	8	boonpok	boonpok	PROPN
ejpam-4736	511	9	.	.	PUNCT
ejpam-4736	512	1	sober	sober	ADJ
ejpam-4736	512	2	δp(λ	δp(λ	PROPN
ejpam-4736	512	3	,	,	PUNCT
ejpam-4736	512	4	s)-r0	s)-r0	PRON
ejpam-4736	512	5	spaces	space	VERB
ejpam-4736	512	6	.	.	PUNCT
ejpam-4736	513	1	international	international	ADJ
ejpam-4736	513	2	journal	journal	PROPN
ejpam-4736	513	3	of	of	ADP
ejpam-4736	513	4	mathematics	mathematic	NOUN
ejpam-4736	513	5	and	and	CCONJ
ejpam-4736	513	6	computer	computer	NOUN
ejpam-4736	513	7	science	science	NOUN
ejpam-4736	513	8	,	,	PUNCT
ejpam-4736	513	9	18(4):761–765	18(4):761–765	NUM
ejpam-4736	513	10	,	,	PUNCT
ejpam-4736	513	11	2023	2023	NUM
ejpam-4736	513	12	.	.	PUNCT
ejpam-4736	514	1	[	[	X
ejpam-4736	514	2	18	18	NUM
ejpam-4736	514	3	]	]	PUNCT
ejpam-4736	514	4	p.	p.	NOUN
ejpam-4736	514	5	torton	torton	PROPN
ejpam-4736	514	6	,	,	PUNCT
ejpam-4736	514	7	c.	c.	PROPN
ejpam-4736	514	8	viriyapong	viriyapong	PROPN
ejpam-4736	514	9	,	,	PUNCT
ejpam-4736	514	10	and	and	CCONJ
ejpam-4736	514	11	c.	c.	PROPN
ejpam-4736	514	12	boonpok	boonpok	PROPN
ejpam-4736	514	13	.	.	PUNCT
ejpam-4736	515	1	some	some	DET
ejpam-4736	515	2	separation	separation	NOUN
ejpam-4736	515	3	axioms	axiom	VERB
ejpam-4736	515	4	in	in	ADP
ejpam-4736	515	5	bigeneralized	bigeneralize	VERB
ejpam-4736	515	6	topological	topological	ADJ
ejpam-4736	515	7	spaces	space	NOUN
ejpam-4736	515	8	.	.	PUNCT
ejpam-4736	516	1	international	international	ADJ
ejpam-4736	516	2	journal	journal	PROPN
ejpam-4736	516	3	of	of	ADP
ejpam-4736	516	4	mathematical	mathematical	ADJ
ejpam-4736	516	5	analysis	analysis	NOUN
ejpam-4736	516	6	,	,	PUNCT
ejpam-4736	516	7	6(56):2789–2796	6(56):2789–2796	NOUN
ejpam-4736	516	8	,	,	PUNCT
ejpam-4736	516	9	2012	2012	NUM
ejpam-4736	516	10	.	.	PUNCT
ejpam-4736	517	1	references	reference	NOUN
ejpam-4736	517	2	2596	2596	NUM
ejpam-4736	517	3	[	[	X
ejpam-4736	517	4	19	19	NUM
ejpam-4736	517	5	]	]	X
ejpam-4736	517	6	n.	n.	NOUN
ejpam-4736	517	7	v.	v.	PROPN
ejpam-4736	517	8	veličko	veličko	PROPN
ejpam-4736	517	9	.	.	PUNCT
ejpam-4736	518	1	h	h	NOUN
ejpam-4736	518	2	-	-	PUNCT
ejpam-4736	518	3	closed	close	VERB
ejpam-4736	518	4	topological	topological	ADJ
ejpam-4736	518	5	spaces	space	NOUN
ejpam-4736	518	6	.	.	PUNCT
ejpam-4736	519	1	american	american	PROPN
ejpam-4736	519	2	mathematical	mathematical	ADJ
ejpam-4736	519	3	society	society	NOUN
ejpam-4736	519	4	translations	translation	NOUN
ejpam-4736	519	5	,	,	PUNCT
ejpam-4736	519	6	78(2):102–118	78(2):102–118	NUM
ejpam-4736	519	7	,	,	PUNCT
ejpam-4736	519	8	1968	1968	NUM
ejpam-4736	519	9	.	.	PUNCT
ejpam-4736	520	1	[	[	X
ejpam-4736	520	2	20	20	NUM
ejpam-4736	520	3	]	]	X
ejpam-4736	520	4	c.	c.	PROPN
ejpam-4736	520	5	viriyapong	viriyapong	PROPN
ejpam-4736	520	6	and	and	CCONJ
ejpam-4736	520	7	c.	c.	PROPN
ejpam-4736	520	8	boonpok	boonpok	PROPN
ejpam-4736	520	9	.	.	PUNCT
ejpam-4736	521	1	on	on	ADP
ejpam-4736	521	2	generalized	generalized	ADJ
ejpam-4736	521	3	(	(	PUNCT
ejpam-4736	521	4	λ	λ	PROPN
ejpam-4736	521	5	,	,	PUNCT
ejpam-4736	521	6	p)-closed	p)-close	VERB
ejpam-4736	521	7	sets	set	NOUN
ejpam-4736	521	8	.	.	PUNCT
ejpam-4736	522	1	international	international	ADJ
ejpam-4736	522	2	journal	journal	NOUN
ejpam-4736	522	3	of	of	ADP
ejpam-4736	522	4	mathematics	mathematic	NOUN
ejpam-4736	522	5	and	and	CCONJ
ejpam-4736	522	6	computer	computer	NOUN
ejpam-4736	522	7	science	science	NOUN
ejpam-4736	522	8	,	,	PUNCT
ejpam-4736	522	9	18(1):79–83	18(1):79–83	NUM
ejpam-4736	522	10	,	,	PUNCT
ejpam-4736	522	11	2023	2023	NUM
ejpam-4736	522	12	.	.	PUNCT