id	sid	tid	token	lemma	pos
ejpam-4274	1	1	european	european	PROPN
ejpam-4274	1	2	journal	journal	PROPN
ejpam-4274	1	3	of	of	ADP
ejpam-4274	1	4	pure	pure	ADJ
ejpam-4274	1	5	and	and	CCONJ
ejpam-4274	1	6	applied	apply	VERB
ejpam-4274	1	7	mathematics	mathematic	NOUN
ejpam-4274	1	8	vol	vol	NOUN
ejpam-4274	1	9	.	.	PROPN
ejpam-4274	2	1	15	15	NUM
ejpam-4274	2	2	,	,	PUNCT
ejpam-4274	2	3	no	no	INTJ
ejpam-4274	2	4	.	.	NOUN
ejpam-4274	2	5	2	2	NUM
ejpam-4274	2	6	,	,	PUNCT
ejpam-4274	2	7	2022	2022	NUM
ejpam-4274	2	8	,	,	PUNCT
ejpam-4274	2	9	415	415	NUM
ejpam-4274	2	10	-	-	SYM
ejpam-4274	2	11	436	436	NUM
ejpam-4274	2	12	issn	issn	PROPN
ejpam-4274	2	13	1307	1307	NUM
ejpam-4274	2	14	-	-	SYM
ejpam-4274	2	15	5543	5543	NUM
ejpam-4274	2	16	–	–	PUNCT
ejpam-4274	2	17	ejpam.com	ejpam.com	X
ejpam-4274	2	18	published	publish	VERB
ejpam-4274	2	19	by	by	ADP
ejpam-4274	2	20	new	new	PROPN
ejpam-4274	2	21	york	york	PROPN
ejpam-4274	2	22	business	business	PROPN
ejpam-4274	2	23	global	global	PROPN
ejpam-4274	2	24	on	on	ADP
ejpam-4274	2	25	(	(	PUNCT
ejpam-4274	2	26	λ	λ	PROPN
ejpam-4274	2	27	,	,	PUNCT
ejpam-4274	2	28	p)-closed	p)-close	VERB
ejpam-4274	2	29	sets	set	NOUN
ejpam-4274	2	30	and	and	CCONJ
ejpam-4274	2	31	the	the	DET
ejpam-4274	2	32	related	related	ADJ
ejpam-4274	2	33	notions	notion	NOUN
ejpam-4274	2	34	in	in	ADP
ejpam-4274	2	35	topological	topological	ADJ
ejpam-4274	2	36	spaces	space	NOUN
ejpam-4274	2	37	chawalit	chawalit	VERB
ejpam-4274	2	38	boonpok1	boonpok1	PROPN
ejpam-4274	2	39	,	,	PUNCT
ejpam-4274	2	40	chokchai	chokchai	ADJ
ejpam-4274	2	41	viriyapong1,∗	viriyapong1,∗	NOUN
ejpam-4274	2	42	1	1	NUM
ejpam-4274	2	43	mathematics	mathematic	NOUN
ejpam-4274	2	44	and	and	CCONJ
ejpam-4274	2	45	applied	apply	VERB
ejpam-4274	2	46	mathematics	mathematics	PROPN
ejpam-4274	2	47	research	research	NOUN
ejpam-4274	2	48	unit	unit	NOUN
ejpam-4274	2	49	,	,	PUNCT
ejpam-4274	2	50	department	department	NOUN
ejpam-4274	2	51	of	of	ADP
ejpam-4274	2	52	mathematics	mathematic	NOUN
ejpam-4274	2	53	,	,	PUNCT
ejpam-4274	2	54	faculty	faculty	NOUN
ejpam-4274	2	55	of	of	ADP
ejpam-4274	2	56	science	science	NOUN
ejpam-4274	2	57	,	,	PUNCT
ejpam-4274	2	58	mahasarakham	mahasarakham	PROPN
ejpam-4274	2	59	university	university	PROPN
ejpam-4274	2	60	,	,	PUNCT
ejpam-4274	2	61	maha	maha	PROPN
ejpam-4274	2	62	sarakham	sarakham	PROPN
ejpam-4274	2	63	,	,	PUNCT
ejpam-4274	2	64	44150	44150	NUM
ejpam-4274	2	65	,	,	PUNCT
ejpam-4274	2	66	thailand	thailand	PROPN
ejpam-4274	2	67	abstract	abstract	PROPN
ejpam-4274	2	68	.	.	PUNCT
ejpam-4274	3	1	this	this	DET
ejpam-4274	3	2	article	article	NOUN
ejpam-4274	3	3	deals	deal	VERB
ejpam-4274	3	4	with	with	ADP
ejpam-4274	3	5	the	the	DET
ejpam-4274	3	6	concepts	concept	NOUN
ejpam-4274	3	7	of	of	ADP
ejpam-4274	3	8	λp	λp	NOUN
ejpam-4274	3	9	-	-	PUNCT
ejpam-4274	3	10	sets	set	NOUN
ejpam-4274	3	11	and	and	CCONJ
ejpam-4274	3	12	(	(	PUNCT
ejpam-4274	3	13	λ	λ	PROPN
ejpam-4274	3	14	,	,	PUNCT
ejpam-4274	3	15	p)-closed	p)-close	VERB
ejpam-4274	3	16	sets	set	NOUN
ejpam-4274	3	17	which	which	PRON
ejpam-4274	3	18	are	be	AUX
ejpam-4274	3	19	defined	define	VERB
ejpam-4274	3	20	by	by	ADP
ejpam-4274	3	21	utilizing	utilize	VERB
ejpam-4274	3	22	the	the	DET
ejpam-4274	3	23	notions	notion	NOUN
ejpam-4274	3	24	of	of	ADP
ejpam-4274	3	25	preopen	preopen	ADJ
ejpam-4274	3	26	sets	set	NOUN
ejpam-4274	3	27	and	and	CCONJ
ejpam-4274	3	28	preclosed	preclose	VERB
ejpam-4274	3	29	sets	set	NOUN
ejpam-4274	3	30	.	.	PUNCT
ejpam-4274	4	1	we	we	PRON
ejpam-4274	4	2	also	also	ADV
ejpam-4274	4	3	introduce	introduce	VERB
ejpam-4274	4	4	and	and	CCONJ
ejpam-4274	4	5	characterize	characterize	VERB
ejpam-4274	4	6	some	some	DET
ejpam-4274	4	7	new	new	ADJ
ejpam-4274	4	8	low	low	ADJ
ejpam-4274	4	9	separation	separation	NOUN
ejpam-4274	4	10	axioms	axiom	NOUN
ejpam-4274	4	11	.	.	PUNCT
ejpam-4274	5	1	characterizations	characterization	NOUN
ejpam-4274	5	2	of	of	ADP
ejpam-4274	5	3	λp	λp	PROPN
ejpam-4274	5	4	-	-	PUNCT
ejpam-4274	5	5	r0	r0	NOUN
ejpam-4274	5	6	spaces	space	NOUN
ejpam-4274	5	7	are	be	AUX
ejpam-4274	5	8	given	give	VERB
ejpam-4274	5	9	.	.	PUNCT
ejpam-4274	6	1	moreover	moreover	ADV
ejpam-4274	6	2	,	,	PUNCT
ejpam-4274	6	3	we	we	PRON
ejpam-4274	6	4	introduce	introduce	VERB
ejpam-4274	6	5	the	the	DET
ejpam-4274	6	6	concept	concept	NOUN
ejpam-4274	6	7	of	of	ADP
ejpam-4274	6	8	weakly	weakly	ADJ
ejpam-4274	6	9	(	(	PUNCT
ejpam-4274	6	10	λ	λ	PROPN
ejpam-4274	6	11	,	,	PUNCT
ejpam-4274	6	12	p)-continuous	p)-continuous	ADJ
ejpam-4274	6	13	functions	function	NOUN
ejpam-4274	6	14	.	.	PUNCT
ejpam-4274	7	1	in	in	ADP
ejpam-4274	7	2	particular	particular	ADJ
ejpam-4274	7	3	,	,	PUNCT
ejpam-4274	7	4	several	several	ADJ
ejpam-4274	7	5	characterizations	characterization	NOUN
ejpam-4274	7	6	of	of	ADP
ejpam-4274	7	7	weakly	weakly	ADJ
ejpam-4274	7	8	(	(	PUNCT
ejpam-4274	7	9	λ	λ	NOUN
ejpam-4274	7	10	,	,	PUNCT
ejpam-4274	7	11	p)-continuous	p)-continuous	ADJ
ejpam-4274	7	12	functions	function	NOUN
ejpam-4274	7	13	are	be	AUX
ejpam-4274	7	14	established	establish	VERB
ejpam-4274	7	15	.	.	PUNCT
ejpam-4274	8	1	2020	2020	NUM
ejpam-4274	8	2	mathematics	mathematics	PROPN
ejpam-4274	8	3	subject	subject	NOUN
ejpam-4274	8	4	classifications	classification	NOUN
ejpam-4274	8	5	:	:	PUNCT
ejpam-4274	8	6	54a05	54a05	NUM
ejpam-4274	8	7	,	,	PUNCT
ejpam-4274	8	8	54c08	54c08	NUM
ejpam-4274	8	9	,	,	PUNCT
ejpam-4274	8	10	54d10	54d10	NUM
ejpam-4274	8	11	key	key	ADJ
ejpam-4274	8	12	words	word	NOUN
ejpam-4274	8	13	and	and	CCONJ
ejpam-4274	8	14	phrases	phrase	NOUN
ejpam-4274	8	15	:	:	PUNCT
ejpam-4274	8	16	λp	λp	X
ejpam-4274	8	17	-	-	PUNCT
ejpam-4274	8	18	set	set	NOUN
ejpam-4274	8	19	,	,	PUNCT
ejpam-4274	8	20	(	(	PUNCT
ejpam-4274	8	21	λ	λ	PROPN
ejpam-4274	8	22	,	,	PUNCT
ejpam-4274	8	23	p)-closed	p)-close	VERB
ejpam-4274	8	24	set	set	NOUN
ejpam-4274	8	25	,	,	PUNCT
ejpam-4274	8	26	λp	λp	ADJ
ejpam-4274	8	27	-	-	PUNCT
ejpam-4274	8	28	r0	r0	NOUN
ejpam-4274	8	29	space	space	NOUN
ejpam-4274	8	30	,	,	PUNCT
ejpam-4274	8	31	weakly	weakly	ADJ
ejpam-4274	8	32	(	(	PUNCT
ejpam-4274	8	33	λ	λ	NOUN
ejpam-4274	8	34	,	,	PUNCT
ejpam-4274	8	35	p)-continuous	p)-continuous	ADJ
ejpam-4274	8	36	function	function	NOUN
ejpam-4274	8	37	1	1	NUM
ejpam-4274	8	38	.	.	PUNCT
ejpam-4274	8	39	introduction	introduction	NOUN
ejpam-4274	8	40	in	in	ADP
ejpam-4274	8	41	1982	1982	NUM
ejpam-4274	8	42	,	,	PUNCT
ejpam-4274	8	43	mashhour	mashhour	PROPN
ejpam-4274	8	44	et	et	PROPN
ejpam-4274	8	45	al	al	PROPN
ejpam-4274	8	46	.	.	PUNCT
ejpam-4274	9	1	[	[	X
ejpam-4274	9	2	16	16	NUM
ejpam-4274	9	3	]	]	PUNCT
ejpam-4274	9	4	introduced	introduce	VERB
ejpam-4274	9	5	the	the	DET
ejpam-4274	9	6	notion	notion	NOUN
ejpam-4274	9	7	of	of	ADP
ejpam-4274	9	8	preopen	preopen	ADJ
ejpam-4274	9	9	sets	set	NOUN
ejpam-4274	9	10	which	which	PRON
ejpam-4274	9	11	is	be	AUX
ejpam-4274	9	12	also	also	ADV
ejpam-4274	9	13	known	know	VERB
ejpam-4274	9	14	under	under	ADP
ejpam-4274	9	15	the	the	DET
ejpam-4274	9	16	name	name	NOUN
ejpam-4274	9	17	of	of	ADP
ejpam-4274	9	18	locally	locally	ADV
ejpam-4274	9	19	dense	dense	ADJ
ejpam-4274	9	20	sets	set	NOUN
ejpam-4274	9	21	[	[	X
ejpam-4274	9	22	7	7	X
ejpam-4274	9	23	]	]	PUNCT
ejpam-4274	9	24	in	in	ADP
ejpam-4274	9	25	the	the	DET
ejpam-4274	9	26	literature	literature	NOUN
ejpam-4274	9	27	.	.	PUNCT
ejpam-4274	10	1	since	since	SCONJ
ejpam-4274	10	2	then	then	ADV
ejpam-4274	10	3	,	,	PUNCT
ejpam-4274	10	4	this	this	DET
ejpam-4274	10	5	notion	notion	NOUN
ejpam-4274	10	6	received	receive	VERB
ejpam-4274	10	7	wide	wide	ADJ
ejpam-4274	10	8	usage	usage	NOUN
ejpam-4274	10	9	in	in	ADP
ejpam-4274	10	10	general	general	ADJ
ejpam-4274	10	11	topology	topology	NOUN
ejpam-4274	10	12	.	.	PUNCT
ejpam-4274	11	1	kar	kar	PROPN
ejpam-4274	11	2	and	and	CCONJ
ejpam-4274	11	3	bhattachryya	bhattachryya	NOUN
ejpam-4274	11	4	[	[	X
ejpam-4274	11	5	12	12	NUM
ejpam-4274	11	6	]	]	PUNCT
ejpam-4274	11	7	introduced	introduce	VERB
ejpam-4274	11	8	new	new	ADJ
ejpam-4274	11	9	separation	separation	NOUN
ejpam-4274	11	10	axioms	axiom	VERB
ejpam-4274	11	11	pre	pre	ADJ
ejpam-4274	11	12	-	-	NOUN
ejpam-4274	11	13	t0	t0	ADJ
ejpam-4274	11	14	,	,	PUNCT
ejpam-4274	11	15	pre	pre	ADJ
ejpam-4274	11	16	-	-	NOUN
ejpam-4274	11	17	t1	t1	NOUN
ejpam-4274	11	18	and	and	CCONJ
ejpam-4274	11	19	pre	pre	NOUN
ejpam-4274	11	20	-	-	NOUN
ejpam-4274	11	21	t2	t2	NOUN
ejpam-4274	11	22	by	by	ADP
ejpam-4274	11	23	using	use	VERB
ejpam-4274	11	24	preopen	preopen	ADJ
ejpam-4274	11	25	sets	set	NOUN
ejpam-4274	11	26	due	due	ADP
ejpam-4274	11	27	to	to	ADP
ejpam-4274	11	28	mashhour	mashhour	PROPN
ejpam-4274	11	29	et	et	PROPN
ejpam-4274	11	30	al	al	PROPN
ejpam-4274	11	31	.	.	PUNCT
ejpam-4274	12	1	[	[	X
ejpam-4274	12	2	16	16	NUM
ejpam-4274	12	3	]	]	PUNCT
ejpam-4274	12	4	.	.	PUNCT
ejpam-4274	13	1	caldas	caldas	PROPN
ejpam-4274	13	2	[	[	X
ejpam-4274	13	3	3	3	NUM
ejpam-4274	13	4	]	]	PUNCT
ejpam-4274	13	5	and	and	CCONJ
ejpam-4274	13	6	jafari	jafari	ADJ
ejpam-4274	13	7	[	[	X
ejpam-4274	13	8	11	11	NUM
ejpam-4274	13	9	]	]	PUNCT
ejpam-4274	13	10	introduced	introduce	VERB
ejpam-4274	13	11	independently	independently	ADV
ejpam-4274	13	12	the	the	DET
ejpam-4274	13	13	notions	notion	NOUN
ejpam-4274	13	14	of	of	ADP
ejpam-4274	13	15	p	p	PROPN
ejpam-4274	13	16	-	-	PUNCT
ejpam-4274	13	17	d	d	NOUN
ejpam-4274	13	18	-	-	PUNCT
ejpam-4274	13	19	sets	set	NOUN
ejpam-4274	13	20	and	and	CCONJ
ejpam-4274	13	21	a	a	DET
ejpam-4274	13	22	separation	separation	NOUN
ejpam-4274	13	23	axiom	axiom	NOUN
ejpam-4274	13	24	p	p	NOUN
ejpam-4274	13	25	-	-	PUNCT
ejpam-4274	13	26	d1	d1	NOUN
ejpam-4274	13	27	which	which	PRON
ejpam-4274	13	28	is	be	AUX
ejpam-4274	13	29	strictly	strictly	ADV
ejpam-4274	13	30	between	between	ADP
ejpam-4274	13	31	pre	pre	NOUN
ejpam-4274	13	32	-	-	NOUN
ejpam-4274	13	33	t0	t0	NOUN
ejpam-4274	13	34	and	and	CCONJ
ejpam-4274	13	35	pre	pre	ADJ
ejpam-4274	13	36	-	-	NOUN
ejpam-4274	13	37	t1	t1	NOUN
ejpam-4274	13	38	.	.	PUNCT
ejpam-4274	14	1	caldas	caldas	PROPN
ejpam-4274	14	2	et	et	PROPN
ejpam-4274	14	3	al	al	PROPN
ejpam-4274	14	4	.	.	PUNCT
ejpam-4274	15	1	[	[	X
ejpam-4274	15	2	4	4	X
ejpam-4274	15	3	]	]	PUNCT
ejpam-4274	15	4	introduced	introduce	VERB
ejpam-4274	15	5	two	two	NUM
ejpam-4274	15	6	new	new	ADJ
ejpam-4274	15	7	classes	class	NOUN
ejpam-4274	15	8	of	of	ADP
ejpam-4274	15	9	topological	topological	ADJ
ejpam-4274	15	10	spaces	space	NOUN
ejpam-4274	15	11	called	call	VERB
ejpam-4274	15	12	pre	pre	NOUN
ejpam-4274	15	13	-	-	ADJ
ejpam-4274	15	14	r0	r0	ADJ
ejpam-4274	15	15	and	and	CCONJ
ejpam-4274	15	16	pre	pre	ADJ
ejpam-4274	15	17	-	-	ADJ
ejpam-4274	15	18	r1	r1	ADJ
ejpam-4274	15	19	spaces	space	NOUN
ejpam-4274	15	20	in	in	ADP
ejpam-4274	15	21	terms	term	NOUN
ejpam-4274	15	22	of	of	ADP
ejpam-4274	15	23	concept	concept	NOUN
ejpam-4274	15	24	of	of	ADP
ejpam-4274	15	25	preopen	preopen	ADJ
ejpam-4274	15	26	sets	set	NOUN
ejpam-4274	15	27	and	and	CCONJ
ejpam-4274	15	28	investigated	investigate	VERB
ejpam-4274	15	29	some	some	PRON
ejpam-4274	15	30	of	of	ADP
ejpam-4274	15	31	their	their	PRON
ejpam-4274	15	32	fundamental	fundamental	ADJ
ejpam-4274	15	33	properties	property	NOUN
ejpam-4274	15	34	.	.	PUNCT
ejpam-4274	16	1	mashhour	mashhour	INTJ
ejpam-4274	16	2	et	et	PROPN
ejpam-4274	16	3	al	al	PROPN
ejpam-4274	16	4	.	.	PUNCT
ejpam-4274	17	1	[	[	X
ejpam-4274	17	2	15	15	NUM
ejpam-4274	17	3	]	]	PUNCT
ejpam-4274	17	4	introduced	introduce	VERB
ejpam-4274	17	5	and	and	CCONJ
ejpam-4274	17	6	studied	study	VERB
ejpam-4274	17	7	the	the	DET
ejpam-4274	17	8	concept	concept	NOUN
ejpam-4274	17	9	of	of	ADP
ejpam-4274	17	10	supra	supra	PROPN
ejpam-4274	17	11	topological	topological	ADJ
ejpam-4274	17	12	spaces	space	NOUN
ejpam-4274	17	13	by	by	ADP
ejpam-4274	17	14	dropping	drop	VERB
ejpam-4274	17	15	a	a	DET
ejpam-4274	17	16	finite	finite	ADJ
ejpam-4274	17	17	intersection	intersection	NOUN
ejpam-4274	17	18	condition	condition	NOUN
ejpam-4274	17	19	of	of	ADP
ejpam-4274	17	20	topological	topological	ADJ
ejpam-4274	17	21	spaces	space	NOUN
ejpam-4274	17	22	.	.	PUNCT
ejpam-4274	18	1	el	el	PROPN
ejpam-4274	18	2	-	-	PUNCT
ejpam-4274	18	3	shafei	shafei	NOUN
ejpam-4274	18	4	et	et	PROPN
ejpam-4274	18	5	al	al	PROPN
ejpam-4274	18	6	.	.	PUNCT
ejpam-4274	19	1	[	[	X
ejpam-4274	19	2	9	9	NUM
ejpam-4274	19	3	]	]	PUNCT
ejpam-4274	19	4	defined	define	VERB
ejpam-4274	19	5	some	some	DET
ejpam-4274	19	6	concepts	concept	NOUN
ejpam-4274	19	7	on	on	ADP
ejpam-4274	19	8	supra	supra	PROPN
ejpam-4274	19	9	topological	topological	ADJ
ejpam-4274	19	10	spaces	space	NOUN
ejpam-4274	19	11	using	use	VERB
ejpam-4274	19	12	supra	supra	ADJ
ejpam-4274	19	13	preopen	preopen	ADJ
ejpam-4274	19	14	sets	set	NOUN
ejpam-4274	19	15	and	and	CCONJ
ejpam-4274	19	16	investigated	investigate	VERB
ejpam-4274	19	17	main	main	ADJ
ejpam-4274	19	18	properties	property	NOUN
ejpam-4274	19	19	.	.	PUNCT
ejpam-4274	20	1	al	al	PROPN
ejpam-4274	20	2	-	-	PUNCT
ejpam-4274	20	3	shami	shami	PROPN
ejpam-4274	20	4	et	et	PROPN
ejpam-4274	20	5	al	al	PROPN
ejpam-4274	20	6	.	.	PUNCT
ejpam-4274	21	1	[	[	X
ejpam-4274	21	2	2	2	X
ejpam-4274	21	3	]	]	PUNCT
ejpam-4274	21	4	introduced	introduce	VERB
ejpam-4274	21	5	and	and	CCONJ
ejpam-4274	21	6	investigated	investigate	VERB
ejpam-4274	21	7	new	new	ADJ
ejpam-4274	21	8	separation	separation	NOUN
ejpam-4274	21	9	axioms	axiom	NOUN
ejpam-4274	21	10	,	,	PUNCT
ejpam-4274	21	11	namely	namely	ADV
ejpam-4274	21	12	supra	supra	ADJ
ejpam-4274	21	13	semi	semi	ADV
ejpam-4274	21	14	ti	ti	NOUN
ejpam-4274	21	15	-	-	NOUN
ejpam-4274	21	16	spaces	space	NOUN
ejpam-4274	21	17	(	(	PUNCT
ejpam-4274	21	18	i	i	NOUN
ejpam-4274	21	19	=	=	NOUN
ejpam-4274	21	20	0	0	NUM
ejpam-4274	21	21	,	,	PUNCT
ejpam-4274	21	22	1	1	NUM
ejpam-4274	21	23	,	,	PUNCT
ejpam-4274	21	24	2	2	NUM
ejpam-4274	21	25	,	,	PUNCT
ejpam-4274	21	26	3	3	NUM
ejpam-4274	21	27	,	,	PUNCT
ejpam-4274	21	28	4	4	NUM
ejpam-4274	21	29	)	)	PUNCT
ejpam-4274	21	30	.	.	PUNCT
ejpam-4274	22	1	in	in	ADP
ejpam-4274	22	2	[	[	X
ejpam-4274	22	3	1	1	NUM
ejpam-4274	22	4	]	]	PUNCT
ejpam-4274	22	5	,	,	PUNCT
ejpam-4274	22	6	the	the	DET
ejpam-4274	22	7	present	present	ADJ
ejpam-4274	22	8	author	author	NOUN
ejpam-4274	22	9	introduce	introduce	VERB
ejpam-4274	22	10	the	the	DET
ejpam-4274	22	11	version	version	NOUN
ejpam-4274	22	12	of	of	ADP
ejpam-4274	22	13	complete	complete	ADJ
ejpam-4274	22	14	hausdorffness	hausdorffness	NOUN
ejpam-4274	22	15	and	and	CCONJ
ejpam-4274	22	16	complete	complete	ADJ
ejpam-4274	22	17	regularity	regularity	NOUN
ejpam-4274	22	18	on	on	ADP
ejpam-4274	22	19	supra	supra	PROPN
ejpam-4274	22	20	topological	topological	ADJ
ejpam-4274	22	21	spaces	space	NOUN
ejpam-4274	22	22	and	and	CCONJ
ejpam-4274	22	23	discussed	discuss	VERB
ejpam-4274	22	24	their	their	PRON
ejpam-4274	22	25	∗corresponding	∗corresponding	NOUN
ejpam-4274	22	26	author	author	NOUN
ejpam-4274	22	27	.	.	PUNCT
ejpam-4274	23	1	doi	doi	NOUN
ejpam-4274	23	2	:	:	PUNCT
ejpam-4274	23	3	https://doi.org/10.29020/nybg.ejpam.v15i2.4274	https://doi.org/10.29020/nybg.ejpam.v15i2.4274	NOUN
ejpam-4274	23	4	email	email	NOUN
ejpam-4274	23	5	addresses	address	NOUN
ejpam-4274	23	6	:	:	PUNCT
ejpam-4274	23	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4274	23	8	(	(	PUNCT
ejpam-4274	23	9	c.	c.	PROPN
ejpam-4274	23	10	boonpok	boonpok	PROPN
ejpam-4274	23	11	)	)	PUNCT
ejpam-4274	23	12	,	,	PUNCT
ejpam-4274	23	13	chokchai.v@msu.ac.th	chokchai.v@msu.ac.th	INTJ
ejpam-4274	23	14	(	(	PUNCT
ejpam-4274	23	15	c.	c.	PROPN
ejpam-4274	23	16	viriyapong	viriyapong	PROPN
ejpam-4274	23	17	)	)	PUNCT
ejpam-4274	23	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4274	24	1	415	415	NUM
ejpam-4274	25	1	©	©	NOUN
ejpam-4274	25	2	2022	2022	NUM
ejpam-4274	25	3	ejpam	ejpam	VERB
ejpam-4274	25	4	all	all	DET
ejpam-4274	25	5	rights	right	NOUN
ejpam-4274	25	6	reserved	reserve	VERB
ejpam-4274	25	7	.	.	PUNCT
ejpam-4274	26	1	c.	c.	PROPN
ejpam-4274	26	2	boonpok	boonpok	PROPN
ejpam-4274	26	3	,	,	PUNCT
ejpam-4274	26	4	c.	c.	PROPN
ejpam-4274	26	5	viriyapong	viriyapong	PROPN
ejpam-4274	26	6	/	/	SYM
ejpam-4274	26	7	eur	eur	PROPN
ejpam-4274	26	8	.	.	PUNCT
ejpam-4274	27	1	j.	j.	PROPN
ejpam-4274	27	2	pure	pure	PROPN
ejpam-4274	27	3	appl	appl	PROPN
ejpam-4274	27	4	.	.	PROPN
ejpam-4274	27	5	math	math	PROPN
ejpam-4274	27	6	,	,	PUNCT
ejpam-4274	27	7	15	15	NUM
ejpam-4274	27	8	(	(	PUNCT
ejpam-4274	27	9	2	2	NUM
ejpam-4274	27	10	)	)	PUNCT
ejpam-4274	27	11	(	(	PUNCT
ejpam-4274	27	12	2022	2022	NUM
ejpam-4274	27	13	)	)	PUNCT
ejpam-4274	27	14	,	,	PUNCT
ejpam-4274	27	15	415	415	NUM
ejpam-4274	27	16	-	-	SYM
ejpam-4274	27	17	436	436	NUM
ejpam-4274	27	18	416	416	NUM
ejpam-4274	27	19	fundamental	fundamental	ADJ
ejpam-4274	27	20	properties	property	NOUN
ejpam-4274	27	21	.	.	PUNCT
ejpam-4274	28	1	cammaroto	cammaroto	NOUN
ejpam-4274	28	2	and	and	CCONJ
ejpam-4274	28	3	noiri	noiri	ADV
ejpam-4274	29	1	[	[	X
ejpam-4274	29	2	6	6	NUM
ejpam-4274	29	3	]	]	PUNCT
ejpam-4274	29	4	defined	define	VERB
ejpam-4274	29	5	λm	λm	NOUN
ejpam-4274	29	6	-	-	PUNCT
ejpam-4274	29	7	sets	set	NOUN
ejpam-4274	29	8	and	and	CCONJ
ejpam-4274	29	9	generalized	generalized	ADJ
ejpam-4274	29	10	λmsets	λmset	NOUN
ejpam-4274	29	11	in	in	ADP
ejpam-4274	29	12	an	an	DET
ejpam-4274	29	13	m	m	NOUN
ejpam-4274	29	14	-	-	PUNCT
ejpam-4274	29	15	spaces	space	NOUN
ejpam-4274	29	16	(	(	PUNCT
ejpam-4274	29	17	x	x	X
ejpam-4274	29	18	,	,	PUNCT
ejpam-4274	29	19	m	m	NOUN
ejpam-4274	29	20	)	)	PUNCT
ejpam-4274	29	21	which	which	PRON
ejpam-4274	29	22	is	be	AUX
ejpam-4274	29	23	equivalent	equivalent	ADJ
ejpam-4274	29	24	to	to	ADP
ejpam-4274	29	25	a	a	DET
ejpam-4274	29	26	generalized	generalized	ADJ
ejpam-4274	29	27	topological	topological	ADJ
ejpam-4274	29	28	spaces	space	NOUN
ejpam-4274	29	29	[	[	X
ejpam-4274	29	30	14	14	NUM
ejpam-4274	29	31	]	]	PUNCT
ejpam-4274	29	32	and	and	CCONJ
ejpam-4274	29	33	investigated	investigate	VERB
ejpam-4274	29	34	properties	property	NOUN
ejpam-4274	29	35	of	of	ADP
ejpam-4274	29	36	several	several	ADJ
ejpam-4274	29	37	low	low	ADJ
ejpam-4274	29	38	separation	separation	NOUN
ejpam-4274	29	39	axioms	axiom	NOUN
ejpam-4274	29	40	of	of	ADP
ejpam-4274	29	41	topologies	topology	NOUN
ejpam-4274	29	42	constructed	construct	VERB
ejpam-4274	29	43	by	by	ADP
ejpam-4274	29	44	the	the	DET
ejpam-4274	29	45	families	family	NOUN
ejpam-4274	29	46	of	of	ADP
ejpam-4274	29	47	these	these	DET
ejpam-4274	29	48	sets	set	NOUN
ejpam-4274	29	49	.	.	PUNCT
ejpam-4274	30	1	ganster	ganster	NOUN
ejpam-4274	30	2	et	et	PROPN
ejpam-4274	30	3	al	al	PROPN
ejpam-4274	30	4	.	.	PUNCT
ejpam-4274	31	1	[	[	X
ejpam-4274	31	2	10	10	NUM
ejpam-4274	31	3	]	]	PUNCT
ejpam-4274	31	4	introduced	introduce	VERB
ejpam-4274	31	5	the	the	DET
ejpam-4274	31	6	notions	notion	NOUN
ejpam-4274	31	7	of	of	ADP
ejpam-4274	31	8	a	a	DET
ejpam-4274	31	9	pre	pre	ADJ
ejpam-4274	31	10	-	-	ADJ
ejpam-4274	31	11	λ	λ	NOUN
ejpam-4274	31	12	-	-	NOUN
ejpam-4274	31	13	set	set	VERB
ejpam-4274	31	14	and	and	CCONJ
ejpam-4274	31	15	a	a	DET
ejpam-4274	31	16	pre	pre	ADJ
ejpam-4274	31	17	-	-	NOUN
ejpam-4274	31	18	v	v	ADJ
ejpam-4274	31	19	-set	-set	PUNCT
ejpam-4274	31	20	in	in	ADP
ejpam-4274	31	21	a	a	DET
ejpam-4274	31	22	topological	topological	ADJ
ejpam-4274	31	23	space	space	NOUN
ejpam-4274	31	24	and	and	CCONJ
ejpam-4274	31	25	studied	study	VERB
ejpam-4274	31	26	the	the	DET
ejpam-4274	31	27	fundamental	fundamental	ADJ
ejpam-4274	31	28	properties	property	NOUN
ejpam-4274	31	29	of	of	ADP
ejpam-4274	31	30	pre	pre	NOUN
ejpam-4274	31	31	-	-	NOUN
ejpam-4274	31	32	λsets	λset	NOUN
ejpam-4274	31	33	and	and	CCONJ
ejpam-4274	31	34	pre	pre	ADJ
ejpam-4274	31	35	-	-	ADJ
ejpam-4274	31	36	v	v	ADJ
ejpam-4274	31	37	-sets	-set	NOUN
ejpam-4274	31	38	.	.	PUNCT
ejpam-4274	32	1	caldas	caldas	PROPN
ejpam-4274	32	2	et	et	PROPN
ejpam-4274	32	3	al	al	PROPN
ejpam-4274	32	4	.	.	PUNCT
ejpam-4274	33	1	[	[	X
ejpam-4274	33	2	5	5	NUM
ejpam-4274	33	3	]	]	PUNCT
ejpam-4274	33	4	introduced	introduce	VERB
ejpam-4274	33	5	and	and	CCONJ
ejpam-4274	33	6	studied	study	VERB
ejpam-4274	33	7	two	two	NUM
ejpam-4274	33	8	new	new	ADJ
ejpam-4274	33	9	weak	weak	ADJ
ejpam-4274	33	10	separation	separation	NOUN
ejpam-4274	33	11	axioms	axiom	NOUN
ejpam-4274	33	12	called	call	VERB
ejpam-4274	33	13	λθ	λθ	NOUN
ejpam-4274	33	14	-	-	PUNCT
ejpam-4274	33	15	r0	r0	NOUN
ejpam-4274	33	16	and	and	CCONJ
ejpam-4274	33	17	λθ	λθ	NOUN
ejpam-4274	33	18	-	-	PUNCT
ejpam-4274	33	19	r1	r1	NOUN
ejpam-4274	33	20	spaces	space	NOUN
ejpam-4274	33	21	by	by	ADP
ejpam-4274	33	22	using	use	VERB
ejpam-4274	33	23	the	the	DET
ejpam-4274	33	24	notions	notion	NOUN
ejpam-4274	33	25	of	of	ADP
ejpam-4274	33	26	(	(	PUNCT
ejpam-4274	33	27	λ	λ	PROPN
ejpam-4274	33	28	,	,	PUNCT
ejpam-4274	33	29	θ)-open	θ)-open	VERB
ejpam-4274	33	30	sets	set	NOUN
ejpam-4274	33	31	and	and	CCONJ
ejpam-4274	33	32	(	(	PUNCT
ejpam-4274	33	33	λ	λ	INTJ
ejpam-4274	33	34	,	,	PUNCT
ejpam-4274	33	35	θ)closure	θ)closure	PROPN
ejpam-4274	33	36	operators	operator	NOUN
ejpam-4274	33	37	.	.	PUNCT
ejpam-4274	34	1	the	the	DET
ejpam-4274	34	2	concept	concept	NOUN
ejpam-4274	34	3	of	of	ADP
ejpam-4274	34	4	weak	weak	ADJ
ejpam-4274	34	5	continuity	continuity	NOUN
ejpam-4274	34	6	due	due	ADP
ejpam-4274	34	7	to	to	ADP
ejpam-4274	34	8	levine	levine	PROPN
ejpam-4274	34	9	[	[	X
ejpam-4274	34	10	13	13	NUM
ejpam-4274	34	11	]	]	PUNCT
ejpam-4274	34	12	is	be	AUX
ejpam-4274	34	13	one	one	NUM
ejpam-4274	34	14	of	of	ADP
ejpam-4274	34	15	the	the	DET
ejpam-4274	34	16	most	most	ADV
ejpam-4274	34	17	important	important	ADJ
ejpam-4274	34	18	weak	weak	ADJ
ejpam-4274	34	19	forms	form	NOUN
ejpam-4274	34	20	of	of	ADP
ejpam-4274	34	21	continuity	continuity	NOUN
ejpam-4274	34	22	in	in	ADP
ejpam-4274	34	23	topological	topological	ADJ
ejpam-4274	34	24	spaces	space	NOUN
ejpam-4274	34	25	.	.	PUNCT
ejpam-4274	35	1	rose	rise	VERB
ejpam-4274	36	1	[	[	X
ejpam-4274	36	2	18	18	NUM
ejpam-4274	36	3	]	]	PUNCT
ejpam-4274	36	4	introduced	introduce	VERB
ejpam-4274	36	5	the	the	DET
ejpam-4274	36	6	notion	notion	NOUN
ejpam-4274	36	7	of	of	ADP
ejpam-4274	36	8	subweakly	subweakly	ADJ
ejpam-4274	36	9	continuous	continuous	ADJ
ejpam-4274	36	10	functions	function	NOUN
ejpam-4274	36	11	and	and	CCONJ
ejpam-4274	36	12	investigated	investigate	VERB
ejpam-4274	36	13	the	the	DET
ejpam-4274	36	14	relationships	relationship	NOUN
ejpam-4274	36	15	between	between	ADP
ejpam-4274	36	16	subweak	subweak	NOUN
ejpam-4274	36	17	continuity	continuity	NOUN
ejpam-4274	36	18	and	and	CCONJ
ejpam-4274	36	19	weak	weak	ADJ
ejpam-4274	36	20	continuity	continuity	NOUN
ejpam-4274	36	21	.	.	PUNCT
ejpam-4274	37	1	popa	popa	NOUN
ejpam-4274	37	2	and	and	CCONJ
ejpam-4274	37	3	noiri	noiri	ADV
ejpam-4274	38	1	[	[	X
ejpam-4274	38	2	17	17	NUM
ejpam-4274	38	3	]	]	PUNCT
ejpam-4274	38	4	introduced	introduce	VERB
ejpam-4274	38	5	the	the	DET
ejpam-4274	38	6	concept	concept	NOUN
ejpam-4274	38	7	of	of	ADP
ejpam-4274	38	8	weakly	weakly	ADJ
ejpam-4274	38	9	(	(	PUNCT
ejpam-4274	38	10	τ	τ	PROPN
ejpam-4274	38	11	,	,	PUNCT
ejpam-4274	38	12	m)-continuous	m)-continuous	ADJ
ejpam-4274	38	13	functions	function	NOUN
ejpam-4274	38	14	as	as	ADP
ejpam-4274	38	15	functions	function	NOUN
ejpam-4274	38	16	from	from	ADP
ejpam-4274	38	17	a	a	DET
ejpam-4274	38	18	topological	topological	ADJ
ejpam-4274	38	19	space	space	NOUN
ejpam-4274	38	20	into	into	ADP
ejpam-4274	38	21	a	a	DET
ejpam-4274	38	22	set	set	NOUN
ejpam-4274	38	23	satisfying	satisfy	VERB
ejpam-4274	38	24	some	some	DET
ejpam-4274	38	25	minimal	minimal	ADJ
ejpam-4274	38	26	conditions	condition	NOUN
ejpam-4274	38	27	and	and	CCONJ
ejpam-4274	38	28	investigated	investigate	VERB
ejpam-4274	38	29	several	several	ADJ
ejpam-4274	38	30	characterizations	characterization	NOUN
ejpam-4274	38	31	of	of	ADP
ejpam-4274	38	32	such	such	ADJ
ejpam-4274	38	33	functions	function	NOUN
ejpam-4274	38	34	.	.	PUNCT
ejpam-4274	39	1	the	the	DET
ejpam-4274	39	2	paper	paper	NOUN
ejpam-4274	39	3	is	be	AUX
ejpam-4274	39	4	organized	organize	VERB
ejpam-4274	39	5	as	as	SCONJ
ejpam-4274	39	6	follows	follow	VERB
ejpam-4274	39	7	.	.	PUNCT
ejpam-4274	40	1	in	in	ADP
ejpam-4274	40	2	section	section	NOUN
ejpam-4274	40	3	3	3	NUM
ejpam-4274	40	4	,	,	PUNCT
ejpam-4274	40	5	we	we	PRON
ejpam-4274	40	6	obtain	obtain	VERB
ejpam-4274	40	7	fundamental	fundamental	ADJ
ejpam-4274	40	8	properties	property	NOUN
ejpam-4274	40	9	of	of	ADP
ejpam-4274	40	10	λp	λp	NOUN
ejpam-4274	40	11	-	-	PUNCT
ejpam-4274	40	12	sets	set	NOUN
ejpam-4274	40	13	and	and	CCONJ
ejpam-4274	40	14	investigate	investigate	VERB
ejpam-4274	40	15	low	low	ADJ
ejpam-4274	40	16	separation	separation	NOUN
ejpam-4274	40	17	axioms	axiom	NOUN
ejpam-4274	40	18	of	of	ADP
ejpam-4274	40	19	an	an	DET
ejpam-4274	40	20	alexandorff	alexandorff	ADJ
ejpam-4274	40	21	spaces	space	NOUN
ejpam-4274	40	22	(	(	PUNCT
ejpam-4274	40	23	x	x	X
ejpam-4274	40	24	,	,	PUNCT
ejpam-4274	40	25	λp	λp	PROPN
ejpam-4274	40	26	)	)	PUNCT
ejpam-4274	40	27	.	.	PUNCT
ejpam-4274	41	1	in	in	ADP
ejpam-4274	41	2	section	section	NOUN
ejpam-4274	41	3	4	4	NUM
ejpam-4274	41	4	,	,	PUNCT
ejpam-4274	41	5	we	we	PRON
ejpam-4274	41	6	introduce	introduce	VERB
ejpam-4274	41	7	the	the	DET
ejpam-4274	41	8	concept	concept	NOUN
ejpam-4274	41	9	of	of	ADP
ejpam-4274	41	10	(	(	PUNCT
ejpam-4274	41	11	λ	λ	PROPN
ejpam-4274	41	12	,	,	PUNCT
ejpam-4274	41	13	p)-closed	p)-close	VERB
ejpam-4274	41	14	sets	set	NOUN
ejpam-4274	41	15	and	and	CCONJ
ejpam-4274	41	16	investigate	investigate	VERB
ejpam-4274	41	17	properties	property	NOUN
ejpam-4274	41	18	of	of	ADP
ejpam-4274	41	19	several	several	ADJ
ejpam-4274	41	20	low	low	ADJ
ejpam-4274	41	21	separation	separation	NOUN
ejpam-4274	41	22	axioms	axiom	NOUN
ejpam-4274	41	23	of	of	ADP
ejpam-4274	41	24	topologies	topology	NOUN
ejpam-4274	41	25	constructed	construct	VERB
ejpam-4274	41	26	by	by	ADP
ejpam-4274	41	27	the	the	DET
ejpam-4274	41	28	families	family	NOUN
ejpam-4274	41	29	of	of	ADP
ejpam-4274	41	30	these	these	DET
ejpam-4274	41	31	sets	set	NOUN
ejpam-4274	41	32	.	.	PUNCT
ejpam-4274	42	1	in	in	ADP
ejpam-4274	42	2	section	section	NOUN
ejpam-4274	42	3	5	5	NUM
ejpam-4274	42	4	,	,	PUNCT
ejpam-4274	42	5	we	we	PRON
ejpam-4274	42	6	investigate	investigate	VERB
ejpam-4274	42	7	some	some	DET
ejpam-4274	42	8	characterizations	characterization	NOUN
ejpam-4274	42	9	of	of	ADP
ejpam-4274	42	10	λp	λp	NOUN
ejpam-4274	42	11	-	-	PUNCT
ejpam-4274	42	12	r0	r0	NOUN
ejpam-4274	42	13	spaces	space	NOUN
ejpam-4274	42	14	.	.	PUNCT
ejpam-4274	43	1	in	in	ADP
ejpam-4274	43	2	the	the	DET
ejpam-4274	43	3	last	last	ADJ
ejpam-4274	43	4	section	section	NOUN
ejpam-4274	43	5	,	,	PUNCT
ejpam-4274	43	6	we	we	PRON
ejpam-4274	43	7	introduce	introduce	VERB
ejpam-4274	43	8	the	the	DET
ejpam-4274	43	9	concept	concept	NOUN
ejpam-4274	43	10	of	of	ADP
ejpam-4274	43	11	weakly	weakly	ADJ
ejpam-4274	43	12	(	(	PUNCT
ejpam-4274	43	13	λ	λ	PROPN
ejpam-4274	43	14	,	,	PUNCT
ejpam-4274	43	15	p)-continuous	p)-continuous	ADJ
ejpam-4274	43	16	functions	function	NOUN
ejpam-4274	43	17	and	and	CCONJ
ejpam-4274	43	18	investigate	investigate	VERB
ejpam-4274	43	19	several	several	ADJ
ejpam-4274	43	20	characterizations	characterization	NOUN
ejpam-4274	43	21	of	of	ADP
ejpam-4274	43	22	such	such	ADJ
ejpam-4274	43	23	functions	function	NOUN
ejpam-4274	43	24	.	.	PUNCT
ejpam-4274	44	1	2	2	X
ejpam-4274	44	2	.	.	X
ejpam-4274	44	3	preliminaries	preliminary	NOUN
ejpam-4274	44	4	throughout	throughout	ADP
ejpam-4274	44	5	the	the	DET
ejpam-4274	44	6	present	present	ADJ
ejpam-4274	44	7	paper	paper	NOUN
ejpam-4274	44	8	,	,	PUNCT
ejpam-4274	44	9	spaces	space	NOUN
ejpam-4274	44	10	(	(	PUNCT
ejpam-4274	44	11	x	x	X
ejpam-4274	44	12	,	,	PUNCT
ejpam-4274	44	13	τ	τ	X
ejpam-4274	44	14	)	)	PUNCT
ejpam-4274	44	15	and	and	CCONJ
ejpam-4274	44	16	(	(	PUNCT
ejpam-4274	44	17	y	y	PROPN
ejpam-4274	44	18	,	,	PUNCT
ejpam-4274	44	19	σ	σ	PROPN
ejpam-4274	44	20	)	)	PUNCT
ejpam-4274	44	21	(	(	PUNCT
ejpam-4274	44	22	or	or	CCONJ
ejpam-4274	44	23	simply	simply	ADV
ejpam-4274	44	24	x	x	X
ejpam-4274	44	25	and	and	CCONJ
ejpam-4274	44	26	y	y	PROPN
ejpam-4274	44	27	)	)	PUNCT
ejpam-4274	44	28	always	always	ADV
ejpam-4274	44	29	mean	mean	VERB
ejpam-4274	44	30	topological	topological	ADJ
ejpam-4274	44	31	spaces	space	NOUN
ejpam-4274	44	32	on	on	ADP
ejpam-4274	44	33	which	which	PRON
ejpam-4274	44	34	no	no	DET
ejpam-4274	44	35	separation	separation	NOUN
ejpam-4274	44	36	axioms	axiom	NOUN
ejpam-4274	44	37	are	be	AUX
ejpam-4274	44	38	assumed	assume	VERB
ejpam-4274	44	39	unless	unless	SCONJ
ejpam-4274	44	40	explicitly	explicitly	ADV
ejpam-4274	44	41	stated	state	VERB
ejpam-4274	44	42	.	.	PUNCT
ejpam-4274	45	1	for	for	ADP
ejpam-4274	45	2	a	a	DET
ejpam-4274	45	3	subset	subset	NOUN
ejpam-4274	45	4	a	a	PRON
ejpam-4274	45	5	of	of	ADP
ejpam-4274	45	6	a	a	DET
ejpam-4274	45	7	topological	topological	ADJ
ejpam-4274	45	8	space	space	NOUN
ejpam-4274	45	9	(	(	PUNCT
ejpam-4274	45	10	x	x	X
ejpam-4274	45	11	,	,	PUNCT
ejpam-4274	45	12	τ	τ	PROPN
ejpam-4274	45	13	)	)	PUNCT
ejpam-4274	45	14	,	,	PUNCT
ejpam-4274	45	15	cl(a	cl(a	NUM
ejpam-4274	45	16	)	)	PUNCT
ejpam-4274	45	17	and	and	CCONJ
ejpam-4274	45	18	int(a	int(a	PROPN
ejpam-4274	45	19	)	)	PUNCT
ejpam-4274	45	20	represent	represent	VERB
ejpam-4274	45	21	the	the	DET
ejpam-4274	45	22	closure	closure	NOUN
ejpam-4274	45	23	and	and	CCONJ
ejpam-4274	45	24	the	the	DET
ejpam-4274	45	25	interior	interior	NOUN
ejpam-4274	45	26	of	of	ADP
ejpam-4274	45	27	a	a	PRON
ejpam-4274	45	28	,	,	PUNCT
ejpam-4274	45	29	respectively	respectively	ADV
ejpam-4274	45	30	.	.	PUNCT
ejpam-4274	46	1	a	a	DET
ejpam-4274	46	2	subset	subset	NOUN
ejpam-4274	46	3	a	a	PRON
ejpam-4274	46	4	of	of	ADP
ejpam-4274	46	5	a	a	DET
ejpam-4274	46	6	topological	topological	ADJ
ejpam-4274	46	7	space	space	NOUN
ejpam-4274	46	8	(	(	PUNCT
ejpam-4274	46	9	x	x	X
ejpam-4274	46	10	,	,	PUNCT
ejpam-4274	46	11	τ	τ	X
ejpam-4274	46	12	)	)	PUNCT
ejpam-4274	46	13	is	be	AUX
ejpam-4274	46	14	said	say	VERB
ejpam-4274	46	15	to	to	PART
ejpam-4274	46	16	be	be	AUX
ejpam-4274	46	17	preopen	preopen	ADJ
ejpam-4274	46	18	[	[	X
ejpam-4274	46	19	16	16	NUM
ejpam-4274	46	20	]	]	PUNCT
ejpam-4274	46	21	(	(	PUNCT
ejpam-4274	46	22	resp	resp	NOUN
ejpam-4274	46	23	.	.	PUNCT
ejpam-4274	47	1	preclosed	preclose	VERB
ejpam-4274	47	2	[	[	X
ejpam-4274	47	3	16	16	NUM
ejpam-4274	47	4	]	]	SYM
ejpam-4274	47	5	)	)	PUNCT
ejpam-4274	47	6	if	if	SCONJ
ejpam-4274	47	7	a	a	DET
ejpam-4274	47	8	⊆	⊆	NUM
ejpam-4274	47	9	int(cl(a	int(cl(a	PROPN
ejpam-4274	47	10	)	)	PUNCT
ejpam-4274	47	11	)	)	PUNCT
ejpam-4274	48	1	(	(	PUNCT
ejpam-4274	48	2	resp	resp	NOUN
ejpam-4274	48	3	.	.	PUNCT
ejpam-4274	48	4	cl(int(a	cl(int(a	PROPN
ejpam-4274	48	5	)	)	PUNCT
ejpam-4274	48	6	)	)	PUNCT
ejpam-4274	49	1	⊆	⊆	NUM
ejpam-4274	49	2	a	a	PRON
ejpam-4274	49	3	)	)	PUNCT
ejpam-4274	49	4	.	.	PUNCT
ejpam-4274	50	1	by	by	ADP
ejpam-4274	50	2	po(x	po(x	NUM
ejpam-4274	50	3	,	,	PUNCT
ejpam-4274	50	4	τ	τ	X
ejpam-4274	50	5	)	)	PUNCT
ejpam-4274	50	6	and	and	CCONJ
ejpam-4274	50	7	pc(x	pc(x	NOUN
ejpam-4274	50	8	,	,	PUNCT
ejpam-4274	50	9	τ	τ	PROPN
ejpam-4274	50	10	)	)	PUNCT
ejpam-4274	50	11	,	,	PUNCT
ejpam-4274	50	12	we	we	PRON
ejpam-4274	50	13	denote	denote	VERB
ejpam-4274	50	14	the	the	DET
ejpam-4274	50	15	collection	collection	NOUN
ejpam-4274	50	16	of	of	ADP
ejpam-4274	50	17	all	all	DET
ejpam-4274	50	18	preopen	preopen	ADJ
ejpam-4274	50	19	sets	set	NOUN
ejpam-4274	50	20	and	and	CCONJ
ejpam-4274	50	21	the	the	DET
ejpam-4274	50	22	collection	collection	NOUN
ejpam-4274	50	23	of	of	ADP
ejpam-4274	50	24	all	all	DET
ejpam-4274	50	25	preclosed	preclose	VERB
ejpam-4274	50	26	sets	set	NOUN
ejpam-4274	50	27	of	of	ADP
ejpam-4274	50	28	a	a	DET
ejpam-4274	50	29	topological	topological	ADJ
ejpam-4274	50	30	space	space	NOUN
ejpam-4274	50	31	(	(	PUNCT
ejpam-4274	50	32	x	x	X
ejpam-4274	50	33	,	,	PUNCT
ejpam-4274	50	34	τ	τ	PROPN
ejpam-4274	50	35	)	)	PUNCT
ejpam-4274	50	36	,	,	PUNCT
ejpam-4274	50	37	respectively	respectively	ADV
ejpam-4274	50	38	.	.	PUNCT
ejpam-4274	51	1	the	the	DET
ejpam-4274	51	2	intersection	intersection	NOUN
ejpam-4274	51	3	of	of	ADP
ejpam-4274	51	4	all	all	DET
ejpam-4274	51	5	preclosed	preclose	VERB
ejpam-4274	51	6	sets	set	NOUN
ejpam-4274	51	7	containig	containig	PROPN
ejpam-4274	51	8	a	a	PRON
ejpam-4274	51	9	is	be	AUX
ejpam-4274	51	10	called	call	VERB
ejpam-4274	51	11	the	the	DET
ejpam-4274	51	12	preclosure	preclosure	ADJ
ejpam-4274	51	13	[	[	X
ejpam-4274	51	14	8	8	NUM
ejpam-4274	51	15	]	]	PUNCT
ejpam-4274	51	16	of	of	ADP
ejpam-4274	51	17	a	a	PRON
ejpam-4274	51	18	and	and	CCONJ
ejpam-4274	51	19	is	be	AUX
ejpam-4274	51	20	denoted	denote	VERB
ejpam-4274	51	21	by	by	ADP
ejpam-4274	51	22	pcl(a	pcl(a	PROPN
ejpam-4274	51	23	)	)	PUNCT
ejpam-4274	51	24	.	.	PUNCT
ejpam-4274	52	1	definition	definition	NOUN
ejpam-4274	52	2	1	1	NUM
ejpam-4274	52	3	.	.	PUNCT
ejpam-4274	53	1	a	a	DET
ejpam-4274	53	2	topological	topological	ADJ
ejpam-4274	53	3	space	space	NOUN
ejpam-4274	53	4	(	(	PUNCT
ejpam-4274	53	5	x	x	X
ejpam-4274	53	6	,	,	PUNCT
ejpam-4274	53	7	τ	τ	X
ejpam-4274	53	8	)	)	PUNCT
ejpam-4274	53	9	is	be	AUX
ejpam-4274	53	10	said	say	VERB
ejpam-4274	53	11	to	to	PART
ejpam-4274	53	12	be	be	AUX
ejpam-4274	53	13	:	:	PUNCT
ejpam-4274	53	14	(	(	PUNCT
ejpam-4274	53	15	1	1	X
ejpam-4274	53	16	)	)	PUNCT
ejpam-4274	53	17	pre	pre	NOUN
ejpam-4274	53	18	-	-	NOUN
ejpam-4274	53	19	t0	t0	NOUN
ejpam-4274	53	20	[	[	X
ejpam-4274	53	21	12	12	NUM
ejpam-4274	53	22	]	]	PUNCT
ejpam-4274	53	23	if	if	SCONJ
ejpam-4274	53	24	,	,	PUNCT
ejpam-4274	53	25	for	for	ADP
ejpam-4274	53	26	each	each	DET
ejpam-4274	53	27	pair	pair	NOUN
ejpam-4274	53	28	of	of	ADP
ejpam-4274	53	29	distinct	distinct	ADJ
ejpam-4274	53	30	points	point	NOUN
ejpam-4274	53	31	of	of	ADP
ejpam-4274	53	32	x	x	NOUN
ejpam-4274	53	33	,	,	PUNCT
ejpam-4274	53	34	there	there	PRON
ejpam-4274	53	35	exists	exist	VERB
ejpam-4274	53	36	a	a	DET
ejpam-4274	53	37	preopen	preopen	ADJ
ejpam-4274	53	38	set	set	NOUN
ejpam-4274	53	39	containing	contain	VERB
ejpam-4274	53	40	one	one	NUM
ejpam-4274	53	41	of	of	ADP
ejpam-4274	53	42	the	the	DET
ejpam-4274	53	43	points	point	NOUN
ejpam-4274	53	44	but	but	CCONJ
ejpam-4274	53	45	not	not	PART
ejpam-4274	53	46	the	the	DET
ejpam-4274	53	47	other	other	ADJ
ejpam-4274	53	48	;	;	PUNCT
ejpam-4274	53	49	(	(	PUNCT
ejpam-4274	53	50	2	2	X
ejpam-4274	53	51	)	)	PUNCT
ejpam-4274	53	52	pre	pre	NOUN
ejpam-4274	53	53	-	-	NOUN
ejpam-4274	53	54	t1	t1	NOUN
ejpam-4274	53	55	[	[	X
ejpam-4274	53	56	12	12	NUM
ejpam-4274	53	57	]	]	PUNCT
ejpam-4274	53	58	if	if	SCONJ
ejpam-4274	53	59	,	,	PUNCT
ejpam-4274	53	60	for	for	ADP
ejpam-4274	53	61	each	each	DET
ejpam-4274	53	62	pair	pair	NOUN
ejpam-4274	53	63	of	of	ADP
ejpam-4274	53	64	distinct	distinct	ADJ
ejpam-4274	53	65	points	point	NOUN
ejpam-4274	53	66	x	x	PUNCT
ejpam-4274	53	67	and	and	CCONJ
ejpam-4274	53	68	y	y	PROPN
ejpam-4274	53	69	of	of	ADP
ejpam-4274	53	70	x	x	PRON
ejpam-4274	53	71	,	,	PUNCT
ejpam-4274	53	72	there	there	PRON
ejpam-4274	53	73	exists	exist	VERB
ejpam-4274	53	74	a	a	DET
ejpam-4274	53	75	pair	pair	NOUN
ejpam-4274	53	76	of	of	ADP
ejpam-4274	53	77	preopen	preopen	ADJ
ejpam-4274	53	78	sets	set	NOUN
ejpam-4274	53	79	one	one	NUM
ejpam-4274	53	80	containing	contain	VERB
ejpam-4274	53	81	x	x	PUNCT
ejpam-4274	53	82	but	but	CCONJ
ejpam-4274	53	83	not	not	PART
ejpam-4274	53	84	y	y	PROPN
ejpam-4274	53	85	and	and	CCONJ
ejpam-4274	53	86	the	the	DET
ejpam-4274	53	87	other	other	ADJ
ejpam-4274	53	88	containing	contain	VERB
ejpam-4274	53	89	y	y	PROPN
ejpam-4274	53	90	but	but	CCONJ
ejpam-4274	53	91	not	not	PART
ejpam-4274	53	92	x	x	NOUN
ejpam-4274	53	93	;	;	PUNCT
ejpam-4274	53	94	(	(	PUNCT
ejpam-4274	53	95	3	3	X
ejpam-4274	53	96	)	)	PUNCT
ejpam-4274	53	97	pre	pre	ADJ
ejpam-4274	53	98	-	-	NOUN
ejpam-4274	53	99	r0	r0	ADJ
ejpam-4274	53	100	[	[	NOUN
ejpam-4274	53	101	4	4	X
ejpam-4274	53	102	]	]	X
ejpam-4274	53	103	if	if	SCONJ
ejpam-4274	53	104	every	every	DET
ejpam-4274	53	105	preopen	preopen	ADJ
ejpam-4274	53	106	set	set	NOUN
ejpam-4274	53	107	contains	contain	VERB
ejpam-4274	53	108	the	the	DET
ejpam-4274	53	109	preclosure	preclosure	NOUN
ejpam-4274	53	110	of	of	ADP
ejpam-4274	53	111	each	each	PRON
ejpam-4274	53	112	of	of	ADP
ejpam-4274	53	113	its	its	PRON
ejpam-4274	53	114	singletons	singleton	NOUN
ejpam-4274	53	115	.	.	PUNCT
ejpam-4274	54	1	definition	definition	NOUN
ejpam-4274	54	2	2	2	NUM
ejpam-4274	54	3	.	.	PUNCT
ejpam-4274	54	4	let	let	VERB
ejpam-4274	54	5	a	a	DET
ejpam-4274	54	6	be	be	AUX
ejpam-4274	54	7	a	a	DET
ejpam-4274	54	8	subset	subset	NOUN
ejpam-4274	54	9	of	of	ADP
ejpam-4274	54	10	a	a	DET
ejpam-4274	54	11	topological	topological	ADJ
ejpam-4274	54	12	space	space	NOUN
ejpam-4274	54	13	(	(	PUNCT
ejpam-4274	54	14	x	x	X
ejpam-4274	54	15	,	,	PUNCT
ejpam-4274	54	16	τ	τ	PROPN
ejpam-4274	54	17	)	)	PUNCT
ejpam-4274	54	18	.	.	PUNCT
ejpam-4274	55	1	a	a	DET
ejpam-4274	55	2	subset	subset	NOUN
ejpam-4274	55	3	λp(a	λp(a	NOUN
ejpam-4274	55	4	)	)	PUNCT
ejpam-4274	56	1	[	[	X
ejpam-4274	56	2	10	10	NUM
ejpam-4274	56	3	]	]	PUNCT
ejpam-4274	56	4	is	be	AUX
ejpam-4274	56	5	defined	define	VERB
ejpam-4274	56	6	as	as	SCONJ
ejpam-4274	56	7	follows	follow	VERB
ejpam-4274	56	8	:	:	PUNCT
ejpam-4274	56	9	λp(a	λp(a	NUM
ejpam-4274	56	10	)	)	PUNCT
ejpam-4274	57	1	=	=	PUNCT
ejpam-4274	58	1	∩{o	∩{o	NUM
ejpam-4274	58	2	∈	∈	PROPN
ejpam-4274	58	3	po(x	po(x	VERB
ejpam-4274	58	4	,	,	PUNCT
ejpam-4274	58	5	τ)|a	τ)|a	VERB
ejpam-4274	58	6	⊆	⊆	NUM
ejpam-4274	58	7	o	o	NOUN
ejpam-4274	58	8	}	}	PUNCT
ejpam-4274	58	9	.	.	PUNCT
ejpam-4274	59	1	c.	c.	PROPN
ejpam-4274	59	2	boonpok	boonpok	PROPN
ejpam-4274	59	3	,	,	PUNCT
ejpam-4274	59	4	c.	c.	PROPN
ejpam-4274	59	5	viriyapong	viriyapong	PROPN
ejpam-4274	59	6	/	/	SYM
ejpam-4274	59	7	eur	eur	PROPN
ejpam-4274	59	8	.	.	PUNCT
ejpam-4274	60	1	j.	j.	PROPN
ejpam-4274	60	2	pure	pure	PROPN
ejpam-4274	60	3	appl	appl	PROPN
ejpam-4274	60	4	.	.	PROPN
ejpam-4274	60	5	math	math	PROPN
ejpam-4274	60	6	,	,	PUNCT
ejpam-4274	60	7	15	15	NUM
ejpam-4274	60	8	(	(	PUNCT
ejpam-4274	60	9	2	2	NUM
ejpam-4274	60	10	)	)	PUNCT
ejpam-4274	60	11	(	(	PUNCT
ejpam-4274	60	12	2022	2022	NUM
ejpam-4274	60	13	)	)	PUNCT
ejpam-4274	60	14	,	,	PUNCT
ejpam-4274	60	15	415	415	NUM
ejpam-4274	60	16	-	-	SYM
ejpam-4274	60	17	436	436	NUM
ejpam-4274	60	18	417	417	NUM
ejpam-4274	60	19	lemma	lemma	PROPN
ejpam-4274	60	20	1	1	NUM
ejpam-4274	60	21	.	.	PUNCT
ejpam-4274	61	1	[	[	X
ejpam-4274	61	2	10	10	NUM
ejpam-4274	61	3	]	]	PUNCT
ejpam-4274	61	4	for	for	ADP
ejpam-4274	61	5	subsets	subset	NOUN
ejpam-4274	61	6	a	a	PRON
ejpam-4274	61	7	,	,	PUNCT
ejpam-4274	61	8	b	b	NOUN
ejpam-4274	61	9	and	and	CCONJ
ejpam-4274	61	10	ai(i	ai(i	CCONJ
ejpam-4274	61	11	∈	∈	PROPN
ejpam-4274	61	12	i	i	PROPN
ejpam-4274	61	13	)	)	PUNCT
ejpam-4274	61	14	of	of	ADP
ejpam-4274	61	15	a	a	DET
ejpam-4274	61	16	topological	topological	ADJ
ejpam-4274	61	17	space	space	NOUN
ejpam-4274	61	18	(	(	PUNCT
ejpam-4274	61	19	x	x	X
ejpam-4274	61	20	,	,	PUNCT
ejpam-4274	61	21	τ	τ	PROPN
ejpam-4274	61	22	)	)	PUNCT
ejpam-4274	61	23	,	,	PUNCT
ejpam-4274	61	24	the	the	DET
ejpam-4274	61	25	following	follow	VERB
ejpam-4274	61	26	properties	property	NOUN
ejpam-4274	61	27	hold	hold	VERB
ejpam-4274	61	28	:	:	PUNCT
ejpam-4274	61	29	(	(	PUNCT
ejpam-4274	61	30	1	1	X
ejpam-4274	61	31	)	)	PUNCT
ejpam-4274	61	32	a	a	DET
ejpam-4274	61	33	⊆	⊆	NUM
ejpam-4274	61	34	λp(a	λp(a	NUM
ejpam-4274	61	35	)	)	PUNCT
ejpam-4274	61	36	.	.	PUNCT
ejpam-4274	62	1	(	(	PUNCT
ejpam-4274	62	2	2	2	X
ejpam-4274	62	3	)	)	PUNCT
ejpam-4274	62	4	if	if	SCONJ
ejpam-4274	62	5	a	a	DET
ejpam-4274	62	6	⊆	⊆	NUM
ejpam-4274	62	7	b	b	NOUN
ejpam-4274	62	8	,	,	PUNCT
ejpam-4274	62	9	then	then	ADV
ejpam-4274	62	10	λp(a	λp(a	NUM
ejpam-4274	62	11	)	)	PUNCT
ejpam-4274	62	12	⊆	⊆	NUM
ejpam-4274	62	13	λp(b	λp(b	NOUN
ejpam-4274	62	14	)	)	PUNCT
ejpam-4274	62	15	.	.	PUNCT
ejpam-4274	63	1	(	(	PUNCT
ejpam-4274	63	2	3	3	X
ejpam-4274	63	3	)	)	PUNCT
ejpam-4274	63	4	λp(λp(a	λp(λp(a	NUM
ejpam-4274	63	5	)	)	PUNCT
ejpam-4274	63	6	)	)	PUNCT
ejpam-4274	64	1	=	=	SYM
ejpam-4274	64	2	λp(a	λp(a	NOUN
ejpam-4274	64	3	)	)	PUNCT
ejpam-4274	64	4	.	.	PUNCT
ejpam-4274	65	1	(	(	PUNCT
ejpam-4274	65	2	4	4	X
ejpam-4274	65	3	)	)	PUNCT
ejpam-4274	65	4	λp(∩{ai	λp(∩{ai	PROPN
ejpam-4274	66	1	|	|	ADV
ejpam-4274	66	2	i	i	PRON
ejpam-4274	66	3	∈	∈	PROPN
ejpam-4274	66	4	i	i	PRON
ejpam-4274	66	5	}	}	PUNCT
ejpam-4274	66	6	)	)	PUNCT
ejpam-4274	67	1	⊆	⊆	NUM
ejpam-4274	67	2	∩{λp(ai	∩{λp(ai	PROPN
ejpam-4274	67	3	)	)	PUNCT
ejpam-4274	68	1	|	|	ADV
ejpam-4274	68	2	i	i	PRON
ejpam-4274	68	3	∈	∈	VERB
ejpam-4274	68	4	i	i	PRON
ejpam-4274	68	5	}	}	PUNCT
ejpam-4274	68	6	.	.	PUNCT
ejpam-4274	69	1	(	(	PUNCT
ejpam-4274	69	2	5	5	X
ejpam-4274	69	3	)	)	PUNCT
ejpam-4274	69	4	λp(∪{ai	λp(∪{ai	NOUN
ejpam-4274	70	1	|	|	INTJ
ejpam-4274	70	2	i	i	PRON
ejpam-4274	70	3	∈	∈	VERB
ejpam-4274	70	4	i	i	PRON
ejpam-4274	70	5	}	}	PUNCT
ejpam-4274	70	6	)	)	PUNCT
ejpam-4274	71	1	=	=	SYM
ejpam-4274	71	2	∪{λp(ai	∪{λp(ai	X
ejpam-4274	71	3	)	)	PUNCT
ejpam-4274	71	4	|	|	ADV
ejpam-4274	71	5	i	i	PRON
ejpam-4274	71	6	∈	∈	VERB
ejpam-4274	71	7	i	i	PRON
ejpam-4274	71	8	}	}	PUNCT
ejpam-4274	71	9	.	.	PUNCT
ejpam-4274	72	1	3	3	X
ejpam-4274	72	2	.	.	X
ejpam-4274	72	3	λp	λp	NOUN
ejpam-4274	72	4	-	-	PUNCT
ejpam-4274	72	5	sets	set	NOUN
ejpam-4274	72	6	and	and	CCONJ
ejpam-4274	72	7	a	a	DET
ejpam-4274	72	8	topological	topological	ADJ
ejpam-4274	72	9	space	space	NOUN
ejpam-4274	72	10	(	(	PUNCT
ejpam-4274	72	11	x	x	X
ejpam-4274	72	12	,	,	PUNCT
ejpam-4274	72	13	λp	λp	PROPN
ejpam-4274	72	14	)	)	PUNCT
ejpam-4274	72	15	in	in	ADP
ejpam-4274	72	16	this	this	DET
ejpam-4274	72	17	section	section	NOUN
ejpam-4274	72	18	,	,	PUNCT
ejpam-4274	72	19	we	we	PRON
ejpam-4274	72	20	obtain	obtain	VERB
ejpam-4274	72	21	fundamental	fundamental	ADJ
ejpam-4274	72	22	properties	property	NOUN
ejpam-4274	72	23	of	of	ADP
ejpam-4274	72	24	λp	λp	NOUN
ejpam-4274	72	25	-	-	PUNCT
ejpam-4274	72	26	sets	set	NOUN
ejpam-4274	72	27	and	and	CCONJ
ejpam-4274	72	28	investigate	investigate	VERB
ejpam-4274	72	29	low	low	ADJ
ejpam-4274	72	30	separation	separation	NOUN
ejpam-4274	72	31	axioms	axiom	NOUN
ejpam-4274	72	32	of	of	ADP
ejpam-4274	72	33	an	an	DET
ejpam-4274	72	34	alexandorff	alexandorff	ADJ
ejpam-4274	72	35	space	space	NOUN
ejpam-4274	72	36	(	(	PUNCT
ejpam-4274	72	37	x	x	X
ejpam-4274	72	38	,	,	PUNCT
ejpam-4274	72	39	λp	λp	PROPN
ejpam-4274	72	40	)	)	PUNCT
ejpam-4274	72	41	.	.	PUNCT
ejpam-4274	73	1	definition	definition	NOUN
ejpam-4274	73	2	3	3	NUM
ejpam-4274	73	3	.	.	PUNCT
ejpam-4274	74	1	a	a	DET
ejpam-4274	74	2	subset	subset	NOUN
ejpam-4274	74	3	a	a	PRON
ejpam-4274	74	4	of	of	ADP
ejpam-4274	74	5	a	a	DET
ejpam-4274	74	6	topological	topological	ADJ
ejpam-4274	74	7	space	space	NOUN
ejpam-4274	74	8	(	(	PUNCT
ejpam-4274	74	9	x	x	X
ejpam-4274	74	10	,	,	PUNCT
ejpam-4274	74	11	τ	τ	X
ejpam-4274	74	12	)	)	PUNCT
ejpam-4274	74	13	is	be	AUX
ejpam-4274	74	14	called	call	VERB
ejpam-4274	74	15	a	a	DET
ejpam-4274	74	16	λp	λp	NOUN
ejpam-4274	74	17	-	-	PUNCT
ejpam-4274	74	18	set	set	VERB
ejpam-4274	74	19	(	(	PUNCT
ejpam-4274	74	20	pre	pre	ADJ
ejpam-4274	74	21	-	-	ADJ
ejpam-4274	74	22	λ	λ	NOUN
ejpam-4274	74	23	-	-	NOUN
ejpam-4274	74	24	set	set	NOUN
ejpam-4274	74	25	[	[	X
ejpam-4274	74	26	10	10	NUM
ejpam-4274	74	27	]	]	SYM
ejpam-4274	74	28	)	)	PUNCT
ejpam-4274	74	29	if	if	SCONJ
ejpam-4274	74	30	a	a	DET
ejpam-4274	74	31	=	=	NOUN
ejpam-4274	74	32	λp(a	λp(a	NUM
ejpam-4274	74	33	)	)	PUNCT
ejpam-4274	74	34	.	.	PUNCT
ejpam-4274	75	1	the	the	DET
ejpam-4274	75	2	family	family	NOUN
ejpam-4274	75	3	of	of	ADP
ejpam-4274	75	4	all	all	DET
ejpam-4274	75	5	λp	λp	NOUN
ejpam-4274	75	6	-	-	PUNCT
ejpam-4274	75	7	sets	set	NOUN
ejpam-4274	75	8	of	of	ADP
ejpam-4274	75	9	(	(	PUNCT
ejpam-4274	75	10	x	x	NOUN
ejpam-4274	75	11	,	,	PUNCT
ejpam-4274	75	12	τ	τ	X
ejpam-4274	75	13	)	)	PUNCT
ejpam-4274	75	14	is	be	AUX
ejpam-4274	75	15	denoted	denote	VERB
ejpam-4274	75	16	by	by	ADP
ejpam-4274	75	17	λp(x	λp(x	NOUN
ejpam-4274	75	18	,	,	PUNCT
ejpam-4274	75	19	τ	τ	X
ejpam-4274	75	20	)	)	PUNCT
ejpam-4274	75	21	(	(	PUNCT
ejpam-4274	75	22	or	or	CCONJ
ejpam-4274	75	23	simply	simply	ADV
ejpam-4274	75	24	λp	λp	PROPN
ejpam-4274	75	25	)	)	PUNCT
ejpam-4274	75	26	.	.	PUNCT
ejpam-4274	76	1	lemma	lemma	PROPN
ejpam-4274	76	2	2	2	NUM
ejpam-4274	76	3	.	.	X
ejpam-4274	77	1	for	for	ADP
ejpam-4274	77	2	a	a	DET
ejpam-4274	77	3	subset	subset	NOUN
ejpam-4274	77	4	a	a	PRON
ejpam-4274	77	5	of	of	ADP
ejpam-4274	77	6	a	a	DET
ejpam-4274	77	7	topological	topological	ADJ
ejpam-4274	77	8	space	space	NOUN
ejpam-4274	77	9	(	(	PUNCT
ejpam-4274	77	10	x	x	X
ejpam-4274	77	11	,	,	PUNCT
ejpam-4274	77	12	τ	τ	PROPN
ejpam-4274	77	13	)	)	PUNCT
ejpam-4274	77	14	,	,	PUNCT
ejpam-4274	77	15	the	the	DET
ejpam-4274	77	16	following	follow	VERB
ejpam-4274	77	17	properties	property	NOUN
ejpam-4274	77	18	hold	hold	VERB
ejpam-4274	77	19	:	:	PUNCT
ejpam-4274	77	20	(	(	PUNCT
ejpam-4274	77	21	1	1	X
ejpam-4274	77	22	)	)	PUNCT
ejpam-4274	77	23	λp(a	λp(a	NUM
ejpam-4274	77	24	)	)	PUNCT
ejpam-4274	77	25	is	be	AUX
ejpam-4274	77	26	a	a	DET
ejpam-4274	77	27	λp	λp	NOUN
ejpam-4274	77	28	-	-	PUNCT
ejpam-4274	77	29	set	set	NOUN
ejpam-4274	77	30	.	.	PUNCT
ejpam-4274	78	1	(	(	PUNCT
ejpam-4274	78	2	2	2	X
ejpam-4274	78	3	)	)	PUNCT
ejpam-4274	78	4	if	if	SCONJ
ejpam-4274	78	5	a	a	PRON
ejpam-4274	78	6	is	be	AUX
ejpam-4274	78	7	preopen	preopen	ADJ
ejpam-4274	78	8	,	,	PUNCT
ejpam-4274	78	9	then	then	ADV
ejpam-4274	78	10	a	a	PRON
ejpam-4274	78	11	is	be	AUX
ejpam-4274	78	12	a	a	DET
ejpam-4274	78	13	λp	λp	NOUN
ejpam-4274	78	14	-	-	PUNCT
ejpam-4274	78	15	set	set	NOUN
ejpam-4274	78	16	.	.	PUNCT
ejpam-4274	79	1	proof	proof	NOUN
ejpam-4274	79	2	.	.	PUNCT
ejpam-4274	80	1	this	this	PRON
ejpam-4274	80	2	follows	follow	VERB
ejpam-4274	80	3	readily	readily	ADV
ejpam-4274	80	4	from	from	ADP
ejpam-4274	80	5	lemma	lemma	PROPN
ejpam-4274	80	6	1	1	NUM
ejpam-4274	80	7	.	.	PUNCT
ejpam-4274	81	1	lemma	lemma	PROPN
ejpam-4274	81	2	3	3	X
ejpam-4274	81	3	.	.	PUNCT
ejpam-4274	82	1	[	[	X
ejpam-4274	82	2	10	10	NUM
ejpam-4274	82	3	]	]	PUNCT
ejpam-4274	82	4	for	for	ADP
ejpam-4274	82	5	subsets	subset	NOUN
ejpam-4274	82	6	a	a	PRON
ejpam-4274	82	7	and	and	CCONJ
ejpam-4274	82	8	ai(i	ai(i	SYM
ejpam-4274	82	9	∈	∈	PROPN
ejpam-4274	82	10	i	i	PROPN
ejpam-4274	82	11	)	)	PUNCT
ejpam-4274	82	12	of	of	ADP
ejpam-4274	82	13	a	a	DET
ejpam-4274	82	14	topological	topological	ADJ
ejpam-4274	82	15	space	space	NOUN
ejpam-4274	82	16	(	(	PUNCT
ejpam-4274	82	17	x	x	X
ejpam-4274	82	18	,	,	PUNCT
ejpam-4274	82	19	τ	τ	PROPN
ejpam-4274	82	20	)	)	PUNCT
ejpam-4274	82	21	,	,	PUNCT
ejpam-4274	82	22	the	the	DET
ejpam-4274	82	23	following	follow	VERB
ejpam-4274	82	24	properties	property	NOUN
ejpam-4274	82	25	hold	hold	VERB
ejpam-4274	82	26	:	:	PUNCT
ejpam-4274	82	27	(	(	PUNCT
ejpam-4274	82	28	1	1	X
ejpam-4274	82	29	)	)	PUNCT
ejpam-4274	82	30	∅	∅	NOUN
ejpam-4274	82	31	and	and	CCONJ
ejpam-4274	82	32	x	x	X
ejpam-4274	82	33	are	be	AUX
ejpam-4274	82	34	pre	pre	ADJ
ejpam-4274	82	35	-	-	ADJ
ejpam-4274	82	36	λ	λ	NOUN
ejpam-4274	82	37	-	-	PUNCT
ejpam-4274	82	38	sets	set	NOUN
ejpam-4274	82	39	.	.	PUNCT
ejpam-4274	83	1	(	(	PUNCT
ejpam-4274	83	2	2	2	X
ejpam-4274	83	3	)	)	PUNCT
ejpam-4274	83	4	every	every	DET
ejpam-4274	83	5	union	union	NOUN
ejpam-4274	83	6	of	of	ADP
ejpam-4274	83	7	pre	pre	ADJ
ejpam-4274	83	8	-	-	ADJ
ejpam-4274	83	9	λ	λ	NOUN
ejpam-4274	83	10	-	-	PUNCT
ejpam-4274	83	11	sets	set	NOUN
ejpam-4274	83	12	is	be	AUX
ejpam-4274	83	13	a	a	DET
ejpam-4274	83	14	pre	pre	ADJ
ejpam-4274	83	15	-	-	ADJ
ejpam-4274	83	16	λp	λp	NOUN
ejpam-4274	83	17	-	-	PUNCT
ejpam-4274	83	18	set	set	NOUN
ejpam-4274	83	19	.	.	PUNCT
ejpam-4274	84	1	(	(	PUNCT
ejpam-4274	84	2	3	3	X
ejpam-4274	84	3	)	)	PUNCT
ejpam-4274	84	4	every	every	DET
ejpam-4274	84	5	intersection	intersection	NOUN
ejpam-4274	84	6	of	of	ADP
ejpam-4274	84	7	pre	pre	ADJ
ejpam-4274	84	8	-	-	ADJ
ejpam-4274	84	9	λ	λ	NOUN
ejpam-4274	84	10	-	-	PUNCT
ejpam-4274	84	11	sets	set	NOUN
ejpam-4274	84	12	is	be	AUX
ejpam-4274	84	13	a	a	DET
ejpam-4274	84	14	pre	pre	ADJ
ejpam-4274	84	15	-	-	ADJ
ejpam-4274	84	16	λ	λ	NOUN
ejpam-4274	84	17	-	-	NOUN
ejpam-4274	84	18	set	set	NOUN
ejpam-4274	84	19	.	.	PUNCT
ejpam-4274	85	1	theorem	theorem	NOUN
ejpam-4274	85	2	1	1	NUM
ejpam-4274	85	3	.	.	X
ejpam-4274	86	1	for	for	ADP
ejpam-4274	86	2	a	a	DET
ejpam-4274	86	3	topological	topological	ADJ
ejpam-4274	86	4	space	space	NOUN
ejpam-4274	86	5	(	(	PUNCT
ejpam-4274	86	6	x	x	X
ejpam-4274	86	7	,	,	PUNCT
ejpam-4274	86	8	τ	τ	PROPN
ejpam-4274	86	9	)	)	PUNCT
ejpam-4274	86	10	,	,	PUNCT
ejpam-4274	86	11	the	the	DET
ejpam-4274	86	12	pair	pair	NOUN
ejpam-4274	86	13	(	(	PUNCT
ejpam-4274	86	14	x	x	X
ejpam-4274	86	15	,	,	PUNCT
ejpam-4274	86	16	λp	λp	X
ejpam-4274	86	17	)	)	PUNCT
ejpam-4274	86	18	is	be	AUX
ejpam-4274	86	19	an	an	DET
ejpam-4274	86	20	alexandroff	alexandroff	ADJ
ejpam-4274	86	21	space	space	NOUN
ejpam-4274	86	22	.	.	PUNCT
ejpam-4274	87	1	proof	proof	NOUN
ejpam-4274	87	2	.	.	PUNCT
ejpam-4274	88	1	this	this	PRON
ejpam-4274	88	2	is	be	AUX
ejpam-4274	88	3	an	an	DET
ejpam-4274	88	4	immediate	immediate	ADJ
ejpam-4274	88	5	consequence	consequence	NOUN
ejpam-4274	88	6	of	of	ADP
ejpam-4274	88	7	lemma	lemma	PROPN
ejpam-4274	88	8	3	3	NUM
ejpam-4274	88	9	.	.	PUNCT
ejpam-4274	88	10	theorem	theorem	NOUN
ejpam-4274	88	11	2	2	NUM
ejpam-4274	88	12	.	.	X
ejpam-4274	89	1	let	let	VERB
ejpam-4274	89	2	(	(	PUNCT
ejpam-4274	89	3	x	x	NOUN
ejpam-4274	89	4	,	,	PUNCT
ejpam-4274	89	5	τ	τ	X
ejpam-4274	89	6	)	)	PUNCT
ejpam-4274	89	7	be	be	VERB
ejpam-4274	89	8	a	a	DET
ejpam-4274	89	9	topological	topological	ADJ
ejpam-4274	89	10	space	space	NOUN
ejpam-4274	89	11	.	.	PUNCT
ejpam-4274	90	1	then	then	ADV
ejpam-4274	90	2	,	,	PUNCT
ejpam-4274	90	3	λp	λp	PROPN
ejpam-4274	90	4	=	=	SYM
ejpam-4274	90	5	λλp	λλp	PROPN
ejpam-4274	90	6	.	.	PUNCT
ejpam-4274	91	1	proof	proof	NOUN
ejpam-4274	91	2	.	.	PUNCT
ejpam-4274	92	1	by	by	ADP
ejpam-4274	92	2	lemma	lemma	PROPN
ejpam-4274	92	3	2	2	NUM
ejpam-4274	92	4	,	,	PUNCT
ejpam-4274	92	5	po(x	po(x	NUM
ejpam-4274	92	6	,	,	PUNCT
ejpam-4274	92	7	τ	τ	X
ejpam-4274	92	8	)	)	PUNCT
ejpam-4274	92	9	⊆	⊆	NUM
ejpam-4274	92	10	λp	λp	PROPN
ejpam-4274	92	11	.	.	PUNCT
ejpam-4274	92	12	let	let	VERB
ejpam-4274	92	13	a	a	DET
ejpam-4274	92	14	be	be	AUX
ejpam-4274	92	15	any	any	DET
ejpam-4274	92	16	subset	subset	NOUN
ejpam-4274	92	17	of	of	ADP
ejpam-4274	92	18	x.	x.	NOUN
ejpam-4274	92	19	then	then	ADV
ejpam-4274	92	20	,	,	PUNCT
ejpam-4274	92	21	λλp	λλp	PROPN
ejpam-4274	93	1	=	=	SYM
ejpam-4274	93	2	∩{u	∩{u	PROPN
ejpam-4274	94	1	|	|	ADV
ejpam-4274	94	2	a	a	DET
ejpam-4274	94	3	⊆	⊆	NUM
ejpam-4274	94	4	u	u	NOUN
ejpam-4274	94	5	,	,	PUNCT
ejpam-4274	94	6	u	u	PROPN
ejpam-4274	94	7	∈	∈	PROPN
ejpam-4274	94	8	λp	λp	PROPN
ejpam-4274	94	9	}	}	PUNCT
ejpam-4274	94	10	⊆	⊆	NUM
ejpam-4274	94	11	{	{	PUNCT
ejpam-4274	94	12	u	u	NOUN
ejpam-4274	94	13	|	|	ADV
ejpam-4274	94	14	a	a	DET
ejpam-4274	94	15	⊆	⊆	NUM
ejpam-4274	94	16	u	u	NOUN
ejpam-4274	94	17	,	,	PUNCT
ejpam-4274	94	18	u	u	PROPN
ejpam-4274	94	19	∈	∈	PROPN
ejpam-4274	94	20	po(x	po(x	NOUN
ejpam-4274	94	21	,	,	PUNCT
ejpam-4274	94	22	τ	τ	NOUN
ejpam-4274	94	23	)	)	PUNCT
ejpam-4274	94	24	}	}	PUNCT
ejpam-4274	94	25	=	=	SYM
ejpam-4274	94	26	λp(a	λp(a	NOUN
ejpam-4274	94	27	)	)	PUNCT
ejpam-4274	94	28	.	.	PUNCT
ejpam-4274	95	1	thus	thus	ADV
ejpam-4274	95	2	,	,	PUNCT
ejpam-4274	95	3	λλp(a	λλp(a	PROPN
ejpam-4274	95	4	)	)	PUNCT
ejpam-4274	95	5	⊆	⊆	NUM
ejpam-4274	95	6	λp(a	λp(a	NUM
ejpam-4274	95	7	)	)	PUNCT
ejpam-4274	95	8	.	.	PUNCT
ejpam-4274	96	1	now	now	ADV
ejpam-4274	96	2	,	,	PUNCT
ejpam-4274	96	3	we	we	PRON
ejpam-4274	96	4	suppose	suppose	VERB
ejpam-4274	96	5	that	that	SCONJ
ejpam-4274	96	6	x	x	PROPN
ejpam-4274	96	7	6∈	6∈	NOUN
ejpam-4274	96	8	λλp(a	λλp(a	PROPN
ejpam-4274	96	9	)	)	PUNCT
ejpam-4274	96	10	.	.	PUNCT
ejpam-4274	97	1	then	then	ADV
ejpam-4274	97	2	,	,	PUNCT
ejpam-4274	97	3	there	there	PRON
ejpam-4274	97	4	exists	exist	VERB
ejpam-4274	97	5	u	u	PROPN
ejpam-4274	97	6	∈	∈	PROPN
ejpam-4274	97	7	λp	λp	ADP
ejpam-4274	97	8	such	such	ADJ
ejpam-4274	97	9	that	that	SCONJ
ejpam-4274	97	10	a	a	DET
ejpam-4274	97	11	⊆	⊆	NUM
ejpam-4274	97	12	u	u	NOUN
ejpam-4274	97	13	and	and	CCONJ
ejpam-4274	97	14	x	x	SYM
ejpam-4274	97	15	6∈	6∈	PROPN
ejpam-4274	97	16	u	u	NOUN
ejpam-4274	97	17	.	.	PUNCT
ejpam-4274	98	1	since	since	SCONJ
ejpam-4274	98	2	x	x	PROPN
ejpam-4274	98	3	6∈	6∈	PROPN
ejpam-4274	98	4	u	u	NOUN
ejpam-4274	98	5	,	,	PUNCT
ejpam-4274	98	6	there	there	PRON
ejpam-4274	98	7	exists	exist	VERB
ejpam-4274	98	8	v	v	ADP
ejpam-4274	98	9	∈	∈	PROPN
ejpam-4274	98	10	po(x	po(x	NOUN
ejpam-4274	98	11	,	,	PUNCT
ejpam-4274	98	12	τ	τ	X
ejpam-4274	98	13	)	)	PUNCT
ejpam-4274	98	14	such	such	ADJ
ejpam-4274	98	15	that	that	SCONJ
ejpam-4274	98	16	u	u	PROPN
ejpam-4274	98	17	⊆	⊆	NUM
ejpam-4274	98	18	v	v	NOUN
ejpam-4274	98	19	and	and	CCONJ
ejpam-4274	98	20	x	x	NOUN
ejpam-4274	98	21	6∈	6∈	NOUN
ejpam-4274	98	22	v	v	NOUN
ejpam-4274	98	23	.	.	PUNCT
ejpam-4274	99	1	therefore	therefore	ADV
ejpam-4274	99	2	,	,	PUNCT
ejpam-4274	99	3	x	x	PROPN
ejpam-4274	99	4	6∈	6∈	NOUN
ejpam-4274	99	5	λp(a	λp(a	NUM
ejpam-4274	99	6	)	)	PUNCT
ejpam-4274	99	7	.	.	PUNCT
ejpam-4274	100	1	this	this	PRON
ejpam-4274	100	2	shows	show	VERB
ejpam-4274	100	3	that	that	SCONJ
ejpam-4274	100	4	λλp(a	λλp(a	NOUN
ejpam-4274	100	5	)	)	PUNCT
ejpam-4274	100	6	⊇	⊇	NOUN
ejpam-4274	100	7	λp(a	λp(a	NUM
ejpam-4274	100	8	)	)	PUNCT
ejpam-4274	100	9	and	and	CCONJ
ejpam-4274	100	10	hence	hence	ADV
ejpam-4274	100	11	λp(a	λp(a	NUM
ejpam-4274	100	12	)	)	PUNCT
ejpam-4274	100	13	=	=	SYM
ejpam-4274	101	1	λλp(a	λλp(a	PROPN
ejpam-4274	101	2	)	)	PUNCT
ejpam-4274	101	3	.	.	PUNCT
ejpam-4274	102	1	c.	c.	PROPN
ejpam-4274	102	2	boonpok	boonpok	PROPN
ejpam-4274	102	3	,	,	PUNCT
ejpam-4274	102	4	c.	c.	PROPN
ejpam-4274	102	5	viriyapong	viriyapong	PROPN
ejpam-4274	102	6	/	/	SYM
ejpam-4274	102	7	eur	eur	PROPN
ejpam-4274	102	8	.	.	PUNCT
ejpam-4274	103	1	j.	j.	PROPN
ejpam-4274	103	2	pure	pure	PROPN
ejpam-4274	103	3	appl	appl	PROPN
ejpam-4274	103	4	.	.	PROPN
ejpam-4274	103	5	math	math	PROPN
ejpam-4274	103	6	,	,	PUNCT
ejpam-4274	103	7	15	15	NUM
ejpam-4274	103	8	(	(	PUNCT
ejpam-4274	103	9	2	2	NUM
ejpam-4274	103	10	)	)	PUNCT
ejpam-4274	103	11	(	(	PUNCT
ejpam-4274	103	12	2022	2022	NUM
ejpam-4274	103	13	)	)	PUNCT
ejpam-4274	103	14	,	,	PUNCT
ejpam-4274	103	15	415	415	NUM
ejpam-4274	103	16	-	-	SYM
ejpam-4274	103	17	436	436	NUM
ejpam-4274	103	18	418	418	NUM
ejpam-4274	103	19	theorem	theorem	VERB
ejpam-4274	103	20	3	3	NUM
ejpam-4274	103	21	.	.	PUNCT
ejpam-4274	104	1	a	a	DET
ejpam-4274	104	2	topological	topological	ADJ
ejpam-4274	104	3	space	space	NOUN
ejpam-4274	104	4	(	(	PUNCT
ejpam-4274	104	5	x	x	X
ejpam-4274	104	6	,	,	PUNCT
ejpam-4274	104	7	τ	τ	X
ejpam-4274	104	8	)	)	PUNCT
ejpam-4274	104	9	is	be	AUX
ejpam-4274	104	10	pre	pre	ADJ
ejpam-4274	104	11	-	-	ADJ
ejpam-4274	104	12	r0	r0	ADJ
ejpam-4274	104	13	if	if	SCONJ
ejpam-4274	104	14	and	and	CCONJ
ejpam-4274	104	15	only	only	ADV
ejpam-4274	104	16	if	if	SCONJ
ejpam-4274	104	17	the	the	DET
ejpam-4274	104	18	topological	topological	ADJ
ejpam-4274	104	19	space	space	NOUN
ejpam-4274	104	20	(	(	PUNCT
ejpam-4274	104	21	x	x	X
ejpam-4274	104	22	,	,	PUNCT
ejpam-4274	104	23	λp	λp	X
ejpam-4274	104	24	)	)	PUNCT
ejpam-4274	104	25	is	be	AUX
ejpam-4274	104	26	r0	r0	NOUN
ejpam-4274	104	27	.	.	PUNCT
ejpam-4274	105	1	proof	proof	NOUN
ejpam-4274	105	2	.	.	PUNCT
ejpam-4274	106	1	let	let	VERB
ejpam-4274	106	2	v	v	X
ejpam-4274	106	3	∈	∈	VERB
ejpam-4274	106	4	λp	λp	X
ejpam-4274	106	5	and	and	CCONJ
ejpam-4274	106	6	let	let	VERB
ejpam-4274	106	7	x	x	SYM
ejpam-4274	106	8	∈	∈	PROPN
ejpam-4274	106	9	v	v	NOUN
ejpam-4274	106	10	.	.	PUNCT
ejpam-4274	107	1	then	then	ADV
ejpam-4274	107	2	,	,	PUNCT
ejpam-4274	107	3	x	x	PUNCT
ejpam-4274	107	4	∈	∈	NOUN
ejpam-4274	107	5	λp(v	λp(v	X
ejpam-4274	107	6	)	)	PUNCT
ejpam-4274	108	1	=	=	PUNCT
ejpam-4274	109	1	∩{u	∩{u	PROPN
ejpam-4274	109	2	|	|	ADV
ejpam-4274	109	3	v	v	X
ejpam-4274	109	4	⊆	⊆	NUM
ejpam-4274	109	5	u	u	NOUN
ejpam-4274	109	6	,	,	PUNCT
ejpam-4274	109	7	u	u	PROPN
ejpam-4274	109	8	∈	∈	PROPN
ejpam-4274	109	9	po(x	po(x	NOUN
ejpam-4274	109	10	,	,	PUNCT
ejpam-4274	109	11	τ	τ	PROPN
ejpam-4274	109	12	)	)	PUNCT
ejpam-4274	109	13	}	}	PUNCT
ejpam-4274	109	14	and	and	CCONJ
ejpam-4274	109	15	x	x	PUNCT
ejpam-4274	109	16	∈	∈	PROPN
ejpam-4274	109	17	u	u	NOUN
ejpam-4274	109	18	for	for	ADP
ejpam-4274	109	19	any	any	DET
ejpam-4274	109	20	u	u	PROPN
ejpam-4274	109	21	∈	∈	PROPN
ejpam-4274	109	22	po(x	po(x	NOUN
ejpam-4274	109	23	,	,	PUNCT
ejpam-4274	109	24	τ	τ	X
ejpam-4274	109	25	)	)	PUNCT
ejpam-4274	109	26	containing	contain	VERB
ejpam-4274	109	27	v	v	NOUN
ejpam-4274	109	28	.	.	PUNCT
ejpam-4274	110	1	since	since	SCONJ
ejpam-4274	110	2	(	(	PUNCT
ejpam-4274	110	3	x	x	X
ejpam-4274	110	4	,	,	PUNCT
ejpam-4274	110	5	τ	τ	X
ejpam-4274	110	6	)	)	PUNCT
ejpam-4274	110	7	is	be	AUX
ejpam-4274	110	8	pre	pre	ADJ
ejpam-4274	110	9	-	-	ADJ
ejpam-4274	110	10	r0	r0	ADJ
ejpam-4274	110	11	,	,	PUNCT
ejpam-4274	110	12	pcl({x	pcl({x	NOUN
ejpam-4274	110	13	}	}	PUNCT
ejpam-4274	110	14	)	)	PUNCT
ejpam-4274	111	1	⊆	⊆	NUM
ejpam-4274	111	2	u	u	NOUN
ejpam-4274	111	3	for	for	ADP
ejpam-4274	111	4	every	every	DET
ejpam-4274	111	5	u	u	PROPN
ejpam-4274	111	6	∈	∈	PROPN
ejpam-4274	111	7	po(x	po(x	NOUN
ejpam-4274	111	8	,	,	PUNCT
ejpam-4274	111	9	τ	τ	X
ejpam-4274	111	10	)	)	PUNCT
ejpam-4274	111	11	containing	contain	VERB
ejpam-4274	111	12	v	v	NOUN
ejpam-4274	111	13	.	.	PUNCT
ejpam-4274	112	1	thus	thus	ADV
ejpam-4274	112	2	,	,	PUNCT
ejpam-4274	112	3	pcl({x	pcl({x	NOUN
ejpam-4274	112	4	}	}	PUNCT
ejpam-4274	112	5	)	)	PUNCT
ejpam-4274	113	1	⊆	⊆	NUM
ejpam-4274	113	2	∩{u	∩{u	NUM
ejpam-4274	113	3	|	|	ADV
ejpam-4274	113	4	v	v	VERB
ejpam-4274	113	5	⊆	⊆	NUM
ejpam-4274	113	6	u	u	NOUN
ejpam-4274	113	7	,	,	PUNCT
ejpam-4274	113	8	u	u	PROPN
ejpam-4274	113	9	∈	∈	PROPN
ejpam-4274	113	10	po(x	po(x	NOUN
ejpam-4274	113	11	,	,	PUNCT
ejpam-4274	113	12	τ	τ	NOUN
ejpam-4274	113	13	)	)	PUNCT
ejpam-4274	113	14	}	}	PUNCT
ejpam-4274	113	15	=	=	SYM
ejpam-4274	113	16	λp(v	λp(v	X
ejpam-4274	113	17	)	)	PUNCT
ejpam-4274	114	1	=	=	PUNCT
ejpam-4274	114	2	v.	v.	CCONJ
ejpam-4274	114	3	since	since	SCONJ
ejpam-4274	114	4	po(x	po(x	NUM
ejpam-4274	114	5	,	,	PUNCT
ejpam-4274	114	6	τ	τ	X
ejpam-4274	114	7	)	)	PUNCT
ejpam-4274	114	8	⊆	⊆	NUM
ejpam-4274	114	9	λp	λp	PROPN
ejpam-4274	114	10	,	,	PUNCT
ejpam-4274	114	11	λp	λp	PROPN
ejpam-4274	114	12	-	-	PUNCT
ejpam-4274	114	13	cl({x	cl({x	NOUN
ejpam-4274	114	14	}	}	PUNCT
ejpam-4274	114	15	)	)	PUNCT
ejpam-4274	114	16	⊆	⊆	NUM
ejpam-4274	114	17	pcl({x	pcl({x	NOUN
ejpam-4274	114	18	}	}	PUNCT
ejpam-4274	114	19	)	)	PUNCT
ejpam-4274	114	20	⊆	⊆	NUM
ejpam-4274	114	21	v	v	NOUN
ejpam-4274	114	22	,	,	PUNCT
ejpam-4274	114	23	where	where	SCONJ
ejpam-4274	114	24	λp	λp	NOUN
ejpam-4274	114	25	-	-	PUNCT
ejpam-4274	114	26	cl({x	cl({x	NOUN
ejpam-4274	114	27	}	}	PUNCT
ejpam-4274	114	28	)	)	PUNCT
ejpam-4274	114	29	denotes	denote	VERB
ejpam-4274	114	30	the	the	DET
ejpam-4274	114	31	closure	closure	NOUN
ejpam-4274	114	32	of	of	ADP
ejpam-4274	114	33	the	the	DET
ejpam-4274	114	34	singleton	singleton	NOUN
ejpam-4274	114	35	{	{	PUNCT
ejpam-4274	114	36	x	x	NOUN
ejpam-4274	114	37	}	}	PUNCT
ejpam-4274	114	38	in	in	ADP
ejpam-4274	114	39	the	the	DET
ejpam-4274	114	40	topological	topological	ADJ
ejpam-4274	114	41	space	space	NOUN
ejpam-4274	114	42	(	(	PUNCT
ejpam-4274	114	43	x	x	X
ejpam-4274	114	44	,	,	PUNCT
ejpam-4274	114	45	λp	λp	PROPN
ejpam-4274	114	46	)	)	PUNCT
ejpam-4274	114	47	.	.	PUNCT
ejpam-4274	115	1	this	this	PRON
ejpam-4274	115	2	shows	show	VERB
ejpam-4274	115	3	that	that	SCONJ
ejpam-4274	115	4	(	(	PUNCT
ejpam-4274	115	5	x	x	X
ejpam-4274	115	6	,	,	PUNCT
ejpam-4274	115	7	λp	λp	X
ejpam-4274	115	8	)	)	PUNCT
ejpam-4274	115	9	is	be	AUX
ejpam-4274	115	10	r0	r0	NOUN
ejpam-4274	115	11	.	.	PUNCT
ejpam-4274	116	1	conversely	conversely	ADV
ejpam-4274	116	2	,	,	PUNCT
ejpam-4274	116	3	suppose	suppose	VERB
ejpam-4274	116	4	that	that	SCONJ
ejpam-4274	116	5	(	(	PUNCT
ejpam-4274	116	6	x	x	X
ejpam-4274	116	7	,	,	PUNCT
ejpam-4274	116	8	λp	λp	X
ejpam-4274	116	9	)	)	PUNCT
ejpam-4274	116	10	is	be	AUX
ejpam-4274	116	11	r0	r0	NOUN
ejpam-4274	116	12	.	.	PUNCT
ejpam-4274	117	1	let	let	VERB
ejpam-4274	117	2	v	v	X
ejpam-4274	117	3	∈	∈	VERB
ejpam-4274	118	1	λp	λp	X
ejpam-4274	119	1	and	and	CCONJ
ejpam-4274	119	2	x	x	PROPN
ejpam-4274	119	3	∈	∈	NOUN
ejpam-4274	119	4	v	v	NOUN
ejpam-4274	119	5	.	.	PUNCT
ejpam-4274	120	1	since	since	SCONJ
ejpam-4274	120	2	po(x	po(x	NUM
ejpam-4274	120	3	,	,	PUNCT
ejpam-4274	120	4	τ	τ	X
ejpam-4274	120	5	)	)	PUNCT
ejpam-4274	120	6	⊆	⊆	NUM
ejpam-4274	120	7	λp	λp	NOUN
ejpam-4274	120	8	,	,	PUNCT
ejpam-4274	120	9	we	we	PRON
ejpam-4274	120	10	have	have	VERB
ejpam-4274	120	11	λp	λp	NOUN
ejpam-4274	120	12	-	-	PUNCT
ejpam-4274	120	13	cl({x	cl({x	NOUN
ejpam-4274	120	14	}	}	PUNCT
ejpam-4274	120	15	)	)	PUNCT
ejpam-4274	121	1	⊆	⊆	NUM
ejpam-4274	121	2	v	v	NOUN
ejpam-4274	121	3	.	.	PUNCT
ejpam-4274	122	1	since	since	SCONJ
ejpam-4274	122	2	x	x	INTJ
ejpam-4274	122	3	−	−	PROPN
ejpam-4274	122	4	λp	λp	NOUN
ejpam-4274	122	5	-	-	PUNCT
ejpam-4274	122	6	cl({x	cl({x	NOUN
ejpam-4274	122	7	}	}	PUNCT
ejpam-4274	122	8	)	)	PUNCT
ejpam-4274	122	9	∈	∈	PROPN
ejpam-4274	122	10	λp	λp	PROPN
ejpam-4274	122	11	,	,	PUNCT
ejpam-4274	122	12	x	x	PROPN
ejpam-4274	122	13	−	−	NOUN
ejpam-4274	122	14	λp	λp	NOUN
ejpam-4274	122	15	-	-	PUNCT
ejpam-4274	122	16	cl({x	cl({x	NOUN
ejpam-4274	122	17	}	}	PUNCT
ejpam-4274	122	18	)	)	PUNCT
ejpam-4274	123	1	=	=	PUNCT
ejpam-4274	124	1	∩{u	∩{u	PROPN
ejpam-4274	125	1	|	|	ADV
ejpam-4274	125	2	x	x	PUNCT
ejpam-4274	125	3	−	−	NOUN
ejpam-4274	125	4	λp	λp	NOUN
ejpam-4274	125	5	-	-	PUNCT
ejpam-4274	125	6	cl({x	cl({x	NOUN
ejpam-4274	125	7	}	}	PUNCT
ejpam-4274	125	8	)	)	PUNCT
ejpam-4274	125	9	⊆	⊆	NUM
ejpam-4274	125	10	u	u	NOUN
ejpam-4274	125	11	,	,	PUNCT
ejpam-4274	125	12	u	u	PROPN
ejpam-4274	125	13	∈	∈	PROPN
ejpam-4274	125	14	po(x	po(x	NOUN
ejpam-4274	125	15	,	,	PUNCT
ejpam-4274	125	16	τ	τ	NOUN
ejpam-4274	125	17	)	)	PUNCT
ejpam-4274	125	18	}	}	PUNCT
ejpam-4274	125	19	.	.	PUNCT
ejpam-4274	126	1	then	then	ADV
ejpam-4274	126	2	,	,	PUNCT
ejpam-4274	126	3	there	there	PRON
ejpam-4274	126	4	exists	exist	VERB
ejpam-4274	126	5	u	u	PROPN
ejpam-4274	126	6	∈	∈	PROPN
ejpam-4274	126	7	po(x	po(x	NOUN
ejpam-4274	126	8	,	,	PUNCT
ejpam-4274	126	9	τ	τ	X
ejpam-4274	126	10	)	)	PUNCT
ejpam-4274	126	11	such	such	ADJ
ejpam-4274	126	12	that	that	SCONJ
ejpam-4274	126	13	x	x	X
ejpam-4274	126	14	−	−	NOUN
ejpam-4274	126	15	λp	λp	NOUN
ejpam-4274	126	16	-	-	PUNCT
ejpam-4274	126	17	cl({x	cl({x	NOUN
ejpam-4274	126	18	}	}	PUNCT
ejpam-4274	126	19	)	)	PUNCT
ejpam-4274	127	1	⊆	⊆	NUM
ejpam-4274	127	2	u	u	NOUN
ejpam-4274	127	3	and	and	CCONJ
ejpam-4274	127	4	x	x	SYM
ejpam-4274	127	5	6∈	6∈	PROPN
ejpam-4274	127	6	u	u	PROPN
ejpam-4274	127	7	.	.	PUNCT
ejpam-4274	128	1	thus	thus	ADV
ejpam-4274	128	2	,	,	PUNCT
ejpam-4274	128	3	x	x	PUNCT
ejpam-4274	128	4	∈	∈	NOUN
ejpam-4274	128	5	x	x	X
ejpam-4274	128	6	−	−	PUNCT
ejpam-4274	128	7	u	u	NOUN
ejpam-4274	128	8	⊆	⊆	NUM
ejpam-4274	128	9	λp	λp	NOUN
ejpam-4274	128	10	-	-	PUNCT
ejpam-4274	128	11	cl({x	cl({x	NOUN
ejpam-4274	128	12	}	}	PUNCT
ejpam-4274	128	13	)	)	PUNCT
ejpam-4274	128	14	⊆	⊆	NUM
ejpam-4274	128	15	v	v	NOUN
ejpam-4274	128	16	.	.	PUNCT
ejpam-4274	129	1	since	since	SCONJ
ejpam-4274	129	2	x	x	X
ejpam-4274	129	3	−	−	PROPN
ejpam-4274	129	4	u	u	NOUN
ejpam-4274	129	5	is	be	AUX
ejpam-4274	129	6	preclosed	preclose	VERB
ejpam-4274	129	7	,	,	PUNCT
ejpam-4274	129	8	pcl({x	pcl({x	NOUN
ejpam-4274	129	9	}	}	PUNCT
ejpam-4274	129	10	)	)	PUNCT
ejpam-4274	130	1	⊆	⊆	NUM
ejpam-4274	130	2	x	x	SYM
ejpam-4274	130	3	−	−	NUM
ejpam-4274	130	4	u	u	NOUN
ejpam-4274	130	5	⊆	⊆	NUM
ejpam-4274	130	6	v	v	NOUN
ejpam-4274	130	7	.	.	PUNCT
ejpam-4274	131	1	consequently	consequently	ADV
ejpam-4274	131	2	,	,	PUNCT
ejpam-4274	131	3	we	we	PRON
ejpam-4274	131	4	obtain	obtain	VERB
ejpam-4274	131	5	(	(	PUNCT
ejpam-4274	131	6	x	x	NOUN
ejpam-4274	131	7	,	,	PUNCT
ejpam-4274	131	8	τ	τ	X
ejpam-4274	131	9	)	)	PUNCT
ejpam-4274	131	10	is	be	AUX
ejpam-4274	131	11	pre	pre	ADJ
ejpam-4274	131	12	-	-	ADJ
ejpam-4274	131	13	r0	r0	ADJ
ejpam-4274	131	14	.	.	PUNCT
ejpam-4274	132	1	theorem	theorem	VERB
ejpam-4274	132	2	4	4	NUM
ejpam-4274	132	3	.	.	PUNCT
ejpam-4274	133	1	a	a	DET
ejpam-4274	133	2	topological	topological	ADJ
ejpam-4274	133	3	space	space	NOUN
ejpam-4274	133	4	(	(	PUNCT
ejpam-4274	133	5	x	x	X
ejpam-4274	133	6	,	,	PUNCT
ejpam-4274	133	7	τ	τ	X
ejpam-4274	133	8	)	)	PUNCT
ejpam-4274	133	9	is	be	AUX
ejpam-4274	133	10	pre	pre	ADJ
ejpam-4274	133	11	-	-	NOUN
ejpam-4274	133	12	t0	t0	NOUN
ejpam-4274	133	13	if	if	SCONJ
ejpam-4274	133	14	and	and	CCONJ
ejpam-4274	133	15	only	only	ADV
ejpam-4274	133	16	if	if	SCONJ
ejpam-4274	133	17	the	the	DET
ejpam-4274	133	18	topological	topological	ADJ
ejpam-4274	133	19	space	space	NOUN
ejpam-4274	133	20	(	(	PUNCT
ejpam-4274	133	21	x	x	X
ejpam-4274	133	22	,	,	PUNCT
ejpam-4274	133	23	λp	λp	X
ejpam-4274	133	24	)	)	PUNCT
ejpam-4274	133	25	is	be	AUX
ejpam-4274	133	26	t0	t0	NOUN
ejpam-4274	133	27	.	.	PUNCT
ejpam-4274	134	1	proof	proof	NOUN
ejpam-4274	134	2	.	.	PUNCT
ejpam-4274	135	1	this	this	PRON
ejpam-4274	135	2	is	be	AUX
ejpam-4274	135	3	obvious	obvious	ADJ
ejpam-4274	135	4	since	since	SCONJ
ejpam-4274	135	5	po(x	po(x	NUM
ejpam-4274	135	6	,	,	PUNCT
ejpam-4274	135	7	τ	τ	X
ejpam-4274	135	8	)	)	PUNCT
ejpam-4274	135	9	⊆	⊆	NUM
ejpam-4274	135	10	λp	λp	X
ejpam-4274	135	11	.	.	PUNCT
ejpam-4274	136	1	conversely	conversely	ADV
ejpam-4274	136	2	,	,	PUNCT
ejpam-4274	136	3	let	let	VERB
ejpam-4274	136	4	x	x	PRON
ejpam-4274	136	5	and	and	CCONJ
ejpam-4274	136	6	y	y	PROPN
ejpam-4274	136	7	be	be	AUX
ejpam-4274	136	8	any	any	DET
ejpam-4274	136	9	pair	pair	NOUN
ejpam-4274	136	10	of	of	ADP
ejpam-4274	136	11	distinct	distinct	ADJ
ejpam-4274	136	12	points	point	NOUN
ejpam-4274	136	13	of	of	ADP
ejpam-4274	136	14	x.	x.	NOUN
ejpam-4274	136	15	since	since	SCONJ
ejpam-4274	136	16	(	(	PUNCT
ejpam-4274	136	17	x	x	X
ejpam-4274	136	18	,	,	PUNCT
ejpam-4274	136	19	λp	λp	X
ejpam-4274	136	20	)	)	PUNCT
ejpam-4274	136	21	is	be	AUX
ejpam-4274	136	22	t0	t0	NUM
ejpam-4274	136	23	,	,	PUNCT
ejpam-4274	136	24	there	there	PRON
ejpam-4274	136	25	exists	exist	VERB
ejpam-4274	136	26	v	v	ADP
ejpam-4274	136	27	∈	∈	PROPN
ejpam-4274	136	28	λp	λp	ADP
ejpam-4274	136	29	such	such	ADJ
ejpam-4274	136	30	that	that	SCONJ
ejpam-4274	136	31	either	either	CCONJ
ejpam-4274	136	32	x	x	SYM
ejpam-4274	136	33	∈	∈	PROPN
ejpam-4274	136	34	v	v	NOUN
ejpam-4274	136	35	and	and	CCONJ
ejpam-4274	136	36	y	y	PROPN
ejpam-4274	136	37	6∈	6∈	PROPN
ejpam-4274	136	38	v	v	NOUN
ejpam-4274	136	39	or	or	CCONJ
ejpam-4274	136	40	x	x	ADJ
ejpam-4274	136	41	6∈	6∈	NOUN
ejpam-4274	136	42	v	v	NOUN
ejpam-4274	136	43	and	and	CCONJ
ejpam-4274	136	44	y	y	PROPN
ejpam-4274	136	45	∈	∈	PROPN
ejpam-4274	136	46	v	v	NOUN
ejpam-4274	136	47	.	.	PUNCT
ejpam-4274	137	1	in	in	ADP
ejpam-4274	137	2	case	case	NOUN
ejpam-4274	137	3	x	x	X
ejpam-4274	137	4	∈	∈	PROPN
ejpam-4274	137	5	v	v	NOUN
ejpam-4274	137	6	and	and	CCONJ
ejpam-4274	137	7	y	y	PROPN
ejpam-4274	137	8	6∈	6∈	PROPN
ejpam-4274	137	9	v	v	NOUN
ejpam-4274	137	10	,	,	PUNCT
ejpam-4274	137	11	there	there	PRON
ejpam-4274	137	12	exists	exist	VERB
ejpam-4274	137	13	u	u	PROPN
ejpam-4274	137	14	∈	∈	PROPN
ejpam-4274	137	15	po(x	po(x	NOUN
ejpam-4274	137	16	,	,	PUNCT
ejpam-4274	137	17	τ	τ	X
ejpam-4274	137	18	)	)	PUNCT
ejpam-4274	137	19	such	such	ADJ
ejpam-4274	137	20	that	that	PRON
ejpam-4274	137	21	v	v	ADP
ejpam-4274	137	22	⊆	⊆	NUM
ejpam-4274	137	23	u	u	NOUN
ejpam-4274	137	24	and	and	CCONJ
ejpam-4274	137	25	y	y	PROPN
ejpam-4274	137	26	6∈	6∈	PROPN
ejpam-4274	137	27	u	u	PROPN
ejpam-4274	137	28	.	.	PUNCT
ejpam-4274	138	1	however	however	ADV
ejpam-4274	138	2	,	,	PUNCT
ejpam-4274	138	3	since	since	SCONJ
ejpam-4274	138	4	x	x	PROPN
ejpam-4274	138	5	∈	∈	PROPN
ejpam-4274	138	6	v	v	NOUN
ejpam-4274	138	7	,	,	PUNCT
ejpam-4274	138	8	x	x	SYM
ejpam-4274	138	9	∈	∈	PROPN
ejpam-4274	138	10	u	u	NOUN
ejpam-4274	138	11	.	.	PUNCT
ejpam-4274	139	1	in	in	ADP
ejpam-4274	139	2	case	case	NOUN
ejpam-4274	139	3	x	x	PUNCT
ejpam-4274	139	4	6∈	6∈	NOUN
ejpam-4274	139	5	v	v	NOUN
ejpam-4274	139	6	and	and	CCONJ
ejpam-4274	139	7	y	y	PROPN
ejpam-4274	139	8	∈	∈	PROPN
ejpam-4274	139	9	v	v	NOUN
ejpam-4274	139	10	,	,	PUNCT
ejpam-4274	139	11	similarly	similarly	ADV
ejpam-4274	139	12	there	there	PRON
ejpam-4274	139	13	exists	exist	VERB
ejpam-4274	139	14	u	u	PROPN
ejpam-4274	139	15	∈	∈	PROPN
ejpam-4274	139	16	po(x	po(x	NOUN
ejpam-4274	139	17	,	,	PUNCT
ejpam-4274	139	18	τ	τ	X
ejpam-4274	139	19	)	)	PUNCT
ejpam-4274	139	20	such	such	ADJ
ejpam-4274	139	21	that	that	SCONJ
ejpam-4274	139	22	x	x	SYM
ejpam-4274	139	23	6∈	6∈	NOUN
ejpam-4274	139	24	u	u	NOUN
ejpam-4274	139	25	and	and	CCONJ
ejpam-4274	139	26	y	y	PROPN
ejpam-4274	139	27	∈	∈	PROPN
ejpam-4274	139	28	u	u	PROPN
ejpam-4274	139	29	.	.	PUNCT
ejpam-4274	140	1	thus	thus	ADV
ejpam-4274	140	2	,	,	PUNCT
ejpam-4274	140	3	(	(	PUNCT
ejpam-4274	140	4	x	x	X
ejpam-4274	140	5	,	,	PUNCT
ejpam-4274	140	6	τ	τ	X
ejpam-4274	140	7	)	)	PUNCT
ejpam-4274	140	8	is	be	AUX
ejpam-4274	140	9	pre	pre	ADJ
ejpam-4274	140	10	-	-	NOUN
ejpam-4274	140	11	t0	t0	NOUN
ejpam-4274	140	12	.	.	PUNCT
ejpam-4274	141	1	lemma	lemma	PROPN
ejpam-4274	141	2	4	4	NUM
ejpam-4274	141	3	.	.	PUNCT
ejpam-4274	142	1	for	for	ADP
ejpam-4274	142	2	a	a	DET
ejpam-4274	142	3	topological	topological	ADJ
ejpam-4274	142	4	space	space	NOUN
ejpam-4274	142	5	(	(	PUNCT
ejpam-4274	142	6	x	x	X
ejpam-4274	142	7	,	,	PUNCT
ejpam-4274	142	8	τ	τ	PROPN
ejpam-4274	142	9	)	)	PUNCT
ejpam-4274	142	10	,	,	PUNCT
ejpam-4274	142	11	the	the	DET
ejpam-4274	142	12	following	follow	VERB
ejpam-4274	142	13	properties	property	NOUN
ejpam-4274	142	14	are	be	AUX
ejpam-4274	142	15	equivalent	equivalent	ADJ
ejpam-4274	142	16	:	:	PUNCT
ejpam-4274	142	17	(	(	PUNCT
ejpam-4274	142	18	1	1	X
ejpam-4274	142	19	)	)	PUNCT
ejpam-4274	142	20	(	(	PUNCT
ejpam-4274	142	21	x	x	X
ejpam-4274	142	22	,	,	PUNCT
ejpam-4274	142	23	τ	τ	X
ejpam-4274	142	24	)	)	PUNCT
ejpam-4274	142	25	is	be	AUX
ejpam-4274	142	26	pre	pre	ADJ
ejpam-4274	142	27	-	-	NOUN
ejpam-4274	142	28	t1	t1	ADJ
ejpam-4274	142	29	;	;	PUNCT
ejpam-4274	142	30	(	(	PUNCT
ejpam-4274	142	31	2	2	X
ejpam-4274	142	32	)	)	PUNCT
ejpam-4274	142	33	for	for	ADP
ejpam-4274	142	34	each	each	DET
ejpam-4274	142	35	x	x	SYM
ejpam-4274	142	36	∈	∈	PROPN
ejpam-4274	142	37	x	x	NOUN
ejpam-4274	142	38	,	,	PUNCT
ejpam-4274	142	39	the	the	DET
ejpam-4274	142	40	singleton	singleton	NOUN
ejpam-4274	142	41	{	{	PUNCT
ejpam-4274	142	42	x	x	NOUN
ejpam-4274	142	43	}	}	PUNCT
ejpam-4274	142	44	is	be	AUX
ejpam-4274	142	45	preclosed	preclose	VERB
ejpam-4274	142	46	in	in	ADP
ejpam-4274	142	47	(	(	PUNCT
ejpam-4274	142	48	x	x	X
ejpam-4274	142	49	,	,	PUNCT
ejpam-4274	142	50	τ	τ	PROPN
ejpam-4274	142	51	)	)	PUNCT
ejpam-4274	142	52	.	.	PUNCT
ejpam-4274	143	1	(	(	PUNCT
ejpam-4274	143	2	3	3	X
ejpam-4274	143	3	)	)	PUNCT
ejpam-4274	143	4	for	for	ADP
ejpam-4274	143	5	each	each	DET
ejpam-4274	143	6	x	x	SYM
ejpam-4274	143	7	∈	∈	PROPN
ejpam-4274	143	8	x	x	NOUN
ejpam-4274	143	9	,	,	PUNCT
ejpam-4274	143	10	the	the	DET
ejpam-4274	143	11	singleton	singleton	NOUN
ejpam-4274	143	12	{	{	PUNCT
ejpam-4274	143	13	x	x	NOUN
ejpam-4274	143	14	}	}	PUNCT
ejpam-4274	143	15	is	be	AUX
ejpam-4274	143	16	a	a	DET
ejpam-4274	143	17	λp	λp	NOUN
ejpam-4274	143	18	-	-	PUNCT
ejpam-4274	143	19	set	set	NOUN
ejpam-4274	143	20	.	.	PUNCT
ejpam-4274	144	1	proof	proof	NOUN
ejpam-4274	144	2	.	.	PUNCT
ejpam-4274	145	1	(	(	PUNCT
ejpam-4274	145	2	1	1	X
ejpam-4274	145	3	)	)	PUNCT
ejpam-4274	145	4	⇒	⇒	NOUN
ejpam-4274	145	5	(	(	PUNCT
ejpam-4274	145	6	2	2	NUM
ejpam-4274	145	7	):	):	PUNCT
ejpam-4274	145	8	let	let	VERB
ejpam-4274	145	9	y	y	PRON
ejpam-4274	145	10	be	be	AUX
ejpam-4274	145	11	any	any	DET
ejpam-4274	145	12	point	point	NOUN
ejpam-4274	145	13	of	of	ADP
ejpam-4274	145	14	x	x	PUNCT
ejpam-4274	145	15	and	and	CCONJ
ejpam-4274	145	16	let	let	VERB
ejpam-4274	145	17	x	x	SYM
ejpam-4274	145	18	∈	∈	PROPN
ejpam-4274	145	19	x	x	X
ejpam-4274	145	20	−	−	PROPN
ejpam-4274	145	21	{	{	PUNCT
ejpam-4274	145	22	y	y	NOUN
ejpam-4274	145	23	}	}	PUNCT
ejpam-4274	145	24	.	.	PUNCT
ejpam-4274	146	1	then	then	ADV
ejpam-4274	146	2	,	,	PUNCT
ejpam-4274	146	3	there	there	PRON
ejpam-4274	146	4	exists	exist	VERB
ejpam-4274	146	5	a	a	DET
ejpam-4274	146	6	preopen	preopen	ADJ
ejpam-4274	146	7	set	set	NOUN
ejpam-4274	146	8	vx	vx	ADP
ejpam-4274	146	9	such	such	ADJ
ejpam-4274	146	10	that	that	SCONJ
ejpam-4274	146	11	x	x	SYM
ejpam-4274	146	12	∈	∈	PROPN
ejpam-4274	146	13	vx	vx	PROPN
ejpam-4274	146	14	and	and	CCONJ
ejpam-4274	146	15	y	y	PROPN
ejpam-4274	146	16	6∈	6∈	PROPN
ejpam-4274	146	17	vx	vx	PROPN
ejpam-4274	146	18	.	.	PROPN
ejpam-4274	146	19	thus	thus	ADV
ejpam-4274	146	20	,	,	PUNCT
ejpam-4274	146	21	x	x	SYM
ejpam-4274	146	22	−{y	−{y	NOUN
ejpam-4274	146	23	}	}	PUNCT
ejpam-4274	146	24	=	=	SYM
ejpam-4274	147	1	∪x∈x−{y}vx	∪x∈x−{y}vx	PROPN
ejpam-4274	147	2	and	and	CCONJ
ejpam-4274	147	3	hence	hence	ADV
ejpam-4274	147	4	the	the	DET
ejpam-4274	147	5	singleton	singleton	NOUN
ejpam-4274	147	6	{	{	PUNCT
ejpam-4274	147	7	y	y	NOUN
ejpam-4274	147	8	}	}	PUNCT
ejpam-4274	147	9	is	be	AUX
ejpam-4274	147	10	preclosed	preclose	VERB
ejpam-4274	147	11	in	in	ADP
ejpam-4274	147	12	x.	x.	NOUN
ejpam-4274	147	13	(	(	PUNCT
ejpam-4274	147	14	2	2	NUM
ejpam-4274	147	15	)	)	PUNCT
ejpam-4274	147	16	⇒	⇒	NOUN
ejpam-4274	147	17	(	(	PUNCT
ejpam-4274	147	18	3	3	NUM
ejpam-4274	147	19	):	):	PUNCT
ejpam-4274	147	20	let	let	VERB
ejpam-4274	147	21	x	x	PRON
ejpam-4274	147	22	be	be	AUX
ejpam-4274	147	23	any	any	DET
ejpam-4274	147	24	point	point	NOUN
ejpam-4274	147	25	of	of	ADP
ejpam-4274	147	26	x	x	PUNCT
ejpam-4274	147	27	and	and	CCONJ
ejpam-4274	147	28	let	let	VERB
ejpam-4274	147	29	y	y	PROPN
ejpam-4274	147	30	∈	∈	PROPN
ejpam-4274	147	31	x	x	PUNCT
ejpam-4274	147	32	−	−	PROPN
ejpam-4274	147	33	{	{	PUNCT
ejpam-4274	147	34	x	x	NOUN
ejpam-4274	147	35	}	}	PUNCT
ejpam-4274	147	36	.	.	PUNCT
ejpam-4274	148	1	then	then	ADV
ejpam-4274	148	2	,	,	PUNCT
ejpam-4274	148	3	x	x	PUNCT
ejpam-4274	148	4	∈	∈	PROPN
ejpam-4274	148	5	(	(	PUNCT
ejpam-4274	148	6	x	x	PART
ejpam-4274	148	7	−	−	PROPN
ejpam-4274	148	8	{	{	PUNCT
ejpam-4274	148	9	y	y	NOUN
ejpam-4274	148	10	}	}	PUNCT
ejpam-4274	148	11	)	)	PUNCT
ejpam-4274	148	12	∈	∈	PROPN
ejpam-4274	148	13	po(x	po(x	NOUN
ejpam-4274	148	14	,	,	PUNCT
ejpam-4274	148	15	τ	τ	X
ejpam-4274	148	16	)	)	PUNCT
ejpam-4274	148	17	and	and	CCONJ
ejpam-4274	148	18	λp({x	λp({x	ADV
ejpam-4274	148	19	}	}	PUNCT
ejpam-4274	148	20	)	)	PUNCT
ejpam-4274	148	21	⊆	⊆	NUM
ejpam-4274	148	22	x	x	SYM
ejpam-4274	148	23	−	−	PROPN
ejpam-4274	148	24	{	{	PUNCT
ejpam-4274	148	25	y	y	NOUN
ejpam-4274	148	26	}	}	PUNCT
ejpam-4274	148	27	.	.	PUNCT
ejpam-4274	149	1	therefore	therefore	ADV
ejpam-4274	149	2	,	,	PUNCT
ejpam-4274	149	3	y	y	PROPN
ejpam-4274	149	4	6∈	6∈	PROPN
ejpam-4274	149	5	λp({x	λp({x	ADV
ejpam-4274	149	6	}	}	PUNCT
ejpam-4274	149	7	)	)	PUNCT
ejpam-4274	149	8	and	and	CCONJ
ejpam-4274	149	9	hence	hence	ADV
ejpam-4274	149	10	λp({x	λp({x	ADV
ejpam-4274	149	11	}	}	PUNCT
ejpam-4274	149	12	)	)	PUNCT
ejpam-4274	149	13	⊆	⊆	NUM
ejpam-4274	149	14	{	{	PUNCT
ejpam-4274	149	15	x	x	NOUN
ejpam-4274	149	16	}	}	PUNCT
ejpam-4274	149	17	.	.	PUNCT
ejpam-4274	150	1	thus	thus	ADV
ejpam-4274	150	2	,	,	PUNCT
ejpam-4274	150	3	λp({x	λp({x	ADV
ejpam-4274	150	4	}	}	PUNCT
ejpam-4274	150	5	)	)	PUNCT
ejpam-4274	150	6	=	=	PRON
ejpam-4274	150	7	{	{	PUNCT
ejpam-4274	150	8	x	x	NOUN
ejpam-4274	150	9	}	}	PUNCT
ejpam-4274	150	10	.	.	PUNCT
ejpam-4274	151	1	this	this	PRON
ejpam-4274	151	2	shows	show	VERB
ejpam-4274	151	3	that	that	SCONJ
ejpam-4274	151	4	{	{	PUNCT
ejpam-4274	151	5	x	x	X
ejpam-4274	151	6	}	}	PUNCT
ejpam-4274	151	7	is	be	AUX
ejpam-4274	151	8	a	a	DET
ejpam-4274	151	9	λp	λp	NOUN
ejpam-4274	151	10	-	-	PUNCT
ejpam-4274	151	11	set	set	NOUN
ejpam-4274	151	12	.	.	PUNCT
ejpam-4274	152	1	c.	c.	PROPN
ejpam-4274	152	2	boonpok	boonpok	PROPN
ejpam-4274	152	3	,	,	PUNCT
ejpam-4274	152	4	c.	c.	PROPN
ejpam-4274	152	5	viriyapong	viriyapong	PROPN
ejpam-4274	152	6	/	/	SYM
ejpam-4274	152	7	eur	eur	PROPN
ejpam-4274	152	8	.	.	PUNCT
ejpam-4274	153	1	j.	j.	PROPN
ejpam-4274	153	2	pure	pure	PROPN
ejpam-4274	153	3	appl	appl	PROPN
ejpam-4274	153	4	.	.	PROPN
ejpam-4274	153	5	math	math	PROPN
ejpam-4274	153	6	,	,	PUNCT
ejpam-4274	153	7	15	15	NUM
ejpam-4274	153	8	(	(	PUNCT
ejpam-4274	153	9	2	2	NUM
ejpam-4274	153	10	)	)	PUNCT
ejpam-4274	153	11	(	(	PUNCT
ejpam-4274	153	12	2022	2022	NUM
ejpam-4274	153	13	)	)	PUNCT
ejpam-4274	153	14	,	,	PUNCT
ejpam-4274	153	15	415	415	NUM
ejpam-4274	153	16	-	-	SYM
ejpam-4274	153	17	436	436	NUM
ejpam-4274	153	18	419	419	NUM
ejpam-4274	153	19	(	(	PUNCT
ejpam-4274	153	20	3	3	NUM
ejpam-4274	153	21	)	)	PUNCT
ejpam-4274	153	22	⇒	⇒	NOUN
ejpam-4274	153	23	(	(	PUNCT
ejpam-4274	153	24	1	1	NUM
ejpam-4274	153	25	):	):	PUNCT
ejpam-4274	153	26	suppose	suppose	VERB
ejpam-4274	153	27	that	that	SCONJ
ejpam-4274	153	28	the	the	DET
ejpam-4274	153	29	singleton	singleton	PROPN
ejpam-4274	153	30	{	{	PUNCT
ejpam-4274	153	31	x	x	NOUN
ejpam-4274	153	32	}	}	PUNCT
ejpam-4274	153	33	is	be	AUX
ejpam-4274	153	34	a	a	DET
ejpam-4274	153	35	λp	λp	NOUN
ejpam-4274	153	36	-	-	PUNCT
ejpam-4274	153	37	set	set	NOUN
ejpam-4274	153	38	for	for	ADP
ejpam-4274	153	39	each	each	DET
ejpam-4274	153	40	x	x	SYM
ejpam-4274	153	41	∈	∈	PROPN
ejpam-4274	153	42	x.	x.	NOUN
ejpam-4274	153	43	let	let	VERB
ejpam-4274	153	44	x	x	PRON
ejpam-4274	153	45	and	and	CCONJ
ejpam-4274	153	46	y	y	PROPN
ejpam-4274	153	47	be	be	AUX
ejpam-4274	153	48	any	any	DET
ejpam-4274	153	49	distinct	distinct	ADJ
ejpam-4274	153	50	points	point	NOUN
ejpam-4274	153	51	.	.	PUNCT
ejpam-4274	154	1	then	then	ADV
ejpam-4274	154	2	,	,	PUNCT
ejpam-4274	154	3	y	y	PROPN
ejpam-4274	154	4	6∈	6∈	PROPN
ejpam-4274	154	5	λp({x	λp({x	ADV
ejpam-4274	154	6	}	}	PUNCT
ejpam-4274	154	7	)	)	PUNCT
ejpam-4274	154	8	and	and	CCONJ
ejpam-4274	154	9	there	there	PRON
ejpam-4274	154	10	exists	exist	VERB
ejpam-4274	154	11	a	a	DET
ejpam-4274	154	12	preopen	preopen	ADJ
ejpam-4274	154	13	set	set	VERB
ejpam-4274	154	14	ux	ux	ADP
ejpam-4274	154	15	such	such	ADJ
ejpam-4274	154	16	that	that	SCONJ
ejpam-4274	154	17	x	x	SYM
ejpam-4274	154	18	∈	∈	NOUN
ejpam-4274	154	19	ux	ux	NOUN
ejpam-4274	154	20	and	and	CCONJ
ejpam-4274	154	21	y	y	PROPN
ejpam-4274	154	22	6∈	6∈	PROPN
ejpam-4274	154	23	ux	ux	PROPN
ejpam-4274	154	24	.	.	PUNCT
ejpam-4274	155	1	similarly	similarly	ADV
ejpam-4274	155	2	,	,	PUNCT
ejpam-4274	155	3	x	x	X
ejpam-4274	155	4	6∈	6∈	NOUN
ejpam-4274	155	5	λp({y	λp({y	ADJ
ejpam-4274	155	6	}	}	PUNCT
ejpam-4274	155	7	)	)	PUNCT
ejpam-4274	155	8	and	and	CCONJ
ejpam-4274	155	9	there	there	PRON
ejpam-4274	155	10	exists	exist	VERB
ejpam-4274	155	11	a	a	DET
ejpam-4274	155	12	preopen	preopen	ADJ
ejpam-4274	155	13	set	set	VERB
ejpam-4274	155	14	uy	uy	ADP
ejpam-4274	155	15	such	such	ADJ
ejpam-4274	155	16	that	that	SCONJ
ejpam-4274	155	17	y	y	PROPN
ejpam-4274	155	18	∈	∈	PROPN
ejpam-4274	155	19	uy	uy	PROPN
ejpam-4274	155	20	and	and	CCONJ
ejpam-4274	155	21	x	x	SYM
ejpam-4274	155	22	6∈	6∈	NOUN
ejpam-4274	155	23	uy	uy	INTJ
ejpam-4274	155	24	.	.	PUNCT
ejpam-4274	156	1	this	this	PRON
ejpam-4274	156	2	shows	show	VERB
ejpam-4274	156	3	that	that	SCONJ
ejpam-4274	156	4	(	(	PUNCT
ejpam-4274	156	5	x	x	X
ejpam-4274	156	6	,	,	PUNCT
ejpam-4274	156	7	τ	τ	X
ejpam-4274	156	8	)	)	PUNCT
ejpam-4274	156	9	is	be	AUX
ejpam-4274	156	10	pre	pre	ADJ
ejpam-4274	156	11	-	-	ADJ
ejpam-4274	156	12	t1	t1	NOUN
ejpam-4274	156	13	.	.	PUNCT
ejpam-4274	157	1	theorem	theorem	NOUN
ejpam-4274	157	2	5	5	NUM
ejpam-4274	157	3	.	.	PUNCT
ejpam-4274	158	1	a	a	DET
ejpam-4274	158	2	topological	topological	ADJ
ejpam-4274	158	3	space	space	NOUN
ejpam-4274	158	4	(	(	PUNCT
ejpam-4274	158	5	x	x	X
ejpam-4274	158	6	,	,	PUNCT
ejpam-4274	158	7	τ	τ	X
ejpam-4274	158	8	)	)	PUNCT
ejpam-4274	158	9	is	be	AUX
ejpam-4274	158	10	pre	pre	ADJ
ejpam-4274	158	11	-	-	NOUN
ejpam-4274	158	12	t1	t1	ADJ
ejpam-4274	158	13	if	if	SCONJ
ejpam-4274	158	14	and	and	CCONJ
ejpam-4274	158	15	only	only	ADV
ejpam-4274	158	16	if	if	SCONJ
ejpam-4274	158	17	the	the	DET
ejpam-4274	158	18	topological	topological	ADJ
ejpam-4274	158	19	space	space	NOUN
ejpam-4274	158	20	(	(	PUNCT
ejpam-4274	158	21	x	x	X
ejpam-4274	158	22	,	,	PUNCT
ejpam-4274	158	23	λp	λp	X
ejpam-4274	158	24	)	)	PUNCT
ejpam-4274	158	25	is	be	AUX
ejpam-4274	158	26	discrete	discrete	ADJ
ejpam-4274	158	27	.	.	PUNCT
ejpam-4274	159	1	proof	proof	NOUN
ejpam-4274	159	2	.	.	PUNCT
ejpam-4274	160	1	suppose	suppose	VERB
ejpam-4274	160	2	that	that	SCONJ
ejpam-4274	160	3	(	(	PUNCT
ejpam-4274	160	4	x	x	X
ejpam-4274	160	5	,	,	PUNCT
ejpam-4274	160	6	τ	τ	X
ejpam-4274	160	7	)	)	PUNCT
ejpam-4274	160	8	is	be	AUX
ejpam-4274	160	9	pre	pre	ADJ
ejpam-4274	160	10	-	-	ADJ
ejpam-4274	160	11	t1	t1	NOUN
ejpam-4274	160	12	.	.	PUNCT
ejpam-4274	161	1	let	let	VERB
ejpam-4274	161	2	x	x	SYM
ejpam-4274	161	3	∈	∈	PROPN
ejpam-4274	161	4	x.	x.	NOUN
ejpam-4274	161	5	by	by	ADP
ejpam-4274	161	6	lemma	lemma	PROPN
ejpam-4274	161	7	4	4	NUM
ejpam-4274	161	8	,	,	PUNCT
ejpam-4274	161	9	{	{	PUNCT
ejpam-4274	161	10	x	x	X
ejpam-4274	161	11	}	}	PUNCT
ejpam-4274	161	12	is	be	AUX
ejpam-4274	161	13	a	a	DET
ejpam-4274	161	14	λp	λp	ADV
ejpam-4274	161	15	-	-	PUNCT
ejpam-4274	161	16	set	set	NOUN
ejpam-4274	161	17	and	and	CCONJ
ejpam-4274	161	18	hence	hence	ADV
ejpam-4274	161	19	{	{	PUNCT
ejpam-4274	161	20	x	x	X
ejpam-4274	161	21	}	}	PUNCT
ejpam-4274	161	22	is	be	AUX
ejpam-4274	161	23	open	open	ADJ
ejpam-4274	161	24	in	in	ADP
ejpam-4274	161	25	(	(	PUNCT
ejpam-4274	161	26	x	x	X
ejpam-4274	161	27	,	,	PUNCT
ejpam-4274	161	28	λp	λp	PROPN
ejpam-4274	161	29	)	)	PUNCT
ejpam-4274	161	30	.	.	PUNCT
ejpam-4274	162	1	thus	thus	ADV
ejpam-4274	162	2	,	,	PUNCT
ejpam-4274	162	3	every	every	DET
ejpam-4274	162	4	subset	subset	NOUN
ejpam-4274	162	5	of	of	ADP
ejpam-4274	162	6	x	x	PUNCT
ejpam-4274	162	7	is	be	AUX
ejpam-4274	162	8	open	open	ADJ
ejpam-4274	162	9	in	in	ADP
ejpam-4274	162	10	(	(	PUNCT
ejpam-4274	162	11	x	x	X
ejpam-4274	162	12	,	,	PUNCT
ejpam-4274	162	13	λp	λp	PROPN
ejpam-4274	162	14	)	)	PUNCT
ejpam-4274	162	15	.	.	PUNCT
ejpam-4274	163	1	this	this	PRON
ejpam-4274	163	2	shows	show	VERB
ejpam-4274	163	3	that	that	SCONJ
ejpam-4274	163	4	(	(	PUNCT
ejpam-4274	163	5	x	x	X
ejpam-4274	163	6	,	,	PUNCT
ejpam-4274	163	7	λp	λp	X
ejpam-4274	163	8	)	)	PUNCT
ejpam-4274	163	9	is	be	AUX
ejpam-4274	163	10	discrete	discrete	ADJ
ejpam-4274	163	11	.	.	PUNCT
ejpam-4274	164	1	conversely	conversely	ADV
ejpam-4274	164	2	,	,	PUNCT
ejpam-4274	164	3	suppose	suppose	VERB
ejpam-4274	164	4	that	that	SCONJ
ejpam-4274	164	5	a	a	DET
ejpam-4274	164	6	topological	topological	ADJ
ejpam-4274	164	7	space	space	NOUN
ejpam-4274	164	8	(	(	PUNCT
ejpam-4274	164	9	x	x	X
ejpam-4274	164	10	,	,	PUNCT
ejpam-4274	164	11	λp	λp	X
ejpam-4274	164	12	)	)	PUNCT
ejpam-4274	164	13	is	be	AUX
ejpam-4274	164	14	discrete	discrete	ADJ
ejpam-4274	164	15	.	.	PUNCT
ejpam-4274	165	1	for	for	ADP
ejpam-4274	165	2	any	any	DET
ejpam-4274	165	3	point	point	NOUN
ejpam-4274	165	4	x	x	X
ejpam-4274	165	5	∈	∈	NOUN
ejpam-4274	165	6	x	x	X
ejpam-4274	165	7	,	,	PUNCT
ejpam-4274	165	8	{	{	PUNCT
ejpam-4274	165	9	x	x	NOUN
ejpam-4274	165	10	}	}	PUNCT
ejpam-4274	165	11	is	be	AUX
ejpam-4274	165	12	open	open	ADJ
ejpam-4274	165	13	in	in	ADP
ejpam-4274	165	14	(	(	PUNCT
ejpam-4274	165	15	x	x	X
ejpam-4274	165	16	,	,	PUNCT
ejpam-4274	165	17	λp	λp	PROPN
ejpam-4274	165	18	)	)	PUNCT
ejpam-4274	165	19	and	and	CCONJ
ejpam-4274	165	20	hence	hence	ADV
ejpam-4274	165	21	{	{	PUNCT
ejpam-4274	165	22	x	x	X
ejpam-4274	165	23	}	}	PUNCT
ejpam-4274	165	24	is	be	AUX
ejpam-4274	165	25	a	a	DET
ejpam-4274	165	26	λp	λp	NOUN
ejpam-4274	165	27	-	-	PUNCT
ejpam-4274	165	28	set	set	NOUN
ejpam-4274	165	29	,	,	PUNCT
ejpam-4274	165	30	by	by	ADP
ejpam-4274	165	31	lemma	lemma	PROPN
ejpam-4274	165	32	4	4	NUM
ejpam-4274	165	33	,	,	PUNCT
ejpam-4274	165	34	we	we	PRON
ejpam-4274	165	35	have	have	AUX
ejpam-4274	165	36	(	(	PUNCT
ejpam-4274	165	37	x	x	NOUN
ejpam-4274	165	38	,	,	PUNCT
ejpam-4274	165	39	τ	τ	X
ejpam-4274	165	40	)	)	PUNCT
ejpam-4274	165	41	is	be	AUX
ejpam-4274	165	42	pre	pre	ADJ
ejpam-4274	165	43	-	-	ADJ
ejpam-4274	165	44	t1	t1	ADJ
ejpam-4274	165	45	.	.	PUNCT
ejpam-4274	166	1	corollary	corollary	ADJ
ejpam-4274	166	2	1	1	NUM
ejpam-4274	166	3	.	.	PUNCT
ejpam-4274	167	1	for	for	ADP
ejpam-4274	167	2	a	a	DET
ejpam-4274	167	3	topological	topological	ADJ
ejpam-4274	167	4	space	space	NOUN
ejpam-4274	167	5	(	(	PUNCT
ejpam-4274	167	6	x	x	X
ejpam-4274	167	7	,	,	PUNCT
ejpam-4274	167	8	τ	τ	PROPN
ejpam-4274	167	9	)	)	PUNCT
ejpam-4274	167	10	,	,	PUNCT
ejpam-4274	167	11	the	the	DET
ejpam-4274	167	12	following	follow	VERB
ejpam-4274	167	13	properties	property	NOUN
ejpam-4274	167	14	are	be	AUX
ejpam-4274	167	15	equivalent	equivalent	ADJ
ejpam-4274	167	16	:	:	PUNCT
ejpam-4274	167	17	(	(	PUNCT
ejpam-4274	167	18	1	1	X
ejpam-4274	167	19	)	)	PUNCT
ejpam-4274	167	20	(	(	PUNCT
ejpam-4274	167	21	x	x	X
ejpam-4274	167	22	,	,	PUNCT
ejpam-4274	167	23	τ	τ	X
ejpam-4274	167	24	)	)	PUNCT
ejpam-4274	167	25	is	be	AUX
ejpam-4274	167	26	pre	pre	ADJ
ejpam-4274	167	27	-	-	NOUN
ejpam-4274	167	28	t1	t1	ADJ
ejpam-4274	167	29	;	;	PUNCT
ejpam-4274	167	30	(	(	PUNCT
ejpam-4274	167	31	2	2	X
ejpam-4274	167	32	)	)	PUNCT
ejpam-4274	167	33	(	(	PUNCT
ejpam-4274	167	34	x	x	X
ejpam-4274	167	35	,	,	PUNCT
ejpam-4274	167	36	τ	τ	X
ejpam-4274	167	37	)	)	PUNCT
ejpam-4274	167	38	is	be	AUX
ejpam-4274	167	39	pre	pre	ADJ
ejpam-4274	167	40	-	-	ADJ
ejpam-4274	167	41	r0	r0	ADJ
ejpam-4274	167	42	and	and	CCONJ
ejpam-4274	167	43	pre	pre	NOUN
ejpam-4274	167	44	-	-	NOUN
ejpam-4274	167	45	t0	t0	NOUN
ejpam-4274	167	46	;	;	PUNCT
ejpam-4274	167	47	(	(	PUNCT
ejpam-4274	167	48	3	3	X
ejpam-4274	167	49	)	)	PUNCT
ejpam-4274	167	50	(	(	PUNCT
ejpam-4274	167	51	x	x	X
ejpam-4274	167	52	,	,	PUNCT
ejpam-4274	167	53	λp	λp	X
ejpam-4274	167	54	)	)	PUNCT
ejpam-4274	167	55	is	be	AUX
ejpam-4274	167	56	r0	r0	NOUN
ejpam-4274	167	57	and	and	CCONJ
ejpam-4274	167	58	t0	t0	PRON
ejpam-4274	167	59	;	;	PUNCT
ejpam-4274	167	60	(	(	PUNCT
ejpam-4274	167	61	4	4	NUM
ejpam-4274	167	62	)	)	PUNCT
ejpam-4274	167	63	(	(	PUNCT
ejpam-4274	167	64	x	x	X
ejpam-4274	167	65	,	,	PUNCT
ejpam-4274	167	66	λp	λp	X
ejpam-4274	167	67	)	)	PUNCT
ejpam-4274	167	68	is	be	AUX
ejpam-4274	167	69	t1	t1	NOUN
ejpam-4274	167	70	;	;	PUNCT
ejpam-4274	167	71	(	(	PUNCT
ejpam-4274	167	72	5	5	NUM
ejpam-4274	167	73	)	)	PUNCT
ejpam-4274	167	74	(	(	PUNCT
ejpam-4274	167	75	x	x	X
ejpam-4274	167	76	,	,	PUNCT
ejpam-4274	167	77	λp	λp	X
ejpam-4274	167	78	)	)	PUNCT
ejpam-4274	167	79	is	be	AUX
ejpam-4274	167	80	discrete	discrete	ADJ
ejpam-4274	167	81	.	.	PUNCT
ejpam-4274	168	1	proof	proof	NOUN
ejpam-4274	168	2	.	.	PUNCT
ejpam-4274	169	1	(	(	PUNCT
ejpam-4274	169	2	1	1	X
ejpam-4274	169	3	)	)	PUNCT
ejpam-4274	169	4	⇒	⇒	NOUN
ejpam-4274	169	5	(	(	PUNCT
ejpam-4274	169	6	2	2	NUM
ejpam-4274	169	7	):	):	PUNCT
ejpam-4274	169	8	by	by	ADP
ejpam-4274	169	9	lemma	lemma	PROPN
ejpam-4274	169	10	4	4	NUM
ejpam-4274	169	11	,	,	PUNCT
ejpam-4274	169	12	every	every	DET
ejpam-4274	169	13	pre	pre	ADJ
ejpam-4274	169	14	-	-	ADJ
ejpam-4274	169	15	t1	t1	ADJ
ejpam-4274	169	16	space	space	NOUN
ejpam-4274	169	17	is	be	AUX
ejpam-4274	169	18	pre	pre	ADJ
ejpam-4274	169	19	-	-	ADJ
ejpam-4274	169	20	r0	r0	ADJ
ejpam-4274	169	21	and	and	CCONJ
ejpam-4274	169	22	pre	pre	NOUN
ejpam-4274	169	23	-	-	NOUN
ejpam-4274	169	24	t0	t0	NOUN
ejpam-4274	169	25	.	.	PUNCT
ejpam-4274	170	1	(	(	PUNCT
ejpam-4274	170	2	2	2	X
ejpam-4274	170	3	)	)	PUNCT
ejpam-4274	170	4	⇒	⇒	NOUN
ejpam-4274	170	5	(	(	PUNCT
ejpam-4274	170	6	1	1	NUM
ejpam-4274	170	7	):	):	PUNCT
ejpam-4274	170	8	since	since	SCONJ
ejpam-4274	170	9	(	(	PUNCT
ejpam-4274	170	10	x	x	X
ejpam-4274	170	11	,	,	PUNCT
ejpam-4274	170	12	τ	τ	X
ejpam-4274	170	13	)	)	PUNCT
ejpam-4274	170	14	is	be	AUX
ejpam-4274	170	15	pre	pre	ADJ
ejpam-4274	170	16	-	-	ADJ
ejpam-4274	170	17	t0	t0	NOUN
ejpam-4274	170	18	,	,	PUNCT
ejpam-4274	170	19	for	for	ADP
ejpam-4274	170	20	any	any	DET
ejpam-4274	170	21	distinct	distinct	ADJ
ejpam-4274	170	22	point	point	NOUN
ejpam-4274	170	23	x	x	NOUN
ejpam-4274	170	24	,	,	PUNCT
ejpam-4274	170	25	y	y	PROPN
ejpam-4274	170	26	of	of	ADP
ejpam-4274	170	27	x	x	PRON
ejpam-4274	170	28	,	,	PUNCT
ejpam-4274	170	29	there	there	PRON
ejpam-4274	170	30	exists	exist	VERB
ejpam-4274	170	31	a	a	DET
ejpam-4274	170	32	preopen	preopen	ADJ
ejpam-4274	170	33	set	set	VERB
ejpam-4274	170	34	u	u	NOUN
ejpam-4274	170	35	of	of	ADP
ejpam-4274	170	36	x	x	SYM
ejpam-4274	170	37	such	such	ADJ
ejpam-4274	170	38	that	that	SCONJ
ejpam-4274	170	39	x	x	SYM
ejpam-4274	170	40	∈	∈	PROPN
ejpam-4274	170	41	u	u	NOUN
ejpam-4274	170	42	and	and	CCONJ
ejpam-4274	170	43	y	y	PROPN
ejpam-4274	170	44	6∈	6∈	PROPN
ejpam-4274	170	45	u	u	PROPN
ejpam-4274	170	46	.	.	PUNCT
ejpam-4274	171	1	hence	hence	ADV
ejpam-4274	171	2	,	,	PUNCT
ejpam-4274	171	3	pcl({x	pcl({x	NOUN
ejpam-4274	171	4	}	}	PUNCT
ejpam-4274	171	5	)	)	PUNCT
ejpam-4274	172	1	⊆	⊆	NUM
ejpam-4274	172	2	u	u	NOUN
ejpam-4274	172	3	since	since	SCONJ
ejpam-4274	172	4	(	(	PUNCT
ejpam-4274	172	5	x	x	NOUN
ejpam-4274	172	6	,	,	PUNCT
ejpam-4274	172	7	τ	τ	X
ejpam-4274	172	8	)	)	PUNCT
ejpam-4274	172	9	is	be	AUX
ejpam-4274	172	10	pre	pre	ADJ
ejpam-4274	172	11	-	-	NOUN
ejpam-4274	172	12	r0	r0	NOUN
ejpam-4274	172	13	.	.	PUNCT
ejpam-4274	173	1	thus	thus	ADV
ejpam-4274	173	2	,	,	PUNCT
ejpam-4274	173	3	x	x	PROPN
ejpam-4274	173	4	6∈	6∈	NOUN
ejpam-4274	173	5	x	x	NOUN
ejpam-4274	173	6	−	−	PUNCT
ejpam-4274	173	7	pcl({x	pcl({x	NOUN
ejpam-4274	173	8	}	}	PUNCT
ejpam-4274	173	9	)	)	PUNCT
ejpam-4274	173	10	and	and	CCONJ
ejpam-4274	173	11	hence	hence	ADV
ejpam-4274	173	12	y	y	PROPN
ejpam-4274	173	13	∈	∈	PROPN
ejpam-4274	173	14	x	x	PUNCT
ejpam-4274	173	15	−	−	PUNCT
ejpam-4274	173	16	u	u	NOUN
ejpam-4274	173	17	⊆	⊆	NUM
ejpam-4274	173	18	x	x	SYM
ejpam-4274	173	19	−	−	NOUN
ejpam-4274	173	20	pcl({x	pcl({x	NOUN
ejpam-4274	173	21	}	}	PUNCT
ejpam-4274	173	22	)	)	PUNCT
ejpam-4274	173	23	∈	∈	PROPN
ejpam-4274	173	24	po(x	po(x	NOUN
ejpam-4274	173	25	,	,	PUNCT
ejpam-4274	173	26	τ	τ	PROPN
ejpam-4274	173	27	)	)	PUNCT
ejpam-4274	173	28	.	.	PUNCT
ejpam-4274	174	1	this	this	PRON
ejpam-4274	174	2	shows	show	VERB
ejpam-4274	174	3	that	that	SCONJ
ejpam-4274	174	4	(	(	PUNCT
ejpam-4274	174	5	x	x	X
ejpam-4274	174	6	,	,	PUNCT
ejpam-4274	174	7	τ	τ	X
ejpam-4274	174	8	)	)	PUNCT
ejpam-4274	174	9	is	be	AUX
ejpam-4274	174	10	pre	pre	ADJ
ejpam-4274	174	11	-	-	ADJ
ejpam-4274	174	12	t1	t1	ADJ
ejpam-4274	174	13	.	.	PUNCT
ejpam-4274	175	1	(	(	PUNCT
ejpam-4274	175	2	2	2	X
ejpam-4274	175	3	)	)	PUNCT
ejpam-4274	175	4	⇔	⇔	X
ejpam-4274	175	5	(	(	PUNCT
ejpam-4274	175	6	3	3	NUM
ejpam-4274	175	7	):	):	PUNCT
ejpam-4274	175	8	this	this	PRON
ejpam-4274	175	9	is	be	AUX
ejpam-4274	175	10	an	an	DET
ejpam-4274	175	11	immediate	immediate	ADJ
ejpam-4274	175	12	consequence	consequence	NOUN
ejpam-4274	175	13	of	of	ADP
ejpam-4274	175	14	theorem	theorem	ADJ
ejpam-4274	175	15	3	3	NUM
ejpam-4274	175	16	and	and	CCONJ
ejpam-4274	175	17	theorem	theorem	VERB
ejpam-4274	175	18	4	4	NUM
ejpam-4274	175	19	.	.	PUNCT
ejpam-4274	175	20	(	(	PUNCT
ejpam-4274	175	21	3	3	X
ejpam-4274	175	22	)	)	PUNCT
ejpam-4274	175	23	⇔	⇔	X
ejpam-4274	175	24	(	(	PUNCT
ejpam-4274	175	25	4	4	NUM
ejpam-4274	175	26	):	):	PUNCT
ejpam-4274	175	27	this	this	DET
ejpam-4274	175	28	proof	proof	NOUN
ejpam-4274	175	29	is	be	AUX
ejpam-4274	175	30	obvious	obvious	ADJ
ejpam-4274	175	31	.	.	PUNCT
ejpam-4274	176	1	(	(	PUNCT
ejpam-4274	176	2	5	5	X
ejpam-4274	176	3	)	)	PUNCT
ejpam-4274	176	4	⇔	⇔	X
ejpam-4274	176	5	(	(	PUNCT
ejpam-4274	176	6	1	1	NUM
ejpam-4274	176	7	):	):	PUNCT
ejpam-4274	176	8	this	this	PRON
ejpam-4274	176	9	is	be	AUX
ejpam-4274	176	10	an	an	DET
ejpam-4274	176	11	immediate	immediate	ADJ
ejpam-4274	176	12	consequence	consequence	NOUN
ejpam-4274	176	13	of	of	ADP
ejpam-4274	176	14	theorem	theorem	NOUN
ejpam-4274	176	15	5	5	NUM
ejpam-4274	176	16	.	.	NOUN
ejpam-4274	176	17	4	4	NUM
ejpam-4274	176	18	.	.	PUNCT
ejpam-4274	177	1	(	(	PUNCT
ejpam-4274	177	2	λ	λ	X
ejpam-4274	177	3	,	,	PUNCT
ejpam-4274	177	4	p)-closed	p)-close	VERB
ejpam-4274	177	5	sets	set	NOUN
ejpam-4274	177	6	in	in	ADP
ejpam-4274	177	7	this	this	DET
ejpam-4274	177	8	section	section	NOUN
ejpam-4274	177	9	,	,	PUNCT
ejpam-4274	177	10	we	we	PRON
ejpam-4274	177	11	introduce	introduce	VERB
ejpam-4274	177	12	the	the	DET
ejpam-4274	177	13	notion	notion	NOUN
ejpam-4274	177	14	of	of	ADP
ejpam-4274	177	15	(	(	PUNCT
ejpam-4274	177	16	λ	λ	PROPN
ejpam-4274	177	17	,	,	PUNCT
ejpam-4274	177	18	p)-closed	p)-close	VERB
ejpam-4274	177	19	sets	set	NOUN
ejpam-4274	177	20	in	in	ADP
ejpam-4274	177	21	topological	topological	ADJ
ejpam-4274	177	22	spaces	space	NOUN
ejpam-4274	177	23	.	.	PUNCT
ejpam-4274	178	1	moreover	moreover	ADV
ejpam-4274	178	2	,	,	PUNCT
ejpam-4274	178	3	some	some	DET
ejpam-4274	178	4	properties	property	NOUN
ejpam-4274	178	5	of	of	ADP
ejpam-4274	178	6	(	(	PUNCT
ejpam-4274	178	7	λ	λ	PROPN
ejpam-4274	178	8	,	,	PUNCT
ejpam-4274	178	9	p)-closed	p)-close	VERB
ejpam-4274	178	10	sets	set	NOUN
ejpam-4274	178	11	are	be	AUX
ejpam-4274	178	12	discussed	discuss	VERB
ejpam-4274	178	13	.	.	PUNCT
ejpam-4274	179	1	definition	definition	NOUN
ejpam-4274	179	2	4	4	NUM
ejpam-4274	179	3	.	.	PUNCT
ejpam-4274	180	1	a	a	DET
ejpam-4274	180	2	subset	subset	NOUN
ejpam-4274	180	3	a	a	PRON
ejpam-4274	180	4	of	of	ADP
ejpam-4274	180	5	a	a	DET
ejpam-4274	180	6	topological	topological	ADJ
ejpam-4274	180	7	space	space	NOUN
ejpam-4274	180	8	(	(	PUNCT
ejpam-4274	180	9	x	x	X
ejpam-4274	180	10	,	,	PUNCT
ejpam-4274	180	11	τ	τ	X
ejpam-4274	180	12	)	)	PUNCT
ejpam-4274	180	13	is	be	AUX
ejpam-4274	180	14	called	call	VERB
ejpam-4274	180	15	(	(	PUNCT
ejpam-4274	180	16	λ	λ	X
ejpam-4274	180	17	,	,	PUNCT
ejpam-4274	180	18	p)-closed	p)-close	VERB
ejpam-4274	180	19	if	if	SCONJ
ejpam-4274	180	20	a	a	DET
ejpam-4274	180	21	=	=	X
ejpam-4274	180	22	t	t	NOUN
ejpam-4274	180	23	∩c	∩c	NOUN
ejpam-4274	180	24	,	,	PUNCT
ejpam-4274	180	25	where	where	SCONJ
ejpam-4274	180	26	t	t	PROPN
ejpam-4274	180	27	is	be	AUX
ejpam-4274	180	28	a	a	DET
ejpam-4274	180	29	λp	λp	ADV
ejpam-4274	180	30	-	-	PUNCT
ejpam-4274	180	31	set	set	NOUN
ejpam-4274	180	32	and	and	CCONJ
ejpam-4274	180	33	c	c	NOUN
ejpam-4274	180	34	is	be	AUX
ejpam-4274	180	35	a	a	DET
ejpam-4274	180	36	preclosed	preclose	VERB
ejpam-4274	180	37	set	set	NOUN
ejpam-4274	180	38	.	.	PUNCT
ejpam-4274	181	1	the	the	DET
ejpam-4274	181	2	collection	collection	NOUN
ejpam-4274	181	3	of	of	ADP
ejpam-4274	181	4	all	all	DET
ejpam-4274	181	5	(	(	PUNCT
ejpam-4274	181	6	λ	λ	PROPN
ejpam-4274	181	7	,	,	PUNCT
ejpam-4274	181	8	p)-closed	p)-close	VERB
ejpam-4274	181	9	sets	set	NOUN
ejpam-4274	181	10	in	in	ADP
ejpam-4274	181	11	a	a	DET
ejpam-4274	181	12	topological	topological	ADJ
ejpam-4274	181	13	space	space	NOUN
ejpam-4274	181	14	(	(	PUNCT
ejpam-4274	181	15	x	x	X
ejpam-4274	181	16	,	,	PUNCT
ejpam-4274	181	17	τ	τ	X
ejpam-4274	181	18	)	)	PUNCT
ejpam-4274	181	19	is	be	AUX
ejpam-4274	181	20	denoted	denote	VERB
ejpam-4274	181	21	by	by	ADP
ejpam-4274	181	22	λpc(x	λpc(x	PROPN
ejpam-4274	181	23	,	,	PUNCT
ejpam-4274	181	24	τ	τ	PROPN
ejpam-4274	181	25	)	)	PUNCT
ejpam-4274	181	26	.	.	PUNCT
ejpam-4274	182	1	theorem	theorem	VERB
ejpam-4274	182	2	6	6	NUM
ejpam-4274	182	3	.	.	PUNCT
ejpam-4274	182	4	for	for	ADP
ejpam-4274	182	5	a	a	DET
ejpam-4274	182	6	subset	subset	NOUN
ejpam-4274	182	7	a	a	PRON
ejpam-4274	182	8	of	of	ADP
ejpam-4274	182	9	a	a	DET
ejpam-4274	182	10	topological	topological	ADJ
ejpam-4274	182	11	space	space	NOUN
ejpam-4274	182	12	(	(	PUNCT
ejpam-4274	182	13	x	x	X
ejpam-4274	182	14	,	,	PUNCT
ejpam-4274	182	15	τ	τ	PROPN
ejpam-4274	182	16	)	)	PUNCT
ejpam-4274	182	17	,	,	PUNCT
ejpam-4274	182	18	the	the	DET
ejpam-4274	182	19	following	follow	VERB
ejpam-4274	182	20	properties	property	NOUN
ejpam-4274	182	21	are	be	AUX
ejpam-4274	182	22	equivalent	equivalent	ADJ
ejpam-4274	182	23	:	:	PUNCT
ejpam-4274	182	24	c.	c.	PROPN
ejpam-4274	182	25	boonpok	boonpok	PROPN
ejpam-4274	182	26	,	,	PUNCT
ejpam-4274	182	27	c.	c.	PROPN
ejpam-4274	182	28	viriyapong	viriyapong	PROPN
ejpam-4274	182	29	/	/	SYM
ejpam-4274	182	30	eur	eur	PROPN
ejpam-4274	182	31	.	.	PUNCT
ejpam-4274	183	1	j.	j.	PROPN
ejpam-4274	183	2	pure	pure	PROPN
ejpam-4274	183	3	appl	appl	PROPN
ejpam-4274	183	4	.	.	PROPN
ejpam-4274	183	5	math	math	PROPN
ejpam-4274	183	6	,	,	PUNCT
ejpam-4274	183	7	15	15	NUM
ejpam-4274	183	8	(	(	PUNCT
ejpam-4274	183	9	2	2	NUM
ejpam-4274	183	10	)	)	PUNCT
ejpam-4274	183	11	(	(	PUNCT
ejpam-4274	183	12	2022	2022	NUM
ejpam-4274	183	13	)	)	PUNCT
ejpam-4274	183	14	,	,	PUNCT
ejpam-4274	183	15	415	415	NUM
ejpam-4274	183	16	-	-	SYM
ejpam-4274	183	17	436	436	NUM
ejpam-4274	183	18	420	420	NUM
ejpam-4274	183	19	(	(	PUNCT
ejpam-4274	183	20	1	1	NUM
ejpam-4274	183	21	)	)	PUNCT
ejpam-4274	184	1	a	a	PRON
ejpam-4274	184	2	is	be	AUX
ejpam-4274	184	3	(	(	PUNCT
ejpam-4274	184	4	λ	λ	X
ejpam-4274	184	5	,	,	PUNCT
ejpam-4274	184	6	p)-closed	p)-close	VERB
ejpam-4274	184	7	;	;	PUNCT
ejpam-4274	184	8	(	(	PUNCT
ejpam-4274	184	9	2	2	X
ejpam-4274	184	10	)	)	PUNCT
ejpam-4274	184	11	a	a	DET
ejpam-4274	184	12	=	=	SYM
ejpam-4274	184	13	t	t	PROPN
ejpam-4274	184	14	∩	∩	ADJ
ejpam-4274	184	15	pcl(a	pcl(a	PROPN
ejpam-4274	184	16	)	)	PUNCT
ejpam-4274	184	17	,	,	PUNCT
ejpam-4274	184	18	where	where	SCONJ
ejpam-4274	184	19	t	t	PROPN
ejpam-4274	184	20	is	be	AUX
ejpam-4274	184	21	a	a	DET
ejpam-4274	184	22	λp	λp	NOUN
ejpam-4274	184	23	-	-	PUNCT
ejpam-4274	184	24	set	set	NOUN
ejpam-4274	184	25	;	;	PUNCT
ejpam-4274	184	26	(	(	PUNCT
ejpam-4274	184	27	3	3	X
ejpam-4274	184	28	)	)	PUNCT
ejpam-4274	184	29	a	a	DET
ejpam-4274	184	30	=	=	NOUN
ejpam-4274	184	31	λp(a	λp(a	NOUN
ejpam-4274	184	32	)	)	PUNCT
ejpam-4274	184	33	∩	∩	NOUN
ejpam-4274	184	34	pcl(a	pcl(a	NOUN
ejpam-4274	184	35	)	)	PUNCT
ejpam-4274	184	36	.	.	PUNCT
ejpam-4274	185	1	proof	proof	NOUN
ejpam-4274	185	2	.	.	PUNCT
ejpam-4274	186	1	(	(	PUNCT
ejpam-4274	186	2	1	1	X
ejpam-4274	186	3	)	)	PUNCT
ejpam-4274	186	4	⇒	⇒	NOUN
ejpam-4274	186	5	(	(	PUNCT
ejpam-4274	186	6	2	2	NUM
ejpam-4274	186	7	):	):	PUNCT
ejpam-4274	186	8	let	let	VERB
ejpam-4274	186	9	a	a	DET
ejpam-4274	186	10	=	=	SYM
ejpam-4274	186	11	t	t	PROPN
ejpam-4274	186	12	∩	∩	ADJ
ejpam-4274	186	13	c	c	NOUN
ejpam-4274	186	14	,	,	PUNCT
ejpam-4274	186	15	where	where	SCONJ
ejpam-4274	186	16	t	t	PROPN
ejpam-4274	186	17	is	be	AUX
ejpam-4274	186	18	a	a	DET
ejpam-4274	186	19	λp	λp	ADV
ejpam-4274	186	20	-	-	PUNCT
ejpam-4274	186	21	set	set	NOUN
ejpam-4274	186	22	and	and	CCONJ
ejpam-4274	186	23	c	c	NOUN
ejpam-4274	186	24	is	be	AUX
ejpam-4274	186	25	a	a	DET
ejpam-4274	186	26	preclosed	preclose	VERB
ejpam-4274	186	27	set	set	NOUN
ejpam-4274	186	28	.	.	PUNCT
ejpam-4274	187	1	since	since	SCONJ
ejpam-4274	187	2	a	a	DET
ejpam-4274	187	3	⊆	⊆	NUM
ejpam-4274	187	4	c	c	NOUN
ejpam-4274	187	5	,	,	PUNCT
ejpam-4274	187	6	we	we	PRON
ejpam-4274	187	7	have	have	VERB
ejpam-4274	187	8	pcl(a	pcl(a	NOUN
ejpam-4274	187	9	)	)	PUNCT
ejpam-4274	187	10	⊆	⊆	NUM
ejpam-4274	187	11	c	c	NOUN
ejpam-4274	187	12	and	and	CCONJ
ejpam-4274	187	13	hence	hence	ADV
ejpam-4274	187	14	a	a	DET
ejpam-4274	187	15	=	=	SYM
ejpam-4274	187	16	t	t	PROPN
ejpam-4274	187	17	∩	∩	PROPN
ejpam-4274	187	18	c	c	PROPN
ejpam-4274	187	19	⊇	⊇	PROPN
ejpam-4274	187	20	t	t	PROPN
ejpam-4274	187	21	∩	∩	ADJ
ejpam-4274	187	22	pcl(a	pcl(a	PROPN
ejpam-4274	187	23	)	)	PUNCT
ejpam-4274	187	24	⊇	⊇	PROPN
ejpam-4274	187	25	a.	a.	NOUN
ejpam-4274	187	26	consequently	consequently	ADV
ejpam-4274	187	27	,	,	PUNCT
ejpam-4274	187	28	we	we	PRON
ejpam-4274	187	29	obtain	obtain	VERB
ejpam-4274	187	30	a	a	DET
ejpam-4274	187	31	=	=	SYM
ejpam-4274	187	32	t	t	NOUN
ejpam-4274	187	33	∩	∩	ADJ
ejpam-4274	187	34	pcl(a	pcl(a	NOUN
ejpam-4274	187	35	)	)	PUNCT
ejpam-4274	187	36	.	.	PUNCT
ejpam-4274	188	1	(	(	PUNCT
ejpam-4274	188	2	2	2	X
ejpam-4274	188	3	)	)	PUNCT
ejpam-4274	188	4	⇒	⇒	NOUN
ejpam-4274	188	5	(	(	PUNCT
ejpam-4274	188	6	3	3	NUM
ejpam-4274	188	7	):	):	PUNCT
ejpam-4274	188	8	let	let	VERB
ejpam-4274	188	9	a	a	DET
ejpam-4274	188	10	=	=	SYM
ejpam-4274	188	11	t	t	NOUN
ejpam-4274	188	12	∩pcl(a	∩pcl(a	NUM
ejpam-4274	188	13	)	)	PUNCT
ejpam-4274	188	14	,	,	PUNCT
ejpam-4274	188	15	where	where	SCONJ
ejpam-4274	188	16	t	t	PROPN
ejpam-4274	188	17	is	be	AUX
ejpam-4274	188	18	a	a	DET
ejpam-4274	188	19	λp	λp	NOUN
ejpam-4274	188	20	-	-	PUNCT
ejpam-4274	188	21	set	set	NOUN
ejpam-4274	188	22	.	.	PUNCT
ejpam-4274	189	1	since	since	SCONJ
ejpam-4274	189	2	a	a	DET
ejpam-4274	189	3	⊆	⊆	NUM
ejpam-4274	189	4	t	t	NOUN
ejpam-4274	189	5	,	,	PUNCT
ejpam-4274	189	6	λp(a	λp(a	NUM
ejpam-4274	189	7	)	)	PUNCT
ejpam-4274	190	1	⊆	⊆	NUM
ejpam-4274	190	2	λp(t	λp(t	NOUN
ejpam-4274	190	3	)	)	PUNCT
ejpam-4274	190	4	=	=	SYM
ejpam-4274	190	5	t	t	NOUN
ejpam-4274	190	6	and	and	CCONJ
ejpam-4274	190	7	hence	hence	ADV
ejpam-4274	190	8	a	a	DET
ejpam-4274	190	9	⊆	⊆	NUM
ejpam-4274	190	10	λp(a	λp(a	NUM
ejpam-4274	190	11	)	)	PUNCT
ejpam-4274	190	12	∩	∩	ADJ
ejpam-4274	190	13	pcl(a	pcl(a	X
ejpam-4274	190	14	)	)	PUNCT
ejpam-4274	191	1	⊆	⊆	NUM
ejpam-4274	191	2	t	t	NOUN
ejpam-4274	191	3	∩	∩	ADJ
ejpam-4274	191	4	pcl(a	pcl(a	X
ejpam-4274	191	5	)	)	PUNCT
ejpam-4274	191	6	=	=	SYM
ejpam-4274	191	7	a.	a.	NOUN
ejpam-4274	191	8	thus	thus	ADV
ejpam-4274	191	9	,	,	PUNCT
ejpam-4274	191	10	a	a	DET
ejpam-4274	191	11	=	=	NOUN
ejpam-4274	191	12	λp(a	λp(a	NOUN
ejpam-4274	191	13	)	)	PUNCT
ejpam-4274	191	14	∩	∩	NOUN
ejpam-4274	191	15	pcl(a	pcl(a	NOUN
ejpam-4274	191	16	)	)	PUNCT
ejpam-4274	191	17	.	.	PUNCT
ejpam-4274	192	1	(	(	PUNCT
ejpam-4274	192	2	3	3	X
ejpam-4274	192	3	)	)	PUNCT
ejpam-4274	192	4	⇒	⇒	NOUN
ejpam-4274	192	5	(	(	PUNCT
ejpam-4274	192	6	1	1	NUM
ejpam-4274	192	7	):	):	PUNCT
ejpam-4274	192	8	since	since	SCONJ
ejpam-4274	192	9	λp(a	λp(a	NUM
ejpam-4274	192	10	)	)	PUNCT
ejpam-4274	192	11	is	be	AUX
ejpam-4274	192	12	a	a	DET
ejpam-4274	192	13	λp	λp	NOUN
ejpam-4274	192	14	-	-	PUNCT
ejpam-4274	192	15	set	set	VERB
ejpam-4274	192	16	,	,	PUNCT
ejpam-4274	192	17	pcl(a	pcl(a	PROPN
ejpam-4274	192	18	)	)	PUNCT
ejpam-4274	192	19	is	be	AUX
ejpam-4274	192	20	preclosed	preclose	VERB
ejpam-4274	192	21	and	and	CCONJ
ejpam-4274	192	22	a	a	DET
ejpam-4274	192	23	=	=	X
ejpam-4274	192	24	λp(a)∩	λp(a)∩	X
ejpam-4274	192	25	pcl(a	pcl(a	PROPN
ejpam-4274	192	26	)	)	PUNCT
ejpam-4274	192	27	.	.	PUNCT
ejpam-4274	193	1	this	this	PRON
ejpam-4274	193	2	shows	show	VERB
ejpam-4274	193	3	that	that	SCONJ
ejpam-4274	193	4	a	a	PRON
ejpam-4274	193	5	is	be	AUX
ejpam-4274	193	6	(	(	PUNCT
ejpam-4274	193	7	λ	λ	X
ejpam-4274	193	8	,	,	PUNCT
ejpam-4274	193	9	p)-closed	p)-close	VERB
ejpam-4274	193	10	.	.	PUNCT
ejpam-4274	194	1	definition	definition	NOUN
ejpam-4274	194	2	5	5	NUM
ejpam-4274	194	3	.	.	PUNCT
ejpam-4274	195	1	a	a	DET
ejpam-4274	195	2	subset	subset	NOUN
ejpam-4274	195	3	a	a	PRON
ejpam-4274	195	4	of	of	ADP
ejpam-4274	195	5	a	a	DET
ejpam-4274	195	6	topological	topological	ADJ
ejpam-4274	195	7	space	space	NOUN
ejpam-4274	195	8	(	(	PUNCT
ejpam-4274	195	9	x	x	X
ejpam-4274	195	10	,	,	PUNCT
ejpam-4274	195	11	τ	τ	X
ejpam-4274	195	12	)	)	PUNCT
ejpam-4274	195	13	is	be	AUX
ejpam-4274	195	14	said	say	VERB
ejpam-4274	195	15	to	to	PART
ejpam-4274	195	16	be	be	AUX
ejpam-4274	195	17	(	(	PUNCT
ejpam-4274	195	18	λ	λ	X
ejpam-4274	195	19	,	,	PUNCT
ejpam-4274	195	20	p)-open	p)-open	VERB
ejpam-4274	195	21	if	if	SCONJ
ejpam-4274	195	22	the	the	DET
ejpam-4274	195	23	complement	complement	NOUN
ejpam-4274	195	24	of	of	ADP
ejpam-4274	195	25	a	a	DET
ejpam-4274	195	26	is	is	NOUN
ejpam-4274	195	27	(	(	PUNCT
ejpam-4274	195	28	λ	λ	X
ejpam-4274	195	29	,	,	PUNCT
ejpam-4274	195	30	p)-closed	p)-close	VERB
ejpam-4274	195	31	.	.	PUNCT
ejpam-4274	196	1	the	the	DET
ejpam-4274	196	2	collection	collection	NOUN
ejpam-4274	196	3	of	of	ADP
ejpam-4274	196	4	all	all	DET
ejpam-4274	196	5	(	(	PUNCT
ejpam-4274	196	6	λ	λ	NOUN
ejpam-4274	196	7	,	,	PUNCT
ejpam-4274	196	8	p)-open	p)-open	VERB
ejpam-4274	196	9	sets	set	NOUN
ejpam-4274	196	10	in	in	ADP
ejpam-4274	196	11	a	a	DET
ejpam-4274	196	12	topological	topological	ADJ
ejpam-4274	196	13	space	space	NOUN
ejpam-4274	196	14	(	(	PUNCT
ejpam-4274	196	15	x	x	X
ejpam-4274	196	16	,	,	PUNCT
ejpam-4274	196	17	τ	τ	X
ejpam-4274	196	18	)	)	PUNCT
ejpam-4274	196	19	is	be	AUX
ejpam-4274	196	20	denoted	denote	VERB
ejpam-4274	196	21	by	by	ADP
ejpam-4274	196	22	λpo(x	λpo(x	PROPN
ejpam-4274	196	23	,	,	PUNCT
ejpam-4274	196	24	τ	τ	PROPN
ejpam-4274	196	25	)	)	PUNCT
ejpam-4274	196	26	.	.	PUNCT
ejpam-4274	197	1	theorem	theorem	VERB
ejpam-4274	197	2	7	7	NUM
ejpam-4274	197	3	.	.	X
ejpam-4274	197	4	for	for	ADP
ejpam-4274	197	5	a	a	DET
ejpam-4274	197	6	subset	subset	NOUN
ejpam-4274	197	7	aγ(γ	aγ(γ	X
ejpam-4274	197	8	∈	∈	PROPN
ejpam-4274	197	9	γ	γ	X
ejpam-4274	197	10	)	)	PUNCT
ejpam-4274	197	11	of	of	ADP
ejpam-4274	197	12	a	a	DET
ejpam-4274	197	13	topological	topological	ADJ
ejpam-4274	197	14	space	space	NOUN
ejpam-4274	197	15	(	(	PUNCT
ejpam-4274	197	16	x	x	X
ejpam-4274	197	17	,	,	PUNCT
ejpam-4274	197	18	τ	τ	PROPN
ejpam-4274	197	19	)	)	PUNCT
ejpam-4274	197	20	,	,	PUNCT
ejpam-4274	197	21	the	the	DET
ejpam-4274	197	22	following	follow	VERB
ejpam-4274	197	23	properties	property	NOUN
ejpam-4274	197	24	hold	hold	VERB
ejpam-4274	197	25	:	:	PUNCT
ejpam-4274	197	26	(	(	PUNCT
ejpam-4274	197	27	1	1	X
ejpam-4274	197	28	)	)	PUNCT
ejpam-4274	197	29	if	if	SCONJ
ejpam-4274	197	30	aγ	aγ	PRON
ejpam-4274	197	31	is	be	AUX
ejpam-4274	197	32	(	(	PUNCT
ejpam-4274	197	33	λ	λ	X
ejpam-4274	197	34	,	,	PUNCT
ejpam-4274	197	35	p)-closed	p)-close	VERB
ejpam-4274	197	36	for	for	ADP
ejpam-4274	197	37	each	each	DET
ejpam-4274	197	38	γ	γ	PROPN
ejpam-4274	197	39	∈	∈	PROPN
ejpam-4274	197	40	γ	γ	X
ejpam-4274	197	41	,	,	PUNCT
ejpam-4274	197	42	then	then	ADV
ejpam-4274	197	43	∩{aγ	∩{aγ	VERB
ejpam-4274	197	44	|	|	ADV
ejpam-4274	197	45	γ	γ	PROPN
ejpam-4274	197	46	∈	∈	PROPN
ejpam-4274	197	47	γ	γ	X
ejpam-4274	197	48	}	}	PUNCT
ejpam-4274	197	49	is	be	AUX
ejpam-4274	197	50	(	(	PUNCT
ejpam-4274	197	51	λ	λ	X
ejpam-4274	197	52	,	,	PUNCT
ejpam-4274	197	53	p)-closed	p)-close	VERB
ejpam-4274	197	54	.	.	PUNCT
ejpam-4274	198	1	(	(	PUNCT
ejpam-4274	198	2	2	2	X
ejpam-4274	198	3	)	)	PUNCT
ejpam-4274	198	4	if	if	SCONJ
ejpam-4274	198	5	aγ	aγ	PRON
ejpam-4274	198	6	is	be	AUX
ejpam-4274	198	7	(	(	PUNCT
ejpam-4274	198	8	λ	λ	X
ejpam-4274	198	9	,	,	PUNCT
ejpam-4274	198	10	p)-open	p)-open	VERB
ejpam-4274	198	11	for	for	ADP
ejpam-4274	198	12	each	each	DET
ejpam-4274	198	13	γ	γ	PROPN
ejpam-4274	198	14	∈	∈	PROPN
ejpam-4274	198	15	γ	γ	X
ejpam-4274	198	16	,	,	PUNCT
ejpam-4274	198	17	then	then	ADV
ejpam-4274	198	18	∪{aγ	∪{aγ	PROPN
ejpam-4274	198	19	|	|	ADV
ejpam-4274	198	20	γ	γ	PROPN
ejpam-4274	198	21	∈	∈	PROPN
ejpam-4274	198	22	γ	γ	X
ejpam-4274	198	23	}	}	PUNCT
ejpam-4274	198	24	is	be	AUX
ejpam-4274	198	25	(	(	PUNCT
ejpam-4274	198	26	λ	λ	X
ejpam-4274	198	27	,	,	PUNCT
ejpam-4274	198	28	p)-open	p)-open	ADJ
ejpam-4274	198	29	.	.	PUNCT
ejpam-4274	199	1	proof	proof	NOUN
ejpam-4274	199	2	.	.	PUNCT
ejpam-4274	200	1	(	(	PUNCT
ejpam-4274	200	2	1	1	X
ejpam-4274	200	3	)	)	PUNCT
ejpam-4274	200	4	suppose	suppose	VERB
ejpam-4274	200	5	that	that	SCONJ
ejpam-4274	200	6	aγ	aγ	PRON
ejpam-4274	200	7	is	be	AUX
ejpam-4274	200	8	(	(	PUNCT
ejpam-4274	200	9	λ	λ	X
ejpam-4274	200	10	,	,	PUNCT
ejpam-4274	200	11	p)-closed	p)-close	VERB
ejpam-4274	200	12	for	for	ADP
ejpam-4274	200	13	each	each	DET
ejpam-4274	200	14	γ	γ	PROPN
ejpam-4274	200	15	∈	∈	PROPN
ejpam-4274	200	16	γ	γ	X
ejpam-4274	200	17	.	.	PROPN
ejpam-4274	201	1	then	then	ADV
ejpam-4274	201	2	,	,	PUNCT
ejpam-4274	201	3	for	for	ADP
ejpam-4274	201	4	each	each	DET
ejpam-4274	201	5	γ	γ	NOUN
ejpam-4274	201	6	,	,	PUNCT
ejpam-4274	201	7	there	there	PRON
ejpam-4274	201	8	exist	exist	VERB
ejpam-4274	201	9	a	a	DET
ejpam-4274	201	10	λp	λp	ADV
ejpam-4274	201	11	-	-	PUNCT
ejpam-4274	201	12	set	set	VERB
ejpam-4274	201	13	tγ	tγ	NOUN
ejpam-4274	201	14	and	and	CCONJ
ejpam-4274	201	15	a	a	DET
ejpam-4274	201	16	preclosed	preclose	VERB
ejpam-4274	201	17	set	set	NOUN
ejpam-4274	201	18	cγ	cγ	NOUN
ejpam-4274	201	19	such	such	ADJ
ejpam-4274	201	20	that	that	SCONJ
ejpam-4274	201	21	aγ	aγ	PRON
ejpam-4274	201	22	=	=	SYM
ejpam-4274	201	23	tγ	tγ	PROPN
ejpam-4274	201	24	∩	∩	ADJ
ejpam-4274	201	25	cγ	cγ	NOUN
ejpam-4274	201	26	.	.	PUNCT
ejpam-4274	202	1	thus	thus	ADV
ejpam-4274	202	2	,	,	PUNCT
ejpam-4274	202	3	∩γ∈γaγ	∩γ∈γaγ	PUNCT
ejpam-4274	202	4	=	=	SYM
ejpam-4274	202	5	∩γ∈γ(tγ	∩γ∈γ(tγ	NOUN
ejpam-4274	202	6	∩	∩	NOUN
ejpam-4274	202	7	cγ	cγ	NOUN
ejpam-4274	202	8	)	)	PUNCT
ejpam-4274	202	9	=	=	SYM
ejpam-4274	202	10	(	(	PUNCT
ejpam-4274	202	11	∩γ∈γtγ	∩γ∈γtγ	NOUN
ejpam-4274	202	12	)	)	PUNCT
ejpam-4274	202	13	∩	∩	NOUN
ejpam-4274	202	14	(	(	PUNCT
ejpam-4274	202	15	∩γ∈γcγ	∩γ∈γcγ	NOUN
ejpam-4274	202	16	)	)	PUNCT
ejpam-4274	202	17	.	.	PUNCT
ejpam-4274	203	1	since	since	SCONJ
ejpam-4274	203	2	∩γ∈γcγ	∩γ∈γcγ	PROPN
ejpam-4274	203	3	is	be	AUX
ejpam-4274	203	4	a	a	DET
ejpam-4274	203	5	preclosed	preclose	VERB
ejpam-4274	203	6	set	set	NOUN
ejpam-4274	203	7	and	and	CCONJ
ejpam-4274	203	8	by	by	ADP
ejpam-4274	203	9	lemma	lemma	PROPN
ejpam-4274	203	10	3	3	NUM
ejpam-4274	203	11	,	,	PUNCT
ejpam-4274	203	12	we	we	PRON
ejpam-4274	203	13	have	have	VERB
ejpam-4274	203	14	∩γ∈γtγ	∩γ∈γtγ	X
ejpam-4274	203	15	is	be	AUX
ejpam-4274	203	16	a	a	DET
ejpam-4274	203	17	λp	λp	NOUN
ejpam-4274	203	18	-	-	PUNCT
ejpam-4274	203	19	set	set	NOUN
ejpam-4274	203	20	.	.	PUNCT
ejpam-4274	204	1	this	this	PRON
ejpam-4274	204	2	shows	show	VERB
ejpam-4274	204	3	that	that	SCONJ
ejpam-4274	204	4	∩γ∈γaγ	∩γ∈γaγ	VERB
ejpam-4274	204	5	is	be	AUX
ejpam-4274	204	6	(	(	PUNCT
ejpam-4274	204	7	λ	λ	X
ejpam-4274	204	8	,	,	PUNCT
ejpam-4274	204	9	p)-closed	p)-close	VERB
ejpam-4274	204	10	.	.	PUNCT
ejpam-4274	205	1	(	(	PUNCT
ejpam-4274	205	2	2	2	X
ejpam-4274	205	3	)	)	PUNCT
ejpam-4274	205	4	let	let	VERB
ejpam-4274	205	5	aγ	aγ	PRON
ejpam-4274	205	6	be	be	AUX
ejpam-4274	205	7	(	(	PUNCT
ejpam-4274	205	8	λ	λ	X
ejpam-4274	205	9	,	,	PUNCT
ejpam-4274	205	10	p)-open	p)-open	VERB
ejpam-4274	205	11	for	for	ADP
ejpam-4274	205	12	each	each	DET
ejpam-4274	205	13	γ	γ	PROPN
ejpam-4274	205	14	∈	∈	PROPN
ejpam-4274	205	15	γ	γ	X
ejpam-4274	205	16	.	.	PUNCT
ejpam-4274	206	1	then	then	ADV
ejpam-4274	206	2	,	,	PUNCT
ejpam-4274	206	3	x	x	PUNCT
ejpam-4274	206	4	−	−	NOUN
ejpam-4274	207	1	aγ	aγ	PRON
ejpam-4274	207	2	is	be	AUX
ejpam-4274	207	3	(	(	PUNCT
ejpam-4274	207	4	λ	λ	X
ejpam-4274	207	5	,	,	PUNCT
ejpam-4274	207	6	p)-closed	p)-close	VERB
ejpam-4274	207	7	,	,	PUNCT
ejpam-4274	207	8	by	by	ADP
ejpam-4274	207	9	(	(	PUNCT
ejpam-4274	207	10	1	1	NUM
ejpam-4274	207	11	)	)	PUNCT
ejpam-4274	207	12	,	,	PUNCT
ejpam-4274	207	13	we	we	PRON
ejpam-4274	207	14	have	have	VERB
ejpam-4274	207	15	x	x	PART
ejpam-4274	207	16	−	−	NOUN
ejpam-4274	207	17	∪γ∈γaγ	∪γ∈γaγ	NOUN
ejpam-4274	207	18	=	=	SYM
ejpam-4274	207	19	∩γ∈γ(x	∩γ∈γ(x	NOUN
ejpam-4274	207	20	−aγ	−aγ	PROPN
ejpam-4274	207	21	)	)	PUNCT
ejpam-4274	207	22	is	be	AUX
ejpam-4274	207	23	(	(	PUNCT
ejpam-4274	207	24	λ	λ	X
ejpam-4274	207	25	,	,	PUNCT
ejpam-4274	207	26	p)-closed	p)-close	VERB
ejpam-4274	207	27	and	and	CCONJ
ejpam-4274	207	28	hence	hence	ADV
ejpam-4274	207	29	∪γ∈γaγ	∪γ∈γaγ	ADV
ejpam-4274	207	30	is	be	AUX
ejpam-4274	207	31	(	(	PUNCT
ejpam-4274	207	32	λ	λ	INTJ
ejpam-4274	207	33	,	,	PUNCT
ejpam-4274	207	34	p)-open	p)-open	NOUN
ejpam-4274	207	35	.	.	PUNCT
ejpam-4274	208	1	theorem	theorem	NOUN
ejpam-4274	208	2	8	8	NUM
ejpam-4274	208	3	.	.	PUNCT
ejpam-4274	209	1	let	let	AUX
ejpam-4274	209	2	(	(	PUNCT
ejpam-4274	209	3	x	x	NOUN
ejpam-4274	209	4	,	,	PUNCT
ejpam-4274	209	5	τ	τ	X
ejpam-4274	209	6	)	)	PUNCT
ejpam-4274	209	7	be	be	AUX
ejpam-4274	209	8	a	a	DET
ejpam-4274	209	9	pre	pre	ADJ
ejpam-4274	209	10	-	-	ADJ
ejpam-4274	209	11	r0	r0	ADJ
ejpam-4274	209	12	space	space	NOUN
ejpam-4274	209	13	.	.	PUNCT
ejpam-4274	210	1	for	for	SCONJ
ejpam-4274	210	2	each	each	DET
ejpam-4274	210	3	x	x	SYM
ejpam-4274	210	4	∈	∈	PROPN
ejpam-4274	210	5	x	x	X
ejpam-4274	210	6	,	,	PUNCT
ejpam-4274	210	7	{	{	PUNCT
ejpam-4274	210	8	x	x	X
ejpam-4274	210	9	}	}	PUNCT
ejpam-4274	210	10	is	be	AUX
ejpam-4274	210	11	(	(	PUNCT
ejpam-4274	210	12	λ	λ	X
ejpam-4274	210	13	,	,	PUNCT
ejpam-4274	210	14	p)-closed	p)-close	VERB
ejpam-4274	210	15	if	if	SCONJ
ejpam-4274	210	16	and	and	CCONJ
ejpam-4274	210	17	only	only	ADV
ejpam-4274	210	18	if	if	SCONJ
ejpam-4274	210	19	{	{	PUNCT
ejpam-4274	210	20	x	x	NOUN
ejpam-4274	210	21	}	}	PUNCT
ejpam-4274	210	22	is	be	AUX
ejpam-4274	210	23	preclosed	preclose	VERB
ejpam-4274	210	24	.	.	PUNCT
ejpam-4274	211	1	proof	proof	NOUN
ejpam-4274	211	2	.	.	PUNCT
ejpam-4274	212	1	suppose	suppose	VERB
ejpam-4274	212	2	that	that	SCONJ
ejpam-4274	212	3	{	{	PUNCT
ejpam-4274	212	4	x	x	X
ejpam-4274	212	5	}	}	PUNCT
ejpam-4274	212	6	is	be	AUX
ejpam-4274	212	7	a	a	DET
ejpam-4274	212	8	(	(	PUNCT
ejpam-4274	212	9	λ	λ	PROPN
ejpam-4274	212	10	,	,	PUNCT
ejpam-4274	212	11	p)-closed	p)-close	VERB
ejpam-4274	212	12	set	set	NOUN
ejpam-4274	212	13	.	.	PUNCT
ejpam-4274	213	1	by	by	ADP
ejpam-4274	213	2	theorem	theorem	NOUN
ejpam-4274	213	3	6	6	NUM
ejpam-4274	213	4	,	,	PUNCT
ejpam-4274	213	5	{	{	PUNCT
ejpam-4274	213	6	x	x	NOUN
ejpam-4274	213	7	}	}	PUNCT
ejpam-4274	213	8	=	=	PUNCT
ejpam-4274	213	9	λp({x	λp({x	ADJ
ejpam-4274	213	10	}	}	PUNCT
ejpam-4274	213	11	)	)	PUNCT
ejpam-4274	213	12	∩	∩	ADJ
ejpam-4274	213	13	pcl({x	pcl({x	NOUN
ejpam-4274	213	14	}	}	PUNCT
ejpam-4274	213	15	)	)	PUNCT
ejpam-4274	213	16	.	.	PUNCT
ejpam-4274	214	1	for	for	ADP
ejpam-4274	214	2	any	any	DET
ejpam-4274	214	3	preopen	preopen	ADJ
ejpam-4274	214	4	set	set	VERB
ejpam-4274	214	5	u	u	NOUN
ejpam-4274	214	6	containing	contain	VERB
ejpam-4274	214	7	x	x	PRON
ejpam-4274	214	8	,	,	PUNCT
ejpam-4274	214	9	pcl({x	pcl({x	NOUN
ejpam-4274	214	10	}	}	PUNCT
ejpam-4274	214	11	)	)	PUNCT
ejpam-4274	214	12	⊆	⊆	NUM
ejpam-4274	214	13	u	u	NOUN
ejpam-4274	214	14	and	and	CCONJ
ejpam-4274	214	15	hence	hence	ADV
ejpam-4274	214	16	pcl({x	pcl({x	NOUN
ejpam-4274	214	17	}	}	PUNCT
ejpam-4274	214	18	)	)	PUNCT
ejpam-4274	214	19	⊆	⊆	NUM
ejpam-4274	214	20	λp({x	λp({x	NOUN
ejpam-4274	214	21	}	}	PUNCT
ejpam-4274	214	22	)	)	PUNCT
ejpam-4274	214	23	.	.	PUNCT
ejpam-4274	215	1	thus	thus	ADV
ejpam-4274	215	2	,	,	PUNCT
ejpam-4274	215	3	{	{	PUNCT
ejpam-4274	215	4	x	x	X
ejpam-4274	215	5	}	}	PUNCT
ejpam-4274	215	6	=	=	PUNCT
ejpam-4274	215	7	λp({x	λp({x	ADJ
ejpam-4274	215	8	}	}	PUNCT
ejpam-4274	215	9	)	)	PUNCT
ejpam-4274	215	10	∩	∩	ADJ
ejpam-4274	215	11	pcl({x	pcl({x	NOUN
ejpam-4274	215	12	}	}	PUNCT
ejpam-4274	215	13	)	)	PUNCT
ejpam-4274	215	14	⊇	⊇	PROPN
ejpam-4274	215	15	pcl({x	pcl({x	NOUN
ejpam-4274	215	16	}	}	PUNCT
ejpam-4274	215	17	)	)	PUNCT
ejpam-4274	215	18	.	.	PUNCT
ejpam-4274	216	1	this	this	PRON
ejpam-4274	216	2	shows	show	VERB
ejpam-4274	216	3	that	that	SCONJ
ejpam-4274	216	4	{	{	PUNCT
ejpam-4274	216	5	x	x	X
ejpam-4274	216	6	}	}	PUNCT
ejpam-4274	216	7	is	be	AUX
ejpam-4274	216	8	preclosed	preclose	VERB
ejpam-4274	216	9	.	.	PUNCT
ejpam-4274	217	1	conversely	conversely	ADV
ejpam-4274	217	2	,	,	PUNCT
ejpam-4274	217	3	suppose	suppose	VERB
ejpam-4274	217	4	that	that	SCONJ
ejpam-4274	217	5	{	{	PUNCT
ejpam-4274	217	6	x	x	X
ejpam-4274	217	7	}	}	PUNCT
ejpam-4274	217	8	is	be	AUX
ejpam-4274	217	9	a	a	DET
ejpam-4274	217	10	preclosed	preclose	VERB
ejpam-4274	217	11	set	set	NOUN
ejpam-4274	217	12	.	.	PUNCT
ejpam-4274	218	1	since	since	SCONJ
ejpam-4274	218	2	{	{	PUNCT
ejpam-4274	218	3	x	x	NOUN
ejpam-4274	218	4	}	}	PUNCT
ejpam-4274	218	5	⊆	⊆	NUM
ejpam-4274	218	6	λp({x	λp({x	NOUN
ejpam-4274	218	7	}	}	PUNCT
ejpam-4274	218	8	)	)	PUNCT
ejpam-4274	218	9	,	,	PUNCT
ejpam-4274	218	10	we	we	PRON
ejpam-4274	218	11	have	have	VERB
ejpam-4274	218	12	λp({x	λp({x	ADV
ejpam-4274	218	13	}	}	PUNCT
ejpam-4274	218	14	)	)	PUNCT
ejpam-4274	218	15	∩	∩	ADJ
ejpam-4274	218	16	pcl({x	pcl({x	NOUN
ejpam-4274	218	17	}	}	PUNCT
ejpam-4274	218	18	)	)	PUNCT
ejpam-4274	218	19	=	=	PUNCT
ejpam-4274	219	1	λp({x	λp({x	ADJ
ejpam-4274	219	2	}	}	PUNCT
ejpam-4274	219	3	)	)	PUNCT
ejpam-4274	219	4	∩	∩	NOUN
ejpam-4274	219	5	{	{	PUNCT
ejpam-4274	219	6	x	x	NOUN
ejpam-4274	219	7	}	}	PUNCT
ejpam-4274	219	8	=	=	SYM
ejpam-4274	219	9	{	{	PUNCT
ejpam-4274	219	10	x	x	NOUN
ejpam-4274	219	11	}	}	PUNCT
ejpam-4274	219	12	,	,	PUNCT
ejpam-4274	219	13	by	by	ADP
ejpam-4274	219	14	theorem	theorem	NOUN
ejpam-4274	219	15	6	6	NUM
ejpam-4274	219	16	,	,	PUNCT
ejpam-4274	219	17	{	{	PUNCT
ejpam-4274	219	18	x	x	NOUN
ejpam-4274	219	19	}	}	PUNCT
ejpam-4274	219	20	is	be	AUX
ejpam-4274	219	21	(	(	PUNCT
ejpam-4274	219	22	λ	λ	X
ejpam-4274	219	23	,	,	PUNCT
ejpam-4274	219	24	p)-closed	p)-close	VERB
ejpam-4274	219	25	.	.	PUNCT
ejpam-4274	220	1	c.	c.	PROPN
ejpam-4274	220	2	boonpok	boonpok	PROPN
ejpam-4274	220	3	,	,	PUNCT
ejpam-4274	220	4	c.	c.	PROPN
ejpam-4274	220	5	viriyapong	viriyapong	PROPN
ejpam-4274	220	6	/	/	SYM
ejpam-4274	220	7	eur	eur	PROPN
ejpam-4274	220	8	.	.	PUNCT
ejpam-4274	221	1	j.	j.	PROPN
ejpam-4274	221	2	pure	pure	PROPN
ejpam-4274	221	3	appl	appl	PROPN
ejpam-4274	221	4	.	.	PROPN
ejpam-4274	221	5	math	math	PROPN
ejpam-4274	221	6	,	,	PUNCT
ejpam-4274	221	7	15	15	NUM
ejpam-4274	221	8	(	(	PUNCT
ejpam-4274	221	9	2	2	NUM
ejpam-4274	221	10	)	)	PUNCT
ejpam-4274	221	11	(	(	PUNCT
ejpam-4274	221	12	2022	2022	NUM
ejpam-4274	221	13	)	)	PUNCT
ejpam-4274	221	14	,	,	PUNCT
ejpam-4274	221	15	415	415	NUM
ejpam-4274	221	16	-	-	SYM
ejpam-4274	221	17	436	436	NUM
ejpam-4274	221	18	421	421	NUM
ejpam-4274	221	19	theorem	theorem	NOUN
ejpam-4274	221	20	9	9	NUM
ejpam-4274	221	21	.	.	PUNCT
ejpam-4274	222	1	a	a	DET
ejpam-4274	222	2	topological	topological	ADJ
ejpam-4274	222	3	space	space	NOUN
ejpam-4274	222	4	(	(	PUNCT
ejpam-4274	222	5	x	x	X
ejpam-4274	222	6	,	,	PUNCT
ejpam-4274	222	7	τ	τ	X
ejpam-4274	222	8	)	)	PUNCT
ejpam-4274	222	9	is	be	AUX
ejpam-4274	222	10	pre	pre	ADJ
ejpam-4274	222	11	-	-	NOUN
ejpam-4274	222	12	t0	t0	NOUN
ejpam-4274	222	13	if	if	SCONJ
ejpam-4274	222	14	and	and	CCONJ
ejpam-4274	222	15	only	only	ADV
ejpam-4274	222	16	if	if	SCONJ
ejpam-4274	222	17	for	for	SCONJ
ejpam-4274	222	18	each	each	DET
ejpam-4274	222	19	x	x	SYM
ejpam-4274	222	20	∈	∈	PROPN
ejpam-4274	222	21	x	x	NOUN
ejpam-4274	222	22	,	,	PUNCT
ejpam-4274	222	23	the	the	DET
ejpam-4274	222	24	singleton	singleton	NOUN
ejpam-4274	222	25	{	{	PUNCT
ejpam-4274	222	26	x	x	NOUN
ejpam-4274	222	27	}	}	PUNCT
ejpam-4274	222	28	is	be	AUX
ejpam-4274	222	29	(	(	PUNCT
ejpam-4274	222	30	λ	λ	X
ejpam-4274	222	31	,	,	PUNCT
ejpam-4274	222	32	p)-closed	p)-close	VERB
ejpam-4274	222	33	.	.	PUNCT
ejpam-4274	223	1	proof	proof	NOUN
ejpam-4274	223	2	.	.	PUNCT
ejpam-4274	224	1	suppose	suppose	VERB
ejpam-4274	224	2	that	that	SCONJ
ejpam-4274	224	3	(	(	PUNCT
ejpam-4274	224	4	x	x	X
ejpam-4274	224	5	,	,	PUNCT
ejpam-4274	224	6	τ	τ	X
ejpam-4274	224	7	)	)	PUNCT
ejpam-4274	224	8	is	be	AUX
ejpam-4274	224	9	pre	pre	ADJ
ejpam-4274	224	10	-	-	NOUN
ejpam-4274	224	11	t0	t0	NOUN
ejpam-4274	224	12	.	.	PUNCT
ejpam-4274	225	1	for	for	ADP
ejpam-4274	225	2	each	each	DET
ejpam-4274	225	3	x	x	SYM
ejpam-4274	225	4	∈	∈	PROPN
ejpam-4274	225	5	x	x	X
ejpam-4274	225	6	,	,	PUNCT
ejpam-4274	225	7	it	it	PRON
ejpam-4274	225	8	is	be	AUX
ejpam-4274	225	9	obvious	obvious	ADJ
ejpam-4274	225	10	that	that	SCONJ
ejpam-4274	225	11	{	{	PUNCT
ejpam-4274	225	12	x	x	NOUN
ejpam-4274	225	13	}	}	PUNCT
ejpam-4274	225	14	⊆	⊆	NUM
ejpam-4274	225	15	λp({x	λp({x	NOUN
ejpam-4274	225	16	}	}	PUNCT
ejpam-4274	225	17	)	)	PUNCT
ejpam-4274	225	18	∩	∩	ADJ
ejpam-4274	225	19	pcl({x	pcl({x	NOUN
ejpam-4274	225	20	}	}	PUNCT
ejpam-4274	225	21	)	)	PUNCT
ejpam-4274	225	22	.	.	PUNCT
ejpam-4274	226	1	if	if	SCONJ
ejpam-4274	226	2	y	y	PROPN
ejpam-4274	226	3	6=	6=	PROPN
ejpam-4274	226	4	x	x	PROPN
ejpam-4274	226	5	,	,	PUNCT
ejpam-4274	226	6	(	(	PUNCT
ejpam-4274	226	7	i	i	NOUN
ejpam-4274	226	8	)	)	PUNCT
ejpam-4274	226	9	there	there	PRON
ejpam-4274	226	10	exists	exist	VERB
ejpam-4274	226	11	a	a	DET
ejpam-4274	226	12	preopen	preopen	ADJ
ejpam-4274	226	13	set	set	NOUN
ejpam-4274	226	14	vx	vx	ADP
ejpam-4274	226	15	such	such	ADJ
ejpam-4274	226	16	that	that	SCONJ
ejpam-4274	226	17	y	y	PROPN
ejpam-4274	226	18	6∈	6∈	PROPN
ejpam-4274	226	19	vx	vx	PROPN
ejpam-4274	226	20	and	and	CCONJ
ejpam-4274	226	21	x	x	SYM
ejpam-4274	226	22	∈	∈	PROPN
ejpam-4274	226	23	vx	vx	PROPN
ejpam-4274	226	24	or	or	CCONJ
ejpam-4274	226	25	(	(	PUNCT
ejpam-4274	226	26	ii	ii	NOUN
ejpam-4274	226	27	)	)	PUNCT
ejpam-4274	226	28	there	there	PRON
ejpam-4274	226	29	exists	exist	VERB
ejpam-4274	226	30	a	a	DET
ejpam-4274	226	31	preopen	preopen	ADJ
ejpam-4274	226	32	set	set	NOUN
ejpam-4274	226	33	vy	vy	ADP
ejpam-4274	226	34	such	such	ADJ
ejpam-4274	226	35	that	that	SCONJ
ejpam-4274	226	36	x	x	SYM
ejpam-4274	226	37	6∈	6∈	NOUN
ejpam-4274	226	38	vy	vy	NOUN
ejpam-4274	226	39	and	and	CCONJ
ejpam-4274	226	40	y	y	PROPN
ejpam-4274	226	41	∈	∈	PROPN
ejpam-4274	226	42	vy	vy	NOUN
ejpam-4274	226	43	.	.	PUNCT
ejpam-4274	227	1	in	in	ADP
ejpam-4274	227	2	case	case	NOUN
ejpam-4274	227	3	of	of	ADP
ejpam-4274	227	4	(	(	PUNCT
ejpam-4274	227	5	i	i	PROPN
ejpam-4274	227	6	)	)	PUNCT
ejpam-4274	227	7	,	,	PUNCT
ejpam-4274	227	8	y	y	PROPN
ejpam-4274	227	9	6∈	6∈	PROPN
ejpam-4274	227	10	λp({x	λp({x	ADV
ejpam-4274	227	11	}	}	PUNCT
ejpam-4274	227	12	)	)	PUNCT
ejpam-4274	227	13	and	and	CCONJ
ejpam-4274	227	14	y	y	PROPN
ejpam-4274	227	15	6∈	6∈	PROPN
ejpam-4274	227	16	λp({x	λp({x	NOUN
ejpam-4274	227	17	}	}	PUNCT
ejpam-4274	227	18	)	)	PUNCT
ejpam-4274	227	19	∩	∩	ADJ
ejpam-4274	227	20	pcl({x	pcl({x	NOUN
ejpam-4274	227	21	}	}	PUNCT
ejpam-4274	227	22	)	)	PUNCT
ejpam-4274	227	23	.	.	PUNCT
ejpam-4274	228	1	thus	thus	ADV
ejpam-4274	228	2	,	,	PUNCT
ejpam-4274	228	3	{	{	PUNCT
ejpam-4274	228	4	x	x	NOUN
ejpam-4274	228	5	}	}	PUNCT
ejpam-4274	228	6	⊇	⊇	NOUN
ejpam-4274	228	7	λp({x	λp({x	NOUN
ejpam-4274	228	8	}	}	PUNCT
ejpam-4274	228	9	)	)	PUNCT
ejpam-4274	228	10	∩	∩	ADJ
ejpam-4274	228	11	pcl({x	pcl({x	NOUN
ejpam-4274	228	12	}	}	PUNCT
ejpam-4274	228	13	)	)	PUNCT
ejpam-4274	228	14	.	.	PUNCT
ejpam-4274	229	1	in	in	ADP
ejpam-4274	229	2	case	case	NOUN
ejpam-4274	229	3	(	(	PUNCT
ejpam-4274	229	4	ii	ii	NOUN
ejpam-4274	229	5	)	)	PUNCT
ejpam-4274	229	6	,	,	PUNCT
ejpam-4274	229	7	y	y	PROPN
ejpam-4274	229	8	6∈	6∈	PROPN
ejpam-4274	229	9	pcl({x	pcl({x	PROPN
ejpam-4274	229	10	}	}	PUNCT
ejpam-4274	229	11	)	)	PUNCT
ejpam-4274	229	12	and	and	CCONJ
ejpam-4274	229	13	y	y	PROPN
ejpam-4274	229	14	6∈	6∈	PROPN
ejpam-4274	229	15	λp({x	λp({x	NOUN
ejpam-4274	229	16	}	}	PUNCT
ejpam-4274	229	17	)	)	PUNCT
ejpam-4274	229	18	∩	∩	ADJ
ejpam-4274	229	19	pcl({x	pcl({x	NOUN
ejpam-4274	229	20	}	}	PUNCT
ejpam-4274	229	21	)	)	PUNCT
ejpam-4274	229	22	.	.	PUNCT
ejpam-4274	230	1	this	this	PRON
ejpam-4274	230	2	shows	show	VERB
ejpam-4274	230	3	that	that	SCONJ
ejpam-4274	230	4	{	{	PUNCT
ejpam-4274	230	5	x	x	NOUN
ejpam-4274	230	6	}	}	PUNCT
ejpam-4274	230	7	⊇	⊇	NOUN
ejpam-4274	230	8	λp({x	λp({x	NOUN
ejpam-4274	230	9	}	}	PUNCT
ejpam-4274	230	10	)	)	PUNCT
ejpam-4274	230	11	∩	∩	ADJ
ejpam-4274	230	12	pcl({x	pcl({x	NOUN
ejpam-4274	230	13	}	}	PUNCT
ejpam-4274	230	14	)	)	PUNCT
ejpam-4274	230	15	.	.	PUNCT
ejpam-4274	231	1	consequently	consequently	ADV
ejpam-4274	231	2	,	,	PUNCT
ejpam-4274	231	3	we	we	PRON
ejpam-4274	231	4	obtain	obtain	VERB
ejpam-4274	231	5	{	{	PUNCT
ejpam-4274	231	6	x	x	NOUN
ejpam-4274	231	7	}	}	PUNCT
ejpam-4274	231	8	=	=	PUNCT
ejpam-4274	231	9	λp({x	λp({x	ADJ
ejpam-4274	231	10	}	}	PUNCT
ejpam-4274	231	11	)	)	PUNCT
ejpam-4274	231	12	∩	∩	ADJ
ejpam-4274	231	13	pcl({x	pcl({x	NOUN
ejpam-4274	231	14	}	}	PUNCT
ejpam-4274	231	15	)	)	PUNCT
ejpam-4274	231	16	.	.	PUNCT
ejpam-4274	232	1	conversely	conversely	ADV
ejpam-4274	232	2	,	,	PUNCT
ejpam-4274	232	3	suppose	suppose	VERB
ejpam-4274	232	4	that	that	SCONJ
ejpam-4274	232	5	(	(	PUNCT
ejpam-4274	232	6	x	x	X
ejpam-4274	232	7	,	,	PUNCT
ejpam-4274	232	8	τ	τ	X
ejpam-4274	232	9	)	)	PUNCT
ejpam-4274	232	10	is	be	AUX
ejpam-4274	232	11	not	not	PART
ejpam-4274	232	12	pre	pre	ADJ
ejpam-4274	232	13	-	-	NOUN
ejpam-4274	232	14	t0	t0	NOUN
ejpam-4274	232	15	.	.	PUNCT
ejpam-4274	233	1	there	there	PRON
ejpam-4274	233	2	exist	exist	VERB
ejpam-4274	233	3	two	two	NUM
ejpam-4274	233	4	distinct	distinct	ADJ
ejpam-4274	233	5	points	point	NOUN
ejpam-4274	233	6	x	x	X
ejpam-4274	233	7	,	,	PUNCT
ejpam-4274	233	8	y	y	PROPN
ejpam-4274	233	9	such	such	ADJ
ejpam-4274	233	10	that	that	SCONJ
ejpam-4274	233	11	(	(	PUNCT
ejpam-4274	233	12	i	i	NOUN
ejpam-4274	233	13	)	)	PUNCT
ejpam-4274	233	14	y	y	PROPN
ejpam-4274	233	15	∈	∈	PROPN
ejpam-4274	233	16	vx	vx	PROPN
ejpam-4274	233	17	for	for	ADP
ejpam-4274	233	18	every	every	DET
ejpam-4274	233	19	preopen	preopen	ADJ
ejpam-4274	233	20	set	set	VERB
ejpam-4274	233	21	vx	vx	ADP
ejpam-4274	233	22	containing	contain	VERB
ejpam-4274	233	23	x	x	PROPN
ejpam-4274	233	24	and	and	CCONJ
ejpam-4274	233	25	(	(	PUNCT
ejpam-4274	233	26	ii	ii	NOUN
ejpam-4274	233	27	)	)	PUNCT
ejpam-4274	233	28	x	x	SYM
ejpam-4274	233	29	∈	∈	PROPN
ejpam-4274	233	30	vy	vy	NOUN
ejpam-4274	233	31	for	for	ADP
ejpam-4274	233	32	every	every	DET
ejpam-4274	233	33	preopen	preopen	ADJ
ejpam-4274	233	34	set	set	VERB
ejpam-4274	233	35	vy	vy	NOUN
ejpam-4274	233	36	containing	contain	VERB
ejpam-4274	233	37	y.	y.	PROPN
ejpam-4274	233	38	from	from	ADP
ejpam-4274	233	39	(	(	PUNCT
ejpam-4274	233	40	i	i	NOUN
ejpam-4274	233	41	)	)	PUNCT
ejpam-4274	233	42	and	and	CCONJ
ejpam-4274	233	43	(	(	PUNCT
ejpam-4274	233	44	ii	ii	NOUN
ejpam-4274	233	45	)	)	PUNCT
ejpam-4274	233	46	,	,	PUNCT
ejpam-4274	233	47	we	we	PRON
ejpam-4274	233	48	obtain	obtain	VERB
ejpam-4274	233	49	y	y	PROPN
ejpam-4274	233	50	∈	∈	PROPN
ejpam-4274	233	51	λp({x	λp({x	NOUN
ejpam-4274	233	52	}	}	PUNCT
ejpam-4274	233	53	)	)	PUNCT
ejpam-4274	233	54	and	and	CCONJ
ejpam-4274	233	55	y	y	PROPN
ejpam-4274	233	56	∈	∈	PROPN
ejpam-4274	233	57	pcl({x	pcl({x	PROPN
ejpam-4274	233	58	}	}	PUNCT
ejpam-4274	233	59	)	)	PUNCT
ejpam-4274	233	60	,	,	PUNCT
ejpam-4274	233	61	respectively	respectively	ADV
ejpam-4274	233	62	.	.	PUNCT
ejpam-4274	234	1	therefore	therefore	ADV
ejpam-4274	234	2	,	,	PUNCT
ejpam-4274	234	3	we	we	PRON
ejpam-4274	234	4	have	have	VERB
ejpam-4274	234	5	y	y	PROPN
ejpam-4274	234	6	∈	∈	PROPN
ejpam-4274	234	7	λp({x	λp({x	NOUN
ejpam-4274	234	8	}	}	PUNCT
ejpam-4274	234	9	)	)	PUNCT
ejpam-4274	234	10	∩	∩	ADJ
ejpam-4274	234	11	pcl({x	pcl({x	NOUN
ejpam-4274	234	12	}	}	PUNCT
ejpam-4274	234	13	)	)	PUNCT
ejpam-4274	234	14	.	.	PUNCT
ejpam-4274	235	1	by	by	ADP
ejpam-4274	235	2	theorem	theorem	NOUN
ejpam-4274	235	3	6	6	NUM
ejpam-4274	235	4	,	,	PUNCT
ejpam-4274	235	5	{	{	PUNCT
ejpam-4274	235	6	x	x	NOUN
ejpam-4274	235	7	}	}	PUNCT
ejpam-4274	235	8	=	=	PUNCT
ejpam-4274	235	9	λp({x	λp({x	ADJ
ejpam-4274	235	10	}	}	PUNCT
ejpam-4274	235	11	)	)	PUNCT
ejpam-4274	235	12	∩	∩	ADJ
ejpam-4274	235	13	pcl({x	pcl({x	NOUN
ejpam-4274	235	14	}	}	PUNCT
ejpam-4274	235	15	)	)	PUNCT
ejpam-4274	235	16	since	since	SCONJ
ejpam-4274	235	17	{	{	PUNCT
ejpam-4274	235	18	x	x	X
ejpam-4274	235	19	}	}	PUNCT
ejpam-4274	235	20	is	be	AUX
ejpam-4274	235	21	(	(	PUNCT
ejpam-4274	235	22	λ	λ	X
ejpam-4274	235	23	,	,	PUNCT
ejpam-4274	235	24	p)-closed	p)-close	VERB
ejpam-4274	235	25	.	.	PUNCT
ejpam-4274	236	1	this	this	PRON
ejpam-4274	236	2	is	be	AUX
ejpam-4274	236	3	contrary	contrary	ADJ
ejpam-4274	236	4	to	to	ADP
ejpam-4274	236	5	x	x	SYM
ejpam-4274	236	6	6=	6=	ADP
ejpam-4274	236	7	y.	y.	NOUN
ejpam-4274	236	8	definition	definition	NOUN
ejpam-4274	236	9	6	6	NUM
ejpam-4274	236	10	.	.	PUNCT
ejpam-4274	237	1	let	let	VERB
ejpam-4274	237	2	a	a	DET
ejpam-4274	237	3	be	be	AUX
ejpam-4274	237	4	a	a	DET
ejpam-4274	237	5	subset	subset	NOUN
ejpam-4274	237	6	of	of	ADP
ejpam-4274	237	7	a	a	DET
ejpam-4274	237	8	topological	topological	ADJ
ejpam-4274	237	9	space	space	NOUN
ejpam-4274	237	10	(	(	PUNCT
ejpam-4274	237	11	x	x	X
ejpam-4274	237	12	,	,	PUNCT
ejpam-4274	237	13	τ	τ	PROPN
ejpam-4274	237	14	)	)	PUNCT
ejpam-4274	237	15	.	.	PUNCT
ejpam-4274	238	1	a	a	DET
ejpam-4274	238	2	point	point	NOUN
ejpam-4274	238	3	x	x	X
ejpam-4274	238	4	∈	∈	NOUN
ejpam-4274	238	5	x	x	PUNCT
ejpam-4274	238	6	is	be	AUX
ejpam-4274	238	7	called	call	VERB
ejpam-4274	238	8	a	a	DET
ejpam-4274	238	9	(	(	PUNCT
ejpam-4274	238	10	λ	λ	NOUN
ejpam-4274	238	11	,	,	PUNCT
ejpam-4274	238	12	p)-cluster	p)-cluster	NOUN
ejpam-4274	238	13	point	point	NOUN
ejpam-4274	238	14	of	of	ADP
ejpam-4274	238	15	a	a	PRON
ejpam-4274	238	16	if	if	SCONJ
ejpam-4274	238	17	a	a	DET
ejpam-4274	238	18	∩	∩	ADJ
ejpam-4274	238	19	u	u	NOUN
ejpam-4274	238	20	6=	6=	NOUN
ejpam-4274	238	21	∅	∅	NOUN
ejpam-4274	238	22	for	for	ADP
ejpam-4274	238	23	every	every	DET
ejpam-4274	238	24	(	(	PUNCT
ejpam-4274	238	25	λ	λ	NOUN
ejpam-4274	238	26	,	,	PUNCT
ejpam-4274	238	27	p)-open	p)-open	VERB
ejpam-4274	238	28	set	set	VERB
ejpam-4274	238	29	u	u	NOUN
ejpam-4274	238	30	of	of	ADP
ejpam-4274	238	31	x	x	SYM
ejpam-4274	238	32	containing	contain	VERB
ejpam-4274	238	33	x.	x.	NOUN
ejpam-4274	238	34	the	the	DET
ejpam-4274	238	35	set	set	NOUN
ejpam-4274	238	36	of	of	ADP
ejpam-4274	238	37	all	all	DET
ejpam-4274	238	38	(	(	PUNCT
ejpam-4274	238	39	λ	λ	NOUN
ejpam-4274	238	40	,	,	PUNCT
ejpam-4274	238	41	p)-cluster	p)-cluster	VERB
ejpam-4274	238	42	points	point	NOUN
ejpam-4274	238	43	of	of	ADP
ejpam-4274	238	44	a	a	PRON
ejpam-4274	238	45	is	be	AUX
ejpam-4274	238	46	called	call	VERB
ejpam-4274	238	47	the	the	DET
ejpam-4274	238	48	(	(	PUNCT
ejpam-4274	238	49	λ	λ	PROPN
ejpam-4274	238	50	,	,	PUNCT
ejpam-4274	238	51	p)-closure	p)-closure	NOUN
ejpam-4274	238	52	of	of	ADP
ejpam-4274	238	53	a	a	PRON
ejpam-4274	238	54	and	and	CCONJ
ejpam-4274	238	55	is	be	AUX
ejpam-4274	238	56	denoted	denote	VERB
ejpam-4274	238	57	by	by	ADP
ejpam-4274	238	58	a(λ	a(λ	PROPN
ejpam-4274	238	59	,	,	PUNCT
ejpam-4274	238	60	p	p	NOUN
ejpam-4274	238	61	)	)	PUNCT
ejpam-4274	238	62	.	.	PUNCT
ejpam-4274	239	1	lemma	lemma	PROPN
ejpam-4274	239	2	5	5	NUM
ejpam-4274	239	3	.	.	PUNCT
ejpam-4274	240	1	for	for	ADP
ejpam-4274	240	2	subsets	subset	NOUN
ejpam-4274	240	3	a	a	DET
ejpam-4274	240	4	,	,	PUNCT
ejpam-4274	240	5	b	b	PROPN
ejpam-4274	240	6	of	of	ADP
ejpam-4274	240	7	a	a	DET
ejpam-4274	240	8	topological	topological	ADJ
ejpam-4274	240	9	space	space	NOUN
ejpam-4274	240	10	(	(	PUNCT
ejpam-4274	240	11	x	x	X
ejpam-4274	240	12	,	,	PUNCT
ejpam-4274	240	13	τ	τ	PROPN
ejpam-4274	240	14	)	)	PUNCT
ejpam-4274	240	15	,	,	PUNCT
ejpam-4274	240	16	the	the	DET
ejpam-4274	240	17	following	follow	VERB
ejpam-4274	240	18	properties	property	NOUN
ejpam-4274	240	19	hold	hold	VERB
ejpam-4274	240	20	:	:	PUNCT
ejpam-4274	240	21	(	(	PUNCT
ejpam-4274	240	22	1	1	X
ejpam-4274	240	23	)	)	PUNCT
ejpam-4274	240	24	a	a	DET
ejpam-4274	240	25	⊆	⊆	NUM
ejpam-4274	240	26	a(λ	a(λ	ADJ
ejpam-4274	240	27	,	,	PUNCT
ejpam-4274	240	28	p	p	NOUN
ejpam-4274	240	29	)	)	PUNCT
ejpam-4274	240	30	and	and	CCONJ
ejpam-4274	240	31	[	[	X
ejpam-4274	240	32	a(λ	a(λ	ADV
ejpam-4274	240	33	,	,	PUNCT
ejpam-4274	240	34	p)](λ	p)](λ	X
ejpam-4274	240	35	,	,	PUNCT
ejpam-4274	240	36	p	p	NOUN
ejpam-4274	240	37	)	)	PUNCT
ejpam-4274	240	38	=	=	PUNCT
ejpam-4274	241	1	a(λ	a(λ	PROPN
ejpam-4274	241	2	,	,	PUNCT
ejpam-4274	241	3	p	p	NOUN
ejpam-4274	241	4	)	)	PUNCT
ejpam-4274	241	5	.	.	PUNCT
ejpam-4274	242	1	(	(	PUNCT
ejpam-4274	242	2	2	2	X
ejpam-4274	242	3	)	)	PUNCT
ejpam-4274	242	4	if	if	SCONJ
ejpam-4274	242	5	a	a	DET
ejpam-4274	242	6	⊆	⊆	NUM
ejpam-4274	242	7	b	b	NOUN
ejpam-4274	242	8	,	,	PUNCT
ejpam-4274	242	9	then	then	ADV
ejpam-4274	242	10	a(λ	a(λ	ADV
ejpam-4274	242	11	,	,	PUNCT
ejpam-4274	242	12	p	p	NOUN
ejpam-4274	242	13	)	)	PUNCT
ejpam-4274	242	14	⊆	⊆	NUM
ejpam-4274	242	15	b(λ	b(λ	NOUN
ejpam-4274	242	16	,	,	PUNCT
ejpam-4274	242	17	p	p	NOUN
ejpam-4274	242	18	)	)	PUNCT
ejpam-4274	242	19	.	.	PUNCT
ejpam-4274	243	1	(	(	PUNCT
ejpam-4274	243	2	3	3	X
ejpam-4274	243	3	)	)	PUNCT
ejpam-4274	243	4	a(λ	a(λ	ADV
ejpam-4274	243	5	,	,	PUNCT
ejpam-4274	243	6	p	p	NOUN
ejpam-4274	243	7	)	)	PUNCT
ejpam-4274	243	8	=	=	SYM
ejpam-4274	243	9	∩{f	∩{f	NOUN
ejpam-4274	243	10	|a	|a	VERB
ejpam-4274	243	11	⊆	⊆	NUM
ejpam-4274	243	12	f	f	PROPN
ejpam-4274	243	13	and	and	CCONJ
ejpam-4274	243	14	f	f	PROPN
ejpam-4274	243	15	is	be	AUX
ejpam-4274	243	16	(	(	PUNCT
ejpam-4274	243	17	λ	λ	X
ejpam-4274	243	18	,	,	PUNCT
ejpam-4274	243	19	p)-closed	p)-close	VERB
ejpam-4274	243	20	}	}	PUNCT
ejpam-4274	243	21	.	.	PUNCT
ejpam-4274	244	1	(	(	PUNCT
ejpam-4274	244	2	4	4	NUM
ejpam-4274	244	3	)	)	PUNCT
ejpam-4274	244	4	a(λ	a(λ	ADV
ejpam-4274	244	5	,	,	PUNCT
ejpam-4274	244	6	p	p	NOUN
ejpam-4274	244	7	)	)	PUNCT
ejpam-4274	244	8	is	be	AUX
ejpam-4274	244	9	(	(	PUNCT
ejpam-4274	244	10	λ	λ	X
ejpam-4274	244	11	,	,	PUNCT
ejpam-4274	244	12	p)-closed	p)-close	VERB
ejpam-4274	244	13	.	.	PUNCT
ejpam-4274	245	1	(	(	PUNCT
ejpam-4274	245	2	5	5	X
ejpam-4274	245	3	)	)	PUNCT
ejpam-4274	245	4	a	a	PRON
ejpam-4274	245	5	is	be	AUX
ejpam-4274	245	6	(	(	PUNCT
ejpam-4274	245	7	λ	λ	X
ejpam-4274	245	8	,	,	PUNCT
ejpam-4274	245	9	p)-closed	p)-close	VERB
ejpam-4274	245	10	if	if	SCONJ
ejpam-4274	245	11	and	and	CCONJ
ejpam-4274	245	12	only	only	ADV
ejpam-4274	245	13	if	if	SCONJ
ejpam-4274	245	14	a	a	PRON
ejpam-4274	245	15	=	=	X
ejpam-4274	245	16	a(λ	a(λ	ADJ
ejpam-4274	245	17	,	,	PUNCT
ejpam-4274	245	18	p	p	NOUN
ejpam-4274	245	19	)	)	PUNCT
ejpam-4274	245	20	.	.	PUNCT
ejpam-4274	246	1	remark	remark	PROPN
ejpam-4274	246	2	1	1	NUM
ejpam-4274	246	3	.	.	PUNCT
ejpam-4274	247	1	every	every	DET
ejpam-4274	247	2	λp	λp	NOUN
ejpam-4274	247	3	-	-	PUNCT
ejpam-4274	247	4	set	set	NOUN
ejpam-4274	247	5	is	be	AUX
ejpam-4274	247	6	(	(	PUNCT
ejpam-4274	247	7	λ	λ	X
ejpam-4274	247	8	,	,	PUNCT
ejpam-4274	247	9	p)-closed	p)-close	VERB
ejpam-4274	247	10	.	.	PUNCT
ejpam-4274	248	1	the	the	DET
ejpam-4274	248	2	converse	converse	NOUN
ejpam-4274	248	3	of	of	ADP
ejpam-4274	248	4	remark	remark	NOUN
ejpam-4274	248	5	1	1	NUM
ejpam-4274	248	6	is	be	AUX
ejpam-4274	248	7	not	not	PART
ejpam-4274	248	8	true	true	ADJ
ejpam-4274	248	9	in	in	ADP
ejpam-4274	248	10	general	general	ADJ
ejpam-4274	248	11	as	as	SCONJ
ejpam-4274	248	12	shown	show	VERB
ejpam-4274	248	13	by	by	ADP
ejpam-4274	248	14	the	the	DET
ejpam-4274	248	15	following	follow	VERB
ejpam-4274	248	16	example	example	NOUN
ejpam-4274	248	17	.	.	PUNCT
ejpam-4274	249	1	example	example	NOUN
ejpam-4274	250	1	1	1	NUM
ejpam-4274	250	2	.	.	PUNCT
ejpam-4274	250	3	let	let	VERB
ejpam-4274	250	4	x	x	PUNCT
ejpam-4274	250	5	=	=	NOUN
ejpam-4274	250	6	{	{	PUNCT
ejpam-4274	250	7	−2,−1	−2,−1	ADV
ejpam-4274	250	8	}	}	PUNCT
ejpam-4274	250	9	with	with	ADP
ejpam-4274	250	10	a	a	DET
ejpam-4274	250	11	topology	topology	NOUN
ejpam-4274	250	12	τ	τ	X
ejpam-4274	250	13	=	=	SYM
ejpam-4274	250	14	{	{	PUNCT
ejpam-4274	250	15	∅	∅	NOUN
ejpam-4274	250	16	,	,	PUNCT
ejpam-4274	250	17	{	{	PUNCT
ejpam-4274	250	18	−2	−2	NOUN
ejpam-4274	250	19	}	}	PUNCT
ejpam-4274	250	20	,	,	PUNCT
ejpam-4274	250	21	x	x	NOUN
ejpam-4274	250	22	}	}	PUNCT
ejpam-4274	250	23	.	.	PUNCT
ejpam-4274	251	1	then	then	ADV
ejpam-4274	251	2	,	,	PUNCT
ejpam-4274	251	3	{	{	PUNCT
ejpam-4274	251	4	−1	−1	NOUN
ejpam-4274	251	5	}	}	PUNCT
ejpam-4274	251	6	is	be	AUX
ejpam-4274	251	7	a	a	DET
ejpam-4274	251	8	(	(	PUNCT
ejpam-4274	251	9	λ	λ	PROPN
ejpam-4274	251	10	,	,	PUNCT
ejpam-4274	251	11	p)-closed	p)-close	VERB
ejpam-4274	251	12	set	set	NOUN
ejpam-4274	251	13	,	,	PUNCT
ejpam-4274	251	14	but	but	CCONJ
ejpam-4274	251	15	{	{	PUNCT
ejpam-4274	251	16	−1	−1	NOUN
ejpam-4274	251	17	}	}	PUNCT
ejpam-4274	251	18	is	be	AUX
ejpam-4274	251	19	not	not	PART
ejpam-4274	251	20	a	a	DET
ejpam-4274	251	21	λp	λp	NOUN
ejpam-4274	251	22	-	-	PUNCT
ejpam-4274	251	23	set	set	NOUN
ejpam-4274	251	24	.	.	PUNCT
ejpam-4274	252	1	lemma	lemma	PROPN
ejpam-4274	252	2	6	6	NUM
ejpam-4274	252	3	.	.	PUNCT
ejpam-4274	253	1	for	for	ADP
ejpam-4274	253	2	a	a	DET
ejpam-4274	253	3	subset	subset	NOUN
ejpam-4274	253	4	a	a	PRON
ejpam-4274	253	5	of	of	ADP
ejpam-4274	253	6	a	a	DET
ejpam-4274	253	7	topological	topological	ADJ
ejpam-4274	253	8	space	space	NOUN
ejpam-4274	253	9	(	(	PUNCT
ejpam-4274	253	10	x	x	X
ejpam-4274	253	11	,	,	PUNCT
ejpam-4274	253	12	τ	τ	PROPN
ejpam-4274	253	13	)	)	PUNCT
ejpam-4274	253	14	,	,	PUNCT
ejpam-4274	253	15	the	the	DET
ejpam-4274	253	16	following	follow	VERB
ejpam-4274	253	17	properties	property	NOUN
ejpam-4274	253	18	hold	hold	VERB
ejpam-4274	253	19	:	:	PUNCT
ejpam-4274	253	20	(	(	PUNCT
ejpam-4274	253	21	1	1	X
ejpam-4274	253	22	)	)	PUNCT
ejpam-4274	253	23	if	if	SCONJ
ejpam-4274	253	24	a	a	PRON
ejpam-4274	253	25	is	be	AUX
ejpam-4274	253	26	preclosed	preclose	VERB
ejpam-4274	253	27	,	,	PUNCT
ejpam-4274	253	28	then	then	ADV
ejpam-4274	253	29	a	a	PRON
ejpam-4274	253	30	is	be	AUX
ejpam-4274	253	31	(	(	PUNCT
ejpam-4274	253	32	λ	λ	X
ejpam-4274	253	33	,	,	PUNCT
ejpam-4274	253	34	p)-closed	p)-close	VERB
ejpam-4274	253	35	.	.	PUNCT
ejpam-4274	254	1	(	(	PUNCT
ejpam-4274	254	2	2	2	X
ejpam-4274	254	3	)	)	PUNCT
ejpam-4274	254	4	a	a	PRON
ejpam-4274	254	5	is	be	AUX
ejpam-4274	254	6	(	(	PUNCT
ejpam-4274	254	7	λ	λ	X
ejpam-4274	254	8	,	,	PUNCT
ejpam-4274	254	9	p)-closed	p)-close	VERB
ejpam-4274	254	10	if	if	SCONJ
ejpam-4274	254	11	and	and	CCONJ
ejpam-4274	254	12	only	only	ADV
ejpam-4274	254	13	if	if	SCONJ
ejpam-4274	254	14	a	a	DET
ejpam-4274	254	15	=	=	NOUN
ejpam-4274	254	16	λp(a	λp(a	NOUN
ejpam-4274	254	17	)	)	PUNCT
ejpam-4274	254	18	∩a(λ	∩a(λ	NOUN
ejpam-4274	254	19	,	,	PUNCT
ejpam-4274	254	20	p	p	NOUN
ejpam-4274	254	21	)	)	PUNCT
ejpam-4274	254	22	.	.	PUNCT
ejpam-4274	255	1	c.	c.	PROPN
ejpam-4274	255	2	boonpok	boonpok	PROPN
ejpam-4274	255	3	,	,	PUNCT
ejpam-4274	255	4	c.	c.	PROPN
ejpam-4274	255	5	viriyapong	viriyapong	PROPN
ejpam-4274	255	6	/	/	SYM
ejpam-4274	255	7	eur	eur	PROPN
ejpam-4274	255	8	.	.	PUNCT
ejpam-4274	256	1	j.	j.	PROPN
ejpam-4274	256	2	pure	pure	PROPN
ejpam-4274	256	3	appl	appl	PROPN
ejpam-4274	256	4	.	.	PROPN
ejpam-4274	256	5	math	math	PROPN
ejpam-4274	256	6	,	,	PUNCT
ejpam-4274	256	7	15	15	NUM
ejpam-4274	256	8	(	(	PUNCT
ejpam-4274	256	9	2	2	NUM
ejpam-4274	256	10	)	)	PUNCT
ejpam-4274	256	11	(	(	PUNCT
ejpam-4274	256	12	2022	2022	NUM
ejpam-4274	256	13	)	)	PUNCT
ejpam-4274	256	14	,	,	PUNCT
ejpam-4274	256	15	415	415	NUM
ejpam-4274	256	16	-	-	SYM
ejpam-4274	256	17	436	436	NUM
ejpam-4274	256	18	422	422	NUM
ejpam-4274	256	19	proof	proof	NOUN
ejpam-4274	256	20	.	.	PUNCT
ejpam-4274	257	1	(	(	PUNCT
ejpam-4274	257	2	1	1	X
ejpam-4274	257	3	)	)	PUNCT
ejpam-4274	257	4	it	it	PRON
ejpam-4274	257	5	is	be	AUX
ejpam-4274	257	6	sufficient	sufficient	ADJ
ejpam-4274	257	7	to	to	PART
ejpam-4274	257	8	observe	observe	VERB
ejpam-4274	257	9	that	that	SCONJ
ejpam-4274	257	10	a	a	DET
ejpam-4274	257	11	=	=	SYM
ejpam-4274	257	12	x∩a	x∩a	PROPN
ejpam-4274	257	13	,	,	PUNCT
ejpam-4274	257	14	where	where	SCONJ
ejpam-4274	257	15	the	the	DET
ejpam-4274	257	16	whole	whole	ADJ
ejpam-4274	257	17	set	set	NOUN
ejpam-4274	257	18	x	x	PUNCT
ejpam-4274	257	19	is	be	AUX
ejpam-4274	257	20	a	a	DET
ejpam-4274	257	21	λp	λp	NOUN
ejpam-4274	257	22	-	-	PUNCT
ejpam-4274	257	23	set	set	NOUN
ejpam-4274	257	24	.	.	PUNCT
ejpam-4274	258	1	(	(	PUNCT
ejpam-4274	258	2	2	2	X
ejpam-4274	258	3	)	)	PUNCT
ejpam-4274	258	4	let	let	VERB
ejpam-4274	258	5	a	a	PRON
ejpam-4274	258	6	be	be	AUX
ejpam-4274	258	7	a	a	DET
ejpam-4274	258	8	(	(	PUNCT
ejpam-4274	258	9	λ	λ	PROPN
ejpam-4274	258	10	,	,	PUNCT
ejpam-4274	258	11	p)-closed	p)-close	VERB
ejpam-4274	258	12	set	set	NOUN
ejpam-4274	258	13	.	.	PUNCT
ejpam-4274	259	1	then	then	ADV
ejpam-4274	259	2	,	,	PUNCT
ejpam-4274	259	3	there	there	PRON
ejpam-4274	259	4	exist	exist	VERB
ejpam-4274	259	5	a	a	DET
ejpam-4274	259	6	λp	λp	ADV
ejpam-4274	259	7	-	-	PUNCT
ejpam-4274	259	8	set	set	VERB
ejpam-4274	259	9	t	t	NOUN
ejpam-4274	259	10	and	and	CCONJ
ejpam-4274	259	11	a	a	DET
ejpam-4274	259	12	preclosed	preclose	VERB
ejpam-4274	259	13	set	set	NOUN
ejpam-4274	259	14	c	c	PROPN
ejpam-4274	259	15	such	such	ADJ
ejpam-4274	259	16	that	that	SCONJ
ejpam-4274	259	17	a	a	DET
ejpam-4274	259	18	=	=	SYM
ejpam-4274	259	19	t	t	PROPN
ejpam-4274	259	20	∩	∩	ADJ
ejpam-4274	259	21	c.	c.	PROPN
ejpam-4274	259	22	since	since	SCONJ
ejpam-4274	259	23	a	a	DET
ejpam-4274	259	24	⊆	⊆	NUM
ejpam-4274	259	25	t	t	NOUN
ejpam-4274	259	26	,	,	PUNCT
ejpam-4274	259	27	we	we	PRON
ejpam-4274	259	28	have	have	VERB
ejpam-4274	259	29	a	a	DET
ejpam-4274	259	30	⊆	⊆	NUM
ejpam-4274	259	31	λp(a	λp(a	NUM
ejpam-4274	259	32	)	)	PUNCT
ejpam-4274	259	33	⊆	⊆	NUM
ejpam-4274	259	34	λp(t	λp(t	NOUN
ejpam-4274	259	35	)	)	PUNCT
ejpam-4274	259	36	=	=	SYM
ejpam-4274	259	37	t	t	PROPN
ejpam-4274	259	38	.	.	PUNCT
ejpam-4274	260	1	since	since	SCONJ
ejpam-4274	260	2	c	c	PROPN
ejpam-4274	260	3	is	be	AUX
ejpam-4274	260	4	preclosed	preclose	VERB
ejpam-4274	260	5	,	,	PUNCT
ejpam-4274	260	6	by	by	ADP
ejpam-4274	260	7	(	(	PUNCT
ejpam-4274	260	8	1	1	NUM
ejpam-4274	260	9	)	)	PUNCT
ejpam-4274	260	10	,	,	PUNCT
ejpam-4274	260	11	c	c	PROPN
ejpam-4274	260	12	is	be	AUX
ejpam-4274	260	13	(	(	PUNCT
ejpam-4274	260	14	λ	λ	X
ejpam-4274	260	15	,	,	PUNCT
ejpam-4274	260	16	p)-closed	p)-close	VERB
ejpam-4274	260	17	.	.	PUNCT
ejpam-4274	261	1	since	since	SCONJ
ejpam-4274	261	2	a	a	DET
ejpam-4274	261	3	⊆	⊆	NUM
ejpam-4274	261	4	c	c	NOUN
ejpam-4274	261	5	,	,	PUNCT
ejpam-4274	261	6	a	a	DET
ejpam-4274	261	7	⊆	⊆	NUM
ejpam-4274	261	8	a(λ	a(λ	ADJ
ejpam-4274	261	9	,	,	PUNCT
ejpam-4274	261	10	p	p	NOUN
ejpam-4274	261	11	)	)	PUNCT
ejpam-4274	261	12	⊆	⊆	NUM
ejpam-4274	261	13	c(λ	c(λ	PROPN
ejpam-4274	261	14	,	,	PUNCT
ejpam-4274	261	15	p	p	X
ejpam-4274	261	16	)	)	PUNCT
ejpam-4274	261	17	=	=	SYM
ejpam-4274	261	18	c	c	NOUN
ejpam-4274	261	19	and	and	CCONJ
ejpam-4274	261	20	hence	hence	ADV
ejpam-4274	261	21	a	a	DET
ejpam-4274	261	22	⊆	⊆	NUM
ejpam-4274	261	23	λp(a	λp(a	NUM
ejpam-4274	261	24	)	)	PUNCT
ejpam-4274	261	25	∩a(λ	∩a(λ	NOUN
ejpam-4274	261	26	,	,	PUNCT
ejpam-4274	261	27	p	p	NOUN
ejpam-4274	261	28	)	)	PUNCT
ejpam-4274	261	29	⊆	⊆	NUM
ejpam-4274	261	30	t	t	NOUN
ejpam-4274	261	31	∩	∩	NOUN
ejpam-4274	261	32	c	c	NOUN
ejpam-4274	261	33	=	=	PUNCT
ejpam-4274	261	34	a.	a.	NOUN
ejpam-4274	261	35	thus	thus	ADV
ejpam-4274	261	36	,	,	PUNCT
ejpam-4274	261	37	a	a	DET
ejpam-4274	261	38	=	=	NOUN
ejpam-4274	261	39	λp(a	λp(a	NOUN
ejpam-4274	261	40	)	)	PUNCT
ejpam-4274	261	41	∩a(λ	∩a(λ	NOUN
ejpam-4274	261	42	,	,	PUNCT
ejpam-4274	261	43	p	p	NOUN
ejpam-4274	261	44	)	)	PUNCT
ejpam-4274	261	45	.	.	PUNCT
ejpam-4274	262	1	conversely	conversely	ADV
ejpam-4274	262	2	,	,	PUNCT
ejpam-4274	262	3	let	let	VERB
ejpam-4274	262	4	a	a	DET
ejpam-4274	262	5	=	=	PUNCT
ejpam-4274	262	6	λp(a	λp(a	NOUN
ejpam-4274	262	7	)	)	PUNCT
ejpam-4274	262	8	∩	∩	NOUN
ejpam-4274	262	9	a(λ	a(λ	PROPN
ejpam-4274	262	10	,	,	PUNCT
ejpam-4274	262	11	p	p	NOUN
ejpam-4274	262	12	)	)	PUNCT
ejpam-4274	262	13	.	.	PUNCT
ejpam-4274	263	1	since	since	SCONJ
ejpam-4274	263	2	λp(a	λp(a	NUM
ejpam-4274	263	3	)	)	PUNCT
ejpam-4274	263	4	is	be	AUX
ejpam-4274	263	5	a	a	DET
ejpam-4274	263	6	λp	λp	NOUN
ejpam-4274	263	7	-	-	PUNCT
ejpam-4274	263	8	set	set	NOUN
ejpam-4274	263	9	,	,	PUNCT
ejpam-4274	263	10	by	by	ADP
ejpam-4274	263	11	remark	remark	NOUN
ejpam-4274	263	12	1	1	NUM
ejpam-4274	263	13	,	,	PUNCT
ejpam-4274	263	14	λp(a	λp(a	NUM
ejpam-4274	263	15	)	)	PUNCT
ejpam-4274	264	1	is	be	AUX
ejpam-4274	264	2	(	(	PUNCT
ejpam-4274	264	3	λ	λ	X
ejpam-4274	264	4	,	,	PUNCT
ejpam-4274	264	5	p)-closed	p)-close	VERB
ejpam-4274	264	6	.	.	PUNCT
ejpam-4274	265	1	since	since	SCONJ
ejpam-4274	265	2	a(λ	a(λ	PROPN
ejpam-4274	265	3	,	,	PUNCT
ejpam-4274	265	4	p	p	NOUN
ejpam-4274	265	5	)	)	PUNCT
ejpam-4274	265	6	is	be	AUX
ejpam-4274	265	7	(	(	PUNCT
ejpam-4274	265	8	λ	λ	X
ejpam-4274	265	9	,	,	PUNCT
ejpam-4274	265	10	p)-closed	p)-close	VERB
ejpam-4274	265	11	,	,	PUNCT
ejpam-4274	265	12	by	by	ADP
ejpam-4274	265	13	theroem	theroem	NOUN
ejpam-4274	265	14	7(1	7(1	NUM
ejpam-4274	265	15	)	)	PUNCT
ejpam-4274	265	16	,	,	PUNCT
ejpam-4274	265	17	λp(a)∩a(λ	λp(a)∩a(λ	NOUN
ejpam-4274	265	18	,	,	PUNCT
ejpam-4274	265	19	p	p	NOUN
ejpam-4274	265	20	)	)	PUNCT
ejpam-4274	265	21	is	be	AUX
ejpam-4274	265	22	(	(	PUNCT
ejpam-4274	265	23	λ	λ	X
ejpam-4274	265	24	,	,	PUNCT
ejpam-4274	265	25	p)-closed	p)-close	VERB
ejpam-4274	265	26	and	and	CCONJ
ejpam-4274	265	27	hence	hence	ADV
ejpam-4274	265	28	a	a	PRON
ejpam-4274	265	29	is	be	AUX
ejpam-4274	265	30	(	(	PUNCT
ejpam-4274	265	31	λ	λ	X
ejpam-4274	265	32	,	,	PUNCT
ejpam-4274	265	33	p)-closed	p)-close	VERB
ejpam-4274	265	34	.	.	PUNCT
ejpam-4274	266	1	the	the	DET
ejpam-4274	266	2	following	follow	VERB
ejpam-4274	266	3	example	example	NOUN
ejpam-4274	266	4	shows	show	VERB
ejpam-4274	266	5	that	that	SCONJ
ejpam-4274	266	6	the	the	DET
ejpam-4274	266	7	converse	converse	NOUN
ejpam-4274	266	8	of	of	ADP
ejpam-4274	266	9	lemma	lemma	PROPN
ejpam-4274	266	10	6(1	6(1	PROPN
ejpam-4274	266	11	)	)	PUNCT
ejpam-4274	266	12	is	be	AUX
ejpam-4274	266	13	not	not	PART
ejpam-4274	266	14	true	true	ADJ
ejpam-4274	266	15	in	in	ADP
ejpam-4274	266	16	general	general	ADJ
ejpam-4274	266	17	.	.	PUNCT
ejpam-4274	266	18	example	example	NOUN
ejpam-4274	267	1	2	2	NUM
ejpam-4274	267	2	.	.	PUNCT
ejpam-4274	267	3	let	let	VERB
ejpam-4274	267	4	x	x	PUNCT
ejpam-4274	267	5	=	=	NOUN
ejpam-4274	267	6	{	{	PUNCT
ejpam-4274	267	7	−2,−1	−2,−1	PROPN
ejpam-4274	267	8	,	,	PUNCT
ejpam-4274	267	9	0	0	NUM
ejpam-4274	267	10	,	,	PUNCT
ejpam-4274	267	11	1	1	NUM
ejpam-4274	267	12	,	,	PUNCT
ejpam-4274	267	13	2	2	NUM
ejpam-4274	267	14	}	}	PUNCT
ejpam-4274	267	15	with	with	ADP
ejpam-4274	267	16	a	a	DET
ejpam-4274	267	17	topology	topology	NOUN
ejpam-4274	267	18	τ	τ	X
ejpam-4274	267	19	=	=	SYM
ejpam-4274	267	20	{	{	PUNCT
ejpam-4274	267	21	∅	∅	NOUN
ejpam-4274	267	22	,	,	PUNCT
ejpam-4274	267	23	{	{	PUNCT
ejpam-4274	267	24	−2	−2	NOUN
ejpam-4274	267	25	}	}	PUNCT
ejpam-4274	267	26	,	,	PUNCT
ejpam-4274	267	27	{	{	PUNCT
ejpam-4274	267	28	2	2	NUM
ejpam-4274	267	29	}	}	PUNCT
ejpam-4274	267	30	,	,	PUNCT
ejpam-4274	267	31	{	{	PUNCT
ejpam-4274	267	32	−2	−2	NOUN
ejpam-4274	267	33	,	,	PUNCT
ejpam-4274	267	34	2	2	NUM
ejpam-4274	267	35	}	}	PUNCT
ejpam-4274	267	36	,	,	PUNCT
ejpam-4274	267	37	x	x	NOUN
ejpam-4274	267	38	}	}	PUNCT
ejpam-4274	267	39	.	.	PUNCT
ejpam-4274	268	1	then	then	ADV
ejpam-4274	268	2	,	,	PUNCT
ejpam-4274	268	3	{	{	PUNCT
ejpam-4274	268	4	−2	−2	NOUN
ejpam-4274	268	5	,	,	PUNCT
ejpam-4274	268	6	2	2	NUM
ejpam-4274	268	7	}	}	PUNCT
ejpam-4274	268	8	is	be	AUX
ejpam-4274	268	9	(	(	PUNCT
ejpam-4274	268	10	λ	λ	X
ejpam-4274	268	11	,	,	PUNCT
ejpam-4274	268	12	p)-closed	p)-close	VERB
ejpam-4274	268	13	,	,	PUNCT
ejpam-4274	268	14	but	but	CCONJ
ejpam-4274	268	15	{	{	PUNCT
ejpam-4274	268	16	−2	−2	NOUN
ejpam-4274	268	17	,	,	PUNCT
ejpam-4274	268	18	2	2	NUM
ejpam-4274	268	19	}	}	PUNCT
ejpam-4274	268	20	is	be	AUX
ejpam-4274	268	21	not	not	PART
ejpam-4274	268	22	preclosed	preclose	VERB
ejpam-4274	268	23	.	.	PUNCT
ejpam-4274	269	1	definition	definition	NOUN
ejpam-4274	269	2	7	7	NUM
ejpam-4274	269	3	.	.	PUNCT
ejpam-4274	270	1	let	let	VERB
ejpam-4274	270	2	a	a	DET
ejpam-4274	270	3	be	be	AUX
ejpam-4274	270	4	a	a	DET
ejpam-4274	270	5	subset	subset	NOUN
ejpam-4274	270	6	of	of	ADP
ejpam-4274	270	7	a	a	DET
ejpam-4274	270	8	topological	topological	ADJ
ejpam-4274	270	9	space	space	NOUN
ejpam-4274	270	10	(	(	PUNCT
ejpam-4274	270	11	x	x	X
ejpam-4274	270	12	,	,	PUNCT
ejpam-4274	270	13	τ	τ	PROPN
ejpam-4274	270	14	)	)	PUNCT
ejpam-4274	270	15	.	.	PUNCT
ejpam-4274	271	1	a	a	DET
ejpam-4274	271	2	subset	subset	NOUN
ejpam-4274	271	3	λ(λ	λ(λ	ADP
ejpam-4274	271	4	,	,	PUNCT
ejpam-4274	271	5	p)(a	p)(a	PROPN
ejpam-4274	271	6	)	)	PUNCT
ejpam-4274	271	7	is	be	AUX
ejpam-4274	271	8	defined	define	VERB
ejpam-4274	271	9	as	as	SCONJ
ejpam-4274	271	10	follows	follow	VERB
ejpam-4274	271	11	:	:	PUNCT
ejpam-4274	272	1	λ(λ	λ(λ	ADV
ejpam-4274	272	2	,	,	PUNCT
ejpam-4274	272	3	p)(a	p)(a	NOUN
ejpam-4274	272	4	)	)	PUNCT
ejpam-4274	272	5	=	=	PUNCT
ejpam-4274	273	1	∩{u	∩{u	PROPN
ejpam-4274	273	2	∈	∈	PROPN
ejpam-4274	273	3	λpo(x	λpo(x	PROPN
ejpam-4274	273	4	,	,	PUNCT
ejpam-4274	273	5	τ	τ	X
ejpam-4274	273	6	)	)	PUNCT
ejpam-4274	273	7	|	|	ADV
ejpam-4274	273	8	a	a	DET
ejpam-4274	273	9	⊆	⊆	NUM
ejpam-4274	273	10	u	u	NOUN
ejpam-4274	273	11	}	}	PUNCT
ejpam-4274	273	12	.	.	PUNCT
ejpam-4274	274	1	lemma	lemma	PROPN
ejpam-4274	274	2	7	7	NUM
ejpam-4274	274	3	.	.	X
ejpam-4274	274	4	for	for	ADP
ejpam-4274	274	5	subsets	subset	NOUN
ejpam-4274	274	6	a	a	DET
ejpam-4274	274	7	,	,	PUNCT
ejpam-4274	274	8	b	b	PROPN
ejpam-4274	274	9	of	of	ADP
ejpam-4274	274	10	a	a	DET
ejpam-4274	274	11	topological	topological	ADJ
ejpam-4274	274	12	space	space	NOUN
ejpam-4274	274	13	(	(	PUNCT
ejpam-4274	274	14	x	x	X
ejpam-4274	274	15	,	,	PUNCT
ejpam-4274	274	16	τ	τ	PROPN
ejpam-4274	274	17	)	)	PUNCT
ejpam-4274	274	18	,	,	PUNCT
ejpam-4274	274	19	the	the	DET
ejpam-4274	274	20	following	follow	VERB
ejpam-4274	274	21	properties	property	NOUN
ejpam-4274	274	22	hold	hold	VERB
ejpam-4274	274	23	:	:	PUNCT
ejpam-4274	274	24	(	(	PUNCT
ejpam-4274	274	25	1	1	X
ejpam-4274	274	26	)	)	PUNCT
ejpam-4274	274	27	a	a	DET
ejpam-4274	274	28	⊆	⊆	NUM
ejpam-4274	274	29	λ(λ	λ(λ	NOUN
ejpam-4274	274	30	,	,	PUNCT
ejpam-4274	274	31	p)(a	p)(a	NOUN
ejpam-4274	274	32	)	)	PUNCT
ejpam-4274	274	33	.	.	PUNCT
ejpam-4274	275	1	(	(	PUNCT
ejpam-4274	275	2	2	2	X
ejpam-4274	275	3	)	)	PUNCT
ejpam-4274	275	4	if	if	SCONJ
ejpam-4274	275	5	a	a	DET
ejpam-4274	275	6	⊆	⊆	NUM
ejpam-4274	275	7	b	b	NOUN
ejpam-4274	275	8	,	,	PUNCT
ejpam-4274	275	9	then	then	ADV
ejpam-4274	275	10	λ(λ	λ(λ	PROPN
ejpam-4274	275	11	,	,	PUNCT
ejpam-4274	275	12	p)(a	p)(a	NOUN
ejpam-4274	275	13	)	)	PUNCT
ejpam-4274	275	14	⊆	⊆	NUM
ejpam-4274	275	15	λ(λ	λ(λ	ADP
ejpam-4274	275	16	,	,	PUNCT
ejpam-4274	275	17	p)(b	p)(b	ADJ
ejpam-4274	275	18	)	)	PUNCT
ejpam-4274	275	19	.	.	PUNCT
ejpam-4274	276	1	(	(	PUNCT
ejpam-4274	276	2	3	3	X
ejpam-4274	276	3	)	)	PUNCT
ejpam-4274	276	4	λ(λ	λ(λ	ADV
ejpam-4274	276	5	,	,	PUNCT
ejpam-4274	276	6	p)[λ(λ	p)[λ(λ	NOUN
ejpam-4274	276	7	,	,	PUNCT
ejpam-4274	276	8	p)(a	p)(a	NOUN
ejpam-4274	276	9	)	)	PUNCT
ejpam-4274	276	10	]	]	PUNCT
ejpam-4274	277	1	=	=	PUNCT
ejpam-4274	277	2	λ(λ	λ(λ	PROPN
ejpam-4274	277	3	,	,	PUNCT
ejpam-4274	277	4	p)(a	p)(a	PROPN
ejpam-4274	277	5	)	)	PUNCT
ejpam-4274	277	6	;	;	PUNCT
ejpam-4274	277	7	(	(	PUNCT
ejpam-4274	277	8	4	4	X
ejpam-4274	277	9	)	)	PUNCT
ejpam-4274	277	10	if	if	SCONJ
ejpam-4274	277	11	a	a	PRON
ejpam-4274	277	12	is	be	AUX
ejpam-4274	277	13	(	(	PUNCT
ejpam-4274	277	14	λ	λ	NOUN
ejpam-4274	277	15	,	,	PUNCT
ejpam-4274	277	16	p)-open	p)-open	ADJ
ejpam-4274	277	17	,	,	PUNCT
ejpam-4274	277	18	then	then	ADV
ejpam-4274	277	19	λ(λ	λ(λ	PROPN
ejpam-4274	277	20	,	,	PUNCT
ejpam-4274	277	21	p)(a	p)(a	NOUN
ejpam-4274	277	22	)	)	PUNCT
ejpam-4274	277	23	=	=	PUNCT
ejpam-4274	277	24	a.	a.	NOUN
ejpam-4274	277	25	lemma	lemma	PROPN
ejpam-4274	277	26	8	8	X
ejpam-4274	277	27	.	.	PUNCT
ejpam-4274	278	1	let	let	VERB
ejpam-4274	278	2	(	(	PUNCT
ejpam-4274	278	3	x	x	NOUN
ejpam-4274	278	4	,	,	PUNCT
ejpam-4274	278	5	τ	τ	X
ejpam-4274	278	6	)	)	PUNCT
ejpam-4274	278	7	be	be	VERB
ejpam-4274	278	8	a	a	DET
ejpam-4274	278	9	topological	topological	ADJ
ejpam-4274	278	10	space	space	NOUN
ejpam-4274	278	11	and	and	CCONJ
ejpam-4274	278	12	let	let	VERB
ejpam-4274	278	13	x	x	PRON
ejpam-4274	278	14	,	,	PUNCT
ejpam-4274	278	15	y	y	PROPN
ejpam-4274	278	16	∈	∈	PROPN
ejpam-4274	278	17	x.	x.	NOUN
ejpam-4274	279	1	then	then	ADV
ejpam-4274	279	2	,	,	PUNCT
ejpam-4274	279	3	y	y	PROPN
ejpam-4274	279	4	∈	∈	PROPN
ejpam-4274	279	5	λ(λ	λ(λ	PROPN
ejpam-4274	279	6	,	,	PUNCT
ejpam-4274	279	7	p)({x	p)({x	NOUN
ejpam-4274	279	8	}	}	PUNCT
ejpam-4274	279	9	)	)	PUNCT
ejpam-4274	279	10	if	if	SCONJ
ejpam-4274	279	11	and	and	CCONJ
ejpam-4274	279	12	only	only	ADV
ejpam-4274	279	13	if	if	SCONJ
ejpam-4274	279	14	x	x	SYM
ejpam-4274	279	15	∈	∈	PROPN
ejpam-4274	279	16	{	{	PUNCT
ejpam-4274	279	17	y}(λ	y}(λ	PROPN
ejpam-4274	279	18	,	,	PUNCT
ejpam-4274	279	19	p	p	NOUN
ejpam-4274	279	20	)	)	PUNCT
ejpam-4274	279	21	.	.	PUNCT
ejpam-4274	280	1	proof	proof	NOUN
ejpam-4274	280	2	.	.	PUNCT
ejpam-4274	281	1	let	let	VERB
ejpam-4274	281	2	y	y	PROPN
ejpam-4274	281	3	6∈	6∈	PROPN
ejpam-4274	281	4	λ(λ	λ(λ	ADV
ejpam-4274	281	5	,	,	PUNCT
ejpam-4274	281	6	p)({x	p)({x	NOUN
ejpam-4274	281	7	}	}	PUNCT
ejpam-4274	281	8	)	)	PUNCT
ejpam-4274	281	9	.	.	PUNCT
ejpam-4274	282	1	then	then	ADV
ejpam-4274	282	2	,	,	PUNCT
ejpam-4274	282	3	there	there	PRON
ejpam-4274	282	4	exists	exist	VERB
ejpam-4274	282	5	a	a	DET
ejpam-4274	282	6	(	(	PUNCT
ejpam-4274	282	7	λ	λ	NOUN
ejpam-4274	282	8	,	,	PUNCT
ejpam-4274	282	9	p)-open	p)-open	VERB
ejpam-4274	282	10	set	set	VERB
ejpam-4274	282	11	v	v	NOUN
ejpam-4274	282	12	containing	contain	VERB
ejpam-4274	282	13	x	x	PUNCT
ejpam-4274	282	14	such	such	ADJ
ejpam-4274	282	15	that	that	SCONJ
ejpam-4274	282	16	y	y	PROPN
ejpam-4274	282	17	6∈	6∈	PROPN
ejpam-4274	282	18	v	v	NOUN
ejpam-4274	282	19	.	.	PUNCT
ejpam-4274	283	1	hence	hence	ADV
ejpam-4274	283	2	,	,	PUNCT
ejpam-4274	283	3	x	x	PROPN
ejpam-4274	283	4	6∈	6∈	PROPN
ejpam-4274	283	5	{	{	PUNCT
ejpam-4274	283	6	y}(λ	y}(λ	PROPN
ejpam-4274	283	7	,	,	PUNCT
ejpam-4274	283	8	p	p	NOUN
ejpam-4274	283	9	)	)	PUNCT
ejpam-4274	283	10	.	.	PUNCT
ejpam-4274	284	1	the	the	DET
ejpam-4274	284	2	converse	converse	NOUN
ejpam-4274	284	3	is	be	AUX
ejpam-4274	284	4	similarly	similarly	ADV
ejpam-4274	284	5	shown	show	VERB
ejpam-4274	284	6	.	.	PUNCT
ejpam-4274	285	1	a	a	DET
ejpam-4274	285	2	subset	subset	NOUN
ejpam-4274	285	3	nx	nx	NOUN
ejpam-4274	285	4	of	of	ADP
ejpam-4274	285	5	a	a	DET
ejpam-4274	285	6	topological	topological	ADJ
ejpam-4274	285	7	space	space	NOUN
ejpam-4274	285	8	(	(	PUNCT
ejpam-4274	285	9	x	x	X
ejpam-4274	285	10	,	,	PUNCT
ejpam-4274	285	11	τ	τ	X
ejpam-4274	285	12	)	)	PUNCT
ejpam-4274	285	13	is	be	AUX
ejpam-4274	285	14	said	say	VERB
ejpam-4274	285	15	to	to	PART
ejpam-4274	285	16	be	be	AUX
ejpam-4274	285	17	(	(	PUNCT
ejpam-4274	285	18	λ	λ	X
ejpam-4274	285	19	,	,	PUNCT
ejpam-4274	285	20	p)-neighbourhood	p)-neighbourhood	NOUN
ejpam-4274	285	21	of	of	ADP
ejpam-4274	285	22	a	a	DET
ejpam-4274	285	23	point	point	NOUN
ejpam-4274	285	24	x	x	X
ejpam-4274	285	25	∈	∈	NOUN
ejpam-4274	285	26	x	x	INTJ
ejpam-4274	285	27	if	if	SCONJ
ejpam-4274	285	28	there	there	PRON
ejpam-4274	285	29	exists	exist	VERB
ejpam-4274	285	30	a	a	DET
ejpam-4274	285	31	(	(	PUNCT
ejpam-4274	285	32	λ	λ	NOUN
ejpam-4274	285	33	,	,	PUNCT
ejpam-4274	285	34	p)-open	p)-open	VERB
ejpam-4274	285	35	set	set	VERB
ejpam-4274	285	36	u	u	PRON
ejpam-4274	285	37	such	such	ADJ
ejpam-4274	285	38	that	that	SCONJ
ejpam-4274	285	39	x	x	SYM
ejpam-4274	285	40	∈	∈	PROPN
ejpam-4274	285	41	u	u	NOUN
ejpam-4274	285	42	⊆	⊆	NUM
ejpam-4274	285	43	nx	nx	X
ejpam-4274	285	44	.	.	PUNCT
ejpam-4274	286	1	lemma	lemma	PROPN
ejpam-4274	286	2	9	9	NUM
ejpam-4274	286	3	.	.	PUNCT
ejpam-4274	287	1	a	a	DET
ejpam-4274	287	2	subset	subset	NOUN
ejpam-4274	287	3	a	a	PRON
ejpam-4274	287	4	of	of	ADP
ejpam-4274	287	5	a	a	DET
ejpam-4274	287	6	topological	topological	ADJ
ejpam-4274	287	7	space	space	NOUN
ejpam-4274	287	8	(	(	PUNCT
ejpam-4274	287	9	x	x	X
ejpam-4274	287	10	,	,	PUNCT
ejpam-4274	287	11	τ	τ	X
ejpam-4274	287	12	)	)	PUNCT
ejpam-4274	287	13	is	be	AUX
ejpam-4274	287	14	(	(	PUNCT
ejpam-4274	287	15	λ	λ	X
ejpam-4274	287	16	,	,	PUNCT
ejpam-4274	287	17	p)-open	p)-open	VERB
ejpam-4274	287	18	if	if	SCONJ
ejpam-4274	287	19	and	and	CCONJ
ejpam-4274	287	20	only	only	ADV
ejpam-4274	287	21	if	if	SCONJ
ejpam-4274	287	22	a	a	PRON
ejpam-4274	287	23	is	be	AUX
ejpam-4274	287	24	(	(	PUNCT
ejpam-4274	287	25	λ	λ	PROPN
ejpam-4274	287	26	,	,	PUNCT
ejpam-4274	287	27	p)-neighbourhood	p)-neighbourhood	NOUN
ejpam-4274	287	28	of	of	ADP
ejpam-4274	287	29	each	each	DET
ejpam-4274	287	30	x	x	SYM
ejpam-4274	287	31	∈	∈	PROPN
ejpam-4274	287	32	a.	a.	NOUN
ejpam-4274	287	33	definition	definition	NOUN
ejpam-4274	287	34	8	8	NUM
ejpam-4274	287	35	.	.	PUNCT
ejpam-4274	288	1	let	let	VERB
ejpam-4274	288	2	a	a	DET
ejpam-4274	288	3	be	be	AUX
ejpam-4274	288	4	a	a	DET
ejpam-4274	288	5	subset	subset	NOUN
ejpam-4274	288	6	of	of	ADP
ejpam-4274	288	7	a	a	DET
ejpam-4274	288	8	topological	topological	ADJ
ejpam-4274	288	9	space	space	NOUN
ejpam-4274	288	10	(	(	PUNCT
ejpam-4274	288	11	x	x	X
ejpam-4274	288	12	,	,	PUNCT
ejpam-4274	288	13	τ	τ	PROPN
ejpam-4274	288	14	)	)	PUNCT
ejpam-4274	288	15	.	.	PUNCT
ejpam-4274	289	1	a	a	DET
ejpam-4274	289	2	subset	subset	NOUN
ejpam-4274	289	3	〈	〈	PROPN
ejpam-4274	289	4	x〉p	x〉p	PROPN
ejpam-4274	289	5	is	be	AUX
ejpam-4274	289	6	defined	define	VERB
ejpam-4274	289	7	as	as	SCONJ
ejpam-4274	289	8	follows	follow	VERB
ejpam-4274	289	9	:	:	PUNCT
ejpam-4274	289	10	〈	〈	PROPN
ejpam-4274	289	11	x〉p	x〉p	NOUN
ejpam-4274	289	12	=	=	SYM
ejpam-4274	289	13	λ(λ	λ(λ	PROPN
ejpam-4274	289	14	,	,	PUNCT
ejpam-4274	289	15	p)({x	p)({x	NOUN
ejpam-4274	289	16	}	}	PUNCT
ejpam-4274	289	17	)	)	PUNCT
ejpam-4274	289	18	∩	∩	NOUN
ejpam-4274	289	19	{	{	PUNCT
ejpam-4274	289	20	x}(λ	x}(λ	PROPN
ejpam-4274	289	21	,	,	PUNCT
ejpam-4274	289	22	p	p	NOUN
ejpam-4274	289	23	)	)	PUNCT
ejpam-4274	289	24	.	.	PUNCT
ejpam-4274	290	1	theorem	theorem	ADJ
ejpam-4274	290	2	10	10	NUM
ejpam-4274	290	3	.	.	PUNCT
ejpam-4274	291	1	for	for	ADP
ejpam-4274	291	2	a	a	DET
ejpam-4274	291	3	topological	topological	ADJ
ejpam-4274	291	4	space	space	NOUN
ejpam-4274	291	5	(	(	PUNCT
ejpam-4274	291	6	x	x	X
ejpam-4274	291	7	,	,	PUNCT
ejpam-4274	291	8	τ	τ	PROPN
ejpam-4274	291	9	)	)	PUNCT
ejpam-4274	291	10	,	,	PUNCT
ejpam-4274	291	11	the	the	DET
ejpam-4274	291	12	following	follow	VERB
ejpam-4274	291	13	properties	property	NOUN
ejpam-4274	291	14	hold	hold	VERB
ejpam-4274	291	15	:	:	PUNCT
ejpam-4274	291	16	(	(	PUNCT
ejpam-4274	291	17	1	1	X
ejpam-4274	291	18	)	)	PUNCT
ejpam-4274	291	19	λ(λ	λ(λ	ADV
ejpam-4274	291	20	,	,	PUNCT
ejpam-4274	291	21	p)(a	p)(a	NOUN
ejpam-4274	291	22	)	)	PUNCT
ejpam-4274	292	1	=	=	PRON
ejpam-4274	292	2	{	{	PUNCT
ejpam-4274	292	3	x	x	PUNCT
ejpam-4274	292	4	∈	∈	NOUN
ejpam-4274	292	5	x	x	PUNCT
ejpam-4274	292	6	|	|	ADV
ejpam-4274	292	7	a	a	DET
ejpam-4274	292	8	∩	∩	NOUN
ejpam-4274	292	9	{	{	PUNCT
ejpam-4274	292	10	x}(λ	x}(λ	PROPN
ejpam-4274	292	11	,	,	PUNCT
ejpam-4274	292	12	p	p	NOUN
ejpam-4274	292	13	)	)	PUNCT
ejpam-4274	292	14	6=	6=	ADP
ejpam-4274	292	15	∅	∅	NOUN
ejpam-4274	292	16	}	}	PUNCT
ejpam-4274	292	17	for	for	ADP
ejpam-4274	292	18	each	each	PRON
ejpam-4274	292	19	subset	subset	VERB
ejpam-4274	292	20	a	a	PRON
ejpam-4274	292	21	of	of	ADP
ejpam-4274	292	22	x.	x.	NOUN
ejpam-4274	292	23	(	(	PUNCT
ejpam-4274	292	24	2	2	NUM
ejpam-4274	292	25	)	)	PUNCT
ejpam-4274	292	26	for	for	ADP
ejpam-4274	292	27	each	each	DET
ejpam-4274	292	28	x	x	SYM
ejpam-4274	292	29	∈	∈	PROPN
ejpam-4274	292	30	x	x	SYM
ejpam-4274	292	31	,	,	PUNCT
ejpam-4274	292	32	λ(λ	λ(λ	PROPN
ejpam-4274	292	33	,	,	PUNCT
ejpam-4274	292	34	p)(〈x〉p	p)(〈x〉p	NOUN
ejpam-4274	292	35	)	)	PUNCT
ejpam-4274	292	36	=	=	SYM
ejpam-4274	293	1	λ(λ	λ(λ	PROPN
ejpam-4274	293	2	,	,	PUNCT
ejpam-4274	293	3	p)({x	p)({x	NOUN
ejpam-4274	293	4	}	}	PUNCT
ejpam-4274	293	5	)	)	PUNCT
ejpam-4274	293	6	.	.	PUNCT
ejpam-4274	294	1	c.	c.	PROPN
ejpam-4274	294	2	boonpok	boonpok	PROPN
ejpam-4274	294	3	,	,	PUNCT
ejpam-4274	294	4	c.	c.	PROPN
ejpam-4274	294	5	viriyapong	viriyapong	PROPN
ejpam-4274	294	6	/	/	SYM
ejpam-4274	294	7	eur	eur	PROPN
ejpam-4274	294	8	.	.	PUNCT
ejpam-4274	295	1	j.	j.	PROPN
ejpam-4274	295	2	pure	pure	PROPN
ejpam-4274	295	3	appl	appl	PROPN
ejpam-4274	295	4	.	.	PROPN
ejpam-4274	295	5	math	math	PROPN
ejpam-4274	295	6	,	,	PUNCT
ejpam-4274	295	7	15	15	NUM
ejpam-4274	295	8	(	(	PUNCT
ejpam-4274	295	9	2	2	NUM
ejpam-4274	295	10	)	)	PUNCT
ejpam-4274	295	11	(	(	PUNCT
ejpam-4274	295	12	2022	2022	NUM
ejpam-4274	295	13	)	)	PUNCT
ejpam-4274	295	14	,	,	PUNCT
ejpam-4274	295	15	415	415	NUM
ejpam-4274	295	16	-	-	SYM
ejpam-4274	295	17	436	436	NUM
ejpam-4274	295	18	423	423	NUM
ejpam-4274	295	19	(	(	PUNCT
ejpam-4274	295	20	3	3	NUM
ejpam-4274	295	21	)	)	PUNCT
ejpam-4274	295	22	for	for	ADP
ejpam-4274	295	23	each	each	DET
ejpam-4274	295	24	x	x	SYM
ejpam-4274	295	25	∈	∈	PROPN
ejpam-4274	295	26	x	x	X
ejpam-4274	295	27	,	,	PUNCT
ejpam-4274	295	28	(	(	PUNCT
ejpam-4274	295	29	〈	〈	PROPN
ejpam-4274	295	30	x〉p)(λ	x〉p)(λ	PRON
ejpam-4274	295	31	,	,	PUNCT
ejpam-4274	295	32	p	p	NOUN
ejpam-4274	295	33	)	)	PUNCT
ejpam-4274	295	34	=	=	SYM
ejpam-4274	295	35	{	{	PUNCT
ejpam-4274	295	36	x}(λ	x}(λ	PROPN
ejpam-4274	295	37	,	,	PUNCT
ejpam-4274	295	38	p	p	NOUN
ejpam-4274	295	39	)	)	PUNCT
ejpam-4274	295	40	.	.	PUNCT
ejpam-4274	296	1	(	(	PUNCT
ejpam-4274	296	2	4	4	X
ejpam-4274	296	3	)	)	PUNCT
ejpam-4274	296	4	if	if	SCONJ
ejpam-4274	296	5	u	u	PRON
ejpam-4274	296	6	is	be	AUX
ejpam-4274	296	7	(	(	PUNCT
ejpam-4274	296	8	λ	λ	X
ejpam-4274	296	9	,	,	PUNCT
ejpam-4274	296	10	p)-open	p)-open	VERB
ejpam-4274	296	11	and	and	CCONJ
ejpam-4274	296	12	x	x	SYM
ejpam-4274	296	13	∈	∈	PROPN
ejpam-4274	296	14	u	u	NOUN
ejpam-4274	296	15	,	,	PUNCT
ejpam-4274	296	16	then	then	ADV
ejpam-4274	296	17	〈	〈	PROPN
ejpam-4274	296	18	x〉p	x〉p	ADJ
ejpam-4274	296	19	⊆	⊆	NUM
ejpam-4274	296	20	u	u	NOUN
ejpam-4274	296	21	.	.	PUNCT
ejpam-4274	297	1	(	(	PUNCT
ejpam-4274	297	2	5	5	NUM
ejpam-4274	297	3	)	)	PUNCT
ejpam-4274	297	4	if	if	SCONJ
ejpam-4274	297	5	f	f	PROPN
ejpam-4274	297	6	is	be	AUX
ejpam-4274	297	7	(	(	PUNCT
ejpam-4274	297	8	λ	λ	X
ejpam-4274	297	9	,	,	PUNCT
ejpam-4274	297	10	p)-closed	p)-close	VERB
ejpam-4274	297	11	and	and	CCONJ
ejpam-4274	297	12	x	x	SYM
ejpam-4274	297	13	∈	∈	PROPN
ejpam-4274	297	14	f	f	NOUN
ejpam-4274	297	15	,	,	PUNCT
ejpam-4274	297	16	then	then	ADV
ejpam-4274	297	17	〈	〈	PROPN
ejpam-4274	297	18	x〉p	x〉p	VERB
ejpam-4274	297	19	⊆	⊆	NUM
ejpam-4274	297	20	f	f	NOUN
ejpam-4274	297	21	.	.	PUNCT
ejpam-4274	298	1	proof	proof	NOUN
ejpam-4274	298	2	.	.	PUNCT
ejpam-4274	299	1	(	(	PUNCT
ejpam-4274	299	2	1	1	X
ejpam-4274	299	3	)	)	PUNCT
ejpam-4274	299	4	suppose	suppose	VERB
ejpam-4274	299	5	that	that	SCONJ
ejpam-4274	299	6	a	a	DET
ejpam-4274	299	7	∩	∩	NOUN
ejpam-4274	299	8	{	{	PUNCT
ejpam-4274	299	9	x}(λ	x}(λ	PROPN
ejpam-4274	299	10	,	,	PUNCT
ejpam-4274	299	11	p	p	NOUN
ejpam-4274	299	12	)	)	PUNCT
ejpam-4274	299	13	=	=	PUNCT
ejpam-4274	299	14	∅.	∅.	NOUN
ejpam-4274	299	15	then	then	ADV
ejpam-4274	299	16	,	,	PUNCT
ejpam-4274	299	17	we	we	PRON
ejpam-4274	299	18	have	have	VERB
ejpam-4274	299	19	x	x	PROPN
ejpam-4274	299	20	6∈	6∈	NOUN
ejpam-4274	299	21	x	x	NOUN
ejpam-4274	299	22	−	−	PROPN
ejpam-4274	299	23	{	{	PUNCT
ejpam-4274	299	24	x}(λ	x}(λ	PROPN
ejpam-4274	299	25	,	,	PUNCT
ejpam-4274	299	26	p	p	NOUN
ejpam-4274	299	27	)	)	PUNCT
ejpam-4274	299	28	which	which	PRON
ejpam-4274	299	29	is	be	AUX
ejpam-4274	299	30	a	a	DET
ejpam-4274	299	31	(	(	PUNCT
ejpam-4274	299	32	λ	λ	NOUN
ejpam-4274	299	33	,	,	PUNCT
ejpam-4274	299	34	p)-open	p)-open	VERB
ejpam-4274	299	35	set	set	VERB
ejpam-4274	299	36	containing	contain	VERB
ejpam-4274	299	37	a.	a.	NOUN
ejpam-4274	299	38	therefore	therefore	ADV
ejpam-4274	299	39	,	,	PUNCT
ejpam-4274	299	40	x	x	PROPN
ejpam-4274	299	41	6∈	6∈	NOUN
ejpam-4274	299	42	λ(λ	λ(λ	ADV
ejpam-4274	299	43	,	,	PUNCT
ejpam-4274	299	44	p)(a	p)(a	PROPN
ejpam-4274	299	45	)	)	PUNCT
ejpam-4274	299	46	.	.	PUNCT
ejpam-4274	300	1	consequently	consequently	ADV
ejpam-4274	300	2	,	,	PUNCT
ejpam-4274	300	3	we	we	PRON
ejpam-4274	300	4	have	have	VERB
ejpam-4274	300	5	λ(λ	λ(λ	NOUN
ejpam-4274	300	6	,	,	PUNCT
ejpam-4274	300	7	p)(a	p)(a	NOUN
ejpam-4274	300	8	)	)	PUNCT
ejpam-4274	301	1	⊆	⊆	NUM
ejpam-4274	301	2	{	{	PUNCT
ejpam-4274	301	3	x	x	SYM
ejpam-4274	301	4	∈	∈	PROPN
ejpam-4274	301	5	x	x	X
ejpam-4274	301	6	|	|	ADV
ejpam-4274	301	7	a	a	DET
ejpam-4274	301	8	∩	∩	NOUN
ejpam-4274	301	9	{	{	PUNCT
ejpam-4274	301	10	x}(λ	x}(λ	PROPN
ejpam-4274	301	11	,	,	PUNCT
ejpam-4274	301	12	p	p	NOUN
ejpam-4274	301	13	)	)	PUNCT
ejpam-4274	301	14	6=	6=	ADP
ejpam-4274	301	15	∅	∅	NOUN
ejpam-4274	301	16	}	}	PUNCT
ejpam-4274	301	17	.	.	PUNCT
ejpam-4274	302	1	next	next	ADV
ejpam-4274	302	2	,	,	PUNCT
ejpam-4274	302	3	let	let	VERB
ejpam-4274	302	4	x	x	PUNCT
ejpam-4274	302	5	∈	∈	PROPN
ejpam-4274	302	6	x	x	X
ejpam-4274	302	7	such	such	ADJ
ejpam-4274	302	8	that	that	SCONJ
ejpam-4274	302	9	a	a	DET
ejpam-4274	302	10	∩	∩	NOUN
ejpam-4274	302	11	{	{	PUNCT
ejpam-4274	302	12	x}(λ	x}(λ	PROPN
ejpam-4274	302	13	,	,	PUNCT
ejpam-4274	302	14	p	p	NOUN
ejpam-4274	302	15	)	)	PUNCT
ejpam-4274	302	16	6=	6=	NOUN
ejpam-4274	302	17	∅	∅	NOUN
ejpam-4274	302	18	and	and	CCONJ
ejpam-4274	302	19	suppose	suppose	VERB
ejpam-4274	302	20	that	that	SCONJ
ejpam-4274	302	21	x	x	PROPN
ejpam-4274	302	22	6∈	6∈	NOUN
ejpam-4274	302	23	λ(λ	λ(λ	ADV
ejpam-4274	302	24	,	,	PUNCT
ejpam-4274	302	25	p)(a	p)(a	PROPN
ejpam-4274	302	26	)	)	PUNCT
ejpam-4274	302	27	.	.	PUNCT
ejpam-4274	303	1	then	then	ADV
ejpam-4274	303	2	,	,	PUNCT
ejpam-4274	303	3	there	there	PRON
ejpam-4274	303	4	exists	exist	VERB
ejpam-4274	303	5	a	a	DET
ejpam-4274	303	6	(	(	PUNCT
ejpam-4274	303	7	λ	λ	NOUN
ejpam-4274	303	8	,	,	PUNCT
ejpam-4274	303	9	p)-open	p)-open	VERB
ejpam-4274	303	10	set	set	VERB
ejpam-4274	303	11	u	u	NOUN
ejpam-4274	303	12	containing	contain	VERB
ejpam-4274	303	13	a	a	DET
ejpam-4274	303	14	and	and	CCONJ
ejpam-4274	303	15	x	x	SYM
ejpam-4274	303	16	6∈	6∈	NOUN
ejpam-4274	303	17	u	u	NOUN
ejpam-4274	303	18	.	.	PUNCT
ejpam-4274	304	1	let	let	VERB
ejpam-4274	304	2	y	y	PROPN
ejpam-4274	304	3	∈	∈	PROPN
ejpam-4274	304	4	a	a	DET
ejpam-4274	304	5	∩	∩	NOUN
ejpam-4274	304	6	{	{	PUNCT
ejpam-4274	304	7	x}(λ	x}(λ	PROPN
ejpam-4274	304	8	,	,	PUNCT
ejpam-4274	304	9	p	p	NOUN
ejpam-4274	304	10	)	)	PUNCT
ejpam-4274	304	11	.	.	PUNCT
ejpam-4274	305	1	hence	hence	ADV
ejpam-4274	305	2	,	,	PUNCT
ejpam-4274	305	3	u	u	PROPN
ejpam-4274	305	4	is	be	AUX
ejpam-4274	305	5	a	a	DET
ejpam-4274	305	6	(	(	PUNCT
ejpam-4274	305	7	λ	λ	PROPN
ejpam-4274	305	8	,	,	PUNCT
ejpam-4274	305	9	p)-neighbourhood	p)-neighbourhood	NOUN
ejpam-4274	305	10	of	of	ADP
ejpam-4274	305	11	y	y	PRON
ejpam-4274	305	12	which	which	PRON
ejpam-4274	305	13	does	do	AUX
ejpam-4274	305	14	not	not	PART
ejpam-4274	305	15	contain	contain	VERB
ejpam-4274	305	16	x.	x.	NOUN
ejpam-4274	305	17	by	by	ADP
ejpam-4274	305	18	this	this	DET
ejpam-4274	305	19	contradiction	contradiction	NOUN
ejpam-4274	305	20	x	x	X
ejpam-4274	305	21	∈	∈	PROPN
ejpam-4274	305	22	λ(λ	λ(λ	PROPN
ejpam-4274	305	23	,	,	PUNCT
ejpam-4274	305	24	p)(a	p)(a	PROPN
ejpam-4274	305	25	)	)	PUNCT
ejpam-4274	305	26	.	.	PUNCT
ejpam-4274	306	1	(	(	PUNCT
ejpam-4274	306	2	2	2	X
ejpam-4274	306	3	)	)	PUNCT
ejpam-4274	306	4	let	let	VERB
ejpam-4274	306	5	x	x	SYM
ejpam-4274	306	6	∈	∈	PROPN
ejpam-4274	306	7	x.	x.	NOUN
ejpam-4274	306	8	then	then	ADV
ejpam-4274	306	9	,	,	PUNCT
ejpam-4274	306	10	we	we	PRON
ejpam-4274	306	11	have	have	VERB
ejpam-4274	306	12	{	{	PUNCT
ejpam-4274	306	13	x	x	NOUN
ejpam-4274	306	14	}	}	PUNCT
ejpam-4274	306	15	⊆	⊆	NUM
ejpam-4274	306	16	{	{	PUNCT
ejpam-4274	306	17	x}(λ	x}(λ	PROPN
ejpam-4274	306	18	,	,	PUNCT
ejpam-4274	306	19	p	p	NOUN
ejpam-4274	306	20	)	)	PUNCT
ejpam-4274	306	21	∩	∩	PROPN
ejpam-4274	306	22	λ(λ	λ(λ	PROPN
ejpam-4274	306	23	,	,	PUNCT
ejpam-4274	306	24	p)({x	p)({x	NOUN
ejpam-4274	306	25	}	}	PUNCT
ejpam-4274	306	26	)	)	PUNCT
ejpam-4274	306	27	=	=	PUNCT
ejpam-4274	307	1	〈	〈	PROPN
ejpam-4274	307	2	x〉p	x〉p	PROPN
ejpam-4274	307	3	.	.	PUNCT
ejpam-4274	308	1	by	by	ADP
ejpam-4274	308	2	lemma	lemma	PROPN
ejpam-4274	308	3	7	7	NUM
ejpam-4274	308	4	,	,	PUNCT
ejpam-4274	308	5	λ(λ	λ(λ	PROPN
ejpam-4274	308	6	,	,	PUNCT
ejpam-4274	308	7	p)({x	p)({x	NOUN
ejpam-4274	308	8	}	}	PUNCT
ejpam-4274	308	9	)	)	PUNCT
ejpam-4274	309	1	⊆	⊆	NUM
ejpam-4274	309	2	λ(λ	λ(λ	ADP
ejpam-4274	309	3	,	,	PUNCT
ejpam-4274	309	4	p)(〈x〉p	p)(〈x〉p	PROPN
ejpam-4274	309	5	)	)	PUNCT
ejpam-4274	309	6	.	.	PUNCT
ejpam-4274	310	1	next	next	ADV
ejpam-4274	310	2	,	,	PUNCT
ejpam-4274	310	3	we	we	PRON
ejpam-4274	310	4	show	show	VERB
ejpam-4274	310	5	the	the	DET
ejpam-4274	310	6	opposite	opposite	ADJ
ejpam-4274	310	7	implication	implication	NOUN
ejpam-4274	310	8	.	.	PUNCT
ejpam-4274	310	9	suppose	suppose	VERB
ejpam-4274	310	10	that	that	SCONJ
ejpam-4274	310	11	y	y	PROPN
ejpam-4274	310	12	6∈	6∈	PROPN
ejpam-4274	310	13	λ(λ	λ(λ	ADV
ejpam-4274	310	14	,	,	PUNCT
ejpam-4274	310	15	p)({x	p)({x	NOUN
ejpam-4274	310	16	}	}	PUNCT
ejpam-4274	310	17	)	)	PUNCT
ejpam-4274	310	18	.	.	PUNCT
ejpam-4274	311	1	then	then	ADV
ejpam-4274	311	2	,	,	PUNCT
ejpam-4274	311	3	there	there	PRON
ejpam-4274	311	4	exists	exist	VERB
ejpam-4274	311	5	a	a	DET
ejpam-4274	311	6	(	(	PUNCT
ejpam-4274	311	7	λ	λ	NOUN
ejpam-4274	311	8	,	,	PUNCT
ejpam-4274	311	9	p)-open	p)-open	VERB
ejpam-4274	311	10	set	set	VERB
ejpam-4274	311	11	v	v	ADP
ejpam-4274	311	12	such	such	ADJ
ejpam-4274	311	13	that	that	SCONJ
ejpam-4274	311	14	x	x	SYM
ejpam-4274	311	15	∈	∈	PROPN
ejpam-4274	311	16	v	v	NOUN
ejpam-4274	311	17	and	and	CCONJ
ejpam-4274	311	18	y	y	PROPN
ejpam-4274	311	19	6∈	6∈	PROPN
ejpam-4274	311	20	v	v	NOUN
ejpam-4274	311	21	.	.	PUNCT
ejpam-4274	312	1	since	since	SCONJ
ejpam-4274	312	2	〈	〈	PROPN
ejpam-4274	312	3	x〉p	x〉p	NOUN
ejpam-4274	312	4	⊆	⊆	NUM
ejpam-4274	312	5	λ(λ	λ(λ	PROPN
ejpam-4274	312	6	,	,	PUNCT
ejpam-4274	312	7	p)({x	p)({x	NOUN
ejpam-4274	312	8	}	}	PUNCT
ejpam-4274	312	9	)	)	PUNCT
ejpam-4274	313	1	⊆	⊆	NUM
ejpam-4274	313	2	λ(λ	λ(λ	ADP
ejpam-4274	313	3	,	,	PUNCT
ejpam-4274	313	4	p)(v	p)(v	X
ejpam-4274	313	5	)	)	PUNCT
ejpam-4274	314	1	=	=	SYM
ejpam-4274	314	2	v	v	X
ejpam-4274	314	3	,	,	PUNCT
ejpam-4274	314	4	we	we	PRON
ejpam-4274	314	5	have	have	VERB
ejpam-4274	314	6	λ(λ	λ(λ	PROPN
ejpam-4274	314	7	,	,	PUNCT
ejpam-4274	314	8	p)(〈x〉p	p)(〈x〉p	NOUN
ejpam-4274	314	9	)	)	PUNCT
ejpam-4274	314	10	⊆	⊆	NUM
ejpam-4274	314	11	v	v	NOUN
ejpam-4274	314	12	.	.	PUNCT
ejpam-4274	315	1	since	since	SCONJ
ejpam-4274	315	2	y	y	PROPN
ejpam-4274	315	3	6∈	6∈	PROPN
ejpam-4274	315	4	v	v	NUM
ejpam-4274	315	5	,	,	PUNCT
ejpam-4274	315	6	we	we	PRON
ejpam-4274	315	7	have	have	VERB
ejpam-4274	315	8	y	y	PROPN
ejpam-4274	315	9	6∈	6∈	PROPN
ejpam-4274	315	10	λ(λ	λ(λ	PROPN
ejpam-4274	315	11	,	,	PUNCT
ejpam-4274	315	12	p)(〈x〉p	p)(〈x〉p	PROPN
ejpam-4274	315	13	)	)	PUNCT
ejpam-4274	315	14	.	.	PUNCT
ejpam-4274	316	1	thus	thus	ADV
ejpam-4274	316	2	,	,	PUNCT
ejpam-4274	316	3	λ(λ	λ(λ	ADV
ejpam-4274	316	4	,	,	PUNCT
ejpam-4274	316	5	p)(〈x〉p	p)(〈x〉p	NOUN
ejpam-4274	316	6	)	)	PUNCT
ejpam-4274	316	7	⊆	⊆	NUM
ejpam-4274	316	8	λ(λ	λ(λ	PROPN
ejpam-4274	316	9	,	,	PUNCT
ejpam-4274	316	10	p)({x	p)({x	NOUN
ejpam-4274	316	11	}	}	PUNCT
ejpam-4274	316	12	)	)	PUNCT
ejpam-4274	316	13	and	and	CCONJ
ejpam-4274	316	14	hence	hence	ADV
ejpam-4274	316	15	λ(λ	λ(λ	PROPN
ejpam-4274	316	16	,	,	PUNCT
ejpam-4274	316	17	p)({x	p)({x	NOUN
ejpam-4274	316	18	}	}	PUNCT
ejpam-4274	316	19	)	)	PUNCT
ejpam-4274	316	20	=	=	SYM
ejpam-4274	317	1	λ(λ	λ(λ	PROPN
ejpam-4274	317	2	,	,	PUNCT
ejpam-4274	317	3	p)(〈x〉p	p)(〈x〉p	PROPN
ejpam-4274	317	4	)	)	PUNCT
ejpam-4274	317	5	.	.	PUNCT
ejpam-4274	318	1	(	(	PUNCT
ejpam-4274	318	2	3	3	X
ejpam-4274	318	3	)	)	PUNCT
ejpam-4274	318	4	by	by	ADP
ejpam-4274	318	5	the	the	DET
ejpam-4274	318	6	definition	definition	NOUN
ejpam-4274	318	7	of	of	ADP
ejpam-4274	318	8	〈	〈	PROPN
ejpam-4274	318	9	x〉p	x〉p	PROPN
ejpam-4274	318	10	,	,	PUNCT
ejpam-4274	318	11	we	we	PRON
ejpam-4274	318	12	have	have	VERB
ejpam-4274	318	13	{	{	PUNCT
ejpam-4274	318	14	x	x	NOUN
ejpam-4274	318	15	}	}	PUNCT
ejpam-4274	318	16	⊆	⊆	NUM
ejpam-4274	318	17	〈	〈	ADP
ejpam-4274	318	18	x〉p	x〉p	PROPN
ejpam-4274	318	19	and	and	CCONJ
ejpam-4274	318	20	{	{	PUNCT
ejpam-4274	318	21	x}(λ	x}(λ	PROPN
ejpam-4274	318	22	,	,	PUNCT
ejpam-4274	318	23	p	p	NOUN
ejpam-4274	318	24	)	)	PUNCT
ejpam-4274	318	25	⊆	⊆	NUM
ejpam-4274	318	26	(	(	PUNCT
ejpam-4274	318	27	〈	〈	PROPN
ejpam-4274	318	28	x〉p)(λ	x〉p)(λ	PRON
ejpam-4274	318	29	,	,	PUNCT
ejpam-4274	318	30	p	p	NOUN
ejpam-4274	318	31	)	)	PUNCT
ejpam-4274	318	32	by	by	ADP
ejpam-4274	318	33	lemma	lemma	PROPN
ejpam-4274	318	34	5	5	NUM
ejpam-4274	318	35	.	.	PUNCT
ejpam-4274	319	1	on	on	ADP
ejpam-4274	319	2	the	the	DET
ejpam-4274	319	3	other	other	ADJ
ejpam-4274	319	4	hand	hand	NOUN
ejpam-4274	319	5	,	,	PUNCT
ejpam-4274	319	6	we	we	PRON
ejpam-4274	319	7	have	have	VERB
ejpam-4274	319	8	〈	〈	VERB
ejpam-4274	319	9	x〉p	x〉p	ADJ
ejpam-4274	319	10	⊆	⊆	X
ejpam-4274	319	11	{	{	PUNCT
ejpam-4274	319	12	x}(λ	x}(λ	PROPN
ejpam-4274	319	13	,	,	PUNCT
ejpam-4274	319	14	p	p	NOUN
ejpam-4274	319	15	)	)	PUNCT
ejpam-4274	319	16	and	and	CCONJ
ejpam-4274	319	17	(	(	PUNCT
ejpam-4274	319	18	〈	〈	PROPN
ejpam-4274	319	19	x〉p)(λ	x〉p)(λ	PROPN
ejpam-4274	319	20	,	,	PUNCT
ejpam-4274	319	21	p	p	NOUN
ejpam-4274	319	22	)	)	PUNCT
ejpam-4274	319	23	⊆	⊆	NUM
ejpam-4274	319	24	(	(	PUNCT
ejpam-4274	319	25	{	{	PUNCT
ejpam-4274	319	26	x}(λ	x}(λ	PROPN
ejpam-4274	319	27	,	,	PUNCT
ejpam-4274	319	28	p))(λ	p))(λ	PROPN
ejpam-4274	319	29	,	,	PUNCT
ejpam-4274	319	30	p	p	NOUN
ejpam-4274	319	31	)	)	PUNCT
ejpam-4274	319	32	=	=	SYM
ejpam-4274	319	33	{	{	PUNCT
ejpam-4274	319	34	x}(λ	x}(λ	PROPN
ejpam-4274	319	35	,	,	PUNCT
ejpam-4274	319	36	p	p	NOUN
ejpam-4274	319	37	)	)	PUNCT
ejpam-4274	319	38	.	.	PUNCT
ejpam-4274	320	1	thus	thus	ADV
ejpam-4274	320	2	,	,	PUNCT
ejpam-4274	320	3	(	(	PUNCT
ejpam-4274	320	4	〈	〈	PROPN
ejpam-4274	320	5	x〉p)(λ	x〉p)(λ	PRON
ejpam-4274	320	6	,	,	PUNCT
ejpam-4274	320	7	p	p	NOUN
ejpam-4274	320	8	)	)	PUNCT
ejpam-4274	320	9	=	=	SYM
ejpam-4274	320	10	{	{	PUNCT
ejpam-4274	320	11	x}(λ	x}(λ	PROPN
ejpam-4274	320	12	,	,	PUNCT
ejpam-4274	320	13	p	p	NOUN
ejpam-4274	320	14	)	)	PUNCT
ejpam-4274	320	15	.	.	PUNCT
ejpam-4274	321	1	(	(	PUNCT
ejpam-4274	321	2	4	4	X
ejpam-4274	321	3	)	)	PUNCT
ejpam-4274	321	4	let	let	VERB
ejpam-4274	321	5	u	u	PRON
ejpam-4274	321	6	be	be	AUX
ejpam-4274	321	7	a	a	DET
ejpam-4274	321	8	(	(	PUNCT
ejpam-4274	321	9	λ	λ	NOUN
ejpam-4274	321	10	,	,	PUNCT
ejpam-4274	321	11	p)-open	p)-open	VERB
ejpam-4274	321	12	set	set	VERB
ejpam-4274	321	13	and	and	CCONJ
ejpam-4274	321	14	let	let	VERB
ejpam-4274	321	15	x	x	PUNCT
ejpam-4274	321	16	∈	∈	PROPN
ejpam-4274	321	17	u	u	NOUN
ejpam-4274	321	18	.	.	PUNCT
ejpam-4274	322	1	by	by	ADP
ejpam-4274	322	2	lemma	lemma	PROPN
ejpam-4274	322	3	7	7	NUM
ejpam-4274	322	4	,	,	PUNCT
ejpam-4274	322	5	λ(λ	λ(λ	PROPN
ejpam-4274	322	6	,	,	PUNCT
ejpam-4274	322	7	p)({x	p)({x	NOUN
ejpam-4274	322	8	}	}	PUNCT
ejpam-4274	322	9	)	)	PUNCT
ejpam-4274	323	1	⊆	⊆	NUM
ejpam-4274	323	2	u	u	NOUN
ejpam-4274	323	3	and	and	CCONJ
ejpam-4274	323	4	hence	hence	ADV
ejpam-4274	323	5	〈	〈	ADV
ejpam-4274	323	6	x〉p	x〉p	ADJ
ejpam-4274	323	7	⊆	⊆	NUM
ejpam-4274	323	8	u	u	NOUN
ejpam-4274	323	9	.	.	PUNCT
ejpam-4274	324	1	(	(	PUNCT
ejpam-4274	324	2	5	5	X
ejpam-4274	324	3	)	)	PUNCT
ejpam-4274	324	4	let	let	VERB
ejpam-4274	324	5	f	f	PRON
ejpam-4274	324	6	be	be	AUX
ejpam-4274	324	7	a	a	DET
ejpam-4274	324	8	(	(	PUNCT
ejpam-4274	324	9	λ	λ	PROPN
ejpam-4274	324	10	,	,	PUNCT
ejpam-4274	324	11	p)-closed	p)-close	VERB
ejpam-4274	324	12	set	set	NOUN
ejpam-4274	324	13	and	and	CCONJ
ejpam-4274	324	14	let	let	VERB
ejpam-4274	324	15	x	x	SYM
ejpam-4274	324	16	∈	∈	PROPN
ejpam-4274	324	17	f	f	X
ejpam-4274	324	18	.	.	PUNCT
ejpam-4274	325	1	by	by	ADP
ejpam-4274	325	2	lemma	lemma	PROPN
ejpam-4274	325	3	5	5	NUM
ejpam-4274	325	4	,	,	PUNCT
ejpam-4274	325	5	we	we	PRON
ejpam-4274	325	6	have	have	VERB
ejpam-4274	325	7	〈	〈	PROPN
ejpam-4274	325	8	x〉p	x〉p	NOUN
ejpam-4274	325	9	=	=	SYM
ejpam-4274	325	10	{	{	PUNCT
ejpam-4274	325	11	x}(λ	x}(λ	PROPN
ejpam-4274	325	12	,	,	PUNCT
ejpam-4274	325	13	p	p	NOUN
ejpam-4274	325	14	)	)	PUNCT
ejpam-4274	325	15	∩	∩	PROPN
ejpam-4274	325	16	λ(λ	λ(λ	PROPN
ejpam-4274	325	17	,	,	PUNCT
ejpam-4274	325	18	p)({x	p)({x	NOUN
ejpam-4274	325	19	}	}	PUNCT
ejpam-4274	325	20	)	)	PUNCT
ejpam-4274	325	21	⊆	⊆	X
ejpam-4274	325	22	{	{	PUNCT
ejpam-4274	325	23	x}(λ	x}(λ	PROPN
ejpam-4274	325	24	,	,	PUNCT
ejpam-4274	325	25	p	p	NOUN
ejpam-4274	325	26	)	)	PUNCT
ejpam-4274	325	27	⊆	⊆	NUM
ejpam-4274	325	28	f	f	X
ejpam-4274	325	29	(	(	PUNCT
ejpam-4274	325	30	λ	λ	PROPN
ejpam-4274	325	31	,	,	PUNCT
ejpam-4274	325	32	p	p	NOUN
ejpam-4274	325	33	)	)	PUNCT
ejpam-4274	326	1	=	=	SYM
ejpam-4274	326	2	f.	f.	PROPN
ejpam-4274	326	3	lemma	lemma	PROPN
ejpam-4274	326	4	10	10	NUM
ejpam-4274	326	5	.	.	PUNCT
ejpam-4274	327	1	for	for	ADP
ejpam-4274	327	2	any	any	DET
ejpam-4274	327	3	points	point	NOUN
ejpam-4274	327	4	x	x	PUNCT
ejpam-4274	327	5	and	and	CCONJ
ejpam-4274	327	6	y	y	PROPN
ejpam-4274	327	7	in	in	ADP
ejpam-4274	327	8	a	a	DET
ejpam-4274	327	9	topological	topological	ADJ
ejpam-4274	327	10	space	space	NOUN
ejpam-4274	327	11	(	(	PUNCT
ejpam-4274	327	12	x	x	X
ejpam-4274	327	13	,	,	PUNCT
ejpam-4274	327	14	τ	τ	PROPN
ejpam-4274	327	15	)	)	PUNCT
ejpam-4274	327	16	,	,	PUNCT
ejpam-4274	327	17	the	the	DET
ejpam-4274	327	18	following	follow	VERB
ejpam-4274	327	19	properties	property	NOUN
ejpam-4274	327	20	are	be	AUX
ejpam-4274	327	21	equivalent	equivalent	ADJ
ejpam-4274	327	22	:	:	PUNCT
ejpam-4274	327	23	(	(	PUNCT
ejpam-4274	327	24	1	1	X
ejpam-4274	327	25	)	)	PUNCT
ejpam-4274	327	26	λ(λ	λ(λ	ADV
ejpam-4274	327	27	,	,	PUNCT
ejpam-4274	327	28	p)({x	p)({x	NOUN
ejpam-4274	327	29	}	}	PUNCT
ejpam-4274	327	30	)	)	PUNCT
ejpam-4274	327	31	6=	6=	ADP
ejpam-4274	328	1	λ(λ	λ(λ	ADP
ejpam-4274	328	2	,	,	PUNCT
ejpam-4274	328	3	p)({y	p)({y	PROPN
ejpam-4274	328	4	}	}	PUNCT
ejpam-4274	328	5	)	)	PUNCT
ejpam-4274	328	6	;	;	PUNCT
ejpam-4274	328	7	(	(	PUNCT
ejpam-4274	328	8	2	2	X
ejpam-4274	328	9	)	)	PUNCT
ejpam-4274	328	10	{	{	PUNCT
ejpam-4274	328	11	x}(λ	x}(λ	PROPN
ejpam-4274	328	12	,	,	PUNCT
ejpam-4274	328	13	p	p	NOUN
ejpam-4274	328	14	)	)	PUNCT
ejpam-4274	328	15	6=	6=	ADP
ejpam-4274	328	16	{	{	PUNCT
ejpam-4274	328	17	y}(λ	y}(λ	PROPN
ejpam-4274	328	18	,	,	PUNCT
ejpam-4274	328	19	p	p	NOUN
ejpam-4274	328	20	)	)	PUNCT
ejpam-4274	328	21	.	.	PUNCT
ejpam-4274	329	1	proof	proof	NOUN
ejpam-4274	329	2	.	.	PUNCT
ejpam-4274	330	1	(	(	PUNCT
ejpam-4274	330	2	1	1	X
ejpam-4274	330	3	)	)	PUNCT
ejpam-4274	330	4	⇒	⇒	NOUN
ejpam-4274	330	5	(	(	PUNCT
ejpam-4274	330	6	2	2	NUM
ejpam-4274	330	7	):	):	PUNCT
ejpam-4274	330	8	suppose	suppose	VERB
ejpam-4274	330	9	that	that	SCONJ
ejpam-4274	330	10	λ(λ	λ(λ	PROPN
ejpam-4274	330	11	,	,	PUNCT
ejpam-4274	330	12	p)({x	p)({x	NOUN
ejpam-4274	330	13	}	}	PUNCT
ejpam-4274	330	14	)	)	PUNCT
ejpam-4274	330	15	6=	6=	ADP
ejpam-4274	330	16	λ(λ	λ(λ	ADP
ejpam-4274	330	17	,	,	PUNCT
ejpam-4274	330	18	p)({y	p)({y	PROPN
ejpam-4274	330	19	}	}	PUNCT
ejpam-4274	330	20	)	)	PUNCT
ejpam-4274	330	21	.	.	PUNCT
ejpam-4274	331	1	there	there	PRON
ejpam-4274	331	2	exists	exist	VERB
ejpam-4274	331	3	a	a	DET
ejpam-4274	331	4	point	point	NOUN
ejpam-4274	331	5	z	z	NOUN
ejpam-4274	331	6	∈	∈	PROPN
ejpam-4274	331	7	x	x	PUNCT
ejpam-4274	331	8	such	such	ADJ
ejpam-4274	331	9	that	that	SCONJ
ejpam-4274	331	10	z	z	PROPN
ejpam-4274	331	11	∈	∈	PROPN
ejpam-4274	331	12	λ(λ	λ(λ	PROPN
ejpam-4274	331	13	,	,	PUNCT
ejpam-4274	331	14	p)({x	p)({x	NOUN
ejpam-4274	331	15	}	}	PUNCT
ejpam-4274	331	16	)	)	PUNCT
ejpam-4274	331	17	and	and	CCONJ
ejpam-4274	331	18	z	z	PROPN
ejpam-4274	331	19	6∈	6∈	PROPN
ejpam-4274	332	1	λ(λ	λ(λ	ADV
ejpam-4274	332	2	,	,	PUNCT
ejpam-4274	332	3	p)({y	p)({y	PROPN
ejpam-4274	332	4	}	}	PUNCT
ejpam-4274	332	5	)	)	PUNCT
ejpam-4274	332	6	or	or	CCONJ
ejpam-4274	332	7	z	z	NOUN
ejpam-4274	332	8	∈	∈	PROPN
ejpam-4274	332	9	λ(λ	λ(λ	PROPN
ejpam-4274	332	10	,	,	PUNCT
ejpam-4274	332	11	p)({y	p)({y	PROPN
ejpam-4274	332	12	}	}	PUNCT
ejpam-4274	332	13	)	)	PUNCT
ejpam-4274	332	14	and	and	CCONJ
ejpam-4274	332	15	z	z	PROPN
ejpam-4274	332	16	6∈	6∈	PROPN
ejpam-4274	333	1	λ(λ	λ(λ	ADV
ejpam-4274	333	2	,	,	PUNCT
ejpam-4274	333	3	p)({x	p)({x	NOUN
ejpam-4274	333	4	}	}	PUNCT
ejpam-4274	333	5	)	)	PUNCT
ejpam-4274	333	6	.	.	PUNCT
ejpam-4274	334	1	we	we	PRON
ejpam-4274	334	2	prove	prove	VERB
ejpam-4274	334	3	only	only	ADV
ejpam-4274	334	4	the	the	DET
ejpam-4274	334	5	first	first	ADJ
ejpam-4274	334	6	case	case	NOUN
ejpam-4274	334	7	being	be	AUX
ejpam-4274	334	8	the	the	DET
ejpam-4274	334	9	second	second	ADJ
ejpam-4274	334	10	analogous	analogous	NOUN
ejpam-4274	334	11	.	.	PUNCT
ejpam-4274	335	1	from	from	ADP
ejpam-4274	335	2	z	z	PROPN
ejpam-4274	335	3	∈	∈	PROPN
ejpam-4274	335	4	λ(λ	λ(λ	PROPN
ejpam-4274	335	5	,	,	PUNCT
ejpam-4274	335	6	p)({x	p)({x	PROPN
ejpam-4274	335	7	}	}	PUNCT
ejpam-4274	335	8	)	)	PUNCT
ejpam-4274	335	9	it	it	PRON
ejpam-4274	335	10	follows	follow	VERB
ejpam-4274	335	11	that	that	SCONJ
ejpam-4274	335	12	{	{	PUNCT
ejpam-4274	335	13	x	x	NOUN
ejpam-4274	335	14	}	}	PUNCT
ejpam-4274	335	15	∩	∩	NOUN
ejpam-4274	335	16	{	{	PUNCT
ejpam-4274	335	17	z}(λ	z}(λ	PROPN
ejpam-4274	335	18	,	,	PUNCT
ejpam-4274	335	19	p	p	NOUN
ejpam-4274	335	20	)	)	PUNCT
ejpam-4274	335	21	6=	6=	ADP
ejpam-4274	335	22	∅	∅	NOUN
ejpam-4274	335	23	which	which	PRON
ejpam-4274	335	24	implies	imply	VERB
ejpam-4274	335	25	x	x	X
ejpam-4274	335	26	∈	∈	PROPN
ejpam-4274	335	27	{	{	PUNCT
ejpam-4274	335	28	z}(λ	z}(λ	PROPN
ejpam-4274	335	29	,	,	PUNCT
ejpam-4274	335	30	p	p	NOUN
ejpam-4274	335	31	)	)	PUNCT
ejpam-4274	335	32	.	.	PUNCT
ejpam-4274	336	1	by	by	ADP
ejpam-4274	336	2	z	z	PROPN
ejpam-4274	336	3	6∈	6∈	PROPN
ejpam-4274	336	4	λ(λ	λ(λ	PROPN
ejpam-4274	336	5	,	,	PUNCT
ejpam-4274	336	6	p)({y	p)({y	PROPN
ejpam-4274	336	7	}	}	PUNCT
ejpam-4274	336	8	)	)	PUNCT
ejpam-4274	336	9	,	,	PUNCT
ejpam-4274	336	10	{	{	PUNCT
ejpam-4274	336	11	y	y	NOUN
ejpam-4274	336	12	}	}	PUNCT
ejpam-4274	336	13	∩	∩	NOUN
ejpam-4274	336	14	{	{	PUNCT
ejpam-4274	336	15	z}(λ	z}(λ	PROPN
ejpam-4274	336	16	,	,	PUNCT
ejpam-4274	336	17	p	p	NOUN
ejpam-4274	336	18	)	)	PUNCT
ejpam-4274	336	19	=	=	PUNCT
ejpam-4274	336	20	∅.	∅.	NOUN
ejpam-4274	336	21	since	since	SCONJ
ejpam-4274	336	22	x	x	PROPN
ejpam-4274	336	23	∈	∈	PROPN
ejpam-4274	336	24	{	{	PUNCT
ejpam-4274	336	25	z}(λ	z}(λ	PROPN
ejpam-4274	336	26	,	,	PUNCT
ejpam-4274	336	27	p	p	NOUN
ejpam-4274	336	28	)	)	PUNCT
ejpam-4274	336	29	,	,	PUNCT
ejpam-4274	336	30	{	{	PUNCT
ejpam-4274	336	31	x}(λ	x}(λ	PROPN
ejpam-4274	336	32	,	,	PUNCT
ejpam-4274	336	33	p	p	NOUN
ejpam-4274	336	34	)	)	PUNCT
ejpam-4274	336	35	⊆	⊆	NUM
ejpam-4274	336	36	{	{	PUNCT
ejpam-4274	336	37	z}(λ	z}(λ	PROPN
ejpam-4274	336	38	,	,	PUNCT
ejpam-4274	336	39	p	p	NOUN
ejpam-4274	336	40	)	)	PUNCT
ejpam-4274	336	41	and	and	CCONJ
ejpam-4274	336	42	{	{	PUNCT
ejpam-4274	336	43	y	y	NOUN
ejpam-4274	336	44	}	}	PUNCT
ejpam-4274	336	45	∩	∩	NOUN
ejpam-4274	336	46	{	{	PUNCT
ejpam-4274	336	47	x}(λ	x}(λ	PROPN
ejpam-4274	336	48	,	,	PUNCT
ejpam-4274	336	49	p	p	NOUN
ejpam-4274	336	50	)	)	PUNCT
ejpam-4274	336	51	=	=	PUNCT
ejpam-4274	336	52	∅.	∅.	VERB
ejpam-4274	336	53	therefore	therefore	ADV
ejpam-4274	336	54	,	,	PUNCT
ejpam-4274	336	55	it	it	PRON
ejpam-4274	336	56	follows	follow	VERB
ejpam-4274	336	57	that	that	SCONJ
ejpam-4274	336	58	{	{	PUNCT
ejpam-4274	336	59	x}(λ	x}(λ	PROPN
ejpam-4274	336	60	,	,	PUNCT
ejpam-4274	336	61	p	p	NOUN
ejpam-4274	336	62	)	)	PUNCT
ejpam-4274	336	63	6=	6=	ADP
ejpam-4274	336	64	{	{	PUNCT
ejpam-4274	336	65	y}(λ	y}(λ	PROPN
ejpam-4274	336	66	,	,	PUNCT
ejpam-4274	336	67	p	p	NOUN
ejpam-4274	336	68	)	)	PUNCT
ejpam-4274	336	69	.	.	PUNCT
ejpam-4274	337	1	thus	thus	ADV
ejpam-4274	337	2	,	,	PUNCT
ejpam-4274	337	3	λ(λ	λ(λ	PROPN
ejpam-4274	337	4	,	,	PUNCT
ejpam-4274	337	5	p)({x	p)({x	NOUN
ejpam-4274	337	6	}	}	PUNCT
ejpam-4274	337	7	)	)	PUNCT
ejpam-4274	337	8	6=	6=	ADP
ejpam-4274	337	9	λ(λ	λ(λ	ADP
ejpam-4274	337	10	,	,	PUNCT
ejpam-4274	337	11	p)({y	p)({y	PROPN
ejpam-4274	337	12	}	}	PUNCT
ejpam-4274	337	13	)	)	PUNCT
ejpam-4274	337	14	implies	imply	VERB
ejpam-4274	337	15	that	that	SCONJ
ejpam-4274	337	16	{	{	PUNCT
ejpam-4274	337	17	x}(λ	x}(λ	PROPN
ejpam-4274	337	18	,	,	PUNCT
ejpam-4274	337	19	p	p	NOUN
ejpam-4274	337	20	)	)	PUNCT
ejpam-4274	337	21	6=	6=	ADP
ejpam-4274	337	22	{	{	PUNCT
ejpam-4274	337	23	y}(λ	y}(λ	PROPN
ejpam-4274	337	24	,	,	PUNCT
ejpam-4274	337	25	p	p	NOUN
ejpam-4274	337	26	)	)	PUNCT
ejpam-4274	337	27	.	.	PUNCT
ejpam-4274	338	1	(	(	PUNCT
ejpam-4274	338	2	2	2	X
ejpam-4274	338	3	)	)	PUNCT
ejpam-4274	338	4	⇒	⇒	NOUN
ejpam-4274	338	5	(	(	PUNCT
ejpam-4274	338	6	1	1	NUM
ejpam-4274	338	7	):	):	PUNCT
ejpam-4274	338	8	suppose	suppose	VERB
ejpam-4274	338	9	that	that	SCONJ
ejpam-4274	338	10	{	{	PUNCT
ejpam-4274	338	11	x}(λ	x}(λ	PROPN
ejpam-4274	338	12	,	,	PUNCT
ejpam-4274	338	13	p	p	NOUN
ejpam-4274	338	14	)	)	PUNCT
ejpam-4274	338	15	6=	6=	ADP
ejpam-4274	338	16	{	{	PUNCT
ejpam-4274	338	17	y}(λ	y}(λ	PROPN
ejpam-4274	338	18	,	,	PUNCT
ejpam-4274	338	19	p	p	NOUN
ejpam-4274	338	20	)	)	PUNCT
ejpam-4274	338	21	.	.	PUNCT
ejpam-4274	339	1	then	then	ADV
ejpam-4274	339	2	,	,	PUNCT
ejpam-4274	339	3	there	there	PRON
ejpam-4274	339	4	exists	exist	VERB
ejpam-4274	339	5	a	a	DET
ejpam-4274	339	6	point	point	NOUN
ejpam-4274	339	7	z	z	NOUN
ejpam-4274	339	8	∈	∈	PROPN
ejpam-4274	339	9	x	x	PUNCT
ejpam-4274	339	10	such	such	ADJ
ejpam-4274	339	11	that	that	SCONJ
ejpam-4274	339	12	z	z	PROPN
ejpam-4274	339	13	∈	∈	PROPN
ejpam-4274	339	14	{	{	PUNCT
ejpam-4274	339	15	x}(λ	x}(λ	PROPN
ejpam-4274	339	16	,	,	PUNCT
ejpam-4274	339	17	p	p	NOUN
ejpam-4274	339	18	)	)	PUNCT
ejpam-4274	339	19	and	and	CCONJ
ejpam-4274	339	20	z	z	PROPN
ejpam-4274	339	21	6∈	6∈	PROPN
ejpam-4274	339	22	{	{	PUNCT
ejpam-4274	339	23	y}(λ	y}(λ	PROPN
ejpam-4274	339	24	,	,	PUNCT
ejpam-4274	339	25	p	p	NOUN
ejpam-4274	339	26	)	)	PUNCT
ejpam-4274	339	27	or	or	CCONJ
ejpam-4274	339	28	z	z	NOUN
ejpam-4274	339	29	∈	∈	PROPN
ejpam-4274	339	30	{	{	PUNCT
ejpam-4274	339	31	y}(λ	y}(λ	PROPN
ejpam-4274	339	32	,	,	PUNCT
ejpam-4274	339	33	p	p	NOUN
ejpam-4274	339	34	)	)	PUNCT
ejpam-4274	339	35	and	and	CCONJ
ejpam-4274	339	36	z	z	PROPN
ejpam-4274	339	37	6∈	6∈	PROPN
ejpam-4274	339	38	{	{	PUNCT
ejpam-4274	339	39	x}(λ	x}(λ	PROPN
ejpam-4274	339	40	,	,	PUNCT
ejpam-4274	339	41	p	p	NOUN
ejpam-4274	339	42	)	)	PUNCT
ejpam-4274	339	43	.	.	PUNCT
ejpam-4274	340	1	we	we	PRON
ejpam-4274	340	2	prove	prove	VERB
ejpam-4274	340	3	only	only	ADV
ejpam-4274	340	4	the	the	DET
ejpam-4274	340	5	first	first	ADJ
ejpam-4274	340	6	case	case	NOUN
ejpam-4274	340	7	being	be	AUX
ejpam-4274	340	8	the	the	DET
ejpam-4274	340	9	second	second	ADJ
ejpam-4274	340	10	analogous	analogous	NOUN
ejpam-4274	340	11	.	.	PUNCT
ejpam-4274	341	1	it	it	PRON
ejpam-4274	341	2	follows	follow	VERB
ejpam-4274	341	3	that	that	SCONJ
ejpam-4274	341	4	there	there	PRON
ejpam-4274	341	5	exists	exist	VERB
ejpam-4274	341	6	a	a	DET
ejpam-4274	341	7	(	(	PUNCT
ejpam-4274	341	8	λ	λ	NOUN
ejpam-4274	341	9	,	,	PUNCT
ejpam-4274	341	10	p)-open	p)-open	VERB
ejpam-4274	341	11	set	set	VERB
ejpam-4274	341	12	containing	contain	VERB
ejpam-4274	341	13	z	z	NOUN
ejpam-4274	341	14	and	and	CCONJ
ejpam-4274	341	15	therefore	therefore	ADV
ejpam-4274	341	16	x	x	X
ejpam-4274	341	17	but	but	CCONJ
ejpam-4274	341	18	not	not	PART
ejpam-4274	341	19	y	y	NOUN
ejpam-4274	341	20	,	,	PUNCT
ejpam-4274	341	21	namely	namely	ADV
ejpam-4274	341	22	,	,	PUNCT
ejpam-4274	341	23	y	y	PROPN
ejpam-4274	341	24	6∈	6∈	PROPN
ejpam-4274	341	25	λ(λ	λ(λ	ADV
ejpam-4274	341	26	,	,	PUNCT
ejpam-4274	341	27	p)({x	p)({x	NOUN
ejpam-4274	341	28	}	}	PUNCT
ejpam-4274	341	29	)	)	PUNCT
ejpam-4274	341	30	and	and	CCONJ
ejpam-4274	341	31	hence	hence	ADV
ejpam-4274	341	32	λ(λ	λ(λ	PROPN
ejpam-4274	341	33	,	,	PUNCT
ejpam-4274	341	34	p)({x	p)({x	NOUN
ejpam-4274	341	35	}	}	PUNCT
ejpam-4274	341	36	)	)	PUNCT
ejpam-4274	341	37	6=	6=	ADP
ejpam-4274	342	1	λ(λ	λ(λ	ADP
ejpam-4274	342	2	,	,	PUNCT
ejpam-4274	342	3	p)({y	p)({y	PROPN
ejpam-4274	342	4	}	}	PUNCT
ejpam-4274	342	5	)	)	PUNCT
ejpam-4274	342	6	.	.	PUNCT
ejpam-4274	343	1	c.	c.	PROPN
ejpam-4274	343	2	boonpok	boonpok	PROPN
ejpam-4274	343	3	,	,	PUNCT
ejpam-4274	343	4	c.	c.	PROPN
ejpam-4274	343	5	viriyapong	viriyapong	PROPN
ejpam-4274	343	6	/	/	SYM
ejpam-4274	343	7	eur	eur	PROPN
ejpam-4274	343	8	.	.	PUNCT
ejpam-4274	344	1	j.	j.	PROPN
ejpam-4274	344	2	pure	pure	PROPN
ejpam-4274	344	3	appl	appl	PROPN
ejpam-4274	344	4	.	.	PROPN
ejpam-4274	344	5	math	math	PROPN
ejpam-4274	344	6	,	,	PUNCT
ejpam-4274	344	7	15	15	NUM
ejpam-4274	344	8	(	(	PUNCT
ejpam-4274	344	9	2	2	NUM
ejpam-4274	344	10	)	)	PUNCT
ejpam-4274	344	11	(	(	PUNCT
ejpam-4274	344	12	2022	2022	NUM
ejpam-4274	344	13	)	)	PUNCT
ejpam-4274	344	14	,	,	PUNCT
ejpam-4274	344	15	415	415	NUM
ejpam-4274	344	16	-	-	SYM
ejpam-4274	344	17	436	436	NUM
ejpam-4274	344	18	424	424	NUM
ejpam-4274	344	19	lemma	lemma	PROPN
ejpam-4274	344	20	11	11	NUM
ejpam-4274	344	21	.	.	PUNCT
ejpam-4274	345	1	for	for	ADP
ejpam-4274	345	2	any	any	DET
ejpam-4274	345	3	points	point	NOUN
ejpam-4274	345	4	x	x	PUNCT
ejpam-4274	345	5	and	and	CCONJ
ejpam-4274	345	6	y	y	PROPN
ejpam-4274	345	7	in	in	ADP
ejpam-4274	345	8	a	a	DET
ejpam-4274	345	9	topological	topological	ADJ
ejpam-4274	345	10	space	space	NOUN
ejpam-4274	345	11	(	(	PUNCT
ejpam-4274	345	12	x	x	X
ejpam-4274	345	13	,	,	PUNCT
ejpam-4274	345	14	τ	τ	PROPN
ejpam-4274	345	15	)	)	PUNCT
ejpam-4274	345	16	,	,	PUNCT
ejpam-4274	345	17	the	the	DET
ejpam-4274	345	18	following	follow	VERB
ejpam-4274	345	19	properties	property	NOUN
ejpam-4274	345	20	hold	hold	VERB
ejpam-4274	345	21	:	:	PUNCT
ejpam-4274	345	22	(	(	PUNCT
ejpam-4274	345	23	1	1	X
ejpam-4274	345	24	)	)	PUNCT
ejpam-4274	345	25	y	y	PROPN
ejpam-4274	345	26	∈	∈	PROPN
ejpam-4274	346	1	λ(λ	λ(λ	PROPN
ejpam-4274	346	2	,	,	PUNCT
ejpam-4274	346	3	p)({x	p)({x	NOUN
ejpam-4274	346	4	}	}	PUNCT
ejpam-4274	346	5	)	)	PUNCT
ejpam-4274	347	1	if	if	SCONJ
ejpam-4274	347	2	and	and	CCONJ
ejpam-4274	347	3	only	only	ADV
ejpam-4274	347	4	if	if	SCONJ
ejpam-4274	347	5	x	x	SYM
ejpam-4274	347	6	∈	∈	PROPN
ejpam-4274	347	7	{	{	PUNCT
ejpam-4274	347	8	y}(λ	y}(λ	PROPN
ejpam-4274	347	9	,	,	PUNCT
ejpam-4274	347	10	p	p	NOUN
ejpam-4274	347	11	)	)	PUNCT
ejpam-4274	347	12	;	;	PUNCT
ejpam-4274	347	13	(	(	PUNCT
ejpam-4274	347	14	2	2	X
ejpam-4274	347	15	)	)	PUNCT
ejpam-4274	347	16	λ(λ	λ(λ	PROPN
ejpam-4274	347	17	,	,	PUNCT
ejpam-4274	347	18	p)({x	p)({x	NOUN
ejpam-4274	347	19	}	}	PUNCT
ejpam-4274	347	20	)	)	PUNCT
ejpam-4274	347	21	=	=	SYM
ejpam-4274	347	22	λ(λ	λ(λ	PROPN
ejpam-4274	347	23	,	,	PUNCT
ejpam-4274	347	24	p)({y	p)({y	PROPN
ejpam-4274	347	25	}	}	PUNCT
ejpam-4274	347	26	)	)	PUNCT
ejpam-4274	347	27	if	if	SCONJ
ejpam-4274	347	28	and	and	CCONJ
ejpam-4274	347	29	only	only	ADV
ejpam-4274	347	30	if	if	SCONJ
ejpam-4274	347	31	{	{	PUNCT
ejpam-4274	347	32	x}(λ	x}(λ	PROPN
ejpam-4274	347	33	,	,	PUNCT
ejpam-4274	347	34	p	p	NOUN
ejpam-4274	347	35	)	)	PUNCT
ejpam-4274	347	36	=	=	SYM
ejpam-4274	347	37	{	{	PUNCT
ejpam-4274	347	38	y}(λ	y}(λ	PROPN
ejpam-4274	347	39	,	,	PUNCT
ejpam-4274	347	40	p	p	NOUN
ejpam-4274	347	41	)	)	PUNCT
ejpam-4274	347	42	.	.	PUNCT
ejpam-4274	348	1	proof	proof	NOUN
ejpam-4274	348	2	.	.	PUNCT
ejpam-4274	349	1	(	(	PUNCT
ejpam-4274	349	2	1	1	X
ejpam-4274	349	3	)	)	PUNCT
ejpam-4274	349	4	let	let	VERB
ejpam-4274	349	5	x	x	SYM
ejpam-4274	349	6	6∈	6∈	PROPN
ejpam-4274	349	7	{	{	PUNCT
ejpam-4274	349	8	y}(λ	y}(λ	PROPN
ejpam-4274	349	9	,	,	PUNCT
ejpam-4274	349	10	p	p	NOUN
ejpam-4274	349	11	)	)	PUNCT
ejpam-4274	349	12	.	.	PUNCT
ejpam-4274	350	1	then	then	ADV
ejpam-4274	350	2	,	,	PUNCT
ejpam-4274	350	3	there	there	PRON
ejpam-4274	350	4	exists	exist	VERB
ejpam-4274	350	5	a	a	DET
ejpam-4274	350	6	(	(	PUNCT
ejpam-4274	350	7	λ	λ	NOUN
ejpam-4274	350	8	,	,	PUNCT
ejpam-4274	350	9	p)-open	p)-open	VERB
ejpam-4274	350	10	set	set	VERB
ejpam-4274	350	11	u	u	PRON
ejpam-4274	350	12	such	such	ADJ
ejpam-4274	350	13	that	that	SCONJ
ejpam-4274	350	14	x	x	SYM
ejpam-4274	350	15	∈	∈	PROPN
ejpam-4274	350	16	u	u	NOUN
ejpam-4274	350	17	and	and	CCONJ
ejpam-4274	350	18	y	y	PROPN
ejpam-4274	350	19	6∈	6∈	PROPN
ejpam-4274	350	20	u	u	PROPN
ejpam-4274	350	21	.	.	PUNCT
ejpam-4274	351	1	therefore	therefore	ADV
ejpam-4274	351	2	,	,	PUNCT
ejpam-4274	351	3	y	y	PROPN
ejpam-4274	351	4	6∈	6∈	PROPN
ejpam-4274	351	5	λ(λ	λ(λ	ADV
ejpam-4274	351	6	,	,	PUNCT
ejpam-4274	351	7	p)({x	p)({x	NOUN
ejpam-4274	351	8	}	}	PUNCT
ejpam-4274	351	9	)	)	PUNCT
ejpam-4274	351	10	.	.	PUNCT
ejpam-4274	352	1	the	the	DET
ejpam-4274	352	2	converse	converse	NOUN
ejpam-4274	352	3	is	be	AUX
ejpam-4274	352	4	similarly	similarly	ADV
ejpam-4274	352	5	shown	show	VERB
ejpam-4274	352	6	.	.	PUNCT
ejpam-4274	353	1	(	(	PUNCT
ejpam-4274	353	2	2	2	X
ejpam-4274	353	3	)	)	PUNCT
ejpam-4274	353	4	suppose	suppose	VERB
ejpam-4274	353	5	that	that	SCONJ
ejpam-4274	353	6	λ(λ	λ(λ	PROPN
ejpam-4274	353	7	,	,	PUNCT
ejpam-4274	353	8	p)({x	p)({x	NOUN
ejpam-4274	353	9	}	}	PUNCT
ejpam-4274	353	10	)	)	PUNCT
ejpam-4274	354	1	=	=	SYM
ejpam-4274	354	2	λ(λ	λ(λ	PROPN
ejpam-4274	354	3	,	,	PUNCT
ejpam-4274	354	4	p)({y	p)({y	PROPN
ejpam-4274	354	5	}	}	PUNCT
ejpam-4274	354	6	)	)	PUNCT
ejpam-4274	354	7	for	for	ADP
ejpam-4274	354	8	any	any	DET
ejpam-4274	354	9	points	point	NOUN
ejpam-4274	354	10	x	x	PUNCT
ejpam-4274	354	11	and	and	CCONJ
ejpam-4274	354	12	y	y	PROPN
ejpam-4274	354	13	in	in	ADP
ejpam-4274	354	14	x.	x.	NOUN
ejpam-4274	354	15	since	since	SCONJ
ejpam-4274	354	16	x	x	PROPN
ejpam-4274	354	17	∈	∈	PROPN
ejpam-4274	354	18	λ(λ	λ(λ	PROPN
ejpam-4274	354	19	,	,	PUNCT
ejpam-4274	354	20	p)({x	p)({x	NOUN
ejpam-4274	354	21	}	}	PUNCT
ejpam-4274	354	22	)	)	PUNCT
ejpam-4274	354	23	,	,	PUNCT
ejpam-4274	354	24	x	x	PUNCT
ejpam-4274	354	25	∈	∈	PROPN
ejpam-4274	354	26	λ(λ	λ(λ	PROPN
ejpam-4274	354	27	,	,	PUNCT
ejpam-4274	354	28	p)({y	p)({y	PROPN
ejpam-4274	354	29	}	}	PUNCT
ejpam-4274	354	30	)	)	PUNCT
ejpam-4274	354	31	and	and	CCONJ
ejpam-4274	354	32	by	by	ADP
ejpam-4274	354	33	(	(	PUNCT
ejpam-4274	354	34	1	1	NUM
ejpam-4274	354	35	)	)	PUNCT
ejpam-4274	354	36	,	,	PUNCT
ejpam-4274	354	37	y	y	PROPN
ejpam-4274	354	38	∈	∈	PROPN
ejpam-4274	354	39	{	{	PUNCT
ejpam-4274	354	40	x}(λ	x}(λ	PROPN
ejpam-4274	354	41	,	,	PUNCT
ejpam-4274	354	42	p	p	NOUN
ejpam-4274	354	43	)	)	PUNCT
ejpam-4274	354	44	.	.	PUNCT
ejpam-4274	355	1	by	by	ADP
ejpam-4274	355	2	lemma	lemma	PROPN
ejpam-4274	355	3	5	5	NUM
ejpam-4274	355	4	,	,	PUNCT
ejpam-4274	355	5	{	{	PUNCT
ejpam-4274	355	6	y}(λ	y}(λ	PROPN
ejpam-4274	355	7	,	,	PUNCT
ejpam-4274	355	8	p	p	NOUN
ejpam-4274	355	9	)	)	PUNCT
ejpam-4274	355	10	⊆	⊆	NUM
ejpam-4274	355	11	{	{	PUNCT
ejpam-4274	355	12	x}(λ	x}(λ	PROPN
ejpam-4274	355	13	,	,	PUNCT
ejpam-4274	355	14	p	p	NOUN
ejpam-4274	355	15	)	)	PUNCT
ejpam-4274	355	16	.	.	PUNCT
ejpam-4274	356	1	similarly	similarly	ADV
ejpam-4274	356	2	,	,	PUNCT
ejpam-4274	356	3	we	we	PRON
ejpam-4274	356	4	have	have	VERB
ejpam-4274	356	5	{	{	PUNCT
ejpam-4274	356	6	x}(λ	x}(λ	PROPN
ejpam-4274	356	7	,	,	PUNCT
ejpam-4274	356	8	p	p	NOUN
ejpam-4274	356	9	)	)	PUNCT
ejpam-4274	356	10	⊆	⊆	NUM
ejpam-4274	356	11	{	{	PUNCT
ejpam-4274	356	12	y}(λ	y}(λ	PROPN
ejpam-4274	356	13	,	,	PUNCT
ejpam-4274	356	14	p	p	NOUN
ejpam-4274	356	15	)	)	PUNCT
ejpam-4274	356	16	and	and	CCONJ
ejpam-4274	356	17	hence	hence	ADV
ejpam-4274	356	18	{	{	PUNCT
ejpam-4274	356	19	x}(λ	x}(λ	PROPN
ejpam-4274	356	20	,	,	PUNCT
ejpam-4274	356	21	p	p	NOUN
ejpam-4274	356	22	)	)	PUNCT
ejpam-4274	356	23	=	=	SYM
ejpam-4274	356	24	{	{	PUNCT
ejpam-4274	356	25	y}(λ	y}(λ	PROPN
ejpam-4274	356	26	,	,	PUNCT
ejpam-4274	356	27	p	p	NOUN
ejpam-4274	356	28	)	)	PUNCT
ejpam-4274	356	29	.	.	PUNCT
ejpam-4274	357	1	conversely	conversely	ADV
ejpam-4274	357	2	,	,	PUNCT
ejpam-4274	357	3	suppose	suppose	VERB
ejpam-4274	357	4	that	that	SCONJ
ejpam-4274	357	5	{	{	PUNCT
ejpam-4274	357	6	x}(λ	x}(λ	PROPN
ejpam-4274	357	7	,	,	PUNCT
ejpam-4274	357	8	p	p	NOUN
ejpam-4274	357	9	)	)	PUNCT
ejpam-4274	357	10	=	=	SYM
ejpam-4274	357	11	{	{	PUNCT
ejpam-4274	357	12	y}(λ	y}(λ	PROPN
ejpam-4274	357	13	,	,	PUNCT
ejpam-4274	357	14	p	p	NOUN
ejpam-4274	357	15	)	)	PUNCT
ejpam-4274	357	16	.	.	PUNCT
ejpam-4274	358	1	since	since	SCONJ
ejpam-4274	358	2	x	x	PROPN
ejpam-4274	358	3	∈	∈	PROPN
ejpam-4274	358	4	{	{	PUNCT
ejpam-4274	358	5	x}(λ	x}(λ	PROPN
ejpam-4274	358	6	,	,	PUNCT
ejpam-4274	358	7	p	p	NOUN
ejpam-4274	358	8	)	)	PUNCT
ejpam-4274	358	9	,	,	PUNCT
ejpam-4274	358	10	x	x	PUNCT
ejpam-4274	358	11	∈	∈	PROPN
ejpam-4274	358	12	{	{	PUNCT
ejpam-4274	358	13	y}(λ	y}(λ	PROPN
ejpam-4274	358	14	,	,	PUNCT
ejpam-4274	358	15	p	p	NOUN
ejpam-4274	358	16	)	)	PUNCT
ejpam-4274	358	17	and	and	CCONJ
ejpam-4274	358	18	by	by	ADP
ejpam-4274	358	19	(	(	PUNCT
ejpam-4274	358	20	1	1	NUM
ejpam-4274	358	21	)	)	PUNCT
ejpam-4274	358	22	,	,	PUNCT
ejpam-4274	358	23	y	y	PROPN
ejpam-4274	358	24	∈	∈	PROPN
ejpam-4274	358	25	λ(λ	λ(λ	PROPN
ejpam-4274	358	26	,	,	PUNCT
ejpam-4274	358	27	p)({x	p)({x	NOUN
ejpam-4274	358	28	}	}	PUNCT
ejpam-4274	358	29	)	)	PUNCT
ejpam-4274	358	30	.	.	PUNCT
ejpam-4274	359	1	by	by	ADP
ejpam-4274	359	2	lemma	lemma	PROPN
ejpam-4274	359	3	7	7	NUM
ejpam-4274	359	4	,	,	PUNCT
ejpam-4274	359	5	λ(λ	λ(λ	PROPN
ejpam-4274	359	6	,	,	PUNCT
ejpam-4274	359	7	p)({y	p)({y	PROPN
ejpam-4274	359	8	}	}	PUNCT
ejpam-4274	359	9	)	)	PUNCT
ejpam-4274	359	10	⊆	⊆	NUM
ejpam-4274	359	11	λ(λ	λ(λ	ADP
ejpam-4274	359	12	,	,	PUNCT
ejpam-4274	359	13	p)(λ(λ	p)(λ(λ	NOUN
ejpam-4274	359	14	,	,	PUNCT
ejpam-4274	359	15	p)({x	p)({x	NOUN
ejpam-4274	359	16	}	}	PUNCT
ejpam-4274	359	17	)	)	PUNCT
ejpam-4274	359	18	)	)	PUNCT
ejpam-4274	360	1	=	=	SYM
ejpam-4274	360	2	λ(λ	λ(λ	PROPN
ejpam-4274	360	3	,	,	PUNCT
ejpam-4274	360	4	p)({x	p)({x	NOUN
ejpam-4274	360	5	}	}	PUNCT
ejpam-4274	360	6	)	)	PUNCT
ejpam-4274	360	7	.	.	PUNCT
ejpam-4274	361	1	similarly	similarly	ADV
ejpam-4274	361	2	,	,	PUNCT
ejpam-4274	361	3	we	we	PRON
ejpam-4274	361	4	have	have	VERB
ejpam-4274	361	5	λ(λ	λ(λ	PROPN
ejpam-4274	361	6	,	,	PUNCT
ejpam-4274	361	7	p)({x	p)({x	NOUN
ejpam-4274	361	8	}	}	PUNCT
ejpam-4274	361	9	)	)	PUNCT
ejpam-4274	362	1	⊆	⊆	NUM
ejpam-4274	362	2	λ(λ	λ(λ	PROPN
ejpam-4274	362	3	,	,	PUNCT
ejpam-4274	362	4	p)({y	p)({y	PROPN
ejpam-4274	362	5	}	}	PUNCT
ejpam-4274	362	6	)	)	PUNCT
ejpam-4274	362	7	and	and	CCONJ
ejpam-4274	362	8	hence	hence	ADV
ejpam-4274	362	9	λ(λ	λ(λ	PROPN
ejpam-4274	362	10	,	,	PUNCT
ejpam-4274	362	11	p)({x	p)({x	NOUN
ejpam-4274	362	12	}	}	PUNCT
ejpam-4274	362	13	)	)	PUNCT
ejpam-4274	362	14	=	=	SYM
ejpam-4274	362	15	λ(λ	λ(λ	PROPN
ejpam-4274	362	16	,	,	PUNCT
ejpam-4274	362	17	p)({y	p)({y	PROPN
ejpam-4274	362	18	}	}	PUNCT
ejpam-4274	362	19	)	)	PUNCT
ejpam-4274	362	20	.	.	PUNCT
ejpam-4274	363	1	5	5	X
ejpam-4274	363	2	.	.	PUNCT
ejpam-4274	363	3	characterizations	characterization	NOUN
ejpam-4274	363	4	of	of	ADP
ejpam-4274	363	5	λp	λp	PROPN
ejpam-4274	363	6	-	-	PUNCT
ejpam-4274	363	7	r0	r0	NOUN
ejpam-4274	363	8	spaces	space	NOUN
ejpam-4274	363	9	in	in	ADP
ejpam-4274	363	10	this	this	DET
ejpam-4274	363	11	section	section	NOUN
ejpam-4274	363	12	,	,	PUNCT
ejpam-4274	363	13	we	we	PRON
ejpam-4274	363	14	introduce	introduce	VERB
ejpam-4274	363	15	the	the	DET
ejpam-4274	363	16	concept	concept	NOUN
ejpam-4274	363	17	of	of	ADP
ejpam-4274	363	18	λp	λp	PROPN
ejpam-4274	363	19	-	-	PUNCT
ejpam-4274	363	20	r0	r0	NOUN
ejpam-4274	363	21	spaces	space	NOUN
ejpam-4274	363	22	.	.	PUNCT
ejpam-4274	364	1	moreover	moreover	ADV
ejpam-4274	364	2	,	,	PUNCT
ejpam-4274	364	3	some	some	DET
ejpam-4274	364	4	characterizations	characterization	NOUN
ejpam-4274	364	5	of	of	ADP
ejpam-4274	364	6	λp	λp	PROPN
ejpam-4274	364	7	-	-	PUNCT
ejpam-4274	364	8	r0	r0	NOUN
ejpam-4274	364	9	spaces	space	NOUN
ejpam-4274	364	10	are	be	AUX
ejpam-4274	364	11	investigated	investigate	VERB
ejpam-4274	364	12	.	.	PUNCT
ejpam-4274	365	1	definition	definition	NOUN
ejpam-4274	365	2	9	9	NUM
ejpam-4274	365	3	.	.	PUNCT
ejpam-4274	366	1	a	a	DET
ejpam-4274	366	2	topological	topological	ADJ
ejpam-4274	366	3	space	space	NOUN
ejpam-4274	366	4	(	(	PUNCT
ejpam-4274	366	5	x	x	X
ejpam-4274	366	6	,	,	PUNCT
ejpam-4274	366	7	τ	τ	X
ejpam-4274	366	8	)	)	PUNCT
ejpam-4274	366	9	is	be	AUX
ejpam-4274	366	10	called	call	VERB
ejpam-4274	366	11	a	a	DET
ejpam-4274	366	12	λp	λp	PROPN
ejpam-4274	366	13	-	-	PUNCT
ejpam-4274	366	14	r0	r0	NOUN
ejpam-4274	366	15	space	space	NOUN
ejpam-4274	366	16	if	if	SCONJ
ejpam-4274	366	17	,	,	PUNCT
ejpam-4274	366	18	for	for	SCONJ
ejpam-4274	366	19	each	each	DET
ejpam-4274	366	20	(	(	PUNCT
ejpam-4274	366	21	λ	λ	PROPN
ejpam-4274	366	22	,	,	PUNCT
ejpam-4274	366	23	p)-open	p)-open	VERB
ejpam-4274	366	24	set	set	VERB
ejpam-4274	366	25	u	u	NOUN
ejpam-4274	366	26	and	and	CCONJ
ejpam-4274	366	27	each	each	DET
ejpam-4274	366	28	x	x	SYM
ejpam-4274	366	29	∈	∈	PROPN
ejpam-4274	366	30	u	u	NOUN
ejpam-4274	366	31	,	,	PUNCT
ejpam-4274	366	32	{	{	PUNCT
ejpam-4274	366	33	x}(λ	x}(λ	PROPN
ejpam-4274	366	34	,	,	PUNCT
ejpam-4274	366	35	p	p	NOUN
ejpam-4274	366	36	)	)	PUNCT
ejpam-4274	366	37	⊆	⊆	NUM
ejpam-4274	366	38	u	u	NOUN
ejpam-4274	366	39	.	.	PUNCT
ejpam-4274	367	1	theorem	theorem	VERB
ejpam-4274	367	2	11	11	NUM
ejpam-4274	367	3	.	.	PUNCT
ejpam-4274	368	1	for	for	ADP
ejpam-4274	368	2	a	a	DET
ejpam-4274	368	3	topological	topological	ADJ
ejpam-4274	368	4	space	space	NOUN
ejpam-4274	368	5	(	(	PUNCT
ejpam-4274	368	6	x	x	X
ejpam-4274	368	7	,	,	PUNCT
ejpam-4274	368	8	τ	τ	PROPN
ejpam-4274	368	9	)	)	PUNCT
ejpam-4274	368	10	,	,	PUNCT
ejpam-4274	368	11	the	the	DET
ejpam-4274	368	12	following	follow	VERB
ejpam-4274	368	13	properties	property	NOUN
ejpam-4274	368	14	are	be	AUX
ejpam-4274	368	15	equivalent	equivalent	ADJ
ejpam-4274	368	16	:	:	PUNCT
ejpam-4274	368	17	(	(	PUNCT
ejpam-4274	368	18	1	1	X
ejpam-4274	368	19	)	)	PUNCT
ejpam-4274	368	20	(	(	PUNCT
ejpam-4274	368	21	x	x	X
ejpam-4274	368	22	,	,	PUNCT
ejpam-4274	368	23	τ	τ	X
ejpam-4274	368	24	)	)	PUNCT
ejpam-4274	368	25	is	be	AUX
ejpam-4274	368	26	λp	λp	PROPN
ejpam-4274	368	27	-	-	PUNCT
ejpam-4274	368	28	r0	r0	NOUN
ejpam-4274	368	29	;	;	PUNCT
ejpam-4274	368	30	(	(	PUNCT
ejpam-4274	368	31	2	2	X
ejpam-4274	368	32	)	)	PUNCT
ejpam-4274	368	33	for	for	SCONJ
ejpam-4274	368	34	each	each	DET
ejpam-4274	368	35	(	(	PUNCT
ejpam-4274	368	36	λ	λ	PROPN
ejpam-4274	368	37	,	,	PUNCT
ejpam-4274	368	38	p)-closed	p)-close	VERB
ejpam-4274	368	39	set	set	VERB
ejpam-4274	368	40	f	f	PROPN
ejpam-4274	368	41	and	and	CCONJ
ejpam-4274	368	42	each	each	DET
ejpam-4274	368	43	x	x	SYM
ejpam-4274	368	44	∈	∈	PROPN
ejpam-4274	368	45	x	x	X
ejpam-4274	368	46	−	−	PROPN
ejpam-4274	368	47	f	f	NOUN
ejpam-4274	368	48	,	,	PUNCT
ejpam-4274	368	49	there	there	PRON
ejpam-4274	368	50	exists	exist	VERB
ejpam-4274	368	51	a	a	DET
ejpam-4274	368	52	(	(	PUNCT
ejpam-4274	368	53	λ	λ	NOUN
ejpam-4274	368	54	,	,	PUNCT
ejpam-4274	368	55	p)-open	p)-open	VERB
ejpam-4274	368	56	set	set	VERB
ejpam-4274	368	57	u	u	PRON
ejpam-4274	368	58	such	such	ADJ
ejpam-4274	368	59	that	that	SCONJ
ejpam-4274	368	60	f	f	PROPN
ejpam-4274	368	61	⊆	⊆	NUM
ejpam-4274	368	62	u	u	NOUN
ejpam-4274	368	63	and	and	CCONJ
ejpam-4274	368	64	x	x	SYM
ejpam-4274	368	65	6∈	6∈	NOUN
ejpam-4274	368	66	u	u	NOUN
ejpam-4274	368	67	;	;	PUNCT
ejpam-4274	368	68	(	(	PUNCT
ejpam-4274	368	69	3	3	X
ejpam-4274	368	70	)	)	PUNCT
ejpam-4274	368	71	for	for	ADP
ejpam-4274	368	72	each	each	DET
ejpam-4274	368	73	(	(	PUNCT
ejpam-4274	368	74	λ	λ	PROPN
ejpam-4274	368	75	,	,	PUNCT
ejpam-4274	368	76	p)-closed	p)-close	VERB
ejpam-4274	368	77	set	set	VERB
ejpam-4274	368	78	f	f	PROPN
ejpam-4274	368	79	and	and	CCONJ
ejpam-4274	368	80	each	each	DET
ejpam-4274	368	81	x	x	SYM
ejpam-4274	368	82	∈	∈	PROPN
ejpam-4274	368	83	x	x	X
ejpam-4274	369	1	−	−	PROPN
ejpam-4274	369	2	f	f	PROPN
ejpam-4274	369	3	,	,	PUNCT
ejpam-4274	369	4	f	f	PROPN
ejpam-4274	369	5	∩	∩	PROPN
ejpam-4274	369	6	{	{	PUNCT
ejpam-4274	369	7	x}(λ	x}(λ	PROPN
ejpam-4274	369	8	,	,	PUNCT
ejpam-4274	369	9	p	p	NOUN
ejpam-4274	369	10	)	)	PUNCT
ejpam-4274	369	11	=	=	NOUN
ejpam-4274	369	12	∅	∅	NOUN
ejpam-4274	369	13	;	;	PUNCT
ejpam-4274	369	14	(	(	PUNCT
ejpam-4274	369	15	4	4	X
ejpam-4274	369	16	)	)	PUNCT
ejpam-4274	369	17	for	for	ADP
ejpam-4274	369	18	each	each	DET
ejpam-4274	369	19	x	x	NOUN
ejpam-4274	369	20	,	,	PUNCT
ejpam-4274	369	21	y	y	PROPN
ejpam-4274	369	22	∈	∈	PROPN
ejpam-4274	369	23	x	x	X
ejpam-4274	369	24	,	,	PUNCT
ejpam-4274	369	25	{	{	PUNCT
ejpam-4274	369	26	x}(λ	x}(λ	PROPN
ejpam-4274	369	27	,	,	PUNCT
ejpam-4274	369	28	p	p	NOUN
ejpam-4274	369	29	)	)	PUNCT
ejpam-4274	369	30	=	=	SYM
ejpam-4274	369	31	{	{	PUNCT
ejpam-4274	369	32	y}(λ	y}(λ	PROPN
ejpam-4274	369	33	,	,	PUNCT
ejpam-4274	369	34	p	p	NOUN
ejpam-4274	369	35	)	)	PUNCT
ejpam-4274	369	36	or	or	CCONJ
ejpam-4274	369	37	{	{	PUNCT
ejpam-4274	369	38	x}(λ	x}(λ	PROPN
ejpam-4274	369	39	,	,	PUNCT
ejpam-4274	369	40	p	p	NOUN
ejpam-4274	369	41	)	)	PUNCT
ejpam-4274	369	42	∩	∩	NOUN
ejpam-4274	369	43	{	{	PUNCT
ejpam-4274	369	44	y}(λ	y}(λ	PROPN
ejpam-4274	369	45	,	,	PUNCT
ejpam-4274	369	46	p	p	NOUN
ejpam-4274	369	47	)	)	PUNCT
ejpam-4274	369	48	=	=	PUNCT
ejpam-4274	369	49	∅.	∅.	NOUN
ejpam-4274	369	50	proof	proof	NOUN
ejpam-4274	369	51	.	.	PUNCT
ejpam-4274	370	1	(	(	PUNCT
ejpam-4274	370	2	1	1	X
ejpam-4274	370	3	)	)	PUNCT
ejpam-4274	370	4	⇒	⇒	NOUN
ejpam-4274	370	5	(	(	PUNCT
ejpam-4274	370	6	2	2	NUM
ejpam-4274	370	7	):	):	PUNCT
ejpam-4274	370	8	let	let	VERB
ejpam-4274	370	9	f	f	PRON
ejpam-4274	370	10	be	be	AUX
ejpam-4274	370	11	a	a	DET
ejpam-4274	370	12	(	(	PUNCT
ejpam-4274	370	13	λ	λ	PROPN
ejpam-4274	370	14	,	,	PUNCT
ejpam-4274	370	15	p)-closed	p)-close	VERB
ejpam-4274	370	16	set	set	NOUN
ejpam-4274	370	17	and	and	CCONJ
ejpam-4274	370	18	let	let	VERB
ejpam-4274	370	19	x	x	SYM
ejpam-4274	370	20	∈	∈	PROPN
ejpam-4274	370	21	x	x	X
ejpam-4274	370	22	−	−	PROPN
ejpam-4274	370	23	f	f	X
ejpam-4274	370	24	.	.	PUNCT
ejpam-4274	371	1	then	then	ADV
ejpam-4274	371	2	,	,	PUNCT
ejpam-4274	371	3	we	we	PRON
ejpam-4274	371	4	have	have	VERB
ejpam-4274	371	5	{	{	PUNCT
ejpam-4274	371	6	x}(λ	x}(λ	PROPN
ejpam-4274	371	7	,	,	PUNCT
ejpam-4274	371	8	p	p	NOUN
ejpam-4274	371	9	)	)	PUNCT
ejpam-4274	371	10	⊆	⊆	NUM
ejpam-4274	371	11	x	x	SYM
ejpam-4274	371	12	−f	−f	NOUN
ejpam-4274	371	13	.	.	PUNCT
ejpam-4274	372	1	let	let	VERB
ejpam-4274	372	2	u	u	PRON
ejpam-4274	372	3	=	=	NOUN
ejpam-4274	372	4	x	x	SYM
ejpam-4274	372	5	−{x}(λ	−{x}(λ	VERB
ejpam-4274	372	6	,	,	PUNCT
ejpam-4274	372	7	p	p	NOUN
ejpam-4274	372	8	)	)	PUNCT
ejpam-4274	372	9	,	,	PUNCT
ejpam-4274	372	10	then	then	ADV
ejpam-4274	372	11	u	u	NOUN
ejpam-4274	372	12	is	be	AUX
ejpam-4274	372	13	a	a	DET
ejpam-4274	372	14	(	(	PUNCT
ejpam-4274	372	15	λ	λ	NOUN
ejpam-4274	372	16	,	,	PUNCT
ejpam-4274	372	17	p)-open	p)-open	VERB
ejpam-4274	372	18	set	set	VERB
ejpam-4274	372	19	such	such	ADJ
ejpam-4274	372	20	that	that	SCONJ
ejpam-4274	372	21	f	f	PROPN
ejpam-4274	372	22	⊆	⊆	NUM
ejpam-4274	372	23	u	u	NOUN
ejpam-4274	372	24	and	and	CCONJ
ejpam-4274	372	25	x	x	SYM
ejpam-4274	372	26	6∈	6∈	PROPN
ejpam-4274	372	27	u	u	PROPN
ejpam-4274	372	28	.	.	PUNCT
ejpam-4274	373	1	(	(	PUNCT
ejpam-4274	373	2	2	2	X
ejpam-4274	373	3	)	)	PUNCT
ejpam-4274	373	4	⇒	⇒	NOUN
ejpam-4274	373	5	(	(	PUNCT
ejpam-4274	373	6	3	3	NUM
ejpam-4274	373	7	):	):	PUNCT
ejpam-4274	373	8	let	let	VERB
ejpam-4274	373	9	f	f	PRON
ejpam-4274	373	10	be	be	AUX
ejpam-4274	373	11	a	a	DET
ejpam-4274	373	12	(	(	PUNCT
ejpam-4274	373	13	λ	λ	PROPN
ejpam-4274	373	14	,	,	PUNCT
ejpam-4274	373	15	p)-closed	p)-close	VERB
ejpam-4274	373	16	set	set	NOUN
ejpam-4274	373	17	and	and	CCONJ
ejpam-4274	373	18	let	let	VERB
ejpam-4274	373	19	x	x	SYM
ejpam-4274	373	20	∈	∈	PROPN
ejpam-4274	373	21	x	x	X
ejpam-4274	373	22	−	−	PROPN
ejpam-4274	373	23	f	f	X
ejpam-4274	373	24	.	.	PUNCT
ejpam-4274	374	1	there	there	PRON
ejpam-4274	374	2	exists	exist	VERB
ejpam-4274	374	3	a	a	DET
ejpam-4274	374	4	(	(	PUNCT
ejpam-4274	374	5	λ	λ	NOUN
ejpam-4274	374	6	,	,	PUNCT
ejpam-4274	374	7	p)-open	p)-open	VERB
ejpam-4274	374	8	set	set	VERB
ejpam-4274	374	9	u	u	PRON
ejpam-4274	374	10	such	such	ADJ
ejpam-4274	374	11	that	that	SCONJ
ejpam-4274	374	12	f	f	PROPN
ejpam-4274	374	13	⊆	⊆	NUM
ejpam-4274	374	14	u	u	NOUN
ejpam-4274	374	15	and	and	CCONJ
ejpam-4274	374	16	x	x	SYM
ejpam-4274	374	17	6∈	6∈	PROPN
ejpam-4274	374	18	u	u	PROPN
ejpam-4274	374	19	.	.	PUNCT
ejpam-4274	375	1	thus	thus	ADV
ejpam-4274	375	2	,	,	PUNCT
ejpam-4274	375	3	u	u	PROPN
ejpam-4274	375	4	∩	∩	NOUN
ejpam-4274	375	5	{	{	PUNCT
ejpam-4274	375	6	x}(λ	x}(λ	PROPN
ejpam-4274	375	7	,	,	PUNCT
ejpam-4274	375	8	p	p	NOUN
ejpam-4274	375	9	)	)	PUNCT
ejpam-4274	375	10	=	=	NOUN
ejpam-4274	375	11	∅	∅	NOUN
ejpam-4274	375	12	and	and	CCONJ
ejpam-4274	375	13	hence	hence	ADV
ejpam-4274	375	14	f	f	PROPN
ejpam-4274	375	15	∩	∩	PROPN
ejpam-4274	375	16	{	{	PUNCT
ejpam-4274	375	17	x}(λ	x}(λ	PROPN
ejpam-4274	375	18	,	,	PUNCT
ejpam-4274	375	19	p	p	NOUN
ejpam-4274	375	20	)	)	PUNCT
ejpam-4274	375	21	=	=	SYM
ejpam-4274	375	22	∅.	∅.	X
ejpam-4274	375	23	(	(	PUNCT
ejpam-4274	375	24	3	3	NUM
ejpam-4274	375	25	)	)	PUNCT
ejpam-4274	375	26	⇒	⇒	NOUN
ejpam-4274	375	27	(	(	PUNCT
ejpam-4274	375	28	4	4	NUM
ejpam-4274	375	29	):	):	PUNCT
ejpam-4274	375	30	let	let	VERB
ejpam-4274	375	31	x	x	PRON
ejpam-4274	375	32	,	,	PUNCT
ejpam-4274	375	33	y	y	PROPN
ejpam-4274	375	34	be	be	VERB
ejpam-4274	375	35	distinct	distinct	ADJ
ejpam-4274	375	36	points	point	NOUN
ejpam-4274	375	37	of	of	ADP
ejpam-4274	375	38	x.	x.	NOUN
ejpam-4274	375	39	suppose	suppose	VERB
ejpam-4274	375	40	that	that	SCONJ
ejpam-4274	375	41	{	{	PUNCT
ejpam-4274	375	42	x}(λ	x}(λ	PROPN
ejpam-4274	375	43	,	,	PUNCT
ejpam-4274	375	44	p	p	NOUN
ejpam-4274	375	45	)	)	PUNCT
ejpam-4274	375	46	6=	6=	ADP
ejpam-4274	375	47	{	{	PUNCT
ejpam-4274	375	48	y}(λ	y}(λ	PROPN
ejpam-4274	375	49	,	,	PUNCT
ejpam-4274	375	50	p	p	NOUN
ejpam-4274	375	51	)	)	PUNCT
ejpam-4274	375	52	.	.	PUNCT
ejpam-4274	376	1	by	by	ADP
ejpam-4274	376	2	(	(	PUNCT
ejpam-4274	376	3	3	3	NUM
ejpam-4274	376	4	)	)	PUNCT
ejpam-4274	376	5	,	,	PUNCT
ejpam-4274	376	6	x	x	PUNCT
ejpam-4274	376	7	∈	∈	PROPN
ejpam-4274	376	8	{	{	PUNCT
ejpam-4274	376	9	y}(λ	y}(λ	PROPN
ejpam-4274	376	10	,	,	PUNCT
ejpam-4274	376	11	p	p	NOUN
ejpam-4274	376	12	)	)	PUNCT
ejpam-4274	376	13	and	and	CCONJ
ejpam-4274	376	14	y	y	PROPN
ejpam-4274	376	15	∈	∈	PROPN
ejpam-4274	376	16	{	{	PUNCT
ejpam-4274	376	17	x}(λ	x}(λ	PROPN
ejpam-4274	376	18	,	,	PUNCT
ejpam-4274	376	19	p	p	NOUN
ejpam-4274	376	20	)	)	PUNCT
ejpam-4274	376	21	.	.	PUNCT
ejpam-4274	377	1	thus	thus	ADV
ejpam-4274	377	2	,	,	PUNCT
ejpam-4274	377	3	{	{	PUNCT
ejpam-4274	377	4	x}(λ	x}(λ	PROPN
ejpam-4274	377	5	,	,	PUNCT
ejpam-4274	377	6	p	p	NOUN
ejpam-4274	377	7	)	)	PUNCT
ejpam-4274	377	8	⊆	⊆	NUM
ejpam-4274	377	9	{	{	PUNCT
ejpam-4274	377	10	y}(λ	y}(λ	PROPN
ejpam-4274	377	11	,	,	PUNCT
ejpam-4274	377	12	p	p	NOUN
ejpam-4274	377	13	)	)	PUNCT
ejpam-4274	377	14	⊆	⊆	NUM
ejpam-4274	377	15	{	{	PUNCT
ejpam-4274	377	16	x}(λ	x}(λ	PROPN
ejpam-4274	377	17	,	,	PUNCT
ejpam-4274	377	18	p	p	NOUN
ejpam-4274	377	19	)	)	PUNCT
ejpam-4274	377	20	and	and	CCONJ
ejpam-4274	377	21	hence	hence	ADV
ejpam-4274	377	22	{	{	PUNCT
ejpam-4274	377	23	x}(λ	x}(λ	PROPN
ejpam-4274	377	24	,	,	PUNCT
ejpam-4274	377	25	p	p	NOUN
ejpam-4274	377	26	)	)	PUNCT
ejpam-4274	377	27	=	=	SYM
ejpam-4274	377	28	{	{	PUNCT
ejpam-4274	377	29	y}(λ	y}(λ	PROPN
ejpam-4274	377	30	,	,	PUNCT
ejpam-4274	377	31	p	p	NOUN
ejpam-4274	377	32	)	)	PUNCT
ejpam-4274	377	33	.	.	PUNCT
ejpam-4274	378	1	c.	c.	PROPN
ejpam-4274	378	2	boonpok	boonpok	PROPN
ejpam-4274	378	3	,	,	PUNCT
ejpam-4274	378	4	c.	c.	PROPN
ejpam-4274	378	5	viriyapong	viriyapong	PROPN
ejpam-4274	378	6	/	/	SYM
ejpam-4274	378	7	eur	eur	PROPN
ejpam-4274	378	8	.	.	PUNCT
ejpam-4274	379	1	j.	j.	PROPN
ejpam-4274	379	2	pure	pure	PROPN
ejpam-4274	379	3	appl	appl	PROPN
ejpam-4274	379	4	.	.	PROPN
ejpam-4274	379	5	math	math	PROPN
ejpam-4274	379	6	,	,	PUNCT
ejpam-4274	379	7	15	15	NUM
ejpam-4274	379	8	(	(	PUNCT
ejpam-4274	379	9	2	2	NUM
ejpam-4274	379	10	)	)	PUNCT
ejpam-4274	379	11	(	(	PUNCT
ejpam-4274	379	12	2022	2022	NUM
ejpam-4274	379	13	)	)	PUNCT
ejpam-4274	379	14	,	,	PUNCT
ejpam-4274	379	15	415	415	NUM
ejpam-4274	379	16	-	-	SYM
ejpam-4274	379	17	436	436	NUM
ejpam-4274	379	18	425	425	NUM
ejpam-4274	379	19	(	(	PUNCT
ejpam-4274	379	20	4	4	NUM
ejpam-4274	379	21	)	)	PUNCT
ejpam-4274	379	22	⇒	⇒	NOUN
ejpam-4274	379	23	(	(	PUNCT
ejpam-4274	379	24	1	1	NUM
ejpam-4274	379	25	):	):	PUNCT
ejpam-4274	379	26	let	let	VERB
ejpam-4274	379	27	u	u	PRON
ejpam-4274	379	28	be	be	AUX
ejpam-4274	379	29	a	a	DET
ejpam-4274	379	30	(	(	PUNCT
ejpam-4274	379	31	λ	λ	NOUN
ejpam-4274	379	32	,	,	PUNCT
ejpam-4274	379	33	p)-open	p)-open	VERB
ejpam-4274	379	34	set	set	VERB
ejpam-4274	379	35	and	and	CCONJ
ejpam-4274	379	36	let	let	VERB
ejpam-4274	379	37	x	x	PUNCT
ejpam-4274	379	38	∈	∈	PROPN
ejpam-4274	379	39	u	u	NOUN
ejpam-4274	379	40	.	.	PUNCT
ejpam-4274	380	1	for	for	ADP
ejpam-4274	380	2	each	each	DET
ejpam-4274	380	3	y	y	PROPN
ejpam-4274	380	4	6∈	6∈	PROPN
ejpam-4274	380	5	u	u	NOUN
ejpam-4274	380	6	,	,	PUNCT
ejpam-4274	380	7	we	we	PRON
ejpam-4274	380	8	have	have	VERB
ejpam-4274	380	9	u	u	NOUN
ejpam-4274	380	10	∩	∩	NOUN
ejpam-4274	380	11	{	{	PUNCT
ejpam-4274	380	12	y}(λ	y}(λ	PROPN
ejpam-4274	380	13	,	,	PUNCT
ejpam-4274	380	14	p	p	NOUN
ejpam-4274	380	15	)	)	PUNCT
ejpam-4274	380	16	=	=	NOUN
ejpam-4274	380	17	∅	∅	NOUN
ejpam-4274	380	18	and	and	CCONJ
ejpam-4274	380	19	hence	hence	ADV
ejpam-4274	380	20	x	x	PROPN
ejpam-4274	380	21	6∈	6∈	PROPN
ejpam-4274	380	22	{	{	PUNCT
ejpam-4274	380	23	y}(λ	y}(λ	PROPN
ejpam-4274	380	24	,	,	PUNCT
ejpam-4274	380	25	p	p	NOUN
ejpam-4274	380	26	)	)	PUNCT
ejpam-4274	380	27	.	.	PUNCT
ejpam-4274	381	1	therefore	therefore	ADV
ejpam-4274	381	2	,	,	PUNCT
ejpam-4274	381	3	{	{	PUNCT
ejpam-4274	381	4	y}(λ	y}(λ	PROPN
ejpam-4274	381	5	,	,	PUNCT
ejpam-4274	381	6	p	p	NOUN
ejpam-4274	381	7	)	)	PUNCT
ejpam-4274	381	8	6=	6=	ADP
ejpam-4274	381	9	{	{	PUNCT
ejpam-4274	381	10	x}(λ	x}(λ	PROPN
ejpam-4274	381	11	,	,	PUNCT
ejpam-4274	381	12	p	p	NOUN
ejpam-4274	381	13	)	)	PUNCT
ejpam-4274	381	14	.	.	PUNCT
ejpam-4274	382	1	by	by	ADP
ejpam-4274	382	2	(	(	PUNCT
ejpam-4274	382	3	4	4	NUM
ejpam-4274	382	4	)	)	PUNCT
ejpam-4274	382	5	,	,	PUNCT
ejpam-4274	382	6	{	{	PUNCT
ejpam-4274	382	7	x}(λ	x}(λ	PROPN
ejpam-4274	382	8	,	,	PUNCT
ejpam-4274	382	9	p	p	NOUN
ejpam-4274	382	10	)	)	PUNCT
ejpam-4274	382	11	∩	∩	NOUN
ejpam-4274	382	12	{	{	PUNCT
ejpam-4274	382	13	y}(λ	y}(λ	PROPN
ejpam-4274	382	14	,	,	PUNCT
ejpam-4274	382	15	p	p	NOUN
ejpam-4274	382	16	)	)	PUNCT
ejpam-4274	382	17	=	=	PUNCT
ejpam-4274	382	18	∅.	∅.	NOUN
ejpam-4274	382	19	since	since	SCONJ
ejpam-4274	382	20	x	x	SYM
ejpam-4274	382	21	−	−	PROPN
ejpam-4274	382	22	u	u	NOUN
ejpam-4274	382	23	is	be	AUX
ejpam-4274	382	24	(	(	PUNCT
ejpam-4274	382	25	λ	λ	X
ejpam-4274	382	26	,	,	PUNCT
ejpam-4274	382	27	p)-closed	p)-close	VERB
ejpam-4274	382	28	,	,	PUNCT
ejpam-4274	382	29	y	y	PROPN
ejpam-4274	382	30	∈	∈	PROPN
ejpam-4274	382	31	{	{	PUNCT
ejpam-4274	382	32	y}(λ	y}(λ	PROPN
ejpam-4274	382	33	,	,	PUNCT
ejpam-4274	382	34	p	p	NOUN
ejpam-4274	382	35	)	)	PUNCT
ejpam-4274	382	36	⊆	⊆	NUM
ejpam-4274	382	37	x	x	SYM
ejpam-4274	382	38	−	−	NOUN
ejpam-4274	382	39	u	u	NOUN
ejpam-4274	382	40	and	and	CCONJ
ejpam-4274	382	41	∪y∈x−u{y}(λ	∪y∈x−u{y}(λ	NOUN
ejpam-4274	382	42	,	,	PUNCT
ejpam-4274	382	43	p	p	NOUN
ejpam-4274	382	44	)	)	PUNCT
ejpam-4274	382	45	=	=	PUNCT
ejpam-4274	383	1	x	x	PUNCT
ejpam-4274	383	2	−	−	PROPN
ejpam-4274	383	3	u	u	NOUN
ejpam-4274	383	4	.	.	PUNCT
ejpam-4274	384	1	thus	thus	ADV
ejpam-4274	384	2	,	,	PUNCT
ejpam-4274	384	3	{	{	PUNCT
ejpam-4274	384	4	x}(λ	x}(λ	PROPN
ejpam-4274	384	5	,	,	PUNCT
ejpam-4274	384	6	p	p	NOUN
ejpam-4274	384	7	)	)	PUNCT
ejpam-4274	384	8	∩	∩	NOUN
ejpam-4274	384	9	(	(	PUNCT
ejpam-4274	384	10	x	x	SYM
ejpam-4274	384	11	−	−	PROPN
ejpam-4274	384	12	u	u	NOUN
ejpam-4274	384	13	)	)	PUNCT
ejpam-4274	384	14	=	=	SYM
ejpam-4274	384	15	{	{	PUNCT
ejpam-4274	384	16	x}(λ	x}(λ	PROPN
ejpam-4274	384	17	,	,	PUNCT
ejpam-4274	384	18	p	p	NOUN
ejpam-4274	384	19	)	)	PUNCT
ejpam-4274	384	20	∩	∩	NOUN
ejpam-4274	384	21	[	[	X
ejpam-4274	384	22	∪y∈x−u{y}(λ	∪y∈x−u{y}(λ	NOUN
ejpam-4274	384	23	,	,	PUNCT
ejpam-4274	384	24	p	p	NOUN
ejpam-4274	384	25	)	)	PUNCT
ejpam-4274	384	26	]	]	PUNCT
ejpam-4274	384	27	=	=	PUNCT
ejpam-4274	384	28	∪y∈x−u	∪y∈x−u	NOUN
ejpam-4274	384	29	[	[	X
ejpam-4274	384	30	{	{	PUNCT
ejpam-4274	384	31	x}(λ	x}(λ	PROPN
ejpam-4274	384	32	,	,	PUNCT
ejpam-4274	384	33	p	p	NOUN
ejpam-4274	384	34	)	)	PUNCT
ejpam-4274	384	35	∩	∩	NOUN
ejpam-4274	384	36	{	{	PUNCT
ejpam-4274	384	37	y}(λ	y}(λ	PROPN
ejpam-4274	384	38	,	,	PUNCT
ejpam-4274	384	39	p	p	NOUN
ejpam-4274	384	40	)	)	PUNCT
ejpam-4274	384	41	]	]	PUNCT
ejpam-4274	384	42	=	=	PUNCT
ejpam-4274	384	43	∅	∅	NOUN
ejpam-4274	384	44	and	and	CCONJ
ejpam-4274	384	45	hence	hence	ADV
ejpam-4274	384	46	{	{	PUNCT
ejpam-4274	384	47	x}(λ	x}(λ	PROPN
ejpam-4274	384	48	,	,	PUNCT
ejpam-4274	384	49	p	p	NOUN
ejpam-4274	384	50	)	)	PUNCT
ejpam-4274	384	51	⊆	⊆	NUM
ejpam-4274	384	52	u	u	NOUN
ejpam-4274	384	53	.	.	PUNCT
ejpam-4274	385	1	this	this	PRON
ejpam-4274	385	2	shows	show	VERB
ejpam-4274	385	3	that	that	SCONJ
ejpam-4274	385	4	(	(	PUNCT
ejpam-4274	385	5	x	x	X
ejpam-4274	385	6	,	,	PUNCT
ejpam-4274	385	7	τ	τ	X
ejpam-4274	385	8	)	)	PUNCT
ejpam-4274	385	9	is	be	AUX
ejpam-4274	385	10	λp	λp	PROPN
ejpam-4274	385	11	-	-	PUNCT
ejpam-4274	385	12	r0	r0	NOUN
ejpam-4274	385	13	.	.	PUNCT
ejpam-4274	386	1	corollary	corollary	ADJ
ejpam-4274	386	2	2	2	NUM
ejpam-4274	386	3	.	.	PUNCT
ejpam-4274	387	1	a	a	DET
ejpam-4274	387	2	topological	topological	ADJ
ejpam-4274	387	3	space	space	NOUN
ejpam-4274	387	4	(	(	PUNCT
ejpam-4274	387	5	x	x	X
ejpam-4274	387	6	,	,	PUNCT
ejpam-4274	387	7	τ	τ	X
ejpam-4274	387	8	)	)	PUNCT
ejpam-4274	387	9	is	be	AUX
ejpam-4274	387	10	λp	λp	NOUN
ejpam-4274	387	11	-	-	PUNCT
ejpam-4274	387	12	r0	r0	NOUN
ejpam-4274	387	13	if	if	SCONJ
ejpam-4274	387	14	and	and	CCONJ
ejpam-4274	387	15	only	only	ADV
ejpam-4274	387	16	if	if	SCONJ
ejpam-4274	387	17	,	,	PUNCT
ejpam-4274	387	18	for	for	ADP
ejpam-4274	387	19	each	each	DET
ejpam-4274	387	20	x	x	NOUN
ejpam-4274	387	21	,	,	PUNCT
ejpam-4274	387	22	y	y	PROPN
ejpam-4274	387	23	∈	∈	PROPN
ejpam-4274	387	24	x	x	X
ejpam-4274	387	25	,	,	PUNCT
ejpam-4274	387	26	{	{	PUNCT
ejpam-4274	387	27	x}(λ	x}(λ	PROPN
ejpam-4274	387	28	,	,	PUNCT
ejpam-4274	387	29	p	p	NOUN
ejpam-4274	387	30	)	)	PUNCT
ejpam-4274	387	31	6=	6=	ADP
ejpam-4274	387	32	{	{	PUNCT
ejpam-4274	387	33	y}(λ	y}(λ	PROPN
ejpam-4274	387	34	,	,	PUNCT
ejpam-4274	387	35	p	p	NOUN
ejpam-4274	387	36	)	)	PUNCT
ejpam-4274	387	37	implies	imply	VERB
ejpam-4274	387	38	{	{	PUNCT
ejpam-4274	387	39	x}(λ	x}(λ	PROPN
ejpam-4274	387	40	,	,	PUNCT
ejpam-4274	387	41	p	p	NOUN
ejpam-4274	387	42	)	)	PUNCT
ejpam-4274	387	43	∩	∩	NOUN
ejpam-4274	387	44	{	{	PUNCT
ejpam-4274	387	45	y}(λ	y}(λ	PROPN
ejpam-4274	387	46	,	,	PUNCT
ejpam-4274	387	47	p	p	NOUN
ejpam-4274	387	48	)	)	PUNCT
ejpam-4274	387	49	=	=	PUNCT
ejpam-4274	387	50	∅.	∅.	NOUN
ejpam-4274	387	51	proof	proof	NOUN
ejpam-4274	387	52	.	.	PUNCT
ejpam-4274	388	1	this	this	PRON
ejpam-4274	388	2	is	be	AUX
ejpam-4274	388	3	obvious	obvious	ADJ
ejpam-4274	388	4	by	by	ADP
ejpam-4274	388	5	theorem	theorem	NOUN
ejpam-4274	388	6	11	11	NUM
ejpam-4274	388	7	.	.	PUNCT
ejpam-4274	389	1	conversely	conversely	ADV
ejpam-4274	389	2	,	,	PUNCT
ejpam-4274	389	3	let	let	VERB
ejpam-4274	389	4	u	u	PRON
ejpam-4274	389	5	be	be	AUX
ejpam-4274	389	6	a	a	DET
ejpam-4274	389	7	(	(	PUNCT
ejpam-4274	389	8	λ	λ	NOUN
ejpam-4274	389	9	,	,	PUNCT
ejpam-4274	389	10	p)-open	p)-open	VERB
ejpam-4274	389	11	set	set	VERB
ejpam-4274	389	12	and	and	CCONJ
ejpam-4274	389	13	let	let	VERB
ejpam-4274	389	14	x	x	PUNCT
ejpam-4274	389	15	∈	∈	PROPN
ejpam-4274	389	16	u	u	NOUN
ejpam-4274	389	17	.	.	PUNCT
ejpam-4274	390	1	if	if	SCONJ
ejpam-4274	390	2	y	y	PROPN
ejpam-4274	390	3	6∈	6∈	PROPN
ejpam-4274	390	4	u	u	PROPN
ejpam-4274	390	5	,	,	PUNCT
ejpam-4274	390	6	then	then	ADV
ejpam-4274	390	7	u	u	NOUN
ejpam-4274	390	8	∩	∩	NOUN
ejpam-4274	390	9	{	{	PUNCT
ejpam-4274	390	10	y}(λ	y}(λ	PROPN
ejpam-4274	390	11	,	,	PUNCT
ejpam-4274	390	12	p	p	NOUN
ejpam-4274	390	13	)	)	PUNCT
ejpam-4274	390	14	=	=	PUNCT
ejpam-4274	390	15	∅.	∅.	ADP
ejpam-4274	390	16	thus	thus	ADV
ejpam-4274	390	17	,	,	PUNCT
ejpam-4274	390	18	x	x	PROPN
ejpam-4274	390	19	6∈	6∈	PROPN
ejpam-4274	390	20	{	{	PUNCT
ejpam-4274	390	21	y}(λ	y}(λ	PROPN
ejpam-4274	390	22	,	,	PUNCT
ejpam-4274	390	23	p	p	NOUN
ejpam-4274	390	24	)	)	PUNCT
ejpam-4274	390	25	and	and	CCONJ
ejpam-4274	390	26	{	{	PUNCT
ejpam-4274	390	27	x}(λ	x}(λ	PROPN
ejpam-4274	390	28	,	,	PUNCT
ejpam-4274	390	29	p	p	NOUN
ejpam-4274	390	30	)	)	PUNCT
ejpam-4274	390	31	6=	6=	ADP
ejpam-4274	390	32	{	{	PUNCT
ejpam-4274	390	33	y}(λ	y}(λ	PROPN
ejpam-4274	390	34	,	,	PUNCT
ejpam-4274	390	35	p	p	NOUN
ejpam-4274	390	36	)	)	PUNCT
ejpam-4274	390	37	.	.	PUNCT
ejpam-4274	391	1	by	by	ADP
ejpam-4274	391	2	the	the	DET
ejpam-4274	391	3	hypothesis	hypothesis	NOUN
ejpam-4274	391	4	,	,	PUNCT
ejpam-4274	391	5	{	{	PUNCT
ejpam-4274	391	6	x}(λ	x}(λ	PROPN
ejpam-4274	391	7	,	,	PUNCT
ejpam-4274	391	8	p	p	NOUN
ejpam-4274	391	9	)	)	PUNCT
ejpam-4274	391	10	∩	∩	NOUN
ejpam-4274	391	11	{	{	PUNCT
ejpam-4274	391	12	y}(λ	y}(λ	PROPN
ejpam-4274	391	13	,	,	PUNCT
ejpam-4274	391	14	p	p	NOUN
ejpam-4274	391	15	)	)	PUNCT
ejpam-4274	391	16	=	=	NOUN
ejpam-4274	391	17	∅	∅	NOUN
ejpam-4274	391	18	and	and	CCONJ
ejpam-4274	391	19	hence	hence	ADV
ejpam-4274	391	20	y	y	PROPN
ejpam-4274	391	21	6∈	6∈	PROPN
ejpam-4274	391	22	{	{	PUNCT
ejpam-4274	391	23	x}(λ	x}(λ	PROPN
ejpam-4274	391	24	,	,	PUNCT
ejpam-4274	391	25	p	p	NOUN
ejpam-4274	391	26	)	)	PUNCT
ejpam-4274	391	27	.	.	PUNCT
ejpam-4274	392	1	therefore	therefore	ADV
ejpam-4274	392	2	,	,	PUNCT
ejpam-4274	392	3	{	{	PUNCT
ejpam-4274	392	4	x}(λ	x}(λ	PROPN
ejpam-4274	392	5	,	,	PUNCT
ejpam-4274	392	6	p	p	NOUN
ejpam-4274	392	7	)	)	PUNCT
ejpam-4274	392	8	⊆	⊆	NUM
ejpam-4274	392	9	u	u	NOUN
ejpam-4274	392	10	.	.	PUNCT
ejpam-4274	393	1	thus	thus	ADV
ejpam-4274	393	2	,	,	PUNCT
ejpam-4274	393	3	(	(	PUNCT
ejpam-4274	393	4	x	x	X
ejpam-4274	393	5	,	,	PUNCT
ejpam-4274	393	6	τ	τ	X
ejpam-4274	393	7	)	)	PUNCT
ejpam-4274	393	8	is	be	AUX
ejpam-4274	393	9	λp	λp	PROPN
ejpam-4274	393	10	-	-	PUNCT
ejpam-4274	393	11	r0	r0	NOUN
ejpam-4274	393	12	.	.	PUNCT
ejpam-4274	394	1	theorem	theorem	NOUN
ejpam-4274	394	2	12	12	NUM
ejpam-4274	394	3	.	.	PUNCT
ejpam-4274	395	1	a	a	DET
ejpam-4274	395	2	topological	topological	ADJ
ejpam-4274	395	3	space	space	NOUN
ejpam-4274	395	4	(	(	PUNCT
ejpam-4274	395	5	x	x	X
ejpam-4274	395	6	,	,	PUNCT
ejpam-4274	395	7	τ	τ	X
ejpam-4274	395	8	)	)	PUNCT
ejpam-4274	395	9	is	be	AUX
ejpam-4274	395	10	λp	λp	NOUN
ejpam-4274	395	11	-	-	PUNCT
ejpam-4274	395	12	r0	r0	NOUN
ejpam-4274	395	13	if	if	SCONJ
ejpam-4274	395	14	and	and	CCONJ
ejpam-4274	395	15	only	only	ADV
ejpam-4274	395	16	if	if	SCONJ
ejpam-4274	395	17	,	,	PUNCT
ejpam-4274	395	18	for	for	ADP
ejpam-4274	395	19	each	each	DET
ejpam-4274	395	20	x	x	NOUN
ejpam-4274	395	21	,	,	PUNCT
ejpam-4274	395	22	y	y	PROPN
ejpam-4274	395	23	∈	∈	PROPN
ejpam-4274	395	24	x	x	PRON
ejpam-4274	395	25	,	,	PUNCT
ejpam-4274	395	26	λ(λ	λ(λ	PROPN
ejpam-4274	395	27	,	,	PUNCT
ejpam-4274	395	28	p)({x	p)({x	NOUN
ejpam-4274	395	29	}	}	PUNCT
ejpam-4274	395	30	)	)	PUNCT
ejpam-4274	395	31	6=	6=	ADP
ejpam-4274	396	1	λ(λ	λ(λ	ADP
ejpam-4274	396	2	,	,	PUNCT
ejpam-4274	396	3	p)({y	p)({y	PROPN
ejpam-4274	396	4	}	}	PUNCT
ejpam-4274	396	5	)	)	PUNCT
ejpam-4274	396	6	implies	imply	VERB
ejpam-4274	396	7	λ(λ	λ(λ	PROPN
ejpam-4274	396	8	,	,	PUNCT
ejpam-4274	396	9	p)({x	p)({x	NOUN
ejpam-4274	396	10	}	}	PUNCT
ejpam-4274	396	11	)	)	PUNCT
ejpam-4274	396	12	∩	∩	NOUN
ejpam-4274	396	13	λ(λ	λ(λ	PROPN
ejpam-4274	396	14	,	,	PUNCT
ejpam-4274	396	15	p)({y	p)({y	PROPN
ejpam-4274	396	16	}	}	PUNCT
ejpam-4274	396	17	)	)	PUNCT
ejpam-4274	397	1	=	=	PUNCT
ejpam-4274	397	2	∅.	∅.	NOUN
ejpam-4274	397	3	proof	proof	NOUN
ejpam-4274	397	4	.	.	PUNCT
ejpam-4274	397	5	suppose	suppose	VERB
ejpam-4274	397	6	that	that	SCONJ
ejpam-4274	397	7	λ(λ	λ(λ	PROPN
ejpam-4274	397	8	,	,	PUNCT
ejpam-4274	397	9	p)({x	p)({x	NOUN
ejpam-4274	397	10	}	}	PUNCT
ejpam-4274	397	11	)	)	PUNCT
ejpam-4274	397	12	∩	∩	NOUN
ejpam-4274	397	13	λ(λ	λ(λ	PROPN
ejpam-4274	397	14	,	,	PUNCT
ejpam-4274	397	15	p)({y	p)({y	PROPN
ejpam-4274	397	16	}	}	PUNCT
ejpam-4274	397	17	)	)	PUNCT
ejpam-4274	397	18	6=	6=	ADP
ejpam-4274	397	19	∅.	∅.	AUX
ejpam-4274	397	20	let	let	VERB
ejpam-4274	397	21	z	z	PROPN
ejpam-4274	397	22	∈	∈	PROPN
ejpam-4274	397	23	λ(λ	λ(λ	PROPN
ejpam-4274	397	24	,	,	PUNCT
ejpam-4274	397	25	p)({x	p)({x	NOUN
ejpam-4274	397	26	}	}	PUNCT
ejpam-4274	397	27	)	)	PUNCT
ejpam-4274	397	28	∩	∩	NOUN
ejpam-4274	397	29	λ(λ	λ(λ	ADP
ejpam-4274	397	30	,	,	PUNCT
ejpam-4274	397	31	p)({y	p)({y	PROPN
ejpam-4274	397	32	}	}	PUNCT
ejpam-4274	397	33	)	)	PUNCT
ejpam-4274	397	34	.	.	PUNCT
ejpam-4274	398	1	then	then	ADV
ejpam-4274	398	2	,	,	PUNCT
ejpam-4274	398	3	z	z	PROPN
ejpam-4274	398	4	∈	∈	PROPN
ejpam-4274	398	5	λ(λ	λ(λ	PROPN
ejpam-4274	398	6	,	,	PUNCT
ejpam-4274	398	7	p)({x	p)({x	NOUN
ejpam-4274	398	8	}	}	PUNCT
ejpam-4274	398	9	)	)	PUNCT
ejpam-4274	398	10	and	and	CCONJ
ejpam-4274	398	11	by	by	ADP
ejpam-4274	398	12	lemma	lemma	PROPN
ejpam-4274	398	13	11	11	NUM
ejpam-4274	398	14	,	,	PUNCT
ejpam-4274	398	15	x	x	SYM
ejpam-4274	398	16	∈	∈	PROPN
ejpam-4274	398	17	{	{	PUNCT
ejpam-4274	398	18	z}(λ	z}(λ	PROPN
ejpam-4274	398	19	,	,	PUNCT
ejpam-4274	398	20	p	p	NOUN
ejpam-4274	398	21	)	)	PUNCT
ejpam-4274	398	22	.	.	PUNCT
ejpam-4274	399	1	thus	thus	ADV
ejpam-4274	399	2	,	,	PUNCT
ejpam-4274	399	3	x	x	SYM
ejpam-4274	399	4	∈	∈	PROPN
ejpam-4274	399	5	{	{	PUNCT
ejpam-4274	399	6	z}(λ	z}(λ	PROPN
ejpam-4274	399	7	,	,	PUNCT
ejpam-4274	399	8	p	p	NOUN
ejpam-4274	399	9	)	)	PUNCT
ejpam-4274	399	10	∩	∩	NOUN
ejpam-4274	399	11	{	{	PUNCT
ejpam-4274	399	12	x}(λ	x}(λ	PROPN
ejpam-4274	399	13	,	,	PUNCT
ejpam-4274	399	14	p	p	NOUN
ejpam-4274	399	15	)	)	PUNCT
ejpam-4274	399	16	and	and	CCONJ
ejpam-4274	399	17	by	by	ADP
ejpam-4274	399	18	corollary	corollary	ADJ
ejpam-4274	399	19	2	2	NUM
ejpam-4274	399	20	,	,	PUNCT
ejpam-4274	399	21	{	{	PUNCT
ejpam-4274	399	22	z}(λ	z}(λ	PROPN
ejpam-4274	399	23	,	,	PUNCT
ejpam-4274	399	24	p	p	NOUN
ejpam-4274	399	25	)	)	PUNCT
ejpam-4274	399	26	=	=	SYM
ejpam-4274	399	27	{	{	PUNCT
ejpam-4274	399	28	x}(λ	x}(λ	PROPN
ejpam-4274	399	29	,	,	PUNCT
ejpam-4274	399	30	p	p	NOUN
ejpam-4274	399	31	)	)	PUNCT
ejpam-4274	399	32	.	.	PUNCT
ejpam-4274	400	1	similarly	similarly	ADV
ejpam-4274	400	2	,	,	PUNCT
ejpam-4274	400	3	we	we	PRON
ejpam-4274	400	4	have	have	AUX
ejpam-4274	400	5	{	{	PUNCT
ejpam-4274	400	6	z}(λ	z}(λ	PROPN
ejpam-4274	400	7	,	,	PUNCT
ejpam-4274	400	8	p	p	NOUN
ejpam-4274	400	9	)	)	PUNCT
ejpam-4274	400	10	=	=	SYM
ejpam-4274	400	11	{	{	PUNCT
ejpam-4274	400	12	y}(λ	y}(λ	PROPN
ejpam-4274	400	13	,	,	PUNCT
ejpam-4274	400	14	p	p	NOUN
ejpam-4274	400	15	)	)	PUNCT
ejpam-4274	400	16	and	and	CCONJ
ejpam-4274	400	17	by	by	ADP
ejpam-4274	400	18	lemma	lemma	PROPN
ejpam-4274	400	19	11	11	NUM
ejpam-4274	400	20	,	,	PUNCT
ejpam-4274	400	21	λ(λ	λ(λ	PROPN
ejpam-4274	400	22	,	,	PUNCT
ejpam-4274	400	23	p)({x	p)({x	NOUN
ejpam-4274	400	24	}	}	PUNCT
ejpam-4274	400	25	)	)	PUNCT
ejpam-4274	401	1	=	=	SYM
ejpam-4274	401	2	λ(λ	λ(λ	PROPN
ejpam-4274	401	3	,	,	PUNCT
ejpam-4274	401	4	p)({y	p)({y	PROPN
ejpam-4274	401	5	}	}	PUNCT
ejpam-4274	401	6	)	)	PUNCT
ejpam-4274	401	7	.	.	PUNCT
ejpam-4274	402	1	conversely	conversely	ADV
ejpam-4274	402	2	,	,	PUNCT
ejpam-4274	402	3	we	we	PRON
ejpam-4274	402	4	shows	show	VERB
ejpam-4274	402	5	the	the	DET
ejpam-4274	402	6	sufficiency	sufficiency	NOUN
ejpam-4274	402	7	by	by	ADP
ejpam-4274	402	8	using	use	VERB
ejpam-4274	402	9	corollary	corollary	ADJ
ejpam-4274	402	10	2	2	NUM
ejpam-4274	402	11	.	.	PUNCT
ejpam-4274	402	12	suppose	suppose	VERB
ejpam-4274	402	13	that	that	SCONJ
ejpam-4274	402	14	{	{	PUNCT
ejpam-4274	402	15	x}(λ	x}(λ	PROPN
ejpam-4274	402	16	,	,	PUNCT
ejpam-4274	402	17	p	p	NOUN
ejpam-4274	402	18	)	)	PUNCT
ejpam-4274	402	19	6=	6=	ADP
ejpam-4274	402	20	{	{	PUNCT
ejpam-4274	402	21	y}(λ	y}(λ	PROPN
ejpam-4274	402	22	,	,	PUNCT
ejpam-4274	402	23	p	p	NOUN
ejpam-4274	402	24	)	)	PUNCT
ejpam-4274	402	25	.	.	PUNCT
ejpam-4274	403	1	by	by	ADP
ejpam-4274	403	2	lemma	lemma	PROPN
ejpam-4274	403	3	11	11	NUM
ejpam-4274	403	4	,	,	PUNCT
ejpam-4274	403	5	λ(λ	λ(λ	PROPN
ejpam-4274	403	6	,	,	PUNCT
ejpam-4274	403	7	p)({x	p)({x	NOUN
ejpam-4274	403	8	}	}	PUNCT
ejpam-4274	403	9	)	)	PUNCT
ejpam-4274	403	10	6=	6=	ADP
ejpam-4274	403	11	λ(λ	λ(λ	ADP
ejpam-4274	403	12	,	,	PUNCT
ejpam-4274	403	13	p)({y	p)({y	PROPN
ejpam-4274	403	14	}	}	PUNCT
ejpam-4274	403	15	)	)	PUNCT
ejpam-4274	403	16	and	and	CCONJ
ejpam-4274	403	17	hence	hence	ADV
ejpam-4274	403	18	λ(λ	λ(λ	PROPN
ejpam-4274	403	19	,	,	PUNCT
ejpam-4274	403	20	p)({x	p)({x	NOUN
ejpam-4274	403	21	}	}	PUNCT
ejpam-4274	403	22	)	)	PUNCT
ejpam-4274	403	23	∩	∩	NOUN
ejpam-4274	403	24	λ(λ	λ(λ	PROPN
ejpam-4274	403	25	,	,	PUNCT
ejpam-4274	403	26	p)({y	p)({y	PROPN
ejpam-4274	403	27	}	}	PUNCT
ejpam-4274	403	28	)	)	PUNCT
ejpam-4274	403	29	=	=	PUNCT
ejpam-4274	403	30	∅.	∅.	VERB
ejpam-4274	403	31	therefore	therefore	ADV
ejpam-4274	403	32	,	,	PUNCT
ejpam-4274	403	33	{	{	PUNCT
ejpam-4274	403	34	x}(λ	x}(λ	PROPN
ejpam-4274	403	35	,	,	PUNCT
ejpam-4274	403	36	p	p	NOUN
ejpam-4274	403	37	)	)	PUNCT
ejpam-4274	403	38	∩	∩	NOUN
ejpam-4274	403	39	{	{	PUNCT
ejpam-4274	403	40	y}(λ	y}(λ	PROPN
ejpam-4274	403	41	,	,	PUNCT
ejpam-4274	403	42	p	p	NOUN
ejpam-4274	403	43	)	)	PUNCT
ejpam-4274	403	44	=	=	PUNCT
ejpam-4274	403	45	∅.	∅.	NOUN
ejpam-4274	403	46	in	in	ADP
ejpam-4274	403	47	fact	fact	NOUN
ejpam-4274	403	48	,	,	PUNCT
ejpam-4274	403	49	assume	assume	VERB
ejpam-4274	403	50	z	z	PROPN
ejpam-4274	403	51	∈	∈	PROPN
ejpam-4274	403	52	{	{	PUNCT
ejpam-4274	403	53	x}(λ	x}(λ	PROPN
ejpam-4274	403	54	,	,	PUNCT
ejpam-4274	403	55	p	p	NOUN
ejpam-4274	403	56	)	)	PUNCT
ejpam-4274	403	57	∩	∩	NOUN
ejpam-4274	403	58	{	{	PUNCT
ejpam-4274	403	59	y}(λ	y}(λ	PROPN
ejpam-4274	403	60	,	,	PUNCT
ejpam-4274	403	61	p	p	NOUN
ejpam-4274	403	62	)	)	PUNCT
ejpam-4274	403	63	.	.	PUNCT
ejpam-4274	404	1	then	then	ADV
ejpam-4274	404	2	,	,	PUNCT
ejpam-4274	404	3	z	z	PROPN
ejpam-4274	404	4	∈	∈	PROPN
ejpam-4274	404	5	{	{	PUNCT
ejpam-4274	404	6	x}(λ	x}(λ	PROPN
ejpam-4274	404	7	,	,	PUNCT
ejpam-4274	404	8	p	p	NOUN
ejpam-4274	404	9	)	)	PUNCT
ejpam-4274	404	10	implies	imply	VERB
ejpam-4274	404	11	x	x	X
ejpam-4274	404	12	∈	∈	PROPN
ejpam-4274	404	13	λ(λ	λ(λ	PROPN
ejpam-4274	404	14	,	,	PUNCT
ejpam-4274	404	15	p)({z	p)({z	PROPN
ejpam-4274	404	16	}	}	PUNCT
ejpam-4274	404	17	)	)	PUNCT
ejpam-4274	404	18	and	and	CCONJ
ejpam-4274	404	19	hence	hence	ADV
ejpam-4274	404	20	x	x	X
ejpam-4274	404	21	∈	∈	PROPN
ejpam-4274	404	22	λ(λ	λ(λ	PROPN
ejpam-4274	404	23	,	,	PUNCT
ejpam-4274	404	24	p)({z	p)({z	PROPN
ejpam-4274	404	25	}	}	PUNCT
ejpam-4274	404	26	)	)	PUNCT
ejpam-4274	404	27	∩	∩	NOUN
ejpam-4274	404	28	λ(λ	λ(λ	PROPN
ejpam-4274	404	29	,	,	PUNCT
ejpam-4274	404	30	p)({x	p)({x	NOUN
ejpam-4274	404	31	}	}	PUNCT
ejpam-4274	404	32	)	)	PUNCT
ejpam-4274	404	33	.	.	PUNCT
ejpam-4274	405	1	by	by	ADP
ejpam-4274	405	2	the	the	DET
ejpam-4274	405	3	hypothesis	hypothesis	NOUN
ejpam-4274	405	4	,	,	PUNCT
ejpam-4274	405	5	λ(λ	λ(λ	ADV
ejpam-4274	405	6	,	,	PUNCT
ejpam-4274	405	7	p)({z	p)({z	PROPN
ejpam-4274	405	8	}	}	PUNCT
ejpam-4274	405	9	)	)	PUNCT
ejpam-4274	405	10	=	=	SYM
ejpam-4274	405	11	λ(λ	λ(λ	PROPN
ejpam-4274	405	12	,	,	PUNCT
ejpam-4274	405	13	p)({x	p)({x	NOUN
ejpam-4274	405	14	}	}	PUNCT
ejpam-4274	405	15	)	)	PUNCT
ejpam-4274	405	16	and	and	CCONJ
ejpam-4274	405	17	by	by	ADP
ejpam-4274	405	18	lemma	lemma	PROPN
ejpam-4274	405	19	11	11	NUM
ejpam-4274	405	20	,	,	PUNCT
ejpam-4274	405	21	{	{	PUNCT
ejpam-4274	405	22	z}(λ	z}(λ	PROPN
ejpam-4274	405	23	,	,	PUNCT
ejpam-4274	405	24	p	p	NOUN
ejpam-4274	405	25	)	)	PUNCT
ejpam-4274	405	26	=	=	SYM
ejpam-4274	405	27	{	{	PUNCT
ejpam-4274	405	28	x}(λ	x}(λ	PROPN
ejpam-4274	405	29	,	,	PUNCT
ejpam-4274	405	30	p	p	NOUN
ejpam-4274	405	31	)	)	PUNCT
ejpam-4274	405	32	.	.	PUNCT
ejpam-4274	406	1	similarly	similarly	ADV
ejpam-4274	406	2	,	,	PUNCT
ejpam-4274	406	3	we	we	PRON
ejpam-4274	406	4	have	have	AUX
ejpam-4274	406	5	{	{	PUNCT
ejpam-4274	406	6	z}(λ	z}(λ	PROPN
ejpam-4274	406	7	,	,	PUNCT
ejpam-4274	406	8	p	p	NOUN
ejpam-4274	406	9	)	)	PUNCT
ejpam-4274	406	10	=	=	SYM
ejpam-4274	406	11	{	{	PUNCT
ejpam-4274	406	12	y}(λ	y}(λ	PROPN
ejpam-4274	406	13	,	,	PUNCT
ejpam-4274	406	14	p	p	NOUN
ejpam-4274	406	15	)	)	PUNCT
ejpam-4274	406	16	and	and	CCONJ
ejpam-4274	406	17	hence	hence	ADV
ejpam-4274	406	18	{	{	PUNCT
ejpam-4274	406	19	x}(λ	x}(λ	PROPN
ejpam-4274	406	20	,	,	PUNCT
ejpam-4274	406	21	p	p	NOUN
ejpam-4274	406	22	)	)	PUNCT
ejpam-4274	406	23	=	=	SYM
ejpam-4274	406	24	{	{	PUNCT
ejpam-4274	406	25	y}(λ	y}(λ	PROPN
ejpam-4274	406	26	,	,	PUNCT
ejpam-4274	406	27	p	p	NOUN
ejpam-4274	406	28	)	)	PUNCT
ejpam-4274	406	29	.	.	PUNCT
ejpam-4274	407	1	this	this	PRON
ejpam-4274	407	2	contradicts	contradict	VERB
ejpam-4274	407	3	that	that	SCONJ
ejpam-4274	407	4	{	{	PUNCT
ejpam-4274	407	5	x}(λ	x}(λ	PROPN
ejpam-4274	407	6	,	,	PUNCT
ejpam-4274	407	7	p	p	NOUN
ejpam-4274	407	8	)	)	PUNCT
ejpam-4274	407	9	6=	6=	ADP
ejpam-4274	407	10	{	{	PUNCT
ejpam-4274	407	11	y}(λ	y}(λ	PROPN
ejpam-4274	407	12	,	,	PUNCT
ejpam-4274	407	13	p	p	NOUN
ejpam-4274	407	14	)	)	PUNCT
ejpam-4274	407	15	.	.	PUNCT
ejpam-4274	408	1	thus	thus	ADV
ejpam-4274	408	2	,	,	PUNCT
ejpam-4274	408	3	{	{	PUNCT
ejpam-4274	408	4	x}(λ	x}(λ	PROPN
ejpam-4274	408	5	,	,	PUNCT
ejpam-4274	408	6	p	p	NOUN
ejpam-4274	408	7	)	)	PUNCT
ejpam-4274	408	8	∩	∩	NOUN
ejpam-4274	408	9	{	{	PUNCT
ejpam-4274	408	10	y}(λ	y}(λ	PROPN
ejpam-4274	408	11	,	,	PUNCT
ejpam-4274	408	12	p	p	NOUN
ejpam-4274	408	13	)	)	PUNCT
ejpam-4274	408	14	=	=	PUNCT
ejpam-4274	408	15	∅.	∅.	ADP
ejpam-4274	408	16	this	this	PRON
ejpam-4274	408	17	shows	show	VERB
ejpam-4274	408	18	that	that	SCONJ
ejpam-4274	408	19	(	(	PUNCT
ejpam-4274	408	20	x	x	X
ejpam-4274	408	21	,	,	PUNCT
ejpam-4274	408	22	τ	τ	X
ejpam-4274	408	23	)	)	PUNCT
ejpam-4274	408	24	is	be	AUX
ejpam-4274	408	25	λp	λp	PROPN
ejpam-4274	408	26	-	-	PUNCT
ejpam-4274	408	27	r0	r0	NOUN
ejpam-4274	408	28	.	.	PUNCT
ejpam-4274	409	1	theorem	theorem	VERB
ejpam-4274	409	2	13	13	NUM
ejpam-4274	409	3	.	.	PUNCT
ejpam-4274	410	1	for	for	ADP
ejpam-4274	410	2	a	a	DET
ejpam-4274	410	3	topological	topological	ADJ
ejpam-4274	410	4	space	space	NOUN
ejpam-4274	410	5	(	(	PUNCT
ejpam-4274	410	6	x	x	X
ejpam-4274	410	7	,	,	PUNCT
ejpam-4274	410	8	τ	τ	PROPN
ejpam-4274	410	9	)	)	PUNCT
ejpam-4274	410	10	,	,	PUNCT
ejpam-4274	410	11	the	the	DET
ejpam-4274	410	12	following	follow	VERB
ejpam-4274	410	13	properties	property	NOUN
ejpam-4274	410	14	are	be	AUX
ejpam-4274	410	15	equivalent	equivalent	ADJ
ejpam-4274	410	16	:	:	PUNCT
ejpam-4274	410	17	(	(	PUNCT
ejpam-4274	410	18	1	1	X
ejpam-4274	410	19	)	)	PUNCT
ejpam-4274	410	20	(	(	PUNCT
ejpam-4274	410	21	x	x	X
ejpam-4274	410	22	,	,	PUNCT
ejpam-4274	410	23	τ	τ	X
ejpam-4274	410	24	)	)	PUNCT
ejpam-4274	410	25	is	be	AUX
ejpam-4274	410	26	λp	λp	PROPN
ejpam-4274	410	27	-	-	PUNCT
ejpam-4274	410	28	r0	r0	NOUN
ejpam-4274	410	29	;	;	PUNCT
ejpam-4274	410	30	(	(	PUNCT
ejpam-4274	410	31	2	2	X
ejpam-4274	410	32	)	)	PUNCT
ejpam-4274	410	33	x	x	SYM
ejpam-4274	410	34	∈	∈	PROPN
ejpam-4274	410	35	{	{	PUNCT
ejpam-4274	410	36	y}(λ	y}(λ	PROPN
ejpam-4274	410	37	,	,	PUNCT
ejpam-4274	410	38	p	p	NOUN
ejpam-4274	410	39	)	)	PUNCT
ejpam-4274	410	40	if	if	SCONJ
ejpam-4274	411	1	and	and	CCONJ
ejpam-4274	411	2	only	only	ADV
ejpam-4274	411	3	if	if	SCONJ
ejpam-4274	411	4	y	y	PROPN
ejpam-4274	411	5	∈	∈	PROPN
ejpam-4274	411	6	{	{	PUNCT
ejpam-4274	411	7	x}(λ	x}(λ	PROPN
ejpam-4274	411	8	,	,	PUNCT
ejpam-4274	411	9	p	p	NOUN
ejpam-4274	411	10	)	)	PUNCT
ejpam-4274	411	11	.	.	PUNCT
ejpam-4274	412	1	proof	proof	NOUN
ejpam-4274	412	2	.	.	PUNCT
ejpam-4274	413	1	(	(	PUNCT
ejpam-4274	413	2	1	1	X
ejpam-4274	413	3	)	)	PUNCT
ejpam-4274	413	4	⇒	⇒	NOUN
ejpam-4274	413	5	(	(	PUNCT
ejpam-4274	413	6	2	2	NUM
ejpam-4274	413	7	):	):	PUNCT
ejpam-4274	413	8	suppose	suppose	VERB
ejpam-4274	413	9	that	that	SCONJ
ejpam-4274	413	10	(	(	PUNCT
ejpam-4274	413	11	x	x	X
ejpam-4274	413	12	,	,	PUNCT
ejpam-4274	413	13	τ	τ	X
ejpam-4274	413	14	)	)	PUNCT
ejpam-4274	413	15	is	be	AUX
ejpam-4274	413	16	λp	λp	PROPN
ejpam-4274	413	17	-	-	PUNCT
ejpam-4274	413	18	r0	r0	NOUN
ejpam-4274	413	19	.	.	PUNCT
ejpam-4274	414	1	let	let	VERB
ejpam-4274	414	2	x	x	PRON
ejpam-4274	414	3	∈	∈	PROPN
ejpam-4274	414	4	{	{	PUNCT
ejpam-4274	414	5	y}(λ	y}(λ	PROPN
ejpam-4274	414	6	,	,	PUNCT
ejpam-4274	414	7	p	p	NOUN
ejpam-4274	414	8	)	)	PUNCT
ejpam-4274	414	9	.	.	PUNCT
ejpam-4274	415	1	by	by	ADP
ejpam-4274	415	2	lemma	lemma	PROPN
ejpam-4274	415	3	11	11	NUM
ejpam-4274	415	4	,	,	PUNCT
ejpam-4274	415	5	y	y	PROPN
ejpam-4274	415	6	∈	∈	PROPN
ejpam-4274	415	7	λ(λ	λ(λ	PROPN
ejpam-4274	415	8	,	,	PUNCT
ejpam-4274	415	9	p)({x	p)({x	NOUN
ejpam-4274	415	10	}	}	PUNCT
ejpam-4274	415	11	)	)	PUNCT
ejpam-4274	415	12	and	and	CCONJ
ejpam-4274	415	13	hence	hence	ADV
ejpam-4274	415	14	λ(λ	λ(λ	PROPN
ejpam-4274	415	15	,	,	PUNCT
ejpam-4274	415	16	p)({x	p)({x	NOUN
ejpam-4274	415	17	}	}	PUNCT
ejpam-4274	415	18	)	)	PUNCT
ejpam-4274	415	19	∩	∩	NOUN
ejpam-4274	415	20	λ(λ	λ(λ	PROPN
ejpam-4274	415	21	,	,	PUNCT
ejpam-4274	415	22	p)({y	p)({y	PROPN
ejpam-4274	415	23	}	}	PUNCT
ejpam-4274	415	24	)	)	PUNCT
ejpam-4274	415	25	6=	6=	ADP
ejpam-4274	416	1	∅.	∅.	VERB
ejpam-4274	416	2	by	by	ADP
ejpam-4274	416	3	theorem	theorem	NOUN
ejpam-4274	416	4	12	12	NUM
ejpam-4274	416	5	,	,	PUNCT
ejpam-4274	416	6	we	we	PRON
ejpam-4274	416	7	have	have	VERB
ejpam-4274	416	8	λ(λ	λ(λ	PROPN
ejpam-4274	416	9	,	,	PUNCT
ejpam-4274	416	10	p)({x	p)({x	NOUN
ejpam-4274	416	11	}	}	PUNCT
ejpam-4274	416	12	)	)	PUNCT
ejpam-4274	417	1	=	=	SYM
ejpam-4274	417	2	λ(λ	λ(λ	PROPN
ejpam-4274	417	3	,	,	PUNCT
ejpam-4274	417	4	p)({y	p)({y	PROPN
ejpam-4274	417	5	}	}	PUNCT
ejpam-4274	417	6	)	)	PUNCT
ejpam-4274	417	7	c.	c.	PROPN
ejpam-4274	417	8	boonpok	boonpok	PROPN
ejpam-4274	417	9	,	,	PUNCT
ejpam-4274	417	10	c.	c.	PROPN
ejpam-4274	417	11	viriyapong	viriyapong	PROPN
ejpam-4274	417	12	/	/	SYM
ejpam-4274	417	13	eur	eur	PROPN
ejpam-4274	417	14	.	.	PUNCT
ejpam-4274	418	1	j.	j.	PROPN
ejpam-4274	418	2	pure	pure	PROPN
ejpam-4274	418	3	appl	appl	PROPN
ejpam-4274	418	4	.	.	PROPN
ejpam-4274	418	5	math	math	PROPN
ejpam-4274	418	6	,	,	PUNCT
ejpam-4274	418	7	15	15	NUM
ejpam-4274	418	8	(	(	PUNCT
ejpam-4274	418	9	2	2	NUM
ejpam-4274	418	10	)	)	PUNCT
ejpam-4274	418	11	(	(	PUNCT
ejpam-4274	418	12	2022	2022	NUM
ejpam-4274	418	13	)	)	PUNCT
ejpam-4274	418	14	,	,	PUNCT
ejpam-4274	418	15	415	415	NUM
ejpam-4274	418	16	-	-	SYM
ejpam-4274	418	17	436	436	NUM
ejpam-4274	418	18	426	426	NUM
ejpam-4274	418	19	and	and	CCONJ
ejpam-4274	418	20	hence	hence	ADV
ejpam-4274	418	21	x	x	X
ejpam-4274	418	22	∈	∈	PROPN
ejpam-4274	418	23	λ(λ	λ(λ	PROPN
ejpam-4274	418	24	,	,	PUNCT
ejpam-4274	418	25	p)({y	p)({y	PROPN
ejpam-4274	418	26	}	}	PUNCT
ejpam-4274	418	27	)	)	PUNCT
ejpam-4274	418	28	.	.	PUNCT
ejpam-4274	419	1	thus	thus	ADV
ejpam-4274	419	2	,	,	PUNCT
ejpam-4274	419	3	y	y	PROPN
ejpam-4274	419	4	∈	∈	PROPN
ejpam-4274	419	5	{	{	PUNCT
ejpam-4274	419	6	x}(λ	x}(λ	PROPN
ejpam-4274	419	7	,	,	PUNCT
ejpam-4274	419	8	p	p	NOUN
ejpam-4274	419	9	)	)	PUNCT
ejpam-4274	419	10	by	by	ADP
ejpam-4274	419	11	lemma	lemma	PROPN
ejpam-4274	419	12	11	11	NUM
ejpam-4274	419	13	.	.	PUNCT
ejpam-4274	420	1	the	the	DET
ejpam-4274	420	2	converse	converse	NOUN
ejpam-4274	420	3	is	be	AUX
ejpam-4274	420	4	similarly	similarly	ADV
ejpam-4274	420	5	shown	show	VERB
ejpam-4274	420	6	.	.	PUNCT
ejpam-4274	421	1	(	(	PUNCT
ejpam-4274	421	2	2	2	X
ejpam-4274	421	3	)	)	PUNCT
ejpam-4274	421	4	⇒	⇒	NOUN
ejpam-4274	421	5	(	(	PUNCT
ejpam-4274	421	6	1	1	NUM
ejpam-4274	421	7	):	):	PUNCT
ejpam-4274	421	8	let	let	VERB
ejpam-4274	421	9	u	u	PRON
ejpam-4274	421	10	be	be	AUX
ejpam-4274	421	11	a	a	DET
ejpam-4274	421	12	(	(	PUNCT
ejpam-4274	421	13	λ	λ	NOUN
ejpam-4274	421	14	,	,	PUNCT
ejpam-4274	421	15	p)-open	p)-open	VERB
ejpam-4274	421	16	set	set	VERB
ejpam-4274	421	17	and	and	CCONJ
ejpam-4274	421	18	let	let	VERB
ejpam-4274	421	19	x	x	PUNCT
ejpam-4274	421	20	∈	∈	PROPN
ejpam-4274	421	21	u	u	NOUN
ejpam-4274	421	22	.	.	PUNCT
ejpam-4274	422	1	if	if	SCONJ
ejpam-4274	422	2	y	y	PROPN
ejpam-4274	422	3	6∈	6∈	PROPN
ejpam-4274	422	4	u	u	PROPN
ejpam-4274	422	5	,	,	PUNCT
ejpam-4274	422	6	then	then	ADV
ejpam-4274	422	7	{	{	PUNCT
ejpam-4274	422	8	y}(λ	y}(λ	PROPN
ejpam-4274	422	9	,	,	PUNCT
ejpam-4274	422	10	p	p	NOUN
ejpam-4274	422	11	)	)	PUNCT
ejpam-4274	422	12	∩	∩	NOUN
ejpam-4274	422	13	u	u	NOUN
ejpam-4274	422	14	=	=	PUNCT
ejpam-4274	422	15	∅.	∅.	VERB
ejpam-4274	422	16	thus	thus	ADV
ejpam-4274	422	17	,	,	PUNCT
ejpam-4274	422	18	x	x	PROPN
ejpam-4274	422	19	6∈	6∈	PROPN
ejpam-4274	422	20	{	{	PUNCT
ejpam-4274	422	21	y}(λ	y}(λ	PROPN
ejpam-4274	422	22	,	,	PUNCT
ejpam-4274	422	23	p	p	NOUN
ejpam-4274	422	24	)	)	PUNCT
ejpam-4274	422	25	and	and	CCONJ
ejpam-4274	422	26	hence	hence	ADV
ejpam-4274	422	27	y	y	PROPN
ejpam-4274	422	28	6∈	6∈	PROPN
ejpam-4274	422	29	{	{	PUNCT
ejpam-4274	422	30	x}(λ	x}(λ	PROPN
ejpam-4274	422	31	,	,	PUNCT
ejpam-4274	422	32	p	p	NOUN
ejpam-4274	422	33	)	)	PUNCT
ejpam-4274	422	34	.	.	PUNCT
ejpam-4274	423	1	this	this	PRON
ejpam-4274	423	2	implies	imply	VERB
ejpam-4274	423	3	that	that	SCONJ
ejpam-4274	423	4	{	{	PUNCT
ejpam-4274	423	5	x}(λ	x}(λ	PROPN
ejpam-4274	423	6	,	,	PUNCT
ejpam-4274	423	7	p	p	NOUN
ejpam-4274	423	8	)	)	PUNCT
ejpam-4274	423	9	⊆	⊆	NUM
ejpam-4274	423	10	u	u	NOUN
ejpam-4274	423	11	.	.	PUNCT
ejpam-4274	424	1	therefore	therefore	ADV
ejpam-4274	424	2	,	,	PUNCT
ejpam-4274	424	3	(	(	PUNCT
ejpam-4274	424	4	x	x	X
ejpam-4274	424	5	,	,	PUNCT
ejpam-4274	424	6	τ	τ	X
ejpam-4274	424	7	)	)	PUNCT
ejpam-4274	424	8	is	be	AUX
ejpam-4274	424	9	λp	λp	PROPN
ejpam-4274	424	10	-	-	PUNCT
ejpam-4274	424	11	r0	r0	NOUN
ejpam-4274	424	12	.	.	PUNCT
ejpam-4274	425	1	theorem	theorem	PROPN
ejpam-4274	425	2	14	14	NUM
ejpam-4274	425	3	.	.	PUNCT
ejpam-4274	426	1	for	for	ADP
ejpam-4274	426	2	a	a	DET
ejpam-4274	426	3	topological	topological	ADJ
ejpam-4274	426	4	space	space	NOUN
ejpam-4274	426	5	(	(	PUNCT
ejpam-4274	426	6	x	x	X
ejpam-4274	426	7	,	,	PUNCT
ejpam-4274	426	8	τ	τ	PROPN
ejpam-4274	426	9	)	)	PUNCT
ejpam-4274	426	10	,	,	PUNCT
ejpam-4274	426	11	the	the	DET
ejpam-4274	426	12	following	follow	VERB
ejpam-4274	426	13	properties	property	NOUN
ejpam-4274	426	14	are	be	AUX
ejpam-4274	426	15	equivalent	equivalent	ADJ
ejpam-4274	426	16	:	:	PUNCT
ejpam-4274	426	17	(	(	PUNCT
ejpam-4274	426	18	1	1	X
ejpam-4274	426	19	)	)	PUNCT
ejpam-4274	426	20	(	(	PUNCT
ejpam-4274	426	21	x	x	X
ejpam-4274	426	22	,	,	PUNCT
ejpam-4274	426	23	τ	τ	X
ejpam-4274	426	24	)	)	PUNCT
ejpam-4274	426	25	is	be	AUX
ejpam-4274	426	26	λp	λp	PROPN
ejpam-4274	426	27	-	-	PUNCT
ejpam-4274	426	28	r0	r0	NOUN
ejpam-4274	426	29	;	;	PUNCT
ejpam-4274	426	30	(	(	PUNCT
ejpam-4274	426	31	2	2	X
ejpam-4274	426	32	)	)	PUNCT
ejpam-4274	426	33	for	for	ADP
ejpam-4274	426	34	each	each	DET
ejpam-4274	426	35	nonempty	nonempty	NOUN
ejpam-4274	426	36	subset	subset	VERB
ejpam-4274	426	37	a	a	PRON
ejpam-4274	426	38	of	of	ADP
ejpam-4274	426	39	x	x	X
ejpam-4274	426	40	and	and	CCONJ
ejpam-4274	426	41	each	each	PRON
ejpam-4274	426	42	(	(	PUNCT
ejpam-4274	426	43	λ	λ	NOUN
ejpam-4274	426	44	,	,	PUNCT
ejpam-4274	426	45	p)-open	p)-open	VERB
ejpam-4274	426	46	set	set	VERB
ejpam-4274	426	47	u	u	PRON
ejpam-4274	426	48	such	such	ADJ
ejpam-4274	426	49	that	that	SCONJ
ejpam-4274	426	50	a	a	DET
ejpam-4274	426	51	∩	∩	ADJ
ejpam-4274	426	52	u	u	NOUN
ejpam-4274	426	53	6=	6=	NOUN
ejpam-4274	426	54	∅	∅	NOUN
ejpam-4274	426	55	,	,	PUNCT
ejpam-4274	426	56	there	there	PRON
ejpam-4274	426	57	exists	exist	VERB
ejpam-4274	426	58	a	a	DET
ejpam-4274	426	59	(	(	PUNCT
ejpam-4274	426	60	λ	λ	PROPN
ejpam-4274	426	61	,	,	PUNCT
ejpam-4274	426	62	p)-closed	p)-close	VERB
ejpam-4274	426	63	set	set	NOUN
ejpam-4274	426	64	f	f	PROPN
ejpam-4274	427	1	such	such	ADJ
ejpam-4274	427	2	that	that	SCONJ
ejpam-4274	427	3	a	a	DET
ejpam-4274	427	4	∩	∩	ADJ
ejpam-4274	427	5	f	f	PROPN
ejpam-4274	427	6	6=	6=	PROPN
ejpam-4274	427	7	∅	∅	NOUN
ejpam-4274	427	8	and	and	CCONJ
ejpam-4274	427	9	f	f	PROPN
ejpam-4274	427	10	⊆	⊆	NUM
ejpam-4274	427	11	u	u	NOUN
ejpam-4274	427	12	;	;	PUNCT
ejpam-4274	427	13	(	(	PUNCT
ejpam-4274	427	14	3	3	X
ejpam-4274	427	15	)	)	PUNCT
ejpam-4274	427	16	f	f	NOUN
ejpam-4274	427	17	=	=	SYM
ejpam-4274	427	18	λ(λ	λ(λ	PROPN
ejpam-4274	427	19	,	,	PUNCT
ejpam-4274	427	20	p)(f	p)(f	PROPN
ejpam-4274	427	21	)	)	PUNCT
ejpam-4274	427	22	for	for	SCONJ
ejpam-4274	427	23	each	each	PRON
ejpam-4274	427	24	(	(	PUNCT
ejpam-4274	427	25	λ	λ	PROPN
ejpam-4274	427	26	,	,	PUNCT
ejpam-4274	427	27	p)-closed	p)-close	VERB
ejpam-4274	427	28	set	set	VERB
ejpam-4274	427	29	f	f	X
ejpam-4274	427	30	;	;	PUNCT
ejpam-4274	427	31	(	(	PUNCT
ejpam-4274	427	32	4	4	X
ejpam-4274	427	33	)	)	PUNCT
ejpam-4274	427	34	{	{	PUNCT
ejpam-4274	427	35	x}(λ	x}(λ	PROPN
ejpam-4274	427	36	,	,	PUNCT
ejpam-4274	427	37	p	p	NOUN
ejpam-4274	427	38	)	)	PUNCT
ejpam-4274	427	39	=	=	SYM
ejpam-4274	427	40	λ(λ	λ(λ	PROPN
ejpam-4274	427	41	,	,	PUNCT
ejpam-4274	427	42	p)({x	p)({x	NOUN
ejpam-4274	427	43	}	}	PUNCT
ejpam-4274	427	44	)	)	PUNCT
ejpam-4274	427	45	for	for	ADP
ejpam-4274	427	46	each	each	DET
ejpam-4274	427	47	x	x	SYM
ejpam-4274	427	48	∈	∈	PROPN
ejpam-4274	427	49	x.	x.	NOUN
ejpam-4274	427	50	proof	proof	NOUN
ejpam-4274	427	51	.	.	PUNCT
ejpam-4274	428	1	(	(	PUNCT
ejpam-4274	428	2	1	1	X
ejpam-4274	428	3	)	)	PUNCT
ejpam-4274	428	4	⇒	⇒	NOUN
ejpam-4274	428	5	(	(	PUNCT
ejpam-4274	428	6	2	2	NUM
ejpam-4274	428	7	):	):	PUNCT
ejpam-4274	428	8	let	let	VERB
ejpam-4274	428	9	a	a	PRON
ejpam-4274	428	10	be	be	AUX
ejpam-4274	428	11	a	a	DET
ejpam-4274	428	12	nonempty	nonempty	ADJ
ejpam-4274	428	13	subset	subset	NOUN
ejpam-4274	428	14	of	of	ADP
ejpam-4274	428	15	x	x	PUNCT
ejpam-4274	428	16	and	and	CCONJ
ejpam-4274	428	17	let	let	VERB
ejpam-4274	428	18	u	u	PRON
ejpam-4274	428	19	∈	∈	PROPN
ejpam-4274	428	20	λpo(x	λpo(x	PROPN
ejpam-4274	428	21	,	,	PUNCT
ejpam-4274	428	22	τ	τ	X
ejpam-4274	428	23	)	)	PUNCT
ejpam-4274	428	24	such	such	ADJ
ejpam-4274	428	25	that	that	SCONJ
ejpam-4274	428	26	a	a	DET
ejpam-4274	428	27	∩u	∩u	NOUN
ejpam-4274	428	28	6=	6=	ADP
ejpam-4274	428	29	∅.	∅.	NOUN
ejpam-4274	428	30	then	then	ADV
ejpam-4274	428	31	,	,	PUNCT
ejpam-4274	428	32	there	there	PRON
ejpam-4274	428	33	exists	exist	VERB
ejpam-4274	428	34	x	x	X
ejpam-4274	428	35	∈	∈	PROPN
ejpam-4274	428	36	a	a	DET
ejpam-4274	428	37	∩u	∩u	NOUN
ejpam-4274	428	38	and	and	CCONJ
ejpam-4274	428	39	hence	hence	ADV
ejpam-4274	428	40	{	{	PUNCT
ejpam-4274	428	41	x}(λ	x}(λ	PROPN
ejpam-4274	428	42	,	,	PUNCT
ejpam-4274	428	43	p	p	NOUN
ejpam-4274	428	44	)	)	PUNCT
ejpam-4274	428	45	⊆	⊆	NUM
ejpam-4274	428	46	u	u	NOUN
ejpam-4274	428	47	.	.	PUNCT
ejpam-4274	429	1	put	put	VERB
ejpam-4274	429	2	f	f	X
ejpam-4274	429	3	=	=	SYM
ejpam-4274	429	4	{	{	PUNCT
ejpam-4274	429	5	x}(λ	x}(λ	PROPN
ejpam-4274	429	6	,	,	PUNCT
ejpam-4274	429	7	p	p	NOUN
ejpam-4274	429	8	)	)	PUNCT
ejpam-4274	429	9	,	,	PUNCT
ejpam-4274	429	10	then	then	ADV
ejpam-4274	429	11	f	f	PROPN
ejpam-4274	429	12	is	be	AUX
ejpam-4274	429	13	(	(	PUNCT
ejpam-4274	429	14	λ	λ	X
ejpam-4274	429	15	,	,	PUNCT
ejpam-4274	429	16	p)-closed	p)-close	VERB
ejpam-4274	429	17	,	,	PUNCT
ejpam-4274	429	18	a	a	DET
ejpam-4274	429	19	∩	∩	ADJ
ejpam-4274	429	20	f	f	PROPN
ejpam-4274	429	21	6=	6=	PROPN
ejpam-4274	429	22	∅	∅	NOUN
ejpam-4274	429	23	and	and	CCONJ
ejpam-4274	429	24	f	f	PROPN
ejpam-4274	429	25	⊆	⊆	NUM
ejpam-4274	429	26	u	u	NOUN
ejpam-4274	429	27	.	.	PUNCT
ejpam-4274	430	1	(	(	PUNCT
ejpam-4274	430	2	2	2	X
ejpam-4274	430	3	)	)	PUNCT
ejpam-4274	430	4	⇒	⇒	NOUN
ejpam-4274	430	5	(	(	PUNCT
ejpam-4274	430	6	3	3	NUM
ejpam-4274	430	7	):	):	PUNCT
ejpam-4274	430	8	let	let	VERB
ejpam-4274	430	9	f	f	PRON
ejpam-4274	430	10	be	be	AUX
ejpam-4274	430	11	a	a	DET
ejpam-4274	430	12	(	(	PUNCT
ejpam-4274	430	13	λ	λ	PROPN
ejpam-4274	430	14	,	,	PUNCT
ejpam-4274	430	15	p)-closed	p)-close	VERB
ejpam-4274	430	16	set	set	NOUN
ejpam-4274	430	17	.	.	PUNCT
ejpam-4274	431	1	by	by	ADP
ejpam-4274	431	2	lemma	lemma	PROPN
ejpam-4274	431	3	7	7	NUM
ejpam-4274	431	4	,	,	PUNCT
ejpam-4274	431	5	we	we	PRON
ejpam-4274	431	6	have	have	VERB
ejpam-4274	431	7	f	f	PROPN
ejpam-4274	431	8	⊆	⊆	NUM
ejpam-4274	431	9	λ(λ	λ(λ	PROPN
ejpam-4274	431	10	,	,	PUNCT
ejpam-4274	431	11	p)(f	p)(f	PROPN
ejpam-4274	431	12	)	)	PUNCT
ejpam-4274	431	13	.	.	PUNCT
ejpam-4274	432	1	next	next	ADV
ejpam-4274	432	2	,	,	PUNCT
ejpam-4274	432	3	we	we	PRON
ejpam-4274	432	4	show	show	VERB
ejpam-4274	432	5	f	f	PROPN
ejpam-4274	432	6	⊇	⊇	PROPN
ejpam-4274	432	7	λ(λ	λ(λ	PROPN
ejpam-4274	432	8	,	,	PUNCT
ejpam-4274	432	9	p)(f	p)(f	PROPN
ejpam-4274	432	10	)	)	PUNCT
ejpam-4274	432	11	.	.	PUNCT
ejpam-4274	433	1	let	let	VERB
ejpam-4274	433	2	x	x	SYM
ejpam-4274	433	3	6∈	6∈	NOUN
ejpam-4274	433	4	f	f	X
ejpam-4274	433	5	.	.	PUNCT
ejpam-4274	434	1	then	then	ADV
ejpam-4274	434	2	,	,	PUNCT
ejpam-4274	434	3	x	x	PUNCT
ejpam-4274	434	4	∈	∈	PROPN
ejpam-4274	434	5	(	(	PUNCT
ejpam-4274	434	6	x	x	NOUN
ejpam-4274	434	7	−	−	PROPN
ejpam-4274	434	8	f	f	X
ejpam-4274	434	9	)	)	PUNCT
ejpam-4274	434	10	∈	∈	PROPN
ejpam-4274	435	1	λpo(x	λpo(x	PROPN
ejpam-4274	435	2	,	,	PUNCT
ejpam-4274	435	3	τ	τ	X
ejpam-4274	435	4	)	)	PUNCT
ejpam-4274	435	5	and	and	CCONJ
ejpam-4274	435	6	by	by	ADP
ejpam-4274	435	7	(	(	PUNCT
ejpam-4274	435	8	2	2	NUM
ejpam-4274	435	9	)	)	PUNCT
ejpam-4274	435	10	,	,	PUNCT
ejpam-4274	435	11	there	there	PRON
ejpam-4274	435	12	exists	exist	VERB
ejpam-4274	435	13	a	a	DET
ejpam-4274	435	14	(	(	PUNCT
ejpam-4274	435	15	λ	λ	PROPN
ejpam-4274	435	16	,	,	PUNCT
ejpam-4274	435	17	p)-closed	p)-close	VERB
ejpam-4274	435	18	set	set	NOUN
ejpam-4274	435	19	k	k	ADP
ejpam-4274	435	20	such	such	ADJ
ejpam-4274	435	21	that	that	SCONJ
ejpam-4274	435	22	x	x	SYM
ejpam-4274	435	23	∈	∈	PROPN
ejpam-4274	435	24	k	k	PROPN
ejpam-4274	435	25	and	and	CCONJ
ejpam-4274	435	26	k	k	PROPN
ejpam-4274	435	27	⊆	⊆	NUM
ejpam-4274	435	28	x−f	x−f	PROPN
ejpam-4274	435	29	.	.	PUNCT
ejpam-4274	436	1	now	now	ADV
ejpam-4274	436	2	,	,	PUNCT
ejpam-4274	436	3	put	put	VERB
ejpam-4274	436	4	u	u	NOUN
ejpam-4274	436	5	=	=	NOUN
ejpam-4274	436	6	x−k	x−k	PROPN
ejpam-4274	436	7	.	.	PUNCT
ejpam-4274	437	1	then	then	ADV
ejpam-4274	437	2	,	,	PUNCT
ejpam-4274	437	3	f	f	PROPN
ejpam-4274	437	4	⊆	⊆	NUM
ejpam-4274	437	5	u	u	X
ejpam-4274	437	6	∈	∈	PROPN
ejpam-4274	437	7	λpo(x	λpo(x	PROPN
ejpam-4274	437	8	,	,	PUNCT
ejpam-4274	437	9	τ	τ	X
ejpam-4274	437	10	)	)	PUNCT
ejpam-4274	437	11	and	and	CCONJ
ejpam-4274	437	12	x	x	PUNCT
ejpam-4274	437	13	6∈	6∈	PROPN
ejpam-4274	437	14	u	u	NOUN
ejpam-4274	437	15	.	.	PUNCT
ejpam-4274	438	1	thus	thus	ADV
ejpam-4274	438	2	,	,	PUNCT
ejpam-4274	438	3	x	x	PROPN
ejpam-4274	438	4	6∈	6∈	PROPN
ejpam-4274	438	5	λ(λ	λ(λ	PROPN
ejpam-4274	438	6	,	,	PUNCT
ejpam-4274	438	7	p)(f	p)(f	PROPN
ejpam-4274	438	8	)	)	PUNCT
ejpam-4274	438	9	.	.	PUNCT
ejpam-4274	439	1	this	this	PRON
ejpam-4274	439	2	shows	show	VERB
ejpam-4274	439	3	that	that	SCONJ
ejpam-4274	439	4	f	f	PROPN
ejpam-4274	439	5	⊇	⊇	PROPN
ejpam-4274	439	6	λ(λ	λ(λ	PROPN
ejpam-4274	439	7	,	,	PUNCT
ejpam-4274	439	8	p)(f	p)(f	PROPN
ejpam-4274	439	9	)	)	PUNCT
ejpam-4274	439	10	.	.	PUNCT
ejpam-4274	440	1	(	(	PUNCT
ejpam-4274	440	2	3	3	X
ejpam-4274	440	3	)	)	PUNCT
ejpam-4274	440	4	⇒	⇒	NOUN
ejpam-4274	440	5	(	(	PUNCT
ejpam-4274	440	6	4	4	NUM
ejpam-4274	440	7	):	):	PUNCT
ejpam-4274	440	8	let	let	VERB
ejpam-4274	440	9	x	x	PUNCT
ejpam-4274	440	10	∈	∈	PROPN
ejpam-4274	440	11	x	x	PUNCT
ejpam-4274	440	12	and	and	CCONJ
ejpam-4274	440	13	let	let	VERB
ejpam-4274	440	14	y	y	PROPN
ejpam-4274	440	15	6∈	6∈	PROPN
ejpam-4274	440	16	λ(λ	λ(λ	ADV
ejpam-4274	440	17	,	,	PUNCT
ejpam-4274	440	18	p)({x	p)({x	NOUN
ejpam-4274	440	19	}	}	PUNCT
ejpam-4274	440	20	)	)	PUNCT
ejpam-4274	440	21	.	.	PUNCT
ejpam-4274	441	1	there	there	PRON
ejpam-4274	441	2	exists	exist	VERB
ejpam-4274	441	3	a	a	DET
ejpam-4274	441	4	(	(	PUNCT
ejpam-4274	441	5	λ	λ	NOUN
ejpam-4274	441	6	,	,	PUNCT
ejpam-4274	441	7	p)-open	p)-open	VERB
ejpam-4274	441	8	set	set	VERB
ejpam-4274	441	9	u	u	PRON
ejpam-4274	441	10	such	such	ADJ
ejpam-4274	441	11	that	that	SCONJ
ejpam-4274	441	12	x	x	SYM
ejpam-4274	441	13	∈	∈	PROPN
ejpam-4274	441	14	u	u	NOUN
ejpam-4274	441	15	and	and	CCONJ
ejpam-4274	441	16	y	y	PROPN
ejpam-4274	441	17	6∈	6∈	PROPN
ejpam-4274	441	18	u	u	PROPN
ejpam-4274	441	19	.	.	PUNCT
ejpam-4274	442	1	therefore	therefore	ADV
ejpam-4274	442	2	,	,	PUNCT
ejpam-4274	442	3	{	{	PUNCT
ejpam-4274	442	4	y}(λ	y}(λ	PROPN
ejpam-4274	442	5	,	,	PUNCT
ejpam-4274	442	6	p	p	NOUN
ejpam-4274	442	7	)	)	PUNCT
ejpam-4274	442	8	∩	∩	NOUN
ejpam-4274	442	9	u	u	NOUN
ejpam-4274	442	10	=	=	PUNCT
ejpam-4274	442	11	∅.	∅.	X
ejpam-4274	442	12	by	by	ADP
ejpam-4274	442	13	(	(	PUNCT
ejpam-4274	442	14	3	3	NUM
ejpam-4274	442	15	)	)	PUNCT
ejpam-4274	442	16	,	,	PUNCT
ejpam-4274	442	17	we	we	PRON
ejpam-4274	442	18	have	have	VERB
ejpam-4274	442	19	λ(λ	λ(λ	PROPN
ejpam-4274	442	20	,	,	PUNCT
ejpam-4274	442	21	p)({y}(λ	p)({y}(λ	NOUN
ejpam-4274	442	22	,	,	PUNCT
ejpam-4274	442	23	p	p	NOUN
ejpam-4274	442	24	)	)	PUNCT
ejpam-4274	442	25	)	)	PUNCT
ejpam-4274	442	26	∩	∩	NOUN
ejpam-4274	442	27	u	u	NOUN
ejpam-4274	442	28	=	=	PUNCT
ejpam-4274	442	29	∅.	∅.	NOUN
ejpam-4274	442	30	since	since	SCONJ
ejpam-4274	442	31	x	x	PROPN
ejpam-4274	442	32	6∈	6∈	PROPN
ejpam-4274	442	33	λ(λ	λ(λ	PROPN
ejpam-4274	442	34	,	,	PUNCT
ejpam-4274	442	35	p)({y}(λ	p)({y}(λ	PROPN
ejpam-4274	442	36	,	,	PUNCT
ejpam-4274	442	37	p	p	NOUN
ejpam-4274	442	38	)	)	PUNCT
ejpam-4274	442	39	)	)	PUNCT
ejpam-4274	442	40	,	,	PUNCT
ejpam-4274	442	41	there	there	PRON
ejpam-4274	442	42	exists	exist	VERB
ejpam-4274	442	43	a	a	DET
ejpam-4274	442	44	(	(	PUNCT
ejpam-4274	442	45	λ	λ	NOUN
ejpam-4274	442	46	,	,	PUNCT
ejpam-4274	442	47	p)-open	p)-open	VERB
ejpam-4274	442	48	set	set	VERB
ejpam-4274	442	49	g	g	PRON
ejpam-4274	442	50	such	such	ADJ
ejpam-4274	442	51	that	that	SCONJ
ejpam-4274	442	52	{	{	PUNCT
ejpam-4274	442	53	y}(λ	y}(λ	PROPN
ejpam-4274	442	54	,	,	PUNCT
ejpam-4274	442	55	p	p	NOUN
ejpam-4274	442	56	)	)	PUNCT
ejpam-4274	442	57	⊆	⊆	NUM
ejpam-4274	442	58	g	g	NOUN
ejpam-4274	442	59	and	and	CCONJ
ejpam-4274	442	60	x	x	SYM
ejpam-4274	442	61	6∈	6∈	PROPN
ejpam-4274	442	62	g.	g.	PROPN
ejpam-4274	442	63	hence	hence	ADV
ejpam-4274	442	64	,	,	PUNCT
ejpam-4274	442	65	{	{	PUNCT
ejpam-4274	442	66	x}(λ	x}(λ	PROPN
ejpam-4274	442	67	,	,	PUNCT
ejpam-4274	442	68	p	p	NOUN
ejpam-4274	442	69	)	)	PUNCT
ejpam-4274	442	70	∩	∩	NOUN
ejpam-4274	442	71	g	g	NOUN
ejpam-4274	442	72	=	=	PUNCT
ejpam-4274	442	73	∅.	∅.	NOUN
ejpam-4274	442	74	since	since	SCONJ
ejpam-4274	442	75	y	y	PROPN
ejpam-4274	442	76	∈	∈	PROPN
ejpam-4274	442	77	g	g	PROPN
ejpam-4274	442	78	,	,	PUNCT
ejpam-4274	442	79	we	we	PRON
ejpam-4274	442	80	have	have	VERB
ejpam-4274	442	81	y	y	PROPN
ejpam-4274	442	82	6∈	6∈	PROPN
ejpam-4274	442	83	{	{	PUNCT
ejpam-4274	442	84	x}(λ	x}(λ	PROPN
ejpam-4274	442	85	,	,	PUNCT
ejpam-4274	442	86	p	p	NOUN
ejpam-4274	442	87	)	)	PUNCT
ejpam-4274	442	88	and	and	CCONJ
ejpam-4274	442	89	hence	hence	ADV
ejpam-4274	442	90	{	{	PUNCT
ejpam-4274	442	91	x}(λ	x}(λ	PROPN
ejpam-4274	442	92	,	,	PUNCT
ejpam-4274	442	93	p	p	NOUN
ejpam-4274	442	94	)	)	PUNCT
ejpam-4274	442	95	⊆	⊆	NUM
ejpam-4274	442	96	λ(λ	λ(λ	PROPN
ejpam-4274	442	97	,	,	PUNCT
ejpam-4274	442	98	p)({x	p)({x	NOUN
ejpam-4274	442	99	}	}	PUNCT
ejpam-4274	442	100	)	)	PUNCT
ejpam-4274	442	101	.	.	PUNCT
ejpam-4274	443	1	moreover	moreover	ADV
ejpam-4274	443	2	,	,	PUNCT
ejpam-4274	443	3	{	{	PUNCT
ejpam-4274	443	4	x}(λ	x}(λ	PROPN
ejpam-4274	443	5	,	,	PUNCT
ejpam-4274	443	6	p	p	NOUN
ejpam-4274	443	7	)	)	PUNCT
ejpam-4274	443	8	⊆	⊆	NUM
ejpam-4274	443	9	λ(λ	λ(λ	PROPN
ejpam-4274	443	10	,	,	PUNCT
ejpam-4274	443	11	p)({x	p)({x	NOUN
ejpam-4274	443	12	}	}	PUNCT
ejpam-4274	443	13	)	)	PUNCT
ejpam-4274	443	14	⊆	⊆	NUM
ejpam-4274	443	15	λ(λ	λ(λ	PROPN
ejpam-4274	443	16	,	,	PUNCT
ejpam-4274	443	17	p)({x}(λ	p)({x}(λ	NOUN
ejpam-4274	443	18	,	,	PUNCT
ejpam-4274	443	19	p	p	NOUN
ejpam-4274	443	20	)	)	PUNCT
ejpam-4274	443	21	)	)	PUNCT
ejpam-4274	444	1	=	=	PRON
ejpam-4274	444	2	{	{	PUNCT
ejpam-4274	444	3	x}(λ	x}(λ	PROPN
ejpam-4274	444	4	,	,	PUNCT
ejpam-4274	444	5	p	p	NOUN
ejpam-4274	444	6	)	)	PUNCT
ejpam-4274	444	7	.	.	PUNCT
ejpam-4274	445	1	consequently	consequently	ADV
ejpam-4274	445	2	,	,	PUNCT
ejpam-4274	445	3	we	we	PRON
ejpam-4274	445	4	obtain	obtain	VERB
ejpam-4274	445	5	{	{	PUNCT
ejpam-4274	445	6	x}(λ	x}(λ	PROPN
ejpam-4274	445	7	,	,	PUNCT
ejpam-4274	445	8	p	p	NOUN
ejpam-4274	445	9	)	)	PUNCT
ejpam-4274	445	10	=	=	SYM
ejpam-4274	445	11	λ(λ	λ(λ	PROPN
ejpam-4274	445	12	,	,	PUNCT
ejpam-4274	445	13	p)({x	p)({x	NOUN
ejpam-4274	445	14	}	}	PUNCT
ejpam-4274	445	15	)	)	PUNCT
ejpam-4274	445	16	.	.	PUNCT
ejpam-4274	446	1	(	(	PUNCT
ejpam-4274	446	2	4	4	X
ejpam-4274	446	3	)	)	PUNCT
ejpam-4274	446	4	⇒	⇒	NOUN
ejpam-4274	446	5	(	(	PUNCT
ejpam-4274	446	6	5	5	NUM
ejpam-4274	446	7	):	):	PUNCT
ejpam-4274	446	8	the	the	DET
ejpam-4274	446	9	proof	proof	NOUN
ejpam-4274	446	10	is	be	AUX
ejpam-4274	446	11	obvious	obvious	ADJ
ejpam-4274	446	12	.	.	PUNCT
ejpam-4274	447	1	(	(	PUNCT
ejpam-4274	447	2	5	5	X
ejpam-4274	447	3	)	)	PUNCT
ejpam-4274	447	4	⇒	⇒	NOUN
ejpam-4274	447	5	(	(	PUNCT
ejpam-4274	447	6	1	1	NUM
ejpam-4274	447	7	):	):	PUNCT
ejpam-4274	447	8	let	let	VERB
ejpam-4274	447	9	u	u	PRON
ejpam-4274	447	10	∈	∈	PROPN
ejpam-4274	447	11	λpo(x	λpo(x	PROPN
ejpam-4274	447	12	,	,	PUNCT
ejpam-4274	447	13	τ	τ	X
ejpam-4274	447	14	)	)	PUNCT
ejpam-4274	447	15	and	and	CCONJ
ejpam-4274	447	16	let	let	VERB
ejpam-4274	447	17	x	x	PUNCT
ejpam-4274	447	18	∈	∈	PROPN
ejpam-4274	447	19	u	u	NOUN
ejpam-4274	447	20	.	.	PUNCT
ejpam-4274	448	1	if	if	SCONJ
ejpam-4274	448	2	y	y	PROPN
ejpam-4274	448	3	6∈	6∈	PROPN
ejpam-4274	448	4	u	u	PROPN
ejpam-4274	448	5	,	,	PUNCT
ejpam-4274	448	6	then	then	ADV
ejpam-4274	448	7	{	{	PUNCT
ejpam-4274	448	8	y}(λ	y}(λ	PROPN
ejpam-4274	448	9	,	,	PUNCT
ejpam-4274	448	10	p	p	NOUN
ejpam-4274	448	11	)	)	PUNCT
ejpam-4274	448	12	∩	∩	ADJ
ejpam-4274	448	13	u	u	NOUN
ejpam-4274	448	14	=	=	NOUN
ejpam-4274	448	15	∅	∅	NOUN
ejpam-4274	448	16	and	and	CCONJ
ejpam-4274	448	17	x	x	SYM
ejpam-4274	448	18	6∈	6∈	PROPN
ejpam-4274	448	19	{	{	PUNCT
ejpam-4274	448	20	y}(λ	y}(λ	PROPN
ejpam-4274	448	21	,	,	PUNCT
ejpam-4274	448	22	p	p	NOUN
ejpam-4274	448	23	)	)	PUNCT
ejpam-4274	448	24	.	.	PUNCT
ejpam-4274	449	1	by	by	ADP
ejpam-4274	449	2	lemma	lemma	PROPN
ejpam-4274	449	3	11	11	NUM
ejpam-4274	449	4	,	,	PUNCT
ejpam-4274	449	5	y	y	PROPN
ejpam-4274	449	6	6∈	6∈	PROPN
ejpam-4274	449	7	λ(λ	λ(λ	ADV
ejpam-4274	449	8	,	,	PUNCT
ejpam-4274	449	9	p)({x	p)({x	NOUN
ejpam-4274	449	10	}	}	PUNCT
ejpam-4274	449	11	)	)	PUNCT
ejpam-4274	449	12	and	and	CCONJ
ejpam-4274	449	13	by	by	ADP
ejpam-4274	449	14	(	(	PUNCT
ejpam-4274	449	15	5	5	NUM
ejpam-4274	449	16	)	)	PUNCT
ejpam-4274	449	17	,	,	PUNCT
ejpam-4274	449	18	y	y	PROPN
ejpam-4274	449	19	6∈	6∈	PROPN
ejpam-4274	449	20	{	{	PUNCT
ejpam-4274	449	21	x}(λ	x}(λ	PROPN
ejpam-4274	449	22	,	,	PUNCT
ejpam-4274	449	23	p	p	NOUN
ejpam-4274	449	24	)	)	PUNCT
ejpam-4274	449	25	.	.	PUNCT
ejpam-4274	450	1	thus	thus	ADV
ejpam-4274	450	2	,	,	PUNCT
ejpam-4274	450	3	{	{	PUNCT
ejpam-4274	450	4	x}(λ	x}(λ	PROPN
ejpam-4274	450	5	,	,	PUNCT
ejpam-4274	450	6	p	p	NOUN
ejpam-4274	450	7	)	)	PUNCT
ejpam-4274	450	8	⊆	⊆	NUM
ejpam-4274	450	9	u	u	NOUN
ejpam-4274	450	10	and	and	CCONJ
ejpam-4274	450	11	hence	hence	ADV
ejpam-4274	450	12	(	(	PUNCT
ejpam-4274	450	13	x	x	X
ejpam-4274	450	14	,	,	PUNCT
ejpam-4274	450	15	τ	τ	X
ejpam-4274	450	16	)	)	PUNCT
ejpam-4274	450	17	is	be	AUX
ejpam-4274	450	18	λp	λp	PROPN
ejpam-4274	450	19	-	-	PUNCT
ejpam-4274	450	20	r0	r0	NOUN
ejpam-4274	450	21	.	.	PUNCT
ejpam-4274	451	1	corollary	corollary	ADJ
ejpam-4274	451	2	3	3	NUM
ejpam-4274	451	3	.	.	PUNCT
ejpam-4274	452	1	a	a	DET
ejpam-4274	452	2	topological	topological	ADJ
ejpam-4274	452	3	space	space	NOUN
ejpam-4274	452	4	(	(	PUNCT
ejpam-4274	452	5	x	x	X
ejpam-4274	452	6	,	,	PUNCT
ejpam-4274	452	7	τ	τ	X
ejpam-4274	452	8	)	)	PUNCT
ejpam-4274	452	9	is	be	AUX
ejpam-4274	452	10	λp	λp	NOUN
ejpam-4274	452	11	-	-	PUNCT
ejpam-4274	452	12	r0	r0	NOUN
ejpam-4274	452	13	if	if	SCONJ
ejpam-4274	452	14	and	and	CCONJ
ejpam-4274	452	15	only	only	ADV
ejpam-4274	452	16	if	if	SCONJ
ejpam-4274	452	17	{	{	PUNCT
ejpam-4274	452	18	x}(λ	x}(λ	PROPN
ejpam-4274	452	19	,	,	PUNCT
ejpam-4274	452	20	p	p	NOUN
ejpam-4274	452	21	)	)	PUNCT
ejpam-4274	452	22	⊆	⊆	NUM
ejpam-4274	452	23	λ(λ	λ(λ	PROPN
ejpam-4274	452	24	,	,	PUNCT
ejpam-4274	452	25	p)({x	p)({x	NOUN
ejpam-4274	452	26	}	}	PUNCT
ejpam-4274	452	27	)	)	PUNCT
ejpam-4274	452	28	for	for	ADP
ejpam-4274	452	29	each	each	DET
ejpam-4274	452	30	x	x	SYM
ejpam-4274	452	31	∈	∈	PROPN
ejpam-4274	452	32	x.	x.	NOUN
ejpam-4274	452	33	proof	proof	NOUN
ejpam-4274	452	34	.	.	PUNCT
ejpam-4274	453	1	this	this	PRON
ejpam-4274	453	2	is	be	AUX
ejpam-4274	453	3	obvious	obvious	ADJ
ejpam-4274	453	4	by	by	ADP
ejpam-4274	453	5	theorem	theorem	NOUN
ejpam-4274	453	6	14	14	NUM
ejpam-4274	453	7	.	.	PUNCT
ejpam-4274	454	1	conversely	conversely	ADV
ejpam-4274	454	2	,	,	PUNCT
ejpam-4274	454	3	let	let	VERB
ejpam-4274	454	4	x	x	X
ejpam-4274	454	5	∈	∈	PROPN
ejpam-4274	454	6	{	{	PUNCT
ejpam-4274	454	7	y}(λ	y}(λ	PROPN
ejpam-4274	454	8	,	,	PUNCT
ejpam-4274	454	9	p	p	NOUN
ejpam-4274	454	10	)	)	PUNCT
ejpam-4274	454	11	.	.	PUNCT
ejpam-4274	455	1	by	by	ADP
ejpam-4274	455	2	lemma	lemma	PROPN
ejpam-4274	455	3	11	11	NUM
ejpam-4274	455	4	,	,	PUNCT
ejpam-4274	455	5	we	we	PRON
ejpam-4274	455	6	have	have	VERB
ejpam-4274	455	7	y	y	PROPN
ejpam-4274	455	8	∈	∈	PROPN
ejpam-4274	455	9	λ(λ	λ(λ	PROPN
ejpam-4274	455	10	,	,	PUNCT
ejpam-4274	455	11	p)({x	p)({x	NOUN
ejpam-4274	455	12	}	}	PUNCT
ejpam-4274	455	13	)	)	PUNCT
ejpam-4274	455	14	and	and	CCONJ
ejpam-4274	455	15	hence	hence	ADV
ejpam-4274	455	16	y	y	PROPN
ejpam-4274	455	17	∈	∈	PROPN
ejpam-4274	455	18	{	{	PUNCT
ejpam-4274	455	19	x}(λ	x}(λ	PROPN
ejpam-4274	455	20	,	,	PUNCT
ejpam-4274	455	21	p	p	NOUN
ejpam-4274	455	22	)	)	PUNCT
ejpam-4274	455	23	.	.	PUNCT
ejpam-4274	456	1	similarly	similarly	ADV
ejpam-4274	456	2	,	,	PUNCT
ejpam-4274	456	3	if	if	SCONJ
ejpam-4274	456	4	y	y	PROPN
ejpam-4274	456	5	∈	∈	PROPN
ejpam-4274	456	6	{	{	PUNCT
ejpam-4274	456	7	x}(λ	x}(λ	PROPN
ejpam-4274	456	8	,	,	PUNCT
ejpam-4274	456	9	p	p	NOUN
ejpam-4274	456	10	)	)	PUNCT
ejpam-4274	456	11	,	,	PUNCT
ejpam-4274	456	12	then	then	ADV
ejpam-4274	456	13	x	x	SYM
ejpam-4274	456	14	∈	∈	PROPN
ejpam-4274	456	15	{	{	PUNCT
ejpam-4274	456	16	y}(λ	y}(λ	PROPN
ejpam-4274	456	17	,	,	PUNCT
ejpam-4274	456	18	p	p	NOUN
ejpam-4274	456	19	)	)	PUNCT
ejpam-4274	456	20	.	.	PUNCT
ejpam-4274	457	1	it	it	PRON
ejpam-4274	457	2	follows	follow	VERB
ejpam-4274	457	3	from	from	ADP
ejpam-4274	457	4	theorem	theorem	ADJ
ejpam-4274	457	5	13	13	NUM
ejpam-4274	457	6	that	that	PRON
ejpam-4274	457	7	(	(	PUNCT
ejpam-4274	457	8	x	x	X
ejpam-4274	457	9	,	,	PUNCT
ejpam-4274	457	10	τ	τ	X
ejpam-4274	457	11	)	)	PUNCT
ejpam-4274	457	12	is	be	AUX
ejpam-4274	457	13	λp	λp	PROPN
ejpam-4274	457	14	-	-	PUNCT
ejpam-4274	457	15	r0	r0	NOUN
ejpam-4274	457	16	.	.	PUNCT
ejpam-4274	458	1	c.	c.	PROPN
ejpam-4274	458	2	boonpok	boonpok	PROPN
ejpam-4274	458	3	,	,	PUNCT
ejpam-4274	458	4	c.	c.	PROPN
ejpam-4274	458	5	viriyapong	viriyapong	PROPN
ejpam-4274	458	6	/	/	SYM
ejpam-4274	458	7	eur	eur	PROPN
ejpam-4274	458	8	.	.	PUNCT
ejpam-4274	459	1	j.	j.	PROPN
ejpam-4274	459	2	pure	pure	PROPN
ejpam-4274	459	3	appl	appl	PROPN
ejpam-4274	459	4	.	.	PROPN
ejpam-4274	459	5	math	math	PROPN
ejpam-4274	459	6	,	,	PUNCT
ejpam-4274	459	7	15	15	NUM
ejpam-4274	459	8	(	(	PUNCT
ejpam-4274	459	9	2	2	NUM
ejpam-4274	459	10	)	)	PUNCT
ejpam-4274	459	11	(	(	PUNCT
ejpam-4274	459	12	2022	2022	NUM
ejpam-4274	459	13	)	)	PUNCT
ejpam-4274	459	14	,	,	PUNCT
ejpam-4274	459	15	415	415	NUM
ejpam-4274	459	16	-	-	SYM
ejpam-4274	459	17	436	436	NUM
ejpam-4274	459	18	427	427	NUM
ejpam-4274	459	19	theorem	theorem	NOUN
ejpam-4274	459	20	15	15	NUM
ejpam-4274	459	21	.	.	PUNCT
ejpam-4274	460	1	for	for	ADP
ejpam-4274	460	2	a	a	DET
ejpam-4274	460	3	topological	topological	ADJ
ejpam-4274	460	4	space	space	NOUN
ejpam-4274	460	5	(	(	PUNCT
ejpam-4274	460	6	x	x	X
ejpam-4274	460	7	,	,	PUNCT
ejpam-4274	460	8	τ	τ	PROPN
ejpam-4274	460	9	)	)	PUNCT
ejpam-4274	460	10	,	,	PUNCT
ejpam-4274	460	11	the	the	DET
ejpam-4274	460	12	following	follow	VERB
ejpam-4274	460	13	properties	property	NOUN
ejpam-4274	460	14	are	be	AUX
ejpam-4274	460	15	equivalent	equivalent	ADJ
ejpam-4274	460	16	:	:	PUNCT
ejpam-4274	460	17	(	(	PUNCT
ejpam-4274	460	18	1	1	X
ejpam-4274	460	19	)	)	PUNCT
ejpam-4274	460	20	(	(	PUNCT
ejpam-4274	460	21	x	x	X
ejpam-4274	460	22	,	,	PUNCT
ejpam-4274	460	23	τ	τ	X
ejpam-4274	460	24	)	)	PUNCT
ejpam-4274	460	25	is	be	AUX
ejpam-4274	460	26	λp	λp	PROPN
ejpam-4274	460	27	-	-	PUNCT
ejpam-4274	460	28	r0	r0	NOUN
ejpam-4274	460	29	;	;	PUNCT
ejpam-4274	460	30	(	(	PUNCT
ejpam-4274	460	31	2	2	X
ejpam-4274	460	32	)	)	PUNCT
ejpam-4274	460	33	〈	〈	PROPN
ejpam-4274	460	34	x〉p	x〉p	NOUN
ejpam-4274	460	35	=	=	SYM
ejpam-4274	460	36	{	{	PUNCT
ejpam-4274	460	37	x}(λ	x}(λ	PROPN
ejpam-4274	460	38	,	,	PUNCT
ejpam-4274	460	39	p	p	NOUN
ejpam-4274	460	40	)	)	PUNCT
ejpam-4274	460	41	for	for	ADP
ejpam-4274	460	42	each	each	DET
ejpam-4274	460	43	x	x	SYM
ejpam-4274	460	44	∈	∈	PROPN
ejpam-4274	460	45	x	x	X
ejpam-4274	460	46	;	;	PUNCT
ejpam-4274	460	47	(	(	PUNCT
ejpam-4274	460	48	3	3	X
ejpam-4274	460	49	)	)	PUNCT
ejpam-4274	460	50	〈	〈	PROPN
ejpam-4274	460	51	x〉p	x〉p	NOUN
ejpam-4274	460	52	is	be	AUX
ejpam-4274	460	53	(	(	PUNCT
ejpam-4274	460	54	λ	λ	X
ejpam-4274	460	55	,	,	PUNCT
ejpam-4274	460	56	p)-closed	p)-close	VERB
ejpam-4274	460	57	for	for	ADP
ejpam-4274	460	58	each	each	DET
ejpam-4274	460	59	x	x	SYM
ejpam-4274	460	60	∈	∈	PROPN
ejpam-4274	460	61	x.	x.	NOUN
ejpam-4274	460	62	proof	proof	NOUN
ejpam-4274	460	63	.	.	PUNCT
ejpam-4274	461	1	(	(	PUNCT
ejpam-4274	461	2	1	1	X
ejpam-4274	461	3	)	)	PUNCT
ejpam-4274	461	4	⇒	⇒	NOUN
ejpam-4274	461	5	(	(	PUNCT
ejpam-4274	461	6	2	2	NUM
ejpam-4274	461	7	):	):	PUNCT
ejpam-4274	461	8	by	by	ADP
ejpam-4274	461	9	theorem	theorem	NOUN
ejpam-4274	461	10	14	14	NUM
ejpam-4274	461	11	,	,	PUNCT
ejpam-4274	461	12	{	{	PUNCT
ejpam-4274	461	13	x}(λ	x}(λ	PROPN
ejpam-4274	461	14	,	,	PUNCT
ejpam-4274	461	15	p	p	NOUN
ejpam-4274	461	16	)	)	PUNCT
ejpam-4274	461	17	=	=	SYM
ejpam-4274	461	18	λ(λ	λ(λ	PROPN
ejpam-4274	461	19	,	,	PUNCT
ejpam-4274	461	20	p)({x	p)({x	NOUN
ejpam-4274	461	21	}	}	PUNCT
ejpam-4274	461	22	)	)	PUNCT
ejpam-4274	461	23	for	for	ADP
ejpam-4274	461	24	each	each	DET
ejpam-4274	461	25	x	x	SYM
ejpam-4274	461	26	∈	∈	PROPN
ejpam-4274	461	27	x	x	X
ejpam-4274	461	28	and	and	CCONJ
ejpam-4274	461	29	hence	hence	ADV
ejpam-4274	461	30	{	{	PUNCT
ejpam-4274	461	31	x}(λ	x}(λ	PROPN
ejpam-4274	461	32	,	,	PUNCT
ejpam-4274	461	33	p	p	NOUN
ejpam-4274	461	34	)	)	PUNCT
ejpam-4274	461	35	=	=	SYM
ejpam-4274	461	36	{	{	PUNCT
ejpam-4274	461	37	x}(λ	x}(λ	PROPN
ejpam-4274	461	38	,	,	PUNCT
ejpam-4274	461	39	p	p	NOUN
ejpam-4274	461	40	)	)	PUNCT
ejpam-4274	461	41	∩	∩	PROPN
ejpam-4274	461	42	λ(λ	λ(λ	PROPN
ejpam-4274	461	43	,	,	PUNCT
ejpam-4274	461	44	p)({x	p)({x	NOUN
ejpam-4274	461	45	}	}	PUNCT
ejpam-4274	461	46	)	)	PUNCT
ejpam-4274	461	47	=	=	PUNCT
ejpam-4274	462	1	〈	〈	PROPN
ejpam-4274	462	2	x〉p	x〉p	NOUN
ejpam-4274	462	3	.	.	PUNCT
ejpam-4274	463	1	(	(	PUNCT
ejpam-4274	463	2	2	2	X
ejpam-4274	463	3	)	)	PUNCT
ejpam-4274	463	4	⇒	⇒	NOUN
ejpam-4274	463	5	(	(	PUNCT
ejpam-4274	463	6	1	1	NUM
ejpam-4274	463	7	):	):	PUNCT
ejpam-4274	463	8	since	since	SCONJ
ejpam-4274	463	9	{	{	PUNCT
ejpam-4274	463	10	x}(λ	x}(λ	PROPN
ejpam-4274	463	11	,	,	PUNCT
ejpam-4274	463	12	p	p	NOUN
ejpam-4274	463	13	)	)	PUNCT
ejpam-4274	463	14	=	=	PUNCT
ejpam-4274	464	1	〈	〈	NOUN
ejpam-4274	464	2	x〉p	x〉p	NOUN
ejpam-4274	464	3	for	for	ADP
ejpam-4274	464	4	each	each	DET
ejpam-4274	464	5	x	x	SYM
ejpam-4274	464	6	∈	∈	PROPN
ejpam-4274	464	7	x	x	X
ejpam-4274	464	8	,	,	PUNCT
ejpam-4274	464	9	we	we	PRON
ejpam-4274	464	10	have	have	VERB
ejpam-4274	464	11	{	{	PUNCT
ejpam-4274	464	12	x}(λ	x}(λ	PROPN
ejpam-4274	464	13	,	,	PUNCT
ejpam-4274	464	14	p	p	NOUN
ejpam-4274	464	15	)	)	PUNCT
ejpam-4274	464	16	⊆	⊆	NUM
ejpam-4274	464	17	λ(λ	λ(λ	PROPN
ejpam-4274	464	18	,	,	PUNCT
ejpam-4274	464	19	p)({x	p)({x	NOUN
ejpam-4274	464	20	}	}	PUNCT
ejpam-4274	464	21	)	)	PUNCT
ejpam-4274	464	22	.	.	PUNCT
ejpam-4274	465	1	by	by	ADP
ejpam-4274	465	2	corollary	corollary	ADJ
ejpam-4274	465	3	3	3	NUM
ejpam-4274	465	4	,	,	PUNCT
ejpam-4274	465	5	(	(	PUNCT
ejpam-4274	465	6	x	x	X
ejpam-4274	465	7	,	,	PUNCT
ejpam-4274	465	8	τ	τ	X
ejpam-4274	465	9	)	)	PUNCT
ejpam-4274	465	10	is	be	AUX
ejpam-4274	465	11	λp	λp	PROPN
ejpam-4274	465	12	-	-	PUNCT
ejpam-4274	465	13	r0	r0	NOUN
ejpam-4274	465	14	.	.	PUNCT
ejpam-4274	466	1	(	(	PUNCT
ejpam-4274	466	2	2	2	X
ejpam-4274	466	3	)	)	PUNCT
ejpam-4274	466	4	⇔	⇔	X
ejpam-4274	466	5	(	(	PUNCT
ejpam-4274	466	6	3	3	NUM
ejpam-4274	466	7	):	):	PUNCT
ejpam-4274	466	8	this	this	PRON
ejpam-4274	466	9	is	be	AUX
ejpam-4274	466	10	a	a	DET
ejpam-4274	466	11	consequence	consequence	NOUN
ejpam-4274	466	12	of	of	ADP
ejpam-4274	466	13	lemma	lemma	PROPN
ejpam-4274	466	14	7	7	NUM
ejpam-4274	466	15	.	.	NOUN
ejpam-4274	466	16	6	6	NUM
ejpam-4274	466	17	.	.	PUNCT
ejpam-4274	466	18	characterizations	characterization	NOUN
ejpam-4274	466	19	of	of	ADP
ejpam-4274	466	20	weakly	weakly	ADJ
ejpam-4274	466	21	(	(	PUNCT
ejpam-4274	466	22	λ	λ	PROPN
ejpam-4274	466	23	,	,	PUNCT
ejpam-4274	466	24	p)-continuous	p)-continuous	ADJ
ejpam-4274	466	25	functions	function	NOUN
ejpam-4274	466	26	in	in	ADP
ejpam-4274	466	27	this	this	DET
ejpam-4274	466	28	section	section	NOUN
ejpam-4274	466	29	,	,	PUNCT
ejpam-4274	466	30	we	we	PRON
ejpam-4274	466	31	introduce	introduce	VERB
ejpam-4274	466	32	the	the	DET
ejpam-4274	466	33	notion	notion	NOUN
ejpam-4274	466	34	of	of	ADP
ejpam-4274	466	35	weakly	weakly	ADJ
ejpam-4274	466	36	(	(	PUNCT
ejpam-4274	466	37	λ	λ	PROPN
ejpam-4274	466	38	,	,	PUNCT
ejpam-4274	466	39	p)-continuous	p)-continuous	ADJ
ejpam-4274	466	40	functions	function	NOUN
ejpam-4274	466	41	and	and	CCONJ
ejpam-4274	466	42	obtain	obtain	VERB
ejpam-4274	466	43	several	several	ADJ
ejpam-4274	466	44	characterizations	characterization	NOUN
ejpam-4274	466	45	of	of	ADP
ejpam-4274	466	46	weakly	weakly	ADJ
ejpam-4274	466	47	(	(	PUNCT
ejpam-4274	466	48	λ	λ	PROPN
ejpam-4274	466	49	,	,	PUNCT
ejpam-4274	466	50	p)-continuous	p)-continuous	ADJ
ejpam-4274	466	51	functions	function	NOUN
ejpam-4274	466	52	.	.	PUNCT
ejpam-4274	467	1	definition	definition	NOUN
ejpam-4274	467	2	10	10	NUM
ejpam-4274	467	3	.	.	PUNCT
ejpam-4274	468	1	let	let	VERB
ejpam-4274	468	2	a	a	DET
ejpam-4274	468	3	be	be	AUX
ejpam-4274	468	4	a	a	DET
ejpam-4274	468	5	subset	subset	NOUN
ejpam-4274	468	6	of	of	ADP
ejpam-4274	468	7	a	a	DET
ejpam-4274	468	8	topological	topological	ADJ
ejpam-4274	468	9	space	space	NOUN
ejpam-4274	468	10	(	(	PUNCT
ejpam-4274	468	11	x	x	X
ejpam-4274	468	12	,	,	PUNCT
ejpam-4274	468	13	τ	τ	PROPN
ejpam-4274	468	14	)	)	PUNCT
ejpam-4274	468	15	.	.	PUNCT
ejpam-4274	469	1	the	the	DET
ejpam-4274	469	2	union	union	NOUN
ejpam-4274	469	3	of	of	ADP
ejpam-4274	469	4	all	all	DET
ejpam-4274	469	5	(	(	PUNCT
ejpam-4274	469	6	λ	λ	X
ejpam-4274	469	7	,	,	PUNCT
ejpam-4274	469	8	p)-open	p)-open	VERB
ejpam-4274	469	9	sets	set	NOUN
ejpam-4274	469	10	contained	contain	VERB
ejpam-4274	469	11	in	in	ADP
ejpam-4274	469	12	a	a	PRON
ejpam-4274	469	13	is	be	AUX
ejpam-4274	469	14	called	call	VERB
ejpam-4274	469	15	the	the	DET
ejpam-4274	469	16	(	(	PUNCT
ejpam-4274	469	17	λ	λ	PROPN
ejpam-4274	469	18	,	,	PUNCT
ejpam-4274	469	19	p)-interior	p)-interior	ADJ
ejpam-4274	469	20	of	of	ADP
ejpam-4274	469	21	a	a	PRON
ejpam-4274	469	22	and	and	CCONJ
ejpam-4274	469	23	is	be	AUX
ejpam-4274	469	24	denoted	denote	VERB
ejpam-4274	469	25	by	by	ADP
ejpam-4274	469	26	a(λ	a(λ	PROPN
ejpam-4274	469	27	,	,	PUNCT
ejpam-4274	469	28	p	p	NOUN
ejpam-4274	469	29	)	)	PUNCT
ejpam-4274	469	30	.	.	PUNCT
ejpam-4274	470	1	lemma	lemma	PROPN
ejpam-4274	470	2	12	12	NUM
ejpam-4274	470	3	.	.	PUNCT
ejpam-4274	471	1	let	let	VERB
ejpam-4274	471	2	a	a	PRON
ejpam-4274	471	3	and	and	CCONJ
ejpam-4274	471	4	b	b	NOUN
ejpam-4274	471	5	be	be	AUX
ejpam-4274	471	6	subsets	subset	NOUN
ejpam-4274	471	7	of	of	ADP
ejpam-4274	471	8	a	a	DET
ejpam-4274	471	9	topological	topological	ADJ
ejpam-4274	471	10	space	space	NOUN
ejpam-4274	471	11	(	(	PUNCT
ejpam-4274	471	12	x	x	X
ejpam-4274	471	13	,	,	PUNCT
ejpam-4274	471	14	τ	τ	PROPN
ejpam-4274	471	15	)	)	PUNCT
ejpam-4274	471	16	.	.	PUNCT
ejpam-4274	472	1	for	for	ADP
ejpam-4274	472	2	the	the	DET
ejpam-4274	472	3	(	(	PUNCT
ejpam-4274	472	4	λ	λ	PROPN
ejpam-4274	472	5	,	,	PUNCT
ejpam-4274	472	6	p)-interior	p)-interior	ADJ
ejpam-4274	472	7	,	,	PUNCT
ejpam-4274	472	8	the	the	DET
ejpam-4274	472	9	following	follow	VERB
ejpam-4274	472	10	properties	property	NOUN
ejpam-4274	472	11	hold	hold	VERB
ejpam-4274	472	12	:	:	PUNCT
ejpam-4274	472	13	(	(	PUNCT
ejpam-4274	472	14	1	1	X
ejpam-4274	472	15	)	)	PUNCT
ejpam-4274	472	16	a(λ	a(λ	ADV
ejpam-4274	472	17	,	,	PUNCT
ejpam-4274	472	18	p	p	X
ejpam-4274	472	19	)	)	PUNCT
ejpam-4274	472	20	⊆	⊆	NUM
ejpam-4274	472	21	a	a	PRON
ejpam-4274	472	22	and	and	CCONJ
ejpam-4274	472	23	[	[	X
ejpam-4274	472	24	a(λ	a(λ	ADV
ejpam-4274	472	25	,	,	PUNCT
ejpam-4274	472	26	p)](λ	p)](λ	X
ejpam-4274	472	27	,	,	PUNCT
ejpam-4274	472	28	p	p	NOUN
ejpam-4274	472	29	)	)	PUNCT
ejpam-4274	472	30	=	=	PUNCT
ejpam-4274	473	1	a(λ	a(λ	PROPN
ejpam-4274	473	2	,	,	PUNCT
ejpam-4274	473	3	p	p	NOUN
ejpam-4274	473	4	)	)	PUNCT
ejpam-4274	473	5	.	.	PUNCT
ejpam-4274	474	1	(	(	PUNCT
ejpam-4274	474	2	2	2	X
ejpam-4274	474	3	)	)	PUNCT
ejpam-4274	474	4	if	if	SCONJ
ejpam-4274	474	5	a	a	DET
ejpam-4274	474	6	⊆	⊆	NUM
ejpam-4274	474	7	b	b	NOUN
ejpam-4274	474	8	,	,	PUNCT
ejpam-4274	474	9	then	then	ADV
ejpam-4274	474	10	a(λ	a(λ	ADV
ejpam-4274	474	11	,	,	PUNCT
ejpam-4274	474	12	p	p	NOUN
ejpam-4274	474	13	)	)	PUNCT
ejpam-4274	474	14	⊆	⊆	NUM
ejpam-4274	474	15	b(λ	b(λ	NOUN
ejpam-4274	474	16	,	,	PUNCT
ejpam-4274	474	17	p	p	NOUN
ejpam-4274	474	18	)	)	PUNCT
ejpam-4274	474	19	.	.	PUNCT
ejpam-4274	475	1	(	(	PUNCT
ejpam-4274	475	2	3	3	X
ejpam-4274	475	3	)	)	PUNCT
ejpam-4274	475	4	a(λ	a(λ	ADV
ejpam-4274	475	5	,	,	PUNCT
ejpam-4274	475	6	p	p	NOUN
ejpam-4274	475	7	)	)	PUNCT
ejpam-4274	475	8	=	=	SYM
ejpam-4274	475	9	∪{g	∪{g	PROPN
ejpam-4274	475	10	|	|	ADV
ejpam-4274	475	11	g	g	NOUN
ejpam-4274	475	12	⊆	⊆	NUM
ejpam-4274	475	13	a	a	PRON
ejpam-4274	475	14	and	and	CCONJ
ejpam-4274	475	15	g	g	NOUN
ejpam-4274	475	16	is	be	AUX
ejpam-4274	475	17	(	(	PUNCT
ejpam-4274	475	18	λ	λ	X
ejpam-4274	475	19	,	,	PUNCT
ejpam-4274	475	20	p)-open	p)-open	ADJ
ejpam-4274	475	21	}	}	PUNCT
ejpam-4274	475	22	.	.	PUNCT
ejpam-4274	476	1	(	(	PUNCT
ejpam-4274	476	2	4	4	NUM
ejpam-4274	476	3	)	)	PUNCT
ejpam-4274	476	4	a(λ	a(λ	ADV
ejpam-4274	476	5	,	,	PUNCT
ejpam-4274	476	6	p	p	NOUN
ejpam-4274	476	7	)	)	PUNCT
ejpam-4274	476	8	is	be	AUX
ejpam-4274	476	9	(	(	PUNCT
ejpam-4274	476	10	λ	λ	X
ejpam-4274	476	11	,	,	PUNCT
ejpam-4274	476	12	p)-open	p)-open	ADJ
ejpam-4274	476	13	.	.	PUNCT
ejpam-4274	477	1	(	(	PUNCT
ejpam-4274	477	2	5	5	X
ejpam-4274	477	3	)	)	PUNCT
ejpam-4274	477	4	a	a	PRON
ejpam-4274	477	5	is	be	AUX
ejpam-4274	477	6	(	(	PUNCT
ejpam-4274	477	7	λ	λ	X
ejpam-4274	477	8	,	,	PUNCT
ejpam-4274	477	9	p)-open	p)-open	VERB
ejpam-4274	477	10	if	if	SCONJ
ejpam-4274	477	11	and	and	CCONJ
ejpam-4274	477	12	only	only	ADV
ejpam-4274	477	13	if	if	SCONJ
ejpam-4274	477	14	a(λ	a(λ	ADV
ejpam-4274	477	15	,	,	PUNCT
ejpam-4274	477	16	p	p	NOUN
ejpam-4274	477	17	)	)	PUNCT
ejpam-4274	477	18	=	=	SYM
ejpam-4274	477	19	a.	a.	NOUN
ejpam-4274	477	20	(	(	PUNCT
ejpam-4274	477	21	6	6	NUM
ejpam-4274	477	22	)	)	PUNCT
ejpam-4274	478	1	[	[	X
ejpam-4274	478	2	x	x	X
ejpam-4274	478	3	−a](λ	−a](λ	PROPN
ejpam-4274	478	4	,	,	PUNCT
ejpam-4274	478	5	p	p	NOUN
ejpam-4274	478	6	)	)	PUNCT
ejpam-4274	478	7	=	=	SYM
ejpam-4274	478	8	x	x	SYM
ejpam-4274	478	9	−a(λ	−a(λ	NOUN
ejpam-4274	478	10	,	,	PUNCT
ejpam-4274	478	11	p	p	NOUN
ejpam-4274	478	12	)	)	PUNCT
ejpam-4274	478	13	.	.	PUNCT
ejpam-4274	479	1	definition	definition	NOUN
ejpam-4274	479	2	11	11	NUM
ejpam-4274	479	3	.	.	PUNCT
ejpam-4274	480	1	a	a	DET
ejpam-4274	480	2	function	function	NOUN
ejpam-4274	480	3	f	f	NOUN
ejpam-4274	480	4	:	:	PUNCT
ejpam-4274	480	5	(	(	PUNCT
ejpam-4274	480	6	x	x	X
ejpam-4274	480	7	,	,	PUNCT
ejpam-4274	480	8	τ	τ	X
ejpam-4274	480	9	)	)	PUNCT
ejpam-4274	480	10	→	→	SYM
ejpam-4274	480	11	(	(	PUNCT
ejpam-4274	480	12	y	y	PROPN
ejpam-4274	480	13	,	,	PUNCT
ejpam-4274	480	14	σ	σ	PROPN
ejpam-4274	480	15	)	)	PUNCT
ejpam-4274	480	16	is	be	AUX
ejpam-4274	480	17	said	say	VERB
ejpam-4274	480	18	to	to	PART
ejpam-4274	480	19	be	be	AUX
ejpam-4274	480	20	weakly	weakly	ADJ
ejpam-4274	480	21	(	(	PUNCT
ejpam-4274	480	22	λ	λ	X
ejpam-4274	480	23	,	,	PUNCT
ejpam-4274	480	24	p)-continuous	p)-continuous	ADJ
ejpam-4274	480	25	at	at	ADP
ejpam-4274	480	26	a	a	DET
ejpam-4274	480	27	point	point	NOUN
ejpam-4274	480	28	x	x	SYM
ejpam-4274	480	29	∈	∈	NOUN
ejpam-4274	480	30	x	x	INTJ
ejpam-4274	480	31	if	if	SCONJ
ejpam-4274	480	32	,	,	PUNCT
ejpam-4274	480	33	for	for	ADP
ejpam-4274	480	34	each	each	DET
ejpam-4274	480	35	(	(	PUNCT
ejpam-4274	480	36	λ	λ	PROPN
ejpam-4274	480	37	,	,	PUNCT
ejpam-4274	480	38	p)-open	p)-open	VERB
ejpam-4274	480	39	set	set	VERB
ejpam-4274	480	40	v	v	NOUN
ejpam-4274	480	41	containing	contain	VERB
ejpam-4274	480	42	f(x	f(x	PROPN
ejpam-4274	480	43	)	)	PUNCT
ejpam-4274	480	44	,	,	PUNCT
ejpam-4274	480	45	there	there	PRON
ejpam-4274	480	46	exists	exist	VERB
ejpam-4274	480	47	a	a	DET
ejpam-4274	480	48	(	(	PUNCT
ejpam-4274	480	49	λ	λ	NOUN
ejpam-4274	480	50	,	,	PUNCT
ejpam-4274	480	51	p)-open	p)-open	VERB
ejpam-4274	480	52	set	set	VERB
ejpam-4274	480	53	u	u	NOUN
ejpam-4274	480	54	containing	contain	VERB
ejpam-4274	480	55	x	x	PUNCT
ejpam-4274	480	56	such	such	ADJ
ejpam-4274	480	57	that	that	DET
ejpam-4274	480	58	f(u	f(u	PROPN
ejpam-4274	480	59	)	)	PUNCT
ejpam-4274	480	60	⊆	⊆	NUM
ejpam-4274	480	61	v	v	NOUN
ejpam-4274	480	62	(	(	PUNCT
ejpam-4274	480	63	λ	λ	PROPN
ejpam-4274	480	64	,	,	PUNCT
ejpam-4274	480	65	p	p	NOUN
ejpam-4274	480	66	)	)	PUNCT
ejpam-4274	480	67	.	.	PUNCT
ejpam-4274	481	1	a	a	DET
ejpam-4274	481	2	function	function	NOUN
ejpam-4274	481	3	f	f	NOUN
ejpam-4274	481	4	:	:	PUNCT
ejpam-4274	481	5	(	(	PUNCT
ejpam-4274	481	6	x	x	X
ejpam-4274	481	7	,	,	PUNCT
ejpam-4274	481	8	τ	τ	X
ejpam-4274	481	9	)	)	PUNCT
ejpam-4274	481	10	→	→	SYM
ejpam-4274	481	11	(	(	PUNCT
ejpam-4274	481	12	y	y	PROPN
ejpam-4274	481	13	,	,	PUNCT
ejpam-4274	481	14	σ	σ	PROPN
ejpam-4274	481	15	)	)	PUNCT
ejpam-4274	481	16	is	be	AUX
ejpam-4274	481	17	said	say	VERB
ejpam-4274	481	18	to	to	PART
ejpam-4274	481	19	be	be	AUX
ejpam-4274	481	20	(	(	PUNCT
ejpam-4274	481	21	λ	λ	X
ejpam-4274	481	22	,	,	PUNCT
ejpam-4274	481	23	p)-continuous	p)-continuous	ADJ
ejpam-4274	481	24	if	if	SCONJ
ejpam-4274	481	25	f	f	PROPN
ejpam-4274	481	26	has	have	VERB
ejpam-4274	481	27	this	this	DET
ejpam-4274	481	28	property	property	NOUN
ejpam-4274	481	29	at	at	ADP
ejpam-4274	481	30	each	each	DET
ejpam-4274	481	31	point	point	NOUN
ejpam-4274	481	32	x	x	X
ejpam-4274	481	33	∈	∈	PROPN
ejpam-4274	481	34	x.	x.	NOUN
ejpam-4274	481	35	theorem	theorem	VERB
ejpam-4274	481	36	16	16	NUM
ejpam-4274	481	37	.	.	PUNCT
ejpam-4274	482	1	a	a	DET
ejpam-4274	482	2	function	function	NOUN
ejpam-4274	482	3	f	f	NOUN
ejpam-4274	482	4	:	:	PUNCT
ejpam-4274	482	5	(	(	PUNCT
ejpam-4274	482	6	x	x	X
ejpam-4274	482	7	,	,	PUNCT
ejpam-4274	482	8	τ	τ	X
ejpam-4274	482	9	)	)	PUNCT
ejpam-4274	482	10	→	→	SYM
ejpam-4274	482	11	(	(	PUNCT
ejpam-4274	482	12	y	y	PROPN
ejpam-4274	482	13	,	,	PUNCT
ejpam-4274	482	14	σ	σ	PROPN
ejpam-4274	482	15	)	)	PUNCT
ejpam-4274	482	16	is	be	AUX
ejpam-4274	482	17	weakly	weakly	ADJ
ejpam-4274	482	18	(	(	PUNCT
ejpam-4274	482	19	λ	λ	X
ejpam-4274	482	20	,	,	PUNCT
ejpam-4274	482	21	p)-continuous	p)-continuous	ADJ
ejpam-4274	482	22	at	at	ADP
ejpam-4274	482	23	x	x	X
ejpam-4274	482	24	∈	∈	PROPN
ejpam-4274	482	25	x	x	SYM
ejpam-4274	482	26	if	if	SCONJ
ejpam-4274	482	27	and	and	CCONJ
ejpam-4274	482	28	only	only	ADV
ejpam-4274	482	29	if	if	SCONJ
ejpam-4274	482	30	for	for	ADP
ejpam-4274	482	31	each	each	DET
ejpam-4274	482	32	(	(	PUNCT
ejpam-4274	482	33	λ	λ	PROPN
ejpam-4274	482	34	,	,	PUNCT
ejpam-4274	482	35	p)-open	p)-open	VERB
ejpam-4274	482	36	set	set	VERB
ejpam-4274	482	37	v	v	NOUN
ejpam-4274	482	38	containing	contain	VERB
ejpam-4274	482	39	f(x	f(x	PROPN
ejpam-4274	482	40	)	)	PUNCT
ejpam-4274	482	41	,	,	PUNCT
ejpam-4274	483	1	x	x	PUNCT
ejpam-4274	483	2	∈	∈	PROPN
ejpam-4274	484	1	[	[	X
ejpam-4274	484	2	f−1(v	f−1(v	NOUN
ejpam-4274	484	3	(	(	PUNCT
ejpam-4274	484	4	λ	λ	PROPN
ejpam-4274	484	5	,	,	PUNCT
ejpam-4274	484	6	p))](λ	p))](λ	PRON
ejpam-4274	484	7	,	,	PUNCT
ejpam-4274	484	8	p	p	NOUN
ejpam-4274	484	9	)	)	PUNCT
ejpam-4274	484	10	.	.	PUNCT
ejpam-4274	485	1	c.	c.	PROPN
ejpam-4274	485	2	boonpok	boonpok	PROPN
ejpam-4274	485	3	,	,	PUNCT
ejpam-4274	485	4	c.	c.	PROPN
ejpam-4274	485	5	viriyapong	viriyapong	PROPN
ejpam-4274	485	6	/	/	SYM
ejpam-4274	485	7	eur	eur	PROPN
ejpam-4274	485	8	.	.	PUNCT
ejpam-4274	486	1	j.	j.	PROPN
ejpam-4274	486	2	pure	pure	PROPN
ejpam-4274	486	3	appl	appl	PROPN
ejpam-4274	486	4	.	.	PROPN
ejpam-4274	486	5	math	math	PROPN
ejpam-4274	486	6	,	,	PUNCT
ejpam-4274	486	7	15	15	NUM
ejpam-4274	486	8	(	(	PUNCT
ejpam-4274	486	9	2	2	NUM
ejpam-4274	486	10	)	)	PUNCT
ejpam-4274	486	11	(	(	PUNCT
ejpam-4274	486	12	2022	2022	NUM
ejpam-4274	486	13	)	)	PUNCT
ejpam-4274	486	14	,	,	PUNCT
ejpam-4274	486	15	415	415	NUM
ejpam-4274	486	16	-	-	SYM
ejpam-4274	486	17	436	436	NUM
ejpam-4274	486	18	428	428	NUM
ejpam-4274	486	19	proof	proof	NOUN
ejpam-4274	486	20	.	.	PUNCT
ejpam-4274	487	1	let	let	VERB
ejpam-4274	487	2	v	v	PART
ejpam-4274	487	3	be	be	AUX
ejpam-4274	487	4	a	a	DET
ejpam-4274	487	5	(	(	PUNCT
ejpam-4274	487	6	λ	λ	NOUN
ejpam-4274	487	7	,	,	PUNCT
ejpam-4274	487	8	p)-open	p)-open	VERB
ejpam-4274	487	9	set	set	VERB
ejpam-4274	487	10	containing	contain	VERB
ejpam-4274	487	11	f(x	f(x	PROPN
ejpam-4274	487	12	)	)	PUNCT
ejpam-4274	487	13	.	.	PUNCT
ejpam-4274	488	1	then	then	ADV
ejpam-4274	488	2	,	,	PUNCT
ejpam-4274	488	3	there	there	PRON
ejpam-4274	488	4	exists	exist	VERB
ejpam-4274	488	5	a	a	DET
ejpam-4274	488	6	(	(	PUNCT
ejpam-4274	488	7	λ	λ	NOUN
ejpam-4274	488	8	,	,	PUNCT
ejpam-4274	488	9	p)-open	p)-open	VERB
ejpam-4274	488	10	set	set	VERB
ejpam-4274	488	11	u	u	NOUN
ejpam-4274	488	12	containing	contain	VERB
ejpam-4274	488	13	x	x	PUNCT
ejpam-4274	488	14	such	such	ADJ
ejpam-4274	488	15	that	that	DET
ejpam-4274	488	16	f(u	f(u	PROPN
ejpam-4274	488	17	)	)	PUNCT
ejpam-4274	488	18	⊆	⊆	NUM
ejpam-4274	488	19	v	v	NOUN
ejpam-4274	488	20	(	(	PUNCT
ejpam-4274	488	21	λ	λ	PROPN
ejpam-4274	488	22	,	,	PUNCT
ejpam-4274	488	23	p	p	NOUN
ejpam-4274	488	24	)	)	PUNCT
ejpam-4274	488	25	and	and	CCONJ
ejpam-4274	488	26	hence	hence	ADV
ejpam-4274	488	27	x	x	PART
ejpam-4274	488	28	∈	∈	PROPN
ejpam-4274	488	29	u	u	NOUN
ejpam-4274	488	30	⊆	⊆	NUM
ejpam-4274	488	31	f−1(v	f−1(v	NOUN
ejpam-4274	488	32	(	(	PUNCT
ejpam-4274	488	33	λ	λ	PROPN
ejpam-4274	488	34	,	,	PUNCT
ejpam-4274	488	35	p	p	NOUN
ejpam-4274	488	36	)	)	PUNCT
ejpam-4274	488	37	)	)	PUNCT
ejpam-4274	488	38	.	.	PUNCT
ejpam-4274	489	1	thus	thus	ADV
ejpam-4274	489	2	,	,	PUNCT
ejpam-4274	489	3	x	x	PUNCT
ejpam-4274	489	4	∈	∈	PROPN
ejpam-4274	490	1	[	[	X
ejpam-4274	490	2	f−1(v	f−1(v	NOUN
ejpam-4274	490	3	(	(	PUNCT
ejpam-4274	490	4	λ	λ	PROPN
ejpam-4274	490	5	,	,	PUNCT
ejpam-4274	490	6	p))](λ	p))](λ	PRON
ejpam-4274	490	7	,	,	PUNCT
ejpam-4274	490	8	p	p	NOUN
ejpam-4274	490	9	)	)	PUNCT
ejpam-4274	490	10	.	.	PUNCT
ejpam-4274	491	1	conversely	conversely	ADV
ejpam-4274	491	2	,	,	PUNCT
ejpam-4274	491	3	let	let	VERB
ejpam-4274	491	4	v	v	PART
ejpam-4274	491	5	be	be	AUX
ejpam-4274	491	6	a	a	DET
ejpam-4274	491	7	(	(	PUNCT
ejpam-4274	491	8	λ	λ	NOUN
ejpam-4274	491	9	,	,	PUNCT
ejpam-4274	491	10	p)-open	p)-open	VERB
ejpam-4274	491	11	set	set	VERB
ejpam-4274	491	12	containing	contain	VERB
ejpam-4274	491	13	f(x	f(x	PROPN
ejpam-4274	491	14	)	)	PUNCT
ejpam-4274	491	15	.	.	PUNCT
ejpam-4274	492	1	by	by	ADP
ejpam-4274	492	2	the	the	DET
ejpam-4274	492	3	hypothesis	hypothesis	NOUN
ejpam-4274	492	4	,	,	PUNCT
ejpam-4274	492	5	we	we	PRON
ejpam-4274	492	6	have	have	VERB
ejpam-4274	492	7	x	x	PROPN
ejpam-4274	492	8	∈	∈	PROPN
ejpam-4274	493	1	[	[	X
ejpam-4274	493	2	f−1(v	f−1(v	NOUN
ejpam-4274	493	3	(	(	PUNCT
ejpam-4274	493	4	λ	λ	PROPN
ejpam-4274	493	5	,	,	PUNCT
ejpam-4274	493	6	p))](λ	p))](λ	PRON
ejpam-4274	493	7	,	,	PUNCT
ejpam-4274	493	8	p	p	NOUN
ejpam-4274	493	9	)	)	PUNCT
ejpam-4274	493	10	.	.	PUNCT
ejpam-4274	494	1	there	there	PRON
ejpam-4274	494	2	exists	exist	VERB
ejpam-4274	494	3	a	a	DET
ejpam-4274	494	4	(	(	PUNCT
ejpam-4274	494	5	λ	λ	NOUN
ejpam-4274	494	6	,	,	PUNCT
ejpam-4274	494	7	p)-open	p)-open	VERB
ejpam-4274	494	8	set	set	VERB
ejpam-4274	494	9	u	u	PRON
ejpam-4274	494	10	such	such	ADJ
ejpam-4274	494	11	that	that	SCONJ
ejpam-4274	494	12	x	x	SYM
ejpam-4274	494	13	∈	∈	PROPN
ejpam-4274	494	14	u	u	NOUN
ejpam-4274	494	15	⊆	⊆	NUM
ejpam-4274	494	16	f−1(v	f−1(v	NOUN
ejpam-4274	494	17	(	(	PUNCT
ejpam-4274	494	18	λ	λ	PROPN
ejpam-4274	494	19	,	,	PUNCT
ejpam-4274	494	20	p	p	NOUN
ejpam-4274	494	21	)	)	PUNCT
ejpam-4274	494	22	)	)	PUNCT
ejpam-4274	494	23	;	;	PUNCT
ejpam-4274	494	24	hence	hence	ADV
ejpam-4274	494	25	f(u	f(u	PROPN
ejpam-4274	494	26	)	)	PUNCT
ejpam-4274	494	27	⊆	⊆	NUM
ejpam-4274	494	28	v	v	NOUN
ejpam-4274	494	29	(	(	PUNCT
ejpam-4274	494	30	λ	λ	PROPN
ejpam-4274	494	31	,	,	PUNCT
ejpam-4274	494	32	p	p	NOUN
ejpam-4274	494	33	)	)	PUNCT
ejpam-4274	494	34	.	.	PUNCT
ejpam-4274	495	1	this	this	PRON
ejpam-4274	495	2	shows	show	VERB
ejpam-4274	495	3	that	that	SCONJ
ejpam-4274	495	4	f	f	PROPN
ejpam-4274	495	5	is	be	AUX
ejpam-4274	495	6	weakly	weakly	ADJ
ejpam-4274	495	7	(	(	PUNCT
ejpam-4274	495	8	λ	λ	X
ejpam-4274	495	9	,	,	PUNCT
ejpam-4274	495	10	p)-continuous	p)-continuous	ADJ
ejpam-4274	495	11	at	at	ADP
ejpam-4274	495	12	x	x	PROPN
ejpam-4274	495	13	∈	∈	PROPN
ejpam-4274	495	14	x.	x.	NOUN
ejpam-4274	495	15	theorem	theorem	VERB
ejpam-4274	495	16	17	17	NUM
ejpam-4274	495	17	.	.	PUNCT
ejpam-4274	496	1	a	a	DET
ejpam-4274	496	2	function	function	NOUN
ejpam-4274	496	3	f	f	NOUN
ejpam-4274	496	4	:	:	PUNCT
ejpam-4274	496	5	(	(	PUNCT
ejpam-4274	496	6	x	x	X
ejpam-4274	496	7	,	,	PUNCT
ejpam-4274	496	8	τ	τ	X
ejpam-4274	496	9	)	)	PUNCT
ejpam-4274	496	10	→	→	SYM
ejpam-4274	496	11	(	(	PUNCT
ejpam-4274	496	12	y	y	PROPN
ejpam-4274	496	13	,	,	PUNCT
ejpam-4274	496	14	σ	σ	PROPN
ejpam-4274	496	15	)	)	PUNCT
ejpam-4274	496	16	is	be	AUX
ejpam-4274	496	17	weakly	weakly	ADJ
ejpam-4274	496	18	(	(	PUNCT
ejpam-4274	496	19	λ	λ	NOUN
ejpam-4274	496	20	,	,	PUNCT
ejpam-4274	496	21	p)-continuous	p)-continuous	ADJ
ejpam-4274	496	22	if	if	SCONJ
ejpam-4274	496	23	and	and	CCONJ
ejpam-4274	496	24	only	only	ADV
ejpam-4274	496	25	if	if	SCONJ
ejpam-4274	496	26	f−1(v	f−1(v	PROPN
ejpam-4274	496	27	)	)	PUNCT
ejpam-4274	497	1	⊆	⊆	NUM
ejpam-4274	498	1	[	[	X
ejpam-4274	498	2	f−1(v	f−1(v	NOUN
ejpam-4274	498	3	(	(	PUNCT
ejpam-4274	498	4	λ	λ	PROPN
ejpam-4274	498	5	,	,	PUNCT
ejpam-4274	498	6	p))](λ	p))](λ	PRON
ejpam-4274	498	7	,	,	PUNCT
ejpam-4274	498	8	p	p	NOUN
ejpam-4274	498	9	)	)	PUNCT
ejpam-4274	498	10	for	for	ADP
ejpam-4274	498	11	every	every	DET
ejpam-4274	498	12	(	(	PUNCT
ejpam-4274	498	13	λ	λ	NOUN
ejpam-4274	498	14	,	,	PUNCT
ejpam-4274	498	15	p)-open	p)-open	VERB
ejpam-4274	498	16	set	set	VERB
ejpam-4274	498	17	v	v	NOUN
ejpam-4274	498	18	of	of	ADP
ejpam-4274	498	19	y	y	PROPN
ejpam-4274	498	20	.	.	PUNCT
ejpam-4274	499	1	proof	proof	NOUN
ejpam-4274	499	2	.	.	PUNCT
ejpam-4274	500	1	let	let	VERB
ejpam-4274	500	2	v	v	PART
ejpam-4274	500	3	be	be	AUX
ejpam-4274	500	4	any	any	DET
ejpam-4274	500	5	(	(	PUNCT
ejpam-4274	500	6	λ	λ	NOUN
ejpam-4274	500	7	,	,	PUNCT
ejpam-4274	500	8	p)-open	p)-open	VERB
ejpam-4274	500	9	set	set	VERB
ejpam-4274	500	10	of	of	ADP
ejpam-4274	500	11	y	y	PROPN
ejpam-4274	500	12	and	and	CCONJ
ejpam-4274	500	13	let	let	VERB
ejpam-4274	500	14	x	x	X
ejpam-4274	500	15	∈	∈	PROPN
ejpam-4274	500	16	f−1(v	f−1(v	NOUN
ejpam-4274	500	17	)	)	PUNCT
ejpam-4274	500	18	.	.	PUNCT
ejpam-4274	501	1	then	then	ADV
ejpam-4274	501	2	f(x	f(x	PROPN
ejpam-4274	501	3	)	)	PUNCT
ejpam-4274	501	4	∈	∈	PROPN
ejpam-4274	501	5	v	v	NOUN
ejpam-4274	501	6	.	.	PUNCT
ejpam-4274	502	1	since	since	SCONJ
ejpam-4274	502	2	f	f	PROPN
ejpam-4274	502	3	is	be	AUX
ejpam-4274	502	4	weakly	weakly	ADJ
ejpam-4274	502	5	(	(	PUNCT
ejpam-4274	502	6	λ	λ	X
ejpam-4274	502	7	,	,	PUNCT
ejpam-4274	502	8	p)-continuous	p)-continuous	ADJ
ejpam-4274	502	9	at	at	ADP
ejpam-4274	502	10	x	x	X
ejpam-4274	502	11	,	,	PUNCT
ejpam-4274	502	12	by	by	ADP
ejpam-4274	502	13	theorem	theorem	NOUN
ejpam-4274	502	14	16	16	NUM
ejpam-4274	502	15	,	,	PUNCT
ejpam-4274	502	16	x	x	SYM
ejpam-4274	502	17	∈	∈	PROPN
ejpam-4274	503	1	[	[	X
ejpam-4274	503	2	f−1(v	f−1(v	NOUN
ejpam-4274	503	3	(	(	PUNCT
ejpam-4274	503	4	λ	λ	PROPN
ejpam-4274	503	5	,	,	PUNCT
ejpam-4274	503	6	p))](λ	p))](λ	PRON
ejpam-4274	503	7	,	,	PUNCT
ejpam-4274	503	8	p	p	NOUN
ejpam-4274	503	9	)	)	PUNCT
ejpam-4274	503	10	and	and	CCONJ
ejpam-4274	503	11	hence	hence	ADV
ejpam-4274	503	12	f−1(v	f−1(v	NOUN
ejpam-4274	503	13	)	)	PUNCT
ejpam-4274	503	14	⊆	⊆	NUM
ejpam-4274	503	15	[	[	X
ejpam-4274	503	16	f−1(v	f−1(v	NOUN
ejpam-4274	503	17	(	(	PUNCT
ejpam-4274	503	18	λ	λ	PROPN
ejpam-4274	503	19	,	,	PUNCT
ejpam-4274	503	20	p))](λ	p))](λ	PRON
ejpam-4274	503	21	,	,	PUNCT
ejpam-4274	503	22	p	p	NOUN
ejpam-4274	503	23	)	)	PUNCT
ejpam-4274	503	24	.	.	PUNCT
ejpam-4274	504	1	conversely	conversely	ADV
ejpam-4274	504	2	,	,	PUNCT
ejpam-4274	504	3	let	let	VERB
ejpam-4274	504	4	x	x	X
ejpam-4274	504	5	∈	∈	NOUN
ejpam-4274	504	6	x	x	PUNCT
ejpam-4274	504	7	and	and	CCONJ
ejpam-4274	504	8	let	let	VERB
ejpam-4274	504	9	v	v	PART
ejpam-4274	504	10	be	be	AUX
ejpam-4274	504	11	any	any	DET
ejpam-4274	504	12	(	(	PUNCT
ejpam-4274	504	13	λ	λ	NOUN
ejpam-4274	504	14	,	,	PUNCT
ejpam-4274	504	15	p)-open	p)-open	VERB
ejpam-4274	504	16	set	set	VERB
ejpam-4274	504	17	of	of	ADP
ejpam-4274	504	18	y	y	PROPN
ejpam-4274	504	19	containing	contain	VERB
ejpam-4274	504	20	f(x	f(x	PROPN
ejpam-4274	504	21	)	)	PUNCT
ejpam-4274	504	22	.	.	PUNCT
ejpam-4274	505	1	then	then	ADV
ejpam-4274	505	2	,	,	PUNCT
ejpam-4274	505	3	we	we	PRON
ejpam-4274	505	4	have	have	VERB
ejpam-4274	505	5	x	x	X
ejpam-4274	505	6	∈	∈	PROPN
ejpam-4274	505	7	f−1(v	f−1(v	NOUN
ejpam-4274	505	8	)	)	PUNCT
ejpam-4274	505	9	⊆	⊆	NUM
ejpam-4274	506	1	[	[	X
ejpam-4274	506	2	f−1(v	f−1(v	NOUN
ejpam-4274	506	3	(	(	PUNCT
ejpam-4274	506	4	λ	λ	PROPN
ejpam-4274	506	5	,	,	PUNCT
ejpam-4274	506	6	p))](λ	p))](λ	PRON
ejpam-4274	506	7	,	,	PUNCT
ejpam-4274	506	8	p	p	NOUN
ejpam-4274	506	9	)	)	PUNCT
ejpam-4274	506	10	and	and	CCONJ
ejpam-4274	506	11	hence	hence	ADV
ejpam-4274	506	12	x	x	X
ejpam-4274	506	13	∈	∈	PROPN
ejpam-4274	506	14	[	[	X
ejpam-4274	506	15	f−1(v	f−1(v	NOUN
ejpam-4274	506	16	(	(	PUNCT
ejpam-4274	506	17	λ	λ	PROPN
ejpam-4274	506	18	,	,	PUNCT
ejpam-4274	506	19	p))](λ	p))](λ	PRON
ejpam-4274	506	20	,	,	PUNCT
ejpam-4274	506	21	p	p	NOUN
ejpam-4274	506	22	)	)	PUNCT
ejpam-4274	506	23	.	.	PUNCT
ejpam-4274	507	1	thus	thus	ADV
ejpam-4274	507	2	,	,	PUNCT
ejpam-4274	507	3	f	f	PROPN
ejpam-4274	507	4	is	be	AUX
ejpam-4274	507	5	weakly	weakly	ADJ
ejpam-4274	507	6	(	(	PUNCT
ejpam-4274	507	7	λ	λ	NOUN
ejpam-4274	507	8	,	,	PUNCT
ejpam-4274	507	9	p)-continuous	p)-continuous	ADJ
ejpam-4274	507	10	by	by	ADP
ejpam-4274	507	11	theorem	theorem	NOUN
ejpam-4274	507	12	16	16	NUM
ejpam-4274	507	13	.	.	PUNCT
ejpam-4274	508	1	theorem	theorem	VERB
ejpam-4274	508	2	18	18	NUM
ejpam-4274	508	3	.	.	PUNCT
ejpam-4274	509	1	a	a	DET
ejpam-4274	509	2	function	function	NOUN
ejpam-4274	509	3	f	f	NOUN
ejpam-4274	509	4	:	:	PUNCT
ejpam-4274	509	5	(	(	PUNCT
ejpam-4274	509	6	x	x	X
ejpam-4274	509	7	,	,	PUNCT
ejpam-4274	509	8	τ	τ	X
ejpam-4274	509	9	)	)	PUNCT
ejpam-4274	509	10	→	→	SYM
ejpam-4274	509	11	(	(	PUNCT
ejpam-4274	509	12	y	y	PROPN
ejpam-4274	509	13	,	,	PUNCT
ejpam-4274	509	14	σ	σ	PROPN
ejpam-4274	509	15	)	)	PUNCT
ejpam-4274	509	16	is	be	AUX
ejpam-4274	509	17	weakly	weakly	ADJ
ejpam-4274	509	18	(	(	PUNCT
ejpam-4274	509	19	λ	λ	NOUN
ejpam-4274	509	20	,	,	PUNCT
ejpam-4274	509	21	p)-continuous	p)-continuous	ADJ
ejpam-4274	509	22	if	if	SCONJ
ejpam-4274	509	23	and	and	CCONJ
ejpam-4274	509	24	only	only	ADV
ejpam-4274	509	25	if	if	SCONJ
ejpam-4274	509	26	[	[	X
ejpam-4274	509	27	f−1(v	f−1(v	NOUN
ejpam-4274	509	28	)	)	PUNCT
ejpam-4274	509	29	]	]	PUNCT
ejpam-4274	509	30	(	(	PUNCT
ejpam-4274	509	31	λ	λ	X
ejpam-4274	509	32	,	,	PUNCT
ejpam-4274	509	33	p	p	NOUN
ejpam-4274	509	34	)	)	PUNCT
ejpam-4274	509	35	⊆	⊆	NUM
ejpam-4274	509	36	f−1(v	f−1(v	NOUN
ejpam-4274	509	37	(	(	PUNCT
ejpam-4274	509	38	λ	λ	PROPN
ejpam-4274	509	39	,	,	PUNCT
ejpam-4274	509	40	p	p	NOUN
ejpam-4274	509	41	)	)	PUNCT
ejpam-4274	509	42	)	)	PUNCT
ejpam-4274	509	43	for	for	SCONJ
ejpam-4274	509	44	every	every	DET
ejpam-4274	509	45	(	(	PUNCT
ejpam-4274	509	46	λ	λ	NOUN
ejpam-4274	509	47	,	,	PUNCT
ejpam-4274	509	48	p)-open	p)-open	VERB
ejpam-4274	509	49	set	set	VERB
ejpam-4274	509	50	v	v	NOUN
ejpam-4274	509	51	of	of	ADP
ejpam-4274	509	52	y	y	PROPN
ejpam-4274	509	53	.	.	PUNCT
ejpam-4274	510	1	proof	proof	NOUN
ejpam-4274	510	2	.	.	PUNCT
ejpam-4274	511	1	let	let	VERB
ejpam-4274	511	2	v	v	PART
ejpam-4274	511	3	be	be	AUX
ejpam-4274	511	4	any	any	DET
ejpam-4274	511	5	(	(	PUNCT
ejpam-4274	511	6	λ	λ	NOUN
ejpam-4274	511	7	,	,	PUNCT
ejpam-4274	511	8	p)-open	p)-open	PUNCT
ejpam-4274	511	9	subset	subset	NOUN
ejpam-4274	511	10	of	of	ADP
ejpam-4274	511	11	y	y	PROPN
ejpam-4274	511	12	and	and	CCONJ
ejpam-4274	511	13	let	let	VERB
ejpam-4274	511	14	x	x	X
ejpam-4274	511	15	∈	∈	PROPN
ejpam-4274	511	16	f−1(v	f−1(v	NOUN
ejpam-4274	511	17	)	)	PUNCT
ejpam-4274	511	18	.	.	PUNCT
ejpam-4274	512	1	there	there	PRON
ejpam-4274	512	2	exists	exist	VERB
ejpam-4274	512	3	a	a	DET
ejpam-4274	512	4	(	(	PUNCT
ejpam-4274	512	5	λ	λ	NOUN
ejpam-4274	512	6	,	,	PUNCT
ejpam-4274	512	7	p)-open	p)-open	VERB
ejpam-4274	512	8	set	set	VERB
ejpam-4274	512	9	u	u	NOUN
ejpam-4274	512	10	containing	contain	VERB
ejpam-4274	512	11	x	x	PUNCT
ejpam-4274	512	12	such	such	ADJ
ejpam-4274	512	13	that	that	DET
ejpam-4274	512	14	f(u	f(u	PROPN
ejpam-4274	512	15	)	)	PUNCT
ejpam-4274	512	16	⊆	⊆	NUM
ejpam-4274	512	17	v	v	NOUN
ejpam-4274	512	18	(	(	PUNCT
ejpam-4274	512	19	λ	λ	PROPN
ejpam-4274	512	20	,	,	PUNCT
ejpam-4274	512	21	p	p	NOUN
ejpam-4274	512	22	)	)	PUNCT
ejpam-4274	512	23	.	.	PUNCT
ejpam-4274	513	1	since	since	SCONJ
ejpam-4274	513	2	x	x	PROPN
ejpam-4274	513	3	∈	∈	PROPN
ejpam-4274	513	4	u	u	NOUN
ejpam-4274	513	5	⊆	⊆	NUM
ejpam-4274	513	6	f−1(v	f−1(v	NOUN
ejpam-4274	513	7	(	(	PUNCT
ejpam-4274	513	8	λ	λ	PROPN
ejpam-4274	513	9	,	,	PUNCT
ejpam-4274	513	10	p	p	NOUN
ejpam-4274	513	11	)	)	PUNCT
ejpam-4274	513	12	)	)	PUNCT
ejpam-4274	513	13	,	,	PUNCT
ejpam-4274	513	14	we	we	PRON
ejpam-4274	513	15	have	have	VERB
ejpam-4274	513	16	x	x	PROPN
ejpam-4274	513	17	∈	∈	PROPN
ejpam-4274	514	1	[	[	X
ejpam-4274	514	2	f−1(v	f−1(v	NOUN
ejpam-4274	514	3	(	(	PUNCT
ejpam-4274	514	4	λ	λ	PROPN
ejpam-4274	514	5	,	,	PUNCT
ejpam-4274	514	6	p))](λ	p))](λ	PRON
ejpam-4274	514	7	,	,	PUNCT
ejpam-4274	514	8	p	p	NOUN
ejpam-4274	514	9	)	)	PUNCT
ejpam-4274	514	10	and	and	CCONJ
ejpam-4274	514	11	hence	hence	ADV
ejpam-4274	514	12	f−1(v	f−1(v	NOUN
ejpam-4274	514	13	)	)	PUNCT
ejpam-4274	514	14	⊆	⊆	NUM
ejpam-4274	514	15	[	[	X
ejpam-4274	514	16	f−1(v	f−1(v	NOUN
ejpam-4274	514	17	(	(	PUNCT
ejpam-4274	514	18	λ	λ	PROPN
ejpam-4274	514	19	,	,	PUNCT
ejpam-4274	514	20	p))](λ	p))](λ	PRON
ejpam-4274	514	21	,	,	PUNCT
ejpam-4274	514	22	p	p	NOUN
ejpam-4274	514	23	)	)	PUNCT
ejpam-4274	514	24	.	.	PUNCT
ejpam-4274	515	1	conversely	conversely	ADV
ejpam-4274	515	2	,	,	PUNCT
ejpam-4274	515	3	let	let	VERB
ejpam-4274	515	4	x	x	X
ejpam-4274	515	5	∈	∈	NOUN
ejpam-4274	515	6	x	x	PUNCT
ejpam-4274	515	7	and	and	CCONJ
ejpam-4274	515	8	let	let	VERB
ejpam-4274	515	9	v	v	PART
ejpam-4274	515	10	be	be	AUX
ejpam-4274	515	11	any	any	DET
ejpam-4274	515	12	(	(	PUNCT
ejpam-4274	515	13	λ	λ	NOUN
ejpam-4274	515	14	,	,	PUNCT
ejpam-4274	515	15	p)-open	p)-open	VERB
ejpam-4274	515	16	set	set	VERB
ejpam-4274	515	17	containing	contain	VERB
ejpam-4274	515	18	f(x	f(x	PROPN
ejpam-4274	515	19	)	)	PUNCT
ejpam-4274	515	20	.	.	PUNCT
ejpam-4274	516	1	since	since	SCONJ
ejpam-4274	516	2	v	v	ADP
ejpam-4274	516	3	∩	∩	NOUN
ejpam-4274	516	4	[	[	X
ejpam-4274	516	5	y	y	PROPN
ejpam-4274	516	6	−	−	PROPN
ejpam-4274	516	7	v	v	NOUN
ejpam-4274	516	8	(	(	PUNCT
ejpam-4274	516	9	λ	λ	PROPN
ejpam-4274	516	10	,	,	PUNCT
ejpam-4274	516	11	p	p	NOUN
ejpam-4274	516	12	)	)	PUNCT
ejpam-4274	516	13	]	]	PUNCT
ejpam-4274	516	14	=	=	SYM
ejpam-4274	516	15	∅	∅	NOUN
ejpam-4274	516	16	,	,	PUNCT
ejpam-4274	516	17	f(x	f(x	PROPN
ejpam-4274	516	18	)	)	PUNCT
ejpam-4274	516	19	6∈	6∈	NOUN
ejpam-4274	517	1	[	[	X
ejpam-4274	517	2	y	y	PROPN
ejpam-4274	517	3	−	−	PROPN
ejpam-4274	517	4	v	v	PROPN
ejpam-4274	517	5	(	(	PUNCT
ejpam-4274	517	6	λ	λ	PROPN
ejpam-4274	517	7	,	,	PUNCT
ejpam-4274	517	8	p)](λ	p)](λ	ADJ
ejpam-4274	517	9	,	,	PUNCT
ejpam-4274	517	10	p	p	NOUN
ejpam-4274	517	11	)	)	PUNCT
ejpam-4274	517	12	and	and	CCONJ
ejpam-4274	517	13	hence	hence	ADV
ejpam-4274	517	14	x	x	ADP
ejpam-4274	517	15	6∈	6∈	NOUN
ejpam-4274	517	16	f−1([y	f−1([y	NOUN
ejpam-4274	517	17	−	−	PROPN
ejpam-4274	517	18	v	v	NOUN
ejpam-4274	517	19	(	(	PUNCT
ejpam-4274	517	20	λ	λ	PROPN
ejpam-4274	517	21	,	,	PUNCT
ejpam-4274	517	22	p)](λ	p)](λ	ADJ
ejpam-4274	517	23	,	,	PUNCT
ejpam-4274	517	24	p	p	NOUN
ejpam-4274	517	25	)	)	PUNCT
ejpam-4274	517	26	)	)	PUNCT
ejpam-4274	517	27	.	.	PUNCT
ejpam-4274	517	28	by	by	ADP
ejpam-4274	517	29	the	the	DET
ejpam-4274	517	30	hypothesis	hypothesis	NOUN
ejpam-4274	517	31	,	,	PUNCT
ejpam-4274	517	32	x	x	PROPN
ejpam-4274	517	33	6∈	6∈	PROPN
ejpam-4274	518	1	[	[	X
ejpam-4274	518	2	f−1(y	f−1(y	PROPN
ejpam-4274	518	3	−	−	PROPN
ejpam-4274	518	4	v	v	NOUN
ejpam-4274	518	5	(	(	PUNCT
ejpam-4274	518	6	λ	λ	PROPN
ejpam-4274	518	7	,	,	PUNCT
ejpam-4274	518	8	p))](λ	p))](λ	PRON
ejpam-4274	518	9	,	,	PUNCT
ejpam-4274	518	10	p	p	X
ejpam-4274	518	11	)	)	PUNCT
ejpam-4274	518	12	=	=	PUNCT
ejpam-4274	519	1	[	[	X
ejpam-4274	519	2	x	x	X
ejpam-4274	519	3	−	−	X
ejpam-4274	519	4	f−1(v	f−1(v	NOUN
ejpam-4274	519	5	(	(	PUNCT
ejpam-4274	519	6	λ	λ	PROPN
ejpam-4274	519	7	,	,	PUNCT
ejpam-4274	519	8	p))](λ	p))](λ	PRON
ejpam-4274	519	9	,	,	PUNCT
ejpam-4274	519	10	p	p	NOUN
ejpam-4274	519	11	)	)	PUNCT
ejpam-4274	519	12	and	and	CCONJ
ejpam-4274	519	13	there	there	PRON
ejpam-4274	519	14	exists	exist	VERB
ejpam-4274	519	15	a	a	DET
ejpam-4274	519	16	(	(	PUNCT
ejpam-4274	519	17	λ	λ	NOUN
ejpam-4274	519	18	,	,	PUNCT
ejpam-4274	519	19	p)-open	p)-open	VERB
ejpam-4274	519	20	set	set	VERB
ejpam-4274	519	21	u	u	NOUN
ejpam-4274	519	22	containing	contain	VERB
ejpam-4274	519	23	x	x	PUNCT
ejpam-4274	519	24	such	such	ADJ
ejpam-4274	519	25	that	that	SCONJ
ejpam-4274	519	26	u	u	NOUN
ejpam-4274	519	27	∩	∩	NOUN
ejpam-4274	519	28	[	[	X
ejpam-4274	519	29	x	x	X
ejpam-4274	519	30	−	−	PROPN
ejpam-4274	519	31	f−1(v	f−1(v	NOUN
ejpam-4274	519	32	(	(	PUNCT
ejpam-4274	519	33	λ	λ	PROPN
ejpam-4274	519	34	,	,	PUNCT
ejpam-4274	519	35	p	p	NOUN
ejpam-4274	519	36	)	)	PUNCT
ejpam-4274	519	37	)	)	PUNCT
ejpam-4274	519	38	]	]	PUNCT
ejpam-4274	520	1	=	=	PUNCT
ejpam-4274	520	2	∅.	∅.	VERB
ejpam-4274	520	3	thus	thus	ADV
ejpam-4274	520	4	,	,	PUNCT
ejpam-4274	520	5	f(u	f(u	PROPN
ejpam-4274	520	6	)	)	PUNCT
ejpam-4274	520	7	⊆	⊆	NUM
ejpam-4274	520	8	v	v	NOUN
ejpam-4274	520	9	(	(	PUNCT
ejpam-4274	520	10	λ	λ	PROPN
ejpam-4274	520	11	,	,	PUNCT
ejpam-4274	520	12	p	p	NOUN
ejpam-4274	520	13	)	)	PUNCT
ejpam-4274	520	14	.	.	PUNCT
ejpam-4274	521	1	this	this	PRON
ejpam-4274	521	2	shows	show	VERB
ejpam-4274	521	3	that	that	SCONJ
ejpam-4274	521	4	f	f	PROPN
ejpam-4274	521	5	is	be	AUX
ejpam-4274	521	6	is	be	AUX
ejpam-4274	521	7	weakly	weakly	ADJ
ejpam-4274	521	8	(	(	PUNCT
ejpam-4274	521	9	λ	λ	NOUN
ejpam-4274	521	10	,	,	PUNCT
ejpam-4274	521	11	p)-continuous	p)-continuous	ADJ
ejpam-4274	521	12	.	.	PUNCT
ejpam-4274	521	13	theorem	theorem	VERB
ejpam-4274	521	14	19	19	NUM
ejpam-4274	521	15	.	.	PUNCT
ejpam-4274	522	1	for	for	ADP
ejpam-4274	522	2	a	a	DET
ejpam-4274	522	3	function	function	NOUN
ejpam-4274	522	4	f	f	NOUN
ejpam-4274	522	5	:	:	PUNCT
ejpam-4274	522	6	(	(	PUNCT
ejpam-4274	522	7	x	x	X
ejpam-4274	522	8	,	,	PUNCT
ejpam-4274	522	9	τ	τ	X
ejpam-4274	522	10	)	)	PUNCT
ejpam-4274	522	11	→	→	SYM
ejpam-4274	522	12	(	(	PUNCT
ejpam-4274	522	13	y	y	PROPN
ejpam-4274	522	14	,	,	PUNCT
ejpam-4274	522	15	σ	σ	PROPN
ejpam-4274	522	16	)	)	PUNCT
ejpam-4274	522	17	,	,	PUNCT
ejpam-4274	522	18	the	the	DET
ejpam-4274	522	19	following	follow	VERB
ejpam-4274	522	20	properties	property	NOUN
ejpam-4274	522	21	are	be	AUX
ejpam-4274	522	22	equivalent	equivalent	ADJ
ejpam-4274	522	23	:	:	PUNCT
ejpam-4274	522	24	(	(	PUNCT
ejpam-4274	522	25	1	1	X
ejpam-4274	522	26	)	)	PUNCT
ejpam-4274	522	27	f	f	PROPN
ejpam-4274	522	28	is	be	AUX
ejpam-4274	522	29	weakly	weakly	ADJ
ejpam-4274	522	30	(	(	PUNCT
ejpam-4274	522	31	λ	λ	NOUN
ejpam-4274	522	32	,	,	PUNCT
ejpam-4274	522	33	p)-continuous	p)-continuous	ADJ
ejpam-4274	522	34	;	;	PUNCT
ejpam-4274	522	35	(	(	PUNCT
ejpam-4274	522	36	2	2	X
ejpam-4274	522	37	)	)	PUNCT
ejpam-4274	522	38	f−1(u	f−1(u	NOUN
ejpam-4274	522	39	)	)	PUNCT
ejpam-4274	522	40	⊆	⊆	NUM
ejpam-4274	523	1	[	[	X
ejpam-4274	523	2	f−1(u	f−1(u	NOUN
ejpam-4274	523	3	(	(	PUNCT
ejpam-4274	523	4	λ	λ	PROPN
ejpam-4274	523	5	,	,	PUNCT
ejpam-4274	523	6	p))](λ	p))](λ	PRON
ejpam-4274	523	7	,	,	PUNCT
ejpam-4274	523	8	p	p	NOUN
ejpam-4274	523	9	)	)	PUNCT
ejpam-4274	523	10	for	for	ADP
ejpam-4274	523	11	every	every	DET
ejpam-4274	523	12	(	(	PUNCT
ejpam-4274	523	13	λ	λ	NOUN
ejpam-4274	523	14	,	,	PUNCT
ejpam-4274	523	15	p)-open	p)-open	VERB
ejpam-4274	523	16	subset	subset	VERB
ejpam-4274	523	17	u	u	NOUN
ejpam-4274	523	18	of	of	ADP
ejpam-4274	523	19	y	y	PROPN
ejpam-4274	523	20	;	;	PUNCT
ejpam-4274	523	21	(	(	PUNCT
ejpam-4274	523	22	3	3	X
ejpam-4274	523	23	)	)	PUNCT
ejpam-4274	523	24	[	[	X
ejpam-4274	523	25	f−1(f(λ	f−1(f(λ	X
ejpam-4274	523	26	,	,	PUNCT
ejpam-4274	523	27	p	p	NOUN
ejpam-4274	523	28	)	)	PUNCT
ejpam-4274	523	29	)	)	PUNCT
ejpam-4274	523	30	]	]	PUNCT
ejpam-4274	524	1	(	(	PUNCT
ejpam-4274	524	2	λ	λ	X
ejpam-4274	524	3	,	,	PUNCT
ejpam-4274	524	4	p	p	NOUN
ejpam-4274	524	5	)	)	PUNCT
ejpam-4274	524	6	⊆	⊆	NUM
ejpam-4274	524	7	f−1(f	f−1(f	PROPN
ejpam-4274	524	8	)	)	PUNCT
ejpam-4274	524	9	for	for	ADP
ejpam-4274	524	10	every	every	DET
ejpam-4274	524	11	(	(	PUNCT
ejpam-4274	524	12	λ	λ	PROPN
ejpam-4274	524	13	,	,	PUNCT
ejpam-4274	524	14	p)-closed	p)-close	VERB
ejpam-4274	524	15	subset	subset	NOUN
ejpam-4274	524	16	f	f	PROPN
ejpam-4274	524	17	of	of	ADP
ejpam-4274	524	18	y	y	PROPN
ejpam-4274	524	19	;	;	PUNCT
ejpam-4274	524	20	(	(	PUNCT
ejpam-4274	524	21	4	4	X
ejpam-4274	524	22	)	)	PUNCT
ejpam-4274	525	1	[	[	X
ejpam-4274	525	2	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4274	525	3	,	,	PUNCT
ejpam-4274	525	4	p)](λ	p)](λ	X
ejpam-4274	525	5	,	,	PUNCT
ejpam-4274	525	6	p	p	NOUN
ejpam-4274	525	7	)	)	PUNCT
ejpam-4274	525	8	)	)	PUNCT
ejpam-4274	525	9	]	]	PUNCT
ejpam-4274	526	1	(	(	PUNCT
ejpam-4274	526	2	λ	λ	X
ejpam-4274	526	3	,	,	PUNCT
ejpam-4274	526	4	p	p	NOUN
ejpam-4274	526	5	)	)	PUNCT
ejpam-4274	526	6	⊆	⊆	NUM
ejpam-4274	526	7	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4274	526	8	,	,	PUNCT
ejpam-4274	526	9	p	p	NOUN
ejpam-4274	526	10	)	)	PUNCT
ejpam-4274	526	11	)	)	PUNCT
ejpam-4274	526	12	for	for	ADP
ejpam-4274	526	13	every	every	DET
ejpam-4274	526	14	subset	subset	NOUN
ejpam-4274	526	15	a	a	PRON
ejpam-4274	526	16	of	of	ADP
ejpam-4274	526	17	y	y	PROPN
ejpam-4274	526	18	;	;	PUNCT
ejpam-4274	526	19	(	(	PUNCT
ejpam-4274	526	20	5	5	X
ejpam-4274	526	21	)	)	PUNCT
ejpam-4274	526	22	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4274	526	23	,	,	PUNCT
ejpam-4274	526	24	p	p	NOUN
ejpam-4274	526	25	)	)	PUNCT
ejpam-4274	526	26	)	)	PUNCT
ejpam-4274	527	1	⊆	⊆	NUM
ejpam-4274	527	2	[	[	X
ejpam-4274	527	3	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4274	527	4	,	,	PUNCT
ejpam-4274	527	5	p	p	NOUN
ejpam-4274	527	6	)	)	PUNCT
ejpam-4274	527	7	]	]	PUNCT
ejpam-4274	527	8	(	(	PUNCT
ejpam-4274	527	9	λ	λ	X
ejpam-4274	527	10	,	,	PUNCT
ejpam-4274	527	11	p))](λ	p))](λ	PRON
ejpam-4274	527	12	,	,	PUNCT
ejpam-4274	527	13	p	p	NOUN
ejpam-4274	527	14	)	)	PUNCT
ejpam-4274	527	15	for	for	ADP
ejpam-4274	527	16	every	every	DET
ejpam-4274	527	17	subset	subset	NOUN
ejpam-4274	527	18	a	a	PRON
ejpam-4274	527	19	of	of	ADP
ejpam-4274	527	20	y	y	PROPN
ejpam-4274	527	21	;	;	PUNCT
ejpam-4274	527	22	c.	c.	PROPN
ejpam-4274	527	23	boonpok	boonpok	PROPN
ejpam-4274	527	24	,	,	PUNCT
ejpam-4274	527	25	c.	c.	PROPN
ejpam-4274	527	26	viriyapong	viriyapong	PROPN
ejpam-4274	527	27	/	/	SYM
ejpam-4274	527	28	eur	eur	PROPN
ejpam-4274	527	29	.	.	PUNCT
ejpam-4274	528	1	j.	j.	PROPN
ejpam-4274	528	2	pure	pure	PROPN
ejpam-4274	528	3	appl	appl	PROPN
ejpam-4274	528	4	.	.	PROPN
ejpam-4274	528	5	math	math	PROPN
ejpam-4274	528	6	,	,	PUNCT
ejpam-4274	528	7	15	15	NUM
ejpam-4274	528	8	(	(	PUNCT
ejpam-4274	528	9	2	2	NUM
ejpam-4274	528	10	)	)	PUNCT
ejpam-4274	528	11	(	(	PUNCT
ejpam-4274	528	12	2022	2022	NUM
ejpam-4274	528	13	)	)	PUNCT
ejpam-4274	528	14	,	,	PUNCT
ejpam-4274	528	15	415	415	NUM
ejpam-4274	528	16	-	-	SYM
ejpam-4274	528	17	436	436	NUM
ejpam-4274	528	18	429	429	NUM
ejpam-4274	528	19	(	(	PUNCT
ejpam-4274	528	20	6	6	NUM
ejpam-4274	528	21	)	)	PUNCT
ejpam-4274	529	1	[	[	X
ejpam-4274	529	2	f−1(u)](λ	f−1(u)](λ	X
ejpam-4274	529	3	,	,	PUNCT
ejpam-4274	529	4	p	p	NOUN
ejpam-4274	529	5	)	)	PUNCT
ejpam-4274	529	6	⊆	⊆	NUM
ejpam-4274	529	7	f−1(u	f−1(u	NOUN
ejpam-4274	529	8	(	(	PUNCT
ejpam-4274	529	9	λ	λ	PROPN
ejpam-4274	529	10	,	,	PUNCT
ejpam-4274	529	11	p	p	NOUN
ejpam-4274	529	12	)	)	PUNCT
ejpam-4274	529	13	)	)	PUNCT
ejpam-4274	529	14	for	for	ADP
ejpam-4274	529	15	every	every	DET
ejpam-4274	529	16	(	(	PUNCT
ejpam-4274	529	17	λ	λ	NOUN
ejpam-4274	529	18	,	,	PUNCT
ejpam-4274	529	19	p)-open	p)-open	VERB
ejpam-4274	529	20	subset	subset	VERB
ejpam-4274	529	21	u	u	NOUN
ejpam-4274	529	22	of	of	ADP
ejpam-4274	529	23	y	y	PROPN
ejpam-4274	529	24	.	.	PUNCT
ejpam-4274	530	1	proof	proof	NOUN
ejpam-4274	530	2	.	.	PUNCT
ejpam-4274	531	1	(	(	PUNCT
ejpam-4274	531	2	1	1	X
ejpam-4274	531	3	)	)	PUNCT
ejpam-4274	531	4	⇒	⇒	NOUN
ejpam-4274	531	5	(	(	PUNCT
ejpam-4274	531	6	2	2	NUM
ejpam-4274	531	7	):	):	PUNCT
ejpam-4274	531	8	it	it	PRON
ejpam-4274	531	9	follows	follow	VERB
ejpam-4274	531	10	from	from	ADP
ejpam-4274	531	11	theorem	theorem	ADJ
ejpam-4274	531	12	17	17	NUM
ejpam-4274	531	13	.	.	PUNCT
ejpam-4274	532	1	(	(	PUNCT
ejpam-4274	532	2	2	2	X
ejpam-4274	532	3	)	)	PUNCT
ejpam-4274	532	4	⇒	⇒	NOUN
ejpam-4274	532	5	(	(	PUNCT
ejpam-4274	532	6	3	3	NUM
ejpam-4274	532	7	):	):	PUNCT
ejpam-4274	532	8	let	let	VERB
ejpam-4274	532	9	f	f	PRON
ejpam-4274	532	10	be	be	AUX
ejpam-4274	532	11	any	any	DET
ejpam-4274	532	12	(	(	PUNCT
ejpam-4274	532	13	λ	λ	PROPN
ejpam-4274	532	14	,	,	PUNCT
ejpam-4274	532	15	p)-closed	p)-close	VERB
ejpam-4274	532	16	subset	subset	NOUN
ejpam-4274	532	17	of	of	ADP
ejpam-4274	532	18	y	y	PROPN
ejpam-4274	532	19	.	.	PUNCT
ejpam-4274	533	1	then	then	ADV
ejpam-4274	533	2	,	,	PUNCT
ejpam-4274	533	3	y	y	PROPN
ejpam-4274	533	4	−f	−f	PROPN
ejpam-4274	533	5	is	be	AUX
ejpam-4274	533	6	(	(	PUNCT
ejpam-4274	533	7	λ	λ	INTJ
ejpam-4274	533	8	,	,	PUNCT
ejpam-4274	533	9	p)-open	p)-open	ADJ
ejpam-4274	533	10	,	,	PUNCT
ejpam-4274	533	11	by	by	ADP
ejpam-4274	533	12	(	(	PUNCT
ejpam-4274	533	13	2	2	NUM
ejpam-4274	533	14	)	)	PUNCT
ejpam-4274	533	15	,	,	PUNCT
ejpam-4274	533	16	f−1(y	f−1(y	PROPN
ejpam-4274	533	17	−	−	PROPN
ejpam-4274	533	18	f	f	PROPN
ejpam-4274	533	19	)	)	PUNCT
ejpam-4274	534	1	⊆	⊆	NUM
ejpam-4274	534	2	[	[	X
ejpam-4274	534	3	f−1([y	f−1([y	NOUN
ejpam-4274	534	4	−	−	ADP
ejpam-4274	534	5	f	f	X
ejpam-4274	534	6	]	]	X
ejpam-4274	534	7	(	(	PUNCT
ejpam-4274	534	8	λ	λ	NOUN
ejpam-4274	534	9	,	,	PUNCT
ejpam-4274	534	10	p))](λ	p))](λ	PRON
ejpam-4274	534	11	,	,	PUNCT
ejpam-4274	534	12	p	p	X
ejpam-4274	534	13	)	)	PUNCT
ejpam-4274	534	14	=	=	PUNCT
ejpam-4274	535	1	[	[	X
ejpam-4274	535	2	f−1(y	f−1(y	NOUN
ejpam-4274	535	3	−	−	PROPN
ejpam-4274	535	4	f(λ	f(λ	PROPN
ejpam-4274	535	5	,	,	PUNCT
ejpam-4274	535	6	p))](λ	p))](λ	PRON
ejpam-4274	535	7	,	,	PUNCT
ejpam-4274	535	8	p	p	X
ejpam-4274	535	9	)	)	PUNCT
ejpam-4274	535	10	=	=	PUNCT
ejpam-4274	535	11	x	x	X
ejpam-4274	535	12	−	−	PROPN
ejpam-4274	536	1	[	[	X
ejpam-4274	536	2	f−1(f(λ	f−1(f(λ	X
ejpam-4274	536	3	,	,	PUNCT
ejpam-4274	536	4	p	p	NOUN
ejpam-4274	536	5	)	)	PUNCT
ejpam-4274	536	6	)	)	PUNCT
ejpam-4274	536	7	]	]	PUNCT
ejpam-4274	536	8	(	(	PUNCT
ejpam-4274	536	9	λ	λ	X
ejpam-4274	536	10	,	,	PUNCT
ejpam-4274	536	11	p	p	NOUN
ejpam-4274	536	12	)	)	PUNCT
ejpam-4274	536	13	.	.	PUNCT
ejpam-4274	537	1	thus	thus	ADV
ejpam-4274	537	2	,	,	PUNCT
ejpam-4274	537	3	[	[	X
ejpam-4274	537	4	f−1(f(λ	f−1(f(λ	X
ejpam-4274	537	5	,	,	PUNCT
ejpam-4274	537	6	p	p	NOUN
ejpam-4274	537	7	)	)	PUNCT
ejpam-4274	537	8	)	)	PUNCT
ejpam-4274	537	9	]	]	PUNCT
ejpam-4274	538	1	(	(	PUNCT
ejpam-4274	538	2	λ	λ	X
ejpam-4274	538	3	,	,	PUNCT
ejpam-4274	538	4	p	p	NOUN
ejpam-4274	538	5	)	)	PUNCT
ejpam-4274	538	6	⊆	⊆	NUM
ejpam-4274	538	7	f−1(f	f−1(f	PROPN
ejpam-4274	538	8	)	)	PUNCT
ejpam-4274	538	9	.	.	PUNCT
ejpam-4274	539	1	(	(	PUNCT
ejpam-4274	539	2	3	3	X
ejpam-4274	539	3	)	)	PUNCT
ejpam-4274	539	4	⇒	⇒	NOUN
ejpam-4274	539	5	(	(	PUNCT
ejpam-4274	539	6	4	4	NUM
ejpam-4274	539	7	):	):	PUNCT
ejpam-4274	539	8	let	let	VERB
ejpam-4274	539	9	a	a	DET
ejpam-4274	539	10	be	be	AUX
ejpam-4274	539	11	any	any	DET
ejpam-4274	539	12	subset	subset	NOUN
ejpam-4274	539	13	of	of	ADP
ejpam-4274	539	14	y	y	PROPN
ejpam-4274	539	15	.	.	PUNCT
ejpam-4274	540	1	since	since	SCONJ
ejpam-4274	540	2	a(λ	a(λ	PROPN
ejpam-4274	540	3	,	,	PUNCT
ejpam-4274	540	4	p	p	NOUN
ejpam-4274	540	5	)	)	PUNCT
ejpam-4274	540	6	is	be	AUX
ejpam-4274	540	7	(	(	PUNCT
ejpam-4274	540	8	λ	λ	X
ejpam-4274	540	9	,	,	PUNCT
ejpam-4274	540	10	p)-closed	p)-close	VERB
ejpam-4274	540	11	,	,	PUNCT
ejpam-4274	540	12	by	by	ADP
ejpam-4274	540	13	(	(	PUNCT
ejpam-4274	540	14	3	3	NUM
ejpam-4274	540	15	)	)	PUNCT
ejpam-4274	540	16	,	,	PUNCT
ejpam-4274	541	1	[	[	X
ejpam-4274	541	2	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4274	541	3	,	,	PUNCT
ejpam-4274	541	4	p)](λ	p)](λ	X
ejpam-4274	541	5	,	,	PUNCT
ejpam-4274	541	6	p	p	NOUN
ejpam-4274	541	7	)	)	PUNCT
ejpam-4274	541	8	)	)	PUNCT
ejpam-4274	541	9	]	]	PUNCT
ejpam-4274	542	1	(	(	PUNCT
ejpam-4274	542	2	λ	λ	X
ejpam-4274	542	3	,	,	PUNCT
ejpam-4274	542	4	p	p	NOUN
ejpam-4274	542	5	)	)	PUNCT
ejpam-4274	542	6	⊆	⊆	NUM
ejpam-4274	542	7	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4274	542	8	,	,	PUNCT
ejpam-4274	542	9	p	p	NOUN
ejpam-4274	542	10	)	)	PUNCT
ejpam-4274	542	11	)	)	PUNCT
ejpam-4274	542	12	.	.	PUNCT
ejpam-4274	543	1	(	(	PUNCT
ejpam-4274	543	2	4	4	X
ejpam-4274	543	3	)	)	PUNCT
ejpam-4274	543	4	⇒	⇒	NOUN
ejpam-4274	543	5	(	(	PUNCT
ejpam-4274	543	6	5	5	NUM
ejpam-4274	543	7	):	):	PUNCT
ejpam-4274	543	8	let	let	VERB
ejpam-4274	543	9	a	a	PRON
ejpam-4274	543	10	be	be	AUX
ejpam-4274	543	11	any	any	DET
ejpam-4274	543	12	subset	subset	NOUN
ejpam-4274	543	13	of	of	ADP
ejpam-4274	543	14	y	y	PROPN
ejpam-4274	543	15	.	.	PUNCT
ejpam-4274	544	1	by	by	ADP
ejpam-4274	544	2	(	(	PUNCT
ejpam-4274	544	3	4	4	NUM
ejpam-4274	544	4	)	)	PUNCT
ejpam-4274	544	5	,	,	PUNCT
ejpam-4274	544	6	we	we	PRON
ejpam-4274	544	7	have	have	VERB
ejpam-4274	544	8	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4274	544	9	,	,	PUNCT
ejpam-4274	544	10	p	p	NOUN
ejpam-4274	544	11	)	)	PUNCT
ejpam-4274	544	12	)	)	PUNCT
ejpam-4274	545	1	=	=	PUNCT
ejpam-4274	545	2	x	x	X
ejpam-4274	546	1	−	−	NOUN
ejpam-4274	546	2	f−1([y	f−1([y	NOUN
ejpam-4274	546	3	−a](λ	−a](λ	NOUN
ejpam-4274	546	4	,	,	PUNCT
ejpam-4274	546	5	p	p	NOUN
ejpam-4274	546	6	)	)	PUNCT
ejpam-4274	546	7	)	)	PUNCT
ejpam-4274	547	1	⊆	⊆	NUM
ejpam-4274	547	2	x	x	SYM
ejpam-4274	547	3	−	−	NOUN
ejpam-4274	548	1	[	[	X
ejpam-4274	548	2	f−1([[y	f−1([[y	NOUN
ejpam-4274	548	3	−a](λ	−a](λ	NUM
ejpam-4274	548	4	,	,	PUNCT
ejpam-4274	548	5	p)](λ	p)](λ	X
ejpam-4274	548	6	,	,	PUNCT
ejpam-4274	548	7	p	p	NOUN
ejpam-4274	548	8	)	)	PUNCT
ejpam-4274	548	9	)	)	PUNCT
ejpam-4274	548	10	]	]	PUNCT
ejpam-4274	549	1	(	(	PUNCT
ejpam-4274	549	2	λ	λ	X
ejpam-4274	549	3	,	,	PUNCT
ejpam-4274	549	4	p	p	NOUN
ejpam-4274	549	5	)	)	PUNCT
ejpam-4274	549	6	=	=	NOUN
ejpam-4274	550	1	[	[	X
ejpam-4274	550	2	f−1([a(λ	f−1([a(λ	X
ejpam-4274	550	3	,	,	PUNCT
ejpam-4274	550	4	p	p	NOUN
ejpam-4274	550	5	)	)	PUNCT
ejpam-4274	550	6	]	]	PUNCT
ejpam-4274	550	7	(	(	PUNCT
ejpam-4274	550	8	λ	λ	X
ejpam-4274	550	9	,	,	PUNCT
ejpam-4274	550	10	p))](λ	p))](λ	PRON
ejpam-4274	550	11	,	,	PUNCT
ejpam-4274	550	12	p	p	NOUN
ejpam-4274	550	13	)	)	PUNCT
ejpam-4274	550	14	.	.	PUNCT
ejpam-4274	551	1	thus	thus	ADV
ejpam-4274	551	2	,	,	PUNCT
ejpam-4274	551	3	we	we	PRON
ejpam-4274	551	4	get	get	VERB
ejpam-4274	551	5	the	the	DET
ejpam-4274	551	6	result	result	NOUN
ejpam-4274	551	7	.	.	PUNCT
ejpam-4274	552	1	(	(	PUNCT
ejpam-4274	552	2	5	5	X
ejpam-4274	552	3	)	)	PUNCT
ejpam-4274	552	4	⇒	⇒	NOUN
ejpam-4274	552	5	(	(	PUNCT
ejpam-4274	552	6	6	6	NUM
ejpam-4274	552	7	):	):	PUNCT
ejpam-4274	552	8	let	let	VERB
ejpam-4274	552	9	u	u	PRON
ejpam-4274	552	10	be	be	AUX
ejpam-4274	552	11	any	any	DET
ejpam-4274	552	12	(	(	PUNCT
ejpam-4274	552	13	λ	λ	NOUN
ejpam-4274	552	14	,	,	PUNCT
ejpam-4274	552	15	p)-open	p)-open	PUNCT
ejpam-4274	552	16	subset	subset	NOUN
ejpam-4274	552	17	of	of	ADP
ejpam-4274	552	18	y	y	PROPN
ejpam-4274	552	19	.	.	PUNCT
ejpam-4274	552	20	suppose	suppose	VERB
ejpam-4274	552	21	that	that	SCONJ
ejpam-4274	552	22	x	x	SYM
ejpam-4274	552	23	6∈	6∈	PROPN
ejpam-4274	552	24	f−1(u	f−1(u	NOUN
ejpam-4274	552	25	(	(	PUNCT
ejpam-4274	552	26	λ	λ	PROPN
ejpam-4274	552	27	,	,	PUNCT
ejpam-4274	552	28	p	p	NOUN
ejpam-4274	552	29	)	)	PUNCT
ejpam-4274	552	30	)	)	PUNCT
ejpam-4274	552	31	.	.	PUNCT
ejpam-4274	553	1	then	then	ADV
ejpam-4274	553	2	,	,	PUNCT
ejpam-4274	553	3	f(x	f(x	PROPN
ejpam-4274	553	4	)	)	PUNCT
ejpam-4274	553	5	6∈	6∈	PROPN
ejpam-4274	553	6	u	u	NOUN
ejpam-4274	553	7	(	(	PUNCT
ejpam-4274	553	8	λ	λ	PROPN
ejpam-4274	553	9	,	,	PUNCT
ejpam-4274	553	10	p	p	NOUN
ejpam-4274	553	11	)	)	PUNCT
ejpam-4274	553	12	and	and	CCONJ
ejpam-4274	553	13	so	so	ADV
ejpam-4274	553	14	there	there	PRON
ejpam-4274	553	15	exists	exist	VERB
ejpam-4274	553	16	a	a	DET
ejpam-4274	553	17	(	(	PUNCT
ejpam-4274	553	18	λ	λ	NOUN
ejpam-4274	553	19	,	,	PUNCT
ejpam-4274	553	20	p)-open	p)-open	VERB
ejpam-4274	553	21	set	set	VERB
ejpam-4274	553	22	v	v	NOUN
ejpam-4274	553	23	containing	contain	VERB
ejpam-4274	553	24	x	x	PUNCT
ejpam-4274	553	25	such	such	ADJ
ejpam-4274	553	26	that	that	SCONJ
ejpam-4274	553	27	u	u	PROPN
ejpam-4274	553	28	∩	∩	NOUN
ejpam-4274	553	29	v	v	ADP
ejpam-4274	553	30	=	=	PUNCT
ejpam-4274	553	31	∅.	∅.	VERB
ejpam-4274	553	32	thus	thus	ADV
ejpam-4274	553	33	,	,	PUNCT
ejpam-4274	553	34	u	u	PROPN
ejpam-4274	553	35	∩	∩	X
ejpam-4274	553	36	v	v	X
ejpam-4274	553	37	(	(	PUNCT
ejpam-4274	553	38	λ	λ	PROPN
ejpam-4274	553	39	,	,	PUNCT
ejpam-4274	553	40	p	p	NOUN
ejpam-4274	553	41	)	)	PUNCT
ejpam-4274	553	42	=	=	PUNCT
ejpam-4274	553	43	∅.	∅.	X
ejpam-4274	553	44	by	by	ADP
ejpam-4274	553	45	(	(	PUNCT
ejpam-4274	553	46	5	5	NUM
ejpam-4274	553	47	)	)	PUNCT
ejpam-4274	553	48	,	,	PUNCT
ejpam-4274	553	49	x	x	PUNCT
ejpam-4274	553	50	∈	∈	PROPN
ejpam-4274	553	51	f−1(v	f−1(v	NOUN
ejpam-4274	553	52	)	)	PUNCT
ejpam-4274	553	53	⊆	⊆	NUM
ejpam-4274	554	1	[	[	X
ejpam-4274	554	2	f−1(v	f−1(v	NOUN
ejpam-4274	554	3	(	(	PUNCT
ejpam-4274	554	4	λ	λ	PROPN
ejpam-4274	554	5	,	,	PUNCT
ejpam-4274	554	6	p))](λ	p))](λ	PRON
ejpam-4274	554	7	,	,	PUNCT
ejpam-4274	554	8	p	p	NOUN
ejpam-4274	554	9	)	)	PUNCT
ejpam-4274	554	10	.	.	PUNCT
ejpam-4274	555	1	there	there	PRON
ejpam-4274	555	2	exists	exist	VERB
ejpam-4274	555	3	a	a	DET
ejpam-4274	555	4	(	(	PUNCT
ejpam-4274	555	5	λ	λ	PROPN
ejpam-4274	555	6	,	,	PUNCT
ejpam-4274	555	7	p)open	p)open	PROPN
ejpam-4274	555	8	set	set	NOUN
ejpam-4274	555	9	w	w	NOUN
ejpam-4274	555	10	containing	contain	VERB
ejpam-4274	555	11	x	x	PUNCT
ejpam-4274	555	12	such	such	ADJ
ejpam-4274	555	13	that	that	SCONJ
ejpam-4274	555	14	x	x	SYM
ejpam-4274	555	15	∈	∈	PROPN
ejpam-4274	555	16	w	w	NOUN
ejpam-4274	555	17	⊆	⊆	NUM
ejpam-4274	555	18	f−1(v	f−1(v	NOUN
ejpam-4274	555	19	(	(	PUNCT
ejpam-4274	555	20	λ	λ	PROPN
ejpam-4274	555	21	,	,	PUNCT
ejpam-4274	555	22	p	p	NOUN
ejpam-4274	555	23	)	)	PUNCT
ejpam-4274	555	24	)	)	PUNCT
ejpam-4274	555	25	.	.	PUNCT
ejpam-4274	556	1	since	since	SCONJ
ejpam-4274	556	2	u	u	PROPN
ejpam-4274	556	3	∩	∩	PROPN
ejpam-4274	556	4	v	v	X
ejpam-4274	556	5	(	(	PUNCT
ejpam-4274	556	6	λ	λ	PROPN
ejpam-4274	556	7	,	,	PUNCT
ejpam-4274	556	8	p	p	NOUN
ejpam-4274	556	9	)	)	PUNCT
ejpam-4274	556	10	=	=	NOUN
ejpam-4274	556	11	∅	∅	NOUN
ejpam-4274	556	12	and	and	CCONJ
ejpam-4274	556	13	f(w	f(w	PROPN
ejpam-4274	556	14	)	)	PUNCT
ejpam-4274	556	15	⊆	⊆	NUM
ejpam-4274	556	16	v	v	X
ejpam-4274	556	17	(	(	PUNCT
ejpam-4274	556	18	λ	λ	PROPN
ejpam-4274	556	19	,	,	PUNCT
ejpam-4274	556	20	p	p	NOUN
ejpam-4274	556	21	)	)	PUNCT
ejpam-4274	556	22	,	,	PUNCT
ejpam-4274	556	23	we	we	PRON
ejpam-4274	556	24	have	have	VERB
ejpam-4274	556	25	w	w	NOUN
ejpam-4274	556	26	∩	∩	ADJ
ejpam-4274	556	27	f−1(u	f−1(u	NOUN
ejpam-4274	556	28	)	)	PUNCT
ejpam-4274	556	29	=	=	NOUN
ejpam-4274	556	30	∅	∅	NOUN
ejpam-4274	556	31	and	and	CCONJ
ejpam-4274	556	32	hence	hence	ADV
ejpam-4274	556	33	x	x	X
ejpam-4274	556	34	6∈	6∈	PROPN
ejpam-4274	557	1	[	[	X
ejpam-4274	557	2	f−1(u)](λ	f−1(u)](λ	PROPN
ejpam-4274	557	3	,	,	PUNCT
ejpam-4274	557	4	p	p	NOUN
ejpam-4274	557	5	)	)	PUNCT
ejpam-4274	557	6	.	.	PUNCT
ejpam-4274	558	1	this	this	PRON
ejpam-4274	558	2	shows	show	VERB
ejpam-4274	558	3	that	that	SCONJ
ejpam-4274	558	4	[	[	X
ejpam-4274	558	5	f−1(u)](λ	f−1(u)](λ	X
ejpam-4274	558	6	,	,	PUNCT
ejpam-4274	558	7	p	p	NOUN
ejpam-4274	558	8	)	)	PUNCT
ejpam-4274	558	9	⊆	⊆	NUM
ejpam-4274	558	10	f−1(u	f−1(u	NOUN
ejpam-4274	558	11	(	(	PUNCT
ejpam-4274	558	12	λ	λ	PROPN
ejpam-4274	558	13	,	,	PUNCT
ejpam-4274	558	14	p	p	NOUN
ejpam-4274	558	15	)	)	PUNCT
ejpam-4274	558	16	)	)	PUNCT
ejpam-4274	558	17	.	.	PUNCT
ejpam-4274	559	1	(	(	PUNCT
ejpam-4274	559	2	6	6	X
ejpam-4274	559	3	)	)	PUNCT
ejpam-4274	559	4	⇒	⇒	NOUN
ejpam-4274	559	5	(	(	PUNCT
ejpam-4274	559	6	1	1	NUM
ejpam-4274	559	7	):	):	PUNCT
ejpam-4274	559	8	this	this	PRON
ejpam-4274	559	9	is	be	AUX
ejpam-4274	559	10	obvious	obvious	ADJ
ejpam-4274	559	11	from	from	ADP
ejpam-4274	559	12	theorem	theorem	ADJ
ejpam-4274	559	13	18	18	NUM
ejpam-4274	559	14	.	.	PUNCT
ejpam-4274	560	1	definition	definition	NOUN
ejpam-4274	560	2	12	12	NUM
ejpam-4274	560	3	.	.	PUNCT
ejpam-4274	561	1	a	a	DET
ejpam-4274	561	2	subset	subset	NOUN
ejpam-4274	561	3	a	a	PRON
ejpam-4274	561	4	of	of	ADP
ejpam-4274	561	5	a	a	DET
ejpam-4274	561	6	topological	topological	ADJ
ejpam-4274	561	7	space	space	NOUN
ejpam-4274	561	8	(	(	PUNCT
ejpam-4274	561	9	x	x	X
ejpam-4274	561	10	,	,	PUNCT
ejpam-4274	561	11	τ	τ	X
ejpam-4274	561	12	)	)	PUNCT
ejpam-4274	561	13	is	be	AUX
ejpam-4274	561	14	said	say	VERB
ejpam-4274	561	15	to	to	PART
ejpam-4274	561	16	be	be	AUX
ejpam-4274	561	17	:	:	PUNCT
ejpam-4274	561	18	(	(	PUNCT
ejpam-4274	561	19	i	i	NOUN
ejpam-4274	561	20	)	)	PUNCT
ejpam-4274	561	21	s(λ	s(λ	PROPN
ejpam-4274	561	22	,	,	PUNCT
ejpam-4274	561	23	p)-open	p)-open	VERB
ejpam-4274	561	24	if	if	SCONJ
ejpam-4274	561	25	a	a	DET
ejpam-4274	561	26	⊆	⊆	NUM
ejpam-4274	561	27	[	[	X
ejpam-4274	561	28	a(λ	a(λ	ADV
ejpam-4274	561	29	,	,	PUNCT
ejpam-4274	561	30	p	p	NOUN
ejpam-4274	561	31	)	)	PUNCT
ejpam-4274	561	32	]	]	PUNCT
ejpam-4274	561	33	(	(	PUNCT
ejpam-4274	561	34	λ	λ	X
ejpam-4274	561	35	,	,	PUNCT
ejpam-4274	561	36	p	p	NOUN
ejpam-4274	561	37	)	)	PUNCT
ejpam-4274	561	38	;	;	PUNCT
ejpam-4274	561	39	(	(	PUNCT
ejpam-4274	561	40	ii	ii	NOUN
ejpam-4274	561	41	)	)	PUNCT
ejpam-4274	561	42	p(λ	p(λ	NOUN
ejpam-4274	561	43	,	,	PUNCT
ejpam-4274	561	44	p)-open	p)-open	VERB
ejpam-4274	561	45	if	if	SCONJ
ejpam-4274	561	46	a	a	DET
ejpam-4274	561	47	⊆	⊆	NUM
ejpam-4274	561	48	[	[	X
ejpam-4274	561	49	a(λ	a(λ	ADV
ejpam-4274	561	50	,	,	PUNCT
ejpam-4274	561	51	p)](λ	p)](λ	X
ejpam-4274	561	52	,	,	PUNCT
ejpam-4274	561	53	p	p	NOUN
ejpam-4274	561	54	)	)	PUNCT
ejpam-4274	561	55	;	;	PUNCT
ejpam-4274	561	56	(	(	PUNCT
ejpam-4274	561	57	iii	iii	NOUN
ejpam-4274	561	58	)	)	PUNCT
ejpam-4274	561	59	β(λ	β(λ	NOUN
ejpam-4274	561	60	,	,	PUNCT
ejpam-4274	561	61	p)-open	p)-open	VERB
ejpam-4274	561	62	if	if	SCONJ
ejpam-4274	561	63	a	a	DET
ejpam-4274	561	64	⊆	⊆	NUM
ejpam-4274	561	65	[	[	X
ejpam-4274	561	66	[	[	X
ejpam-4274	561	67	a(λ	a(λ	ADJ
ejpam-4274	561	68	,	,	PUNCT
ejpam-4274	561	69	p)](λ	p)](λ	X
ejpam-4274	561	70	,	,	PUNCT
ejpam-4274	561	71	p	p	NOUN
ejpam-4274	561	72	)	)	PUNCT
ejpam-4274	561	73	]	]	PUNCT
ejpam-4274	562	1	(	(	PUNCT
ejpam-4274	562	2	λ	λ	X
ejpam-4274	562	3	,	,	PUNCT
ejpam-4274	562	4	p	p	NOUN
ejpam-4274	562	5	)	)	PUNCT
ejpam-4274	562	6	;	;	PUNCT
ejpam-4274	562	7	(	(	PUNCT
ejpam-4274	562	8	iv	iv	X
ejpam-4274	562	9	)	)	PUNCT
ejpam-4274	562	10	r(λ	r(λ	NOUN
ejpam-4274	562	11	,	,	PUNCT
ejpam-4274	562	12	p)-open	p)-open	VERB
ejpam-4274	562	13	if	if	SCONJ
ejpam-4274	562	14	a	a	PRON
ejpam-4274	562	15	=	=	X
ejpam-4274	562	16	[	[	X
ejpam-4274	562	17	a(λ	a(λ	ADV
ejpam-4274	562	18	,	,	PUNCT
ejpam-4274	562	19	p)](λ	p)](λ	X
ejpam-4274	562	20	,	,	PUNCT
ejpam-4274	562	21	p	p	NOUN
ejpam-4274	562	22	)	)	PUNCT
ejpam-4274	562	23	.	.	PUNCT
ejpam-4274	563	1	the	the	DET
ejpam-4274	563	2	complement	complement	NOUN
ejpam-4274	563	3	of	of	ADP
ejpam-4274	563	4	a	a	DET
ejpam-4274	563	5	s(λ	s(λ	PROPN
ejpam-4274	563	6	,	,	PUNCT
ejpam-4274	563	7	p)-open	p)-open	ADJ
ejpam-4274	563	8	(	(	PUNCT
ejpam-4274	563	9	resp	resp	NOUN
ejpam-4274	563	10	.	.	PUNCT
ejpam-4274	564	1	p(λ	p(λ	NOUN
ejpam-4274	564	2	,	,	PUNCT
ejpam-4274	564	3	p)-open	p)-open	ADJ
ejpam-4274	564	4	,	,	PUNCT
ejpam-4274	564	5	β(λ	β(λ	X
ejpam-4274	564	6	,	,	PUNCT
ejpam-4274	564	7	p)-open	p)-open	ADJ
ejpam-4274	564	8	,	,	PUNCT
ejpam-4274	564	9	r(λ	r(λ	NOUN
ejpam-4274	564	10	,	,	PUNCT
ejpam-4274	564	11	p)-open	p)-open	ADJ
ejpam-4274	564	12	)	)	PUNCT
ejpam-4274	564	13	set	set	NOUN
ejpam-4274	564	14	is	be	AUX
ejpam-4274	564	15	called	call	VERB
ejpam-4274	564	16	s(λ	s(λ	PROPN
ejpam-4274	564	17	,	,	PUNCT
ejpam-4274	564	18	p)-closed	p)-close	VERB
ejpam-4274	564	19	(	(	PUNCT
ejpam-4274	564	20	resp	resp	NOUN
ejpam-4274	564	21	.	.	PUNCT
ejpam-4274	565	1	p(λ	p(λ	NOUN
ejpam-4274	565	2	,	,	PUNCT
ejpam-4274	565	3	p)-closed	p)-close	VERB
ejpam-4274	565	4	,	,	PUNCT
ejpam-4274	565	5	β(λ	β(λ	PROPN
ejpam-4274	565	6	,	,	PUNCT
ejpam-4274	565	7	p)-closed	p)-close	VERB
ejpam-4274	565	8	,	,	PUNCT
ejpam-4274	565	9	r(λ	r(λ	PROPN
ejpam-4274	565	10	,	,	PUNCT
ejpam-4274	565	11	p)-closed	p)-close	VERB
ejpam-4274	565	12	)	)	PUNCT
ejpam-4274	565	13	.	.	PUNCT
ejpam-4274	566	1	theorem	theorem	NOUN
ejpam-4274	566	2	20	20	NUM
ejpam-4274	566	3	.	.	PUNCT
ejpam-4274	567	1	for	for	ADP
ejpam-4274	567	2	a	a	DET
ejpam-4274	567	3	function	function	NOUN
ejpam-4274	567	4	f	f	NOUN
ejpam-4274	567	5	:	:	PUNCT
ejpam-4274	567	6	(	(	PUNCT
ejpam-4274	567	7	x	x	X
ejpam-4274	567	8	,	,	PUNCT
ejpam-4274	567	9	τ	τ	X
ejpam-4274	567	10	)	)	PUNCT
ejpam-4274	567	11	→	→	SYM
ejpam-4274	567	12	(	(	PUNCT
ejpam-4274	567	13	y	y	PROPN
ejpam-4274	567	14	,	,	PUNCT
ejpam-4274	567	15	σ	σ	PROPN
ejpam-4274	567	16	)	)	PUNCT
ejpam-4274	567	17	,	,	PUNCT
ejpam-4274	567	18	the	the	DET
ejpam-4274	567	19	following	follow	VERB
ejpam-4274	567	20	properties	property	NOUN
ejpam-4274	567	21	are	be	AUX
ejpam-4274	567	22	equivalent	equivalent	ADJ
ejpam-4274	567	23	:	:	PUNCT
ejpam-4274	567	24	(	(	PUNCT
ejpam-4274	567	25	1	1	X
ejpam-4274	567	26	)	)	PUNCT
ejpam-4274	567	27	f	f	PROPN
ejpam-4274	567	28	is	be	AUX
ejpam-4274	567	29	weakly	weakly	ADJ
ejpam-4274	567	30	(	(	PUNCT
ejpam-4274	567	31	λ	λ	NOUN
ejpam-4274	567	32	,	,	PUNCT
ejpam-4274	567	33	p)-continuous	p)-continuous	ADJ
ejpam-4274	567	34	;	;	PUNCT
ejpam-4274	567	35	(	(	PUNCT
ejpam-4274	567	36	2	2	X
ejpam-4274	567	37	)	)	PUNCT
ejpam-4274	567	38	[	[	X
ejpam-4274	567	39	f−1(f(λ	f−1(f(λ	X
ejpam-4274	567	40	,	,	PUNCT
ejpam-4274	567	41	p	p	NOUN
ejpam-4274	567	42	)	)	PUNCT
ejpam-4274	567	43	)	)	PUNCT
ejpam-4274	567	44	]	]	PUNCT
ejpam-4274	568	1	(	(	PUNCT
ejpam-4274	568	2	λ	λ	X
ejpam-4274	568	3	,	,	PUNCT
ejpam-4274	568	4	p	p	NOUN
ejpam-4274	568	5	)	)	PUNCT
ejpam-4274	568	6	⊆	⊆	NUM
ejpam-4274	568	7	f−1(f	f−1(f	PROPN
ejpam-4274	568	8	)	)	PUNCT
ejpam-4274	568	9	for	for	ADP
ejpam-4274	568	10	every	every	DET
ejpam-4274	568	11	r(λ	r(λ	NOUN
ejpam-4274	568	12	,	,	PUNCT
ejpam-4274	568	13	p)-closed	p)-close	VERB
ejpam-4274	568	14	subset	subset	NOUN
ejpam-4274	568	15	f	f	PROPN
ejpam-4274	568	16	of	of	ADP
ejpam-4274	568	17	y	y	PROPN
ejpam-4274	568	18	;	;	PUNCT
ejpam-4274	568	19	(	(	PUNCT
ejpam-4274	568	20	3	3	X
ejpam-4274	568	21	)	)	PUNCT
ejpam-4274	568	22	[	[	X
ejpam-4274	568	23	f−1([u	f−1([u	INTJ
ejpam-4274	568	24	(	(	PUNCT
ejpam-4274	568	25	λ	λ	PROPN
ejpam-4274	568	26	,	,	PUNCT
ejpam-4274	568	27	p)](λ	p)](λ	ADJ
ejpam-4274	568	28	,	,	PUNCT
ejpam-4274	568	29	p	p	NOUN
ejpam-4274	568	30	)	)	PUNCT
ejpam-4274	568	31	)	)	PUNCT
ejpam-4274	568	32	]	]	PUNCT
ejpam-4274	569	1	(	(	PUNCT
ejpam-4274	569	2	λ	λ	X
ejpam-4274	569	3	,	,	PUNCT
ejpam-4274	569	4	p	p	NOUN
ejpam-4274	569	5	)	)	PUNCT
ejpam-4274	569	6	⊆	⊆	NUM
ejpam-4274	569	7	f−1(u	f−1(u	NOUN
ejpam-4274	569	8	(	(	PUNCT
ejpam-4274	569	9	λ	λ	PROPN
ejpam-4274	569	10	,	,	PUNCT
ejpam-4274	569	11	p	p	NOUN
ejpam-4274	569	12	)	)	PUNCT
ejpam-4274	569	13	)	)	PUNCT
ejpam-4274	569	14	for	for	ADP
ejpam-4274	569	15	every	every	DET
ejpam-4274	569	16	β(λ	β(λ	NOUN
ejpam-4274	569	17	,	,	PUNCT
ejpam-4274	569	18	p)-open	p)-open	PUNCT
ejpam-4274	569	19	subset	subset	VERB
ejpam-4274	569	20	u	u	NOUN
ejpam-4274	569	21	of	of	ADP
ejpam-4274	569	22	y	y	PROPN
ejpam-4274	569	23	;	;	PUNCT
ejpam-4274	569	24	c.	c.	PROPN
ejpam-4274	569	25	boonpok	boonpok	PROPN
ejpam-4274	569	26	,	,	PUNCT
ejpam-4274	569	27	c.	c.	PROPN
ejpam-4274	569	28	viriyapong	viriyapong	PROPN
ejpam-4274	569	29	/	/	SYM
ejpam-4274	569	30	eur	eur	PROPN
ejpam-4274	569	31	.	.	PUNCT
ejpam-4274	570	1	j.	j.	PROPN
ejpam-4274	570	2	pure	pure	PROPN
ejpam-4274	570	3	appl	appl	PROPN
ejpam-4274	570	4	.	.	PROPN
ejpam-4274	570	5	math	math	PROPN
ejpam-4274	570	6	,	,	PUNCT
ejpam-4274	570	7	15	15	NUM
ejpam-4274	570	8	(	(	PUNCT
ejpam-4274	570	9	2	2	NUM
ejpam-4274	570	10	)	)	PUNCT
ejpam-4274	570	11	(	(	PUNCT
ejpam-4274	570	12	2022	2022	NUM
ejpam-4274	570	13	)	)	PUNCT
ejpam-4274	570	14	,	,	PUNCT
ejpam-4274	570	15	415	415	NUM
ejpam-4274	570	16	-	-	SYM
ejpam-4274	570	17	436	436	NUM
ejpam-4274	570	18	430	430	NUM
ejpam-4274	570	19	(	(	PUNCT
ejpam-4274	570	20	4	4	NUM
ejpam-4274	570	21	)	)	PUNCT
ejpam-4274	571	1	[	[	X
ejpam-4274	571	2	f−1([u	f−1([u	INTJ
ejpam-4274	571	3	(	(	PUNCT
ejpam-4274	571	4	λ	λ	PROPN
ejpam-4274	571	5	,	,	PUNCT
ejpam-4274	571	6	p)](λ	p)](λ	ADJ
ejpam-4274	571	7	,	,	PUNCT
ejpam-4274	571	8	p	p	NOUN
ejpam-4274	571	9	)	)	PUNCT
ejpam-4274	571	10	)	)	PUNCT
ejpam-4274	571	11	]	]	PUNCT
ejpam-4274	572	1	(	(	PUNCT
ejpam-4274	572	2	λ	λ	X
ejpam-4274	572	3	,	,	PUNCT
ejpam-4274	572	4	p	p	NOUN
ejpam-4274	572	5	)	)	PUNCT
ejpam-4274	572	6	⊆	⊆	NUM
ejpam-4274	572	7	f−1(u	f−1(u	NOUN
ejpam-4274	572	8	(	(	PUNCT
ejpam-4274	572	9	λ	λ	PROPN
ejpam-4274	572	10	,	,	PUNCT
ejpam-4274	572	11	p	p	NOUN
ejpam-4274	572	12	)	)	PUNCT
ejpam-4274	572	13	)	)	PUNCT
ejpam-4274	572	14	for	for	ADP
ejpam-4274	572	15	every	every	DET
ejpam-4274	572	16	s(λ	s(λ	PROPN
ejpam-4274	572	17	,	,	PUNCT
ejpam-4274	572	18	p)-open	p)-open	VERB
ejpam-4274	572	19	subset	subset	VERB
ejpam-4274	572	20	u	u	NOUN
ejpam-4274	572	21	of	of	ADP
ejpam-4274	572	22	y	y	PROPN
ejpam-4274	572	23	.	.	PUNCT
ejpam-4274	573	1	proof	proof	NOUN
ejpam-4274	573	2	.	.	PUNCT
ejpam-4274	574	1	(	(	PUNCT
ejpam-4274	574	2	1	1	X
ejpam-4274	574	3	)	)	PUNCT
ejpam-4274	574	4	⇒	⇒	NOUN
ejpam-4274	574	5	(	(	PUNCT
ejpam-4274	574	6	2	2	NUM
ejpam-4274	574	7	):	):	PUNCT
ejpam-4274	574	8	let	let	VERB
ejpam-4274	574	9	f	f	PRON
ejpam-4274	574	10	be	be	AUX
ejpam-4274	574	11	any	any	DET
ejpam-4274	574	12	r(λ	r(λ	NOUN
ejpam-4274	574	13	,	,	PUNCT
ejpam-4274	574	14	p)-closed	p)-close	VERB
ejpam-4274	574	15	subset	subset	NOUN
ejpam-4274	574	16	of	of	ADP
ejpam-4274	574	17	y	y	PROPN
ejpam-4274	574	18	.	.	PUNCT
ejpam-4274	575	1	then	then	ADV
ejpam-4274	575	2	,	,	PUNCT
ejpam-4274	575	3	f(λ	f(λ	PROPN
ejpam-4274	575	4	,	,	PUNCT
ejpam-4274	575	5	p	p	NOUN
ejpam-4274	575	6	)	)	PUNCT
ejpam-4274	575	7	is	be	AUX
ejpam-4274	575	8	(	(	PUNCT
ejpam-4274	575	9	λ	λ	X
ejpam-4274	575	10	,	,	PUNCT
ejpam-4274	575	11	p)-open	p)-open	ADJ
ejpam-4274	575	12	,	,	PUNCT
ejpam-4274	575	13	by	by	ADP
ejpam-4274	575	14	theorem	theorem	NOUN
ejpam-4274	575	15	19	19	NUM
ejpam-4274	575	16	,	,	PUNCT
ejpam-4274	575	17	[	[	X
ejpam-4274	575	18	f−1(f(λ	f−1(f(λ	X
ejpam-4274	575	19	,	,	PUNCT
ejpam-4274	575	20	p	p	NOUN
ejpam-4274	575	21	)	)	PUNCT
ejpam-4274	575	22	)	)	PUNCT
ejpam-4274	575	23	]	]	PUNCT
ejpam-4274	576	1	(	(	PUNCT
ejpam-4274	576	2	λ	λ	X
ejpam-4274	576	3	,	,	PUNCT
ejpam-4274	576	4	p	p	NOUN
ejpam-4274	576	5	)	)	PUNCT
ejpam-4274	576	6	⊆	⊆	NUM
ejpam-4274	576	7	f−1([f(λ	f−1([f(λ	NOUN
ejpam-4274	576	8	,	,	PUNCT
ejpam-4274	576	9	p	p	NOUN
ejpam-4274	576	10	)	)	PUNCT
ejpam-4274	576	11	]	]	PUNCT
ejpam-4274	576	12	(	(	PUNCT
ejpam-4274	576	13	λ	λ	X
ejpam-4274	576	14	,	,	PUNCT
ejpam-4274	576	15	p	p	NOUN
ejpam-4274	576	16	)	)	PUNCT
ejpam-4274	576	17	)	)	PUNCT
ejpam-4274	576	18	.	.	PUNCT
ejpam-4274	577	1	since	since	SCONJ
ejpam-4274	577	2	f	f	PROPN
ejpam-4274	577	3	is	be	AUX
ejpam-4274	577	4	r(λ	r(λ	PROPN
ejpam-4274	577	5	,	,	PUNCT
ejpam-4274	577	6	p)-closed	p)-close	VERB
ejpam-4274	577	7	,	,	PUNCT
ejpam-4274	577	8	we	we	PRON
ejpam-4274	577	9	have	have	VERB
ejpam-4274	577	10	[	[	X
ejpam-4274	577	11	f−1(f(λ	f−1(f(λ	X
ejpam-4274	577	12	,	,	PUNCT
ejpam-4274	577	13	p	p	NOUN
ejpam-4274	577	14	)	)	PUNCT
ejpam-4274	577	15	)	)	PUNCT
ejpam-4274	577	16	]	]	PUNCT
ejpam-4274	578	1	(	(	PUNCT
ejpam-4274	578	2	λ	λ	X
ejpam-4274	578	3	,	,	PUNCT
ejpam-4274	578	4	p	p	NOUN
ejpam-4274	578	5	)	)	PUNCT
ejpam-4274	578	6	⊆	⊆	NUM
ejpam-4274	578	7	f−1([f(λ	f−1([f(λ	NOUN
ejpam-4274	578	8	,	,	PUNCT
ejpam-4274	578	9	p	p	NOUN
ejpam-4274	578	10	)	)	PUNCT
ejpam-4274	578	11	]	]	PUNCT
ejpam-4274	578	12	(	(	PUNCT
ejpam-4274	578	13	λ	λ	X
ejpam-4274	578	14	,	,	PUNCT
ejpam-4274	578	15	p	p	NOUN
ejpam-4274	578	16	)	)	PUNCT
ejpam-4274	578	17	)	)	PUNCT
ejpam-4274	578	18	=	=	SYM
ejpam-4274	578	19	f−1(f	f−1(f	PROPN
ejpam-4274	578	20	)	)	PUNCT
ejpam-4274	578	21	.	.	PUNCT
ejpam-4274	579	1	(	(	PUNCT
ejpam-4274	579	2	2	2	X
ejpam-4274	579	3	)	)	PUNCT
ejpam-4274	579	4	⇒	⇒	NOUN
ejpam-4274	579	5	(	(	PUNCT
ejpam-4274	579	6	3	3	NUM
ejpam-4274	579	7	):	):	PUNCT
ejpam-4274	579	8	let	let	VERB
ejpam-4274	579	9	u	u	PRON
ejpam-4274	579	10	be	be	AUX
ejpam-4274	579	11	any	any	DET
ejpam-4274	579	12	β(λ	β(λ	NOUN
ejpam-4274	579	13	,	,	PUNCT
ejpam-4274	579	14	p)-open	p)-open	PUNCT
ejpam-4274	579	15	set	set	NOUN
ejpam-4274	579	16	.	.	PUNCT
ejpam-4274	580	1	then	then	ADV
ejpam-4274	580	2	,	,	PUNCT
ejpam-4274	580	3	u	u	PROPN
ejpam-4274	580	4	(	(	PUNCT
ejpam-4274	580	5	λ	λ	PROPN
ejpam-4274	580	6	,	,	PUNCT
ejpam-4274	580	7	p	p	NOUN
ejpam-4274	580	8	)	)	PUNCT
ejpam-4274	580	9	⊆	⊆	NUM
ejpam-4274	581	1	[	[	X
ejpam-4274	581	2	[	[	X
ejpam-4274	581	3	u	u	X
ejpam-4274	581	4	(	(	PUNCT
ejpam-4274	581	5	λ	λ	PROPN
ejpam-4274	581	6	,	,	PUNCT
ejpam-4274	581	7	p)](λ	p)](λ	ADJ
ejpam-4274	581	8	,	,	PUNCT
ejpam-4274	581	9	p	p	NOUN
ejpam-4274	581	10	)	)	PUNCT
ejpam-4274	581	11	]	]	PUNCT
ejpam-4274	581	12	(	(	PUNCT
ejpam-4274	581	13	λ	λ	X
ejpam-4274	581	14	,	,	PUNCT
ejpam-4274	581	15	p	p	NOUN
ejpam-4274	581	16	)	)	PUNCT
ejpam-4274	581	17	⊆	⊆	NUM
ejpam-4274	581	18	u	u	NOUN
ejpam-4274	581	19	(	(	PUNCT
ejpam-4274	581	20	λ	λ	PROPN
ejpam-4274	581	21	,	,	PUNCT
ejpam-4274	581	22	p	p	NOUN
ejpam-4274	581	23	)	)	PUNCT
ejpam-4274	581	24	and	and	CCONJ
ejpam-4274	581	25	hence	hence	ADV
ejpam-4274	581	26	u	u	NOUN
ejpam-4274	581	27	(	(	PUNCT
ejpam-4274	581	28	λ	λ	PROPN
ejpam-4274	581	29	,	,	PUNCT
ejpam-4274	581	30	p	p	NOUN
ejpam-4274	581	31	)	)	PUNCT
ejpam-4274	581	32	is	be	AUX
ejpam-4274	581	33	r(λ	r(λ	NOUN
ejpam-4274	581	34	,	,	PUNCT
ejpam-4274	581	35	p)-closed	p)-close	VERB
ejpam-4274	581	36	.	.	PUNCT
ejpam-4274	582	1	by	by	ADP
ejpam-4274	582	2	(	(	PUNCT
ejpam-4274	582	3	2	2	NUM
ejpam-4274	582	4	)	)	PUNCT
ejpam-4274	582	5	,	,	PUNCT
ejpam-4274	582	6	[	[	X
ejpam-4274	582	7	f−1([u	f−1([u	INTJ
ejpam-4274	582	8	(	(	PUNCT
ejpam-4274	582	9	λ	λ	PROPN
ejpam-4274	582	10	,	,	PUNCT
ejpam-4274	582	11	p)](λ	p)](λ	ADJ
ejpam-4274	582	12	,	,	PUNCT
ejpam-4274	582	13	p	p	NOUN
ejpam-4274	582	14	)	)	PUNCT
ejpam-4274	582	15	)	)	PUNCT
ejpam-4274	582	16	]	]	PUNCT
ejpam-4274	582	17	(	(	PUNCT
ejpam-4274	582	18	λ	λ	X
ejpam-4274	582	19	,	,	PUNCT
ejpam-4274	582	20	p	p	NOUN
ejpam-4274	582	21	)	)	PUNCT
ejpam-4274	582	22	⊆	⊆	NUM
ejpam-4274	582	23	f−1(u	f−1(u	NOUN
ejpam-4274	582	24	(	(	PUNCT
ejpam-4274	582	25	λ	λ	PROPN
ejpam-4274	582	26	,	,	PUNCT
ejpam-4274	582	27	p	p	NOUN
ejpam-4274	582	28	)	)	PUNCT
ejpam-4274	582	29	)	)	PUNCT
ejpam-4274	582	30	.	.	PUNCT
ejpam-4274	583	1	(	(	PUNCT
ejpam-4274	583	2	3	3	X
ejpam-4274	583	3	)	)	PUNCT
ejpam-4274	583	4	⇒	⇒	NOUN
ejpam-4274	583	5	(	(	PUNCT
ejpam-4274	583	6	4	4	NUM
ejpam-4274	583	7	):	):	PUNCT
ejpam-4274	583	8	the	the	DET
ejpam-4274	583	9	proof	proof	NOUN
ejpam-4274	583	10	is	be	AUX
ejpam-4274	583	11	obvious	obvious	ADJ
ejpam-4274	583	12	.	.	PUNCT
ejpam-4274	584	1	(	(	PUNCT
ejpam-4274	584	2	4	4	X
ejpam-4274	584	3	)	)	PUNCT
ejpam-4274	584	4	⇒	⇒	NOUN
ejpam-4274	584	5	(	(	PUNCT
ejpam-4274	584	6	1	1	NUM
ejpam-4274	584	7	):	):	PUNCT
ejpam-4274	584	8	let	let	VERB
ejpam-4274	584	9	u	u	PRON
ejpam-4274	584	10	be	be	AUX
ejpam-4274	584	11	any	any	DET
ejpam-4274	584	12	(	(	PUNCT
ejpam-4274	584	13	λ	λ	NOUN
ejpam-4274	584	14	,	,	PUNCT
ejpam-4274	584	15	p)-open	p)-open	PUNCT
ejpam-4274	584	16	subset	subset	NOUN
ejpam-4274	584	17	of	of	ADP
ejpam-4274	584	18	y	y	PROPN
ejpam-4274	584	19	.	.	PUNCT
ejpam-4274	585	1	then	then	ADV
ejpam-4274	585	2	,	,	PUNCT
ejpam-4274	585	3	we	we	PRON
ejpam-4274	585	4	have	have	VERB
ejpam-4274	585	5	u	u	NOUN
ejpam-4274	585	6	is	be	AUX
ejpam-4274	585	7	s(λ	s(λ	NOUN
ejpam-4274	585	8	,	,	PUNCT
ejpam-4274	585	9	p)-open	p)-open	ADJ
ejpam-4274	585	10	and	and	CCONJ
ejpam-4274	585	11	by	by	ADP
ejpam-4274	585	12	(	(	PUNCT
ejpam-4274	585	13	4	4	NUM
ejpam-4274	585	14	)	)	PUNCT
ejpam-4274	585	15	,	,	PUNCT
ejpam-4274	586	1	[	[	X
ejpam-4274	586	2	f−1(u)](λ	f−1(u)](λ	X
ejpam-4274	586	3	,	,	PUNCT
ejpam-4274	586	4	p	p	NOUN
ejpam-4274	586	5	)	)	PUNCT
ejpam-4274	586	6	⊆	⊆	NUM
ejpam-4274	587	1	[	[	X
ejpam-4274	587	2	f−1([u(λ	f−1([u(λ	NOUN
ejpam-4274	587	3	,	,	PUNCT
ejpam-4274	587	4	p	p	NOUN
ejpam-4274	587	5	)	)	PUNCT
ejpam-4274	587	6	]	]	PUNCT
ejpam-4274	587	7	(	(	PUNCT
ejpam-4274	587	8	λ	λ	X
ejpam-4274	587	9	,	,	PUNCT
ejpam-4274	587	10	p))](λ	p))](λ	PRON
ejpam-4274	587	11	,	,	PUNCT
ejpam-4274	587	12	p	p	X
ejpam-4274	587	13	)	)	PUNCT
ejpam-4274	587	14	⊆	⊆	NUM
ejpam-4274	587	15	f−1(u	f−1(u	NOUN
ejpam-4274	587	16	(	(	PUNCT
ejpam-4274	587	17	λ	λ	PROPN
ejpam-4274	587	18	,	,	PUNCT
ejpam-4274	587	19	p	p	NOUN
ejpam-4274	587	20	)	)	PUNCT
ejpam-4274	587	21	)	)	PUNCT
ejpam-4274	587	22	.	.	PUNCT
ejpam-4274	588	1	thus	thus	ADV
ejpam-4274	588	2	,	,	PUNCT
ejpam-4274	588	3	f	f	PROPN
ejpam-4274	588	4	is	be	AUX
ejpam-4274	588	5	weakly	weakly	ADJ
ejpam-4274	588	6	(	(	PUNCT
ejpam-4274	588	7	λ	λ	NOUN
ejpam-4274	588	8	,	,	PUNCT
ejpam-4274	588	9	p)continuous	p)continuous	ADJ
ejpam-4274	588	10	by	by	ADP
ejpam-4274	588	11	theorem	theorem	NOUN
ejpam-4274	588	12	19	19	NUM
ejpam-4274	588	13	.	.	PUNCT
ejpam-4274	588	14	theorem	theorem	NOUN
ejpam-4274	588	15	21	21	NUM
ejpam-4274	588	16	.	.	PUNCT
ejpam-4274	589	1	for	for	ADP
ejpam-4274	589	2	a	a	DET
ejpam-4274	589	3	function	function	NOUN
ejpam-4274	589	4	f	f	NOUN
ejpam-4274	589	5	:	:	PUNCT
ejpam-4274	589	6	(	(	PUNCT
ejpam-4274	589	7	x	x	X
ejpam-4274	589	8	,	,	PUNCT
ejpam-4274	589	9	τ	τ	X
ejpam-4274	589	10	)	)	PUNCT
ejpam-4274	589	11	→	→	SYM
ejpam-4274	589	12	(	(	PUNCT
ejpam-4274	589	13	y	y	PROPN
ejpam-4274	589	14	,	,	PUNCT
ejpam-4274	589	15	σ	σ	PROPN
ejpam-4274	589	16	)	)	PUNCT
ejpam-4274	589	17	,	,	PUNCT
ejpam-4274	589	18	the	the	DET
ejpam-4274	589	19	following	follow	VERB
ejpam-4274	589	20	properties	property	NOUN
ejpam-4274	589	21	are	be	AUX
ejpam-4274	589	22	equivalent	equivalent	ADJ
ejpam-4274	589	23	:	:	PUNCT
ejpam-4274	589	24	(	(	PUNCT
ejpam-4274	589	25	1	1	X
ejpam-4274	589	26	)	)	PUNCT
ejpam-4274	589	27	f	f	PROPN
ejpam-4274	589	28	is	be	AUX
ejpam-4274	589	29	weakly	weakly	ADJ
ejpam-4274	589	30	(	(	PUNCT
ejpam-4274	589	31	λ	λ	NOUN
ejpam-4274	589	32	,	,	PUNCT
ejpam-4274	589	33	p)-continuous	p)-continuous	ADJ
ejpam-4274	589	34	;	;	PUNCT
ejpam-4274	589	35	(	(	PUNCT
ejpam-4274	589	36	2	2	X
ejpam-4274	589	37	)	)	PUNCT
ejpam-4274	589	38	[	[	X
ejpam-4274	589	39	f−1([u(λ	f−1([u(λ	X
ejpam-4274	589	40	,	,	PUNCT
ejpam-4274	589	41	p	p	NOUN
ejpam-4274	589	42	)	)	PUNCT
ejpam-4274	589	43	]	]	PUNCT
ejpam-4274	590	1	(	(	PUNCT
ejpam-4274	590	2	λ	λ	X
ejpam-4274	590	3	,	,	PUNCT
ejpam-4274	590	4	p))](λ	p))](λ	PRON
ejpam-4274	590	5	,	,	PUNCT
ejpam-4274	590	6	p	p	X
ejpam-4274	590	7	)	)	PUNCT
ejpam-4274	590	8	⊆	⊆	NUM
ejpam-4274	590	9	f−1(u	f−1(u	NOUN
ejpam-4274	590	10	(	(	PUNCT
ejpam-4274	590	11	λ	λ	PROPN
ejpam-4274	590	12	,	,	PUNCT
ejpam-4274	590	13	p	p	NOUN
ejpam-4274	590	14	)	)	PUNCT
ejpam-4274	590	15	)	)	PUNCT
ejpam-4274	590	16	for	for	ADP
ejpam-4274	590	17	every	every	DET
ejpam-4274	590	18	p(λ	p(λ	NOUN
ejpam-4274	590	19	,	,	PUNCT
ejpam-4274	590	20	p)-open	p)-open	VERB
ejpam-4274	590	21	subset	subset	VERB
ejpam-4274	590	22	u	u	NOUN
ejpam-4274	590	23	of	of	ADP
ejpam-4274	590	24	y	y	PROPN
ejpam-4274	590	25	;	;	PUNCT
ejpam-4274	590	26	(	(	PUNCT
ejpam-4274	590	27	3	3	X
ejpam-4274	590	28	)	)	PUNCT
ejpam-4274	591	1	[	[	X
ejpam-4274	591	2	f−1(u)](λ	f−1(u)](λ	X
ejpam-4274	591	3	,	,	PUNCT
ejpam-4274	591	4	p	p	NOUN
ejpam-4274	591	5	)	)	PUNCT
ejpam-4274	591	6	⊆	⊆	NUM
ejpam-4274	591	7	f−1(u	f−1(u	NOUN
ejpam-4274	591	8	(	(	PUNCT
ejpam-4274	591	9	λ	λ	PROPN
ejpam-4274	591	10	,	,	PUNCT
ejpam-4274	591	11	p	p	NOUN
ejpam-4274	591	12	)	)	PUNCT
ejpam-4274	591	13	)	)	PUNCT
ejpam-4274	591	14	for	for	ADP
ejpam-4274	591	15	every	every	DET
ejpam-4274	591	16	p(λ	p(λ	NOUN
ejpam-4274	591	17	,	,	PUNCT
ejpam-4274	591	18	p)-open	p)-open	VERB
ejpam-4274	591	19	subset	subset	VERB
ejpam-4274	591	20	u	u	NOUN
ejpam-4274	591	21	of	of	ADP
ejpam-4274	591	22	y	y	PROPN
ejpam-4274	591	23	;	;	PUNCT
ejpam-4274	591	24	(	(	PUNCT
ejpam-4274	591	25	4	4	X
ejpam-4274	591	26	)	)	PUNCT
ejpam-4274	591	27	f−1(u	f−1(u	NOUN
ejpam-4274	591	28	)	)	PUNCT
ejpam-4274	591	29	⊆	⊆	NUM
ejpam-4274	592	1	[	[	X
ejpam-4274	592	2	f−1(u	f−1(u	NOUN
ejpam-4274	592	3	(	(	PUNCT
ejpam-4274	592	4	λ	λ	PROPN
ejpam-4274	592	5	,	,	PUNCT
ejpam-4274	592	6	p))](λ	p))](λ	PRON
ejpam-4274	592	7	,	,	PUNCT
ejpam-4274	592	8	p	p	NOUN
ejpam-4274	592	9	)	)	PUNCT
ejpam-4274	592	10	for	for	ADP
ejpam-4274	592	11	every	every	DET
ejpam-4274	592	12	p(λ	p(λ	NOUN
ejpam-4274	592	13	,	,	PUNCT
ejpam-4274	592	14	p)-open	p)-open	VERB
ejpam-4274	592	15	subset	subset	VERB
ejpam-4274	592	16	u	u	NOUN
ejpam-4274	592	17	of	of	ADP
ejpam-4274	592	18	y	y	PROPN
ejpam-4274	592	19	.	.	PUNCT
ejpam-4274	593	1	proof	proof	NOUN
ejpam-4274	593	2	.	.	PUNCT
ejpam-4274	594	1	(	(	PUNCT
ejpam-4274	594	2	1	1	X
ejpam-4274	594	3	)	)	PUNCT
ejpam-4274	594	4	⇒	⇒	NOUN
ejpam-4274	594	5	(	(	PUNCT
ejpam-4274	594	6	2	2	NUM
ejpam-4274	594	7	):	):	PUNCT
ejpam-4274	594	8	let	let	VERB
ejpam-4274	594	9	u	u	PRON
ejpam-4274	594	10	be	be	AUX
ejpam-4274	594	11	any	any	DET
ejpam-4274	594	12	p(λ	p(λ	NOUN
ejpam-4274	594	13	,	,	PUNCT
ejpam-4274	594	14	p)-open	p)-open	PUNCT
ejpam-4274	594	15	subset	subset	NOUN
ejpam-4274	594	16	of	of	ADP
ejpam-4274	594	17	y	y	PROPN
ejpam-4274	594	18	.	.	PUNCT
ejpam-4274	595	1	then	then	ADV
ejpam-4274	595	2	,	,	PUNCT
ejpam-4274	595	3	we	we	PRON
ejpam-4274	595	4	have	have	VERB
ejpam-4274	595	5	u	u	NOUN
ejpam-4274	595	6	(	(	PUNCT
ejpam-4274	595	7	λ	λ	PROPN
ejpam-4274	595	8	,	,	PUNCT
ejpam-4274	595	9	p	p	NOUN
ejpam-4274	595	10	)	)	PUNCT
ejpam-4274	595	11	=	=	NOUN
ejpam-4274	596	1	[	[	X
ejpam-4274	596	2	[	[	X
ejpam-4274	596	3	u	u	X
ejpam-4274	596	4	(	(	PUNCT
ejpam-4274	596	5	λ	λ	PROPN
ejpam-4274	596	6	,	,	PUNCT
ejpam-4274	596	7	p)](λ	p)](λ	ADJ
ejpam-4274	596	8	,	,	PUNCT
ejpam-4274	596	9	p	p	NOUN
ejpam-4274	596	10	)	)	PUNCT
ejpam-4274	596	11	]	]	PUNCT
ejpam-4274	596	12	(	(	PUNCT
ejpam-4274	596	13	λ	λ	X
ejpam-4274	596	14	,	,	PUNCT
ejpam-4274	596	15	p	p	NOUN
ejpam-4274	596	16	)	)	PUNCT
ejpam-4274	596	17	and	and	CCONJ
ejpam-4274	596	18	hence	hence	ADV
ejpam-4274	596	19	u	u	NOUN
ejpam-4274	596	20	(	(	PUNCT
ejpam-4274	596	21	λ	λ	PROPN
ejpam-4274	596	22	,	,	PUNCT
ejpam-4274	596	23	p	p	NOUN
ejpam-4274	596	24	)	)	PUNCT
ejpam-4274	596	25	is	be	AUX
ejpam-4274	596	26	r(λ	r(λ	NOUN
ejpam-4274	596	27	,	,	PUNCT
ejpam-4274	596	28	p)-closed	p)-close	VERB
ejpam-4274	596	29	.	.	PUNCT
ejpam-4274	597	1	by	by	ADP
ejpam-4274	597	2	theorem	theorem	NOUN
ejpam-4274	597	3	20	20	NUM
ejpam-4274	597	4	,	,	PUNCT
ejpam-4274	597	5	[	[	X
ejpam-4274	597	6	f−1([u	f−1([u	INTJ
ejpam-4274	597	7	(	(	PUNCT
ejpam-4274	597	8	λ	λ	PROPN
ejpam-4274	597	9	,	,	PUNCT
ejpam-4274	597	10	p)](λ	p)](λ	ADJ
ejpam-4274	597	11	,	,	PUNCT
ejpam-4274	597	12	p	p	NOUN
ejpam-4274	597	13	)	)	PUNCT
ejpam-4274	597	14	)	)	PUNCT
ejpam-4274	597	15	]	]	PUNCT
ejpam-4274	597	16	(	(	PUNCT
ejpam-4274	597	17	λ	λ	X
ejpam-4274	597	18	,	,	PUNCT
ejpam-4274	597	19	p	p	NOUN
ejpam-4274	597	20	)	)	PUNCT
ejpam-4274	597	21	⊆	⊆	NUM
ejpam-4274	597	22	f−1(u	f−1(u	NOUN
ejpam-4274	597	23	(	(	PUNCT
ejpam-4274	597	24	λ	λ	PROPN
ejpam-4274	597	25	,	,	PUNCT
ejpam-4274	597	26	p	p	NOUN
ejpam-4274	597	27	)	)	PUNCT
ejpam-4274	597	28	)	)	PUNCT
ejpam-4274	597	29	.	.	PUNCT
ejpam-4274	598	1	(	(	PUNCT
ejpam-4274	598	2	2	2	X
ejpam-4274	598	3	)	)	PUNCT
ejpam-4274	598	4	⇒	⇒	NOUN
ejpam-4274	598	5	(	(	PUNCT
ejpam-4274	598	6	3	3	NUM
ejpam-4274	598	7	):	):	PUNCT
ejpam-4274	598	8	let	let	VERB
ejpam-4274	598	9	u	u	PRON
ejpam-4274	598	10	be	be	AUX
ejpam-4274	598	11	any	any	DET
ejpam-4274	598	12	p(λ	p(λ	NOUN
ejpam-4274	598	13	,	,	PUNCT
ejpam-4274	598	14	p)-open	p)-open	PUNCT
ejpam-4274	598	15	subset	subset	NOUN
ejpam-4274	598	16	of	of	ADP
ejpam-4274	598	17	y	y	PROPN
ejpam-4274	598	18	.	.	PUNCT
ejpam-4274	599	1	then	then	ADV
ejpam-4274	599	2	,	,	PUNCT
ejpam-4274	599	3	u	u	NOUN
ejpam-4274	599	4	⊆	⊆	NUM
ejpam-4274	599	5	[	[	X
ejpam-4274	599	6	u	u	X
ejpam-4274	599	7	(	(	PUNCT
ejpam-4274	599	8	λ	λ	PROPN
ejpam-4274	599	9	,	,	PUNCT
ejpam-4274	599	10	p)](λ	p)](λ	ADJ
ejpam-4274	599	11	,	,	PUNCT
ejpam-4274	599	12	p	p	NOUN
ejpam-4274	599	13	)	)	PUNCT
ejpam-4274	599	14	and	and	CCONJ
ejpam-4274	599	15	by	by	ADP
ejpam-4274	599	16	(	(	PUNCT
ejpam-4274	599	17	2	2	NUM
ejpam-4274	599	18	)	)	PUNCT
ejpam-4274	599	19	,	,	PUNCT
ejpam-4274	599	20	we	we	PRON
ejpam-4274	599	21	have	have	VERB
ejpam-4274	599	22	[	[	X
ejpam-4274	599	23	f−1(u)](λ	f−1(u)](λ	X
ejpam-4274	599	24	,	,	PUNCT
ejpam-4274	599	25	p	p	NOUN
ejpam-4274	599	26	)	)	PUNCT
ejpam-4274	599	27	⊆	⊆	NUM
ejpam-4274	599	28	[	[	X
ejpam-4274	599	29	f−1([u	f−1([u	INTJ
ejpam-4274	599	30	(	(	PUNCT
ejpam-4274	599	31	λ	λ	PROPN
ejpam-4274	599	32	,	,	PUNCT
ejpam-4274	599	33	p)](λ	p)](λ	ADJ
ejpam-4274	599	34	,	,	PUNCT
ejpam-4274	599	35	p	p	NOUN
ejpam-4274	599	36	)	)	PUNCT
ejpam-4274	599	37	)	)	PUNCT
ejpam-4274	599	38	]	]	PUNCT
ejpam-4274	600	1	(	(	PUNCT
ejpam-4274	600	2	λ	λ	X
ejpam-4274	600	3	,	,	PUNCT
ejpam-4274	600	4	p	p	NOUN
ejpam-4274	600	5	)	)	PUNCT
ejpam-4274	600	6	⊆	⊆	NUM
ejpam-4274	600	7	f−1(u	f−1(u	NOUN
ejpam-4274	600	8	(	(	PUNCT
ejpam-4274	600	9	λ	λ	PROPN
ejpam-4274	600	10	,	,	PUNCT
ejpam-4274	600	11	p	p	NOUN
ejpam-4274	600	12	)	)	PUNCT
ejpam-4274	600	13	)	)	PUNCT
ejpam-4274	600	14	.	.	PUNCT
ejpam-4274	601	1	(	(	PUNCT
ejpam-4274	601	2	3	3	X
ejpam-4274	601	3	)	)	PUNCT
ejpam-4274	601	4	⇒	⇒	NOUN
ejpam-4274	601	5	(	(	PUNCT
ejpam-4274	601	6	4	4	NUM
ejpam-4274	601	7	):	):	PUNCT
ejpam-4274	601	8	let	let	VERB
ejpam-4274	601	9	u	u	PRON
ejpam-4274	601	10	be	be	AUX
ejpam-4274	601	11	any	any	DET
ejpam-4274	601	12	p(λ	p(λ	NOUN
ejpam-4274	601	13	,	,	PUNCT
ejpam-4274	601	14	p)-open	p)-open	PUNCT
ejpam-4274	601	15	subset	subset	NOUN
ejpam-4274	601	16	of	of	ADP
ejpam-4274	601	17	y	y	PROPN
ejpam-4274	601	18	.	.	PUNCT
ejpam-4274	602	1	by	by	ADP
ejpam-4274	602	2	(	(	PUNCT
ejpam-4274	602	3	3	3	NUM
ejpam-4274	602	4	)	)	PUNCT
ejpam-4274	602	5	,	,	PUNCT
ejpam-4274	602	6	we	we	PRON
ejpam-4274	602	7	have	have	VERB
ejpam-4274	602	8	f−1(u	f−1(u	NOUN
ejpam-4274	602	9	)	)	PUNCT
ejpam-4274	602	10	⊆	⊆	NUM
ejpam-4274	602	11	f−1([u	f−1([u	NOUN
ejpam-4274	602	12	(	(	PUNCT
ejpam-4274	602	13	λ	λ	PROPN
ejpam-4274	602	14	,	,	PUNCT
ejpam-4274	602	15	p)](λ	p)](λ	ADJ
ejpam-4274	602	16	,	,	PUNCT
ejpam-4274	602	17	p	p	NOUN
ejpam-4274	602	18	)	)	PUNCT
ejpam-4274	602	19	)	)	PUNCT
ejpam-4274	603	1	=	=	PUNCT
ejpam-4274	603	2	x	x	X
ejpam-4274	604	1	−	−	NOUN
ejpam-4274	604	2	f−1([y	f−1([y	NOUN
ejpam-4274	604	3	−	−	NOUN
ejpam-4274	604	4	u	u	NOUN
ejpam-4274	604	5	(	(	PUNCT
ejpam-4274	604	6	λ	λ	PROPN
ejpam-4274	604	7	,	,	PUNCT
ejpam-4274	604	8	p)](λ	p)](λ	ADJ
ejpam-4274	604	9	,	,	PUNCT
ejpam-4274	604	10	p	p	NOUN
ejpam-4274	604	11	)	)	PUNCT
ejpam-4274	604	12	)	)	PUNCT
ejpam-4274	605	1	=	=	PUNCT
ejpam-4274	605	2	x	x	X
ejpam-4274	606	1	−	−	PROPN
ejpam-4274	607	1	[	[	X
ejpam-4274	607	2	f−1(y	f−1(y	PROPN
ejpam-4274	607	3	−	−	PROPN
ejpam-4274	607	4	u	u	NOUN
ejpam-4274	607	5	(	(	PUNCT
ejpam-4274	607	6	λ	λ	PROPN
ejpam-4274	607	7	,	,	PUNCT
ejpam-4274	607	8	p))](λ	p))](λ	PRON
ejpam-4274	607	9	,	,	PUNCT
ejpam-4274	607	10	p	p	X
ejpam-4274	607	11	)	)	PUNCT
ejpam-4274	607	12	=	=	NOUN
ejpam-4274	608	1	[	[	X
ejpam-4274	608	2	f−1(u	f−1(u	X
ejpam-4274	608	3	(	(	PUNCT
ejpam-4274	608	4	λ	λ	PROPN
ejpam-4274	608	5	,	,	PUNCT
ejpam-4274	608	6	p))](λ	p))](λ	PRON
ejpam-4274	608	7	,	,	PUNCT
ejpam-4274	608	8	p	p	NOUN
ejpam-4274	608	9	)	)	PUNCT
ejpam-4274	608	10	.	.	PUNCT
ejpam-4274	609	1	(	(	PUNCT
ejpam-4274	609	2	4	4	X
ejpam-4274	609	3	)	)	PUNCT
ejpam-4274	609	4	⇒	⇒	NOUN
ejpam-4274	609	5	(	(	PUNCT
ejpam-4274	609	6	1	1	NUM
ejpam-4274	609	7	):	):	PUNCT
ejpam-4274	609	8	since	since	SCONJ
ejpam-4274	609	9	every	every	DET
ejpam-4274	609	10	(	(	PUNCT
ejpam-4274	609	11	λ	λ	NOUN
ejpam-4274	609	12	,	,	PUNCT
ejpam-4274	609	13	p)-open	p)-open	VERB
ejpam-4274	609	14	set	set	VERB
ejpam-4274	609	15	is	be	AUX
ejpam-4274	609	16	p(λ	p(λ	NOUN
ejpam-4274	609	17	,	,	PUNCT
ejpam-4274	609	18	p)-open	p)-open	ADJ
ejpam-4274	609	19	,	,	PUNCT
ejpam-4274	609	20	by	by	ADP
ejpam-4274	609	21	(	(	PUNCT
ejpam-4274	609	22	4	4	NUM
ejpam-4274	609	23	)	)	PUNCT
ejpam-4274	609	24	and	and	CCONJ
ejpam-4274	609	25	theorem	theorem	VERB
ejpam-4274	609	26	19	19	NUM
ejpam-4274	609	27	,	,	PUNCT
ejpam-4274	609	28	it	it	PRON
ejpam-4274	609	29	follows	follow	VERB
ejpam-4274	609	30	that	that	SCONJ
ejpam-4274	609	31	f	f	PROPN
ejpam-4274	609	32	is	be	AUX
ejpam-4274	609	33	weakly	weakly	ADJ
ejpam-4274	609	34	(	(	PUNCT
ejpam-4274	609	35	λ	λ	NOUN
ejpam-4274	609	36	,	,	PUNCT
ejpam-4274	609	37	p)-continuous	p)-continuous	ADJ
ejpam-4274	609	38	.	.	PUNCT
ejpam-4274	610	1	theorem	theorem	VERB
ejpam-4274	610	2	22	22	NUM
ejpam-4274	610	3	.	.	PUNCT
ejpam-4274	611	1	for	for	ADP
ejpam-4274	611	2	a	a	DET
ejpam-4274	611	3	function	function	NOUN
ejpam-4274	611	4	f	f	NOUN
ejpam-4274	611	5	:	:	PUNCT
ejpam-4274	611	6	(	(	PUNCT
ejpam-4274	611	7	x	x	X
ejpam-4274	611	8	,	,	PUNCT
ejpam-4274	611	9	τ	τ	X
ejpam-4274	611	10	)	)	PUNCT
ejpam-4274	611	11	→	→	SYM
ejpam-4274	611	12	(	(	PUNCT
ejpam-4274	611	13	y	y	PROPN
ejpam-4274	611	14	,	,	PUNCT
ejpam-4274	611	15	σ	σ	PROPN
ejpam-4274	611	16	)	)	PUNCT
ejpam-4274	611	17	,	,	PUNCT
ejpam-4274	611	18	the	the	DET
ejpam-4274	611	19	following	follow	VERB
ejpam-4274	611	20	properties	property	NOUN
ejpam-4274	611	21	are	be	AUX
ejpam-4274	611	22	equivalent	equivalent	ADJ
ejpam-4274	611	23	:	:	PUNCT
ejpam-4274	611	24	(	(	PUNCT
ejpam-4274	611	25	1	1	X
ejpam-4274	611	26	)	)	PUNCT
ejpam-4274	611	27	f	f	PROPN
ejpam-4274	611	28	is	be	AUX
ejpam-4274	611	29	weakly	weakly	ADJ
ejpam-4274	611	30	(	(	PUNCT
ejpam-4274	611	31	λ	λ	NOUN
ejpam-4274	611	32	,	,	PUNCT
ejpam-4274	611	33	p)-continuous	p)-continuous	ADJ
ejpam-4274	611	34	;	;	PUNCT
ejpam-4274	611	35	(	(	PUNCT
ejpam-4274	611	36	2	2	X
ejpam-4274	611	37	)	)	PUNCT
ejpam-4274	611	38	[	[	X
ejpam-4274	611	39	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4274	611	40	,	,	PUNCT
ejpam-4274	611	41	p)](λ	p)](λ	X
ejpam-4274	611	42	,	,	PUNCT
ejpam-4274	611	43	p	p	NOUN
ejpam-4274	611	44	)	)	PUNCT
ejpam-4274	611	45	)	)	PUNCT
ejpam-4274	611	46	]	]	PUNCT
ejpam-4274	612	1	(	(	PUNCT
ejpam-4274	612	2	λ	λ	X
ejpam-4274	612	3	,	,	PUNCT
ejpam-4274	612	4	p	p	NOUN
ejpam-4274	612	5	)	)	PUNCT
ejpam-4274	612	6	⊆	⊆	NUM
ejpam-4274	612	7	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4274	612	8	,	,	PUNCT
ejpam-4274	612	9	p	p	NOUN
ejpam-4274	612	10	)	)	PUNCT
ejpam-4274	612	11	)	)	PUNCT
ejpam-4274	612	12	for	for	ADP
ejpam-4274	612	13	every	every	DET
ejpam-4274	612	14	subset	subset	NOUN
ejpam-4274	612	15	a	a	PRON
ejpam-4274	612	16	of	of	ADP
ejpam-4274	612	17	y	y	PROPN
ejpam-4274	612	18	;	;	PUNCT
ejpam-4274	612	19	c.	c.	PROPN
ejpam-4274	612	20	boonpok	boonpok	PROPN
ejpam-4274	612	21	,	,	PUNCT
ejpam-4274	612	22	c.	c.	PROPN
ejpam-4274	612	23	viriyapong	viriyapong	PROPN
ejpam-4274	612	24	/	/	SYM
ejpam-4274	612	25	eur	eur	PROPN
ejpam-4274	612	26	.	.	PUNCT
ejpam-4274	613	1	j.	j.	PROPN
ejpam-4274	613	2	pure	pure	PROPN
ejpam-4274	613	3	appl	appl	PROPN
ejpam-4274	613	4	.	.	PROPN
ejpam-4274	613	5	math	math	PROPN
ejpam-4274	613	6	,	,	PUNCT
ejpam-4274	613	7	15	15	NUM
ejpam-4274	613	8	(	(	PUNCT
ejpam-4274	613	9	2	2	NUM
ejpam-4274	613	10	)	)	PUNCT
ejpam-4274	613	11	(	(	PUNCT
ejpam-4274	613	12	2022	2022	NUM
ejpam-4274	613	13	)	)	PUNCT
ejpam-4274	613	14	,	,	PUNCT
ejpam-4274	613	15	415	415	NUM
ejpam-4274	613	16	-	-	SYM
ejpam-4274	613	17	436	436	NUM
ejpam-4274	613	18	431	431	NUM
ejpam-4274	613	19	(	(	PUNCT
ejpam-4274	613	20	3	3	NUM
ejpam-4274	613	21	)	)	PUNCT
ejpam-4274	613	22	[	[	X
ejpam-4274	613	23	f−1(f(λ	f−1(f(λ	X
ejpam-4274	613	24	,	,	PUNCT
ejpam-4274	613	25	p	p	NOUN
ejpam-4274	613	26	)	)	PUNCT
ejpam-4274	613	27	)	)	PUNCT
ejpam-4274	613	28	]	]	PUNCT
ejpam-4274	614	1	(	(	PUNCT
ejpam-4274	614	2	λ	λ	X
ejpam-4274	614	3	,	,	PUNCT
ejpam-4274	614	4	p	p	NOUN
ejpam-4274	614	5	)	)	PUNCT
ejpam-4274	614	6	⊆	⊆	NUM
ejpam-4274	614	7	f−1(f	f−1(f	PROPN
ejpam-4274	614	8	)	)	PUNCT
ejpam-4274	614	9	for	for	ADP
ejpam-4274	614	10	every	every	DET
ejpam-4274	614	11	r(λ	r(λ	NOUN
ejpam-4274	614	12	,	,	PUNCT
ejpam-4274	614	13	p)-closed	p)-close	VERB
ejpam-4274	614	14	subset	subset	NOUN
ejpam-4274	614	15	f	f	PROPN
ejpam-4274	614	16	of	of	ADP
ejpam-4274	614	17	y	y	PROPN
ejpam-4274	614	18	;	;	PUNCT
ejpam-4274	614	19	(	(	PUNCT
ejpam-4274	614	20	4	4	X
ejpam-4274	614	21	)	)	PUNCT
ejpam-4274	615	1	[	[	X
ejpam-4274	615	2	f−1(u)](λ	f−1(u)](λ	X
ejpam-4274	615	3	,	,	PUNCT
ejpam-4274	615	4	p	p	NOUN
ejpam-4274	615	5	)	)	PUNCT
ejpam-4274	615	6	⊆	⊆	NUM
ejpam-4274	615	7	f−1(u	f−1(u	NOUN
ejpam-4274	615	8	(	(	PUNCT
ejpam-4274	615	9	λ	λ	PROPN
ejpam-4274	615	10	,	,	PUNCT
ejpam-4274	615	11	p	p	NOUN
ejpam-4274	615	12	)	)	PUNCT
ejpam-4274	615	13	)	)	PUNCT
ejpam-4274	615	14	for	for	ADP
ejpam-4274	615	15	every	every	DET
ejpam-4274	615	16	(	(	PUNCT
ejpam-4274	615	17	λ	λ	NOUN
ejpam-4274	615	18	,	,	PUNCT
ejpam-4274	615	19	p)-open	p)-open	VERB
ejpam-4274	615	20	subset	subset	VERB
ejpam-4274	615	21	u	u	NOUN
ejpam-4274	615	22	of	of	ADP
ejpam-4274	615	23	y	y	PROPN
ejpam-4274	615	24	;	;	PUNCT
ejpam-4274	615	25	(	(	PUNCT
ejpam-4274	615	26	5	5	X
ejpam-4274	615	27	)	)	PUNCT
ejpam-4274	615	28	f−1(u	f−1(u	NOUN
ejpam-4274	615	29	)	)	PUNCT
ejpam-4274	615	30	⊆	⊆	NUM
ejpam-4274	616	1	[	[	X
ejpam-4274	616	2	f−1(u	f−1(u	NOUN
ejpam-4274	616	3	(	(	PUNCT
ejpam-4274	616	4	λ	λ	PROPN
ejpam-4274	616	5	,	,	PUNCT
ejpam-4274	616	6	p))](λ	p))](λ	PRON
ejpam-4274	616	7	,	,	PUNCT
ejpam-4274	616	8	p	p	NOUN
ejpam-4274	616	9	)	)	PUNCT
ejpam-4274	616	10	for	for	ADP
ejpam-4274	616	11	every	every	DET
ejpam-4274	616	12	(	(	PUNCT
ejpam-4274	616	13	λ	λ	NOUN
ejpam-4274	616	14	,	,	PUNCT
ejpam-4274	616	15	p)-open	p)-open	VERB
ejpam-4274	616	16	subset	subset	VERB
ejpam-4274	616	17	u	u	NOUN
ejpam-4274	616	18	of	of	ADP
ejpam-4274	616	19	y	y	PROPN
ejpam-4274	616	20	;	;	PUNCT
ejpam-4274	616	21	(	(	PUNCT
ejpam-4274	616	22	6	6	X
ejpam-4274	616	23	)	)	PUNCT
ejpam-4274	617	1	[	[	X
ejpam-4274	617	2	f−1(u)](λ	f−1(u)](λ	X
ejpam-4274	617	3	,	,	PUNCT
ejpam-4274	617	4	p	p	NOUN
ejpam-4274	617	5	)	)	PUNCT
ejpam-4274	617	6	⊆	⊆	NUM
ejpam-4274	617	7	f−1(u	f−1(u	NOUN
ejpam-4274	617	8	(	(	PUNCT
ejpam-4274	617	9	λ	λ	PROPN
ejpam-4274	617	10	,	,	PUNCT
ejpam-4274	617	11	p	p	NOUN
ejpam-4274	617	12	)	)	PUNCT
ejpam-4274	617	13	)	)	PUNCT
ejpam-4274	617	14	for	for	ADP
ejpam-4274	617	15	every	every	DET
ejpam-4274	617	16	p(λ	p(λ	NOUN
ejpam-4274	617	17	,	,	PUNCT
ejpam-4274	617	18	p)-open	p)-open	VERB
ejpam-4274	617	19	subset	subset	VERB
ejpam-4274	617	20	u	u	NOUN
ejpam-4274	617	21	of	of	ADP
ejpam-4274	617	22	y	y	PROPN
ejpam-4274	617	23	;	;	PUNCT
ejpam-4274	617	24	(	(	PUNCT
ejpam-4274	617	25	7	7	X
ejpam-4274	617	26	)	)	PUNCT
ejpam-4274	617	27	f−1(u	f−1(u	NOUN
ejpam-4274	617	28	)	)	PUNCT
ejpam-4274	617	29	⊆	⊆	NUM
ejpam-4274	618	1	[	[	X
ejpam-4274	618	2	f−1(u	f−1(u	NOUN
ejpam-4274	618	3	(	(	PUNCT
ejpam-4274	618	4	λ	λ	PROPN
ejpam-4274	618	5	,	,	PUNCT
ejpam-4274	618	6	p))](λ	p))](λ	PRON
ejpam-4274	618	7	,	,	PUNCT
ejpam-4274	618	8	p	p	NOUN
ejpam-4274	618	9	)	)	PUNCT
ejpam-4274	618	10	for	for	ADP
ejpam-4274	618	11	every	every	DET
ejpam-4274	618	12	p(λ	p(λ	NOUN
ejpam-4274	618	13	,	,	PUNCT
ejpam-4274	618	14	p)-open	p)-open	VERB
ejpam-4274	618	15	subset	subset	VERB
ejpam-4274	618	16	u	u	NOUN
ejpam-4274	618	17	of	of	ADP
ejpam-4274	618	18	y	y	PROPN
ejpam-4274	618	19	.	.	PUNCT
ejpam-4274	619	1	proof	proof	NOUN
ejpam-4274	619	2	.	.	PUNCT
ejpam-4274	620	1	(	(	PUNCT
ejpam-4274	620	2	1	1	X
ejpam-4274	620	3	)	)	PUNCT
ejpam-4274	620	4	⇒	⇒	NOUN
ejpam-4274	620	5	(	(	PUNCT
ejpam-4274	620	6	2	2	NUM
ejpam-4274	620	7	):	):	PUNCT
ejpam-4274	620	8	let	let	VERB
ejpam-4274	620	9	a	a	DET
ejpam-4274	620	10	be	be	AUX
ejpam-4274	620	11	any	any	DET
ejpam-4274	620	12	subset	subset	NOUN
ejpam-4274	620	13	of	of	ADP
ejpam-4274	620	14	y	y	PROPN
ejpam-4274	620	15	and	and	CCONJ
ejpam-4274	620	16	let	let	VERB
ejpam-4274	620	17	x	x	SYM
ejpam-4274	620	18	∈	∈	PROPN
ejpam-4274	620	19	x	x	X
ejpam-4274	620	20	−	−	PROPN
ejpam-4274	620	21	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4274	620	22	,	,	PUNCT
ejpam-4274	620	23	p	p	NOUN
ejpam-4274	620	24	)	)	PUNCT
ejpam-4274	620	25	)	)	PUNCT
ejpam-4274	620	26	.	.	PUNCT
ejpam-4274	621	1	then	then	ADV
ejpam-4274	621	2	,	,	PUNCT
ejpam-4274	621	3	f(x	f(x	PROPN
ejpam-4274	621	4	)	)	PUNCT
ejpam-4274	621	5	∈	∈	PROPN
ejpam-4274	621	6	y	y	PROPN
ejpam-4274	621	7	−a(λ	−a(λ	PROPN
ejpam-4274	621	8	,	,	PUNCT
ejpam-4274	621	9	p	p	NOUN
ejpam-4274	621	10	)	)	PUNCT
ejpam-4274	621	11	and	and	CCONJ
ejpam-4274	621	12	there	there	PRON
ejpam-4274	621	13	exists	exist	VERB
ejpam-4274	621	14	a	a	DET
ejpam-4274	621	15	(	(	PUNCT
ejpam-4274	621	16	λ	λ	NOUN
ejpam-4274	621	17	,	,	PUNCT
ejpam-4274	621	18	p)-open	p)-open	VERB
ejpam-4274	621	19	set	set	VERB
ejpam-4274	621	20	u	u	NOUN
ejpam-4274	621	21	containing	contain	VERB
ejpam-4274	621	22	f(x	f(x	PROPN
ejpam-4274	621	23	)	)	PUNCT
ejpam-4274	621	24	such	such	ADJ
ejpam-4274	621	25	that	that	SCONJ
ejpam-4274	621	26	u	u	NOUN
ejpam-4274	621	27	∩a	∩a	NOUN
ejpam-4274	621	28	=	=	NOUN
ejpam-4274	621	29	∅	∅	NOUN
ejpam-4274	621	30	and	and	CCONJ
ejpam-4274	621	31	hence	hence	ADV
ejpam-4274	621	32	u	u	NOUN
ejpam-4274	621	33	(	(	PUNCT
ejpam-4274	621	34	λ	λ	PROPN
ejpam-4274	621	35	,	,	PUNCT
ejpam-4274	621	36	p	p	NOUN
ejpam-4274	621	37	)	)	PUNCT
ejpam-4274	621	38	∩	∩	NOUN
ejpam-4274	621	39	[	[	X
ejpam-4274	621	40	a(λ	a(λ	ADJ
ejpam-4274	621	41	,	,	PUNCT
ejpam-4274	621	42	p)](λ	p)](λ	X
ejpam-4274	621	43	,	,	PUNCT
ejpam-4274	621	44	p	p	NOUN
ejpam-4274	621	45	)	)	PUNCT
ejpam-4274	621	46	=	=	PUNCT
ejpam-4274	621	47	∅.	∅.	NOUN
ejpam-4274	621	48	since	since	SCONJ
ejpam-4274	621	49	f	f	PROPN
ejpam-4274	621	50	is	be	AUX
ejpam-4274	621	51	weakly	weakly	ADJ
ejpam-4274	621	52	(	(	PUNCT
ejpam-4274	621	53	λ	λ	NOUN
ejpam-4274	621	54	,	,	PUNCT
ejpam-4274	621	55	p)-continuous	p)-continuous	ADJ
ejpam-4274	621	56	,	,	PUNCT
ejpam-4274	621	57	there	there	PRON
ejpam-4274	621	58	exists	exist	VERB
ejpam-4274	621	59	a	a	DET
ejpam-4274	621	60	(	(	PUNCT
ejpam-4274	621	61	λ	λ	NOUN
ejpam-4274	621	62	,	,	PUNCT
ejpam-4274	621	63	p)-open	p)-open	VERB
ejpam-4274	621	64	set	set	VERB
ejpam-4274	621	65	w	w	NOUN
ejpam-4274	621	66	containing	contain	VERB
ejpam-4274	621	67	x	x	PUNCT
ejpam-4274	621	68	such	such	ADJ
ejpam-4274	621	69	that	that	SCONJ
ejpam-4274	621	70	f(w	f(w	PROPN
ejpam-4274	621	71	)	)	PUNCT
ejpam-4274	622	1	⊆	⊆	NUM
ejpam-4274	622	2	u	u	NOUN
ejpam-4274	622	3	(	(	PUNCT
ejpam-4274	622	4	λ	λ	PROPN
ejpam-4274	622	5	,	,	PUNCT
ejpam-4274	622	6	p	p	NOUN
ejpam-4274	622	7	)	)	PUNCT
ejpam-4274	622	8	.	.	PUNCT
ejpam-4274	623	1	then	then	ADV
ejpam-4274	623	2	w	w	PROPN
ejpam-4274	623	3	∩	∩	PROPN
ejpam-4274	623	4	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4274	623	5	,	,	PUNCT
ejpam-4274	623	6	p)](λ	p)](λ	X
ejpam-4274	623	7	,	,	PUNCT
ejpam-4274	623	8	p	p	NOUN
ejpam-4274	623	9	)	)	PUNCT
ejpam-4274	623	10	)	)	PUNCT
ejpam-4274	624	1	=	=	NOUN
ejpam-4274	624	2	∅	∅	NOUN
ejpam-4274	624	3	and	and	CCONJ
ejpam-4274	624	4	hence	hence	ADV
ejpam-4274	624	5	x	x	X
ejpam-4274	624	6	∈	∈	NOUN
ejpam-4274	624	7	x	x	X
ejpam-4274	624	8	−	−	PROPN
ejpam-4274	625	1	[	[	X
ejpam-4274	625	2	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4274	625	3	,	,	PUNCT
ejpam-4274	625	4	p)](λ	p)](λ	X
ejpam-4274	625	5	,	,	PUNCT
ejpam-4274	625	6	p	p	NOUN
ejpam-4274	625	7	)	)	PUNCT
ejpam-4274	625	8	)	)	PUNCT
ejpam-4274	625	9	]	]	PUNCT
ejpam-4274	626	1	(	(	PUNCT
ejpam-4274	626	2	λ	λ	X
ejpam-4274	626	3	,	,	PUNCT
ejpam-4274	626	4	p	p	NOUN
ejpam-4274	626	5	)	)	PUNCT
ejpam-4274	626	6	.	.	PUNCT
ejpam-4274	627	1	this	this	PRON
ejpam-4274	627	2	shows	show	VERB
ejpam-4274	627	3	that	that	SCONJ
ejpam-4274	627	4	[	[	X
ejpam-4274	627	5	f−1([a(λ	f−1([a(λ	NOUN
ejpam-4274	627	6	,	,	PUNCT
ejpam-4274	627	7	p)](λ	p)](λ	X
ejpam-4274	627	8	,	,	PUNCT
ejpam-4274	627	9	p	p	NOUN
ejpam-4274	627	10	)	)	PUNCT
ejpam-4274	627	11	)	)	PUNCT
ejpam-4274	627	12	]	]	PUNCT
ejpam-4274	628	1	(	(	PUNCT
ejpam-4274	628	2	λ	λ	X
ejpam-4274	628	3	,	,	PUNCT
ejpam-4274	628	4	p	p	NOUN
ejpam-4274	628	5	)	)	PUNCT
ejpam-4274	628	6	⊆	⊆	NUM
ejpam-4274	628	7	f−1(a(λ	f−1(a(λ	NOUN
ejpam-4274	628	8	,	,	PUNCT
ejpam-4274	628	9	p	p	NOUN
ejpam-4274	628	10	)	)	PUNCT
ejpam-4274	628	11	)	)	PUNCT
ejpam-4274	628	12	.	.	PUNCT
ejpam-4274	629	1	(	(	PUNCT
ejpam-4274	629	2	2	2	X
ejpam-4274	629	3	)	)	PUNCT
ejpam-4274	629	4	⇒	⇒	NOUN
ejpam-4274	629	5	(	(	PUNCT
ejpam-4274	629	6	3	3	NUM
ejpam-4274	629	7	):	):	PUNCT
ejpam-4274	629	8	let	let	VERB
ejpam-4274	629	9	f	f	PRON
ejpam-4274	629	10	be	be	AUX
ejpam-4274	629	11	any	any	DET
ejpam-4274	629	12	r(λ	r(λ	NOUN
ejpam-4274	629	13	,	,	PUNCT
ejpam-4274	629	14	p)-closed	p)-close	VERB
ejpam-4274	629	15	subset	subset	NOUN
ejpam-4274	629	16	of	of	ADP
ejpam-4274	629	17	y	y	PROPN
ejpam-4274	629	18	.	.	PUNCT
ejpam-4274	630	1	by	by	ADP
ejpam-4274	630	2	(	(	PUNCT
ejpam-4274	630	3	2	2	NUM
ejpam-4274	630	4	)	)	PUNCT
ejpam-4274	630	5	,	,	PUNCT
ejpam-4274	630	6	we	we	PRON
ejpam-4274	630	7	have	have	VERB
ejpam-4274	630	8	[	[	X
ejpam-4274	630	9	f−1(f(λ	f−1(f(λ	X
ejpam-4274	630	10	,	,	PUNCT
ejpam-4274	630	11	p	p	NOUN
ejpam-4274	630	12	)	)	PUNCT
ejpam-4274	630	13	)	)	PUNCT
ejpam-4274	630	14	]	]	PUNCT
ejpam-4274	631	1	(	(	PUNCT
ejpam-4274	631	2	λ	λ	X
ejpam-4274	631	3	,	,	PUNCT
ejpam-4274	631	4	p	p	NOUN
ejpam-4274	631	5	)	)	PUNCT
ejpam-4274	631	6	=	=	NOUN
ejpam-4274	632	1	[	[	X
ejpam-4274	632	2	f−1([[f(λ	f−1([[f(λ	NOUN
ejpam-4274	632	3	,	,	PUNCT
ejpam-4274	632	4	p	p	NOUN
ejpam-4274	632	5	)	)	PUNCT
ejpam-4274	632	6	]	]	PUNCT
ejpam-4274	632	7	(	(	PUNCT
ejpam-4274	632	8	λ	λ	X
ejpam-4274	632	9	,	,	PUNCT
ejpam-4274	632	10	p)](λ	p)](λ	ADJ
ejpam-4274	632	11	,	,	PUNCT
ejpam-4274	632	12	p	p	NOUN
ejpam-4274	632	13	)	)	PUNCT
ejpam-4274	632	14	)	)	PUNCT
ejpam-4274	632	15	]	]	PUNCT
ejpam-4274	632	16	(	(	PUNCT
ejpam-4274	632	17	λ	λ	X
ejpam-4274	632	18	,	,	PUNCT
ejpam-4274	632	19	p	p	NOUN
ejpam-4274	632	20	)	)	PUNCT
ejpam-4274	632	21	⊆	⊆	NUM
ejpam-4274	632	22	f−1([f(λ	f−1([f(λ	NOUN
ejpam-4274	632	23	,	,	PUNCT
ejpam-4274	632	24	p	p	NOUN
ejpam-4274	632	25	)	)	PUNCT
ejpam-4274	632	26	]	]	PUNCT
ejpam-4274	632	27	(	(	PUNCT
ejpam-4274	632	28	λ	λ	X
ejpam-4274	632	29	,	,	PUNCT
ejpam-4274	632	30	p	p	NOUN
ejpam-4274	632	31	)	)	PUNCT
ejpam-4274	632	32	)	)	PUNCT
ejpam-4274	632	33	=	=	SYM
ejpam-4274	632	34	f−1(f	f−1(f	PROPN
ejpam-4274	632	35	)	)	PUNCT
ejpam-4274	632	36	.	.	PUNCT
ejpam-4274	633	1	(	(	PUNCT
ejpam-4274	633	2	3	3	X
ejpam-4274	633	3	)	)	PUNCT
ejpam-4274	633	4	⇒	⇒	NOUN
ejpam-4274	633	5	(	(	PUNCT
ejpam-4274	633	6	4	4	NUM
ejpam-4274	633	7	):	):	PUNCT
ejpam-4274	633	8	let	let	VERB
ejpam-4274	633	9	u	u	PRON
ejpam-4274	633	10	be	be	AUX
ejpam-4274	633	11	any	any	DET
ejpam-4274	633	12	(	(	PUNCT
ejpam-4274	633	13	λ	λ	NOUN
ejpam-4274	633	14	,	,	PUNCT
ejpam-4274	633	15	p)-open	p)-open	PUNCT
ejpam-4274	633	16	subset	subset	NOUN
ejpam-4274	633	17	of	of	ADP
ejpam-4274	633	18	y	y	PROPN
ejpam-4274	633	19	.	.	PUNCT
ejpam-4274	634	1	since	since	SCONJ
ejpam-4274	634	2	u	u	PRON
ejpam-4274	634	3	(	(	PUNCT
ejpam-4274	634	4	λ	λ	PROPN
ejpam-4274	634	5	,	,	PUNCT
ejpam-4274	634	6	p	p	NOUN
ejpam-4274	634	7	)	)	PUNCT
ejpam-4274	634	8	is	be	AUX
ejpam-4274	634	9	r(λ	r(λ	NOUN
ejpam-4274	634	10	,	,	PUNCT
ejpam-4274	634	11	p)-closed	p)-close	VERB
ejpam-4274	634	12	and	and	CCONJ
ejpam-4274	634	13	by	by	ADP
ejpam-4274	634	14	(	(	PUNCT
ejpam-4274	634	15	3	3	NUM
ejpam-4274	634	16	)	)	PUNCT
ejpam-4274	634	17	,	,	PUNCT
ejpam-4274	635	1	[	[	X
ejpam-4274	635	2	f−1(u)](λ	f−1(u)](λ	X
ejpam-4274	635	3	,	,	PUNCT
ejpam-4274	635	4	p	p	NOUN
ejpam-4274	635	5	)	)	PUNCT
ejpam-4274	635	6	⊆	⊆	NUM
ejpam-4274	635	7	[	[	X
ejpam-4274	635	8	f−1([u	f−1([u	INTJ
ejpam-4274	635	9	(	(	PUNCT
ejpam-4274	635	10	λ	λ	PROPN
ejpam-4274	635	11	,	,	PUNCT
ejpam-4274	635	12	p)](λ	p)](λ	ADJ
ejpam-4274	635	13	,	,	PUNCT
ejpam-4274	635	14	p	p	NOUN
ejpam-4274	635	15	)	)	PUNCT
ejpam-4274	635	16	)	)	PUNCT
ejpam-4274	635	17	]	]	PUNCT
ejpam-4274	636	1	(	(	PUNCT
ejpam-4274	636	2	λ	λ	X
ejpam-4274	636	3	,	,	PUNCT
ejpam-4274	636	4	p	p	NOUN
ejpam-4274	636	5	)	)	PUNCT
ejpam-4274	636	6	⊆	⊆	NUM
ejpam-4274	636	7	f−1(u	f−1(u	NOUN
ejpam-4274	636	8	(	(	PUNCT
ejpam-4274	636	9	λ	λ	PROPN
ejpam-4274	636	10	,	,	PUNCT
ejpam-4274	636	11	p	p	NOUN
ejpam-4274	636	12	)	)	PUNCT
ejpam-4274	636	13	)	)	PUNCT
ejpam-4274	636	14	.	.	PUNCT
ejpam-4274	637	1	(	(	PUNCT
ejpam-4274	637	2	4	4	X
ejpam-4274	637	3	)	)	PUNCT
ejpam-4274	637	4	⇒	⇒	NOUN
ejpam-4274	637	5	(	(	PUNCT
ejpam-4274	637	6	5	5	NUM
ejpam-4274	637	7	):	):	PUNCT
ejpam-4274	637	8	let	let	VERB
ejpam-4274	637	9	u	u	PRON
ejpam-4274	637	10	be	be	AUX
ejpam-4274	637	11	any	any	DET
ejpam-4274	637	12	(	(	PUNCT
ejpam-4274	637	13	λ	λ	NOUN
ejpam-4274	637	14	,	,	PUNCT
ejpam-4274	637	15	p)-open	p)-open	PUNCT
ejpam-4274	637	16	subset	subset	NOUN
ejpam-4274	637	17	of	of	ADP
ejpam-4274	637	18	y	y	PROPN
ejpam-4274	637	19	.	.	PUNCT
ejpam-4274	638	1	since	since	SCONJ
ejpam-4274	638	2	y	y	PROPN
ejpam-4274	638	3	−	−	PROPN
ejpam-4274	638	4	u	u	PROPN
ejpam-4274	638	5	(	(	PUNCT
ejpam-4274	638	6	λ	λ	PROPN
ejpam-4274	638	7	,	,	PUNCT
ejpam-4274	638	8	p	p	NOUN
ejpam-4274	638	9	)	)	PUNCT
ejpam-4274	638	10	is	be	AUX
ejpam-4274	638	11	(	(	PUNCT
ejpam-4274	638	12	λ	λ	X
ejpam-4274	638	13	,	,	PUNCT
ejpam-4274	638	14	p)-open	p)-open	ADJ
ejpam-4274	638	15	,	,	PUNCT
ejpam-4274	638	16	by	by	ADP
ejpam-4274	638	17	(	(	PUNCT
ejpam-4274	638	18	4	4	NUM
ejpam-4274	638	19	)	)	PUNCT
ejpam-4274	638	20	,	,	PUNCT
ejpam-4274	638	21	x	x	X
ejpam-4274	639	1	−	−	NOUN
ejpam-4274	639	2	[	[	X
ejpam-4274	639	3	f−1(u	f−1(u	X
ejpam-4274	639	4	(	(	PUNCT
ejpam-4274	639	5	λ	λ	PROPN
ejpam-4274	639	6	,	,	PUNCT
ejpam-4274	639	7	p))](λ	p))](λ	PRON
ejpam-4274	639	8	,	,	PUNCT
ejpam-4274	639	9	p	p	X
ejpam-4274	639	10	)	)	PUNCT
ejpam-4274	639	11	=	=	NOUN
ejpam-4274	640	1	[	[	X
ejpam-4274	640	2	f−1(y	f−1(y	X
ejpam-4274	640	3	−u	−u	PROPN
ejpam-4274	640	4	(	(	PUNCT
ejpam-4274	640	5	λ	λ	PROPN
ejpam-4274	640	6	,	,	PUNCT
ejpam-4274	640	7	p))](λ	p))](λ	PRON
ejpam-4274	640	8	,	,	PUNCT
ejpam-4274	640	9	p	p	X
ejpam-4274	640	10	)	)	PUNCT
ejpam-4274	640	11	⊆	⊆	NUM
ejpam-4274	640	12	f−1([y	f−1([y	NOUN
ejpam-4274	640	13	−u	−u	NOUN
ejpam-4274	640	14	(	(	PUNCT
ejpam-4274	640	15	λ	λ	PROPN
ejpam-4274	640	16	,	,	PUNCT
ejpam-4274	640	17	p)](λ	p)](λ	ADJ
ejpam-4274	640	18	,	,	PUNCT
ejpam-4274	640	19	p	p	NOUN
ejpam-4274	640	20	)	)	PUNCT
ejpam-4274	640	21	)	)	PUNCT
ejpam-4274	641	1	⊆	⊆	NUM
ejpam-4274	641	2	x	x	SYM
ejpam-4274	641	3	−	−	NUM
ejpam-4274	641	4	f−1(u	f−1(u	PROPN
ejpam-4274	641	5	)	)	PUNCT
ejpam-4274	641	6	and	and	CCONJ
ejpam-4274	641	7	hence	hence	ADV
ejpam-4274	641	8	f−1(u	f−1(u	PROPN
ejpam-4274	641	9	)	)	PUNCT
ejpam-4274	642	1	⊆	⊆	NUM
ejpam-4274	642	2	[	[	X
ejpam-4274	642	3	f−1(u	f−1(u	NOUN
ejpam-4274	642	4	(	(	PUNCT
ejpam-4274	642	5	λ	λ	PROPN
ejpam-4274	642	6	,	,	PUNCT
ejpam-4274	642	7	p))](λ	p))](λ	PRON
ejpam-4274	642	8	,	,	PUNCT
ejpam-4274	642	9	p	p	NOUN
ejpam-4274	642	10	)	)	PUNCT
ejpam-4274	642	11	.	.	PUNCT
ejpam-4274	643	1	(	(	PUNCT
ejpam-4274	643	2	5	5	X
ejpam-4274	643	3	)	)	PUNCT
ejpam-4274	643	4	⇒	⇒	NOUN
ejpam-4274	643	5	(	(	PUNCT
ejpam-4274	643	6	1	1	NUM
ejpam-4274	643	7	):	):	PUNCT
ejpam-4274	643	8	let	let	VERB
ejpam-4274	643	9	x	x	PUNCT
ejpam-4274	643	10	∈	∈	PROPN
ejpam-4274	643	11	x	x	PUNCT
ejpam-4274	643	12	and	and	CCONJ
ejpam-4274	643	13	let	let	VERB
ejpam-4274	643	14	u	u	PRON
ejpam-4274	643	15	be	be	AUX
ejpam-4274	643	16	any	any	DET
ejpam-4274	643	17	(	(	PUNCT
ejpam-4274	643	18	λ	λ	NOUN
ejpam-4274	643	19	,	,	PUNCT
ejpam-4274	643	20	p)-open	p)-open	PUNCT
ejpam-4274	643	21	subset	subset	NOUN
ejpam-4274	643	22	of	of	ADP
ejpam-4274	643	23	y	y	PROPN
ejpam-4274	643	24	containing	contain	VERB
ejpam-4274	643	25	f(x	f(x	PROPN
ejpam-4274	643	26	)	)	PUNCT
ejpam-4274	643	27	.	.	PUNCT
ejpam-4274	644	1	by	by	ADP
ejpam-4274	644	2	(	(	PUNCT
ejpam-4274	644	3	5	5	NUM
ejpam-4274	644	4	)	)	PUNCT
ejpam-4274	644	5	,	,	PUNCT
ejpam-4274	644	6	x	x	PUNCT
ejpam-4274	644	7	∈	∈	PROPN
ejpam-4274	644	8	f−1(u	f−1(u	PROPN
ejpam-4274	644	9	)	)	PUNCT
ejpam-4274	644	10	⊆	⊆	NUM
ejpam-4274	645	1	[	[	X
ejpam-4274	645	2	f−1(u	f−1(u	NOUN
ejpam-4274	645	3	(	(	PUNCT
ejpam-4274	645	4	λ	λ	PROPN
ejpam-4274	645	5	,	,	PUNCT
ejpam-4274	645	6	p))](λ	p))](λ	PRON
ejpam-4274	645	7	,	,	PUNCT
ejpam-4274	645	8	p	p	NOUN
ejpam-4274	645	9	)	)	PUNCT
ejpam-4274	645	10	.	.	PUNCT
ejpam-4274	646	1	put	put	VERB
ejpam-4274	646	2	w	w	NOUN
ejpam-4274	646	3	=	=	PUNCT
ejpam-4274	647	1	[	[	X
ejpam-4274	647	2	f−1(u	f−1(u	X
ejpam-4274	647	3	(	(	PUNCT
ejpam-4274	647	4	λ	λ	PROPN
ejpam-4274	647	5	,	,	PUNCT
ejpam-4274	647	6	p))](λ	p))](λ	PRON
ejpam-4274	647	7	,	,	PUNCT
ejpam-4274	647	8	p	p	NOUN
ejpam-4274	647	9	)	)	PUNCT
ejpam-4274	647	10	.	.	PUNCT
ejpam-4274	648	1	thus	thus	ADV
ejpam-4274	648	2	,	,	PUNCT
ejpam-4274	648	3	f(w	f(w	PROPN
ejpam-4274	648	4	)	)	PUNCT
ejpam-4274	648	5	⊆	⊆	NUM
ejpam-4274	648	6	u	u	NOUN
ejpam-4274	648	7	(	(	PUNCT
ejpam-4274	648	8	λ	λ	PROPN
ejpam-4274	648	9	,	,	PUNCT
ejpam-4274	648	10	p	p	NOUN
ejpam-4274	648	11	)	)	PUNCT
ejpam-4274	648	12	and	and	CCONJ
ejpam-4274	648	13	hence	hence	ADV
ejpam-4274	648	14	f	f	PROPN
ejpam-4274	648	15	is	be	AUX
ejpam-4274	648	16	weakly	weakly	ADJ
ejpam-4274	648	17	(	(	PUNCT
ejpam-4274	648	18	λ	λ	X
ejpam-4274	648	19	,	,	PUNCT
ejpam-4274	648	20	p)-continuous	p)-continuous	ADJ
ejpam-4274	648	21	at	at	ADP
ejpam-4274	648	22	x.	x.	NOUN
ejpam-4274	648	23	this	this	PRON
ejpam-4274	648	24	shows	show	VERB
ejpam-4274	648	25	that	that	SCONJ
ejpam-4274	648	26	f	f	PROPN
ejpam-4274	648	27	is	be	AUX
ejpam-4274	648	28	weakly	weakly	ADJ
ejpam-4274	648	29	(	(	PUNCT
ejpam-4274	648	30	λ	λ	NOUN
ejpam-4274	648	31	,	,	PUNCT
ejpam-4274	648	32	p)-continuous	p)-continuous	ADJ
ejpam-4274	648	33	.	.	PUNCT
ejpam-4274	649	1	(	(	PUNCT
ejpam-4274	649	2	1	1	X
ejpam-4274	649	3	)	)	PUNCT
ejpam-4274	649	4	⇒	⇒	NOUN
ejpam-4274	649	5	(	(	PUNCT
ejpam-4274	649	6	6	6	NUM
ejpam-4274	649	7	):	):	PUNCT
ejpam-4274	649	8	let	let	VERB
ejpam-4274	649	9	u	u	PRON
ejpam-4274	649	10	be	be	AUX
ejpam-4274	649	11	any	any	DET
ejpam-4274	649	12	p(λ	p(λ	NOUN
ejpam-4274	649	13	,	,	PUNCT
ejpam-4274	649	14	p)-open	p)-open	PUNCT
ejpam-4274	649	15	subset	subset	NOUN
ejpam-4274	649	16	of	of	ADP
ejpam-4274	649	17	y	y	PROPN
ejpam-4274	649	18	and	and	CCONJ
ejpam-4274	649	19	let	let	VERB
ejpam-4274	649	20	x	x	SYM
ejpam-4274	649	21	∈	∈	PROPN
ejpam-4274	649	22	x	x	INTJ
ejpam-4274	649	23	−	−	PROPN
ejpam-4274	649	24	f−1(u	f−1(u	NOUN
ejpam-4274	649	25	(	(	PUNCT
ejpam-4274	649	26	λ	λ	PROPN
ejpam-4274	649	27	,	,	PUNCT
ejpam-4274	649	28	p	p	NOUN
ejpam-4274	649	29	)	)	PUNCT
ejpam-4274	649	30	)	)	PUNCT
ejpam-4274	649	31	.	.	PUNCT
ejpam-4274	650	1	there	there	PRON
ejpam-4274	650	2	exists	exist	VERB
ejpam-4274	650	3	a	a	DET
ejpam-4274	650	4	(	(	PUNCT
ejpam-4274	650	5	λ	λ	NOUN
ejpam-4274	650	6	,	,	PUNCT
ejpam-4274	650	7	p)-open	p)-open	VERB
ejpam-4274	650	8	set	set	VERB
ejpam-4274	650	9	v	v	NOUN
ejpam-4274	650	10	containing	contain	VERB
ejpam-4274	650	11	f(x	f(x	PROPN
ejpam-4274	650	12	)	)	PUNCT
ejpam-4274	650	13	such	such	ADJ
ejpam-4274	650	14	that	that	DET
ejpam-4274	650	15	v	v	NOUN
ejpam-4274	650	16	∩u	∩u	NOUN
ejpam-4274	650	17	=	=	PUNCT
ejpam-4274	650	18	∅	∅	NOUN
ejpam-4274	650	19	and	and	CCONJ
ejpam-4274	650	20	hence	hence	ADV
ejpam-4274	651	1	[	[	X
ejpam-4274	651	2	v	v	X
ejpam-4274	651	3	∩u	∩u	NOUN
ejpam-4274	651	4	]	]	X
ejpam-4274	651	5	(	(	PUNCT
ejpam-4274	651	6	λ	λ	X
ejpam-4274	651	7	,	,	PUNCT
ejpam-4274	651	8	p	p	NOUN
ejpam-4274	651	9	)	)	PUNCT
ejpam-4274	651	10	=	=	PUNCT
ejpam-4274	651	11	∅.	∅.	NOUN
ejpam-4274	651	12	since	since	SCONJ
ejpam-4274	651	13	u	u	NOUN
ejpam-4274	651	14	is	be	AUX
ejpam-4274	651	15	(	(	PUNCT
ejpam-4274	651	16	λ	λ	INTJ
ejpam-4274	651	17	,	,	PUNCT
ejpam-4274	651	18	p)-open	p)-open	VERB
ejpam-4274	651	19	,	,	PUNCT
ejpam-4274	651	20	we	we	PRON
ejpam-4274	651	21	have	have	VERB
ejpam-4274	651	22	u	u	NOUN
ejpam-4274	651	23	∩	∩	NOUN
ejpam-4274	651	24	v	v	X
ejpam-4274	651	25	(	(	PUNCT
ejpam-4274	651	26	λ	λ	PROPN
ejpam-4274	651	27	,	,	PUNCT
ejpam-4274	651	28	p	p	NOUN
ejpam-4274	651	29	)	)	PUNCT
ejpam-4274	651	30	⊆	⊆	NUM
ejpam-4274	652	1	[	[	X
ejpam-4274	652	2	u	u	NOUN
ejpam-4274	652	3	∩	∩	NOUN
ejpam-4274	652	4	v	v	ADP
ejpam-4274	652	5	]	]	PUNCT
ejpam-4274	652	6	(	(	PUNCT
ejpam-4274	652	7	λ	λ	X
ejpam-4274	652	8	,	,	PUNCT
ejpam-4274	652	9	p	p	NOUN
ejpam-4274	652	10	)	)	PUNCT
ejpam-4274	652	11	=	=	PUNCT
ejpam-4274	652	12	∅.	∅.	NOUN
ejpam-4274	652	13	since	since	SCONJ
ejpam-4274	652	14	f	f	PROPN
ejpam-4274	652	15	is	be	AUX
ejpam-4274	652	16	weakly	weakly	ADJ
ejpam-4274	652	17	(	(	PUNCT
ejpam-4274	652	18	λ	λ	NOUN
ejpam-4274	652	19	,	,	PUNCT
ejpam-4274	652	20	p)continuous	p)continuous	ADJ
ejpam-4274	652	21	and	and	CCONJ
ejpam-4274	652	22	v	v	NOUN
ejpam-4274	652	23	is	be	AUX
ejpam-4274	652	24	a	a	DET
ejpam-4274	652	25	(	(	PUNCT
ejpam-4274	652	26	λ	λ	NOUN
ejpam-4274	652	27	,	,	PUNCT
ejpam-4274	652	28	p)-open	p)-open	VERB
ejpam-4274	652	29	set	set	VERB
ejpam-4274	652	30	containing	contain	VERB
ejpam-4274	652	31	f(x	f(x	PROPN
ejpam-4274	652	32	)	)	PUNCT
ejpam-4274	652	33	,	,	PUNCT
ejpam-4274	652	34	there	there	PRON
ejpam-4274	652	35	exists	exist	VERB
ejpam-4274	652	36	a	a	DET
ejpam-4274	652	37	(	(	PUNCT
ejpam-4274	652	38	λ	λ	NOUN
ejpam-4274	652	39	,	,	PUNCT
ejpam-4274	652	40	p)-open	p)-open	VERB
ejpam-4274	652	41	set	set	VERB
ejpam-4274	652	42	w	w	NOUN
ejpam-4274	652	43	containing	contain	VERB
ejpam-4274	652	44	x	x	PUNCT
ejpam-4274	652	45	such	such	ADJ
ejpam-4274	652	46	that	that	SCONJ
ejpam-4274	652	47	f(w	f(w	PROPN
ejpam-4274	652	48	)	)	PUNCT
ejpam-4274	653	1	⊆	⊆	NUM
ejpam-4274	653	2	v	v	X
ejpam-4274	653	3	(	(	PUNCT
ejpam-4274	653	4	λ	λ	PROPN
ejpam-4274	653	5	,	,	PUNCT
ejpam-4274	653	6	p	p	NOUN
ejpam-4274	653	7	)	)	PUNCT
ejpam-4274	653	8	.	.	PUNCT
ejpam-4274	654	1	then	then	ADV
ejpam-4274	654	2	,	,	PUNCT
ejpam-4274	654	3	f(w	f(w	PROPN
ejpam-4274	654	4	)	)	PUNCT
ejpam-4274	654	5	∩	∩	NOUN
ejpam-4274	654	6	u	u	NOUN
ejpam-4274	654	7	=	=	NOUN
ejpam-4274	654	8	∅	∅	NOUN
ejpam-4274	654	9	and	and	CCONJ
ejpam-4274	654	10	hence	hence	ADV
ejpam-4274	654	11	w	w	PROPN
ejpam-4274	654	12	∩	∩	ADJ
ejpam-4274	654	13	f−1(u	f−1(u	PROPN
ejpam-4274	654	14	)	)	PUNCT
ejpam-4274	654	15	=	=	PUNCT
ejpam-4274	654	16	∅.	∅.	ADP
ejpam-4274	654	17	thus	thus	ADV
ejpam-4274	654	18	,	,	PUNCT
ejpam-4274	654	19	x	x	PUNCT
ejpam-4274	654	20	∈	∈	NOUN
ejpam-4274	654	21	x	x	X
ejpam-4274	654	22	−	−	PROPN
ejpam-4274	655	1	[	[	X
ejpam-4274	655	2	f−1(u)](λ	f−1(u)](λ	X
ejpam-4274	655	3	,	,	PUNCT
ejpam-4274	655	4	p	p	NOUN
ejpam-4274	655	5	)	)	PUNCT
ejpam-4274	655	6	.	.	PUNCT
ejpam-4274	656	1	this	this	PRON
ejpam-4274	656	2	shows	show	VERB
ejpam-4274	656	3	that	that	SCONJ
ejpam-4274	656	4	[	[	X
ejpam-4274	656	5	f−1(u)](λ	f−1(u)](λ	X
ejpam-4274	656	6	,	,	PUNCT
ejpam-4274	656	7	p	p	NOUN
ejpam-4274	656	8	)	)	PUNCT
ejpam-4274	656	9	⊆	⊆	NUM
ejpam-4274	656	10	f−1(u	f−1(u	NOUN
ejpam-4274	656	11	(	(	PUNCT
ejpam-4274	656	12	λ	λ	PROPN
ejpam-4274	656	13	,	,	PUNCT
ejpam-4274	656	14	p	p	NOUN
ejpam-4274	656	15	)	)	PUNCT
ejpam-4274	656	16	)	)	PUNCT
ejpam-4274	656	17	.	.	PUNCT
ejpam-4274	657	1	(	(	PUNCT
ejpam-4274	657	2	6	6	X
ejpam-4274	657	3	)	)	PUNCT
ejpam-4274	657	4	⇒	⇒	NOUN
ejpam-4274	657	5	(	(	PUNCT
ejpam-4274	657	6	7	7	NUM
ejpam-4274	657	7	):	):	PUNCT
ejpam-4274	657	8	let	let	VERB
ejpam-4274	657	9	u	u	PRON
ejpam-4274	657	10	be	be	AUX
ejpam-4274	657	11	any	any	DET
ejpam-4274	657	12	p(λ	p(λ	NOUN
ejpam-4274	657	13	,	,	PUNCT
ejpam-4274	657	14	p)-open	p)-open	PUNCT
ejpam-4274	657	15	subset	subset	NOUN
ejpam-4274	657	16	of	of	ADP
ejpam-4274	657	17	y	y	PROPN
ejpam-4274	657	18	.	.	PUNCT
ejpam-4274	658	1	since	since	SCONJ
ejpam-4274	658	2	y	y	PROPN
ejpam-4274	658	3	−u	−u	PROPN
ejpam-4274	658	4	(	(	PUNCT
ejpam-4274	658	5	λ	λ	PROPN
ejpam-4274	658	6	,	,	PUNCT
ejpam-4274	658	7	p	p	NOUN
ejpam-4274	658	8	)	)	PUNCT
ejpam-4274	658	9	is	be	AUX
ejpam-4274	658	10	(	(	PUNCT
ejpam-4274	658	11	λ	λ	X
ejpam-4274	658	12	,	,	PUNCT
ejpam-4274	658	13	p)-open	p)-open	ADJ
ejpam-4274	658	14	and	and	CCONJ
ejpam-4274	658	15	by	by	ADP
ejpam-4274	658	16	(	(	PUNCT
ejpam-4274	658	17	6	6	NUM
ejpam-4274	658	18	)	)	PUNCT
ejpam-4274	658	19	,	,	PUNCT
ejpam-4274	658	20	we	we	PRON
ejpam-4274	658	21	have	have	VERB
ejpam-4274	658	22	x	x	X
ejpam-4274	659	1	−	−	PROPN
ejpam-4274	660	1	[	[	X
ejpam-4274	660	2	f−1(u	f−1(u	X
ejpam-4274	660	3	(	(	PUNCT
ejpam-4274	660	4	λ	λ	PROPN
ejpam-4274	660	5	,	,	PUNCT
ejpam-4274	660	6	p))](λ	p))](λ	PRON
ejpam-4274	660	7	,	,	PUNCT
ejpam-4274	660	8	p	p	X
ejpam-4274	660	9	)	)	PUNCT
ejpam-4274	660	10	=	=	PUNCT
ejpam-4274	661	1	[	[	X
ejpam-4274	661	2	f−1(y	f−1(y	NOUN
ejpam-4274	661	3	−	−	PROPN
ejpam-4274	661	4	u	u	NOUN
ejpam-4274	661	5	(	(	PUNCT
ejpam-4274	661	6	λ	λ	PROPN
ejpam-4274	661	7	,	,	PUNCT
ejpam-4274	661	8	p))](λ	p))](λ	PRON
ejpam-4274	661	9	,	,	PUNCT
ejpam-4274	661	10	p	p	NOUN
ejpam-4274	661	11	)	)	PUNCT
ejpam-4274	661	12	⊆	⊆	NUM
ejpam-4274	661	13	f−1([y	f−1([y	NOUN
ejpam-4274	661	14	−	−	NOUN
ejpam-4274	661	15	u	u	NOUN
ejpam-4274	661	16	(	(	PUNCT
ejpam-4274	661	17	λ	λ	PROPN
ejpam-4274	661	18	,	,	PUNCT
ejpam-4274	661	19	p)](λ	p)](λ	ADJ
ejpam-4274	661	20	,	,	PUNCT
ejpam-4274	661	21	p	p	NOUN
ejpam-4274	661	22	)	)	PUNCT
ejpam-4274	661	23	)	)	PUNCT
ejpam-4274	662	1	⊆	⊆	NUM
ejpam-4274	662	2	x	x	SYM
ejpam-4274	662	3	−	−	NUM
ejpam-4274	662	4	f−1(u	f−1(u	PROPN
ejpam-4274	662	5	)	)	PUNCT
ejpam-4274	662	6	and	and	CCONJ
ejpam-4274	662	7	hence	hence	ADV
ejpam-4274	662	8	f−1(u	f−1(u	PROPN
ejpam-4274	662	9	)	)	PUNCT
ejpam-4274	663	1	⊆	⊆	NUM
ejpam-4274	663	2	[	[	X
ejpam-4274	663	3	f−1(u	f−1(u	NOUN
ejpam-4274	663	4	(	(	PUNCT
ejpam-4274	663	5	λ	λ	PROPN
ejpam-4274	663	6	,	,	PUNCT
ejpam-4274	663	7	p))](λ	p))](λ	PRON
ejpam-4274	663	8	,	,	PUNCT
ejpam-4274	663	9	p	p	NOUN
ejpam-4274	663	10	)	)	PUNCT
ejpam-4274	663	11	.	.	PUNCT
ejpam-4274	664	1	(	(	PUNCT
ejpam-4274	664	2	7	7	X
ejpam-4274	664	3	)	)	PUNCT
ejpam-4274	664	4	⇒	⇒	NOUN
ejpam-4274	664	5	(	(	PUNCT
ejpam-4274	664	6	1	1	NUM
ejpam-4274	664	7	):	):	PUNCT
ejpam-4274	664	8	let	let	VERB
ejpam-4274	664	9	x	x	PUNCT
ejpam-4274	664	10	∈	∈	PROPN
ejpam-4274	664	11	x	x	PUNCT
ejpam-4274	664	12	and	and	CCONJ
ejpam-4274	664	13	let	let	VERB
ejpam-4274	664	14	v	v	PART
ejpam-4274	664	15	be	be	AUX
ejpam-4274	664	16	any	any	DET
ejpam-4274	664	17	(	(	PUNCT
ejpam-4274	664	18	λ	λ	NOUN
ejpam-4274	664	19	,	,	PUNCT
ejpam-4274	664	20	p)-open	p)-open	PUNCT
ejpam-4274	664	21	subset	subset	NOUN
ejpam-4274	664	22	of	of	ADP
ejpam-4274	664	23	y	y	PROPN
ejpam-4274	664	24	containing	contain	VERB
ejpam-4274	664	25	f(x	f(x	PROPN
ejpam-4274	664	26	)	)	PUNCT
ejpam-4274	664	27	.	.	PUNCT
ejpam-4274	665	1	then	then	ADV
ejpam-4274	665	2	,	,	PUNCT
ejpam-4274	665	3	v	v	NOUN
ejpam-4274	665	4	is	be	AUX
ejpam-4274	665	5	p(λ	p(λ	NOUN
ejpam-4274	665	6	,	,	PUNCT
ejpam-4274	665	7	p)-open	p)-open	ADJ
ejpam-4274	665	8	,	,	PUNCT
ejpam-4274	665	9	by	by	ADP
ejpam-4274	665	10	(	(	PUNCT
ejpam-4274	665	11	7	7	NUM
ejpam-4274	665	12	)	)	PUNCT
ejpam-4274	665	13	,	,	PUNCT
ejpam-4274	665	14	x	x	PUNCT
ejpam-4274	665	15	∈	∈	PROPN
ejpam-4274	665	16	f−1(v	f−1(v	NOUN
ejpam-4274	665	17	)	)	PUNCT
ejpam-4274	666	1	⊆	⊆	NUM
ejpam-4274	667	1	[	[	X
ejpam-4274	667	2	f−1(v	f−1(v	NOUN
ejpam-4274	667	3	(	(	PUNCT
ejpam-4274	667	4	λ	λ	PROPN
ejpam-4274	667	5	,	,	PUNCT
ejpam-4274	667	6	p))](λ	p))](λ	PRON
ejpam-4274	667	7	,	,	PUNCT
ejpam-4274	667	8	p	p	NOUN
ejpam-4274	667	9	)	)	PUNCT
ejpam-4274	667	10	.	.	PUNCT
ejpam-4274	668	1	put	put	VERB
ejpam-4274	668	2	u	u	NOUN
ejpam-4274	668	3	=	=	PUNCT
ejpam-4274	669	1	[	[	X
ejpam-4274	669	2	f−1(v	f−1(v	NOUN
ejpam-4274	669	3	(	(	PUNCT
ejpam-4274	669	4	λ	λ	PROPN
ejpam-4274	669	5	,	,	PUNCT
ejpam-4274	669	6	p))](λ	p))](λ	PRON
ejpam-4274	669	7	,	,	PUNCT
ejpam-4274	669	8	p	p	NOUN
ejpam-4274	669	9	)	)	PUNCT
ejpam-4274	669	10	.	.	PUNCT
ejpam-4274	670	1	then	then	ADV
ejpam-4274	670	2	,	,	PUNCT
ejpam-4274	670	3	f(u	f(u	PROPN
ejpam-4274	670	4	)	)	PUNCT
ejpam-4274	670	5	⊆	⊆	NUM
ejpam-4274	670	6	v	v	NOUN
ejpam-4274	670	7	(	(	PUNCT
ejpam-4274	670	8	λ	λ	PROPN
ejpam-4274	670	9	,	,	PUNCT
ejpam-4274	670	10	p	p	NOUN
ejpam-4274	670	11	)	)	PUNCT
ejpam-4274	670	12	and	and	CCONJ
ejpam-4274	670	13	hence	hence	ADV
ejpam-4274	670	14	f	f	PROPN
ejpam-4274	670	15	is	be	AUX
ejpam-4274	670	16	weakly	weakly	ADJ
ejpam-4274	670	17	(	(	PUNCT
ejpam-4274	670	18	λ	λ	X
ejpam-4274	670	19	,	,	PUNCT
ejpam-4274	670	20	p)-continuous	p)-continuous	ADJ
ejpam-4274	670	21	at	at	ADP
ejpam-4274	670	22	x.	x.	PROPN
ejpam-4274	670	23	thus	thus	ADV
ejpam-4274	670	24	,	,	PUNCT
ejpam-4274	670	25	f	f	PROPN
ejpam-4274	670	26	is	be	AUX
ejpam-4274	670	27	weakly	weakly	ADJ
ejpam-4274	670	28	(	(	PUNCT
ejpam-4274	670	29	λ	λ	NOUN
ejpam-4274	670	30	,	,	PUNCT
ejpam-4274	670	31	p)-continuous	p)-continuous	ADJ
ejpam-4274	670	32	.	.	PUNCT
ejpam-4274	671	1	c.	c.	PROPN
ejpam-4274	671	2	boonpok	boonpok	PROPN
ejpam-4274	671	3	,	,	PUNCT
ejpam-4274	671	4	c.	c.	PROPN
ejpam-4274	671	5	viriyapong	viriyapong	PROPN
ejpam-4274	671	6	/	/	SYM
ejpam-4274	671	7	eur	eur	PROPN
ejpam-4274	671	8	.	.	PUNCT
ejpam-4274	672	1	j.	j.	PROPN
ejpam-4274	672	2	pure	pure	PROPN
ejpam-4274	672	3	appl	appl	PROPN
ejpam-4274	672	4	.	.	PROPN
ejpam-4274	672	5	math	math	PROPN
ejpam-4274	672	6	,	,	PUNCT
ejpam-4274	672	7	15	15	NUM
ejpam-4274	672	8	(	(	PUNCT
ejpam-4274	672	9	2	2	NUM
ejpam-4274	672	10	)	)	PUNCT
ejpam-4274	672	11	(	(	PUNCT
ejpam-4274	672	12	2022	2022	NUM
ejpam-4274	672	13	)	)	PUNCT
ejpam-4274	672	14	,	,	PUNCT
ejpam-4274	672	15	415	415	NUM
ejpam-4274	672	16	-	-	SYM
ejpam-4274	672	17	436	436	NUM
ejpam-4274	672	18	432	432	NUM
ejpam-4274	672	19	definition	definition	NOUN
ejpam-4274	672	20	13	13	NUM
ejpam-4274	672	21	.	.	PUNCT
ejpam-4274	673	1	a	a	DET
ejpam-4274	673	2	topological	topological	ADJ
ejpam-4274	673	3	space	space	NOUN
ejpam-4274	673	4	(	(	PUNCT
ejpam-4274	673	5	x	x	X
ejpam-4274	673	6	,	,	PUNCT
ejpam-4274	673	7	τ	τ	X
ejpam-4274	673	8	)	)	PUNCT
ejpam-4274	673	9	is	be	AUX
ejpam-4274	673	10	said	say	VERB
ejpam-4274	673	11	to	to	PART
ejpam-4274	673	12	be	be	AUX
ejpam-4274	673	13	λp	λp	NOUN
ejpam-4274	673	14	-	-	PUNCT
ejpam-4274	673	15	t2	t2	NOUN
ejpam-4274	673	16	if	if	SCONJ
ejpam-4274	673	17	,	,	PUNCT
ejpam-4274	673	18	for	for	ADP
ejpam-4274	673	19	any	any	DET
ejpam-4274	673	20	disjoint	disjoint	ADJ
ejpam-4274	673	21	pair	pair	NOUN
ejpam-4274	673	22	of	of	ADP
ejpam-4274	673	23	points	point	NOUN
ejpam-4274	673	24	x	x	PUNCT
ejpam-4274	673	25	and	and	CCONJ
ejpam-4274	673	26	y	y	PROPN
ejpam-4274	673	27	in	in	ADP
ejpam-4274	673	28	x	x	SYM
ejpam-4274	673	29	,	,	PUNCT
ejpam-4274	673	30	there	there	PRON
ejpam-4274	673	31	exist	exist	VERB
ejpam-4274	673	32	(	(	PUNCT
ejpam-4274	673	33	λ	λ	X
ejpam-4274	673	34	,	,	PUNCT
ejpam-4274	673	35	p)-open	p)-open	VERB
ejpam-4274	673	36	sets	set	VERB
ejpam-4274	673	37	u	u	NOUN
ejpam-4274	673	38	and	and	CCONJ
ejpam-4274	673	39	v	v	ADP
ejpam-4274	673	40	such	such	ADJ
ejpam-4274	673	41	that	that	SCONJ
ejpam-4274	673	42	x	x	SYM
ejpam-4274	673	43	∈	∈	PROPN
ejpam-4274	673	44	u	u	NOUN
ejpam-4274	673	45	,	,	PUNCT
ejpam-4274	673	46	y	y	PROPN
ejpam-4274	673	47	∈	∈	PROPN
ejpam-4274	673	48	v	v	NOUN
ejpam-4274	673	49	and	and	CCONJ
ejpam-4274	673	50	u	u	NOUN
ejpam-4274	673	51	∩	∩	NOUN
ejpam-4274	673	52	v	v	NOUN
ejpam-4274	673	53	=	=	PUNCT
ejpam-4274	673	54	∅.	∅.	NOUN
ejpam-4274	673	55	definition	definition	NOUN
ejpam-4274	673	56	14	14	NUM
ejpam-4274	673	57	.	.	PUNCT
ejpam-4274	674	1	a	a	DET
ejpam-4274	674	2	topological	topological	ADJ
ejpam-4274	674	3	space	space	NOUN
ejpam-4274	674	4	(	(	PUNCT
ejpam-4274	674	5	x	x	X
ejpam-4274	674	6	,	,	PUNCT
ejpam-4274	674	7	τ	τ	X
ejpam-4274	674	8	)	)	PUNCT
ejpam-4274	674	9	is	be	AUX
ejpam-4274	674	10	said	say	VERB
ejpam-4274	674	11	to	to	PART
ejpam-4274	674	12	be	be	AUX
ejpam-4274	674	13	λp	λp	PROPN
ejpam-4274	674	14	-	-	ADJ
ejpam-4274	674	15	urysohn	urysohn	ADJ
ejpam-4274	674	16	if	if	SCONJ
ejpam-4274	674	17	,	,	PUNCT
ejpam-4274	674	18	for	for	ADP
ejpam-4274	674	19	each	each	DET
ejpam-4274	674	20	distinct	distinct	ADJ
ejpam-4274	674	21	points	point	NOUN
ejpam-4274	674	22	x	x	NOUN
ejpam-4274	674	23	,	,	PUNCT
ejpam-4274	674	24	y	y	PROPN
ejpam-4274	674	25	∈	∈	PROPN
ejpam-4274	674	26	x	x	PRON
ejpam-4274	674	27	,	,	PUNCT
ejpam-4274	674	28	there	there	PRON
ejpam-4274	674	29	exist	exist	VERB
ejpam-4274	674	30	(	(	PUNCT
ejpam-4274	674	31	λ	λ	X
ejpam-4274	674	32	,	,	PUNCT
ejpam-4274	674	33	p)-open	p)-open	VERB
ejpam-4274	674	34	sets	set	VERB
ejpam-4274	674	35	u	u	NOUN
ejpam-4274	674	36	and	and	CCONJ
ejpam-4274	674	37	v	v	ADP
ejpam-4274	674	38	containing	contain	VERB
ejpam-4274	674	39	x	x	PROPN
ejpam-4274	674	40	and	and	CCONJ
ejpam-4274	674	41	y	y	PROPN
ejpam-4274	674	42	,	,	PUNCT
ejpam-4274	674	43	respectively	respectively	ADV
ejpam-4274	674	44	,	,	PUNCT
ejpam-4274	674	45	such	such	ADJ
ejpam-4274	674	46	that	that	SCONJ
ejpam-4274	674	47	u	u	PROPN
ejpam-4274	674	48	(	(	PUNCT
ejpam-4274	674	49	λ	λ	PROPN
ejpam-4274	674	50	,	,	PUNCT
ejpam-4274	674	51	p	p	NOUN
ejpam-4274	674	52	)	)	PUNCT
ejpam-4274	674	53	∩	∩	ADJ
ejpam-4274	674	54	v	v	X
ejpam-4274	674	55	(	(	PUNCT
ejpam-4274	674	56	λ	λ	PROPN
ejpam-4274	674	57	,	,	PUNCT
ejpam-4274	674	58	p	p	NOUN
ejpam-4274	674	59	)	)	PUNCT
ejpam-4274	674	60	=	=	SYM
ejpam-4274	674	61	∅.	∅.	NOUN
ejpam-4274	674	62	theorem	theorem	VERB
ejpam-4274	674	63	23	23	NUM
ejpam-4274	674	64	.	.	PUNCT
ejpam-4274	675	1	if	if	SCONJ
ejpam-4274	675	2	f	f	PROPN
ejpam-4274	675	3	:	:	PUNCT
ejpam-4274	675	4	(	(	PUNCT
ejpam-4274	675	5	x	x	X
ejpam-4274	675	6	,	,	PUNCT
ejpam-4274	675	7	τ	τ	X
ejpam-4274	675	8	)	)	PUNCT
ejpam-4274	675	9	→	→	SYM
ejpam-4274	675	10	(	(	PUNCT
ejpam-4274	675	11	y	y	PROPN
ejpam-4274	675	12	,	,	PUNCT
ejpam-4274	675	13	σ	σ	PROPN
ejpam-4274	675	14	)	)	PUNCT
ejpam-4274	675	15	is	be	AUX
ejpam-4274	675	16	a	a	DET
ejpam-4274	675	17	weakly	weakly	ADJ
ejpam-4274	675	18	(	(	PUNCT
ejpam-4274	675	19	λ	λ	NOUN
ejpam-4274	675	20	,	,	PUNCT
ejpam-4274	675	21	p)-continuous	p)-continuous	ADJ
ejpam-4274	675	22	injection	injection	NOUN
ejpam-4274	675	23	and	and	CCONJ
ejpam-4274	675	24	(	(	PUNCT
ejpam-4274	675	25	y	y	PROPN
ejpam-4274	675	26	,	,	PUNCT
ejpam-4274	675	27	σ	σ	PROPN
ejpam-4274	675	28	)	)	PUNCT
ejpam-4274	675	29	is	be	AUX
ejpam-4274	675	30	λp	λp	NOUN
ejpam-4274	675	31	-	-	ADJ
ejpam-4274	675	32	urysohn	urysohn	ADJ
ejpam-4274	675	33	,	,	PUNCT
ejpam-4274	675	34	then	then	ADV
ejpam-4274	675	35	(	(	PUNCT
ejpam-4274	675	36	x	x	X
ejpam-4274	675	37	,	,	PUNCT
ejpam-4274	675	38	τ	τ	X
ejpam-4274	675	39	)	)	PUNCT
ejpam-4274	675	40	is	be	AUX
ejpam-4274	675	41	λp	λp	PROPN
ejpam-4274	675	42	-	-	PUNCT
ejpam-4274	675	43	t2	t2	NOUN
ejpam-4274	675	44	.	.	PUNCT
ejpam-4274	676	1	proof	proof	NOUN
ejpam-4274	676	2	.	.	PUNCT
ejpam-4274	677	1	let	let	VERB
ejpam-4274	677	2	x	x	PRON
ejpam-4274	677	3	,	,	PUNCT
ejpam-4274	677	4	y	y	PROPN
ejpam-4274	677	5	be	be	VERB
ejpam-4274	677	6	distinct	distinct	ADJ
ejpam-4274	677	7	points	point	NOUN
ejpam-4274	677	8	of	of	ADP
ejpam-4274	677	9	x.	x.	NOUN
ejpam-4274	677	10	then	then	ADV
ejpam-4274	677	11	,	,	PUNCT
ejpam-4274	677	12	f(x	f(x	PROPN
ejpam-4274	677	13	)	)	PUNCT
ejpam-4274	677	14	6=	6=	SYM
ejpam-4274	678	1	f(y	f(y	NOUN
ejpam-4274	678	2	)	)	PUNCT
ejpam-4274	678	3	.	.	PUNCT
ejpam-4274	679	1	since	since	SCONJ
ejpam-4274	679	2	(	(	PUNCT
ejpam-4274	679	3	y	y	PROPN
ejpam-4274	679	4	,	,	PUNCT
ejpam-4274	679	5	σ	σ	PROPN
ejpam-4274	679	6	)	)	PUNCT
ejpam-4274	679	7	is	be	AUX
ejpam-4274	679	8	λp	λp	NOUN
ejpam-4274	679	9	-	-	ADJ
ejpam-4274	679	10	urysohn	urysohn	ADJ
ejpam-4274	679	11	,	,	PUNCT
ejpam-4274	679	12	there	there	PRON
ejpam-4274	679	13	exist	exist	VERB
ejpam-4274	679	14	(	(	PUNCT
ejpam-4274	679	15	λ	λ	X
ejpam-4274	679	16	,	,	PUNCT
ejpam-4274	679	17	p)-open	p)-open	VERB
ejpam-4274	679	18	sets	set	VERB
ejpam-4274	679	19	u	u	NOUN
ejpam-4274	679	20	and	and	CCONJ
ejpam-4274	679	21	v	v	ADP
ejpam-4274	679	22	containing	contain	VERB
ejpam-4274	679	23	f(x	f(x	PROPN
ejpam-4274	679	24	)	)	PUNCT
ejpam-4274	679	25	and	and	CCONJ
ejpam-4274	679	26	f(y	f(y	NOUN
ejpam-4274	679	27	)	)	PUNCT
ejpam-4274	679	28	,	,	PUNCT
ejpam-4274	679	29	respectively	respectively	ADV
ejpam-4274	679	30	,	,	PUNCT
ejpam-4274	679	31	such	such	ADJ
ejpam-4274	679	32	that	that	SCONJ
ejpam-4274	679	33	u	u	PROPN
ejpam-4274	679	34	(	(	PUNCT
ejpam-4274	679	35	λ	λ	PROPN
ejpam-4274	679	36	,	,	PUNCT
ejpam-4274	679	37	p)∩v	p)∩v	PROPN
ejpam-4274	679	38	(	(	PUNCT
ejpam-4274	679	39	λ	λ	PROPN
ejpam-4274	679	40	,	,	PUNCT
ejpam-4274	679	41	p	p	NOUN
ejpam-4274	679	42	)	)	PUNCT
ejpam-4274	679	43	=	=	PUNCT
ejpam-4274	679	44	∅.	∅.	NOUN
ejpam-4274	679	45	since	since	SCONJ
ejpam-4274	679	46	f	f	PROPN
ejpam-4274	679	47	is	be	AUX
ejpam-4274	679	48	weakly	weakly	ADJ
ejpam-4274	679	49	(	(	PUNCT
ejpam-4274	679	50	λ	λ	NOUN
ejpam-4274	679	51	,	,	PUNCT
ejpam-4274	679	52	p)-continuous	p)-continuous	ADJ
ejpam-4274	679	53	,	,	PUNCT
ejpam-4274	679	54	there	there	PRON
ejpam-4274	679	55	exist	exist	VERB
ejpam-4274	679	56	(	(	PUNCT
ejpam-4274	679	57	λ	λ	X
ejpam-4274	679	58	,	,	PUNCT
ejpam-4274	679	59	p)-open	p)-open	VERB
ejpam-4274	679	60	sets	set	VERB
ejpam-4274	679	61	g	g	NOUN
ejpam-4274	679	62	and	and	CCONJ
ejpam-4274	679	63	w	w	PROPN
ejpam-4274	679	64	containing	contain	VERB
ejpam-4274	679	65	x	x	PROPN
ejpam-4274	679	66	and	and	CCONJ
ejpam-4274	679	67	y	y	PROPN
ejpam-4274	679	68	,	,	PUNCT
ejpam-4274	679	69	respectively	respectively	ADV
ejpam-4274	679	70	,	,	PUNCT
ejpam-4274	679	71	such	such	ADJ
ejpam-4274	679	72	that	that	DET
ejpam-4274	679	73	f(g	f(g	NOUN
ejpam-4274	679	74	)	)	PUNCT
ejpam-4274	679	75	⊆	⊆	NUM
ejpam-4274	679	76	u	u	NOUN
ejpam-4274	679	77	(	(	PUNCT
ejpam-4274	679	78	λ	λ	PROPN
ejpam-4274	679	79	,	,	PUNCT
ejpam-4274	679	80	p	p	NOUN
ejpam-4274	679	81	)	)	PUNCT
ejpam-4274	679	82	and	and	CCONJ
ejpam-4274	679	83	f(w	f(w	PROPN
ejpam-4274	679	84	)	)	PUNCT
ejpam-4274	679	85	⊆	⊆	NUM
ejpam-4274	679	86	v	v	X
ejpam-4274	679	87	(	(	PUNCT
ejpam-4274	679	88	λ	λ	PROPN
ejpam-4274	679	89	,	,	PUNCT
ejpam-4274	679	90	p	p	NOUN
ejpam-4274	679	91	)	)	PUNCT
ejpam-4274	679	92	.	.	PUNCT
ejpam-4274	680	1	this	this	PRON
ejpam-4274	680	2	shows	show	VERB
ejpam-4274	680	3	that	that	SCONJ
ejpam-4274	680	4	g	g	NOUN
ejpam-4274	680	5	∩w	∩w	NOUN
ejpam-4274	680	6	=	=	PUNCT
ejpam-4274	680	7	∅.	∅.	VERB
ejpam-4274	680	8	thus	thus	ADV
ejpam-4274	680	9	,	,	PUNCT
ejpam-4274	680	10	(	(	PUNCT
ejpam-4274	680	11	x	x	X
ejpam-4274	680	12	,	,	PUNCT
ejpam-4274	680	13	τ	τ	X
ejpam-4274	680	14	)	)	PUNCT
ejpam-4274	680	15	is	be	AUX
ejpam-4274	680	16	λp	λp	PROPN
ejpam-4274	680	17	-	-	PUNCT
ejpam-4274	680	18	t2	t2	NOUN
ejpam-4274	680	19	.	.	PUNCT
ejpam-4274	681	1	theorem	theorem	NOUN
ejpam-4274	681	2	24	24	NUM
ejpam-4274	681	3	.	.	PUNCT
ejpam-4274	682	1	if	if	SCONJ
ejpam-4274	682	2	f	f	PROPN
ejpam-4274	682	3	:	:	PUNCT
ejpam-4274	682	4	(	(	PUNCT
ejpam-4274	682	5	x	x	X
ejpam-4274	682	6	,	,	PUNCT
ejpam-4274	682	7	τ	τ	X
ejpam-4274	682	8	)	)	PUNCT
ejpam-4274	682	9	→	→	SYM
ejpam-4274	682	10	(	(	PUNCT
ejpam-4274	682	11	y	y	PROPN
ejpam-4274	682	12	,	,	PUNCT
ejpam-4274	682	13	σ	σ	PROPN
ejpam-4274	682	14	)	)	PUNCT
ejpam-4274	682	15	is	be	AUX
ejpam-4274	682	16	weakly	weakly	ADJ
ejpam-4274	682	17	(	(	PUNCT
ejpam-4274	682	18	λ.p)-continuous	λ.p)-continuous	ADJ
ejpam-4274	682	19	and	and	CCONJ
ejpam-4274	682	20	(	(	PUNCT
ejpam-4274	682	21	y	y	PROPN
ejpam-4274	682	22	,	,	PUNCT
ejpam-4274	682	23	σ	σ	PROPN
ejpam-4274	682	24	)	)	PUNCT
ejpam-4274	682	25	is	be	AUX
ejpam-4274	682	26	λp	λp	PROPN
ejpam-4274	682	27	-	-	PUNCT
ejpam-4274	682	28	t2	t2	NOUN
ejpam-4274	682	29	,	,	PUNCT
ejpam-4274	682	30	then	then	ADV
ejpam-4274	682	31	f	f	PROPN
ejpam-4274	682	32	has	have	VERB
ejpam-4274	682	33	(	(	PUNCT
ejpam-4274	682	34	λ	λ	X
ejpam-4274	682	35	,	,	PUNCT
ejpam-4274	682	36	p)-closed	p)-close	VERB
ejpam-4274	682	37	point	point	NOUN
ejpam-4274	682	38	inverses	inverse	NOUN
ejpam-4274	682	39	.	.	PUNCT
ejpam-4274	683	1	proof	proof	NOUN
ejpam-4274	683	2	.	.	PUNCT
ejpam-4274	684	1	let	let	VERB
ejpam-4274	684	2	y	y	PROPN
ejpam-4274	684	3	∈	∈	PROPN
ejpam-4274	684	4	y	y	PROPN
ejpam-4274	684	5	.	.	PUNCT
ejpam-4274	685	1	we	we	PRON
ejpam-4274	685	2	show	show	VERB
ejpam-4274	685	3	that	that	SCONJ
ejpam-4274	685	4	f−1(y	f−1(y	PROPN
ejpam-4274	685	5	)	)	PUNCT
ejpam-4274	686	1	=	=	PRON
ejpam-4274	686	2	{	{	PUNCT
ejpam-4274	686	3	x	x	PUNCT
ejpam-4274	686	4	∈	∈	PROPN
ejpam-4274	686	5	x	x	X
ejpam-4274	686	6	|	|	ADV
ejpam-4274	686	7	f(x	f(x	PROPN
ejpam-4274	686	8	)	)	PUNCT
ejpam-4274	686	9	=	=	SYM
ejpam-4274	687	1	y	y	X
ejpam-4274	687	2	}	}	PUNCT
ejpam-4274	687	3	is	be	AUX
ejpam-4274	687	4	(	(	PUNCT
ejpam-4274	687	5	λ	λ	X
ejpam-4274	687	6	,	,	PUNCT
ejpam-4274	687	7	p)-closed	p)-close	VERB
ejpam-4274	687	8	,	,	PUNCT
ejpam-4274	687	9	or	or	CCONJ
ejpam-4274	687	10	equivalently	equivalently	ADV
ejpam-4274	687	11	g	g	NOUN
ejpam-4274	687	12	=	=	SYM
ejpam-4274	687	13	{	{	PUNCT
ejpam-4274	687	14	x	x	PUNCT
ejpam-4274	687	15	∈	∈	PROPN
ejpam-4274	687	16	x	x	X
ejpam-4274	687	17	|	|	ADV
ejpam-4274	687	18	f(x	f(x	PROPN
ejpam-4274	687	19	)	)	PUNCT
ejpam-4274	687	20	6=	6=	ADP
ejpam-4274	688	1	y	y	PROPN
ejpam-4274	688	2	}	}	PUNCT
ejpam-4274	688	3	is	be	AUX
ejpam-4274	688	4	(	(	PUNCT
ejpam-4274	688	5	λ	λ	X
ejpam-4274	688	6	,	,	PUNCT
ejpam-4274	688	7	p)-open	p)-open	ADJ
ejpam-4274	688	8	.	.	PUNCT
ejpam-4274	689	1	let	let	VERB
ejpam-4274	689	2	x	x	SYM
ejpam-4274	689	3	∈	∈	PROPN
ejpam-4274	689	4	g.	g.	NOUN
ejpam-4274	689	5	since	since	SCONJ
ejpam-4274	689	6	f(x	f(x	PROPN
ejpam-4274	689	7	)	)	PUNCT
ejpam-4274	689	8	6=	6=	ADP
ejpam-4274	690	1	y	y	PROPN
ejpam-4274	690	2	and	and	CCONJ
ejpam-4274	690	3	(	(	PUNCT
ejpam-4274	690	4	y	y	PROPN
ejpam-4274	690	5	,	,	PUNCT
ejpam-4274	690	6	σ	σ	PROPN
ejpam-4274	690	7	)	)	PUNCT
ejpam-4274	690	8	is	be	AUX
ejpam-4274	690	9	λp	λp	PROPN
ejpam-4274	690	10	-	-	PUNCT
ejpam-4274	690	11	t2	t2	NOUN
ejpam-4274	690	12	,	,	PUNCT
ejpam-4274	690	13	there	there	PRON
ejpam-4274	690	14	exist	exist	VERB
ejpam-4274	690	15	disjoint	disjoint	NOUN
ejpam-4274	690	16	(	(	PUNCT
ejpam-4274	690	17	λ	λ	NOUN
ejpam-4274	690	18	,	,	PUNCT
ejpam-4274	690	19	p)-open	p)-open	VERB
ejpam-4274	690	20	sets	set	VERB
ejpam-4274	690	21	u	u	NOUN
ejpam-4274	690	22	and	and	CCONJ
ejpam-4274	690	23	v	v	ADP
ejpam-4274	690	24	such	such	ADJ
ejpam-4274	690	25	that	that	DET
ejpam-4274	690	26	f(x	f(x	PROPN
ejpam-4274	690	27	)	)	PUNCT
ejpam-4274	690	28	∈	∈	PROPN
ejpam-4274	690	29	u	u	NOUN
ejpam-4274	690	30	and	and	CCONJ
ejpam-4274	690	31	y	y	PROPN
ejpam-4274	690	32	∈	∈	PROPN
ejpam-4274	690	33	v	v	NOUN
ejpam-4274	690	34	.	.	PUNCT
ejpam-4274	691	1	since	since	SCONJ
ejpam-4274	691	2	u	u	PRON
ejpam-4274	691	3	∩v	∩v	NOUN
ejpam-4274	691	4	=	=	SYM
ejpam-4274	691	5	∅	∅	NOUN
ejpam-4274	691	6	,	,	PUNCT
ejpam-4274	691	7	u	u	NOUN
ejpam-4274	691	8	(	(	PUNCT
ejpam-4274	691	9	λ	λ	PROPN
ejpam-4274	691	10	,	,	PUNCT
ejpam-4274	691	11	p)∩v	p)∩v	NOUN
ejpam-4274	691	12	=	=	SYM
ejpam-4274	691	13	∅	∅	NOUN
ejpam-4274	691	14	and	and	CCONJ
ejpam-4274	691	15	hence	hence	ADV
ejpam-4274	691	16	y	y	PROPN
ejpam-4274	691	17	6∈	6∈	PROPN
ejpam-4274	691	18	u	u	PROPN
ejpam-4274	691	19	(	(	PUNCT
ejpam-4274	691	20	λ	λ	PROPN
ejpam-4274	691	21	,	,	PUNCT
ejpam-4274	691	22	p	p	NOUN
ejpam-4274	691	23	)	)	PUNCT
ejpam-4274	691	24	.	.	PUNCT
ejpam-4274	692	1	since	since	SCONJ
ejpam-4274	692	2	f	f	PROPN
ejpam-4274	692	3	is	be	AUX
ejpam-4274	692	4	weakly	weakly	ADJ
ejpam-4274	692	5	(	(	PUNCT
ejpam-4274	692	6	λ	λ	NOUN
ejpam-4274	692	7	,	,	PUNCT
ejpam-4274	692	8	p)-continuous	p)-continuous	ADJ
ejpam-4274	692	9	,	,	PUNCT
ejpam-4274	692	10	there	there	PRON
ejpam-4274	692	11	exists	exist	VERB
ejpam-4274	692	12	a	a	DET
ejpam-4274	692	13	(	(	PUNCT
ejpam-4274	692	14	λ	λ	NOUN
ejpam-4274	692	15	,	,	PUNCT
ejpam-4274	692	16	p)-open	p)-open	VERB
ejpam-4274	692	17	set	set	VERB
ejpam-4274	692	18	w	w	NOUN
ejpam-4274	692	19	containing	contain	VERB
ejpam-4274	692	20	x	x	PUNCT
ejpam-4274	692	21	such	such	ADJ
ejpam-4274	692	22	that	that	SCONJ
ejpam-4274	692	23	f(w	f(w	PROPN
ejpam-4274	692	24	)	)	PUNCT
ejpam-4274	693	1	⊆	⊆	NUM
ejpam-4274	693	2	u	u	NOUN
ejpam-4274	693	3	(	(	PUNCT
ejpam-4274	693	4	λ	λ	PROPN
ejpam-4274	693	5	,	,	PUNCT
ejpam-4274	693	6	p	p	NOUN
ejpam-4274	693	7	)	)	PUNCT
ejpam-4274	693	8	.	.	PUNCT
ejpam-4274	694	1	now	now	ADV
ejpam-4274	694	2	,	,	PUNCT
ejpam-4274	694	3	suppose	suppose	VERB
ejpam-4274	694	4	that	that	SCONJ
ejpam-4274	694	5	w	w	NOUN
ejpam-4274	694	6	is	be	AUX
ejpam-4274	694	7	not	not	PART
ejpam-4274	694	8	contained	contain	VERB
ejpam-4274	694	9	in	in	ADP
ejpam-4274	694	10	g.	g.	PROPN
ejpam-4274	694	11	then	then	ADV
ejpam-4274	694	12	,	,	PUNCT
ejpam-4274	694	13	there	there	PRON
ejpam-4274	694	14	exists	exist	VERB
ejpam-4274	694	15	a	a	DET
ejpam-4274	694	16	point	point	NOUN
ejpam-4274	694	17	z	z	NOUN
ejpam-4274	694	18	∈	∈	PROPN
ejpam-4274	694	19	w	w	ADP
ejpam-4274	694	20	such	such	ADJ
ejpam-4274	694	21	that	that	DET
ejpam-4274	694	22	f(z	f(z	PROPN
ejpam-4274	694	23	)	)	PUNCT
ejpam-4274	694	24	=	=	PUNCT
ejpam-4274	695	1	y.	y.	NOUN
ejpam-4274	695	2	since	since	SCONJ
ejpam-4274	695	3	f(w	f(w	PROPN
ejpam-4274	695	4	)	)	PUNCT
ejpam-4274	695	5	⊆	⊆	NUM
ejpam-4274	695	6	u	u	NOUN
ejpam-4274	695	7	(	(	PUNCT
ejpam-4274	695	8	λ	λ	PROPN
ejpam-4274	695	9	,	,	PUNCT
ejpam-4274	695	10	p	p	NOUN
ejpam-4274	695	11	)	)	PUNCT
ejpam-4274	695	12	,	,	PUNCT
ejpam-4274	695	13	y	y	PROPN
ejpam-4274	695	14	=	=	PUNCT
ejpam-4274	695	15	f(z	f(z	PROPN
ejpam-4274	695	16	)	)	PUNCT
ejpam-4274	695	17	∈	∈	PROPN
ejpam-4274	695	18	u	u	NOUN
ejpam-4274	695	19	(	(	PUNCT
ejpam-4274	695	20	λ	λ	PROPN
ejpam-4274	695	21	,	,	PUNCT
ejpam-4274	695	22	p	p	NOUN
ejpam-4274	695	23	)	)	PUNCT
ejpam-4274	695	24	.	.	PUNCT
ejpam-4274	696	1	this	this	PRON
ejpam-4274	696	2	is	be	AUX
ejpam-4274	696	3	a	a	DET
ejpam-4274	696	4	contradiction	contradiction	NOUN
ejpam-4274	696	5	.	.	PUNCT
ejpam-4274	697	1	therefore	therefore	ADV
ejpam-4274	697	2	,	,	PUNCT
ejpam-4274	697	3	w	w	ADP
ejpam-4274	697	4	⊆	⊆	NUM
ejpam-4274	697	5	g	g	NOUN
ejpam-4274	697	6	and	and	CCONJ
ejpam-4274	697	7	by	by	ADP
ejpam-4274	697	8	lemma	lemma	PROPN
ejpam-4274	697	9	9	9	NUM
ejpam-4274	697	10	,	,	PUNCT
ejpam-4274	697	11	g	g	NOUN
ejpam-4274	697	12	is	be	AUX
ejpam-4274	697	13	(	(	PUNCT
ejpam-4274	697	14	λ	λ	INTJ
ejpam-4274	697	15	,	,	PUNCT
ejpam-4274	697	16	p)-open	p)-open	NOUN
ejpam-4274	697	17	.	.	PUNCT
ejpam-4274	698	1	theorem	theorem	PROPN
ejpam-4274	698	2	25	25	NUM
ejpam-4274	698	3	.	.	PUNCT
ejpam-4274	699	1	let	let	AUX
ejpam-4274	699	2	(	(	PUNCT
ejpam-4274	699	3	x	x	NOUN
ejpam-4274	699	4	,	,	PUNCT
ejpam-4274	699	5	τ	τ	X
ejpam-4274	699	6	)	)	PUNCT
ejpam-4274	699	7	be	be	VERB
ejpam-4274	699	8	a	a	DET
ejpam-4274	699	9	topological	topological	ADJ
ejpam-4274	699	10	space	space	NOUN
ejpam-4274	699	11	.	.	PUNCT
ejpam-4274	700	1	if	if	SCONJ
ejpam-4274	700	2	for	for	ADP
ejpam-4274	700	3	each	each	DET
ejpam-4274	700	4	pair	pair	NOUN
ejpam-4274	700	5	of	of	ADP
ejpam-4274	700	6	distinct	distinct	ADJ
ejpam-4274	700	7	points	point	NOUN
ejpam-4274	700	8	x1	x1	PROPN
ejpam-4274	700	9	and	and	CCONJ
ejpam-4274	700	10	x2	x2	PROPN
ejpam-4274	700	11	in	in	ADP
ejpam-4274	700	12	x	x	NOUN
ejpam-4274	700	13	,	,	PUNCT
ejpam-4274	700	14	there	there	PRON
ejpam-4274	700	15	exists	exist	VERB
ejpam-4274	700	16	a	a	DET
ejpam-4274	700	17	function	function	NOUN
ejpam-4274	700	18	f	f	NOUN
ejpam-4274	700	19	:	:	PUNCT
ejpam-4274	700	20	(	(	PUNCT
ejpam-4274	700	21	x	x	X
ejpam-4274	700	22	,	,	PUNCT
ejpam-4274	700	23	τ	τ	X
ejpam-4274	700	24	)	)	PUNCT
ejpam-4274	700	25	→	→	SYM
ejpam-4274	700	26	(	(	PUNCT
ejpam-4274	700	27	y	y	PROPN
ejpam-4274	700	28	,	,	PUNCT
ejpam-4274	700	29	σ	σ	PROPN
ejpam-4274	700	30	)	)	PUNCT
ejpam-4274	700	31	such	such	ADJ
ejpam-4274	700	32	that	that	SCONJ
ejpam-4274	700	33	(	(	PUNCT
ejpam-4274	700	34	1	1	NUM
ejpam-4274	700	35	)	)	PUNCT
ejpam-4274	700	36	(	(	PUNCT
ejpam-4274	700	37	y	y	PROPN
ejpam-4274	700	38	,	,	PUNCT
ejpam-4274	700	39	σ	σ	PROPN
ejpam-4274	700	40	)	)	PUNCT
ejpam-4274	700	41	is	be	AUX
ejpam-4274	700	42	λp	λp	NOUN
ejpam-4274	700	43	-	-	ADJ
ejpam-4274	700	44	urysohn	urysohn	ADJ
ejpam-4274	700	45	,	,	PUNCT
ejpam-4274	700	46	(	(	PUNCT
ejpam-4274	700	47	2	2	NUM
ejpam-4274	700	48	)	)	PUNCT
ejpam-4274	700	49	f(x1	f(x1	NOUN
ejpam-4274	700	50	)	)	PUNCT
ejpam-4274	700	51	6=	6=	ADP
ejpam-4274	700	52	f(x2	f(x2	NOUN
ejpam-4274	700	53	)	)	PUNCT
ejpam-4274	700	54	and	and	CCONJ
ejpam-4274	700	55	(	(	PUNCT
ejpam-4274	700	56	3	3	X
ejpam-4274	700	57	)	)	PUNCT
ejpam-4274	700	58	f	f	PROPN
ejpam-4274	700	59	is	be	AUX
ejpam-4274	700	60	weakly	weakly	ADJ
ejpam-4274	700	61	(	(	PUNCT
ejpam-4274	700	62	λ	λ	X
ejpam-4274	700	63	,	,	PUNCT
ejpam-4274	700	64	p)-continuous	p)-continuous	ADJ
ejpam-4274	700	65	at	at	ADP
ejpam-4274	700	66	x1	x1	PROPN
ejpam-4274	700	67	and	and	CCONJ
ejpam-4274	700	68	x2	x2	PROPN
ejpam-4274	700	69	,	,	PUNCT
ejpam-4274	700	70	then	then	ADV
ejpam-4274	700	71	(	(	PUNCT
ejpam-4274	700	72	x	x	X
ejpam-4274	700	73	,	,	PUNCT
ejpam-4274	700	74	τ	τ	X
ejpam-4274	700	75	)	)	PUNCT
ejpam-4274	700	76	is	be	AUX
ejpam-4274	700	77	λp	λp	PROPN
ejpam-4274	700	78	-	-	PUNCT
ejpam-4274	700	79	t2	t2	NOUN
ejpam-4274	700	80	.	.	PUNCT
ejpam-4274	701	1	proof	proof	NOUN
ejpam-4274	701	2	.	.	PUNCT
ejpam-4274	702	1	let	let	VERB
ejpam-4274	702	2	x1	x1	NUM
ejpam-4274	702	3	,	,	PUNCT
ejpam-4274	702	4	x2	x2	PRON
ejpam-4274	702	5	be	be	VERB
ejpam-4274	702	6	any	any	DET
ejpam-4274	702	7	distinct	distinct	ADJ
ejpam-4274	702	8	points	point	NOUN
ejpam-4274	702	9	of	of	ADP
ejpam-4274	702	10	x.	x.	NOUN
ejpam-4274	702	11	by	by	ADP
ejpam-4274	702	12	the	the	DET
ejpam-4274	702	13	hypothesis	hypothesis	NOUN
ejpam-4274	702	14	,	,	PUNCT
ejpam-4274	702	15	there	there	PRON
ejpam-4274	702	16	exists	exist	VERB
ejpam-4274	702	17	a	a	DET
ejpam-4274	702	18	function	function	NOUN
ejpam-4274	702	19	f	f	NOUN
ejpam-4274	702	20	:	:	PUNCT
ejpam-4274	702	21	(	(	PUNCT
ejpam-4274	702	22	x	x	X
ejpam-4274	702	23	,	,	PUNCT
ejpam-4274	702	24	τ	τ	X
ejpam-4274	702	25	)	)	PUNCT
ejpam-4274	702	26	→	→	SYM
ejpam-4274	702	27	(	(	PUNCT
ejpam-4274	702	28	y	y	PROPN
ejpam-4274	702	29	,	,	PUNCT
ejpam-4274	702	30	σ	σ	PROPN
ejpam-4274	702	31	)	)	PUNCT
ejpam-4274	702	32	which	which	PRON
ejpam-4274	702	33	satisfies	satisfy	VERB
ejpam-4274	702	34	the	the	DET
ejpam-4274	702	35	conditions	condition	NOUN
ejpam-4274	702	36	(	(	PUNCT
ejpam-4274	702	37	1	1	NUM
ejpam-4274	702	38	)	)	PUNCT
ejpam-4274	702	39	,	,	PUNCT
ejpam-4274	702	40	(	(	PUNCT
ejpam-4274	702	41	2	2	X
ejpam-4274	702	42	)	)	PUNCT
ejpam-4274	702	43	and	and	CCONJ
ejpam-4274	702	44	(	(	PUNCT
ejpam-4274	702	45	3	3	NUM
ejpam-4274	702	46	)	)	PUNCT
ejpam-4274	702	47	.	.	PUNCT
ejpam-4274	703	1	let	let	VERB
ejpam-4274	703	2	yi	yi	X
ejpam-4274	703	3	=	=	SYM
ejpam-4274	703	4	f(xi	f(xi	PROPN
ejpam-4274	703	5	)	)	PUNCT
ejpam-4274	703	6	for	for	ADP
ejpam-4274	703	7	i	i	PROPN
ejpam-4274	703	8	=	=	NOUN
ejpam-4274	703	9	1	1	NUM
ejpam-4274	703	10	,	,	PUNCT
ejpam-4274	703	11	2	2	NUM
ejpam-4274	703	12	.	.	PUNCT
ejpam-4274	704	1	then	then	ADV
ejpam-4274	704	2	,	,	PUNCT
ejpam-4274	704	3	y1	y1	INTJ
ejpam-4274	704	4	6=	6=	NUM
ejpam-4274	704	5	y2	y2	PROPN
ejpam-4274	704	6	.	.	PUNCT
ejpam-4274	705	1	since	since	SCONJ
ejpam-4274	705	2	(	(	PUNCT
ejpam-4274	705	3	y	y	PROPN
ejpam-4274	705	4	,	,	PUNCT
ejpam-4274	705	5	σ	σ	PROPN
ejpam-4274	705	6	)	)	PUNCT
ejpam-4274	705	7	is	be	AUX
ejpam-4274	705	8	λp	λp	NOUN
ejpam-4274	705	9	-	-	ADJ
ejpam-4274	705	10	urysohn	urysohn	ADJ
ejpam-4274	705	11	,	,	PUNCT
ejpam-4274	705	12	there	there	PRON
ejpam-4274	705	13	exist	exist	VERB
ejpam-4274	705	14	(	(	PUNCT
ejpam-4274	705	15	λ	λ	X
ejpam-4274	705	16	,	,	PUNCT
ejpam-4274	705	17	p)-open	p)-open	VERB
ejpam-4274	705	18	sets	set	VERB
ejpam-4274	705	19	vi	vi	NOUN
ejpam-4274	705	20	in	in	ADP
ejpam-4274	705	21	(	(	PUNCT
ejpam-4274	705	22	y	y	PROPN
ejpam-4274	705	23	,	,	PUNCT
ejpam-4274	705	24	σ	σ	PROPN
ejpam-4274	705	25	)	)	PUNCT
ejpam-4274	705	26	containing	contain	VERB
ejpam-4274	705	27	yi	yi	PRON
ejpam-4274	705	28	such	such	ADJ
ejpam-4274	705	29	that	that	DET
ejpam-4274	705	30	v	v	NOUN
ejpam-4274	705	31	(	(	PUNCT
ejpam-4274	705	32	λ	λ	PROPN
ejpam-4274	705	33	,	,	PUNCT
ejpam-4274	705	34	p	p	NOUN
ejpam-4274	705	35	)	)	PUNCT
ejpam-4274	705	36	1	1	NUM
ejpam-4274	705	37	∩	∩	X
ejpam-4274	705	38	v	v	NOUN
ejpam-4274	705	39	(	(	PUNCT
ejpam-4274	705	40	λ	λ	PROPN
ejpam-4274	705	41	,	,	PUNCT
ejpam-4274	705	42	p	p	NOUN
ejpam-4274	705	43	)	)	PUNCT
ejpam-4274	705	44	2	2	NUM
ejpam-4274	705	45	=	=	PUNCT
ejpam-4274	705	46	∅.	∅.	NOUN
ejpam-4274	705	47	since	since	SCONJ
ejpam-4274	705	48	f	f	PROPN
ejpam-4274	705	49	is	be	AUX
ejpam-4274	705	50	weakly	weakly	ADJ
ejpam-4274	705	51	(	(	PUNCT
ejpam-4274	705	52	λ	λ	X
ejpam-4274	705	53	,	,	PUNCT
ejpam-4274	705	54	p)-continuous	p)-continuous	ADJ
ejpam-4274	705	55	at	at	ADP
ejpam-4274	705	56	x1	x1	PROPN
ejpam-4274	705	57	and	and	CCONJ
ejpam-4274	705	58	x2	x2	PROPN
ejpam-4274	705	59	,	,	PUNCT
ejpam-4274	705	60	for	for	ADP
ejpam-4274	705	61	i	i	PROPN
ejpam-4274	705	62	=	=	SYM
ejpam-4274	705	63	1	1	NUM
ejpam-4274	705	64	,	,	PUNCT
ejpam-4274	705	65	2	2	NUM
ejpam-4274	705	66	,	,	PUNCT
ejpam-4274	705	67	there	there	PRON
ejpam-4274	705	68	exist	exist	VERB
ejpam-4274	705	69	(	(	PUNCT
ejpam-4274	705	70	λ	λ	X
ejpam-4274	705	71	,	,	PUNCT
ejpam-4274	705	72	p)-open	p)-open	VERB
ejpam-4274	705	73	sets	set	VERB
ejpam-4274	705	74	ui	ui	NOUN
ejpam-4274	705	75	in	in	ADP
ejpam-4274	705	76	(	(	PUNCT
ejpam-4274	705	77	x	x	NOUN
ejpam-4274	705	78	,	,	PUNCT
ejpam-4274	705	79	τ	τ	X
ejpam-4274	705	80	)	)	PUNCT
ejpam-4274	705	81	containing	contain	VERB
ejpam-4274	705	82	xi	xi	ADP
ejpam-4274	705	83	such	such	ADJ
ejpam-4274	705	84	that	that	PRON
ejpam-4274	705	85	f(ui	f(ui	PROPN
ejpam-4274	705	86	)	)	PUNCT
ejpam-4274	705	87	⊆	⊆	NUM
ejpam-4274	705	88	v	v	NOUN
ejpam-4274	705	89	(	(	PUNCT
ejpam-4274	705	90	λ	λ	PROPN
ejpam-4274	705	91	,	,	PUNCT
ejpam-4274	705	92	p	p	NOUN
ejpam-4274	705	93	)	)	PUNCT
ejpam-4274	705	94	i	i	PRON
ejpam-4274	705	95	.	.	PUNCT
ejpam-4274	706	1	hence	hence	ADV
ejpam-4274	706	2	,	,	PUNCT
ejpam-4274	706	3	we	we	PRON
ejpam-4274	706	4	get	get	VERB
ejpam-4274	706	5	u1	u1	NOUN
ejpam-4274	706	6	∩	∩	NOUN
ejpam-4274	706	7	u2	u2	NOUN
ejpam-4274	706	8	=	=	PUNCT
ejpam-4274	706	9	∅.	∅.	NOUN
ejpam-4274	706	10	this	this	PRON
ejpam-4274	706	11	shows	show	VERB
ejpam-4274	706	12	that	that	SCONJ
ejpam-4274	706	13	(	(	PUNCT
ejpam-4274	706	14	x	x	X
ejpam-4274	706	15	,	,	PUNCT
ejpam-4274	706	16	τ	τ	X
ejpam-4274	706	17	)	)	PUNCT
ejpam-4274	706	18	is	be	AUX
ejpam-4274	706	19	λp	λp	PROPN
ejpam-4274	706	20	-	-	PUNCT
ejpam-4274	706	21	t2	t2	NOUN
ejpam-4274	706	22	.	.	PUNCT
ejpam-4274	707	1	c.	c.	PROPN
ejpam-4274	707	2	boonpok	boonpok	PROPN
ejpam-4274	707	3	,	,	PUNCT
ejpam-4274	707	4	c.	c.	PROPN
ejpam-4274	707	5	viriyapong	viriyapong	PROPN
ejpam-4274	707	6	/	/	SYM
ejpam-4274	707	7	eur	eur	PROPN
ejpam-4274	707	8	.	.	PUNCT
ejpam-4274	708	1	j.	j.	PROPN
ejpam-4274	708	2	pure	pure	PROPN
ejpam-4274	708	3	appl	appl	PROPN
ejpam-4274	708	4	.	.	PROPN
ejpam-4274	708	5	math	math	PROPN
ejpam-4274	708	6	,	,	PUNCT
ejpam-4274	708	7	15	15	NUM
ejpam-4274	708	8	(	(	PUNCT
ejpam-4274	708	9	2	2	NUM
ejpam-4274	708	10	)	)	PUNCT
ejpam-4274	708	11	(	(	PUNCT
ejpam-4274	708	12	2022	2022	NUM
ejpam-4274	708	13	)	)	PUNCT
ejpam-4274	708	14	,	,	PUNCT
ejpam-4274	708	15	415	415	NUM
ejpam-4274	708	16	-	-	SYM
ejpam-4274	708	17	436	436	NUM
ejpam-4274	708	18	433	433	NUM
ejpam-4274	708	19	corollary	corollary	ADJ
ejpam-4274	708	20	4	4	NUM
ejpam-4274	708	21	.	.	PUNCT
ejpam-4274	709	1	if	if	SCONJ
ejpam-4274	709	2	f	f	PROPN
ejpam-4274	709	3	:	:	PUNCT
ejpam-4274	709	4	(	(	PUNCT
ejpam-4274	709	5	x	x	X
ejpam-4274	709	6	,	,	PUNCT
ejpam-4274	709	7	τ	τ	X
ejpam-4274	709	8	)	)	PUNCT
ejpam-4274	709	9	→	→	SYM
ejpam-4274	709	10	(	(	PUNCT
ejpam-4274	709	11	y	y	PROPN
ejpam-4274	709	12	,	,	PUNCT
ejpam-4274	709	13	σ	σ	PROPN
ejpam-4274	709	14	)	)	PUNCT
ejpam-4274	709	15	is	be	AUX
ejpam-4274	709	16	a	a	DET
ejpam-4274	709	17	weakly	weakly	ADJ
ejpam-4274	709	18	(	(	PUNCT
ejpam-4274	709	19	λ	λ	NOUN
ejpam-4274	709	20	,	,	PUNCT
ejpam-4274	709	21	p)-continuous	p)-continuous	ADJ
ejpam-4274	709	22	injection	injection	NOUN
ejpam-4274	709	23	and	and	CCONJ
ejpam-4274	709	24	(	(	PUNCT
ejpam-4274	709	25	y	y	PROPN
ejpam-4274	709	26	,	,	PUNCT
ejpam-4274	709	27	σ	σ	PROPN
ejpam-4274	709	28	)	)	PUNCT
ejpam-4274	709	29	is	be	AUX
ejpam-4274	709	30	λp	λp	NOUN
ejpam-4274	709	31	-	-	ADJ
ejpam-4274	709	32	urysohn	urysohn	ADJ
ejpam-4274	709	33	,	,	PUNCT
ejpam-4274	709	34	then	then	ADV
ejpam-4274	709	35	(	(	PUNCT
ejpam-4274	709	36	x	x	X
ejpam-4274	709	37	,	,	PUNCT
ejpam-4274	709	38	τ	τ	X
ejpam-4274	709	39	)	)	PUNCT
ejpam-4274	709	40	is	be	AUX
ejpam-4274	709	41	λp	λp	PROPN
ejpam-4274	709	42	-	-	PUNCT
ejpam-4274	709	43	t2	t2	NOUN
ejpam-4274	709	44	.	.	PUNCT
ejpam-4274	710	1	definition	definition	NOUN
ejpam-4274	710	2	15	15	NUM
ejpam-4274	710	3	.	.	PUNCT
ejpam-4274	711	1	let	let	VERB
ejpam-4274	711	2	a	a	DET
ejpam-4274	711	3	be	be	AUX
ejpam-4274	711	4	a	a	DET
ejpam-4274	711	5	subset	subset	NOUN
ejpam-4274	711	6	of	of	ADP
ejpam-4274	711	7	a	a	DET
ejpam-4274	711	8	topological	topological	ADJ
ejpam-4274	711	9	space	space	NOUN
ejpam-4274	711	10	(	(	PUNCT
ejpam-4274	711	11	x	x	X
ejpam-4274	711	12	,	,	PUNCT
ejpam-4274	711	13	τ	τ	PROPN
ejpam-4274	711	14	)	)	PUNCT
ejpam-4274	711	15	.	.	PUNCT
ejpam-4274	712	1	the	the	DET
ejpam-4274	712	2	θ(λ	θ(λ	PROPN
ejpam-4274	712	3	,	,	PUNCT
ejpam-4274	712	4	p)-closure	p)-closure	NOUN
ejpam-4274	712	5	of	of	ADP
ejpam-4274	712	6	a	a	DET
ejpam-4274	712	7	,	,	PUNCT
ejpam-4274	712	8	aθ(λ	aθ(λ	ADJ
ejpam-4274	712	9	,	,	PUNCT
ejpam-4274	712	10	p	p	NOUN
ejpam-4274	712	11	)	)	PUNCT
ejpam-4274	712	12	,	,	PUNCT
ejpam-4274	712	13	is	be	AUX
ejpam-4274	712	14	defined	define	VERB
ejpam-4274	712	15	as	as	SCONJ
ejpam-4274	712	16	follows	follow	VERB
ejpam-4274	712	17	:	:	PUNCT
ejpam-4274	712	18	aθ(λ	aθ(λ	NOUN
ejpam-4274	712	19	,	,	PUNCT
ejpam-4274	712	20	p	p	NOUN
ejpam-4274	712	21	)	)	PUNCT
ejpam-4274	712	22	=	=	SYM
ejpam-4274	713	1	{	{	PUNCT
ejpam-4274	713	2	x	x	PUNCT
ejpam-4274	713	3	∈	∈	NOUN
ejpam-4274	713	4	x	x	PUNCT
ejpam-4274	713	5	|	|	ADV
ejpam-4274	713	6	a	a	DET
ejpam-4274	713	7	∩	∩	ADJ
ejpam-4274	713	8	u	u	NOUN
ejpam-4274	713	9	(	(	PUNCT
ejpam-4274	713	10	λ	λ	PROPN
ejpam-4274	713	11	,	,	PUNCT
ejpam-4274	713	12	p	p	NOUN
ejpam-4274	713	13	)	)	PUNCT
ejpam-4274	713	14	6=	6=	ADP
ejpam-4274	713	15	∅	∅	NOUN
ejpam-4274	713	16	for	for	ADP
ejpam-4274	713	17	each	each	DET
ejpam-4274	713	18	(	(	PUNCT
ejpam-4274	713	19	λ	λ	PROPN
ejpam-4274	713	20	,	,	PUNCT
ejpam-4274	713	21	p)-open	p)-open	VERB
ejpam-4274	713	22	set	set	VERB
ejpam-4274	713	23	u	u	NOUN
ejpam-4274	713	24	containing	contain	VERB
ejpam-4274	713	25	x	x	X
ejpam-4274	713	26	}	}	PUNCT
ejpam-4274	713	27	.	.	PUNCT
ejpam-4274	714	1	a	a	DET
ejpam-4274	714	2	subset	subset	NOUN
ejpam-4274	714	3	a	a	PRON
ejpam-4274	714	4	of	of	ADP
ejpam-4274	714	5	a	a	DET
ejpam-4274	714	6	topological	topological	ADJ
ejpam-4274	714	7	space	space	NOUN
ejpam-4274	714	8	(	(	PUNCT
ejpam-4274	714	9	x	x	X
ejpam-4274	714	10	,	,	PUNCT
ejpam-4274	714	11	τ	τ	X
ejpam-4274	714	12	)	)	PUNCT
ejpam-4274	714	13	is	be	AUX
ejpam-4274	714	14	called	call	VERB
ejpam-4274	714	15	θ(λ	θ(λ	PROPN
ejpam-4274	714	16	,	,	PUNCT
ejpam-4274	714	17	p)-closed	p)-close	VERB
ejpam-4274	714	18	if	if	SCONJ
ejpam-4274	714	19	a	a	DET
ejpam-4274	714	20	=	=	NOUN
ejpam-4274	714	21	aθ(λ	aθ(λ	NOUN
ejpam-4274	714	22	,	,	PUNCT
ejpam-4274	714	23	p	p	NOUN
ejpam-4274	714	24	)	)	PUNCT
ejpam-4274	714	25	.	.	PUNCT
ejpam-4274	715	1	the	the	DET
ejpam-4274	715	2	complement	complement	NOUN
ejpam-4274	715	3	of	of	ADP
ejpam-4274	715	4	a	a	DET
ejpam-4274	715	5	θ(λ	θ(λ	PROPN
ejpam-4274	715	6	,	,	PUNCT
ejpam-4274	715	7	p)-closed	p)-close	VERB
ejpam-4274	715	8	set	set	NOUN
ejpam-4274	715	9	is	be	AUX
ejpam-4274	715	10	said	say	VERB
ejpam-4274	715	11	to	to	PART
ejpam-4274	715	12	be	be	AUX
ejpam-4274	715	13	θ(λ	θ(λ	PROPN
ejpam-4274	715	14	,	,	PUNCT
ejpam-4274	715	15	p)-open	p)-open	NOUN
ejpam-4274	715	16	.	.	PUNCT
ejpam-4274	716	1	lemma	lemma	PROPN
ejpam-4274	716	2	13	13	NUM
ejpam-4274	716	3	.	.	PUNCT
ejpam-4274	717	1	let	let	VERB
ejpam-4274	717	2	a	a	DET
ejpam-4274	717	3	be	be	AUX
ejpam-4274	717	4	a	a	DET
ejpam-4274	717	5	subset	subset	NOUN
ejpam-4274	717	6	of	of	ADP
ejpam-4274	717	7	a	a	DET
ejpam-4274	717	8	topological	topological	ADJ
ejpam-4274	717	9	space	space	NOUN
ejpam-4274	717	10	(	(	PUNCT
ejpam-4274	717	11	x	x	X
ejpam-4274	717	12	,	,	PUNCT
ejpam-4274	717	13	τ	τ	PROPN
ejpam-4274	717	14	)	)	PUNCT
ejpam-4274	717	15	.	.	PUNCT
ejpam-4274	718	1	then	then	ADV
ejpam-4274	718	2	,	,	PUNCT
ejpam-4274	718	3	x	x	PUNCT
ejpam-4274	718	4	∈	∈	PROPN
ejpam-4274	718	5	a(λ	a(λ	ADV
ejpam-4274	718	6	,	,	PUNCT
ejpam-4274	718	7	p	p	NOUN
ejpam-4274	718	8	)	)	PUNCT
ejpam-4274	718	9	if	if	SCONJ
ejpam-4274	718	10	and	and	CCONJ
ejpam-4274	718	11	only	only	ADV
ejpam-4274	718	12	if	if	SCONJ
ejpam-4274	718	13	u	u	PROPN
ejpam-4274	718	14	∩a	∩a	PROPN
ejpam-4274	718	15	6=	6=	ADP
ejpam-4274	718	16	∅	∅	NOUN
ejpam-4274	718	17	for	for	ADP
ejpam-4274	718	18	every	every	DET
ejpam-4274	718	19	(	(	PUNCT
ejpam-4274	718	20	λ	λ	NOUN
ejpam-4274	718	21	,	,	PUNCT
ejpam-4274	718	22	p)-open	p)-open	VERB
ejpam-4274	718	23	set	set	VERB
ejpam-4274	718	24	u	u	NOUN
ejpam-4274	718	25	containing	contain	VERB
ejpam-4274	718	26	x.	x.	NOUN
ejpam-4274	718	27	lemma	lemma	PROPN
ejpam-4274	718	28	14	14	NUM
ejpam-4274	718	29	.	.	PUNCT
ejpam-4274	719	1	for	for	ADP
ejpam-4274	719	2	a	a	DET
ejpam-4274	719	3	subset	subset	NOUN
ejpam-4274	719	4	a	a	PRON
ejpam-4274	719	5	of	of	ADP
ejpam-4274	719	6	a	a	DET
ejpam-4274	719	7	topological	topological	ADJ
ejpam-4274	719	8	space	space	NOUN
ejpam-4274	719	9	(	(	PUNCT
ejpam-4274	719	10	x	x	X
ejpam-4274	719	11	,	,	PUNCT
ejpam-4274	719	12	τ	τ	PROPN
ejpam-4274	719	13	)	)	PUNCT
ejpam-4274	719	14	,	,	PUNCT
ejpam-4274	719	15	the	the	DET
ejpam-4274	719	16	following	follow	VERB
ejpam-4274	719	17	properties	property	NOUN
ejpam-4274	719	18	hold	hold	VERB
ejpam-4274	719	19	:	:	PUNCT
ejpam-4274	719	20	(	(	PUNCT
ejpam-4274	719	21	1	1	X
ejpam-4274	719	22	)	)	PUNCT
ejpam-4274	719	23	if	if	SCONJ
ejpam-4274	719	24	a	a	PRON
ejpam-4274	719	25	is	be	AUX
ejpam-4274	719	26	(	(	PUNCT
ejpam-4274	719	27	λ	λ	X
ejpam-4274	719	28	,	,	PUNCT
ejpam-4274	719	29	p)-open	p)-open	VERB
ejpam-4274	719	30	in	in	ADP
ejpam-4274	719	31	(	(	PUNCT
ejpam-4274	719	32	x	x	NOUN
ejpam-4274	719	33	,	,	PUNCT
ejpam-4274	719	34	τ	τ	PROPN
ejpam-4274	719	35	)	)	PUNCT
ejpam-4274	719	36	,	,	PUNCT
ejpam-4274	719	37	then	then	ADV
ejpam-4274	719	38	a(λ	a(λ	ADV
ejpam-4274	719	39	,	,	PUNCT
ejpam-4274	719	40	p	p	X
ejpam-4274	719	41	)	)	PUNCT
ejpam-4274	719	42	=	=	PUNCT
ejpam-4274	719	43	aθ(λ	aθ(λ	NOUN
ejpam-4274	719	44	,	,	PUNCT
ejpam-4274	719	45	p	p	NOUN
ejpam-4274	719	46	)	)	PUNCT
ejpam-4274	719	47	.	.	PUNCT
ejpam-4274	720	1	(	(	PUNCT
ejpam-4274	720	2	2	2	X
ejpam-4274	720	3	)	)	PUNCT
ejpam-4274	720	4	aθ(λ	aθ(λ	NOUN
ejpam-4274	720	5	,	,	PUNCT
ejpam-4274	720	6	p	p	NOUN
ejpam-4274	720	7	)	)	PUNCT
ejpam-4274	720	8	is	be	AUX
ejpam-4274	720	9	(	(	PUNCT
ejpam-4274	720	10	λ	λ	X
ejpam-4274	720	11	,	,	PUNCT
ejpam-4274	720	12	p)-closed	p)-close	VERB
ejpam-4274	720	13	for	for	ADP
ejpam-4274	720	14	every	every	DET
ejpam-4274	720	15	subset	subset	NOUN
ejpam-4274	720	16	a	a	PRON
ejpam-4274	720	17	of	of	ADP
ejpam-4274	720	18	x.	x.	NOUN
ejpam-4274	720	19	proof	proof	NOUN
ejpam-4274	720	20	.	.	PUNCT
ejpam-4274	721	1	(	(	PUNCT
ejpam-4274	721	2	1	1	X
ejpam-4274	721	3	)	)	PUNCT
ejpam-4274	721	4	in	in	ADP
ejpam-4274	721	5	general	general	ADJ
ejpam-4274	721	6	,	,	PUNCT
ejpam-4274	721	7	we	we	PRON
ejpam-4274	721	8	have	have	VERB
ejpam-4274	721	9	a(λ	a(λ	ADV
ejpam-4274	721	10	,	,	PUNCT
ejpam-4274	721	11	p	p	X
ejpam-4274	721	12	)	)	PUNCT
ejpam-4274	722	1	⊆	⊆	NUM
ejpam-4274	722	2	aθ(λ	aθ(λ	NOUN
ejpam-4274	722	3	,	,	PUNCT
ejpam-4274	722	4	p	p	NOUN
ejpam-4274	722	5	)	)	PUNCT
ejpam-4274	722	6	.	.	PUNCT
ejpam-4274	723	1	suppose	suppose	VERB
ejpam-4274	723	2	that	that	SCONJ
ejpam-4274	723	3	x	x	PROPN
ejpam-4274	723	4	6∈	6∈	NOUN
ejpam-4274	723	5	a(λ	a(λ	ADV
ejpam-4274	723	6	,	,	PUNCT
ejpam-4274	723	7	p	p	NOUN
ejpam-4274	723	8	)	)	PUNCT
ejpam-4274	723	9	.	.	PUNCT
ejpam-4274	724	1	by	by	ADP
ejpam-4274	724	2	lemma	lemma	PROPN
ejpam-4274	724	3	13	13	NUM
ejpam-4274	724	4	,	,	PUNCT
ejpam-4274	724	5	there	there	PRON
ejpam-4274	724	6	exists	exist	VERB
ejpam-4274	724	7	a	a	DET
ejpam-4274	724	8	(	(	PUNCT
ejpam-4274	724	9	λ	λ	NOUN
ejpam-4274	724	10	,	,	PUNCT
ejpam-4274	724	11	p)-open	p)-open	VERB
ejpam-4274	724	12	set	set	VERB
ejpam-4274	724	13	u	u	NOUN
ejpam-4274	724	14	containing	contain	VERB
ejpam-4274	724	15	x	x	PUNCT
ejpam-4274	724	16	such	such	ADJ
ejpam-4274	724	17	that	that	SCONJ
ejpam-4274	724	18	u	u	NOUN
ejpam-4274	724	19	∩a	∩a	NOUN
ejpam-4274	724	20	=	=	NOUN
ejpam-4274	724	21	∅	∅	NOUN
ejpam-4274	724	22	;	;	PUNCT
ejpam-4274	724	23	hence	hence	ADV
ejpam-4274	724	24	a∩u	a∩u	PROPN
ejpam-4274	724	25	(	(	PUNCT
ejpam-4274	724	26	λ	λ	X
ejpam-4274	724	27	,	,	PUNCT
ejpam-4274	724	28	p	p	NOUN
ejpam-4274	724	29	)	)	PUNCT
ejpam-4274	724	30	=	=	NOUN
ejpam-4274	724	31	∅	∅	NOUN
ejpam-4274	724	32	since	since	SCONJ
ejpam-4274	724	33	a	a	DET
ejpam-4274	724	34	is	is	NOUN
ejpam-4274	724	35	(	(	PUNCT
ejpam-4274	724	36	λ	λ	NOUN
ejpam-4274	724	37	,	,	PUNCT
ejpam-4274	724	38	p)-open	p)-open	ADJ
ejpam-4274	724	39	.	.	PUNCT
ejpam-4274	725	1	thus	thus	ADV
ejpam-4274	725	2	,	,	PUNCT
ejpam-4274	725	3	x	x	PROPN
ejpam-4274	725	4	6∈	6∈	NOUN
ejpam-4274	725	5	aθ(λ	aθ(λ	NOUN
ejpam-4274	725	6	,	,	PUNCT
ejpam-4274	725	7	p	p	NOUN
ejpam-4274	725	8	)	)	PUNCT
ejpam-4274	725	9	.	.	PUNCT
ejpam-4274	726	1	consequently	consequently	ADV
ejpam-4274	726	2	,	,	PUNCT
ejpam-4274	726	3	we	we	PRON
ejpam-4274	726	4	obtain	obtain	VERB
ejpam-4274	726	5	a(λ	a(λ	ADV
ejpam-4274	726	6	,	,	PUNCT
ejpam-4274	726	7	p	p	X
ejpam-4274	726	8	)	)	PUNCT
ejpam-4274	726	9	=	=	PUNCT
ejpam-4274	726	10	aθ(λ	aθ(λ	NOUN
ejpam-4274	726	11	,	,	PUNCT
ejpam-4274	726	12	p	p	NOUN
ejpam-4274	726	13	)	)	PUNCT
ejpam-4274	726	14	.	.	PUNCT
ejpam-4274	727	1	(	(	PUNCT
ejpam-4274	727	2	2	2	X
ejpam-4274	727	3	)	)	PUNCT
ejpam-4274	727	4	let	let	VERB
ejpam-4274	727	5	x	x	SYM
ejpam-4274	727	6	∈	∈	PROPN
ejpam-4274	727	7	x	x	X
ejpam-4274	727	8	−	−	NOUN
ejpam-4274	727	9	aθ(λ	aθ(λ	NOUN
ejpam-4274	727	10	,	,	PUNCT
ejpam-4274	727	11	p	p	NOUN
ejpam-4274	727	12	)	)	PUNCT
ejpam-4274	727	13	.	.	PUNCT
ejpam-4274	728	1	then	then	ADV
ejpam-4274	728	2	,	,	PUNCT
ejpam-4274	728	3	we	we	PRON
ejpam-4274	728	4	have	have	VERB
ejpam-4274	728	5	x	x	SYM
ejpam-4274	728	6	6∈	6∈	PROPN
ejpam-4274	728	7	aθ(λ	aθ(λ	NOUN
ejpam-4274	728	8	,	,	PUNCT
ejpam-4274	728	9	p	p	NOUN
ejpam-4274	728	10	)	)	PUNCT
ejpam-4274	728	11	.	.	PUNCT
ejpam-4274	729	1	there	there	PRON
ejpam-4274	729	2	exists	exist	VERB
ejpam-4274	729	3	a	a	DET
ejpam-4274	729	4	(	(	PUNCT
ejpam-4274	729	5	λ	λ	NOUN
ejpam-4274	729	6	,	,	PUNCT
ejpam-4274	729	7	p)-open	p)-open	VERB
ejpam-4274	729	8	set	set	VERB
ejpam-4274	729	9	ux	ux	NOUN
ejpam-4274	729	10	containing	contain	VERB
ejpam-4274	729	11	x	x	PUNCT
ejpam-4274	729	12	such	such	ADJ
ejpam-4274	729	13	that	that	SCONJ
ejpam-4274	729	14	a	a	DET
ejpam-4274	729	15	∩	∩	ADJ
ejpam-4274	729	16	u	u	NOUN
ejpam-4274	729	17	(	(	PUNCT
ejpam-4274	729	18	λ	λ	PROPN
ejpam-4274	729	19	,	,	PUNCT
ejpam-4274	729	20	p	p	NOUN
ejpam-4274	729	21	)	)	PUNCT
ejpam-4274	729	22	x	x	SYM
ejpam-4274	729	23	=	=	NOUN
ejpam-4274	729	24	∅	∅	NOUN
ejpam-4274	729	25	and	and	CCONJ
ejpam-4274	729	26	hence	hence	ADV
ejpam-4274	729	27	ux	ux	ADV
ejpam-4274	729	28	∩	∩	NOUN
ejpam-4274	729	29	aθ(λ	aθ(λ	NOUN
ejpam-4274	729	30	,	,	PUNCT
ejpam-4274	729	31	p	p	NOUN
ejpam-4274	729	32	)	)	PUNCT
ejpam-4274	729	33	=	=	PUNCT
ejpam-4274	729	34	∅.	∅.	VERB
ejpam-4274	729	35	therefore	therefore	ADV
ejpam-4274	729	36	,	,	PUNCT
ejpam-4274	729	37	x	x	PUNCT
ejpam-4274	729	38	∈	∈	NOUN
ejpam-4274	729	39	ux	ux	NOUN
ejpam-4274	730	1	⊆	⊆	NUM
ejpam-4274	730	2	x	x	SYM
ejpam-4274	730	3	−	−	NOUN
ejpam-4274	730	4	aθ(λ	aθ(λ	NOUN
ejpam-4274	730	5	,	,	PUNCT
ejpam-4274	730	6	p	p	NOUN
ejpam-4274	730	7	)	)	PUNCT
ejpam-4274	730	8	.	.	PUNCT
ejpam-4274	731	1	thus	thus	ADV
ejpam-4274	731	2	,	,	PUNCT
ejpam-4274	731	3	x	x	PUNCT
ejpam-4274	731	4	−	−	NOUN
ejpam-4274	731	5	aθ(λ	aθ(λ	NOUN
ejpam-4274	731	6	,	,	PUNCT
ejpam-4274	731	7	p	p	NOUN
ejpam-4274	731	8	)	)	PUNCT
ejpam-4274	731	9	=	=	SYM
ejpam-4274	731	10	∪x∈x−aθ(λ	∪x∈x−aθ(λ	ADJ
ejpam-4274	731	11	,	,	PUNCT
ejpam-4274	731	12	p)ux	p)ux	PROPN
ejpam-4274	731	13	and	and	CCONJ
ejpam-4274	731	14	hence	hence	ADV
ejpam-4274	731	15	x	x	ADP
ejpam-4274	731	16	−	−	NOUN
ejpam-4274	731	17	aθ(λ	aθ(λ	NOUN
ejpam-4274	731	18	,	,	PUNCT
ejpam-4274	731	19	p	p	NOUN
ejpam-4274	731	20	)	)	PUNCT
ejpam-4274	731	21	is	be	AUX
ejpam-4274	731	22	(	(	PUNCT
ejpam-4274	731	23	λ	λ	X
ejpam-4274	731	24	,	,	PUNCT
ejpam-4274	731	25	p)-open	p)-open	ADJ
ejpam-4274	731	26	.	.	PUNCT
ejpam-4274	732	1	this	this	PRON
ejpam-4274	732	2	shows	show	VERB
ejpam-4274	732	3	that	that	SCONJ
ejpam-4274	732	4	aθ(λ	aθ(λ	NOUN
ejpam-4274	732	5	,	,	PUNCT
ejpam-4274	732	6	p	p	NOUN
ejpam-4274	732	7	)	)	PUNCT
ejpam-4274	732	8	is	be	AUX
ejpam-4274	732	9	(	(	PUNCT
ejpam-4274	732	10	λ	λ	X
ejpam-4274	732	11	,	,	PUNCT
ejpam-4274	732	12	p)-closed	p)-close	VERB
ejpam-4274	732	13	.	.	PUNCT
ejpam-4274	733	1	theorem	theorem	VERB
ejpam-4274	733	2	26	26	NUM
ejpam-4274	733	3	.	.	PUNCT
ejpam-4274	734	1	for	for	ADP
ejpam-4274	734	2	a	a	DET
ejpam-4274	734	3	function	function	NOUN
ejpam-4274	734	4	f	f	NOUN
ejpam-4274	734	5	:	:	PUNCT
ejpam-4274	734	6	(	(	PUNCT
ejpam-4274	734	7	x	x	X
ejpam-4274	734	8	,	,	PUNCT
ejpam-4274	734	9	τ	τ	X
ejpam-4274	734	10	)	)	PUNCT
ejpam-4274	734	11	→	→	SYM
ejpam-4274	734	12	(	(	PUNCT
ejpam-4274	734	13	y	y	PROPN
ejpam-4274	734	14	,	,	PUNCT
ejpam-4274	734	15	σ	σ	PROPN
ejpam-4274	734	16	)	)	PUNCT
ejpam-4274	734	17	,	,	PUNCT
ejpam-4274	734	18	the	the	DET
ejpam-4274	734	19	following	follow	VERB
ejpam-4274	734	20	properties	property	NOUN
ejpam-4274	734	21	are	be	AUX
ejpam-4274	734	22	equivalent	equivalent	ADJ
ejpam-4274	734	23	:	:	PUNCT
ejpam-4274	734	24	(	(	PUNCT
ejpam-4274	734	25	1	1	X
ejpam-4274	734	26	)	)	PUNCT
ejpam-4274	734	27	f	f	PROPN
ejpam-4274	734	28	is	be	AUX
ejpam-4274	734	29	weakly	weakly	ADJ
ejpam-4274	734	30	(	(	PUNCT
ejpam-4274	734	31	λ	λ	NOUN
ejpam-4274	734	32	,	,	PUNCT
ejpam-4274	734	33	p)-continuous	p)-continuous	ADJ
ejpam-4274	734	34	;	;	PUNCT
ejpam-4274	734	35	(	(	PUNCT
ejpam-4274	734	36	2	2	X
ejpam-4274	734	37	)	)	PUNCT
ejpam-4274	734	38	f(a(λ	f(a(λ	NOUN
ejpam-4274	734	39	,	,	PUNCT
ejpam-4274	734	40	p	p	NOUN
ejpam-4274	734	41	)	)	PUNCT
ejpam-4274	734	42	)	)	PUNCT
ejpam-4274	735	1	⊆	⊆	NUM
ejpam-4274	735	2	[	[	X
ejpam-4274	735	3	f(a)]θ(λ	f(a)]θ(λ	X
ejpam-4274	735	4	,	,	PUNCT
ejpam-4274	735	5	p	p	NOUN
ejpam-4274	735	6	)	)	PUNCT
ejpam-4274	735	7	for	for	ADP
ejpam-4274	735	8	every	every	DET
ejpam-4274	735	9	subset	subset	NOUN
ejpam-4274	735	10	a	a	PRON
ejpam-4274	735	11	of	of	ADP
ejpam-4274	735	12	x	x	PRON
ejpam-4274	735	13	;	;	PUNCT
ejpam-4274	735	14	(	(	PUNCT
ejpam-4274	735	15	3	3	X
ejpam-4274	735	16	)	)	PUNCT
ejpam-4274	736	1	[	[	X
ejpam-4274	736	2	f−1(b)](λ	f−1(b)](λ	X
ejpam-4274	736	3	,	,	PUNCT
ejpam-4274	736	4	p	p	NOUN
ejpam-4274	736	5	)	)	PUNCT
ejpam-4274	736	6	⊆	⊆	NUM
ejpam-4274	736	7	f−1(bθ(λ	f−1(bθ(λ	NOUN
ejpam-4274	736	8	,	,	PUNCT
ejpam-4274	736	9	p	p	NOUN
ejpam-4274	736	10	)	)	PUNCT
ejpam-4274	736	11	)	)	PUNCT
ejpam-4274	736	12	for	for	ADP
ejpam-4274	736	13	every	every	DET
ejpam-4274	736	14	subset	subset	NOUN
ejpam-4274	736	15	b	b	PROPN
ejpam-4274	736	16	of	of	ADP
ejpam-4274	736	17	y	y	PROPN
ejpam-4274	736	18	;	;	PUNCT
ejpam-4274	736	19	(	(	PUNCT
ejpam-4274	736	20	4	4	X
ejpam-4274	736	21	)	)	PUNCT
ejpam-4274	737	1	[	[	X
ejpam-4274	737	2	f−1(v	f−1(v	NOUN
ejpam-4274	737	3	)	)	PUNCT
ejpam-4274	737	4	]	]	PUNCT
ejpam-4274	737	5	(	(	PUNCT
ejpam-4274	737	6	λ	λ	X
ejpam-4274	737	7	,	,	PUNCT
ejpam-4274	737	8	p	p	NOUN
ejpam-4274	737	9	)	)	PUNCT
ejpam-4274	737	10	⊆	⊆	NUM
ejpam-4274	737	11	f−1(v	f−1(v	NOUN
ejpam-4274	737	12	(	(	PUNCT
ejpam-4274	737	13	λ	λ	PROPN
ejpam-4274	737	14	,	,	PUNCT
ejpam-4274	737	15	p	p	NOUN
ejpam-4274	737	16	)	)	PUNCT
ejpam-4274	737	17	)	)	PUNCT
ejpam-4274	737	18	for	for	ADP
ejpam-4274	737	19	every	every	DET
ejpam-4274	737	20	(	(	PUNCT
ejpam-4274	737	21	λ	λ	NOUN
ejpam-4274	737	22	,	,	PUNCT
ejpam-4274	737	23	p)-open	p)-open	PUNCT
ejpam-4274	737	24	subset	subset	VERB
ejpam-4274	737	25	v	v	NOUN
ejpam-4274	737	26	of	of	ADP
ejpam-4274	737	27	y	y	PROPN
ejpam-4274	737	28	.	.	PUNCT
ejpam-4274	738	1	proof	proof	NOUN
ejpam-4274	738	2	.	.	PUNCT
ejpam-4274	739	1	(	(	PUNCT
ejpam-4274	739	2	1	1	X
ejpam-4274	739	3	)	)	PUNCT
ejpam-4274	739	4	⇒	⇒	NOUN
ejpam-4274	739	5	(	(	PUNCT
ejpam-4274	739	6	2	2	NUM
ejpam-4274	739	7	):	):	PUNCT
ejpam-4274	739	8	let	let	VERB
ejpam-4274	739	9	a	a	DET
ejpam-4274	739	10	be	be	AUX
ejpam-4274	739	11	any	any	DET
ejpam-4274	739	12	subset	subset	NOUN
ejpam-4274	739	13	of	of	ADP
ejpam-4274	739	14	x.	x.	NOUN
ejpam-4274	739	15	let	let	VERB
ejpam-4274	739	16	x	x	X
ejpam-4274	739	17	∈	∈	PROPN
ejpam-4274	739	18	a(λ	a(λ	PROPN
ejpam-4274	739	19	,	,	PUNCT
ejpam-4274	739	20	p	p	NOUN
ejpam-4274	739	21	)	)	PUNCT
ejpam-4274	739	22	and	and	CCONJ
ejpam-4274	739	23	v	v	AUX
ejpam-4274	739	24	be	be	AUX
ejpam-4274	739	25	any	any	DET
ejpam-4274	739	26	(	(	PUNCT
ejpam-4274	739	27	λ	λ	NOUN
ejpam-4274	739	28	,	,	PUNCT
ejpam-4274	739	29	p)-open	p)-open	VERB
ejpam-4274	739	30	set	set	VERB
ejpam-4274	739	31	containing	contain	VERB
ejpam-4274	739	32	f(x	f(x	PROPN
ejpam-4274	739	33	)	)	PUNCT
ejpam-4274	739	34	.	.	PUNCT
ejpam-4274	740	1	since	since	SCONJ
ejpam-4274	740	2	f	f	PROPN
ejpam-4274	740	3	is	be	AUX
ejpam-4274	740	4	weakly	weakly	ADJ
ejpam-4274	740	5	(	(	PUNCT
ejpam-4274	740	6	λ	λ	NOUN
ejpam-4274	740	7	,	,	PUNCT
ejpam-4274	740	8	p)-continuous	p)-continuous	ADJ
ejpam-4274	740	9	,	,	PUNCT
ejpam-4274	740	10	there	there	PRON
ejpam-4274	740	11	exists	exist	VERB
ejpam-4274	740	12	a	a	DET
ejpam-4274	740	13	(	(	PUNCT
ejpam-4274	740	14	λ	λ	NOUN
ejpam-4274	740	15	,	,	PUNCT
ejpam-4274	740	16	p)-open	p)-open	VERB
ejpam-4274	740	17	set	set	VERB
ejpam-4274	740	18	u	u	NOUN
ejpam-4274	740	19	containing	contain	VERB
ejpam-4274	740	20	x	x	PUNCT
ejpam-4274	740	21	such	such	ADJ
ejpam-4274	740	22	that	that	DET
ejpam-4274	740	23	f(u	f(u	PROPN
ejpam-4274	740	24	)	)	PUNCT
ejpam-4274	740	25	⊆	⊆	NUM
ejpam-4274	740	26	v	v	NOUN
ejpam-4274	740	27	(	(	PUNCT
ejpam-4274	740	28	λ	λ	PROPN
ejpam-4274	740	29	,	,	PUNCT
ejpam-4274	740	30	p	p	NOUN
ejpam-4274	740	31	)	)	PUNCT
ejpam-4274	740	32	.	.	PUNCT
ejpam-4274	741	1	since	since	SCONJ
ejpam-4274	741	2	x	x	PROPN
ejpam-4274	741	3	∈	∈	PROPN
ejpam-4274	741	4	a(λ	a(λ	PROPN
ejpam-4274	741	5	,	,	PUNCT
ejpam-4274	741	6	p	p	NOUN
ejpam-4274	741	7	)	)	PUNCT
ejpam-4274	741	8	,	,	PUNCT
ejpam-4274	741	9	we	we	PRON
ejpam-4274	741	10	have	have	VERB
ejpam-4274	741	11	u	u	NOUN
ejpam-4274	741	12	∩a	∩a	PROPN
ejpam-4274	741	13	6=	6=	ADP
ejpam-4274	741	14	∅.	∅.	PROPN
ejpam-4274	741	15	it	it	PRON
ejpam-4274	741	16	follows	follow	VERB
ejpam-4274	741	17	that	that	DET
ejpam-4274	741	18	∅	∅	NOUN
ejpam-4274	741	19	6=	6=	ADP
ejpam-4274	741	20	f(u	f(u	ADJ
ejpam-4274	741	21	)	)	PUNCT
ejpam-4274	741	22	∩	∩	ADJ
ejpam-4274	741	23	f(a	f(a	NOUN
ejpam-4274	741	24	)	)	PUNCT
ejpam-4274	741	25	⊆	⊆	NUM
ejpam-4274	741	26	v	v	NOUN
ejpam-4274	741	27	(	(	PUNCT
ejpam-4274	741	28	λ	λ	PROPN
ejpam-4274	741	29	,	,	PUNCT
ejpam-4274	741	30	p	p	NOUN
ejpam-4274	741	31	)	)	PUNCT
ejpam-4274	741	32	∩	∩	ADJ
ejpam-4274	741	33	f(a	f(a	NOUN
ejpam-4274	741	34	)	)	PUNCT
ejpam-4274	741	35	and	and	CCONJ
ejpam-4274	741	36	hence	hence	ADV
ejpam-4274	741	37	v	v	NOUN
ejpam-4274	741	38	(	(	PUNCT
ejpam-4274	741	39	λ	λ	PROPN
ejpam-4274	741	40	,	,	PUNCT
ejpam-4274	741	41	p	p	NOUN
ejpam-4274	741	42	)	)	PUNCT
ejpam-4274	741	43	∩	∩	ADJ
ejpam-4274	741	44	f(a	f(a	NOUN
ejpam-4274	741	45	)	)	PUNCT
ejpam-4274	741	46	6=	6=	ADP
ejpam-4274	741	47	∅.	∅.	ADP
ejpam-4274	741	48	thus	thus	ADV
ejpam-4274	741	49	,	,	PUNCT
ejpam-4274	741	50	f(x	f(x	PROPN
ejpam-4274	741	51	)	)	PUNCT
ejpam-4274	741	52	∈	∈	PROPN
ejpam-4274	742	1	[	[	X
ejpam-4274	742	2	f(a)]θ(λ	f(a)]θ(λ	PROPN
ejpam-4274	742	3	,	,	PUNCT
ejpam-4274	742	4	p	p	NOUN
ejpam-4274	742	5	)	)	PUNCT
ejpam-4274	742	6	.	.	PUNCT
ejpam-4274	743	1	consequently	consequently	ADV
ejpam-4274	743	2	,	,	PUNCT
ejpam-4274	743	3	we	we	PRON
ejpam-4274	743	4	obtain	obtain	VERB
ejpam-4274	743	5	f(a(λ	f(a(λ	NOUN
ejpam-4274	743	6	,	,	PUNCT
ejpam-4274	743	7	p	p	NOUN
ejpam-4274	743	8	)	)	PUNCT
ejpam-4274	743	9	)	)	PUNCT
ejpam-4274	744	1	⊆	⊆	NUM
ejpam-4274	744	2	[	[	X
ejpam-4274	744	3	f(a)]θ(λ	f(a)]θ(λ	X
ejpam-4274	744	4	,	,	PUNCT
ejpam-4274	744	5	p	p	NOUN
ejpam-4274	744	6	)	)	PUNCT
ejpam-4274	744	7	.	.	PUNCT
ejpam-4274	745	1	(	(	PUNCT
ejpam-4274	745	2	2	2	X
ejpam-4274	745	3	)	)	PUNCT
ejpam-4274	745	4	⇒	⇒	NOUN
ejpam-4274	745	5	(	(	PUNCT
ejpam-4274	745	6	3	3	NUM
ejpam-4274	745	7	):	):	PUNCT
ejpam-4274	745	8	let	let	VERB
ejpam-4274	745	9	b	b	X
ejpam-4274	745	10	be	be	AUX
ejpam-4274	745	11	any	any	DET
ejpam-4274	745	12	subset	subset	NOUN
ejpam-4274	745	13	of	of	ADP
ejpam-4274	745	14	y	y	PROPN
ejpam-4274	745	15	.	.	PUNCT
ejpam-4274	746	1	by	by	ADP
ejpam-4274	746	2	(	(	PUNCT
ejpam-4274	746	3	2	2	NUM
ejpam-4274	746	4	)	)	PUNCT
ejpam-4274	746	5	,	,	PUNCT
ejpam-4274	746	6	we	we	PRON
ejpam-4274	746	7	have	have	VERB
ejpam-4274	746	8	f([f−1(b)](λ	f([f−1(b)](λ	PROPN
ejpam-4274	746	9	,	,	PUNCT
ejpam-4274	746	10	p	p	NOUN
ejpam-4274	746	11	)	)	PUNCT
ejpam-4274	746	12	)	)	PUNCT
ejpam-4274	747	1	⊆	⊆	NUM
ejpam-4274	747	2	[	[	X
ejpam-4274	747	3	f(f−1(b))]θ(λ	f(f−1(b))]θ(λ	PROPN
ejpam-4274	747	4	,	,	PUNCT
ejpam-4274	747	5	p	p	NOUN
ejpam-4274	747	6	)	)	PUNCT
ejpam-4274	747	7	⊆	⊆	NUM
ejpam-4274	747	8	bθ(λ	bθ(λ	NOUN
ejpam-4274	747	9	,	,	PUNCT
ejpam-4274	747	10	p	p	NOUN
ejpam-4274	747	11	)	)	PUNCT
ejpam-4274	747	12	and	and	CCONJ
ejpam-4274	747	13	hence	hence	ADV
ejpam-4274	747	14	[	[	X
ejpam-4274	747	15	f−1(b)](λ	f−1(b)](λ	X
ejpam-4274	747	16	,	,	PUNCT
ejpam-4274	747	17	p	p	NOUN
ejpam-4274	747	18	)	)	PUNCT
ejpam-4274	747	19	⊆	⊆	NUM
ejpam-4274	747	20	f−1(bθ(λ	f−1(bθ(λ	NOUN
ejpam-4274	747	21	,	,	PUNCT
ejpam-4274	747	22	p	p	NOUN
ejpam-4274	747	23	)	)	PUNCT
ejpam-4274	747	24	)	)	PUNCT
ejpam-4274	747	25	.	.	PUNCT
ejpam-4274	748	1	c.	c.	PROPN
ejpam-4274	748	2	boonpok	boonpok	PROPN
ejpam-4274	748	3	,	,	PUNCT
ejpam-4274	748	4	c.	c.	PROPN
ejpam-4274	748	5	viriyapong	viriyapong	PROPN
ejpam-4274	748	6	/	/	SYM
ejpam-4274	748	7	eur	eur	PROPN
ejpam-4274	748	8	.	.	PUNCT
ejpam-4274	749	1	j.	j.	PROPN
ejpam-4274	749	2	pure	pure	PROPN
ejpam-4274	749	3	appl	appl	PROPN
ejpam-4274	749	4	.	.	PROPN
ejpam-4274	749	5	math	math	PROPN
ejpam-4274	749	6	,	,	PUNCT
ejpam-4274	749	7	15	15	NUM
ejpam-4274	749	8	(	(	PUNCT
ejpam-4274	749	9	2	2	NUM
ejpam-4274	749	10	)	)	PUNCT
ejpam-4274	749	11	(	(	PUNCT
ejpam-4274	749	12	2022	2022	NUM
ejpam-4274	749	13	)	)	PUNCT
ejpam-4274	749	14	,	,	PUNCT
ejpam-4274	749	15	415	415	NUM
ejpam-4274	749	16	-	-	SYM
ejpam-4274	749	17	436	436	NUM
ejpam-4274	749	18	434	434	NUM
ejpam-4274	749	19	(	(	PUNCT
ejpam-4274	749	20	3	3	NUM
ejpam-4274	749	21	)	)	PUNCT
ejpam-4274	749	22	⇒	⇒	NOUN
ejpam-4274	749	23	(	(	PUNCT
ejpam-4274	749	24	4	4	NUM
ejpam-4274	749	25	):	):	PUNCT
ejpam-4274	749	26	let	let	VERB
ejpam-4274	749	27	v	v	PART
ejpam-4274	749	28	be	be	AUX
ejpam-4274	749	29	any	any	DET
ejpam-4274	749	30	(	(	PUNCT
ejpam-4274	749	31	λ	λ	NOUN
ejpam-4274	749	32	,	,	PUNCT
ejpam-4274	749	33	p)-open	p)-open	PUNCT
ejpam-4274	749	34	subset	subset	NOUN
ejpam-4274	749	35	of	of	ADP
ejpam-4274	749	36	y	y	PROPN
ejpam-4274	749	37	.	.	PUNCT
ejpam-4274	750	1	by	by	ADP
ejpam-4274	750	2	lemma	lemma	PROPN
ejpam-4274	750	3	14	14	NUM
ejpam-4274	750	4	,	,	PUNCT
ejpam-4274	750	5	v	v	NOUN
ejpam-4274	750	6	(	(	PUNCT
ejpam-4274	750	7	λ	λ	PROPN
ejpam-4274	750	8	,	,	PUNCT
ejpam-4274	750	9	p	p	NOUN
ejpam-4274	750	10	)	)	PUNCT
ejpam-4274	750	11	=	=	SYM
ejpam-4274	750	12	v	v	ADP
ejpam-4274	750	13	θ(λ	θ(λ	PROPN
ejpam-4274	750	14	,	,	PUNCT
ejpam-4274	750	15	p	p	NOUN
ejpam-4274	750	16	)	)	PUNCT
ejpam-4274	750	17	.	.	PUNCT
ejpam-4274	751	1	thus	thus	ADV
ejpam-4274	751	2	,	,	PUNCT
ejpam-4274	751	3	the	the	DET
ejpam-4274	751	4	proof	proof	NOUN
ejpam-4274	751	5	is	be	AUX
ejpam-4274	751	6	obvious	obvious	ADJ
ejpam-4274	751	7	.	.	PUNCT
ejpam-4274	752	1	(	(	PUNCT
ejpam-4274	752	2	4	4	X
ejpam-4274	752	3	)	)	PUNCT
ejpam-4274	752	4	⇒	⇒	NOUN
ejpam-4274	752	5	(	(	PUNCT
ejpam-4274	752	6	1	1	NUM
ejpam-4274	752	7	):	):	PUNCT
ejpam-4274	752	8	let	let	VERB
ejpam-4274	752	9	v	v	PART
ejpam-4274	752	10	be	be	AUX
ejpam-4274	752	11	any	any	DET
ejpam-4274	752	12	(	(	PUNCT
ejpam-4274	752	13	λ	λ	NOUN
ejpam-4274	752	14	,	,	PUNCT
ejpam-4274	752	15	p)-open	p)-open	VERB
ejpam-4274	752	16	set	set	VERB
ejpam-4274	752	17	containing	contain	VERB
ejpam-4274	752	18	f(x	f(x	PROPN
ejpam-4274	752	19	)	)	PUNCT
ejpam-4274	752	20	.	.	PUNCT
ejpam-4274	753	1	since	since	SCONJ
ejpam-4274	753	2	v	v	ADP
ejpam-4274	753	3	∩	∩	NOUN
ejpam-4274	753	4	[	[	X
ejpam-4274	753	5	y	y	PROPN
ejpam-4274	753	6	−	−	PROPN
ejpam-4274	753	7	v	v	NOUN
ejpam-4274	753	8	(	(	PUNCT
ejpam-4274	753	9	λ	λ	PROPN
ejpam-4274	753	10	,	,	PUNCT
ejpam-4274	753	11	p	p	NOUN
ejpam-4274	753	12	)	)	PUNCT
ejpam-4274	753	13	]	]	PUNCT
ejpam-4274	754	1	=	=	SYM
ejpam-4274	754	2	∅	∅	NOUN
ejpam-4274	754	3	,	,	PUNCT
ejpam-4274	754	4	we	we	PRON
ejpam-4274	754	5	have	have	VERB
ejpam-4274	754	6	f(x	f(x	PROPN
ejpam-4274	754	7	)	)	PUNCT
ejpam-4274	754	8	6∈	6∈	NOUN
ejpam-4274	755	1	[	[	X
ejpam-4274	755	2	y	y	PROPN
ejpam-4274	755	3	−	−	PROPN
ejpam-4274	755	4	v	v	PROPN
ejpam-4274	755	5	(	(	PUNCT
ejpam-4274	755	6	λ	λ	PROPN
ejpam-4274	755	7	,	,	PUNCT
ejpam-4274	755	8	p)](λ	p)](λ	ADJ
ejpam-4274	755	9	,	,	PUNCT
ejpam-4274	755	10	p	p	NOUN
ejpam-4274	755	11	)	)	PUNCT
ejpam-4274	755	12	and	and	CCONJ
ejpam-4274	755	13	hence	hence	ADV
ejpam-4274	755	14	x	x	ADP
ejpam-4274	755	15	6∈	6∈	NOUN
ejpam-4274	755	16	f−1([y	f−1([y	NOUN
ejpam-4274	755	17	−	−	PROPN
ejpam-4274	755	18	v	v	NOUN
ejpam-4274	755	19	(	(	PUNCT
ejpam-4274	755	20	λ	λ	PROPN
ejpam-4274	755	21	,	,	PUNCT
ejpam-4274	755	22	p)](λ	p)](λ	ADJ
ejpam-4274	755	23	,	,	PUNCT
ejpam-4274	755	24	p	p	NOUN
ejpam-4274	755	25	)	)	PUNCT
ejpam-4274	755	26	)	)	PUNCT
ejpam-4274	755	27	.	.	PUNCT
ejpam-4274	756	1	since	since	SCONJ
ejpam-4274	756	2	y	y	PROPN
ejpam-4274	756	3	−	−	PROPN
ejpam-4274	756	4	v	v	PROPN
ejpam-4274	756	5	(	(	PUNCT
ejpam-4274	756	6	λ	λ	PROPN
ejpam-4274	756	7	,	,	PUNCT
ejpam-4274	756	8	p	p	NOUN
ejpam-4274	756	9	)	)	PUNCT
ejpam-4274	756	10	is	be	AUX
ejpam-4274	756	11	(	(	PUNCT
ejpam-4274	756	12	λ	λ	X
ejpam-4274	756	13	,	,	PUNCT
ejpam-4274	756	14	p)-open	p)-open	ADJ
ejpam-4274	756	15	,	,	PUNCT
ejpam-4274	756	16	by	by	ADP
ejpam-4274	756	17	(	(	PUNCT
ejpam-4274	756	18	4	4	NUM
ejpam-4274	756	19	)	)	PUNCT
ejpam-4274	756	20	,	,	PUNCT
ejpam-4274	756	21	x	x	X
ejpam-4274	756	22	6∈	6∈	NOUN
ejpam-4274	757	1	[	[	X
ejpam-4274	757	2	f−1([y	f−1([y	NOUN
ejpam-4274	757	3	−	−	NOUN
ejpam-4274	757	4	v	v	NOUN
ejpam-4274	757	5	(	(	PUNCT
ejpam-4274	757	6	λ	λ	PROPN
ejpam-4274	757	7	,	,	PUNCT
ejpam-4274	757	8	p)])](λ	p)])](λ	PROPN
ejpam-4274	757	9	,	,	PUNCT
ejpam-4274	757	10	p	p	NOUN
ejpam-4274	757	11	)	)	PUNCT
ejpam-4274	757	12	and	and	CCONJ
ejpam-4274	757	13	there	there	PRON
ejpam-4274	757	14	exists	exist	VERB
ejpam-4274	757	15	a	a	DET
ejpam-4274	757	16	(	(	PUNCT
ejpam-4274	757	17	λ	λ	NOUN
ejpam-4274	757	18	,	,	PUNCT
ejpam-4274	757	19	p)-open	p)-open	VERB
ejpam-4274	757	20	set	set	VERB
ejpam-4274	757	21	u	u	NOUN
ejpam-4274	757	22	containing	contain	VERB
ejpam-4274	757	23	x	x	PUNCT
ejpam-4274	757	24	such	such	ADJ
ejpam-4274	757	25	that	that	SCONJ
ejpam-4274	757	26	u	u	PRON
ejpam-4274	757	27	∩f−1(y	∩f−1(y	PROPN
ejpam-4274	757	28	−v	−v	NOUN
ejpam-4274	757	29	(	(	PUNCT
ejpam-4274	757	30	λ	λ	X
ejpam-4274	757	31	,	,	PUNCT
ejpam-4274	757	32	p	p	NOUN
ejpam-4274	757	33	)	)	PUNCT
ejpam-4274	757	34	)	)	PUNCT
ejpam-4274	758	1	=	=	NOUN
ejpam-4274	758	2	∅	∅	NOUN
ejpam-4274	758	3	;	;	PUNCT
ejpam-4274	758	4	hence	hence	ADV
ejpam-4274	758	5	f(u)∩	f(u)∩	PROPN
ejpam-4274	758	6	[	[	X
ejpam-4274	758	7	y	y	PROPN
ejpam-4274	758	8	−v	−v	NOUN
ejpam-4274	758	9	(	(	PUNCT
ejpam-4274	758	10	λ	λ	X
ejpam-4274	758	11	,	,	PUNCT
ejpam-4274	758	12	p	p	NOUN
ejpam-4274	758	13	)	)	PUNCT
ejpam-4274	758	14	]	]	PUNCT
ejpam-4274	759	1	=	=	PUNCT
ejpam-4274	759	2	∅.	∅.	VERB
ejpam-4274	759	3	this	this	PRON
ejpam-4274	759	4	shows	show	VERB
ejpam-4274	759	5	that	that	SCONJ
ejpam-4274	759	6	f(u	f(u	PROPN
ejpam-4274	759	7	)	)	PUNCT
ejpam-4274	759	8	⊆	⊆	NUM
ejpam-4274	759	9	v	v	NOUN
ejpam-4274	759	10	(	(	PUNCT
ejpam-4274	759	11	λ	λ	PROPN
ejpam-4274	759	12	,	,	PUNCT
ejpam-4274	759	13	p	p	NOUN
ejpam-4274	759	14	)	)	PUNCT
ejpam-4274	759	15	.	.	PUNCT
ejpam-4274	760	1	thus	thus	ADV
ejpam-4274	760	2	,	,	PUNCT
ejpam-4274	760	3	f	f	PROPN
ejpam-4274	760	4	is	be	AUX
ejpam-4274	760	5	weakly	weakly	ADJ
ejpam-4274	760	6	(	(	PUNCT
ejpam-4274	760	7	λ	λ	NOUN
ejpam-4274	760	8	,	,	PUNCT
ejpam-4274	760	9	p)-continuous	p)-continuous	ADJ
ejpam-4274	760	10	.	.	PUNCT
ejpam-4274	761	1	definition	definition	NOUN
ejpam-4274	761	2	16	16	NUM
ejpam-4274	761	3	.	.	PUNCT
ejpam-4274	762	1	a	a	DET
ejpam-4274	762	2	topological	topological	ADJ
ejpam-4274	762	3	space	space	NOUN
ejpam-4274	762	4	(	(	PUNCT
ejpam-4274	762	5	x	x	X
ejpam-4274	762	6	,	,	PUNCT
ejpam-4274	762	7	τ	τ	X
ejpam-4274	762	8	)	)	PUNCT
ejpam-4274	762	9	is	be	AUX
ejpam-4274	762	10	said	say	VERB
ejpam-4274	762	11	to	to	PART
ejpam-4274	762	12	be	be	AUX
ejpam-4274	762	13	λp	λp	NOUN
ejpam-4274	762	14	-	-	ADJ
ejpam-4274	762	15	regular	regular	ADJ
ejpam-4274	762	16	if	if	SCONJ
ejpam-4274	762	17	,	,	PUNCT
ejpam-4274	762	18	for	for	ADP
ejpam-4274	762	19	each	each	PRON
ejpam-4274	762	20	(	(	PUNCT
ejpam-4274	762	21	λ	λ	PROPN
ejpam-4274	762	22	,	,	PUNCT
ejpam-4274	762	23	p)-closed	p)-close	VERB
ejpam-4274	762	24	set	set	VERB
ejpam-4274	762	25	f	f	PROPN
ejpam-4274	762	26	and	and	CCONJ
ejpam-4274	762	27	each	each	DET
ejpam-4274	762	28	x	x	SYM
ejpam-4274	762	29	6∈	6∈	PROPN
ejpam-4274	762	30	f	f	NOUN
ejpam-4274	762	31	,	,	PUNCT
ejpam-4274	762	32	there	there	PRON
ejpam-4274	762	33	exist	exist	VERB
ejpam-4274	762	34	disjoint	disjoint	NOUN
ejpam-4274	762	35	(	(	PUNCT
ejpam-4274	762	36	λ	λ	NOUN
ejpam-4274	762	37	,	,	PUNCT
ejpam-4274	762	38	p)-open	p)-open	VERB
ejpam-4274	762	39	sets	set	VERB
ejpam-4274	762	40	u	u	NOUN
ejpam-4274	762	41	and	and	CCONJ
ejpam-4274	762	42	v	v	ADP
ejpam-4274	762	43	such	such	ADJ
ejpam-4274	762	44	that	that	SCONJ
ejpam-4274	762	45	x	x	SYM
ejpam-4274	762	46	∈	∈	PROPN
ejpam-4274	762	47	u	u	NOUN
ejpam-4274	762	48	and	and	CCONJ
ejpam-4274	762	49	f	f	PROPN
ejpam-4274	762	50	⊆	⊆	NUM
ejpam-4274	762	51	v	v	NOUN
ejpam-4274	762	52	.	.	PUNCT
ejpam-4274	763	1	lemma	lemma	PROPN
ejpam-4274	763	2	15	15	NUM
ejpam-4274	763	3	.	.	PUNCT
ejpam-4274	764	1	a	a	DET
ejpam-4274	764	2	topological	topological	ADJ
ejpam-4274	764	3	space	space	NOUN
ejpam-4274	764	4	(	(	PUNCT
ejpam-4274	764	5	x	x	X
ejpam-4274	764	6	,	,	PUNCT
ejpam-4274	764	7	τ	τ	X
ejpam-4274	764	8	)	)	PUNCT
ejpam-4274	764	9	is	be	AUX
ejpam-4274	764	10	λp	λp	NOUN
ejpam-4274	764	11	-	-	ADJ
ejpam-4274	764	12	regular	regular	ADJ
ejpam-4274	764	13	if	if	SCONJ
ejpam-4274	764	14	and	and	CCONJ
ejpam-4274	764	15	only	only	ADV
ejpam-4274	764	16	if	if	SCONJ
ejpam-4274	764	17	for	for	ADP
ejpam-4274	764	18	each	each	DET
ejpam-4274	764	19	x	x	SYM
ejpam-4274	764	20	∈	∈	PROPN
ejpam-4274	764	21	x	x	X
ejpam-4274	764	22	and	and	CCONJ
ejpam-4274	764	23	each	each	PRON
ejpam-4274	764	24	(	(	PUNCT
ejpam-4274	764	25	λ	λ	NOUN
ejpam-4274	764	26	,	,	PUNCT
ejpam-4274	764	27	p)-open	p)-open	VERB
ejpam-4274	764	28	set	set	VERB
ejpam-4274	764	29	u	u	NOUN
ejpam-4274	764	30	containing	contain	VERB
ejpam-4274	764	31	x	x	PRON
ejpam-4274	764	32	,	,	PUNCT
ejpam-4274	764	33	there	there	PRON
ejpam-4274	764	34	exists	exist	VERB
ejpam-4274	764	35	a	a	DET
ejpam-4274	764	36	(	(	PUNCT
ejpam-4274	764	37	λ	λ	NOUN
ejpam-4274	764	38	,	,	PUNCT
ejpam-4274	764	39	p)-open	p)-open	VERB
ejpam-4274	764	40	set	set	VERB
ejpam-4274	764	41	v	v	ADP
ejpam-4274	764	42	such	such	ADJ
ejpam-4274	764	43	that	that	SCONJ
ejpam-4274	764	44	x	x	SYM
ejpam-4274	764	45	∈	∈	NOUN
ejpam-4274	764	46	v	v	ADP
ejpam-4274	764	47	⊆	⊆	NUM
ejpam-4274	764	48	v	v	NOUN
ejpam-4274	764	49	(	(	PUNCT
ejpam-4274	764	50	λ	λ	PROPN
ejpam-4274	764	51	,	,	PUNCT
ejpam-4274	764	52	p	p	NOUN
ejpam-4274	764	53	)	)	PUNCT
ejpam-4274	764	54	⊆	⊆	NUM
ejpam-4274	764	55	u	u	NOUN
ejpam-4274	764	56	.	.	PUNCT
ejpam-4274	765	1	proof	proof	NOUN
ejpam-4274	765	2	.	.	PUNCT
ejpam-4274	766	1	let	let	VERB
ejpam-4274	766	2	x	x	PUNCT
ejpam-4274	766	3	∈	∈	PROPN
ejpam-4274	766	4	x	x	PUNCT
ejpam-4274	766	5	and	and	CCONJ
ejpam-4274	766	6	let	let	VERB
ejpam-4274	766	7	u	u	PRON
ejpam-4274	766	8	be	be	AUX
ejpam-4274	766	9	a	a	DET
ejpam-4274	766	10	(	(	PUNCT
ejpam-4274	766	11	λ	λ	NOUN
ejpam-4274	766	12	,	,	PUNCT
ejpam-4274	766	13	p)-open	p)-open	VERB
ejpam-4274	766	14	set	set	VERB
ejpam-4274	766	15	containing	contain	VERB
ejpam-4274	766	16	x.	x.	NOUN
ejpam-4274	766	17	then	then	ADV
ejpam-4274	766	18	,	,	PUNCT
ejpam-4274	766	19	x	x	X
ejpam-4274	766	20	6∈	6∈	NOUN
ejpam-4274	766	21	x	x	NOUN
ejpam-4274	766	22	−	−	NOUN
ejpam-4274	766	23	u	u	NOUN
ejpam-4274	766	24	and	and	CCONJ
ejpam-4274	766	25	x	x	SYM
ejpam-4274	766	26	−	−	NOUN
ejpam-4274	766	27	u	u	NOUN
ejpam-4274	766	28	is	be	AUX
ejpam-4274	766	29	(	(	PUNCT
ejpam-4274	766	30	λ	λ	X
ejpam-4274	766	31	,	,	PUNCT
ejpam-4274	766	32	p)-closed	p)-close	VERB
ejpam-4274	766	33	.	.	PUNCT
ejpam-4274	767	1	there	there	PRON
ejpam-4274	767	2	exist	exist	VERB
ejpam-4274	767	3	disjoint	disjoint	NOUN
ejpam-4274	767	4	(	(	PUNCT
ejpam-4274	767	5	λ	λ	NOUN
ejpam-4274	767	6	,	,	PUNCT
ejpam-4274	767	7	p)-open	p)-open	VERB
ejpam-4274	767	8	sets	set	VERB
ejpam-4274	767	9	v	v	ADP
ejpam-4274	767	10	and	and	CCONJ
ejpam-4274	767	11	w	w	ADP
ejpam-4274	767	12	such	such	ADJ
ejpam-4274	767	13	that	that	SCONJ
ejpam-4274	767	14	x	x	SYM
ejpam-4274	767	15	∈	∈	PROPN
ejpam-4274	767	16	v	v	NOUN
ejpam-4274	767	17	and	and	CCONJ
ejpam-4274	767	18	x	x	NOUN
ejpam-4274	767	19	−	−	PROPN
ejpam-4274	767	20	u	u	NOUN
ejpam-4274	767	21	⊆	⊆	NUM
ejpam-4274	767	22	w	w	NOUN
ejpam-4274	767	23	.	.	PUNCT
ejpam-4274	768	1	thus	thus	ADV
ejpam-4274	768	2	,	,	PUNCT
ejpam-4274	768	3	v	v	ADP
ejpam-4274	768	4	⊆	⊆	NUM
ejpam-4274	768	5	x	x	SYM
ejpam-4274	768	6	−	−	PROPN
ejpam-4274	768	7	w	w	NOUN
ejpam-4274	768	8	⊆	⊆	NUM
ejpam-4274	768	9	u	u	NOUN
ejpam-4274	768	10	.	.	PUNCT
ejpam-4274	769	1	since	since	SCONJ
ejpam-4274	769	2	x	x	X
ejpam-4274	769	3	−	−	PROPN
ejpam-4274	769	4	w	w	NOUN
ejpam-4274	769	5	is	be	AUX
ejpam-4274	769	6	(	(	PUNCT
ejpam-4274	769	7	λ	λ	X
ejpam-4274	769	8	,	,	PUNCT
ejpam-4274	769	9	p)-closed	p)-close	VERB
ejpam-4274	769	10	,	,	PUNCT
ejpam-4274	769	11	we	we	PRON
ejpam-4274	769	12	have	have	VERB
ejpam-4274	769	13	v	v	NUM
ejpam-4274	769	14	(	(	PUNCT
ejpam-4274	769	15	λ	λ	PROPN
ejpam-4274	769	16	,	,	PUNCT
ejpam-4274	769	17	p	p	NOUN
ejpam-4274	769	18	)	)	PUNCT
ejpam-4274	769	19	⊆	⊆	NUM
ejpam-4274	769	20	x	x	PUNCT
ejpam-4274	769	21	−w	−w	ADV
ejpam-4274	769	22	⊆	⊆	NUM
ejpam-4274	769	23	u	u	NOUN
ejpam-4274	769	24	and	and	CCONJ
ejpam-4274	769	25	hence	hence	ADV
ejpam-4274	769	26	x	x	ADP
ejpam-4274	769	27	∈	∈	NOUN
ejpam-4274	769	28	v	v	ADP
ejpam-4274	769	29	⊆	⊆	NUM
ejpam-4274	769	30	v	v	NOUN
ejpam-4274	769	31	(	(	PUNCT
ejpam-4274	769	32	λ	λ	PROPN
ejpam-4274	769	33	,	,	PUNCT
ejpam-4274	769	34	p	p	NOUN
ejpam-4274	769	35	)	)	PUNCT
ejpam-4274	769	36	⊆	⊆	NUM
ejpam-4274	769	37	u	u	NOUN
ejpam-4274	769	38	.	.	PUNCT
ejpam-4274	770	1	conversely	conversely	ADV
ejpam-4274	770	2	,	,	PUNCT
ejpam-4274	770	3	let	let	VERB
ejpam-4274	770	4	f	f	PRON
ejpam-4274	770	5	be	be	AUX
ejpam-4274	770	6	a	a	DET
ejpam-4274	770	7	(	(	PUNCT
ejpam-4274	770	8	λ	λ	PROPN
ejpam-4274	770	9	,	,	PUNCT
ejpam-4274	770	10	p)-closed	p)-close	VERB
ejpam-4274	770	11	set	set	NOUN
ejpam-4274	770	12	and	and	CCONJ
ejpam-4274	770	13	let	let	VERB
ejpam-4274	770	14	x	x	SYM
ejpam-4274	770	15	6∈	6∈	PROPN
ejpam-4274	770	16	f	f	X
ejpam-4274	770	17	.	.	PUNCT
ejpam-4274	771	1	then	then	ADV
ejpam-4274	771	2	,	,	PUNCT
ejpam-4274	771	3	x	x	PUNCT
ejpam-4274	771	4	∈	∈	NOUN
ejpam-4274	771	5	x	x	X
ejpam-4274	771	6	−f	−f	NOUN
ejpam-4274	771	7	.	.	PUNCT
ejpam-4274	772	1	since	since	SCONJ
ejpam-4274	772	2	x	x	PROPN
ejpam-4274	772	3	−f	−f	PROPN
ejpam-4274	772	4	is	be	AUX
ejpam-4274	772	5	(	(	PUNCT
ejpam-4274	772	6	λ	λ	INTJ
ejpam-4274	772	7	,	,	PUNCT
ejpam-4274	772	8	p)-open	p)-open	VERB
ejpam-4274	772	9	,	,	PUNCT
ejpam-4274	772	10	there	there	PRON
ejpam-4274	772	11	exists	exist	VERB
ejpam-4274	772	12	a	a	DET
ejpam-4274	772	13	(	(	PUNCT
ejpam-4274	772	14	λ	λ	NOUN
ejpam-4274	772	15	,	,	PUNCT
ejpam-4274	772	16	p)-open	p)-open	VERB
ejpam-4274	772	17	set	set	VERB
ejpam-4274	772	18	v	v	ADP
ejpam-4274	772	19	such	such	ADJ
ejpam-4274	772	20	that	that	SCONJ
ejpam-4274	772	21	x	x	SYM
ejpam-4274	772	22	∈	∈	NOUN
ejpam-4274	772	23	v	v	ADP
ejpam-4274	772	24	⊆	⊆	NUM
ejpam-4274	772	25	v	v	NOUN
ejpam-4274	772	26	(	(	PUNCT
ejpam-4274	772	27	λ	λ	PROPN
ejpam-4274	772	28	,	,	PUNCT
ejpam-4274	772	29	p	p	NOUN
ejpam-4274	772	30	)	)	PUNCT
ejpam-4274	772	31	⊆	⊆	NUM
ejpam-4274	772	32	x	x	SYM
ejpam-4274	772	33	−	−	PROPN
ejpam-4274	772	34	f	f	NOUN
ejpam-4274	772	35	and	and	CCONJ
ejpam-4274	772	36	hence	hence	ADV
ejpam-4274	772	37	f	f	PROPN
ejpam-4274	772	38	⊆	⊆	NUM
ejpam-4274	772	39	x	x	SYM
ejpam-4274	772	40	−	−	NUM
ejpam-4274	772	41	v	v	NOUN
ejpam-4274	772	42	(	(	PUNCT
ejpam-4274	772	43	λ	λ	PROPN
ejpam-4274	772	44	,	,	PUNCT
ejpam-4274	772	45	p	p	NOUN
ejpam-4274	772	46	)	)	PUNCT
ejpam-4274	772	47	.	.	PUNCT
ejpam-4274	773	1	this	this	PRON
ejpam-4274	773	2	shows	show	VERB
ejpam-4274	773	3	that	that	SCONJ
ejpam-4274	773	4	(	(	PUNCT
ejpam-4274	773	5	x	x	X
ejpam-4274	773	6	,	,	PUNCT
ejpam-4274	773	7	τ	τ	X
ejpam-4274	773	8	)	)	PUNCT
ejpam-4274	773	9	is	be	AUX
ejpam-4274	773	10	λp	λp	NOUN
ejpam-4274	773	11	-	-	PUNCT
ejpam-4274	773	12	regular	regular	ADJ
ejpam-4274	773	13	.	.	PUNCT
ejpam-4274	774	1	lemma	lemma	PROPN
ejpam-4274	774	2	16	16	NUM
ejpam-4274	774	3	.	.	PUNCT
ejpam-4274	775	1	let	let	AUX
ejpam-4274	775	2	(	(	PUNCT
ejpam-4274	775	3	x	x	NOUN
ejpam-4274	775	4	,	,	PUNCT
ejpam-4274	775	5	τ	τ	X
ejpam-4274	775	6	)	)	PUNCT
ejpam-4274	775	7	be	be	VERB
ejpam-4274	775	8	a	a	DET
ejpam-4274	775	9	λp	λp	NOUN
ejpam-4274	775	10	-	-	PUNCT
ejpam-4274	775	11	regular	regular	ADJ
ejpam-4274	775	12	space	space	NOUN
ejpam-4274	775	13	.	.	PUNCT
ejpam-4274	776	1	then	then	ADV
ejpam-4274	776	2	,	,	PUNCT
ejpam-4274	776	3	the	the	DET
ejpam-4274	776	4	following	follow	VERB
ejpam-4274	776	5	properties	property	NOUN
ejpam-4274	776	6	hold	hold	VERB
ejpam-4274	776	7	:	:	PUNCT
ejpam-4274	776	8	(	(	PUNCT
ejpam-4274	776	9	1	1	X
ejpam-4274	776	10	)	)	PUNCT
ejpam-4274	776	11	a(λ	a(λ	ADV
ejpam-4274	776	12	,	,	PUNCT
ejpam-4274	776	13	p	p	X
ejpam-4274	776	14	)	)	PUNCT
ejpam-4274	776	15	=	=	PUNCT
ejpam-4274	776	16	aθ(λ	aθ(λ	NOUN
ejpam-4274	776	17	,	,	PUNCT
ejpam-4274	776	18	p	p	NOUN
ejpam-4274	776	19	)	)	PUNCT
ejpam-4274	776	20	for	for	ADP
ejpam-4274	776	21	every	every	DET
ejpam-4274	776	22	subset	subset	NOUN
ejpam-4274	776	23	a	a	PRON
ejpam-4274	776	24	of	of	ADP
ejpam-4274	776	25	x.	x.	NOUN
ejpam-4274	776	26	(	(	PUNCT
ejpam-4274	776	27	2	2	X
ejpam-4274	776	28	)	)	PUNCT
ejpam-4274	776	29	every	every	PRON
ejpam-4274	776	30	(	(	PUNCT
ejpam-4274	776	31	λ	λ	NOUN
ejpam-4274	776	32	,	,	PUNCT
ejpam-4274	776	33	p)-open	p)-open	VERB
ejpam-4274	776	34	set	set	VERB
ejpam-4274	776	35	is	be	AUX
ejpam-4274	776	36	θ(λ	θ(λ	PROPN
ejpam-4274	776	37	,	,	PUNCT
ejpam-4274	776	38	p)-open	p)-open	NOUN
ejpam-4274	776	39	.	.	PUNCT
ejpam-4274	777	1	proof	proof	NOUN
ejpam-4274	777	2	.	.	PUNCT
ejpam-4274	778	1	(	(	PUNCT
ejpam-4274	778	2	1	1	X
ejpam-4274	778	3	)	)	PUNCT
ejpam-4274	778	4	in	in	ADP
ejpam-4274	778	5	general	general	ADJ
ejpam-4274	778	6	,	,	PUNCT
ejpam-4274	778	7	we	we	PRON
ejpam-4274	778	8	have	have	VERB
ejpam-4274	778	9	a(λ	a(λ	ADV
ejpam-4274	778	10	,	,	PUNCT
ejpam-4274	778	11	p	p	X
ejpam-4274	778	12	)	)	PUNCT
ejpam-4274	779	1	⊆	⊆	NUM
ejpam-4274	779	2	aθ(λ	aθ(λ	NOUN
ejpam-4274	779	3	,	,	PUNCT
ejpam-4274	779	4	p	p	NOUN
ejpam-4274	779	5	)	)	PUNCT
ejpam-4274	779	6	for	for	ADP
ejpam-4274	779	7	every	every	DET
ejpam-4274	779	8	subset	subset	NOUN
ejpam-4274	779	9	a	a	PRON
ejpam-4274	779	10	of	of	ADP
ejpam-4274	779	11	x.	x.	NOUN
ejpam-4274	779	12	next	next	ADV
ejpam-4274	779	13	,	,	PUNCT
ejpam-4274	779	14	we	we	PRON
ejpam-4274	779	15	show	show	VERB
ejpam-4274	779	16	that	that	SCONJ
ejpam-4274	779	17	aθ(λ	aθ(λ	NOUN
ejpam-4274	779	18	,	,	PUNCT
ejpam-4274	779	19	p	p	NOUN
ejpam-4274	779	20	)	)	PUNCT
ejpam-4274	779	21	⊆	⊆	NUM
ejpam-4274	779	22	a(λ	a(λ	ADV
ejpam-4274	779	23	,	,	PUNCT
ejpam-4274	779	24	p	p	NOUN
ejpam-4274	779	25	)	)	PUNCT
ejpam-4274	779	26	.	.	PUNCT
ejpam-4274	780	1	let	let	VERB
ejpam-4274	780	2	x	x	PUNCT
ejpam-4274	780	3	∈	∈	PROPN
ejpam-4274	780	4	aθ(λ	aθ(λ	NOUN
ejpam-4274	780	5	,	,	PUNCT
ejpam-4274	780	6	p	p	NOUN
ejpam-4274	780	7	)	)	PUNCT
ejpam-4274	780	8	and	and	CCONJ
ejpam-4274	780	9	u	u	PRON
ejpam-4274	780	10	be	be	VERB
ejpam-4274	780	11	any	any	DET
ejpam-4274	780	12	(	(	PUNCT
ejpam-4274	780	13	λ	λ	NOUN
ejpam-4274	780	14	,	,	PUNCT
ejpam-4274	780	15	p)-open	p)-open	VERB
ejpam-4274	780	16	set	set	VERB
ejpam-4274	780	17	containing	contain	VERB
ejpam-4274	780	18	x.	x.	NOUN
ejpam-4274	780	19	by	by	ADP
ejpam-4274	780	20	lemma	lemma	PROPN
ejpam-4274	780	21	15	15	NUM
ejpam-4274	780	22	,	,	PUNCT
ejpam-4274	780	23	there	there	PRON
ejpam-4274	780	24	exists	exist	VERB
ejpam-4274	780	25	a	a	DET
ejpam-4274	780	26	(	(	PUNCT
ejpam-4274	780	27	λ	λ	NOUN
ejpam-4274	780	28	,	,	PUNCT
ejpam-4274	780	29	p)-open	p)-open	VERB
ejpam-4274	780	30	set	set	VERB
ejpam-4274	780	31	v	v	ADP
ejpam-4274	780	32	such	such	ADJ
ejpam-4274	780	33	that	that	SCONJ
ejpam-4274	780	34	x	x	SYM
ejpam-4274	780	35	∈	∈	NOUN
ejpam-4274	780	36	v	v	ADP
ejpam-4274	780	37	⊆	⊆	NUM
ejpam-4274	780	38	v	v	NOUN
ejpam-4274	780	39	(	(	PUNCT
ejpam-4274	780	40	λ	λ	PROPN
ejpam-4274	780	41	,	,	PUNCT
ejpam-4274	780	42	p	p	NOUN
ejpam-4274	780	43	)	)	PUNCT
ejpam-4274	780	44	⊆	⊆	NUM
ejpam-4274	780	45	u	u	NOUN
ejpam-4274	780	46	.	.	PUNCT
ejpam-4274	781	1	since	since	SCONJ
ejpam-4274	781	2	x	x	PROPN
ejpam-4274	781	3	∈	∈	PROPN
ejpam-4274	781	4	aθ(λ	aθ(λ	NOUN
ejpam-4274	781	5	,	,	PUNCT
ejpam-4274	781	6	p	p	NOUN
ejpam-4274	781	7	)	)	PUNCT
ejpam-4274	781	8	,	,	PUNCT
ejpam-4274	781	9	it	it	PRON
ejpam-4274	781	10	follows	follow	VERB
ejpam-4274	781	11	that	that	SCONJ
ejpam-4274	781	12	a	a	DET
ejpam-4274	781	13	∩	∩	ADJ
ejpam-4274	781	14	v	v	NOUN
ejpam-4274	781	15	(	(	PUNCT
ejpam-4274	781	16	λ	λ	PROPN
ejpam-4274	781	17	,	,	PUNCT
ejpam-4274	781	18	p	p	NOUN
ejpam-4274	781	19	)	)	PUNCT
ejpam-4274	781	20	6=	6=	NOUN
ejpam-4274	781	21	∅	∅	NOUN
ejpam-4274	781	22	and	and	CCONJ
ejpam-4274	781	23	hence	hence	ADV
ejpam-4274	781	24	u	u	NOUN
ejpam-4274	781	25	∩	∩	NOUN
ejpam-4274	781	26	a	a	DET
ejpam-4274	781	27	6=	6=	NUM
ejpam-4274	781	28	∅.	∅.	ADP
ejpam-4274	781	29	thus	thus	ADV
ejpam-4274	781	30	,	,	PUNCT
ejpam-4274	781	31	x	x	SYM
ejpam-4274	781	32	∈	∈	PROPN
ejpam-4274	781	33	a(λ	a(λ	ADV
ejpam-4274	781	34	,	,	PUNCT
ejpam-4274	781	35	p	p	NOUN
ejpam-4274	781	36	)	)	PUNCT
ejpam-4274	781	37	.	.	PUNCT
ejpam-4274	782	1	consequently	consequently	ADV
ejpam-4274	782	2	,	,	PUNCT
ejpam-4274	782	3	we	we	PRON
ejpam-4274	782	4	obtain	obtain	VERB
ejpam-4274	782	5	aθ(λ	aθ(λ	NOUN
ejpam-4274	782	6	,	,	PUNCT
ejpam-4274	782	7	p	p	NOUN
ejpam-4274	782	8	)	)	PUNCT
ejpam-4274	782	9	⊆	⊆	NUM
ejpam-4274	782	10	a(λ	a(λ	ADV
ejpam-4274	782	11	,	,	PUNCT
ejpam-4274	782	12	p	p	NOUN
ejpam-4274	782	13	)	)	PUNCT
ejpam-4274	782	14	.	.	PUNCT
ejpam-4274	783	1	(	(	PUNCT
ejpam-4274	783	2	2	2	X
ejpam-4274	783	3	)	)	PUNCT
ejpam-4274	783	4	let	let	VERB
ejpam-4274	783	5	v	v	PART
ejpam-4274	783	6	be	be	AUX
ejpam-4274	783	7	a	a	DET
ejpam-4274	783	8	(	(	PUNCT
ejpam-4274	783	9	λ	λ	NOUN
ejpam-4274	783	10	,	,	PUNCT
ejpam-4274	783	11	p)-open	p)-open	ADJ
ejpam-4274	783	12	set	set	VERB
ejpam-4274	783	13	.	.	PUNCT
ejpam-4274	784	1	by	by	ADP
ejpam-4274	784	2	(	(	PUNCT
ejpam-4274	784	3	1	1	NUM
ejpam-4274	784	4	)	)	PUNCT
ejpam-4274	784	5	,	,	PUNCT
ejpam-4274	784	6	we	we	PRON
ejpam-4274	784	7	have	have	VERB
ejpam-4274	784	8	x	x	X
ejpam-4274	784	9	−	−	NOUN
ejpam-4274	784	10	v	v	NOUN
ejpam-4274	784	11	=	=	PUNCT
ejpam-4274	785	1	[	[	X
ejpam-4274	785	2	x	x	X
ejpam-4274	785	3	−	−	NOUN
ejpam-4274	785	4	v	v	NOUN
ejpam-4274	785	5	]	]	X
ejpam-4274	785	6	(	(	PUNCT
ejpam-4274	785	7	λ	λ	X
ejpam-4274	785	8	,	,	PUNCT
ejpam-4274	785	9	p	p	NOUN
ejpam-4274	785	10	)	)	PUNCT
ejpam-4274	785	11	=	=	PUNCT
ejpam-4274	786	1	[	[	X
ejpam-4274	786	2	x	x	X
ejpam-4274	786	3	−	−	X
ejpam-4274	786	4	v	v	X
ejpam-4274	786	5	]	]	X
ejpam-4274	786	6	θ(λ	θ(λ	PROPN
ejpam-4274	786	7	,	,	PUNCT
ejpam-4274	786	8	p	p	NOUN
ejpam-4274	786	9	)	)	PUNCT
ejpam-4274	786	10	and	and	CCONJ
ejpam-4274	786	11	hence	hence	ADV
ejpam-4274	786	12	x	x	X
ejpam-4274	787	1	−	−	PROPN
ejpam-4274	787	2	v	v	NOUN
ejpam-4274	787	3	is	be	AUX
ejpam-4274	787	4	θ(λ	θ(λ	PROPN
ejpam-4274	787	5	,	,	PUNCT
ejpam-4274	787	6	p)-closed	p)-close	VERB
ejpam-4274	787	7	.	.	PUNCT
ejpam-4274	788	1	thus	thus	ADV
ejpam-4274	788	2	,	,	PUNCT
ejpam-4274	788	3	v	v	NOUN
ejpam-4274	788	4	is	be	AUX
ejpam-4274	788	5	θ(λ	θ(λ	PROPN
ejpam-4274	788	6	,	,	PUNCT
ejpam-4274	788	7	p)-open	p)-open	NOUN
ejpam-4274	788	8	.	.	PUNCT
ejpam-4274	789	1	theorem	theorem	PROPN
ejpam-4274	789	2	27	27	NUM
ejpam-4274	789	3	.	.	PUNCT
ejpam-4274	790	1	let	let	AUX
ejpam-4274	790	2	(	(	PUNCT
ejpam-4274	790	3	y	y	PROPN
ejpam-4274	790	4	,	,	PUNCT
ejpam-4274	790	5	σ	σ	PROPN
ejpam-4274	790	6	)	)	PUNCT
ejpam-4274	790	7	be	be	VERB
ejpam-4274	790	8	a	a	DET
ejpam-4274	790	9	λp	λp	NOUN
ejpam-4274	790	10	-	-	PUNCT
ejpam-4274	790	11	regular	regular	ADJ
ejpam-4274	790	12	space	space	NOUN
ejpam-4274	790	13	.	.	PUNCT
ejpam-4274	791	1	for	for	ADP
ejpam-4274	791	2	a	a	DET
ejpam-4274	791	3	function	function	NOUN
ejpam-4274	791	4	f	f	NOUN
ejpam-4274	791	5	:	:	PUNCT
ejpam-4274	791	6	(	(	PUNCT
ejpam-4274	791	7	x	x	X
ejpam-4274	791	8	,	,	PUNCT
ejpam-4274	791	9	τ	τ	X
ejpam-4274	791	10	)	)	PUNCT
ejpam-4274	791	11	→	→	SYM
ejpam-4274	791	12	(	(	PUNCT
ejpam-4274	791	13	y	y	PROPN
ejpam-4274	791	14	,	,	PUNCT
ejpam-4274	791	15	σ	σ	PROPN
ejpam-4274	791	16	)	)	PUNCT
ejpam-4274	791	17	,	,	PUNCT
ejpam-4274	791	18	the	the	DET
ejpam-4274	791	19	following	follow	VERB
ejpam-4274	791	20	properties	property	NOUN
ejpam-4274	791	21	are	be	AUX
ejpam-4274	791	22	equivalent	equivalent	ADJ
ejpam-4274	791	23	:	:	PUNCT
ejpam-4274	791	24	(	(	PUNCT
ejpam-4274	791	25	1	1	X
ejpam-4274	791	26	)	)	PUNCT
ejpam-4274	791	27	f−1(bθ(λ	f−1(bθ(λ	NOUN
ejpam-4274	791	28	,	,	PUNCT
ejpam-4274	791	29	p	p	NOUN
ejpam-4274	791	30	)	)	PUNCT
ejpam-4274	791	31	)	)	PUNCT
ejpam-4274	791	32	is	be	AUX
ejpam-4274	791	33	θ(λ	θ(λ	PROPN
ejpam-4274	791	34	,	,	PUNCT
ejpam-4274	791	35	p)-closed	p)-close	VERB
ejpam-4274	791	36	in	in	ADP
ejpam-4274	791	37	x	x	PUNCT
ejpam-4274	791	38	for	for	ADP
ejpam-4274	791	39	every	every	DET
ejpam-4274	791	40	subset	subset	NOUN
ejpam-4274	791	41	b	b	PROPN
ejpam-4274	791	42	of	of	ADP
ejpam-4274	791	43	y	y	PROPN
ejpam-4274	791	44	;	;	PUNCT
ejpam-4274	791	45	(	(	PUNCT
ejpam-4274	791	46	2	2	X
ejpam-4274	791	47	)	)	PUNCT
ejpam-4274	791	48	f	f	PROPN
ejpam-4274	791	49	is	be	AUX
ejpam-4274	791	50	weakly	weakly	ADJ
ejpam-4274	791	51	(	(	PUNCT
ejpam-4274	791	52	λ	λ	NOUN
ejpam-4274	791	53	,	,	PUNCT
ejpam-4274	791	54	p)-continuous	p)-continuous	ADJ
ejpam-4274	791	55	;	;	PUNCT
ejpam-4274	791	56	(	(	PUNCT
ejpam-4274	791	57	3	3	X
ejpam-4274	791	58	)	)	PUNCT
ejpam-4274	791	59	f−1(f	f−1(f	NOUN
ejpam-4274	791	60	)	)	PUNCT
ejpam-4274	791	61	is	be	AUX
ejpam-4274	791	62	(	(	PUNCT
ejpam-4274	791	63	λ	λ	X
ejpam-4274	791	64	,	,	PUNCT
ejpam-4274	791	65	p)-closed	p)-close	VERB
ejpam-4274	791	66	in	in	ADP
ejpam-4274	791	67	x	x	PUNCT
ejpam-4274	791	68	for	for	ADP
ejpam-4274	791	69	every	every	DET
ejpam-4274	791	70	θ(λ	θ(λ	PROPN
ejpam-4274	791	71	,	,	PUNCT
ejpam-4274	791	72	p)-closed	p)-close	VERB
ejpam-4274	791	73	subset	subset	NOUN
ejpam-4274	791	74	f	f	PROPN
ejpam-4274	791	75	of	of	ADP
ejpam-4274	791	76	y	y	PROPN
ejpam-4274	791	77	;	;	PUNCT
ejpam-4274	791	78	references	reference	NOUN
ejpam-4274	791	79	435	435	NUM
ejpam-4274	791	80	(	(	PUNCT
ejpam-4274	791	81	4	4	NUM
ejpam-4274	791	82	)	)	PUNCT
ejpam-4274	791	83	f−1(v	f−1(v	NOUN
ejpam-4274	791	84	)	)	PUNCT
ejpam-4274	791	85	is	be	AUX
ejpam-4274	791	86	(	(	PUNCT
ejpam-4274	791	87	λ	λ	X
ejpam-4274	791	88	,	,	PUNCT
ejpam-4274	791	89	p)-open	p)-open	VERB
ejpam-4274	791	90	in	in	ADP
ejpam-4274	791	91	x	x	PUNCT
ejpam-4274	791	92	for	for	ADP
ejpam-4274	791	93	every	every	DET
ejpam-4274	791	94	θ(λ	θ(λ	PROPN
ejpam-4274	791	95	,	,	PUNCT
ejpam-4274	791	96	p)-open	p)-open	VERB
ejpam-4274	791	97	subset	subset	VERB
ejpam-4274	791	98	v	v	NOUN
ejpam-4274	791	99	of	of	ADP
ejpam-4274	791	100	y	y	PROPN
ejpam-4274	791	101	.	.	PUNCT
ejpam-4274	792	1	proof	proof	NOUN
ejpam-4274	792	2	.	.	PUNCT
ejpam-4274	793	1	(	(	PUNCT
ejpam-4274	793	2	1	1	X
ejpam-4274	793	3	)	)	PUNCT
ejpam-4274	793	4	⇒	⇒	NOUN
ejpam-4274	793	5	(	(	PUNCT
ejpam-4274	793	6	2	2	NUM
ejpam-4274	793	7	):	):	PUNCT
ejpam-4274	793	8	let	let	VERB
ejpam-4274	793	9	b	b	X
ejpam-4274	793	10	be	be	AUX
ejpam-4274	793	11	any	any	DET
ejpam-4274	793	12	subset	subset	NOUN
ejpam-4274	793	13	of	of	ADP
ejpam-4274	793	14	y	y	PROPN
ejpam-4274	793	15	.	.	PUNCT
ejpam-4274	794	1	then	then	ADV
ejpam-4274	794	2	,	,	PUNCT
ejpam-4274	794	3	[	[	X
ejpam-4274	794	4	f−1(b)](λ	f−1(b)](λ	X
ejpam-4274	794	5	,	,	PUNCT
ejpam-4274	794	6	p	p	NOUN
ejpam-4274	794	7	)	)	PUNCT
ejpam-4274	794	8	⊆	⊆	NUM
ejpam-4274	794	9	[	[	X
ejpam-4274	794	10	f−1(bθ(λ	f−1(bθ(λ	NOUN
ejpam-4274	794	11	,	,	PUNCT
ejpam-4274	794	12	p))](λ	p))](λ	PRON
ejpam-4274	794	13	,	,	PUNCT
ejpam-4274	794	14	p	p	NOUN
ejpam-4274	794	15	)	)	PUNCT
ejpam-4274	794	16	=	=	SYM
ejpam-4274	794	17	f−1(bθ(λ	f−1(bθ(λ	NOUN
ejpam-4274	794	18	,	,	PUNCT
ejpam-4274	794	19	p	p	NOUN
ejpam-4274	794	20	)	)	PUNCT
ejpam-4274	794	21	)	)	PUNCT
ejpam-4274	794	22	,	,	PUNCT
ejpam-4274	794	23	by	by	ADP
ejpam-4274	794	24	theorem	theorem	NOUN
ejpam-4274	794	25	26	26	NUM
ejpam-4274	794	26	,	,	PUNCT
ejpam-4274	794	27	f	f	PROPN
ejpam-4274	794	28	is	be	AUX
ejpam-4274	794	29	is	be	AUX
ejpam-4274	794	30	weakly	weakly	ADJ
ejpam-4274	794	31	(	(	PUNCT
ejpam-4274	794	32	λ	λ	NOUN
ejpam-4274	794	33	,	,	PUNCT
ejpam-4274	794	34	p)-continuous	p)-continuous	ADJ
ejpam-4274	794	35	.	.	PUNCT
ejpam-4274	795	1	(	(	PUNCT
ejpam-4274	795	2	2	2	X
ejpam-4274	795	3	)	)	PUNCT
ejpam-4274	795	4	⇒	⇒	NOUN
ejpam-4274	795	5	(	(	PUNCT
ejpam-4274	795	6	3	3	NUM
ejpam-4274	795	7	):	):	PUNCT
ejpam-4274	795	8	let	let	VERB
ejpam-4274	795	9	f	f	PRON
ejpam-4274	795	10	be	be	AUX
ejpam-4274	795	11	any	any	DET
ejpam-4274	795	12	θ(λ	θ(λ	PROPN
ejpam-4274	795	13	,	,	PUNCT
ejpam-4274	795	14	p)-closed	p)-close	VERB
ejpam-4274	795	15	subset	subset	NOUN
ejpam-4274	795	16	of	of	ADP
ejpam-4274	795	17	y	y	PROPN
ejpam-4274	795	18	.	.	PUNCT
ejpam-4274	796	1	by	by	ADP
ejpam-4274	796	2	theorem	theorem	NOUN
ejpam-4274	796	3	26	26	NUM
ejpam-4274	796	4	,	,	PUNCT
ejpam-4274	796	5	we	we	PRON
ejpam-4274	796	6	have	have	VERB
ejpam-4274	796	7	[	[	X
ejpam-4274	796	8	f−1(f	f−1(f	PROPN
ejpam-4274	796	9	)	)	PUNCT
ejpam-4274	796	10	]	]	PUNCT
ejpam-4274	796	11	(	(	PUNCT
ejpam-4274	796	12	λ	λ	X
ejpam-4274	796	13	,	,	PUNCT
ejpam-4274	796	14	p	p	NOUN
ejpam-4274	796	15	)	)	PUNCT
ejpam-4274	796	16	⊆	⊆	NUM
ejpam-4274	796	17	f−1(f	f−1(f	PROPN
ejpam-4274	796	18	θ(λ	θ(λ	PROPN
ejpam-4274	796	19	,	,	PUNCT
ejpam-4274	796	20	p	p	NOUN
ejpam-4274	796	21	)	)	PUNCT
ejpam-4274	796	22	)	)	PUNCT
ejpam-4274	797	1	=	=	SYM
ejpam-4274	797	2	f−1(f	f−1(f	PROPN
ejpam-4274	797	3	)	)	PUNCT
ejpam-4274	797	4	and	and	CCONJ
ejpam-4274	797	5	hence	hence	ADV
ejpam-4274	797	6	f−1(f	f−1(f	PROPN
ejpam-4274	797	7	)	)	PUNCT
ejpam-4274	797	8	is	be	AUX
ejpam-4274	797	9	(	(	PUNCT
ejpam-4274	797	10	λ	λ	X
ejpam-4274	797	11	,	,	PUNCT
ejpam-4274	797	12	p)-closed	p)-close	VERB
ejpam-4274	797	13	in	in	ADP
ejpam-4274	797	14	x.	x.	NOUN
ejpam-4274	797	15	(	(	PUNCT
ejpam-4274	797	16	3	3	NUM
ejpam-4274	797	17	)	)	PUNCT
ejpam-4274	797	18	⇒	⇒	NOUN
ejpam-4274	797	19	(	(	PUNCT
ejpam-4274	797	20	4	4	NUM
ejpam-4274	797	21	):	):	PUNCT
ejpam-4274	797	22	let	let	VERB
ejpam-4274	797	23	v	v	PART
ejpam-4274	797	24	be	be	AUX
ejpam-4274	797	25	any	any	DET
ejpam-4274	797	26	θ(λ	θ(λ	PROPN
ejpam-4274	797	27	,	,	PUNCT
ejpam-4274	797	28	p)-open	p)-open	PUNCT
ejpam-4274	797	29	subset	subset	NOUN
ejpam-4274	797	30	of	of	ADP
ejpam-4274	797	31	y	y	PROPN
ejpam-4274	797	32	.	.	PUNCT
ejpam-4274	798	1	then	then	ADV
ejpam-4274	798	2	,	,	PUNCT
ejpam-4274	798	3	y	y	PROPN
ejpam-4274	798	4	−	−	PROPN
ejpam-4274	798	5	v	v	PROPN
ejpam-4274	798	6	is	be	AUX
ejpam-4274	798	7	θ(λ	θ(λ	PROPN
ejpam-4274	798	8	,	,	PUNCT
ejpam-4274	798	9	p)-closed	p)-close	VERB
ejpam-4274	798	10	,	,	PUNCT
ejpam-4274	798	11	by	by	ADP
ejpam-4274	798	12	(	(	PUNCT
ejpam-4274	798	13	3	3	NUM
ejpam-4274	798	14	)	)	PUNCT
ejpam-4274	798	15	,	,	PUNCT
ejpam-4274	798	16	x	x	PUNCT
ejpam-4274	798	17	−	−	PROPN
ejpam-4274	798	18	f−1(v	f−1(v	NOUN
ejpam-4274	798	19	)	)	PUNCT
ejpam-4274	799	1	=	=	PUNCT
ejpam-4274	799	2	f−1(y	f−1(y	PROPN
ejpam-4274	799	3	−	−	PROPN
ejpam-4274	799	4	v	v	NOUN
ejpam-4274	799	5	)	)	PUNCT
ejpam-4274	799	6	is	be	AUX
ejpam-4274	799	7	(	(	PUNCT
ejpam-4274	799	8	λ	λ	X
ejpam-4274	799	9	,	,	PUNCT
ejpam-4274	799	10	p)-closed	p)-close	VERB
ejpam-4274	799	11	in	in	ADP
ejpam-4274	799	12	x.	x.	NOUN
ejpam-4274	799	13	thus	thus	ADV
ejpam-4274	799	14	,	,	PUNCT
ejpam-4274	799	15	f−1(v	f−1(v	PROPN
ejpam-4274	799	16	)	)	PUNCT
ejpam-4274	799	17	is	be	AUX
ejpam-4274	799	18	(	(	PUNCT
ejpam-4274	799	19	λ	λ	X
ejpam-4274	799	20	,	,	PUNCT
ejpam-4274	799	21	p)-open	p)-open	ADJ
ejpam-4274	799	22	.	.	PUNCT
ejpam-4274	800	1	(	(	PUNCT
ejpam-4274	800	2	4	4	X
ejpam-4274	800	3	)	)	PUNCT
ejpam-4274	800	4	⇒	⇒	NOUN
ejpam-4274	800	5	(	(	PUNCT
ejpam-4274	800	6	1	1	NUM
ejpam-4274	800	7	):	):	PUNCT
ejpam-4274	800	8	let	let	VERB
ejpam-4274	800	9	b	b	X
ejpam-4274	800	10	be	be	AUX
ejpam-4274	800	11	any	any	DET
ejpam-4274	800	12	subset	subset	NOUN
ejpam-4274	800	13	of	of	ADP
ejpam-4274	800	14	y	y	PROPN
ejpam-4274	800	15	.	.	PUNCT
ejpam-4274	801	1	by	by	ADP
ejpam-4274	801	2	lemma	lemma	PROPN
ejpam-4274	801	3	14	14	NUM
ejpam-4274	801	4	,	,	PUNCT
ejpam-4274	801	5	bθ(λ	bθ(λ	X
ejpam-4274	801	6	,	,	PUNCT
ejpam-4274	801	7	p	p	NOUN
ejpam-4274	801	8	)	)	PUNCT
ejpam-4274	801	9	is	be	AUX
ejpam-4274	801	10	(	(	PUNCT
ejpam-4274	801	11	λ	λ	X
ejpam-4274	801	12	,	,	PUNCT
ejpam-4274	801	13	p)-closed	p)-close	VERB
ejpam-4274	801	14	in	in	ADP
ejpam-4274	801	15	y	y	PROPN
ejpam-4274	801	16	and	and	CCONJ
ejpam-4274	801	17	by	by	ADP
ejpam-4274	801	18	lemma	lemma	PROPN
ejpam-4274	801	19	16	16	NUM
ejpam-4274	801	20	,	,	PUNCT
ejpam-4274	801	21	y	y	PROPN
ejpam-4274	801	22	−bθ(λ	−bθ(λ	PROPN
ejpam-4274	801	23	,	,	PUNCT
ejpam-4274	801	24	p	p	NOUN
ejpam-4274	801	25	)	)	PUNCT
ejpam-4274	801	26	is	be	AUX
ejpam-4274	801	27	θ(λ	θ(λ	PROPN
ejpam-4274	801	28	,	,	PUNCT
ejpam-4274	801	29	p)-open	p)-open	VERB
ejpam-4274	801	30	in	in	ADP
ejpam-4274	801	31	y	y	PROPN
ejpam-4274	801	32	.	.	PUNCT
ejpam-4274	802	1	thus	thus	ADV
ejpam-4274	802	2	,	,	PUNCT
ejpam-4274	802	3	by	by	ADP
ejpam-4274	802	4	(	(	PUNCT
ejpam-4274	802	5	4	4	NUM
ejpam-4274	802	6	)	)	PUNCT
ejpam-4274	802	7	,	,	PUNCT
ejpam-4274	802	8	we	we	PRON
ejpam-4274	802	9	have	have	VERB
ejpam-4274	802	10	x	x	X
ejpam-4274	802	11	−	−	PRON
ejpam-4274	802	12	f−1(bθ(λ	f−1(bθ(λ	NOUN
ejpam-4274	802	13	,	,	PUNCT
ejpam-4274	802	14	p	p	NOUN
ejpam-4274	802	15	)	)	PUNCT
ejpam-4274	802	16	)	)	PUNCT
ejpam-4274	803	1	=	=	SYM
ejpam-4274	803	2	f−1(y	f−1(y	PROPN
ejpam-4274	804	1	−bθ(λ	−bθ(λ	PROPN
ejpam-4274	804	2	,	,	PUNCT
ejpam-4274	804	3	p	p	NOUN
ejpam-4274	804	4	)	)	PUNCT
ejpam-4274	804	5	)	)	PUNCT
ejpam-4274	804	6	is	be	AUX
ejpam-4274	804	7	(	(	PUNCT
ejpam-4274	804	8	λ	λ	X
ejpam-4274	804	9	,	,	PUNCT
ejpam-4274	804	10	p)-open	p)-open	VERB
ejpam-4274	804	11	in	in	ADP
ejpam-4274	804	12	x	x	PUNCT
ejpam-4274	804	13	and	and	CCONJ
ejpam-4274	804	14	hence	hence	ADV
ejpam-4274	804	15	f−1(bθ(λ	f−1(bθ(λ	ADJ
ejpam-4274	804	16	,	,	PUNCT
ejpam-4274	804	17	p	p	NOUN
ejpam-4274	804	18	)	)	PUNCT
ejpam-4274	804	19	)	)	PUNCT
ejpam-4274	805	1	is	be	AUX
ejpam-4274	805	2	θ(λ	θ(λ	PROPN
ejpam-4274	805	3	,	,	PUNCT
ejpam-4274	805	4	p)-closed	p)-close	VERB
ejpam-4274	805	5	.	.	PUNCT
ejpam-4274	806	1	7	7	X
ejpam-4274	806	2	.	.	X
ejpam-4274	806	3	conclusion	conclusion	NOUN
ejpam-4274	806	4	closedness	closedness	NOUN
ejpam-4274	806	5	and	and	CCONJ
ejpam-4274	806	6	openness	openness	NOUN
ejpam-4274	806	7	are	be	AUX
ejpam-4274	806	8	fundamental	fundamental	ADJ
ejpam-4274	806	9	with	with	ADP
ejpam-4274	806	10	respect	respect	NOUN
ejpam-4274	806	11	to	to	ADP
ejpam-4274	806	12	the	the	DET
ejpam-4274	806	13	investigation	investigation	NOUN
ejpam-4274	806	14	of	of	ADP
ejpam-4274	806	15	general	general	ADJ
ejpam-4274	806	16	topological	topological	ADJ
ejpam-4274	806	17	spaces	space	NOUN
ejpam-4274	806	18	.	.	PUNCT
ejpam-4274	807	1	various	various	ADJ
ejpam-4274	807	2	types	type	NOUN
ejpam-4274	807	3	of	of	ADP
ejpam-4274	807	4	generalizations	generalization	NOUN
ejpam-4274	807	5	of	of	ADP
ejpam-4274	807	6	closed	closed	ADJ
ejpam-4274	807	7	sets	set	NOUN
ejpam-4274	807	8	and	and	CCONJ
ejpam-4274	807	9	open	open	ADJ
ejpam-4274	807	10	sets	set	NOUN
ejpam-4274	807	11	in	in	ADP
ejpam-4274	807	12	topological	topological	ADJ
ejpam-4274	807	13	spaces	space	NOUN
ejpam-4274	807	14	have	have	AUX
ejpam-4274	807	15	been	be	AUX
ejpam-4274	807	16	researched	research	VERB
ejpam-4274	807	17	by	by	ADP
ejpam-4274	807	18	many	many	ADJ
ejpam-4274	807	19	mathematicians	mathematician	NOUN
ejpam-4274	807	20	.	.	PUNCT
ejpam-4274	808	1	this	this	DET
ejpam-4274	808	2	article	article	NOUN
ejpam-4274	808	3	is	be	AUX
ejpam-4274	808	4	devoted	devote	VERB
ejpam-4274	808	5	to	to	ADP
ejpam-4274	808	6	introducing	introduce	VERB
ejpam-4274	808	7	and	and	CCONJ
ejpam-4274	808	8	discussing	discuss	VERB
ejpam-4274	808	9	the	the	DET
ejpam-4274	808	10	concepts	concept	NOUN
ejpam-4274	808	11	of	of	ADP
ejpam-4274	808	12	(	(	PUNCT
ejpam-4274	808	13	λ	λ	PROPN
ejpam-4274	808	14	,	,	PUNCT
ejpam-4274	808	15	p)-closed	p)-close	VERB
ejpam-4274	808	16	sets	set	NOUN
ejpam-4274	808	17	and	and	CCONJ
ejpam-4274	808	18	(	(	PUNCT
ejpam-4274	808	19	λ	λ	NOUN
ejpam-4274	808	20	,	,	PUNCT
ejpam-4274	808	21	p)-open	p)-open	VERB
ejpam-4274	808	22	sets	set	NOUN
ejpam-4274	808	23	.	.	PUNCT
ejpam-4274	809	1	moreover	moreover	ADV
ejpam-4274	809	2	,	,	PUNCT
ejpam-4274	809	3	some	some	DET
ejpam-4274	809	4	characterizations	characterization	NOUN
ejpam-4274	809	5	of	of	ADP
ejpam-4274	809	6	λp	λp	PROPN
ejpam-4274	809	7	-	-	PUNCT
ejpam-4274	809	8	r0	r0	NOUN
ejpam-4274	809	9	spaces	space	NOUN
ejpam-4274	809	10	are	be	AUX
ejpam-4274	809	11	explored	explore	VERB
ejpam-4274	809	12	.	.	PUNCT
ejpam-4274	810	1	additionally	additionally	ADV
ejpam-4274	810	2	,	,	PUNCT
ejpam-4274	810	3	several	several	ADJ
ejpam-4274	810	4	characterizations	characterization	NOUN
ejpam-4274	810	5	of	of	ADP
ejpam-4274	810	6	weakly	weakly	ADJ
ejpam-4274	810	7	(	(	PUNCT
ejpam-4274	810	8	λ	λ	NOUN
ejpam-4274	810	9	,	,	PUNCT
ejpam-4274	810	10	p)-continuous	p)-continuous	ADJ
ejpam-4274	810	11	functions	function	NOUN
ejpam-4274	810	12	are	be	AUX
ejpam-4274	810	13	established	establish	VERB
ejpam-4274	810	14	.	.	PUNCT
ejpam-4274	811	1	the	the	DET
ejpam-4274	811	2	ideas	idea	NOUN
ejpam-4274	811	3	and	and	CCONJ
ejpam-4274	811	4	results	result	NOUN
ejpam-4274	811	5	of	of	ADP
ejpam-4274	811	6	this	this	DET
ejpam-4274	811	7	article	article	NOUN
ejpam-4274	811	8	may	may	AUX
ejpam-4274	811	9	motivate	motivate	VERB
ejpam-4274	811	10	further	further	ADJ
ejpam-4274	811	11	research	research	NOUN
ejpam-4274	811	12	.	.	PUNCT
ejpam-4274	812	1	acknowledgements	acknowledgement	NOUN
ejpam-4274	812	2	this	this	DET
ejpam-4274	812	3	research	research	NOUN
ejpam-4274	812	4	project	project	NOUN
ejpam-4274	812	5	was	be	AUX
ejpam-4274	812	6	financially	financially	ADV
ejpam-4274	812	7	supported	support	VERB
ejpam-4274	812	8	by	by	ADP
ejpam-4274	812	9	mahasarakham	mahasarakham	PROPN
ejpam-4274	812	10	university	university	PROPN
ejpam-4274	812	11	.	.	PUNCT
ejpam-4274	813	1	references	reference	NOUN
ejpam-4274	813	2	[	[	X
ejpam-4274	813	3	1	1	NUM
ejpam-4274	813	4	]	]	PUNCT
ejpam-4274	813	5	t.	t.	PROPN
ejpam-4274	813	6	m.	m.	PROPN
ejpam-4274	813	7	al	al	PROPN
ejpam-4274	813	8	-	-	PUNCT
ejpam-4274	813	9	shami	shami	PROPN
ejpam-4274	813	10	.	.	PUNCT
ejpam-4274	814	1	complete	complete	ADJ
ejpam-4274	814	2	hausdorffness	hausdorffness	NOUN
ejpam-4274	814	3	and	and	CCONJ
ejpam-4274	814	4	complete	complete	ADJ
ejpam-4274	814	5	regularity	regularity	NOUN
ejpam-4274	814	6	on	on	ADP
ejpam-4274	814	7	supra	supra	PROPN
ejpam-4274	814	8	topological	topological	ADJ
ejpam-4274	814	9	spaces	space	NOUN
ejpam-4274	814	10	.	.	PUNCT
ejpam-4274	815	1	journal	journal	NOUN
ejpam-4274	815	2	of	of	ADP
ejpam-4274	815	3	applied	apply	VERB
ejpam-4274	815	4	mathematics	mathematic	NOUN
ejpam-4274	815	5	,	,	PUNCT
ejpam-4274	815	6	2021	2021	NUM
ejpam-4274	815	7	:	:	PUNCT
ejpam-4274	815	8	article	article	NOUN
ejpam-4274	815	9	i	i	PROPN
ejpam-4274	815	10	d	d	PROPN
ejpam-4274	815	11	5517702	5517702	NUM
ejpam-4274	815	12	,	,	PUNCT
ejpam-4274	815	13	7	7	NUM
ejpam-4274	815	14	pages	page	NOUN
ejpam-4274	815	15	,	,	PUNCT
ejpam-4274	815	16	2021	2021	NUM
ejpam-4274	815	17	.	.	PUNCT
ejpam-4274	816	1	[	[	X
ejpam-4274	816	2	2	2	X
ejpam-4274	816	3	]	]	PUNCT
ejpam-4274	816	4	t.	t.	PROPN
ejpam-4274	816	5	m.	m.	PROPN
ejpam-4274	816	6	al	al	PROPN
ejpam-4274	816	7	-	-	PUNCT
ejpam-4274	816	8	shami	shami	PROPN
ejpam-4274	816	9	,	,	PUNCT
ejpam-4274	816	10	e.	e.	PROPN
ejpam-4274	816	11	a.	a.	PROPN
ejpam-4274	816	12	abo	abo	PROPN
ejpam-4274	816	13	-	-	PUNCT
ejpam-4274	816	14	tabl	tabl	NOUN
ejpam-4274	816	15	,	,	PUNCT
ejpam-4274	816	16	b.	b.	PROPN
ejpam-4274	816	17	a.	a.	PROPN
ejpam-4274	816	18	asaad	asaad	PROPN
ejpam-4274	816	19	,	,	PUNCT
ejpam-4274	816	20	and	and	CCONJ
ejpam-4274	816	21	m.	m.	NOUN
ejpam-4274	816	22	a.	a.	NOUN
ejpam-4274	816	23	arahet	arahet	PROPN
ejpam-4274	816	24	.	.	PUNCT
ejpam-4274	817	1	limit	limit	NOUN
ejpam-4274	817	2	points	point	NOUN
ejpam-4274	817	3	and	and	CCONJ
ejpam-4274	817	4	separation	separation	NOUN
ejpam-4274	817	5	axioms	axiom	NOUN
ejpam-4274	817	6	with	with	ADP
ejpam-4274	817	7	respect	respect	NOUN
ejpam-4274	817	8	to	to	ADP
ejpam-4274	817	9	supra	supra	PROPN
ejpam-4274	817	10	semi	semi	ADJ
ejpam-4274	817	11	-	-	ADJ
ejpam-4274	817	12	open	open	ADJ
ejpam-4274	817	13	sets	set	NOUN
ejpam-4274	817	14	.	.	PUNCT
ejpam-4274	818	1	european	european	ADJ
ejpam-4274	818	2	journal	journal	PROPN
ejpam-4274	818	3	of	of	ADP
ejpam-4274	818	4	pure	pure	ADJ
ejpam-4274	818	5	and	and	CCONJ
ejpam-4274	818	6	applied	applied	ADJ
ejpam-4274	818	7	mathematics	mathematic	NOUN
ejpam-4274	818	8	,	,	PUNCT
ejpam-4274	818	9	13(3):427–443	13(3):427–443	PROPN
ejpam-4274	818	10	,	,	PUNCT
ejpam-4274	818	11	2020	2020	NUM
ejpam-4274	818	12	.	.	PUNCT
ejpam-4274	819	1	[	[	X
ejpam-4274	819	2	3	3	NUM
ejpam-4274	819	3	]	]	PUNCT
ejpam-4274	819	4	m.	m.	NOUN
ejpam-4274	819	5	caldas	caldas	PROPN
ejpam-4274	819	6	.	.	PUNCT
ejpam-4274	820	1	a	a	DET
ejpam-4274	820	2	separation	separation	NOUN
ejpam-4274	820	3	axiom	axiom	NOUN
ejpam-4274	820	4	between	between	ADP
ejpam-4274	820	5	pre	pre	NOUN
ejpam-4274	820	6	-	-	NOUN
ejpam-4274	820	7	t0	t0	NOUN
ejpam-4274	820	8	and	and	CCONJ
ejpam-4274	820	9	pre	pre	ADJ
ejpam-4274	820	10	-	-	NOUN
ejpam-4274	820	11	t1	t1	NOUN
ejpam-4274	820	12	.	.	PUNCT
ejpam-4274	821	1	east	east	PROPN
ejpam-4274	821	2	-	-	PUNCT
ejpam-4274	821	3	west	west	PROPN
ejpam-4274	821	4	journal	journal	PROPN
ejpam-4274	821	5	of	of	ADP
ejpam-4274	821	6	mathematics	mathematic	NOUN
ejpam-4274	821	7	,	,	PUNCT
ejpam-4274	821	8	3:171–177	3:171–177	NUM
ejpam-4274	821	9	,	,	PUNCT
ejpam-4274	821	10	2001	2001	NUM
ejpam-4274	821	11	.	.	PUNCT
ejpam-4274	822	1	references	reference	NOUN
ejpam-4274	822	2	436	436	NUM
ejpam-4274	823	1	[	[	X
ejpam-4274	823	2	4	4	NUM
ejpam-4274	823	3	]	]	PUNCT
ejpam-4274	823	4	m.	m.	NOUN
ejpam-4274	823	5	caldas	caldas	PROPN
ejpam-4274	823	6	,	,	PUNCT
ejpam-4274	823	7	s.	s.	PROPN
ejpam-4274	823	8	jafari	jafari	PROPN
ejpam-4274	823	9	,	,	PUNCT
ejpam-4274	823	10	and	and	CCONJ
ejpam-4274	823	11	t.	t.	PROPN
ejpam-4274	823	12	noiri	noiri	PROPN
ejpam-4274	823	13	.	.	PUNCT
ejpam-4274	824	1	characterizations	characterization	NOUN
ejpam-4274	824	2	of	of	ADP
ejpam-4274	824	3	pre	pre	ADJ
ejpam-4274	824	4	-	-	ADJ
ejpam-4274	824	5	r0	r0	ADJ
ejpam-4274	824	6	and	and	CCONJ
ejpam-4274	824	7	pre	pre	ADJ
ejpam-4274	824	8	-	-	ADJ
ejpam-4274	824	9	r1	r1	ADJ
ejpam-4274	824	10	topological	topological	ADJ
ejpam-4274	824	11	spaces	space	NOUN
ejpam-4274	824	12	.	.	PUNCT
ejpam-4274	825	1	topology	topology	NOUN
ejpam-4274	825	2	proceedings	proceeding	NOUN
ejpam-4274	825	3	,	,	PUNCT
ejpam-4274	825	4	25:17–30	25:17–30	PROPN
ejpam-4274	825	5	,	,	PUNCT
ejpam-4274	825	6	2000	2000	NUM
ejpam-4274	825	7	.	.	PUNCT
ejpam-4274	826	1	[	[	X
ejpam-4274	826	2	5	5	NUM
ejpam-4274	826	3	]	]	PUNCT
ejpam-4274	826	4	m.	m.	NOUN
ejpam-4274	826	5	caldas	caldas	PROPN
ejpam-4274	826	6	,	,	PUNCT
ejpam-4274	826	7	s.	s.	PROPN
ejpam-4274	826	8	jafari	jafari	PROPN
ejpam-4274	826	9	,	,	PUNCT
ejpam-4274	826	10	and	and	CCONJ
ejpam-4274	826	11	t.	t.	PROPN
ejpam-4274	826	12	noiri	noiri	PROPN
ejpam-4274	826	13	.	.	PUNCT
ejpam-4274	827	1	characterizations	characterization	NOUN
ejpam-4274	827	2	of	of	ADP
ejpam-4274	827	3	λθ	λθ	NOUN
ejpam-4274	827	4	-	-	PUNCT
ejpam-4274	827	5	r0	r0	NOUN
ejpam-4274	827	6	and	and	CCONJ
ejpam-4274	827	7	λθ	λθ	NOUN
ejpam-4274	827	8	-	-	PUNCT
ejpam-4274	827	9	r1	r1	NOUN
ejpam-4274	827	10	topological	topological	ADJ
ejpam-4274	827	11	spaces	space	NOUN
ejpam-4274	827	12	.	.	PUNCT
ejpam-4274	828	1	acta	acta	PROPN
ejpam-4274	828	2	mathematica	mathematica	PROPN
ejpam-4274	828	3	hungarica	hungarica	PROPN
ejpam-4274	828	4	,	,	PUNCT
ejpam-4274	828	5	103:85–95	103:85–95	NUM
ejpam-4274	828	6	,	,	PUNCT
ejpam-4274	828	7	2004	2004	NUM
ejpam-4274	828	8	.	.	PUNCT
ejpam-4274	829	1	[	[	X
ejpam-4274	829	2	6	6	NUM
ejpam-4274	829	3	]	]	PUNCT
ejpam-4274	829	4	f.	f.	NOUN
ejpam-4274	829	5	cammarato	cammarato	PROPN
ejpam-4274	829	6	and	and	CCONJ
ejpam-4274	829	7	t.	t.	PROPN
ejpam-4274	829	8	noiri	noiri	PROPN
ejpam-4274	829	9	.	.	PUNCT
ejpam-4274	830	1	on	on	ADP
ejpam-4274	830	2	λm	λm	NOUN
ejpam-4274	830	3	-	-	PUNCT
ejpam-4274	830	4	sets	set	NOUN
ejpam-4274	830	5	and	and	CCONJ
ejpam-4274	830	6	related	relate	VERB
ejpam-4274	830	7	topological	topological	ADJ
ejpam-4274	830	8	spaces	space	NOUN
ejpam-4274	830	9	.	.	PUNCT
ejpam-4274	831	1	acta	acta	PROPN
ejpam-4274	831	2	mathematica	mathematica	PROPN
ejpam-4274	831	3	hungarica	hungarica	PROPN
ejpam-4274	831	4	,	,	PUNCT
ejpam-4274	831	5	109:261–279	109:261–279	NUM
ejpam-4274	831	6	,	,	PUNCT
ejpam-4274	831	7	2005	2005	NUM
ejpam-4274	831	8	.	.	PUNCT
ejpam-4274	832	1	[	[	X
ejpam-4274	832	2	7	7	X
ejpam-4274	832	3	]	]	X
ejpam-4274	832	4	h.	h.	NOUN
ejpam-4274	832	5	corson	corson	PROPN
ejpam-4274	832	6	and	and	CCONJ
ejpam-4274	832	7	e.	e.	PROPN
ejpam-4274	832	8	michael	michael	PROPN
ejpam-4274	832	9	.	.	PUNCT
ejpam-4274	833	1	metrizability	metrizability	NOUN
ejpam-4274	833	2	of	of	ADP
ejpam-4274	833	3	certain	certain	ADJ
ejpam-4274	833	4	countable	countable	ADJ
ejpam-4274	833	5	unions	union	NOUN
ejpam-4274	833	6	.	.	PUNCT
ejpam-4274	834	1	illinois	illinois	PROPN
ejpam-4274	834	2	journal	journal	PROPN
ejpam-4274	834	3	of	of	ADP
ejpam-4274	834	4	mathematics	mathematic	NOUN
ejpam-4274	834	5	,	,	PUNCT
ejpam-4274	834	6	8:351–360	8:351–360	NUM
ejpam-4274	834	7	,	,	PUNCT
ejpam-4274	834	8	1964	1964	NUM
ejpam-4274	834	9	.	.	PUNCT
ejpam-4274	835	1	[	[	X
ejpam-4274	835	2	8	8	X
ejpam-4274	835	3	]	]	PUNCT
ejpam-4274	835	4	s.	s.	PROPN
ejpam-4274	835	5	n.	n.	PROPN
ejpam-4274	835	6	el	el	PROPN
ejpam-4274	835	7	-	-	PROPN
ejpam-4274	835	8	deeb	deeb	PROPN
ejpam-4274	835	9	,	,	PUNCT
ejpam-4274	835	10	i.	i.	PROPN
ejpam-4274	835	11	a.	a.	PROPN
ejpam-4274	835	12	hasanein	hasanein	PROPN
ejpam-4274	835	13	,	,	PUNCT
ejpam-4274	835	14	a.	a.	PROPN
ejpam-4274	835	15	s.	s.	PROPN
ejpam-4274	835	16	mashhour	mashhour	PROPN
ejpam-4274	835	17	,	,	PUNCT
ejpam-4274	835	18	and	and	CCONJ
ejpam-4274	835	19	t.	t.	PROPN
ejpam-4274	835	20	noiri	noiri	PROPN
ejpam-4274	835	21	.	.	PUNCT
ejpam-4274	836	1	on	on	ADP
ejpam-4274	836	2	p	p	NOUN
ejpam-4274	836	3	-	-	PUNCT
ejpam-4274	836	4	regular	regular	ADJ
ejpam-4274	836	5	spaces	space	NOUN
ejpam-4274	836	6	.	.	PUNCT
ejpam-4274	837	1	bulletin	bulletin	PROPN
ejpam-4274	837	2	mathématique	mathématique	PROPN
ejpam-4274	837	3	de	de	X
ejpam-4274	837	4	la	la	PROPN
ejpam-4274	837	5	société	société	PROPN
ejpam-4274	837	6	des	des	PROPN
ejpam-4274	837	7	sciences	sciences	PROPN
ejpam-4274	837	8	mathématiques	mathématiques	PROPN
ejpam-4274	837	9	de	de	PROPN
ejpam-4274	837	10	roumanie	roumanie	PROPN
ejpam-4274	837	11	,	,	PUNCT
ejpam-4274	837	12	27:311–315	27:311–315	PROPN
ejpam-4274	837	13	,	,	PUNCT
ejpam-4274	837	14	1983	1983	NUM
ejpam-4274	837	15	.	.	PUNCT
ejpam-4274	838	1	[	[	X
ejpam-4274	838	2	9	9	NUM
ejpam-4274	838	3	]	]	PUNCT
ejpam-4274	838	4	m.	m.	NOUN
ejpam-4274	838	5	e.	e.	PROPN
ejpam-4274	838	6	el	el	PROPN
ejpam-4274	838	7	-	-	PROPN
ejpam-4274	838	8	shafei	shafei	PROPN
ejpam-4274	838	9	,	,	PUNCT
ejpam-4274	838	10	a.	a.	NOUN
ejpam-4274	838	11	h.	h.	PROPN
ejpam-4274	838	12	zakari	zakari	PROPN
ejpam-4274	838	13	,	,	PUNCT
ejpam-4274	838	14	and	and	CCONJ
ejpam-4274	838	15	t.	t.	PROPN
ejpam-4274	838	16	m.	m.	PROPN
ejpam-4274	838	17	al	al	PROPN
ejpam-4274	838	18	-	-	PUNCT
ejpam-4274	838	19	shami	shami	PROPN
ejpam-4274	838	20	.	.	PUNCT
ejpam-4274	839	1	some	some	DET
ejpam-4274	839	2	applications	application	NOUN
ejpam-4274	839	3	of	of	ADP
ejpam-4274	839	4	supra	supra	ADJ
ejpam-4274	839	5	preopen	preopen	ADJ
ejpam-4274	839	6	sets	set	NOUN
ejpam-4274	839	7	.	.	PUNCT
ejpam-4274	840	1	journal	journal	NOUN
ejpam-4274	840	2	of	of	ADP
ejpam-4274	840	3	mathematics	mathematic	NOUN
ejpam-4274	840	4	,	,	PUNCT
ejpam-4274	840	5	2020	2020	NUM
ejpam-4274	840	6	:	:	PUNCT
ejpam-4274	840	7	article	article	NOUN
ejpam-4274	840	8	i	i	PROPN
ejpam-4274	840	9	d	d	PROPN
ejpam-4274	840	10	9634206	9634206	NUM
ejpam-4274	840	11	,	,	PUNCT
ejpam-4274	840	12	11	11	NUM
ejpam-4274	840	13	pages	page	NOUN
ejpam-4274	840	14	,	,	PUNCT
ejpam-4274	840	15	2020	2020	NUM
ejpam-4274	840	16	.	.	PUNCT
ejpam-4274	841	1	[	[	X
ejpam-4274	841	2	10	10	NUM
ejpam-4274	841	3	]	]	PUNCT
ejpam-4274	841	4	m.	m.	NOUN
ejpam-4274	841	5	ganster	ganster	NOUN
ejpam-4274	841	6	,	,	PUNCT
ejpam-4274	841	7	s.	s.	PROPN
ejpam-4274	841	8	jafari	jafari	PROPN
ejpam-4274	841	9	,	,	PUNCT
ejpam-4274	841	10	and	and	CCONJ
ejpam-4274	841	11	t.	t.	PROPN
ejpam-4274	841	12	noiri	noiri	PROPN
ejpam-4274	841	13	.	.	PUNCT
ejpam-4274	842	1	on	on	ADP
ejpam-4274	842	2	pre	pre	ADJ
ejpam-4274	842	3	-	-	ADJ
ejpam-4274	842	4	λ	λ	NOUN
ejpam-4274	842	5	-	-	NOUN
ejpam-4274	842	6	sets	set	NOUN
ejpam-4274	842	7	and	and	CCONJ
ejpam-4274	842	8	pre	pre	ADJ
ejpam-4274	842	9	-	-	ADJ
ejpam-4274	842	10	v	v	ADJ
ejpam-4274	842	11	-sets	-set	NOUN
ejpam-4274	842	12	.	.	PUNCT
ejpam-4274	843	1	acta	acta	PROPN
ejpam-4274	843	2	mathematica	mathematica	PROPN
ejpam-4274	843	3	hungarica	hungarica	PROPN
ejpam-4274	843	4	,	,	PUNCT
ejpam-4274	843	5	95:337–343	95:337–343	PROPN
ejpam-4274	843	6	,	,	PUNCT
ejpam-4274	843	7	2002	2002	NUM
ejpam-4274	843	8	.	.	PUNCT
ejpam-4274	844	1	[	[	X
ejpam-4274	844	2	11	11	NUM
ejpam-4274	844	3	]	]	PUNCT
ejpam-4274	844	4	s.	s.	PROPN
ejpam-4274	844	5	jafari	jafari	PROPN
ejpam-4274	844	6	.	.	PUNCT
ejpam-4274	845	1	on	on	ADP
ejpam-4274	845	2	a	a	DET
ejpam-4274	845	3	weak	weak	ADJ
ejpam-4274	845	4	separation	separation	NOUN
ejpam-4274	845	5	axiom	axiom	NOUN
ejpam-4274	845	6	.	.	PUNCT
ejpam-4274	846	1	far	far	PROPN
ejpam-4274	846	2	east	east	PROPN
ejpam-4274	846	3	journal	journal	PROPN
ejpam-4274	846	4	of	of	ADP
ejpam-4274	846	5	mathematical	mathematical	ADJ
ejpam-4274	846	6	sciences	sciences	PROPN
ejpam-4274	846	7	,	,	PUNCT
ejpam-4274	846	8	3:779–787	3:779–787	NUM
ejpam-4274	846	9	,	,	PUNCT
ejpam-4274	846	10	2001	2001	NUM
ejpam-4274	846	11	.	.	PUNCT
ejpam-4274	847	1	[	[	X
ejpam-4274	847	2	12	12	NUM
ejpam-4274	847	3	]	]	PUNCT
ejpam-4274	847	4	a.	a.	NOUN
ejpam-4274	847	5	kar	kar	PROPN
ejpam-4274	847	6	and	and	CCONJ
ejpam-4274	847	7	p.	p.	NOUN
ejpam-4274	847	8	bhattacharry	bhattacharry	NOUN
ejpam-4274	847	9	.	.	PUNCT
ejpam-4274	848	1	some	some	DET
ejpam-4274	848	2	weak	weak	ADJ
ejpam-4274	848	3	separation	separation	NOUN
ejpam-4274	848	4	axioms	axiom	NOUN
ejpam-4274	848	5	.	.	PUNCT
ejpam-4274	849	1	bulletin	bulletin	NOUN
ejpam-4274	849	2	of	of	ADP
ejpam-4274	849	3	the	the	DET
ejpam-4274	849	4	calcutta	calcutta	PROPN
ejpam-4274	849	5	mathematical	mathematical	ADJ
ejpam-4274	849	6	society	society	NOUN
ejpam-4274	849	7	,	,	PUNCT
ejpam-4274	849	8	82:415–422	82:415–422	NUM
ejpam-4274	849	9	,	,	PUNCT
ejpam-4274	849	10	1990	1990	NUM
ejpam-4274	849	11	.	.	PUNCT
ejpam-4274	850	1	[	[	X
ejpam-4274	850	2	13	13	NUM
ejpam-4274	850	3	]	]	X
ejpam-4274	850	4	n.	n.	PROPN
ejpam-4274	850	5	levine	levine	PROPN
ejpam-4274	850	6	.	.	PUNCT
ejpam-4274	851	1	a	a	DET
ejpam-4274	851	2	decomposition	decomposition	NOUN
ejpam-4274	851	3	of	of	ADP
ejpam-4274	851	4	continuity	continuity	NOUN
ejpam-4274	851	5	in	in	ADP
ejpam-4274	851	6	topological	topological	ADJ
ejpam-4274	851	7	spaces	space	NOUN
ejpam-4274	851	8	.	.	PUNCT
ejpam-4274	852	1	the	the	DET
ejpam-4274	852	2	american	american	PROPN
ejpam-4274	852	3	mathematical	mathematical	PROPN
ejpam-4274	852	4	monthly	monthly	ADV
ejpam-4274	852	5	,	,	PUNCT
ejpam-4274	852	6	68:44–46	68:44–46	NUM
ejpam-4274	852	7	,	,	PUNCT
ejpam-4274	852	8	1961	1961	NUM
ejpam-4274	852	9	.	.	PUNCT
ejpam-4274	853	1	[	[	X
ejpam-4274	853	2	14	14	NUM
ejpam-4274	853	3	]	]	X
ejpam-4274	853	4	s.	s.	PROPN
ejpam-4274	853	5	lugojan	lugojan	PROPN
ejpam-4274	853	6	.	.	PUNCT
ejpam-4274	854	1	generalized	generalized	ADJ
ejpam-4274	854	2	topology	topology	NOUN
ejpam-4274	854	3	.	.	PUNCT
ejpam-4274	855	1	studii	studii	PROPN
ejpam-4274	855	2	şi	şi	PROPN
ejpam-4274	855	3	cercetări	cercetări	PROPN
ejpam-4274	855	4	de	de	X
ejpam-4274	855	5	matematică	matematică	PROPN
ejpam-4274	855	6	,	,	PUNCT
ejpam-4274	855	7	34:348–360	34:348–360	PROPN
ejpam-4274	855	8	,	,	PUNCT
ejpam-4274	855	9	1982	1982	NUM
ejpam-4274	855	10	.	.	PUNCT
ejpam-4274	856	1	[	[	X
ejpam-4274	856	2	15	15	NUM
ejpam-4274	856	3	]	]	X
ejpam-4274	856	4	a.	a.	NOUN
ejpam-4274	856	5	s.	s.	PROPN
ejpam-4274	856	6	mashhour	mashhour	PROPN
ejpam-4274	856	7	,	,	PUNCT
ejpam-4274	856	8	a.	a.	PROPN
ejpam-4274	856	9	a.	a.	PROPN
ejpam-4274	856	10	allam	allam	PROPN
ejpam-4274	856	11	,	,	PUNCT
ejpam-4274	856	12	f.	f.	PROPN
ejpam-4274	856	13	s.	s.	PROPN
ejpam-4274	856	14	mahmous	mahmous	PROPN
ejpam-4274	856	15	,	,	PUNCT
ejpam-4274	856	16	and	and	CCONJ
ejpam-4274	856	17	f.	f.	PROPN
ejpam-4274	856	18	h.	h.	PROPN
ejpam-4274	856	19	kheder	kheder	PROPN
ejpam-4274	856	20	.	.	PUNCT
ejpam-4274	857	1	on	on	ADP
ejpam-4274	857	2	supra	supra	PROPN
ejpam-4274	857	3	topological	topological	ADJ
ejpam-4274	857	4	spaces	space	NOUN
ejpam-4274	857	5	.	.	PUNCT
ejpam-4274	858	1	indian	indian	ADJ
ejpam-4274	858	2	journal	journal	PROPN
ejpam-4274	858	3	of	of	ADP
ejpam-4274	858	4	pure	pure	ADJ
ejpam-4274	858	5	and	and	CCONJ
ejpam-4274	858	6	applied	applied	ADJ
ejpam-4274	858	7	mathematics	mathematic	NOUN
ejpam-4274	858	8	,	,	PUNCT
ejpam-4274	858	9	14(4):502–510	14(4):502–510	PROPN
ejpam-4274	858	10	,	,	PUNCT
ejpam-4274	858	11	1983	1983	NUM
ejpam-4274	858	12	.	.	PUNCT
ejpam-4274	859	1	[	[	X
ejpam-4274	859	2	16	16	NUM
ejpam-4274	859	3	]	]	PUNCT
ejpam-4274	859	4	a.	a.	NOUN
ejpam-4274	859	5	s.	s.	PROPN
ejpam-4274	859	6	mashhour	mashhour	PROPN
ejpam-4274	859	7	,	,	PUNCT
ejpam-4274	859	8	m.	m.	PROPN
ejpam-4274	859	9	e.	e.	PROPN
ejpam-4274	859	10	abd	abd	PROPN
ejpam-4274	859	11	el	el	PROPN
ejpam-4274	859	12	-	-	PROPN
ejpam-4274	859	13	monsef	monsef	ADJ
ejpam-4274	859	14	,	,	PUNCT
ejpam-4274	859	15	and	and	CCONJ
ejpam-4274	859	16	s.	s.	PROPN
ejpam-4274	859	17	n.	n.	PROPN
ejpam-4274	859	18	el	el	PROPN
ejpam-4274	859	19	-	-	PROPN
ejpam-4274	859	20	deeb	deeb	PROPN
ejpam-4274	859	21	.	.	PUNCT
ejpam-4274	860	1	on	on	ADP
ejpam-4274	860	2	precontinuous	precontinuous	ADJ
ejpam-4274	860	3	and	and	CCONJ
ejpam-4274	860	4	weak	weak	ADJ
ejpam-4274	860	5	precontinuous	precontinuous	ADJ
ejpam-4274	860	6	mappings	mapping	NOUN
ejpam-4274	860	7	.	.	PUNCT
ejpam-4274	861	1	proceedings	proceeding	NOUN
ejpam-4274	861	2	of	of	ADP
ejpam-4274	861	3	the	the	DET
ejpam-4274	861	4	mathematical	mathematical	ADJ
ejpam-4274	861	5	and	and	CCONJ
ejpam-4274	861	6	physical	physical	ADJ
ejpam-4274	861	7	society	society	NOUN
ejpam-4274	861	8	of	of	ADP
ejpam-4274	861	9	egypt	egypt	PROPN
ejpam-4274	861	10	,	,	PUNCT
ejpam-4274	861	11	53:47–53	53:47–53	NUM
ejpam-4274	861	12	,	,	PUNCT
ejpam-4274	861	13	1982	1982	NUM
ejpam-4274	861	14	.	.	PUNCT
ejpam-4274	862	1	[	[	X
ejpam-4274	862	2	17	17	NUM
ejpam-4274	862	3	]	]	PUNCT
ejpam-4274	862	4	v.	v.	CCONJ
ejpam-4274	862	5	popa	popa	NOUN
ejpam-4274	862	6	and	and	CCONJ
ejpam-4274	862	7	t.	t.	PROPN
ejpam-4274	862	8	noiri	noiri	PROPN
ejpam-4274	862	9	.	.	PUNCT
ejpam-4274	863	1	on	on	ADP
ejpam-4274	863	2	weakly	weakly	ADJ
ejpam-4274	863	3	(	(	PUNCT
ejpam-4274	863	4	τ	τ	PROPN
ejpam-4274	863	5	,	,	PUNCT
ejpam-4274	863	6	m)continuous	m)continuous	ADJ
ejpam-4274	863	7	functions	function	NOUN
ejpam-4274	863	8	.	.	PUNCT
ejpam-4274	864	1	rendiconti	rendiconti	VERB
ejpam-4274	864	2	del	del	PROPN
ejpam-4274	864	3	circolo	circolo	PROPN
ejpam-4274	864	4	matematico	matematico	NOUN
ejpam-4274	864	5	de	de	X
ejpam-4274	864	6	palermo	palermo	NOUN
ejpam-4274	864	7	,	,	PUNCT
ejpam-4274	864	8	51:295–316	51:295–316	PROPN
ejpam-4274	864	9	,	,	PUNCT
ejpam-4274	864	10	2002	2002	NUM
ejpam-4274	864	11	.	.	PUNCT
ejpam-4274	865	1	[	[	X
ejpam-4274	865	2	18	18	NUM
ejpam-4274	865	3	]	]	X
ejpam-4274	865	4	d.	d.	PROPN
ejpam-4274	865	5	a.	a.	PROPN
ejpam-4274	865	6	rose	rise	VERB
ejpam-4274	865	7	.	.	PUNCT
ejpam-4274	866	1	weak	weak	ADJ
ejpam-4274	866	2	continuity	continuity	NOUN
ejpam-4274	866	3	and	and	CCONJ
ejpam-4274	866	4	almost	almost	ADV
ejpam-4274	866	5	continuity	continuity	NOUN
ejpam-4274	866	6	.	.	PUNCT
ejpam-4274	867	1	international	international	ADJ
ejpam-4274	867	2	journal	journal	PROPN
ejpam-4274	867	3	of	of	ADP
ejpam-4274	867	4	mathematics	mathematics	PROPN
ejpam-4274	867	5	and	and	CCONJ
ejpam-4274	867	6	mathematical	mathematical	ADJ
ejpam-4274	867	7	sciences	science	NOUN
ejpam-4274	867	8	,	,	PUNCT
ejpam-4274	867	9	7:311–318	7:311–318	PROPN
ejpam-4274	867	10	,	,	PUNCT
ejpam-4274	867	11	1984	1984	NUM
ejpam-4274	867	12	.	.	PUNCT