id	sid	tid	token	lemma	pos
ejpam-5732	1	1	european	european	PROPN
ejpam-5732	1	2	journal	journal	PROPN
ejpam-5732	1	3	of	of	ADP
ejpam-5732	1	4	pure	pure	ADJ
ejpam-5732	1	5	and	and	CCONJ
ejpam-5732	1	6	applied	applied	ADJ
ejpam-5732	1	7	mathematics	mathematic	NOUN
ejpam-5732	1	8	2025	2025	NUM
ejpam-5732	1	9	,	,	PUNCT
ejpam-5732	1	10	vol	vol	NOUN
ejpam-5732	1	11	.	.	PROPN
ejpam-5732	1	12	18	18	NUM
ejpam-5732	1	13	,	,	PUNCT
ejpam-5732	1	14	issue	issue	NOUN
ejpam-5732	1	15	1	1	NUM
ejpam-5732	1	16	,	,	PUNCT
ejpam-5732	1	17	article	article	NOUN
ejpam-5732	1	18	number	number	NOUN
ejpam-5732	1	19	5732	5732	NUM
ejpam-5732	1	20	issn	issn	VERB
ejpam-5732	1	21	1307	1307	NUM
ejpam-5732	1	22	-	-	SYM
ejpam-5732	1	23	5543	5543	NUM
ejpam-5732	1	24	–	–	PUNCT
ejpam-5732	1	25	ejpam.com	ejpam.com	X
ejpam-5732	1	26	published	publish	VERB
ejpam-5732	1	27	by	by	ADP
ejpam-5732	1	28	new	new	PROPN
ejpam-5732	1	29	york	york	PROPN
ejpam-5732	1	30	business	business	PROPN
ejpam-5732	1	31	global	global	ADJ
ejpam-5732	1	32	characterizations	characterization	NOUN
ejpam-5732	1	33	of	of	ADP
ejpam-5732	1	34	δ1	δ1	NOUN
ejpam-5732	1	35	-	-	PUNCT
ejpam-5732	1	36	βi	βi	NOUN
ejpam-5732	1	37	-	-	NOUN
ejpam-5732	1	38	paracompactness	paracompactness	NOUN
ejpam-5732	1	39	concerning	concern	VERB
ejpam-5732	1	40	an	an	DET
ejpam-5732	1	41	ideal	ideal	ADJ
ejpam-5732	1	42	chawalit	chawalit	VERB
ejpam-5732	1	43	boonpok1	boonpok1	NOUN
ejpam-5732	1	44	,	,	PUNCT
ejpam-5732	1	45	areeyuth	areeyuth	NOUN
ejpam-5732	1	46	sama	sama	NOUN
ejpam-5732	1	47	-	-	PUNCT
ejpam-5732	1	48	ae2,∗	ae2,∗	VERB
ejpam-5732	1	49	1	1	NUM
ejpam-5732	1	50	mathematics	mathematic	NOUN
ejpam-5732	1	51	and	and	CCONJ
ejpam-5732	1	52	applied	apply	VERB
ejpam-5732	1	53	mathematics	mathematics	PROPN
ejpam-5732	1	54	research	research	NOUN
ejpam-5732	1	55	unit	unit	NOUN
ejpam-5732	1	56	,	,	PUNCT
ejpam-5732	1	57	department	department	NOUN
ejpam-5732	1	58	of	of	ADP
ejpam-5732	1	59	mathematics	mathematic	NOUN
ejpam-5732	1	60	,	,	PUNCT
ejpam-5732	1	61	faculty	faculty	NOUN
ejpam-5732	1	62	of	of	ADP
ejpam-5732	1	63	science	science	NOUN
ejpam-5732	1	64	,	,	PUNCT
ejpam-5732	1	65	mahasarakham	mahasarakham	PROPN
ejpam-5732	1	66	university	university	PROPN
ejpam-5732	1	67	,	,	PUNCT
ejpam-5732	1	68	maha	maha	PROPN
ejpam-5732	1	69	sarakham	sarakham	PROPN
ejpam-5732	1	70	,	,	PUNCT
ejpam-5732	1	71	44150	44150	NUM
ejpam-5732	1	72	,	,	PUNCT
ejpam-5732	1	73	thailand	thailand	PROPN
ejpam-5732	1	74	2	2	NUM
ejpam-5732	1	75	department	department	NOUN
ejpam-5732	1	76	of	of	ADP
ejpam-5732	1	77	mathematics	mathematic	NOUN
ejpam-5732	1	78	and	and	CCONJ
ejpam-5732	1	79	computer	computer	NOUN
ejpam-5732	1	80	science	science	NOUN
ejpam-5732	1	81	,	,	PUNCT
ejpam-5732	1	82	faculty	faculty	NOUN
ejpam-5732	1	83	of	of	ADP
ejpam-5732	1	84	science	science	NOUN
ejpam-5732	1	85	and	and	CCONJ
ejpam-5732	1	86	technology	technology	NOUN
ejpam-5732	1	87	,	,	PUNCT
ejpam-5732	1	88	prince	prince	NOUN
ejpam-5732	1	89	of	of	ADP
ejpam-5732	1	90	songkla	songkla	PROPN
ejpam-5732	1	91	university	university	PROPN
ejpam-5732	1	92	,	,	PUNCT
ejpam-5732	1	93	pattani	pattani	NOUN
ejpam-5732	1	94	campus	campus	NOUN
ejpam-5732	1	95	,	,	PUNCT
ejpam-5732	1	96	pattani	pattani	NOUN
ejpam-5732	1	97	,	,	PUNCT
ejpam-5732	1	98	94000	94000	NUM
ejpam-5732	1	99	,	,	PUNCT
ejpam-5732	1	100	thailand	thailand	PROPN
ejpam-5732	1	101	abstract	abstract	PROPN
ejpam-5732	1	102	.	.	PUNCT
ejpam-5732	2	1	al	al	PROPN
ejpam-5732	2	2	-	-	PUNCT
ejpam-5732	2	3	jarrah	jarrah	PROPN
ejpam-5732	2	4	presented	present	VERB
ejpam-5732	2	5	and	and	CCONJ
ejpam-5732	2	6	examined	examine	VERB
ejpam-5732	2	7	the	the	DET
ejpam-5732	2	8	idea	idea	NOUN
ejpam-5732	2	9	of	of	ADP
ejpam-5732	2	10	β1	β1	PROPN
ejpam-5732	2	11	-	-	PUNCT
ejpam-5732	2	12	paracompactness	paracompactness	NOUN
ejpam-5732	2	13	in	in	ADP
ejpam-5732	2	14	topological	topological	ADJ
ejpam-5732	2	15	spaces	space	NOUN
ejpam-5732	2	16	,	,	PUNCT
ejpam-5732	2	17	whereas	whereas	SCONJ
ejpam-5732	2	18	qahis	qahis	PRON
ejpam-5732	2	19	extended	extend	VERB
ejpam-5732	2	20	the	the	DET
ejpam-5732	2	21	original	original	ADJ
ejpam-5732	2	22	idea	idea	NOUN
ejpam-5732	2	23	of	of	ADP
ejpam-5732	2	24	β1	β1	NOUN
ejpam-5732	2	25	-	-	PUNCT
ejpam-5732	2	26	paracompact	paracompact	NOUN
ejpam-5732	2	27	spaces	space	NOUN
ejpam-5732	2	28	by	by	ADP
ejpam-5732	2	29	further	far	ADV
ejpam-5732	2	30	developing	develop	VERB
ejpam-5732	2	31	and	and	CCONJ
ejpam-5732	2	32	investigating	investigate	VERB
ejpam-5732	2	33	the	the	DET
ejpam-5732	2	34	idea	idea	NOUN
ejpam-5732	2	35	of	of	ADP
ejpam-5732	2	36	β1	β1	NOUN
ejpam-5732	2	37	-	-	PUNCT
ejpam-5732	2	38	paracompact	paracompact	NOUN
ejpam-5732	2	39	spaces	space	NOUN
ejpam-5732	2	40	with	with	ADP
ejpam-5732	2	41	respect	respect	NOUN
ejpam-5732	2	42	to	to	ADP
ejpam-5732	2	43	an	an	DET
ejpam-5732	2	44	ideal	ideal	NOUN
ejpam-5732	2	45	.	.	PUNCT
ejpam-5732	3	1	this	this	DET
ejpam-5732	3	2	work	work	NOUN
ejpam-5732	3	3	analyzes	analyze	VERB
ejpam-5732	3	4	the	the	DET
ejpam-5732	3	5	properties	property	NOUN
ejpam-5732	3	6	,	,	PUNCT
ejpam-5732	3	7	subsets	subset	NOUN
ejpam-5732	3	8	,	,	PUNCT
ejpam-5732	3	9	and	and	CCONJ
ejpam-5732	3	10	subspaces	subspace	NOUN
ejpam-5732	3	11	of	of	ADP
ejpam-5732	3	12	δ1	δ1	NOUN
ejpam-5732	3	13	-	-	PUNCT
ejpam-5732	3	14	βi	βi	PRON
ejpam-5732	3	15	-	-	PUNCT
ejpam-5732	3	16	paracompact	paracompact	ADJ
ejpam-5732	3	17	spaces	space	NOUN
ejpam-5732	3	18	,	,	PUNCT
ejpam-5732	3	19	which	which	PRON
ejpam-5732	3	20	are	be	AUX
ejpam-5732	3	21	wider	wide	ADJ
ejpam-5732	3	22	in	in	ADP
ejpam-5732	3	23	scope	scope	NOUN
ejpam-5732	3	24	than	than	ADP
ejpam-5732	3	25	the	the	DET
ejpam-5732	3	26	β1	β1	NOUN
ejpam-5732	3	27	-	-	PUNCT
ejpam-5732	3	28	paracompact	paracompact	NOUN
ejpam-5732	3	29	spaces	space	NOUN
ejpam-5732	3	30	delineated	delineate	VERB
ejpam-5732	3	31	by	by	ADP
ejpam-5732	3	32	qahis	qahis	NOUN
ejpam-5732	3	33	.	.	PUNCT
ejpam-5732	4	1	furthermore	furthermore	ADV
ejpam-5732	4	2	,	,	PUNCT
ejpam-5732	4	3	we	we	PRON
ejpam-5732	4	4	investigate	investigate	VERB
ejpam-5732	4	5	the	the	DET
ejpam-5732	4	6	invariants	invariant	NOUN
ejpam-5732	4	7	of	of	ADP
ejpam-5732	4	8	δ1	δ1	NOUN
ejpam-5732	4	9	-	-	PUNCT
ejpam-5732	4	10	βi	βi	PRON
ejpam-5732	4	11	-	-	PUNCT
ejpam-5732	4	12	paracompact	paracompact	ADJ
ejpam-5732	4	13	spaces	space	NOUN
ejpam-5732	4	14	via	via	ADP
ejpam-5732	4	15	the	the	DET
ejpam-5732	4	16	view	view	NOUN
ejpam-5732	4	17	of	of	ADP
ejpam-5732	4	18	functions	function	NOUN
ejpam-5732	4	19	.	.	PUNCT
ejpam-5732	5	1	2020	2020	NUM
ejpam-5732	5	2	mathematics	mathematic	NOUN
ejpam-5732	5	3	subject	subject	NOUN
ejpam-5732	5	4	classifications	classification	NOUN
ejpam-5732	5	5	:	:	PUNCT
ejpam-5732	5	6	54a05	54a05	NUM
ejpam-5732	5	7	,	,	PUNCT
ejpam-5732	5	8	54b05	54b05	NUM
ejpam-5732	5	9	,	,	PUNCT
ejpam-5732	5	10	54c08	54c08	NUM
ejpam-5732	5	11	key	key	ADJ
ejpam-5732	5	12	words	word	NOUN
ejpam-5732	5	13	and	and	CCONJ
ejpam-5732	5	14	phrases	phrase	NOUN
ejpam-5732	5	15	:	:	PUNCT
ejpam-5732	5	16	ideal	ideal	ADJ
ejpam-5732	5	17	topological	topological	ADJ
ejpam-5732	5	18	space	space	NOUN
ejpam-5732	5	19	,	,	PUNCT
ejpam-5732	5	20	δ1	δ1	NOUN
ejpam-5732	5	21	-	-	PUNCT
ejpam-5732	5	22	βi	βi	PRON
ejpam-5732	5	23	-	-	PUNCT
ejpam-5732	5	24	paracompact	paracompact	ADJ
ejpam-5732	5	25	,	,	PUNCT
ejpam-5732	5	26	δ	δ	PROPN
ejpam-5732	5	27	-	-	PUNCT
ejpam-5732	5	28	βi	βi	ADV
ejpam-5732	5	29	-	-	PUNCT
ejpam-5732	5	30	open	open	ADJ
ejpam-5732	5	31	1	1	NUM
ejpam-5732	5	32	.	.	PUNCT
ejpam-5732	5	33	introduction	introduction	NOUN
ejpam-5732	5	34	paracompact	paracompact	NOUN
ejpam-5732	5	35	spaces	space	NOUN
ejpam-5732	5	36	,	,	PUNCT
ejpam-5732	5	37	established	establish	VERB
ejpam-5732	5	38	considerably	considerably	ADV
ejpam-5732	5	39	later	later	ADV
ejpam-5732	5	40	than	than	ADP
ejpam-5732	5	41	the	the	DET
ejpam-5732	5	42	two	two	NUM
ejpam-5732	5	43	earlier	early	ADJ
ejpam-5732	5	44	classes	class	NOUN
ejpam-5732	5	45	,	,	PUNCT
ejpam-5732	5	46	are	be	AUX
ejpam-5732	5	47	seen	see	VERB
ejpam-5732	5	48	to	to	PART
ejpam-5732	5	49	be	be	AUX
ejpam-5732	5	50	one	one	NUM
ejpam-5732	5	51	of	of	ADP
ejpam-5732	5	52	the	the	DET
ejpam-5732	5	53	most	most	ADV
ejpam-5732	5	54	important	important	ADJ
ejpam-5732	5	55	classes	class	NOUN
ejpam-5732	5	56	of	of	ADP
ejpam-5732	5	57	topological	topological	ADJ
ejpam-5732	5	58	spaces	space	NOUN
ejpam-5732	5	59	,	,	PUNCT
ejpam-5732	5	60	concurrently	concurrently	ADV
ejpam-5732	5	61	generalizing	generalize	VERB
ejpam-5732	5	62	both	both	CCONJ
ejpam-5732	5	63	metrizable	metrizable	ADJ
ejpam-5732	5	64	and	and	CCONJ
ejpam-5732	5	65	compact	compact	ADJ
ejpam-5732	5	66	spaces	space	NOUN
ejpam-5732	5	67	.	.	PUNCT
ejpam-5732	6	1	paracompact	paracompact	ADJ
ejpam-5732	6	2	spaces	space	NOUN
ejpam-5732	6	3	were	be	AUX
ejpam-5732	6	4	promptly	promptly	ADV
ejpam-5732	6	5	recognized	recognize	VERB
ejpam-5732	6	6	by	by	ADP
ejpam-5732	6	7	topologists	topologist	NOUN
ejpam-5732	6	8	and	and	CCONJ
ejpam-5732	6	9	analysts	analyst	NOUN
ejpam-5732	6	10	.	.	PUNCT
ejpam-5732	7	1	a	a	DET
ejpam-5732	7	2	paracompact	paracompact	ADJ
ejpam-5732	7	3	space	space	NOUN
ejpam-5732	7	4	in	in	ADP
ejpam-5732	7	5	mathematics	mathematics	PROPN
ejpam-5732	7	6	is	be	AUX
ejpam-5732	7	7	a	a	DET
ejpam-5732	7	8	topological	topological	ADJ
ejpam-5732	7	9	space	space	NOUN
ejpam-5732	7	10	where	where	SCONJ
ejpam-5732	7	11	every	every	DET
ejpam-5732	7	12	open	open	ADJ
ejpam-5732	7	13	cover	cover	NOUN
ejpam-5732	7	14	has	have	VERB
ejpam-5732	7	15	an	an	DET
ejpam-5732	7	16	open	open	ADJ
ejpam-5732	7	17	refinement	refinement	NOUN
ejpam-5732	7	18	that	that	PRON
ejpam-5732	7	19	is	be	AUX
ejpam-5732	7	20	locally	locally	ADV
ejpam-5732	7	21	finite	finite	ADJ
ejpam-5732	7	22	.	.	PUNCT
ejpam-5732	8	1	the	the	DET
ejpam-5732	8	2	notion	notion	NOUN
ejpam-5732	8	3	of	of	ADP
ejpam-5732	8	4	spaces	space	NOUN
ejpam-5732	8	5	was	be	AUX
ejpam-5732	8	6	initially	initially	ADV
ejpam-5732	8	7	developed	develop	VERB
ejpam-5732	8	8	by	by	ADP
ejpam-5732	8	9	dieudonné	dieudonné	NOUN
ejpam-5732	8	10	[	[	X
ejpam-5732	8	11	10	10	NUM
ejpam-5732	8	12	]	]	PUNCT
ejpam-5732	8	13	in	in	ADP
ejpam-5732	8	14	1944	1944	NUM
ejpam-5732	8	15	.	.	PUNCT
ejpam-5732	9	1	in	in	ADP
ejpam-5732	9	2	1969	1969	NUM
ejpam-5732	9	3	,	,	PUNCT
ejpam-5732	9	4	singal	singal	NOUN
ejpam-5732	9	5	and	and	CCONJ
ejpam-5732	9	6	arya	arya	NOUN
ejpam-5732	9	7	[	[	X
ejpam-5732	9	8	24	24	NUM
ejpam-5732	9	9	]	]	PUNCT
ejpam-5732	9	10	introduced	introduce	VERB
ejpam-5732	9	11	a	a	DET
ejpam-5732	9	12	novel	novel	ADJ
ejpam-5732	9	13	notion	notion	NOUN
ejpam-5732	9	14	of	of	ADP
ejpam-5732	9	15	paracompactness	paracompactness	NOUN
ejpam-5732	9	16	termed	term	VERB
ejpam-5732	9	17	almost	almost	ADV
ejpam-5732	9	18	paracompactness	paracompactness	NOUN
ejpam-5732	9	19	,	,	PUNCT
ejpam-5732	9	20	which	which	PRON
ejpam-5732	9	21	is	be	AUX
ejpam-5732	9	22	a	a	DET
ejpam-5732	9	23	weaker	weak	ADJ
ejpam-5732	9	24	form	form	NOUN
ejpam-5732	9	25	of	of	ADP
ejpam-5732	9	26	standard	standard	ADJ
ejpam-5732	9	27	paracompactness	paracompactness	NOUN
ejpam-5732	9	28	that	that	PRON
ejpam-5732	9	29	defines	define	VERB
ejpam-5732	9	30	its	its	PRON
ejpam-5732	9	31	fundamental	fundamental	ADJ
ejpam-5732	9	32	topological	topological	ADJ
ejpam-5732	9	33	properties	property	NOUN
ejpam-5732	9	34	.	.	PUNCT
ejpam-5732	10	1	a	a	DET
ejpam-5732	10	2	hausdorff	hausdorff	NOUN
ejpam-5732	10	3	space	space	NOUN
ejpam-5732	10	4	is	be	AUX
ejpam-5732	10	5	paracompact	paracompact	ADJ
ejpam-5732	10	6	if	if	SCONJ
ejpam-5732	10	7	and	and	CCONJ
ejpam-5732	10	8	only	only	ADV
ejpam-5732	10	9	if	if	SCONJ
ejpam-5732	10	10	it	it	PRON
ejpam-5732	10	11	allows	allow	VERB
ejpam-5732	10	12	partitions	partition	NOUN
ejpam-5732	10	13	of	of	ADP
ejpam-5732	10	14	unity	unity	NOUN
ejpam-5732	10	15	that	that	PRON
ejpam-5732	10	16	are	be	AUX
ejpam-5732	10	17	subservient	subservient	ADJ
ejpam-5732	10	18	to	to	ADP
ejpam-5732	10	19	any	any	DET
ejpam-5732	10	20	open	open	ADJ
ejpam-5732	10	21	cover	cover	NOUN
ejpam-5732	10	22	.	.	PUNCT
ejpam-5732	11	1	every	every	DET
ejpam-5732	11	2	paracompact	paracompact	ADJ
ejpam-5732	11	3	hausdorff	hausdorff	NOUN
ejpam-5732	11	4	space	space	NOUN
ejpam-5732	11	5	is	be	AUX
ejpam-5732	11	6	normal	normal	ADJ
ejpam-5732	11	7	,	,	PUNCT
ejpam-5732	11	8	as	as	SCONJ
ejpam-5732	11	9	referenced	reference	VERB
ejpam-5732	11	10	in	in	ADP
ejpam-5732	11	11	[	[	X
ejpam-5732	11	12	12	12	NUM
ejpam-5732	11	13	]	]	PUNCT
ejpam-5732	11	14	.	.	PUNCT
ejpam-5732	12	1	various	various	ADJ
ejpam-5732	12	2	forms	form	NOUN
ejpam-5732	12	3	of	of	ADP
ejpam-5732	12	4	generalized	generalized	ADJ
ejpam-5732	12	5	paracompactness	paracompactness	NOUN
ejpam-5732	12	6	in	in	ADP
ejpam-5732	12	7	literature	literature	NOUN
ejpam-5732	12	8	,	,	PUNCT
ejpam-5732	12	9	including	include	VERB
ejpam-5732	12	10	s	s	NOUN
ejpam-5732	12	11	-	-	NOUN
ejpam-5732	12	12	paracompactness	paracompactness	NOUN
ejpam-5732	12	13	[	[	X
ejpam-5732	12	14	4	4	NUM
ejpam-5732	12	15	]	]	PUNCT
ejpam-5732	12	16	,	,	PUNCT
ejpam-5732	12	17	p3	p3	PROPN
ejpam-5732	12	18	-	-	NOUN
ejpam-5732	12	19	paracompactness	paracompactness	NOUN
ejpam-5732	12	20	[	[	X
ejpam-5732	12	21	9	9	NUM
ejpam-5732	12	22	]	]	PUNCT
ejpam-5732	12	23	,	,	PUNCT
ejpam-5732	12	24	and	and	CCONJ
ejpam-5732	12	25	β	β	X
ejpam-5732	12	26	-	-	NOUN
ejpam-5732	12	27	paracompactness	paracompactness	NOUN
ejpam-5732	12	28	[	[	X
ejpam-5732	12	29	5	5	NUM
ejpam-5732	12	30	]	]	PUNCT
ejpam-5732	12	31	,	,	PUNCT
ejpam-5732	12	32	are	be	AUX
ejpam-5732	12	33	evaluated	evaluate	VERB
ejpam-5732	12	34	.	.	PUNCT
ejpam-5732	13	1	in	in	ADP
ejpam-5732	13	2	2006	2006	NUM
ejpam-5732	13	3	,	,	PUNCT
ejpam-5732	13	4	al	al	PROPN
ejpam-5732	13	5	-	-	PROPN
ejpam-5732	13	6	zoubi	zoubi	PROPN
ejpam-5732	13	7	[	[	X
ejpam-5732	13	8	4	4	NUM
ejpam-5732	13	9	]	]	PUNCT
ejpam-5732	13	10	employed	employ	VERB
ejpam-5732	13	11	semi	semi	ADJ
ejpam-5732	13	12	-	-	ADJ
ejpam-5732	13	13	open	open	ADJ
ejpam-5732	13	14	sets	set	NOUN
ejpam-5732	13	15	to	to	PART
ejpam-5732	13	16	characterize	characterize	VERB
ejpam-5732	13	17	s	s	NOUN
ejpam-5732	13	18	-	-	PUNCT
ejpam-5732	13	19	paracompact	paracompact	ADJ
ejpam-5732	13	20	spaces	space	NOUN
ejpam-5732	13	21	,	,	PUNCT
ejpam-5732	13	22	a	a	DET
ejpam-5732	13	23	generalization	generalization	NOUN
ejpam-5732	13	24	of	of	ADP
ejpam-5732	13	25	paracompact	paracompact	ADJ
ejpam-5732	13	26	spaces	space	NOUN
ejpam-5732	13	27	,	,	PUNCT
ejpam-5732	13	28	and	and	CCONJ
ejpam-5732	13	29	analyzed	analyze	VERB
ejpam-5732	13	30	the	the	DET
ejpam-5732	13	31	relationships	relationship	NOUN
ejpam-5732	13	32	between	between	ADP
ejpam-5732	13	33	these	these	DET
ejpam-5732	13	34	spaces	space	NOUN
ejpam-5732	13	35	.	.	PUNCT
ejpam-5732	14	1	li	li	PROPN
ejpam-5732	14	2	and	and	CCONJ
ejpam-5732	14	3	song	song	NOUN
ejpam-5732	14	4	[	[	X
ejpam-5732	14	5	18	18	NUM
ejpam-5732	14	6	]	]	PUNCT
ejpam-5732	14	7	∗corresponding	∗corresponde	VERB
ejpam-5732	14	8	author	author	NOUN
ejpam-5732	14	9	.	.	PUNCT
ejpam-5732	15	1	doi	doi	NOUN
ejpam-5732	15	2	:	:	PUNCT
ejpam-5732	15	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5732	https://doi.org/10.29020/nybg.ejpam.v18i1.5732	PRON
ejpam-5732	15	4	email	email	NOUN
ejpam-5732	15	5	addresses	address	NOUN
ejpam-5732	15	6	:	:	PUNCT
ejpam-5732	15	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-5732	15	8	(	(	PUNCT
ejpam-5732	15	9	c.	c.	PROPN
ejpam-5732	15	10	boonpok	boonpok	PROPN
ejpam-5732	15	11	)	)	PUNCT
ejpam-5732	15	12	,	,	PUNCT
ejpam-5732	15	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-5732	15	14	(	(	PUNCT
ejpam-5732	15	15	a.	a.	PROPN
ejpam-5732	15	16	sama	sama	PROPN
ejpam-5732	15	17	-	-	PUNCT
ejpam-5732	15	18	ae	ae	PROPN
ejpam-5732	15	19	)	)	PUNCT
ejpam-5732	15	20	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5732	15	21	1	1	NUM
ejpam-5732	15	22	copyright	copyright	NOUN
ejpam-5732	15	23	:	:	PUNCT
ejpam-5732	16	1	©	©	PROPN
ejpam-5732	16	2	2025	2025	NUM
ejpam-5732	16	3	the	the	DET
ejpam-5732	16	4	author(s	author(s	NOUN
ejpam-5732	16	5	)	)	PUNCT
ejpam-5732	16	6	.	.	PUNCT
ejpam-5732	17	1	(	(	PUNCT
ejpam-5732	17	2	cc	cc	NOUN
ejpam-5732	17	3	by	by	ADP
ejpam-5732	17	4	-	-	PUNCT
ejpam-5732	17	5	nc	nc	PROPN
ejpam-5732	17	6	4.0	4.0	NUM
ejpam-5732	17	7	)	)	PUNCT
ejpam-5732	17	8	c.	c.	NOUN
ejpam-5732	17	9	boonpok	boonpok	PROPN
ejpam-5732	17	10	,	,	PUNCT
ejpam-5732	17	11	a.	a.	PROPN
ejpam-5732	17	12	sama	sama	PROPN
ejpam-5732	17	13	-	-	PUNCT
ejpam-5732	17	14	ae	ae	PROPN
ejpam-5732	17	15	/	/	SYM
ejpam-5732	17	16	eur	eur	PROPN
ejpam-5732	17	17	.	.	PUNCT
ejpam-5732	18	1	j.	j.	PROPN
ejpam-5732	18	2	pure	pure	PROPN
ejpam-5732	18	3	appl	appl	PROPN
ejpam-5732	18	4	.	.	PROPN
ejpam-5732	18	5	math	math	PROPN
ejpam-5732	18	6	,	,	PUNCT
ejpam-5732	18	7	18	18	NUM
ejpam-5732	18	8	(	(	PUNCT
ejpam-5732	18	9	1	1	NUM
ejpam-5732	18	10	)	)	PUNCT
ejpam-5732	18	11	(	(	PUNCT
ejpam-5732	18	12	2025	2025	NUM
ejpam-5732	18	13	)	)	PUNCT
ejpam-5732	18	14	,	,	PUNCT
ejpam-5732	18	15	5732	5732	NUM
ejpam-5732	18	16	2	2	NUM
ejpam-5732	18	17	of	of	ADP
ejpam-5732	18	18	13	13	NUM
ejpam-5732	18	19	developed	develop	VERB
ejpam-5732	18	20	a	a	DET
ejpam-5732	18	21	hausdorff	hausdorff	NOUN
ejpam-5732	18	22	s	s	NOUN
ejpam-5732	18	23	-	-	PUNCT
ejpam-5732	18	24	paracompact	paracompact	ADJ
ejpam-5732	18	25	space	space	NOUN
ejpam-5732	18	26	that	that	PRON
ejpam-5732	18	27	is	be	AUX
ejpam-5732	18	28	not	not	PART
ejpam-5732	18	29	paracompact	paracompact	ADJ
ejpam-5732	18	30	and	and	CCONJ
ejpam-5732	18	31	looked	look	VERB
ejpam-5732	18	32	at	at	ADP
ejpam-5732	18	33	other	other	ADJ
ejpam-5732	18	34	characterizations	characterization	NOUN
ejpam-5732	18	35	of	of	ADP
ejpam-5732	18	36	s	s	NOUN
ejpam-5732	18	37	-	-	PUNCT
ejpam-5732	18	38	paracompact	paracompact	ADJ
ejpam-5732	18	39	spaces	space	NOUN
ejpam-5732	18	40	.	.	PUNCT
ejpam-5732	19	1	in	in	ADP
ejpam-5732	19	2	2013	2013	NUM
ejpam-5732	19	3	,	,	PUNCT
ejpam-5732	19	4	demir	demir	PROPN
ejpam-5732	19	5	and	and	CCONJ
ejpam-5732	19	6	ozbakir	ozbakir	VERB
ejpam-5732	19	7	[	[	X
ejpam-5732	19	8	9	9	NUM
ejpam-5732	19	9	]	]	PUNCT
ejpam-5732	19	10	proposed	propose	VERB
ejpam-5732	19	11	a	a	DET
ejpam-5732	19	12	diminished	diminished	ADJ
ejpam-5732	19	13	variant	variant	NOUN
ejpam-5732	19	14	of	of	ADP
ejpam-5732	19	15	expandable	expandable	ADJ
ejpam-5732	19	16	and	and	CCONJ
ejpam-5732	19	17	paracompact	paracompact	ADJ
ejpam-5732	19	18	spaces	space	NOUN
ejpam-5732	19	19	,	,	PUNCT
ejpam-5732	19	20	termed	term	VERB
ejpam-5732	19	21	β	β	NOUN
ejpam-5732	19	22	-	-	ADJ
ejpam-5732	19	23	expandable	expandable	ADJ
ejpam-5732	19	24	spaces	space	NOUN
ejpam-5732	19	25	and	and	CCONJ
ejpam-5732	19	26	β	β	NOUN
ejpam-5732	19	27	-	-	ADJ
ejpam-5732	19	28	paracompact	paracompact	ADJ
ejpam-5732	19	29	spaces	space	NOUN
ejpam-5732	19	30	,	,	PUNCT
ejpam-5732	19	31	respectively	respectively	ADV
ejpam-5732	19	32	.	.	PUNCT
ejpam-5732	20	1	every	every	DET
ejpam-5732	20	2	β	β	X
ejpam-5732	20	3	-	-	ADJ
ejpam-5732	20	4	paracompact	paracompact	ADJ
ejpam-5732	20	5	space	space	NOUN
ejpam-5732	20	6	is	be	AUX
ejpam-5732	20	7	a	a	DET
ejpam-5732	20	8	β	β	NOUN
ejpam-5732	20	9	-	-	ADJ
ejpam-5732	20	10	expandable	expandable	ADJ
ejpam-5732	20	11	space	space	NOUN
ejpam-5732	20	12	,	,	PUNCT
ejpam-5732	20	13	as	as	SCONJ
ejpam-5732	20	14	demonstrated	demonstrate	VERB
ejpam-5732	20	15	in	in	ADP
ejpam-5732	20	16	a	a	DET
ejpam-5732	20	17	provided	provide	VERB
ejpam-5732	20	18	argument	argument	NOUN
ejpam-5732	20	19	.	.	PUNCT
ejpam-5732	21	1	yildirim	yildirim	PROPN
ejpam-5732	21	2	et	et	PROPN
ejpam-5732	21	3	al	al	PROPN
ejpam-5732	21	4	.	.	PUNCT
ejpam-5732	22	1	[	[	X
ejpam-5732	22	2	25	25	NUM
ejpam-5732	22	3	]	]	PUNCT
ejpam-5732	22	4	came	come	VERB
ejpam-5732	22	5	up	up	ADP
ejpam-5732	22	6	with	with	ADP
ejpam-5732	22	7	the	the	DET
ejpam-5732	22	8	idea	idea	NOUN
ejpam-5732	22	9	of	of	ADP
ejpam-5732	22	10	β	β	NOUN
ejpam-5732	22	11	-	-	NOUN
ejpam-5732	22	12	paracompactness	paracompactness	NOUN
ejpam-5732	22	13	in	in	ADP
ejpam-5732	22	14	an	an	DET
ejpam-5732	22	15	ideal	ideal	ADJ
ejpam-5732	22	16	topological	topological	ADJ
ejpam-5732	22	17	space	space	NOUN
ejpam-5732	22	18	and	and	CCONJ
ejpam-5732	22	19	compared	compare	VERB
ejpam-5732	22	20	it	it	PRON
ejpam-5732	22	21	to	to	ADP
ejpam-5732	22	22	other	other	ADJ
ejpam-5732	22	23	types	type	NOUN
ejpam-5732	22	24	of	of	ADP
ejpam-5732	22	25	paracompactness	paracompactness	NOUN
ejpam-5732	22	26	that	that	PRON
ejpam-5732	22	27	are	be	AUX
ejpam-5732	22	28	already	already	ADV
ejpam-5732	22	29	known	know	VERB
ejpam-5732	22	30	.	.	PUNCT
ejpam-5732	23	1	in	in	ADP
ejpam-5732	23	2	2024	2024	NUM
ejpam-5732	23	3	,	,	PUNCT
ejpam-5732	23	4	alrababah	alrababah	NOUN
ejpam-5732	23	5	et	et	PROPN
ejpam-5732	23	6	al	al	PROPN
ejpam-5732	23	7	.	.	PUNCT
ejpam-5732	24	1	[	[	X
ejpam-5732	24	2	6	6	NUM
ejpam-5732	24	3	]	]	PUNCT
ejpam-5732	24	4	studied	study	VERB
ejpam-5732	24	5	the	the	DET
ejpam-5732	24	6	notion	notion	NOUN
ejpam-5732	24	7	,	,	PUNCT
ejpam-5732	24	8	attributes	attribute	NOUN
ejpam-5732	24	9	,	,	PUNCT
ejpam-5732	24	10	and	and	CCONJ
ejpam-5732	24	11	theorems	theorem	NOUN
ejpam-5732	24	12	associated	associate	VERB
ejpam-5732	24	13	with	with	ADP
ejpam-5732	24	14	dparacompact	dparacompact	NOUN
ejpam-5732	24	15	spaces	space	NOUN
ejpam-5732	24	16	.	.	PUNCT
ejpam-5732	25	1	in	in	ADP
ejpam-5732	25	2	1930	1930	NUM
ejpam-5732	25	3	,	,	PUNCT
ejpam-5732	25	4	kuratowski	kuratowski	PROPN
ejpam-5732	25	5	proposed	propose	VERB
ejpam-5732	25	6	the	the	DET
ejpam-5732	25	7	concept	concept	NOUN
ejpam-5732	25	8	of	of	ADP
ejpam-5732	25	9	an	an	DET
ejpam-5732	25	10	ideal	ideal	ADJ
ejpam-5732	25	11	topological	topological	ADJ
ejpam-5732	25	12	space	space	NOUN
ejpam-5732	25	13	[	[	X
ejpam-5732	25	14	17	17	NUM
ejpam-5732	25	15	]	]	PUNCT
ejpam-5732	25	16	.	.	PUNCT
ejpam-5732	26	1	jankovic	jankovic	PROPN
ejpam-5732	26	2	and	and	CCONJ
ejpam-5732	26	3	hamlett	hamlett	PROPN
ejpam-5732	27	1	[	[	X
ejpam-5732	27	2	15	15	NUM
ejpam-5732	27	3	]	]	PUNCT
ejpam-5732	27	4	have	have	AUX
ejpam-5732	27	5	performed	perform	VERB
ejpam-5732	27	6	an	an	DET
ejpam-5732	27	7	investigation	investigation	NOUN
ejpam-5732	27	8	and	and	CCONJ
ejpam-5732	27	9	offered	offer	VERB
ejpam-5732	27	10	a	a	DET
ejpam-5732	27	11	thorough	thorough	ADJ
ejpam-5732	27	12	explanation	explanation	NOUN
ejpam-5732	27	13	of	of	ADP
ejpam-5732	27	14	the	the	DET
ejpam-5732	27	15	basic	basic	ADJ
ejpam-5732	27	16	characteristics	characteristic	NOUN
ejpam-5732	27	17	related	relate	VERB
ejpam-5732	27	18	to	to	ADP
ejpam-5732	27	19	ideal	ideal	ADJ
ejpam-5732	27	20	topological	topological	ADJ
ejpam-5732	27	21	spaces	space	NOUN
ejpam-5732	27	22	.	.	PUNCT
ejpam-5732	28	1	in	in	ADP
ejpam-5732	28	2	addition	addition	NOUN
ejpam-5732	28	3	to	to	ADP
ejpam-5732	28	4	elucidating	elucidate	VERB
ejpam-5732	28	5	the	the	DET
ejpam-5732	28	6	concept	concept	NOUN
ejpam-5732	28	7	of	of	ADP
ejpam-5732	28	8	i	i	NOUN
ejpam-5732	28	9	-	-	PUNCT
ejpam-5732	28	10	open	open	ADJ
ejpam-5732	28	11	sets	set	NOUN
ejpam-5732	28	12	,	,	PUNCT
ejpam-5732	28	13	they	they	PRON
ejpam-5732	28	14	conducted	conduct	VERB
ejpam-5732	28	15	exhaustive	exhaustive	ADJ
ejpam-5732	28	16	research	research	NOUN
ejpam-5732	28	17	on	on	ADP
ejpam-5732	28	18	topologies	topology	NOUN
ejpam-5732	28	19	that	that	PRON
ejpam-5732	28	20	make	make	VERB
ejpam-5732	28	21	use	use	NOUN
ejpam-5732	28	22	of	of	ADP
ejpam-5732	28	23	ideals	ideal	NOUN
ejpam-5732	28	24	.	.	PUNCT
ejpam-5732	29	1	a	a	DET
ejpam-5732	29	2	thorough	thorough	ADJ
ejpam-5732	29	3	investigation	investigation	NOUN
ejpam-5732	29	4	of	of	ADP
ejpam-5732	29	5	the	the	DET
ejpam-5732	29	6	idea	idea	NOUN
ejpam-5732	29	7	of	of	ADP
ejpam-5732	29	8	i	i	NOUN
ejpam-5732	29	9	-	-	PUNCT
ejpam-5732	29	10	open	open	ADJ
ejpam-5732	29	11	sets	set	NOUN
ejpam-5732	29	12	was	be	AUX
ejpam-5732	29	13	carried	carry	VERB
ejpam-5732	29	14	out	out	ADP
ejpam-5732	29	15	by	by	ADP
ejpam-5732	29	16	abd	abd	PROPN
ejpam-5732	29	17	.	.	PUNCT
ejpam-5732	30	1	el	el	PROPN
ejpam-5732	30	2	-	-	PUNCT
ejpam-5732	30	3	monsef	monsef	PROPN
ejpam-5732	30	4	et	et	PROPN
ejpam-5732	30	5	al	al	PROPN
ejpam-5732	30	6	.	.	PUNCT
ejpam-5732	31	1	[	[	X
ejpam-5732	31	2	1	1	NUM
ejpam-5732	31	3	]	]	PUNCT
ejpam-5732	31	4	.	.	PUNCT
ejpam-5732	32	1	the	the	DET
ejpam-5732	32	2	concept	concept	NOUN
ejpam-5732	32	3	of	of	ADP
ejpam-5732	32	4	ig	ig	PROPN
ejpam-5732	32	5	-	-	PUNCT
ejpam-5732	32	6	closed	closed	ADJ
ejpam-5732	32	7	sets	set	NOUN
ejpam-5732	32	8	was	be	AUX
ejpam-5732	32	9	originally	originally	ADV
ejpam-5732	32	10	presented	present	VERB
ejpam-5732	32	11	by	by	ADP
ejpam-5732	32	12	dontchev	dontchev	PROPN
ejpam-5732	32	13	et	et	PROPN
ejpam-5732	32	14	al	al	PROPN
ejpam-5732	32	15	.	.	PUNCT
ejpam-5732	33	1	[	[	X
ejpam-5732	33	2	11	11	NUM
ejpam-5732	33	3	]	]	PUNCT
ejpam-5732	33	4	in	in	ADP
ejpam-5732	33	5	the	the	DET
ejpam-5732	33	6	year	year	NOUN
ejpam-5732	33	7	1999	1999	NUM
ejpam-5732	33	8	.	.	PUNCT
ejpam-5732	34	1	there	there	PRON
ejpam-5732	34	2	was	be	VERB
ejpam-5732	34	3	an	an	DET
ejpam-5732	34	4	early	early	ADJ
ejpam-5732	34	5	proposal	proposal	NOUN
ejpam-5732	34	6	made	make	VERB
ejpam-5732	34	7	by	by	ADP
ejpam-5732	34	8	abd	abd	PROPN
ejpam-5732	34	9	.	.	PUNCT
ejpam-5732	35	1	el	el	PROPN
ejpam-5732	35	2	-	-	PUNCT
ejpam-5732	35	3	monsef	monsef	PROPN
ejpam-5732	35	4	et	et	PROPN
ejpam-5732	35	5	al	al	PROPN
ejpam-5732	35	6	.	.	PUNCT
ejpam-5732	36	1	[	[	X
ejpam-5732	36	2	2	2	X
ejpam-5732	36	3	]	]	PUNCT
ejpam-5732	36	4	about	about	ADP
ejpam-5732	36	5	the	the	DET
ejpam-5732	36	6	concept	concept	NOUN
ejpam-5732	36	7	of	of	ADP
ejpam-5732	36	8	the	the	DET
ejpam-5732	36	9	s	s	NOUN
ejpam-5732	36	10	-	-	ADJ
ejpam-5732	36	11	local	local	ADJ
ejpam-5732	36	12	function	function	NOUN
ejpam-5732	36	13	.	.	PUNCT
ejpam-5732	37	1	khan	khan	PROPN
ejpam-5732	37	2	and	and	CCONJ
ejpam-5732	37	3	noiri	noiri	ADV
ejpam-5732	38	1	[	[	X
ejpam-5732	38	2	16	16	NUM
ejpam-5732	38	3	]	]	PUNCT
ejpam-5732	38	4	subsequently	subsequently	ADV
ejpam-5732	38	5	performed	perform	VERB
ejpam-5732	38	6	an	an	DET
ejpam-5732	38	7	analysis	analysis	NOUN
ejpam-5732	38	8	on	on	ADP
ejpam-5732	38	9	this	this	DET
ejpam-5732	38	10	concept	concept	NOUN
ejpam-5732	38	11	.	.	PUNCT
ejpam-5732	39	1	the	the	DET
ejpam-5732	39	2	concept	concept	NOUN
ejpam-5732	39	3	of	of	ADP
ejpam-5732	39	4	paracompactness	paracompactness	NOUN
ejpam-5732	39	5	in	in	ADP
ejpam-5732	39	6	respect	respect	NOUN
ejpam-5732	39	7	to	to	ADP
ejpam-5732	39	8	an	an	DET
ejpam-5732	39	9	ideal	ideal	NOUN
ejpam-5732	39	10	was	be	AUX
ejpam-5732	39	11	originally	originally	ADV
ejpam-5732	39	12	presented	present	VERB
ejpam-5732	39	13	by	by	ADP
ejpam-5732	39	14	zahid	zahid	PROPN
ejpam-5732	39	15	[	[	X
ejpam-5732	39	16	27	27	NUM
ejpam-5732	39	17	]	]	PUNCT
ejpam-5732	39	18	,	,	PUNCT
ejpam-5732	39	19	and	and	CCONJ
ejpam-5732	39	20	it	it	PRON
ejpam-5732	39	21	was	be	AUX
ejpam-5732	39	22	subsequently	subsequently	ADV
ejpam-5732	39	23	investigated	investigate	VERB
ejpam-5732	39	24	by	by	ADP
ejpam-5732	39	25	hamlett	hamlett	PROPN
ejpam-5732	39	26	et	et	PROPN
ejpam-5732	39	27	al	al	PROPN
ejpam-5732	39	28	.	.	PUNCT
ejpam-5732	40	1	[	[	X
ejpam-5732	40	2	13	13	NUM
ejpam-5732	40	3	]	]	PUNCT
ejpam-5732	40	4	.	.	PUNCT
ejpam-5732	41	1	an	an	DET
ejpam-5732	41	2	investigation	investigation	NOUN
ejpam-5732	41	3	was	be	AUX
ejpam-5732	41	4	studied	study	VERB
ejpam-5732	41	5	by	by	ADP
ejpam-5732	41	6	sathiyasundari	sathiyasundari	PROPN
ejpam-5732	41	7	and	and	CCONJ
ejpam-5732	41	8	renukadevi	renukadevi	ADJ
ejpam-5732	41	9	[	[	X
ejpam-5732	41	10	23	23	NUM
ejpam-5732	41	11	]	]	PUNCT
ejpam-5732	41	12	to	to	PART
ejpam-5732	41	13	investigate	investigate	VERB
ejpam-5732	41	14	the	the	DET
ejpam-5732	41	15	idea	idea	NOUN
ejpam-5732	41	16	of	of	ADP
ejpam-5732	41	17	i	i	PROPN
ejpam-5732	41	18	-	-	PROPN
ejpam-5732	41	19	paracompactness	paracompactness	PROPN
ejpam-5732	41	20	and	and	CCONJ
ejpam-5732	41	21	to	to	PART
ejpam-5732	41	22	assess	assess	VERB
ejpam-5732	41	23	its	its	PRON
ejpam-5732	41	24	properties	property	NOUN
ejpam-5732	41	25	.	.	PUNCT
ejpam-5732	42	1	the	the	DET
ejpam-5732	42	2	concept	concept	NOUN
ejpam-5732	42	3	of	of	ADP
ejpam-5732	42	4	i	i	NOUN
ejpam-5732	42	5	-	-	PUNCT
ejpam-5732	42	6	paracompact	paracompact	ADJ
ejpam-5732	42	7	spaces	space	NOUN
ejpam-5732	42	8	has	have	AUX
ejpam-5732	42	9	been	be	AUX
ejpam-5732	42	10	expanded	expand	VERB
ejpam-5732	42	11	to	to	PART
ejpam-5732	42	12	include	include	VERB
ejpam-5732	42	13	some	some	DET
ejpam-5732	42	14	conclusions	conclusion	NOUN
ejpam-5732	42	15	that	that	PRON
ejpam-5732	42	16	were	be	AUX
ejpam-5732	42	17	obtained	obtain	VERB
ejpam-5732	42	18	from	from	ADP
ejpam-5732	42	19	the	the	DET
ejpam-5732	42	20	concept	concept	NOUN
ejpam-5732	42	21	of	of	ADP
ejpam-5732	42	22	paracompact	paracompact	ADJ
ejpam-5732	42	23	spaces	space	NOUN
ejpam-5732	42	24	.	.	PUNCT
ejpam-5732	43	1	within	within	ADP
ejpam-5732	43	2	the	the	DET
ejpam-5732	43	3	context	context	NOUN
ejpam-5732	43	4	of	of	ADP
ejpam-5732	43	5	ideal	ideal	ADJ
ejpam-5732	43	6	topological	topological	ADJ
ejpam-5732	43	7	spaces	space	NOUN
ejpam-5732	43	8	,	,	PUNCT
ejpam-5732	43	9	sanabria	sanabria	PROPN
ejpam-5732	43	10	et	et	PROPN
ejpam-5732	43	11	al	al	PROPN
ejpam-5732	43	12	.	.	PUNCT
ejpam-5732	44	1	[	[	X
ejpam-5732	44	2	22	22	NUM
ejpam-5732	44	3	]	]	PUNCT
ejpam-5732	44	4	conducted	conduct	VERB
ejpam-5732	44	5	an	an	DET
ejpam-5732	44	6	investigation	investigation	NOUN
ejpam-5732	44	7	into	into	ADP
ejpam-5732	44	8	the	the	DET
ejpam-5732	44	9	concept	concept	NOUN
ejpam-5732	44	10	of	of	ADP
ejpam-5732	44	11	s	s	NOUN
ejpam-5732	44	12	-	-	NOUN
ejpam-5732	44	13	paracompactness	paracompactness	NOUN
ejpam-5732	44	14	.	.	PUNCT
ejpam-5732	45	1	the	the	DET
ejpam-5732	45	2	focus	focus	NOUN
ejpam-5732	45	3	of	of	ADP
ejpam-5732	45	4	their	their	PRON
ejpam-5732	45	5	research	research	NOUN
ejpam-5732	45	6	was	be	AUX
ejpam-5732	45	7	on	on	ADP
ejpam-5732	45	8	the	the	DET
ejpam-5732	45	9	development	development	NOUN
ejpam-5732	45	10	and	and	CCONJ
ejpam-5732	45	11	investigation	investigation	NOUN
ejpam-5732	45	12	of	of	ADP
ejpam-5732	45	13	a	a	DET
ejpam-5732	45	14	new	new	ADJ
ejpam-5732	45	15	category	category	NOUN
ejpam-5732	45	16	of	of	ADP
ejpam-5732	45	17	spaces	space	NOUN
ejpam-5732	45	18	,	,	PUNCT
ejpam-5732	45	19	which	which	PRON
ejpam-5732	45	20	they	they	PRON
ejpam-5732	45	21	referred	refer	VERB
ejpam-5732	45	22	to	to	ADP
ejpam-5732	45	23	as	as	SCONJ
ejpam-5732	45	24	i	i	PROPN
ejpam-5732	45	25	-	-	PUNCT
ejpam-5732	45	26	s	s	NOUN
ejpam-5732	45	27	-	-	PUNCT
ejpam-5732	45	28	paracompact	paracompact	ADJ
ejpam-5732	45	29	spaces	space	NOUN
ejpam-5732	45	30	.	.	PUNCT
ejpam-5732	46	1	these	these	DET
ejpam-5732	46	2	spaces	space	NOUN
ejpam-5732	46	3	were	be	AUX
ejpam-5732	46	4	constructed	construct	VERB
ejpam-5732	46	5	inside	inside	ADP
ejpam-5732	46	6	the	the	DET
ejpam-5732	46	7	framework	framework	NOUN
ejpam-5732	46	8	of	of	ADP
ejpam-5732	46	9	an	an	DET
ejpam-5732	46	10	ideal	ideal	ADJ
ejpam-5732	46	11	topological	topological	ADJ
ejpam-5732	46	12	space	space	NOUN
ejpam-5732	46	13	.	.	PUNCT
ejpam-5732	47	1	spaces	space	NOUN
ejpam-5732	47	2	that	that	PRON
ejpam-5732	47	3	are	be	AUX
ejpam-5732	47	4	both	both	PRON
ejpam-5732	47	5	s	s	NOUN
ejpam-5732	47	6	-	-	NOUN
ejpam-5732	47	7	paracompact	paracompact	ADJ
ejpam-5732	47	8	and	and	CCONJ
ejpam-5732	47	9	iparacompact	iparacompact	ADJ
ejpam-5732	47	10	are	be	AUX
ejpam-5732	47	11	included	include	VERB
ejpam-5732	47	12	in	in	ADP
ejpam-5732	47	13	this	this	DET
ejpam-5732	47	14	class	class	NOUN
ejpam-5732	47	15	.	.	PUNCT
ejpam-5732	48	1	in	in	ADP
ejpam-5732	48	2	2016	2016	NUM
ejpam-5732	48	3	,	,	PUNCT
ejpam-5732	48	4	al	al	PROPN
ejpam-5732	48	5	-	-	PUNCT
ejpam-5732	48	6	jarrah	jarrah	PROPN
ejpam-5732	48	7	[	[	X
ejpam-5732	48	8	3	3	NUM
ejpam-5732	48	9	]	]	PUNCT
ejpam-5732	48	10	introduced	introduce	VERB
ejpam-5732	48	11	the	the	DET
ejpam-5732	48	12	concept	concept	NOUN
ejpam-5732	48	13	of	of	ADP
ejpam-5732	48	14	β1	β1	PROPN
ejpam-5732	48	15	-	-	PUNCT
ejpam-5732	48	16	paracompactness	paracompactness	NOUN
ejpam-5732	48	17	,	,	PUNCT
ejpam-5732	48	18	employing	employ	VERB
ejpam-5732	48	19	the	the	DET
ejpam-5732	48	20	definition	definition	NOUN
ejpam-5732	48	21	of	of	ADP
ejpam-5732	48	22	β	β	NOUN
ejpam-5732	48	23	-	-	VERB
ejpam-5732	48	24	open	open	ADJ
ejpam-5732	48	25	as	as	SCONJ
ejpam-5732	48	26	follows	follow	VERB
ejpam-5732	48	27	:	:	PUNCT
ejpam-5732	48	28	a	a	DET
ejpam-5732	48	29	topological	topological	ADJ
ejpam-5732	48	30	space	space	NOUN
ejpam-5732	48	31	(	(	PUNCT
ejpam-5732	48	32	x	x	X
ejpam-5732	48	33	,	,	PUNCT
ejpam-5732	48	34	τ	τ	X
ejpam-5732	48	35	)	)	PUNCT
ejpam-5732	48	36	is	be	AUX
ejpam-5732	48	37	β1	β1	NOUN
ejpam-5732	48	38	-	-	PUNCT
ejpam-5732	48	39	paracompact	paracompact	NOUN
ejpam-5732	48	40	if	if	SCONJ
ejpam-5732	48	41	every	every	DET
ejpam-5732	48	42	β	β	NOUN
ejpam-5732	48	43	-	-	ADJ
ejpam-5732	48	44	open	open	ADJ
ejpam-5732	48	45	cover	cover	NOUN
ejpam-5732	48	46	of	of	ADP
ejpam-5732	48	47	x	x	PUNCT
ejpam-5732	48	48	has	have	VERB
ejpam-5732	48	49	a	a	DET
ejpam-5732	48	50	locally	locally	ADV
ejpam-5732	48	51	finite	finite	ADJ
ejpam-5732	48	52	open	open	ADJ
ejpam-5732	48	53	refinement	refinement	NOUN
ejpam-5732	48	54	.	.	PUNCT
ejpam-5732	49	1	in	in	ADP
ejpam-5732	49	2	2019	2019	NUM
ejpam-5732	49	3	,	,	PUNCT
ejpam-5732	49	4	qahis	qahis	PRON
ejpam-5732	49	5	[	[	X
ejpam-5732	49	6	21	21	NUM
ejpam-5732	49	7	]	]	PUNCT
ejpam-5732	49	8	developed	develop	VERB
ejpam-5732	49	9	a	a	DET
ejpam-5732	49	10	novel	novel	ADJ
ejpam-5732	49	11	class	class	NOUN
ejpam-5732	49	12	of	of	ADP
ejpam-5732	49	13	β1	β1	NOUN
ejpam-5732	49	14	-	-	PUNCT
ejpam-5732	49	15	paracompact	paracompact	NOUN
ejpam-5732	49	16	spaces	space	NOUN
ejpam-5732	49	17	related	relate	VERB
ejpam-5732	49	18	to	to	ADP
ejpam-5732	49	19	an	an	DET
ejpam-5732	49	20	ideal	ideal	NOUN
ejpam-5732	49	21	,	,	PUNCT
ejpam-5732	49	22	analyzing	analyze	VERB
ejpam-5732	49	23	their	their	PRON
ejpam-5732	49	24	characterizations	characterization	NOUN
ejpam-5732	49	25	and	and	CCONJ
ejpam-5732	49	26	exploring	explore	VERB
ejpam-5732	49	27	the	the	DET
ejpam-5732	49	28	corresponding	corresponding	ADJ
ejpam-5732	49	29	invariants	invariant	NOUN
ejpam-5732	49	30	.	.	PUNCT
ejpam-5732	50	1	this	this	DET
ejpam-5732	50	2	study	study	NOUN
ejpam-5732	50	3	presents	present	VERB
ejpam-5732	50	4	a	a	DET
ejpam-5732	50	5	new	new	ADJ
ejpam-5732	50	6	classification	classification	NOUN
ejpam-5732	50	7	of	of	ADP
ejpam-5732	50	8	β1	β1	NOUN
ejpam-5732	50	9	-	-	PUNCT
ejpam-5732	50	10	paracompact	paracompact	NOUN
ejpam-5732	50	11	spaces	space	NOUN
ejpam-5732	50	12	that	that	PRON
ejpam-5732	50	13	is	be	AUX
ejpam-5732	50	14	wider	wide	ADJ
ejpam-5732	50	15	than	than	ADP
ejpam-5732	50	16	that	that	PRON
ejpam-5732	50	17	suggested	suggest	VERB
ejpam-5732	50	18	by	by	ADP
ejpam-5732	50	19	qahis	qahis	NOUN
ejpam-5732	50	20	.	.	PROPN
ejpam-5732	51	1	2	2	X
ejpam-5732	51	2	.	.	NUM
ejpam-5732	51	3	preliminaries	preliminary	NOUN
ejpam-5732	51	4	in	in	ADP
ejpam-5732	51	5	this	this	DET
ejpam-5732	51	6	article	article	NOUN
ejpam-5732	51	7	,	,	PUNCT
ejpam-5732	51	8	the	the	DET
ejpam-5732	51	9	notation	notation	NOUN
ejpam-5732	51	10	(	(	PUNCT
ejpam-5732	51	11	x	x	X
ejpam-5732	51	12	,	,	PUNCT
ejpam-5732	51	13	τ	τ	X
ejpam-5732	51	14	)	)	PUNCT
ejpam-5732	51	15	denotes	denote	VERB
ejpam-5732	51	16	a	a	DET
ejpam-5732	51	17	topological	topological	ADJ
ejpam-5732	51	18	space	space	NOUN
ejpam-5732	51	19	without	without	ADP
ejpam-5732	51	20	any	any	DET
ejpam-5732	51	21	assumptions	assumption	NOUN
ejpam-5732	51	22	on	on	ADP
ejpam-5732	51	23	separation	separation	NOUN
ejpam-5732	51	24	axioms	axiom	NOUN
ejpam-5732	51	25	.	.	PUNCT
ejpam-5732	52	1	for	for	ADP
ejpam-5732	52	2	a	a	DET
ejpam-5732	52	3	subset	subset	NOUN
ejpam-5732	52	4	a	a	PRON
ejpam-5732	52	5	of	of	ADP
ejpam-5732	52	6	a	a	DET
ejpam-5732	52	7	topological	topological	ADJ
ejpam-5732	52	8	space	space	NOUN
ejpam-5732	52	9	(	(	PUNCT
ejpam-5732	52	10	x	x	X
ejpam-5732	52	11	,	,	PUNCT
ejpam-5732	52	12	τ	τ	PROPN
ejpam-5732	52	13	)	)	PUNCT
ejpam-5732	52	14	,	,	PUNCT
ejpam-5732	52	15	ci(a	ci(a	NOUN
ejpam-5732	52	16	)	)	PUNCT
ejpam-5732	52	17	represents	represent	VERB
ejpam-5732	52	18	the	the	DET
ejpam-5732	52	19	closure	closure	NOUN
ejpam-5732	52	20	of	of	ADP
ejpam-5732	52	21	a	a	DET
ejpam-5732	52	22	in	in	ADP
ejpam-5732	52	23	(	(	PUNCT
ejpam-5732	52	24	x	x	NOUN
ejpam-5732	52	25	,	,	PUNCT
ejpam-5732	52	26	τ	τ	PROPN
ejpam-5732	52	27	)	)	PUNCT
ejpam-5732	52	28	,	,	PUNCT
ejpam-5732	52	29	whereas	whereas	SCONJ
ejpam-5732	52	30	int(a	int(a	NOUN
ejpam-5732	52	31	)	)	PUNCT
ejpam-5732	52	32	signifies	signify	VERB
ejpam-5732	52	33	the	the	DET
ejpam-5732	52	34	interior	interior	NOUN
ejpam-5732	52	35	of	of	ADP
ejpam-5732	52	36	a	a	DET
ejpam-5732	52	37	in	in	ADP
ejpam-5732	52	38	(	(	PUNCT
ejpam-5732	52	39	x	x	NOUN
ejpam-5732	52	40	,	,	PUNCT
ejpam-5732	52	41	τ	τ	PROPN
ejpam-5732	52	42	)	)	PUNCT
ejpam-5732	52	43	.	.	PUNCT
ejpam-5732	53	1	an	an	DET
ejpam-5732	53	2	ideal	ideal	NOUN
ejpam-5732	53	3	i	i	PRON
ejpam-5732	53	4	on	on	ADP
ejpam-5732	53	5	a	a	DET
ejpam-5732	53	6	topological	topological	ADJ
ejpam-5732	53	7	space	space	NOUN
ejpam-5732	53	8	(	(	PUNCT
ejpam-5732	53	9	x	x	X
ejpam-5732	53	10	,	,	PUNCT
ejpam-5732	53	11	τ	τ	X
ejpam-5732	53	12	)	)	PUNCT
ejpam-5732	53	13	is	be	AUX
ejpam-5732	53	14	a	a	DET
ejpam-5732	53	15	nonempty	nonempty	ADJ
ejpam-5732	53	16	collection	collection	NOUN
ejpam-5732	53	17	of	of	ADP
ejpam-5732	53	18	subsets	subset	NOUN
ejpam-5732	53	19	of	of	ADP
ejpam-5732	53	20	x	x	PRON
ejpam-5732	53	21	that	that	PRON
ejpam-5732	53	22	fulfills	fulfill	VERB
ejpam-5732	53	23	the	the	DET
ejpam-5732	53	24	following	follow	VERB
ejpam-5732	53	25	criteria	criterion	NOUN
ejpam-5732	53	26	:	:	PUNCT
ejpam-5732	53	27	(	(	PUNCT
ejpam-5732	53	28	i	i	NOUN
ejpam-5732	53	29	)	)	PUNCT
ejpam-5732	54	1	a	a	PRON
ejpam-5732	54	2	∈	∈	NOUN
ejpam-5732	55	1	i	i	PRON
ejpam-5732	55	2	and	and	CCONJ
ejpam-5732	55	3	b	b	PROPN
ejpam-5732	55	4	⊂	⊂	PROPN
ejpam-5732	55	5	a	a	PRON
ejpam-5732	55	6	implies	imply	VERB
ejpam-5732	55	7	b	b	X
ejpam-5732	55	8	∈	∈	PROPN
ejpam-5732	55	9	i	i	PRON
ejpam-5732	55	10	,	,	PUNCT
ejpam-5732	55	11	and	and	CCONJ
ejpam-5732	55	12	c.	c.	PROPN
ejpam-5732	55	13	boonpok	boonpok	PROPN
ejpam-5732	55	14	,	,	PUNCT
ejpam-5732	55	15	a.	a.	PROPN
ejpam-5732	55	16	sama	sama	PROPN
ejpam-5732	55	17	-	-	PUNCT
ejpam-5732	55	18	ae	ae	PROPN
ejpam-5732	55	19	/	/	SYM
ejpam-5732	55	20	eur	eur	PROPN
ejpam-5732	55	21	.	.	PUNCT
ejpam-5732	56	1	j.	j.	PROPN
ejpam-5732	56	2	pure	pure	PROPN
ejpam-5732	56	3	appl	appl	PROPN
ejpam-5732	56	4	.	.	PROPN
ejpam-5732	56	5	math	math	PROPN
ejpam-5732	56	6	,	,	PUNCT
ejpam-5732	56	7	18	18	NUM
ejpam-5732	56	8	(	(	PUNCT
ejpam-5732	56	9	1	1	NUM
ejpam-5732	56	10	)	)	PUNCT
ejpam-5732	56	11	(	(	PUNCT
ejpam-5732	56	12	2025	2025	NUM
ejpam-5732	56	13	)	)	PUNCT
ejpam-5732	56	14	,	,	PUNCT
ejpam-5732	56	15	5732	5732	NUM
ejpam-5732	56	16	3	3	NUM
ejpam-5732	56	17	of	of	ADP
ejpam-5732	56	18	13	13	NUM
ejpam-5732	56	19	(	(	PUNCT
ejpam-5732	56	20	ii	ii	NOUN
ejpam-5732	56	21	)	)	PUNCT
ejpam-5732	57	1	a	a	DET
ejpam-5732	57	2	∈	∈	PROPN
ejpam-5732	57	3	i	i	PRON
ejpam-5732	57	4	and	and	CCONJ
ejpam-5732	57	5	b	b	X
ejpam-5732	57	6	∈	∈	PROPN
ejpam-5732	57	7	i	i	PRON
ejpam-5732	57	8	implies	imply	VERB
ejpam-5732	57	9	a	a	DET
ejpam-5732	57	10	∪b	∪b	PUNCT
ejpam-5732	57	11	∈	∈	PROPN
ejpam-5732	57	12	i.	i.	NOUN
ejpam-5732	57	13	an	an	DET
ejpam-5732	57	14	ideal	ideal	ADJ
ejpam-5732	57	15	topological	topological	ADJ
ejpam-5732	57	16	space	space	NOUN
ejpam-5732	57	17	(	(	PUNCT
ejpam-5732	57	18	x	x	X
ejpam-5732	57	19	,	,	PUNCT
ejpam-5732	57	20	τ	τ	PROPN
ejpam-5732	57	21	,	,	PUNCT
ejpam-5732	57	22	i	i	PROPN
ejpam-5732	57	23	)	)	PUNCT
ejpam-5732	57	24	is	be	AUX
ejpam-5732	57	25	characterized	characterize	VERB
ejpam-5732	57	26	as	as	ADP
ejpam-5732	57	27	a	a	DET
ejpam-5732	57	28	topological	topological	ADJ
ejpam-5732	57	29	space	space	NOUN
ejpam-5732	57	30	(	(	PUNCT
ejpam-5732	57	31	x	x	X
ejpam-5732	57	32	,	,	PUNCT
ejpam-5732	57	33	τ	τ	X
ejpam-5732	57	34	)	)	PUNCT
ejpam-5732	57	35	paired	pair	VERB
ejpam-5732	57	36	with	with	ADP
ejpam-5732	57	37	an	an	DET
ejpam-5732	57	38	ideal	ideal	NOUN
ejpam-5732	57	39	i	i	PRON
ejpam-5732	57	40	on	on	ADP
ejpam-5732	57	41	the	the	DET
ejpam-5732	57	42	set	set	NOUN
ejpam-5732	57	43	x.	x.	NOUN
ejpam-5732	58	1	the	the	DET
ejpam-5732	58	2	collection	collection	NOUN
ejpam-5732	58	3	of	of	ADP
ejpam-5732	58	4	all	all	DET
ejpam-5732	58	5	subsets	subset	NOUN
ejpam-5732	58	6	of	of	ADP
ejpam-5732	58	7	x	x	SYM
ejpam-5732	58	8	is	be	AUX
ejpam-5732	58	9	represented	represent	VERB
ejpam-5732	58	10	as	as	ADP
ejpam-5732	58	11	p	p	PROPN
ejpam-5732	58	12	(	(	PUNCT
ejpam-5732	58	13	x	x	NOUN
ejpam-5732	58	14	)	)	PUNCT
ejpam-5732	58	15	.	.	PUNCT
ejpam-5732	59	1	a	a	DET
ejpam-5732	59	2	set	set	NOUN
ejpam-5732	59	3	operator	operator	NOUN
ejpam-5732	59	4	(	(	PUNCT
ejpam-5732	59	5	.)∗	.)∗	X
ejpam-5732	59	6	:	:	PUNCT
ejpam-5732	59	7	p	p	X
ejpam-5732	59	8	(	(	PUNCT
ejpam-5732	59	9	x	x	NOUN
ejpam-5732	59	10	)	)	PUNCT
ejpam-5732	59	11	→	→	SYM
ejpam-5732	59	12	p	p	X
ejpam-5732	59	13	(	(	PUNCT
ejpam-5732	59	14	x	x	NOUN
ejpam-5732	59	15	)	)	PUNCT
ejpam-5732	59	16	,	,	PUNCT
ejpam-5732	59	17	characterized	characterize	VERB
ejpam-5732	59	18	as	as	ADP
ejpam-5732	59	19	a	a	DET
ejpam-5732	59	20	local	local	ADJ
ejpam-5732	59	21	function	function	NOUN
ejpam-5732	59	22	[	[	X
ejpam-5732	59	23	17	17	NUM
ejpam-5732	59	24	]	]	PUNCT
ejpam-5732	59	25	,	,	PUNCT
ejpam-5732	59	26	is	be	AUX
ejpam-5732	59	27	defined	define	VERB
ejpam-5732	59	28	with	with	ADP
ejpam-5732	59	29	respect	respect	NOUN
ejpam-5732	59	30	to	to	ADP
ejpam-5732	59	31	τ	τ	PROPN
ejpam-5732	59	32	and	and	CCONJ
ejpam-5732	59	33	i	i	PRON
ejpam-5732	59	34	:	:	PUNCT
ejpam-5732	59	35	for	for	ADP
ejpam-5732	59	36	a	a	DET
ejpam-5732	59	37	⊂	⊂	PROPN
ejpam-5732	59	38	x	x	SYM
ejpam-5732	59	39	,	,	PUNCT
ejpam-5732	59	40	a∗(i	a∗(i	PROPN
ejpam-5732	59	41	,	,	PUNCT
ejpam-5732	59	42	τ	τ	X
ejpam-5732	59	43	)	)	PUNCT
ejpam-5732	59	44	=	=	PRON
ejpam-5732	60	1	{	{	PUNCT
ejpam-5732	60	2	x	x	PUNCT
ejpam-5732	60	3	∈	∈	PROPN
ejpam-5732	60	4	x	x	X
ejpam-5732	60	5	:	:	PUNCT
ejpam-5732	60	6	u	u	NOUN
ejpam-5732	60	7	∩	∩	NOUN
ejpam-5732	60	8	a	a	DET
ejpam-5732	60	9	̸∈	̸∈	PROPN
ejpam-5732	60	10	i	i	PROPN
ejpam-5732	60	11	for	for	ADP
ejpam-5732	60	12	all	all	DET
ejpam-5732	60	13	u	u	PROPN
ejpam-5732	60	14	∈	∈	NOUN
ejpam-5732	60	15	τ(x	τ(x	NOUN
ejpam-5732	60	16	)	)	PUNCT
ejpam-5732	60	17	}	}	PUNCT
ejpam-5732	60	18	,	,	PUNCT
ejpam-5732	60	19	where	where	SCONJ
ejpam-5732	60	20	τ(x	τ(x	NOUN
ejpam-5732	60	21	)	)	PUNCT
ejpam-5732	60	22	=	=	PRON
ejpam-5732	60	23	{	{	PUNCT
ejpam-5732	60	24	u	u	X
ejpam-5732	60	25	∈	∈	PROPN
ejpam-5732	60	26	τ	τ	X
ejpam-5732	60	27	:	:	PUNCT
ejpam-5732	60	28	x	x	SYM
ejpam-5732	60	29	∈	∈	PROPN
ejpam-5732	60	30	u	u	NOUN
ejpam-5732	60	31	}	}	PUNCT
ejpam-5732	60	32	.	.	PUNCT
ejpam-5732	61	1	we	we	PRON
ejpam-5732	61	2	simply	simply	ADV
ejpam-5732	61	3	write	write	VERB
ejpam-5732	61	4	a∗	a∗	PROPN
ejpam-5732	61	5	instead	instead	ADV
ejpam-5732	61	6	of	of	ADP
ejpam-5732	61	7	a∗(i	a∗(i	PROPN
ejpam-5732	61	8	,	,	PUNCT
ejpam-5732	61	9	τ	τ	PROPN
ejpam-5732	61	10	)	)	PUNCT
ejpam-5732	61	11	.	.	PUNCT
ejpam-5732	62	1	a	a	DET
ejpam-5732	62	2	topology	topology	NOUN
ejpam-5732	62	3	τ∗(i	τ∗(i	PROPN
ejpam-5732	62	4	)	)	PUNCT
ejpam-5732	62	5	,	,	PUNCT
ejpam-5732	62	6	or	or	CCONJ
ejpam-5732	62	7	τ∗	τ∗	NOUN
ejpam-5732	62	8	for	for	ADP
ejpam-5732	62	9	brevity	brevity	NOUN
ejpam-5732	62	10	that	that	PRON
ejpam-5732	62	11	is	be	AUX
ejpam-5732	62	12	finer	fine	ADJ
ejpam-5732	62	13	than	than	SCONJ
ejpam-5732	62	14	τ	τ	PROPN
ejpam-5732	62	15	may	may	AUX
ejpam-5732	62	16	be	be	AUX
ejpam-5732	62	17	created	create	VERB
ejpam-5732	62	18	for	for	ADP
ejpam-5732	62	19	any	any	DET
ejpam-5732	62	20	ideal	ideal	ADJ
ejpam-5732	62	21	topological	topological	ADJ
ejpam-5732	62	22	space	space	NOUN
ejpam-5732	62	23	,	,	PUNCT
ejpam-5732	62	24	defined	define	VERB
ejpam-5732	62	25	by	by	ADP
ejpam-5732	62	26	β(i	β(i	NOUN
ejpam-5732	62	27	,	,	PUNCT
ejpam-5732	62	28	τ	τ	X
ejpam-5732	62	29	)	)	PUNCT
ejpam-5732	62	30	=	=	PRON
ejpam-5732	62	31	{	{	PUNCT
ejpam-5732	62	32	u	u	NOUN
ejpam-5732	62	33	−	−	PROPN
ejpam-5732	63	1	i	i	PRON
ejpam-5732	63	2	:	:	PUNCT
ejpam-5732	63	3	u	u	PROPN
ejpam-5732	63	4	∈	∈	PROPN
ejpam-5732	63	5	τ	τ	X
ejpam-5732	63	6	and	and	CCONJ
ejpam-5732	63	7	i	i	PRON
ejpam-5732	63	8	∈	∈	PROPN
ejpam-5732	63	9	i	i	PRON
ejpam-5732	63	10	}	}	PUNCT
ejpam-5732	63	11	.	.	PUNCT
ejpam-5732	64	1	nevertheless	nevertheless	ADV
ejpam-5732	64	2	,	,	PUNCT
ejpam-5732	64	3	β(i	β(i	PRON
ejpam-5732	64	4	,	,	PUNCT
ejpam-5732	64	5	τ	τ	X
ejpam-5732	64	6	)	)	PUNCT
ejpam-5732	64	7	does	do	AUX
ejpam-5732	64	8	not	not	PART
ejpam-5732	64	9	uniformly	uniformly	ADV
ejpam-5732	64	10	define	define	VERB
ejpam-5732	64	11	a	a	DET
ejpam-5732	64	12	topology	topology	NOUN
ejpam-5732	64	13	.	.	PUNCT
ejpam-5732	65	1	furthermore	furthermore	ADV
ejpam-5732	65	2	,	,	PUNCT
ejpam-5732	65	3	cl∗(a	cl∗(a	NOUN
ejpam-5732	65	4	)	)	PUNCT
ejpam-5732	65	5	=	=	PUNCT
ejpam-5732	65	6	a	a	DET
ejpam-5732	65	7	∪	∪	ADJ
ejpam-5732	65	8	a∗	a∗	NOUN
ejpam-5732	65	9	provides	provide	VERB
ejpam-5732	65	10	a	a	DET
ejpam-5732	65	11	kuratowski	kuratowski	ADJ
ejpam-5732	65	12	closure	closure	NOUN
ejpam-5732	65	13	operator	operator	NOUN
ejpam-5732	65	14	for	for	ADP
ejpam-5732	65	15	τ∗.	τ∗.	NOUN
ejpam-5732	65	16	we	we	PRON
ejpam-5732	65	17	simply	simply	ADV
ejpam-5732	65	18	write	write	VERB
ejpam-5732	65	19	for	for	ADP
ejpam-5732	65	20	τ∗	τ∗	NOUN
ejpam-5732	65	21	for	for	ADP
ejpam-5732	65	22	τ∗(i	τ∗(i	PROPN
ejpam-5732	65	23	,	,	PUNCT
ejpam-5732	65	24	τ	τ	PROPN
ejpam-5732	65	25	)	)	PUNCT
ejpam-5732	65	26	.	.	PUNCT
ejpam-5732	66	1	if	if	SCONJ
ejpam-5732	66	2	β(i	β(i	PRON
ejpam-5732	66	3	,	,	PUNCT
ejpam-5732	66	4	τ	τ	X
ejpam-5732	66	5	)	)	PUNCT
ejpam-5732	66	6	=	=	SYM
ejpam-5732	66	7	τ∗	τ∗	NOUN
ejpam-5732	66	8	,	,	PUNCT
ejpam-5732	66	9	then	then	ADV
ejpam-5732	66	10	we	we	PRON
ejpam-5732	66	11	say	say	VERB
ejpam-5732	66	12	i	i	PRON
ejpam-5732	66	13	is	be	AUX
ejpam-5732	66	14	τ	τ	PROPN
ejpam-5732	66	15	-simple	-simple	X
ejpam-5732	67	1	[	[	X
ejpam-5732	67	2	15	15	NUM
ejpam-5732	67	3	]	]	PUNCT
ejpam-5732	67	4	.	.	PUNCT
ejpam-5732	68	1	if	if	SCONJ
ejpam-5732	68	2	(	(	PUNCT
ejpam-5732	68	3	x	x	X
ejpam-5732	68	4	,	,	PUNCT
ejpam-5732	68	5	τ	τ	PROPN
ejpam-5732	68	6	,	,	PUNCT
ejpam-5732	68	7	i	i	NOUN
ejpam-5732	68	8	)	)	PUNCT
ejpam-5732	68	9	satisfies	satisfy	VERB
ejpam-5732	68	10	this	this	DET
ejpam-5732	68	11	condition	condition	NOUN
ejpam-5732	68	12	,	,	PUNCT
ejpam-5732	68	13	then	then	ADV
ejpam-5732	68	14	i	i	PRON
ejpam-5732	68	15	is	be	AUX
ejpam-5732	68	16	said	say	VERB
ejpam-5732	68	17	to	to	PART
ejpam-5732	68	18	be	be	AUX
ejpam-5732	68	19	compatible	compatible	ADJ
ejpam-5732	68	20	[	[	X
ejpam-5732	68	21	15	15	NUM
ejpam-5732	68	22	]	]	PUNCT
ejpam-5732	68	23	or	or	CCONJ
ejpam-5732	68	24	i	i	PRON
ejpam-5732	68	25	is	be	AUX
ejpam-5732	68	26	said	say	VERB
ejpam-5732	68	27	to	to	PART
ejpam-5732	68	28	be	be	AUX
ejpam-5732	68	29	τ	τ	PROPN
ejpam-5732	68	30	-local	-local	PROPN
ejpam-5732	68	31	.	.	PUNCT
ejpam-5732	69	1	given	give	VERB
ejpam-5732	69	2	an	an	DET
ejpam-5732	69	3	ideal	ideal	ADJ
ejpam-5732	69	4	topological	topological	ADJ
ejpam-5732	69	5	space	space	NOUN
ejpam-5732	69	6	(	(	PUNCT
ejpam-5732	69	7	x	x	X
ejpam-5732	69	8	,	,	PUNCT
ejpam-5732	69	9	τ	τ	PROPN
ejpam-5732	69	10	,	,	PUNCT
ejpam-5732	69	11	i	i	PROPN
ejpam-5732	69	12	)	)	PUNCT
ejpam-5732	69	13	,	,	PUNCT
ejpam-5732	69	14	we	we	PRON
ejpam-5732	69	15	say	say	VERB
ejpam-5732	69	16	i	i	PRON
ejpam-5732	69	17	is	be	AUX
ejpam-5732	69	18	i	i	NOUN
ejpam-5732	69	19	-	-	PUNCT
ejpam-5732	69	20	codense	codense	NOUN
ejpam-5732	69	21	if	if	SCONJ
ejpam-5732	69	22	i	i	PRON
ejpam-5732	69	23	∩	∩	NOUN
ejpam-5732	69	24	τ	τ	X
ejpam-5732	69	25	=	=	PUNCT
ejpam-5732	69	26	{	{	PUNCT
ejpam-5732	69	27	∅	∅	NOUN
ejpam-5732	69	28	}	}	PUNCT
ejpam-5732	69	29	.	.	PUNCT
ejpam-5732	70	1	definition	definition	NOUN
ejpam-5732	70	2	1	1	NUM
ejpam-5732	70	3	.	.	PUNCT
ejpam-5732	71	1	[	[	X
ejpam-5732	71	2	14	14	NUM
ejpam-5732	71	3	]	]	PUNCT
ejpam-5732	71	4	let	let	VERB
ejpam-5732	71	5	a	a	PRON
ejpam-5732	71	6	be	be	AUX
ejpam-5732	71	7	a	a	DET
ejpam-5732	71	8	subset	subset	NOUN
ejpam-5732	71	9	of	of	ADP
ejpam-5732	71	10	an	an	DET
ejpam-5732	71	11	ideal	ideal	ADJ
ejpam-5732	71	12	topological	topological	ADJ
ejpam-5732	71	13	space	space	NOUN
ejpam-5732	71	14	(	(	PUNCT
ejpam-5732	71	15	x	x	X
ejpam-5732	71	16	,	,	PUNCT
ejpam-5732	71	17	τ	τ	PROPN
ejpam-5732	71	18	,	,	PUNCT
ejpam-5732	71	19	i	i	PROPN
ejpam-5732	71	20	)	)	PUNCT
ejpam-5732	71	21	.	.	PUNCT
ejpam-5732	72	1	a	a	DET
ejpam-5732	72	2	point	point	NOUN
ejpam-5732	72	3	x	x	X
ejpam-5732	72	4	∈	∈	NOUN
ejpam-5732	72	5	x	x	PUNCT
ejpam-5732	72	6	is	be	AUX
ejpam-5732	72	7	called	call	VERB
ejpam-5732	72	8	a	a	DET
ejpam-5732	72	9	δi	δi	NOUN
ejpam-5732	72	10	-	-	PUNCT
ejpam-5732	72	11	cluster	cluster	NOUN
ejpam-5732	72	12	point	point	NOUN
ejpam-5732	72	13	of	of	ADP
ejpam-5732	72	14	a	a	DET
ejpam-5732	72	15	if	if	NOUN
ejpam-5732	72	16	int(cl∗(u	int(cl∗(u	NOUN
ejpam-5732	72	17	)	)	PUNCT
ejpam-5732	72	18	)	)	PUNCT
ejpam-5732	72	19	∩	∩	NOUN
ejpam-5732	72	20	a	a	DET
ejpam-5732	72	21	̸=	̸=	PROPN
ejpam-5732	72	22	∅	∅	NOUN
ejpam-5732	72	23	for	for	ADP
ejpam-5732	72	24	every	every	DET
ejpam-5732	72	25	neighborhood	neighborhood	NOUN
ejpam-5732	72	26	u	u	NOUN
ejpam-5732	72	27	of	of	ADP
ejpam-5732	72	28	x.	x.	NOUN
ejpam-5732	72	29	the	the	DET
ejpam-5732	72	30	δcli(a	δcli(a	NOUN
ejpam-5732	72	31	)	)	PUNCT
ejpam-5732	72	32	represents	represent	VERB
ejpam-5732	72	33	the	the	DET
ejpam-5732	72	34	δi	δi	NOUN
ejpam-5732	72	35	-	-	PUNCT
ejpam-5732	72	36	closure	closure	NOUN
ejpam-5732	72	37	of	of	ADP
ejpam-5732	72	38	a	a	PRON
ejpam-5732	72	39	,	,	PUNCT
ejpam-5732	72	40	which	which	PRON
ejpam-5732	72	41	is	be	AUX
ejpam-5732	72	42	the	the	DET
ejpam-5732	72	43	set	set	NOUN
ejpam-5732	72	44	of	of	ADP
ejpam-5732	72	45	all	all	DET
ejpam-5732	72	46	δi	δi	NOUN
ejpam-5732	72	47	-	-	PUNCT
ejpam-5732	72	48	cluster	cluster	NOUN
ejpam-5732	72	49	points	point	NOUN
ejpam-5732	72	50	of	of	ADP
ejpam-5732	72	51	a.	a.	NOUN
ejpam-5732	72	52	a	a	DET
ejpam-5732	72	53	subset	subset	VERB
ejpam-5732	72	54	a	a	PRON
ejpam-5732	72	55	of	of	ADP
ejpam-5732	72	56	x	x	PRON
ejpam-5732	72	57	is	be	AUX
ejpam-5732	72	58	called	call	VERB
ejpam-5732	72	59	a	a	DET
ejpam-5732	72	60	δi	δi	ADV
ejpam-5732	72	61	-	-	PUNCT
ejpam-5732	72	62	closed	closed	ADJ
ejpam-5732	72	63	[	[	X
ejpam-5732	72	64	26	26	NUM
ejpam-5732	72	65	]	]	X
ejpam-5732	72	66	if	if	SCONJ
ejpam-5732	72	67	δcli(a	δcli(a	ADJ
ejpam-5732	72	68	)	)	PUNCT
ejpam-5732	72	69	=	=	SYM
ejpam-5732	72	70	a	a	NOUN
ejpam-5732	72	71	,	,	PUNCT
ejpam-5732	72	72	and	and	CCONJ
ejpam-5732	72	73	the	the	DET
ejpam-5732	72	74	complement	complement	NOUN
ejpam-5732	72	75	of	of	ADP
ejpam-5732	72	76	a	a	DET
ejpam-5732	72	77	δi	δi	ADV
ejpam-5732	72	78	-	-	PUNCT
ejpam-5732	72	79	closed	close	VERB
ejpam-5732	72	80	set	set	NOUN
ejpam-5732	72	81	is	be	AUX
ejpam-5732	72	82	called	call	VERB
ejpam-5732	72	83	a	a	DET
ejpam-5732	72	84	δi	δi	ADV
ejpam-5732	72	85	-	-	PUNCT
ejpam-5732	72	86	open	open	ADJ
ejpam-5732	72	87	set	set	NOUN
ejpam-5732	72	88	.	.	PUNCT
ejpam-5732	73	1	the	the	DET
ejpam-5732	73	2	union	union	NOUN
ejpam-5732	73	3	of	of	ADP
ejpam-5732	73	4	all	all	DET
ejpam-5732	73	5	δi	δi	ADV
ejpam-5732	73	6	-	-	PUNCT
ejpam-5732	73	7	open	open	ADJ
ejpam-5732	73	8	sets	set	NOUN
ejpam-5732	73	9	included	include	VERB
ejpam-5732	73	10	in	in	ADP
ejpam-5732	73	11	a	a	PRON
ejpam-5732	73	12	is	be	AUX
ejpam-5732	73	13	the	the	DET
ejpam-5732	73	14	δi	δi	NOUN
ejpam-5732	73	15	-	-	PUNCT
ejpam-5732	73	16	interior	interior	NOUN
ejpam-5732	73	17	of	of	ADP
ejpam-5732	73	18	a	a	PRON
ejpam-5732	73	19	,	,	PUNCT
ejpam-5732	73	20	which	which	PRON
ejpam-5732	73	21	will	will	AUX
ejpam-5732	73	22	be	be	AUX
ejpam-5732	73	23	represented	represent	VERB
ejpam-5732	73	24	by	by	ADP
ejpam-5732	73	25	δinti(a	δinti(a	PROPN
ejpam-5732	73	26	)	)	PUNCT
ejpam-5732	73	27	.	.	PUNCT
ejpam-5732	74	1	lemma	lemma	PROPN
ejpam-5732	74	2	1	1	NUM
ejpam-5732	74	3	.	.	PUNCT
ejpam-5732	75	1	[	[	X
ejpam-5732	75	2	14	14	NUM
ejpam-5732	75	3	]	]	PUNCT
ejpam-5732	75	4	let	let	VERB
ejpam-5732	75	5	a	a	PRON
ejpam-5732	75	6	and	and	CCONJ
ejpam-5732	75	7	b	b	NOUN
ejpam-5732	75	8	be	be	AUX
ejpam-5732	75	9	subsets	subset	NOUN
ejpam-5732	75	10	of	of	ADP
ejpam-5732	75	11	an	an	DET
ejpam-5732	75	12	ideal	ideal	ADJ
ejpam-5732	75	13	topological	topological	ADJ
ejpam-5732	75	14	space	space	NOUN
ejpam-5732	75	15	(	(	PUNCT
ejpam-5732	75	16	x	x	X
ejpam-5732	75	17	,	,	PUNCT
ejpam-5732	75	18	τ	τ	PROPN
ejpam-5732	75	19	,	,	PUNCT
ejpam-5732	75	20	i	i	PROPN
ejpam-5732	75	21	)	)	PUNCT
ejpam-5732	75	22	.	.	PUNCT
ejpam-5732	76	1	the	the	DET
ejpam-5732	76	2	following	follow	VERB
ejpam-5732	76	3	statements	statement	NOUN
ejpam-5732	76	4	are	be	AUX
ejpam-5732	76	5	true	true	ADJ
ejpam-5732	76	6	:	:	PUNCT
ejpam-5732	76	7	(	(	PUNCT
ejpam-5732	76	8	i	i	NOUN
ejpam-5732	76	9	)	)	PUNCT
ejpam-5732	76	10	if	if	SCONJ
ejpam-5732	76	11	a	a	DET
ejpam-5732	76	12	⊂	⊂	X
ejpam-5732	76	13	b	b	X
ejpam-5732	76	14	then	then	ADV
ejpam-5732	76	15	δcli(a	δcli(a	PROPN
ejpam-5732	76	16	)	)	PUNCT
ejpam-5732	76	17	⊂	⊂	PROPN
ejpam-5732	76	18	δcli(b	δcli(b	PROPN
ejpam-5732	76	19	)	)	PUNCT
ejpam-5732	76	20	.	.	PUNCT
ejpam-5732	77	1	(	(	PUNCT
ejpam-5732	77	2	ii	ii	NOUN
ejpam-5732	77	3	)	)	PUNCT
ejpam-5732	77	4	if	if	SCONJ
ejpam-5732	77	5	a	a	PRON
ejpam-5732	77	6	is	be	AUX
ejpam-5732	77	7	an	an	DET
ejpam-5732	77	8	open	open	ADJ
ejpam-5732	77	9	set	set	NOUN
ejpam-5732	77	10	,	,	PUNCT
ejpam-5732	77	11	then	then	ADV
ejpam-5732	77	12	δcli(a	δcli(a	ADJ
ejpam-5732	77	13	)	)	PUNCT
ejpam-5732	77	14	=	=	SYM
ejpam-5732	77	15	a.	a.	NOUN
ejpam-5732	77	16	(	(	PUNCT
ejpam-5732	77	17	iii	iii	NOUN
ejpam-5732	77	18	)	)	PUNCT
ejpam-5732	77	19	if	if	SCONJ
ejpam-5732	77	20	a	a	PRON
ejpam-5732	77	21	is	be	AUX
ejpam-5732	77	22	a	a	DET
ejpam-5732	77	23	closed	closed	ADJ
ejpam-5732	77	24	set	set	NOUN
ejpam-5732	77	25	,	,	PUNCT
ejpam-5732	77	26	then	then	ADV
ejpam-5732	77	27	δinti(a	δinti(a	PROPN
ejpam-5732	77	28	)	)	PUNCT
ejpam-5732	77	29	=	=	PUNCT
ejpam-5732	77	30	a.	a.	NOUN
ejpam-5732	77	31	definition	definition	NOUN
ejpam-5732	77	32	2	2	NUM
ejpam-5732	77	33	.	.	PUNCT
ejpam-5732	78	1	[	[	X
ejpam-5732	78	2	14	14	NUM
ejpam-5732	78	3	]	]	PUNCT
ejpam-5732	78	4	a	a	DET
ejpam-5732	78	5	subset	subset	NOUN
ejpam-5732	78	6	a	a	PRON
ejpam-5732	78	7	of	of	ADP
ejpam-5732	78	8	an	an	DET
ejpam-5732	78	9	ideal	ideal	ADJ
ejpam-5732	78	10	topological	topological	ADJ
ejpam-5732	78	11	space	space	NOUN
ejpam-5732	78	12	(	(	PUNCT
ejpam-5732	78	13	x	x	X
ejpam-5732	78	14	,	,	PUNCT
ejpam-5732	78	15	τ	τ	PROPN
ejpam-5732	78	16	,	,	PUNCT
ejpam-5732	78	17	i	i	PROPN
ejpam-5732	78	18	)	)	PUNCT
ejpam-5732	78	19	is	be	AUX
ejpam-5732	78	20	called	call	VERB
ejpam-5732	78	21	δ	δ	PROPN
ejpam-5732	78	22	-	-	PUNCT
ejpam-5732	78	23	βi	βi	ADV
ejpam-5732	78	24	-	-	PUNCT
ejpam-5732	78	25	open	open	ADJ
ejpam-5732	78	26	if	if	SCONJ
ejpam-5732	78	27	a	a	DET
ejpam-5732	78	28	⊂	⊂	PROPN
ejpam-5732	78	29	cl(int(δcli(a	cl(int(δcli(a	PROPN
ejpam-5732	78	30	)	)	PUNCT
ejpam-5732	78	31	)	)	PUNCT
ejpam-5732	78	32	)	)	PUNCT
ejpam-5732	78	33	and	and	CCONJ
ejpam-5732	78	34	is	be	AUX
ejpam-5732	78	35	called	call	VERB
ejpam-5732	78	36	δ	δ	PROPN
ejpam-5732	78	37	-	-	PUNCT
ejpam-5732	78	38	βi	βi	ADV
ejpam-5732	78	39	-	-	PUNCT
ejpam-5732	78	40	closed	closed	ADJ
ejpam-5732	78	41	if	if	SCONJ
ejpam-5732	78	42	int(cl(δinti(a	int(cl(δinti(a	NOUN
ejpam-5732	78	43	)	)	PUNCT
ejpam-5732	78	44	)	)	PUNCT
ejpam-5732	78	45	)	)	PUNCT
ejpam-5732	79	1	⊂	⊂	PROPN
ejpam-5732	79	2	a.	a.	NOUN
ejpam-5732	79	3	definition	definition	NOUN
ejpam-5732	79	4	3	3	NUM
ejpam-5732	79	5	.	.	PUNCT
ejpam-5732	80	1	[	[	X
ejpam-5732	80	2	14	14	NUM
ejpam-5732	80	3	]	]	X
ejpam-5732	80	4	let	let	AUX
ejpam-5732	80	5	(	(	PUNCT
ejpam-5732	80	6	x	x	NOUN
ejpam-5732	80	7	,	,	PUNCT
ejpam-5732	80	8	τ	τ	PROPN
ejpam-5732	80	9	,	,	PUNCT
ejpam-5732	80	10	i	i	PRON
ejpam-5732	80	11	)	)	PUNCT
ejpam-5732	80	12	be	be	VERB
ejpam-5732	80	13	an	an	DET
ejpam-5732	80	14	ideal	ideal	ADJ
ejpam-5732	80	15	topological	topological	ADJ
ejpam-5732	80	16	space	space	NOUN
ejpam-5732	80	17	.	.	PUNCT
ejpam-5732	81	1	the	the	DET
ejpam-5732	81	2	union	union	NOUN
ejpam-5732	81	3	of	of	ADP
ejpam-5732	81	4	all	all	DET
ejpam-5732	81	5	δ	δ	PROPN
ejpam-5732	81	6	-	-	PUNCT
ejpam-5732	81	7	βiopen	βiopen	ADJ
ejpam-5732	81	8	sets	set	NOUN
ejpam-5732	81	9	contained	contain	VERB
ejpam-5732	81	10	in	in	ADP
ejpam-5732	81	11	a	a	PRON
ejpam-5732	81	12	is	be	AUX
ejpam-5732	81	13	called	call	VERB
ejpam-5732	81	14	the	the	DET
ejpam-5732	81	15	δ	δ	PROPN
ejpam-5732	81	16	-	-	PUNCT
ejpam-5732	81	17	βi	βi	PROPN
ejpam-5732	81	18	-	-	NOUN
ejpam-5732	81	19	interior	interior	NOUN
ejpam-5732	81	20	of	of	ADP
ejpam-5732	81	21	a	a	DET
ejpam-5732	81	22	denoted	denote	VERB
ejpam-5732	81	23	by	by	ADP
ejpam-5732	81	24	δ	δ	PROPN
ejpam-5732	81	25	-	-	PROPN
ejpam-5732	81	26	βinti(a	βinti(a	NOUN
ejpam-5732	81	27	)	)	PUNCT
ejpam-5732	81	28	.	.	PUNCT
ejpam-5732	82	1	the	the	DET
ejpam-5732	82	2	intersection	intersection	NOUN
ejpam-5732	82	3	of	of	ADP
ejpam-5732	82	4	all	all	DET
ejpam-5732	82	5	δ	δ	PROPN
ejpam-5732	82	6	-	-	PUNCT
ejpam-5732	82	7	βi	βi	ADV
ejpam-5732	82	8	-	-	PUNCT
ejpam-5732	82	9	closed	close	VERB
ejpam-5732	82	10	sets	set	NOUN
ejpam-5732	82	11	containing	contain	VERB
ejpam-5732	82	12	a	a	PRON
ejpam-5732	82	13	is	be	AUX
ejpam-5732	82	14	called	call	VERB
ejpam-5732	82	15	the	the	DET
ejpam-5732	82	16	δ	δ	PROPN
ejpam-5732	82	17	-	-	PUNCT
ejpam-5732	82	18	βi	βi	NOUN
ejpam-5732	82	19	-	-	PUNCT
ejpam-5732	82	20	closure	closure	NOUN
ejpam-5732	82	21	of	of	ADP
ejpam-5732	82	22	a	a	DET
ejpam-5732	82	23	denoted	denote	VERB
ejpam-5732	82	24	by	by	ADP
ejpam-5732	82	25	δ	δ	PROPN
ejpam-5732	82	26	-	-	PUNCT
ejpam-5732	82	27	βcli(a	βcli(a	NOUN
ejpam-5732	82	28	)	)	PUNCT
ejpam-5732	82	29	.	.	PUNCT
ejpam-5732	83	1	the	the	DET
ejpam-5732	83	2	following	follow	VERB
ejpam-5732	83	3	lemma	lemma	PROPN
ejpam-5732	83	4	is	be	AUX
ejpam-5732	83	5	easily	easily	ADV
ejpam-5732	83	6	derived	derive	VERB
ejpam-5732	83	7	from	from	ADP
ejpam-5732	83	8	the	the	DET
ejpam-5732	83	9	definition	definition	NOUN
ejpam-5732	83	10	3	3	NUM
ejpam-5732	83	11	.	.	PUNCT
ejpam-5732	84	1	lemma	lemma	PROPN
ejpam-5732	84	2	2	2	X
ejpam-5732	84	3	.	.	PUNCT
ejpam-5732	84	4	let	let	VERB
ejpam-5732	84	5	a	a	DET
ejpam-5732	84	6	be	be	AUX
ejpam-5732	84	7	a	a	DET
ejpam-5732	84	8	subset	subset	NOUN
ejpam-5732	84	9	of	of	ADP
ejpam-5732	84	10	an	an	DET
ejpam-5732	84	11	ideal	ideal	ADJ
ejpam-5732	84	12	topological	topological	ADJ
ejpam-5732	84	13	space	space	NOUN
ejpam-5732	84	14	(	(	PUNCT
ejpam-5732	84	15	x	x	X
ejpam-5732	84	16	,	,	PUNCT
ejpam-5732	84	17	τ	τ	PROPN
ejpam-5732	84	18	,	,	PUNCT
ejpam-5732	84	19	i	i	PROPN
ejpam-5732	84	20	)	)	PUNCT
ejpam-5732	84	21	.	.	PUNCT
ejpam-5732	85	1	the	the	DET
ejpam-5732	85	2	following	follow	VERB
ejpam-5732	85	3	statements	statement	NOUN
ejpam-5732	85	4	are	be	AUX
ejpam-5732	85	5	true	true	ADJ
ejpam-5732	85	6	:	:	PUNCT
ejpam-5732	85	7	(	(	PUNCT
ejpam-5732	85	8	i	i	NOUN
ejpam-5732	85	9	)	)	PUNCT
ejpam-5732	85	10	δ	δ	PROPN
ejpam-5732	85	11	-	-	PUNCT
ejpam-5732	85	12	βcli(a	βcli(a	NOUN
ejpam-5732	85	13	)	)	PUNCT
ejpam-5732	85	14	⊂	⊂	PROPN
ejpam-5732	85	15	cl(a	cl(a	X
ejpam-5732	85	16	)	)	PUNCT
ejpam-5732	85	17	.	.	PUNCT
ejpam-5732	86	1	(	(	PUNCT
ejpam-5732	86	2	ii	ii	NOUN
ejpam-5732	86	3	)	)	PUNCT
ejpam-5732	86	4	if	if	SCONJ
ejpam-5732	86	5	a	a	PRON
ejpam-5732	86	6	is	be	AUX
ejpam-5732	86	7	open	open	ADJ
ejpam-5732	86	8	,	,	PUNCT
ejpam-5732	86	9	then	then	ADV
ejpam-5732	86	10	a	a	PRON
ejpam-5732	86	11	is	be	AUX
ejpam-5732	86	12	δ	δ	PROPN
ejpam-5732	86	13	-	-	PUNCT
ejpam-5732	86	14	βi	βi	ADV
ejpam-5732	86	15	-	-	PUNCT
ejpam-5732	86	16	open	open	ADJ
ejpam-5732	86	17	.	.	PUNCT
ejpam-5732	87	1	(	(	PUNCT
ejpam-5732	87	2	iii	iii	X
ejpam-5732	87	3	)	)	PUNCT
ejpam-5732	87	4	if	if	SCONJ
ejpam-5732	87	5	a	a	PRON
ejpam-5732	87	6	is	be	AUX
ejpam-5732	87	7	closed	closed	ADJ
ejpam-5732	87	8	,	,	PUNCT
ejpam-5732	87	9	then	then	ADV
ejpam-5732	87	10	a	a	PRON
ejpam-5732	87	11	is	be	AUX
ejpam-5732	87	12	δ	δ	PROPN
ejpam-5732	87	13	-	-	PUNCT
ejpam-5732	87	14	βi	βi	ADV
ejpam-5732	87	15	-	-	PUNCT
ejpam-5732	87	16	closed	close	VERB
ejpam-5732	87	17	and	and	CCONJ
ejpam-5732	87	18	δ	δ	PROPN
ejpam-5732	87	19	-	-	PUNCT
ejpam-5732	87	20	βcli(a	βcli(a	NOUN
ejpam-5732	87	21	)	)	PUNCT
ejpam-5732	87	22	=	=	PUNCT
ejpam-5732	87	23	a.	a.	NOUN
ejpam-5732	87	24	c.	c.	PROPN
ejpam-5732	87	25	boonpok	boonpok	PROPN
ejpam-5732	87	26	,	,	PUNCT
ejpam-5732	87	27	a.	a.	PROPN
ejpam-5732	87	28	sama	sama	PROPN
ejpam-5732	87	29	-	-	PUNCT
ejpam-5732	87	30	ae	ae	PROPN
ejpam-5732	87	31	/	/	SYM
ejpam-5732	87	32	eur	eur	PROPN
ejpam-5732	87	33	.	.	PUNCT
ejpam-5732	88	1	j.	j.	PROPN
ejpam-5732	88	2	pure	pure	PROPN
ejpam-5732	88	3	appl	appl	PROPN
ejpam-5732	88	4	.	.	PROPN
ejpam-5732	88	5	math	math	PROPN
ejpam-5732	88	6	,	,	PUNCT
ejpam-5732	88	7	18	18	NUM
ejpam-5732	88	8	(	(	PUNCT
ejpam-5732	88	9	1	1	NUM
ejpam-5732	88	10	)	)	PUNCT
ejpam-5732	88	11	(	(	PUNCT
ejpam-5732	88	12	2025	2025	NUM
ejpam-5732	88	13	)	)	PUNCT
ejpam-5732	88	14	,	,	PUNCT
ejpam-5732	88	15	5732	5732	NUM
ejpam-5732	88	16	4	4	NUM
ejpam-5732	88	17	of	of	ADP
ejpam-5732	88	18	13	13	NUM
ejpam-5732	88	19	a	a	DET
ejpam-5732	88	20	collection	collection	NOUN
ejpam-5732	88	21	v	v	NOUN
ejpam-5732	88	22	of	of	ADP
ejpam-5732	88	23	subsets	subset	NOUN
ejpam-5732	88	24	of	of	ADP
ejpam-5732	88	25	a	a	DET
ejpam-5732	88	26	topological	topological	ADJ
ejpam-5732	88	27	space	space	NOUN
ejpam-5732	88	28	x	x	PRON
ejpam-5732	88	29	is	be	AUX
ejpam-5732	88	30	said	say	VERB
ejpam-5732	88	31	to	to	PART
ejpam-5732	88	32	be	be	AUX
ejpam-5732	88	33	locally	locally	ADV
ejpam-5732	88	34	finite	finite	ADJ
ejpam-5732	88	35	if	if	SCONJ
ejpam-5732	88	36	each	each	DET
ejpam-5732	88	37	point	point	NOUN
ejpam-5732	88	38	x	x	X
ejpam-5732	88	39	∈	∈	NOUN
ejpam-5732	88	40	x	x	PUNCT
ejpam-5732	88	41	has	have	VERB
ejpam-5732	88	42	a	a	DET
ejpam-5732	88	43	neighborhood	neighborhood	NOUN
ejpam-5732	88	44	u	u	NOUN
ejpam-5732	88	45	that	that	PRON
ejpam-5732	88	46	contains	contain	VERB
ejpam-5732	88	47	x	x	PUNCT
ejpam-5732	88	48	and	and	CCONJ
ejpam-5732	88	49	u	u	NOUN
ejpam-5732	88	50	intersects	intersect	NOUN
ejpam-5732	88	51	only	only	ADV
ejpam-5732	88	52	finitely	finitely	ADV
ejpam-5732	88	53	many	many	ADJ
ejpam-5732	88	54	of	of	ADP
ejpam-5732	88	55	the	the	DET
ejpam-5732	88	56	sets	set	NOUN
ejpam-5732	88	57	in	in	ADP
ejpam-5732	88	58	the	the	DET
ejpam-5732	88	59	collection	collection	NOUN
ejpam-5732	88	60	v.	v.	ADP
ejpam-5732	88	61	the	the	DET
ejpam-5732	88	62	upcoming	upcoming	ADJ
ejpam-5732	88	63	theorems	theorem	NOUN
ejpam-5732	88	64	will	will	AUX
ejpam-5732	88	65	use	use	VERB
ejpam-5732	88	66	the	the	DET
ejpam-5732	88	67	following	follow	VERB
ejpam-5732	88	68	lemmas	lemmas	PROPN
ejpam-5732	88	69	.	.	PUNCT
ejpam-5732	89	1	lemma	lemma	PROPN
ejpam-5732	89	2	3	3	X
ejpam-5732	89	3	.	.	PUNCT
ejpam-5732	90	1	[	[	X
ejpam-5732	90	2	7	7	X
ejpam-5732	90	3	]	]	X
ejpam-5732	90	4	the	the	DET
ejpam-5732	90	5	union	union	NOUN
ejpam-5732	90	6	of	of	ADP
ejpam-5732	90	7	a	a	DET
ejpam-5732	90	8	finite	finite	ADJ
ejpam-5732	90	9	family	family	NOUN
ejpam-5732	90	10	of	of	ADP
ejpam-5732	90	11	locally	locally	ADV
ejpam-5732	90	12	finite	finite	ADJ
ejpam-5732	90	13	collection	collection	NOUN
ejpam-5732	90	14	of	of	ADP
ejpam-5732	90	15	sets	set	NOUN
ejpam-5732	90	16	in	in	ADP
ejpam-5732	90	17	a	a	DET
ejpam-5732	90	18	topological	topological	ADJ
ejpam-5732	90	19	space	space	NOUN
ejpam-5732	90	20	is	be	AUX
ejpam-5732	90	21	a	a	DET
ejpam-5732	90	22	locally	locally	ADV
ejpam-5732	90	23	finite	finite	ADJ
ejpam-5732	90	24	family	family	NOUN
ejpam-5732	90	25	of	of	ADP
ejpam-5732	90	26	sets	set	NOUN
ejpam-5732	90	27	.	.	PUNCT
ejpam-5732	91	1	lemma	lemma	PROPN
ejpam-5732	91	2	4	4	NUM
ejpam-5732	91	3	.	.	PUNCT
ejpam-5732	92	1	[	[	X
ejpam-5732	92	2	8	8	X
ejpam-5732	92	3	]	]	X
ejpam-5732	92	4	if	if	SCONJ
ejpam-5732	92	5	{	{	PUNCT
ejpam-5732	92	6	uα	uα	X
ejpam-5732	92	7	:	:	PUNCT
ejpam-5732	92	8	α	α	PROPN
ejpam-5732	92	9	∈	∈	PROPN
ejpam-5732	92	10	λ	λ	PROPN
ejpam-5732	92	11	}	}	PUNCT
ejpam-5732	92	12	is	be	AUX
ejpam-5732	92	13	a	a	DET
ejpam-5732	92	14	locally	locally	ADV
ejpam-5732	92	15	finite	finite	ADJ
ejpam-5732	92	16	family	family	NOUN
ejpam-5732	92	17	of	of	ADP
ejpam-5732	92	18	subsets	subset	NOUN
ejpam-5732	92	19	in	in	ADP
ejpam-5732	92	20	a	a	DET
ejpam-5732	92	21	topological	topological	ADJ
ejpam-5732	92	22	space	space	NOUN
ejpam-5732	92	23	x	x	PUNCT
ejpam-5732	92	24	and	and	CCONJ
ejpam-5732	92	25	if	if	SCONJ
ejpam-5732	92	26	vα	vα	X
ejpam-5732	92	27	⊂	⊂	X
ejpam-5732	92	28	uα	uα	PROPN
ejpam-5732	92	29	for	for	ADP
ejpam-5732	92	30	all	all	DET
ejpam-5732	92	31	α	α	PRON
ejpam-5732	92	32	∈	∈	PROPN
ejpam-5732	92	33	λ	λ	PROPN
ejpam-5732	92	34	,	,	PUNCT
ejpam-5732	92	35	then	then	ADV
ejpam-5732	92	36	the	the	DET
ejpam-5732	92	37	family	family	NOUN
ejpam-5732	92	38	{	{	PUNCT
ejpam-5732	92	39	vα	vα	X
ejpam-5732	92	40	:	:	PUNCT
ejpam-5732	92	41	α	α	PROPN
ejpam-5732	92	42	∈	∈	PROPN
ejpam-5732	92	43	λ	λ	PROPN
ejpam-5732	92	44	}	}	PUNCT
ejpam-5732	92	45	is	be	AUX
ejpam-5732	92	46	a	a	DET
ejpam-5732	92	47	locally	locally	ADV
ejpam-5732	92	48	finite	finite	NOUN
ejpam-5732	92	49	in	in	ADP
ejpam-5732	92	50	x.	x.	PROPN
ejpam-5732	92	51	lemma	lemma	PROPN
ejpam-5732	93	1	5	5	NUM
ejpam-5732	93	2	.	.	PUNCT
ejpam-5732	94	1	[	[	X
ejpam-5732	94	2	13	13	NUM
ejpam-5732	94	3	]	]	PUNCT
ejpam-5732	94	4	let	let	AUX
ejpam-5732	94	5	(	(	PUNCT
ejpam-5732	94	6	x	x	NOUN
ejpam-5732	94	7	,	,	PUNCT
ejpam-5732	94	8	τ	τ	X
ejpam-5732	94	9	)	)	PUNCT
ejpam-5732	94	10	and	and	CCONJ
ejpam-5732	94	11	(	(	PUNCT
ejpam-5732	94	12	y	y	PROPN
ejpam-5732	94	13	,	,	PUNCT
ejpam-5732	94	14	σ	σ	PROPN
ejpam-5732	94	15	)	)	PUNCT
ejpam-5732	94	16	be	be	VERB
ejpam-5732	94	17	topological	topological	ADJ
ejpam-5732	94	18	spaces	space	NOUN
ejpam-5732	94	19	.	.	PUNCT
ejpam-5732	95	1	if	if	SCONJ
ejpam-5732	95	2	f	f	PROPN
ejpam-5732	95	3	:	:	PUNCT
ejpam-5732	95	4	(	(	PUNCT
ejpam-5732	95	5	x	x	X
ejpam-5732	95	6	,	,	PUNCT
ejpam-5732	95	7	τ	τ	X
ejpam-5732	95	8	)	)	PUNCT
ejpam-5732	95	9	→	→	SYM
ejpam-5732	95	10	(	(	PUNCT
ejpam-5732	95	11	y	y	PROPN
ejpam-5732	95	12	,	,	PUNCT
ejpam-5732	95	13	σ	σ	PROPN
ejpam-5732	95	14	)	)	PUNCT
ejpam-5732	95	15	is	be	AUX
ejpam-5732	95	16	a	a	DET
ejpam-5732	95	17	continuous	continuous	ADJ
ejpam-5732	95	18	surjective	surjective	ADJ
ejpam-5732	95	19	function	function	NOUN
ejpam-5732	95	20	and	and	CCONJ
ejpam-5732	95	21	{	{	PUNCT
ejpam-5732	95	22	uα	uα	X
ejpam-5732	95	23	:	:	PUNCT
ejpam-5732	95	24	α	α	PROPN
ejpam-5732	95	25	∈	∈	PROPN
ejpam-5732	95	26	λ	λ	PROPN
ejpam-5732	95	27	}	}	PUNCT
ejpam-5732	95	28	is	be	AUX
ejpam-5732	95	29	a	a	DET
ejpam-5732	95	30	locally	locally	ADV
ejpam-5732	95	31	finite	finite	NOUN
ejpam-5732	95	32	in	in	ADP
ejpam-5732	95	33	y	y	PROPN
ejpam-5732	95	34	,	,	PUNCT
ejpam-5732	95	35	then	then	ADV
ejpam-5732	95	36	{	{	PUNCT
ejpam-5732	95	37	f−1(uα	f−1(uα	PROPN
ejpam-5732	95	38	)	)	PUNCT
ejpam-5732	95	39	:	:	PUNCT
ejpam-5732	96	1	α	α	PROPN
ejpam-5732	96	2	∈	∈	PROPN
ejpam-5732	96	3	λ	λ	PROPN
ejpam-5732	96	4	}	}	PUNCT
ejpam-5732	96	5	is	be	AUX
ejpam-5732	96	6	a	a	DET
ejpam-5732	96	7	locally	locally	ADV
ejpam-5732	96	8	finite	finite	NOUN
ejpam-5732	96	9	in	in	ADP
ejpam-5732	96	10	x.	x.	NOUN
ejpam-5732	96	11	let	let	VERB
ejpam-5732	96	12	(	(	PUNCT
ejpam-5732	96	13	x	x	NOUN
ejpam-5732	96	14	,	,	PUNCT
ejpam-5732	96	15	τ	τ	X
ejpam-5732	96	16	)	)	PUNCT
ejpam-5732	96	17	and	and	CCONJ
ejpam-5732	96	18	(	(	PUNCT
ejpam-5732	96	19	y	y	PROPN
ejpam-5732	96	20	,	,	PUNCT
ejpam-5732	96	21	σ	σ	PROPN
ejpam-5732	96	22	)	)	PUNCT
ejpam-5732	96	23	denote	denote	VERB
ejpam-5732	96	24	topological	topological	ADJ
ejpam-5732	96	25	spaces	space	NOUN
ejpam-5732	96	26	.	.	PUNCT
ejpam-5732	97	1	a	a	DET
ejpam-5732	97	2	function	function	NOUN
ejpam-5732	97	3	f	f	NOUN
ejpam-5732	97	4	:	:	PUNCT
ejpam-5732	97	5	(	(	PUNCT
ejpam-5732	97	6	x	x	X
ejpam-5732	97	7	,	,	PUNCT
ejpam-5732	97	8	τ	τ	X
ejpam-5732	97	9	)	)	PUNCT
ejpam-5732	97	10	→	→	SYM
ejpam-5732	97	11	(	(	PUNCT
ejpam-5732	97	12	y	y	PROPN
ejpam-5732	97	13	,	,	PUNCT
ejpam-5732	97	14	σ	σ	PROPN
ejpam-5732	97	15	)	)	PUNCT
ejpam-5732	97	16	is	be	AUX
ejpam-5732	97	17	called	call	VERB
ejpam-5732	97	18	almost	almost	ADV
ejpam-5732	97	19	closed	closed	ADJ
ejpam-5732	97	20	[	[	PUNCT
ejpam-5732	97	21	20	20	NUM
ejpam-5732	97	22	]	]	X
ejpam-5732	97	23	if	if	SCONJ
ejpam-5732	97	24	for	for	ADP
ejpam-5732	97	25	any	any	DET
ejpam-5732	97	26	regular	regular	ADJ
ejpam-5732	97	27	closed	closed	ADJ
ejpam-5732	97	28	set	set	VERB
ejpam-5732	97	29	f	f	PROPN
ejpam-5732	97	30	in	in	ADP
ejpam-5732	97	31	x	x	PROPN
ejpam-5732	97	32	,	,	PUNCT
ejpam-5732	97	33	the	the	DET
ejpam-5732	97	34	image	image	NOUN
ejpam-5732	97	35	f(f	f(f	PROPN
ejpam-5732	97	36	)	)	PUNCT
ejpam-5732	97	37	is	be	AUX
ejpam-5732	97	38	closed	close	VERB
ejpam-5732	97	39	in	in	ADP
ejpam-5732	97	40	y	y	PROPN
ejpam-5732	97	41	.	.	PUNCT
ejpam-5732	98	1	a	a	DET
ejpam-5732	98	2	subset	subset	NOUN
ejpam-5732	98	3	k	k	PROPN
ejpam-5732	98	4	of	of	ADP
ejpam-5732	98	5	the	the	DET
ejpam-5732	98	6	space	space	NOUN
ejpam-5732	98	7	x	x	PUNCT
ejpam-5732	98	8	is	be	AUX
ejpam-5732	98	9	defined	define	VERB
ejpam-5732	98	10	as	as	ADP
ejpam-5732	98	11	n	n	X
ejpam-5732	98	12	-closed	-close	VERB
ejpam-5732	98	13	relative	relative	ADJ
ejpam-5732	98	14	to	to	ADP
ejpam-5732	98	15	x	x	PROPN
ejpam-5732	99	1	[	[	X
ejpam-5732	99	2	20	20	NUM
ejpam-5732	99	3	]	]	PUNCT
ejpam-5732	99	4	if	if	SCONJ
ejpam-5732	99	5	every	every	DET
ejpam-5732	99	6	cover	cover	NOUN
ejpam-5732	99	7	of	of	ADP
ejpam-5732	99	8	k	k	X
ejpam-5732	99	9	by	by	ADP
ejpam-5732	99	10	regular	regular	ADJ
ejpam-5732	99	11	open	open	ADJ
ejpam-5732	99	12	sets	set	NOUN
ejpam-5732	99	13	of	of	ADP
ejpam-5732	99	14	x	x	PRON
ejpam-5732	99	15	possesses	possess	VERB
ejpam-5732	99	16	a	a	DET
ejpam-5732	99	17	finite	finite	PROPN
ejpam-5732	99	18	subcover	subcover	PROPN
ejpam-5732	99	19	.	.	PUNCT
ejpam-5732	100	1	lemma	lemma	PROPN
ejpam-5732	100	2	6	6	NUM
ejpam-5732	100	3	.	.	PUNCT
ejpam-5732	101	1	[	[	X
ejpam-5732	101	2	19	19	NUM
ejpam-5732	101	3	]	]	X
ejpam-5732	101	4	let	let	AUX
ejpam-5732	101	5	(	(	PUNCT
ejpam-5732	101	6	x	x	NOUN
ejpam-5732	101	7	,	,	PUNCT
ejpam-5732	101	8	τ	τ	X
ejpam-5732	101	9	)	)	PUNCT
ejpam-5732	101	10	and	and	CCONJ
ejpam-5732	101	11	(	(	PUNCT
ejpam-5732	101	12	y	y	PROPN
ejpam-5732	101	13	,	,	PUNCT
ejpam-5732	101	14	σ	σ	PROPN
ejpam-5732	101	15	)	)	PUNCT
ejpam-5732	101	16	be	be	VERB
ejpam-5732	101	17	topological	topological	ADJ
ejpam-5732	101	18	spaces	space	NOUN
ejpam-5732	101	19	and	and	CCONJ
ejpam-5732	101	20	f	f	NOUN
ejpam-5732	101	21	:	:	PUNCT
ejpam-5732	101	22	(	(	PUNCT
ejpam-5732	101	23	x	x	X
ejpam-5732	101	24	,	,	PUNCT
ejpam-5732	101	25	τ	τ	X
ejpam-5732	101	26	)	)	PUNCT
ejpam-5732	101	27	→	→	SYM
ejpam-5732	101	28	(	(	PUNCT
ejpam-5732	101	29	y	y	PROPN
ejpam-5732	101	30	,	,	PUNCT
ejpam-5732	101	31	σ	σ	PROPN
ejpam-5732	101	32	)	)	PUNCT
ejpam-5732	101	33	be	be	VERB
ejpam-5732	101	34	almost	almost	ADV
ejpam-5732	101	35	closed	close	VERB
ejpam-5732	101	36	surjection	surjection	NOUN
ejpam-5732	101	37	with	with	ADP
ejpam-5732	101	38	n	n	ADV
ejpam-5732	101	39	-closed	-closed	ADJ
ejpam-5732	101	40	point	point	NOUN
ejpam-5732	101	41	inverse	inverse	NOUN
ejpam-5732	101	42	.	.	PUNCT
ejpam-5732	102	1	if	if	SCONJ
ejpam-5732	102	2	{	{	PUNCT
ejpam-5732	102	3	uα	uα	X
ejpam-5732	102	4	:	:	PUNCT
ejpam-5732	102	5	α	α	PROPN
ejpam-5732	102	6	∈	∈	PROPN
ejpam-5732	102	7	λ	λ	PROPN
ejpam-5732	102	8	}	}	PUNCT
ejpam-5732	102	9	is	be	AUX
ejpam-5732	102	10	a	a	DET
ejpam-5732	102	11	locally	locally	ADV
ejpam-5732	102	12	finite	finite	ADJ
ejpam-5732	102	13	open	open	ADJ
ejpam-5732	102	14	cover	cover	NOUN
ejpam-5732	102	15	of	of	ADP
ejpam-5732	102	16	x	x	PRON
ejpam-5732	102	17	,	,	PUNCT
ejpam-5732	102	18	then	then	ADV
ejpam-5732	102	19	{	{	PUNCT
ejpam-5732	102	20	f(uα	f(uα	NOUN
ejpam-5732	102	21	)	)	PUNCT
ejpam-5732	102	22	:	:	PUNCT
ejpam-5732	102	23	α	α	PROPN
ejpam-5732	102	24	∈	∈	PROPN
ejpam-5732	102	25	λ	λ	PROPN
ejpam-5732	102	26	}	}	PUNCT
ejpam-5732	102	27	is	be	AUX
ejpam-5732	102	28	a	a	DET
ejpam-5732	102	29	locally	locally	ADV
ejpam-5732	102	30	finite	finite	ADJ
ejpam-5732	102	31	cover	cover	NOUN
ejpam-5732	102	32	of	of	ADP
ejpam-5732	102	33	y	y	PROPN
ejpam-5732	102	34	.	.	PUNCT
ejpam-5732	103	1	3	3	X
ejpam-5732	103	2	.	.	X
ejpam-5732	103	3	δ1	δ1	NOUN
ejpam-5732	103	4	-	-	PUNCT
ejpam-5732	103	5	βi	βi	NOUN
ejpam-5732	103	6	-	-	PUNCT
ejpam-5732	103	7	paracompactness	paracompactness	NOUN
ejpam-5732	103	8	of	of	ADP
ejpam-5732	103	9	spaces	space	NOUN
ejpam-5732	103	10	and	and	CCONJ
ejpam-5732	103	11	subsets	subset	NOUN
ejpam-5732	103	12	this	this	DET
ejpam-5732	103	13	section	section	NOUN
ejpam-5732	103	14	discusses	discuss	VERB
ejpam-5732	103	15	the	the	DET
ejpam-5732	103	16	concept	concept	NOUN
ejpam-5732	103	17	of	of	ADP
ejpam-5732	103	18	δ1	δ1	NOUN
ejpam-5732	103	19	-	-	PUNCT
ejpam-5732	103	20	βi	βi	NOUN
ejpam-5732	103	21	-	-	NOUN
ejpam-5732	103	22	paracompactness	paracompactness	NOUN
ejpam-5732	103	23	,	,	PUNCT
ejpam-5732	103	24	a	a	DET
ejpam-5732	103	25	less	less	ADV
ejpam-5732	103	26	strict	strict	ADJ
ejpam-5732	103	27	form	form	NOUN
ejpam-5732	103	28	of	of	ADP
ejpam-5732	103	29	β1paracompactness	β1paracompactness	NOUN
ejpam-5732	103	30	examined	examine	VERB
ejpam-5732	103	31	by	by	ADP
ejpam-5732	103	32	qahis	qahis	PRON
ejpam-5732	104	1	[	[	X
ejpam-5732	104	2	21	21	NUM
ejpam-5732	104	3	]	]	PUNCT
ejpam-5732	104	4	,	,	PUNCT
ejpam-5732	104	5	followed	follow	VERB
ejpam-5732	104	6	by	by	ADP
ejpam-5732	104	7	an	an	DET
ejpam-5732	104	8	exploration	exploration	NOUN
ejpam-5732	104	9	of	of	ADP
ejpam-5732	104	10	its	its	PRON
ejpam-5732	104	11	characterization	characterization	NOUN
ejpam-5732	104	12	.	.	PUNCT
ejpam-5732	105	1	al	al	PROPN
ejpam-5732	105	2	-	-	PUNCT
ejpam-5732	105	3	jarrah	jarrah	PROPN
ejpam-5732	105	4	[	[	X
ejpam-5732	105	5	3	3	NUM
ejpam-5732	105	6	]	]	PUNCT
ejpam-5732	105	7	defined	define	VERB
ejpam-5732	105	8	β1	β1	PROPN
ejpam-5732	105	9	-	-	PUNCT
ejpam-5732	105	10	paracompactness	paracompactness	PROPN
ejpam-5732	105	11	as	as	SCONJ
ejpam-5732	105	12	follows	follow	VERB
ejpam-5732	105	13	:	:	PUNCT
ejpam-5732	105	14	a	a	DET
ejpam-5732	105	15	topological	topological	ADJ
ejpam-5732	105	16	space	space	NOUN
ejpam-5732	105	17	(	(	PUNCT
ejpam-5732	105	18	x	x	X
ejpam-5732	105	19	,	,	PUNCT
ejpam-5732	105	20	τ	τ	X
ejpam-5732	105	21	)	)	PUNCT
ejpam-5732	105	22	is	be	AUX
ejpam-5732	105	23	called	call	VERB
ejpam-5732	105	24	β1	β1	NOUN
ejpam-5732	105	25	-	-	PUNCT
ejpam-5732	105	26	paracompact	paracompact	NOUN
ejpam-5732	105	27	if	if	SCONJ
ejpam-5732	105	28	every	every	DET
ejpam-5732	105	29	β	β	NOUN
ejpam-5732	105	30	-	-	ADJ
ejpam-5732	105	31	open	open	ADJ
ejpam-5732	105	32	cover	cover	NOUN
ejpam-5732	105	33	of	of	ADP
ejpam-5732	105	34	x	x	PUNCT
ejpam-5732	105	35	has	have	VERB
ejpam-5732	105	36	a	a	DET
ejpam-5732	105	37	locally	locally	ADV
ejpam-5732	105	38	finite	finite	ADJ
ejpam-5732	105	39	open	open	ADJ
ejpam-5732	105	40	refinement	refinement	NOUN
ejpam-5732	105	41	.	.	PUNCT
ejpam-5732	106	1	qahis	qahis	PRON
ejpam-5732	107	1	[	[	X
ejpam-5732	107	2	21	21	NUM
ejpam-5732	107	3	]	]	PUNCT
ejpam-5732	107	4	expanded	expand	VERB
ejpam-5732	107	5	the	the	DET
ejpam-5732	107	6	notion	notion	NOUN
ejpam-5732	107	7	of	of	ADP
ejpam-5732	107	8	β1	β1	PROPN
ejpam-5732	107	9	-	-	PUNCT
ejpam-5732	107	10	paracompactness	paracompactness	NOUN
ejpam-5732	107	11	to	to	ADP
ejpam-5732	107	12	β1	β1	PROPN
ejpam-5732	107	13	-	-	PUNCT
ejpam-5732	107	14	paracompactness	paracompactness	NOUN
ejpam-5732	107	15	concerning	concern	VERB
ejpam-5732	107	16	an	an	DET
ejpam-5732	107	17	ideal	ideal	NOUN
ejpam-5732	107	18	as	as	SCONJ
ejpam-5732	107	19	follows	follow	VERB
ejpam-5732	107	20	:	:	PUNCT
ejpam-5732	107	21	an	an	DET
ejpam-5732	107	22	ideal	ideal	ADJ
ejpam-5732	107	23	topological	topological	ADJ
ejpam-5732	107	24	space	space	NOUN
ejpam-5732	107	25	(	(	PUNCT
ejpam-5732	107	26	x	x	X
ejpam-5732	107	27	,	,	PUNCT
ejpam-5732	107	28	τ	τ	PROPN
ejpam-5732	107	29	,	,	PUNCT
ejpam-5732	107	30	i	i	PROPN
ejpam-5732	107	31	)	)	PUNCT
ejpam-5732	107	32	is	be	AUX
ejpam-5732	107	33	said	say	VERB
ejpam-5732	107	34	to	to	PART
ejpam-5732	107	35	be	be	AUX
ejpam-5732	107	36	β1iparacompact	β1iparacompact	PROPN
ejpam-5732	107	37	if	if	SCONJ
ejpam-5732	107	38	every	every	DET
ejpam-5732	107	39	β	β	NOUN
ejpam-5732	107	40	-	-	ADJ
ejpam-5732	107	41	open	open	ADJ
ejpam-5732	107	42	cover	cover	NOUN
ejpam-5732	107	43	u	u	NOUN
ejpam-5732	107	44	of	of	ADP
ejpam-5732	107	45	x	x	PUNCT
ejpam-5732	107	46	has	have	VERB
ejpam-5732	107	47	a	a	DET
ejpam-5732	107	48	locally	locally	ADV
ejpam-5732	107	49	finite	finite	ADJ
ejpam-5732	107	50	open	open	ADJ
ejpam-5732	107	51	refinement	refinement	NOUN
ejpam-5732	107	52	v	v	ADP
ejpam-5732	107	53	such	such	ADJ
ejpam-5732	107	54	that	that	PRON
ejpam-5732	107	55	x	x	X
ejpam-5732	108	1	−	−	NOUN
ejpam-5732	108	2	∪{v	∪{v	NOUN
ejpam-5732	108	3	:	:	PUNCT
ejpam-5732	108	4	v	v	NUM
ejpam-5732	108	5	∈	∈	PROPN
ejpam-5732	108	6	v	v	NOUN
ejpam-5732	108	7	}	}	PUNCT
ejpam-5732	108	8	∈	∈	PROPN
ejpam-5732	108	9	i.	i.	NOUN
ejpam-5732	108	10	utilizing	utilize	VERB
ejpam-5732	108	11	δ	δ	PROPN
ejpam-5732	108	12	-	-	PUNCT
ejpam-5732	108	13	βi	βi	PRON
ejpam-5732	108	14	-	-	PUNCT
ejpam-5732	108	15	open	open	ADJ
ejpam-5732	108	16	sets	set	NOUN
ejpam-5732	108	17	,	,	PUNCT
ejpam-5732	108	18	we	we	PRON
ejpam-5732	108	19	establish	establish	VERB
ejpam-5732	108	20	a	a	DET
ejpam-5732	108	21	new	new	ADJ
ejpam-5732	108	22	type	type	NOUN
ejpam-5732	108	23	of	of	ADP
ejpam-5732	108	24	paracompactness	paracompactness	NOUN
ejpam-5732	108	25	that	that	PRON
ejpam-5732	108	26	is	be	AUX
ejpam-5732	108	27	weaker	weak	ADJ
ejpam-5732	108	28	than	than	ADP
ejpam-5732	108	29	the	the	DET
ejpam-5732	108	30	one	one	NOUN
ejpam-5732	108	31	that	that	PRON
ejpam-5732	108	32	qahis	qahis	PROPN
ejpam-5732	108	33	developed	develop	VERB
ejpam-5732	108	34	.	.	PUNCT
ejpam-5732	109	1	definition	definition	NOUN
ejpam-5732	109	2	4	4	NUM
ejpam-5732	109	3	.	.	PUNCT
ejpam-5732	110	1	an	an	DET
ejpam-5732	110	2	ideal	ideal	ADJ
ejpam-5732	110	3	topological	topological	ADJ
ejpam-5732	110	4	space	space	NOUN
ejpam-5732	110	5	(	(	PUNCT
ejpam-5732	110	6	x	x	X
ejpam-5732	110	7	,	,	PUNCT
ejpam-5732	110	8	τ	τ	PROPN
ejpam-5732	110	9	,	,	PUNCT
ejpam-5732	110	10	i	i	PROPN
ejpam-5732	110	11	)	)	PUNCT
ejpam-5732	110	12	is	be	AUX
ejpam-5732	110	13	said	say	VERB
ejpam-5732	110	14	to	to	PART
ejpam-5732	110	15	be	be	AUX
ejpam-5732	110	16	δ1	δ1	VERB
ejpam-5732	110	17	-	-	PUNCT
ejpam-5732	110	18	βi	βi	NOUN
ejpam-5732	110	19	-	-	NOUN
ejpam-5732	110	20	paracompact	paracompact	NOUN
ejpam-5732	110	21	if	if	SCONJ
ejpam-5732	110	22	every	every	DET
ejpam-5732	110	23	δ	δ	PROPN
ejpam-5732	110	24	-	-	PUNCT
ejpam-5732	110	25	βi	βi	ADV
ejpam-5732	110	26	-	-	PUNCT
ejpam-5732	110	27	open	open	ADJ
ejpam-5732	110	28	cover	cover	NOUN
ejpam-5732	110	29	u	u	NOUN
ejpam-5732	110	30	of	of	ADP
ejpam-5732	110	31	x	x	PUNCT
ejpam-5732	110	32	has	have	VERB
ejpam-5732	110	33	a	a	DET
ejpam-5732	110	34	locally	locally	ADV
ejpam-5732	110	35	finite	finite	ADJ
ejpam-5732	110	36	open	open	ADJ
ejpam-5732	110	37	refinement	refinement	PROPN
ejpam-5732	110	38	v	v	NOUN
ejpam-5732	110	39	(	(	PUNCT
ejpam-5732	110	40	not	not	PART
ejpam-5732	110	41	necessarily	necessarily	ADV
ejpam-5732	110	42	a	a	DET
ejpam-5732	110	43	cover	cover	NOUN
ejpam-5732	110	44	)	)	PUNCT
ejpam-5732	110	45	such	such	ADJ
ejpam-5732	110	46	that	that	SCONJ
ejpam-5732	110	47	x	x	X
ejpam-5732	111	1	−	−	NOUN
ejpam-5732	112	1	∪{v	∪{v	NOUN
ejpam-5732	112	2	:	:	PUNCT
ejpam-5732	112	3	v	v	NUM
ejpam-5732	112	4	∈	∈	PROPN
ejpam-5732	112	5	v	v	NOUN
ejpam-5732	112	6	}	}	PUNCT
ejpam-5732	112	7	∈	∈	PROPN
ejpam-5732	112	8	i.	i.	NOUN
ejpam-5732	112	9	the	the	DET
ejpam-5732	112	10	collection	collection	NOUN
ejpam-5732	112	11	v	v	NOUN
ejpam-5732	112	12	of	of	ADP
ejpam-5732	112	13	subsets	subset	NOUN
ejpam-5732	112	14	of	of	ADP
ejpam-5732	112	15	x	x	PUNCT
ejpam-5732	112	16	such	such	ADJ
ejpam-5732	112	17	that	that	SCONJ
ejpam-5732	112	18	x	x	X
ejpam-5732	113	1	−	−	NOUN
ejpam-5732	113	2	∪{v	∪{v	NOUN
ejpam-5732	113	3	:	:	PUNCT
ejpam-5732	113	4	v	v	NUM
ejpam-5732	113	5	∈	∈	PROPN
ejpam-5732	113	6	v	v	NOUN
ejpam-5732	113	7	}	}	PUNCT
ejpam-5732	113	8	∈	∈	NOUN
ejpam-5732	113	9	i	i	PRON
ejpam-5732	113	10	is	be	AUX
ejpam-5732	113	11	called	call	VERB
ejpam-5732	113	12	an	an	DET
ejpam-5732	113	13	i	i	NOUN
ejpam-5732	113	14	-	-	PUNCT
ejpam-5732	113	15	cover	cover	NOUN
ejpam-5732	113	16	of	of	ADP
ejpam-5732	113	17	x.	x.	NOUN
ejpam-5732	113	18	a	a	DET
ejpam-5732	113	19	subset	subset	NOUN
ejpam-5732	113	20	a	a	PRON
ejpam-5732	113	21	of	of	ADP
ejpam-5732	113	22	an	an	DET
ejpam-5732	113	23	ideal	ideal	ADJ
ejpam-5732	113	24	topological	topological	ADJ
ejpam-5732	113	25	space	space	NOUN
ejpam-5732	113	26	(	(	PUNCT
ejpam-5732	113	27	x	x	X
ejpam-5732	113	28	,	,	PUNCT
ejpam-5732	113	29	τ	τ	PROPN
ejpam-5732	113	30	,	,	PUNCT
ejpam-5732	113	31	i	i	PROPN
ejpam-5732	113	32	)	)	PUNCT
ejpam-5732	113	33	is	be	AUX
ejpam-5732	113	34	said	say	VERB
ejpam-5732	113	35	to	to	PART
ejpam-5732	113	36	be	be	AUX
ejpam-5732	113	37	δ1	δ1	VERB
ejpam-5732	113	38	-	-	PUNCT
ejpam-5732	113	39	βi	βi	PRON
ejpam-5732	113	40	-	-	PUNCT
ejpam-5732	113	41	paracompact	paracompact	NOUN
ejpam-5732	113	42	relative	relative	ADJ
ejpam-5732	113	43	to	to	ADP
ejpam-5732	113	44	x	x	PRON
ejpam-5732	113	45	if	if	SCONJ
ejpam-5732	113	46	for	for	SCONJ
ejpam-5732	113	47	every	every	DET
ejpam-5732	113	48	δ	δ	PROPN
ejpam-5732	113	49	-	-	PUNCT
ejpam-5732	113	50	βi	βi	ADV
ejpam-5732	113	51	-	-	PUNCT
ejpam-5732	113	52	open	open	ADJ
ejpam-5732	113	53	cover	cover	NOUN
ejpam-5732	113	54	u	u	NOUN
ejpam-5732	113	55	of	of	ADP
ejpam-5732	113	56	a	a	PRON
ejpam-5732	113	57	has	have	VERB
ejpam-5732	113	58	a	a	DET
ejpam-5732	113	59	locally	locally	ADV
ejpam-5732	113	60	finite	finite	ADJ
ejpam-5732	113	61	open	open	ADJ
ejpam-5732	113	62	refinement	refinement	NOUN
ejpam-5732	113	63	v	v	ADP
ejpam-5732	113	64	such	such	ADJ
ejpam-5732	113	65	that	that	DET
ejpam-5732	113	66	a−	a−	PROPN
ejpam-5732	113	67	∪{v	∪{v	PROPN
ejpam-5732	113	68	:	:	PUNCT
ejpam-5732	113	69	v	v	NUM
ejpam-5732	113	70	∈	∈	PROPN
ejpam-5732	113	71	v	v	NOUN
ejpam-5732	113	72	}	}	PUNCT
ejpam-5732	113	73	∈	∈	PROPN
ejpam-5732	113	74	i.	i.	NOUN
ejpam-5732	113	75	we	we	PRON
ejpam-5732	113	76	have	have	VERB
ejpam-5732	113	77	the	the	DET
ejpam-5732	113	78	following	follow	VERB
ejpam-5732	113	79	result	result	NOUN
ejpam-5732	113	80	based	base	VERB
ejpam-5732	113	81	on	on	ADP
ejpam-5732	113	82	the	the	DET
ejpam-5732	113	83	definition	definition	NOUN
ejpam-5732	113	84	that	that	PRON
ejpam-5732	113	85	was	be	AUX
ejpam-5732	113	86	previously	previously	ADV
ejpam-5732	113	87	given	give	VERB
ejpam-5732	113	88	.	.	PUNCT
ejpam-5732	114	1	theorem	theorem	NOUN
ejpam-5732	114	2	1	1	NUM
ejpam-5732	114	3	.	.	PUNCT
ejpam-5732	115	1	if	if	SCONJ
ejpam-5732	115	2	an	an	DET
ejpam-5732	115	3	ideal	ideal	ADJ
ejpam-5732	115	4	topological	topological	ADJ
ejpam-5732	115	5	space	space	NOUN
ejpam-5732	115	6	(	(	PUNCT
ejpam-5732	115	7	x	x	X
ejpam-5732	115	8	,	,	PUNCT
ejpam-5732	115	9	τ	τ	PROPN
ejpam-5732	115	10	,	,	PUNCT
ejpam-5732	115	11	i	i	PROPN
ejpam-5732	115	12	)	)	PUNCT
ejpam-5732	115	13	is	be	AUX
ejpam-5732	115	14	δ1	δ1	NOUN
ejpam-5732	115	15	-	-	PUNCT
ejpam-5732	115	16	βi	βi	PRON
ejpam-5732	115	17	-	-	PUNCT
ejpam-5732	115	18	paracompact	paracompact	ADJ
ejpam-5732	115	19	,	,	PUNCT
ejpam-5732	115	20	then	then	ADV
ejpam-5732	115	21	it	it	PRON
ejpam-5732	115	22	is	be	AUX
ejpam-5732	115	23	β1iparacompact	β1iparacompact	PROPN
ejpam-5732	115	24	.	.	PUNCT
ejpam-5732	116	1	c.	c.	PROPN
ejpam-5732	116	2	boonpok	boonpok	PROPN
ejpam-5732	116	3	,	,	PUNCT
ejpam-5732	116	4	a.	a.	PROPN
ejpam-5732	116	5	sama	sama	PROPN
ejpam-5732	116	6	-	-	PUNCT
ejpam-5732	116	7	ae	ae	PROPN
ejpam-5732	116	8	/	/	SYM
ejpam-5732	116	9	eur	eur	PROPN
ejpam-5732	116	10	.	.	PUNCT
ejpam-5732	117	1	j.	j.	PROPN
ejpam-5732	117	2	pure	pure	PROPN
ejpam-5732	117	3	appl	appl	PROPN
ejpam-5732	117	4	.	.	PROPN
ejpam-5732	117	5	math	math	PROPN
ejpam-5732	117	6	,	,	PUNCT
ejpam-5732	117	7	18	18	NUM
ejpam-5732	117	8	(	(	PUNCT
ejpam-5732	117	9	1	1	NUM
ejpam-5732	117	10	)	)	PUNCT
ejpam-5732	117	11	(	(	PUNCT
ejpam-5732	117	12	2025	2025	NUM
ejpam-5732	117	13	)	)	PUNCT
ejpam-5732	117	14	,	,	PUNCT
ejpam-5732	117	15	5732	5732	NUM
ejpam-5732	117	16	5	5	NUM
ejpam-5732	117	17	of	of	ADP
ejpam-5732	117	18	13	13	NUM
ejpam-5732	117	19	proof	proof	NOUN
ejpam-5732	117	20	.	.	PUNCT
ejpam-5732	118	1	the	the	DET
ejpam-5732	118	2	theorem	theorem	NOUN
ejpam-5732	118	3	is	be	AUX
ejpam-5732	118	4	established	establish	VERB
ejpam-5732	118	5	as	as	ADP
ejpam-5732	118	6	every	every	DET
ejpam-5732	118	7	β	β	NOUN
ejpam-5732	118	8	-	-	ADJ
ejpam-5732	118	9	open	open	ADJ
ejpam-5732	118	10	cover	cover	NOUN
ejpam-5732	118	11	of	of	ADP
ejpam-5732	118	12	x	x	PRON
ejpam-5732	118	13	serves	serve	VERB
ejpam-5732	118	14	as	as	ADP
ejpam-5732	118	15	a	a	DET
ejpam-5732	118	16	δ	δ	PROPN
ejpam-5732	118	17	-	-	PUNCT
ejpam-5732	118	18	βi	βi	ADV
ejpam-5732	118	19	-	-	PUNCT
ejpam-5732	118	20	open	open	ADJ
ejpam-5732	118	21	cover	cover	NOUN
ejpam-5732	118	22	.	.	PUNCT
ejpam-5732	119	1	corollary	corollary	ADJ
ejpam-5732	119	2	1	1	NUM
ejpam-5732	119	3	.	.	PUNCT
ejpam-5732	120	1	if	if	SCONJ
ejpam-5732	120	2	an	an	DET
ejpam-5732	120	3	ideal	ideal	ADJ
ejpam-5732	120	4	topological	topological	ADJ
ejpam-5732	120	5	space	space	NOUN
ejpam-5732	120	6	(	(	PUNCT
ejpam-5732	120	7	x	x	X
ejpam-5732	120	8	,	,	PUNCT
ejpam-5732	120	9	τ	τ	PROPN
ejpam-5732	120	10	,	,	PUNCT
ejpam-5732	120	11	i	i	PROPN
ejpam-5732	120	12	)	)	PUNCT
ejpam-5732	120	13	is	be	AUX
ejpam-5732	120	14	δ1	δ1	NOUN
ejpam-5732	120	15	-	-	PUNCT
ejpam-5732	120	16	βi	βi	PRON
ejpam-5732	120	17	-	-	PUNCT
ejpam-5732	120	18	paracompact	paracompact	ADJ
ejpam-5732	120	19	,	,	PUNCT
ejpam-5732	120	20	then	then	ADV
ejpam-5732	120	21	it	it	PRON
ejpam-5732	120	22	is	be	AUX
ejpam-5732	120	23	paracompact	paracompact	ADJ
ejpam-5732	120	24	.	.	PUNCT
ejpam-5732	121	1	proof	proof	NOUN
ejpam-5732	121	2	.	.	PUNCT
ejpam-5732	122	1	since	since	SCONJ
ejpam-5732	122	2	any	any	DET
ejpam-5732	122	3	β1i	β1i	ADJ
ejpam-5732	122	4	-	-	ADJ
ejpam-5732	122	5	paracompact	paracompact	ADJ
ejpam-5732	122	6	space	space	NOUN
ejpam-5732	122	7	is	be	AUX
ejpam-5732	122	8	always	always	ADV
ejpam-5732	122	9	a	a	DET
ejpam-5732	122	10	paracompact	paracompact	ADJ
ejpam-5732	122	11	space	space	NOUN
ejpam-5732	122	12	,	,	PUNCT
ejpam-5732	122	13	the	the	DET
ejpam-5732	122	14	corollary	corollary	NOUN
ejpam-5732	122	15	is	be	AUX
ejpam-5732	122	16	obtained	obtain	VERB
ejpam-5732	122	17	.	.	PUNCT
ejpam-5732	123	1	consider	consider	VERB
ejpam-5732	123	2	the	the	DET
ejpam-5732	123	3	ideal	ideal	ADJ
ejpam-5732	123	4	topological	topological	ADJ
ejpam-5732	123	5	space	space	NOUN
ejpam-5732	123	6	(	(	PUNCT
ejpam-5732	123	7	x	x	X
ejpam-5732	123	8	,	,	PUNCT
ejpam-5732	123	9	τ	τ	PROPN
ejpam-5732	123	10	,	,	PUNCT
ejpam-5732	123	11	i	i	PROPN
ejpam-5732	123	12	)	)	PUNCT
ejpam-5732	123	13	,	,	PUNCT
ejpam-5732	123	14	where	where	SCONJ
ejpam-5732	123	15	x	x	X
ejpam-5732	123	16	=	=	PRON
ejpam-5732	123	17	{	{	PUNCT
ejpam-5732	123	18	1	1	NUM
ejpam-5732	123	19	,	,	PUNCT
ejpam-5732	123	20	2	2	NUM
ejpam-5732	123	21	,	,	PUNCT
ejpam-5732	123	22	3	3	NUM
ejpam-5732	123	23	}	}	PUNCT
ejpam-5732	123	24	,	,	PUNCT
ejpam-5732	123	25	τ	τ	X
ejpam-5732	123	26	=	=	PUNCT
ejpam-5732	123	27	{	{	PUNCT
ejpam-5732	123	28	∅	∅	NOUN
ejpam-5732	123	29	,	,	PUNCT
ejpam-5732	123	30	x	x	X
ejpam-5732	123	31	,	,	PUNCT
ejpam-5732	123	32	{	{	PUNCT
ejpam-5732	123	33	1	1	NUM
ejpam-5732	123	34	}	}	PUNCT
ejpam-5732	123	35	}	}	PUNCT
ejpam-5732	123	36	and	and	CCONJ
ejpam-5732	123	37	i	i	PRON
ejpam-5732	123	38	=	=	PUNCT
ejpam-5732	123	39	{	{	PUNCT
ejpam-5732	123	40	∅	∅	NOUN
ejpam-5732	123	41	,	,	PUNCT
ejpam-5732	123	42	{	{	PUNCT
ejpam-5732	123	43	2	2	NUM
ejpam-5732	123	44	}	}	PUNCT
ejpam-5732	123	45	,	,	PUNCT
ejpam-5732	123	46	{	{	PUNCT
ejpam-5732	123	47	3	3	NUM
ejpam-5732	123	48	}	}	PUNCT
ejpam-5732	123	49	,	,	PUNCT
ejpam-5732	123	50	{	{	PUNCT
ejpam-5732	123	51	2	2	NUM
ejpam-5732	123	52	,	,	PUNCT
ejpam-5732	123	53	3	3	NUM
ejpam-5732	123	54	}	}	PUNCT
ejpam-5732	123	55	}	}	PUNCT
ejpam-5732	123	56	.	.	PUNCT
ejpam-5732	124	1	hence	hence	ADV
ejpam-5732	124	2	,	,	PUNCT
ejpam-5732	124	3	the	the	DET
ejpam-5732	124	4	set	set	NOUN
ejpam-5732	124	5	of	of	ADP
ejpam-5732	124	6	all	all	DET
ejpam-5732	124	7	δ	δ	PROPN
ejpam-5732	124	8	-	-	PUNCT
ejpam-5732	124	9	βi	βi	ADV
ejpam-5732	124	10	-	-	PUNCT
ejpam-5732	124	11	open	open	ADJ
ejpam-5732	124	12	sets	set	NOUN
ejpam-5732	124	13	of	of	ADP
ejpam-5732	124	14	x	x	SYM
ejpam-5732	124	15	is	be	AUX
ejpam-5732	124	16	{	{	PUNCT
ejpam-5732	124	17	a	a	X
ejpam-5732	124	18	:	:	PUNCT
ejpam-5732	124	19	a	a	DET
ejpam-5732	124	20	⊂	⊂	X
ejpam-5732	124	21	x	x	X
ejpam-5732	124	22	}	}	PUNCT
ejpam-5732	124	23	.	.	PUNCT
ejpam-5732	125	1	every	every	DET
ejpam-5732	125	2	δ	δ	PROPN
ejpam-5732	125	3	-	-	PUNCT
ejpam-5732	125	4	βi	βi	ADV
ejpam-5732	125	5	-	-	PUNCT
ejpam-5732	125	6	open	open	ADJ
ejpam-5732	125	7	cover	cover	NOUN
ejpam-5732	125	8	u	u	NOUN
ejpam-5732	125	9	of	of	ADP
ejpam-5732	125	10	x	x	PART
ejpam-5732	125	11	possesses	possess	VERB
ejpam-5732	125	12	a	a	DET
ejpam-5732	125	13	locally	locally	ADV
ejpam-5732	125	14	finite	finite	ADJ
ejpam-5732	125	15	open	open	ADJ
ejpam-5732	125	16	refinement	refinement	NOUN
ejpam-5732	125	17	{	{	PUNCT
ejpam-5732	125	18	{	{	PUNCT
ejpam-5732	125	19	1	1	NUM
ejpam-5732	125	20	}	}	PUNCT
ejpam-5732	125	21	}	}	PUNCT
ejpam-5732	125	22	,	,	PUNCT
ejpam-5732	125	23	such	such	ADJ
ejpam-5732	125	24	that	that	SCONJ
ejpam-5732	125	25	x	x	X
ejpam-5732	125	26	−	−	NOUN
ejpam-5732	125	27	{	{	PUNCT
ejpam-5732	125	28	1	1	NUM
ejpam-5732	125	29	}	}	PUNCT
ejpam-5732	125	30	=	=	NOUN
ejpam-5732	125	31	{	{	PUNCT
ejpam-5732	125	32	2	2	NUM
ejpam-5732	125	33	,	,	PUNCT
ejpam-5732	125	34	3	3	NUM
ejpam-5732	125	35	}	}	SYM
ejpam-5732	125	36	∈	∈	PROPN
ejpam-5732	125	37	i.	i.	NOUN
ejpam-5732	125	38	consequently	consequently	ADV
ejpam-5732	125	39	,	,	PUNCT
ejpam-5732	125	40	(	(	PUNCT
ejpam-5732	125	41	x	x	X
ejpam-5732	125	42	,	,	PUNCT
ejpam-5732	125	43	τ	τ	PROPN
ejpam-5732	125	44	,	,	PUNCT
ejpam-5732	125	45	i	i	PROPN
ejpam-5732	125	46	)	)	PUNCT
ejpam-5732	125	47	is	be	AUX
ejpam-5732	125	48	δ1	δ1	NOUN
ejpam-5732	125	49	-	-	PUNCT
ejpam-5732	125	50	βi	βi	PRON
ejpam-5732	125	51	-	-	PUNCT
ejpam-5732	125	52	paracompact	paracompact	ADJ
ejpam-5732	125	53	;	;	PUNCT
ejpam-5732	125	54	but	but	CCONJ
ejpam-5732	125	55	,	,	PUNCT
ejpam-5732	125	56	(	(	PUNCT
ejpam-5732	125	57	x	x	X
ejpam-5732	125	58	,	,	PUNCT
ejpam-5732	125	59	τ	τ	X
ejpam-5732	125	60	)	)	PUNCT
ejpam-5732	125	61	is	be	AUX
ejpam-5732	125	62	not	not	PART
ejpam-5732	125	63	β1	β1	NOUN
ejpam-5732	125	64	-	-	PUNCT
ejpam-5732	125	65	paracompact	paracompact	ADJ
ejpam-5732	125	66	,	,	PUNCT
ejpam-5732	125	67	as	as	SCONJ
ejpam-5732	125	68	there	there	PRON
ejpam-5732	125	69	exists	exist	VERB
ejpam-5732	125	70	a	a	DET
ejpam-5732	125	71	β	β	NOUN
ejpam-5732	125	72	-	-	ADJ
ejpam-5732	125	73	open	open	ADJ
ejpam-5732	125	74	cover	cover	NOUN
ejpam-5732	125	75	{	{	PUNCT
ejpam-5732	125	76	{	{	PUNCT
ejpam-5732	125	77	1	1	NUM
ejpam-5732	125	78	,	,	PUNCT
ejpam-5732	125	79	2	2	NUM
ejpam-5732	125	80	}	}	PUNCT
ejpam-5732	125	81	,	,	PUNCT
ejpam-5732	125	82	{	{	PUNCT
ejpam-5732	125	83	1	1	NUM
ejpam-5732	125	84	,	,	PUNCT
ejpam-5732	125	85	3	3	NUM
ejpam-5732	125	86	}	}	PUNCT
ejpam-5732	125	87	}	}	PUNCT
ejpam-5732	125	88	of	of	ADP
ejpam-5732	125	89	(	(	PUNCT
ejpam-5732	125	90	x	x	NOUN
ejpam-5732	125	91	,	,	PUNCT
ejpam-5732	125	92	τ	τ	X
ejpam-5732	125	93	)	)	PUNCT
ejpam-5732	125	94	that	that	PRON
ejpam-5732	125	95	lacks	lack	VERB
ejpam-5732	125	96	a	a	DET
ejpam-5732	125	97	locally	locally	ADV
ejpam-5732	125	98	finite	finite	ADJ
ejpam-5732	125	99	open	open	ADJ
ejpam-5732	125	100	refinement	refinement	NOUN
ejpam-5732	126	1	[	[	X
ejpam-5732	126	2	21	21	NUM
ejpam-5732	126	3	]	]	PUNCT
ejpam-5732	126	4	.	.	PUNCT
ejpam-5732	127	1	in	in	ADP
ejpam-5732	127	2	the	the	DET
ejpam-5732	127	3	subsequent	subsequent	ADJ
ejpam-5732	127	4	theorem	theorem	NOUN
ejpam-5732	127	5	,	,	PUNCT
ejpam-5732	127	6	we	we	PRON
ejpam-5732	127	7	discuss	discuss	VERB
ejpam-5732	127	8	a	a	DET
ejpam-5732	127	9	space	space	NOUN
ejpam-5732	127	10	endowed	endow	VERB
ejpam-5732	127	11	with	with	ADP
ejpam-5732	127	12	two	two	NUM
ejpam-5732	127	13	topologies	topology	NOUN
ejpam-5732	127	14	;	;	PUNCT
ejpam-5732	127	15	hence	hence	ADV
ejpam-5732	127	16	,	,	PUNCT
ejpam-5732	127	17	to	to	PART
ejpam-5732	127	18	avoid	avoid	VERB
ejpam-5732	127	19	ambiguity	ambiguity	NOUN
ejpam-5732	127	20	,	,	PUNCT
ejpam-5732	127	21	we	we	PRON
ejpam-5732	127	22	must	must	AUX
ejpam-5732	127	23	redefine	redefine	VERB
ejpam-5732	127	24	the	the	DET
ejpam-5732	127	25	concept	concept	NOUN
ejpam-5732	127	26	of	of	ADP
ejpam-5732	127	27	local	local	ADJ
ejpam-5732	127	28	finiteness	finiteness	NOUN
ejpam-5732	127	29	.	.	PUNCT
ejpam-5732	128	1	a	a	DET
ejpam-5732	128	2	collection	collection	NOUN
ejpam-5732	128	3	v	v	NOUN
ejpam-5732	128	4	of	of	ADP
ejpam-5732	128	5	subsets	subset	NOUN
ejpam-5732	128	6	of	of	ADP
ejpam-5732	128	7	an	an	DET
ejpam-5732	128	8	ideal	ideal	ADJ
ejpam-5732	128	9	topological	topological	ADJ
ejpam-5732	128	10	space	space	NOUN
ejpam-5732	128	11	(	(	PUNCT
ejpam-5732	128	12	x	x	X
ejpam-5732	128	13	,	,	PUNCT
ejpam-5732	128	14	τ	τ	PROPN
ejpam-5732	128	15	,	,	PUNCT
ejpam-5732	128	16	i	i	PROPN
ejpam-5732	128	17	)	)	PUNCT
ejpam-5732	128	18	is	be	AUX
ejpam-5732	128	19	said	say	VERB
ejpam-5732	128	20	to	to	PART
ejpam-5732	128	21	be	be	AUX
ejpam-5732	128	22	τ	τ	PROPN
ejpam-5732	128	23	-locally	-locally	ADV
ejpam-5732	128	24	finite	finite	ADJ
ejpam-5732	128	25	if	if	SCONJ
ejpam-5732	128	26	for	for	ADP
ejpam-5732	128	27	each	each	DET
ejpam-5732	128	28	x	x	SYM
ejpam-5732	128	29	∈	∈	PROPN
ejpam-5732	128	30	x	x	X
ejpam-5732	128	31	,	,	PUNCT
ejpam-5732	128	32	there	there	PRON
ejpam-5732	128	33	exists	exist	VERB
ejpam-5732	128	34	an	an	DET
ejpam-5732	128	35	open	open	ADJ
ejpam-5732	128	36	set	set	NOUN
ejpam-5732	128	37	u	u	NOUN
ejpam-5732	128	38	∈	∈	PROPN
ejpam-5732	128	39	τ	τ	X
ejpam-5732	128	40	such	such	ADJ
ejpam-5732	128	41	that	that	SCONJ
ejpam-5732	128	42	x	x	SYM
ejpam-5732	128	43	∈	∈	PROPN
ejpam-5732	128	44	u	u	NOUN
ejpam-5732	128	45	and	and	CCONJ
ejpam-5732	128	46	u	u	NOUN
ejpam-5732	128	47	intersects	intersect	NOUN
ejpam-5732	128	48	with	with	ADP
ejpam-5732	128	49	at	at	ADV
ejpam-5732	128	50	most	most	ADV
ejpam-5732	128	51	finitely	finitely	ADV
ejpam-5732	128	52	many	many	ADJ
ejpam-5732	128	53	elements	element	NOUN
ejpam-5732	128	54	of	of	ADP
ejpam-5732	128	55	v.	v.	INTJ
ejpam-5732	128	56	as	as	SCONJ
ejpam-5732	128	57	stated	state	VERB
ejpam-5732	128	58	in	in	ADP
ejpam-5732	128	59	[	[	X
ejpam-5732	128	60	14	14	NUM
ejpam-5732	128	61	]	]	PUNCT
ejpam-5732	128	62	,	,	PUNCT
ejpam-5732	128	63	the	the	DET
ejpam-5732	128	64	intersection	intersection	NOUN
ejpam-5732	128	65	of	of	ADP
ejpam-5732	128	66	any	any	DET
ejpam-5732	128	67	two	two	NUM
ejpam-5732	128	68	δ	δ	PROPN
ejpam-5732	128	69	-	-	PUNCT
ejpam-5732	128	70	βi	βi	PRON
ejpam-5732	128	71	-	-	PUNCT
ejpam-5732	128	72	open	open	ADJ
ejpam-5732	128	73	sets	set	NOUN
ejpam-5732	128	74	is	be	AUX
ejpam-5732	128	75	not	not	PART
ejpam-5732	128	76	necessarily	necessarily	ADV
ejpam-5732	128	77	a	a	DET
ejpam-5732	128	78	δ	δ	NOUN
ejpam-5732	128	79	-	-	PUNCT
ejpam-5732	128	80	βi	βi	ADV
ejpam-5732	128	81	-	-	PUNCT
ejpam-5732	128	82	open	open	ADJ
ejpam-5732	128	83	set	set	NOUN
ejpam-5732	128	84	;	;	PUNCT
ejpam-5732	128	85	therefore	therefore	ADV
ejpam-5732	128	86	,	,	PUNCT
ejpam-5732	128	87	this	this	DET
ejpam-5732	128	88	assumption	assumption	NOUN
ejpam-5732	128	89	must	must	AUX
ejpam-5732	128	90	be	be	AUX
ejpam-5732	128	91	made	make	VERB
ejpam-5732	128	92	in	in	ADP
ejpam-5732	128	93	the	the	DET
ejpam-5732	128	94	subsequent	subsequent	ADJ
ejpam-5732	128	95	theorem	theorem	NOUN
ejpam-5732	128	96	.	.	PUNCT
ejpam-5732	128	97	theorem	theorem	NOUN
ejpam-5732	128	98	2	2	NUM
ejpam-5732	128	99	.	.	PUNCT
ejpam-5732	129	1	let	let	VERB
ejpam-5732	129	2	(	(	PUNCT
ejpam-5732	129	3	x	x	X
ejpam-5732	129	4	,	,	PUNCT
ejpam-5732	129	5	τ	τ	PROPN
ejpam-5732	129	6	,	,	PUNCT
ejpam-5732	129	7	i	i	PRON
ejpam-5732	129	8	)	)	PUNCT
ejpam-5732	129	9	be	be	VERB
ejpam-5732	129	10	an	an	DET
ejpam-5732	129	11	ideal	ideal	ADJ
ejpam-5732	129	12	topological	topological	ADJ
ejpam-5732	129	13	space	space	NOUN
ejpam-5732	129	14	.	.	PUNCT
ejpam-5732	130	1	if	if	SCONJ
ejpam-5732	130	2	i	i	PRON
ejpam-5732	130	3	is	be	AUX
ejpam-5732	130	4	codense	codense	NOUN
ejpam-5732	130	5	and	and	CCONJ
ejpam-5732	130	6	τ	τ	PROPN
ejpam-5732	130	7	-simple	-simple	PROPN
ejpam-5732	130	8	,	,	PUNCT
ejpam-5732	130	9	(	(	PUNCT
ejpam-5732	130	10	x	x	NOUN
ejpam-5732	130	11	,	,	PUNCT
ejpam-5732	130	12	τ∗	τ∗	PROPN
ejpam-5732	130	13	,	,	PUNCT
ejpam-5732	130	14	i	i	NOUN
ejpam-5732	130	15	)	)	PUNCT
ejpam-5732	130	16	is	be	AUX
ejpam-5732	130	17	δ1	δ1	NOUN
ejpam-5732	130	18	-	-	PUNCT
ejpam-5732	130	19	βi	βi	PRON
ejpam-5732	130	20	-	-	PUNCT
ejpam-5732	130	21	paracompact	paracompact	ADJ
ejpam-5732	130	22	,	,	PUNCT
ejpam-5732	130	23	any	any	DET
ejpam-5732	130	24	δ	δ	PROPN
ejpam-5732	130	25	-	-	PUNCT
ejpam-5732	130	26	βi	βi	ADV
ejpam-5732	130	27	-	-	PUNCT
ejpam-5732	130	28	open	open	NOUN
ejpam-5732	130	29	set	set	NOUN
ejpam-5732	130	30	in	in	ADP
ejpam-5732	130	31	(	(	PUNCT
ejpam-5732	130	32	x	x	NOUN
ejpam-5732	130	33	,	,	PUNCT
ejpam-5732	130	34	τ	τ	PROPN
ejpam-5732	130	35	,	,	PUNCT
ejpam-5732	130	36	i	i	PROPN
ejpam-5732	130	37	)	)	PUNCT
ejpam-5732	130	38	is	be	AUX
ejpam-5732	130	39	a	a	DET
ejpam-5732	130	40	δ	δ	PROPN
ejpam-5732	130	41	-	-	PUNCT
ejpam-5732	130	42	βi	βi	ADV
ejpam-5732	130	43	-	-	PUNCT
ejpam-5732	130	44	open	open	NOUN
ejpam-5732	130	45	set	set	NOUN
ejpam-5732	130	46	in	in	ADP
ejpam-5732	130	47	(	(	PUNCT
ejpam-5732	130	48	x	x	NOUN
ejpam-5732	130	49	,	,	PUNCT
ejpam-5732	130	50	τ∗	τ∗	PROPN
ejpam-5732	130	51	,	,	PUNCT
ejpam-5732	130	52	i	i	NOUN
ejpam-5732	130	53	)	)	PUNCT
ejpam-5732	130	54	,	,	PUNCT
ejpam-5732	130	55	and	and	CCONJ
ejpam-5732	130	56	the	the	DET
ejpam-5732	130	57	intersection	intersection	NOUN
ejpam-5732	130	58	of	of	ADP
ejpam-5732	130	59	two	two	NUM
ejpam-5732	130	60	δ	δ	PROPN
ejpam-5732	130	61	-	-	PUNCT
ejpam-5732	130	62	βi	βi	ADV
ejpam-5732	130	63	-	-	PUNCT
ejpam-5732	130	64	open	open	ADJ
ejpam-5732	130	65	sets	set	NOUN
ejpam-5732	130	66	in	in	ADP
ejpam-5732	130	67	(	(	PUNCT
ejpam-5732	130	68	x	x	NOUN
ejpam-5732	130	69	,	,	PUNCT
ejpam-5732	130	70	τ	τ	PROPN
ejpam-5732	130	71	,	,	PUNCT
ejpam-5732	130	72	i	i	PRON
ejpam-5732	130	73	)	)	PUNCT
ejpam-5732	130	74	remains	remain	VERB
ejpam-5732	130	75	a	a	DET
ejpam-5732	130	76	δ	δ	PROPN
ejpam-5732	130	77	-	-	PUNCT
ejpam-5732	130	78	βi	βi	ADV
ejpam-5732	130	79	-	-	PUNCT
ejpam-5732	130	80	open	open	NOUN
ejpam-5732	130	81	set	set	NOUN
ejpam-5732	130	82	in	in	ADP
ejpam-5732	130	83	(	(	PUNCT
ejpam-5732	130	84	x	x	NOUN
ejpam-5732	130	85	,	,	PUNCT
ejpam-5732	130	86	τ	τ	PROPN
ejpam-5732	130	87	,	,	PUNCT
ejpam-5732	130	88	i	i	PROPN
ejpam-5732	130	89	)	)	PUNCT
ejpam-5732	130	90	,	,	PUNCT
ejpam-5732	130	91	then	then	ADV
ejpam-5732	130	92	every	every	DET
ejpam-5732	130	93	δ	δ	PROPN
ejpam-5732	130	94	-	-	PUNCT
ejpam-5732	130	95	βi	βi	ADV
ejpam-5732	130	96	-	-	PUNCT
ejpam-5732	130	97	open	open	ADJ
ejpam-5732	130	98	cover	cover	NOUN
ejpam-5732	130	99	of	of	ADP
ejpam-5732	130	100	(	(	PUNCT
ejpam-5732	130	101	x	x	X
ejpam-5732	130	102	,	,	PUNCT
ejpam-5732	130	103	τ	τ	PROPN
ejpam-5732	130	104	,	,	PUNCT
ejpam-5732	130	105	i	i	NOUN
ejpam-5732	130	106	)	)	PUNCT
ejpam-5732	130	107	has	have	VERB
ejpam-5732	130	108	a	a	DET
ejpam-5732	130	109	locally	locally	ADV
ejpam-5732	130	110	finite	finite	ADJ
ejpam-5732	130	111	δ	δ	PROPN
ejpam-5732	130	112	-	-	PUNCT
ejpam-5732	130	113	βi	βi	ADV
ejpam-5732	130	114	-	-	PUNCT
ejpam-5732	130	115	open	open	ADJ
ejpam-5732	130	116	i	i	NOUN
ejpam-5732	130	117	-	-	PUNCT
ejpam-5732	130	118	cover	cover	NOUN
ejpam-5732	130	119	refinement	refinement	NOUN
ejpam-5732	130	120	.	.	PUNCT
ejpam-5732	131	1	proof	proof	NOUN
ejpam-5732	131	2	.	.	PUNCT
ejpam-5732	132	1	let	let	VERB
ejpam-5732	132	2	u	u	PRON
ejpam-5732	132	3	=	=	PUNCT
ejpam-5732	132	4	{	{	PUNCT
ejpam-5732	132	5	uα	uα	X
ejpam-5732	132	6	:	:	PUNCT
ejpam-5732	132	7	α	α	PROPN
ejpam-5732	132	8	∈	∈	PROPN
ejpam-5732	132	9	λ1	λ1	PROPN
ejpam-5732	132	10	}	}	PUNCT
ejpam-5732	132	11	be	be	VERB
ejpam-5732	132	12	a	a	DET
ejpam-5732	132	13	δ	δ	PROPN
ejpam-5732	132	14	-	-	PUNCT
ejpam-5732	132	15	βi	βi	ADV
ejpam-5732	132	16	-	-	PUNCT
ejpam-5732	132	17	open	open	ADJ
ejpam-5732	132	18	cover	cover	NOUN
ejpam-5732	132	19	of	of	ADP
ejpam-5732	132	20	(	(	PUNCT
ejpam-5732	132	21	x	x	X
ejpam-5732	132	22	,	,	PUNCT
ejpam-5732	132	23	τ	τ	PROPN
ejpam-5732	132	24	,	,	PUNCT
ejpam-5732	132	25	i	i	PROPN
ejpam-5732	132	26	)	)	PUNCT
ejpam-5732	132	27	.	.	PUNCT
ejpam-5732	133	1	then	then	ADV
ejpam-5732	133	2	,	,	PUNCT
ejpam-5732	133	3	u	u	NOUN
ejpam-5732	133	4	is	be	AUX
ejpam-5732	133	5	a	a	DET
ejpam-5732	133	6	δ	δ	PROPN
ejpam-5732	133	7	-	-	PUNCT
ejpam-5732	133	8	βi	βi	ADV
ejpam-5732	133	9	-	-	PUNCT
ejpam-5732	133	10	open	open	ADJ
ejpam-5732	133	11	cover	cover	NOUN
ejpam-5732	133	12	of	of	ADP
ejpam-5732	133	13	(	(	PUNCT
ejpam-5732	133	14	x	x	NOUN
ejpam-5732	133	15	,	,	PUNCT
ejpam-5732	133	16	τ∗	τ∗	PROPN
ejpam-5732	133	17	,	,	PUNCT
ejpam-5732	133	18	i	i	PROPN
ejpam-5732	133	19	)	)	PUNCT
ejpam-5732	133	20	.	.	PUNCT
ejpam-5732	134	1	since	since	SCONJ
ejpam-5732	134	2	(	(	PUNCT
ejpam-5732	134	3	x	x	NOUN
ejpam-5732	134	4	,	,	PUNCT
ejpam-5732	134	5	τ∗	τ∗	PROPN
ejpam-5732	134	6	,	,	PUNCT
ejpam-5732	134	7	i	i	NOUN
ejpam-5732	134	8	)	)	PUNCT
ejpam-5732	134	9	is	be	AUX
ejpam-5732	134	10	δ1	δ1	NOUN
ejpam-5732	134	11	-	-	PUNCT
ejpam-5732	134	12	βi	βi	PRON
ejpam-5732	134	13	-	-	PUNCT
ejpam-5732	134	14	paracompact	paracompact	ADJ
ejpam-5732	134	15	,	,	PUNCT
ejpam-5732	134	16	u	u	NOUN
ejpam-5732	134	17	has	have	VERB
ejpam-5732	134	18	τ∗-locally	τ∗-locally	ADV
ejpam-5732	134	19	finite	finite	ADJ
ejpam-5732	134	20	refinement	refinement	NOUN
ejpam-5732	134	21	v	v	NOUN
ejpam-5732	134	22	=	=	PUNCT
ejpam-5732	134	23	{	{	PUNCT
ejpam-5732	134	24	gλ	gλ	NOUN
ejpam-5732	134	25	:	:	PUNCT
ejpam-5732	134	26	λ	λ	PROPN
ejpam-5732	134	27	∈	∈	PROPN
ejpam-5732	134	28	λ2	λ2	NOUN
ejpam-5732	134	29	}	}	PUNCT
ejpam-5732	134	30	of	of	ADP
ejpam-5732	134	31	open	open	ADJ
ejpam-5732	134	32	sets	set	NOUN
ejpam-5732	134	33	in	in	ADP
ejpam-5732	134	34	(	(	PUNCT
ejpam-5732	134	35	x	x	NOUN
ejpam-5732	134	36	,	,	PUNCT
ejpam-5732	134	37	τ∗	τ∗	PROPN
ejpam-5732	134	38	,	,	PUNCT
ejpam-5732	134	39	i	i	NOUN
ejpam-5732	134	40	)	)	PUNCT
ejpam-5732	134	41	such	such	ADJ
ejpam-5732	134	42	that	that	SCONJ
ejpam-5732	134	43	x	x	PRON
ejpam-5732	135	1	−	−	NOUN
ejpam-5732	135	2	∪{gλ	∪{gλ	NOUN
ejpam-5732	135	3	:	:	PUNCT
ejpam-5732	135	4	λ	λ	PROPN
ejpam-5732	135	5	∈	∈	PROPN
ejpam-5732	135	6	λ2	λ2	PROPN
ejpam-5732	135	7	}	}	PUNCT
ejpam-5732	135	8	∈	∈	PROPN
ejpam-5732	135	9	i	i	PRON
ejpam-5732	135	10	,	,	PUNCT
ejpam-5732	135	11	where	where	SCONJ
ejpam-5732	135	12	gλ	gλ	NOUN
ejpam-5732	135	13	=	=	PUNCT
ejpam-5732	135	14	vλ	vλ	ADP
ejpam-5732	135	15	−	−	PROPN
ejpam-5732	135	16	iλ	iλ	NOUN
ejpam-5732	135	17	,	,	PUNCT
ejpam-5732	135	18	vλ	vλ	INTJ
ejpam-5732	135	19	∈	∈	PROPN
ejpam-5732	135	20	τ	τ	X
ejpam-5732	135	21	and	and	CCONJ
ejpam-5732	135	22	iλ	iλ	PROPN
ejpam-5732	135	23	∈	∈	PROPN
ejpam-5732	135	24	i	i	PRON
ejpam-5732	135	25	for	for	ADP
ejpam-5732	135	26	all	all	DET
ejpam-5732	135	27	λ	λ	PROPN
ejpam-5732	135	28	∈	∈	PROPN
ejpam-5732	135	29	λ2	λ2	NOUN
ejpam-5732	135	30	.	.	PUNCT
ejpam-5732	136	1	because	because	SCONJ
ejpam-5732	136	2	v	v	NOUN
ejpam-5732	136	3	is	be	AUX
ejpam-5732	136	4	τ∗-locally	τ∗-locally	ADV
ejpam-5732	136	5	finite	finite	ADJ
ejpam-5732	136	6	,	,	PUNCT
ejpam-5732	136	7	for	for	ADP
ejpam-5732	136	8	every	every	DET
ejpam-5732	136	9	x	x	SYM
ejpam-5732	136	10	∈	∈	PROPN
ejpam-5732	136	11	x	x	NOUN
ejpam-5732	136	12	,	,	PUNCT
ejpam-5732	136	13	there	there	PRON
ejpam-5732	136	14	exists	exist	VERB
ejpam-5732	136	15	h	h	NOUN
ejpam-5732	136	16	∈	∈	PROPN
ejpam-5732	136	17	τ∗	τ∗	NOUN
ejpam-5732	136	18	containing	contain	VERB
ejpam-5732	136	19	x	x	PUNCT
ejpam-5732	136	20	such	such	ADJ
ejpam-5732	136	21	that	that	SCONJ
ejpam-5732	136	22	h	h	NOUN
ejpam-5732	136	23	∩	∩	ADJ
ejpam-5732	136	24	gλ	gλ	NOUN
ejpam-5732	136	25	=	=	PUNCT
ejpam-5732	136	26	∅	∅	NOUN
ejpam-5732	136	27	for	for	ADP
ejpam-5732	136	28	λ	λ	PROPN
ejpam-5732	136	29	̸=	̸=	PROPN
ejpam-5732	136	30	λ1	λ1	PROPN
ejpam-5732	136	31	,	,	PUNCT
ejpam-5732	136	32	λ2	λ2	NOUN
ejpam-5732	136	33	,	,	PUNCT
ejpam-5732	136	34	...	...	PUNCT
ejpam-5732	136	35	,	,	PUNCT
ejpam-5732	136	36	λn	λn	NOUN
ejpam-5732	136	37	.	.	PUNCT
ejpam-5732	137	1	given	give	VERB
ejpam-5732	137	2	that	that	SCONJ
ejpam-5732	137	3	i	i	PRON
ejpam-5732	137	4	is	be	AUX
ejpam-5732	137	5	τ	τ	PROPN
ejpam-5732	137	6	-simple	-simple	NUM
ejpam-5732	137	7	,	,	PUNCT
ejpam-5732	137	8	h	h	NOUN
ejpam-5732	137	9	=	=	SYM
ejpam-5732	137	10	u	u	NOUN
ejpam-5732	137	11	−	−	PROPN
ejpam-5732	138	1	i	i	PRON
ejpam-5732	138	2	,	,	PUNCT
ejpam-5732	138	3	for	for	ADP
ejpam-5732	138	4	some	some	DET
ejpam-5732	138	5	u	u	NOUN
ejpam-5732	138	6	∈	∈	PROPN
ejpam-5732	138	7	τ	τ	X
ejpam-5732	138	8	and	and	CCONJ
ejpam-5732	138	9	i	i	PROPN
ejpam-5732	138	10	∈	∈	PROPN
ejpam-5732	138	11	i.	i.	NOUN
ejpam-5732	138	12	hence	hence	ADV
ejpam-5732	138	13	,	,	PUNCT
ejpam-5732	138	14	(	(	PUNCT
ejpam-5732	138	15	u	u	NOUN
ejpam-5732	138	16	−	−	PROPN
ejpam-5732	138	17	i)∩gλ	i)∩gλ	ADJ
ejpam-5732	138	18	=	=	SYM
ejpam-5732	138	19	∅	∅	NOUN
ejpam-5732	138	20	for	for	ADP
ejpam-5732	138	21	λ	λ	PROPN
ejpam-5732	138	22	̸=	̸=	PROPN
ejpam-5732	138	23	λ1	λ1	PROPN
ejpam-5732	138	24	,	,	PUNCT
ejpam-5732	138	25	λ2	λ2	NOUN
ejpam-5732	138	26	,	,	PUNCT
ejpam-5732	138	27	...	...	PUNCT
ejpam-5732	138	28	,	,	PUNCT
ejpam-5732	138	29	λn	λn	NOUN
ejpam-5732	138	30	,	,	PUNCT
ejpam-5732	138	31	which	which	PRON
ejpam-5732	138	32	implies	imply	VERB
ejpam-5732	138	33	that	that	SCONJ
ejpam-5732	138	34	(	(	PUNCT
ejpam-5732	138	35	u	u	NOUN
ejpam-5732	138	36	∩	∩	NOUN
ejpam-5732	138	37	vλ	vλ	ADJ
ejpam-5732	138	38	)	)	PUNCT
ejpam-5732	138	39	−	−	PROPN
ejpam-5732	139	1	(	(	PUNCT
ejpam-5732	139	2	i	i	PRON
ejpam-5732	139	3	∪	∪	VERB
ejpam-5732	139	4	iλ	iλ	NOUN
ejpam-5732	139	5	)	)	PUNCT
ejpam-5732	139	6	=	=	NOUN
ejpam-5732	139	7	∅	∅	NOUN
ejpam-5732	139	8	for	for	ADP
ejpam-5732	139	9	λ	λ	PROPN
ejpam-5732	139	10	̸=	̸=	PROPN
ejpam-5732	139	11	λ1	λ1	PROPN
ejpam-5732	139	12	,	,	PUNCT
ejpam-5732	139	13	λ2	λ2	NOUN
ejpam-5732	139	14	,	,	PUNCT
ejpam-5732	139	15	...	...	PUNCT
ejpam-5732	139	16	,	,	PUNCT
ejpam-5732	139	17	λn	λn	NOUN
ejpam-5732	139	18	.	.	PUNCT
ejpam-5732	140	1	as	as	SCONJ
ejpam-5732	140	2	i	i	PRON
ejpam-5732	140	3	is	be	AUX
ejpam-5732	140	4	codense	codense	NOUN
ejpam-5732	140	5	,	,	PUNCT
ejpam-5732	140	6	u	u	NOUN
ejpam-5732	140	7	∩	∩	NOUN
ejpam-5732	140	8	vλ	vλ	ADP
ejpam-5732	140	9	=	=	NOUN
ejpam-5732	140	10	∅	∅	NOUN
ejpam-5732	140	11	for	for	ADP
ejpam-5732	140	12	λ	λ	PROPN
ejpam-5732	140	13	̸=	̸=	PROPN
ejpam-5732	140	14	λ1	λ1	PROPN
ejpam-5732	140	15	,	,	PUNCT
ejpam-5732	140	16	λ2	λ2	NOUN
ejpam-5732	140	17	,	,	PUNCT
ejpam-5732	140	18	...	...	PUNCT
ejpam-5732	140	19	,	,	PUNCT
ejpam-5732	140	20	λn	λn	NOUN
ejpam-5732	140	21	,	,	PUNCT
ejpam-5732	140	22	and	and	CCONJ
ejpam-5732	140	23	therefore	therefore	ADV
ejpam-5732	140	24	u	u	NOUN
ejpam-5732	140	25	∩	∩	NOUN
ejpam-5732	140	26	(	(	PUNCT
ejpam-5732	140	27	vλ	vλ	ADP
ejpam-5732	140	28	∩	∩	NOUN
ejpam-5732	140	29	uα	uα	NOUN
ejpam-5732	140	30	)	)	PUNCT
ejpam-5732	140	31	=	=	NOUN
ejpam-5732	140	32	∅	∅	NOUN
ejpam-5732	140	33	for	for	ADP
ejpam-5732	140	34	λ	λ	PROPN
ejpam-5732	140	35	̸=	̸=	PROPN
ejpam-5732	140	36	λ1	λ1	PROPN
ejpam-5732	140	37	,	,	PUNCT
ejpam-5732	140	38	λ2	λ2	PROPN
ejpam-5732	140	39	,	,	PUNCT
ejpam-5732	140	40	...	...	PUNCT
ejpam-5732	140	41	,	,	PUNCT
ejpam-5732	140	42	λn	λn	PROPN
ejpam-5732	140	43	and	and	CCONJ
ejpam-5732	140	44	for	for	ADP
ejpam-5732	140	45	all	all	DET
ejpam-5732	140	46	α	α	DET
ejpam-5732	140	47	∈	∈	PROPN
ejpam-5732	140	48	λ1	λ1	PROPN
ejpam-5732	140	49	.	.	PUNCT
ejpam-5732	141	1	consequently	consequently	ADV
ejpam-5732	141	2	,	,	PUNCT
ejpam-5732	141	3	w	w	PROPN
ejpam-5732	141	4	=	=	X
ejpam-5732	141	5	{	{	PUNCT
ejpam-5732	141	6	vλ	vλ	ADP
ejpam-5732	141	7	∩	∩	NOUN
ejpam-5732	141	8	uα	uα	NOUN
ejpam-5732	141	9	:	:	PUNCT
ejpam-5732	141	10	α	α	PROPN
ejpam-5732	141	11	∈	∈	PROPN
ejpam-5732	141	12	λ1	λ1	PROPN
ejpam-5732	141	13	,	,	PUNCT
ejpam-5732	141	14	λ	λ	PROPN
ejpam-5732	141	15	∈	∈	PROPN
ejpam-5732	141	16	λ2	λ2	NOUN
ejpam-5732	141	17	}	}	PUNCT
ejpam-5732	141	18	is	be	AUX
ejpam-5732	141	19	τ	τ	PROPN
ejpam-5732	141	20	-locally	-locally	ADV
ejpam-5732	141	21	finite	finite	ADJ
ejpam-5732	141	22	,	,	PUNCT
ejpam-5732	141	23	and	and	CCONJ
ejpam-5732	141	24	w	w	NOUN
ejpam-5732	141	25	is	be	AUX
ejpam-5732	141	26	a	a	DET
ejpam-5732	141	27	δ	δ	PROPN
ejpam-5732	141	28	-	-	PUNCT
ejpam-5732	141	29	βi	βi	ADV
ejpam-5732	141	30	-	-	PUNCT
ejpam-5732	141	31	open	open	ADJ
ejpam-5732	141	32	refinement	refinement	NOUN
ejpam-5732	141	33	of	of	ADP
ejpam-5732	141	34	u	u	PROPN
ejpam-5732	141	35	,	,	PUNCT
ejpam-5732	141	36	following	follow	VERB
ejpam-5732	141	37	assumption	assumption	NOUN
ejpam-5732	141	38	.	.	PUNCT
ejpam-5732	142	1	next	next	ADV
ejpam-5732	142	2	,	,	PUNCT
ejpam-5732	142	3	we	we	PRON
ejpam-5732	142	4	will	will	AUX
ejpam-5732	142	5	show	show	VERB
ejpam-5732	142	6	that	that	SCONJ
ejpam-5732	142	7	w	w	NOUN
ejpam-5732	142	8	refines	refine	VERB
ejpam-5732	142	9	u	u	NOUN
ejpam-5732	142	10	.	.	PUNCT
ejpam-5732	143	1	because	because	SCONJ
ejpam-5732	143	2	v	v	NOUN
ejpam-5732	143	3	refines	refine	VERB
ejpam-5732	143	4	u	u	NOUN
ejpam-5732	143	5	,	,	PUNCT
ejpam-5732	143	6	for	for	ADP
ejpam-5732	143	7	every	every	DET
ejpam-5732	143	8	gλ	gλ	NOUN
ejpam-5732	143	9	∈	∈	PROPN
ejpam-5732	143	10	v	v	NOUN
ejpam-5732	143	11	,	,	PUNCT
ejpam-5732	143	12	there	there	PRON
ejpam-5732	143	13	exists	exist	VERB
ejpam-5732	143	14	uα	uα	PROPN
ejpam-5732	143	15	∈	∈	PROPN
ejpam-5732	143	16	u	u	NOUN
ejpam-5732	143	17	such	such	ADJ
ejpam-5732	143	18	that	that	SCONJ
ejpam-5732	143	19	gλ	gλ	PROPN
ejpam-5732	143	20	⊂	⊂	PROPN
ejpam-5732	143	21	uα	uα	PROPN
ejpam-5732	143	22	.	.	PUNCT
ejpam-5732	144	1	therefore	therefore	ADV
ejpam-5732	144	2	gλ	gλ	PROPN
ejpam-5732	144	3	=	=	SYM
ejpam-5732	144	4	uα∩gλ	uα∩gλ	PROPN
ejpam-5732	144	5	=	=	SYM
ejpam-5732	144	6	uα∩(vλ−iλ	uα∩(vλ−iλ	PROPN
ejpam-5732	144	7	)	)	PUNCT
ejpam-5732	144	8	=	=	PUNCT
ejpam-5732	144	9	(	(	PUNCT
ejpam-5732	144	10	vλ∩uα)−iλ	vλ∩uα)−iλ	NOUN
ejpam-5732	144	11	⊂	⊂	PROPN
ejpam-5732	144	12	vλ∩uα	vλ∩uα	X
ejpam-5732	144	13	⊂	⊂	PROPN
ejpam-5732	145	1	uα	uα	PROPN
ejpam-5732	145	2	.	.	PUNCT
ejpam-5732	146	1	it	it	PRON
ejpam-5732	146	2	implies	imply	VERB
ejpam-5732	146	3	that	that	SCONJ
ejpam-5732	146	4	x−∪{vλ∩uα	x−∪{vλ∩uα	PROPN
ejpam-5732	146	5	:	:	PUNCT
ejpam-5732	146	6	α	α	PROPN
ejpam-5732	146	7	∈	∈	PROPN
ejpam-5732	146	8	λ1	λ1	PROPN
ejpam-5732	146	9	,	,	PUNCT
ejpam-5732	146	10	λ	λ	PROPN
ejpam-5732	146	11	∈	∈	PROPN
ejpam-5732	146	12	λ2	λ2	PROPN
ejpam-5732	146	13	}	}	PUNCT
ejpam-5732	146	14	⊂	⊂	PROPN
ejpam-5732	146	15	x−∪{gλ	x−∪{gλ	NOUN
ejpam-5732	146	16	:	:	PUNCT
ejpam-5732	147	1	λ	λ	X
ejpam-5732	147	2	∈	∈	PROPN
ejpam-5732	147	3	λ2	λ2	PROPN
ejpam-5732	147	4	}	}	PUNCT
ejpam-5732	147	5	∈	∈	PROPN
ejpam-5732	147	6	i	i	PRON
ejpam-5732	147	7	,	,	PUNCT
ejpam-5732	147	8	which	which	PRON
ejpam-5732	147	9	implies	imply	VERB
ejpam-5732	147	10	that	that	SCONJ
ejpam-5732	147	11	x	x	X
ejpam-5732	148	1	−	−	PRON
ejpam-5732	148	2	∪{vλ	∪{vλ	NUM
ejpam-5732	148	3	∩	∩	X
ejpam-5732	148	4	uα	uα	NOUN
ejpam-5732	148	5	:	:	PUNCT
ejpam-5732	148	6	α	α	PROPN
ejpam-5732	148	7	∈	∈	PROPN
ejpam-5732	148	8	λ1	λ1	PROPN
ejpam-5732	148	9	,	,	PUNCT
ejpam-5732	148	10	λ	λ	PROPN
ejpam-5732	148	11	∈	∈	PROPN
ejpam-5732	148	12	λ2	λ2	PROPN
ejpam-5732	148	13	}	}	PUNCT
ejpam-5732	148	14	∈	∈	PROPN
ejpam-5732	148	15	i.	i.	NOUN
ejpam-5732	148	16	the	the	DET
ejpam-5732	148	17	following	follow	VERB
ejpam-5732	148	18	lemma	lemma	PROPN
ejpam-5732	148	19	is	be	AUX
ejpam-5732	148	20	required	require	VERB
ejpam-5732	148	21	for	for	ADP
ejpam-5732	148	22	the	the	DET
ejpam-5732	148	23	proof	proof	NOUN
ejpam-5732	148	24	of	of	ADP
ejpam-5732	148	25	proposition	proposition	NOUN
ejpam-5732	148	26	1	1	NUM
ejpam-5732	148	27	.	.	PUNCT
ejpam-5732	149	1	lemma	lemma	PROPN
ejpam-5732	149	2	7	7	X
ejpam-5732	149	3	.	.	PUNCT
ejpam-5732	150	1	let	let	VERB
ejpam-5732	150	2	(	(	PUNCT
ejpam-5732	150	3	x	x	X
ejpam-5732	150	4	,	,	PUNCT
ejpam-5732	150	5	τ	τ	PROPN
ejpam-5732	150	6	,	,	PUNCT
ejpam-5732	150	7	i	i	PRON
ejpam-5732	150	8	)	)	PUNCT
ejpam-5732	150	9	be	be	VERB
ejpam-5732	150	10	an	an	DET
ejpam-5732	150	11	ideal	ideal	ADJ
ejpam-5732	150	12	topological	topological	ADJ
ejpam-5732	150	13	space	space	NOUN
ejpam-5732	150	14	and	and	CCONJ
ejpam-5732	150	15	a	a	DET
ejpam-5732	150	16	⊂	⊂	PROPN
ejpam-5732	150	17	x.	x.	NOUN
ejpam-5732	151	1	the	the	DET
ejpam-5732	151	2	following	follow	VERB
ejpam-5732	151	3	statements	statement	NOUN
ejpam-5732	151	4	are	be	AUX
ejpam-5732	151	5	true	true	ADJ
ejpam-5732	151	6	:	:	PUNCT
ejpam-5732	151	7	c.	c.	PROPN
ejpam-5732	151	8	boonpok	boonpok	PROPN
ejpam-5732	151	9	,	,	PUNCT
ejpam-5732	151	10	a.	a.	PROPN
ejpam-5732	151	11	sama	sama	PROPN
ejpam-5732	151	12	-	-	PUNCT
ejpam-5732	151	13	ae	ae	PROPN
ejpam-5732	151	14	/	/	SYM
ejpam-5732	151	15	eur	eur	PROPN
ejpam-5732	151	16	.	.	PUNCT
ejpam-5732	152	1	j.	j.	PROPN
ejpam-5732	152	2	pure	pure	PROPN
ejpam-5732	152	3	appl	appl	PROPN
ejpam-5732	152	4	.	.	PROPN
ejpam-5732	152	5	math	math	PROPN
ejpam-5732	152	6	,	,	PUNCT
ejpam-5732	152	7	18	18	NUM
ejpam-5732	152	8	(	(	PUNCT
ejpam-5732	152	9	1	1	NUM
ejpam-5732	152	10	)	)	PUNCT
ejpam-5732	152	11	(	(	PUNCT
ejpam-5732	152	12	2025	2025	NUM
ejpam-5732	152	13	)	)	PUNCT
ejpam-5732	152	14	,	,	PUNCT
ejpam-5732	152	15	5732	5732	NUM
ejpam-5732	152	16	6	6	NUM
ejpam-5732	152	17	of	of	ADP
ejpam-5732	152	18	13	13	NUM
ejpam-5732	152	19	(	(	PUNCT
ejpam-5732	152	20	i	i	NOUN
ejpam-5732	152	21	)	)	PUNCT
ejpam-5732	152	22	x	x	SYM
ejpam-5732	153	1	∈	∈	PROPN
ejpam-5732	153	2	δ	δ	PROPN
ejpam-5732	153	3	-	-	PUNCT
ejpam-5732	153	4	βcli(a	βcli(a	NOUN
ejpam-5732	153	5	)	)	PUNCT
ejpam-5732	153	6	if	if	SCONJ
ejpam-5732	153	7	and	and	CCONJ
ejpam-5732	153	8	only	only	ADV
ejpam-5732	153	9	if	if	SCONJ
ejpam-5732	153	10	g	g	PROPN
ejpam-5732	153	11	∩a	∩a	PROPN
ejpam-5732	153	12	̸=	̸=	PROPN
ejpam-5732	153	13	∅	∅	NOUN
ejpam-5732	153	14	for	for	ADP
ejpam-5732	153	15	every	every	DET
ejpam-5732	153	16	δ	δ	PROPN
ejpam-5732	153	17	-	-	PUNCT
ejpam-5732	153	18	βi	βi	ADV
ejpam-5732	153	19	-	-	PUNCT
ejpam-5732	153	20	open	open	NOUN
ejpam-5732	153	21	set	set	VERB
ejpam-5732	153	22	g	g	NOUN
ejpam-5732	153	23	containing	contain	VERB
ejpam-5732	153	24	x.	x.	NOUN
ejpam-5732	153	25	(	(	PUNCT
ejpam-5732	153	26	ii	ii	PROPN
ejpam-5732	153	27	)	)	PUNCT
ejpam-5732	153	28	g	g	PROPN
ejpam-5732	153	29	∩	∩	PROPN
ejpam-5732	153	30	δ	δ	PROPN
ejpam-5732	153	31	-	-	PUNCT
ejpam-5732	153	32	βcli(a	βcli(a	NOUN
ejpam-5732	153	33	)	)	PUNCT
ejpam-5732	153	34	=	=	NOUN
ejpam-5732	153	35	∅	∅	NOUN
ejpam-5732	153	36	if	if	SCONJ
ejpam-5732	153	37	and	and	CCONJ
ejpam-5732	153	38	only	only	ADV
ejpam-5732	153	39	if	if	SCONJ
ejpam-5732	153	40	g	g	PROPN
ejpam-5732	153	41	∩a	∩a	NOUN
ejpam-5732	153	42	=	=	PUNCT
ejpam-5732	153	43	∅	∅	NOUN
ejpam-5732	153	44	,	,	PUNCT
ejpam-5732	153	45	for	for	ADP
ejpam-5732	153	46	every	every	DET
ejpam-5732	153	47	δ	δ	PROPN
ejpam-5732	153	48	-	-	PUNCT
ejpam-5732	153	49	βi	βi	ADV
ejpam-5732	153	50	-	-	PUNCT
ejpam-5732	153	51	open	open	ADJ
ejpam-5732	153	52	set	set	VERB
ejpam-5732	153	53	g.	g.	NOUN
ejpam-5732	153	54	proof	proof	NOUN
ejpam-5732	153	55	.	.	PUNCT
ejpam-5732	154	1	(	(	PUNCT
ejpam-5732	154	2	i	i	NOUN
ejpam-5732	154	3	)	)	PUNCT
ejpam-5732	154	4	it	it	PRON
ejpam-5732	154	5	is	be	AUX
ejpam-5732	154	6	derived	derive	VERB
ejpam-5732	154	7	from	from	ADP
ejpam-5732	154	8	theorem	theorem	NOUN
ejpam-5732	154	9	1	1	NUM
ejpam-5732	154	10	in	in	ADP
ejpam-5732	154	11	[	[	X
ejpam-5732	154	12	14	14	NUM
ejpam-5732	154	13	]	]	PUNCT
ejpam-5732	154	14	.	.	PUNCT
ejpam-5732	155	1	(	(	PUNCT
ejpam-5732	155	2	ii	ii	X
ejpam-5732	155	3	)	)	PUNCT
ejpam-5732	155	4	this	this	PRON
ejpam-5732	155	5	is	be	AUX
ejpam-5732	155	6	a	a	DET
ejpam-5732	155	7	consequence	consequence	NOUN
ejpam-5732	155	8	of	of	ADP
ejpam-5732	155	9	(	(	PUNCT
ejpam-5732	155	10	i	i	NOUN
ejpam-5732	155	11	)	)	PUNCT
ejpam-5732	155	12	.	.	PUNCT
ejpam-5732	156	1	proposition	proposition	NOUN
ejpam-5732	156	2	1	1	NUM
ejpam-5732	156	3	.	.	PUNCT
ejpam-5732	157	1	let	let	VERB
ejpam-5732	157	2	(	(	PUNCT
ejpam-5732	157	3	x	x	X
ejpam-5732	157	4	,	,	PUNCT
ejpam-5732	157	5	τ	τ	PROPN
ejpam-5732	157	6	,	,	PUNCT
ejpam-5732	157	7	i	i	PRON
ejpam-5732	157	8	)	)	PUNCT
ejpam-5732	157	9	be	be	VERB
ejpam-5732	157	10	a	a	DET
ejpam-5732	157	11	δ1	δ1	VERB
ejpam-5732	157	12	-	-	PUNCT
ejpam-5732	157	13	βi	βi	PRON
ejpam-5732	157	14	-	-	PUNCT
ejpam-5732	157	15	paracompact	paracompact	ADJ
ejpam-5732	157	16	space	space	NOUN
ejpam-5732	157	17	.	.	PUNCT
ejpam-5732	158	1	if	if	SCONJ
ejpam-5732	158	2	for	for	ADP
ejpam-5732	158	3	any	any	DET
ejpam-5732	158	4	δ	δ	PROPN
ejpam-5732	158	5	-	-	PUNCT
ejpam-5732	158	6	βi	βi	ADV
ejpam-5732	158	7	-	-	PUNCT
ejpam-5732	158	8	open	open	ADJ
ejpam-5732	158	9	set	set	NOUN
ejpam-5732	158	10	u	u	NOUN
ejpam-5732	158	11	that	that	PRON
ejpam-5732	158	12	contains	contain	VERB
ejpam-5732	158	13	x	x	PRON
ejpam-5732	158	14	,	,	PUNCT
ejpam-5732	158	15	there	there	PRON
ejpam-5732	158	16	exists	exist	VERB
ejpam-5732	158	17	a	a	DET
ejpam-5732	158	18	δ	δ	PROPN
ejpam-5732	158	19	-	-	PUNCT
ejpam-5732	158	20	βi	βi	ADV
ejpam-5732	158	21	-	-	PUNCT
ejpam-5732	158	22	open	open	NOUN
ejpam-5732	158	23	set	set	VERB
ejpam-5732	158	24	v	v	ADP
ejpam-5732	158	25	such	such	ADJ
ejpam-5732	158	26	that	that	SCONJ
ejpam-5732	158	27	x	x	SYM
ejpam-5732	158	28	∈	∈	PROPN
ejpam-5732	158	29	v	v	ADP
ejpam-5732	158	30	⊂	⊂	PROPN
ejpam-5732	158	31	δ	δ	PROPN
ejpam-5732	158	32	-	-	PUNCT
ejpam-5732	158	33	βcli(v	βcli(v	PROPN
ejpam-5732	158	34	)	)	PUNCT
ejpam-5732	159	1	⊂	⊂	PROPN
ejpam-5732	159	2	u	u	PROPN
ejpam-5732	159	3	,	,	PUNCT
ejpam-5732	159	4	then	then	ADV
ejpam-5732	159	5	every	every	DET
ejpam-5732	159	6	δ	δ	PROPN
ejpam-5732	159	7	-	-	PUNCT
ejpam-5732	159	8	βi	βi	ADV
ejpam-5732	159	9	-	-	PUNCT
ejpam-5732	159	10	open	open	ADJ
ejpam-5732	159	11	cover	cover	NOUN
ejpam-5732	159	12	of	of	ADP
ejpam-5732	159	13	x	x	PUNCT
ejpam-5732	159	14	has	have	VERB
ejpam-5732	159	15	a	a	DET
ejpam-5732	159	16	δ	δ	PROPN
ejpam-5732	159	17	-	-	PUNCT
ejpam-5732	159	18	βi	βi	ADV
ejpam-5732	159	19	-	-	PUNCT
ejpam-5732	159	20	locally	locally	ADV
ejpam-5732	159	21	finite	finite	PROPN
ejpam-5732	159	22	δ	δ	PROPN
ejpam-5732	159	23	-	-	ADJ
ejpam-5732	159	24	βi	βi	ADV
ejpam-5732	159	25	-	-	PUNCT
ejpam-5732	159	26	closed	close	VERB
ejpam-5732	159	27	i	i	NOUN
ejpam-5732	159	28	-	-	PUNCT
ejpam-5732	159	29	cover	cover	NOUN
ejpam-5732	159	30	refinement	refinement	NOUN
ejpam-5732	159	31	.	.	PUNCT
ejpam-5732	160	1	proof	proof	NOUN
ejpam-5732	160	2	.	.	PUNCT
ejpam-5732	161	1	let	let	VERB
ejpam-5732	161	2	u	u	PRON
ejpam-5732	161	3	be	be	AUX
ejpam-5732	161	4	a	a	DET
ejpam-5732	161	5	δ	δ	PROPN
ejpam-5732	161	6	-	-	PUNCT
ejpam-5732	161	7	βi	βi	ADV
ejpam-5732	161	8	-	-	PUNCT
ejpam-5732	161	9	open	open	ADJ
ejpam-5732	161	10	cover	cover	NOUN
ejpam-5732	161	11	of	of	ADP
ejpam-5732	161	12	x.	x.	NOUN
ejpam-5732	161	13	for	for	ADP
ejpam-5732	161	14	each	each	DET
ejpam-5732	161	15	x	x	SYM
ejpam-5732	161	16	∈	∈	PROPN
ejpam-5732	161	17	x	x	NOUN
ejpam-5732	161	18	,	,	PUNCT
ejpam-5732	161	19	let	let	VERB
ejpam-5732	161	20	ux	ux	PRON
ejpam-5732	161	21	be	be	AUX
ejpam-5732	161	22	a	a	DET
ejpam-5732	161	23	δ	δ	PROPN
ejpam-5732	161	24	-	-	PUNCT
ejpam-5732	161	25	βi	βi	ADV
ejpam-5732	161	26	-	-	PUNCT
ejpam-5732	161	27	open	open	ADJ
ejpam-5732	161	28	set	set	NOUN
ejpam-5732	161	29	in	in	ADP
ejpam-5732	161	30	u	u	NOUN
ejpam-5732	161	31	containing	contain	VERB
ejpam-5732	161	32	x.	x.	NOUN
ejpam-5732	161	33	by	by	ADP
ejpam-5732	161	34	assumption	assumption	NOUN
ejpam-5732	161	35	,	,	PUNCT
ejpam-5732	161	36	there	there	PRON
ejpam-5732	161	37	exists	exist	VERB
ejpam-5732	161	38	a	a	DET
ejpam-5732	161	39	δ	δ	PROPN
ejpam-5732	161	40	-	-	PUNCT
ejpam-5732	161	41	βi	βi	ADV
ejpam-5732	161	42	-	-	PUNCT
ejpam-5732	161	43	open	open	NOUN
ejpam-5732	161	44	set	set	NOUN
ejpam-5732	161	45	vx	vx	ADP
ejpam-5732	161	46	such	such	ADJ
ejpam-5732	161	47	that	that	SCONJ
ejpam-5732	161	48	x	x	SYM
ejpam-5732	161	49	∈	∈	PROPN
ejpam-5732	161	50	vx	vx	PROPN
ejpam-5732	161	51	⊂	⊂	PROPN
ejpam-5732	161	52	δ	δ	PROPN
ejpam-5732	161	53	-	-	PUNCT
ejpam-5732	161	54	βcli(vx	βcli(vx	ADJ
ejpam-5732	161	55	)	)	PUNCT
ejpam-5732	161	56	⊂	⊂	PROPN
ejpam-5732	161	57	ux	ux	PROPN
ejpam-5732	161	58	.	.	PUNCT
ejpam-5732	162	1	thus	thus	ADV
ejpam-5732	162	2	v	v	X
ejpam-5732	162	3	=	=	SYM
ejpam-5732	162	4	{	{	PUNCT
ejpam-5732	162	5	vx	vx	X
ejpam-5732	162	6	:	:	PUNCT
ejpam-5732	162	7	x	x	SYM
ejpam-5732	162	8	∈	∈	PROPN
ejpam-5732	162	9	x	x	PRON
ejpam-5732	162	10	}	}	PUNCT
ejpam-5732	162	11	is	be	AUX
ejpam-5732	162	12	a	a	DET
ejpam-5732	162	13	δ	δ	PROPN
ejpam-5732	162	14	-	-	PUNCT
ejpam-5732	162	15	βi	βi	ADV
ejpam-5732	162	16	-	-	PUNCT
ejpam-5732	162	17	open	open	ADJ
ejpam-5732	162	18	cover	cover	NOUN
ejpam-5732	162	19	refinement	refinement	NOUN
ejpam-5732	162	20	of	of	ADP
ejpam-5732	162	21	u	u	PROPN
ejpam-5732	162	22	.	.	PUNCT
ejpam-5732	163	1	as	as	SCONJ
ejpam-5732	163	2	(	(	PUNCT
ejpam-5732	163	3	x	x	NOUN
ejpam-5732	163	4	,	,	PUNCT
ejpam-5732	163	5	τ	τ	PROPN
ejpam-5732	163	6	,	,	PUNCT
ejpam-5732	163	7	i	i	PROPN
ejpam-5732	163	8	)	)	PUNCT
ejpam-5732	163	9	is	be	AUX
ejpam-5732	163	10	δ1	δ1	NOUN
ejpam-5732	163	11	-	-	PUNCT
ejpam-5732	163	12	βi	βi	PRON
ejpam-5732	163	13	-	-	PUNCT
ejpam-5732	163	14	paracompact	paracompact	ADJ
ejpam-5732	163	15	,	,	PUNCT
ejpam-5732	163	16	there	there	PRON
ejpam-5732	163	17	exists	exist	VERB
ejpam-5732	163	18	a	a	DET
ejpam-5732	163	19	locally	locally	ADV
ejpam-5732	163	20	finite	finite	ADJ
ejpam-5732	163	21	open	open	ADJ
ejpam-5732	163	22	refinement	refinement	NOUN
ejpam-5732	163	23	h	h	NOUN
ejpam-5732	163	24	=	=	PRON
ejpam-5732	163	25	{	{	PUNCT
ejpam-5732	163	26	hα	hα	X
ejpam-5732	163	27	:	:	PUNCT
ejpam-5732	163	28	α	α	PROPN
ejpam-5732	163	29	∈	∈	PROPN
ejpam-5732	163	30	λ	λ	X
ejpam-5732	163	31	}	}	PUNCT
ejpam-5732	163	32	which	which	PRON
ejpam-5732	163	33	refines	refine	VERB
ejpam-5732	163	34	v	v	ADP
ejpam-5732	163	35	and	and	CCONJ
ejpam-5732	163	36	x	x	NOUN
ejpam-5732	163	37	−	−	ADP
ejpam-5732	163	38	∪{hα	∪{hα	PROPN
ejpam-5732	163	39	:	:	PUNCT
ejpam-5732	163	40	α	α	PROPN
ejpam-5732	163	41	∈	∈	PROPN
ejpam-5732	163	42	λ	λ	PROPN
ejpam-5732	163	43	}	}	PUNCT
ejpam-5732	163	44	∈	∈	PROPN
ejpam-5732	163	45	i.	i.	NOUN
ejpam-5732	163	46	by	by	ADP
ejpam-5732	163	47	(	(	PUNCT
ejpam-5732	163	48	ii	ii	NOUN
ejpam-5732	163	49	)	)	PUNCT
ejpam-5732	163	50	of	of	ADP
ejpam-5732	163	51	lemma	lemma	PROPN
ejpam-5732	163	52	7	7	NUM
ejpam-5732	163	53	,	,	PUNCT
ejpam-5732	163	54	h1	h1	NOUN
ejpam-5732	163	55	=	=	SYM
ejpam-5732	163	56	{	{	PUNCT
ejpam-5732	163	57	δ	δ	NOUN
ejpam-5732	163	58	-	-	NOUN
ejpam-5732	163	59	βcli(hα	βcli(hα	NOUN
ejpam-5732	163	60	)	)	PUNCT
ejpam-5732	163	61	:	:	PUNCT
ejpam-5732	163	62	α	α	PROPN
ejpam-5732	163	63	∈	∈	PROPN
ejpam-5732	163	64	λ	λ	PROPN
ejpam-5732	163	65	}	}	PUNCT
ejpam-5732	163	66	is	be	AUX
ejpam-5732	163	67	δ	δ	PROPN
ejpam-5732	163	68	-	-	PUNCT
ejpam-5732	163	69	βi	βi	ADV
ejpam-5732	163	70	-	-	PUNCT
ejpam-5732	163	71	locally	locally	ADV
ejpam-5732	163	72	finite	finite	NOUN
ejpam-5732	163	73	.	.	PUNCT
ejpam-5732	164	1	as	as	SCONJ
ejpam-5732	164	2	hα	hα	ADP
ejpam-5732	164	3	⊂	⊂	PROPN
ejpam-5732	164	4	δ	δ	PROPN
ejpam-5732	164	5	-	-	PUNCT
ejpam-5732	164	6	βcli(hα	βcli(hα	NOUN
ejpam-5732	164	7	)	)	PUNCT
ejpam-5732	164	8	for	for	ADP
ejpam-5732	164	9	all	all	DET
ejpam-5732	164	10	α	α	PRON
ejpam-5732	164	11	∈	∈	PROPN
ejpam-5732	164	12	λ	λ	PROPN
ejpam-5732	164	13	,	,	PUNCT
ejpam-5732	164	14	x	x	NOUN
ejpam-5732	164	15	−	−	ADP
ejpam-5732	164	16	∪{δ	∪{δ	PROPN
ejpam-5732	164	17	-	-	NOUN
ejpam-5732	164	18	βcli(hα	βcli(hα	NOUN
ejpam-5732	164	19	)	)	PUNCT
ejpam-5732	164	20	:	:	PUNCT
ejpam-5732	165	1	α	α	PROPN
ejpam-5732	165	2	∈	∈	PROPN
ejpam-5732	165	3	λ	λ	PROPN
ejpam-5732	165	4	}	}	PUNCT
ejpam-5732	165	5	⊂	⊂	PROPN
ejpam-5732	165	6	x	x	PUNCT
ejpam-5732	165	7	−	−	PUNCT
ejpam-5732	165	8	∪{hα	∪{hα	NUM
ejpam-5732	165	9	:	:	PUNCT
ejpam-5732	165	10	α	α	PROPN
ejpam-5732	165	11	∈	∈	PROPN
ejpam-5732	165	12	λ	λ	NOUN
ejpam-5732	165	13	}	}	PUNCT
ejpam-5732	165	14	,	,	PUNCT
ejpam-5732	165	15	and	and	CCONJ
ejpam-5732	165	16	it	it	PRON
ejpam-5732	165	17	therefore	therefore	ADV
ejpam-5732	165	18	implies	imply	VERB
ejpam-5732	165	19	that	that	SCONJ
ejpam-5732	165	20	x	x	X
ejpam-5732	165	21	−	−	PRON
ejpam-5732	165	22	∪{δ	∪{δ	PROPN
ejpam-5732	165	23	-	-	NOUN
ejpam-5732	165	24	βcli(hα	βcli(hα	NOUN
ejpam-5732	165	25	)	)	PUNCT
ejpam-5732	165	26	:	:	PUNCT
ejpam-5732	165	27	α	α	PROPN
ejpam-5732	165	28	∈	∈	PROPN
ejpam-5732	165	29	λ	λ	PROPN
ejpam-5732	165	30	}	}	PUNCT
ejpam-5732	165	31	∈	∈	PROPN
ejpam-5732	165	32	i.	i.	NOUN
ejpam-5732	165	33	hence	hence	ADV
ejpam-5732	165	34	,	,	PUNCT
ejpam-5732	165	35	h1	h1	PROPN
ejpam-5732	165	36	is	be	AUX
ejpam-5732	165	37	an	an	DET
ejpam-5732	165	38	i	i	NOUN
ejpam-5732	165	39	-	-	NOUN
ejpam-5732	165	40	cover	cover	NOUN
ejpam-5732	165	41	.	.	PUNCT
ejpam-5732	166	1	next	next	ADV
ejpam-5732	166	2	,	,	PUNCT
ejpam-5732	166	3	we	we	PRON
ejpam-5732	166	4	shall	shall	AUX
ejpam-5732	166	5	verify	verify	VERB
ejpam-5732	166	6	that	that	SCONJ
ejpam-5732	166	7	h1	h1	NOUN
ejpam-5732	166	8	refines	refine	VERB
ejpam-5732	166	9	u	u	PRON
ejpam-5732	166	10	.	.	PUNCT
ejpam-5732	167	1	let	let	VERB
ejpam-5732	167	2	δ	δ	NOUN
ejpam-5732	167	3	-	-	PUNCT
ejpam-5732	167	4	βcli(hα	βcli(hα	VERB
ejpam-5732	167	5	)	)	PUNCT
ejpam-5732	167	6	∈	∈	PROPN
ejpam-5732	167	7	h1	h1	PROPN
ejpam-5732	167	8	.	.	PUNCT
ejpam-5732	168	1	since	since	SCONJ
ejpam-5732	168	2	h	h	NOUN
ejpam-5732	168	3	refines	refine	VERB
ejpam-5732	168	4	v	v	ADP
ejpam-5732	168	5	,	,	PUNCT
ejpam-5732	168	6	there	there	PRON
ejpam-5732	168	7	exists	exist	VERB
ejpam-5732	168	8	vx	vx	PROPN
ejpam-5732	168	9	∈	∈	PROPN
ejpam-5732	168	10	v	v	ADP
ejpam-5732	168	11	such	such	ADJ
ejpam-5732	168	12	that	that	SCONJ
ejpam-5732	168	13	hα	hα	ADP
ejpam-5732	168	14	⊂	⊂	PROPN
ejpam-5732	168	15	vx	vx	PROPN
ejpam-5732	168	16	,	,	PUNCT
ejpam-5732	168	17	it	it	PRON
ejpam-5732	168	18	implies	imply	VERB
ejpam-5732	168	19	that	that	SCONJ
ejpam-5732	168	20	δ	δ	PROPN
ejpam-5732	168	21	-	-	PUNCT
ejpam-5732	168	22	βcli(hα	βcli(hα	NOUN
ejpam-5732	168	23	)	)	PUNCT
ejpam-5732	168	24	⊂	⊂	PROPN
ejpam-5732	168	25	δ	δ	PROPN
ejpam-5732	168	26	-	-	PUNCT
ejpam-5732	168	27	βcli(vx	βcli(vx	ADJ
ejpam-5732	168	28	)	)	PUNCT
ejpam-5732	168	29	⊂	⊂	PROPN
ejpam-5732	168	30	ux	ux	PROPN
ejpam-5732	168	31	.	.	PUNCT
ejpam-5732	168	32	consequenty	consequenty	PROPN
ejpam-5732	168	33	,	,	PUNCT
ejpam-5732	168	34	h1	h1	AUX
ejpam-5732	168	35	refines	refine	VERB
ejpam-5732	168	36	u	u	PRON
ejpam-5732	168	37	.	.	PUNCT
ejpam-5732	169	1	therefore	therefore	ADV
ejpam-5732	169	2	,	,	PUNCT
ejpam-5732	169	3	the	the	DET
ejpam-5732	169	4	proposition	proposition	NOUN
ejpam-5732	169	5	has	have	AUX
ejpam-5732	169	6	been	be	AUX
ejpam-5732	169	7	established	establish	VERB
ejpam-5732	169	8	.	.	PUNCT
ejpam-5732	170	1	theorem	theorem	NOUN
ejpam-5732	170	2	3	3	NUM
ejpam-5732	170	3	.	.	PUNCT
ejpam-5732	171	1	if	if	SCONJ
ejpam-5732	171	2	an	an	DET
ejpam-5732	171	3	ideal	ideal	ADJ
ejpam-5732	171	4	topological	topological	ADJ
ejpam-5732	171	5	space	space	NOUN
ejpam-5732	171	6	(	(	PUNCT
ejpam-5732	171	7	x	x	X
ejpam-5732	171	8	,	,	PUNCT
ejpam-5732	171	9	τ	τ	PROPN
ejpam-5732	171	10	,	,	PUNCT
ejpam-5732	171	11	i	i	PROPN
ejpam-5732	171	12	)	)	PUNCT
ejpam-5732	171	13	is	be	AUX
ejpam-5732	171	14	δ1	δ1	NOUN
ejpam-5732	171	15	-	-	PUNCT
ejpam-5732	171	16	βi	βi	PRON
ejpam-5732	171	17	-	-	PUNCT
ejpam-5732	171	18	paracompact	paracompact	ADJ
ejpam-5732	171	19	,	,	PUNCT
ejpam-5732	171	20	then	then	ADV
ejpam-5732	171	21	(	(	PUNCT
ejpam-5732	171	22	x	x	NOUN
ejpam-5732	171	23	,	,	PUNCT
ejpam-5732	171	24	τ∗	τ∗	PROPN
ejpam-5732	171	25	,	,	PUNCT
ejpam-5732	171	26	i	i	NOUN
ejpam-5732	171	27	)	)	PUNCT
ejpam-5732	171	28	is	be	AUX
ejpam-5732	171	29	δ1	δ1	NOUN
ejpam-5732	171	30	-	-	PUNCT
ejpam-5732	171	31	βi	βi	PRON
ejpam-5732	171	32	-	-	PUNCT
ejpam-5732	171	33	paracompact	paracompact	ADJ
ejpam-5732	171	34	.	.	PUNCT
ejpam-5732	172	1	proof	proof	NOUN
ejpam-5732	172	2	.	.	PUNCT
ejpam-5732	173	1	let	let	VERB
ejpam-5732	173	2	u	u	PRON
ejpam-5732	173	3	=	=	PUNCT
ejpam-5732	173	4	{	{	PUNCT
ejpam-5732	173	5	uα	uα	X
ejpam-5732	173	6	:	:	PUNCT
ejpam-5732	173	7	α	α	PROPN
ejpam-5732	173	8	∈	∈	PROPN
ejpam-5732	173	9	λ1	λ1	PROPN
ejpam-5732	173	10	}	}	PUNCT
ejpam-5732	173	11	be	be	VERB
ejpam-5732	173	12	a	a	DET
ejpam-5732	173	13	δ	δ	PROPN
ejpam-5732	173	14	-	-	PUNCT
ejpam-5732	173	15	βi	βi	ADV
ejpam-5732	173	16	-	-	PUNCT
ejpam-5732	173	17	open	open	ADJ
ejpam-5732	173	18	cover	cover	NOUN
ejpam-5732	173	19	of	of	ADP
ejpam-5732	173	20	(	(	PUNCT
ejpam-5732	173	21	x	x	NOUN
ejpam-5732	173	22	,	,	PUNCT
ejpam-5732	173	23	τ∗	τ∗	PROPN
ejpam-5732	173	24	,	,	PUNCT
ejpam-5732	173	25	i	i	PROPN
ejpam-5732	173	26	)	)	PUNCT
ejpam-5732	173	27	,	,	PUNCT
ejpam-5732	173	28	where	where	SCONJ
ejpam-5732	173	29	uα	uα	NOUN
ejpam-5732	173	30	=	=	PUNCT
ejpam-5732	173	31	vα	vα	INTJ
ejpam-5732	173	32	−	−	PROPN
ejpam-5732	173	33	iα	iα	PROPN
ejpam-5732	173	34	,	,	PUNCT
ejpam-5732	173	35	vα	vα	ADP
ejpam-5732	173	36	∈	∈	PROPN
ejpam-5732	173	37	τ	τ	X
ejpam-5732	173	38	and	and	CCONJ
ejpam-5732	173	39	iα	iα	ADP
ejpam-5732	173	40	∈	∈	PROPN
ejpam-5732	173	41	i	i	PRON
ejpam-5732	173	42	for	for	ADP
ejpam-5732	173	43	all	all	DET
ejpam-5732	173	44	α	α	DET
ejpam-5732	173	45	∈	∈	PROPN
ejpam-5732	173	46	λ1	λ1	PROPN
ejpam-5732	173	47	.	.	PUNCT
ejpam-5732	174	1	then	then	ADV
ejpam-5732	174	2	,	,	PUNCT
ejpam-5732	174	3	v	v	NOUN
ejpam-5732	174	4	=	=	SYM
ejpam-5732	174	5	{	{	PUNCT
ejpam-5732	174	6	vα	vα	X
ejpam-5732	174	7	:	:	PUNCT
ejpam-5732	174	8	α	α	PROPN
ejpam-5732	174	9	∈	∈	PROPN
ejpam-5732	174	10	λ1	λ1	PROPN
ejpam-5732	174	11	}	}	PUNCT
ejpam-5732	174	12	is	be	AUX
ejpam-5732	174	13	a	a	DET
ejpam-5732	174	14	δ	δ	PROPN
ejpam-5732	174	15	-	-	PUNCT
ejpam-5732	174	16	βi	βi	ADV
ejpam-5732	174	17	-	-	PUNCT
ejpam-5732	174	18	open	open	ADJ
ejpam-5732	174	19	cover	cover	NOUN
ejpam-5732	174	20	of	of	ADP
ejpam-5732	174	21	(	(	PUNCT
ejpam-5732	174	22	x	x	X
ejpam-5732	174	23	,	,	PUNCT
ejpam-5732	174	24	τ	τ	PROPN
ejpam-5732	174	25	,	,	PUNCT
ejpam-5732	174	26	i	i	PROPN
ejpam-5732	174	27	)	)	PUNCT
ejpam-5732	174	28	and	and	CCONJ
ejpam-5732	174	29	therefore	therefore	ADV
ejpam-5732	174	30	there	there	PRON
ejpam-5732	174	31	exists	exist	VERB
ejpam-5732	174	32	a	a	DET
ejpam-5732	174	33	τ	τ	PROPN
ejpam-5732	174	34	-locally	-locally	ADV
ejpam-5732	174	35	finite	finite	ADJ
ejpam-5732	174	36	open	open	ADJ
ejpam-5732	174	37	refinement	refinement	NOUN
ejpam-5732	175	1	w	w	PROPN
ejpam-5732	175	2	=	=	PRON
ejpam-5732	175	3	{	{	PUNCT
ejpam-5732	175	4	wλ	wλ	NOUN
ejpam-5732	175	5	:	:	PUNCT
ejpam-5732	175	6	λ	λ	PROPN
ejpam-5732	175	7	∈	∈	PROPN
ejpam-5732	175	8	λ2	λ2	NOUN
ejpam-5732	175	9	}	}	PUNCT
ejpam-5732	175	10	such	such	ADJ
ejpam-5732	175	11	that	that	SCONJ
ejpam-5732	175	12	x	x	PRON
ejpam-5732	175	13	−	−	PRON
ejpam-5732	175	14	∪{wλ	∪{wλ	NOUN
ejpam-5732	175	15	:	:	PUNCT
ejpam-5732	175	16	λ	λ	PROPN
ejpam-5732	175	17	∈	∈	PROPN
ejpam-5732	175	18	λ2	λ2	PROPN
ejpam-5732	175	19	}	}	PUNCT
ejpam-5732	175	20	∈	∈	PROPN
ejpam-5732	175	21	i	i	PRON
ejpam-5732	175	22	,	,	PUNCT
ejpam-5732	175	23	as	as	ADP
ejpam-5732	175	24	(	(	PUNCT
ejpam-5732	175	25	x	x	NOUN
ejpam-5732	175	26	,	,	PUNCT
ejpam-5732	175	27	τ	τ	PROPN
ejpam-5732	175	28	,	,	PUNCT
ejpam-5732	175	29	i	i	PROPN
ejpam-5732	175	30	)	)	PUNCT
ejpam-5732	175	31	is	be	AUX
ejpam-5732	175	32	δ1	δ1	NOUN
ejpam-5732	175	33	-	-	PUNCT
ejpam-5732	175	34	βi	βi	PRON
ejpam-5732	175	35	-	-	PUNCT
ejpam-5732	175	36	paracompact	paracompact	ADJ
ejpam-5732	175	37	.	.	PUNCT
ejpam-5732	176	1	now	now	ADV
ejpam-5732	176	2	,	,	PUNCT
ejpam-5732	176	3	we	we	PRON
ejpam-5732	176	4	have	have	VERB
ejpam-5732	176	5	{	{	PUNCT
ejpam-5732	176	6	wλ	wλ	NOUN
ejpam-5732	176	7	∩	∩	X
ejpam-5732	176	8	iα′	iα′	PROPN
ejpam-5732	176	9	:	:	PUNCT
ejpam-5732	176	10	λ	λ	X
ejpam-5732	176	11	∈	∈	NOUN
ejpam-5732	176	12	λ2	λ2	PROPN
ejpam-5732	176	13	}	}	PUNCT
ejpam-5732	176	14	,	,	PUNCT
ejpam-5732	176	15	for	for	ADP
ejpam-5732	176	16	some	some	DET
ejpam-5732	176	17	α′	α′	NUM
ejpam-5732	176	18	∈	∈	PROPN
ejpam-5732	176	19	λ1	λ1	NOUN
ejpam-5732	176	20	,	,	PUNCT
ejpam-5732	176	21	is	be	AUX
ejpam-5732	176	22	a	a	DET
ejpam-5732	176	23	set	set	NOUN
ejpam-5732	176	24	of	of	ADP
ejpam-5732	176	25	subset	subset	NOUN
ejpam-5732	176	26	of	of	ADP
ejpam-5732	176	27	i	i	PRON
ejpam-5732	176	28	and	and	CCONJ
ejpam-5732	176	29	hence	hence	ADV
ejpam-5732	176	30	,	,	PUNCT
ejpam-5732	176	31	by	by	ADP
ejpam-5732	176	32	assumption	assumption	NOUN
ejpam-5732	176	33	,	,	PUNCT
ejpam-5732	176	34	∪λ∈λ2(wλ	∪λ∈λ2(wλ	PROPN
ejpam-5732	176	35	∩	∩	ADJ
ejpam-5732	176	36	iα′	iα′	NOUN
ejpam-5732	176	37	)	)	PUNCT
ejpam-5732	176	38	∈	∈	PROPN
ejpam-5732	176	39	i.	i.	NOUN
ejpam-5732	176	40	hence	hence	ADV
ejpam-5732	176	41	,	,	PUNCT
ejpam-5732	176	42	x	x	PUNCT
ejpam-5732	176	43	−	−	NOUN
ejpam-5732	176	44	∪λ∈λ2(wλ	∪λ∈λ2(wλ	PROPN
ejpam-5732	176	45	−	−	NOUN
ejpam-5732	176	46	iα′	iα′	PROPN
ejpam-5732	176	47	)	)	PUNCT
ejpam-5732	177	1	⊂	⊂	PROPN
ejpam-5732	177	2	(	(	PUNCT
ejpam-5732	177	3	x	x	X
ejpam-5732	177	4	−	−	ADP
ejpam-5732	177	5	∪λ∈λ2wλ	∪λ∈λ2wλ	NOUN
ejpam-5732	177	6	)	)	PUNCT
ejpam-5732	177	7	∪	∪	NOUN
ejpam-5732	177	8	(	(	PUNCT
ejpam-5732	177	9	∪λ∈λ2wλ	∪λ∈λ2wλ	ADJ
ejpam-5732	177	10	∩	∩	ADJ
ejpam-5732	177	11	iα′	iα′	X
ejpam-5732	177	12	)	)	PUNCT
ejpam-5732	177	13	∈	∈	PROPN
ejpam-5732	178	1	i	i	PRON
ejpam-5732	178	2	,	,	PUNCT
ejpam-5732	178	3	which	which	PRON
ejpam-5732	178	4	implies	imply	VERB
ejpam-5732	178	5	that	that	SCONJ
ejpam-5732	178	6	x	x	PUNCT
ejpam-5732	178	7	−	−	NOUN
ejpam-5732	178	8	∪λ∈λ2(wλ	∪λ∈λ2(wλ	NUM
ejpam-5732	178	9	−	−	NOUN
ejpam-5732	178	10	iα′	iα′	PROPN
ejpam-5732	178	11	)	)	PUNCT
ejpam-5732	178	12	∈	∈	PROPN
ejpam-5732	178	13	i.	i.	NOUN
ejpam-5732	178	14	as	as	SCONJ
ejpam-5732	178	15	w	w	PROPN
ejpam-5732	178	16	is	be	AUX
ejpam-5732	178	17	τ	τ	PROPN
ejpam-5732	178	18	-locally	-locally	ADV
ejpam-5732	178	19	finite	finite	ADJ
ejpam-5732	178	20	,	,	PUNCT
ejpam-5732	178	21	w	w	NOUN
ejpam-5732	178	22	′	′	NOUN
ejpam-5732	178	23	=	=	PUNCT
ejpam-5732	178	24	{	{	PUNCT
ejpam-5732	178	25	wλ	wλ	NOUN
ejpam-5732	178	26	−	−	PROPN
ejpam-5732	178	27	iα′	iα′	PROPN
ejpam-5732	178	28	:	:	PUNCT
ejpam-5732	178	29	λ	λ	PROPN
ejpam-5732	178	30	∈	∈	PROPN
ejpam-5732	178	31	λ2	λ2	PROPN
ejpam-5732	178	32	}	}	PUNCT
ejpam-5732	178	33	is	be	AUX
ejpam-5732	178	34	τ	τ	PROPN
ejpam-5732	178	35	-locally	-locally	ADV
ejpam-5732	178	36	finite	finite	ADJ
ejpam-5732	178	37	.	.	PUNCT
ejpam-5732	179	1	because	because	SCONJ
ejpam-5732	179	2	τ∗	τ∗	NOUN
ejpam-5732	179	3	is	be	AUX
ejpam-5732	179	4	finer	fine	ADJ
ejpam-5732	179	5	than	than	ADP
ejpam-5732	179	6	τ	τ	PROPN
ejpam-5732	179	7	,	,	PUNCT
ejpam-5732	179	8	w	w	NOUN
ejpam-5732	179	9	′	′	NUM
ejpam-5732	179	10	is	be	AUX
ejpam-5732	179	11	a	a	DET
ejpam-5732	179	12	τ∗-locally	τ∗-locally	ADV
ejpam-5732	179	13	finite	finite	NOUN
ejpam-5732	179	14	τ∗-open	τ∗-open	PUNCT
ejpam-5732	179	15	which	which	PRON
ejpam-5732	179	16	refines	refine	VERB
ejpam-5732	179	17	u	u	PRON
ejpam-5732	179	18	.	.	PUNCT
ejpam-5732	180	1	consequently	consequently	ADV
ejpam-5732	180	2	,	,	PUNCT
ejpam-5732	180	3	(	(	PUNCT
ejpam-5732	180	4	x	x	NOUN
ejpam-5732	180	5	,	,	PUNCT
ejpam-5732	180	6	τ∗	τ∗	PROPN
ejpam-5732	180	7	,	,	PUNCT
ejpam-5732	180	8	i	i	NOUN
ejpam-5732	180	9	)	)	PUNCT
ejpam-5732	180	10	is	be	AUX
ejpam-5732	180	11	δ1	δ1	NOUN
ejpam-5732	180	12	-	-	PUNCT
ejpam-5732	180	13	βi	βi	PRON
ejpam-5732	180	14	-	-	PUNCT
ejpam-5732	180	15	paracompact	paracompact	ADJ
ejpam-5732	180	16	.	.	PUNCT
ejpam-5732	181	1	a	a	DET
ejpam-5732	181	2	topological	topological	ADJ
ejpam-5732	181	3	space	space	NOUN
ejpam-5732	181	4	(	(	PUNCT
ejpam-5732	181	5	x	x	X
ejpam-5732	181	6	,	,	PUNCT
ejpam-5732	181	7	τ	τ	X
ejpam-5732	181	8	)	)	PUNCT
ejpam-5732	181	9	is	be	AUX
ejpam-5732	181	10	a	a	DET
ejpam-5732	181	11	t2	t2	NOUN
ejpam-5732	181	12	space	space	NOUN
ejpam-5732	181	13	if	if	SCONJ
ejpam-5732	181	14	any	any	DET
ejpam-5732	181	15	two	two	NUM
ejpam-5732	181	16	distinct	distinct	ADJ
ejpam-5732	181	17	points	point	NOUN
ejpam-5732	181	18	x	x	PUNCT
ejpam-5732	181	19	and	and	CCONJ
ejpam-5732	181	20	y	y	PROPN
ejpam-5732	181	21	in	in	ADP
ejpam-5732	181	22	x	x	SYM
ejpam-5732	181	23	,	,	PUNCT
ejpam-5732	181	24	there	there	PRON
ejpam-5732	181	25	exist	exist	VERB
ejpam-5732	181	26	disjoint	disjoint	ADJ
ejpam-5732	181	27	open	open	ADJ
ejpam-5732	181	28	neighborhoods	neighborhood	NOUN
ejpam-5732	181	29	u	u	NOUN
ejpam-5732	181	30	and	and	CCONJ
ejpam-5732	181	31	v	v	ADP
ejpam-5732	181	32	such	such	ADJ
ejpam-5732	181	33	that	that	SCONJ
ejpam-5732	181	34	x	x	SYM
ejpam-5732	181	35	∈	∈	PROPN
ejpam-5732	181	36	u	u	NOUN
ejpam-5732	181	37	and	and	CCONJ
ejpam-5732	181	38	y	y	PROPN
ejpam-5732	181	39	∈	∈	PROPN
ejpam-5732	181	40	v	v	NOUN
ejpam-5732	181	41	.	.	PUNCT
ejpam-5732	182	1	the	the	DET
ejpam-5732	182	2	following	follow	VERB
ejpam-5732	182	3	theorem	theorem	NOUN
ejpam-5732	182	4	establishes	establish	VERB
ejpam-5732	182	5	a	a	DET
ejpam-5732	182	6	characteristic	characteristic	NOUN
ejpam-5732	182	7	of	of	ADP
ejpam-5732	182	8	a	a	DET
ejpam-5732	182	9	δ1	δ1	NOUN
ejpam-5732	182	10	-	-	PUNCT
ejpam-5732	182	11	βi	βi	PRON
ejpam-5732	182	12	-	-	PUNCT
ejpam-5732	182	13	paracompact	paracompact	NOUN
ejpam-5732	182	14	subset	subset	NOUN
ejpam-5732	182	15	within	within	ADP
ejpam-5732	182	16	a	a	DET
ejpam-5732	182	17	t2	t2	NOUN
ejpam-5732	182	18	ideal	ideal	ADJ
ejpam-5732	182	19	topological	topological	ADJ
ejpam-5732	182	20	space	space	NOUN
ejpam-5732	182	21	x.	x.	NOUN
ejpam-5732	182	22	theorem	theorem	VERB
ejpam-5732	182	23	4	4	NUM
ejpam-5732	182	24	.	.	PUNCT
ejpam-5732	183	1	if	if	SCONJ
ejpam-5732	183	2	an	an	DET
ejpam-5732	183	3	ideal	ideal	ADJ
ejpam-5732	183	4	topological	topological	ADJ
ejpam-5732	183	5	space	space	NOUN
ejpam-5732	183	6	(	(	PUNCT
ejpam-5732	183	7	x	x	X
ejpam-5732	183	8	,	,	PUNCT
ejpam-5732	183	9	τ	τ	PROPN
ejpam-5732	183	10	,	,	PUNCT
ejpam-5732	183	11	i	i	PROPN
ejpam-5732	183	12	)	)	PUNCT
ejpam-5732	183	13	is	be	AUX
ejpam-5732	183	14	a	a	DET
ejpam-5732	183	15	t2	t2	NOUN
ejpam-5732	183	16	space	space	NOUN
ejpam-5732	183	17	and	and	CCONJ
ejpam-5732	183	18	a	a	DET
ejpam-5732	183	19	is	be	AUX
ejpam-5732	183	20	δ1	δ1	NOUN
ejpam-5732	183	21	-	-	PUNCT
ejpam-5732	183	22	βi	βi	PRON
ejpam-5732	183	23	-	-	PUNCT
ejpam-5732	183	24	paracompact	paracompact	NOUN
ejpam-5732	183	25	relative	relative	NOUN
ejpam-5732	183	26	to	to	ADP
ejpam-5732	183	27	x	x	PRON
ejpam-5732	183	28	,	,	PUNCT
ejpam-5732	183	29	then	then	ADV
ejpam-5732	183	30	a	a	PRON
ejpam-5732	183	31	is	be	AUX
ejpam-5732	183	32	closed	close	VERB
ejpam-5732	183	33	in	in	ADP
ejpam-5732	183	34	(	(	PUNCT
ejpam-5732	183	35	x	x	NOUN
ejpam-5732	183	36	,	,	PUNCT
ejpam-5732	183	37	τ∗	τ∗	PROPN
ejpam-5732	183	38	,	,	PUNCT
ejpam-5732	183	39	i	i	PROPN
ejpam-5732	183	40	)	)	PUNCT
ejpam-5732	183	41	.	.	PUNCT
ejpam-5732	184	1	c.	c.	PROPN
ejpam-5732	184	2	boonpok	boonpok	PROPN
ejpam-5732	184	3	,	,	PUNCT
ejpam-5732	184	4	a.	a.	PROPN
ejpam-5732	184	5	sama	sama	PROPN
ejpam-5732	184	6	-	-	PUNCT
ejpam-5732	184	7	ae	ae	PROPN
ejpam-5732	184	8	/	/	SYM
ejpam-5732	184	9	eur	eur	PROPN
ejpam-5732	184	10	.	.	PUNCT
ejpam-5732	185	1	j.	j.	PROPN
ejpam-5732	185	2	pure	pure	PROPN
ejpam-5732	185	3	appl	appl	PROPN
ejpam-5732	185	4	.	.	PROPN
ejpam-5732	185	5	math	math	PROPN
ejpam-5732	185	6	,	,	PUNCT
ejpam-5732	185	7	18	18	NUM
ejpam-5732	185	8	(	(	PUNCT
ejpam-5732	185	9	1	1	NUM
ejpam-5732	185	10	)	)	PUNCT
ejpam-5732	185	11	(	(	PUNCT
ejpam-5732	185	12	2025	2025	NUM
ejpam-5732	185	13	)	)	PUNCT
ejpam-5732	185	14	,	,	PUNCT
ejpam-5732	185	15	5732	5732	NUM
ejpam-5732	185	16	7	7	NUM
ejpam-5732	185	17	of	of	ADP
ejpam-5732	185	18	13	13	NUM
ejpam-5732	185	19	proof	proof	NOUN
ejpam-5732	185	20	.	.	PUNCT
ejpam-5732	186	1	we	we	PRON
ejpam-5732	186	2	shall	shall	AUX
ejpam-5732	186	3	verify	verify	VERB
ejpam-5732	186	4	that	that	SCONJ
ejpam-5732	186	5	a∗	a∗	PROPN
ejpam-5732	186	6	⊂	⊂	PROPN
ejpam-5732	186	7	a.	a.	NOUN
ejpam-5732	186	8	suppose	suppose	VERB
ejpam-5732	186	9	x	x	PUNCT
ejpam-5732	186	10	̸∈	̸∈	PROPN
ejpam-5732	186	11	a.	a.	NOUN
ejpam-5732	186	12	as	as	ADP
ejpam-5732	186	13	(	(	PUNCT
ejpam-5732	186	14	x	x	NOUN
ejpam-5732	186	15	,	,	PUNCT
ejpam-5732	186	16	τ	τ	PROPN
ejpam-5732	186	17	,	,	PUNCT
ejpam-5732	186	18	i	i	PROPN
ejpam-5732	186	19	)	)	PUNCT
ejpam-5732	186	20	is	be	AUX
ejpam-5732	186	21	t2	t2	NOUN
ejpam-5732	186	22	,	,	PUNCT
ejpam-5732	186	23	for	for	ADP
ejpam-5732	186	24	each	each	DET
ejpam-5732	186	25	y	y	PROPN
ejpam-5732	186	26	∈	∈	PROPN
ejpam-5732	186	27	a	a	PRON
ejpam-5732	186	28	,	,	PUNCT
ejpam-5732	186	29	there	there	PRON
ejpam-5732	186	30	exists	exist	VERB
ejpam-5732	186	31	an	an	DET
ejpam-5732	186	32	open	open	ADJ
ejpam-5732	186	33	set	set	NOUN
ejpam-5732	186	34	uy	uy	ADP
ejpam-5732	186	35	such	such	ADJ
ejpam-5732	186	36	that	that	SCONJ
ejpam-5732	186	37	y	y	PROPN
ejpam-5732	186	38	∈	∈	PROPN
ejpam-5732	186	39	uy	uy	PROPN
ejpam-5732	186	40	and	and	CCONJ
ejpam-5732	186	41	x	x	X
ejpam-5732	186	42	̸∈	̸∈	PROPN
ejpam-5732	186	43	cl(uy	cl(uy	PROPN
ejpam-5732	186	44	)	)	PUNCT
ejpam-5732	186	45	.	.	PUNCT
ejpam-5732	187	1	thus	thus	ADV
ejpam-5732	187	2	,	,	PUNCT
ejpam-5732	187	3	the	the	DET
ejpam-5732	187	4	family	family	NOUN
ejpam-5732	187	5	u	u	NOUN
ejpam-5732	187	6	=	=	PUNCT
ejpam-5732	187	7	{	{	PUNCT
ejpam-5732	187	8	uy	uy	NOUN
ejpam-5732	187	9	:	:	PUNCT
ejpam-5732	187	10	y	y	PROPN
ejpam-5732	187	11	∈	∈	PROPN
ejpam-5732	187	12	a	a	PRON
ejpam-5732	187	13	}	}	PUNCT
ejpam-5732	187	14	is	be	AUX
ejpam-5732	187	15	an	an	DET
ejpam-5732	187	16	open	open	ADJ
ejpam-5732	187	17	cover	cover	NOUN
ejpam-5732	187	18	of	of	ADP
ejpam-5732	187	19	a	a	PRON
ejpam-5732	187	20	,	,	PUNCT
ejpam-5732	187	21	and	and	CCONJ
ejpam-5732	187	22	hence	hence	ADV
ejpam-5732	187	23	it	it	PRON
ejpam-5732	187	24	is	be	AUX
ejpam-5732	187	25	a	a	DET
ejpam-5732	187	26	δ	δ	PROPN
ejpam-5732	187	27	-	-	PUNCT
ejpam-5732	187	28	βi	βi	ADV
ejpam-5732	187	29	-	-	PUNCT
ejpam-5732	187	30	open	open	ADJ
ejpam-5732	187	31	cover	cover	NOUN
ejpam-5732	187	32	of	of	ADP
ejpam-5732	187	33	a.	a.	NOUN
ejpam-5732	187	34	as	as	ADP
ejpam-5732	187	35	a	a	DET
ejpam-5732	187	36	is	be	AUX
ejpam-5732	187	37	δ1	δ1	NOUN
ejpam-5732	187	38	-	-	PUNCT
ejpam-5732	187	39	βi	βi	PRON
ejpam-5732	187	40	-	-	PUNCT
ejpam-5732	187	41	paracompact	paracompact	ADJ
ejpam-5732	187	42	,	,	PUNCT
ejpam-5732	187	43	u	u	NOUN
ejpam-5732	187	44	has	have	VERB
ejpam-5732	187	45	a	a	DET
ejpam-5732	187	46	τ	τ	PROPN
ejpam-5732	187	47	-locally	-locally	ADV
ejpam-5732	187	48	finite	finite	ADJ
ejpam-5732	187	49	open	open	ADJ
ejpam-5732	187	50	refinement	refinement	NOUN
ejpam-5732	187	51	v	v	X
ejpam-5732	187	52	=	=	PUNCT
ejpam-5732	187	53	{	{	PUNCT
ejpam-5732	187	54	vα	vα	X
ejpam-5732	187	55	:	:	PUNCT
ejpam-5732	187	56	α	α	PROPN
ejpam-5732	187	57	∈	∈	PROPN
ejpam-5732	187	58	λ	λ	NOUN
ejpam-5732	187	59	}	}	PUNCT
ejpam-5732	187	60	of	of	ADP
ejpam-5732	187	61	u	u	PRON
ejpam-5732	187	62	such	such	ADJ
ejpam-5732	187	63	that	that	SCONJ
ejpam-5732	187	64	a−∪{vα	a−∪{vα	VERB
ejpam-5732	187	65	:	:	PUNCT
ejpam-5732	187	66	α	α	PROPN
ejpam-5732	187	67	∈	∈	PROPN
ejpam-5732	187	68	λ	λ	PROPN
ejpam-5732	187	69	}	}	PUNCT
ejpam-5732	187	70	∈	∈	PROPN
ejpam-5732	187	71	i.	i.	NOUN
ejpam-5732	187	72	now	now	ADV
ejpam-5732	187	73	x	x	X
ejpam-5732	187	74	̸∈	̸∈	PROPN
ejpam-5732	187	75	cl(vα	cl(vα	PROPN
ejpam-5732	187	76	)	)	PUNCT
ejpam-5732	187	77	for	for	ADP
ejpam-5732	187	78	all	all	DET
ejpam-5732	187	79	α	α	PROPN
ejpam-5732	187	80	implies	imply	VERB
ejpam-5732	187	81	that	that	SCONJ
ejpam-5732	187	82	x	x	PROPN
ejpam-5732	187	83	̸∈	̸∈	PROPN
ejpam-5732	187	84	∪{cl(vα	∪{cl(vα	PROPN
ejpam-5732	187	85	)	)	PUNCT
ejpam-5732	187	86	:	:	PUNCT
ejpam-5732	188	1	α	α	PROPN
ejpam-5732	188	2	∈	∈	PROPN
ejpam-5732	188	3	λ	λ	NOUN
ejpam-5732	188	4	}	}	PUNCT
ejpam-5732	188	5	.	.	PUNCT
ejpam-5732	189	1	because	because	SCONJ
ejpam-5732	189	2	of	of	ADP
ejpam-5732	189	3	∪{cl(vα	∪{cl(vα	NOUN
ejpam-5732	189	4	)	)	PUNCT
ejpam-5732	189	5	:	:	PUNCT
ejpam-5732	189	6	α	α	PROPN
ejpam-5732	189	7	∈	∈	PROPN
ejpam-5732	189	8	λ	λ	NOUN
ejpam-5732	189	9	}	}	PUNCT
ejpam-5732	189	10	=	=	NOUN
ejpam-5732	189	11	cl(∪{vα	cl(∪{vα	NOUN
ejpam-5732	189	12	:	:	PUNCT
ejpam-5732	189	13	α	α	PROPN
ejpam-5732	189	14	∈	∈	PROPN
ejpam-5732	189	15	λ	λ	NOUN
ejpam-5732	189	16	}	}	PUNCT
ejpam-5732	189	17	)	)	PUNCT
ejpam-5732	189	18	,	,	PUNCT
ejpam-5732	189	19	x	x	PRON
ejpam-5732	189	20	is	be	AUX
ejpam-5732	189	21	not	not	PART
ejpam-5732	189	22	in	in	ADP
ejpam-5732	189	23	cl(∪{vα	cl(∪{vα	PROPN
ejpam-5732	189	24	:	:	PUNCT
ejpam-5732	189	25	α	α	PROPN
ejpam-5732	189	26	∈	∈	PROPN
ejpam-5732	189	27	λ	λ	NOUN
ejpam-5732	189	28	}	}	PUNCT
ejpam-5732	189	29	)	)	PUNCT
ejpam-5732	189	30	.	.	PUNCT
ejpam-5732	190	1	let	let	VERB
ejpam-5732	190	2	g1	g1	PROPN
ejpam-5732	190	3	=	=	PUNCT
ejpam-5732	190	4	x	x	SYM
ejpam-5732	190	5	−	−	PROPN
ejpam-5732	190	6	cl(∪{vα	cl(∪{vα	NOUN
ejpam-5732	190	7	:	:	PUNCT
ejpam-5732	190	8	α	α	PROPN
ejpam-5732	190	9	∈	∈	PROPN
ejpam-5732	190	10	λ	λ	NOUN
ejpam-5732	190	11	}	}	PUNCT
ejpam-5732	190	12	)	)	PUNCT
ejpam-5732	190	13	and	and	CCONJ
ejpam-5732	190	14	g2	g2	PROPN
ejpam-5732	190	15	=	=	PUNCT
ejpam-5732	191	1	a	a	DET
ejpam-5732	191	2	−	−	PROPN
ejpam-5732	191	3	cl(∪{vα	cl(∪{vα	NOUN
ejpam-5732	191	4	:	:	PUNCT
ejpam-5732	191	5	α	α	PROPN
ejpam-5732	191	6	∈	∈	PROPN
ejpam-5732	191	7	λ	λ	NOUN
ejpam-5732	191	8	}	}	PUNCT
ejpam-5732	191	9	)	)	PUNCT
ejpam-5732	191	10	.	.	PUNCT
ejpam-5732	192	1	we	we	PRON
ejpam-5732	192	2	know	know	VERB
ejpam-5732	192	3	that	that	SCONJ
ejpam-5732	192	4	g1	g1	PROPN
ejpam-5732	192	5	∈	∈	PROPN
ejpam-5732	192	6	τ	τ	X
ejpam-5732	192	7	,	,	PUNCT
ejpam-5732	192	8	g2	g2	PROPN
ejpam-5732	192	9	∈	∈	PROPN
ejpam-5732	193	1	i	i	PRON
ejpam-5732	193	2	,	,	PUNCT
ejpam-5732	193	3	(	(	PUNCT
ejpam-5732	193	4	g1	g1	PROPN
ejpam-5732	193	5	−g2	−g2	ADJ
ejpam-5732	193	6	)	)	PUNCT
ejpam-5732	193	7	∩a	∩a	NOUN
ejpam-5732	193	8	=	=	NOUN
ejpam-5732	193	9	∅	∅	NOUN
ejpam-5732	193	10	,	,	PUNCT
ejpam-5732	193	11	and	and	CCONJ
ejpam-5732	193	12	x	x	X
ejpam-5732	193	13	∈	∈	PROPN
ejpam-5732	193	14	g1	g1	PROPN
ejpam-5732	193	15	−g2	−g2	PROPN
ejpam-5732	193	16	∈	∈	PROPN
ejpam-5732	193	17	τ∗	τ∗	NOUN
ejpam-5732	193	18	,	,	PUNCT
ejpam-5732	193	19	it	it	PRON
ejpam-5732	193	20	follows	follow	VERB
ejpam-5732	193	21	that	that	SCONJ
ejpam-5732	193	22	x	x	SYM
ejpam-5732	193	23	̸∈	̸∈	PROPN
ejpam-5732	193	24	a∗.	a∗.	NOUN
ejpam-5732	193	25	consequently	consequently	ADV
ejpam-5732	193	26	,	,	PUNCT
ejpam-5732	193	27	a∗	a∗	PROPN
ejpam-5732	193	28	is	be	AUX
ejpam-5732	193	29	closed	close	VERB
ejpam-5732	193	30	in	in	ADP
ejpam-5732	193	31	the	the	DET
ejpam-5732	193	32	topological	topological	ADJ
ejpam-5732	193	33	space	space	NOUN
ejpam-5732	193	34	(	(	PUNCT
ejpam-5732	193	35	x	x	NOUN
ejpam-5732	193	36	,	,	PUNCT
ejpam-5732	193	37	τ∗	τ∗	NOUN
ejpam-5732	193	38	)	)	PUNCT
ejpam-5732	193	39	.	.	PUNCT
ejpam-5732	194	1	the	the	DET
ejpam-5732	194	2	theorem	theorem	NOUN
ejpam-5732	194	3	has	have	AUX
ejpam-5732	194	4	been	be	AUX
ejpam-5732	194	5	shown	show	VERB
ejpam-5732	194	6	.	.	PUNCT
ejpam-5732	195	1	theorem	theorem	ADJ
ejpam-5732	195	2	5	5	NUM
ejpam-5732	195	3	.	.	PUNCT
ejpam-5732	196	1	if	if	SCONJ
ejpam-5732	196	2	an	an	DET
ejpam-5732	196	3	ideal	ideal	ADJ
ejpam-5732	196	4	topological	topological	ADJ
ejpam-5732	196	5	space	space	NOUN
ejpam-5732	196	6	(	(	PUNCT
ejpam-5732	196	7	x	x	X
ejpam-5732	196	8	,	,	PUNCT
ejpam-5732	196	9	τ	τ	PROPN
ejpam-5732	196	10	,	,	PUNCT
ejpam-5732	196	11	i	i	PROPN
ejpam-5732	196	12	)	)	PUNCT
ejpam-5732	196	13	is	be	AUX
ejpam-5732	196	14	δ1	δ1	NOUN
ejpam-5732	196	15	-	-	PUNCT
ejpam-5732	196	16	βi	βi	NOUN
ejpam-5732	196	17	-	-	PUNCT
ejpam-5732	196	18	paracompact	paracompact	NOUN
ejpam-5732	196	19	and	and	CCONJ
ejpam-5732	196	20	a	a	DET
ejpam-5732	196	21	⊂	⊂	X
ejpam-5732	196	22	x	x	X
ejpam-5732	196	23	is	be	AUX
ejpam-5732	196	24	δ	δ	PROPN
ejpam-5732	196	25	-	-	PUNCT
ejpam-5732	196	26	βi	βi	ADV
ejpam-5732	196	27	-	-	PUNCT
ejpam-5732	196	28	closed	close	VERB
ejpam-5732	196	29	in	in	ADP
ejpam-5732	196	30	x	x	NOUN
ejpam-5732	196	31	,	,	PUNCT
ejpam-5732	196	32	then	then	ADV
ejpam-5732	196	33	a	a	PRON
ejpam-5732	196	34	is	be	AUX
ejpam-5732	196	35	δ1	δ1	NOUN
ejpam-5732	196	36	-	-	PUNCT
ejpam-5732	196	37	βi	βi	PRON
ejpam-5732	196	38	-	-	PUNCT
ejpam-5732	196	39	paracompact	paracompact	ADJ
ejpam-5732	196	40	.	.	PUNCT
ejpam-5732	197	1	proof	proof	NOUN
ejpam-5732	197	2	.	.	PUNCT
ejpam-5732	198	1	let	let	VERB
ejpam-5732	198	2	u	u	PRON
ejpam-5732	198	3	=	=	PUNCT
ejpam-5732	198	4	{	{	PUNCT
ejpam-5732	198	5	uα	uα	X
ejpam-5732	198	6	:	:	PUNCT
ejpam-5732	198	7	α	α	PROPN
ejpam-5732	198	8	∈	∈	PROPN
ejpam-5732	198	9	λ	λ	NOUN
ejpam-5732	198	10	}	}	PUNCT
ejpam-5732	198	11	be	be	VERB
ejpam-5732	198	12	a	a	DET
ejpam-5732	198	13	δ	δ	PROPN
ejpam-5732	198	14	-	-	PUNCT
ejpam-5732	198	15	βi	βi	ADV
ejpam-5732	198	16	-	-	PUNCT
ejpam-5732	198	17	open	open	ADJ
ejpam-5732	198	18	cover	cover	NOUN
ejpam-5732	198	19	of	of	ADP
ejpam-5732	198	20	a.	a.	NOUN
ejpam-5732	198	21	as	as	ADP
ejpam-5732	198	22	x	x	X
ejpam-5732	198	23	−	−	PROPN
ejpam-5732	198	24	a	a	PRON
ejpam-5732	198	25	is	be	AUX
ejpam-5732	198	26	a	a	DET
ejpam-5732	198	27	δ	δ	PROPN
ejpam-5732	198	28	-	-	PUNCT
ejpam-5732	198	29	βi	βi	ADV
ejpam-5732	198	30	-	-	PUNCT
ejpam-5732	198	31	open	open	ADJ
ejpam-5732	198	32	subset	subset	NOUN
ejpam-5732	198	33	of	of	ADP
ejpam-5732	198	34	x	x	PROPN
ejpam-5732	198	35	,	,	PUNCT
ejpam-5732	198	36	h	h	NOUN
ejpam-5732	198	37	=	=	PRON
ejpam-5732	198	38	{	{	PUNCT
ejpam-5732	198	39	uα	uα	X
ejpam-5732	198	40	:	:	PUNCT
ejpam-5732	198	41	α	α	PROPN
ejpam-5732	198	42	∈	∈	PROPN
ejpam-5732	198	43	λ	λ	PROPN
ejpam-5732	198	44	}	}	PUNCT
ejpam-5732	198	45	∪	∪	NOUN
ejpam-5732	198	46	{	{	PUNCT
ejpam-5732	198	47	x	x	NOUN
ejpam-5732	198	48	−	−	PROPN
ejpam-5732	198	49	a	a	PRON
ejpam-5732	198	50	}	}	PUNCT
ejpam-5732	198	51	is	be	AUX
ejpam-5732	198	52	a	a	DET
ejpam-5732	198	53	δ	δ	PROPN
ejpam-5732	198	54	-	-	PUNCT
ejpam-5732	198	55	βi	βi	ADV
ejpam-5732	198	56	-	-	PUNCT
ejpam-5732	198	57	open	open	ADJ
ejpam-5732	198	58	cover	cover	NOUN
ejpam-5732	198	59	of	of	ADP
ejpam-5732	198	60	x.	x.	NOUN
ejpam-5732	198	61	by	by	ADP
ejpam-5732	198	62	assumption	assumption	NOUN
ejpam-5732	198	63	,	,	PUNCT
ejpam-5732	198	64	h	h	NOUN
ejpam-5732	198	65	has	have	VERB
ejpam-5732	198	66	a	a	DET
ejpam-5732	198	67	locally	locally	ADV
ejpam-5732	198	68	finite	finite	ADJ
ejpam-5732	198	69	open	open	ADJ
ejpam-5732	198	70	refinement	refinement	NOUN
ejpam-5732	198	71	v	v	X
ejpam-5732	198	72	=	=	PUNCT
ejpam-5732	198	73	{	{	PUNCT
ejpam-5732	198	74	vλ	vλ	INTJ
ejpam-5732	198	75	:	:	PUNCT
ejpam-5732	198	76	λ	λ	PROPN
ejpam-5732	198	77	∈	∈	PROPN
ejpam-5732	198	78	λ1	λ1	PROPN
ejpam-5732	198	79	}	}	PUNCT
ejpam-5732	198	80	∪	∪	NOUN
ejpam-5732	198	81	{	{	PUNCT
ejpam-5732	198	82	v	v	NOUN
ejpam-5732	198	83	}	}	PUNCT
ejpam-5732	198	84	such	such	ADJ
ejpam-5732	198	85	that	that	PRON
ejpam-5732	198	86	for	for	ADP
ejpam-5732	198	87	each	each	DET
ejpam-5732	198	88	λ	λ	PROPN
ejpam-5732	198	89	∈	∈	PROPN
ejpam-5732	198	90	λ1	λ1	PROPN
ejpam-5732	198	91	,	,	PUNCT
ejpam-5732	198	92	vλ	vλ	ADP
ejpam-5732	198	93	⊂	⊂	PROPN
ejpam-5732	198	94	uα	uα	PROPN
ejpam-5732	198	95	for	for	ADP
ejpam-5732	198	96	some	some	DET
ejpam-5732	198	97	α	α	NOUN
ejpam-5732	198	98	∈	∈	PROPN
ejpam-5732	198	99	λ	λ	PROPN
ejpam-5732	198	100	,	,	PUNCT
ejpam-5732	198	101	v	v	ADP
ejpam-5732	198	102	⊂	⊂	PROPN
ejpam-5732	198	103	x	x	PUNCT
ejpam-5732	199	1	−	−	NOUN
ejpam-5732	199	2	a	a	PRON
ejpam-5732	199	3	and	and	CCONJ
ejpam-5732	199	4	x	x	SYM
ejpam-5732	199	5	−	−	PROPN
ejpam-5732	199	6	(	(	PUNCT
ejpam-5732	199	7	∪{vλ	∪{vλ	NUM
ejpam-5732	199	8	:	:	PUNCT
ejpam-5732	199	9	λ	λ	PROPN
ejpam-5732	199	10	∈	∈	PROPN
ejpam-5732	199	11	λ1	λ1	PROPN
ejpam-5732	199	12	}	}	PUNCT
ejpam-5732	199	13	∪	∪	NOUN
ejpam-5732	199	14	{	{	PUNCT
ejpam-5732	199	15	v	v	NOUN
ejpam-5732	199	16	}	}	PUNCT
ejpam-5732	199	17	)	)	PUNCT
ejpam-5732	199	18	∈	∈	PROPN
ejpam-5732	199	19	i.	i.	NOUN
ejpam-5732	199	20	as	as	ADP
ejpam-5732	199	21	a	a	DET
ejpam-5732	199	22	−	−	PROPN
ejpam-5732	199	23	∪{vλ	∪{vλ	NUM
ejpam-5732	199	24	:	:	PUNCT
ejpam-5732	199	25	λ	λ	PROPN
ejpam-5732	199	26	∈	∈	PROPN
ejpam-5732	199	27	λ1	λ1	PROPN
ejpam-5732	199	28	}	}	PUNCT
ejpam-5732	199	29	=	=	PUNCT
ejpam-5732	199	30	a	a	DET
ejpam-5732	199	31	∩	∩	X
ejpam-5732	199	32	x	x	X
ejpam-5732	199	33	−	−	NOUN
ejpam-5732	199	34	∪{vλ	∪{vλ	NUM
ejpam-5732	199	35	:	:	PUNCT
ejpam-5732	199	36	λ	λ	PROPN
ejpam-5732	199	37	∈	∈	PROPN
ejpam-5732	199	38	λ1	λ1	PROPN
ejpam-5732	199	39	}	}	PUNCT
ejpam-5732	199	40	=	=	PUNCT
ejpam-5732	199	41	a	a	DET
ejpam-5732	199	42	∩	∩	X
ejpam-5732	199	43	x	x	X
ejpam-5732	199	44	−	−	NOUN
ejpam-5732	199	45	∪({vλ	∪({vλ	ADP
ejpam-5732	199	46	:	:	PUNCT
ejpam-5732	199	47	λ	λ	PROPN
ejpam-5732	199	48	∈	∈	PROPN
ejpam-5732	199	49	λ1	λ1	PROPN
ejpam-5732	199	50	}	}	PUNCT
ejpam-5732	199	51	∪	∪	NOUN
ejpam-5732	199	52	{	{	PUNCT
ejpam-5732	199	53	v	v	NOUN
ejpam-5732	199	54	}	}	PUNCT
ejpam-5732	199	55	)	)	PUNCT
ejpam-5732	200	1	⊂	⊂	PROPN
ejpam-5732	200	2	x	x	PUNCT
ejpam-5732	201	1	−	−	ADP
ejpam-5732	201	2	∪({vλ	∪({vλ	ADP
ejpam-5732	201	3	:	:	PUNCT
ejpam-5732	201	4	λ	λ	PROPN
ejpam-5732	201	5	∈	∈	PROPN
ejpam-5732	201	6	λ1	λ1	PROPN
ejpam-5732	201	7	}	}	PUNCT
ejpam-5732	201	8	∪	∪	NOUN
ejpam-5732	201	9	{	{	PUNCT
ejpam-5732	201	10	v	v	NOUN
ejpam-5732	201	11	}	}	PUNCT
ejpam-5732	201	12	)	)	PUNCT
ejpam-5732	201	13	,	,	PUNCT
ejpam-5732	201	14	we	we	PRON
ejpam-5732	201	15	have	have	VERB
ejpam-5732	201	16	a	a	DET
ejpam-5732	201	17	−	−	NOUN
ejpam-5732	201	18	∪{vλ	∪{vλ	PROPN
ejpam-5732	201	19	:	:	PUNCT
ejpam-5732	201	20	λ	λ	PROPN
ejpam-5732	201	21	∈	∈	PROPN
ejpam-5732	201	22	λ1	λ1	PROPN
ejpam-5732	201	23	}	}	PUNCT
ejpam-5732	201	24	∈	∈	PROPN
ejpam-5732	201	25	i.	i.	NOUN
ejpam-5732	201	26	for	for	ADP
ejpam-5732	201	27	any	any	DET
ejpam-5732	201	28	λ	λ	PROPN
ejpam-5732	201	29	∈	∈	PROPN
ejpam-5732	201	30	λ1	λ1	NOUN
ejpam-5732	201	31	,	,	PUNCT
ejpam-5732	201	32	there	there	PRON
ejpam-5732	201	33	exists	exist	VERB
ejpam-5732	201	34	α	α	PRON
ejpam-5732	201	35	∈	∈	PROPN
ejpam-5732	201	36	λ	λ	NOUN
ejpam-5732	201	37	such	such	ADJ
ejpam-5732	201	38	that	that	SCONJ
ejpam-5732	201	39	vλ	vλ	ADP
ejpam-5732	201	40	⊂	⊂	PROPN
ejpam-5732	201	41	uα	uα	PROPN
ejpam-5732	201	42	,	,	PUNCT
ejpam-5732	201	43	showing	show	VERB
ejpam-5732	201	44	that	that	SCONJ
ejpam-5732	201	45	{	{	PUNCT
ejpam-5732	201	46	vλ	vλ	INTJ
ejpam-5732	201	47	:	:	PUNCT
ejpam-5732	201	48	λ	λ	PROPN
ejpam-5732	201	49	∈	∈	PROPN
ejpam-5732	201	50	λ1	λ1	PROPN
ejpam-5732	201	51	}	}	PUNCT
ejpam-5732	201	52	represents	represent	VERB
ejpam-5732	201	53	a	a	DET
ejpam-5732	201	54	locally	locally	ADV
ejpam-5732	201	55	finite	finite	ADJ
ejpam-5732	201	56	open	open	ADJ
ejpam-5732	201	57	refinement	refinement	NOUN
ejpam-5732	201	58	of	of	ADP
ejpam-5732	201	59	u	u	PROPN
ejpam-5732	201	60	.	.	PUNCT
ejpam-5732	202	1	this	this	PRON
ejpam-5732	202	2	demonstrates	demonstrate	VERB
ejpam-5732	202	3	that	that	SCONJ
ejpam-5732	202	4	a	a	PRON
ejpam-5732	202	5	is	be	AUX
ejpam-5732	202	6	δ1	δ1	NOUN
ejpam-5732	202	7	-	-	PUNCT
ejpam-5732	202	8	βi	βi	PRON
ejpam-5732	202	9	-	-	PUNCT
ejpam-5732	202	10	paracompact	paracompact	NOUN
ejpam-5732	202	11	.	.	PUNCT
ejpam-5732	203	1	theorem	theorem	NOUN
ejpam-5732	203	2	6	6	NUM
ejpam-5732	203	3	.	.	PUNCT
ejpam-5732	204	1	let	let	VERB
ejpam-5732	204	2	a	a	PRON
ejpam-5732	204	3	and	and	CCONJ
ejpam-5732	204	4	b	b	NOUN
ejpam-5732	204	5	be	be	AUX
ejpam-5732	204	6	subsets	subset	NOUN
ejpam-5732	204	7	of	of	ADP
ejpam-5732	204	8	an	an	DET
ejpam-5732	204	9	ideal	ideal	ADJ
ejpam-5732	204	10	topological	topological	ADJ
ejpam-5732	204	11	space	space	NOUN
ejpam-5732	204	12	(	(	PUNCT
ejpam-5732	204	13	x	x	X
ejpam-5732	204	14	,	,	PUNCT
ejpam-5732	204	15	τ	τ	PROPN
ejpam-5732	204	16	,	,	PUNCT
ejpam-5732	204	17	i	i	PROPN
ejpam-5732	204	18	)	)	PUNCT
ejpam-5732	204	19	.	.	PUNCT
ejpam-5732	205	1	if	if	SCONJ
ejpam-5732	205	2	a	a	PRON
ejpam-5732	205	3	is	be	AUX
ejpam-5732	205	4	δ1	δ1	NOUN
ejpam-5732	205	5	-	-	PUNCT
ejpam-5732	205	6	βiparacompact	βiparacompact	NOUN
ejpam-5732	205	7	and	and	CCONJ
ejpam-5732	205	8	b	b	NOUN
ejpam-5732	205	9	is	be	AUX
ejpam-5732	205	10	δ	δ	PROPN
ejpam-5732	205	11	-	-	PUNCT
ejpam-5732	205	12	βi	βi	ADV
ejpam-5732	205	13	-	-	PUNCT
ejpam-5732	205	14	closed	close	VERB
ejpam-5732	205	15	in	in	ADP
ejpam-5732	205	16	x	x	NOUN
ejpam-5732	205	17	,	,	PUNCT
ejpam-5732	205	18	then	then	ADV
ejpam-5732	205	19	a	a	DET
ejpam-5732	205	20	∩b	∩b	NOUN
ejpam-5732	205	21	is	be	AUX
ejpam-5732	205	22	δ1	δ1	NOUN
ejpam-5732	205	23	-	-	PUNCT
ejpam-5732	205	24	βi	βi	PRON
ejpam-5732	205	25	-	-	PUNCT
ejpam-5732	205	26	paracompact	paracompact	ADJ
ejpam-5732	205	27	.	.	PUNCT
ejpam-5732	206	1	proof	proof	NOUN
ejpam-5732	206	2	.	.	PUNCT
ejpam-5732	207	1	let	let	VERB
ejpam-5732	207	2	u	u	PRON
ejpam-5732	207	3	=	=	PUNCT
ejpam-5732	207	4	{	{	PUNCT
ejpam-5732	207	5	uα	uα	X
ejpam-5732	207	6	:	:	PUNCT
ejpam-5732	207	7	α	α	PROPN
ejpam-5732	207	8	∈	∈	PROPN
ejpam-5732	207	9	λ	λ	NOUN
ejpam-5732	207	10	}	}	PUNCT
ejpam-5732	207	11	be	be	VERB
ejpam-5732	207	12	a	a	DET
ejpam-5732	207	13	δ	δ	PROPN
ejpam-5732	207	14	-	-	PUNCT
ejpam-5732	207	15	βi	βi	ADV
ejpam-5732	207	16	-	-	PUNCT
ejpam-5732	207	17	open	open	ADJ
ejpam-5732	207	18	cover	cover	NOUN
ejpam-5732	207	19	of	of	ADP
ejpam-5732	207	20	a∩b	a∩b	PROPN
ejpam-5732	207	21	.	.	PUNCT
ejpam-5732	208	1	since	since	SCONJ
ejpam-5732	208	2	x−b	x−b	NOUN
ejpam-5732	208	3	is	be	AUX
ejpam-5732	208	4	a	a	DET
ejpam-5732	208	5	δ	δ	PROPN
ejpam-5732	208	6	-	-	PUNCT
ejpam-5732	208	7	βi	βi	ADV
ejpam-5732	208	8	-	-	PUNCT
ejpam-5732	208	9	open	open	NOUN
ejpam-5732	208	10	subset	subset	NOUN
ejpam-5732	208	11	in	in	ADP
ejpam-5732	208	12	x	x	NOUN
ejpam-5732	208	13	,	,	PUNCT
ejpam-5732	208	14	u1	u1	NOUN
ejpam-5732	208	15	=	=	SYM
ejpam-5732	208	16	{	{	PUNCT
ejpam-5732	208	17	uα	uα	X
ejpam-5732	208	18	:	:	PUNCT
ejpam-5732	208	19	α	α	PROPN
ejpam-5732	208	20	∈	∈	PROPN
ejpam-5732	208	21	λ	λ	PROPN
ejpam-5732	208	22	}	}	PUNCT
ejpam-5732	208	23	∪	∪	NOUN
ejpam-5732	208	24	{	{	PUNCT
ejpam-5732	208	25	x	x	NOUN
ejpam-5732	208	26	−	−	PROPN
ejpam-5732	208	27	b	b	X
ejpam-5732	208	28	}	}	PUNCT
ejpam-5732	208	29	is	be	AUX
ejpam-5732	208	30	a	a	DET
ejpam-5732	208	31	δ	δ	PROPN
ejpam-5732	208	32	-	-	PUNCT
ejpam-5732	208	33	βi	βi	ADV
ejpam-5732	208	34	-	-	PUNCT
ejpam-5732	208	35	open	open	ADJ
ejpam-5732	208	36	cover	cover	NOUN
ejpam-5732	208	37	of	of	ADP
ejpam-5732	208	38	a.	a.	NOUN
ejpam-5732	208	39	since	since	SCONJ
ejpam-5732	208	40	a	a	PRON
ejpam-5732	208	41	is	be	AUX
ejpam-5732	208	42	δ1	δ1	NOUN
ejpam-5732	208	43	-	-	PUNCT
ejpam-5732	208	44	βiparacompact	βiparacompact	ADJ
ejpam-5732	208	45	,	,	PUNCT
ejpam-5732	208	46	u1	u1	NOUN
ejpam-5732	208	47	has	have	VERB
ejpam-5732	208	48	a	a	DET
ejpam-5732	208	49	locally	locally	ADV
ejpam-5732	208	50	finite	finite	ADJ
ejpam-5732	208	51	open	open	ADJ
ejpam-5732	208	52	refinement	refinement	NOUN
ejpam-5732	208	53	v	v	X
ejpam-5732	208	54	=	=	PUNCT
ejpam-5732	208	55	{	{	PUNCT
ejpam-5732	208	56	vλ	vλ	INTJ
ejpam-5732	208	57	:	:	PUNCT
ejpam-5732	208	58	λ	λ	PROPN
ejpam-5732	208	59	∈	∈	PROPN
ejpam-5732	208	60	λ1	λ1	PROPN
ejpam-5732	208	61	}	}	PUNCT
ejpam-5732	208	62	∪	∪	NOUN
ejpam-5732	208	63	{	{	PUNCT
ejpam-5732	208	64	v	v	NOUN
ejpam-5732	208	65	}	}	PUNCT
ejpam-5732	208	66	such	such	ADJ
ejpam-5732	208	67	that	that	PRON
ejpam-5732	208	68	for	for	ADP
ejpam-5732	208	69	each	each	DET
ejpam-5732	208	70	λ	λ	PROPN
ejpam-5732	208	71	∈	∈	PROPN
ejpam-5732	208	72	λ1	λ1	PROPN
ejpam-5732	208	73	,	,	PUNCT
ejpam-5732	208	74	vλ	vλ	ADP
ejpam-5732	208	75	⊂	⊂	PROPN
ejpam-5732	208	76	uα	uα	PROPN
ejpam-5732	208	77	for	for	ADP
ejpam-5732	208	78	some	some	DET
ejpam-5732	208	79	α	α	NOUN
ejpam-5732	208	80	∈	∈	PROPN
ejpam-5732	208	81	λ	λ	PROPN
ejpam-5732	208	82	,	,	PUNCT
ejpam-5732	208	83	v	v	ADP
ejpam-5732	208	84	⊂	⊂	PROPN
ejpam-5732	208	85	x−b	x−b	NOUN
ejpam-5732	208	86	and	and	CCONJ
ejpam-5732	208	87	a−(∪{vλ	a−(∪{vλ	PROPN
ejpam-5732	208	88	:	:	PUNCT
ejpam-5732	208	89	λ	λ	X
ejpam-5732	208	90	∈	∈	PROPN
ejpam-5732	208	91	λ1}∪{v	λ1}∪{v	NOUN
ejpam-5732	208	92	}	}	PUNCT
ejpam-5732	208	93	)	)	PUNCT
ejpam-5732	208	94	∈	∈	PROPN
ejpam-5732	208	95	i.	i.	NOUN
ejpam-5732	208	96	as	as	ADP
ejpam-5732	208	97	a∩b−∪{vλ	a∩b−∪{vλ	PROPN
ejpam-5732	208	98	:	:	PUNCT
ejpam-5732	208	99	λ	λ	PROPN
ejpam-5732	208	100	∈	∈	PROPN
ejpam-5732	208	101	λ1	λ1	PROPN
ejpam-5732	208	102	}	}	PUNCT
ejpam-5732	208	103	=	=	PUNCT
ejpam-5732	208	104	a∩b−∪({vλ	a∩b−∪({vλ	NOUN
ejpam-5732	208	105	:	:	PUNCT
ejpam-5732	208	106	λ	λ	X
ejpam-5732	208	107	∈	∈	PROPN
ejpam-5732	208	108	λ1}∪{v	λ1}∪{v	NOUN
ejpam-5732	208	109	}	}	PUNCT
ejpam-5732	208	110	)	)	PUNCT
ejpam-5732	209	1	⊂	⊂	PROPN
ejpam-5732	209	2	a−∪({vλ	a−∪({vλ	NOUN
ejpam-5732	209	3	:	:	PUNCT
ejpam-5732	209	4	λ	λ	X
ejpam-5732	209	5	∈	∈	PROPN
ejpam-5732	209	6	λ1}∪{v	λ1}∪{v	NOUN
ejpam-5732	209	7	}	}	PUNCT
ejpam-5732	209	8	)	)	PUNCT
ejpam-5732	209	9	,	,	PUNCT
ejpam-5732	209	10	we	we	PRON
ejpam-5732	209	11	have	have	VERB
ejpam-5732	209	12	a	a	DET
ejpam-5732	209	13	∩	∩	ADJ
ejpam-5732	209	14	b	b	NOUN
ejpam-5732	209	15	−	−	PROPN
ejpam-5732	209	16	∪{vλ	∪{vλ	NUM
ejpam-5732	209	17	:	:	PUNCT
ejpam-5732	209	18	λ	λ	PROPN
ejpam-5732	209	19	∈	∈	PROPN
ejpam-5732	209	20	λ1	λ1	PROPN
ejpam-5732	209	21	}	}	PUNCT
ejpam-5732	209	22	∈	∈	PROPN
ejpam-5732	209	23	i.	i.	NOUN
ejpam-5732	209	24	for	for	ADP
ejpam-5732	209	25	any	any	DET
ejpam-5732	209	26	λ	λ	PROPN
ejpam-5732	209	27	∈	∈	PROPN
ejpam-5732	209	28	λ1	λ1	NOUN
ejpam-5732	209	29	,	,	PUNCT
ejpam-5732	209	30	there	there	PRON
ejpam-5732	209	31	exists	exist	VERB
ejpam-5732	209	32	a	a	DET
ejpam-5732	209	33	α	α	NOUN
ejpam-5732	209	34	∈	∈	PROPN
ejpam-5732	209	35	λ	λ	NOUN
ejpam-5732	209	36	such	such	ADJ
ejpam-5732	209	37	that	that	SCONJ
ejpam-5732	209	38	vλ	vλ	ADP
ejpam-5732	209	39	⊂	⊂	PROPN
ejpam-5732	209	40	uα	uα	PROPN
ejpam-5732	209	41	,	,	PUNCT
ejpam-5732	209	42	indicating	indicate	VERB
ejpam-5732	209	43	that	that	SCONJ
ejpam-5732	209	44	the	the	DET
ejpam-5732	209	45	collection	collection	NOUN
ejpam-5732	209	46	{	{	PUNCT
ejpam-5732	209	47	vλ	vλ	INTJ
ejpam-5732	209	48	:	:	PUNCT
ejpam-5732	209	49	λ	λ	PROPN
ejpam-5732	209	50	∈	∈	PROPN
ejpam-5732	209	51	λ1	λ1	PROPN
ejpam-5732	209	52	}	}	PUNCT
ejpam-5732	209	53	constitutes	constitute	VERB
ejpam-5732	209	54	a	a	DET
ejpam-5732	209	55	locally	locally	ADV
ejpam-5732	209	56	finite	finite	ADJ
ejpam-5732	209	57	open	open	ADJ
ejpam-5732	209	58	refinement	refinement	NOUN
ejpam-5732	209	59	of	of	ADP
ejpam-5732	209	60	u	u	PROPN
ejpam-5732	209	61	.	.	PUNCT
ejpam-5732	210	1	this	this	PRON
ejpam-5732	210	2	suggests	suggest	VERB
ejpam-5732	210	3	that	that	SCONJ
ejpam-5732	210	4	a	a	DET
ejpam-5732	210	5	∩b	∩b	NOUN
ejpam-5732	210	6	is	be	AUX
ejpam-5732	210	7	δ1	δ1	NOUN
ejpam-5732	210	8	-	-	PUNCT
ejpam-5732	210	9	βi	βi	PRON
ejpam-5732	210	10	-	-	PUNCT
ejpam-5732	210	11	paracompact	paracompact	NOUN
ejpam-5732	210	12	.	.	PUNCT
ejpam-5732	211	1	theorem	theorem	NOUN
ejpam-5732	211	2	7	7	NUM
ejpam-5732	211	3	.	.	PUNCT
ejpam-5732	212	1	let	let	VERB
ejpam-5732	212	2	a	a	PRON
ejpam-5732	212	3	and	and	CCONJ
ejpam-5732	212	4	b	b	NOUN
ejpam-5732	212	5	be	be	AUX
ejpam-5732	212	6	subsets	subset	NOUN
ejpam-5732	212	7	of	of	ADP
ejpam-5732	212	8	an	an	DET
ejpam-5732	212	9	ideal	ideal	ADJ
ejpam-5732	212	10	topological	topological	ADJ
ejpam-5732	212	11	space	space	NOUN
ejpam-5732	212	12	(	(	PUNCT
ejpam-5732	212	13	x	x	X
ejpam-5732	212	14	,	,	PUNCT
ejpam-5732	212	15	τ	τ	PROPN
ejpam-5732	212	16	,	,	PUNCT
ejpam-5732	212	17	i	i	PROPN
ejpam-5732	212	18	)	)	PUNCT
ejpam-5732	212	19	.	.	PUNCT
ejpam-5732	213	1	if	if	SCONJ
ejpam-5732	213	2	a	a	PRON
ejpam-5732	213	3	and	and	CCONJ
ejpam-5732	213	4	b	b	NOUN
ejpam-5732	213	5	are	be	AUX
ejpam-5732	213	6	δ1	δ1	NOUN
ejpam-5732	213	7	-	-	PUNCT
ejpam-5732	213	8	βi	βi	NOUN
ejpam-5732	213	9	-	-	NOUN
ejpam-5732	213	10	paracompact	paracompact	NOUN
ejpam-5732	213	11	in	in	ADP
ejpam-5732	213	12	x	x	PRON
ejpam-5732	213	13	,	,	PUNCT
ejpam-5732	213	14	then	then	ADV
ejpam-5732	213	15	a	a	DET
ejpam-5732	213	16	∪b	∪b	PRON
ejpam-5732	213	17	is	be	AUX
ejpam-5732	213	18	δ1	δ1	NOUN
ejpam-5732	213	19	-	-	PUNCT
ejpam-5732	213	20	βi	βi	PRON
ejpam-5732	213	21	-	-	PUNCT
ejpam-5732	213	22	paracompact	paracompact	ADJ
ejpam-5732	213	23	.	.	PUNCT
ejpam-5732	214	1	proof	proof	NOUN
ejpam-5732	214	2	.	.	PUNCT
ejpam-5732	215	1	let	let	VERB
ejpam-5732	215	2	u	u	PRON
ejpam-5732	215	3	=	=	PUNCT
ejpam-5732	215	4	{	{	PUNCT
ejpam-5732	215	5	uα	uα	X
ejpam-5732	215	6	:	:	PUNCT
ejpam-5732	215	7	α	α	PROPN
ejpam-5732	215	8	∈	∈	PROPN
ejpam-5732	215	9	λ	λ	NOUN
ejpam-5732	215	10	}	}	PUNCT
ejpam-5732	215	11	be	be	VERB
ejpam-5732	215	12	a	a	DET
ejpam-5732	215	13	δ	δ	PROPN
ejpam-5732	215	14	-	-	PUNCT
ejpam-5732	215	15	βi	βi	ADV
ejpam-5732	215	16	-	-	PUNCT
ejpam-5732	215	17	open	open	ADJ
ejpam-5732	215	18	cover	cover	NOUN
ejpam-5732	215	19	of	of	ADP
ejpam-5732	215	20	a	a	DET
ejpam-5732	215	21	∪	∪	X
ejpam-5732	215	22	b.	b.	NOUN
ejpam-5732	215	23	it	it	PRON
ejpam-5732	215	24	follows	follow	VERB
ejpam-5732	215	25	that	that	SCONJ
ejpam-5732	215	26	u	u	NOUN
ejpam-5732	215	27	is	be	AUX
ejpam-5732	215	28	an	an	DET
ejpam-5732	215	29	δ	δ	PROPN
ejpam-5732	215	30	-	-	PUNCT
ejpam-5732	215	31	βi	βi	ADV
ejpam-5732	215	32	-	-	PUNCT
ejpam-5732	215	33	open	open	ADJ
ejpam-5732	215	34	cover	cover	NOUN
ejpam-5732	215	35	of	of	ADP
ejpam-5732	215	36	a	a	PRON
ejpam-5732	215	37	and	and	CCONJ
ejpam-5732	215	38	b.	b.	PROPN
ejpam-5732	215	39	by	by	ADP
ejpam-5732	215	40	hypothesis	hypothesis	NOUN
ejpam-5732	215	41	,	,	PUNCT
ejpam-5732	215	42	there	there	PRON
ejpam-5732	215	43	are	be	VERB
ejpam-5732	215	44	locally	locally	ADV
ejpam-5732	215	45	finite	finite	ADJ
ejpam-5732	215	46	open	open	ADJ
ejpam-5732	215	47	families	family	NOUN
ejpam-5732	215	48	v	v	NOUN
ejpam-5732	215	49	=	=	PUNCT
ejpam-5732	215	50	{	{	PUNCT
ejpam-5732	215	51	vλ	vλ	INTJ
ejpam-5732	215	52	:	:	PUNCT
ejpam-5732	215	53	λ	λ	PROPN
ejpam-5732	215	54	∈	∈	PROPN
ejpam-5732	215	55	λ1	λ1	PROPN
ejpam-5732	215	56	}	}	PUNCT
ejpam-5732	215	57	of	of	ADP
ejpam-5732	215	58	a	a	PRON
ejpam-5732	215	59	and	and	CCONJ
ejpam-5732	215	60	w	w	NOUN
ejpam-5732	215	61	=	=	PUNCT
ejpam-5732	215	62	{	{	PUNCT
ejpam-5732	215	63	wµ	wµ	NOUN
ejpam-5732	215	64	:	:	PUNCT
ejpam-5732	215	65	µ	µ	X
ejpam-5732	215	66	∈	∈	PROPN
ejpam-5732	215	67	λ2	λ2	PROPN
ejpam-5732	215	68	}	}	PUNCT
ejpam-5732	215	69	of	of	ADP
ejpam-5732	215	70	b	b	NUM
ejpam-5732	215	71	which	which	PRON
ejpam-5732	215	72	refine	refine	VERB
ejpam-5732	215	73	u	u	PRON
ejpam-5732	215	74	such	such	ADJ
ejpam-5732	215	75	that	that	DET
ejpam-5732	215	76	a−∪{vλ	a−∪{vλ	NOUN
ejpam-5732	215	77	:	:	PUNCT
ejpam-5732	215	78	λ	λ	PROPN
ejpam-5732	215	79	∈	∈	PROPN
ejpam-5732	215	80	λ1	λ1	PROPN
ejpam-5732	215	81	}	}	PUNCT
ejpam-5732	215	82	∈	∈	PROPN
ejpam-5732	215	83	i	i	PROPN
ejpam-5732	215	84	and	and	CCONJ
ejpam-5732	215	85	b	b	NOUN
ejpam-5732	215	86	−	−	X
ejpam-5732	215	87	∪{wµ	∪{wµ	NOUN
ejpam-5732	215	88	:	:	PUNCT
ejpam-5732	215	89	µ	µ	PROPN
ejpam-5732	215	90	∈	∈	PROPN
ejpam-5732	215	91	λ2	λ2	PROPN
ejpam-5732	215	92	}	}	PUNCT
ejpam-5732	215	93	∈	∈	PROPN
ejpam-5732	215	94	i.	i.	NOUN
ejpam-5732	215	95	it	it	PRON
ejpam-5732	215	96	suggests	suggest	VERB
ejpam-5732	215	97	that	that	SCONJ
ejpam-5732	215	98	a−	a−	PROPN
ejpam-5732	215	99	∪{vλ	∪{vλ	PROPN
ejpam-5732	215	100	:	:	PUNCT
ejpam-5732	215	101	λ	λ	PROPN
ejpam-5732	215	102	∈	∈	PROPN
ejpam-5732	215	103	λ1	λ1	PROPN
ejpam-5732	215	104	}	}	PUNCT
ejpam-5732	215	105	=	=	SYM
ejpam-5732	215	106	i1	i1	PROPN
ejpam-5732	215	107	and	and	CCONJ
ejpam-5732	215	108	b	b	NOUN
ejpam-5732	215	109	−	−	X
ejpam-5732	215	110	∪{wµ	∪{wµ	NOUN
ejpam-5732	215	111	:	:	PUNCT
ejpam-5732	215	112	µ	µ	PROPN
ejpam-5732	215	113	∈	∈	PROPN
ejpam-5732	215	114	λ2	λ2	NOUN
ejpam-5732	215	115	}	}	PUNCT
ejpam-5732	215	116	=	=	SYM
ejpam-5732	215	117	i2	i2	PROPN
ejpam-5732	215	118	,	,	PUNCT
ejpam-5732	215	119	for	for	ADP
ejpam-5732	215	120	some	some	DET
ejpam-5732	215	121	i1	i1	NOUN
ejpam-5732	215	122	,	,	PUNCT
ejpam-5732	215	123	i2	i2	PROPN
ejpam-5732	215	124	∈	∈	PROPN
ejpam-5732	215	125	i.	i.	NOUN
ejpam-5732	215	126	hence	hence	ADV
ejpam-5732	215	127	,	,	PUNCT
ejpam-5732	215	128	a	a	DET
ejpam-5732	215	129	∪b	∪b	X
ejpam-5732	215	130	⊂	⊂	PROPN
ejpam-5732	215	131	(	(	PUNCT
ejpam-5732	215	132	∪{vλ	∪{vλ	NUM
ejpam-5732	215	133	:	:	PUNCT
ejpam-5732	215	134	λ	λ	PROPN
ejpam-5732	215	135	∈	∈	PROPN
ejpam-5732	215	136	λ1	λ1	PROPN
ejpam-5732	215	137	}	}	PUNCT
ejpam-5732	215	138	∪	∪	NOUN
ejpam-5732	215	139	i1	i1	PROPN
ejpam-5732	215	140	)	)	PUNCT
ejpam-5732	215	141	∪	∪	ADP
ejpam-5732	215	142	(	(	PUNCT
ejpam-5732	215	143	∪{wµ	∪{wµ	NOUN
ejpam-5732	215	144	:	:	PUNCT
ejpam-5732	215	145	µ	µ	PROPN
ejpam-5732	215	146	∈	∈	PROPN
ejpam-5732	215	147	λ2	λ2	PROPN
ejpam-5732	215	148	}	}	PUNCT
ejpam-5732	215	149	∪	∪	NOUN
ejpam-5732	215	150	i2	i2	PROPN
ejpam-5732	215	151	)	)	PUNCT
ejpam-5732	215	152	c.	c.	PROPN
ejpam-5732	215	153	boonpok	boonpok	PROPN
ejpam-5732	215	154	,	,	PUNCT
ejpam-5732	215	155	a.	a.	PROPN
ejpam-5732	215	156	sama	sama	PROPN
ejpam-5732	215	157	-	-	PUNCT
ejpam-5732	215	158	ae	ae	PROPN
ejpam-5732	215	159	/	/	SYM
ejpam-5732	215	160	eur	eur	PROPN
ejpam-5732	215	161	.	.	PUNCT
ejpam-5732	216	1	j.	j.	PROPN
ejpam-5732	216	2	pure	pure	PROPN
ejpam-5732	216	3	appl	appl	PROPN
ejpam-5732	216	4	.	.	PROPN
ejpam-5732	216	5	math	math	PROPN
ejpam-5732	216	6	,	,	PUNCT
ejpam-5732	216	7	18	18	NUM
ejpam-5732	216	8	(	(	PUNCT
ejpam-5732	216	9	1	1	NUM
ejpam-5732	216	10	)	)	PUNCT
ejpam-5732	216	11	(	(	PUNCT
ejpam-5732	216	12	2025	2025	NUM
ejpam-5732	216	13	)	)	PUNCT
ejpam-5732	216	14	,	,	PUNCT
ejpam-5732	216	15	5732	5732	NUM
ejpam-5732	216	16	8	8	NUM
ejpam-5732	216	17	of	of	ADP
ejpam-5732	216	18	13	13	NUM
ejpam-5732	216	19	=	=	SYM
ejpam-5732	216	20	∪{vλ	∪{vλ	X
ejpam-5732	216	21	∪wµ	∪wµ	X
ejpam-5732	216	22	:	:	PUNCT
ejpam-5732	216	23	λ	λ	PROPN
ejpam-5732	216	24	∈	∈	PROPN
ejpam-5732	216	25	λ1	λ1	PROPN
ejpam-5732	216	26	,	,	PUNCT
ejpam-5732	216	27	µ	µ	PROPN
ejpam-5732	216	28	∈	∈	PROPN
ejpam-5732	216	29	λ2	λ2	NOUN
ejpam-5732	216	30	}	}	PUNCT
ejpam-5732	216	31	∪	∪	NOUN
ejpam-5732	216	32	(	(	PUNCT
ejpam-5732	216	33	i1	i1	PROPN
ejpam-5732	216	34	∪	∪	PROPN
ejpam-5732	216	35	i2	i2	PROPN
ejpam-5732	216	36	)	)	PUNCT
ejpam-5732	216	37	,	,	PUNCT
ejpam-5732	216	38	which	which	PRON
ejpam-5732	216	39	implies	imply	VERB
ejpam-5732	216	40	that	that	SCONJ
ejpam-5732	216	41	a	a	DET
ejpam-5732	216	42	∪	∪	X
ejpam-5732	216	43	b	b	NOUN
ejpam-5732	216	44	⊂	⊂	X
ejpam-5732	216	45	∪{vλ	∪{vλ	X
ejpam-5732	216	46	∪wµ	∪wµ	X
ejpam-5732	216	47	:	:	PUNCT
ejpam-5732	216	48	λ	λ	PROPN
ejpam-5732	216	49	∈	∈	PROPN
ejpam-5732	216	50	λ1	λ1	PROPN
ejpam-5732	216	51	,	,	PUNCT
ejpam-5732	216	52	µ	µ	PROPN
ejpam-5732	216	53	∈	∈	PROPN
ejpam-5732	216	54	λ2	λ2	PROPN
ejpam-5732	216	55	}	}	PUNCT
ejpam-5732	216	56	∈	∈	PROPN
ejpam-5732	216	57	i.	i.	NOUN
ejpam-5732	216	58	as	as	ADP
ejpam-5732	216	59	v	v	NUM
ejpam-5732	216	60	and	and	CCONJ
ejpam-5732	216	61	w	w	NOUN
ejpam-5732	216	62	are	be	AUX
ejpam-5732	216	63	locally	locally	ADV
ejpam-5732	216	64	finite	finite	ADJ
ejpam-5732	216	65	,	,	PUNCT
ejpam-5732	216	66	for	for	ADP
ejpam-5732	216	67	any	any	DET
ejpam-5732	216	68	point	point	NOUN
ejpam-5732	216	69	x	x	X
ejpam-5732	216	70	∈	∈	NOUN
ejpam-5732	216	71	x	x	PUNCT
ejpam-5732	216	72	there	there	PRON
ejpam-5732	216	73	exist	exist	VERB
ejpam-5732	216	74	δ	δ	PROPN
ejpam-5732	216	75	-	-	PUNCT
ejpam-5732	216	76	βi	βi	PRON
ejpam-5732	216	77	-	-	PUNCT
ejpam-5732	216	78	open	open	ADJ
ejpam-5732	216	79	sets	set	NOUN
ejpam-5732	216	80	g1	g1	NOUN
ejpam-5732	216	81	and	and	CCONJ
ejpam-5732	216	82	g2	g2	PROPN
ejpam-5732	216	83	such	such	ADJ
ejpam-5732	216	84	that	that	SCONJ
ejpam-5732	216	85	both	both	CCONJ
ejpam-5732	216	86	g1	g1	PROPN
ejpam-5732	216	87	and	and	CCONJ
ejpam-5732	216	88	g2	g2	PROPN
ejpam-5732	216	89	intersect	intersect	ADJ
ejpam-5732	216	90	at	at	ADP
ejpam-5732	216	91	most	most	ADV
ejpam-5732	216	92	finitely	finitely	ADJ
ejpam-5732	216	93	members	member	NOUN
ejpam-5732	216	94	of	of	ADP
ejpam-5732	216	95	u	u	NOUN
ejpam-5732	216	96	and	and	CCONJ
ejpam-5732	216	97	v	v	NOUN
ejpam-5732	216	98	,	,	PUNCT
ejpam-5732	216	99	respectively	respectively	ADV
ejpam-5732	216	100	.	.	PUNCT
ejpam-5732	217	1	thus	thus	ADV
ejpam-5732	217	2	,	,	PUNCT
ejpam-5732	217	3	at	at	ADP
ejpam-5732	217	4	most	most	ADV
ejpam-5732	217	5	finitely	finitely	ADV
ejpam-5732	217	6	many	many	ADJ
ejpam-5732	217	7	members	member	NOUN
ejpam-5732	217	8	of	of	ADP
ejpam-5732	217	9	{	{	PUNCT
ejpam-5732	217	10	vλ	vλ	X
ejpam-5732	217	11	∪wµ	∪wµ	PROPN
ejpam-5732	217	12	:	:	PUNCT
ejpam-5732	217	13	λ	λ	PROPN
ejpam-5732	217	14	∈	∈	PROPN
ejpam-5732	217	15	λ1	λ1	PROPN
ejpam-5732	217	16	,	,	PUNCT
ejpam-5732	217	17	µ	µ	PROPN
ejpam-5732	217	18	∈	∈	NOUN
ejpam-5732	217	19	λ2	λ2	NOUN
ejpam-5732	217	20	}	}	PUNCT
ejpam-5732	217	21	,	,	PUNCT
ejpam-5732	217	22	does	do	AUX
ejpam-5732	217	23	g1	g1	VERB
ejpam-5732	217	24	∩g2	∩g2	PROPN
ejpam-5732	217	25	intersect	intersect	PROPN
ejpam-5732	217	26	.	.	PUNCT
ejpam-5732	218	1	therefore	therefore	ADV
ejpam-5732	218	2	,	,	PUNCT
ejpam-5732	218	3	a	a	DET
ejpam-5732	218	4	∪	∪	NOUN
ejpam-5732	218	5	b	b	NOUN
ejpam-5732	218	6	is	be	AUX
ejpam-5732	218	7	δ1	δ1	NOUN
ejpam-5732	218	8	-	-	PUNCT
ejpam-5732	218	9	βi	βi	PRON
ejpam-5732	218	10	-	-	PUNCT
ejpam-5732	218	11	paracompact	paracompact	ADJ
ejpam-5732	218	12	.	.	PUNCT
ejpam-5732	219	1	4	4	X
ejpam-5732	219	2	.	.	X
ejpam-5732	219	3	preserving	preserve	VERB
ejpam-5732	219	4	δ1	δ1	VERB
ejpam-5732	219	5	-	-	PUNCT
ejpam-5732	219	6	βi	βi	NOUN
ejpam-5732	219	7	-	-	NOUN
ejpam-5732	219	8	paracompactness	paracompactness	NOUN
ejpam-5732	219	9	in	in	ADP
ejpam-5732	219	10	this	this	DET
ejpam-5732	219	11	section	section	NOUN
ejpam-5732	219	12	,	,	PUNCT
ejpam-5732	219	13	we	we	PRON
ejpam-5732	219	14	will	will	AUX
ejpam-5732	219	15	illustrate	illustrate	VERB
ejpam-5732	219	16	how	how	SCONJ
ejpam-5732	219	17	δ1	δ1	VERB
ejpam-5732	219	18	-	-	PUNCT
ejpam-5732	219	19	βi	βi	PRON
ejpam-5732	219	20	-	-	NOUN
ejpam-5732	219	21	paracompactness	paracompactness	NOUN
ejpam-5732	219	22	is	be	AUX
ejpam-5732	219	23	maintained	maintain	VERB
ejpam-5732	219	24	under	under	ADP
ejpam-5732	219	25	specific	specific	ADJ
ejpam-5732	219	26	conditions	condition	NOUN
ejpam-5732	219	27	.	.	PUNCT
ejpam-5732	220	1	we	we	PRON
ejpam-5732	220	2	start	start	VERB
ejpam-5732	220	3	by	by	ADP
ejpam-5732	220	4	introducing	introduce	VERB
ejpam-5732	220	5	the	the	DET
ejpam-5732	220	6	subsequent	subsequent	ADJ
ejpam-5732	220	7	definition	definition	NOUN
ejpam-5732	220	8	.	.	PUNCT
ejpam-5732	221	1	definition	definition	NOUN
ejpam-5732	221	2	5	5	NUM
ejpam-5732	221	3	.	.	PUNCT
ejpam-5732	222	1	let	let	VERB
ejpam-5732	222	2	(	(	PUNCT
ejpam-5732	222	3	x	x	X
ejpam-5732	222	4	,	,	PUNCT
ejpam-5732	222	5	τ	τ	PROPN
ejpam-5732	222	6	,	,	PUNCT
ejpam-5732	222	7	i	i	PROPN
ejpam-5732	222	8	)	)	PUNCT
ejpam-5732	222	9	,	,	PUNCT
ejpam-5732	222	10	and	and	CCONJ
ejpam-5732	222	11	(	(	PUNCT
ejpam-5732	222	12	y	y	PROPN
ejpam-5732	222	13	,	,	PUNCT
ejpam-5732	222	14	τ	τ	PROPN
ejpam-5732	222	15	′,j	′,j	NOUN
ejpam-5732	222	16	)	)	PUNCT
ejpam-5732	222	17	be	be	AUX
ejpam-5732	222	18	ideal	ideal	ADJ
ejpam-5732	222	19	topological	topological	ADJ
ejpam-5732	222	20	spaces	space	NOUN
ejpam-5732	222	21	,	,	PUNCT
ejpam-5732	222	22	and	and	CCONJ
ejpam-5732	222	23	f	f	X
ejpam-5732	222	24	:	:	PUNCT
ejpam-5732	222	25	x	x	X
ejpam-5732	222	26	→	→	SYM
ejpam-5732	222	27	y	y	X
ejpam-5732	222	28	be	be	AUX
ejpam-5732	222	29	a	a	DET
ejpam-5732	222	30	function	function	NOUN
ejpam-5732	222	31	.	.	PUNCT
ejpam-5732	223	1	(	(	PUNCT
ejpam-5732	223	2	i	i	NOUN
ejpam-5732	223	3	)	)	PUNCT
ejpam-5732	223	4	f	f	PROPN
ejpam-5732	223	5	is	be	AUX
ejpam-5732	223	6	called	call	VERB
ejpam-5732	223	7	δ	δ	PROPN
ejpam-5732	223	8	-	-	PUNCT
ejpam-5732	223	9	βi	βi	ADV
ejpam-5732	223	10	-	-	PUNCT
ejpam-5732	223	11	irresolute	irresolute	ADJ
ejpam-5732	223	12	if	if	SCONJ
ejpam-5732	223	13	f−1(v	f−1(v	PROPN
ejpam-5732	223	14	)	)	PUNCT
ejpam-5732	223	15	is	be	AUX
ejpam-5732	223	16	a	a	DET
ejpam-5732	223	17	δ	δ	PROPN
ejpam-5732	223	18	-	-	PUNCT
ejpam-5732	223	19	βi	βi	ADV
ejpam-5732	223	20	-	-	PUNCT
ejpam-5732	223	21	open	open	NOUN
ejpam-5732	223	22	set	set	NOUN
ejpam-5732	223	23	in	in	ADP
ejpam-5732	223	24	x	x	PUNCT
ejpam-5732	223	25	for	for	SCONJ
ejpam-5732	223	26	every	every	DET
ejpam-5732	223	27	δ	δ	PROPN
ejpam-5732	223	28	-	-	PUNCT
ejpam-5732	223	29	βj	βj	PUNCT
ejpam-5732	223	30	-open	-open	NOUN
ejpam-5732	223	31	set	set	VERB
ejpam-5732	223	32	v	v	NOUN
ejpam-5732	223	33	in	in	ADP
ejpam-5732	223	34	y	y	PROPN
ejpam-5732	223	35	.	.	PUNCT
ejpam-5732	224	1	(	(	PUNCT
ejpam-5732	224	2	ii	ii	X
ejpam-5732	224	3	)	)	PUNCT
ejpam-5732	224	4	f	f	PROPN
ejpam-5732	224	5	is	be	AUX
ejpam-5732	224	6	called	call	VERB
ejpam-5732	224	7	δ	δ	PROPN
ejpam-5732	224	8	-	-	PUNCT
ejpam-5732	224	9	βi	βi	ADV
ejpam-5732	224	10	-	-	PUNCT
ejpam-5732	224	11	open	open	ADJ
ejpam-5732	224	12	if	if	SCONJ
ejpam-5732	224	13	f(u	f(u	PROPN
ejpam-5732	224	14	)	)	PUNCT
ejpam-5732	224	15	is	be	AUX
ejpam-5732	224	16	a	a	DET
ejpam-5732	224	17	δ	δ	PROPN
ejpam-5732	224	18	-	-	PUNCT
ejpam-5732	224	19	βj	βj	PUNCT
ejpam-5732	224	20	-open	-open	NOUN
ejpam-5732	224	21	set	set	VERB
ejpam-5732	224	22	in	in	ADP
ejpam-5732	224	23	y	y	PROPN
ejpam-5732	224	24	for	for	ADP
ejpam-5732	224	25	every	every	DET
ejpam-5732	224	26	δ	δ	PROPN
ejpam-5732	224	27	-	-	PUNCT
ejpam-5732	224	28	βi	βi	ADV
ejpam-5732	224	29	-	-	PUNCT
ejpam-5732	224	30	open	open	ADJ
ejpam-5732	224	31	set	set	NOUN
ejpam-5732	224	32	u	u	NOUN
ejpam-5732	224	33	in	in	ADP
ejpam-5732	224	34	x.	x.	PROPN
ejpam-5732	224	35	(	(	PUNCT
ejpam-5732	224	36	iii	iii	X
ejpam-5732	224	37	)	)	PUNCT
ejpam-5732	224	38	f	f	PROPN
ejpam-5732	224	39	is	be	AUX
ejpam-5732	224	40	called	call	VERB
ejpam-5732	224	41	δ	δ	PROPN
ejpam-5732	224	42	-	-	PUNCT
ejpam-5732	224	43	βi	βi	ADV
ejpam-5732	224	44	-	-	PUNCT
ejpam-5732	224	45	closed	closed	ADJ
ejpam-5732	224	46	if	if	SCONJ
ejpam-5732	224	47	f(u	f(u	PROPN
ejpam-5732	224	48	)	)	PUNCT
ejpam-5732	224	49	is	be	AUX
ejpam-5732	224	50	a	a	DET
ejpam-5732	224	51	δ	δ	PROPN
ejpam-5732	224	52	-	-	PUNCT
ejpam-5732	224	53	βj	βj	PUNCT
ejpam-5732	224	54	-closed	-close	VERB
ejpam-5732	224	55	set	set	NOUN
ejpam-5732	224	56	in	in	ADP
ejpam-5732	224	57	y	y	PROPN
ejpam-5732	224	58	for	for	ADP
ejpam-5732	224	59	every	every	DET
ejpam-5732	224	60	δ	δ	PROPN
ejpam-5732	224	61	-	-	PUNCT
ejpam-5732	224	62	βi	βi	ADV
ejpam-5732	224	63	-	-	PUNCT
ejpam-5732	224	64	closed	close	VERB
ejpam-5732	224	65	set	set	NOUN
ejpam-5732	224	66	u	u	NOUN
ejpam-5732	224	67	in	in	ADP
ejpam-5732	224	68	x.	x.	PROPN
ejpam-5732	224	69	note	note	VERB
ejpam-5732	224	70	that	that	SCONJ
ejpam-5732	224	71	f−1(j	f−1(j	NOUN
ejpam-5732	224	72	)	)	PUNCT
ejpam-5732	224	73	is	be	AUX
ejpam-5732	224	74	an	an	DET
ejpam-5732	224	75	ideal	ideal	NOUN
ejpam-5732	224	76	on	on	ADP
ejpam-5732	224	77	x	x	SYM
ejpam-5732	224	78	if	if	SCONJ
ejpam-5732	224	79	f	f	X
ejpam-5732	224	80	:	:	PUNCT
ejpam-5732	224	81	x	x	X
ejpam-5732	224	82	→	→	SYM
ejpam-5732	224	83	y	y	PROPN
ejpam-5732	224	84	is	be	AUX
ejpam-5732	224	85	a	a	DET
ejpam-5732	224	86	function	function	NOUN
ejpam-5732	224	87	,	,	PUNCT
ejpam-5732	224	88	(	(	PUNCT
ejpam-5732	224	89	x	x	X
ejpam-5732	224	90	,	,	PUNCT
ejpam-5732	224	91	τ	τ	X
ejpam-5732	224	92	)	)	PUNCT
ejpam-5732	224	93	is	be	AUX
ejpam-5732	224	94	a	a	DET
ejpam-5732	224	95	topological	topological	ADJ
ejpam-5732	224	96	space	space	NOUN
ejpam-5732	224	97	,	,	PUNCT
ejpam-5732	224	98	and	and	CCONJ
ejpam-5732	224	99	(	(	PUNCT
ejpam-5732	224	100	y	y	PROPN
ejpam-5732	224	101	,	,	PUNCT
ejpam-5732	224	102	τ	τ	PROPN
ejpam-5732	224	103	′	′	NUM
ejpam-5732	224	104	)	)	PUNCT
ejpam-5732	224	105	is	be	AUX
ejpam-5732	224	106	a	a	DET
ejpam-5732	224	107	topological	topological	ADJ
ejpam-5732	224	108	space	space	NOUN
ejpam-5732	224	109	with	with	ADP
ejpam-5732	224	110	an	an	DET
ejpam-5732	224	111	ideal	ideal	ADJ
ejpam-5732	224	112	j	j	PROPN
ejpam-5732	224	113	.	.	PUNCT
ejpam-5732	225	1	furthermore	furthermore	ADV
ejpam-5732	225	2	,	,	PUNCT
ejpam-5732	225	3	given	give	VERB
ejpam-5732	225	4	that	that	SCONJ
ejpam-5732	225	5	f	f	PROPN
ejpam-5732	225	6	is	be	AUX
ejpam-5732	225	7	surjective	surjective	ADJ
ejpam-5732	225	8	and	and	CCONJ
ejpam-5732	225	9	x	x	PRON
ejpam-5732	225	10	possesses	possess	VERB
ejpam-5732	225	11	an	an	DET
ejpam-5732	225	12	ideal	ideal	NOUN
ejpam-5732	225	13	i	i	PRON
ejpam-5732	225	14	,	,	PUNCT
ejpam-5732	225	15	f(i	f(i	PROPN
ejpam-5732	225	16	)	)	PUNCT
ejpam-5732	225	17	is	be	AUX
ejpam-5732	225	18	an	an	DET
ejpam-5732	225	19	ideal	ideal	NOUN
ejpam-5732	225	20	on	on	ADP
ejpam-5732	225	21	y	y	PROPN
ejpam-5732	225	22	.	.	PUNCT
ejpam-5732	226	1	the	the	DET
ejpam-5732	226	2	proof	proof	NOUN
ejpam-5732	226	3	of	of	ADP
ejpam-5732	226	4	theorem	theorem	ADJ
ejpam-5732	226	5	8	8	NUM
ejpam-5732	226	6	requires	require	VERB
ejpam-5732	226	7	the	the	DET
ejpam-5732	226	8	following	follow	VERB
ejpam-5732	226	9	lemma	lemma	PROPN
ejpam-5732	226	10	.	.	PUNCT
ejpam-5732	227	1	lemma	lemma	PROPN
ejpam-5732	227	2	8	8	NUM
ejpam-5732	227	3	.	.	PUNCT
ejpam-5732	228	1	let	let	VERB
ejpam-5732	228	2	(	(	PUNCT
ejpam-5732	228	3	x	x	X
ejpam-5732	228	4	,	,	PUNCT
ejpam-5732	228	5	τ	τ	PROPN
ejpam-5732	228	6	,	,	PUNCT
ejpam-5732	228	7	i	i	PROPN
ejpam-5732	228	8	)	)	PUNCT
ejpam-5732	228	9	and	and	CCONJ
ejpam-5732	228	10	(	(	PUNCT
ejpam-5732	228	11	y	y	PROPN
ejpam-5732	228	12	,	,	PUNCT
ejpam-5732	228	13	τ	τ	PROPN
ejpam-5732	228	14	′,j	′,j	NOUN
ejpam-5732	228	15	)	)	PUNCT
ejpam-5732	228	16	be	be	AUX
ejpam-5732	228	17	ideal	ideal	ADJ
ejpam-5732	228	18	topological	topological	ADJ
ejpam-5732	228	19	spaces	space	NOUN
ejpam-5732	228	20	,	,	PUNCT
ejpam-5732	228	21	and	and	CCONJ
ejpam-5732	228	22	f	f	X
ejpam-5732	228	23	:	:	PUNCT
ejpam-5732	228	24	x	x	X
ejpam-5732	228	25	→	→	SYM
ejpam-5732	228	26	y	y	PROPN
ejpam-5732	228	27	be	be	AUX
ejpam-5732	228	28	surjective	surjective	ADJ
ejpam-5732	228	29	.	.	PUNCT
ejpam-5732	229	1	then	then	ADV
ejpam-5732	229	2	f	f	PROPN
ejpam-5732	229	3	is	be	AUX
ejpam-5732	229	4	δ	δ	PROPN
ejpam-5732	229	5	-	-	PUNCT
ejpam-5732	229	6	βi	βi	ADV
ejpam-5732	229	7	-	-	PUNCT
ejpam-5732	229	8	closed	closed	ADJ
ejpam-5732	229	9	if	if	SCONJ
ejpam-5732	229	10	and	and	CCONJ
ejpam-5732	229	11	only	only	ADV
ejpam-5732	229	12	if	if	SCONJ
ejpam-5732	229	13	for	for	ADP
ejpam-5732	229	14	every	every	DET
ejpam-5732	229	15	y	y	PROPN
ejpam-5732	229	16	∈	∈	PROPN
ejpam-5732	229	17	y	y	PROPN
ejpam-5732	229	18	and	and	CCONJ
ejpam-5732	229	19	for	for	ADP
ejpam-5732	229	20	every	every	DET
ejpam-5732	229	21	δ	δ	PROPN
ejpam-5732	229	22	-	-	PUNCT
ejpam-5732	229	23	βi	βi	ADV
ejpam-5732	229	24	-	-	PUNCT
ejpam-5732	229	25	open	open	ADJ
ejpam-5732	229	26	set	set	VERB
ejpam-5732	229	27	u	u	NOUN
ejpam-5732	229	28	in	in	ADP
ejpam-5732	229	29	x	x	SYM
ejpam-5732	229	30	that	that	PRON
ejpam-5732	229	31	contains	contain	VERB
ejpam-5732	229	32	{	{	PUNCT
ejpam-5732	229	33	f−1(y	f−1(y	PROPN
ejpam-5732	229	34	)	)	PUNCT
ejpam-5732	229	35	}	}	PUNCT
ejpam-5732	229	36	,	,	PUNCT
ejpam-5732	229	37	there	there	PRON
ejpam-5732	229	38	exists	exist	VERB
ejpam-5732	229	39	a	a	DET
ejpam-5732	229	40	δ	δ	PROPN
ejpam-5732	229	41	-	-	PUNCT
ejpam-5732	229	42	βj	βj	PUNCT
ejpam-5732	229	43	-open	-open	NOUN
ejpam-5732	229	44	set	set	VERB
ejpam-5732	229	45	v	v	NOUN
ejpam-5732	229	46	containing	contain	VERB
ejpam-5732	229	47	y	y	NOUN
ejpam-5732	229	48	in	in	ADP
ejpam-5732	229	49	y	y	PROPN
ejpam-5732	229	50	such	such	ADJ
ejpam-5732	229	51	that	that	DET
ejpam-5732	229	52	f−1(v	f−1(v	PROPN
ejpam-5732	229	53	)	)	PUNCT
ejpam-5732	230	1	⊂	⊂	PROPN
ejpam-5732	230	2	u	u	PROPN
ejpam-5732	230	3	.	.	PUNCT
ejpam-5732	231	1	proof	proof	NOUN
ejpam-5732	231	2	.	.	PUNCT
ejpam-5732	232	1	let	let	VERB
ejpam-5732	232	2	y	y	PROPN
ejpam-5732	232	3	∈	∈	PROPN
ejpam-5732	232	4	y	y	PROPN
ejpam-5732	232	5	and	and	CCONJ
ejpam-5732	232	6	u	u	PRON
ejpam-5732	232	7	be	be	VERB
ejpam-5732	232	8	a	a	DET
ejpam-5732	232	9	δ	δ	PROPN
ejpam-5732	232	10	-	-	PUNCT
ejpam-5732	232	11	βi	βi	ADV
ejpam-5732	232	12	-	-	PUNCT
ejpam-5732	232	13	open	open	ADJ
ejpam-5732	232	14	set	set	NOUN
ejpam-5732	232	15	inx	inx	VERB
ejpam-5732	232	16	such	such	ADJ
ejpam-5732	232	17	that	that	SCONJ
ejpam-5732	232	18	{	{	PUNCT
ejpam-5732	232	19	f−1(y	f−1(y	PROPN
ejpam-5732	232	20	)	)	PUNCT
ejpam-5732	232	21	}	}	PUNCT
ejpam-5732	233	1	⊂	⊂	PROPN
ejpam-5732	233	2	u	u	NOUN
ejpam-5732	233	3	.	.	PUNCT
ejpam-5732	234	1	we	we	PRON
ejpam-5732	234	2	have	have	VERB
ejpam-5732	234	3	that	that	PRON
ejpam-5732	234	4	v	v	NOUN
ejpam-5732	234	5	=	=	SYM
ejpam-5732	234	6	y	y	NOUN
ejpam-5732	234	7	−f(x−u	−f(x−u	NOUN
ejpam-5732	234	8	)	)	PUNCT
ejpam-5732	234	9	is	be	AUX
ejpam-5732	234	10	a	a	DET
ejpam-5732	234	11	δ	δ	PROPN
ejpam-5732	234	12	-	-	PUNCT
ejpam-5732	234	13	βi	βi	ADV
ejpam-5732	234	14	-	-	PUNCT
ejpam-5732	234	15	open	open	NOUN
ejpam-5732	234	16	set	set	VERB
ejpam-5732	234	17	such	such	ADJ
ejpam-5732	234	18	that	that	SCONJ
ejpam-5732	234	19	y	y	PROPN
ejpam-5732	234	20	∈	∈	PROPN
ejpam-5732	234	21	v	v	NOUN
ejpam-5732	234	22	and	and	CCONJ
ejpam-5732	234	23	f−1(v	f−1(v	PROPN
ejpam-5732	234	24	)	)	PUNCT
ejpam-5732	235	1	⊂	⊂	PROPN
ejpam-5732	235	2	u	u	PROPN
ejpam-5732	235	3	.	.	PUNCT
ejpam-5732	236	1	subsequently	subsequently	ADV
ejpam-5732	236	2	,	,	PUNCT
ejpam-5732	236	3	the	the	DET
ejpam-5732	236	4	necessity	necessity	NOUN
ejpam-5732	236	5	is	be	AUX
ejpam-5732	236	6	verified	verify	VERB
ejpam-5732	236	7	.	.	PUNCT
ejpam-5732	237	1	sufficiency	sufficiency	PROPN
ejpam-5732	237	2	will	will	AUX
ejpam-5732	237	3	now	now	ADV
ejpam-5732	237	4	be	be	AUX
ejpam-5732	237	5	demonstrated	demonstrate	VERB
ejpam-5732	237	6	.	.	PUNCT
ejpam-5732	238	1	let	let	VERB
ejpam-5732	238	2	f	f	PRON
ejpam-5732	238	3	be	be	AUX
ejpam-5732	238	4	a	a	DET
ejpam-5732	238	5	δ	δ	PROPN
ejpam-5732	238	6	-	-	PUNCT
ejpam-5732	238	7	βi	βi	ADV
ejpam-5732	238	8	-	-	PUNCT
ejpam-5732	238	9	closed	close	VERB
ejpam-5732	238	10	subset	subset	NOUN
ejpam-5732	238	11	of	of	ADP
ejpam-5732	238	12	x	x	X
ejpam-5732	238	13	and	and	CCONJ
ejpam-5732	238	14	y	y	PROPN
ejpam-5732	238	15	∈	∈	PROPN
ejpam-5732	239	1	y	y	PROPN
ejpam-5732	239	2	−	−	PROPN
ejpam-5732	239	3	f(f	f(f	PROPN
ejpam-5732	239	4	)	)	PUNCT
ejpam-5732	239	5	.	.	PUNCT
ejpam-5732	240	1	consequently	consequently	ADV
ejpam-5732	240	2	,	,	PUNCT
ejpam-5732	240	3	{	{	PUNCT
ejpam-5732	240	4	f−1(y	f−1(y	PROPN
ejpam-5732	240	5	)	)	PUNCT
ejpam-5732	240	6	}	}	PUNCT
ejpam-5732	241	1	⊂	⊂	X
ejpam-5732	241	2	x	x	X
ejpam-5732	242	1	−	−	PROPN
ejpam-5732	242	2	f	f	X
ejpam-5732	242	3	.	.	PUNCT
ejpam-5732	243	1	according	accord	VERB
ejpam-5732	243	2	to	to	ADP
ejpam-5732	243	3	the	the	DET
ejpam-5732	243	4	hypothesis	hypothesis	NOUN
ejpam-5732	243	5	,	,	PUNCT
ejpam-5732	243	6	there	there	PRON
ejpam-5732	243	7	exists	exist	VERB
ejpam-5732	243	8	a	a	DET
ejpam-5732	243	9	δ	δ	PROPN
ejpam-5732	243	10	-	-	PUNCT
ejpam-5732	243	11	βj	βj	PUNCT
ejpam-5732	243	12	-open	-open	ADJ
ejpam-5732	243	13	set	set	NOUN
ejpam-5732	243	14	vy	vy	ADP
ejpam-5732	243	15	such	such	ADJ
ejpam-5732	243	16	that	that	SCONJ
ejpam-5732	243	17	f−1(vy	f−1(vy	PROPN
ejpam-5732	243	18	)	)	PUNCT
ejpam-5732	243	19	⊂	⊂	PROPN
ejpam-5732	244	1	x	x	X
ejpam-5732	245	1	−	−	PROPN
ejpam-5732	245	2	f	f	X
ejpam-5732	245	3	,	,	PUNCT
ejpam-5732	245	4	thereby	thereby	ADV
ejpam-5732	245	5	suggesting	suggest	VERB
ejpam-5732	245	6	that	that	SCONJ
ejpam-5732	245	7	y	y	PROPN
ejpam-5732	245	8	∈	∈	PROPN
ejpam-5732	245	9	vy	vy	NOUN
ejpam-5732	245	10	⊂	⊂	PROPN
ejpam-5732	245	11	y	y	PROPN
ejpam-5732	245	12	−	−	PROPN
ejpam-5732	245	13	f(f	f(f	PROPN
ejpam-5732	245	14	)	)	PUNCT
ejpam-5732	245	15	.	.	PUNCT
ejpam-5732	246	1	therefore	therefore	ADV
ejpam-5732	246	2	,	,	PUNCT
ejpam-5732	246	3	y	y	PROPN
ejpam-5732	246	4	−	−	PROPN
ejpam-5732	246	5	f(f	f(f	PROPN
ejpam-5732	246	6	)	)	PUNCT
ejpam-5732	247	1	=	=	PUNCT
ejpam-5732	248	1	∪{vy	∪{vy	PROPN
ejpam-5732	248	2	:	:	PUNCT
ejpam-5732	248	3	y	y	PROPN
ejpam-5732	248	4	∈	∈	PROPN
ejpam-5732	248	5	y	y	PROPN
ejpam-5732	248	6	}	}	PUNCT
ejpam-5732	248	7	forms	form	VERB
ejpam-5732	248	8	a	a	DET
ejpam-5732	248	9	δ	δ	PROPN
ejpam-5732	248	10	-	-	PUNCT
ejpam-5732	248	11	βi	βi	ADV
ejpam-5732	248	12	-	-	PUNCT
ejpam-5732	248	13	open	open	ADJ
ejpam-5732	248	14	set	set	NOUN
ejpam-5732	248	15	in	in	ADP
ejpam-5732	248	16	y	y	PROPN
ejpam-5732	248	17	.	.	PUNCT
ejpam-5732	249	1	thus	thus	ADV
ejpam-5732	249	2	,	,	PUNCT
ejpam-5732	249	3	f(f	f(f	PROPN
ejpam-5732	249	4	)	)	PUNCT
ejpam-5732	249	5	represents	represent	VERB
ejpam-5732	249	6	a	a	DET
ejpam-5732	249	7	δ	δ	PROPN
ejpam-5732	249	8	-	-	PUNCT
ejpam-5732	249	9	βi	βi	ADV
ejpam-5732	249	10	-	-	PUNCT
ejpam-5732	249	11	closed	close	VERB
ejpam-5732	249	12	set	set	NOUN
ejpam-5732	249	13	.	.	PUNCT
ejpam-5732	250	1	we	we	PRON
ejpam-5732	250	2	conclude	conclude	VERB
ejpam-5732	250	3	that	that	SCONJ
ejpam-5732	250	4	f	f	PROPN
ejpam-5732	250	5	is	be	AUX
ejpam-5732	250	6	δ	δ	PROPN
ejpam-5732	250	7	-	-	PUNCT
ejpam-5732	250	8	βi	βi	ADV
ejpam-5732	250	9	-	-	PUNCT
ejpam-5732	250	10	closed	closed	ADJ
ejpam-5732	250	11	.	.	PUNCT
ejpam-5732	251	1	next	next	ADV
ejpam-5732	251	2	,	,	PUNCT
ejpam-5732	251	3	we	we	PRON
ejpam-5732	251	4	provide	provide	VERB
ejpam-5732	251	5	the	the	DET
ejpam-5732	251	6	characteristics	characteristic	NOUN
ejpam-5732	251	7	of	of	ADP
ejpam-5732	251	8	a	a	DET
ejpam-5732	251	9	function	function	NOUN
ejpam-5732	251	10	that	that	PRON
ejpam-5732	251	11	maps	map	VERB
ejpam-5732	251	12	between	between	ADP
ejpam-5732	251	13	two	two	NUM
ejpam-5732	251	14	ideal	ideal	ADJ
ejpam-5732	251	15	topological	topological	ADJ
ejpam-5732	251	16	spaces	space	NOUN
ejpam-5732	251	17	,	,	PUNCT
ejpam-5732	251	18	where	where	SCONJ
ejpam-5732	251	19	one	one	NUM
ejpam-5732	251	20	space	space	NOUN
ejpam-5732	251	21	conveys	convey	VERB
ejpam-5732	251	22	identical	identical	ADJ
ejpam-5732	251	23	properties	property	NOUN
ejpam-5732	251	24	to	to	ADP
ejpam-5732	251	25	the	the	DET
ejpam-5732	251	26	other	other	ADJ
ejpam-5732	251	27	.	.	PUNCT
ejpam-5732	252	1	initially	initially	ADV
ejpam-5732	252	2	,	,	PUNCT
ejpam-5732	252	3	we	we	PRON
ejpam-5732	252	4	define	define	VERB
ejpam-5732	252	5	the	the	DET
ejpam-5732	252	6	concept	concept	NOUN
ejpam-5732	252	7	of	of	ADP
ejpam-5732	252	8	δ	δ	PROPN
ejpam-5732	252	9	-	-	PUNCT
ejpam-5732	252	10	βi	βi	NOUN
ejpam-5732	252	11	-	-	PUNCT
ejpam-5732	252	12	compactness	compactness	NOUN
ejpam-5732	252	13	and	and	CCONJ
ejpam-5732	252	14	offer	offer	VERB
ejpam-5732	252	15	a	a	DET
ejpam-5732	252	16	lemma	lemma	PROPN
ejpam-5732	252	17	utilized	utilize	VERB
ejpam-5732	252	18	in	in	ADP
ejpam-5732	252	19	the	the	DET
ejpam-5732	252	20	proof	proof	NOUN
ejpam-5732	252	21	of	of	ADP
ejpam-5732	252	22	the	the	DET
ejpam-5732	252	23	theorem	theorem	ADJ
ejpam-5732	252	24	8	8	NUM
ejpam-5732	252	25	.	.	PUNCT
ejpam-5732	252	26	c.	c.	PROPN
ejpam-5732	252	27	boonpok	boonpok	PROPN
ejpam-5732	252	28	,	,	PUNCT
ejpam-5732	252	29	a.	a.	PROPN
ejpam-5732	252	30	sama	sama	PROPN
ejpam-5732	252	31	-	-	PUNCT
ejpam-5732	252	32	ae	ae	PROPN
ejpam-5732	252	33	/	/	SYM
ejpam-5732	252	34	eur	eur	PROPN
ejpam-5732	252	35	.	.	PUNCT
ejpam-5732	253	1	j.	j.	PROPN
ejpam-5732	253	2	pure	pure	PROPN
ejpam-5732	253	3	appl	appl	PROPN
ejpam-5732	253	4	.	.	PROPN
ejpam-5732	253	5	math	math	PROPN
ejpam-5732	253	6	,	,	PUNCT
ejpam-5732	253	7	18	18	NUM
ejpam-5732	253	8	(	(	PUNCT
ejpam-5732	253	9	1	1	NUM
ejpam-5732	253	10	)	)	PUNCT
ejpam-5732	253	11	(	(	PUNCT
ejpam-5732	253	12	2025	2025	NUM
ejpam-5732	253	13	)	)	PUNCT
ejpam-5732	253	14	,	,	PUNCT
ejpam-5732	253	15	5732	5732	NUM
ejpam-5732	253	16	9	9	NUM
ejpam-5732	253	17	of	of	ADP
ejpam-5732	253	18	13	13	NUM
ejpam-5732	253	19	definition	definition	NOUN
ejpam-5732	253	20	6	6	NUM
ejpam-5732	253	21	.	.	PUNCT
ejpam-5732	254	1	an	an	DET
ejpam-5732	254	2	ideal	ideal	ADJ
ejpam-5732	254	3	topological	topological	ADJ
ejpam-5732	254	4	space	space	NOUN
ejpam-5732	254	5	(	(	PUNCT
ejpam-5732	254	6	x	x	X
ejpam-5732	254	7	,	,	PUNCT
ejpam-5732	254	8	τ	τ	PROPN
ejpam-5732	254	9	,	,	PUNCT
ejpam-5732	254	10	i	i	PROPN
ejpam-5732	254	11	)	)	PUNCT
ejpam-5732	254	12	is	be	AUX
ejpam-5732	254	13	said	say	VERB
ejpam-5732	254	14	to	to	PART
ejpam-5732	254	15	be	be	AUX
ejpam-5732	254	16	δ	δ	PROPN
ejpam-5732	254	17	-	-	PUNCT
ejpam-5732	254	18	βi	βi	ADV
ejpam-5732	254	19	-	-	ADJ
ejpam-5732	254	20	compact	compact	ADJ
ejpam-5732	254	21	if	if	SCONJ
ejpam-5732	254	22	every	every	DET
ejpam-5732	254	23	cover	cover	NOUN
ejpam-5732	254	24	v	v	NOUN
ejpam-5732	254	25	of	of	ADP
ejpam-5732	254	26	δ	δ	PROPN
ejpam-5732	254	27	-	-	PUNCT
ejpam-5732	254	28	βi	βi	ADV
ejpam-5732	254	29	-	-	PUNCT
ejpam-5732	254	30	open	open	ADJ
ejpam-5732	254	31	sets	set	NOUN
ejpam-5732	254	32	of	of	ADP
ejpam-5732	254	33	x	x	PUNCT
ejpam-5732	254	34	has	have	AUX
ejpam-5732	254	35	v1	v1	NOUN
ejpam-5732	254	36	,	,	PUNCT
ejpam-5732	254	37	v2	v2	PROPN
ejpam-5732	254	38	,	,	PUNCT
ejpam-5732	254	39	...	...	PUNCT
ejpam-5732	254	40	,	,	PUNCT
ejpam-5732	254	41	vn	vn	PROPN
ejpam-5732	254	42	∈	∈	PROPN
ejpam-5732	254	43	v	v	ADP
ejpam-5732	254	44	such	such	ADJ
ejpam-5732	254	45	that	that	SCONJ
ejpam-5732	254	46	x	x	SYM
ejpam-5732	254	47	⊂	⊂	PROPN
ejpam-5732	254	48	v1	v1	VERB
ejpam-5732	254	49	∪	∪	ADJ
ejpam-5732	254	50	v2	v2	PROPN
ejpam-5732	254	51	∪	∪	X
ejpam-5732	254	52	·	·	PUNCT
ejpam-5732	254	53	·	·	PUNCT
ejpam-5732	254	54	·	·	PUNCT
ejpam-5732	254	55	∪	∪	ADP
ejpam-5732	254	56	vn	vn	PROPN
ejpam-5732	254	57	.	.	PUNCT
ejpam-5732	255	1	the	the	DET
ejpam-5732	255	2	next	next	ADJ
ejpam-5732	255	3	theorem	theorem	NOUN
ejpam-5732	255	4	describe	describe	VERB
ejpam-5732	255	5	characterizations	characterization	NOUN
ejpam-5732	255	6	of	of	ADP
ejpam-5732	255	7	a	a	DET
ejpam-5732	255	8	function	function	NOUN
ejpam-5732	255	9	that	that	PRON
ejpam-5732	255	10	maps	map	VERB
ejpam-5732	255	11	from	from	ADP
ejpam-5732	255	12	a	a	DET
ejpam-5732	255	13	δ1	δ1	NOUN
ejpam-5732	255	14	-	-	PUNCT
ejpam-5732	255	15	βiparacompact	βiparacompact	NOUN
ejpam-5732	255	16	ideal	ideal	ADJ
ejpam-5732	255	17	topological	topological	ADJ
ejpam-5732	255	18	space	space	NOUN
ejpam-5732	255	19	(	(	PUNCT
ejpam-5732	255	20	x	x	X
ejpam-5732	255	21	,	,	PUNCT
ejpam-5732	255	22	τ	τ	PROPN
ejpam-5732	255	23	,	,	PUNCT
ejpam-5732	255	24	i	i	PROPN
ejpam-5732	255	25	)	)	PUNCT
ejpam-5732	255	26	to	to	ADP
ejpam-5732	255	27	an	an	DET
ejpam-5732	255	28	ideal	ideal	ADJ
ejpam-5732	255	29	topological	topological	ADJ
ejpam-5732	255	30	space	space	NOUN
ejpam-5732	255	31	(	(	PUNCT
ejpam-5732	255	32	y	y	PROPN
ejpam-5732	255	33	,	,	PUNCT
ejpam-5732	255	34	τ	τ	PROPN
ejpam-5732	255	35	′,j	′,j	NOUN
ejpam-5732	255	36	)	)	PUNCT
ejpam-5732	255	37	,	,	PUNCT
ejpam-5732	255	38	ensuring	ensure	VERB
ejpam-5732	255	39	that	that	SCONJ
ejpam-5732	255	40	y	y	PROPN
ejpam-5732	255	41	has	have	VERB
ejpam-5732	255	42	the	the	DET
ejpam-5732	255	43	same	same	ADJ
ejpam-5732	255	44	properties	property	NOUN
ejpam-5732	255	45	as	as	SCONJ
ejpam-5732	255	46	x.	x.	NOUN
ejpam-5732	255	47	theorem	theorem	VERB
ejpam-5732	255	48	8	8	NUM
ejpam-5732	255	49	.	.	PUNCT
ejpam-5732	256	1	let	let	VERB
ejpam-5732	256	2	(	(	PUNCT
ejpam-5732	256	3	x	x	X
ejpam-5732	256	4	,	,	PUNCT
ejpam-5732	256	5	τ	τ	PROPN
ejpam-5732	256	6	,	,	PUNCT
ejpam-5732	256	7	i	i	PROPN
ejpam-5732	256	8	)	)	PUNCT
ejpam-5732	256	9	and	and	CCONJ
ejpam-5732	256	10	(	(	PUNCT
ejpam-5732	256	11	y	y	PROPN
ejpam-5732	256	12	,	,	PUNCT
ejpam-5732	256	13	τ	τ	PROPN
ejpam-5732	256	14	′,j	′,j	NOUN
ejpam-5732	256	15	)	)	PUNCT
ejpam-5732	256	16	be	be	AUX
ejpam-5732	256	17	ideal	ideal	ADJ
ejpam-5732	256	18	topological	topological	ADJ
ejpam-5732	256	19	spaces	space	NOUN
ejpam-5732	256	20	.	.	PUNCT
ejpam-5732	257	1	suppose	suppose	VERB
ejpam-5732	257	2	that	that	SCONJ
ejpam-5732	257	3	f	f	X
ejpam-5732	257	4	:	:	PUNCT
ejpam-5732	257	5	x	x	X
ejpam-5732	257	6	→	→	SYM
ejpam-5732	257	7	y	y	PROPN
ejpam-5732	257	8	satisfies	satisfy	VERB
ejpam-5732	257	9	the	the	DET
ejpam-5732	257	10	following	following	ADJ
ejpam-5732	257	11	statements	statement	NOUN
ejpam-5732	257	12	:	:	PUNCT
ejpam-5732	257	13	(	(	PUNCT
ejpam-5732	257	14	i	i	NOUN
ejpam-5732	257	15	)	)	PUNCT
ejpam-5732	257	16	f	f	PROPN
ejpam-5732	257	17	is	be	AUX
ejpam-5732	257	18	open	open	ADJ
ejpam-5732	257	19	;	;	PUNCT
ejpam-5732	257	20	(	(	PUNCT
ejpam-5732	257	21	ii	ii	NOUN
ejpam-5732	257	22	)	)	PUNCT
ejpam-5732	257	23	f	f	PROPN
ejpam-5732	257	24	is	be	AUX
ejpam-5732	257	25	δ	δ	PROPN
ejpam-5732	257	26	-	-	PUNCT
ejpam-5732	257	27	βi	βi	ADV
ejpam-5732	257	28	-	-	PUNCT
ejpam-5732	257	29	irresolute	irresolute	ADJ
ejpam-5732	257	30	;	;	PUNCT
ejpam-5732	257	31	(	(	PUNCT
ejpam-5732	257	32	iii	iii	X
ejpam-5732	257	33	)	)	PUNCT
ejpam-5732	257	34	f	f	PROPN
ejpam-5732	257	35	is	be	AUX
ejpam-5732	257	36	δ	δ	PROPN
ejpam-5732	257	37	-	-	PUNCT
ejpam-5732	257	38	βi	βi	ADV
ejpam-5732	257	39	-	-	PUNCT
ejpam-5732	257	40	closed	closed	ADJ
ejpam-5732	257	41	;	;	PUNCT
ejpam-5732	257	42	(	(	PUNCT
ejpam-5732	257	43	iv	iv	X
ejpam-5732	257	44	)	)	PUNCT
ejpam-5732	257	45	f	f	PROPN
ejpam-5732	257	46	is	be	AUX
ejpam-5732	257	47	a	a	DET
ejpam-5732	257	48	surjective	surjective	ADJ
ejpam-5732	257	49	function	function	NOUN
ejpam-5732	257	50	with	with	ADP
ejpam-5732	257	51	{	{	PUNCT
ejpam-5732	257	52	f−1(y	f−1(y	PROPN
ejpam-5732	257	53	)	)	PUNCT
ejpam-5732	257	54	}	}	PUNCT
ejpam-5732	257	55	is	be	AUX
ejpam-5732	257	56	δ	δ	PROPN
ejpam-5732	257	57	-	-	PUNCT
ejpam-5732	257	58	βi	βi	ADV
ejpam-5732	257	59	-	-	ADJ
ejpam-5732	257	60	compact	compact	ADJ
ejpam-5732	257	61	for	for	ADP
ejpam-5732	257	62	every	every	DET
ejpam-5732	257	63	y	y	PROPN
ejpam-5732	257	64	∈	∈	PROPN
ejpam-5732	257	65	y	y	PROPN
ejpam-5732	257	66	;	;	PUNCT
ejpam-5732	257	67	and	and	CCONJ
ejpam-5732	257	68	(	(	PUNCT
ejpam-5732	257	69	v	v	NOUN
ejpam-5732	257	70	)	)	PUNCT
ejpam-5732	257	71	f(i	f(i	NOUN
ejpam-5732	257	72	)	)	PUNCT
ejpam-5732	258	1	⊂	⊂	PROPN
ejpam-5732	258	2	j	j	PROPN
ejpam-5732	258	3	.	.	PUNCT
ejpam-5732	259	1	if	if	SCONJ
ejpam-5732	259	2	(	(	PUNCT
ejpam-5732	259	3	x	x	X
ejpam-5732	259	4	,	,	PUNCT
ejpam-5732	259	5	τ	τ	PROPN
ejpam-5732	259	6	,	,	PUNCT
ejpam-5732	259	7	i	i	PROPN
ejpam-5732	259	8	)	)	PUNCT
ejpam-5732	259	9	is	be	AUX
ejpam-5732	259	10	δ1	δ1	NOUN
ejpam-5732	259	11	-	-	PUNCT
ejpam-5732	259	12	βi	βi	PRON
ejpam-5732	259	13	-	-	PUNCT
ejpam-5732	259	14	paracompact	paracompact	ADJ
ejpam-5732	259	15	,	,	PUNCT
ejpam-5732	259	16	then	then	ADV
ejpam-5732	259	17	(	(	PUNCT
ejpam-5732	259	18	y	y	PROPN
ejpam-5732	259	19	,	,	PUNCT
ejpam-5732	259	20	τ	τ	PROPN
ejpam-5732	259	21	′,j	′,j	NOUN
ejpam-5732	259	22	)	)	PUNCT
ejpam-5732	259	23	is	be	AUX
ejpam-5732	259	24	δ1	δ1	NOUN
ejpam-5732	259	25	-	-	PUNCT
ejpam-5732	259	26	βj	βj	NOUN
ejpam-5732	259	27	-paracompact	-paracompact	ADJ
ejpam-5732	259	28	.	.	PUNCT
ejpam-5732	260	1	proof	proof	NOUN
ejpam-5732	260	2	.	.	PUNCT
ejpam-5732	261	1	let	let	VERB
ejpam-5732	261	2	u	u	PRON
ejpam-5732	261	3	=	=	PUNCT
ejpam-5732	261	4	{	{	PUNCT
ejpam-5732	261	5	uλ	uλ	X
ejpam-5732	261	6	:	:	PUNCT
ejpam-5732	261	7	λ	λ	X
ejpam-5732	261	8	∈	∈	PROPN
ejpam-5732	261	9	λ	λ	PROPN
ejpam-5732	261	10	}	}	PUNCT
ejpam-5732	261	11	be	be	VERB
ejpam-5732	261	12	a	a	DET
ejpam-5732	261	13	δ	δ	PROPN
ejpam-5732	261	14	-	-	PUNCT
ejpam-5732	261	15	βi	βi	ADV
ejpam-5732	261	16	-	-	PUNCT
ejpam-5732	261	17	open	open	ADJ
ejpam-5732	261	18	cover	cover	NOUN
ejpam-5732	261	19	of	of	ADP
ejpam-5732	261	20	y	y	PROPN
ejpam-5732	261	21	.	.	PUNCT
ejpam-5732	262	1	assuming	assume	VERB
ejpam-5732	262	2	that	that	SCONJ
ejpam-5732	262	3	f	f	PROPN
ejpam-5732	262	4	is	be	AUX
ejpam-5732	262	5	δ	δ	PROPN
ejpam-5732	262	6	-	-	NOUN
ejpam-5732	262	7	βiirresolute	βiirresolute	NOUN
ejpam-5732	262	8	,	,	PUNCT
ejpam-5732	262	9	it	it	PRON
ejpam-5732	262	10	follows	follow	VERB
ejpam-5732	262	11	that	that	SCONJ
ejpam-5732	262	12	h	h	NOUN
ejpam-5732	262	13	=	=	PRON
ejpam-5732	262	14	{	{	PUNCT
ejpam-5732	262	15	f−1(uλ	f−1(uλ	PROPN
ejpam-5732	262	16	)	)	PUNCT
ejpam-5732	262	17	:	:	PUNCT
ejpam-5732	263	1	λ	λ	X
ejpam-5732	263	2	∈	∈	PROPN
ejpam-5732	263	3	λ	λ	PROPN
ejpam-5732	263	4	}	}	PUNCT
ejpam-5732	263	5	forms	form	VERB
ejpam-5732	263	6	a	a	DET
ejpam-5732	263	7	δ	δ	PROPN
ejpam-5732	263	8	-	-	PUNCT
ejpam-5732	263	9	βi	βi	ADV
ejpam-5732	263	10	-	-	PUNCT
ejpam-5732	263	11	open	open	ADJ
ejpam-5732	263	12	cover	cover	NOUN
ejpam-5732	263	13	of	of	ADP
ejpam-5732	263	14	x.	x.	NOUN
ejpam-5732	263	15	given	give	VERB
ejpam-5732	263	16	that	that	SCONJ
ejpam-5732	263	17	x	x	PRON
ejpam-5732	263	18	is	be	AUX
ejpam-5732	263	19	δ1	δ1	NOUN
ejpam-5732	263	20	-	-	PUNCT
ejpam-5732	263	21	βi	βi	PRON
ejpam-5732	263	22	-	-	PUNCT
ejpam-5732	263	23	paracompact	paracompact	ADJ
ejpam-5732	263	24	,	,	PUNCT
ejpam-5732	263	25	the	the	DET
ejpam-5732	263	26	collection	collection	NOUN
ejpam-5732	263	27	h	h	NOUN
ejpam-5732	263	28	possesses	possess	VERB
ejpam-5732	263	29	a	a	DET
ejpam-5732	263	30	τ	τ	PROPN
ejpam-5732	263	31	-locally	-locally	ADV
ejpam-5732	263	32	finite	finite	ADJ
ejpam-5732	263	33	refinement	refinement	NOUN
ejpam-5732	263	34	v	v	ADP
ejpam-5732	263	35	=	=	PUNCT
ejpam-5732	263	36	{	{	PUNCT
ejpam-5732	263	37	vα	vα	X
ejpam-5732	263	38	:	:	PUNCT
ejpam-5732	263	39	α	α	PROPN
ejpam-5732	263	40	∈	∈	PROPN
ejpam-5732	263	41	λ1	λ1	PROPN
ejpam-5732	263	42	}	}	PUNCT
ejpam-5732	263	43	such	such	ADJ
ejpam-5732	263	44	that	that	SCONJ
ejpam-5732	263	45	x−∪{vα	x−∪{vα	PROPN
ejpam-5732	263	46	:	:	PUNCT
ejpam-5732	264	1	α	α	PROPN
ejpam-5732	264	2	∈	∈	PROPN
ejpam-5732	264	3	λ1	λ1	PROPN
ejpam-5732	264	4	}	}	PUNCT
ejpam-5732	264	5	∈	∈	PROPN
ejpam-5732	264	6	i.	i.	NOUN
ejpam-5732	264	7	as	as	SCONJ
ejpam-5732	264	8	f	f	PROPN
ejpam-5732	264	9	is	be	AUX
ejpam-5732	264	10	open	open	ADJ
ejpam-5732	264	11	,	,	PUNCT
ejpam-5732	264	12	f(v	f(v	NOUN
ejpam-5732	264	13	)	)	PUNCT
ejpam-5732	264	14	=	=	PRON
ejpam-5732	264	15	{	{	PUNCT
ejpam-5732	264	16	f(vα	f(vα	PROPN
ejpam-5732	264	17	)	)	PUNCT
ejpam-5732	264	18	:	:	PUNCT
ejpam-5732	264	19	α	α	PROPN
ejpam-5732	264	20	∈	∈	PROPN
ejpam-5732	264	21	λ1	λ1	PROPN
ejpam-5732	264	22	}	}	PUNCT
ejpam-5732	264	23	is	be	AUX
ejpam-5732	264	24	an	an	DET
ejpam-5732	264	25	open	open	ADJ
ejpam-5732	264	26	refinement	refinement	NOUN
ejpam-5732	264	27	of	of	ADP
ejpam-5732	264	28	u	u	PROPN
ejpam-5732	264	29	and	and	CCONJ
ejpam-5732	264	30	y	y	PROPN
ejpam-5732	264	31	−∪{f(vα	−∪{f(vα	NUM
ejpam-5732	264	32	)	)	PUNCT
ejpam-5732	264	33	:	:	PUNCT
ejpam-5732	265	1	α	α	PROPN
ejpam-5732	265	2	∈	∈	PROPN
ejpam-5732	265	3	λ1	λ1	PROPN
ejpam-5732	265	4	}	}	PUNCT
ejpam-5732	265	5	∈	∈	PROPN
ejpam-5732	265	6	j	j	PROPN
ejpam-5732	265	7	.	.	PUNCT
ejpam-5732	266	1	next	next	ADV
ejpam-5732	266	2	,	,	PUNCT
ejpam-5732	266	3	we	we	PRON
ejpam-5732	266	4	shall	shall	AUX
ejpam-5732	266	5	verify	verify	VERB
ejpam-5732	266	6	that	that	DET
ejpam-5732	266	7	f(v	f(v	NOUN
ejpam-5732	266	8	)	)	PUNCT
ejpam-5732	266	9	is	be	AUX
ejpam-5732	266	10	τ	τ	PROPN
ejpam-5732	266	11	′-locally	′-locally	ADV
ejpam-5732	266	12	finite	finite	VERB
ejpam-5732	266	13	.	.	PUNCT
ejpam-5732	267	1	let	let	VERB
ejpam-5732	267	2	y	y	PROPN
ejpam-5732	267	3	∈	∈	PROPN
ejpam-5732	267	4	y	y	PROPN
ejpam-5732	267	5	.	.	PUNCT
ejpam-5732	268	1	since	since	SCONJ
ejpam-5732	268	2	v	v	NOUN
ejpam-5732	268	3	is	be	AUX
ejpam-5732	268	4	τ	τ	PROPN
ejpam-5732	268	5	-locally	-locally	ADV
ejpam-5732	268	6	finite	finite	ADJ
ejpam-5732	268	7	,	,	PUNCT
ejpam-5732	268	8	for	for	ADP
ejpam-5732	268	9	x	x	PROPN
ejpam-5732	268	10	∈	∈	PROPN
ejpam-5732	268	11	{	{	PUNCT
ejpam-5732	268	12	f−1(y	f−1(y	PROPN
ejpam-5732	268	13	)	)	PUNCT
ejpam-5732	268	14	}	}	PUNCT
ejpam-5732	268	15	,	,	PUNCT
ejpam-5732	268	16	there	there	PRON
ejpam-5732	268	17	exists	exist	VERB
ejpam-5732	268	18	an	an	DET
ejpam-5732	268	19	open	open	ADJ
ejpam-5732	268	20	set	set	VERB
ejpam-5732	268	21	gx	gx	PROPN
ejpam-5732	268	22	containing	contain	VERB
ejpam-5732	268	23	x	x	PUNCT
ejpam-5732	268	24	such	such	ADJ
ejpam-5732	268	25	that	that	SCONJ
ejpam-5732	268	26	gx	gx	PROPN
ejpam-5732	268	27	intersects	intersect	NOUN
ejpam-5732	268	28	at	at	ADV
ejpam-5732	268	29	most	most	ADV
ejpam-5732	268	30	finitely	finitely	ADV
ejpam-5732	268	31	many	many	ADJ
ejpam-5732	268	32	members	member	NOUN
ejpam-5732	268	33	of	of	ADP
ejpam-5732	268	34	v.	v.	ADV
ejpam-5732	268	35	because	because	SCONJ
ejpam-5732	268	36	{	{	PUNCT
ejpam-5732	268	37	f−1(y	f−1(y	PROPN
ejpam-5732	268	38	)	)	PUNCT
ejpam-5732	268	39	}	}	PUNCT
ejpam-5732	268	40	is	be	AUX
ejpam-5732	268	41	δ	δ	PROPN
ejpam-5732	268	42	-	-	PUNCT
ejpam-5732	268	43	βi	βi	ADV
ejpam-5732	268	44	-	-	ADJ
ejpam-5732	268	45	compact	compact	ADJ
ejpam-5732	268	46	and	and	CCONJ
ejpam-5732	268	47	{	{	PUNCT
ejpam-5732	268	48	gx	gx	PROPN
ejpam-5732	268	49	:	:	PUNCT
ejpam-5732	268	50	f(x	f(x	PROPN
ejpam-5732	268	51	)	)	PUNCT
ejpam-5732	269	1	=	=	SYM
ejpam-5732	269	2	y	y	PROPN
ejpam-5732	269	3	}	}	PUNCT
ejpam-5732	269	4	forms	form	VERB
ejpam-5732	269	5	an	an	DET
ejpam-5732	269	6	open	open	ADJ
ejpam-5732	269	7	cover	cover	NOUN
ejpam-5732	269	8	of	of	ADP
ejpam-5732	269	9	{	{	PUNCT
ejpam-5732	269	10	f−1(y	f−1(y	PROPN
ejpam-5732	269	11	)	)	PUNCT
ejpam-5732	269	12	}	}	PUNCT
ejpam-5732	269	13	,	,	PUNCT
ejpam-5732	269	14	there	there	PRON
ejpam-5732	269	15	exists	exist	VERB
ejpam-5732	269	16	a	a	DET
ejpam-5732	269	17	finite	finite	ADJ
ejpam-5732	269	18	subcollection	subcollection	NOUN
ejpam-5732	269	19	hy	hy	PROPN
ejpam-5732	269	20	such	such	ADJ
ejpam-5732	269	21	that	that	SCONJ
ejpam-5732	269	22	{	{	PUNCT
ejpam-5732	269	23	f−1(y	f−1(y	PROPN
ejpam-5732	269	24	)	)	PUNCT
ejpam-5732	269	25	}	}	PUNCT
ejpam-5732	269	26	⊂	⊂	PROPN
ejpam-5732	269	27	∪hy	∪hy	NOUN
ejpam-5732	269	28	,	,	PUNCT
ejpam-5732	269	29	and	and	CCONJ
ejpam-5732	269	30	∪hy	∪hy	NOUN
ejpam-5732	269	31	intersects	intersect	NOUN
ejpam-5732	269	32	at	at	ADP
ejpam-5732	269	33	most	most	ADV
ejpam-5732	269	34	finitely	finitely	ADV
ejpam-5732	269	35	many	many	ADJ
ejpam-5732	269	36	members	member	NOUN
ejpam-5732	269	37	of	of	ADP
ejpam-5732	269	38	v.	v.	ADP
ejpam-5732	269	39	assuming	assume	VERB
ejpam-5732	269	40	f	f	PROPN
ejpam-5732	269	41	is	be	AUX
ejpam-5732	269	42	δ	δ	PROPN
ejpam-5732	269	43	-	-	PUNCT
ejpam-5732	269	44	βi	βi	ADV
ejpam-5732	269	45	-	-	PUNCT
ejpam-5732	269	46	closed	closed	ADJ
ejpam-5732	269	47	,	,	PUNCT
ejpam-5732	269	48	by	by	ADP
ejpam-5732	269	49	applying	apply	VERB
ejpam-5732	269	50	lemma	lemma	PROPN
ejpam-5732	269	51	8	8	NUM
ejpam-5732	269	52	,	,	PUNCT
ejpam-5732	269	53	there	there	PRON
ejpam-5732	269	54	exists	exist	VERB
ejpam-5732	269	55	a	a	DET
ejpam-5732	269	56	δ	δ	PROPN
ejpam-5732	269	57	-	-	PUNCT
ejpam-5732	269	58	βj	βj	PUNCT
ejpam-5732	269	59	-open	-open	NOUN
ejpam-5732	269	60	set	set	VERB
ejpam-5732	269	61	wy	wy	PROPN
ejpam-5732	269	62	containing	contain	VERB
ejpam-5732	269	63	y	y	PRON
ejpam-5732	269	64	such	such	ADJ
ejpam-5732	269	65	that	that	SCONJ
ejpam-5732	269	66	f−1(wy	f−1(wy	PROPN
ejpam-5732	269	67	)	)	PUNCT
ejpam-5732	269	68	⊂	⊂	PROPN
ejpam-5732	269	69	∪hy	∪hy	NOUN
ejpam-5732	269	70	.	.	PUNCT
ejpam-5732	270	1	hence	hence	ADV
ejpam-5732	270	2	,	,	PUNCT
ejpam-5732	270	3	f	f	PROPN
ejpam-5732	270	4	−1(wy	−1(wy	NOUN
ejpam-5732	270	5	)	)	PUNCT
ejpam-5732	270	6	intersects	intersect	NOUN
ejpam-5732	270	7	at	at	ADP
ejpam-5732	270	8	most	most	ADV
ejpam-5732	270	9	finitely	finitely	ADV
ejpam-5732	270	10	many	many	ADJ
ejpam-5732	270	11	members	member	NOUN
ejpam-5732	270	12	of	of	ADP
ejpam-5732	270	13	v.	v.	ADP
ejpam-5732	270	14	this	this	PRON
ejpam-5732	270	15	indicates	indicate	VERB
ejpam-5732	270	16	that	that	SCONJ
ejpam-5732	270	17	wy	wy	PROPN
ejpam-5732	270	18	intersects	intersect	NOUN
ejpam-5732	270	19	at	at	ADV
ejpam-5732	270	20	most	most	ADV
ejpam-5732	270	21	finitely	finitely	ADV
ejpam-5732	270	22	many	many	ADJ
ejpam-5732	270	23	members	member	NOUN
ejpam-5732	270	24	of	of	ADP
ejpam-5732	270	25	f(v	f(v	NOUN
ejpam-5732	270	26	)	)	PUNCT
ejpam-5732	270	27	.	.	PUNCT
ejpam-5732	271	1	consequently	consequently	ADV
ejpam-5732	271	2	,	,	PUNCT
ejpam-5732	271	3	since	since	SCONJ
ejpam-5732	271	4	f(v	f(v	NOUN
ejpam-5732	271	5	)	)	PUNCT
ejpam-5732	271	6	is	be	AUX
ejpam-5732	271	7	a	a	DET
ejpam-5732	271	8	τ	τ	PROPN
ejpam-5732	271	9	′-locally	′-locally	ADV
ejpam-5732	271	10	finite	finite	VERB
ejpam-5732	271	11	in	in	ADP
ejpam-5732	271	12	y	y	PROPN
ejpam-5732	271	13	,	,	PUNCT
ejpam-5732	271	14	it	it	PRON
ejpam-5732	271	15	implies	imply	VERB
ejpam-5732	271	16	that	that	SCONJ
ejpam-5732	271	17	(	(	PUNCT
ejpam-5732	271	18	y	y	PROPN
ejpam-5732	271	19	,	,	PUNCT
ejpam-5732	271	20	τ	τ	PROPN
ejpam-5732	271	21	′,j	′,j	NOUN
ejpam-5732	271	22	)	)	PUNCT
ejpam-5732	271	23	is	be	AUX
ejpam-5732	271	24	δ1	δ1	NOUN
ejpam-5732	271	25	-	-	PUNCT
ejpam-5732	271	26	βj	βj	NOUN
ejpam-5732	271	27	-paracompact	-paracompact	NOUN
ejpam-5732	271	28	.	.	PUNCT
ejpam-5732	272	1	the	the	DET
ejpam-5732	272	2	subsequent	subsequent	ADJ
ejpam-5732	272	3	theorem	theorem	NOUN
ejpam-5732	272	4	and	and	CCONJ
ejpam-5732	272	5	corollaries	corollary	NOUN
ejpam-5732	272	6	establish	establish	VERB
ejpam-5732	272	7	characterizations	characterization	NOUN
ejpam-5732	272	8	of	of	ADP
ejpam-5732	272	9	a	a	DET
ejpam-5732	272	10	function	function	NOUN
ejpam-5732	272	11	that	that	PRON
ejpam-5732	272	12	maps	map	VERB
ejpam-5732	272	13	from	from	ADP
ejpam-5732	272	14	a	a	DET
ejpam-5732	272	15	δ1	δ1	NOUN
ejpam-5732	272	16	-	-	PUNCT
ejpam-5732	272	17	βi	βi	PRON
ejpam-5732	272	18	-	-	PUNCT
ejpam-5732	272	19	paracompact	paracompact	ADJ
ejpam-5732	272	20	ideal	ideal	ADJ
ejpam-5732	272	21	topological	topological	ADJ
ejpam-5732	272	22	space	space	NOUN
ejpam-5732	272	23	(	(	PUNCT
ejpam-5732	272	24	x	x	X
ejpam-5732	272	25	,	,	PUNCT
ejpam-5732	272	26	τ	τ	PROPN
ejpam-5732	272	27	,	,	PUNCT
ejpam-5732	272	28	i	i	PROPN
ejpam-5732	272	29	)	)	PUNCT
ejpam-5732	272	30	to	to	ADP
ejpam-5732	272	31	a	a	DET
ejpam-5732	272	32	topological	topological	ADJ
ejpam-5732	272	33	space	space	NOUN
ejpam-5732	272	34	(	(	PUNCT
ejpam-5732	272	35	y	y	PROPN
ejpam-5732	272	36	,	,	PUNCT
ejpam-5732	272	37	τ	τ	PROPN
ejpam-5732	272	38	′	′	NUM
ejpam-5732	272	39	)	)	PUNCT
ejpam-5732	272	40	,	,	PUNCT
ejpam-5732	272	41	guaranteeing	guarantee	VERB
ejpam-5732	272	42	that	that	SCONJ
ejpam-5732	272	43	y	y	PROPN
ejpam-5732	272	44	possesses	possess	VERB
ejpam-5732	272	45	the	the	DET
ejpam-5732	272	46	same	same	ADJ
ejpam-5732	272	47	characteristics	characteristic	NOUN
ejpam-5732	272	48	as	as	SCONJ
ejpam-5732	272	49	x.	x.	NOUN
ejpam-5732	272	50	theorem	theorem	VERB
ejpam-5732	272	51	9	9	NUM
ejpam-5732	272	52	.	.	PUNCT
ejpam-5732	273	1	let	let	VERB
ejpam-5732	273	2	(	(	PUNCT
ejpam-5732	273	3	x	x	X
ejpam-5732	273	4	,	,	PUNCT
ejpam-5732	273	5	τ	τ	PROPN
ejpam-5732	273	6	,	,	PUNCT
ejpam-5732	273	7	i	i	PRON
ejpam-5732	273	8	)	)	PUNCT
ejpam-5732	273	9	be	be	VERB
ejpam-5732	273	10	an	an	DET
ejpam-5732	273	11	ideal	ideal	ADJ
ejpam-5732	273	12	topological	topological	ADJ
ejpam-5732	273	13	space	space	NOUN
ejpam-5732	273	14	and	and	CCONJ
ejpam-5732	273	15	(	(	PUNCT
ejpam-5732	273	16	y	y	PROPN
ejpam-5732	273	17	,	,	PUNCT
ejpam-5732	273	18	τ	τ	PROPN
ejpam-5732	273	19	′	′	NUM
ejpam-5732	273	20	)	)	PUNCT
ejpam-5732	273	21	be	be	AUX
ejpam-5732	273	22	a	a	DET
ejpam-5732	273	23	topological	topological	ADJ
ejpam-5732	273	24	space	space	NOUN
ejpam-5732	273	25	.	.	PUNCT
ejpam-5732	274	1	suppose	suppose	VERB
ejpam-5732	274	2	that	that	SCONJ
ejpam-5732	274	3	f	f	X
ejpam-5732	274	4	:	:	PUNCT
ejpam-5732	274	5	x	x	X
ejpam-5732	274	6	→	→	SYM
ejpam-5732	274	7	y	y	PROPN
ejpam-5732	274	8	satisfies	satisfy	VERB
ejpam-5732	274	9	the	the	DET
ejpam-5732	274	10	following	following	ADJ
ejpam-5732	274	11	statements	statement	NOUN
ejpam-5732	274	12	:	:	PUNCT
ejpam-5732	274	13	(	(	PUNCT
ejpam-5732	274	14	i	i	NOUN
ejpam-5732	274	15	)	)	PUNCT
ejpam-5732	274	16	f	f	PROPN
ejpam-5732	274	17	is	be	AUX
ejpam-5732	274	18	open	open	ADJ
ejpam-5732	274	19	;	;	PUNCT
ejpam-5732	274	20	(	(	PUNCT
ejpam-5732	274	21	ii	ii	NOUN
ejpam-5732	274	22	)	)	PUNCT
ejpam-5732	274	23	f	f	PROPN
ejpam-5732	274	24	is	be	AUX
ejpam-5732	274	25	δ	δ	PROPN
ejpam-5732	274	26	-	-	PUNCT
ejpam-5732	274	27	βi	βi	ADV
ejpam-5732	274	28	-	-	PUNCT
ejpam-5732	274	29	irresolute	irresolute	ADJ
ejpam-5732	274	30	;	;	PUNCT
ejpam-5732	274	31	(	(	PUNCT
ejpam-5732	274	32	iii	iii	X
ejpam-5732	274	33	)	)	PUNCT
ejpam-5732	274	34	f	f	PROPN
ejpam-5732	274	35	is	be	AUX
ejpam-5732	274	36	almost	almost	ADV
ejpam-5732	274	37	closed	closed	ADJ
ejpam-5732	274	38	;	;	PUNCT
ejpam-5732	274	39	and	and	CCONJ
ejpam-5732	274	40	c.	c.	PROPN
ejpam-5732	274	41	boonpok	boonpok	PROPN
ejpam-5732	274	42	,	,	PUNCT
ejpam-5732	274	43	a.	a.	PROPN
ejpam-5732	274	44	sama	sama	PROPN
ejpam-5732	274	45	-	-	PUNCT
ejpam-5732	274	46	ae	ae	PROPN
ejpam-5732	274	47	/	/	SYM
ejpam-5732	274	48	eur	eur	PROPN
ejpam-5732	274	49	.	.	PUNCT
ejpam-5732	275	1	j.	j.	PROPN
ejpam-5732	275	2	pure	pure	PROPN
ejpam-5732	275	3	appl	appl	PROPN
ejpam-5732	275	4	.	.	PROPN
ejpam-5732	275	5	math	math	PROPN
ejpam-5732	275	6	,	,	PUNCT
ejpam-5732	275	7	18	18	NUM
ejpam-5732	275	8	(	(	PUNCT
ejpam-5732	275	9	1	1	NUM
ejpam-5732	275	10	)	)	PUNCT
ejpam-5732	275	11	(	(	PUNCT
ejpam-5732	275	12	2025	2025	NUM
ejpam-5732	275	13	)	)	PUNCT
ejpam-5732	275	14	,	,	PUNCT
ejpam-5732	275	15	5732	5732	NUM
ejpam-5732	275	16	10	10	NUM
ejpam-5732	275	17	of	of	ADP
ejpam-5732	275	18	13	13	NUM
ejpam-5732	275	19	(	(	PUNCT
ejpam-5732	275	20	iv	iv	X
ejpam-5732	275	21	)	)	PUNCT
ejpam-5732	275	22	f	f	PROPN
ejpam-5732	275	23	is	be	AUX
ejpam-5732	275	24	a	a	DET
ejpam-5732	275	25	surjective	surjective	ADJ
ejpam-5732	275	26	function	function	NOUN
ejpam-5732	275	27	with	with	ADP
ejpam-5732	275	28	n	n	ADV
ejpam-5732	275	29	-closed	-closed	ADJ
ejpam-5732	275	30	point	point	NOUN
ejpam-5732	275	31	inverse	inverse	NOUN
ejpam-5732	275	32	.	.	PUNCT
ejpam-5732	276	1	if	if	SCONJ
ejpam-5732	276	2	(	(	PUNCT
ejpam-5732	276	3	x	x	X
ejpam-5732	276	4	,	,	PUNCT
ejpam-5732	276	5	τ	τ	PROPN
ejpam-5732	276	6	,	,	PUNCT
ejpam-5732	276	7	i	i	PROPN
ejpam-5732	276	8	)	)	PUNCT
ejpam-5732	276	9	is	be	AUX
ejpam-5732	276	10	δ1	δ1	NOUN
ejpam-5732	276	11	-	-	PUNCT
ejpam-5732	276	12	βi	βi	PRON
ejpam-5732	276	13	-	-	PUNCT
ejpam-5732	276	14	paracompact	paracompact	ADJ
ejpam-5732	276	15	,	,	PUNCT
ejpam-5732	276	16	then	then	ADV
ejpam-5732	276	17	(	(	PUNCT
ejpam-5732	276	18	y	y	PROPN
ejpam-5732	276	19	,	,	PUNCT
ejpam-5732	276	20	τ	τ	PROPN
ejpam-5732	276	21	′	′	NUM
ejpam-5732	276	22	,	,	PUNCT
ejpam-5732	276	23	f(i	f(i	PROPN
ejpam-5732	276	24	)	)	PUNCT
ejpam-5732	276	25	)	)	PUNCT
ejpam-5732	276	26	is	be	AUX
ejpam-5732	276	27	δ1	δ1	NOUN
ejpam-5732	276	28	-	-	PUNCT
ejpam-5732	276	29	βf(i)-paracompact	βf(i)-paracompact	PROPN
ejpam-5732	276	30	.	.	PUNCT
ejpam-5732	277	1	proof	proof	NOUN
ejpam-5732	277	2	.	.	PUNCT
ejpam-5732	278	1	as	as	ADP
ejpam-5732	278	2	f	f	PROPN
ejpam-5732	278	3	:	:	PUNCT
ejpam-5732	278	4	x	x	X
ejpam-5732	278	5	→	→	SYM
ejpam-5732	278	6	y	y	PROPN
ejpam-5732	278	7	is	be	AUX
ejpam-5732	278	8	surjective	surjective	ADJ
ejpam-5732	278	9	,	,	PUNCT
ejpam-5732	278	10	f(i	f(i	NUM
ejpam-5732	278	11	)	)	PUNCT
ejpam-5732	278	12	is	be	AUX
ejpam-5732	278	13	an	an	DET
ejpam-5732	278	14	ideal	ideal	NOUN
ejpam-5732	278	15	on	on	ADP
ejpam-5732	278	16	y	y	PROPN
ejpam-5732	278	17	.	.	PUNCT
ejpam-5732	279	1	let	let	VERB
ejpam-5732	279	2	u	u	PRON
ejpam-5732	279	3	=	=	PUNCT
ejpam-5732	279	4	{	{	PUNCT
ejpam-5732	279	5	uλ	uλ	X
ejpam-5732	279	6	:	:	PUNCT
ejpam-5732	279	7	λ	λ	X
ejpam-5732	279	8	∈	∈	PROPN
ejpam-5732	279	9	λ	λ	PROPN
ejpam-5732	279	10	}	}	PUNCT
ejpam-5732	279	11	be	be	VERB
ejpam-5732	279	12	a	a	DET
ejpam-5732	279	13	δ	δ	NOUN
ejpam-5732	279	14	-	-	PUNCT
ejpam-5732	279	15	βf(i)-open	βf(i)-open	ADJ
ejpam-5732	279	16	cover	cover	NOUN
ejpam-5732	279	17	of	of	ADP
ejpam-5732	279	18	y	y	PROPN
ejpam-5732	279	19	.	.	PUNCT
ejpam-5732	280	1	as	as	SCONJ
ejpam-5732	280	2	f	f	PROPN
ejpam-5732	280	3	is	be	AUX
ejpam-5732	280	4	δ	δ	PROPN
ejpam-5732	280	5	-	-	PUNCT
ejpam-5732	280	6	βi	βi	ADV
ejpam-5732	280	7	-	-	PUNCT
ejpam-5732	280	8	irresolute	irresolute	ADJ
ejpam-5732	280	9	,	,	PUNCT
ejpam-5732	280	10	h	h	NOUN
ejpam-5732	280	11	=	=	PRON
ejpam-5732	280	12	{	{	PUNCT
ejpam-5732	280	13	f−1(uλ	f−1(uλ	PROPN
ejpam-5732	280	14	)	)	PUNCT
ejpam-5732	280	15	:	:	PUNCT
ejpam-5732	281	1	λ	λ	X
ejpam-5732	281	2	∈	∈	PROPN
ejpam-5732	281	3	λ	λ	PROPN
ejpam-5732	281	4	}	}	PUNCT
ejpam-5732	281	5	is	be	AUX
ejpam-5732	281	6	a	a	DET
ejpam-5732	281	7	δ	δ	PROPN
ejpam-5732	281	8	-	-	PUNCT
ejpam-5732	281	9	βi	βi	ADV
ejpam-5732	281	10	-	-	PUNCT
ejpam-5732	281	11	open	open	ADJ
ejpam-5732	281	12	cover	cover	NOUN
ejpam-5732	281	13	of	of	ADP
ejpam-5732	281	14	x.	x.	NOUN
ejpam-5732	281	15	since	since	SCONJ
ejpam-5732	281	16	x	x	PROPN
ejpam-5732	281	17	is	be	AUX
ejpam-5732	281	18	δ1	δ1	NOUN
ejpam-5732	281	19	-	-	PUNCT
ejpam-5732	281	20	βi	βi	PRON
ejpam-5732	281	21	-	-	PUNCT
ejpam-5732	281	22	paracompact	paracompact	ADJ
ejpam-5732	281	23	,	,	PUNCT
ejpam-5732	281	24	h	h	NOUN
ejpam-5732	281	25	has	have	VERB
ejpam-5732	281	26	a	a	DET
ejpam-5732	281	27	locally	locally	ADV
ejpam-5732	281	28	finite	finite	ADJ
ejpam-5732	281	29	open	open	ADJ
ejpam-5732	281	30	refinement	refinement	NOUN
ejpam-5732	281	31	h1	h1	PROPN
ejpam-5732	281	32	=	=	PRON
ejpam-5732	281	33	{	{	PUNCT
ejpam-5732	281	34	hα	hα	X
ejpam-5732	281	35	:	:	PUNCT
ejpam-5732	281	36	α	α	PROPN
ejpam-5732	281	37	∈	∈	PROPN
ejpam-5732	281	38	λ1	λ1	PROPN
ejpam-5732	281	39	}	}	PUNCT
ejpam-5732	281	40	such	such	ADJ
ejpam-5732	281	41	that	that	SCONJ
ejpam-5732	281	42	x	x	PRON
ejpam-5732	281	43	−∪{hα	−∪{hα	NOUN
ejpam-5732	281	44	:	:	PUNCT
ejpam-5732	281	45	α	α	PROPN
ejpam-5732	281	46	∈	∈	PROPN
ejpam-5732	281	47	λ1	λ1	PROPN
ejpam-5732	281	48	}	}	PUNCT
ejpam-5732	281	49	∈	∈	PROPN
ejpam-5732	281	50	i.	i.	NOUN
ejpam-5732	281	51	thus	thus	ADV
ejpam-5732	281	52	,	,	PUNCT
ejpam-5732	281	53	f(x	f(x	PROPN
ejpam-5732	281	54	−∪{hα	−∪{hα	PROPN
ejpam-5732	281	55	:	:	PUNCT
ejpam-5732	281	56	α	α	PROPN
ejpam-5732	281	57	∈	∈	PROPN
ejpam-5732	281	58	λ1	λ1	PROPN
ejpam-5732	281	59	}	}	PUNCT
ejpam-5732	281	60	)	)	PUNCT
ejpam-5732	281	61	∈	∈	PROPN
ejpam-5732	281	62	f(i	f(i	PROPN
ejpam-5732	281	63	)	)	PUNCT
ejpam-5732	281	64	.	.	PUNCT
ejpam-5732	282	1	we	we	PRON
ejpam-5732	282	2	know	know	VERB
ejpam-5732	282	3	that	that	SCONJ
ejpam-5732	282	4	y	y	PROPN
ejpam-5732	282	5	−	−	PROPN
ejpam-5732	282	6	∪{f(hα	∪{f(hα	NUM
ejpam-5732	282	7	)	)	PUNCT
ejpam-5732	282	8	:	:	PUNCT
ejpam-5732	283	1	α	α	PROPN
ejpam-5732	283	2	∈	∈	PROPN
ejpam-5732	283	3	λ1	λ1	PROPN
ejpam-5732	283	4	}	}	PUNCT
ejpam-5732	283	5	⊂	⊂	PROPN
ejpam-5732	283	6	f(x	f(x	PROPN
ejpam-5732	283	7	−	−	PROPN
ejpam-5732	284	1	∪{hα	∪{hα	PROPN
ejpam-5732	284	2	:	:	PUNCT
ejpam-5732	284	3	α	α	PROPN
ejpam-5732	284	4	∈	∈	PROPN
ejpam-5732	284	5	λ1	λ1	PROPN
ejpam-5732	284	6	}	}	PUNCT
ejpam-5732	284	7	)	)	PUNCT
ejpam-5732	284	8	,	,	PUNCT
ejpam-5732	284	9	we	we	PRON
ejpam-5732	284	10	therefore	therefore	ADV
ejpam-5732	284	11	have	have	VERB
ejpam-5732	284	12	that	that	PRON
ejpam-5732	284	13	y	y	PROPN
ejpam-5732	284	14	−	−	PROPN
ejpam-5732	284	15	∪{f(hα	∪{f(hα	NUM
ejpam-5732	284	16	)	)	PUNCT
ejpam-5732	284	17	:	:	PUNCT
ejpam-5732	285	1	α	α	PROPN
ejpam-5732	285	2	∈	∈	PROPN
ejpam-5732	285	3	λ1	λ1	PROPN
ejpam-5732	285	4	}	}	PUNCT
ejpam-5732	285	5	∈	∈	PROPN
ejpam-5732	285	6	f(i	f(i	PROPN
ejpam-5732	285	7	)	)	PUNCT
ejpam-5732	285	8	.	.	PUNCT
ejpam-5732	286	1	since	since	SCONJ
ejpam-5732	286	2	f	f	PROPN
ejpam-5732	286	3	is	be	AUX
ejpam-5732	286	4	open	open	ADJ
ejpam-5732	286	5	,	,	PUNCT
ejpam-5732	286	6	almost	almost	ADV
ejpam-5732	286	7	closed	closed	ADJ
ejpam-5732	286	8	,	,	PUNCT
ejpam-5732	286	9	surjective	surjective	ADJ
ejpam-5732	286	10	with	with	ADP
ejpam-5732	286	11	n	n	ADV
ejpam-5732	286	12	-closed	-closed	ADJ
ejpam-5732	286	13	point	point	NOUN
ejpam-5732	286	14	inverse	inverse	NOUN
ejpam-5732	286	15	,	,	PUNCT
ejpam-5732	286	16	andh1	andh1	NOUN
ejpam-5732	286	17	is	be	AUX
ejpam-5732	286	18	locally	locally	ADV
ejpam-5732	286	19	finite	finite	ADJ
ejpam-5732	286	20	,	,	PUNCT
ejpam-5732	286	21	f(h1	f(h1	NOUN
ejpam-5732	286	22	)	)	PUNCT
ejpam-5732	286	23	=	=	PRON
ejpam-5732	286	24	{	{	PUNCT
ejpam-5732	286	25	f(hα	f(hα	PROPN
ejpam-5732	286	26	)	)	PUNCT
ejpam-5732	286	27	:	:	PUNCT
ejpam-5732	287	1	α	α	PROPN
ejpam-5732	287	2	∈	∈	PROPN
ejpam-5732	287	3	λ1	λ1	PROPN
ejpam-5732	287	4	}	}	PUNCT
ejpam-5732	287	5	is	be	AUX
ejpam-5732	287	6	locally	locally	ADV
ejpam-5732	287	7	finite	finite	ADJ
ejpam-5732	287	8	by	by	ADP
ejpam-5732	287	9	lemma	lemma	PROPN
ejpam-5732	287	10	6	6	NUM
ejpam-5732	287	11	.	.	PUNCT
ejpam-5732	288	1	next	next	ADV
ejpam-5732	288	2	,	,	PUNCT
ejpam-5732	288	3	we	we	PRON
ejpam-5732	288	4	shall	shall	AUX
ejpam-5732	288	5	verify	verify	VERB
ejpam-5732	288	6	that	that	DET
ejpam-5732	288	7	f(h1	f(h1	NOUN
ejpam-5732	288	8	)	)	PUNCT
ejpam-5732	288	9	refines	refine	VERB
ejpam-5732	288	10	u	u	PRON
ejpam-5732	288	11	.	.	PUNCT
ejpam-5732	289	1	let	let	VERB
ejpam-5732	289	2	f(hα	f(hα	NOUN
ejpam-5732	289	3	)	)	PUNCT
ejpam-5732	289	4	∈	∈	PROPN
ejpam-5732	289	5	f(h1	f(h1	NOUN
ejpam-5732	289	6	)	)	PUNCT
ejpam-5732	289	7	.	.	PUNCT
ejpam-5732	290	1	then	then	ADV
ejpam-5732	290	2	hα	hα	ADP
ejpam-5732	290	3	∈	∈	PROPN
ejpam-5732	290	4	h1	h1	PROPN
ejpam-5732	290	5	.	.	PUNCT
ejpam-5732	291	1	as	as	SCONJ
ejpam-5732	291	2	h1	h1	PROPN
ejpam-5732	291	3	refines	refine	VERB
ejpam-5732	291	4	h	h	NOUN
ejpam-5732	291	5	,	,	PUNCT
ejpam-5732	291	6	there	there	PRON
ejpam-5732	291	7	exists	exist	VERB
ejpam-5732	291	8	f−1(uλ	f−1(uλ	PROPN
ejpam-5732	291	9	)	)	PUNCT
ejpam-5732	291	10	∈	∈	PROPN
ejpam-5732	291	11	h	h	NOUN
ejpam-5732	291	12	such	such	ADJ
ejpam-5732	291	13	that	that	SCONJ
ejpam-5732	291	14	hα	hα	ADP
ejpam-5732	291	15	⊂	⊂	PROPN
ejpam-5732	291	16	f−1(uλ	f−1(uλ	PROPN
ejpam-5732	291	17	)	)	PUNCT
ejpam-5732	291	18	for	for	ADP
ejpam-5732	291	19	some	some	DET
ejpam-5732	291	20	λ	λ	PROPN
ejpam-5732	291	21	∈	∈	PROPN
ejpam-5732	291	22	λ	λ	PROPN
ejpam-5732	291	23	.	.	PUNCT
ejpam-5732	291	24	therefore	therefore	ADV
ejpam-5732	291	25	,	,	PUNCT
ejpam-5732	291	26	f(hα	f(hα	PROPN
ejpam-5732	291	27	)	)	PUNCT
ejpam-5732	291	28	⊂	⊂	PROPN
ejpam-5732	291	29	f(f−1uλ	f(f−1uλ	NOUN
ejpam-5732	291	30	)	)	PUNCT
ejpam-5732	291	31	)	)	PUNCT
ejpam-5732	292	1	⊂	⊂	PROPN
ejpam-5732	292	2	uλ	uλ	PROPN
ejpam-5732	292	3	.	.	PUNCT
ejpam-5732	293	1	this	this	PRON
ejpam-5732	293	2	shows	show	VERB
ejpam-5732	293	3	that	that	SCONJ
ejpam-5732	293	4	(	(	PUNCT
ejpam-5732	293	5	y	y	PROPN
ejpam-5732	293	6	,	,	PUNCT
ejpam-5732	293	7	τ	τ	PROPN
ejpam-5732	293	8	′	′	NUM
ejpam-5732	293	9	,	,	PUNCT
ejpam-5732	293	10	f(i	f(i	PROPN
ejpam-5732	293	11	)	)	PUNCT
ejpam-5732	293	12	)	)	PUNCT
ejpam-5732	293	13	is	be	AUX
ejpam-5732	293	14	δ1	δ1	NOUN
ejpam-5732	293	15	-	-	PUNCT
ejpam-5732	293	16	βf(i)-paracompact	βf(i)-paracompact	PROPN
ejpam-5732	293	17	.	.	PUNCT
ejpam-5732	294	1	as	as	SCONJ
ejpam-5732	294	2	any	any	DET
ejpam-5732	294	3	compact	compact	ADJ
ejpam-5732	294	4	set	set	NOUN
ejpam-5732	294	5	is	be	AUX
ejpam-5732	294	6	a	a	DET
ejpam-5732	294	7	n	n	ADV
ejpam-5732	294	8	-closed	-close	VERB
ejpam-5732	294	9	set	set	NOUN
ejpam-5732	294	10	and	and	CCONJ
ejpam-5732	294	11	any	any	DET
ejpam-5732	294	12	closed	closed	ADJ
ejpam-5732	294	13	map	map	NOUN
ejpam-5732	294	14	is	be	AUX
ejpam-5732	294	15	an	an	DET
ejpam-5732	294	16	almost	almost	ADV
ejpam-5732	294	17	closed	close	VERB
ejpam-5732	294	18	map	map	NOUN
ejpam-5732	294	19	,	,	PUNCT
ejpam-5732	294	20	by	by	ADP
ejpam-5732	294	21	theorem	theorem	NOUN
ejpam-5732	294	22	9	9	NUM
ejpam-5732	294	23	,	,	PUNCT
ejpam-5732	294	24	we	we	PRON
ejpam-5732	294	25	have	have	VERB
ejpam-5732	294	26	the	the	DET
ejpam-5732	294	27	following	follow	VERB
ejpam-5732	294	28	corollary	corollary	NOUN
ejpam-5732	294	29	.	.	PUNCT
ejpam-5732	295	1	corollary	corollary	ADJ
ejpam-5732	295	2	2	2	NUM
ejpam-5732	295	3	.	.	PUNCT
ejpam-5732	296	1	let	let	VERB
ejpam-5732	296	2	a	a	DET
ejpam-5732	296	3	function	function	NOUN
ejpam-5732	296	4	f	f	NOUN
ejpam-5732	296	5	:	:	PUNCT
ejpam-5732	296	6	(	(	PUNCT
ejpam-5732	296	7	x	x	X
ejpam-5732	296	8	,	,	PUNCT
ejpam-5732	296	9	τ	τ	PROPN
ejpam-5732	296	10	,	,	PUNCT
ejpam-5732	296	11	i	i	NOUN
ejpam-5732	296	12	)	)	PUNCT
ejpam-5732	296	13	→	→	SYM
ejpam-5732	296	14	(	(	PUNCT
ejpam-5732	296	15	y	y	PROPN
ejpam-5732	296	16	,	,	PUNCT
ejpam-5732	296	17	τ	τ	PROPN
ejpam-5732	296	18	′	′	NUM
ejpam-5732	296	19	)	)	PUNCT
ejpam-5732	296	20	be	be	AUX
ejpam-5732	296	21	open	open	ADJ
ejpam-5732	296	22	,	,	PUNCT
ejpam-5732	296	23	δ	δ	PROPN
ejpam-5732	296	24	-	-	PUNCT
ejpam-5732	296	25	βi	βi	PRON
ejpam-5732	296	26	-	-	PUNCT
ejpam-5732	296	27	irresolute	irresolute	ADJ
ejpam-5732	296	28	,	,	PUNCT
ejpam-5732	296	29	closed	closed	ADJ
ejpam-5732	296	30	,	,	PUNCT
ejpam-5732	296	31	and	and	CCONJ
ejpam-5732	296	32	surjective	surjective	VERB
ejpam-5732	296	33	with	with	ADP
ejpam-5732	296	34	compact	compact	ADJ
ejpam-5732	296	35	point	point	NOUN
ejpam-5732	296	36	inverse	inverse	NOUN
ejpam-5732	296	37	.	.	PUNCT
ejpam-5732	297	1	if	if	SCONJ
ejpam-5732	297	2	(	(	PUNCT
ejpam-5732	297	3	x	x	X
ejpam-5732	297	4	,	,	PUNCT
ejpam-5732	297	5	τ	τ	PROPN
ejpam-5732	297	6	,	,	PUNCT
ejpam-5732	297	7	i	i	PROPN
ejpam-5732	297	8	)	)	PUNCT
ejpam-5732	297	9	is	be	AUX
ejpam-5732	297	10	δ1	δ1	NOUN
ejpam-5732	297	11	-	-	PUNCT
ejpam-5732	297	12	βi	βi	PRON
ejpam-5732	297	13	-	-	PUNCT
ejpam-5732	297	14	paracompact	paracompact	ADJ
ejpam-5732	297	15	,	,	PUNCT
ejpam-5732	297	16	then	then	ADV
ejpam-5732	297	17	(	(	PUNCT
ejpam-5732	297	18	y	y	PROPN
ejpam-5732	297	19	,	,	PUNCT
ejpam-5732	297	20	τ	τ	PROPN
ejpam-5732	297	21	′	′	NUM
ejpam-5732	297	22	,	,	PUNCT
ejpam-5732	297	23	f(i	f(i	PROPN
ejpam-5732	297	24	)	)	PUNCT
ejpam-5732	297	25	)	)	PUNCT
ejpam-5732	297	26	is	be	AUX
ejpam-5732	297	27	δ1	δ1	NOUN
ejpam-5732	297	28	-	-	PUNCT
ejpam-5732	297	29	βf(i)-paracompact	βf(i)-paracompact	PROPN
ejpam-5732	297	30	.	.	PUNCT
ejpam-5732	298	1	by	by	ADP
ejpam-5732	298	2	observing	observe	VERB
ejpam-5732	298	3	the	the	DET
ejpam-5732	298	4	proof	proof	NOUN
ejpam-5732	298	5	of	of	ADP
ejpam-5732	298	6	theorem	theorem	NOUN
ejpam-5732	298	7	9	9	NUM
ejpam-5732	298	8	,	,	PUNCT
ejpam-5732	298	9	we	we	PRON
ejpam-5732	298	10	will	will	AUX
ejpam-5732	298	11	obtain	obtain	VERB
ejpam-5732	298	12	the	the	DET
ejpam-5732	298	13	following	follow	VERB
ejpam-5732	298	14	corollary	corollary	NOUN
ejpam-5732	298	15	.	.	PUNCT
ejpam-5732	299	1	corollary	corollary	ADJ
ejpam-5732	299	2	3	3	NUM
ejpam-5732	299	3	.	.	PUNCT
ejpam-5732	300	1	let	let	VERB
ejpam-5732	300	2	a	a	DET
ejpam-5732	300	3	function	function	NOUN
ejpam-5732	300	4	f	f	NOUN
ejpam-5732	300	5	:	:	PUNCT
ejpam-5732	300	6	(	(	PUNCT
ejpam-5732	300	7	x	x	X
ejpam-5732	300	8	,	,	PUNCT
ejpam-5732	300	9	τ	τ	PROPN
ejpam-5732	300	10	,	,	PUNCT
ejpam-5732	300	11	i	i	NOUN
ejpam-5732	300	12	)	)	PUNCT
ejpam-5732	300	13	→	→	SYM
ejpam-5732	300	14	(	(	PUNCT
ejpam-5732	300	15	y	y	PROPN
ejpam-5732	300	16	,	,	PUNCT
ejpam-5732	300	17	τ	τ	PROPN
ejpam-5732	300	18	′	′	NUM
ejpam-5732	300	19	)	)	PUNCT
ejpam-5732	300	20	be	be	AUX
ejpam-5732	300	21	open	open	ADJ
ejpam-5732	300	22	,	,	PUNCT
ejpam-5732	300	23	δ	δ	PROPN
ejpam-5732	300	24	-	-	PUNCT
ejpam-5732	300	25	βi	βi	PRON
ejpam-5732	300	26	-	-	PUNCT
ejpam-5732	300	27	irresolute	irresolute	ADJ
ejpam-5732	300	28	,	,	PUNCT
ejpam-5732	300	29	almost	almost	ADV
ejpam-5732	300	30	closed	closed	ADJ
ejpam-5732	300	31	,	,	PUNCT
ejpam-5732	300	32	surjective	surjective	ADJ
ejpam-5732	300	33	,	,	PUNCT
ejpam-5732	300	34	and	and	CCONJ
ejpam-5732	300	35	f(v	f(v	NOUN
ejpam-5732	300	36	)	)	PUNCT
ejpam-5732	300	37	is	be	AUX
ejpam-5732	300	38	a	a	DET
ejpam-5732	300	39	τ	τ	PROPN
ejpam-5732	300	40	′-locally	′-locally	ADV
ejpam-5732	300	41	finite	finite	VERB
ejpam-5732	300	42	in	in	ADP
ejpam-5732	300	43	y	y	PROPN
ejpam-5732	300	44	for	for	ADP
ejpam-5732	300	45	every	every	DET
ejpam-5732	300	46	τ	τ	PROPN
ejpam-5732	300	47	-locally	-locally	ADV
ejpam-5732	300	48	finite	finite	ADJ
ejpam-5732	300	49	v	v	NOUN
ejpam-5732	300	50	in	in	ADP
ejpam-5732	300	51	x.	x.	NOUN
ejpam-5732	301	1	if	if	SCONJ
ejpam-5732	301	2	(	(	PUNCT
ejpam-5732	301	3	x	x	NOUN
ejpam-5732	301	4	,	,	PUNCT
ejpam-5732	301	5	τ	τ	PROPN
ejpam-5732	301	6	,	,	PUNCT
ejpam-5732	301	7	i	i	PROPN
ejpam-5732	301	8	)	)	PUNCT
ejpam-5732	301	9	is	be	AUX
ejpam-5732	301	10	δ1	δ1	NOUN
ejpam-5732	301	11	-	-	PUNCT
ejpam-5732	301	12	βi	βi	PRON
ejpam-5732	301	13	-	-	PUNCT
ejpam-5732	301	14	paracompact	paracompact	ADJ
ejpam-5732	301	15	,	,	PUNCT
ejpam-5732	301	16	then	then	ADV
ejpam-5732	301	17	(	(	PUNCT
ejpam-5732	301	18	y	y	PROPN
ejpam-5732	301	19	,	,	PUNCT
ejpam-5732	301	20	τ	τ	PROPN
ejpam-5732	301	21	′	′	NUM
ejpam-5732	301	22	,	,	PUNCT
ejpam-5732	301	23	f(i	f(i	PROPN
ejpam-5732	301	24	)	)	PUNCT
ejpam-5732	301	25	)	)	PUNCT
ejpam-5732	301	26	is	be	AUX
ejpam-5732	301	27	δ1	δ1	NOUN
ejpam-5732	301	28	-	-	PUNCT
ejpam-5732	301	29	βf(i)-paracompact	βf(i)-paracompact	PROPN
ejpam-5732	301	30	.	.	PUNCT
ejpam-5732	302	1	the	the	DET
ejpam-5732	302	2	following	follow	VERB
ejpam-5732	302	3	theorem	theorem	NOUN
ejpam-5732	302	4	provides	provide	VERB
ejpam-5732	302	5	properties	property	NOUN
ejpam-5732	302	6	of	of	ADP
ejpam-5732	302	7	a	a	DET
ejpam-5732	302	8	function	function	NOUN
ejpam-5732	302	9	that	that	PRON
ejpam-5732	302	10	maps	map	VERB
ejpam-5732	302	11	from	from	ADP
ejpam-5732	302	12	a	a	DET
ejpam-5732	302	13	topological	topological	ADJ
ejpam-5732	302	14	space	space	NOUN
ejpam-5732	302	15	x	x	PUNCT
ejpam-5732	302	16	to	to	ADP
ejpam-5732	302	17	a	a	DET
ejpam-5732	302	18	δ1	δ1	VERB
ejpam-5732	302	19	-	-	PUNCT
ejpam-5732	302	20	βi	βi	PRON
ejpam-5732	302	21	-	-	PUNCT
ejpam-5732	302	22	paracompact	paracompact	ADJ
ejpam-5732	302	23	ideal	ideal	ADJ
ejpam-5732	302	24	topological	topological	ADJ
ejpam-5732	302	25	space	space	NOUN
ejpam-5732	302	26	y	y	PROPN
ejpam-5732	302	27	guarantees	guarantee	VERB
ejpam-5732	302	28	that	that	SCONJ
ejpam-5732	302	29	x	x	PRON
ejpam-5732	302	30	exhibits	exhibit	VERB
ejpam-5732	302	31	identical	identical	ADJ
ejpam-5732	302	32	characteristics	characteristic	NOUN
ejpam-5732	302	33	to	to	ADP
ejpam-5732	302	34	y	y	PROPN
ejpam-5732	302	35	.	.	PUNCT
ejpam-5732	303	1	theorem	theorem	ADJ
ejpam-5732	303	2	10	10	NUM
ejpam-5732	303	3	.	.	PUNCT
ejpam-5732	304	1	let	let	VERB
ejpam-5732	304	2	(	(	PUNCT
ejpam-5732	304	3	x	x	NOUN
ejpam-5732	304	4	,	,	PUNCT
ejpam-5732	304	5	τ	τ	X
ejpam-5732	304	6	)	)	PUNCT
ejpam-5732	304	7	be	be	VERB
ejpam-5732	304	8	a	a	DET
ejpam-5732	304	9	topological	topological	ADJ
ejpam-5732	304	10	space	space	NOUN
ejpam-5732	304	11	and	and	CCONJ
ejpam-5732	304	12	(	(	PUNCT
ejpam-5732	304	13	y	y	PROPN
ejpam-5732	304	14	,	,	PUNCT
ejpam-5732	304	15	τ	τ	PROPN
ejpam-5732	304	16	′,j	′,j	NOUN
ejpam-5732	304	17	)	)	PUNCT
ejpam-5732	304	18	be	be	AUX
ejpam-5732	304	19	an	an	DET
ejpam-5732	304	20	ideal	ideal	ADJ
ejpam-5732	304	21	topological	topological	ADJ
ejpam-5732	304	22	space	space	NOUN
ejpam-5732	304	23	.	.	PUNCT
ejpam-5732	305	1	suppose	suppose	VERB
ejpam-5732	305	2	that	that	SCONJ
ejpam-5732	305	3	f	f	X
ejpam-5732	305	4	:	:	PUNCT
ejpam-5732	305	5	x	x	X
ejpam-5732	305	6	→	→	SYM
ejpam-5732	305	7	y	y	PROPN
ejpam-5732	305	8	satisfies	satisfy	VERB
ejpam-5732	305	9	the	the	DET
ejpam-5732	305	10	following	following	ADJ
ejpam-5732	305	11	statements	statement	NOUN
ejpam-5732	305	12	:	:	PUNCT
ejpam-5732	305	13	(	(	PUNCT
ejpam-5732	305	14	i	i	NOUN
ejpam-5732	305	15	)	)	PUNCT
ejpam-5732	305	16	f	f	PROPN
ejpam-5732	305	17	is	be	AUX
ejpam-5732	305	18	δ	δ	PROPN
ejpam-5732	305	19	-	-	PUNCT
ejpam-5732	305	20	βi	βi	ADV
ejpam-5732	305	21	-	-	ADJ
ejpam-5732	305	22	open	open	ADJ
ejpam-5732	305	23	;	;	PUNCT
ejpam-5732	305	24	(	(	PUNCT
ejpam-5732	305	25	ii	ii	NOUN
ejpam-5732	305	26	)	)	PUNCT
ejpam-5732	305	27	f	f	PROPN
ejpam-5732	305	28	is	be	AUX
ejpam-5732	305	29	continuous	continuous	ADJ
ejpam-5732	305	30	;	;	PUNCT
ejpam-5732	305	31	and	and	CCONJ
ejpam-5732	305	32	(	(	PUNCT
ejpam-5732	305	33	iii	iii	X
ejpam-5732	305	34	)	)	PUNCT
ejpam-5732	305	35	f	f	PROPN
ejpam-5732	305	36	is	be	AUX
ejpam-5732	305	37	bijective	bijective	ADJ
ejpam-5732	305	38	.	.	PUNCT
ejpam-5732	306	1	if	if	SCONJ
ejpam-5732	306	2	(	(	PUNCT
ejpam-5732	306	3	y	y	PROPN
ejpam-5732	306	4	,	,	PUNCT
ejpam-5732	306	5	τ	τ	PROPN
ejpam-5732	306	6	′,j	′,j	NOUN
ejpam-5732	306	7	)	)	PUNCT
ejpam-5732	306	8	is	be	AUX
ejpam-5732	306	9	δ1	δ1	NOUN
ejpam-5732	306	10	-	-	PUNCT
ejpam-5732	306	11	βj	βj	NOUN
ejpam-5732	306	12	-paracompact	-paracompact	NOUN
ejpam-5732	306	13	,	,	PUNCT
ejpam-5732	306	14	then	then	ADV
ejpam-5732	306	15	(	(	PUNCT
ejpam-5732	306	16	x	x	X
ejpam-5732	306	17	,	,	PUNCT
ejpam-5732	306	18	τ	τ	PROPN
ejpam-5732	306	19	,	,	PUNCT
ejpam-5732	306	20	f−1(j	f−1(j	NOUN
ejpam-5732	306	21	)	)	PUNCT
ejpam-5732	306	22	)	)	PUNCT
ejpam-5732	306	23	is	be	AUX
ejpam-5732	306	24	δ1	δ1	NOUN
ejpam-5732	306	25	-	-	PUNCT
ejpam-5732	306	26	βf−1(j	βf−1(j	NOUN
ejpam-5732	306	27	)	)	PUNCT
ejpam-5732	306	28	-paracompact	-paracompact	ADJ
ejpam-5732	306	29	.	.	PUNCT
ejpam-5732	307	1	proof	proof	NOUN
ejpam-5732	307	2	.	.	PUNCT
ejpam-5732	308	1	let	let	VERB
ejpam-5732	308	2	i	i	PRON
ejpam-5732	308	3	=	=	PUNCT
ejpam-5732	308	4	f−1(j	f−1(j	PROPN
ejpam-5732	308	5	)	)	PUNCT
ejpam-5732	308	6	.	.	PUNCT
ejpam-5732	309	1	let	let	VERB
ejpam-5732	309	2	u	u	PRON
ejpam-5732	309	3	=	=	PUNCT
ejpam-5732	309	4	{	{	PUNCT
ejpam-5732	309	5	uα	uα	X
ejpam-5732	309	6	:	:	PUNCT
ejpam-5732	309	7	α	α	PROPN
ejpam-5732	309	8	∈	∈	PROPN
ejpam-5732	309	9	λ1	λ1	PROPN
ejpam-5732	309	10	}	}	PUNCT
ejpam-5732	309	11	be	be	VERB
ejpam-5732	309	12	a	a	DET
ejpam-5732	309	13	δ	δ	PROPN
ejpam-5732	309	14	-	-	PUNCT
ejpam-5732	309	15	βi	βi	ADV
ejpam-5732	309	16	-	-	PUNCT
ejpam-5732	309	17	open	open	ADJ
ejpam-5732	309	18	cover	cover	NOUN
ejpam-5732	309	19	of	of	ADP
ejpam-5732	309	20	x.	x.	NOUN
ejpam-5732	309	21	as	as	SCONJ
ejpam-5732	309	22	f	f	PROPN
ejpam-5732	309	23	is	be	AUX
ejpam-5732	309	24	δ	δ	PROPN
ejpam-5732	309	25	-	-	PUNCT
ejpam-5732	309	26	βi	βi	ADV
ejpam-5732	309	27	-	-	PUNCT
ejpam-5732	309	28	open	open	ADJ
ejpam-5732	309	29	,	,	PUNCT
ejpam-5732	309	30	f(u	f(u	PROPN
ejpam-5732	309	31	)	)	PUNCT
ejpam-5732	310	1	=	=	PRON
ejpam-5732	310	2	{	{	PUNCT
ejpam-5732	310	3	f(uα	f(uα	NOUN
ejpam-5732	310	4	)	)	PUNCT
ejpam-5732	310	5	:	:	PUNCT
ejpam-5732	310	6	α	α	PROPN
ejpam-5732	310	7	∈	∈	PROPN
ejpam-5732	310	8	λ1	λ1	PROPN
ejpam-5732	310	9	}	}	PUNCT
ejpam-5732	310	10	is	be	AUX
ejpam-5732	310	11	a	a	DET
ejpam-5732	310	12	δ	δ	PROPN
ejpam-5732	310	13	-	-	PUNCT
ejpam-5732	310	14	βj	βj	PUNCT
ejpam-5732	310	15	-open	-open	ADJ
ejpam-5732	310	16	cover	cover	NOUN
ejpam-5732	310	17	of	of	ADP
ejpam-5732	310	18	y	y	PROPN
ejpam-5732	310	19	.	.	PUNCT
ejpam-5732	311	1	by	by	ADP
ejpam-5732	311	2	hypothesis	hypothesis	NOUN
ejpam-5732	311	3	,	,	PUNCT
ejpam-5732	311	4	f(u	f(u	PROPN
ejpam-5732	311	5	)	)	PUNCT
ejpam-5732	311	6	has	have	VERB
ejpam-5732	311	7	a	a	DET
ejpam-5732	311	8	locally	locally	ADV
ejpam-5732	311	9	finite	finite	ADJ
ejpam-5732	311	10	open	open	ADJ
ejpam-5732	311	11	refinement	refinement	NOUN
ejpam-5732	311	12	h	h	NOUN
ejpam-5732	311	13	=	=	PRON
ejpam-5732	311	14	{	{	PUNCT
ejpam-5732	311	15	hλ	hλ	X
ejpam-5732	311	16	:	:	PUNCT
ejpam-5732	311	17	λ	λ	PROPN
ejpam-5732	311	18	∈	∈	PROPN
ejpam-5732	311	19	λ2	λ2	NOUN
ejpam-5732	311	20	}	}	PUNCT
ejpam-5732	311	21	such	such	ADJ
ejpam-5732	311	22	that	that	SCONJ
ejpam-5732	311	23	y	y	PROPN
ejpam-5732	311	24	−	−	PROPN
ejpam-5732	311	25	∪{hλ	∪{hλ	PROPN
ejpam-5732	311	26	:	:	PUNCT
ejpam-5732	311	27	λ	λ	PROPN
ejpam-5732	311	28	∈	∈	PROPN
ejpam-5732	311	29	λ2	λ2	PROPN
ejpam-5732	311	30	}	}	PUNCT
ejpam-5732	311	31	∈	∈	PROPN
ejpam-5732	311	32	j	j	PROPN
ejpam-5732	311	33	.	.	PUNCT
ejpam-5732	312	1	it	it	PRON
ejpam-5732	312	2	implies	imply	VERB
ejpam-5732	312	3	that	that	SCONJ
ejpam-5732	312	4	y	y	PROPN
ejpam-5732	312	5	−	−	PROPN
ejpam-5732	312	6	∪{hλ	∪{hλ	PROPN
ejpam-5732	312	7	:	:	PUNCT
ejpam-5732	312	8	λ	λ	PROPN
ejpam-5732	312	9	∈	∈	PROPN
ejpam-5732	312	10	λ2	λ2	PROPN
ejpam-5732	312	11	}	}	PUNCT
ejpam-5732	312	12	=	=	SYM
ejpam-5732	312	13	j	j	PROPN
ejpam-5732	312	14	for	for	ADP
ejpam-5732	312	15	some	some	DET
ejpam-5732	312	16	j	j	PROPN
ejpam-5732	312	17	∈	∈	PROPN
ejpam-5732	312	18	j	j	PROPN
ejpam-5732	312	19	,	,	PUNCT
ejpam-5732	312	20	which	which	PRON
ejpam-5732	312	21	follows	follow	VERB
ejpam-5732	312	22	that	that	SCONJ
ejpam-5732	312	23	y	y	PROPN
ejpam-5732	312	24	=	=	SYM
ejpam-5732	312	25	∪{hλ	∪{hλ	PROPN
ejpam-5732	312	26	:	:	PUNCT
ejpam-5732	312	27	λ	λ	PROPN
ejpam-5732	312	28	∈	∈	PROPN
ejpam-5732	312	29	λ2	λ2	PROPN
ejpam-5732	312	30	}	}	PUNCT
ejpam-5732	312	31	∪	∪	PROPN
ejpam-5732	312	32	j	j	PROPN
ejpam-5732	312	33	.	.	PUNCT
ejpam-5732	313	1	hence	hence	ADV
ejpam-5732	313	2	,	,	PUNCT
ejpam-5732	313	3	x	x	PUNCT
ejpam-5732	313	4	=	=	PUNCT
ejpam-5732	313	5	f−1(y	f−1(y	PROPN
ejpam-5732	313	6	)	)	PUNCT
ejpam-5732	314	1	=	=	PUNCT
ejpam-5732	314	2	f−1(∪{hλ	f−1(∪{hλ	ADJ
ejpam-5732	314	3	:	:	PUNCT
ejpam-5732	314	4	λ	λ	PROPN
ejpam-5732	314	5	∈	∈	PROPN
ejpam-5732	314	6	λ2	λ2	PROPN
ejpam-5732	314	7	}	}	PUNCT
ejpam-5732	314	8	∪	∪	PROPN
ejpam-5732	314	9	j	j	PROPN
ejpam-5732	314	10	)	)	PUNCT
ejpam-5732	314	11	=	=	PROPN
ejpam-5732	314	12	c.	c.	PROPN
ejpam-5732	314	13	boonpok	boonpok	PROPN
ejpam-5732	314	14	,	,	PUNCT
ejpam-5732	314	15	a.	a.	PROPN
ejpam-5732	314	16	sama	sama	PROPN
ejpam-5732	314	17	-	-	PUNCT
ejpam-5732	314	18	ae	ae	PROPN
ejpam-5732	314	19	/	/	SYM
ejpam-5732	314	20	eur	eur	PROPN
ejpam-5732	314	21	.	.	PUNCT
ejpam-5732	315	1	j.	j.	PROPN
ejpam-5732	315	2	pure	pure	PROPN
ejpam-5732	315	3	appl	appl	PROPN
ejpam-5732	315	4	.	.	PROPN
ejpam-5732	315	5	math	math	PROPN
ejpam-5732	315	6	,	,	PUNCT
ejpam-5732	315	7	18	18	NUM
ejpam-5732	315	8	(	(	PUNCT
ejpam-5732	315	9	1	1	NUM
ejpam-5732	315	10	)	)	PUNCT
ejpam-5732	315	11	(	(	PUNCT
ejpam-5732	315	12	2025	2025	NUM
ejpam-5732	315	13	)	)	PUNCT
ejpam-5732	315	14	,	,	PUNCT
ejpam-5732	315	15	5732	5732	NUM
ejpam-5732	315	16	11	11	NUM
ejpam-5732	315	17	of	of	ADP
ejpam-5732	315	18	13	13	NUM
ejpam-5732	315	19	(	(	PUNCT
ejpam-5732	315	20	∪{f−1(hλ	∪{f−1(hλ	PROPN
ejpam-5732	315	21	)	)	PUNCT
ejpam-5732	315	22	:	:	PUNCT
ejpam-5732	316	1	λ	λ	X
ejpam-5732	316	2	∈	∈	NOUN
ejpam-5732	316	3	λ2	λ2	NOUN
ejpam-5732	316	4	}	}	PUNCT
ejpam-5732	316	5	)	)	PUNCT
ejpam-5732	316	6	∪	∪	ADP
ejpam-5732	316	7	f−1(j	f−1(j	NOUN
ejpam-5732	316	8	)	)	PUNCT
ejpam-5732	316	9	.	.	PUNCT
ejpam-5732	317	1	as	as	ADP
ejpam-5732	317	2	f−1(j	f−1(j	NOUN
ejpam-5732	317	3	)	)	PUNCT
ejpam-5732	317	4	∈	∈	PROPN
ejpam-5732	318	1	i	i	PRON
ejpam-5732	318	2	,	,	PUNCT
ejpam-5732	318	3	x	x	PROPN
ejpam-5732	318	4	−	−	PROPN
ejpam-5732	318	5	∪{f−1(hλ	∪{f−1(hλ	PROPN
ejpam-5732	318	6	)	)	PUNCT
ejpam-5732	318	7	:	:	PUNCT
ejpam-5732	319	1	λ	λ	X
ejpam-5732	319	2	∈	∈	PROPN
ejpam-5732	319	3	λ2	λ2	PROPN
ejpam-5732	319	4	}	}	PUNCT
ejpam-5732	319	5	∈	∈	PROPN
ejpam-5732	319	6	i.	i.	NOUN
ejpam-5732	319	7	consequently	consequently	ADV
ejpam-5732	319	8	,	,	PUNCT
ejpam-5732	319	9	v	v	NOUN
ejpam-5732	319	10	=	=	SYM
ejpam-5732	319	11	{	{	PUNCT
ejpam-5732	319	12	f−1(hλ	f−1(hλ	PROPN
ejpam-5732	319	13	)	)	PUNCT
ejpam-5732	319	14	:	:	PUNCT
ejpam-5732	320	1	λ	λ	X
ejpam-5732	320	2	∈	∈	PROPN
ejpam-5732	320	3	λ2	λ2	NOUN
ejpam-5732	320	4	}	}	PUNCT
ejpam-5732	320	5	is	be	AUX
ejpam-5732	320	6	locally	locally	ADV
ejpam-5732	320	7	finite	finite	ADJ
ejpam-5732	320	8	,	,	PUNCT
ejpam-5732	320	9	as	as	SCONJ
ejpam-5732	320	10	demonstrated	demonstrate	VERB
ejpam-5732	320	11	by	by	ADP
ejpam-5732	320	12	lemma	lemma	PROPN
ejpam-5732	320	13	5	5	NUM
ejpam-5732	320	14	.	.	PUNCT
ejpam-5732	321	1	now	now	ADV
ejpam-5732	321	2	we	we	PRON
ejpam-5732	321	3	will	will	AUX
ejpam-5732	321	4	verify	verify	VERB
ejpam-5732	321	5	that	that	SCONJ
ejpam-5732	321	6	v	v	NOUN
ejpam-5732	321	7	refines	refine	VERB
ejpam-5732	321	8	u	u	PRON
ejpam-5732	321	9	.	.	PUNCT
ejpam-5732	322	1	let	let	VERB
ejpam-5732	322	2	f−1(hλ	f−1(hλ	PROPN
ejpam-5732	322	3	)	)	PUNCT
ejpam-5732	322	4	∈	∈	PROPN
ejpam-5732	323	1	v.	v.	CCONJ
ejpam-5732	323	2	hence	hence	ADV
ejpam-5732	323	3	hλ	hλ	ADP
ejpam-5732	323	4	∈	∈	PROPN
ejpam-5732	323	5	h.	h.	PROPN
ejpam-5732	323	6	given	give	VERB
ejpam-5732	323	7	that	that	SCONJ
ejpam-5732	323	8	h	h	NOUN
ejpam-5732	323	9	refines	refine	VERB
ejpam-5732	323	10	f(u	f(u	PROPN
ejpam-5732	323	11	)	)	PUNCT
ejpam-5732	323	12	,	,	PUNCT
ejpam-5732	323	13	there	there	PRON
ejpam-5732	323	14	exists	exist	VERB
ejpam-5732	323	15	an	an	DET
ejpam-5732	323	16	element	element	NOUN
ejpam-5732	323	17	f(uλ	f(uλ	NOUN
ejpam-5732	323	18	)	)	PUNCT
ejpam-5732	323	19	∈	∈	PROPN
ejpam-5732	323	20	f(u	f(u	PROPN
ejpam-5732	323	21	)	)	PUNCT
ejpam-5732	323	22	such	such	ADJ
ejpam-5732	323	23	that	that	PRON
ejpam-5732	323	24	hλ	hλ	ADP
ejpam-5732	323	25	⊂	⊂	PROPN
ejpam-5732	323	26	f(uλ	f(uλ	PROPN
ejpam-5732	323	27	)	)	PUNCT
ejpam-5732	323	28	.	.	PUNCT
ejpam-5732	324	1	this	this	PRON
ejpam-5732	324	2	implies	imply	VERB
ejpam-5732	324	3	that	that	SCONJ
ejpam-5732	324	4	f−1(hλ	f−1(hλ	PROPN
ejpam-5732	324	5	)	)	PUNCT
ejpam-5732	324	6	⊂	⊂	PROPN
ejpam-5732	324	7	f−1(f(uλ	f−1(f(uλ	PROPN
ejpam-5732	324	8	)	)	PUNCT
ejpam-5732	324	9	)	)	PUNCT
ejpam-5732	325	1	=	=	PUNCT
ejpam-5732	326	1	uλ	uλ	ADP
ejpam-5732	326	2	∈	∈	PROPN
ejpam-5732	326	3	u	u	PROPN
ejpam-5732	326	4	.	.	PUNCT
ejpam-5732	327	1	therefore	therefore	ADV
ejpam-5732	327	2	,	,	PUNCT
ejpam-5732	327	3	x	x	X
ejpam-5732	327	4	is	be	AUX
ejpam-5732	327	5	δ1	δ1	NOUN
ejpam-5732	327	6	-	-	PUNCT
ejpam-5732	327	7	βi	βi	PRON
ejpam-5732	327	8	-	-	PUNCT
ejpam-5732	327	9	paracompact	paracompact	ADJ
ejpam-5732	327	10	.	.	PUNCT
ejpam-5732	328	1	the	the	DET
ejpam-5732	328	2	final	final	ADJ
ejpam-5732	328	3	result	result	NOUN
ejpam-5732	328	4	of	of	ADP
ejpam-5732	328	5	this	this	DET
ejpam-5732	328	6	section	section	NOUN
ejpam-5732	328	7	presents	present	VERB
ejpam-5732	328	8	attributes	attribute	NOUN
ejpam-5732	328	9	of	of	ADP
ejpam-5732	328	10	a	a	DET
ejpam-5732	328	11	function	function	NOUN
ejpam-5732	328	12	whereby	whereby	SCONJ
ejpam-5732	328	13	the	the	DET
ejpam-5732	328	14	inverse	inverse	ADJ
ejpam-5732	328	15	image	image	NOUN
ejpam-5732	328	16	of	of	ADP
ejpam-5732	328	17	a	a	DET
ejpam-5732	328	18	δ1	δ1	NOUN
ejpam-5732	328	19	-	-	PUNCT
ejpam-5732	328	20	βj	βj	PUNCT
ejpam-5732	328	21	-paracompact	-paracompact	NOUN
ejpam-5732	328	22	subset	subset	NOUN
ejpam-5732	328	23	is	be	AUX
ejpam-5732	328	24	a	a	DET
ejpam-5732	328	25	δ1	δ1	VERB
ejpam-5732	328	26	-	-	PUNCT
ejpam-5732	328	27	βi	βi	NOUN
ejpam-5732	328	28	-	-	PUNCT
ejpam-5732	328	29	paracompact	paracompact	NOUN
ejpam-5732	328	30	subset	subset	NOUN
ejpam-5732	328	31	.	.	PUNCT
ejpam-5732	329	1	theorem	theorem	NOUN
ejpam-5732	329	2	11	11	NUM
ejpam-5732	329	3	.	.	PUNCT
ejpam-5732	330	1	let	let	VERB
ejpam-5732	330	2	(	(	PUNCT
ejpam-5732	330	3	x	x	X
ejpam-5732	330	4	,	,	PUNCT
ejpam-5732	330	5	τ	τ	PROPN
ejpam-5732	330	6	,	,	PUNCT
ejpam-5732	330	7	i	i	PROPN
ejpam-5732	330	8	)	)	PUNCT
ejpam-5732	330	9	and	and	CCONJ
ejpam-5732	330	10	(	(	PUNCT
ejpam-5732	330	11	y	y	PROPN
ejpam-5732	330	12	,	,	PUNCT
ejpam-5732	330	13	τ	τ	PROPN
ejpam-5732	330	14	′,j	′,j	NOUN
ejpam-5732	330	15	)	)	PUNCT
ejpam-5732	330	16	be	be	AUX
ejpam-5732	330	17	ideal	ideal	ADJ
ejpam-5732	330	18	topological	topological	ADJ
ejpam-5732	330	19	spaces	space	NOUN
ejpam-5732	330	20	.	.	PUNCT
ejpam-5732	331	1	suppose	suppose	VERB
ejpam-5732	331	2	that	that	SCONJ
ejpam-5732	331	3	f	f	X
ejpam-5732	331	4	:	:	PUNCT
ejpam-5732	331	5	x	x	X
ejpam-5732	331	6	→	→	SYM
ejpam-5732	331	7	y	y	PROPN
ejpam-5732	331	8	is	be	AUX
ejpam-5732	331	9	δ	δ	PROPN
ejpam-5732	331	10	-	-	PUNCT
ejpam-5732	331	11	βi	βi	ADV
ejpam-5732	331	12	-	-	PUNCT
ejpam-5732	331	13	open	open	ADJ
ejpam-5732	331	14	,	,	PUNCT
ejpam-5732	331	15	continuous	continuous	ADJ
ejpam-5732	331	16	,	,	PUNCT
ejpam-5732	331	17	and	and	CCONJ
ejpam-5732	331	18	bijective	bijective	ADJ
ejpam-5732	331	19	with	with	ADP
ejpam-5732	331	20	f(i	f(i	PROPN
ejpam-5732	331	21	)	)	PUNCT
ejpam-5732	332	1	=	=	SYM
ejpam-5732	332	2	j	j	PROPN
ejpam-5732	332	3	.	.	PUNCT
ejpam-5732	333	1	if	if	SCONJ
ejpam-5732	333	2	a	a	DET
ejpam-5732	333	3	⊂	⊂	PROPN
ejpam-5732	333	4	y	y	PROPN
ejpam-5732	333	5	is	be	AUX
ejpam-5732	333	6	a	a	DET
ejpam-5732	333	7	δ1	δ1	NOUN
ejpam-5732	333	8	-	-	PUNCT
ejpam-5732	333	9	βj	βj	NOUN
ejpam-5732	333	10	paracompact	paracompact	NOUN
ejpam-5732	333	11	subset	subset	NOUN
ejpam-5732	333	12	of	of	ADP
ejpam-5732	333	13	y	y	PROPN
ejpam-5732	333	14	,	,	PUNCT
ejpam-5732	333	15	then	then	ADV
ejpam-5732	333	16	f−1(a	f−1(a	PROPN
ejpam-5732	333	17	)	)	PUNCT
ejpam-5732	333	18	is	be	AUX
ejpam-5732	333	19	a	a	DET
ejpam-5732	333	20	δ1	δ1	VERB
ejpam-5732	333	21	-	-	PUNCT
ejpam-5732	333	22	βi	βi	NOUN
ejpam-5732	333	23	-	-	PUNCT
ejpam-5732	333	24	paracompact	paracompact	NOUN
ejpam-5732	333	25	subset	subset	NOUN
ejpam-5732	333	26	of	of	ADP
ejpam-5732	333	27	x.	x.	NOUN
ejpam-5732	333	28	proof	proof	PROPN
ejpam-5732	333	29	.	.	PUNCT
ejpam-5732	334	1	let	let	VERB
ejpam-5732	334	2	u	u	PRON
ejpam-5732	334	3	=	=	PUNCT
ejpam-5732	334	4	{	{	PUNCT
ejpam-5732	334	5	uλ	uλ	X
ejpam-5732	334	6	:	:	PUNCT
ejpam-5732	334	7	λ	λ	X
ejpam-5732	334	8	∈	∈	PROPN
ejpam-5732	334	9	λ	λ	PROPN
ejpam-5732	334	10	}	}	PUNCT
ejpam-5732	334	11	be	be	VERB
ejpam-5732	334	12	a	a	DET
ejpam-5732	334	13	δ	δ	PROPN
ejpam-5732	334	14	-	-	PUNCT
ejpam-5732	334	15	βi	βi	ADV
ejpam-5732	334	16	-	-	PUNCT
ejpam-5732	334	17	open	open	ADJ
ejpam-5732	334	18	cover	cover	NOUN
ejpam-5732	334	19	of	of	ADP
ejpam-5732	334	20	f−1(a	f−1(a	NOUN
ejpam-5732	334	21	)	)	PUNCT
ejpam-5732	334	22	in	in	ADP
ejpam-5732	334	23	x.	x.	NOUN
ejpam-5732	334	24	since	since	SCONJ
ejpam-5732	334	25	f	f	PROPN
ejpam-5732	334	26	is	be	AUX
ejpam-5732	334	27	δ	δ	PROPN
ejpam-5732	334	28	-	-	PUNCT
ejpam-5732	334	29	βi	βi	ADV
ejpam-5732	334	30	-	-	PUNCT
ejpam-5732	334	31	open	open	ADJ
ejpam-5732	334	32	,	,	PUNCT
ejpam-5732	334	33	f(u	f(u	PROPN
ejpam-5732	334	34	)	)	PUNCT
ejpam-5732	334	35	=	=	PRON
ejpam-5732	334	36	{	{	PUNCT
ejpam-5732	334	37	f(uλ	f(uλ	PROPN
ejpam-5732	334	38	)	)	PUNCT
ejpam-5732	334	39	:	:	PUNCT
ejpam-5732	335	1	λ	λ	X
ejpam-5732	335	2	∈	∈	PROPN
ejpam-5732	335	3	λ	λ	PROPN
ejpam-5732	335	4	}	}	PUNCT
ejpam-5732	335	5	is	be	AUX
ejpam-5732	335	6	a	a	DET
ejpam-5732	335	7	δ	δ	PROPN
ejpam-5732	335	8	-	-	PUNCT
ejpam-5732	335	9	βi	βi	ADV
ejpam-5732	335	10	-	-	PUNCT
ejpam-5732	335	11	open	open	ADJ
ejpam-5732	335	12	cover	cover	NOUN
ejpam-5732	335	13	of	of	ADP
ejpam-5732	335	14	a	a	PRON
ejpam-5732	335	15	in	in	ADP
ejpam-5732	335	16	y	y	PROPN
ejpam-5732	335	17	.	.	PUNCT
ejpam-5732	336	1	by	by	ADP
ejpam-5732	336	2	hypothesis	hypothesis	NOUN
ejpam-5732	336	3	,	,	PUNCT
ejpam-5732	336	4	f(u	f(u	PROPN
ejpam-5732	336	5	)	)	PUNCT
ejpam-5732	336	6	has	have	VERB
ejpam-5732	336	7	a	a	DET
ejpam-5732	336	8	locally	locally	ADV
ejpam-5732	336	9	finite	finite	ADJ
ejpam-5732	336	10	open	open	ADJ
ejpam-5732	336	11	refinement	refinement	NOUN
ejpam-5732	336	12	h	h	NOUN
ejpam-5732	336	13	=	=	PRON
ejpam-5732	336	14	{	{	PUNCT
ejpam-5732	336	15	vα	vα	X
ejpam-5732	336	16	:	:	PUNCT
ejpam-5732	336	17	α	α	PROPN
ejpam-5732	336	18	∈	∈	PROPN
ejpam-5732	336	19	λ1	λ1	PROPN
ejpam-5732	336	20	}	}	PUNCT
ejpam-5732	336	21	of	of	ADP
ejpam-5732	336	22	a	a	DET
ejpam-5732	336	23	such	such	ADJ
ejpam-5732	336	24	that	that	SCONJ
ejpam-5732	336	25	a	a	DET
ejpam-5732	336	26	−	−	NOUN
ejpam-5732	336	27	∪{vα	∪{vα	NUM
ejpam-5732	336	28	:	:	PUNCT
ejpam-5732	336	29	α	α	PROPN
ejpam-5732	336	30	∈	∈	PROPN
ejpam-5732	336	31	λ1	λ1	PROPN
ejpam-5732	336	32	}	}	PUNCT
ejpam-5732	336	33	∈	∈	PROPN
ejpam-5732	336	34	j	j	PROPN
ejpam-5732	336	35	.	.	PUNCT
ejpam-5732	337	1	then	then	ADV
ejpam-5732	337	2	,	,	PUNCT
ejpam-5732	337	3	f−1(a	f−1(a	PROPN
ejpam-5732	337	4	)	)	PUNCT
ejpam-5732	338	1	−	−	PROPN
ejpam-5732	338	2	∪{f−1(vα	∪{f−1(vα	PROPN
ejpam-5732	338	3	)	)	PUNCT
ejpam-5732	338	4	:	:	PUNCT
ejpam-5732	338	5	α	α	PROPN
ejpam-5732	338	6	∈	∈	PROPN
ejpam-5732	338	7	λ1	λ1	PROPN
ejpam-5732	338	8	}	}	PUNCT
ejpam-5732	338	9	∈	∈	PROPN
ejpam-5732	338	10	f−1(j	f−1(j	NOUN
ejpam-5732	338	11	)	)	PUNCT
ejpam-5732	338	12	=	=	SYM
ejpam-5732	338	13	i.	i.	NOUN
ejpam-5732	338	14	as	as	SCONJ
ejpam-5732	338	15	f	f	PROPN
ejpam-5732	338	16	is	be	AUX
ejpam-5732	338	17	continuous	continuous	ADJ
ejpam-5732	338	18	,	,	PUNCT
ejpam-5732	338	19	v	v	NOUN
ejpam-5732	338	20	=	=	SYM
ejpam-5732	338	21	{	{	PUNCT
ejpam-5732	338	22	f−1(vα	f−1(vα	PROPN
ejpam-5732	338	23	)	)	PUNCT
ejpam-5732	338	24	:	:	PUNCT
ejpam-5732	338	25	α	α	PROPN
ejpam-5732	338	26	∈	∈	PROPN
ejpam-5732	338	27	λ1	λ1	PROPN
ejpam-5732	338	28	}	}	PUNCT
ejpam-5732	338	29	is	be	AUX
ejpam-5732	338	30	locally	locally	ADV
ejpam-5732	338	31	finite	finite	ADJ
ejpam-5732	338	32	.	.	PUNCT
ejpam-5732	339	1	next	next	ADV
ejpam-5732	339	2	,	,	PUNCT
ejpam-5732	339	3	we	we	PRON
ejpam-5732	339	4	will	will	AUX
ejpam-5732	339	5	prove	prove	VERB
ejpam-5732	339	6	that	that	SCONJ
ejpam-5732	339	7	v	v	NOUN
ejpam-5732	339	8	refines	refine	VERB
ejpam-5732	339	9	u	u	PRON
ejpam-5732	339	10	.	.	PUNCT
ejpam-5732	340	1	let	let	VERB
ejpam-5732	340	2	f−1(vα	f−1(vα	VERB
ejpam-5732	340	3	)	)	PUNCT
ejpam-5732	340	4	∈	∈	PROPN
ejpam-5732	340	5	v.	v.	CCONJ
ejpam-5732	340	6	as	as	SCONJ
ejpam-5732	340	7	h	h	NOUN
ejpam-5732	340	8	refines	refine	NOUN
ejpam-5732	340	9	f(u	f(u	PROPN
ejpam-5732	340	10	)	)	PUNCT
ejpam-5732	340	11	,	,	PUNCT
ejpam-5732	340	12	there	there	PRON
ejpam-5732	340	13	exists	exist	VERB
ejpam-5732	340	14	f(uλ	f(uλ	PROPN
ejpam-5732	340	15	)	)	PUNCT
ejpam-5732	340	16	∈	∈	PROPN
ejpam-5732	340	17	f(u	f(u	PROPN
ejpam-5732	340	18	)	)	PUNCT
ejpam-5732	340	19	such	such	ADJ
ejpam-5732	340	20	that	that	SCONJ
ejpam-5732	340	21	vα	vα	PROPN
ejpam-5732	340	22	⊂	⊂	X
ejpam-5732	340	23	f(uλ	f(uλ	PROPN
ejpam-5732	340	24	)	)	PUNCT
ejpam-5732	340	25	.	.	PUNCT
ejpam-5732	341	1	it	it	PRON
ejpam-5732	341	2	follows	follow	VERB
ejpam-5732	341	3	that	that	SCONJ
ejpam-5732	341	4	f−1(vα	f−1(vα	PROPN
ejpam-5732	341	5	)	)	PUNCT
ejpam-5732	341	6	⊂	⊂	PROPN
ejpam-5732	341	7	f−1(f(uλ	f−1(f(uλ	PROPN
ejpam-5732	341	8	)	)	PUNCT
ejpam-5732	341	9	)	)	PUNCT
ejpam-5732	342	1	=	=	SYM
ejpam-5732	342	2	uλ	uλ	PROPN
ejpam-5732	342	3	.	.	PUNCT
ejpam-5732	343	1	therefore	therefore	ADV
ejpam-5732	343	2	f−1(a	f−1(a	PROPN
ejpam-5732	343	3	)	)	PUNCT
ejpam-5732	343	4	is	be	AUX
ejpam-5732	343	5	a	a	DET
ejpam-5732	343	6	δ1	δ1	VERB
ejpam-5732	343	7	-	-	PUNCT
ejpam-5732	343	8	βi	βi	NOUN
ejpam-5732	343	9	-	-	PUNCT
ejpam-5732	343	10	paracompact	paracompact	NOUN
ejpam-5732	343	11	subset	subset	NOUN
ejpam-5732	343	12	of	of	ADP
ejpam-5732	343	13	x.	x.	NOUN
ejpam-5732	343	14	5	5	NUM
ejpam-5732	343	15	.	.	PUNCT
ejpam-5732	343	16	conclusion	conclusion	NOUN
ejpam-5732	343	17	this	this	DET
ejpam-5732	343	18	paper	paper	NOUN
ejpam-5732	343	19	examines	examine	VERB
ejpam-5732	343	20	the	the	DET
ejpam-5732	343	21	characteristics	characteristic	NOUN
ejpam-5732	343	22	of	of	ADP
ejpam-5732	343	23	δ1	δ1	NOUN
ejpam-5732	343	24	-	-	PUNCT
ejpam-5732	343	25	βi	βi	PRON
ejpam-5732	343	26	-	-	PUNCT
ejpam-5732	343	27	paracompact	paracompact	ADJ
ejpam-5732	343	28	spaces	space	NOUN
ejpam-5732	343	29	,	,	PUNCT
ejpam-5732	343	30	which	which	PRON
ejpam-5732	343	31	are	be	AUX
ejpam-5732	343	32	broader	broad	ADJ
ejpam-5732	343	33	in	in	ADP
ejpam-5732	343	34	scope	scope	NOUN
ejpam-5732	343	35	than	than	ADP
ejpam-5732	343	36	the	the	DET
ejpam-5732	343	37	β1	β1	NOUN
ejpam-5732	343	38	-	-	PUNCT
ejpam-5732	343	39	paracompact	paracompact	NOUN
ejpam-5732	343	40	spaces	space	NOUN
ejpam-5732	343	41	defined	define	VERB
ejpam-5732	343	42	by	by	ADP
ejpam-5732	343	43	qahis	qahis	PRON
ejpam-5732	344	1	[	[	X
ejpam-5732	344	2	21	21	NUM
ejpam-5732	344	3	]	]	PUNCT
ejpam-5732	344	4	.	.	PUNCT
ejpam-5732	345	1	additionally	additionally	ADV
ejpam-5732	345	2	,	,	PUNCT
ejpam-5732	345	3	we	we	PRON
ejpam-5732	345	4	investigate	investigate	VERB
ejpam-5732	345	5	the	the	DET
ejpam-5732	345	6	invariants	invariant	NOUN
ejpam-5732	345	7	of	of	ADP
ejpam-5732	345	8	δ1	δ1	NOUN
ejpam-5732	345	9	-	-	PUNCT
ejpam-5732	345	10	βi	βi	NOUN
ejpam-5732	345	11	-	-	NOUN
ejpam-5732	345	12	paracompactness	paracompactness	NOUN
ejpam-5732	345	13	via	via	ADP
ejpam-5732	345	14	functions	function	NOUN
ejpam-5732	345	15	.	.	PUNCT
ejpam-5732	346	1	we	we	PRON
ejpam-5732	346	2	established	establish	VERB
ejpam-5732	346	3	that	that	SCONJ
ejpam-5732	346	4	if	if	SCONJ
ejpam-5732	346	5	(	(	PUNCT
ejpam-5732	346	6	x	x	NOUN
ejpam-5732	346	7	,	,	PUNCT
ejpam-5732	346	8	τ	τ	PROPN
ejpam-5732	346	9	,	,	PUNCT
ejpam-5732	346	10	i	i	PROPN
ejpam-5732	346	11	)	)	PUNCT
ejpam-5732	346	12	is	be	AUX
ejpam-5732	346	13	δ1	δ1	NOUN
ejpam-5732	346	14	-	-	PUNCT
ejpam-5732	346	15	βi	βi	PRON
ejpam-5732	346	16	-	-	PUNCT
ejpam-5732	346	17	paracompact	paracompact	ADJ
ejpam-5732	346	18	,	,	PUNCT
ejpam-5732	346	19	then	then	ADV
ejpam-5732	346	20	(	(	PUNCT
ejpam-5732	346	21	x	x	NOUN
ejpam-5732	346	22	,	,	PUNCT
ejpam-5732	346	23	τ∗	τ∗	PROPN
ejpam-5732	346	24	,	,	PUNCT
ejpam-5732	346	25	i	i	NOUN
ejpam-5732	346	26	)	)	PUNCT
ejpam-5732	346	27	is	be	AUX
ejpam-5732	346	28	δ1	δ1	NOUN
ejpam-5732	346	29	-	-	PUNCT
ejpam-5732	346	30	βi	βi	PRON
ejpam-5732	346	31	-	-	PUNCT
ejpam-5732	346	32	paracompact	paracompact	ADJ
ejpam-5732	346	33	,	,	PUNCT
ejpam-5732	346	34	and	and	CCONJ
ejpam-5732	346	35	that	that	SCONJ
ejpam-5732	346	36	every	every	DET
ejpam-5732	346	37	δ1	δ1	NOUN
ejpam-5732	346	38	-	-	PUNCT
ejpam-5732	346	39	βi	βi	PRON
ejpam-5732	346	40	-	-	PUNCT
ejpam-5732	346	41	closed	close	VERB
ejpam-5732	346	42	subset	subset	NOUN
ejpam-5732	346	43	of	of	ADP
ejpam-5732	346	44	a	a	DET
ejpam-5732	346	45	δ1	δ1	VERB
ejpam-5732	346	46	-	-	PUNCT
ejpam-5732	346	47	βi	βi	PRON
ejpam-5732	346	48	-	-	PUNCT
ejpam-5732	346	49	paracompact	paracompact	NOUN
ejpam-5732	346	50	space	space	NOUN
ejpam-5732	346	51	is	be	AUX
ejpam-5732	346	52	a	a	DET
ejpam-5732	346	53	δ1	δ1	VERB
ejpam-5732	346	54	-	-	PUNCT
ejpam-5732	346	55	βi	βi	NOUN
ejpam-5732	346	56	-	-	PUNCT
ejpam-5732	346	57	paracompact	paracompact	NOUN
ejpam-5732	346	58	subset	subset	NOUN
ejpam-5732	346	59	.	.	PUNCT
ejpam-5732	347	1	the	the	DET
ejpam-5732	347	2	union	union	NOUN
ejpam-5732	347	3	of	of	ADP
ejpam-5732	347	4	two	two	NUM
ejpam-5732	347	5	δ1	δ1	VERB
ejpam-5732	347	6	-	-	PUNCT
ejpam-5732	347	7	βi	βi	PRON
ejpam-5732	347	8	-	-	PUNCT
ejpam-5732	347	9	paracompact	paracompact	ADJ
ejpam-5732	347	10	subsets	subset	NOUN
ejpam-5732	347	11	is	be	AUX
ejpam-5732	347	12	a	a	DET
ejpam-5732	347	13	δ1	δ1	VERB
ejpam-5732	347	14	-	-	PUNCT
ejpam-5732	347	15	βi	βi	NOUN
ejpam-5732	347	16	-	-	PUNCT
ejpam-5732	347	17	paracompact	paracompact	NOUN
ejpam-5732	347	18	subset	subset	NOUN
ejpam-5732	347	19	,	,	PUNCT
ejpam-5732	347	20	and	and	CCONJ
ejpam-5732	347	21	the	the	DET
ejpam-5732	347	22	intersection	intersection	NOUN
ejpam-5732	347	23	of	of	ADP
ejpam-5732	347	24	a	a	DET
ejpam-5732	347	25	δ1	δ1	NOUN
ejpam-5732	347	26	-	-	PUNCT
ejpam-5732	347	27	βiparacompact	βiparacompact	NOUN
ejpam-5732	347	28	subset	subset	NOUN
ejpam-5732	347	29	with	with	ADP
ejpam-5732	347	30	a	a	DET
ejpam-5732	347	31	δ1	δ1	VERB
ejpam-5732	347	32	-	-	PUNCT
ejpam-5732	347	33	βi	βi	PRON
ejpam-5732	347	34	-	-	PUNCT
ejpam-5732	347	35	closed	close	VERB
ejpam-5732	347	36	set	set	NOUN
ejpam-5732	347	37	is	be	AUX
ejpam-5732	347	38	δ1	δ1	NOUN
ejpam-5732	347	39	-	-	PUNCT
ejpam-5732	347	40	βi	βi	PRON
ejpam-5732	347	41	-	-	PUNCT
ejpam-5732	347	42	paracompact	paracompact	NOUN
ejpam-5732	347	43	subset	subset	NOUN
ejpam-5732	347	44	.	.	PUNCT
ejpam-5732	348	1	furthermore	furthermore	ADV
ejpam-5732	348	2	,	,	PUNCT
ejpam-5732	348	3	we	we	PRON
ejpam-5732	348	4	demonstrate	demonstrate	VERB
ejpam-5732	348	5	that	that	SCONJ
ejpam-5732	348	6	δ1	δ1	NOUN
ejpam-5732	348	7	-	-	PUNCT
ejpam-5732	348	8	βi	βi	PRON
ejpam-5732	348	9	-	-	NOUN
ejpam-5732	348	10	paracompactness	paracompactness	NOUN
ejpam-5732	348	11	is	be	AUX
ejpam-5732	348	12	preserved	preserve	VERB
ejpam-5732	348	13	under	under	ADP
ejpam-5732	348	14	specific	specific	ADJ
ejpam-5732	348	15	conditions	condition	NOUN
ejpam-5732	348	16	:	:	PUNCT
ejpam-5732	348	17	if	if	SCONJ
ejpam-5732	348	18	f	f	X
ejpam-5732	348	19	:	:	PUNCT
ejpam-5732	348	20	x	x	X
ejpam-5732	348	21	→	→	SYM
ejpam-5732	348	22	y	y	PROPN
ejpam-5732	348	23	is	be	AUX
ejpam-5732	348	24	open	open	ADJ
ejpam-5732	348	25	,	,	PUNCT
ejpam-5732	348	26	δ	δ	PROPN
ejpam-5732	348	27	-	-	PUNCT
ejpam-5732	348	28	βi	βi	PRON
ejpam-5732	348	29	-	-	PUNCT
ejpam-5732	348	30	irresolute	irresolute	ADJ
ejpam-5732	348	31	,	,	PUNCT
ejpam-5732	348	32	almost	almost	ADV
ejpam-5732	348	33	closed	closed	ADJ
ejpam-5732	348	34	,	,	PUNCT
ejpam-5732	348	35	surjective	surjective	ADJ
ejpam-5732	348	36	with	with	ADP
ejpam-5732	348	37	n	n	ADV
ejpam-5732	348	38	-closed	-closed	ADJ
ejpam-5732	348	39	point	point	NOUN
ejpam-5732	348	40	inverse	inverse	NOUN
ejpam-5732	348	41	,	,	PUNCT
ejpam-5732	348	42	and	and	CCONJ
ejpam-5732	348	43	if	if	SCONJ
ejpam-5732	348	44	(	(	PUNCT
ejpam-5732	348	45	x	x	NOUN
ejpam-5732	348	46	,	,	PUNCT
ejpam-5732	348	47	τ	τ	PROPN
ejpam-5732	348	48	,	,	PUNCT
ejpam-5732	348	49	i	i	PROPN
ejpam-5732	348	50	)	)	PUNCT
ejpam-5732	348	51	is	be	AUX
ejpam-5732	348	52	δ1	δ1	NOUN
ejpam-5732	348	53	-	-	PUNCT
ejpam-5732	348	54	βi	βi	PRON
ejpam-5732	348	55	-	-	PUNCT
ejpam-5732	348	56	paracompact	paracompact	ADJ
ejpam-5732	348	57	,	,	PUNCT
ejpam-5732	348	58	then	then	ADV
ejpam-5732	348	59	(	(	PUNCT
ejpam-5732	348	60	y	y	PROPN
ejpam-5732	348	61	,	,	PUNCT
ejpam-5732	348	62	τ	τ	PROPN
ejpam-5732	348	63	′	′	NUM
ejpam-5732	348	64	,	,	PUNCT
ejpam-5732	348	65	f(i	f(i	PROPN
ejpam-5732	348	66	)	)	PUNCT
ejpam-5732	348	67	)	)	PUNCT
ejpam-5732	348	68	is	be	AUX
ejpam-5732	348	69	δ1	δ1	NOUN
ejpam-5732	348	70	-	-	PUNCT
ejpam-5732	348	71	βf(i)-paracompact	βf(i)-paracompact	PROPN
ejpam-5732	348	72	.	.	PUNCT
ejpam-5732	349	1	if	if	SCONJ
ejpam-5732	349	2	f	f	PROPN
ejpam-5732	349	3	:	:	PUNCT
ejpam-5732	349	4	x	x	X
ejpam-5732	349	5	→	→	SYM
ejpam-5732	349	6	y	y	PROPN
ejpam-5732	349	7	is	be	AUX
ejpam-5732	349	8	δ	δ	PROPN
ejpam-5732	349	9	-	-	PUNCT
ejpam-5732	349	10	βi	βi	ADV
ejpam-5732	349	11	-	-	PUNCT
ejpam-5732	349	12	open	open	ADJ
ejpam-5732	349	13	,	,	PUNCT
ejpam-5732	349	14	continuous	continuous	ADJ
ejpam-5732	349	15	,	,	PUNCT
ejpam-5732	349	16	bijective	bijective	ADJ
ejpam-5732	349	17	,	,	PUNCT
ejpam-5732	349	18	and	and	CCONJ
ejpam-5732	349	19	if	if	SCONJ
ejpam-5732	349	20	(	(	PUNCT
ejpam-5732	349	21	y	y	PROPN
ejpam-5732	349	22	,	,	PUNCT
ejpam-5732	349	23	τ	τ	PROPN
ejpam-5732	349	24	′,j	′,j	NOUN
ejpam-5732	349	25	)	)	PUNCT
ejpam-5732	349	26	is	be	AUX
ejpam-5732	349	27	δ1	δ1	NOUN
ejpam-5732	349	28	-	-	PUNCT
ejpam-5732	349	29	βj	βj	NOUN
ejpam-5732	349	30	-paracompact	-paracompact	NOUN
ejpam-5732	349	31	,	,	PUNCT
ejpam-5732	349	32	then	then	ADV
ejpam-5732	349	33	(	(	PUNCT
ejpam-5732	349	34	x	x	X
ejpam-5732	349	35	,	,	PUNCT
ejpam-5732	349	36	τ	τ	PROPN
ejpam-5732	349	37	,	,	PUNCT
ejpam-5732	349	38	f−1(j	f−1(j	NOUN
ejpam-5732	349	39	)	)	PUNCT
ejpam-5732	349	40	)	)	PUNCT
ejpam-5732	349	41	is	be	AUX
ejpam-5732	349	42	δ1	δ1	NOUN
ejpam-5732	349	43	-	-	PUNCT
ejpam-5732	349	44	βf−1(j	βf−1(j	NOUN
ejpam-5732	349	45	)	)	PUNCT
ejpam-5732	349	46	-paracompact	-paracompact	NOUN
ejpam-5732	349	47	.	.	PUNCT
ejpam-5732	350	1	acknowledgements	acknowledgement	NOUN
ejpam-5732	350	2	the	the	DET
ejpam-5732	350	3	authors	author	NOUN
ejpam-5732	350	4	would	would	AUX
ejpam-5732	350	5	like	like	VERB
ejpam-5732	350	6	to	to	PART
ejpam-5732	350	7	express	express	VERB
ejpam-5732	350	8	their	their	PRON
ejpam-5732	350	9	gratitude	gratitude	NOUN
ejpam-5732	350	10	to	to	ADP
ejpam-5732	350	11	the	the	DET
ejpam-5732	350	12	reviewers	reviewer	NOUN
ejpam-5732	350	13	for	for	ADP
ejpam-5732	350	14	their	their	PRON
ejpam-5732	350	15	helpful	helpful	ADJ
ejpam-5732	350	16	comments	comment	NOUN
ejpam-5732	350	17	and	and	CCONJ
ejpam-5732	350	18	suggestions	suggestion	NOUN
ejpam-5732	350	19	that	that	PRON
ejpam-5732	350	20	assisted	assist	VERB
ejpam-5732	350	21	enhance	enhance	VERB
ejpam-5732	350	22	this	this	DET
ejpam-5732	350	23	paper	paper	NOUN
ejpam-5732	350	24	better	well	ADV
ejpam-5732	350	25	.	.	PUNCT
ejpam-5732	351	1	this	this	DET
ejpam-5732	351	2	research	research	NOUN
ejpam-5732	351	3	was	be	AUX
ejpam-5732	351	4	supported	support	VERB
ejpam-5732	351	5	by	by	ADP
ejpam-5732	351	6	the	the	DET
ejpam-5732	351	7	national	national	ADJ
ejpam-5732	351	8	science	science	NOUN
ejpam-5732	351	9	,	,	PUNCT
ejpam-5732	351	10	research	research	NOUN
ejpam-5732	351	11	,	,	PUNCT
ejpam-5732	351	12	and	and	CCONJ
ejpam-5732	351	13	innovation	innovation	NOUN
ejpam-5732	351	14	fund	fund	NOUN
ejpam-5732	351	15	(	(	PUNCT
ejpam-5732	351	16	nsrf	nsrf	NOUN
ejpam-5732	351	17	)	)	PUNCT
ejpam-5732	351	18	and	and	CCONJ
ejpam-5732	351	19	prince	prince	NOUN
ejpam-5732	351	20	of	of	ADP
ejpam-5732	351	21	songkla	songkla	PROPN
ejpam-5732	351	22	university	university	PROPN
ejpam-5732	351	23	(	(	PUNCT
ejpam-5732	351	24	ref	ref	NOUN
ejpam-5732	351	25	.	.	PUNCT
ejpam-5732	352	1	no	no	INTJ
ejpam-5732	352	2	.	.	PUNCT
ejpam-5732	352	3	sat6701343s	sat6701343s	PROPN
ejpam-5732	352	4	)	)	PUNCT
ejpam-5732	352	5	.	.	PUNCT
ejpam-5732	353	1	c.	c.	PROPN
ejpam-5732	353	2	boonpok	boonpok	PROPN
ejpam-5732	353	3	,	,	PUNCT
ejpam-5732	353	4	a.	a.	PROPN
ejpam-5732	353	5	sama	sama	PROPN
ejpam-5732	353	6	-	-	PUNCT
ejpam-5732	353	7	ae	ae	PROPN
ejpam-5732	353	8	/	/	SYM
ejpam-5732	353	9	eur	eur	PROPN
ejpam-5732	353	10	.	.	PUNCT
ejpam-5732	354	1	j.	j.	PROPN
ejpam-5732	354	2	pure	pure	PROPN
ejpam-5732	354	3	appl	appl	PROPN
ejpam-5732	354	4	.	.	PROPN
ejpam-5732	354	5	math	math	PROPN
ejpam-5732	354	6	,	,	PUNCT
ejpam-5732	354	7	18	18	NUM
ejpam-5732	354	8	(	(	PUNCT
ejpam-5732	354	9	1	1	NUM
ejpam-5732	354	10	)	)	PUNCT
ejpam-5732	354	11	(	(	PUNCT
ejpam-5732	354	12	2025	2025	NUM
ejpam-5732	354	13	)	)	PUNCT
ejpam-5732	354	14	,	,	PUNCT
ejpam-5732	354	15	5732	5732	NUM
ejpam-5732	354	16	12	12	NUM
ejpam-5732	354	17	of	of	ADP
ejpam-5732	354	18	13	13	NUM
ejpam-5732	354	19	references	reference	NOUN
ejpam-5732	354	20	[	[	X
ejpam-5732	354	21	1	1	NUM
ejpam-5732	354	22	]	]	PUNCT
ejpam-5732	354	23	m.	m.	PROPN
ejpam-5732	354	24	e.	e.	PROPN
ejpam-5732	354	25	abd.el	abd.el	PROPN
ejpam-5732	354	26	-	-	PUNCT
ejpam-5732	354	27	monsef	monsef	ADJ
ejpam-5732	354	28	,	,	PUNCT
ejpam-5732	354	29	e.	e.	PROPN
ejpam-5732	354	30	f.	f.	PROPN
ejpam-5732	354	31	lashien	lashien	PROPN
ejpam-5732	354	32	,	,	PUNCT
ejpam-5732	354	33	and	and	CCONJ
ejpam-5732	354	34	a.	a.	NOUN
ejpam-5732	354	35	a.	a.	NOUN
ejpam-5732	354	36	nasef	nasef	PROPN
ejpam-5732	354	37	.	.	PUNCT
ejpam-5732	355	1	on	on	ADP
ejpam-5732	355	2	i	i	NOUN
ejpam-5732	355	3	-	-	PUNCT
ejpam-5732	355	4	open	open	ADJ
ejpam-5732	355	5	sets	set	NOUN
ejpam-5732	355	6	and	and	CCONJ
ejpam-5732	355	7	icontinuous	icontinuous	ADJ
ejpam-5732	355	8	functions	function	NOUN
ejpam-5732	355	9	.	.	PUNCT
ejpam-5732	356	1	kyungpook	kyungpook	PROPN
ejpam-5732	356	2	mathematical	mathematical	PROPN
ejpam-5732	356	3	journal	journal	PROPN
ejpam-5732	356	4	,	,	PUNCT
ejpam-5732	356	5	32(2):21–30	32(2):21–30	NUM
ejpam-5732	356	6	,	,	PUNCT
ejpam-5732	356	7	1992	1992	NUM
ejpam-5732	356	8	.	.	PUNCT
ejpam-5732	357	1	[	[	X
ejpam-5732	357	2	2	2	NUM
ejpam-5732	357	3	]	]	PUNCT
ejpam-5732	357	4	m.	m.	NOUN
ejpam-5732	357	5	e.	e.	PROPN
ejpam-5732	357	6	abd.el	abd.el	PROPN
ejpam-5732	357	7	-	-	PUNCT
ejpam-5732	357	8	monsef	monsef	ADJ
ejpam-5732	357	9	,	,	PUNCT
ejpam-5732	357	10	e.	e.	PROPN
ejpam-5732	357	11	f.	f.	PROPN
ejpam-5732	357	12	lashien	lashien	PROPN
ejpam-5732	357	13	,	,	PUNCT
ejpam-5732	357	14	and	and	CCONJ
ejpam-5732	357	15	a.	a.	NOUN
ejpam-5732	357	16	a.	a.	NOUN
ejpam-5732	357	17	nasef	nasef	PROPN
ejpam-5732	357	18	.	.	PUNCT
ejpam-5732	358	1	some	some	DET
ejpam-5732	358	2	topological	topological	ADJ
ejpam-5732	358	3	operators	operator	NOUN
ejpam-5732	358	4	via	via	ADP
ejpam-5732	358	5	ideals	ideal	NOUN
ejpam-5732	358	6	.	.	PUNCT
ejpam-5732	359	1	kyungpook	kyungpook	PROPN
ejpam-5732	359	2	mathematical	mathematical	PROPN
ejpam-5732	359	3	journal	journal	PROPN
ejpam-5732	359	4	,	,	PUNCT
ejpam-5732	359	5	32(2):273284	32(2):273284	NUM
ejpam-5732	359	6	,	,	PUNCT
ejpam-5732	359	7	1992	1992	NUM
ejpam-5732	359	8	.	.	PUNCT
ejpam-5732	360	1	[	[	X
ejpam-5732	360	2	3	3	X
ejpam-5732	360	3	]	]	X
ejpam-5732	360	4	h.	h.	PROPN
ejpam-5732	360	5	h.	h.	PROPN
ejpam-5732	360	6	al	al	PROPN
ejpam-5732	360	7	-	-	PUNCT
ejpam-5732	360	8	jarrah	jarrah	PROPN
ejpam-5732	360	9	.	.	PUNCT
ejpam-5732	361	1	β1	β1	NOUN
ejpam-5732	361	2	-	-	PUNCT
ejpam-5732	361	3	paracompact	paracompact	NOUN
ejpam-5732	361	4	spaces	space	NOUN
ejpam-5732	361	5	.	.	PUNCT
ejpam-5732	362	1	journal	journal	PROPN
ejpam-5732	362	2	of	of	ADP
ejpam-5732	362	3	nonlinear	nonlinear	PROPN
ejpam-5732	362	4	sciences	sciences	PROPN
ejpam-5732	362	5	and	and	CCONJ
ejpam-5732	362	6	applications	application	NOUN
ejpam-5732	362	7	,	,	PUNCT
ejpam-5732	362	8	9:1728–1734	9:1728–1734	NUM
ejpam-5732	362	9	,	,	PUNCT
ejpam-5732	362	10	2016	2016	NUM
ejpam-5732	362	11	.	.	PUNCT
ejpam-5732	363	1	[	[	X
ejpam-5732	363	2	4	4	X
ejpam-5732	363	3	]	]	PUNCT
ejpam-5732	363	4	k.	k.	PROPN
ejpam-5732	363	5	al	al	PROPN
ejpam-5732	363	6	-	-	PROPN
ejpam-5732	363	7	zoubi	zoubi	PROPN
ejpam-5732	363	8	.	.	PUNCT
ejpam-5732	364	1	s	s	X
ejpam-5732	364	2	-	-	PUNCT
ejpam-5732	364	3	paracompact	paracompact	ADJ
ejpam-5732	364	4	spaces	space	NOUN
ejpam-5732	364	5	.	.	PUNCT
ejpam-5732	365	1	acta	acta	PROPN
ejpam-5732	365	2	mathematica	mathematica	PROPN
ejpam-5732	365	3	hungarica	hungarica	PROPN
ejpam-5732	365	4	,	,	PUNCT
ejpam-5732	365	5	110:203–212	110:203–212	NUM
ejpam-5732	365	6	,	,	PUNCT
ejpam-5732	365	7	2006	2006	NUM
ejpam-5732	365	8	.	.	PUNCT
ejpam-5732	366	1	[	[	X
ejpam-5732	366	2	5	5	X
ejpam-5732	366	3	]	]	PUNCT
ejpam-5732	366	4	k.	k.	PROPN
ejpam-5732	366	5	al	al	PROPN
ejpam-5732	366	6	-	-	PROPN
ejpam-5732	366	7	zoubi	zoubi	PROPN
ejpam-5732	366	8	and	and	CCONJ
ejpam-5732	366	9	s.	s.	PROPN
ejpam-5732	366	10	al	al	PROPN
ejpam-5732	366	11	-	-	PROPN
ejpam-5732	366	12	ghour	ghour	PROPN
ejpam-5732	366	13	.	.	PUNCT
ejpam-5732	367	1	on	on	ADP
ejpam-5732	367	2	p3	p3	NOUN
ejpam-5732	367	3	-	-	PUNCT
ejpam-5732	367	4	paracompact	paracompact	ADJ
ejpam-5732	367	5	spaces	space	NOUN
ejpam-5732	367	6	.	.	PUNCT
ejpam-5732	368	1	international	international	ADJ
ejpam-5732	368	2	journal	journal	PROPN
ejpam-5732	368	3	of	of	ADP
ejpam-5732	368	4	mathematics	mathematics	PROPN
ejpam-5732	368	5	and	and	CCONJ
ejpam-5732	368	6	mathematical	mathematical	ADJ
ejpam-5732	368	7	sciences	science	NOUN
ejpam-5732	368	8	,	,	PUNCT
ejpam-5732	368	9	2007:1–16	2007:1–16	NUM
ejpam-5732	368	10	,	,	PUNCT
ejpam-5732	368	11	2007	2007	NUM
ejpam-5732	368	12	.	.	PUNCT
ejpam-5732	369	1	[	[	X
ejpam-5732	369	2	6	6	NUM
ejpam-5732	369	3	]	]	PUNCT
ejpam-5732	369	4	r.	r.	PROPN
ejpam-5732	369	5	alrababah	alrababah	PROPN
ejpam-5732	369	6	,	,	PUNCT
ejpam-5732	369	7	a.	a.	PROPN
ejpam-5732	369	8	amourah	amourah	PROPN
ejpam-5732	369	9	,	,	PUNCT
ejpam-5732	369	10	j.	j.	PROPN
ejpam-5732	369	11	salah	salah	PROPN
ejpam-5732	369	12	,	,	PUNCT
ejpam-5732	369	13	and	and	CCONJ
ejpam-5732	369	14	r.	r.	PROPN
ejpam-5732	369	15	ahmad	ahmad	PROPN
ejpam-5732	369	16	.	.	PUNCT
ejpam-5732	370	1	paracompactness	paracompactness	PROPN
ejpam-5732	370	2	in	in	ADP
ejpam-5732	370	3	topological	topological	ADJ
ejpam-5732	370	4	spaces	space	NOUN
ejpam-5732	370	5	.	.	PUNCT
ejpam-5732	371	1	european	european	ADJ
ejpam-5732	371	2	journal	journal	PROPN
ejpam-5732	371	3	of	of	ADP
ejpam-5732	371	4	pure	pure	ADJ
ejpam-5732	371	5	and	and	CCONJ
ejpam-5732	371	6	applied	applied	ADJ
ejpam-5732	371	7	mathematic	mathematic	ADJ
ejpam-5732	371	8	,	,	PUNCT
ejpam-5732	371	9	17(4):2990–3003	17(4):2990–3003	NUM
ejpam-5732	371	10	,	,	PUNCT
ejpam-5732	371	11	2024	2024	NUM
ejpam-5732	371	12	.	.	PUNCT
ejpam-5732	372	1	[	[	X
ejpam-5732	372	2	7	7	NUM
ejpam-5732	372	3	]	]	X
ejpam-5732	372	4	a.	a.	NOUN
ejpam-5732	372	5	v.	v.	ADP
ejpam-5732	372	6	arkhangelskii	arkhangelskii	PROPN
ejpam-5732	372	7	and	and	CCONJ
ejpam-5732	372	8	v.	v.	ADP
ejpam-5732	372	9	i.	i.	PROPN
ejpam-5732	372	10	ponomarev	ponomarev	PROPN
ejpam-5732	372	11	.	.	PUNCT
ejpam-5732	373	1	fundamentals	fundamental	NOUN
ejpam-5732	373	2	of	of	ADP
ejpam-5732	373	3	general	general	ADJ
ejpam-5732	373	4	topology	topology	NOUN
ejpam-5732	373	5	:	:	PUNCT
ejpam-5732	373	6	problems	problem	NOUN
ejpam-5732	373	7	and	and	CCONJ
ejpam-5732	373	8	exercises	exercise	NOUN
ejpam-5732	373	9	.	.	PUNCT
ejpam-5732	374	1	international	international	ADJ
ejpam-5732	374	2	hindustan	hindustan	PROPN
ejpam-5732	374	3	publishing	publishing	NOUN
ejpam-5732	374	4	,	,	PUNCT
ejpam-5732	374	5	india	india	PROPN
ejpam-5732	374	6	,	,	PUNCT
ejpam-5732	374	7	1984	1984	NUM
ejpam-5732	374	8	.	.	PUNCT
ejpam-5732	375	1	[	[	X
ejpam-5732	375	2	8	8	NUM
ejpam-5732	375	3	]	]	X
ejpam-5732	375	4	n.	n.	NOUN
ejpam-5732	375	5	bourbaki	bourbaki	PROPN
ejpam-5732	375	6	.	.	PUNCT
ejpam-5732	376	1	general	general	ADJ
ejpam-5732	376	2	topology	topology	PROPN
ejpam-5732	376	3	.	.	PUNCT
ejpam-5732	377	1	hermann	hermann	PROPN
ejpam-5732	377	2	addison	addison	PROPN
ejpam-5732	377	3	wesley	wesley	PROPN
ejpam-5732	377	4	,	,	PUNCT
ejpam-5732	377	5	massachusets	massachusets	PROPN
ejpam-5732	377	6	,	,	PUNCT
ejpam-5732	377	7	1966	1966	NUM
ejpam-5732	377	8	.	.	PUNCT
ejpam-5732	378	1	[	[	X
ejpam-5732	378	2	9	9	NUM
ejpam-5732	378	3	]	]	SYM
ejpam-5732	378	4	i.	i.	PROPN
ejpam-5732	378	5	demir	demir	PROPN
ejpam-5732	378	6	and	and	CCONJ
ejpam-5732	378	7	o.b	o.b	PROPN
ejpam-5732	378	8	.	.	PROPN
ejpam-5732	378	9	ozbakir	ozbakir	PROPN
ejpam-5732	378	10	.	.	PUNCT
ejpam-5732	379	1	on	on	ADP
ejpam-5732	379	2	β	β	ADJ
ejpam-5732	379	3	-	-	ADJ
ejpam-5732	379	4	paracompact	paracompact	ADJ
ejpam-5732	379	5	spaces	space	NOUN
ejpam-5732	379	6	.	.	PUNCT
ejpam-5732	380	1	filomat	filomat	NOUN
ejpam-5732	380	2	,	,	PUNCT
ejpam-5732	380	3	27(6):971–976	27(6):971–976	PROPN
ejpam-5732	380	4	,	,	PUNCT
ejpam-5732	380	5	2013	2013	NUM
ejpam-5732	380	6	.	.	PUNCT
ejpam-5732	381	1	[	[	X
ejpam-5732	381	2	10	10	NUM
ejpam-5732	381	3	]	]	PUNCT
ejpam-5732	381	4	j.	j.	PROPN
ejpam-5732	381	5	dieudonné.	dieudonné.	PROPN
ejpam-5732	381	6	une	une	PROPN
ejpam-5732	381	7	généralisation	généralisation	PROPN
ejpam-5732	381	8	des	des	PROPN
ejpam-5732	381	9	espaces	espace	NOUN
ejpam-5732	381	10	compacts	compact	NOUN
ejpam-5732	381	11	.	.	PUNCT
ejpam-5732	382	1	journal	journal	PROPN
ejpam-5732	382	2	de	de	PROPN
ejpam-5732	382	3	mathématiques	mathématiques	PROPN
ejpam-5732	382	4	pures	pure	NOUN
ejpam-5732	382	5	et	et	NOUN
ejpam-5732	382	6	appliquées	appliquée	NOUN
ejpam-5732	382	7	,	,	PUNCT
ejpam-5732	382	8	23(9):65–76	23(9):65–76	NUM
ejpam-5732	382	9	,	,	PUNCT
ejpam-5732	382	10	1944	1944	NUM
ejpam-5732	382	11	.	.	PUNCT
ejpam-5732	383	1	[	[	X
ejpam-5732	383	2	11	11	NUM
ejpam-5732	383	3	]	]	PUNCT
ejpam-5732	383	4	j.	j.	PROPN
ejpam-5732	383	5	dontchev	dontchev	PROPN
ejpam-5732	383	6	,	,	PUNCT
ejpam-5732	383	7	m.	m.	NOUN
ejpam-5732	383	8	ganster	ganster	NOUN
ejpam-5732	383	9	,	,	PUNCT
ejpam-5732	383	10	and	and	CCONJ
ejpam-5732	383	11	t.	t.	PROPN
ejpam-5732	383	12	noiri	noiri	PROPN
ejpam-5732	383	13	.	.	PUNCT
ejpam-5732	384	1	unified	unified	ADJ
ejpam-5732	384	2	operation	operation	NOUN
ejpam-5732	384	3	approach	approach	NOUN
ejpam-5732	384	4	of	of	ADP
ejpam-5732	384	5	generalized	generalized	ADJ
ejpam-5732	384	6	closed	close	VERB
ejpam-5732	384	7	sets	set	NOUN
ejpam-5732	384	8	via	via	ADP
ejpam-5732	384	9	topological	topological	ADJ
ejpam-5732	384	10	ideals	ideal	NOUN
ejpam-5732	384	11	.	.	PUNCT
ejpam-5732	385	1	mathematica	mathematica	PROPN
ejpam-5732	385	2	japonica	japonica	PROPN
ejpam-5732	385	3	,	,	PUNCT
ejpam-5732	385	4	49:395–402	49:395–402	PROPN
ejpam-5732	385	5	,	,	PUNCT
ejpam-5732	385	6	1999	1999	NUM
ejpam-5732	385	7	.	.	PUNCT
ejpam-5732	386	1	[	[	X
ejpam-5732	386	2	12	12	NUM
ejpam-5732	386	3	]	]	PUNCT
ejpam-5732	386	4	j.	j.	PROPN
ejpam-5732	386	5	dugundji	dugundji	PROPN
ejpam-5732	386	6	.	.	PUNCT
ejpam-5732	387	1	topology	topology	PROPN
ejpam-5732	387	2	.	.	PUNCT
ejpam-5732	388	1	allyn	allyn	PROPN
ejpam-5732	388	2	and	and	CCONJ
ejpam-5732	388	3	bacon	bacon	PROPN
ejpam-5732	388	4	,	,	PUNCT
ejpam-5732	388	5	boston	boston	PROPN
ejpam-5732	388	6	,	,	PUNCT
ejpam-5732	388	7	1966	1966	NUM
ejpam-5732	388	8	.	.	PUNCT
ejpam-5732	389	1	[	[	X
ejpam-5732	389	2	13	13	NUM
ejpam-5732	389	3	]	]	PUNCT
ejpam-5732	389	4	t.	t.	PROPN
ejpam-5732	389	5	r.	r.	PROPN
ejpam-5732	389	6	hamlett	hamlett	PROPN
ejpam-5732	389	7	,	,	PUNCT
ejpam-5732	389	8	d.	d.	PROPN
ejpam-5732	389	9	rose	rise	VERB
ejpam-5732	389	10	,	,	PUNCT
ejpam-5732	389	11	and	and	CCONJ
ejpam-5732	389	12	d.	d.	PROPN
ejpam-5732	389	13	jankovic	jankovic	PROPN
ejpam-5732	389	14	.	.	PUNCT
ejpam-5732	390	1	paracompactness	paracompactness	PROPN
ejpam-5732	390	2	with	with	ADP
ejpam-5732	390	3	respect	respect	NOUN
ejpam-5732	390	4	to	to	ADP
ejpam-5732	390	5	an	an	DET
ejpam-5732	390	6	ideal	ideal	NOUN
ejpam-5732	390	7	.	.	PUNCT
ejpam-5732	391	1	international	international	ADJ
ejpam-5732	391	2	journal	journal	PROPN
ejpam-5732	391	3	of	of	ADP
ejpam-5732	391	4	mathematics	mathematics	PROPN
ejpam-5732	391	5	and	and	CCONJ
ejpam-5732	391	6	mathematical	mathematical	ADJ
ejpam-5732	391	7	sciences	science	NOUN
ejpam-5732	391	8	,	,	PUNCT
ejpam-5732	391	9	20:433–442	20:433–442	NUM
ejpam-5732	391	10	,	,	PUNCT
ejpam-5732	391	11	1997	1997	NUM
ejpam-5732	391	12	.	.	PUNCT
ejpam-5732	392	1	[	[	X
ejpam-5732	392	2	14	14	NUM
ejpam-5732	392	3	]	]	X
ejpam-5732	392	4	e.	e.	PROPN
ejpam-5732	392	5	hatir	hatir	PROPN
ejpam-5732	392	6	.	.	PUNCT
ejpam-5732	393	1	on	on	ADP
ejpam-5732	393	2	decompositions	decomposition	NOUN
ejpam-5732	393	3	of	of	ADP
ejpam-5732	393	4	continuity	continuity	NOUN
ejpam-5732	393	5	and	and	CCONJ
ejpam-5732	393	6	complete	complete	ADJ
ejpam-5732	393	7	continuity	continuity	NOUN
ejpam-5732	393	8	in	in	ADP
ejpam-5732	393	9	ideal	ideal	ADJ
ejpam-5732	393	10	topological	topological	ADJ
ejpam-5732	393	11	spaces	space	NOUN
ejpam-5732	393	12	.	.	PUNCT
ejpam-5732	394	1	european	european	ADJ
ejpam-5732	394	2	journal	journal	PROPN
ejpam-5732	394	3	of	of	ADP
ejpam-5732	394	4	pure	pure	ADJ
ejpam-5732	394	5	and	and	CCONJ
ejpam-5732	394	6	applied	applied	ADJ
ejpam-5732	394	7	mathematics	mathematic	NOUN
ejpam-5732	394	8	,	,	PUNCT
ejpam-5732	394	9	6(3):352–362	6(3):352–362	NUM
ejpam-5732	394	10	,	,	PUNCT
ejpam-5732	394	11	2013	2013	NUM
ejpam-5732	394	12	.	.	PUNCT
ejpam-5732	395	1	[	[	X
ejpam-5732	395	2	15	15	NUM
ejpam-5732	395	3	]	]	X
ejpam-5732	395	4	d.	d.	PROPN
ejpam-5732	395	5	janković	janković	PROPN
ejpam-5732	395	6	and	and	CCONJ
ejpam-5732	395	7	t.	t.	PROPN
ejpam-5732	395	8	r.	r.	PROPN
ejpam-5732	395	9	hamlett	hamlett	PROPN
ejpam-5732	395	10	.	.	PUNCT
ejpam-5732	396	1	new	new	ADJ
ejpam-5732	396	2	topologies	topology	NOUN
ejpam-5732	396	3	from	from	ADP
ejpam-5732	396	4	old	old	ADJ
ejpam-5732	396	5	via	via	ADP
ejpam-5732	396	6	ideals	ideal	NOUN
ejpam-5732	396	7	.	.	PUNCT
ejpam-5732	397	1	the	the	DET
ejpam-5732	397	2	american	american	PROPN
ejpam-5732	397	3	mathematical	mathematical	PROPN
ejpam-5732	397	4	monthly	monthly	PROPN
ejpam-5732	397	5	,	,	PUNCT
ejpam-5732	397	6	97(4):295–310	97(4):295–310	PROPN
ejpam-5732	397	7	,	,	PUNCT
ejpam-5732	397	8	1990	1990	NUM
ejpam-5732	397	9	.	.	PUNCT
ejpam-5732	398	1	[	[	X
ejpam-5732	398	2	16	16	NUM
ejpam-5732	398	3	]	]	X
ejpam-5732	398	4	m.	m.	NOUN
ejpam-5732	398	5	khan	khan	PROPN
ejpam-5732	398	6	and	and	CCONJ
ejpam-5732	398	7	t.	t.	PROPN
ejpam-5732	398	8	noiri	noiri	PROPN
ejpam-5732	398	9	.	.	PUNCT
ejpam-5732	399	1	semi	semi	ADJ
ejpam-5732	399	2	-	-	ADJ
ejpam-5732	399	3	local	local	ADJ
ejpam-5732	399	4	functions	function	NOUN
ejpam-5732	399	5	in	in	ADP
ejpam-5732	399	6	ideal	ideal	ADJ
ejpam-5732	399	7	topological	topological	ADJ
ejpam-5732	399	8	spaces	space	NOUN
ejpam-5732	399	9	.	.	PUNCT
ejpam-5732	400	1	journal	journal	NOUN
ejpam-5732	400	2	of	of	ADP
ejpam-5732	400	3	advanced	advanced	ADJ
ejpam-5732	400	4	research	research	NOUN
ejpam-5732	400	5	in	in	ADP
ejpam-5732	400	6	pure	pure	ADJ
ejpam-5732	400	7	mathematics	mathematic	NOUN
ejpam-5732	400	8	,	,	PUNCT
ejpam-5732	400	9	2(1):36–42	2(1):36–42	NUM
ejpam-5732	400	10	,	,	PUNCT
ejpam-5732	400	11	2010	2010	NUM
ejpam-5732	400	12	.	.	PUNCT
ejpam-5732	401	1	[	[	X
ejpam-5732	401	2	17	17	NUM
ejpam-5732	401	3	]	]	PUNCT
ejpam-5732	401	4	k.	k.	PROPN
ejpam-5732	401	5	kuratowski	kuratowski	PROPN
ejpam-5732	401	6	.	.	PUNCT
ejpam-5732	402	1	topology	topology	PROPN
ejpam-5732	402	2	i.	i.	PROPN
ejpam-5732	402	3	państwowe	państwowe	PROPN
ejpam-5732	402	4	wydawnictwo	wydawnictwo	PROPN
ejpam-5732	402	5	naukowe	naukowe	PROPN
ejpam-5732	402	6	,	,	PUNCT
ejpam-5732	402	7	warszawa	warszawa	PROPN
ejpam-5732	402	8	,	,	PUNCT
ejpam-5732	402	9	1933	1933	NUM
ejpam-5732	402	10	.	.	PUNCT
ejpam-5732	403	1	[	[	X
ejpam-5732	403	2	18	18	NUM
ejpam-5732	403	3	]	]	PUNCT
ejpam-5732	403	4	p.	p.	PROPN
ejpam-5732	403	5	y.	y.	PROPN
ejpam-5732	403	6	li	li	PROPN
ejpam-5732	403	7	and	and	CCONJ
ejpam-5732	403	8	y.	y.	PROPN
ejpam-5732	403	9	k.	k.	PROPN
ejpam-5732	403	10	song	song	PROPN
ejpam-5732	403	11	.	.	PUNCT
ejpam-5732	404	1	some	some	DET
ejpam-5732	404	2	remarks	remark	NOUN
ejpam-5732	404	3	on	on	ADP
ejpam-5732	404	4	s	s	NOUN
ejpam-5732	404	5	-	-	PUNCT
ejpam-5732	404	6	paracompact	paracompact	ADJ
ejpam-5732	404	7	spaces	space	NOUN
ejpam-5732	404	8	.	.	PUNCT
ejpam-5732	405	1	acta	acta	PROPN
ejpam-5732	405	2	mathematica	mathematica	PROPN
ejpam-5732	405	3	hungarica	hungarica	PROPN
ejpam-5732	405	4	,	,	PUNCT
ejpam-5732	405	5	118:345–355	118:345–355	NUM
ejpam-5732	405	6	,	,	PUNCT
ejpam-5732	405	7	2008	2008	NUM
ejpam-5732	405	8	.	.	PUNCT
ejpam-5732	406	1	[	[	X
ejpam-5732	406	2	19	19	NUM
ejpam-5732	406	3	]	]	PUNCT
ejpam-5732	406	4	t.	t.	PROPN
ejpam-5732	406	5	noiri	noiri	PROPN
ejpam-5732	406	6	.	.	PUNCT
ejpam-5732	407	1	completely	completely	ADV
ejpam-5732	407	2	continuous	continuous	ADJ
ejpam-5732	407	3	image	image	NOUN
ejpam-5732	407	4	of	of	ADP
ejpam-5732	407	5	nearly	nearly	ADV
ejpam-5732	407	6	paracompact	paracompact	ADJ
ejpam-5732	407	7	space	space	NOUN
ejpam-5732	407	8	.	.	PUNCT
ejpam-5732	408	1	matematichki	matematichki	PROPN
ejpam-5732	408	2	vesnik	vesnik	PROPN
ejpam-5732	408	3	,	,	PUNCT
ejpam-5732	408	4	29:59–64	29:59–64	PROPN
ejpam-5732	408	5	,	,	PUNCT
ejpam-5732	408	6	1977	1977	NUM
ejpam-5732	408	7	.	.	PUNCT
ejpam-5732	409	1	[	[	X
ejpam-5732	409	2	20	20	NUM
ejpam-5732	409	3	]	]	PUNCT
ejpam-5732	409	4	t.	t.	PROPN
ejpam-5732	409	5	noiri	noiri	PROPN
ejpam-5732	409	6	.	.	PUNCT
ejpam-5732	410	1	a	a	DET
ejpam-5732	410	2	note	note	NOUN
ejpam-5732	410	3	on	on	ADP
ejpam-5732	410	4	inverse	inverse	NOUN
ejpam-5732	410	5	-	-	PUNCT
ejpam-5732	410	6	preservations	preservation	NOUN
ejpam-5732	410	7	of	of	ADP
ejpam-5732	410	8	regular	regular	ADJ
ejpam-5732	410	9	open	open	ADJ
ejpam-5732	410	10	sets	set	NOUN
ejpam-5732	410	11	.	.	PUNCT
ejpam-5732	411	1	publications	publication	NOUN
ejpam-5732	411	2	de	de	X
ejpam-5732	411	3	l’institut	l’institut	X
ejpam-5732	411	4	mathématique	mathématique	PROPN
ejpam-5732	411	5	.	.	PUNCT
ejpam-5732	411	6	nouvelle	nouvelle	PROPN
ejpam-5732	411	7	série	série	PROPN
ejpam-5732	411	8	,	,	PUNCT
ejpam-5732	411	9	50:99–102	50:99–102	NUM
ejpam-5732	411	10	,	,	PUNCT
ejpam-5732	411	11	1984	1984	NUM
ejpam-5732	411	12	.	.	PUNCT
ejpam-5732	412	1	[	[	X
ejpam-5732	412	2	21	21	NUM
ejpam-5732	412	3	]	]	PUNCT
ejpam-5732	412	4	a.	a.	NOUN
ejpam-5732	412	5	qahis	qahis	PROPN
ejpam-5732	412	6	.	.	PUNCT
ejpam-5732	413	1	β1	β1	NOUN
ejpam-5732	413	2	-	-	PUNCT
ejpam-5732	413	3	paracompact	paracompact	NOUN
ejpam-5732	413	4	spaces	space	NOUN
ejpam-5732	413	5	with	with	ADP
ejpam-5732	413	6	respect	respect	NOUN
ejpam-5732	413	7	to	to	ADP
ejpam-5732	413	8	an	an	DET
ejpam-5732	413	9	ideal	ideal	NOUN
ejpam-5732	413	10	.	.	PUNCT
ejpam-5732	414	1	european	european	ADJ
ejpam-5732	414	2	journal	journal	PROPN
ejpam-5732	414	3	of	of	ADP
ejpam-5732	414	4	pure	pure	ADJ
ejpam-5732	414	5	and	and	CCONJ
ejpam-5732	414	6	applied	applied	ADJ
ejpam-5732	414	7	mathematics	mathematic	NOUN
ejpam-5732	414	8	,	,	PUNCT
ejpam-5732	414	9	12(1):135–145	12(1):135–145	PROPN
ejpam-5732	414	10	,	,	PUNCT
ejpam-5732	414	11	2019	2019	NUM
ejpam-5732	414	12	.	.	PUNCT
ejpam-5732	415	1	[	[	X
ejpam-5732	415	2	22	22	NUM
ejpam-5732	415	3	]	]	X
ejpam-5732	415	4	j.	j.	PROPN
ejpam-5732	415	5	sanabria	sanabria	PROPN
ejpam-5732	415	6	,	,	PUNCT
ejpam-5732	415	7	e.	e.	PROPN
ejpam-5732	415	8	rosas	rosas	PROPN
ejpam-5732	415	9	,	,	PUNCT
ejpam-5732	415	10	c.	c.	PROPN
ejpam-5732	415	11	carpintero	carpintero	PROPN
ejpam-5732	415	12	,	,	PUNCT
ejpam-5732	415	13	m.	m.	NOUN
ejpam-5732	415	14	salas	salas	PROPN
ejpam-5732	415	15	-	-	PUNCT
ejpam-5732	415	16	brown	brown	PROPN
ejpam-5732	415	17	,	,	PUNCT
ejpam-5732	415	18	and	and	CCONJ
ejpam-5732	415	19	o.	o.	PROPN
ejpam-5732	415	20	garćıa	garćıa	PROPN
ejpam-5732	415	21	.	.	PROPN
ejpam-5732	415	22	sparacompactness	sparacompactness	PROPN
ejpam-5732	415	23	in	in	ADP
ejpam-5732	415	24	ideal	ideal	ADJ
ejpam-5732	415	25	topological	topological	ADJ
ejpam-5732	415	26	spaces	space	NOUN
ejpam-5732	415	27	.	.	PUNCT
ejpam-5732	416	1	matematički	matematički	PROPN
ejpam-5732	416	2	vesnik	vesnik	PROPN
ejpam-5732	416	3	,	,	PUNCT
ejpam-5732	416	4	68(3):192–203	68(3):192–203	PROPN
ejpam-5732	416	5	,	,	PUNCT
ejpam-5732	416	6	c.	c.	PROPN
ejpam-5732	416	7	boonpok	boonpok	PROPN
ejpam-5732	416	8	,	,	PUNCT
ejpam-5732	416	9	a.	a.	PROPN
ejpam-5732	416	10	sama	sama	PROPN
ejpam-5732	416	11	-	-	PUNCT
ejpam-5732	416	12	ae	ae	PROPN
ejpam-5732	416	13	/	/	SYM
ejpam-5732	416	14	eur	eur	PROPN
ejpam-5732	416	15	.	.	PUNCT
ejpam-5732	417	1	j.	j.	PROPN
ejpam-5732	417	2	pure	pure	PROPN
ejpam-5732	417	3	appl	appl	PROPN
ejpam-5732	417	4	.	.	PROPN
ejpam-5732	417	5	math	math	PROPN
ejpam-5732	417	6	,	,	PUNCT
ejpam-5732	417	7	18	18	NUM
ejpam-5732	417	8	(	(	PUNCT
ejpam-5732	417	9	1	1	NUM
ejpam-5732	417	10	)	)	PUNCT
ejpam-5732	417	11	(	(	PUNCT
ejpam-5732	417	12	2025	2025	NUM
ejpam-5732	417	13	)	)	PUNCT
ejpam-5732	417	14	,	,	PUNCT
ejpam-5732	417	15	5732	5732	NUM
ejpam-5732	417	16	13	13	NUM
ejpam-5732	417	17	of	of	ADP
ejpam-5732	417	18	13	13	NUM
ejpam-5732	417	19	2016	2016	NUM
ejpam-5732	417	20	.	.	PUNCT
ejpam-5732	418	1	[	[	X
ejpam-5732	418	2	23	23	NUM
ejpam-5732	418	3	]	]	X
ejpam-5732	418	4	n.	n.	NOUN
ejpam-5732	418	5	sathiyasundari	sathiyasundari	PROPN
ejpam-5732	418	6	and	and	CCONJ
ejpam-5732	418	7	v.	v.	ADP
ejpam-5732	418	8	renukadevi	renukadevi	NOUN
ejpam-5732	418	9	.	.	PUNCT
ejpam-5732	419	1	paracompactness	paracompactness	NOUN
ejpam-5732	419	2	with	with	ADP
ejpam-5732	419	3	respect	respect	NOUN
ejpam-5732	419	4	to	to	ADP
ejpam-5732	419	5	an	an	DET
ejpam-5732	419	6	ideal	ideal	NOUN
ejpam-5732	419	7	.	.	PUNCT
ejpam-5732	420	1	filomat	filomat	NOUN
ejpam-5732	420	2	,	,	PUNCT
ejpam-5732	420	3	20(2):333–339	20(2):333–339	NOUN
ejpam-5732	420	4	,	,	PUNCT
ejpam-5732	420	5	2013	2013	NUM
ejpam-5732	420	6	.	.	PUNCT
ejpam-5732	421	1	[	[	X
ejpam-5732	421	2	24	24	NUM
ejpam-5732	421	3	]	]	PUNCT
ejpam-5732	421	4	m.	m.	NOUN
ejpam-5732	421	5	k.	k.	PROPN
ejpam-5732	421	6	singal	singal	PROPN
ejpam-5732	421	7	and	and	CCONJ
ejpam-5732	421	8	s.	s.	PROPN
ejpam-5732	421	9	p.	p.	PROPN
ejpam-5732	421	10	arya	arya	PROPN
ejpam-5732	421	11	.	.	PUNCT
ejpam-5732	422	1	on	on	ADP
ejpam-5732	422	2	nearly	nearly	ADV
ejpam-5732	422	3	paracompact	paracompact	ADJ
ejpam-5732	422	4	spaces	space	NOUN
ejpam-5732	422	5	.	.	PUNCT
ejpam-5732	423	1	matematički	matematički	PROPN
ejpam-5732	423	2	vesnik	vesnik	PROPN
ejpam-5732	423	3	,	,	PUNCT
ejpam-5732	423	4	6(47):3–16	6(47):3–16	NOUN
ejpam-5732	423	5	,	,	PUNCT
ejpam-5732	423	6	1969	1969	NUM
ejpam-5732	423	7	.	.	PUNCT
ejpam-5732	424	1	[	[	X
ejpam-5732	424	2	25	25	NUM
ejpam-5732	424	3	]	]	X
ejpam-5732	424	4	e.	e.	PROPN
ejpam-5732	424	5	d.	d.	PROPN
ejpam-5732	424	6	yildirim	yildirim	PROPN
ejpam-5732	424	7	,	,	PUNCT
ejpam-5732	424	8	o.	o.	PROPN
ejpam-5732	424	9	b.	b.	PROPN
ejpam-5732	424	10	ozbakir	ozbakir	PROPN
ejpam-5732	424	11	,	,	PUNCT
ejpam-5732	424	12	and	and	CCONJ
ejpam-5732	424	13	a.	a.	PROPN
ejpam-5732	424	14	c.	c.	PROPN
ejpam-5732	424	15	guler	guler	NOUN
ejpam-5732	424	16	.	.	PUNCT
ejpam-5732	425	1	β	β	X
ejpam-5732	425	2	-	-	PUNCT
ejpam-5732	425	3	paracompactness	paracompactness	NOUN
ejpam-5732	425	4	in	in	ADP
ejpam-5732	425	5	ideal	ideal	ADJ
ejpam-5732	425	6	topological	topological	ADJ
ejpam-5732	425	7	space	space	NOUN
ejpam-5732	425	8	.	.	PUNCT
ejpam-5732	426	1	european	european	ADJ
ejpam-5732	426	2	journal	journal	PROPN
ejpam-5732	426	3	of	of	ADP
ejpam-5732	426	4	pure	pure	ADJ
ejpam-5732	426	5	and	and	CCONJ
ejpam-5732	426	6	applied	applied	ADJ
ejpam-5732	426	7	mathematics	mathematic	NOUN
ejpam-5732	426	8	,	,	PUNCT
ejpam-5732	426	9	12(2):270–278	12(2):270–278	PROPN
ejpam-5732	426	10	,	,	PUNCT
ejpam-5732	426	11	2019	2019	NUM
ejpam-5732	426	12	.	.	PUNCT
ejpam-5732	427	1	[	[	X
ejpam-5732	427	2	26	26	NUM
ejpam-5732	427	3	]	]	PUNCT
ejpam-5732	427	4	s.	s.	PROPN
ejpam-5732	427	5	yuksel	yuksel	PROPN
ejpam-5732	427	6	,	,	PUNCT
ejpam-5732	427	7	a.	a.	PROPN
ejpam-5732	427	8	acikgoz	acikgoz	PROPN
ejpam-5732	427	9	,	,	PUNCT
ejpam-5732	427	10	and	and	CCONJ
ejpam-5732	427	11	t.	t.	PROPN
ejpam-5732	427	12	noiri	noiri	PROPN
ejpam-5732	427	13	.	.	PUNCT
ejpam-5732	428	1	on	on	ADP
ejpam-5732	428	2	δ	δ	PROPN
ejpam-5732	428	3	-	-	PUNCT
ejpam-5732	428	4	i	i	NOUN
ejpam-5732	428	5	-	-	PUNCT
ejpam-5732	428	6	continuous	continuous	ADJ
ejpam-5732	428	7	functions	function	NOUN
ejpam-5732	428	8	.	.	PUNCT
ejpam-5732	429	1	acta	acta	PROPN
ejpam-5732	429	2	mathematica	mathematica	PROPN
ejpam-5732	429	3	hungarica	hungarica	PROPN
ejpam-5732	429	4	,	,	PUNCT
ejpam-5732	429	5	29:39–51	29:39–51	NUM
ejpam-5732	429	6	,	,	PUNCT
ejpam-5732	429	7	2005	2005	NUM
ejpam-5732	429	8	.	.	PUNCT
ejpam-5732	430	1	[	[	X
ejpam-5732	430	2	27	27	NUM
ejpam-5732	430	3	]	]	PUNCT
ejpam-5732	430	4	m.	m.	NOUN
ejpam-5732	430	5	i.	i.	PROPN
ejpam-5732	430	6	zahid	zahid	PROPN
ejpam-5732	430	7	.	.	PUNCT
ejpam-5732	431	1	para	para	PROPN
ejpam-5732	431	2	h	h	NOUN
ejpam-5732	431	3	-	-	PUNCT
ejpam-5732	431	4	closed	closed	ADJ
ejpam-5732	431	5	spaces	space	NOUN
ejpam-5732	431	6	,	,	PUNCT
ejpam-5732	431	7	locally	locally	ADV
ejpam-5732	431	8	para	para	ADJ
ejpam-5732	431	9	h	h	NOUN
ejpam-5732	431	10	-	-	PUNCT
ejpam-5732	431	11	closed	closed	ADJ
ejpam-5732	431	12	spaces	space	NOUN
ejpam-5732	431	13	and	and	CCONJ
ejpam-5732	431	14	their	their	PRON
ejpam-5732	431	15	minimal	minimal	ADJ
ejpam-5732	431	16	topologies	topology	NOUN
ejpam-5732	431	17	.	.	PUNCT
ejpam-5732	432	1	phd	phd	NOUN
ejpam-5732	432	2	thesis	thesis	PROPN
ejpam-5732	432	3	,	,	PUNCT
ejpam-5732	432	4	university	university	PROPN
ejpam-5732	432	5	of	of	ADP
ejpam-5732	432	6	pittsburgh	pittsburgh	PROPN
ejpam-5732	432	7	,	,	PUNCT
ejpam-5732	432	8	1981	1981	NUM
ejpam-5732	432	9	.	.	PUNCT