id	sid	tid	token	lemma	pos
ejpam-4276	1	1	european	european	PROPN
ejpam-4276	1	2	journal	journal	PROPN
ejpam-4276	1	3	of	of	ADP
ejpam-4276	1	4	pure	pure	ADJ
ejpam-4276	1	5	and	and	CCONJ
ejpam-4276	1	6	applied	apply	VERB
ejpam-4276	1	7	mathematics	mathematic	NOUN
ejpam-4276	1	8	vol	vol	NOUN
ejpam-4276	1	9	.	.	PROPN
ejpam-4276	2	1	15	15	NUM
ejpam-4276	2	2	,	,	PUNCT
ejpam-4276	2	3	no	no	INTJ
ejpam-4276	2	4	.	.	NOUN
ejpam-4276	2	5	2	2	NUM
ejpam-4276	2	6	,	,	PUNCT
ejpam-4276	2	7	2022	2022	NUM
ejpam-4276	2	8	,	,	PUNCT
ejpam-4276	2	9	572	572	NUM
ejpam-4276	2	10	-	-	SYM
ejpam-4276	2	11	588	588	NUM
ejpam-4276	2	12	issn	issn	PROPN
ejpam-4276	2	13	1307	1307	NUM
ejpam-4276	2	14	-	-	SYM
ejpam-4276	2	15	5543	5543	NUM
ejpam-4276	2	16	–	–	PUNCT
ejpam-4276	2	17	ejpam.com	ejpam.com	X
ejpam-4276	2	18	published	publish	VERB
ejpam-4276	2	19	by	by	ADP
ejpam-4276	2	20	new	new	PROPN
ejpam-4276	2	21	york	york	PROPN
ejpam-4276	2	22	business	business	PROPN
ejpam-4276	2	23	global	global	PROPN
ejpam-4276	2	24	(	(	PUNCT
ejpam-4276	2	25	λ	λ	PROPN
ejpam-4276	2	26	,	,	PUNCT
ejpam-4276	2	27	sp)-open	sp)-open	ADJ
ejpam-4276	2	28	sets	set	NOUN
ejpam-4276	2	29	in	in	ADP
ejpam-4276	2	30	topological	topological	ADJ
ejpam-4276	2	31	spaces	space	NOUN
ejpam-4276	2	32	chawalit	chawalit	VERB
ejpam-4276	2	33	boonpok1	boonpok1	PROPN
ejpam-4276	2	34	,	,	PUNCT
ejpam-4276	2	35	jeeranunt	jeeranunt	PROPN
ejpam-4276	2	36	khampakdee1,∗	khampakdee1,∗	PROPN
ejpam-4276	2	37	1	1	NUM
ejpam-4276	2	38	mathematics	mathematic	NOUN
ejpam-4276	2	39	and	and	CCONJ
ejpam-4276	2	40	applied	apply	VERB
ejpam-4276	2	41	mathematics	mathematics	PROPN
ejpam-4276	2	42	research	research	NOUN
ejpam-4276	2	43	unit	unit	NOUN
ejpam-4276	2	44	,	,	PUNCT
ejpam-4276	2	45	department	department	NOUN
ejpam-4276	2	46	of	of	ADP
ejpam-4276	2	47	mathematics	mathematic	NOUN
ejpam-4276	2	48	,	,	PUNCT
ejpam-4276	2	49	faculty	faculty	NOUN
ejpam-4276	2	50	of	of	ADP
ejpam-4276	2	51	science	science	NOUN
ejpam-4276	2	52	,	,	PUNCT
ejpam-4276	2	53	mahasarakham	mahasarakham	PROPN
ejpam-4276	2	54	university	university	PROPN
ejpam-4276	2	55	,	,	PUNCT
ejpam-4276	2	56	maha	maha	PROPN
ejpam-4276	2	57	sarakham	sarakham	PROPN
ejpam-4276	2	58	,	,	PUNCT
ejpam-4276	2	59	44150	44150	NUM
ejpam-4276	2	60	,	,	PUNCT
ejpam-4276	2	61	thailand	thailand	PROPN
ejpam-4276	2	62	abstract	abstract	PROPN
ejpam-4276	2	63	.	.	PUNCT
ejpam-4276	3	1	this	this	DET
ejpam-4276	3	2	paper	paper	NOUN
ejpam-4276	3	3	is	be	AUX
ejpam-4276	3	4	concerned	concern	VERB
ejpam-4276	3	5	with	with	ADP
ejpam-4276	3	6	the	the	DET
ejpam-4276	3	7	concepts	concept	NOUN
ejpam-4276	3	8	of	of	ADP
ejpam-4276	3	9	s(λ	s(λ	PROPN
ejpam-4276	3	10	,	,	PUNCT
ejpam-4276	3	11	sp)-open	sp)-open	ADJ
ejpam-4276	3	12	sets	set	NOUN
ejpam-4276	3	13	,	,	PUNCT
ejpam-4276	3	14	p(λ	p(λ	NOUN
ejpam-4276	3	15	,	,	PUNCT
ejpam-4276	3	16	sp)-open	sp)-open	ADJ
ejpam-4276	3	17	sets	set	NOUN
ejpam-4276	3	18	,	,	PUNCT
ejpam-4276	3	19	α(λ	α(λ	PROPN
ejpam-4276	3	20	,	,	PUNCT
ejpam-4276	3	21	sp)-open	sp)-open	ADJ
ejpam-4276	3	22	sets	set	NOUN
ejpam-4276	3	23	,	,	PUNCT
ejpam-4276	3	24	β(λ	β(λ	X
ejpam-4276	3	25	,	,	PUNCT
ejpam-4276	3	26	sp)-open	sp)-open	ADJ
ejpam-4276	3	27	sets	set	NOUN
ejpam-4276	3	28	and	and	CCONJ
ejpam-4276	3	29	b(λ	b(λ	NOUN
ejpam-4276	3	30	,	,	PUNCT
ejpam-4276	3	31	sp)-open	sp)-open	ADJ
ejpam-4276	3	32	sets	set	NOUN
ejpam-4276	3	33	.	.	PUNCT
ejpam-4276	4	1	some	some	DET
ejpam-4276	4	2	properties	property	NOUN
ejpam-4276	4	3	of	of	ADP
ejpam-4276	4	4	s(λ	s(λ	PROPN
ejpam-4276	4	5	,	,	PUNCT
ejpam-4276	4	6	sp)-open	sp)-open	ADJ
ejpam-4276	4	7	sets	set	NOUN
ejpam-4276	4	8	,	,	PUNCT
ejpam-4276	4	9	p(λ	p(λ	NOUN
ejpam-4276	4	10	,	,	PUNCT
ejpam-4276	4	11	sp)-open	sp)-open	ADJ
ejpam-4276	4	12	sets	set	NOUN
ejpam-4276	4	13	,	,	PUNCT
ejpam-4276	4	14	α(λ	α(λ	PROPN
ejpam-4276	4	15	,	,	PUNCT
ejpam-4276	4	16	sp)-open	sp)-open	ADJ
ejpam-4276	4	17	sets	set	NOUN
ejpam-4276	4	18	,	,	PUNCT
ejpam-4276	4	19	β(λ	β(λ	X
ejpam-4276	4	20	,	,	PUNCT
ejpam-4276	4	21	sp)-open	sp)-open	ADJ
ejpam-4276	4	22	sets	set	NOUN
ejpam-4276	4	23	and	and	CCONJ
ejpam-4276	4	24	b(λ	b(λ	NOUN
ejpam-4276	4	25	,	,	PUNCT
ejpam-4276	4	26	sp)-open	sp)-open	ADJ
ejpam-4276	4	27	sets	set	NOUN
ejpam-4276	4	28	are	be	AUX
ejpam-4276	4	29	discussed	discuss	VERB
ejpam-4276	4	30	.	.	PUNCT
ejpam-4276	5	1	in	in	ADP
ejpam-4276	5	2	particular	particular	ADJ
ejpam-4276	5	3	,	,	PUNCT
ejpam-4276	5	4	the	the	DET
ejpam-4276	5	5	relationships	relationship	NOUN
ejpam-4276	5	6	between	between	ADP
ejpam-4276	5	7	s(λ	s(λ	PROPN
ejpam-4276	5	8	,	,	PUNCT
ejpam-4276	5	9	sp)-open	sp)-open	ADJ
ejpam-4276	5	10	sets	set	NOUN
ejpam-4276	5	11	,	,	PUNCT
ejpam-4276	5	12	p(λ	p(λ	NOUN
ejpam-4276	5	13	,	,	PUNCT
ejpam-4276	5	14	sp)-open	sp)-open	ADJ
ejpam-4276	5	15	sets	set	NOUN
ejpam-4276	5	16	,	,	PUNCT
ejpam-4276	5	17	α(λ	α(λ	PROPN
ejpam-4276	5	18	,	,	PUNCT
ejpam-4276	5	19	sp)-open	sp)-open	ADJ
ejpam-4276	5	20	sets	set	NOUN
ejpam-4276	5	21	,	,	PUNCT
ejpam-4276	5	22	β(λ	β(λ	X
ejpam-4276	5	23	,	,	PUNCT
ejpam-4276	5	24	sp)-open	sp)-open	ADJ
ejpam-4276	5	25	sets	set	NOUN
ejpam-4276	5	26	,	,	PUNCT
ejpam-4276	5	27	b(λ	b(λ	PROPN
ejpam-4276	5	28	,	,	PUNCT
ejpam-4276	5	29	sp)-open	sp)-open	ADJ
ejpam-4276	5	30	sets	set	NOUN
ejpam-4276	5	31	and	and	CCONJ
ejpam-4276	5	32	other	other	ADJ
ejpam-4276	5	33	related	related	ADJ
ejpam-4276	5	34	sets	set	NOUN
ejpam-4276	5	35	are	be	AUX
ejpam-4276	5	36	established	establish	VERB
ejpam-4276	5	37	.	.	PUNCT
ejpam-4276	6	1	moreover	moreover	ADV
ejpam-4276	6	2	,	,	PUNCT
ejpam-4276	6	3	several	several	ADJ
ejpam-4276	6	4	characterizations	characterization	NOUN
ejpam-4276	6	5	of	of	ADP
ejpam-4276	6	6	λsp	λsp	NOUN
ejpam-4276	6	7	-	-	PUNCT
ejpam-4276	6	8	extremally	extremally	ADV
ejpam-4276	6	9	disconnected	disconnected	ADJ
ejpam-4276	6	10	spaces	space	NOUN
ejpam-4276	6	11	are	be	AUX
ejpam-4276	6	12	investigated	investigate	VERB
ejpam-4276	6	13	.	.	PUNCT
ejpam-4276	7	1	2020	2020	NUM
ejpam-4276	7	2	mathematics	mathematic	NOUN
ejpam-4276	7	3	subject	subject	NOUN
ejpam-4276	7	4	classifications	classification	NOUN
ejpam-4276	7	5	:	:	PUNCT
ejpam-4276	7	6	54a05	54a05	NUM
ejpam-4276	7	7	,	,	PUNCT
ejpam-4276	7	8	54g05	54g05	NUM
ejpam-4276	7	9	key	key	ADJ
ejpam-4276	7	10	words	word	NOUN
ejpam-4276	7	11	and	and	CCONJ
ejpam-4276	7	12	phrases	phrase	NOUN
ejpam-4276	7	13	:	:	PUNCT
ejpam-4276	7	14	(	(	PUNCT
ejpam-4276	7	15	λ	λ	X
ejpam-4276	7	16	,	,	PUNCT
ejpam-4276	7	17	sp)-closed	sp)-close	VERB
ejpam-4276	7	18	set	set	VERB
ejpam-4276	7	19	,	,	PUNCT
ejpam-4276	7	20	(	(	PUNCT
ejpam-4276	7	21	λ	λ	NOUN
ejpam-4276	7	22	,	,	PUNCT
ejpam-4276	7	23	sp)-open	sp)-open	ADJ
ejpam-4276	7	24	set	set	NOUN
ejpam-4276	7	25	,	,	PUNCT
ejpam-4276	7	26	λsp	λsp	PROPN
ejpam-4276	7	27	-	-	PUNCT
ejpam-4276	7	28	extremally	extremally	ADV
ejpam-4276	7	29	disconnected	disconnected	ADJ
ejpam-4276	7	30	space	space	NOUN
ejpam-4276	7	31	1	1	NUM
ejpam-4276	7	32	.	.	PUNCT
ejpam-4276	8	1	introduction	introduction	NOUN
ejpam-4276	8	2	semi	semi	ADJ
ejpam-4276	8	3	-	-	ADJ
ejpam-4276	8	4	open	open	ADJ
ejpam-4276	8	5	sets	set	NOUN
ejpam-4276	8	6	,	,	PUNCT
ejpam-4276	8	7	preopen	preopen	ADJ
ejpam-4276	8	8	sets	set	NOUN
ejpam-4276	8	9	,	,	PUNCT
ejpam-4276	8	10	α	α	NOUN
ejpam-4276	8	11	-	-	ADJ
ejpam-4276	8	12	open	open	ADJ
ejpam-4276	8	13	sets	set	NOUN
ejpam-4276	8	14	,	,	PUNCT
ejpam-4276	8	15	b	b	X
ejpam-4276	8	16	-	-	PUNCT
ejpam-4276	8	17	open	open	ADJ
ejpam-4276	8	18	sets	set	NOUN
ejpam-4276	8	19	and	and	CCONJ
ejpam-4276	8	20	β	β	NOUN
ejpam-4276	8	21	-	-	ADJ
ejpam-4276	8	22	open	open	ADJ
ejpam-4276	8	23	sets	set	NOUN
ejpam-4276	8	24	play	play	VERB
ejpam-4276	8	25	an	an	DET
ejpam-4276	8	26	important	important	ADJ
ejpam-4276	8	27	for	for	ADP
ejpam-4276	8	28	the	the	DET
ejpam-4276	8	29	study	study	NOUN
ejpam-4276	8	30	and	and	CCONJ
ejpam-4276	8	31	investigation	investigation	NOUN
ejpam-4276	8	32	in	in	ADP
ejpam-4276	8	33	topological	topological	ADJ
ejpam-4276	8	34	spaces	space	NOUN
ejpam-4276	8	35	.	.	PUNCT
ejpam-4276	9	1	in	in	ADP
ejpam-4276	9	2	1963	1963	NUM
ejpam-4276	9	3	,	,	PUNCT
ejpam-4276	9	4	levine	levine	PROPN
ejpam-4276	9	5	[	[	X
ejpam-4276	9	6	6	6	NUM
ejpam-4276	9	7	]	]	PUNCT
ejpam-4276	9	8	introduced	introduce	VERB
ejpam-4276	9	9	the	the	DET
ejpam-4276	9	10	concept	concept	NOUN
ejpam-4276	9	11	of	of	ADP
ejpam-4276	9	12	semi	semi	ADJ
ejpam-4276	9	13	-	-	ADJ
ejpam-4276	9	14	open	open	ADJ
ejpam-4276	9	15	sets	set	NOUN
ejpam-4276	9	16	in	in	ADP
ejpam-4276	9	17	topological	topological	ADJ
ejpam-4276	9	18	spaces	space	NOUN
ejpam-4276	9	19	.	.	PUNCT
ejpam-4276	10	1	after	after	ADP
ejpam-4276	10	2	the	the	DET
ejpam-4276	10	3	work	work	NOUN
ejpam-4276	10	4	of	of	ADP
ejpam-4276	10	5	levine	levine	PROPN
ejpam-4276	10	6	on	on	ADP
ejpam-4276	10	7	semiopen	semiopen	ADJ
ejpam-4276	10	8	sets	set	NOUN
ejpam-4276	10	9	,	,	PUNCT
ejpam-4276	10	10	several	several	ADJ
ejpam-4276	10	11	mathematicians	mathematician	NOUN
ejpam-4276	10	12	turned	turn	VERB
ejpam-4276	10	13	their	their	PRON
ejpam-4276	10	14	attention	attention	NOUN
ejpam-4276	10	15	to	to	ADP
ejpam-4276	10	16	the	the	DET
ejpam-4276	10	17	generalizations	generalization	NOUN
ejpam-4276	10	18	of	of	ADP
ejpam-4276	10	19	various	various	ADJ
ejpam-4276	10	20	concepts	concept	NOUN
ejpam-4276	10	21	of	of	ADP
ejpam-4276	10	22	topology	topology	NOUN
ejpam-4276	10	23	by	by	ADP
ejpam-4276	10	24	considering	consider	VERB
ejpam-4276	10	25	semi	semi	ADJ
ejpam-4276	10	26	-	-	ADJ
ejpam-4276	10	27	open	open	ADJ
ejpam-4276	10	28	sets	set	NOUN
ejpam-4276	10	29	instead	instead	ADV
ejpam-4276	10	30	of	of	ADP
ejpam-4276	10	31	open	open	ADJ
ejpam-4276	10	32	sets	set	NOUN
ejpam-4276	10	33	.	.	PUNCT
ejpam-4276	11	1	while	while	SCONJ
ejpam-4276	11	2	open	open	ADJ
ejpam-4276	11	3	sets	set	NOUN
ejpam-4276	11	4	are	be	AUX
ejpam-4276	11	5	replaced	replace	VERB
ejpam-4276	11	6	by	by	ADP
ejpam-4276	11	7	semi	semi	ADJ
ejpam-4276	11	8	-	-	ADJ
ejpam-4276	11	9	open	open	ADJ
ejpam-4276	11	10	sets	set	NOUN
ejpam-4276	11	11	,	,	PUNCT
ejpam-4276	11	12	new	new	ADJ
ejpam-4276	11	13	results	result	NOUN
ejpam-4276	11	14	are	be	AUX
ejpam-4276	11	15	obtained	obtain	VERB
ejpam-4276	11	16	in	in	ADP
ejpam-4276	11	17	some	some	DET
ejpam-4276	11	18	occasions	occasion	NOUN
ejpam-4276	11	19	and	and	CCONJ
ejpam-4276	11	20	in	in	ADP
ejpam-4276	11	21	other	other	ADJ
ejpam-4276	11	22	occasions	occasion	NOUN
ejpam-4276	11	23	substantial	substantial	ADJ
ejpam-4276	11	24	generalizations	generalization	NOUN
ejpam-4276	11	25	are	be	AUX
ejpam-4276	11	26	exhibited	exhibit	VERB
ejpam-4276	11	27	.	.	PUNCT
ejpam-4276	12	1	in	in	ADP
ejpam-4276	12	2	this	this	DET
ejpam-4276	12	3	direction	direction	NOUN
ejpam-4276	12	4	,	,	PUNCT
ejpam-4276	12	5	in	in	ADP
ejpam-4276	12	6	1975	1975	NUM
ejpam-4276	12	7	,	,	PUNCT
ejpam-4276	12	8	maheshwari	maheshwari	NOUN
ejpam-4276	12	9	and	and	CCONJ
ejpam-4276	12	10	prasad	prasad	PROPN
ejpam-4276	13	1	[	[	X
ejpam-4276	13	2	7	7	NUM
ejpam-4276	13	3	]	]	PUNCT
ejpam-4276	13	4	,	,	PUNCT
ejpam-4276	13	5	used	use	VERB
ejpam-4276	13	6	semi	semi	ADJ
ejpam-4276	13	7	-	-	ADJ
ejpam-4276	13	8	open	open	ADJ
ejpam-4276	13	9	sets	set	NOUN
ejpam-4276	13	10	to	to	PART
ejpam-4276	13	11	define	define	VERB
ejpam-4276	13	12	and	and	CCONJ
ejpam-4276	13	13	investigate	investigate	VERB
ejpam-4276	13	14	three	three	NUM
ejpam-4276	13	15	new	new	ADJ
ejpam-4276	13	16	separation	separation	NOUN
ejpam-4276	13	17	axioms	axiom	NOUN
ejpam-4276	13	18	called	call	VERB
ejpam-4276	13	19	semi	semi	ADJ
ejpam-4276	13	20	-	-	ADJ
ejpam-4276	13	21	t0	t0	ADJ
ejpam-4276	13	22	,	,	PUNCT
ejpam-4276	13	23	semi	semi	ADJ
ejpam-4276	13	24	-	-	NOUN
ejpam-4276	13	25	t1	t1	NOUN
ejpam-4276	13	26	and	and	CCONJ
ejpam-4276	13	27	semi	semi	ADJ
ejpam-4276	13	28	-	-	NOUN
ejpam-4276	13	29	t2	t2	NOUN
ejpam-4276	13	30	.	.	PUNCT
ejpam-4276	14	1	later	later	ADV
ejpam-4276	14	2	,	,	PUNCT
ejpam-4276	14	3	in	in	ADP
ejpam-4276	14	4	1987	1987	NUM
ejpam-4276	14	5	,	,	PUNCT
ejpam-4276	14	6	bhattacharya	bhattacharya	NOUN
ejpam-4276	14	7	and	and	CCONJ
ejpam-4276	14	8	lahiri	lahiri	PROPN
ejpam-4276	15	1	[	[	X
ejpam-4276	15	2	2	2	NUM
ejpam-4276	15	3	]	]	PUNCT
ejpam-4276	15	4	generalized	generalize	VERB
ejpam-4276	15	5	the	the	DET
ejpam-4276	15	6	concept	concept	NOUN
ejpam-4276	15	7	of	of	ADP
ejpam-4276	15	8	closed	closed	ADJ
ejpam-4276	15	9	sets	set	NOUN
ejpam-4276	15	10	to	to	ADP
ejpam-4276	15	11	semi	semi	ADJ
ejpam-4276	15	12	-	-	ADJ
ejpam-4276	15	13	generalized	generalized	ADJ
ejpam-4276	15	14	closed	closed	ADJ
ejpam-4276	15	15	sets	set	NOUN
ejpam-4276	15	16	with	with	ADP
ejpam-4276	15	17	the	the	DET
ejpam-4276	15	18	help	help	NOUN
ejpam-4276	15	19	of	of	ADP
ejpam-4276	15	20	semi	semi	ADJ
ejpam-4276	15	21	-	-	NOUN
ejpam-4276	15	22	openness	openness	NOUN
ejpam-4276	15	23	.	.	PUNCT
ejpam-4276	16	1	the	the	DET
ejpam-4276	16	2	notion	notion	NOUN
ejpam-4276	16	3	of	of	ADP
ejpam-4276	16	4	α	α	NOUN
ejpam-4276	16	5	-	-	ADJ
ejpam-4276	16	6	open	open	ADJ
ejpam-4276	16	7	sets	set	NOUN
ejpam-4276	16	8	(	(	PUNCT
ejpam-4276	16	9	originally	originally	ADV
ejpam-4276	16	10	called	call	VERB
ejpam-4276	16	11	α	α	NOUN
ejpam-4276	16	12	-	-	PUNCT
ejpam-4276	16	13	sets	set	NOUN
ejpam-4276	16	14	)	)	PUNCT
ejpam-4276	16	15	in	in	ADP
ejpam-4276	16	16	topological	topological	ADJ
ejpam-4276	16	17	spaces	space	NOUN
ejpam-4276	16	18	was	be	AUX
ejpam-4276	16	19	introduced	introduce	VERB
ejpam-4276	16	20	by	by	ADP
ejpam-4276	16	21	njåstad	njåstad	NOUN
ejpam-4276	16	22	[	[	X
ejpam-4276	16	23	10	10	NUM
ejpam-4276	16	24	]	]	PUNCT
ejpam-4276	16	25	in	in	ADP
ejpam-4276	16	26	1965	1965	NUM
ejpam-4276	16	27	.	.	PUNCT
ejpam-4276	17	1	by	by	ADP
ejpam-4276	17	2	using	use	VERB
ejpam-4276	17	3	α	α	NUM
ejpam-4276	17	4	-	-	ADJ
ejpam-4276	17	5	open	open	ADJ
ejpam-4276	17	6	sets	set	NOUN
ejpam-4276	17	7	,	,	PUNCT
ejpam-4276	17	8	mashhour	mashhour	PROPN
ejpam-4276	17	9	et	et	NOUN
ejpam-4276	17	10	al	al	PROPN
ejpam-4276	17	11	.	.	PUNCT
ejpam-4276	18	1	[	[	X
ejpam-4276	18	2	9	9	NUM
ejpam-4276	18	3	]	]	PUNCT
ejpam-4276	18	4	defined	define	VERB
ejpam-4276	18	5	and	and	CCONJ
ejpam-4276	18	6	studied	study	VERB
ejpam-4276	18	7	the	the	DET
ejpam-4276	18	8	notions	notion	NOUN
ejpam-4276	18	9	of	of	ADP
ejpam-4276	18	10	α	α	NOUN
ejpam-4276	18	11	-	-	PUNCT
ejpam-4276	18	12	continuity	continuity	NOUN
ejpam-4276	18	13	and	and	CCONJ
ejpam-4276	18	14	α	α	NOUN
ejpam-4276	18	15	-	-	NOUN
ejpam-4276	18	16	openness	openness	NOUN
ejpam-4276	18	17	in	in	ADP
ejpam-4276	18	18	topological	topological	ADJ
ejpam-4276	18	19	spaces	space	NOUN
ejpam-4276	18	20	.	.	PUNCT
ejpam-4276	19	1	in	in	ADP
ejpam-4276	19	2	1982	1982	NUM
ejpam-4276	19	3	,	,	PUNCT
ejpam-4276	19	4	mashhour	mashhour	PROPN
ejpam-4276	19	5	et	et	PROPN
ejpam-4276	19	6	al	al	PROPN
ejpam-4276	19	7	.	.	PUNCT
ejpam-4276	20	1	[	[	X
ejpam-4276	20	2	8	8	NUM
ejpam-4276	20	3	]	]	PUNCT
ejpam-4276	20	4	introduced	introduce	VERB
ejpam-4276	20	5	and	and	CCONJ
ejpam-4276	20	6	investigated	investigate	VERB
ejpam-4276	20	7	the	the	DET
ejpam-4276	20	8	concepts	concept	NOUN
ejpam-4276	20	9	of	of	ADP
ejpam-4276	20	10	preopen	preopen	ADJ
ejpam-4276	20	11	sets	set	NOUN
ejpam-4276	20	12	and	and	CCONJ
ejpam-4276	20	13	precontinuous	precontinuous	ADJ
ejpam-4276	20	14	functions	function	NOUN
ejpam-4276	20	15	in	in	ADP
ejpam-4276	20	16	topological	topological	ADJ
ejpam-4276	20	17	spaces	space	NOUN
ejpam-4276	20	18	.	.	PUNCT
ejpam-4276	21	1	in	in	ADP
ejpam-4276	21	2	1983	1983	NUM
ejpam-4276	21	3	,	,	PUNCT
ejpam-4276	21	4	abd	abd	PROPN
ejpam-4276	21	5	el	el	PROPN
ejpam-4276	21	6	-	-	PROPN
ejpam-4276	21	7	monsef	monsef	PROPN
ejpam-4276	21	8	et	et	PROPN
ejpam-4276	21	9	al	al	PROPN
ejpam-4276	21	10	.	.	PUNCT
ejpam-4276	22	1	[	[	X
ejpam-4276	22	2	4	4	X
ejpam-4276	22	3	]	]	PUNCT
ejpam-4276	22	4	introduced	introduce	VERB
ejpam-4276	22	5	a	a	DET
ejpam-4276	22	6	weak	weak	ADJ
ejpam-4276	22	7	form	form	NOUN
ejpam-4276	22	8	of	of	ADP
ejpam-4276	22	9	open	open	ADJ
ejpam-4276	22	10	sets	set	NOUN
ejpam-4276	22	11	called	call	VERB
ejpam-4276	22	12	β	β	NOUN
ejpam-4276	22	13	-	-	ADJ
ejpam-4276	22	14	open	open	ADJ
ejpam-4276	22	15	sets	set	NOUN
ejpam-4276	22	16	.	.	PUNCT
ejpam-4276	23	1	the	the	DET
ejpam-4276	23	2	concept	concept	NOUN
ejpam-4276	23	3	of	of	ADP
ejpam-4276	23	4	β	β	ADJ
ejpam-4276	23	5	-	-	ADJ
ejpam-4276	23	6	open	open	ADJ
ejpam-4276	23	7	sets	set	NOUN
ejpam-4276	23	8	is	be	AUX
ejpam-4276	23	9	equivalent	equivalent	ADJ
ejpam-4276	23	10	to	to	ADP
ejpam-4276	23	11	that	that	DET
ejpam-4276	23	12	∗corresponding	∗corresponde	VERB
ejpam-4276	23	13	author	author	NOUN
ejpam-4276	23	14	.	.	PUNCT
ejpam-4276	24	1	doi	doi	NOUN
ejpam-4276	24	2	:	:	PUNCT
ejpam-4276	24	3	https://doi.org/10.29020/nybg.ejpam.v15i2.4276	https://doi.org/10.29020/nybg.ejpam.v15i2.4276	NUM
ejpam-4276	24	4	email	email	NOUN
ejpam-4276	24	5	addresses	address	NOUN
ejpam-4276	24	6	:	:	PUNCT
ejpam-4276	25	1	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4276	25	2	(	(	PUNCT
ejpam-4276	25	3	c.	c.	PROPN
ejpam-4276	25	4	boonpok	boonpok	PROPN
ejpam-4276	25	5	)	)	PUNCT
ejpam-4276	25	6	,	,	PUNCT
ejpam-4276	25	7	jeeranunt.k@msu.ac.th	jeeranunt.k@msu.ac.th	INTJ
ejpam-4276	25	8	(	(	PUNCT
ejpam-4276	25	9	j.	j.	PROPN
ejpam-4276	25	10	khampakdee	khampakdee	PROPN
ejpam-4276	25	11	)	)	PUNCT
ejpam-4276	25	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4276	26	1	572	572	NUM
ejpam-4276	26	2	©	©	PROPN
ejpam-4276	26	3	2022	2022	NUM
ejpam-4276	26	4	ejpam	ejpam	VERB
ejpam-4276	26	5	all	all	DET
ejpam-4276	26	6	rights	right	NOUN
ejpam-4276	26	7	reserved	reserve	VERB
ejpam-4276	26	8	.	.	PUNCT
ejpam-4276	27	1	c.	c.	PROPN
ejpam-4276	27	2	boonpok	boonpok	PROPN
ejpam-4276	27	3	,	,	PUNCT
ejpam-4276	27	4	j.	j.	PROPN
ejpam-4276	27	5	khampakdee	khampakdee	PROPN
ejpam-4276	27	6	/	/	PUNCT
ejpam-4276	27	7	eur	eur	PROPN
ejpam-4276	27	8	.	.	PUNCT
ejpam-4276	28	1	j.	j.	PROPN
ejpam-4276	28	2	pure	pure	PROPN
ejpam-4276	28	3	appl	appl	PROPN
ejpam-4276	28	4	.	.	PROPN
ejpam-4276	28	5	math	math	PROPN
ejpam-4276	28	6	,	,	PUNCT
ejpam-4276	28	7	15	15	NUM
ejpam-4276	28	8	(	(	PUNCT
ejpam-4276	28	9	2	2	NUM
ejpam-4276	28	10	)	)	PUNCT
ejpam-4276	28	11	(	(	PUNCT
ejpam-4276	28	12	2022	2022	NUM
ejpam-4276	28	13	)	)	PUNCT
ejpam-4276	28	14	,	,	PUNCT
ejpam-4276	28	15	572	572	NUM
ejpam-4276	28	16	-	-	SYM
ejpam-4276	28	17	588	588	NUM
ejpam-4276	28	18	573	573	NUM
ejpam-4276	28	19	of	of	ADP
ejpam-4276	28	20	semi	semi	ADJ
ejpam-4276	28	21	-	-	ADJ
ejpam-4276	28	22	preopen	preopen	ADJ
ejpam-4276	28	23	sets	set	NOUN
ejpam-4276	28	24	[	[	X
ejpam-4276	28	25	1	1	NUM
ejpam-4276	28	26	]	]	PUNCT
ejpam-4276	28	27	.	.	PUNCT
ejpam-4276	29	1	in	in	ADP
ejpam-4276	29	2	1996	1996	NUM
ejpam-4276	29	3	,	,	PUNCT
ejpam-4276	29	4	andrijević	andrijević	VERB
ejpam-4276	30	1	[	[	X
ejpam-4276	30	2	1	1	X
ejpam-4276	30	3	]	]	PUNCT
ejpam-4276	30	4	introduced	introduce	VERB
ejpam-4276	30	5	a	a	DET
ejpam-4276	30	6	class	class	NOUN
ejpam-4276	30	7	of	of	ADP
ejpam-4276	30	8	generalized	generalized	ADJ
ejpam-4276	30	9	open	open	ADJ
ejpam-4276	30	10	sets	set	NOUN
ejpam-4276	30	11	in	in	ADP
ejpam-4276	30	12	a	a	DET
ejpam-4276	30	13	topological	topological	ADJ
ejpam-4276	30	14	space	space	NOUN
ejpam-4276	30	15	,	,	PUNCT
ejpam-4276	30	16	the	the	DET
ejpam-4276	30	17	so	so	ADV
ejpam-4276	30	18	-	-	PUNCT
ejpam-4276	30	19	called	call	VERB
ejpam-4276	30	20	b	b	NOUN
ejpam-4276	30	21	-	-	PUNCT
ejpam-4276	30	22	open	open	ADJ
ejpam-4276	30	23	sets	set	NOUN
ejpam-4276	30	24	.	.	PUNCT
ejpam-4276	31	1	the	the	DET
ejpam-4276	31	2	class	class	NOUN
ejpam-4276	31	3	of	of	ADP
ejpam-4276	31	4	b	b	NOUN
ejpam-4276	31	5	-	-	PUNCT
ejpam-4276	31	6	open	open	ADJ
ejpam-4276	31	7	sets	set	NOUN
ejpam-4276	31	8	is	be	AUX
ejpam-4276	31	9	contained	contain	VERB
ejpam-4276	31	10	in	in	ADP
ejpam-4276	31	11	the	the	DET
ejpam-4276	31	12	class	class	NOUN
ejpam-4276	31	13	of	of	ADP
ejpam-4276	31	14	β	β	ADJ
ejpam-4276	31	15	-	-	ADJ
ejpam-4276	31	16	open	open	ADJ
ejpam-4276	31	17	sets	set	NOUN
ejpam-4276	31	18	and	and	CCONJ
ejpam-4276	31	19	contains	contain	VERB
ejpam-4276	31	20	all	all	DET
ejpam-4276	31	21	semi	semi	ADJ
ejpam-4276	31	22	-	-	ADJ
ejpam-4276	31	23	open	open	ADJ
ejpam-4276	31	24	sets	set	NOUN
ejpam-4276	31	25	and	and	CCONJ
ejpam-4276	31	26	preopen	preopen	ADJ
ejpam-4276	31	27	sets	set	NOUN
ejpam-4276	31	28	.	.	PUNCT
ejpam-4276	32	1	the	the	DET
ejpam-4276	32	2	concept	concept	NOUN
ejpam-4276	32	3	of	of	ADP
ejpam-4276	32	4	extremally	extremally	ADV
ejpam-4276	32	5	disconnected	disconnected	ADJ
ejpam-4276	32	6	topological	topological	ADJ
ejpam-4276	32	7	spaces	space	NOUN
ejpam-4276	32	8	was	be	AUX
ejpam-4276	32	9	first	first	ADV
ejpam-4276	32	10	introduced	introduce	VERB
ejpam-4276	32	11	by	by	ADP
ejpam-4276	32	12	gillman	gillman	PROPN
ejpam-4276	32	13	and	and	CCONJ
ejpam-4276	32	14	jerison	jerison	NOUN
ejpam-4276	33	1	[	[	X
ejpam-4276	33	2	5	5	NUM
ejpam-4276	33	3	]	]	PUNCT
ejpam-4276	33	4	.	.	PUNCT
ejpam-4276	34	1	a	a	DET
ejpam-4276	34	2	topological	topological	ADJ
ejpam-4276	34	3	space	space	NOUN
ejpam-4276	34	4	is	be	AUX
ejpam-4276	34	5	called	call	VERB
ejpam-4276	34	6	extremally	extremally	ADV
ejpam-4276	34	7	disconnected	disconnected	ADJ
ejpam-4276	34	8	if	if	SCONJ
ejpam-4276	34	9	the	the	DET
ejpam-4276	34	10	closure	closure	NOUN
ejpam-4276	34	11	of	of	ADP
ejpam-4276	34	12	every	every	DET
ejpam-4276	34	13	open	open	ADJ
ejpam-4276	34	14	set	set	NOUN
ejpam-4276	34	15	is	be	AUX
ejpam-4276	34	16	open	open	ADJ
ejpam-4276	34	17	.	.	PUNCT
ejpam-4276	35	1	sivaraj	sivaraj	PROPN
ejpam-4276	35	2	[	[	X
ejpam-4276	35	3	13	13	NUM
ejpam-4276	35	4	]	]	PUNCT
ejpam-4276	35	5	investigated	investigate	VERB
ejpam-4276	35	6	some	some	DET
ejpam-4276	35	7	characterizations	characterization	NOUN
ejpam-4276	35	8	of	of	ADP
ejpam-4276	35	9	extremally	extremally	ADV
ejpam-4276	35	10	disconnected	disconnect	VERB
ejpam-4276	35	11	spaces	space	NOUN
ejpam-4276	35	12	by	by	ADP
ejpam-4276	35	13	utilizing	utilize	VERB
ejpam-4276	35	14	semi	semi	ADJ
ejpam-4276	35	15	-	-	ADJ
ejpam-4276	35	16	open	open	ADJ
ejpam-4276	35	17	sets	set	NOUN
ejpam-4276	35	18	due	due	ADP
ejpam-4276	35	19	to	to	ADP
ejpam-4276	35	20	levine	levine	PROPN
ejpam-4276	35	21	[	[	X
ejpam-4276	35	22	6	6	NUM
ejpam-4276	35	23	]	]	PUNCT
ejpam-4276	35	24	.	.	PUNCT
ejpam-4276	36	1	noiri	noiri	PROPN
ejpam-4276	37	1	[	[	X
ejpam-4276	37	2	11	11	NUM
ejpam-4276	37	3	]	]	PUNCT
ejpam-4276	37	4	obtained	obtain	VERB
ejpam-4276	37	5	several	several	ADJ
ejpam-4276	37	6	characterizations	characterization	NOUN
ejpam-4276	37	7	of	of	ADP
ejpam-4276	37	8	extremally	extremally	ADV
ejpam-4276	37	9	disconnected	disconnect	VERB
ejpam-4276	37	10	spaces	space	NOUN
ejpam-4276	37	11	by	by	ADP
ejpam-4276	37	12	utilizing	utilize	VERB
ejpam-4276	37	13	preopen	preopen	ADJ
ejpam-4276	37	14	sets	set	NOUN
ejpam-4276	37	15	and	and	CCONJ
ejpam-4276	37	16	semi	semi	ADJ
ejpam-4276	37	17	-	-	ADJ
ejpam-4276	37	18	preopen	preopen	ADJ
ejpam-4276	37	19	sets	set	NOUN
ejpam-4276	37	20	.	.	PUNCT
ejpam-4276	38	1	in	in	ADP
ejpam-4276	38	2	2004	2004	NUM
ejpam-4276	38	3	,	,	PUNCT
ejpam-4276	38	4	noiri	noiri	PROPN
ejpam-4276	38	5	and	and	CCONJ
ejpam-4276	38	6	hatir	hatir	NOUN
ejpam-4276	38	7	[	[	X
ejpam-4276	38	8	12	12	NUM
ejpam-4276	38	9	]	]	PUNCT
ejpam-4276	38	10	introduced	introduce	VERB
ejpam-4276	38	11	the	the	DET
ejpam-4276	38	12	notion	notion	NOUN
ejpam-4276	38	13	of	of	ADP
ejpam-4276	38	14	λsp	λsp	NOUN
ejpam-4276	38	15	-	-	PUNCT
ejpam-4276	38	16	sets	set	NOUN
ejpam-4276	38	17	in	in	ADP
ejpam-4276	38	18	terms	term	NOUN
ejpam-4276	38	19	of	of	ADP
ejpam-4276	38	20	the	the	DET
ejpam-4276	38	21	concept	concept	NOUN
ejpam-4276	38	22	of	of	ADP
ejpam-4276	38	23	β	β	ADJ
ejpam-4276	38	24	-	-	ADJ
ejpam-4276	38	25	open	open	ADJ
ejpam-4276	38	26	sets	set	NOUN
ejpam-4276	38	27	and	and	CCONJ
ejpam-4276	38	28	investigated	investigate	VERB
ejpam-4276	38	29	the	the	DET
ejpam-4276	38	30	notion	notion	NOUN
ejpam-4276	38	31	of	of	ADP
ejpam-4276	38	32	λsp	λsp	NOUN
ejpam-4276	38	33	-	-	PUNCT
ejpam-4276	38	34	closed	close	VERB
ejpam-4276	38	35	sets	set	NOUN
ejpam-4276	38	36	by	by	ADP
ejpam-4276	38	37	using	use	VERB
ejpam-4276	38	38	λspsets	λspset	NOUN
ejpam-4276	38	39	.	.	PUNCT
ejpam-4276	39	1	in	in	ADP
ejpam-4276	39	2	[	[	X
ejpam-4276	39	3	3	3	NUM
ejpam-4276	39	4	]	]	PUNCT
ejpam-4276	39	5	,	,	PUNCT
ejpam-4276	39	6	the	the	DET
ejpam-4276	39	7	author	author	NOUN
ejpam-4276	39	8	introduced	introduce	VERB
ejpam-4276	39	9	the	the	DET
ejpam-4276	39	10	concepts	concept	NOUN
ejpam-4276	39	11	of	of	ADP
ejpam-4276	39	12	(	(	PUNCT
ejpam-4276	39	13	λ	λ	PROPN
ejpam-4276	39	14	,	,	PUNCT
ejpam-4276	39	15	sp)-open	sp)-open	ADJ
ejpam-4276	39	16	sets	set	NOUN
ejpam-4276	39	17	and	and	CCONJ
ejpam-4276	39	18	(	(	PUNCT
ejpam-4276	39	19	λ	λ	PROPN
ejpam-4276	39	20	,	,	PUNCT
ejpam-4276	39	21	sp)-closed	sp)-close	VERB
ejpam-4276	39	22	sets	set	NOUN
ejpam-4276	39	23	which	which	PRON
ejpam-4276	39	24	are	be	AUX
ejpam-4276	39	25	defined	define	VERB
ejpam-4276	39	26	by	by	ADP
ejpam-4276	39	27	utilizing	utilize	VERB
ejpam-4276	39	28	the	the	DET
ejpam-4276	39	29	notions	notion	NOUN
ejpam-4276	39	30	of	of	ADP
ejpam-4276	39	31	λsp	λsp	NOUN
ejpam-4276	39	32	-	-	PUNCT
ejpam-4276	39	33	sets	set	NOUN
ejpam-4276	39	34	and	and	CCONJ
ejpam-4276	39	35	β	β	NOUN
ejpam-4276	39	36	-	-	ADJ
ejpam-4276	39	37	closed	closed	ADJ
ejpam-4276	39	38	sets	set	NOUN
ejpam-4276	39	39	.	.	PUNCT
ejpam-4276	40	1	the	the	DET
ejpam-4276	40	2	purpose	purpose	NOUN
ejpam-4276	40	3	of	of	ADP
ejpam-4276	40	4	the	the	DET
ejpam-4276	40	5	present	present	ADJ
ejpam-4276	40	6	paper	paper	NOUN
ejpam-4276	40	7	is	be	AUX
ejpam-4276	40	8	to	to	PART
ejpam-4276	40	9	investigate	investigate	VERB
ejpam-4276	40	10	some	some	DET
ejpam-4276	40	11	properties	property	NOUN
ejpam-4276	40	12	of	of	ADP
ejpam-4276	40	13	s(λ	s(λ	PROPN
ejpam-4276	40	14	,	,	PUNCT
ejpam-4276	40	15	sp)-open	sp)-open	ADJ
ejpam-4276	40	16	sets	set	NOUN
ejpam-4276	40	17	,	,	PUNCT
ejpam-4276	40	18	p(λ	p(λ	NOUN
ejpam-4276	40	19	,	,	PUNCT
ejpam-4276	40	20	sp)-open	sp)-open	ADJ
ejpam-4276	40	21	sets	set	NOUN
ejpam-4276	40	22	,	,	PUNCT
ejpam-4276	40	23	α(λ	α(λ	PROPN
ejpam-4276	40	24	,	,	PUNCT
ejpam-4276	40	25	sp)-open	sp)-open	ADJ
ejpam-4276	40	26	sets	set	NOUN
ejpam-4276	40	27	,	,	PUNCT
ejpam-4276	40	28	β(λ	β(λ	X
ejpam-4276	40	29	,	,	PUNCT
ejpam-4276	40	30	sp)-open	sp)-open	ADJ
ejpam-4276	40	31	sets	set	NOUN
ejpam-4276	40	32	and	and	CCONJ
ejpam-4276	40	33	b(λ	b(λ	NOUN
ejpam-4276	40	34	,	,	PUNCT
ejpam-4276	40	35	sp)-open	sp)-open	ADJ
ejpam-4276	40	36	sets	set	NOUN
ejpam-4276	40	37	.	.	PUNCT
ejpam-4276	41	1	in	in	ADP
ejpam-4276	41	2	particular	particular	ADJ
ejpam-4276	41	3	,	,	PUNCT
ejpam-4276	41	4	the	the	DET
ejpam-4276	41	5	relationships	relationship	NOUN
ejpam-4276	41	6	between	between	ADP
ejpam-4276	41	7	s(λ	s(λ	PROPN
ejpam-4276	41	8	,	,	PUNCT
ejpam-4276	41	9	sp)-open	sp)-open	ADJ
ejpam-4276	41	10	sets	set	NOUN
ejpam-4276	41	11	,	,	PUNCT
ejpam-4276	41	12	p(λ	p(λ	NOUN
ejpam-4276	41	13	,	,	PUNCT
ejpam-4276	41	14	sp)-open	sp)-open	ADJ
ejpam-4276	41	15	sets	set	NOUN
ejpam-4276	41	16	,	,	PUNCT
ejpam-4276	41	17	α(λ	α(λ	PROPN
ejpam-4276	41	18	,	,	PUNCT
ejpam-4276	41	19	sp)-open	sp)-open	ADJ
ejpam-4276	41	20	sets	set	NOUN
ejpam-4276	41	21	,	,	PUNCT
ejpam-4276	41	22	β(λ	β(λ	PROPN
ejpam-4276	41	23	,	,	PUNCT
ejpam-4276	41	24	sp)open	sp)open	NOUN
ejpam-4276	41	25	sets	set	NOUN
ejpam-4276	41	26	,	,	PUNCT
ejpam-4276	41	27	b(λ	b(λ	PROPN
ejpam-4276	41	28	,	,	PUNCT
ejpam-4276	41	29	sp)-open	sp)-open	ADJ
ejpam-4276	41	30	sets	set	NOUN
ejpam-4276	41	31	and	and	CCONJ
ejpam-4276	41	32	other	other	ADJ
ejpam-4276	41	33	related	related	ADJ
ejpam-4276	41	34	sets	set	NOUN
ejpam-4276	41	35	are	be	AUX
ejpam-4276	41	36	explored	explore	VERB
ejpam-4276	41	37	.	.	PUNCT
ejpam-4276	42	1	furthermore	furthermore	ADV
ejpam-4276	42	2	,	,	PUNCT
ejpam-4276	42	3	some	some	DET
ejpam-4276	42	4	characterizations	characterization	NOUN
ejpam-4276	42	5	of	of	ADP
ejpam-4276	42	6	λsp	λsp	NOUN
ejpam-4276	42	7	-	-	PUNCT
ejpam-4276	42	8	extremally	extremally	ADV
ejpam-4276	42	9	disconnected	disconnected	ADJ
ejpam-4276	42	10	spaces	space	NOUN
ejpam-4276	42	11	are	be	AUX
ejpam-4276	42	12	discussed	discuss	VERB
ejpam-4276	42	13	.	.	PUNCT
ejpam-4276	43	1	2	2	X
ejpam-4276	43	2	.	.	X
ejpam-4276	43	3	preliminaries	preliminary	NOUN
ejpam-4276	43	4	throughout	throughout	ADP
ejpam-4276	43	5	the	the	DET
ejpam-4276	43	6	paper	paper	NOUN
ejpam-4276	43	7	,	,	PUNCT
ejpam-4276	43	8	spaces	space	NOUN
ejpam-4276	43	9	(	(	PUNCT
ejpam-4276	43	10	x	x	X
ejpam-4276	43	11	,	,	PUNCT
ejpam-4276	43	12	τ	τ	X
ejpam-4276	43	13	)	)	PUNCT
ejpam-4276	43	14	and	and	CCONJ
ejpam-4276	43	15	(	(	PUNCT
ejpam-4276	43	16	y	y	PROPN
ejpam-4276	43	17	,	,	PUNCT
ejpam-4276	43	18	σ	σ	PROPN
ejpam-4276	43	19	)	)	PUNCT
ejpam-4276	43	20	(	(	PUNCT
ejpam-4276	43	21	or	or	CCONJ
ejpam-4276	43	22	simply	simply	ADV
ejpam-4276	43	23	x	x	X
ejpam-4276	43	24	and	and	CCONJ
ejpam-4276	43	25	y	y	PROPN
ejpam-4276	43	26	)	)	PUNCT
ejpam-4276	43	27	always	always	ADV
ejpam-4276	43	28	mean	mean	VERB
ejpam-4276	43	29	topological	topological	ADJ
ejpam-4276	43	30	spaces	space	NOUN
ejpam-4276	43	31	on	on	ADP
ejpam-4276	43	32	which	which	PRON
ejpam-4276	43	33	no	no	DET
ejpam-4276	43	34	separation	separation	NOUN
ejpam-4276	43	35	axioms	axiom	NOUN
ejpam-4276	43	36	are	be	AUX
ejpam-4276	43	37	assumed	assume	VERB
ejpam-4276	43	38	unless	unless	SCONJ
ejpam-4276	43	39	explicitly	explicitly	ADV
ejpam-4276	43	40	stated	state	VERB
ejpam-4276	43	41	.	.	PUNCT
ejpam-4276	44	1	let	let	VERB
ejpam-4276	44	2	a	a	DET
ejpam-4276	44	3	be	be	AUX
ejpam-4276	44	4	a	a	DET
ejpam-4276	44	5	subset	subset	NOUN
ejpam-4276	44	6	of	of	ADP
ejpam-4276	44	7	a	a	DET
ejpam-4276	44	8	topological	topological	ADJ
ejpam-4276	44	9	space	space	NOUN
ejpam-4276	44	10	(	(	PUNCT
ejpam-4276	44	11	x	x	X
ejpam-4276	44	12	,	,	PUNCT
ejpam-4276	44	13	τ	τ	PROPN
ejpam-4276	44	14	)	)	PUNCT
ejpam-4276	44	15	.	.	PUNCT
ejpam-4276	45	1	the	the	DET
ejpam-4276	45	2	closure	closure	NOUN
ejpam-4276	45	3	of	of	ADP
ejpam-4276	45	4	a	a	PRON
ejpam-4276	45	5	and	and	CCONJ
ejpam-4276	45	6	the	the	DET
ejpam-4276	45	7	interior	interior	NOUN
ejpam-4276	45	8	of	of	ADP
ejpam-4276	45	9	a	a	PRON
ejpam-4276	45	10	are	be	AUX
ejpam-4276	45	11	denoted	denote	VERB
ejpam-4276	45	12	by	by	ADP
ejpam-4276	45	13	cl(a	cl(a	NOUN
ejpam-4276	45	14	)	)	PUNCT
ejpam-4276	45	15	and	and	CCONJ
ejpam-4276	45	16	int(a	int(a	PROPN
ejpam-4276	45	17	)	)	PUNCT
ejpam-4276	45	18	,	,	PUNCT
ejpam-4276	45	19	respectively	respectively	ADV
ejpam-4276	45	20	.	.	PUNCT
ejpam-4276	46	1	a	a	DET
ejpam-4276	46	2	subset	subset	NOUN
ejpam-4276	46	3	a	a	PRON
ejpam-4276	46	4	is	be	AUX
ejpam-4276	46	5	said	say	VERB
ejpam-4276	46	6	to	to	PART
ejpam-4276	46	7	be	be	AUX
ejpam-4276	46	8	β	β	X
ejpam-4276	46	9	-	-	ADJ
ejpam-4276	46	10	open	open	ADJ
ejpam-4276	46	11	[	[	X
ejpam-4276	46	12	4	4	NUM
ejpam-4276	46	13	]	]	X
ejpam-4276	46	14	if	if	SCONJ
ejpam-4276	46	15	a	a	DET
ejpam-4276	46	16	⊆	⊆	NUM
ejpam-4276	46	17	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-4276	46	18	)	)	PUNCT
ejpam-4276	46	19	)	)	PUNCT
ejpam-4276	46	20	)	)	PUNCT
ejpam-4276	46	21	.	.	PUNCT
ejpam-4276	47	1	the	the	DET
ejpam-4276	47	2	complement	complement	NOUN
ejpam-4276	47	3	of	of	ADP
ejpam-4276	47	4	a	a	DET
ejpam-4276	47	5	β	β	NOUN
ejpam-4276	47	6	-	-	ADJ
ejpam-4276	47	7	open	open	ADJ
ejpam-4276	47	8	set	set	NOUN
ejpam-4276	47	9	a	a	PRON
ejpam-4276	47	10	is	be	AUX
ejpam-4276	47	11	called	call	VERB
ejpam-4276	47	12	β	β	NOUN
ejpam-4276	47	13	-	-	VERB
ejpam-4276	47	14	closed	closed	ADJ
ejpam-4276	47	15	.	.	PUNCT
ejpam-4276	48	1	the	the	DET
ejpam-4276	48	2	family	family	NOUN
ejpam-4276	48	3	of	of	ADP
ejpam-4276	48	4	all	all	DET
ejpam-4276	48	5	β	β	ADJ
ejpam-4276	48	6	-	-	ADJ
ejpam-4276	48	7	open	open	ADJ
ejpam-4276	48	8	sets	set	NOUN
ejpam-4276	48	9	of	of	ADP
ejpam-4276	48	10	a	a	DET
ejpam-4276	48	11	topological	topological	ADJ
ejpam-4276	48	12	space	space	NOUN
ejpam-4276	48	13	(	(	PUNCT
ejpam-4276	48	14	x	x	X
ejpam-4276	48	15	,	,	PUNCT
ejpam-4276	48	16	τ	τ	X
ejpam-4276	48	17	)	)	PUNCT
ejpam-4276	48	18	is	be	AUX
ejpam-4276	48	19	denoted	denote	VERB
ejpam-4276	48	20	by	by	ADP
ejpam-4276	48	21	β(x	β(x	PROPN
ejpam-4276	48	22	,	,	PUNCT
ejpam-4276	48	23	τ	τ	PROPN
ejpam-4276	48	24	)	)	PUNCT
ejpam-4276	48	25	.	.	PUNCT
ejpam-4276	49	1	a	a	DET
ejpam-4276	49	2	subset	subset	NOUN
ejpam-4276	49	3	λsp(a	λsp(a	NOUN
ejpam-4276	49	4	)	)	PUNCT
ejpam-4276	50	1	[	[	X
ejpam-4276	50	2	12	12	NUM
ejpam-4276	50	3	]	]	PUNCT
ejpam-4276	50	4	is	be	AUX
ejpam-4276	50	5	defined	define	VERB
ejpam-4276	50	6	as	as	SCONJ
ejpam-4276	50	7	follows	follow	VERB
ejpam-4276	50	8	:	:	PUNCT
ejpam-4276	50	9	λsp(a	λsp(a	NUM
ejpam-4276	50	10	)	)	PUNCT
ejpam-4276	50	11	=	=	PUNCT
ejpam-4276	51	1	∩{u	∩{u	PROPN
ejpam-4276	51	2	|	|	ADV
ejpam-4276	51	3	a	a	DET
ejpam-4276	51	4	⊆	⊆	NUM
ejpam-4276	51	5	u	u	NOUN
ejpam-4276	51	6	,	,	PUNCT
ejpam-4276	51	7	u	u	NOUN
ejpam-4276	51	8	∈	∈	PROPN
ejpam-4276	51	9	β(x	β(x	PROPN
ejpam-4276	51	10	,	,	PUNCT
ejpam-4276	51	11	τ	τ	X
ejpam-4276	51	12	)	)	PUNCT
ejpam-4276	51	13	}	}	PUNCT
ejpam-4276	51	14	.	.	PUNCT
ejpam-4276	52	1	lemma	lemma	PROPN
ejpam-4276	52	2	1	1	NUM
ejpam-4276	52	3	.	.	PUNCT
ejpam-4276	53	1	[	[	X
ejpam-4276	53	2	12	12	NUM
ejpam-4276	53	3	]	]	PUNCT
ejpam-4276	53	4	for	for	ADP
ejpam-4276	53	5	subsets	subset	NOUN
ejpam-4276	53	6	a	a	PRON
ejpam-4276	53	7	,	,	PUNCT
ejpam-4276	53	8	b	b	PROPN
ejpam-4276	53	9	and	and	CCONJ
ejpam-4276	53	10	aα(α	aα(α	NOUN
ejpam-4276	53	11	∈	∈	PROPN
ejpam-4276	53	12	∇	∇	NOUN
ejpam-4276	53	13	)	)	PUNCT
ejpam-4276	53	14	of	of	ADP
ejpam-4276	53	15	a	a	DET
ejpam-4276	53	16	topological	topological	ADJ
ejpam-4276	53	17	space	space	NOUN
ejpam-4276	53	18	(	(	PUNCT
ejpam-4276	53	19	x	x	X
ejpam-4276	53	20	,	,	PUNCT
ejpam-4276	53	21	τ	τ	PROPN
ejpam-4276	53	22	)	)	PUNCT
ejpam-4276	53	23	,	,	PUNCT
ejpam-4276	53	24	the	the	DET
ejpam-4276	53	25	following	follow	VERB
ejpam-4276	53	26	hold	hold	NOUN
ejpam-4276	53	27	:	:	PUNCT
ejpam-4276	53	28	(	(	PUNCT
ejpam-4276	53	29	1	1	X
ejpam-4276	53	30	)	)	PUNCT
ejpam-4276	53	31	a	a	DET
ejpam-4276	53	32	⊆	⊆	NUM
ejpam-4276	53	33	λsp(a	λsp(a	NOUN
ejpam-4276	53	34	)	)	PUNCT
ejpam-4276	53	35	.	.	PUNCT
ejpam-4276	54	1	(	(	PUNCT
ejpam-4276	54	2	2	2	X
ejpam-4276	54	3	)	)	PUNCT
ejpam-4276	54	4	if	if	SCONJ
ejpam-4276	54	5	a	a	DET
ejpam-4276	54	6	⊆	⊆	NUM
ejpam-4276	54	7	b	b	NOUN
ejpam-4276	54	8	,	,	PUNCT
ejpam-4276	54	9	then	then	ADV
ejpam-4276	54	10	λsp(a	λsp(a	PROPN
ejpam-4276	54	11	)	)	PUNCT
ejpam-4276	54	12	⊆	⊆	NUM
ejpam-4276	54	13	λsp(b	λsp(b	PROPN
ejpam-4276	54	14	)	)	PUNCT
ejpam-4276	54	15	.	.	PUNCT
ejpam-4276	55	1	(	(	PUNCT
ejpam-4276	55	2	3	3	X
ejpam-4276	55	3	)	)	PUNCT
ejpam-4276	55	4	λsp(λsp(a	λsp(λsp(a	NUM
ejpam-4276	55	5	)	)	PUNCT
ejpam-4276	55	6	)	)	PUNCT
ejpam-4276	56	1	=	=	SYM
ejpam-4276	56	2	λsp(a	λsp(a	PROPN
ejpam-4276	56	3	)	)	PUNCT
ejpam-4276	56	4	.	.	PUNCT
ejpam-4276	57	1	(	(	PUNCT
ejpam-4276	57	2	4	4	X
ejpam-4276	57	3	)	)	PUNCT
ejpam-4276	57	4	if	if	SCONJ
ejpam-4276	57	5	u	u	PROPN
ejpam-4276	57	6	∈	∈	PROPN
ejpam-4276	57	7	β(x	β(x	PROPN
ejpam-4276	57	8	,	,	PUNCT
ejpam-4276	57	9	τ	τ	PROPN
ejpam-4276	57	10	)	)	PUNCT
ejpam-4276	57	11	,	,	PUNCT
ejpam-4276	57	12	then	then	ADV
ejpam-4276	57	13	λsp(u	λsp(u	X
ejpam-4276	57	14	)	)	PUNCT
ejpam-4276	57	15	=	=	SYM
ejpam-4276	57	16	u	u	NOUN
ejpam-4276	57	17	.	.	PUNCT
ejpam-4276	58	1	(	(	PUNCT
ejpam-4276	58	2	5	5	NUM
ejpam-4276	58	3	)	)	PUNCT
ejpam-4276	58	4	λsp(∩{aα|α	λsp(∩{aα|α	ADV
ejpam-4276	58	5	∈	∈	NOUN
ejpam-4276	58	6	∇	∇	NOUN
ejpam-4276	58	7	}	}	PUNCT
ejpam-4276	58	8	)	)	PUNCT
ejpam-4276	58	9	⊆	⊆	NUM
ejpam-4276	58	10	∩{λsp(aα)|α	∩{λsp(aα)|α	PROPN
ejpam-4276	58	11	∈	∈	NOUN
ejpam-4276	58	12	∇	∇	X
ejpam-4276	58	13	}	}	PUNCT
ejpam-4276	58	14	.	.	PUNCT
ejpam-4276	59	1	(	(	PUNCT
ejpam-4276	59	2	6	6	NUM
ejpam-4276	59	3	)	)	PUNCT
ejpam-4276	59	4	λsp(∪{aα|α	λsp(∪{aα|α	PROPN
ejpam-4276	59	5	∈	∈	PROPN
ejpam-4276	59	6	∇	∇	X
ejpam-4276	59	7	}	}	PUNCT
ejpam-4276	59	8	)	)	PUNCT
ejpam-4276	59	9	=	=	SYM
ejpam-4276	60	1	∪{λsp(aα)|α	∪{λsp(aα)|α	PROPN
ejpam-4276	60	2	∈	∈	NOUN
ejpam-4276	60	3	∇	∇	NOUN
ejpam-4276	60	4	}	}	PUNCT
ejpam-4276	60	5	.	.	PUNCT
ejpam-4276	61	1	a	a	DET
ejpam-4276	61	2	subset	subset	NOUN
ejpam-4276	61	3	a	a	PRON
ejpam-4276	61	4	of	of	ADP
ejpam-4276	61	5	a	a	DET
ejpam-4276	61	6	topological	topological	ADJ
ejpam-4276	61	7	space	space	NOUN
ejpam-4276	61	8	(	(	PUNCT
ejpam-4276	61	9	x	x	X
ejpam-4276	61	10	,	,	PUNCT
ejpam-4276	61	11	τ	τ	X
ejpam-4276	61	12	)	)	PUNCT
ejpam-4276	61	13	is	be	AUX
ejpam-4276	61	14	called	call	VERB
ejpam-4276	61	15	a	a	DET
ejpam-4276	61	16	λsp	λsp	NOUN
ejpam-4276	61	17	-	-	PUNCT
ejpam-4276	61	18	set	set	NOUN
ejpam-4276	61	19	[	[	X
ejpam-4276	61	20	12	12	NUM
ejpam-4276	61	21	]	]	X
ejpam-4276	61	22	if	if	SCONJ
ejpam-4276	61	23	a	a	DET
ejpam-4276	61	24	=	=	NOUN
ejpam-4276	61	25	λsp(a	λsp(a	NOUN
ejpam-4276	61	26	)	)	PUNCT
ejpam-4276	61	27	.	.	PUNCT
ejpam-4276	62	1	the	the	DET
ejpam-4276	62	2	family	family	NOUN
ejpam-4276	62	3	of	of	ADP
ejpam-4276	62	4	all	all	DET
ejpam-4276	62	5	λsp	λsp	NOUN
ejpam-4276	62	6	-	-	PUNCT
ejpam-4276	62	7	sets	set	NOUN
ejpam-4276	62	8	of	of	ADP
ejpam-4276	62	9	a	a	DET
ejpam-4276	62	10	topological	topological	ADJ
ejpam-4276	62	11	space	space	NOUN
ejpam-4276	62	12	(	(	PUNCT
ejpam-4276	62	13	x	x	X
ejpam-4276	62	14	,	,	PUNCT
ejpam-4276	62	15	τ	τ	X
ejpam-4276	62	16	)	)	PUNCT
ejpam-4276	62	17	is	be	AUX
ejpam-4276	62	18	denoted	denote	VERB
ejpam-4276	62	19	by	by	ADP
ejpam-4276	62	20	λsp(x	λsp(x	PROPN
ejpam-4276	62	21	,	,	PUNCT
ejpam-4276	62	22	τ	τ	X
ejpam-4276	62	23	)	)	PUNCT
ejpam-4276	62	24	(	(	PUNCT
ejpam-4276	62	25	or	or	CCONJ
ejpam-4276	62	26	simply	simply	ADV
ejpam-4276	62	27	λsp	λsp	PROPN
ejpam-4276	62	28	)	)	PUNCT
ejpam-4276	62	29	.	.	PUNCT
ejpam-4276	63	1	c.	c.	PROPN
ejpam-4276	63	2	boonpok	boonpok	PROPN
ejpam-4276	63	3	,	,	PUNCT
ejpam-4276	63	4	j.	j.	PROPN
ejpam-4276	63	5	khampakdee	khampakdee	PROPN
ejpam-4276	63	6	/	/	PUNCT
ejpam-4276	63	7	eur	eur	PROPN
ejpam-4276	63	8	.	.	PUNCT
ejpam-4276	64	1	j.	j.	PROPN
ejpam-4276	64	2	pure	pure	PROPN
ejpam-4276	64	3	appl	appl	PROPN
ejpam-4276	64	4	.	.	PROPN
ejpam-4276	64	5	math	math	PROPN
ejpam-4276	64	6	,	,	PUNCT
ejpam-4276	64	7	15	15	NUM
ejpam-4276	64	8	(	(	PUNCT
ejpam-4276	64	9	2	2	NUM
ejpam-4276	64	10	)	)	PUNCT
ejpam-4276	64	11	(	(	PUNCT
ejpam-4276	64	12	2022	2022	NUM
ejpam-4276	64	13	)	)	PUNCT
ejpam-4276	64	14	,	,	PUNCT
ejpam-4276	64	15	572	572	NUM
ejpam-4276	64	16	-	-	SYM
ejpam-4276	64	17	588	588	NUM
ejpam-4276	64	18	574	574	NUM
ejpam-4276	64	19	lemma	lemma	PROPN
ejpam-4276	64	20	2	2	NUM
ejpam-4276	64	21	.	.	PUNCT
ejpam-4276	65	1	[	[	X
ejpam-4276	65	2	12	12	NUM
ejpam-4276	65	3	]	]	PUNCT
ejpam-4276	65	4	for	for	ADP
ejpam-4276	65	5	subsets	subset	NOUN
ejpam-4276	65	6	a	a	PRON
ejpam-4276	65	7	and	and	CCONJ
ejpam-4276	65	8	aα(α	aα(α	NOUN
ejpam-4276	65	9	∈	∈	NOUN
ejpam-4276	65	10	∇	∇	NOUN
ejpam-4276	65	11	)	)	PUNCT
ejpam-4276	65	12	of	of	ADP
ejpam-4276	65	13	a	a	DET
ejpam-4276	65	14	topological	topological	ADJ
ejpam-4276	65	15	space	space	NOUN
ejpam-4276	65	16	(	(	PUNCT
ejpam-4276	65	17	x	x	X
ejpam-4276	65	18	,	,	PUNCT
ejpam-4276	65	19	τ	τ	PROPN
ejpam-4276	65	20	)	)	PUNCT
ejpam-4276	65	21	,	,	PUNCT
ejpam-4276	65	22	the	the	DET
ejpam-4276	65	23	following	follow	VERB
ejpam-4276	65	24	hold	hold	NOUN
ejpam-4276	65	25	:	:	PUNCT
ejpam-4276	65	26	(	(	PUNCT
ejpam-4276	65	27	1	1	X
ejpam-4276	65	28	)	)	PUNCT
ejpam-4276	65	29	λsp(a	λsp(a	NOUN
ejpam-4276	65	30	)	)	PUNCT
ejpam-4276	65	31	is	be	AUX
ejpam-4276	65	32	a	a	DET
ejpam-4276	65	33	λsp	λsp	NOUN
ejpam-4276	65	34	-	-	PUNCT
ejpam-4276	65	35	set	set	NOUN
ejpam-4276	65	36	.	.	PUNCT
ejpam-4276	66	1	(	(	PUNCT
ejpam-4276	66	2	2	2	X
ejpam-4276	66	3	)	)	PUNCT
ejpam-4276	66	4	if	if	SCONJ
ejpam-4276	66	5	a	a	PRON
ejpam-4276	66	6	is	be	AUX
ejpam-4276	66	7	β	β	NOUN
ejpam-4276	66	8	-	-	ADJ
ejpam-4276	66	9	open	open	ADJ
ejpam-4276	66	10	,	,	PUNCT
ejpam-4276	66	11	then	then	ADV
ejpam-4276	66	12	a	a	PRON
ejpam-4276	66	13	is	be	AUX
ejpam-4276	66	14	a	a	DET
ejpam-4276	66	15	λsp	λsp	NOUN
ejpam-4276	66	16	-	-	PUNCT
ejpam-4276	66	17	set	set	NOUN
ejpam-4276	66	18	.	.	PUNCT
ejpam-4276	67	1	(	(	PUNCT
ejpam-4276	67	2	3	3	X
ejpam-4276	67	3	)	)	PUNCT
ejpam-4276	67	4	if	if	SCONJ
ejpam-4276	67	5	aα	aα	NOUN
ejpam-4276	67	6	is	be	AUX
ejpam-4276	67	7	a	a	DET
ejpam-4276	67	8	λsp	λsp	NOUN
ejpam-4276	67	9	-	-	PUNCT
ejpam-4276	67	10	set	set	VERB
ejpam-4276	67	11	for	for	ADP
ejpam-4276	67	12	each	each	DET
ejpam-4276	67	13	α	α	PROPN
ejpam-4276	67	14	∈	∈	PROPN
ejpam-4276	67	15	∇	∇	NOUN
ejpam-4276	67	16	,	,	PUNCT
ejpam-4276	67	17	then	then	ADV
ejpam-4276	67	18	∩α∈∇aα	∩α∈∇aα	PROPN
ejpam-4276	67	19	is	be	AUX
ejpam-4276	67	20	a	a	DET
ejpam-4276	67	21	λsp	λsp	NOUN
ejpam-4276	67	22	-	-	PUNCT
ejpam-4276	67	23	set	set	NOUN
ejpam-4276	67	24	.	.	PUNCT
ejpam-4276	68	1	(	(	PUNCT
ejpam-4276	68	2	4	4	X
ejpam-4276	68	3	)	)	PUNCT
ejpam-4276	68	4	if	if	SCONJ
ejpam-4276	68	5	aα	aα	NOUN
ejpam-4276	68	6	is	be	AUX
ejpam-4276	68	7	a	a	DET
ejpam-4276	68	8	λsp	λsp	NOUN
ejpam-4276	68	9	-	-	PUNCT
ejpam-4276	68	10	set	set	VERB
ejpam-4276	68	11	for	for	ADP
ejpam-4276	68	12	each	each	DET
ejpam-4276	68	13	α	α	PROPN
ejpam-4276	68	14	∈	∈	PROPN
ejpam-4276	68	15	∇	∇	NOUN
ejpam-4276	68	16	,	,	PUNCT
ejpam-4276	68	17	then	then	ADV
ejpam-4276	68	18	∪α∈∇aα	∪α∈∇aα	VERB
ejpam-4276	68	19	is	be	AUX
ejpam-4276	68	20	a	a	DET
ejpam-4276	68	21	λsp	λsp	NOUN
ejpam-4276	68	22	-	-	PUNCT
ejpam-4276	68	23	set	set	NOUN
ejpam-4276	68	24	.	.	PUNCT
ejpam-4276	69	1	a	a	DET
ejpam-4276	69	2	subset	subset	NOUN
ejpam-4276	69	3	a	a	PRON
ejpam-4276	69	4	of	of	ADP
ejpam-4276	69	5	a	a	DET
ejpam-4276	69	6	topological	topological	ADJ
ejpam-4276	69	7	space	space	NOUN
ejpam-4276	69	8	(	(	PUNCT
ejpam-4276	69	9	x	x	X
ejpam-4276	69	10	,	,	PUNCT
ejpam-4276	69	11	τ	τ	X
ejpam-4276	69	12	)	)	PUNCT
ejpam-4276	69	13	is	be	AUX
ejpam-4276	69	14	called	call	VERB
ejpam-4276	69	15	(	(	PUNCT
ejpam-4276	69	16	λ	λ	X
ejpam-4276	69	17	,	,	PUNCT
ejpam-4276	69	18	sp)-closed	sp)-close	VERB
ejpam-4276	69	19	[	[	PUNCT
ejpam-4276	69	20	3	3	X
ejpam-4276	69	21	]	]	X
ejpam-4276	69	22	if	if	SCONJ
ejpam-4276	69	23	a	a	DET
ejpam-4276	69	24	=	=	X
ejpam-4276	69	25	t	t	NOUN
ejpam-4276	69	26	∩c	∩c	NOUN
ejpam-4276	69	27	,	,	PUNCT
ejpam-4276	69	28	where	where	SCONJ
ejpam-4276	69	29	t	t	PROPN
ejpam-4276	69	30	is	be	AUX
ejpam-4276	69	31	a	a	DET
ejpam-4276	69	32	λsp	λsp	NOUN
ejpam-4276	69	33	-	-	PUNCT
ejpam-4276	69	34	set	set	VERB
ejpam-4276	69	35	and	and	CCONJ
ejpam-4276	69	36	c	c	NOUN
ejpam-4276	69	37	is	be	AUX
ejpam-4276	69	38	a	a	DET
ejpam-4276	69	39	β	β	NOUN
ejpam-4276	69	40	-	-	ADJ
ejpam-4276	69	41	closed	closed	ADJ
ejpam-4276	69	42	set	set	NOUN
ejpam-4276	69	43	.	.	PUNCT
ejpam-4276	70	1	the	the	DET
ejpam-4276	70	2	complement	complement	NOUN
ejpam-4276	70	3	of	of	ADP
ejpam-4276	70	4	a	a	DET
ejpam-4276	70	5	(	(	PUNCT
ejpam-4276	70	6	λ	λ	PROPN
ejpam-4276	70	7	,	,	PUNCT
ejpam-4276	70	8	sp)-closed	sp)-close	VERB
ejpam-4276	70	9	set	set	VERB
ejpam-4276	70	10	is	be	AUX
ejpam-4276	70	11	called	call	VERB
ejpam-4276	70	12	(	(	PUNCT
ejpam-4276	70	13	λ	λ	NOUN
ejpam-4276	70	14	,	,	PUNCT
ejpam-4276	70	15	sp)-open	sp)-open	NOUN
ejpam-4276	70	16	.	.	PUNCT
ejpam-4276	71	1	the	the	DET
ejpam-4276	71	2	family	family	NOUN
ejpam-4276	71	3	of	of	ADP
ejpam-4276	71	4	all	all	DET
ejpam-4276	71	5	(	(	PUNCT
ejpam-4276	71	6	λ	λ	NOUN
ejpam-4276	71	7	,	,	PUNCT
ejpam-4276	71	8	sp)-open	sp)-open	ADJ
ejpam-4276	71	9	(	(	PUNCT
ejpam-4276	71	10	resp	resp	NOUN
ejpam-4276	71	11	.	.	PUNCT
ejpam-4276	72	1	(	(	PUNCT
ejpam-4276	72	2	λ	λ	X
ejpam-4276	72	3	,	,	PUNCT
ejpam-4276	72	4	sp)-closed	sp)-closed	ADJ
ejpam-4276	72	5	)	)	PUNCT
ejpam-4276	72	6	sets	set	NOUN
ejpam-4276	72	7	of	of	ADP
ejpam-4276	72	8	a	a	DET
ejpam-4276	72	9	topological	topological	ADJ
ejpam-4276	72	10	space	space	NOUN
ejpam-4276	72	11	(	(	PUNCT
ejpam-4276	72	12	x	x	X
ejpam-4276	72	13	,	,	PUNCT
ejpam-4276	72	14	τ	τ	X
ejpam-4276	72	15	)	)	PUNCT
ejpam-4276	72	16	is	be	AUX
ejpam-4276	72	17	denoted	denote	VERB
ejpam-4276	72	18	by	by	ADP
ejpam-4276	72	19	λspo(x	λspo(x	PROPN
ejpam-4276	72	20	,	,	PUNCT
ejpam-4276	72	21	τ	τ	PROPN
ejpam-4276	72	22	)	)	PUNCT
ejpam-4276	72	23	(	(	PUNCT
ejpam-4276	72	24	resp	resp	NOUN
ejpam-4276	72	25	.	.	PUNCT
ejpam-4276	73	1	λspc(x	λspc(x	NOUN
ejpam-4276	73	2	,	,	PUNCT
ejpam-4276	73	3	τ	τ	PROPN
ejpam-4276	73	4	)	)	PUNCT
ejpam-4276	73	5	)	)	PUNCT
ejpam-4276	73	6	.	.	PUNCT
ejpam-4276	74	1	let	let	VERB
ejpam-4276	74	2	a	a	PRON
ejpam-4276	74	3	be	be	AUX
ejpam-4276	74	4	a	a	DET
ejpam-4276	74	5	subsets	subset	NOUN
ejpam-4276	74	6	of	of	ADP
ejpam-4276	74	7	a	a	DET
ejpam-4276	74	8	topological	topological	ADJ
ejpam-4276	74	9	space	space	NOUN
ejpam-4276	74	10	(	(	PUNCT
ejpam-4276	74	11	x	x	X
ejpam-4276	74	12	,	,	PUNCT
ejpam-4276	74	13	τ	τ	PROPN
ejpam-4276	74	14	)	)	PUNCT
ejpam-4276	74	15	.	.	PUNCT
ejpam-4276	75	1	a	a	DET
ejpam-4276	75	2	point	point	NOUN
ejpam-4276	75	3	x	x	X
ejpam-4276	75	4	∈	∈	NOUN
ejpam-4276	75	5	x	x	PUNCT
ejpam-4276	75	6	is	be	AUX
ejpam-4276	75	7	called	call	VERB
ejpam-4276	75	8	a	a	DET
ejpam-4276	75	9	(	(	PUNCT
ejpam-4276	75	10	λ	λ	NOUN
ejpam-4276	75	11	,	,	PUNCT
ejpam-4276	75	12	sp)-cluster	sp)-cluster	NOUN
ejpam-4276	75	13	point	point	NOUN
ejpam-4276	75	14	[	[	X
ejpam-4276	75	15	3	3	X
ejpam-4276	75	16	]	]	PUNCT
ejpam-4276	75	17	of	of	ADP
ejpam-4276	75	18	a	a	DET
ejpam-4276	75	19	if	if	SCONJ
ejpam-4276	75	20	a∩u	a∩u	PROPN
ejpam-4276	75	21	6=	6=	NOUN
ejpam-4276	75	22	∅	∅	NOUN
ejpam-4276	75	23	for	for	ADP
ejpam-4276	75	24	every	every	DET
ejpam-4276	75	25	(	(	PUNCT
ejpam-4276	75	26	λ	λ	NOUN
ejpam-4276	75	27	,	,	PUNCT
ejpam-4276	75	28	sp)-open	sp)-open	NOUN
ejpam-4276	75	29	set	set	VERB
ejpam-4276	75	30	u	u	NOUN
ejpam-4276	75	31	of	of	ADP
ejpam-4276	75	32	x	x	SYM
ejpam-4276	75	33	containing	contain	VERB
ejpam-4276	75	34	x.	x.	NOUN
ejpam-4276	75	35	the	the	DET
ejpam-4276	75	36	set	set	NOUN
ejpam-4276	75	37	of	of	ADP
ejpam-4276	75	38	all	all	DET
ejpam-4276	75	39	(	(	PUNCT
ejpam-4276	75	40	λ	λ	PROPN
ejpam-4276	75	41	,	,	PUNCT
ejpam-4276	75	42	sp)-cluster	sp)-cluster	NOUN
ejpam-4276	75	43	points	point	NOUN
ejpam-4276	75	44	of	of	ADP
ejpam-4276	75	45	a	a	PRON
ejpam-4276	75	46	is	be	AUX
ejpam-4276	75	47	called	call	VERB
ejpam-4276	75	48	the	the	DET
ejpam-4276	75	49	(	(	PUNCT
ejpam-4276	75	50	λ	λ	PROPN
ejpam-4276	75	51	,	,	PUNCT
ejpam-4276	75	52	sp)-closure	sp)-closure	NOUN
ejpam-4276	75	53	of	of	ADP
ejpam-4276	75	54	a	a	PRON
ejpam-4276	75	55	and	and	CCONJ
ejpam-4276	75	56	is	be	AUX
ejpam-4276	75	57	denoted	denote	VERB
ejpam-4276	75	58	by	by	ADP
ejpam-4276	75	59	a(λ	a(λ	ADV
ejpam-4276	75	60	,	,	PUNCT
ejpam-4276	75	61	sp	sp	NOUN
ejpam-4276	75	62	)	)	PUNCT
ejpam-4276	75	63	.	.	PUNCT
ejpam-4276	76	1	lemma	lemma	PROPN
ejpam-4276	76	2	3	3	X
ejpam-4276	76	3	.	.	PUNCT
ejpam-4276	77	1	[	[	X
ejpam-4276	77	2	3	3	X
ejpam-4276	77	3	]	]	PUNCT
ejpam-4276	77	4	let	let	VERB
ejpam-4276	77	5	a	a	PRON
ejpam-4276	77	6	and	and	CCONJ
ejpam-4276	77	7	b	b	NOUN
ejpam-4276	77	8	be	be	AUX
ejpam-4276	77	9	subsets	subset	NOUN
ejpam-4276	77	10	of	of	ADP
ejpam-4276	77	11	a	a	DET
ejpam-4276	77	12	topological	topological	ADJ
ejpam-4276	77	13	space	space	NOUN
ejpam-4276	77	14	(	(	PUNCT
ejpam-4276	77	15	x	x	X
ejpam-4276	77	16	,	,	PUNCT
ejpam-4276	77	17	τ	τ	PROPN
ejpam-4276	77	18	)	)	PUNCT
ejpam-4276	77	19	.	.	PUNCT
ejpam-4276	78	1	for	for	ADP
ejpam-4276	78	2	the	the	DET
ejpam-4276	78	3	(	(	PUNCT
ejpam-4276	78	4	λ	λ	PROPN
ejpam-4276	78	5	,	,	PUNCT
ejpam-4276	78	6	sp)-closure	sp)-closure	NOUN
ejpam-4276	78	7	,	,	PUNCT
ejpam-4276	78	8	the	the	DET
ejpam-4276	78	9	following	follow	VERB
ejpam-4276	78	10	properties	property	NOUN
ejpam-4276	78	11	hold	hold	VERB
ejpam-4276	78	12	:	:	PUNCT
ejpam-4276	78	13	(	(	PUNCT
ejpam-4276	78	14	1	1	X
ejpam-4276	78	15	)	)	PUNCT
ejpam-4276	78	16	a	a	DET
ejpam-4276	78	17	⊆	⊆	NUM
ejpam-4276	78	18	a(λ	a(λ	ADJ
ejpam-4276	78	19	,	,	PUNCT
ejpam-4276	78	20	sp	sp	NOUN
ejpam-4276	78	21	)	)	PUNCT
ejpam-4276	78	22	and	and	CCONJ
ejpam-4276	78	23	[	[	X
ejpam-4276	78	24	a(λ	a(λ	ADV
ejpam-4276	78	25	,	,	PUNCT
ejpam-4276	78	26	sp)](λ	sp)](λ	PROPN
ejpam-4276	78	27	,	,	PUNCT
ejpam-4276	78	28	sp	sp	NOUN
ejpam-4276	78	29	)	)	PUNCT
ejpam-4276	78	30	=	=	PUNCT
ejpam-4276	78	31	a(λ	a(λ	ADV
ejpam-4276	78	32	,	,	PUNCT
ejpam-4276	78	33	sp	sp	NOUN
ejpam-4276	78	34	)	)	PUNCT
ejpam-4276	78	35	.	.	PUNCT
ejpam-4276	79	1	(	(	PUNCT
ejpam-4276	79	2	2	2	X
ejpam-4276	79	3	)	)	PUNCT
ejpam-4276	79	4	if	if	SCONJ
ejpam-4276	79	5	a	a	DET
ejpam-4276	79	6	⊆	⊆	NUM
ejpam-4276	79	7	b	b	NOUN
ejpam-4276	79	8	,	,	PUNCT
ejpam-4276	79	9	then	then	ADV
ejpam-4276	79	10	a(λ	a(λ	ADV
ejpam-4276	79	11	,	,	PUNCT
ejpam-4276	79	12	sp	sp	NOUN
ejpam-4276	79	13	)	)	PUNCT
ejpam-4276	79	14	⊆	⊆	NUM
ejpam-4276	79	15	b(λ	b(λ	NOUN
ejpam-4276	79	16	,	,	PUNCT
ejpam-4276	79	17	sp	sp	NOUN
ejpam-4276	79	18	)	)	PUNCT
ejpam-4276	79	19	.	.	PUNCT
ejpam-4276	80	1	(	(	PUNCT
ejpam-4276	80	2	3	3	X
ejpam-4276	80	3	)	)	PUNCT
ejpam-4276	80	4	a(λ	a(λ	ADV
ejpam-4276	80	5	,	,	PUNCT
ejpam-4276	80	6	sp	sp	NOUN
ejpam-4276	80	7	)	)	PUNCT
ejpam-4276	80	8	is	be	AUX
ejpam-4276	80	9	(	(	PUNCT
ejpam-4276	80	10	λ	λ	X
ejpam-4276	80	11	,	,	PUNCT
ejpam-4276	80	12	sp)-closed	sp)-close	VERB
ejpam-4276	80	13	.	.	PUNCT
ejpam-4276	81	1	(	(	PUNCT
ejpam-4276	81	2	4	4	X
ejpam-4276	81	3	)	)	PUNCT
ejpam-4276	81	4	a	a	PRON
ejpam-4276	81	5	is	be	AUX
ejpam-4276	81	6	(	(	PUNCT
ejpam-4276	81	7	λ	λ	X
ejpam-4276	81	8	,	,	PUNCT
ejpam-4276	81	9	sp)-closed	sp)-close	VERB
ejpam-4276	81	10	if	if	SCONJ
ejpam-4276	81	11	and	and	CCONJ
ejpam-4276	81	12	only	only	ADV
ejpam-4276	81	13	if	if	SCONJ
ejpam-4276	81	14	a(λ	a(λ	ADV
ejpam-4276	81	15	,	,	PUNCT
ejpam-4276	81	16	sp	sp	NOUN
ejpam-4276	81	17	)	)	PUNCT
ejpam-4276	81	18	=	=	VERB
ejpam-4276	81	19	a.	a.	NOUN
ejpam-4276	81	20	let	let	VERB
ejpam-4276	81	21	a	a	PRON
ejpam-4276	81	22	be	be	AUX
ejpam-4276	81	23	a	a	DET
ejpam-4276	81	24	subset	subset	NOUN
ejpam-4276	81	25	of	of	ADP
ejpam-4276	81	26	a	a	DET
ejpam-4276	81	27	topological	topological	ADJ
ejpam-4276	81	28	space	space	NOUN
ejpam-4276	81	29	(	(	PUNCT
ejpam-4276	81	30	x	x	X
ejpam-4276	81	31	,	,	PUNCT
ejpam-4276	81	32	τ	τ	PROPN
ejpam-4276	81	33	)	)	PUNCT
ejpam-4276	81	34	.	.	PUNCT
ejpam-4276	82	1	the	the	DET
ejpam-4276	82	2	union	union	NOUN
ejpam-4276	82	3	of	of	ADP
ejpam-4276	82	4	all	all	DET
ejpam-4276	82	5	(	(	PUNCT
ejpam-4276	82	6	λ	λ	NOUN
ejpam-4276	82	7	,	,	PUNCT
ejpam-4276	82	8	sp)-open	sp)-open	ADJ
ejpam-4276	82	9	sets	set	NOUN
ejpam-4276	82	10	contained	contain	VERB
ejpam-4276	82	11	in	in	ADP
ejpam-4276	82	12	a	a	PRON
ejpam-4276	82	13	is	be	AUX
ejpam-4276	82	14	called	call	VERB
ejpam-4276	82	15	the	the	DET
ejpam-4276	82	16	(	(	PUNCT
ejpam-4276	82	17	λ	λ	PROPN
ejpam-4276	82	18	,	,	PUNCT
ejpam-4276	82	19	sp)-interior	sp)-interior	NOUN
ejpam-4276	82	20	[	[	X
ejpam-4276	82	21	3	3	NUM
ejpam-4276	82	22	]	]	PUNCT
ejpam-4276	82	23	of	of	ADP
ejpam-4276	82	24	a	a	PRON
ejpam-4276	82	25	and	and	CCONJ
ejpam-4276	82	26	is	be	AUX
ejpam-4276	82	27	denoted	denote	VERB
ejpam-4276	82	28	by	by	ADP
ejpam-4276	82	29	a(λ	a(λ	ADV
ejpam-4276	82	30	,	,	PUNCT
ejpam-4276	82	31	sp	sp	NOUN
ejpam-4276	82	32	)	)	PUNCT
ejpam-4276	82	33	.	.	PUNCT
ejpam-4276	83	1	lemma	lemma	PROPN
ejpam-4276	83	2	4	4	NUM
ejpam-4276	83	3	.	.	PUNCT
ejpam-4276	84	1	[	[	X
ejpam-4276	84	2	3	3	X
ejpam-4276	84	3	]	]	PUNCT
ejpam-4276	84	4	for	for	ADP
ejpam-4276	84	5	subsets	subset	NOUN
ejpam-4276	84	6	a	a	PRON
ejpam-4276	84	7	and	and	CCONJ
ejpam-4276	84	8	b	b	NOUN
ejpam-4276	84	9	of	of	ADP
ejpam-4276	84	10	a	a	DET
ejpam-4276	84	11	topological	topological	ADJ
ejpam-4276	84	12	space	space	NOUN
ejpam-4276	84	13	(	(	PUNCT
ejpam-4276	84	14	x	x	X
ejpam-4276	84	15	,	,	PUNCT
ejpam-4276	84	16	τ	τ	PROPN
ejpam-4276	84	17	)	)	PUNCT
ejpam-4276	84	18	,	,	PUNCT
ejpam-4276	84	19	the	the	DET
ejpam-4276	84	20	following	follow	VERB
ejpam-4276	84	21	properties	property	NOUN
ejpam-4276	84	22	hold	hold	VERB
ejpam-4276	84	23	:	:	PUNCT
ejpam-4276	84	24	(	(	PUNCT
ejpam-4276	84	25	1	1	X
ejpam-4276	84	26	)	)	PUNCT
ejpam-4276	84	27	a(λ	a(λ	ADV
ejpam-4276	84	28	,	,	PUNCT
ejpam-4276	84	29	sp	sp	NOUN
ejpam-4276	84	30	)	)	PUNCT
ejpam-4276	84	31	⊆	⊆	NUM
ejpam-4276	84	32	a	a	DET
ejpam-4276	84	33	and	and	CCONJ
ejpam-4276	84	34	[	[	X
ejpam-4276	84	35	a(λ	a(λ	ADV
ejpam-4276	84	36	,	,	PUNCT
ejpam-4276	84	37	sp)](λ	sp)](λ	PROPN
ejpam-4276	84	38	,	,	PUNCT
ejpam-4276	84	39	sp	sp	NOUN
ejpam-4276	84	40	)	)	PUNCT
ejpam-4276	84	41	=	=	PUNCT
ejpam-4276	84	42	a(λ	a(λ	ADV
ejpam-4276	84	43	,	,	PUNCT
ejpam-4276	84	44	sp	sp	NOUN
ejpam-4276	84	45	)	)	PUNCT
ejpam-4276	84	46	.	.	PUNCT
ejpam-4276	85	1	(	(	PUNCT
ejpam-4276	85	2	2	2	X
ejpam-4276	85	3	)	)	PUNCT
ejpam-4276	85	4	if	if	SCONJ
ejpam-4276	85	5	a	a	DET
ejpam-4276	85	6	⊆	⊆	NUM
ejpam-4276	85	7	b	b	NOUN
ejpam-4276	85	8	,	,	PUNCT
ejpam-4276	85	9	then	then	ADV
ejpam-4276	85	10	a(λ	a(λ	ADV
ejpam-4276	85	11	,	,	PUNCT
ejpam-4276	85	12	sp	sp	NOUN
ejpam-4276	85	13	)	)	PUNCT
ejpam-4276	85	14	⊆	⊆	NUM
ejpam-4276	85	15	b(λ	b(λ	NOUN
ejpam-4276	85	16	,	,	PUNCT
ejpam-4276	85	17	sp	sp	NOUN
ejpam-4276	85	18	)	)	PUNCT
ejpam-4276	85	19	.	.	PUNCT
ejpam-4276	86	1	(	(	PUNCT
ejpam-4276	86	2	3	3	X
ejpam-4276	86	3	)	)	PUNCT
ejpam-4276	86	4	a(λ	a(λ	ADV
ejpam-4276	86	5	,	,	PUNCT
ejpam-4276	86	6	sp	sp	NOUN
ejpam-4276	86	7	)	)	PUNCT
ejpam-4276	86	8	is	be	AUX
ejpam-4276	86	9	(	(	PUNCT
ejpam-4276	86	10	λ	λ	INTJ
ejpam-4276	86	11	,	,	PUNCT
ejpam-4276	86	12	sp)-open	sp)-open	NOUN
ejpam-4276	86	13	.	.	PUNCT
ejpam-4276	87	1	(	(	PUNCT
ejpam-4276	87	2	4	4	X
ejpam-4276	87	3	)	)	PUNCT
ejpam-4276	87	4	a	a	DET
ejpam-4276	87	5	is	be	AUX
ejpam-4276	87	6	(	(	PUNCT
ejpam-4276	87	7	λ	λ	NOUN
ejpam-4276	87	8	,	,	PUNCT
ejpam-4276	87	9	sp)-open	sp)-open	ADJ
ejpam-4276	87	10	if	if	SCONJ
ejpam-4276	87	11	and	and	CCONJ
ejpam-4276	87	12	only	only	ADV
ejpam-4276	87	13	if	if	SCONJ
ejpam-4276	87	14	a(λ	a(λ	ADV
ejpam-4276	87	15	,	,	PUNCT
ejpam-4276	87	16	sp	sp	NOUN
ejpam-4276	87	17	)	)	PUNCT
ejpam-4276	87	18	=	=	SYM
ejpam-4276	87	19	a.	a.	NOUN
ejpam-4276	87	20	(	(	PUNCT
ejpam-4276	87	21	5	5	NUM
ejpam-4276	87	22	)	)	PUNCT
ejpam-4276	88	1	[	[	X
ejpam-4276	88	2	x	x	X
ejpam-4276	88	3	−a](λ	−a](λ	PROPN
ejpam-4276	88	4	,	,	PUNCT
ejpam-4276	88	5	sp	sp	NOUN
ejpam-4276	88	6	)	)	PUNCT
ejpam-4276	88	7	=	=	SYM
ejpam-4276	88	8	x	x	SYM
ejpam-4276	88	9	−a(λ	−a(λ	NOUN
ejpam-4276	88	10	,	,	PUNCT
ejpam-4276	88	11	sp	sp	NOUN
ejpam-4276	88	12	)	)	PUNCT
ejpam-4276	88	13	.	.	PUNCT
ejpam-4276	89	1	(	(	PUNCT
ejpam-4276	89	2	6	6	NUM
ejpam-4276	89	3	)	)	PUNCT
ejpam-4276	90	1	[	[	X
ejpam-4276	90	2	x	x	X
ejpam-4276	90	3	−a](λ	−a](λ	PROPN
ejpam-4276	90	4	,	,	PUNCT
ejpam-4276	90	5	sp	sp	NOUN
ejpam-4276	90	6	)	)	PUNCT
ejpam-4276	90	7	=	=	SYM
ejpam-4276	90	8	x	x	SYM
ejpam-4276	90	9	−a(λ	−a(λ	NOUN
ejpam-4276	90	10	,	,	PUNCT
ejpam-4276	90	11	sp	sp	NOUN
ejpam-4276	90	12	)	)	PUNCT
ejpam-4276	90	13	.	.	PUNCT
ejpam-4276	91	1	c.	c.	PROPN
ejpam-4276	91	2	boonpok	boonpok	PROPN
ejpam-4276	91	3	,	,	PUNCT
ejpam-4276	91	4	j.	j.	PROPN
ejpam-4276	91	5	khampakdee	khampakdee	PROPN
ejpam-4276	91	6	/	/	PUNCT
ejpam-4276	91	7	eur	eur	PROPN
ejpam-4276	91	8	.	.	PUNCT
ejpam-4276	92	1	j.	j.	PROPN
ejpam-4276	92	2	pure	pure	PROPN
ejpam-4276	92	3	appl	appl	PROPN
ejpam-4276	92	4	.	.	PROPN
ejpam-4276	92	5	math	math	PROPN
ejpam-4276	92	6	,	,	PUNCT
ejpam-4276	92	7	15	15	NUM
ejpam-4276	92	8	(	(	PUNCT
ejpam-4276	92	9	2	2	NUM
ejpam-4276	92	10	)	)	PUNCT
ejpam-4276	92	11	(	(	PUNCT
ejpam-4276	92	12	2022	2022	NUM
ejpam-4276	92	13	)	)	PUNCT
ejpam-4276	92	14	,	,	PUNCT
ejpam-4276	92	15	572	572	NUM
ejpam-4276	92	16	-	-	SYM
ejpam-4276	92	17	588	588	NUM
ejpam-4276	92	18	575	575	NUM
ejpam-4276	92	19	3	3	NUM
ejpam-4276	92	20	.	.	PUNCT
ejpam-4276	93	1	generalized	generalize	VERB
ejpam-4276	93	2	(	(	PUNCT
ejpam-4276	93	3	λ	λ	NOUN
ejpam-4276	93	4	,	,	PUNCT
ejpam-4276	93	5	sp)-open	sp)-open	ADJ
ejpam-4276	93	6	sets	set	NOUN
ejpam-4276	93	7	in	in	ADP
ejpam-4276	93	8	this	this	DET
ejpam-4276	93	9	section	section	NOUN
ejpam-4276	93	10	,	,	PUNCT
ejpam-4276	93	11	we	we	PRON
ejpam-4276	93	12	investigate	investigate	VERB
ejpam-4276	93	13	some	some	DET
ejpam-4276	93	14	properties	property	NOUN
ejpam-4276	93	15	of	of	ADP
ejpam-4276	93	16	s(λ	s(λ	PROPN
ejpam-4276	93	17	,	,	PUNCT
ejpam-4276	93	18	sp)-open	sp)-open	ADJ
ejpam-4276	93	19	sets	set	NOUN
ejpam-4276	93	20	,	,	PUNCT
ejpam-4276	93	21	p(λ	p(λ	NOUN
ejpam-4276	93	22	,	,	PUNCT
ejpam-4276	93	23	sp)-open	sp)-open	ADJ
ejpam-4276	93	24	sets	set	NOUN
ejpam-4276	93	25	,	,	PUNCT
ejpam-4276	93	26	α(λ	α(λ	PROPN
ejpam-4276	93	27	,	,	PUNCT
ejpam-4276	93	28	sp)-open	sp)-open	ADJ
ejpam-4276	93	29	sets	set	NOUN
ejpam-4276	93	30	,	,	PUNCT
ejpam-4276	93	31	β(λ	β(λ	X
ejpam-4276	93	32	,	,	PUNCT
ejpam-4276	93	33	sp)-open	sp)-open	ADJ
ejpam-4276	93	34	sets	set	NOUN
ejpam-4276	93	35	and	and	CCONJ
ejpam-4276	93	36	b(λ	b(λ	NOUN
ejpam-4276	93	37	,	,	PUNCT
ejpam-4276	93	38	sp)-open	sp)-open	ADJ
ejpam-4276	93	39	sets	set	NOUN
ejpam-4276	93	40	.	.	PUNCT
ejpam-4276	94	1	definition	definition	NOUN
ejpam-4276	94	2	1	1	NUM
ejpam-4276	94	3	.	.	PUNCT
ejpam-4276	95	1	[	[	X
ejpam-4276	95	2	3	3	X
ejpam-4276	95	3	]	]	PUNCT
ejpam-4276	95	4	a	a	DET
ejpam-4276	95	5	subset	subset	NOUN
ejpam-4276	95	6	a	a	PRON
ejpam-4276	95	7	of	of	ADP
ejpam-4276	95	8	a	a	DET
ejpam-4276	95	9	topological	topological	ADJ
ejpam-4276	95	10	space	space	NOUN
ejpam-4276	95	11	(	(	PUNCT
ejpam-4276	95	12	x	x	X
ejpam-4276	95	13	,	,	PUNCT
ejpam-4276	95	14	τ	τ	X
ejpam-4276	95	15	)	)	PUNCT
ejpam-4276	95	16	is	be	AUX
ejpam-4276	95	17	said	say	VERB
ejpam-4276	95	18	to	to	PART
ejpam-4276	95	19	be	be	AUX
ejpam-4276	95	20	:	:	PUNCT
ejpam-4276	95	21	(	(	PUNCT
ejpam-4276	95	22	i	i	NOUN
ejpam-4276	95	23	)	)	PUNCT
ejpam-4276	95	24	s(λ	s(λ	PROPN
ejpam-4276	95	25	,	,	PUNCT
ejpam-4276	95	26	sp)-open	sp)-open	VERB
ejpam-4276	95	27	if	if	SCONJ
ejpam-4276	95	28	a	a	DET
ejpam-4276	95	29	⊆	⊆	NUM
ejpam-4276	95	30	[	[	X
ejpam-4276	95	31	a(λ	a(λ	ADV
ejpam-4276	95	32	,	,	PUNCT
ejpam-4276	95	33	sp	sp	NOUN
ejpam-4276	95	34	)	)	PUNCT
ejpam-4276	95	35	]	]	PUNCT
ejpam-4276	95	36	(	(	PUNCT
ejpam-4276	95	37	λ	λ	NOUN
ejpam-4276	95	38	,	,	PUNCT
ejpam-4276	95	39	sp	sp	NOUN
ejpam-4276	95	40	)	)	PUNCT
ejpam-4276	95	41	;	;	PUNCT
ejpam-4276	95	42	(	(	PUNCT
ejpam-4276	95	43	ii	ii	NOUN
ejpam-4276	95	44	)	)	PUNCT
ejpam-4276	95	45	p(λ	p(λ	NOUN
ejpam-4276	95	46	,	,	PUNCT
ejpam-4276	95	47	sp)-open	sp)-open	VERB
ejpam-4276	95	48	if	if	SCONJ
ejpam-4276	95	49	a	a	DET
ejpam-4276	95	50	⊆	⊆	NUM
ejpam-4276	95	51	[	[	X
ejpam-4276	95	52	a(λ	a(λ	ADJ
ejpam-4276	95	53	,	,	PUNCT
ejpam-4276	95	54	sp)](λ	sp)](λ	PROPN
ejpam-4276	95	55	,	,	PUNCT
ejpam-4276	95	56	sp	sp	NOUN
ejpam-4276	95	57	)	)	PUNCT
ejpam-4276	95	58	;	;	PUNCT
ejpam-4276	95	59	(	(	PUNCT
ejpam-4276	95	60	iii	iii	X
ejpam-4276	95	61	)	)	PUNCT
ejpam-4276	95	62	α(λ	α(λ	PROPN
ejpam-4276	95	63	,	,	PUNCT
ejpam-4276	95	64	sp)-open	sp)-open	VERB
ejpam-4276	95	65	if	if	SCONJ
ejpam-4276	95	66	a	a	DET
ejpam-4276	95	67	⊆	⊆	NUM
ejpam-4276	95	68	[	[	X
ejpam-4276	95	69	[	[	X
ejpam-4276	95	70	a(λ	a(λ	ADJ
ejpam-4276	95	71	,	,	PUNCT
ejpam-4276	95	72	sp	sp	NOUN
ejpam-4276	95	73	)	)	PUNCT
ejpam-4276	95	74	]	]	PUNCT
ejpam-4276	95	75	(	(	PUNCT
ejpam-4276	95	76	λ	λ	X
ejpam-4276	95	77	,	,	PUNCT
ejpam-4276	95	78	sp)](λ	sp)](λ	PROPN
ejpam-4276	95	79	,	,	PUNCT
ejpam-4276	95	80	sp	sp	NOUN
ejpam-4276	95	81	)	)	PUNCT
ejpam-4276	95	82	;	;	PUNCT
ejpam-4276	95	83	(	(	PUNCT
ejpam-4276	95	84	iv	iv	X
ejpam-4276	95	85	)	)	PUNCT
ejpam-4276	95	86	β(λ	β(λ	NOUN
ejpam-4276	95	87	,	,	PUNCT
ejpam-4276	95	88	sp)-open	sp)-open	VERB
ejpam-4276	95	89	if	if	SCONJ
ejpam-4276	95	90	a	a	DET
ejpam-4276	95	91	⊆	⊆	NUM
ejpam-4276	95	92	[	[	X
ejpam-4276	95	93	[	[	X
ejpam-4276	95	94	a(λ	a(λ	ADJ
ejpam-4276	95	95	,	,	PUNCT
ejpam-4276	95	96	sp)](λ	sp)](λ	PROPN
ejpam-4276	95	97	,	,	PUNCT
ejpam-4276	95	98	sp	sp	NOUN
ejpam-4276	95	99	)	)	PUNCT
ejpam-4276	95	100	]	]	PUNCT
ejpam-4276	95	101	(	(	PUNCT
ejpam-4276	95	102	λ	λ	NOUN
ejpam-4276	95	103	,	,	PUNCT
ejpam-4276	95	104	sp	sp	NOUN
ejpam-4276	95	105	)	)	PUNCT
ejpam-4276	95	106	.	.	PUNCT
ejpam-4276	96	1	the	the	DET
ejpam-4276	96	2	family	family	NOUN
ejpam-4276	96	3	of	of	ADP
ejpam-4276	96	4	all	all	DET
ejpam-4276	96	5	s(λ	s(λ	NOUN
ejpam-4276	96	6	,	,	PUNCT
ejpam-4276	96	7	sp)-open	sp)-open	ADJ
ejpam-4276	96	8	(	(	PUNCT
ejpam-4276	96	9	resp	resp	NOUN
ejpam-4276	96	10	.	.	PUNCT
ejpam-4276	97	1	p(λ	p(λ	NOUN
ejpam-4276	97	2	,	,	PUNCT
ejpam-4276	97	3	sp)-open	sp)-open	NOUN
ejpam-4276	97	4	,	,	PUNCT
ejpam-4276	97	5	α(λ	α(λ	PROPN
ejpam-4276	97	6	,	,	PUNCT
ejpam-4276	97	7	sp)-open	sp)-open	NOUN
ejpam-4276	97	8	,	,	PUNCT
ejpam-4276	97	9	β(λ	β(λ	X
ejpam-4276	97	10	,	,	PUNCT
ejpam-4276	97	11	sp)-open	sp)-open	NOUN
ejpam-4276	97	12	)	)	PUNCT
ejpam-4276	97	13	sets	set	NOUN
ejpam-4276	97	14	in	in	ADP
ejpam-4276	97	15	a	a	DET
ejpam-4276	97	16	topological	topological	ADJ
ejpam-4276	97	17	space	space	NOUN
ejpam-4276	97	18	(	(	PUNCT
ejpam-4276	97	19	x	x	X
ejpam-4276	97	20	,	,	PUNCT
ejpam-4276	97	21	τ	τ	X
ejpam-4276	97	22	)	)	PUNCT
ejpam-4276	97	23	is	be	AUX
ejpam-4276	97	24	denoted	denote	VERB
ejpam-4276	97	25	by	by	ADP
ejpam-4276	97	26	sλspo(x	sλspo(x	PROPN
ejpam-4276	97	27	,	,	PUNCT
ejpam-4276	97	28	τ	τ	PROPN
ejpam-4276	97	29	)	)	PUNCT
ejpam-4276	97	30	(	(	PUNCT
ejpam-4276	97	31	resp	resp	NOUN
ejpam-4276	97	32	.	.	PUNCT
ejpam-4276	98	1	pλspo(x	pλspo(x	ADJ
ejpam-4276	98	2	,	,	PUNCT
ejpam-4276	98	3	τ	τ	PROPN
ejpam-4276	98	4	)	)	PUNCT
ejpam-4276	98	5	,	,	PUNCT
ejpam-4276	98	6	αλspo(x	αλspo(x	NOUN
ejpam-4276	98	7	,	,	PUNCT
ejpam-4276	98	8	τ	τ	PROPN
ejpam-4276	98	9	)	)	PUNCT
ejpam-4276	98	10	,	,	PUNCT
ejpam-4276	98	11	βλspo(x	βλspo(x	PROPN
ejpam-4276	98	12	,	,	PUNCT
ejpam-4276	98	13	τ	τ	PROPN
ejpam-4276	98	14	)	)	PUNCT
ejpam-4276	98	15	)	)	PUNCT
ejpam-4276	98	16	.	.	PUNCT
ejpam-4276	99	1	the	the	DET
ejpam-4276	99	2	complement	complement	NOUN
ejpam-4276	99	3	of	of	ADP
ejpam-4276	99	4	a	a	DET
ejpam-4276	99	5	s(λ	s(λ	PROPN
ejpam-4276	99	6	,	,	PUNCT
ejpam-4276	99	7	sp)-open	sp)-open	ADJ
ejpam-4276	99	8	(	(	PUNCT
ejpam-4276	99	9	resp	resp	NOUN
ejpam-4276	99	10	.	.	PUNCT
ejpam-4276	100	1	p(λ	p(λ	NOUN
ejpam-4276	100	2	,	,	PUNCT
ejpam-4276	100	3	sp)-open	sp)-open	NOUN
ejpam-4276	100	4	,	,	PUNCT
ejpam-4276	100	5	α(λ	α(λ	PROPN
ejpam-4276	100	6	,	,	PUNCT
ejpam-4276	100	7	sp)-open	sp)-open	NOUN
ejpam-4276	100	8	,	,	PUNCT
ejpam-4276	100	9	β(λ	β(λ	X
ejpam-4276	100	10	,	,	PUNCT
ejpam-4276	100	11	sp)-open	sp)-open	NOUN
ejpam-4276	100	12	)	)	PUNCT
ejpam-4276	100	13	set	set	NOUN
ejpam-4276	100	14	is	be	AUX
ejpam-4276	100	15	called	call	VERB
ejpam-4276	100	16	s(λ	s(λ	PROPN
ejpam-4276	100	17	,	,	PUNCT
ejpam-4276	100	18	sp)-closed	sp)-close	VERB
ejpam-4276	100	19	(	(	PUNCT
ejpam-4276	100	20	resp	resp	NOUN
ejpam-4276	100	21	.	.	PUNCT
ejpam-4276	101	1	p(λ	p(λ	NOUN
ejpam-4276	101	2	,	,	PUNCT
ejpam-4276	101	3	sp)-closed	sp)-close	VERB
ejpam-4276	101	4	,	,	PUNCT
ejpam-4276	101	5	α(λ	α(λ	PROPN
ejpam-4276	101	6	,	,	PUNCT
ejpam-4276	101	7	sp)-closed	sp)-close	VERB
ejpam-4276	101	8	,	,	PUNCT
ejpam-4276	101	9	β(λ	β(λ	X
ejpam-4276	101	10	,	,	PUNCT
ejpam-4276	101	11	sp)closed	sp)close	VERB
ejpam-4276	101	12	)	)	PUNCT
ejpam-4276	101	13	.	.	PUNCT
ejpam-4276	102	1	the	the	DET
ejpam-4276	102	2	family	family	NOUN
ejpam-4276	102	3	of	of	ADP
ejpam-4276	102	4	all	all	DET
ejpam-4276	102	5	s(λ	s(λ	NOUN
ejpam-4276	102	6	,	,	PUNCT
ejpam-4276	102	7	sp)-closed	sp)-close	VERB
ejpam-4276	102	8	(	(	PUNCT
ejpam-4276	102	9	resp	resp	NOUN
ejpam-4276	102	10	.	.	PUNCT
ejpam-4276	103	1	p(λ	p(λ	NOUN
ejpam-4276	103	2	,	,	PUNCT
ejpam-4276	103	3	sp)-closed	sp)-close	VERB
ejpam-4276	103	4	,	,	PUNCT
ejpam-4276	103	5	α(λ	α(λ	PROPN
ejpam-4276	103	6	,	,	PUNCT
ejpam-4276	103	7	sp)-closed	sp)-close	VERB
ejpam-4276	103	8	,	,	PUNCT
ejpam-4276	103	9	β(λ	β(λ	X
ejpam-4276	103	10	,	,	PUNCT
ejpam-4276	103	11	sp)closed	sp)close	VERB
ejpam-4276	103	12	)	)	PUNCT
ejpam-4276	103	13	sets	set	NOUN
ejpam-4276	103	14	in	in	ADP
ejpam-4276	103	15	a	a	DET
ejpam-4276	103	16	topological	topological	ADJ
ejpam-4276	103	17	space	space	NOUN
ejpam-4276	103	18	(	(	PUNCT
ejpam-4276	103	19	x	x	X
ejpam-4276	103	20	,	,	PUNCT
ejpam-4276	103	21	τ	τ	X
ejpam-4276	103	22	)	)	PUNCT
ejpam-4276	103	23	is	be	AUX
ejpam-4276	103	24	denoted	denote	VERB
ejpam-4276	103	25	by	by	ADP
ejpam-4276	103	26	sλspc(x	sλspc(x	PROPN
ejpam-4276	103	27	,	,	PUNCT
ejpam-4276	103	28	τ	τ	PROPN
ejpam-4276	103	29	)	)	PUNCT
ejpam-4276	103	30	(	(	PUNCT
ejpam-4276	103	31	resp	resp	NOUN
ejpam-4276	103	32	.	.	PUNCT
ejpam-4276	104	1	pλspc(x	pλspc(x	NOUN
ejpam-4276	104	2	,	,	PUNCT
ejpam-4276	104	3	τ	τ	PROPN
ejpam-4276	104	4	)	)	PUNCT
ejpam-4276	104	5	,	,	PUNCT
ejpam-4276	104	6	αλspc(x	αλspc(x	NOUN
ejpam-4276	104	7	,	,	PUNCT
ejpam-4276	104	8	τ	τ	PROPN
ejpam-4276	104	9	)	)	PUNCT
ejpam-4276	104	10	,	,	PUNCT
ejpam-4276	104	11	βλspc(x	βλspc(x	PROPN
ejpam-4276	104	12	,	,	PUNCT
ejpam-4276	104	13	τ	τ	PROPN
ejpam-4276	104	14	)	)	PUNCT
ejpam-4276	104	15	)	)	PUNCT
ejpam-4276	104	16	.	.	PUNCT
ejpam-4276	105	1	proposition	proposition	NOUN
ejpam-4276	105	2	1	1	NUM
ejpam-4276	105	3	.	.	PUNCT
ejpam-4276	106	1	for	for	ADP
ejpam-4276	106	2	a	a	DET
ejpam-4276	106	3	topological	topological	ADJ
ejpam-4276	106	4	space	space	NOUN
ejpam-4276	106	5	(	(	PUNCT
ejpam-4276	106	6	x	x	X
ejpam-4276	106	7	,	,	PUNCT
ejpam-4276	106	8	τ	τ	PROPN
ejpam-4276	106	9	)	)	PUNCT
ejpam-4276	106	10	,	,	PUNCT
ejpam-4276	106	11	the	the	DET
ejpam-4276	106	12	following	follow	VERB
ejpam-4276	106	13	properties	property	NOUN
ejpam-4276	106	14	hold	hold	VERB
ejpam-4276	106	15	:	:	PUNCT
ejpam-4276	106	16	(	(	PUNCT
ejpam-4276	106	17	1	1	X
ejpam-4276	106	18	)	)	PUNCT
ejpam-4276	106	19	λspo(x	λspo(x	NOUN
ejpam-4276	106	20	,	,	PUNCT
ejpam-4276	106	21	τ	τ	PROPN
ejpam-4276	106	22	)	)	PUNCT
ejpam-4276	106	23	⊆	⊆	NUM
ejpam-4276	106	24	αλspo(x	αλspo(x	NOUN
ejpam-4276	106	25	,	,	PUNCT
ejpam-4276	106	26	τ	τ	PROPN
ejpam-4276	106	27	)	)	PUNCT
ejpam-4276	106	28	⊆	⊆	NUM
ejpam-4276	106	29	sλspo(x	sλspo(x	PROPN
ejpam-4276	106	30	,	,	PUNCT
ejpam-4276	106	31	τ	τ	PROPN
ejpam-4276	106	32	)	)	PUNCT
ejpam-4276	106	33	⊆	⊆	NUM
ejpam-4276	106	34	βλspo(x	βλspo(x	NUM
ejpam-4276	106	35	,	,	PUNCT
ejpam-4276	106	36	τ	τ	PROPN
ejpam-4276	106	37	)	)	PUNCT
ejpam-4276	106	38	.	.	PUNCT
ejpam-4276	107	1	(	(	PUNCT
ejpam-4276	107	2	2	2	X
ejpam-4276	107	3	)	)	PUNCT
ejpam-4276	107	4	αλspo(x	αλspo(x	NOUN
ejpam-4276	107	5	,	,	PUNCT
ejpam-4276	107	6	τ	τ	PROPN
ejpam-4276	107	7	)	)	PUNCT
ejpam-4276	107	8	⊆	⊆	NUM
ejpam-4276	107	9	pλspo(x	pλspo(x	NOUN
ejpam-4276	107	10	,	,	PUNCT
ejpam-4276	107	11	τ	τ	X
ejpam-4276	107	12	)	)	PUNCT
ejpam-4276	107	13	⊆	⊆	NUM
ejpam-4276	107	14	βλspo(x	βλspo(x	NUM
ejpam-4276	107	15	,	,	PUNCT
ejpam-4276	107	16	τ	τ	PROPN
ejpam-4276	107	17	)	)	PUNCT
ejpam-4276	107	18	.	.	PUNCT
ejpam-4276	108	1	(	(	PUNCT
ejpam-4276	108	2	3	3	X
ejpam-4276	108	3	)	)	PUNCT
ejpam-4276	108	4	αλspo(x	αλspo(x	NOUN
ejpam-4276	108	5	,	,	PUNCT
ejpam-4276	108	6	τ	τ	X
ejpam-4276	108	7	)	)	PUNCT
ejpam-4276	109	1	=	=	SYM
ejpam-4276	109	2	sλspo(x	sλspo(x	PROPN
ejpam-4276	109	3	,	,	PUNCT
ejpam-4276	109	4	τ	τ	NOUN
ejpam-4276	109	5	)	)	PUNCT
ejpam-4276	109	6	∩	∩	X
ejpam-4276	109	7	pλspo(x	pλspo(x	ADJ
ejpam-4276	109	8	,	,	PUNCT
ejpam-4276	109	9	τ	τ	PROPN
ejpam-4276	109	10	)	)	PUNCT
ejpam-4276	109	11	.	.	PUNCT
ejpam-4276	110	1	proof	proof	NOUN
ejpam-4276	110	2	.	.	PUNCT
ejpam-4276	111	1	(	(	PUNCT
ejpam-4276	111	2	1	1	X
ejpam-4276	111	3	)	)	PUNCT
ejpam-4276	111	4	let	let	VERB
ejpam-4276	111	5	v	v	NOUN
ejpam-4276	111	6	∈	∈	PROPN
ejpam-4276	111	7	λspo(x	λspo(x	NOUN
ejpam-4276	111	8	,	,	PUNCT
ejpam-4276	111	9	τ	τ	PROPN
ejpam-4276	111	10	)	)	PUNCT
ejpam-4276	111	11	.	.	PUNCT
ejpam-4276	112	1	then	then	ADV
ejpam-4276	112	2	,	,	PUNCT
ejpam-4276	112	3	we	we	PRON
ejpam-4276	112	4	have	have	VERB
ejpam-4276	112	5	v	v	NOUN
ejpam-4276	112	6	=	=	SYM
ejpam-4276	112	7	v(λ	v(λ	PROPN
ejpam-4276	112	8	,	,	PUNCT
ejpam-4276	112	9	sp	sp	NOUN
ejpam-4276	112	10	)	)	PUNCT
ejpam-4276	112	11	⊆	⊆	NUM
ejpam-4276	113	1	[	[	X
ejpam-4276	113	2	[	[	X
ejpam-4276	113	3	v(λ	v(λ	PROPN
ejpam-4276	113	4	,	,	PUNCT
ejpam-4276	113	5	sp	sp	NOUN
ejpam-4276	113	6	)	)	PUNCT
ejpam-4276	113	7	]	]	PUNCT
ejpam-4276	114	1	(	(	PUNCT
ejpam-4276	114	2	λ	λ	X
ejpam-4276	114	3	,	,	PUNCT
ejpam-4276	114	4	sp)](λ	sp)](λ	PROPN
ejpam-4276	114	5	,	,	PUNCT
ejpam-4276	114	6	sp	sp	NOUN
ejpam-4276	114	7	)	)	PUNCT
ejpam-4276	114	8	⊆	⊆	NUM
ejpam-4276	114	9	[	[	X
ejpam-4276	114	10	v	v	X
ejpam-4276	114	11	(	(	PUNCT
ejpam-4276	114	12	λ	λ	PROPN
ejpam-4276	114	13	,	,	PUNCT
ejpam-4276	114	14	sp)](λ	sp)](λ	PROPN
ejpam-4276	114	15	,	,	PUNCT
ejpam-4276	114	16	sp	sp	NOUN
ejpam-4276	114	17	)	)	PUNCT
ejpam-4276	114	18	⊆	⊆	NUM
ejpam-4276	115	1	[	[	X
ejpam-4276	115	2	[	[	X
ejpam-4276	115	3	v	v	X
ejpam-4276	115	4	(	(	PUNCT
ejpam-4276	115	5	λ	λ	PROPN
ejpam-4276	115	6	,	,	PUNCT
ejpam-4276	115	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	115	8	,	,	PUNCT
ejpam-4276	115	9	sp	sp	NOUN
ejpam-4276	115	10	)	)	PUNCT
ejpam-4276	115	11	]	]	PUNCT
ejpam-4276	115	12	(	(	PUNCT
ejpam-4276	115	13	λ	λ	NOUN
ejpam-4276	115	14	,	,	PUNCT
ejpam-4276	115	15	sp	sp	NOUN
ejpam-4276	115	16	)	)	PUNCT
ejpam-4276	115	17	.	.	PUNCT
ejpam-4276	116	1	thus	thus	ADV
ejpam-4276	116	2	,	,	PUNCT
ejpam-4276	116	3	λspo(x	λspo(x	PROPN
ejpam-4276	116	4	,	,	PUNCT
ejpam-4276	116	5	τ	τ	PROPN
ejpam-4276	116	6	)	)	PUNCT
ejpam-4276	116	7	⊆	⊆	NUM
ejpam-4276	116	8	αλspo(x	αλspo(x	NOUN
ejpam-4276	116	9	,	,	PUNCT
ejpam-4276	116	10	τ	τ	PROPN
ejpam-4276	116	11	)	)	PUNCT
ejpam-4276	116	12	⊆	⊆	NUM
ejpam-4276	116	13	sλspo(x	sλspo(x	PROPN
ejpam-4276	116	14	,	,	PUNCT
ejpam-4276	116	15	τ	τ	PROPN
ejpam-4276	116	16	)	)	PUNCT
ejpam-4276	116	17	⊆	⊆	NUM
ejpam-4276	116	18	βλspo(x	βλspo(x	NUM
ejpam-4276	116	19	,	,	PUNCT
ejpam-4276	116	20	τ	τ	PROPN
ejpam-4276	116	21	)	)	PUNCT
ejpam-4276	116	22	.	.	PUNCT
ejpam-4276	117	1	(	(	PUNCT
ejpam-4276	117	2	2	2	X
ejpam-4276	117	3	)	)	PUNCT
ejpam-4276	117	4	let	let	VERB
ejpam-4276	117	5	v	v	NUM
ejpam-4276	117	6	∈	∈	PROPN
ejpam-4276	117	7	αλspo(x	αλspo(x	NOUN
ejpam-4276	117	8	,	,	PUNCT
ejpam-4276	117	9	τ	τ	PROPN
ejpam-4276	117	10	)	)	PUNCT
ejpam-4276	117	11	.	.	PUNCT
ejpam-4276	118	1	then	then	ADV
ejpam-4276	118	2	,	,	PUNCT
ejpam-4276	118	3	v	v	ADP
ejpam-4276	118	4	⊆	⊆	NUM
ejpam-4276	118	5	[	[	X
ejpam-4276	118	6	v	v	X
ejpam-4276	118	7	(	(	PUNCT
ejpam-4276	118	8	λ	λ	PROPN
ejpam-4276	118	9	,	,	PUNCT
ejpam-4276	118	10	sp)](λ	sp)](λ	PROPN
ejpam-4276	118	11	,	,	PUNCT
ejpam-4276	118	12	sp	sp	NOUN
ejpam-4276	118	13	)	)	PUNCT
ejpam-4276	118	14	⊆	⊆	NUM
ejpam-4276	119	1	[	[	X
ejpam-4276	119	2	[	[	X
ejpam-4276	119	3	v	v	X
ejpam-4276	119	4	(	(	PUNCT
ejpam-4276	119	5	λ	λ	PROPN
ejpam-4276	119	6	,	,	PUNCT
ejpam-4276	119	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	119	8	,	,	PUNCT
ejpam-4276	119	9	sp	sp	NOUN
ejpam-4276	119	10	)	)	PUNCT
ejpam-4276	119	11	]	]	PUNCT
ejpam-4276	119	12	(	(	PUNCT
ejpam-4276	119	13	λ	λ	NOUN
ejpam-4276	119	14	,	,	PUNCT
ejpam-4276	119	15	sp	sp	NOUN
ejpam-4276	119	16	)	)	PUNCT
ejpam-4276	119	17	and	and	CCONJ
ejpam-4276	119	18	hence	hence	ADV
ejpam-4276	119	19	αλspo(x	αλspo(x	NUM
ejpam-4276	119	20	,	,	PUNCT
ejpam-4276	119	21	τ	τ	PROPN
ejpam-4276	119	22	)	)	PUNCT
ejpam-4276	119	23	⊆	⊆	NUM
ejpam-4276	119	24	pλspo(x	pλspo(x	NOUN
ejpam-4276	119	25	,	,	PUNCT
ejpam-4276	119	26	τ	τ	X
ejpam-4276	119	27	)	)	PUNCT
ejpam-4276	119	28	⊆	⊆	NUM
ejpam-4276	119	29	βλspo(x	βλspo(x	NUM
ejpam-4276	119	30	,	,	PUNCT
ejpam-4276	119	31	τ	τ	PROPN
ejpam-4276	119	32	)	)	PUNCT
ejpam-4276	119	33	.	.	PUNCT
ejpam-4276	120	1	(	(	PUNCT
ejpam-4276	120	2	3	3	X
ejpam-4276	120	3	)	)	PUNCT
ejpam-4276	120	4	by	by	ADP
ejpam-4276	120	5	(	(	PUNCT
ejpam-4276	120	6	1	1	NUM
ejpam-4276	120	7	)	)	PUNCT
ejpam-4276	120	8	and	and	CCONJ
ejpam-4276	120	9	(	(	PUNCT
ejpam-4276	120	10	2	2	NUM
ejpam-4276	120	11	)	)	PUNCT
ejpam-4276	120	12	,	,	PUNCT
ejpam-4276	120	13	we	we	PRON
ejpam-4276	120	14	have	have	VERB
ejpam-4276	120	15	αλspo(x	αλspo(x	NOUN
ejpam-4276	120	16	,	,	PUNCT
ejpam-4276	120	17	τ	τ	PROPN
ejpam-4276	120	18	)	)	PUNCT
ejpam-4276	120	19	⊆	⊆	NUM
ejpam-4276	120	20	sλspo(x	sλspo(x	PROPN
ejpam-4276	120	21	,	,	PUNCT
ejpam-4276	120	22	τ	τ	NOUN
ejpam-4276	120	23	)	)	PUNCT
ejpam-4276	120	24	∩	∩	X
ejpam-4276	120	25	pλspo(x	pλspo(x	ADJ
ejpam-4276	120	26	,	,	PUNCT
ejpam-4276	120	27	τ	τ	PROPN
ejpam-4276	120	28	)	)	PUNCT
ejpam-4276	120	29	.	.	PUNCT
ejpam-4276	121	1	let	let	VERB
ejpam-4276	121	2	v	v	X
ejpam-4276	121	3	∈	∈	PROPN
ejpam-4276	121	4	sλspo(x	sλspo(x	PROPN
ejpam-4276	121	5	,	,	PUNCT
ejpam-4276	121	6	τ	τ	NOUN
ejpam-4276	121	7	)	)	PUNCT
ejpam-4276	121	8	∩	∩	X
ejpam-4276	121	9	pλspo(x	pλspo(x	ADJ
ejpam-4276	121	10	,	,	PUNCT
ejpam-4276	121	11	τ	τ	PROPN
ejpam-4276	121	12	)	)	PUNCT
ejpam-4276	121	13	.	.	PUNCT
ejpam-4276	122	1	then	then	ADV
ejpam-4276	122	2	,	,	PUNCT
ejpam-4276	122	3	v	v	PROPN
ejpam-4276	122	4	∈	∈	PROPN
ejpam-4276	122	5	sλspo(x	sλspo(x	PROPN
ejpam-4276	122	6	,	,	PUNCT
ejpam-4276	122	7	τ	τ	X
ejpam-4276	122	8	)	)	PUNCT
ejpam-4276	122	9	and	and	CCONJ
ejpam-4276	122	10	v	v	ADP
ejpam-4276	122	11	∈	∈	PROPN
ejpam-4276	122	12	pλspo(x	pλspo(x	NOUN
ejpam-4276	122	13	,	,	PUNCT
ejpam-4276	122	14	τ	τ	PROPN
ejpam-4276	122	15	)	)	PUNCT
ejpam-4276	122	16	.	.	PUNCT
ejpam-4276	123	1	therefore	therefore	ADV
ejpam-4276	123	2	,	,	PUNCT
ejpam-4276	123	3	v	v	ADP
ejpam-4276	123	4	⊆	⊆	NUM
ejpam-4276	123	5	[	[	X
ejpam-4276	123	6	v(λ	v(λ	PROPN
ejpam-4276	123	7	,	,	PUNCT
ejpam-4276	123	8	sp	sp	NOUN
ejpam-4276	123	9	)	)	PUNCT
ejpam-4276	123	10	]	]	PUNCT
ejpam-4276	123	11	(	(	PUNCT
ejpam-4276	123	12	λ	λ	NOUN
ejpam-4276	123	13	,	,	PUNCT
ejpam-4276	123	14	sp	sp	NOUN
ejpam-4276	123	15	)	)	PUNCT
ejpam-4276	123	16	and	and	CCONJ
ejpam-4276	123	17	v	v	ADP
ejpam-4276	123	18	⊆	⊆	NUM
ejpam-4276	123	19	[	[	X
ejpam-4276	123	20	v	v	X
ejpam-4276	123	21	(	(	PUNCT
ejpam-4276	123	22	λ	λ	PROPN
ejpam-4276	123	23	,	,	PUNCT
ejpam-4276	123	24	sp)](λ	sp)](λ	PROPN
ejpam-4276	123	25	,	,	PUNCT
ejpam-4276	123	26	sp	sp	NOUN
ejpam-4276	123	27	)	)	PUNCT
ejpam-4276	123	28	.	.	PUNCT
ejpam-4276	124	1	thus	thus	ADV
ejpam-4276	124	2	,	,	PUNCT
ejpam-4276	124	3	v	v	ADP
ejpam-4276	124	4	⊆	⊆	NUM
ejpam-4276	124	5	[	[	X
ejpam-4276	124	6	v	v	X
ejpam-4276	124	7	(	(	PUNCT
ejpam-4276	124	8	λ	λ	PROPN
ejpam-4276	124	9	,	,	PUNCT
ejpam-4276	124	10	sp)](λ	sp)](λ	PROPN
ejpam-4276	124	11	,	,	PUNCT
ejpam-4276	124	12	sp	sp	NOUN
ejpam-4276	124	13	)	)	PUNCT
ejpam-4276	124	14	⊆	⊆	NUM
ejpam-4276	124	15	[	[	X
ejpam-4276	124	16	[	[	X
ejpam-4276	124	17	v(λ	v(λ	PROPN
ejpam-4276	124	18	,	,	PUNCT
ejpam-4276	124	19	sp	sp	NOUN
ejpam-4276	124	20	)	)	PUNCT
ejpam-4276	124	21	]	]	PUNCT
ejpam-4276	124	22	(	(	PUNCT
ejpam-4276	124	23	λ	λ	X
ejpam-4276	124	24	,	,	PUNCT
ejpam-4276	124	25	sp)](λ	sp)](λ	PROPN
ejpam-4276	124	26	,	,	PUNCT
ejpam-4276	124	27	sp	sp	NOUN
ejpam-4276	124	28	)	)	PUNCT
ejpam-4276	124	29	and	and	CCONJ
ejpam-4276	124	30	hence	hence	ADV
ejpam-4276	124	31	v	v	ADP
ejpam-4276	124	32	∈	∈	PROPN
ejpam-4276	124	33	αλspo(x	αλspo(x	NOUN
ejpam-4276	124	34	,	,	PUNCT
ejpam-4276	124	35	τ	τ	PROPN
ejpam-4276	124	36	)	)	PUNCT
ejpam-4276	124	37	.	.	PUNCT
ejpam-4276	125	1	consequently	consequently	ADV
ejpam-4276	125	2	,	,	PUNCT
ejpam-4276	125	3	we	we	PRON
ejpam-4276	125	4	obtain	obtain	VERB
ejpam-4276	125	5	sλspo(x	sλspo(x	PROPN
ejpam-4276	125	6	,	,	PUNCT
ejpam-4276	125	7	τ	τ	NOUN
ejpam-4276	125	8	)	)	PUNCT
ejpam-4276	125	9	∩	∩	X
ejpam-4276	125	10	pλspo(x	pλspo(x	ADJ
ejpam-4276	125	11	,	,	PUNCT
ejpam-4276	125	12	τ	τ	X
ejpam-4276	125	13	)	)	PUNCT
ejpam-4276	125	14	⊆	⊆	NUM
ejpam-4276	125	15	αλspo(x	αλspo(x	NOUN
ejpam-4276	125	16	,	,	PUNCT
ejpam-4276	125	17	τ	τ	PROPN
ejpam-4276	125	18	)	)	PUNCT
ejpam-4276	125	19	.	.	PUNCT
ejpam-4276	126	1	this	this	PRON
ejpam-4276	126	2	shows	show	VERB
ejpam-4276	126	3	that	that	SCONJ
ejpam-4276	126	4	αλspo(x	αλspo(x	NOUN
ejpam-4276	126	5	,	,	PUNCT
ejpam-4276	126	6	τ	τ	X
ejpam-4276	126	7	)	)	PUNCT
ejpam-4276	126	8	=	=	SYM
ejpam-4276	126	9	sλspo(x	sλspo(x	PROPN
ejpam-4276	126	10	,	,	PUNCT
ejpam-4276	126	11	τ	τ	NOUN
ejpam-4276	126	12	)	)	PUNCT
ejpam-4276	126	13	∩	∩	X
ejpam-4276	126	14	pλspo(x	pλspo(x	ADJ
ejpam-4276	126	15	,	,	PUNCT
ejpam-4276	126	16	τ	τ	PROPN
ejpam-4276	126	17	)	)	PUNCT
ejpam-4276	126	18	.	.	PUNCT
ejpam-4276	127	1	definition	definition	NOUN
ejpam-4276	127	2	2	2	NUM
ejpam-4276	127	3	.	.	PUNCT
ejpam-4276	128	1	a	a	DET
ejpam-4276	128	2	subset	subset	NOUN
ejpam-4276	128	3	a	a	PRON
ejpam-4276	128	4	of	of	ADP
ejpam-4276	128	5	a	a	DET
ejpam-4276	128	6	topological	topological	ADJ
ejpam-4276	128	7	space	space	NOUN
ejpam-4276	128	8	(	(	PUNCT
ejpam-4276	128	9	x	x	X
ejpam-4276	128	10	,	,	PUNCT
ejpam-4276	128	11	τ	τ	X
ejpam-4276	128	12	)	)	PUNCT
ejpam-4276	128	13	is	be	AUX
ejpam-4276	128	14	said	say	VERB
ejpam-4276	128	15	to	to	PART
ejpam-4276	128	16	be	be	AUX
ejpam-4276	128	17	r(λ	r(λ	NOUN
ejpam-4276	128	18	,	,	PUNCT
ejpam-4276	128	19	sp)-open	sp)-open	ADJ
ejpam-4276	128	20	if	if	SCONJ
ejpam-4276	128	21	a	a	PRON
ejpam-4276	128	22	=	=	X
ejpam-4276	129	1	[	[	X
ejpam-4276	129	2	a(λ	a(λ	PROPN
ejpam-4276	129	3	,	,	PUNCT
ejpam-4276	129	4	sp)](λ	sp)](λ	PROPN
ejpam-4276	129	5	,	,	PUNCT
ejpam-4276	129	6	sp	sp	NOUN
ejpam-4276	129	7	)	)	PUNCT
ejpam-4276	129	8	.	.	PUNCT
ejpam-4276	130	1	the	the	DET
ejpam-4276	130	2	complement	complement	NOUN
ejpam-4276	130	3	of	of	ADP
ejpam-4276	130	4	a	a	DET
ejpam-4276	130	5	r(λ	r(λ	NOUN
ejpam-4276	130	6	,	,	PUNCT
ejpam-4276	130	7	sp)-open	sp)-open	ADJ
ejpam-4276	130	8	set	set	NOUN
ejpam-4276	130	9	is	be	AUX
ejpam-4276	130	10	said	say	VERB
ejpam-4276	130	11	to	to	PART
ejpam-4276	130	12	be	be	AUX
ejpam-4276	130	13	r(λ	r(λ	NOUN
ejpam-4276	130	14	,	,	PUNCT
ejpam-4276	130	15	sp)-closed	sp)-closed	ADJ
ejpam-4276	130	16	.	.	PUNCT
ejpam-4276	131	1	c.	c.	PROPN
ejpam-4276	131	2	boonpok	boonpok	PROPN
ejpam-4276	131	3	,	,	PUNCT
ejpam-4276	131	4	j.	j.	PROPN
ejpam-4276	131	5	khampakdee	khampakdee	PROPN
ejpam-4276	131	6	/	/	PUNCT
ejpam-4276	131	7	eur	eur	PROPN
ejpam-4276	131	8	.	.	PUNCT
ejpam-4276	132	1	j.	j.	PROPN
ejpam-4276	132	2	pure	pure	PROPN
ejpam-4276	132	3	appl	appl	PROPN
ejpam-4276	132	4	.	.	PROPN
ejpam-4276	132	5	math	math	PROPN
ejpam-4276	132	6	,	,	PUNCT
ejpam-4276	132	7	15	15	NUM
ejpam-4276	132	8	(	(	PUNCT
ejpam-4276	132	9	2	2	NUM
ejpam-4276	132	10	)	)	PUNCT
ejpam-4276	132	11	(	(	PUNCT
ejpam-4276	132	12	2022	2022	NUM
ejpam-4276	132	13	)	)	PUNCT
ejpam-4276	132	14	,	,	PUNCT
ejpam-4276	132	15	572	572	NUM
ejpam-4276	132	16	-	-	SYM
ejpam-4276	132	17	588	588	NUM
ejpam-4276	132	18	576	576	NUM
ejpam-4276	132	19	the	the	DET
ejpam-4276	132	20	family	family	NOUN
ejpam-4276	132	21	of	of	ADP
ejpam-4276	132	22	all	all	DET
ejpam-4276	132	23	r(λ	r(λ	NOUN
ejpam-4276	132	24	,	,	PUNCT
ejpam-4276	132	25	sp)-open	sp)-open	ADJ
ejpam-4276	132	26	(	(	PUNCT
ejpam-4276	132	27	resp	resp	NOUN
ejpam-4276	132	28	.	.	PUNCT
ejpam-4276	133	1	r(λ	r(λ	NOUN
ejpam-4276	133	2	,	,	PUNCT
ejpam-4276	133	3	sp)-closed	sp)-close	VERB
ejpam-4276	133	4	)	)	PUNCT
ejpam-4276	133	5	sets	set	NOUN
ejpam-4276	133	6	in	in	ADP
ejpam-4276	133	7	a	a	DET
ejpam-4276	133	8	topological	topological	ADJ
ejpam-4276	133	9	space	space	NOUN
ejpam-4276	133	10	(	(	PUNCT
ejpam-4276	133	11	x	x	X
ejpam-4276	133	12	,	,	PUNCT
ejpam-4276	133	13	τ	τ	X
ejpam-4276	133	14	)	)	PUNCT
ejpam-4276	133	15	is	be	AUX
ejpam-4276	133	16	denoted	denote	VERB
ejpam-4276	133	17	by	by	ADP
ejpam-4276	133	18	rλspo(x	rλspo(x	PROPN
ejpam-4276	133	19	,	,	PUNCT
ejpam-4276	133	20	τ	τ	PROPN
ejpam-4276	133	21	)	)	PUNCT
ejpam-4276	133	22	(	(	PUNCT
ejpam-4276	133	23	resp	resp	NOUN
ejpam-4276	133	24	.	.	PUNCT
ejpam-4276	134	1	rλspc(x	rλspc(x	PROPN
ejpam-4276	134	2	,	,	PUNCT
ejpam-4276	134	3	τ	τ	PROPN
ejpam-4276	134	4	)	)	PUNCT
ejpam-4276	134	5	)	)	PUNCT
ejpam-4276	134	6	.	.	PUNCT
ejpam-4276	135	1	proposition	proposition	NOUN
ejpam-4276	135	2	2	2	NUM
ejpam-4276	135	3	.	.	PUNCT
ejpam-4276	135	4	let	let	VERB
ejpam-4276	135	5	a	a	DET
ejpam-4276	135	6	be	be	AUX
ejpam-4276	135	7	a	a	DET
ejpam-4276	135	8	subset	subset	NOUN
ejpam-4276	135	9	of	of	ADP
ejpam-4276	135	10	a	a	DET
ejpam-4276	135	11	topological	topological	ADJ
ejpam-4276	135	12	space	space	NOUN
ejpam-4276	135	13	(	(	PUNCT
ejpam-4276	135	14	x	x	X
ejpam-4276	135	15	,	,	PUNCT
ejpam-4276	135	16	τ	τ	PROPN
ejpam-4276	135	17	)	)	PUNCT
ejpam-4276	135	18	,	,	PUNCT
ejpam-4276	135	19	the	the	DET
ejpam-4276	135	20	following	follow	VERB
ejpam-4276	135	21	properties	property	NOUN
ejpam-4276	135	22	hold	hold	VERB
ejpam-4276	135	23	:	:	PUNCT
ejpam-4276	135	24	(	(	PUNCT
ejpam-4276	135	25	1	1	X
ejpam-4276	135	26	)	)	PUNCT
ejpam-4276	135	27	a	a	PRON
ejpam-4276	135	28	is	be	AUX
ejpam-4276	135	29	r(λ	r(λ	NOUN
ejpam-4276	135	30	,	,	PUNCT
ejpam-4276	135	31	sp)-open	sp)-open	ADJ
ejpam-4276	135	32	if	if	SCONJ
ejpam-4276	135	33	and	and	CCONJ
ejpam-4276	135	34	only	only	ADV
ejpam-4276	135	35	if	if	SCONJ
ejpam-4276	135	36	a	a	DET
ejpam-4276	135	37	=	=	NOUN
ejpam-4276	135	38	f(λ	f(λ	NOUN
ejpam-4276	135	39	,	,	PUNCT
ejpam-4276	135	40	sp	sp	NOUN
ejpam-4276	135	41	)	)	PUNCT
ejpam-4276	135	42	for	for	ADP
ejpam-4276	135	43	some	some	PRON
ejpam-4276	135	44	(	(	PUNCT
ejpam-4276	135	45	λ	λ	PROPN
ejpam-4276	135	46	,	,	PUNCT
ejpam-4276	135	47	sp)-closed	sp)-close	VERB
ejpam-4276	135	48	set	set	VERB
ejpam-4276	135	49	f	f	PROPN
ejpam-4276	135	50	.	.	PUNCT
ejpam-4276	136	1	(	(	PUNCT
ejpam-4276	136	2	2	2	X
ejpam-4276	136	3	)	)	PUNCT
ejpam-4276	136	4	a	a	PRON
ejpam-4276	136	5	is	be	AUX
ejpam-4276	136	6	r(λ	r(λ	NOUN
ejpam-4276	136	7	,	,	PUNCT
ejpam-4276	136	8	sp)-closed	sp)-close	VERB
ejpam-4276	136	9	if	if	SCONJ
ejpam-4276	136	10	and	and	CCONJ
ejpam-4276	136	11	only	only	ADV
ejpam-4276	136	12	if	if	SCONJ
ejpam-4276	136	13	a	a	DET
ejpam-4276	136	14	=	=	X
ejpam-4276	136	15	u	u	X
ejpam-4276	136	16	(	(	PUNCT
ejpam-4276	136	17	λ	λ	PROPN
ejpam-4276	136	18	,	,	PUNCT
ejpam-4276	136	19	sp	sp	NOUN
ejpam-4276	136	20	)	)	PUNCT
ejpam-4276	136	21	for	for	ADP
ejpam-4276	136	22	some	some	PRON
ejpam-4276	136	23	(	(	PUNCT
ejpam-4276	136	24	λ	λ	NOUN
ejpam-4276	136	25	,	,	PUNCT
ejpam-4276	136	26	sp)-open	sp)-open	ADJ
ejpam-4276	136	27	set	set	NOUN
ejpam-4276	136	28	u	u	NOUN
ejpam-4276	136	29	.	.	PUNCT
ejpam-4276	137	1	proposition	proposition	NOUN
ejpam-4276	137	2	3	3	X
ejpam-4276	137	3	.	.	PUNCT
ejpam-4276	138	1	let	let	VERB
ejpam-4276	138	2	a	a	DET
ejpam-4276	138	3	be	be	AUX
ejpam-4276	138	4	a	a	DET
ejpam-4276	138	5	subset	subset	NOUN
ejpam-4276	138	6	of	of	ADP
ejpam-4276	138	7	a	a	DET
ejpam-4276	138	8	topological	topological	ADJ
ejpam-4276	138	9	space	space	NOUN
ejpam-4276	138	10	(	(	PUNCT
ejpam-4276	138	11	x	x	X
ejpam-4276	138	12	,	,	PUNCT
ejpam-4276	138	13	τ	τ	PROPN
ejpam-4276	138	14	)	)	PUNCT
ejpam-4276	138	15	,	,	PUNCT
ejpam-4276	138	16	the	the	DET
ejpam-4276	138	17	following	follow	VERB
ejpam-4276	138	18	properties	property	NOUN
ejpam-4276	138	19	hold	hold	VERB
ejpam-4276	138	20	:	:	PUNCT
ejpam-4276	138	21	(	(	PUNCT
ejpam-4276	138	22	1	1	X
ejpam-4276	138	23	)	)	PUNCT
ejpam-4276	138	24	a	a	PRON
ejpam-4276	138	25	is	be	AUX
ejpam-4276	138	26	s(λ	s(λ	NOUN
ejpam-4276	138	27	,	,	PUNCT
ejpam-4276	138	28	sp)-closed	sp)-close	VERB
ejpam-4276	138	29	if	if	SCONJ
ejpam-4276	138	30	and	and	CCONJ
ejpam-4276	138	31	only	only	ADV
ejpam-4276	138	32	if	if	SCONJ
ejpam-4276	138	33	[	[	X
ejpam-4276	138	34	a(λ	a(λ	ADV
ejpam-4276	138	35	,	,	PUNCT
ejpam-4276	138	36	sp)](λ	sp)](λ	PROPN
ejpam-4276	138	37	,	,	PUNCT
ejpam-4276	138	38	sp	sp	NOUN
ejpam-4276	138	39	)	)	PUNCT
ejpam-4276	138	40	⊆	⊆	NUM
ejpam-4276	138	41	a.	a.	NOUN
ejpam-4276	138	42	(	(	PUNCT
ejpam-4276	138	43	2	2	NUM
ejpam-4276	138	44	)	)	PUNCT
ejpam-4276	138	45	a	a	PRON
ejpam-4276	138	46	is	be	AUX
ejpam-4276	138	47	p(λ	p(λ	NOUN
ejpam-4276	138	48	,	,	PUNCT
ejpam-4276	138	49	sp)-closed	sp)-close	VERB
ejpam-4276	138	50	if	if	SCONJ
ejpam-4276	138	51	and	and	CCONJ
ejpam-4276	138	52	only	only	ADV
ejpam-4276	138	53	if	if	SCONJ
ejpam-4276	138	54	[	[	X
ejpam-4276	138	55	a(λ	a(λ	ADV
ejpam-4276	138	56	,	,	PUNCT
ejpam-4276	138	57	sp	sp	NOUN
ejpam-4276	138	58	)	)	PUNCT
ejpam-4276	138	59	]	]	PUNCT
ejpam-4276	139	1	(	(	PUNCT
ejpam-4276	139	2	λ	λ	NOUN
ejpam-4276	139	3	,	,	PUNCT
ejpam-4276	139	4	sp	sp	NOUN
ejpam-4276	139	5	)	)	PUNCT
ejpam-4276	139	6	⊆	⊆	NUM
ejpam-4276	139	7	a.	a.	NOUN
ejpam-4276	139	8	(	(	PUNCT
ejpam-4276	139	9	3	3	NUM
ejpam-4276	139	10	)	)	PUNCT
ejpam-4276	139	11	a	a	PRON
ejpam-4276	139	12	is	be	AUX
ejpam-4276	139	13	α(λ	α(λ	PROPN
ejpam-4276	139	14	,	,	PUNCT
ejpam-4276	139	15	sp)-closed	sp)-close	VERB
ejpam-4276	139	16	if	if	SCONJ
ejpam-4276	139	17	and	and	CCONJ
ejpam-4276	139	18	only	only	ADV
ejpam-4276	139	19	if	if	SCONJ
ejpam-4276	139	20	[	[	X
ejpam-4276	139	21	[	[	X
ejpam-4276	139	22	a(λ	a(λ	ADJ
ejpam-4276	139	23	,	,	PUNCT
ejpam-4276	139	24	sp)](λ	sp)](λ	PROPN
ejpam-4276	139	25	,	,	PUNCT
ejpam-4276	139	26	sp	sp	NOUN
ejpam-4276	139	27	)	)	PUNCT
ejpam-4276	139	28	]	]	PUNCT
ejpam-4276	139	29	(	(	PUNCT
ejpam-4276	139	30	λ	λ	NOUN
ejpam-4276	139	31	,	,	PUNCT
ejpam-4276	139	32	sp	sp	NOUN
ejpam-4276	139	33	)	)	PUNCT
ejpam-4276	139	34	⊆	⊆	NUM
ejpam-4276	139	35	a.	a.	NOUN
ejpam-4276	139	36	(	(	PUNCT
ejpam-4276	139	37	4	4	NUM
ejpam-4276	139	38	)	)	PUNCT
ejpam-4276	139	39	a	a	PRON
ejpam-4276	139	40	is	be	AUX
ejpam-4276	139	41	β(λ	β(λ	NOUN
ejpam-4276	139	42	,	,	PUNCT
ejpam-4276	139	43	sp)-closed	sp)-close	VERB
ejpam-4276	139	44	if	if	SCONJ
ejpam-4276	139	45	and	and	CCONJ
ejpam-4276	139	46	only	only	ADV
ejpam-4276	139	47	if	if	SCONJ
ejpam-4276	139	48	[	[	X
ejpam-4276	139	49	[	[	X
ejpam-4276	139	50	a(λ	a(λ	ADJ
ejpam-4276	139	51	,	,	PUNCT
ejpam-4276	139	52	sp	sp	NOUN
ejpam-4276	139	53	)	)	PUNCT
ejpam-4276	139	54	]	]	PUNCT
ejpam-4276	140	1	(	(	PUNCT
ejpam-4276	140	2	λ	λ	X
ejpam-4276	140	3	,	,	PUNCT
ejpam-4276	140	4	sp)](λ	sp)](λ	PROPN
ejpam-4276	140	5	,	,	PUNCT
ejpam-4276	140	6	sp	sp	NOUN
ejpam-4276	140	7	)	)	PUNCT
ejpam-4276	140	8	⊆	⊆	NUM
ejpam-4276	140	9	a.	a.	NOUN
ejpam-4276	140	10	proposition	proposition	NOUN
ejpam-4276	140	11	4	4	NUM
ejpam-4276	140	12	.	.	X
ejpam-4276	141	1	for	for	ADP
ejpam-4276	141	2	a	a	DET
ejpam-4276	141	3	subset	subset	NOUN
ejpam-4276	141	4	a	a	PRON
ejpam-4276	141	5	of	of	ADP
ejpam-4276	141	6	a	a	DET
ejpam-4276	141	7	topological	topological	ADJ
ejpam-4276	141	8	space	space	NOUN
ejpam-4276	141	9	(	(	PUNCT
ejpam-4276	141	10	x	x	X
ejpam-4276	141	11	,	,	PUNCT
ejpam-4276	141	12	τ	τ	PROPN
ejpam-4276	141	13	)	)	PUNCT
ejpam-4276	141	14	,	,	PUNCT
ejpam-4276	141	15	the	the	DET
ejpam-4276	141	16	following	follow	VERB
ejpam-4276	141	17	properties	property	NOUN
ejpam-4276	141	18	are	be	AUX
ejpam-4276	141	19	equivalent	equivalent	ADJ
ejpam-4276	141	20	:	:	PUNCT
ejpam-4276	141	21	(	(	PUNCT
ejpam-4276	141	22	1	1	X
ejpam-4276	141	23	)	)	PUNCT
ejpam-4276	141	24	a	a	PRON
ejpam-4276	141	25	is	be	AUX
ejpam-4276	141	26	r(λ	r(λ	NOUN
ejpam-4276	141	27	,	,	PUNCT
ejpam-4276	141	28	sp)-open	sp)-open	NOUN
ejpam-4276	141	29	.	.	PUNCT
ejpam-4276	142	1	(	(	PUNCT
ejpam-4276	142	2	2	2	X
ejpam-4276	142	3	)	)	PUNCT
ejpam-4276	142	4	a	a	PRON
ejpam-4276	142	5	is	be	AUX
ejpam-4276	142	6	(	(	PUNCT
ejpam-4276	142	7	λ	λ	NOUN
ejpam-4276	142	8	,	,	PUNCT
ejpam-4276	142	9	sp)-open	sp)-open	NOUN
ejpam-4276	142	10	and	and	CCONJ
ejpam-4276	142	11	s(λ	s(λ	NOUN
ejpam-4276	142	12	,	,	PUNCT
ejpam-4276	142	13	sp)-closed	sp)-close	VERB
ejpam-4276	142	14	.	.	PUNCT
ejpam-4276	143	1	(	(	PUNCT
ejpam-4276	143	2	3	3	X
ejpam-4276	143	3	)	)	PUNCT
ejpam-4276	143	4	a	a	PRON
ejpam-4276	143	5	is	be	AUX
ejpam-4276	143	6	α(λ	α(λ	PROPN
ejpam-4276	143	7	,	,	PUNCT
ejpam-4276	143	8	sp)-open	sp)-open	NOUN
ejpam-4276	143	9	and	and	CCONJ
ejpam-4276	143	10	s(λ	s(λ	NOUN
ejpam-4276	143	11	,	,	PUNCT
ejpam-4276	143	12	sp)-closed	sp)-close	VERB
ejpam-4276	143	13	.	.	PUNCT
ejpam-4276	144	1	(	(	PUNCT
ejpam-4276	144	2	4	4	X
ejpam-4276	144	3	)	)	PUNCT
ejpam-4276	144	4	a	a	PRON
ejpam-4276	144	5	is	be	AUX
ejpam-4276	144	6	p(λ	p(λ	NOUN
ejpam-4276	144	7	,	,	PUNCT
ejpam-4276	144	8	sp)-open	sp)-open	NOUN
ejpam-4276	144	9	and	and	CCONJ
ejpam-4276	144	10	s(λ	s(λ	NOUN
ejpam-4276	144	11	,	,	PUNCT
ejpam-4276	144	12	sp)-closed	sp)-close	VERB
ejpam-4276	144	13	.	.	PUNCT
ejpam-4276	145	1	(	(	PUNCT
ejpam-4276	145	2	5	5	X
ejpam-4276	145	3	)	)	PUNCT
ejpam-4276	145	4	a	a	PRON
ejpam-4276	145	5	is	be	AUX
ejpam-4276	145	6	(	(	PUNCT
ejpam-4276	145	7	λ	λ	NOUN
ejpam-4276	145	8	,	,	PUNCT
ejpam-4276	145	9	sp)-open	sp)-open	ADJ
ejpam-4276	145	10	and	and	CCONJ
ejpam-4276	145	11	β(λ	β(λ	NOUN
ejpam-4276	145	12	,	,	PUNCT
ejpam-4276	145	13	sp)-closed	sp)-close	VERB
ejpam-4276	145	14	.	.	PUNCT
ejpam-4276	146	1	(	(	PUNCT
ejpam-4276	146	2	6	6	NUM
ejpam-4276	146	3	)	)	PUNCT
ejpam-4276	146	4	a	a	PRON
ejpam-4276	146	5	is	be	AUX
ejpam-4276	146	6	α(λ	α(λ	PROPN
ejpam-4276	146	7	,	,	PUNCT
ejpam-4276	146	8	sp)-open	sp)-open	ADJ
ejpam-4276	146	9	and	and	CCONJ
ejpam-4276	146	10	β(λ	β(λ	NOUN
ejpam-4276	146	11	,	,	PUNCT
ejpam-4276	146	12	sp)-closed	sp)-close	VERB
ejpam-4276	146	13	.	.	PUNCT
ejpam-4276	147	1	proof	proof	NOUN
ejpam-4276	147	2	.	.	PUNCT
ejpam-4276	148	1	(	(	PUNCT
ejpam-4276	148	2	1	1	X
ejpam-4276	148	3	)	)	PUNCT
ejpam-4276	148	4	⇒	⇒	NOUN
ejpam-4276	148	5	(	(	PUNCT
ejpam-4276	148	6	2	2	NUM
ejpam-4276	148	7	)	)	PUNCT
ejpam-4276	148	8	⇒	⇒	NOUN
ejpam-4276	148	9	(	(	PUNCT
ejpam-4276	148	10	3	3	NUM
ejpam-4276	148	11	)	)	PUNCT
ejpam-4276	148	12	⇒	⇒	NOUN
ejpam-4276	148	13	(	(	PUNCT
ejpam-4276	148	14	4	4	NUM
ejpam-4276	148	15	):	):	PUNCT
ejpam-4276	148	16	obvious	obvious	ADJ
ejpam-4276	148	17	.	.	PUNCT
ejpam-4276	149	1	(	(	PUNCT
ejpam-4276	149	2	4	4	X
ejpam-4276	149	3	)	)	PUNCT
ejpam-4276	149	4	⇒	⇒	NOUN
ejpam-4276	149	5	(	(	PUNCT
ejpam-4276	149	6	5	5	NUM
ejpam-4276	149	7	):	):	PUNCT
ejpam-4276	149	8	let	let	VERB
ejpam-4276	149	9	a	a	DET
ejpam-4276	149	10	be	be	AUX
ejpam-4276	149	11	(	(	PUNCT
ejpam-4276	149	12	λ	λ	NOUN
ejpam-4276	149	13	,	,	PUNCT
ejpam-4276	149	14	sp)-open	sp)-open	NOUN
ejpam-4276	149	15	and	and	CCONJ
ejpam-4276	149	16	s(λ	s(λ	NOUN
ejpam-4276	149	17	,	,	PUNCT
ejpam-4276	149	18	sp)-closed	sp)-close	VERB
ejpam-4276	149	19	.	.	PUNCT
ejpam-4276	150	1	then	then	ADV
ejpam-4276	150	2	,	,	PUNCT
ejpam-4276	150	3	a	a	DET
ejpam-4276	150	4	⊆	⊆	NUM
ejpam-4276	150	5	[	[	X
ejpam-4276	150	6	a(λ	a(λ	ADJ
ejpam-4276	150	7	,	,	PUNCT
ejpam-4276	150	8	sp)](λ	sp)](λ	PROPN
ejpam-4276	150	9	,	,	PUNCT
ejpam-4276	150	10	sp	sp	NOUN
ejpam-4276	150	11	)	)	PUNCT
ejpam-4276	150	12	and	and	CCONJ
ejpam-4276	150	13	[	[	X
ejpam-4276	150	14	a(λ	a(λ	ADV
ejpam-4276	150	15	,	,	PUNCT
ejpam-4276	150	16	sp)](λ	sp)](λ	PROPN
ejpam-4276	150	17	,	,	PUNCT
ejpam-4276	150	18	sp	sp	NOUN
ejpam-4276	150	19	)	)	PUNCT
ejpam-4276	150	20	⊆	⊆	NUM
ejpam-4276	150	21	a.	a.	NOUN
ejpam-4276	150	22	this	this	PRON
ejpam-4276	150	23	implies	imply	VERB
ejpam-4276	150	24	that	that	SCONJ
ejpam-4276	150	25	a	a	DET
ejpam-4276	150	26	=	=	X
ejpam-4276	150	27	[	[	X
ejpam-4276	150	28	a(λ	a(λ	PROPN
ejpam-4276	150	29	,	,	PUNCT
ejpam-4276	150	30	sp)](λ	sp)](λ	PROPN
ejpam-4276	150	31	,	,	PUNCT
ejpam-4276	150	32	sp	sp	NOUN
ejpam-4276	150	33	)	)	PUNCT
ejpam-4276	150	34	.	.	PUNCT
ejpam-4276	151	1	therefore	therefore	ADV
ejpam-4276	151	2	,	,	PUNCT
ejpam-4276	151	3	a	a	PRON
ejpam-4276	151	4	is	be	AUX
ejpam-4276	151	5	r(λ	r(λ	NOUN
ejpam-4276	151	6	,	,	PUNCT
ejpam-4276	151	7	sp)-open	sp)-open	ADJ
ejpam-4276	151	8	and	and	CCONJ
ejpam-4276	151	9	hence	hence	ADV
ejpam-4276	151	10	a	a	PRON
ejpam-4276	151	11	is	be	AUX
ejpam-4276	151	12	(	(	PUNCT
ejpam-4276	151	13	λ	λ	NOUN
ejpam-4276	151	14	,	,	PUNCT
ejpam-4276	151	15	sp)-open	sp)-open	NOUN
ejpam-4276	151	16	.	.	PUNCT
ejpam-4276	152	1	since	since	SCONJ
ejpam-4276	152	2	every	every	DET
ejpam-4276	152	3	s(λ	s(λ	PROPN
ejpam-4276	152	4	,	,	PUNCT
ejpam-4276	152	5	sp)-closed	sp)-close	VERB
ejpam-4276	152	6	set	set	VERB
ejpam-4276	152	7	is	be	AUX
ejpam-4276	152	8	β(λ	β(λ	NOUN
ejpam-4276	152	9	,	,	PUNCT
ejpam-4276	152	10	sp)-closed	sp)-close	VERB
ejpam-4276	152	11	.	.	PUNCT
ejpam-4276	153	1	thus	thus	ADV
ejpam-4276	153	2	,	,	PUNCT
ejpam-4276	153	3	a	a	PRON
ejpam-4276	153	4	is	be	AUX
ejpam-4276	153	5	(	(	PUNCT
ejpam-4276	153	6	λ	λ	NOUN
ejpam-4276	153	7	,	,	PUNCT
ejpam-4276	153	8	sp)-open	sp)-open	ADJ
ejpam-4276	153	9	and	and	CCONJ
ejpam-4276	153	10	β(λ	β(λ	NOUN
ejpam-4276	153	11	,	,	PUNCT
ejpam-4276	153	12	sp)-closed	sp)-close	VERB
ejpam-4276	153	13	.	.	PUNCT
ejpam-4276	154	1	(	(	PUNCT
ejpam-4276	154	2	5	5	X
ejpam-4276	154	3	)	)	PUNCT
ejpam-4276	154	4	⇒	⇒	NOUN
ejpam-4276	154	5	(	(	PUNCT
ejpam-4276	154	6	6	6	NUM
ejpam-4276	154	7	):	):	PUNCT
ejpam-4276	154	8	the	the	DET
ejpam-4276	154	9	proof	proof	NOUN
ejpam-4276	154	10	is	be	AUX
ejpam-4276	154	11	obvious	obvious	ADJ
ejpam-4276	154	12	.	.	PUNCT
ejpam-4276	155	1	(	(	PUNCT
ejpam-4276	155	2	6	6	NUM
ejpam-4276	155	3	)	)	PUNCT
ejpam-4276	155	4	⇒	⇒	NOUN
ejpam-4276	155	5	(	(	PUNCT
ejpam-4276	155	6	1	1	NUM
ejpam-4276	155	7	):	):	PUNCT
ejpam-4276	155	8	let	let	VERB
ejpam-4276	155	9	a	a	DET
ejpam-4276	155	10	be	be	AUX
ejpam-4276	155	11	α(λ	α(λ	PROPN
ejpam-4276	155	12	,	,	PUNCT
ejpam-4276	155	13	sp)-open	sp)-open	ADJ
ejpam-4276	155	14	and	and	CCONJ
ejpam-4276	155	15	β(λ	β(λ	NOUN
ejpam-4276	155	16	,	,	PUNCT
ejpam-4276	155	17	sp)-closed	sp)-close	VERB
ejpam-4276	155	18	.	.	PUNCT
ejpam-4276	156	1	then	then	ADV
ejpam-4276	156	2	,	,	PUNCT
ejpam-4276	156	3	a	a	DET
ejpam-4276	156	4	⊆	⊆	NUM
ejpam-4276	156	5	[	[	X
ejpam-4276	156	6	[	[	X
ejpam-4276	156	7	a(λ	a(λ	ADJ
ejpam-4276	156	8	,	,	PUNCT
ejpam-4276	156	9	sp	sp	NOUN
ejpam-4276	156	10	)	)	PUNCT
ejpam-4276	156	11	]	]	PUNCT
ejpam-4276	157	1	(	(	PUNCT
ejpam-4276	157	2	λ	λ	X
ejpam-4276	157	3	,	,	PUNCT
ejpam-4276	157	4	sp)](λ	sp)](λ	PROPN
ejpam-4276	157	5	,	,	PUNCT
ejpam-4276	157	6	sp	sp	NOUN
ejpam-4276	157	7	)	)	PUNCT
ejpam-4276	157	8	and	and	CCONJ
ejpam-4276	157	9	[	[	X
ejpam-4276	157	10	[	[	X
ejpam-4276	157	11	a(λ	a(λ	ADJ
ejpam-4276	157	12	,	,	PUNCT
ejpam-4276	157	13	sp	sp	NOUN
ejpam-4276	157	14	)	)	PUNCT
ejpam-4276	157	15	]	]	PUNCT
ejpam-4276	157	16	(	(	PUNCT
ejpam-4276	157	17	λ	λ	X
ejpam-4276	157	18	,	,	PUNCT
ejpam-4276	157	19	sp)](λ	sp)](λ	PROPN
ejpam-4276	157	20	,	,	PUNCT
ejpam-4276	157	21	sp	sp	NOUN
ejpam-4276	157	22	)	)	PUNCT
ejpam-4276	157	23	⊆	⊆	NUM
ejpam-4276	157	24	a.	a.	NOUN
ejpam-4276	157	25	thus	thus	ADV
ejpam-4276	157	26	,	,	PUNCT
ejpam-4276	157	27	a	a	PRON
ejpam-4276	157	28	=	=	X
ejpam-4276	158	1	[	[	X
ejpam-4276	158	2	[	[	X
ejpam-4276	158	3	a(λ	a(λ	ADJ
ejpam-4276	158	4	,	,	PUNCT
ejpam-4276	158	5	sp	sp	NOUN
ejpam-4276	158	6	)	)	PUNCT
ejpam-4276	158	7	]	]	PUNCT
ejpam-4276	158	8	(	(	PUNCT
ejpam-4276	158	9	λ	λ	X
ejpam-4276	158	10	,	,	PUNCT
ejpam-4276	158	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	158	12	,	,	PUNCT
ejpam-4276	158	13	sp	sp	NOUN
ejpam-4276	158	14	)	)	PUNCT
ejpam-4276	158	15	and	and	CCONJ
ejpam-4276	158	16	hence	hence	ADV
ejpam-4276	158	17	a(λ	a(λ	ADV
ejpam-4276	158	18	,	,	PUNCT
ejpam-4276	158	19	sp	sp	NOUN
ejpam-4276	158	20	)	)	PUNCT
ejpam-4276	158	21	=	=	PUNCT
ejpam-4276	159	1	[	[	X
ejpam-4276	159	2	[	[	X
ejpam-4276	159	3	a(λ	a(λ	ADJ
ejpam-4276	159	4	,	,	PUNCT
ejpam-4276	159	5	sp	sp	NOUN
ejpam-4276	159	6	)	)	PUNCT
ejpam-4276	159	7	]	]	PUNCT
ejpam-4276	159	8	(	(	PUNCT
ejpam-4276	159	9	λ	λ	X
ejpam-4276	159	10	,	,	PUNCT
ejpam-4276	159	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	159	12	,	,	PUNCT
ejpam-4276	159	13	sp	sp	NOUN
ejpam-4276	159	14	)	)	PUNCT
ejpam-4276	159	15	=	=	SYM
ejpam-4276	160	1	a.	a.	NOUN
ejpam-4276	160	2	therefore	therefore	ADV
ejpam-4276	160	3	,	,	PUNCT
ejpam-4276	160	4	[	[	X
ejpam-4276	160	5	a(λ	a(λ	ADV
ejpam-4276	160	6	,	,	PUNCT
ejpam-4276	160	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	160	8	,	,	PUNCT
ejpam-4276	160	9	sp	sp	NOUN
ejpam-4276	160	10	)	)	PUNCT
ejpam-4276	160	11	=	=	PUNCT
ejpam-4276	161	1	[	[	X
ejpam-4276	161	2	[	[	X
ejpam-4276	161	3	a(λ	a(λ	ADJ
ejpam-4276	161	4	,	,	PUNCT
ejpam-4276	161	5	sp	sp	NOUN
ejpam-4276	161	6	)	)	PUNCT
ejpam-4276	161	7	]	]	PUNCT
ejpam-4276	161	8	(	(	PUNCT
ejpam-4276	161	9	λ	λ	X
ejpam-4276	161	10	,	,	PUNCT
ejpam-4276	161	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	161	12	,	,	PUNCT
ejpam-4276	161	13	sp	sp	NOUN
ejpam-4276	161	14	)	)	PUNCT
ejpam-4276	161	15	=	=	SYM
ejpam-4276	162	1	a.	a.	NOUN
ejpam-4276	162	2	this	this	PRON
ejpam-4276	162	3	shows	show	VERB
ejpam-4276	162	4	that	that	SCONJ
ejpam-4276	162	5	a	a	PRON
ejpam-4276	162	6	is	be	AUX
ejpam-4276	162	7	r(λ	r(λ	NOUN
ejpam-4276	162	8	,	,	PUNCT
ejpam-4276	162	9	sp)-open	sp)-open	NOUN
ejpam-4276	162	10	.	.	PUNCT
ejpam-4276	163	1	c.	c.	PROPN
ejpam-4276	163	2	boonpok	boonpok	PROPN
ejpam-4276	163	3	,	,	PUNCT
ejpam-4276	163	4	j.	j.	PROPN
ejpam-4276	163	5	khampakdee	khampakdee	PROPN
ejpam-4276	163	6	/	/	PUNCT
ejpam-4276	163	7	eur	eur	PROPN
ejpam-4276	163	8	.	.	PUNCT
ejpam-4276	164	1	j.	j.	PROPN
ejpam-4276	164	2	pure	pure	PROPN
ejpam-4276	164	3	appl	appl	PROPN
ejpam-4276	164	4	.	.	PROPN
ejpam-4276	164	5	math	math	PROPN
ejpam-4276	164	6	,	,	PUNCT
ejpam-4276	164	7	15	15	NUM
ejpam-4276	164	8	(	(	PUNCT
ejpam-4276	164	9	2	2	NUM
ejpam-4276	164	10	)	)	PUNCT
ejpam-4276	164	11	(	(	PUNCT
ejpam-4276	164	12	2022	2022	NUM
ejpam-4276	164	13	)	)	PUNCT
ejpam-4276	164	14	,	,	PUNCT
ejpam-4276	164	15	572	572	NUM
ejpam-4276	164	16	-	-	SYM
ejpam-4276	164	17	588	588	NUM
ejpam-4276	164	18	577	577	NUM
ejpam-4276	164	19	corollary	corollary	ADJ
ejpam-4276	164	20	1	1	NUM
ejpam-4276	164	21	.	.	PUNCT
ejpam-4276	165	1	for	for	ADP
ejpam-4276	165	2	a	a	DET
ejpam-4276	165	3	subset	subset	NOUN
ejpam-4276	165	4	a	a	PRON
ejpam-4276	165	5	of	of	ADP
ejpam-4276	165	6	a	a	DET
ejpam-4276	165	7	topological	topological	ADJ
ejpam-4276	165	8	space	space	NOUN
ejpam-4276	165	9	(	(	PUNCT
ejpam-4276	165	10	x	x	X
ejpam-4276	165	11	,	,	PUNCT
ejpam-4276	165	12	τ	τ	PROPN
ejpam-4276	165	13	)	)	PUNCT
ejpam-4276	165	14	,	,	PUNCT
ejpam-4276	165	15	the	the	DET
ejpam-4276	165	16	following	follow	VERB
ejpam-4276	165	17	properties	property	NOUN
ejpam-4276	165	18	are	be	AUX
ejpam-4276	165	19	equivalent	equivalent	ADJ
ejpam-4276	165	20	:	:	PUNCT
ejpam-4276	165	21	(	(	PUNCT
ejpam-4276	165	22	1	1	X
ejpam-4276	165	23	)	)	PUNCT
ejpam-4276	165	24	a	a	PRON
ejpam-4276	165	25	is	be	AUX
ejpam-4276	165	26	r(λ	r(λ	NOUN
ejpam-4276	165	27	,	,	PUNCT
ejpam-4276	165	28	sp)-closed	sp)-close	VERB
ejpam-4276	165	29	.	.	PUNCT
ejpam-4276	166	1	(	(	PUNCT
ejpam-4276	166	2	2	2	X
ejpam-4276	166	3	)	)	PUNCT
ejpam-4276	166	4	a	a	PRON
ejpam-4276	166	5	is	be	AUX
ejpam-4276	166	6	(	(	PUNCT
ejpam-4276	166	7	λ	λ	X
ejpam-4276	166	8	,	,	PUNCT
ejpam-4276	166	9	sp)-closed	sp)-closed	ADJ
ejpam-4276	166	10	and	and	CCONJ
ejpam-4276	166	11	s(λ	s(λ	NOUN
ejpam-4276	166	12	,	,	PUNCT
ejpam-4276	166	13	sp)-open	sp)-open	NOUN
ejpam-4276	166	14	.	.	PUNCT
ejpam-4276	167	1	(	(	PUNCT
ejpam-4276	167	2	3	3	X
ejpam-4276	167	3	)	)	PUNCT
ejpam-4276	167	4	a	a	PRON
ejpam-4276	167	5	is	be	AUX
ejpam-4276	167	6	α(λ	α(λ	PROPN
ejpam-4276	167	7	,	,	PUNCT
ejpam-4276	167	8	sp)-closed	sp)-close	VERB
ejpam-4276	167	9	and	and	CCONJ
ejpam-4276	167	10	s(λ	s(λ	NOUN
ejpam-4276	167	11	,	,	PUNCT
ejpam-4276	167	12	sp)-open	sp)-open	NOUN
ejpam-4276	167	13	.	.	PUNCT
ejpam-4276	168	1	(	(	PUNCT
ejpam-4276	168	2	4	4	X
ejpam-4276	168	3	)	)	PUNCT
ejpam-4276	168	4	a	a	PRON
ejpam-4276	168	5	is	be	AUX
ejpam-4276	168	6	p(λ	p(λ	NOUN
ejpam-4276	168	7	,	,	PUNCT
ejpam-4276	168	8	sp)-closed	sp)-close	VERB
ejpam-4276	168	9	and	and	CCONJ
ejpam-4276	168	10	s(λ	s(λ	NOUN
ejpam-4276	168	11	,	,	PUNCT
ejpam-4276	168	12	sp)-open	sp)-open	NOUN
ejpam-4276	168	13	.	.	PUNCT
ejpam-4276	169	1	(	(	PUNCT
ejpam-4276	169	2	5	5	X
ejpam-4276	169	3	)	)	PUNCT
ejpam-4276	169	4	a	a	PRON
ejpam-4276	169	5	is	be	AUX
ejpam-4276	169	6	(	(	PUNCT
ejpam-4276	169	7	λ	λ	X
ejpam-4276	169	8	,	,	PUNCT
ejpam-4276	169	9	sp)-closed	sp)-closed	ADJ
ejpam-4276	169	10	and	and	CCONJ
ejpam-4276	169	11	β(λ	β(λ	PROPN
ejpam-4276	169	12	,	,	PUNCT
ejpam-4276	169	13	sp)-open	sp)-open	NOUN
ejpam-4276	169	14	.	.	PUNCT
ejpam-4276	170	1	(	(	PUNCT
ejpam-4276	170	2	6	6	NUM
ejpam-4276	170	3	)	)	PUNCT
ejpam-4276	170	4	a	a	PRON
ejpam-4276	170	5	is	be	AUX
ejpam-4276	170	6	α(λ	α(λ	PROPN
ejpam-4276	170	7	,	,	PUNCT
ejpam-4276	170	8	sp)-closed	sp)-close	VERB
ejpam-4276	170	9	and	and	CCONJ
ejpam-4276	170	10	β(λ	β(λ	PROPN
ejpam-4276	170	11	,	,	PUNCT
ejpam-4276	170	12	sp)-open	sp)-open	NOUN
ejpam-4276	170	13	.	.	PUNCT
ejpam-4276	171	1	proposition	proposition	NOUN
ejpam-4276	171	2	5	5	NUM
ejpam-4276	171	3	.	.	PUNCT
ejpam-4276	172	1	for	for	ADP
ejpam-4276	172	2	a	a	DET
ejpam-4276	172	3	subset	subset	NOUN
ejpam-4276	172	4	a	a	PRON
ejpam-4276	172	5	of	of	ADP
ejpam-4276	172	6	a	a	DET
ejpam-4276	172	7	topological	topological	ADJ
ejpam-4276	172	8	space	space	NOUN
ejpam-4276	172	9	(	(	PUNCT
ejpam-4276	172	10	x	x	X
ejpam-4276	172	11	,	,	PUNCT
ejpam-4276	172	12	τ	τ	PROPN
ejpam-4276	172	13	)	)	PUNCT
ejpam-4276	172	14	,	,	PUNCT
ejpam-4276	172	15	the	the	DET
ejpam-4276	172	16	following	follow	VERB
ejpam-4276	172	17	properties	property	NOUN
ejpam-4276	172	18	hold	hold	VERB
ejpam-4276	172	19	:	:	PUNCT
ejpam-4276	172	20	(	(	PUNCT
ejpam-4276	172	21	1	1	X
ejpam-4276	172	22	)	)	PUNCT
ejpam-4276	173	1	[	[	X
ejpam-4276	173	2	[	[	X
ejpam-4276	173	3	[	[	X
ejpam-4276	173	4	a(λ	a(λ	ADJ
ejpam-4276	173	5	,	,	PUNCT
ejpam-4276	173	6	sp	sp	NOUN
ejpam-4276	173	7	)	)	PUNCT
ejpam-4276	173	8	]	]	PUNCT
ejpam-4276	174	1	(	(	PUNCT
ejpam-4276	174	2	λ	λ	X
ejpam-4276	174	3	,	,	PUNCT
ejpam-4276	174	4	sp)](λ	sp)](λ	PROPN
ejpam-4276	174	5	,	,	PUNCT
ejpam-4276	174	6	sp	sp	NOUN
ejpam-4276	174	7	)	)	PUNCT
ejpam-4276	174	8	]	]	PUNCT
ejpam-4276	174	9	(	(	PUNCT
ejpam-4276	174	10	λ	λ	NOUN
ejpam-4276	174	11	,	,	PUNCT
ejpam-4276	174	12	sp	sp	NOUN
ejpam-4276	174	13	)	)	PUNCT
ejpam-4276	174	14	=	=	PUNCT
ejpam-4276	175	1	[	[	X
ejpam-4276	175	2	a(λ	a(λ	ADV
ejpam-4276	175	3	,	,	PUNCT
ejpam-4276	175	4	sp	sp	NOUN
ejpam-4276	175	5	)	)	PUNCT
ejpam-4276	175	6	]	]	PUNCT
ejpam-4276	175	7	(	(	PUNCT
ejpam-4276	175	8	λ	λ	NOUN
ejpam-4276	175	9	,	,	PUNCT
ejpam-4276	175	10	sp	sp	NOUN
ejpam-4276	175	11	)	)	PUNCT
ejpam-4276	175	12	.	.	PUNCT
ejpam-4276	176	1	(	(	PUNCT
ejpam-4276	176	2	2	2	X
ejpam-4276	176	3	)	)	PUNCT
ejpam-4276	177	1	[	[	X
ejpam-4276	177	2	[	[	X
ejpam-4276	177	3	[	[	X
ejpam-4276	177	4	a(λ	a(λ	ADJ
ejpam-4276	177	5	,	,	PUNCT
ejpam-4276	177	6	sp)](λ	sp)](λ	PROPN
ejpam-4276	177	7	,	,	PUNCT
ejpam-4276	177	8	sp	sp	NOUN
ejpam-4276	177	9	)	)	PUNCT
ejpam-4276	177	10	]	]	PUNCT
ejpam-4276	177	11	(	(	PUNCT
ejpam-4276	177	12	λ	λ	X
ejpam-4276	177	13	,	,	PUNCT
ejpam-4276	177	14	sp)](λ	sp)](λ	PROPN
ejpam-4276	177	15	,	,	PUNCT
ejpam-4276	177	16	sp	sp	NOUN
ejpam-4276	177	17	)	)	PUNCT
ejpam-4276	177	18	=	=	PUNCT
ejpam-4276	178	1	[	[	X
ejpam-4276	178	2	a(λ	a(λ	ADV
ejpam-4276	178	3	,	,	PUNCT
ejpam-4276	178	4	sp)](λ	sp)](λ	PROPN
ejpam-4276	178	5	,	,	PUNCT
ejpam-4276	178	6	sp	sp	NOUN
ejpam-4276	178	7	)	)	PUNCT
ejpam-4276	178	8	.	.	PUNCT
ejpam-4276	179	1	definition	definition	NOUN
ejpam-4276	179	2	3	3	NUM
ejpam-4276	179	3	.	.	PUNCT
ejpam-4276	180	1	a	a	DET
ejpam-4276	180	2	subset	subset	NOUN
ejpam-4276	180	3	a	a	PRON
ejpam-4276	180	4	of	of	ADP
ejpam-4276	180	5	a	a	DET
ejpam-4276	180	6	topological	topological	ADJ
ejpam-4276	180	7	space	space	NOUN
ejpam-4276	180	8	(	(	PUNCT
ejpam-4276	180	9	x	x	X
ejpam-4276	180	10	,	,	PUNCT
ejpam-4276	180	11	τ	τ	X
ejpam-4276	180	12	)	)	PUNCT
ejpam-4276	180	13	is	be	AUX
ejpam-4276	180	14	called	call	VERB
ejpam-4276	180	15	(	(	PUNCT
ejpam-4276	180	16	λ	λ	PROPN
ejpam-4276	180	17	,	,	PUNCT
ejpam-4276	180	18	sp)-clopen	sp)-clopen	ADJ
ejpam-4276	180	19	if	if	SCONJ
ejpam-4276	180	20	a	a	PRON
ejpam-4276	180	21	is	be	AUX
ejpam-4276	180	22	both	both	PRON
ejpam-4276	180	23	(	(	PUNCT
ejpam-4276	180	24	λ	λ	NOUN
ejpam-4276	180	25	,	,	PUNCT
ejpam-4276	180	26	sp)-open	sp)-open	ADJ
ejpam-4276	180	27	and	and	CCONJ
ejpam-4276	180	28	(	(	PUNCT
ejpam-4276	180	29	λ	λ	PROPN
ejpam-4276	180	30	,	,	PUNCT
ejpam-4276	180	31	sp)-closed	sp)-close	VERB
ejpam-4276	180	32	.	.	PUNCT
ejpam-4276	181	1	proposition	proposition	NOUN
ejpam-4276	181	2	6	6	NUM
ejpam-4276	181	3	.	.	PUNCT
ejpam-4276	182	1	for	for	ADP
ejpam-4276	182	2	a	a	DET
ejpam-4276	182	3	subset	subset	NOUN
ejpam-4276	182	4	a	a	PRON
ejpam-4276	182	5	of	of	ADP
ejpam-4276	182	6	a	a	DET
ejpam-4276	182	7	topological	topological	ADJ
ejpam-4276	182	8	space	space	NOUN
ejpam-4276	182	9	(	(	PUNCT
ejpam-4276	182	10	x	x	X
ejpam-4276	182	11	,	,	PUNCT
ejpam-4276	182	12	τ	τ	PROPN
ejpam-4276	182	13	)	)	PUNCT
ejpam-4276	182	14	,	,	PUNCT
ejpam-4276	182	15	the	the	DET
ejpam-4276	182	16	following	follow	VERB
ejpam-4276	182	17	properties	property	NOUN
ejpam-4276	182	18	are	be	AUX
ejpam-4276	182	19	equivalent	equivalent	ADJ
ejpam-4276	182	20	:	:	PUNCT
ejpam-4276	182	21	(	(	PUNCT
ejpam-4276	182	22	1	1	X
ejpam-4276	182	23	)	)	PUNCT
ejpam-4276	182	24	a	a	PRON
ejpam-4276	182	25	is	be	AUX
ejpam-4276	182	26	(	(	PUNCT
ejpam-4276	182	27	λ	λ	PROPN
ejpam-4276	182	28	,	,	PUNCT
ejpam-4276	182	29	sp)-clopen	sp)-clopen	NOUN
ejpam-4276	182	30	.	.	PUNCT
ejpam-4276	183	1	(	(	PUNCT
ejpam-4276	183	2	2	2	X
ejpam-4276	183	3	)	)	PUNCT
ejpam-4276	183	4	a	a	PRON
ejpam-4276	183	5	is	be	AUX
ejpam-4276	183	6	r(λ	r(λ	NOUN
ejpam-4276	183	7	,	,	PUNCT
ejpam-4276	183	8	sp)-open	sp)-open	NOUN
ejpam-4276	183	9	and	and	CCONJ
ejpam-4276	183	10	r(λ	r(λ	NOUN
ejpam-4276	183	11	,	,	PUNCT
ejpam-4276	183	12	sp)-closed	sp)-close	VERB
ejpam-4276	183	13	.	.	PUNCT
ejpam-4276	184	1	(	(	PUNCT
ejpam-4276	184	2	3	3	X
ejpam-4276	184	3	)	)	PUNCT
ejpam-4276	184	4	a	a	PRON
ejpam-4276	184	5	is	be	AUX
ejpam-4276	184	6	(	(	PUNCT
ejpam-4276	184	7	λ	λ	NOUN
ejpam-4276	184	8	,	,	PUNCT
ejpam-4276	184	9	sp)-open	sp)-open	NOUN
ejpam-4276	184	10	and	and	CCONJ
ejpam-4276	184	11	α(λ	α(λ	PROPN
ejpam-4276	184	12	,	,	PUNCT
ejpam-4276	184	13	sp)-closed	sp)-close	VERB
ejpam-4276	184	14	.	.	PUNCT
ejpam-4276	185	1	(	(	PUNCT
ejpam-4276	185	2	4	4	X
ejpam-4276	185	3	)	)	PUNCT
ejpam-4276	185	4	a	a	PRON
ejpam-4276	185	5	is	be	AUX
ejpam-4276	185	6	(	(	PUNCT
ejpam-4276	185	7	λ	λ	NOUN
ejpam-4276	185	8	,	,	PUNCT
ejpam-4276	185	9	sp)-open	sp)-open	NOUN
ejpam-4276	185	10	and	and	CCONJ
ejpam-4276	185	11	p(λ	p(λ	NOUN
ejpam-4276	185	12	,	,	PUNCT
ejpam-4276	185	13	sp)-closed	sp)-close	VERB
ejpam-4276	185	14	.	.	PUNCT
ejpam-4276	186	1	(	(	PUNCT
ejpam-4276	186	2	5	5	X
ejpam-4276	186	3	)	)	PUNCT
ejpam-4276	186	4	a	a	PRON
ejpam-4276	186	5	is	be	AUX
ejpam-4276	186	6	α(λ	α(λ	PROPN
ejpam-4276	186	7	,	,	PUNCT
ejpam-4276	186	8	sp)-open	sp)-open	NOUN
ejpam-4276	186	9	and	and	CCONJ
ejpam-4276	186	10	p(λ	p(λ	NOUN
ejpam-4276	186	11	,	,	PUNCT
ejpam-4276	186	12	sp)-closed	sp)-close	VERB
ejpam-4276	186	13	.	.	PUNCT
ejpam-4276	187	1	(	(	PUNCT
ejpam-4276	187	2	6	6	NUM
ejpam-4276	187	3	)	)	PUNCT
ejpam-4276	187	4	a	a	PRON
ejpam-4276	187	5	is	be	AUX
ejpam-4276	187	6	α(λ	α(λ	PROPN
ejpam-4276	187	7	,	,	PUNCT
ejpam-4276	187	8	sp)-open	sp)-open	ADJ
ejpam-4276	187	9	and	and	CCONJ
ejpam-4276	187	10	(	(	PUNCT
ejpam-4276	187	11	λ	λ	PROPN
ejpam-4276	187	12	,	,	PUNCT
ejpam-4276	187	13	sp)-closed	sp)-close	VERB
ejpam-4276	187	14	.	.	PUNCT
ejpam-4276	188	1	(	(	PUNCT
ejpam-4276	188	2	7	7	X
ejpam-4276	188	3	)	)	PUNCT
ejpam-4276	188	4	a	a	PRON
ejpam-4276	188	5	is	be	AUX
ejpam-4276	188	6	p(λ	p(λ	NOUN
ejpam-4276	188	7	,	,	PUNCT
ejpam-4276	188	8	sp)-open	sp)-open	ADJ
ejpam-4276	188	9	and	and	CCONJ
ejpam-4276	188	10	(	(	PUNCT
ejpam-4276	188	11	λ	λ	PROPN
ejpam-4276	188	12	,	,	PUNCT
ejpam-4276	188	13	sp)-closed	sp)-close	VERB
ejpam-4276	188	14	.	.	PUNCT
ejpam-4276	189	1	(	(	PUNCT
ejpam-4276	189	2	8)	8)	NUM
ejpam-4276	189	3	a	a	PRON
ejpam-4276	189	4	is	be	AUX
ejpam-4276	189	5	β(λ	β(λ	NOUN
ejpam-4276	189	6	,	,	PUNCT
ejpam-4276	189	7	sp)-open	sp)-open	ADJ
ejpam-4276	189	8	and	and	CCONJ
ejpam-4276	189	9	α(λ	α(λ	PROPN
ejpam-4276	189	10	,	,	PUNCT
ejpam-4276	189	11	sp)-closed	sp)-close	VERB
ejpam-4276	189	12	.	.	PUNCT
ejpam-4276	190	1	proof	proof	NOUN
ejpam-4276	190	2	.	.	PUNCT
ejpam-4276	191	1	(	(	PUNCT
ejpam-4276	191	2	1	1	X
ejpam-4276	191	3	)	)	PUNCT
ejpam-4276	191	4	⇒	⇒	NOUN
ejpam-4276	191	5	(	(	PUNCT
ejpam-4276	191	6	2	2	NUM
ejpam-4276	191	7	)	)	PUNCT
ejpam-4276	191	8	⇒	⇒	NOUN
ejpam-4276	191	9	(	(	PUNCT
ejpam-4276	191	10	3	3	NUM
ejpam-4276	191	11	)	)	PUNCT
ejpam-4276	191	12	⇒	⇒	NOUN
ejpam-4276	191	13	(	(	PUNCT
ejpam-4276	191	14	4	4	NUM
ejpam-4276	191	15	)	)	PUNCT
ejpam-4276	191	16	⇒	⇒	NOUN
ejpam-4276	191	17	(	(	PUNCT
ejpam-4276	191	18	5	5	NUM
ejpam-4276	191	19	):	):	PUNCT
ejpam-4276	191	20	obvious	obvious	ADJ
ejpam-4276	191	21	.	.	PUNCT
ejpam-4276	192	1	(	(	PUNCT
ejpam-4276	192	2	5	5	X
ejpam-4276	192	3	)	)	PUNCT
ejpam-4276	192	4	⇒	⇒	NOUN
ejpam-4276	192	5	(	(	PUNCT
ejpam-4276	192	6	6	6	NUM
ejpam-4276	192	7	):	):	PUNCT
ejpam-4276	192	8	let	let	VERB
ejpam-4276	192	9	a	a	DET
ejpam-4276	192	10	be	be	AUX
ejpam-4276	192	11	α(λ	α(λ	PROPN
ejpam-4276	192	12	,	,	PUNCT
ejpam-4276	192	13	sp)-open	sp)-open	NOUN
ejpam-4276	192	14	and	and	CCONJ
ejpam-4276	192	15	p(λ	p(λ	NOUN
ejpam-4276	192	16	,	,	PUNCT
ejpam-4276	192	17	sp)-closed	sp)-close	VERB
ejpam-4276	192	18	.	.	PUNCT
ejpam-4276	193	1	then	then	ADV
ejpam-4276	193	2	,	,	PUNCT
ejpam-4276	193	3	a	a	DET
ejpam-4276	193	4	⊆	⊆	NUM
ejpam-4276	193	5	[	[	X
ejpam-4276	193	6	[	[	X
ejpam-4276	193	7	a(λ	a(λ	ADJ
ejpam-4276	193	8	,	,	PUNCT
ejpam-4276	193	9	sp	sp	NOUN
ejpam-4276	193	10	)	)	PUNCT
ejpam-4276	193	11	]	]	PUNCT
ejpam-4276	194	1	(	(	PUNCT
ejpam-4276	194	2	λ	λ	X
ejpam-4276	194	3	,	,	PUNCT
ejpam-4276	194	4	sp)](λ	sp)](λ	PROPN
ejpam-4276	194	5	,	,	PUNCT
ejpam-4276	194	6	sp	sp	NOUN
ejpam-4276	194	7	)	)	PUNCT
ejpam-4276	194	8	and	and	CCONJ
ejpam-4276	194	9	[	[	X
ejpam-4276	194	10	[	[	X
ejpam-4276	194	11	a(λ	a(λ	ADJ
ejpam-4276	194	12	,	,	PUNCT
ejpam-4276	194	13	sp	sp	NOUN
ejpam-4276	194	14	)	)	PUNCT
ejpam-4276	194	15	]	]	PUNCT
ejpam-4276	194	16	(	(	PUNCT
ejpam-4276	194	17	λ	λ	X
ejpam-4276	194	18	,	,	PUNCT
ejpam-4276	194	19	sp)](λ	sp)](λ	PROPN
ejpam-4276	194	20	,	,	PUNCT
ejpam-4276	194	21	sp	sp	NOUN
ejpam-4276	194	22	)	)	PUNCT
ejpam-4276	194	23	⊆	⊆	NUM
ejpam-4276	194	24	a.	a.	NOUN
ejpam-4276	194	25	thus	thus	ADV
ejpam-4276	194	26	,	,	PUNCT
ejpam-4276	194	27	a	a	PRON
ejpam-4276	194	28	=	=	X
ejpam-4276	195	1	[	[	X
ejpam-4276	195	2	[	[	X
ejpam-4276	195	3	a(λ	a(λ	ADJ
ejpam-4276	195	4	,	,	PUNCT
ejpam-4276	195	5	sp	sp	NOUN
ejpam-4276	195	6	)	)	PUNCT
ejpam-4276	195	7	]	]	PUNCT
ejpam-4276	195	8	(	(	PUNCT
ejpam-4276	195	9	λ	λ	X
ejpam-4276	195	10	,	,	PUNCT
ejpam-4276	195	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	195	12	,	,	PUNCT
ejpam-4276	195	13	sp	sp	NOUN
ejpam-4276	195	14	)	)	PUNCT
ejpam-4276	195	15	and	and	CCONJ
ejpam-4276	195	16	hence	hence	ADV
ejpam-4276	195	17	a(λ	a(λ	ADV
ejpam-4276	195	18	,	,	PUNCT
ejpam-4276	195	19	sp	sp	NOUN
ejpam-4276	195	20	)	)	PUNCT
ejpam-4276	195	21	=	=	PUNCT
ejpam-4276	196	1	[	[	X
ejpam-4276	196	2	[	[	X
ejpam-4276	196	3	[	[	X
ejpam-4276	196	4	a(λ	a(λ	ADJ
ejpam-4276	196	5	,	,	PUNCT
ejpam-4276	196	6	sp	sp	NOUN
ejpam-4276	196	7	)	)	PUNCT
ejpam-4276	196	8	]	]	PUNCT
ejpam-4276	196	9	(	(	PUNCT
ejpam-4276	196	10	λ	λ	X
ejpam-4276	196	11	,	,	PUNCT
ejpam-4276	196	12	sp)](λ	sp)](λ	PROPN
ejpam-4276	196	13	,	,	PUNCT
ejpam-4276	196	14	sp	sp	NOUN
ejpam-4276	196	15	)	)	PUNCT
ejpam-4276	196	16	]	]	PUNCT
ejpam-4276	196	17	(	(	PUNCT
ejpam-4276	196	18	λ	λ	NOUN
ejpam-4276	196	19	,	,	PUNCT
ejpam-4276	196	20	sp	sp	NOUN
ejpam-4276	196	21	)	)	PUNCT
ejpam-4276	196	22	.	.	PUNCT
ejpam-4276	197	1	by	by	ADP
ejpam-4276	197	2	proposition	proposition	NOUN
ejpam-4276	197	3	5	5	NUM
ejpam-4276	197	4	,	,	PUNCT
ejpam-4276	197	5	a(λ	a(λ	ADV
ejpam-4276	197	6	,	,	PUNCT
ejpam-4276	197	7	sp	sp	NOUN
ejpam-4276	197	8	)	)	PUNCT
ejpam-4276	197	9	=	=	PUNCT
ejpam-4276	198	1	[	[	X
ejpam-4276	198	2	a(λ	a(λ	ADV
ejpam-4276	198	3	,	,	PUNCT
ejpam-4276	198	4	sp	sp	NOUN
ejpam-4276	198	5	)	)	PUNCT
ejpam-4276	198	6	]	]	PUNCT
ejpam-4276	198	7	(	(	PUNCT
ejpam-4276	198	8	λ	λ	NOUN
ejpam-4276	198	9	,	,	PUNCT
ejpam-4276	198	10	sp	sp	NOUN
ejpam-4276	198	11	)	)	PUNCT
ejpam-4276	198	12	.	.	PUNCT
ejpam-4276	199	1	since	since	SCONJ
ejpam-4276	199	2	[	[	X
ejpam-4276	199	3	a(λ	a(λ	ADV
ejpam-4276	199	4	,	,	PUNCT
ejpam-4276	199	5	sp	sp	NOUN
ejpam-4276	199	6	)	)	PUNCT
ejpam-4276	199	7	]	]	PUNCT
ejpam-4276	199	8	(	(	PUNCT
ejpam-4276	199	9	λ	λ	NOUN
ejpam-4276	199	10	,	,	PUNCT
ejpam-4276	199	11	sp	sp	NOUN
ejpam-4276	199	12	)	)	PUNCT
ejpam-4276	199	13	⊆	⊆	PROPN
ejpam-4276	199	14	a	a	PRON
ejpam-4276	199	15	,	,	PUNCT
ejpam-4276	199	16	we	we	PRON
ejpam-4276	199	17	have	have	AUX
ejpam-4276	199	18	a(λ	a(λ	ADV
ejpam-4276	199	19	,	,	PUNCT
ejpam-4276	199	20	sp	sp	NOUN
ejpam-4276	199	21	)	)	PUNCT
ejpam-4276	199	22	⊆	⊆	NUM
ejpam-4276	199	23	a	a	PRON
ejpam-4276	199	24	and	and	CCONJ
ejpam-4276	199	25	hence	hence	ADV
ejpam-4276	199	26	a(λ	a(λ	ADV
ejpam-4276	199	27	,	,	PUNCT
ejpam-4276	199	28	sp	sp	NOUN
ejpam-4276	199	29	)	)	PUNCT
ejpam-4276	199	30	=	=	SYM
ejpam-4276	199	31	a.	a.	NOUN
ejpam-4276	199	32	this	this	PRON
ejpam-4276	199	33	shows	show	VERB
ejpam-4276	199	34	that	that	SCONJ
ejpam-4276	199	35	a	a	PRON
ejpam-4276	199	36	is	be	AUX
ejpam-4276	199	37	(	(	PUNCT
ejpam-4276	199	38	λ	λ	NOUN
ejpam-4276	199	39	,	,	PUNCT
ejpam-4276	199	40	sp)-closed	sp)-close	VERB
ejpam-4276	199	41	.	.	PUNCT
ejpam-4276	200	1	c.	c.	PROPN
ejpam-4276	200	2	boonpok	boonpok	PROPN
ejpam-4276	200	3	,	,	PUNCT
ejpam-4276	200	4	j.	j.	PROPN
ejpam-4276	200	5	khampakdee	khampakdee	PROPN
ejpam-4276	200	6	/	/	PUNCT
ejpam-4276	200	7	eur	eur	PROPN
ejpam-4276	200	8	.	.	PUNCT
ejpam-4276	201	1	j.	j.	PROPN
ejpam-4276	201	2	pure	pure	PROPN
ejpam-4276	201	3	appl	appl	PROPN
ejpam-4276	201	4	.	.	PROPN
ejpam-4276	201	5	math	math	PROPN
ejpam-4276	201	6	,	,	PUNCT
ejpam-4276	201	7	15	15	NUM
ejpam-4276	201	8	(	(	PUNCT
ejpam-4276	201	9	2	2	NUM
ejpam-4276	201	10	)	)	PUNCT
ejpam-4276	201	11	(	(	PUNCT
ejpam-4276	201	12	2022	2022	NUM
ejpam-4276	201	13	)	)	PUNCT
ejpam-4276	201	14	,	,	PUNCT
ejpam-4276	201	15	572	572	NUM
ejpam-4276	201	16	-	-	SYM
ejpam-4276	201	17	588	588	NUM
ejpam-4276	201	18	578	578	NUM
ejpam-4276	201	19	(	(	PUNCT
ejpam-4276	201	20	6	6	NUM
ejpam-4276	201	21	)	)	PUNCT
ejpam-4276	201	22	⇒	⇒	NOUN
ejpam-4276	201	23	(	(	PUNCT
ejpam-4276	201	24	7	7	NUM
ejpam-4276	201	25	)	)	PUNCT
ejpam-4276	201	26	⇒	⇒	NOUN
ejpam-4276	201	27	(	(	PUNCT
ejpam-4276	201	28	8)	8)	NUM
ejpam-4276	201	29	:	:	SYM
ejpam-4276	201	30	obvious	obvious	ADJ
ejpam-4276	201	31	.	.	PUNCT
ejpam-4276	202	1	(	(	PUNCT
ejpam-4276	202	2	8)	8)	NUM
ejpam-4276	202	3	⇒	⇒	NOUN
ejpam-4276	202	4	(	(	PUNCT
ejpam-4276	202	5	1	1	NUM
ejpam-4276	202	6	):	):	PUNCT
ejpam-4276	202	7	let	let	VERB
ejpam-4276	202	8	a	a	DET
ejpam-4276	202	9	be	be	AUX
ejpam-4276	202	10	β(λ	β(λ	X
ejpam-4276	202	11	,	,	PUNCT
ejpam-4276	202	12	sp)-open	sp)-open	ADJ
ejpam-4276	202	13	and	and	CCONJ
ejpam-4276	202	14	α(λ	α(λ	PROPN
ejpam-4276	202	15	,	,	PUNCT
ejpam-4276	202	16	sp)-closed	sp)-close	VERB
ejpam-4276	202	17	.	.	PUNCT
ejpam-4276	203	1	then	then	ADV
ejpam-4276	203	2	,	,	PUNCT
ejpam-4276	203	3	a	a	DET
ejpam-4276	203	4	⊆	⊆	NUM
ejpam-4276	203	5	[	[	X
ejpam-4276	203	6	a(λ	a(λ	ADJ
ejpam-4276	203	7	,	,	PUNCT
ejpam-4276	203	8	sp)](λ	sp)](λ	PROPN
ejpam-4276	203	9	,	,	PUNCT
ejpam-4276	203	10	sp	sp	NOUN
ejpam-4276	203	11	)	)	PUNCT
ejpam-4276	203	12	and	and	CCONJ
ejpam-4276	204	1	[	[	X
ejpam-4276	204	2	[	[	X
ejpam-4276	204	3	a(λ	a(λ	ADJ
ejpam-4276	204	4	,	,	PUNCT
ejpam-4276	204	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	204	6	,	,	PUNCT
ejpam-4276	204	7	sp	sp	NOUN
ejpam-4276	204	8	)	)	PUNCT
ejpam-4276	204	9	]	]	PUNCT
ejpam-4276	204	10	(	(	PUNCT
ejpam-4276	204	11	λ	λ	NOUN
ejpam-4276	204	12	,	,	PUNCT
ejpam-4276	204	13	sp	sp	NOUN
ejpam-4276	204	14	)	)	PUNCT
ejpam-4276	204	15	⊆	⊆	NUM
ejpam-4276	204	16	a.	a.	NOUN
ejpam-4276	204	17	thus	thus	ADV
ejpam-4276	204	18	,	,	PUNCT
ejpam-4276	204	19	a(λ	a(λ	ADV
ejpam-4276	204	20	,	,	PUNCT
ejpam-4276	204	21	sp	sp	NOUN
ejpam-4276	204	22	)	)	PUNCT
ejpam-4276	204	23	⊆	⊆	NUM
ejpam-4276	205	1	[	[	X
ejpam-4276	205	2	[	[	X
ejpam-4276	205	3	a(λ	a(λ	ADJ
ejpam-4276	205	4	,	,	PUNCT
ejpam-4276	205	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	205	6	,	,	PUNCT
ejpam-4276	205	7	sp	sp	NOUN
ejpam-4276	205	8	)	)	PUNCT
ejpam-4276	205	9	]	]	PUNCT
ejpam-4276	206	1	(	(	PUNCT
ejpam-4276	206	2	λ	λ	NOUN
ejpam-4276	206	3	,	,	PUNCT
ejpam-4276	206	4	sp	sp	NOUN
ejpam-4276	206	5	)	)	PUNCT
ejpam-4276	206	6	⊆	⊆	NUM
ejpam-4276	206	7	a	a	PRON
ejpam-4276	206	8	and	and	CCONJ
ejpam-4276	206	9	hence	hence	ADV
ejpam-4276	206	10	a(λ	a(λ	ADV
ejpam-4276	206	11	,	,	PUNCT
ejpam-4276	206	12	sp	sp	NOUN
ejpam-4276	206	13	)	)	PUNCT
ejpam-4276	206	14	⊆	⊆	NUM
ejpam-4276	206	15	a.	a.	NOUN
ejpam-4276	206	16	therefore	therefore	ADV
ejpam-4276	206	17	,	,	PUNCT
ejpam-4276	206	18	a	a	PRON
ejpam-4276	206	19	is	be	AUX
ejpam-4276	206	20	(	(	PUNCT
ejpam-4276	206	21	λ	λ	X
ejpam-4276	206	22	,	,	PUNCT
ejpam-4276	206	23	sp)-closed	sp)-close	VERB
ejpam-4276	206	24	.	.	PUNCT
ejpam-4276	207	1	since	since	SCONJ
ejpam-4276	207	2	[	[	X
ejpam-4276	207	3	[	[	X
ejpam-4276	207	4	a(λ	a(λ	ADJ
ejpam-4276	207	5	,	,	PUNCT
ejpam-4276	207	6	sp)](λ	sp)](λ	PROPN
ejpam-4276	207	7	,	,	PUNCT
ejpam-4276	207	8	sp	sp	NOUN
ejpam-4276	207	9	)	)	PUNCT
ejpam-4276	207	10	]	]	PUNCT
ejpam-4276	207	11	(	(	PUNCT
ejpam-4276	207	12	λ	λ	NOUN
ejpam-4276	207	13	,	,	PUNCT
ejpam-4276	207	14	sp	sp	NOUN
ejpam-4276	207	15	)	)	PUNCT
ejpam-4276	207	16	⊆	⊆	PROPN
ejpam-4276	207	17	a	a	PRON
ejpam-4276	207	18	,	,	PUNCT
ejpam-4276	207	19	we	we	PRON
ejpam-4276	207	20	have	have	VERB
ejpam-4276	207	21	[	[	X
ejpam-4276	207	22	[	[	X
ejpam-4276	207	23	[	[	X
ejpam-4276	207	24	a(λ	a(λ	ADJ
ejpam-4276	207	25	,	,	PUNCT
ejpam-4276	207	26	sp)](λ	sp)](λ	PROPN
ejpam-4276	207	27	,	,	PUNCT
ejpam-4276	207	28	sp	sp	NOUN
ejpam-4276	207	29	)	)	PUNCT
ejpam-4276	207	30	]	]	PUNCT
ejpam-4276	207	31	(	(	PUNCT
ejpam-4276	207	32	λ	λ	X
ejpam-4276	207	33	,	,	PUNCT
ejpam-4276	207	34	sp)](λ	sp)](λ	PROPN
ejpam-4276	207	35	,	,	PUNCT
ejpam-4276	207	36	sp	sp	NOUN
ejpam-4276	207	37	)	)	PUNCT
ejpam-4276	207	38	⊆	⊆	NUM
ejpam-4276	207	39	a(λ	a(λ	ADV
ejpam-4276	207	40	,	,	PUNCT
ejpam-4276	207	41	sp	sp	NOUN
ejpam-4276	207	42	)	)	PUNCT
ejpam-4276	207	43	,	,	PUNCT
ejpam-4276	207	44	by	by	ADP
ejpam-4276	207	45	proposition	proposition	NOUN
ejpam-4276	207	46	5	5	NUM
ejpam-4276	207	47	,	,	PUNCT
ejpam-4276	207	48	a	a	DET
ejpam-4276	207	49	⊆	⊆	NUM
ejpam-4276	207	50	[	[	X
ejpam-4276	207	51	a(λ	a(λ	ADJ
ejpam-4276	207	52	,	,	PUNCT
ejpam-4276	207	53	sp)](λ	sp)](λ	PROPN
ejpam-4276	207	54	,	,	PUNCT
ejpam-4276	207	55	sp	sp	NOUN
ejpam-4276	207	56	)	)	PUNCT
ejpam-4276	207	57	⊆	⊆	NUM
ejpam-4276	207	58	a(λ	a(λ	ADV
ejpam-4276	207	59	,	,	PUNCT
ejpam-4276	207	60	sp	sp	NOUN
ejpam-4276	207	61	)	)	PUNCT
ejpam-4276	207	62	and	and	CCONJ
ejpam-4276	207	63	hence	hence	ADV
ejpam-4276	207	64	a	a	DET
ejpam-4276	207	65	⊆	⊆	NUM
ejpam-4276	207	66	a(λ	a(λ	ADJ
ejpam-4276	207	67	,	,	PUNCT
ejpam-4276	207	68	sp	sp	NOUN
ejpam-4276	207	69	)	)	PUNCT
ejpam-4276	207	70	.	.	PUNCT
ejpam-4276	208	1	thus	thus	ADV
ejpam-4276	208	2	,	,	PUNCT
ejpam-4276	208	3	a	a	DET
ejpam-4276	208	4	is	be	AUX
ejpam-4276	208	5	(	(	PUNCT
ejpam-4276	208	6	λ	λ	INTJ
ejpam-4276	208	7	,	,	PUNCT
ejpam-4276	208	8	sp)open	sp)open	VERB
ejpam-4276	208	9	.	.	PUNCT
ejpam-4276	209	1	therefore	therefore	ADV
ejpam-4276	209	2	,	,	PUNCT
ejpam-4276	209	3	a	a	PRON
ejpam-4276	209	4	is	be	AUX
ejpam-4276	209	5	(	(	PUNCT
ejpam-4276	209	6	λ	λ	PROPN
ejpam-4276	209	7	,	,	PUNCT
ejpam-4276	209	8	sp)-clopen	sp)-clopen	NOUN
ejpam-4276	209	9	.	.	PUNCT
ejpam-4276	210	1	definition	definition	NOUN
ejpam-4276	210	2	4	4	NUM
ejpam-4276	210	3	.	.	PUNCT
ejpam-4276	211	1	a	a	DET
ejpam-4276	211	2	subset	subset	NOUN
ejpam-4276	211	3	a	a	PRON
ejpam-4276	211	4	of	of	ADP
ejpam-4276	211	5	a	a	DET
ejpam-4276	211	6	topological	topological	ADJ
ejpam-4276	211	7	space	space	NOUN
ejpam-4276	211	8	(	(	PUNCT
ejpam-4276	211	9	x	x	X
ejpam-4276	211	10	,	,	PUNCT
ejpam-4276	211	11	τ	τ	X
ejpam-4276	211	12	)	)	PUNCT
ejpam-4276	211	13	is	be	AUX
ejpam-4276	211	14	said	say	VERB
ejpam-4276	211	15	to	to	PART
ejpam-4276	211	16	be	be	AUX
ejpam-4276	211	17	:	:	PUNCT
ejpam-4276	211	18	(	(	PUNCT
ejpam-4276	211	19	i	i	NOUN
ejpam-4276	211	20	)	)	PUNCT
ejpam-4276	211	21	α(λ	α(λ	PROPN
ejpam-4276	211	22	,	,	PUNCT
ejpam-4276	211	23	sp)-regular	sp)-regular	ADJ
ejpam-4276	211	24	if	if	SCONJ
ejpam-4276	211	25	a	a	PRON
ejpam-4276	211	26	=	=	X
ejpam-4276	212	1	[	[	X
ejpam-4276	212	2	[	[	X
ejpam-4276	212	3	a(λ	a(λ	ADJ
ejpam-4276	212	4	,	,	PUNCT
ejpam-4276	212	5	sp	sp	NOUN
ejpam-4276	212	6	)	)	PUNCT
ejpam-4276	212	7	]	]	PUNCT
ejpam-4276	212	8	(	(	PUNCT
ejpam-4276	212	9	λ	λ	X
ejpam-4276	212	10	,	,	PUNCT
ejpam-4276	212	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	212	12	,	,	PUNCT
ejpam-4276	212	13	sp	sp	NOUN
ejpam-4276	212	14	)	)	PUNCT
ejpam-4276	212	15	;	;	PUNCT
ejpam-4276	212	16	(	(	PUNCT
ejpam-4276	212	17	ii	ii	NOUN
ejpam-4276	212	18	)	)	PUNCT
ejpam-4276	212	19	β(λ	β(λ	PUNCT
ejpam-4276	212	20	,	,	PUNCT
ejpam-4276	212	21	sp)-regular	sp)-regular	ADJ
ejpam-4276	212	22	if	if	SCONJ
ejpam-4276	212	23	a	a	PRON
ejpam-4276	212	24	=	=	X
ejpam-4276	213	1	[	[	X
ejpam-4276	213	2	[	[	X
ejpam-4276	213	3	a(λ	a(λ	ADJ
ejpam-4276	213	4	,	,	PUNCT
ejpam-4276	213	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	213	6	,	,	PUNCT
ejpam-4276	213	7	sp	sp	NOUN
ejpam-4276	213	8	)	)	PUNCT
ejpam-4276	213	9	]	]	PUNCT
ejpam-4276	214	1	(	(	PUNCT
ejpam-4276	214	2	λ	λ	NOUN
ejpam-4276	214	3	,	,	PUNCT
ejpam-4276	214	4	sp	sp	NOUN
ejpam-4276	214	5	)	)	PUNCT
ejpam-4276	214	6	.	.	PUNCT
ejpam-4276	215	1	proposition	proposition	NOUN
ejpam-4276	215	2	7	7	NUM
ejpam-4276	215	3	.	.	PUNCT
ejpam-4276	215	4	let	let	VERB
ejpam-4276	215	5	a	a	DET
ejpam-4276	215	6	be	be	AUX
ejpam-4276	215	7	a	a	DET
ejpam-4276	215	8	subset	subset	NOUN
ejpam-4276	215	9	of	of	ADP
ejpam-4276	215	10	a	a	DET
ejpam-4276	215	11	topological	topological	ADJ
ejpam-4276	215	12	space	space	NOUN
ejpam-4276	215	13	(	(	PUNCT
ejpam-4276	215	14	x	x	X
ejpam-4276	215	15	,	,	PUNCT
ejpam-4276	215	16	τ	τ	PROPN
ejpam-4276	215	17	)	)	PUNCT
ejpam-4276	215	18	.	.	PUNCT
ejpam-4276	216	1	then	then	ADV
ejpam-4276	216	2	,	,	PUNCT
ejpam-4276	216	3	a	a	PRON
ejpam-4276	216	4	is	be	AUX
ejpam-4276	216	5	r(λ	r(λ	NOUN
ejpam-4276	216	6	,	,	PUNCT
ejpam-4276	216	7	sp)-open	sp)-open	ADJ
ejpam-4276	216	8	if	if	SCONJ
ejpam-4276	216	9	and	and	CCONJ
ejpam-4276	216	10	only	only	ADV
ejpam-4276	216	11	if	if	SCONJ
ejpam-4276	216	12	a	a	PRON
ejpam-4276	216	13	is	be	AUX
ejpam-4276	216	14	α(λ	α(λ	PROPN
ejpam-4276	216	15	,	,	PUNCT
ejpam-4276	216	16	sp)-regular	sp)-regular	NOUN
ejpam-4276	216	17	.	.	PUNCT
ejpam-4276	217	1	proof	proof	NOUN
ejpam-4276	217	2	.	.	PUNCT
ejpam-4276	218	1	suppose	suppose	VERB
ejpam-4276	218	2	that	that	SCONJ
ejpam-4276	218	3	a	a	PRON
ejpam-4276	218	4	is	be	AUX
ejpam-4276	218	5	a	a	DET
ejpam-4276	218	6	r(λ	r(λ	NOUN
ejpam-4276	218	7	,	,	PUNCT
ejpam-4276	218	8	sp)-open	sp)-open	ADJ
ejpam-4276	218	9	set	set	NOUN
ejpam-4276	218	10	.	.	PUNCT
ejpam-4276	219	1	then	then	ADV
ejpam-4276	219	2	,	,	PUNCT
ejpam-4276	219	3	a	a	DET
ejpam-4276	219	4	=	=	X
ejpam-4276	219	5	[	[	X
ejpam-4276	219	6	a(λ	a(λ	PROPN
ejpam-4276	219	7	,	,	PUNCT
ejpam-4276	219	8	sp)](λ	sp)](λ	PROPN
ejpam-4276	219	9	,	,	PUNCT
ejpam-4276	219	10	sp	sp	NOUN
ejpam-4276	219	11	)	)	PUNCT
ejpam-4276	219	12	.	.	PUNCT
ejpam-4276	220	1	this	this	PRON
ejpam-4276	220	2	implies	imply	VERB
ejpam-4276	220	3	that	that	SCONJ
ejpam-4276	220	4	a	a	PRON
ejpam-4276	220	5	is	be	AUX
ejpam-4276	220	6	(	(	PUNCT
ejpam-4276	220	7	λ	λ	NOUN
ejpam-4276	220	8	,	,	PUNCT
ejpam-4276	220	9	sp)-open	sp)-open	ADJ
ejpam-4276	220	10	and	and	CCONJ
ejpam-4276	220	11	so	so	ADV
ejpam-4276	220	12	a	a	PRON
ejpam-4276	220	13	=	=	X
ejpam-4276	221	1	[	[	X
ejpam-4276	221	2	[	[	X
ejpam-4276	221	3	a(λ	a(λ	ADJ
ejpam-4276	221	4	,	,	PUNCT
ejpam-4276	221	5	sp	sp	NOUN
ejpam-4276	221	6	)	)	PUNCT
ejpam-4276	221	7	]	]	PUNCT
ejpam-4276	221	8	(	(	PUNCT
ejpam-4276	221	9	λ	λ	X
ejpam-4276	221	10	,	,	PUNCT
ejpam-4276	221	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	221	12	,	,	PUNCT
ejpam-4276	221	13	sp	sp	NOUN
ejpam-4276	221	14	)	)	PUNCT
ejpam-4276	221	15	.	.	PUNCT
ejpam-4276	222	1	thus	thus	ADV
ejpam-4276	222	2	,	,	PUNCT
ejpam-4276	222	3	a	a	PRON
ejpam-4276	222	4	is	be	AUX
ejpam-4276	222	5	α(λ	α(λ	PROPN
ejpam-4276	222	6	,	,	PUNCT
ejpam-4276	222	7	sp)-regular	sp)-regular	NOUN
ejpam-4276	222	8	.	.	PUNCT
ejpam-4276	223	1	conversely	conversely	ADV
ejpam-4276	223	2	,	,	PUNCT
ejpam-4276	223	3	suppose	suppose	VERB
ejpam-4276	223	4	that	that	SCONJ
ejpam-4276	223	5	a	a	PRON
ejpam-4276	223	6	is	be	AUX
ejpam-4276	223	7	an	an	DET
ejpam-4276	223	8	α(λ	α(λ	PROPN
ejpam-4276	223	9	,	,	PUNCT
ejpam-4276	223	10	sp)-regular	sp)-regular	ADJ
ejpam-4276	223	11	set	set	NOUN
ejpam-4276	223	12	.	.	PUNCT
ejpam-4276	224	1	then	then	ADV
ejpam-4276	224	2	,	,	PUNCT
ejpam-4276	224	3	a	a	PRON
ejpam-4276	224	4	=	=	X
ejpam-4276	225	1	[	[	X
ejpam-4276	225	2	[	[	X
ejpam-4276	225	3	a(λ	a(λ	ADJ
ejpam-4276	225	4	,	,	PUNCT
ejpam-4276	225	5	sp	sp	NOUN
ejpam-4276	225	6	)	)	PUNCT
ejpam-4276	225	7	]	]	PUNCT
ejpam-4276	225	8	(	(	PUNCT
ejpam-4276	225	9	λ	λ	X
ejpam-4276	225	10	,	,	PUNCT
ejpam-4276	225	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	225	12	,	,	PUNCT
ejpam-4276	225	13	sp	sp	NOUN
ejpam-4276	225	14	)	)	PUNCT
ejpam-4276	225	15	.	.	PUNCT
ejpam-4276	226	1	therefore	therefore	ADV
ejpam-4276	226	2	,	,	PUNCT
ejpam-4276	226	3	a	a	PRON
ejpam-4276	226	4	=	=	X
ejpam-4276	227	1	[	[	X
ejpam-4276	227	2	[	[	X
ejpam-4276	227	3	[	[	X
ejpam-4276	227	4	[	[	X
ejpam-4276	227	5	a(λ	a(λ	ADJ
ejpam-4276	227	6	,	,	PUNCT
ejpam-4276	227	7	sp	sp	NOUN
ejpam-4276	227	8	)	)	PUNCT
ejpam-4276	227	9	]	]	PUNCT
ejpam-4276	227	10	(	(	PUNCT
ejpam-4276	227	11	λ	λ	X
ejpam-4276	227	12	,	,	PUNCT
ejpam-4276	227	13	sp)](λ	sp)](λ	PROPN
ejpam-4276	227	14	,	,	PUNCT
ejpam-4276	227	15	sp)](λ	sp)](λ	PROPN
ejpam-4276	227	16	,	,	PUNCT
ejpam-4276	227	17	sp)](λ	sp)](λ	PROPN
ejpam-4276	227	18	,	,	PUNCT
ejpam-4276	227	19	sp	sp	NOUN
ejpam-4276	227	20	)	)	PUNCT
ejpam-4276	227	21	=	=	PUNCT
ejpam-4276	228	1	[	[	X
ejpam-4276	228	2	[	[	X
ejpam-4276	228	3	a(λ	a(λ	ADJ
ejpam-4276	228	4	,	,	PUNCT
ejpam-4276	228	5	sp	sp	NOUN
ejpam-4276	228	6	)	)	PUNCT
ejpam-4276	228	7	]	]	PUNCT
ejpam-4276	228	8	(	(	PUNCT
ejpam-4276	228	9	λ	λ	X
ejpam-4276	228	10	,	,	PUNCT
ejpam-4276	228	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	228	12	,	,	PUNCT
ejpam-4276	228	13	sp	sp	NOUN
ejpam-4276	228	14	)	)	PUNCT
ejpam-4276	228	15	=	=	PUNCT
ejpam-4276	229	1	a	a	PRON
ejpam-4276	229	2	and	and	CCONJ
ejpam-4276	229	3	hence	hence	ADV
ejpam-4276	229	4	a	a	PRON
ejpam-4276	229	5	=	=	X
ejpam-4276	230	1	[	[	X
ejpam-4276	230	2	a(λ	a(λ	PROPN
ejpam-4276	230	3	,	,	PUNCT
ejpam-4276	230	4	sp)](λ	sp)](λ	PROPN
ejpam-4276	230	5	,	,	PUNCT
ejpam-4276	230	6	sp	sp	NOUN
ejpam-4276	230	7	)	)	PUNCT
ejpam-4276	230	8	.	.	PUNCT
ejpam-4276	231	1	thus	thus	ADV
ejpam-4276	231	2	,	,	PUNCT
ejpam-4276	231	3	a	a	PRON
ejpam-4276	231	4	is	be	AUX
ejpam-4276	231	5	(	(	PUNCT
ejpam-4276	231	6	λ	λ	NOUN
ejpam-4276	231	7	,	,	PUNCT
ejpam-4276	231	8	sp)-open	sp)-open	NOUN
ejpam-4276	231	9	.	.	PUNCT
ejpam-4276	232	1	proposition	proposition	NOUN
ejpam-4276	232	2	8	8	NUM
ejpam-4276	232	3	.	.	PUNCT
ejpam-4276	233	1	let	let	VERB
ejpam-4276	233	2	a	a	DET
ejpam-4276	233	3	be	be	AUX
ejpam-4276	233	4	a	a	DET
ejpam-4276	233	5	subset	subset	NOUN
ejpam-4276	233	6	of	of	ADP
ejpam-4276	233	7	a	a	DET
ejpam-4276	233	8	topological	topological	ADJ
ejpam-4276	233	9	space	space	NOUN
ejpam-4276	233	10	(	(	PUNCT
ejpam-4276	233	11	x	x	X
ejpam-4276	233	12	,	,	PUNCT
ejpam-4276	233	13	τ	τ	PROPN
ejpam-4276	233	14	)	)	PUNCT
ejpam-4276	233	15	.	.	PUNCT
ejpam-4276	234	1	then	then	ADV
ejpam-4276	234	2	,	,	PUNCT
ejpam-4276	234	3	a	a	PRON
ejpam-4276	234	4	is	be	AUX
ejpam-4276	234	5	r(λ	r(λ	NOUN
ejpam-4276	234	6	,	,	PUNCT
ejpam-4276	234	7	sp)-closed	sp)-close	VERB
ejpam-4276	234	8	if	if	SCONJ
ejpam-4276	234	9	and	and	CCONJ
ejpam-4276	234	10	only	only	ADV
ejpam-4276	234	11	if	if	SCONJ
ejpam-4276	234	12	a	a	PRON
ejpam-4276	234	13	is	be	AUX
ejpam-4276	234	14	β(λ	β(λ	NOUN
ejpam-4276	234	15	,	,	PUNCT
ejpam-4276	234	16	sp)-regular	sp)-regular	NOUN
ejpam-4276	234	17	.	.	PUNCT
ejpam-4276	235	1	proof	proof	NOUN
ejpam-4276	235	2	.	.	PUNCT
ejpam-4276	236	1	suppose	suppose	VERB
ejpam-4276	236	2	that	that	SCONJ
ejpam-4276	236	3	a	a	PRON
ejpam-4276	236	4	is	be	AUX
ejpam-4276	236	5	a	a	DET
ejpam-4276	236	6	r(λ	r(λ	NOUN
ejpam-4276	236	7	,	,	PUNCT
ejpam-4276	236	8	sp)-closed	sp)-close	VERB
ejpam-4276	236	9	set	set	NOUN
ejpam-4276	236	10	.	.	PUNCT
ejpam-4276	237	1	then	then	ADV
ejpam-4276	237	2	,	,	PUNCT
ejpam-4276	237	3	we	we	PRON
ejpam-4276	237	4	have	have	VERB
ejpam-4276	237	5	a	a	DET
ejpam-4276	237	6	=	=	X
ejpam-4276	237	7	[	[	X
ejpam-4276	237	8	a(λ	a(λ	ADV
ejpam-4276	237	9	,	,	PUNCT
ejpam-4276	237	10	sp	sp	NOUN
ejpam-4276	237	11	)	)	PUNCT
ejpam-4276	237	12	]	]	PUNCT
ejpam-4276	237	13	(	(	PUNCT
ejpam-4276	237	14	λ	λ	NOUN
ejpam-4276	237	15	,	,	PUNCT
ejpam-4276	237	16	sp	sp	NOUN
ejpam-4276	237	17	)	)	PUNCT
ejpam-4276	237	18	and	and	CCONJ
ejpam-4276	237	19	so	so	ADV
ejpam-4276	237	20	a	a	PRON
ejpam-4276	237	21	is	be	AUX
ejpam-4276	237	22	(	(	PUNCT
ejpam-4276	237	23	λ	λ	NOUN
ejpam-4276	237	24	,	,	PUNCT
ejpam-4276	237	25	sp)-closed	sp)-close	VERB
ejpam-4276	237	26	.	.	PUNCT
ejpam-4276	238	1	therefore	therefore	ADV
ejpam-4276	238	2	,	,	PUNCT
ejpam-4276	238	3	a	a	DET
ejpam-4276	238	4	=	=	X
ejpam-4276	238	5	[	[	X
ejpam-4276	238	6	a(λ	a(λ	ADV
ejpam-4276	238	7	,	,	PUNCT
ejpam-4276	238	8	sp	sp	NOUN
ejpam-4276	238	9	)	)	PUNCT
ejpam-4276	238	10	]	]	PUNCT
ejpam-4276	239	1	(	(	PUNCT
ejpam-4276	239	2	λ	λ	NOUN
ejpam-4276	239	3	,	,	PUNCT
ejpam-4276	239	4	sp	sp	NOUN
ejpam-4276	239	5	)	)	PUNCT
ejpam-4276	239	6	=	=	PUNCT
ejpam-4276	240	1	[	[	X
ejpam-4276	240	2	[	[	X
ejpam-4276	240	3	a(λ	a(λ	ADJ
ejpam-4276	240	4	,	,	PUNCT
ejpam-4276	240	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	240	6	,	,	PUNCT
ejpam-4276	240	7	sp	sp	NOUN
ejpam-4276	240	8	)	)	PUNCT
ejpam-4276	240	9	]	]	PUNCT
ejpam-4276	240	10	(	(	PUNCT
ejpam-4276	240	11	λ	λ	NOUN
ejpam-4276	240	12	,	,	PUNCT
ejpam-4276	240	13	sp	sp	NOUN
ejpam-4276	240	14	)	)	PUNCT
ejpam-4276	240	15	.	.	PUNCT
ejpam-4276	241	1	this	this	PRON
ejpam-4276	241	2	shows	show	VERB
ejpam-4276	241	3	that	that	SCONJ
ejpam-4276	241	4	a	a	PRON
ejpam-4276	241	5	is	be	AUX
ejpam-4276	241	6	β(λ	β(λ	NOUN
ejpam-4276	241	7	,	,	PUNCT
ejpam-4276	241	8	sp)-regular	sp)-regular	NOUN
ejpam-4276	241	9	.	.	PUNCT
ejpam-4276	242	1	conversely	conversely	ADV
ejpam-4276	242	2	,	,	PUNCT
ejpam-4276	242	3	suppose	suppose	VERB
ejpam-4276	242	4	that	that	SCONJ
ejpam-4276	242	5	a	a	PRON
ejpam-4276	242	6	is	be	AUX
ejpam-4276	242	7	a	a	DET
ejpam-4276	242	8	β(λ	β(λ	NOUN
ejpam-4276	242	9	,	,	PUNCT
ejpam-4276	242	10	sp)-regular	sp)-regular	ADJ
ejpam-4276	242	11	set	set	NOUN
ejpam-4276	242	12	.	.	PUNCT
ejpam-4276	243	1	then	then	ADV
ejpam-4276	243	2	,	,	PUNCT
ejpam-4276	243	3	a	a	PRON
ejpam-4276	243	4	=	=	X
ejpam-4276	244	1	[	[	X
ejpam-4276	244	2	[	[	X
ejpam-4276	244	3	a(λ	a(λ	ADJ
ejpam-4276	244	4	,	,	PUNCT
ejpam-4276	244	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	244	6	,	,	PUNCT
ejpam-4276	244	7	sp	sp	NOUN
ejpam-4276	244	8	)	)	PUNCT
ejpam-4276	244	9	]	]	PUNCT
ejpam-4276	245	1	(	(	PUNCT
ejpam-4276	245	2	λ	λ	NOUN
ejpam-4276	245	3	,	,	PUNCT
ejpam-4276	245	4	sp	sp	NOUN
ejpam-4276	245	5	)	)	PUNCT
ejpam-4276	245	6	=	=	PUNCT
ejpam-4276	246	1	[	[	X
ejpam-4276	246	2	a(λ	a(λ	ADV
ejpam-4276	246	3	,	,	PUNCT
ejpam-4276	246	4	sp	sp	NOUN
ejpam-4276	246	5	)	)	PUNCT
ejpam-4276	246	6	]	]	PUNCT
ejpam-4276	246	7	(	(	PUNCT
ejpam-4276	246	8	λ	λ	NOUN
ejpam-4276	246	9	,	,	PUNCT
ejpam-4276	246	10	sp	sp	NOUN
ejpam-4276	246	11	)	)	PUNCT
ejpam-4276	246	12	.	.	PUNCT
ejpam-4276	247	1	thus	thus	ADV
ejpam-4276	247	2	,	,	PUNCT
ejpam-4276	247	3	a	a	PRON
ejpam-4276	247	4	is	be	AUX
ejpam-4276	247	5	r(λ	r(λ	NOUN
ejpam-4276	247	6	,	,	PUNCT
ejpam-4276	247	7	sp)-closed	sp)-close	VERB
ejpam-4276	247	8	.	.	PUNCT
ejpam-4276	248	1	proposition	proposition	NOUN
ejpam-4276	248	2	9	9	NUM
ejpam-4276	248	3	.	.	PUNCT
ejpam-4276	249	1	for	for	ADP
ejpam-4276	249	2	a	a	DET
ejpam-4276	249	3	subset	subset	NOUN
ejpam-4276	249	4	a	a	PRON
ejpam-4276	249	5	of	of	ADP
ejpam-4276	249	6	a	a	DET
ejpam-4276	249	7	topological	topological	ADJ
ejpam-4276	249	8	space	space	NOUN
ejpam-4276	249	9	(	(	PUNCT
ejpam-4276	249	10	x	x	X
ejpam-4276	249	11	,	,	PUNCT
ejpam-4276	249	12	τ	τ	PROPN
ejpam-4276	249	13	)	)	PUNCT
ejpam-4276	249	14	,	,	PUNCT
ejpam-4276	249	15	the	the	DET
ejpam-4276	249	16	following	follow	VERB
ejpam-4276	249	17	properties	property	NOUN
ejpam-4276	249	18	are	be	AUX
ejpam-4276	249	19	equivalent	equivalent	ADJ
ejpam-4276	249	20	:	:	PUNCT
ejpam-4276	249	21	(	(	PUNCT
ejpam-4276	249	22	1	1	X
ejpam-4276	249	23	)	)	PUNCT
ejpam-4276	249	24	a	a	PRON
ejpam-4276	249	25	is	be	AUX
ejpam-4276	249	26	β(λ	β(λ	NOUN
ejpam-4276	249	27	,	,	PUNCT
ejpam-4276	249	28	sp)-regular	sp)-regular	NOUN
ejpam-4276	249	29	.	.	PUNCT
ejpam-4276	250	1	(	(	PUNCT
ejpam-4276	250	2	2	2	X
ejpam-4276	250	3	)	)	PUNCT
ejpam-4276	250	4	a	a	PRON
ejpam-4276	250	5	is	be	AUX
ejpam-4276	250	6	β(λ	β(λ	NOUN
ejpam-4276	250	7	,	,	PUNCT
ejpam-4276	250	8	sp)-open	sp)-open	ADJ
ejpam-4276	250	9	and	and	CCONJ
ejpam-4276	250	10	(	(	PUNCT
ejpam-4276	250	11	λ	λ	PROPN
ejpam-4276	250	12	,	,	PUNCT
ejpam-4276	250	13	sp)-closed	sp)-close	VERB
ejpam-4276	250	14	.	.	PUNCT
ejpam-4276	251	1	(	(	PUNCT
ejpam-4276	251	2	3	3	X
ejpam-4276	251	3	)	)	PUNCT
ejpam-4276	251	4	a	a	PRON
ejpam-4276	251	5	is	be	AUX
ejpam-4276	251	6	β(λ	β(λ	NOUN
ejpam-4276	251	7	,	,	PUNCT
ejpam-4276	251	8	sp)-open	sp)-open	ADJ
ejpam-4276	251	9	and	and	CCONJ
ejpam-4276	251	10	α(λ	α(λ	PROPN
ejpam-4276	251	11	,	,	PUNCT
ejpam-4276	251	12	sp)-closed	sp)-close	VERB
ejpam-4276	251	13	.	.	PUNCT
ejpam-4276	252	1	proposition	proposition	NOUN
ejpam-4276	252	2	10	10	NUM
ejpam-4276	252	3	.	.	PUNCT
ejpam-4276	253	1	for	for	ADP
ejpam-4276	253	2	a	a	DET
ejpam-4276	253	3	subset	subset	NOUN
ejpam-4276	253	4	a	a	PRON
ejpam-4276	253	5	of	of	ADP
ejpam-4276	253	6	a	a	DET
ejpam-4276	253	7	topological	topological	ADJ
ejpam-4276	253	8	space	space	NOUN
ejpam-4276	253	9	(	(	PUNCT
ejpam-4276	253	10	x	x	X
ejpam-4276	253	11	,	,	PUNCT
ejpam-4276	253	12	τ	τ	PROPN
ejpam-4276	253	13	)	)	PUNCT
ejpam-4276	253	14	,	,	PUNCT
ejpam-4276	253	15	the	the	DET
ejpam-4276	253	16	following	follow	VERB
ejpam-4276	253	17	properties	property	NOUN
ejpam-4276	253	18	are	be	AUX
ejpam-4276	253	19	equivalent	equivalent	ADJ
ejpam-4276	253	20	:	:	PUNCT
ejpam-4276	253	21	c.	c.	PROPN
ejpam-4276	253	22	boonpok	boonpok	PROPN
ejpam-4276	253	23	,	,	PUNCT
ejpam-4276	253	24	j.	j.	PROPN
ejpam-4276	253	25	khampakdee	khampakdee	PROPN
ejpam-4276	253	26	/	/	PUNCT
ejpam-4276	253	27	eur	eur	PROPN
ejpam-4276	253	28	.	.	PUNCT
ejpam-4276	254	1	j.	j.	PROPN
ejpam-4276	254	2	pure	pure	PROPN
ejpam-4276	254	3	appl	appl	PROPN
ejpam-4276	254	4	.	.	PROPN
ejpam-4276	254	5	math	math	PROPN
ejpam-4276	254	6	,	,	PUNCT
ejpam-4276	254	7	15	15	NUM
ejpam-4276	254	8	(	(	PUNCT
ejpam-4276	254	9	2	2	NUM
ejpam-4276	254	10	)	)	PUNCT
ejpam-4276	254	11	(	(	PUNCT
ejpam-4276	254	12	2022	2022	NUM
ejpam-4276	254	13	)	)	PUNCT
ejpam-4276	254	14	,	,	PUNCT
ejpam-4276	254	15	572	572	NUM
ejpam-4276	254	16	-	-	SYM
ejpam-4276	254	17	588	588	NUM
ejpam-4276	254	18	579	579	NUM
ejpam-4276	254	19	(	(	PUNCT
ejpam-4276	254	20	1	1	X
ejpam-4276	254	21	)	)	PUNCT
ejpam-4276	254	22	a	a	PRON
ejpam-4276	254	23	is	be	AUX
ejpam-4276	254	24	α(λ	α(λ	PROPN
ejpam-4276	254	25	,	,	PUNCT
ejpam-4276	254	26	sp)-regular	sp)-regular	NOUN
ejpam-4276	254	27	.	.	PUNCT
ejpam-4276	255	1	(	(	PUNCT
ejpam-4276	255	2	2	2	X
ejpam-4276	255	3	)	)	PUNCT
ejpam-4276	255	4	a	a	PRON
ejpam-4276	255	5	is	be	AUX
ejpam-4276	255	6	α(λ	α(λ	PROPN
ejpam-4276	255	7	,	,	PUNCT
ejpam-4276	255	8	sp)-open	sp)-open	ADJ
ejpam-4276	255	9	and	and	CCONJ
ejpam-4276	255	10	β(λ	β(λ	NOUN
ejpam-4276	255	11	,	,	PUNCT
ejpam-4276	255	12	sp)-closed	sp)-close	VERB
ejpam-4276	255	13	.	.	PUNCT
ejpam-4276	256	1	definition	definition	NOUN
ejpam-4276	256	2	5	5	NUM
ejpam-4276	256	3	.	.	PUNCT
ejpam-4276	257	1	a	a	DET
ejpam-4276	257	2	subset	subset	NOUN
ejpam-4276	257	3	a	a	PRON
ejpam-4276	257	4	of	of	ADP
ejpam-4276	257	5	a	a	DET
ejpam-4276	257	6	topological	topological	ADJ
ejpam-4276	257	7	space	space	NOUN
ejpam-4276	257	8	(	(	PUNCT
ejpam-4276	257	9	x	x	X
ejpam-4276	257	10	,	,	PUNCT
ejpam-4276	257	11	τ	τ	X
ejpam-4276	257	12	)	)	PUNCT
ejpam-4276	257	13	is	be	AUX
ejpam-4276	257	14	said	say	VERB
ejpam-4276	257	15	to	to	PART
ejpam-4276	257	16	be	be	AUX
ejpam-4276	257	17	b(λ	b(λ	NOUN
ejpam-4276	257	18	,	,	PUNCT
ejpam-4276	257	19	sp)-open	sp)-open	ADJ
ejpam-4276	257	20	if	if	SCONJ
ejpam-4276	257	21	a	a	DET
ejpam-4276	257	22	⊆	⊆	NUM
ejpam-4276	257	23	[	[	X
ejpam-4276	257	24	a(λ	a(λ	ADV
ejpam-4276	257	25	,	,	PUNCT
ejpam-4276	257	26	sp	sp	NOUN
ejpam-4276	257	27	)	)	PUNCT
ejpam-4276	257	28	]	]	PUNCT
ejpam-4276	257	29	(	(	PUNCT
ejpam-4276	257	30	λ	λ	NOUN
ejpam-4276	257	31	,	,	PUNCT
ejpam-4276	257	32	sp	sp	NOUN
ejpam-4276	257	33	)	)	PUNCT
ejpam-4276	257	34	∪	∪	ADP
ejpam-4276	257	35	[	[	X
ejpam-4276	257	36	a(λ	a(λ	ADJ
ejpam-4276	257	37	,	,	PUNCT
ejpam-4276	257	38	sp)](λ	sp)](λ	PROPN
ejpam-4276	257	39	,	,	PUNCT
ejpam-4276	257	40	sp	sp	NOUN
ejpam-4276	257	41	)	)	PUNCT
ejpam-4276	257	42	.	.	PUNCT
ejpam-4276	258	1	the	the	DET
ejpam-4276	258	2	complement	complement	NOUN
ejpam-4276	258	3	of	of	ADP
ejpam-4276	258	4	a	a	DET
ejpam-4276	258	5	b(λ	b(λ	NOUN
ejpam-4276	258	6	,	,	PUNCT
ejpam-4276	258	7	sp)-open	sp)-open	ADJ
ejpam-4276	258	8	set	set	NOUN
ejpam-4276	258	9	is	be	AUX
ejpam-4276	258	10	said	say	VERB
ejpam-4276	258	11	to	to	PART
ejpam-4276	258	12	be	be	AUX
ejpam-4276	258	13	b(λ	b(λ	NOUN
ejpam-4276	258	14	,	,	PUNCT
ejpam-4276	258	15	sp)-closed	sp)-close	VERB
ejpam-4276	258	16	.	.	PUNCT
ejpam-4276	259	1	the	the	DET
ejpam-4276	259	2	family	family	NOUN
ejpam-4276	259	3	of	of	ADP
ejpam-4276	259	4	all	all	DET
ejpam-4276	259	5	b(λ	b(λ	NOUN
ejpam-4276	259	6	,	,	PUNCT
ejpam-4276	259	7	sp)-open	sp)-open	ADJ
ejpam-4276	259	8	(	(	PUNCT
ejpam-4276	259	9	resp	resp	NOUN
ejpam-4276	259	10	.	.	PUNCT
ejpam-4276	260	1	b(λ	b(λ	NOUN
ejpam-4276	260	2	,	,	PUNCT
ejpam-4276	260	3	sp)-closed	sp)-close	VERB
ejpam-4276	260	4	)	)	PUNCT
ejpam-4276	260	5	sets	set	NOUN
ejpam-4276	260	6	in	in	ADP
ejpam-4276	260	7	a	a	DET
ejpam-4276	260	8	topological	topological	ADJ
ejpam-4276	260	9	space	space	NOUN
ejpam-4276	260	10	(	(	PUNCT
ejpam-4276	260	11	x	x	X
ejpam-4276	260	12	,	,	PUNCT
ejpam-4276	260	13	τ	τ	X
ejpam-4276	260	14	)	)	PUNCT
ejpam-4276	260	15	is	be	AUX
ejpam-4276	260	16	denoted	denote	VERB
ejpam-4276	260	17	by	by	ADP
ejpam-4276	260	18	bλspo(x	bλspo(x	PROPN
ejpam-4276	260	19	,	,	PUNCT
ejpam-4276	260	20	τ	τ	X
ejpam-4276	260	21	)	)	PUNCT
ejpam-4276	260	22	(	(	PUNCT
ejpam-4276	260	23	resp	resp	NOUN
ejpam-4276	260	24	.	.	PUNCT
ejpam-4276	261	1	bλspc(x	bλspc(x	NOUN
ejpam-4276	261	2	,	,	PUNCT
ejpam-4276	261	3	τ	τ	PROPN
ejpam-4276	261	4	)	)	PUNCT
ejpam-4276	261	5	)	)	PUNCT
ejpam-4276	261	6	.	.	PUNCT
ejpam-4276	262	1	remark	remark	PROPN
ejpam-4276	262	2	1	1	NUM
ejpam-4276	262	3	.	.	PUNCT
ejpam-4276	263	1	it	it	PRON
ejpam-4276	263	2	is	be	AUX
ejpam-4276	263	3	easy	easy	ADJ
ejpam-4276	263	4	to	to	PART
ejpam-4276	263	5	see	see	VERB
ejpam-4276	263	6	that	that	PRON
ejpam-4276	263	7	for	for	ADP
ejpam-4276	263	8	a	a	DET
ejpam-4276	263	9	topological	topological	ADJ
ejpam-4276	263	10	space	space	NOUN
ejpam-4276	263	11	(	(	PUNCT
ejpam-4276	263	12	x	x	X
ejpam-4276	263	13	,	,	PUNCT
ejpam-4276	263	14	τ	τ	PROPN
ejpam-4276	263	15	)	)	PUNCT
ejpam-4276	263	16	,	,	PUNCT
ejpam-4276	263	17	sλspo(x	sλspo(x	PROPN
ejpam-4276	263	18	,	,	PUNCT
ejpam-4276	263	19	τ	τ	PROPN
ejpam-4276	263	20	)	)	PUNCT
ejpam-4276	263	21	∪	∪	X
ejpam-4276	263	22	pλspo(x	pλspo(x	PROPN
ejpam-4276	263	23	,	,	PUNCT
ejpam-4276	263	24	τ	τ	PROPN
ejpam-4276	263	25	)	)	PUNCT
ejpam-4276	263	26	⊆	⊆	NUM
ejpam-4276	263	27	bλspo(x	bλspo(x	NOUN
ejpam-4276	263	28	,	,	PUNCT
ejpam-4276	263	29	τ	τ	PROPN
ejpam-4276	263	30	)	)	PUNCT
ejpam-4276	263	31	⊆	⊆	NUM
ejpam-4276	263	32	βλspo(x	βλspo(x	NUM
ejpam-4276	263	33	,	,	PUNCT
ejpam-4276	263	34	τ	τ	PROPN
ejpam-4276	263	35	)	)	PUNCT
ejpam-4276	263	36	.	.	PUNCT
ejpam-4276	264	1	proposition	proposition	NOUN
ejpam-4276	264	2	11	11	NUM
ejpam-4276	264	3	.	.	PUNCT
ejpam-4276	265	1	let	let	VERB
ejpam-4276	265	2	a	a	DET
ejpam-4276	265	3	be	be	AUX
ejpam-4276	265	4	a	a	DET
ejpam-4276	265	5	subset	subset	NOUN
ejpam-4276	265	6	of	of	ADP
ejpam-4276	265	7	a	a	DET
ejpam-4276	265	8	topological	topological	ADJ
ejpam-4276	265	9	space	space	NOUN
ejpam-4276	265	10	(	(	PUNCT
ejpam-4276	265	11	x	x	X
ejpam-4276	265	12	,	,	PUNCT
ejpam-4276	265	13	τ	τ	PROPN
ejpam-4276	265	14	)	)	PUNCT
ejpam-4276	265	15	.	.	PUNCT
ejpam-4276	266	1	if	if	SCONJ
ejpam-4276	266	2	a	a	DET
ejpam-4276	266	3	=	=	X
ejpam-4276	266	4	b	b	X
ejpam-4276	266	5	∪	∪	X
ejpam-4276	266	6	c	c	NOUN
ejpam-4276	266	7	,	,	PUNCT
ejpam-4276	266	8	where	where	SCONJ
ejpam-4276	266	9	a	a	PRON
ejpam-4276	266	10	is	be	AUX
ejpam-4276	266	11	s(λ	s(λ	NOUN
ejpam-4276	266	12	,	,	PUNCT
ejpam-4276	266	13	sp)-open	sp)-open	ADJ
ejpam-4276	266	14	and	and	CCONJ
ejpam-4276	266	15	c	c	PROPN
ejpam-4276	266	16	is	be	AUX
ejpam-4276	266	17	p(λ	p(λ	PROPN
ejpam-4276	266	18	,	,	PUNCT
ejpam-4276	266	19	sp)-open	sp)-open	NOUN
ejpam-4276	266	20	,	,	PUNCT
ejpam-4276	266	21	then	then	ADV
ejpam-4276	266	22	a	a	PRON
ejpam-4276	266	23	is	be	AUX
ejpam-4276	266	24	b(λ	b(λ	NOUN
ejpam-4276	266	25	,	,	PUNCT
ejpam-4276	266	26	sp)-open	sp)-open	NOUN
ejpam-4276	266	27	.	.	PUNCT
ejpam-4276	267	1	the	the	DET
ejpam-4276	267	2	following	following	ADJ
ejpam-4276	267	3	result	result	NOUN
ejpam-4276	267	4	is	be	AUX
ejpam-4276	267	5	an	an	DET
ejpam-4276	267	6	immediate	immediate	ADJ
ejpam-4276	267	7	consequence	consequence	NOUN
ejpam-4276	267	8	of	of	ADP
ejpam-4276	267	9	proposition	proposition	NOUN
ejpam-4276	267	10	5	5	NUM
ejpam-4276	267	11	and	and	CCONJ
ejpam-4276	267	12	remark	remark	NOUN
ejpam-4276	267	13	1	1	NUM
ejpam-4276	267	14	.	.	PUNCT
ejpam-4276	267	15	corollary	corollary	ADJ
ejpam-4276	267	16	2	2	NUM
ejpam-4276	267	17	.	.	PUNCT
ejpam-4276	267	18	for	for	ADP
ejpam-4276	267	19	a	a	DET
ejpam-4276	267	20	subset	subset	NOUN
ejpam-4276	267	21	a	a	PRON
ejpam-4276	267	22	of	of	ADP
ejpam-4276	267	23	a	a	DET
ejpam-4276	267	24	topological	topological	ADJ
ejpam-4276	267	25	space	space	NOUN
ejpam-4276	267	26	(	(	PUNCT
ejpam-4276	267	27	x	x	X
ejpam-4276	267	28	,	,	PUNCT
ejpam-4276	267	29	τ	τ	PROPN
ejpam-4276	267	30	)	)	PUNCT
ejpam-4276	267	31	,	,	PUNCT
ejpam-4276	267	32	the	the	DET
ejpam-4276	267	33	following	follow	VERB
ejpam-4276	267	34	properties	property	NOUN
ejpam-4276	267	35	are	be	AUX
ejpam-4276	267	36	equivalent	equivalent	ADJ
ejpam-4276	267	37	:	:	PUNCT
ejpam-4276	267	38	(	(	PUNCT
ejpam-4276	267	39	1	1	X
ejpam-4276	267	40	)	)	PUNCT
ejpam-4276	267	41	a	a	PRON
ejpam-4276	267	42	is	be	AUX
ejpam-4276	267	43	r(λ	r(λ	NOUN
ejpam-4276	267	44	,	,	PUNCT
ejpam-4276	267	45	sp)-open	sp)-open	NOUN
ejpam-4276	267	46	.	.	PUNCT
ejpam-4276	268	1	(	(	PUNCT
ejpam-4276	268	2	2	2	X
ejpam-4276	268	3	)	)	PUNCT
ejpam-4276	268	4	a	a	PRON
ejpam-4276	268	5	is	be	AUX
ejpam-4276	268	6	(	(	PUNCT
ejpam-4276	268	7	λ	λ	NOUN
ejpam-4276	268	8	,	,	PUNCT
ejpam-4276	268	9	sp)-open	sp)-open	NOUN
ejpam-4276	268	10	and	and	CCONJ
ejpam-4276	268	11	b(λ	b(λ	NOUN
ejpam-4276	268	12	,	,	PUNCT
ejpam-4276	268	13	sp)-closed	sp)-close	VERB
ejpam-4276	268	14	.	.	PUNCT
ejpam-4276	269	1	(	(	PUNCT
ejpam-4276	269	2	3	3	X
ejpam-4276	269	3	)	)	PUNCT
ejpam-4276	269	4	a	a	PRON
ejpam-4276	269	5	is	be	AUX
ejpam-4276	269	6	α(λ	α(λ	PROPN
ejpam-4276	269	7	,	,	PUNCT
ejpam-4276	269	8	sp)-open	sp)-open	NOUN
ejpam-4276	269	9	and	and	CCONJ
ejpam-4276	269	10	b(λ	b(λ	NOUN
ejpam-4276	269	11	,	,	PUNCT
ejpam-4276	269	12	sp)-closed	sp)-close	VERB
ejpam-4276	269	13	.	.	PUNCT
ejpam-4276	270	1	lemma	lemma	PROPN
ejpam-4276	270	2	5	5	X
ejpam-4276	270	3	.	.	PUNCT
ejpam-4276	271	1	let	let	VERB
ejpam-4276	271	2	a	a	DET
ejpam-4276	271	3	be	be	AUX
ejpam-4276	271	4	a	a	DET
ejpam-4276	271	5	subset	subset	NOUN
ejpam-4276	271	6	of	of	ADP
ejpam-4276	271	7	a	a	DET
ejpam-4276	271	8	topological	topological	ADJ
ejpam-4276	271	9	space	space	NOUN
ejpam-4276	271	10	(	(	PUNCT
ejpam-4276	271	11	x	x	X
ejpam-4276	271	12	,	,	PUNCT
ejpam-4276	271	13	τ	τ	PROPN
ejpam-4276	271	14	)	)	PUNCT
ejpam-4276	271	15	.	.	PUNCT
ejpam-4276	272	1	if	if	SCONJ
ejpam-4276	272	2	a	a	PRON
ejpam-4276	272	3	is	be	AUX
ejpam-4276	272	4	s(λ	s(λ	NOUN
ejpam-4276	272	5	,	,	PUNCT
ejpam-4276	272	6	sp)-closed	sp)-closed	ADJ
ejpam-4276	272	7	and	and	CCONJ
ejpam-4276	272	8	β(λ	β(λ	PROPN
ejpam-4276	272	9	,	,	PUNCT
ejpam-4276	272	10	sp)-open	sp)-open	NOUN
ejpam-4276	272	11	,	,	PUNCT
ejpam-4276	272	12	then	then	ADV
ejpam-4276	272	13	a	a	PRON
ejpam-4276	272	14	is	be	AUX
ejpam-4276	272	15	s(λ	s(λ	NOUN
ejpam-4276	272	16	,	,	PUNCT
ejpam-4276	272	17	sp)-open	sp)-open	NOUN
ejpam-4276	272	18	.	.	PUNCT
ejpam-4276	273	1	proof	proof	NOUN
ejpam-4276	273	2	.	.	PUNCT
ejpam-4276	274	1	since	since	SCONJ
ejpam-4276	274	2	a	a	PRON
ejpam-4276	274	3	is	be	AUX
ejpam-4276	274	4	s(λ	s(λ	PROPN
ejpam-4276	274	5	,	,	PUNCT
ejpam-4276	274	6	sp)-closed	sp)-close	VERB
ejpam-4276	274	7	,	,	PUNCT
ejpam-4276	274	8	it	it	PRON
ejpam-4276	274	9	follows	follow	VERB
ejpam-4276	274	10	from	from	ADP
ejpam-4276	274	11	proposition	proposition	NOUN
ejpam-4276	274	12	3	3	NUM
ejpam-4276	274	13	that	that	PRON
ejpam-4276	274	14	[	[	X
ejpam-4276	274	15	a(λ	a(λ	ADV
ejpam-4276	274	16	,	,	PUNCT
ejpam-4276	274	17	sp)](λ	sp)](λ	PROPN
ejpam-4276	274	18	,	,	PUNCT
ejpam-4276	274	19	sp	sp	NOUN
ejpam-4276	274	20	)	)	PUNCT
ejpam-4276	274	21	⊆	⊆	NUM
ejpam-4276	274	22	a.	a.	NOUN
ejpam-4276	274	23	since	since	SCONJ
ejpam-4276	274	24	a	a	PRON
ejpam-4276	274	25	is	be	AUX
ejpam-4276	274	26	β(λ	β(λ	NOUN
ejpam-4276	274	27	,	,	PUNCT
ejpam-4276	274	28	sp)-open	sp)-open	ADJ
ejpam-4276	274	29	,	,	PUNCT
ejpam-4276	274	30	[	[	X
ejpam-4276	274	31	a(λ	a(λ	ADV
ejpam-4276	274	32	,	,	PUNCT
ejpam-4276	274	33	sp)](λ	sp)](λ	PROPN
ejpam-4276	274	34	,	,	PUNCT
ejpam-4276	274	35	sp	sp	NOUN
ejpam-4276	274	36	)	)	PUNCT
ejpam-4276	274	37	⊆	⊆	NUM
ejpam-4276	274	38	a	a	DET
ejpam-4276	274	39	⊆	⊆	NUM
ejpam-4276	275	1	[	[	X
ejpam-4276	275	2	[	[	X
ejpam-4276	275	3	a(λ	a(λ	ADJ
ejpam-4276	275	4	,	,	PUNCT
ejpam-4276	275	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	275	6	,	,	PUNCT
ejpam-4276	275	7	sp	sp	NOUN
ejpam-4276	275	8	)	)	PUNCT
ejpam-4276	275	9	]	]	PUNCT
ejpam-4276	276	1	(	(	PUNCT
ejpam-4276	276	2	λ	λ	NOUN
ejpam-4276	276	3	,	,	PUNCT
ejpam-4276	276	4	sp	sp	NOUN
ejpam-4276	276	5	)	)	PUNCT
ejpam-4276	276	6	.	.	PUNCT
ejpam-4276	277	1	thus	thus	ADV
ejpam-4276	277	2	,	,	PUNCT
ejpam-4276	277	3	[	[	X
ejpam-4276	277	4	a(λ	a(λ	ADV
ejpam-4276	277	5	,	,	PUNCT
ejpam-4276	277	6	sp)](λ	sp)](λ	PROPN
ejpam-4276	277	7	,	,	PUNCT
ejpam-4276	277	8	sp	sp	NOUN
ejpam-4276	277	9	)	)	PUNCT
ejpam-4276	277	10	⊆	⊆	NUM
ejpam-4276	277	11	a(λ	a(λ	ADV
ejpam-4276	277	12	,	,	PUNCT
ejpam-4276	277	13	sp	sp	NOUN
ejpam-4276	277	14	)	)	PUNCT
ejpam-4276	277	15	.	.	PUNCT
ejpam-4276	278	1	therefore	therefore	ADV
ejpam-4276	278	2	,	,	PUNCT
ejpam-4276	278	3	[	[	X
ejpam-4276	278	4	[	[	X
ejpam-4276	278	5	a(λ	a(λ	ADJ
ejpam-4276	278	6	,	,	PUNCT
ejpam-4276	278	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	278	8	,	,	PUNCT
ejpam-4276	278	9	sp	sp	NOUN
ejpam-4276	278	10	)	)	PUNCT
ejpam-4276	278	11	]	]	PUNCT
ejpam-4276	279	1	(	(	PUNCT
ejpam-4276	279	2	λ	λ	NOUN
ejpam-4276	279	3	,	,	PUNCT
ejpam-4276	279	4	sp	sp	NOUN
ejpam-4276	279	5	)	)	PUNCT
ejpam-4276	279	6	⊆	⊆	NUM
ejpam-4276	279	7	[	[	X
ejpam-4276	279	8	a(λ	a(λ	ADV
ejpam-4276	279	9	,	,	PUNCT
ejpam-4276	279	10	sp	sp	NOUN
ejpam-4276	279	11	)	)	PUNCT
ejpam-4276	279	12	]	]	PUNCT
ejpam-4276	279	13	(	(	PUNCT
ejpam-4276	279	14	λ	λ	NOUN
ejpam-4276	279	15	,	,	PUNCT
ejpam-4276	279	16	sp	sp	NOUN
ejpam-4276	279	17	)	)	PUNCT
ejpam-4276	279	18	and	and	CCONJ
ejpam-4276	279	19	hence	hence	ADV
ejpam-4276	279	20	a	a	PRON
ejpam-4276	279	21	is	be	AUX
ejpam-4276	279	22	s(λ	s(λ	PROPN
ejpam-4276	279	23	,	,	PUNCT
ejpam-4276	279	24	sp)-open	sp)-open	NOUN
ejpam-4276	279	25	.	.	PUNCT
ejpam-4276	280	1	proposition	proposition	NOUN
ejpam-4276	280	2	12	12	NUM
ejpam-4276	280	3	.	.	PUNCT
ejpam-4276	281	1	let	let	VERB
ejpam-4276	281	2	a	a	DET
ejpam-4276	281	3	be	be	AUX
ejpam-4276	281	4	a	a	DET
ejpam-4276	281	5	subset	subset	NOUN
ejpam-4276	281	6	of	of	ADP
ejpam-4276	281	7	a	a	DET
ejpam-4276	281	8	topological	topological	ADJ
ejpam-4276	281	9	space	space	NOUN
ejpam-4276	281	10	(	(	PUNCT
ejpam-4276	281	11	x	x	X
ejpam-4276	281	12	,	,	PUNCT
ejpam-4276	281	13	τ	τ	PROPN
ejpam-4276	281	14	)	)	PUNCT
ejpam-4276	281	15	.	.	PUNCT
ejpam-4276	282	1	if	if	SCONJ
ejpam-4276	282	2	a	a	PRON
ejpam-4276	282	3	is	be	AUX
ejpam-4276	282	4	b(λ	b(λ	NOUN
ejpam-4276	282	5	,	,	PUNCT
ejpam-4276	282	6	sp)-open	sp)-open	ADJ
ejpam-4276	282	7	,	,	PUNCT
ejpam-4276	282	8	then	then	ADV
ejpam-4276	282	9	a(λ	a(λ	ADV
ejpam-4276	282	10	,	,	PUNCT
ejpam-4276	282	11	sp	sp	NOUN
ejpam-4276	282	12	)	)	PUNCT
ejpam-4276	282	13	is	be	AUX
ejpam-4276	282	14	r(λ	r(λ	NOUN
ejpam-4276	282	15	,	,	PUNCT
ejpam-4276	282	16	sp)-closed	sp)-closed	ADJ
ejpam-4276	282	17	.	.	PUNCT
ejpam-4276	283	1	proof	proof	NOUN
ejpam-4276	283	2	.	.	PUNCT
ejpam-4276	284	1	since	since	SCONJ
ejpam-4276	284	2	a	a	PRON
ejpam-4276	284	3	is	be	AUX
ejpam-4276	284	4	b(λ	b(λ	NOUN
ejpam-4276	284	5	,	,	PUNCT
ejpam-4276	284	6	sp)-open	sp)-open	ADJ
ejpam-4276	284	7	,	,	PUNCT
ejpam-4276	284	8	we	we	PRON
ejpam-4276	284	9	have	have	VERB
ejpam-4276	284	10	a	a	DET
ejpam-4276	284	11	⊆	⊆	NUM
ejpam-4276	284	12	[	[	X
ejpam-4276	284	13	a(λ	a(λ	ADV
ejpam-4276	284	14	,	,	PUNCT
ejpam-4276	284	15	sp	sp	NOUN
ejpam-4276	284	16	)	)	PUNCT
ejpam-4276	284	17	]	]	PUNCT
ejpam-4276	284	18	(	(	PUNCT
ejpam-4276	284	19	λ	λ	NOUN
ejpam-4276	284	20	,	,	PUNCT
ejpam-4276	284	21	sp	sp	NOUN
ejpam-4276	284	22	)	)	PUNCT
ejpam-4276	284	23	∪	∪	ADP
ejpam-4276	284	24	[	[	X
ejpam-4276	284	25	a(λ	a(λ	ADJ
ejpam-4276	284	26	,	,	PUNCT
ejpam-4276	284	27	sp)](λ	sp)](λ	PROPN
ejpam-4276	284	28	,	,	PUNCT
ejpam-4276	284	29	sp	sp	NOUN
ejpam-4276	284	30	)	)	PUNCT
ejpam-4276	284	31	and	and	CCONJ
ejpam-4276	284	32	hence	hence	ADV
ejpam-4276	284	33	a(λ	a(λ	ADV
ejpam-4276	284	34	,	,	PUNCT
ejpam-4276	284	35	sp	sp	NOUN
ejpam-4276	284	36	)	)	PUNCT
ejpam-4276	284	37	⊆	⊆	NUM
ejpam-4276	285	1	[	[	X
ejpam-4276	285	2	[	[	X
ejpam-4276	285	3	a(λ	a(λ	ADJ
ejpam-4276	285	4	,	,	PUNCT
ejpam-4276	285	5	sp	sp	NOUN
ejpam-4276	285	6	)	)	PUNCT
ejpam-4276	285	7	]	]	PUNCT
ejpam-4276	286	1	(	(	PUNCT
ejpam-4276	286	2	λ	λ	NOUN
ejpam-4276	286	3	,	,	PUNCT
ejpam-4276	286	4	sp	sp	NOUN
ejpam-4276	286	5	)	)	PUNCT
ejpam-4276	286	6	∪	∪	ADP
ejpam-4276	286	7	[	[	X
ejpam-4276	286	8	a(λ	a(λ	ADJ
ejpam-4276	286	9	,	,	PUNCT
ejpam-4276	286	10	sp)](λ	sp)](λ	PROPN
ejpam-4276	286	11	,	,	PUNCT
ejpam-4276	286	12	sp	sp	NOUN
ejpam-4276	286	13	)	)	PUNCT
ejpam-4276	286	14	]	]	PUNCT
ejpam-4276	286	15	(	(	PUNCT
ejpam-4276	286	16	λ	λ	NOUN
ejpam-4276	286	17	,	,	PUNCT
ejpam-4276	286	18	sp	sp	NOUN
ejpam-4276	286	19	)	)	PUNCT
ejpam-4276	286	20	⊆	⊆	NUM
ejpam-4276	287	1	[	[	X
ejpam-4276	287	2	[	[	X
ejpam-4276	287	3	a(λ	a(λ	ADJ
ejpam-4276	287	4	,	,	PUNCT
ejpam-4276	287	5	sp	sp	NOUN
ejpam-4276	287	6	)	)	PUNCT
ejpam-4276	287	7	]	]	PUNCT
ejpam-4276	287	8	(	(	PUNCT
ejpam-4276	287	9	λ	λ	X
ejpam-4276	287	10	,	,	PUNCT
ejpam-4276	287	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	287	12	,	,	PUNCT
ejpam-4276	287	13	sp	sp	NOUN
ejpam-4276	287	14	)	)	PUNCT
ejpam-4276	287	15	∪	∪	ADP
ejpam-4276	288	1	[	[	X
ejpam-4276	288	2	[	[	X
ejpam-4276	288	3	a(λ	a(λ	ADJ
ejpam-4276	288	4	,	,	PUNCT
ejpam-4276	288	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	288	6	,	,	PUNCT
ejpam-4276	288	7	sp	sp	NOUN
ejpam-4276	288	8	)	)	PUNCT
ejpam-4276	288	9	]	]	PUNCT
ejpam-4276	289	1	(	(	PUNCT
ejpam-4276	289	2	λ	λ	NOUN
ejpam-4276	289	3	,	,	PUNCT
ejpam-4276	289	4	sp	sp	NOUN
ejpam-4276	289	5	)	)	PUNCT
ejpam-4276	289	6	=	=	PUNCT
ejpam-4276	290	1	[	[	X
ejpam-4276	290	2	[	[	X
ejpam-4276	290	3	a(λ	a(λ	ADJ
ejpam-4276	290	4	,	,	PUNCT
ejpam-4276	290	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	290	6	,	,	PUNCT
ejpam-4276	290	7	sp	sp	NOUN
ejpam-4276	290	8	)	)	PUNCT
ejpam-4276	290	9	]	]	PUNCT
ejpam-4276	291	1	(	(	PUNCT
ejpam-4276	291	2	λ	λ	NOUN
ejpam-4276	291	3	,	,	PUNCT
ejpam-4276	291	4	sp	sp	NOUN
ejpam-4276	291	5	)	)	PUNCT
ejpam-4276	291	6	⊆	⊆	NUM
ejpam-4276	291	7	a(λ	a(λ	ADV
ejpam-4276	291	8	,	,	PUNCT
ejpam-4276	291	9	sp	sp	NOUN
ejpam-4276	291	10	)	)	PUNCT
ejpam-4276	291	11	.	.	PUNCT
ejpam-4276	292	1	thus	thus	ADV
ejpam-4276	292	2	,	,	PUNCT
ejpam-4276	292	3	a(λ	a(λ	ADV
ejpam-4276	292	4	,	,	PUNCT
ejpam-4276	292	5	sp	sp	NOUN
ejpam-4276	292	6	)	)	PUNCT
ejpam-4276	292	7	=	=	PUNCT
ejpam-4276	293	1	[	[	X
ejpam-4276	293	2	[	[	X
ejpam-4276	293	3	a(λ	a(λ	ADJ
ejpam-4276	293	4	,	,	PUNCT
ejpam-4276	293	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	293	6	,	,	PUNCT
ejpam-4276	293	7	sp	sp	NOUN
ejpam-4276	293	8	)	)	PUNCT
ejpam-4276	293	9	]	]	PUNCT
ejpam-4276	293	10	(	(	PUNCT
ejpam-4276	293	11	λ	λ	NOUN
ejpam-4276	293	12	,	,	PUNCT
ejpam-4276	293	13	sp	sp	NOUN
ejpam-4276	293	14	)	)	PUNCT
ejpam-4276	293	15	.	.	PUNCT
ejpam-4276	294	1	this	this	PRON
ejpam-4276	294	2	shows	show	VERB
ejpam-4276	294	3	that	that	SCONJ
ejpam-4276	294	4	a(λ	a(λ	ADV
ejpam-4276	294	5	,	,	PUNCT
ejpam-4276	294	6	sp	sp	NOUN
ejpam-4276	294	7	)	)	PUNCT
ejpam-4276	294	8	is	be	AUX
ejpam-4276	294	9	r(λ	r(λ	NOUN
ejpam-4276	294	10	,	,	PUNCT
ejpam-4276	294	11	sp)-closed	sp)-closed	ADJ
ejpam-4276	294	12	.	.	PUNCT
ejpam-4276	295	1	c.	c.	PROPN
ejpam-4276	295	2	boonpok	boonpok	PROPN
ejpam-4276	295	3	,	,	PUNCT
ejpam-4276	295	4	j.	j.	PROPN
ejpam-4276	295	5	khampakdee	khampakdee	PROPN
ejpam-4276	295	6	/	/	PUNCT
ejpam-4276	295	7	eur	eur	PROPN
ejpam-4276	295	8	.	.	PUNCT
ejpam-4276	296	1	j.	j.	PROPN
ejpam-4276	296	2	pure	pure	PROPN
ejpam-4276	296	3	appl	appl	PROPN
ejpam-4276	296	4	.	.	PROPN
ejpam-4276	296	5	math	math	PROPN
ejpam-4276	296	6	,	,	PUNCT
ejpam-4276	296	7	15	15	NUM
ejpam-4276	296	8	(	(	PUNCT
ejpam-4276	296	9	2	2	NUM
ejpam-4276	296	10	)	)	PUNCT
ejpam-4276	296	11	(	(	PUNCT
ejpam-4276	296	12	2022	2022	NUM
ejpam-4276	296	13	)	)	PUNCT
ejpam-4276	296	14	,	,	PUNCT
ejpam-4276	296	15	572	572	NUM
ejpam-4276	296	16	-	-	SYM
ejpam-4276	296	17	588	588	NUM
ejpam-4276	296	18	580	580	NUM
ejpam-4276	296	19	corollary	corollary	ADJ
ejpam-4276	296	20	3	3	NUM
ejpam-4276	296	21	.	.	PUNCT
ejpam-4276	297	1	for	for	ADP
ejpam-4276	297	2	a	a	DET
ejpam-4276	297	3	subset	subset	NOUN
ejpam-4276	297	4	a	a	PRON
ejpam-4276	297	5	of	of	ADP
ejpam-4276	297	6	a	a	DET
ejpam-4276	297	7	topological	topological	ADJ
ejpam-4276	297	8	space	space	NOUN
ejpam-4276	297	9	(	(	PUNCT
ejpam-4276	297	10	x	x	X
ejpam-4276	297	11	,	,	PUNCT
ejpam-4276	297	12	τ	τ	PROPN
ejpam-4276	297	13	)	)	PUNCT
ejpam-4276	297	14	,	,	PUNCT
ejpam-4276	297	15	the	the	DET
ejpam-4276	297	16	following	follow	VERB
ejpam-4276	297	17	properties	property	NOUN
ejpam-4276	297	18	hold	hold	VERB
ejpam-4276	297	19	:	:	PUNCT
ejpam-4276	297	20	(	(	PUNCT
ejpam-4276	297	21	1	1	X
ejpam-4276	297	22	)	)	PUNCT
ejpam-4276	297	23	if	if	SCONJ
ejpam-4276	297	24	a	a	PRON
ejpam-4276	297	25	is	be	AUX
ejpam-4276	297	26	s(λ	s(λ	PROPN
ejpam-4276	297	27	,	,	PUNCT
ejpam-4276	297	28	sp)-open	sp)-open	ADJ
ejpam-4276	297	29	,	,	PUNCT
ejpam-4276	297	30	then	then	ADV
ejpam-4276	297	31	a(λ	a(λ	ADV
ejpam-4276	297	32	,	,	PUNCT
ejpam-4276	297	33	sp	sp	NOUN
ejpam-4276	297	34	)	)	PUNCT
ejpam-4276	297	35	is	be	AUX
ejpam-4276	297	36	r(λ	r(λ	NOUN
ejpam-4276	297	37	,	,	PUNCT
ejpam-4276	297	38	sp)-closed	sp)-close	VERB
ejpam-4276	297	39	.	.	PUNCT
ejpam-4276	298	1	(	(	PUNCT
ejpam-4276	298	2	2	2	X
ejpam-4276	298	3	)	)	PUNCT
ejpam-4276	298	4	if	if	SCONJ
ejpam-4276	298	5	a	a	PRON
ejpam-4276	298	6	is	be	AUX
ejpam-4276	298	7	p(λ	p(λ	NOUN
ejpam-4276	298	8	,	,	PUNCT
ejpam-4276	298	9	sp)-open	sp)-open	NOUN
ejpam-4276	298	10	,	,	PUNCT
ejpam-4276	298	11	then	then	ADV
ejpam-4276	298	12	a(λ	a(λ	ADV
ejpam-4276	298	13	,	,	PUNCT
ejpam-4276	298	14	sp	sp	NOUN
ejpam-4276	298	15	)	)	PUNCT
ejpam-4276	298	16	is	be	AUX
ejpam-4276	298	17	r(λ	r(λ	NOUN
ejpam-4276	298	18	,	,	PUNCT
ejpam-4276	298	19	sp)-closed	sp)-close	VERB
ejpam-4276	298	20	.	.	PUNCT
ejpam-4276	299	1	(	(	PUNCT
ejpam-4276	299	2	3	3	X
ejpam-4276	299	3	)	)	PUNCT
ejpam-4276	299	4	if	if	SCONJ
ejpam-4276	299	5	a	a	PRON
ejpam-4276	299	6	is	be	AUX
ejpam-4276	299	7	α(λ	α(λ	PROPN
ejpam-4276	299	8	,	,	PUNCT
ejpam-4276	299	9	sp)-open	sp)-open	ADJ
ejpam-4276	299	10	,	,	PUNCT
ejpam-4276	299	11	then	then	ADV
ejpam-4276	299	12	a(λ	a(λ	ADV
ejpam-4276	299	13	,	,	PUNCT
ejpam-4276	299	14	sp	sp	NOUN
ejpam-4276	299	15	)	)	PUNCT
ejpam-4276	299	16	is	be	AUX
ejpam-4276	299	17	r(λ	r(λ	NOUN
ejpam-4276	299	18	,	,	PUNCT
ejpam-4276	299	19	sp)-closed	sp)-close	VERB
ejpam-4276	299	20	.	.	PUNCT
ejpam-4276	300	1	proposition	proposition	NOUN
ejpam-4276	300	2	13	13	NUM
ejpam-4276	300	3	.	.	PUNCT
ejpam-4276	301	1	for	for	ADP
ejpam-4276	301	2	a	a	DET
ejpam-4276	301	3	subset	subset	NOUN
ejpam-4276	301	4	a	a	PRON
ejpam-4276	301	5	of	of	ADP
ejpam-4276	301	6	a	a	DET
ejpam-4276	301	7	topological	topological	ADJ
ejpam-4276	301	8	space	space	NOUN
ejpam-4276	301	9	(	(	PUNCT
ejpam-4276	301	10	x	x	X
ejpam-4276	301	11	,	,	PUNCT
ejpam-4276	301	12	τ	τ	PROPN
ejpam-4276	301	13	)	)	PUNCT
ejpam-4276	301	14	,	,	PUNCT
ejpam-4276	301	15	the	the	DET
ejpam-4276	301	16	following	follow	VERB
ejpam-4276	301	17	properties	property	NOUN
ejpam-4276	301	18	are	be	AUX
ejpam-4276	301	19	equivalent	equivalent	ADJ
ejpam-4276	301	20	:	:	PUNCT
ejpam-4276	301	21	(	(	PUNCT
ejpam-4276	301	22	1	1	X
ejpam-4276	301	23	)	)	PUNCT
ejpam-4276	301	24	a	a	DET
ejpam-4276	301	25	∈	∈	PROPN
ejpam-4276	301	26	βλspo(x	βλspo(x	NOUN
ejpam-4276	301	27	,	,	PUNCT
ejpam-4276	301	28	τ	τ	PROPN
ejpam-4276	301	29	)	)	PUNCT
ejpam-4276	301	30	.	.	PUNCT
ejpam-4276	302	1	(	(	PUNCT
ejpam-4276	302	2	2	2	NUM
ejpam-4276	302	3	)	)	PUNCT
ejpam-4276	302	4	a(λ	a(λ	ADV
ejpam-4276	302	5	,	,	PUNCT
ejpam-4276	302	6	sp	sp	NOUN
ejpam-4276	302	7	)	)	PUNCT
ejpam-4276	302	8	∈	∈	PROPN
ejpam-4276	302	9	rλspc(x	rλspc(x	PROPN
ejpam-4276	302	10	,	,	PUNCT
ejpam-4276	302	11	τ	τ	PROPN
ejpam-4276	302	12	)	)	PUNCT
ejpam-4276	302	13	.	.	PUNCT
ejpam-4276	303	1	(	(	PUNCT
ejpam-4276	303	2	3	3	X
ejpam-4276	303	3	)	)	PUNCT
ejpam-4276	303	4	a(λ	a(λ	ADV
ejpam-4276	303	5	,	,	PUNCT
ejpam-4276	303	6	sp	sp	NOUN
ejpam-4276	303	7	)	)	PUNCT
ejpam-4276	303	8	∈	∈	PROPN
ejpam-4276	303	9	βλspo(x	βλspo(x	PROPN
ejpam-4276	303	10	,	,	PUNCT
ejpam-4276	303	11	τ	τ	PROPN
ejpam-4276	303	12	)	)	PUNCT
ejpam-4276	303	13	.	.	PUNCT
ejpam-4276	304	1	(	(	PUNCT
ejpam-4276	304	2	4	4	NUM
ejpam-4276	304	3	)	)	PUNCT
ejpam-4276	304	4	a(λ	a(λ	ADV
ejpam-4276	304	5	,	,	PUNCT
ejpam-4276	304	6	sp	sp	NOUN
ejpam-4276	304	7	)	)	PUNCT
ejpam-4276	304	8	∈	∈	PROPN
ejpam-4276	304	9	sλspo(x	sλspo(x	PROPN
ejpam-4276	304	10	,	,	PUNCT
ejpam-4276	304	11	τ	τ	PROPN
ejpam-4276	304	12	)	)	PUNCT
ejpam-4276	304	13	.	.	PUNCT
ejpam-4276	305	1	(	(	PUNCT
ejpam-4276	305	2	5	5	NUM
ejpam-4276	305	3	)	)	PUNCT
ejpam-4276	305	4	a(λ	a(λ	ADV
ejpam-4276	305	5	,	,	PUNCT
ejpam-4276	305	6	sp	sp	NOUN
ejpam-4276	305	7	)	)	PUNCT
ejpam-4276	305	8	∈	∈	PROPN
ejpam-4276	305	9	bλspo(x	bλspo(x	NOUN
ejpam-4276	305	10	,	,	PUNCT
ejpam-4276	305	11	τ	τ	PROPN
ejpam-4276	305	12	)	)	PUNCT
ejpam-4276	305	13	.	.	PUNCT
ejpam-4276	306	1	proof	proof	NOUN
ejpam-4276	306	2	.	.	PUNCT
ejpam-4276	307	1	(	(	PUNCT
ejpam-4276	307	2	1	1	X
ejpam-4276	307	3	)	)	PUNCT
ejpam-4276	307	4	⇒	⇒	NOUN
ejpam-4276	307	5	(	(	PUNCT
ejpam-4276	307	6	2	2	NUM
ejpam-4276	307	7	):	):	PUNCT
ejpam-4276	307	8	let	let	VERB
ejpam-4276	307	9	a	a	DET
ejpam-4276	307	10	∈	∈	PROPN
ejpam-4276	307	11	βλspo(x	βλspo(x	NOUN
ejpam-4276	307	12	,	,	PUNCT
ejpam-4276	307	13	τ	τ	PROPN
ejpam-4276	307	14	)	)	PUNCT
ejpam-4276	307	15	.	.	PUNCT
ejpam-4276	308	1	then	then	ADV
ejpam-4276	308	2	,	,	PUNCT
ejpam-4276	308	3	we	we	PRON
ejpam-4276	308	4	have	have	VERB
ejpam-4276	308	5	a	a	DET
ejpam-4276	308	6	⊆	⊆	NUM
ejpam-4276	308	7	[	[	X
ejpam-4276	308	8	[	[	X
ejpam-4276	308	9	a(λ	a(λ	ADJ
ejpam-4276	308	10	,	,	PUNCT
ejpam-4276	308	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	308	12	,	,	PUNCT
ejpam-4276	308	13	sp	sp	NOUN
ejpam-4276	308	14	)	)	PUNCT
ejpam-4276	308	15	]	]	PUNCT
ejpam-4276	308	16	(	(	PUNCT
ejpam-4276	308	17	λ	λ	NOUN
ejpam-4276	308	18	,	,	PUNCT
ejpam-4276	308	19	sp	sp	NOUN
ejpam-4276	308	20	)	)	PUNCT
ejpam-4276	308	21	and	and	CCONJ
ejpam-4276	308	22	hence	hence	ADV
ejpam-4276	308	23	a(λ	a(λ	ADV
ejpam-4276	308	24	,	,	PUNCT
ejpam-4276	308	25	sp	sp	NOUN
ejpam-4276	308	26	)	)	PUNCT
ejpam-4276	308	27	⊆	⊆	NUM
ejpam-4276	309	1	[	[	X
ejpam-4276	309	2	[	[	X
ejpam-4276	309	3	a(λ	a(λ	ADJ
ejpam-4276	309	4	,	,	PUNCT
ejpam-4276	309	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	309	6	,	,	PUNCT
ejpam-4276	309	7	sp	sp	NOUN
ejpam-4276	309	8	)	)	PUNCT
ejpam-4276	309	9	]	]	PUNCT
ejpam-4276	310	1	(	(	PUNCT
ejpam-4276	310	2	λ	λ	NOUN
ejpam-4276	310	3	,	,	PUNCT
ejpam-4276	310	4	sp	sp	NOUN
ejpam-4276	310	5	)	)	PUNCT
ejpam-4276	310	6	⊆	⊆	NUM
ejpam-4276	310	7	a(λ	a(λ	ADV
ejpam-4276	310	8	,	,	PUNCT
ejpam-4276	310	9	sp	sp	NOUN
ejpam-4276	310	10	)	)	PUNCT
ejpam-4276	310	11	.	.	PUNCT
ejpam-4276	311	1	thus	thus	ADV
ejpam-4276	311	2	,	,	PUNCT
ejpam-4276	311	3	a(λ	a(λ	ADV
ejpam-4276	311	4	,	,	PUNCT
ejpam-4276	311	5	sp	sp	NOUN
ejpam-4276	311	6	)	)	PUNCT
ejpam-4276	311	7	=	=	PUNCT
ejpam-4276	312	1	[	[	X
ejpam-4276	312	2	[	[	X
ejpam-4276	312	3	a(λ	a(λ	ADJ
ejpam-4276	312	4	,	,	PUNCT
ejpam-4276	312	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	312	6	,	,	PUNCT
ejpam-4276	312	7	sp	sp	NOUN
ejpam-4276	312	8	)	)	PUNCT
ejpam-4276	312	9	]	]	PUNCT
ejpam-4276	312	10	(	(	PUNCT
ejpam-4276	312	11	λ	λ	NOUN
ejpam-4276	312	12	,	,	PUNCT
ejpam-4276	312	13	sp	sp	NOUN
ejpam-4276	312	14	)	)	PUNCT
ejpam-4276	312	15	.	.	PUNCT
ejpam-4276	313	1	therefore	therefore	ADV
ejpam-4276	313	2	,	,	PUNCT
ejpam-4276	313	3	a(λ	a(λ	ADV
ejpam-4276	313	4	,	,	PUNCT
ejpam-4276	313	5	sp	sp	NOUN
ejpam-4276	313	6	)	)	PUNCT
ejpam-4276	313	7	∈	∈	PROPN
ejpam-4276	313	8	rλspc(x	rλspc(x	PROPN
ejpam-4276	313	9	,	,	PUNCT
ejpam-4276	313	10	τ	τ	PROPN
ejpam-4276	313	11	)	)	PUNCT
ejpam-4276	313	12	.	.	PUNCT
ejpam-4276	314	1	(	(	PUNCT
ejpam-4276	314	2	2	2	X
ejpam-4276	314	3	)	)	PUNCT
ejpam-4276	314	4	⇒	⇒	NOUN
ejpam-4276	314	5	(	(	PUNCT
ejpam-4276	314	6	3	3	NUM
ejpam-4276	314	7	)	)	PUNCT
ejpam-4276	314	8	⇒	⇒	NOUN
ejpam-4276	314	9	(	(	PUNCT
ejpam-4276	314	10	4	4	NUM
ejpam-4276	314	11	)	)	PUNCT
ejpam-4276	314	12	⇒	⇒	NOUN
ejpam-4276	314	13	(	(	PUNCT
ejpam-4276	314	14	5	5	NUM
ejpam-4276	314	15	):	):	PUNCT
ejpam-4276	314	16	obvious	obvious	ADJ
ejpam-4276	314	17	.	.	PUNCT
ejpam-4276	315	1	(	(	PUNCT
ejpam-4276	315	2	5	5	X
ejpam-4276	315	3	)	)	PUNCT
ejpam-4276	315	4	⇒	⇒	NOUN
ejpam-4276	315	5	(	(	PUNCT
ejpam-4276	315	6	1	1	NUM
ejpam-4276	315	7	):	):	PUNCT
ejpam-4276	315	8	let	let	VERB
ejpam-4276	315	9	a(λ	a(λ	ADV
ejpam-4276	315	10	,	,	PUNCT
ejpam-4276	315	11	sp	sp	NOUN
ejpam-4276	315	12	)	)	PUNCT
ejpam-4276	315	13	∈	∈	PROPN
ejpam-4276	315	14	bλspo(x	bλspo(x	NOUN
ejpam-4276	315	15	,	,	PUNCT
ejpam-4276	315	16	τ	τ	PROPN
ejpam-4276	315	17	)	)	PUNCT
ejpam-4276	315	18	.	.	PUNCT
ejpam-4276	316	1	then	then	ADV
ejpam-4276	316	2	,	,	PUNCT
ejpam-4276	316	3	a(λ	a(λ	ADV
ejpam-4276	316	4	,	,	PUNCT
ejpam-4276	316	5	sp	sp	NOUN
ejpam-4276	316	6	)	)	PUNCT
ejpam-4276	316	7	⊆	⊆	NUM
ejpam-4276	317	1	[	[	X
ejpam-4276	317	2	[	[	X
ejpam-4276	317	3	a(λ	a(λ	ADJ
ejpam-4276	317	4	,	,	PUNCT
ejpam-4276	317	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	317	6	,	,	PUNCT
ejpam-4276	317	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	317	8	,	,	PUNCT
ejpam-4276	317	9	sp	sp	NOUN
ejpam-4276	317	10	)	)	PUNCT
ejpam-4276	317	11	∪	∪	ADP
ejpam-4276	317	12	[	[	X
ejpam-4276	317	13	[	[	X
ejpam-4276	317	14	a(λ	a(λ	ADJ
ejpam-4276	317	15	,	,	PUNCT
ejpam-4276	317	16	sp)](λ	sp)](λ	PROPN
ejpam-4276	317	17	,	,	PUNCT
ejpam-4276	317	18	sp	sp	NOUN
ejpam-4276	317	19	)	)	PUNCT
ejpam-4276	317	20	]	]	PUNCT
ejpam-4276	317	21	(	(	PUNCT
ejpam-4276	317	22	λ	λ	NOUN
ejpam-4276	317	23	,	,	PUNCT
ejpam-4276	317	24	sp	sp	NOUN
ejpam-4276	317	25	)	)	PUNCT
ejpam-4276	317	26	=	=	PUNCT
ejpam-4276	318	1	[	[	X
ejpam-4276	318	2	a(λ	a(λ	ADV
ejpam-4276	318	3	,	,	PUNCT
ejpam-4276	318	4	sp)](λ	sp)](λ	PROPN
ejpam-4276	318	5	,	,	PUNCT
ejpam-4276	318	6	sp	sp	NOUN
ejpam-4276	318	7	)	)	PUNCT
ejpam-4276	318	8	∪	∪	ADP
ejpam-4276	318	9	[	[	X
ejpam-4276	318	10	[	[	X
ejpam-4276	318	11	a(λ	a(λ	ADJ
ejpam-4276	318	12	,	,	PUNCT
ejpam-4276	318	13	sp)](λ	sp)](λ	PROPN
ejpam-4276	318	14	,	,	PUNCT
ejpam-4276	318	15	sp	sp	NOUN
ejpam-4276	318	16	)	)	PUNCT
ejpam-4276	318	17	]	]	PUNCT
ejpam-4276	318	18	(	(	PUNCT
ejpam-4276	318	19	λ	λ	NOUN
ejpam-4276	318	20	,	,	PUNCT
ejpam-4276	318	21	sp	sp	NOUN
ejpam-4276	318	22	)	)	PUNCT
ejpam-4276	318	23	=	=	PUNCT
ejpam-4276	319	1	[	[	X
ejpam-4276	319	2	[	[	X
ejpam-4276	319	3	a(λ	a(λ	ADJ
ejpam-4276	319	4	,	,	PUNCT
ejpam-4276	319	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	319	6	,	,	PUNCT
ejpam-4276	319	7	sp	sp	NOUN
ejpam-4276	319	8	)	)	PUNCT
ejpam-4276	319	9	]	]	PUNCT
ejpam-4276	319	10	(	(	PUNCT
ejpam-4276	319	11	λ	λ	NOUN
ejpam-4276	319	12	,	,	PUNCT
ejpam-4276	319	13	sp	sp	NOUN
ejpam-4276	319	14	)	)	PUNCT
ejpam-4276	319	15	and	and	CCONJ
ejpam-4276	319	16	hence	hence	ADV
ejpam-4276	319	17	a	a	DET
ejpam-4276	319	18	⊆	⊆	NUM
ejpam-4276	320	1	[	[	X
ejpam-4276	320	2	[	[	X
ejpam-4276	320	3	a(λ	a(λ	ADJ
ejpam-4276	320	4	,	,	PUNCT
ejpam-4276	320	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	320	6	,	,	PUNCT
ejpam-4276	320	7	sp	sp	NOUN
ejpam-4276	320	8	)	)	PUNCT
ejpam-4276	320	9	]	]	PUNCT
ejpam-4276	321	1	(	(	PUNCT
ejpam-4276	321	2	λ	λ	NOUN
ejpam-4276	321	3	,	,	PUNCT
ejpam-4276	321	4	sp	sp	NOUN
ejpam-4276	321	5	)	)	PUNCT
ejpam-4276	321	6	.	.	PUNCT
ejpam-4276	322	1	thus	thus	ADV
ejpam-4276	322	2	,	,	PUNCT
ejpam-4276	322	3	a	a	DET
ejpam-4276	322	4	∈	∈	PROPN
ejpam-4276	322	5	βλspo(x	βλspo(x	NOUN
ejpam-4276	322	6	,	,	PUNCT
ejpam-4276	322	7	τ	τ	PROPN
ejpam-4276	322	8	)	)	PUNCT
ejpam-4276	322	9	.	.	PUNCT
ejpam-4276	323	1	corollary	corollary	ADJ
ejpam-4276	323	2	4	4	NUM
ejpam-4276	323	3	.	.	PUNCT
ejpam-4276	324	1	for	for	ADP
ejpam-4276	324	2	a	a	DET
ejpam-4276	324	3	subset	subset	NOUN
ejpam-4276	324	4	a	a	PRON
ejpam-4276	324	5	of	of	ADP
ejpam-4276	324	6	a	a	DET
ejpam-4276	324	7	topological	topological	ADJ
ejpam-4276	324	8	space	space	NOUN
ejpam-4276	324	9	(	(	PUNCT
ejpam-4276	324	10	x	x	X
ejpam-4276	324	11	,	,	PUNCT
ejpam-4276	324	12	τ	τ	PROPN
ejpam-4276	324	13	)	)	PUNCT
ejpam-4276	324	14	,	,	PUNCT
ejpam-4276	324	15	the	the	DET
ejpam-4276	324	16	following	follow	VERB
ejpam-4276	324	17	properties	property	NOUN
ejpam-4276	324	18	are	be	AUX
ejpam-4276	324	19	equivalent	equivalent	ADJ
ejpam-4276	324	20	:	:	PUNCT
ejpam-4276	324	21	(	(	PUNCT
ejpam-4276	324	22	1	1	X
ejpam-4276	324	23	)	)	PUNCT
ejpam-4276	324	24	a	a	DET
ejpam-4276	324	25	∈	∈	PROPN
ejpam-4276	324	26	βλspc(x	βλspc(x	PROPN
ejpam-4276	324	27	,	,	PUNCT
ejpam-4276	324	28	τ	τ	PROPN
ejpam-4276	324	29	)	)	PUNCT
ejpam-4276	324	30	.	.	PUNCT
ejpam-4276	325	1	(	(	PUNCT
ejpam-4276	325	2	2	2	NUM
ejpam-4276	325	3	)	)	PUNCT
ejpam-4276	325	4	a(λ	a(λ	ADV
ejpam-4276	325	5	,	,	PUNCT
ejpam-4276	325	6	sp	sp	NOUN
ejpam-4276	325	7	)	)	PUNCT
ejpam-4276	325	8	∈	∈	PROPN
ejpam-4276	325	9	rλspo(x	rλspo(x	NOUN
ejpam-4276	325	10	,	,	PUNCT
ejpam-4276	325	11	τ	τ	PROPN
ejpam-4276	325	12	)	)	PUNCT
ejpam-4276	325	13	.	.	PUNCT
ejpam-4276	326	1	(	(	PUNCT
ejpam-4276	326	2	3	3	X
ejpam-4276	326	3	)	)	PUNCT
ejpam-4276	326	4	a(λ	a(λ	ADV
ejpam-4276	326	5	,	,	PUNCT
ejpam-4276	326	6	sp	sp	NOUN
ejpam-4276	326	7	)	)	PUNCT
ejpam-4276	326	8	∈	∈	PROPN
ejpam-4276	326	9	βλspc(x	βλspc(x	PROPN
ejpam-4276	326	10	,	,	PUNCT
ejpam-4276	326	11	τ	τ	PROPN
ejpam-4276	326	12	)	)	PUNCT
ejpam-4276	326	13	.	.	PUNCT
ejpam-4276	327	1	(	(	PUNCT
ejpam-4276	327	2	4	4	NUM
ejpam-4276	327	3	)	)	PUNCT
ejpam-4276	327	4	a(λ	a(λ	ADV
ejpam-4276	327	5	,	,	PUNCT
ejpam-4276	327	6	sp	sp	NOUN
ejpam-4276	327	7	)	)	PUNCT
ejpam-4276	327	8	∈	∈	PROPN
ejpam-4276	327	9	sλspc(x	sλspc(x	NOUN
ejpam-4276	327	10	,	,	PUNCT
ejpam-4276	327	11	τ	τ	PROPN
ejpam-4276	327	12	)	)	PUNCT
ejpam-4276	327	13	.	.	PUNCT
ejpam-4276	328	1	(	(	PUNCT
ejpam-4276	328	2	5	5	NUM
ejpam-4276	328	3	)	)	PUNCT
ejpam-4276	328	4	a(λ	a(λ	ADV
ejpam-4276	328	5	,	,	PUNCT
ejpam-4276	328	6	sp	sp	NOUN
ejpam-4276	328	7	)	)	PUNCT
ejpam-4276	328	8	∈	∈	NOUN
ejpam-4276	328	9	bλspc(x	bλspc(x	NOUN
ejpam-4276	328	10	,	,	PUNCT
ejpam-4276	328	11	τ	τ	PROPN
ejpam-4276	328	12	)	)	PUNCT
ejpam-4276	328	13	.	.	PUNCT
ejpam-4276	329	1	definition	definition	NOUN
ejpam-4276	329	2	6	6	NUM
ejpam-4276	329	3	.	.	PUNCT
ejpam-4276	330	1	a	a	DET
ejpam-4276	330	2	subset	subset	NOUN
ejpam-4276	330	3	a	a	PRON
ejpam-4276	330	4	of	of	ADP
ejpam-4276	330	5	a	a	DET
ejpam-4276	330	6	topological	topological	ADJ
ejpam-4276	330	7	space	space	NOUN
ejpam-4276	330	8	(	(	PUNCT
ejpam-4276	330	9	x	x	X
ejpam-4276	330	10	,	,	PUNCT
ejpam-4276	330	11	τ	τ	X
ejpam-4276	330	12	)	)	PUNCT
ejpam-4276	330	13	is	be	AUX
ejpam-4276	330	14	called	call	VERB
ejpam-4276	330	15	rs(λ	rs(λ	NOUN
ejpam-4276	330	16	,	,	PUNCT
ejpam-4276	330	17	sp)-open	sp)-open	VERB
ejpam-4276	330	18	if	if	SCONJ
ejpam-4276	330	19	there	there	PRON
ejpam-4276	330	20	exists	exist	VERB
ejpam-4276	330	21	a	a	DET
ejpam-4276	330	22	r(λ	r(λ	NOUN
ejpam-4276	330	23	,	,	PUNCT
ejpam-4276	330	24	sp)-open	sp)-open	NOUN
ejpam-4276	330	25	set	set	VERB
ejpam-4276	330	26	u	u	PRON
ejpam-4276	330	27	such	such	ADJ
ejpam-4276	330	28	that	that	SCONJ
ejpam-4276	330	29	u	u	NOUN
ejpam-4276	330	30	⊆	⊆	NUM
ejpam-4276	330	31	a	a	DET
ejpam-4276	330	32	⊆	⊆	NUM
ejpam-4276	330	33	u	u	NOUN
ejpam-4276	330	34	(	(	PUNCT
ejpam-4276	330	35	λ	λ	PROPN
ejpam-4276	330	36	,	,	PUNCT
ejpam-4276	330	37	sp	sp	NOUN
ejpam-4276	330	38	)	)	PUNCT
ejpam-4276	330	39	.	.	PUNCT
ejpam-4276	331	1	the	the	DET
ejpam-4276	331	2	complement	complement	NOUN
ejpam-4276	331	3	of	of	ADP
ejpam-4276	331	4	a	a	DET
ejpam-4276	331	5	rs(λ	rs(λ	NOUN
ejpam-4276	331	6	,	,	PUNCT
ejpam-4276	331	7	sp)-open	sp)-open	ADJ
ejpam-4276	331	8	set	set	NOUN
ejpam-4276	331	9	is	be	AUX
ejpam-4276	331	10	called	call	VERB
ejpam-4276	331	11	rs(λ	rs(λ	NOUN
ejpam-4276	331	12	,	,	PUNCT
ejpam-4276	331	13	sp)-closed	sp)-closed	ADJ
ejpam-4276	331	14	.	.	PUNCT
ejpam-4276	332	1	c.	c.	PROPN
ejpam-4276	332	2	boonpok	boonpok	PROPN
ejpam-4276	332	3	,	,	PUNCT
ejpam-4276	332	4	j.	j.	PROPN
ejpam-4276	332	5	khampakdee	khampakdee	PROPN
ejpam-4276	332	6	/	/	PUNCT
ejpam-4276	332	7	eur	eur	PROPN
ejpam-4276	332	8	.	.	PUNCT
ejpam-4276	333	1	j.	j.	PROPN
ejpam-4276	333	2	pure	pure	PROPN
ejpam-4276	333	3	appl	appl	PROPN
ejpam-4276	333	4	.	.	PROPN
ejpam-4276	333	5	math	math	PROPN
ejpam-4276	333	6	,	,	PUNCT
ejpam-4276	333	7	15	15	NUM
ejpam-4276	333	8	(	(	PUNCT
ejpam-4276	333	9	2	2	NUM
ejpam-4276	333	10	)	)	PUNCT
ejpam-4276	333	11	(	(	PUNCT
ejpam-4276	333	12	2022	2022	NUM
ejpam-4276	333	13	)	)	PUNCT
ejpam-4276	333	14	,	,	PUNCT
ejpam-4276	333	15	572	572	NUM
ejpam-4276	333	16	-	-	SYM
ejpam-4276	333	17	588	588	NUM
ejpam-4276	333	18	581	581	NUM
ejpam-4276	333	19	the	the	DET
ejpam-4276	333	20	family	family	NOUN
ejpam-4276	333	21	of	of	ADP
ejpam-4276	333	22	all	all	DET
ejpam-4276	333	23	rs(λ	rs(λ	NUM
ejpam-4276	333	24	,	,	PUNCT
ejpam-4276	333	25	sp)-open	sp)-open	ADJ
ejpam-4276	333	26	(	(	PUNCT
ejpam-4276	333	27	resp	resp	NOUN
ejpam-4276	333	28	.	.	PUNCT
ejpam-4276	334	1	rs(λ	rs(λ	PROPN
ejpam-4276	334	2	,	,	PUNCT
ejpam-4276	334	3	sp)-closed	sp)-close	VERB
ejpam-4276	334	4	)	)	PUNCT
ejpam-4276	334	5	sets	set	NOUN
ejpam-4276	334	6	in	in	ADP
ejpam-4276	334	7	a	a	DET
ejpam-4276	334	8	topological	topological	ADJ
ejpam-4276	334	9	space	space	NOUN
ejpam-4276	334	10	(	(	PUNCT
ejpam-4276	334	11	x	x	X
ejpam-4276	334	12	,	,	PUNCT
ejpam-4276	334	13	τ	τ	X
ejpam-4276	334	14	)	)	PUNCT
ejpam-4276	334	15	is	be	AUX
ejpam-4276	334	16	denoted	denote	VERB
ejpam-4276	334	17	by	by	ADP
ejpam-4276	334	18	rsλspo(x	rsλspo(x	NOUN
ejpam-4276	334	19	,	,	PUNCT
ejpam-4276	334	20	τ	τ	X
ejpam-4276	334	21	)	)	PUNCT
ejpam-4276	334	22	(	(	PUNCT
ejpam-4276	334	23	resp	resp	NOUN
ejpam-4276	334	24	.	.	PUNCT
ejpam-4276	335	1	rsλspc(x	rsλspc(x	PROPN
ejpam-4276	335	2	,	,	PUNCT
ejpam-4276	335	3	τ	τ	PROPN
ejpam-4276	335	4	)	)	PUNCT
ejpam-4276	335	5	)	)	PUNCT
ejpam-4276	335	6	.	.	PUNCT
ejpam-4276	336	1	remark	remark	NOUN
ejpam-4276	336	2	2	2	NUM
ejpam-4276	336	3	.	.	PUNCT
ejpam-4276	337	1	it	it	PRON
ejpam-4276	337	2	is	be	AUX
ejpam-4276	337	3	clear	clear	ADJ
ejpam-4276	337	4	that	that	SCONJ
ejpam-4276	337	5	every	every	DET
ejpam-4276	337	6	r(λ	r(λ	NOUN
ejpam-4276	337	7	,	,	PUNCT
ejpam-4276	337	8	sp)-open	sp)-open	ADJ
ejpam-4276	337	9	set	set	NOUN
ejpam-4276	337	10	is	be	AUX
ejpam-4276	337	11	rs(λ	rs(λ	NUM
ejpam-4276	337	12	,	,	PUNCT
ejpam-4276	337	13	sp)-open	sp)-open	NOUN
ejpam-4276	337	14	.	.	PUNCT
ejpam-4276	338	1	proposition	proposition	NOUN
ejpam-4276	338	2	14	14	NUM
ejpam-4276	338	3	.	.	PUNCT
ejpam-4276	339	1	for	for	ADP
ejpam-4276	339	2	a	a	DET
ejpam-4276	339	3	subset	subset	NOUN
ejpam-4276	339	4	a	a	PRON
ejpam-4276	339	5	of	of	ADP
ejpam-4276	339	6	a	a	DET
ejpam-4276	339	7	topological	topological	ADJ
ejpam-4276	339	8	space	space	NOUN
ejpam-4276	339	9	(	(	PUNCT
ejpam-4276	339	10	x	x	X
ejpam-4276	339	11	,	,	PUNCT
ejpam-4276	339	12	τ	τ	PROPN
ejpam-4276	339	13	)	)	PUNCT
ejpam-4276	339	14	,	,	PUNCT
ejpam-4276	339	15	the	the	DET
ejpam-4276	339	16	following	follow	VERB
ejpam-4276	339	17	properties	property	NOUN
ejpam-4276	339	18	are	be	AUX
ejpam-4276	339	19	equivalent	equivalent	ADJ
ejpam-4276	339	20	:	:	PUNCT
ejpam-4276	339	21	(	(	PUNCT
ejpam-4276	339	22	1	1	X
ejpam-4276	339	23	)	)	PUNCT
ejpam-4276	339	24	a	a	PRON
ejpam-4276	339	25	is	be	AUX
ejpam-4276	339	26	rs(λ	rs(λ	NUM
ejpam-4276	339	27	,	,	PUNCT
ejpam-4276	339	28	sp)-open	sp)-open	NOUN
ejpam-4276	339	29	.	.	PUNCT
ejpam-4276	340	1	(	(	PUNCT
ejpam-4276	340	2	2	2	X
ejpam-4276	340	3	)	)	PUNCT
ejpam-4276	340	4	a	a	PRON
ejpam-4276	340	5	is	be	AUX
ejpam-4276	340	6	s(λ	s(λ	NOUN
ejpam-4276	340	7	,	,	PUNCT
ejpam-4276	340	8	sp)-open	sp)-open	ADJ
ejpam-4276	340	9	and	and	CCONJ
ejpam-4276	340	10	s(λ	s(λ	NOUN
ejpam-4276	340	11	,	,	PUNCT
ejpam-4276	340	12	sp)-closed	sp)-close	VERB
ejpam-4276	340	13	.	.	PUNCT
ejpam-4276	341	1	(	(	PUNCT
ejpam-4276	341	2	3	3	X
ejpam-4276	341	3	)	)	PUNCT
ejpam-4276	341	4	a	a	PRON
ejpam-4276	341	5	is	be	AUX
ejpam-4276	341	6	b(λ	b(λ	NOUN
ejpam-4276	341	7	,	,	PUNCT
ejpam-4276	341	8	sp)-open	sp)-open	NOUN
ejpam-4276	341	9	and	and	CCONJ
ejpam-4276	341	10	s(λ	s(λ	NOUN
ejpam-4276	341	11	,	,	PUNCT
ejpam-4276	341	12	sp)-closed	sp)-close	VERB
ejpam-4276	341	13	.	.	PUNCT
ejpam-4276	342	1	(	(	PUNCT
ejpam-4276	342	2	4	4	X
ejpam-4276	342	3	)	)	PUNCT
ejpam-4276	342	4	a	a	PRON
ejpam-4276	342	5	is	be	AUX
ejpam-4276	342	6	β(λ	β(λ	NOUN
ejpam-4276	342	7	,	,	PUNCT
ejpam-4276	342	8	sp)-open	sp)-open	ADJ
ejpam-4276	342	9	and	and	CCONJ
ejpam-4276	342	10	s(λ	s(λ	NOUN
ejpam-4276	342	11	,	,	PUNCT
ejpam-4276	342	12	sp)-closed	sp)-close	VERB
ejpam-4276	342	13	.	.	PUNCT
ejpam-4276	343	1	(	(	PUNCT
ejpam-4276	343	2	5	5	X
ejpam-4276	343	3	)	)	PUNCT
ejpam-4276	343	4	a	a	PRON
ejpam-4276	343	5	is	be	AUX
ejpam-4276	343	6	s(λ	s(λ	NOUN
ejpam-4276	343	7	,	,	PUNCT
ejpam-4276	343	8	sp)-open	sp)-open	ADJ
ejpam-4276	343	9	and	and	CCONJ
ejpam-4276	343	10	β(λ	β(λ	NOUN
ejpam-4276	343	11	,	,	PUNCT
ejpam-4276	343	12	sp)-closed	sp)-close	VERB
ejpam-4276	343	13	.	.	PUNCT
ejpam-4276	344	1	(	(	PUNCT
ejpam-4276	344	2	6	6	NUM
ejpam-4276	344	3	)	)	PUNCT
ejpam-4276	344	4	a	a	PRON
ejpam-4276	344	5	is	be	AUX
ejpam-4276	344	6	s(λ	s(λ	NOUN
ejpam-4276	344	7	,	,	PUNCT
ejpam-4276	344	8	sp)-open	sp)-open	ADJ
ejpam-4276	344	9	and	and	CCONJ
ejpam-4276	344	10	β(λ	β(λ	NOUN
ejpam-4276	344	11	,	,	PUNCT
ejpam-4276	344	12	sp)-closed	sp)-close	VERB
ejpam-4276	344	13	.	.	PUNCT
ejpam-4276	345	1	proof	proof	NOUN
ejpam-4276	345	2	.	.	PUNCT
ejpam-4276	346	1	(	(	PUNCT
ejpam-4276	346	2	1	1	X
ejpam-4276	346	3	)	)	PUNCT
ejpam-4276	346	4	⇒	⇒	NOUN
ejpam-4276	346	5	(	(	PUNCT
ejpam-4276	346	6	2	2	NUM
ejpam-4276	346	7	):	):	PUNCT
ejpam-4276	346	8	suppose	suppose	VERB
ejpam-4276	346	9	that	that	SCONJ
ejpam-4276	346	10	a	a	PRON
ejpam-4276	346	11	is	be	AUX
ejpam-4276	346	12	a	a	DET
ejpam-4276	346	13	rs(λ	rs(λ	NUM
ejpam-4276	346	14	,	,	PUNCT
ejpam-4276	346	15	sp)-open	sp)-open	ADJ
ejpam-4276	346	16	set	set	NOUN
ejpam-4276	346	17	.	.	PUNCT
ejpam-4276	347	1	there	there	PRON
ejpam-4276	347	2	exists	exist	VERB
ejpam-4276	347	3	a	a	DET
ejpam-4276	347	4	r(λ	r(λ	NOUN
ejpam-4276	347	5	,	,	PUNCT
ejpam-4276	347	6	sp)open	sp)open	NOUN
ejpam-4276	347	7	set	set	VERB
ejpam-4276	347	8	u	u	PRON
ejpam-4276	347	9	such	such	ADJ
ejpam-4276	347	10	that	that	SCONJ
ejpam-4276	347	11	u	u	NOUN
ejpam-4276	347	12	⊆	⊆	NUM
ejpam-4276	347	13	a	a	DET
ejpam-4276	347	14	⊆	⊆	NUM
ejpam-4276	347	15	u	u	NOUN
ejpam-4276	347	16	(	(	PUNCT
ejpam-4276	347	17	λ	λ	PROPN
ejpam-4276	347	18	,	,	PUNCT
ejpam-4276	347	19	sp	sp	NOUN
ejpam-4276	347	20	)	)	PUNCT
ejpam-4276	347	21	.	.	PUNCT
ejpam-4276	348	1	then	then	ADV
ejpam-4276	348	2	,	,	PUNCT
ejpam-4276	348	3	u	u	NOUN
ejpam-4276	348	4	⊆	⊆	NUM
ejpam-4276	348	5	a(λ	a(λ	ADV
ejpam-4276	348	6	,	,	PUNCT
ejpam-4276	348	7	sp	sp	NOUN
ejpam-4276	348	8	)	)	PUNCT
ejpam-4276	348	9	and	and	CCONJ
ejpam-4276	348	10	hence	hence	ADV
ejpam-4276	348	11	a	a	DET
ejpam-4276	348	12	⊆	⊆	NUM
ejpam-4276	348	13	u	u	NOUN
ejpam-4276	348	14	(	(	PUNCT
ejpam-4276	348	15	λ	λ	PROPN
ejpam-4276	348	16	,	,	PUNCT
ejpam-4276	348	17	sp	sp	NOUN
ejpam-4276	348	18	)	)	PUNCT
ejpam-4276	348	19	⊆	⊆	NUM
ejpam-4276	348	20	[	[	X
ejpam-4276	348	21	a(λ	a(λ	ADV
ejpam-4276	348	22	,	,	PUNCT
ejpam-4276	348	23	sp)](λ	sp)](λ	PROPN
ejpam-4276	348	24	,	,	PUNCT
ejpam-4276	348	25	sp	sp	NOUN
ejpam-4276	348	26	)	)	PUNCT
ejpam-4276	348	27	.	.	PUNCT
ejpam-4276	349	1	therefore	therefore	ADV
ejpam-4276	349	2	,	,	PUNCT
ejpam-4276	349	3	a	a	PRON
ejpam-4276	349	4	is	be	AUX
ejpam-4276	349	5	s(λ	s(λ	NOUN
ejpam-4276	349	6	,	,	PUNCT
ejpam-4276	349	7	sp)-open	sp)-open	NOUN
ejpam-4276	349	8	.	.	PUNCT
ejpam-4276	350	1	on	on	ADP
ejpam-4276	350	2	the	the	DET
ejpam-4276	350	3	other	other	ADJ
ejpam-4276	350	4	hand	hand	NOUN
ejpam-4276	350	5	,	,	PUNCT
ejpam-4276	350	6	since	since	SCONJ
ejpam-4276	350	7	u	u	PRON
ejpam-4276	350	8	(	(	PUNCT
ejpam-4276	350	9	λ	λ	PROPN
ejpam-4276	350	10	,	,	PUNCT
ejpam-4276	350	11	sp	sp	NOUN
ejpam-4276	350	12	)	)	PUNCT
ejpam-4276	350	13	=	=	PUNCT
ejpam-4276	350	14	a(λ	a(λ	ADV
ejpam-4276	350	15	,	,	PUNCT
ejpam-4276	350	16	sp	sp	NOUN
ejpam-4276	350	17	)	)	PUNCT
ejpam-4276	350	18	and	and	CCONJ
ejpam-4276	350	19	u	u	NOUN
ejpam-4276	350	20	is	be	AUX
ejpam-4276	350	21	r(λ	r(λ	NOUN
ejpam-4276	350	22	,	,	PUNCT
ejpam-4276	350	23	sp)-open	sp)-open	ADJ
ejpam-4276	350	24	,	,	PUNCT
ejpam-4276	350	25	[	[	X
ejpam-4276	350	26	a(λ	a(λ	ADV
ejpam-4276	350	27	,	,	PUNCT
ejpam-4276	350	28	sp)](λ	sp)](λ	PROPN
ejpam-4276	350	29	,	,	PUNCT
ejpam-4276	350	30	sp	sp	NOUN
ejpam-4276	350	31	)	)	PUNCT
ejpam-4276	350	32	=	=	PUNCT
ejpam-4276	351	1	[	[	X
ejpam-4276	351	2	u	u	X
ejpam-4276	351	3	(	(	PUNCT
ejpam-4276	351	4	λ	λ	PROPN
ejpam-4276	351	5	,	,	PUNCT
ejpam-4276	351	6	sp)](λ	sp)](λ	PROPN
ejpam-4276	351	7	,	,	PUNCT
ejpam-4276	351	8	sp	sp	NOUN
ejpam-4276	351	9	)	)	PUNCT
ejpam-4276	351	10	=	=	SYM
ejpam-4276	351	11	u	u	NOUN
ejpam-4276	351	12	⊆	⊆	NUM
ejpam-4276	351	13	a.	a.	NOUN
ejpam-4276	351	14	thus	thus	ADV
ejpam-4276	351	15	,	,	PUNCT
ejpam-4276	351	16	by	by	ADP
ejpam-4276	351	17	proposition	proposition	NOUN
ejpam-4276	351	18	3	3	NUM
ejpam-4276	351	19	,	,	PUNCT
ejpam-4276	351	20	a	a	PRON
ejpam-4276	351	21	is	be	AUX
ejpam-4276	351	22	s(λ	s(λ	PROPN
ejpam-4276	351	23	,	,	PUNCT
ejpam-4276	351	24	sp)-closed	sp)-close	VERB
ejpam-4276	351	25	.	.	PUNCT
ejpam-4276	352	1	(	(	PUNCT
ejpam-4276	352	2	2	2	X
ejpam-4276	352	3	)	)	PUNCT
ejpam-4276	352	4	⇒	⇒	NOUN
ejpam-4276	352	5	(	(	PUNCT
ejpam-4276	352	6	3	3	NUM
ejpam-4276	352	7	)	)	PUNCT
ejpam-4276	352	8	and	and	CCONJ
ejpam-4276	352	9	(	(	PUNCT
ejpam-4276	352	10	3	3	X
ejpam-4276	352	11	)	)	PUNCT
ejpam-4276	352	12	⇒	⇒	NOUN
ejpam-4276	352	13	(	(	PUNCT
ejpam-4276	352	14	4	4	NUM
ejpam-4276	352	15	):	):	PUNCT
ejpam-4276	352	16	the	the	DET
ejpam-4276	352	17	proofs	proof	NOUN
ejpam-4276	352	18	are	be	AUX
ejpam-4276	352	19	obvious	obvious	ADJ
ejpam-4276	352	20	.	.	PUNCT
ejpam-4276	353	1	(	(	PUNCT
ejpam-4276	353	2	4	4	X
ejpam-4276	353	3	)	)	PUNCT
ejpam-4276	353	4	⇒	⇒	NOUN
ejpam-4276	353	5	(	(	PUNCT
ejpam-4276	353	6	5	5	NUM
ejpam-4276	353	7	):	):	PUNCT
ejpam-4276	353	8	follows	follow	VERB
ejpam-4276	353	9	from	from	ADP
ejpam-4276	353	10	lemma	lemma	PROPN
ejpam-4276	353	11	5	5	NUM
ejpam-4276	353	12	and	and	CCONJ
ejpam-4276	353	13	since	since	SCONJ
ejpam-4276	353	14	sλspo(x	sλspo(x	PROPN
ejpam-4276	353	15	,	,	PUNCT
ejpam-4276	353	16	τ	τ	PROPN
ejpam-4276	353	17	)	)	PUNCT
ejpam-4276	353	18	⊆	⊆	NUM
ejpam-4276	353	19	bλspo(x	bλspo(x	NOUN
ejpam-4276	353	20	,	,	PUNCT
ejpam-4276	353	21	τ	τ	PROPN
ejpam-4276	353	22	)	)	PUNCT
ejpam-4276	353	23	.	.	PUNCT
ejpam-4276	354	1	(	(	PUNCT
ejpam-4276	354	2	5	5	X
ejpam-4276	354	3	)	)	PUNCT
ejpam-4276	354	4	⇒	⇒	NOUN
ejpam-4276	354	5	(	(	PUNCT
ejpam-4276	354	6	6	6	NUM
ejpam-4276	354	7	):	):	PUNCT
ejpam-4276	354	8	the	the	DET
ejpam-4276	354	9	proof	proof	NOUN
ejpam-4276	354	10	is	be	AUX
ejpam-4276	354	11	obvious	obvious	ADJ
ejpam-4276	354	12	.	.	PUNCT
ejpam-4276	355	1	(	(	PUNCT
ejpam-4276	355	2	6	6	NUM
ejpam-4276	355	3	)	)	PUNCT
ejpam-4276	355	4	⇒	⇒	NOUN
ejpam-4276	355	5	(	(	PUNCT
ejpam-4276	355	6	1	1	NUM
ejpam-4276	355	7	):	):	PUNCT
ejpam-4276	355	8	since	since	SCONJ
ejpam-4276	355	9	a	a	PRON
ejpam-4276	355	10	is	be	AUX
ejpam-4276	355	11	s(λ	s(λ	NOUN
ejpam-4276	355	12	,	,	PUNCT
ejpam-4276	355	13	sp)-open	sp)-open	ADJ
ejpam-4276	355	14	and	and	CCONJ
ejpam-4276	355	15	β(λ	β(λ	NOUN
ejpam-4276	355	16	,	,	PUNCT
ejpam-4276	355	17	sp)-closed	sp)-close	VERB
ejpam-4276	355	18	,	,	PUNCT
ejpam-4276	355	19	it	it	PRON
ejpam-4276	355	20	follows	follow	VERB
ejpam-4276	355	21	from	from	ADP
ejpam-4276	355	22	lemma	lemma	PROPN
ejpam-4276	355	23	5	5	NUM
ejpam-4276	355	24	that	that	SCONJ
ejpam-4276	355	25	a	a	PRON
ejpam-4276	355	26	is	be	AUX
ejpam-4276	355	27	s(λ	s(λ	PROPN
ejpam-4276	355	28	,	,	PUNCT
ejpam-4276	355	29	sp)-closed	sp)-close	VERB
ejpam-4276	355	30	.	.	PUNCT
ejpam-4276	356	1	thus	thus	ADV
ejpam-4276	356	2	,	,	PUNCT
ejpam-4276	356	3	by	by	ADP
ejpam-4276	356	4	proposition	proposition	NOUN
ejpam-4276	356	5	3	3	NUM
ejpam-4276	356	6	,	,	PUNCT
ejpam-4276	356	7	[	[	X
ejpam-4276	356	8	a(λ	a(λ	ADV
ejpam-4276	356	9	,	,	PUNCT
ejpam-4276	356	10	sp)](λ	sp)](λ	PROPN
ejpam-4276	356	11	,	,	PUNCT
ejpam-4276	356	12	sp	sp	NOUN
ejpam-4276	356	13	)	)	PUNCT
ejpam-4276	356	14	⊆	⊆	NUM
ejpam-4276	356	15	a	a	DET
ejpam-4276	356	16	⊆	⊆	NUM
ejpam-4276	356	17	[	[	X
ejpam-4276	356	18	a(λ	a(λ	ADV
ejpam-4276	356	19	,	,	PUNCT
ejpam-4276	356	20	sp	sp	NOUN
ejpam-4276	356	21	)	)	PUNCT
ejpam-4276	356	22	]	]	PUNCT
ejpam-4276	356	23	(	(	PUNCT
ejpam-4276	356	24	λ	λ	NOUN
ejpam-4276	356	25	,	,	PUNCT
ejpam-4276	356	26	sp	sp	NOUN
ejpam-4276	356	27	)	)	PUNCT
ejpam-4276	356	28	⊆	⊆	NUM
ejpam-4276	357	1	[	[	X
ejpam-4276	357	2	[	[	X
ejpam-4276	357	3	a(λ	a(λ	ADJ
ejpam-4276	357	4	,	,	PUNCT
ejpam-4276	357	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	357	6	,	,	PUNCT
ejpam-4276	357	7	sp	sp	NOUN
ejpam-4276	357	8	)	)	PUNCT
ejpam-4276	357	9	]	]	PUNCT
ejpam-4276	358	1	(	(	PUNCT
ejpam-4276	358	2	λ	λ	NOUN
ejpam-4276	358	3	,	,	PUNCT
ejpam-4276	358	4	sp	sp	NOUN
ejpam-4276	358	5	)	)	PUNCT
ejpam-4276	358	6	.	.	PUNCT
ejpam-4276	359	1	let	let	VERB
ejpam-4276	359	2	u	u	PRON
ejpam-4276	359	3	=	=	PUNCT
ejpam-4276	360	1	[	[	X
ejpam-4276	360	2	a(λ	a(λ	PROPN
ejpam-4276	360	3	,	,	PUNCT
ejpam-4276	360	4	sp)](λ	sp)](λ	PROPN
ejpam-4276	360	5	,	,	PUNCT
ejpam-4276	360	6	sp	sp	NOUN
ejpam-4276	360	7	)	)	PUNCT
ejpam-4276	360	8	.	.	PUNCT
ejpam-4276	361	1	then	then	ADV
ejpam-4276	361	2	,	,	PUNCT
ejpam-4276	361	3	u	u	NOUN
ejpam-4276	361	4	is	be	AUX
ejpam-4276	361	5	r(λ	r(λ	NOUN
ejpam-4276	361	6	,	,	PUNCT
ejpam-4276	361	7	sp)-open	sp)-open	ADJ
ejpam-4276	361	8	and	and	CCONJ
ejpam-4276	361	9	hence	hence	ADV
ejpam-4276	361	10	u	u	NOUN
ejpam-4276	361	11	⊆	⊆	NUM
ejpam-4276	361	12	a	a	DET
ejpam-4276	361	13	⊆	⊆	NUM
ejpam-4276	361	14	u	u	NOUN
ejpam-4276	361	15	(	(	PUNCT
ejpam-4276	361	16	λ	λ	PROPN
ejpam-4276	361	17	,	,	PUNCT
ejpam-4276	361	18	sp	sp	NOUN
ejpam-4276	361	19	)	)	PUNCT
ejpam-4276	361	20	.	.	PUNCT
ejpam-4276	362	1	thus	thus	ADV
ejpam-4276	362	2	,	,	PUNCT
ejpam-4276	362	3	a	a	PRON
ejpam-4276	362	4	is	be	AUX
ejpam-4276	362	5	rs(λ	rs(λ	NUM
ejpam-4276	362	6	,	,	PUNCT
ejpam-4276	362	7	sp)-open	sp)-open	NOUN
ejpam-4276	362	8	.	.	PUNCT
ejpam-4276	363	1	remark	remark	NOUN
ejpam-4276	363	2	3	3	NUM
ejpam-4276	363	3	.	.	PUNCT
ejpam-4276	364	1	it	it	PRON
ejpam-4276	364	2	is	be	AUX
ejpam-4276	364	3	clear	clear	ADJ
ejpam-4276	364	4	from	from	ADP
ejpam-4276	364	5	proposition	proposition	NOUN
ejpam-4276	364	6	14	14	NUM
ejpam-4276	364	7	that	that	SCONJ
ejpam-4276	364	8	if	if	SCONJ
ejpam-4276	364	9	a	a	PRON
ejpam-4276	364	10	is	be	AUX
ejpam-4276	364	11	a	a	DET
ejpam-4276	364	12	rs(λ	rs(λ	NUM
ejpam-4276	364	13	,	,	PUNCT
ejpam-4276	364	14	sp)-open	sp)-open	ADJ
ejpam-4276	364	15	set	set	NOUN
ejpam-4276	364	16	of	of	ADP
ejpam-4276	364	17	a	a	DET
ejpam-4276	364	18	topological	topological	ADJ
ejpam-4276	364	19	space	space	NOUN
ejpam-4276	364	20	(	(	PUNCT
ejpam-4276	364	21	x	x	X
ejpam-4276	364	22	,	,	PUNCT
ejpam-4276	364	23	τ	τ	PROPN
ejpam-4276	364	24	)	)	PUNCT
ejpam-4276	364	25	,	,	PUNCT
ejpam-4276	364	26	then	then	ADV
ejpam-4276	364	27	x	x	PUNCT
ejpam-4276	364	28	−a	−a	NOUN
ejpam-4276	364	29	is	be	AUX
ejpam-4276	364	30	rs(λ	rs(λ	NUM
ejpam-4276	364	31	,	,	PUNCT
ejpam-4276	364	32	sp)-open	sp)-open	NOUN
ejpam-4276	364	33	.	.	PUNCT
ejpam-4276	365	1	proposition	proposition	NOUN
ejpam-4276	365	2	15	15	NUM
ejpam-4276	365	3	.	.	PUNCT
ejpam-4276	366	1	let	let	VERB
ejpam-4276	366	2	(	(	PUNCT
ejpam-4276	366	3	x	x	NOUN
ejpam-4276	366	4	,	,	PUNCT
ejpam-4276	366	5	τ	τ	X
ejpam-4276	366	6	)	)	PUNCT
ejpam-4276	366	7	be	be	VERB
ejpam-4276	366	8	a	a	DET
ejpam-4276	366	9	topological	topological	ADJ
ejpam-4276	366	10	space	space	NOUN
ejpam-4276	366	11	and	and	CCONJ
ejpam-4276	366	12	x	x	PUNCT
ejpam-4276	366	13	∈	∈	PROPN
ejpam-4276	366	14	x.	x.	NOUN
ejpam-4276	366	15	then	then	ADV
ejpam-4276	366	16	,	,	PUNCT
ejpam-4276	366	17	{	{	PUNCT
ejpam-4276	366	18	x	x	X
ejpam-4276	366	19	}	}	PUNCT
ejpam-4276	366	20	is	be	AUX
ejpam-4276	366	21	(	(	PUNCT
ejpam-4276	366	22	λ	λ	X
ejpam-4276	366	23	,	,	PUNCT
ejpam-4276	366	24	sp)-open	sp)-open	ADJ
ejpam-4276	366	25	if	if	SCONJ
ejpam-4276	366	26	and	and	CCONJ
ejpam-4276	366	27	only	only	ADV
ejpam-4276	366	28	if	if	SCONJ
ejpam-4276	366	29	{	{	PUNCT
ejpam-4276	366	30	x	x	NOUN
ejpam-4276	366	31	}	}	PUNCT
ejpam-4276	366	32	is	be	AUX
ejpam-4276	366	33	s(λ	s(λ	PROPN
ejpam-4276	366	34	,	,	PUNCT
ejpam-4276	366	35	sp)-open	sp)-open	NOUN
ejpam-4276	366	36	.	.	PUNCT
ejpam-4276	367	1	proof	proof	NOUN
ejpam-4276	367	2	.	.	PUNCT
ejpam-4276	368	1	the	the	DET
ejpam-4276	368	2	necessity	necessity	NOUN
ejpam-4276	368	3	is	be	AUX
ejpam-4276	368	4	clear	clear	ADJ
ejpam-4276	368	5	.	.	PUNCT
ejpam-4276	369	1	suppose	suppose	VERB
ejpam-4276	369	2	that	that	SCONJ
ejpam-4276	369	3	{	{	PUNCT
ejpam-4276	369	4	x	x	X
ejpam-4276	369	5	}	}	PUNCT
ejpam-4276	369	6	is	be	AUX
ejpam-4276	369	7	s(λ	s(λ	PROPN
ejpam-4276	369	8	,	,	PUNCT
ejpam-4276	369	9	sp)-open	sp)-open	NOUN
ejpam-4276	369	10	.	.	PUNCT
ejpam-4276	370	1	then	then	ADV
ejpam-4276	370	2	,	,	PUNCT
ejpam-4276	370	3	{	{	PUNCT
ejpam-4276	370	4	x	x	X
ejpam-4276	370	5	}	}	PUNCT
ejpam-4276	370	6	⊆	⊆	NUM
ejpam-4276	370	7	[	[	X
ejpam-4276	370	8	{	{	PUNCT
ejpam-4276	370	9	x}(λ	x}(λ	PROPN
ejpam-4276	370	10	,	,	PUNCT
ejpam-4276	370	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	370	12	,	,	PUNCT
ejpam-4276	370	13	sp	sp	NOUN
ejpam-4276	370	14	)	)	PUNCT
ejpam-4276	370	15	.	.	PUNCT
ejpam-4276	371	1	now	now	ADV
ejpam-4276	371	2	{	{	PUNCT
ejpam-4276	371	3	x}(λ	x}(λ	PROPN
ejpam-4276	371	4	,	,	PUNCT
ejpam-4276	371	5	sp	sp	NOUN
ejpam-4276	371	6	)	)	PUNCT
ejpam-4276	371	7	is	be	AUX
ejpam-4276	371	8	either	either	CCONJ
ejpam-4276	371	9	{	{	PUNCT
ejpam-4276	371	10	x	x	NOUN
ejpam-4276	371	11	}	}	PUNCT
ejpam-4276	371	12	or	or	CCONJ
ejpam-4276	371	13	∅.	∅.	VERB
ejpam-4276	371	14	since	since	SCONJ
ejpam-4276	371	15	∅(λ	∅(λ	NOUN
ejpam-4276	371	16	,	,	PUNCT
ejpam-4276	371	17	sp	sp	NOUN
ejpam-4276	371	18	)	)	PUNCT
ejpam-4276	371	19	=	=	NOUN
ejpam-4276	371	20	∅	∅	NOUN
ejpam-4276	371	21	and	and	CCONJ
ejpam-4276	371	22	{	{	PUNCT
ejpam-4276	371	23	x	x	NOUN
ejpam-4276	371	24	}	}	PUNCT
ejpam-4276	371	25	⊆	⊆	NUM
ejpam-4276	371	26	[	[	X
ejpam-4276	371	27	{	{	PUNCT
ejpam-4276	371	28	x}(λ	x}(λ	PROPN
ejpam-4276	371	29	,	,	PUNCT
ejpam-4276	371	30	sp)](λ	sp)](λ	PROPN
ejpam-4276	371	31	,	,	PUNCT
ejpam-4276	371	32	sp	sp	NOUN
ejpam-4276	371	33	)	)	PUNCT
ejpam-4276	371	34	,	,	PUNCT
ejpam-4276	371	35	{	{	PUNCT
ejpam-4276	371	36	x}(λ	x}(λ	PROPN
ejpam-4276	371	37	,	,	PUNCT
ejpam-4276	371	38	sp	sp	NOUN
ejpam-4276	371	39	)	)	PUNCT
ejpam-4276	371	40	6=	6=	ADP
ejpam-4276	371	41	∅.	∅.	ADP
ejpam-4276	371	42	therefore	therefore	ADV
ejpam-4276	371	43	,	,	PUNCT
ejpam-4276	371	44	{	{	PUNCT
ejpam-4276	371	45	x}(λ	x}(λ	PROPN
ejpam-4276	371	46	,	,	PUNCT
ejpam-4276	371	47	sp	sp	NOUN
ejpam-4276	371	48	)	)	PUNCT
ejpam-4276	371	49	=	=	SYM
ejpam-4276	371	50	{	{	PUNCT
ejpam-4276	371	51	x	x	NOUN
ejpam-4276	371	52	}	}	PUNCT
ejpam-4276	371	53	and	and	CCONJ
ejpam-4276	371	54	by	by	ADP
ejpam-4276	371	55	lemma	lemma	PROPN
ejpam-4276	371	56	4	4	NUM
ejpam-4276	371	57	,	,	PUNCT
ejpam-4276	371	58	{	{	PUNCT
ejpam-4276	371	59	x	x	NOUN
ejpam-4276	371	60	}	}	PUNCT
ejpam-4276	371	61	is	be	AUX
ejpam-4276	371	62	(	(	PUNCT
ejpam-4276	371	63	λ	λ	NOUN
ejpam-4276	371	64	,	,	PUNCT
ejpam-4276	371	65	sp)-open	sp)-open	NOUN
ejpam-4276	371	66	.	.	PUNCT
ejpam-4276	372	1	lemma	lemma	PROPN
ejpam-4276	372	2	6	6	NUM
ejpam-4276	372	3	.	.	PUNCT
ejpam-4276	373	1	let	let	VERB
ejpam-4276	373	2	(	(	PUNCT
ejpam-4276	373	3	x	x	NOUN
ejpam-4276	373	4	,	,	PUNCT
ejpam-4276	373	5	τ	τ	X
ejpam-4276	373	6	)	)	PUNCT
ejpam-4276	373	7	be	be	VERB
ejpam-4276	373	8	a	a	DET
ejpam-4276	373	9	topological	topological	ADJ
ejpam-4276	373	10	space	space	NOUN
ejpam-4276	373	11	and	and	CCONJ
ejpam-4276	373	12	a	a	DET
ejpam-4276	373	13	⊆	⊆	NUM
ejpam-4276	373	14	x.	x.	NOUN
ejpam-4276	374	1	if	if	SCONJ
ejpam-4276	374	2	u	u	PROPN
ejpam-4276	374	3	∈	∈	PROPN
ejpam-4276	374	4	λspo(x	λspo(x	PROPN
ejpam-4276	374	5	,	,	PUNCT
ejpam-4276	374	6	τ	τ	PROPN
ejpam-4276	374	7	)	)	PUNCT
ejpam-4276	374	8	and	and	CCONJ
ejpam-4276	374	9	u∩a	u∩a	NOUN
ejpam-4276	374	10	=	=	NOUN
ejpam-4276	374	11	∅	∅	NOUN
ejpam-4276	374	12	,	,	PUNCT
ejpam-4276	374	13	then	then	ADV
ejpam-4276	374	14	u	u	PROPN
ejpam-4276	374	15	∩a(λ	∩a(λ	NOUN
ejpam-4276	374	16	,	,	PUNCT
ejpam-4276	374	17	sp	sp	NOUN
ejpam-4276	374	18	)	)	PUNCT
ejpam-4276	374	19	=	=	PUNCT
ejpam-4276	374	20	∅.	∅.	NOUN
ejpam-4276	374	21	proposition	proposition	NOUN
ejpam-4276	374	22	16	16	NUM
ejpam-4276	374	23	.	.	PUNCT
ejpam-4276	375	1	let	let	VERB
ejpam-4276	375	2	(	(	PUNCT
ejpam-4276	375	3	x	x	NOUN
ejpam-4276	375	4	,	,	PUNCT
ejpam-4276	375	5	τ	τ	X
ejpam-4276	375	6	)	)	PUNCT
ejpam-4276	375	7	be	be	VERB
ejpam-4276	375	8	a	a	DET
ejpam-4276	375	9	topological	topological	ADJ
ejpam-4276	375	10	space	space	NOUN
ejpam-4276	375	11	and	and	CCONJ
ejpam-4276	376	1	x	x	PUNCT
ejpam-4276	376	2	∈	∈	PROPN
ejpam-4276	376	3	x.	x.	NOUN
ejpam-4276	376	4	then	then	ADV
ejpam-4276	376	5	,	,	PUNCT
ejpam-4276	376	6	the	the	DET
ejpam-4276	376	7	following	follow	VERB
ejpam-4276	376	8	properties	property	NOUN
ejpam-4276	376	9	are	be	AUX
ejpam-4276	376	10	equivalent	equivalent	ADJ
ejpam-4276	376	11	:	:	PUNCT
ejpam-4276	376	12	c.	c.	PROPN
ejpam-4276	376	13	boonpok	boonpok	PROPN
ejpam-4276	376	14	,	,	PUNCT
ejpam-4276	376	15	j.	j.	PROPN
ejpam-4276	376	16	khampakdee	khampakdee	PROPN
ejpam-4276	376	17	/	/	PUNCT
ejpam-4276	376	18	eur	eur	PROPN
ejpam-4276	376	19	.	.	PUNCT
ejpam-4276	377	1	j.	j.	PROPN
ejpam-4276	377	2	pure	pure	PROPN
ejpam-4276	377	3	appl	appl	PROPN
ejpam-4276	377	4	.	.	PROPN
ejpam-4276	377	5	math	math	PROPN
ejpam-4276	377	6	,	,	PUNCT
ejpam-4276	377	7	15	15	NUM
ejpam-4276	377	8	(	(	PUNCT
ejpam-4276	377	9	2	2	NUM
ejpam-4276	377	10	)	)	PUNCT
ejpam-4276	377	11	(	(	PUNCT
ejpam-4276	377	12	2022	2022	NUM
ejpam-4276	377	13	)	)	PUNCT
ejpam-4276	377	14	,	,	PUNCT
ejpam-4276	377	15	572	572	NUM
ejpam-4276	377	16	-	-	SYM
ejpam-4276	377	17	588	588	NUM
ejpam-4276	377	18	582	582	NUM
ejpam-4276	377	19	(	(	PUNCT
ejpam-4276	377	20	1	1	NUM
ejpam-4276	377	21	)	)	PUNCT
ejpam-4276	377	22	{	{	PUNCT
ejpam-4276	377	23	x	x	X
ejpam-4276	377	24	}	}	PUNCT
ejpam-4276	377	25	is	be	AUX
ejpam-4276	377	26	p(λ	p(λ	NOUN
ejpam-4276	377	27	,	,	PUNCT
ejpam-4276	377	28	sp)-open	sp)-open	NOUN
ejpam-4276	377	29	.	.	PUNCT
ejpam-4276	378	1	(	(	PUNCT
ejpam-4276	378	2	2	2	NUM
ejpam-4276	378	3	)	)	PUNCT
ejpam-4276	378	4	{	{	PUNCT
ejpam-4276	378	5	x	x	NOUN
ejpam-4276	378	6	}	}	PUNCT
ejpam-4276	378	7	is	be	AUX
ejpam-4276	378	8	b(λ	b(λ	NOUN
ejpam-4276	378	9	,	,	PUNCT
ejpam-4276	378	10	sp)-open	sp)-open	NOUN
ejpam-4276	378	11	.	.	PUNCT
ejpam-4276	379	1	(	(	PUNCT
ejpam-4276	379	2	3	3	X
ejpam-4276	379	3	)	)	PUNCT
ejpam-4276	379	4	{	{	PUNCT
ejpam-4276	379	5	x	x	NOUN
ejpam-4276	379	6	}	}	PUNCT
ejpam-4276	379	7	is	be	AUX
ejpam-4276	379	8	β(λ	β(λ	X
ejpam-4276	379	9	,	,	PUNCT
ejpam-4276	379	10	sp)-open	sp)-open	NOUN
ejpam-4276	379	11	.	.	PUNCT
ejpam-4276	380	1	proof	proof	NOUN
ejpam-4276	380	2	.	.	PUNCT
ejpam-4276	381	1	(	(	PUNCT
ejpam-4276	381	2	1	1	X
ejpam-4276	381	3	)	)	PUNCT
ejpam-4276	381	4	⇒	⇒	NOUN
ejpam-4276	381	5	(	(	PUNCT
ejpam-4276	381	6	2	2	NUM
ejpam-4276	381	7	)	)	PUNCT
ejpam-4276	381	8	and	and	CCONJ
ejpam-4276	381	9	(	(	PUNCT
ejpam-4276	381	10	2	2	X
ejpam-4276	381	11	)	)	PUNCT
ejpam-4276	381	12	⇒	⇒	NOUN
ejpam-4276	381	13	(	(	PUNCT
ejpam-4276	381	14	3	3	X
ejpam-4276	381	15	)	)	PUNCT
ejpam-4276	381	16	follows	follow	VERB
ejpam-4276	381	17	from	from	ADP
ejpam-4276	381	18	remark	remark	NOUN
ejpam-4276	381	19	1	1	NUM
ejpam-4276	381	20	.	.	PUNCT
ejpam-4276	382	1	(	(	PUNCT
ejpam-4276	382	2	3	3	X
ejpam-4276	382	3	)	)	PUNCT
ejpam-4276	382	4	⇒	⇒	NOUN
ejpam-4276	382	5	(	(	PUNCT
ejpam-4276	382	6	1	1	NUM
ejpam-4276	382	7	):	):	PUNCT
ejpam-4276	382	8	let	let	VERB
ejpam-4276	382	9	{	{	PUNCT
ejpam-4276	382	10	x	x	VERB
ejpam-4276	382	11	}	}	PUNCT
ejpam-4276	382	12	be	be	AUX
ejpam-4276	382	13	β(λ	β(λ	X
ejpam-4276	382	14	,	,	PUNCT
ejpam-4276	382	15	sp)-open	sp)-open	NOUN
ejpam-4276	382	16	.	.	PUNCT
ejpam-4276	383	1	assume	assume	VERB
ejpam-4276	383	2	that	that	SCONJ
ejpam-4276	383	3	{	{	PUNCT
ejpam-4276	383	4	x	x	X
ejpam-4276	383	5	}	}	PUNCT
ejpam-4276	383	6	is	be	AUX
ejpam-4276	383	7	not	not	PART
ejpam-4276	383	8	p(λ	p(λ	NOUN
ejpam-4276	383	9	,	,	PUNCT
ejpam-4276	383	10	sp)-open	sp)-open	NOUN
ejpam-4276	383	11	.	.	PUNCT
ejpam-4276	384	1	then	then	ADV
ejpam-4276	384	2	,	,	PUNCT
ejpam-4276	384	3	{	{	PUNCT
ejpam-4276	384	4	x	x	X
ejpam-4276	384	5	}	}	PUNCT
ejpam-4276	384	6	*	*	PUNCT
ejpam-4276	385	1	[	[	X
ejpam-4276	385	2	{	{	PUNCT
ejpam-4276	385	3	x}(λ	x}(λ	PROPN
ejpam-4276	385	4	,	,	PUNCT
ejpam-4276	385	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	385	6	,	,	PUNCT
ejpam-4276	385	7	sp	sp	NOUN
ejpam-4276	385	8	)	)	PUNCT
ejpam-4276	385	9	,	,	PUNCT
ejpam-4276	385	10	that	that	ADV
ejpam-4276	385	11	is	be	AUX
ejpam-4276	385	12	{	{	PUNCT
ejpam-4276	385	13	x	x	NOUN
ejpam-4276	385	14	}	}	PUNCT
ejpam-4276	385	15	∩	∩	NOUN
ejpam-4276	385	16	[	[	X
ejpam-4276	385	17	{	{	PUNCT
ejpam-4276	385	18	x}(λ	x}(λ	PROPN
ejpam-4276	385	19	,	,	PUNCT
ejpam-4276	385	20	sp)](λ	sp)](λ	PROPN
ejpam-4276	385	21	,	,	PUNCT
ejpam-4276	385	22	sp	sp	NOUN
ejpam-4276	385	23	)	)	PUNCT
ejpam-4276	385	24	=	=	PUNCT
ejpam-4276	385	25	∅.	∅.	ADV
ejpam-4276	385	26	since	since	SCONJ
ejpam-4276	385	27	[	[	X
ejpam-4276	385	28	{	{	PUNCT
ejpam-4276	385	29	x}(λ	x}(λ	PROPN
ejpam-4276	385	30	,	,	PUNCT
ejpam-4276	385	31	sp)](λ	sp)](λ	PROPN
ejpam-4276	385	32	,	,	PUNCT
ejpam-4276	385	33	sp	sp	NOUN
ejpam-4276	385	34	)	)	PUNCT
ejpam-4276	385	35	is	be	AUX
ejpam-4276	385	36	(	(	PUNCT
ejpam-4276	385	37	λ	λ	INTJ
ejpam-4276	385	38	,	,	PUNCT
ejpam-4276	385	39	sp)open	sp)open	VERB
ejpam-4276	385	40	,	,	PUNCT
ejpam-4276	385	41	it	it	PRON
ejpam-4276	385	42	follows	follow	VERB
ejpam-4276	385	43	from	from	ADP
ejpam-4276	385	44	lemma	lemma	PROPN
ejpam-4276	385	45	6	6	NUM
ejpam-4276	385	46	that	that	SCONJ
ejpam-4276	385	47	{	{	PUNCT
ejpam-4276	385	48	x}(λ	x}(λ	PROPN
ejpam-4276	385	49	,	,	PUNCT
ejpam-4276	385	50	sp)∩[{x}(λ	sp)∩[{x}(λ	NOUN
ejpam-4276	385	51	,	,	PUNCT
ejpam-4276	385	52	sp)](λ	sp)](λ	PROPN
ejpam-4276	385	53	,	,	PUNCT
ejpam-4276	385	54	sp	sp	NOUN
ejpam-4276	385	55	)	)	PUNCT
ejpam-4276	385	56	=	=	NOUN
ejpam-4276	385	57	∅.	∅.	ADP
ejpam-4276	385	58	thus	thus	ADV
ejpam-4276	385	59	,	,	PUNCT
ejpam-4276	385	60	[	[	X
ejpam-4276	385	61	{	{	PUNCT
ejpam-4276	385	62	x}(λ	x}(λ	PROPN
ejpam-4276	385	63	,	,	PUNCT
ejpam-4276	385	64	sp)](λ	sp)](λ	PROPN
ejpam-4276	385	65	,	,	PUNCT
ejpam-4276	385	66	sp	sp	NOUN
ejpam-4276	385	67	)	)	PUNCT
ejpam-4276	385	68	=	=	NOUN
ejpam-4276	385	69	∅	∅	NOUN
ejpam-4276	385	70	and	and	CCONJ
ejpam-4276	385	71	hence	hence	ADV
ejpam-4276	385	72	[	[	X
ejpam-4276	385	73	[	[	X
ejpam-4276	385	74	{	{	PUNCT
ejpam-4276	385	75	x}(λ	x}(λ	PROPN
ejpam-4276	385	76	,	,	PUNCT
ejpam-4276	385	77	sp)](λ	sp)](λ	PROPN
ejpam-4276	385	78	,	,	PUNCT
ejpam-4276	385	79	sp)](λ	sp)](λ	PROPN
ejpam-4276	385	80	,	,	PUNCT
ejpam-4276	385	81	sp	sp	NOUN
ejpam-4276	385	82	)	)	PUNCT
ejpam-4276	385	83	=	=	SYM
ejpam-4276	385	84	∅(λ	∅(λ	NOUN
ejpam-4276	385	85	,	,	PUNCT
ejpam-4276	385	86	sp	sp	NOUN
ejpam-4276	385	87	)	)	PUNCT
ejpam-4276	385	88	=	=	PUNCT
ejpam-4276	385	89	∅.	∅.	NOUN
ejpam-4276	385	90	this	this	PRON
ejpam-4276	385	91	is	be	AUX
ejpam-4276	385	92	a	a	DET
ejpam-4276	385	93	contradiction	contradiction	NOUN
ejpam-4276	385	94	.	.	PUNCT
ejpam-4276	386	1	proposition	proposition	NOUN
ejpam-4276	386	2	17	17	NUM
ejpam-4276	386	3	.	.	PUNCT
ejpam-4276	387	1	let	let	AUX
ejpam-4276	387	2	(	(	PUNCT
ejpam-4276	387	3	x	x	NOUN
ejpam-4276	387	4	,	,	PUNCT
ejpam-4276	387	5	τ	τ	X
ejpam-4276	387	6	)	)	PUNCT
ejpam-4276	387	7	be	be	VERB
ejpam-4276	387	8	a	a	DET
ejpam-4276	387	9	topological	topological	ADJ
ejpam-4276	387	10	space	space	NOUN
ejpam-4276	387	11	and	and	CCONJ
ejpam-4276	387	12	x	x	PUNCT
ejpam-4276	387	13	∈	∈	PROPN
ejpam-4276	387	14	x.	x.	NOUN
ejpam-4276	387	15	then	then	ADV
ejpam-4276	387	16	,	,	PUNCT
ejpam-4276	387	17	{	{	PUNCT
ejpam-4276	387	18	x	x	X
ejpam-4276	387	19	}	}	PUNCT
ejpam-4276	387	20	is	be	AUX
ejpam-4276	387	21	p(λ	p(λ	NOUN
ejpam-4276	387	22	,	,	PUNCT
ejpam-4276	387	23	sp)-open	sp)-open	ADJ
ejpam-4276	387	24	or	or	CCONJ
ejpam-4276	387	25	{	{	PUNCT
ejpam-4276	387	26	x	x	X
ejpam-4276	387	27	}	}	PUNCT
ejpam-4276	387	28	is	be	AUX
ejpam-4276	387	29	α(λ	α(λ	PROPN
ejpam-4276	387	30	,	,	PUNCT
ejpam-4276	387	31	sp)-closed	sp)-close	VERB
ejpam-4276	387	32	.	.	PUNCT
ejpam-4276	388	1	proof	proof	NOUN
ejpam-4276	388	2	.	.	PUNCT
ejpam-4276	389	1	assume	assume	VERB
ejpam-4276	389	2	that	that	SCONJ
ejpam-4276	389	3	{	{	PUNCT
ejpam-4276	389	4	x	x	X
ejpam-4276	389	5	}	}	PUNCT
ejpam-4276	389	6	is	be	AUX
ejpam-4276	389	7	not	not	PART
ejpam-4276	389	8	p(λ	p(λ	NOUN
ejpam-4276	389	9	,	,	PUNCT
ejpam-4276	389	10	sp)-open	sp)-open	NOUN
ejpam-4276	389	11	.	.	PUNCT
ejpam-4276	390	1	then	then	ADV
ejpam-4276	390	2	,	,	PUNCT
ejpam-4276	390	3	{	{	PUNCT
ejpam-4276	390	4	x	x	X
ejpam-4276	390	5	}	}	PUNCT
ejpam-4276	390	6	*	*	PUNCT
ejpam-4276	391	1	[	[	X
ejpam-4276	391	2	{	{	PUNCT
ejpam-4276	391	3	x}(λ	x}(λ	PROPN
ejpam-4276	391	4	,	,	PUNCT
ejpam-4276	391	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	391	6	,	,	PUNCT
ejpam-4276	391	7	sp	sp	NOUN
ejpam-4276	391	8	)	)	PUNCT
ejpam-4276	391	9	and	and	CCONJ
ejpam-4276	391	10	hence	hence	ADV
ejpam-4276	391	11	{	{	PUNCT
ejpam-4276	391	12	x	x	NOUN
ejpam-4276	391	13	}	}	PUNCT
ejpam-4276	391	14	∩	∩	NOUN
ejpam-4276	391	15	[	[	X
ejpam-4276	391	16	{	{	PUNCT
ejpam-4276	391	17	x}(λ	x}(λ	PROPN
ejpam-4276	391	18	,	,	PUNCT
ejpam-4276	391	19	sp)](λ	sp)](λ	PROPN
ejpam-4276	391	20	,	,	PUNCT
ejpam-4276	391	21	sp	sp	NOUN
ejpam-4276	391	22	)	)	PUNCT
ejpam-4276	391	23	=	=	PUNCT
ejpam-4276	391	24	∅.	∅.	ADV
ejpam-4276	391	25	since	since	SCONJ
ejpam-4276	391	26	[	[	X
ejpam-4276	391	27	{	{	PUNCT
ejpam-4276	391	28	x}(λ	x}(λ	PROPN
ejpam-4276	391	29	,	,	PUNCT
ejpam-4276	391	30	sp)](λ	sp)](λ	PROPN
ejpam-4276	391	31	,	,	PUNCT
ejpam-4276	391	32	sp	sp	NOUN
ejpam-4276	391	33	)	)	PUNCT
ejpam-4276	391	34	is	be	AUX
ejpam-4276	391	35	(	(	PUNCT
ejpam-4276	391	36	λ	λ	INTJ
ejpam-4276	391	37	,	,	PUNCT
ejpam-4276	391	38	sp)-open	sp)-open	ADJ
ejpam-4276	391	39	,	,	PUNCT
ejpam-4276	391	40	it	it	PRON
ejpam-4276	391	41	follows	follow	VERB
ejpam-4276	391	42	from	from	ADP
ejpam-4276	391	43	lemma	lemma	PROPN
ejpam-4276	391	44	6	6	NUM
ejpam-4276	391	45	that	that	SCONJ
ejpam-4276	391	46	{	{	PUNCT
ejpam-4276	391	47	x}(λ	x}(λ	PROPN
ejpam-4276	391	48	,	,	PUNCT
ejpam-4276	391	49	sp	sp	NOUN
ejpam-4276	391	50	)	)	PUNCT
ejpam-4276	391	51	∩	∩	NOUN
ejpam-4276	391	52	[	[	X
ejpam-4276	391	53	{	{	PUNCT
ejpam-4276	391	54	x}(λ	x}(λ	PROPN
ejpam-4276	391	55	,	,	PUNCT
ejpam-4276	391	56	sp)](λ	sp)](λ	PROPN
ejpam-4276	391	57	,	,	PUNCT
ejpam-4276	391	58	sp	sp	NOUN
ejpam-4276	391	59	)	)	PUNCT
ejpam-4276	391	60	=	=	PUNCT
ejpam-4276	391	61	∅.	∅.	VERB
ejpam-4276	391	62	therefore	therefore	ADV
ejpam-4276	391	63	,	,	PUNCT
ejpam-4276	391	64	[	[	X
ejpam-4276	391	65	{	{	PUNCT
ejpam-4276	391	66	x}(λ	x}(λ	PROPN
ejpam-4276	391	67	,	,	PUNCT
ejpam-4276	391	68	sp)](λ	sp)](λ	PROPN
ejpam-4276	391	69	,	,	PUNCT
ejpam-4276	391	70	sp	sp	NOUN
ejpam-4276	391	71	)	)	PUNCT
ejpam-4276	391	72	=	=	PUNCT
ejpam-4276	391	73	∅.	∅.	ADP
ejpam-4276	391	74	this	this	PRON
ejpam-4276	391	75	implies	imply	VERB
ejpam-4276	391	76	that	that	SCONJ
ejpam-4276	391	77	[	[	X
ejpam-4276	391	78	[	[	X
ejpam-4276	391	79	{	{	PUNCT
ejpam-4276	391	80	x}(λ	x}(λ	PROPN
ejpam-4276	391	81	,	,	PUNCT
ejpam-4276	391	82	sp)](λ	sp)](λ	PROPN
ejpam-4276	391	83	,	,	PUNCT
ejpam-4276	391	84	sp)](λ	sp)](λ	PROPN
ejpam-4276	391	85	,	,	PUNCT
ejpam-4276	391	86	sp	sp	NOUN
ejpam-4276	391	87	)	)	PUNCT
ejpam-4276	392	1	=	=	SYM
ejpam-4276	392	2	∅(λ	∅(λ	NOUN
ejpam-4276	392	3	,	,	PUNCT
ejpam-4276	392	4	sp	sp	NOUN
ejpam-4276	392	5	)	)	PUNCT
ejpam-4276	392	6	=	=	NOUN
ejpam-4276	392	7	∅.	∅.	VERB
ejpam-4276	392	8	thus	thus	ADV
ejpam-4276	392	9	,	,	PUNCT
ejpam-4276	392	10	by	by	ADP
ejpam-4276	392	11	proposition	proposition	NOUN
ejpam-4276	392	12	3	3	NUM
ejpam-4276	392	13	,	,	PUNCT
ejpam-4276	392	14	{	{	PUNCT
ejpam-4276	392	15	x	x	X
ejpam-4276	392	16	}	}	PUNCT
ejpam-4276	392	17	is	be	AUX
ejpam-4276	392	18	α(λ	α(λ	PROPN
ejpam-4276	392	19	,	,	PUNCT
ejpam-4276	392	20	sp)-closed	sp)-close	VERB
ejpam-4276	392	21	.	.	PUNCT
ejpam-4276	393	1	proposition	proposition	NOUN
ejpam-4276	393	2	18	18	NUM
ejpam-4276	393	3	.	.	PUNCT
ejpam-4276	394	1	let	let	VERB
ejpam-4276	394	2	a	a	DET
ejpam-4276	394	3	be	be	AUX
ejpam-4276	394	4	a	a	DET
ejpam-4276	394	5	subset	subset	NOUN
ejpam-4276	394	6	of	of	ADP
ejpam-4276	394	7	a	a	DET
ejpam-4276	394	8	topological	topological	ADJ
ejpam-4276	394	9	space	space	NOUN
ejpam-4276	394	10	(	(	PUNCT
ejpam-4276	394	11	x	x	X
ejpam-4276	394	12	,	,	PUNCT
ejpam-4276	394	13	τ	τ	PROPN
ejpam-4276	394	14	)	)	PUNCT
ejpam-4276	394	15	.	.	PUNCT
ejpam-4276	395	1	then	then	ADV
ejpam-4276	395	2	,	,	PUNCT
ejpam-4276	395	3	a	a	PRON
ejpam-4276	395	4	is	be	AUX
ejpam-4276	395	5	s(λ	s(λ	NOUN
ejpam-4276	395	6	,	,	PUNCT
ejpam-4276	395	7	sp)-open	sp)-open	ADJ
ejpam-4276	395	8	if	if	SCONJ
ejpam-4276	395	9	and	and	CCONJ
ejpam-4276	395	10	only	only	ADV
ejpam-4276	395	11	if	if	SCONJ
ejpam-4276	395	12	there	there	PRON
ejpam-4276	395	13	exists	exist	VERB
ejpam-4276	395	14	a	a	DET
ejpam-4276	395	15	(	(	PUNCT
ejpam-4276	395	16	λ	λ	NOUN
ejpam-4276	395	17	,	,	PUNCT
ejpam-4276	395	18	sp)-open	sp)-open	NOUN
ejpam-4276	395	19	set	set	VERB
ejpam-4276	395	20	u	u	PRON
ejpam-4276	395	21	such	such	ADJ
ejpam-4276	395	22	that	that	SCONJ
ejpam-4276	395	23	u	u	NOUN
ejpam-4276	395	24	⊆	⊆	NUM
ejpam-4276	395	25	a	a	DET
ejpam-4276	395	26	⊆	⊆	NUM
ejpam-4276	395	27	u	u	NOUN
ejpam-4276	395	28	(	(	PUNCT
ejpam-4276	395	29	λ	λ	PROPN
ejpam-4276	395	30	,	,	PUNCT
ejpam-4276	395	31	sp	sp	NOUN
ejpam-4276	395	32	)	)	PUNCT
ejpam-4276	395	33	.	.	PUNCT
ejpam-4276	396	1	proof	proof	NOUN
ejpam-4276	396	2	.	.	PUNCT
ejpam-4276	397	1	suppose	suppose	VERB
ejpam-4276	397	2	that	that	SCONJ
ejpam-4276	397	3	a	a	PRON
ejpam-4276	397	4	is	be	AUX
ejpam-4276	397	5	s(λ	s(λ	NOUN
ejpam-4276	397	6	,	,	PUNCT
ejpam-4276	397	7	sp)-open	sp)-open	NOUN
ejpam-4276	397	8	.	.	PUNCT
ejpam-4276	398	1	then	then	ADV
ejpam-4276	398	2	,	,	PUNCT
ejpam-4276	398	3	a	a	DET
ejpam-4276	398	4	⊆	⊆	NUM
ejpam-4276	398	5	[	[	X
ejpam-4276	398	6	a(λ	a(λ	ADV
ejpam-4276	398	7	,	,	PUNCT
ejpam-4276	398	8	sp	sp	NOUN
ejpam-4276	398	9	)	)	PUNCT
ejpam-4276	398	10	]	]	PUNCT
ejpam-4276	399	1	(	(	PUNCT
ejpam-4276	399	2	λ	λ	NOUN
ejpam-4276	399	3	,	,	PUNCT
ejpam-4276	399	4	sp	sp	NOUN
ejpam-4276	399	5	)	)	PUNCT
ejpam-4276	399	6	.	.	PUNCT
ejpam-4276	400	1	let	let	VERB
ejpam-4276	400	2	u	u	PRON
ejpam-4276	400	3	=	=	PUNCT
ejpam-4276	400	4	a(λ	a(λ	ADV
ejpam-4276	400	5	,	,	PUNCT
ejpam-4276	400	6	sp	sp	NOUN
ejpam-4276	400	7	)	)	PUNCT
ejpam-4276	400	8	.	.	PUNCT
ejpam-4276	401	1	thus	thus	ADV
ejpam-4276	401	2	,	,	PUNCT
ejpam-4276	401	3	we	we	PRON
ejpam-4276	401	4	obtain	obtain	VERB
ejpam-4276	401	5	u	u	NOUN
ejpam-4276	401	6	⊆	⊆	NUM
ejpam-4276	401	7	a	a	DET
ejpam-4276	401	8	⊆	⊆	NUM
ejpam-4276	401	9	u	u	NOUN
ejpam-4276	401	10	(	(	PUNCT
ejpam-4276	401	11	λ	λ	PROPN
ejpam-4276	401	12	,	,	PUNCT
ejpam-4276	401	13	sp	sp	NOUN
ejpam-4276	401	14	)	)	PUNCT
ejpam-4276	401	15	.	.	PUNCT
ejpam-4276	402	1	conversely	conversely	ADV
ejpam-4276	402	2	,	,	PUNCT
ejpam-4276	402	3	assume	assume	VERB
ejpam-4276	402	4	that	that	SCONJ
ejpam-4276	402	5	there	there	PRON
ejpam-4276	402	6	exists	exist	VERB
ejpam-4276	402	7	a	a	DET
ejpam-4276	402	8	(	(	PUNCT
ejpam-4276	402	9	λ	λ	NOUN
ejpam-4276	402	10	,	,	PUNCT
ejpam-4276	402	11	sp)-open	sp)-open	NOUN
ejpam-4276	402	12	set	set	VERB
ejpam-4276	402	13	u	u	PRON
ejpam-4276	402	14	such	such	ADJ
ejpam-4276	402	15	that	that	SCONJ
ejpam-4276	402	16	u	u	NOUN
ejpam-4276	402	17	⊆	⊆	NUM
ejpam-4276	402	18	a	a	DET
ejpam-4276	402	19	⊆	⊆	NUM
ejpam-4276	402	20	u	u	NOUN
ejpam-4276	402	21	(	(	PUNCT
ejpam-4276	402	22	λ	λ	PROPN
ejpam-4276	402	23	,	,	PUNCT
ejpam-4276	402	24	sp	sp	NOUN
ejpam-4276	402	25	)	)	PUNCT
ejpam-4276	402	26	.	.	PUNCT
ejpam-4276	403	1	then	then	ADV
ejpam-4276	403	2	,	,	PUNCT
ejpam-4276	403	3	u	u	NOUN
ejpam-4276	403	4	⊆	⊆	NUM
ejpam-4276	403	5	a(λ	a(λ	ADV
ejpam-4276	403	6	,	,	PUNCT
ejpam-4276	403	7	sp	sp	NOUN
ejpam-4276	403	8	)	)	PUNCT
ejpam-4276	403	9	and	and	CCONJ
ejpam-4276	403	10	hence	hence	ADV
ejpam-4276	403	11	u	u	NOUN
ejpam-4276	403	12	(	(	PUNCT
ejpam-4276	403	13	λ	λ	PROPN
ejpam-4276	403	14	,	,	PUNCT
ejpam-4276	403	15	sp	sp	NOUN
ejpam-4276	403	16	)	)	PUNCT
ejpam-4276	403	17	⊆	⊆	NUM
ejpam-4276	403	18	[	[	X
ejpam-4276	403	19	a(λ	a(λ	ADV
ejpam-4276	403	20	,	,	PUNCT
ejpam-4276	403	21	sp	sp	NOUN
ejpam-4276	403	22	)	)	PUNCT
ejpam-4276	403	23	]	]	PUNCT
ejpam-4276	403	24	(	(	PUNCT
ejpam-4276	403	25	λ	λ	NOUN
ejpam-4276	403	26	,	,	PUNCT
ejpam-4276	403	27	sp	sp	NOUN
ejpam-4276	403	28	)	)	PUNCT
ejpam-4276	403	29	.	.	PUNCT
ejpam-4276	404	1	since	since	SCONJ
ejpam-4276	404	2	a	a	DET
ejpam-4276	404	3	⊆	⊆	NUM
ejpam-4276	404	4	u	u	NOUN
ejpam-4276	404	5	(	(	PUNCT
ejpam-4276	404	6	λ	λ	PROPN
ejpam-4276	404	7	,	,	PUNCT
ejpam-4276	404	8	sp	sp	NOUN
ejpam-4276	404	9	)	)	PUNCT
ejpam-4276	404	10	,	,	PUNCT
ejpam-4276	404	11	a	a	DET
ejpam-4276	404	12	⊆	⊆	NUM
ejpam-4276	404	13	[	[	X
ejpam-4276	404	14	a(λ	a(λ	ADV
ejpam-4276	404	15	,	,	PUNCT
ejpam-4276	404	16	sp	sp	NOUN
ejpam-4276	404	17	)	)	PUNCT
ejpam-4276	404	18	]	]	PUNCT
ejpam-4276	404	19	(	(	PUNCT
ejpam-4276	404	20	λ	λ	NOUN
ejpam-4276	404	21	,	,	PUNCT
ejpam-4276	404	22	sp	sp	NOUN
ejpam-4276	404	23	)	)	PUNCT
ejpam-4276	404	24	.	.	PUNCT
ejpam-4276	405	1	thus	thus	ADV
ejpam-4276	405	2	,	,	PUNCT
ejpam-4276	405	3	a	a	PRON
ejpam-4276	405	4	is	be	AUX
ejpam-4276	405	5	s(λ	s(λ	PROPN
ejpam-4276	405	6	,	,	PUNCT
ejpam-4276	405	7	sp)-open	sp)-open	NOUN
ejpam-4276	405	8	.	.	PUNCT
ejpam-4276	406	1	proposition	proposition	NOUN
ejpam-4276	406	2	19	19	NUM
ejpam-4276	406	3	.	.	PUNCT
ejpam-4276	407	1	let	let	VERB
ejpam-4276	407	2	a	a	DET
ejpam-4276	407	3	be	be	AUX
ejpam-4276	407	4	a	a	DET
ejpam-4276	407	5	subset	subset	NOUN
ejpam-4276	407	6	of	of	ADP
ejpam-4276	407	7	a	a	DET
ejpam-4276	407	8	topological	topological	ADJ
ejpam-4276	407	9	space	space	NOUN
ejpam-4276	407	10	(	(	PUNCT
ejpam-4276	407	11	x	x	X
ejpam-4276	407	12	,	,	PUNCT
ejpam-4276	407	13	τ	τ	PROPN
ejpam-4276	407	14	)	)	PUNCT
ejpam-4276	407	15	.	.	PUNCT
ejpam-4276	408	1	if	if	SCONJ
ejpam-4276	408	2	there	there	PRON
ejpam-4276	408	3	exists	exist	VERB
ejpam-4276	408	4	a	a	DET
ejpam-4276	408	5	p(λ	p(λ	NOUN
ejpam-4276	408	6	,	,	PUNCT
ejpam-4276	408	7	sp)open	sp)open	NOUN
ejpam-4276	408	8	set	set	VERB
ejpam-4276	408	9	u	u	PRON
ejpam-4276	408	10	such	such	ADJ
ejpam-4276	408	11	that	that	SCONJ
ejpam-4276	408	12	u	u	NOUN
ejpam-4276	408	13	⊆	⊆	NUM
ejpam-4276	408	14	a	a	DET
ejpam-4276	408	15	⊆	⊆	NUM
ejpam-4276	408	16	u	u	NOUN
ejpam-4276	408	17	(	(	PUNCT
ejpam-4276	408	18	λ	λ	PROPN
ejpam-4276	408	19	,	,	PUNCT
ejpam-4276	408	20	sp	sp	NOUN
ejpam-4276	408	21	)	)	PUNCT
ejpam-4276	408	22	,	,	PUNCT
ejpam-4276	408	23	then	then	ADV
ejpam-4276	408	24	a	a	PRON
ejpam-4276	408	25	is	be	AUX
ejpam-4276	408	26	β(λ	β(λ	NOUN
ejpam-4276	408	27	,	,	PUNCT
ejpam-4276	408	28	sp)-open	sp)-open	NOUN
ejpam-4276	408	29	.	.	PUNCT
ejpam-4276	409	1	proof	proof	NOUN
ejpam-4276	409	2	.	.	PUNCT
ejpam-4276	410	1	since	since	SCONJ
ejpam-4276	410	2	u	u	PROPN
ejpam-4276	410	3	⊆	⊆	NUM
ejpam-4276	410	4	a	a	DET
ejpam-4276	410	5	⊆	⊆	NUM
ejpam-4276	410	6	u	u	NOUN
ejpam-4276	410	7	(	(	PUNCT
ejpam-4276	410	8	λ	λ	PROPN
ejpam-4276	410	9	,	,	PUNCT
ejpam-4276	410	10	sp	sp	NOUN
ejpam-4276	410	11	)	)	PUNCT
ejpam-4276	410	12	,	,	PUNCT
ejpam-4276	410	13	we	we	PRON
ejpam-4276	410	14	have	have	VERB
ejpam-4276	410	15	a(λ	a(λ	ADV
ejpam-4276	410	16	,	,	PUNCT
ejpam-4276	410	17	sp	sp	NOUN
ejpam-4276	410	18	)	)	PUNCT
ejpam-4276	410	19	=	=	SYM
ejpam-4276	410	20	u	u	NOUN
ejpam-4276	410	21	(	(	PUNCT
ejpam-4276	410	22	λ	λ	PROPN
ejpam-4276	410	23	,	,	PUNCT
ejpam-4276	410	24	sp	sp	NOUN
ejpam-4276	410	25	)	)	PUNCT
ejpam-4276	410	26	and	and	CCONJ
ejpam-4276	410	27	hence	hence	ADV
ejpam-4276	410	28	[	[	X
ejpam-4276	410	29	a(λ	a(λ	PROPN
ejpam-4276	410	30	,	,	PUNCT
ejpam-4276	410	31	sp)](λ	sp)](λ	PROPN
ejpam-4276	410	32	,	,	PUNCT
ejpam-4276	410	33	sp	sp	NOUN
ejpam-4276	410	34	)	)	PUNCT
ejpam-4276	410	35	=	=	PUNCT
ejpam-4276	411	1	[	[	X
ejpam-4276	411	2	u	u	X
ejpam-4276	411	3	(	(	PUNCT
ejpam-4276	411	4	λ	λ	PROPN
ejpam-4276	411	5	,	,	PUNCT
ejpam-4276	411	6	sp)](λ	sp)](λ	PROPN
ejpam-4276	411	7	,	,	PUNCT
ejpam-4276	411	8	sp	sp	NOUN
ejpam-4276	411	9	)	)	PUNCT
ejpam-4276	411	10	.	.	PUNCT
ejpam-4276	412	1	since	since	SCONJ
ejpam-4276	412	2	u	u	NOUN
ejpam-4276	412	3	is	be	AUX
ejpam-4276	412	4	p(λ	p(λ	NOUN
ejpam-4276	412	5	,	,	PUNCT
ejpam-4276	412	6	sp)-open	sp)-open	NOUN
ejpam-4276	412	7	,	,	PUNCT
ejpam-4276	412	8	we	we	PRON
ejpam-4276	412	9	have	have	VERB
ejpam-4276	412	10	u	u	NOUN
ejpam-4276	412	11	⊆	⊆	NUM
ejpam-4276	412	12	[	[	X
ejpam-4276	412	13	a(λ	a(λ	ADJ
ejpam-4276	412	14	,	,	PUNCT
ejpam-4276	412	15	sp)](λ	sp)](λ	PROPN
ejpam-4276	412	16	,	,	PUNCT
ejpam-4276	412	17	sp	sp	NOUN
ejpam-4276	412	18	)	)	PUNCT
ejpam-4276	412	19	.	.	PUNCT
ejpam-4276	413	1	thus	thus	ADV
ejpam-4276	413	2	,	,	PUNCT
ejpam-4276	413	3	a	a	DET
ejpam-4276	413	4	⊆	⊆	NUM
ejpam-4276	413	5	u	u	NOUN
ejpam-4276	413	6	(	(	PUNCT
ejpam-4276	413	7	λ	λ	PROPN
ejpam-4276	413	8	,	,	PUNCT
ejpam-4276	413	9	sp	sp	NOUN
ejpam-4276	413	10	)	)	PUNCT
ejpam-4276	413	11	and	and	CCONJ
ejpam-4276	413	12	hence	hence	ADV
ejpam-4276	413	13	a	a	DET
ejpam-4276	413	14	⊆	⊆	NUM
ejpam-4276	413	15	[	[	X
ejpam-4276	413	16	[	[	X
ejpam-4276	413	17	a(λ	a(λ	ADJ
ejpam-4276	413	18	,	,	PUNCT
ejpam-4276	413	19	sp)](λ	sp)](λ	PROPN
ejpam-4276	413	20	,	,	PUNCT
ejpam-4276	413	21	sp	sp	NOUN
ejpam-4276	413	22	)	)	PUNCT
ejpam-4276	413	23	]	]	PUNCT
ejpam-4276	413	24	(	(	PUNCT
ejpam-4276	413	25	λ	λ	NOUN
ejpam-4276	413	26	,	,	PUNCT
ejpam-4276	413	27	sp	sp	NOUN
ejpam-4276	413	28	)	)	PUNCT
ejpam-4276	413	29	.	.	PUNCT
ejpam-4276	414	1	this	this	PRON
ejpam-4276	414	2	shows	show	VERB
ejpam-4276	414	3	that	that	SCONJ
ejpam-4276	414	4	a	a	PRON
ejpam-4276	414	5	is	be	AUX
ejpam-4276	414	6	β(λ	β(λ	NOUN
ejpam-4276	414	7	,	,	PUNCT
ejpam-4276	414	8	sp)-open	sp)-open	NOUN
ejpam-4276	414	9	.	.	PUNCT
ejpam-4276	415	1	a	a	DET
ejpam-4276	415	2	subset	subset	NOUN
ejpam-4276	415	3	d	d	NOUN
ejpam-4276	415	4	of	of	ADP
ejpam-4276	415	5	a	a	DET
ejpam-4276	415	6	topological	topological	ADJ
ejpam-4276	415	7	space	space	NOUN
ejpam-4276	415	8	(	(	PUNCT
ejpam-4276	415	9	x	x	X
ejpam-4276	415	10	,	,	PUNCT
ejpam-4276	415	11	τ	τ	X
ejpam-4276	415	12	)	)	PUNCT
ejpam-4276	415	13	is	be	AUX
ejpam-4276	415	14	called	call	VERB
ejpam-4276	415	15	λsp	λsp	ADV
ejpam-4276	415	16	-	-	PUNCT
ejpam-4276	415	17	dense	dense	ADJ
ejpam-4276	415	18	[	[	X
ejpam-4276	415	19	3	3	NUM
ejpam-4276	415	20	]	]	X
ejpam-4276	415	21	if	if	SCONJ
ejpam-4276	415	22	d(λ	d(λ	PROPN
ejpam-4276	415	23	,	,	PUNCT
ejpam-4276	415	24	sp	sp	NOUN
ejpam-4276	415	25	)	)	PUNCT
ejpam-4276	415	26	=	=	PUNCT
ejpam-4276	416	1	x.	x.	PUNCT
ejpam-4276	416	2	d	d	PROPN
ejpam-4276	416	3	is	be	AUX
ejpam-4276	416	4	called	call	VERB
ejpam-4276	416	5	λsp	λsp	NOUN
ejpam-4276	416	6	-	-	NOUN
ejpam-4276	416	7	codense	codense	NOUN
ejpam-4276	416	8	[	[	X
ejpam-4276	416	9	3	3	X
ejpam-4276	416	10	]	]	X
ejpam-4276	416	11	if	if	SCONJ
ejpam-4276	416	12	x	x	PROPN
ejpam-4276	416	13	−d	−d	PROPN
ejpam-4276	416	14	is	be	AUX
ejpam-4276	416	15	λsp	λsp	NOUN
ejpam-4276	416	16	-	-	PUNCT
ejpam-4276	416	17	dense	dense	ADJ
ejpam-4276	416	18	.	.	PUNCT
ejpam-4276	417	1	proposition	proposition	NOUN
ejpam-4276	417	2	20	20	NUM
ejpam-4276	417	3	.	.	PUNCT
ejpam-4276	418	1	let	let	VERB
ejpam-4276	418	2	(	(	PUNCT
ejpam-4276	418	3	x	x	NOUN
ejpam-4276	418	4	,	,	PUNCT
ejpam-4276	418	5	τ	τ	X
ejpam-4276	418	6	)	)	PUNCT
ejpam-4276	418	7	be	be	VERB
ejpam-4276	418	8	a	a	DET
ejpam-4276	418	9	topological	topological	ADJ
ejpam-4276	418	10	space	space	NOUN
ejpam-4276	418	11	and	and	CCONJ
ejpam-4276	419	1	d	d	PROPN
ejpam-4276	419	2	⊆	⊆	NUM
ejpam-4276	419	3	x.	x.	NOUN
ejpam-4276	419	4	then	then	ADV
ejpam-4276	419	5	,	,	PUNCT
ejpam-4276	419	6	the	the	DET
ejpam-4276	419	7	following	follow	VERB
ejpam-4276	419	8	properties	property	NOUN
ejpam-4276	419	9	are	be	AUX
ejpam-4276	419	10	equivalent	equivalent	ADJ
ejpam-4276	419	11	:	:	PUNCT
ejpam-4276	419	12	(	(	PUNCT
ejpam-4276	419	13	1	1	X
ejpam-4276	419	14	)	)	PUNCT
ejpam-4276	419	15	d	d	NOUN
ejpam-4276	419	16	is	be	AUX
ejpam-4276	419	17	λsp	λsp	ADV
ejpam-4276	419	18	-	-	PUNCT
ejpam-4276	419	19	dense	dense	ADJ
ejpam-4276	419	20	.	.	PUNCT
ejpam-4276	420	1	c.	c.	PROPN
ejpam-4276	420	2	boonpok	boonpok	PROPN
ejpam-4276	420	3	,	,	PUNCT
ejpam-4276	420	4	j.	j.	PROPN
ejpam-4276	420	5	khampakdee	khampakdee	PROPN
ejpam-4276	420	6	/	/	PUNCT
ejpam-4276	420	7	eur	eur	PROPN
ejpam-4276	420	8	.	.	PUNCT
ejpam-4276	421	1	j.	j.	PROPN
ejpam-4276	421	2	pure	pure	PROPN
ejpam-4276	421	3	appl	appl	PROPN
ejpam-4276	421	4	.	.	PROPN
ejpam-4276	421	5	math	math	PROPN
ejpam-4276	421	6	,	,	PUNCT
ejpam-4276	421	7	15	15	NUM
ejpam-4276	421	8	(	(	PUNCT
ejpam-4276	421	9	2	2	NUM
ejpam-4276	421	10	)	)	PUNCT
ejpam-4276	421	11	(	(	PUNCT
ejpam-4276	421	12	2022	2022	NUM
ejpam-4276	421	13	)	)	PUNCT
ejpam-4276	421	14	,	,	PUNCT
ejpam-4276	421	15	572	572	NUM
ejpam-4276	421	16	-	-	SYM
ejpam-4276	421	17	588	588	NUM
ejpam-4276	421	18	583	583	NUM
ejpam-4276	421	19	(	(	PUNCT
ejpam-4276	421	20	2	2	NUM
ejpam-4276	421	21	)	)	PUNCT
ejpam-4276	421	22	if	if	SCONJ
ejpam-4276	421	23	f	f	PROPN
ejpam-4276	421	24	is	be	AUX
ejpam-4276	421	25	any	any	DET
ejpam-4276	421	26	(	(	PUNCT
ejpam-4276	421	27	λ	λ	PROPN
ejpam-4276	421	28	,	,	PUNCT
ejpam-4276	421	29	sp)-closed	sp)-close	VERB
ejpam-4276	421	30	set	set	ADJ
ejpam-4276	421	31	and	and	CCONJ
ejpam-4276	421	32	d	d	NOUN
ejpam-4276	421	33	⊆	⊆	NUM
ejpam-4276	421	34	f	f	NOUN
ejpam-4276	421	35	,	,	PUNCT
ejpam-4276	421	36	then	then	ADV
ejpam-4276	421	37	f	f	PROPN
ejpam-4276	421	38	=	=	PUNCT
ejpam-4276	421	39	x.	x.	NOUN
ejpam-4276	421	40	(	(	PUNCT
ejpam-4276	421	41	3	3	X
ejpam-4276	421	42	)	)	PUNCT
ejpam-4276	421	43	each	each	PRON
ejpam-4276	421	44	nonempty	nonempty	ADJ
ejpam-4276	421	45	(	(	PUNCT
ejpam-4276	421	46	λ	λ	NOUN
ejpam-4276	421	47	,	,	PUNCT
ejpam-4276	421	48	sp)-open	sp)-open	ADJ
ejpam-4276	421	49	set	set	NOUN
ejpam-4276	421	50	contains	contain	VERB
ejpam-4276	421	51	an	an	DET
ejpam-4276	421	52	element	element	NOUN
ejpam-4276	421	53	of	of	ADP
ejpam-4276	421	54	d.	d.	PROPN
ejpam-4276	421	55	(	(	PUNCT
ejpam-4276	421	56	4	4	NUM
ejpam-4276	421	57	)	)	PUNCT
ejpam-4276	421	58	the	the	DET
ejpam-4276	421	59	complement	complement	NOUN
ejpam-4276	421	60	of	of	ADP
ejpam-4276	421	61	d	d	PROPN
ejpam-4276	421	62	has	have	AUX
ejpam-4276	421	63	empty	empty	ADJ
ejpam-4276	421	64	(	(	PUNCT
ejpam-4276	421	65	λ	λ	NOUN
ejpam-4276	421	66	,	,	PUNCT
ejpam-4276	421	67	sp)-interior	sp)-interior	NOUN
ejpam-4276	421	68	.	.	PUNCT
ejpam-4276	422	1	proof	proof	NOUN
ejpam-4276	422	2	.	.	PUNCT
ejpam-4276	423	1	(	(	PUNCT
ejpam-4276	423	2	1	1	X
ejpam-4276	423	3	)	)	PUNCT
ejpam-4276	423	4	⇒	⇒	NOUN
ejpam-4276	423	5	(	(	PUNCT
ejpam-4276	423	6	2	2	NUM
ejpam-4276	423	7	):	):	PUNCT
ejpam-4276	423	8	let	let	VERB
ejpam-4276	423	9	f	f	PRON
ejpam-4276	423	10	be	be	AUX
ejpam-4276	423	11	a	a	DET
ejpam-4276	423	12	(	(	PUNCT
ejpam-4276	423	13	λ	λ	NOUN
ejpam-4276	423	14	,	,	PUNCT
ejpam-4276	423	15	sp)-closed	sp)-close	VERB
ejpam-4276	423	16	set	set	VERB
ejpam-4276	423	17	such	such	ADJ
ejpam-4276	423	18	that	that	SCONJ
ejpam-4276	423	19	d	d	PROPN
ejpam-4276	423	20	⊆	⊆	NUM
ejpam-4276	423	21	f	f	NOUN
ejpam-4276	423	22	.	.	PUNCT
ejpam-4276	424	1	then	then	ADV
ejpam-4276	424	2	,	,	PUNCT
ejpam-4276	424	3	x	x	SYM
ejpam-4276	424	4	=	=	SYM
ejpam-4276	424	5	d(λ	d(λ	PROPN
ejpam-4276	424	6	,	,	PUNCT
ejpam-4276	424	7	sp	sp	NOUN
ejpam-4276	424	8	)	)	PUNCT
ejpam-4276	424	9	⊆	⊆	NUM
ejpam-4276	424	10	f	f	X
ejpam-4276	424	11	(	(	PUNCT
ejpam-4276	424	12	λ	λ	PROPN
ejpam-4276	424	13	,	,	PUNCT
ejpam-4276	424	14	sp	sp	NOUN
ejpam-4276	424	15	)	)	PUNCT
ejpam-4276	424	16	=	=	SYM
ejpam-4276	424	17	f	f	PROPN
ejpam-4276	424	18	.	.	PUNCT
ejpam-4276	425	1	(	(	PUNCT
ejpam-4276	425	2	2	2	X
ejpam-4276	425	3	)	)	PUNCT
ejpam-4276	425	4	⇒	⇒	NOUN
ejpam-4276	425	5	(	(	PUNCT
ejpam-4276	425	6	3	3	NUM
ejpam-4276	425	7	):	):	PUNCT
ejpam-4276	425	8	let	let	VERB
ejpam-4276	425	9	u	u	PRON
ejpam-4276	425	10	be	be	AUX
ejpam-4276	425	11	a	a	DET
ejpam-4276	425	12	nonempty	nonempty	ADJ
ejpam-4276	425	13	(	(	PUNCT
ejpam-4276	425	14	λ	λ	NOUN
ejpam-4276	425	15	,	,	PUNCT
ejpam-4276	425	16	sp)-open	sp)-open	NOUN
ejpam-4276	425	17	set	set	VERB
ejpam-4276	425	18	such	such	ADJ
ejpam-4276	425	19	that	that	SCONJ
ejpam-4276	425	20	u	u	NOUN
ejpam-4276	425	21	∩	∩	NOUN
ejpam-4276	425	22	d	d	NOUN
ejpam-4276	425	23	=	=	SYM
ejpam-4276	425	24	∅	∅	NOUN
ejpam-4276	425	25	;	;	PUNCT
ejpam-4276	425	26	then	then	ADV
ejpam-4276	425	27	d	d	PROPN
ejpam-4276	425	28	⊆	⊆	NUM
ejpam-4276	425	29	x	x	SYM
ejpam-4276	425	30	−	−	PROPN
ejpam-4276	425	31	u	u	PROPN
ejpam-4276	425	32	6=	6=	PROPN
ejpam-4276	425	33	x	x	PROPN
ejpam-4276	425	34	,	,	PUNCT
ejpam-4276	425	35	which	which	PRON
ejpam-4276	425	36	contradicts	contradict	VERB
ejpam-4276	425	37	(	(	PUNCT
ejpam-4276	425	38	2	2	NUM
ejpam-4276	425	39	)	)	PUNCT
ejpam-4276	425	40	,	,	PUNCT
ejpam-4276	425	41	since	since	SCONJ
ejpam-4276	425	42	x	x	NUM
ejpam-4276	425	43	−	−	PROPN
ejpam-4276	425	44	u	u	NOUN
ejpam-4276	425	45	is	be	AUX
ejpam-4276	425	46	(	(	PUNCT
ejpam-4276	425	47	λ	λ	X
ejpam-4276	425	48	,	,	PUNCT
ejpam-4276	425	49	sp)-closed	sp)-close	VERB
ejpam-4276	425	50	.	.	PUNCT
ejpam-4276	426	1	(	(	PUNCT
ejpam-4276	426	2	3	3	X
ejpam-4276	426	3	)	)	PUNCT
ejpam-4276	426	4	⇒	⇒	NOUN
ejpam-4276	426	5	(	(	PUNCT
ejpam-4276	426	6	4	4	NUM
ejpam-4276	426	7	):	):	PUNCT
ejpam-4276	426	8	assume	assume	VERB
ejpam-4276	426	9	that	that	SCONJ
ejpam-4276	426	10	[	[	X
ejpam-4276	426	11	x−d](λ	x−d](λ	PROPN
ejpam-4276	426	12	,	,	PUNCT
ejpam-4276	426	13	sp	sp	NOUN
ejpam-4276	426	14	)	)	PUNCT
ejpam-4276	426	15	6=	6=	NOUN
ejpam-4276	426	16	∅	∅	NOUN
ejpam-4276	426	17	;	;	PUNCT
ejpam-4276	426	18	since	since	SCONJ
ejpam-4276	426	19	[	[	X
ejpam-4276	426	20	x−d](λ	x−d](λ	PROPN
ejpam-4276	426	21	,	,	PUNCT
ejpam-4276	426	22	sp	sp	NOUN
ejpam-4276	426	23	)	)	PUNCT
ejpam-4276	426	24	is	be	AUX
ejpam-4276	426	25	(	(	PUNCT
ejpam-4276	426	26	λ	λ	INTJ
ejpam-4276	426	27	,	,	PUNCT
ejpam-4276	426	28	sp)-open	sp)-open	ADJ
ejpam-4276	426	29	,	,	PUNCT
ejpam-4276	426	30	there	there	PRON
ejpam-4276	426	31	is	be	VERB
ejpam-4276	426	32	a	a	DET
ejpam-4276	426	33	nonempty	nonempty	ADJ
ejpam-4276	426	34	(	(	PUNCT
ejpam-4276	426	35	λ	λ	NOUN
ejpam-4276	426	36	,	,	PUNCT
ejpam-4276	426	37	sp)-open	sp)-open	NOUN
ejpam-4276	426	38	set	set	VERB
ejpam-4276	426	39	u	u	PRON
ejpam-4276	426	40	such	such	ADJ
ejpam-4276	426	41	that	that	SCONJ
ejpam-4276	426	42	u	u	PROPN
ejpam-4276	426	43	⊆	⊆	NUM
ejpam-4276	426	44	[	[	X
ejpam-4276	426	45	x−d](λ	x−d](λ	PROPN
ejpam-4276	426	46	,	,	PUNCT
ejpam-4276	426	47	sp	sp	NOUN
ejpam-4276	426	48	)	)	PUNCT
ejpam-4276	426	49	,	,	PUNCT
ejpam-4276	426	50	and	and	CCONJ
ejpam-4276	426	51	since	since	SCONJ
ejpam-4276	426	52	[	[	X
ejpam-4276	426	53	x−d](λ	x−d](λ	PROPN
ejpam-4276	426	54	,	,	PUNCT
ejpam-4276	426	55	sp	sp	NOUN
ejpam-4276	426	56	)	)	PUNCT
ejpam-4276	426	57	⊆	⊆	NUM
ejpam-4276	426	58	x−d	x−d	PROPN
ejpam-4276	426	59	,	,	PUNCT
ejpam-4276	426	60	u	u	NOUN
ejpam-4276	426	61	contains	contain	VERB
ejpam-4276	426	62	no	no	DET
ejpam-4276	426	63	point	point	NOUN
ejpam-4276	426	64	of	of	ADP
ejpam-4276	426	65	d.	d.	PROPN
ejpam-4276	426	66	(	(	PUNCT
ejpam-4276	426	67	3	3	NUM
ejpam-4276	426	68	)	)	PUNCT
ejpam-4276	426	69	⇒	⇒	NOUN
ejpam-4276	426	70	(	(	PUNCT
ejpam-4276	426	71	4	4	NUM
ejpam-4276	426	72	):	):	PUNCT
ejpam-4276	426	73	[	[	X
ejpam-4276	426	74	x	x	X
ejpam-4276	426	75	−d](λ	−d](λ	NOUN
ejpam-4276	426	76	,	,	PUNCT
ejpam-4276	426	77	sp	sp	NOUN
ejpam-4276	426	78	)	)	PUNCT
ejpam-4276	426	79	=	=	PUNCT
ejpam-4276	427	1	x	x	SYM
ejpam-4276	427	2	−d(λ	−d(λ	NOUN
ejpam-4276	427	3	,	,	PUNCT
ejpam-4276	427	4	sp	sp	NOUN
ejpam-4276	427	5	)	)	PUNCT
ejpam-4276	427	6	=	=	NOUN
ejpam-4276	427	7	∅	∅	NOUN
ejpam-4276	427	8	so	so	SCONJ
ejpam-4276	427	9	that	that	SCONJ
ejpam-4276	427	10	d(λ	d(λ	PROPN
ejpam-4276	427	11	,	,	PUNCT
ejpam-4276	427	12	sp	sp	NOUN
ejpam-4276	427	13	)	)	PUNCT
ejpam-4276	427	14	=	=	SYM
ejpam-4276	427	15	x.	x.	NOUN
ejpam-4276	427	16	remark	remark	VERB
ejpam-4276	427	17	4	4	NUM
ejpam-4276	427	18	.	.	PUNCT
ejpam-4276	428	1	let	let	VERB
ejpam-4276	428	2	a	a	DET
ejpam-4276	428	3	be	be	AUX
ejpam-4276	428	4	a	a	DET
ejpam-4276	428	5	subset	subset	NOUN
ejpam-4276	428	6	of	of	ADP
ejpam-4276	428	7	a	a	DET
ejpam-4276	428	8	topological	topological	ADJ
ejpam-4276	428	9	space	space	NOUN
ejpam-4276	428	10	(	(	PUNCT
ejpam-4276	428	11	x	x	X
ejpam-4276	428	12	,	,	PUNCT
ejpam-4276	428	13	τ	τ	PROPN
ejpam-4276	428	14	)	)	PUNCT
ejpam-4276	428	15	.	.	PUNCT
ejpam-4276	429	1	if	if	SCONJ
ejpam-4276	429	2	a	a	PRON
ejpam-4276	429	3	is	be	AUX
ejpam-4276	429	4	λsp	λsp	NOUN
ejpam-4276	429	5	-	-	PUNCT
ejpam-4276	429	6	dense	dense	ADJ
ejpam-4276	429	7	,	,	PUNCT
ejpam-4276	429	8	then	then	ADV
ejpam-4276	429	9	a	a	PRON
ejpam-4276	429	10	is	be	AUX
ejpam-4276	429	11	p(λ	p(λ	NOUN
ejpam-4276	429	12	,	,	PUNCT
ejpam-4276	429	13	sp)-open	sp)-open	NOUN
ejpam-4276	429	14	.	.	PUNCT
ejpam-4276	430	1	proposition	proposition	NOUN
ejpam-4276	430	2	21	21	NUM
ejpam-4276	430	3	.	.	PUNCT
ejpam-4276	431	1	let	let	VERB
ejpam-4276	431	2	a	a	DET
ejpam-4276	431	3	be	be	AUX
ejpam-4276	431	4	a	a	DET
ejpam-4276	431	5	subset	subset	NOUN
ejpam-4276	431	6	of	of	ADP
ejpam-4276	431	7	a	a	DET
ejpam-4276	431	8	topological	topological	ADJ
ejpam-4276	431	9	space	space	NOUN
ejpam-4276	431	10	(	(	PUNCT
ejpam-4276	431	11	x	x	X
ejpam-4276	431	12	,	,	PUNCT
ejpam-4276	431	13	τ	τ	PROPN
ejpam-4276	431	14	)	)	PUNCT
ejpam-4276	431	15	.	.	PUNCT
ejpam-4276	432	1	if	if	SCONJ
ejpam-4276	432	2	a	a	PRON
ejpam-4276	432	3	is	be	AUX
ejpam-4276	432	4	p(λ	p(λ	NOUN
ejpam-4276	432	5	,	,	PUNCT
ejpam-4276	432	6	sp)-open	sp)-open	NOUN
ejpam-4276	432	7	,	,	PUNCT
ejpam-4276	432	8	then	then	ADV
ejpam-4276	432	9	a	a	PRON
ejpam-4276	432	10	is	be	AUX
ejpam-4276	432	11	the	the	DET
ejpam-4276	432	12	intersection	intersection	NOUN
ejpam-4276	432	13	of	of	ADP
ejpam-4276	432	14	a	a	DET
ejpam-4276	432	15	r(λ	r(λ	NOUN
ejpam-4276	432	16	,	,	PUNCT
ejpam-4276	432	17	sp)-open	sp)-open	NOUN
ejpam-4276	432	18	set	set	NOUN
ejpam-4276	432	19	and	and	CCONJ
ejpam-4276	432	20	a	a	DET
ejpam-4276	432	21	λsp	λsp	ADV
ejpam-4276	432	22	-	-	PUNCT
ejpam-4276	432	23	dense	dense	ADJ
ejpam-4276	432	24	set	set	NOUN
ejpam-4276	432	25	.	.	PUNCT
ejpam-4276	433	1	proof	proof	NOUN
ejpam-4276	433	2	.	.	PUNCT
ejpam-4276	434	1	suppose	suppose	VERB
ejpam-4276	434	2	that	that	SCONJ
ejpam-4276	434	3	a	a	PRON
ejpam-4276	434	4	is	be	AUX
ejpam-4276	434	5	a	a	DET
ejpam-4276	434	6	p(λ	p(λ	NOUN
ejpam-4276	434	7	,	,	PUNCT
ejpam-4276	434	8	sp)-open	sp)-open	ADJ
ejpam-4276	434	9	set	set	NOUN
ejpam-4276	434	10	.	.	PUNCT
ejpam-4276	435	1	then	then	ADV
ejpam-4276	435	2	,	,	PUNCT
ejpam-4276	435	3	we	we	PRON
ejpam-4276	435	4	have	have	VERB
ejpam-4276	435	5	a	a	DET
ejpam-4276	435	6	⊆	⊆	NUM
ejpam-4276	435	7	[	[	X
ejpam-4276	435	8	a(λ	a(λ	ADJ
ejpam-4276	435	9	,	,	PUNCT
ejpam-4276	435	10	sp)](λ	sp)](λ	PROPN
ejpam-4276	435	11	,	,	PUNCT
ejpam-4276	435	12	sp	sp	NOUN
ejpam-4276	435	13	)	)	PUNCT
ejpam-4276	435	14	and	and	CCONJ
ejpam-4276	435	15	hence	hence	ADV
ejpam-4276	435	16	a	a	PRON
ejpam-4276	435	17	=	=	X
ejpam-4276	436	1	[	[	X
ejpam-4276	436	2	a	a	DET
ejpam-4276	436	3	∪	∪	ADJ
ejpam-4276	436	4	[	[	X
ejpam-4276	436	5	x	x	X
ejpam-4276	436	6	−	−	NOUN
ejpam-4276	436	7	a(λ	a(λ	ADV
ejpam-4276	436	8	,	,	PUNCT
ejpam-4276	436	9	sp	sp	NOUN
ejpam-4276	436	10	)	)	PUNCT
ejpam-4276	436	11	]	]	PUNCT
ejpam-4276	436	12	]	]	X
ejpam-4276	436	13	∩	∩	NOUN
ejpam-4276	436	14	[	[	X
ejpam-4276	436	15	a(λ	a(λ	PROPN
ejpam-4276	436	16	,	,	PUNCT
ejpam-4276	436	17	sp)](λ	sp)](λ	PROPN
ejpam-4276	436	18	,	,	PUNCT
ejpam-4276	436	19	sp	sp	NOUN
ejpam-4276	436	20	)	)	PUNCT
ejpam-4276	436	21	.	.	PUNCT
ejpam-4276	437	1	let	let	VERB
ejpam-4276	437	2	c	c	NOUN
ejpam-4276	438	1	=	=	PUNCT
ejpam-4276	439	1	[	[	X
ejpam-4276	439	2	a(λ	a(λ	ADV
ejpam-4276	439	3	,	,	PUNCT
ejpam-4276	439	4	sp)](λ	sp)](λ	PROPN
ejpam-4276	439	5	,	,	PUNCT
ejpam-4276	439	6	sp	sp	NOUN
ejpam-4276	439	7	)	)	PUNCT
ejpam-4276	439	8	and	and	CCONJ
ejpam-4276	439	9	d	d	NOUN
ejpam-4276	439	10	=	=	NOUN
ejpam-4276	439	11	a	a	DET
ejpam-4276	439	12	∪	∪	ADJ
ejpam-4276	439	13	[	[	X
ejpam-4276	439	14	x	x	X
ejpam-4276	439	15	−	−	NOUN
ejpam-4276	439	16	a(λ	a(λ	ADV
ejpam-4276	439	17	,	,	PUNCT
ejpam-4276	439	18	sp	sp	NOUN
ejpam-4276	439	19	)	)	PUNCT
ejpam-4276	439	20	]	]	PUNCT
ejpam-4276	439	21	.	.	PUNCT
ejpam-4276	440	1	then	then	ADV
ejpam-4276	440	2	,	,	PUNCT
ejpam-4276	440	3	c	c	PROPN
ejpam-4276	440	4	is	be	AUX
ejpam-4276	440	5	r(λ	r(λ	NOUN
ejpam-4276	440	6	,	,	PUNCT
ejpam-4276	440	7	sp)-open	sp)-open	ADJ
ejpam-4276	440	8	,	,	PUNCT
ejpam-4276	440	9	by	by	ADP
ejpam-4276	440	10	proposition	proposition	NOUN
ejpam-4276	440	11	2	2	NUM
ejpam-4276	440	12	,	,	PUNCT
ejpam-4276	440	13	a(λ	a(λ	ADV
ejpam-4276	440	14	,	,	PUNCT
ejpam-4276	440	15	sp	sp	NOUN
ejpam-4276	440	16	)	)	PUNCT
ejpam-4276	440	17	⊆	⊆	NUM
ejpam-4276	440	18	d(λ	d(λ	PROPN
ejpam-4276	440	19	,	,	PUNCT
ejpam-4276	440	20	sp	sp	NOUN
ejpam-4276	440	21	)	)	PUNCT
ejpam-4276	440	22	since	since	SCONJ
ejpam-4276	440	23	a	a	DET
ejpam-4276	440	24	⊆	⊆	NUM
ejpam-4276	440	25	d	d	PROPN
ejpam-4276	440	26	and	and	CCONJ
ejpam-4276	440	27	x	x	SYM
ejpam-4276	440	28	−a(λ	−a(λ	NOUN
ejpam-4276	440	29	,	,	PUNCT
ejpam-4276	440	30	sp	sp	NOUN
ejpam-4276	440	31	)	)	PUNCT
ejpam-4276	440	32	⊆	⊆	NUM
ejpam-4276	440	33	d	d	NOUN
ejpam-4276	440	34	⊆	⊆	NUM
ejpam-4276	440	35	d(λ	d(λ	PROPN
ejpam-4276	440	36	,	,	PUNCT
ejpam-4276	440	37	sp	sp	NOUN
ejpam-4276	440	38	)	)	PUNCT
ejpam-4276	440	39	.	.	PUNCT
ejpam-4276	441	1	thus	thus	ADV
ejpam-4276	441	2	,	,	PUNCT
ejpam-4276	441	3	d(λ	d(λ	PROPN
ejpam-4276	441	4	,	,	PUNCT
ejpam-4276	441	5	sp	sp	NOUN
ejpam-4276	441	6	)	)	PUNCT
ejpam-4276	441	7	=	=	SYM
ejpam-4276	441	8	x.	x.	NOUN
ejpam-4276	441	9	corollary	corollary	NOUN
ejpam-4276	441	10	5	5	X
ejpam-4276	441	11	.	.	PUNCT
ejpam-4276	442	1	let	let	VERB
ejpam-4276	442	2	a	a	DET
ejpam-4276	442	3	be	be	AUX
ejpam-4276	442	4	a	a	DET
ejpam-4276	442	5	subset	subset	NOUN
ejpam-4276	442	6	of	of	ADP
ejpam-4276	442	7	a	a	DET
ejpam-4276	442	8	topological	topological	ADJ
ejpam-4276	442	9	space	space	NOUN
ejpam-4276	442	10	(	(	PUNCT
ejpam-4276	442	11	x	x	X
ejpam-4276	442	12	,	,	PUNCT
ejpam-4276	442	13	τ	τ	PROPN
ejpam-4276	442	14	)	)	PUNCT
ejpam-4276	442	15	.	.	PUNCT
ejpam-4276	443	1	if	if	SCONJ
ejpam-4276	443	2	a	a	PRON
ejpam-4276	443	3	is	be	AUX
ejpam-4276	443	4	p(λ	p(λ	NOUN
ejpam-4276	443	5	,	,	PUNCT
ejpam-4276	443	6	sp)-closed	sp)-close	VERB
ejpam-4276	443	7	,	,	PUNCT
ejpam-4276	443	8	then	then	ADV
ejpam-4276	443	9	a	a	PRON
ejpam-4276	443	10	is	be	AUX
ejpam-4276	443	11	the	the	DET
ejpam-4276	443	12	union	union	NOUN
ejpam-4276	443	13	of	of	ADP
ejpam-4276	443	14	a	a	DET
ejpam-4276	443	15	r(λ	r(λ	NOUN
ejpam-4276	443	16	,	,	PUNCT
ejpam-4276	443	17	sp)-closed	sp)-close	VERB
ejpam-4276	443	18	set	set	ADJ
ejpam-4276	443	19	and	and	CCONJ
ejpam-4276	443	20	a	a	DET
ejpam-4276	443	21	set	set	NOUN
ejpam-4276	443	22	has	have	VERB
ejpam-4276	443	23	empty	empty	ADJ
ejpam-4276	443	24	(	(	PUNCT
ejpam-4276	443	25	λ	λ	NOUN
ejpam-4276	443	26	,	,	PUNCT
ejpam-4276	443	27	sp)-interior	sp)-interior	NOUN
ejpam-4276	443	28	.	.	PUNCT
ejpam-4276	444	1	proposition	proposition	NOUN
ejpam-4276	444	2	22	22	NUM
ejpam-4276	444	3	.	.	PUNCT
ejpam-4276	445	1	let	let	VERB
ejpam-4276	445	2	a	a	DET
ejpam-4276	445	3	be	be	AUX
ejpam-4276	445	4	a	a	DET
ejpam-4276	445	5	subset	subset	NOUN
ejpam-4276	445	6	of	of	ADP
ejpam-4276	445	7	a	a	DET
ejpam-4276	445	8	topological	topological	ADJ
ejpam-4276	445	9	space	space	NOUN
ejpam-4276	445	10	(	(	PUNCT
ejpam-4276	445	11	x	x	X
ejpam-4276	445	12	,	,	PUNCT
ejpam-4276	445	13	τ	τ	PROPN
ejpam-4276	445	14	)	)	PUNCT
ejpam-4276	445	15	.	.	PUNCT
ejpam-4276	446	1	if	if	SCONJ
ejpam-4276	446	2	a	a	PRON
ejpam-4276	446	3	is	be	AUX
ejpam-4276	446	4	s(λ	s(λ	PROPN
ejpam-4276	446	5	,	,	PUNCT
ejpam-4276	446	6	sp)-open	sp)-open	ADJ
ejpam-4276	446	7	,	,	PUNCT
ejpam-4276	446	8	then	then	ADV
ejpam-4276	446	9	a	a	PRON
ejpam-4276	446	10	is	be	AUX
ejpam-4276	446	11	the	the	DET
ejpam-4276	446	12	intersection	intersection	NOUN
ejpam-4276	446	13	of	of	ADP
ejpam-4276	446	14	a	a	DET
ejpam-4276	446	15	r(λ	r(λ	NOUN
ejpam-4276	446	16	,	,	PUNCT
ejpam-4276	446	17	sp)-closed	sp)-close	VERB
ejpam-4276	446	18	set	set	VERB
ejpam-4276	446	19	f	f	PROPN
ejpam-4276	446	20	and	and	CCONJ
ejpam-4276	446	21	a	a	DET
ejpam-4276	446	22	set	set	NOUN
ejpam-4276	446	23	c	c	NOUN
ejpam-4276	446	24	such	such	ADJ
ejpam-4276	446	25	that	that	SCONJ
ejpam-4276	446	26	c(λ	c(λ	PROPN
ejpam-4276	446	27	,	,	PUNCT
ejpam-4276	446	28	sp	sp	NOUN
ejpam-4276	446	29	)	)	PUNCT
ejpam-4276	446	30	is	be	AUX
ejpam-4276	446	31	λsp	λsp	NOUN
ejpam-4276	446	32	-	-	PUNCT
ejpam-4276	446	33	dense	dense	ADJ
ejpam-4276	446	34	.	.	PUNCT
ejpam-4276	447	1	proof	proof	NOUN
ejpam-4276	447	2	.	.	PUNCT
ejpam-4276	448	1	suppose	suppose	VERB
ejpam-4276	448	2	that	that	SCONJ
ejpam-4276	448	3	a	a	PRON
ejpam-4276	448	4	is	be	AUX
ejpam-4276	448	5	s(λ	s(λ	NOUN
ejpam-4276	448	6	,	,	PUNCT
ejpam-4276	448	7	sp)-open	sp)-open	NOUN
ejpam-4276	448	8	.	.	PUNCT
ejpam-4276	449	1	then	then	ADV
ejpam-4276	449	2	,	,	PUNCT
ejpam-4276	449	3	we	we	PRON
ejpam-4276	449	4	have	have	VERB
ejpam-4276	449	5	a	a	DET
ejpam-4276	449	6	⊆	⊆	NUM
ejpam-4276	449	7	[	[	X
ejpam-4276	449	8	a(λ	a(λ	ADV
ejpam-4276	449	9	,	,	PUNCT
ejpam-4276	449	10	sp	sp	NOUN
ejpam-4276	449	11	)	)	PUNCT
ejpam-4276	449	12	]	]	PUNCT
ejpam-4276	449	13	(	(	PUNCT
ejpam-4276	449	14	λ	λ	NOUN
ejpam-4276	449	15	,	,	PUNCT
ejpam-4276	449	16	sp	sp	NOUN
ejpam-4276	449	17	)	)	PUNCT
ejpam-4276	449	18	and	and	CCONJ
ejpam-4276	449	19	hence	hence	ADV
ejpam-4276	449	20	a	a	PRON
ejpam-4276	449	21	=	=	X
ejpam-4276	450	1	[	[	X
ejpam-4276	450	2	a	a	DET
ejpam-4276	450	3	∪	∪	ADJ
ejpam-4276	450	4	[	[	X
ejpam-4276	450	5	x	x	X
ejpam-4276	450	6	−	−	PROPN
ejpam-4276	450	7	[	[	X
ejpam-4276	450	8	a(λ	a(λ	ADV
ejpam-4276	450	9	,	,	PUNCT
ejpam-4276	450	10	sp	sp	NOUN
ejpam-4276	450	11	)	)	PUNCT
ejpam-4276	450	12	]	]	PUNCT
ejpam-4276	450	13	(	(	PUNCT
ejpam-4276	450	14	λ	λ	NOUN
ejpam-4276	450	15	,	,	PUNCT
ejpam-4276	450	16	sp	sp	NOUN
ejpam-4276	450	17	)	)	PUNCT
ejpam-4276	450	18	]	]	PUNCT
ejpam-4276	450	19	]	]	X
ejpam-4276	450	20	∩	∩	NOUN
ejpam-4276	450	21	[	[	X
ejpam-4276	450	22	a(λ	a(λ	ADV
ejpam-4276	450	23	,	,	PUNCT
ejpam-4276	450	24	sp	sp	NOUN
ejpam-4276	450	25	)	)	PUNCT
ejpam-4276	450	26	]	]	PUNCT
ejpam-4276	450	27	(	(	PUNCT
ejpam-4276	450	28	λ	λ	NOUN
ejpam-4276	450	29	,	,	PUNCT
ejpam-4276	450	30	sp	sp	NOUN
ejpam-4276	450	31	)	)	PUNCT
ejpam-4276	450	32	.	.	PUNCT
ejpam-4276	451	1	let	let	VERB
ejpam-4276	451	2	f	f	NOUN
ejpam-4276	451	3	=	=	PUNCT
ejpam-4276	452	1	[	[	X
ejpam-4276	452	2	a(λ	a(λ	ADV
ejpam-4276	452	3	,	,	PUNCT
ejpam-4276	452	4	sp	sp	NOUN
ejpam-4276	452	5	)	)	PUNCT
ejpam-4276	452	6	]	]	PUNCT
ejpam-4276	453	1	(	(	PUNCT
ejpam-4276	453	2	λ	λ	NOUN
ejpam-4276	453	3	,	,	PUNCT
ejpam-4276	453	4	sp	sp	NOUN
ejpam-4276	453	5	)	)	PUNCT
ejpam-4276	453	6	and	and	CCONJ
ejpam-4276	453	7	c	c	X
ejpam-4276	453	8	=	=	NOUN
ejpam-4276	453	9	a	a	DET
ejpam-4276	453	10	∪	∪	X
ejpam-4276	453	11	[	[	X
ejpam-4276	453	12	x	x	X
ejpam-4276	453	13	−	−	PROPN
ejpam-4276	454	1	[	[	X
ejpam-4276	454	2	a(λ	a(λ	ADV
ejpam-4276	454	3	,	,	PUNCT
ejpam-4276	454	4	sp	sp	NOUN
ejpam-4276	454	5	)	)	PUNCT
ejpam-4276	454	6	]	]	PUNCT
ejpam-4276	455	1	(	(	PUNCT
ejpam-4276	455	2	λ	λ	NOUN
ejpam-4276	455	3	,	,	PUNCT
ejpam-4276	455	4	sp	sp	NOUN
ejpam-4276	455	5	)	)	PUNCT
ejpam-4276	455	6	]	]	PUNCT
ejpam-4276	455	7	.	.	PUNCT
ejpam-4276	456	1	then	then	ADV
ejpam-4276	456	2	,	,	PUNCT
ejpam-4276	456	3	f	f	PROPN
ejpam-4276	456	4	is	be	AUX
ejpam-4276	456	5	r(λ	r(λ	NOUN
ejpam-4276	456	6	,	,	PUNCT
ejpam-4276	456	7	sp)-closed	sp)-close	VERB
ejpam-4276	456	8	,	,	PUNCT
ejpam-4276	456	9	by	by	ADP
ejpam-4276	456	10	proposition	proposition	NOUN
ejpam-4276	456	11	2	2	NUM
ejpam-4276	456	12	,	,	PUNCT
ejpam-4276	456	13	we	we	PRON
ejpam-4276	456	14	have	have	VERB
ejpam-4276	456	15	[	[	X
ejpam-4276	456	16	a(λ	a(λ	ADV
ejpam-4276	456	17	,	,	PUNCT
ejpam-4276	456	18	sp	sp	NOUN
ejpam-4276	456	19	)	)	PUNCT
ejpam-4276	456	20	]	]	PUNCT
ejpam-4276	457	1	(	(	PUNCT
ejpam-4276	457	2	λ	λ	NOUN
ejpam-4276	457	3	,	,	PUNCT
ejpam-4276	457	4	sp	sp	NOUN
ejpam-4276	457	5	)	)	PUNCT
ejpam-4276	457	6	⊆	⊆	NUM
ejpam-4276	458	1	[	[	X
ejpam-4276	458	2	c(λ	c(λ	PROPN
ejpam-4276	458	3	,	,	PUNCT
ejpam-4276	458	4	sp	sp	NOUN
ejpam-4276	458	5	)	)	PUNCT
ejpam-4276	458	6	]	]	PUNCT
ejpam-4276	459	1	(	(	PUNCT
ejpam-4276	459	2	λ	λ	NOUN
ejpam-4276	459	3	,	,	PUNCT
ejpam-4276	459	4	sp	sp	NOUN
ejpam-4276	459	5	)	)	PUNCT
ejpam-4276	459	6	.	.	PUNCT
ejpam-4276	460	1	since	since	SCONJ
ejpam-4276	460	2	x−	x−	PROPN
ejpam-4276	460	3	[	[	X
ejpam-4276	460	4	a(λ	a(λ	ADV
ejpam-4276	460	5	,	,	PUNCT
ejpam-4276	460	6	sp	sp	NOUN
ejpam-4276	460	7	)	)	PUNCT
ejpam-4276	460	8	]	]	PUNCT
ejpam-4276	460	9	(	(	PUNCT
ejpam-4276	460	10	λ	λ	NOUN
ejpam-4276	460	11	,	,	PUNCT
ejpam-4276	460	12	sp	sp	NOUN
ejpam-4276	460	13	)	)	PUNCT
ejpam-4276	460	14	⊆	⊆	NUM
ejpam-4276	460	15	c	c	NOUN
ejpam-4276	460	16	and	and	CCONJ
ejpam-4276	460	17	x−	x−	NOUN
ejpam-4276	460	18	[	[	X
ejpam-4276	460	19	a(λ	a(λ	ADV
ejpam-4276	460	20	,	,	PUNCT
ejpam-4276	460	21	sp	sp	NOUN
ejpam-4276	460	22	)	)	PUNCT
ejpam-4276	460	23	]	]	PUNCT
ejpam-4276	460	24	(	(	PUNCT
ejpam-4276	460	25	λ	λ	NOUN
ejpam-4276	460	26	,	,	PUNCT
ejpam-4276	460	27	sp	sp	NOUN
ejpam-4276	460	28	)	)	PUNCT
ejpam-4276	460	29	is	be	AUX
ejpam-4276	460	30	(	(	PUNCT
ejpam-4276	460	31	λ	λ	INTJ
ejpam-4276	460	32	,	,	PUNCT
ejpam-4276	460	33	sp)-open	sp)-open	ADJ
ejpam-4276	460	34	,	,	PUNCT
ejpam-4276	460	35	x−	x−	PROPN
ejpam-4276	461	1	[	[	X
ejpam-4276	461	2	a(λ	a(λ	ADV
ejpam-4276	461	3	,	,	PUNCT
ejpam-4276	461	4	sp	sp	NOUN
ejpam-4276	461	5	)	)	PUNCT
ejpam-4276	461	6	]	]	PUNCT
ejpam-4276	461	7	(	(	PUNCT
ejpam-4276	461	8	λ	λ	NOUN
ejpam-4276	461	9	,	,	PUNCT
ejpam-4276	461	10	sp	sp	NOUN
ejpam-4276	461	11	)	)	PUNCT
ejpam-4276	461	12	⊆	⊆	NUM
ejpam-4276	461	13	c(λ	c(λ	PROPN
ejpam-4276	461	14	,	,	PUNCT
ejpam-4276	461	15	sp	sp	NOUN
ejpam-4276	461	16	)	)	PUNCT
ejpam-4276	461	17	⊆	⊆	NUM
ejpam-4276	462	1	[	[	X
ejpam-4276	462	2	c(λ	c(λ	PROPN
ejpam-4276	462	3	,	,	PUNCT
ejpam-4276	462	4	sp	sp	NOUN
ejpam-4276	462	5	)	)	PUNCT
ejpam-4276	462	6	]	]	PUNCT
ejpam-4276	463	1	(	(	PUNCT
ejpam-4276	463	2	λ	λ	NOUN
ejpam-4276	463	3	,	,	PUNCT
ejpam-4276	463	4	sp	sp	NOUN
ejpam-4276	463	5	)	)	PUNCT
ejpam-4276	463	6	.	.	PUNCT
ejpam-4276	464	1	thus	thus	ADV
ejpam-4276	464	2	,	,	PUNCT
ejpam-4276	464	3	[	[	X
ejpam-4276	464	4	c(λ	c(λ	PROPN
ejpam-4276	464	5	,	,	PUNCT
ejpam-4276	464	6	sp	sp	NOUN
ejpam-4276	464	7	)	)	PUNCT
ejpam-4276	464	8	]	]	PUNCT
ejpam-4276	465	1	(	(	PUNCT
ejpam-4276	465	2	λ	λ	NOUN
ejpam-4276	465	3	,	,	PUNCT
ejpam-4276	465	4	sp	sp	NOUN
ejpam-4276	465	5	)	)	PUNCT
ejpam-4276	465	6	=	=	SYM
ejpam-4276	465	7	x.	x.	NOUN
ejpam-4276	465	8	corollary	corollary	NOUN
ejpam-4276	465	9	6	6	NUM
ejpam-4276	465	10	.	.	PUNCT
ejpam-4276	466	1	let	let	VERB
ejpam-4276	466	2	a	a	DET
ejpam-4276	466	3	be	be	AUX
ejpam-4276	466	4	a	a	DET
ejpam-4276	466	5	subset	subset	NOUN
ejpam-4276	466	6	of	of	ADP
ejpam-4276	466	7	a	a	DET
ejpam-4276	466	8	topological	topological	ADJ
ejpam-4276	466	9	space	space	NOUN
ejpam-4276	466	10	(	(	PUNCT
ejpam-4276	466	11	x	x	X
ejpam-4276	466	12	,	,	PUNCT
ejpam-4276	466	13	τ	τ	PROPN
ejpam-4276	466	14	)	)	PUNCT
ejpam-4276	466	15	.	.	PUNCT
ejpam-4276	467	1	if	if	SCONJ
ejpam-4276	467	2	a	a	PRON
ejpam-4276	467	3	is	be	AUX
ejpam-4276	467	4	s(λ	s(λ	PROPN
ejpam-4276	467	5	,	,	PUNCT
ejpam-4276	467	6	sp)-closed	sp)-close	VERB
ejpam-4276	467	7	,	,	PUNCT
ejpam-4276	467	8	then	then	ADV
ejpam-4276	467	9	a	a	PRON
ejpam-4276	467	10	is	be	AUX
ejpam-4276	467	11	the	the	DET
ejpam-4276	467	12	union	union	NOUN
ejpam-4276	467	13	of	of	ADP
ejpam-4276	467	14	a	a	DET
ejpam-4276	467	15	r(λ	r(λ	NOUN
ejpam-4276	467	16	,	,	PUNCT
ejpam-4276	467	17	sp)-open	sp)-open	NOUN
ejpam-4276	467	18	set	set	NOUN
ejpam-4276	467	19	and	and	CCONJ
ejpam-4276	467	20	a	a	DET
ejpam-4276	467	21	set	set	NOUN
ejpam-4276	467	22	whose	whose	DET
ejpam-4276	467	23	(	(	PUNCT
ejpam-4276	467	24	λ	λ	PROPN
ejpam-4276	467	25	,	,	PUNCT
ejpam-4276	467	26	sp)-closure	sp)-closure	NOUN
ejpam-4276	467	27	has	have	AUX
ejpam-4276	467	28	empty	empty	ADJ
ejpam-4276	467	29	(	(	PUNCT
ejpam-4276	467	30	λ	λ	NOUN
ejpam-4276	467	31	,	,	PUNCT
ejpam-4276	467	32	sp)-interior	sp)-interior	NOUN
ejpam-4276	467	33	.	.	PUNCT
ejpam-4276	468	1	c.	c.	PROPN
ejpam-4276	468	2	boonpok	boonpok	PROPN
ejpam-4276	468	3	,	,	PUNCT
ejpam-4276	468	4	j.	j.	PROPN
ejpam-4276	468	5	khampakdee	khampakdee	PROPN
ejpam-4276	468	6	/	/	PUNCT
ejpam-4276	468	7	eur	eur	PROPN
ejpam-4276	468	8	.	.	PUNCT
ejpam-4276	469	1	j.	j.	PROPN
ejpam-4276	469	2	pure	pure	PROPN
ejpam-4276	469	3	appl	appl	PROPN
ejpam-4276	469	4	.	.	PROPN
ejpam-4276	469	5	math	math	PROPN
ejpam-4276	469	6	,	,	PUNCT
ejpam-4276	469	7	15	15	NUM
ejpam-4276	469	8	(	(	PUNCT
ejpam-4276	469	9	2	2	NUM
ejpam-4276	469	10	)	)	PUNCT
ejpam-4276	469	11	(	(	PUNCT
ejpam-4276	469	12	2022	2022	NUM
ejpam-4276	469	13	)	)	PUNCT
ejpam-4276	469	14	,	,	PUNCT
ejpam-4276	469	15	572	572	NUM
ejpam-4276	469	16	-	-	SYM
ejpam-4276	469	17	588	588	NUM
ejpam-4276	469	18	584	584	NUM
ejpam-4276	469	19	proposition	proposition	NOUN
ejpam-4276	469	20	23	23	NUM
ejpam-4276	469	21	.	.	PUNCT
ejpam-4276	470	1	let	let	VERB
ejpam-4276	470	2	a	a	DET
ejpam-4276	470	3	be	be	AUX
ejpam-4276	470	4	a	a	DET
ejpam-4276	470	5	subset	subset	NOUN
ejpam-4276	470	6	of	of	ADP
ejpam-4276	470	7	a	a	DET
ejpam-4276	470	8	topological	topological	ADJ
ejpam-4276	470	9	space	space	NOUN
ejpam-4276	470	10	(	(	PUNCT
ejpam-4276	470	11	x	x	X
ejpam-4276	470	12	,	,	PUNCT
ejpam-4276	470	13	τ	τ	PROPN
ejpam-4276	470	14	)	)	PUNCT
ejpam-4276	470	15	.	.	PUNCT
ejpam-4276	471	1	if	if	SCONJ
ejpam-4276	471	2	a	a	PRON
ejpam-4276	471	3	is	be	AUX
ejpam-4276	471	4	β(λ	β(λ	NOUN
ejpam-4276	471	5	,	,	PUNCT
ejpam-4276	471	6	sp)-open	sp)-open	ADJ
ejpam-4276	471	7	,	,	PUNCT
ejpam-4276	471	8	then	then	ADV
ejpam-4276	471	9	a	a	PRON
ejpam-4276	471	10	is	be	AUX
ejpam-4276	471	11	the	the	DET
ejpam-4276	471	12	intersection	intersection	NOUN
ejpam-4276	471	13	of	of	ADP
ejpam-4276	471	14	a	a	DET
ejpam-4276	471	15	r(λ	r(λ	NOUN
ejpam-4276	471	16	,	,	PUNCT
ejpam-4276	471	17	sp)-closed	sp)-close	VERB
ejpam-4276	471	18	set	set	VERB
ejpam-4276	471	19	f	f	PROPN
ejpam-4276	471	20	and	and	CCONJ
ejpam-4276	471	21	a	a	DET
ejpam-4276	471	22	λsp	λsp	ADV
ejpam-4276	471	23	-	-	PUNCT
ejpam-4276	471	24	dense	dense	ADJ
ejpam-4276	471	25	set	set	VERB
ejpam-4276	471	26	d.	d.	PROPN
ejpam-4276	471	27	proof	proof	NOUN
ejpam-4276	471	28	.	.	PUNCT
ejpam-4276	472	1	suppose	suppose	VERB
ejpam-4276	472	2	that	that	SCONJ
ejpam-4276	472	3	a	a	PRON
ejpam-4276	472	4	is	be	AUX
ejpam-4276	472	5	β(λ	β(λ	NOUN
ejpam-4276	472	6	,	,	PUNCT
ejpam-4276	472	7	sp)-open	sp)-open	NOUN
ejpam-4276	472	8	.	.	PUNCT
ejpam-4276	473	1	then	then	ADV
ejpam-4276	473	2	,	,	PUNCT
ejpam-4276	473	3	we	we	PRON
ejpam-4276	473	4	have	have	VERB
ejpam-4276	473	5	a	a	DET
ejpam-4276	473	6	⊆	⊆	NUM
ejpam-4276	473	7	[	[	X
ejpam-4276	473	8	[	[	X
ejpam-4276	473	9	a(λ	a(λ	ADJ
ejpam-4276	473	10	,	,	PUNCT
ejpam-4276	473	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	473	12	,	,	PUNCT
ejpam-4276	473	13	sp	sp	NOUN
ejpam-4276	473	14	)	)	PUNCT
ejpam-4276	473	15	]	]	PUNCT
ejpam-4276	473	16	(	(	PUNCT
ejpam-4276	473	17	λ	λ	NOUN
ejpam-4276	473	18	,	,	PUNCT
ejpam-4276	473	19	sp	sp	NOUN
ejpam-4276	473	20	)	)	PUNCT
ejpam-4276	473	21	and	and	CCONJ
ejpam-4276	473	22	hence	hence	ADV
ejpam-4276	473	23	a	a	PRON
ejpam-4276	473	24	=	=	X
ejpam-4276	474	1	[	[	X
ejpam-4276	474	2	a	a	DET
ejpam-4276	474	3	∪	∪	ADJ
ejpam-4276	474	4	[	[	X
ejpam-4276	474	5	x	x	X
ejpam-4276	474	6	−	−	NOUN
ejpam-4276	474	7	a(λ	a(λ	ADV
ejpam-4276	474	8	,	,	PUNCT
ejpam-4276	474	9	sp	sp	NOUN
ejpam-4276	474	10	)	)	PUNCT
ejpam-4276	474	11	]	]	PUNCT
ejpam-4276	474	12	]	]	PUNCT
ejpam-4276	474	13	∩	∩	NOUN
ejpam-4276	474	14	[	[	X
ejpam-4276	474	15	[	[	X
ejpam-4276	474	16	a(λ	a(λ	ADJ
ejpam-4276	474	17	,	,	PUNCT
ejpam-4276	474	18	sp)](λ	sp)](λ	PROPN
ejpam-4276	474	19	,	,	PUNCT
ejpam-4276	474	20	sp	sp	NOUN
ejpam-4276	474	21	)	)	PUNCT
ejpam-4276	474	22	]	]	PUNCT
ejpam-4276	474	23	(	(	PUNCT
ejpam-4276	474	24	λ	λ	NOUN
ejpam-4276	474	25	,	,	PUNCT
ejpam-4276	474	26	sp	sp	NOUN
ejpam-4276	474	27	)	)	PUNCT
ejpam-4276	474	28	.	.	PUNCT
ejpam-4276	475	1	let	let	VERB
ejpam-4276	475	2	f	f	NOUN
ejpam-4276	476	1	=	=	PUNCT
ejpam-4276	477	1	[	[	X
ejpam-4276	477	2	[	[	X
ejpam-4276	477	3	a(λ	a(λ	ADJ
ejpam-4276	477	4	,	,	PUNCT
ejpam-4276	477	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	477	6	,	,	PUNCT
ejpam-4276	477	7	sp	sp	NOUN
ejpam-4276	477	8	)	)	PUNCT
ejpam-4276	477	9	]	]	PUNCT
ejpam-4276	478	1	(	(	PUNCT
ejpam-4276	478	2	λ	λ	NOUN
ejpam-4276	478	3	,	,	PUNCT
ejpam-4276	478	4	sp	sp	NOUN
ejpam-4276	478	5	)	)	PUNCT
ejpam-4276	478	6	and	and	CCONJ
ejpam-4276	478	7	d	d	NOUN
ejpam-4276	478	8	=	=	SYM
ejpam-4276	478	9	a∪	a∪	PROPN
ejpam-4276	479	1	[	[	X
ejpam-4276	479	2	x	x	X
ejpam-4276	479	3	−a(λ	−a(λ	NOUN
ejpam-4276	479	4	,	,	PUNCT
ejpam-4276	479	5	sp	sp	NOUN
ejpam-4276	479	6	)	)	PUNCT
ejpam-4276	479	7	]	]	PUNCT
ejpam-4276	479	8	.	.	PUNCT
ejpam-4276	480	1	then	then	ADV
ejpam-4276	480	2	,	,	PUNCT
ejpam-4276	480	3	f	f	PROPN
ejpam-4276	480	4	is	be	AUX
ejpam-4276	480	5	r(λ	r(λ	PROPN
ejpam-4276	480	6	,	,	PUNCT
ejpam-4276	480	7	sp)-closed	sp)-close	VERB
ejpam-4276	480	8	by	by	ADP
ejpam-4276	480	9	proposition	proposition	NOUN
ejpam-4276	480	10	2	2	NUM
ejpam-4276	480	11	,	,	PUNCT
ejpam-4276	480	12	also	also	ADV
ejpam-4276	480	13	a(λ	a(λ	ADV
ejpam-4276	480	14	,	,	PUNCT
ejpam-4276	480	15	sp	sp	NOUN
ejpam-4276	480	16	)	)	PUNCT
ejpam-4276	480	17	⊆	⊆	NUM
ejpam-4276	480	18	d(λ	d(λ	PROPN
ejpam-4276	480	19	,	,	PUNCT
ejpam-4276	480	20	sp	sp	NOUN
ejpam-4276	480	21	)	)	PUNCT
ejpam-4276	480	22	.	.	PUNCT
ejpam-4276	481	1	since	since	SCONJ
ejpam-4276	481	2	x	x	PROPN
ejpam-4276	481	3	−a(λ	−a(λ	NOUN
ejpam-4276	481	4	,	,	PUNCT
ejpam-4276	481	5	sp	sp	NOUN
ejpam-4276	481	6	)	)	PUNCT
ejpam-4276	481	7	⊆	⊆	NUM
ejpam-4276	481	8	d	d	NOUN
ejpam-4276	481	9	⊆	⊆	NUM
ejpam-4276	481	10	d(λ	d(λ	PROPN
ejpam-4276	481	11	,	,	PUNCT
ejpam-4276	481	12	sp	sp	NOUN
ejpam-4276	481	13	)	)	PUNCT
ejpam-4276	481	14	,	,	PUNCT
ejpam-4276	481	15	we	we	PRON
ejpam-4276	481	16	have	have	VERB
ejpam-4276	481	17	d(λ	d(λ	PROPN
ejpam-4276	481	18	,	,	PUNCT
ejpam-4276	481	19	sp	sp	NOUN
ejpam-4276	481	20	)	)	PUNCT
ejpam-4276	481	21	=	=	SYM
ejpam-4276	481	22	x.	x.	NOUN
ejpam-4276	481	23	corollary	corollary	NOUN
ejpam-4276	481	24	7	7	X
ejpam-4276	481	25	.	.	PUNCT
ejpam-4276	482	1	let	let	VERB
ejpam-4276	482	2	a	a	DET
ejpam-4276	482	3	be	be	AUX
ejpam-4276	482	4	a	a	DET
ejpam-4276	482	5	subset	subset	NOUN
ejpam-4276	482	6	of	of	ADP
ejpam-4276	482	7	a	a	DET
ejpam-4276	482	8	topological	topological	ADJ
ejpam-4276	482	9	space	space	NOUN
ejpam-4276	482	10	(	(	PUNCT
ejpam-4276	482	11	x	x	X
ejpam-4276	482	12	,	,	PUNCT
ejpam-4276	482	13	τ	τ	PROPN
ejpam-4276	482	14	)	)	PUNCT
ejpam-4276	482	15	.	.	PUNCT
ejpam-4276	483	1	if	if	SCONJ
ejpam-4276	483	2	a	a	PRON
ejpam-4276	483	3	is	be	AUX
ejpam-4276	483	4	β(λ	β(λ	NOUN
ejpam-4276	483	5	,	,	PUNCT
ejpam-4276	483	6	sp)-closed	sp)-close	VERB
ejpam-4276	483	7	,	,	PUNCT
ejpam-4276	483	8	then	then	ADV
ejpam-4276	483	9	a	a	PRON
ejpam-4276	483	10	is	be	AUX
ejpam-4276	483	11	the	the	DET
ejpam-4276	483	12	union	union	NOUN
ejpam-4276	483	13	of	of	ADP
ejpam-4276	483	14	a	a	DET
ejpam-4276	483	15	r(λ	r(λ	NOUN
ejpam-4276	483	16	,	,	PUNCT
ejpam-4276	483	17	sp)-open	sp)-open	NOUN
ejpam-4276	483	18	set	set	NOUN
ejpam-4276	483	19	and	and	CCONJ
ejpam-4276	483	20	a	a	DET
ejpam-4276	483	21	set	set	NOUN
ejpam-4276	483	22	has	have	VERB
ejpam-4276	483	23	empty	empty	ADJ
ejpam-4276	483	24	(	(	PUNCT
ejpam-4276	483	25	λ	λ	NOUN
ejpam-4276	483	26	,	,	PUNCT
ejpam-4276	483	27	sp)-interior	sp)-interior	NOUN
ejpam-4276	483	28	.	.	PUNCT
ejpam-4276	484	1	lemma	lemma	PROPN
ejpam-4276	484	2	7	7	X
ejpam-4276	484	3	.	.	PUNCT
ejpam-4276	484	4	let	let	VERB
ejpam-4276	484	5	a	a	DET
ejpam-4276	484	6	be	be	AUX
ejpam-4276	484	7	a	a	DET
ejpam-4276	484	8	subset	subset	NOUN
ejpam-4276	484	9	of	of	ADP
ejpam-4276	484	10	a	a	DET
ejpam-4276	484	11	topological	topological	ADJ
ejpam-4276	484	12	space	space	NOUN
ejpam-4276	484	13	(	(	PUNCT
ejpam-4276	484	14	x	x	X
ejpam-4276	484	15	,	,	PUNCT
ejpam-4276	484	16	τ	τ	PROPN
ejpam-4276	484	17	)	)	PUNCT
ejpam-4276	484	18	.	.	PUNCT
ejpam-4276	485	1	if	if	SCONJ
ejpam-4276	485	2	a	a	PRON
ejpam-4276	485	3	is	be	AUX
ejpam-4276	485	4	(	(	PUNCT
ejpam-4276	485	5	λ	λ	X
ejpam-4276	485	6	,	,	PUNCT
ejpam-4276	485	7	sp)-closed	sp)-closed	ADJ
ejpam-4276	485	8	and	and	CCONJ
ejpam-4276	485	9	p(λ	p(λ	NOUN
ejpam-4276	485	10	,	,	PUNCT
ejpam-4276	485	11	sp)-open	sp)-open	NOUN
ejpam-4276	485	12	,	,	PUNCT
ejpam-4276	485	13	then	then	ADV
ejpam-4276	485	14	a	a	PRON
ejpam-4276	485	15	is	be	AUX
ejpam-4276	485	16	(	(	PUNCT
ejpam-4276	485	17	λ	λ	NOUN
ejpam-4276	485	18	,	,	PUNCT
ejpam-4276	485	19	sp)-open	sp)-open	NOUN
ejpam-4276	485	20	.	.	PUNCT
ejpam-4276	486	1	theorem	theorem	NOUN
ejpam-4276	486	2	1	1	NUM
ejpam-4276	486	3	.	.	X
ejpam-4276	487	1	for	for	ADP
ejpam-4276	487	2	a	a	DET
ejpam-4276	487	3	topological	topological	ADJ
ejpam-4276	487	4	space	space	NOUN
ejpam-4276	487	5	(	(	PUNCT
ejpam-4276	487	6	x	x	X
ejpam-4276	487	7	,	,	PUNCT
ejpam-4276	487	8	τ	τ	PROPN
ejpam-4276	487	9	)	)	PUNCT
ejpam-4276	487	10	,	,	PUNCT
ejpam-4276	487	11	the	the	DET
ejpam-4276	487	12	following	follow	VERB
ejpam-4276	487	13	properties	property	NOUN
ejpam-4276	487	14	are	be	AUX
ejpam-4276	487	15	equivalent	equivalent	ADJ
ejpam-4276	487	16	:	:	PUNCT
ejpam-4276	487	17	(	(	PUNCT
ejpam-4276	487	18	1	1	X
ejpam-4276	487	19	)	)	PUNCT
ejpam-4276	487	20	every	every	DET
ejpam-4276	487	21	s(λ	s(λ	PROPN
ejpam-4276	487	22	,	,	PUNCT
ejpam-4276	487	23	sp)-open	sp)-open	ADJ
ejpam-4276	487	24	set	set	NOUN
ejpam-4276	487	25	of	of	ADP
ejpam-4276	487	26	x	x	PUNCT
ejpam-4276	487	27	is	be	AUX
ejpam-4276	487	28	α(λ	α(λ	PROPN
ejpam-4276	487	29	,	,	PUNCT
ejpam-4276	487	30	sp)-open	sp)-open	NOUN
ejpam-4276	487	31	.	.	PUNCT
ejpam-4276	488	1	(	(	PUNCT
ejpam-4276	488	2	2	2	X
ejpam-4276	488	3	)	)	PUNCT
ejpam-4276	488	4	every	every	DET
ejpam-4276	488	5	s(λ	s(λ	PROPN
ejpam-4276	488	6	,	,	PUNCT
ejpam-4276	488	7	sp)-open	sp)-open	ADJ
ejpam-4276	488	8	set	set	NOUN
ejpam-4276	488	9	of	of	ADP
ejpam-4276	488	10	x	x	PROPN
ejpam-4276	488	11	is	be	AUX
ejpam-4276	488	12	p(λ	p(λ	NOUN
ejpam-4276	488	13	,	,	PUNCT
ejpam-4276	488	14	sp)-open	sp)-open	NOUN
ejpam-4276	488	15	.	.	PUNCT
ejpam-4276	489	1	(	(	PUNCT
ejpam-4276	489	2	3	3	X
ejpam-4276	489	3	)	)	PUNCT
ejpam-4276	489	4	every	every	DET
ejpam-4276	489	5	β(λ	β(λ	NOUN
ejpam-4276	489	6	,	,	PUNCT
ejpam-4276	489	7	sp)-open	sp)-open	ADJ
ejpam-4276	489	8	set	set	NOUN
ejpam-4276	489	9	of	of	ADP
ejpam-4276	489	10	x	x	PROPN
ejpam-4276	489	11	is	be	AUX
ejpam-4276	489	12	p(λ	p(λ	NOUN
ejpam-4276	489	13	,	,	PUNCT
ejpam-4276	489	14	sp)-open	sp)-open	NOUN
ejpam-4276	489	15	.	.	PUNCT
ejpam-4276	490	1	(	(	PUNCT
ejpam-4276	490	2	4	4	X
ejpam-4276	490	3	)	)	PUNCT
ejpam-4276	490	4	every	every	DET
ejpam-4276	490	5	b(λ	b(λ	NOUN
ejpam-4276	490	6	,	,	PUNCT
ejpam-4276	490	7	sp)-open	sp)-open	ADJ
ejpam-4276	490	8	set	set	NOUN
ejpam-4276	490	9	of	of	ADP
ejpam-4276	490	10	x	x	PROPN
ejpam-4276	490	11	is	be	AUX
ejpam-4276	490	12	p(λ	p(λ	NOUN
ejpam-4276	490	13	,	,	PUNCT
ejpam-4276	490	14	sp)-open	sp)-open	NOUN
ejpam-4276	490	15	.	.	PUNCT
ejpam-4276	491	1	(	(	PUNCT
ejpam-4276	491	2	5	5	X
ejpam-4276	491	3	)	)	PUNCT
ejpam-4276	491	4	every	every	DET
ejpam-4276	491	5	rs(λ	rs(λ	NOUN
ejpam-4276	491	6	,	,	PUNCT
ejpam-4276	491	7	sp)-open	sp)-open	VERB
ejpam-4276	491	8	set	set	NOUN
ejpam-4276	491	9	of	of	ADP
ejpam-4276	491	10	x	x	PROPN
ejpam-4276	491	11	is	be	AUX
ejpam-4276	491	12	p(λ	p(λ	NOUN
ejpam-4276	491	13	,	,	PUNCT
ejpam-4276	491	14	sp)-open	sp)-open	NOUN
ejpam-4276	491	15	.	.	PUNCT
ejpam-4276	492	1	(	(	PUNCT
ejpam-4276	492	2	6	6	NUM
ejpam-4276	492	3	)	)	PUNCT
ejpam-4276	492	4	every	every	DET
ejpam-4276	492	5	rs(λ	rs(λ	NOUN
ejpam-4276	492	6	,	,	PUNCT
ejpam-4276	492	7	sp)-open	sp)-open	VERB
ejpam-4276	492	8	set	set	NOUN
ejpam-4276	492	9	of	of	ADP
ejpam-4276	492	10	x	x	PUNCT
ejpam-4276	492	11	is	be	AUX
ejpam-4276	492	12	r(λ	r(λ	NOUN
ejpam-4276	492	13	,	,	PUNCT
ejpam-4276	492	14	sp)-open	sp)-open	NOUN
ejpam-4276	492	15	.	.	PUNCT
ejpam-4276	493	1	(	(	PUNCT
ejpam-4276	493	2	7	7	X
ejpam-4276	493	3	)	)	PUNCT
ejpam-4276	493	4	every	every	DET
ejpam-4276	493	5	r(λ	r(λ	NOUN
ejpam-4276	493	6	,	,	PUNCT
ejpam-4276	493	7	sp)-closed	sp)-close	VERB
ejpam-4276	493	8	set	set	NOUN
ejpam-4276	493	9	of	of	ADP
ejpam-4276	493	10	x	x	PROPN
ejpam-4276	493	11	is	be	AUX
ejpam-4276	493	12	p(λ	p(λ	NOUN
ejpam-4276	493	13	,	,	PUNCT
ejpam-4276	493	14	sp)-open	sp)-open	NOUN
ejpam-4276	493	15	.	.	PUNCT
ejpam-4276	494	1	(	(	PUNCT
ejpam-4276	494	2	8)	8)	NUM
ejpam-4276	494	3	every	every	DET
ejpam-4276	494	4	r(λ	r(λ	NOUN
ejpam-4276	494	5	,	,	PUNCT
ejpam-4276	494	6	sp)-closed	sp)-close	VERB
ejpam-4276	494	7	set	set	NOUN
ejpam-4276	494	8	of	of	ADP
ejpam-4276	494	9	x	x	PUNCT
ejpam-4276	494	10	is	be	AUX
ejpam-4276	494	11	(	(	PUNCT
ejpam-4276	494	12	λ	λ	NOUN
ejpam-4276	494	13	,	,	PUNCT
ejpam-4276	494	14	sp)-open	sp)-open	NOUN
ejpam-4276	494	15	.	.	PUNCT
ejpam-4276	495	1	proof	proof	NOUN
ejpam-4276	495	2	.	.	PUNCT
ejpam-4276	496	1	(	(	PUNCT
ejpam-4276	496	2	1	1	X
ejpam-4276	496	3	)	)	PUNCT
ejpam-4276	496	4	⇒	⇒	NOUN
ejpam-4276	496	5	(	(	PUNCT
ejpam-4276	496	6	2	2	NUM
ejpam-4276	496	7	):	):	PUNCT
ejpam-4276	496	8	this	this	PRON
ejpam-4276	496	9	is	be	AUX
ejpam-4276	496	10	obvious	obvious	ADJ
ejpam-4276	496	11	since	since	SCONJ
ejpam-4276	496	12	αλspo(x	αλspo(x	NOUN
ejpam-4276	496	13	,	,	PUNCT
ejpam-4276	496	14	τ	τ	PROPN
ejpam-4276	496	15	)	)	PUNCT
ejpam-4276	496	16	⊆	⊆	NUM
ejpam-4276	496	17	sλspo(x	sλspo(x	PROPN
ejpam-4276	496	18	,	,	PUNCT
ejpam-4276	496	19	τ	τ	PROPN
ejpam-4276	496	20	)	)	PUNCT
ejpam-4276	496	21	.	.	PUNCT
ejpam-4276	497	1	(	(	PUNCT
ejpam-4276	497	2	2	2	X
ejpam-4276	497	3	)	)	PUNCT
ejpam-4276	497	4	⇒	⇒	NOUN
ejpam-4276	497	5	(	(	PUNCT
ejpam-4276	497	6	3	3	NUM
ejpam-4276	497	7	):	):	PUNCT
ejpam-4276	497	8	let	let	VERB
ejpam-4276	497	9	a	a	PRON
ejpam-4276	497	10	be	be	AUX
ejpam-4276	497	11	a	a	DET
ejpam-4276	497	12	β(λ	β(λ	NOUN
ejpam-4276	497	13	,	,	PUNCT
ejpam-4276	497	14	sp)-open	sp)-open	ADJ
ejpam-4276	497	15	set	set	NOUN
ejpam-4276	497	16	.	.	PUNCT
ejpam-4276	498	1	then	then	ADV
ejpam-4276	498	2	,	,	PUNCT
ejpam-4276	498	3	a	a	DET
ejpam-4276	498	4	⊆	⊆	NUM
ejpam-4276	498	5	[	[	X
ejpam-4276	498	6	[	[	X
ejpam-4276	498	7	a(λ	a(λ	ADJ
ejpam-4276	498	8	,	,	PUNCT
ejpam-4276	498	9	sp)](λ	sp)](λ	PROPN
ejpam-4276	498	10	,	,	PUNCT
ejpam-4276	498	11	sp	sp	NOUN
ejpam-4276	498	12	)	)	PUNCT
ejpam-4276	498	13	]	]	PUNCT
ejpam-4276	499	1	(	(	PUNCT
ejpam-4276	499	2	λ	λ	NOUN
ejpam-4276	499	3	,	,	PUNCT
ejpam-4276	499	4	sp	sp	NOUN
ejpam-4276	499	5	)	)	PUNCT
ejpam-4276	499	6	.	.	PUNCT
ejpam-4276	500	1	it	it	PRON
ejpam-4276	500	2	follows	follow	VERB
ejpam-4276	500	3	from	from	ADP
ejpam-4276	500	4	proposition	proposition	NOUN
ejpam-4276	500	5	2	2	NUM
ejpam-4276	500	6	that	that	PRON
ejpam-4276	500	7	b	b	X
ejpam-4276	501	1	=	=	PUNCT
ejpam-4276	502	1	[	[	X
ejpam-4276	502	2	[	[	X
ejpam-4276	502	3	a(λ	a(λ	ADJ
ejpam-4276	502	4	,	,	PUNCT
ejpam-4276	502	5	sp)](λ	sp)](λ	PROPN
ejpam-4276	502	6	,	,	PUNCT
ejpam-4276	502	7	sp	sp	NOUN
ejpam-4276	502	8	)	)	PUNCT
ejpam-4276	502	9	]	]	PUNCT
ejpam-4276	503	1	(	(	PUNCT
ejpam-4276	503	2	λ	λ	NOUN
ejpam-4276	503	3	,	,	PUNCT
ejpam-4276	503	4	sp	sp	NOUN
ejpam-4276	503	5	)	)	PUNCT
ejpam-4276	503	6	is	be	AUX
ejpam-4276	503	7	r(λ	r(λ	NOUN
ejpam-4276	503	8	,	,	PUNCT
ejpam-4276	503	9	sp)-closed	sp)-close	VERB
ejpam-4276	503	10	and	and	CCONJ
ejpam-4276	503	11	thus	thus	ADV
ejpam-4276	503	12	s(λ	s(λ	NOUN
ejpam-4276	503	13	,	,	PUNCT
ejpam-4276	503	14	sp)-open	sp)-open	NOUN
ejpam-4276	503	15	.	.	PUNCT
ejpam-4276	504	1	by	by	ADP
ejpam-4276	504	2	(	(	PUNCT
ejpam-4276	504	3	2	2	NUM
ejpam-4276	504	4	)	)	PUNCT
ejpam-4276	504	5	,	,	PUNCT
ejpam-4276	504	6	b	b	PROPN
ejpam-4276	504	7	is	be	AUX
ejpam-4276	504	8	p(λ	p(λ	NOUN
ejpam-4276	504	9	,	,	PUNCT
ejpam-4276	504	10	sp)-open	sp)-open	ADJ
ejpam-4276	504	11	and	and	CCONJ
ejpam-4276	504	12	hence	hence	ADV
ejpam-4276	504	13	a	a	DET
ejpam-4276	504	14	⊆	⊆	NUM
ejpam-4276	504	15	b	b	SYM
ejpam-4276	504	16	⊆	⊆	NUM
ejpam-4276	504	17	[	[	X
ejpam-4276	504	18	b(λ	b(λ	PROPN
ejpam-4276	504	19	,	,	PUNCT
ejpam-4276	504	20	sp)](λ	sp)](λ	PROPN
ejpam-4276	504	21	,	,	PUNCT
ejpam-4276	504	22	sp	sp	NOUN
ejpam-4276	504	23	)	)	PUNCT
ejpam-4276	504	24	=	=	SYM
ejpam-4276	504	25	b(λ	b(λ	NOUN
ejpam-4276	504	26	,	,	PUNCT
ejpam-4276	504	27	sp	sp	NOUN
ejpam-4276	504	28	)	)	PUNCT
ejpam-4276	504	29	.	.	PUNCT
ejpam-4276	505	1	also	also	ADV
ejpam-4276	505	2	it	it	PRON
ejpam-4276	505	3	is	be	AUX
ejpam-4276	505	4	clear	clear	ADJ
ejpam-4276	505	5	that	that	SCONJ
ejpam-4276	505	6	b	b	X
ejpam-4276	505	7	⊆	⊆	NUM
ejpam-4276	505	8	a(λ	a(λ	ADV
ejpam-4276	505	9	,	,	PUNCT
ejpam-4276	505	10	sp	sp	NOUN
ejpam-4276	505	11	)	)	PUNCT
ejpam-4276	505	12	and	and	CCONJ
ejpam-4276	505	13	thus	thus	ADV
ejpam-4276	505	14	b(λ	b(λ	NOUN
ejpam-4276	505	15	,	,	PUNCT
ejpam-4276	505	16	sp	sp	NOUN
ejpam-4276	505	17	)	)	PUNCT
ejpam-4276	505	18	⊆	⊆	NUM
ejpam-4276	505	19	[	[	X
ejpam-4276	505	20	a(λ	a(λ	ADV
ejpam-4276	505	21	,	,	PUNCT
ejpam-4276	505	22	sp)](λ	sp)](λ	PROPN
ejpam-4276	505	23	,	,	PUNCT
ejpam-4276	505	24	sp	sp	NOUN
ejpam-4276	505	25	)	)	PUNCT
ejpam-4276	505	26	.	.	PUNCT
ejpam-4276	506	1	therefore	therefore	ADV
ejpam-4276	506	2	,	,	PUNCT
ejpam-4276	506	3	a	a	DET
ejpam-4276	506	4	⊆	⊆	NUM
ejpam-4276	506	5	[	[	X
ejpam-4276	506	6	a(λ	a(λ	ADJ
ejpam-4276	506	7	,	,	PUNCT
ejpam-4276	506	8	sp)](λ	sp)](λ	PROPN
ejpam-4276	506	9	,	,	PUNCT
ejpam-4276	506	10	sp	sp	NOUN
ejpam-4276	506	11	)	)	PUNCT
ejpam-4276	506	12	.	.	PUNCT
ejpam-4276	507	1	this	this	PRON
ejpam-4276	507	2	shows	show	VERB
ejpam-4276	507	3	that	that	SCONJ
ejpam-4276	507	4	a	a	PRON
ejpam-4276	507	5	is	be	AUX
ejpam-4276	507	6	p(λ	p(λ	NOUN
ejpam-4276	507	7	,	,	PUNCT
ejpam-4276	507	8	sp)-open	sp)-open	NOUN
ejpam-4276	507	9	.	.	PUNCT
ejpam-4276	508	1	(	(	PUNCT
ejpam-4276	508	2	3	3	X
ejpam-4276	508	3	)	)	PUNCT
ejpam-4276	508	4	⇒	⇒	NOUN
ejpam-4276	508	5	(	(	PUNCT
ejpam-4276	508	6	4	4	NUM
ejpam-4276	508	7	):	):	PUNCT
ejpam-4276	508	8	this	this	PRON
ejpam-4276	508	9	is	be	AUX
ejpam-4276	508	10	obvious	obvious	ADJ
ejpam-4276	508	11	since	since	SCONJ
ejpam-4276	508	12	bλspo(x	bλspo(x	NOUN
ejpam-4276	508	13	,	,	PUNCT
ejpam-4276	508	14	τ	τ	PROPN
ejpam-4276	508	15	)	)	PUNCT
ejpam-4276	508	16	⊆	⊆	NUM
ejpam-4276	508	17	βλspo(x	βλspo(x	NUM
ejpam-4276	508	18	,	,	PUNCT
ejpam-4276	508	19	τ	τ	PROPN
ejpam-4276	508	20	)	)	PUNCT
ejpam-4276	508	21	.	.	PUNCT
ejpam-4276	509	1	(	(	PUNCT
ejpam-4276	509	2	4	4	X
ejpam-4276	509	3	)	)	PUNCT
ejpam-4276	509	4	⇒	⇒	NOUN
ejpam-4276	509	5	(	(	PUNCT
ejpam-4276	509	6	5	5	NUM
ejpam-4276	509	7	):	):	PUNCT
ejpam-4276	509	8	it	it	PRON
ejpam-4276	509	9	follows	follow	VERB
ejpam-4276	509	10	from	from	ADP
ejpam-4276	509	11	proposition	proposition	NOUN
ejpam-4276	509	12	14	14	NUM
ejpam-4276	509	13	that	that	PRON
ejpam-4276	509	14	rsλspo(x	rsλspo(x	NOUN
ejpam-4276	509	15	,	,	PUNCT
ejpam-4276	509	16	τ	τ	X
ejpam-4276	509	17	)	)	PUNCT
ejpam-4276	509	18	⊆	⊆	NUM
ejpam-4276	509	19	sλspo(x	sλspo(x	PROPN
ejpam-4276	509	20	,	,	PUNCT
ejpam-4276	509	21	τ	τ	PROPN
ejpam-4276	509	22	)	)	PUNCT
ejpam-4276	509	23	.	.	PUNCT
ejpam-4276	510	1	since	since	SCONJ
ejpam-4276	510	2	sλspo(x	sλspo(x	PROPN
ejpam-4276	510	3	,	,	PUNCT
ejpam-4276	510	4	τ	τ	PROPN
ejpam-4276	510	5	)	)	PUNCT
ejpam-4276	510	6	⊆	⊆	NUM
ejpam-4276	510	7	bλspo(x	bλspo(x	NOUN
ejpam-4276	510	8	,	,	PUNCT
ejpam-4276	510	9	τ	τ	PROPN
ejpam-4276	510	10	)	)	PUNCT
ejpam-4276	510	11	,	,	PUNCT
ejpam-4276	510	12	rsλspo(x	rsλspo(x	NOUN
ejpam-4276	510	13	,	,	PUNCT
ejpam-4276	510	14	τ	τ	PROPN
ejpam-4276	510	15	)	)	PUNCT
ejpam-4276	510	16	⊆	⊆	NUM
ejpam-4276	510	17	bλspo(x	bλspo(x	NOUN
ejpam-4276	510	18	,	,	PUNCT
ejpam-4276	510	19	τ	τ	PROPN
ejpam-4276	510	20	)	)	PUNCT
ejpam-4276	510	21	.	.	PUNCT
ejpam-4276	511	1	thus	thus	ADV
ejpam-4276	511	2	,	,	PUNCT
ejpam-4276	511	3	the	the	DET
ejpam-4276	511	4	result	result	NOUN
ejpam-4276	511	5	follows	follow	VERB
ejpam-4276	511	6	from	from	ADP
ejpam-4276	511	7	(	(	PUNCT
ejpam-4276	511	8	3	3	NUM
ejpam-4276	511	9	)	)	PUNCT
ejpam-4276	511	10	.	.	PUNCT
ejpam-4276	512	1	(	(	PUNCT
ejpam-4276	512	2	5	5	X
ejpam-4276	512	3	)	)	PUNCT
ejpam-4276	512	4	⇒	⇒	NOUN
ejpam-4276	512	5	(	(	PUNCT
ejpam-4276	512	6	6	6	NUM
ejpam-4276	512	7	):	):	PUNCT
ejpam-4276	512	8	since	since	SCONJ
ejpam-4276	512	9	every	every	DET
ejpam-4276	512	10	rs(λ	rs(λ	NOUN
ejpam-4276	512	11	,	,	PUNCT
ejpam-4276	512	12	sp)-open	sp)-open	ADJ
ejpam-4276	512	13	set	set	NOUN
ejpam-4276	512	14	is	be	AUX
ejpam-4276	512	15	s(λ	s(λ	PROPN
ejpam-4276	512	16	,	,	PUNCT
ejpam-4276	512	17	sp)-closed	sp)-close	VERB
ejpam-4276	512	18	,	,	PUNCT
ejpam-4276	512	19	it	it	PRON
ejpam-4276	512	20	follows	follow	VERB
ejpam-4276	512	21	from	from	ADP
ejpam-4276	512	22	(	(	PUNCT
ejpam-4276	512	23	4	4	NUM
ejpam-4276	512	24	)	)	PUNCT
ejpam-4276	512	25	that	that	PRON
ejpam-4276	512	26	a	a	DET
ejpam-4276	512	27	rs(λ	rs(λ	NUM
ejpam-4276	512	28	,	,	PUNCT
ejpam-4276	512	29	sp)-open	sp)-open	ADJ
ejpam-4276	512	30	set	set	NOUN
ejpam-4276	512	31	is	be	AUX
ejpam-4276	512	32	both	both	DET
ejpam-4276	512	33	s(λ	s(λ	NOUN
ejpam-4276	512	34	,	,	PUNCT
ejpam-4276	512	35	sp)-closed	sp)-close	VERB
ejpam-4276	512	36	and	and	CCONJ
ejpam-4276	512	37	p(λ	p(λ	NOUN
ejpam-4276	512	38	,	,	PUNCT
ejpam-4276	512	39	sp)-open	sp)-open	NOUN
ejpam-4276	512	40	.	.	PUNCT
ejpam-4276	513	1	thus	thus	ADV
ejpam-4276	513	2	,	,	PUNCT
ejpam-4276	513	3	by	by	ADP
ejpam-4276	513	4	proposition	proposition	NOUN
ejpam-4276	513	5	4	4	NUM
ejpam-4276	513	6	,	,	PUNCT
ejpam-4276	513	7	follows	follow	VERB
ejpam-4276	513	8	.	.	PUNCT
ejpam-4276	514	1	(	(	PUNCT
ejpam-4276	514	2	6	6	NUM
ejpam-4276	514	3	)	)	PUNCT
ejpam-4276	514	4	⇒	⇒	NOUN
ejpam-4276	514	5	(	(	PUNCT
ejpam-4276	514	6	7	7	NUM
ejpam-4276	514	7	):	):	PUNCT
ejpam-4276	514	8	follows	follow	VERB
ejpam-4276	514	9	from	from	ADP
ejpam-4276	514	10	proposition	proposition	NOUN
ejpam-4276	514	11	4	4	NUM
ejpam-4276	514	12	and	and	CCONJ
ejpam-4276	514	13	remark	remark	NOUN
ejpam-4276	514	14	3	3	NUM
ejpam-4276	514	15	.	.	PUNCT
ejpam-4276	514	16	(	(	PUNCT
ejpam-4276	514	17	7	7	X
ejpam-4276	514	18	)	)	PUNCT
ejpam-4276	514	19	⇒	⇒	NOUN
ejpam-4276	514	20	(	(	PUNCT
ejpam-4276	514	21	8)	8)	NUM
ejpam-4276	514	22	:	:	PUNCT
ejpam-4276	514	23	follows	follow	VERB
ejpam-4276	514	24	from	from	ADP
ejpam-4276	514	25	lemma	lemma	PROPN
ejpam-4276	514	26	7	7	NUM
ejpam-4276	514	27	.	.	PUNCT
ejpam-4276	514	28	c.	c.	PROPN
ejpam-4276	514	29	boonpok	boonpok	PROPN
ejpam-4276	514	30	,	,	PUNCT
ejpam-4276	514	31	j.	j.	PROPN
ejpam-4276	514	32	khampakdee	khampakdee	PROPN
ejpam-4276	514	33	/	/	PUNCT
ejpam-4276	514	34	eur	eur	PROPN
ejpam-4276	514	35	.	.	PUNCT
ejpam-4276	515	1	j.	j.	PROPN
ejpam-4276	515	2	pure	pure	PROPN
ejpam-4276	515	3	appl	appl	PROPN
ejpam-4276	515	4	.	.	PROPN
ejpam-4276	515	5	math	math	PROPN
ejpam-4276	515	6	,	,	PUNCT
ejpam-4276	515	7	15	15	NUM
ejpam-4276	515	8	(	(	PUNCT
ejpam-4276	515	9	2	2	NUM
ejpam-4276	515	10	)	)	PUNCT
ejpam-4276	515	11	(	(	PUNCT
ejpam-4276	515	12	2022	2022	NUM
ejpam-4276	515	13	)	)	PUNCT
ejpam-4276	515	14	,	,	PUNCT
ejpam-4276	515	15	572	572	NUM
ejpam-4276	515	16	-	-	SYM
ejpam-4276	515	17	588	588	NUM
ejpam-4276	515	18	585	585	NUM
ejpam-4276	515	19	(	(	PUNCT
ejpam-4276	515	20	8)	8)	NUM
ejpam-4276	515	21	⇒	⇒	NOUN
ejpam-4276	515	22	(	(	PUNCT
ejpam-4276	515	23	1	1	NUM
ejpam-4276	515	24	):	):	PUNCT
ejpam-4276	515	25	let	let	VERB
ejpam-4276	515	26	a	a	PRON
ejpam-4276	515	27	be	be	AUX
ejpam-4276	515	28	a	a	DET
ejpam-4276	515	29	s(λ	s(λ	PROPN
ejpam-4276	515	30	,	,	PUNCT
ejpam-4276	515	31	sp)-open	sp)-open	ADJ
ejpam-4276	515	32	set	set	NOUN
ejpam-4276	515	33	.	.	PUNCT
ejpam-4276	516	1	then	then	ADV
ejpam-4276	516	2	,	,	PUNCT
ejpam-4276	516	3	by	by	ADP
ejpam-4276	516	4	corollary	corollary	ADJ
ejpam-4276	516	5	3	3	NUM
ejpam-4276	516	6	,	,	PUNCT
ejpam-4276	516	7	a(λ	a(λ	ADV
ejpam-4276	516	8	,	,	PUNCT
ejpam-4276	516	9	sp	sp	NOUN
ejpam-4276	516	10	)	)	PUNCT
ejpam-4276	516	11	is	be	AUX
ejpam-4276	516	12	r(λ	r(λ	NOUN
ejpam-4276	516	13	,	,	PUNCT
ejpam-4276	516	14	sp)closed	sp)close	VERB
ejpam-4276	516	15	.	.	PUNCT
ejpam-4276	517	1	by	by	ADP
ejpam-4276	517	2	(	(	PUNCT
ejpam-4276	517	3	8)	8)	NUM
ejpam-4276	517	4	,	,	PUNCT
ejpam-4276	517	5	a(λ	a(λ	ADV
ejpam-4276	517	6	,	,	PUNCT
ejpam-4276	517	7	sp	sp	NOUN
ejpam-4276	517	8	)	)	PUNCT
ejpam-4276	517	9	is	be	AUX
ejpam-4276	517	10	(	(	PUNCT
ejpam-4276	517	11	λ	λ	X
ejpam-4276	517	12	,	,	PUNCT
ejpam-4276	517	13	sp)-open	sp)-open	ADJ
ejpam-4276	517	14	and	and	CCONJ
ejpam-4276	517	15	hence	hence	ADV
ejpam-4276	517	16	a(λ	a(λ	ADV
ejpam-4276	517	17	,	,	PUNCT
ejpam-4276	517	18	sp	sp	NOUN
ejpam-4276	517	19	)	)	PUNCT
ejpam-4276	517	20	⊆	⊆	NUM
ejpam-4276	517	21	[	[	X
ejpam-4276	517	22	a(λ	a(λ	ADV
ejpam-4276	517	23	,	,	PUNCT
ejpam-4276	517	24	sp)](λ	sp)](λ	PROPN
ejpam-4276	517	25	,	,	PUNCT
ejpam-4276	517	26	sp	sp	NOUN
ejpam-4276	517	27	)	)	PUNCT
ejpam-4276	517	28	.	.	PUNCT
ejpam-4276	518	1	therefore	therefore	ADV
ejpam-4276	518	2	,	,	PUNCT
ejpam-4276	518	3	a	a	PRON
ejpam-4276	518	4	is	be	AUX
ejpam-4276	518	5	p(λ	p(λ	NOUN
ejpam-4276	518	6	,	,	PUNCT
ejpam-4276	518	7	sp)-open	sp)-open	NOUN
ejpam-4276	518	8	.	.	PUNCT
ejpam-4276	519	1	since	since	SCONJ
ejpam-4276	519	2	a	a	DET
ejpam-4276	519	3	∈	∈	PROPN
ejpam-4276	519	4	sλspo(x	sλspo(x	PROPN
ejpam-4276	519	5	,	,	PUNCT
ejpam-4276	519	6	τ	τ	NOUN
ejpam-4276	519	7	)	)	PUNCT
ejpam-4276	519	8	∩	∩	X
ejpam-4276	519	9	pλspo(x	pλspo(x	ADJ
ejpam-4276	519	10	,	,	PUNCT
ejpam-4276	519	11	τ	τ	X
ejpam-4276	519	12	)	)	PUNCT
ejpam-4276	519	13	=	=	SYM
ejpam-4276	519	14	αλspo(x	αλspo(x	PROPN
ejpam-4276	519	15	,	,	PUNCT
ejpam-4276	519	16	τ	τ	PROPN
ejpam-4276	519	17	)	)	PUNCT
ejpam-4276	519	18	,	,	PUNCT
ejpam-4276	519	19	(	(	PUNCT
ejpam-4276	519	20	1	1	X
ejpam-4276	519	21	)	)	PUNCT
ejpam-4276	519	22	follows	follow	VERB
ejpam-4276	519	23	.	.	PUNCT
ejpam-4276	520	1	corollary	corollary	ADJ
ejpam-4276	520	2	8	8	NUM
ejpam-4276	520	3	.	.	PUNCT
ejpam-4276	521	1	for	for	ADP
ejpam-4276	521	2	a	a	DET
ejpam-4276	521	3	topological	topological	ADJ
ejpam-4276	521	4	space	space	NOUN
ejpam-4276	521	5	(	(	PUNCT
ejpam-4276	521	6	x	x	X
ejpam-4276	521	7	,	,	PUNCT
ejpam-4276	521	8	τ	τ	PROPN
ejpam-4276	521	9	)	)	PUNCT
ejpam-4276	521	10	,	,	PUNCT
ejpam-4276	521	11	the	the	DET
ejpam-4276	521	12	following	follow	VERB
ejpam-4276	521	13	properties	property	NOUN
ejpam-4276	521	14	are	be	AUX
ejpam-4276	521	15	equivalent	equivalent	ADJ
ejpam-4276	521	16	:	:	PUNCT
ejpam-4276	521	17	(	(	PUNCT
ejpam-4276	521	18	1	1	X
ejpam-4276	521	19	)	)	PUNCT
ejpam-4276	521	20	αλspo(x	αλspo(x	NOUN
ejpam-4276	521	21	,	,	PUNCT
ejpam-4276	521	22	τ	τ	X
ejpam-4276	521	23	)	)	PUNCT
ejpam-4276	521	24	=	=	SYM
ejpam-4276	521	25	sλspo(x	sλspo(x	PROPN
ejpam-4276	521	26	,	,	PUNCT
ejpam-4276	521	27	τ	τ	PROPN
ejpam-4276	521	28	)	)	PUNCT
ejpam-4276	521	29	.	.	PUNCT
ejpam-4276	522	1	(	(	PUNCT
ejpam-4276	522	2	2	2	X
ejpam-4276	522	3	)	)	PUNCT
ejpam-4276	522	4	every	every	DET
ejpam-4276	522	5	rs(λ	rs(λ	NOUN
ejpam-4276	522	6	,	,	PUNCT
ejpam-4276	522	7	sp)-open	sp)-open	VERB
ejpam-4276	522	8	set	set	NOUN
ejpam-4276	522	9	of	of	ADP
ejpam-4276	522	10	x	x	PROPN
ejpam-4276	522	11	is	be	AUX
ejpam-4276	522	12	p(λ	p(λ	NOUN
ejpam-4276	522	13	,	,	PUNCT
ejpam-4276	522	14	sp)-closed	sp)-close	VERB
ejpam-4276	522	15	.	.	PUNCT
ejpam-4276	523	1	(	(	PUNCT
ejpam-4276	523	2	3	3	X
ejpam-4276	523	3	)	)	PUNCT
ejpam-4276	523	4	every	every	DET
ejpam-4276	523	5	rs(λ	rs(λ	NOUN
ejpam-4276	523	6	,	,	PUNCT
ejpam-4276	523	7	sp)-open	sp)-open	VERB
ejpam-4276	523	8	set	set	NOUN
ejpam-4276	523	9	of	of	ADP
ejpam-4276	523	10	x	x	PROPN
ejpam-4276	523	11	is	be	AUX
ejpam-4276	523	12	r(λ	r(λ	NOUN
ejpam-4276	523	13	,	,	PUNCT
ejpam-4276	523	14	sp)-closed	sp)-closed	ADJ
ejpam-4276	523	15	.	.	PUNCT
ejpam-4276	524	1	proof	proof	NOUN
ejpam-4276	524	2	.	.	PUNCT
ejpam-4276	525	1	follows	follow	VERB
ejpam-4276	525	2	from	from	ADP
ejpam-4276	525	3	remark	remark	NOUN
ejpam-4276	525	4	3	3	NUM
ejpam-4276	525	5	and	and	CCONJ
ejpam-4276	525	6	theorem	theorem	VERB
ejpam-4276	525	7	1	1	NUM
ejpam-4276	525	8	.	.	PUNCT
ejpam-4276	525	9	definition	definition	NOUN
ejpam-4276	525	10	7	7	NUM
ejpam-4276	525	11	.	.	PUNCT
ejpam-4276	526	1	a	a	DET
ejpam-4276	526	2	subset	subset	NOUN
ejpam-4276	526	3	a	a	PRON
ejpam-4276	526	4	of	of	ADP
ejpam-4276	526	5	a	a	DET
ejpam-4276	526	6	topological	topological	ADJ
ejpam-4276	526	7	space	space	NOUN
ejpam-4276	526	8	(	(	PUNCT
ejpam-4276	526	9	x	x	X
ejpam-4276	526	10	,	,	PUNCT
ejpam-4276	526	11	τ	τ	X
ejpam-4276	526	12	)	)	PUNCT
ejpam-4276	526	13	is	be	AUX
ejpam-4276	526	14	called	call	VERB
ejpam-4276	526	15	p(λ	p(λ	PROPN
ejpam-4276	526	16	,	,	PUNCT
ejpam-4276	526	17	sp)-clopen	sp)-clopen	ADJ
ejpam-4276	526	18	if	if	SCONJ
ejpam-4276	526	19	a	a	PRON
ejpam-4276	526	20	is	be	AUX
ejpam-4276	526	21	both	both	DET
ejpam-4276	526	22	p(λ	p(λ	NOUN
ejpam-4276	526	23	,	,	PUNCT
ejpam-4276	526	24	sp)-open	sp)-open	NOUN
ejpam-4276	526	25	and	and	CCONJ
ejpam-4276	526	26	p(λ	p(λ	NOUN
ejpam-4276	526	27	,	,	PUNCT
ejpam-4276	526	28	sp)-closed	sp)-close	VERB
ejpam-4276	526	29	.	.	PUNCT
ejpam-4276	527	1	corollary	corollary	ADJ
ejpam-4276	527	2	9	9	NUM
ejpam-4276	527	3	.	.	PUNCT
ejpam-4276	528	1	for	for	ADP
ejpam-4276	528	2	a	a	DET
ejpam-4276	528	3	topological	topological	ADJ
ejpam-4276	528	4	space	space	NOUN
ejpam-4276	528	5	(	(	PUNCT
ejpam-4276	528	6	x	x	X
ejpam-4276	528	7	,	,	PUNCT
ejpam-4276	528	8	τ	τ	PROPN
ejpam-4276	528	9	)	)	PUNCT
ejpam-4276	528	10	,	,	PUNCT
ejpam-4276	528	11	the	the	DET
ejpam-4276	528	12	following	follow	VERB
ejpam-4276	528	13	properties	property	NOUN
ejpam-4276	528	14	are	be	AUX
ejpam-4276	528	15	equivalent	equivalent	ADJ
ejpam-4276	528	16	:	:	PUNCT
ejpam-4276	528	17	(	(	PUNCT
ejpam-4276	528	18	1	1	X
ejpam-4276	528	19	)	)	PUNCT
ejpam-4276	528	20	αλspo(x	αλspo(x	NOUN
ejpam-4276	528	21	,	,	PUNCT
ejpam-4276	528	22	τ	τ	X
ejpam-4276	528	23	)	)	PUNCT
ejpam-4276	528	24	=	=	SYM
ejpam-4276	528	25	sλspo(x	sλspo(x	PROPN
ejpam-4276	528	26	,	,	PUNCT
ejpam-4276	528	27	τ	τ	PROPN
ejpam-4276	528	28	)	)	PUNCT
ejpam-4276	528	29	.	.	PUNCT
ejpam-4276	529	1	(	(	PUNCT
ejpam-4276	529	2	2	2	X
ejpam-4276	529	3	)	)	PUNCT
ejpam-4276	529	4	every	every	DET
ejpam-4276	529	5	rs(λ	rs(λ	NOUN
ejpam-4276	529	6	,	,	PUNCT
ejpam-4276	529	7	sp)-open	sp)-open	VERB
ejpam-4276	529	8	set	set	NOUN
ejpam-4276	529	9	of	of	ADP
ejpam-4276	529	10	x	x	PROPN
ejpam-4276	529	11	is	be	AUX
ejpam-4276	529	12	p(λ	p(λ	NOUN
ejpam-4276	529	13	,	,	PUNCT
ejpam-4276	529	14	sp)-clopen	sp)-clopen	NOUN
ejpam-4276	529	15	.	.	PUNCT
ejpam-4276	530	1	(	(	PUNCT
ejpam-4276	530	2	3	3	X
ejpam-4276	530	3	)	)	PUNCT
ejpam-4276	530	4	every	every	DET
ejpam-4276	530	5	rs(λ	rs(λ	NOUN
ejpam-4276	530	6	,	,	PUNCT
ejpam-4276	530	7	sp)-open	sp)-open	VERB
ejpam-4276	530	8	set	set	NOUN
ejpam-4276	530	9	of	of	ADP
ejpam-4276	530	10	x	x	PUNCT
ejpam-4276	530	11	is	be	AUX
ejpam-4276	530	12	(	(	PUNCT
ejpam-4276	530	13	λ	λ	PROPN
ejpam-4276	530	14	,	,	PUNCT
ejpam-4276	530	15	sp)-clopen	sp)-clopen	NOUN
ejpam-4276	530	16	.	.	PUNCT
ejpam-4276	531	1	proof	proof	NOUN
ejpam-4276	531	2	.	.	PUNCT
ejpam-4276	532	1	follows	follow	VERB
ejpam-4276	532	2	from	from	ADP
ejpam-4276	532	3	theorem	theorem	ADJ
ejpam-4276	532	4	1	1	NUM
ejpam-4276	532	5	and	and	CCONJ
ejpam-4276	532	6	corollary	corollary	ADJ
ejpam-4276	532	7	8	8	NUM
ejpam-4276	532	8	.	.	PUNCT
ejpam-4276	533	1	proposition	proposition	NOUN
ejpam-4276	533	2	24	24	NUM
ejpam-4276	533	3	.	.	PUNCT
ejpam-4276	534	1	for	for	ADP
ejpam-4276	534	2	a	a	DET
ejpam-4276	534	3	topological	topological	ADJ
ejpam-4276	534	4	space	space	NOUN
ejpam-4276	534	5	(	(	PUNCT
ejpam-4276	534	6	x	x	X
ejpam-4276	534	7	,	,	PUNCT
ejpam-4276	534	8	τ	τ	PROPN
ejpam-4276	534	9	)	)	PUNCT
ejpam-4276	534	10	,	,	PUNCT
ejpam-4276	534	11	the	the	DET
ejpam-4276	534	12	following	follow	VERB
ejpam-4276	534	13	properties	property	NOUN
ejpam-4276	534	14	are	be	AUX
ejpam-4276	534	15	equivalent	equivalent	ADJ
ejpam-4276	534	16	:	:	PUNCT
ejpam-4276	534	17	(	(	PUNCT
ejpam-4276	534	18	1	1	X
ejpam-4276	534	19	)	)	PUNCT
ejpam-4276	534	20	every	every	DET
ejpam-4276	534	21	p(λ	p(λ	NOUN
ejpam-4276	534	22	,	,	PUNCT
ejpam-4276	534	23	sp)-open	sp)-open	ADJ
ejpam-4276	534	24	set	set	NOUN
ejpam-4276	534	25	of	of	ADP
ejpam-4276	534	26	x	x	PUNCT
ejpam-4276	534	27	is	be	AUX
ejpam-4276	534	28	α(λ	α(λ	PROPN
ejpam-4276	534	29	,	,	PUNCT
ejpam-4276	534	30	sp)-open	sp)-open	NOUN
ejpam-4276	534	31	.	.	PUNCT
ejpam-4276	535	1	(	(	PUNCT
ejpam-4276	535	2	2	2	X
ejpam-4276	535	3	)	)	PUNCT
ejpam-4276	535	4	every	every	DET
ejpam-4276	535	5	p(λ	p(λ	NOUN
ejpam-4276	535	6	,	,	PUNCT
ejpam-4276	535	7	sp)-open	sp)-open	ADJ
ejpam-4276	535	8	set	set	NOUN
ejpam-4276	535	9	of	of	ADP
ejpam-4276	535	10	x	x	PUNCT
ejpam-4276	535	11	is	be	AUX
ejpam-4276	535	12	s(λ	s(λ	PROPN
ejpam-4276	535	13	,	,	PUNCT
ejpam-4276	535	14	sp)-open	sp)-open	NOUN
ejpam-4276	535	15	.	.	PUNCT
ejpam-4276	536	1	proof	proof	NOUN
ejpam-4276	536	2	.	.	PUNCT
ejpam-4276	537	1	follows	follow	VERB
ejpam-4276	537	2	from	from	ADP
ejpam-4276	537	3	proposition	proposition	NOUN
ejpam-4276	537	4	1	1	NUM
ejpam-4276	537	5	.	.	NOUN
ejpam-4276	537	6	4	4	NUM
ejpam-4276	537	7	.	.	X
ejpam-4276	538	1	some	some	DET
ejpam-4276	538	2	characterizations	characterization	NOUN
ejpam-4276	538	3	of	of	ADP
ejpam-4276	538	4	λsp	λsp	NOUN
ejpam-4276	538	5	-	-	PUNCT
ejpam-4276	538	6	extremally	extremally	ADV
ejpam-4276	538	7	disconnected	disconnected	ADJ
ejpam-4276	538	8	spaces	space	NOUN
ejpam-4276	538	9	in	in	ADP
ejpam-4276	538	10	this	this	DET
ejpam-4276	538	11	section	section	NOUN
ejpam-4276	538	12	,	,	PUNCT
ejpam-4276	538	13	we	we	PRON
ejpam-4276	538	14	investigate	investigate	VERB
ejpam-4276	538	15	some	some	DET
ejpam-4276	538	16	characterizations	characterization	NOUN
ejpam-4276	538	17	of	of	ADP
ejpam-4276	538	18	λsp	λsp	NOUN
ejpam-4276	538	19	-	-	PUNCT
ejpam-4276	538	20	extremally	extremally	ADV
ejpam-4276	538	21	disconnected	disconnected	ADJ
ejpam-4276	538	22	spaces	space	NOUN
ejpam-4276	538	23	.	.	PUNCT
ejpam-4276	539	1	definition	definition	NOUN
ejpam-4276	539	2	8	8	NUM
ejpam-4276	539	3	.	.	PUNCT
ejpam-4276	540	1	[	[	X
ejpam-4276	540	2	3	3	X
ejpam-4276	540	3	]	]	PUNCT
ejpam-4276	540	4	a	a	DET
ejpam-4276	540	5	topological	topological	ADJ
ejpam-4276	540	6	space	space	NOUN
ejpam-4276	540	7	(	(	PUNCT
ejpam-4276	540	8	x	x	X
ejpam-4276	540	9	,	,	PUNCT
ejpam-4276	540	10	τ	τ	X
ejpam-4276	540	11	)	)	PUNCT
ejpam-4276	540	12	is	be	AUX
ejpam-4276	540	13	called	call	VERB
ejpam-4276	540	14	λsp	λsp	INTJ
ejpam-4276	540	15	-	-	PUNCT
ejpam-4276	540	16	extremally	extremally	ADV
ejpam-4276	540	17	disconnected	disconnected	ADJ
ejpam-4276	540	18	if	if	SCONJ
ejpam-4276	540	19	u	u	PROPN
ejpam-4276	540	20	(	(	PUNCT
ejpam-4276	540	21	λ	λ	PROPN
ejpam-4276	540	22	,	,	PUNCT
ejpam-4276	540	23	sp	sp	NOUN
ejpam-4276	540	24	)	)	PUNCT
ejpam-4276	540	25	is	be	AUX
ejpam-4276	540	26	(	(	PUNCT
ejpam-4276	540	27	λ	λ	X
ejpam-4276	540	28	,	,	PUNCT
ejpam-4276	540	29	sp)-open	sp)-open	ADJ
ejpam-4276	540	30	in	in	ADP
ejpam-4276	540	31	x	x	PUNCT
ejpam-4276	540	32	for	for	SCONJ
ejpam-4276	540	33	every	every	DET
ejpam-4276	540	34	(	(	PUNCT
ejpam-4276	540	35	λ	λ	NOUN
ejpam-4276	540	36	,	,	PUNCT
ejpam-4276	540	37	sp)-open	sp)-open	NOUN
ejpam-4276	540	38	set	set	VERB
ejpam-4276	540	39	u	u	NOUN
ejpam-4276	540	40	of	of	ADP
ejpam-4276	540	41	x.	x.	PROPN
ejpam-4276	540	42	theorem	theorem	VERB
ejpam-4276	540	43	2	2	NUM
ejpam-4276	540	44	.	.	X
ejpam-4276	540	45	for	for	ADP
ejpam-4276	540	46	a	a	DET
ejpam-4276	540	47	topological	topological	ADJ
ejpam-4276	540	48	space	space	NOUN
ejpam-4276	540	49	(	(	PUNCT
ejpam-4276	540	50	x	x	X
ejpam-4276	540	51	,	,	PUNCT
ejpam-4276	540	52	τ	τ	PROPN
ejpam-4276	540	53	)	)	PUNCT
ejpam-4276	540	54	,	,	PUNCT
ejpam-4276	540	55	the	the	DET
ejpam-4276	540	56	following	follow	VERB
ejpam-4276	540	57	properties	property	NOUN
ejpam-4276	540	58	are	be	AUX
ejpam-4276	540	59	equivalent	equivalent	ADJ
ejpam-4276	540	60	:	:	PUNCT
ejpam-4276	540	61	(	(	PUNCT
ejpam-4276	540	62	1	1	X
ejpam-4276	540	63	)	)	PUNCT
ejpam-4276	540	64	(	(	PUNCT
ejpam-4276	540	65	x	x	X
ejpam-4276	540	66	,	,	PUNCT
ejpam-4276	540	67	τ	τ	X
ejpam-4276	540	68	)	)	PUNCT
ejpam-4276	540	69	is	be	AUX
ejpam-4276	540	70	λsp	λsp	VERB
ejpam-4276	540	71	-	-	PUNCT
ejpam-4276	540	72	extremally	extremally	ADV
ejpam-4276	540	73	disconnected	disconnected	ADJ
ejpam-4276	540	74	.	.	PUNCT
ejpam-4276	541	1	(	(	PUNCT
ejpam-4276	541	2	2	2	X
ejpam-4276	541	3	)	)	PUNCT
ejpam-4276	541	4	for	for	ADP
ejpam-4276	541	5	each	each	DET
ejpam-4276	541	6	v	v	ADP
ejpam-4276	541	7	∈	∈	PROPN
ejpam-4276	541	8	βλspo(x	βλspo(x	PROPN
ejpam-4276	541	9	,	,	PUNCT
ejpam-4276	541	10	τ	τ	PROPN
ejpam-4276	541	11	)	)	PUNCT
ejpam-4276	541	12	,	,	PUNCT
ejpam-4276	541	13	v	v	NOUN
ejpam-4276	541	14	(	(	PUNCT
ejpam-4276	541	15	λ	λ	NOUN
ejpam-4276	541	16	,	,	PUNCT
ejpam-4276	541	17	sp	sp	NOUN
ejpam-4276	541	18	)	)	PUNCT
ejpam-4276	541	19	∈	∈	PROPN
ejpam-4276	541	20	rλspo(x	rλspo(x	NOUN
ejpam-4276	541	21	,	,	PUNCT
ejpam-4276	541	22	τ	τ	PROPN
ejpam-4276	541	23	)	)	PUNCT
ejpam-4276	541	24	.	.	PUNCT
ejpam-4276	542	1	c.	c.	PROPN
ejpam-4276	542	2	boonpok	boonpok	PROPN
ejpam-4276	542	3	,	,	PUNCT
ejpam-4276	542	4	j.	j.	PROPN
ejpam-4276	542	5	khampakdee	khampakdee	PROPN
ejpam-4276	542	6	/	/	PUNCT
ejpam-4276	542	7	eur	eur	PROPN
ejpam-4276	542	8	.	.	PUNCT
ejpam-4276	543	1	j.	j.	PROPN
ejpam-4276	543	2	pure	pure	PROPN
ejpam-4276	543	3	appl	appl	PROPN
ejpam-4276	543	4	.	.	PROPN
ejpam-4276	543	5	math	math	PROPN
ejpam-4276	543	6	,	,	PUNCT
ejpam-4276	543	7	15	15	NUM
ejpam-4276	543	8	(	(	PUNCT
ejpam-4276	543	9	2	2	NUM
ejpam-4276	543	10	)	)	PUNCT
ejpam-4276	543	11	(	(	PUNCT
ejpam-4276	543	12	2022	2022	NUM
ejpam-4276	543	13	)	)	PUNCT
ejpam-4276	543	14	,	,	PUNCT
ejpam-4276	543	15	572	572	NUM
ejpam-4276	543	16	-	-	SYM
ejpam-4276	543	17	588	588	NUM
ejpam-4276	543	18	586	586	NUM
ejpam-4276	543	19	(	(	PUNCT
ejpam-4276	543	20	3	3	NUM
ejpam-4276	543	21	)	)	PUNCT
ejpam-4276	543	22	for	for	ADP
ejpam-4276	543	23	each	each	DET
ejpam-4276	543	24	v	v	ADP
ejpam-4276	543	25	∈	∈	PROPN
ejpam-4276	543	26	bλspo(x	bλspo(x	NOUN
ejpam-4276	543	27	,	,	PUNCT
ejpam-4276	543	28	τ	τ	PROPN
ejpam-4276	543	29	)	)	PUNCT
ejpam-4276	543	30	,	,	PUNCT
ejpam-4276	543	31	v	v	NOUN
ejpam-4276	543	32	(	(	PUNCT
ejpam-4276	543	33	λ	λ	NOUN
ejpam-4276	543	34	,	,	PUNCT
ejpam-4276	543	35	sp	sp	NOUN
ejpam-4276	543	36	)	)	PUNCT
ejpam-4276	543	37	∈	∈	PROPN
ejpam-4276	543	38	rλspo(x	rλspo(x	NOUN
ejpam-4276	543	39	,	,	PUNCT
ejpam-4276	543	40	τ	τ	PROPN
ejpam-4276	543	41	)	)	PUNCT
ejpam-4276	543	42	.	.	PUNCT
ejpam-4276	544	1	(	(	PUNCT
ejpam-4276	544	2	4	4	X
ejpam-4276	544	3	)	)	PUNCT
ejpam-4276	544	4	for	for	ADP
ejpam-4276	544	5	each	each	DET
ejpam-4276	544	6	v	v	ADP
ejpam-4276	544	7	∈	∈	PROPN
ejpam-4276	544	8	sλspo(x	sλspo(x	PROPN
ejpam-4276	544	9	,	,	PUNCT
ejpam-4276	544	10	τ	τ	PROPN
ejpam-4276	544	11	)	)	PUNCT
ejpam-4276	544	12	,	,	PUNCT
ejpam-4276	544	13	v	v	NOUN
ejpam-4276	544	14	(	(	PUNCT
ejpam-4276	544	15	λ	λ	NOUN
ejpam-4276	544	16	,	,	PUNCT
ejpam-4276	544	17	sp	sp	NOUN
ejpam-4276	544	18	)	)	PUNCT
ejpam-4276	544	19	∈	∈	PROPN
ejpam-4276	544	20	rλspo(x	rλspo(x	NOUN
ejpam-4276	544	21	,	,	PUNCT
ejpam-4276	544	22	τ	τ	PROPN
ejpam-4276	544	23	)	)	PUNCT
ejpam-4276	544	24	.	.	PUNCT
ejpam-4276	545	1	(	(	PUNCT
ejpam-4276	545	2	5	5	X
ejpam-4276	545	3	)	)	PUNCT
ejpam-4276	545	4	for	for	ADP
ejpam-4276	545	5	each	each	DET
ejpam-4276	545	6	v	v	ADP
ejpam-4276	545	7	∈	∈	PROPN
ejpam-4276	545	8	αλspo(x	αλspo(x	NOUN
ejpam-4276	545	9	,	,	PUNCT
ejpam-4276	545	10	τ	τ	PROPN
ejpam-4276	545	11	)	)	PUNCT
ejpam-4276	545	12	,	,	PUNCT
ejpam-4276	545	13	v	v	NOUN
ejpam-4276	545	14	(	(	PUNCT
ejpam-4276	545	15	λ	λ	NOUN
ejpam-4276	545	16	,	,	PUNCT
ejpam-4276	545	17	sp	sp	NOUN
ejpam-4276	545	18	)	)	PUNCT
ejpam-4276	545	19	∈	∈	PROPN
ejpam-4276	545	20	rλspo(x	rλspo(x	NOUN
ejpam-4276	545	21	,	,	PUNCT
ejpam-4276	545	22	τ	τ	PROPN
ejpam-4276	545	23	)	)	PUNCT
ejpam-4276	545	24	.	.	PUNCT
ejpam-4276	546	1	(	(	PUNCT
ejpam-4276	546	2	6	6	NUM
ejpam-4276	546	3	)	)	PUNCT
ejpam-4276	546	4	for	for	ADP
ejpam-4276	546	5	each	each	DET
ejpam-4276	546	6	v	v	X
ejpam-4276	546	7	∈	∈	PROPN
ejpam-4276	546	8	λspo(x	λspo(x	NOUN
ejpam-4276	546	9	,	,	PUNCT
ejpam-4276	546	10	τ	τ	PROPN
ejpam-4276	546	11	)	)	PUNCT
ejpam-4276	546	12	,	,	PUNCT
ejpam-4276	546	13	v	v	NOUN
ejpam-4276	546	14	(	(	PUNCT
ejpam-4276	546	15	λ	λ	NOUN
ejpam-4276	546	16	,	,	PUNCT
ejpam-4276	546	17	sp	sp	NOUN
ejpam-4276	546	18	)	)	PUNCT
ejpam-4276	546	19	∈	∈	PROPN
ejpam-4276	546	20	rλspo(x	rλspo(x	NOUN
ejpam-4276	546	21	,	,	PUNCT
ejpam-4276	546	22	τ	τ	PROPN
ejpam-4276	546	23	)	)	PUNCT
ejpam-4276	546	24	.	.	PUNCT
ejpam-4276	547	1	(	(	PUNCT
ejpam-4276	547	2	7	7	X
ejpam-4276	547	3	)	)	PUNCT
ejpam-4276	547	4	for	for	ADP
ejpam-4276	547	5	each	each	DET
ejpam-4276	547	6	v	v	NUM
ejpam-4276	547	7	∈	∈	PROPN
ejpam-4276	547	8	rλspo(x	rλspo(x	NOUN
ejpam-4276	547	9	,	,	PUNCT
ejpam-4276	547	10	τ	τ	PROPN
ejpam-4276	547	11	)	)	PUNCT
ejpam-4276	547	12	,	,	PUNCT
ejpam-4276	547	13	v	v	NOUN
ejpam-4276	547	14	(	(	PUNCT
ejpam-4276	547	15	λ	λ	NOUN
ejpam-4276	547	16	,	,	PUNCT
ejpam-4276	547	17	sp	sp	NOUN
ejpam-4276	547	18	)	)	PUNCT
ejpam-4276	547	19	∈	∈	PROPN
ejpam-4276	547	20	rλspo(x	rλspo(x	NOUN
ejpam-4276	547	21	,	,	PUNCT
ejpam-4276	547	22	τ	τ	PROPN
ejpam-4276	547	23	)	)	PUNCT
ejpam-4276	547	24	.	.	PUNCT
ejpam-4276	548	1	(	(	PUNCT
ejpam-4276	548	2	8)	8)	NUM
ejpam-4276	548	3	for	for	ADP
ejpam-4276	548	4	each	each	DET
ejpam-4276	548	5	v	v	NUM
ejpam-4276	548	6	∈	∈	PROPN
ejpam-4276	548	7	pλspo(x	pλspo(x	NOUN
ejpam-4276	548	8	,	,	PUNCT
ejpam-4276	548	9	τ	τ	PROPN
ejpam-4276	548	10	)	)	PUNCT
ejpam-4276	548	11	,	,	PUNCT
ejpam-4276	548	12	v	v	NOUN
ejpam-4276	548	13	(	(	PUNCT
ejpam-4276	548	14	λ	λ	NOUN
ejpam-4276	548	15	,	,	PUNCT
ejpam-4276	548	16	sp	sp	NOUN
ejpam-4276	548	17	)	)	PUNCT
ejpam-4276	548	18	∈	∈	PROPN
ejpam-4276	548	19	rλspo(x	rλspo(x	NOUN
ejpam-4276	548	20	,	,	PUNCT
ejpam-4276	548	21	τ	τ	PROPN
ejpam-4276	548	22	)	)	PUNCT
ejpam-4276	548	23	.	.	PUNCT
ejpam-4276	549	1	proof	proof	NOUN
ejpam-4276	549	2	.	.	PUNCT
ejpam-4276	550	1	(	(	PUNCT
ejpam-4276	550	2	1	1	X
ejpam-4276	550	3	)	)	PUNCT
ejpam-4276	550	4	⇒	⇒	NOUN
ejpam-4276	550	5	(	(	PUNCT
ejpam-4276	550	6	2	2	NUM
ejpam-4276	550	7	):	):	PUNCT
ejpam-4276	550	8	let	let	VERB
ejpam-4276	550	9	v	v	NUM
ejpam-4276	550	10	∈	∈	PROPN
ejpam-4276	550	11	βλspo(x	βλspo(x	PROPN
ejpam-4276	550	12	,	,	PUNCT
ejpam-4276	550	13	τ	τ	PROPN
ejpam-4276	550	14	)	)	PUNCT
ejpam-4276	550	15	.	.	PUNCT
ejpam-4276	551	1	by	by	ADP
ejpam-4276	551	2	(	(	PUNCT
ejpam-4276	551	3	1	1	NUM
ejpam-4276	551	4	)	)	PUNCT
ejpam-4276	551	5	and	and	CCONJ
ejpam-4276	551	6	proposition	proposition	NOUN
ejpam-4276	551	7	13	13	NUM
ejpam-4276	551	8	,	,	PUNCT
ejpam-4276	551	9	we	we	PRON
ejpam-4276	551	10	have	have	VERB
ejpam-4276	551	11	v	v	NUM
ejpam-4276	551	12	(	(	PUNCT
ejpam-4276	551	13	λ	λ	NOUN
ejpam-4276	551	14	,	,	PUNCT
ejpam-4276	551	15	sp	sp	NOUN
ejpam-4276	551	16	)	)	PUNCT
ejpam-4276	551	17	=	=	PUNCT
ejpam-4276	552	1	[	[	X
ejpam-4276	552	2	[	[	X
ejpam-4276	552	3	v	v	X
ejpam-4276	552	4	(	(	PUNCT
ejpam-4276	552	5	λ	λ	PROPN
ejpam-4276	552	6	,	,	PUNCT
ejpam-4276	552	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	552	8	,	,	PUNCT
ejpam-4276	552	9	sp	sp	NOUN
ejpam-4276	552	10	)	)	PUNCT
ejpam-4276	552	11	]	]	PUNCT
ejpam-4276	552	12	(	(	PUNCT
ejpam-4276	552	13	λ	λ	NOUN
ejpam-4276	552	14	,	,	PUNCT
ejpam-4276	552	15	sp	sp	NOUN
ejpam-4276	552	16	)	)	PUNCT
ejpam-4276	552	17	=	=	PUNCT
ejpam-4276	553	1	[	[	X
ejpam-4276	553	2	[	[	X
ejpam-4276	553	3	[	[	X
ejpam-4276	553	4	v	v	X
ejpam-4276	553	5	(	(	PUNCT
ejpam-4276	553	6	λ	λ	PROPN
ejpam-4276	553	7	,	,	PUNCT
ejpam-4276	553	8	sp)](λ	sp)](λ	PROPN
ejpam-4276	553	9	,	,	PUNCT
ejpam-4276	553	10	sp	sp	NOUN
ejpam-4276	553	11	)	)	PUNCT
ejpam-4276	553	12	]	]	PUNCT
ejpam-4276	553	13	(	(	PUNCT
ejpam-4276	553	14	λ	λ	X
ejpam-4276	553	15	,	,	PUNCT
ejpam-4276	553	16	sp)](λ	sp)](λ	PROPN
ejpam-4276	553	17	,	,	PUNCT
ejpam-4276	553	18	sp	sp	NOUN
ejpam-4276	553	19	)	)	PUNCT
ejpam-4276	553	20	=	=	PUNCT
ejpam-4276	554	1	[	[	X
ejpam-4276	554	2	v	v	X
ejpam-4276	554	3	(	(	PUNCT
ejpam-4276	554	4	λ	λ	PROPN
ejpam-4276	554	5	,	,	PUNCT
ejpam-4276	554	6	sp)](λ	sp)](λ	PROPN
ejpam-4276	554	7	,	,	PUNCT
ejpam-4276	554	8	sp	sp	NOUN
ejpam-4276	554	9	)	)	PUNCT
ejpam-4276	554	10	and	and	CCONJ
ejpam-4276	554	11	hence	hence	ADV
ejpam-4276	554	12	v	v	NOUN
ejpam-4276	554	13	(	(	PUNCT
ejpam-4276	554	14	λ	λ	NOUN
ejpam-4276	554	15	,	,	PUNCT
ejpam-4276	554	16	sp	sp	NOUN
ejpam-4276	554	17	)	)	PUNCT
ejpam-4276	554	18	=	=	PUNCT
ejpam-4276	555	1	[	[	X
ejpam-4276	555	2	v	v	X
ejpam-4276	555	3	(	(	PUNCT
ejpam-4276	555	4	λ	λ	PROPN
ejpam-4276	555	5	,	,	PUNCT
ejpam-4276	555	6	sp)](λ	sp)](λ	PROPN
ejpam-4276	555	7	,	,	PUNCT
ejpam-4276	555	8	sp	sp	NOUN
ejpam-4276	555	9	)	)	PUNCT
ejpam-4276	555	10	=	=	NOUN
ejpam-4276	556	1	[	[	X
ejpam-4276	556	2	[	[	X
ejpam-4276	556	3	v	v	X
ejpam-4276	556	4	(	(	PUNCT
ejpam-4276	556	5	λ	λ	PROPN
ejpam-4276	556	6	,	,	PUNCT
ejpam-4276	556	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	556	8	,	,	PUNCT
ejpam-4276	556	9	sp)](λ	sp)](λ	PROPN
ejpam-4276	556	10	,	,	PUNCT
ejpam-4276	556	11	sp	sp	NOUN
ejpam-4276	556	12	)	)	PUNCT
ejpam-4276	556	13	.	.	PUNCT
ejpam-4276	557	1	thus	thus	ADV
ejpam-4276	557	2	,	,	PUNCT
ejpam-4276	557	3	v	v	INTJ
ejpam-4276	557	4	(	(	PUNCT
ejpam-4276	557	5	λ	λ	NOUN
ejpam-4276	557	6	,	,	PUNCT
ejpam-4276	557	7	sp	sp	NOUN
ejpam-4276	557	8	)	)	PUNCT
ejpam-4276	557	9	∈	∈	PROPN
ejpam-4276	557	10	rλspo(x	rλspo(x	NOUN
ejpam-4276	557	11	,	,	PUNCT
ejpam-4276	557	12	τ	τ	PROPN
ejpam-4276	557	13	)	)	PUNCT
ejpam-4276	557	14	.	.	PUNCT
ejpam-4276	558	1	(	(	PUNCT
ejpam-4276	558	2	2	2	X
ejpam-4276	558	3	)	)	PUNCT
ejpam-4276	558	4	⇒	⇒	NOUN
ejpam-4276	558	5	(	(	PUNCT
ejpam-4276	558	6	3	3	NUM
ejpam-4276	558	7	)	)	PUNCT
ejpam-4276	558	8	⇒	⇒	NOUN
ejpam-4276	558	9	(	(	PUNCT
ejpam-4276	558	10	4	4	NUM
ejpam-4276	558	11	)	)	PUNCT
ejpam-4276	558	12	⇒	⇒	NOUN
ejpam-4276	558	13	(	(	PUNCT
ejpam-4276	558	14	5	5	NUM
ejpam-4276	558	15	)	)	PUNCT
ejpam-4276	558	16	⇒	⇒	NOUN
ejpam-4276	558	17	(	(	PUNCT
ejpam-4276	558	18	6	6	NUM
ejpam-4276	558	19	)	)	PUNCT
ejpam-4276	558	20	⇒	⇒	NOUN
ejpam-4276	558	21	(	(	PUNCT
ejpam-4276	558	22	7	7	NUM
ejpam-4276	558	23	):	):	PUNCT
ejpam-4276	558	24	obvious	obvious	ADJ
ejpam-4276	558	25	.	.	PUNCT
ejpam-4276	559	1	(	(	PUNCT
ejpam-4276	559	2	7	7	X
ejpam-4276	559	3	)	)	PUNCT
ejpam-4276	559	4	⇒	⇒	NOUN
ejpam-4276	559	5	(	(	PUNCT
ejpam-4276	559	6	8)	8)	NUM
ejpam-4276	559	7	:	:	PUNCT
ejpam-4276	559	8	let	let	VERB
ejpam-4276	559	9	v	v	NUM
ejpam-4276	559	10	∈	∈	PROPN
ejpam-4276	559	11	pλspo(x	pλspo(x	NOUN
ejpam-4276	559	12	,	,	PUNCT
ejpam-4276	559	13	τ	τ	PROPN
ejpam-4276	559	14	)	)	PUNCT
ejpam-4276	559	15	.	.	PUNCT
ejpam-4276	560	1	then	then	ADV
ejpam-4276	560	2	,	,	PUNCT
ejpam-4276	560	3	we	we	PRON
ejpam-4276	560	4	have	have	VERB
ejpam-4276	560	5	[	[	X
ejpam-4276	560	6	v	v	X
ejpam-4276	560	7	(	(	PUNCT
ejpam-4276	560	8	λ	λ	PROPN
ejpam-4276	560	9	,	,	PUNCT
ejpam-4276	560	10	sp)](λ	sp)](λ	PROPN
ejpam-4276	560	11	,	,	PUNCT
ejpam-4276	560	12	sp	sp	NOUN
ejpam-4276	560	13	)	)	PUNCT
ejpam-4276	560	14	is	be	AUX
ejpam-4276	560	15	r(λ	r(λ	NOUN
ejpam-4276	560	16	,	,	PUNCT
ejpam-4276	560	17	sp)-open	sp)-open	ADJ
ejpam-4276	560	18	,	,	PUNCT
ejpam-4276	560	19	by	by	ADP
ejpam-4276	560	20	(	(	PUNCT
ejpam-4276	560	21	7	7	NUM
ejpam-4276	560	22	)	)	PUNCT
ejpam-4276	560	23	,	,	PUNCT
ejpam-4276	561	1	[	[	X
ejpam-4276	561	2	[	[	X
ejpam-4276	561	3	v	v	X
ejpam-4276	561	4	(	(	PUNCT
ejpam-4276	561	5	λ	λ	PROPN
ejpam-4276	561	6	,	,	PUNCT
ejpam-4276	561	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	561	8	,	,	PUNCT
ejpam-4276	561	9	sp	sp	NOUN
ejpam-4276	561	10	)	)	PUNCT
ejpam-4276	561	11	]	]	PUNCT
ejpam-4276	561	12	(	(	PUNCT
ejpam-4276	561	13	λ	λ	NOUN
ejpam-4276	561	14	,	,	PUNCT
ejpam-4276	561	15	sp	sp	NOUN
ejpam-4276	561	16	)	)	PUNCT
ejpam-4276	561	17	is	be	AUX
ejpam-4276	561	18	r(λ	r(λ	NOUN
ejpam-4276	561	19	,	,	PUNCT
ejpam-4276	561	20	sp)-open	sp)-open	NOUN
ejpam-4276	561	21	.	.	PUNCT
ejpam-4276	562	1	therefore	therefore	ADV
ejpam-4276	562	2	,	,	PUNCT
ejpam-4276	562	3	v	v	INTJ
ejpam-4276	562	4	(	(	PUNCT
ejpam-4276	562	5	λ	λ	NOUN
ejpam-4276	562	6	,	,	PUNCT
ejpam-4276	562	7	sp	sp	NOUN
ejpam-4276	562	8	)	)	PUNCT
ejpam-4276	562	9	=	=	PUNCT
ejpam-4276	563	1	[	[	X
ejpam-4276	563	2	[	[	X
ejpam-4276	563	3	v	v	X
ejpam-4276	563	4	(	(	PUNCT
ejpam-4276	563	5	λ	λ	PROPN
ejpam-4276	563	6	,	,	PUNCT
ejpam-4276	563	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	563	8	,	,	PUNCT
ejpam-4276	563	9	sp	sp	NOUN
ejpam-4276	563	10	)	)	PUNCT
ejpam-4276	563	11	]	]	PUNCT
ejpam-4276	563	12	(	(	PUNCT
ejpam-4276	563	13	λ	λ	NOUN
ejpam-4276	563	14	,	,	PUNCT
ejpam-4276	563	15	sp	sp	NOUN
ejpam-4276	563	16	)	)	PUNCT
ejpam-4276	563	17	=	=	PUNCT
ejpam-4276	564	1	[	[	X
ejpam-4276	564	2	[	[	X
ejpam-4276	564	3	[	[	X
ejpam-4276	564	4	[	[	X
ejpam-4276	564	5	v	v	X
ejpam-4276	564	6	(	(	PUNCT
ejpam-4276	564	7	λ	λ	PROPN
ejpam-4276	564	8	,	,	PUNCT
ejpam-4276	564	9	sp)](λ	sp)](λ	PROPN
ejpam-4276	564	10	,	,	PUNCT
ejpam-4276	564	11	sp	sp	NOUN
ejpam-4276	564	12	)	)	PUNCT
ejpam-4276	564	13	]	]	PUNCT
ejpam-4276	564	14	(	(	PUNCT
ejpam-4276	564	15	λ	λ	X
ejpam-4276	564	16	,	,	PUNCT
ejpam-4276	564	17	sp)](λ	sp)](λ	PROPN
ejpam-4276	564	18	,	,	PUNCT
ejpam-4276	564	19	sp)](λ	sp)](λ	PROPN
ejpam-4276	564	20	,	,	PUNCT
ejpam-4276	564	21	sp	sp	NOUN
ejpam-4276	564	22	)	)	PUNCT
ejpam-4276	564	23	=	=	PUNCT
ejpam-4276	565	1	[	[	X
ejpam-4276	565	2	v	v	X
ejpam-4276	565	3	(	(	PUNCT
ejpam-4276	565	4	λ	λ	PROPN
ejpam-4276	565	5	,	,	PUNCT
ejpam-4276	565	6	sp)](λ	sp)](λ	PROPN
ejpam-4276	565	7	,	,	PUNCT
ejpam-4276	565	8	sp	sp	NOUN
ejpam-4276	565	9	)	)	PUNCT
ejpam-4276	565	10	=	=	NOUN
ejpam-4276	566	1	[	[	X
ejpam-4276	566	2	[	[	X
ejpam-4276	566	3	v	v	X
ejpam-4276	566	4	(	(	PUNCT
ejpam-4276	566	5	λ	λ	PROPN
ejpam-4276	566	6	,	,	PUNCT
ejpam-4276	566	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	566	8	,	,	PUNCT
ejpam-4276	566	9	sp)](λ	sp)](λ	PROPN
ejpam-4276	566	10	,	,	PUNCT
ejpam-4276	566	11	sp	sp	NOUN
ejpam-4276	566	12	)	)	PUNCT
ejpam-4276	566	13	.	.	PUNCT
ejpam-4276	567	1	thus	thus	ADV
ejpam-4276	567	2	,	,	PUNCT
ejpam-4276	567	3	v	v	INTJ
ejpam-4276	567	4	(	(	PUNCT
ejpam-4276	567	5	λ	λ	NOUN
ejpam-4276	567	6	,	,	PUNCT
ejpam-4276	567	7	sp	sp	NOUN
ejpam-4276	567	8	)	)	PUNCT
ejpam-4276	567	9	∈	∈	PROPN
ejpam-4276	567	10	rλspo(x	rλspo(x	NOUN
ejpam-4276	567	11	,	,	PUNCT
ejpam-4276	567	12	τ	τ	PROPN
ejpam-4276	567	13	)	)	PUNCT
ejpam-4276	567	14	.	.	PUNCT
ejpam-4276	568	1	(	(	PUNCT
ejpam-4276	568	2	8)	8)	NUM
ejpam-4276	568	3	⇒	⇒	NOUN
ejpam-4276	568	4	(	(	PUNCT
ejpam-4276	568	5	1	1	NUM
ejpam-4276	568	6	):	):	PUNCT
ejpam-4276	568	7	the	the	DET
ejpam-4276	568	8	proof	proof	NOUN
ejpam-4276	568	9	is	be	AUX
ejpam-4276	568	10	obvious	obvious	ADJ
ejpam-4276	568	11	.	.	PUNCT
ejpam-4276	569	1	theorem	theorem	NOUN
ejpam-4276	569	2	3	3	NUM
ejpam-4276	569	3	.	.	X
ejpam-4276	569	4	for	for	ADP
ejpam-4276	569	5	a	a	DET
ejpam-4276	569	6	topological	topological	ADJ
ejpam-4276	569	7	space	space	NOUN
ejpam-4276	569	8	(	(	PUNCT
ejpam-4276	569	9	x	x	X
ejpam-4276	569	10	,	,	PUNCT
ejpam-4276	569	11	τ	τ	PROPN
ejpam-4276	569	12	)	)	PUNCT
ejpam-4276	569	13	,	,	PUNCT
ejpam-4276	569	14	the	the	DET
ejpam-4276	569	15	following	follow	VERB
ejpam-4276	569	16	properties	property	NOUN
ejpam-4276	569	17	are	be	AUX
ejpam-4276	569	18	equivalent	equivalent	ADJ
ejpam-4276	569	19	:	:	PUNCT
ejpam-4276	569	20	(	(	PUNCT
ejpam-4276	569	21	1	1	X
ejpam-4276	569	22	)	)	PUNCT
ejpam-4276	569	23	(	(	PUNCT
ejpam-4276	569	24	x	x	X
ejpam-4276	569	25	,	,	PUNCT
ejpam-4276	569	26	τ	τ	X
ejpam-4276	569	27	)	)	PUNCT
ejpam-4276	569	28	is	be	AUX
ejpam-4276	569	29	λsp	λsp	VERB
ejpam-4276	569	30	-	-	PUNCT
ejpam-4276	569	31	extremally	extremally	ADV
ejpam-4276	569	32	disconnected	disconnected	ADJ
ejpam-4276	569	33	.	.	PUNCT
ejpam-4276	570	1	(	(	PUNCT
ejpam-4276	570	2	2	2	X
ejpam-4276	570	3	)	)	PUNCT
ejpam-4276	570	4	rλspc(x	rλspc(x	PROPN
ejpam-4276	570	5	,	,	PUNCT
ejpam-4276	570	6	τ	τ	PROPN
ejpam-4276	570	7	)	)	PUNCT
ejpam-4276	570	8	⊆	⊆	NUM
ejpam-4276	570	9	λspo(x	λspo(x	PROPN
ejpam-4276	570	10	,	,	PUNCT
ejpam-4276	570	11	τ	τ	PROPN
ejpam-4276	570	12	)	)	PUNCT
ejpam-4276	570	13	.	.	PUNCT
ejpam-4276	571	1	(	(	PUNCT
ejpam-4276	571	2	3	3	X
ejpam-4276	571	3	)	)	PUNCT
ejpam-4276	571	4	rλspc(x	rλspc(x	PROPN
ejpam-4276	571	5	,	,	PUNCT
ejpam-4276	571	6	τ	τ	PROPN
ejpam-4276	571	7	)	)	PUNCT
ejpam-4276	571	8	⊆	⊆	NUM
ejpam-4276	571	9	αλspo(x	αλspo(x	NOUN
ejpam-4276	571	10	,	,	PUNCT
ejpam-4276	571	11	τ	τ	PROPN
ejpam-4276	571	12	)	)	PUNCT
ejpam-4276	571	13	.	.	PUNCT
ejpam-4276	572	1	(	(	PUNCT
ejpam-4276	572	2	4	4	X
ejpam-4276	572	3	)	)	PUNCT
ejpam-4276	572	4	rλspc(x	rλspc(x	PROPN
ejpam-4276	572	5	,	,	PUNCT
ejpam-4276	572	6	τ	τ	PROPN
ejpam-4276	572	7	)	)	PUNCT
ejpam-4276	572	8	⊆	⊆	NUM
ejpam-4276	572	9	pλspo(x	pλspo(x	NOUN
ejpam-4276	572	10	,	,	PUNCT
ejpam-4276	572	11	τ	τ	PROPN
ejpam-4276	572	12	)	)	PUNCT
ejpam-4276	572	13	.	.	PUNCT
ejpam-4276	573	1	(	(	PUNCT
ejpam-4276	573	2	5	5	X
ejpam-4276	573	3	)	)	PUNCT
ejpam-4276	573	4	sλspo(x	sλspo(x	PROPN
ejpam-4276	573	5	,	,	PUNCT
ejpam-4276	573	6	τ	τ	PROPN
ejpam-4276	573	7	)	)	PUNCT
ejpam-4276	573	8	⊆	⊆	NUM
ejpam-4276	573	9	αλspo(x	αλspo(x	NOUN
ejpam-4276	573	10	,	,	PUNCT
ejpam-4276	573	11	τ	τ	PROPN
ejpam-4276	573	12	)	)	PUNCT
ejpam-4276	573	13	.	.	PUNCT
ejpam-4276	574	1	(	(	PUNCT
ejpam-4276	574	2	6	6	X
ejpam-4276	574	3	)	)	PUNCT
ejpam-4276	574	4	sλspc(x	sλspc(x	NOUN
ejpam-4276	574	5	,	,	PUNCT
ejpam-4276	574	6	τ	τ	PROPN
ejpam-4276	574	7	)	)	PUNCT
ejpam-4276	574	8	⊆	⊆	NUM
ejpam-4276	574	9	αλspc(x	αλspc(x	NOUN
ejpam-4276	574	10	,	,	PUNCT
ejpam-4276	574	11	τ	τ	PROPN
ejpam-4276	574	12	)	)	PUNCT
ejpam-4276	574	13	.	.	PUNCT
ejpam-4276	575	1	(	(	PUNCT
ejpam-4276	575	2	7	7	X
ejpam-4276	575	3	)	)	PUNCT
ejpam-4276	575	4	sλspc(x	sλspc(x	NOUN
ejpam-4276	575	5	,	,	PUNCT
ejpam-4276	575	6	τ	τ	PROPN
ejpam-4276	575	7	)	)	PUNCT
ejpam-4276	575	8	⊆	⊆	NUM
ejpam-4276	575	9	pλspc(x	pλspc(x	NOUN
ejpam-4276	575	10	,	,	PUNCT
ejpam-4276	575	11	τ	τ	PROPN
ejpam-4276	575	12	)	)	PUNCT
ejpam-4276	575	13	.	.	PUNCT
ejpam-4276	576	1	(	(	PUNCT
ejpam-4276	576	2	8)	8)	NUM
ejpam-4276	576	3	sλspo(x	sλspo(x	PROPN
ejpam-4276	576	4	,	,	PUNCT
ejpam-4276	576	5	τ	τ	PROPN
ejpam-4276	576	6	)	)	PUNCT
ejpam-4276	576	7	⊆	⊆	NUM
ejpam-4276	576	8	pλspo(x	pλspo(x	NOUN
ejpam-4276	576	9	,	,	PUNCT
ejpam-4276	576	10	τ	τ	PROPN
ejpam-4276	576	11	)	)	PUNCT
ejpam-4276	576	12	.	.	PUNCT
ejpam-4276	577	1	(	(	PUNCT
ejpam-4276	577	2	9	9	X
ejpam-4276	577	3	)	)	PUNCT
ejpam-4276	577	4	βλspo(x	βλspo(x	PROPN
ejpam-4276	577	5	,	,	PUNCT
ejpam-4276	577	6	τ	τ	PROPN
ejpam-4276	577	7	)	)	PUNCT
ejpam-4276	577	8	⊆	⊆	NUM
ejpam-4276	577	9	pλspo(x	pλspo(x	NOUN
ejpam-4276	577	10	,	,	PUNCT
ejpam-4276	577	11	τ	τ	PROPN
ejpam-4276	577	12	)	)	PUNCT
ejpam-4276	577	13	.	.	PUNCT
ejpam-4276	578	1	c.	c.	PROPN
ejpam-4276	578	2	boonpok	boonpok	PROPN
ejpam-4276	578	3	,	,	PUNCT
ejpam-4276	578	4	j.	j.	PROPN
ejpam-4276	578	5	khampakdee	khampakdee	PROPN
ejpam-4276	578	6	/	/	PUNCT
ejpam-4276	578	7	eur	eur	PROPN
ejpam-4276	578	8	.	.	PUNCT
ejpam-4276	579	1	j.	j.	PROPN
ejpam-4276	579	2	pure	pure	PROPN
ejpam-4276	579	3	appl	appl	PROPN
ejpam-4276	579	4	.	.	PROPN
ejpam-4276	579	5	math	math	PROPN
ejpam-4276	579	6	,	,	PUNCT
ejpam-4276	579	7	15	15	NUM
ejpam-4276	579	8	(	(	PUNCT
ejpam-4276	579	9	2	2	NUM
ejpam-4276	579	10	)	)	PUNCT
ejpam-4276	579	11	(	(	PUNCT
ejpam-4276	579	12	2022	2022	NUM
ejpam-4276	579	13	)	)	PUNCT
ejpam-4276	579	14	,	,	PUNCT
ejpam-4276	579	15	572	572	NUM
ejpam-4276	579	16	-	-	SYM
ejpam-4276	579	17	588	588	NUM
ejpam-4276	579	18	587	587	NUM
ejpam-4276	579	19	(	(	PUNCT
ejpam-4276	579	20	10	10	NUM
ejpam-4276	579	21	)	)	PUNCT
ejpam-4276	579	22	βλspc(x	βλspc(x	PROPN
ejpam-4276	579	23	,	,	PUNCT
ejpam-4276	579	24	τ	τ	PROPN
ejpam-4276	579	25	)	)	PUNCT
ejpam-4276	579	26	⊆	⊆	NUM
ejpam-4276	579	27	pλspc(x	pλspc(x	NOUN
ejpam-4276	579	28	,	,	PUNCT
ejpam-4276	579	29	τ	τ	PROPN
ejpam-4276	579	30	)	)	PUNCT
ejpam-4276	579	31	.	.	PUNCT
ejpam-4276	580	1	(	(	PUNCT
ejpam-4276	580	2	11	11	X
ejpam-4276	580	3	)	)	PUNCT
ejpam-4276	580	4	bλspc(x	bλspc(x	NOUN
ejpam-4276	580	5	,	,	PUNCT
ejpam-4276	580	6	τ	τ	PROPN
ejpam-4276	580	7	)	)	PUNCT
ejpam-4276	580	8	⊆	⊆	NUM
ejpam-4276	580	9	pλspc(x	pλspc(x	NOUN
ejpam-4276	580	10	,	,	PUNCT
ejpam-4276	580	11	τ	τ	PROPN
ejpam-4276	580	12	)	)	PUNCT
ejpam-4276	580	13	.	.	PUNCT
ejpam-4276	581	1	(	(	PUNCT
ejpam-4276	581	2	12	12	NUM
ejpam-4276	581	3	)	)	PUNCT
ejpam-4276	581	4	bλspo(x	bλspo(x	NOUN
ejpam-4276	581	5	,	,	PUNCT
ejpam-4276	581	6	τ	τ	PROPN
ejpam-4276	581	7	)	)	PUNCT
ejpam-4276	581	8	⊆	⊆	NUM
ejpam-4276	581	9	pλspo(x	pλspo(x	NOUN
ejpam-4276	581	10	,	,	PUNCT
ejpam-4276	581	11	τ	τ	PROPN
ejpam-4276	581	12	)	)	PUNCT
ejpam-4276	581	13	.	.	PUNCT
ejpam-4276	582	1	(	(	PUNCT
ejpam-4276	582	2	13	13	X
ejpam-4276	582	3	)	)	PUNCT
ejpam-4276	582	4	rλspo(x	rλspo(x	NOUN
ejpam-4276	582	5	,	,	PUNCT
ejpam-4276	582	6	τ	τ	PROPN
ejpam-4276	582	7	)	)	PUNCT
ejpam-4276	582	8	⊆	⊆	NUM
ejpam-4276	582	9	pλspc(x	pλspc(x	NOUN
ejpam-4276	582	10	,	,	PUNCT
ejpam-4276	582	11	τ	τ	PROPN
ejpam-4276	582	12	)	)	PUNCT
ejpam-4276	582	13	.	.	PUNCT
ejpam-4276	583	1	(	(	PUNCT
ejpam-4276	583	2	14	14	X
ejpam-4276	583	3	)	)	PUNCT
ejpam-4276	583	4	rλspo(x	rλspo(x	NOUN
ejpam-4276	583	5	,	,	PUNCT
ejpam-4276	583	6	τ	τ	PROPN
ejpam-4276	583	7	)	)	PUNCT
ejpam-4276	583	8	⊆	⊆	NUM
ejpam-4276	583	9	λspc(x	λspc(x	PROPN
ejpam-4276	583	10	,	,	PUNCT
ejpam-4276	583	11	τ	τ	PROPN
ejpam-4276	583	12	)	)	PUNCT
ejpam-4276	583	13	.	.	PUNCT
ejpam-4276	584	1	(	(	PUNCT
ejpam-4276	584	2	15	15	X
ejpam-4276	584	3	)	)	PUNCT
ejpam-4276	584	4	rλspo(x	rλspo(x	NOUN
ejpam-4276	584	5	,	,	PUNCT
ejpam-4276	584	6	τ	τ	PROPN
ejpam-4276	584	7	)	)	PUNCT
ejpam-4276	584	8	⊆	⊆	NUM
ejpam-4276	584	9	αλspc(x	αλspc(x	NOUN
ejpam-4276	584	10	,	,	PUNCT
ejpam-4276	584	11	τ	τ	PROPN
ejpam-4276	584	12	)	)	PUNCT
ejpam-4276	584	13	.	.	PUNCT
ejpam-4276	585	1	proof	proof	NOUN
ejpam-4276	585	2	.	.	PUNCT
ejpam-4276	586	1	(	(	PUNCT
ejpam-4276	586	2	1	1	X
ejpam-4276	586	3	)	)	PUNCT
ejpam-4276	586	4	⇒	⇒	NOUN
ejpam-4276	586	5	(	(	PUNCT
ejpam-4276	586	6	2	2	NUM
ejpam-4276	586	7	):	):	PUNCT
ejpam-4276	586	8	let	let	VERB
ejpam-4276	586	9	v	v	NUM
ejpam-4276	586	10	∈	∈	PROPN
ejpam-4276	586	11	rλspc(x	rλspc(x	PROPN
ejpam-4276	586	12	,	,	PUNCT
ejpam-4276	586	13	τ	τ	PROPN
ejpam-4276	586	14	)	)	PUNCT
ejpam-4276	586	15	.	.	PUNCT
ejpam-4276	587	1	then	then	ADV
ejpam-4276	587	2	,	,	PUNCT
ejpam-4276	587	3	we	we	PRON
ejpam-4276	587	4	have	have	VERB
ejpam-4276	587	5	v	v	NOUN
ejpam-4276	587	6	=	=	SYM
ejpam-4276	588	1	[	[	X
ejpam-4276	588	2	v(λ	v(λ	PROPN
ejpam-4276	588	3	,	,	PUNCT
ejpam-4276	588	4	sp	sp	NOUN
ejpam-4276	588	5	)	)	PUNCT
ejpam-4276	588	6	]	]	PUNCT
ejpam-4276	588	7	(	(	PUNCT
ejpam-4276	588	8	λ	λ	NOUN
ejpam-4276	588	9	,	,	PUNCT
ejpam-4276	588	10	sp	sp	NOUN
ejpam-4276	588	11	)	)	PUNCT
ejpam-4276	588	12	.	.	PUNCT
ejpam-4276	589	1	since	since	SCONJ
ejpam-4276	589	2	(	(	PUNCT
ejpam-4276	589	3	x	x	X
ejpam-4276	589	4	,	,	PUNCT
ejpam-4276	589	5	τ	τ	X
ejpam-4276	589	6	)	)	PUNCT
ejpam-4276	589	7	is	be	AUX
ejpam-4276	589	8	λsp	λsp	VERB
ejpam-4276	589	9	-	-	PUNCT
ejpam-4276	589	10	extremally	extremally	ADV
ejpam-4276	589	11	disconnected	disconnect	VERB
ejpam-4276	589	12	,	,	PUNCT
ejpam-4276	589	13	v(λ	v(λ	PROPN
ejpam-4276	589	14	,	,	PUNCT
ejpam-4276	589	15	sp	sp	NOUN
ejpam-4276	589	16	)	)	PUNCT
ejpam-4276	589	17	=	=	PUNCT
ejpam-4276	590	1	[	[	X
ejpam-4276	590	2	[	[	X
ejpam-4276	590	3	v(λ	v(λ	PROPN
ejpam-4276	590	4	,	,	PUNCT
ejpam-4276	590	5	sp	sp	NOUN
ejpam-4276	590	6	)	)	PUNCT
ejpam-4276	590	7	]	]	PUNCT
ejpam-4276	590	8	(	(	PUNCT
ejpam-4276	590	9	λ	λ	X
ejpam-4276	590	10	,	,	PUNCT
ejpam-4276	590	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	590	12	,	,	PUNCT
ejpam-4276	590	13	sp	sp	NOUN
ejpam-4276	590	14	)	)	PUNCT
ejpam-4276	590	15	=	=	NOUN
ejpam-4276	591	1	[	[	X
ejpam-4276	591	2	v(λ	v(λ	PROPN
ejpam-4276	591	3	,	,	PUNCT
ejpam-4276	591	4	sp	sp	NOUN
ejpam-4276	591	5	)	)	PUNCT
ejpam-4276	591	6	]	]	PUNCT
ejpam-4276	591	7	(	(	PUNCT
ejpam-4276	591	8	λ	λ	NOUN
ejpam-4276	591	9	,	,	PUNCT
ejpam-4276	591	10	sp	sp	NOUN
ejpam-4276	591	11	)	)	PUNCT
ejpam-4276	591	12	=	=	SYM
ejpam-4276	591	13	v	v	NOUN
ejpam-4276	591	14	and	and	CCONJ
ejpam-4276	591	15	hence	hence	ADV
ejpam-4276	591	16	v	v	NOUN
ejpam-4276	591	17	∈	∈	ADJ
ejpam-4276	591	18	λspo(x	λspo(x	NOUN
ejpam-4276	591	19	,	,	PUNCT
ejpam-4276	591	20	τ	τ	PROPN
ejpam-4276	591	21	)	)	PUNCT
ejpam-4276	591	22	.	.	PUNCT
ejpam-4276	592	1	consequently	consequently	ADV
ejpam-4276	592	2	,	,	PUNCT
ejpam-4276	592	3	we	we	PRON
ejpam-4276	592	4	obtain	obtain	VERB
ejpam-4276	592	5	rλspc(x	rλspc(x	PROPN
ejpam-4276	592	6	,	,	PUNCT
ejpam-4276	592	7	τ	τ	PROPN
ejpam-4276	592	8	)	)	PUNCT
ejpam-4276	592	9	⊆	⊆	NUM
ejpam-4276	592	10	λspo(x	λspo(x	PROPN
ejpam-4276	592	11	,	,	PUNCT
ejpam-4276	592	12	τ	τ	PROPN
ejpam-4276	592	13	)	)	PUNCT
ejpam-4276	592	14	.	.	PUNCT
ejpam-4276	593	1	(	(	PUNCT
ejpam-4276	593	2	2	2	X
ejpam-4276	593	3	)	)	PUNCT
ejpam-4276	593	4	⇒	⇒	NOUN
ejpam-4276	593	5	(	(	PUNCT
ejpam-4276	593	6	3	3	NUM
ejpam-4276	593	7	)	)	PUNCT
ejpam-4276	593	8	⇒	⇒	NOUN
ejpam-4276	593	9	(	(	PUNCT
ejpam-4276	593	10	4	4	NUM
ejpam-4276	593	11	):	):	PUNCT
ejpam-4276	593	12	obvious	obvious	ADJ
ejpam-4276	593	13	.	.	PUNCT
ejpam-4276	594	1	(	(	PUNCT
ejpam-4276	594	2	4	4	X
ejpam-4276	594	3	)	)	PUNCT
ejpam-4276	594	4	⇒	⇒	NOUN
ejpam-4276	594	5	(	(	PUNCT
ejpam-4276	594	6	5	5	NUM
ejpam-4276	594	7	):	):	PUNCT
ejpam-4276	594	8	let	let	VERB
ejpam-4276	594	9	v	v	NUM
ejpam-4276	594	10	∈	∈	PROPN
ejpam-4276	594	11	sλspo(x	sλspo(x	PROPN
ejpam-4276	594	12	,	,	PUNCT
ejpam-4276	594	13	τ	τ	PROPN
ejpam-4276	594	14	)	)	PUNCT
ejpam-4276	594	15	.	.	PUNCT
ejpam-4276	595	1	then	then	ADV
ejpam-4276	595	2	,	,	PUNCT
ejpam-4276	595	3	we	we	PRON
ejpam-4276	595	4	have	have	VERB
ejpam-4276	595	5	v	v	ADP
ejpam-4276	595	6	⊆	⊆	NUM
ejpam-4276	595	7	[	[	X
ejpam-4276	595	8	v(λ	v(λ	PROPN
ejpam-4276	595	9	,	,	PUNCT
ejpam-4276	595	10	sp	sp	NOUN
ejpam-4276	595	11	)	)	PUNCT
ejpam-4276	595	12	]	]	PUNCT
ejpam-4276	595	13	(	(	PUNCT
ejpam-4276	595	14	λ	λ	NOUN
ejpam-4276	595	15	,	,	PUNCT
ejpam-4276	595	16	sp	sp	NOUN
ejpam-4276	595	17	)	)	PUNCT
ejpam-4276	595	18	.	.	PUNCT
ejpam-4276	596	1	since	since	SCONJ
ejpam-4276	596	2	[	[	X
ejpam-4276	596	3	v(λ	v(λ	PROPN
ejpam-4276	596	4	,	,	PUNCT
ejpam-4276	596	5	sp	sp	NOUN
ejpam-4276	596	6	)	)	PUNCT
ejpam-4276	596	7	]	]	PUNCT
ejpam-4276	596	8	(	(	PUNCT
ejpam-4276	596	9	λ	λ	NOUN
ejpam-4276	596	10	,	,	PUNCT
ejpam-4276	596	11	sp	sp	NOUN
ejpam-4276	596	12	)	)	PUNCT
ejpam-4276	596	13	is	be	AUX
ejpam-4276	596	14	r(λ	r(λ	NOUN
ejpam-4276	596	15	,	,	PUNCT
ejpam-4276	596	16	sp)-closed	sp)-close	VERB
ejpam-4276	596	17	,	,	PUNCT
ejpam-4276	596	18	by	by	ADP
ejpam-4276	596	19	(	(	PUNCT
ejpam-4276	596	20	4	4	NUM
ejpam-4276	596	21	)	)	PUNCT
ejpam-4276	596	22	,	,	PUNCT
ejpam-4276	597	1	[	[	X
ejpam-4276	597	2	v(λ	v(λ	PROPN
ejpam-4276	597	3	,	,	PUNCT
ejpam-4276	597	4	sp	sp	NOUN
ejpam-4276	597	5	)	)	PUNCT
ejpam-4276	597	6	]	]	PUNCT
ejpam-4276	598	1	(	(	PUNCT
ejpam-4276	598	2	λ	λ	NOUN
ejpam-4276	598	3	,	,	PUNCT
ejpam-4276	598	4	sp	sp	NOUN
ejpam-4276	598	5	)	)	PUNCT
ejpam-4276	598	6	is	be	AUX
ejpam-4276	598	7	p(λ	p(λ	NOUN
ejpam-4276	598	8	,	,	PUNCT
ejpam-4276	598	9	sp)-open	sp)-open	NOUN
ejpam-4276	598	10	and	and	CCONJ
ejpam-4276	598	11	hence	hence	ADV
ejpam-4276	598	12	[	[	X
ejpam-4276	598	13	v(λ	v(λ	PROPN
ejpam-4276	598	14	,	,	PUNCT
ejpam-4276	598	15	sp	sp	NOUN
ejpam-4276	598	16	)	)	PUNCT
ejpam-4276	598	17	]	]	PUNCT
ejpam-4276	599	1	(	(	PUNCT
ejpam-4276	599	2	λ	λ	NOUN
ejpam-4276	599	3	,	,	PUNCT
ejpam-4276	599	4	sp	sp	NOUN
ejpam-4276	599	5	)	)	PUNCT
ejpam-4276	599	6	⊆	⊆	NUM
ejpam-4276	600	1	[	[	X
ejpam-4276	600	2	[	[	X
ejpam-4276	600	3	[	[	X
ejpam-4276	600	4	v(λ	v(λ	PROPN
ejpam-4276	600	5	,	,	PUNCT
ejpam-4276	600	6	sp	sp	NOUN
ejpam-4276	600	7	)	)	PUNCT
ejpam-4276	600	8	]	]	PUNCT
ejpam-4276	600	9	(	(	PUNCT
ejpam-4276	600	10	λ	λ	X
ejpam-4276	600	11	,	,	PUNCT
ejpam-4276	600	12	sp)](λ	sp)](λ	PROPN
ejpam-4276	600	13	,	,	PUNCT
ejpam-4276	600	14	sp)](λ	sp)](λ	PROPN
ejpam-4276	600	15	,	,	PUNCT
ejpam-4276	600	16	sp	sp	NOUN
ejpam-4276	600	17	)	)	PUNCT
ejpam-4276	600	18	=	=	PUNCT
ejpam-4276	601	1	[	[	X
ejpam-4276	601	2	[	[	X
ejpam-4276	601	3	v(λ	v(λ	PROPN
ejpam-4276	601	4	,	,	PUNCT
ejpam-4276	601	5	sp	sp	NOUN
ejpam-4276	601	6	)	)	PUNCT
ejpam-4276	601	7	]	]	PUNCT
ejpam-4276	601	8	(	(	PUNCT
ejpam-4276	601	9	λ	λ	X
ejpam-4276	601	10	,	,	PUNCT
ejpam-4276	601	11	sp)](λ	sp)](λ	PROPN
ejpam-4276	601	12	,	,	PUNCT
ejpam-4276	601	13	sp	sp	NOUN
ejpam-4276	601	14	)	)	PUNCT
ejpam-4276	601	15	.	.	PUNCT
ejpam-4276	602	1	this	this	PRON
ejpam-4276	602	2	implies	imply	VERB
ejpam-4276	602	3	that	that	SCONJ
ejpam-4276	602	4	v	v	ADP
ejpam-4276	602	5	⊆	⊆	NUM
ejpam-4276	602	6	[	[	X
ejpam-4276	602	7	[	[	X
ejpam-4276	602	8	v(λ	v(λ	PROPN
ejpam-4276	602	9	,	,	PUNCT
ejpam-4276	602	10	sp	sp	NOUN
ejpam-4276	602	11	)	)	PUNCT
ejpam-4276	602	12	]	]	PUNCT
ejpam-4276	602	13	(	(	PUNCT
ejpam-4276	602	14	λ	λ	X
ejpam-4276	602	15	,	,	PUNCT
ejpam-4276	602	16	sp)](λ	sp)](λ	PROPN
ejpam-4276	602	17	,	,	PUNCT
ejpam-4276	602	18	sp	sp	NOUN
ejpam-4276	602	19	)	)	PUNCT
ejpam-4276	602	20	and	and	CCONJ
ejpam-4276	602	21	hence	hence	ADV
ejpam-4276	602	22	v	v	ADP
ejpam-4276	602	23	∈	∈	PROPN
ejpam-4276	602	24	αλspo(x	αλspo(x	NOUN
ejpam-4276	602	25	,	,	PUNCT
ejpam-4276	602	26	τ	τ	PROPN
ejpam-4276	602	27	)	)	PUNCT
ejpam-4276	602	28	.	.	PUNCT
ejpam-4276	603	1	thus	thus	ADV
ejpam-4276	603	2	,	,	PUNCT
ejpam-4276	603	3	sλspo(x	sλspo(x	PROPN
ejpam-4276	603	4	,	,	PUNCT
ejpam-4276	603	5	τ	τ	PROPN
ejpam-4276	603	6	)	)	PUNCT
ejpam-4276	603	7	⊆	⊆	NUM
ejpam-4276	603	8	αλspo(x	αλspo(x	NOUN
ejpam-4276	603	9	,	,	PUNCT
ejpam-4276	603	10	τ	τ	PROPN
ejpam-4276	603	11	)	)	PUNCT
ejpam-4276	603	12	.	.	PUNCT
ejpam-4276	604	1	(	(	PUNCT
ejpam-4276	604	2	5	5	X
ejpam-4276	604	3	)	)	PUNCT
ejpam-4276	604	4	⇒	⇒	NOUN
ejpam-4276	604	5	(	(	PUNCT
ejpam-4276	604	6	6	6	NUM
ejpam-4276	604	7	)	)	PUNCT
ejpam-4276	604	8	⇒	⇒	NOUN
ejpam-4276	604	9	(	(	PUNCT
ejpam-4276	604	10	7	7	NUM
ejpam-4276	604	11	)	)	PUNCT
ejpam-4276	604	12	⇒	⇒	NOUN
ejpam-4276	604	13	(	(	PUNCT
ejpam-4276	604	14	8)	8)	NUM
ejpam-4276	604	15	:	:	SYM
ejpam-4276	604	16	obvious	obvious	ADJ
ejpam-4276	604	17	.	.	PUNCT
ejpam-4276	605	1	(	(	PUNCT
ejpam-4276	605	2	8)	8)	NUM
ejpam-4276	605	3	⇒	⇒	NOUN
ejpam-4276	605	4	(	(	PUNCT
ejpam-4276	605	5	9	9	NUM
ejpam-4276	605	6	):	):	PUNCT
ejpam-4276	605	7	let	let	VERB
ejpam-4276	605	8	v	v	PRON
ejpam-4276	605	9	∈	∈	PROPN
ejpam-4276	605	10	βλspo(x	βλspo(x	PROPN
ejpam-4276	605	11	,	,	PUNCT
ejpam-4276	605	12	τ	τ	PROPN
ejpam-4276	605	13	)	)	PUNCT
ejpam-4276	605	14	.	.	PUNCT
ejpam-4276	606	1	by	by	ADP
ejpam-4276	606	2	proposition	proposition	NOUN
ejpam-4276	606	3	13	13	NUM
ejpam-4276	606	4	,	,	PUNCT
ejpam-4276	606	5	v	v	NOUN
ejpam-4276	606	6	(	(	PUNCT
ejpam-4276	606	7	λ	λ	NOUN
ejpam-4276	606	8	,	,	PUNCT
ejpam-4276	606	9	sp	sp	NOUN
ejpam-4276	606	10	)	)	PUNCT
ejpam-4276	606	11	is	be	AUX
ejpam-4276	606	12	s(λ	s(λ	PROPN
ejpam-4276	606	13	,	,	PUNCT
ejpam-4276	606	14	sp)-open	sp)-open	ADJ
ejpam-4276	606	15	,	,	PUNCT
ejpam-4276	606	16	by	by	ADP
ejpam-4276	606	17	(	(	PUNCT
ejpam-4276	606	18	8)	8)	NUM
ejpam-4276	606	19	,	,	PUNCT
ejpam-4276	606	20	v	v	NOUN
ejpam-4276	606	21	(	(	PUNCT
ejpam-4276	606	22	λ	λ	NOUN
ejpam-4276	606	23	,	,	PUNCT
ejpam-4276	606	24	sp	sp	NOUN
ejpam-4276	606	25	)	)	PUNCT
ejpam-4276	606	26	is	be	AUX
ejpam-4276	606	27	p(λ	p(λ	NOUN
ejpam-4276	606	28	,	,	PUNCT
ejpam-4276	606	29	sp)-open	sp)-open	NOUN
ejpam-4276	606	30	.	.	PUNCT
ejpam-4276	607	1	thus	thus	ADV
ejpam-4276	607	2	,	,	PUNCT
ejpam-4276	607	3	v	v	INTJ
ejpam-4276	607	4	(	(	PUNCT
ejpam-4276	607	5	λ	λ	NOUN
ejpam-4276	607	6	,	,	PUNCT
ejpam-4276	607	7	sp	sp	NOUN
ejpam-4276	607	8	)	)	PUNCT
ejpam-4276	607	9	⊆	⊆	NUM
ejpam-4276	608	1	[	[	X
ejpam-4276	608	2	[	[	X
ejpam-4276	608	3	v	v	X
ejpam-4276	608	4	(	(	PUNCT
ejpam-4276	608	5	λ	λ	PROPN
ejpam-4276	608	6	,	,	PUNCT
ejpam-4276	608	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	608	8	,	,	PUNCT
ejpam-4276	608	9	sp)](λ	sp)](λ	PROPN
ejpam-4276	608	10	,	,	PUNCT
ejpam-4276	608	11	sp	sp	NOUN
ejpam-4276	608	12	)	)	PUNCT
ejpam-4276	608	13	=	=	PUNCT
ejpam-4276	609	1	[	[	X
ejpam-4276	609	2	v	v	X
ejpam-4276	609	3	(	(	PUNCT
ejpam-4276	609	4	λ	λ	PROPN
ejpam-4276	609	5	,	,	PUNCT
ejpam-4276	609	6	sp)](λ	sp)](λ	PROPN
ejpam-4276	609	7	,	,	PUNCT
ejpam-4276	609	8	sp	sp	NOUN
ejpam-4276	609	9	)	)	PUNCT
ejpam-4276	609	10	and	and	CCONJ
ejpam-4276	609	11	hence	hence	ADV
ejpam-4276	609	12	v	v	ADP
ejpam-4276	609	13	⊆	⊆	NUM
ejpam-4276	609	14	[	[	X
ejpam-4276	609	15	v	v	X
ejpam-4276	609	16	(	(	PUNCT
ejpam-4276	609	17	λ	λ	PROPN
ejpam-4276	609	18	,	,	PUNCT
ejpam-4276	609	19	sp)](λ	sp)](λ	PROPN
ejpam-4276	609	20	,	,	PUNCT
ejpam-4276	609	21	sp	sp	NOUN
ejpam-4276	609	22	)	)	PUNCT
ejpam-4276	609	23	.	.	PUNCT
ejpam-4276	610	1	therefore	therefore	ADV
ejpam-4276	610	2	,	,	PUNCT
ejpam-4276	610	3	v	v	PROPN
ejpam-4276	610	4	∈	∈	PROPN
ejpam-4276	610	5	pλspo(x	pλspo(x	NOUN
ejpam-4276	610	6	,	,	PUNCT
ejpam-4276	610	7	τ	τ	PROPN
ejpam-4276	610	8	)	)	PUNCT
ejpam-4276	610	9	.	.	PUNCT
ejpam-4276	611	1	this	this	PRON
ejpam-4276	611	2	shows	show	VERB
ejpam-4276	611	3	that	that	SCONJ
ejpam-4276	611	4	βλspo(x	βλspo(x	PROPN
ejpam-4276	611	5	,	,	PUNCT
ejpam-4276	611	6	τ	τ	PROPN
ejpam-4276	611	7	)	)	PUNCT
ejpam-4276	611	8	⊆	⊆	NUM
ejpam-4276	611	9	pλspo(x	pλspo(x	NOUN
ejpam-4276	611	10	,	,	PUNCT
ejpam-4276	611	11	τ	τ	PROPN
ejpam-4276	611	12	)	)	PUNCT
ejpam-4276	611	13	.	.	PUNCT
ejpam-4276	612	1	(	(	PUNCT
ejpam-4276	612	2	9	9	X
ejpam-4276	612	3	)	)	PUNCT
ejpam-4276	612	4	⇒	⇒	NOUN
ejpam-4276	612	5	(	(	PUNCT
ejpam-4276	612	6	10	10	NUM
ejpam-4276	612	7	)	)	PUNCT
ejpam-4276	612	8	⇒	⇒	NOUN
ejpam-4276	612	9	(	(	PUNCT
ejpam-4276	612	10	11	11	NUM
ejpam-4276	612	11	)	)	PUNCT
ejpam-4276	612	12	⇒	⇒	NOUN
ejpam-4276	612	13	(	(	PUNCT
ejpam-4276	612	14	12	12	NUM
ejpam-4276	612	15	):	):	PUNCT
ejpam-4276	612	16	obvious	obvious	ADJ
ejpam-4276	612	17	.	.	PUNCT
ejpam-4276	613	1	(	(	PUNCT
ejpam-4276	613	2	12	12	NUM
ejpam-4276	613	3	)	)	PUNCT
ejpam-4276	613	4	⇒	⇒	NOUN
ejpam-4276	613	5	(	(	PUNCT
ejpam-4276	613	6	13	13	NUM
ejpam-4276	613	7	):	):	PUNCT
ejpam-4276	613	8	let	let	VERB
ejpam-4276	613	9	v	v	NUM
ejpam-4276	613	10	∈	∈	PROPN
ejpam-4276	613	11	rλspo(x	rλspo(x	NOUN
ejpam-4276	613	12	,	,	PUNCT
ejpam-4276	613	13	τ	τ	PROPN
ejpam-4276	613	14	)	)	PUNCT
ejpam-4276	613	15	.	.	PUNCT
ejpam-4276	614	1	then	then	ADV
ejpam-4276	614	2	,	,	PUNCT
ejpam-4276	614	3	v	v	NOUN
ejpam-4276	614	4	is	be	AUX
ejpam-4276	614	5	β(λ	β(λ	X
ejpam-4276	614	6	,	,	PUNCT
ejpam-4276	614	7	sp)-open	sp)-open	ADJ
ejpam-4276	614	8	,	,	PUNCT
ejpam-4276	614	9	by	by	ADP
ejpam-4276	614	10	proposition	proposition	NOUN
ejpam-4276	614	11	13	13	NUM
ejpam-4276	614	12	,	,	PUNCT
ejpam-4276	614	13	v	v	NOUN
ejpam-4276	614	14	(	(	PUNCT
ejpam-4276	614	15	λ	λ	NOUN
ejpam-4276	614	16	,	,	PUNCT
ejpam-4276	614	17	sp	sp	NOUN
ejpam-4276	614	18	)	)	PUNCT
ejpam-4276	614	19	is	be	AUX
ejpam-4276	614	20	b(λ	b(λ	NOUN
ejpam-4276	614	21	,	,	PUNCT
ejpam-4276	614	22	sp)-open	sp)-open	ADJ
ejpam-4276	614	23	and	and	CCONJ
ejpam-4276	614	24	by	by	ADP
ejpam-4276	614	25	(	(	PUNCT
ejpam-4276	614	26	12	12	NUM
ejpam-4276	614	27	)	)	PUNCT
ejpam-4276	614	28	,	,	PUNCT
ejpam-4276	614	29	v	v	X
ejpam-4276	614	30	(	(	PUNCT
ejpam-4276	614	31	λ	λ	NOUN
ejpam-4276	614	32	,	,	PUNCT
ejpam-4276	614	33	sp	sp	NOUN
ejpam-4276	614	34	)	)	PUNCT
ejpam-4276	614	35	is	be	AUX
ejpam-4276	614	36	p(λ	p(λ	NOUN
ejpam-4276	614	37	,	,	PUNCT
ejpam-4276	614	38	sp)-open	sp)-open	NOUN
ejpam-4276	614	39	.	.	PUNCT
ejpam-4276	615	1	thus	thus	ADV
ejpam-4276	615	2	,	,	PUNCT
ejpam-4276	615	3	v	v	INTJ
ejpam-4276	615	4	(	(	PUNCT
ejpam-4276	615	5	λ	λ	NOUN
ejpam-4276	615	6	,	,	PUNCT
ejpam-4276	615	7	sp	sp	NOUN
ejpam-4276	615	8	)	)	PUNCT
ejpam-4276	615	9	⊆	⊆	NUM
ejpam-4276	616	1	[	[	X
ejpam-4276	616	2	[	[	X
ejpam-4276	616	3	v	v	X
ejpam-4276	616	4	(	(	PUNCT
ejpam-4276	616	5	λ	λ	PROPN
ejpam-4276	616	6	,	,	PUNCT
ejpam-4276	616	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	616	8	,	,	PUNCT
ejpam-4276	616	9	sp)](λ	sp)](λ	PROPN
ejpam-4276	616	10	,	,	PUNCT
ejpam-4276	616	11	sp	sp	NOUN
ejpam-4276	616	12	)	)	PUNCT
ejpam-4276	616	13	=	=	PUNCT
ejpam-4276	617	1	[	[	X
ejpam-4276	617	2	v	v	X
ejpam-4276	617	3	(	(	PUNCT
ejpam-4276	617	4	λ	λ	PROPN
ejpam-4276	617	5	,	,	PUNCT
ejpam-4276	617	6	sp)](λ	sp)](λ	PROPN
ejpam-4276	617	7	,	,	PUNCT
ejpam-4276	617	8	sp	sp	NOUN
ejpam-4276	617	9	)	)	PUNCT
ejpam-4276	617	10	=	=	VERB
ejpam-4276	617	11	v.	v.	CCONJ
ejpam-4276	617	12	therefore	therefore	ADV
ejpam-4276	617	13	,	,	PUNCT
ejpam-4276	617	14	v	v	NOUN
ejpam-4276	617	15	is	be	AUX
ejpam-4276	617	16	p(λ	p(λ	NOUN
ejpam-4276	617	17	,	,	PUNCT
ejpam-4276	617	18	sp)-closed	sp)-close	VERB
ejpam-4276	617	19	and	and	CCONJ
ejpam-4276	617	20	hence	hence	ADV
ejpam-4276	617	21	rλspo(x	rλspo(x	PROPN
ejpam-4276	617	22	,	,	PUNCT
ejpam-4276	617	23	τ	τ	PROPN
ejpam-4276	617	24	)	)	PUNCT
ejpam-4276	617	25	⊆	⊆	NUM
ejpam-4276	617	26	pλspc(x	pλspc(x	NOUN
ejpam-4276	617	27	,	,	PUNCT
ejpam-4276	617	28	τ	τ	PROPN
ejpam-4276	617	29	)	)	PUNCT
ejpam-4276	617	30	.	.	PUNCT
ejpam-4276	618	1	(	(	PUNCT
ejpam-4276	618	2	13	13	NUM
ejpam-4276	618	3	)	)	PUNCT
ejpam-4276	618	4	⇒	⇒	NOUN
ejpam-4276	618	5	(	(	PUNCT
ejpam-4276	618	6	14	14	NUM
ejpam-4276	618	7	):	):	PUNCT
ejpam-4276	618	8	let	let	VERB
ejpam-4276	618	9	v	v	NUM
ejpam-4276	618	10	∈	∈	PROPN
ejpam-4276	618	11	rλspo(x	rλspo(x	NOUN
ejpam-4276	618	12	,	,	PUNCT
ejpam-4276	618	13	τ	τ	PROPN
ejpam-4276	618	14	)	)	PUNCT
ejpam-4276	618	15	.	.	PUNCT
ejpam-4276	619	1	by	by	ADP
ejpam-4276	619	2	(	(	PUNCT
ejpam-4276	619	3	13	13	NUM
ejpam-4276	619	4	)	)	PUNCT
ejpam-4276	619	5	,	,	PUNCT
ejpam-4276	619	6	we	we	PRON
ejpam-4276	619	7	have	have	VERB
ejpam-4276	619	8	v	v	NOUN
ejpam-4276	619	9	is	be	AUX
ejpam-4276	619	10	p(λ	p(λ	NOUN
ejpam-4276	619	11	,	,	PUNCT
ejpam-4276	619	12	sp)-closed	sp)-close	VERB
ejpam-4276	619	13	and	and	CCONJ
ejpam-4276	619	14	hence	hence	ADV
ejpam-4276	619	15	[	[	X
ejpam-4276	619	16	v(λ	v(λ	PROPN
ejpam-4276	619	17	,	,	PUNCT
ejpam-4276	619	18	sp	sp	NOUN
ejpam-4276	619	19	)	)	PUNCT
ejpam-4276	619	20	]	]	PUNCT
ejpam-4276	620	1	(	(	PUNCT
ejpam-4276	620	2	λ	λ	NOUN
ejpam-4276	620	3	,	,	PUNCT
ejpam-4276	620	4	sp	sp	NOUN
ejpam-4276	620	5	)	)	PUNCT
ejpam-4276	620	6	⊆	⊆	NUM
ejpam-4276	620	7	v	v	NOUN
ejpam-4276	620	8	.	.	PUNCT
ejpam-4276	621	1	since	since	SCONJ
ejpam-4276	621	2	v	v	NOUN
ejpam-4276	621	3	is	be	AUX
ejpam-4276	621	4	(	(	PUNCT
ejpam-4276	621	5	λ	λ	NOUN
ejpam-4276	621	6	,	,	PUNCT
ejpam-4276	621	7	sp)-open	sp)-open	ADJ
ejpam-4276	621	8	,	,	PUNCT
ejpam-4276	621	9	v	v	NOUN
ejpam-4276	621	10	(	(	PUNCT
ejpam-4276	621	11	λ	λ	NOUN
ejpam-4276	621	12	,	,	PUNCT
ejpam-4276	621	13	sp	sp	NOUN
ejpam-4276	621	14	)	)	PUNCT
ejpam-4276	621	15	⊆	⊆	NUM
ejpam-4276	621	16	v	v	NOUN
ejpam-4276	621	17	.	.	PUNCT
ejpam-4276	622	1	thus	thus	ADV
ejpam-4276	622	2	,	,	PUNCT
ejpam-4276	622	3	v	v	PROPN
ejpam-4276	622	4	∈	∈	PROPN
ejpam-4276	622	5	λspc(x	λspc(x	PROPN
ejpam-4276	622	6	,	,	PUNCT
ejpam-4276	622	7	τ	τ	PROPN
ejpam-4276	622	8	)	)	PUNCT
ejpam-4276	622	9	.	.	PUNCT
ejpam-4276	623	1	consequently	consequently	ADV
ejpam-4276	623	2	,	,	PUNCT
ejpam-4276	623	3	we	we	PRON
ejpam-4276	623	4	obtain	obtain	VERB
ejpam-4276	623	5	rλspo(x	rλspo(x	NOUN
ejpam-4276	623	6	,	,	PUNCT
ejpam-4276	623	7	τ	τ	PROPN
ejpam-4276	623	8	)	)	PUNCT
ejpam-4276	623	9	⊆	⊆	NUM
ejpam-4276	623	10	λspc(x	λspc(x	PROPN
ejpam-4276	623	11	,	,	PUNCT
ejpam-4276	623	12	τ	τ	PROPN
ejpam-4276	623	13	)	)	PUNCT
ejpam-4276	623	14	.	.	PUNCT
ejpam-4276	624	1	(	(	PUNCT
ejpam-4276	624	2	14	14	NUM
ejpam-4276	624	3	)	)	PUNCT
ejpam-4276	624	4	⇒	⇒	NOUN
ejpam-4276	624	5	(	(	PUNCT
ejpam-4276	624	6	15	15	NUM
ejpam-4276	624	7	):	):	PUNCT
ejpam-4276	624	8	the	the	DET
ejpam-4276	624	9	proof	proof	NOUN
ejpam-4276	624	10	is	be	AUX
ejpam-4276	624	11	obvious	obvious	ADJ
ejpam-4276	624	12	.	.	PUNCT
ejpam-4276	625	1	(	(	PUNCT
ejpam-4276	625	2	15	15	NUM
ejpam-4276	625	3	)	)	PUNCT
ejpam-4276	625	4	⇒	⇒	NOUN
ejpam-4276	625	5	(	(	PUNCT
ejpam-4276	625	6	1	1	NUM
ejpam-4276	625	7	):	):	PUNCT
ejpam-4276	625	8	let	let	VERB
ejpam-4276	625	9	v	v	PART
ejpam-4276	625	10	be	be	AUX
ejpam-4276	625	11	a	a	DET
ejpam-4276	625	12	(	(	PUNCT
ejpam-4276	625	13	λ	λ	NOUN
ejpam-4276	625	14	,	,	PUNCT
ejpam-4276	625	15	sp)-open	sp)-open	ADJ
ejpam-4276	625	16	set	set	NOUN
ejpam-4276	625	17	.	.	PUNCT
ejpam-4276	626	1	then	then	ADV
ejpam-4276	626	2	,	,	PUNCT
ejpam-4276	626	3	[	[	X
ejpam-4276	626	4	v	v	X
ejpam-4276	626	5	(	(	PUNCT
ejpam-4276	626	6	λ	λ	PROPN
ejpam-4276	626	7	,	,	PUNCT
ejpam-4276	626	8	sp)](λ	sp)](λ	PROPN
ejpam-4276	626	9	,	,	PUNCT
ejpam-4276	626	10	sp	sp	NOUN
ejpam-4276	626	11	)	)	PUNCT
ejpam-4276	626	12	is	be	AUX
ejpam-4276	626	13	r(λ	r(λ	NOUN
ejpam-4276	626	14	,	,	PUNCT
ejpam-4276	626	15	sp)-open	sp)-open	ADJ
ejpam-4276	626	16	,	,	PUNCT
ejpam-4276	626	17	by	by	ADP
ejpam-4276	626	18	(	(	PUNCT
ejpam-4276	626	19	15	15	NUM
ejpam-4276	626	20	)	)	PUNCT
ejpam-4276	626	21	,	,	PUNCT
ejpam-4276	627	1	[	[	X
ejpam-4276	627	2	v	v	X
ejpam-4276	627	3	(	(	PUNCT
ejpam-4276	627	4	λ	λ	PROPN
ejpam-4276	627	5	,	,	PUNCT
ejpam-4276	627	6	sp)](λ	sp)](λ	PROPN
ejpam-4276	627	7	,	,	PUNCT
ejpam-4276	627	8	sp	sp	NOUN
ejpam-4276	627	9	)	)	PUNCT
ejpam-4276	627	10	is	be	AUX
ejpam-4276	627	11	α(λ	α(λ	PROPN
ejpam-4276	627	12	,	,	PUNCT
ejpam-4276	627	13	sp)-closed	sp)-close	VERB
ejpam-4276	627	14	.	.	PUNCT
ejpam-4276	628	1	therefore	therefore	ADV
ejpam-4276	628	2	,	,	PUNCT
ejpam-4276	628	3	v	v	INTJ
ejpam-4276	628	4	(	(	PUNCT
ejpam-4276	628	5	λ	λ	NOUN
ejpam-4276	628	6	,	,	PUNCT
ejpam-4276	628	7	sp	sp	NOUN
ejpam-4276	628	8	)	)	PUNCT
ejpam-4276	628	9	⊆	⊆	NUM
ejpam-4276	629	1	[	[	X
ejpam-4276	629	2	[	[	X
ejpam-4276	629	3	v	v	X
ejpam-4276	629	4	(	(	PUNCT
ejpam-4276	629	5	λ	λ	PROPN
ejpam-4276	629	6	,	,	PUNCT
ejpam-4276	629	7	sp)](λ	sp)](λ	PROPN
ejpam-4276	629	8	,	,	PUNCT
ejpam-4276	629	9	sp	sp	NOUN
ejpam-4276	629	10	)	)	PUNCT
ejpam-4276	629	11	]	]	PUNCT
ejpam-4276	629	12	(	(	PUNCT
ejpam-4276	629	13	λ	λ	NOUN
ejpam-4276	629	14	,	,	PUNCT
ejpam-4276	629	15	sp	sp	NOUN
ejpam-4276	629	16	)	)	PUNCT
ejpam-4276	629	17	=	=	PUNCT
ejpam-4276	630	1	[	[	X
ejpam-4276	630	2	[	[	X
ejpam-4276	630	3	[	[	X
ejpam-4276	630	4	[	[	X
ejpam-4276	630	5	v	v	X
ejpam-4276	630	6	(	(	PUNCT
ejpam-4276	630	7	λ	λ	PROPN
ejpam-4276	630	8	,	,	PUNCT
ejpam-4276	630	9	sp)](λ	sp)](λ	PROPN
ejpam-4276	630	10	,	,	PUNCT
ejpam-4276	630	11	sp	sp	NOUN
ejpam-4276	630	12	)	)	PUNCT
ejpam-4276	630	13	]	]	PUNCT
ejpam-4276	630	14	(	(	PUNCT
ejpam-4276	630	15	λ	λ	X
ejpam-4276	630	16	,	,	PUNCT
ejpam-4276	630	17	sp)](λ	sp)](λ	PROPN
ejpam-4276	630	18	,	,	PUNCT
ejpam-4276	630	19	sp	sp	NOUN
ejpam-4276	630	20	)	)	PUNCT
ejpam-4276	630	21	]	]	PUNCT
ejpam-4276	630	22	(	(	PUNCT
ejpam-4276	630	23	λ	λ	NOUN
ejpam-4276	630	24	,	,	PUNCT
ejpam-4276	630	25	sp	sp	NOUN
ejpam-4276	630	26	)	)	PUNCT
ejpam-4276	630	27	⊆	⊆	NUM
ejpam-4276	630	28	[	[	X
ejpam-4276	630	29	v	v	X
ejpam-4276	630	30	(	(	PUNCT
ejpam-4276	630	31	λ	λ	PROPN
ejpam-4276	630	32	,	,	PUNCT
ejpam-4276	630	33	sp)](λ	sp)](λ	PROPN
ejpam-4276	630	34	,	,	PUNCT
ejpam-4276	630	35	sp	sp	NOUN
ejpam-4276	630	36	)	)	PUNCT
ejpam-4276	630	37	.	.	PUNCT
ejpam-4276	631	1	thus	thus	ADV
ejpam-4276	631	2	,	,	PUNCT
ejpam-4276	631	3	v	v	INTJ
ejpam-4276	631	4	(	(	PUNCT
ejpam-4276	631	5	λ	λ	NOUN
ejpam-4276	631	6	,	,	PUNCT
ejpam-4276	631	7	sp	sp	NOUN
ejpam-4276	631	8	)	)	PUNCT
ejpam-4276	631	9	is	be	AUX
ejpam-4276	631	10	(	(	PUNCT
ejpam-4276	631	11	λ	λ	INTJ
ejpam-4276	631	12	,	,	PUNCT
ejpam-4276	631	13	sp)-open	sp)-open	NOUN
ejpam-4276	631	14	.	.	PUNCT
ejpam-4276	632	1	this	this	PRON
ejpam-4276	632	2	shows	show	VERB
ejpam-4276	632	3	that	that	SCONJ
ejpam-4276	632	4	(	(	PUNCT
ejpam-4276	632	5	x	x	X
ejpam-4276	632	6	,	,	PUNCT
ejpam-4276	632	7	τ	τ	X
ejpam-4276	632	8	)	)	PUNCT
ejpam-4276	632	9	is	be	AUX
ejpam-4276	632	10	λsp	λsp	VERB
ejpam-4276	632	11	-	-	PUNCT
ejpam-4276	632	12	extremally	extremally	ADV
ejpam-4276	632	13	disconnected	disconnect	VERB
ejpam-4276	632	14	.	.	PUNCT
ejpam-4276	633	1	references	reference	NOUN
ejpam-4276	633	2	588	588	NUM
ejpam-4276	633	3	acknowledgements	acknowledgement	NOUN
ejpam-4276	633	4	this	this	DET
ejpam-4276	633	5	research	research	NOUN
ejpam-4276	633	6	project	project	NOUN
ejpam-4276	633	7	was	be	AUX
ejpam-4276	633	8	financially	financially	ADV
ejpam-4276	633	9	supported	support	VERB
ejpam-4276	633	10	by	by	ADP
ejpam-4276	633	11	mahasarakham	mahasarakham	PROPN
ejpam-4276	633	12	university	university	PROPN
ejpam-4276	633	13	.	.	PUNCT
ejpam-4276	634	1	references	reference	NOUN
ejpam-4276	634	2	[	[	X
ejpam-4276	634	3	1	1	X
ejpam-4276	634	4	]	]	PUNCT
ejpam-4276	634	5	d.	d.	PROPN
ejpam-4276	634	6	andrijević	andrijević	PROPN
ejpam-4276	634	7	.	.	PUNCT
ejpam-4276	635	1	on	on	ADP
ejpam-4276	635	2	b	b	X
ejpam-4276	635	3	-	-	PUNCT
ejpam-4276	635	4	open	open	ADJ
ejpam-4276	635	5	sets	set	NOUN
ejpam-4276	635	6	.	.	PUNCT
ejpam-4276	636	1	matematički	matematički	PROPN
ejpam-4276	636	2	vesnik	vesnik	PROPN
ejpam-4276	636	3	,	,	PUNCT
ejpam-4276	636	4	48:59–64	48:59–64	PROPN
ejpam-4276	636	5	,	,	PUNCT
ejpam-4276	636	6	1996	1996	NUM
ejpam-4276	636	7	.	.	PUNCT
ejpam-4276	637	1	[	[	X
ejpam-4276	637	2	2	2	NUM
ejpam-4276	637	3	]	]	PUNCT
ejpam-4276	637	4	p.	p.	NOUN
ejpam-4276	637	5	bhattacharya	bhattacharya	PROPN
ejpam-4276	637	6	and	and	CCONJ
ejpam-4276	637	7	b.	b.	PROPN
ejpam-4276	637	8	k.	k.	PROPN
ejpam-4276	637	9	lahiri	lahiri	PROPN
ejpam-4276	637	10	.	.	PUNCT
ejpam-4276	638	1	semi	semi	ADJ
ejpam-4276	638	2	-	-	ADJ
ejpam-4276	638	3	generalized	generalized	ADJ
ejpam-4276	638	4	closed	closed	ADJ
ejpam-4276	638	5	sets	set	NOUN
ejpam-4276	638	6	in	in	ADP
ejpam-4276	638	7	topology	topology	NOUN
ejpam-4276	638	8	.	.	PUNCT
ejpam-4276	639	1	indian	indian	PROPN
ejpam-4276	639	2	journal	journal	PROPN
ejpam-4276	639	3	of	of	ADP
ejpam-4276	639	4	mathematics	mathematic	NOUN
ejpam-4276	639	5	,	,	PUNCT
ejpam-4276	639	6	29:375–382	29:375–382	NUM
ejpam-4276	639	7	,	,	PUNCT
ejpam-4276	639	8	1987	1987	NUM
ejpam-4276	639	9	.	.	PUNCT
ejpam-4276	640	1	[	[	X
ejpam-4276	640	2	3	3	X
ejpam-4276	640	3	]	]	PUNCT
ejpam-4276	640	4	c.	c.	PROPN
ejpam-4276	640	5	boonpok	boonpok	PROPN
ejpam-4276	640	6	.	.	PUNCT
ejpam-4276	641	1	(	(	PUNCT
ejpam-4276	641	2	λ	λ	NOUN
ejpam-4276	641	3	,	,	PUNCT
ejpam-4276	641	4	sp)-closed	sp)-close	VERB
ejpam-4276	641	5	sets	set	NOUN
ejpam-4276	641	6	and	and	CCONJ
ejpam-4276	641	7	related	related	ADJ
ejpam-4276	641	8	topics	topic	NOUN
ejpam-4276	641	9	in	in	ADP
ejpam-4276	641	10	topological	topological	ADJ
ejpam-4276	641	11	spaces	space	NOUN
ejpam-4276	641	12	.	.	PUNCT
ejpam-4276	642	1	wseas	wseas	VERB
ejpam-4276	642	2	transactions	transaction	NOUN
ejpam-4276	642	3	on	on	ADP
ejpam-4276	642	4	mathematics	mathematic	NOUN
ejpam-4276	642	5	,	,	PUNCT
ejpam-4276	642	6	19:321–322	19:321–322	PROPN
ejpam-4276	642	7	,	,	PUNCT
ejpam-4276	642	8	2020	2020	NUM
ejpam-4276	642	9	.	.	PUNCT
ejpam-4276	643	1	[	[	X
ejpam-4276	643	2	4	4	X
ejpam-4276	643	3	]	]	PUNCT
ejpam-4276	643	4	m.	m.	NOUN
ejpam-4276	643	5	e.	e.	PROPN
ejpam-4276	643	6	abd	abd	PROPN
ejpam-4276	644	1	el	el	PROPN
ejpam-4276	644	2	-	-	PROPN
ejpam-4276	644	3	monsef	monsef	PROPN
ejpam-4276	644	4	,	,	PUNCT
ejpam-4276	644	5	s.	s.	PROPN
ejpam-4276	644	6	n.	n.	PROPN
ejpam-4276	644	7	el	el	PROPN
ejpam-4276	644	8	-	-	PROPN
ejpam-4276	644	9	deeb	deeb	PROPN
ejpam-4276	644	10	,	,	PUNCT
ejpam-4276	644	11	and	and	CCONJ
ejpam-4276	644	12	r.	r.	PROPN
ejpam-4276	644	13	a.	a.	PROPN
ejpam-4276	644	14	mahmoud	mahmoud	PROPN
ejpam-4276	644	15	.	.	PUNCT
ejpam-4276	645	1	β	β	X
ejpam-4276	645	2	-	-	ADJ
ejpam-4276	645	3	open	open	ADJ
ejpam-4276	645	4	sets	set	NOUN
ejpam-4276	645	5	and	and	CCONJ
ejpam-4276	645	6	βcontinuous	βcontinuous	ADJ
ejpam-4276	645	7	mappings	mapping	NOUN
ejpam-4276	645	8	.	.	PUNCT
ejpam-4276	646	1	bulletin	bulletin	NOUN
ejpam-4276	646	2	of	of	ADP
ejpam-4276	646	3	the	the	DET
ejpam-4276	646	4	faculty	faculty	NOUN
ejpam-4276	646	5	of	of	ADP
ejpam-4276	646	6	science	science	NOUN
ejpam-4276	646	7	.	.	PUNCT
ejpam-4276	647	1	assiut	assiut	PROPN
ejpam-4276	647	2	university	university	PROPN
ejpam-4276	647	3	.	.	PUNCT
ejpam-4276	647	4	,	,	PUNCT
ejpam-4276	647	5	12:77–90	12:77–90	NUM
ejpam-4276	647	6	,	,	PUNCT
ejpam-4276	647	7	1983	1983	NUM
ejpam-4276	647	8	.	.	PUNCT
ejpam-4276	648	1	[	[	X
ejpam-4276	648	2	5	5	NUM
ejpam-4276	648	3	]	]	PUNCT
ejpam-4276	648	4	l.	l.	PROPN
ejpam-4276	648	5	gillman	gillman	PROPN
ejpam-4276	648	6	and	and	CCONJ
ejpam-4276	648	7	m.	m.	PROPN
ejpam-4276	648	8	jerison	jerison	PROPN
ejpam-4276	648	9	.	.	PUNCT
ejpam-4276	649	1	rings	ring	NOUN
ejpam-4276	649	2	of	of	ADP
ejpam-4276	649	3	continuous	continuous	ADJ
ejpam-4276	649	4	functions	function	NOUN
ejpam-4276	649	5	.	.	PUNCT
ejpam-4276	650	1	the	the	DET
ejpam-4276	650	2	university	university	NOUN
ejpam-4276	650	3	series	series	NOUN
ejpam-4276	650	4	in	in	ADP
ejpam-4276	650	5	higher	high	ADJ
ejpam-4276	650	6	mathematics	mathematic	NOUN
ejpam-4276	650	7	,	,	PUNCT
ejpam-4276	650	8	van	van	PROPN
ejpam-4276	650	9	nostrand	nostrand	PROPN
ejpam-4276	650	10	,	,	PUNCT
ejpam-4276	650	11	princeton	princeton	PROPN
ejpam-4276	650	12	,	,	PUNCT
ejpam-4276	650	13	new	new	PROPN
ejpam-4276	650	14	york	york	PROPN
ejpam-4276	650	15	,	,	PUNCT
ejpam-4276	650	16	1960	1960	NUM
ejpam-4276	650	17	.	.	PUNCT
ejpam-4276	651	1	[	[	X
ejpam-4276	651	2	6	6	NUM
ejpam-4276	651	3	]	]	X
ejpam-4276	651	4	n.	n.	PROPN
ejpam-4276	651	5	levine	levine	PROPN
ejpam-4276	651	6	.	.	PUNCT
ejpam-4276	652	1	semi	semi	ADJ
ejpam-4276	652	2	-	-	ADJ
ejpam-4276	652	3	open	open	ADJ
ejpam-4276	652	4	sets	set	NOUN
ejpam-4276	652	5	and	and	CCONJ
ejpam-4276	652	6	semi	semi	ADJ
ejpam-4276	652	7	-	-	NOUN
ejpam-4276	652	8	continuity	continuity	NOUN
ejpam-4276	652	9	in	in	ADP
ejpam-4276	652	10	topological	topological	ADJ
ejpam-4276	652	11	spaces	space	NOUN
ejpam-4276	652	12	.	.	PUNCT
ejpam-4276	653	1	the	the	DET
ejpam-4276	653	2	american	american	PROPN
ejpam-4276	653	3	mathematical	mathematical	PROPN
ejpam-4276	653	4	monthly	monthly	ADV
ejpam-4276	653	5	,	,	PUNCT
ejpam-4276	653	6	70:36–41	70:36–41	NUM
ejpam-4276	653	7	,	,	PUNCT
ejpam-4276	653	8	1963	1963	NUM
ejpam-4276	653	9	.	.	PUNCT
ejpam-4276	654	1	[	[	X
ejpam-4276	654	2	7	7	X
ejpam-4276	654	3	]	]	PUNCT
ejpam-4276	654	4	s.	s.	PROPN
ejpam-4276	654	5	n.	n.	PROPN
ejpam-4276	654	6	maheshwari	maheshwari	PROPN
ejpam-4276	654	7	and	and	CCONJ
ejpam-4276	654	8	r.	r.	PROPN
ejpam-4276	654	9	prasad	prasad	PROPN
ejpam-4276	654	10	.	.	PUNCT
ejpam-4276	655	1	some	some	DET
ejpam-4276	655	2	new	new	ADJ
ejpam-4276	655	3	separation	separation	NOUN
ejpam-4276	655	4	axioms	axiom	VERB
ejpam-4276	655	5	.	.	PUNCT
ejpam-4276	656	1	annales	annales	PROPN
ejpam-4276	656	2	de	de	PROPN
ejpam-4276	656	3	la	la	PROPN
ejpam-4276	656	4	société	société	PROPN
ejpam-4276	656	5	scientifique	scientifique	PROPN
ejpam-4276	656	6	bruxelles	bruxelles	PROPN
ejpam-4276	656	7	,	,	PUNCT
ejpam-4276	656	8	89:395–402	89:395–402	PROPN
ejpam-4276	656	9	,	,	PUNCT
ejpam-4276	656	10	1975	1975	NUM
ejpam-4276	656	11	.	.	PUNCT
ejpam-4276	657	1	[	[	X
ejpam-4276	657	2	8	8	NUM
ejpam-4276	657	3	]	]	PUNCT
ejpam-4276	657	4	a.	a.	NOUN
ejpam-4276	657	5	s.	s.	PROPN
ejpam-4276	657	6	mashhour	mashhour	PROPN
ejpam-4276	657	7	,	,	PUNCT
ejpam-4276	657	8	m.	m.	PROPN
ejpam-4276	657	9	e.	e.	PROPN
ejpam-4276	657	10	abd	abd	PROPN
ejpam-4276	658	1	el	el	PROPN
ejpam-4276	658	2	-	-	PROPN
ejpam-4276	658	3	monsef	monsef	ADJ
ejpam-4276	658	4	,	,	PUNCT
ejpam-4276	658	5	and	and	CCONJ
ejpam-4276	658	6	s.	s.	PROPN
ejpam-4276	658	7	n.	n.	PROPN
ejpam-4276	658	8	el	el	PROPN
ejpam-4276	658	9	-	-	PROPN
ejpam-4276	658	10	deeb	deeb	PROPN
ejpam-4276	658	11	.	.	PUNCT
ejpam-4276	659	1	on	on	ADP
ejpam-4276	659	2	precontinuous	precontinuous	ADJ
ejpam-4276	659	3	and	and	CCONJ
ejpam-4276	659	4	weak	weak	ADJ
ejpam-4276	659	5	precontinuous	precontinuous	ADJ
ejpam-4276	659	6	functions	function	NOUN
ejpam-4276	659	7	.	.	PUNCT
ejpam-4276	660	1	proceedings	proceeding	NOUN
ejpam-4276	660	2	of	of	ADP
ejpam-4276	660	3	the	the	DET
ejpam-4276	660	4	mathematical	mathematical	ADJ
ejpam-4276	660	5	and	and	CCONJ
ejpam-4276	660	6	physical	physical	ADJ
ejpam-4276	660	7	society	society	PROPN
ejpam-4276	660	8	egypt	egypt	PROPN
ejpam-4276	660	9	,	,	PUNCT
ejpam-4276	660	10	53:47–53	53:47–53	NUM
ejpam-4276	660	11	,	,	PUNCT
ejpam-4276	660	12	1982	1982	NUM
ejpam-4276	660	13	.	.	PUNCT
ejpam-4276	661	1	[	[	X
ejpam-4276	661	2	9	9	NUM
ejpam-4276	661	3	]	]	PUNCT
ejpam-4276	661	4	a.	a.	NOUN
ejpam-4276	661	5	s.	s.	PROPN
ejpam-4276	661	6	mashhour	mashhour	PROPN
ejpam-4276	661	7	,	,	PUNCT
ejpam-4276	661	8	i.	i.	PROPN
ejpam-4276	661	9	n.	n.	PROPN
ejpam-4276	661	10	hasanein	hasanein	PROPN
ejpam-4276	661	11	,	,	PUNCT
ejpam-4276	661	12	and	and	CCONJ
ejpam-4276	661	13	s.	s.	PROPN
ejpam-4276	661	14	n.	n.	PROPN
ejpam-4276	661	15	el	el	PROPN
ejpam-4276	661	16	-	-	PROPN
ejpam-4276	661	17	deeb	deeb	PROPN
ejpam-4276	661	18	.	.	PUNCT
ejpam-4276	662	1	α	α	X
ejpam-4276	662	2	-	-	ADJ
ejpam-4276	662	3	continuous	continuous	ADJ
ejpam-4276	662	4	and	and	CCONJ
ejpam-4276	662	5	α	α	NOUN
ejpam-4276	662	6	-	-	ADJ
ejpam-4276	662	7	open	open	ADJ
ejpam-4276	662	8	mappings	mapping	NOUN
ejpam-4276	662	9	.	.	PUNCT
ejpam-4276	663	1	acta	acta	PROPN
ejpam-4276	663	2	mathematica	mathematica	PROPN
ejpam-4276	663	3	hungarica	hungarica	PROPN
ejpam-4276	663	4	,	,	PUNCT
ejpam-4276	663	5	41:213–218	41:213–218	PROPN
ejpam-4276	663	6	,	,	PUNCT
ejpam-4276	663	7	1983	1983	NUM
ejpam-4276	663	8	.	.	PUNCT
ejpam-4276	664	1	[	[	X
ejpam-4276	664	2	10	10	NUM
ejpam-4276	664	3	]	]	X
ejpam-4276	664	4	o.	o.	NOUN
ejpam-4276	664	5	njåstad	njåstad	PROPN
ejpam-4276	664	6	.	.	PUNCT
ejpam-4276	665	1	on	on	ADP
ejpam-4276	665	2	some	some	DET
ejpam-4276	665	3	classes	class	NOUN
ejpam-4276	665	4	of	of	ADP
ejpam-4276	665	5	nearly	nearly	ADV
ejpam-4276	665	6	open	open	ADJ
ejpam-4276	665	7	sets	set	NOUN
ejpam-4276	665	8	.	.	PUNCT
ejpam-4276	666	1	pasific	pasific	PROPN
ejpam-4276	666	2	journal	journal	PROPN
ejpam-4276	666	3	of	of	ADP
ejpam-4276	666	4	mathematics	mathematic	NOUN
ejpam-4276	666	5	,	,	PUNCT
ejpam-4276	666	6	15:961–970	15:961–970	PROPN
ejpam-4276	666	7	,	,	PUNCT
ejpam-4276	666	8	1965	1965	NUM
ejpam-4276	666	9	.	.	PUNCT
ejpam-4276	667	1	[	[	X
ejpam-4276	667	2	11	11	NUM
ejpam-4276	667	3	]	]	PUNCT
ejpam-4276	667	4	t.	t.	PROPN
ejpam-4276	667	5	noiri	noiri	PROPN
ejpam-4276	667	6	.	.	PUNCT
ejpam-4276	668	1	characterizations	characterization	NOUN
ejpam-4276	668	2	of	of	ADP
ejpam-4276	668	3	extremally	extremally	ADV
ejpam-4276	668	4	disconnected	disconnected	ADJ
ejpam-4276	668	5	spaces	space	NOUN
ejpam-4276	668	6	.	.	PUNCT
ejpam-4276	669	1	indian	indian	ADJ
ejpam-4276	669	2	journal	journal	PROPN
ejpam-4276	669	3	of	of	ADP
ejpam-4276	669	4	pure	pure	ADJ
ejpam-4276	669	5	and	and	CCONJ
ejpam-4276	669	6	applied	applied	ADJ
ejpam-4276	669	7	mathematics	mathematic	NOUN
ejpam-4276	669	8	,	,	PUNCT
ejpam-4276	669	9	19:325–329	19:325–329	NUM
ejpam-4276	669	10	,	,	PUNCT
ejpam-4276	669	11	1988	1988	NUM
ejpam-4276	669	12	.	.	PUNCT
ejpam-4276	670	1	[	[	X
ejpam-4276	670	2	12	12	NUM
ejpam-4276	670	3	]	]	PUNCT
ejpam-4276	670	4	t.	t.	PROPN
ejpam-4276	670	5	noiri	noiri	PROPN
ejpam-4276	670	6	and	and	CCONJ
ejpam-4276	670	7	e.	e.	PROPN
ejpam-4276	670	8	hatir	hatir	PROPN
ejpam-4276	670	9	.	.	PUNCT
ejpam-4276	671	1	λsp	λsp	NOUN
ejpam-4276	671	2	-	-	PUNCT
ejpam-4276	671	3	sets	set	NOUN
ejpam-4276	671	4	and	and	CCONJ
ejpam-4276	671	5	some	some	DET
ejpam-4276	671	6	weak	weak	ADJ
ejpam-4276	671	7	separation	separation	NOUN
ejpam-4276	671	8	axioms	axiom	NOUN
ejpam-4276	671	9	.	.	PUNCT
ejpam-4276	672	1	acta	acta	PROPN
ejpam-4276	672	2	mathematica	mathematica	PROPN
ejpam-4276	672	3	hungarica	hungarica	PROPN
ejpam-4276	672	4	,	,	PUNCT
ejpam-4276	672	5	103:225–232	103:225–232	NUM
ejpam-4276	672	6	,	,	PUNCT
ejpam-4276	672	7	2004	2004	NUM
ejpam-4276	672	8	.	.	PUNCT
ejpam-4276	673	1	[	[	X
ejpam-4276	673	2	13	13	NUM
ejpam-4276	673	3	]	]	PUNCT
ejpam-4276	673	4	d.	d.	NOUN
ejpam-4276	673	5	sivaraj	sivaraj	PROPN
ejpam-4276	673	6	.	.	PUNCT
ejpam-4276	674	1	a	a	DET
ejpam-4276	674	2	note	note	NOUN
ejpam-4276	674	3	on	on	ADP
ejpam-4276	674	4	extremally	extremally	ADV
ejpam-4276	674	5	disconnected	disconnected	ADJ
ejpam-4276	674	6	spaces	space	NOUN
ejpam-4276	674	7	.	.	PUNCT
ejpam-4276	675	1	indian	indian	ADJ
ejpam-4276	675	2	journal	journal	PROPN
ejpam-4276	675	3	of	of	ADP
ejpam-4276	675	4	pure	pure	ADJ
ejpam-4276	675	5	and	and	CCONJ
ejpam-4276	675	6	applied	applied	ADJ
ejpam-4276	675	7	mathematics	mathematic	NOUN
ejpam-4276	675	8	,	,	PUNCT
ejpam-4276	675	9	17:1373–1375	17:1373–1375	PROPN
ejpam-4276	675	10	,	,	PUNCT
ejpam-4276	675	11	1986	1986	NUM
ejpam-4276	675	12	.	.	PUNCT