id	sid	tid	token	lemma	pos
ejpam-6297	1	1	european	european	PROPN
ejpam-6297	1	2	journal	journal	PROPN
ejpam-6297	1	3	of	of	ADP
ejpam-6297	1	4	pure	pure	ADJ
ejpam-6297	1	5	and	and	CCONJ
ejpam-6297	1	6	applied	applied	ADJ
ejpam-6297	1	7	mathematics	mathematic	NOUN
ejpam-6297	1	8	2025	2025	NUM
ejpam-6297	1	9	,	,	PUNCT
ejpam-6297	1	10	vol	vol	NOUN
ejpam-6297	1	11	.	.	PROPN
ejpam-6297	1	12	18	18	NUM
ejpam-6297	1	13	,	,	PUNCT
ejpam-6297	1	14	issue	issue	NOUN
ejpam-6297	1	15	3	3	NUM
ejpam-6297	1	16	,	,	PUNCT
ejpam-6297	1	17	article	article	NOUN
ejpam-6297	1	18	number	number	NOUN
ejpam-6297	1	19	6297	6297	NUM
ejpam-6297	1	20	issn	issn	PROPN
ejpam-6297	1	21	1307	1307	NUM
ejpam-6297	1	22	-	-	SYM
ejpam-6297	1	23	5543	5543	NUM
ejpam-6297	1	24	–	–	PUNCT
ejpam-6297	1	25	ejpam.com	ejpam.com	X
ejpam-6297	1	26	published	publish	VERB
ejpam-6297	1	27	by	by	ADP
ejpam-6297	1	28	new	new	PROPN
ejpam-6297	1	29	york	york	PROPN
ejpam-6297	1	30	business	business	PROPN
ejpam-6297	1	31	global	global	ADJ
ejpam-6297	1	32	strong	strong	ADJ
ejpam-6297	1	33	β	β	X
ejpam-6297	1	34	-	-	ADJ
ejpam-6297	1	35	i	i	NOUN
ejpam-6297	1	36	-	-	PUNCT
ejpam-6297	1	37	submaximality	submaximality	PROPN
ejpam-6297	1	38	and	and	CCONJ
ejpam-6297	1	39	β	β	X
ejpam-6297	1	40	-	-	ADJ
ejpam-6297	1	41	i	i	NOUN
ejpam-6297	1	42	-	-	NOUN
ejpam-6297	1	43	paracompactness	paracompactness	PROPN
ejpam-6297	1	44	in	in	ADP
ejpam-6297	1	45	ideal	ideal	ADJ
ejpam-6297	1	46	topological	topological	ADJ
ejpam-6297	1	47	spaces	space	NOUN
ejpam-6297	1	48	chawalit	chawalit	VERB
ejpam-6297	1	49	boonpok1	boonpok1	PROPN
ejpam-6297	1	50	,	,	PUNCT
ejpam-6297	1	51	palin	palin	PROPN
ejpam-6297	1	52	raktaow2	raktaow2	PROPN
ejpam-6297	1	53	,	,	PUNCT
ejpam-6297	1	54	areeyuth	areeyuth	NOUN
ejpam-6297	1	55	sama	sama	NOUN
ejpam-6297	1	56	-	-	PUNCT
ejpam-6297	1	57	ae3,∗	ae3,∗	ADJ
ejpam-6297	1	58	1	1	NUM
ejpam-6297	1	59	mathematics	mathematic	NOUN
ejpam-6297	1	60	and	and	CCONJ
ejpam-6297	1	61	applied	apply	VERB
ejpam-6297	1	62	mathematics	mathematics	PROPN
ejpam-6297	1	63	research	research	NOUN
ejpam-6297	1	64	unit	unit	NOUN
ejpam-6297	1	65	,	,	PUNCT
ejpam-6297	1	66	department	department	NOUN
ejpam-6297	1	67	of	of	ADP
ejpam-6297	1	68	mathematics	mathematic	NOUN
ejpam-6297	1	69	,	,	PUNCT
ejpam-6297	1	70	faculty	faculty	NOUN
ejpam-6297	1	71	of	of	ADP
ejpam-6297	1	72	science	science	NOUN
ejpam-6297	1	73	,	,	PUNCT
ejpam-6297	1	74	mahasarakham	mahasarakham	PROPN
ejpam-6297	1	75	university	university	PROPN
ejpam-6297	1	76	,	,	PUNCT
ejpam-6297	1	77	maha	maha	PROPN
ejpam-6297	1	78	sarakham	sarakham	PROPN
ejpam-6297	1	79	,	,	PUNCT
ejpam-6297	1	80	44150	44150	NUM
ejpam-6297	1	81	,	,	PUNCT
ejpam-6297	1	82	thailand	thailand	PROPN
ejpam-6297	1	83	2	2	NUM
ejpam-6297	1	84	applied	apply	VERB
ejpam-6297	1	85	mathematics	mathematic	NOUN
ejpam-6297	1	86	and	and	CCONJ
ejpam-6297	1	87	innovation	innovation	NOUN
ejpam-6297	1	88	of	of	ADP
ejpam-6297	1	89	mathematics	mathematic	NOUN
ejpam-6297	1	90	teaching	teaching	NOUN
ejpam-6297	1	91	,	,	PUNCT
ejpam-6297	1	92	faculty	faculty	NOUN
ejpam-6297	1	93	of	of	ADP
ejpam-6297	1	94	science	science	NOUN
ejpam-6297	1	95	and	and	CCONJ
ejpam-6297	1	96	technology	technology	NOUN
ejpam-6297	1	97	,	,	PUNCT
ejpam-6297	1	98	prince	prince	NOUN
ejpam-6297	1	99	of	of	ADP
ejpam-6297	1	100	songkla	songkla	PROPN
ejpam-6297	1	101	university	university	PROPN
ejpam-6297	1	102	,	,	PUNCT
ejpam-6297	1	103	pattani	pattani	NOUN
ejpam-6297	1	104	campus	campus	NOUN
ejpam-6297	1	105	,	,	PUNCT
ejpam-6297	1	106	pattani	pattani	NOUN
ejpam-6297	1	107	,	,	PUNCT
ejpam-6297	1	108	94000	94000	NUM
ejpam-6297	1	109	,	,	PUNCT
ejpam-6297	1	110	thailand	thailand	PROPN
ejpam-6297	1	111	3	3	NUM
ejpam-6297	1	112	department	department	NOUN
ejpam-6297	1	113	of	of	ADP
ejpam-6297	1	114	mathematics	mathematic	NOUN
ejpam-6297	1	115	and	and	CCONJ
ejpam-6297	1	116	computer	computer	NOUN
ejpam-6297	1	117	science	science	NOUN
ejpam-6297	1	118	,	,	PUNCT
ejpam-6297	1	119	faculty	faculty	NOUN
ejpam-6297	1	120	of	of	ADP
ejpam-6297	1	121	science	science	NOUN
ejpam-6297	1	122	and	and	CCONJ
ejpam-6297	1	123	technology	technology	NOUN
ejpam-6297	1	124	,	,	PUNCT
ejpam-6297	1	125	prince	prince	NOUN
ejpam-6297	1	126	of	of	ADP
ejpam-6297	1	127	songkla	songkla	PROPN
ejpam-6297	1	128	university	university	PROPN
ejpam-6297	1	129	,	,	PUNCT
ejpam-6297	1	130	pattani	pattani	NOUN
ejpam-6297	1	131	campus	campus	NOUN
ejpam-6297	1	132	,	,	PUNCT
ejpam-6297	1	133	pattani	pattani	NOUN
ejpam-6297	1	134	,	,	PUNCT
ejpam-6297	1	135	94000	94000	NUM
ejpam-6297	1	136	,	,	PUNCT
ejpam-6297	1	137	thailand	thailand	PROPN
ejpam-6297	1	138	abstract	abstract	PROPN
ejpam-6297	1	139	.	.	PUNCT
ejpam-6297	2	1	this	this	DET
ejpam-6297	2	2	work	work	NOUN
ejpam-6297	2	3	introduces	introduce	NOUN
ejpam-6297	2	4	and	and	CCONJ
ejpam-6297	2	5	examines	examine	VERB
ejpam-6297	2	6	the	the	DET
ejpam-6297	2	7	concepts	concept	NOUN
ejpam-6297	2	8	of	of	ADP
ejpam-6297	2	9	strong	strong	ADJ
ejpam-6297	2	10	β	β	X
ejpam-6297	2	11	-	-	ADJ
ejpam-6297	2	12	i	i	NOUN
ejpam-6297	2	13	-	-	PUNCT
ejpam-6297	2	14	submaximality	submaximality	NOUN
ejpam-6297	2	15	and	and	CCONJ
ejpam-6297	2	16	strong	strong	ADJ
ejpam-6297	2	17	β	β	X
ejpam-6297	2	18	-	-	ADJ
ejpam-6297	2	19	i	i	NOUN
ejpam-6297	2	20	-	-	NOUN
ejpam-6297	2	21	paracompactness	paracompactness	PROPN
ejpam-6297	2	22	in	in	ADP
ejpam-6297	2	23	ideal	ideal	ADJ
ejpam-6297	2	24	topological	topological	ADJ
ejpam-6297	2	25	spaces	space	NOUN
ejpam-6297	2	26	,	,	PUNCT
ejpam-6297	2	27	presenting	present	VERB
ejpam-6297	2	28	them	they	PRON
ejpam-6297	2	29	as	as	ADP
ejpam-6297	2	30	natural	natural	ADJ
ejpam-6297	2	31	extensions	extension	NOUN
ejpam-6297	2	32	of	of	ADP
ejpam-6297	2	33	the	the	DET
ejpam-6297	2	34	classical	classical	ADJ
ejpam-6297	2	35	notions	notion	NOUN
ejpam-6297	2	36	of	of	ADP
ejpam-6297	2	37	submaximality	submaximality	NOUN
ejpam-6297	2	38	and	and	CCONJ
ejpam-6297	2	39	paracompactness	paracompactness	NOUN
ejpam-6297	2	40	.	.	PUNCT
ejpam-6297	3	1	the	the	DET
ejpam-6297	3	2	study	study	NOUN
ejpam-6297	3	3	emphasizes	emphasize	VERB
ejpam-6297	3	4	the	the	DET
ejpam-6297	3	5	analysis	analysis	NOUN
ejpam-6297	3	6	of	of	ADP
ejpam-6297	3	7	submaximal	submaximal	ADJ
ejpam-6297	3	8	spaces	space	NOUN
ejpam-6297	3	9	through	through	ADP
ejpam-6297	3	10	the	the	DET
ejpam-6297	3	11	lens	lens	NOUN
ejpam-6297	3	12	of	of	ADP
ejpam-6297	3	13	strong	strong	ADJ
ejpam-6297	3	14	β	β	X
ejpam-6297	3	15	-	-	ADJ
ejpam-6297	3	16	i	i	NOUN
ejpam-6297	3	17	-	-	PUNCT
ejpam-6297	3	18	open	open	ADJ
ejpam-6297	3	19	sets	set	NOUN
ejpam-6297	3	20	.	.	PUNCT
ejpam-6297	4	1	additionally	additionally	ADV
ejpam-6297	4	2	,	,	PUNCT
ejpam-6297	4	3	it	it	PRON
ejpam-6297	4	4	offers	offer	VERB
ejpam-6297	4	5	several	several	ADJ
ejpam-6297	4	6	characterizations	characterization	NOUN
ejpam-6297	4	7	of	of	ADP
ejpam-6297	4	8	strong	strong	ADJ
ejpam-6297	4	9	β	β	X
ejpam-6297	4	10	-	-	ADJ
ejpam-6297	4	11	i	i	NOUN
ejpam-6297	4	12	-	-	PUNCT
ejpam-6297	4	13	paracompact	paracompact	ADJ
ejpam-6297	4	14	spaces	space	NOUN
ejpam-6297	4	15	and	and	CCONJ
ejpam-6297	4	16	investigates	investigate	VERB
ejpam-6297	4	17	the	the	DET
ejpam-6297	4	18	preservation	preservation	NOUN
ejpam-6297	4	19	of	of	ADP
ejpam-6297	4	20	this	this	DET
ejpam-6297	4	21	property	property	NOUN
ejpam-6297	4	22	under	under	ADP
ejpam-6297	4	23	mappings	mapping	NOUN
ejpam-6297	4	24	.	.	PUNCT
ejpam-6297	5	1	2020	2020	NUM
ejpam-6297	5	2	mathematics	mathematic	NOUN
ejpam-6297	5	3	subject	subject	NOUN
ejpam-6297	5	4	classifications	classification	NOUN
ejpam-6297	5	5	:	:	PUNCT
ejpam-6297	5	6	54a05	54a05	NUM
ejpam-6297	5	7	,	,	PUNCT
ejpam-6297	5	8	54b05	54b05	NUM
ejpam-6297	5	9	,	,	PUNCT
ejpam-6297	5	10	54c08	54c08	NUM
ejpam-6297	5	11	key	key	ADJ
ejpam-6297	5	12	words	word	NOUN
ejpam-6297	5	13	and	and	CCONJ
ejpam-6297	5	14	phrases	phrase	NOUN
ejpam-6297	5	15	:	:	PUNCT
ejpam-6297	5	16	ideal	ideal	ADJ
ejpam-6297	5	17	topological	topological	ADJ
ejpam-6297	5	18	space	space	NOUN
ejpam-6297	5	19	,	,	PUNCT
ejpam-6297	5	20	strong	strong	ADJ
ejpam-6297	5	21	β	β	X
ejpam-6297	5	22	-	-	ADJ
ejpam-6297	5	23	i	i	PRON
ejpam-6297	5	24	-	-	PUNCT
ejpam-6297	5	25	open	open	ADJ
ejpam-6297	5	26	,	,	PUNCT
ejpam-6297	5	27	strong	strong	ADJ
ejpam-6297	5	28	β	β	X
ejpam-6297	5	29	-	-	ADJ
ejpam-6297	5	30	i	i	NOUN
ejpam-6297	5	31	-	-	PUNCT
ejpam-6297	5	32	submaximal	submaximal	ADJ
ejpam-6297	5	33	,	,	PUNCT
ejpam-6297	5	34	strong	strong	ADJ
ejpam-6297	5	35	β	β	X
ejpam-6297	5	36	-	-	ADJ
ejpam-6297	5	37	i	i	NOUN
ejpam-6297	5	38	-	-	PUNCT
ejpam-6297	5	39	paracompactness	paracompactness	NOUN
ejpam-6297	5	40	1	1	NUM
ejpam-6297	5	41	.	.	PUNCT
ejpam-6297	6	1	introduction	introduction	NOUN
ejpam-6297	6	2	and	and	CCONJ
ejpam-6297	6	3	preliminaries	preliminary	NOUN
ejpam-6297	6	4	general	general	ADJ
ejpam-6297	6	5	topology	topology	NOUN
ejpam-6297	6	6	has	have	AUX
ejpam-6297	6	7	demonstrated	demonstrate	VERB
ejpam-6297	6	8	its	its	PRON
ejpam-6297	6	9	efficacy	efficacy	NOUN
ejpam-6297	6	10	in	in	ADP
ejpam-6297	6	11	both	both	CCONJ
ejpam-6297	6	12	theoretical	theoretical	ADJ
ejpam-6297	6	13	and	and	CCONJ
ejpam-6297	6	14	practical	practical	ADJ
ejpam-6297	6	15	domains	domain	NOUN
ejpam-6297	6	16	.	.	PUNCT
ejpam-6297	7	1	topology	topology	NOUN
ejpam-6297	7	2	’s	’s	PART
ejpam-6297	7	3	significance	significance	NOUN
ejpam-6297	7	4	has	have	AUX
ejpam-6297	7	5	emerged	emerge	VERB
ejpam-6297	7	6	in	in	ADP
ejpam-6297	7	7	various	various	ADJ
ejpam-6297	7	8	domains	domain	NOUN
ejpam-6297	7	9	,	,	PUNCT
ejpam-6297	7	10	including	include	VERB
ejpam-6297	7	11	computational	computational	ADJ
ejpam-6297	7	12	topology	topology	NOUN
ejpam-6297	7	13	,	,	PUNCT
ejpam-6297	7	14	geometric	geometric	ADJ
ejpam-6297	7	15	design	design	NOUN
ejpam-6297	7	16	,	,	PUNCT
ejpam-6297	7	17	computer	computer	NOUN
ejpam-6297	7	18	-	-	PUNCT
ejpam-6297	7	19	aided	aid	VERB
ejpam-6297	7	20	design	design	NOUN
ejpam-6297	7	21	,	,	PUNCT
ejpam-6297	7	22	and	and	CCONJ
ejpam-6297	7	23	engineering	engineering	NOUN
ejpam-6297	7	24	.	.	PUNCT
ejpam-6297	8	1	khalimsky	khalimsky	PROPN
ejpam-6297	8	2	et	et	PROPN
ejpam-6297	8	3	al	al	PROPN
ejpam-6297	8	4	.	.	PUNCT
ejpam-6297	9	1	[	[	X
ejpam-6297	9	2	1	1	X
ejpam-6297	9	3	]	]	PUNCT
ejpam-6297	9	4	and	and	CCONJ
ejpam-6297	9	5	kong	kong	PROPN
ejpam-6297	9	6	and	and	CCONJ
ejpam-6297	9	7	kopperman	kopperman	NOUN
ejpam-6297	9	8	[	[	X
ejpam-6297	9	9	2	2	NUM
ejpam-6297	9	10	]	]	PUNCT
ejpam-6297	9	11	advanced	advanced	ADJ
ejpam-6297	9	12	digital	digital	ADJ
ejpam-6297	9	13	topology	topology	NOUN
ejpam-6297	9	14	and	and	CCONJ
ejpam-6297	9	15	computer	computer	NOUN
ejpam-6297	9	16	graphics	graphic	NOUN
ejpam-6297	9	17	by	by	ADP
ejpam-6297	9	18	the	the	DET
ejpam-6297	9	19	application	application	NOUN
ejpam-6297	9	20	of	of	ADP
ejpam-6297	9	21	connected	connected	ADJ
ejpam-6297	9	22	topologies	topology	NOUN
ejpam-6297	9	23	on	on	ADP
ejpam-6297	9	24	finite	finite	NOUN
ejpam-6297	9	25	ordered	order	VERB
ejpam-6297	9	26	sets	set	NOUN
ejpam-6297	9	27	.	.	PUNCT
ejpam-6297	10	1	moore	moore	PROPN
ejpam-6297	10	2	and	and	CCONJ
ejpam-6297	10	3	peters	peters	PROPN
ejpam-6297	10	4	[	[	X
ejpam-6297	10	5	3	3	NUM
ejpam-6297	10	6	]	]	PUNCT
ejpam-6297	10	7	examined	examine	VERB
ejpam-6297	10	8	computational	computational	ADJ
ejpam-6297	10	9	topology	topology	NOUN
ejpam-6297	10	10	in	in	ADP
ejpam-6297	10	11	geometric	geometric	ADJ
ejpam-6297	10	12	and	and	CCONJ
ejpam-6297	10	13	molecular	molecular	ADJ
ejpam-6297	10	14	design	design	NOUN
ejpam-6297	10	15	,	,	PUNCT
ejpam-6297	10	16	whereas	whereas	SCONJ
ejpam-6297	10	17	rosen	rosen	PROPN
ejpam-6297	10	18	and	and	CCONJ
ejpam-6297	10	19	peters	peters	PROPN
ejpam-6297	10	20	[	[	X
ejpam-6297	10	21	4	4	NUM
ejpam-6297	10	22	]	]	PUNCT
ejpam-6297	10	23	employed	employ	VERB
ejpam-6297	10	24	topological	topological	ADJ
ejpam-6297	10	25	approaches	approach	NOUN
ejpam-6297	10	26	in	in	ADP
ejpam-6297	10	27	engineering	engineering	NOUN
ejpam-6297	10	28	design	design	NOUN
ejpam-6297	10	29	research	research	NOUN
ejpam-6297	10	30	.	.	PUNCT
ejpam-6297	11	1	the	the	DET
ejpam-6297	11	2	concept	concept	NOUN
ejpam-6297	11	3	of	of	ADP
ejpam-6297	11	4	submaximality	submaximality	NOUN
ejpam-6297	11	5	in	in	ADP
ejpam-6297	11	6	general	general	ADJ
ejpam-6297	11	7	topological	topological	ADJ
ejpam-6297	11	8	spaces	space	NOUN
ejpam-6297	11	9	was	be	AUX
ejpam-6297	11	10	first	first	ADV
ejpam-6297	11	11	introduced	introduce	VERB
ejpam-6297	11	12	by	by	ADP
ejpam-6297	11	13	hewitt	hewitt	PROPN
ejpam-6297	12	1	[	[	X
ejpam-6297	12	2	5	5	NUM
ejpam-6297	12	3	]	]	PUNCT
ejpam-6297	12	4	,	,	PUNCT
ejpam-6297	12	5	who	who	PRON
ejpam-6297	12	6	defined	define	VERB
ejpam-6297	12	7	a	a	DET
ejpam-6297	12	8	space	space	NOUN
ejpam-6297	12	9	as	as	ADP
ejpam-6297	12	10	submaximal	submaximal	ADJ
ejpam-6297	12	11	if	if	SCONJ
ejpam-6297	12	12	every	every	DET
ejpam-6297	12	13	dense	dense	ADJ
ejpam-6297	12	14	subset	subset	NOUN
ejpam-6297	12	15	is	be	AUX
ejpam-6297	12	16	open	open	ADJ
ejpam-6297	12	17	.	.	PUNCT
ejpam-6297	13	1	this	this	DET
ejpam-6297	13	2	property	property	NOUN
ejpam-6297	13	3	plays	play	VERB
ejpam-6297	13	4	a	a	DET
ejpam-6297	13	5	significant	significant	ADJ
ejpam-6297	13	6	role	role	NOUN
ejpam-6297	13	7	in	in	ADP
ejpam-6297	13	8	topology	topology	NOUN
ejpam-6297	13	9	,	,	PUNCT
ejpam-6297	13	10	often	often	ADV
ejpam-6297	13	11	serving	serve	VERB
ejpam-6297	13	12	as	as	ADP
ejpam-6297	13	13	a	a	DET
ejpam-6297	13	14	crucial	crucial	ADJ
ejpam-6297	13	15	condition	condition	NOUN
ejpam-6297	13	16	in	in	ADP
ejpam-6297	13	17	the	the	DET
ejpam-6297	13	18	study	study	NOUN
ejpam-6297	13	19	of	of	ADP
ejpam-6297	13	20	∗corresponding	∗corresponde	VERB
ejpam-6297	13	21	author	author	NOUN
ejpam-6297	13	22	.	.	PUNCT
ejpam-6297	14	1	doi	doi	NOUN
ejpam-6297	14	2	:	:	PUNCT
ejpam-6297	14	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6297	https://doi.org/10.29020/nybg.ejpam.v18i3.6297	NOUN
ejpam-6297	14	4	email	email	NOUN
ejpam-6297	14	5	addresses	address	NOUN
ejpam-6297	14	6	:	:	PUNCT
ejpam-6297	14	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-6297	14	8	(	(	PUNCT
ejpam-6297	14	9	c.	c.	PROPN
ejpam-6297	14	10	boonpok	boonpok	PROPN
ejpam-6297	14	11	)	)	PUNCT
ejpam-6297	14	12	,	,	PUNCT
ejpam-6297	14	13	palin.rt@gmail.com	palin.rt@gmail.com	PROPN
ejpam-6297	14	14	(	(	PUNCT
ejpam-6297	14	15	p.	p.	NOUN
ejpam-6297	14	16	raktaow	raktaow	PROPN
ejpam-6297	14	17	)	)	PUNCT
ejpam-6297	14	18	,	,	PUNCT
ejpam-6297	14	19	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-6297	14	20	(	(	PUNCT
ejpam-6297	14	21	a.	a.	PROPN
ejpam-6297	14	22	sama	sama	PROPN
ejpam-6297	14	23	-	-	PUNCT
ejpam-6297	14	24	ae	ae	PROPN
ejpam-6297	14	25	)	)	PUNCT
ejpam-6297	14	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6297	15	1	1	1	NUM
ejpam-6297	15	2	copyright	copyright	NOUN
ejpam-6297	15	3	:	:	PUNCT
ejpam-6297	15	4	©	©	PROPN
ejpam-6297	15	5	2025	2025	NUM
ejpam-6297	15	6	the	the	DET
ejpam-6297	15	7	author(s	author(s	NOUN
ejpam-6297	15	8	)	)	PUNCT
ejpam-6297	15	9	.	.	PUNCT
ejpam-6297	16	1	(	(	PUNCT
ejpam-6297	16	2	cc	cc	NOUN
ejpam-6297	16	3	by	by	ADP
ejpam-6297	16	4	-	-	PUNCT
ejpam-6297	16	5	nc	nc	PROPN
ejpam-6297	16	6	4.0	4.0	NUM
ejpam-6297	16	7	)	)	PUNCT
ejpam-6297	16	8	c.	c.	NOUN
ejpam-6297	16	9	boonpok	boonpok	PROPN
ejpam-6297	16	10	,	,	PUNCT
ejpam-6297	16	11	p.	p.	PROPN
ejpam-6297	16	12	raktaow	raktaow	NOUN
ejpam-6297	16	13	,	,	PUNCT
ejpam-6297	16	14	a.	a.	PROPN
ejpam-6297	16	15	sama	sama	PROPN
ejpam-6297	16	16	-	-	PUNCT
ejpam-6297	16	17	ae	ae	PROPN
ejpam-6297	16	18	/	/	SYM
ejpam-6297	16	19	eur	eur	PROPN
ejpam-6297	16	20	.	.	PUNCT
ejpam-6297	17	1	j.	j.	PROPN
ejpam-6297	17	2	pure	pure	PROPN
ejpam-6297	17	3	appl	appl	PROPN
ejpam-6297	17	4	.	.	PROPN
ejpam-6297	17	5	math	math	PROPN
ejpam-6297	17	6	,	,	PUNCT
ejpam-6297	17	7	18	18	NUM
ejpam-6297	17	8	(	(	PUNCT
ejpam-6297	17	9	3	3	NUM
ejpam-6297	17	10	)	)	PUNCT
ejpam-6297	17	11	(	(	PUNCT
ejpam-6297	17	12	2025	2025	NUM
ejpam-6297	17	13	)	)	PUNCT
ejpam-6297	17	14	,	,	PUNCT
ejpam-6297	17	15	6297	6297	NUM
ejpam-6297	17	16	2	2	NUM
ejpam-6297	17	17	of	of	ADP
ejpam-6297	17	18	23	23	NUM
ejpam-6297	17	19	maximal	maximal	ADJ
ejpam-6297	17	20	topologies	topology	NOUN
ejpam-6297	17	21	related	relate	VERB
ejpam-6297	17	22	to	to	ADP
ejpam-6297	17	23	various	various	ADJ
ejpam-6297	17	24	topological	topological	ADJ
ejpam-6297	17	25	invariants	invariant	NOUN
ejpam-6297	17	26	.	.	PUNCT
ejpam-6297	18	1	the	the	DET
ejpam-6297	18	2	original	original	ADJ
ejpam-6297	18	3	idea	idea	NOUN
ejpam-6297	18	4	of	of	ADP
ejpam-6297	18	5	submaximal	submaximal	ADJ
ejpam-6297	18	6	spaces	space	NOUN
ejpam-6297	18	7	can	can	AUX
ejpam-6297	18	8	be	be	AUX
ejpam-6297	18	9	traced	trace	VERB
ejpam-6297	18	10	back	back	ADV
ejpam-6297	18	11	to	to	PART
ejpam-6297	18	12	bourbaki	bourbaki	VERB
ejpam-6297	18	13	[	[	X
ejpam-6297	18	14	6	6	NUM
ejpam-6297	18	15	]	]	PUNCT
ejpam-6297	18	16	.	.	PUNCT
ejpam-6297	19	1	hewitt	hewitt	PROPN
ejpam-6297	19	2	’s	’s	PART
ejpam-6297	19	3	foundational	foundational	ADJ
ejpam-6297	19	4	work	work	NOUN
ejpam-6297	19	5	laid	lay	VERB
ejpam-6297	19	6	the	the	DET
ejpam-6297	19	7	basis	basis	NOUN
ejpam-6297	19	8	for	for	ADP
ejpam-6297	19	9	further	further	ADJ
ejpam-6297	19	10	developments	development	NOUN
ejpam-6297	19	11	by	by	ADP
ejpam-6297	19	12	arhangel’skii	arhangel’skii	ADJ
ejpam-6297	19	13	and	and	CCONJ
ejpam-6297	19	14	collins	collin	VERB
ejpam-6297	20	1	[	[	X
ejpam-6297	20	2	7	7	NUM
ejpam-6297	20	3	]	]	PUNCT
ejpam-6297	20	4	,	,	PUNCT
ejpam-6297	20	5	who	who	PRON
ejpam-6297	20	6	provided	provide	VERB
ejpam-6297	20	7	necessary	necessary	ADJ
ejpam-6297	20	8	and	and	CCONJ
ejpam-6297	20	9	sufficient	sufficient	ADJ
ejpam-6297	20	10	conditions	condition	NOUN
ejpam-6297	20	11	for	for	ADP
ejpam-6297	20	12	submaximality	submaximality	NOUN
ejpam-6297	20	13	and	and	CCONJ
ejpam-6297	20	14	showed	show	VERB
ejpam-6297	20	15	that	that	SCONJ
ejpam-6297	20	16	all	all	DET
ejpam-6297	20	17	such	such	ADJ
ejpam-6297	20	18	spaces	space	NOUN
ejpam-6297	20	19	are	be	AUX
ejpam-6297	20	20	left	leave	VERB
ejpam-6297	20	21	-	-	PUNCT
ejpam-6297	20	22	separated	separate	VERB
ejpam-6297	20	23	.	.	PUNCT
ejpam-6297	21	1	their	their	PRON
ejpam-6297	21	2	contributions	contribution	NOUN
ejpam-6297	21	3	also	also	ADV
ejpam-6297	21	4	sparked	spark	VERB
ejpam-6297	21	5	interest	interest	NOUN
ejpam-6297	21	6	in	in	ADP
ejpam-6297	21	7	whether	whether	SCONJ
ejpam-6297	21	8	every	every	DET
ejpam-6297	21	9	submaximal	submaximal	ADJ
ejpam-6297	21	10	space	space	NOUN
ejpam-6297	21	11	is	be	AUX
ejpam-6297	21	12	σ	σ	NOUN
ejpam-6297	21	13	-	-	NOUN
ejpam-6297	21	14	discrete	discrete	NOUN
ejpam-6297	21	15	.	.	PUNCT
ejpam-6297	22	1	additionally	additionally	ADV
ejpam-6297	22	2	,	,	PUNCT
ejpam-6297	22	3	they	they	PRON
ejpam-6297	22	4	established	establish	VERB
ejpam-6297	22	5	that	that	SCONJ
ejpam-6297	22	6	any	any	DET
ejpam-6297	22	7	connected	connected	ADJ
ejpam-6297	22	8	hausdorff	hausdorff	NOUN
ejpam-6297	22	9	space	space	NOUN
ejpam-6297	22	10	that	that	PRON
ejpam-6297	22	11	does	do	AUX
ejpam-6297	22	12	not	not	PART
ejpam-6297	22	13	admit	admit	VERB
ejpam-6297	22	14	a	a	DET
ejpam-6297	22	15	strictly	strictly	ADV
ejpam-6297	22	16	finer	fine	ADJ
ejpam-6297	22	17	connected	connected	ADJ
ejpam-6297	22	18	topology	topology	NOUN
ejpam-6297	22	19	is	be	AUX
ejpam-6297	22	20	necessarily	necessarily	ADV
ejpam-6297	22	21	submaximal	submaximal	ADJ
ejpam-6297	22	22	[	[	X
ejpam-6297	22	23	7	7	NUM
ejpam-6297	22	24	]	]	PUNCT
ejpam-6297	22	25	.	.	PUNCT
ejpam-6297	23	1	dontchev	dontchev	PROPN
ejpam-6297	24	1	[	[	X
ejpam-6297	24	2	8	8	NUM
ejpam-6297	24	3	]	]	PUNCT
ejpam-6297	24	4	characterized	characterize	VERB
ejpam-6297	24	5	submaximal	submaximal	ADJ
ejpam-6297	24	6	spaces	space	NOUN
ejpam-6297	24	7	using	use	VERB
ejpam-6297	24	8	various	various	ADJ
ejpam-6297	24	9	topological	topological	ADJ
ejpam-6297	24	10	notions	notion	NOUN
ejpam-6297	24	11	studied	study	VERB
ejpam-6297	24	12	the	the	DET
ejpam-6297	24	13	connections	connection	NOUN
ejpam-6297	24	14	between	between	ADP
ejpam-6297	24	15	submaximal	submaximal	ADJ
ejpam-6297	24	16	and	and	CCONJ
ejpam-6297	24	17	related	related	ADJ
ejpam-6297	24	18	spaces	space	NOUN
ejpam-6297	24	19	.	.	PUNCT
ejpam-6297	25	1	tokgoz	tokgoz	PROPN
ejpam-6297	25	2	and	and	CCONJ
ejpam-6297	25	3	yalvac	yalvac	NOUN
ejpam-6297	26	1	[	[	X
ejpam-6297	26	2	9	9	NUM
ejpam-6297	26	3	]	]	PUNCT
ejpam-6297	26	4	developed	develop	VERB
ejpam-6297	26	5	the	the	DET
ejpam-6297	26	6	concept	concept	NOUN
ejpam-6297	26	7	of	of	ADP
ejpam-6297	26	8	submaximality	submaximality	NOUN
ejpam-6297	26	9	and	and	CCONJ
ejpam-6297	26	10	offer	offer	VERB
ejpam-6297	26	11	some	some	DET
ejpam-6297	26	12	new	new	ADJ
ejpam-6297	26	13	results	result	NOUN
ejpam-6297	26	14	.	.	PUNCT
ejpam-6297	27	1	paracompact	paracompact	ADJ
ejpam-6297	27	2	spaces	space	NOUN
ejpam-6297	27	3	are	be	AUX
ejpam-6297	27	4	considered	consider	VERB
ejpam-6297	27	5	one	one	NUM
ejpam-6297	27	6	of	of	ADP
ejpam-6297	27	7	the	the	DET
ejpam-6297	27	8	key	key	ADJ
ejpam-6297	27	9	classes	class	NOUN
ejpam-6297	27	10	of	of	ADP
ejpam-6297	27	11	topological	topological	ADJ
ejpam-6297	27	12	spaces	space	NOUN
ejpam-6297	27	13	,	,	PUNCT
ejpam-6297	27	14	as	as	SCONJ
ejpam-6297	27	15	they	they	PRON
ejpam-6297	27	16	generalize	generalize	VERB
ejpam-6297	27	17	both	both	CCONJ
ejpam-6297	27	18	metrizable	metrizable	ADJ
ejpam-6297	27	19	and	and	CCONJ
ejpam-6297	27	20	compact	compact	ADJ
ejpam-6297	27	21	spaces	space	NOUN
ejpam-6297	27	22	.	.	PUNCT
ejpam-6297	28	1	these	these	DET
ejpam-6297	28	2	spaces	space	NOUN
ejpam-6297	28	3	were	be	AUX
ejpam-6297	28	4	quickly	quickly	ADV
ejpam-6297	28	5	recognized	recognize	VERB
ejpam-6297	28	6	by	by	ADP
ejpam-6297	28	7	topologists	topologist	NOUN
ejpam-6297	28	8	and	and	CCONJ
ejpam-6297	28	9	analysts	analyst	NOUN
ejpam-6297	28	10	.	.	PUNCT
ejpam-6297	29	1	a	a	DET
ejpam-6297	29	2	paracompact	paracompact	ADJ
ejpam-6297	29	3	space	space	NOUN
ejpam-6297	29	4	is	be	AUX
ejpam-6297	29	5	defined	define	VERB
ejpam-6297	29	6	as	as	ADP
ejpam-6297	29	7	a	a	DET
ejpam-6297	29	8	topological	topological	ADJ
ejpam-6297	29	9	space	space	NOUN
ejpam-6297	29	10	in	in	ADP
ejpam-6297	29	11	which	which	PRON
ejpam-6297	29	12	every	every	DET
ejpam-6297	29	13	open	open	ADJ
ejpam-6297	29	14	cover	cover	NOUN
ejpam-6297	29	15	has	have	VERB
ejpam-6297	29	16	an	an	DET
ejpam-6297	29	17	open	open	ADJ
ejpam-6297	29	18	refinement	refinement	NOUN
ejpam-6297	29	19	that	that	PRON
ejpam-6297	29	20	is	be	AUX
ejpam-6297	29	21	locally	locally	ADV
ejpam-6297	29	22	finite	finite	ADJ
ejpam-6297	29	23	.	.	PUNCT
ejpam-6297	30	1	this	this	DET
ejpam-6297	30	2	concept	concept	NOUN
ejpam-6297	30	3	was	be	AUX
ejpam-6297	30	4	first	first	ADV
ejpam-6297	30	5	introduced	introduce	VERB
ejpam-6297	30	6	by	by	ADP
ejpam-6297	30	7	dieudonné	dieudonné	NOUN
ejpam-6297	30	8	[	[	X
ejpam-6297	30	9	10	10	NUM
ejpam-6297	30	10	]	]	PUNCT
ejpam-6297	30	11	in	in	ADP
ejpam-6297	30	12	1944	1944	NUM
ejpam-6297	30	13	.	.	PUNCT
ejpam-6297	31	1	a	a	DET
ejpam-6297	31	2	hausdorff	hausdorff	NOUN
ejpam-6297	31	3	space	space	NOUN
ejpam-6297	31	4	is	be	AUX
ejpam-6297	31	5	paracompact	paracompact	ADJ
ejpam-6297	31	6	if	if	SCONJ
ejpam-6297	31	7	and	and	CCONJ
ejpam-6297	31	8	only	only	ADV
ejpam-6297	31	9	if	if	SCONJ
ejpam-6297	31	10	it	it	PRON
ejpam-6297	31	11	admits	admit	VERB
ejpam-6297	31	12	partitions	partition	NOUN
ejpam-6297	31	13	of	of	ADP
ejpam-6297	31	14	unity	unity	NOUN
ejpam-6297	31	15	that	that	PRON
ejpam-6297	31	16	are	be	AUX
ejpam-6297	31	17	subordinate	subordinate	ADJ
ejpam-6297	31	18	to	to	ADP
ejpam-6297	31	19	any	any	DET
ejpam-6297	31	20	open	open	ADJ
ejpam-6297	31	21	cover	cover	NOUN
ejpam-6297	31	22	.	.	PUNCT
ejpam-6297	32	1	additionally	additionally	ADV
ejpam-6297	32	2	,	,	PUNCT
ejpam-6297	32	3	all	all	DET
ejpam-6297	32	4	paracompact	paracompact	ADJ
ejpam-6297	32	5	hausdorff	hausdorff	NOUN
ejpam-6297	32	6	spaces	space	NOUN
ejpam-6297	32	7	are	be	AUX
ejpam-6297	32	8	normal	normal	ADJ
ejpam-6297	32	9	,	,	PUNCT
ejpam-6297	32	10	as	as	SCONJ
ejpam-6297	32	11	shown	show	VERB
ejpam-6297	32	12	in	in	ADP
ejpam-6297	32	13	[	[	X
ejpam-6297	32	14	11	11	NUM
ejpam-6297	32	15	]	]	PUNCT
ejpam-6297	32	16	.	.	PUNCT
ejpam-6297	33	1	various	various	ADJ
ejpam-6297	33	2	generalized	generalized	ADJ
ejpam-6297	33	3	forms	form	NOUN
ejpam-6297	33	4	of	of	ADP
ejpam-6297	33	5	paracompactness	paracompactness	NOUN
ejpam-6297	33	6	,	,	PUNCT
ejpam-6297	33	7	such	such	ADJ
ejpam-6297	33	8	as	as	ADP
ejpam-6297	33	9	s	s	NOUN
ejpam-6297	33	10	-	-	NOUN
ejpam-6297	33	11	paracompactness	paracompactness	NOUN
ejpam-6297	33	12	[	[	X
ejpam-6297	33	13	12	12	NUM
ejpam-6297	33	14	]	]	PUNCT
ejpam-6297	33	15	,	,	PUNCT
ejpam-6297	33	16	p3	p3	PROPN
ejpam-6297	33	17	-	-	NOUN
ejpam-6297	33	18	paracompactness	paracompactness	NOUN
ejpam-6297	33	19	[	[	X
ejpam-6297	33	20	13	13	NUM
ejpam-6297	33	21	]	]	PUNCT
ejpam-6297	33	22	,	,	PUNCT
ejpam-6297	33	23	and	and	CCONJ
ejpam-6297	33	24	β	β	X
ejpam-6297	33	25	-	-	NOUN
ejpam-6297	33	26	paracompactness	paracompactness	NOUN
ejpam-6297	33	27	[	[	X
ejpam-6297	33	28	14	14	NUM
ejpam-6297	33	29	]	]	PUNCT
ejpam-6297	33	30	,	,	PUNCT
ejpam-6297	33	31	have	have	AUX
ejpam-6297	33	32	been	be	AUX
ejpam-6297	33	33	explored	explore	VERB
ejpam-6297	33	34	in	in	ADP
ejpam-6297	33	35	the	the	DET
ejpam-6297	33	36	literature	literature	NOUN
ejpam-6297	33	37	.	.	PUNCT
ejpam-6297	34	1	in	in	ADP
ejpam-6297	34	2	2006	2006	NUM
ejpam-6297	34	3	,	,	PUNCT
ejpam-6297	34	4	al	al	PROPN
ejpam-6297	34	5	-	-	PROPN
ejpam-6297	34	6	zoubi	zoubi	PROPN
ejpam-6297	34	7	[	[	X
ejpam-6297	34	8	12	12	NUM
ejpam-6297	34	9	]	]	PUNCT
ejpam-6297	34	10	used	use	VERB
ejpam-6297	34	11	semi	semi	ADJ
ejpam-6297	34	12	-	-	ADJ
ejpam-6297	34	13	open	open	ADJ
ejpam-6297	34	14	sets	set	NOUN
ejpam-6297	34	15	to	to	PART
ejpam-6297	34	16	define	define	VERB
ejpam-6297	34	17	s	s	NOUN
ejpam-6297	34	18	-	-	PUNCT
ejpam-6297	34	19	paracompact	paracompact	ADJ
ejpam-6297	34	20	spaces	space	NOUN
ejpam-6297	34	21	,	,	PUNCT
ejpam-6297	34	22	a	a	DET
ejpam-6297	34	23	generalization	generalization	NOUN
ejpam-6297	34	24	of	of	ADP
ejpam-6297	34	25	paracompact	paracompact	ADJ
ejpam-6297	34	26	spaces	space	NOUN
ejpam-6297	34	27	,	,	PUNCT
ejpam-6297	34	28	and	and	CCONJ
ejpam-6297	34	29	studied	study	VERB
ejpam-6297	34	30	their	their	PRON
ejpam-6297	34	31	relationships	relationship	NOUN
ejpam-6297	34	32	.	.	PUNCT
ejpam-6297	35	1	li	li	PROPN
ejpam-6297	35	2	and	and	CCONJ
ejpam-6297	35	3	song	song	NOUN
ejpam-6297	35	4	[	[	X
ejpam-6297	35	5	15	15	NUM
ejpam-6297	35	6	]	]	PUNCT
ejpam-6297	35	7	constructed	construct	VERB
ejpam-6297	35	8	a	a	DET
ejpam-6297	35	9	hausdorff	hausdorff	NOUN
ejpam-6297	35	10	s	s	NOUN
ejpam-6297	35	11	-	-	PUNCT
ejpam-6297	35	12	paracompact	paracompact	ADJ
ejpam-6297	35	13	space	space	NOUN
ejpam-6297	35	14	that	that	PRON
ejpam-6297	35	15	is	be	AUX
ejpam-6297	35	16	not	not	PART
ejpam-6297	35	17	paracompact	paracompact	ADJ
ejpam-6297	35	18	and	and	CCONJ
ejpam-6297	35	19	further	far	ADV
ejpam-6297	35	20	investigated	investigate	VERB
ejpam-6297	35	21	the	the	DET
ejpam-6297	35	22	characterizations	characterization	NOUN
ejpam-6297	35	23	of	of	ADP
ejpam-6297	35	24	s	s	NOUN
ejpam-6297	35	25	-	-	PUNCT
ejpam-6297	35	26	paracompact	paracompact	ADJ
ejpam-6297	35	27	spaces	space	NOUN
ejpam-6297	35	28	.	.	PUNCT
ejpam-6297	36	1	the	the	DET
ejpam-6297	36	2	concept	concept	NOUN
ejpam-6297	36	3	of	of	ADP
ejpam-6297	36	4	ideal	ideal	ADJ
ejpam-6297	36	5	topological	topological	ADJ
ejpam-6297	36	6	spaces	space	NOUN
ejpam-6297	36	7	was	be	AUX
ejpam-6297	36	8	first	first	ADV
ejpam-6297	36	9	introduced	introduce	VERB
ejpam-6297	36	10	by	by	ADP
ejpam-6297	36	11	kuratowski	kuratowski	NOUN
ejpam-6297	36	12	[	[	X
ejpam-6297	36	13	16	16	NUM
ejpam-6297	36	14	]	]	PUNCT
ejpam-6297	36	15	and	and	CCONJ
ejpam-6297	36	16	vaidyanathaswamy	vaidyanathaswamy	VERB
ejpam-6297	36	17	[	[	X
ejpam-6297	36	18	17	17	NUM
ejpam-6297	36	19	]	]	PUNCT
ejpam-6297	36	20	.	.	PUNCT
ejpam-6297	37	1	the	the	DET
ejpam-6297	37	2	integration	integration	NOUN
ejpam-6297	37	3	of	of	ADP
ejpam-6297	37	4	ideals	ideal	NOUN
ejpam-6297	37	5	into	into	ADP
ejpam-6297	37	6	topological	topological	ADJ
ejpam-6297	37	7	structures	structure	NOUN
ejpam-6297	37	8	has	have	AUX
ejpam-6297	37	9	led	lead	VERB
ejpam-6297	37	10	to	to	ADP
ejpam-6297	37	11	the	the	DET
ejpam-6297	37	12	generalization	generalization	NOUN
ejpam-6297	37	13	of	of	ADP
ejpam-6297	37	14	certain	certain	ADJ
ejpam-6297	37	15	classical	classical	ADJ
ejpam-6297	37	16	topological	topological	ADJ
ejpam-6297	37	17	ideas	idea	NOUN
ejpam-6297	37	18	.	.	PUNCT
ejpam-6297	38	1	the	the	DET
ejpam-6297	38	2	topology	topology	NOUN
ejpam-6297	38	3	τ	τ	PROPN
ejpam-6297	38	4	of	of	ADP
ejpam-6297	38	5	a	a	DET
ejpam-6297	38	6	space	space	NOUN
ejpam-6297	38	7	can	can	AUX
ejpam-6297	38	8	be	be	AUX
ejpam-6297	38	9	augmented	augment	VERB
ejpam-6297	38	10	to	to	ADP
ejpam-6297	38	11	a	a	DET
ejpam-6297	38	12	topology	topology	NOUN
ejpam-6297	38	13	τ⋆	τ⋆	NOUN
ejpam-6297	38	14	through	through	ADP
ejpam-6297	38	15	an	an	DET
ejpam-6297	38	16	ideal	ideal	NOUN
ejpam-6297	38	17	i	i	PROPN
ejpam-6297	38	18	,	,	PUNCT
ejpam-6297	38	19	resulting	result	VERB
ejpam-6297	38	20	in	in	ADP
ejpam-6297	38	21	the	the	DET
ejpam-6297	38	22	creation	creation	NOUN
ejpam-6297	38	23	of	of	ADP
ejpam-6297	38	24	an	an	DET
ejpam-6297	38	25	ideal	ideal	ADJ
ejpam-6297	38	26	topological	topological	ADJ
ejpam-6297	38	27	space	space	NOUN
ejpam-6297	38	28	.	.	PUNCT
ejpam-6297	39	1	this	this	DET
ejpam-6297	39	2	framework	framework	NOUN
ejpam-6297	39	3	has	have	AUX
ejpam-6297	39	4	been	be	AUX
ejpam-6297	39	5	utilized	utilize	VERB
ejpam-6297	39	6	in	in	ADP
ejpam-6297	39	7	areas	area	NOUN
ejpam-6297	39	8	such	such	ADJ
ejpam-6297	39	9	as	as	ADP
ejpam-6297	39	10	ideal	ideal	ADJ
ejpam-6297	39	11	resolvability	resolvability	NOUN
ejpam-6297	39	12	[	[	X
ejpam-6297	39	13	18	18	NUM
ejpam-6297	39	14	]	]	PUNCT
ejpam-6297	39	15	,	,	PUNCT
ejpam-6297	39	16	paracompactness	paracompactness	NOUN
ejpam-6297	39	17	concerning	concern	VERB
ejpam-6297	39	18	ideals	ideal	NOUN
ejpam-6297	39	19	[	[	X
ejpam-6297	39	20	19	19	NUM
ejpam-6297	39	21	]	]	PUNCT
ejpam-6297	39	22	,	,	PUNCT
ejpam-6297	39	23	and	and	CCONJ
ejpam-6297	39	24	continuity	continuity	NOUN
ejpam-6297	39	25	decomposition	decomposition	NOUN
ejpam-6297	39	26	[	[	X
ejpam-6297	39	27	20	20	NUM
ejpam-6297	39	28	]	]	PUNCT
ejpam-6297	39	29	.	.	PUNCT
ejpam-6297	40	1	additionally	additionally	ADV
ejpam-6297	40	2	,	,	PUNCT
ejpam-6297	40	3	novel	novel	ADJ
ejpam-6297	40	4	topological	topological	ADJ
ejpam-6297	40	5	constructs	construct	NOUN
ejpam-6297	40	6	founded	found	VERB
ejpam-6297	40	7	on	on	ADP
ejpam-6297	40	8	ideals	ideal	NOUN
ejpam-6297	40	9	were	be	AUX
ejpam-6297	40	10	presented	present	VERB
ejpam-6297	40	11	in	in	ADP
ejpam-6297	40	12	[	[	X
ejpam-6297	40	13	21	21	NUM
ejpam-6297	40	14	]	]	PUNCT
ejpam-6297	40	15	.	.	PUNCT
ejpam-6297	41	1	jankovic	jankovic	PROPN
ejpam-6297	41	2	and	and	CCONJ
ejpam-6297	41	3	hamlett	hamlett	PROPN
ejpam-6297	41	4	[	[	X
ejpam-6297	41	5	21	21	NUM
ejpam-6297	41	6	]	]	PUNCT
ejpam-6297	41	7	examined	examine	VERB
ejpam-6297	41	8	and	and	CCONJ
ejpam-6297	41	9	elucidated	elucidate	VERB
ejpam-6297	41	10	the	the	DET
ejpam-6297	41	11	basic	basic	ADJ
ejpam-6297	41	12	characteristics	characteristic	NOUN
ejpam-6297	41	13	of	of	ADP
ejpam-6297	41	14	these	these	DET
ejpam-6297	41	15	spaces	space	NOUN
ejpam-6297	41	16	,	,	PUNCT
ejpam-6297	41	17	proposing	propose	VERB
ejpam-6297	41	18	the	the	DET
ejpam-6297	41	19	notion	notion	NOUN
ejpam-6297	41	20	of	of	ADP
ejpam-6297	41	21	i	i	NOUN
ejpam-6297	41	22	-	-	PUNCT
ejpam-6297	41	23	open	open	ADJ
ejpam-6297	41	24	sets	set	NOUN
ejpam-6297	41	25	and	and	CCONJ
ejpam-6297	41	26	performing	perform	VERB
ejpam-6297	41	27	comprehensive	comprehensive	ADJ
ejpam-6297	41	28	analyses	analysis	NOUN
ejpam-6297	41	29	of	of	ADP
ejpam-6297	41	30	topologies	topology	NOUN
ejpam-6297	41	31	employing	employ	VERB
ejpam-6297	41	32	ideals	ideal	NOUN
ejpam-6297	41	33	.	.	PUNCT
ejpam-6297	42	1	abd	abd	PROPN
ejpam-6297	42	2	.	.	PUNCT
ejpam-6297	43	1	el	el	PROPN
ejpam-6297	43	2	-	-	PUNCT
ejpam-6297	43	3	monsef	monsef	PROPN
ejpam-6297	43	4	et	et	PROPN
ejpam-6297	43	5	al	al	PROPN
ejpam-6297	43	6	.	.	PUNCT
ejpam-6297	44	1	[	[	X
ejpam-6297	44	2	22	22	NUM
ejpam-6297	44	3	]	]	PUNCT
ejpam-6297	44	4	conducted	conduct	VERB
ejpam-6297	44	5	a	a	DET
ejpam-6297	44	6	comprehensive	comprehensive	ADJ
ejpam-6297	44	7	analysis	analysis	NOUN
ejpam-6297	44	8	of	of	ADP
ejpam-6297	44	9	i	i	NOUN
ejpam-6297	44	10	-	-	PUNCT
ejpam-6297	44	11	open	open	ADJ
ejpam-6297	44	12	sets	set	NOUN
ejpam-6297	44	13	.	.	PUNCT
ejpam-6297	45	1	the	the	DET
ejpam-6297	45	2	notion	notion	NOUN
ejpam-6297	45	3	of	of	ADP
ejpam-6297	45	4	ig	ig	PROPN
ejpam-6297	45	5	-	-	PUNCT
ejpam-6297	45	6	closed	closed	ADJ
ejpam-6297	45	7	sets	set	NOUN
ejpam-6297	45	8	was	be	AUX
ejpam-6297	45	9	established	establish	VERB
ejpam-6297	45	10	by	by	ADP
ejpam-6297	45	11	dontchev	dontchev	PROPN
ejpam-6297	45	12	et	et	PROPN
ejpam-6297	45	13	al	al	PROPN
ejpam-6297	45	14	.	.	PROPN
ejpam-6297	46	1	in	in	ADP
ejpam-6297	46	2	1999	1999	NUM
ejpam-6297	46	3	[	[	X
ejpam-6297	46	4	18	18	NUM
ejpam-6297	46	5	]	]	PUNCT
ejpam-6297	46	6	.	.	PUNCT
ejpam-6297	47	1	furthermore	furthermore	ADV
ejpam-6297	47	2	,	,	PUNCT
ejpam-6297	47	3	abd	abd	PROPN
ejpam-6297	47	4	el	el	PROPN
ejpam-6297	47	5	-	-	PROPN
ejpam-6297	47	6	monsef	monsef	PROPN
ejpam-6297	47	7	et	et	PROPN
ejpam-6297	47	8	al	al	PROPN
ejpam-6297	47	9	.	.	PUNCT
ejpam-6297	48	1	[	[	X
ejpam-6297	48	2	23	23	NUM
ejpam-6297	48	3	]	]	PUNCT
ejpam-6297	48	4	introduced	introduce	VERB
ejpam-6297	48	5	the	the	DET
ejpam-6297	48	6	concept	concept	NOUN
ejpam-6297	48	7	of	of	ADP
ejpam-6297	48	8	the	the	DET
ejpam-6297	48	9	s	s	ADJ
ejpam-6297	48	10	-	-	ADJ
ejpam-6297	48	11	local	local	ADJ
ejpam-6297	48	12	function	function	NOUN
ejpam-6297	48	13	,	,	PUNCT
ejpam-6297	48	14	which	which	PRON
ejpam-6297	48	15	was	be	AUX
ejpam-6297	48	16	subsequently	subsequently	ADV
ejpam-6297	48	17	examined	examine	VERB
ejpam-6297	48	18	by	by	ADP
ejpam-6297	48	19	khan	khan	PROPN
ejpam-6297	48	20	and	and	CCONJ
ejpam-6297	48	21	noiri	noiri	ADV
ejpam-6297	48	22	[	[	X
ejpam-6297	48	23	24	24	NUM
ejpam-6297	48	24	]	]	PUNCT
ejpam-6297	48	25	.	.	PUNCT
ejpam-6297	49	1	ekici	ekici	NOUN
ejpam-6297	49	2	and	and	CCONJ
ejpam-6297	49	3	noiri	noiri	ADV
ejpam-6297	50	1	[	[	X
ejpam-6297	50	2	25	25	NUM
ejpam-6297	50	3	]	]	PUNCT
ejpam-6297	50	4	introduced	introduce	VERB
ejpam-6297	50	5	the	the	DET
ejpam-6297	50	6	notion	notion	NOUN
ejpam-6297	50	7	of	of	ADP
ejpam-6297	50	8	i	i	NOUN
ejpam-6297	50	9	-	-	PUNCT
ejpam-6297	50	10	submaximal	submaximal	ADJ
ejpam-6297	50	11	ideal	ideal	ADJ
ejpam-6297	50	12	topological	topological	ADJ
ejpam-6297	50	13	spaces	space	NOUN
ejpam-6297	50	14	and	and	CCONJ
ejpam-6297	50	15	studied	study	VERB
ejpam-6297	50	16	several	several	ADJ
ejpam-6297	50	17	characterizations	characterization	NOUN
ejpam-6297	50	18	and	and	CCONJ
ejpam-6297	50	19	further	further	ADJ
ejpam-6297	50	20	properties	property	NOUN
ejpam-6297	50	21	of	of	ADP
ejpam-6297	50	22	i	i	PRON
ejpam-6297	50	23	-	-	PUNCT
ejpam-6297	50	24	submaximal	submaximal	ADJ
ejpam-6297	50	25	.	.	PUNCT
ejpam-6297	51	1	recently	recently	ADV
ejpam-6297	51	2	,	,	PUNCT
ejpam-6297	51	3	boonpok	boonpok	PROPN
ejpam-6297	51	4	[	[	X
ejpam-6297	51	5	26	26	NUM
ejpam-6297	51	6	]	]	PUNCT
ejpam-6297	51	7	presented	present	VERB
ejpam-6297	51	8	the	the	DET
ejpam-6297	51	9	notion	notion	NOUN
ejpam-6297	51	10	of	of	ADP
ejpam-6297	51	11	semi	semi	ADJ
ejpam-6297	51	12	-	-	ADJ
ejpam-6297	51	13	i	i	ADV
ejpam-6297	51	14	-	-	PUNCT
ejpam-6297	51	15	submaximal	submaximal	ADJ
ejpam-6297	51	16	ideal	ideal	ADJ
ejpam-6297	51	17	topological	topological	ADJ
ejpam-6297	51	18	spaces	space	NOUN
ejpam-6297	51	19	and	and	CCONJ
ejpam-6297	51	20	examined	examine	VERB
ejpam-6297	51	21	their	their	PRON
ejpam-6297	51	22	characterizations	characterization	NOUN
ejpam-6297	51	23	.	.	PUNCT
ejpam-6297	52	1	inspired	inspire	VERB
ejpam-6297	52	2	by	by	ADP
ejpam-6297	52	3	these	these	DET
ejpam-6297	52	4	advancements	advancement	NOUN
ejpam-6297	52	5	,	,	PUNCT
ejpam-6297	52	6	this	this	DET
ejpam-6297	52	7	work	work	NOUN
ejpam-6297	52	8	aims	aim	VERB
ejpam-6297	52	9	to	to	PART
ejpam-6297	52	10	offer	offer	VERB
ejpam-6297	52	11	the	the	DET
ejpam-6297	52	12	concept	concept	NOUN
ejpam-6297	52	13	of	of	ADP
ejpam-6297	52	14	strong	strong	ADJ
ejpam-6297	52	15	β	β	X
ejpam-6297	52	16	-	-	ADJ
ejpam-6297	52	17	i	i	NOUN
ejpam-6297	52	18	-	-	PUNCT
ejpam-6297	52	19	submaximal	submaximal	ADJ
ejpam-6297	52	20	ideal	ideal	ADJ
ejpam-6297	52	21	topological	topological	ADJ
ejpam-6297	52	22	spaces	space	NOUN
ejpam-6297	52	23	,	,	PUNCT
ejpam-6297	52	24	which	which	PRON
ejpam-6297	52	25	serves	serve	VERB
ejpam-6297	52	26	as	as	ADP
ejpam-6297	52	27	a	a	DET
ejpam-6297	52	28	natural	natural	ADJ
ejpam-6297	52	29	extension	extension	NOUN
ejpam-6297	52	30	of	of	ADP
ejpam-6297	52	31	the	the	DET
ejpam-6297	52	32	previously	previously	ADV
ejpam-6297	52	33	examined	examine	VERB
ejpam-6297	52	34	semi	semi	ADJ
ejpam-6297	52	35	-	-	ADJ
ejpam-6297	52	36	i	i	ADV
ejpam-6297	52	37	-	-	PUNCT
ejpam-6297	52	38	submaximal	submaximal	ADJ
ejpam-6297	52	39	spaces	space	NOUN
ejpam-6297	52	40	.	.	PUNCT
ejpam-6297	53	1	the	the	DET
ejpam-6297	53	2	notion	notion	NOUN
ejpam-6297	53	3	of	of	ADP
ejpam-6297	53	4	paracompactness	paracompactness	NOUN
ejpam-6297	53	5	with	with	ADP
ejpam-6297	53	6	respect	respect	NOUN
ejpam-6297	53	7	to	to	ADP
ejpam-6297	53	8	an	an	DET
ejpam-6297	53	9	ideal	ideal	NOUN
ejpam-6297	53	10	was	be	AUX
ejpam-6297	53	11	first	first	ADV
ejpam-6297	53	12	introduced	introduce	VERB
ejpam-6297	53	13	by	by	ADP
ejpam-6297	53	14	zahid	zahid	PROPN
ejpam-6297	53	15	[	[	X
ejpam-6297	53	16	27	27	NUM
ejpam-6297	53	17	]	]	PUNCT
ejpam-6297	53	18	and	and	CCONJ
ejpam-6297	53	19	later	later	ADV
ejpam-6297	53	20	examined	examine	VERB
ejpam-6297	53	21	by	by	ADP
ejpam-6297	53	22	hamlett	hamlett	PROPN
ejpam-6297	53	23	et	et	PROPN
ejpam-6297	53	24	al	al	PROPN
ejpam-6297	53	25	.	.	PUNCT
ejpam-6297	54	1	[	[	X
ejpam-6297	54	2	19	19	NUM
ejpam-6297	54	3	]	]	PUNCT
ejpam-6297	54	4	.	.	PUNCT
ejpam-6297	55	1	sathiyasundari	sathiyasundari	PROPN
ejpam-6297	55	2	and	and	CCONJ
ejpam-6297	55	3	renukadevi	renukadevi	ADJ
ejpam-6297	55	4	[	[	X
ejpam-6297	55	5	28	28	NUM
ejpam-6297	55	6	]	]	X
ejpam-6297	55	7	c.	c.	PROPN
ejpam-6297	55	8	boonpok	boonpok	PROPN
ejpam-6297	55	9	,	,	PUNCT
ejpam-6297	55	10	p.	p.	PROPN
ejpam-6297	55	11	raktaow	raktaow	NOUN
ejpam-6297	55	12	,	,	PUNCT
ejpam-6297	55	13	a.	a.	PROPN
ejpam-6297	55	14	sama	sama	PROPN
ejpam-6297	55	15	-	-	PUNCT
ejpam-6297	55	16	ae	ae	PROPN
ejpam-6297	55	17	/	/	SYM
ejpam-6297	55	18	eur	eur	PROPN
ejpam-6297	55	19	.	.	PUNCT
ejpam-6297	56	1	j.	j.	PROPN
ejpam-6297	56	2	pure	pure	PROPN
ejpam-6297	56	3	appl	appl	PROPN
ejpam-6297	56	4	.	.	PROPN
ejpam-6297	56	5	math	math	PROPN
ejpam-6297	56	6	,	,	PUNCT
ejpam-6297	56	7	18	18	NUM
ejpam-6297	56	8	(	(	PUNCT
ejpam-6297	56	9	3	3	NUM
ejpam-6297	56	10	)	)	PUNCT
ejpam-6297	56	11	(	(	PUNCT
ejpam-6297	56	12	2025	2025	NUM
ejpam-6297	56	13	)	)	PUNCT
ejpam-6297	56	14	,	,	PUNCT
ejpam-6297	56	15	6297	6297	NUM
ejpam-6297	56	16	3	3	NUM
ejpam-6297	56	17	of	of	ADP
ejpam-6297	56	18	23	23	NUM
ejpam-6297	56	19	investigated	investigate	VERB
ejpam-6297	56	20	i	i	PROPN
ejpam-6297	56	21	-	-	PROPN
ejpam-6297	56	22	paracompactness	paracompactness	PROPN
ejpam-6297	56	23	and	and	CCONJ
ejpam-6297	56	24	its	its	PRON
ejpam-6297	56	25	properties	property	NOUN
ejpam-6297	56	26	,	,	PUNCT
ejpam-6297	56	27	extending	extend	VERB
ejpam-6297	56	28	certain	certain	ADJ
ejpam-6297	56	29	results	result	NOUN
ejpam-6297	56	30	from	from	ADP
ejpam-6297	56	31	paracompact	paracompact	ADJ
ejpam-6297	56	32	spaces	space	NOUN
ejpam-6297	56	33	to	to	ADP
ejpam-6297	56	34	i	i	NOUN
ejpam-6297	56	35	-	-	PUNCT
ejpam-6297	56	36	paracompact	paracompact	ADJ
ejpam-6297	56	37	spaces	space	NOUN
ejpam-6297	56	38	.	.	PUNCT
ejpam-6297	57	1	the	the	DET
ejpam-6297	57	2	concept	concept	NOUN
ejpam-6297	57	3	of	of	ADP
ejpam-6297	57	4	s	s	NOUN
ejpam-6297	57	5	-	-	NOUN
ejpam-6297	57	6	paracompactness	paracompactness	NOUN
ejpam-6297	57	7	in	in	ADP
ejpam-6297	57	8	ideal	ideal	ADJ
ejpam-6297	57	9	topological	topological	ADJ
ejpam-6297	57	10	spaces	space	NOUN
ejpam-6297	57	11	was	be	AUX
ejpam-6297	57	12	studied	study	VERB
ejpam-6297	57	13	by	by	ADP
ejpam-6297	57	14	sanabria	sanabria	PROPN
ejpam-6297	57	15	et	et	PROPN
ejpam-6297	57	16	al	al	PROPN
ejpam-6297	57	17	.	.	PUNCT
ejpam-6297	58	1	[	[	X
ejpam-6297	58	2	29	29	NUM
ejpam-6297	58	3	]	]	PUNCT
ejpam-6297	58	4	,	,	PUNCT
ejpam-6297	58	5	who	who	PRON
ejpam-6297	58	6	introduced	introduce	VERB
ejpam-6297	58	7	i	i	PROPN
ejpam-6297	58	8	-	-	PUNCT
ejpam-6297	58	9	s	s	NOUN
ejpam-6297	58	10	-	-	PUNCT
ejpam-6297	58	11	paracompact	paracompact	ADJ
ejpam-6297	58	12	spaces	space	NOUN
ejpam-6297	58	13	,	,	PUNCT
ejpam-6297	58	14	a	a	DET
ejpam-6297	58	15	new	new	ADJ
ejpam-6297	58	16	type	type	NOUN
ejpam-6297	58	17	of	of	ADP
ejpam-6297	58	18	space	space	NOUN
ejpam-6297	58	19	that	that	PRON
ejpam-6297	58	20	includes	include	VERB
ejpam-6297	58	21	both	both	DET
ejpam-6297	58	22	s	s	NOUN
ejpam-6297	58	23	-	-	NOUN
ejpam-6297	58	24	paracompact	paracompact	ADJ
ejpam-6297	58	25	and	and	CCONJ
ejpam-6297	58	26	i	i	NOUN
ejpam-6297	58	27	-	-	PUNCT
ejpam-6297	58	28	paracompact	paracompact	ADJ
ejpam-6297	58	29	spaces	space	NOUN
ejpam-6297	58	30	.	.	PUNCT
ejpam-6297	59	1	in	in	ADP
ejpam-6297	59	2	2013	2013	NUM
ejpam-6297	59	3	,	,	PUNCT
ejpam-6297	59	4	demir	demir	PROPN
ejpam-6297	59	5	and	and	CCONJ
ejpam-6297	59	6	ozbakir	ozbakir	VERB
ejpam-6297	59	7	[	[	X
ejpam-6297	59	8	14	14	NUM
ejpam-6297	59	9	]	]	PUNCT
ejpam-6297	59	10	proposed	propose	VERB
ejpam-6297	59	11	a	a	DET
ejpam-6297	59	12	modified	modify	VERB
ejpam-6297	59	13	version	version	NOUN
ejpam-6297	59	14	of	of	ADP
ejpam-6297	59	15	expandable	expandable	ADJ
ejpam-6297	59	16	and	and	CCONJ
ejpam-6297	59	17	paracompact	paracompact	ADJ
ejpam-6297	59	18	spaces	space	NOUN
ejpam-6297	59	19	,	,	PUNCT
ejpam-6297	59	20	termed	term	VERB
ejpam-6297	59	21	β	β	NOUN
ejpam-6297	59	22	-	-	NOUN
ejpam-6297	59	23	expandable	expandable	ADJ
ejpam-6297	59	24	and	and	CCONJ
ejpam-6297	59	25	β	β	NOUN
ejpam-6297	59	26	-	-	ADJ
ejpam-6297	59	27	paracompact	paracompact	ADJ
ejpam-6297	59	28	spaces	space	NOUN
ejpam-6297	59	29	,	,	PUNCT
ejpam-6297	59	30	respectively	respectively	ADV
ejpam-6297	59	31	.	.	PUNCT
ejpam-6297	60	1	they	they	PRON
ejpam-6297	60	2	demonstrated	demonstrate	VERB
ejpam-6297	60	3	that	that	SCONJ
ejpam-6297	60	4	every	every	DET
ejpam-6297	60	5	β	β	X
ejpam-6297	60	6	-	-	ADJ
ejpam-6297	60	7	paracompact	paracompact	ADJ
ejpam-6297	60	8	space	space	NOUN
ejpam-6297	60	9	is	be	AUX
ejpam-6297	60	10	essentially	essentially	ADV
ejpam-6297	60	11	a	a	DET
ejpam-6297	60	12	β	β	NOUN
ejpam-6297	60	13	-	-	ADJ
ejpam-6297	60	14	expandable	expandable	ADJ
ejpam-6297	60	15	space	space	NOUN
ejpam-6297	60	16	.	.	PUNCT
ejpam-6297	61	1	yildirim	yildirim	PROPN
ejpam-6297	61	2	et	et	PROPN
ejpam-6297	61	3	al	al	PROPN
ejpam-6297	61	4	.	.	PUNCT
ejpam-6297	62	1	[	[	X
ejpam-6297	62	2	30	30	NUM
ejpam-6297	62	3	]	]	PUNCT
ejpam-6297	62	4	introduced	introduce	VERB
ejpam-6297	62	5	the	the	DET
ejpam-6297	62	6	concept	concept	NOUN
ejpam-6297	62	7	of	of	ADP
ejpam-6297	62	8	β	β	NOUN
ejpam-6297	62	9	-	-	NOUN
ejpam-6297	62	10	paracompactness	paracompactness	NOUN
ejpam-6297	62	11	within	within	ADP
ejpam-6297	62	12	ideal	ideal	ADJ
ejpam-6297	62	13	topological	topological	ADJ
ejpam-6297	62	14	spaces	space	NOUN
ejpam-6297	62	15	and	and	CCONJ
ejpam-6297	62	16	compared	compare	VERB
ejpam-6297	62	17	it	it	PRON
ejpam-6297	62	18	with	with	ADP
ejpam-6297	62	19	existing	exist	VERB
ejpam-6297	62	20	forms	form	NOUN
ejpam-6297	62	21	of	of	ADP
ejpam-6297	62	22	paracompactness	paracompactness	NOUN
ejpam-6297	62	23	.	.	PUNCT
ejpam-6297	63	1	recently	recently	ADV
ejpam-6297	63	2	,	,	PUNCT
ejpam-6297	63	3	boonpok	boonpok	PROPN
ejpam-6297	63	4	et	et	PROPN
ejpam-6297	63	5	al	al	PROPN
ejpam-6297	63	6	.	.	PUNCT
ejpam-6297	64	1	[	[	X
ejpam-6297	64	2	31	31	NUM
ejpam-6297	64	3	]	]	PUNCT
ejpam-6297	64	4	proposed	propose	VERB
ejpam-6297	64	5	the	the	DET
ejpam-6297	64	6	concept	concept	NOUN
ejpam-6297	64	7	of	of	ADP
ejpam-6297	64	8	δ	δ	PROPN
ejpam-6297	64	9	-	-	PUNCT
ejpam-6297	64	10	βi	βi	PROPN
ejpam-6297	64	11	-	-	NOUN
ejpam-6297	64	12	paracompactness	paracompactness	NOUN
ejpam-6297	64	13	in	in	ADP
ejpam-6297	64	14	the	the	DET
ejpam-6297	64	15	context	context	NOUN
ejpam-6297	64	16	of	of	ADP
ejpam-6297	64	17	ideal	ideal	ADJ
ejpam-6297	64	18	topological	topological	ADJ
ejpam-6297	64	19	spaces	space	NOUN
ejpam-6297	64	20	as	as	ADP
ejpam-6297	64	21	a	a	DET
ejpam-6297	64	22	weaker	weak	ADJ
ejpam-6297	64	23	variant	variant	NOUN
ejpam-6297	64	24	of	of	ADP
ejpam-6297	64	25	β	β	NOUN
ejpam-6297	64	26	-	-	NOUN
ejpam-6297	64	27	paracompactness	paracompactness	NOUN
ejpam-6297	64	28	.	.	PUNCT
ejpam-6297	65	1	additionally	additionally	ADV
ejpam-6297	65	2	,	,	PUNCT
ejpam-6297	65	3	boonpok	boonpok	PROPN
ejpam-6297	65	4	and	and	CCONJ
ejpam-6297	65	5	sama	sama	PROPN
ejpam-6297	65	6	-	-	PUNCT
ejpam-6297	65	7	ae	ae	PROPN
ejpam-6297	65	8	[	[	X
ejpam-6297	65	9	32	32	NUM
ejpam-6297	65	10	]	]	PUNCT
ejpam-6297	65	11	provided	provide	VERB
ejpam-6297	65	12	characterizations	characterization	NOUN
ejpam-6297	65	13	of	of	ADP
ejpam-6297	65	14	δ1	δ1	NOUN
ejpam-6297	65	15	-	-	PUNCT
ejpam-6297	65	16	βi	βi	NOUN
ejpam-6297	65	17	-	-	NOUN
ejpam-6297	65	18	paracompactness	paracompactness	NOUN
ejpam-6297	65	19	with	with	ADP
ejpam-6297	65	20	respect	respect	NOUN
ejpam-6297	65	21	to	to	ADP
ejpam-6297	65	22	an	an	DET
ejpam-6297	65	23	ideal	ideal	NOUN
ejpam-6297	65	24	.	.	PUNCT
ejpam-6297	66	1	this	this	DET
ejpam-6297	66	2	paper	paper	NOUN
ejpam-6297	66	3	constructs	construct	VERB
ejpam-6297	66	4	strong	strong	ADJ
ejpam-6297	66	5	β	β	X
ejpam-6297	66	6	-	-	ADJ
ejpam-6297	66	7	i	i	NOUN
ejpam-6297	66	8	-	-	PUNCT
ejpam-6297	66	9	paracompact	paracompact	ADJ
ejpam-6297	66	10	spaces	space	NOUN
ejpam-6297	66	11	using	use	VERB
ejpam-6297	66	12	strong	strong	ADJ
ejpam-6297	66	13	β	β	NOUN
ejpam-6297	66	14	-	-	ADJ
ejpam-6297	66	15	i	i	NOUN
ejpam-6297	66	16	-	-	PUNCT
ejpam-6297	66	17	open	open	ADJ
ejpam-6297	66	18	sets	set	NOUN
ejpam-6297	66	19	,	,	PUNCT
ejpam-6297	66	20	and	and	CCONJ
ejpam-6297	66	21	the	the	DET
ejpam-6297	66	22	spaces	space	NOUN
ejpam-6297	66	23	under	under	ADP
ejpam-6297	66	24	study	study	NOUN
ejpam-6297	66	25	are	be	AUX
ejpam-6297	66	26	examined	examine	VERB
ejpam-6297	66	27	in	in	ADP
ejpam-6297	66	28	detail	detail	NOUN
ejpam-6297	66	29	with	with	ADP
ejpam-6297	66	30	respect	respect	NOUN
ejpam-6297	66	31	to	to	ADP
ejpam-6297	66	32	β	β	NOUN
ejpam-6297	66	33	-	-	NOUN
ejpam-6297	66	34	paracompactness	paracompactness	NOUN
ejpam-6297	66	35	,	,	PUNCT
ejpam-6297	66	36	as	as	SCONJ
ejpam-6297	66	37	described	describe	VERB
ejpam-6297	66	38	in	in	ADP
ejpam-6297	66	39	reference	reference	NOUN
ejpam-6297	66	40	[	[	X
ejpam-6297	66	41	30	30	NUM
ejpam-6297	66	42	]	]	PUNCT
ejpam-6297	66	43	.	.	PUNCT
ejpam-6297	67	1	multiple	multiple	ADJ
ejpam-6297	67	2	characterizations	characterization	NOUN
ejpam-6297	67	3	of	of	ADP
ejpam-6297	67	4	strong	strong	ADJ
ejpam-6297	67	5	β	β	X
ejpam-6297	67	6	-	-	ADJ
ejpam-6297	67	7	i	i	NOUN
ejpam-6297	67	8	-	-	PUNCT
ejpam-6297	67	9	paracompactness	paracompactness	PROPN
ejpam-6297	67	10	are	be	AUX
ejpam-6297	67	11	presented	present	VERB
ejpam-6297	67	12	to	to	PART
ejpam-6297	67	13	enhance	enhance	VERB
ejpam-6297	67	14	the	the	DET
ejpam-6297	67	15	theory	theory	NOUN
ejpam-6297	67	16	of	of	ADP
ejpam-6297	67	17	ideal	ideal	ADJ
ejpam-6297	67	18	topology	topology	NOUN
ejpam-6297	67	19	and	and	CCONJ
ejpam-6297	67	20	give	give	VERB
ejpam-6297	67	21	a	a	DET
ejpam-6297	67	22	wider	wide	ADJ
ejpam-6297	67	23	framework	framework	NOUN
ejpam-6297	67	24	for	for	ADP
ejpam-6297	67	25	future	future	ADJ
ejpam-6297	67	26	research	research	NOUN
ejpam-6297	67	27	.	.	PUNCT
ejpam-6297	68	1	throughout	throughout	ADP
ejpam-6297	68	2	this	this	DET
ejpam-6297	68	3	paper	paper	NOUN
ejpam-6297	68	4	,	,	PUNCT
ejpam-6297	68	5	unless	unless	SCONJ
ejpam-6297	68	6	otherwise	otherwise	ADV
ejpam-6297	68	7	specified	specify	VERB
ejpam-6297	68	8	,	,	PUNCT
ejpam-6297	68	9	the	the	DET
ejpam-6297	68	10	symbols	symbol	NOUN
ejpam-6297	68	11	(	(	PUNCT
ejpam-6297	68	12	x	x	X
ejpam-6297	68	13	,	,	PUNCT
ejpam-6297	68	14	τ	τ	PROPN
ejpam-6297	68	15	)	)	PUNCT
ejpam-6297	68	16	,	,	PUNCT
ejpam-6297	68	17	or	or	CCONJ
ejpam-6297	68	18	simply	simply	ADV
ejpam-6297	68	19	x	x	X
ejpam-6297	68	20	,	,	PUNCT
ejpam-6297	68	21	refer	refer	VERB
ejpam-6297	68	22	to	to	ADP
ejpam-6297	68	23	a	a	DET
ejpam-6297	68	24	general	general	ADJ
ejpam-6297	68	25	topological	topological	ADJ
ejpam-6297	68	26	space	space	NOUN
ejpam-6297	68	27	without	without	ADP
ejpam-6297	68	28	assuming	assume	VERB
ejpam-6297	68	29	any	any	DET
ejpam-6297	68	30	separation	separation	NOUN
ejpam-6297	68	31	axioms	axiom	VERB
ejpam-6297	68	32	.	.	PUNCT
ejpam-6297	69	1	for	for	ADP
ejpam-6297	69	2	a	a	DET
ejpam-6297	69	3	subset	subset	NOUN
ejpam-6297	69	4	a	a	PRON
ejpam-6297	69	5	of	of	ADP
ejpam-6297	69	6	a	a	DET
ejpam-6297	69	7	topological	topological	ADJ
ejpam-6297	69	8	space	space	NOUN
ejpam-6297	69	9	(	(	PUNCT
ejpam-6297	69	10	x	x	X
ejpam-6297	69	11	,	,	PUNCT
ejpam-6297	69	12	τ	τ	PROPN
ejpam-6297	69	13	)	)	PUNCT
ejpam-6297	69	14	,	,	PUNCT
ejpam-6297	69	15	the	the	DET
ejpam-6297	69	16	closure	closure	NOUN
ejpam-6297	69	17	and	and	CCONJ
ejpam-6297	69	18	interior	interior	NOUN
ejpam-6297	69	19	of	of	ADP
ejpam-6297	69	20	a	a	PRON
ejpam-6297	69	21	are	be	AUX
ejpam-6297	69	22	denoted	denote	VERB
ejpam-6297	69	23	by	by	ADP
ejpam-6297	69	24	cl(a	cl(a	NOUN
ejpam-6297	69	25	)	)	PUNCT
ejpam-6297	69	26	and	and	CCONJ
ejpam-6297	69	27	int(a	int(a	PROPN
ejpam-6297	69	28	)	)	PUNCT
ejpam-6297	69	29	,	,	PUNCT
ejpam-6297	69	30	respectively	respectively	ADV
ejpam-6297	69	31	.	.	PUNCT
ejpam-6297	70	1	an	an	DET
ejpam-6297	70	2	ideal	ideal	NOUN
ejpam-6297	70	3	i	i	PRON
ejpam-6297	70	4	on	on	ADP
ejpam-6297	70	5	a	a	DET
ejpam-6297	70	6	set	set	NOUN
ejpam-6297	70	7	x	x	PUNCT
ejpam-6297	70	8	is	be	AUX
ejpam-6297	70	9	a	a	DET
ejpam-6297	70	10	nonempty	nonempty	ADJ
ejpam-6297	70	11	family	family	NOUN
ejpam-6297	70	12	of	of	ADP
ejpam-6297	70	13	subsets	subset	NOUN
ejpam-6297	70	14	of	of	ADP
ejpam-6297	70	15	x	x	PUNCT
ejpam-6297	70	16	such	such	ADJ
ejpam-6297	70	17	that	that	SCONJ
ejpam-6297	70	18	if	if	SCONJ
ejpam-6297	70	19	a	a	DET
ejpam-6297	70	20	∈	∈	X
ejpam-6297	70	21	i	i	PRON
ejpam-6297	70	22	and	and	CCONJ
ejpam-6297	70	23	b	b	X
ejpam-6297	70	24	⊆	⊆	NUM
ejpam-6297	70	25	a	a	PRON
ejpam-6297	70	26	,	,	PUNCT
ejpam-6297	70	27	then	then	ADV
ejpam-6297	70	28	b	b	X
ejpam-6297	70	29	∈	∈	PROPN
ejpam-6297	70	30	i	i	PRON
ejpam-6297	70	31	,	,	PUNCT
ejpam-6297	70	32	and	and	CCONJ
ejpam-6297	70	33	if	if	SCONJ
ejpam-6297	70	34	a	a	DET
ejpam-6297	70	35	∈	∈	X
ejpam-6297	70	36	i	i	PRON
ejpam-6297	70	37	and	and	CCONJ
ejpam-6297	70	38	b	b	X
ejpam-6297	70	39	∈	∈	PROPN
ejpam-6297	70	40	i	i	PRON
ejpam-6297	70	41	,	,	PUNCT
ejpam-6297	70	42	then	then	ADV
ejpam-6297	70	43	a	a	DET
ejpam-6297	70	44	∪	∪	X
ejpam-6297	70	45	b	b	PROPN
ejpam-6297	70	46	∈	∈	PROPN
ejpam-6297	70	47	i.	i.	NOUN
ejpam-6297	70	48	a	a	DET
ejpam-6297	70	49	topological	topological	ADJ
ejpam-6297	70	50	space	space	NOUN
ejpam-6297	70	51	(	(	PUNCT
ejpam-6297	70	52	x	x	X
ejpam-6297	70	53	,	,	PUNCT
ejpam-6297	70	54	τ	τ	X
ejpam-6297	70	55	)	)	PUNCT
ejpam-6297	70	56	together	together	ADV
ejpam-6297	70	57	with	with	ADP
ejpam-6297	70	58	an	an	DET
ejpam-6297	70	59	ideal	ideal	NOUN
ejpam-6297	70	60	i	i	PRON
ejpam-6297	70	61	is	be	AUX
ejpam-6297	70	62	called	call	VERB
ejpam-6297	70	63	an	an	DET
ejpam-6297	70	64	ideal	ideal	ADJ
ejpam-6297	70	65	topological	topological	ADJ
ejpam-6297	70	66	space	space	NOUN
ejpam-6297	70	67	and	and	CCONJ
ejpam-6297	70	68	denoted	denote	VERB
ejpam-6297	70	69	by	by	ADP
ejpam-6297	70	70	(	(	PUNCT
ejpam-6297	70	71	x	x	NOUN
ejpam-6297	70	72	,	,	PUNCT
ejpam-6297	70	73	τ	τ	PROPN
ejpam-6297	70	74	,	,	PUNCT
ejpam-6297	70	75	i	i	PROPN
ejpam-6297	70	76	)	)	PUNCT
ejpam-6297	70	77	.	.	PUNCT
ejpam-6297	71	1	the	the	DET
ejpam-6297	71	2	set	set	NOUN
ejpam-6297	71	3	of	of	ADP
ejpam-6297	71	4	all	all	DET
ejpam-6297	71	5	subsets	subset	NOUN
ejpam-6297	71	6	of	of	ADP
ejpam-6297	71	7	x	x	SYM
ejpam-6297	71	8	is	be	AUX
ejpam-6297	71	9	denoted	denote	VERB
ejpam-6297	71	10	as	as	ADP
ejpam-6297	71	11	p	p	PROPN
ejpam-6297	71	12	(	(	PUNCT
ejpam-6297	71	13	x	x	NOUN
ejpam-6297	71	14	)	)	PUNCT
ejpam-6297	71	15	.	.	PUNCT
ejpam-6297	72	1	a	a	DET
ejpam-6297	72	2	set	set	NOUN
ejpam-6297	72	3	operator	operator	NOUN
ejpam-6297	72	4	(	(	PUNCT
ejpam-6297	72	5	·	·	PUNCT
ejpam-6297	72	6	)	)	PUNCT
ejpam-6297	72	7	∗	∗	NOUN
ejpam-6297	72	8	:	:	PUNCT
ejpam-6297	72	9	p	p	X
ejpam-6297	72	10	(	(	PUNCT
ejpam-6297	72	11	x	x	NOUN
ejpam-6297	72	12	)	)	PUNCT
ejpam-6297	72	13	→	→	SYM
ejpam-6297	72	14	p	p	X
ejpam-6297	72	15	(	(	PUNCT
ejpam-6297	72	16	x	x	NOUN
ejpam-6297	72	17	)	)	PUNCT
ejpam-6297	72	18	,	,	PUNCT
ejpam-6297	72	19	known	know	VERB
ejpam-6297	72	20	as	as	ADP
ejpam-6297	72	21	a	a	DET
ejpam-6297	72	22	local	local	ADJ
ejpam-6297	72	23	function	function	NOUN
ejpam-6297	72	24	[	[	X
ejpam-6297	72	25	16	16	NUM
ejpam-6297	72	26	]	]	PUNCT
ejpam-6297	72	27	,	,	PUNCT
ejpam-6297	72	28	is	be	AUX
ejpam-6297	72	29	defined	define	VERB
ejpam-6297	72	30	with	with	ADP
ejpam-6297	72	31	respect	respect	NOUN
ejpam-6297	72	32	to	to	ADP
ejpam-6297	72	33	a	a	DET
ejpam-6297	72	34	topology	topology	NOUN
ejpam-6297	72	35	τ	τ	PROPN
ejpam-6297	72	36	and	and	CCONJ
ejpam-6297	72	37	an	an	DET
ejpam-6297	72	38	ideal	ideal	ADJ
ejpam-6297	72	39	i.	i.	NOUN
ejpam-6297	72	40	for	for	ADP
ejpam-6297	72	41	any	any	DET
ejpam-6297	72	42	subset	subset	NOUN
ejpam-6297	72	43	a	a	DET
ejpam-6297	72	44	⊆	⊆	NUM
ejpam-6297	72	45	x	x	SYM
ejpam-6297	72	46	,	,	PUNCT
ejpam-6297	72	47	it	it	PRON
ejpam-6297	72	48	is	be	AUX
ejpam-6297	72	49	given	give	VERB
ejpam-6297	72	50	by	by	ADP
ejpam-6297	72	51	a∗(i	a∗(i	PROPN
ejpam-6297	72	52	,	,	PUNCT
ejpam-6297	72	53	τ	τ	X
ejpam-6297	72	54	)	)	PUNCT
ejpam-6297	72	55	=	=	PRON
ejpam-6297	73	1	{	{	PUNCT
ejpam-6297	73	2	x	x	PUNCT
ejpam-6297	73	3	∈	∈	PROPN
ejpam-6297	73	4	x	x	X
ejpam-6297	73	5	:	:	PUNCT
ejpam-6297	73	6	u	u	NOUN
ejpam-6297	73	7	∩a	∩a	NOUN
ejpam-6297	73	8	/∈	/∈	PUNCT
ejpam-6297	74	1	i	i	PRON
ejpam-6297	74	2	for	for	ADP
ejpam-6297	74	3	every	every	DET
ejpam-6297	74	4	u	u	PROPN
ejpam-6297	74	5	∈	∈	PROPN
ejpam-6297	74	6	τ(x	τ(x	NOUN
ejpam-6297	74	7	)	)	PUNCT
ejpam-6297	74	8	}	}	PUNCT
ejpam-6297	74	9	,	,	PUNCT
ejpam-6297	74	10	where	where	SCONJ
ejpam-6297	74	11	τ(x	τ(x	NOUN
ejpam-6297	74	12	)	)	PUNCT
ejpam-6297	74	13	=	=	PRON
ejpam-6297	74	14	{	{	PUNCT
ejpam-6297	74	15	u	u	X
ejpam-6297	74	16	∈	∈	PROPN
ejpam-6297	74	17	τ	τ	X
ejpam-6297	74	18	:	:	PUNCT
ejpam-6297	74	19	x	x	X
ejpam-6297	74	20	∈	∈	PROPN
ejpam-6297	74	21	u	u	NOUN
ejpam-6297	74	22	}	}	PUNCT
ejpam-6297	74	23	represents	represent	VERB
ejpam-6297	74	24	the	the	DET
ejpam-6297	74	25	set	set	NOUN
ejpam-6297	74	26	of	of	ADP
ejpam-6297	74	27	open	open	ADJ
ejpam-6297	74	28	neighborhoods	neighborhood	NOUN
ejpam-6297	74	29	of	of	ADP
ejpam-6297	74	30	x	x	PUNCT
ejpam-6297	74	31	in	in	ADP
ejpam-6297	74	32	the	the	DET
ejpam-6297	74	33	topology	topology	NOUN
ejpam-6297	74	34	τ	τ	PROPN
ejpam-6297	74	35	.	.	PUNCT
ejpam-6297	75	1	this	this	DET
ejpam-6297	75	2	operator	operator	NOUN
ejpam-6297	75	3	induces	induce	VERB
ejpam-6297	75	4	a	a	DET
ejpam-6297	75	5	finer	fine	ADJ
ejpam-6297	75	6	topology	topology	NOUN
ejpam-6297	75	7	on	on	ADP
ejpam-6297	75	8	x	x	PRON
ejpam-6297	75	9	,	,	PUNCT
ejpam-6297	75	10	called	call	VERB
ejpam-6297	75	11	the	the	DET
ejpam-6297	75	12	∗-topology	∗-topology	NOUN
ejpam-6297	75	13	,	,	PUNCT
ejpam-6297	75	14	denoted	denote	VERB
ejpam-6297	75	15	by	by	ADP
ejpam-6297	75	16	τ∗(i	τ∗(i	PROPN
ejpam-6297	75	17	)	)	PUNCT
ejpam-6297	75	18	.	.	PUNCT
ejpam-6297	76	1	the	the	DET
ejpam-6297	76	2	∗-topology	∗-topology	NOUN
ejpam-6297	76	3	is	be	AUX
ejpam-6297	76	4	generated	generate	VERB
ejpam-6297	76	5	by	by	ADP
ejpam-6297	76	6	the	the	DET
ejpam-6297	76	7	kuratowski	kuratowski	ADJ
ejpam-6297	76	8	closure	closure	NOUN
ejpam-6297	76	9	operator	operator	NOUN
ejpam-6297	76	10	,	,	PUNCT
ejpam-6297	76	11	which	which	PRON
ejpam-6297	76	12	is	be	AUX
ejpam-6297	76	13	defined	define	VERB
ejpam-6297	76	14	as	as	ADP
ejpam-6297	76	15	cl∗(a	cl∗(a	NOUN
ejpam-6297	76	16	)	)	PUNCT
ejpam-6297	76	17	=	=	PUNCT
ejpam-6297	77	1	a∪a∗	a∪a∗	PROPN
ejpam-6297	77	2	[	[	PUNCT
ejpam-6297	77	3	21	21	NUM
ejpam-6297	77	4	]	]	PUNCT
ejpam-6297	77	5	.	.	PUNCT
ejpam-6297	78	1	for	for	ADP
ejpam-6297	78	2	any	any	DET
ejpam-6297	78	3	ideal	ideal	ADJ
ejpam-6297	78	4	topological	topological	ADJ
ejpam-6297	78	5	space	space	NOUN
ejpam-6297	78	6	,	,	PUNCT
ejpam-6297	78	7	a	a	DET
ejpam-6297	78	8	finer	fine	ADJ
ejpam-6297	78	9	topology	topology	NOUN
ejpam-6297	78	10	τ∗(i	τ∗(i	PROPN
ejpam-6297	78	11	)	)	PUNCT
ejpam-6297	78	12	,	,	PUNCT
ejpam-6297	78	13	or	or	CCONJ
ejpam-6297	78	14	simply	simply	ADV
ejpam-6297	78	15	τ∗	τ∗	ADJ
ejpam-6297	78	16	,	,	PUNCT
ejpam-6297	78	17	always	always	ADV
ejpam-6297	78	18	exists	exist	VERB
ejpam-6297	78	19	.	.	PUNCT
ejpam-6297	79	1	it	it	PRON
ejpam-6297	79	2	is	be	AUX
ejpam-6297	79	3	generated	generate	VERB
ejpam-6297	79	4	by	by	ADP
ejpam-6297	79	5	the	the	DET
ejpam-6297	79	6	subbasis	subbasis	NOUN
ejpam-6297	79	7	β(i	β(i	SYM
ejpam-6297	79	8	,	,	PUNCT
ejpam-6297	79	9	τ	τ	X
ejpam-6297	79	10	)	)	PUNCT
ejpam-6297	79	11	=	=	PRON
ejpam-6297	79	12	{	{	PUNCT
ejpam-6297	79	13	u	u	NOUN
ejpam-6297	79	14	−	−	PROPN
ejpam-6297	80	1	i	i	PRON
ejpam-6297	80	2	:	:	PUNCT
ejpam-6297	80	3	u	u	PROPN
ejpam-6297	80	4	∈	∈	PROPN
ejpam-6297	80	5	τ	τ	PROPN
ejpam-6297	80	6	,	,	PUNCT
ejpam-6297	80	7	i	i	PRON
ejpam-6297	80	8	∈	∈	PROPN
ejpam-6297	80	9	i	i	X
ejpam-6297	80	10	}	}	PUNCT
ejpam-6297	80	11	.	.	PUNCT
ejpam-6297	81	1	however	however	ADV
ejpam-6297	81	2	,	,	PUNCT
ejpam-6297	81	3	β(i	β(i	PRON
ejpam-6297	81	4	,	,	PUNCT
ejpam-6297	81	5	τ	τ	X
ejpam-6297	81	6	)	)	PUNCT
ejpam-6297	81	7	does	do	AUX
ejpam-6297	81	8	not	not	PART
ejpam-6297	81	9	necessarily	necessarily	ADV
ejpam-6297	81	10	form	form	VERB
ejpam-6297	81	11	a	a	DET
ejpam-6297	81	12	topology	topology	NOUN
ejpam-6297	81	13	in	in	ADP
ejpam-6297	81	14	general	general	ADJ
ejpam-6297	81	15	[	[	X
ejpam-6297	81	16	21	21	NUM
ejpam-6297	81	17	]	]	PUNCT
ejpam-6297	81	18	.	.	PUNCT
ejpam-6297	82	1	it	it	PRON
ejpam-6297	82	2	is	be	AUX
ejpam-6297	82	3	clear	clear	ADJ
ejpam-6297	82	4	that	that	SCONJ
ejpam-6297	82	5	a∗	a∗	ADJ
ejpam-6297	82	6	⊆	⊆	NUM
ejpam-6297	82	7	b∗	b∗	ADJ
ejpam-6297	82	8	and	and	CCONJ
ejpam-6297	82	9	cl∗(a	cl∗(a	ADJ
ejpam-6297	82	10	)	)	PUNCT
ejpam-6297	82	11	⊆	⊆	NUM
ejpam-6297	82	12	cl∗(b	cl∗(b	NOUN
ejpam-6297	82	13	)	)	PUNCT
ejpam-6297	82	14	if	if	SCONJ
ejpam-6297	82	15	a	a	DET
ejpam-6297	82	16	⊆	⊆	NUM
ejpam-6297	82	17	b.	b.	NOUN
ejpam-6297	82	18	definition	definition	NOUN
ejpam-6297	82	19	1	1	NUM
ejpam-6297	82	20	.	.	PUNCT
ejpam-6297	83	1	[	[	X
ejpam-6297	83	2	33	33	NUM
ejpam-6297	83	3	]	]	PUNCT
ejpam-6297	83	4	a	a	DET
ejpam-6297	83	5	subset	subset	NOUN
ejpam-6297	83	6	a	a	PRON
ejpam-6297	83	7	of	of	ADP
ejpam-6297	83	8	an	an	DET
ejpam-6297	83	9	ideal	ideal	ADJ
ejpam-6297	83	10	topological	topological	ADJ
ejpam-6297	83	11	space	space	NOUN
ejpam-6297	83	12	(	(	PUNCT
ejpam-6297	83	13	x	x	X
ejpam-6297	83	14	,	,	PUNCT
ejpam-6297	83	15	τ	τ	PROPN
ejpam-6297	83	16	,	,	PUNCT
ejpam-6297	83	17	i	i	PROPN
ejpam-6297	83	18	)	)	PUNCT
ejpam-6297	83	19	is	be	AUX
ejpam-6297	83	20	called	call	VERB
ejpam-6297	83	21	strong	strong	ADJ
ejpam-6297	83	22	β	β	NOUN
ejpam-6297	83	23	-	-	VERB
ejpam-6297	83	24	iopen	iopen	VERB
ejpam-6297	83	25	if	if	SCONJ
ejpam-6297	83	26	a	a	DET
ejpam-6297	83	27	⊆	⊆	NUM
ejpam-6297	83	28	cl∗(int(cl∗(a	cl∗(int(cl∗(a	NOUN
ejpam-6297	83	29	)	)	PUNCT
ejpam-6297	83	30	)	)	PUNCT
ejpam-6297	83	31	)	)	PUNCT
ejpam-6297	83	32	.	.	PUNCT
ejpam-6297	84	1	the	the	DET
ejpam-6297	84	2	complement	complement	NOUN
ejpam-6297	84	3	of	of	ADP
ejpam-6297	84	4	a	a	DET
ejpam-6297	84	5	strong	strong	ADJ
ejpam-6297	84	6	β	β	X
ejpam-6297	84	7	-	-	ADJ
ejpam-6297	84	8	i	i	PRON
ejpam-6297	84	9	-	-	PUNCT
ejpam-6297	84	10	open	open	ADJ
ejpam-6297	84	11	set	set	NOUN
ejpam-6297	84	12	is	be	AUX
ejpam-6297	84	13	called	call	VERB
ejpam-6297	84	14	a	a	DET
ejpam-6297	84	15	strong	strong	ADJ
ejpam-6297	84	16	β	β	X
ejpam-6297	84	17	-	-	ADJ
ejpam-6297	84	18	i	i	NOUN
ejpam-6297	84	19	-	-	PUNCT
ejpam-6297	84	20	closed	close	VERB
ejpam-6297	84	21	set	set	NOUN
ejpam-6297	84	22	.	.	PUNCT
ejpam-6297	85	1	according	accord	VERB
ejpam-6297	85	2	to	to	ADP
ejpam-6297	85	3	definition	definition	NOUN
ejpam-6297	85	4	1	1	NUM
ejpam-6297	85	5	,	,	PUNCT
ejpam-6297	85	6	we	we	PRON
ejpam-6297	85	7	have	have	VERB
ejpam-6297	85	8	the	the	DET
ejpam-6297	85	9	following	follow	VERB
ejpam-6297	85	10	lemma	lemma	PROPN
ejpam-6297	85	11	.	.	PUNCT
ejpam-6297	86	1	lemma	lemma	PROPN
ejpam-6297	86	2	1	1	NUM
ejpam-6297	86	3	.	.	PUNCT
ejpam-6297	87	1	in	in	ADP
ejpam-6297	87	2	an	an	DET
ejpam-6297	87	3	ideal	ideal	ADJ
ejpam-6297	87	4	topological	topological	ADJ
ejpam-6297	87	5	space	space	NOUN
ejpam-6297	87	6	,	,	PUNCT
ejpam-6297	87	7	the	the	DET
ejpam-6297	87	8	following	follow	VERB
ejpam-6297	87	9	properties	property	NOUN
ejpam-6297	87	10	are	be	AUX
ejpam-6297	87	11	satisfied	satisfied	ADJ
ejpam-6297	87	12	:	:	PUNCT
ejpam-6297	87	13	c.	c.	PROPN
ejpam-6297	87	14	boonpok	boonpok	PROPN
ejpam-6297	87	15	,	,	PUNCT
ejpam-6297	87	16	p.	p.	PROPN
ejpam-6297	87	17	raktaow	raktaow	NOUN
ejpam-6297	87	18	,	,	PUNCT
ejpam-6297	87	19	a.	a.	PROPN
ejpam-6297	87	20	sama	sama	PROPN
ejpam-6297	87	21	-	-	PUNCT
ejpam-6297	87	22	ae	ae	PROPN
ejpam-6297	87	23	/	/	SYM
ejpam-6297	87	24	eur	eur	PROPN
ejpam-6297	87	25	.	.	PUNCT
ejpam-6297	88	1	j.	j.	PROPN
ejpam-6297	88	2	pure	pure	PROPN
ejpam-6297	88	3	appl	appl	PROPN
ejpam-6297	88	4	.	.	PROPN
ejpam-6297	88	5	math	math	PROPN
ejpam-6297	88	6	,	,	PUNCT
ejpam-6297	88	7	18	18	NUM
ejpam-6297	88	8	(	(	PUNCT
ejpam-6297	88	9	3	3	NUM
ejpam-6297	88	10	)	)	PUNCT
ejpam-6297	88	11	(	(	PUNCT
ejpam-6297	88	12	2025	2025	NUM
ejpam-6297	88	13	)	)	PUNCT
ejpam-6297	88	14	,	,	PUNCT
ejpam-6297	88	15	6297	6297	NUM
ejpam-6297	88	16	4	4	NUM
ejpam-6297	88	17	of	of	ADP
ejpam-6297	88	18	23	23	NUM
ejpam-6297	88	19	(	(	PUNCT
ejpam-6297	88	20	1	1	NUM
ejpam-6297	88	21	)	)	PUNCT
ejpam-6297	88	22	the	the	DET
ejpam-6297	88	23	arbitrary	arbitrary	ADJ
ejpam-6297	88	24	union	union	NOUN
ejpam-6297	88	25	of	of	ADP
ejpam-6297	88	26	strong	strong	ADJ
ejpam-6297	88	27	β	β	X
ejpam-6297	88	28	-	-	ADJ
ejpam-6297	88	29	i	i	NOUN
ejpam-6297	88	30	-	-	PUNCT
ejpam-6297	88	31	open	open	ADJ
ejpam-6297	88	32	sets	set	NOUN
ejpam-6297	88	33	is	be	AUX
ejpam-6297	88	34	itself	itself	PRON
ejpam-6297	88	35	a	a	DET
ejpam-6297	88	36	strong	strong	ADJ
ejpam-6297	88	37	β	β	X
ejpam-6297	88	38	-	-	ADJ
ejpam-6297	88	39	i	i	NOUN
ejpam-6297	88	40	-	-	PUNCT
ejpam-6297	88	41	open	open	ADJ
ejpam-6297	88	42	set	set	NOUN
ejpam-6297	88	43	;	;	PUNCT
ejpam-6297	88	44	and	and	CCONJ
ejpam-6297	88	45	(	(	PUNCT
ejpam-6297	88	46	2	2	X
ejpam-6297	88	47	)	)	PUNCT
ejpam-6297	88	48	the	the	DET
ejpam-6297	88	49	arbitrary	arbitrary	ADJ
ejpam-6297	88	50	intersection	intersection	NOUN
ejpam-6297	88	51	of	of	ADP
ejpam-6297	88	52	strong	strong	ADJ
ejpam-6297	88	53	β	β	X
ejpam-6297	88	54	-	-	ADJ
ejpam-6297	88	55	i	i	NOUN
ejpam-6297	88	56	-	-	PUNCT
ejpam-6297	88	57	closed	close	VERB
ejpam-6297	88	58	sets	set	NOUN
ejpam-6297	88	59	remains	remain	VERB
ejpam-6297	88	60	a	a	DET
ejpam-6297	88	61	strong	strong	ADJ
ejpam-6297	88	62	β	β	X
ejpam-6297	88	63	-	-	ADJ
ejpam-6297	88	64	i	i	NOUN
ejpam-6297	88	65	-	-	PUNCT
ejpam-6297	88	66	closed	close	VERB
ejpam-6297	88	67	set	set	NOUN
ejpam-6297	88	68	.	.	PUNCT
ejpam-6297	89	1	definition	definition	NOUN
ejpam-6297	89	2	2	2	NUM
ejpam-6297	89	3	.	.	PUNCT
ejpam-6297	90	1	[	[	X
ejpam-6297	90	2	33	33	NUM
ejpam-6297	90	3	]	]	PUNCT
ejpam-6297	90	4	the	the	DET
ejpam-6297	90	5	strong	strong	ADJ
ejpam-6297	90	6	β	β	X
ejpam-6297	90	7	-	-	ADJ
ejpam-6297	90	8	i	i	NOUN
ejpam-6297	90	9	-	-	PUNCT
ejpam-6297	90	10	closure	closure	NOUN
ejpam-6297	90	11	of	of	ADP
ejpam-6297	90	12	a	a	DET
ejpam-6297	90	13	subset	subset	NOUN
ejpam-6297	90	14	a	a	PRON
ejpam-6297	90	15	of	of	ADP
ejpam-6297	90	16	an	an	DET
ejpam-6297	90	17	ideal	ideal	ADJ
ejpam-6297	90	18	topological	topological	ADJ
ejpam-6297	90	19	space	space	NOUN
ejpam-6297	90	20	(	(	PUNCT
ejpam-6297	90	21	x	x	X
ejpam-6297	90	22	,	,	PUNCT
ejpam-6297	90	23	τ	τ	PROPN
ejpam-6297	90	24	,	,	PUNCT
ejpam-6297	90	25	i	i	PROPN
ejpam-6297	90	26	)	)	PUNCT
ejpam-6297	90	27	,	,	PUNCT
ejpam-6297	90	28	denoted	denote	VERB
ejpam-6297	90	29	by	by	ADP
ejpam-6297	90	30	sβ	sβ	PROPN
ejpam-6297	90	31	cli(a	cli(a	PROPN
ejpam-6297	90	32	)	)	PUNCT
ejpam-6297	90	33	,	,	PUNCT
ejpam-6297	90	34	is	be	AUX
ejpam-6297	90	35	defined	define	VERB
ejpam-6297	90	36	as	as	ADP
ejpam-6297	90	37	the	the	DET
ejpam-6297	90	38	intersection	intersection	NOUN
ejpam-6297	90	39	of	of	ADP
ejpam-6297	90	40	all	all	DET
ejpam-6297	90	41	strong	strong	ADJ
ejpam-6297	90	42	β	β	X
ejpam-6297	90	43	-	-	ADJ
ejpam-6297	90	44	i	i	NOUN
ejpam-6297	90	45	-	-	PUNCT
ejpam-6297	90	46	closed	close	VERB
ejpam-6297	90	47	sets	set	NOUN
ejpam-6297	90	48	containing	contain	VERB
ejpam-6297	90	49	a	a	PRON
ejpam-6297	90	50	,	,	PUNCT
ejpam-6297	90	51	while	while	SCONJ
ejpam-6297	90	52	the	the	DET
ejpam-6297	90	53	strong	strong	ADJ
ejpam-6297	90	54	β	β	X
ejpam-6297	90	55	-	-	ADJ
ejpam-6297	90	56	i	i	NOUN
ejpam-6297	90	57	-	-	NOUN
ejpam-6297	90	58	interior	interior	NOUN
ejpam-6297	90	59	of	of	ADP
ejpam-6297	90	60	a	a	PRON
ejpam-6297	90	61	,	,	PUNCT
ejpam-6297	90	62	denoted	denote	VERB
ejpam-6297	90	63	by	by	ADP
ejpam-6297	90	64	sβ	sβ	NUM
ejpam-6297	90	65	inti(a	inti(a	PROPN
ejpam-6297	90	66	)	)	PUNCT
ejpam-6297	90	67	,	,	PUNCT
ejpam-6297	90	68	is	be	AUX
ejpam-6297	90	69	defined	define	VERB
ejpam-6297	90	70	as	as	ADP
ejpam-6297	90	71	the	the	DET
ejpam-6297	90	72	union	union	NOUN
ejpam-6297	90	73	of	of	ADP
ejpam-6297	90	74	all	all	DET
ejpam-6297	90	75	strong	strong	ADJ
ejpam-6297	90	76	β	β	X
ejpam-6297	90	77	-	-	ADJ
ejpam-6297	90	78	i	i	NOUN
ejpam-6297	90	79	-	-	PUNCT
ejpam-6297	90	80	open	open	ADJ
ejpam-6297	90	81	sets	set	NOUN
ejpam-6297	90	82	contained	contain	VERB
ejpam-6297	90	83	in	in	ADP
ejpam-6297	90	84	a.	a.	NOUN
ejpam-6297	90	85	utilizing	utilize	VERB
ejpam-6297	90	86	definition	definition	NOUN
ejpam-6297	90	87	2	2	NUM
ejpam-6297	90	88	,	,	PUNCT
ejpam-6297	90	89	we	we	PRON
ejpam-6297	90	90	have	have	VERB
ejpam-6297	90	91	the	the	DET
ejpam-6297	90	92	following	follow	VERB
ejpam-6297	90	93	lemma	lemma	PROPN
ejpam-6297	90	94	.	.	PUNCT
ejpam-6297	91	1	lemma	lemma	PROPN
ejpam-6297	91	2	2	2	X
ejpam-6297	91	3	.	.	PUNCT
ejpam-6297	92	1	let	let	VERB
ejpam-6297	92	2	a	a	PRON
ejpam-6297	92	3	and	and	CCONJ
ejpam-6297	92	4	b	b	NOUN
ejpam-6297	92	5	be	be	AUX
ejpam-6297	92	6	subsets	subset	NOUN
ejpam-6297	92	7	of	of	ADP
ejpam-6297	92	8	an	an	DET
ejpam-6297	92	9	ideal	ideal	ADJ
ejpam-6297	92	10	topological	topological	ADJ
ejpam-6297	92	11	space	space	NOUN
ejpam-6297	92	12	(	(	PUNCT
ejpam-6297	92	13	x	x	X
ejpam-6297	92	14	,	,	PUNCT
ejpam-6297	92	15	τ	τ	PROPN
ejpam-6297	92	16	,	,	PUNCT
ejpam-6297	92	17	i	i	PROPN
ejpam-6297	92	18	)	)	PUNCT
ejpam-6297	92	19	.	.	PUNCT
ejpam-6297	93	1	the	the	DET
ejpam-6297	93	2	following	follow	VERB
ejpam-6297	93	3	statements	statement	NOUN
ejpam-6297	93	4	hold	hold	VERB
ejpam-6297	93	5	:	:	PUNCT
ejpam-6297	93	6	(	(	PUNCT
ejpam-6297	93	7	1	1	X
ejpam-6297	93	8	)	)	PUNCT
ejpam-6297	93	9	a	a	DET
ejpam-6297	93	10	⊆	⊆	NUM
ejpam-6297	93	11	sβ	sβ	PROPN
ejpam-6297	93	12	cli(a	cli(a	PROPN
ejpam-6297	93	13	)	)	PUNCT
ejpam-6297	93	14	;	;	PUNCT
ejpam-6297	93	15	(	(	PUNCT
ejpam-6297	93	16	2	2	X
ejpam-6297	93	17	)	)	PUNCT
ejpam-6297	93	18	sβ	sβ	NOUN
ejpam-6297	93	19	inti(a	inti(a	PROPN
ejpam-6297	93	20	)	)	PUNCT
ejpam-6297	93	21	⊆	⊆	NUM
ejpam-6297	93	22	a	a	PRON
ejpam-6297	93	23	;	;	PUNCT
ejpam-6297	93	24	(	(	PUNCT
ejpam-6297	93	25	3	3	X
ejpam-6297	93	26	)	)	PUNCT
ejpam-6297	93	27	if	if	SCONJ
ejpam-6297	93	28	a	a	DET
ejpam-6297	93	29	⊆	⊆	NUM
ejpam-6297	93	30	b	b	NOUN
ejpam-6297	93	31	,	,	PUNCT
ejpam-6297	93	32	then	then	ADV
ejpam-6297	93	33	sβ	sβ	PROPN
ejpam-6297	93	34	inti(a	inti(a	PROPN
ejpam-6297	93	35	)	)	PUNCT
ejpam-6297	93	36	⊆	⊆	NUM
ejpam-6297	93	37	sβ	sβ	PROPN
ejpam-6297	93	38	inti(b	inti(b	PROPN
ejpam-6297	93	39	)	)	PUNCT
ejpam-6297	93	40	and	and	CCONJ
ejpam-6297	93	41	sβ	sβ	PROPN
ejpam-6297	93	42	cli(a	cli(a	PROPN
ejpam-6297	93	43	)	)	PUNCT
ejpam-6297	93	44	⊆	⊆	PROPN
ejpam-6297	93	45	sβ	sβ	NOUN
ejpam-6297	93	46	cli(b	cli(b	PROPN
ejpam-6297	93	47	)	)	PUNCT
ejpam-6297	93	48	;	;	PUNCT
ejpam-6297	93	49	(	(	PUNCT
ejpam-6297	93	50	4	4	X
ejpam-6297	93	51	)	)	PUNCT
ejpam-6297	93	52	x	x	NOUN
ejpam-6297	93	53	−	−	PROPN
ejpam-6297	93	54	sβ	sβ	PROPN
ejpam-6297	93	55	cli(a	cli(a	PROPN
ejpam-6297	93	56	)	)	PUNCT
ejpam-6297	93	57	=	=	PRON
ejpam-6297	93	58	sβ	sβ	PROPN
ejpam-6297	93	59	inti(x	inti(x	NUM
ejpam-6297	93	60	−a	−a	NOUN
ejpam-6297	93	61	)	)	PUNCT
ejpam-6297	93	62	;	;	PUNCT
ejpam-6297	93	63	and	and	CCONJ
ejpam-6297	93	64	(	(	PUNCT
ejpam-6297	93	65	5	5	X
ejpam-6297	93	66	)	)	PUNCT
ejpam-6297	93	67	x	x	NOUN
ejpam-6297	93	68	−	−	PROPN
ejpam-6297	93	69	sβ	sβ	PROPN
ejpam-6297	93	70	inti(a	inti(a	PROPN
ejpam-6297	93	71	)	)	PUNCT
ejpam-6297	93	72	=	=	SYM
ejpam-6297	93	73	sβ	sβ	PROPN
ejpam-6297	93	74	cli(x	cli(x	NOUN
ejpam-6297	93	75	−a	−a	NOUN
ejpam-6297	93	76	)	)	PUNCT
ejpam-6297	93	77	.	.	PUNCT
ejpam-6297	94	1	proof	proof	NOUN
ejpam-6297	94	2	.	.	PUNCT
ejpam-6297	95	1	statements	statement	NOUN
ejpam-6297	95	2	(	(	PUNCT
ejpam-6297	95	3	1	1	NUM
ejpam-6297	95	4	)	)	PUNCT
ejpam-6297	95	5	,	,	PUNCT
ejpam-6297	95	6	(	(	PUNCT
ejpam-6297	95	7	2	2	NUM
ejpam-6297	95	8	)	)	PUNCT
ejpam-6297	95	9	,	,	PUNCT
ejpam-6297	95	10	and	and	CCONJ
ejpam-6297	95	11	(	(	PUNCT
ejpam-6297	95	12	3	3	X
ejpam-6297	95	13	)	)	PUNCT
ejpam-6297	95	14	follow	follow	VERB
ejpam-6297	95	15	directly	directly	ADV
ejpam-6297	95	16	from	from	ADP
ejpam-6297	95	17	the	the	DET
ejpam-6297	95	18	definitions	definition	NOUN
ejpam-6297	95	19	of	of	ADP
ejpam-6297	95	20	sβ	sβ	PROPN
ejpam-6297	95	21	cli(a	cli(a	PROPN
ejpam-6297	95	22	)	)	PUNCT
ejpam-6297	95	23	and	and	CCONJ
ejpam-6297	95	24	sβ	sβ	DET
ejpam-6297	95	25	inti(a	inti(a	PROPN
ejpam-6297	95	26	)	)	PUNCT
ejpam-6297	95	27	.	.	PUNCT
ejpam-6297	96	1	to	to	PART
ejpam-6297	96	2	establish	establish	VERB
ejpam-6297	96	3	statement	statement	NOUN
ejpam-6297	96	4	(	(	PUNCT
ejpam-6297	96	5	4	4	NUM
ejpam-6297	96	6	)	)	PUNCT
ejpam-6297	96	7	,	,	PUNCT
ejpam-6297	96	8	observe	observe	VERB
ejpam-6297	96	9	that	that	SCONJ
ejpam-6297	96	10	x	x	PUNCT
ejpam-6297	96	11	−	−	NOUN
ejpam-6297	96	12	sβ	sβ	PROPN
ejpam-6297	96	13	cli(a	cli(a	PROPN
ejpam-6297	96	14	)	)	PUNCT
ejpam-6297	96	15	=	=	PUNCT
ejpam-6297	97	1	x	x	PUNCT
ejpam-6297	97	2	−	−	NOUN
ejpam-6297	97	3	∩{f	∩{f	NOUN
ejpam-6297	97	4	|	|	ADV
ejpam-6297	97	5	a	a	DET
ejpam-6297	97	6	⊆	⊆	NUM
ejpam-6297	97	7	f	f	NUM
ejpam-6297	97	8	,	,	PUNCT
ejpam-6297	97	9	f	f	PROPN
ejpam-6297	97	10	is	be	AUX
ejpam-6297	97	11	strong	strong	ADJ
ejpam-6297	97	12	β	β	NOUN
ejpam-6297	97	13	-	-	ADJ
ejpam-6297	97	14	i	i	NOUN
ejpam-6297	97	15	-	-	PUNCT
ejpam-6297	97	16	closed	closed	ADJ
ejpam-6297	97	17	}	}	PUNCT
ejpam-6297	97	18	=	=	SYM
ejpam-6297	97	19	∪{x	∪{x	PROPN
ejpam-6297	97	20	−	−	PROPN
ejpam-6297	97	21	f	f	NOUN
ejpam-6297	98	1	|	|	ADV
ejpam-6297	98	2	x	x	NOUN
ejpam-6297	99	1	−	−	NOUN
ejpam-6297	99	2	f	f	NOUN
ejpam-6297	99	3	⊆	⊆	NUM
ejpam-6297	99	4	x	x	SYM
ejpam-6297	99	5	−a	−a	NOUN
ejpam-6297	99	6	,	,	PUNCT
ejpam-6297	99	7	x	x	PUNCT
ejpam-6297	99	8	−	−	PROPN
ejpam-6297	99	9	f	f	PROPN
ejpam-6297	99	10	is	be	AUX
ejpam-6297	99	11	strong	strong	ADJ
ejpam-6297	99	12	β	β	NOUN
ejpam-6297	99	13	-	-	ADJ
ejpam-6297	99	14	i	i	PRON
ejpam-6297	99	15	-	-	PUNCT
ejpam-6297	99	16	open	open	ADJ
ejpam-6297	99	17	}	}	PUNCT
ejpam-6297	99	18	=	=	SYM
ejpam-6297	99	19	∪{g	∪{g	PROPN
ejpam-6297	99	20	|	|	ADV
ejpam-6297	99	21	g	g	NOUN
ejpam-6297	99	22	⊆	⊆	NUM
ejpam-6297	99	23	x	x	SYM
ejpam-6297	99	24	−a	−a	NOUN
ejpam-6297	99	25	,	,	PUNCT
ejpam-6297	99	26	g	g	PROPN
ejpam-6297	99	27	is	be	AUX
ejpam-6297	99	28	strong	strong	ADJ
ejpam-6297	99	29	β	β	NOUN
ejpam-6297	99	30	-	-	ADJ
ejpam-6297	99	31	i	i	PRON
ejpam-6297	99	32	-	-	PUNCT
ejpam-6297	99	33	open	open	ADJ
ejpam-6297	99	34	}	}	PUNCT
ejpam-6297	99	35	=	=	SYM
ejpam-6297	99	36	sβ	sβ	NUM
ejpam-6297	99	37	inti(x	inti(x	NUM
ejpam-6297	99	38	−a	−a	NOUN
ejpam-6297	99	39	)	)	PUNCT
ejpam-6297	99	40	,	,	PUNCT
ejpam-6297	99	41	which	which	PRON
ejpam-6297	99	42	confirms	confirm	VERB
ejpam-6297	99	43	statement	statement	NOUN
ejpam-6297	99	44	(	(	PUNCT
ejpam-6297	99	45	4	4	NUM
ejpam-6297	99	46	)	)	PUNCT
ejpam-6297	99	47	.	.	PUNCT
ejpam-6297	100	1	similarly	similarly	ADV
ejpam-6297	100	2	,	,	PUNCT
ejpam-6297	100	3	for	for	ADP
ejpam-6297	100	4	statement	statement	NOUN
ejpam-6297	100	5	(	(	PUNCT
ejpam-6297	100	6	5	5	NUM
ejpam-6297	100	7	)	)	PUNCT
ejpam-6297	100	8	,	,	PUNCT
ejpam-6297	100	9	note	note	VERB
ejpam-6297	100	10	that	that	SCONJ
ejpam-6297	100	11	x	x	PUNCT
ejpam-6297	100	12	−	−	NOUN
ejpam-6297	100	13	sβ	sβ	NOUN
ejpam-6297	100	14	inti(a	inti(a	PROPN
ejpam-6297	100	15	)	)	PUNCT
ejpam-6297	101	1	=	=	PUNCT
ejpam-6297	101	2	x	x	X
ejpam-6297	102	1	−	−	PROPN
ejpam-6297	102	2	∪{g	∪{g	PROPN
ejpam-6297	102	3	|	|	ADV
ejpam-6297	102	4	g	g	NOUN
ejpam-6297	102	5	⊆	⊆	NUM
ejpam-6297	102	6	a	a	PRON
ejpam-6297	102	7	,	,	PUNCT
ejpam-6297	102	8	g	g	PROPN
ejpam-6297	102	9	is	be	AUX
ejpam-6297	102	10	strong	strong	ADJ
ejpam-6297	102	11	β	β	NOUN
ejpam-6297	102	12	-	-	ADJ
ejpam-6297	102	13	i	i	PRON
ejpam-6297	102	14	-	-	PUNCT
ejpam-6297	102	15	open	open	ADJ
ejpam-6297	102	16	}	}	PUNCT
ejpam-6297	102	17	=	=	SYM
ejpam-6297	102	18	∩{x	∩{x	X
ejpam-6297	102	19	−g	−g	NOUN
ejpam-6297	103	1	|	|	ADV
ejpam-6297	103	2	x	x	PART
ejpam-6297	103	3	−a	−a	VERB
ejpam-6297	103	4	⊆	⊆	NUM
ejpam-6297	103	5	x	x	SYM
ejpam-6297	103	6	−g	−g	NOUN
ejpam-6297	103	7	,	,	PUNCT
ejpam-6297	103	8	x	x	SYM
ejpam-6297	103	9	−g	−g	NOUN
ejpam-6297	103	10	is	be	AUX
ejpam-6297	103	11	strong	strong	ADJ
ejpam-6297	103	12	β	β	NOUN
ejpam-6297	103	13	-	-	ADJ
ejpam-6297	103	14	i	i	NOUN
ejpam-6297	103	15	-	-	PUNCT
ejpam-6297	103	16	closed	closed	ADJ
ejpam-6297	103	17	}	}	PUNCT
ejpam-6297	103	18	=	=	SYM
ejpam-6297	103	19	∩{f	∩{f	NOUN
ejpam-6297	103	20	|	|	ADV
ejpam-6297	103	21	x	x	PART
ejpam-6297	103	22	−a	−a	NOUN
ejpam-6297	103	23	⊆	⊆	NUM
ejpam-6297	103	24	f	f	NUM
ejpam-6297	103	25	,	,	PUNCT
ejpam-6297	103	26	f	f	PROPN
ejpam-6297	103	27	is	be	AUX
ejpam-6297	103	28	strong	strong	ADJ
ejpam-6297	103	29	β	β	NOUN
ejpam-6297	103	30	-	-	ADJ
ejpam-6297	103	31	i	i	NOUN
ejpam-6297	103	32	-	-	PUNCT
ejpam-6297	103	33	closed	closed	ADJ
ejpam-6297	103	34	}	}	PUNCT
ejpam-6297	103	35	=	=	SYM
ejpam-6297	103	36	sβ	sβ	PROPN
ejpam-6297	103	37	cli(x	cli(x	NOUN
ejpam-6297	103	38	−a	−a	NOUN
ejpam-6297	103	39	)	)	PUNCT
ejpam-6297	103	40	,	,	PUNCT
ejpam-6297	103	41	thereby	thereby	ADV
ejpam-6297	103	42	verifying	verify	VERB
ejpam-6297	103	43	statement	statement	NOUN
ejpam-6297	103	44	(	(	PUNCT
ejpam-6297	103	45	5	5	NUM
ejpam-6297	103	46	)	)	PUNCT
ejpam-6297	103	47	.	.	PUNCT
ejpam-6297	104	1	the	the	DET
ejpam-6297	104	2	following	follow	VERB
ejpam-6297	104	3	lemma	lemma	PROPN
ejpam-6297	104	4	outlines	outline	VERB
ejpam-6297	104	5	fundamental	fundamental	ADJ
ejpam-6297	104	6	characteristics	characteristic	NOUN
ejpam-6297	104	7	and	and	CCONJ
ejpam-6297	104	8	conditions	condition	NOUN
ejpam-6297	104	9	related	relate	VERB
ejpam-6297	104	10	to	to	ADP
ejpam-6297	104	11	strong	strong	ADJ
ejpam-6297	104	12	β	β	X
ejpam-6297	104	13	-	-	ADJ
ejpam-6297	104	14	i	i	PRON
ejpam-6297	104	15	-	-	PUNCT
ejpam-6297	104	16	open	open	ADJ
ejpam-6297	104	17	and	and	CCONJ
ejpam-6297	104	18	strong	strong	ADJ
ejpam-6297	104	19	β	β	X
ejpam-6297	104	20	-	-	ADJ
ejpam-6297	104	21	i	i	NOUN
ejpam-6297	104	22	-	-	PUNCT
ejpam-6297	104	23	closed	close	VERB
ejpam-6297	104	24	sets	set	NOUN
ejpam-6297	104	25	within	within	ADP
ejpam-6297	104	26	the	the	DET
ejpam-6297	104	27	context	context	NOUN
ejpam-6297	104	28	of	of	ADP
ejpam-6297	104	29	ideal	ideal	ADJ
ejpam-6297	104	30	topological	topological	ADJ
ejpam-6297	104	31	spaces	space	NOUN
ejpam-6297	104	32	.	.	PUNCT
ejpam-6297	105	1	lemma	lemma	PROPN
ejpam-6297	105	2	3	3	X
ejpam-6297	105	3	.	.	PUNCT
ejpam-6297	106	1	let	let	VERB
ejpam-6297	106	2	a	a	DET
ejpam-6297	106	3	be	be	AUX
ejpam-6297	106	4	a	a	DET
ejpam-6297	106	5	subset	subset	NOUN
ejpam-6297	106	6	of	of	ADP
ejpam-6297	106	7	an	an	DET
ejpam-6297	106	8	ideal	ideal	ADJ
ejpam-6297	106	9	topological	topological	ADJ
ejpam-6297	106	10	space	space	NOUN
ejpam-6297	106	11	(	(	PUNCT
ejpam-6297	106	12	x	x	X
ejpam-6297	106	13	,	,	PUNCT
ejpam-6297	106	14	τ	τ	PROPN
ejpam-6297	106	15	,	,	PUNCT
ejpam-6297	106	16	i	i	PROPN
ejpam-6297	106	17	)	)	PUNCT
ejpam-6297	106	18	.	.	PUNCT
ejpam-6297	107	1	the	the	DET
ejpam-6297	107	2	following	follow	VERB
ejpam-6297	107	3	properties	property	NOUN
ejpam-6297	107	4	hold	hold	VERB
ejpam-6297	107	5	:	:	PUNCT
ejpam-6297	107	6	(	(	PUNCT
ejpam-6297	107	7	1	1	X
ejpam-6297	107	8	)	)	PUNCT
ejpam-6297	107	9	if	if	SCONJ
ejpam-6297	107	10	a	a	PRON
ejpam-6297	107	11	is	be	AUX
ejpam-6297	107	12	open	open	ADJ
ejpam-6297	107	13	,	,	PUNCT
ejpam-6297	107	14	then	then	ADV
ejpam-6297	107	15	a	a	PRON
ejpam-6297	107	16	is	be	AUX
ejpam-6297	107	17	strong	strong	ADJ
ejpam-6297	107	18	β	β	NOUN
ejpam-6297	107	19	-	-	ADJ
ejpam-6297	107	20	i	i	PRON
ejpam-6297	107	21	-	-	PUNCT
ejpam-6297	107	22	open	open	ADJ
ejpam-6297	107	23	;	;	PUNCT
ejpam-6297	107	24	c.	c.	PROPN
ejpam-6297	107	25	boonpok	boonpok	PROPN
ejpam-6297	107	26	,	,	PUNCT
ejpam-6297	107	27	p.	p.	PROPN
ejpam-6297	107	28	raktaow	raktaow	NOUN
ejpam-6297	107	29	,	,	PUNCT
ejpam-6297	107	30	a.	a.	PROPN
ejpam-6297	107	31	sama	sama	PROPN
ejpam-6297	107	32	-	-	PUNCT
ejpam-6297	107	33	ae	ae	PROPN
ejpam-6297	107	34	/	/	SYM
ejpam-6297	107	35	eur	eur	PROPN
ejpam-6297	107	36	.	.	PUNCT
ejpam-6297	108	1	j.	j.	PROPN
ejpam-6297	108	2	pure	pure	PROPN
ejpam-6297	108	3	appl	appl	PROPN
ejpam-6297	108	4	.	.	PROPN
ejpam-6297	108	5	math	math	PROPN
ejpam-6297	108	6	,	,	PUNCT
ejpam-6297	108	7	18	18	NUM
ejpam-6297	108	8	(	(	PUNCT
ejpam-6297	108	9	3	3	NUM
ejpam-6297	108	10	)	)	PUNCT
ejpam-6297	108	11	(	(	PUNCT
ejpam-6297	108	12	2025	2025	NUM
ejpam-6297	108	13	)	)	PUNCT
ejpam-6297	108	14	,	,	PUNCT
ejpam-6297	108	15	6297	6297	NUM
ejpam-6297	108	16	5	5	NUM
ejpam-6297	108	17	of	of	ADP
ejpam-6297	108	18	23	23	NUM
ejpam-6297	108	19	(	(	PUNCT
ejpam-6297	108	20	2	2	NUM
ejpam-6297	108	21	)	)	PUNCT
ejpam-6297	108	22	if	if	SCONJ
ejpam-6297	108	23	a	a	PRON
ejpam-6297	108	24	is	be	AUX
ejpam-6297	108	25	closed	closed	ADJ
ejpam-6297	108	26	,	,	PUNCT
ejpam-6297	108	27	then	then	ADV
ejpam-6297	108	28	a	a	PRON
ejpam-6297	108	29	is	be	AUX
ejpam-6297	108	30	strong	strong	ADJ
ejpam-6297	108	31	β	β	NOUN
ejpam-6297	108	32	-	-	ADJ
ejpam-6297	108	33	i	i	NOUN
ejpam-6297	108	34	-	-	PUNCT
ejpam-6297	108	35	closed	closed	ADJ
ejpam-6297	108	36	;	;	PUNCT
ejpam-6297	108	37	(	(	PUNCT
ejpam-6297	108	38	3	3	X
ejpam-6297	108	39	)	)	PUNCT
ejpam-6297	108	40	sβ	sβ	NOUN
ejpam-6297	108	41	inti(a	inti(a	PROPN
ejpam-6297	108	42	)	)	PUNCT
ejpam-6297	108	43	is	be	AUX
ejpam-6297	108	44	strong	strong	ADJ
ejpam-6297	108	45	β	β	NOUN
ejpam-6297	108	46	-	-	ADJ
ejpam-6297	108	47	i	i	PRON
ejpam-6297	108	48	-	-	PUNCT
ejpam-6297	108	49	open	open	ADJ
ejpam-6297	108	50	,	,	PUNCT
ejpam-6297	108	51	and	and	CCONJ
ejpam-6297	108	52	int(a	int(a	PROPN
ejpam-6297	108	53	)	)	PUNCT
ejpam-6297	108	54	⊆	⊆	NUM
ejpam-6297	108	55	sβ	sβ	NUM
ejpam-6297	108	56	inti(a	inti(a	PROPN
ejpam-6297	108	57	)	)	PUNCT
ejpam-6297	108	58	;	;	PUNCT
ejpam-6297	108	59	(	(	PUNCT
ejpam-6297	108	60	4	4	X
ejpam-6297	108	61	)	)	PUNCT
ejpam-6297	108	62	sβ	sβ	PROPN
ejpam-6297	108	63	cli(a	cli(a	PROPN
ejpam-6297	108	64	)	)	PUNCT
ejpam-6297	108	65	is	be	AUX
ejpam-6297	108	66	strong	strong	ADJ
ejpam-6297	108	67	β	β	NOUN
ejpam-6297	108	68	-	-	ADJ
ejpam-6297	108	69	i	i	NOUN
ejpam-6297	108	70	-	-	PUNCT
ejpam-6297	108	71	closed	closed	ADJ
ejpam-6297	108	72	,	,	PUNCT
ejpam-6297	108	73	and	and	CCONJ
ejpam-6297	108	74	sβ	sβ	PROPN
ejpam-6297	108	75	cli(a	cli(a	PROPN
ejpam-6297	108	76	)	)	PUNCT
ejpam-6297	108	77	⊆	⊆	NUM
ejpam-6297	108	78	cl(a	cl(a	NUM
ejpam-6297	108	79	)	)	PUNCT
ejpam-6297	108	80	;	;	PUNCT
ejpam-6297	108	81	(	(	PUNCT
ejpam-6297	108	82	5	5	X
ejpam-6297	108	83	)	)	PUNCT
ejpam-6297	108	84	a	a	PRON
ejpam-6297	108	85	is	be	AUX
ejpam-6297	108	86	strong	strong	ADJ
ejpam-6297	108	87	β	β	NOUN
ejpam-6297	108	88	-	-	ADJ
ejpam-6297	108	89	i	i	PRON
ejpam-6297	108	90	-	-	PUNCT
ejpam-6297	108	91	open	open	ADJ
ejpam-6297	108	92	if	if	SCONJ
ejpam-6297	108	93	and	and	CCONJ
ejpam-6297	108	94	only	only	ADV
ejpam-6297	108	95	if	if	SCONJ
ejpam-6297	108	96	a	a	PRON
ejpam-6297	108	97	=	=	X
ejpam-6297	108	98	sβ	sβ	NUM
ejpam-6297	108	99	inti(a	inti(a	PROPN
ejpam-6297	108	100	)	)	PUNCT
ejpam-6297	108	101	;	;	PUNCT
ejpam-6297	108	102	(	(	PUNCT
ejpam-6297	108	103	6	6	X
ejpam-6297	108	104	)	)	PUNCT
ejpam-6297	108	105	a	a	PRON
ejpam-6297	108	106	is	be	AUX
ejpam-6297	108	107	strong	strong	ADJ
ejpam-6297	108	108	β	β	NOUN
ejpam-6297	108	109	-	-	ADJ
ejpam-6297	108	110	i	i	NOUN
ejpam-6297	108	111	-	-	PUNCT
ejpam-6297	108	112	closed	close	VERB
ejpam-6297	108	113	if	if	SCONJ
ejpam-6297	108	114	and	and	CCONJ
ejpam-6297	108	115	only	only	ADV
ejpam-6297	108	116	if	if	SCONJ
ejpam-6297	108	117	a	a	PRON
ejpam-6297	108	118	=	=	X
ejpam-6297	108	119	sβ	sβ	PROPN
ejpam-6297	108	120	cli(a	cli(a	PROPN
ejpam-6297	108	121	)	)	PUNCT
ejpam-6297	108	122	;	;	PUNCT
ejpam-6297	108	123	and	and	CCONJ
ejpam-6297	108	124	(	(	PUNCT
ejpam-6297	108	125	7	7	X
ejpam-6297	108	126	)	)	PUNCT
ejpam-6297	108	127	x	x	SYM
ejpam-6297	108	128	∈	∈	PROPN
ejpam-6297	108	129	sβ	sβ	PROPN
ejpam-6297	108	130	cli(a	cli(a	PROPN
ejpam-6297	108	131	)	)	PUNCT
ejpam-6297	109	1	if	if	SCONJ
ejpam-6297	109	2	and	and	CCONJ
ejpam-6297	109	3	only	only	ADV
ejpam-6297	110	1	if	if	SCONJ
ejpam-6297	110	2	u	u	PROPN
ejpam-6297	110	3	∩a	∩a	PROPN
ejpam-6297	110	4	̸=	̸=	PROPN
ejpam-6297	110	5	∅	∅	NOUN
ejpam-6297	110	6	for	for	ADP
ejpam-6297	110	7	every	every	DET
ejpam-6297	110	8	strong	strong	ADJ
ejpam-6297	110	9	β	β	X
ejpam-6297	110	10	-	-	ADJ
ejpam-6297	110	11	i	i	PRON
ejpam-6297	110	12	-	-	PUNCT
ejpam-6297	110	13	open	open	ADJ
ejpam-6297	110	14	set	set	VERB
ejpam-6297	110	15	u	u	NOUN
ejpam-6297	110	16	containing	contain	VERB
ejpam-6297	110	17	x.	x.	NOUN
ejpam-6297	110	18	proof	proof	NOUN
ejpam-6297	110	19	.	.	PUNCT
ejpam-6297	111	1	(	(	PUNCT
ejpam-6297	111	2	1	1	NUM
ejpam-6297	111	3	):	):	PUNCT
ejpam-6297	111	4	let	let	VERB
ejpam-6297	111	5	a	a	PRON
ejpam-6297	111	6	be	be	AUX
ejpam-6297	111	7	an	an	DET
ejpam-6297	111	8	open	open	ADJ
ejpam-6297	111	9	set	set	NOUN
ejpam-6297	111	10	.	.	PUNCT
ejpam-6297	112	1	then	then	ADV
ejpam-6297	112	2	a	a	DET
ejpam-6297	112	3	=	=	SYM
ejpam-6297	112	4	int(a	int(a	NOUN
ejpam-6297	112	5	)	)	PUNCT
ejpam-6297	112	6	,	,	PUNCT
ejpam-6297	112	7	and	and	CCONJ
ejpam-6297	112	8	since	since	SCONJ
ejpam-6297	112	9	a	a	DET
ejpam-6297	112	10	⊆	⊆	NUM
ejpam-6297	112	11	cl∗(a	cl∗(a	NOUN
ejpam-6297	112	12	)	)	PUNCT
ejpam-6297	112	13	,	,	PUNCT
ejpam-6297	112	14	we	we	PRON
ejpam-6297	112	15	have	have	VERB
ejpam-6297	112	16	a	a	DET
ejpam-6297	112	17	=	=	SYM
ejpam-6297	112	18	int(a	int(a	PROPN
ejpam-6297	112	19	)	)	PUNCT
ejpam-6297	112	20	⊆	⊆	NUM
ejpam-6297	112	21	int(cl∗(a	int(cl∗(a	NOUN
ejpam-6297	112	22	)	)	PUNCT
ejpam-6297	112	23	)	)	PUNCT
ejpam-6297	112	24	.	.	PUNCT
ejpam-6297	113	1	it	it	PRON
ejpam-6297	113	2	follows	follow	VERB
ejpam-6297	113	3	that	that	SCONJ
ejpam-6297	113	4	a	a	DET
ejpam-6297	113	5	⊆	⊆	NUM
ejpam-6297	113	6	cl∗(a	cl∗(a	NOUN
ejpam-6297	113	7	)	)	PUNCT
ejpam-6297	113	8	⊆	⊆	NUM
ejpam-6297	113	9	cl∗(int(cl∗(a	cl∗(int(cl∗(a	NOUN
ejpam-6297	113	10	)	)	PUNCT
ejpam-6297	113	11	)	)	PUNCT
ejpam-6297	113	12	)	)	PUNCT
ejpam-6297	113	13	,	,	PUNCT
ejpam-6297	113	14	which	which	PRON
ejpam-6297	113	15	implies	imply	VERB
ejpam-6297	113	16	that	that	SCONJ
ejpam-6297	113	17	a	a	PRON
ejpam-6297	113	18	is	be	AUX
ejpam-6297	113	19	strong	strong	ADJ
ejpam-6297	113	20	β	β	NOUN
ejpam-6297	113	21	-	-	ADJ
ejpam-6297	113	22	i	i	PRON
ejpam-6297	113	23	-	-	PUNCT
ejpam-6297	113	24	open	open	ADJ
ejpam-6297	113	25	.	.	PUNCT
ejpam-6297	114	1	(	(	PUNCT
ejpam-6297	114	2	2	2	NUM
ejpam-6297	114	3	):	):	PUNCT
ejpam-6297	114	4	the	the	DET
ejpam-6297	114	5	result	result	NOUN
ejpam-6297	114	6	follows	follow	VERB
ejpam-6297	114	7	directly	directly	ADV
ejpam-6297	114	8	from	from	ADP
ejpam-6297	114	9	(	(	PUNCT
ejpam-6297	114	10	1	1	NUM
ejpam-6297	114	11	)	)	PUNCT
ejpam-6297	114	12	.	.	PUNCT
ejpam-6297	115	1	(	(	PUNCT
ejpam-6297	115	2	3	3	NUM
ejpam-6297	115	3	):	):	PUNCT
ejpam-6297	115	4	let	let	VERB
ejpam-6297	115	5	uα	uα	PRON
ejpam-6297	115	6	be	be	AUX
ejpam-6297	115	7	any	any	PRON
ejpam-6297	115	8	strong	strong	ADJ
ejpam-6297	115	9	β	β	X
ejpam-6297	115	10	-	-	ADJ
ejpam-6297	115	11	i	i	PRON
ejpam-6297	115	12	-	-	PUNCT
ejpam-6297	115	13	open	open	ADJ
ejpam-6297	115	14	set	set	VERB
ejpam-6297	115	15	with	with	ADP
ejpam-6297	115	16	uα	uα	PROPN
ejpam-6297	115	17	⊆	⊆	NUM
ejpam-6297	115	18	a.	a.	NOUN
ejpam-6297	115	19	by	by	ADP
ejpam-6297	115	20	the	the	DET
ejpam-6297	115	21	definition	definition	NOUN
ejpam-6297	115	22	of	of	ADP
ejpam-6297	115	23	a	a	DET
ejpam-6297	115	24	strong	strong	ADJ
ejpam-6297	115	25	β	β	X
ejpam-6297	115	26	-	-	ADJ
ejpam-6297	115	27	i	i	NOUN
ejpam-6297	115	28	-	-	PUNCT
ejpam-6297	115	29	open	open	ADJ
ejpam-6297	115	30	set	set	NOUN
ejpam-6297	115	31	,	,	PUNCT
ejpam-6297	115	32	we	we	PRON
ejpam-6297	115	33	have	have	VERB
ejpam-6297	115	34	that	that	PRON
ejpam-6297	115	35	uα	uα	PROPN
ejpam-6297	115	36	⊆	⊆	NUM
ejpam-6297	115	37	sβ	sβ	NUM
ejpam-6297	115	38	inti(a	inti(a	PROPN
ejpam-6297	115	39	)	)	PUNCT
ejpam-6297	115	40	.	.	PUNCT
ejpam-6297	116	1	since	since	SCONJ
ejpam-6297	116	2	uα	uα	PROPN
ejpam-6297	116	3	is	be	AUX
ejpam-6297	116	4	strong	strong	ADJ
ejpam-6297	116	5	β	β	NOUN
ejpam-6297	116	6	-	-	ADJ
ejpam-6297	116	7	i	i	PRON
ejpam-6297	116	8	-	-	PUNCT
ejpam-6297	116	9	open	open	ADJ
ejpam-6297	116	10	,	,	PUNCT
ejpam-6297	116	11	we	we	PRON
ejpam-6297	116	12	have	have	VERB
ejpam-6297	116	13	uα	uα	PROPN
ejpam-6297	116	14	⊆	⊆	NUM
ejpam-6297	116	15	cl∗(int(cl∗(uα	cl∗(int(cl∗(uα	NOUN
ejpam-6297	116	16	)	)	PUNCT
ejpam-6297	116	17	)	)	PUNCT
ejpam-6297	116	18	)	)	PUNCT
ejpam-6297	116	19	.	.	PUNCT
ejpam-6297	117	1	then	then	ADV
ejpam-6297	117	2	,	,	PUNCT
ejpam-6297	117	3	sβ	sβ	X
ejpam-6297	117	4	inti(a	inti(a	PROPN
ejpam-6297	117	5	)	)	PUNCT
ejpam-6297	117	6	=	=	VERB
ejpam-6297	117	7	∪αuα	∪αuα	VERB
ejpam-6297	117	8	⊆	⊆	NUM
ejpam-6297	117	9	∪α	∪α	NUM
ejpam-6297	117	10	cl	cl	NOUN
ejpam-6297	117	11	∗(int(cl∗(uα	∗(int(cl∗(uα	NOUN
ejpam-6297	117	12	)	)	PUNCT
ejpam-6297	117	13	)	)	PUNCT
ejpam-6297	117	14	)	)	PUNCT
ejpam-6297	118	1	⊆	⊆	NUM
ejpam-6297	118	2	cl∗(int(cl∗(sβ	cl∗(int(cl∗(sβ	PROPN
ejpam-6297	118	3	inti(a	inti(a	PROPN
ejpam-6297	118	4	)	)	PUNCT
ejpam-6297	118	5	)	)	PUNCT
ejpam-6297	118	6	)	)	PUNCT
ejpam-6297	118	7	)	)	PUNCT
ejpam-6297	118	8	,	,	PUNCT
ejpam-6297	118	9	and	and	CCONJ
ejpam-6297	118	10	therefore	therefore	ADV
ejpam-6297	118	11	sβ	sβ	PROPN
ejpam-6297	118	12	inti(a	inti(a	PROPN
ejpam-6297	118	13	)	)	PUNCT
ejpam-6297	118	14	is	be	AUX
ejpam-6297	118	15	a	a	DET
ejpam-6297	118	16	strong	strong	ADJ
ejpam-6297	118	17	β	β	X
ejpam-6297	118	18	-	-	ADJ
ejpam-6297	118	19	i	i	NOUN
ejpam-6297	118	20	-	-	PUNCT
ejpam-6297	118	21	open	open	ADJ
ejpam-6297	118	22	set	set	NOUN
ejpam-6297	118	23	.	.	PUNCT
ejpam-6297	119	1	from	from	ADP
ejpam-6297	119	2	the	the	DET
ejpam-6297	119	3	definitions	definition	NOUN
ejpam-6297	119	4	of	of	ADP
ejpam-6297	119	5	int(a	int(a	PROPN
ejpam-6297	119	6	)	)	PUNCT
ejpam-6297	119	7	and	and	CCONJ
ejpam-6297	119	8	sβ	sβ	ADP
ejpam-6297	119	9	inti(a	inti(a	PROPN
ejpam-6297	119	10	)	)	PUNCT
ejpam-6297	119	11	,	,	PUNCT
ejpam-6297	119	12	we	we	PRON
ejpam-6297	119	13	deduce	deduce	VERB
ejpam-6297	119	14	that	that	PRON
ejpam-6297	119	15	int(a	int(a	AUX
ejpam-6297	119	16	)	)	PUNCT
ejpam-6297	119	17	⊆	⊆	PROPN
ejpam-6297	119	18	sβ	sβ	NUM
ejpam-6297	119	19	inti(a	inti(a	PROPN
ejpam-6297	119	20	)	)	PUNCT
ejpam-6297	119	21	.	.	PUNCT
ejpam-6297	120	1	(	(	PUNCT
ejpam-6297	120	2	4	4	NUM
ejpam-6297	120	3	):	):	PUNCT
ejpam-6297	120	4	by	by	ADP
ejpam-6297	120	5	part	part	NOUN
ejpam-6297	120	6	(	(	PUNCT
ejpam-6297	120	7	4	4	NUM
ejpam-6297	120	8	)	)	PUNCT
ejpam-6297	120	9	of	of	ADP
ejpam-6297	120	10	lemma	lemma	PROPN
ejpam-6297	120	11	2	2	NUM
ejpam-6297	120	12	,	,	PUNCT
ejpam-6297	120	13	it	it	PRON
ejpam-6297	120	14	follows	follow	VERB
ejpam-6297	120	15	that	that	SCONJ
ejpam-6297	120	16	sβ	sβ	PROPN
ejpam-6297	120	17	cli(a	cli(a	PROPN
ejpam-6297	120	18	)	)	PUNCT
ejpam-6297	120	19	is	be	AUX
ejpam-6297	120	20	strong	strong	ADJ
ejpam-6297	120	21	β	β	NOUN
ejpam-6297	120	22	-	-	ADJ
ejpam-6297	120	23	i	i	NOUN
ejpam-6297	120	24	-	-	PUNCT
ejpam-6297	120	25	closed	closed	ADJ
ejpam-6297	120	26	.	.	PUNCT
ejpam-6297	121	1	moreover	moreover	ADV
ejpam-6297	121	2	,	,	PUNCT
ejpam-6297	121	3	from	from	ADP
ejpam-6297	121	4	the	the	DET
ejpam-6297	121	5	definitions	definition	NOUN
ejpam-6297	121	6	of	of	ADP
ejpam-6297	121	7	sβ	sβ	PROPN
ejpam-6297	121	8	cli(a	cli(a	PROPN
ejpam-6297	121	9	)	)	PUNCT
ejpam-6297	121	10	and	and	CCONJ
ejpam-6297	121	11	cl(a	cl(a	NUM
ejpam-6297	121	12	)	)	PUNCT
ejpam-6297	121	13	,	,	PUNCT
ejpam-6297	121	14	we	we	PRON
ejpam-6297	121	15	have	have	VERB
ejpam-6297	121	16	that	that	PRON
ejpam-6297	121	17	sβ	sβ	PROPN
ejpam-6297	121	18	cli(a	cli(a	PROPN
ejpam-6297	121	19	)	)	PUNCT
ejpam-6297	121	20	⊆	⊆	NUM
ejpam-6297	121	21	cl(a	cl(a	NUM
ejpam-6297	121	22	)	)	PUNCT
ejpam-6297	121	23	.	.	PUNCT
ejpam-6297	122	1	(	(	PUNCT
ejpam-6297	122	2	5	5	NUM
ejpam-6297	122	3	):	):	PUNCT
ejpam-6297	122	4	since	since	SCONJ
ejpam-6297	122	5	sβ	sβ	NUM
ejpam-6297	122	6	inti(a	inti(a	PROPN
ejpam-6297	122	7	)	)	PUNCT
ejpam-6297	122	8	is	be	AUX
ejpam-6297	122	9	the	the	DET
ejpam-6297	122	10	union	union	NOUN
ejpam-6297	122	11	of	of	ADP
ejpam-6297	122	12	all	all	DET
ejpam-6297	122	13	strong	strong	ADJ
ejpam-6297	122	14	β	β	X
ejpam-6297	122	15	-	-	ADJ
ejpam-6297	122	16	i	i	NOUN
ejpam-6297	122	17	-	-	PUNCT
ejpam-6297	122	18	open	open	ADJ
ejpam-6297	122	19	sets	set	NOUN
ejpam-6297	122	20	contained	contain	VERB
ejpam-6297	122	21	in	in	ADP
ejpam-6297	122	22	a	a	PRON
ejpam-6297	122	23	,	,	PUNCT
ejpam-6297	122	24	it	it	PRON
ejpam-6297	122	25	follows	follow	VERB
ejpam-6297	122	26	that	that	SCONJ
ejpam-6297	122	27	sβ	sβ	PROPN
ejpam-6297	122	28	inti(a	inti(a	PROPN
ejpam-6297	122	29	)	)	PUNCT
ejpam-6297	122	30	⊆	⊆	NUM
ejpam-6297	122	31	a.	a.	NOUN
ejpam-6297	122	32	therefore	therefore	ADV
ejpam-6297	122	33	,	,	PUNCT
ejpam-6297	122	34	a	a	PRON
ejpam-6297	122	35	is	be	AUX
ejpam-6297	122	36	strong	strong	ADJ
ejpam-6297	122	37	β	β	NOUN
ejpam-6297	122	38	-	-	ADJ
ejpam-6297	122	39	i	i	PRON
ejpam-6297	122	40	-	-	PUNCT
ejpam-6297	122	41	open	open	ADJ
ejpam-6297	122	42	if	if	SCONJ
ejpam-6297	122	43	and	and	CCONJ
ejpam-6297	122	44	only	only	ADV
ejpam-6297	122	45	if	if	SCONJ
ejpam-6297	122	46	a	a	PRON
ejpam-6297	122	47	=	=	X
ejpam-6297	122	48	sβ	sβ	NUM
ejpam-6297	122	49	inti(a	inti(a	NOUN
ejpam-6297	122	50	)	)	PUNCT
ejpam-6297	122	51	.	.	PUNCT
ejpam-6297	123	1	(	(	PUNCT
ejpam-6297	123	2	6	6	NUM
ejpam-6297	123	3	):	):	PUNCT
ejpam-6297	123	4	as	as	ADP
ejpam-6297	123	5	sβ	sβ	PROPN
ejpam-6297	123	6	cli(a	cli(a	PROPN
ejpam-6297	123	7	)	)	PUNCT
ejpam-6297	123	8	is	be	AUX
ejpam-6297	123	9	the	the	DET
ejpam-6297	123	10	intersection	intersection	NOUN
ejpam-6297	123	11	of	of	ADP
ejpam-6297	123	12	all	all	DET
ejpam-6297	123	13	strong	strong	ADJ
ejpam-6297	123	14	β	β	X
ejpam-6297	123	15	-	-	ADJ
ejpam-6297	123	16	i	i	NOUN
ejpam-6297	123	17	-	-	PUNCT
ejpam-6297	123	18	closed	close	VERB
ejpam-6297	123	19	sets	set	NOUN
ejpam-6297	123	20	containing	contain	VERB
ejpam-6297	123	21	a	a	PRON
ejpam-6297	123	22	,	,	PUNCT
ejpam-6297	123	23	it	it	PRON
ejpam-6297	123	24	implies	imply	VERB
ejpam-6297	123	25	that	that	SCONJ
ejpam-6297	123	26	a	a	DET
ejpam-6297	123	27	⊆	⊆	NUM
ejpam-6297	123	28	sβ	sβ	PROPN
ejpam-6297	123	29	cli(a	cli(a	PROPN
ejpam-6297	123	30	)	)	PUNCT
ejpam-6297	123	31	.	.	PUNCT
ejpam-6297	124	1	hence	hence	ADV
ejpam-6297	124	2	,	,	PUNCT
ejpam-6297	124	3	a	a	PRON
ejpam-6297	124	4	is	be	AUX
ejpam-6297	124	5	strong	strong	ADJ
ejpam-6297	124	6	β	β	NOUN
ejpam-6297	124	7	-	-	ADJ
ejpam-6297	124	8	i	i	NOUN
ejpam-6297	124	9	-	-	PUNCT
ejpam-6297	124	10	closed	close	VERB
ejpam-6297	124	11	if	if	SCONJ
ejpam-6297	124	12	and	and	CCONJ
ejpam-6297	124	13	only	only	ADV
ejpam-6297	124	14	if	if	SCONJ
ejpam-6297	124	15	a	a	PRON
ejpam-6297	124	16	=	=	X
ejpam-6297	124	17	sβ	sβ	PROPN
ejpam-6297	124	18	cli(a	cli(a	PROPN
ejpam-6297	124	19	)	)	PUNCT
ejpam-6297	124	20	.	.	PUNCT
ejpam-6297	125	1	(	(	PUNCT
ejpam-6297	125	2	7	7	NUM
ejpam-6297	125	3	):	):	PUNCT
ejpam-6297	125	4	assume	assume	VERB
ejpam-6297	125	5	that	that	SCONJ
ejpam-6297	125	6	x	x	SYM
ejpam-6297	125	7	∈	∈	PROPN
ejpam-6297	125	8	sβ	sβ	PROPN
ejpam-6297	125	9	cli(a	cli(a	PROPN
ejpam-6297	125	10	)	)	PUNCT
ejpam-6297	125	11	.	.	PUNCT
ejpam-6297	126	1	suppose	suppose	VERB
ejpam-6297	126	2	there	there	PRON
ejpam-6297	126	3	exists	exist	VERB
ejpam-6297	126	4	a	a	DET
ejpam-6297	126	5	strong	strong	ADJ
ejpam-6297	126	6	β	β	X
ejpam-6297	126	7	-	-	ADJ
ejpam-6297	126	8	i	i	PRON
ejpam-6297	126	9	-	-	PUNCT
ejpam-6297	126	10	open	open	ADJ
ejpam-6297	126	11	set	set	NOUN
ejpam-6297	126	12	u	u	NOUN
ejpam-6297	126	13	containing	contain	VERB
ejpam-6297	126	14	x	x	PUNCT
ejpam-6297	126	15	such	such	ADJ
ejpam-6297	126	16	that	that	SCONJ
ejpam-6297	126	17	u	u	PROPN
ejpam-6297	126	18	∩	∩	NOUN
ejpam-6297	126	19	a	a	DET
ejpam-6297	126	20	=	=	PUNCT
ejpam-6297	126	21	∅.	∅.	NOUN
ejpam-6297	126	22	then	then	ADV
ejpam-6297	126	23	it	it	PRON
ejpam-6297	126	24	follows	follow	VERB
ejpam-6297	126	25	that	that	SCONJ
ejpam-6297	126	26	a	a	DET
ejpam-6297	126	27	⊆	⊆	NUM
ejpam-6297	126	28	x	x	SYM
ejpam-6297	126	29	−	−	PROPN
ejpam-6297	126	30	u	u	NOUN
ejpam-6297	126	31	.	.	PUNCT
ejpam-6297	127	1	since	since	SCONJ
ejpam-6297	127	2	x	x	X
ejpam-6297	127	3	−	−	PROPN
ejpam-6297	127	4	u	u	NOUN
ejpam-6297	127	5	is	be	AUX
ejpam-6297	127	6	strong	strong	ADJ
ejpam-6297	127	7	β	β	NOUN
ejpam-6297	127	8	-	-	ADJ
ejpam-6297	127	9	i	i	NOUN
ejpam-6297	127	10	-	-	PUNCT
ejpam-6297	127	11	closed	closed	ADJ
ejpam-6297	127	12	,	,	PUNCT
ejpam-6297	127	13	this	this	PRON
ejpam-6297	127	14	would	would	AUX
ejpam-6297	127	15	imply	imply	VERB
ejpam-6297	127	16	x	x	PROPN
ejpam-6297	127	17	/∈	/∈	PUNCT
ejpam-6297	127	18	sβ	sβ	PROPN
ejpam-6297	127	19	cli(a	cli(a	PROPN
ejpam-6297	127	20	)	)	PUNCT
ejpam-6297	127	21	,	,	PUNCT
ejpam-6297	127	22	which	which	PRON
ejpam-6297	127	23	contradicts	contradict	VERB
ejpam-6297	127	24	the	the	DET
ejpam-6297	127	25	assumption	assumption	NOUN
ejpam-6297	127	26	.	.	PUNCT
ejpam-6297	128	1	conversely	conversely	ADV
ejpam-6297	128	2	,	,	PUNCT
ejpam-6297	128	3	suppose	suppose	VERB
ejpam-6297	128	4	that	that	SCONJ
ejpam-6297	128	5	for	for	ADP
ejpam-6297	128	6	every	every	DET
ejpam-6297	128	7	strong	strong	ADJ
ejpam-6297	128	8	β	β	X
ejpam-6297	128	9	-	-	ADJ
ejpam-6297	128	10	i	i	PRON
ejpam-6297	128	11	-	-	PUNCT
ejpam-6297	128	12	open	open	ADJ
ejpam-6297	128	13	set	set	NOUN
ejpam-6297	128	14	u	u	NOUN
ejpam-6297	128	15	containing	contain	VERB
ejpam-6297	128	16	x	x	PRON
ejpam-6297	128	17	,	,	PUNCT
ejpam-6297	128	18	we	we	PRON
ejpam-6297	128	19	have	have	VERB
ejpam-6297	128	20	u∩a	u∩a	PROPN
ejpam-6297	128	21	̸=	̸=	PROPN
ejpam-6297	128	22	∅.	∅.	AUX
ejpam-6297	128	23	assume	assume	VERB
ejpam-6297	128	24	,	,	PUNCT
ejpam-6297	128	25	for	for	ADP
ejpam-6297	128	26	the	the	DET
ejpam-6297	128	27	sake	sake	NOUN
ejpam-6297	128	28	of	of	ADP
ejpam-6297	128	29	contradiction	contradiction	NOUN
ejpam-6297	128	30	,	,	PUNCT
ejpam-6297	128	31	that	that	SCONJ
ejpam-6297	128	32	x	x	X
ejpam-6297	128	33	/∈	/∈	PUNCT
ejpam-6297	128	34	sβ	sβ	PROPN
ejpam-6297	128	35	cli(a	cli(a	PROPN
ejpam-6297	128	36	)	)	PUNCT
ejpam-6297	128	37	.	.	PUNCT
ejpam-6297	129	1	then	then	ADV
ejpam-6297	129	2	there	there	PRON
ejpam-6297	129	3	exists	exist	VERB
ejpam-6297	129	4	a	a	DET
ejpam-6297	129	5	strong	strong	ADJ
ejpam-6297	129	6	β	β	X
ejpam-6297	129	7	-	-	ADJ
ejpam-6297	129	8	i	i	NOUN
ejpam-6297	129	9	-	-	PUNCT
ejpam-6297	129	10	closed	close	VERB
ejpam-6297	129	11	set	set	NOUN
ejpam-6297	129	12	f	f	PROPN
ejpam-6297	129	13	such	such	ADJ
ejpam-6297	129	14	that	that	SCONJ
ejpam-6297	129	15	a	a	DET
ejpam-6297	129	16	⊆	⊆	NUM
ejpam-6297	129	17	f	f	PROPN
ejpam-6297	129	18	and	and	CCONJ
ejpam-6297	129	19	x	x	PROPN
ejpam-6297	129	20	/∈	/∈	PROPN
ejpam-6297	130	1	f	f	PROPN
ejpam-6297	130	2	.	.	PUNCT
ejpam-6297	131	1	consequently	consequently	ADV
ejpam-6297	131	2	,	,	PUNCT
ejpam-6297	131	3	x	x	PUNCT
ejpam-6297	131	4	∈	∈	NOUN
ejpam-6297	131	5	x	x	X
ejpam-6297	131	6	−	−	PROPN
ejpam-6297	131	7	f	f	X
ejpam-6297	131	8	,	,	PUNCT
ejpam-6297	131	9	which	which	PRON
ejpam-6297	131	10	is	be	AUX
ejpam-6297	131	11	a	a	DET
ejpam-6297	131	12	strong	strong	ADJ
ejpam-6297	131	13	β	β	X
ejpam-6297	131	14	-	-	ADJ
ejpam-6297	131	15	i	i	PRON
ejpam-6297	131	16	-	-	PUNCT
ejpam-6297	131	17	open	open	ADJ
ejpam-6297	131	18	set	set	VERB
ejpam-6297	131	19	disjoint	disjoint	NOUN
ejpam-6297	131	20	from	from	ADP
ejpam-6297	131	21	a	a	PRON
ejpam-6297	131	22	,	,	PUNCT
ejpam-6297	131	23	leading	lead	VERB
ejpam-6297	131	24	to	to	ADP
ejpam-6297	131	25	a	a	DET
ejpam-6297	131	26	contradiction	contradiction	NOUN
ejpam-6297	131	27	.	.	PUNCT
ejpam-6297	132	1	therefore	therefore	ADV
ejpam-6297	132	2	,	,	PUNCT
ejpam-6297	132	3	it	it	PRON
ejpam-6297	132	4	follows	follow	VERB
ejpam-6297	132	5	that	that	SCONJ
ejpam-6297	132	6	x	x	PUNCT
ejpam-6297	132	7	∈	∈	PROPN
ejpam-6297	132	8	sβ	sβ	PROPN
ejpam-6297	132	9	cli(a	cli(a	PROPN
ejpam-6297	132	10	)	)	PUNCT
ejpam-6297	132	11	.	.	PUNCT
ejpam-6297	133	1	2	2	X
ejpam-6297	133	2	.	.	X
ejpam-6297	133	3	strong	strong	ADJ
ejpam-6297	133	4	β	β	X
ejpam-6297	133	5	-	-	ADJ
ejpam-6297	133	6	i	i	NOUN
ejpam-6297	133	7	-	-	PUNCT
ejpam-6297	133	8	submaximality	submaximality	NOUN
ejpam-6297	133	9	in	in	ADP
ejpam-6297	133	10	this	this	DET
ejpam-6297	133	11	section	section	NOUN
ejpam-6297	133	12	,	,	PUNCT
ejpam-6297	133	13	we	we	PRON
ejpam-6297	133	14	explore	explore	VERB
ejpam-6297	133	15	a	a	DET
ejpam-6297	133	16	collection	collection	NOUN
ejpam-6297	133	17	of	of	ADP
ejpam-6297	133	18	equivalent	equivalent	ADJ
ejpam-6297	133	19	conditions	condition	NOUN
ejpam-6297	133	20	that	that	PRON
ejpam-6297	133	21	provide	provide	VERB
ejpam-6297	133	22	a	a	DET
ejpam-6297	133	23	thorough	thorough	ADJ
ejpam-6297	133	24	characterization	characterization	NOUN
ejpam-6297	133	25	of	of	ADP
ejpam-6297	133	26	when	when	SCONJ
ejpam-6297	133	27	an	an	DET
ejpam-6297	133	28	ideal	ideal	ADJ
ejpam-6297	133	29	topological	topological	ADJ
ejpam-6297	133	30	space	space	NOUN
ejpam-6297	133	31	(	(	PUNCT
ejpam-6297	133	32	x	x	X
ejpam-6297	133	33	,	,	PUNCT
ejpam-6297	133	34	τ	τ	PROPN
ejpam-6297	133	35	,	,	PUNCT
ejpam-6297	133	36	i	i	PROPN
ejpam-6297	133	37	)	)	PUNCT
ejpam-6297	133	38	can	can	AUX
ejpam-6297	133	39	be	be	AUX
ejpam-6297	133	40	regarded	regard	VERB
ejpam-6297	133	41	as	as	ADP
ejpam-6297	133	42	strong	strong	ADJ
ejpam-6297	133	43	β	β	X
ejpam-6297	133	44	-	-	ADJ
ejpam-6297	133	45	i	i	NOUN
ejpam-6297	133	46	-	-	PUNCT
ejpam-6297	133	47	submaximal	submaximal	ADJ
ejpam-6297	133	48	,	,	PUNCT
ejpam-6297	133	49	thereby	thereby	ADV
ejpam-6297	133	50	establishing	establish	VERB
ejpam-6297	133	51	a	a	DET
ejpam-6297	133	52	foundational	foundational	ADJ
ejpam-6297	133	53	understanding	understanding	NOUN
ejpam-6297	133	54	of	of	ADP
ejpam-6297	133	55	the	the	DET
ejpam-6297	133	56	underlying	underlie	VERB
ejpam-6297	133	57	properties	property	NOUN
ejpam-6297	133	58	that	that	PRON
ejpam-6297	133	59	define	define	VERB
ejpam-6297	133	60	this	this	DET
ejpam-6297	133	61	class	class	NOUN
ejpam-6297	133	62	of	of	ADP
ejpam-6297	133	63	spaces	space	NOUN
ejpam-6297	133	64	.	.	PUNCT
ejpam-6297	134	1	c.	c.	PROPN
ejpam-6297	134	2	boonpok	boonpok	PROPN
ejpam-6297	134	3	,	,	PUNCT
ejpam-6297	134	4	p.	p.	PROPN
ejpam-6297	134	5	raktaow	raktaow	NOUN
ejpam-6297	134	6	,	,	PUNCT
ejpam-6297	134	7	a.	a.	PROPN
ejpam-6297	134	8	sama	sama	PROPN
ejpam-6297	134	9	-	-	PUNCT
ejpam-6297	134	10	ae	ae	PROPN
ejpam-6297	134	11	/	/	SYM
ejpam-6297	134	12	eur	eur	PROPN
ejpam-6297	134	13	.	.	PUNCT
ejpam-6297	135	1	j.	j.	PROPN
ejpam-6297	135	2	pure	pure	PROPN
ejpam-6297	135	3	appl	appl	PROPN
ejpam-6297	135	4	.	.	PROPN
ejpam-6297	135	5	math	math	PROPN
ejpam-6297	135	6	,	,	PUNCT
ejpam-6297	135	7	18	18	NUM
ejpam-6297	135	8	(	(	PUNCT
ejpam-6297	135	9	3	3	NUM
ejpam-6297	135	10	)	)	PUNCT
ejpam-6297	135	11	(	(	PUNCT
ejpam-6297	135	12	2025	2025	NUM
ejpam-6297	135	13	)	)	PUNCT
ejpam-6297	135	14	,	,	PUNCT
ejpam-6297	135	15	6297	6297	NUM
ejpam-6297	135	16	6	6	NUM
ejpam-6297	135	17	of	of	ADP
ejpam-6297	135	18	23	23	NUM
ejpam-6297	135	19	definition	definition	NOUN
ejpam-6297	135	20	3	3	NUM
ejpam-6297	135	21	.	.	PUNCT
ejpam-6297	136	1	let	let	VERB
ejpam-6297	136	2	(	(	PUNCT
ejpam-6297	136	3	x	x	X
ejpam-6297	136	4	,	,	PUNCT
ejpam-6297	136	5	τ	τ	PROPN
ejpam-6297	136	6	,	,	PUNCT
ejpam-6297	136	7	i	i	PRON
ejpam-6297	136	8	)	)	PUNCT
ejpam-6297	136	9	be	be	VERB
ejpam-6297	136	10	an	an	DET
ejpam-6297	136	11	ideal	ideal	ADJ
ejpam-6297	136	12	topological	topological	ADJ
ejpam-6297	136	13	space	space	NOUN
ejpam-6297	136	14	.	.	PUNCT
ejpam-6297	137	1	a	a	DET
ejpam-6297	137	2	subset	subset	NOUN
ejpam-6297	137	3	a	a	DET
ejpam-6297	137	4	⊆	⊆	NUM
ejpam-6297	137	5	x	x	SYM
ejpam-6297	137	6	is	be	AUX
ejpam-6297	137	7	said	say	VERB
ejpam-6297	137	8	to	to	PART
ejpam-6297	137	9	be	be	AUX
ejpam-6297	137	10	:	:	PUNCT
ejpam-6297	137	11	(	(	PUNCT
ejpam-6297	137	12	1	1	X
ejpam-6297	137	13	)	)	PUNCT
ejpam-6297	137	14	strong	strong	ADJ
ejpam-6297	137	15	β	β	X
ejpam-6297	137	16	-	-	ADJ
ejpam-6297	137	17	i	i	NOUN
ejpam-6297	137	18	-	-	PUNCT
ejpam-6297	137	19	dense	dense	ADJ
ejpam-6297	137	20	if	if	SCONJ
ejpam-6297	137	21	sβ	sβ	PROPN
ejpam-6297	137	22	cli(a	cli(a	PROPN
ejpam-6297	137	23	)	)	PUNCT
ejpam-6297	137	24	=	=	SYM
ejpam-6297	138	1	x	x	NOUN
ejpam-6297	138	2	;	;	PUNCT
ejpam-6297	138	3	and	and	CCONJ
ejpam-6297	138	4	(	(	PUNCT
ejpam-6297	138	5	2	2	X
ejpam-6297	138	6	)	)	PUNCT
ejpam-6297	138	7	strong	strong	ADJ
ejpam-6297	138	8	β	β	X
ejpam-6297	138	9	-	-	ADJ
ejpam-6297	138	10	i	i	NOUN
ejpam-6297	138	11	-	-	PUNCT
ejpam-6297	138	12	codense	codense	NOUN
ejpam-6297	138	13	if	if	SCONJ
ejpam-6297	138	14	x	x	PRON
ejpam-6297	138	15	−a	−a	NOUN
ejpam-6297	138	16	is	be	AUX
ejpam-6297	138	17	strong	strong	ADJ
ejpam-6297	138	18	β	β	NOUN
ejpam-6297	138	19	-	-	ADJ
ejpam-6297	138	20	i	i	NOUN
ejpam-6297	138	21	-	-	PUNCT
ejpam-6297	138	22	dense	dense	ADJ
ejpam-6297	138	23	.	.	PUNCT
ejpam-6297	139	1	definition	definition	NOUN
ejpam-6297	139	2	4	4	NUM
ejpam-6297	139	3	.	.	PUNCT
ejpam-6297	140	1	an	an	DET
ejpam-6297	140	2	ideal	ideal	ADJ
ejpam-6297	140	3	topological	topological	ADJ
ejpam-6297	140	4	space	space	NOUN
ejpam-6297	140	5	(	(	PUNCT
ejpam-6297	140	6	x	x	X
ejpam-6297	140	7	,	,	PUNCT
ejpam-6297	140	8	τ	τ	PROPN
ejpam-6297	140	9	,	,	PUNCT
ejpam-6297	140	10	i	i	PROPN
ejpam-6297	140	11	)	)	PUNCT
ejpam-6297	140	12	is	be	AUX
ejpam-6297	140	13	called	call	VERB
ejpam-6297	140	14	strong	strong	ADJ
ejpam-6297	140	15	β	β	X
ejpam-6297	140	16	-	-	ADJ
ejpam-6297	140	17	i	i	PRON
ejpam-6297	140	18	-	-	PUNCT
ejpam-6297	140	19	submaximal	submaximal	ADJ
ejpam-6297	140	20	if	if	SCONJ
ejpam-6297	140	21	each	each	DET
ejpam-6297	140	22	strong	strong	ADJ
ejpam-6297	140	23	β	β	X
ejpam-6297	140	24	-	-	ADJ
ejpam-6297	140	25	i	i	NOUN
ejpam-6297	140	26	-	-	PUNCT
ejpam-6297	140	27	dense	dense	ADJ
ejpam-6297	140	28	subset	subset	NOUN
ejpam-6297	140	29	of	of	ADP
ejpam-6297	140	30	x	x	PUNCT
ejpam-6297	140	31	is	be	AUX
ejpam-6297	140	32	strong	strong	ADJ
ejpam-6297	140	33	β	β	NOUN
ejpam-6297	140	34	-	-	ADJ
ejpam-6297	140	35	i	i	PRON
ejpam-6297	140	36	-	-	PUNCT
ejpam-6297	140	37	open	open	ADJ
ejpam-6297	140	38	.	.	PUNCT
ejpam-6297	141	1	let	let	VERB
ejpam-6297	141	2	x	x	PUNCT
ejpam-6297	141	3	=	=	PRON
ejpam-6297	141	4	{	{	PUNCT
ejpam-6297	141	5	a	a	DET
ejpam-6297	141	6	,	,	PUNCT
ejpam-6297	141	7	b	b	NOUN
ejpam-6297	141	8	,	,	PUNCT
ejpam-6297	141	9	c	c	NOUN
ejpam-6297	141	10	}	}	PUNCT
ejpam-6297	141	11	,	,	PUNCT
ejpam-6297	141	12	τ	τ	X
ejpam-6297	141	13	=	=	PUNCT
ejpam-6297	141	14	{	{	PUNCT
ejpam-6297	141	15	∅	∅	NOUN
ejpam-6297	141	16	,	,	PUNCT
ejpam-6297	141	17	{	{	PUNCT
ejpam-6297	141	18	a	a	X
ejpam-6297	141	19	}	}	PUNCT
ejpam-6297	141	20	,	,	PUNCT
ejpam-6297	141	21	{	{	PUNCT
ejpam-6297	141	22	a	a	DET
ejpam-6297	141	23	,	,	PUNCT
ejpam-6297	141	24	b	b	NOUN
ejpam-6297	141	25	}	}	PUNCT
ejpam-6297	141	26	,	,	PUNCT
ejpam-6297	141	27	x	x	NOUN
ejpam-6297	141	28	}	}	PUNCT
ejpam-6297	141	29	,	,	PUNCT
ejpam-6297	141	30	and	and	CCONJ
ejpam-6297	141	31	i	i	PRON
ejpam-6297	141	32	=	=	PUNCT
ejpam-6297	141	33	{	{	PUNCT
ejpam-6297	141	34	∅	∅	NOUN
ejpam-6297	141	35	,	,	PUNCT
ejpam-6297	141	36	{	{	PUNCT
ejpam-6297	141	37	c	c	NOUN
ejpam-6297	141	38	}	}	PUNCT
ejpam-6297	141	39	}	}	PUNCT
ejpam-6297	141	40	.	.	PUNCT
ejpam-6297	142	1	it	it	PRON
ejpam-6297	142	2	is	be	AUX
ejpam-6297	142	3	clear	clear	ADJ
ejpam-6297	142	4	that	that	SCONJ
ejpam-6297	142	5	the	the	DET
ejpam-6297	142	6	only	only	ADJ
ejpam-6297	142	7	strong	strong	ADJ
ejpam-6297	142	8	β	β	X
ejpam-6297	142	9	-	-	ADJ
ejpam-6297	142	10	i	i	NOUN
ejpam-6297	142	11	-	-	PUNCT
ejpam-6297	142	12	dense	dense	ADJ
ejpam-6297	142	13	sets	set	NOUN
ejpam-6297	142	14	are	be	AUX
ejpam-6297	142	15	{	{	PUNCT
ejpam-6297	142	16	a	a	NOUN
ejpam-6297	142	17	}	}	PUNCT
ejpam-6297	142	18	,	,	PUNCT
ejpam-6297	142	19	{	{	PUNCT
ejpam-6297	142	20	a	a	DET
ejpam-6297	142	21	,	,	PUNCT
ejpam-6297	142	22	b	b	NOUN
ejpam-6297	142	23	}	}	PUNCT
ejpam-6297	142	24	,	,	PUNCT
ejpam-6297	142	25	and	and	CCONJ
ejpam-6297	142	26	{	{	PUNCT
ejpam-6297	142	27	a	a	PRON
ejpam-6297	142	28	,	,	PUNCT
ejpam-6297	142	29	b	b	NOUN
ejpam-6297	142	30	,	,	PUNCT
ejpam-6297	142	31	c	c	NOUN
ejpam-6297	142	32	}	}	PUNCT
ejpam-6297	142	33	,	,	PUNCT
ejpam-6297	142	34	all	all	PRON
ejpam-6297	142	35	of	of	ADP
ejpam-6297	142	36	which	which	PRON
ejpam-6297	142	37	are	be	AUX
ejpam-6297	142	38	also	also	ADV
ejpam-6297	142	39	strong	strong	ADJ
ejpam-6297	142	40	β	β	X
ejpam-6297	142	41	-	-	ADJ
ejpam-6297	142	42	i	i	PRON
ejpam-6297	142	43	-	-	PUNCT
ejpam-6297	142	44	open	open	ADJ
ejpam-6297	142	45	.	.	PUNCT
ejpam-6297	143	1	hence	hence	ADV
ejpam-6297	143	2	,	,	PUNCT
ejpam-6297	143	3	the	the	DET
ejpam-6297	143	4	space	space	NOUN
ejpam-6297	143	5	(	(	PUNCT
ejpam-6297	143	6	x	x	X
ejpam-6297	143	7	,	,	PUNCT
ejpam-6297	143	8	τ	τ	PROPN
ejpam-6297	143	9	,	,	PUNCT
ejpam-6297	143	10	i	i	PROPN
ejpam-6297	143	11	)	)	PUNCT
ejpam-6297	143	12	is	be	AUX
ejpam-6297	143	13	strong	strong	ADJ
ejpam-6297	143	14	β	β	NOUN
ejpam-6297	143	15	-	-	ADJ
ejpam-6297	143	16	i	i	NOUN
ejpam-6297	143	17	-	-	PUNCT
ejpam-6297	143	18	submaximal	submaximal	ADJ
ejpam-6297	143	19	.	.	PUNCT
ejpam-6297	144	1	definition	definition	NOUN
ejpam-6297	144	2	5	5	NUM
ejpam-6297	144	3	.	.	PUNCT
ejpam-6297	145	1	let	let	VERB
ejpam-6297	145	2	(	(	PUNCT
ejpam-6297	145	3	x	x	X
ejpam-6297	145	4	,	,	PUNCT
ejpam-6297	145	5	τ	τ	PROPN
ejpam-6297	145	6	,	,	PUNCT
ejpam-6297	145	7	i	i	PRON
ejpam-6297	145	8	)	)	PUNCT
ejpam-6297	145	9	be	be	VERB
ejpam-6297	145	10	an	an	DET
ejpam-6297	145	11	ideal	ideal	ADJ
ejpam-6297	145	12	topological	topological	ADJ
ejpam-6297	145	13	space	space	NOUN
ejpam-6297	145	14	.	.	PUNCT
ejpam-6297	146	1	a	a	DET
ejpam-6297	146	2	subset	subset	NOUN
ejpam-6297	146	3	a	a	DET
ejpam-6297	146	4	⊆	⊆	NUM
ejpam-6297	146	5	x	x	SYM
ejpam-6297	146	6	is	be	AUX
ejpam-6297	146	7	said	say	VERB
ejpam-6297	146	8	to	to	PART
ejpam-6297	146	9	be	be	AUX
ejpam-6297	146	10	:	:	PUNCT
ejpam-6297	146	11	(	(	PUNCT
ejpam-6297	146	12	1	1	X
ejpam-6297	146	13	)	)	PUNCT
ejpam-6297	146	14	locally	locally	ADV
ejpam-6297	146	15	strong	strong	ADJ
ejpam-6297	146	16	β	β	X
ejpam-6297	146	17	-	-	ADJ
ejpam-6297	146	18	i	i	NOUN
ejpam-6297	146	19	-	-	PUNCT
ejpam-6297	146	20	closed	close	VERB
ejpam-6297	146	21	if	if	SCONJ
ejpam-6297	146	22	a	a	PRON
ejpam-6297	146	23	is	be	AUX
ejpam-6297	146	24	the	the	DET
ejpam-6297	146	25	intersection	intersection	NOUN
ejpam-6297	146	26	of	of	ADP
ejpam-6297	146	27	a	a	DET
ejpam-6297	146	28	strong	strong	ADJ
ejpam-6297	146	29	β	β	X
ejpam-6297	146	30	-	-	ADJ
ejpam-6297	146	31	i	i	PRON
ejpam-6297	146	32	-	-	PUNCT
ejpam-6297	146	33	open	open	ADJ
ejpam-6297	146	34	set	set	NOUN
ejpam-6297	146	35	and	and	CCONJ
ejpam-6297	146	36	a	a	DET
ejpam-6297	146	37	strong	strong	ADJ
ejpam-6297	146	38	β	β	X
ejpam-6297	146	39	-	-	ADJ
ejpam-6297	146	40	i	i	NOUN
ejpam-6297	146	41	-	-	PUNCT
ejpam-6297	146	42	closed	close	VERB
ejpam-6297	146	43	set	set	NOUN
ejpam-6297	146	44	;	;	PUNCT
ejpam-6297	146	45	and	and	CCONJ
ejpam-6297	146	46	(	(	PUNCT
ejpam-6297	146	47	2	2	X
ejpam-6297	146	48	)	)	PUNCT
ejpam-6297	146	49	co	co	NOUN
ejpam-6297	146	50	-	-	ADJ
ejpam-6297	146	51	locally	locally	ADV
ejpam-6297	146	52	strong	strong	ADJ
ejpam-6297	146	53	β	β	NOUN
ejpam-6297	146	54	-	-	ADJ
ejpam-6297	146	55	i	i	NOUN
ejpam-6297	146	56	-	-	PUNCT
ejpam-6297	146	57	closed	close	VERB
ejpam-6297	146	58	if	if	SCONJ
ejpam-6297	146	59	a	a	PRON
ejpam-6297	146	60	is	be	AUX
ejpam-6297	146	61	the	the	DET
ejpam-6297	146	62	union	union	NOUN
ejpam-6297	146	63	of	of	ADP
ejpam-6297	146	64	a	a	DET
ejpam-6297	146	65	strong	strong	ADJ
ejpam-6297	146	66	β	β	X
ejpam-6297	146	67	-	-	ADJ
ejpam-6297	146	68	i	i	PRON
ejpam-6297	146	69	-	-	PUNCT
ejpam-6297	146	70	open	open	ADJ
ejpam-6297	146	71	set	set	NOUN
ejpam-6297	146	72	and	and	CCONJ
ejpam-6297	146	73	a	a	DET
ejpam-6297	146	74	strong	strong	ADJ
ejpam-6297	146	75	β	β	X
ejpam-6297	146	76	-	-	ADJ
ejpam-6297	146	77	i	i	NOUN
ejpam-6297	146	78	-	-	PUNCT
ejpam-6297	146	79	closed	close	VERB
ejpam-6297	146	80	set	set	NOUN
ejpam-6297	146	81	.	.	PUNCT
ejpam-6297	147	1	theorem	theorem	VERB
ejpam-6297	147	2	1	1	NUM
ejpam-6297	147	3	presents	present	NOUN
ejpam-6297	147	4	five	five	NUM
ejpam-6297	147	5	equivalent	equivalent	ADJ
ejpam-6297	147	6	conditions	condition	NOUN
ejpam-6297	147	7	that	that	PRON
ejpam-6297	147	8	characterize	characterize	VERB
ejpam-6297	147	9	when	when	SCONJ
ejpam-6297	147	10	a	a	DET
ejpam-6297	147	11	subset	subset	NOUN
ejpam-6297	147	12	a	a	PRON
ejpam-6297	147	13	of	of	ADP
ejpam-6297	147	14	an	an	DET
ejpam-6297	147	15	ideal	ideal	ADJ
ejpam-6297	147	16	topological	topological	ADJ
ejpam-6297	147	17	space	space	NOUN
ejpam-6297	147	18	is	be	AUX
ejpam-6297	147	19	locally	locally	ADV
ejpam-6297	147	20	strong	strong	ADJ
ejpam-6297	147	21	β	β	X
ejpam-6297	147	22	-	-	ADJ
ejpam-6297	147	23	i	i	NOUN
ejpam-6297	147	24	-	-	PUNCT
ejpam-6297	147	25	closed	closed	ADJ
ejpam-6297	147	26	.	.	PUNCT
ejpam-6297	148	1	theorem	theorem	NOUN
ejpam-6297	148	2	1	1	NUM
ejpam-6297	148	3	.	.	X
ejpam-6297	148	4	for	for	ADP
ejpam-6297	148	5	a	a	DET
ejpam-6297	148	6	subset	subset	NOUN
ejpam-6297	148	7	a	a	PRON
ejpam-6297	148	8	of	of	ADP
ejpam-6297	148	9	an	an	DET
ejpam-6297	148	10	ideal	ideal	ADJ
ejpam-6297	148	11	topological	topological	ADJ
ejpam-6297	148	12	space	space	NOUN
ejpam-6297	148	13	(	(	PUNCT
ejpam-6297	148	14	x	x	X
ejpam-6297	148	15	,	,	PUNCT
ejpam-6297	148	16	τ	τ	PROPN
ejpam-6297	148	17	,	,	PUNCT
ejpam-6297	148	18	i	i	PROPN
ejpam-6297	148	19	)	)	PUNCT
ejpam-6297	148	20	,	,	PUNCT
ejpam-6297	148	21	the	the	DET
ejpam-6297	148	22	following	follow	VERB
ejpam-6297	148	23	properties	property	NOUN
ejpam-6297	148	24	are	be	AUX
ejpam-6297	148	25	equivalent	equivalent	ADJ
ejpam-6297	148	26	:	:	PUNCT
ejpam-6297	148	27	(	(	PUNCT
ejpam-6297	148	28	1	1	X
ejpam-6297	148	29	)	)	PUNCT
ejpam-6297	148	30	a	a	PRON
ejpam-6297	148	31	is	be	AUX
ejpam-6297	148	32	locally	locally	ADV
ejpam-6297	148	33	strong	strong	ADJ
ejpam-6297	148	34	β	β	X
ejpam-6297	148	35	-	-	ADJ
ejpam-6297	148	36	i	i	NOUN
ejpam-6297	148	37	-	-	PUNCT
ejpam-6297	148	38	closed	closed	ADJ
ejpam-6297	148	39	;	;	PUNCT
ejpam-6297	148	40	(	(	PUNCT
ejpam-6297	148	41	2	2	X
ejpam-6297	148	42	)	)	PUNCT
ejpam-6297	148	43	a	a	DET
ejpam-6297	148	44	=	=	X
ejpam-6297	148	45	u	u	NOUN
ejpam-6297	148	46	∩	∩	X
ejpam-6297	148	47	sβ	sβ	PROPN
ejpam-6297	148	48	cli(a	cli(a	PROPN
ejpam-6297	148	49	)	)	PUNCT
ejpam-6297	148	50	for	for	ADP
ejpam-6297	148	51	some	some	DET
ejpam-6297	148	52	strong	strong	ADJ
ejpam-6297	148	53	β	β	X
ejpam-6297	148	54	-	-	ADJ
ejpam-6297	148	55	i	i	PRON
ejpam-6297	148	56	-	-	PUNCT
ejpam-6297	148	57	open	open	ADJ
ejpam-6297	148	58	set	set	NOUN
ejpam-6297	148	59	u	u	NOUN
ejpam-6297	148	60	;	;	PUNCT
ejpam-6297	148	61	(	(	PUNCT
ejpam-6297	148	62	3	3	X
ejpam-6297	148	63	)	)	PUNCT
ejpam-6297	148	64	sβ	sβ	NOUN
ejpam-6297	148	65	cli(a)−a	cli(a)−a	NOUN
ejpam-6297	148	66	is	be	AUX
ejpam-6297	148	67	strong	strong	ADJ
ejpam-6297	148	68	β	β	NOUN
ejpam-6297	148	69	-	-	ADJ
ejpam-6297	148	70	i	i	NOUN
ejpam-6297	148	71	-	-	PUNCT
ejpam-6297	148	72	closed	closed	ADJ
ejpam-6297	148	73	;	;	PUNCT
ejpam-6297	148	74	(	(	PUNCT
ejpam-6297	148	75	4	4	X
ejpam-6297	148	76	)	)	PUNCT
ejpam-6297	148	77	a	a	DET
ejpam-6297	148	78	∪	∪	NOUN
ejpam-6297	148	79	(	(	PUNCT
ejpam-6297	148	80	x	x	X
ejpam-6297	148	81	−	−	PROPN
ejpam-6297	148	82	sβ	sβ	PROPN
ejpam-6297	148	83	cli(a	cli(a	PROPN
ejpam-6297	148	84	)	)	PUNCT
ejpam-6297	148	85	)	)	PUNCT
ejpam-6297	148	86	is	be	AUX
ejpam-6297	148	87	strong	strong	ADJ
ejpam-6297	148	88	β	β	NOUN
ejpam-6297	148	89	-	-	ADJ
ejpam-6297	148	90	i	i	PRON
ejpam-6297	148	91	-	-	PUNCT
ejpam-6297	148	92	open	open	ADJ
ejpam-6297	148	93	;	;	PUNCT
ejpam-6297	148	94	and	and	CCONJ
ejpam-6297	148	95	(	(	PUNCT
ejpam-6297	148	96	5	5	X
ejpam-6297	148	97	)	)	PUNCT
ejpam-6297	148	98	a	a	DET
ejpam-6297	148	99	⊆	⊆	NUM
ejpam-6297	148	100	sβ	sβ	PROPN
ejpam-6297	148	101	inti(a	inti(a	PROPN
ejpam-6297	148	102	∪	∪	ADV
ejpam-6297	148	103	(	(	PUNCT
ejpam-6297	148	104	x	x	X
ejpam-6297	148	105	−	−	PROPN
ejpam-6297	148	106	sβ	sβ	PROPN
ejpam-6297	148	107	cli(a	cli(a	PROPN
ejpam-6297	148	108	)	)	PUNCT
ejpam-6297	148	109	)	)	PUNCT
ejpam-6297	148	110	)	)	PUNCT
ejpam-6297	148	111	.	.	PUNCT
ejpam-6297	149	1	proof	proof	NOUN
ejpam-6297	149	2	.	.	PUNCT
ejpam-6297	150	1	(	(	PUNCT
ejpam-6297	150	2	1	1	X
ejpam-6297	150	3	)	)	PUNCT
ejpam-6297	150	4	⇒	⇒	NOUN
ejpam-6297	150	5	(	(	PUNCT
ejpam-6297	150	6	2	2	NUM
ejpam-6297	150	7	):	):	PUNCT
ejpam-6297	150	8	suppose	suppose	VERB
ejpam-6297	150	9	a	a	PRON
ejpam-6297	150	10	is	be	AUX
ejpam-6297	150	11	locally	locally	ADV
ejpam-6297	150	12	strong	strong	ADJ
ejpam-6297	150	13	β	β	X
ejpam-6297	150	14	-	-	ADJ
ejpam-6297	150	15	i	i	NOUN
ejpam-6297	150	16	-	-	PUNCT
ejpam-6297	150	17	closed	closed	ADJ
ejpam-6297	150	18	.	.	PUNCT
ejpam-6297	151	1	then	then	ADV
ejpam-6297	151	2	there	there	PRON
ejpam-6297	151	3	exists	exist	VERB
ejpam-6297	151	4	a	a	DET
ejpam-6297	151	5	strong	strong	ADJ
ejpam-6297	151	6	β	β	X
ejpam-6297	151	7	-	-	ADJ
ejpam-6297	151	8	i	i	PRON
ejpam-6297	151	9	-	-	PUNCT
ejpam-6297	151	10	open	open	ADJ
ejpam-6297	151	11	set	set	NOUN
ejpam-6297	151	12	u	u	NOUN
ejpam-6297	151	13	and	and	CCONJ
ejpam-6297	151	14	a	a	DET
ejpam-6297	151	15	strong	strong	ADJ
ejpam-6297	151	16	β	β	X
ejpam-6297	151	17	-	-	ADJ
ejpam-6297	151	18	i	i	NOUN
ejpam-6297	151	19	-	-	PUNCT
ejpam-6297	151	20	closed	close	VERB
ejpam-6297	151	21	set	set	NOUN
ejpam-6297	151	22	f	f	PROPN
ejpam-6297	151	23	such	such	ADJ
ejpam-6297	151	24	that	that	SCONJ
ejpam-6297	151	25	a	a	DET
ejpam-6297	151	26	=	=	X
ejpam-6297	151	27	u	u	NOUN
ejpam-6297	151	28	∩f	∩f	NOUN
ejpam-6297	151	29	.	.	PUNCT
ejpam-6297	152	1	given	give	VERB
ejpam-6297	152	2	that	that	SCONJ
ejpam-6297	152	3	f	f	PROPN
ejpam-6297	152	4	is	be	AUX
ejpam-6297	152	5	strong	strong	ADJ
ejpam-6297	152	6	β	β	NOUN
ejpam-6297	152	7	-	-	ADJ
ejpam-6297	152	8	i	i	NOUN
ejpam-6297	152	9	-	-	PUNCT
ejpam-6297	152	10	closed	closed	ADJ
ejpam-6297	152	11	,	,	PUNCT
ejpam-6297	152	12	it	it	PRON
ejpam-6297	152	13	follows	follow	VERB
ejpam-6297	152	14	that	that	SCONJ
ejpam-6297	152	15	sβ	sβ	PROPN
ejpam-6297	152	16	cli(a	cli(a	PROPN
ejpam-6297	152	17	)	)	PUNCT
ejpam-6297	152	18	⊆	⊆	NUM
ejpam-6297	152	19	sβ	sβ	NOUN
ejpam-6297	152	20	cli(f	cli(f	NOUN
ejpam-6297	152	21	)	)	PUNCT
ejpam-6297	153	1	=	=	SYM
ejpam-6297	153	2	f	f	PROPN
ejpam-6297	153	3	,	,	PUNCT
ejpam-6297	153	4	so	so	ADV
ejpam-6297	153	5	a	a	DET
ejpam-6297	153	6	⊆	⊆	NUM
ejpam-6297	153	7	u	u	NOUN
ejpam-6297	153	8	∩	∩	X
ejpam-6297	153	9	sβ	sβ	PROPN
ejpam-6297	153	10	cli(a	cli(a	PROPN
ejpam-6297	153	11	)	)	PUNCT
ejpam-6297	153	12	⊆	⊆	NUM
ejpam-6297	153	13	u	u	NOUN
ejpam-6297	153	14	∩f	∩f	NOUN
ejpam-6297	153	15	=	=	PUNCT
ejpam-6297	154	1	a.	a.	NOUN
ejpam-6297	154	2	consequently	consequently	ADV
ejpam-6297	154	3	,	,	PUNCT
ejpam-6297	154	4	a	a	DET
ejpam-6297	154	5	=	=	X
ejpam-6297	154	6	u	u	NOUN
ejpam-6297	154	7	∩	∩	X
ejpam-6297	154	8	sβ	sβ	PROPN
ejpam-6297	154	9	cli(a	cli(a	PROPN
ejpam-6297	154	10	)	)	PUNCT
ejpam-6297	154	11	.	.	PUNCT
ejpam-6297	155	1	(	(	PUNCT
ejpam-6297	155	2	2	2	X
ejpam-6297	155	3	)	)	PUNCT
ejpam-6297	155	4	⇒	⇒	NOUN
ejpam-6297	155	5	(	(	PUNCT
ejpam-6297	155	6	3	3	NUM
ejpam-6297	155	7	):	):	PUNCT
ejpam-6297	155	8	suppose	suppose	VERB
ejpam-6297	155	9	a	a	DET
ejpam-6297	155	10	=	=	X
ejpam-6297	155	11	u	u	NOUN
ejpam-6297	155	12	∩	∩	X
ejpam-6297	155	13	sβ	sβ	PROPN
ejpam-6297	155	14	cli(a	cli(a	PROPN
ejpam-6297	155	15	)	)	PUNCT
ejpam-6297	155	16	for	for	ADP
ejpam-6297	155	17	some	some	DET
ejpam-6297	155	18	strong	strong	ADJ
ejpam-6297	155	19	β	β	X
ejpam-6297	155	20	-	-	ADJ
ejpam-6297	155	21	i	i	PRON
ejpam-6297	155	22	-	-	PUNCT
ejpam-6297	155	23	open	open	ADJ
ejpam-6297	155	24	set	set	NOUN
ejpam-6297	155	25	u	u	PROPN
ejpam-6297	155	26	.	.	PUNCT
ejpam-6297	156	1	since	since	SCONJ
ejpam-6297	156	2	sβ	sβ	NUM
ejpam-6297	156	3	cli(a)−a	cli(a)−a	NOUN
ejpam-6297	156	4	=	=	SYM
ejpam-6297	156	5	sβ	sβ	PROPN
ejpam-6297	156	6	cli(a	cli(a	PROPN
ejpam-6297	156	7	)	)	PUNCT
ejpam-6297	156	8	∩	∩	NOUN
ejpam-6297	156	9	(	(	PUNCT
ejpam-6297	156	10	x	x	NOUN
ejpam-6297	156	11	−a	−a	NOUN
ejpam-6297	156	12	)	)	PUNCT
ejpam-6297	156	13	=	=	PUNCT
ejpam-6297	157	1	(	(	PUNCT
ejpam-6297	157	2	x	x	SYM
ejpam-6297	157	3	−	−	PROPN
ejpam-6297	157	4	(	(	PUNCT
ejpam-6297	157	5	u	u	NOUN
ejpam-6297	157	6	∩	∩	X
ejpam-6297	157	7	sβ	sβ	PROPN
ejpam-6297	157	8	cli(a	cli(a	PROPN
ejpam-6297	157	9	)	)	PUNCT
ejpam-6297	157	10	)	)	PUNCT
ejpam-6297	157	11	)	)	PUNCT
ejpam-6297	157	12	∩	∩	PROPN
ejpam-6297	157	13	sβ	sβ	PROPN
ejpam-6297	157	14	cli(a	cli(a	PROPN
ejpam-6297	157	15	)	)	PUNCT
ejpam-6297	157	16	=	=	PUNCT
ejpam-6297	157	17	(	(	PUNCT
ejpam-6297	157	18	x	x	X
ejpam-6297	157	19	−	−	PROPN
ejpam-6297	157	20	u	u	NOUN
ejpam-6297	157	21	)	)	PUNCT
ejpam-6297	157	22	∩	∩	NOUN
ejpam-6297	157	23	sβ	sβ	PROPN
ejpam-6297	157	24	cli(a	cli(a	PROPN
ejpam-6297	157	25	)	)	PUNCT
ejpam-6297	157	26	,	,	PUNCT
ejpam-6297	157	27	sβ	sβ	PROPN
ejpam-6297	157	28	cli(a)−a	cli(a)−a	PROPN
ejpam-6297	157	29	is	be	AUX
ejpam-6297	157	30	strong	strong	ADJ
ejpam-6297	157	31	β	β	NOUN
ejpam-6297	157	32	-	-	ADJ
ejpam-6297	157	33	i	i	NOUN
ejpam-6297	157	34	-	-	PUNCT
ejpam-6297	157	35	closed	closed	ADJ
ejpam-6297	157	36	.	.	PUNCT
ejpam-6297	158	1	c.	c.	PROPN
ejpam-6297	158	2	boonpok	boonpok	PROPN
ejpam-6297	158	3	,	,	PUNCT
ejpam-6297	158	4	p.	p.	PROPN
ejpam-6297	158	5	raktaow	raktaow	NOUN
ejpam-6297	158	6	,	,	PUNCT
ejpam-6297	158	7	a.	a.	PROPN
ejpam-6297	158	8	sama	sama	PROPN
ejpam-6297	158	9	-	-	PUNCT
ejpam-6297	158	10	ae	ae	PROPN
ejpam-6297	158	11	/	/	SYM
ejpam-6297	158	12	eur	eur	PROPN
ejpam-6297	158	13	.	.	PUNCT
ejpam-6297	159	1	j.	j.	PROPN
ejpam-6297	159	2	pure	pure	PROPN
ejpam-6297	159	3	appl	appl	PROPN
ejpam-6297	159	4	.	.	PROPN
ejpam-6297	159	5	math	math	PROPN
ejpam-6297	159	6	,	,	PUNCT
ejpam-6297	159	7	18	18	NUM
ejpam-6297	159	8	(	(	PUNCT
ejpam-6297	159	9	3	3	NUM
ejpam-6297	159	10	)	)	PUNCT
ejpam-6297	159	11	(	(	PUNCT
ejpam-6297	159	12	2025	2025	NUM
ejpam-6297	159	13	)	)	PUNCT
ejpam-6297	159	14	,	,	PUNCT
ejpam-6297	159	15	6297	6297	NUM
ejpam-6297	159	16	7	7	NUM
ejpam-6297	159	17	of	of	ADP
ejpam-6297	159	18	23	23	NUM
ejpam-6297	159	19	(	(	PUNCT
ejpam-6297	159	20	3	3	NUM
ejpam-6297	159	21	)	)	PUNCT
ejpam-6297	159	22	⇒	⇒	NOUN
ejpam-6297	159	23	(	(	PUNCT
ejpam-6297	159	24	4	4	NUM
ejpam-6297	159	25	):	):	PUNCT
ejpam-6297	159	26	assume	assume	VERB
ejpam-6297	159	27	that	that	SCONJ
ejpam-6297	159	28	the	the	DET
ejpam-6297	159	29	set	set	NOUN
ejpam-6297	159	30	sβ	sβ	PROPN
ejpam-6297	159	31	cli(a	cli(a	PROPN
ejpam-6297	159	32	)	)	PUNCT
ejpam-6297	159	33	−	−	PROPN
ejpam-6297	159	34	a	a	PRON
ejpam-6297	159	35	is	be	AUX
ejpam-6297	159	36	strong	strong	ADJ
ejpam-6297	159	37	β	β	NOUN
ejpam-6297	159	38	-	-	ADJ
ejpam-6297	159	39	i	i	NOUN
ejpam-6297	159	40	-	-	PUNCT
ejpam-6297	159	41	closed	closed	ADJ
ejpam-6297	159	42	.	.	PUNCT
ejpam-6297	160	1	given	give	VERB
ejpam-6297	160	2	that	that	PRON
ejpam-6297	160	3	x	x	PRON
ejpam-6297	160	4	−	−	PROPN
ejpam-6297	160	5	(	(	PUNCT
ejpam-6297	160	6	sβ	sβ	PROPN
ejpam-6297	160	7	cli(a	cli(a	PROPN
ejpam-6297	160	8	)	)	PUNCT
ejpam-6297	160	9	−	−	PROPN
ejpam-6297	160	10	a	a	X
ejpam-6297	160	11	)	)	PUNCT
ejpam-6297	160	12	=	=	AUX
ejpam-6297	160	13	(	(	PUNCT
ejpam-6297	160	14	x	x	X
ejpam-6297	160	15	−	−	NOUN
ejpam-6297	160	16	sβ	sβ	PROPN
ejpam-6297	160	17	cli(a	cli(a	PROPN
ejpam-6297	160	18	)	)	PUNCT
ejpam-6297	160	19	)	)	PUNCT
ejpam-6297	160	20	∪	∪	ADP
ejpam-6297	160	21	a	a	X
ejpam-6297	160	22	,	,	PUNCT
ejpam-6297	160	23	it	it	PRON
ejpam-6297	160	24	follows	follow	VERB
ejpam-6297	160	25	that	that	SCONJ
ejpam-6297	160	26	a	a	DET
ejpam-6297	160	27	∪	∪	NOUN
ejpam-6297	160	28	(	(	PUNCT
ejpam-6297	160	29	x	x	NOUN
ejpam-6297	160	30	−	−	PROPN
ejpam-6297	160	31	sβ	sβ	PROPN
ejpam-6297	160	32	cli(a	cli(a	PROPN
ejpam-6297	160	33	)	)	PUNCT
ejpam-6297	160	34	)	)	PUNCT
ejpam-6297	160	35	is	be	AUX
ejpam-6297	160	36	strong	strong	ADJ
ejpam-6297	160	37	β	β	NOUN
ejpam-6297	160	38	-	-	ADJ
ejpam-6297	160	39	i	i	PRON
ejpam-6297	160	40	-	-	PUNCT
ejpam-6297	160	41	open	open	ADJ
ejpam-6297	160	42	.	.	PUNCT
ejpam-6297	161	1	(	(	PUNCT
ejpam-6297	161	2	4	4	X
ejpam-6297	161	3	)	)	PUNCT
ejpam-6297	161	4	⇒	⇒	NOUN
ejpam-6297	161	5	(	(	PUNCT
ejpam-6297	161	6	5	5	NUM
ejpam-6297	161	7	):	):	PUNCT
ejpam-6297	161	8	the	the	DET
ejpam-6297	161	9	evidence	evidence	NOUN
ejpam-6297	161	10	is	be	AUX
ejpam-6297	161	11	clear	clear	ADJ
ejpam-6297	161	12	.	.	PUNCT
ejpam-6297	162	1	(	(	PUNCT
ejpam-6297	162	2	5	5	X
ejpam-6297	162	3	)	)	PUNCT
ejpam-6297	162	4	⇒	⇒	NOUN
ejpam-6297	162	5	(	(	PUNCT
ejpam-6297	162	6	1	1	NUM
ejpam-6297	162	7	):	):	PUNCT
ejpam-6297	162	8	suppose	suppose	VERB
ejpam-6297	162	9	that	that	SCONJ
ejpam-6297	162	10	a	a	DET
ejpam-6297	162	11	⊆	⊆	NUM
ejpam-6297	162	12	sβ	sβ	PROPN
ejpam-6297	162	13	inti(a	inti(a	PROPN
ejpam-6297	162	14	∪	∪	ADV
ejpam-6297	162	15	(	(	PUNCT
ejpam-6297	162	16	x	x	X
ejpam-6297	162	17	−	−	PROPN
ejpam-6297	162	18	sβ	sβ	PROPN
ejpam-6297	162	19	cli(a	cli(a	PROPN
ejpam-6297	162	20	)	)	PUNCT
ejpam-6297	162	21	)	)	PUNCT
ejpam-6297	162	22	)	)	PUNCT
ejpam-6297	162	23	.	.	PUNCT
ejpam-6297	163	1	since	since	SCONJ
ejpam-6297	163	2	x	x	PRON
ejpam-6297	163	3	−	−	PROPN
ejpam-6297	163	4	sβ	sβ	PROPN
ejpam-6297	163	5	cli(a	cli(a	PROPN
ejpam-6297	163	6	)	)	PUNCT
ejpam-6297	163	7	is	be	AUX
ejpam-6297	163	8	strong	strong	ADJ
ejpam-6297	163	9	β	β	NOUN
ejpam-6297	163	10	-	-	ADJ
ejpam-6297	163	11	i	i	PRON
ejpam-6297	163	12	-	-	PUNCT
ejpam-6297	163	13	open	open	ADJ
ejpam-6297	163	14	,	,	PUNCT
ejpam-6297	163	15	we	we	PRON
ejpam-6297	163	16	have	have	VERB
ejpam-6297	163	17	x	x	INTJ
ejpam-6297	163	18	−	−	PROPN
ejpam-6297	163	19	sβ	sβ	PROPN
ejpam-6297	163	20	cli(a	cli(a	PROPN
ejpam-6297	163	21	)	)	PUNCT
ejpam-6297	163	22	=	=	PUNCT
ejpam-6297	164	1	sβ	sβ	PROPN
ejpam-6297	164	2	inti(x	inti(x	INTJ
ejpam-6297	164	3	−	−	PROPN
ejpam-6297	164	4	sβ	sβ	PROPN
ejpam-6297	164	5	cli(a	cli(a	PROPN
ejpam-6297	164	6	)	)	PUNCT
ejpam-6297	164	7	)	)	PUNCT
ejpam-6297	164	8	.	.	PUNCT
ejpam-6297	165	1	it	it	PRON
ejpam-6297	165	2	follows	follow	VERB
ejpam-6297	165	3	that	that	SCONJ
ejpam-6297	165	4	x	x	PUNCT
ejpam-6297	165	5	−	−	NOUN
ejpam-6297	165	6	sβ	sβ	PROPN
ejpam-6297	165	7	cli(a	cli(a	PROPN
ejpam-6297	165	8	)	)	PUNCT
ejpam-6297	165	9	=	=	PUNCT
ejpam-6297	166	1	sβ	sβ	PROPN
ejpam-6297	166	2	inti(x	inti(x	INTJ
ejpam-6297	166	3	−	−	PROPN
ejpam-6297	166	4	sβ	sβ	PROPN
ejpam-6297	166	5	cli(a	cli(a	PROPN
ejpam-6297	166	6	)	)	PUNCT
ejpam-6297	166	7	)	)	PUNCT
ejpam-6297	167	1	⊆	⊆	X
ejpam-6297	167	2	sβ	sβ	PROPN
ejpam-6297	167	3	inti(a	inti(a	PROPN
ejpam-6297	167	4	∪	∪	ADV
ejpam-6297	167	5	(	(	PUNCT
ejpam-6297	167	6	x	x	X
ejpam-6297	167	7	−	−	PROPN
ejpam-6297	167	8	sβ	sβ	PROPN
ejpam-6297	167	9	cli(a	cli(a	PROPN
ejpam-6297	167	10	)	)	PUNCT
ejpam-6297	167	11	)	)	PUNCT
ejpam-6297	167	12	)	)	PUNCT
ejpam-6297	167	13	.	.	PUNCT
ejpam-6297	168	1	therefore	therefore	ADV
ejpam-6297	168	2	,	,	PUNCT
ejpam-6297	168	3	a	a	DET
ejpam-6297	168	4	∪	∪	ADJ
ejpam-6297	168	5	(	(	PUNCT
ejpam-6297	168	6	x	x	NOUN
ejpam-6297	168	7	−	−	PROPN
ejpam-6297	168	8	sβ	sβ	PROPN
ejpam-6297	168	9	cli(a	cli(a	PROPN
ejpam-6297	168	10	)	)	PUNCT
ejpam-6297	168	11	)	)	PUNCT
ejpam-6297	169	1	⊆	⊆	X
ejpam-6297	169	2	sβ	sβ	PROPN
ejpam-6297	169	3	inti(a	inti(a	PROPN
ejpam-6297	169	4	∪	∪	ADV
ejpam-6297	169	5	(	(	PUNCT
ejpam-6297	169	6	x	x	X
ejpam-6297	169	7	−	−	PROPN
ejpam-6297	169	8	sβ	sβ	PROPN
ejpam-6297	169	9	cli(a	cli(a	PROPN
ejpam-6297	169	10	)	)	PUNCT
ejpam-6297	169	11	)	)	PUNCT
ejpam-6297	169	12	)	)	PUNCT
ejpam-6297	169	13	,	,	PUNCT
ejpam-6297	169	14	which	which	PRON
ejpam-6297	169	15	shows	show	VERB
ejpam-6297	169	16	that	that	SCONJ
ejpam-6297	169	17	a	a	DET
ejpam-6297	169	18	∪	∪	NOUN
ejpam-6297	169	19	(	(	PUNCT
ejpam-6297	169	20	x	x	NOUN
ejpam-6297	169	21	−	−	PROPN
ejpam-6297	169	22	sβ	sβ	PROPN
ejpam-6297	169	23	cli(a	cli(a	PROPN
ejpam-6297	169	24	)	)	PUNCT
ejpam-6297	169	25	)	)	PUNCT
ejpam-6297	169	26	is	be	AUX
ejpam-6297	169	27	strong	strong	ADJ
ejpam-6297	169	28	β	β	NOUN
ejpam-6297	169	29	-	-	ADJ
ejpam-6297	169	30	i	i	PRON
ejpam-6297	169	31	-	-	PUNCT
ejpam-6297	169	32	open	open	ADJ
ejpam-6297	169	33	.	.	PUNCT
ejpam-6297	170	1	since	since	SCONJ
ejpam-6297	170	2	a	a	PRON
ejpam-6297	170	3	=	=	X
ejpam-6297	170	4	(	(	PUNCT
ejpam-6297	170	5	a	a	DET
ejpam-6297	170	6	∪	∪	X
ejpam-6297	170	7	(	(	PUNCT
ejpam-6297	170	8	x	x	SYM
ejpam-6297	170	9	−	−	PROPN
ejpam-6297	170	10	sβ	sβ	PROPN
ejpam-6297	170	11	cli(a	cli(a	PROPN
ejpam-6297	170	12	)	)	PUNCT
ejpam-6297	170	13	)	)	PUNCT
ejpam-6297	170	14	)	)	PUNCT
ejpam-6297	170	15	∩	∩	PROPN
ejpam-6297	170	16	sβ	sβ	PROPN
ejpam-6297	170	17	cli(a	cli(a	PROPN
ejpam-6297	170	18	)	)	PUNCT
ejpam-6297	170	19	,	,	PUNCT
ejpam-6297	170	20	it	it	PRON
ejpam-6297	170	21	follows	follow	VERB
ejpam-6297	170	22	that	that	SCONJ
ejpam-6297	170	23	a	a	PRON
ejpam-6297	170	24	is	be	AUX
ejpam-6297	170	25	locally	locally	ADV
ejpam-6297	170	26	strong	strong	ADJ
ejpam-6297	170	27	β	β	X
ejpam-6297	170	28	-	-	ADJ
ejpam-6297	170	29	i	i	NOUN
ejpam-6297	170	30	-	-	PUNCT
ejpam-6297	170	31	closed	closed	ADJ
ejpam-6297	170	32	.	.	PUNCT
ejpam-6297	171	1	the	the	DET
ejpam-6297	171	2	subsequent	subsequent	ADJ
ejpam-6297	171	3	theorem	theorem	NOUN
ejpam-6297	171	4	presents	present	VERB
ejpam-6297	171	5	five	five	NUM
ejpam-6297	171	6	mutually	mutually	ADV
ejpam-6297	171	7	equivalent	equivalent	ADJ
ejpam-6297	171	8	conditions	condition	NOUN
ejpam-6297	171	9	,	,	PUNCT
ejpam-6297	171	10	each	each	PRON
ejpam-6297	171	11	involving	involve	VERB
ejpam-6297	171	12	strong	strong	ADJ
ejpam-6297	171	13	β	β	X
ejpam-6297	171	14	-	-	ADJ
ejpam-6297	171	15	i	i	NOUN
ejpam-6297	171	16	-	-	PUNCT
ejpam-6297	171	17	closed	close	VERB
ejpam-6297	171	18	sets	set	NOUN
ejpam-6297	171	19	,	,	PUNCT
ejpam-6297	171	20	strong	strong	ADJ
ejpam-6297	171	21	β	β	X
ejpam-6297	171	22	-	-	ADJ
ejpam-6297	171	23	i	i	NOUN
ejpam-6297	171	24	-	-	PUNCT
ejpam-6297	171	25	dense	dense	ADJ
ejpam-6297	171	26	sets	set	NOUN
ejpam-6297	171	27	,	,	PUNCT
ejpam-6297	171	28	and	and	CCONJ
ejpam-6297	171	29	b	b	X
ejpam-6297	171	30	-	-	PUNCT
ejpam-6297	171	31	sisets	siset	NOUN
ejpam-6297	171	32	,	,	PUNCT
ejpam-6297	171	33	which	which	PRON
ejpam-6297	171	34	together	together	ADV
ejpam-6297	171	35	provide	provide	VERB
ejpam-6297	171	36	a	a	DET
ejpam-6297	171	37	comprehensive	comprehensive	ADJ
ejpam-6297	171	38	characterization	characterization	NOUN
ejpam-6297	171	39	of	of	ADP
ejpam-6297	171	40	when	when	SCONJ
ejpam-6297	171	41	an	an	DET
ejpam-6297	171	42	ideal	ideal	ADJ
ejpam-6297	171	43	topological	topological	ADJ
ejpam-6297	171	44	space	space	NOUN
ejpam-6297	171	45	qualifies	qualifie	NOUN
ejpam-6297	171	46	as	as	ADP
ejpam-6297	171	47	strong	strong	ADJ
ejpam-6297	171	48	β	β	NOUN
ejpam-6297	171	49	-	-	NOUN
ejpam-6297	171	50	isubmaximal	isubmaximal	ADJ
ejpam-6297	171	51	.	.	PUNCT
ejpam-6297	172	1	to	to	PART
ejpam-6297	172	2	proceed	proceed	VERB
ejpam-6297	172	3	,	,	PUNCT
ejpam-6297	172	4	we	we	PRON
ejpam-6297	172	5	begin	begin	VERB
ejpam-6297	172	6	by	by	ADP
ejpam-6297	172	7	introducing	introduce	VERB
ejpam-6297	172	8	the	the	DET
ejpam-6297	172	9	concept	concept	NOUN
ejpam-6297	172	10	of	of	ADP
ejpam-6297	172	11	a	a	DET
ejpam-6297	172	12	b	b	NOUN
ejpam-6297	172	13	-	-	PUNCT
ejpam-6297	172	14	si	si	NOUN
ejpam-6297	172	15	set	set	NOUN
ejpam-6297	172	16	within	within	ADP
ejpam-6297	172	17	the	the	DET
ejpam-6297	172	18	context	context	NOUN
ejpam-6297	172	19	of	of	ADP
ejpam-6297	172	20	an	an	DET
ejpam-6297	172	21	ideal	ideal	ADJ
ejpam-6297	172	22	topological	topological	ADJ
ejpam-6297	172	23	space	space	NOUN
ejpam-6297	172	24	.	.	PUNCT
ejpam-6297	173	1	definition	definition	NOUN
ejpam-6297	173	2	6	6	NUM
ejpam-6297	173	3	.	.	PUNCT
ejpam-6297	174	1	a	a	DET
ejpam-6297	174	2	subset	subset	NOUN
ejpam-6297	174	3	a	a	PRON
ejpam-6297	174	4	of	of	ADP
ejpam-6297	174	5	an	an	DET
ejpam-6297	174	6	ideal	ideal	ADJ
ejpam-6297	174	7	topological	topological	ADJ
ejpam-6297	174	8	space	space	NOUN
ejpam-6297	174	9	(	(	PUNCT
ejpam-6297	174	10	x	x	X
ejpam-6297	174	11	,	,	PUNCT
ejpam-6297	174	12	τ	τ	PROPN
ejpam-6297	174	13	,	,	PUNCT
ejpam-6297	174	14	i	i	PROPN
ejpam-6297	174	15	)	)	PUNCT
ejpam-6297	174	16	is	be	AUX
ejpam-6297	174	17	called	call	VERB
ejpam-6297	174	18	a	a	DET
ejpam-6297	174	19	b	b	NOUN
ejpam-6297	174	20	-	-	PUNCT
ejpam-6297	174	21	si	si	NOUN
ejpam-6297	174	22	set	set	NOUN
ejpam-6297	174	23	if	if	SCONJ
ejpam-6297	174	24	it	it	PRON
ejpam-6297	174	25	can	can	AUX
ejpam-6297	174	26	be	be	AUX
ejpam-6297	174	27	written	write	VERB
ejpam-6297	174	28	as	as	ADP
ejpam-6297	174	29	a	a	DET
ejpam-6297	174	30	=	=	SYM
ejpam-6297	174	31	u	u	NOUN
ejpam-6297	174	32	∩	∩	NOUN
ejpam-6297	174	33	v	v	NOUN
ejpam-6297	174	34	,	,	PUNCT
ejpam-6297	174	35	where	where	SCONJ
ejpam-6297	174	36	u	u	NOUN
ejpam-6297	174	37	is	be	AUX
ejpam-6297	174	38	a	a	DET
ejpam-6297	174	39	strong	strong	ADJ
ejpam-6297	174	40	β	β	X
ejpam-6297	174	41	-	-	ADJ
ejpam-6297	174	42	i	i	NOUN
ejpam-6297	174	43	-	-	PUNCT
ejpam-6297	174	44	open	open	ADJ
ejpam-6297	174	45	set	set	NOUN
ejpam-6297	174	46	,	,	PUNCT
ejpam-6297	174	47	and	and	CCONJ
ejpam-6297	174	48	v	v	NOUN
ejpam-6297	174	49	is	be	AUX
ejpam-6297	174	50	a	a	DET
ejpam-6297	174	51	set	set	NOUN
ejpam-6297	174	52	that	that	PRON
ejpam-6297	174	53	satisfies	satisfy	VERB
ejpam-6297	174	54	the	the	DET
ejpam-6297	174	55	condition	condition	NOUN
ejpam-6297	174	56	sβ	sβ	X
ejpam-6297	174	57	inti(v	inti(v	NOUN
ejpam-6297	174	58	)	)	PUNCT
ejpam-6297	175	1	=	=	SYM
ejpam-6297	175	2	sβ	sβ	PROPN
ejpam-6297	175	3	inti(sβ	inti(sβ	PROPN
ejpam-6297	175	4	cli(v	cli(v	PROPN
ejpam-6297	175	5	)	)	PUNCT
ejpam-6297	175	6	)	)	PUNCT
ejpam-6297	175	7	.	.	PUNCT
ejpam-6297	176	1	theorem	theorem	NOUN
ejpam-6297	176	2	2	2	NUM
ejpam-6297	176	3	.	.	X
ejpam-6297	176	4	for	for	ADP
ejpam-6297	176	5	an	an	DET
ejpam-6297	176	6	ideal	ideal	ADJ
ejpam-6297	176	7	topological	topological	ADJ
ejpam-6297	176	8	space	space	NOUN
ejpam-6297	176	9	(	(	PUNCT
ejpam-6297	176	10	x	x	X
ejpam-6297	176	11	,	,	PUNCT
ejpam-6297	176	12	τ	τ	PROPN
ejpam-6297	176	13	,	,	PUNCT
ejpam-6297	176	14	i	i	PROPN
ejpam-6297	176	15	)	)	PUNCT
ejpam-6297	176	16	,	,	PUNCT
ejpam-6297	176	17	the	the	DET
ejpam-6297	176	18	following	follow	VERB
ejpam-6297	176	19	properties	property	NOUN
ejpam-6297	176	20	are	be	AUX
ejpam-6297	176	21	equivalent	equivalent	ADJ
ejpam-6297	176	22	:	:	PUNCT
ejpam-6297	176	23	(	(	PUNCT
ejpam-6297	176	24	1	1	X
ejpam-6297	176	25	)	)	PUNCT
ejpam-6297	176	26	(	(	PUNCT
ejpam-6297	176	27	x	x	X
ejpam-6297	176	28	,	,	PUNCT
ejpam-6297	176	29	τ	τ	PROPN
ejpam-6297	176	30	,	,	PUNCT
ejpam-6297	176	31	i	i	PROPN
ejpam-6297	176	32	)	)	PUNCT
ejpam-6297	176	33	is	be	AUX
ejpam-6297	176	34	strong	strong	ADJ
ejpam-6297	176	35	β	β	NOUN
ejpam-6297	176	36	-	-	ADJ
ejpam-6297	176	37	i	i	NOUN
ejpam-6297	176	38	-	-	PUNCT
ejpam-6297	176	39	submaximal	submaximal	ADJ
ejpam-6297	176	40	;	;	PUNCT
ejpam-6297	176	41	(	(	PUNCT
ejpam-6297	176	42	2	2	X
ejpam-6297	176	43	)	)	PUNCT
ejpam-6297	176	44	for	for	ADP
ejpam-6297	176	45	every	every	DET
ejpam-6297	176	46	subset	subset	NOUN
ejpam-6297	176	47	a	a	PRON
ejpam-6297	176	48	of	of	ADP
ejpam-6297	176	49	x	x	PRON
ejpam-6297	176	50	,	,	PUNCT
ejpam-6297	176	51	sβ	sβ	PROPN
ejpam-6297	176	52	cli(a)−a	cli(a)−a	PROPN
ejpam-6297	176	53	is	be	AUX
ejpam-6297	176	54	strong	strong	ADJ
ejpam-6297	176	55	β	β	NOUN
ejpam-6297	176	56	-	-	ADJ
ejpam-6297	176	57	i	i	NOUN
ejpam-6297	176	58	-	-	PUNCT
ejpam-6297	176	59	closed	closed	ADJ
ejpam-6297	176	60	;	;	PUNCT
ejpam-6297	176	61	(	(	PUNCT
ejpam-6297	176	62	3	3	X
ejpam-6297	176	63	)	)	PUNCT
ejpam-6297	176	64	every	every	DET
ejpam-6297	176	65	subset	subset	NOUN
ejpam-6297	176	66	of	of	ADP
ejpam-6297	176	67	x	x	PUNCT
ejpam-6297	176	68	is	be	AUX
ejpam-6297	176	69	locally	locally	ADV
ejpam-6297	176	70	strong	strong	ADJ
ejpam-6297	176	71	β	β	X
ejpam-6297	176	72	-	-	ADJ
ejpam-6297	176	73	i	i	NOUN
ejpam-6297	176	74	-	-	PUNCT
ejpam-6297	176	75	closed	closed	ADJ
ejpam-6297	176	76	;	;	PUNCT
ejpam-6297	176	77	(	(	PUNCT
ejpam-6297	176	78	4	4	X
ejpam-6297	176	79	)	)	PUNCT
ejpam-6297	176	80	every	every	DET
ejpam-6297	176	81	subset	subset	NOUN
ejpam-6297	176	82	of	of	ADP
ejpam-6297	176	83	x	x	PUNCT
ejpam-6297	176	84	is	be	AUX
ejpam-6297	176	85	a	a	DET
ejpam-6297	176	86	b	b	PROPN
ejpam-6297	176	87	-	-	PUNCT
ejpam-6297	176	88	si	si	NOUN
ejpam-6297	176	89	set	set	NOUN
ejpam-6297	176	90	;	;	PUNCT
ejpam-6297	176	91	and	and	CCONJ
ejpam-6297	176	92	(	(	PUNCT
ejpam-6297	176	93	5	5	X
ejpam-6297	176	94	)	)	PUNCT
ejpam-6297	176	95	every	every	PRON
ejpam-6297	176	96	strong	strong	ADJ
ejpam-6297	176	97	β	β	X
ejpam-6297	176	98	-	-	ADJ
ejpam-6297	176	99	i	i	NOUN
ejpam-6297	176	100	-	-	PUNCT
ejpam-6297	176	101	dense	dense	ADJ
ejpam-6297	176	102	subset	subset	NOUN
ejpam-6297	176	103	of	of	ADP
ejpam-6297	176	104	x	x	PUNCT
ejpam-6297	176	105	is	be	AUX
ejpam-6297	176	106	a	a	DET
ejpam-6297	176	107	b	b	PROPN
ejpam-6297	176	108	-	-	PUNCT
ejpam-6297	176	109	si	si	NOUN
ejpam-6297	176	110	set	set	NOUN
ejpam-6297	176	111	.	.	PUNCT
ejpam-6297	177	1	proof	proof	NOUN
ejpam-6297	177	2	.	.	PUNCT
ejpam-6297	178	1	(	(	PUNCT
ejpam-6297	178	2	1	1	X
ejpam-6297	178	3	)	)	PUNCT
ejpam-6297	178	4	⇒	⇒	NOUN
ejpam-6297	178	5	(	(	PUNCT
ejpam-6297	178	6	2	2	NUM
ejpam-6297	178	7	):	):	PUNCT
ejpam-6297	178	8	let	let	VERB
ejpam-6297	178	9	(	(	PUNCT
ejpam-6297	178	10	x	x	NOUN
ejpam-6297	178	11	,	,	PUNCT
ejpam-6297	178	12	τ	τ	PROPN
ejpam-6297	178	13	,	,	PUNCT
ejpam-6297	178	14	i	i	PRON
ejpam-6297	178	15	)	)	PUNCT
ejpam-6297	178	16	be	be	AUX
ejpam-6297	178	17	strong	strong	ADJ
ejpam-6297	178	18	β	β	NOUN
ejpam-6297	178	19	-	-	ADJ
ejpam-6297	178	20	i	i	PRON
ejpam-6297	178	21	-	-	PUNCT
ejpam-6297	178	22	submaximal	submaximal	ADJ
ejpam-6297	178	23	and	and	CCONJ
ejpam-6297	178	24	let	let	VERB
ejpam-6297	178	25	a	a	DET
ejpam-6297	178	26	⊆	⊆	NUM
ejpam-6297	178	27	x.	x.	NOUN
ejpam-6297	178	28	as	as	ADP
ejpam-6297	178	29	x	x	PROPN
ejpam-6297	178	30	=	=	VERB
ejpam-6297	178	31	sβ	sβ	PROPN
ejpam-6297	178	32	cli(a	cli(a	PROPN
ejpam-6297	178	33	)	)	PUNCT
ejpam-6297	178	34	∪	∪	NOUN
ejpam-6297	178	35	(	(	PUNCT
ejpam-6297	178	36	x	x	SYM
ejpam-6297	178	37	−	−	PROPN
ejpam-6297	178	38	sβ	sβ	PROPN
ejpam-6297	178	39	cli(a	cli(a	PROPN
ejpam-6297	178	40	)	)	PUNCT
ejpam-6297	178	41	)	)	PUNCT
ejpam-6297	179	1	⊆	⊆	X
ejpam-6297	179	2	sβ	sβ	PROPN
ejpam-6297	179	3	cli(a	cli(a	PROPN
ejpam-6297	179	4	)	)	PUNCT
ejpam-6297	179	5	∪	∪	NOUN
ejpam-6297	179	6	(	(	PUNCT
ejpam-6297	179	7	x	x	X
ejpam-6297	179	8	−	−	NOUN
ejpam-6297	179	9	sβ	sβ	PROPN
ejpam-6297	179	10	inti(sβ	inti(sβ	PROPN
ejpam-6297	179	11	cli(a	cli(a	PROPN
ejpam-6297	179	12	)	)	PUNCT
ejpam-6297	179	13	)	)	PUNCT
ejpam-6297	179	14	c.	c.	PROPN
ejpam-6297	179	15	boonpok	boonpok	PROPN
ejpam-6297	179	16	,	,	PUNCT
ejpam-6297	179	17	p.	p.	PROPN
ejpam-6297	179	18	raktaow	raktaow	NOUN
ejpam-6297	179	19	,	,	PUNCT
ejpam-6297	179	20	a.	a.	PROPN
ejpam-6297	179	21	sama	sama	PROPN
ejpam-6297	179	22	-	-	PUNCT
ejpam-6297	179	23	ae	ae	PROPN
ejpam-6297	179	24	/	/	SYM
ejpam-6297	179	25	eur	eur	PROPN
ejpam-6297	179	26	.	.	PUNCT
ejpam-6297	180	1	j.	j.	PROPN
ejpam-6297	180	2	pure	pure	PROPN
ejpam-6297	180	3	appl	appl	PROPN
ejpam-6297	180	4	.	.	PROPN
ejpam-6297	180	5	math	math	PROPN
ejpam-6297	180	6	,	,	PUNCT
ejpam-6297	180	7	18	18	NUM
ejpam-6297	180	8	(	(	PUNCT
ejpam-6297	180	9	3	3	NUM
ejpam-6297	180	10	)	)	PUNCT
ejpam-6297	180	11	(	(	PUNCT
ejpam-6297	180	12	2025	2025	NUM
ejpam-6297	180	13	)	)	PUNCT
ejpam-6297	180	14	,	,	PUNCT
ejpam-6297	180	15	6297	6297	NUM
ejpam-6297	180	16	8	8	NUM
ejpam-6297	180	17	of	of	ADP
ejpam-6297	180	18	23	23	NUM
ejpam-6297	180	19	=	=	SYM
ejpam-6297	180	20	sβ	sβ	PROPN
ejpam-6297	180	21	cli(a	cli(a	PROPN
ejpam-6297	180	22	)	)	PUNCT
ejpam-6297	180	23	∪	∪	NOUN
ejpam-6297	180	24	sβ	sβ	PRON
ejpam-6297	180	25	cli(x	cli(x	NOUN
ejpam-6297	180	26	−	−	PROPN
ejpam-6297	180	27	sβ	sβ	PROPN
ejpam-6297	180	28	cli(a	cli(a	PROPN
ejpam-6297	180	29	)	)	PUNCT
ejpam-6297	180	30	)	)	PUNCT
ejpam-6297	181	1	⊆	⊆	X
ejpam-6297	181	2	sβ	sβ	PRON
ejpam-6297	181	3	cli(a	cli(a	PROPN
ejpam-6297	181	4	∪	∪	ADV
ejpam-6297	182	1	(	(	PUNCT
ejpam-6297	182	2	x	x	X
ejpam-6297	182	3	−	−	PROPN
ejpam-6297	182	4	sβ	sβ	PROPN
ejpam-6297	182	5	cli(a	cli(a	PROPN
ejpam-6297	182	6	)	)	PUNCT
ejpam-6297	182	7	)	)	PUNCT
ejpam-6297	182	8	)	)	PUNCT
ejpam-6297	183	1	=	=	PRON
ejpam-6297	183	2	sβ	sβ	PROPN
ejpam-6297	183	3	cli(x	cli(x	NOUN
ejpam-6297	183	4	−	−	PROPN
ejpam-6297	183	5	(	(	PUNCT
ejpam-6297	183	6	sβ	sβ	NOUN
ejpam-6297	183	7	cli(a)−a	cli(a)−a	NUM
ejpam-6297	183	8	)	)	PUNCT
ejpam-6297	183	9	)	)	PUNCT
ejpam-6297	183	10	,	,	PUNCT
ejpam-6297	183	11	we	we	PRON
ejpam-6297	183	12	have	have	VERB
ejpam-6297	183	13	sβ	sβ	NUM
ejpam-6297	183	14	cli(x	cli(x	NOUN
ejpam-6297	184	1	−	−	PROPN
ejpam-6297	184	2	(	(	PUNCT
ejpam-6297	184	3	sβ	sβ	PROPN
ejpam-6297	184	4	cli(a	cli(a	PROPN
ejpam-6297	184	5	)	)	PUNCT
ejpam-6297	184	6	−	−	PROPN
ejpam-6297	184	7	a	a	NOUN
ejpam-6297	184	8	)	)	PUNCT
ejpam-6297	184	9	)	)	PUNCT
ejpam-6297	185	1	=	=	PUNCT
ejpam-6297	185	2	x	x	PUNCT
ejpam-6297	185	3	and	and	CCONJ
ejpam-6297	185	4	hence	hence	ADV
ejpam-6297	185	5	x	x	X
ejpam-6297	185	6	−	−	PROPN
ejpam-6297	185	7	(	(	PUNCT
ejpam-6297	185	8	sβ	sβ	PROPN
ejpam-6297	185	9	cli(a	cli(a	PROPN
ejpam-6297	185	10	)	)	PUNCT
ejpam-6297	185	11	−	−	PROPN
ejpam-6297	185	12	a	a	PRON
ejpam-6297	185	13	)	)	PUNCT
ejpam-6297	185	14	is	be	AUX
ejpam-6297	185	15	strong	strong	ADJ
ejpam-6297	185	16	β	β	NOUN
ejpam-6297	185	17	-	-	NOUN
ejpam-6297	185	18	idense	idense	NOUN
ejpam-6297	185	19	.	.	PUNCT
ejpam-6297	186	1	according	accord	VERB
ejpam-6297	186	2	to	to	ADP
ejpam-6297	186	3	the	the	DET
ejpam-6297	186	4	hypothesis	hypothesis	NOUN
ejpam-6297	186	5	,	,	PUNCT
ejpam-6297	186	6	x−(sβ	x−(sβ	PROPN
ejpam-6297	186	7	cli(a)−a	cli(a)−a	PROPN
ejpam-6297	186	8	)	)	PUNCT
ejpam-6297	186	9	is	be	AUX
ejpam-6297	186	10	strong	strong	ADJ
ejpam-6297	186	11	β	β	NOUN
ejpam-6297	186	12	-	-	ADJ
ejpam-6297	186	13	i	i	PRON
ejpam-6297	186	14	-	-	PUNCT
ejpam-6297	186	15	open	open	ADJ
ejpam-6297	186	16	.	.	PUNCT
ejpam-6297	187	1	consequently	consequently	ADV
ejpam-6297	187	2	,	,	PUNCT
ejpam-6297	187	3	sβ	sβ	PROPN
ejpam-6297	187	4	cli(a)−a	cli(a)−a	PROPN
ejpam-6297	187	5	is	be	AUX
ejpam-6297	187	6	strong	strong	ADJ
ejpam-6297	187	7	β	β	NOUN
ejpam-6297	187	8	-	-	ADJ
ejpam-6297	187	9	i	i	NOUN
ejpam-6297	187	10	-	-	PUNCT
ejpam-6297	187	11	closed	closed	ADJ
ejpam-6297	187	12	.	.	PUNCT
ejpam-6297	188	1	(	(	PUNCT
ejpam-6297	188	2	2	2	X
ejpam-6297	188	3	)	)	PUNCT
ejpam-6297	188	4	⇒	⇒	NOUN
ejpam-6297	188	5	(	(	PUNCT
ejpam-6297	188	6	3	3	NUM
ejpam-6297	188	7	):	):	PUNCT
ejpam-6297	188	8	by	by	ADP
ejpam-6297	188	9	theorem	theorem	NOUN
ejpam-6297	188	10	1	1	NUM
ejpam-6297	188	11	,	,	PUNCT
ejpam-6297	188	12	a	a	DET
ejpam-6297	188	13	subset	subset	NOUN
ejpam-6297	188	14	a	a	PRON
ejpam-6297	188	15	is	be	AUX
ejpam-6297	188	16	locally	locally	ADV
ejpam-6297	188	17	strong	strong	ADJ
ejpam-6297	188	18	β	β	X
ejpam-6297	188	19	-	-	ADJ
ejpam-6297	188	20	i	i	NOUN
ejpam-6297	188	21	-	-	PUNCT
ejpam-6297	188	22	closed	close	VERB
ejpam-6297	188	23	if	if	SCONJ
ejpam-6297	189	1	and	and	CCONJ
ejpam-6297	189	2	only	only	ADV
ejpam-6297	189	3	if	if	SCONJ
ejpam-6297	189	4	sβ	sβ	NUM
ejpam-6297	189	5	cli(a)−a	cli(a)−a	NOUN
ejpam-6297	189	6	is	be	AUX
ejpam-6297	189	7	strong	strong	ADJ
ejpam-6297	189	8	β	β	NOUN
ejpam-6297	189	9	-	-	ADJ
ejpam-6297	189	10	i	i	NOUN
ejpam-6297	189	11	-	-	PUNCT
ejpam-6297	189	12	closed	closed	ADJ
ejpam-6297	189	13	.	.	PUNCT
ejpam-6297	190	1	(	(	PUNCT
ejpam-6297	190	2	3	3	X
ejpam-6297	190	3	)	)	PUNCT
ejpam-6297	190	4	⇒	⇒	NOUN
ejpam-6297	190	5	(	(	PUNCT
ejpam-6297	190	6	4	4	NUM
ejpam-6297	190	7	):	):	PUNCT
ejpam-6297	190	8	if	if	SCONJ
ejpam-6297	190	9	every	every	DET
ejpam-6297	190	10	subset	subset	NOUN
ejpam-6297	190	11	of	of	ADP
ejpam-6297	190	12	x	x	PUNCT
ejpam-6297	190	13	is	be	AUX
ejpam-6297	190	14	locally	locally	ADV
ejpam-6297	190	15	strong	strong	ADJ
ejpam-6297	190	16	β	β	X
ejpam-6297	190	17	-	-	ADJ
ejpam-6297	190	18	i	i	NOUN
ejpam-6297	190	19	-	-	PUNCT
ejpam-6297	190	20	closed	closed	ADJ
ejpam-6297	190	21	,	,	PUNCT
ejpam-6297	190	22	then	then	ADV
ejpam-6297	190	23	for	for	ADP
ejpam-6297	190	24	any	any	DET
ejpam-6297	190	25	a	a	DET
ejpam-6297	190	26	⊆	⊆	NUM
ejpam-6297	190	27	x	x	SYM
ejpam-6297	190	28	,	,	PUNCT
ejpam-6297	190	29	we	we	PRON
ejpam-6297	190	30	can	can	AUX
ejpam-6297	190	31	write	write	VERB
ejpam-6297	190	32	a	a	DET
ejpam-6297	190	33	=	=	X
ejpam-6297	190	34	u	u	NOUN
ejpam-6297	190	35	∩	∩	NOUN
ejpam-6297	190	36	v	v	ADP
ejpam-6297	190	37	where	where	SCONJ
ejpam-6297	190	38	u	u	NOUN
ejpam-6297	190	39	is	be	AUX
ejpam-6297	190	40	strong	strong	ADJ
ejpam-6297	190	41	β	β	NOUN
ejpam-6297	190	42	-	-	ADJ
ejpam-6297	190	43	i	i	PRON
ejpam-6297	190	44	-	-	PUNCT
ejpam-6297	190	45	open	open	ADJ
ejpam-6297	190	46	and	and	CCONJ
ejpam-6297	190	47	v	v	NOUN
ejpam-6297	190	48	is	be	AUX
ejpam-6297	190	49	strong	strong	ADJ
ejpam-6297	190	50	β	β	NOUN
ejpam-6297	190	51	-	-	ADJ
ejpam-6297	190	52	i	i	NOUN
ejpam-6297	190	53	-	-	PUNCT
ejpam-6297	190	54	closed	closed	ADJ
ejpam-6297	190	55	.	.	PUNCT
ejpam-6297	191	1	since	since	SCONJ
ejpam-6297	191	2	v	v	NUM
ejpam-6297	191	3	=	=	SYM
ejpam-6297	191	4	sβ	sβ	NOUN
ejpam-6297	191	5	cli(v	cli(v	PROPN
ejpam-6297	191	6	)	)	PUNCT
ejpam-6297	191	7	,	,	PUNCT
ejpam-6297	191	8	we	we	PRON
ejpam-6297	191	9	get	get	VERB
ejpam-6297	191	10	sβ	sβ	NOUN
ejpam-6297	191	11	inti(v	inti(v	NOUN
ejpam-6297	191	12	)	)	PUNCT
ejpam-6297	192	1	=	=	SYM
ejpam-6297	192	2	sβ	sβ	PROPN
ejpam-6297	192	3	inti(sβ	inti(sβ	PROPN
ejpam-6297	192	4	cli(v	cli(v	PROPN
ejpam-6297	192	5	)	)	PUNCT
ejpam-6297	192	6	)	)	PUNCT
ejpam-6297	192	7	.	.	PUNCT
ejpam-6297	193	1	therefore	therefore	ADV
ejpam-6297	193	2	a	a	PRON
ejpam-6297	193	3	is	be	AUX
ejpam-6297	193	4	a	a	DET
ejpam-6297	193	5	b	b	PROPN
ejpam-6297	193	6	-	-	PUNCT
ejpam-6297	193	7	si	si	NOUN
ejpam-6297	193	8	set	set	NOUN
ejpam-6297	193	9	.	.	PUNCT
ejpam-6297	194	1	(	(	PUNCT
ejpam-6297	194	2	4	4	X
ejpam-6297	194	3	)	)	PUNCT
ejpam-6297	194	4	⇒	⇒	NOUN
ejpam-6297	194	5	(	(	PUNCT
ejpam-6297	194	6	5	5	NUM
ejpam-6297	194	7	):	):	PUNCT
ejpam-6297	194	8	it	it	PRON
ejpam-6297	194	9	is	be	AUX
ejpam-6297	194	10	obvious	obvious	ADJ
ejpam-6297	194	11	.	.	PUNCT
ejpam-6297	195	1	(	(	PUNCT
ejpam-6297	195	2	5	5	X
ejpam-6297	195	3	)	)	PUNCT
ejpam-6297	195	4	⇒	⇒	NOUN
ejpam-6297	195	5	(	(	PUNCT
ejpam-6297	195	6	1	1	NUM
ejpam-6297	195	7	):	):	PUNCT
ejpam-6297	195	8	let	let	VERB
ejpam-6297	195	9	a	a	PRON
ejpam-6297	195	10	be	be	AUX
ejpam-6297	195	11	a	a	DET
ejpam-6297	195	12	strong	strong	ADJ
ejpam-6297	195	13	β	β	NOUN
ejpam-6297	195	14	-	-	ADJ
ejpam-6297	195	15	i	i	NOUN
ejpam-6297	195	16	-	-	PUNCT
ejpam-6297	195	17	dense	dense	ADJ
ejpam-6297	195	18	subset	subset	NOUN
ejpam-6297	195	19	,	,	PUNCT
ejpam-6297	195	20	and	and	CCONJ
ejpam-6297	195	21	assume	assume	VERB
ejpam-6297	195	22	that	that	SCONJ
ejpam-6297	195	23	every	every	DET
ejpam-6297	195	24	strong	strong	ADJ
ejpam-6297	195	25	β	β	NOUN
ejpam-6297	195	26	-	-	PUNCT
ejpam-6297	195	27	idense	idense	NOUN
ejpam-6297	195	28	subset	subset	NOUN
ejpam-6297	195	29	of	of	ADP
ejpam-6297	195	30	x	x	PUNCT
ejpam-6297	195	31	is	be	AUX
ejpam-6297	195	32	a	a	DET
ejpam-6297	195	33	b	b	PROPN
ejpam-6297	195	34	-	-	PUNCT
ejpam-6297	195	35	si	si	ADJ
ejpam-6297	195	36	set	set	NOUN
ejpam-6297	195	37	.	.	PUNCT
ejpam-6297	196	1	then	then	ADV
ejpam-6297	196	2	a	a	PRON
ejpam-6297	196	3	can	can	AUX
ejpam-6297	196	4	be	be	AUX
ejpam-6297	196	5	expressed	express	VERB
ejpam-6297	196	6	as	as	ADP
ejpam-6297	196	7	a	a	DET
ejpam-6297	196	8	=	=	SYM
ejpam-6297	196	9	u	u	NOUN
ejpam-6297	196	10	∩	∩	NOUN
ejpam-6297	196	11	v	v	NOUN
ejpam-6297	196	12	,	,	PUNCT
ejpam-6297	196	13	where	where	SCONJ
ejpam-6297	196	14	u	u	NOUN
ejpam-6297	196	15	is	be	AUX
ejpam-6297	196	16	a	a	DET
ejpam-6297	196	17	strong	strong	ADJ
ejpam-6297	196	18	β	β	X
ejpam-6297	196	19	-	-	ADJ
ejpam-6297	196	20	i	i	PRON
ejpam-6297	196	21	-	-	PUNCT
ejpam-6297	196	22	open	open	ADJ
ejpam-6297	196	23	set	set	NOUN
ejpam-6297	196	24	and	and	CCONJ
ejpam-6297	196	25	v	v	NOUN
ejpam-6297	196	26	is	be	AUX
ejpam-6297	196	27	a	a	DET
ejpam-6297	196	28	set	set	NOUN
ejpam-6297	196	29	satisfying	satisfy	VERB
ejpam-6297	196	30	sβ	sβ	X
ejpam-6297	196	31	inti(v	inti(v	NOUN
ejpam-6297	196	32	)	)	PUNCT
ejpam-6297	196	33	=	=	SYM
ejpam-6297	197	1	sβ	sβ	PROPN
ejpam-6297	197	2	inti(sβ	inti(sβ	PROPN
ejpam-6297	197	3	cli(v	cli(v	PROPN
ejpam-6297	197	4	)	)	PUNCT
ejpam-6297	197	5	)	)	PUNCT
ejpam-6297	197	6	.	.	PUNCT
ejpam-6297	198	1	given	give	VERB
ejpam-6297	198	2	that	that	SCONJ
ejpam-6297	198	3	a	a	DET
ejpam-6297	198	4	⊆	⊆	NUM
ejpam-6297	198	5	v	v	NOUN
ejpam-6297	198	6	,	,	PUNCT
ejpam-6297	198	7	it	it	PRON
ejpam-6297	198	8	implies	imply	VERB
ejpam-6297	198	9	that	that	SCONJ
ejpam-6297	198	10	x	x	X
ejpam-6297	198	11	=	=	PRON
ejpam-6297	198	12	sβ	sβ	PROPN
ejpam-6297	198	13	cli(a	cli(a	PROPN
ejpam-6297	198	14	)	)	PUNCT
ejpam-6297	198	15	⊆	⊆	NUM
ejpam-6297	198	16	sβ	sβ	NUM
ejpam-6297	198	17	cli(v	cli(v	PROPN
ejpam-6297	198	18	)	)	PUNCT
ejpam-6297	198	19	,	,	PUNCT
ejpam-6297	198	20	and	and	CCONJ
ejpam-6297	198	21	therefore	therefore	ADV
ejpam-6297	198	22	,	,	PUNCT
ejpam-6297	198	23	x	x	X
ejpam-6297	198	24	=	=	PUNCT
ejpam-6297	198	25	sβ	sβ	NUM
ejpam-6297	198	26	cli(v	cli(v	PROPN
ejpam-6297	198	27	)	)	PUNCT
ejpam-6297	198	28	.	.	PUNCT
ejpam-6297	199	1	thus	thus	ADV
ejpam-6297	199	2	x	x	X
ejpam-6297	199	3	=	=	SYM
ejpam-6297	199	4	sβ	sβ	NUM
ejpam-6297	199	5	inti(x	inti(x	NOUN
ejpam-6297	199	6	)	)	PUNCT
ejpam-6297	199	7	=	=	SYM
ejpam-6297	200	1	sβ	sβ	PROPN
ejpam-6297	200	2	inti(sβ	inti(sβ	PROPN
ejpam-6297	200	3	cli(v	cli(v	PROPN
ejpam-6297	200	4	)	)	PUNCT
ejpam-6297	200	5	)	)	PUNCT
ejpam-6297	201	1	=	=	SYM
ejpam-6297	201	2	sβ	sβ	NUM
ejpam-6297	201	3	inti(v	inti(v	NOUN
ejpam-6297	201	4	)	)	PUNCT
ejpam-6297	201	5	.	.	PUNCT
ejpam-6297	202	1	this	this	PRON
ejpam-6297	202	2	indicates	indicate	VERB
ejpam-6297	202	3	that	that	SCONJ
ejpam-6297	202	4	v	v	NOUN
ejpam-6297	202	5	=	=	SYM
ejpam-6297	202	6	x	x	NOUN
ejpam-6297	202	7	,	,	PUNCT
ejpam-6297	202	8	hence	hence	ADV
ejpam-6297	202	9	a	a	DET
ejpam-6297	202	10	=	=	X
ejpam-6297	202	11	u	u	NOUN
ejpam-6297	202	12	∩	∩	NOUN
ejpam-6297	202	13	v	v	NOUN
ejpam-6297	202	14	=	=	SYM
ejpam-6297	202	15	u	u	NOUN
ejpam-6297	202	16	∩	∩	NOUN
ejpam-6297	202	17	x	x	X
ejpam-6297	202	18	=	=	SYM
ejpam-6297	202	19	u	u	PROPN
ejpam-6297	202	20	.	.	PUNCT
ejpam-6297	203	1	therefore	therefore	ADV
ejpam-6297	203	2	,	,	PUNCT
ejpam-6297	203	3	a	a	PRON
ejpam-6297	203	4	is	be	AUX
ejpam-6297	203	5	classified	classify	VERB
ejpam-6297	203	6	as	as	ADP
ejpam-6297	203	7	strong	strong	ADJ
ejpam-6297	203	8	β	β	X
ejpam-6297	203	9	-	-	ADJ
ejpam-6297	203	10	i	i	PRON
ejpam-6297	203	11	-	-	PUNCT
ejpam-6297	203	12	open	open	ADJ
ejpam-6297	203	13	.	.	PUNCT
ejpam-6297	204	1	accordingly	accordingly	ADV
ejpam-6297	204	2	,	,	PUNCT
ejpam-6297	204	3	(	(	PUNCT
ejpam-6297	204	4	x	x	X
ejpam-6297	204	5	,	,	PUNCT
ejpam-6297	204	6	τ	τ	PROPN
ejpam-6297	204	7	,	,	PUNCT
ejpam-6297	204	8	i	i	PROPN
ejpam-6297	204	9	)	)	PUNCT
ejpam-6297	204	10	is	be	AUX
ejpam-6297	204	11	defined	define	VERB
ejpam-6297	204	12	as	as	ADP
ejpam-6297	204	13	strong	strong	ADJ
ejpam-6297	204	14	β	β	X
ejpam-6297	204	15	-	-	ADJ
ejpam-6297	204	16	i	i	NOUN
ejpam-6297	204	17	-	-	PUNCT
ejpam-6297	204	18	submaximal	submaximal	ADJ
ejpam-6297	204	19	.	.	PUNCT
ejpam-6297	205	1	theorem	theorem	VERB
ejpam-6297	205	2	3	3	NUM
ejpam-6297	205	3	presents	present	NOUN
ejpam-6297	205	4	three	three	NUM
ejpam-6297	205	5	equivalent	equivalent	ADJ
ejpam-6297	205	6	conditions	condition	NOUN
ejpam-6297	205	7	involving	involve	VERB
ejpam-6297	205	8	co	co	ADJ
ejpam-6297	205	9	-	-	ADJ
ejpam-6297	205	10	locally	locally	ADV
ejpam-6297	205	11	strong	strong	ADJ
ejpam-6297	205	12	β	β	NOUN
ejpam-6297	205	13	-	-	ADJ
ejpam-6297	205	14	i	i	NOUN
ejpam-6297	205	15	-	-	PUNCT
ejpam-6297	205	16	closed	close	VERB
ejpam-6297	205	17	sets	set	NOUN
ejpam-6297	205	18	and	and	CCONJ
ejpam-6297	205	19	strong	strong	ADJ
ejpam-6297	205	20	β	β	X
ejpam-6297	205	21	-	-	ADJ
ejpam-6297	205	22	i	i	NOUN
ejpam-6297	205	23	-	-	PUNCT
ejpam-6297	205	24	closed	close	VERB
ejpam-6297	205	25	sets	set	NOUN
ejpam-6297	205	26	that	that	PRON
ejpam-6297	205	27	characterize	characterize	VERB
ejpam-6297	205	28	when	when	SCONJ
ejpam-6297	205	29	a	a	DET
ejpam-6297	205	30	topological	topological	ADJ
ejpam-6297	205	31	space	space	NOUN
ejpam-6297	205	32	is	be	AUX
ejpam-6297	205	33	strong	strong	ADJ
ejpam-6297	205	34	β	β	NOUN
ejpam-6297	205	35	-	-	NOUN
ejpam-6297	205	36	isubmaximal	isubmaximal	ADJ
ejpam-6297	205	37	.	.	PUNCT
ejpam-6297	206	1	theorem	theorem	VERB
ejpam-6297	206	2	3	3	NUM
ejpam-6297	206	3	.	.	X
ejpam-6297	206	4	for	for	ADP
ejpam-6297	206	5	an	an	DET
ejpam-6297	206	6	ideal	ideal	ADJ
ejpam-6297	206	7	topological	topological	ADJ
ejpam-6297	206	8	space	space	NOUN
ejpam-6297	206	9	(	(	PUNCT
ejpam-6297	206	10	x	x	X
ejpam-6297	206	11	,	,	PUNCT
ejpam-6297	206	12	τ	τ	PROPN
ejpam-6297	206	13	,	,	PUNCT
ejpam-6297	206	14	i	i	PROPN
ejpam-6297	206	15	)	)	PUNCT
ejpam-6297	206	16	,	,	PUNCT
ejpam-6297	206	17	the	the	DET
ejpam-6297	206	18	following	follow	VERB
ejpam-6297	206	19	properties	property	NOUN
ejpam-6297	206	20	are	be	AUX
ejpam-6297	206	21	equivalent	equivalent	ADJ
ejpam-6297	206	22	:	:	PUNCT
ejpam-6297	206	23	(	(	PUNCT
ejpam-6297	206	24	1	1	X
ejpam-6297	206	25	)	)	PUNCT
ejpam-6297	206	26	(	(	PUNCT
ejpam-6297	206	27	x	x	X
ejpam-6297	206	28	,	,	PUNCT
ejpam-6297	206	29	τ	τ	PROPN
ejpam-6297	206	30	,	,	PUNCT
ejpam-6297	206	31	i	i	PROPN
ejpam-6297	206	32	)	)	PUNCT
ejpam-6297	206	33	is	be	AUX
ejpam-6297	206	34	strong	strong	ADJ
ejpam-6297	206	35	β	β	NOUN
ejpam-6297	206	36	-	-	ADJ
ejpam-6297	206	37	i	i	NOUN
ejpam-6297	206	38	-	-	PUNCT
ejpam-6297	206	39	submaximal	submaximal	ADJ
ejpam-6297	206	40	;	;	PUNCT
ejpam-6297	206	41	(	(	PUNCT
ejpam-6297	206	42	2	2	X
ejpam-6297	206	43	)	)	PUNCT
ejpam-6297	206	44	every	every	DET
ejpam-6297	206	45	subset	subset	NOUN
ejpam-6297	206	46	of	of	ADP
ejpam-6297	206	47	x	x	PUNCT
ejpam-6297	206	48	is	be	AUX
ejpam-6297	206	49	co	co	ADJ
ejpam-6297	206	50	-	-	ADJ
ejpam-6297	206	51	locally	locally	ADV
ejpam-6297	206	52	strong	strong	ADJ
ejpam-6297	206	53	β	β	X
ejpam-6297	206	54	-	-	ADJ
ejpam-6297	206	55	i	i	NOUN
ejpam-6297	206	56	-	-	PUNCT
ejpam-6297	206	57	closed	closed	ADJ
ejpam-6297	206	58	;	;	PUNCT
ejpam-6297	206	59	(	(	PUNCT
ejpam-6297	206	60	3	3	X
ejpam-6297	206	61	)	)	PUNCT
ejpam-6297	206	62	for	for	ADP
ejpam-6297	206	63	every	every	DET
ejpam-6297	206	64	subset	subset	NOUN
ejpam-6297	206	65	a	a	PRON
ejpam-6297	206	66	of	of	ADP
ejpam-6297	206	67	x	x	SYM
ejpam-6297	206	68	such	such	ADJ
ejpam-6297	206	69	that	that	SCONJ
ejpam-6297	206	70	sβ	sβ	NOUN
ejpam-6297	206	71	inti(a	inti(a	NOUN
ejpam-6297	206	72	)	)	PUNCT
ejpam-6297	207	1	=	=	NOUN
ejpam-6297	207	2	∅	∅	NOUN
ejpam-6297	207	3	is	be	AUX
ejpam-6297	207	4	strong	strong	ADJ
ejpam-6297	207	5	β	β	NOUN
ejpam-6297	207	6	-	-	ADJ
ejpam-6297	207	7	i	i	NOUN
ejpam-6297	207	8	-	-	PUNCT
ejpam-6297	207	9	closed	closed	ADJ
ejpam-6297	207	10	;	;	PUNCT
ejpam-6297	207	11	proof	proof	NOUN
ejpam-6297	207	12	.	.	PUNCT
ejpam-6297	208	1	(	(	PUNCT
ejpam-6297	208	2	1	1	X
ejpam-6297	208	3	)	)	PUNCT
ejpam-6297	208	4	⇒	⇒	NOUN
ejpam-6297	208	5	(	(	PUNCT
ejpam-6297	208	6	2	2	NUM
ejpam-6297	208	7	):	):	PUNCT
ejpam-6297	208	8	assume	assume	VERB
ejpam-6297	208	9	that	that	SCONJ
ejpam-6297	208	10	(	(	PUNCT
ejpam-6297	208	11	x	x	X
ejpam-6297	208	12	,	,	PUNCT
ejpam-6297	208	13	τ	τ	PROPN
ejpam-6297	208	14	,	,	PUNCT
ejpam-6297	208	15	i	i	PROPN
ejpam-6297	208	16	)	)	PUNCT
ejpam-6297	208	17	is	be	AUX
ejpam-6297	208	18	a	a	DET
ejpam-6297	208	19	strong	strong	ADJ
ejpam-6297	208	20	β	β	X
ejpam-6297	208	21	-	-	ADJ
ejpam-6297	208	22	i	i	NOUN
ejpam-6297	208	23	-	-	PUNCT
ejpam-6297	208	24	submaximal	submaximal	ADJ
ejpam-6297	208	25	space	space	NOUN
ejpam-6297	208	26	.	.	PUNCT
ejpam-6297	209	1	for	for	ADP
ejpam-6297	209	2	any	any	DET
ejpam-6297	209	3	subset	subset	NOUN
ejpam-6297	209	4	a	a	DET
ejpam-6297	209	5	⊆	⊆	NUM
ejpam-6297	209	6	x	x	NOUN
ejpam-6297	209	7	,	,	PUNCT
ejpam-6297	209	8	theorem	theorem	VERB
ejpam-6297	209	9	2	2	NUM
ejpam-6297	209	10	guarantees	guarantee	VERB
ejpam-6297	209	11	the	the	DET
ejpam-6297	209	12	existence	existence	NOUN
ejpam-6297	209	13	of	of	ADP
ejpam-6297	209	14	a	a	DET
ejpam-6297	209	15	strong	strong	ADJ
ejpam-6297	209	16	β	β	X
ejpam-6297	209	17	-	-	ADJ
ejpam-6297	209	18	i	i	PRON
ejpam-6297	209	19	-	-	PUNCT
ejpam-6297	209	20	open	open	ADJ
ejpam-6297	209	21	set	set	NOUN
ejpam-6297	209	22	u	u	NOUN
ejpam-6297	209	23	and	and	CCONJ
ejpam-6297	209	24	a	a	DET
ejpam-6297	209	25	strong	strong	ADJ
ejpam-6297	209	26	β	β	X
ejpam-6297	209	27	-	-	ADJ
ejpam-6297	209	28	i	i	NOUN
ejpam-6297	209	29	-	-	PUNCT
ejpam-6297	209	30	closed	close	VERB
ejpam-6297	209	31	set	set	VERB
ejpam-6297	209	32	v	v	ADP
ejpam-6297	209	33	such	such	ADJ
ejpam-6297	209	34	that	that	SCONJ
ejpam-6297	209	35	x	x	SYM
ejpam-6297	209	36	−a	−a	NOUN
ejpam-6297	209	37	=	=	SYM
ejpam-6297	209	38	u	u	NOUN
ejpam-6297	209	39	∩	∩	NOUN
ejpam-6297	209	40	v	v	NOUN
ejpam-6297	209	41	.	.	PUNCT
ejpam-6297	210	1	it	it	PRON
ejpam-6297	210	2	follows	follow	VERB
ejpam-6297	210	3	that	that	SCONJ
ejpam-6297	210	4	a	a	DET
ejpam-6297	210	5	=	=	X
ejpam-6297	210	6	(	(	PUNCT
ejpam-6297	210	7	x	x	SYM
ejpam-6297	210	8	−	−	PROPN
ejpam-6297	210	9	u	u	NOUN
ejpam-6297	210	10	)	)	PUNCT
ejpam-6297	210	11	∪	∪	ADV
ejpam-6297	210	12	(	(	PUNCT
ejpam-6297	210	13	x	x	NOUN
ejpam-6297	210	14	−	−	PROPN
ejpam-6297	210	15	v	v	NOUN
ejpam-6297	210	16	)	)	PUNCT
ejpam-6297	210	17	,	,	PUNCT
ejpam-6297	210	18	where	where	SCONJ
ejpam-6297	210	19	x	x	PUNCT
ejpam-6297	210	20	−	−	PROPN
ejpam-6297	210	21	u	u	NOUN
ejpam-6297	210	22	is	be	AUX
ejpam-6297	210	23	strong	strong	ADJ
ejpam-6297	210	24	β	β	NOUN
ejpam-6297	210	25	-	-	ADJ
ejpam-6297	210	26	i	i	NOUN
ejpam-6297	210	27	-	-	PUNCT
ejpam-6297	210	28	closed	closed	ADJ
ejpam-6297	210	29	and	and	CCONJ
ejpam-6297	210	30	x	x	SYM
ejpam-6297	210	31	−	−	NOUN
ejpam-6297	210	32	v	v	NOUN
ejpam-6297	210	33	is	be	AUX
ejpam-6297	210	34	strong	strong	ADJ
ejpam-6297	210	35	β	β	NOUN
ejpam-6297	210	36	-	-	ADJ
ejpam-6297	210	37	i	i	PRON
ejpam-6297	210	38	-	-	PUNCT
ejpam-6297	210	39	open	open	ADJ
ejpam-6297	210	40	.	.	PUNCT
ejpam-6297	211	1	therefore	therefore	ADV
ejpam-6297	211	2	,	,	PUNCT
ejpam-6297	211	3	a	a	PRON
ejpam-6297	211	4	can	can	AUX
ejpam-6297	211	5	be	be	AUX
ejpam-6297	211	6	expressed	express	VERB
ejpam-6297	211	7	as	as	ADP
ejpam-6297	211	8	the	the	DET
ejpam-6297	211	9	union	union	NOUN
ejpam-6297	211	10	of	of	ADP
ejpam-6297	211	11	a	a	DET
ejpam-6297	211	12	strong	strong	ADJ
ejpam-6297	211	13	β	β	X
ejpam-6297	211	14	-	-	ADJ
ejpam-6297	211	15	i	i	NOUN
ejpam-6297	211	16	-	-	PUNCT
ejpam-6297	211	17	closed	close	VERB
ejpam-6297	211	18	set	set	NOUN
ejpam-6297	211	19	and	and	CCONJ
ejpam-6297	211	20	a	a	DET
ejpam-6297	211	21	strong	strong	ADJ
ejpam-6297	211	22	β	β	X
ejpam-6297	211	23	-	-	ADJ
ejpam-6297	211	24	i	i	NOUN
ejpam-6297	211	25	-	-	PUNCT
ejpam-6297	211	26	open	open	ADJ
ejpam-6297	211	27	set	set	NOUN
ejpam-6297	211	28	,	,	PUNCT
ejpam-6297	211	29	which	which	PRON
ejpam-6297	211	30	means	mean	VERB
ejpam-6297	211	31	a	a	PRON
ejpam-6297	211	32	is	be	AUX
ejpam-6297	211	33	co	co	ADJ
ejpam-6297	211	34	-	-	ADJ
ejpam-6297	211	35	locally	locally	ADV
ejpam-6297	211	36	strong	strong	ADJ
ejpam-6297	211	37	β	β	X
ejpam-6297	211	38	-	-	ADJ
ejpam-6297	211	39	i	i	NOUN
ejpam-6297	211	40	-	-	PUNCT
ejpam-6297	211	41	closed	closed	ADJ
ejpam-6297	211	42	.	.	PUNCT
ejpam-6297	212	1	(	(	PUNCT
ejpam-6297	212	2	2	2	X
ejpam-6297	212	3	)	)	PUNCT
ejpam-6297	212	4	⇒	⇒	NOUN
ejpam-6297	212	5	(	(	PUNCT
ejpam-6297	212	6	3	3	NUM
ejpam-6297	212	7	):	):	PUNCT
ejpam-6297	212	8	suppose	suppose	VERB
ejpam-6297	212	9	that	that	SCONJ
ejpam-6297	212	10	every	every	DET
ejpam-6297	212	11	subset	subset	NOUN
ejpam-6297	212	12	of	of	ADP
ejpam-6297	212	13	x	x	PUNCT
ejpam-6297	212	14	is	be	AUX
ejpam-6297	212	15	co	co	ADJ
ejpam-6297	212	16	-	-	ADJ
ejpam-6297	212	17	locally	locally	ADV
ejpam-6297	212	18	strong	strong	ADJ
ejpam-6297	212	19	β	β	X
ejpam-6297	212	20	-	-	ADJ
ejpam-6297	212	21	i	i	NOUN
ejpam-6297	212	22	-	-	PUNCT
ejpam-6297	212	23	closed	closed	ADJ
ejpam-6297	212	24	.	.	PUNCT
ejpam-6297	213	1	let	let	VERB
ejpam-6297	213	2	a	a	DET
ejpam-6297	213	3	⊆	⊆	NUM
ejpam-6297	213	4	x	x	SYM
ejpam-6297	213	5	be	be	AUX
ejpam-6297	213	6	such	such	ADJ
ejpam-6297	213	7	that	that	SCONJ
ejpam-6297	213	8	sβ	sβ	NOUN
ejpam-6297	213	9	inti(a	inti(a	PROPN
ejpam-6297	213	10	)	)	PUNCT
ejpam-6297	213	11	=	=	PUNCT
ejpam-6297	213	12	∅.	∅.	ADP
ejpam-6297	213	13	then	then	ADV
ejpam-6297	213	14	,	,	PUNCT
ejpam-6297	213	15	by	by	ADP
ejpam-6297	213	16	assumption	assumption	NOUN
ejpam-6297	213	17	,	,	PUNCT
ejpam-6297	213	18	a	a	PRON
ejpam-6297	213	19	can	can	AUX
ejpam-6297	213	20	be	be	AUX
ejpam-6297	213	21	written	write	VERB
ejpam-6297	213	22	as	as	ADP
ejpam-6297	213	23	a	a	DET
ejpam-6297	213	24	=	=	SYM
ejpam-6297	213	25	u	u	NOUN
ejpam-6297	213	26	∪v	∪v	PUNCT
ejpam-6297	213	27	,	,	PUNCT
ejpam-6297	213	28	where	where	SCONJ
ejpam-6297	213	29	u	u	NOUN
ejpam-6297	213	30	is	be	AUX
ejpam-6297	213	31	strong	strong	ADJ
ejpam-6297	213	32	β	β	NOUN
ejpam-6297	213	33	-	-	ADJ
ejpam-6297	213	34	i	i	PRON
ejpam-6297	213	35	-	-	PUNCT
ejpam-6297	213	36	open	open	ADJ
ejpam-6297	213	37	and	and	CCONJ
ejpam-6297	213	38	v	v	NOUN
ejpam-6297	213	39	is	be	AUX
ejpam-6297	213	40	strong	strong	ADJ
ejpam-6297	213	41	β	β	NOUN
ejpam-6297	213	42	-	-	ADJ
ejpam-6297	213	43	i	i	NOUN
ejpam-6297	213	44	-	-	PUNCT
ejpam-6297	213	45	closed	closed	ADJ
ejpam-6297	213	46	.	.	PUNCT
ejpam-6297	214	1	since	since	SCONJ
ejpam-6297	214	2	u	u	PROPN
ejpam-6297	214	3	⊆	⊆	NUM
ejpam-6297	214	4	a	a	PRON
ejpam-6297	214	5	,	,	PUNCT
ejpam-6297	214	6	it	it	PRON
ejpam-6297	214	7	follows	follow	VERB
ejpam-6297	214	8	that	that	SCONJ
ejpam-6297	214	9	u	u	NOUN
ejpam-6297	214	10	=	=	NOUN
ejpam-6297	214	11	sβ	sβ	PROPN
ejpam-6297	214	12	inti(u	inti(u	PROPN
ejpam-6297	214	13	)	)	PUNCT
ejpam-6297	214	14	⊆	⊆	PROPN
ejpam-6297	214	15	sβ	sβ	NUM
ejpam-6297	214	16	inti(a	inti(a	PROPN
ejpam-6297	214	17	)	)	PUNCT
ejpam-6297	214	18	=	=	NOUN
ejpam-6297	214	19	∅	∅	NOUN
ejpam-6297	214	20	,	,	PUNCT
ejpam-6297	214	21	c.	c.	PROPN
ejpam-6297	214	22	boonpok	boonpok	PROPN
ejpam-6297	214	23	,	,	PUNCT
ejpam-6297	214	24	p.	p.	PROPN
ejpam-6297	214	25	raktaow	raktaow	NOUN
ejpam-6297	214	26	,	,	PUNCT
ejpam-6297	214	27	a.	a.	PROPN
ejpam-6297	214	28	sama	sama	PROPN
ejpam-6297	214	29	-	-	PUNCT
ejpam-6297	214	30	ae	ae	PROPN
ejpam-6297	214	31	/	/	SYM
ejpam-6297	214	32	eur	eur	PROPN
ejpam-6297	214	33	.	.	PUNCT
ejpam-6297	215	1	j.	j.	PROPN
ejpam-6297	215	2	pure	pure	PROPN
ejpam-6297	215	3	appl	appl	PROPN
ejpam-6297	215	4	.	.	PROPN
ejpam-6297	215	5	math	math	PROPN
ejpam-6297	215	6	,	,	PUNCT
ejpam-6297	215	7	18	18	NUM
ejpam-6297	215	8	(	(	PUNCT
ejpam-6297	215	9	3	3	NUM
ejpam-6297	215	10	)	)	PUNCT
ejpam-6297	215	11	(	(	PUNCT
ejpam-6297	215	12	2025	2025	NUM
ejpam-6297	215	13	)	)	PUNCT
ejpam-6297	215	14	,	,	PUNCT
ejpam-6297	215	15	6297	6297	NUM
ejpam-6297	215	16	9	9	NUM
ejpam-6297	215	17	of	of	ADP
ejpam-6297	215	18	23	23	NUM
ejpam-6297	215	19	which	which	PRON
ejpam-6297	215	20	implies	imply	VERB
ejpam-6297	215	21	u	u	NOUN
ejpam-6297	215	22	=	=	PROPN
ejpam-6297	215	23	∅.	∅.	VERB
ejpam-6297	215	24	thus	thus	ADV
ejpam-6297	215	25	,	,	PUNCT
ejpam-6297	215	26	a	a	DET
ejpam-6297	215	27	=	=	NOUN
ejpam-6297	215	28	v	v	NOUN
ejpam-6297	215	29	,	,	PUNCT
ejpam-6297	215	30	and	and	CCONJ
ejpam-6297	215	31	since	since	SCONJ
ejpam-6297	215	32	v	v	NOUN
ejpam-6297	215	33	is	be	AUX
ejpam-6297	215	34	strong	strong	ADJ
ejpam-6297	215	35	β	β	NOUN
ejpam-6297	215	36	-	-	ADJ
ejpam-6297	215	37	i	i	NOUN
ejpam-6297	215	38	-	-	PUNCT
ejpam-6297	215	39	closed	closed	ADJ
ejpam-6297	215	40	,	,	PUNCT
ejpam-6297	215	41	it	it	PRON
ejpam-6297	215	42	follows	follow	VERB
ejpam-6297	215	43	that	that	SCONJ
ejpam-6297	215	44	a	a	PRON
ejpam-6297	215	45	is	be	AUX
ejpam-6297	215	46	strong	strong	ADJ
ejpam-6297	215	47	β	β	NOUN
ejpam-6297	215	48	-	-	ADJ
ejpam-6297	215	49	i	i	NOUN
ejpam-6297	215	50	-	-	PUNCT
ejpam-6297	215	51	closed	closed	ADJ
ejpam-6297	215	52	.	.	PUNCT
ejpam-6297	216	1	(	(	PUNCT
ejpam-6297	216	2	3	3	X
ejpam-6297	216	3	)	)	PUNCT
ejpam-6297	216	4	⇒	⇒	NOUN
ejpam-6297	216	5	(	(	PUNCT
ejpam-6297	216	6	1	1	NUM
ejpam-6297	216	7	):	):	PUNCT
ejpam-6297	216	8	let	let	VERB
ejpam-6297	216	9	a	a	DET
ejpam-6297	216	10	⊆	⊆	NUM
ejpam-6297	216	11	x	x	PUNCT
ejpam-6297	216	12	be	be	AUX
ejpam-6297	216	13	a	a	DET
ejpam-6297	216	14	strong	strong	ADJ
ejpam-6297	216	15	-	-	PUNCT
ejpam-6297	216	16	i	i	NOUN
ejpam-6297	216	17	-	-	PUNCT
ejpam-6297	216	18	dense	dense	ADJ
ejpam-6297	216	19	subset	subset	NOUN
ejpam-6297	216	20	.	.	PUNCT
ejpam-6297	217	1	we	we	PRON
ejpam-6297	217	2	aim	aim	VERB
ejpam-6297	217	3	to	to	PART
ejpam-6297	217	4	show	show	VERB
ejpam-6297	217	5	that	that	SCONJ
ejpam-6297	217	6	a	a	PRON
ejpam-6297	217	7	is	be	AUX
ejpam-6297	217	8	strong	strong	ADJ
ejpam-6297	217	9	β	β	NOUN
ejpam-6297	217	10	-	-	ADJ
ejpam-6297	217	11	i	i	PRON
ejpam-6297	217	12	-	-	PUNCT
ejpam-6297	217	13	open	open	ADJ
ejpam-6297	217	14	.	.	PUNCT
ejpam-6297	218	1	since	since	SCONJ
ejpam-6297	218	2	x	x	PRON
ejpam-6297	218	3	−	−	PROPN
ejpam-6297	218	4	sβ	sβ	PROPN
ejpam-6297	218	5	cli(a	cli(a	PROPN
ejpam-6297	218	6	)	)	PUNCT
ejpam-6297	218	7	=	=	PRON
ejpam-6297	218	8	sβ	sβ	PROPN
ejpam-6297	218	9	inti(x	inti(x	NOUN
ejpam-6297	218	10	−a	−a	NOUN
ejpam-6297	218	11	)	)	PUNCT
ejpam-6297	219	1	=	=	PUNCT
ejpam-6297	219	2	∅	∅	NOUN
ejpam-6297	219	3	,	,	PUNCT
ejpam-6297	219	4	it	it	PRON
ejpam-6297	219	5	follows	follow	VERB
ejpam-6297	219	6	that	that	SCONJ
ejpam-6297	219	7	the	the	DET
ejpam-6297	219	8	complement	complement	NOUN
ejpam-6297	219	9	x−a	x−a	NOUN
ejpam-6297	219	10	has	have	AUX
ejpam-6297	219	11	empty	empty	ADJ
ejpam-6297	219	12	strong	strong	ADJ
ejpam-6297	219	13	β	β	X
ejpam-6297	219	14	-	-	ADJ
ejpam-6297	219	15	i	i	NOUN
ejpam-6297	219	16	-	-	NOUN
ejpam-6297	219	17	interior	interior	NOUN
ejpam-6297	219	18	.	.	PUNCT
ejpam-6297	220	1	by	by	ADP
ejpam-6297	220	2	assumption	assumption	NOUN
ejpam-6297	220	3	,	,	PUNCT
ejpam-6297	220	4	such	such	DET
ejpam-6297	220	5	a	a	DET
ejpam-6297	220	6	subset	subset	NOUN
ejpam-6297	220	7	must	must	AUX
ejpam-6297	220	8	be	be	AUX
ejpam-6297	220	9	strong	strong	ADJ
ejpam-6297	220	10	β	β	NOUN
ejpam-6297	220	11	-	-	ADJ
ejpam-6297	220	12	i	i	NOUN
ejpam-6297	220	13	-	-	PUNCT
ejpam-6297	220	14	closed	closed	ADJ
ejpam-6297	220	15	.	.	PUNCT
ejpam-6297	221	1	hence	hence	ADV
ejpam-6297	221	2	,	,	PUNCT
ejpam-6297	221	3	x	x	PUNCT
ejpam-6297	221	4	−	−	NOUN
ejpam-6297	221	5	a	a	PRON
ejpam-6297	221	6	is	be	AUX
ejpam-6297	221	7	strong	strong	ADJ
ejpam-6297	221	8	β	β	NOUN
ejpam-6297	221	9	-	-	ADJ
ejpam-6297	221	10	i	i	NOUN
ejpam-6297	221	11	-	-	PUNCT
ejpam-6297	221	12	closed	closed	ADJ
ejpam-6297	221	13	,	,	PUNCT
ejpam-6297	221	14	and	and	CCONJ
ejpam-6297	221	15	therefore	therefore	ADV
ejpam-6297	221	16	a	a	PRON
ejpam-6297	221	17	is	be	AUX
ejpam-6297	221	18	strong	strong	ADJ
ejpam-6297	221	19	β	β	NOUN
ejpam-6297	221	20	-	-	ADJ
ejpam-6297	221	21	i	i	PRON
ejpam-6297	221	22	-	-	PUNCT
ejpam-6297	221	23	open	open	ADJ
ejpam-6297	221	24	.	.	PUNCT
ejpam-6297	222	1	definition	definition	NOUN
ejpam-6297	222	2	7	7	NUM
ejpam-6297	222	3	.	.	PUNCT
ejpam-6297	223	1	a	a	DET
ejpam-6297	223	2	subset	subset	NOUN
ejpam-6297	223	3	a	a	PRON
ejpam-6297	223	4	of	of	ADP
ejpam-6297	223	5	an	an	DET
ejpam-6297	223	6	ideal	ideal	ADJ
ejpam-6297	223	7	topological	topological	ADJ
ejpam-6297	223	8	space	space	NOUN
ejpam-6297	223	9	(	(	PUNCT
ejpam-6297	223	10	x	x	X
ejpam-6297	223	11	,	,	PUNCT
ejpam-6297	223	12	τ	τ	PROPN
ejpam-6297	223	13	,	,	PUNCT
ejpam-6297	223	14	i	i	PROPN
ejpam-6297	223	15	)	)	PUNCT
ejpam-6297	223	16	is	be	AUX
ejpam-6297	223	17	referred	refer	VERB
ejpam-6297	223	18	to	to	ADP
ejpam-6297	223	19	as	as	ADP
ejpam-6297	223	20	strong	strong	ADJ
ejpam-6297	223	21	β	β	X
ejpam-6297	223	22	-	-	ADJ
ejpam-6297	223	23	i	i	PRON
ejpam-6297	223	24	-	-	NOUN
ejpam-6297	223	25	discrete	discrete	ADJ
ejpam-6297	223	26	if	if	SCONJ
ejpam-6297	223	27	for	for	ADP
ejpam-6297	223	28	every	every	DET
ejpam-6297	223	29	point	point	NOUN
ejpam-6297	223	30	x	x	X
ejpam-6297	223	31	∈	∈	PROPN
ejpam-6297	223	32	a	a	PRON
ejpam-6297	223	33	,	,	PUNCT
ejpam-6297	223	34	the	the	DET
ejpam-6297	223	35	singleton	singleton	PROPN
ejpam-6297	223	36	set	set	NOUN
ejpam-6297	223	37	{	{	PUNCT
ejpam-6297	223	38	x	x	NOUN
ejpam-6297	223	39	}	}	PUNCT
ejpam-6297	223	40	is	be	AUX
ejpam-6297	223	41	strong	strong	ADJ
ejpam-6297	223	42	β	β	NOUN
ejpam-6297	223	43	-	-	ADJ
ejpam-6297	223	44	i	i	NOUN
ejpam-6297	223	45	-	-	PUNCT
ejpam-6297	223	46	closed	closed	ADJ
ejpam-6297	223	47	.	.	PUNCT
ejpam-6297	224	1	theorem	theorem	NOUN
ejpam-6297	224	2	4	4	NUM
ejpam-6297	224	3	establishes	establish	VERB
ejpam-6297	224	4	five	five	NUM
ejpam-6297	224	5	equivalent	equivalent	ADJ
ejpam-6297	224	6	conditions	condition	NOUN
ejpam-6297	224	7	involving	involve	VERB
ejpam-6297	224	8	strong	strong	ADJ
ejpam-6297	224	9	β	β	X
ejpam-6297	224	10	-	-	ADJ
ejpam-6297	224	11	i	i	NOUN
ejpam-6297	224	12	-	-	PUNCT
ejpam-6297	224	13	discreteness	discreteness	PROPN
ejpam-6297	224	14	and	and	CCONJ
ejpam-6297	224	15	strong	strong	ADJ
ejpam-6297	224	16	β	β	X
ejpam-6297	224	17	-	-	ADJ
ejpam-6297	224	18	i	i	NOUN
ejpam-6297	224	19	-	-	PUNCT
ejpam-6297	224	20	closed	close	VERB
ejpam-6297	224	21	sets	set	NOUN
ejpam-6297	224	22	,	,	PUNCT
ejpam-6297	224	23	which	which	PRON
ejpam-6297	224	24	collectively	collectively	ADV
ejpam-6297	224	25	characterize	characterize	VERB
ejpam-6297	224	26	when	when	SCONJ
ejpam-6297	224	27	an	an	DET
ejpam-6297	224	28	ideal	ideal	ADJ
ejpam-6297	224	29	topological	topological	ADJ
ejpam-6297	224	30	space	space	NOUN
ejpam-6297	224	31	(	(	PUNCT
ejpam-6297	224	32	x	x	X
ejpam-6297	224	33	,	,	PUNCT
ejpam-6297	224	34	τ	τ	PROPN
ejpam-6297	224	35	,	,	PUNCT
ejpam-6297	224	36	i	i	PROPN
ejpam-6297	224	37	)	)	PUNCT
ejpam-6297	224	38	can	can	AUX
ejpam-6297	224	39	be	be	AUX
ejpam-6297	224	40	regarded	regard	VERB
ejpam-6297	224	41	as	as	ADP
ejpam-6297	224	42	strong	strong	ADJ
ejpam-6297	224	43	β	β	X
ejpam-6297	224	44	-	-	ADJ
ejpam-6297	224	45	i	i	NOUN
ejpam-6297	224	46	-	-	PUNCT
ejpam-6297	224	47	submaximal	submaximal	ADJ
ejpam-6297	224	48	.	.	PUNCT
ejpam-6297	225	1	the	the	DET
ejpam-6297	225	2	proof	proof	NOUN
ejpam-6297	225	3	of	of	ADP
ejpam-6297	225	4	theorem	theorem	ADJ
ejpam-6297	225	5	4	4	NUM
ejpam-6297	225	6	relies	rely	VERB
ejpam-6297	225	7	on	on	ADP
ejpam-6297	225	8	the	the	DET
ejpam-6297	225	9	following	follow	VERB
ejpam-6297	225	10	lemma	lemma	PROPN
ejpam-6297	225	11	.	.	PUNCT
ejpam-6297	226	1	lemma	lemma	PROPN
ejpam-6297	226	2	4	4	X
ejpam-6297	226	3	.	.	PUNCT
ejpam-6297	226	4	let	let	VERB
ejpam-6297	226	5	a	a	DET
ejpam-6297	226	6	be	be	AUX
ejpam-6297	226	7	a	a	DET
ejpam-6297	226	8	subset	subset	NOUN
ejpam-6297	226	9	of	of	ADP
ejpam-6297	226	10	an	an	DET
ejpam-6297	226	11	ideal	ideal	ADJ
ejpam-6297	226	12	topological	topological	ADJ
ejpam-6297	226	13	space	space	NOUN
ejpam-6297	226	14	(	(	PUNCT
ejpam-6297	226	15	x	x	X
ejpam-6297	226	16	,	,	PUNCT
ejpam-6297	226	17	τ	τ	PROPN
ejpam-6297	226	18	,	,	PUNCT
ejpam-6297	226	19	i	i	PROPN
ejpam-6297	226	20	)	)	PUNCT
ejpam-6297	226	21	.	.	PUNCT
ejpam-6297	227	1	then	then	ADV
ejpam-6297	227	2	,	,	PUNCT
ejpam-6297	227	3	sβ	sβ	PROPN
ejpam-6297	227	4	inti(sβ	inti(sβ	PROPN
ejpam-6297	227	5	cli(a)−a	cli(a)−a	PROPN
ejpam-6297	227	6	)	)	PUNCT
ejpam-6297	227	7	=	=	PUNCT
ejpam-6297	227	8	∅.	∅.	NOUN
ejpam-6297	227	9	proof	proof	NOUN
ejpam-6297	227	10	.	.	PUNCT
ejpam-6297	228	1	let	let	VERB
ejpam-6297	228	2	a	a	DET
ejpam-6297	228	3	be	be	AUX
ejpam-6297	228	4	a	a	DET
ejpam-6297	228	5	subset	subset	NOUN
ejpam-6297	228	6	of	of	ADP
ejpam-6297	228	7	x.	x.	NOUN
ejpam-6297	228	8	since	since	SCONJ
ejpam-6297	228	9	sβ	sβ	PROPN
ejpam-6297	228	10	inti(x	inti(x	NUM
ejpam-6297	228	11	−a	−a	NOUN
ejpam-6297	228	12	)	)	PUNCT
ejpam-6297	229	1	=	=	PUNCT
ejpam-6297	230	1	x	x	X
ejpam-6297	230	2	−	−	NOUN
ejpam-6297	230	3	sβ	sβ	PROPN
ejpam-6297	230	4	cli(a	cli(a	PROPN
ejpam-6297	230	5	)	)	PUNCT
ejpam-6297	230	6	,	,	PUNCT
ejpam-6297	230	7	we	we	PRON
ejpam-6297	230	8	have	have	VERB
ejpam-6297	230	9	sβ	sβ	NUM
ejpam-6297	230	10	inti(sβ	inti(sβ	PROPN
ejpam-6297	230	11	cli(a)−a	cli(a)−a	PROPN
ejpam-6297	230	12	)	)	PUNCT
ejpam-6297	230	13	=	=	PUNCT
ejpam-6297	230	14	sβ	sβ	PROPN
ejpam-6297	230	15	inti(sβ	inti(sβ	PROPN
ejpam-6297	230	16	cli(a	cli(a	PROPN
ejpam-6297	230	17	)	)	PUNCT
ejpam-6297	230	18	∩	∩	NOUN
ejpam-6297	230	19	(	(	PUNCT
ejpam-6297	230	20	x	x	NOUN
ejpam-6297	230	21	−a	−a	NOUN
ejpam-6297	230	22	)	)	PUNCT
ejpam-6297	230	23	)	)	PUNCT
ejpam-6297	231	1	⊆	⊆	NUM
ejpam-6297	231	2	sβ	sβ	NUM
ejpam-6297	231	3	inti(sβ	inti(sβ	PROPN
ejpam-6297	231	4	cli(a	cli(a	PROPN
ejpam-6297	231	5	)	)	PUNCT
ejpam-6297	231	6	)	)	PUNCT
ejpam-6297	231	7	∩	∩	NOUN
ejpam-6297	231	8	sβ	sβ	PRON
ejpam-6297	231	9	inti(x	inti(x	PROPN
ejpam-6297	231	10	−a	−a	NOUN
ejpam-6297	231	11	)	)	PUNCT
ejpam-6297	231	12	=	=	SYM
ejpam-6297	231	13	sβ	sβ	PROPN
ejpam-6297	231	14	inti(sβ	inti(sβ	PROPN
ejpam-6297	231	15	cli(a	cli(a	PROPN
ejpam-6297	231	16	)	)	PUNCT
ejpam-6297	231	17	)	)	PUNCT
ejpam-6297	231	18	∩	∩	NOUN
ejpam-6297	231	19	(	(	PUNCT
ejpam-6297	231	20	x	x	SYM
ejpam-6297	231	21	−	−	NOUN
ejpam-6297	231	22	sβ	sβ	PROPN
ejpam-6297	231	23	cli(a	cli(a	PROPN
ejpam-6297	231	24	)	)	PUNCT
ejpam-6297	231	25	)	)	PUNCT
ejpam-6297	232	1	⊆	⊆	X
ejpam-6297	232	2	sβ	sβ	PROPN
ejpam-6297	232	3	cli(a	cli(a	PROPN
ejpam-6297	232	4	)	)	PUNCT
ejpam-6297	232	5	∩	∩	NOUN
ejpam-6297	232	6	(	(	PUNCT
ejpam-6297	232	7	x	x	SYM
ejpam-6297	232	8	−	−	NOUN
ejpam-6297	232	9	sβ	sβ	PROPN
ejpam-6297	232	10	cli(a	cli(a	PROPN
ejpam-6297	232	11	)	)	PUNCT
ejpam-6297	232	12	)	)	PUNCT
ejpam-6297	233	1	=	=	PUNCT
ejpam-6297	233	2	∅.	∅.	NOUN
ejpam-6297	233	3	theorem	theorem	VERB
ejpam-6297	233	4	4	4	NUM
ejpam-6297	233	5	.	.	X
ejpam-6297	233	6	for	for	ADP
ejpam-6297	233	7	an	an	DET
ejpam-6297	233	8	ideal	ideal	ADJ
ejpam-6297	233	9	topological	topological	ADJ
ejpam-6297	233	10	space	space	NOUN
ejpam-6297	233	11	(	(	PUNCT
ejpam-6297	233	12	x	x	X
ejpam-6297	233	13	,	,	PUNCT
ejpam-6297	233	14	τ	τ	PROPN
ejpam-6297	233	15	,	,	PUNCT
ejpam-6297	233	16	i	i	PROPN
ejpam-6297	233	17	)	)	PUNCT
ejpam-6297	233	18	,	,	PUNCT
ejpam-6297	233	19	the	the	DET
ejpam-6297	233	20	following	follow	VERB
ejpam-6297	233	21	properties	property	NOUN
ejpam-6297	233	22	are	be	AUX
ejpam-6297	233	23	equivalent	equivalent	ADJ
ejpam-6297	233	24	:	:	PUNCT
ejpam-6297	233	25	(	(	PUNCT
ejpam-6297	233	26	1	1	X
ejpam-6297	233	27	)	)	PUNCT
ejpam-6297	233	28	(	(	PUNCT
ejpam-6297	233	29	x	x	X
ejpam-6297	233	30	,	,	PUNCT
ejpam-6297	233	31	τ	τ	PROPN
ejpam-6297	233	32	,	,	PUNCT
ejpam-6297	233	33	i	i	PROPN
ejpam-6297	233	34	)	)	PUNCT
ejpam-6297	233	35	is	be	AUX
ejpam-6297	233	36	strong	strong	ADJ
ejpam-6297	233	37	β	β	NOUN
ejpam-6297	233	38	-	-	ADJ
ejpam-6297	233	39	i	i	NOUN
ejpam-6297	233	40	-	-	PUNCT
ejpam-6297	233	41	submaximal	submaximal	ADJ
ejpam-6297	233	42	;	;	PUNCT
ejpam-6297	233	43	(	(	PUNCT
ejpam-6297	233	44	2	2	X
ejpam-6297	233	45	)	)	PUNCT
ejpam-6297	233	46	for	for	ADP
ejpam-6297	233	47	every	every	DET
ejpam-6297	233	48	subset	subset	NOUN
ejpam-6297	233	49	a	a	PRON
ejpam-6297	233	50	of	of	ADP
ejpam-6297	233	51	x	x	PUNCT
ejpam-6297	233	52	with	with	ADP
ejpam-6297	233	53	sβ	sβ	NOUN
ejpam-6297	233	54	inti(a	inti(a	NOUN
ejpam-6297	233	55	)	)	PUNCT
ejpam-6297	234	1	=	=	NOUN
ejpam-6297	234	2	∅	∅	NOUN
ejpam-6297	234	3	is	be	AUX
ejpam-6297	234	4	strong	strong	ADJ
ejpam-6297	234	5	β	β	NOUN
ejpam-6297	234	6	-	-	ADJ
ejpam-6297	234	7	i	i	NOUN
ejpam-6297	234	8	-	-	PUNCT
ejpam-6297	234	9	closed	closed	ADJ
ejpam-6297	234	10	and	and	CCONJ
ejpam-6297	234	11	strong	strong	ADJ
ejpam-6297	234	12	β	β	NOUN
ejpam-6297	234	13	-	-	NOUN
ejpam-6297	234	14	idiscrete	idiscrete	NOUN
ejpam-6297	234	15	;	;	PUNCT
ejpam-6297	234	16	(	(	PUNCT
ejpam-6297	234	17	3	3	X
ejpam-6297	234	18	)	)	PUNCT
ejpam-6297	234	19	for	for	ADP
ejpam-6297	234	20	every	every	DET
ejpam-6297	234	21	subset	subset	NOUN
ejpam-6297	234	22	a	a	PRON
ejpam-6297	234	23	of	of	ADP
ejpam-6297	234	24	x	x	PRON
ejpam-6297	234	25	,	,	PUNCT
ejpam-6297	234	26	the	the	DET
ejpam-6297	234	27	set	set	NOUN
ejpam-6297	234	28	sβ	sβ	PROPN
ejpam-6297	234	29	cli(a	cli(a	PROPN
ejpam-6297	234	30	)	)	PUNCT
ejpam-6297	234	31	−	−	PROPN
ejpam-6297	235	1	a	a	PRON
ejpam-6297	235	2	is	be	AUX
ejpam-6297	235	3	strong	strong	ADJ
ejpam-6297	235	4	β	β	NOUN
ejpam-6297	235	5	-	-	ADJ
ejpam-6297	235	6	i	i	NOUN
ejpam-6297	235	7	-	-	PUNCT
ejpam-6297	235	8	closed	closed	ADJ
ejpam-6297	235	9	and	and	CCONJ
ejpam-6297	235	10	strong	strong	ADJ
ejpam-6297	235	11	β	β	X
ejpam-6297	235	12	-	-	ADJ
ejpam-6297	235	13	i	i	NOUN
ejpam-6297	235	14	-	-	NOUN
ejpam-6297	235	15	discrete	discrete	ADJ
ejpam-6297	235	16	;	;	PUNCT
ejpam-6297	235	17	(	(	PUNCT
ejpam-6297	235	18	4	4	X
ejpam-6297	235	19	)	)	PUNCT
ejpam-6297	235	20	every	every	PRON
ejpam-6297	235	21	strong	strong	ADJ
ejpam-6297	235	22	β	β	X
ejpam-6297	235	23	-	-	ADJ
ejpam-6297	235	24	i	i	NOUN
ejpam-6297	235	25	-	-	PUNCT
ejpam-6297	235	26	codense	codense	NOUN
ejpam-6297	235	27	subset	subset	NOUN
ejpam-6297	235	28	of	of	ADP
ejpam-6297	235	29	x	x	PUNCT
ejpam-6297	235	30	is	be	AUX
ejpam-6297	235	31	strong	strong	ADJ
ejpam-6297	235	32	β	β	NOUN
ejpam-6297	235	33	-	-	ADJ
ejpam-6297	235	34	i	i	NOUN
ejpam-6297	235	35	-	-	PUNCT
ejpam-6297	235	36	closed	closed	ADJ
ejpam-6297	235	37	and	and	CCONJ
ejpam-6297	235	38	strong	strong	ADJ
ejpam-6297	235	39	β	β	X
ejpam-6297	235	40	-	-	ADJ
ejpam-6297	235	41	i	i	NOUN
ejpam-6297	235	42	-	-	NOUN
ejpam-6297	235	43	discrete	discrete	ADJ
ejpam-6297	235	44	;	;	PUNCT
ejpam-6297	235	45	and	and	CCONJ
ejpam-6297	235	46	(	(	PUNCT
ejpam-6297	235	47	5	5	X
ejpam-6297	235	48	)	)	PUNCT
ejpam-6297	235	49	every	every	PRON
ejpam-6297	235	50	strong	strong	ADJ
ejpam-6297	235	51	β	β	X
ejpam-6297	235	52	-	-	ADJ
ejpam-6297	235	53	i	i	NOUN
ejpam-6297	235	54	-	-	PUNCT
ejpam-6297	235	55	codense	codense	NOUN
ejpam-6297	235	56	subset	subset	NOUN
ejpam-6297	235	57	of	of	ADP
ejpam-6297	235	58	x	x	PUNCT
ejpam-6297	235	59	is	be	AUX
ejpam-6297	235	60	strong	strong	ADJ
ejpam-6297	235	61	β	β	NOUN
ejpam-6297	235	62	-	-	ADJ
ejpam-6297	235	63	i	i	NOUN
ejpam-6297	235	64	-	-	PUNCT
ejpam-6297	235	65	closed	closed	ADJ
ejpam-6297	235	66	.	.	PUNCT
ejpam-6297	236	1	c.	c.	PROPN
ejpam-6297	236	2	boonpok	boonpok	PROPN
ejpam-6297	236	3	,	,	PUNCT
ejpam-6297	236	4	p.	p.	PROPN
ejpam-6297	236	5	raktaow	raktaow	NOUN
ejpam-6297	236	6	,	,	PUNCT
ejpam-6297	236	7	a.	a.	PROPN
ejpam-6297	236	8	sama	sama	PROPN
ejpam-6297	236	9	-	-	PUNCT
ejpam-6297	236	10	ae	ae	PROPN
ejpam-6297	236	11	/	/	SYM
ejpam-6297	236	12	eur	eur	PROPN
ejpam-6297	236	13	.	.	PUNCT
ejpam-6297	237	1	j.	j.	PROPN
ejpam-6297	237	2	pure	pure	PROPN
ejpam-6297	237	3	appl	appl	PROPN
ejpam-6297	237	4	.	.	PROPN
ejpam-6297	237	5	math	math	PROPN
ejpam-6297	237	6	,	,	PUNCT
ejpam-6297	237	7	18	18	NUM
ejpam-6297	237	8	(	(	PUNCT
ejpam-6297	237	9	3	3	NUM
ejpam-6297	237	10	)	)	PUNCT
ejpam-6297	237	11	(	(	PUNCT
ejpam-6297	237	12	2025	2025	NUM
ejpam-6297	237	13	)	)	PUNCT
ejpam-6297	237	14	,	,	PUNCT
ejpam-6297	237	15	6297	6297	NUM
ejpam-6297	237	16	10	10	NUM
ejpam-6297	237	17	of	of	ADP
ejpam-6297	237	18	23	23	NUM
ejpam-6297	237	19	proof	proof	NOUN
ejpam-6297	237	20	.	.	PUNCT
ejpam-6297	238	1	(	(	PUNCT
ejpam-6297	238	2	1	1	X
ejpam-6297	238	3	)	)	PUNCT
ejpam-6297	238	4	⇒	⇒	NOUN
ejpam-6297	238	5	(	(	PUNCT
ejpam-6297	238	6	2	2	NUM
ejpam-6297	238	7	):	):	PUNCT
ejpam-6297	238	8	assume	assume	VERB
ejpam-6297	238	9	that	that	SCONJ
ejpam-6297	238	10	(	(	PUNCT
ejpam-6297	238	11	x	x	X
ejpam-6297	238	12	,	,	PUNCT
ejpam-6297	238	13	τ	τ	PROPN
ejpam-6297	238	14	,	,	PUNCT
ejpam-6297	238	15	i	i	PROPN
ejpam-6297	238	16	)	)	PUNCT
ejpam-6297	238	17	is	be	AUX
ejpam-6297	238	18	strong	strong	ADJ
ejpam-6297	238	19	β	β	NOUN
ejpam-6297	238	20	-	-	ADJ
ejpam-6297	238	21	i	i	NOUN
ejpam-6297	238	22	-	-	PUNCT
ejpam-6297	238	23	submaximal	submaximal	ADJ
ejpam-6297	238	24	.	.	PUNCT
ejpam-6297	239	1	let	let	VERB
ejpam-6297	239	2	a	a	PRON
ejpam-6297	239	3	be	be	AUX
ejpam-6297	239	4	a	a	DET
ejpam-6297	239	5	subset	subset	NOUN
ejpam-6297	239	6	of	of	ADP
ejpam-6297	239	7	x	x	SYM
ejpam-6297	239	8	such	such	ADJ
ejpam-6297	239	9	that	that	SCONJ
ejpam-6297	239	10	sβ	sβ	NOUN
ejpam-6297	239	11	inti(a	inti(a	PROPN
ejpam-6297	239	12	)	)	PUNCT
ejpam-6297	239	13	=	=	PUNCT
ejpam-6297	239	14	∅.	∅.	NOUN
ejpam-6297	239	15	then	then	ADV
ejpam-6297	239	16	,	,	PUNCT
ejpam-6297	239	17	sβ	sβ	PROPN
ejpam-6297	239	18	cli(x	cli(x	NOUN
ejpam-6297	239	19	−a	−a	NOUN
ejpam-6297	239	20	)	)	PUNCT
ejpam-6297	239	21	=	=	PUNCT
ejpam-6297	240	1	x	x	X
ejpam-6297	240	2	−	−	NOUN
ejpam-6297	240	3	sβ	sβ	NOUN
ejpam-6297	240	4	inti(a	inti(a	PROPN
ejpam-6297	240	5	)	)	PUNCT
ejpam-6297	240	6	=	=	SYM
ejpam-6297	241	1	x	x	X
ejpam-6297	241	2	,	,	PUNCT
ejpam-6297	241	3	so	so	ADV
ejpam-6297	241	4	x	x	PUNCT
ejpam-6297	241	5	−	−	NOUN
ejpam-6297	241	6	a	a	PRON
ejpam-6297	241	7	is	be	AUX
ejpam-6297	241	8	strong	strong	ADJ
ejpam-6297	241	9	β	β	NOUN
ejpam-6297	241	10	-	-	ADJ
ejpam-6297	241	11	i	i	NOUN
ejpam-6297	241	12	-	-	PUNCT
ejpam-6297	241	13	dense	dense	ADJ
ejpam-6297	241	14	.	.	PUNCT
ejpam-6297	242	1	by	by	ADP
ejpam-6297	242	2	assumption	assumption	NOUN
ejpam-6297	242	3	,	,	PUNCT
ejpam-6297	242	4	x	x	PUNCT
ejpam-6297	242	5	−	−	NOUN
ejpam-6297	242	6	a	a	PRON
ejpam-6297	242	7	is	be	AUX
ejpam-6297	242	8	strong	strong	ADJ
ejpam-6297	242	9	β	β	NOUN
ejpam-6297	242	10	-	-	ADJ
ejpam-6297	242	11	i	i	PRON
ejpam-6297	242	12	-	-	PUNCT
ejpam-6297	242	13	open	open	ADJ
ejpam-6297	242	14	.	.	PUNCT
ejpam-6297	243	1	thus	thus	ADV
ejpam-6297	243	2	,	,	PUNCT
ejpam-6297	243	3	a	a	PRON
ejpam-6297	243	4	is	be	AUX
ejpam-6297	243	5	strong	strong	ADJ
ejpam-6297	243	6	β	β	NOUN
ejpam-6297	243	7	-	-	ADJ
ejpam-6297	243	8	i	i	NOUN
ejpam-6297	243	9	-	-	PUNCT
ejpam-6297	243	10	closed	closed	ADJ
ejpam-6297	243	11	.	.	PUNCT
ejpam-6297	244	1	next	next	ADV
ejpam-6297	244	2	,	,	PUNCT
ejpam-6297	244	3	we	we	PRON
ejpam-6297	244	4	verify	verify	VERB
ejpam-6297	244	5	that	that	SCONJ
ejpam-6297	244	6	a	a	PRON
ejpam-6297	244	7	is	be	AUX
ejpam-6297	244	8	strong	strong	ADJ
ejpam-6297	244	9	β	β	NOUN
ejpam-6297	244	10	-	-	ADJ
ejpam-6297	244	11	i	i	NOUN
ejpam-6297	244	12	-	-	PUNCT
ejpam-6297	244	13	discrete	discrete	ADJ
ejpam-6297	244	14	.	.	PUNCT
ejpam-6297	245	1	let	let	VERB
ejpam-6297	245	2	x	x	PUNCT
ejpam-6297	245	3	∈	∈	VERB
ejpam-6297	245	4	a.	a.	NOUN
ejpam-6297	245	5	since	since	SCONJ
ejpam-6297	245	6	{	{	PUNCT
ejpam-6297	245	7	x	x	NOUN
ejpam-6297	245	8	}	}	PUNCT
ejpam-6297	245	9	⊆	⊆	NUM
ejpam-6297	245	10	a	a	PRON
ejpam-6297	245	11	,	,	PUNCT
ejpam-6297	245	12	we	we	PRON
ejpam-6297	245	13	have	have	VERB
ejpam-6297	245	14	sβ	sβ	VERB
ejpam-6297	245	15	inti({x	inti({x	PRON
ejpam-6297	245	16	}	}	PUNCT
ejpam-6297	245	17	)	)	PUNCT
ejpam-6297	246	1	⊆	⊆	NUM
ejpam-6297	246	2	sβ	sβ	NUM
ejpam-6297	246	3	inti(a	inti(a	PROPN
ejpam-6297	246	4	)	)	PUNCT
ejpam-6297	246	5	=	=	SYM
ejpam-6297	246	6	∅	∅	NOUN
ejpam-6297	246	7	,	,	PUNCT
ejpam-6297	246	8	which	which	PRON
ejpam-6297	246	9	implies	imply	VERB
ejpam-6297	246	10	sβ	sβ	NUM
ejpam-6297	246	11	inti({x	inti({x	PRON
ejpam-6297	246	12	}	}	PUNCT
ejpam-6297	246	13	)	)	PUNCT
ejpam-6297	246	14	=	=	PUNCT
ejpam-6297	246	15	∅.	∅.	NOUN
ejpam-6297	246	16	then	then	ADV
ejpam-6297	246	17	,	,	PUNCT
ejpam-6297	246	18	sβ	sβ	X
ejpam-6297	246	19	cli(x	cli(x	NOUN
ejpam-6297	246	20	−	−	PROPN
ejpam-6297	246	21	{	{	PUNCT
ejpam-6297	246	22	x	x	NOUN
ejpam-6297	246	23	}	}	PUNCT
ejpam-6297	246	24	)	)	PUNCT
ejpam-6297	246	25	=	=	PUNCT
ejpam-6297	247	1	x	x	X
ejpam-6297	247	2	−	−	NOUN
ejpam-6297	247	3	sβ	sβ	VERB
ejpam-6297	247	4	inti({x	inti({x	PRON
ejpam-6297	247	5	}	}	PUNCT
ejpam-6297	247	6	)	)	PUNCT
ejpam-6297	248	1	=	=	SYM
ejpam-6297	248	2	x	x	X
ejpam-6297	248	3	,	,	PUNCT
ejpam-6297	248	4	so	so	ADV
ejpam-6297	248	5	x	x	NOUN
ejpam-6297	248	6	−{x	−{x	NUM
ejpam-6297	248	7	}	}	PUNCT
ejpam-6297	248	8	is	be	AUX
ejpam-6297	248	9	strong	strong	ADJ
ejpam-6297	248	10	β	β	NOUN
ejpam-6297	248	11	-	-	ADJ
ejpam-6297	248	12	i	i	PRON
ejpam-6297	248	13	-	-	PUNCT
ejpam-6297	248	14	open	open	ADJ
ejpam-6297	248	15	,	,	PUNCT
ejpam-6297	248	16	which	which	PRON
ejpam-6297	248	17	means	mean	VERB
ejpam-6297	248	18	{	{	PUNCT
ejpam-6297	248	19	x	x	NOUN
ejpam-6297	248	20	}	}	PUNCT
ejpam-6297	248	21	is	be	AUX
ejpam-6297	248	22	strong	strong	ADJ
ejpam-6297	248	23	β	β	NOUN
ejpam-6297	248	24	-	-	ADJ
ejpam-6297	248	25	i	i	NOUN
ejpam-6297	248	26	-	-	PUNCT
ejpam-6297	248	27	closed	closed	ADJ
ejpam-6297	248	28	.	.	PUNCT
ejpam-6297	249	1	since	since	SCONJ
ejpam-6297	249	2	this	this	PRON
ejpam-6297	249	3	holds	hold	VERB
ejpam-6297	249	4	for	for	ADP
ejpam-6297	249	5	each	each	DET
ejpam-6297	249	6	x	x	SYM
ejpam-6297	249	7	∈	∈	PROPN
ejpam-6297	249	8	a	a	X
ejpam-6297	249	9	,	,	PUNCT
ejpam-6297	249	10	it	it	PRON
ejpam-6297	249	11	follows	follow	VERB
ejpam-6297	249	12	that	that	SCONJ
ejpam-6297	249	13	a	a	PRON
ejpam-6297	249	14	is	be	AUX
ejpam-6297	249	15	strong	strong	ADJ
ejpam-6297	249	16	β	β	NOUN
ejpam-6297	249	17	-	-	ADJ
ejpam-6297	249	18	i	i	NOUN
ejpam-6297	249	19	-	-	PUNCT
ejpam-6297	249	20	discrete	discrete	NOUN
ejpam-6297	249	21	.	.	PUNCT
ejpam-6297	250	1	(	(	PUNCT
ejpam-6297	250	2	2	2	X
ejpam-6297	250	3	)	)	PUNCT
ejpam-6297	250	4	⇒	⇒	NOUN
ejpam-6297	250	5	(	(	PUNCT
ejpam-6297	250	6	3	3	NUM
ejpam-6297	250	7	):	):	PUNCT
ejpam-6297	250	8	suppose	suppose	VERB
ejpam-6297	250	9	every	every	DET
ejpam-6297	250	10	subset	subset	NOUN
ejpam-6297	250	11	a	a	PRON
ejpam-6297	250	12	of	of	ADP
ejpam-6297	250	13	x	x	PUNCT
ejpam-6297	250	14	with	with	ADP
ejpam-6297	250	15	sβ	sβ	NOUN
ejpam-6297	250	16	inti(a	inti(a	NOUN
ejpam-6297	250	17	)	)	PUNCT
ejpam-6297	250	18	=	=	NOUN
ejpam-6297	250	19	∅	∅	NOUN
ejpam-6297	250	20	is	be	AUX
ejpam-6297	250	21	strong	strong	ADJ
ejpam-6297	250	22	β	β	NOUN
ejpam-6297	250	23	-	-	ADJ
ejpam-6297	250	24	i	i	NOUN
ejpam-6297	250	25	-	-	PUNCT
ejpam-6297	250	26	closed	closed	ADJ
ejpam-6297	250	27	and	and	CCONJ
ejpam-6297	250	28	strong	strong	ADJ
ejpam-6297	250	29	β	β	X
ejpam-6297	250	30	-	-	ADJ
ejpam-6297	250	31	i	i	NOUN
ejpam-6297	250	32	-	-	PUNCT
ejpam-6297	250	33	discrete	discrete	NOUN
ejpam-6297	250	34	.	.	PUNCT
ejpam-6297	251	1	by	by	ADP
ejpam-6297	251	2	lemma	lemma	PROPN
ejpam-6297	251	3	4	4	NUM
ejpam-6297	251	4	,	,	PUNCT
ejpam-6297	251	5	we	we	PRON
ejpam-6297	251	6	have	have	VERB
ejpam-6297	251	7	sβ	sβ	NUM
ejpam-6297	251	8	inti(sβ	inti(sβ	PROPN
ejpam-6297	251	9	cli(a)−a	cli(a)−a	PROPN
ejpam-6297	251	10	)	)	PUNCT
ejpam-6297	251	11	=	=	PUNCT
ejpam-6297	251	12	∅.	∅.	NOUN
ejpam-6297	251	13	by	by	ADP
ejpam-6297	251	14	the	the	DET
ejpam-6297	251	15	hypothesis	hypothesis	NOUN
ejpam-6297	251	16	,	,	PUNCT
ejpam-6297	251	17	this	this	PRON
ejpam-6297	251	18	implies	imply	VERB
ejpam-6297	251	19	that	that	SCONJ
ejpam-6297	251	20	the	the	DET
ejpam-6297	251	21	set	set	NOUN
ejpam-6297	251	22	sβ	sβ	NOUN
ejpam-6297	251	23	cli(a)−a	cli(a)−a	NOUN
ejpam-6297	251	24	is	be	AUX
ejpam-6297	251	25	strong	strong	ADJ
ejpam-6297	251	26	β	β	NOUN
ejpam-6297	251	27	-	-	ADJ
ejpam-6297	251	28	i	i	NOUN
ejpam-6297	251	29	-	-	PUNCT
ejpam-6297	251	30	closed	closed	ADJ
ejpam-6297	251	31	and	and	CCONJ
ejpam-6297	251	32	strong	strong	ADJ
ejpam-6297	251	33	β	β	X
ejpam-6297	251	34	-	-	ADJ
ejpam-6297	251	35	i	i	NOUN
ejpam-6297	251	36	-	-	PUNCT
ejpam-6297	251	37	discrete	discrete	NOUN
ejpam-6297	251	38	.	.	PUNCT
ejpam-6297	252	1	(	(	PUNCT
ejpam-6297	252	2	3	3	X
ejpam-6297	252	3	)	)	PUNCT
ejpam-6297	252	4	⇒	⇒	NOUN
ejpam-6297	252	5	(	(	PUNCT
ejpam-6297	252	6	4	4	NUM
ejpam-6297	252	7	):	):	PUNCT
ejpam-6297	252	8	suppose	suppose	VERB
ejpam-6297	252	9	every	every	DET
ejpam-6297	252	10	subset	subset	NOUN
ejpam-6297	252	11	a	a	PRON
ejpam-6297	252	12	of	of	ADP
ejpam-6297	252	13	x	x	NOUN
ejpam-6297	252	14	satisfies	satisfie	NOUN
ejpam-6297	252	15	that	that	PRON
ejpam-6297	252	16	sβ	sβ	PROPN
ejpam-6297	252	17	cli(a)−a	cli(a)−a	NOUN
ejpam-6297	252	18	is	be	AUX
ejpam-6297	252	19	strong	strong	ADJ
ejpam-6297	252	20	β	β	NOUN
ejpam-6297	252	21	-	-	ADJ
ejpam-6297	252	22	i	i	NOUN
ejpam-6297	252	23	-	-	PUNCT
ejpam-6297	252	24	closed	closed	ADJ
ejpam-6297	252	25	and	and	CCONJ
ejpam-6297	252	26	strong	strong	ADJ
ejpam-6297	252	27	β	β	X
ejpam-6297	252	28	-	-	ADJ
ejpam-6297	252	29	i	i	NOUN
ejpam-6297	252	30	-	-	PUNCT
ejpam-6297	252	31	discrete	discrete	ADJ
ejpam-6297	252	32	.	.	PUNCT
ejpam-6297	253	1	let	let	VERB
ejpam-6297	253	2	a	a	DET
ejpam-6297	253	3	be	be	AUX
ejpam-6297	253	4	a	a	DET
ejpam-6297	253	5	strong	strong	ADJ
ejpam-6297	253	6	β	β	NOUN
ejpam-6297	253	7	-	-	ADJ
ejpam-6297	253	8	i	i	NOUN
ejpam-6297	253	9	-	-	PUNCT
ejpam-6297	253	10	codense	codense	NOUN
ejpam-6297	253	11	subset	subset	NOUN
ejpam-6297	253	12	of	of	ADP
ejpam-6297	253	13	x.	x.	NOUN
ejpam-6297	253	14	then	then	ADV
ejpam-6297	253	15	x	x	PART
ejpam-6297	253	16	−a	−a	NOUN
ejpam-6297	253	17	is	be	AUX
ejpam-6297	253	18	strong	strong	ADJ
ejpam-6297	253	19	β	β	NOUN
ejpam-6297	253	20	-	-	ADJ
ejpam-6297	253	21	i	i	NOUN
ejpam-6297	253	22	-	-	PUNCT
ejpam-6297	253	23	dense	dense	ADJ
ejpam-6297	253	24	,	,	PUNCT
ejpam-6297	253	25	i.e.	i.e.	X
ejpam-6297	253	26	,	,	PUNCT
ejpam-6297	253	27	x	x	SYM
ejpam-6297	253	28	=	=	SYM
ejpam-6297	253	29	sβ	sβ	PROPN
ejpam-6297	253	30	cli(x	cli(x	NOUN
ejpam-6297	253	31	−a	−a	NOUN
ejpam-6297	253	32	)	)	PUNCT
ejpam-6297	253	33	.	.	PUNCT
ejpam-6297	254	1	hence	hence	ADV
ejpam-6297	254	2	,	,	PUNCT
ejpam-6297	254	3	a	a	DET
ejpam-6297	254	4	=	=	X
ejpam-6297	254	5	x	x	SYM
ejpam-6297	254	6	−	−	PROPN
ejpam-6297	254	7	(	(	PUNCT
ejpam-6297	254	8	x	x	NOUN
ejpam-6297	254	9	−a	−a	NOUN
ejpam-6297	254	10	)	)	PUNCT
ejpam-6297	254	11	=	=	SYM
ejpam-6297	254	12	sβ	sβ	PROPN
ejpam-6297	254	13	cli(x	cli(x	NOUN
ejpam-6297	254	14	−a)−	−a)−	PROPN
ejpam-6297	254	15	(	(	PUNCT
ejpam-6297	254	16	x	x	NOUN
ejpam-6297	254	17	−a	−a	NOUN
ejpam-6297	254	18	)	)	PUNCT
ejpam-6297	254	19	.	.	PUNCT
ejpam-6297	255	1	by	by	ADP
ejpam-6297	255	2	the	the	DET
ejpam-6297	255	3	assumption	assumption	NOUN
ejpam-6297	255	4	,	,	PUNCT
ejpam-6297	255	5	a	a	PRON
ejpam-6297	255	6	is	be	AUX
ejpam-6297	255	7	strong	strong	ADJ
ejpam-6297	255	8	β	β	NOUN
ejpam-6297	255	9	-	-	ADJ
ejpam-6297	255	10	i	i	NOUN
ejpam-6297	255	11	-	-	PUNCT
ejpam-6297	255	12	closed	closed	ADJ
ejpam-6297	255	13	and	and	CCONJ
ejpam-6297	255	14	strong	strong	ADJ
ejpam-6297	255	15	β	β	X
ejpam-6297	255	16	-	-	ADJ
ejpam-6297	255	17	i	i	NOUN
ejpam-6297	255	18	-	-	PUNCT
ejpam-6297	255	19	discrete	discrete	NOUN
ejpam-6297	255	20	.	.	PUNCT
ejpam-6297	256	1	(	(	PUNCT
ejpam-6297	256	2	4	4	X
ejpam-6297	256	3	)	)	PUNCT
ejpam-6297	256	4	⇒	⇒	NOUN
ejpam-6297	256	5	(	(	PUNCT
ejpam-6297	256	6	5	5	NUM
ejpam-6297	256	7	):	):	PUNCT
ejpam-6297	256	8	this	this	PRON
ejpam-6297	256	9	is	be	AUX
ejpam-6297	256	10	obvious	obvious	ADJ
ejpam-6297	256	11	.	.	PUNCT
ejpam-6297	257	1	(	(	PUNCT
ejpam-6297	257	2	5)⇒	5)⇒	NUM
ejpam-6297	257	3	(	(	PUNCT
ejpam-6297	257	4	1	1	NUM
ejpam-6297	257	5	):	):	PUNCT
ejpam-6297	257	6	suppose	suppose	VERB
ejpam-6297	257	7	that	that	SCONJ
ejpam-6297	257	8	every	every	DET
ejpam-6297	257	9	strong	strong	ADJ
ejpam-6297	257	10	β	β	X
ejpam-6297	257	11	-	-	ADJ
ejpam-6297	257	12	i	i	NOUN
ejpam-6297	257	13	-	-	PUNCT
ejpam-6297	257	14	codense	codense	NOUN
ejpam-6297	257	15	subset	subset	NOUN
ejpam-6297	257	16	ofx	ofx	PROPN
ejpam-6297	257	17	is	be	AUX
ejpam-6297	257	18	strong	strong	ADJ
ejpam-6297	257	19	β	β	NOUN
ejpam-6297	257	20	-	-	ADJ
ejpam-6297	257	21	i	i	NOUN
ejpam-6297	257	22	-	-	PUNCT
ejpam-6297	257	23	closed	closed	ADJ
ejpam-6297	257	24	.	.	PUNCT
ejpam-6297	258	1	let	let	VERB
ejpam-6297	258	2	a	a	DET
ejpam-6297	258	3	be	be	AUX
ejpam-6297	258	4	a	a	DET
ejpam-6297	258	5	strong	strong	ADJ
ejpam-6297	258	6	β	β	NOUN
ejpam-6297	258	7	-	-	ADJ
ejpam-6297	258	8	i	i	NOUN
ejpam-6297	258	9	-	-	PUNCT
ejpam-6297	258	10	dense	dense	ADJ
ejpam-6297	258	11	subset	subset	NOUN
ejpam-6297	258	12	of	of	ADP
ejpam-6297	258	13	x.	x.	PROPN
ejpam-6297	258	14	then	then	ADV
ejpam-6297	258	15	its	its	PRON
ejpam-6297	258	16	complement	complement	NOUN
ejpam-6297	258	17	x	x	PUNCT
ejpam-6297	258	18	−a	−a	NOUN
ejpam-6297	258	19	is	be	AUX
ejpam-6297	258	20	strong	strong	ADJ
ejpam-6297	258	21	β	β	NOUN
ejpam-6297	258	22	-	-	ADJ
ejpam-6297	258	23	i	i	NOUN
ejpam-6297	258	24	-	-	PUNCT
ejpam-6297	258	25	codense	codense	NOUN
ejpam-6297	258	26	.	.	PUNCT
ejpam-6297	259	1	by	by	ADP
ejpam-6297	259	2	the	the	DET
ejpam-6297	259	3	hypothesis	hypothesis	NOUN
ejpam-6297	259	4	,	,	PUNCT
ejpam-6297	259	5	x	x	PUNCT
ejpam-6297	259	6	−	−	NOUN
ejpam-6297	259	7	a	a	PRON
ejpam-6297	259	8	is	be	AUX
ejpam-6297	259	9	strong	strong	ADJ
ejpam-6297	259	10	β	β	NOUN
ejpam-6297	259	11	-	-	ADJ
ejpam-6297	259	12	i	i	NOUN
ejpam-6297	259	13	-	-	PUNCT
ejpam-6297	259	14	closed	closed	ADJ
ejpam-6297	259	15	.	.	PUNCT
ejpam-6297	260	1	thus	thus	ADV
ejpam-6297	260	2	,	,	PUNCT
ejpam-6297	260	3	a	a	PRON
ejpam-6297	260	4	is	be	AUX
ejpam-6297	260	5	strong	strong	ADJ
ejpam-6297	260	6	β	β	NOUN
ejpam-6297	260	7	-	-	ADJ
ejpam-6297	260	8	i	i	PRON
ejpam-6297	260	9	-	-	PUNCT
ejpam-6297	260	10	open	open	ADJ
ejpam-6297	260	11	.	.	PUNCT
ejpam-6297	261	1	therefore	therefore	ADV
ejpam-6297	261	2	,	,	PUNCT
ejpam-6297	261	3	the	the	DET
ejpam-6297	261	4	space	space	NOUN
ejpam-6297	261	5	(	(	PUNCT
ejpam-6297	261	6	x	x	X
ejpam-6297	261	7	,	,	PUNCT
ejpam-6297	261	8	τ	τ	PROPN
ejpam-6297	261	9	,	,	PUNCT
ejpam-6297	261	10	i	i	PROPN
ejpam-6297	261	11	)	)	PUNCT
ejpam-6297	261	12	is	be	AUX
ejpam-6297	261	13	strong	strong	ADJ
ejpam-6297	261	14	β	β	NOUN
ejpam-6297	261	15	-	-	ADJ
ejpam-6297	261	16	i	i	NOUN
ejpam-6297	261	17	-	-	PUNCT
ejpam-6297	261	18	submaximal	submaximal	ADJ
ejpam-6297	261	19	.	.	PUNCT
ejpam-6297	262	1	the	the	DET
ejpam-6297	262	2	final	final	ADJ
ejpam-6297	262	3	theorem	theorem	NOUN
ejpam-6297	262	4	in	in	ADP
ejpam-6297	262	5	this	this	DET
ejpam-6297	262	6	section	section	NOUN
ejpam-6297	262	7	outlines	outline	VERB
ejpam-6297	262	8	three	three	NUM
ejpam-6297	262	9	equivalent	equivalent	ADJ
ejpam-6297	262	10	conditions	condition	NOUN
ejpam-6297	262	11	that	that	PRON
ejpam-6297	262	12	determine	determine	VERB
ejpam-6297	262	13	when	when	SCONJ
ejpam-6297	262	14	an	an	DET
ejpam-6297	262	15	ideal	ideal	ADJ
ejpam-6297	262	16	topological	topological	ADJ
ejpam-6297	262	17	space	space	NOUN
ejpam-6297	262	18	(	(	PUNCT
ejpam-6297	262	19	x	x	X
ejpam-6297	262	20	,	,	PUNCT
ejpam-6297	262	21	τ	τ	PROPN
ejpam-6297	262	22	,	,	PUNCT
ejpam-6297	262	23	i	i	PROPN
ejpam-6297	262	24	)	)	PUNCT
ejpam-6297	262	25	is	be	AUX
ejpam-6297	262	26	strong	strong	ADJ
ejpam-6297	262	27	β	β	NOUN
ejpam-6297	262	28	-	-	ADJ
ejpam-6297	262	29	i	i	NOUN
ejpam-6297	262	30	-	-	PUNCT
ejpam-6297	262	31	submaximal	submaximal	ADJ
ejpam-6297	262	32	.	.	PUNCT
ejpam-6297	263	1	theorem	theorem	NOUN
ejpam-6297	263	2	5	5	NUM
ejpam-6297	263	3	.	.	X
ejpam-6297	263	4	for	for	ADP
ejpam-6297	263	5	an	an	DET
ejpam-6297	263	6	ideal	ideal	ADJ
ejpam-6297	263	7	topological	topological	ADJ
ejpam-6297	263	8	space	space	NOUN
ejpam-6297	263	9	(	(	PUNCT
ejpam-6297	263	10	x	x	X
ejpam-6297	263	11	,	,	PUNCT
ejpam-6297	263	12	τ	τ	PROPN
ejpam-6297	263	13	,	,	PUNCT
ejpam-6297	263	14	i	i	PROPN
ejpam-6297	263	15	)	)	PUNCT
ejpam-6297	263	16	,	,	PUNCT
ejpam-6297	263	17	the	the	DET
ejpam-6297	263	18	following	follow	VERB
ejpam-6297	263	19	properties	property	NOUN
ejpam-6297	263	20	are	be	AUX
ejpam-6297	263	21	equivalent	equivalent	ADJ
ejpam-6297	263	22	:	:	PUNCT
ejpam-6297	263	23	(	(	PUNCT
ejpam-6297	263	24	1	1	X
ejpam-6297	263	25	)	)	PUNCT
ejpam-6297	263	26	(	(	PUNCT
ejpam-6297	263	27	x	x	X
ejpam-6297	263	28	,	,	PUNCT
ejpam-6297	263	29	τ	τ	PROPN
ejpam-6297	263	30	,	,	PUNCT
ejpam-6297	263	31	i	i	PROPN
ejpam-6297	263	32	)	)	PUNCT
ejpam-6297	263	33	is	be	AUX
ejpam-6297	263	34	strong	strong	ADJ
ejpam-6297	263	35	β	β	NOUN
ejpam-6297	263	36	-	-	ADJ
ejpam-6297	263	37	i	i	NOUN
ejpam-6297	263	38	-	-	PUNCT
ejpam-6297	263	39	submaximal	submaximal	ADJ
ejpam-6297	263	40	;	;	PUNCT
ejpam-6297	263	41	c.	c.	PROPN
ejpam-6297	263	42	boonpok	boonpok	PROPN
ejpam-6297	263	43	,	,	PUNCT
ejpam-6297	263	44	p.	p.	PROPN
ejpam-6297	263	45	raktaow	raktaow	NOUN
ejpam-6297	263	46	,	,	PUNCT
ejpam-6297	263	47	a.	a.	PROPN
ejpam-6297	263	48	sama	sama	PROPN
ejpam-6297	263	49	-	-	PUNCT
ejpam-6297	263	50	ae	ae	PROPN
ejpam-6297	263	51	/	/	SYM
ejpam-6297	263	52	eur	eur	PROPN
ejpam-6297	263	53	.	.	PUNCT
ejpam-6297	264	1	j.	j.	PROPN
ejpam-6297	264	2	pure	pure	PROPN
ejpam-6297	264	3	appl	appl	PROPN
ejpam-6297	264	4	.	.	PROPN
ejpam-6297	264	5	math	math	PROPN
ejpam-6297	264	6	,	,	PUNCT
ejpam-6297	264	7	18	18	NUM
ejpam-6297	264	8	(	(	PUNCT
ejpam-6297	264	9	3	3	NUM
ejpam-6297	264	10	)	)	PUNCT
ejpam-6297	264	11	(	(	PUNCT
ejpam-6297	264	12	2025	2025	NUM
ejpam-6297	264	13	)	)	PUNCT
ejpam-6297	264	14	,	,	PUNCT
ejpam-6297	264	15	6297	6297	NUM
ejpam-6297	264	16	11	11	NUM
ejpam-6297	264	17	of	of	ADP
ejpam-6297	264	18	23	23	NUM
ejpam-6297	264	19	(	(	PUNCT
ejpam-6297	264	20	2	2	NUM
ejpam-6297	264	21	)	)	PUNCT
ejpam-6297	264	22	every	every	DET
ejpam-6297	264	23	subset	subset	NOUN
ejpam-6297	264	24	of	of	ADP
ejpam-6297	264	25	x	x	PUNCT
ejpam-6297	264	26	is	be	AUX
ejpam-6297	264	27	locally	locally	ADV
ejpam-6297	264	28	strong	strong	ADJ
ejpam-6297	264	29	β	β	X
ejpam-6297	264	30	-	-	ADJ
ejpam-6297	264	31	i	i	NOUN
ejpam-6297	264	32	-	-	PUNCT
ejpam-6297	264	33	closed	closed	ADJ
ejpam-6297	264	34	;	;	PUNCT
ejpam-6297	264	35	and	and	CCONJ
ejpam-6297	264	36	(	(	PUNCT
ejpam-6297	264	37	3	3	X
ejpam-6297	264	38	)	)	PUNCT
ejpam-6297	264	39	every	every	PRON
ejpam-6297	264	40	strong	strong	ADJ
ejpam-6297	264	41	β	β	X
ejpam-6297	264	42	-	-	ADJ
ejpam-6297	264	43	i	i	NOUN
ejpam-6297	264	44	-	-	PUNCT
ejpam-6297	264	45	dense	dense	ADJ
ejpam-6297	264	46	subset	subset	NOUN
ejpam-6297	264	47	of	of	ADP
ejpam-6297	264	48	x	x	PUNCT
ejpam-6297	264	49	is	be	AUX
ejpam-6297	264	50	locally	locally	ADV
ejpam-6297	264	51	strong	strong	ADJ
ejpam-6297	264	52	β	β	X
ejpam-6297	264	53	-	-	ADJ
ejpam-6297	264	54	i	i	NOUN
ejpam-6297	264	55	-	-	PUNCT
ejpam-6297	264	56	closed	closed	ADJ
ejpam-6297	264	57	.	.	PUNCT
ejpam-6297	265	1	proof	proof	NOUN
ejpam-6297	265	2	.	.	PUNCT
ejpam-6297	266	1	(	(	PUNCT
ejpam-6297	266	2	1	1	X
ejpam-6297	266	3	)	)	PUNCT
ejpam-6297	266	4	⇒	⇒	NOUN
ejpam-6297	266	5	(	(	PUNCT
ejpam-6297	266	6	2	2	NUM
ejpam-6297	266	7	):	):	PUNCT
ejpam-6297	266	8	assume	assume	VERB
ejpam-6297	266	9	that	that	SCONJ
ejpam-6297	266	10	(	(	PUNCT
ejpam-6297	266	11	x	x	X
ejpam-6297	266	12	,	,	PUNCT
ejpam-6297	266	13	τ	τ	PROPN
ejpam-6297	266	14	,	,	PUNCT
ejpam-6297	266	15	i	i	PROPN
ejpam-6297	266	16	)	)	PUNCT
ejpam-6297	266	17	is	be	AUX
ejpam-6297	266	18	strong	strong	ADJ
ejpam-6297	266	19	β	β	NOUN
ejpam-6297	266	20	-	-	ADJ
ejpam-6297	266	21	i	i	NOUN
ejpam-6297	266	22	-	-	PUNCT
ejpam-6297	266	23	submaximal	submaximal	ADJ
ejpam-6297	266	24	.	.	PUNCT
ejpam-6297	267	1	let	let	VERB
ejpam-6297	267	2	a	a	DET
ejpam-6297	267	3	⊆	⊆	NUM
ejpam-6297	267	4	x.	x.	NOUN
ejpam-6297	267	5	as	as	ADP
ejpam-6297	267	6	the	the	DET
ejpam-6297	267	7	same	same	ADJ
ejpam-6297	267	8	proof	proof	NOUN
ejpam-6297	267	9	in	in	ADP
ejpam-6297	267	10	theorem	theorem	NOUN
ejpam-6297	267	11	2	2	NUM
ejpam-6297	267	12	,	,	PUNCT
ejpam-6297	267	13	we	we	PRON
ejpam-6297	267	14	have	have	VERB
ejpam-6297	267	15	sβ	sβ	NUM
ejpam-6297	267	16	cli(x	cli(x	NOUN
ejpam-6297	268	1	−	−	PROPN
ejpam-6297	268	2	(	(	PUNCT
ejpam-6297	268	3	sβ	sβ	PROPN
ejpam-6297	268	4	cli(a	cli(a	PROPN
ejpam-6297	268	5	)	)	PUNCT
ejpam-6297	268	6	−	−	PROPN
ejpam-6297	268	7	a	a	NOUN
ejpam-6297	268	8	)	)	PUNCT
ejpam-6297	268	9	)	)	PUNCT
ejpam-6297	269	1	=	=	PUNCT
ejpam-6297	270	1	x	x	PUNCT
ejpam-6297	270	2	and	and	CCONJ
ejpam-6297	270	3	hence	hence	ADV
ejpam-6297	270	4	x−	x−	PROPN
ejpam-6297	270	5	(	(	PUNCT
ejpam-6297	270	6	sβ	sβ	NOUN
ejpam-6297	270	7	cli(a)−a	cli(a)−a	NOUN
ejpam-6297	270	8	)	)	PUNCT
ejpam-6297	270	9	is	be	AUX
ejpam-6297	270	10	strong	strong	ADJ
ejpam-6297	270	11	β	β	NOUN
ejpam-6297	270	12	-	-	ADJ
ejpam-6297	270	13	i	i	NOUN
ejpam-6297	270	14	-	-	PUNCT
ejpam-6297	270	15	dense	dense	ADJ
ejpam-6297	270	16	.	.	PUNCT
ejpam-6297	271	1	by	by	ADP
ejpam-6297	271	2	the	the	DET
ejpam-6297	271	3	hypothesis	hypothesis	NOUN
ejpam-6297	271	4	,	,	PUNCT
ejpam-6297	271	5	x−	x−	PROPN
ejpam-6297	271	6	(	(	PUNCT
ejpam-6297	271	7	sβ	sβ	NOUN
ejpam-6297	271	8	cli(a)−a	cli(a)−a	NOUN
ejpam-6297	271	9	)	)	PUNCT
ejpam-6297	271	10	is	be	AUX
ejpam-6297	271	11	strong	strong	ADJ
ejpam-6297	271	12	β	β	NOUN
ejpam-6297	271	13	-	-	ADJ
ejpam-6297	271	14	i	i	PRON
ejpam-6297	271	15	-	-	PUNCT
ejpam-6297	271	16	open	open	ADJ
ejpam-6297	271	17	.	.	PUNCT
ejpam-6297	272	1	therefore	therefore	ADV
ejpam-6297	272	2	,	,	PUNCT
ejpam-6297	272	3	sβ	sβ	PROPN
ejpam-6297	272	4	cli(a)−a	cli(a)−a	PROPN
ejpam-6297	272	5	is	be	AUX
ejpam-6297	272	6	strong	strong	ADJ
ejpam-6297	272	7	β	β	NOUN
ejpam-6297	272	8	-	-	ADJ
ejpam-6297	272	9	i	i	NOUN
ejpam-6297	272	10	-	-	PUNCT
ejpam-6297	272	11	closed	closed	ADJ
ejpam-6297	272	12	.	.	PUNCT
ejpam-6297	273	1	by	by	ADP
ejpam-6297	273	2	theorem	theorem	NOUN
ejpam-6297	273	3	1	1	NUM
ejpam-6297	273	4	,	,	PUNCT
ejpam-6297	273	5	a	a	PRON
ejpam-6297	273	6	is	be	AUX
ejpam-6297	273	7	locally	locally	ADV
ejpam-6297	273	8	strong	strong	ADJ
ejpam-6297	273	9	β	β	X
ejpam-6297	273	10	-	-	ADJ
ejpam-6297	273	11	i	i	NOUN
ejpam-6297	273	12	-	-	PUNCT
ejpam-6297	273	13	closed	closed	ADJ
ejpam-6297	273	14	.	.	PUNCT
ejpam-6297	274	1	(	(	PUNCT
ejpam-6297	274	2	2	2	X
ejpam-6297	274	3	)	)	PUNCT
ejpam-6297	274	4	⇒	⇒	NOUN
ejpam-6297	274	5	(	(	PUNCT
ejpam-6297	274	6	3	3	NUM
ejpam-6297	274	7	):	):	PUNCT
ejpam-6297	274	8	it	it	PRON
ejpam-6297	274	9	is	be	AUX
ejpam-6297	274	10	clear	clear	ADJ
ejpam-6297	274	11	.	.	PUNCT
ejpam-6297	275	1	(	(	PUNCT
ejpam-6297	275	2	3	3	X
ejpam-6297	275	3	)	)	PUNCT
ejpam-6297	275	4	⇒	⇒	NOUN
ejpam-6297	275	5	(	(	PUNCT
ejpam-6297	275	6	1	1	NUM
ejpam-6297	275	7	):	):	PUNCT
ejpam-6297	275	8	assume	assume	VERB
ejpam-6297	275	9	that	that	SCONJ
ejpam-6297	275	10	every	every	DET
ejpam-6297	275	11	strong	strong	ADJ
ejpam-6297	275	12	β	β	X
ejpam-6297	275	13	-	-	ADJ
ejpam-6297	275	14	i	i	NOUN
ejpam-6297	275	15	-	-	PUNCT
ejpam-6297	275	16	dense	dense	ADJ
ejpam-6297	275	17	subset	subset	NOUN
ejpam-6297	275	18	of	of	ADP
ejpam-6297	275	19	x	x	PUNCT
ejpam-6297	275	20	is	be	AUX
ejpam-6297	275	21	locally	locally	ADV
ejpam-6297	275	22	strong	strong	ADJ
ejpam-6297	275	23	β	β	X
ejpam-6297	275	24	-	-	PUNCT
ejpam-6297	275	25	iclosed	iclose	VERB
ejpam-6297	275	26	.	.	PUNCT
ejpam-6297	276	1	let	let	VERB
ejpam-6297	276	2	a	a	DET
ejpam-6297	276	3	be	be	AUX
ejpam-6297	276	4	a	a	DET
ejpam-6297	276	5	strong	strong	ADJ
ejpam-6297	276	6	β	β	NOUN
ejpam-6297	276	7	-	-	ADJ
ejpam-6297	276	8	i	i	NOUN
ejpam-6297	276	9	-	-	PUNCT
ejpam-6297	276	10	dense	dense	ADJ
ejpam-6297	276	11	subset	subset	NOUN
ejpam-6297	276	12	of	of	ADP
ejpam-6297	276	13	x.	x.	NOUN
ejpam-6297	276	14	this	this	PRON
ejpam-6297	276	15	implies	imply	VERB
ejpam-6297	276	16	sβ	sβ	PROPN
ejpam-6297	276	17	cli(a	cli(a	PROPN
ejpam-6297	276	18	)	)	PUNCT
ejpam-6297	276	19	=	=	PUNCT
ejpam-6297	277	1	x.	x.	NOUN
ejpam-6297	277	2	by	by	ADP
ejpam-6297	277	3	assumption	assumption	NOUN
ejpam-6297	277	4	,	,	PUNCT
ejpam-6297	277	5	a	a	PRON
ejpam-6297	277	6	is	be	AUX
ejpam-6297	277	7	locally	locally	ADV
ejpam-6297	277	8	strong	strong	ADJ
ejpam-6297	277	9	β	β	X
ejpam-6297	277	10	-	-	ADJ
ejpam-6297	277	11	i	i	NOUN
ejpam-6297	277	12	-	-	PUNCT
ejpam-6297	277	13	closed	closed	ADJ
ejpam-6297	277	14	,	,	PUNCT
ejpam-6297	277	15	meaning	mean	VERB
ejpam-6297	277	16	there	there	PRON
ejpam-6297	277	17	exist	exist	VERB
ejpam-6297	277	18	a	a	DET
ejpam-6297	277	19	strong	strong	ADJ
ejpam-6297	277	20	β	β	X
ejpam-6297	277	21	-	-	ADJ
ejpam-6297	277	22	i	i	PRON
ejpam-6297	277	23	-	-	PUNCT
ejpam-6297	277	24	open	open	ADJ
ejpam-6297	277	25	set	set	NOUN
ejpam-6297	277	26	u	u	NOUN
ejpam-6297	277	27	and	and	CCONJ
ejpam-6297	277	28	a	a	DET
ejpam-6297	277	29	strong	strong	ADJ
ejpam-6297	277	30	β	β	X
ejpam-6297	277	31	-	-	ADJ
ejpam-6297	277	32	i	i	NOUN
ejpam-6297	277	33	-	-	PUNCT
ejpam-6297	277	34	closed	close	VERB
ejpam-6297	277	35	set	set	VERB
ejpam-6297	277	36	v	v	ADP
ejpam-6297	277	37	such	such	DET
ejpam-6297	277	38	that	that	SCONJ
ejpam-6297	277	39	a	a	DET
ejpam-6297	277	40	=	=	X
ejpam-6297	277	41	u	u	NOUN
ejpam-6297	277	42	∩	∩	NOUN
ejpam-6297	277	43	v	v	NOUN
ejpam-6297	277	44	.	.	PUNCT
ejpam-6297	278	1	since	since	SCONJ
ejpam-6297	278	2	a	a	DET
ejpam-6297	278	3	⊆	⊆	NUM
ejpam-6297	278	4	v	v	NOUN
ejpam-6297	278	5	,	,	PUNCT
ejpam-6297	278	6	we	we	PRON
ejpam-6297	278	7	have	have	VERB
ejpam-6297	278	8	that	that	PRON
ejpam-6297	278	9	x	x	X
ejpam-6297	278	10	=	=	PRON
ejpam-6297	278	11	sβ	sβ	PROPN
ejpam-6297	278	12	cli(a	cli(a	PROPN
ejpam-6297	278	13	)	)	PUNCT
ejpam-6297	278	14	⊆	⊆	NUM
ejpam-6297	278	15	sβ	sβ	NUM
ejpam-6297	278	16	cli(v	cli(v	NOUN
ejpam-6297	278	17	)	)	PUNCT
ejpam-6297	279	1	=	=	SYM
ejpam-6297	279	2	v	v	NOUN
ejpam-6297	279	3	,	,	PUNCT
ejpam-6297	279	4	which	which	PRON
ejpam-6297	279	5	gives	give	VERB
ejpam-6297	279	6	v	v	NOUN
ejpam-6297	279	7	=	=	PUNCT
ejpam-6297	279	8	x.	x.	NOUN
ejpam-6297	279	9	thus	thus	ADV
ejpam-6297	279	10	,	,	PUNCT
ejpam-6297	279	11	a	a	DET
ejpam-6297	279	12	=	=	X
ejpam-6297	279	13	u	u	NOUN
ejpam-6297	279	14	∩v	∩v	NOUN
ejpam-6297	279	15	=	=	PUNCT
ejpam-6297	279	16	u	u	NOUN
ejpam-6297	279	17	∩x	∩x	NOUN
ejpam-6297	279	18	=	=	SYM
ejpam-6297	279	19	u	u	NOUN
ejpam-6297	279	20	and	and	CCONJ
ejpam-6297	279	21	so	so	ADV
ejpam-6297	279	22	a	a	PRON
ejpam-6297	279	23	is	be	AUX
ejpam-6297	279	24	strong	strong	ADJ
ejpam-6297	279	25	β	β	NOUN
ejpam-6297	279	26	-	-	ADJ
ejpam-6297	279	27	i	i	PRON
ejpam-6297	279	28	-	-	PUNCT
ejpam-6297	279	29	open	open	ADJ
ejpam-6297	279	30	.	.	PUNCT
ejpam-6297	280	1	consequently	consequently	ADV
ejpam-6297	280	2	,	,	PUNCT
ejpam-6297	280	3	(	(	PUNCT
ejpam-6297	280	4	x	x	X
ejpam-6297	280	5	,	,	PUNCT
ejpam-6297	280	6	τ	τ	PROPN
ejpam-6297	280	7	,	,	PUNCT
ejpam-6297	280	8	i	i	PROPN
ejpam-6297	280	9	)	)	PUNCT
ejpam-6297	280	10	is	be	AUX
ejpam-6297	280	11	strong	strong	ADJ
ejpam-6297	280	12	β	β	NOUN
ejpam-6297	280	13	-	-	ADJ
ejpam-6297	280	14	i	i	NOUN
ejpam-6297	280	15	-	-	PUNCT
ejpam-6297	280	16	submaximal	submaximal	ADJ
ejpam-6297	280	17	.	.	PUNCT
ejpam-6297	281	1	3	3	X
ejpam-6297	281	2	.	.	X
ejpam-6297	281	3	stong	stong	ADJ
ejpam-6297	281	4	β	β	PROPN
ejpam-6297	281	5	-	-	PROPN
ejpam-6297	281	6	i	i	NOUN
ejpam-6297	281	7	-	-	PUNCT
ejpam-6297	281	8	paracompactness	paracompactness	PROPN
ejpam-6297	281	9	and	and	CCONJ
ejpam-6297	281	10	characterizations	characterization	NOUN
ejpam-6297	281	11	this	this	DET
ejpam-6297	281	12	section	section	NOUN
ejpam-6297	281	13	explores	explore	VERB
ejpam-6297	281	14	the	the	DET
ejpam-6297	281	15	notion	notion	NOUN
ejpam-6297	281	16	of	of	ADP
ejpam-6297	281	17	sβ	sβ	PROPN
ejpam-6297	281	18	-	-	PUNCT
ejpam-6297	281	19	i	i	NOUN
ejpam-6297	281	20	-	-	PUNCT
ejpam-6297	281	21	paracompactness	paracompactness	NOUN
ejpam-6297	281	22	,	,	PUNCT
ejpam-6297	281	23	which	which	PRON
ejpam-6297	281	24	is	be	AUX
ejpam-6297	281	25	a	a	DET
ejpam-6297	281	26	variant	variant	NOUN
ejpam-6297	281	27	of	of	ADP
ejpam-6297	281	28	the	the	DET
ejpam-6297	281	29	i	i	PROPN
ejpam-6297	281	30	-	-	PUNCT
ejpam-6297	281	31	β	β	NOUN
ejpam-6297	281	32	-	-	ADJ
ejpam-6297	281	33	paracompactness	paracompactness	NOUN
ejpam-6297	281	34	concept	concept	NOUN
ejpam-6297	281	35	introduced	introduce	VERB
ejpam-6297	281	36	by	by	ADP
ejpam-6297	281	37	yildirim	yildirim	PROPN
ejpam-6297	281	38	et	et	PROPN
ejpam-6297	281	39	al	al	PROPN
ejpam-6297	281	40	.	.	PUNCT
ejpam-6297	282	1	[	[	X
ejpam-6297	282	2	30	30	NUM
ejpam-6297	282	3	]	]	PUNCT
ejpam-6297	282	4	.	.	PUNCT
ejpam-6297	283	1	we	we	PRON
ejpam-6297	283	2	aim	aim	VERB
ejpam-6297	283	3	to	to	PART
ejpam-6297	283	4	present	present	VERB
ejpam-6297	283	5	a	a	DET
ejpam-6297	283	6	formal	formal	ADJ
ejpam-6297	283	7	description	description	NOUN
ejpam-6297	283	8	of	of	ADP
ejpam-6297	283	9	this	this	DET
ejpam-6297	283	10	concept	concept	NOUN
ejpam-6297	283	11	.	.	PUNCT
ejpam-6297	284	1	let	let	VERB
ejpam-6297	284	2	a	a	DET
ejpam-6297	284	3	=	=	X
ejpam-6297	284	4	{	{	PUNCT
ejpam-6297	284	5	uα	uα	X
ejpam-6297	284	6	:	:	PUNCT
ejpam-6297	284	7	α	α	PROPN
ejpam-6297	284	8	∈	∈	PROPN
ejpam-6297	284	9	λ1	λ1	PROPN
ejpam-6297	284	10	}	}	PUNCT
ejpam-6297	284	11	and	and	CCONJ
ejpam-6297	284	12	b	b	X
ejpam-6297	284	13	=	=	PRON
ejpam-6297	284	14	{	{	PUNCT
ejpam-6297	284	15	vµ	vµ	X
ejpam-6297	284	16	:	:	PUNCT
ejpam-6297	284	17	µ	µ	PROPN
ejpam-6297	284	18	∈	∈	PROPN
ejpam-6297	284	19	λ2	λ2	PROPN
ejpam-6297	284	20	}	}	PUNCT
ejpam-6297	284	21	be	be	VERB
ejpam-6297	284	22	two	two	NUM
ejpam-6297	284	23	families	family	NOUN
ejpam-6297	284	24	of	of	ADP
ejpam-6297	284	25	subsets	subset	NOUN
ejpam-6297	284	26	in	in	ADP
ejpam-6297	284	27	a	a	DET
ejpam-6297	284	28	topological	topological	ADJ
ejpam-6297	284	29	space	space	NOUN
ejpam-6297	284	30	x.	x.	NOUN
ejpam-6297	285	1	we	we	PRON
ejpam-6297	285	2	say	say	VERB
ejpam-6297	285	3	that	that	SCONJ
ejpam-6297	285	4	a	a	PRON
ejpam-6297	285	5	is	be	AUX
ejpam-6297	285	6	a	a	DET
ejpam-6297	285	7	refinement	refinement	NOUN
ejpam-6297	285	8	of	of	ADP
ejpam-6297	285	9	b	b	NOUN
ejpam-6297	285	10	if	if	SCONJ
ejpam-6297	285	11	for	for	ADP
ejpam-6297	285	12	every	every	DET
ejpam-6297	285	13	α	α	PROPN
ejpam-6297	285	14	∈	∈	PROPN
ejpam-6297	285	15	λ1	λ1	NOUN
ejpam-6297	285	16	,	,	PUNCT
ejpam-6297	285	17	there	there	PRON
ejpam-6297	285	18	exists	exist	VERB
ejpam-6297	285	19	µ	µ	PRON
ejpam-6297	285	20	∈	∈	NOUN
ejpam-6297	285	21	λ2	λ2	NOUN
ejpam-6297	285	22	such	such	ADJ
ejpam-6297	285	23	that	that	SCONJ
ejpam-6297	285	24	uα	uα	PROPN
ejpam-6297	285	25	⊆	⊆	NUM
ejpam-6297	285	26	vµ.	vµ.	VERB
ejpam-6297	285	27	a	a	DET
ejpam-6297	285	28	family	family	NOUN
ejpam-6297	285	29	a	a	PRON
ejpam-6297	285	30	of	of	ADP
ejpam-6297	285	31	subsets	subset	NOUN
ejpam-6297	285	32	of	of	ADP
ejpam-6297	285	33	a	a	DET
ejpam-6297	285	34	topological	topological	ADJ
ejpam-6297	285	35	space	space	NOUN
ejpam-6297	285	36	(	(	PUNCT
ejpam-6297	285	37	x	x	X
ejpam-6297	285	38	,	,	PUNCT
ejpam-6297	285	39	τ	τ	X
ejpam-6297	285	40	)	)	PUNCT
ejpam-6297	285	41	is	be	AUX
ejpam-6297	285	42	called	call	VERB
ejpam-6297	285	43	β	β	ADJ
ejpam-6297	285	44	-	-	PUNCT
ejpam-6297	285	45	locally	locally	ADV
ejpam-6297	285	46	finite	finite	NOUN
ejpam-6297	285	47	[	[	X
ejpam-6297	285	48	14	14	NUM
ejpam-6297	285	49	]	]	PUNCT
ejpam-6297	285	50	if	if	SCONJ
ejpam-6297	285	51	,	,	PUNCT
ejpam-6297	285	52	for	for	ADP
ejpam-6297	285	53	each	each	DET
ejpam-6297	285	54	point	point	NOUN
ejpam-6297	285	55	x	x	X
ejpam-6297	285	56	∈	∈	NOUN
ejpam-6297	285	57	x	x	NOUN
ejpam-6297	285	58	,	,	PUNCT
ejpam-6297	285	59	there	there	PRON
ejpam-6297	285	60	exists	exist	VERB
ejpam-6297	285	61	a	a	DET
ejpam-6297	285	62	β	β	NOUN
ejpam-6297	285	63	-	-	ADJ
ejpam-6297	285	64	open	open	ADJ
ejpam-6297	285	65	neighborhood	neighborhood	NOUN
ejpam-6297	285	66	u	u	NOUN
ejpam-6297	285	67	of	of	ADP
ejpam-6297	285	68	x	x	PRON
ejpam-6297	285	69	that	that	PRON
ejpam-6297	285	70	intersects	intersect	VERB
ejpam-6297	285	71	only	only	ADV
ejpam-6297	285	72	finitely	finitely	ADV
ejpam-6297	285	73	many	many	ADJ
ejpam-6297	285	74	sets	set	NOUN
ejpam-6297	285	75	from	from	ADP
ejpam-6297	285	76	a.	a.	PROPN
ejpam-6297	285	77	yildirim	yildirim	PROPN
ejpam-6297	285	78	et	et	PROPN
ejpam-6297	285	79	al	al	PROPN
ejpam-6297	285	80	.	.	PUNCT
ejpam-6297	286	1	[	[	X
ejpam-6297	286	2	30	30	NUM
ejpam-6297	286	3	]	]	PUNCT
ejpam-6297	286	4	defined	define	VERB
ejpam-6297	286	5	an	an	DET
ejpam-6297	286	6	ideal	ideal	ADJ
ejpam-6297	286	7	topological	topological	ADJ
ejpam-6297	286	8	space	space	NOUN
ejpam-6297	286	9	(	(	PUNCT
ejpam-6297	286	10	x	x	X
ejpam-6297	286	11	,	,	PUNCT
ejpam-6297	286	12	τ	τ	PROPN
ejpam-6297	286	13	,	,	PUNCT
ejpam-6297	286	14	i	i	PROPN
ejpam-6297	286	15	)	)	PUNCT
ejpam-6297	286	16	to	to	PART
ejpam-6297	286	17	be	be	AUX
ejpam-6297	286	18	i	i	NOUN
ejpam-6297	286	19	-	-	PUNCT
ejpam-6297	286	20	β	β	NOUN
ejpam-6297	286	21	-	-	NOUN
ejpam-6297	286	22	paracompact	paracompact	ADJ
ejpam-6297	286	23	if	if	SCONJ
ejpam-6297	286	24	every	every	DET
ejpam-6297	286	25	open	open	ADJ
ejpam-6297	286	26	cover	cover	NOUN
ejpam-6297	286	27	of	of	ADP
ejpam-6297	286	28	x	x	PUNCT
ejpam-6297	286	29	has	have	VERB
ejpam-6297	286	30	a	a	DET
ejpam-6297	286	31	β	β	NOUN
ejpam-6297	286	32	-	-	ADJ
ejpam-6297	286	33	locally	locally	ADV
ejpam-6297	286	34	finite	finite	PROPN
ejpam-6297	286	35	refinement	refinement	NOUN
ejpam-6297	286	36	v	v	ADP
ejpam-6297	286	37	consisting	consist	VERB
ejpam-6297	286	38	of	of	ADP
ejpam-6297	286	39	β	β	ADJ
ejpam-6297	286	40	-	-	ADJ
ejpam-6297	286	41	open	open	ADJ
ejpam-6297	286	42	sets	set	NOUN
ejpam-6297	286	43	,	,	PUNCT
ejpam-6297	286	44	and	and	CCONJ
ejpam-6297	286	45	the	the	DET
ejpam-6297	286	46	set	set	NOUN
ejpam-6297	286	47	x	x	PUNCT
ejpam-6297	286	48	−	−	PUNCT
ejpam-6297	286	49	∪{v	∪{v	NOUN
ejpam-6297	286	50	:	:	PUNCT
ejpam-6297	286	51	v	v	NUM
ejpam-6297	286	52	∈	∈	PROPN
ejpam-6297	286	53	v	v	NOUN
ejpam-6297	286	54	}	}	PUNCT
ejpam-6297	286	55	is	be	AUX
ejpam-6297	286	56	an	an	DET
ejpam-6297	286	57	element	element	NOUN
ejpam-6297	286	58	of	of	ADP
ejpam-6297	286	59	i.	i.	PROPN
ejpam-6297	286	60	definition	definition	NOUN
ejpam-6297	286	61	8	8	NUM
ejpam-6297	286	62	.	.	PUNCT
ejpam-6297	287	1	a	a	DET
ejpam-6297	287	2	collection	collection	NOUN
ejpam-6297	287	3	a	a	PRON
ejpam-6297	287	4	of	of	ADP
ejpam-6297	287	5	subsets	subset	NOUN
ejpam-6297	287	6	of	of	ADP
ejpam-6297	287	7	an	an	DET
ejpam-6297	287	8	ideal	ideal	ADJ
ejpam-6297	287	9	topological	topological	ADJ
ejpam-6297	287	10	space	space	NOUN
ejpam-6297	287	11	(	(	PUNCT
ejpam-6297	287	12	x	x	X
ejpam-6297	287	13	,	,	PUNCT
ejpam-6297	287	14	τ	τ	PROPN
ejpam-6297	287	15	,	,	PUNCT
ejpam-6297	287	16	i	i	PROPN
ejpam-6297	287	17	)	)	PUNCT
ejpam-6297	287	18	is	be	AUX
ejpam-6297	287	19	said	say	VERB
ejpam-6297	287	20	to	to	PART
ejpam-6297	287	21	be	be	AUX
ejpam-6297	287	22	sβ	sβ	NOUN
ejpam-6297	287	23	-	-	PUNCT
ejpam-6297	287	24	i	i	PRON
ejpam-6297	287	25	-	-	PUNCT
ejpam-6297	287	26	locally	locally	ADV
ejpam-6297	287	27	finite	finite	NOUN
ejpam-6297	287	28	if	if	SCONJ
ejpam-6297	287	29	for	for	ADP
ejpam-6297	287	30	each	each	DET
ejpam-6297	287	31	x	x	SYM
ejpam-6297	287	32	∈	∈	PROPN
ejpam-6297	287	33	x	x	X
ejpam-6297	287	34	,	,	PUNCT
ejpam-6297	287	35	there	there	PRON
ejpam-6297	287	36	exists	exist	VERB
ejpam-6297	287	37	a	a	DET
ejpam-6297	287	38	strong	strong	ADJ
ejpam-6297	287	39	β	β	X
ejpam-6297	287	40	-	-	ADJ
ejpam-6297	287	41	i	i	PRON
ejpam-6297	287	42	-	-	PUNCT
ejpam-6297	287	43	open	open	ADJ
ejpam-6297	287	44	set	set	NOUN
ejpam-6297	287	45	u	u	NOUN
ejpam-6297	287	46	containing	contain	VERB
ejpam-6297	287	47	x	x	X
ejpam-6297	287	48	and	and	CCONJ
ejpam-6297	287	49	u	u	NOUN
ejpam-6297	287	50	intersects	intersect	NOUN
ejpam-6297	287	51	at	at	ADV
ejpam-6297	287	52	most	most	ADV
ejpam-6297	287	53	finitely	finitely	ADV
ejpam-6297	287	54	many	many	ADJ
ejpam-6297	287	55	members	member	NOUN
ejpam-6297	287	56	of	of	ADP
ejpam-6297	287	57	a.	a.	PROPN
ejpam-6297	287	58	lemma	lemma	PROPN
ejpam-6297	287	59	5	5	X
ejpam-6297	287	60	.	.	PUNCT
ejpam-6297	287	61	let	let	VERB
ejpam-6297	287	62	a	a	PRON
ejpam-6297	287	63	be	be	AUX
ejpam-6297	287	64	a	a	DET
ejpam-6297	287	65	collection	collection	NOUN
ejpam-6297	287	66	of	of	ADP
ejpam-6297	287	67	subsets	subset	NOUN
ejpam-6297	287	68	of	of	ADP
ejpam-6297	287	69	an	an	DET
ejpam-6297	287	70	ideal	ideal	ADJ
ejpam-6297	287	71	topological	topological	ADJ
ejpam-6297	287	72	space	space	NOUN
ejpam-6297	287	73	(	(	PUNCT
ejpam-6297	287	74	x	x	X
ejpam-6297	287	75	,	,	PUNCT
ejpam-6297	287	76	τ	τ	PROPN
ejpam-6297	287	77	,	,	PUNCT
ejpam-6297	287	78	i	i	PROPN
ejpam-6297	287	79	)	)	PUNCT
ejpam-6297	287	80	.	.	PUNCT
ejpam-6297	288	1	if	if	SCONJ
ejpam-6297	288	2	a	a	PRON
ejpam-6297	288	3	is	be	AUX
ejpam-6297	288	4	sβ	sβ	NOUN
ejpam-6297	288	5	-	-	PUNCT
ejpam-6297	288	6	i	i	NOUN
ejpam-6297	288	7	-	-	PUNCT
ejpam-6297	288	8	locally	locally	ADV
ejpam-6297	288	9	finite	finite	NOUN
ejpam-6297	288	10	,	,	PUNCT
ejpam-6297	288	11	then	then	ADV
ejpam-6297	288	12	it	it	PRON
ejpam-6297	288	13	is	be	AUX
ejpam-6297	288	14	β	β	ADJ
ejpam-6297	288	15	-	-	ADJ
ejpam-6297	288	16	locally	locally	ADV
ejpam-6297	288	17	finite	finite	NOUN
ejpam-6297	288	18	.	.	PUNCT
ejpam-6297	289	1	proof	proof	NOUN
ejpam-6297	289	2	.	.	PUNCT
ejpam-6297	290	1	assume	assume	VERB
ejpam-6297	290	2	that	that	SCONJ
ejpam-6297	290	3	a	a	PRON
ejpam-6297	290	4	is	be	AUX
ejpam-6297	290	5	sβ	sβ	NOUN
ejpam-6297	290	6	-	-	PUNCT
ejpam-6297	290	7	i	i	NOUN
ejpam-6297	290	8	-	-	PUNCT
ejpam-6297	290	9	locally	locally	ADV
ejpam-6297	290	10	finite	finite	NOUN
ejpam-6297	290	11	.	.	PUNCT
ejpam-6297	291	1	we	we	PRON
ejpam-6297	291	2	aim	aim	VERB
ejpam-6297	291	3	to	to	PART
ejpam-6297	291	4	show	show	VERB
ejpam-6297	291	5	that	that	SCONJ
ejpam-6297	291	6	a	a	PRON
ejpam-6297	291	7	is	be	AUX
ejpam-6297	291	8	β	β	X
ejpam-6297	291	9	-	-	ADJ
ejpam-6297	291	10	locally	locally	ADV
ejpam-6297	291	11	finite	finite	NOUN
ejpam-6297	291	12	.	.	PUNCT
ejpam-6297	292	1	let	let	VERB
ejpam-6297	292	2	x	x	SYM
ejpam-6297	292	3	∈	∈	PROPN
ejpam-6297	292	4	x.	x.	NOUN
ejpam-6297	292	5	since	since	SCONJ
ejpam-6297	292	6	a	a	PRON
ejpam-6297	292	7	is	be	AUX
ejpam-6297	292	8	sβ	sβ	NOUN
ejpam-6297	292	9	-	-	PUNCT
ejpam-6297	292	10	i	i	NOUN
ejpam-6297	292	11	-	-	PUNCT
ejpam-6297	292	12	locally	locally	ADV
ejpam-6297	292	13	finite	finite	NOUN
ejpam-6297	292	14	,	,	PUNCT
ejpam-6297	292	15	there	there	PRON
ejpam-6297	292	16	exists	exist	VERB
ejpam-6297	292	17	a	a	DET
ejpam-6297	292	18	strong	strong	ADJ
ejpam-6297	292	19	β	β	X
ejpam-6297	292	20	-	-	ADJ
ejpam-6297	292	21	i	i	PRON
ejpam-6297	292	22	-	-	PUNCT
ejpam-6297	292	23	open	open	ADV
ejpam-6297	292	24	set	set	VERB
ejpam-6297	292	25	gx	gx	PROPN
ejpam-6297	292	26	containing	contain	VERB
ejpam-6297	292	27	x	x	PUNCT
ejpam-6297	292	28	that	that	DET
ejpam-6297	292	29	intersects	intersect	VERB
ejpam-6297	292	30	only	only	ADV
ejpam-6297	292	31	finitely	finitely	ADV
ejpam-6297	292	32	many	many	ADJ
ejpam-6297	292	33	members	member	NOUN
ejpam-6297	292	34	of	of	ADP
ejpam-6297	292	35	a.	a.	NOUN
ejpam-6297	292	36	because	because	SCONJ
ejpam-6297	292	37	every	every	DET
ejpam-6297	292	38	strong	strong	ADJ
ejpam-6297	292	39	β	β	X
ejpam-6297	292	40	-	-	ADJ
ejpam-6297	292	41	i	i	NOUN
ejpam-6297	292	42	-	-	PUNCT
ejpam-6297	292	43	open	open	ADJ
ejpam-6297	292	44	set	set	NOUN
ejpam-6297	292	45	is	be	AUX
ejpam-6297	292	46	also	also	ADV
ejpam-6297	292	47	β	β	NOUN
ejpam-6297	292	48	-	-	ADJ
ejpam-6297	292	49	open	open	ADJ
ejpam-6297	292	50	,	,	PUNCT
ejpam-6297	292	51	gx	gx	PROPN
ejpam-6297	292	52	is	be	AUX
ejpam-6297	292	53	a	a	DET
ejpam-6297	292	54	β	β	NOUN
ejpam-6297	292	55	-	-	ADJ
ejpam-6297	292	56	open	open	ADJ
ejpam-6297	292	57	neighborhood	neighborhood	NOUN
ejpam-6297	292	58	of	of	ADP
ejpam-6297	292	59	x	x	PUNCT
ejpam-6297	292	60	with	with	ADP
ejpam-6297	292	61	the	the	DET
ejpam-6297	292	62	same	same	ADJ
ejpam-6297	292	63	finiteness	finiteness	NOUN
ejpam-6297	292	64	property	property	NOUN
ejpam-6297	292	65	.	.	PUNCT
ejpam-6297	293	1	thus	thus	ADV
ejpam-6297	293	2	,	,	PUNCT
ejpam-6297	293	3	a	a	PRON
ejpam-6297	293	4	is	be	AUX
ejpam-6297	293	5	β	β	X
ejpam-6297	293	6	-	-	ADJ
ejpam-6297	293	7	locally	locally	ADV
ejpam-6297	293	8	finite	finite	NOUN
ejpam-6297	293	9	.	.	PUNCT
ejpam-6297	294	1	c.	c.	PROPN
ejpam-6297	294	2	boonpok	boonpok	PROPN
ejpam-6297	294	3	,	,	PUNCT
ejpam-6297	294	4	p.	p.	PROPN
ejpam-6297	294	5	raktaow	raktaow	NOUN
ejpam-6297	294	6	,	,	PUNCT
ejpam-6297	294	7	a.	a.	PROPN
ejpam-6297	294	8	sama	sama	PROPN
ejpam-6297	294	9	-	-	PUNCT
ejpam-6297	294	10	ae	ae	PROPN
ejpam-6297	294	11	/	/	SYM
ejpam-6297	294	12	eur	eur	PROPN
ejpam-6297	294	13	.	.	PUNCT
ejpam-6297	295	1	j.	j.	PROPN
ejpam-6297	295	2	pure	pure	PROPN
ejpam-6297	295	3	appl	appl	PROPN
ejpam-6297	295	4	.	.	PROPN
ejpam-6297	295	5	math	math	PROPN
ejpam-6297	295	6	,	,	PUNCT
ejpam-6297	295	7	18	18	NUM
ejpam-6297	295	8	(	(	PUNCT
ejpam-6297	295	9	3	3	NUM
ejpam-6297	295	10	)	)	PUNCT
ejpam-6297	295	11	(	(	PUNCT
ejpam-6297	295	12	2025	2025	NUM
ejpam-6297	295	13	)	)	PUNCT
ejpam-6297	295	14	,	,	PUNCT
ejpam-6297	295	15	6297	6297	NUM
ejpam-6297	295	16	12	12	NUM
ejpam-6297	295	17	of	of	ADP
ejpam-6297	295	18	23	23	NUM
ejpam-6297	295	19	definition	definition	NOUN
ejpam-6297	295	20	9	9	NUM
ejpam-6297	295	21	.	.	PUNCT
ejpam-6297	296	1	an	an	DET
ejpam-6297	296	2	ideal	ideal	ADJ
ejpam-6297	296	3	topological	topological	ADJ
ejpam-6297	296	4	space	space	NOUN
ejpam-6297	296	5	(	(	PUNCT
ejpam-6297	296	6	x	x	X
ejpam-6297	296	7	,	,	PUNCT
ejpam-6297	296	8	τ	τ	PROPN
ejpam-6297	296	9	,	,	PUNCT
ejpam-6297	296	10	i	i	PROPN
ejpam-6297	296	11	)	)	PUNCT
ejpam-6297	296	12	is	be	AUX
ejpam-6297	296	13	said	say	VERB
ejpam-6297	296	14	to	to	PART
ejpam-6297	296	15	be	be	AUX
ejpam-6297	296	16	sβ	sβ	NOUN
ejpam-6297	296	17	-	-	PUNCT
ejpam-6297	296	18	i	i	NOUN
ejpam-6297	296	19	-	-	NOUN
ejpam-6297	296	20	paracompact	paracompact	ADJ
ejpam-6297	296	21	if	if	SCONJ
ejpam-6297	296	22	every	every	DET
ejpam-6297	296	23	open	open	ADJ
ejpam-6297	296	24	cover	cover	NOUN
ejpam-6297	296	25	of	of	ADP
ejpam-6297	296	26	x	x	PUNCT
ejpam-6297	296	27	has	have	VERB
ejpam-6297	296	28	an	an	DET
ejpam-6297	296	29	sβ	sβ	NOUN
ejpam-6297	296	30	-	-	PUNCT
ejpam-6297	296	31	i	i	NOUN
ejpam-6297	296	32	-	-	PUNCT
ejpam-6297	296	33	locally	locally	ADV
ejpam-6297	296	34	finite	finite	PROPN
ejpam-6297	296	35	refinement	refinement	NOUN
ejpam-6297	296	36	a	a	DET
ejpam-6297	296	37	consisting	consisting	NOUN
ejpam-6297	296	38	of	of	ADP
ejpam-6297	296	39	strong	strong	ADJ
ejpam-6297	296	40	β	β	X
ejpam-6297	296	41	-	-	ADJ
ejpam-6297	296	42	i	i	NOUN
ejpam-6297	296	43	-	-	PUNCT
ejpam-6297	296	44	open	open	ADJ
ejpam-6297	296	45	sets	set	NOUN
ejpam-6297	296	46	(	(	PUNCT
ejpam-6297	296	47	not	not	PART
ejpam-6297	296	48	necessarily	necessarily	ADV
ejpam-6297	296	49	a	a	DET
ejpam-6297	296	50	cover	cover	NOUN
ejpam-6297	296	51	)	)	PUNCT
ejpam-6297	296	52	such	such	ADJ
ejpam-6297	296	53	that	that	SCONJ
ejpam-6297	296	54	x	x	X
ejpam-6297	297	1	−	−	NOUN
ejpam-6297	297	2	∪{v	∪{v	NOUN
ejpam-6297	297	3	:	:	PUNCT
ejpam-6297	297	4	v	v	NUM
ejpam-6297	297	5	∈	∈	PROPN
ejpam-6297	297	6	a	a	DET
ejpam-6297	297	7	}	}	PUNCT
ejpam-6297	297	8	∈	∈	PROPN
ejpam-6297	297	9	i.	i.	NOUN
ejpam-6297	297	10	the	the	DET
ejpam-6297	297	11	collection	collection	NOUN
ejpam-6297	297	12	a	a	PRON
ejpam-6297	297	13	of	of	ADP
ejpam-6297	297	14	subsets	subset	NOUN
ejpam-6297	297	15	of	of	ADP
ejpam-6297	297	16	x	x	PUNCT
ejpam-6297	297	17	such	such	ADJ
ejpam-6297	297	18	that	that	SCONJ
ejpam-6297	297	19	x	x	X
ejpam-6297	298	1	−	−	NOUN
ejpam-6297	298	2	∪{v	∪{v	NOUN
ejpam-6297	298	3	:	:	PUNCT
ejpam-6297	298	4	v	v	NUM
ejpam-6297	298	5	∈	∈	PROPN
ejpam-6297	298	6	a	a	DET
ejpam-6297	298	7	}	}	PUNCT
ejpam-6297	298	8	∈	∈	NOUN
ejpam-6297	298	9	i	i	PRON
ejpam-6297	298	10	is	be	AUX
ejpam-6297	298	11	called	call	VERB
ejpam-6297	298	12	an	an	DET
ejpam-6297	298	13	i	i	NOUN
ejpam-6297	298	14	-	-	NOUN
ejpam-6297	298	15	cover	cover	NOUN
ejpam-6297	298	16	.	.	PUNCT
ejpam-6297	299	1	a	a	DET
ejpam-6297	299	2	subset	subset	NOUN
ejpam-6297	299	3	a	a	PRON
ejpam-6297	299	4	of	of	ADP
ejpam-6297	299	5	an	an	DET
ejpam-6297	299	6	ideal	ideal	ADJ
ejpam-6297	299	7	topological	topological	ADJ
ejpam-6297	299	8	space	space	NOUN
ejpam-6297	299	9	(	(	PUNCT
ejpam-6297	299	10	x	x	X
ejpam-6297	299	11	,	,	PUNCT
ejpam-6297	299	12	τ	τ	PROPN
ejpam-6297	299	13	,	,	PUNCT
ejpam-6297	299	14	i	i	PROPN
ejpam-6297	299	15	)	)	PUNCT
ejpam-6297	299	16	is	be	AUX
ejpam-6297	299	17	said	say	VERB
ejpam-6297	299	18	to	to	PART
ejpam-6297	299	19	be	be	AUX
ejpam-6297	299	20	sβ	sβ	NOUN
ejpam-6297	299	21	-	-	PUNCT
ejpam-6297	299	22	i	i	NOUN
ejpam-6297	299	23	-	-	NOUN
ejpam-6297	299	24	paracompact	paracompact	NOUN
ejpam-6297	299	25	if	if	SCONJ
ejpam-6297	299	26	for	for	ADP
ejpam-6297	299	27	any	any	DET
ejpam-6297	299	28	open	open	ADJ
ejpam-6297	299	29	cover	cover	NOUN
ejpam-6297	299	30	of	of	ADP
ejpam-6297	299	31	a	a	PRON
ejpam-6297	299	32	has	have	VERB
ejpam-6297	299	33	an	an	DET
ejpam-6297	299	34	sβ	sβ	NOUN
ejpam-6297	299	35	-	-	PUNCT
ejpam-6297	299	36	i	i	NOUN
ejpam-6297	299	37	-	-	PUNCT
ejpam-6297	299	38	locally	locally	ADV
ejpam-6297	299	39	finite	finite	PROPN
ejpam-6297	299	40	refinement	refinement	NOUN
ejpam-6297	299	41	a	a	DET
ejpam-6297	299	42	consisting	consisting	NOUN
ejpam-6297	299	43	of	of	ADP
ejpam-6297	299	44	strong	strong	ADJ
ejpam-6297	299	45	β	β	X
ejpam-6297	299	46	-	-	ADJ
ejpam-6297	299	47	i	i	NOUN
ejpam-6297	299	48	-	-	PUNCT
ejpam-6297	299	49	open	open	ADJ
ejpam-6297	299	50	sets	set	VERB
ejpam-6297	299	51	such	such	ADJ
ejpam-6297	299	52	that	that	DET
ejpam-6297	299	53	a−	a−	PROPN
ejpam-6297	299	54	∪{v	∪{v	PROPN
ejpam-6297	299	55	:	:	PUNCT
ejpam-6297	299	56	v	v	NUM
ejpam-6297	299	57	∈	∈	PROPN
ejpam-6297	299	58	a	a	DET
ejpam-6297	299	59	}	}	PUNCT
ejpam-6297	299	60	∈	∈	PROPN
ejpam-6297	299	61	i.	i.	NOUN
ejpam-6297	299	62	the	the	DET
ejpam-6297	299	63	two	two	NUM
ejpam-6297	299	64	theorems	theorem	NOUN
ejpam-6297	299	65	that	that	PRON
ejpam-6297	299	66	follow	follow	VERB
ejpam-6297	299	67	arise	arise	NOUN
ejpam-6297	299	68	from	from	ADP
ejpam-6297	299	69	the	the	DET
ejpam-6297	299	70	fact	fact	NOUN
ejpam-6297	299	71	that	that	SCONJ
ejpam-6297	299	72	every	every	DET
ejpam-6297	299	73	open	open	ADJ
ejpam-6297	299	74	set	set	NOUN
ejpam-6297	299	75	is	be	AUX
ejpam-6297	299	76	strong	strong	ADJ
ejpam-6297	299	77	β	β	NOUN
ejpam-6297	299	78	-	-	ADJ
ejpam-6297	299	79	i	i	PRON
ejpam-6297	299	80	-	-	PUNCT
ejpam-6297	299	81	open	open	ADJ
ejpam-6297	299	82	,	,	PUNCT
ejpam-6297	299	83	every	every	DET
ejpam-6297	299	84	strong	strong	ADJ
ejpam-6297	299	85	β	β	X
ejpam-6297	299	86	-	-	ADJ
ejpam-6297	299	87	i	i	NOUN
ejpam-6297	299	88	-	-	PUNCT
ejpam-6297	299	89	open	open	ADJ
ejpam-6297	299	90	set	set	NOUN
ejpam-6297	299	91	is	be	AUX
ejpam-6297	299	92	β	β	NOUN
ejpam-6297	299	93	-	-	ADJ
ejpam-6297	299	94	open	open	ADJ
ejpam-6297	299	95	and	and	CCONJ
ejpam-6297	299	96	∅	∅	NOUN
ejpam-6297	299	97	is	be	AUX
ejpam-6297	299	98	in	in	ADP
ejpam-6297	299	99	any	any	DET
ejpam-6297	299	100	ideal	ideal	NOUN
ejpam-6297	299	101	.	.	PUNCT
ejpam-6297	300	1	theorem	theorem	NOUN
ejpam-6297	300	2	6	6	NUM
ejpam-6297	300	3	.	.	PUNCT
ejpam-6297	301	1	if	if	SCONJ
ejpam-6297	301	2	a	a	DET
ejpam-6297	301	3	topological	topological	ADJ
ejpam-6297	301	4	space	space	NOUN
ejpam-6297	301	5	(	(	PUNCT
ejpam-6297	301	6	x	x	X
ejpam-6297	301	7	,	,	PUNCT
ejpam-6297	301	8	τ	τ	X
ejpam-6297	301	9	)	)	PUNCT
ejpam-6297	301	10	is	be	AUX
ejpam-6297	301	11	paracompact	paracompact	ADJ
ejpam-6297	301	12	,	,	PUNCT
ejpam-6297	301	13	then	then	ADV
ejpam-6297	301	14	(	(	PUNCT
ejpam-6297	301	15	x	x	X
ejpam-6297	301	16	,	,	PUNCT
ejpam-6297	301	17	τ	τ	PROPN
ejpam-6297	301	18	,	,	PUNCT
ejpam-6297	301	19	i	i	PROPN
ejpam-6297	301	20	)	)	PUNCT
ejpam-6297	301	21	is	be	AUX
ejpam-6297	301	22	sβ	sβ	PROPN
ejpam-6297	301	23	-	-	PUNCT
ejpam-6297	301	24	i	i	NOUN
ejpam-6297	301	25	-	-	PUNCT
ejpam-6297	301	26	paracompact	paracompact	ADJ
ejpam-6297	301	27	.	.	PUNCT
ejpam-6297	302	1	proof	proof	NOUN
ejpam-6297	302	2	.	.	PUNCT
ejpam-6297	303	1	it	it	PRON
ejpam-6297	303	2	is	be	AUX
ejpam-6297	303	3	evident	evident	ADJ
ejpam-6297	303	4	,	,	PUNCT
ejpam-6297	303	5	as	as	SCONJ
ejpam-6297	303	6	∅	∅	NOUN
ejpam-6297	303	7	∈	∈	PROPN
ejpam-6297	303	8	i.	i.	NOUN
ejpam-6297	303	9	theorem	theorem	VERB
ejpam-6297	303	10	7	7	NUM
ejpam-6297	303	11	.	.	PUNCT
ejpam-6297	304	1	if	if	SCONJ
ejpam-6297	304	2	(	(	PUNCT
ejpam-6297	304	3	x	x	X
ejpam-6297	304	4	,	,	PUNCT
ejpam-6297	304	5	τ	τ	PROPN
ejpam-6297	304	6	,	,	PUNCT
ejpam-6297	304	7	i	i	PROPN
ejpam-6297	304	8	)	)	PUNCT
ejpam-6297	304	9	is	be	AUX
ejpam-6297	304	10	sβ	sβ	PROPN
ejpam-6297	304	11	-	-	PUNCT
ejpam-6297	304	12	i	i	NOUN
ejpam-6297	304	13	-	-	NOUN
ejpam-6297	304	14	paracompact	paracompact	NOUN
ejpam-6297	304	15	then	then	ADV
ejpam-6297	304	16	it	it	PRON
ejpam-6297	304	17	is	be	AUX
ejpam-6297	304	18	i	i	PROPN
ejpam-6297	304	19	-	-	PUNCT
ejpam-6297	304	20	β	β	NOUN
ejpam-6297	304	21	-	-	NOUN
ejpam-6297	304	22	paracompact	paracompact	ADJ
ejpam-6297	304	23	.	.	PUNCT
ejpam-6297	305	1	proof	proof	NOUN
ejpam-6297	305	2	.	.	PUNCT
ejpam-6297	306	1	every	every	DET
ejpam-6297	306	2	sβ	sβ	PROPN
ejpam-6297	306	3	-	-	PUNCT
ejpam-6297	306	4	i	i	NOUN
ejpam-6297	306	5	-	-	PUNCT
ejpam-6297	306	6	locally	locally	ADV
ejpam-6297	306	7	finite	finite	ADJ
ejpam-6297	306	8	collection	collection	NOUN
ejpam-6297	306	9	of	of	ADP
ejpam-6297	306	10	subsets	subset	NOUN
ejpam-6297	306	11	of	of	ADP
ejpam-6297	306	12	x	x	PUNCT
ejpam-6297	306	13	is	be	AUX
ejpam-6297	306	14	β	β	X
ejpam-6297	306	15	-	-	ADJ
ejpam-6297	306	16	locally	locally	ADV
ejpam-6297	306	17	finite	finite	NOUN
ejpam-6297	306	18	,	,	PUNCT
ejpam-6297	306	19	as	as	SCONJ
ejpam-6297	306	20	demonstrated	demonstrate	VERB
ejpam-6297	306	21	by	by	ADP
ejpam-6297	306	22	lemma	lemma	PROPN
ejpam-6297	306	23	5	5	NUM
ejpam-6297	306	24	.	.	PUNCT
ejpam-6297	307	1	furthermore	furthermore	ADV
ejpam-6297	307	2	,	,	PUNCT
ejpam-6297	307	3	each	each	DET
ejpam-6297	307	4	strong	strong	ADJ
ejpam-6297	307	5	β	β	X
ejpam-6297	307	6	-	-	ADJ
ejpam-6297	307	7	i	i	NOUN
ejpam-6297	307	8	-	-	PUNCT
ejpam-6297	307	9	open	open	ADJ
ejpam-6297	307	10	set	set	NOUN
ejpam-6297	307	11	is	be	AUX
ejpam-6297	307	12	a	a	DET
ejpam-6297	307	13	β	β	NOUN
ejpam-6297	307	14	-	-	ADJ
ejpam-6297	307	15	open	open	ADJ
ejpam-6297	307	16	set	set	NOUN
ejpam-6297	307	17	.	.	PUNCT
ejpam-6297	308	1	we	we	PRON
ejpam-6297	308	2	can	can	AUX
ejpam-6297	308	3	continue	continue	VERB
ejpam-6297	308	4	with	with	ADP
ejpam-6297	308	5	the	the	DET
ejpam-6297	308	6	proof	proof	NOUN
ejpam-6297	308	7	by	by	ADP
ejpam-6297	308	8	following	follow	VERB
ejpam-6297	308	9	to	to	ADP
ejpam-6297	308	10	the	the	DET
ejpam-6297	308	11	definitions	definition	NOUN
ejpam-6297	308	12	of	of	ADP
ejpam-6297	308	13	β	β	NOUN
ejpam-6297	308	14	-	-	ADJ
ejpam-6297	308	15	i	i	NOUN
ejpam-6297	308	16	-	-	PUNCT
ejpam-6297	308	17	paracompactness	paracompactness	PROPN
ejpam-6297	308	18	and	and	CCONJ
ejpam-6297	308	19	sβ	sβ	NOUN
ejpam-6297	308	20	-	-	PUNCT
ejpam-6297	308	21	iparacompactness	iparacompactness	NOUN
ejpam-6297	308	22	.	.	PUNCT
ejpam-6297	309	1	consider	consider	VERB
ejpam-6297	309	2	the	the	DET
ejpam-6297	309	3	set	set	NOUN
ejpam-6297	309	4	n	n	PROPN
ejpam-6297	309	5	of	of	ADP
ejpam-6297	309	6	all	all	DET
ejpam-6297	309	7	positive	positive	ADJ
ejpam-6297	309	8	integers	integer	NOUN
ejpam-6297	309	9	.	.	PUNCT
ejpam-6297	310	1	define	define	VERB
ejpam-6297	310	2	a	a	DET
ejpam-6297	310	3	topology	topology	NOUN
ejpam-6297	310	4	τ	τ	PROPN
ejpam-6297	310	5	on	on	ADP
ejpam-6297	310	6	n	n	X
ejpam-6297	310	7	by	by	ADP
ejpam-6297	310	8	τ	τ	X
ejpam-6297	310	9	=	=	SYM
ejpam-6297	310	10	{	{	PUNCT
ejpam-6297	310	11	∅	∅	NOUN
ejpam-6297	310	12	}	}	PUNCT
ejpam-6297	310	13	∪	∪	NOUN
ejpam-6297	310	14	{	{	PUNCT
ejpam-6297	310	15	n	n	CCONJ
ejpam-6297	310	16	}	}	PUNCT
ejpam-6297	310	17	∪	∪	X
ejpam-6297	310	18	{	{	PUNCT
ejpam-6297	310	19	{	{	PUNCT
ejpam-6297	310	20	1	1	NUM
ejpam-6297	310	21	,	,	PUNCT
ejpam-6297	310	22	2	2	NUM
ejpam-6297	310	23	,	,	PUNCT
ejpam-6297	310	24	.	.	PUNCT
ejpam-6297	310	25	.	.	PUNCT
ejpam-6297	311	1	.	.	PUNCT
ejpam-6297	312	1	,	,	PUNCT
ejpam-6297	312	2	n	n	CCONJ
ejpam-6297	312	3	}	}	PUNCT
ejpam-6297	312	4	:	:	PUNCT
ejpam-6297	312	5	n	n	X
ejpam-6297	312	6	∈	∈	PROPN
ejpam-6297	312	7	n	n	CCONJ
ejpam-6297	312	8	}	}	PUNCT
ejpam-6297	312	9	.	.	PUNCT
ejpam-6297	313	1	let	let	VERB
ejpam-6297	313	2	i	i	PRON
ejpam-6297	313	3	=	=	PRON
ejpam-6297	313	4	{	{	PUNCT
ejpam-6297	313	5	a	a	DET
ejpam-6297	313	6	⊆	⊆	NUM
ejpam-6297	313	7	n	n	NOUN
ejpam-6297	313	8	:	:	PUNCT
ejpam-6297	313	9	1	1	NUM
ejpam-6297	313	10	/∈	/∈	SYM
ejpam-6297	313	11	a	a	X
ejpam-6297	313	12	}	}	PUNCT
ejpam-6297	313	13	.	.	PUNCT
ejpam-6297	314	1	it	it	PRON
ejpam-6297	314	2	is	be	AUX
ejpam-6297	314	3	easy	easy	ADJ
ejpam-6297	314	4	to	to	PART
ejpam-6297	314	5	see	see	VERB
ejpam-6297	314	6	that	that	SCONJ
ejpam-6297	314	7	i	i	PRON
ejpam-6297	314	8	is	be	AUX
ejpam-6297	314	9	an	an	DET
ejpam-6297	314	10	ideal	ideal	NOUN
ejpam-6297	314	11	on	on	ADP
ejpam-6297	314	12	n.	n.	NOUN
ejpam-6297	314	13	consider	consider	VERB
ejpam-6297	314	14	the	the	DET
ejpam-6297	314	15	open	open	ADJ
ejpam-6297	314	16	cover	cover	NOUN
ejpam-6297	314	17	u	u	NOUN
ejpam-6297	314	18	=	=	PUNCT
ejpam-6297	314	19	{	{	PUNCT
ejpam-6297	314	20	{	{	PUNCT
ejpam-6297	314	21	1	1	NUM
ejpam-6297	314	22	,	,	PUNCT
ejpam-6297	314	23	2	2	NUM
ejpam-6297	314	24	,	,	PUNCT
ejpam-6297	314	25	3	3	NUM
ejpam-6297	314	26	,	,	PUNCT
ejpam-6297	314	27	.	.	PUNCT
ejpam-6297	314	28	.	.	PUNCT
ejpam-6297	315	1	.	.	PUNCT
ejpam-6297	316	1	,	,	PUNCT
ejpam-6297	316	2	n	n	CCONJ
ejpam-6297	316	3	}	}	PUNCT
ejpam-6297	316	4	:	:	PUNCT
ejpam-6297	316	5	n	n	X
ejpam-6297	316	6	∈	∈	PROPN
ejpam-6297	316	7	n	n	CCONJ
ejpam-6297	316	8	}	}	PUNCT
ejpam-6297	316	9	of	of	ADP
ejpam-6297	316	10	n.	n.	NOUN
ejpam-6297	316	11	it	it	PRON
ejpam-6297	316	12	is	be	AUX
ejpam-6297	316	13	evident	evident	ADJ
ejpam-6297	316	14	that	that	SCONJ
ejpam-6297	316	15	there	there	PRON
ejpam-6297	316	16	does	do	AUX
ejpam-6297	316	17	not	not	PART
ejpam-6297	316	18	exist	exist	VERB
ejpam-6297	316	19	a	a	DET
ejpam-6297	316	20	locally	locally	ADV
ejpam-6297	316	21	finite	finite	ADJ
ejpam-6297	316	22	open	open	ADJ
ejpam-6297	316	23	refinement	refinement	PROPN
ejpam-6297	316	24	v	v	PROPN
ejpam-6297	316	25	of	of	ADP
ejpam-6297	316	26	u	u	PRON
ejpam-6297	316	27	that	that	PRON
ejpam-6297	316	28	still	still	ADV
ejpam-6297	316	29	covers	cover	VERB
ejpam-6297	316	30	n.	n.	NOUN
ejpam-6297	316	31	therefore	therefore	ADV
ejpam-6297	316	32	,	,	PUNCT
ejpam-6297	316	33	the	the	DET
ejpam-6297	316	34	space	space	NOUN
ejpam-6297	316	35	(	(	PUNCT
ejpam-6297	316	36	n	n	CCONJ
ejpam-6297	316	37	,	,	PUNCT
ejpam-6297	316	38	τ	τ	X
ejpam-6297	316	39	)	)	PUNCT
ejpam-6297	316	40	is	be	AUX
ejpam-6297	316	41	not	not	PART
ejpam-6297	316	42	paracompact	paracompact	ADJ
ejpam-6297	316	43	.	.	PUNCT
ejpam-6297	317	1	nevertheless	nevertheless	ADV
ejpam-6297	317	2	,	,	PUNCT
ejpam-6297	317	3	n	n	PRON
ejpam-6297	317	4	is	be	AUX
ejpam-6297	317	5	an	an	DET
ejpam-6297	317	6	sβ	sβ	PROPN
ejpam-6297	317	7	-	-	PUNCT
ejpam-6297	317	8	i	i	NOUN
ejpam-6297	317	9	-	-	PUNCT
ejpam-6297	317	10	paracompact	paracompact	ADJ
ejpam-6297	317	11	space	space	NOUN
ejpam-6297	317	12	.	.	PUNCT
ejpam-6297	318	1	for	for	ADP
ejpam-6297	318	2	any	any	DET
ejpam-6297	318	3	open	open	ADJ
ejpam-6297	318	4	cover	cover	NOUN
ejpam-6297	318	5	u	u	NOUN
ejpam-6297	318	6	of	of	ADP
ejpam-6297	318	7	n	n	CCONJ
ejpam-6297	318	8	,	,	PUNCT
ejpam-6297	318	9	there	there	PRON
ejpam-6297	318	10	exists	exist	VERB
ejpam-6297	318	11	refinement	refinement	NOUN
ejpam-6297	318	12	v	v	ADP
ejpam-6297	318	13	=	=	PRON
ejpam-6297	318	14	{	{	PUNCT
ejpam-6297	318	15	{	{	PUNCT
ejpam-6297	318	16	1	1	NUM
ejpam-6297	318	17	}	}	PUNCT
ejpam-6297	318	18	}	}	PUNCT
ejpam-6297	318	19	of	of	ADP
ejpam-6297	318	20	a	a	DET
ejpam-6297	318	21	strong	strong	ADJ
ejpam-6297	318	22	β	β	X
ejpam-6297	318	23	-	-	ADJ
ejpam-6297	318	24	i	i	PRON
ejpam-6297	318	25	-	-	PUNCT
ejpam-6297	318	26	open	open	ADJ
ejpam-6297	318	27	set	set	NOUN
ejpam-6297	318	28	{	{	PUNCT
ejpam-6297	318	29	1	1	NUM
ejpam-6297	318	30	}	}	PUNCT
ejpam-6297	318	31	that	that	SCONJ
ejpam-6297	318	32	v	v	NOUN
ejpam-6297	318	33	is	be	AUX
ejpam-6297	318	34	sβ	sβ	NOUN
ejpam-6297	318	35	-	-	PUNCT
ejpam-6297	318	36	i	i	NOUN
ejpam-6297	318	37	-	-	PUNCT
ejpam-6297	318	38	locally	locally	ADV
ejpam-6297	318	39	finite	finite	NOUN
ejpam-6297	318	40	.	.	PUNCT
ejpam-6297	319	1	this	this	PRON
ejpam-6297	319	2	holds	hold	VERB
ejpam-6297	319	3	because	because	SCONJ
ejpam-6297	319	4	the	the	DET
ejpam-6297	319	5	complement	complement	NOUN
ejpam-6297	319	6	n−	n−	NOUN
ejpam-6297	319	7	{	{	PUNCT
ejpam-6297	319	8	1	1	NUM
ejpam-6297	319	9	}	}	PUNCT
ejpam-6297	319	10	=	=	PUNCT
ejpam-6297	319	11	{	{	PUNCT
ejpam-6297	319	12	2	2	NUM
ejpam-6297	319	13	,	,	PUNCT
ejpam-6297	319	14	3	3	NUM
ejpam-6297	319	15	,	,	PUNCT
ejpam-6297	319	16	.	.	PUNCT
ejpam-6297	319	17	.	.	PUNCT
ejpam-6297	319	18	.	.	PUNCT
ejpam-6297	320	1	}	}	PUNCT
ejpam-6297	320	2	∈	∈	PROPN
ejpam-6297	321	1	i	i	PRON
ejpam-6297	321	2	,	,	PUNCT
ejpam-6297	321	3	satisfying	satisfy	VERB
ejpam-6297	321	4	the	the	DET
ejpam-6297	321	5	required	required	ADJ
ejpam-6297	321	6	condition	condition	NOUN
ejpam-6297	321	7	.	.	PUNCT
ejpam-6297	322	1	theorem	theorem	ADJ
ejpam-6297	322	2	8	8	NUM
ejpam-6297	322	3	.	.	PUNCT
ejpam-6297	323	1	let	let	VERB
ejpam-6297	323	2	(	(	PUNCT
ejpam-6297	323	3	x	x	X
ejpam-6297	323	4	,	,	PUNCT
ejpam-6297	323	5	τ	τ	PROPN
ejpam-6297	323	6	,	,	PUNCT
ejpam-6297	323	7	i	i	PRON
ejpam-6297	323	8	)	)	PUNCT
ejpam-6297	323	9	be	be	VERB
ejpam-6297	323	10	an	an	DET
ejpam-6297	323	11	ideal	ideal	ADJ
ejpam-6297	323	12	topological	topological	ADJ
ejpam-6297	323	13	space	space	NOUN
ejpam-6297	323	14	and	and	CCONJ
ejpam-6297	323	15	a	a	DET
ejpam-6297	323	16	⊆	⊆	NUM
ejpam-6297	323	17	x.	x.	NOUN
ejpam-6297	323	18	then	then	ADV
ejpam-6297	323	19	g∩sβ	g∩sβ	PROPN
ejpam-6297	323	20	cli(a	cli(a	PROPN
ejpam-6297	323	21	)	)	PUNCT
ejpam-6297	323	22	=	=	PUNCT
ejpam-6297	323	23	∅	∅	NOUN
ejpam-6297	323	24	if	if	SCONJ
ejpam-6297	323	25	and	and	CCONJ
ejpam-6297	323	26	only	only	ADV
ejpam-6297	323	27	if	if	SCONJ
ejpam-6297	323	28	g	g	PROPN
ejpam-6297	323	29	∩a	∩a	NOUN
ejpam-6297	323	30	=	=	NOUN
ejpam-6297	323	31	∅	∅	NOUN
ejpam-6297	323	32	,	,	PUNCT
ejpam-6297	323	33	for	for	ADP
ejpam-6297	323	34	all	all	DET
ejpam-6297	323	35	strong	strong	ADJ
ejpam-6297	323	36	β	β	X
ejpam-6297	323	37	-	-	ADJ
ejpam-6297	323	38	i	i	NOUN
ejpam-6297	323	39	-	-	PUNCT
ejpam-6297	323	40	open	open	NOUN
ejpam-6297	323	41	subset	subset	NOUN
ejpam-6297	323	42	g	g	NOUN
ejpam-6297	323	43	of	of	ADP
ejpam-6297	323	44	x.	x.	NOUN
ejpam-6297	323	45	proof	proof	NOUN
ejpam-6297	323	46	.	.	PUNCT
ejpam-6297	324	1	it	it	PRON
ejpam-6297	324	2	follows	follow	VERB
ejpam-6297	324	3	from	from	ADP
ejpam-6297	324	4	part	part	NOUN
ejpam-6297	324	5	(	(	PUNCT
ejpam-6297	324	6	7	7	NUM
ejpam-6297	324	7	)	)	PUNCT
ejpam-6297	324	8	of	of	ADP
ejpam-6297	324	9	lemma	lemma	PROPN
ejpam-6297	324	10	3	3	NUM
ejpam-6297	324	11	,	,	PUNCT
ejpam-6297	324	12	together	together	ADV
ejpam-6297	324	13	with	with	ADP
ejpam-6297	324	14	the	the	DET
ejpam-6297	324	15	fact	fact	NOUN
ejpam-6297	324	16	that	that	SCONJ
ejpam-6297	324	17	a	a	DET
ejpam-6297	324	18	⊆	⊆	NUM
ejpam-6297	324	19	sβ	sβ	PROPN
ejpam-6297	324	20	cli(a	cli(a	PROPN
ejpam-6297	324	21	)	)	PUNCT
ejpam-6297	324	22	.	.	PUNCT
ejpam-6297	325	1	theorem	theorem	NOUN
ejpam-6297	325	2	9	9	NUM
ejpam-6297	325	3	.	.	PUNCT
ejpam-6297	326	1	let	let	VERB
ejpam-6297	326	2	a	a	PRON
ejpam-6297	326	3	=	=	X
ejpam-6297	326	4	{	{	PUNCT
ejpam-6297	326	5	vλ	vλ	INTJ
ejpam-6297	326	6	:	:	PUNCT
ejpam-6297	326	7	λ	λ	PROPN
ejpam-6297	326	8	∈	∈	PROPN
ejpam-6297	326	9	λ	λ	PROPN
ejpam-6297	326	10	}	}	PUNCT
ejpam-6297	326	11	be	be	VERB
ejpam-6297	326	12	a	a	DET
ejpam-6297	326	13	collection	collection	NOUN
ejpam-6297	326	14	of	of	ADP
ejpam-6297	326	15	subsets	subset	NOUN
ejpam-6297	326	16	of	of	ADP
ejpam-6297	326	17	an	an	DET
ejpam-6297	326	18	ideal	ideal	ADJ
ejpam-6297	326	19	topological	topological	ADJ
ejpam-6297	326	20	space	space	NOUN
ejpam-6297	326	21	(	(	PUNCT
ejpam-6297	326	22	x	x	X
ejpam-6297	326	23	,	,	PUNCT
ejpam-6297	326	24	τ	τ	PROPN
ejpam-6297	326	25	,	,	PUNCT
ejpam-6297	326	26	i	i	PROPN
ejpam-6297	326	27	)	)	PUNCT
ejpam-6297	326	28	.	.	PUNCT
ejpam-6297	327	1	the	the	DET
ejpam-6297	327	2	following	follow	VERB
ejpam-6297	327	3	statements	statement	NOUN
ejpam-6297	327	4	are	be	AUX
ejpam-6297	327	5	true	true	ADJ
ejpam-6297	327	6	.	.	PUNCT
ejpam-6297	328	1	(	(	PUNCT
ejpam-6297	328	2	1	1	X
ejpam-6297	328	3	)	)	PUNCT
ejpam-6297	328	4	if	if	SCONJ
ejpam-6297	328	5	a	a	PRON
ejpam-6297	328	6	is	be	AUX
ejpam-6297	328	7	sβ	sβ	NOUN
ejpam-6297	328	8	-	-	PUNCT
ejpam-6297	328	9	i	i	NOUN
ejpam-6297	328	10	-	-	PUNCT
ejpam-6297	328	11	locally	locally	ADV
ejpam-6297	328	12	finite	finite	NOUN
ejpam-6297	328	13	and	and	CCONJ
ejpam-6297	328	14	hλ	hλ	ADV
ejpam-6297	328	15	⊆	⊆	NUM
ejpam-6297	328	16	vλ	vλ	ADJ
ejpam-6297	328	17	for	for	ADP
ejpam-6297	328	18	all	all	DET
ejpam-6297	328	19	λ	λ	PROPN
ejpam-6297	328	20	∈	∈	PROPN
ejpam-6297	328	21	λ	λ	PROPN
ejpam-6297	328	22	,	,	PUNCT
ejpam-6297	328	23	then	then	ADV
ejpam-6297	328	24	b	b	X
ejpam-6297	328	25	=	=	PRON
ejpam-6297	328	26	{	{	PUNCT
ejpam-6297	328	27	hλ	hλ	X
ejpam-6297	328	28	:	:	PUNCT
ejpam-6297	328	29	λ	λ	PROPN
ejpam-6297	328	30	∈	∈	PROPN
ejpam-6297	328	31	λ	λ	PROPN
ejpam-6297	328	32	}	}	PUNCT
ejpam-6297	328	33	is	be	AUX
ejpam-6297	328	34	sβ	sβ	NOUN
ejpam-6297	328	35	-	-	PUNCT
ejpam-6297	328	36	i	i	NOUN
ejpam-6297	328	37	-	-	PUNCT
ejpam-6297	328	38	locally	locally	ADV
ejpam-6297	328	39	finite	finite	NOUN
ejpam-6297	328	40	.	.	PUNCT
ejpam-6297	329	1	(	(	PUNCT
ejpam-6297	329	2	2	2	X
ejpam-6297	329	3	)	)	PUNCT
ejpam-6297	329	4	a	a	PRON
ejpam-6297	329	5	is	be	AUX
ejpam-6297	329	6	sβ	sβ	NOUN
ejpam-6297	329	7	-	-	PUNCT
ejpam-6297	329	8	i	i	NOUN
ejpam-6297	329	9	-	-	PUNCT
ejpam-6297	329	10	locally	locally	ADV
ejpam-6297	329	11	finite	finite	VERB
ejpam-6297	329	12	if	if	SCONJ
ejpam-6297	329	13	and	and	CCONJ
ejpam-6297	329	14	only	only	ADV
ejpam-6297	329	15	if	if	SCONJ
ejpam-6297	329	16	{	{	PUNCT
ejpam-6297	329	17	sβ	sβ	NOUN
ejpam-6297	329	18	cli(vλ	cli(vλ	NOUN
ejpam-6297	329	19	)	)	PUNCT
ejpam-6297	329	20	:	:	PUNCT
ejpam-6297	330	1	λ	λ	X
ejpam-6297	330	2	∈	∈	PROPN
ejpam-6297	330	3	λ	λ	PROPN
ejpam-6297	330	4	}	}	PUNCT
ejpam-6297	330	5	is	be	AUX
ejpam-6297	330	6	sβ	sβ	NOUN
ejpam-6297	330	7	-	-	PUNCT
ejpam-6297	330	8	i	i	NOUN
ejpam-6297	330	9	-	-	PUNCT
ejpam-6297	330	10	locally	locally	ADV
ejpam-6297	330	11	finite	finite	NOUN
ejpam-6297	330	12	.	.	PUNCT
ejpam-6297	331	1	c.	c.	PROPN
ejpam-6297	331	2	boonpok	boonpok	PROPN
ejpam-6297	331	3	,	,	PUNCT
ejpam-6297	331	4	p.	p.	PROPN
ejpam-6297	331	5	raktaow	raktaow	NOUN
ejpam-6297	331	6	,	,	PUNCT
ejpam-6297	331	7	a.	a.	PROPN
ejpam-6297	331	8	sama	sama	PROPN
ejpam-6297	331	9	-	-	PUNCT
ejpam-6297	331	10	ae	ae	PROPN
ejpam-6297	331	11	/	/	SYM
ejpam-6297	331	12	eur	eur	PROPN
ejpam-6297	331	13	.	.	PUNCT
ejpam-6297	332	1	j.	j.	PROPN
ejpam-6297	332	2	pure	pure	PROPN
ejpam-6297	332	3	appl	appl	PROPN
ejpam-6297	332	4	.	.	PROPN
ejpam-6297	332	5	math	math	PROPN
ejpam-6297	332	6	,	,	PUNCT
ejpam-6297	332	7	18	18	NUM
ejpam-6297	332	8	(	(	PUNCT
ejpam-6297	332	9	3	3	NUM
ejpam-6297	332	10	)	)	PUNCT
ejpam-6297	332	11	(	(	PUNCT
ejpam-6297	332	12	2025	2025	NUM
ejpam-6297	332	13	)	)	PUNCT
ejpam-6297	332	14	,	,	PUNCT
ejpam-6297	332	15	6297	6297	NUM
ejpam-6297	332	16	13	13	NUM
ejpam-6297	332	17	of	of	ADP
ejpam-6297	332	18	23	23	NUM
ejpam-6297	332	19	proof	proof	NOUN
ejpam-6297	332	20	.	.	PUNCT
ejpam-6297	333	1	(	(	PUNCT
ejpam-6297	333	2	1	1	NUM
ejpam-6297	333	3	):	):	PUNCT
ejpam-6297	333	4	let	let	VERB
ejpam-6297	333	5	x	x	PUNCT
ejpam-6297	333	6	∈	∈	PROPN
ejpam-6297	333	7	x.	x.	NOUN
ejpam-6297	333	8	since	since	SCONJ
ejpam-6297	333	9	a	a	PRON
ejpam-6297	333	10	is	be	AUX
ejpam-6297	333	11	sβ	sβ	NOUN
ejpam-6297	333	12	-	-	PUNCT
ejpam-6297	333	13	i	i	NOUN
ejpam-6297	333	14	-	-	PUNCT
ejpam-6297	333	15	locally	locally	ADV
ejpam-6297	333	16	finite	finite	NOUN
ejpam-6297	333	17	,	,	PUNCT
ejpam-6297	333	18	there	there	PRON
ejpam-6297	333	19	exists	exist	VERB
ejpam-6297	333	20	a	a	DET
ejpam-6297	333	21	strong	strong	ADJ
ejpam-6297	333	22	β	β	X
ejpam-6297	333	23	-	-	ADJ
ejpam-6297	333	24	i	i	PRON
ejpam-6297	333	25	-	-	PUNCT
ejpam-6297	333	26	open	open	ADJ
ejpam-6297	333	27	set	set	NOUN
ejpam-6297	333	28	u	u	NOUN
ejpam-6297	333	29	containing	contain	VERB
ejpam-6297	333	30	x	x	X
ejpam-6297	333	31	,	,	PUNCT
ejpam-6297	333	32	which	which	PRON
ejpam-6297	333	33	intersects	intersect	VERB
ejpam-6297	333	34	at	at	ADP
ejpam-6297	333	35	most	most	ADV
ejpam-6297	333	36	finitely	finitely	ADV
ejpam-6297	333	37	many	many	ADJ
ejpam-6297	333	38	elements	element	NOUN
ejpam-6297	333	39	of	of	ADP
ejpam-6297	333	40	a.	a.	NOUN
ejpam-6297	333	41	as	as	ADP
ejpam-6297	333	42	hλ	hλ	NOUN
ejpam-6297	333	43	⊆	⊆	NUM
ejpam-6297	333	44	vλ	vλ	ADJ
ejpam-6297	333	45	for	for	ADP
ejpam-6297	333	46	all	all	DET
ejpam-6297	333	47	λ	λ	PROPN
ejpam-6297	333	48	∈	∈	PROPN
ejpam-6297	333	49	λ	λ	PROPN
ejpam-6297	333	50	,	,	PUNCT
ejpam-6297	333	51	it	it	PRON
ejpam-6297	333	52	follows	follow	VERB
ejpam-6297	333	53	that	that	SCONJ
ejpam-6297	333	54	u	u	PRON
ejpam-6297	333	55	intersects	intersect	VERB
ejpam-6297	333	56	at	at	ADV
ejpam-6297	333	57	most	most	ADV
ejpam-6297	333	58	finitely	finitely	ADV
ejpam-6297	333	59	many	many	ADJ
ejpam-6297	333	60	of	of	ADP
ejpam-6297	333	61	the	the	DET
ejpam-6297	333	62	sets	set	NOUN
ejpam-6297	333	63	in	in	ADP
ejpam-6297	333	64	b	b	NOUN
ejpam-6297	333	65	=	=	PRON
ejpam-6297	333	66	{	{	PUNCT
ejpam-6297	333	67	hλ	hλ	X
ejpam-6297	333	68	:	:	PUNCT
ejpam-6297	333	69	λ	λ	PROPN
ejpam-6297	333	70	∈	∈	PROPN
ejpam-6297	333	71	λ	λ	NOUN
ejpam-6297	333	72	}	}	PUNCT
ejpam-6297	333	73	.	.	PUNCT
ejpam-6297	334	1	hence	hence	ADV
ejpam-6297	334	2	,	,	PUNCT
ejpam-6297	334	3	b	b	X
ejpam-6297	334	4	=	=	PRON
ejpam-6297	334	5	{	{	PUNCT
ejpam-6297	334	6	hλ	hλ	X
ejpam-6297	334	7	:	:	PUNCT
ejpam-6297	334	8	λ	λ	PROPN
ejpam-6297	334	9	∈	∈	PROPN
ejpam-6297	334	10	λ	λ	PROPN
ejpam-6297	334	11	}	}	PUNCT
ejpam-6297	334	12	is	be	AUX
ejpam-6297	334	13	sβ	sβ	NOUN
ejpam-6297	334	14	-	-	PUNCT
ejpam-6297	334	15	i	i	NOUN
ejpam-6297	334	16	-	-	PUNCT
ejpam-6297	334	17	locally	locally	ADV
ejpam-6297	334	18	finite	finite	NOUN
ejpam-6297	334	19	.	.	PUNCT
ejpam-6297	335	1	(	(	PUNCT
ejpam-6297	335	2	2	2	NUM
ejpam-6297	335	3	):	):	PUNCT
ejpam-6297	335	4	let	let	VERB
ejpam-6297	335	5	a	a	PRON
ejpam-6297	335	6	be	be	AUX
ejpam-6297	335	7	sβ	sβ	VERB
ejpam-6297	335	8	-	-	PUNCT
ejpam-6297	335	9	i	i	NOUN
ejpam-6297	335	10	-	-	PUNCT
ejpam-6297	335	11	locally	locally	ADV
ejpam-6297	335	12	finite	finite	NOUN
ejpam-6297	335	13	and	and	CCONJ
ejpam-6297	335	14	let	let	VERB
ejpam-6297	335	15	x	x	X
ejpam-6297	335	16	∈	∈	PROPN
ejpam-6297	335	17	x.	x.	NOUN
ejpam-6297	335	18	then	then	ADV
ejpam-6297	335	19	,	,	PUNCT
ejpam-6297	335	20	there	there	PRON
ejpam-6297	335	21	exists	exist	VERB
ejpam-6297	335	22	a	a	DET
ejpam-6297	335	23	strong	strong	ADJ
ejpam-6297	335	24	β	β	X
ejpam-6297	335	25	-	-	ADJ
ejpam-6297	335	26	i	i	PRON
ejpam-6297	335	27	-	-	PUNCT
ejpam-6297	335	28	open	open	ADJ
ejpam-6297	335	29	set	set	VERB
ejpam-6297	335	30	g	g	NOUN
ejpam-6297	335	31	containing	contain	VERB
ejpam-6297	335	32	x	x	PUNCT
ejpam-6297	335	33	that	that	PRON
ejpam-6297	335	34	satisfies	satisfy	VERB
ejpam-6297	335	35	g∩vλ	g∩vλ	PROPN
ejpam-6297	335	36	=	=	NOUN
ejpam-6297	335	37	∅	∅	NOUN
ejpam-6297	335	38	for	for	ADP
ejpam-6297	335	39	every	every	DET
ejpam-6297	335	40	λ	λ	PROPN
ejpam-6297	335	41	̸=	̸=	PROPN
ejpam-6297	335	42	λ1	λ1	PROPN
ejpam-6297	335	43	,	,	PUNCT
ejpam-6297	335	44	λ2	λ2	NOUN
ejpam-6297	335	45	,	,	PUNCT
ejpam-6297	335	46	.	.	PUNCT
ejpam-6297	335	47	.	.	PUNCT
ejpam-6297	336	1	.	.	PUNCT
ejpam-6297	337	1	,	,	PUNCT
ejpam-6297	337	2	λn	λn	NOUN
ejpam-6297	337	3	.	.	PUNCT
ejpam-6297	338	1	by	by	ADP
ejpam-6297	338	2	theorem	theorem	NOUN
ejpam-6297	338	3	8	8	NUM
ejpam-6297	338	4	,	,	PUNCT
ejpam-6297	338	5	we	we	PRON
ejpam-6297	338	6	obtain	obtain	VERB
ejpam-6297	338	7	that	that	SCONJ
ejpam-6297	338	8	g∩sβ	g∩sβ	PROPN
ejpam-6297	338	9	cli(vλ	cli(vλ	NOUN
ejpam-6297	338	10	)	)	PUNCT
ejpam-6297	338	11	=	=	NOUN
ejpam-6297	338	12	∅	∅	NOUN
ejpam-6297	338	13	for	for	ADP
ejpam-6297	338	14	every	every	DET
ejpam-6297	338	15	λ	λ	PROPN
ejpam-6297	338	16	̸=	̸=	PROPN
ejpam-6297	338	17	λ1	λ1	PROPN
ejpam-6297	338	18	,	,	PUNCT
ejpam-6297	338	19	λ2	λ2	NOUN
ejpam-6297	338	20	,	,	PUNCT
ejpam-6297	338	21	.	.	PUNCT
ejpam-6297	338	22	.	.	PUNCT
ejpam-6297	338	23	.	.	PUNCT
ejpam-6297	339	1	,	,	PUNCT
ejpam-6297	339	2	λn	λn	PROPN
ejpam-6297	339	3	.	.	PUNCT
ejpam-6297	340	1	therefore	therefore	ADV
ejpam-6297	340	2	,	,	PUNCT
ejpam-6297	340	3	{	{	PUNCT
ejpam-6297	340	4	sβ	sβ	NOUN
ejpam-6297	340	5	cli(vλ	cli(vλ	NOUN
ejpam-6297	340	6	)	)	PUNCT
ejpam-6297	340	7	:	:	PUNCT
ejpam-6297	341	1	λ	λ	X
ejpam-6297	341	2	∈	∈	PROPN
ejpam-6297	341	3	λ	λ	PROPN
ejpam-6297	341	4	}	}	PUNCT
ejpam-6297	341	5	is	be	AUX
ejpam-6297	341	6	sβ	sβ	NOUN
ejpam-6297	341	7	-	-	PUNCT
ejpam-6297	341	8	i	i	NOUN
ejpam-6297	341	9	-	-	PUNCT
ejpam-6297	341	10	locally	locally	ADV
ejpam-6297	341	11	finite	finite	NOUN
ejpam-6297	341	12	.	.	PUNCT
ejpam-6297	342	1	the	the	DET
ejpam-6297	342	2	converse	converse	NOUN
ejpam-6297	342	3	follows	follow	VERB
ejpam-6297	342	4	from	from	ADP
ejpam-6297	342	5	(	(	PUNCT
ejpam-6297	342	6	1	1	NUM
ejpam-6297	342	7	)	)	PUNCT
ejpam-6297	342	8	.	.	PUNCT
ejpam-6297	343	1	theorem	theorem	ADJ
ejpam-6297	343	2	10	10	NUM
ejpam-6297	343	3	.	.	PUNCT
ejpam-6297	344	1	if	if	SCONJ
ejpam-6297	344	2	(	(	PUNCT
ejpam-6297	344	3	x	x	X
ejpam-6297	344	4	,	,	PUNCT
ejpam-6297	344	5	τ	τ	PROPN
ejpam-6297	344	6	,	,	PUNCT
ejpam-6297	344	7	i	i	PROPN
ejpam-6297	344	8	)	)	PUNCT
ejpam-6297	344	9	is	be	AUX
ejpam-6297	344	10	sβ	sβ	PROPN
ejpam-6297	344	11	-	-	PUNCT
ejpam-6297	344	12	i	i	NOUN
ejpam-6297	344	13	-	-	PUNCT
ejpam-6297	344	14	paracompact	paracompact	PROPN
ejpam-6297	344	15	and	and	CCONJ
ejpam-6297	344	16	j	j	PROPN
ejpam-6297	344	17	is	be	AUX
ejpam-6297	344	18	an	an	DET
ejpam-6297	344	19	ideal	ideal	NOUN
ejpam-6297	344	20	on	on	ADP
ejpam-6297	344	21	x	x	PUNCT
ejpam-6297	344	22	with	with	ADP
ejpam-6297	344	23	i	i	PRON
ejpam-6297	344	24	⊆	⊆	NUM
ejpam-6297	344	25	j	j	PROPN
ejpam-6297	344	26	,	,	PUNCT
ejpam-6297	344	27	then	then	ADV
ejpam-6297	344	28	(	(	PUNCT
ejpam-6297	344	29	x	x	X
ejpam-6297	344	30	,	,	PUNCT
ejpam-6297	344	31	τ	τ	PROPN
ejpam-6297	344	32	,	,	PUNCT
ejpam-6297	344	33	j	j	PROPN
ejpam-6297	344	34	)	)	PUNCT
ejpam-6297	344	35	is	be	AUX
ejpam-6297	344	36	sβ	sβ	PROPN
ejpam-6297	344	37	-	-	PUNCT
ejpam-6297	344	38	j	j	NOUN
ejpam-6297	344	39	-paracompact	-paracompact	NOUN
ejpam-6297	344	40	.	.	PUNCT
ejpam-6297	345	1	proof	proof	NOUN
ejpam-6297	345	2	.	.	PUNCT
ejpam-6297	346	1	let	let	VERB
ejpam-6297	346	2	(	(	PUNCT
ejpam-6297	346	3	x	x	X
ejpam-6297	346	4	,	,	PUNCT
ejpam-6297	346	5	τ	τ	PROPN
ejpam-6297	346	6	,	,	PUNCT
ejpam-6297	346	7	i	i	PRON
ejpam-6297	346	8	)	)	PUNCT
ejpam-6297	346	9	be	be	VERB
ejpam-6297	346	10	sβ	sβ	VERB
ejpam-6297	346	11	-	-	PUNCT
ejpam-6297	346	12	i	i	NOUN
ejpam-6297	346	13	-	-	NOUN
ejpam-6297	346	14	paracompact	paracompact	ADJ
ejpam-6297	346	15	,	,	PUNCT
ejpam-6297	346	16	and	and	CCONJ
ejpam-6297	346	17	suppose	suppose	VERB
ejpam-6297	346	18	i	i	PRON
ejpam-6297	346	19	⊆	⊆	NUM
ejpam-6297	346	20	j	j	PROPN
ejpam-6297	346	21	.	.	PUNCT
ejpam-6297	347	1	let	let	VERB
ejpam-6297	347	2	a	a	PRON
ejpam-6297	347	3	=	=	X
ejpam-6297	347	4	{	{	PUNCT
ejpam-6297	347	5	uα	uα	X
ejpam-6297	347	6	:	:	PUNCT
ejpam-6297	347	7	α	α	PROPN
ejpam-6297	347	8	∈	∈	PROPN
ejpam-6297	347	9	λ	λ	NOUN
ejpam-6297	347	10	}	}	PUNCT
ejpam-6297	347	11	be	be	VERB
ejpam-6297	347	12	an	an	DET
ejpam-6297	347	13	open	open	ADJ
ejpam-6297	347	14	cover	cover	NOUN
ejpam-6297	347	15	of	of	ADP
ejpam-6297	347	16	x.	x.	NOUN
ejpam-6297	347	17	since	since	SCONJ
ejpam-6297	347	18	(	(	PUNCT
ejpam-6297	347	19	x	x	X
ejpam-6297	347	20	,	,	PUNCT
ejpam-6297	347	21	τ	τ	PROPN
ejpam-6297	347	22	,	,	PUNCT
ejpam-6297	347	23	i	i	PROPN
ejpam-6297	347	24	)	)	PUNCT
ejpam-6297	347	25	is	be	AUX
ejpam-6297	347	26	sβ	sβ	PROPN
ejpam-6297	347	27	-	-	PUNCT
ejpam-6297	347	28	i	i	NOUN
ejpam-6297	347	29	-	-	PUNCT
ejpam-6297	347	30	paracompact	paracompact	ADJ
ejpam-6297	347	31	,	,	PUNCT
ejpam-6297	347	32	by	by	ADP
ejpam-6297	347	33	definition	definition	NOUN
ejpam-6297	347	34	,	,	PUNCT
ejpam-6297	347	35	there	there	PRON
ejpam-6297	347	36	exists	exist	VERB
ejpam-6297	347	37	an	an	DET
ejpam-6297	347	38	sβ	sβ	NOUN
ejpam-6297	347	39	-	-	PUNCT
ejpam-6297	347	40	i	i	NOUN
ejpam-6297	347	41	-	-	PUNCT
ejpam-6297	347	42	locally	locally	ADV
ejpam-6297	347	43	finite	finite	NOUN
ejpam-6297	347	44	refinement	refinement	NOUN
ejpam-6297	347	45	a′	a′	PROPN
ejpam-6297	347	46	of	of	ADP
ejpam-6297	347	47	a	a	DET
ejpam-6297	347	48	consisting	consisting	NOUN
ejpam-6297	347	49	of	of	ADP
ejpam-6297	347	50	strong	strong	ADJ
ejpam-6297	347	51	β	β	X
ejpam-6297	347	52	-	-	ADJ
ejpam-6297	347	53	i	i	NOUN
ejpam-6297	347	54	-	-	PUNCT
ejpam-6297	347	55	open	open	ADJ
ejpam-6297	347	56	sets	set	VERB
ejpam-6297	347	57	such	such	ADJ
ejpam-6297	347	58	that	that	PRON
ejpam-6297	347	59	:	:	PUNCT
ejpam-6297	347	60	x	x	X
ejpam-6297	348	1	−	−	X
ejpam-6297	348	2	∪{v	∪{v	NOUN
ejpam-6297	348	3	:	:	PUNCT
ejpam-6297	348	4	v	v	NUM
ejpam-6297	348	5	∈	∈	NOUN
ejpam-6297	348	6	a′	a′	PROPN
ejpam-6297	348	7	}	}	PUNCT
ejpam-6297	348	8	∈	∈	PROPN
ejpam-6297	348	9	i.	i.	NOUN
ejpam-6297	348	10	because	because	SCONJ
ejpam-6297	348	11	i	i	PROPN
ejpam-6297	348	12	⊆	⊆	NUM
ejpam-6297	348	13	j	j	PROPN
ejpam-6297	348	14	,	,	PUNCT
ejpam-6297	348	15	it	it	PRON
ejpam-6297	348	16	follows	follow	VERB
ejpam-6297	348	17	that	that	SCONJ
ejpam-6297	348	18	:	:	PUNCT
ejpam-6297	348	19	x	x	X
ejpam-6297	348	20	−	−	X
ejpam-6297	348	21	∪{v	∪{v	NOUN
ejpam-6297	348	22	:	:	PUNCT
ejpam-6297	348	23	v	v	NUM
ejpam-6297	348	24	∈	∈	NOUN
ejpam-6297	348	25	a′	a′	PROPN
ejpam-6297	348	26	}	}	PUNCT
ejpam-6297	348	27	∈	∈	PROPN
ejpam-6297	348	28	j	j	PROPN
ejpam-6297	348	29	.	.	PUNCT
ejpam-6297	349	1	thus	thus	ADV
ejpam-6297	349	2	,	,	PUNCT
ejpam-6297	349	3	(	(	PUNCT
ejpam-6297	349	4	x	x	X
ejpam-6297	349	5	,	,	PUNCT
ejpam-6297	349	6	τ	τ	PROPN
ejpam-6297	349	7	,	,	PUNCT
ejpam-6297	349	8	j	j	PROPN
ejpam-6297	349	9	)	)	PUNCT
ejpam-6297	349	10	is	be	AUX
ejpam-6297	349	11	sβ	sβ	PROPN
ejpam-6297	349	12	-	-	PUNCT
ejpam-6297	349	13	j	j	NOUN
ejpam-6297	349	14	-paracompact	-paracompact	PROPN
ejpam-6297	349	15	.	.	PUNCT
ejpam-6297	350	1	lemma	lemma	PROPN
ejpam-6297	350	2	6	6	NUM
ejpam-6297	350	3	.	.	PUNCT
ejpam-6297	351	1	if	if	SCONJ
ejpam-6297	351	2	an	an	DET
ejpam-6297	351	3	open	open	ADJ
ejpam-6297	351	4	cover	cover	VERB
ejpam-6297	351	5	a	a	DET
ejpam-6297	351	6	=	=	X
ejpam-6297	351	7	{	{	PUNCT
ejpam-6297	351	8	uλ	uλ	NOUN
ejpam-6297	351	9	:	:	PUNCT
ejpam-6297	351	10	λ	λ	X
ejpam-6297	351	11	∈	∈	PROPN
ejpam-6297	351	12	λ	λ	NOUN
ejpam-6297	351	13	}	}	PUNCT
ejpam-6297	351	14	of	of	ADP
ejpam-6297	351	15	an	an	DET
ejpam-6297	351	16	ideal	ideal	ADJ
ejpam-6297	351	17	topological	topological	ADJ
ejpam-6297	351	18	space	space	NOUN
ejpam-6297	351	19	(	(	PUNCT
ejpam-6297	351	20	x	x	X
ejpam-6297	351	21	,	,	PUNCT
ejpam-6297	351	22	τ	τ	PROPN
ejpam-6297	351	23	,	,	PUNCT
ejpam-6297	351	24	i	i	NOUN
ejpam-6297	351	25	)	)	PUNCT
ejpam-6297	351	26	has	have	VERB
ejpam-6297	351	27	an	an	DET
ejpam-6297	351	28	sβ	sβ	NOUN
ejpam-6297	351	29	-	-	PUNCT
ejpam-6297	351	30	i	i	NOUN
ejpam-6297	351	31	-	-	PUNCT
ejpam-6297	351	32	locally	locally	ADV
ejpam-6297	351	33	finite	finite	VERB
ejpam-6297	351	34	strong	strong	ADJ
ejpam-6297	351	35	β	β	X
ejpam-6297	351	36	-	-	ADJ
ejpam-6297	351	37	i	i	NOUN
ejpam-6297	351	38	-	-	PUNCT
ejpam-6297	351	39	open	open	ADJ
ejpam-6297	351	40	refinement	refinement	NOUN
ejpam-6297	351	41	b	b	NOUN
ejpam-6297	351	42	such	such	ADJ
ejpam-6297	351	43	that	that	DET
ejpam-6297	351	44	x−∪{v	x−∪{v	NOUN
ejpam-6297	351	45	:	:	PUNCT
ejpam-6297	352	1	v	v	NUM
ejpam-6297	352	2	∈	∈	PROPN
ejpam-6297	352	3	b	b	X
ejpam-6297	352	4	}	}	PUNCT
ejpam-6297	352	5	∈	∈	PROPN
ejpam-6297	352	6	i	i	PRON
ejpam-6297	352	7	,	,	PUNCT
ejpam-6297	352	8	then	then	ADV
ejpam-6297	352	9	there	there	PRON
ejpam-6297	352	10	exists	exist	VERB
ejpam-6297	352	11	a	a	DET
ejpam-6297	352	12	precise	precise	ADJ
ejpam-6297	352	13	sβ	sβ	NOUN
ejpam-6297	352	14	-	-	PUNCT
ejpam-6297	352	15	i	i	NOUN
ejpam-6297	352	16	-	-	PUNCT
ejpam-6297	352	17	locally	locally	ADV
ejpam-6297	352	18	finite	finite	VERB
ejpam-6297	352	19	strong	strong	ADJ
ejpam-6297	352	20	β	β	X
ejpam-6297	352	21	-	-	ADJ
ejpam-6297	352	22	i	i	NOUN
ejpam-6297	352	23	-	-	PUNCT
ejpam-6297	352	24	open	open	ADJ
ejpam-6297	352	25	refinement	refinement	NOUN
ejpam-6297	352	26	c	c	NOUN
ejpam-6297	352	27	=	=	PRON
ejpam-6297	352	28	{	{	PUNCT
ejpam-6297	352	29	hλ	hλ	X
ejpam-6297	352	30	:	:	PUNCT
ejpam-6297	352	31	λ	λ	PROPN
ejpam-6297	352	32	∈	∈	PROPN
ejpam-6297	352	33	λ	λ	NOUN
ejpam-6297	352	34	}	}	PUNCT
ejpam-6297	352	35	of	of	ADP
ejpam-6297	352	36	a	a	DET
ejpam-6297	352	37	such	such	ADJ
ejpam-6297	352	38	that	that	SCONJ
ejpam-6297	352	39	x	x	PROPN
ejpam-6297	352	40	−	−	PROPN
ejpam-6297	352	41	∪{hλ	∪{hλ	PROPN
ejpam-6297	352	42	:	:	PUNCT
ejpam-6297	352	43	λ	λ	PROPN
ejpam-6297	352	44	∈	∈	PROPN
ejpam-6297	352	45	λ	λ	PROPN
ejpam-6297	352	46	}	}	PUNCT
ejpam-6297	352	47	∈	∈	PROPN
ejpam-6297	352	48	i.	i.	NOUN
ejpam-6297	352	49	proof	proof	NOUN
ejpam-6297	352	50	.	.	PUNCT
ejpam-6297	353	1	a	a	DET
ejpam-6297	353	2	similar	similar	ADJ
ejpam-6297	353	3	technique	technique	NOUN
ejpam-6297	353	4	is	be	AUX
ejpam-6297	353	5	employed	employ	VERB
ejpam-6297	353	6	as	as	ADP
ejpam-6297	353	7	in	in	ADP
ejpam-6297	353	8	the	the	DET
ejpam-6297	353	9	proof	proof	NOUN
ejpam-6297	353	10	of	of	ADP
ejpam-6297	353	11	lemma	lemma	PROPN
ejpam-6297	353	12	1.3	1.3	NUM
ejpam-6297	353	13	in	in	ADP
ejpam-6297	353	14	[	[	X
ejpam-6297	353	15	29	29	NUM
ejpam-6297	353	16	]	]	PUNCT
ejpam-6297	353	17	.	.	PUNCT
ejpam-6297	354	1	definition	definition	NOUN
ejpam-6297	354	2	10	10	NUM
ejpam-6297	354	3	.	.	PUNCT
ejpam-6297	355	1	an	an	DET
ejpam-6297	355	2	ideal	ideal	ADJ
ejpam-6297	355	3	topological	topological	ADJ
ejpam-6297	355	4	space	space	NOUN
ejpam-6297	355	5	(	(	PUNCT
ejpam-6297	355	6	x	x	X
ejpam-6297	355	7	,	,	PUNCT
ejpam-6297	355	8	τ	τ	PROPN
ejpam-6297	355	9	,	,	PUNCT
ejpam-6297	355	10	i	i	PROPN
ejpam-6297	355	11	)	)	PUNCT
ejpam-6297	355	12	is	be	AUX
ejpam-6297	355	13	sβ	sβ	PROPN
ejpam-6297	355	14	-	-	PUNCT
ejpam-6297	355	15	i	i	NOUN
ejpam-6297	355	16	-	-	PUNCT
ejpam-6297	355	17	regular	regular	ADJ
ejpam-6297	355	18	if	if	SCONJ
ejpam-6297	355	19	for	for	ADP
ejpam-6297	355	20	any	any	DET
ejpam-6297	355	21	closed	closed	ADJ
ejpam-6297	355	22	subset	subset	NOUN
ejpam-6297	355	23	f	f	PROPN
ejpam-6297	355	24	of	of	ADP
ejpam-6297	355	25	x	x	X
ejpam-6297	355	26	and	and	CCONJ
ejpam-6297	355	27	x	x	PUNCT
ejpam-6297	355	28	̸∈	̸∈	PROPN
ejpam-6297	355	29	f	f	PROPN
ejpam-6297	355	30	,	,	PUNCT
ejpam-6297	355	31	there	there	PRON
ejpam-6297	355	32	exist	exist	VERB
ejpam-6297	355	33	disjoint	disjoint	NOUN
ejpam-6297	355	34	strong	strong	ADJ
ejpam-6297	355	35	β	β	X
ejpam-6297	355	36	-	-	ADJ
ejpam-6297	355	37	i	i	NOUN
ejpam-6297	355	38	-	-	PUNCT
ejpam-6297	355	39	open	open	ADJ
ejpam-6297	355	40	sets	set	VERB
ejpam-6297	355	41	u	u	NOUN
ejpam-6297	355	42	and	and	CCONJ
ejpam-6297	355	43	v	v	ADP
ejpam-6297	355	44	such	such	ADJ
ejpam-6297	355	45	that	that	SCONJ
ejpam-6297	355	46	x	x	SYM
ejpam-6297	355	47	∈	∈	PROPN
ejpam-6297	355	48	u	u	NOUN
ejpam-6297	355	49	and	and	CCONJ
ejpam-6297	355	50	f	f	PROPN
ejpam-6297	355	51	−	−	PROPN
ejpam-6297	355	52	v	v	PROPN
ejpam-6297	355	53	∈	∈	PROPN
ejpam-6297	355	54	i.	i.	NOUN
ejpam-6297	355	55	theorem	theorem	VERB
ejpam-6297	355	56	11	11	NUM
ejpam-6297	355	57	.	.	PUNCT
ejpam-6297	356	1	if	if	SCONJ
ejpam-6297	356	2	(	(	PUNCT
ejpam-6297	356	3	x	x	X
ejpam-6297	356	4	,	,	PUNCT
ejpam-6297	356	5	τ	τ	PROPN
ejpam-6297	356	6	,	,	PUNCT
ejpam-6297	356	7	i	i	PROPN
ejpam-6297	356	8	)	)	PUNCT
ejpam-6297	356	9	is	be	AUX
ejpam-6297	356	10	sβ	sβ	PROPN
ejpam-6297	356	11	-	-	PUNCT
ejpam-6297	356	12	i	i	NOUN
ejpam-6297	356	13	-	-	PUNCT
ejpam-6297	356	14	paracompact	paracompact	NOUN
ejpam-6297	356	15	and	and	CCONJ
ejpam-6297	356	16	hausdorff	hausdorff	NOUN
ejpam-6297	356	17	,	,	PUNCT
ejpam-6297	356	18	then	then	ADV
ejpam-6297	356	19	(	(	PUNCT
ejpam-6297	356	20	x	x	X
ejpam-6297	356	21	,	,	PUNCT
ejpam-6297	356	22	τ	τ	PROPN
ejpam-6297	356	23	,	,	PUNCT
ejpam-6297	356	24	i	i	PROPN
ejpam-6297	356	25	)	)	PUNCT
ejpam-6297	356	26	is	be	AUX
ejpam-6297	356	27	sβ	sβ	ADJ
ejpam-6297	356	28	-	-	PUNCT
ejpam-6297	356	29	iregular	iregular	ADJ
ejpam-6297	356	30	.	.	PUNCT
ejpam-6297	357	1	proof	proof	NOUN
ejpam-6297	357	2	.	.	PUNCT
ejpam-6297	358	1	let	let	VERB
ejpam-6297	358	2	f	f	PRON
ejpam-6297	358	3	be	be	AUX
ejpam-6297	358	4	a	a	DET
ejpam-6297	358	5	closed	closed	ADJ
ejpam-6297	358	6	subset	subset	NOUN
ejpam-6297	358	7	of	of	ADP
ejpam-6297	358	8	x	x	X
ejpam-6297	358	9	,	,	PUNCT
ejpam-6297	358	10	and	and	CCONJ
ejpam-6297	358	11	let	let	VERB
ejpam-6297	358	12	x	x	PROPN
ejpam-6297	358	13	̸∈	̸∈	PROPN
ejpam-6297	358	14	f	f	PROPN
ejpam-6297	358	15	.	.	PUNCT
ejpam-6297	359	1	for	for	ADP
ejpam-6297	359	2	each	each	DET
ejpam-6297	359	3	y	y	PROPN
ejpam-6297	359	4	∈	∈	PROPN
ejpam-6297	359	5	f	f	PROPN
ejpam-6297	359	6	,	,	PUNCT
ejpam-6297	359	7	since	since	SCONJ
ejpam-6297	359	8	x	x	PRON
ejpam-6297	359	9	is	be	AUX
ejpam-6297	359	10	hausdorff	hausdorff	NOUN
ejpam-6297	359	11	,	,	PUNCT
ejpam-6297	359	12	there	there	PRON
ejpam-6297	359	13	exist	exist	VERB
ejpam-6297	359	14	disjoint	disjoint	ADJ
ejpam-6297	359	15	open	open	ADJ
ejpam-6297	359	16	sets	set	NOUN
ejpam-6297	359	17	vx	vx	PUNCT
ejpam-6297	359	18	and	and	CCONJ
ejpam-6297	359	19	oxy	oxy	ADJ
ejpam-6297	359	20	such	such	ADJ
ejpam-6297	359	21	that	that	SCONJ
ejpam-6297	359	22	x	x	SYM
ejpam-6297	359	23	∈	∈	PROPN
ejpam-6297	359	24	vx	vx	PROPN
ejpam-6297	359	25	and	and	CCONJ
ejpam-6297	359	26	y	y	PROPN
ejpam-6297	359	27	∈	∈	PROPN
ejpam-6297	359	28	oxy	oxy	PROPN
ejpam-6297	359	29	.	.	PUNCT
ejpam-6297	360	1	this	this	PRON
ejpam-6297	360	2	implies	imply	VERB
ejpam-6297	360	3	that	that	SCONJ
ejpam-6297	360	4	y	y	PROPN
ejpam-6297	360	5	̸∈	̸∈	PROPN
ejpam-6297	360	6	cl(vx	cl(vx	PROPN
ejpam-6297	360	7	)	)	PUNCT
ejpam-6297	360	8	.	.	PUNCT
ejpam-6297	361	1	now	now	ADV
ejpam-6297	361	2	,	,	PUNCT
ejpam-6297	361	3	consider	consider	VERB
ejpam-6297	361	4	the	the	DET
ejpam-6297	361	5	family	family	NOUN
ejpam-6297	361	6	a	a	X
ejpam-6297	361	7	=	=	X
ejpam-6297	361	8	{	{	PUNCT
ejpam-6297	361	9	oxy	oxy	NOUN
ejpam-6297	361	10	:	:	PUNCT
ejpam-6297	361	11	y	y	PROPN
ejpam-6297	361	12	∈	∈	PROPN
ejpam-6297	361	13	f	f	X
ejpam-6297	361	14	}	}	PUNCT
ejpam-6297	361	15	∪	∪	NOUN
ejpam-6297	361	16	{	{	PUNCT
ejpam-6297	361	17	x	x	NOUN
ejpam-6297	361	18	−	−	PROPN
ejpam-6297	361	19	f	f	X
ejpam-6297	361	20	}	}	PUNCT
ejpam-6297	361	21	,	,	PUNCT
ejpam-6297	361	22	which	which	PRON
ejpam-6297	361	23	is	be	AUX
ejpam-6297	361	24	an	an	DET
ejpam-6297	361	25	open	open	ADJ
ejpam-6297	361	26	cover	cover	NOUN
ejpam-6297	361	27	of	of	ADP
ejpam-6297	361	28	x.	x.	NOUN
ejpam-6297	361	29	by	by	ADP
ejpam-6297	361	30	assumption	assumption	NOUN
ejpam-6297	361	31	,	,	PUNCT
ejpam-6297	361	32	there	there	PRON
ejpam-6297	361	33	exists	exist	VERB
ejpam-6297	361	34	an	an	DET
ejpam-6297	361	35	sβ	sβ	NOUN
ejpam-6297	361	36	-	-	PUNCT
ejpam-6297	361	37	i	i	NOUN
ejpam-6297	361	38	-	-	PUNCT
ejpam-6297	361	39	locally	locally	ADV
ejpam-6297	361	40	finite	finite	VERB
ejpam-6297	361	41	strong	strong	ADJ
ejpam-6297	361	42	β	β	X
ejpam-6297	361	43	-	-	ADJ
ejpam-6297	361	44	i	i	NOUN
ejpam-6297	361	45	-	-	PUNCT
ejpam-6297	361	46	open	open	ADJ
ejpam-6297	362	1	refinement	refinement	NOUN
ejpam-6297	362	2	b	b	PROPN
ejpam-6297	362	3	=	=	PUNCT
ejpam-6297	362	4	{	{	PUNCT
ejpam-6297	362	5	hxy	hxy	NOUN
ejpam-6297	362	6	:	:	PUNCT
ejpam-6297	362	7	y	y	PROPN
ejpam-6297	362	8	∈	∈	PROPN
ejpam-6297	362	9	f	f	X
ejpam-6297	362	10	}	}	PUNCT
ejpam-6297	362	11	∪	∪	X
ejpam-6297	362	12	{	{	PUNCT
ejpam-6297	362	13	w	w	NOUN
ejpam-6297	362	14	}	}	PUNCT
ejpam-6297	362	15	such	such	ADJ
ejpam-6297	362	16	that	that	SCONJ
ejpam-6297	362	17	:	:	PUNCT
ejpam-6297	362	18	hxy	hxy	NOUN
ejpam-6297	362	19	⊆	⊆	NUM
ejpam-6297	362	20	oxy	oxy	NOUN
ejpam-6297	362	21	for	for	ADP
ejpam-6297	362	22	each	each	DET
ejpam-6297	362	23	y	y	PROPN
ejpam-6297	362	24	∈	∈	PROPN
ejpam-6297	362	25	f	f	PROPN
ejpam-6297	362	26	,	,	PUNCT
ejpam-6297	362	27	w	w	PROPN
ejpam-6297	362	28	⊆	⊆	NUM
ejpam-6297	362	29	x	x	SYM
ejpam-6297	362	30	−	−	PROPN
ejpam-6297	362	31	f	f	NOUN
ejpam-6297	362	32	,	,	PUNCT
ejpam-6297	362	33	and	and	CCONJ
ejpam-6297	362	34	x	x	X
ejpam-6297	362	35	−	−	PROPN
ejpam-6297	362	36	(	(	PUNCT
ejpam-6297	362	37	∪{hxy	∪{hxy	NOUN
ejpam-6297	362	38	:	:	PUNCT
ejpam-6297	362	39	y	y	PROPN
ejpam-6297	362	40	∈	∈	PROPN
ejpam-6297	362	41	f	f	X
ejpam-6297	362	42	}	}	PUNCT
ejpam-6297	362	43	∪	∪	X
ejpam-6297	362	44	{	{	PUNCT
ejpam-6297	362	45	w	w	NOUN
ejpam-6297	362	46	}	}	PUNCT
ejpam-6297	362	47	)	)	PUNCT
ejpam-6297	362	48	∈	∈	PROPN
ejpam-6297	362	49	i.	i.	PROPN
ejpam-6297	362	50	c.	c.	PROPN
ejpam-6297	362	51	boonpok	boonpok	PROPN
ejpam-6297	362	52	,	,	PUNCT
ejpam-6297	362	53	p.	p.	PROPN
ejpam-6297	362	54	raktaow	raktaow	NOUN
ejpam-6297	362	55	,	,	PUNCT
ejpam-6297	362	56	a.	a.	PROPN
ejpam-6297	362	57	sama	sama	PROPN
ejpam-6297	362	58	-	-	PUNCT
ejpam-6297	362	59	ae	ae	PROPN
ejpam-6297	362	60	/	/	SYM
ejpam-6297	362	61	eur	eur	PROPN
ejpam-6297	362	62	.	.	PUNCT
ejpam-6297	363	1	j.	j.	PROPN
ejpam-6297	363	2	pure	pure	PROPN
ejpam-6297	363	3	appl	appl	PROPN
ejpam-6297	363	4	.	.	PROPN
ejpam-6297	363	5	math	math	PROPN
ejpam-6297	363	6	,	,	PUNCT
ejpam-6297	363	7	18	18	NUM
ejpam-6297	363	8	(	(	PUNCT
ejpam-6297	363	9	3	3	NUM
ejpam-6297	363	10	)	)	PUNCT
ejpam-6297	363	11	(	(	PUNCT
ejpam-6297	363	12	2025	2025	NUM
ejpam-6297	363	13	)	)	PUNCT
ejpam-6297	363	14	,	,	PUNCT
ejpam-6297	363	15	6297	6297	NUM
ejpam-6297	363	16	14	14	NUM
ejpam-6297	363	17	of	of	ADP
ejpam-6297	363	18	23	23	NUM
ejpam-6297	363	19	next	next	ADJ
ejpam-6297	363	20	,	,	PUNCT
ejpam-6297	363	21	let	let	VERB
ejpam-6297	363	22	us	we	PRON
ejpam-6297	363	23	define	define	VERB
ejpam-6297	363	24	the	the	DET
ejpam-6297	363	25	sets	set	NOUN
ejpam-6297	363	26	v	v	ADP
ejpam-6297	363	27	=	=	SYM
ejpam-6297	363	28	∪{hxy	∪{hxy	NOUN
ejpam-6297	363	29	:	:	PUNCT
ejpam-6297	363	30	y	y	PROPN
ejpam-6297	363	31	∈	∈	PROPN
ejpam-6297	364	1	f	f	AUX
ejpam-6297	364	2	}	}	PUNCT
ejpam-6297	364	3	and	and	CCONJ
ejpam-6297	364	4	u	u	X
ejpam-6297	364	5	=	=	PROPN
ejpam-6297	364	6	x−{sβ	x−{sβ	PROPN
ejpam-6297	364	7	cli(∪(hxy	cli(∪(hxy	NOUN
ejpam-6297	364	8	)	)	PUNCT
ejpam-6297	364	9	)	)	PUNCT
ejpam-6297	364	10	:	:	PUNCT
ejpam-6297	365	1	y	y	PROPN
ejpam-6297	365	2	∈	∈	PROPN
ejpam-6297	365	3	f	f	X
ejpam-6297	365	4	}	}	PUNCT
ejpam-6297	365	5	.	.	PUNCT
ejpam-6297	366	1	we	we	PRON
ejpam-6297	366	2	assert	assert	VERB
ejpam-6297	366	3	that	that	SCONJ
ejpam-6297	366	4	u	u	PROPN
ejpam-6297	366	5	and	and	CCONJ
ejpam-6297	366	6	v	v	NOUN
ejpam-6297	366	7	are	be	AUX
ejpam-6297	366	8	disjoint	disjoint	ADV
ejpam-6297	366	9	strong	strong	ADJ
ejpam-6297	366	10	β	β	X
ejpam-6297	366	11	-	-	ADJ
ejpam-6297	366	12	i	i	NOUN
ejpam-6297	366	13	-	-	PUNCT
ejpam-6297	366	14	open	open	ADJ
ejpam-6297	366	15	subsets	subset	NOUN
ejpam-6297	366	16	of	of	ADP
ejpam-6297	366	17	x.	x.	NOUN
ejpam-6297	366	18	given	give	VERB
ejpam-6297	366	19	that	that	DET
ejpam-6297	366	20	hxy	hxy	NOUN
ejpam-6297	366	21	⊆	⊆	NUM
ejpam-6297	366	22	oxy	oxy	NOUN
ejpam-6297	366	23	and	and	CCONJ
ejpam-6297	366	24	sβ	sβ	PROPN
ejpam-6297	366	25	cli(hxy	cli(hxy	NOUN
ejpam-6297	366	26	)	)	PUNCT
ejpam-6297	366	27	⊆	⊆	NUM
ejpam-6297	366	28	cl(hxy	cl(hxy	NOUN
ejpam-6297	366	29	)	)	PUNCT
ejpam-6297	366	30	,	,	PUNCT
ejpam-6297	366	31	it	it	PRON
ejpam-6297	366	32	follows	follow	VERB
ejpam-6297	366	33	that	that	SCONJ
ejpam-6297	366	34	sβ	sβ	ADP
ejpam-6297	366	35	cli(hxy	cli(hxy	NOUN
ejpam-6297	366	36	)	)	PUNCT
ejpam-6297	366	37	⊆	⊆	NUM
ejpam-6297	366	38	cl(oxy	cl(oxy	PROPN
ejpam-6297	366	39	)	)	PUNCT
ejpam-6297	366	40	.	.	PUNCT
ejpam-6297	367	1	as	as	ADP
ejpam-6297	367	2	x	x	PROPN
ejpam-6297	367	3	/∈	/∈	PROPN
ejpam-6297	367	4	cl(oxy	cl(oxy	PROPN
ejpam-6297	367	5	)	)	PUNCT
ejpam-6297	367	6	,	,	PUNCT
ejpam-6297	367	7	it	it	PRON
ejpam-6297	367	8	consequently	consequently	ADV
ejpam-6297	367	9	follows	follow	VERB
ejpam-6297	367	10	that	that	SCONJ
ejpam-6297	367	11	x	x	PROPN
ejpam-6297	367	12	/∈	/∈	PUNCT
ejpam-6297	367	13	sβ	sβ	PROPN
ejpam-6297	367	14	cli(hxy	cli(hxy	NOUN
ejpam-6297	367	15	)	)	PUNCT
ejpam-6297	367	16	,	,	PUNCT
ejpam-6297	367	17	and	and	CCONJ
ejpam-6297	367	18	thus	thus	ADV
ejpam-6297	367	19	x	x	PART
ejpam-6297	367	20	∈	∈	PROPN
ejpam-6297	367	21	u	u	NOUN
ejpam-6297	367	22	.	.	PUNCT
ejpam-6297	368	1	furthermore	furthermore	ADV
ejpam-6297	368	2	,	,	PUNCT
ejpam-6297	368	3	we	we	PRON
ejpam-6297	368	4	observe	observe	VERB
ejpam-6297	368	5	that	that	SCONJ
ejpam-6297	368	6	f	f	PROPN
ejpam-6297	369	1	−	−	NOUN
ejpam-6297	369	2	v	v	NOUN
ejpam-6297	369	3	=	=	SYM
ejpam-6297	369	4	f	f	PROPN
ejpam-6297	369	5	−	−	PROPN
ejpam-6297	370	1	∪{hxy	∪{hxy	NOUN
ejpam-6297	370	2	:	:	PUNCT
ejpam-6297	370	3	y	y	PROPN
ejpam-6297	370	4	∈	∈	PROPN
ejpam-6297	370	5	f	f	X
ejpam-6297	370	6	}	}	PUNCT
ejpam-6297	370	7	⊆	⊆	NUM
ejpam-6297	370	8	x	x	SYM
ejpam-6297	370	9	−	−	PROPN
ejpam-6297	370	10	(	(	PUNCT
ejpam-6297	370	11	∪{hxy	∪{hxy	NOUN
ejpam-6297	370	12	:	:	PUNCT
ejpam-6297	370	13	y	y	PROPN
ejpam-6297	370	14	∈	∈	PROPN
ejpam-6297	370	15	f	f	X
ejpam-6297	370	16	}	}	PUNCT
ejpam-6297	370	17	∪w	∪w	NUM
ejpam-6297	370	18	)	)	PUNCT
ejpam-6297	370	19	∈	∈	PROPN
ejpam-6297	370	20	i.	i.	NOUN
ejpam-6297	370	21	thus	thus	ADV
ejpam-6297	370	22	,	,	PUNCT
ejpam-6297	370	23	u	u	NOUN
ejpam-6297	370	24	and	and	CCONJ
ejpam-6297	370	25	v	v	NOUN
ejpam-6297	370	26	are	be	AUX
ejpam-6297	370	27	indeed	indeed	ADV
ejpam-6297	370	28	disjoint	disjoint	ADJ
ejpam-6297	370	29	sβ	sβ	PROPN
ejpam-6297	370	30	-	-	PUNCT
ejpam-6297	370	31	i	i	NOUN
ejpam-6297	370	32	-	-	PUNCT
ejpam-6297	370	33	open	open	ADJ
ejpam-6297	370	34	sets	set	NOUN
ejpam-6297	370	35	,	,	PUNCT
ejpam-6297	370	36	satisfying	satisfy	VERB
ejpam-6297	370	37	x	x	X
ejpam-6297	370	38	∈	∈	PROPN
ejpam-6297	370	39	u	u	NOUN
ejpam-6297	370	40	and	and	CCONJ
ejpam-6297	370	41	f	f	PROPN
ejpam-6297	370	42	−	−	PROPN
ejpam-6297	370	43	v	v	PROPN
ejpam-6297	370	44	∈	∈	PROPN
ejpam-6297	370	45	i.	i.	NOUN
ejpam-6297	370	46	this	this	PRON
ejpam-6297	370	47	confirms	confirm	VERB
ejpam-6297	370	48	that	that	SCONJ
ejpam-6297	370	49	the	the	DET
ejpam-6297	370	50	space	space	NOUN
ejpam-6297	370	51	(	(	PUNCT
ejpam-6297	370	52	x	x	X
ejpam-6297	370	53	,	,	PUNCT
ejpam-6297	370	54	τ	τ	PROPN
ejpam-6297	370	55	,	,	PUNCT
ejpam-6297	370	56	i	i	PROPN
ejpam-6297	370	57	)	)	PUNCT
ejpam-6297	370	58	is	be	AUX
ejpam-6297	370	59	sβ	sβ	PROPN
ejpam-6297	370	60	-	-	PUNCT
ejpam-6297	370	61	i	i	NOUN
ejpam-6297	370	62	-	-	PUNCT
ejpam-6297	370	63	regular	regular	ADJ
ejpam-6297	370	64	.	.	PUNCT
ejpam-6297	371	1	theorem	theorem	NOUN
ejpam-6297	371	2	12	12	NUM
ejpam-6297	371	3	.	.	PUNCT
ejpam-6297	372	1	let	let	VERB
ejpam-6297	372	2	(	(	PUNCT
ejpam-6297	372	3	x	x	X
ejpam-6297	372	4	,	,	PUNCT
ejpam-6297	372	5	τ	τ	PROPN
ejpam-6297	372	6	,	,	PUNCT
ejpam-6297	372	7	i	i	PRON
ejpam-6297	372	8	)	)	PUNCT
ejpam-6297	372	9	be	be	VERB
ejpam-6297	372	10	an	an	DET
ejpam-6297	372	11	ideal	ideal	ADJ
ejpam-6297	372	12	topological	topological	ADJ
ejpam-6297	372	13	space	space	NOUN
ejpam-6297	372	14	.	.	PUNCT
ejpam-6297	373	1	the	the	DET
ejpam-6297	373	2	following	follow	VERB
ejpam-6297	373	3	statements	statement	NOUN
ejpam-6297	373	4	are	be	AUX
ejpam-6297	373	5	equivalent	equivalent	ADJ
ejpam-6297	373	6	:	:	PUNCT
ejpam-6297	373	7	(	(	PUNCT
ejpam-6297	373	8	1	1	X
ejpam-6297	373	9	)	)	PUNCT
ejpam-6297	373	10	for	for	ADP
ejpam-6297	373	11	every	every	DET
ejpam-6297	373	12	closed	close	VERB
ejpam-6297	373	13	subset	subset	NOUN
ejpam-6297	373	14	f	f	PROPN
ejpam-6297	373	15	of	of	ADP
ejpam-6297	373	16	x	x	PUNCT
ejpam-6297	373	17	and	and	CCONJ
ejpam-6297	373	18	every	every	DET
ejpam-6297	373	19	x	x	X
ejpam-6297	373	20	̸∈	̸∈	PROPN
ejpam-6297	373	21	f	f	PROPN
ejpam-6297	373	22	,	,	PUNCT
ejpam-6297	373	23	there	there	PRON
ejpam-6297	373	24	exist	exist	VERB
ejpam-6297	373	25	disjoint	disjoint	NOUN
ejpam-6297	373	26	strong	strong	ADJ
ejpam-6297	373	27	β	β	X
ejpam-6297	373	28	-	-	ADJ
ejpam-6297	373	29	i	i	NOUN
ejpam-6297	373	30	-	-	PUNCT
ejpam-6297	373	31	open	open	ADJ
ejpam-6297	373	32	sets	set	VERB
ejpam-6297	373	33	u	u	NOUN
ejpam-6297	373	34	and	and	CCONJ
ejpam-6297	373	35	v	v	ADP
ejpam-6297	373	36	such	such	ADJ
ejpam-6297	373	37	that	that	SCONJ
ejpam-6297	373	38	x	x	SYM
ejpam-6297	373	39	∈	∈	PROPN
ejpam-6297	373	40	u	u	NOUN
ejpam-6297	373	41	and	and	CCONJ
ejpam-6297	373	42	f	f	PROPN
ejpam-6297	373	43	−	−	PROPN
ejpam-6297	373	44	v	v	PROPN
ejpam-6297	373	45	∈	∈	PROPN
ejpam-6297	373	46	i.	i.	NOUN
ejpam-6297	373	47	(	(	PUNCT
ejpam-6297	373	48	2	2	NUM
ejpam-6297	373	49	)	)	PUNCT
ejpam-6297	373	50	for	for	ADP
ejpam-6297	373	51	every	every	DET
ejpam-6297	373	52	open	open	NOUN
ejpam-6297	373	53	subset	subset	NOUN
ejpam-6297	373	54	g	g	NOUN
ejpam-6297	373	55	of	of	ADP
ejpam-6297	373	56	x	x	PUNCT
ejpam-6297	373	57	and	and	CCONJ
ejpam-6297	373	58	every	every	DET
ejpam-6297	373	59	x	x	PROPN
ejpam-6297	373	60	∈	∈	PROPN
ejpam-6297	373	61	g	g	NOUN
ejpam-6297	373	62	,	,	PUNCT
ejpam-6297	373	63	there	there	PRON
ejpam-6297	373	64	exists	exist	VERB
ejpam-6297	373	65	a	a	DET
ejpam-6297	373	66	strong	strong	ADJ
ejpam-6297	373	67	β	β	X
ejpam-6297	373	68	-	-	ADJ
ejpam-6297	373	69	i	i	PRON
ejpam-6297	373	70	-	-	PUNCT
ejpam-6297	373	71	open	open	ADJ
ejpam-6297	373	72	set	set	VERB
ejpam-6297	373	73	u	u	PRON
ejpam-6297	373	74	such	such	ADJ
ejpam-6297	373	75	that	that	SCONJ
ejpam-6297	373	76	x	x	SYM
ejpam-6297	373	77	∈	∈	PROPN
ejpam-6297	373	78	u	u	NOUN
ejpam-6297	373	79	and	and	CCONJ
ejpam-6297	373	80	sβ	sβ	PRON
ejpam-6297	373	81	cli(u)−g	cli(u)−g	PROPN
ejpam-6297	373	82	∈	∈	PROPN
ejpam-6297	373	83	i.	i.	NOUN
ejpam-6297	373	84	proof	proof	NOUN
ejpam-6297	373	85	.	.	PUNCT
ejpam-6297	374	1	(	(	PUNCT
ejpam-6297	374	2	1	1	X
ejpam-6297	374	3	)	)	PUNCT
ejpam-6297	374	4	⇒	⇒	NOUN
ejpam-6297	374	5	(	(	PUNCT
ejpam-6297	374	6	2	2	NUM
ejpam-6297	374	7	):	):	PUNCT
ejpam-6297	374	8	let	let	VERB
ejpam-6297	374	9	g	g	PRON
ejpam-6297	374	10	be	be	AUX
ejpam-6297	374	11	an	an	DET
ejpam-6297	374	12	open	open	ADJ
ejpam-6297	374	13	set	set	NOUN
ejpam-6297	374	14	and	and	CCONJ
ejpam-6297	374	15	x	x	SYM
ejpam-6297	374	16	∈	∈	PROPN
ejpam-6297	374	17	g.	g.	NOUN
ejpam-6297	375	1	then	then	ADV
ejpam-6297	375	2	x	x	X
ejpam-6297	376	1	−	−	PROPN
ejpam-6297	376	2	g	g	NOUN
ejpam-6297	376	3	is	be	AUX
ejpam-6297	376	4	closed	close	VERB
ejpam-6297	376	5	,	,	PUNCT
ejpam-6297	376	6	and	and	CCONJ
ejpam-6297	376	7	since	since	SCONJ
ejpam-6297	376	8	x	x	PROPN
ejpam-6297	376	9	/∈	/∈	PUNCT
ejpam-6297	376	10	x	x	NOUN
ejpam-6297	376	11	−g	−g	NOUN
ejpam-6297	376	12	,	,	PUNCT
ejpam-6297	376	13	by	by	ADP
ejpam-6297	376	14	assumption	assumption	NOUN
ejpam-6297	376	15	,	,	PUNCT
ejpam-6297	376	16	there	there	PRON
ejpam-6297	376	17	exist	exist	VERB
ejpam-6297	376	18	disjoint	disjoint	NOUN
ejpam-6297	376	19	strong	strong	ADJ
ejpam-6297	376	20	β	β	X
ejpam-6297	376	21	-	-	ADJ
ejpam-6297	376	22	i	i	NOUN
ejpam-6297	376	23	-	-	PUNCT
ejpam-6297	376	24	open	open	ADJ
ejpam-6297	376	25	sets	set	VERB
ejpam-6297	376	26	u	u	NOUN
ejpam-6297	376	27	and	and	CCONJ
ejpam-6297	376	28	v	v	ADP
ejpam-6297	376	29	such	such	ADJ
ejpam-6297	376	30	that	that	SCONJ
ejpam-6297	376	31	x	x	SYM
ejpam-6297	376	32	∈	∈	PROPN
ejpam-6297	376	33	u	u	NOUN
ejpam-6297	376	34	and	and	CCONJ
ejpam-6297	376	35	(	(	PUNCT
ejpam-6297	376	36	x	x	PUNCT
ejpam-6297	376	37	−	−	PROPN
ejpam-6297	376	38	g	g	NOUN
ejpam-6297	376	39	)	)	PUNCT
ejpam-6297	376	40	−	−	PROPN
ejpam-6297	376	41	v	v	PROPN
ejpam-6297	376	42	∈	∈	PROPN
ejpam-6297	376	43	i.	i.	NOUN
ejpam-6297	376	44	since	since	SCONJ
ejpam-6297	376	45	u	u	PROPN
ejpam-6297	376	46	and	and	CCONJ
ejpam-6297	376	47	v	v	NOUN
ejpam-6297	376	48	are	be	AUX
ejpam-6297	376	49	disjoint	disjoint	ADJ
ejpam-6297	376	50	,	,	PUNCT
ejpam-6297	376	51	by	by	ADP
ejpam-6297	376	52	theorem	theorem	NOUN
ejpam-6297	376	53	8	8	NUM
ejpam-6297	376	54	,	,	PUNCT
ejpam-6297	376	55	we	we	PRON
ejpam-6297	376	56	have	have	AUX
ejpam-6297	376	57	sβ	sβ	VERB
ejpam-6297	376	58	cli(u	cli(u	PROPN
ejpam-6297	376	59	)	)	PUNCT
ejpam-6297	376	60	⊆	⊆	NUM
ejpam-6297	376	61	x	x	SYM
ejpam-6297	376	62	−	−	NUM
ejpam-6297	376	63	v	v	NOUN
ejpam-6297	376	64	.	.	PUNCT
ejpam-6297	377	1	therefore	therefore	ADV
ejpam-6297	377	2	,	,	PUNCT
ejpam-6297	377	3	sβ	sβ	PRON
ejpam-6297	377	4	cli(u	cli(u	PROPN
ejpam-6297	377	5	)	)	PUNCT
ejpam-6297	377	6	∩	∩	NOUN
ejpam-6297	377	7	(	(	PUNCT
ejpam-6297	377	8	x	x	NOUN
ejpam-6297	377	9	−g	−g	NOUN
ejpam-6297	377	10	)	)	PUNCT
ejpam-6297	377	11	⊆	⊆	NUM
ejpam-6297	377	12	(	(	PUNCT
ejpam-6297	377	13	x	x	SYM
ejpam-6297	377	14	−g)−	−g)−	PROPN
ejpam-6297	377	15	v	v	NOUN
ejpam-6297	377	16	.	.	PUNCT
ejpam-6297	378	1	hence	hence	ADV
ejpam-6297	378	2	,	,	PUNCT
ejpam-6297	378	3	we	we	PRON
ejpam-6297	378	4	conclude	conclude	VERB
ejpam-6297	378	5	that	that	SCONJ
ejpam-6297	378	6	sβ	sβ	PROPN
ejpam-6297	378	7	cli(u	cli(u	PROPN
ejpam-6297	378	8	)	)	PUNCT
ejpam-6297	378	9	∩	∩	NOUN
ejpam-6297	378	10	(	(	PUNCT
ejpam-6297	378	11	x	x	NOUN
ejpam-6297	378	12	−g	−g	NOUN
ejpam-6297	378	13	)	)	PUNCT
ejpam-6297	378	14	=	=	PUNCT
ejpam-6297	379	1	sβ	sβ	AUX
ejpam-6297	379	2	cli(u)−g	cli(u)−g	PROPN
ejpam-6297	379	3	∈	∈	PROPN
ejpam-6297	379	4	i.	i.	NOUN
ejpam-6297	379	5	(	(	PUNCT
ejpam-6297	379	6	2	2	NUM
ejpam-6297	379	7	)	)	PUNCT
ejpam-6297	379	8	⇒	⇒	NOUN
ejpam-6297	379	9	(	(	PUNCT
ejpam-6297	379	10	1	1	NUM
ejpam-6297	379	11	):	):	PUNCT
ejpam-6297	379	12	let	let	VERB
ejpam-6297	379	13	f	f	PRON
ejpam-6297	379	14	be	be	AUX
ejpam-6297	379	15	a	a	DET
ejpam-6297	379	16	closed	closed	ADJ
ejpam-6297	379	17	set	set	NOUN
ejpam-6297	379	18	and	and	CCONJ
ejpam-6297	379	19	x	x	SYM
ejpam-6297	379	20	/∈	/∈	PROPN
ejpam-6297	380	1	f	f	PROPN
ejpam-6297	380	2	.	.	PUNCT
ejpam-6297	381	1	this	this	PRON
ejpam-6297	381	2	implies	imply	VERB
ejpam-6297	381	3	that	that	SCONJ
ejpam-6297	381	4	x	x	PUNCT
ejpam-6297	382	1	−	−	NOUN
ejpam-6297	382	2	f	f	PROPN
ejpam-6297	382	3	is	be	AUX
ejpam-6297	382	4	open	open	ADJ
ejpam-6297	382	5	,	,	PUNCT
ejpam-6297	382	6	and	and	CCONJ
ejpam-6297	382	7	x	x	X
ejpam-6297	382	8	∈	∈	NOUN
ejpam-6297	382	9	x	x	X
ejpam-6297	382	10	−	−	PROPN
ejpam-6297	382	11	f	f	X
ejpam-6297	382	12	.	.	PUNCT
ejpam-6297	383	1	by	by	ADP
ejpam-6297	383	2	assumption	assumption	NOUN
ejpam-6297	383	3	,	,	PUNCT
ejpam-6297	383	4	there	there	PRON
ejpam-6297	383	5	exists	exist	VERB
ejpam-6297	383	6	a	a	DET
ejpam-6297	383	7	strong	strong	ADJ
ejpam-6297	383	8	β	β	X
ejpam-6297	383	9	-	-	ADJ
ejpam-6297	383	10	i	i	PRON
ejpam-6297	383	11	-	-	PUNCT
ejpam-6297	383	12	open	open	ADJ
ejpam-6297	383	13	set	set	VERB
ejpam-6297	383	14	u	u	PRON
ejpam-6297	383	15	such	such	ADJ
ejpam-6297	383	16	that	that	SCONJ
ejpam-6297	383	17	x	x	SYM
ejpam-6297	383	18	∈	∈	PROPN
ejpam-6297	383	19	u	u	NOUN
ejpam-6297	383	20	and	and	CCONJ
ejpam-6297	383	21	sβ	sβ	PROPN
ejpam-6297	383	22	cli(u)−(x−f	cli(u)−(x−f	PROPN
ejpam-6297	383	23	)	)	PUNCT
ejpam-6297	383	24	∈	∈	PROPN
ejpam-6297	383	25	i.	i.	NOUN
ejpam-6297	383	26	thus	thus	ADV
ejpam-6297	383	27	,	,	PUNCT
ejpam-6297	383	28	we	we	PRON
ejpam-6297	383	29	define	define	VERB
ejpam-6297	383	30	v	v	NOUN
ejpam-6297	383	31	=	=	SYM
ejpam-6297	383	32	x−sβ	x−sβ	PROPN
ejpam-6297	383	33	cli(u	cli(u	PROPN
ejpam-6297	383	34	)	)	PUNCT
ejpam-6297	383	35	,	,	PUNCT
ejpam-6297	383	36	which	which	PRON
ejpam-6297	383	37	is	be	AUX
ejpam-6297	383	38	a	a	DET
ejpam-6297	383	39	strong	strong	ADJ
ejpam-6297	383	40	β	β	X
ejpam-6297	383	41	-	-	ADJ
ejpam-6297	383	42	i	i	NOUN
ejpam-6297	383	43	-	-	PUNCT
ejpam-6297	383	44	open	open	ADJ
ejpam-6297	383	45	set	set	NOUN
ejpam-6297	383	46	.	.	PUNCT
ejpam-6297	384	1	since	since	SCONJ
ejpam-6297	384	2	u	u	PROPN
ejpam-6297	384	3	and	and	CCONJ
ejpam-6297	384	4	v	v	NOUN
ejpam-6297	384	5	are	be	AUX
ejpam-6297	384	6	disjoint	disjoint	ADJ
ejpam-6297	384	7	,	,	PUNCT
ejpam-6297	384	8	we	we	PRON
ejpam-6297	384	9	have	have	AUX
ejpam-6297	384	10	f−v	f−v	VERB
ejpam-6297	384	11	=	=	SYM
ejpam-6297	384	12	f−(x−sβ	f−(x−sβ	DET
ejpam-6297	384	13	cli(u	cli(u	PROPN
ejpam-6297	384	14	)	)	PUNCT
ejpam-6297	384	15	)	)	PUNCT
ejpam-6297	385	1	=	=	PRON
ejpam-6297	385	2	sβ	sβ	PROPN
ejpam-6297	385	3	cli(u)−(x−f	cli(u)−(x−f	PROPN
ejpam-6297	385	4	)	)	PUNCT
ejpam-6297	385	5	∈	∈	PROPN
ejpam-6297	385	6	i.	i.	NOUN
ejpam-6297	385	7	by	by	ADP
ejpam-6297	385	8	theorem	theorem	ADJ
ejpam-6297	385	9	11	11	NUM
ejpam-6297	385	10	and	and	CCONJ
ejpam-6297	385	11	theorem	theorem	VERB
ejpam-6297	385	12	12	12	NUM
ejpam-6297	385	13	,	,	PUNCT
ejpam-6297	385	14	we	we	PRON
ejpam-6297	385	15	have	have	VERB
ejpam-6297	385	16	the	the	DET
ejpam-6297	385	17	following	follow	VERB
ejpam-6297	385	18	colollary	colollary	NOUN
ejpam-6297	385	19	.	.	PUNCT
ejpam-6297	386	1	corollary	corollary	ADJ
ejpam-6297	386	2	1	1	NUM
ejpam-6297	386	3	.	.	PUNCT
ejpam-6297	387	1	an	an	DET
ejpam-6297	387	2	ideal	ideal	ADJ
ejpam-6297	387	3	topological	topological	ADJ
ejpam-6297	387	4	space	space	NOUN
ejpam-6297	387	5	(	(	PUNCT
ejpam-6297	387	6	x	x	X
ejpam-6297	387	7	,	,	PUNCT
ejpam-6297	387	8	τ	τ	PROPN
ejpam-6297	387	9	,	,	PUNCT
ejpam-6297	387	10	i	i	PROPN
ejpam-6297	387	11	)	)	PUNCT
ejpam-6297	387	12	is	be	AUX
ejpam-6297	387	13	sβ	sβ	PROPN
ejpam-6297	387	14	-	-	PUNCT
ejpam-6297	387	15	i	i	NOUN
ejpam-6297	387	16	-	-	PUNCT
ejpam-6297	387	17	paracompact	paracompact	ADJ
ejpam-6297	387	18	and	and	CCONJ
ejpam-6297	387	19	hausdorff	hausdorff	NOUN
ejpam-6297	387	20	if	if	SCONJ
ejpam-6297	387	21	and	and	CCONJ
ejpam-6297	387	22	only	only	ADV
ejpam-6297	387	23	if	if	SCONJ
ejpam-6297	387	24	for	for	ADP
ejpam-6297	387	25	any	any	DET
ejpam-6297	387	26	open	open	ADJ
ejpam-6297	387	27	set	set	NOUN
ejpam-6297	387	28	g	g	PROPN
ejpam-6297	387	29	⊆	⊆	NUM
ejpam-6297	387	30	x	x	NOUN
ejpam-6297	387	31	and	and	CCONJ
ejpam-6297	387	32	for	for	ADP
ejpam-6297	387	33	every	every	DET
ejpam-6297	387	34	point	point	NOUN
ejpam-6297	387	35	x	x	X
ejpam-6297	387	36	∈	∈	NOUN
ejpam-6297	387	37	g	g	NOUN
ejpam-6297	387	38	,	,	PUNCT
ejpam-6297	387	39	there	there	PRON
ejpam-6297	387	40	exists	exist	VERB
ejpam-6297	387	41	a	a	DET
ejpam-6297	387	42	strong	strong	ADJ
ejpam-6297	387	43	β	β	X
ejpam-6297	387	44	-	-	ADJ
ejpam-6297	387	45	i	i	PRON
ejpam-6297	387	46	-	-	PUNCT
ejpam-6297	387	47	open	open	ADJ
ejpam-6297	387	48	set	set	VERB
ejpam-6297	387	49	u	u	PRON
ejpam-6297	387	50	such	such	ADJ
ejpam-6297	387	51	that	that	SCONJ
ejpam-6297	387	52	x	x	SYM
ejpam-6297	387	53	∈	∈	PROPN
ejpam-6297	387	54	u	u	NOUN
ejpam-6297	387	55	and	and	CCONJ
ejpam-6297	387	56	sβ	sβ	PRON
ejpam-6297	387	57	cli(u)−g	cli(u)−g	PROPN
ejpam-6297	387	58	∈	∈	PROPN
ejpam-6297	387	59	i.	i.	PROPN
ejpam-6297	387	60	theorem	theorem	VERB
ejpam-6297	387	61	13	13	NUM
ejpam-6297	387	62	.	.	PUNCT
ejpam-6297	388	1	if	if	SCONJ
ejpam-6297	388	2	an	an	DET
ejpam-6297	388	3	ideal	ideal	ADJ
ejpam-6297	388	4	topological	topological	ADJ
ejpam-6297	388	5	space	space	NOUN
ejpam-6297	388	6	(	(	PUNCT
ejpam-6297	388	7	x	x	X
ejpam-6297	388	8	,	,	PUNCT
ejpam-6297	388	9	τ	τ	PROPN
ejpam-6297	388	10	,	,	PUNCT
ejpam-6297	388	11	i	i	PROPN
ejpam-6297	388	12	)	)	PUNCT
ejpam-6297	388	13	is	be	AUX
ejpam-6297	388	14	sβ	sβ	PROPN
ejpam-6297	388	15	-	-	PUNCT
ejpam-6297	388	16	i	i	NOUN
ejpam-6297	388	17	-	-	PUNCT
ejpam-6297	388	18	paracompact	paracompact	ADJ
ejpam-6297	388	19	and	and	CCONJ
ejpam-6297	388	20	regular	regular	ADJ
ejpam-6297	388	21	,	,	PUNCT
ejpam-6297	388	22	then	then	ADV
ejpam-6297	388	23	every	every	DET
ejpam-6297	388	24	open	open	ADJ
ejpam-6297	388	25	cover	cover	NOUN
ejpam-6297	388	26	of	of	ADP
ejpam-6297	388	27	x	x	PUNCT
ejpam-6297	388	28	has	have	VERB
ejpam-6297	388	29	an	an	DET
ejpam-6297	388	30	sβ	sβ	NOUN
ejpam-6297	388	31	-	-	PUNCT
ejpam-6297	388	32	i	i	NOUN
ejpam-6297	388	33	-	-	PUNCT
ejpam-6297	388	34	locally	locally	ADV
ejpam-6297	388	35	finite	finite	NOUN
ejpam-6297	388	36	i	i	NOUN
ejpam-6297	388	37	-	-	PUNCT
ejpam-6297	388	38	cover	cover	NOUN
ejpam-6297	388	39	refinement	refinement	NOUN
ejpam-6297	388	40	of	of	ADP
ejpam-6297	388	41	strong	strong	ADJ
ejpam-6297	388	42	β	β	X
ejpam-6297	388	43	-	-	ADJ
ejpam-6297	388	44	i	i	NOUN
ejpam-6297	388	45	-	-	PUNCT
ejpam-6297	388	46	closed	close	VERB
ejpam-6297	388	47	sets	set	NOUN
ejpam-6297	388	48	.	.	PUNCT
ejpam-6297	389	1	proof	proof	NOUN
ejpam-6297	389	2	.	.	PUNCT
ejpam-6297	390	1	let	let	VERB
ejpam-6297	390	2	a	a	DET
ejpam-6297	390	3	be	be	AUX
ejpam-6297	390	4	an	an	DET
ejpam-6297	390	5	open	open	ADJ
ejpam-6297	390	6	cover	cover	NOUN
ejpam-6297	390	7	of	of	ADP
ejpam-6297	390	8	x.	x.	NOUN
ejpam-6297	390	9	by	by	ADP
ejpam-6297	390	10	the	the	DET
ejpam-6297	390	11	regularity	regularity	NOUN
ejpam-6297	390	12	of	of	ADP
ejpam-6297	390	13	x	x	PRON
ejpam-6297	390	14	,	,	PUNCT
ejpam-6297	390	15	for	for	ADP
ejpam-6297	390	16	each	each	DET
ejpam-6297	390	17	x	x	SYM
ejpam-6297	390	18	∈	∈	PROPN
ejpam-6297	390	19	x	x	X
ejpam-6297	390	20	and	and	CCONJ
ejpam-6297	390	21	ux	ux	PROPN
ejpam-6297	390	22	∈	∈	PROPN
ejpam-6297	390	23	a	a	DET
ejpam-6297	390	24	containing	contain	VERB
ejpam-6297	390	25	x	x	NOUN
ejpam-6297	390	26	,	,	PUNCT
ejpam-6297	390	27	there	there	PRON
ejpam-6297	390	28	exists	exist	VERB
ejpam-6297	390	29	an	an	DET
ejpam-6297	390	30	open	open	ADJ
ejpam-6297	390	31	set	set	NOUN
ejpam-6297	390	32	gx	gx	PROPN
ejpam-6297	390	33	such	such	ADJ
ejpam-6297	390	34	that	that	SCONJ
ejpam-6297	390	35	x	x	SYM
ejpam-6297	390	36	∈	∈	PROPN
ejpam-6297	390	37	gx	gx	PROPN
ejpam-6297	390	38	and	and	CCONJ
ejpam-6297	390	39	cl(gx	cl(gx	PROPN
ejpam-6297	390	40	)	)	PUNCT
ejpam-6297	390	41	⊆	⊆	NUM
ejpam-6297	390	42	ux	ux	NOUN
ejpam-6297	390	43	.	.	PUNCT
ejpam-6297	391	1	thus	thus	ADV
ejpam-6297	391	2	,	,	PUNCT
ejpam-6297	391	3	the	the	DET
ejpam-6297	391	4	family	family	NOUN
ejpam-6297	391	5	a1	a1	NOUN
ejpam-6297	391	6	=	=	SYM
ejpam-6297	391	7	{	{	PUNCT
ejpam-6297	391	8	gx	gx	PROPN
ejpam-6297	391	9	:	:	PUNCT
ejpam-6297	391	10	x	x	SYM
ejpam-6297	391	11	∈	∈	PROPN
ejpam-6297	391	12	x	x	PRON
ejpam-6297	391	13	}	}	PUNCT
ejpam-6297	391	14	is	be	AUX
ejpam-6297	391	15	an	an	DET
ejpam-6297	391	16	open	open	ADJ
ejpam-6297	391	17	cover	cover	NOUN
ejpam-6297	391	18	of	of	ADP
ejpam-6297	391	19	x.	x.	NOUN
ejpam-6297	391	20	as	as	SCONJ
ejpam-6297	391	21	x	x	PROPN
ejpam-6297	391	22	is	be	AUX
ejpam-6297	391	23	sβ	sβ	PROPN
ejpam-6297	391	24	-	-	PUNCT
ejpam-6297	391	25	i	i	NOUN
ejpam-6297	391	26	-	-	PUNCT
ejpam-6297	391	27	paracompact	paracompact	ADJ
ejpam-6297	391	28	,	,	PUNCT
ejpam-6297	391	29	the	the	DET
ejpam-6297	391	30	cover	cover	NOUN
ejpam-6297	391	31	a1	a1	NOUN
ejpam-6297	391	32	has	have	VERB
ejpam-6297	391	33	an	an	DET
ejpam-6297	391	34	sβ	sβ	NOUN
ejpam-6297	391	35	-	-	PUNCT
ejpam-6297	391	36	i	i	NOUN
ejpam-6297	391	37	-	-	PUNCT
ejpam-6297	391	38	locally	locally	ADV
ejpam-6297	391	39	finite	finite	ADJ
ejpam-6297	391	40	refinement	refinement	NOUN
ejpam-6297	391	41	b1	b1	NOUN
ejpam-6297	391	42	=	=	SYM
ejpam-6297	391	43	{	{	PUNCT
ejpam-6297	391	44	vλ	vλ	INTJ
ejpam-6297	391	45	:	:	PUNCT
ejpam-6297	391	46	λ	λ	PROPN
ejpam-6297	391	47	∈	∈	PROPN
ejpam-6297	391	48	λ	λ	PROPN
ejpam-6297	391	49	}	}	PUNCT
ejpam-6297	391	50	consisting	consist	VERB
ejpam-6297	391	51	of	of	ADP
ejpam-6297	391	52	strong	strong	ADJ
ejpam-6297	391	53	β	β	X
ejpam-6297	391	54	-	-	ADJ
ejpam-6297	391	55	i	i	NOUN
ejpam-6297	391	56	-	-	PUNCT
ejpam-6297	391	57	open	open	ADJ
ejpam-6297	391	58	sets	set	VERB
ejpam-6297	391	59	such	such	ADJ
ejpam-6297	391	60	that	that	SCONJ
ejpam-6297	391	61	x	x	X
ejpam-6297	391	62	−	−	NOUN
ejpam-6297	391	63	∪{vλ	∪{vλ	NUM
ejpam-6297	391	64	:	:	PUNCT
ejpam-6297	391	65	λ	λ	PROPN
ejpam-6297	391	66	∈	∈	PROPN
ejpam-6297	391	67	λ	λ	PROPN
ejpam-6297	391	68	}	}	PUNCT
ejpam-6297	391	69	∈	∈	PROPN
ejpam-6297	391	70	i.	i.	NOUN
ejpam-6297	391	71	since	since	SCONJ
ejpam-6297	391	72	vλ	vλ	ADP
ejpam-6297	391	73	⊆	⊆	NUM
ejpam-6297	391	74	sβ	sβ	NUM
ejpam-6297	391	75	cli(vλ	cli(vλ	NOUN
ejpam-6297	391	76	)	)	PUNCT
ejpam-6297	391	77	for	for	ADP
ejpam-6297	391	78	each	each	DET
ejpam-6297	391	79	λ	λ	NOUN
ejpam-6297	391	80	,	,	PUNCT
ejpam-6297	391	81	and	and	CCONJ
ejpam-6297	391	82	i	i	PRON
ejpam-6297	391	83	is	be	AUX
ejpam-6297	391	84	an	an	DET
ejpam-6297	391	85	ideal	ideal	NOUN
ejpam-6297	391	86	,	,	PUNCT
ejpam-6297	391	87	we	we	PRON
ejpam-6297	391	88	conclude	conclude	VERB
ejpam-6297	391	89	that	that	SCONJ
ejpam-6297	391	90	x−∪{sβ	x−∪{sβ	NOUN
ejpam-6297	391	91	cli(vλ	cli(vλ	NOUN
ejpam-6297	391	92	)	)	PUNCT
ejpam-6297	391	93	:	:	PUNCT
ejpam-6297	392	1	λ	λ	X
ejpam-6297	392	2	∈	∈	PROPN
ejpam-6297	392	3	λ	λ	PROPN
ejpam-6297	392	4	}	}	PUNCT
ejpam-6297	392	5	∈	∈	PROPN
ejpam-6297	392	6	i.	i.	NOUN
ejpam-6297	392	7	by	by	ADP
ejpam-6297	392	8	theorem	theorem	NOUN
ejpam-6297	392	9	9	9	NUM
ejpam-6297	392	10	,	,	PUNCT
ejpam-6297	392	11	the	the	DET
ejpam-6297	392	12	collection	collection	NOUN
ejpam-6297	392	13	c.	c.	PROPN
ejpam-6297	392	14	boonpok	boonpok	PROPN
ejpam-6297	392	15	,	,	PUNCT
ejpam-6297	392	16	p.	p.	PROPN
ejpam-6297	392	17	raktaow	raktaow	NOUN
ejpam-6297	392	18	,	,	PUNCT
ejpam-6297	392	19	a.	a.	PROPN
ejpam-6297	392	20	sama	sama	PROPN
ejpam-6297	392	21	-	-	PUNCT
ejpam-6297	392	22	ae	ae	PROPN
ejpam-6297	392	23	/	/	SYM
ejpam-6297	392	24	eur	eur	PROPN
ejpam-6297	392	25	.	.	PUNCT
ejpam-6297	393	1	j.	j.	PROPN
ejpam-6297	393	2	pure	pure	PROPN
ejpam-6297	393	3	appl	appl	PROPN
ejpam-6297	393	4	.	.	PROPN
ejpam-6297	393	5	math	math	PROPN
ejpam-6297	393	6	,	,	PUNCT
ejpam-6297	393	7	18	18	NUM
ejpam-6297	393	8	(	(	PUNCT
ejpam-6297	393	9	3	3	NUM
ejpam-6297	393	10	)	)	PUNCT
ejpam-6297	393	11	(	(	PUNCT
ejpam-6297	393	12	2025	2025	NUM
ejpam-6297	393	13	)	)	PUNCT
ejpam-6297	393	14	,	,	PUNCT
ejpam-6297	393	15	6297	6297	NUM
ejpam-6297	393	16	15	15	NUM
ejpam-6297	393	17	of	of	ADP
ejpam-6297	393	18	23	23	NUM
ejpam-6297	393	19	b	b	NOUN
ejpam-6297	393	20	=	=	SYM
ejpam-6297	393	21	{	{	PUNCT
ejpam-6297	393	22	sβ	sβ	NOUN
ejpam-6297	393	23	cli(vλ	cli(vλ	NOUN
ejpam-6297	393	24	)	)	PUNCT
ejpam-6297	393	25	:	:	PUNCT
ejpam-6297	394	1	vλ	vλ	ADP
ejpam-6297	394	2	∈	∈	PROPN
ejpam-6297	394	3	b1	b1	NOUN
ejpam-6297	394	4	}	}	PUNCT
ejpam-6297	394	5	is	be	AUX
ejpam-6297	394	6	sβ	sβ	NOUN
ejpam-6297	394	7	-	-	PUNCT
ejpam-6297	394	8	i	i	NOUN
ejpam-6297	394	9	-	-	PUNCT
ejpam-6297	394	10	locally	locally	ADV
ejpam-6297	394	11	finite	finite	NOUN
ejpam-6297	394	12	.	.	PUNCT
ejpam-6297	395	1	since	since	SCONJ
ejpam-6297	395	2	b1	b1	NOUN
ejpam-6297	395	3	refines	refine	VERB
ejpam-6297	395	4	a1	a1	NOUN
ejpam-6297	395	5	,	,	PUNCT
ejpam-6297	395	6	for	for	ADP
ejpam-6297	395	7	each	each	DET
ejpam-6297	395	8	λ	λ	PROPN
ejpam-6297	395	9	∈	∈	PROPN
ejpam-6297	395	10	λ	λ	PROPN
ejpam-6297	395	11	,	,	PUNCT
ejpam-6297	395	12	there	there	PRON
ejpam-6297	395	13	exists	exist	VERB
ejpam-6297	395	14	some	some	DET
ejpam-6297	395	15	gx	gx	PROPN
ejpam-6297	395	16	∈	∈	PROPN
ejpam-6297	395	17	a1	a1	NOUN
ejpam-6297	395	18	such	such	ADJ
ejpam-6297	395	19	that	that	SCONJ
ejpam-6297	395	20	vλ	vλ	ADV
ejpam-6297	395	21	⊆	⊆	NUM
ejpam-6297	395	22	gx	gx	PROPN
ejpam-6297	395	23	.	.	PUNCT
ejpam-6297	396	1	therefore	therefore	ADV
ejpam-6297	396	2	,	,	PUNCT
ejpam-6297	396	3	we	we	PRON
ejpam-6297	396	4	have	have	VERB
ejpam-6297	396	5	:	:	PUNCT
ejpam-6297	396	6	sβ	sβ	NUM
ejpam-6297	396	7	cli(vλ	cli(vλ	NOUN
ejpam-6297	396	8	)	)	PUNCT
ejpam-6297	396	9	⊆	⊆	NUM
ejpam-6297	396	10	cl(vλ	cl(vλ	NOUN
ejpam-6297	396	11	)	)	PUNCT
ejpam-6297	396	12	⊆	⊆	NUM
ejpam-6297	396	13	cl(gx	cl(gx	NOUN
ejpam-6297	396	14	)	)	PUNCT
ejpam-6297	396	15	.	.	PUNCT
ejpam-6297	397	1	consequently	consequently	ADV
ejpam-6297	397	2	,	,	PUNCT
ejpam-6297	397	3	sβ	sβ	X
ejpam-6297	397	4	cli(vλ	cli(vλ	NOUN
ejpam-6297	397	5	)	)	PUNCT
ejpam-6297	397	6	⊆	⊆	NUM
ejpam-6297	397	7	ux	ux	NOUN
ejpam-6297	397	8	.	.	PUNCT
ejpam-6297	398	1	this	this	PRON
ejpam-6297	398	2	shows	show	VERB
ejpam-6297	398	3	that	that	SCONJ
ejpam-6297	398	4	b	b	NOUN
ejpam-6297	398	5	refines	refine	VERB
ejpam-6297	398	6	a.	a.	NOUN
ejpam-6297	398	7	thus	thus	ADV
ejpam-6297	398	8	,	,	PUNCT
ejpam-6297	398	9	the	the	DET
ejpam-6297	398	10	collection	collection	NOUN
ejpam-6297	398	11	b	b	PROPN
ejpam-6297	398	12	=	=	X
ejpam-6297	398	13	{	{	PUNCT
ejpam-6297	398	14	sβ	sβ	NOUN
ejpam-6297	398	15	cli(vλ	cli(vλ	NOUN
ejpam-6297	398	16	)	)	PUNCT
ejpam-6297	398	17	:	:	PUNCT
ejpam-6297	399	1	vλ	vλ	ADP
ejpam-6297	399	2	∈	∈	PROPN
ejpam-6297	399	3	b1	b1	NOUN
ejpam-6297	399	4	}	}	PUNCT
ejpam-6297	399	5	is	be	AUX
ejpam-6297	399	6	an	an	DET
ejpam-6297	399	7	sβ	sβ	NOUN
ejpam-6297	399	8	-	-	PUNCT
ejpam-6297	399	9	i	i	NOUN
ejpam-6297	399	10	-	-	PUNCT
ejpam-6297	399	11	locally	locally	ADV
ejpam-6297	399	12	finite	finite	NOUN
ejpam-6297	399	13	i	i	NOUN
ejpam-6297	399	14	-	-	PUNCT
ejpam-6297	399	15	cover	cover	NOUN
ejpam-6297	399	16	refinement	refinement	NOUN
ejpam-6297	399	17	of	of	ADP
ejpam-6297	399	18	strong	strong	ADJ
ejpam-6297	399	19	β	β	X
ejpam-6297	399	20	-	-	ADJ
ejpam-6297	399	21	i	i	NOUN
ejpam-6297	399	22	-	-	PUNCT
ejpam-6297	399	23	closed	close	VERB
ejpam-6297	399	24	sets	set	NOUN
ejpam-6297	399	25	.	.	PUNCT
ejpam-6297	400	1	theorem	theorem	VERB
ejpam-6297	400	2	14	14	NUM
ejpam-6297	400	3	.	.	PUNCT
ejpam-6297	401	1	if	if	SCONJ
ejpam-6297	401	2	an	an	DET
ejpam-6297	401	3	ideal	ideal	ADJ
ejpam-6297	401	4	topological	topological	ADJ
ejpam-6297	401	5	space	space	NOUN
ejpam-6297	401	6	(	(	PUNCT
ejpam-6297	401	7	x	x	X
ejpam-6297	401	8	,	,	PUNCT
ejpam-6297	401	9	τ	τ	PROPN
ejpam-6297	401	10	,	,	PUNCT
ejpam-6297	401	11	i	i	PROPN
ejpam-6297	401	12	)	)	PUNCT
ejpam-6297	401	13	is	be	AUX
ejpam-6297	401	14	hausdorff	hausdorff	NOUN
ejpam-6297	401	15	and	and	CCONJ
ejpam-6297	401	16	a	a	PRON
ejpam-6297	401	17	is	be	AUX
ejpam-6297	401	18	an	an	DET
ejpam-6297	401	19	sβ	sβ	NOUN
ejpam-6297	401	20	-	-	PUNCT
ejpam-6297	401	21	iparacompact	iparacompact	NOUN
ejpam-6297	401	22	subset	subset	NOUN
ejpam-6297	401	23	of	of	ADP
ejpam-6297	401	24	x	x	PRON
ejpam-6297	401	25	,	,	PUNCT
ejpam-6297	401	26	then	then	ADV
ejpam-6297	401	27	a	a	PRON
ejpam-6297	401	28	is	be	AUX
ejpam-6297	401	29	a	a	DET
ejpam-6297	401	30	closed	closed	ADJ
ejpam-6297	401	31	set	set	NOUN
ejpam-6297	401	32	in	in	ADP
ejpam-6297	401	33	(	(	PUNCT
ejpam-6297	401	34	x	x	NOUN
ejpam-6297	401	35	,	,	PUNCT
ejpam-6297	401	36	τ∗	τ∗	NOUN
ejpam-6297	401	37	)	)	PUNCT
ejpam-6297	401	38	.	.	PUNCT
ejpam-6297	402	1	proof	proof	NOUN
ejpam-6297	402	2	.	.	PUNCT
ejpam-6297	403	1	let	let	VERB
ejpam-6297	403	2	x	x	SYM
ejpam-6297	403	3	∈	∈	PROPN
ejpam-6297	403	4	x	x	X
ejpam-6297	403	5	−	−	NOUN
ejpam-6297	403	6	a.	a.	NOUN
ejpam-6297	403	7	since	since	SCONJ
ejpam-6297	403	8	x	x	PROPN
ejpam-6297	403	9	is	be	AUX
ejpam-6297	403	10	hausdorff	hausdorff	NOUN
ejpam-6297	403	11	,	,	PUNCT
ejpam-6297	403	12	for	for	ADP
ejpam-6297	403	13	each	each	DET
ejpam-6297	403	14	y	y	PROPN
ejpam-6297	403	15	∈	∈	PROPN
ejpam-6297	403	16	a	a	PRON
ejpam-6297	403	17	,	,	PUNCT
ejpam-6297	403	18	there	there	PRON
ejpam-6297	403	19	exists	exist	VERB
ejpam-6297	403	20	an	an	DET
ejpam-6297	403	21	open	open	ADJ
ejpam-6297	403	22	set	set	NOUN
ejpam-6297	403	23	gy	gy	PROPN
ejpam-6297	403	24	∈	∈	PROPN
ejpam-6297	403	25	τ	τ	X
ejpam-6297	403	26	such	such	ADJ
ejpam-6297	403	27	that	that	SCONJ
ejpam-6297	403	28	y	y	PROPN
ejpam-6297	403	29	∈	∈	PROPN
ejpam-6297	403	30	gy	gy	PROPN
ejpam-6297	403	31	and	and	CCONJ
ejpam-6297	403	32	x	x	PROPN
ejpam-6297	403	33	̸∈	̸∈	PROPN
ejpam-6297	403	34	gy	gy	PROPN
ejpam-6297	403	35	.	.	PUNCT
ejpam-6297	404	1	thus	thus	ADV
ejpam-6297	404	2	,	,	PUNCT
ejpam-6297	404	3	the	the	DET
ejpam-6297	404	4	family	family	NOUN
ejpam-6297	404	5	u	u	NOUN
ejpam-6297	404	6	=	=	PUNCT
ejpam-6297	404	7	{	{	PUNCT
ejpam-6297	404	8	gy	gy	INTJ
ejpam-6297	404	9	:	:	PUNCT
ejpam-6297	404	10	y	y	PROPN
ejpam-6297	404	11	∈	∈	PROPN
ejpam-6297	404	12	a	a	DET
ejpam-6297	404	13	}	}	PUNCT
ejpam-6297	404	14	forms	form	VERB
ejpam-6297	404	15	an	an	DET
ejpam-6297	404	16	open	open	ADJ
ejpam-6297	404	17	cover	cover	NOUN
ejpam-6297	404	18	of	of	ADP
ejpam-6297	404	19	a.	a.	NOUN
ejpam-6297	404	20	since	since	SCONJ
ejpam-6297	404	21	a	a	PRON
ejpam-6297	404	22	is	be	AUX
ejpam-6297	404	23	an	an	DET
ejpam-6297	404	24	sβ	sβ	NOUN
ejpam-6297	404	25	-	-	PUNCT
ejpam-6297	404	26	i	i	NOUN
ejpam-6297	404	27	-	-	PUNCT
ejpam-6297	404	28	paracompact	paracompact	NOUN
ejpam-6297	404	29	subset	subset	NOUN
ejpam-6297	404	30	of	of	ADP
ejpam-6297	404	31	x	x	PRON
ejpam-6297	404	32	,	,	PUNCT
ejpam-6297	404	33	the	the	DET
ejpam-6297	404	34	cover	cover	NOUN
ejpam-6297	404	35	u	u	NOUN
ejpam-6297	404	36	has	have	VERB
ejpam-6297	404	37	an	an	DET
ejpam-6297	404	38	sβ	sβ	NOUN
ejpam-6297	404	39	-	-	PUNCT
ejpam-6297	404	40	i	i	NOUN
ejpam-6297	404	41	-	-	PUNCT
ejpam-6297	404	42	locally	locally	ADV
ejpam-6297	404	43	finite	finite	VERB
ejpam-6297	404	44	strong	strong	ADJ
ejpam-6297	404	45	β	β	X
ejpam-6297	404	46	-	-	ADJ
ejpam-6297	404	47	i	i	NOUN
ejpam-6297	404	48	-	-	PUNCT
ejpam-6297	404	49	open	open	ADJ
ejpam-6297	404	50	refinement	refinement	NOUN
ejpam-6297	404	51	v	v	NOUN
ejpam-6297	404	52	=	=	PUNCT
ejpam-6297	404	53	{	{	PUNCT
ejpam-6297	404	54	vα	vα	X
ejpam-6297	404	55	:	:	PUNCT
ejpam-6297	404	56	α	α	PROPN
ejpam-6297	404	57	∈	∈	PROPN
ejpam-6297	404	58	λ	λ	NOUN
ejpam-6297	404	59	}	}	PUNCT
ejpam-6297	404	60	such	such	ADJ
ejpam-6297	404	61	that	that	SCONJ
ejpam-6297	404	62	a−	a−	PROPN
ejpam-6297	404	63	∪{vα	∪{vα	PROPN
ejpam-6297	404	64	:	:	PUNCT
ejpam-6297	404	65	α	α	PROPN
ejpam-6297	404	66	∈	∈	PROPN
ejpam-6297	404	67	λ	λ	PROPN
ejpam-6297	404	68	}	}	PUNCT
ejpam-6297	404	69	∈	∈	PROPN
ejpam-6297	404	70	i.	i.	NOUN
ejpam-6297	404	71	since	since	SCONJ
ejpam-6297	404	72	x	x	PROPN
ejpam-6297	404	73	̸∈	̸∈	PROPN
ejpam-6297	404	74	cl(vα	cl(vα	PROPN
ejpam-6297	404	75	)	)	PUNCT
ejpam-6297	404	76	for	for	ADP
ejpam-6297	404	77	all	all	PRON
ejpam-6297	404	78	α	α	PRON
ejpam-6297	404	79	∈	∈	PROPN
ejpam-6297	404	80	λ	λ	PROPN
ejpam-6297	404	81	,	,	PUNCT
ejpam-6297	404	82	it	it	PRON
ejpam-6297	404	83	follows	follow	VERB
ejpam-6297	404	84	that	that	SCONJ
ejpam-6297	404	85	x	x	PROPN
ejpam-6297	404	86	̸∈	̸∈	PROPN
ejpam-6297	404	87	∪{cl(vα	∪{cl(vα	PROPN
ejpam-6297	404	88	)	)	PUNCT
ejpam-6297	404	89	:	:	PUNCT
ejpam-6297	405	1	α	α	PROPN
ejpam-6297	405	2	∈	∈	PROPN
ejpam-6297	405	3	λ	λ	NOUN
ejpam-6297	405	4	}	}	PUNCT
ejpam-6297	405	5	.	.	PUNCT
ejpam-6297	406	1	moreover	moreover	ADV
ejpam-6297	406	2	,	,	PUNCT
ejpam-6297	406	3	since	since	SCONJ
ejpam-6297	406	4	the	the	DET
ejpam-6297	406	5	locally	locally	ADV
ejpam-6297	406	6	finite	finite	ADJ
ejpam-6297	406	7	family	family	NOUN
ejpam-6297	406	8	is	be	AUX
ejpam-6297	406	9	closure	closure	NOUN
ejpam-6297	406	10	-	-	PUNCT
ejpam-6297	406	11	preserving	preserving	NOUN
ejpam-6297	406	12	,	,	PUNCT
ejpam-6297	406	13	we	we	PRON
ejpam-6297	406	14	have	have	VERB
ejpam-6297	406	15	:	:	PUNCT
ejpam-6297	406	16	∪{cl(vα	∪{cl(vα	NUM
ejpam-6297	406	17	)	)	PUNCT
ejpam-6297	406	18	:	:	PUNCT
ejpam-6297	407	1	α	α	PROPN
ejpam-6297	407	2	∈	∈	PROPN
ejpam-6297	407	3	λ	λ	X
ejpam-6297	407	4	}	}	PUNCT
ejpam-6297	407	5	=	=	ADJ
ejpam-6297	407	6	cl	cl	NOUN
ejpam-6297	407	7	(	(	PUNCT
ejpam-6297	407	8	∪{vα	∪{vα	NOUN
ejpam-6297	407	9	:	:	PUNCT
ejpam-6297	407	10	α	α	PROPN
ejpam-6297	407	11	∈	∈	PROPN
ejpam-6297	407	12	λ	λ	NOUN
ejpam-6297	407	13	}	}	PUNCT
ejpam-6297	407	14	)	)	PUNCT
ejpam-6297	407	15	,	,	PUNCT
ejpam-6297	407	16	so	so	SCONJ
ejpam-6297	407	17	that	that	SCONJ
ejpam-6297	407	18	x	x	PUNCT
ejpam-6297	407	19	̸∈	̸∈	PROPN
ejpam-6297	407	20	cl	cl	X
ejpam-6297	407	21	(	(	PUNCT
ejpam-6297	407	22	∪{vα	∪{vα	NOUN
ejpam-6297	407	23	:	:	PUNCT
ejpam-6297	407	24	α	α	PROPN
ejpam-6297	407	25	∈	∈	PROPN
ejpam-6297	407	26	λ	λ	NOUN
ejpam-6297	407	27	}	}	PUNCT
ejpam-6297	407	28	)	)	PUNCT
ejpam-6297	407	29	.	.	PUNCT
ejpam-6297	408	1	let	let	VERB
ejpam-6297	408	2	g	g	NOUN
ejpam-6297	408	3	=	=	PUNCT
ejpam-6297	408	4	x	x	SYM
ejpam-6297	408	5	−	−	PROPN
ejpam-6297	408	6	cl	cl	NOUN
ejpam-6297	408	7	(	(	PUNCT
ejpam-6297	408	8	∪{vα	∪{vα	NOUN
ejpam-6297	408	9	:	:	PUNCT
ejpam-6297	408	10	α	α	PROPN
ejpam-6297	408	11	∈	∈	PROPN
ejpam-6297	408	12	λ	λ	NOUN
ejpam-6297	408	13	}	}	PUNCT
ejpam-6297	408	14	)	)	PUNCT
ejpam-6297	408	15	and	and	CCONJ
ejpam-6297	408	16	j	j	PROPN
ejpam-6297	408	17	=	=	SYM
ejpam-6297	408	18	a−	a−	PROPN
ejpam-6297	408	19	cl	cl	NOUN
ejpam-6297	408	20	(	(	PUNCT
ejpam-6297	408	21	∪{vα	∪{vα	NOUN
ejpam-6297	408	22	:	:	PUNCT
ejpam-6297	408	23	α	α	PROPN
ejpam-6297	408	24	∈	∈	PROPN
ejpam-6297	408	25	λ	λ	NOUN
ejpam-6297	408	26	}	}	PUNCT
ejpam-6297	408	27	)	)	PUNCT
ejpam-6297	408	28	.	.	PUNCT
ejpam-6297	409	1	we	we	PRON
ejpam-6297	409	2	know	know	VERB
ejpam-6297	409	3	that	that	SCONJ
ejpam-6297	409	4	g	g	PROPN
ejpam-6297	409	5	∈	∈	PROPN
ejpam-6297	409	6	τ	τ	X
ejpam-6297	409	7	and	and	CCONJ
ejpam-6297	409	8	j	j	PROPN
ejpam-6297	409	9	⊆	⊆	NUM
ejpam-6297	409	10	a	a	DET
ejpam-6297	409	11	−	−	NOUN
ejpam-6297	409	12	∪{vα	∪{vα	NOUN
ejpam-6297	409	13	:	:	PUNCT
ejpam-6297	409	14	α	α	PROPN
ejpam-6297	409	15	∈	∈	PROPN
ejpam-6297	409	16	λ	λ	PROPN
ejpam-6297	409	17	}	}	PUNCT
ejpam-6297	409	18	∈	∈	PROPN
ejpam-6297	409	19	i	i	PROPN
ejpam-6297	409	20	,	,	PUNCT
ejpam-6297	409	21	and	and	CCONJ
ejpam-6297	409	22	additionally	additionally	ADV
ejpam-6297	409	23	,	,	PUNCT
ejpam-6297	409	24	(	(	PUNCT
ejpam-6297	409	25	g	g	PROPN
ejpam-6297	409	26	−	−	PROPN
ejpam-6297	409	27	j	j	PROPN
ejpam-6297	409	28	)	)	PUNCT
ejpam-6297	409	29	∩	∩	NOUN
ejpam-6297	409	30	a	a	DET
ejpam-6297	409	31	=	=	X
ejpam-6297	409	32	∅.	∅.	NOUN
ejpam-6297	409	33	thus	thus	ADV
ejpam-6297	409	34	,	,	PUNCT
ejpam-6297	409	35	x	x	PROPN
ejpam-6297	409	36	̸∈	̸∈	PROPN
ejpam-6297	409	37	a∗	a∗	PROPN
ejpam-6297	409	38	,	,	PUNCT
ejpam-6297	409	39	confirming	confirm	VERB
ejpam-6297	409	40	that	that	SCONJ
ejpam-6297	409	41	a∗	a∗	NOUN
ejpam-6297	409	42	⊆	⊆	NUM
ejpam-6297	409	43	a.	a.	NOUN
ejpam-6297	409	44	theorem	theorem	NOUN
ejpam-6297	409	45	15	15	NUM
ejpam-6297	409	46	.	.	PUNCT
ejpam-6297	410	1	let	let	VERB
ejpam-6297	410	2	a	a	PRON
ejpam-6297	410	3	and	and	CCONJ
ejpam-6297	410	4	b	b	NOUN
ejpam-6297	410	5	be	be	AUX
ejpam-6297	410	6	subsets	subset	NOUN
ejpam-6297	410	7	of	of	ADP
ejpam-6297	410	8	an	an	DET
ejpam-6297	410	9	ideal	ideal	ADJ
ejpam-6297	410	10	topological	topological	ADJ
ejpam-6297	410	11	space	space	NOUN
ejpam-6297	410	12	(	(	PUNCT
ejpam-6297	410	13	x	x	X
ejpam-6297	410	14	,	,	PUNCT
ejpam-6297	410	15	τ	τ	PROPN
ejpam-6297	410	16	,	,	PUNCT
ejpam-6297	410	17	i	i	PROPN
ejpam-6297	410	18	)	)	PUNCT
ejpam-6297	410	19	.	.	PUNCT
ejpam-6297	411	1	if	if	SCONJ
ejpam-6297	411	2	a	a	PRON
ejpam-6297	411	3	and	and	CCONJ
ejpam-6297	411	4	b	b	NOUN
ejpam-6297	411	5	are	be	AUX
ejpam-6297	411	6	sβ	sβ	ADJ
ejpam-6297	411	7	-	-	PUNCT
ejpam-6297	411	8	i	i	NOUN
ejpam-6297	411	9	-	-	PUNCT
ejpam-6297	411	10	paracompact	paracompact	ADJ
ejpam-6297	411	11	subsets	subset	NOUN
ejpam-6297	411	12	of	of	ADP
ejpam-6297	411	13	x	x	NOUN
ejpam-6297	411	14	,	,	PUNCT
ejpam-6297	411	15	then	then	ADV
ejpam-6297	411	16	a	a	DET
ejpam-6297	411	17	∪b	∪b	PUNCT
ejpam-6297	411	18	is	be	AUX
ejpam-6297	411	19	also	also	ADV
ejpam-6297	411	20	an	an	DET
ejpam-6297	411	21	sβ	sβ	NOUN
ejpam-6297	411	22	-	-	PUNCT
ejpam-6297	411	23	i	i	NOUN
ejpam-6297	411	24	-	-	PUNCT
ejpam-6297	411	25	paracompact	paracompact	NOUN
ejpam-6297	411	26	subset	subset	NOUN
ejpam-6297	411	27	of	of	ADP
ejpam-6297	411	28	x	x	SYM
ejpam-6297	411	29	proof	proof	NOUN
ejpam-6297	411	30	.	.	PUNCT
ejpam-6297	412	1	let	let	VERB
ejpam-6297	412	2	a	a	DET
ejpam-6297	412	3	=	=	X
ejpam-6297	412	4	{	{	PUNCT
ejpam-6297	412	5	uλ	uλ	NOUN
ejpam-6297	412	6	:	:	PUNCT
ejpam-6297	412	7	λ	λ	X
ejpam-6297	412	8	∈	∈	PROPN
ejpam-6297	412	9	λ	λ	PROPN
ejpam-6297	412	10	}	}	PUNCT
ejpam-6297	412	11	be	be	VERB
ejpam-6297	412	12	an	an	DET
ejpam-6297	412	13	open	open	ADJ
ejpam-6297	412	14	cover	cover	NOUN
ejpam-6297	412	15	of	of	ADP
ejpam-6297	412	16	a	a	DET
ejpam-6297	412	17	∪	∪	X
ejpam-6297	412	18	b.	b.	NOUN
ejpam-6297	412	19	this	this	DET
ejpam-6297	412	20	cover	cover	NOUN
ejpam-6297	412	21	a	a	PRON
ejpam-6297	412	22	also	also	ADV
ejpam-6297	412	23	serves	serve	VERB
ejpam-6297	412	24	as	as	ADP
ejpam-6297	412	25	an	an	DET
ejpam-6297	412	26	open	open	ADJ
ejpam-6297	412	27	cover	cover	NOUN
ejpam-6297	412	28	for	for	ADP
ejpam-6297	412	29	both	both	CCONJ
ejpam-6297	412	30	a	a	PRON
ejpam-6297	412	31	and	and	CCONJ
ejpam-6297	412	32	b.	b.	NOUN
ejpam-6297	412	33	by	by	ADP
ejpam-6297	412	34	assumption	assumption	NOUN
ejpam-6297	412	35	,	,	PUNCT
ejpam-6297	412	36	there	there	PRON
ejpam-6297	412	37	exist	exist	VERB
ejpam-6297	412	38	sβ	sβ	NOUN
ejpam-6297	412	39	-	-	PUNCT
ejpam-6297	412	40	i	i	NOUN
ejpam-6297	412	41	-	-	PUNCT
ejpam-6297	412	42	locally	locally	ADV
ejpam-6297	412	43	finite	finite	VERB
ejpam-6297	412	44	strong	strong	ADJ
ejpam-6297	412	45	β	β	X
ejpam-6297	412	46	-	-	ADJ
ejpam-6297	412	47	i	i	NOUN
ejpam-6297	412	48	-	-	PUNCT
ejpam-6297	412	49	open	open	ADJ
ejpam-6297	412	50	families	family	NOUN
ejpam-6297	412	51	b	b	NOUN
ejpam-6297	412	52	=	=	PUNCT
ejpam-6297	412	53	{	{	PUNCT
ejpam-6297	412	54	vα	vα	X
ejpam-6297	412	55	:	:	PUNCT
ejpam-6297	412	56	α	α	PROPN
ejpam-6297	412	57	∈	∈	PROPN
ejpam-6297	412	58	λ1	λ1	PROPN
ejpam-6297	412	59	}	}	PUNCT
ejpam-6297	412	60	for	for	ADP
ejpam-6297	412	61	a	a	PRON
ejpam-6297	412	62	and	and	CCONJ
ejpam-6297	412	63	c	c	NOUN
ejpam-6297	412	64	=	=	SYM
ejpam-6297	412	65	{	{	PUNCT
ejpam-6297	412	66	wµ	wµ	NOUN
ejpam-6297	412	67	:	:	PUNCT
ejpam-6297	412	68	µ	µ	X
ejpam-6297	412	69	∈	∈	PROPN
ejpam-6297	412	70	λ2	λ2	PROPN
ejpam-6297	412	71	}	}	PUNCT
ejpam-6297	412	72	for	for	ADP
ejpam-6297	412	73	b	b	NOUN
ejpam-6297	412	74	,	,	PUNCT
ejpam-6297	412	75	which	which	PRON
ejpam-6297	412	76	refine	refine	VERB
ejpam-6297	412	77	a	a	DET
ejpam-6297	412	78	such	such	ADJ
ejpam-6297	412	79	that	that	SCONJ
ejpam-6297	412	80	:	:	PUNCT
ejpam-6297	412	81	a−	a−	PROPN
ejpam-6297	412	82	∪{vα	∪{vα	NOUN
ejpam-6297	412	83	:	:	PUNCT
ejpam-6297	412	84	α	α	PROPN
ejpam-6297	412	85	∈	∈	PROPN
ejpam-6297	412	86	λ1	λ1	PROPN
ejpam-6297	412	87	}	}	PUNCT
ejpam-6297	412	88	∈	∈	PROPN
ejpam-6297	412	89	i	i	PROPN
ejpam-6297	412	90	,	,	PUNCT
ejpam-6297	412	91	b	b	PROPN
ejpam-6297	412	92	−	−	X
ejpam-6297	412	93	∪{wµ	∪{wµ	NOUN
ejpam-6297	412	94	:	:	PUNCT
ejpam-6297	412	95	µ	µ	PROPN
ejpam-6297	412	96	∈	∈	PROPN
ejpam-6297	412	97	λ2	λ2	PROPN
ejpam-6297	412	98	}	}	PUNCT
ejpam-6297	412	99	∈	∈	PROPN
ejpam-6297	412	100	i.	i.	NOUN
ejpam-6297	412	101	this	this	PRON
ejpam-6297	412	102	implies	imply	VERB
ejpam-6297	412	103	that	that	SCONJ
ejpam-6297	412	104	:	:	PUNCT
ejpam-6297	412	105	a	a	DET
ejpam-6297	412	106	⊆	⊆	NUM
ejpam-6297	412	107	∪{vα	∪{vα	NOUN
ejpam-6297	412	108	:	:	PUNCT
ejpam-6297	412	109	α	α	PROPN
ejpam-6297	412	110	∈	∈	PROPN
ejpam-6297	412	111	λ1	λ1	PROPN
ejpam-6297	412	112	}	}	PUNCT
ejpam-6297	412	113	∪	∪	PROPN
ejpam-6297	412	114	i1	i1	PROPN
ejpam-6297	412	115	,	,	PUNCT
ejpam-6297	412	116	b	b	PROPN
ejpam-6297	412	117	⊆	⊆	NUM
ejpam-6297	412	118	∪{wµ	∪{wµ	NOUN
ejpam-6297	412	119	:	:	PUNCT
ejpam-6297	412	120	µ	µ	PROPN
ejpam-6297	412	121	∈	∈	PROPN
ejpam-6297	412	122	λ2	λ2	PROPN
ejpam-6297	412	123	}	}	PUNCT
ejpam-6297	412	124	∪	∪	NOUN
ejpam-6297	412	125	i2	i2	PROPN
ejpam-6297	412	126	,	,	PUNCT
ejpam-6297	412	127	where	where	SCONJ
ejpam-6297	412	128	i1	i1	PROPN
ejpam-6297	412	129	,	,	PUNCT
ejpam-6297	412	130	i2	i2	PROPN
ejpam-6297	412	131	∈	∈	PROPN
ejpam-6297	412	132	i.	i.	NOUN
ejpam-6297	412	133	therefore	therefore	ADV
ejpam-6297	412	134	,	,	PUNCT
ejpam-6297	412	135	we	we	PRON
ejpam-6297	412	136	have	have	VERB
ejpam-6297	412	137	:	:	PUNCT
ejpam-6297	412	138	a	a	DET
ejpam-6297	412	139	∪b	∪b	NUM
ejpam-6297	412	140	⊆	⊆	NUM
ejpam-6297	412	141	∪({vα	∪({vα	PROPN
ejpam-6297	412	142	:	:	PUNCT
ejpam-6297	412	143	α	α	PROPN
ejpam-6297	412	144	∈	∈	PROPN
ejpam-6297	412	145	λ1	λ1	PROPN
ejpam-6297	412	146	}	}	PUNCT
ejpam-6297	412	147	∪	∪	NOUN
ejpam-6297	412	148	{	{	PUNCT
ejpam-6297	412	149	wµ	wµ	NOUN
ejpam-6297	412	150	:	:	PUNCT
ejpam-6297	412	151	µ	µ	X
ejpam-6297	412	152	∈	∈	PROPN
ejpam-6297	412	153	λ2	λ2	NOUN
ejpam-6297	412	154	}	}	PUNCT
ejpam-6297	412	155	)	)	PUNCT
ejpam-6297	412	156	∪	∪	NOUN
ejpam-6297	412	157	(	(	PUNCT
ejpam-6297	412	158	i1	i1	PROPN
ejpam-6297	412	159	∪	∪	PROPN
ejpam-6297	412	160	i2	i2	PROPN
ejpam-6297	412	161	)	)	PUNCT
ejpam-6297	412	162	.	.	PUNCT
ejpam-6297	413	1	it	it	PRON
ejpam-6297	413	2	follows	follow	VERB
ejpam-6297	413	3	that	that	SCONJ
ejpam-6297	413	4	:	:	PUNCT
ejpam-6297	413	5	a	a	DET
ejpam-6297	413	6	∪b	∪b	PUNCT
ejpam-6297	413	7	−	−	PROPN
ejpam-6297	413	8	∪{vα	∪{vα	X
ejpam-6297	413	9	∪wµ	∪wµ	X
ejpam-6297	413	10	:	:	PUNCT
ejpam-6297	413	11	α	α	PROPN
ejpam-6297	413	12	∈	∈	PROPN
ejpam-6297	413	13	λ1	λ1	PROPN
ejpam-6297	413	14	,	,	PUNCT
ejpam-6297	413	15	µ	µ	PROPN
ejpam-6297	413	16	∈	∈	NOUN
ejpam-6297	413	17	λ2	λ2	NOUN
ejpam-6297	413	18	}	}	PUNCT
ejpam-6297	413	19	⊆	⊆	NUM
ejpam-6297	413	20	i1	i1	PROPN
ejpam-6297	413	21	∪	∪	PROPN
ejpam-6297	413	22	i2	i2	PROPN
ejpam-6297	413	23	∈	∈	PROPN
ejpam-6297	413	24	i.	i.	NOUN
ejpam-6297	413	25	we	we	PRON
ejpam-6297	413	26	see	see	VERB
ejpam-6297	413	27	that	that	SCONJ
ejpam-6297	414	1	the	the	DET
ejpam-6297	414	2	collection	collection	NOUN
ejpam-6297	414	3	d	d	NOUN
ejpam-6297	414	4	=	=	PRON
ejpam-6297	414	5	{	{	PUNCT
ejpam-6297	414	6	vα	vα	X
ejpam-6297	414	7	∪	∪	VERB
ejpam-6297	414	8	wµ	wµ	ADP
ejpam-6297	414	9	:	:	PUNCT
ejpam-6297	414	10	α	α	PROPN
ejpam-6297	414	11	∈	∈	PROPN
ejpam-6297	414	12	λ1	λ1	PROPN
ejpam-6297	414	13	,	,	PUNCT
ejpam-6297	414	14	µ	µ	PROPN
ejpam-6297	414	15	∈	∈	NOUN
ejpam-6297	414	16	λ2	λ2	NOUN
ejpam-6297	414	17	}	}	PUNCT
ejpam-6297	414	18	of	of	ADP
ejpam-6297	414	19	strong	strong	ADJ
ejpam-6297	414	20	β	β	X
ejpam-6297	414	21	-	-	ADJ
ejpam-6297	414	22	i	i	NOUN
ejpam-6297	414	23	-	-	PUNCT
ejpam-6297	414	24	open	open	ADJ
ejpam-6297	414	25	sets	set	NOUN
ejpam-6297	414	26	is	be	AUX
ejpam-6297	414	27	sβ	sβ	NOUN
ejpam-6297	414	28	-	-	PUNCT
ejpam-6297	414	29	i	i	NOUN
ejpam-6297	414	30	-	-	PUNCT
ejpam-6297	414	31	locally	locally	ADV
ejpam-6297	414	32	finite	finite	NOUN
ejpam-6297	414	33	and	and	CCONJ
ejpam-6297	414	34	refines	refine	VERB
ejpam-6297	414	35	a.	a.	NOUN
ejpam-6297	414	36	consequently	consequently	ADV
ejpam-6297	414	37	,	,	PUNCT
ejpam-6297	414	38	a	a	DET
ejpam-6297	414	39	∪	∪	X
ejpam-6297	414	40	b	b	NOUN
ejpam-6297	414	41	is	be	AUX
ejpam-6297	414	42	an	an	DET
ejpam-6297	414	43	sβ	sβ	NOUN
ejpam-6297	414	44	-	-	PUNCT
ejpam-6297	414	45	i	i	NOUN
ejpam-6297	414	46	-	-	PUNCT
ejpam-6297	414	47	paracompact	paracompact	NOUN
ejpam-6297	414	48	subset	subset	NOUN
ejpam-6297	414	49	of	of	ADP
ejpam-6297	414	50	x.	x.	PROPN
ejpam-6297	414	51	c.	c.	PROPN
ejpam-6297	414	52	boonpok	boonpok	PROPN
ejpam-6297	414	53	,	,	PUNCT
ejpam-6297	414	54	p.	p.	PROPN
ejpam-6297	414	55	raktaow	raktaow	NOUN
ejpam-6297	414	56	,	,	PUNCT
ejpam-6297	414	57	a.	a.	PROPN
ejpam-6297	414	58	sama	sama	PROPN
ejpam-6297	414	59	-	-	PUNCT
ejpam-6297	414	60	ae	ae	PROPN
ejpam-6297	414	61	/	/	SYM
ejpam-6297	414	62	eur	eur	PROPN
ejpam-6297	414	63	.	.	PUNCT
ejpam-6297	415	1	j.	j.	PROPN
ejpam-6297	415	2	pure	pure	PROPN
ejpam-6297	415	3	appl	appl	PROPN
ejpam-6297	415	4	.	.	PROPN
ejpam-6297	415	5	math	math	PROPN
ejpam-6297	415	6	,	,	PUNCT
ejpam-6297	415	7	18	18	NUM
ejpam-6297	415	8	(	(	PUNCT
ejpam-6297	415	9	3	3	NUM
ejpam-6297	415	10	)	)	PUNCT
ejpam-6297	415	11	(	(	PUNCT
ejpam-6297	415	12	2025	2025	NUM
ejpam-6297	415	13	)	)	PUNCT
ejpam-6297	415	14	,	,	PUNCT
ejpam-6297	415	15	6297	6297	NUM
ejpam-6297	415	16	16	16	NUM
ejpam-6297	415	17	of	of	ADP
ejpam-6297	415	18	23	23	NUM
ejpam-6297	415	19	theorem	theorem	NOUN
ejpam-6297	415	20	16	16	NUM
ejpam-6297	415	21	.	.	PUNCT
ejpam-6297	416	1	let	let	VERB
ejpam-6297	416	2	(	(	PUNCT
ejpam-6297	416	3	x	x	X
ejpam-6297	416	4	,	,	PUNCT
ejpam-6297	416	5	τ	τ	PROPN
ejpam-6297	416	6	,	,	PUNCT
ejpam-6297	416	7	i	i	PRON
ejpam-6297	416	8	)	)	PUNCT
ejpam-6297	416	9	be	be	VERB
ejpam-6297	416	10	an	an	DET
ejpam-6297	416	11	ideal	ideal	ADJ
ejpam-6297	416	12	topological	topological	ADJ
ejpam-6297	416	13	space	space	NOUN
ejpam-6297	416	14	.	.	PUNCT
ejpam-6297	417	1	if	if	SCONJ
ejpam-6297	417	2	a	a	PRON
ejpam-6297	417	3	is	be	AUX
ejpam-6297	417	4	an	an	DET
ejpam-6297	417	5	sβ	sβ	NOUN
ejpam-6297	417	6	-	-	PUNCT
ejpam-6297	417	7	i	i	NOUN
ejpam-6297	417	8	-	-	PUNCT
ejpam-6297	417	9	paracompact	paracompact	NOUN
ejpam-6297	417	10	subset	subset	NOUN
ejpam-6297	417	11	of	of	ADP
ejpam-6297	417	12	x	x	PUNCT
ejpam-6297	417	13	and	and	CCONJ
ejpam-6297	417	14	b	b	PROPN
ejpam-6297	417	15	is	be	AUX
ejpam-6297	417	16	a	a	DET
ejpam-6297	417	17	closed	closed	ADJ
ejpam-6297	417	18	subset	subset	NOUN
ejpam-6297	417	19	of	of	ADP
ejpam-6297	417	20	x	x	PRON
ejpam-6297	417	21	,	,	PUNCT
ejpam-6297	417	22	then	then	ADV
ejpam-6297	417	23	a∩b	a∩b	PROPN
ejpam-6297	417	24	is	be	AUX
ejpam-6297	417	25	also	also	ADV
ejpam-6297	417	26	an	an	DET
ejpam-6297	417	27	sβ	sβ	NOUN
ejpam-6297	417	28	-	-	PUNCT
ejpam-6297	417	29	i	i	NOUN
ejpam-6297	417	30	-	-	PUNCT
ejpam-6297	417	31	paracompact	paracompact	NOUN
ejpam-6297	417	32	subset	subset	NOUN
ejpam-6297	417	33	of	of	ADP
ejpam-6297	417	34	x.	x.	NOUN
ejpam-6297	417	35	proof	proof	PROPN
ejpam-6297	417	36	.	.	PUNCT
ejpam-6297	418	1	let	let	VERB
ejpam-6297	418	2	a	a	DET
ejpam-6297	418	3	=	=	X
ejpam-6297	418	4	{	{	PUNCT
ejpam-6297	418	5	uλ	uλ	NOUN
ejpam-6297	418	6	:	:	PUNCT
ejpam-6297	418	7	λ	λ	X
ejpam-6297	418	8	∈	∈	PROPN
ejpam-6297	418	9	λ	λ	PROPN
ejpam-6297	418	10	}	}	PUNCT
ejpam-6297	418	11	be	be	VERB
ejpam-6297	418	12	an	an	DET
ejpam-6297	418	13	open	open	ADJ
ejpam-6297	418	14	cover	cover	NOUN
ejpam-6297	418	15	of	of	ADP
ejpam-6297	418	16	a	a	DET
ejpam-6297	418	17	∩	∩	ADJ
ejpam-6297	418	18	b.	b.	NOUN
ejpam-6297	418	19	since	since	SCONJ
ejpam-6297	418	20	x	x	PROPN
ejpam-6297	418	21	−	−	PROPN
ejpam-6297	418	22	b	b	NOUN
ejpam-6297	418	23	is	be	AUX
ejpam-6297	418	24	open	open	ADJ
ejpam-6297	418	25	in	in	ADP
ejpam-6297	418	26	x	x	PRON
ejpam-6297	418	27	,	,	PUNCT
ejpam-6297	418	28	the	the	DET
ejpam-6297	418	29	collection	collection	NOUN
ejpam-6297	418	30	a1	a1	NOUN
ejpam-6297	418	31	=	=	SYM
ejpam-6297	418	32	{	{	PUNCT
ejpam-6297	418	33	uλ	uλ	X
ejpam-6297	418	34	:	:	PUNCT
ejpam-6297	418	35	λ	λ	X
ejpam-6297	418	36	∈	∈	NOUN
ejpam-6297	418	37	λ}∪{x−b	λ}∪{x−b	NOUN
ejpam-6297	418	38	}	}	PUNCT
ejpam-6297	418	39	forms	form	VERB
ejpam-6297	418	40	an	an	DET
ejpam-6297	418	41	open	open	ADJ
ejpam-6297	418	42	cover	cover	NOUN
ejpam-6297	418	43	of	of	ADP
ejpam-6297	418	44	a.	a.	NOUN
ejpam-6297	418	45	by	by	ADP
ejpam-6297	418	46	assumption	assumption	NOUN
ejpam-6297	418	47	and	and	CCONJ
ejpam-6297	418	48	lemma	lemma	PROPN
ejpam-6297	418	49	6	6	NUM
ejpam-6297	418	50	,	,	PUNCT
ejpam-6297	418	51	a1	a1	NOUN
ejpam-6297	418	52	has	have	VERB
ejpam-6297	418	53	a	a	DET
ejpam-6297	418	54	precise	precise	ADJ
ejpam-6297	418	55	sβ	sβ	NOUN
ejpam-6297	418	56	-	-	PUNCT
ejpam-6297	418	57	i	i	NOUN
ejpam-6297	418	58	-	-	PUNCT
ejpam-6297	418	59	locally	locally	ADV
ejpam-6297	418	60	finite	finite	VERB
ejpam-6297	418	61	strong	strong	ADJ
ejpam-6297	418	62	β	β	X
ejpam-6297	418	63	-	-	ADJ
ejpam-6297	418	64	i	i	NOUN
ejpam-6297	418	65	-	-	PUNCT
ejpam-6297	418	66	open	open	ADJ
ejpam-6297	418	67	refinement	refinement	NOUN
ejpam-6297	418	68	b	b	PROPN
ejpam-6297	418	69	=	=	PUNCT
ejpam-6297	418	70	{	{	PUNCT
ejpam-6297	418	71	vλ	vλ	INTJ
ejpam-6297	418	72	:	:	PUNCT
ejpam-6297	418	73	λ	λ	PROPN
ejpam-6297	418	74	∈	∈	PROPN
ejpam-6297	418	75	λ	λ	PROPN
ejpam-6297	418	76	}	}	PUNCT
ejpam-6297	418	77	∪	∪	ADJ
ejpam-6297	418	78	{	{	PUNCT
ejpam-6297	418	79	v	v	NOUN
ejpam-6297	418	80	}	}	PUNCT
ejpam-6297	418	81	such	such	ADJ
ejpam-6297	418	82	that	that	SCONJ
ejpam-6297	418	83	:	:	PUNCT
ejpam-6297	418	84	vλ	vλ	ADP
ejpam-6297	418	85	⊆	⊆	NUM
ejpam-6297	418	86	uλ	uλ	NOUN
ejpam-6297	418	87	for	for	ADP
ejpam-6297	418	88	all	all	DET
ejpam-6297	418	89	λ	λ	PROPN
ejpam-6297	418	90	∈	∈	PROPN
ejpam-6297	418	91	λ	λ	PROPN
ejpam-6297	418	92	,	,	PUNCT
ejpam-6297	418	93	v	v	ADP
ejpam-6297	418	94	⊆	⊆	NUM
ejpam-6297	418	95	x	x	SYM
ejpam-6297	418	96	−b	−b	ADJ
ejpam-6297	418	97	,	,	PUNCT
ejpam-6297	418	98	a−	a−	PROPN
ejpam-6297	418	99	(	(	PUNCT
ejpam-6297	418	100	∪{vλ	∪{vλ	NUM
ejpam-6297	418	101	:	:	PUNCT
ejpam-6297	418	102	λ	λ	PROPN
ejpam-6297	418	103	∈	∈	PROPN
ejpam-6297	418	104	λ	λ	PROPN
ejpam-6297	418	105	}	}	PUNCT
ejpam-6297	418	106	∪	∪	ADJ
ejpam-6297	418	107	{	{	PUNCT
ejpam-6297	418	108	v	v	NOUN
ejpam-6297	418	109	}	}	PUNCT
ejpam-6297	418	110	)	)	PUNCT
ejpam-6297	418	111	∈	∈	PROPN
ejpam-6297	418	112	i.	i.	NOUN
ejpam-6297	418	113	now	now	ADV
ejpam-6297	418	114	,	,	PUNCT
ejpam-6297	418	115	observe	observe	VERB
ejpam-6297	418	116	that	that	SCONJ
ejpam-6297	418	117	:	:	PUNCT
ejpam-6297	419	1	a∩b−∪{vλ	a∩b−∪{vλ	NOUN
ejpam-6297	419	2	:	:	PUNCT
ejpam-6297	419	3	λ	λ	PROPN
ejpam-6297	419	4	∈	∈	PROPN
ejpam-6297	419	5	λ	λ	NOUN
ejpam-6297	419	6	}	}	PUNCT
ejpam-6297	419	7	=	=	PUNCT
ejpam-6297	419	8	a∩b−	a∩b−	ADV
ejpam-6297	419	9	(	(	PUNCT
ejpam-6297	419	10	∪{vλ	∪{vλ	NUM
ejpam-6297	419	11	:	:	PUNCT
ejpam-6297	419	12	λ	λ	PROPN
ejpam-6297	419	13	∈	∈	PROPN
ejpam-6297	419	14	λ}∪{v	λ}∪{v	NOUN
ejpam-6297	419	15	}	}	PUNCT
ejpam-6297	419	16	)	)	PUNCT
ejpam-6297	419	17	⊆	⊆	NUM
ejpam-6297	419	18	a−	a−	NOUN
ejpam-6297	419	19	(	(	PUNCT
ejpam-6297	419	20	∪{vλ	∪{vλ	NUM
ejpam-6297	419	21	:	:	PUNCT
ejpam-6297	419	22	λ	λ	PROPN
ejpam-6297	419	23	∈	∈	PROPN
ejpam-6297	419	24	λ}∪{v	λ}∪{v	NOUN
ejpam-6297	419	25	}	}	PUNCT
ejpam-6297	419	26	)	)	PUNCT
ejpam-6297	419	27	∈	∈	PROPN
ejpam-6297	419	28	i.	i.	NOUN
ejpam-6297	419	29	thus	thus	ADV
ejpam-6297	419	30	,	,	PUNCT
ejpam-6297	419	31	we	we	PRON
ejpam-6297	419	32	have	have	VERB
ejpam-6297	419	33	a∩b−∪{vλ	a∩b−∪{vλ	NOUN
ejpam-6297	419	34	:	:	PUNCT
ejpam-6297	419	35	λ	λ	PROPN
ejpam-6297	419	36	∈	∈	PROPN
ejpam-6297	419	37	λ	λ	PROPN
ejpam-6297	419	38	}	}	PUNCT
ejpam-6297	419	39	∈	∈	PROPN
ejpam-6297	419	40	i.	i.	NOUN
ejpam-6297	419	41	it	it	PRON
ejpam-6297	419	42	follows	follow	VERB
ejpam-6297	419	43	that	that	SCONJ
ejpam-6297	419	44	the	the	DET
ejpam-6297	419	45	collection	collection	NOUN
ejpam-6297	419	46	b1	b1	NOUN
ejpam-6297	419	47	=	=	SYM
ejpam-6297	419	48	{	{	PUNCT
ejpam-6297	419	49	vλ	vλ	INTJ
ejpam-6297	419	50	:	:	PUNCT
ejpam-6297	419	51	λ	λ	PROPN
ejpam-6297	419	52	∈	∈	PROPN
ejpam-6297	419	53	λ	λ	NOUN
ejpam-6297	419	54	}	}	PUNCT
ejpam-6297	419	55	,	,	PUNCT
ejpam-6297	419	56	consisting	consist	VERB
ejpam-6297	419	57	of	of	ADP
ejpam-6297	419	58	strong	strong	ADJ
ejpam-6297	419	59	β	β	X
ejpam-6297	419	60	-	-	ADJ
ejpam-6297	419	61	i	i	NOUN
ejpam-6297	419	62	-	-	PUNCT
ejpam-6297	419	63	open	open	ADJ
ejpam-6297	419	64	sets	set	NOUN
ejpam-6297	419	65	,	,	PUNCT
ejpam-6297	419	66	is	be	AUX
ejpam-6297	419	67	sβ	sβ	NOUN
ejpam-6297	419	68	-	-	PUNCT
ejpam-6297	419	69	i	i	NOUN
ejpam-6297	419	70	-	-	PUNCT
ejpam-6297	419	71	locally	locally	ADV
ejpam-6297	419	72	finite	finite	NOUN
ejpam-6297	419	73	and	and	CCONJ
ejpam-6297	419	74	refines	refine	VERB
ejpam-6297	419	75	a.	a.	NOUN
ejpam-6297	419	76	therefore	therefore	ADV
ejpam-6297	419	77	,	,	PUNCT
ejpam-6297	419	78	a	a	DET
ejpam-6297	419	79	∩	∩	ADJ
ejpam-6297	419	80	b	b	NOUN
ejpam-6297	419	81	is	be	AUX
ejpam-6297	419	82	an	an	DET
ejpam-6297	419	83	sβ	sβ	NOUN
ejpam-6297	419	84	-	-	PUNCT
ejpam-6297	419	85	i	i	NOUN
ejpam-6297	419	86	-	-	PUNCT
ejpam-6297	419	87	paracompact	paracompact	NOUN
ejpam-6297	419	88	subset	subset	NOUN
ejpam-6297	419	89	of	of	ADP
ejpam-6297	419	90	x.	x.	NOUN
ejpam-6297	419	91	as	as	ADP
ejpam-6297	419	92	a	a	DET
ejpam-6297	419	93	consequence	consequence	NOUN
ejpam-6297	419	94	of	of	ADP
ejpam-6297	419	95	theorem	theorem	NOUN
ejpam-6297	419	96	16	16	NUM
ejpam-6297	419	97	,	,	PUNCT
ejpam-6297	419	98	we	we	PRON
ejpam-6297	419	99	obtain	obtain	VERB
ejpam-6297	419	100	the	the	DET
ejpam-6297	419	101	following	follow	VERB
ejpam-6297	419	102	corollaries	corollary	NOUN
ejpam-6297	419	103	.	.	PUNCT
ejpam-6297	420	1	corollary	corollary	ADJ
ejpam-6297	420	2	2	2	NUM
ejpam-6297	420	3	.	.	PUNCT
ejpam-6297	421	1	if	if	SCONJ
ejpam-6297	421	2	a	a	PRON
ejpam-6297	421	3	is	be	AUX
ejpam-6297	421	4	a	a	DET
ejpam-6297	421	5	closed	closed	ADJ
ejpam-6297	421	6	subset	subset	NOUN
ejpam-6297	421	7	of	of	ADP
ejpam-6297	421	8	an	an	DET
ejpam-6297	421	9	sβ	sβ	NOUN
ejpam-6297	421	10	-	-	PUNCT
ejpam-6297	421	11	i	i	NOUN
ejpam-6297	421	12	-	-	PUNCT
ejpam-6297	421	13	paracompact	paracompact	ADJ
ejpam-6297	421	14	space	space	NOUN
ejpam-6297	421	15	(	(	PUNCT
ejpam-6297	421	16	x	x	X
ejpam-6297	421	17	,	,	PUNCT
ejpam-6297	421	18	τ	τ	PROPN
ejpam-6297	421	19	,	,	PUNCT
ejpam-6297	421	20	i	i	PROPN
ejpam-6297	421	21	)	)	PUNCT
ejpam-6297	421	22	,	,	PUNCT
ejpam-6297	421	23	then	then	ADV
ejpam-6297	421	24	a	a	PRON
ejpam-6297	421	25	is	be	AUX
ejpam-6297	421	26	also	also	ADV
ejpam-6297	421	27	an	an	DET
ejpam-6297	421	28	sβ	sβ	NOUN
ejpam-6297	421	29	-	-	PUNCT
ejpam-6297	421	30	i	i	NOUN
ejpam-6297	421	31	-	-	PUNCT
ejpam-6297	421	32	paracompact	paracompact	NOUN
ejpam-6297	421	33	subset	subset	NOUN
ejpam-6297	421	34	of	of	ADP
ejpam-6297	421	35	x.	x.	PROPN
ejpam-6297	421	36	corollary	corollary	PROPN
ejpam-6297	421	37	3	3	X
ejpam-6297	421	38	.	.	PUNCT
ejpam-6297	422	1	let	let	VERB
ejpam-6297	422	2	a	a	PRON
ejpam-6297	422	3	and	and	CCONJ
ejpam-6297	422	4	b	b	NOUN
ejpam-6297	422	5	be	be	AUX
ejpam-6297	422	6	closed	close	VERB
ejpam-6297	422	7	subsets	subset	NOUN
ejpam-6297	422	8	of	of	ADP
ejpam-6297	422	9	an	an	DET
ejpam-6297	422	10	sβ	sβ	NOUN
ejpam-6297	422	11	-	-	PUNCT
ejpam-6297	422	12	i	i	NOUN
ejpam-6297	422	13	-	-	PUNCT
ejpam-6297	422	14	paracompact	paracompact	ADJ
ejpam-6297	422	15	space	space	NOUN
ejpam-6297	422	16	of	of	ADP
ejpam-6297	422	17	(	(	PUNCT
ejpam-6297	422	18	x	x	X
ejpam-6297	422	19	,	,	PUNCT
ejpam-6297	422	20	τ	τ	PROPN
ejpam-6297	422	21	,	,	PUNCT
ejpam-6297	422	22	i	i	PROPN
ejpam-6297	422	23	)	)	PUNCT
ejpam-6297	422	24	,	,	PUNCT
ejpam-6297	422	25	then	then	ADV
ejpam-6297	422	26	a	a	DET
ejpam-6297	422	27	∪b	∪b	PUNCT
ejpam-6297	422	28	is	be	AUX
ejpam-6297	422	29	also	also	ADV
ejpam-6297	422	30	an	an	DET
ejpam-6297	422	31	sβ	sβ	NOUN
ejpam-6297	422	32	-	-	PUNCT
ejpam-6297	422	33	i	i	NOUN
ejpam-6297	422	34	-	-	PUNCT
ejpam-6297	422	35	paracompact	paracompact	NOUN
ejpam-6297	422	36	subset	subset	NOUN
ejpam-6297	422	37	of	of	ADP
ejpam-6297	422	38	x	x	SYM
ejpam-6297	422	39	corollary	corollary	ADJ
ejpam-6297	422	40	4	4	NUM
ejpam-6297	422	41	.	.	PUNCT
ejpam-6297	423	1	if	if	SCONJ
ejpam-6297	423	2	a	a	PRON
ejpam-6297	423	3	is	be	AUX
ejpam-6297	423	4	an	an	DET
ejpam-6297	423	5	sβ	sβ	NOUN
ejpam-6297	423	6	-	-	PUNCT
ejpam-6297	423	7	i	i	NOUN
ejpam-6297	423	8	-	-	PUNCT
ejpam-6297	423	9	paracompact	paracompact	NOUN
ejpam-6297	423	10	subset	subset	NOUN
ejpam-6297	423	11	of	of	ADP
ejpam-6297	423	12	x	x	PUNCT
ejpam-6297	423	13	and	and	CCONJ
ejpam-6297	423	14	b	b	PROPN
ejpam-6297	423	15	is	be	AUX
ejpam-6297	423	16	an	an	DET
ejpam-6297	423	17	open	open	ADJ
ejpam-6297	423	18	set	set	NOUN
ejpam-6297	423	19	contained	contain	VERB
ejpam-6297	423	20	a	a	PRON
ejpam-6297	423	21	,	,	PUNCT
ejpam-6297	423	22	then	then	ADV
ejpam-6297	423	23	a−b	a−b	PROPN
ejpam-6297	423	24	is	be	AUX
ejpam-6297	423	25	an	an	DET
ejpam-6297	423	26	sβ	sβ	VERB
ejpam-6297	423	27	-	-	PUNCT
ejpam-6297	423	28	i	i	NOUN
ejpam-6297	423	29	-	-	PUNCT
ejpam-6297	423	30	paracompact	paracompact	NOUN
ejpam-6297	423	31	subset	subset	NOUN
ejpam-6297	423	32	of	of	ADP
ejpam-6297	423	33	x.	x.	PROPN
ejpam-6297	423	34	lemma	lemma	PROPN
ejpam-6297	423	35	7	7	NUM
ejpam-6297	423	36	.	.	PUNCT
ejpam-6297	424	1	[	[	X
ejpam-6297	424	2	19	19	NUM
ejpam-6297	424	3	]	]	PUNCT
ejpam-6297	424	4	let	let	VERB
ejpam-6297	424	5	i	i	PRON
ejpam-6297	424	6	be	be	AUX
ejpam-6297	424	7	an	an	DET
ejpam-6297	424	8	ideal	ideal	NOUN
ejpam-6297	424	9	on	on	ADP
ejpam-6297	424	10	a	a	DET
ejpam-6297	424	11	topological	topological	ADJ
ejpam-6297	424	12	space	space	NOUN
ejpam-6297	424	13	x.	x.	NOUN
ejpam-6297	425	1	if	if	SCONJ
ejpam-6297	425	2	y	y	PROPN
ejpam-6297	425	3	is	be	AUX
ejpam-6297	425	4	a	a	DET
ejpam-6297	425	5	subset	subset	NOUN
ejpam-6297	425	6	of	of	ADP
ejpam-6297	425	7	x	x	PRON
ejpam-6297	425	8	,	,	PUNCT
ejpam-6297	425	9	then	then	ADV
ejpam-6297	425	10	iy	iy	PROPN
ejpam-6297	425	11	=	=	PUNCT
ejpam-6297	425	12	{	{	PUNCT
ejpam-6297	425	13	i	i	PROPN
ejpam-6297	425	14	∩	∩	NOUN
ejpam-6297	425	15	y	y	NOUN
ejpam-6297	425	16	:	:	PUNCT
ejpam-6297	425	17	i	i	PRON
ejpam-6297	425	18	∈	∈	VERB
ejpam-6297	425	19	i	i	PRON
ejpam-6297	425	20	}	}	PUNCT
ejpam-6297	425	21	is	be	AUX
ejpam-6297	425	22	an	an	DET
ejpam-6297	425	23	ideal	ideal	NOUN
ejpam-6297	425	24	on	on	ADP
ejpam-6297	425	25	y	y	PROPN
ejpam-6297	425	26	.	.	PUNCT
ejpam-6297	426	1	theorem	theorem	PROPN
ejpam-6297	426	2	17	17	NUM
ejpam-6297	426	3	.	.	PUNCT
ejpam-6297	427	1	let	let	VERB
ejpam-6297	427	2	a	a	PRON
ejpam-6297	427	3	and	and	CCONJ
ejpam-6297	427	4	b	b	NOUN
ejpam-6297	427	5	be	be	AUX
ejpam-6297	427	6	subsets	subset	NOUN
ejpam-6297	427	7	of	of	ADP
ejpam-6297	427	8	an	an	DET
ejpam-6297	427	9	ideal	ideal	ADJ
ejpam-6297	427	10	topological	topological	ADJ
ejpam-6297	427	11	space	space	NOUN
ejpam-6297	427	12	(	(	PUNCT
ejpam-6297	427	13	x	x	X
ejpam-6297	427	14	,	,	PUNCT
ejpam-6297	427	15	τ	τ	PROPN
ejpam-6297	427	16	,	,	PUNCT
ejpam-6297	427	17	i	i	PROPN
ejpam-6297	427	18	)	)	PUNCT
ejpam-6297	427	19	.	.	PUNCT
ejpam-6297	428	1	if	if	SCONJ
ejpam-6297	428	2	a	a	PRON
ejpam-6297	428	3	is	be	AUX
ejpam-6297	428	4	an	an	DET
ejpam-6297	428	5	sβ	sβ	NOUN
ejpam-6297	428	6	-	-	PUNCT
ejpam-6297	428	7	ib	ib	NOUN
ejpam-6297	428	8	-	-	PUNCT
ejpam-6297	428	9	paracompact	paracompact	NOUN
ejpam-6297	428	10	subset	subset	NOUN
ejpam-6297	428	11	of	of	ADP
ejpam-6297	428	12	b	b	PROPN
ejpam-6297	428	13	and	and	CCONJ
ejpam-6297	428	14	b	b	PROPN
ejpam-6297	428	15	is	be	AUX
ejpam-6297	428	16	a	a	DET
ejpam-6297	428	17	subset	subset	NOUN
ejpam-6297	428	18	of	of	ADP
ejpam-6297	428	19	x	x	PUNCT
ejpam-6297	428	20	whose	whose	DET
ejpam-6297	428	21	intersection	intersection	NOUN
ejpam-6297	428	22	with	with	ADP
ejpam-6297	428	23	any	any	DET
ejpam-6297	428	24	strong	strong	ADJ
ejpam-6297	428	25	β	β	X
ejpam-6297	428	26	-	-	ADJ
ejpam-6297	428	27	i	i	PRON
ejpam-6297	428	28	-	-	PUNCT
ejpam-6297	428	29	open	open	ADJ
ejpam-6297	428	30	is	be	AUX
ejpam-6297	428	31	again	again	ADV
ejpam-6297	428	32	strong	strong	ADJ
ejpam-6297	428	33	β	β	X
ejpam-6297	428	34	-	-	ADJ
ejpam-6297	428	35	i	i	PRON
ejpam-6297	428	36	-	-	PUNCT
ejpam-6297	428	37	open	open	ADJ
ejpam-6297	428	38	,	,	PUNCT
ejpam-6297	428	39	then	then	ADV
ejpam-6297	428	40	a	a	PRON
ejpam-6297	428	41	is	be	AUX
ejpam-6297	428	42	an	an	DET
ejpam-6297	428	43	sβ	sβ	NOUN
ejpam-6297	428	44	-	-	PUNCT
ejpam-6297	428	45	i	i	NOUN
ejpam-6297	428	46	-	-	PUNCT
ejpam-6297	428	47	paracompact	paracompact	NOUN
ejpam-6297	428	48	subset	subset	NOUN
ejpam-6297	428	49	of	of	ADP
ejpam-6297	428	50	x.	x.	NOUN
ejpam-6297	428	51	proof	proof	PROPN
ejpam-6297	428	52	.	.	PUNCT
ejpam-6297	429	1	let	let	VERB
ejpam-6297	429	2	a	a	PRON
ejpam-6297	429	3	=	=	X
ejpam-6297	429	4	{	{	PUNCT
ejpam-6297	429	5	uα	uα	X
ejpam-6297	429	6	:	:	PUNCT
ejpam-6297	429	7	α	α	PROPN
ejpam-6297	429	8	∈	∈	PROPN
ejpam-6297	429	9	λ	λ	NOUN
ejpam-6297	429	10	}	}	PUNCT
ejpam-6297	429	11	be	be	VERB
ejpam-6297	429	12	an	an	DET
ejpam-6297	429	13	open	open	ADJ
ejpam-6297	429	14	cover	cover	NOUN
ejpam-6297	429	15	of	of	ADP
ejpam-6297	429	16	a	a	DET
ejpam-6297	429	17	inx	inx	NOUN
ejpam-6297	429	18	.	.	PUNCT
ejpam-6297	430	1	then	then	ADV
ejpam-6297	430	2	,	,	PUNCT
ejpam-6297	430	3	ua	ua	PROPN
ejpam-6297	430	4	=	=	PROPN
ejpam-6297	430	5	{	{	PUNCT
ejpam-6297	430	6	uα∩b	uα∩b	NOUN
ejpam-6297	430	7	:	:	PUNCT
ejpam-6297	430	8	α	α	PROPN
ejpam-6297	430	9	∈	∈	PROPN
ejpam-6297	430	10	λ	λ	PROPN
ejpam-6297	430	11	}	}	PUNCT
ejpam-6297	430	12	is	be	AUX
ejpam-6297	430	13	an	an	DET
ejpam-6297	430	14	open	open	ADJ
ejpam-6297	430	15	cover	cover	NOUN
ejpam-6297	430	16	of	of	ADP
ejpam-6297	430	17	a	a	PRON
ejpam-6297	430	18	in	in	ADP
ejpam-6297	430	19	b.	b.	PROPN
ejpam-6297	430	20	since	since	SCONJ
ejpam-6297	430	21	a	a	PRON
ejpam-6297	430	22	is	be	AUX
ejpam-6297	430	23	an	an	DET
ejpam-6297	430	24	sβ	sβ	NOUN
ejpam-6297	430	25	-	-	PUNCT
ejpam-6297	430	26	ib	ib	NOUN
ejpam-6297	430	27	-	-	PUNCT
ejpam-6297	430	28	paracompact	paracompact	NOUN
ejpam-6297	430	29	subset	subset	NOUN
ejpam-6297	430	30	of	of	ADP
ejpam-6297	430	31	b	b	PROPN
ejpam-6297	430	32	,	,	PUNCT
ejpam-6297	430	33	the	the	DET
ejpam-6297	430	34	collection	collection	NOUN
ejpam-6297	430	35	ua	ua	PROPN
ejpam-6297	430	36	has	have	VERB
ejpam-6297	430	37	a	a	DET
ejpam-6297	430	38	precise	precise	ADJ
ejpam-6297	430	39	sβ	sβ	NOUN
ejpam-6297	430	40	-	-	PUNCT
ejpam-6297	430	41	ib	ib	NOUN
ejpam-6297	430	42	-	-	PUNCT
ejpam-6297	430	43	locally	locally	ADV
ejpam-6297	430	44	finite	finite	NOUN
ejpam-6297	430	45	strong	strong	ADJ
ejpam-6297	430	46	β	β	X
ejpam-6297	430	47	-	-	ADJ
ejpam-6297	430	48	i	i	NOUN
ejpam-6297	430	49	-	-	PUNCT
ejpam-6297	430	50	open	open	ADJ
ejpam-6297	430	51	refinement	refinement	NOUN
ejpam-6297	430	52	va	va	PROPN
ejpam-6297	430	53	=	=	PUNCT
ejpam-6297	430	54	{	{	PUNCT
ejpam-6297	430	55	vα	vα	ADP
ejpam-6297	430	56	∩	∩	ADJ
ejpam-6297	430	57	b	b	NOUN
ejpam-6297	430	58	:	:	PUNCT
ejpam-6297	430	59	α	α	PROPN
ejpam-6297	430	60	∈	∈	PROPN
ejpam-6297	430	61	λ	λ	NOUN
ejpam-6297	430	62	}	}	PUNCT
ejpam-6297	430	63	such	such	ADJ
ejpam-6297	430	64	that	that	SCONJ
ejpam-6297	430	65	vα	vα	ADP
ejpam-6297	430	66	⊆	⊆	NUM
ejpam-6297	430	67	uα	uα	NOUN
ejpam-6297	430	68	for	for	ADP
ejpam-6297	430	69	all	all	DET
ejpam-6297	430	70	α	α	PRON
ejpam-6297	430	71	∈	∈	PROPN
ejpam-6297	430	72	λ	λ	NOUN
ejpam-6297	430	73	,	,	PUNCT
ejpam-6297	430	74	and	and	CCONJ
ejpam-6297	430	75	a	a	DET
ejpam-6297	430	76	−	−	NOUN
ejpam-6297	430	77	∪{vα	∪{vα	X
ejpam-6297	430	78	∩	∩	PROPN
ejpam-6297	430	79	b	b	X
ejpam-6297	430	80	:	:	PUNCT
ejpam-6297	430	81	α	α	PROPN
ejpam-6297	430	82	∈	∈	PROPN
ejpam-6297	430	83	λ	λ	PROPN
ejpam-6297	430	84	}	}	PUNCT
ejpam-6297	430	85	∈	∈	PROPN
ejpam-6297	430	86	ib	ib	NOUN
ejpam-6297	430	87	.	.	PUNCT
ejpam-6297	431	1	since	since	SCONJ
ejpam-6297	431	2	vα	vα	PROPN
ejpam-6297	431	3	is	be	AUX
ejpam-6297	431	4	a	a	DET
ejpam-6297	431	5	strong	strong	ADJ
ejpam-6297	431	6	β	β	X
ejpam-6297	431	7	-	-	ADJ
ejpam-6297	431	8	i	i	NOUN
ejpam-6297	431	9	-	-	PUNCT
ejpam-6297	431	10	open	open	ADJ
ejpam-6297	431	11	subset	subset	NOUN
ejpam-6297	431	12	of	of	ADP
ejpam-6297	431	13	x	x	PUNCT
ejpam-6297	431	14	for	for	ADP
ejpam-6297	431	15	all	all	DET
ejpam-6297	431	16	α	α	PRON
ejpam-6297	431	17	∈	∈	PROPN
ejpam-6297	431	18	λ	λ	PROPN
ejpam-6297	431	19	,	,	PUNCT
ejpam-6297	431	20	the	the	DET
ejpam-6297	431	21	collection	collection	NOUN
ejpam-6297	431	22	b	b	NOUN
ejpam-6297	431	23	=	=	PUNCT
ejpam-6297	431	24	{	{	PUNCT
ejpam-6297	431	25	vα	vα	X
ejpam-6297	431	26	:	:	PUNCT
ejpam-6297	431	27	α	α	PROPN
ejpam-6297	431	28	∈	∈	PROPN
ejpam-6297	431	29	λ	λ	NOUN
ejpam-6297	431	30	}	}	PUNCT
ejpam-6297	431	31	of	of	ADP
ejpam-6297	431	32	strong	strong	ADJ
ejpam-6297	431	33	β	β	X
ejpam-6297	431	34	-	-	ADJ
ejpam-6297	431	35	i	i	NOUN
ejpam-6297	431	36	-	-	PUNCT
ejpam-6297	431	37	open	open	ADJ
ejpam-6297	431	38	sets	set	NOUN
ejpam-6297	431	39	of	of	ADP
ejpam-6297	431	40	x	x	PUNCT
ejpam-6297	431	41	is	be	AUX
ejpam-6297	431	42	sβ	sβ	NOUN
ejpam-6297	431	43	-	-	PUNCT
ejpam-6297	431	44	i	i	NOUN
ejpam-6297	431	45	-	-	PUNCT
ejpam-6297	431	46	locally	locally	ADV
ejpam-6297	431	47	finite	finite	NOUN
ejpam-6297	431	48	and	and	CCONJ
ejpam-6297	431	49	refines	refine	VERB
ejpam-6297	431	50	a.	a.	NOUN
ejpam-6297	431	51	now	now	ADV
ejpam-6297	431	52	,	,	PUNCT
ejpam-6297	431	53	we	we	PRON
ejpam-6297	431	54	have	have	VERB
ejpam-6297	431	55	a−	a−	PROPN
ejpam-6297	431	56	∪{vα	∪{vα	NOUN
ejpam-6297	431	57	:	:	PUNCT
ejpam-6297	431	58	α	α	PROPN
ejpam-6297	431	59	∈	∈	PROPN
ejpam-6297	431	60	λ	λ	PROPN
ejpam-6297	431	61	}	}	PUNCT
ejpam-6297	431	62	⊆	⊆	NUM
ejpam-6297	431	63	a−	a−	PROPN
ejpam-6297	431	64	∪{vα	∪{vα	NOUN
ejpam-6297	431	65	∩b	∩b	NOUN
ejpam-6297	431	66	:	:	PUNCT
ejpam-6297	431	67	α	α	PROPN
ejpam-6297	431	68	∈	∈	PROPN
ejpam-6297	431	69	λ	λ	PROPN
ejpam-6297	431	70	}	}	PUNCT
ejpam-6297	431	71	∈	∈	PROPN
ejpam-6297	431	72	ib	ib	NOUN
ejpam-6297	431	73	⊆	⊆	NUM
ejpam-6297	431	74	i.	i.	NOUN
ejpam-6297	431	75	therefore	therefore	ADV
ejpam-6297	431	76	,	,	PUNCT
ejpam-6297	431	77	a	a	PRON
ejpam-6297	431	78	is	be	AUX
ejpam-6297	431	79	an	an	DET
ejpam-6297	431	80	sβ	sβ	NOUN
ejpam-6297	431	81	-	-	PUNCT
ejpam-6297	431	82	i	i	NOUN
ejpam-6297	431	83	-	-	PUNCT
ejpam-6297	431	84	paracompact	paracompact	NOUN
ejpam-6297	431	85	subset	subset	NOUN
ejpam-6297	431	86	of	of	ADP
ejpam-6297	431	87	x.	x.	PROPN
ejpam-6297	431	88	c.	c.	PROPN
ejpam-6297	431	89	boonpok	boonpok	PROPN
ejpam-6297	431	90	,	,	PUNCT
ejpam-6297	431	91	p.	p.	PROPN
ejpam-6297	431	92	raktaow	raktaow	NOUN
ejpam-6297	431	93	,	,	PUNCT
ejpam-6297	431	94	a.	a.	PROPN
ejpam-6297	431	95	sama	sama	PROPN
ejpam-6297	431	96	-	-	PUNCT
ejpam-6297	431	97	ae	ae	PROPN
ejpam-6297	431	98	/	/	SYM
ejpam-6297	431	99	eur	eur	PROPN
ejpam-6297	431	100	.	.	PUNCT
ejpam-6297	432	1	j.	j.	PROPN
ejpam-6297	432	2	pure	pure	PROPN
ejpam-6297	432	3	appl	appl	PROPN
ejpam-6297	432	4	.	.	PROPN
ejpam-6297	432	5	math	math	PROPN
ejpam-6297	432	6	,	,	PUNCT
ejpam-6297	432	7	18	18	NUM
ejpam-6297	432	8	(	(	PUNCT
ejpam-6297	432	9	3	3	NUM
ejpam-6297	432	10	)	)	PUNCT
ejpam-6297	432	11	(	(	PUNCT
ejpam-6297	432	12	2025	2025	NUM
ejpam-6297	432	13	)	)	PUNCT
ejpam-6297	432	14	,	,	PUNCT
ejpam-6297	432	15	6297	6297	NUM
ejpam-6297	432	16	17	17	NUM
ejpam-6297	432	17	of	of	ADP
ejpam-6297	432	18	23	23	NUM
ejpam-6297	432	19	4	4	NUM
ejpam-6297	432	20	.	.	PUNCT
ejpam-6297	433	1	preservation	preservation	NOUN
ejpam-6297	433	2	of	of	ADP
ejpam-6297	433	3	strong	strong	ADJ
ejpam-6297	433	4	β	β	X
ejpam-6297	433	5	-	-	ADJ
ejpam-6297	433	6	i	i	NOUN
ejpam-6297	433	7	-	-	NOUN
ejpam-6297	433	8	paracompactness	paracompactness	NOUN
ejpam-6297	433	9	in	in	ADP
ejpam-6297	433	10	this	this	DET
ejpam-6297	433	11	part	part	NOUN
ejpam-6297	433	12	,	,	PUNCT
ejpam-6297	433	13	we	we	PRON
ejpam-6297	433	14	will	will	AUX
ejpam-6297	433	15	show	show	VERB
ejpam-6297	433	16	that	that	SCONJ
ejpam-6297	433	17	the	the	DET
ejpam-6297	433	18	property	property	NOUN
ejpam-6297	433	19	of	of	ADP
ejpam-6297	433	20	sβ	sβ	PROPN
ejpam-6297	433	21	-	-	PUNCT
ejpam-6297	433	22	i	i	NOUN
ejpam-6297	433	23	-	-	NOUN
ejpam-6297	433	24	paracompactness	paracompactness	PROPN
ejpam-6297	433	25	is	be	AUX
ejpam-6297	433	26	maintained	maintain	VERB
ejpam-6297	433	27	under	under	ADP
ejpam-6297	433	28	specific	specific	ADJ
ejpam-6297	433	29	circumstances	circumstance	NOUN
ejpam-6297	433	30	.	.	PUNCT
ejpam-6297	434	1	we	we	PRON
ejpam-6297	434	2	begin	begin	VERB
ejpam-6297	434	3	by	by	ADP
ejpam-6297	434	4	introducing	introduce	VERB
ejpam-6297	434	5	the	the	DET
ejpam-6297	434	6	following	follow	VERB
ejpam-6297	434	7	definition	definition	NOUN
ejpam-6297	434	8	.	.	PUNCT
ejpam-6297	435	1	definition	definition	NOUN
ejpam-6297	435	2	11	11	NUM
ejpam-6297	435	3	.	.	PUNCT
ejpam-6297	436	1	let	let	VERB
ejpam-6297	436	2	(	(	PUNCT
ejpam-6297	436	3	x	x	X
ejpam-6297	436	4	,	,	PUNCT
ejpam-6297	436	5	τ	τ	PROPN
ejpam-6297	436	6	,	,	PUNCT
ejpam-6297	436	7	i	i	PROPN
ejpam-6297	436	8	)	)	PUNCT
ejpam-6297	436	9	and	and	CCONJ
ejpam-6297	436	10	(	(	PUNCT
ejpam-6297	436	11	y	y	PROPN
ejpam-6297	436	12	,	,	PUNCT
ejpam-6297	436	13	τ	τ	PROPN
ejpam-6297	436	14	′,j	′,j	NOUN
ejpam-6297	436	15	)	)	PUNCT
ejpam-6297	436	16	be	be	AUX
ejpam-6297	436	17	ideal	ideal	ADJ
ejpam-6297	436	18	topological	topological	ADJ
ejpam-6297	436	19	spaces	space	NOUN
ejpam-6297	436	20	,	,	PUNCT
ejpam-6297	436	21	and	and	CCONJ
ejpam-6297	436	22	let	let	VERB
ejpam-6297	436	23	f	f	PRON
ejpam-6297	436	24	:	:	PUNCT
ejpam-6297	436	25	x	x	X
ejpam-6297	436	26	→	→	SYM
ejpam-6297	436	27	y	y	X
ejpam-6297	436	28	be	be	AUX
ejpam-6297	436	29	a	a	DET
ejpam-6297	436	30	function	function	NOUN
ejpam-6297	436	31	.	.	PUNCT
ejpam-6297	437	1	(	(	PUNCT
ejpam-6297	437	2	1	1	X
ejpam-6297	437	3	)	)	PUNCT
ejpam-6297	437	4	the	the	DET
ejpam-6297	437	5	function	function	NOUN
ejpam-6297	437	6	f	f	PROPN
ejpam-6297	437	7	is	be	AUX
ejpam-6297	437	8	said	say	VERB
ejpam-6297	437	9	to	to	PART
ejpam-6297	437	10	be	be	AUX
ejpam-6297	437	11	sβ	sβ	NOUN
ejpam-6297	437	12	-	-	PUNCT
ejpam-6297	437	13	i	i	NOUN
ejpam-6297	437	14	-	-	PUNCT
ejpam-6297	437	15	open	open	ADJ
ejpam-6297	437	16	if	if	SCONJ
ejpam-6297	437	17	the	the	DET
ejpam-6297	437	18	image	image	NOUN
ejpam-6297	437	19	of	of	ADP
ejpam-6297	437	20	every	every	DET
ejpam-6297	437	21	strong	strong	ADJ
ejpam-6297	437	22	β	β	X
ejpam-6297	437	23	-	-	ADJ
ejpam-6297	437	24	i	i	PRON
ejpam-6297	437	25	-	-	PUNCT
ejpam-6297	437	26	open	open	ADJ
ejpam-6297	437	27	set	set	NOUN
ejpam-6297	437	28	in	in	ADP
ejpam-6297	437	29	x	x	PRON
ejpam-6297	437	30	is	be	AUX
ejpam-6297	437	31	a	a	DET
ejpam-6297	437	32	strong	strong	ADJ
ejpam-6297	437	33	β	β	NOUN
ejpam-6297	437	34	-	-	ADJ
ejpam-6297	437	35	j	j	ADJ
ejpam-6297	437	36	-open	-open	NOUN
ejpam-6297	437	37	set	set	VERB
ejpam-6297	437	38	in	in	ADP
ejpam-6297	437	39	y	y	PROPN
ejpam-6297	437	40	.	.	PUNCT
ejpam-6297	438	1	(	(	PUNCT
ejpam-6297	438	2	2	2	X
ejpam-6297	438	3	)	)	PUNCT
ejpam-6297	438	4	the	the	DET
ejpam-6297	438	5	function	function	NOUN
ejpam-6297	438	6	f	f	PROPN
ejpam-6297	438	7	is	be	AUX
ejpam-6297	438	8	said	say	VERB
ejpam-6297	438	9	to	to	PART
ejpam-6297	438	10	be	be	AUX
ejpam-6297	438	11	sβ	sβ	PROPN
ejpam-6297	438	12	-	-	PUNCT
ejpam-6297	438	13	i	i	NOUN
ejpam-6297	438	14	-	-	PUNCT
ejpam-6297	438	15	closed	close	VERB
ejpam-6297	438	16	if	if	SCONJ
ejpam-6297	438	17	the	the	DET
ejpam-6297	438	18	image	image	NOUN
ejpam-6297	438	19	of	of	ADP
ejpam-6297	438	20	every	every	DET
ejpam-6297	438	21	strong	strong	ADJ
ejpam-6297	438	22	β	β	X
ejpam-6297	438	23	-	-	ADJ
ejpam-6297	438	24	i	i	NOUN
ejpam-6297	438	25	-	-	PUNCT
ejpam-6297	438	26	closed	close	VERB
ejpam-6297	438	27	set	set	NOUN
ejpam-6297	438	28	in	in	ADP
ejpam-6297	438	29	x	x	PRON
ejpam-6297	438	30	is	be	AUX
ejpam-6297	438	31	a	a	DET
ejpam-6297	438	32	strong	strong	ADJ
ejpam-6297	438	33	β	β	NOUN
ejpam-6297	438	34	-	-	PUNCT
ejpam-6297	438	35	j	j	NOUN
ejpam-6297	438	36	-closed	-close	VERB
ejpam-6297	438	37	set	set	NOUN
ejpam-6297	438	38	in	in	ADP
ejpam-6297	438	39	y	y	PROPN
ejpam-6297	438	40	.	.	PUNCT
ejpam-6297	439	1	(	(	PUNCT
ejpam-6297	439	2	3	3	X
ejpam-6297	439	3	)	)	PUNCT
ejpam-6297	439	4	the	the	DET
ejpam-6297	439	5	function	function	NOUN
ejpam-6297	439	6	f	f	PROPN
ejpam-6297	439	7	is	be	AUX
ejpam-6297	439	8	said	say	VERB
ejpam-6297	439	9	to	to	PART
ejpam-6297	439	10	be	be	AUX
ejpam-6297	439	11	sβ	sβ	PROPN
ejpam-6297	439	12	-	-	PUNCT
ejpam-6297	439	13	i	i	NOUN
ejpam-6297	439	14	-	-	NOUN
ejpam-6297	439	15	irresolute	irresolute	ADJ
ejpam-6297	439	16	if	if	SCONJ
ejpam-6297	439	17	the	the	DET
ejpam-6297	439	18	preimage	preimage	NOUN
ejpam-6297	439	19	of	of	ADP
ejpam-6297	439	20	every	every	DET
ejpam-6297	439	21	strong	strong	ADJ
ejpam-6297	439	22	β	β	PROPN
ejpam-6297	439	23	-	-	ADJ
ejpam-6297	439	24	j	j	ADJ
ejpam-6297	439	25	-open	-open	NOUN
ejpam-6297	439	26	set	set	VERB
ejpam-6297	439	27	in	in	ADP
ejpam-6297	439	28	y	y	PROPN
ejpam-6297	439	29	is	be	AUX
ejpam-6297	439	30	a	a	DET
ejpam-6297	439	31	strong	strong	ADJ
ejpam-6297	439	32	β	β	X
ejpam-6297	439	33	-	-	ADJ
ejpam-6297	439	34	i	i	PRON
ejpam-6297	439	35	-	-	PUNCT
ejpam-6297	439	36	open	open	ADJ
ejpam-6297	439	37	set	set	NOUN
ejpam-6297	439	38	in	in	ADP
ejpam-6297	439	39	x.	x.	NOUN
ejpam-6297	439	40	observe	observe	VERB
ejpam-6297	439	41	that	that	DET
ejpam-6297	439	42	f−1(j	f−1(j	NOUN
ejpam-6297	439	43	)	)	PUNCT
ejpam-6297	439	44	forms	form	VERB
ejpam-6297	439	45	an	an	DET
ejpam-6297	439	46	ideal	ideal	NOUN
ejpam-6297	439	47	on	on	ADP
ejpam-6297	439	48	x	x	SYM
ejpam-6297	439	49	if	if	SCONJ
ejpam-6297	439	50	f	f	X
ejpam-6297	439	51	:	:	PUNCT
ejpam-6297	439	52	x	x	X
ejpam-6297	439	53	→	→	SYM
ejpam-6297	439	54	y	y	PROPN
ejpam-6297	439	55	is	be	AUX
ejpam-6297	439	56	a	a	DET
ejpam-6297	439	57	function	function	NOUN
ejpam-6297	439	58	and	and	CCONJ
ejpam-6297	439	59	y	y	PROPN
ejpam-6297	439	60	is	be	AUX
ejpam-6297	439	61	a	a	DET
ejpam-6297	439	62	topological	topological	ADJ
ejpam-6297	439	63	space	space	NOUN
ejpam-6297	439	64	endowed	endow	VERB
ejpam-6297	439	65	with	with	ADP
ejpam-6297	439	66	an	an	DET
ejpam-6297	439	67	ideal	ideal	ADJ
ejpam-6297	439	68	j	j	PROPN
ejpam-6297	439	69	.	.	PUNCT
ejpam-6297	440	1	furthermore	furthermore	ADV
ejpam-6297	440	2	,	,	PUNCT
ejpam-6297	440	3	if	if	SCONJ
ejpam-6297	440	4	f	f	PROPN
ejpam-6297	440	5	is	be	AUX
ejpam-6297	440	6	surjective	surjective	ADJ
ejpam-6297	440	7	and	and	CCONJ
ejpam-6297	440	8	x	x	PRON
ejpam-6297	440	9	has	have	VERB
ejpam-6297	440	10	an	an	DET
ejpam-6297	440	11	ideal	ideal	NOUN
ejpam-6297	441	1	i	i	PRON
ejpam-6297	441	2	,	,	PUNCT
ejpam-6297	441	3	it	it	PRON
ejpam-6297	441	4	follows	follow	VERB
ejpam-6297	441	5	that	that	SCONJ
ejpam-6297	441	6	f(i	f(i	PROPN
ejpam-6297	441	7	)	)	PUNCT
ejpam-6297	441	8	forms	form	VERB
ejpam-6297	441	9	an	an	DET
ejpam-6297	441	10	ideal	ideal	NOUN
ejpam-6297	441	11	in	in	ADP
ejpam-6297	441	12	y	y	PROPN
ejpam-6297	441	13	.	.	PUNCT
ejpam-6297	442	1	we	we	PRON
ejpam-6297	442	2	now	now	ADV
ejpam-6297	442	3	present	present	VERB
ejpam-6297	442	4	the	the	DET
ejpam-6297	442	5	characteristics	characteristic	NOUN
ejpam-6297	442	6	of	of	ADP
ejpam-6297	442	7	a	a	DET
ejpam-6297	442	8	function	function	NOUN
ejpam-6297	442	9	mapped	map	VERB
ejpam-6297	442	10	between	between	ADP
ejpam-6297	442	11	two	two	NUM
ejpam-6297	442	12	ideal	ideal	ADJ
ejpam-6297	442	13	topological	topological	ADJ
ejpam-6297	442	14	spaces	space	NOUN
ejpam-6297	442	15	,	,	PUNCT
ejpam-6297	442	16	where	where	SCONJ
ejpam-6297	442	17	one	one	NUM
ejpam-6297	442	18	space	space	NOUN
ejpam-6297	442	19	reflects	reflect	VERB
ejpam-6297	442	20	the	the	DET
ejpam-6297	442	21	same	same	ADJ
ejpam-6297	442	22	properties	property	NOUN
ejpam-6297	442	23	as	as	ADP
ejpam-6297	442	24	the	the	DET
ejpam-6297	442	25	other	other	ADJ
ejpam-6297	442	26	.	.	PUNCT
ejpam-6297	443	1	to	to	PART
ejpam-6297	443	2	begin	begin	VERB
ejpam-6297	443	3	,	,	PUNCT
ejpam-6297	443	4	we	we	PRON
ejpam-6297	443	5	introduce	introduce	VERB
ejpam-6297	443	6	the	the	DET
ejpam-6297	443	7	notion	notion	NOUN
ejpam-6297	443	8	of	of	ADP
ejpam-6297	443	9	sβ	sβ	PROPN
ejpam-6297	443	10	-	-	PUNCT
ejpam-6297	443	11	i	i	NOUN
ejpam-6297	443	12	-	-	PUNCT
ejpam-6297	443	13	compactness	compactness	NOUN
ejpam-6297	443	14	and	and	CCONJ
ejpam-6297	443	15	state	state	NOUN
ejpam-6297	443	16	a	a	DET
ejpam-6297	443	17	lemma	lemma	PROPN
ejpam-6297	443	18	that	that	PRON
ejpam-6297	443	19	will	will	AUX
ejpam-6297	443	20	be	be	AUX
ejpam-6297	443	21	employed	employ	VERB
ejpam-6297	443	22	in	in	ADP
ejpam-6297	443	23	proving	prove	VERB
ejpam-6297	443	24	theorem	theorem	ADJ
ejpam-6297	443	25	18	18	NUM
ejpam-6297	443	26	.	.	PUNCT
ejpam-6297	443	27	definition	definition	NOUN
ejpam-6297	443	28	12	12	NUM
ejpam-6297	443	29	.	.	PUNCT
ejpam-6297	444	1	an	an	DET
ejpam-6297	444	2	ideal	ideal	ADJ
ejpam-6297	444	3	topological	topological	ADJ
ejpam-6297	444	4	space	space	NOUN
ejpam-6297	444	5	(	(	PUNCT
ejpam-6297	444	6	x	x	X
ejpam-6297	444	7	,	,	PUNCT
ejpam-6297	444	8	τ	τ	PROPN
ejpam-6297	444	9	,	,	PUNCT
ejpam-6297	444	10	i	i	PROPN
ejpam-6297	444	11	)	)	PUNCT
ejpam-6297	444	12	is	be	AUX
ejpam-6297	444	13	said	say	VERB
ejpam-6297	444	14	to	to	PART
ejpam-6297	444	15	be	be	AUX
ejpam-6297	444	16	sβ	sβ	VERB
ejpam-6297	444	17	-	-	PUNCT
ejpam-6297	444	18	i	i	NOUN
ejpam-6297	444	19	-	-	NOUN
ejpam-6297	444	20	compact	compact	ADJ
ejpam-6297	444	21	if	if	SCONJ
ejpam-6297	444	22	every	every	PRON
ejpam-6297	444	23	cover	cover	VERB
ejpam-6297	444	24	a	a	PRON
ejpam-6297	444	25	of	of	ADP
ejpam-6297	444	26	strong	strong	ADJ
ejpam-6297	444	27	β	β	X
ejpam-6297	444	28	-	-	ADJ
ejpam-6297	444	29	i	i	NOUN
ejpam-6297	444	30	-	-	PUNCT
ejpam-6297	444	31	open	open	ADJ
ejpam-6297	444	32	subsets	subset	NOUN
ejpam-6297	444	33	of	of	ADP
ejpam-6297	444	34	x	x	PUNCT
ejpam-6297	444	35	has	have	VERB
ejpam-6297	444	36	a	a	DET
ejpam-6297	444	37	finite	finite	ADJ
ejpam-6297	444	38	subcover	subcover	PROPN
ejpam-6297	444	39	,	,	PUNCT
ejpam-6297	444	40	i.e.	i.e.	X
ejpam-6297	444	41	,	,	PUNCT
ejpam-6297	444	42	there	there	PRON
ejpam-6297	444	43	exist	exist	VERB
ejpam-6297	444	44	sets	set	NOUN
ejpam-6297	444	45	a1	a1	NOUN
ejpam-6297	444	46	,	,	PUNCT
ejpam-6297	444	47	a2	a2	PROPN
ejpam-6297	444	48	,	,	PUNCT
ejpam-6297	444	49	.	.	PUNCT
ejpam-6297	444	50	.	.	PUNCT
ejpam-6297	445	1	.	.	PUNCT
ejpam-6297	446	1	,	,	PUNCT
ejpam-6297	446	2	an	an	DET
ejpam-6297	446	3	∈	∈	PROPN
ejpam-6297	446	4	a	a	DET
ejpam-6297	446	5	such	such	ADJ
ejpam-6297	446	6	that	that	SCONJ
ejpam-6297	446	7	x	x	SYM
ejpam-6297	446	8	⊆	⊆	NUM
ejpam-6297	446	9	a1	a1	NOUN
ejpam-6297	446	10	∪a2	∪a2	NOUN
ejpam-6297	446	11	∪	∪	X
ejpam-6297	446	12	·	·	PUNCT
ejpam-6297	446	13	·	·	PUNCT
ejpam-6297	446	14	·	·	PUNCT
ejpam-6297	446	15	∪an	∪an	PROPN
ejpam-6297	446	16	.	.	PUNCT
ejpam-6297	447	1	lemma	lemma	PROPN
ejpam-6297	447	2	8	8	NUM
ejpam-6297	447	3	.	.	PUNCT
ejpam-6297	448	1	let	let	VERB
ejpam-6297	448	2	(	(	PUNCT
ejpam-6297	448	3	x	x	X
ejpam-6297	448	4	,	,	PUNCT
ejpam-6297	448	5	τ	τ	PROPN
ejpam-6297	448	6	,	,	PUNCT
ejpam-6297	448	7	i	i	PROPN
ejpam-6297	448	8	)	)	PUNCT
ejpam-6297	448	9	and	and	CCONJ
ejpam-6297	448	10	(	(	PUNCT
ejpam-6297	448	11	y	y	PROPN
ejpam-6297	448	12	,	,	PUNCT
ejpam-6297	448	13	τ	τ	PROPN
ejpam-6297	448	14	′,j	′,j	NOUN
ejpam-6297	448	15	)	)	PUNCT
ejpam-6297	448	16	be	be	AUX
ejpam-6297	448	17	ideal	ideal	ADJ
ejpam-6297	448	18	topological	topological	ADJ
ejpam-6297	448	19	spaces	space	NOUN
ejpam-6297	448	20	,	,	PUNCT
ejpam-6297	448	21	and	and	CCONJ
ejpam-6297	448	22	f	f	X
ejpam-6297	448	23	:	:	PUNCT
ejpam-6297	448	24	x	x	X
ejpam-6297	448	25	→	→	SYM
ejpam-6297	448	26	y	y	PROPN
ejpam-6297	448	27	be	be	AUX
ejpam-6297	448	28	surjective	surjective	ADJ
ejpam-6297	448	29	.	.	PUNCT
ejpam-6297	449	1	then	then	ADV
ejpam-6297	449	2	f	f	PROPN
ejpam-6297	449	3	is	be	AUX
ejpam-6297	449	4	strong	strong	ADJ
ejpam-6297	449	5	β	β	NOUN
ejpam-6297	449	6	-	-	ADJ
ejpam-6297	449	7	i	i	NOUN
ejpam-6297	449	8	-	-	PUNCT
ejpam-6297	449	9	closed	close	VERB
ejpam-6297	449	10	if	if	SCONJ
ejpam-6297	449	11	and	and	CCONJ
ejpam-6297	449	12	only	only	ADV
ejpam-6297	449	13	if	if	SCONJ
ejpam-6297	449	14	for	for	ADP
ejpam-6297	449	15	every	every	DET
ejpam-6297	449	16	y	y	PROPN
ejpam-6297	449	17	∈	∈	PROPN
ejpam-6297	449	18	y	y	PROPN
ejpam-6297	449	19	and	and	CCONJ
ejpam-6297	449	20	for	for	ADP
ejpam-6297	449	21	every	every	PRON
ejpam-6297	449	22	strong	strong	ADJ
ejpam-6297	449	23	β	β	X
ejpam-6297	449	24	-	-	ADJ
ejpam-6297	449	25	i	i	PRON
ejpam-6297	449	26	-	-	PUNCT
ejpam-6297	449	27	open	open	ADJ
ejpam-6297	449	28	set	set	VERB
ejpam-6297	449	29	u	u	NOUN
ejpam-6297	449	30	in	in	ADP
ejpam-6297	449	31	x	x	SYM
ejpam-6297	449	32	containing	contain	VERB
ejpam-6297	449	33	{	{	PUNCT
ejpam-6297	449	34	f−1(y	f−1(y	PROPN
ejpam-6297	449	35	)	)	PUNCT
ejpam-6297	449	36	}	}	PUNCT
ejpam-6297	449	37	,	,	PUNCT
ejpam-6297	449	38	there	there	PRON
ejpam-6297	449	39	exists	exist	VERB
ejpam-6297	449	40	a	a	DET
ejpam-6297	449	41	strong	strong	ADJ
ejpam-6297	449	42	β	β	NOUN
ejpam-6297	449	43	-	-	ADJ
ejpam-6297	449	44	j	j	ADJ
ejpam-6297	449	45	-open	-open	NOUN
ejpam-6297	449	46	set	set	VERB
ejpam-6297	449	47	v	v	NOUN
ejpam-6297	449	48	containing	contain	VERB
ejpam-6297	449	49	y	y	PRON
ejpam-6297	449	50	such	such	ADJ
ejpam-6297	449	51	that	that	DET
ejpam-6297	449	52	f−1(v	f−1(v	NOUN
ejpam-6297	449	53	)	)	PUNCT
ejpam-6297	450	1	⊆	⊆	NUM
ejpam-6297	450	2	u	u	NOUN
ejpam-6297	450	3	.	.	PUNCT
ejpam-6297	451	1	proof	proof	NOUN
ejpam-6297	451	2	.	.	PUNCT
ejpam-6297	452	1	let	let	VERB
ejpam-6297	452	2	y	y	PROPN
ejpam-6297	452	3	∈	∈	PROPN
ejpam-6297	452	4	y	y	PROPN
ejpam-6297	452	5	and	and	CCONJ
ejpam-6297	452	6	let	let	VERB
ejpam-6297	452	7	u	u	PRON
ejpam-6297	452	8	be	be	AUX
ejpam-6297	452	9	a	a	DET
ejpam-6297	452	10	strong	strong	ADJ
ejpam-6297	452	11	β	β	X
ejpam-6297	452	12	-	-	ADJ
ejpam-6297	452	13	i	i	NOUN
ejpam-6297	452	14	-	-	PUNCT
ejpam-6297	452	15	open	open	ADJ
ejpam-6297	452	16	subset	subset	NOUN
ejpam-6297	452	17	of	of	ADP
ejpam-6297	452	18	x	x	SYM
ejpam-6297	452	19	such	such	ADJ
ejpam-6297	452	20	that	that	SCONJ
ejpam-6297	452	21	{	{	PUNCT
ejpam-6297	452	22	f−1(y	f−1(y	PROPN
ejpam-6297	452	23	)	)	PUNCT
ejpam-6297	452	24	}	}	PUNCT
ejpam-6297	452	25	⊆	⊆	NUM
ejpam-6297	452	26	u	u	NOUN
ejpam-6297	452	27	.	.	PUNCT
ejpam-6297	453	1	define	define	VERB
ejpam-6297	453	2	the	the	DET
ejpam-6297	453	3	set	set	NOUN
ejpam-6297	453	4	v	v	NOUN
ejpam-6297	453	5	=	=	SYM
ejpam-6297	453	6	y	y	PROPN
ejpam-6297	453	7	−	−	PROPN
ejpam-6297	453	8	f(x	f(x	PROPN
ejpam-6297	453	9	−	−	PROPN
ejpam-6297	453	10	u	u	NOUN
ejpam-6297	453	11	)	)	PUNCT
ejpam-6297	453	12	,	,	PUNCT
ejpam-6297	453	13	which	which	PRON
ejpam-6297	453	14	is	be	AUX
ejpam-6297	453	15	strong	strong	ADJ
ejpam-6297	453	16	β	β	NOUN
ejpam-6297	453	17	-	-	PUNCT
ejpam-6297	453	18	j	j	NOUN
ejpam-6297	453	19	-open	-open	NOUN
ejpam-6297	453	20	.	.	PUNCT
ejpam-6297	454	1	clearly	clearly	ADV
ejpam-6297	454	2	,	,	PUNCT
ejpam-6297	454	3	y	y	PROPN
ejpam-6297	454	4	∈	∈	PROPN
ejpam-6297	454	5	v	v	NOUN
ejpam-6297	454	6	and	and	CCONJ
ejpam-6297	454	7	f−1(v	f−1(v	NOUN
ejpam-6297	454	8	)	)	PUNCT
ejpam-6297	455	1	⊆	⊆	NUM
ejpam-6297	455	2	u	u	NOUN
ejpam-6297	455	3	.	.	PUNCT
ejpam-6297	456	1	therefore	therefore	ADV
ejpam-6297	456	2	,	,	PUNCT
ejpam-6297	456	3	the	the	DET
ejpam-6297	456	4	necessity	necessity	NOUN
ejpam-6297	456	5	is	be	AUX
ejpam-6297	456	6	established	establish	VERB
ejpam-6297	456	7	.	.	PUNCT
ejpam-6297	457	1	now	now	ADV
ejpam-6297	457	2	,	,	PUNCT
ejpam-6297	457	3	let	let	VERB
ejpam-6297	457	4	f	f	PRON
ejpam-6297	457	5	be	be	AUX
ejpam-6297	457	6	a	a	DET
ejpam-6297	457	7	strong	strong	ADJ
ejpam-6297	457	8	β	β	X
ejpam-6297	457	9	-	-	ADJ
ejpam-6297	457	10	i	i	NOUN
ejpam-6297	457	11	-	-	PUNCT
ejpam-6297	457	12	closed	close	VERB
ejpam-6297	457	13	subset	subset	NOUN
ejpam-6297	457	14	of	of	ADP
ejpam-6297	457	15	x	x	X
ejpam-6297	457	16	,	,	PUNCT
ejpam-6297	457	17	and	and	CCONJ
ejpam-6297	457	18	let	let	VERB
ejpam-6297	457	19	y	y	PROPN
ejpam-6297	457	20	∈	∈	PROPN
ejpam-6297	457	21	y	y	PROPN
ejpam-6297	457	22	−	−	PROPN
ejpam-6297	457	23	f(f	f(f	PROPN
ejpam-6297	457	24	)	)	PUNCT
ejpam-6297	457	25	.	.	PUNCT
ejpam-6297	458	1	this	this	PRON
ejpam-6297	458	2	implies	imply	VERB
ejpam-6297	458	3	that	that	SCONJ
ejpam-6297	458	4	{	{	PUNCT
ejpam-6297	458	5	f−1(y	f−1(y	PROPN
ejpam-6297	458	6	)	)	PUNCT
ejpam-6297	458	7	}	}	PUNCT
ejpam-6297	458	8	⊆	⊆	NUM
ejpam-6297	458	9	x	x	SYM
ejpam-6297	458	10	−f	−f	NOUN
ejpam-6297	458	11	.	.	PUNCT
ejpam-6297	459	1	by	by	ADP
ejpam-6297	459	2	assumption	assumption	NOUN
ejpam-6297	459	3	,	,	PUNCT
ejpam-6297	459	4	there	there	PRON
ejpam-6297	459	5	exists	exist	VERB
ejpam-6297	459	6	a	a	DET
ejpam-6297	459	7	strong	strong	ADJ
ejpam-6297	459	8	β	β	NOUN
ejpam-6297	459	9	-	-	ADJ
ejpam-6297	459	10	j	j	ADJ
ejpam-6297	459	11	-open	-open	NOUN
ejpam-6297	459	12	set	set	VERB
ejpam-6297	459	13	vy	vy	NOUN
ejpam-6297	459	14	containing	contain	VERB
ejpam-6297	459	15	y	y	PRON
ejpam-6297	459	16	such	such	ADJ
ejpam-6297	459	17	that	that	SCONJ
ejpam-6297	459	18	f−1(vy	f−1(vy	VERB
ejpam-6297	459	19	)	)	PUNCT
ejpam-6297	459	20	⊆	⊆	NUM
ejpam-6297	459	21	x	x	SYM
ejpam-6297	459	22	−	−	PROPN
ejpam-6297	459	23	f	f	NOUN
ejpam-6297	459	24	,	,	PUNCT
ejpam-6297	459	25	and	and	CCONJ
ejpam-6297	459	26	consequently	consequently	ADV
ejpam-6297	459	27	,	,	PUNCT
ejpam-6297	459	28	y	y	PROPN
ejpam-6297	459	29	∈	∈	PROPN
ejpam-6297	459	30	vy	vy	VERB
ejpam-6297	459	31	⊆	⊆	NUM
ejpam-6297	459	32	y	y	PROPN
ejpam-6297	459	33	−	−	PROPN
ejpam-6297	459	34	f(f	f(f	PROPN
ejpam-6297	459	35	)	)	PUNCT
ejpam-6297	459	36	.	.	PUNCT
ejpam-6297	460	1	therefore	therefore	ADV
ejpam-6297	460	2	,	,	PUNCT
ejpam-6297	460	3	the	the	DET
ejpam-6297	460	4	set	set	NOUN
ejpam-6297	460	5	y	y	PROPN
ejpam-6297	460	6	−	−	PROPN
ejpam-6297	460	7	f(f	f(f	PROPN
ejpam-6297	460	8	)	)	PUNCT
ejpam-6297	461	1	=	=	PUNCT
ejpam-6297	462	1	∪{vy	∪{vy	PROPN
ejpam-6297	462	2	:	:	PUNCT
ejpam-6297	462	3	y	y	PROPN
ejpam-6297	462	4	∈	∈	PROPN
ejpam-6297	462	5	y	y	PROPN
ejpam-6297	462	6	}	}	PUNCT
ejpam-6297	462	7	is	be	AUX
ejpam-6297	462	8	a	a	DET
ejpam-6297	462	9	strong	strong	ADJ
ejpam-6297	462	10	β	β	NOUN
ejpam-6297	462	11	-	-	ADJ
ejpam-6297	462	12	j	j	PROPN
ejpam-6297	462	13	-open	-open	PROPN
ejpam-6297	462	14	set	set	NOUN
ejpam-6297	462	15	.	.	PUNCT
ejpam-6297	463	1	hence	hence	ADV
ejpam-6297	463	2	,	,	PUNCT
ejpam-6297	463	3	f(f	f(f	PROPN
ejpam-6297	463	4	)	)	PUNCT
ejpam-6297	463	5	is	be	AUX
ejpam-6297	463	6	a	a	DET
ejpam-6297	463	7	strong	strong	ADJ
ejpam-6297	463	8	β	β	NOUN
ejpam-6297	463	9	-	-	PUNCT
ejpam-6297	463	10	j	j	NOUN
ejpam-6297	463	11	-closed	-close	VERB
ejpam-6297	463	12	set	set	NOUN
ejpam-6297	463	13	in	in	ADP
ejpam-6297	463	14	y	y	PROPN
ejpam-6297	463	15	.	.	PUNCT
ejpam-6297	464	1	theorem	theorem	PROPN
ejpam-6297	464	2	18	18	NUM
ejpam-6297	464	3	.	.	PUNCT
ejpam-6297	465	1	let	let	VERB
ejpam-6297	465	2	(	(	PUNCT
ejpam-6297	465	3	x	x	X
ejpam-6297	465	4	,	,	PUNCT
ejpam-6297	465	5	τ	τ	PROPN
ejpam-6297	465	6	,	,	PUNCT
ejpam-6297	465	7	i	i	PROPN
ejpam-6297	465	8	)	)	PUNCT
ejpam-6297	465	9	and	and	CCONJ
ejpam-6297	465	10	(	(	PUNCT
ejpam-6297	465	11	y	y	PROPN
ejpam-6297	465	12	,	,	PUNCT
ejpam-6297	465	13	τ	τ	PROPN
ejpam-6297	465	14	′,j	′,j	NOUN
ejpam-6297	465	15	)	)	PUNCT
ejpam-6297	465	16	be	be	AUX
ejpam-6297	465	17	ideal	ideal	ADJ
ejpam-6297	465	18	topological	topological	ADJ
ejpam-6297	465	19	spaces	space	NOUN
ejpam-6297	465	20	.	.	PUNCT
ejpam-6297	466	1	suppose	suppose	VERB
ejpam-6297	466	2	that	that	SCONJ
ejpam-6297	466	3	f	f	X
ejpam-6297	466	4	:	:	PUNCT
ejpam-6297	466	5	x	x	X
ejpam-6297	466	6	→	→	SYM
ejpam-6297	466	7	y	y	PROPN
ejpam-6297	466	8	satisfies	satisfy	VERB
ejpam-6297	466	9	the	the	DET
ejpam-6297	466	10	following	following	ADJ
ejpam-6297	466	11	statements	statement	NOUN
ejpam-6297	466	12	:	:	PUNCT
ejpam-6297	466	13	c.	c.	PROPN
ejpam-6297	466	14	boonpok	boonpok	PROPN
ejpam-6297	466	15	,	,	PUNCT
ejpam-6297	466	16	p.	p.	PROPN
ejpam-6297	466	17	raktaow	raktaow	NOUN
ejpam-6297	466	18	,	,	PUNCT
ejpam-6297	466	19	a.	a.	PROPN
ejpam-6297	466	20	sama	sama	PROPN
ejpam-6297	466	21	-	-	PUNCT
ejpam-6297	466	22	ae	ae	PROPN
ejpam-6297	466	23	/	/	SYM
ejpam-6297	466	24	eur	eur	PROPN
ejpam-6297	466	25	.	.	PUNCT
ejpam-6297	467	1	j.	j.	PROPN
ejpam-6297	467	2	pure	pure	PROPN
ejpam-6297	467	3	appl	appl	PROPN
ejpam-6297	467	4	.	.	PROPN
ejpam-6297	467	5	math	math	PROPN
ejpam-6297	467	6	,	,	PUNCT
ejpam-6297	467	7	18	18	NUM
ejpam-6297	467	8	(	(	PUNCT
ejpam-6297	467	9	3	3	NUM
ejpam-6297	467	10	)	)	PUNCT
ejpam-6297	467	11	(	(	PUNCT
ejpam-6297	467	12	2025	2025	NUM
ejpam-6297	467	13	)	)	PUNCT
ejpam-6297	467	14	,	,	PUNCT
ejpam-6297	467	15	6297	6297	NUM
ejpam-6297	467	16	18	18	NUM
ejpam-6297	467	17	of	of	ADP
ejpam-6297	467	18	23	23	NUM
ejpam-6297	467	19	(	(	PUNCT
ejpam-6297	467	20	1	1	NUM
ejpam-6297	467	21	)	)	PUNCT
ejpam-6297	467	22	f	f	PROPN
ejpam-6297	467	23	is	be	AUX
ejpam-6297	467	24	continuous	continuous	ADJ
ejpam-6297	467	25	;	;	PUNCT
ejpam-6297	467	26	(	(	PUNCT
ejpam-6297	467	27	2	2	X
ejpam-6297	467	28	)	)	PUNCT
ejpam-6297	467	29	f	f	PROPN
ejpam-6297	467	30	is	be	AUX
ejpam-6297	467	31	sβ	sβ	PROPN
ejpam-6297	467	32	-	-	PUNCT
ejpam-6297	467	33	i	i	NOUN
ejpam-6297	467	34	-	-	PUNCT
ejpam-6297	467	35	open	open	ADJ
ejpam-6297	467	36	;	;	PUNCT
ejpam-6297	467	37	(	(	PUNCT
ejpam-6297	467	38	3	3	X
ejpam-6297	467	39	)	)	PUNCT
ejpam-6297	467	40	f	f	PROPN
ejpam-6297	467	41	is	be	AUX
ejpam-6297	467	42	sβ	sβ	PROPN
ejpam-6297	467	43	-	-	PUNCT
ejpam-6297	467	44	i	i	NOUN
ejpam-6297	467	45	-	-	PUNCT
ejpam-6297	467	46	closed	closed	ADJ
ejpam-6297	467	47	;	;	PUNCT
ejpam-6297	467	48	(	(	PUNCT
ejpam-6297	467	49	4	4	X
ejpam-6297	467	50	)	)	PUNCT
ejpam-6297	467	51	f	f	PROPN
ejpam-6297	467	52	is	be	AUX
ejpam-6297	467	53	surjective	surjective	ADJ
ejpam-6297	467	54	with	with	ADP
ejpam-6297	467	55	{	{	PUNCT
ejpam-6297	467	56	f−1(y	f−1(y	PROPN
ejpam-6297	467	57	)	)	PUNCT
ejpam-6297	467	58	}	}	PUNCT
ejpam-6297	467	59	being	be	AUX
ejpam-6297	467	60	sβ	sβ	ADJ
ejpam-6297	467	61	-	-	PUNCT
ejpam-6297	467	62	i	i	NOUN
ejpam-6297	467	63	-	-	NOUN
ejpam-6297	467	64	compact	compact	ADJ
ejpam-6297	467	65	for	for	ADP
ejpam-6297	467	66	every	every	DET
ejpam-6297	467	67	y	y	PROPN
ejpam-6297	467	68	∈	∈	PROPN
ejpam-6297	467	69	y	y	PROPN
ejpam-6297	467	70	;	;	PUNCT
ejpam-6297	467	71	and	and	CCONJ
ejpam-6297	467	72	(	(	PUNCT
ejpam-6297	467	73	5	5	X
ejpam-6297	467	74	)	)	PUNCT
ejpam-6297	467	75	f(i	f(i	NUM
ejpam-6297	467	76	)	)	PUNCT
ejpam-6297	468	1	⊆	⊆	NUM
ejpam-6297	468	2	j	j	NOUN
ejpam-6297	468	3	.	.	PUNCT
ejpam-6297	469	1	if	if	SCONJ
ejpam-6297	469	2	x	x	PRON
ejpam-6297	469	3	is	be	AUX
ejpam-6297	469	4	sβ	sβ	AUX
ejpam-6297	469	5	-	-	PUNCT
ejpam-6297	469	6	i	i	NOUN
ejpam-6297	469	7	-	-	PUNCT
ejpam-6297	469	8	paracompact	paracompact	ADJ
ejpam-6297	469	9	,	,	PUNCT
ejpam-6297	469	10	then	then	ADV
ejpam-6297	469	11	y	y	PROPN
ejpam-6297	469	12	is	be	AUX
ejpam-6297	469	13	sβ	sβ	PROPN
ejpam-6297	469	14	-	-	PUNCT
ejpam-6297	469	15	j	j	NOUN
ejpam-6297	469	16	-paracompact	-paracompact	NOUN
ejpam-6297	469	17	.	.	PUNCT
ejpam-6297	470	1	proof	proof	NOUN
ejpam-6297	470	2	.	.	PUNCT
ejpam-6297	471	1	let	let	VERB
ejpam-6297	471	2	a	a	DET
ejpam-6297	471	3	=	=	X
ejpam-6297	471	4	{	{	PUNCT
ejpam-6297	471	5	uλ	uλ	NOUN
ejpam-6297	471	6	:	:	PUNCT
ejpam-6297	471	7	λ	λ	X
ejpam-6297	471	8	∈	∈	PROPN
ejpam-6297	471	9	λ	λ	PROPN
ejpam-6297	471	10	}	}	PUNCT
ejpam-6297	471	11	be	be	VERB
ejpam-6297	471	12	an	an	DET
ejpam-6297	471	13	open	open	ADJ
ejpam-6297	471	14	cover	cover	NOUN
ejpam-6297	471	15	of	of	ADP
ejpam-6297	471	16	y	y	PROPN
ejpam-6297	471	17	.	.	PUNCT
ejpam-6297	472	1	this	this	PRON
ejpam-6297	472	2	implies	imply	VERB
ejpam-6297	472	3	that	that	PRON
ejpam-6297	472	4	b	b	X
ejpam-6297	472	5	=	=	SYM
ejpam-6297	472	6	{	{	PUNCT
ejpam-6297	472	7	f−1(uλ	f−1(uλ	PROPN
ejpam-6297	472	8	)	)	PUNCT
ejpam-6297	472	9	:	:	PUNCT
ejpam-6297	472	10	λ	λ	X
ejpam-6297	472	11	∈	∈	PROPN
ejpam-6297	472	12	λ	λ	PROPN
ejpam-6297	472	13	}	}	PUNCT
ejpam-6297	472	14	is	be	AUX
ejpam-6297	472	15	an	an	DET
ejpam-6297	472	16	open	open	ADJ
ejpam-6297	472	17	cover	cover	NOUN
ejpam-6297	472	18	of	of	ADP
ejpam-6297	472	19	x.	x.	NOUN
ejpam-6297	472	20	since	since	SCONJ
ejpam-6297	472	21	x	x	PROPN
ejpam-6297	472	22	is	be	AUX
ejpam-6297	472	23	sβ	sβ	NOUN
ejpam-6297	472	24	-	-	PUNCT
ejpam-6297	472	25	i	i	NOUN
ejpam-6297	472	26	-	-	PUNCT
ejpam-6297	472	27	paracompact	paracompact	ADJ
ejpam-6297	472	28	,	,	PUNCT
ejpam-6297	472	29	the	the	DET
ejpam-6297	472	30	collection	collection	NOUN
ejpam-6297	472	31	b	b	PROPN
ejpam-6297	472	32	has	have	VERB
ejpam-6297	472	33	a	a	DET
ejpam-6297	472	34	precise	precise	ADJ
ejpam-6297	472	35	sβ	sβ	NOUN
ejpam-6297	472	36	-	-	PUNCT
ejpam-6297	472	37	i	i	NOUN
ejpam-6297	472	38	-	-	PUNCT
ejpam-6297	472	39	locally	locally	ADV
ejpam-6297	472	40	finite	finite	PROPN
ejpam-6297	472	41	refinement	refinement	NOUN
ejpam-6297	472	42	c	c	PROPN
ejpam-6297	473	1	=	=	PUNCT
ejpam-6297	473	2	{	{	PUNCT
ejpam-6297	473	3	vλ	vλ	INTJ
ejpam-6297	473	4	:	:	PUNCT
ejpam-6297	473	5	λ	λ	PROPN
ejpam-6297	473	6	∈	∈	PROPN
ejpam-6297	473	7	λ	λ	NOUN
ejpam-6297	473	8	}	}	PUNCT
ejpam-6297	473	9	,	,	PUNCT
ejpam-6297	473	10	where	where	SCONJ
ejpam-6297	473	11	each	each	DET
ejpam-6297	473	12	vλ	vλ	NOUN
ejpam-6297	473	13	is	be	AUX
ejpam-6297	473	14	strong	strong	ADJ
ejpam-6297	473	15	β	β	NOUN
ejpam-6297	473	16	-	-	ADJ
ejpam-6297	473	17	i	i	PRON
ejpam-6297	473	18	-	-	PUNCT
ejpam-6297	473	19	open	open	ADJ
ejpam-6297	473	20	and	and	CCONJ
ejpam-6297	473	21	x	x	SYM
ejpam-6297	473	22	−	−	PROPN
ejpam-6297	473	23	∪λ∈λvλ	∪λ∈λvλ	PROPN
ejpam-6297	473	24	∈	∈	PROPN
ejpam-6297	473	25	i.	i.	NOUN
ejpam-6297	473	26	since	since	SCONJ
ejpam-6297	473	27	f	f	PROPN
ejpam-6297	473	28	is	be	AUX
ejpam-6297	473	29	sβ	sβ	PROPN
ejpam-6297	473	30	-	-	PUNCT
ejpam-6297	473	31	i	i	NOUN
ejpam-6297	473	32	-	-	PUNCT
ejpam-6297	473	33	open	open	ADJ
ejpam-6297	473	34	,	,	PUNCT
ejpam-6297	473	35	the	the	DET
ejpam-6297	473	36	family	family	NOUN
ejpam-6297	473	37	f(c	f(c	PROPN
ejpam-6297	473	38	)	)	PUNCT
ejpam-6297	473	39	=	=	SYM
ejpam-6297	473	40	{	{	PUNCT
ejpam-6297	473	41	f(vλ	f(vλ	NOUN
ejpam-6297	473	42	)	)	PUNCT
ejpam-6297	473	43	:	:	PUNCT
ejpam-6297	474	1	λ	λ	X
ejpam-6297	474	2	∈	∈	PROPN
ejpam-6297	474	3	λ	λ	PROPN
ejpam-6297	474	4	}	}	PUNCT
ejpam-6297	474	5	consists	consist	VERB
ejpam-6297	474	6	of	of	ADP
ejpam-6297	474	7	strong	strong	ADJ
ejpam-6297	474	8	β	β	NOUN
ejpam-6297	474	9	-	-	ADJ
ejpam-6297	474	10	j	j	ADJ
ejpam-6297	474	11	-open	-open	NOUN
ejpam-6297	474	12	sets	set	NOUN
ejpam-6297	474	13	and	and	CCONJ
ejpam-6297	474	14	is	be	AUX
ejpam-6297	474	15	a	a	DET
ejpam-6297	474	16	refinement	refinement	NOUN
ejpam-6297	474	17	of	of	ADP
ejpam-6297	474	18	a.	a.	NOUN
ejpam-6297	474	19	moreover	moreover	ADV
ejpam-6297	474	20	,	,	PUNCT
ejpam-6297	474	21	we	we	PRON
ejpam-6297	474	22	have	have	VERB
ejpam-6297	474	23	y	y	PROPN
ejpam-6297	474	24	−	−	NOUN
ejpam-6297	474	25	∪λ∈λf(vλ	∪λ∈λf(vλ	NOUN
ejpam-6297	474	26	)	)	PUNCT
ejpam-6297	475	1	∈	∈	PROPN
ejpam-6297	475	2	j	j	PROPN
ejpam-6297	475	3	.	.	PUNCT
ejpam-6297	476	1	next	next	ADV
ejpam-6297	476	2	,	,	PUNCT
ejpam-6297	476	3	we	we	PRON
ejpam-6297	476	4	check	check	VERB
ejpam-6297	476	5	that	that	DET
ejpam-6297	476	6	f(c	f(c	PROPN
ejpam-6297	476	7	)	)	PUNCT
ejpam-6297	476	8	is	be	AUX
ejpam-6297	476	9	sβ	sβ	PROPN
ejpam-6297	476	10	-	-	PUNCT
ejpam-6297	476	11	j	j	NOUN
ejpam-6297	476	12	-locally	-locally	ADV
ejpam-6297	476	13	finite	finite	PROPN
ejpam-6297	476	14	.	.	PUNCT
ejpam-6297	477	1	let	let	VERB
ejpam-6297	477	2	y	y	PROPN
ejpam-6297	477	3	∈	∈	PROPN
ejpam-6297	477	4	y	y	PROPN
ejpam-6297	477	5	.	.	PUNCT
ejpam-6297	478	1	since	since	SCONJ
ejpam-6297	478	2	c	c	PROPN
ejpam-6297	478	3	is	be	AUX
ejpam-6297	478	4	sβ	sβ	NOUN
ejpam-6297	478	5	-	-	PUNCT
ejpam-6297	478	6	i	i	NOUN
ejpam-6297	478	7	-	-	PUNCT
ejpam-6297	478	8	locally	locally	ADV
ejpam-6297	478	9	finite	finite	NOUN
ejpam-6297	478	10	,	,	PUNCT
ejpam-6297	478	11	for	for	ADP
ejpam-6297	478	12	each	each	DET
ejpam-6297	478	13	x	x	SYM
ejpam-6297	478	14	∈	∈	PROPN
ejpam-6297	478	15	{	{	PUNCT
ejpam-6297	478	16	f−1(y	f−1(y	PROPN
ejpam-6297	478	17	)	)	PUNCT
ejpam-6297	478	18	}	}	PUNCT
ejpam-6297	478	19	,	,	PUNCT
ejpam-6297	478	20	there	there	PRON
ejpam-6297	478	21	exists	exist	VERB
ejpam-6297	478	22	a	a	DET
ejpam-6297	478	23	strong	strong	ADJ
ejpam-6297	478	24	β	β	X
ejpam-6297	478	25	-	-	ADJ
ejpam-6297	478	26	i	i	PRON
ejpam-6297	478	27	-	-	PUNCT
ejpam-6297	478	28	open	open	ADV
ejpam-6297	478	29	set	set	VERB
ejpam-6297	478	30	gx	gx	PROPN
ejpam-6297	478	31	containing	contain	VERB
ejpam-6297	478	32	x	x	PUNCT
ejpam-6297	478	33	such	such	ADJ
ejpam-6297	478	34	that	that	SCONJ
ejpam-6297	478	35	gx	gx	PROPN
ejpam-6297	478	36	intersects	intersect	NOUN
ejpam-6297	478	37	at	at	ADV
ejpam-6297	478	38	most	most	ADV
ejpam-6297	478	39	finitely	finitely	ADV
ejpam-6297	478	40	many	many	ADJ
ejpam-6297	478	41	elements	element	NOUN
ejpam-6297	478	42	of	of	ADP
ejpam-6297	478	43	c.	c.	NOUN
ejpam-6297	478	44	because	because	SCONJ
ejpam-6297	478	45	{	{	PUNCT
ejpam-6297	478	46	f−1(y	f−1(y	PROPN
ejpam-6297	478	47	)	)	PUNCT
ejpam-6297	478	48	}	}	PUNCT
ejpam-6297	478	49	is	be	AUX
ejpam-6297	478	50	sβ	sβ	PROPN
ejpam-6297	478	51	-	-	PUNCT
ejpam-6297	478	52	i	i	NOUN
ejpam-6297	478	53	-	-	NOUN
ejpam-6297	478	54	compact	compact	ADJ
ejpam-6297	478	55	,	,	PUNCT
ejpam-6297	478	56	and	and	CCONJ
ejpam-6297	478	57	the	the	DET
ejpam-6297	478	58	family	family	NOUN
ejpam-6297	478	59	{	{	PUNCT
ejpam-6297	478	60	gx	gx	PROPN
ejpam-6297	478	61	:	:	PUNCT
ejpam-6297	478	62	f(x	f(x	PROPN
ejpam-6297	478	63	)	)	PUNCT
ejpam-6297	479	1	=	=	SYM
ejpam-6297	479	2	y	y	PROPN
ejpam-6297	479	3	}	}	PUNCT
ejpam-6297	479	4	forms	form	VERB
ejpam-6297	479	5	a	a	DET
ejpam-6297	479	6	strong	strong	ADJ
ejpam-6297	479	7	β	β	X
ejpam-6297	479	8	-	-	ADJ
ejpam-6297	479	9	i	i	NOUN
ejpam-6297	479	10	-	-	PUNCT
ejpam-6297	479	11	open	open	ADJ
ejpam-6297	479	12	cover	cover	NOUN
ejpam-6297	479	13	of	of	ADP
ejpam-6297	479	14	{	{	PUNCT
ejpam-6297	479	15	f−1(y	f−1(y	PROPN
ejpam-6297	479	16	)	)	PUNCT
ejpam-6297	479	17	}	}	PUNCT
ejpam-6297	479	18	,	,	PUNCT
ejpam-6297	479	19	there	there	PRON
ejpam-6297	479	20	exists	exist	VERB
ejpam-6297	479	21	a	a	DET
ejpam-6297	479	22	finite	finite	ADJ
ejpam-6297	479	23	subcover	subcover	PROPN
ejpam-6297	479	24	{	{	PUNCT
ejpam-6297	479	25	hy	hy	PROPN
ejpam-6297	479	26	}	}	PUNCT
ejpam-6297	479	27	such	such	ADJ
ejpam-6297	479	28	that	that	SCONJ
ejpam-6297	479	29	{	{	PUNCT
ejpam-6297	479	30	f−1(y	f−1(y	PROPN
ejpam-6297	479	31	)	)	PUNCT
ejpam-6297	479	32	}	}	PUNCT
ejpam-6297	479	33	⊆	⊆	NUM
ejpam-6297	479	34	∪hy	∪hy	NOUN
ejpam-6297	479	35	,	,	PUNCT
ejpam-6297	479	36	and	and	CCONJ
ejpam-6297	479	37	∪hy	∪hy	NOUN
ejpam-6297	479	38	intersects	intersect	NOUN
ejpam-6297	479	39	at	at	ADP
ejpam-6297	479	40	most	most	ADV
ejpam-6297	479	41	finitely	finitely	ADV
ejpam-6297	479	42	many	many	ADJ
ejpam-6297	479	43	elements	element	NOUN
ejpam-6297	479	44	of	of	ADP
ejpam-6297	479	45	c.	c.	NOUN
ejpam-6297	479	46	as	as	SCONJ
ejpam-6297	479	47	f	f	PROPN
ejpam-6297	479	48	is	be	AUX
ejpam-6297	479	49	sβ	sβ	PROPN
ejpam-6297	479	50	-	-	PUNCT
ejpam-6297	479	51	i	i	NOUN
ejpam-6297	479	52	-	-	PUNCT
ejpam-6297	479	53	closed	closed	ADJ
ejpam-6297	479	54	,	,	PUNCT
ejpam-6297	479	55	applying	apply	VERB
ejpam-6297	479	56	lemma	lemma	PROPN
ejpam-6297	479	57	8	8	NUM
ejpam-6297	479	58	,	,	PUNCT
ejpam-6297	479	59	there	there	PRON
ejpam-6297	479	60	exists	exist	VERB
ejpam-6297	479	61	a	a	DET
ejpam-6297	479	62	strong	strong	ADJ
ejpam-6297	479	63	β	β	NOUN
ejpam-6297	479	64	-	-	ADJ
ejpam-6297	479	65	j	j	ADJ
ejpam-6297	479	66	-open	-open	NOUN
ejpam-6297	479	67	set	set	VERB
ejpam-6297	479	68	wy	wy	PROPN
ejpam-6297	479	69	containing	contain	VERB
ejpam-6297	479	70	y	y	PRON
ejpam-6297	479	71	such	such	ADJ
ejpam-6297	479	72	that	that	SCONJ
ejpam-6297	479	73	f−1(wy	f−1(wy	PROPN
ejpam-6297	479	74	)	)	PUNCT
ejpam-6297	479	75	⊆	⊆	NUM
ejpam-6297	479	76	∪hy	∪hy	NOUN
ejpam-6297	479	77	.	.	PUNCT
ejpam-6297	480	1	thus	thus	ADV
ejpam-6297	480	2	,	,	PUNCT
ejpam-6297	480	3	f−1(wy	f−1(wy	PROPN
ejpam-6297	480	4	)	)	PUNCT
ejpam-6297	480	5	intersects	intersect	NOUN
ejpam-6297	480	6	at	at	ADP
ejpam-6297	480	7	most	most	ADV
ejpam-6297	480	8	finitely	finitely	ADV
ejpam-6297	480	9	many	many	ADJ
ejpam-6297	480	10	elements	element	NOUN
ejpam-6297	480	11	of	of	ADP
ejpam-6297	480	12	c	c	NOUN
ejpam-6297	480	13	,	,	PUNCT
ejpam-6297	480	14	which	which	PRON
ejpam-6297	480	15	implies	imply	VERB
ejpam-6297	480	16	that	that	SCONJ
ejpam-6297	480	17	wy	wy	PROPN
ejpam-6297	480	18	intersects	intersect	NOUN
ejpam-6297	480	19	at	at	ADV
ejpam-6297	480	20	most	most	ADV
ejpam-6297	480	21	finitely	finitely	ADV
ejpam-6297	480	22	many	many	ADJ
ejpam-6297	480	23	elements	element	NOUN
ejpam-6297	480	24	of	of	ADP
ejpam-6297	480	25	f(c	f(c	PROPN
ejpam-6297	480	26	)	)	PUNCT
ejpam-6297	480	27	.	.	PUNCT
ejpam-6297	481	1	therefore	therefore	ADV
ejpam-6297	481	2	,	,	PUNCT
ejpam-6297	481	3	f(c	f(c	PROPN
ejpam-6297	481	4	)	)	PUNCT
ejpam-6297	481	5	is	be	AUX
ejpam-6297	481	6	sβ	sβ	PROPN
ejpam-6297	481	7	-	-	PUNCT
ejpam-6297	481	8	j	j	NOUN
ejpam-6297	481	9	-locally	-locally	ADV
ejpam-6297	481	10	finite	finite	ADJ
ejpam-6297	481	11	in	in	ADP
ejpam-6297	481	12	y	y	PROPN
ejpam-6297	481	13	.	.	PUNCT
ejpam-6297	482	1	consequently	consequently	ADV
ejpam-6297	482	2	,	,	PUNCT
ejpam-6297	482	3	(	(	PUNCT
ejpam-6297	482	4	y	y	PROPN
ejpam-6297	482	5	,	,	PUNCT
ejpam-6297	482	6	τ	τ	PROPN
ejpam-6297	482	7	′,j	′,j	NOUN
ejpam-6297	482	8	)	)	PUNCT
ejpam-6297	482	9	is	be	AUX
ejpam-6297	482	10	sβ	sβ	PROPN
ejpam-6297	482	11	-	-	PUNCT
ejpam-6297	482	12	j	j	NOUN
ejpam-6297	482	13	-paracompact	-paracompact	PROPN
ejpam-6297	482	14	.	.	PUNCT
ejpam-6297	483	1	the	the	DET
ejpam-6297	483	2	next	next	ADJ
ejpam-6297	483	3	theorem	theorem	NOUN
ejpam-6297	483	4	characterizes	characterize	VERB
ejpam-6297	483	5	a	a	DET
ejpam-6297	483	6	function	function	NOUN
ejpam-6297	483	7	from	from	ADP
ejpam-6297	483	8	an	an	DET
ejpam-6297	483	9	sβ	sβ	NOUN
ejpam-6297	483	10	-	-	PUNCT
ejpam-6297	483	11	i	i	NOUN
ejpam-6297	483	12	-	-	PUNCT
ejpam-6297	483	13	paracompact	paracompact	ADJ
ejpam-6297	483	14	ideal	ideal	ADJ
ejpam-6297	483	15	topological	topological	ADJ
ejpam-6297	483	16	space	space	NOUN
ejpam-6297	483	17	(	(	PUNCT
ejpam-6297	483	18	x	x	X
ejpam-6297	483	19	,	,	PUNCT
ejpam-6297	483	20	τ	τ	PROPN
ejpam-6297	483	21	,	,	PUNCT
ejpam-6297	483	22	i	i	PROPN
ejpam-6297	483	23	)	)	PUNCT
ejpam-6297	483	24	to	to	ADP
ejpam-6297	483	25	a	a	DET
ejpam-6297	483	26	topological	topological	ADJ
ejpam-6297	483	27	space	space	NOUN
ejpam-6297	483	28	(	(	PUNCT
ejpam-6297	483	29	y	y	PROPN
ejpam-6297	483	30	,	,	PUNCT
ejpam-6297	483	31	τ	τ	PROPN
ejpam-6297	483	32	′	′	NUM
ejpam-6297	483	33	)	)	PUNCT
ejpam-6297	483	34	,	,	PUNCT
ejpam-6297	483	35	ensuring	ensure	VERB
ejpam-6297	483	36	that	that	SCONJ
ejpam-6297	483	37	y	y	PROPN
ejpam-6297	483	38	retains	retain	VERB
ejpam-6297	483	39	the	the	DET
ejpam-6297	483	40	same	same	ADJ
ejpam-6297	483	41	structural	structural	ADJ
ejpam-6297	483	42	properties	property	NOUN
ejpam-6297	483	43	as	as	SCONJ
ejpam-6297	483	44	x.	x.	NOUN
ejpam-6297	483	45	theorem	theorem	VERB
ejpam-6297	483	46	19	19	NUM
ejpam-6297	483	47	.	.	PUNCT
ejpam-6297	484	1	let	let	VERB
ejpam-6297	484	2	(	(	PUNCT
ejpam-6297	484	3	x	x	X
ejpam-6297	484	4	,	,	PUNCT
ejpam-6297	484	5	τ	τ	PROPN
ejpam-6297	484	6	,	,	PUNCT
ejpam-6297	484	7	i	i	PRON
ejpam-6297	484	8	)	)	PUNCT
ejpam-6297	484	9	be	be	VERB
ejpam-6297	484	10	an	an	DET
ejpam-6297	484	11	ideal	ideal	ADJ
ejpam-6297	484	12	topological	topological	ADJ
ejpam-6297	484	13	space	space	NOUN
ejpam-6297	484	14	and	and	CCONJ
ejpam-6297	484	15	(	(	PUNCT
ejpam-6297	484	16	y	y	PROPN
ejpam-6297	484	17	,	,	PUNCT
ejpam-6297	484	18	τ	τ	PROPN
ejpam-6297	484	19	′	′	NUM
ejpam-6297	484	20	)	)	PUNCT
ejpam-6297	484	21	a	a	DET
ejpam-6297	484	22	topological	topological	ADJ
ejpam-6297	484	23	space	space	NOUN
ejpam-6297	484	24	.	.	PUNCT
ejpam-6297	485	1	suppose	suppose	VERB
ejpam-6297	485	2	that	that	SCONJ
ejpam-6297	485	3	f	f	X
ejpam-6297	485	4	:	:	PUNCT
ejpam-6297	485	5	x	x	X
ejpam-6297	485	6	→	→	SYM
ejpam-6297	485	7	y	y	PROPN
ejpam-6297	485	8	satisfies	satisfy	VERB
ejpam-6297	485	9	the	the	DET
ejpam-6297	485	10	following	following	ADJ
ejpam-6297	485	11	statements	statement	NOUN
ejpam-6297	485	12	:	:	PUNCT
ejpam-6297	485	13	(	(	PUNCT
ejpam-6297	485	14	1	1	X
ejpam-6297	485	15	)	)	PUNCT
ejpam-6297	485	16	f	f	PROPN
ejpam-6297	485	17	is	be	AUX
ejpam-6297	485	18	sβ	sβ	PROPN
ejpam-6297	485	19	-	-	PUNCT
ejpam-6297	485	20	i	i	NOUN
ejpam-6297	485	21	-	-	PUNCT
ejpam-6297	485	22	irresolute	irresolute	ADJ
ejpam-6297	485	23	;	;	PUNCT
ejpam-6297	485	24	(	(	PUNCT
ejpam-6297	485	25	2	2	X
ejpam-6297	485	26	)	)	PUNCT
ejpam-6297	485	27	f	f	PROPN
ejpam-6297	485	28	is	be	AUX
ejpam-6297	485	29	continuous	continuous	ADJ
ejpam-6297	485	30	;	;	PUNCT
ejpam-6297	485	31	c.	c.	PROPN
ejpam-6297	485	32	boonpok	boonpok	PROPN
ejpam-6297	485	33	,	,	PUNCT
ejpam-6297	485	34	p.	p.	PROPN
ejpam-6297	485	35	raktaow	raktaow	NOUN
ejpam-6297	485	36	,	,	PUNCT
ejpam-6297	485	37	a.	a.	PROPN
ejpam-6297	485	38	sama	sama	PROPN
ejpam-6297	485	39	-	-	PUNCT
ejpam-6297	485	40	ae	ae	PROPN
ejpam-6297	485	41	/	/	SYM
ejpam-6297	485	42	eur	eur	PROPN
ejpam-6297	485	43	.	.	PUNCT
ejpam-6297	486	1	j.	j.	PROPN
ejpam-6297	486	2	pure	pure	PROPN
ejpam-6297	486	3	appl	appl	PROPN
ejpam-6297	486	4	.	.	PROPN
ejpam-6297	486	5	math	math	PROPN
ejpam-6297	486	6	,	,	PUNCT
ejpam-6297	486	7	18	18	NUM
ejpam-6297	486	8	(	(	PUNCT
ejpam-6297	486	9	3	3	NUM
ejpam-6297	486	10	)	)	PUNCT
ejpam-6297	486	11	(	(	PUNCT
ejpam-6297	486	12	2025	2025	NUM
ejpam-6297	486	13	)	)	PUNCT
ejpam-6297	486	14	,	,	PUNCT
ejpam-6297	486	15	6297	6297	NUM
ejpam-6297	486	16	19	19	NUM
ejpam-6297	486	17	of	of	ADP
ejpam-6297	486	18	23	23	NUM
ejpam-6297	486	19	(	(	PUNCT
ejpam-6297	486	20	3	3	NUM
ejpam-6297	486	21	)	)	PUNCT
ejpam-6297	486	22	f	f	PROPN
ejpam-6297	486	23	is	be	AUX
ejpam-6297	486	24	sβ	sβ	PROPN
ejpam-6297	486	25	-	-	PUNCT
ejpam-6297	486	26	i	i	NOUN
ejpam-6297	486	27	-	-	PUNCT
ejpam-6297	486	28	open	open	ADJ
ejpam-6297	486	29	;	;	PUNCT
ejpam-6297	486	30	(	(	PUNCT
ejpam-6297	486	31	4	4	X
ejpam-6297	486	32	)	)	PUNCT
ejpam-6297	486	33	f	f	PROPN
ejpam-6297	486	34	is	be	AUX
ejpam-6297	486	35	surjective	surjective	ADJ
ejpam-6297	486	36	;	;	PUNCT
ejpam-6297	486	37	and	and	CCONJ
ejpam-6297	486	38	(	(	PUNCT
ejpam-6297	486	39	5	5	X
ejpam-6297	486	40	)	)	PUNCT
ejpam-6297	486	41	f(v	f(v	NOUN
ejpam-6297	486	42	)	)	PUNCT
ejpam-6297	486	43	is	be	AUX
ejpam-6297	486	44	sβ	sβ	NOUN
ejpam-6297	486	45	-	-	PUNCT
ejpam-6297	486	46	f(i)-locally	f(i)-locally	ADV
ejpam-6297	486	47	finite	finite	VERB
ejpam-6297	486	48	in	in	ADP
ejpam-6297	486	49	y	y	PROPN
ejpam-6297	486	50	for	for	ADP
ejpam-6297	486	51	every	every	DET
ejpam-6297	486	52	sβ	sβ	PROPN
ejpam-6297	486	53	-	-	PUNCT
ejpam-6297	486	54	i	i	NOUN
ejpam-6297	486	55	-	-	PUNCT
ejpam-6297	486	56	locally	locally	ADV
ejpam-6297	486	57	finite	finite	VERB
ejpam-6297	486	58	v	v	NOUN
ejpam-6297	486	59	in	in	ADP
ejpam-6297	486	60	x.	x.	NOUN
ejpam-6297	487	1	if	if	SCONJ
ejpam-6297	487	2	(	(	PUNCT
ejpam-6297	487	3	x	x	NOUN
ejpam-6297	487	4	,	,	PUNCT
ejpam-6297	487	5	τ	τ	PROPN
ejpam-6297	487	6	,	,	PUNCT
ejpam-6297	487	7	i	i	PROPN
ejpam-6297	487	8	)	)	PUNCT
ejpam-6297	487	9	is	be	AUX
ejpam-6297	487	10	sβ	sβ	PROPN
ejpam-6297	487	11	-	-	PUNCT
ejpam-6297	487	12	i	i	NOUN
ejpam-6297	487	13	-	-	PUNCT
ejpam-6297	487	14	paracompact	paracompact	ADJ
ejpam-6297	487	15	,	,	PUNCT
ejpam-6297	487	16	then	then	ADV
ejpam-6297	487	17	(	(	PUNCT
ejpam-6297	487	18	y	y	PROPN
ejpam-6297	487	19	,	,	PUNCT
ejpam-6297	487	20	τ	τ	PROPN
ejpam-6297	487	21	′	′	NUM
ejpam-6297	487	22	,	,	PUNCT
ejpam-6297	487	23	f(i	f(i	PROPN
ejpam-6297	487	24	)	)	PUNCT
ejpam-6297	487	25	)	)	PUNCT
ejpam-6297	487	26	is	be	AUX
ejpam-6297	487	27	sβ	sβ	PROPN
ejpam-6297	487	28	-	-	PUNCT
ejpam-6297	487	29	f(i)-paracompact	f(i)-paracompact	PROPN
ejpam-6297	487	30	.	.	PUNCT
ejpam-6297	488	1	proof	proof	NOUN
ejpam-6297	488	2	.	.	PUNCT
ejpam-6297	489	1	let	let	VERB
ejpam-6297	489	2	a	a	DET
ejpam-6297	489	3	=	=	X
ejpam-6297	489	4	{	{	PUNCT
ejpam-6297	489	5	uλ	uλ	NOUN
ejpam-6297	489	6	:	:	PUNCT
ejpam-6297	489	7	λ	λ	X
ejpam-6297	489	8	∈	∈	PROPN
ejpam-6297	489	9	λ	λ	PROPN
ejpam-6297	489	10	}	}	PUNCT
ejpam-6297	489	11	be	be	VERB
ejpam-6297	489	12	an	an	DET
ejpam-6297	489	13	open	open	ADJ
ejpam-6297	489	14	cover	cover	NOUN
ejpam-6297	489	15	of	of	ADP
ejpam-6297	489	16	y	y	PROPN
ejpam-6297	489	17	.	.	PUNCT
ejpam-6297	490	1	this	this	PRON
ejpam-6297	490	2	implies	imply	VERB
ejpam-6297	490	3	that	that	PRON
ejpam-6297	490	4	b	b	X
ejpam-6297	490	5	=	=	SYM
ejpam-6297	490	6	{	{	PUNCT
ejpam-6297	490	7	f−1(uλ	f−1(uλ	PROPN
ejpam-6297	490	8	)	)	PUNCT
ejpam-6297	490	9	:	:	PUNCT
ejpam-6297	490	10	λ	λ	X
ejpam-6297	490	11	∈	∈	PROPN
ejpam-6297	490	12	λ	λ	PROPN
ejpam-6297	490	13	}	}	PUNCT
ejpam-6297	490	14	forms	form	VERB
ejpam-6297	490	15	an	an	DET
ejpam-6297	490	16	open	open	ADJ
ejpam-6297	490	17	cover	cover	NOUN
ejpam-6297	490	18	of	of	ADP
ejpam-6297	490	19	x.	x.	NOUN
ejpam-6297	490	20	since	since	SCONJ
ejpam-6297	490	21	x	x	PROPN
ejpam-6297	490	22	is	be	AUX
ejpam-6297	490	23	sβ	sβ	NOUN
ejpam-6297	490	24	-	-	PUNCT
ejpam-6297	490	25	i	i	NOUN
ejpam-6297	490	26	-	-	PUNCT
ejpam-6297	490	27	paracompact	paracompact	ADJ
ejpam-6297	490	28	,	,	PUNCT
ejpam-6297	490	29	the	the	DET
ejpam-6297	490	30	collection	collection	NOUN
ejpam-6297	490	31	b	b	PROPN
ejpam-6297	490	32	has	have	VERB
ejpam-6297	490	33	a	a	DET
ejpam-6297	490	34	precise	precise	ADJ
ejpam-6297	490	35	sβ	sβ	NOUN
ejpam-6297	490	36	-	-	PUNCT
ejpam-6297	490	37	i	i	NOUN
ejpam-6297	490	38	-	-	PUNCT
ejpam-6297	490	39	locally	locally	ADV
ejpam-6297	490	40	finite	finite	VERB
ejpam-6297	490	41	strong	strong	ADJ
ejpam-6297	490	42	β	β	X
ejpam-6297	490	43	-	-	ADJ
ejpam-6297	490	44	i	i	NOUN
ejpam-6297	490	45	-	-	PUNCT
ejpam-6297	490	46	open	open	ADJ
ejpam-6297	490	47	refinement	refinement	NOUN
ejpam-6297	491	1	c	c	NOUN
ejpam-6297	492	1	=	=	PUNCT
ejpam-6297	493	1	{	{	PUNCT
ejpam-6297	494	1	vλ	vλ	INTJ
ejpam-6297	494	2	:	:	PUNCT
ejpam-6297	494	3	λ	λ	PROPN
ejpam-6297	494	4	∈	∈	PROPN
ejpam-6297	494	5	λ	λ	NOUN
ejpam-6297	494	6	}	}	PUNCT
ejpam-6297	494	7	such	such	ADJ
ejpam-6297	494	8	that	that	SCONJ
ejpam-6297	494	9	x	x	PUNCT
ejpam-6297	495	1	−	−	NOUN
ejpam-6297	495	2	∪λ∈λvλ	∪λ∈λvλ	PROPN
ejpam-6297	495	3	∈	∈	PROPN
ejpam-6297	495	4	i.	i.	NOUN
ejpam-6297	495	5	since	since	SCONJ
ejpam-6297	495	6	y	y	PROPN
ejpam-6297	495	7	−	−	PROPN
ejpam-6297	495	8	∪λ∈λf(vλ	∪λ∈λf(vλ	NOUN
ejpam-6297	495	9	)	)	PUNCT
ejpam-6297	496	1	⊆	⊆	NUM
ejpam-6297	496	2	f	f	X
ejpam-6297	496	3	(	(	PUNCT
ejpam-6297	496	4	x	x	SYM
ejpam-6297	496	5	−	−	PROPN
ejpam-6297	496	6	∪λ∈λvλ	∪λ∈λvλ	NOUN
ejpam-6297	496	7	)	)	PUNCT
ejpam-6297	496	8	,	,	PUNCT
ejpam-6297	496	9	and	and	CCONJ
ejpam-6297	496	10	f	f	X
ejpam-6297	496	11	(	(	PUNCT
ejpam-6297	496	12	x	x	SYM
ejpam-6297	496	13	−	−	PROPN
ejpam-6297	496	14	∪λ∈λvλ	∪λ∈λvλ	PROPN
ejpam-6297	496	15	)	)	PUNCT
ejpam-6297	496	16	∈	∈	PROPN
ejpam-6297	496	17	f(i	f(i	PROPN
ejpam-6297	496	18	)	)	PUNCT
ejpam-6297	496	19	,	,	PUNCT
ejpam-6297	496	20	it	it	PRON
ejpam-6297	496	21	follows	follow	VERB
ejpam-6297	496	22	that	that	SCONJ
ejpam-6297	496	23	y	y	PROPN
ejpam-6297	496	24	−	−	NOUN
ejpam-6297	496	25	∪λ∈λf(vλ	∪λ∈λf(vλ	NOUN
ejpam-6297	496	26	)	)	PUNCT
ejpam-6297	497	1	∈	∈	PROPN
ejpam-6297	497	2	f(i	f(i	PROPN
ejpam-6297	497	3	)	)	PUNCT
ejpam-6297	497	4	.	.	PUNCT
ejpam-6297	498	1	because	because	SCONJ
ejpam-6297	498	2	f	f	PROPN
ejpam-6297	498	3	is	be	AUX
ejpam-6297	498	4	surjective	surjective	ADJ
ejpam-6297	498	5	,	,	PUNCT
ejpam-6297	498	6	f(i	f(i	NUM
ejpam-6297	498	7	)	)	PUNCT
ejpam-6297	498	8	is	be	AUX
ejpam-6297	498	9	an	an	DET
ejpam-6297	498	10	ideal	ideal	NOUN
ejpam-6297	498	11	in	in	ADP
ejpam-6297	498	12	y	y	PROPN
ejpam-6297	498	13	.	.	PUNCT
ejpam-6297	499	1	by	by	ADP
ejpam-6297	499	2	assumption	assumption	NOUN
ejpam-6297	499	3	,	,	PUNCT
ejpam-6297	499	4	f(c	f(c	PROPN
ejpam-6297	499	5	)	)	PUNCT
ejpam-6297	499	6	=	=	SYM
ejpam-6297	499	7	{	{	PUNCT
ejpam-6297	499	8	f(vλ	f(vλ	NOUN
ejpam-6297	499	9	)	)	PUNCT
ejpam-6297	499	10	:	:	PUNCT
ejpam-6297	499	11	λ	λ	X
ejpam-6297	499	12	∈	∈	PROPN
ejpam-6297	499	13	λ	λ	PROPN
ejpam-6297	499	14	}	}	PUNCT
ejpam-6297	499	15	is	be	AUX
ejpam-6297	499	16	a	a	DET
ejpam-6297	499	17	precise	precise	ADJ
ejpam-6297	499	18	sβ	sβ	ADJ
ejpam-6297	499	19	-	-	PUNCT
ejpam-6297	499	20	f(i)-open	f(i)-open	PROPN
ejpam-6297	499	21	refinement	refinement	NOUN
ejpam-6297	499	22	of	of	ADP
ejpam-6297	499	23	strong	strong	ADJ
ejpam-6297	499	24	β	β	X
ejpam-6297	499	25	-	-	PUNCT
ejpam-6297	499	26	f(i)-open	f(i)-open	ADJ
ejpam-6297	499	27	sets	set	NOUN
ejpam-6297	499	28	in	in	ADP
ejpam-6297	499	29	y	y	PROPN
ejpam-6297	499	30	.	.	PUNCT
ejpam-6297	500	1	now	now	ADV
ejpam-6297	500	2	,	,	PUNCT
ejpam-6297	500	3	we	we	PRON
ejpam-6297	500	4	proceed	proceed	VERB
ejpam-6297	500	5	to	to	PART
ejpam-6297	500	6	verify	verify	VERB
ejpam-6297	500	7	that	that	DET
ejpam-6297	500	8	f(c	f(c	PROPN
ejpam-6297	500	9	)	)	PUNCT
ejpam-6297	500	10	refines	refine	VERB
ejpam-6297	500	11	a.	a.	NOUN
ejpam-6297	500	12	for	for	ADP
ejpam-6297	500	13	each	each	DET
ejpam-6297	500	14	f(vλ	f(vλ	NOUN
ejpam-6297	500	15	)	)	PUNCT
ejpam-6297	500	16	∈	∈	PROPN
ejpam-6297	500	17	f(c	f(c	PROPN
ejpam-6297	500	18	)	)	PUNCT
ejpam-6297	500	19	,	,	PUNCT
ejpam-6297	500	20	we	we	PRON
ejpam-6297	500	21	have	have	VERB
ejpam-6297	500	22	vλ	vλ	INTJ
ejpam-6297	500	23	∈	∈	PROPN
ejpam-6297	500	24	c	c	NOUN
ejpam-6297	500	25	,	,	PUNCT
ejpam-6297	500	26	and	and	CCONJ
ejpam-6297	500	27	there	there	PRON
ejpam-6297	500	28	exists	exist	VERB
ejpam-6297	500	29	uλ	uλ	ADP
ejpam-6297	500	30	∈	∈	PROPN
ejpam-6297	500	31	a	a	DET
ejpam-6297	500	32	such	such	ADJ
ejpam-6297	500	33	that	that	SCONJ
ejpam-6297	500	34	vλ	vλ	ADP
ejpam-6297	500	35	⊆	⊆	NUM
ejpam-6297	500	36	f−1(uλ	f−1(uλ	PROPN
ejpam-6297	500	37	)	)	PUNCT
ejpam-6297	500	38	,	,	PUNCT
ejpam-6297	500	39	as	as	SCONJ
ejpam-6297	500	40	c	c	PROPN
ejpam-6297	500	41	refines	refine	VERB
ejpam-6297	500	42	b.	b.	PROPN
ejpam-6297	500	43	this	this	PRON
ejpam-6297	500	44	implies	imply	VERB
ejpam-6297	500	45	that	that	SCONJ
ejpam-6297	500	46	f(vλ	f(vλ	NOUN
ejpam-6297	500	47	)	)	PUNCT
ejpam-6297	500	48	⊆	⊆	NUM
ejpam-6297	500	49	f(f−1(uλ	f(f−1(uλ	NOUN
ejpam-6297	500	50	)	)	PUNCT
ejpam-6297	500	51	)	)	PUNCT
ejpam-6297	501	1	⊆	⊆	NUM
ejpam-6297	501	2	uλ	uλ	NOUN
ejpam-6297	501	3	.	.	PUNCT
ejpam-6297	502	1	consequently	consequently	ADV
ejpam-6297	502	2	,	,	PUNCT
ejpam-6297	502	3	(	(	PUNCT
ejpam-6297	502	4	y	y	PROPN
ejpam-6297	502	5	,	,	PUNCT
ejpam-6297	502	6	τ	τ	PROPN
ejpam-6297	502	7	′	′	NUM
ejpam-6297	502	8	,	,	PUNCT
ejpam-6297	502	9	f(i	f(i	PROPN
ejpam-6297	502	10	)	)	PUNCT
ejpam-6297	502	11	)	)	PUNCT
ejpam-6297	502	12	is	be	AUX
ejpam-6297	502	13	sβ	sβ	PROPN
ejpam-6297	502	14	-	-	PUNCT
ejpam-6297	502	15	f(i)-paracompact	f(i)-paracompact	PROPN
ejpam-6297	502	16	.	.	PUNCT
ejpam-6297	503	1	the	the	DET
ejpam-6297	503	2	following	follow	VERB
ejpam-6297	503	3	theorem	theorem	ADJ
ejpam-6297	503	4	outlines	outline	VERB
ejpam-6297	503	5	conditions	condition	NOUN
ejpam-6297	503	6	under	under	ADP
ejpam-6297	503	7	which	which	PRON
ejpam-6297	503	8	a	a	DET
ejpam-6297	503	9	function	function	NOUN
ejpam-6297	503	10	from	from	ADP
ejpam-6297	503	11	a	a	DET
ejpam-6297	503	12	topological	topological	ADJ
ejpam-6297	503	13	space	space	NOUN
ejpam-6297	503	14	x	x	PUNCT
ejpam-6297	503	15	to	to	ADP
ejpam-6297	503	16	an	an	DET
ejpam-6297	503	17	sβ	sβ	PROPN
ejpam-6297	503	18	-	-	PUNCT
ejpam-6297	503	19	j	j	NOUN
ejpam-6297	503	20	-paracompact	-paracompact	PROPN
ejpam-6297	503	21	ideal	ideal	ADJ
ejpam-6297	503	22	topological	topological	ADJ
ejpam-6297	503	23	space	space	NOUN
ejpam-6297	503	24	y	y	PROPN
ejpam-6297	503	25	ensures	ensure	VERB
ejpam-6297	503	26	that	that	SCONJ
ejpam-6297	503	27	x	x	PRON
ejpam-6297	503	28	shares	share	VERB
ejpam-6297	503	29	the	the	DET
ejpam-6297	503	30	same	same	ADJ
ejpam-6297	503	31	properties	property	NOUN
ejpam-6297	503	32	as	as	ADP
ejpam-6297	503	33	y	y	PROPN
ejpam-6297	503	34	.	.	PUNCT
ejpam-6297	504	1	theorem	theorem	VERB
ejpam-6297	504	2	20	20	NUM
ejpam-6297	504	3	.	.	PUNCT
ejpam-6297	505	1	let	let	VERB
ejpam-6297	505	2	(	(	PUNCT
ejpam-6297	505	3	x	x	NOUN
ejpam-6297	505	4	,	,	PUNCT
ejpam-6297	505	5	τ	τ	X
ejpam-6297	505	6	)	)	PUNCT
ejpam-6297	505	7	be	be	VERB
ejpam-6297	505	8	a	a	DET
ejpam-6297	505	9	topological	topological	ADJ
ejpam-6297	505	10	space	space	NOUN
ejpam-6297	505	11	and	and	CCONJ
ejpam-6297	505	12	(	(	PUNCT
ejpam-6297	505	13	y	y	PROPN
ejpam-6297	505	14	,	,	PUNCT
ejpam-6297	505	15	τ	τ	PROPN
ejpam-6297	505	16	′,j	′,j	NOUN
ejpam-6297	505	17	)	)	PUNCT
ejpam-6297	505	18	an	an	DET
ejpam-6297	505	19	ideal	ideal	ADJ
ejpam-6297	505	20	topological	topological	ADJ
ejpam-6297	505	21	space	space	NOUN
ejpam-6297	505	22	.	.	PUNCT
ejpam-6297	506	1	suppose	suppose	VERB
ejpam-6297	506	2	that	that	SCONJ
ejpam-6297	506	3	f	f	X
ejpam-6297	506	4	:	:	PUNCT
ejpam-6297	506	5	x	x	X
ejpam-6297	506	6	→	→	SYM
ejpam-6297	506	7	y	y	PROPN
ejpam-6297	506	8	satisfies	satisfy	VERB
ejpam-6297	506	9	the	the	DET
ejpam-6297	506	10	following	following	ADJ
ejpam-6297	506	11	statements	statement	NOUN
ejpam-6297	506	12	:	:	PUNCT
ejpam-6297	506	13	(	(	PUNCT
ejpam-6297	506	14	1	1	X
ejpam-6297	506	15	)	)	PUNCT
ejpam-6297	506	16	f	f	PROPN
ejpam-6297	506	17	is	be	AUX
ejpam-6297	506	18	open	open	ADJ
ejpam-6297	506	19	;	;	PUNCT
ejpam-6297	506	20	(	(	PUNCT
ejpam-6297	506	21	2	2	X
ejpam-6297	506	22	)	)	PUNCT
ejpam-6297	506	23	f	f	PROPN
ejpam-6297	506	24	is	be	AUX
ejpam-6297	506	25	sβ	sβ	NOUN
ejpam-6297	506	26	-	-	PUNCT
ejpam-6297	506	27	f−1(j	f−1(j	NOUN
ejpam-6297	506	28	)	)	PUNCT
ejpam-6297	506	29	-irresolute	-irresolute	NOUN
ejpam-6297	506	30	;	;	PUNCT
ejpam-6297	506	31	and	and	CCONJ
ejpam-6297	506	32	(	(	PUNCT
ejpam-6297	506	33	3	3	X
ejpam-6297	506	34	)	)	PUNCT
ejpam-6297	506	35	f	f	PROPN
ejpam-6297	506	36	is	be	AUX
ejpam-6297	506	37	bijective	bijective	ADJ
ejpam-6297	506	38	.	.	PUNCT
ejpam-6297	507	1	if	if	SCONJ
ejpam-6297	507	2	(	(	PUNCT
ejpam-6297	507	3	y	y	PROPN
ejpam-6297	507	4	,	,	PUNCT
ejpam-6297	507	5	τ	τ	PROPN
ejpam-6297	507	6	′,j	′,j	NOUN
ejpam-6297	507	7	)	)	PUNCT
ejpam-6297	507	8	is	be	AUX
ejpam-6297	507	9	sβ	sβ	PROPN
ejpam-6297	507	10	-	-	PUNCT
ejpam-6297	507	11	j	j	NOUN
ejpam-6297	507	12	-paracompact	-paracompact	PROPN
ejpam-6297	507	13	,	,	PUNCT
ejpam-6297	507	14	then	then	ADV
ejpam-6297	507	15	(	(	PUNCT
ejpam-6297	507	16	x	x	X
ejpam-6297	507	17	,	,	PUNCT
ejpam-6297	507	18	τ	τ	PROPN
ejpam-6297	507	19	,	,	PUNCT
ejpam-6297	507	20	f−1(j	f−1(j	NOUN
ejpam-6297	507	21	)	)	PUNCT
ejpam-6297	507	22	)	)	PUNCT
ejpam-6297	507	23	is	be	AUX
ejpam-6297	507	24	sβ	sβ	NOUN
ejpam-6297	507	25	-	-	PUNCT
ejpam-6297	507	26	f−1(j	f−1(j	NOUN
ejpam-6297	507	27	)	)	PUNCT
ejpam-6297	507	28	-paracompact	-paracompact	NOUN
ejpam-6297	507	29	.	.	PUNCT
ejpam-6297	508	1	proof	proof	NOUN
ejpam-6297	508	2	.	.	PUNCT
ejpam-6297	509	1	let	let	VERB
ejpam-6297	509	2	a	a	DET
ejpam-6297	509	3	=	=	X
ejpam-6297	509	4	{	{	PUNCT
ejpam-6297	509	5	uλ	uλ	NOUN
ejpam-6297	509	6	:	:	PUNCT
ejpam-6297	509	7	λ	λ	X
ejpam-6297	509	8	∈	∈	PROPN
ejpam-6297	509	9	λ	λ	PROPN
ejpam-6297	509	10	}	}	PUNCT
ejpam-6297	509	11	be	be	VERB
ejpam-6297	509	12	an	an	DET
ejpam-6297	509	13	open	open	ADJ
ejpam-6297	509	14	cover	cover	NOUN
ejpam-6297	509	15	of	of	ADP
ejpam-6297	509	16	x.	x.	NOUN
ejpam-6297	509	17	since	since	SCONJ
ejpam-6297	509	18	f	f	PROPN
ejpam-6297	509	19	is	be	AUX
ejpam-6297	509	20	open	open	ADJ
ejpam-6297	509	21	,	,	PUNCT
ejpam-6297	509	22	the	the	DET
ejpam-6297	509	23	collection	collection	NOUN
ejpam-6297	509	24	f(a	f(a	NOUN
ejpam-6297	509	25	)	)	PUNCT
ejpam-6297	510	1	=	=	PRON
ejpam-6297	510	2	{	{	PUNCT
ejpam-6297	510	3	f(uλ	f(uλ	PROPN
ejpam-6297	510	4	)	)	PUNCT
ejpam-6297	510	5	:	:	PUNCT
ejpam-6297	511	1	λ	λ	X
ejpam-6297	511	2	∈	∈	PROPN
ejpam-6297	511	3	λ	λ	PROPN
ejpam-6297	511	4	}	}	PUNCT
ejpam-6297	511	5	is	be	AUX
ejpam-6297	511	6	an	an	DET
ejpam-6297	511	7	open	open	ADJ
ejpam-6297	511	8	cover	cover	NOUN
ejpam-6297	511	9	of	of	ADP
ejpam-6297	511	10	y	y	PROPN
ejpam-6297	511	11	.	.	PUNCT
ejpam-6297	512	1	by	by	ADP
ejpam-6297	512	2	hypothesis	hypothesis	NOUN
ejpam-6297	512	3	,	,	PUNCT
ejpam-6297	512	4	f(a	f(a	PROPN
ejpam-6297	512	5	)	)	PUNCT
ejpam-6297	512	6	has	have	VERB
ejpam-6297	512	7	a	a	DET
ejpam-6297	512	8	precise	precise	ADJ
ejpam-6297	512	9	sβ	sβ	NOUN
ejpam-6297	512	10	-	-	PUNCT
ejpam-6297	512	11	j	j	NOUN
ejpam-6297	512	12	-locally	-locally	ADV
ejpam-6297	512	13	finite	finite	VERB
ejpam-6297	512	14	strong	strong	ADJ
ejpam-6297	512	15	β	β	PROPN
ejpam-6297	512	16	-	-	ADJ
ejpam-6297	512	17	j	j	PROPN
ejpam-6297	512	18	-open	-open	PROPN
ejpam-6297	512	19	refinement	refinement	PROPN
ejpam-6297	512	20	b	b	PROPN
ejpam-6297	512	21	=	=	PUNCT
ejpam-6297	512	22	{	{	PUNCT
ejpam-6297	512	23	vλ	vλ	INTJ
ejpam-6297	512	24	:	:	PUNCT
ejpam-6297	512	25	λ	λ	PROPN
ejpam-6297	512	26	∈	∈	PROPN
ejpam-6297	512	27	λ	λ	NOUN
ejpam-6297	512	28	}	}	PUNCT
ejpam-6297	512	29	such	such	ADJ
ejpam-6297	512	30	that	that	SCONJ
ejpam-6297	512	31	y	y	PROPN
ejpam-6297	512	32	−	−	PROPN
ejpam-6297	512	33	∪λ∈λvλ	∪λ∈λvλ	PROPN
ejpam-6297	512	34	∈	∈	PROPN
ejpam-6297	512	35	j	j	PROPN
ejpam-6297	512	36	.	.	PUNCT
ejpam-6297	513	1	c.	c.	PROPN
ejpam-6297	513	2	boonpok	boonpok	PROPN
ejpam-6297	513	3	,	,	PUNCT
ejpam-6297	513	4	p.	p.	PROPN
ejpam-6297	513	5	raktaow	raktaow	NOUN
ejpam-6297	513	6	,	,	PUNCT
ejpam-6297	513	7	a.	a.	PROPN
ejpam-6297	513	8	sama	sama	PROPN
ejpam-6297	513	9	-	-	PUNCT
ejpam-6297	513	10	ae	ae	PROPN
ejpam-6297	513	11	/	/	SYM
ejpam-6297	513	12	eur	eur	PROPN
ejpam-6297	513	13	.	.	PUNCT
ejpam-6297	514	1	j.	j.	PROPN
ejpam-6297	514	2	pure	pure	PROPN
ejpam-6297	514	3	appl	appl	PROPN
ejpam-6297	514	4	.	.	PROPN
ejpam-6297	514	5	math	math	PROPN
ejpam-6297	514	6	,	,	PUNCT
ejpam-6297	514	7	18	18	NUM
ejpam-6297	514	8	(	(	PUNCT
ejpam-6297	514	9	3	3	NUM
ejpam-6297	514	10	)	)	PUNCT
ejpam-6297	514	11	(	(	PUNCT
ejpam-6297	514	12	2025	2025	NUM
ejpam-6297	514	13	)	)	PUNCT
ejpam-6297	514	14	,	,	PUNCT
ejpam-6297	514	15	6297	6297	NUM
ejpam-6297	514	16	20	20	NUM
ejpam-6297	514	17	of	of	ADP
ejpam-6297	514	18	23	23	NUM
ejpam-6297	514	19	this	this	PRON
ejpam-6297	514	20	implies	imply	VERB
ejpam-6297	514	21	that	that	SCONJ
ejpam-6297	514	22	y	y	PROPN
ejpam-6297	514	23	−	−	PROPN
ejpam-6297	514	24	∪λ∈λvλ	∪λ∈λvλ	PROPN
ejpam-6297	514	25	=	=	PROPN
ejpam-6297	514	26	j	j	PROPN
ejpam-6297	514	27	for	for	ADP
ejpam-6297	514	28	some	some	DET
ejpam-6297	514	29	j	j	PROPN
ejpam-6297	514	30	∈	∈	PROPN
ejpam-6297	514	31	j	j	PROPN
ejpam-6297	514	32	,	,	PUNCT
ejpam-6297	514	33	which	which	PRON
ejpam-6297	514	34	means	mean	VERB
ejpam-6297	514	35	f−1(y	f−1(y	PROPN
ejpam-6297	514	36	)	)	PUNCT
ejpam-6297	515	1	−	−	PROPN
ejpam-6297	515	2	∪λ∈λf	∪λ∈λf	ADJ
ejpam-6297	515	3	−1(vλ	−1(vλ	NOUN
ejpam-6297	515	4	)	)	PUNCT
ejpam-6297	515	5	=	=	SYM
ejpam-6297	515	6	f−1(y	f−1(y	PROPN
ejpam-6297	515	7	)	)	PUNCT
ejpam-6297	515	8	−	−	PROPN
ejpam-6297	515	9	f−1(∪λ∈λvλ	f−1(∪λ∈λvλ	PROPN
ejpam-6297	515	10	)	)	PUNCT
ejpam-6297	515	11	=	=	SYM
ejpam-6297	515	12	f−1(j	f−1(j	PROPN
ejpam-6297	515	13	)	)	PUNCT
ejpam-6297	515	14	.	.	PUNCT
ejpam-6297	516	1	hence	hence	ADV
ejpam-6297	516	2	,	,	PUNCT
ejpam-6297	516	3	x	x	PUNCT
ejpam-6297	516	4	−	−	PRON
ejpam-6297	516	5	∪λ∈λf	∪λ∈λf	ADJ
ejpam-6297	516	6	−1(vλ	−1(vλ	NOUN
ejpam-6297	516	7	)	)	PUNCT
ejpam-6297	516	8	∈	∈	PROPN
ejpam-6297	516	9	f−1(j	f−1(j	NOUN
ejpam-6297	516	10	)	)	PUNCT
ejpam-6297	516	11	.	.	PUNCT
ejpam-6297	517	1	let	let	VERB
ejpam-6297	517	2	i	i	PRON
ejpam-6297	517	3	=	=	PUNCT
ejpam-6297	517	4	f−1(j	f−1(j	PROPN
ejpam-6297	517	5	)	)	PUNCT
ejpam-6297	517	6	.	.	PUNCT
ejpam-6297	518	1	since	since	SCONJ
ejpam-6297	518	2	f	f	PROPN
ejpam-6297	518	3	is	be	AUX
ejpam-6297	518	4	sβ	sβ	PROPN
ejpam-6297	518	5	-	-	PUNCT
ejpam-6297	518	6	i	i	NOUN
ejpam-6297	518	7	-	-	PUNCT
ejpam-6297	518	8	irresolute	irresolute	PROPN
ejpam-6297	518	9	,	,	PUNCT
ejpam-6297	518	10	the	the	DET
ejpam-6297	518	11	collection	collection	NOUN
ejpam-6297	518	12	c	c	NOUN
ejpam-6297	518	13	=	=	SYM
ejpam-6297	518	14	{	{	PUNCT
ejpam-6297	518	15	f−1(vλ	f−1(vλ	X
ejpam-6297	518	16	)	)	PUNCT
ejpam-6297	518	17	:	:	PUNCT
ejpam-6297	519	1	λ	λ	X
ejpam-6297	519	2	∈	∈	PROPN
ejpam-6297	519	3	λ	λ	PROPN
ejpam-6297	519	4	}	}	PUNCT
ejpam-6297	519	5	forms	form	VERB
ejpam-6297	519	6	an	an	DET
ejpam-6297	519	7	sβ	sβ	NOUN
ejpam-6297	519	8	-	-	PUNCT
ejpam-6297	519	9	i	i	NOUN
ejpam-6297	519	10	-	-	PUNCT
ejpam-6297	519	11	locally	locally	ADV
ejpam-6297	519	12	finite	finite	ADJ
ejpam-6297	519	13	collection	collection	NOUN
ejpam-6297	519	14	of	of	ADP
ejpam-6297	519	15	strong	strong	ADJ
ejpam-6297	519	16	β	β	X
ejpam-6297	519	17	-	-	ADJ
ejpam-6297	519	18	i	i	NOUN
ejpam-6297	519	19	-	-	PUNCT
ejpam-6297	519	20	open	open	ADJ
ejpam-6297	519	21	sets	set	NOUN
ejpam-6297	519	22	.	.	PUNCT
ejpam-6297	520	1	for	for	ADP
ejpam-6297	520	2	each	each	DET
ejpam-6297	520	3	f−1(vλ	f−1(vλ	NOUN
ejpam-6297	520	4	)	)	PUNCT
ejpam-6297	520	5	∈	∈	NOUN
ejpam-6297	520	6	c	c	X
ejpam-6297	520	7	,	,	PUNCT
ejpam-6297	520	8	since	since	SCONJ
ejpam-6297	520	9	vλ	vλ	ADP
ejpam-6297	520	10	∈	∈	PROPN
ejpam-6297	520	11	b	b	NOUN
ejpam-6297	520	12	,	,	PUNCT
ejpam-6297	520	13	there	there	PRON
ejpam-6297	520	14	exists	exist	VERB
ejpam-6297	520	15	uλ	uλ	ADP
ejpam-6297	520	16	∈	∈	PROPN
ejpam-6297	520	17	a	a	DET
ejpam-6297	520	18	such	such	ADJ
ejpam-6297	520	19	that	that	SCONJ
ejpam-6297	520	20	vλ	vλ	ADP
ejpam-6297	520	21	⊆	⊆	NUM
ejpam-6297	520	22	f(uλ	f(uλ	NOUN
ejpam-6297	520	23	)	)	PUNCT
ejpam-6297	520	24	as	as	ADP
ejpam-6297	520	25	b	b	NOUN
ejpam-6297	520	26	refines	refine	VERB
ejpam-6297	520	27	f(a	f(a	NOUN
ejpam-6297	520	28	)	)	PUNCT
ejpam-6297	520	29	.	.	PUNCT
ejpam-6297	521	1	thus	thus	ADV
ejpam-6297	521	2	,	,	PUNCT
ejpam-6297	521	3	f−1(vλ	f−1(vλ	NOUN
ejpam-6297	521	4	)	)	PUNCT
ejpam-6297	521	5	⊆	⊆	NUM
ejpam-6297	521	6	f−1(f(uλ	f−1(f(uλ	NOUN
ejpam-6297	521	7	)	)	PUNCT
ejpam-6297	521	8	)	)	PUNCT
ejpam-6297	521	9	=	=	SYM
ejpam-6297	521	10	uλ	uλ	NOUN
ejpam-6297	521	11	.	.	PUNCT
ejpam-6297	522	1	the	the	DET
ejpam-6297	522	2	refinement	refinement	NOUN
ejpam-6297	522	3	of	of	ADP
ejpam-6297	522	4	a	a	PRON
ejpam-6297	522	5	by	by	ADP
ejpam-6297	522	6	c	c	PROPN
ejpam-6297	522	7	is	be	AUX
ejpam-6297	522	8	then	then	ADV
ejpam-6297	522	9	asserted	assert	VERB
ejpam-6297	522	10	.	.	PUNCT
ejpam-6297	523	1	therefore	therefore	ADV
ejpam-6297	523	2	,	,	PUNCT
ejpam-6297	523	3	(	(	PUNCT
ejpam-6297	523	4	x	x	X
ejpam-6297	523	5	,	,	PUNCT
ejpam-6297	523	6	τ	τ	PROPN
ejpam-6297	523	7	,	,	PUNCT
ejpam-6297	523	8	i	i	PROPN
ejpam-6297	523	9	)	)	PUNCT
ejpam-6297	523	10	is	be	AUX
ejpam-6297	523	11	shown	show	VERB
ejpam-6297	523	12	to	to	PART
ejpam-6297	523	13	be	be	AUX
ejpam-6297	523	14	sβ	sβ	NOUN
ejpam-6297	523	15	-	-	PUNCT
ejpam-6297	523	16	iparacompact	iparacompact	ADJ
ejpam-6297	523	17	.	.	PUNCT
ejpam-6297	524	1	5	5	X
ejpam-6297	524	2	.	.	X
ejpam-6297	524	3	conclusions	conclusion	NOUN
ejpam-6297	524	4	conclusions	conclusion	NOUN
ejpam-6297	524	5	:	:	PUNCT
ejpam-6297	524	6	this	this	DET
ejpam-6297	524	7	paper	paper	NOUN
ejpam-6297	524	8	introduces	introduce	NOUN
ejpam-6297	524	9	and	and	CCONJ
ejpam-6297	524	10	investigates	investigate	VERB
ejpam-6297	524	11	the	the	DET
ejpam-6297	524	12	concept	concept	NOUN
ejpam-6297	524	13	of	of	ADP
ejpam-6297	524	14	strong	strong	ADJ
ejpam-6297	524	15	β	β	X
ejpam-6297	524	16	-	-	ADJ
ejpam-6297	524	17	i	i	NOUN
ejpam-6297	524	18	-	-	PUNCT
ejpam-6297	524	19	submaximal	submaximal	ADJ
ejpam-6297	524	20	ideal	ideal	ADJ
ejpam-6297	524	21	topological	topological	ADJ
ejpam-6297	524	22	spaces	space	NOUN
ejpam-6297	524	23	,	,	PUNCT
ejpam-6297	524	24	which	which	PRON
ejpam-6297	524	25	generalizes	generalize	VERB
ejpam-6297	524	26	submaximality	submaximality	NOUN
ejpam-6297	524	27	in	in	ADP
ejpam-6297	524	28	the	the	DET
ejpam-6297	524	29	context	context	NOUN
ejpam-6297	524	30	of	of	ADP
ejpam-6297	524	31	ideal	ideal	ADJ
ejpam-6297	524	32	topology	topology	NOUN
ejpam-6297	524	33	.	.	PUNCT
ejpam-6297	525	1	we	we	PRON
ejpam-6297	525	2	have	have	AUX
ejpam-6297	525	3	demonstrated	demonstrate	VERB
ejpam-6297	525	4	that	that	SCONJ
ejpam-6297	525	5	the	the	DET
ejpam-6297	525	6	following	follow	VERB
ejpam-6297	525	7	statements	statement	NOUN
ejpam-6297	525	8	are	be	AUX
ejpam-6297	525	9	equivalent	equivalent	ADJ
ejpam-6297	525	10	:	:	PUNCT
ejpam-6297	525	11	•	•	X
ejpam-6297	525	12	(	(	PUNCT
ejpam-6297	525	13	x	x	X
ejpam-6297	525	14	,	,	PUNCT
ejpam-6297	525	15	τ	τ	PROPN
ejpam-6297	525	16	,	,	PUNCT
ejpam-6297	525	17	i	i	NOUN
ejpam-6297	525	18	)	)	PUNCT
ejpam-6297	525	19	possesses	possess	VERB
ejpam-6297	525	20	the	the	DET
ejpam-6297	525	21	property	property	NOUN
ejpam-6297	525	22	of	of	ADP
ejpam-6297	525	23	strong	strong	ADJ
ejpam-6297	525	24	β	β	X
ejpam-6297	525	25	-	-	ADJ
ejpam-6297	525	26	i	i	NOUN
ejpam-6297	525	27	-	-	PUNCT
ejpam-6297	525	28	submaximality	submaximality	NOUN
ejpam-6297	525	29	.	.	PUNCT
ejpam-6297	526	1	•	•	NUM
ejpam-6297	526	2	for	for	ADP
ejpam-6297	526	3	every	every	DET
ejpam-6297	526	4	subset	subset	NOUN
ejpam-6297	526	5	a	a	PRON
ejpam-6297	526	6	of	of	ADP
ejpam-6297	526	7	x	x	PRON
ejpam-6297	526	8	,	,	PUNCT
ejpam-6297	526	9	the	the	DET
ejpam-6297	526	10	set	set	NOUN
ejpam-6297	526	11	sβ	sβ	NOUN
ejpam-6297	526	12	cli(a)−a	cli(a)−a	NOUN
ejpam-6297	526	13	is	be	AUX
ejpam-6297	526	14	strong	strong	ADJ
ejpam-6297	526	15	β	β	NOUN
ejpam-6297	526	16	-	-	ADJ
ejpam-6297	526	17	i	i	NOUN
ejpam-6297	526	18	-	-	PUNCT
ejpam-6297	526	19	closed	closed	ADJ
ejpam-6297	526	20	.	.	PUNCT
ejpam-6297	527	1	•	•	NOUN
ejpam-6297	527	2	each	each	DET
ejpam-6297	527	3	subset	subset	NOUN
ejpam-6297	527	4	of	of	ADP
ejpam-6297	527	5	x	x	PUNCT
ejpam-6297	527	6	is	be	AUX
ejpam-6297	527	7	locally	locally	ADV
ejpam-6297	527	8	strong	strong	ADJ
ejpam-6297	527	9	β	β	X
ejpam-6297	527	10	-	-	ADJ
ejpam-6297	527	11	i	i	NOUN
ejpam-6297	527	12	-	-	PUNCT
ejpam-6297	527	13	closed	closed	ADJ
ejpam-6297	527	14	.	.	PUNCT
ejpam-6297	528	1	•	•	NUM
ejpam-6297	528	2	every	every	DET
ejpam-6297	528	3	subset	subset	NOUN
ejpam-6297	528	4	of	of	ADP
ejpam-6297	528	5	x	x	PUNCT
ejpam-6297	528	6	qualifies	qualifie	NOUN
ejpam-6297	528	7	as	as	ADP
ejpam-6297	528	8	a	a	DET
ejpam-6297	528	9	b	b	NOUN
ejpam-6297	528	10	-	-	PUNCT
ejpam-6297	528	11	si	si	NOUN
ejpam-6297	528	12	set	set	NOUN
ejpam-6297	528	13	.	.	PUNCT
ejpam-6297	529	1	•	•	NUM
ejpam-6297	529	2	any	any	DET
ejpam-6297	529	3	strong	strong	ADJ
ejpam-6297	529	4	β	β	X
ejpam-6297	529	5	-	-	ADJ
ejpam-6297	529	6	i	i	NOUN
ejpam-6297	529	7	-	-	PUNCT
ejpam-6297	529	8	dense	dense	ADJ
ejpam-6297	529	9	subset	subset	NOUN
ejpam-6297	529	10	of	of	ADP
ejpam-6297	529	11	x	x	PUNCT
ejpam-6297	529	12	is	be	AUX
ejpam-6297	529	13	a	a	DET
ejpam-6297	529	14	b	b	PROPN
ejpam-6297	529	15	-	-	PUNCT
ejpam-6297	529	16	si	si	NOUN
ejpam-6297	529	17	set	set	NOUN
ejpam-6297	529	18	.	.	PUNCT
ejpam-6297	530	1	•	•	NUM
ejpam-6297	530	2	every	every	DET
ejpam-6297	530	3	subset	subset	NOUN
ejpam-6297	530	4	of	of	ADP
ejpam-6297	530	5	x	x	PUNCT
ejpam-6297	530	6	is	be	AUX
ejpam-6297	530	7	co	co	ADJ
ejpam-6297	530	8	-	-	ADJ
ejpam-6297	530	9	locally	locally	ADV
ejpam-6297	530	10	strong	strong	ADJ
ejpam-6297	530	11	β	β	X
ejpam-6297	530	12	-	-	ADJ
ejpam-6297	530	13	i	i	NOUN
ejpam-6297	530	14	-	-	PUNCT
ejpam-6297	530	15	closed	closed	ADJ
ejpam-6297	530	16	.	.	PUNCT
ejpam-6297	531	1	•	•	INTJ
ejpam-6297	531	2	if	if	SCONJ
ejpam-6297	531	3	a	a	DET
ejpam-6297	531	4	subset	subset	NOUN
ejpam-6297	531	5	a	a	PRON
ejpam-6297	531	6	of	of	ADP
ejpam-6297	531	7	x	x	NOUN
ejpam-6297	531	8	satisfies	satisfie	NOUN
ejpam-6297	531	9	sβ	sβ	X
ejpam-6297	531	10	inti(a	inti(a	PROPN
ejpam-6297	531	11	)	)	PUNCT
ejpam-6297	531	12	=	=	SYM
ejpam-6297	531	13	∅	∅	NOUN
ejpam-6297	531	14	,	,	PUNCT
ejpam-6297	531	15	then	then	ADV
ejpam-6297	531	16	it	it	PRON
ejpam-6297	531	17	is	be	AUX
ejpam-6297	531	18	strong	strong	ADJ
ejpam-6297	531	19	β	β	NOUN
ejpam-6297	531	20	-	-	ADJ
ejpam-6297	531	21	i	i	NOUN
ejpam-6297	531	22	-	-	PUNCT
ejpam-6297	531	23	closed	closed	ADJ
ejpam-6297	531	24	.	.	PUNCT
ejpam-6297	532	1	•	•	NOUN
ejpam-6297	532	2	all	all	PRON
ejpam-6297	532	3	strong	strong	ADJ
ejpam-6297	532	4	β	β	NOUN
ejpam-6297	532	5	-	-	ADJ
ejpam-6297	532	6	i	i	NOUN
ejpam-6297	532	7	-	-	PUNCT
ejpam-6297	532	8	codense	codense	NOUN
ejpam-6297	532	9	subsets	subset	NOUN
ejpam-6297	532	10	of	of	ADP
ejpam-6297	532	11	x	x	SYM
ejpam-6297	532	12	are	be	AUX
ejpam-6297	532	13	strong	strong	ADJ
ejpam-6297	532	14	β	β	NOUN
ejpam-6297	532	15	-	-	ADJ
ejpam-6297	532	16	i	i	NOUN
ejpam-6297	532	17	-	-	PUNCT
ejpam-6297	532	18	closed	closed	ADJ
ejpam-6297	532	19	.	.	PUNCT
ejpam-6297	533	1	•	•	NOUN
ejpam-6297	533	2	every	every	PRON
ejpam-6297	533	3	subset	subset	VERB
ejpam-6297	533	4	a	a	PRON
ejpam-6297	533	5	of	of	ADP
ejpam-6297	533	6	x	x	PUNCT
ejpam-6297	533	7	for	for	ADP
ejpam-6297	533	8	which	which	PRON
ejpam-6297	533	9	sβ	sβ	NUM
ejpam-6297	533	10	inti(a	inti(a	NOUN
ejpam-6297	533	11	)	)	PUNCT
ejpam-6297	533	12	=	=	NOUN
ejpam-6297	533	13	∅	∅	NOUN
ejpam-6297	533	14	holds	hold	VERB
ejpam-6297	533	15	is	be	AUX
ejpam-6297	533	16	both	both	PRON
ejpam-6297	533	17	strong	strong	ADJ
ejpam-6297	533	18	β	β	X
ejpam-6297	533	19	-	-	ADJ
ejpam-6297	533	20	i	i	NOUN
ejpam-6297	533	21	-	-	PUNCT
ejpam-6297	533	22	closed	closed	ADJ
ejpam-6297	533	23	and	and	CCONJ
ejpam-6297	533	24	strong	strong	ADJ
ejpam-6297	533	25	β	β	X
ejpam-6297	533	26	-	-	ADJ
ejpam-6297	533	27	i	i	NOUN
ejpam-6297	533	28	-	-	PUNCT
ejpam-6297	533	29	discrete	discrete	NOUN
ejpam-6297	533	30	.	.	PUNCT
ejpam-6297	534	1	•	•	NUM
ejpam-6297	534	2	for	for	ADP
ejpam-6297	534	3	any	any	DET
ejpam-6297	534	4	subset	subset	NOUN
ejpam-6297	534	5	a	a	PRON
ejpam-6297	534	6	of	of	ADP
ejpam-6297	534	7	x	x	PRON
ejpam-6297	534	8	,	,	PUNCT
ejpam-6297	534	9	the	the	DET
ejpam-6297	534	10	set	set	NOUN
ejpam-6297	534	11	sβ	sβ	PROPN
ejpam-6297	534	12	cli(a	cli(a	PROPN
ejpam-6297	534	13	)	)	PUNCT
ejpam-6297	534	14	−	−	PROPN
ejpam-6297	535	1	a	a	PRON
ejpam-6297	535	2	is	be	AUX
ejpam-6297	535	3	both	both	PRON
ejpam-6297	535	4	strong	strong	ADJ
ejpam-6297	535	5	β	β	X
ejpam-6297	535	6	-	-	ADJ
ejpam-6297	535	7	i	i	NOUN
ejpam-6297	535	8	-	-	PUNCT
ejpam-6297	535	9	closed	closed	ADJ
ejpam-6297	535	10	and	and	CCONJ
ejpam-6297	535	11	strong	strong	ADJ
ejpam-6297	535	12	β	β	X
ejpam-6297	535	13	-	-	ADJ
ejpam-6297	535	14	i	i	NOUN
ejpam-6297	535	15	-	-	PUNCT
ejpam-6297	535	16	discrete	discrete	NOUN
ejpam-6297	535	17	.	.	PUNCT
ejpam-6297	536	1	•	•	NOUN
ejpam-6297	536	2	each	each	DET
ejpam-6297	536	3	strong	strong	ADJ
ejpam-6297	536	4	β	β	X
ejpam-6297	536	5	-	-	ADJ
ejpam-6297	536	6	i	i	NOUN
ejpam-6297	536	7	-	-	PUNCT
ejpam-6297	536	8	codense	codense	NOUN
ejpam-6297	536	9	subset	subset	NOUN
ejpam-6297	536	10	of	of	ADP
ejpam-6297	536	11	x	x	PUNCT
ejpam-6297	536	12	is	be	AUX
ejpam-6297	536	13	strong	strong	ADJ
ejpam-6297	536	14	β	β	NOUN
ejpam-6297	536	15	-	-	ADJ
ejpam-6297	536	16	i	i	NOUN
ejpam-6297	536	17	-	-	PUNCT
ejpam-6297	536	18	closed	closed	ADJ
ejpam-6297	536	19	and	and	CCONJ
ejpam-6297	536	20	strong	strong	ADJ
ejpam-6297	537	1	β	β	X
ejpam-6297	537	2	-	-	ADJ
ejpam-6297	537	3	i	i	NOUN
ejpam-6297	537	4	-	-	PUNCT
ejpam-6297	537	5	discrete	discrete	NOUN
ejpam-6297	537	6	.	.	PUNCT
ejpam-6297	538	1	c.	c.	PROPN
ejpam-6297	538	2	boonpok	boonpok	PROPN
ejpam-6297	538	3	,	,	PUNCT
ejpam-6297	538	4	p.	p.	PROPN
ejpam-6297	538	5	raktaow	raktaow	NOUN
ejpam-6297	538	6	,	,	PUNCT
ejpam-6297	538	7	a.	a.	PROPN
ejpam-6297	538	8	sama	sama	PROPN
ejpam-6297	538	9	-	-	PUNCT
ejpam-6297	538	10	ae	ae	PROPN
ejpam-6297	538	11	/	/	SYM
ejpam-6297	538	12	eur	eur	PROPN
ejpam-6297	538	13	.	.	PUNCT
ejpam-6297	539	1	j.	j.	PROPN
ejpam-6297	539	2	pure	pure	PROPN
ejpam-6297	539	3	appl	appl	PROPN
ejpam-6297	539	4	.	.	PROPN
ejpam-6297	539	5	math	math	PROPN
ejpam-6297	539	6	,	,	PUNCT
ejpam-6297	539	7	18	18	NUM
ejpam-6297	539	8	(	(	PUNCT
ejpam-6297	539	9	3	3	NUM
ejpam-6297	539	10	)	)	PUNCT
ejpam-6297	539	11	(	(	PUNCT
ejpam-6297	539	12	2025	2025	NUM
ejpam-6297	539	13	)	)	PUNCT
ejpam-6297	539	14	,	,	PUNCT
ejpam-6297	539	15	6297	6297	NUM
ejpam-6297	539	16	21	21	NUM
ejpam-6297	539	17	of	of	ADP
ejpam-6297	539	18	23	23	NUM
ejpam-6297	539	19	this	this	DET
ejpam-6297	539	20	study	study	NOUN
ejpam-6297	539	21	explores	explore	VERB
ejpam-6297	539	22	various	various	ADJ
ejpam-6297	539	23	characterizations	characterization	NOUN
ejpam-6297	539	24	of	of	ADP
ejpam-6297	539	25	sβ	sβ	PROPN
ejpam-6297	539	26	-	-	PUNCT
ejpam-6297	539	27	i	i	NOUN
ejpam-6297	539	28	-	-	NOUN
ejpam-6297	539	29	paracompactness	paracompactness	PROPN
ejpam-6297	539	30	in	in	ADP
ejpam-6297	539	31	ideal	ideal	ADJ
ejpam-6297	539	32	topological	topological	ADJ
ejpam-6297	539	33	spaces	space	NOUN
ejpam-6297	539	34	,	,	PUNCT
ejpam-6297	539	35	highlighting	highlight	VERB
ejpam-6297	539	36	it	it	PRON
ejpam-6297	539	37	as	as	ADP
ejpam-6297	539	38	a	a	DET
ejpam-6297	539	39	stronger	strong	ADJ
ejpam-6297	539	40	variant	variant	NOUN
ejpam-6297	539	41	of	of	ADP
ejpam-6297	539	42	β	β	NOUN
ejpam-6297	539	43	-	-	NOUN
ejpam-6297	539	44	paracompactness	paracompactness	NOUN
ejpam-6297	539	45	.	.	PUNCT
ejpam-6297	540	1	it	it	PRON
ejpam-6297	540	2	is	be	AUX
ejpam-6297	540	3	established	establish	VERB
ejpam-6297	540	4	that	that	SCONJ
ejpam-6297	540	5	every	every	DET
ejpam-6297	540	6	sβ	sβ	PROPN
ejpam-6297	540	7	-	-	PUNCT
ejpam-6297	540	8	i	i	NOUN
ejpam-6297	540	9	-	-	PUNCT
ejpam-6297	540	10	paracompact	paracompact	ADJ
ejpam-6297	540	11	space	space	NOUN
ejpam-6297	540	12	is	be	AUX
ejpam-6297	540	13	necessarily	necessarily	ADV
ejpam-6297	540	14	i	i	NOUN
ejpam-6297	540	15	-	-	PUNCT
ejpam-6297	540	16	β	β	NOUN
ejpam-6297	540	17	-	-	NOUN
ejpam-6297	540	18	paracompact	paracompact	ADJ
ejpam-6297	540	19	,	,	PUNCT
ejpam-6297	540	20	and	and	CCONJ
ejpam-6297	540	21	in	in	ADP
ejpam-6297	540	22	the	the	DET
ejpam-6297	540	23	case	case	NOUN
ejpam-6297	540	24	of	of	ADP
ejpam-6297	540	25	hausdorff	hausdorff	NOUN
ejpam-6297	540	26	spaces	space	NOUN
ejpam-6297	540	27	,	,	PUNCT
ejpam-6297	540	28	sβ	sβ	PROPN
ejpam-6297	540	29	-	-	PUNCT
ejpam-6297	540	30	i	i	PROPN
ejpam-6297	540	31	-	-	PUNCT
ejpam-6297	540	32	paracompactness	paracompactness	PROPN
ejpam-6297	540	33	implies	imply	VERB
ejpam-6297	540	34	sβ	sβ	PROPN
ejpam-6297	540	35	-	-	PUNCT
ejpam-6297	540	36	i	i	NOUN
ejpam-6297	540	37	-	-	PUNCT
ejpam-6297	540	38	regularity	regularity	NOUN
ejpam-6297	540	39	.	.	PUNCT
ejpam-6297	541	1	moreover	moreover	ADV
ejpam-6297	541	2	,	,	PUNCT
ejpam-6297	541	3	it	it	PRON
ejpam-6297	541	4	is	be	AUX
ejpam-6297	541	5	shown	show	VERB
ejpam-6297	541	6	that	that	SCONJ
ejpam-6297	541	7	the	the	DET
ejpam-6297	541	8	union	union	NOUN
ejpam-6297	541	9	of	of	ADP
ejpam-6297	541	10	two	two	NUM
ejpam-6297	541	11	sβ	sβ	PROPN
ejpam-6297	541	12	-	-	PUNCT
ejpam-6297	541	13	i	i	NOUN
ejpam-6297	541	14	-	-	PUNCT
ejpam-6297	541	15	paracompact	paracompact	ADJ
ejpam-6297	541	16	subsets	subset	NOUN
ejpam-6297	541	17	remains	remain	VERB
ejpam-6297	541	18	sβ	sβ	ADJ
ejpam-6297	541	19	-	-	PUNCT
ejpam-6297	541	20	iparacompact	iparacompact	NOUN
ejpam-6297	541	21	,	,	PUNCT
ejpam-6297	541	22	and	and	CCONJ
ejpam-6297	541	23	the	the	DET
ejpam-6297	541	24	intersection	intersection	NOUN
ejpam-6297	541	25	of	of	ADP
ejpam-6297	541	26	an	an	DET
ejpam-6297	541	27	sβ	sβ	NOUN
ejpam-6297	541	28	-	-	PUNCT
ejpam-6297	541	29	i	i	NOUN
ejpam-6297	541	30	-	-	PUNCT
ejpam-6297	541	31	paracompact	paracompact	NOUN
ejpam-6297	541	32	subset	subset	NOUN
ejpam-6297	541	33	with	with	ADP
ejpam-6297	541	34	a	a	DET
ejpam-6297	541	35	closed	closed	ADJ
ejpam-6297	541	36	set	set	NOUN
ejpam-6297	541	37	also	also	ADV
ejpam-6297	541	38	retains	retain	VERB
ejpam-6297	541	39	the	the	DET
ejpam-6297	541	40	property	property	NOUN
ejpam-6297	541	41	.	.	PUNCT
ejpam-6297	542	1	the	the	DET
ejpam-6297	542	2	work	work	NOUN
ejpam-6297	542	3	further	far	ADV
ejpam-6297	542	4	demonstrates	demonstrate	VERB
ejpam-6297	542	5	the	the	DET
ejpam-6297	542	6	preservation	preservation	NOUN
ejpam-6297	542	7	of	of	ADP
ejpam-6297	542	8	sβ	sβ	PROPN
ejpam-6297	542	9	-	-	PUNCT
ejpam-6297	542	10	i	i	NOUN
ejpam-6297	542	11	-	-	NOUN
ejpam-6297	542	12	paracompactness	paracompactness	NOUN
ejpam-6297	542	13	under	under	ADP
ejpam-6297	542	14	certain	certain	ADJ
ejpam-6297	542	15	mappings	mapping	NOUN
ejpam-6297	542	16	.	.	PUNCT
ejpam-6297	543	1	specifically	specifically	ADV
ejpam-6297	543	2	,	,	PUNCT
ejpam-6297	543	3	if	if	SCONJ
ejpam-6297	543	4	a	a	DET
ejpam-6297	543	5	map	map	NOUN
ejpam-6297	543	6	f	f	X
ejpam-6297	543	7	:	:	PUNCT
ejpam-6297	543	8	x	x	X
ejpam-6297	543	9	→	→	SYM
ejpam-6297	543	10	y	y	PROPN
ejpam-6297	543	11	is	be	AUX
ejpam-6297	543	12	continuous	continuous	ADJ
ejpam-6297	543	13	,	,	PUNCT
ejpam-6297	543	14	sβ	sβ	PROPN
ejpam-6297	543	15	-	-	PUNCT
ejpam-6297	543	16	i	i	NOUN
ejpam-6297	543	17	-	-	PUNCT
ejpam-6297	543	18	open	open	ADJ
ejpam-6297	543	19	,	,	PUNCT
ejpam-6297	543	20	sβ	sβ	PROPN
ejpam-6297	543	21	-	-	PUNCT
ejpam-6297	543	22	i	i	NOUN
ejpam-6297	543	23	-	-	PUNCT
ejpam-6297	543	24	closed	closed	ADJ
ejpam-6297	543	25	,	,	PUNCT
ejpam-6297	543	26	and	and	CCONJ
ejpam-6297	543	27	surjective	surjective	ADJ
ejpam-6297	543	28	,	,	PUNCT
ejpam-6297	543	29	with	with	ADP
ejpam-6297	543	30	each	each	DET
ejpam-6297	543	31	fiber	fiber	NOUN
ejpam-6297	543	32	{	{	PUNCT
ejpam-6297	543	33	f−1(y	f−1(y	PROPN
ejpam-6297	543	34	)	)	PUNCT
ejpam-6297	543	35	}	}	PUNCT
ejpam-6297	543	36	being	be	AUX
ejpam-6297	543	37	sβ	sβ	VERB
ejpam-6297	543	38	-	-	PUNCT
ejpam-6297	543	39	i	i	NOUN
ejpam-6297	543	40	-	-	NOUN
ejpam-6297	543	41	compact	compact	ADJ
ejpam-6297	543	42	,	,	PUNCT
ejpam-6297	543	43	and	and	CCONJ
ejpam-6297	543	44	if	if	SCONJ
ejpam-6297	543	45	f(i	f(i	PROPN
ejpam-6297	543	46	)	)	PUNCT
ejpam-6297	543	47	⊆	⊆	NUM
ejpam-6297	543	48	j	j	PROPN
ejpam-6297	543	49	,	,	PUNCT
ejpam-6297	543	50	then	then	ADV
ejpam-6297	543	51	sβ	sβ	PROPN
ejpam-6297	543	52	-	-	PUNCT
ejpam-6297	543	53	i	i	NOUN
ejpam-6297	543	54	-	-	NOUN
ejpam-6297	543	55	paracompactness	paracompactness	NOUN
ejpam-6297	543	56	of	of	ADP
ejpam-6297	543	57	x	x	PROPN
ejpam-6297	543	58	implies	imply	VERB
ejpam-6297	543	59	the	the	DET
ejpam-6297	543	60	same	same	ADJ
ejpam-6297	543	61	for	for	ADP
ejpam-6297	543	62	y	y	PROPN
ejpam-6297	543	63	.	.	PUNCT
ejpam-6297	544	1	similarly	similarly	ADV
ejpam-6297	544	2	,	,	PUNCT
ejpam-6297	544	3	if	if	SCONJ
ejpam-6297	544	4	f	f	PROPN
ejpam-6297	544	5	is	be	AUX
ejpam-6297	544	6	sβ	sβ	PROPN
ejpam-6297	544	7	-	-	PUNCT
ejpam-6297	544	8	i	i	NOUN
ejpam-6297	544	9	-	-	PUNCT
ejpam-6297	544	10	irresolute	irresolute	ADJ
ejpam-6297	544	11	,	,	PUNCT
ejpam-6297	544	12	continuous	continuous	ADJ
ejpam-6297	544	13	,	,	PUNCT
ejpam-6297	544	14	sβ	sβ	PROPN
ejpam-6297	544	15	-	-	PUNCT
ejpam-6297	544	16	i	i	NOUN
ejpam-6297	544	17	-	-	PUNCT
ejpam-6297	544	18	open	open	ADJ
ejpam-6297	544	19	,	,	PUNCT
ejpam-6297	544	20	and	and	CCONJ
ejpam-6297	544	21	surjective	surjective	ADJ
ejpam-6297	544	22	,	,	PUNCT
ejpam-6297	544	23	and	and	CCONJ
ejpam-6297	544	24	for	for	ADP
ejpam-6297	544	25	every	every	DET
ejpam-6297	544	26	sβ	sβ	PROPN
ejpam-6297	544	27	-	-	PUNCT
ejpam-6297	544	28	i	i	NOUN
ejpam-6297	544	29	-	-	PUNCT
ejpam-6297	544	30	locally	locally	ADV
ejpam-6297	544	31	finite	finite	ADJ
ejpam-6297	544	32	family	family	PROPN
ejpam-6297	544	33	v	v	PROPN
ejpam-6297	544	34	,	,	PUNCT
ejpam-6297	544	35	the	the	DET
ejpam-6297	544	36	image	image	NOUN
ejpam-6297	544	37	f(v	f(v	NOUN
ejpam-6297	544	38	)	)	PUNCT
ejpam-6297	544	39	is	be	AUX
ejpam-6297	544	40	sβ	sβ	PROPN
ejpam-6297	544	41	-	-	PUNCT
ejpam-6297	544	42	j	j	NOUN
ejpam-6297	544	43	-locally	-locally	ADV
ejpam-6297	544	44	finite	finite	ADJ
ejpam-6297	544	45	,	,	PUNCT
ejpam-6297	544	46	then	then	ADV
ejpam-6297	544	47	the	the	DET
ejpam-6297	544	48	sβ	sβ	NOUN
ejpam-6297	544	49	-	-	PUNCT
ejpam-6297	544	50	iparacompactness	iparacompactness	NOUN
ejpam-6297	544	51	of	of	ADP
ejpam-6297	544	52	x	x	PUNCT
ejpam-6297	544	53	ensures	ensure	VERB
ejpam-6297	544	54	that	that	SCONJ
ejpam-6297	544	55	y	y	PROPN
ejpam-6297	544	56	is	be	AUX
ejpam-6297	544	57	sβ	sβ	PROPN
ejpam-6297	544	58	-	-	PUNCT
ejpam-6297	544	59	j	j	NOUN
ejpam-6297	544	60	-paracompact	-paracompact	PROPN
ejpam-6297	544	61	.	.	PUNCT
ejpam-6297	545	1	finally	finally	ADV
ejpam-6297	545	2	,	,	PUNCT
ejpam-6297	545	3	if	if	SCONJ
ejpam-6297	545	4	f	f	PROPN
ejpam-6297	545	5	is	be	AUX
ejpam-6297	545	6	open	open	ADJ
ejpam-6297	545	7	,	,	PUNCT
ejpam-6297	545	8	sβ	sβ	PROPN
ejpam-6297	545	9	-	-	PUNCT
ejpam-6297	545	10	i	i	NOUN
ejpam-6297	545	11	-	-	PUNCT
ejpam-6297	545	12	irresolute	irresolute	ADJ
ejpam-6297	545	13	,	,	PUNCT
ejpam-6297	545	14	and	and	CCONJ
ejpam-6297	545	15	bijective	bijective	ADJ
ejpam-6297	545	16	,	,	PUNCT
ejpam-6297	545	17	and	and	CCONJ
ejpam-6297	545	18	y	y	PROPN
ejpam-6297	545	19	is	be	AUX
ejpam-6297	545	20	sβ	sβ	PROPN
ejpam-6297	545	21	-	-	PUNCT
ejpam-6297	545	22	j	j	NOUN
ejpam-6297	545	23	-paracompact	-paracompact	PROPN
ejpam-6297	545	24	,	,	PUNCT
ejpam-6297	545	25	then	then	ADV
ejpam-6297	545	26	it	it	PRON
ejpam-6297	545	27	follows	follow	VERB
ejpam-6297	545	28	that	that	SCONJ
ejpam-6297	545	29	x	x	PRON
ejpam-6297	545	30	is	be	AUX
ejpam-6297	545	31	also	also	ADV
ejpam-6297	545	32	sβ	sβ	PROPN
ejpam-6297	545	33	-	-	PUNCT
ejpam-6297	545	34	i	i	NOUN
ejpam-6297	545	35	-	-	PUNCT
ejpam-6297	545	36	paracompact	paracompact	ADJ
ejpam-6297	545	37	.	.	PUNCT
ejpam-6297	546	1	applications	application	NOUN
ejpam-6297	546	2	:	:	PUNCT
ejpam-6297	546	3	strong	strong	ADJ
ejpam-6297	546	4	β	β	X
ejpam-6297	546	5	-	-	ADJ
ejpam-6297	546	6	i	i	NOUN
ejpam-6297	546	7	-	-	PUNCT
ejpam-6297	546	8	submaximality	submaximality	PROPN
ejpam-6297	546	9	and	and	CCONJ
ejpam-6297	546	10	paracompactness	paracompactness	NOUN
ejpam-6297	546	11	contribute	contribute	VERB
ejpam-6297	546	12	significantly	significantly	ADV
ejpam-6297	546	13	to	to	ADP
ejpam-6297	546	14	fields	field	NOUN
ejpam-6297	546	15	such	such	ADJ
ejpam-6297	546	16	as	as	ADP
ejpam-6297	546	17	digital	digital	ADJ
ejpam-6297	546	18	topology	topology	NOUN
ejpam-6297	546	19	,	,	PUNCT
ejpam-6297	546	20	computational	computational	ADJ
ejpam-6297	546	21	geometry	geometry	NOUN
ejpam-6297	546	22	,	,	PUNCT
ejpam-6297	546	23	and	and	CCONJ
ejpam-6297	546	24	data	data	VERB
ejpam-6297	546	25	analysis	analysis	NOUN
ejpam-6297	546	26	by	by	ADP
ejpam-6297	546	27	utilizing	utilize	VERB
ejpam-6297	546	28	generalized	generalize	VERB
ejpam-6297	546	29	open	open	ADJ
ejpam-6297	546	30	sets	set	NOUN
ejpam-6297	546	31	and	and	CCONJ
ejpam-6297	546	32	locally	locally	ADV
ejpam-6297	546	33	finite	finite	ADJ
ejpam-6297	546	34	covers	cover	NOUN
ejpam-6297	546	35	.	.	PUNCT
ejpam-6297	547	1	these	these	DET
ejpam-6297	547	2	properties	property	NOUN
ejpam-6297	547	3	broaden	broaden	VERB
ejpam-6297	547	4	the	the	DET
ejpam-6297	547	5	scope	scope	NOUN
ejpam-6297	547	6	of	of	ADP
ejpam-6297	547	7	continuity	continuity	NOUN
ejpam-6297	547	8	results	result	NOUN
ejpam-6297	547	9	within	within	ADP
ejpam-6297	547	10	ideal	ideal	ADJ
ejpam-6297	547	11	spaces	space	NOUN
ejpam-6297	547	12	and	and	CCONJ
ejpam-6297	547	13	find	find	VERB
ejpam-6297	547	14	relevance	relevance	NOUN
ejpam-6297	547	15	in	in	ADP
ejpam-6297	547	16	areas	area	NOUN
ejpam-6297	547	17	like	like	ADP
ejpam-6297	547	18	fuzzy	fuzzy	ADJ
ejpam-6297	547	19	and	and	CCONJ
ejpam-6297	547	20	soft	soft	ADJ
ejpam-6297	547	21	topology	topology	NOUN
ejpam-6297	547	22	,	,	PUNCT
ejpam-6297	547	23	effectively	effectively	ADV
ejpam-6297	547	24	linking	link	VERB
ejpam-6297	547	25	abstract	abstract	ADJ
ejpam-6297	547	26	topological	topological	ADJ
ejpam-6297	547	27	theory	theory	NOUN
ejpam-6297	547	28	with	with	ADP
ejpam-6297	547	29	computational	computational	ADJ
ejpam-6297	547	30	approaches	approach	NOUN
ejpam-6297	547	31	to	to	PART
ejpam-6297	547	32	support	support	VERB
ejpam-6297	547	33	better	well	ADJ
ejpam-6297	547	34	approximation	approximation	NOUN
ejpam-6297	547	35	techniques	technique	NOUN
ejpam-6297	547	36	and	and	CCONJ
ejpam-6297	547	37	structural	structural	ADJ
ejpam-6297	547	38	interpretations	interpretation	NOUN
ejpam-6297	547	39	.	.	PUNCT
ejpam-6297	548	1	future	future	ADJ
ejpam-6297	548	2	works	work	NOUN
ejpam-6297	548	3	:	:	PUNCT
ejpam-6297	548	4	potential	potential	ADJ
ejpam-6297	548	5	research	research	NOUN
ejpam-6297	548	6	avenues	avenue	NOUN
ejpam-6297	548	7	include	include	VERB
ejpam-6297	548	8	extending	extend	VERB
ejpam-6297	548	9	the	the	DET
ejpam-6297	548	10	concepts	concept	NOUN
ejpam-6297	548	11	to	to	PART
ejpam-6297	548	12	fuzzy	fuzzy	ADJ
ejpam-6297	548	13	and	and	CCONJ
ejpam-6297	548	14	neutrosophic	neutrosophic	ADJ
ejpam-6297	548	15	ideals	ideal	NOUN
ejpam-6297	548	16	,	,	PUNCT
ejpam-6297	548	17	exploring	explore	VERB
ejpam-6297	548	18	their	their	PRON
ejpam-6297	548	19	impact	impact	NOUN
ejpam-6297	548	20	on	on	ADP
ejpam-6297	548	21	separation	separation	NOUN
ejpam-6297	548	22	axioms	axiom	NOUN
ejpam-6297	548	23	and	and	CCONJ
ejpam-6297	548	24	product	product	NOUN
ejpam-6297	548	25	topologies	topology	NOUN
ejpam-6297	548	26	,	,	PUNCT
ejpam-6297	548	27	investigating	investigate	VERB
ejpam-6297	548	28	connections	connection	NOUN
ejpam-6297	548	29	with	with	ADP
ejpam-6297	548	30	measure	measure	NOUN
ejpam-6297	548	31	theory	theory	NOUN
ejpam-6297	548	32	,	,	PUNCT
ejpam-6297	548	33	analyzing	analyze	VERB
ejpam-6297	548	34	relationships	relationship	NOUN
ejpam-6297	548	35	among	among	ADP
ejpam-6297	548	36	various	various	ADJ
ejpam-6297	548	37	generalized	generalized	ADJ
ejpam-6297	548	38	open	open	ADJ
ejpam-6297	548	39	sets	set	NOUN
ejpam-6297	548	40	,	,	PUNCT
ejpam-6297	548	41	and	and	CCONJ
ejpam-6297	548	42	creating	create	VERB
ejpam-6297	548	43	algorithms	algorithm	NOUN
ejpam-6297	548	44	for	for	ADP
ejpam-6297	548	45	detecting	detect	VERB
ejpam-6297	548	46	these	these	DET
ejpam-6297	548	47	properties	property	NOUN
ejpam-6297	548	48	in	in	ADP
ejpam-6297	548	49	finite	finite	ADJ
ejpam-6297	548	50	spaces	space	NOUN
ejpam-6297	548	51	—	—	PUNCT
ejpam-6297	548	52	thereby	thereby	ADV
ejpam-6297	548	53	enriching	enrich	VERB
ejpam-6297	548	54	both	both	CCONJ
ejpam-6297	548	55	the	the	DET
ejpam-6297	548	56	theoretical	theoretical	ADJ
ejpam-6297	548	57	framework	framework	NOUN
ejpam-6297	548	58	and	and	CCONJ
ejpam-6297	548	59	real	real	ADJ
ejpam-6297	548	60	-	-	PUNCT
ejpam-6297	548	61	world	world	NOUN
ejpam-6297	548	62	applications	application	NOUN
ejpam-6297	548	63	.	.	PUNCT
ejpam-6297	549	1	acknowledgements	acknowledgement	NOUN
ejpam-6297	549	2	we	we	PRON
ejpam-6297	549	3	sincerely	sincerely	ADV
ejpam-6297	549	4	appreciate	appreciate	VERB
ejpam-6297	549	5	all	all	DET
ejpam-6297	549	6	those	those	PRON
ejpam-6297	549	7	who	who	PRON
ejpam-6297	549	8	contributed	contribute	VERB
ejpam-6297	549	9	to	to	ADP
ejpam-6297	549	10	our	our	PRON
ejpam-6297	549	11	research	research	NOUN
ejpam-6297	549	12	.	.	PUNCT
ejpam-6297	550	1	their	their	PRON
ejpam-6297	550	2	direction	direction	NOUN
ejpam-6297	550	3	,	,	PUNCT
ejpam-6297	550	4	collaboration	collaboration	NOUN
ejpam-6297	550	5	,	,	PUNCT
ejpam-6297	550	6	and	and	CCONJ
ejpam-6297	550	7	assistance	assistance	NOUN
ejpam-6297	550	8	have	have	AUX
ejpam-6297	550	9	been	be	AUX
ejpam-6297	550	10	essential	essential	ADJ
ejpam-6297	550	11	to	to	ADP
ejpam-6297	550	12	the	the	DET
ejpam-6297	550	13	success	success	NOUN
ejpam-6297	550	14	of	of	ADP
ejpam-6297	550	15	this	this	DET
ejpam-6297	550	16	project	project	NOUN
ejpam-6297	550	17	.	.	PUNCT
ejpam-6297	551	1	this	this	DET
ejpam-6297	551	2	research	research	NOUN
ejpam-6297	551	3	was	be	AUX
ejpam-6297	551	4	supported	support	VERB
ejpam-6297	551	5	by	by	ADP
ejpam-6297	551	6	the	the	DET
ejpam-6297	551	7	national	national	ADJ
ejpam-6297	551	8	science	science	NOUN
ejpam-6297	551	9	,	,	PUNCT
ejpam-6297	551	10	research	research	NOUN
ejpam-6297	551	11	,	,	PUNCT
ejpam-6297	551	12	and	and	CCONJ
ejpam-6297	551	13	innovation	innovation	NOUN
ejpam-6297	551	14	fund	fund	NOUN
ejpam-6297	551	15	(	(	PUNCT
ejpam-6297	551	16	nsrf	nsrf	NOUN
ejpam-6297	551	17	)	)	PUNCT
ejpam-6297	551	18	and	and	CCONJ
ejpam-6297	551	19	prince	prince	NOUN
ejpam-6297	551	20	of	of	ADP
ejpam-6297	551	21	songkla	songkla	PROPN
ejpam-6297	551	22	university	university	PROPN
ejpam-6297	551	23	(	(	PUNCT
ejpam-6297	551	24	ref	ref	NOUN
ejpam-6297	551	25	.	.	PUNCT
ejpam-6297	552	1	no	no	INTJ
ejpam-6297	552	2	.	.	PUNCT
ejpam-6297	552	3	sat6801325s	sat6801325s	NOUN
ejpam-6297	552	4	)	)	PUNCT
ejpam-6297	552	5	.	.	PUNCT
ejpam-6297	553	1	references	reference	NOUN
ejpam-6297	553	2	[	[	X
ejpam-6297	553	3	1	1	X
ejpam-6297	553	4	]	]	PUNCT
ejpam-6297	553	5	e.	e.	PROPN
ejpam-6297	553	6	d.	d.	PROPN
ejpam-6297	553	7	khalimsky	khalimsky	PROPN
ejpam-6297	553	8	,	,	PUNCT
ejpam-6297	553	9	r.	r.	PROPN
ejpam-6297	553	10	kopperman	kopperman	PROPN
ejpam-6297	553	11	,	,	PUNCT
ejpam-6297	553	12	and	and	CCONJ
ejpam-6297	554	1	p.	p.	PROPN
ejpam-6297	554	2	r.	r.	PROPN
ejpam-6297	554	3	meyer	meyer	PROPN
ejpam-6297	554	4	.	.	PUNCT
ejpam-6297	554	5	computer	computer	NOUN
ejpam-6297	554	6	graphics	graphic	NOUN
ejpam-6297	554	7	and	and	CCONJ
ejpam-6297	554	8	connected	connected	ADJ
ejpam-6297	554	9	topologies	topology	NOUN
ejpam-6297	554	10	on	on	ADP
ejpam-6297	554	11	finite	finite	NOUN
ejpam-6297	554	12	ordered	order	VERB
ejpam-6297	554	13	sets	set	NOUN
ejpam-6297	554	14	.	.	PUNCT
ejpam-6297	555	1	topology	topology	NOUN
ejpam-6297	555	2	and	and	CCONJ
ejpam-6297	555	3	its	its	PRON
ejpam-6297	555	4	applications	application	NOUN
ejpam-6297	555	5	,	,	PUNCT
ejpam-6297	555	6	36:1–17	36:1–17	NUM
ejpam-6297	555	7	,	,	PUNCT
ejpam-6297	555	8	1990	1990	NUM
ejpam-6297	555	9	.	.	PUNCT
ejpam-6297	556	1	[	[	X
ejpam-6297	556	2	2	2	X
ejpam-6297	556	3	]	]	PUNCT
ejpam-6297	556	4	t.	t.	PROPN
ejpam-6297	556	5	y.	y.	PROPN
ejpam-6297	556	6	kong	kong	PROPN
ejpam-6297	556	7	,	,	PUNCT
ejpam-6297	556	8	r.	r.	PROPN
ejpam-6297	556	9	kopperman	kopperman	PROPN
ejpam-6297	556	10	,	,	PUNCT
ejpam-6297	556	11	and	and	CCONJ
ejpam-6297	556	12	p.	p.	PROPN
ejpam-6297	556	13	r.	r.	PROPN
ejpam-6297	556	14	meyer	meyer	PROPN
ejpam-6297	556	15	.	.	PUNCT
ejpam-6297	557	1	a	a	DET
ejpam-6297	557	2	topological	topological	ADJ
ejpam-6297	557	3	approach	approach	NOUN
ejpam-6297	557	4	to	to	ADP
ejpam-6297	557	5	digital	digital	ADJ
ejpam-6297	557	6	topology	topology	NOUN
ejpam-6297	557	7	.	.	PUNCT
ejpam-6297	558	1	american	american	PROPN
ejpam-6297	558	2	mathematical	mathematical	PROPN
ejpam-6297	558	3	monthly	monthly	ADV
ejpam-6297	558	4	,	,	PUNCT
ejpam-6297	558	5	98:901–917	98:901–917	NUM
ejpam-6297	558	6	,	,	PUNCT
ejpam-6297	558	7	1991	1991	NUM
ejpam-6297	558	8	.	.	PUNCT
ejpam-6297	559	1	[	[	X
ejpam-6297	559	2	3	3	X
ejpam-6297	559	3	]	]	X
ejpam-6297	559	4	e.	e.	PROPN
ejpam-6297	559	5	moore	moore	PROPN
ejpam-6297	559	6	and	and	CCONJ
ejpam-6297	559	7	t.	t.	PROPN
ejpam-6297	559	8	petersy	petersy	NOUN
ejpam-6297	559	9	.	.	PUNCT
ejpam-6297	560	1	computational	computational	ADJ
ejpam-6297	560	2	topology	topology	NOUN
ejpam-6297	560	3	for	for	ADP
ejpam-6297	560	4	geometric	geometric	ADJ
ejpam-6297	560	5	design	design	NOUN
ejpam-6297	560	6	and	and	CCONJ
ejpam-6297	560	7	molecular	molecular	ADJ
ejpam-6297	560	8	design	design	NOUN
ejpam-6297	560	9	.	.	PUNCT
ejpam-6297	561	1	siam	siam	ADJ
ejpam-6297	561	2	proceedings	proceeding	NOUN
ejpam-6297	561	3	series	series	NOUN
ejpam-6297	561	4	,	,	PUNCT
ejpam-6297	561	5	1	1	NUM
ejpam-6297	561	6	,	,	PUNCT
ejpam-6297	561	7	2005	2005	NUM
ejpam-6297	561	8	.	.	PUNCT
ejpam-6297	562	1	c.	c.	PROPN
ejpam-6297	562	2	boonpok	boonpok	PROPN
ejpam-6297	562	3	,	,	PUNCT
ejpam-6297	562	4	p.	p.	PROPN
ejpam-6297	562	5	raktaow	raktaow	NOUN
ejpam-6297	562	6	,	,	PUNCT
ejpam-6297	562	7	a.	a.	PROPN
ejpam-6297	562	8	sama	sama	PROPN
ejpam-6297	562	9	-	-	PUNCT
ejpam-6297	562	10	ae	ae	PROPN
ejpam-6297	562	11	/	/	SYM
ejpam-6297	562	12	eur	eur	PROPN
ejpam-6297	562	13	.	.	PUNCT
ejpam-6297	563	1	j.	j.	PROPN
ejpam-6297	563	2	pure	pure	PROPN
ejpam-6297	563	3	appl	appl	PROPN
ejpam-6297	563	4	.	.	PROPN
ejpam-6297	563	5	math	math	PROPN
ejpam-6297	563	6	,	,	PUNCT
ejpam-6297	563	7	18	18	NUM
ejpam-6297	563	8	(	(	PUNCT
ejpam-6297	563	9	3	3	NUM
ejpam-6297	563	10	)	)	PUNCT
ejpam-6297	563	11	(	(	PUNCT
ejpam-6297	563	12	2025	2025	NUM
ejpam-6297	563	13	)	)	PUNCT
ejpam-6297	563	14	,	,	PUNCT
ejpam-6297	563	15	6297	6297	NUM
ejpam-6297	563	16	22	22	NUM
ejpam-6297	563	17	of	of	ADP
ejpam-6297	563	18	23	23	NUM
ejpam-6297	563	19	[	[	SYM
ejpam-6297	563	20	4	4	NUM
ejpam-6297	563	21	]	]	X
ejpam-6297	563	22	d.	d.	PROPN
ejpam-6297	563	23	w.	w.	PROPN
ejpam-6297	563	24	rosen	rosen	PROPN
ejpam-6297	563	25	and	and	CCONJ
ejpam-6297	563	26	t.	t.	PROPN
ejpam-6297	563	27	peters	peters	PROPN
ejpam-6297	563	28	.	.	PUNCT
ejpam-6297	564	1	the	the	DET
ejpam-6297	564	2	role	role	NOUN
ejpam-6297	564	3	of	of	ADP
ejpam-6297	564	4	topology	topology	NOUN
ejpam-6297	564	5	in	in	ADP
ejpam-6297	564	6	engineering	engineering	NOUN
ejpam-6297	564	7	design	design	NOUN
ejpam-6297	564	8	research	research	NOUN
ejpam-6297	564	9	.	.	PUNCT
ejpam-6297	565	1	research	research	NOUN
ejpam-6297	565	2	in	in	ADP
ejpam-6297	565	3	engineering	engineering	NOUN
ejpam-6297	565	4	design	design	NOUN
ejpam-6297	565	5	,	,	PUNCT
ejpam-6297	565	6	2:81–98	2:81–98	NUM
ejpam-6297	565	7	,	,	PUNCT
ejpam-6297	565	8	1996	1996	NUM
ejpam-6297	565	9	.	.	PUNCT
ejpam-6297	566	1	[	[	X
ejpam-6297	566	2	5	5	X
ejpam-6297	566	3	]	]	PUNCT
ejpam-6297	566	4	e.	e.	PROPN
ejpam-6297	566	5	hewitt	hewitt	PROPN
ejpam-6297	566	6	.	.	PUNCT
ejpam-6297	567	1	a	a	DET
ejpam-6297	567	2	problem	problem	NOUN
ejpam-6297	567	3	of	of	ADP
ejpam-6297	567	4	set	set	NOUN
ejpam-6297	567	5	-	-	PUNCT
ejpam-6297	567	6	theoretic	theoretic	NOUN
ejpam-6297	567	7	topology	topology	NOUN
ejpam-6297	567	8	.	.	PUNCT
ejpam-6297	568	1	duke	duke	PROPN
ejpam-6297	568	2	mathematical	mathematical	PROPN
ejpam-6297	568	3	journal	journal	PROPN
ejpam-6297	568	4	,	,	PUNCT
ejpam-6297	568	5	10(2):309–333	10(2):309–333	PROPN
ejpam-6297	568	6	,	,	PUNCT
ejpam-6297	568	7	1943	1943	NUM
ejpam-6297	568	8	.	.	PUNCT
ejpam-6297	569	1	[	[	X
ejpam-6297	569	2	6	6	NUM
ejpam-6297	569	3	]	]	X
ejpam-6297	569	4	n.	n.	NOUN
ejpam-6297	569	5	bourbaki	bourbaki	PROPN
ejpam-6297	569	6	.	.	PUNCT
ejpam-6297	570	1	éléments	éléments	PROPN
ejpam-6297	570	2	de	de	PROPN
ejpam-6297	570	3	mathématique	mathématique	PROPN
ejpam-6297	570	4	.	.	PROPN
ejpam-6297	570	5	topologie	topologie	PROPN
ejpam-6297	570	6	générale	générale	PROPN
ejpam-6297	570	7	(	(	PUNCT
ejpam-6297	570	8	3rd	3rd	ADJ
ejpam-6297	570	9	ed	ed	NOUN
ejpam-6297	570	10	.	.	PUNCT
ejpam-6297	570	11	)	)	PUNCT
ejpam-6297	570	12	,	,	PUNCT
ejpam-6297	570	13	hermann	hermann	PROPN
ejpam-6297	570	14	,	,	PUNCT
ejpam-6297	570	15	1961	1961	NUM
ejpam-6297	570	16	.	.	PUNCT
ejpam-6297	571	1	[	[	X
ejpam-6297	571	2	7	7	NUM
ejpam-6297	571	3	]	]	X
ejpam-6297	571	4	a.	a.	NOUN
ejpam-6297	571	5	v.	v.	PROPN
ejpam-6297	571	6	arhangel’skĭı	arhangel’skĭı	PROPN
ejpam-6297	571	7	and	and	CCONJ
ejpam-6297	571	8	p.	p.	PROPN
ejpam-6297	571	9	j.	j.	PROPN
ejpam-6297	571	10	collins	collins	PROPN
ejpam-6297	571	11	.	.	PUNCT
ejpam-6297	572	1	on	on	ADP
ejpam-6297	572	2	submaximal	submaximal	ADJ
ejpam-6297	572	3	spaces	space	NOUN
ejpam-6297	572	4	.	.	PUNCT
ejpam-6297	573	1	topology	topology	NOUN
ejpam-6297	573	2	and	and	CCONJ
ejpam-6297	573	3	its	its	PRON
ejpam-6297	573	4	applications	application	NOUN
ejpam-6297	573	5	,	,	PUNCT
ejpam-6297	573	6	64(3):219–241	64(3):219–241	PROPN
ejpam-6297	573	7	,	,	PUNCT
ejpam-6297	573	8	1995	1995	NUM
ejpam-6297	573	9	.	.	PUNCT
ejpam-6297	574	1	[	[	X
ejpam-6297	574	2	8	8	X
ejpam-6297	574	3	]	]	PUNCT
ejpam-6297	574	4	j.	j.	PROPN
ejpam-6297	574	5	dontchev	dontchev	PROPN
ejpam-6297	574	6	.	.	PUNCT
ejpam-6297	575	1	on	on	ADP
ejpam-6297	575	2	submaximal	submaximal	ADJ
ejpam-6297	575	3	spaces	space	NOUN
ejpam-6297	575	4	.	.	PUNCT
ejpam-6297	576	1	tamkang	tamkang	PROPN
ejpam-6297	576	2	journal	journal	PROPN
ejpam-6297	576	3	of	of	ADP
ejpam-6297	576	4	mathematics	mathematic	NOUN
ejpam-6297	576	5	,	,	PUNCT
ejpam-6297	576	6	26:243–250	26:243–250	PROPN
ejpam-6297	576	7	,	,	PUNCT
ejpam-6297	576	8	1995	1995	NUM
ejpam-6297	576	9	.	.	PUNCT
ejpam-6297	577	1	[	[	X
ejpam-6297	577	2	9	9	NUM
ejpam-6297	577	3	]	]	PUNCT
ejpam-6297	577	4	s.	s.	PROPN
ejpam-6297	577	5	tokgoz	tokgoz	PROPN
ejpam-6297	577	6	and	and	CCONJ
ejpam-6297	577	7	t.	t.	PROPN
ejpam-6297	577	8	h.	h.	PROPN
ejpam-6297	577	9	yalvac	yalvac	PROPN
ejpam-6297	577	10	.	.	PUNCT
ejpam-6297	578	1	some	some	DET
ejpam-6297	578	2	characterizations	characterization	NOUN
ejpam-6297	578	3	of	of	ADP
ejpam-6297	578	4	submaximality	submaximality	NOUN
ejpam-6297	578	5	.	.	PUNCT
ejpam-6297	579	1	carpathian	carpathian	ADJ
ejpam-6297	579	2	journal	journal	PROPN
ejpam-6297	579	3	of	of	ADP
ejpam-6297	579	4	mathematics	mathematic	NOUN
ejpam-6297	579	5	,	,	PUNCT
ejpam-6297	579	6	25(1):128–137	25(1):128–137	NUM
ejpam-6297	579	7	,	,	PUNCT
ejpam-6297	579	8	2009	2009	NUM
ejpam-6297	579	9	.	.	PUNCT
ejpam-6297	580	1	[	[	X
ejpam-6297	580	2	10	10	NUM
ejpam-6297	580	3	]	]	PUNCT
ejpam-6297	580	4	j.	j.	PROPN
ejpam-6297	580	5	dieudonné.	dieudonné.	PROPN
ejpam-6297	580	6	une	une	PROPN
ejpam-6297	580	7	généralisation	généralisation	PROPN
ejpam-6297	580	8	des	des	PROPN
ejpam-6297	580	9	espaces	espace	NOUN
ejpam-6297	580	10	compacts	compact	NOUN
ejpam-6297	580	11	.	.	PUNCT
ejpam-6297	581	1	journal	journal	PROPN
ejpam-6297	581	2	de	de	PROPN
ejpam-6297	581	3	mathématiques	mathématiques	PROPN
ejpam-6297	581	4	pures	pure	NOUN
ejpam-6297	581	5	et	et	NOUN
ejpam-6297	581	6	appliquées	appliquée	NOUN
ejpam-6297	581	7	,	,	PUNCT
ejpam-6297	581	8	23(9):65–76	23(9):65–76	NUM
ejpam-6297	581	9	,	,	PUNCT
ejpam-6297	581	10	1944	1944	NUM
ejpam-6297	581	11	.	.	PUNCT
ejpam-6297	582	1	[	[	X
ejpam-6297	582	2	11	11	NUM
ejpam-6297	582	3	]	]	PUNCT
ejpam-6297	582	4	j.	j.	PROPN
ejpam-6297	582	5	dugundji	dugundji	PROPN
ejpam-6297	582	6	.	.	PUNCT
ejpam-6297	582	7	topology	topology	PROPN
ejpam-6297	582	8	.	.	PUNCT
ejpam-6297	583	1	allyn	allyn	PROPN
ejpam-6297	583	2	and	and	CCONJ
ejpam-6297	583	3	bacon	bacon	PROPN
ejpam-6297	583	4	,	,	PUNCT
ejpam-6297	583	5	boston	boston	PROPN
ejpam-6297	583	6	,	,	PUNCT
ejpam-6297	583	7	1966	1966	NUM
ejpam-6297	583	8	.	.	PUNCT
ejpam-6297	584	1	[	[	X
ejpam-6297	584	2	12	12	NUM
ejpam-6297	584	3	]	]	PUNCT
ejpam-6297	584	4	k.	k.	PROPN
ejpam-6297	585	1	al	al	PROPN
ejpam-6297	585	2	-	-	PROPN
ejpam-6297	585	3	zoubi	zoubi	PROPN
ejpam-6297	585	4	.	.	PUNCT
ejpam-6297	586	1	s	s	X
ejpam-6297	586	2	-	-	PUNCT
ejpam-6297	586	3	paracompact	paracompact	ADJ
ejpam-6297	586	4	spaces	space	NOUN
ejpam-6297	586	5	.	.	PUNCT
ejpam-6297	587	1	acta	acta	PROPN
ejpam-6297	587	2	mathematica	mathematica	PROPN
ejpam-6297	587	3	hungarica	hungarica	PROPN
ejpam-6297	587	4	,	,	PUNCT
ejpam-6297	587	5	110:203–212	110:203–212	NUM
ejpam-6297	587	6	,	,	PUNCT
ejpam-6297	587	7	2006	2006	NUM
ejpam-6297	587	8	.	.	PUNCT
ejpam-6297	588	1	[	[	X
ejpam-6297	588	2	13	13	NUM
ejpam-6297	588	3	]	]	PUNCT
ejpam-6297	588	4	k.	k.	PROPN
ejpam-6297	589	1	al	al	PROPN
ejpam-6297	589	2	-	-	PROPN
ejpam-6297	589	3	zoubi	zoubi	PROPN
ejpam-6297	589	4	and	and	CCONJ
ejpam-6297	589	5	s.	s.	PROPN
ejpam-6297	589	6	al	al	PROPN
ejpam-6297	589	7	-	-	PROPN
ejpam-6297	589	8	ghour	ghour	PROPN
ejpam-6297	589	9	.	.	PUNCT
ejpam-6297	590	1	on	on	ADP
ejpam-6297	590	2	p3	p3	NOUN
ejpam-6297	590	3	-	-	PUNCT
ejpam-6297	590	4	paracompact	paracompact	ADJ
ejpam-6297	590	5	spaces	space	NOUN
ejpam-6297	590	6	.	.	PUNCT
ejpam-6297	591	1	international	international	ADJ
ejpam-6297	591	2	journal	journal	PROPN
ejpam-6297	591	3	of	of	ADP
ejpam-6297	591	4	mathematics	mathematics	PROPN
ejpam-6297	591	5	and	and	CCONJ
ejpam-6297	591	6	mathematical	mathematical	ADJ
ejpam-6297	591	7	sciences	science	NOUN
ejpam-6297	591	8	,	,	PUNCT
ejpam-6297	591	9	2007:1–16	2007:1–16	NUM
ejpam-6297	591	10	,	,	PUNCT
ejpam-6297	591	11	2007	2007	NUM
ejpam-6297	591	12	.	.	PUNCT
ejpam-6297	592	1	[	[	X
ejpam-6297	592	2	14	14	NUM
ejpam-6297	592	3	]	]	X
ejpam-6297	592	4	i.	i.	PROPN
ejpam-6297	592	5	demir	demir	PROPN
ejpam-6297	592	6	and	and	CCONJ
ejpam-6297	592	7	o.	o.	PROPN
ejpam-6297	592	8	b.	b.	PROPN
ejpam-6297	592	9	ozbakir	ozbakir	PROPN
ejpam-6297	592	10	.	.	PUNCT
ejpam-6297	593	1	on	on	ADP
ejpam-6297	593	2	β	β	ADJ
ejpam-6297	593	3	-	-	ADJ
ejpam-6297	593	4	paracompact	paracompact	ADJ
ejpam-6297	593	5	spaces	space	NOUN
ejpam-6297	593	6	.	.	PUNCT
ejpam-6297	594	1	filomat	filomat	NOUN
ejpam-6297	594	2	,	,	PUNCT
ejpam-6297	594	3	27(6):971–976	27(6):971–976	PROPN
ejpam-6297	594	4	,	,	PUNCT
ejpam-6297	594	5	2013	2013	NUM
ejpam-6297	594	6	.	.	PUNCT
ejpam-6297	595	1	[	[	X
ejpam-6297	595	2	15	15	NUM
ejpam-6297	595	3	]	]	X
ejpam-6297	595	4	p.	p.	PROPN
ejpam-6297	595	5	y.	y.	PROPN
ejpam-6297	595	6	li	li	PROPN
ejpam-6297	595	7	and	and	CCONJ
ejpam-6297	595	8	y.	y.	PROPN
ejpam-6297	595	9	k.	k.	PROPN
ejpam-6297	595	10	song	song	PROPN
ejpam-6297	595	11	.	.	PUNCT
ejpam-6297	596	1	some	some	DET
ejpam-6297	596	2	remarks	remark	NOUN
ejpam-6297	596	3	on	on	ADP
ejpam-6297	596	4	s	s	NOUN
ejpam-6297	596	5	-	-	PUNCT
ejpam-6297	596	6	paracompact	paracompact	ADJ
ejpam-6297	596	7	spaces	space	NOUN
ejpam-6297	596	8	.	.	PUNCT
ejpam-6297	597	1	acta	acta	PROPN
ejpam-6297	597	2	mathematica	mathematica	PROPN
ejpam-6297	597	3	hungarica	hungarica	PROPN
ejpam-6297	597	4	,	,	PUNCT
ejpam-6297	597	5	118:345–355	118:345–355	NUM
ejpam-6297	597	6	,	,	PUNCT
ejpam-6297	597	7	2008	2008	NUM
ejpam-6297	597	8	.	.	PUNCT
ejpam-6297	598	1	[	[	X
ejpam-6297	598	2	16	16	NUM
ejpam-6297	598	3	]	]	PUNCT
ejpam-6297	598	4	k.	k.	PROPN
ejpam-6297	598	5	kuratowski	kuratowski	PROPN
ejpam-6297	598	6	.	.	PUNCT
ejpam-6297	599	1	topology	topology	PROPN
ejpam-6297	599	2	i.	i.	PROPN
ejpam-6297	599	3	panstwowe	panstwowe	PROPN
ejpam-6297	599	4	wydawnictwo	wydawnictwo	PROPN
ejpam-6297	599	5	naukowe	naukowe	PROPN
ejpam-6297	599	6	,	,	PUNCT
ejpam-6297	599	7	warszawa	warszawa	PROPN
ejpam-6297	599	8	,	,	PUNCT
ejpam-6297	599	9	1933	1933	NUM
ejpam-6297	599	10	.	.	PUNCT
ejpam-6297	600	1	[	[	X
ejpam-6297	600	2	17	17	NUM
ejpam-6297	600	3	]	]	X
ejpam-6297	600	4	r.	r.	PROPN
ejpam-6297	600	5	vaidyanathaswamy	vaidyanathaswamy	PROPN
ejpam-6297	600	6	.	.	PUNCT
ejpam-6297	601	1	the	the	DET
ejpam-6297	601	2	localization	localization	NOUN
ejpam-6297	601	3	theory	theory	NOUN
ejpam-6297	601	4	in	in	ADP
ejpam-6297	601	5	set	set	NOUN
ejpam-6297	601	6	-	-	PUNCT
ejpam-6297	601	7	topology	topology	NOUN
ejpam-6297	601	8	.	.	PUNCT
ejpam-6297	602	1	proceedings	proceeding	NOUN
ejpam-6297	602	2	of	of	ADP
ejpam-6297	602	3	the	the	DET
ejpam-6297	602	4	indian	indian	PROPN
ejpam-6297	602	5	academy	academy	PROPN
ejpam-6297	602	6	of	of	ADP
ejpam-6297	602	7	sciences	sciences	PROPN
ejpam-6297	602	8	,	,	PUNCT
ejpam-6297	602	9	20:51–61	20:51–61	NUM
ejpam-6297	602	10	,	,	PUNCT
ejpam-6297	602	11	1945	1945	NUM
ejpam-6297	602	12	.	.	PUNCT
ejpam-6297	603	1	[	[	X
ejpam-6297	603	2	18	18	NUM
ejpam-6297	603	3	]	]	PUNCT
ejpam-6297	603	4	j.	j.	PROPN
ejpam-6297	603	5	dontchev	dontchev	PROPN
ejpam-6297	603	6	,	,	PUNCT
ejpam-6297	603	7	m.	m.	NOUN
ejpam-6297	603	8	ganster	ganster	NOUN
ejpam-6297	603	9	,	,	PUNCT
ejpam-6297	603	10	and	and	CCONJ
ejpam-6297	603	11	t.	t.	PROPN
ejpam-6297	603	12	noiri	noiri	PROPN
ejpam-6297	603	13	.	.	PUNCT
ejpam-6297	604	1	unified	unified	ADJ
ejpam-6297	604	2	operation	operation	NOUN
ejpam-6297	604	3	approach	approach	NOUN
ejpam-6297	604	4	of	of	ADP
ejpam-6297	604	5	generalized	generalized	ADJ
ejpam-6297	604	6	closed	close	VERB
ejpam-6297	604	7	sets	set	NOUN
ejpam-6297	604	8	via	via	ADP
ejpam-6297	604	9	topological	topological	ADJ
ejpam-6297	604	10	ideals	ideal	NOUN
ejpam-6297	604	11	.	.	PUNCT
ejpam-6297	605	1	mathematica	mathematica	PROPN
ejpam-6297	605	2	japonica	japonica	PROPN
ejpam-6297	605	3	,	,	PUNCT
ejpam-6297	605	4	49:395–402	49:395–402	PROPN
ejpam-6297	605	5	,	,	PUNCT
ejpam-6297	605	6	1999	1999	NUM
ejpam-6297	605	7	.	.	PUNCT
ejpam-6297	606	1	[	[	X
ejpam-6297	606	2	19	19	NUM
ejpam-6297	606	3	]	]	PUNCT
ejpam-6297	606	4	t.	t.	PROPN
ejpam-6297	606	5	r.	r.	PROPN
ejpam-6297	606	6	hamlett	hamlett	PROPN
ejpam-6297	606	7	,	,	PUNCT
ejpam-6297	606	8	d.	d.	PROPN
ejpam-6297	606	9	rose	rise	VERB
ejpam-6297	606	10	,	,	PUNCT
ejpam-6297	606	11	and	and	CCONJ
ejpam-6297	606	12	d.	d.	PROPN
ejpam-6297	606	13	jankovic	jankovic	PROPN
ejpam-6297	606	14	.	.	PUNCT
ejpam-6297	607	1	paracompactness	paracompactness	PROPN
ejpam-6297	607	2	with	with	ADP
ejpam-6297	607	3	respect	respect	NOUN
ejpam-6297	607	4	to	to	ADP
ejpam-6297	607	5	an	an	DET
ejpam-6297	607	6	ideal	ideal	NOUN
ejpam-6297	607	7	.	.	PUNCT
ejpam-6297	608	1	international	international	ADJ
ejpam-6297	608	2	journal	journal	PROPN
ejpam-6297	608	3	of	of	ADP
ejpam-6297	608	4	mathematics	mathematics	PROPN
ejpam-6297	608	5	and	and	CCONJ
ejpam-6297	608	6	mathematical	mathematical	ADJ
ejpam-6297	608	7	sciences	science	NOUN
ejpam-6297	608	8	,	,	PUNCT
ejpam-6297	608	9	20:433–442	20:433–442	NUM
ejpam-6297	608	10	,	,	PUNCT
ejpam-6297	608	11	1997	1997	NUM
ejpam-6297	608	12	.	.	PUNCT
ejpam-6297	609	1	[	[	X
ejpam-6297	609	2	20	20	NUM
ejpam-6297	609	3	]	]	PUNCT
ejpam-6297	609	4	e.	e.	PROPN
ejpam-6297	609	5	hatir	hatir	PROPN
ejpam-6297	609	6	.	.	PUNCT
ejpam-6297	610	1	on	on	ADP
ejpam-6297	610	2	decompositions	decomposition	NOUN
ejpam-6297	610	3	of	of	ADP
ejpam-6297	610	4	continuity	continuity	NOUN
ejpam-6297	610	5	and	and	CCONJ
ejpam-6297	610	6	complete	complete	ADJ
ejpam-6297	610	7	continuity	continuity	NOUN
ejpam-6297	610	8	in	in	ADP
ejpam-6297	610	9	ideal	ideal	ADJ
ejpam-6297	610	10	topological	topological	ADJ
ejpam-6297	610	11	spaces	space	NOUN
ejpam-6297	610	12	.	.	PUNCT
ejpam-6297	611	1	european	european	ADJ
ejpam-6297	611	2	journal	journal	PROPN
ejpam-6297	611	3	of	of	ADP
ejpam-6297	611	4	pure	pure	ADJ
ejpam-6297	611	5	and	and	CCONJ
ejpam-6297	611	6	applied	applied	ADJ
ejpam-6297	611	7	mathematics	mathematic	NOUN
ejpam-6297	611	8	,	,	PUNCT
ejpam-6297	611	9	6(3):352–362	6(3):352–362	NUM
ejpam-6297	611	10	,	,	PUNCT
ejpam-6297	611	11	2013	2013	NUM
ejpam-6297	611	12	.	.	PUNCT
ejpam-6297	612	1	[	[	X
ejpam-6297	612	2	21	21	NUM
ejpam-6297	612	3	]	]	X
ejpam-6297	612	4	d.	d.	PROPN
ejpam-6297	612	5	jankovic	jankovic	PROPN
ejpam-6297	612	6	and	and	CCONJ
ejpam-6297	612	7	t.	t.	PROPN
ejpam-6297	612	8	r.	r.	PROPN
ejpam-6297	612	9	hamlett	hamlett	PROPN
ejpam-6297	612	10	.	.	PUNCT
ejpam-6297	613	1	new	new	ADJ
ejpam-6297	613	2	topologies	topology	NOUN
ejpam-6297	613	3	from	from	ADP
ejpam-6297	613	4	old	old	ADJ
ejpam-6297	613	5	via	via	ADP
ejpam-6297	613	6	ideals	ideal	NOUN
ejpam-6297	613	7	.	.	PUNCT
ejpam-6297	614	1	the	the	DET
ejpam-6297	614	2	american	american	PROPN
ejpam-6297	614	3	mathematical	mathematical	PROPN
ejpam-6297	614	4	monthly	monthly	PROPN
ejpam-6297	614	5	,	,	PUNCT
ejpam-6297	614	6	97(4):295–310	97(4):295–310	PROPN
ejpam-6297	614	7	,	,	PUNCT
ejpam-6297	614	8	1990	1990	NUM
ejpam-6297	614	9	.	.	PUNCT
ejpam-6297	615	1	[	[	X
ejpam-6297	615	2	22	22	NUM
ejpam-6297	615	3	]	]	PUNCT
ejpam-6297	615	4	m.	m.	PROPN
ejpam-6297	615	5	e.	e.	PROPN
ejpam-6297	615	6	abd.el	abd.el	PROPN
ejpam-6297	615	7	-	-	PUNCT
ejpam-6297	615	8	monsef	monsef	ADJ
ejpam-6297	615	9	,	,	PUNCT
ejpam-6297	615	10	e.	e.	PROPN
ejpam-6297	615	11	f.	f.	PROPN
ejpam-6297	615	12	lashien	lashien	PROPN
ejpam-6297	615	13	,	,	PUNCT
ejpam-6297	615	14	and	and	CCONJ
ejpam-6297	615	15	a.	a.	NOUN
ejpam-6297	615	16	a.	a.	NOUN
ejpam-6297	615	17	nasef	nasef	PROPN
ejpam-6297	615	18	.	.	PUNCT
ejpam-6297	616	1	on	on	ADP
ejpam-6297	616	2	i	i	NOUN
ejpam-6297	616	3	-	-	PUNCT
ejpam-6297	616	4	open	open	ADJ
ejpam-6297	616	5	sets	set	NOUN
ejpam-6297	616	6	and	and	CCONJ
ejpam-6297	616	7	icontinuous	icontinuous	ADJ
ejpam-6297	616	8	functions	function	NOUN
ejpam-6297	616	9	.	.	PUNCT
ejpam-6297	617	1	kyungpook	kyungpook	PROPN
ejpam-6297	617	2	mathematical	mathematical	PROPN
ejpam-6297	617	3	journal	journal	PROPN
ejpam-6297	617	4	,	,	PUNCT
ejpam-6297	617	5	32(2):21–30	32(2):21–30	NUM
ejpam-6297	617	6	,	,	PUNCT
ejpam-6297	617	7	1992	1992	NUM
ejpam-6297	617	8	.	.	PUNCT
ejpam-6297	618	1	[	[	X
ejpam-6297	618	2	23	23	NUM
ejpam-6297	618	3	]	]	PUNCT
ejpam-6297	618	4	m.	m.	PROPN
ejpam-6297	618	5	e.	e.	PROPN
ejpam-6297	618	6	abd.el	abd.el	PROPN
ejpam-6297	618	7	-	-	PUNCT
ejpam-6297	618	8	monsef	monsef	ADJ
ejpam-6297	618	9	,	,	PUNCT
ejpam-6297	618	10	e.	e.	PROPN
ejpam-6297	618	11	f.	f.	PROPN
ejpam-6297	618	12	lashien	lashien	PROPN
ejpam-6297	618	13	,	,	PUNCT
ejpam-6297	618	14	and	and	CCONJ
ejpam-6297	618	15	a.	a.	NOUN
ejpam-6297	618	16	a.	a.	NOUN
ejpam-6297	618	17	nasef	nasef	PROPN
ejpam-6297	618	18	.	.	PUNCT
ejpam-6297	619	1	some	some	DET
ejpam-6297	619	2	topological	topological	ADJ
ejpam-6297	619	3	operators	operator	NOUN
ejpam-6297	619	4	via	via	ADP
ejpam-6297	619	5	ideals	ideal	NOUN
ejpam-6297	619	6	.	.	PUNCT
ejpam-6297	620	1	kyungpook	kyungpook	PROPN
ejpam-6297	620	2	mathematical	mathematical	PROPN
ejpam-6297	620	3	journal	journal	PROPN
ejpam-6297	620	4	,	,	PUNCT
ejpam-6297	620	5	32(2):273284	32(2):273284	NUM
ejpam-6297	620	6	,	,	PUNCT
ejpam-6297	620	7	1992	1992	NUM
ejpam-6297	620	8	.	.	PUNCT
ejpam-6297	621	1	[	[	X
ejpam-6297	621	2	24	24	NUM
ejpam-6297	621	3	]	]	PUNCT
ejpam-6297	621	4	m.	m.	NOUN
ejpam-6297	621	5	khan	khan	PROPN
ejpam-6297	621	6	and	and	CCONJ
ejpam-6297	621	7	t.	t.	PROPN
ejpam-6297	621	8	noiri	noiri	PROPN
ejpam-6297	621	9	.	.	PUNCT
ejpam-6297	622	1	semi	semi	ADJ
ejpam-6297	622	2	-	-	ADJ
ejpam-6297	622	3	local	local	ADJ
ejpam-6297	622	4	functions	function	NOUN
ejpam-6297	622	5	in	in	ADP
ejpam-6297	622	6	ideal	ideal	ADJ
ejpam-6297	622	7	topological	topological	ADJ
ejpam-6297	622	8	spaces	space	NOUN
ejpam-6297	622	9	.	.	PUNCT
ejpam-6297	623	1	journal	journal	NOUN
ejpam-6297	623	2	of	of	ADP
ejpam-6297	623	3	advanced	advanced	ADJ
ejpam-6297	623	4	research	research	NOUN
ejpam-6297	623	5	in	in	ADP
ejpam-6297	623	6	pure	pure	ADJ
ejpam-6297	623	7	mathematics	mathematic	NOUN
ejpam-6297	623	8	,	,	PUNCT
ejpam-6297	623	9	2(1):36–42	2(1):36–42	NUM
ejpam-6297	623	10	,	,	PUNCT
ejpam-6297	623	11	2010	2010	NUM
ejpam-6297	623	12	.	.	PUNCT
ejpam-6297	624	1	[	[	X
ejpam-6297	624	2	25	25	NUM
ejpam-6297	624	3	]	]	X
ejpam-6297	624	4	e.	e.	PROPN
ejpam-6297	624	5	ekici	ekici	PROPN
ejpam-6297	624	6	and	and	CCONJ
ejpam-6297	624	7	t.	t.	PROPN
ejpam-6297	624	8	noiri	noiri	PROPN
ejpam-6297	624	9	.	.	PUNCT
ejpam-6297	625	1	i	i	PRON
ejpam-6297	625	2	-	-	PUNCT
ejpam-6297	625	3	submaximal	submaximal	ADJ
ejpam-6297	625	4	ideal	ideal	ADJ
ejpam-6297	625	5	topological	topological	ADJ
ejpam-6297	625	6	spaces	space	NOUN
ejpam-6297	625	7	.	.	PUNCT
ejpam-6297	626	1	filomat	filomat	NOUN
ejpam-6297	626	2	,	,	PUNCT
ejpam-6297	626	3	24(4):87–94	24(4):87–94	NUM
ejpam-6297	626	4	,	,	PUNCT
ejpam-6297	626	5	2010	2010	NUM
ejpam-6297	626	6	.	.	PUNCT
ejpam-6297	627	1	[	[	X
ejpam-6297	627	2	26	26	NUM
ejpam-6297	627	3	]	]	PUNCT
ejpam-6297	627	4	c.	c.	PROPN
ejpam-6297	627	5	boonpok	boonpok	PROPN
ejpam-6297	627	6	.	.	PUNCT
ejpam-6297	628	1	semi	semi	ADJ
ejpam-6297	628	2	-	-	PROPN
ejpam-6297	628	3	i	i	NOUN
ejpam-6297	628	4	-	-	PUNCT
ejpam-6297	628	5	submaximality	submaximality	NOUN
ejpam-6297	628	6	.	.	PUNCT
ejpam-6297	629	1	european	european	PROPN
ejpam-6297	629	2	journal	journal	PROPN
ejpam-6297	629	3	of	of	ADP
ejpam-6297	629	4	pure	pure	ADJ
ejpam-6297	629	5	and	and	CCONJ
ejpam-6297	629	6	applied	applied	ADJ
ejpam-6297	629	7	mathematics	mathematic	NOUN
ejpam-6297	629	8	,	,	PUNCT
ejpam-6297	629	9	15(3):938–947	15(3):938–947	PROPN
ejpam-6297	629	10	,	,	PUNCT
ejpam-6297	629	11	2022	2022	NUM
ejpam-6297	629	12	.	.	PUNCT
ejpam-6297	630	1	c.	c.	PROPN
ejpam-6297	630	2	boonpok	boonpok	PROPN
ejpam-6297	630	3	,	,	PUNCT
ejpam-6297	630	4	p.	p.	PROPN
ejpam-6297	630	5	raktaow	raktaow	NOUN
ejpam-6297	630	6	,	,	PUNCT
ejpam-6297	630	7	a.	a.	PROPN
ejpam-6297	630	8	sama	sama	PROPN
ejpam-6297	630	9	-	-	PUNCT
ejpam-6297	630	10	ae	ae	PROPN
ejpam-6297	630	11	/	/	SYM
ejpam-6297	630	12	eur	eur	PROPN
ejpam-6297	630	13	.	.	PUNCT
ejpam-6297	631	1	j.	j.	PROPN
ejpam-6297	631	2	pure	pure	PROPN
ejpam-6297	631	3	appl	appl	PROPN
ejpam-6297	631	4	.	.	PROPN
ejpam-6297	631	5	math	math	PROPN
ejpam-6297	631	6	,	,	PUNCT
ejpam-6297	631	7	18	18	NUM
ejpam-6297	631	8	(	(	PUNCT
ejpam-6297	631	9	3	3	NUM
ejpam-6297	631	10	)	)	PUNCT
ejpam-6297	631	11	(	(	PUNCT
ejpam-6297	631	12	2025	2025	NUM
ejpam-6297	631	13	)	)	PUNCT
ejpam-6297	631	14	,	,	PUNCT
ejpam-6297	631	15	6297	6297	NUM
ejpam-6297	631	16	23	23	NUM
ejpam-6297	631	17	of	of	ADP
ejpam-6297	631	18	23	23	NUM
ejpam-6297	632	1	[	[	SYM
ejpam-6297	632	2	27	27	NUM
ejpam-6297	632	3	]	]	PUNCT
ejpam-6297	632	4	m.	m.	PROPN
ejpam-6297	632	5	i.	i.	PROPN
ejpam-6297	632	6	zahid	zahid	PROPN
ejpam-6297	632	7	.	.	PUNCT
ejpam-6297	633	1	para	para	PROPN
ejpam-6297	633	2	h	h	NOUN
ejpam-6297	633	3	-	-	PUNCT
ejpam-6297	633	4	closed	closed	ADJ
ejpam-6297	633	5	spaces	space	NOUN
ejpam-6297	633	6	,	,	PUNCT
ejpam-6297	633	7	locally	locally	ADV
ejpam-6297	633	8	para	para	ADJ
ejpam-6297	633	9	h	h	NOUN
ejpam-6297	633	10	-	-	PUNCT
ejpam-6297	633	11	closed	closed	ADJ
ejpam-6297	633	12	spaces	space	NOUN
ejpam-6297	633	13	and	and	CCONJ
ejpam-6297	633	14	their	their	PRON
ejpam-6297	633	15	minimal	minimal	ADJ
ejpam-6297	633	16	topologies	topology	NOUN
ejpam-6297	633	17	.	.	PUNCT
ejpam-6297	634	1	phd	phd	NOUN
ejpam-6297	634	2	thesis	thesis	PROPN
ejpam-6297	634	3	,	,	PUNCT
ejpam-6297	634	4	university	university	PROPN
ejpam-6297	634	5	of	of	ADP
ejpam-6297	634	6	pittsburgh	pittsburgh	PROPN
ejpam-6297	634	7	,	,	PUNCT
ejpam-6297	634	8	1981	1981	NUM
ejpam-6297	634	9	.	.	PUNCT
ejpam-6297	635	1	[	[	X
ejpam-6297	635	2	28	28	NUM
ejpam-6297	635	3	]	]	X
ejpam-6297	635	4	n.	n.	NOUN
ejpam-6297	635	5	sathiyasundari	sathiyasundari	PROPN
ejpam-6297	635	6	and	and	CCONJ
ejpam-6297	635	7	v.	v.	ADP
ejpam-6297	635	8	renukadevi	renukadevi	NOUN
ejpam-6297	635	9	.	.	PUNCT
ejpam-6297	636	1	paracompactness	paracompactness	NOUN
ejpam-6297	636	2	with	with	ADP
ejpam-6297	636	3	respect	respect	NOUN
ejpam-6297	636	4	to	to	ADP
ejpam-6297	636	5	an	an	DET
ejpam-6297	636	6	ideal	ideal	NOUN
ejpam-6297	636	7	.	.	PUNCT
ejpam-6297	637	1	filomat	filomat	NOUN
ejpam-6297	637	2	,	,	PUNCT
ejpam-6297	637	3	20(2):333–339	20(2):333–339	NOUN
ejpam-6297	637	4	,	,	PUNCT
ejpam-6297	637	5	2013	2013	NUM
ejpam-6297	637	6	.	.	PUNCT
ejpam-6297	638	1	[	[	X
ejpam-6297	638	2	29	29	NUM
ejpam-6297	638	3	]	]	X
ejpam-6297	638	4	j.	j.	PROPN
ejpam-6297	638	5	sanabria	sanabria	PROPN
ejpam-6297	638	6	,	,	PUNCT
ejpam-6297	638	7	e.	e.	PROPN
ejpam-6297	638	8	rosas	rosas	PROPN
ejpam-6297	638	9	,	,	PUNCT
ejpam-6297	638	10	c.	c.	PROPN
ejpam-6297	638	11	carpintero	carpintero	PROPN
ejpam-6297	638	12	,	,	PUNCT
ejpam-6297	638	13	m.	m.	NOUN
ejpam-6297	638	14	salas	salas	PROPN
ejpam-6297	638	15	-	-	PUNCT
ejpam-6297	638	16	brown	brown	PROPN
ejpam-6297	638	17	,	,	PUNCT
ejpam-6297	638	18	and	and	CCONJ
ejpam-6297	638	19	o.	o.	PROPN
ejpam-6297	638	20	garćıa	garćıa	PROPN
ejpam-6297	638	21	.	.	PROPN
ejpam-6297	638	22	sparacompactness	sparacompactness	PROPN
ejpam-6297	638	23	in	in	ADP
ejpam-6297	638	24	ideal	ideal	ADJ
ejpam-6297	638	25	topological	topological	ADJ
ejpam-6297	638	26	spaces	space	NOUN
ejpam-6297	638	27	.	.	PUNCT
ejpam-6297	639	1	matematički	matematički	PROPN
ejpam-6297	639	2	vesnik	vesnik	PROPN
ejpam-6297	639	3	,	,	PUNCT
ejpam-6297	639	4	68(3):192–203	68(3):192–203	NOUN
ejpam-6297	639	5	,	,	PUNCT
ejpam-6297	639	6	2016	2016	NUM
ejpam-6297	639	7	.	.	PUNCT
ejpam-6297	640	1	[	[	X
ejpam-6297	640	2	30	30	NUM
ejpam-6297	640	3	]	]	X
ejpam-6297	640	4	e.	e.	PROPN
ejpam-6297	640	5	d.	d.	PROPN
ejpam-6297	640	6	yildirim	yildirim	PROPN
ejpam-6297	640	7	,	,	PUNCT
ejpam-6297	640	8	o.	o.	PROPN
ejpam-6297	640	9	b.	b.	PROPN
ejpam-6297	640	10	ozbakir	ozbakir	PROPN
ejpam-6297	640	11	,	,	PUNCT
ejpam-6297	640	12	and	and	CCONJ
ejpam-6297	640	13	a.	a.	PROPN
ejpam-6297	640	14	c.	c.	PROPN
ejpam-6297	640	15	guler	guler	NOUN
ejpam-6297	640	16	.	.	PUNCT
ejpam-6297	641	1	β	β	X
ejpam-6297	641	2	-	-	PUNCT
ejpam-6297	641	3	paracompactness	paracompactness	NOUN
ejpam-6297	641	4	in	in	ADP
ejpam-6297	641	5	ideal	ideal	ADJ
ejpam-6297	641	6	topological	topological	ADJ
ejpam-6297	641	7	space	space	NOUN
ejpam-6297	641	8	.	.	PUNCT
ejpam-6297	642	1	european	european	ADJ
ejpam-6297	642	2	journal	journal	PROPN
ejpam-6297	642	3	of	of	ADP
ejpam-6297	642	4	pure	pure	ADJ
ejpam-6297	642	5	and	and	CCONJ
ejpam-6297	642	6	applied	applied	ADJ
ejpam-6297	642	7	mathematics	mathematic	NOUN
ejpam-6297	642	8	,	,	PUNCT
ejpam-6297	642	9	12(2):270–278	12(2):270–278	PROPN
ejpam-6297	642	10	,	,	PUNCT
ejpam-6297	642	11	2019	2019	NUM
ejpam-6297	642	12	.	.	PUNCT
ejpam-6297	643	1	[	[	X
ejpam-6297	643	2	31	31	NUM
ejpam-6297	643	3	]	]	PUNCT
ejpam-6297	643	4	c.	c.	PROPN
ejpam-6297	643	5	boonpok	boonpok	PROPN
ejpam-6297	643	6	,	,	PUNCT
ejpam-6297	643	7	a.	a.	PROPN
ejpam-6297	643	8	sama	sama	PROPN
ejpam-6297	643	9	-	-	PUNCT
ejpam-6297	643	10	ae	ae	PROPN
ejpam-6297	643	11	,	,	PUNCT
ejpam-6297	643	12	and	and	CCONJ
ejpam-6297	643	13	p.	p.	PROPN
ejpam-6297	643	14	raktaow	raktaow	NOUN
ejpam-6297	643	15	.	.	PUNCT
ejpam-6297	644	1	characterizations	characterization	NOUN
ejpam-6297	644	2	of	of	ADP
ejpam-6297	644	3	generalized	generalized	ADJ
ejpam-6297	644	4	paracompactness	paracompactness	NOUN
ejpam-6297	644	5	in	in	ADP
ejpam-6297	644	6	ideal	ideal	ADJ
ejpam-6297	644	7	topological	topological	ADJ
ejpam-6297	644	8	spaces	space	NOUN
ejpam-6297	644	9	.	.	PUNCT
ejpam-6297	645	1	european	european	ADJ
ejpam-6297	645	2	journal	journal	PROPN
ejpam-6297	645	3	of	of	ADP
ejpam-6297	645	4	pure	pure	ADJ
ejpam-6297	645	5	and	and	CCONJ
ejpam-6297	645	6	applied	applied	ADJ
ejpam-6297	645	7	mathematics	mathematic	NOUN
ejpam-6297	645	8	,	,	PUNCT
ejpam-6297	645	9	18(1):5570	18(1):5570	NUM
ejpam-6297	645	10	,	,	PUNCT
ejpam-6297	645	11	2025	2025	NUM
ejpam-6297	645	12	.	.	PUNCT
ejpam-6297	646	1	[	[	X
ejpam-6297	646	2	32	32	NUM
ejpam-6297	646	3	]	]	PUNCT
ejpam-6297	646	4	c.	c.	PROPN
ejpam-6297	646	5	boonpok	boonpok	PROPN
ejpam-6297	646	6	and	and	CCONJ
ejpam-6297	646	7	a.	a.	PROPN
ejpam-6297	646	8	sama	sama	PROPN
ejpam-6297	646	9	-	-	PUNCT
ejpam-6297	646	10	ae	ae	PROPN
ejpam-6297	646	11	.	.	PUNCT
ejpam-6297	647	1	characterizations	characterization	NOUN
ejpam-6297	647	2	of	of	ADP
ejpam-6297	647	3	δ1	δ1	NOUN
ejpam-6297	647	4	-	-	PUNCT
ejpam-6297	647	5	βi	βi	NOUN
ejpam-6297	647	6	-	-	NOUN
ejpam-6297	647	7	paracompactness	paracompactness	NOUN
ejpam-6297	647	8	concerning	concern	VERB
ejpam-6297	647	9	an	an	DET
ejpam-6297	647	10	ideal	ideal	NOUN
ejpam-6297	647	11	.	.	PUNCT
ejpam-6297	648	1	european	european	ADJ
ejpam-6297	648	2	journal	journal	PROPN
ejpam-6297	648	3	of	of	ADP
ejpam-6297	648	4	pure	pure	ADJ
ejpam-6297	648	5	and	and	CCONJ
ejpam-6297	648	6	applied	applied	ADJ
ejpam-6297	648	7	mathematics	mathematic	NOUN
ejpam-6297	648	8	,	,	PUNCT
ejpam-6297	648	9	18(1):5732	18(1):5732	NUM
ejpam-6297	648	10	,	,	PUNCT
ejpam-6297	648	11	2025	2025	NUM
ejpam-6297	648	12	.	.	PUNCT
ejpam-6297	649	1	[	[	X
ejpam-6297	649	2	33	33	NUM
ejpam-6297	649	3	]	]	PUNCT
ejpam-6297	649	4	m.	m.	NOUN
ejpam-6297	649	5	a.	a.	PROPN
ejpam-6297	649	6	al	al	PROPN
ejpam-6297	649	7	-	-	PUNCT
ejpam-6297	649	8	shumrani	shumrani	PROPN
ejpam-6297	649	9	,	,	PUNCT
ejpam-6297	649	10	c.	c.	PROPN
ejpam-6297	649	11	ozel	ozel	PROPN
ejpam-6297	649	12	,	,	PUNCT
ejpam-6297	649	13	and	and	CCONJ
ejpam-6297	649	14	a.	a.	PROPN
ejpam-6297	649	15	k.	k.	PROPN
ejpam-6297	649	16	kaymakcı	kaymakcı	PROPN
ejpam-6297	649	17	.	.	PUNCT
ejpam-6297	650	1	on	on	ADP
ejpam-6297	650	2	strong	strong	ADJ
ejpam-6297	650	3	β	β	X
ejpam-6297	650	4	-	-	ADJ
ejpam-6297	650	5	i	i	NOUN
ejpam-6297	650	6	-	-	PUNCT
ejpam-6297	650	7	open	open	ADJ
ejpam-6297	650	8	sets	set	NOUN
ejpam-6297	650	9	and	and	CCONJ
ejpam-6297	650	10	decomposition	decomposition	NOUN
ejpam-6297	650	11	of	of	ADP
ejpam-6297	650	12	continuity	continuity	NOUN
ejpam-6297	650	13	in	in	ADP
ejpam-6297	650	14	ideal	ideal	ADJ
ejpam-6297	650	15	topological	topological	ADJ
ejpam-6297	650	16	spaces	space	NOUN
ejpam-6297	650	17	.	.	PUNCT
ejpam-6297	651	1	journal	journal	PROPN
ejpam-6297	651	2	of	of	ADP
ejpam-6297	651	3	mathematical	mathematical	ADJ
ejpam-6297	651	4	analysis	analysis	NOUN
ejpam-6297	651	5	,	,	PUNCT
ejpam-6297	651	6	8(1):143–155	8(1):143–155	NUM
ejpam-6297	651	7	,	,	PUNCT
ejpam-6297	651	8	2017	2017	NUM
ejpam-6297	651	9	.	.	PUNCT