id	sid	tid	token	lemma	pos
ejpam-5570	1	1	european	european	PROPN
ejpam-5570	1	2	journal	journal	PROPN
ejpam-5570	1	3	of	of	ADP
ejpam-5570	1	4	pure	pure	ADJ
ejpam-5570	1	5	and	and	CCONJ
ejpam-5570	1	6	applied	applied	ADJ
ejpam-5570	1	7	mathematics	mathematic	NOUN
ejpam-5570	1	8	2025	2025	NUM
ejpam-5570	1	9	,	,	PUNCT
ejpam-5570	1	10	vol	vol	NOUN
ejpam-5570	1	11	.	.	PROPN
ejpam-5570	1	12	18	18	NUM
ejpam-5570	1	13	,	,	PUNCT
ejpam-5570	1	14	issue	issue	NOUN
ejpam-5570	1	15	1	1	NUM
ejpam-5570	1	16	,	,	PUNCT
ejpam-5570	1	17	article	article	NOUN
ejpam-5570	1	18	number	number	NOUN
ejpam-5570	1	19	5570	5570	NUM
ejpam-5570	1	20	issn	issn	PROPN
ejpam-5570	1	21	1307	1307	NUM
ejpam-5570	1	22	-	-	SYM
ejpam-5570	1	23	5543	5543	NUM
ejpam-5570	1	24	–	–	PUNCT
ejpam-5570	1	25	ejpam.com	ejpam.com	X
ejpam-5570	1	26	published	publish	VERB
ejpam-5570	1	27	by	by	ADP
ejpam-5570	1	28	new	new	PROPN
ejpam-5570	1	29	york	york	PROPN
ejpam-5570	1	30	business	business	PROPN
ejpam-5570	1	31	global	global	ADJ
ejpam-5570	1	32	characterizations	characterization	NOUN
ejpam-5570	1	33	of	of	ADP
ejpam-5570	1	34	generalized	generalized	ADJ
ejpam-5570	1	35	paracompactness	paracompactness	NOUN
ejpam-5570	1	36	in	in	ADP
ejpam-5570	1	37	ideal	ideal	ADJ
ejpam-5570	1	38	topological	topological	ADJ
ejpam-5570	1	39	spaces	space	NOUN
ejpam-5570	1	40	chawalit	chawalit	VERB
ejpam-5570	1	41	boonpok1	boonpok1	PROPN
ejpam-5570	1	42	,	,	PUNCT
ejpam-5570	1	43	areeyuth	areeyuth	NOUN
ejpam-5570	1	44	sama	sama	NOUN
ejpam-5570	1	45	-	-	PUNCT
ejpam-5570	1	46	ae2,∗	ae2,∗	NOUN
ejpam-5570	1	47	,	,	PUNCT
ejpam-5570	1	48	palin	palin	PROPN
ejpam-5570	1	49	raktaow3	raktaow3	NOUN
ejpam-5570	1	50	1	1	NUM
ejpam-5570	1	51	mathematics	mathematic	NOUN
ejpam-5570	1	52	and	and	CCONJ
ejpam-5570	1	53	applied	apply	VERB
ejpam-5570	1	54	mathematics	mathematics	PROPN
ejpam-5570	1	55	research	research	NOUN
ejpam-5570	1	56	unit	unit	NOUN
ejpam-5570	1	57	,	,	PUNCT
ejpam-5570	1	58	department	department	NOUN
ejpam-5570	1	59	of	of	ADP
ejpam-5570	1	60	mathematics	mathematic	NOUN
ejpam-5570	1	61	,	,	PUNCT
ejpam-5570	1	62	faculty	faculty	NOUN
ejpam-5570	1	63	of	of	ADP
ejpam-5570	1	64	science	science	NOUN
ejpam-5570	1	65	,	,	PUNCT
ejpam-5570	1	66	mahasarakham	mahasarakham	PROPN
ejpam-5570	1	67	university	university	PROPN
ejpam-5570	1	68	,	,	PUNCT
ejpam-5570	1	69	maha	maha	PROPN
ejpam-5570	1	70	sarakham	sarakham	PROPN
ejpam-5570	1	71	,	,	PUNCT
ejpam-5570	1	72	44150	44150	NUM
ejpam-5570	1	73	,	,	PUNCT
ejpam-5570	1	74	thailand	thailand	PROPN
ejpam-5570	1	75	2	2	NUM
ejpam-5570	1	76	department	department	NOUN
ejpam-5570	1	77	of	of	ADP
ejpam-5570	1	78	mathematics	mathematic	NOUN
ejpam-5570	1	79	and	and	CCONJ
ejpam-5570	1	80	computer	computer	NOUN
ejpam-5570	1	81	science	science	NOUN
ejpam-5570	1	82	,	,	PUNCT
ejpam-5570	1	83	faculty	faculty	NOUN
ejpam-5570	1	84	of	of	ADP
ejpam-5570	1	85	science	science	NOUN
ejpam-5570	1	86	and	and	CCONJ
ejpam-5570	1	87	technology	technology	NOUN
ejpam-5570	1	88	,	,	PUNCT
ejpam-5570	1	89	prince	prince	NOUN
ejpam-5570	1	90	of	of	ADP
ejpam-5570	1	91	songkla	songkla	PROPN
ejpam-5570	1	92	university	university	PROPN
ejpam-5570	1	93	,	,	PUNCT
ejpam-5570	1	94	pattani	pattani	NOUN
ejpam-5570	1	95	campus	campus	NOUN
ejpam-5570	1	96	,	,	PUNCT
ejpam-5570	1	97	pattani	pattani	NOUN
ejpam-5570	1	98	,	,	PUNCT
ejpam-5570	1	99	94000	94000	NUM
ejpam-5570	1	100	,	,	PUNCT
ejpam-5570	1	101	thailand	thailand	PROPN
ejpam-5570	1	102	3	3	NUM
ejpam-5570	1	103	applied	apply	VERB
ejpam-5570	1	104	mathematics	mathematic	NOUN
ejpam-5570	1	105	and	and	CCONJ
ejpam-5570	1	106	innovation	innovation	NOUN
ejpam-5570	1	107	of	of	ADP
ejpam-5570	1	108	mathematics	mathematic	NOUN
ejpam-5570	1	109	teaching	teaching	NOUN
ejpam-5570	1	110	,	,	PUNCT
ejpam-5570	1	111	faculty	faculty	NOUN
ejpam-5570	1	112	of	of	ADP
ejpam-5570	1	113	science	science	NOUN
ejpam-5570	1	114	and	and	CCONJ
ejpam-5570	1	115	technology	technology	NOUN
ejpam-5570	1	116	,	,	PUNCT
ejpam-5570	1	117	prince	prince	NOUN
ejpam-5570	1	118	of	of	ADP
ejpam-5570	1	119	songkla	songkla	PROPN
ejpam-5570	1	120	university	university	PROPN
ejpam-5570	1	121	,	,	PUNCT
ejpam-5570	1	122	pattani	pattani	NOUN
ejpam-5570	1	123	campus	campus	NOUN
ejpam-5570	1	124	,	,	PUNCT
ejpam-5570	1	125	pattani	pattani	NOUN
ejpam-5570	1	126	,	,	PUNCT
ejpam-5570	1	127	94000	94000	NUM
ejpam-5570	1	128	,	,	PUNCT
ejpam-5570	1	129	thailand	thailand	PROPN
ejpam-5570	1	130	abstract	abstract	PROPN
ejpam-5570	1	131	.	.	PUNCT
ejpam-5570	2	1	the	the	DET
ejpam-5570	2	2	concept	concept	NOUN
ejpam-5570	2	3	of	of	ADP
ejpam-5570	2	4	δ	δ	PROPN
ejpam-5570	2	5	-	-	PUNCT
ejpam-5570	2	6	βi	βi	PROPN
ejpam-5570	2	7	-	-	NOUN
ejpam-5570	2	8	paracompactness	paracompactness	NOUN
ejpam-5570	2	9	in	in	ADP
ejpam-5570	2	10	ideal	ideal	ADJ
ejpam-5570	2	11	topological	topological	ADJ
ejpam-5570	2	12	spaces	space	NOUN
ejpam-5570	2	13	is	be	AUX
ejpam-5570	2	14	introduced	introduce	VERB
ejpam-5570	2	15	as	as	ADP
ejpam-5570	2	16	a	a	DET
ejpam-5570	2	17	weaker	weak	ADJ
ejpam-5570	2	18	form	form	NOUN
ejpam-5570	2	19	of	of	ADP
ejpam-5570	2	20	β	β	NOUN
ejpam-5570	2	21	-	-	NOUN
ejpam-5570	2	22	paracompactness	paracompactness	NOUN
ejpam-5570	2	23	,	,	PUNCT
ejpam-5570	2	24	which	which	PRON
ejpam-5570	2	25	was	be	AUX
ejpam-5570	2	26	looked	look	VERB
ejpam-5570	2	27	at	at	ADP
ejpam-5570	2	28	in	in	ADP
ejpam-5570	2	29	[	[	X
ejpam-5570	2	30	17	17	NUM
ejpam-5570	2	31	]	]	PUNCT
ejpam-5570	2	32	.	.	PUNCT
ejpam-5570	3	1	this	this	DET
ejpam-5570	3	2	study	study	NOUN
ejpam-5570	3	3	examines	examine	VERB
ejpam-5570	3	4	several	several	ADJ
ejpam-5570	3	5	characterizations	characterization	NOUN
ejpam-5570	3	6	of	of	ADP
ejpam-5570	3	7	δ	δ	PROPN
ejpam-5570	3	8	-	-	PUNCT
ejpam-5570	3	9	βi	βi	ADV
ejpam-5570	3	10	-	-	PUNCT
ejpam-5570	3	11	paracompact	paracompact	ADJ
ejpam-5570	3	12	spaces	space	NOUN
ejpam-5570	3	13	and	and	CCONJ
ejpam-5570	3	14	its	its	PRON
ejpam-5570	3	15	subsets	subset	NOUN
ejpam-5570	3	16	.	.	PUNCT
ejpam-5570	4	1	furthermore	furthermore	ADV
ejpam-5570	4	2	,	,	PUNCT
ejpam-5570	4	3	we	we	PRON
ejpam-5570	4	4	investigate	investigate	VERB
ejpam-5570	4	5	the	the	DET
ejpam-5570	4	6	invariants	invariant	NOUN
ejpam-5570	4	7	of	of	ADP
ejpam-5570	4	8	δ	δ	PROPN
ejpam-5570	4	9	-	-	PUNCT
ejpam-5570	4	10	βi	βi	PROPN
ejpam-5570	4	11	-	-	PUNCT
ejpam-5570	4	12	paracompactness	paracompactness	NOUN
ejpam-5570	4	13	through	through	ADP
ejpam-5570	4	14	functions	function	NOUN
ejpam-5570	4	15	.	.	PUNCT
ejpam-5570	5	1	2020	2020	NUM
ejpam-5570	5	2	mathematics	mathematic	NOUN
ejpam-5570	5	3	subject	subject	NOUN
ejpam-5570	5	4	classifications	classification	NOUN
ejpam-5570	5	5	:	:	PUNCT
ejpam-5570	5	6	54a05	54a05	NUM
ejpam-5570	5	7	,	,	PUNCT
ejpam-5570	5	8	54b05	54b05	NUM
ejpam-5570	5	9	,	,	PUNCT
ejpam-5570	5	10	54c08	54c08	NUM
ejpam-5570	5	11	key	key	ADJ
ejpam-5570	5	12	words	word	NOUN
ejpam-5570	5	13	and	and	CCONJ
ejpam-5570	5	14	phrases	phrase	NOUN
ejpam-5570	5	15	:	:	PUNCT
ejpam-5570	5	16	ideal	ideal	ADJ
ejpam-5570	5	17	topological	topological	ADJ
ejpam-5570	5	18	space	space	NOUN
ejpam-5570	5	19	,	,	PUNCT
ejpam-5570	5	20	δ	δ	PROPN
ejpam-5570	5	21	-	-	PUNCT
ejpam-5570	5	22	βi	βi	PROPN
ejpam-5570	5	23	-	-	PUNCT
ejpam-5570	5	24	paracompactness	paracompactness	NOUN
ejpam-5570	5	25	,	,	PUNCT
ejpam-5570	5	26	locally	locally	ADV
ejpam-5570	5	27	finite	finite	VERB
ejpam-5570	5	28	collection	collection	NOUN
ejpam-5570	5	29	1	1	NUM
ejpam-5570	5	30	.	.	PUNCT
ejpam-5570	6	1	introduction	introduction	NOUN
ejpam-5570	6	2	established	establish	VERB
ejpam-5570	6	3	considerably	considerably	ADV
ejpam-5570	6	4	later	later	ADV
ejpam-5570	6	5	than	than	ADP
ejpam-5570	6	6	the	the	DET
ejpam-5570	6	7	two	two	NUM
ejpam-5570	6	8	earlier	early	ADJ
ejpam-5570	6	9	classes	class	NOUN
ejpam-5570	6	10	,	,	PUNCT
ejpam-5570	6	11	paracompact	paracompact	ADJ
ejpam-5570	6	12	spaces	space	NOUN
ejpam-5570	6	13	are	be	AUX
ejpam-5570	6	14	considered	consider	VERB
ejpam-5570	6	15	one	one	NUM
ejpam-5570	6	16	of	of	ADP
ejpam-5570	6	17	the	the	DET
ejpam-5570	6	18	most	most	ADV
ejpam-5570	6	19	important	important	ADJ
ejpam-5570	6	20	classes	class	NOUN
ejpam-5570	6	21	of	of	ADP
ejpam-5570	6	22	topological	topological	ADJ
ejpam-5570	6	23	spaces	space	NOUN
ejpam-5570	6	24	,	,	PUNCT
ejpam-5570	6	25	concurrently	concurrently	ADV
ejpam-5570	6	26	generalizing	generalize	VERB
ejpam-5570	6	27	both	both	CCONJ
ejpam-5570	6	28	metrizable	metrizable	ADJ
ejpam-5570	6	29	and	and	CCONJ
ejpam-5570	6	30	compact	compact	ADJ
ejpam-5570	6	31	spaces	space	NOUN
ejpam-5570	6	32	.	.	PUNCT
ejpam-5570	7	1	topologists	topologist	NOUN
ejpam-5570	7	2	and	and	CCONJ
ejpam-5570	7	3	analysts	analyst	NOUN
ejpam-5570	7	4	quickly	quickly	ADV
ejpam-5570	7	5	acknowledged	acknowledge	VERB
ejpam-5570	7	6	paracompact	paracompact	ADJ
ejpam-5570	7	7	areas	area	NOUN
ejpam-5570	7	8	.	.	PUNCT
ejpam-5570	8	1	the	the	DET
ejpam-5570	8	2	concept	concept	NOUN
ejpam-5570	8	3	of	of	ADP
ejpam-5570	8	4	a	a	DET
ejpam-5570	8	5	paracompact	paracompact	ADJ
ejpam-5570	8	6	space	space	NOUN
ejpam-5570	8	7	in	in	ADP
ejpam-5570	8	8	mathematics	mathematic	NOUN
ejpam-5570	8	9	refers	refer	VERB
ejpam-5570	8	10	to	to	ADP
ejpam-5570	8	11	a	a	DET
ejpam-5570	8	12	topological	topological	ADJ
ejpam-5570	8	13	space	space	NOUN
ejpam-5570	8	14	in	in	ADP
ejpam-5570	8	15	which	which	PRON
ejpam-5570	8	16	each	each	DET
ejpam-5570	8	17	open	open	ADJ
ejpam-5570	8	18	cover	cover	NOUN
ejpam-5570	8	19	possesses	possess	VERB
ejpam-5570	8	20	an	an	DET
ejpam-5570	8	21	open	open	ADJ
ejpam-5570	8	22	refinement	refinement	NOUN
ejpam-5570	8	23	that	that	PRON
ejpam-5570	8	24	is	be	AUX
ejpam-5570	8	25	locally	locally	ADV
ejpam-5570	8	26	finite	finite	ADJ
ejpam-5570	8	27	.	.	PUNCT
ejpam-5570	9	1	this	this	DET
ejpam-5570	9	2	concept	concept	NOUN
ejpam-5570	9	3	of	of	ADP
ejpam-5570	9	4	spaces	space	NOUN
ejpam-5570	9	5	was	be	AUX
ejpam-5570	9	6	first	first	ADV
ejpam-5570	9	7	developed	develop	VERB
ejpam-5570	9	8	by	by	ADP
ejpam-5570	9	9	dieudonné	dieudonné	NOUN
ejpam-5570	9	10	[	[	X
ejpam-5570	9	11	4	4	NUM
ejpam-5570	9	12	]	]	PUNCT
ejpam-5570	9	13	in	in	ADP
ejpam-5570	9	14	1944	1944	NUM
ejpam-5570	9	15	.	.	PUNCT
ejpam-5570	10	1	a	a	DET
ejpam-5570	10	2	hausdorff	hausdorff	NOUN
ejpam-5570	10	3	space	space	NOUN
ejpam-5570	10	4	is	be	AUX
ejpam-5570	10	5	considered	consider	VERB
ejpam-5570	10	6	paracompact	paracompact	ADJ
ejpam-5570	10	7	if	if	SCONJ
ejpam-5570	10	8	and	and	CCONJ
ejpam-5570	10	9	only	only	ADV
ejpam-5570	10	10	if	if	SCONJ
ejpam-5570	10	11	it	it	PRON
ejpam-5570	10	12	allows	allow	VERB
ejpam-5570	10	13	partitions	partition	NOUN
ejpam-5570	10	14	of	of	ADP
ejpam-5570	10	15	unity	unity	NOUN
ejpam-5570	10	16	that	that	PRON
ejpam-5570	10	17	are	be	AUX
ejpam-5570	10	18	subordinate	subordinate	ADJ
ejpam-5570	10	19	to	to	ADP
ejpam-5570	10	20	any	any	DET
ejpam-5570	10	21	open	open	ADJ
ejpam-5570	10	22	cover	cover	NOUN
ejpam-5570	10	23	.	.	PUNCT
ejpam-5570	11	1	all	all	DET
ejpam-5570	11	2	paracompact	paracompact	ADJ
ejpam-5570	11	3	hausdorff	hausdorff	NOUN
ejpam-5570	11	4	spaces	space	NOUN
ejpam-5570	11	5	are	be	AUX
ejpam-5570	11	6	normal	normal	ADJ
ejpam-5570	11	7	;	;	PUNCT
ejpam-5570	11	8	see	see	VERB
ejpam-5570	11	9	[	[	X
ejpam-5570	11	10	6	6	NUM
ejpam-5570	11	11	]	]	PUNCT
ejpam-5570	11	12	.	.	PUNCT
ejpam-5570	12	1	in	in	ADP
ejpam-5570	12	2	literature	literature	NOUN
ejpam-5570	12	3	,	,	PUNCT
ejpam-5570	12	4	different	different	ADJ
ejpam-5570	12	5	kinds	kind	NOUN
ejpam-5570	12	6	of	of	ADP
ejpam-5570	12	7	generalized	generalized	ADJ
ejpam-5570	12	8	paracompactness	paracompactness	NOUN
ejpam-5570	12	9	,	,	PUNCT
ejpam-5570	12	10	such	such	ADJ
ejpam-5570	12	11	as	as	ADP
ejpam-5570	12	12	s	s	NOUN
ejpam-5570	12	13	-	-	NOUN
ejpam-5570	12	14	paracompactness	paracompactness	NOUN
ejpam-5570	12	15	[	[	X
ejpam-5570	12	16	1	1	NUM
ejpam-5570	12	17	]	]	PUNCT
ejpam-5570	12	18	,	,	PUNCT
ejpam-5570	12	19	p3	p3	PROPN
ejpam-5570	12	20	-	-	NOUN
ejpam-5570	12	21	paracompactness	paracompactness	NOUN
ejpam-5570	12	22	[	[	X
ejpam-5570	12	23	3	3	NUM
ejpam-5570	12	24	]	]	PUNCT
ejpam-5570	12	25	,	,	PUNCT
ejpam-5570	12	26	and	and	CCONJ
ejpam-5570	12	27	β	β	X
ejpam-5570	12	28	-	-	NOUN
ejpam-5570	12	29	paracompactness	paracompactness	NOUN
ejpam-5570	12	30	[	[	X
ejpam-5570	12	31	2	2	NUM
ejpam-5570	12	32	]	]	PUNCT
ejpam-5570	12	33	,	,	PUNCT
ejpam-5570	12	34	are	be	AUX
ejpam-5570	12	35	studied	study	VERB
ejpam-5570	12	36	.	.	PUNCT
ejpam-5570	13	1	in	in	ADP
ejpam-5570	13	2	2006	2006	NUM
ejpam-5570	13	3	,	,	PUNCT
ejpam-5570	13	4	al	al	PROPN
ejpam-5570	13	5	-	-	PUNCT
ejpam-5570	13	6	zoubi	zoubi	PROPN
ejpam-5570	13	7	[	[	X
ejpam-5570	13	8	1	1	X
ejpam-5570	13	9	]	]	PUNCT
ejpam-5570	13	10	used	use	VERB
ejpam-5570	13	11	semi	semi	ADJ
ejpam-5570	13	12	-	-	ADJ
ejpam-5570	13	13	open	open	ADJ
ejpam-5570	13	14	sets	set	NOUN
ejpam-5570	13	15	to	to	PART
ejpam-5570	13	16	define	define	VERB
ejpam-5570	13	17	s	s	NOUN
ejpam-5570	13	18	-	-	PUNCT
ejpam-5570	13	19	paracompact	paracompact	ADJ
ejpam-5570	13	20	spaces	space	NOUN
ejpam-5570	13	21	,	,	PUNCT
ejpam-5570	13	22	which	which	PRON
ejpam-5570	13	23	are	be	AUX
ejpam-5570	13	24	a	a	DET
ejpam-5570	13	25	generalization	generalization	NOUN
ejpam-5570	13	26	of	of	ADP
ejpam-5570	13	27	paracompact	paracompact	ADJ
ejpam-5570	13	28	spaces	space	NOUN
ejpam-5570	13	29	,	,	PUNCT
ejpam-5570	13	30	and	and	CCONJ
ejpam-5570	13	31	studied	study	VERB
ejpam-5570	13	32	the	the	DET
ejpam-5570	13	33	relationship	relationship	NOUN
ejpam-5570	13	34	between	between	ADP
ejpam-5570	13	35	the	the	DET
ejpam-5570	13	36	spaces	space	NOUN
ejpam-5570	13	37	.	.	PUNCT
ejpam-5570	14	1	li	li	PROPN
ejpam-5570	14	2	and	and	CCONJ
ejpam-5570	14	3	song	song	NOUN
ejpam-5570	14	4	[	[	X
ejpam-5570	14	5	14	14	NUM
ejpam-5570	14	6	]	]	PUNCT
ejpam-5570	14	7	∗corresponding	∗corresponde	VERB
ejpam-5570	14	8	author	author	NOUN
ejpam-5570	14	9	.	.	PUNCT
ejpam-5570	15	1	doi	doi	NOUN
ejpam-5570	15	2	:	:	PUNCT
ejpam-5570	15	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5570	https://doi.org/10.29020/nybg.ejpam.v18i1.5570	NOUN
ejpam-5570	15	4	email	email	NOUN
ejpam-5570	15	5	addresses	address	NOUN
ejpam-5570	15	6	:	:	PUNCT
ejpam-5570	15	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-5570	15	8	(	(	PUNCT
ejpam-5570	15	9	c.	c.	PROPN
ejpam-5570	15	10	boonpok	boonpok	PROPN
ejpam-5570	15	11	)	)	PUNCT
ejpam-5570	15	12	,	,	PUNCT
ejpam-5570	15	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-5570	15	14	(	(	PUNCT
ejpam-5570	15	15	a.	a.	PROPN
ejpam-5570	15	16	sama	sama	PROPN
ejpam-5570	15	17	-	-	PUNCT
ejpam-5570	15	18	ae	ae	PROPN
ejpam-5570	15	19	)	)	PUNCT
ejpam-5570	15	20	,	,	PUNCT
ejpam-5570	15	21	palin.rt@gmail.com	palin.rt@gmail.com	PROPN
ejpam-5570	15	22	(	(	PUNCT
ejpam-5570	15	23	p.	p.	NOUN
ejpam-5570	15	24	raktaow	raktaow	PROPN
ejpam-5570	15	25	)	)	PUNCT
ejpam-5570	15	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5570	16	1	1	1	NUM
ejpam-5570	16	2	copyright	copyright	NOUN
ejpam-5570	16	3	:	:	PUNCT
ejpam-5570	16	4	©	©	PROPN
ejpam-5570	16	5	2025	2025	NUM
ejpam-5570	16	6	the	the	DET
ejpam-5570	16	7	author(s	author(s	NOUN
ejpam-5570	16	8	)	)	PUNCT
ejpam-5570	16	9	.	.	PUNCT
ejpam-5570	17	1	(	(	PUNCT
ejpam-5570	17	2	cc	cc	NOUN
ejpam-5570	17	3	by	by	ADP
ejpam-5570	17	4	-	-	PUNCT
ejpam-5570	17	5	nc	nc	PROPN
ejpam-5570	17	6	4.0	4.0	NUM
ejpam-5570	17	7	)	)	PUNCT
ejpam-5570	17	8	c.	c.	NOUN
ejpam-5570	17	9	boonpok	boonpok	PROPN
ejpam-5570	17	10	,	,	PUNCT
ejpam-5570	17	11	a.	a.	PROPN
ejpam-5570	17	12	sama	sama	PROPN
ejpam-5570	17	13	-	-	PUNCT
ejpam-5570	17	14	ae	ae	PROPN
ejpam-5570	17	15	,	,	PUNCT
ejpam-5570	17	16	p.	p.	NOUN
ejpam-5570	17	17	raktaow	raktaow	PROPN
ejpam-5570	17	18	/	/	SYM
ejpam-5570	17	19	eur	eur	PROPN
ejpam-5570	17	20	.	.	PUNCT
ejpam-5570	18	1	j.	j.	PROPN
ejpam-5570	18	2	pure	pure	PROPN
ejpam-5570	18	3	appl	appl	PROPN
ejpam-5570	18	4	.	.	PROPN
ejpam-5570	18	5	math	math	PROPN
ejpam-5570	18	6	,	,	PUNCT
ejpam-5570	18	7	18	18	NUM
ejpam-5570	18	8	(	(	PUNCT
ejpam-5570	18	9	1	1	NUM
ejpam-5570	18	10	)	)	PUNCT
ejpam-5570	18	11	(	(	PUNCT
ejpam-5570	18	12	2025	2025	NUM
ejpam-5570	18	13	)	)	PUNCT
ejpam-5570	18	14	,	,	PUNCT
ejpam-5570	18	15	5570	5570	NUM
ejpam-5570	18	16	2	2	NUM
ejpam-5570	18	17	of	of	ADP
ejpam-5570	18	18	12	12	NUM
ejpam-5570	18	19	constructed	construct	VERB
ejpam-5570	18	20	a	a	DET
ejpam-5570	18	21	hausdorff	hausdorff	NOUN
ejpam-5570	18	22	s	s	NOUN
ejpam-5570	18	23	-	-	PUNCT
ejpam-5570	18	24	paracompact	paracompact	ADJ
ejpam-5570	18	25	space	space	NOUN
ejpam-5570	18	26	that	that	PRON
ejpam-5570	18	27	is	be	AUX
ejpam-5570	18	28	not	not	PART
ejpam-5570	18	29	a	a	DET
ejpam-5570	18	30	paracompact	paracompact	ADJ
ejpam-5570	18	31	space	space	NOUN
ejpam-5570	18	32	and	and	CCONJ
ejpam-5570	18	33	studied	study	VERB
ejpam-5570	18	34	more	more	ADJ
ejpam-5570	18	35	characterizations	characterization	NOUN
ejpam-5570	18	36	of	of	ADP
ejpam-5570	18	37	s	s	NOUN
ejpam-5570	18	38	-	-	PUNCT
ejpam-5570	18	39	paracompact	paracompact	ADJ
ejpam-5570	18	40	spaces	space	NOUN
ejpam-5570	18	41	.	.	PUNCT
ejpam-5570	19	1	an	an	DET
ejpam-5570	19	2	ideal	ideal	ADJ
ejpam-5570	19	3	topological	topological	ADJ
ejpam-5570	19	4	space	space	NOUN
ejpam-5570	19	5	was	be	AUX
ejpam-5570	19	6	proposed	propose	VERB
ejpam-5570	19	7	by	by	ADP
ejpam-5570	19	8	kuratowski	kuratowski	PROPN
ejpam-5570	19	9	in	in	ADP
ejpam-5570	19	10	1930	1930	NUM
ejpam-5570	19	11	[	[	X
ejpam-5570	19	12	13	13	NUM
ejpam-5570	19	13	]	]	PUNCT
ejpam-5570	19	14	.	.	PUNCT
ejpam-5570	20	1	moreover	moreover	ADV
ejpam-5570	20	2	,	,	PUNCT
ejpam-5570	20	3	jankovic	jankovic	PROPN
ejpam-5570	20	4	and	and	CCONJ
ejpam-5570	20	5	hamlett	hamlett	PROPN
ejpam-5570	20	6	[	[	X
ejpam-5570	20	7	11	11	NUM
ejpam-5570	20	8	]	]	PUNCT
ejpam-5570	20	9	have	have	AUX
ejpam-5570	20	10	examined	examine	VERB
ejpam-5570	20	11	and	and	CCONJ
ejpam-5570	20	12	described	describe	VERB
ejpam-5570	20	13	the	the	DET
ejpam-5570	20	14	significant	significant	ADJ
ejpam-5570	20	15	properties	property	NOUN
ejpam-5570	20	16	of	of	ADP
ejpam-5570	20	17	ideal	ideal	ADJ
ejpam-5570	20	18	topological	topological	ADJ
ejpam-5570	20	19	spaces	space	NOUN
ejpam-5570	20	20	.	.	PUNCT
ejpam-5570	21	1	they	they	PRON
ejpam-5570	21	2	established	establish	VERB
ejpam-5570	21	3	the	the	DET
ejpam-5570	21	4	concept	concept	NOUN
ejpam-5570	21	5	of	of	ADP
ejpam-5570	21	6	i	i	NOUN
ejpam-5570	21	7	-	-	PUNCT
ejpam-5570	21	8	open	open	ADJ
ejpam-5570	21	9	sets	set	NOUN
ejpam-5570	21	10	and	and	CCONJ
ejpam-5570	21	11	undertook	undertake	VERB
ejpam-5570	21	12	comprehensive	comprehensive	ADJ
ejpam-5570	21	13	investigations	investigation	NOUN
ejpam-5570	21	14	into	into	ADP
ejpam-5570	21	15	topologies	topology	NOUN
ejpam-5570	21	16	utilizing	utilize	VERB
ejpam-5570	21	17	ideals	ideal	NOUN
ejpam-5570	21	18	.	.	PUNCT
ejpam-5570	22	1	abd	abd	PROPN
ejpam-5570	22	2	el	el	PROPN
ejpam-5570	22	3	-	-	PROPN
ejpam-5570	22	4	monsef	monsef	PROPN
ejpam-5570	22	5	et	et	PROPN
ejpam-5570	22	6	al	al	PROPN
ejpam-5570	22	7	.	.	PUNCT
ejpam-5570	23	1	[	[	X
ejpam-5570	23	2	7	7	X
ejpam-5570	23	3	]	]	PUNCT
ejpam-5570	23	4	performed	perform	VERB
ejpam-5570	23	5	an	an	DET
ejpam-5570	23	6	advanced	advanced	ADJ
ejpam-5570	23	7	examination	examination	NOUN
ejpam-5570	23	8	into	into	ADP
ejpam-5570	23	9	the	the	DET
ejpam-5570	23	10	notions	notion	NOUN
ejpam-5570	23	11	of	of	ADP
ejpam-5570	23	12	i	i	NOUN
ejpam-5570	23	13	-	-	PUNCT
ejpam-5570	23	14	open	open	ADJ
ejpam-5570	23	15	sets	set	NOUN
ejpam-5570	23	16	.	.	PUNCT
ejpam-5570	24	1	ig	ig	ADJ
ejpam-5570	24	2	-	-	PUNCT
ejpam-5570	24	3	closed	closed	ADJ
ejpam-5570	24	4	sets	set	NOUN
ejpam-5570	24	5	were	be	AUX
ejpam-5570	24	6	first	first	ADV
ejpam-5570	24	7	introduced	introduce	VERB
ejpam-5570	24	8	by	by	ADP
ejpam-5570	24	9	dontchev	dontchev	PROPN
ejpam-5570	24	10	et	et	PROPN
ejpam-5570	24	11	al	al	PROPN
ejpam-5570	24	12	.	.	PROPN
ejpam-5570	25	1	in	in	ADP
ejpam-5570	25	2	1999	1999	NUM
ejpam-5570	25	3	[	[	X
ejpam-5570	25	4	5	5	NUM
ejpam-5570	25	5	]	]	PUNCT
ejpam-5570	25	6	.	.	PUNCT
ejpam-5570	26	1	abd	abd	PROPN
ejpam-5570	26	2	el	el	PROPN
ejpam-5570	26	3	-	-	PROPN
ejpam-5570	26	4	monsef	monsef	PROPN
ejpam-5570	26	5	et	et	PROPN
ejpam-5570	26	6	al	al	PROPN
ejpam-5570	26	7	.	.	PUNCT
ejpam-5570	27	1	[	[	X
ejpam-5570	27	2	8	8	NUM
ejpam-5570	27	3	]	]	PUNCT
ejpam-5570	27	4	first	first	ADV
ejpam-5570	27	5	introduced	introduce	VERB
ejpam-5570	27	6	the	the	DET
ejpam-5570	27	7	concept	concept	NOUN
ejpam-5570	27	8	of	of	ADP
ejpam-5570	27	9	the	the	DET
ejpam-5570	27	10	s	s	ADJ
ejpam-5570	27	11	-	-	ADJ
ejpam-5570	27	12	local	local	ADJ
ejpam-5570	27	13	function	function	NOUN
ejpam-5570	27	14	,	,	PUNCT
ejpam-5570	27	15	which	which	PRON
ejpam-5570	27	16	was	be	AUX
ejpam-5570	27	17	later	later	ADV
ejpam-5570	27	18	examined	examine	VERB
ejpam-5570	27	19	by	by	ADP
ejpam-5570	27	20	khan	khan	PROPN
ejpam-5570	27	21	and	and	CCONJ
ejpam-5570	27	22	noiri	noiri	ADV
ejpam-5570	27	23	[	[	X
ejpam-5570	27	24	12	12	NUM
ejpam-5570	27	25	]	]	PUNCT
ejpam-5570	27	26	.	.	PUNCT
ejpam-5570	28	1	the	the	DET
ejpam-5570	28	2	concept	concept	NOUN
ejpam-5570	28	3	of	of	ADP
ejpam-5570	28	4	paracompactness	paracompactness	NOUN
ejpam-5570	28	5	with	with	ADP
ejpam-5570	28	6	respect	respect	NOUN
ejpam-5570	28	7	to	to	ADP
ejpam-5570	28	8	an	an	DET
ejpam-5570	28	9	ideal	ideal	NOUN
ejpam-5570	28	10	was	be	AUX
ejpam-5570	28	11	initially	initially	ADV
ejpam-5570	28	12	introduced	introduce	VERB
ejpam-5570	28	13	by	by	ADP
ejpam-5570	28	14	zahid	zahid	PROPN
ejpam-5570	28	15	[	[	X
ejpam-5570	28	16	19	19	NUM
ejpam-5570	28	17	]	]	PUNCT
ejpam-5570	28	18	and	and	CCONJ
ejpam-5570	28	19	later	later	ADV
ejpam-5570	28	20	investigated	investigate	VERB
ejpam-5570	28	21	by	by	ADP
ejpam-5570	28	22	hamlett	hamlett	PROPN
ejpam-5570	28	23	et	et	PROPN
ejpam-5570	28	24	al	al	PROPN
ejpam-5570	28	25	.	.	PUNCT
ejpam-5570	29	1	[	[	X
ejpam-5570	29	2	9	9	NUM
ejpam-5570	29	3	]	]	PUNCT
ejpam-5570	29	4	.	.	PUNCT
ejpam-5570	30	1	in	in	ADP
ejpam-5570	30	2	addition	addition	NOUN
ejpam-5570	30	3	,	,	PUNCT
ejpam-5570	30	4	sathiyasundari	sathiyasundari	NOUN
ejpam-5570	30	5	and	and	CCONJ
ejpam-5570	30	6	renukadevi	renukadevi	ADJ
ejpam-5570	30	7	[	[	X
ejpam-5570	30	8	16	16	NUM
ejpam-5570	30	9	]	]	PUNCT
ejpam-5570	30	10	explored	explore	VERB
ejpam-5570	30	11	the	the	DET
ejpam-5570	30	12	concept	concept	NOUN
ejpam-5570	30	13	of	of	ADP
ejpam-5570	30	14	i	i	NOUN
ejpam-5570	30	15	-	-	PUNCT
ejpam-5570	30	16	paracompact	paracompact	PROPN
ejpam-5570	30	17	and	and	CCONJ
ejpam-5570	30	18	examined	examine	VERB
ejpam-5570	30	19	its	its	PRON
ejpam-5570	30	20	characteristics	characteristic	NOUN
ejpam-5570	30	21	.	.	PUNCT
ejpam-5570	31	1	they	they	PRON
ejpam-5570	31	2	extended	extend	VERB
ejpam-5570	31	3	certain	certain	ADJ
ejpam-5570	31	4	results	result	NOUN
ejpam-5570	31	5	derived	derive	VERB
ejpam-5570	31	6	from	from	ADP
ejpam-5570	31	7	paracompact	paracompact	ADJ
ejpam-5570	31	8	spaces	space	NOUN
ejpam-5570	31	9	to	to	ADP
ejpam-5570	31	10	the	the	DET
ejpam-5570	31	11	notion	notion	NOUN
ejpam-5570	31	12	of	of	ADP
ejpam-5570	31	13	i	i	NOUN
ejpam-5570	31	14	-	-	PUNCT
ejpam-5570	31	15	paracompact	paracompact	ADJ
ejpam-5570	31	16	spaces	space	NOUN
ejpam-5570	31	17	.	.	PUNCT
ejpam-5570	32	1	the	the	DET
ejpam-5570	32	2	notion	notion	NOUN
ejpam-5570	32	3	of	of	ADP
ejpam-5570	32	4	s	s	NOUN
ejpam-5570	32	5	-	-	NOUN
ejpam-5570	32	6	paracompactness	paracompactness	NOUN
ejpam-5570	32	7	in	in	ADP
ejpam-5570	32	8	ideal	ideal	ADJ
ejpam-5570	32	9	topological	topological	ADJ
ejpam-5570	32	10	spaces	space	NOUN
ejpam-5570	32	11	was	be	AUX
ejpam-5570	32	12	studied	study	VERB
ejpam-5570	32	13	by	by	ADP
ejpam-5570	32	14	sanabria	sanabria	PROPN
ejpam-5570	32	15	et	et	PROPN
ejpam-5570	32	16	al	al	PROPN
ejpam-5570	32	17	.	.	PUNCT
ejpam-5570	33	1	[	[	X
ejpam-5570	33	2	15	15	NUM
ejpam-5570	33	3	]	]	PUNCT
ejpam-5570	33	4	.	.	PUNCT
ejpam-5570	34	1	their	their	PRON
ejpam-5570	34	2	work	work	NOUN
ejpam-5570	34	3	involved	involve	VERB
ejpam-5570	34	4	the	the	DET
ejpam-5570	34	5	introduction	introduction	NOUN
ejpam-5570	34	6	and	and	CCONJ
ejpam-5570	34	7	examination	examination	NOUN
ejpam-5570	34	8	of	of	ADP
ejpam-5570	34	9	a	a	DET
ejpam-5570	34	10	new	new	ADJ
ejpam-5570	34	11	kind	kind	NOUN
ejpam-5570	34	12	of	of	ADP
ejpam-5570	34	13	space	space	NOUN
ejpam-5570	34	14	,	,	PUNCT
ejpam-5570	34	15	namely	namely	ADV
ejpam-5570	34	16	i	i	PROPN
ejpam-5570	34	17	-	-	PUNCT
ejpam-5570	34	18	s	s	NOUN
ejpam-5570	34	19	-	-	PUNCT
ejpam-5570	34	20	paracompact	paracompact	ADJ
ejpam-5570	34	21	spaces	space	NOUN
ejpam-5570	34	22	,	,	PUNCT
ejpam-5570	34	23	which	which	PRON
ejpam-5570	34	24	are	be	AUX
ejpam-5570	34	25	defined	define	VERB
ejpam-5570	34	26	on	on	ADP
ejpam-5570	34	27	an	an	DET
ejpam-5570	34	28	ideal	ideal	ADJ
ejpam-5570	34	29	topological	topological	ADJ
ejpam-5570	34	30	space	space	NOUN
ejpam-5570	34	31	.	.	PUNCT
ejpam-5570	35	1	this	this	DET
ejpam-5570	35	2	class	class	NOUN
ejpam-5570	35	3	includes	include	VERB
ejpam-5570	35	4	spaces	space	NOUN
ejpam-5570	35	5	that	that	PRON
ejpam-5570	35	6	are	be	AUX
ejpam-5570	35	7	s	s	NOUN
ejpam-5570	35	8	-	-	NOUN
ejpam-5570	35	9	paracompact	paracompact	ADJ
ejpam-5570	35	10	and	and	CCONJ
ejpam-5570	35	11	i	i	NOUN
ejpam-5570	35	12	-	-	PUNCT
ejpam-5570	35	13	paracompact	paracompact	ADJ
ejpam-5570	35	14	.	.	PUNCT
ejpam-5570	36	1	in	in	ADP
ejpam-5570	36	2	2013	2013	NUM
ejpam-5570	36	3	,	,	PUNCT
ejpam-5570	36	4	demir	demir	PROPN
ejpam-5570	36	5	and	and	CCONJ
ejpam-5570	36	6	ozbakir	ozbakir	VERB
ejpam-5570	36	7	[	[	X
ejpam-5570	36	8	3	3	X
ejpam-5570	36	9	]	]	PUNCT
ejpam-5570	36	10	introduced	introduce	VERB
ejpam-5570	36	11	a	a	DET
ejpam-5570	36	12	diminished	diminished	ADJ
ejpam-5570	36	13	variant	variant	NOUN
ejpam-5570	36	14	of	of	ADP
ejpam-5570	36	15	expandable	expandable	ADJ
ejpam-5570	36	16	and	and	CCONJ
ejpam-5570	36	17	paracompact	paracompact	ADJ
ejpam-5570	36	18	spaces	space	NOUN
ejpam-5570	36	19	,	,	PUNCT
ejpam-5570	36	20	termed	term	VERB
ejpam-5570	36	21	β	β	NOUN
ejpam-5570	36	22	-	-	ADJ
ejpam-5570	36	23	expandable	expandable	ADJ
ejpam-5570	36	24	spaces	space	NOUN
ejpam-5570	36	25	and	and	CCONJ
ejpam-5570	36	26	β	β	NOUN
ejpam-5570	36	27	-	-	ADJ
ejpam-5570	36	28	paracompact	paracompact	ADJ
ejpam-5570	36	29	spaces	space	NOUN
ejpam-5570	36	30	,	,	PUNCT
ejpam-5570	36	31	respectively	respectively	ADV
ejpam-5570	36	32	.	.	PUNCT
ejpam-5570	37	1	the	the	DET
ejpam-5570	37	2	proof	proof	NOUN
ejpam-5570	37	3	was	be	AUX
ejpam-5570	37	4	presented	present	VERB
ejpam-5570	37	5	indicating	indicate	VERB
ejpam-5570	37	6	that	that	SCONJ
ejpam-5570	37	7	any	any	DET
ejpam-5570	37	8	β	β	NOUN
ejpam-5570	37	9	-	-	ADJ
ejpam-5570	37	10	paracompact	paracompact	ADJ
ejpam-5570	37	11	space	space	NOUN
ejpam-5570	37	12	is	be	AUX
ejpam-5570	37	13	essentially	essentially	ADV
ejpam-5570	37	14	a	a	DET
ejpam-5570	37	15	βexpandable	βexpandable	ADJ
ejpam-5570	37	16	space	space	NOUN
ejpam-5570	37	17	.	.	PUNCT
ejpam-5570	38	1	yildirim	yildirim	PROPN
ejpam-5570	38	2	et	et	PROPN
ejpam-5570	38	3	al	al	PROPN
ejpam-5570	38	4	.	.	PUNCT
ejpam-5570	39	1	[	[	X
ejpam-5570	39	2	17	17	NUM
ejpam-5570	39	3	]	]	PUNCT
ejpam-5570	39	4	introduced	introduce	VERB
ejpam-5570	39	5	the	the	DET
ejpam-5570	39	6	notion	notion	NOUN
ejpam-5570	39	7	of	of	ADP
ejpam-5570	39	8	β	β	NOUN
ejpam-5570	39	9	-	-	NOUN
ejpam-5570	39	10	paracompactness	paracompactness	NOUN
ejpam-5570	39	11	inside	inside	ADP
ejpam-5570	39	12	an	an	DET
ejpam-5570	39	13	ideal	ideal	ADJ
ejpam-5570	39	14	topological	topological	ADJ
ejpam-5570	39	15	space	space	NOUN
ejpam-5570	39	16	and	and	CCONJ
ejpam-5570	39	17	performed	perform	VERB
ejpam-5570	39	18	a	a	DET
ejpam-5570	39	19	comparison	comparison	NOUN
ejpam-5570	39	20	study	study	NOUN
ejpam-5570	39	21	with	with	ADP
ejpam-5570	39	22	existing	exist	VERB
ejpam-5570	39	23	types	type	NOUN
ejpam-5570	39	24	of	of	ADP
ejpam-5570	39	25	paracompactness	paracompactness	NOUN
ejpam-5570	39	26	.	.	PUNCT
ejpam-5570	40	1	in	in	ADP
ejpam-5570	40	2	this	this	DET
ejpam-5570	40	3	paper	paper	NOUN
ejpam-5570	40	4	,	,	PUNCT
ejpam-5570	40	5	the	the	DET
ejpam-5570	40	6	δ	δ	PROPN
ejpam-5570	40	7	-	-	PUNCT
ejpam-5570	40	8	βi	βi	PRON
ejpam-5570	40	9	-	-	PUNCT
ejpam-5570	40	10	paracompact	paracompact	ADJ
ejpam-5570	40	11	spaces	space	NOUN
ejpam-5570	40	12	are	be	AUX
ejpam-5570	40	13	constructed	construct	VERB
ejpam-5570	40	14	using	use	VERB
ejpam-5570	40	15	δ	δ	PROPN
ejpam-5570	40	16	-	-	PUNCT
ejpam-5570	40	17	βiopen	βiopen	ADJ
ejpam-5570	40	18	sets	set	NOUN
ejpam-5570	40	19	.	.	PUNCT
ejpam-5570	41	1	the	the	DET
ejpam-5570	41	2	spaces	space	NOUN
ejpam-5570	41	3	under	under	ADP
ejpam-5570	41	4	examination	examination	NOUN
ejpam-5570	41	5	are	be	AUX
ejpam-5570	41	6	an	an	DET
ejpam-5570	41	7	extension	extension	NOUN
ejpam-5570	41	8	of	of	ADP
ejpam-5570	41	9	the	the	DET
ejpam-5570	41	10	β	β	NOUN
ejpam-5570	41	11	-	-	ADJ
ejpam-5570	41	12	paracompact	paracompact	ADJ
ejpam-5570	41	13	spaces	space	NOUN
ejpam-5570	41	14	as	as	ADV
ejpam-5570	41	15	delineated	delineated	ADJ
ejpam-5570	41	16	in	in	ADP
ejpam-5570	41	17	reference	reference	NOUN
ejpam-5570	41	18	[	[	X
ejpam-5570	41	19	17	17	NUM
ejpam-5570	41	20	]	]	SYM
ejpam-5570	41	21	.	.	PUNCT
ejpam-5570	42	1	2	2	X
ejpam-5570	42	2	.	.	X
ejpam-5570	42	3	preliminaries	preliminary	NOUN
ejpam-5570	42	4	throughout	throughout	ADP
ejpam-5570	42	5	this	this	DET
ejpam-5570	42	6	paper	paper	NOUN
ejpam-5570	42	7	,	,	PUNCT
ejpam-5570	42	8	spaces	space	NOUN
ejpam-5570	42	9	(	(	PUNCT
ejpam-5570	42	10	x	x	X
ejpam-5570	42	11	,	,	PUNCT
ejpam-5570	42	12	τ	τ	X
ejpam-5570	42	13	)	)	PUNCT
ejpam-5570	42	14	and	and	CCONJ
ejpam-5570	42	15	(	(	PUNCT
ejpam-5570	42	16	y	y	PROPN
ejpam-5570	42	17	,	,	PUNCT
ejpam-5570	42	18	σ	σ	PROPN
ejpam-5570	42	19	)	)	PUNCT
ejpam-5570	42	20	(	(	PUNCT
ejpam-5570	42	21	or	or	CCONJ
ejpam-5570	42	22	simply	simply	ADV
ejpam-5570	42	23	x	x	X
ejpam-5570	42	24	and	and	CCONJ
ejpam-5570	42	25	y	y	PROPN
ejpam-5570	42	26	)	)	PUNCT
ejpam-5570	42	27	,	,	PUNCT
ejpam-5570	42	28	always	always	ADV
ejpam-5570	42	29	mean	mean	VERB
ejpam-5570	42	30	topological	topological	ADJ
ejpam-5570	42	31	spaces	space	NOUN
ejpam-5570	42	32	on	on	ADP
ejpam-5570	42	33	which	which	PRON
ejpam-5570	42	34	no	no	DET
ejpam-5570	42	35	separation	separation	NOUN
ejpam-5570	42	36	axiom	axiom	NOUN
ejpam-5570	42	37	is	be	AUX
ejpam-5570	42	38	assumed	assume	VERB
ejpam-5570	42	39	.	.	PUNCT
ejpam-5570	43	1	for	for	ADP
ejpam-5570	43	2	a	a	DET
ejpam-5570	43	3	subset	subset	NOUN
ejpam-5570	43	4	a	a	PRON
ejpam-5570	43	5	of	of	ADP
ejpam-5570	43	6	a	a	DET
ejpam-5570	43	7	topological	topological	ADJ
ejpam-5570	43	8	space	space	NOUN
ejpam-5570	43	9	(	(	PUNCT
ejpam-5570	43	10	x	x	X
ejpam-5570	43	11	,	,	PUNCT
ejpam-5570	43	12	τ	τ	PROPN
ejpam-5570	43	13	)	)	PUNCT
ejpam-5570	43	14	,	,	PUNCT
ejpam-5570	43	15	ci(a	ci(a	NOUN
ejpam-5570	43	16	)	)	PUNCT
ejpam-5570	43	17	and	and	CCONJ
ejpam-5570	43	18	int(a	int(a	PROPN
ejpam-5570	43	19	)	)	PUNCT
ejpam-5570	43	20	will	will	AUX
ejpam-5570	43	21	denote	denote	VERB
ejpam-5570	43	22	the	the	DET
ejpam-5570	43	23	closure	closure	NOUN
ejpam-5570	43	24	and	and	CCONJ
ejpam-5570	43	25	interior	interior	NOUN
ejpam-5570	43	26	of	of	ADP
ejpam-5570	43	27	a	a	DET
ejpam-5570	43	28	in	in	ADP
ejpam-5570	43	29	(	(	PUNCT
ejpam-5570	43	30	x	x	NOUN
ejpam-5570	43	31	,	,	PUNCT
ejpam-5570	43	32	τ	τ	PROPN
ejpam-5570	43	33	)	)	PUNCT
ejpam-5570	43	34	,	,	PUNCT
ejpam-5570	43	35	respectively	respectively	ADV
ejpam-5570	43	36	.	.	PUNCT
ejpam-5570	44	1	an	an	DET
ejpam-5570	44	2	ideal	ideal	NOUN
ejpam-5570	44	3	i	i	PRON
ejpam-5570	44	4	on	on	ADP
ejpam-5570	44	5	a	a	DET
ejpam-5570	44	6	topological	topological	ADJ
ejpam-5570	44	7	space	space	NOUN
ejpam-5570	44	8	(	(	PUNCT
ejpam-5570	44	9	x	x	X
ejpam-5570	44	10	,	,	PUNCT
ejpam-5570	44	11	τ	τ	X
ejpam-5570	44	12	)	)	PUNCT
ejpam-5570	44	13	is	be	AUX
ejpam-5570	44	14	a	a	DET
ejpam-5570	44	15	nonempty	nonempty	ADJ
ejpam-5570	44	16	collection	collection	NOUN
ejpam-5570	44	17	of	of	ADP
ejpam-5570	44	18	subsets	subset	NOUN
ejpam-5570	44	19	of	of	ADP
ejpam-5570	44	20	x	x	PRON
ejpam-5570	44	21	which	which	PRON
ejpam-5570	44	22	satisfies	satisfy	VERB
ejpam-5570	44	23	:	:	PUNCT
ejpam-5570	44	24	(	(	PUNCT
ejpam-5570	44	25	i	i	NOUN
ejpam-5570	44	26	)	)	PUNCT
ejpam-5570	45	1	a	a	DET
ejpam-5570	45	2	∈	∈	NOUN
ejpam-5570	45	3	i	i	PRON
ejpam-5570	45	4	and	and	CCONJ
ejpam-5570	45	5	b	b	PROPN
ejpam-5570	45	6	⊂	⊂	PROPN
ejpam-5570	45	7	a	a	PRON
ejpam-5570	45	8	implies	imply	VERB
ejpam-5570	45	9	b	b	X
ejpam-5570	45	10	∈	∈	PROPN
ejpam-5570	45	11	i	i	PRON
ejpam-5570	45	12	,	,	PUNCT
ejpam-5570	45	13	(	(	PUNCT
ejpam-5570	45	14	ii	ii	NOUN
ejpam-5570	45	15	)	)	PUNCT
ejpam-5570	45	16	a	a	DET
ejpam-5570	45	17	∈	∈	PROPN
ejpam-5570	46	1	i	i	PRON
ejpam-5570	46	2	and	and	CCONJ
ejpam-5570	46	3	b	b	X
ejpam-5570	46	4	∈	∈	PROPN
ejpam-5570	46	5	i	i	PRON
ejpam-5570	46	6	implies	imply	VERB
ejpam-5570	46	7	a	a	DET
ejpam-5570	46	8	∪b	∪b	PUNCT
ejpam-5570	46	9	∈	∈	PROPN
ejpam-5570	46	10	i.	i.	NOUN
ejpam-5570	46	11	an	an	DET
ejpam-5570	46	12	ideal	ideal	ADJ
ejpam-5570	46	13	topological	topological	ADJ
ejpam-5570	46	14	space	space	NOUN
ejpam-5570	46	15	(	(	PUNCT
ejpam-5570	46	16	x	x	X
ejpam-5570	46	17	,	,	PUNCT
ejpam-5570	46	18	τ	τ	PROPN
ejpam-5570	46	19	,	,	PUNCT
ejpam-5570	46	20	i	i	PROPN
ejpam-5570	46	21	)	)	PUNCT
ejpam-5570	46	22	is	be	AUX
ejpam-5570	46	23	a	a	DET
ejpam-5570	46	24	topological	topological	ADJ
ejpam-5570	46	25	space	space	NOUN
ejpam-5570	46	26	(	(	PUNCT
ejpam-5570	46	27	x	x	X
ejpam-5570	46	28	,	,	PUNCT
ejpam-5570	46	29	τ	τ	X
ejpam-5570	46	30	)	)	PUNCT
ejpam-5570	46	31	with	with	ADP
ejpam-5570	46	32	an	an	DET
ejpam-5570	46	33	ideal	ideal	NOUN
ejpam-5570	46	34	i	i	PRON
ejpam-5570	46	35	on	on	ADP
ejpam-5570	46	36	x.	x.	NOUN
ejpam-5570	46	37	the	the	DET
ejpam-5570	46	38	set	set	NOUN
ejpam-5570	46	39	of	of	ADP
ejpam-5570	46	40	all	all	DET
ejpam-5570	46	41	subsets	subset	NOUN
ejpam-5570	46	42	of	of	ADP
ejpam-5570	46	43	x	x	SYM
ejpam-5570	46	44	is	be	AUX
ejpam-5570	46	45	denoted	denote	VERB
ejpam-5570	46	46	as	as	ADP
ejpam-5570	46	47	p	p	PROPN
ejpam-5570	46	48	(	(	PUNCT
ejpam-5570	46	49	x	x	NOUN
ejpam-5570	46	50	)	)	PUNCT
ejpam-5570	46	51	.	.	PUNCT
ejpam-5570	47	1	a	a	DET
ejpam-5570	47	2	set	set	NOUN
ejpam-5570	47	3	operator	operator	NOUN
ejpam-5570	47	4	(	(	PUNCT
ejpam-5570	47	5	.)∗	.)∗	X
ejpam-5570	47	6	:	:	PUNCT
ejpam-5570	47	7	p	p	X
ejpam-5570	47	8	(	(	PUNCT
ejpam-5570	47	9	x	x	NOUN
ejpam-5570	47	10	)	)	PUNCT
ejpam-5570	47	11	→	→	SYM
ejpam-5570	47	12	p	p	X
ejpam-5570	47	13	(	(	PUNCT
ejpam-5570	47	14	x	x	NOUN
ejpam-5570	47	15	)	)	PUNCT
ejpam-5570	47	16	,	,	PUNCT
ejpam-5570	47	17	which	which	PRON
ejpam-5570	47	18	is	be	AUX
ejpam-5570	47	19	a	a	DET
ejpam-5570	47	20	local	local	ADJ
ejpam-5570	47	21	function	function	NOUN
ejpam-5570	47	22	[	[	X
ejpam-5570	47	23	13	13	NUM
ejpam-5570	47	24	]	]	PUNCT
ejpam-5570	47	25	,	,	PUNCT
ejpam-5570	47	26	is	be	AUX
ejpam-5570	47	27	defined	define	VERB
ejpam-5570	47	28	with	with	ADP
ejpam-5570	47	29	respect	respect	NOUN
ejpam-5570	47	30	to	to	ADP
ejpam-5570	47	31	τ	τ	PROPN
ejpam-5570	47	32	and	and	CCONJ
ejpam-5570	47	33	i	i	PRON
ejpam-5570	47	34	:	:	PUNCT
ejpam-5570	47	35	for	for	ADP
ejpam-5570	47	36	a	a	DET
ejpam-5570	47	37	⊂	⊂	PROPN
ejpam-5570	47	38	x	x	SYM
ejpam-5570	47	39	,	,	PUNCT
ejpam-5570	47	40	a∗(i	a∗(i	PROPN
ejpam-5570	47	41	,	,	PUNCT
ejpam-5570	47	42	τ	τ	X
ejpam-5570	47	43	)	)	PUNCT
ejpam-5570	47	44	=	=	PRON
ejpam-5570	48	1	{	{	PUNCT
ejpam-5570	48	2	x	x	PUNCT
ejpam-5570	48	3	∈	∈	PROPN
ejpam-5570	48	4	x	x	X
ejpam-5570	48	5	:	:	PUNCT
ejpam-5570	48	6	u	u	X
ejpam-5570	48	7	∩a	∩a	PROPN
ejpam-5570	48	8	̸∈	̸∈	PROPN
ejpam-5570	48	9	i	i	PRON
ejpam-5570	48	10	for	for	ADP
ejpam-5570	48	11	every	every	DET
ejpam-5570	48	12	u	u	PROPN
ejpam-5570	48	13	∈	∈	PROPN
ejpam-5570	48	14	τ(x	τ(x	NOUN
ejpam-5570	48	15	)	)	PUNCT
ejpam-5570	48	16	}	}	PUNCT
ejpam-5570	48	17	,	,	PUNCT
ejpam-5570	48	18	where	where	SCONJ
ejpam-5570	48	19	τ(x	τ(x	NOUN
ejpam-5570	48	20	)	)	PUNCT
ejpam-5570	48	21	=	=	PRON
ejpam-5570	48	22	{	{	PUNCT
ejpam-5570	48	23	u	u	X
ejpam-5570	48	24	∈	∈	PROPN
ejpam-5570	48	25	τ	τ	X
ejpam-5570	48	26	:	:	PUNCT
ejpam-5570	48	27	x	x	SYM
ejpam-5570	48	28	∈	∈	PROPN
ejpam-5570	48	29	u	u	NOUN
ejpam-5570	48	30	}	}	PUNCT
ejpam-5570	48	31	.	.	PUNCT
ejpam-5570	49	1	we	we	PRON
ejpam-5570	49	2	simply	simply	ADV
ejpam-5570	49	3	write	write	VERB
ejpam-5570	49	4	a∗	a∗	PROPN
ejpam-5570	49	5	instead	instead	ADV
ejpam-5570	49	6	of	of	ADP
ejpam-5570	49	7	a∗(i	a∗(i	PROPN
ejpam-5570	49	8	,	,	PUNCT
ejpam-5570	49	9	τ	τ	PROPN
ejpam-5570	49	10	)	)	PUNCT
ejpam-5570	49	11	.	.	PUNCT
ejpam-5570	50	1	x∗	x∗	PROPN
ejpam-5570	50	2	is	be	AUX
ejpam-5570	50	3	often	often	ADV
ejpam-5570	50	4	a	a	DET
ejpam-5570	50	5	proper	proper	ADJ
ejpam-5570	50	6	subset	subset	NOUN
ejpam-5570	50	7	of	of	ADP
ejpam-5570	50	8	x	x	PUNCT
ejpam-5570	50	9	and	and	CCONJ
ejpam-5570	50	10	x	x	SYM
ejpam-5570	50	11	=	=	PUNCT
ejpam-5570	50	12	x∗	x∗	PROPN
ejpam-5570	50	13	if	if	SCONJ
ejpam-5570	50	14	τ	τ	PROPN
ejpam-5570	50	15	∩	∩	X
ejpam-5570	50	16	i	i	PRON
ejpam-5570	50	17	=	=	SYM
ejpam-5570	50	18	{	{	PUNCT
ejpam-5570	50	19	∅	∅	NOUN
ejpam-5570	50	20	}	}	PUNCT
ejpam-5570	50	21	.	.	PUNCT
ejpam-5570	51	1	a	a	DET
ejpam-5570	51	2	topology	topology	NOUN
ejpam-5570	51	3	τ∗(i	τ∗(i	PROPN
ejpam-5570	51	4	)	)	PUNCT
ejpam-5570	51	5	,	,	PUNCT
ejpam-5570	51	6	or	or	CCONJ
ejpam-5570	51	7	more	more	ADV
ejpam-5570	51	8	simply	simply	ADV
ejpam-5570	51	9	τ∗	τ∗	ADJ
ejpam-5570	51	10	,	,	PUNCT
ejpam-5570	51	11	finer	fine	ADJ
ejpam-5570	51	12	than	than	ADP
ejpam-5570	51	13	τ	τ	PROPN
ejpam-5570	51	14	,	,	PUNCT
ejpam-5570	51	15	exists	exist	VERB
ejpam-5570	51	16	for	for	ADP
ejpam-5570	51	17	any	any	DET
ejpam-5570	51	18	ideal	ideal	ADJ
ejpam-5570	51	19	topological	topological	ADJ
ejpam-5570	51	20	space	space	NOUN
ejpam-5570	51	21	and	and	CCONJ
ejpam-5570	51	22	is	be	AUX
ejpam-5570	51	23	generated	generate	VERB
ejpam-5570	51	24	by	by	ADP
ejpam-5570	51	25	β(i	β(i	NOUN
ejpam-5570	51	26	,	,	PUNCT
ejpam-5570	51	27	τ	τ	X
ejpam-5570	51	28	)	)	PUNCT
ejpam-5570	51	29	=	=	PRON
ejpam-5570	51	30	{	{	PUNCT
ejpam-5570	51	31	u	u	NOUN
ejpam-5570	51	32	−	−	PROPN
ejpam-5570	52	1	i	i	PRON
ejpam-5570	52	2	:	:	PUNCT
ejpam-5570	52	3	u	u	PROPN
ejpam-5570	52	4	∈	∈	PROPN
ejpam-5570	52	5	τ	τ	X
ejpam-5570	52	6	and	and	CCONJ
ejpam-5570	52	7	i	i	PRON
ejpam-5570	52	8	∈	∈	PROPN
ejpam-5570	52	9	i	i	X
ejpam-5570	52	10	}	}	PUNCT
ejpam-5570	52	11	.	.	PUNCT
ejpam-5570	53	1	however	however	ADV
ejpam-5570	53	2	,	,	PUNCT
ejpam-5570	53	3	in	in	ADP
ejpam-5570	53	4	general	general	ADJ
ejpam-5570	53	5	,	,	PUNCT
ejpam-5570	53	6	β(i	β(i	PRON
ejpam-5570	53	7	,	,	PUNCT
ejpam-5570	53	8	τ	τ	X
ejpam-5570	53	9	)	)	PUNCT
ejpam-5570	53	10	is	be	AUX
ejpam-5570	53	11	not	not	PART
ejpam-5570	53	12	always	always	ADV
ejpam-5570	53	13	a	a	DET
ejpam-5570	53	14	topology	topology	NOUN
ejpam-5570	53	15	.	.	PUNCT
ejpam-5570	54	1	additionally	additionally	ADV
ejpam-5570	54	2	,	,	PUNCT
ejpam-5570	54	3	cl∗(a	cl∗(a	NOUN
ejpam-5570	54	4	)	)	PUNCT
ejpam-5570	54	5	=	=	PUNCT
ejpam-5570	55	1	a	a	DET
ejpam-5570	55	2	∪	∪	ADJ
ejpam-5570	55	3	a∗	a∗	ADJ
ejpam-5570	55	4	defines	define	VERB
ejpam-5570	55	5	a	a	DET
ejpam-5570	55	6	kuratowski	kuratowski	ADJ
ejpam-5570	55	7	closure	closure	NOUN
ejpam-5570	55	8	operator	operator	NOUN
ejpam-5570	55	9	c.	c.	PROPN
ejpam-5570	55	10	boonpok	boonpok	PROPN
ejpam-5570	55	11	,	,	PUNCT
ejpam-5570	55	12	a.	a.	PROPN
ejpam-5570	55	13	sama	sama	PROPN
ejpam-5570	55	14	-	-	PUNCT
ejpam-5570	55	15	ae	ae	PROPN
ejpam-5570	55	16	,	,	PUNCT
ejpam-5570	55	17	p.	p.	NOUN
ejpam-5570	55	18	raktaow	raktaow	PROPN
ejpam-5570	55	19	/	/	SYM
ejpam-5570	55	20	eur	eur	PROPN
ejpam-5570	55	21	.	.	PUNCT
ejpam-5570	56	1	j.	j.	PROPN
ejpam-5570	56	2	pure	pure	PROPN
ejpam-5570	56	3	appl	appl	PROPN
ejpam-5570	56	4	.	.	PROPN
ejpam-5570	56	5	math	math	PROPN
ejpam-5570	56	6	,	,	PUNCT
ejpam-5570	56	7	18	18	NUM
ejpam-5570	56	8	(	(	PUNCT
ejpam-5570	56	9	1	1	NUM
ejpam-5570	56	10	)	)	PUNCT
ejpam-5570	56	11	(	(	PUNCT
ejpam-5570	56	12	2025	2025	NUM
ejpam-5570	56	13	)	)	PUNCT
ejpam-5570	56	14	,	,	PUNCT
ejpam-5570	56	15	5570	5570	NUM
ejpam-5570	56	16	3	3	NUM
ejpam-5570	56	17	of	of	ADP
ejpam-5570	56	18	12	12	NUM
ejpam-5570	56	19	for	for	ADP
ejpam-5570	56	20	τ∗(i	τ∗(i	PROPN
ejpam-5570	56	21	)	)	PUNCT
ejpam-5570	56	22	.	.	PUNCT
ejpam-5570	57	1	lemma	lemma	PROPN
ejpam-5570	57	2	1	1	NUM
ejpam-5570	57	3	.	.	PUNCT
ejpam-5570	58	1	[	[	X
ejpam-5570	58	2	11	11	NUM
ejpam-5570	58	3	]	]	PUNCT
ejpam-5570	58	4	let	let	VERB
ejpam-5570	58	5	a	a	PRON
ejpam-5570	58	6	and	and	CCONJ
ejpam-5570	58	7	b	b	NOUN
ejpam-5570	58	8	be	be	AUX
ejpam-5570	58	9	subsets	subset	NOUN
ejpam-5570	58	10	of	of	ADP
ejpam-5570	58	11	an	an	DET
ejpam-5570	58	12	ideal	ideal	ADJ
ejpam-5570	58	13	topological	topological	ADJ
ejpam-5570	58	14	space	space	NOUN
ejpam-5570	58	15	(	(	PUNCT
ejpam-5570	58	16	x	x	X
ejpam-5570	58	17	,	,	PUNCT
ejpam-5570	58	18	τ	τ	PROPN
ejpam-5570	58	19	,	,	PUNCT
ejpam-5570	58	20	i	i	PROPN
ejpam-5570	58	21	)	)	PUNCT
ejpam-5570	58	22	.	.	PUNCT
ejpam-5570	59	1	then	then	ADV
ejpam-5570	59	2	the	the	DET
ejpam-5570	59	3	following	follow	VERB
ejpam-5570	59	4	statements	statement	NOUN
ejpam-5570	59	5	are	be	AUX
ejpam-5570	59	6	true	true	ADJ
ejpam-5570	59	7	:	:	PUNCT
ejpam-5570	59	8	(	(	PUNCT
ejpam-5570	59	9	i	i	NOUN
ejpam-5570	59	10	)	)	PUNCT
ejpam-5570	59	11	if	if	SCONJ
ejpam-5570	59	12	a	a	DET
ejpam-5570	59	13	⊂	⊂	PROPN
ejpam-5570	59	14	b	b	PROPN
ejpam-5570	59	15	,	,	PUNCT
ejpam-5570	59	16	then	then	ADV
ejpam-5570	59	17	a∗	a∗	PROPN
ejpam-5570	59	18	⊂	⊂	PROPN
ejpam-5570	59	19	b∗	b∗	PROPN
ejpam-5570	59	20	;	;	PUNCT
ejpam-5570	59	21	(	(	PUNCT
ejpam-5570	59	22	ii	ii	NOUN
ejpam-5570	59	23	)	)	PUNCT
ejpam-5570	59	24	g	g	PROPN
ejpam-5570	59	25	∩a∗	∩a∗	PUNCT
ejpam-5570	59	26	⊂	⊂	PROPN
ejpam-5570	59	27	(	(	PUNCT
ejpam-5570	59	28	g	g	PROPN
ejpam-5570	59	29	∩a)∗	∩a)∗	PROPN
ejpam-5570	59	30	for	for	ADP
ejpam-5570	59	31	all	all	PRON
ejpam-5570	59	32	g	g	PROPN
ejpam-5570	59	33	∈	∈	PROPN
ejpam-5570	59	34	τ	τ	X
ejpam-5570	59	35	;	;	PUNCT
ejpam-5570	59	36	(	(	PUNCT
ejpam-5570	59	37	iii	iii	X
ejpam-5570	59	38	)	)	PUNCT
ejpam-5570	59	39	a∗	a∗	NOUN
ejpam-5570	59	40	=	=	PUNCT
ejpam-5570	59	41	cl(a∗	cl(a∗	PROPN
ejpam-5570	59	42	)	)	PUNCT
ejpam-5570	59	43	⊂	⊂	PROPN
ejpam-5570	59	44	cl(a	cl(a	X
ejpam-5570	59	45	)	)	PUNCT
ejpam-5570	59	46	.	.	PUNCT
ejpam-5570	60	1	definition	definition	NOUN
ejpam-5570	60	2	1	1	NUM
ejpam-5570	60	3	.	.	PUNCT
ejpam-5570	61	1	[	[	X
ejpam-5570	61	2	10	10	NUM
ejpam-5570	61	3	]	]	PUNCT
ejpam-5570	61	4	let	let	VERB
ejpam-5570	61	5	a	a	PRON
ejpam-5570	61	6	be	be	AUX
ejpam-5570	61	7	a	a	DET
ejpam-5570	61	8	subset	subset	NOUN
ejpam-5570	61	9	of	of	ADP
ejpam-5570	61	10	an	an	DET
ejpam-5570	61	11	ideal	ideal	ADJ
ejpam-5570	61	12	topological	topological	ADJ
ejpam-5570	61	13	space	space	NOUN
ejpam-5570	61	14	(	(	PUNCT
ejpam-5570	61	15	x	x	X
ejpam-5570	61	16	,	,	PUNCT
ejpam-5570	61	17	τ	τ	PROPN
ejpam-5570	61	18	,	,	PUNCT
ejpam-5570	61	19	i	i	PROPN
ejpam-5570	61	20	)	)	PUNCT
ejpam-5570	61	21	.	.	PUNCT
ejpam-5570	62	1	a	a	DET
ejpam-5570	62	2	point	point	NOUN
ejpam-5570	62	3	x	x	X
ejpam-5570	62	4	∈	∈	NOUN
ejpam-5570	62	5	x	x	PUNCT
ejpam-5570	62	6	is	be	AUX
ejpam-5570	62	7	called	call	VERB
ejpam-5570	62	8	a	a	DET
ejpam-5570	62	9	δi	δi	NOUN
ejpam-5570	62	10	-	-	PUNCT
ejpam-5570	62	11	cluster	cluster	NOUN
ejpam-5570	62	12	point	point	NOUN
ejpam-5570	62	13	of	of	ADP
ejpam-5570	62	14	a	a	DET
ejpam-5570	62	15	if	if	NOUN
ejpam-5570	62	16	int(cl∗(u	int(cl∗(u	NOUN
ejpam-5570	62	17	)	)	PUNCT
ejpam-5570	62	18	)	)	PUNCT
ejpam-5570	62	19	∩	∩	NOUN
ejpam-5570	62	20	a	a	DET
ejpam-5570	62	21	̸=	̸=	PROPN
ejpam-5570	62	22	∅	∅	NOUN
ejpam-5570	62	23	for	for	ADP
ejpam-5570	62	24	each	each	DET
ejpam-5570	62	25	neighborhood	neighborhood	NOUN
ejpam-5570	62	26	u	u	NOUN
ejpam-5570	62	27	of	of	ADP
ejpam-5570	62	28	x.	x.	NOUN
ejpam-5570	62	29	the	the	DET
ejpam-5570	62	30	set	set	NOUN
ejpam-5570	62	31	of	of	ADP
ejpam-5570	62	32	all	all	DET
ejpam-5570	62	33	δi	δi	NOUN
ejpam-5570	62	34	-	-	PUNCT
ejpam-5570	62	35	cluster	cluster	NOUN
ejpam-5570	62	36	points	point	NOUN
ejpam-5570	62	37	of	of	ADP
ejpam-5570	62	38	a	a	PRON
ejpam-5570	62	39	is	be	AUX
ejpam-5570	62	40	called	call	VERB
ejpam-5570	62	41	the	the	DET
ejpam-5570	62	42	δi	δi	NOUN
ejpam-5570	62	43	-	-	PUNCT
ejpam-5570	62	44	closure	closure	NOUN
ejpam-5570	62	45	of	of	ADP
ejpam-5570	62	46	a	a	PRON
ejpam-5570	62	47	and	and	CCONJ
ejpam-5570	62	48	will	will	AUX
ejpam-5570	62	49	be	be	AUX
ejpam-5570	62	50	denoted	denote	VERB
ejpam-5570	62	51	by	by	ADP
ejpam-5570	62	52	δcli(a	δcli(a	PROPN
ejpam-5570	62	53	)	)	PUNCT
ejpam-5570	62	54	.	.	PUNCT
ejpam-5570	63	1	a	a	PRON
ejpam-5570	63	2	is	be	AUX
ejpam-5570	63	3	said	say	VERB
ejpam-5570	63	4	to	to	PART
ejpam-5570	63	5	be	be	AUX
ejpam-5570	63	6	δi	δi	ADV
ejpam-5570	63	7	-	-	PUNCT
ejpam-5570	63	8	closed	closed	ADJ
ejpam-5570	63	9	[	[	X
ejpam-5570	63	10	18	18	NUM
ejpam-5570	63	11	]	]	X
ejpam-5570	63	12	if	if	SCONJ
ejpam-5570	63	13	δcli(a	δcli(a	ADJ
ejpam-5570	63	14	)	)	PUNCT
ejpam-5570	63	15	=	=	VERB
ejpam-5570	63	16	a.	a.	NOUN
ejpam-5570	63	17	the	the	DET
ejpam-5570	63	18	complement	complement	NOUN
ejpam-5570	63	19	of	of	ADP
ejpam-5570	63	20	a	a	DET
ejpam-5570	63	21	δi	δi	ADV
ejpam-5570	63	22	-	-	PUNCT
ejpam-5570	63	23	closed	close	VERB
ejpam-5570	63	24	set	set	NOUN
ejpam-5570	63	25	is	be	AUX
ejpam-5570	63	26	called	call	VERB
ejpam-5570	63	27	a	a	DET
ejpam-5570	63	28	δi	δi	ADV
ejpam-5570	63	29	-	-	PUNCT
ejpam-5570	63	30	open	open	ADJ
ejpam-5570	63	31	set	set	NOUN
ejpam-5570	63	32	.	.	PUNCT
ejpam-5570	64	1	δi	δi	NOUN
ejpam-5570	64	2	-	-	PUNCT
ejpam-5570	64	3	interior	interior	NOUN
ejpam-5570	64	4	of	of	ADP
ejpam-5570	64	5	a	a	PRON
ejpam-5570	64	6	,	,	PUNCT
ejpam-5570	64	7	will	will	AUX
ejpam-5570	64	8	be	be	AUX
ejpam-5570	64	9	denoted	denote	VERB
ejpam-5570	64	10	by	by	ADP
ejpam-5570	64	11	δinti(a	δinti(a	PROPN
ejpam-5570	64	12	)	)	PUNCT
ejpam-5570	64	13	,	,	PUNCT
ejpam-5570	64	14	is	be	AUX
ejpam-5570	64	15	the	the	DET
ejpam-5570	64	16	union	union	NOUN
ejpam-5570	64	17	of	of	ADP
ejpam-5570	64	18	all	all	DET
ejpam-5570	64	19	δi	δi	ADV
ejpam-5570	64	20	-	-	PUNCT
ejpam-5570	64	21	open	open	ADJ
ejpam-5570	64	22	sets	set	NOUN
ejpam-5570	64	23	contained	contain	VERB
ejpam-5570	64	24	in	in	ADP
ejpam-5570	64	25	a.	a.	PROPN
ejpam-5570	64	26	lemma	lemma	PROPN
ejpam-5570	65	1	2	2	X
ejpam-5570	65	2	.	.	PUNCT
ejpam-5570	66	1	[	[	X
ejpam-5570	66	2	10	10	NUM
ejpam-5570	66	3	]	]	PUNCT
ejpam-5570	66	4	let	let	VERB
ejpam-5570	66	5	a	a	PRON
ejpam-5570	66	6	and	and	CCONJ
ejpam-5570	66	7	b	b	NOUN
ejpam-5570	66	8	be	be	AUX
ejpam-5570	66	9	subsets	subset	NOUN
ejpam-5570	66	10	of	of	ADP
ejpam-5570	66	11	an	an	DET
ejpam-5570	66	12	ideal	ideal	ADJ
ejpam-5570	66	13	topological	topological	ADJ
ejpam-5570	66	14	space	space	NOUN
ejpam-5570	66	15	(	(	PUNCT
ejpam-5570	66	16	x	x	X
ejpam-5570	66	17	,	,	PUNCT
ejpam-5570	66	18	τ	τ	PROPN
ejpam-5570	66	19	,	,	PUNCT
ejpam-5570	66	20	i	i	PROPN
ejpam-5570	66	21	)	)	PUNCT
ejpam-5570	66	22	.	.	PUNCT
ejpam-5570	67	1	then	then	ADV
ejpam-5570	67	2	the	the	DET
ejpam-5570	67	3	following	follow	VERB
ejpam-5570	67	4	statements	statement	NOUN
ejpam-5570	67	5	are	be	AUX
ejpam-5570	67	6	true	true	ADJ
ejpam-5570	67	7	:	:	PUNCT
ejpam-5570	67	8	(	(	PUNCT
ejpam-5570	67	9	i	i	NOUN
ejpam-5570	67	10	)	)	PUNCT
ejpam-5570	67	11	if	if	SCONJ
ejpam-5570	67	12	a	a	DET
ejpam-5570	67	13	⊂	⊂	X
ejpam-5570	67	14	b	b	X
ejpam-5570	67	15	then	then	ADV
ejpam-5570	67	16	δcli(a	δcli(a	PROPN
ejpam-5570	67	17	)	)	PUNCT
ejpam-5570	67	18	⊂	⊂	PROPN
ejpam-5570	67	19	δcli(b	δcli(b	PROPN
ejpam-5570	67	20	)	)	PUNCT
ejpam-5570	67	21	;	;	PUNCT
ejpam-5570	67	22	(	(	PUNCT
ejpam-5570	67	23	ii	ii	NOUN
ejpam-5570	67	24	)	)	PUNCT
ejpam-5570	67	25	if	if	SCONJ
ejpam-5570	67	26	a	a	PRON
ejpam-5570	67	27	is	be	AUX
ejpam-5570	67	28	an	an	DET
ejpam-5570	67	29	open	open	ADJ
ejpam-5570	67	30	set	set	NOUN
ejpam-5570	67	31	,	,	PUNCT
ejpam-5570	67	32	then	then	ADV
ejpam-5570	67	33	δcli(a	δcli(a	ADJ
ejpam-5570	67	34	)	)	PUNCT
ejpam-5570	67	35	=	=	SYM
ejpam-5570	67	36	a	a	X
ejpam-5570	67	37	;	;	PUNCT
ejpam-5570	67	38	(	(	PUNCT
ejpam-5570	67	39	iii	iii	X
ejpam-5570	67	40	)	)	PUNCT
ejpam-5570	67	41	if	if	SCONJ
ejpam-5570	67	42	a	a	PRON
ejpam-5570	67	43	is	be	AUX
ejpam-5570	67	44	a	a	DET
ejpam-5570	67	45	closed	closed	ADJ
ejpam-5570	67	46	set	set	NOUN
ejpam-5570	67	47	,	,	PUNCT
ejpam-5570	67	48	then	then	ADV
ejpam-5570	67	49	δinti(a	δinti(a	PROPN
ejpam-5570	67	50	)	)	PUNCT
ejpam-5570	67	51	=	=	PUNCT
ejpam-5570	67	52	a.	a.	NOUN
ejpam-5570	67	53	definition	definition	NOUN
ejpam-5570	67	54	2	2	NUM
ejpam-5570	67	55	.	.	PUNCT
ejpam-5570	68	1	[	[	X
ejpam-5570	68	2	10	10	NUM
ejpam-5570	68	3	]	]	X
ejpam-5570	68	4	a	a	DET
ejpam-5570	68	5	subset	subset	NOUN
ejpam-5570	68	6	a	a	PRON
ejpam-5570	68	7	of	of	ADP
ejpam-5570	68	8	an	an	DET
ejpam-5570	68	9	ideal	ideal	ADJ
ejpam-5570	68	10	topological	topological	ADJ
ejpam-5570	68	11	space	space	NOUN
ejpam-5570	68	12	(	(	PUNCT
ejpam-5570	68	13	x	x	X
ejpam-5570	68	14	,	,	PUNCT
ejpam-5570	68	15	τ	τ	PROPN
ejpam-5570	68	16	,	,	PUNCT
ejpam-5570	68	17	i	i	PROPN
ejpam-5570	68	18	)	)	PUNCT
ejpam-5570	68	19	is	be	AUX
ejpam-5570	68	20	called	call	VERB
ejpam-5570	68	21	δ	δ	PROPN
ejpam-5570	68	22	-	-	PUNCT
ejpam-5570	68	23	βi	βi	ADV
ejpam-5570	68	24	-	-	PUNCT
ejpam-5570	68	25	open	open	ADJ
ejpam-5570	68	26	if	if	SCONJ
ejpam-5570	68	27	a	a	DET
ejpam-5570	68	28	⊂	⊂	PROPN
ejpam-5570	68	29	cl(int(δcli(a	cl(int(δcli(a	PROPN
ejpam-5570	68	30	)	)	PUNCT
ejpam-5570	68	31	)	)	PUNCT
ejpam-5570	68	32	)	)	PUNCT
ejpam-5570	69	1	and	and	CCONJ
ejpam-5570	69	2	it	it	PRON
ejpam-5570	69	3	is	be	AUX
ejpam-5570	69	4	called	call	VERB
ejpam-5570	69	5	δ	δ	PROPN
ejpam-5570	69	6	-	-	PUNCT
ejpam-5570	69	7	βi	βi	ADV
ejpam-5570	69	8	-	-	PUNCT
ejpam-5570	69	9	closed	closed	ADJ
ejpam-5570	69	10	if	if	SCONJ
ejpam-5570	69	11	int(cl(δinti(a	int(cl(δinti(a	NOUN
ejpam-5570	69	12	)	)	PUNCT
ejpam-5570	69	13	)	)	PUNCT
ejpam-5570	69	14	)	)	PUNCT
ejpam-5570	70	1	⊂	⊂	PROPN
ejpam-5570	70	2	a.	a.	NOUN
ejpam-5570	70	3	definition	definition	NOUN
ejpam-5570	70	4	3	3	NUM
ejpam-5570	70	5	.	.	PUNCT
ejpam-5570	71	1	[	[	X
ejpam-5570	71	2	10	10	NUM
ejpam-5570	71	3	]	]	X
ejpam-5570	71	4	let	let	VERB
ejpam-5570	71	5	(	(	PUNCT
ejpam-5570	71	6	x	x	NOUN
ejpam-5570	71	7	,	,	PUNCT
ejpam-5570	71	8	τ	τ	PROPN
ejpam-5570	71	9	,	,	PUNCT
ejpam-5570	71	10	i	i	PRON
ejpam-5570	71	11	)	)	PUNCT
ejpam-5570	71	12	be	be	VERB
ejpam-5570	71	13	an	an	DET
ejpam-5570	71	14	ideal	ideal	ADJ
ejpam-5570	71	15	topological	topological	ADJ
ejpam-5570	71	16	space	space	NOUN
ejpam-5570	71	17	.	.	PUNCT
ejpam-5570	72	1	the	the	DET
ejpam-5570	72	2	union	union	NOUN
ejpam-5570	72	3	of	of	ADP
ejpam-5570	72	4	all	all	DET
ejpam-5570	72	5	δ	δ	PROPN
ejpam-5570	72	6	-	-	PUNCT
ejpam-5570	72	7	βiopen	βiopen	ADJ
ejpam-5570	72	8	sets	set	NOUN
ejpam-5570	72	9	contained	contain	VERB
ejpam-5570	72	10	in	in	ADP
ejpam-5570	72	11	a	a	PRON
ejpam-5570	72	12	is	be	AUX
ejpam-5570	72	13	called	call	VERB
ejpam-5570	72	14	the	the	DET
ejpam-5570	72	15	δ	δ	PROPN
ejpam-5570	72	16	-	-	PUNCT
ejpam-5570	72	17	βi	βi	PROPN
ejpam-5570	72	18	-	-	NOUN
ejpam-5570	72	19	interior	interior	NOUN
ejpam-5570	72	20	of	of	ADP
ejpam-5570	72	21	a	a	DET
ejpam-5570	72	22	denoted	denote	VERB
ejpam-5570	72	23	by	by	ADP
ejpam-5570	72	24	δ	δ	PROPN
ejpam-5570	72	25	-	-	PROPN
ejpam-5570	72	26	βinti(a	βinti(a	NOUN
ejpam-5570	72	27	)	)	PUNCT
ejpam-5570	72	28	.	.	PUNCT
ejpam-5570	73	1	the	the	DET
ejpam-5570	73	2	intersection	intersection	NOUN
ejpam-5570	73	3	of	of	ADP
ejpam-5570	73	4	all	all	DET
ejpam-5570	73	5	δ	δ	PROPN
ejpam-5570	73	6	-	-	PUNCT
ejpam-5570	73	7	βi	βi	ADV
ejpam-5570	73	8	-	-	PUNCT
ejpam-5570	73	9	closed	close	VERB
ejpam-5570	73	10	sets	set	NOUN
ejpam-5570	73	11	containing	contain	VERB
ejpam-5570	73	12	a	a	PRON
ejpam-5570	73	13	is	be	AUX
ejpam-5570	73	14	called	call	VERB
ejpam-5570	73	15	the	the	DET
ejpam-5570	73	16	δ	δ	PROPN
ejpam-5570	73	17	-	-	PUNCT
ejpam-5570	73	18	βi	βi	NOUN
ejpam-5570	73	19	-	-	PUNCT
ejpam-5570	73	20	closure	closure	NOUN
ejpam-5570	73	21	of	of	ADP
ejpam-5570	73	22	a	a	DET
ejpam-5570	73	23	denoted	denote	VERB
ejpam-5570	73	24	by	by	ADP
ejpam-5570	73	25	δ	δ	PROPN
ejpam-5570	73	26	-	-	PUNCT
ejpam-5570	73	27	βcli(a	βcli(a	NOUN
ejpam-5570	73	28	)	)	PUNCT
ejpam-5570	73	29	.	.	PUNCT
ejpam-5570	74	1	lemma	lemma	PROPN
ejpam-5570	74	2	3	3	X
ejpam-5570	74	3	.	.	PUNCT
ejpam-5570	75	1	let	let	VERB
ejpam-5570	75	2	a	a	DET
ejpam-5570	75	3	be	be	AUX
ejpam-5570	75	4	a	a	DET
ejpam-5570	75	5	subset	subset	NOUN
ejpam-5570	75	6	of	of	ADP
ejpam-5570	75	7	an	an	DET
ejpam-5570	75	8	ideal	ideal	ADJ
ejpam-5570	75	9	topological	topological	ADJ
ejpam-5570	75	10	space	space	NOUN
ejpam-5570	75	11	(	(	PUNCT
ejpam-5570	75	12	x	x	X
ejpam-5570	75	13	,	,	PUNCT
ejpam-5570	75	14	τ	τ	PROPN
ejpam-5570	75	15	,	,	PUNCT
ejpam-5570	75	16	i	i	PROPN
ejpam-5570	75	17	)	)	PUNCT
ejpam-5570	75	18	.	.	PUNCT
ejpam-5570	76	1	then	then	ADV
ejpam-5570	76	2	,	,	PUNCT
ejpam-5570	76	3	(	(	PUNCT
ejpam-5570	76	4	i	i	NOUN
ejpam-5570	76	5	)	)	PUNCT
ejpam-5570	76	6	δ	δ	PROPN
ejpam-5570	76	7	-	-	PUNCT
ejpam-5570	76	8	βcli(a	βcli(a	NOUN
ejpam-5570	76	9	)	)	PUNCT
ejpam-5570	76	10	⊂	⊂	PROPN
ejpam-5570	76	11	cl(a	cl(a	X
ejpam-5570	76	12	)	)	PUNCT
ejpam-5570	76	13	.	.	PUNCT
ejpam-5570	77	1	(	(	PUNCT
ejpam-5570	77	2	ii	ii	NOUN
ejpam-5570	77	3	)	)	PUNCT
ejpam-5570	77	4	if	if	SCONJ
ejpam-5570	77	5	a	a	PRON
ejpam-5570	77	6	is	be	AUX
ejpam-5570	77	7	open	open	ADJ
ejpam-5570	77	8	,	,	PUNCT
ejpam-5570	77	9	then	then	ADV
ejpam-5570	77	10	a	a	PRON
ejpam-5570	77	11	is	be	AUX
ejpam-5570	77	12	δ	δ	PROPN
ejpam-5570	77	13	-	-	PUNCT
ejpam-5570	77	14	βi	βi	ADV
ejpam-5570	77	15	-	-	PUNCT
ejpam-5570	77	16	open	open	ADJ
ejpam-5570	77	17	.	.	PUNCT
ejpam-5570	78	1	(	(	PUNCT
ejpam-5570	78	2	iii	iii	X
ejpam-5570	78	3	)	)	PUNCT
ejpam-5570	78	4	if	if	SCONJ
ejpam-5570	78	5	a	a	PRON
ejpam-5570	78	6	is	be	AUX
ejpam-5570	78	7	closed	closed	ADJ
ejpam-5570	78	8	,	,	PUNCT
ejpam-5570	78	9	then	then	ADV
ejpam-5570	78	10	a	a	PRON
ejpam-5570	78	11	is	be	AUX
ejpam-5570	78	12	δ	δ	PROPN
ejpam-5570	78	13	-	-	PUNCT
ejpam-5570	78	14	βi	βi	ADV
ejpam-5570	78	15	-	-	PUNCT
ejpam-5570	78	16	closed	close	VERB
ejpam-5570	78	17	and	and	CCONJ
ejpam-5570	78	18	δ	δ	PROPN
ejpam-5570	78	19	-	-	PUNCT
ejpam-5570	78	20	βcli(a	βcli(a	NOUN
ejpam-5570	78	21	)	)	PUNCT
ejpam-5570	79	1	=	=	SYM
ejpam-5570	79	2	a.	a.	NOUN
ejpam-5570	79	3	(	(	PUNCT
ejpam-5570	79	4	iv	iv	X
ejpam-5570	79	5	)	)	PUNCT
ejpam-5570	79	6	x	x	SYM
ejpam-5570	80	1	∈	∈	PROPN
ejpam-5570	80	2	δ	δ	PROPN
ejpam-5570	80	3	-	-	PUNCT
ejpam-5570	80	4	βcli(a	βcli(a	NOUN
ejpam-5570	80	5	)	)	PUNCT
ejpam-5570	80	6	if	if	SCONJ
ejpam-5570	80	7	and	and	CCONJ
ejpam-5570	80	8	only	only	ADV
ejpam-5570	80	9	if	if	SCONJ
ejpam-5570	80	10	a	a	DET
ejpam-5570	80	11	∩	∩	NOUN
ejpam-5570	80	12	v	v	ADP
ejpam-5570	80	13	̸=	̸=	PROPN
ejpam-5570	80	14	∅	∅	NOUN
ejpam-5570	80	15	for	for	ADP
ejpam-5570	80	16	every	every	DET
ejpam-5570	80	17	δ	δ	PROPN
ejpam-5570	80	18	-	-	PUNCT
ejpam-5570	80	19	βi	βi	ADV
ejpam-5570	80	20	-	-	PUNCT
ejpam-5570	80	21	open	open	ADJ
ejpam-5570	80	22	set	set	VERB
ejpam-5570	80	23	v	v	NOUN
ejpam-5570	80	24	containing	contain	VERB
ejpam-5570	80	25	x.	x.	NOUN
ejpam-5570	80	26	proof	proof	NOUN
ejpam-5570	80	27	.	.	PUNCT
ejpam-5570	81	1	(	(	PUNCT
ejpam-5570	81	2	i	i	NOUN
ejpam-5570	81	3	)	)	PUNCT
ejpam-5570	81	4	applying	apply	VERB
ejpam-5570	81	5	the	the	DET
ejpam-5570	81	6	closure	closure	NOUN
ejpam-5570	81	7	and	and	CCONJ
ejpam-5570	81	8	δ	δ	PROPN
ejpam-5570	81	9	-	-	PUNCT
ejpam-5570	81	10	βi	βi	PRON
ejpam-5570	81	11	-	-	PUNCT
ejpam-5570	81	12	closure	closure	NOUN
ejpam-5570	81	13	definitions	definition	NOUN
ejpam-5570	81	14	in	in	ADP
ejpam-5570	81	15	x	x	NOUN
ejpam-5570	81	16	,	,	PUNCT
ejpam-5570	81	17	we	we	PRON
ejpam-5570	81	18	obtain	obtain	VERB
ejpam-5570	81	19	(	(	PUNCT
ejpam-5570	81	20	i	i	NOUN
ejpam-5570	81	21	)	)	PUNCT
ejpam-5570	81	22	.	.	PUNCT
ejpam-5570	82	1	(	(	PUNCT
ejpam-5570	82	2	ii	ii	NOUN
ejpam-5570	82	3	)	)	PUNCT
ejpam-5570	82	4	if	if	SCONJ
ejpam-5570	82	5	a	a	PRON
ejpam-5570	82	6	is	be	AUX
ejpam-5570	82	7	open	open	ADJ
ejpam-5570	82	8	,	,	PUNCT
ejpam-5570	82	9	then	then	ADV
ejpam-5570	82	10	a	a	DET
ejpam-5570	82	11	=	=	SYM
ejpam-5570	82	12	int(a	int(a	NOUN
ejpam-5570	82	13	)	)	PUNCT
ejpam-5570	82	14	.	.	PUNCT
ejpam-5570	83	1	consequently	consequently	ADV
ejpam-5570	83	2	,	,	PUNCT
ejpam-5570	83	3	a	a	DET
ejpam-5570	83	4	=	=	PUNCT
ejpam-5570	83	5	δcli(a	δcli(a	NOUN
ejpam-5570	83	6	)	)	PUNCT
ejpam-5570	83	7	as	as	SCONJ
ejpam-5570	83	8	stated	state	VERB
ejpam-5570	83	9	in	in	ADP
ejpam-5570	83	10	lemma	lemma	PROPN
ejpam-5570	83	11	2	2	NUM
ejpam-5570	83	12	.	.	PUNCT
ejpam-5570	84	1	therefore	therefore	ADV
ejpam-5570	84	2	,	,	PUNCT
ejpam-5570	84	3	a	a	DET
ejpam-5570	84	4	=	=	PUNCT
ejpam-5570	84	5	int(a	int(a	NOUN
ejpam-5570	84	6	)	)	PUNCT
ejpam-5570	84	7	=	=	SYM
ejpam-5570	84	8	int(δcli(a	int(δcli(a	NOUN
ejpam-5570	84	9	)	)	PUNCT
ejpam-5570	84	10	)	)	PUNCT
ejpam-5570	85	1	⊂	⊂	PROPN
ejpam-5570	85	2	cl(int(δcli(a	cl(int(δcli(a	PROPN
ejpam-5570	85	3	)	)	PUNCT
ejpam-5570	85	4	)	)	PUNCT
ejpam-5570	85	5	)	)	PUNCT
ejpam-5570	85	6	.	.	PUNCT
ejpam-5570	86	1	thus	thus	ADV
ejpam-5570	86	2	,	,	PUNCT
ejpam-5570	86	3	a	a	PRON
ejpam-5570	86	4	is	be	AUX
ejpam-5570	86	5	characterized	characterize	VERB
ejpam-5570	86	6	as	as	ADP
ejpam-5570	86	7	a	a	DET
ejpam-5570	86	8	δ	δ	PROPN
ejpam-5570	86	9	-	-	PUNCT
ejpam-5570	86	10	βi	βi	ADV
ejpam-5570	86	11	-	-	PUNCT
ejpam-5570	86	12	open	open	ADJ
ejpam-5570	86	13	set	set	NOUN
ejpam-5570	86	14	.	.	PUNCT
ejpam-5570	87	1	c.	c.	PROPN
ejpam-5570	87	2	boonpok	boonpok	PROPN
ejpam-5570	87	3	,	,	PUNCT
ejpam-5570	87	4	a.	a.	PROPN
ejpam-5570	87	5	sama	sama	PROPN
ejpam-5570	87	6	-	-	PUNCT
ejpam-5570	87	7	ae	ae	PROPN
ejpam-5570	87	8	,	,	PUNCT
ejpam-5570	87	9	p.	p.	NOUN
ejpam-5570	87	10	raktaow	raktaow	PROPN
ejpam-5570	87	11	/	/	SYM
ejpam-5570	87	12	eur	eur	PROPN
ejpam-5570	87	13	.	.	PUNCT
ejpam-5570	88	1	j.	j.	PROPN
ejpam-5570	88	2	pure	pure	PROPN
ejpam-5570	88	3	appl	appl	PROPN
ejpam-5570	88	4	.	.	PROPN
ejpam-5570	88	5	math	math	PROPN
ejpam-5570	88	6	,	,	PUNCT
ejpam-5570	88	7	18	18	NUM
ejpam-5570	88	8	(	(	PUNCT
ejpam-5570	88	9	1	1	NUM
ejpam-5570	88	10	)	)	PUNCT
ejpam-5570	88	11	(	(	PUNCT
ejpam-5570	88	12	2025	2025	NUM
ejpam-5570	88	13	)	)	PUNCT
ejpam-5570	88	14	,	,	PUNCT
ejpam-5570	88	15	5570	5570	NUM
ejpam-5570	88	16	4	4	NUM
ejpam-5570	88	17	of	of	ADP
ejpam-5570	88	18	12	12	NUM
ejpam-5570	88	19	(	(	PUNCT
ejpam-5570	88	20	iii	iii	NOUN
ejpam-5570	88	21	)	)	PUNCT
ejpam-5570	88	22	in	in	ADP
ejpam-5570	88	23	order	order	NOUN
ejpam-5570	88	24	to	to	PART
ejpam-5570	88	25	demonstrate	demonstrate	VERB
ejpam-5570	88	26	(	(	PUNCT
ejpam-5570	88	27	iii	iii	NOUN
ejpam-5570	88	28	)	)	PUNCT
ejpam-5570	89	1	,	,	PUNCT
ejpam-5570	89	2	we	we	PRON
ejpam-5570	89	3	assume	assume	VERB
ejpam-5570	89	4	that	that	SCONJ
ejpam-5570	89	5	a	a	PRON
ejpam-5570	89	6	is	be	AUX
ejpam-5570	89	7	closed	closed	ADJ
ejpam-5570	89	8	.	.	PUNCT
ejpam-5570	90	1	this	this	PRON
ejpam-5570	90	2	implies	imply	VERB
ejpam-5570	90	3	that	that	SCONJ
ejpam-5570	90	4	a	a	DET
ejpam-5570	90	5	=	=	NOUN
ejpam-5570	90	6	cl(a	cl(a	X
ejpam-5570	90	7	)	)	PUNCT
ejpam-5570	90	8	,	,	PUNCT
ejpam-5570	90	9	and	and	CCONJ
ejpam-5570	90	10	by	by	ADP
ejpam-5570	90	11	lemma	lemma	PROPN
ejpam-5570	90	12	2	2	NUM
ejpam-5570	90	13	,	,	PUNCT
ejpam-5570	90	14	δinti(a	δinti(a	PROPN
ejpam-5570	90	15	)	)	PUNCT
ejpam-5570	90	16	=	=	SYM
ejpam-5570	90	17	a.	a.	NOUN
ejpam-5570	90	18	therefore	therefore	ADV
ejpam-5570	90	19	,	,	PUNCT
ejpam-5570	90	20	cl(δinti(a	cl(δinti(a	PROPN
ejpam-5570	90	21	)	)	PUNCT
ejpam-5570	90	22	)	)	PUNCT
ejpam-5570	91	1	=	=	SYM
ejpam-5570	91	2	cl(a	cl(a	X
ejpam-5570	91	3	)	)	PUNCT
ejpam-5570	91	4	=	=	SYM
ejpam-5570	91	5	a	a	PRON
ejpam-5570	91	6	,	,	PUNCT
ejpam-5570	91	7	which	which	PRON
ejpam-5570	91	8	implies	imply	VERB
ejpam-5570	91	9	that	that	SCONJ
ejpam-5570	91	10	int(cl(δinti(a	int(cl(δinti(a	NOUN
ejpam-5570	91	11	)	)	PUNCT
ejpam-5570	91	12	)	)	PUNCT
ejpam-5570	91	13	)	)	PUNCT
ejpam-5570	92	1	=	=	PUNCT
ejpam-5570	92	2	int(a	int(a	X
ejpam-5570	92	3	)	)	PUNCT
ejpam-5570	92	4	⊂	⊂	PROPN
ejpam-5570	92	5	a.	a.	PROPN
ejpam-5570	92	6	consequently	consequently	ADV
ejpam-5570	92	7	,	,	PUNCT
ejpam-5570	92	8	a	a	PRON
ejpam-5570	92	9	is	be	AUX
ejpam-5570	92	10	δ	δ	PROPN
ejpam-5570	92	11	-	-	PUNCT
ejpam-5570	92	12	βi	βi	ADV
ejpam-5570	92	13	-	-	PUNCT
ejpam-5570	92	14	closed	closed	ADJ
ejpam-5570	92	15	.	.	PUNCT
ejpam-5570	93	1	it	it	PRON
ejpam-5570	93	2	implies	imply	VERB
ejpam-5570	93	3	that	that	SCONJ
ejpam-5570	93	4	δ	δ	PROPN
ejpam-5570	93	5	-	-	PUNCT
ejpam-5570	93	6	βcli(a	βcli(a	NOUN
ejpam-5570	93	7	)	)	PUNCT
ejpam-5570	93	8	⊂	⊂	PROPN
ejpam-5570	93	9	cl(a	cl(a	X
ejpam-5570	93	10	)	)	PUNCT
ejpam-5570	93	11	=	=	SYM
ejpam-5570	94	1	a.	a.	NOUN
ejpam-5570	94	2	we	we	PRON
ejpam-5570	94	3	deduce	deduce	VERB
ejpam-5570	94	4	that	that	SCONJ
ejpam-5570	94	5	δ	δ	PROPN
ejpam-5570	94	6	-	-	PUNCT
ejpam-5570	94	7	βcli(a	βcli(a	NOUN
ejpam-5570	94	8	)	)	PUNCT
ejpam-5570	94	9	=	=	SYM
ejpam-5570	94	10	a.	a.	NOUN
ejpam-5570	94	11	(	(	PUNCT
ejpam-5570	94	12	iv	iv	X
ejpam-5570	94	13	)	)	PUNCT
ejpam-5570	94	14	let	let	VERB
ejpam-5570	94	15	x	x	X
ejpam-5570	94	16	∈	∈	PROPN
ejpam-5570	94	17	δ	δ	PROPN
ejpam-5570	94	18	-	-	PUNCT
ejpam-5570	94	19	βcli(a	βcli(a	NOUN
ejpam-5570	94	20	)	)	PUNCT
ejpam-5570	94	21	.	.	PUNCT
ejpam-5570	95	1	the	the	DET
ejpam-5570	95	2	point	point	NOUN
ejpam-5570	95	3	x	x	PUNCT
ejpam-5570	95	4	is	be	AUX
ejpam-5570	95	5	then	then	ADV
ejpam-5570	95	6	included	include	VERB
ejpam-5570	95	7	in	in	ADP
ejpam-5570	95	8	all	all	DET
ejpam-5570	95	9	δ	δ	PROPN
ejpam-5570	95	10	-	-	PUNCT
ejpam-5570	95	11	βi	βi	ADV
ejpam-5570	95	12	-	-	PUNCT
ejpam-5570	95	13	closed	close	VERB
ejpam-5570	95	14	sets	set	NOUN
ejpam-5570	95	15	that	that	PRON
ejpam-5570	95	16	contain	contain	VERB
ejpam-5570	95	17	a.	a.	NOUN
ejpam-5570	95	18	assume	assume	VERB
ejpam-5570	96	1	that	that	SCONJ
ejpam-5570	96	2	v	v	ADP
ejpam-5570	96	3	∩	∩	NOUN
ejpam-5570	96	4	a	a	DET
ejpam-5570	96	5	=	=	NOUN
ejpam-5570	96	6	∅	∅	NOUN
ejpam-5570	96	7	for	for	ADP
ejpam-5570	96	8	a	a	DET
ejpam-5570	96	9	δ	δ	PROPN
ejpam-5570	96	10	-	-	PUNCT
ejpam-5570	96	11	βi	βi	ADV
ejpam-5570	96	12	-	-	PUNCT
ejpam-5570	96	13	open	open	NOUN
ejpam-5570	96	14	set	set	VERB
ejpam-5570	96	15	v	v	NOUN
ejpam-5570	96	16	that	that	PRON
ejpam-5570	96	17	contains	contain	VERB
ejpam-5570	96	18	x.	x.	NOUN
ejpam-5570	97	1	it	it	PRON
ejpam-5570	97	2	implies	imply	VERB
ejpam-5570	97	3	that	that	SCONJ
ejpam-5570	97	4	x	x	PUNCT
ejpam-5570	98	1	−	−	NOUN
ejpam-5570	98	2	v	v	NOUN
ejpam-5570	98	3	is	be	AUX
ejpam-5570	98	4	a	a	DET
ejpam-5570	98	5	δ	δ	PROPN
ejpam-5570	98	6	-	-	PUNCT
ejpam-5570	98	7	βi	βi	ADV
ejpam-5570	98	8	-	-	PUNCT
ejpam-5570	98	9	closed	close	VERB
ejpam-5570	98	10	set	set	NOUN
ejpam-5570	98	11	that	that	PRON
ejpam-5570	98	12	contains	contain	VERB
ejpam-5570	98	13	a	a	PRON
ejpam-5570	98	14	but	but	CCONJ
ejpam-5570	98	15	x	x	SYM
ejpam-5570	98	16	̸∈	̸∈	PROPN
ejpam-5570	98	17	x	x	X
ejpam-5570	98	18	−	−	PROPN
ejpam-5570	98	19	v	v	NOUN
ejpam-5570	98	20	.	.	PUNCT
ejpam-5570	99	1	consequently	consequently	ADV
ejpam-5570	99	2	,	,	PUNCT
ejpam-5570	99	3	we	we	PRON
ejpam-5570	99	4	have	have	VERB
ejpam-5570	99	5	a	a	DET
ejpam-5570	99	6	contraction	contraction	NOUN
ejpam-5570	99	7	.	.	PUNCT
ejpam-5570	100	1	as	as	ADP
ejpam-5570	100	2	a	a	DET
ejpam-5570	100	3	result	result	NOUN
ejpam-5570	100	4	,	,	PUNCT
ejpam-5570	100	5	a	a	DET
ejpam-5570	100	6	∩	∩	NOUN
ejpam-5570	100	7	v	v	ADP
ejpam-5570	100	8	̸=	̸=	PROPN
ejpam-5570	100	9	∅	∅	NOUN
ejpam-5570	100	10	for	for	ADP
ejpam-5570	100	11	any	any	DET
ejpam-5570	100	12	δ	δ	PROPN
ejpam-5570	100	13	-	-	PUNCT
ejpam-5570	100	14	βi	βi	ADV
ejpam-5570	100	15	-	-	PUNCT
ejpam-5570	100	16	open	open	NOUN
ejpam-5570	100	17	set	set	VERB
ejpam-5570	100	18	v	v	NOUN
ejpam-5570	100	19	that	that	PRON
ejpam-5570	100	20	contains	contain	VERB
ejpam-5570	100	21	x.	x.	NOUN
ejpam-5570	100	22	according	accord	VERB
ejpam-5570	100	23	to	to	ADP
ejpam-5570	100	24	theorem	theorem	NOUN
ejpam-5570	100	25	1	1	NUM
ejpam-5570	100	26	in	in	ADP
ejpam-5570	100	27	[	[	X
ejpam-5570	100	28	10	10	NUM
ejpam-5570	100	29	]	]	PUNCT
ejpam-5570	100	30	,	,	PUNCT
ejpam-5570	100	31	if	if	SCONJ
ejpam-5570	100	32	a	a	DET
ejpam-5570	100	33	∩	∩	NOUN
ejpam-5570	100	34	v	v	ADP
ejpam-5570	100	35	̸=	̸=	PROPN
ejpam-5570	100	36	∅	∅	NOUN
ejpam-5570	100	37	for	for	ADP
ejpam-5570	100	38	every	every	DET
ejpam-5570	100	39	δ	δ	PROPN
ejpam-5570	100	40	-	-	PUNCT
ejpam-5570	100	41	βi	βi	ADV
ejpam-5570	100	42	-	-	PUNCT
ejpam-5570	100	43	open	open	ADJ
ejpam-5570	100	44	sets	set	NOUN
ejpam-5570	100	45	v	v	ADP
ejpam-5570	100	46	containing	contain	VERB
ejpam-5570	100	47	x	x	NOUN
ejpam-5570	100	48	,	,	PUNCT
ejpam-5570	100	49	then	then	ADV
ejpam-5570	100	50	x	x	SYM
ejpam-5570	100	51	∈	∈	PROPN
ejpam-5570	100	52	δ	δ	PROPN
ejpam-5570	100	53	-	-	PUNCT
ejpam-5570	100	54	βcli(a	βcli(a	NOUN
ejpam-5570	100	55	)	)	PUNCT
ejpam-5570	100	56	.	.	PUNCT
ejpam-5570	101	1	3	3	X
ejpam-5570	101	2	.	.	X
ejpam-5570	101	3	δ	δ	PROPN
ejpam-5570	101	4	-	-	PUNCT
ejpam-5570	101	5	βi	βi	PROPN
ejpam-5570	101	6	-	-	PUNCT
ejpam-5570	101	7	paracompactness	paracompactness	NOUN
ejpam-5570	101	8	and	and	CCONJ
ejpam-5570	101	9	characterizations	characterization	NOUN
ejpam-5570	101	10	this	this	DET
ejpam-5570	101	11	part	part	NOUN
ejpam-5570	101	12	talks	talk	VERB
ejpam-5570	101	13	about	about	ADP
ejpam-5570	101	14	the	the	DET
ejpam-5570	101	15	idea	idea	NOUN
ejpam-5570	101	16	of	of	ADP
ejpam-5570	101	17	δ	δ	PROPN
ejpam-5570	101	18	-	-	PUNCT
ejpam-5570	101	19	βi	βi	PROPN
ejpam-5570	101	20	-	-	PUNCT
ejpam-5570	101	21	paracompactness	paracompactness	NOUN
ejpam-5570	101	22	,	,	PUNCT
ejpam-5570	101	23	which	which	PRON
ejpam-5570	101	24	is	be	AUX
ejpam-5570	101	25	a	a	DET
ejpam-5570	101	26	weaker	weak	ADJ
ejpam-5570	101	27	form	form	NOUN
ejpam-5570	101	28	of	of	ADP
ejpam-5570	101	29	i	i	PROPN
ejpam-5570	101	30	-	-	PUNCT
ejpam-5570	101	31	β	β	NOUN
ejpam-5570	101	32	-	-	NOUN
ejpam-5570	101	33	paracompactness	paracompactness	NOUN
ejpam-5570	101	34	that	that	PRON
ejpam-5570	101	35	was	be	AUX
ejpam-5570	101	36	studied	study	VERB
ejpam-5570	101	37	by	by	ADP
ejpam-5570	101	38	yildirim	yildirim	PROPN
ejpam-5570	101	39	et	et	PROPN
ejpam-5570	101	40	al	al	PROPN
ejpam-5570	101	41	.	.	PUNCT
ejpam-5570	102	1	[	[	X
ejpam-5570	102	2	17	17	NUM
ejpam-5570	102	3	]	]	PUNCT
ejpam-5570	102	4	.	.	PUNCT
ejpam-5570	103	1	we	we	PRON
ejpam-5570	103	2	will	will	AUX
ejpam-5570	103	3	then	then	ADV
ejpam-5570	103	4	look	look	VERB
ejpam-5570	103	5	at	at	ADP
ejpam-5570	103	6	how	how	SCONJ
ejpam-5570	103	7	to	to	PART
ejpam-5570	103	8	describe	describe	VERB
ejpam-5570	103	9	it	it	PRON
ejpam-5570	103	10	.	.	PUNCT
ejpam-5570	104	1	let	let	VERB
ejpam-5570	104	2	u	u	PRON
ejpam-5570	104	3	=	=	X
ejpam-5570	104	4	{	{	PUNCT
ejpam-5570	104	5	uα	uα	X
ejpam-5570	104	6	:	:	PUNCT
ejpam-5570	104	7	α	α	PROPN
ejpam-5570	104	8	∈	∈	PROPN
ejpam-5570	104	9	λ1	λ1	PROPN
ejpam-5570	104	10	}	}	PUNCT
ejpam-5570	104	11	and	and	CCONJ
ejpam-5570	104	12	v	v	NOUN
ejpam-5570	104	13	=	=	PUNCT
ejpam-5570	104	14	{	{	PUNCT
ejpam-5570	104	15	vµ	vµ	X
ejpam-5570	104	16	:	:	PUNCT
ejpam-5570	104	17	µ	µ	PROPN
ejpam-5570	104	18	∈	∈	PROPN
ejpam-5570	104	19	λ2	λ2	PROPN
ejpam-5570	104	20	}	}	PUNCT
ejpam-5570	104	21	be	be	VERB
ejpam-5570	104	22	two	two	NUM
ejpam-5570	104	23	collections	collection	NOUN
ejpam-5570	104	24	of	of	ADP
ejpam-5570	104	25	subsets	subset	NOUN
ejpam-5570	104	26	of	of	ADP
ejpam-5570	104	27	a	a	DET
ejpam-5570	104	28	topological	topological	ADJ
ejpam-5570	104	29	space	space	NOUN
ejpam-5570	104	30	x.	x.	NOUN
ejpam-5570	105	1	the	the	DET
ejpam-5570	105	2	collection	collection	NOUN
ejpam-5570	105	3	u	u	NOUN
ejpam-5570	105	4	is	be	AUX
ejpam-5570	105	5	called	call	VERB
ejpam-5570	105	6	a	a	DET
ejpam-5570	105	7	refinement	refinement	NOUN
ejpam-5570	105	8	of	of	ADP
ejpam-5570	105	9	the	the	DET
ejpam-5570	105	10	collection	collection	NOUN
ejpam-5570	105	11	v	v	NOUN
ejpam-5570	105	12	if	if	SCONJ
ejpam-5570	105	13	for	for	ADP
ejpam-5570	105	14	every	every	DET
ejpam-5570	105	15	α	α	PROPN
ejpam-5570	105	16	∈	∈	PROPN
ejpam-5570	105	17	λ1	λ1	PROPN
ejpam-5570	105	18	there	there	PRON
ejpam-5570	105	19	exists	exist	VERB
ejpam-5570	105	20	µ	µ	PRON
ejpam-5570	105	21	∈	∈	NOUN
ejpam-5570	105	22	λ2	λ2	NOUN
ejpam-5570	105	23	such	such	ADJ
ejpam-5570	105	24	that	that	SCONJ
ejpam-5570	105	25	uα	uα	PROPN
ejpam-5570	105	26	⊂	⊂	PROPN
ejpam-5570	105	27	vµ.	vµ.	VERB
ejpam-5570	105	28	a	a	DET
ejpam-5570	105	29	collection	collection	NOUN
ejpam-5570	105	30	v	v	NOUN
ejpam-5570	105	31	of	of	ADP
ejpam-5570	105	32	subsets	subset	NOUN
ejpam-5570	105	33	of	of	ADP
ejpam-5570	105	34	a	a	DET
ejpam-5570	105	35	topological	topological	ADJ
ejpam-5570	105	36	space	space	NOUN
ejpam-5570	105	37	(	(	PUNCT
ejpam-5570	105	38	x	x	X
ejpam-5570	105	39	,	,	PUNCT
ejpam-5570	105	40	τ	τ	X
ejpam-5570	105	41	)	)	PUNCT
ejpam-5570	105	42	is	be	AUX
ejpam-5570	105	43	said	say	VERB
ejpam-5570	105	44	to	to	PART
ejpam-5570	105	45	be	be	AUX
ejpam-5570	105	46	β	β	X
ejpam-5570	105	47	-	-	ADJ
ejpam-5570	105	48	locally	locally	ADV
ejpam-5570	105	49	finite	finite	NOUN
ejpam-5570	105	50	[	[	X
ejpam-5570	105	51	3	3	NUM
ejpam-5570	105	52	]	]	X
ejpam-5570	105	53	if	if	SCONJ
ejpam-5570	105	54	for	for	ADP
ejpam-5570	105	55	each	each	DET
ejpam-5570	105	56	x	x	SYM
ejpam-5570	105	57	∈	∈	PROPN
ejpam-5570	105	58	x	x	X
ejpam-5570	105	59	,	,	PUNCT
ejpam-5570	105	60	there	there	PRON
ejpam-5570	105	61	exists	exist	VERB
ejpam-5570	105	62	a	a	DET
ejpam-5570	105	63	β	β	NOUN
ejpam-5570	105	64	-	-	ADJ
ejpam-5570	105	65	open	open	ADJ
ejpam-5570	105	66	set	set	NOUN
ejpam-5570	105	67	u	u	NOUN
ejpam-5570	105	68	containing	contain	VERB
ejpam-5570	105	69	x	x	X
ejpam-5570	105	70	and	and	CCONJ
ejpam-5570	105	71	u	u	NOUN
ejpam-5570	105	72	intersects	intersect	NOUN
ejpam-5570	105	73	at	at	ADV
ejpam-5570	105	74	most	most	ADV
ejpam-5570	105	75	finitely	finitely	ADV
ejpam-5570	105	76	many	many	ADJ
ejpam-5570	105	77	members	member	NOUN
ejpam-5570	105	78	of	of	ADP
ejpam-5570	105	79	v.	v.	PROPN
ejpam-5570	105	80	yildirim	yildirim	PROPN
ejpam-5570	105	81	et	et	PROPN
ejpam-5570	105	82	al	al	PROPN
ejpam-5570	105	83	.	.	PUNCT
ejpam-5570	106	1	[	[	X
ejpam-5570	106	2	17	17	NUM
ejpam-5570	106	3	]	]	PUNCT
ejpam-5570	106	4	introduced	introduce	VERB
ejpam-5570	106	5	the	the	DET
ejpam-5570	106	6	concept	concept	NOUN
ejpam-5570	106	7	of	of	ADP
ejpam-5570	106	8	paracompactness	paracompactness	NOUN
ejpam-5570	106	9	in	in	ADP
ejpam-5570	106	10	an	an	DET
ejpam-5570	106	11	ideal	ideal	ADJ
ejpam-5570	106	12	topological	topological	ADJ
ejpam-5570	106	13	space	space	NOUN
ejpam-5570	106	14	as	as	SCONJ
ejpam-5570	106	15	follows	follow	VERB
ejpam-5570	106	16	:	:	PUNCT
ejpam-5570	106	17	an	an	DET
ejpam-5570	106	18	ideal	ideal	ADJ
ejpam-5570	106	19	topological	topological	ADJ
ejpam-5570	106	20	space	space	NOUN
ejpam-5570	106	21	(	(	PUNCT
ejpam-5570	106	22	x	x	X
ejpam-5570	106	23	,	,	PUNCT
ejpam-5570	106	24	τ	τ	PROPN
ejpam-5570	106	25	,	,	PUNCT
ejpam-5570	106	26	i	i	PROPN
ejpam-5570	106	27	)	)	PUNCT
ejpam-5570	106	28	is	be	AUX
ejpam-5570	106	29	said	say	VERB
ejpam-5570	106	30	to	to	PART
ejpam-5570	106	31	be	be	AUX
ejpam-5570	106	32	i	i	NOUN
ejpam-5570	106	33	-	-	PUNCT
ejpam-5570	106	34	β	β	NOUN
ejpam-5570	106	35	-	-	NOUN
ejpam-5570	106	36	paracompact	paracompact	ADJ
ejpam-5570	106	37	if	if	SCONJ
ejpam-5570	106	38	every	every	DET
ejpam-5570	106	39	open	open	ADJ
ejpam-5570	106	40	cover	cover	VERB
ejpam-5570	106	41	u	u	NOUN
ejpam-5570	106	42	of	of	ADP
ejpam-5570	106	43	x	x	PUNCT
ejpam-5570	106	44	has	have	VERB
ejpam-5570	106	45	a	a	DET
ejpam-5570	106	46	β	β	NOUN
ejpam-5570	106	47	-	-	ADJ
ejpam-5570	106	48	locally	locally	ADV
ejpam-5570	106	49	finite	finite	ADJ
ejpam-5570	106	50	β	β	NOUN
ejpam-5570	106	51	-	-	ADJ
ejpam-5570	106	52	open	open	ADJ
ejpam-5570	106	53	refinement	refinement	NOUN
ejpam-5570	106	54	v	v	ADP
ejpam-5570	106	55	such	such	ADJ
ejpam-5570	106	56	that	that	SCONJ
ejpam-5570	106	57	x	x	SYM
ejpam-5570	106	58	−∪{v	−∪{v	NOUN
ejpam-5570	106	59	:	:	PUNCT
ejpam-5570	106	60	v	v	NUM
ejpam-5570	106	61	∈	∈	PROPN
ejpam-5570	106	62	v	v	NOUN
ejpam-5570	106	63	}	}	PUNCT
ejpam-5570	106	64	∈	∈	PROPN
ejpam-5570	106	65	i.	i.	NOUN
ejpam-5570	106	66	definition	definition	NOUN
ejpam-5570	106	67	4	4	NUM
ejpam-5570	106	68	.	.	PUNCT
ejpam-5570	107	1	a	a	DET
ejpam-5570	107	2	collection	collection	NOUN
ejpam-5570	107	3	v	v	NOUN
ejpam-5570	107	4	of	of	ADP
ejpam-5570	107	5	subsets	subset	NOUN
ejpam-5570	107	6	of	of	ADP
ejpam-5570	107	7	an	an	DET
ejpam-5570	107	8	ideal	ideal	ADJ
ejpam-5570	107	9	topological	topological	ADJ
ejpam-5570	107	10	space	space	NOUN
ejpam-5570	107	11	(	(	PUNCT
ejpam-5570	107	12	x	x	X
ejpam-5570	107	13	,	,	PUNCT
ejpam-5570	107	14	τ	τ	PROPN
ejpam-5570	107	15	,	,	PUNCT
ejpam-5570	107	16	i	i	PROPN
ejpam-5570	107	17	)	)	PUNCT
ejpam-5570	107	18	is	be	AUX
ejpam-5570	107	19	said	say	VERB
ejpam-5570	107	20	to	to	PART
ejpam-5570	107	21	be	be	AUX
ejpam-5570	107	22	δ	δ	PROPN
ejpam-5570	107	23	-	-	PUNCT
ejpam-5570	107	24	βi	βi	ADV
ejpam-5570	107	25	-	-	PUNCT
ejpam-5570	107	26	locally	locally	ADV
ejpam-5570	107	27	finite	finite	VERB
ejpam-5570	107	28	if	if	SCONJ
ejpam-5570	107	29	for	for	ADP
ejpam-5570	107	30	each	each	DET
ejpam-5570	107	31	x	x	SYM
ejpam-5570	107	32	∈	∈	PROPN
ejpam-5570	107	33	x	x	X
ejpam-5570	107	34	,	,	PUNCT
ejpam-5570	107	35	there	there	PRON
ejpam-5570	107	36	exists	exist	VERB
ejpam-5570	107	37	a	a	DET
ejpam-5570	107	38	δ	δ	PROPN
ejpam-5570	107	39	-	-	PUNCT
ejpam-5570	107	40	βi	βi	ADV
ejpam-5570	107	41	-	-	PUNCT
ejpam-5570	107	42	open	open	ADJ
ejpam-5570	107	43	set	set	NOUN
ejpam-5570	107	44	u	u	NOUN
ejpam-5570	107	45	containing	contain	VERB
ejpam-5570	107	46	x	x	X
ejpam-5570	107	47	and	and	CCONJ
ejpam-5570	107	48	u	u	NOUN
ejpam-5570	107	49	intersects	intersect	NOUN
ejpam-5570	107	50	at	at	ADV
ejpam-5570	107	51	most	most	ADV
ejpam-5570	107	52	finitely	finitely	ADV
ejpam-5570	107	53	many	many	ADJ
ejpam-5570	107	54	members	member	NOUN
ejpam-5570	107	55	of	of	ADP
ejpam-5570	107	56	v.	v.	ADP
ejpam-5570	107	57	lemma	lemma	PROPN
ejpam-5570	107	58	4	4	X
ejpam-5570	107	59	.	.	PUNCT
ejpam-5570	108	1	let	let	VERB
ejpam-5570	108	2	v	v	PART
ejpam-5570	108	3	be	be	AUX
ejpam-5570	108	4	a	a	DET
ejpam-5570	108	5	collection	collection	NOUN
ejpam-5570	108	6	of	of	ADP
ejpam-5570	108	7	subsets	subset	NOUN
ejpam-5570	108	8	of	of	ADP
ejpam-5570	108	9	an	an	DET
ejpam-5570	108	10	ideal	ideal	ADJ
ejpam-5570	108	11	topological	topological	ADJ
ejpam-5570	108	12	space	space	NOUN
ejpam-5570	108	13	(	(	PUNCT
ejpam-5570	108	14	x	x	X
ejpam-5570	108	15	,	,	PUNCT
ejpam-5570	108	16	τ	τ	PROPN
ejpam-5570	108	17	,	,	PUNCT
ejpam-5570	108	18	i	i	PROPN
ejpam-5570	108	19	)	)	PUNCT
ejpam-5570	108	20	.	.	PUNCT
ejpam-5570	109	1	if	if	SCONJ
ejpam-5570	109	2	v	v	NOUN
ejpam-5570	109	3	is	be	AUX
ejpam-5570	109	4	β	β	X
ejpam-5570	109	5	-	-	ADJ
ejpam-5570	109	6	locally	locally	ADV
ejpam-5570	109	7	finite	finite	NOUN
ejpam-5570	109	8	,	,	PUNCT
ejpam-5570	109	9	then	then	ADV
ejpam-5570	109	10	it	it	PRON
ejpam-5570	109	11	is	be	AUX
ejpam-5570	109	12	δ	δ	PROPN
ejpam-5570	109	13	-	-	PUNCT
ejpam-5570	109	14	βi	βi	ADV
ejpam-5570	109	15	-	-	PUNCT
ejpam-5570	109	16	locally	locally	ADV
ejpam-5570	109	17	finite	finite	NOUN
ejpam-5570	109	18	.	.	PUNCT
ejpam-5570	110	1	proof	proof	NOUN
ejpam-5570	110	2	.	.	PUNCT
ejpam-5570	111	1	let	let	VERB
ejpam-5570	111	2	v	v	PART
ejpam-5570	111	3	be	be	AUX
ejpam-5570	111	4	β	β	X
ejpam-5570	111	5	-	-	ADJ
ejpam-5570	111	6	locally	locally	ADV
ejpam-5570	111	7	finite	finite	NOUN
ejpam-5570	111	8	.	.	PUNCT
ejpam-5570	112	1	we	we	PRON
ejpam-5570	112	2	will	will	AUX
ejpam-5570	112	3	verify	verify	VERB
ejpam-5570	112	4	that	that	PRON
ejpam-5570	112	5	v	v	NOUN
ejpam-5570	112	6	is	be	AUX
ejpam-5570	112	7	δ	δ	PROPN
ejpam-5570	112	8	-	-	PUNCT
ejpam-5570	112	9	βi	βi	ADV
ejpam-5570	112	10	-	-	PUNCT
ejpam-5570	112	11	locally	locally	ADV
ejpam-5570	112	12	finite	finite	NOUN
ejpam-5570	112	13	.	.	PUNCT
ejpam-5570	113	1	let	let	VERB
ejpam-5570	113	2	x	x	SYM
ejpam-5570	113	3	∈	∈	PROPN
ejpam-5570	113	4	x.	x.	NOUN
ejpam-5570	113	5	since	since	SCONJ
ejpam-5570	113	6	v	v	NUM
ejpam-5570	113	7	is	be	AUX
ejpam-5570	113	8	β	β	X
ejpam-5570	113	9	-	-	ADJ
ejpam-5570	113	10	locally	locally	ADV
ejpam-5570	113	11	finite	finite	NOUN
ejpam-5570	113	12	,	,	PUNCT
ejpam-5570	113	13	there	there	PRON
ejpam-5570	113	14	exists	exist	VERB
ejpam-5570	113	15	a	a	DET
ejpam-5570	113	16	β	β	NOUN
ejpam-5570	113	17	-	-	ADJ
ejpam-5570	113	18	open	open	ADJ
ejpam-5570	113	19	set	set	ADJ
ejpam-5570	113	20	gx	gx	PROPN
ejpam-5570	113	21	containing	contain	VERB
ejpam-5570	113	22	x	x	PROPN
ejpam-5570	113	23	,	,	PUNCT
ejpam-5570	113	24	which	which	PRON
ejpam-5570	113	25	intersects	intersect	VERB
ejpam-5570	113	26	at	at	ADP
ejpam-5570	113	27	most	most	ADV
ejpam-5570	113	28	finitely	finitely	ADV
ejpam-5570	113	29	many	many	ADJ
ejpam-5570	113	30	elements	element	NOUN
ejpam-5570	113	31	of	of	ADP
ejpam-5570	113	32	v.	v.	ADV
ejpam-5570	113	33	given	give	VERB
ejpam-5570	113	34	that	that	SCONJ
ejpam-5570	113	35	gx	gx	PROPN
ejpam-5570	113	36	is	be	AUX
ejpam-5570	113	37	β	β	NOUN
ejpam-5570	113	38	-	-	ADJ
ejpam-5570	113	39	open	open	ADJ
ejpam-5570	113	40	,	,	PUNCT
ejpam-5570	113	41	it	it	PRON
ejpam-5570	113	42	follows	follow	VERB
ejpam-5570	113	43	that	that	SCONJ
ejpam-5570	113	44	it	it	PRON
ejpam-5570	113	45	is	be	AUX
ejpam-5570	113	46	δ	δ	PROPN
ejpam-5570	113	47	-	-	PUNCT
ejpam-5570	113	48	βi	βi	ADV
ejpam-5570	113	49	-	-	PUNCT
ejpam-5570	113	50	open	open	ADJ
ejpam-5570	113	51	[	[	X
ejpam-5570	113	52	10	10	NUM
ejpam-5570	113	53	]	]	PUNCT
ejpam-5570	113	54	.	.	PUNCT
ejpam-5570	114	1	consequently	consequently	ADV
ejpam-5570	114	2	,	,	PUNCT
ejpam-5570	114	3	v	v	NOUN
ejpam-5570	114	4	is	be	AUX
ejpam-5570	114	5	δ	δ	PROPN
ejpam-5570	114	6	-	-	PUNCT
ejpam-5570	114	7	βi	βi	ADV
ejpam-5570	114	8	-	-	PUNCT
ejpam-5570	114	9	locally	locally	ADV
ejpam-5570	114	10	finite	finite	NOUN
ejpam-5570	114	11	.	.	PUNCT
ejpam-5570	115	1	definition	definition	NOUN
ejpam-5570	115	2	5	5	NUM
ejpam-5570	115	3	.	.	PUNCT
ejpam-5570	116	1	an	an	DET
ejpam-5570	116	2	ideal	ideal	ADJ
ejpam-5570	116	3	topological	topological	ADJ
ejpam-5570	116	4	space	space	NOUN
ejpam-5570	116	5	(	(	PUNCT
ejpam-5570	116	6	x	x	X
ejpam-5570	116	7	,	,	PUNCT
ejpam-5570	116	8	τ	τ	PROPN
ejpam-5570	116	9	,	,	PUNCT
ejpam-5570	116	10	i	i	PROPN
ejpam-5570	116	11	)	)	PUNCT
ejpam-5570	116	12	is	be	AUX
ejpam-5570	116	13	said	say	VERB
ejpam-5570	116	14	to	to	PART
ejpam-5570	116	15	be	be	AUX
ejpam-5570	116	16	δ	δ	PROPN
ejpam-5570	116	17	-	-	PUNCT
ejpam-5570	116	18	βi	βi	ADV
ejpam-5570	116	19	-	-	NOUN
ejpam-5570	116	20	paracompact	paracompact	NOUN
ejpam-5570	116	21	if	if	SCONJ
ejpam-5570	116	22	every	every	DET
ejpam-5570	116	23	open	open	ADJ
ejpam-5570	116	24	cover	cover	VERB
ejpam-5570	116	25	u	u	NOUN
ejpam-5570	116	26	of	of	ADP
ejpam-5570	116	27	x	x	PUNCT
ejpam-5570	116	28	has	have	VERB
ejpam-5570	116	29	a	a	DET
ejpam-5570	116	30	δ	δ	PROPN
ejpam-5570	116	31	-	-	PUNCT
ejpam-5570	116	32	βi	βi	ADV
ejpam-5570	116	33	-	-	PUNCT
ejpam-5570	116	34	locally	locally	ADV
ejpam-5570	116	35	finite	finite	PROPN
ejpam-5570	116	36	δ	δ	PROPN
ejpam-5570	116	37	-	-	ADJ
ejpam-5570	116	38	βi	βi	ADV
ejpam-5570	116	39	-	-	PUNCT
ejpam-5570	116	40	open	open	ADJ
ejpam-5570	116	41	refinement	refinement	NOUN
ejpam-5570	116	42	v	v	NOUN
ejpam-5570	116	43	(	(	PUNCT
ejpam-5570	116	44	not	not	PART
ejpam-5570	116	45	necessarily	necessarily	ADV
ejpam-5570	116	46	a	a	DET
ejpam-5570	116	47	cover	cover	NOUN
ejpam-5570	116	48	)	)	PUNCT
ejpam-5570	116	49	such	such	ADJ
ejpam-5570	116	50	that	that	SCONJ
ejpam-5570	116	51	x	x	X
ejpam-5570	117	1	−	−	NOUN
ejpam-5570	117	2	∪{v	∪{v	NOUN
ejpam-5570	117	3	:	:	PUNCT
ejpam-5570	117	4	v	v	NUM
ejpam-5570	117	5	∈	∈	PROPN
ejpam-5570	117	6	v	v	NOUN
ejpam-5570	117	7	}	}	PUNCT
ejpam-5570	117	8	∈	∈	PROPN
ejpam-5570	117	9	i.	i.	NOUN
ejpam-5570	117	10	the	the	DET
ejpam-5570	117	11	collection	collection	NOUN
ejpam-5570	117	12	v	v	NOUN
ejpam-5570	117	13	of	of	ADP
ejpam-5570	117	14	subsets	subset	NOUN
ejpam-5570	117	15	of	of	ADP
ejpam-5570	117	16	x	x	PUNCT
ejpam-5570	118	1	such	such	ADJ
ejpam-5570	118	2	that	that	SCONJ
ejpam-5570	118	3	x	x	X
ejpam-5570	118	4	−	−	NOUN
ejpam-5570	118	5	∪{v	∪{v	NOUN
ejpam-5570	118	6	:	:	PUNCT
ejpam-5570	118	7	v	v	NUM
ejpam-5570	118	8	∈	∈	PROPN
ejpam-5570	118	9	v	v	NOUN
ejpam-5570	118	10	}	}	PUNCT
ejpam-5570	118	11	∈	∈	NOUN
ejpam-5570	118	12	i	i	PRON
ejpam-5570	118	13	is	be	AUX
ejpam-5570	118	14	called	call	VERB
ejpam-5570	118	15	an	an	DET
ejpam-5570	118	16	i	i	NOUN
ejpam-5570	118	17	-	-	NOUN
ejpam-5570	118	18	cover	cover	NOUN
ejpam-5570	118	19	.	.	PUNCT
ejpam-5570	119	1	a	a	DET
ejpam-5570	119	2	subset	subset	NOUN
ejpam-5570	119	3	a	a	PRON
ejpam-5570	119	4	of	of	ADP
ejpam-5570	119	5	an	an	DET
ejpam-5570	119	6	ideal	ideal	ADJ
ejpam-5570	119	7	topological	topological	ADJ
ejpam-5570	119	8	space	space	NOUN
ejpam-5570	119	9	(	(	PUNCT
ejpam-5570	119	10	x	x	X
ejpam-5570	119	11	,	,	PUNCT
ejpam-5570	119	12	τ	τ	PROPN
ejpam-5570	119	13	,	,	PUNCT
ejpam-5570	119	14	i	i	PROPN
ejpam-5570	119	15	)	)	PUNCT
ejpam-5570	119	16	is	be	AUX
ejpam-5570	119	17	said	say	VERB
ejpam-5570	119	18	to	to	PART
ejpam-5570	119	19	be	be	AUX
ejpam-5570	119	20	δ	δ	PROPN
ejpam-5570	119	21	-	-	PUNCT
ejpam-5570	119	22	βi	βi	ADV
ejpam-5570	119	23	-	-	NOUN
ejpam-5570	119	24	paracompact	paracompact	NOUN
ejpam-5570	119	25	if	if	SCONJ
ejpam-5570	119	26	for	for	ADP
ejpam-5570	119	27	any	any	DET
ejpam-5570	119	28	open	open	ADJ
ejpam-5570	119	29	cover	cover	NOUN
ejpam-5570	119	30	u	u	NOUN
ejpam-5570	119	31	of	of	ADP
ejpam-5570	119	32	a	a	PRON
ejpam-5570	119	33	has	have	VERB
ejpam-5570	119	34	a	a	DET
ejpam-5570	119	35	δ	δ	PROPN
ejpam-5570	119	36	-	-	PUNCT
ejpam-5570	119	37	βi	βi	ADV
ejpam-5570	119	38	-	-	PUNCT
ejpam-5570	119	39	locally	locally	ADV
ejpam-5570	119	40	finite	finite	PROPN
ejpam-5570	119	41	δ	δ	PROPN
ejpam-5570	119	42	-	-	ADJ
ejpam-5570	119	43	βi	βi	ADV
ejpam-5570	119	44	-	-	PUNCT
ejpam-5570	119	45	open	open	ADJ
ejpam-5570	119	46	refinement	refinement	NOUN
ejpam-5570	119	47	v	v	ADP
ejpam-5570	119	48	such	such	ADJ
ejpam-5570	119	49	that	that	DET
ejpam-5570	119	50	a−	a−	PROPN
ejpam-5570	119	51	∪{v	∪{v	PROPN
ejpam-5570	119	52	:	:	PUNCT
ejpam-5570	119	53	v	v	NUM
ejpam-5570	119	54	∈	∈	PROPN
ejpam-5570	119	55	v	v	NOUN
ejpam-5570	119	56	}	}	PUNCT
ejpam-5570	119	57	∈	∈	PROPN
ejpam-5570	119	58	i.	i.	NOUN
ejpam-5570	119	59	the	the	DET
ejpam-5570	119	60	two	two	NUM
ejpam-5570	119	61	theorems	theorem	NOUN
ejpam-5570	119	62	that	that	PRON
ejpam-5570	119	63	follow	follow	VERB
ejpam-5570	119	64	arise	arise	NOUN
ejpam-5570	119	65	from	from	ADP
ejpam-5570	119	66	the	the	DET
ejpam-5570	119	67	fact	fact	NOUN
ejpam-5570	119	68	that	that	SCONJ
ejpam-5570	119	69	every	every	DET
ejpam-5570	119	70	open	open	ADJ
ejpam-5570	119	71	set	set	NOUN
ejpam-5570	119	72	is	be	AUX
ejpam-5570	119	73	β	β	NOUN
ejpam-5570	119	74	-	-	ADJ
ejpam-5570	119	75	open	open	ADJ
ejpam-5570	119	76	,	,	PUNCT
ejpam-5570	119	77	every	every	DET
ejpam-5570	119	78	β	β	NOUN
ejpam-5570	119	79	-	-	ADJ
ejpam-5570	119	80	open	open	ADJ
ejpam-5570	119	81	is	be	AUX
ejpam-5570	119	82	δ	δ	PROPN
ejpam-5570	119	83	-	-	PUNCT
ejpam-5570	119	84	βi	βi	ADV
ejpam-5570	119	85	-	-	PUNCT
ejpam-5570	119	86	open	open	ADJ
ejpam-5570	119	87	and	and	CCONJ
ejpam-5570	119	88	∅	∅	NOUN
ejpam-5570	119	89	is	be	AUX
ejpam-5570	119	90	in	in	ADP
ejpam-5570	119	91	any	any	DET
ejpam-5570	119	92	ideal	ideal	NOUN
ejpam-5570	119	93	.	.	PUNCT
ejpam-5570	120	1	c.	c.	PROPN
ejpam-5570	120	2	boonpok	boonpok	PROPN
ejpam-5570	120	3	,	,	PUNCT
ejpam-5570	120	4	a.	a.	PROPN
ejpam-5570	120	5	sama	sama	PROPN
ejpam-5570	120	6	-	-	PUNCT
ejpam-5570	120	7	ae	ae	PROPN
ejpam-5570	120	8	,	,	PUNCT
ejpam-5570	120	9	p.	p.	NOUN
ejpam-5570	120	10	raktaow	raktaow	PROPN
ejpam-5570	120	11	/	/	SYM
ejpam-5570	120	12	eur	eur	PROPN
ejpam-5570	120	13	.	.	PUNCT
ejpam-5570	121	1	j.	j.	PROPN
ejpam-5570	121	2	pure	pure	PROPN
ejpam-5570	121	3	appl	appl	PROPN
ejpam-5570	121	4	.	.	PROPN
ejpam-5570	121	5	math	math	PROPN
ejpam-5570	121	6	,	,	PUNCT
ejpam-5570	121	7	18	18	NUM
ejpam-5570	121	8	(	(	PUNCT
ejpam-5570	121	9	1	1	NUM
ejpam-5570	121	10	)	)	PUNCT
ejpam-5570	121	11	(	(	PUNCT
ejpam-5570	121	12	2025	2025	NUM
ejpam-5570	121	13	)	)	PUNCT
ejpam-5570	121	14	,	,	PUNCT
ejpam-5570	121	15	5570	5570	NUM
ejpam-5570	121	16	5	5	NUM
ejpam-5570	121	17	of	of	ADP
ejpam-5570	121	18	12	12	NUM
ejpam-5570	121	19	theorem	theorem	NOUN
ejpam-5570	121	20	1	1	NUM
ejpam-5570	121	21	.	.	PUNCT
ejpam-5570	122	1	if	if	SCONJ
ejpam-5570	122	2	a	a	DET
ejpam-5570	122	3	topological	topological	ADJ
ejpam-5570	122	4	space	space	NOUN
ejpam-5570	122	5	(	(	PUNCT
ejpam-5570	122	6	x	x	X
ejpam-5570	122	7	,	,	PUNCT
ejpam-5570	122	8	τ	τ	X
ejpam-5570	122	9	)	)	PUNCT
ejpam-5570	122	10	is	be	AUX
ejpam-5570	122	11	paracompact	paracompact	ADJ
ejpam-5570	122	12	,	,	PUNCT
ejpam-5570	122	13	then	then	ADV
ejpam-5570	122	14	(	(	PUNCT
ejpam-5570	122	15	x	x	X
ejpam-5570	122	16	,	,	PUNCT
ejpam-5570	122	17	τ	τ	PROPN
ejpam-5570	122	18	,	,	PUNCT
ejpam-5570	122	19	i	i	PROPN
ejpam-5570	122	20	)	)	PUNCT
ejpam-5570	122	21	is	be	AUX
ejpam-5570	122	22	δ	δ	PROPN
ejpam-5570	122	23	-	-	PUNCT
ejpam-5570	122	24	βi	βi	ADV
ejpam-5570	122	25	-	-	PUNCT
ejpam-5570	122	26	paracompact	paracompact	ADJ
ejpam-5570	122	27	.	.	PUNCT
ejpam-5570	123	1	proof	proof	NOUN
ejpam-5570	123	2	.	.	PUNCT
ejpam-5570	124	1	it	it	PRON
ejpam-5570	124	2	is	be	AUX
ejpam-5570	124	3	evident	evident	ADJ
ejpam-5570	124	4	,	,	PUNCT
ejpam-5570	124	5	as	as	SCONJ
ejpam-5570	124	6	∅	∅	NOUN
ejpam-5570	124	7	∈	∈	PROPN
ejpam-5570	124	8	i.	i.	NOUN
ejpam-5570	124	9	theorem	theorem	VERB
ejpam-5570	124	10	2	2	NUM
ejpam-5570	124	11	.	.	PUNCT
ejpam-5570	125	1	if	if	SCONJ
ejpam-5570	125	2	(	(	PUNCT
ejpam-5570	125	3	x	x	X
ejpam-5570	125	4	,	,	PUNCT
ejpam-5570	125	5	τ	τ	PROPN
ejpam-5570	125	6	,	,	PUNCT
ejpam-5570	125	7	i	i	PROPN
ejpam-5570	125	8	)	)	PUNCT
ejpam-5570	125	9	is	be	AUX
ejpam-5570	125	10	i	i	PROPN
ejpam-5570	125	11	-	-	PUNCT
ejpam-5570	125	12	β	β	NOUN
ejpam-5570	125	13	-	-	NOUN
ejpam-5570	125	14	paracompact	paracompact	NOUN
ejpam-5570	125	15	then	then	ADV
ejpam-5570	125	16	it	it	PRON
ejpam-5570	125	17	is	be	AUX
ejpam-5570	125	18	δ	δ	PROPN
ejpam-5570	125	19	-	-	PUNCT
ejpam-5570	125	20	βi	βi	ADV
ejpam-5570	125	21	-	-	PUNCT
ejpam-5570	125	22	paracompact	paracompact	ADJ
ejpam-5570	125	23	.	.	PUNCT
ejpam-5570	126	1	proof	proof	NOUN
ejpam-5570	126	2	.	.	PUNCT
ejpam-5570	127	1	every	every	DET
ejpam-5570	127	2	β	β	X
ejpam-5570	127	3	-	-	ADJ
ejpam-5570	127	4	locally	locally	ADV
ejpam-5570	127	5	finite	finite	ADJ
ejpam-5570	127	6	collection	collection	NOUN
ejpam-5570	127	7	of	of	ADP
ejpam-5570	127	8	subsets	subset	NOUN
ejpam-5570	127	9	of	of	ADP
ejpam-5570	127	10	x	x	PROPN
ejpam-5570	127	11	is	be	AUX
ejpam-5570	127	12	δ	δ	PROPN
ejpam-5570	127	13	-	-	PUNCT
ejpam-5570	127	14	βi	βi	ADV
ejpam-5570	127	15	-	-	PUNCT
ejpam-5570	127	16	locally	locally	ADV
ejpam-5570	127	17	finite	finite	NOUN
ejpam-5570	127	18	,	,	PUNCT
ejpam-5570	127	19	as	as	SCONJ
ejpam-5570	127	20	demonstrated	demonstrate	VERB
ejpam-5570	127	21	by	by	ADP
ejpam-5570	127	22	lemma	lemma	PROPN
ejpam-5570	127	23	4	4	NUM
ejpam-5570	127	24	.	.	PUNCT
ejpam-5570	128	1	furthermore	furthermore	ADV
ejpam-5570	128	2	,	,	PUNCT
ejpam-5570	128	3	each	each	DET
ejpam-5570	128	4	β	β	X
ejpam-5570	128	5	-	-	ADJ
ejpam-5570	128	6	open	open	ADJ
ejpam-5570	128	7	set	set	NOUN
ejpam-5570	128	8	is	be	AUX
ejpam-5570	128	9	δ	δ	PROPN
ejpam-5570	128	10	-	-	PUNCT
ejpam-5570	128	11	βi	βi	ADV
ejpam-5570	128	12	-	-	PUNCT
ejpam-5570	128	13	open	open	ADJ
ejpam-5570	128	14	[	[	X
ejpam-5570	128	15	10	10	NUM
ejpam-5570	128	16	]	]	PUNCT
ejpam-5570	128	17	.	.	PUNCT
ejpam-5570	129	1	we	we	PRON
ejpam-5570	129	2	can	can	AUX
ejpam-5570	129	3	continue	continue	VERB
ejpam-5570	129	4	with	with	ADP
ejpam-5570	129	5	the	the	DET
ejpam-5570	129	6	proof	proof	NOUN
ejpam-5570	129	7	by	by	ADP
ejpam-5570	129	8	following	follow	VERB
ejpam-5570	129	9	to	to	ADP
ejpam-5570	129	10	the	the	DET
ejpam-5570	129	11	definitions	definition	NOUN
ejpam-5570	129	12	of	of	ADP
ejpam-5570	129	13	i	i	PROPN
ejpam-5570	129	14	-	-	PUNCT
ejpam-5570	129	15	β	β	NOUN
ejpam-5570	129	16	-	-	NOUN
ejpam-5570	129	17	paracompactness	paracompactness	NOUN
ejpam-5570	129	18	and	and	CCONJ
ejpam-5570	129	19	δ	δ	PROPN
ejpam-5570	129	20	-	-	PUNCT
ejpam-5570	129	21	βiparacompactness	βiparacompactness	NOUN
ejpam-5570	129	22	.	.	PUNCT
ejpam-5570	130	1	consider	consider	VERB
ejpam-5570	130	2	the	the	DET
ejpam-5570	130	3	set	set	NOUN
ejpam-5570	130	4	x	x	PUNCT
ejpam-5570	130	5	consisting	consist	VERB
ejpam-5570	130	6	of	of	ADP
ejpam-5570	130	7	all	all	DET
ejpam-5570	130	8	positive	positive	ADJ
ejpam-5570	130	9	integers	integer	NOUN
ejpam-5570	130	10	.	.	PUNCT
ejpam-5570	131	1	let	let	VERB
ejpam-5570	131	2	τ	τ	PROPN
ejpam-5570	131	3	=	=	PUNCT
ejpam-5570	131	4	{	{	PUNCT
ejpam-5570	131	5	∅}∪{x}∪{{1	∅}∪{x}∪{{1	PROPN
ejpam-5570	131	6	,	,	PUNCT
ejpam-5570	131	7	2	2	NUM
ejpam-5570	131	8	,	,	PUNCT
ejpam-5570	131	9	...	...	PUNCT
ejpam-5570	131	10	,	,	PUNCT
ejpam-5570	131	11	n	n	CCONJ
ejpam-5570	131	12	}	}	PUNCT
ejpam-5570	131	13	:	:	PUNCT
ejpam-5570	131	14	n	n	X
ejpam-5570	131	15	∈	∈	PROPN
ejpam-5570	131	16	x	x	AUX
ejpam-5570	131	17	}	}	PUNCT
ejpam-5570	131	18	be	be	AUX
ejpam-5570	131	19	a	a	DET
ejpam-5570	131	20	topology	topology	NOUN
ejpam-5570	131	21	on	on	ADP
ejpam-5570	131	22	x.	x.	NOUN
ejpam-5570	131	23	define	define	VERB
ejpam-5570	131	24	i	i	PRON
ejpam-5570	131	25	=	=	PUNCT
ejpam-5570	131	26	{	{	PUNCT
ejpam-5570	131	27	h	h	NOUN
ejpam-5570	131	28	⊂	⊂	PROPN
ejpam-5570	131	29	x	x	X
ejpam-5570	131	30	:	:	PUNCT
ejpam-5570	131	31	1	1	NUM
ejpam-5570	131	32	̸∈	̸∈	PROPN
ejpam-5570	131	33	h	h	PROPN
ejpam-5570	131	34	}	}	PUNCT
ejpam-5570	131	35	as	as	ADP
ejpam-5570	131	36	an	an	DET
ejpam-5570	131	37	ideal	ideal	NOUN
ejpam-5570	131	38	on	on	ADP
ejpam-5570	131	39	the	the	DET
ejpam-5570	131	40	set	set	NOUN
ejpam-5570	131	41	x.	x.	NOUN
ejpam-5570	131	42	an	an	DET
ejpam-5570	131	43	open	open	ADJ
ejpam-5570	131	44	cover	cover	NOUN
ejpam-5570	131	45	of	of	ADP
ejpam-5570	131	46	x	x	PUNCT
ejpam-5570	131	47	is	be	AUX
ejpam-5570	131	48	defined	define	VERB
ejpam-5570	131	49	as	as	ADP
ejpam-5570	131	50	w	w	NOUN
ejpam-5570	131	51	=	=	PRON
ejpam-5570	131	52	{	{	PUNCT
ejpam-5570	131	53	{	{	PUNCT
ejpam-5570	131	54	1	1	NUM
ejpam-5570	131	55	,	,	PUNCT
ejpam-5570	131	56	2	2	NUM
ejpam-5570	131	57	,	,	PUNCT
ejpam-5570	131	58	3	3	NUM
ejpam-5570	131	59	,	,	PUNCT
ejpam-5570	131	60	...	...	PUNCT
ejpam-5570	131	61	,	,	PUNCT
ejpam-5570	131	62	n	n	CCONJ
ejpam-5570	131	63	}	}	PUNCT
ejpam-5570	131	64	:	:	PUNCT
ejpam-5570	132	1	n	n	X
ejpam-5570	132	2	∈	∈	PROPN
ejpam-5570	132	3	x	x	NOUN
ejpam-5570	132	4	}	}	PUNCT
ejpam-5570	132	5	.	.	PUNCT
ejpam-5570	133	1	however	however	ADV
ejpam-5570	133	2	,	,	PUNCT
ejpam-5570	133	3	there	there	PRON
ejpam-5570	133	4	is	be	VERB
ejpam-5570	133	5	no	no	DET
ejpam-5570	133	6	locally	locally	ADV
ejpam-5570	133	7	finite	finite	ADJ
ejpam-5570	133	8	open	open	ADJ
ejpam-5570	133	9	refinement	refinement	PROPN
ejpam-5570	133	10	v	v	NOUN
ejpam-5570	133	11	that	that	PRON
ejpam-5570	133	12	covers	cover	VERB
ejpam-5570	133	13	x.	x.	NOUN
ejpam-5570	133	14	thus	thus	ADV
ejpam-5570	133	15	,	,	PUNCT
ejpam-5570	133	16	(	(	PUNCT
ejpam-5570	133	17	x	x	X
ejpam-5570	133	18	,	,	PUNCT
ejpam-5570	133	19	τ	τ	X
ejpam-5570	133	20	)	)	PUNCT
ejpam-5570	133	21	is	be	AUX
ejpam-5570	133	22	not	not	PART
ejpam-5570	133	23	a	a	DET
ejpam-5570	133	24	paracompact	paracompact	ADJ
ejpam-5570	133	25	space	space	NOUN
ejpam-5570	133	26	.	.	PUNCT
ejpam-5570	134	1	but	but	CCONJ
ejpam-5570	134	2	,	,	PUNCT
ejpam-5570	134	3	x	x	X
ejpam-5570	134	4	is	be	AUX
ejpam-5570	134	5	a	a	DET
ejpam-5570	134	6	δ	δ	PROPN
ejpam-5570	134	7	-	-	PUNCT
ejpam-5570	134	8	βi	βi	PRON
ejpam-5570	134	9	-	-	PUNCT
ejpam-5570	134	10	paracompact	paracompact	ADJ
ejpam-5570	134	11	space	space	NOUN
ejpam-5570	134	12	,	,	PUNCT
ejpam-5570	134	13	as	as	ADP
ejpam-5570	134	14	for	for	ADP
ejpam-5570	134	15	each	each	DET
ejpam-5570	134	16	open	open	ADJ
ejpam-5570	134	17	cover	cover	NOUN
ejpam-5570	134	18	u	u	NOUN
ejpam-5570	134	19	of	of	ADP
ejpam-5570	134	20	x	x	PUNCT
ejpam-5570	134	21	has	have	VERB
ejpam-5570	134	22	a	a	DET
ejpam-5570	134	23	δ	δ	PROPN
ejpam-5570	134	24	-	-	PUNCT
ejpam-5570	134	25	βi	βi	ADV
ejpam-5570	134	26	-	-	PUNCT
ejpam-5570	134	27	locally	locally	ADV
ejpam-5570	134	28	finite	finite	PROPN
ejpam-5570	134	29	δ	δ	PROPN
ejpam-5570	134	30	-	-	ADJ
ejpam-5570	134	31	βi	βi	ADV
ejpam-5570	134	32	-	-	PUNCT
ejpam-5570	134	33	open	open	ADJ
ejpam-5570	134	34	refinement	refinement	NOUN
ejpam-5570	134	35	v	v	NOUN
ejpam-5570	134	36	=	=	PRON
ejpam-5570	134	37	{	{	PUNCT
ejpam-5570	134	38	{	{	PUNCT
ejpam-5570	134	39	1	1	NUM
ejpam-5570	134	40	}	}	PUNCT
ejpam-5570	134	41	}	}	PUNCT
ejpam-5570	134	42	such	such	ADJ
ejpam-5570	134	43	that	that	SCONJ
ejpam-5570	134	44	x	x	X
ejpam-5570	134	45	−	−	NOUN
ejpam-5570	134	46	{	{	PUNCT
ejpam-5570	134	47	1	1	NUM
ejpam-5570	134	48	}	}	PUNCT
ejpam-5570	134	49	=	=	NOUN
ejpam-5570	134	50	{	{	PUNCT
ejpam-5570	134	51	2	2	NUM
ejpam-5570	134	52	,	,	PUNCT
ejpam-5570	134	53	3	3	NUM
ejpam-5570	134	54	,	,	PUNCT
ejpam-5570	134	55	...	...	PUNCT
ejpam-5570	134	56	}	}	PUNCT
ejpam-5570	134	57	∈	∈	PROPN
ejpam-5570	134	58	i.	i.	NOUN
ejpam-5570	134	59	theorem	theorem	VERB
ejpam-5570	134	60	3	3	X
ejpam-5570	134	61	.	.	PUNCT
ejpam-5570	135	1	let	let	VERB
ejpam-5570	135	2	(	(	PUNCT
ejpam-5570	135	3	x	x	X
ejpam-5570	135	4	,	,	PUNCT
ejpam-5570	135	5	τ	τ	PROPN
ejpam-5570	135	6	,	,	PUNCT
ejpam-5570	135	7	i	i	PRON
ejpam-5570	135	8	)	)	PUNCT
ejpam-5570	135	9	be	be	VERB
ejpam-5570	135	10	an	an	DET
ejpam-5570	135	11	ideal	ideal	ADJ
ejpam-5570	135	12	topological	topological	ADJ
ejpam-5570	135	13	space	space	NOUN
ejpam-5570	135	14	and	and	CCONJ
ejpam-5570	135	15	let	let	VERB
ejpam-5570	135	16	g	g	PRON
ejpam-5570	135	17	be	be	AUX
ejpam-5570	135	18	a	a	DET
ejpam-5570	135	19	δ	δ	PROPN
ejpam-5570	135	20	-	-	PUNCT
ejpam-5570	135	21	βi	βi	ADV
ejpam-5570	135	22	-	-	PUNCT
ejpam-5570	135	23	open	open	ADJ
ejpam-5570	135	24	subset	subset	NOUN
ejpam-5570	135	25	of	of	ADP
ejpam-5570	135	26	x.	x.	NOUN
ejpam-5570	135	27	then	then	ADV
ejpam-5570	135	28	g	g	PROPN
ejpam-5570	135	29	∩	∩	PROPN
ejpam-5570	135	30	δ	δ	PROPN
ejpam-5570	135	31	-	-	PUNCT
ejpam-5570	135	32	βcli(a	βcli(a	NOUN
ejpam-5570	135	33	)	)	PUNCT
ejpam-5570	135	34	=	=	NOUN
ejpam-5570	135	35	∅	∅	NOUN
ejpam-5570	135	36	if	if	SCONJ
ejpam-5570	135	37	and	and	CCONJ
ejpam-5570	135	38	only	only	ADV
ejpam-5570	135	39	if	if	SCONJ
ejpam-5570	135	40	g	g	PROPN
ejpam-5570	135	41	∩a	∩a	NOUN
ejpam-5570	135	42	=	=	NOUN
ejpam-5570	135	43	∅	∅	NOUN
ejpam-5570	135	44	,	,	PUNCT
ejpam-5570	135	45	for	for	ADP
ejpam-5570	135	46	all	all	DET
ejpam-5570	135	47	a	a	DET
ejpam-5570	135	48	⊂	⊂	PROPN
ejpam-5570	135	49	x.	x.	NOUN
ejpam-5570	135	50	proof	proof	NOUN
ejpam-5570	135	51	.	.	PUNCT
ejpam-5570	136	1	it	it	PRON
ejpam-5570	136	2	follows	follow	VERB
ejpam-5570	136	3	from	from	ADP
ejpam-5570	136	4	(	(	PUNCT
ejpam-5570	136	5	iv	iv	NOUN
ejpam-5570	136	6	)	)	PUNCT
ejpam-5570	136	7	of	of	ADP
ejpam-5570	136	8	lemma	lemma	PROPN
ejpam-5570	136	9	3	3	NUM
ejpam-5570	136	10	and	and	CCONJ
ejpam-5570	136	11	the	the	DET
ejpam-5570	136	12	fact	fact	NOUN
ejpam-5570	136	13	that	that	SCONJ
ejpam-5570	136	14	a	a	DET
ejpam-5570	136	15	⊂	⊂	PROPN
ejpam-5570	136	16	δ	δ	PROPN
ejpam-5570	136	17	-	-	PUNCT
ejpam-5570	136	18	βcli(a	βcli(a	NOUN
ejpam-5570	136	19	)	)	PUNCT
ejpam-5570	136	20	.	.	PUNCT
ejpam-5570	137	1	theorem	theorem	ADJ
ejpam-5570	137	2	4	4	NUM
ejpam-5570	137	3	.	.	PUNCT
ejpam-5570	138	1	let	let	VERB
ejpam-5570	138	2	v	v	VERB
ejpam-5570	138	3	=	=	PUNCT
ejpam-5570	138	4	{	{	PUNCT
ejpam-5570	138	5	vλ	vλ	INTJ
ejpam-5570	138	6	:	:	PUNCT
ejpam-5570	138	7	λ	λ	PROPN
ejpam-5570	138	8	∈	∈	PROPN
ejpam-5570	138	9	λ	λ	PROPN
ejpam-5570	138	10	}	}	PUNCT
ejpam-5570	138	11	be	be	VERB
ejpam-5570	138	12	a	a	DET
ejpam-5570	138	13	collection	collection	NOUN
ejpam-5570	138	14	of	of	ADP
ejpam-5570	138	15	subsets	subset	NOUN
ejpam-5570	138	16	of	of	ADP
ejpam-5570	138	17	a	a	DET
ejpam-5570	138	18	topological	topological	ADJ
ejpam-5570	138	19	space	space	NOUN
ejpam-5570	138	20	(	(	PUNCT
ejpam-5570	138	21	x	x	X
ejpam-5570	138	22	,	,	PUNCT
ejpam-5570	138	23	τ	τ	PROPN
ejpam-5570	138	24	)	)	PUNCT
ejpam-5570	138	25	.	.	PUNCT
ejpam-5570	139	1	the	the	DET
ejpam-5570	139	2	following	follow	VERB
ejpam-5570	139	3	statements	statement	NOUN
ejpam-5570	139	4	are	be	AUX
ejpam-5570	139	5	true	true	ADJ
ejpam-5570	139	6	.	.	PUNCT
ejpam-5570	140	1	(	(	PUNCT
ejpam-5570	140	2	i	i	NOUN
ejpam-5570	140	3	)	)	PUNCT
ejpam-5570	140	4	if	if	SCONJ
ejpam-5570	140	5	v	v	NOUN
ejpam-5570	140	6	is	be	AUX
ejpam-5570	140	7	δ	δ	PROPN
ejpam-5570	140	8	-	-	PUNCT
ejpam-5570	140	9	βi	βi	ADV
ejpam-5570	140	10	-	-	PUNCT
ejpam-5570	140	11	locally	locally	ADV
ejpam-5570	140	12	finite	finite	NOUN
ejpam-5570	140	13	and	and	CCONJ
ejpam-5570	140	14	hλ	hλ	X
ejpam-5570	140	15	⊂	⊂	PROPN
ejpam-5570	140	16	vλ	vλ	ADJ
ejpam-5570	140	17	for	for	ADP
ejpam-5570	140	18	all	all	DET
ejpam-5570	140	19	λ	λ	PROPN
ejpam-5570	140	20	∈	∈	PROPN
ejpam-5570	140	21	λ	λ	PROPN
ejpam-5570	140	22	,	,	PUNCT
ejpam-5570	140	23	then	then	ADV
ejpam-5570	140	24	h	h	NOUN
ejpam-5570	140	25	=	=	PRON
ejpam-5570	140	26	{	{	PUNCT
ejpam-5570	140	27	hλ	hλ	X
ejpam-5570	140	28	:	:	PUNCT
ejpam-5570	140	29	λ	λ	PROPN
ejpam-5570	140	30	∈	∈	PROPN
ejpam-5570	140	31	λ	λ	PROPN
ejpam-5570	140	32	}	}	PUNCT
ejpam-5570	140	33	is	be	AUX
ejpam-5570	140	34	δ	δ	PROPN
ejpam-5570	140	35	-	-	PUNCT
ejpam-5570	140	36	βi	βi	ADV
ejpam-5570	140	37	-	-	PUNCT
ejpam-5570	140	38	locally	locally	ADV
ejpam-5570	140	39	finite	finite	NOUN
ejpam-5570	140	40	.	.	PUNCT
ejpam-5570	141	1	(	(	PUNCT
ejpam-5570	141	2	ii	ii	NOUN
ejpam-5570	141	3	)	)	PUNCT
ejpam-5570	141	4	v	v	NOUN
ejpam-5570	141	5	is	be	AUX
ejpam-5570	141	6	δ	δ	PROPN
ejpam-5570	141	7	-	-	PUNCT
ejpam-5570	141	8	βi	βi	ADV
ejpam-5570	141	9	-	-	PUNCT
ejpam-5570	141	10	locally	locally	ADV
ejpam-5570	141	11	finite	finite	VERB
ejpam-5570	141	12	if	if	SCONJ
ejpam-5570	141	13	and	and	CCONJ
ejpam-5570	141	14	only	only	ADV
ejpam-5570	141	15	if	if	SCONJ
ejpam-5570	141	16	{	{	PUNCT
ejpam-5570	141	17	δ	δ	NOUN
ejpam-5570	141	18	-	-	PUNCT
ejpam-5570	141	19	βcli(vλ	βcli(vλ	PROPN
ejpam-5570	141	20	)	)	PUNCT
ejpam-5570	141	21	:	:	PUNCT
ejpam-5570	142	1	λ	λ	X
ejpam-5570	142	2	∈	∈	PROPN
ejpam-5570	142	3	λ	λ	PROPN
ejpam-5570	142	4	}	}	PUNCT
ejpam-5570	142	5	is	be	AUX
ejpam-5570	142	6	δ	δ	PROPN
ejpam-5570	142	7	-	-	PUNCT
ejpam-5570	142	8	βi	βi	ADV
ejpam-5570	142	9	-	-	PUNCT
ejpam-5570	142	10	locally	locally	ADV
ejpam-5570	142	11	finite	finite	NOUN
ejpam-5570	142	12	.	.	PUNCT
ejpam-5570	143	1	proof	proof	NOUN
ejpam-5570	143	2	.	.	PUNCT
ejpam-5570	144	1	(	(	PUNCT
ejpam-5570	144	2	i	i	NOUN
ejpam-5570	144	3	)	)	PUNCT
ejpam-5570	144	4	let	let	VERB
ejpam-5570	144	5	x	x	PUNCT
ejpam-5570	144	6	∈	∈	PROPN
ejpam-5570	144	7	x.	x.	NOUN
ejpam-5570	144	8	since	since	SCONJ
ejpam-5570	144	9	v	v	NUM
ejpam-5570	144	10	is	be	AUX
ejpam-5570	144	11	δ	δ	PROPN
ejpam-5570	144	12	-	-	PUNCT
ejpam-5570	144	13	βi	βi	ADV
ejpam-5570	144	14	-	-	PUNCT
ejpam-5570	144	15	locally	locally	ADV
ejpam-5570	144	16	finite	finite	NOUN
ejpam-5570	144	17	,	,	PUNCT
ejpam-5570	144	18	there	there	PRON
ejpam-5570	144	19	exists	exist	VERB
ejpam-5570	144	20	a	a	DET
ejpam-5570	144	21	δ	δ	PROPN
ejpam-5570	144	22	-	-	PUNCT
ejpam-5570	144	23	βi	βi	ADV
ejpam-5570	144	24	-	-	PUNCT
ejpam-5570	144	25	open	open	ADJ
ejpam-5570	144	26	set	set	NOUN
ejpam-5570	144	27	u	u	NOUN
ejpam-5570	144	28	containing	contain	VERB
ejpam-5570	144	29	x	x	X
ejpam-5570	144	30	,	,	PUNCT
ejpam-5570	144	31	which	which	PRON
ejpam-5570	144	32	intersects	intersect	VERB
ejpam-5570	144	33	at	at	ADP
ejpam-5570	144	34	most	most	ADV
ejpam-5570	144	35	finitely	finitely	ADV
ejpam-5570	144	36	many	many	ADJ
ejpam-5570	144	37	elements	element	NOUN
ejpam-5570	144	38	of	of	ADP
ejpam-5570	144	39	v.	v.	INTJ
ejpam-5570	144	40	as	as	ADP
ejpam-5570	144	41	hλ	hλ	X
ejpam-5570	144	42	⊂	⊂	PROPN
ejpam-5570	144	43	vλ	vλ	ADJ
ejpam-5570	144	44	for	for	ADP
ejpam-5570	144	45	all	all	DET
ejpam-5570	144	46	λ	λ	PROPN
ejpam-5570	144	47	∈	∈	PROPN
ejpam-5570	144	48	λ	λ	PROPN
ejpam-5570	144	49	,	,	PUNCT
ejpam-5570	144	50	u	u	NOUN
ejpam-5570	144	51	intersects	intersect	NOUN
ejpam-5570	144	52	with	with	ADP
ejpam-5570	144	53	at	at	ADV
ejpam-5570	144	54	most	most	ADV
ejpam-5570	144	55	finitely	finitely	ADV
ejpam-5570	144	56	many	many	ADJ
ejpam-5570	144	57	of	of	ADP
ejpam-5570	144	58	the	the	DET
ejpam-5570	144	59	elements	element	NOUN
ejpam-5570	144	60	in	in	ADP
ejpam-5570	144	61	h	h	NOUN
ejpam-5570	144	62	=	=	PRON
ejpam-5570	144	63	{	{	PUNCT
ejpam-5570	144	64	hλ	hλ	X
ejpam-5570	144	65	:	:	PUNCT
ejpam-5570	144	66	λ	λ	PROPN
ejpam-5570	144	67	∈	∈	PROPN
ejpam-5570	144	68	λ	λ	NOUN
ejpam-5570	144	69	}	}	PUNCT
ejpam-5570	144	70	.	.	PUNCT
ejpam-5570	145	1	hence	hence	ADV
ejpam-5570	145	2	,	,	PUNCT
ejpam-5570	145	3	h	h	NOUN
ejpam-5570	145	4	=	=	PRON
ejpam-5570	145	5	{	{	PUNCT
ejpam-5570	145	6	hλ	hλ	X
ejpam-5570	145	7	:	:	PUNCT
ejpam-5570	145	8	λ	λ	PROPN
ejpam-5570	145	9	∈	∈	PROPN
ejpam-5570	145	10	λ	λ	PROPN
ejpam-5570	145	11	}	}	PUNCT
ejpam-5570	145	12	is	be	AUX
ejpam-5570	145	13	δ	δ	PROPN
ejpam-5570	145	14	-	-	PUNCT
ejpam-5570	145	15	βi	βi	ADV
ejpam-5570	145	16	-	-	PUNCT
ejpam-5570	145	17	locally	locally	ADV
ejpam-5570	145	18	finite	finite	NOUN
ejpam-5570	145	19	.	.	PUNCT
ejpam-5570	146	1	(	(	PUNCT
ejpam-5570	146	2	ii	ii	NOUN
ejpam-5570	146	3	)	)	PUNCT
ejpam-5570	146	4	let	let	VERB
ejpam-5570	146	5	v	v	PART
ejpam-5570	146	6	be	be	AUX
ejpam-5570	146	7	δ	δ	PROPN
ejpam-5570	146	8	-	-	PUNCT
ejpam-5570	146	9	βi	βi	ADV
ejpam-5570	146	10	-	-	PUNCT
ejpam-5570	146	11	locally	locally	ADV
ejpam-5570	146	12	finite	finite	NOUN
ejpam-5570	146	13	and	and	CCONJ
ejpam-5570	146	14	let	let	VERB
ejpam-5570	146	15	x	x	X
ejpam-5570	146	16	∈	∈	PROPN
ejpam-5570	146	17	x.	x.	NOUN
ejpam-5570	146	18	consequently	consequently	ADV
ejpam-5570	146	19	,	,	PUNCT
ejpam-5570	146	20	there	there	PRON
ejpam-5570	146	21	exists	exist	VERB
ejpam-5570	146	22	a	a	DET
ejpam-5570	146	23	δ	δ	PROPN
ejpam-5570	146	24	-	-	PUNCT
ejpam-5570	146	25	βi	βi	ADV
ejpam-5570	146	26	-	-	PUNCT
ejpam-5570	146	27	open	open	NOUN
ejpam-5570	146	28	set	set	NOUN
ejpam-5570	146	29	g	g	NOUN
ejpam-5570	146	30	that	that	PRON
ejpam-5570	146	31	contains	contain	VERB
ejpam-5570	146	32	x	x	PUNCT
ejpam-5570	146	33	and	and	CCONJ
ejpam-5570	146	34	satisfies	satisfie	NOUN
ejpam-5570	146	35	g	g	PROPN
ejpam-5570	146	36	∩	∩	NOUN
ejpam-5570	146	37	vλ	vλ	ADP
ejpam-5570	146	38	=	=	NOUN
ejpam-5570	146	39	∅	∅	NOUN
ejpam-5570	146	40	for	for	ADP
ejpam-5570	146	41	every	every	DET
ejpam-5570	146	42	λ	λ	PROPN
ejpam-5570	146	43	̸=	̸=	PROPN
ejpam-5570	146	44	λ1	λ1	PROPN
ejpam-5570	146	45	,	,	PUNCT
ejpam-5570	146	46	λ2	λ2	NOUN
ejpam-5570	146	47	,	,	PUNCT
ejpam-5570	146	48	.	.	PUNCT
ejpam-5570	146	49	.	.	PUNCT
ejpam-5570	147	1	.	.	PUNCT
ejpam-5570	148	1	,	,	PUNCT
ejpam-5570	148	2	λn	λn	NOUN
ejpam-5570	148	3	.	.	PUNCT
ejpam-5570	149	1	according	accord	VERB
ejpam-5570	149	2	to	to	ADP
ejpam-5570	149	3	theorem	theorem	NOUN
ejpam-5570	149	4	3	3	NUM
ejpam-5570	149	5	,	,	PUNCT
ejpam-5570	149	6	it	it	PRON
ejpam-5570	149	7	follows	follow	VERB
ejpam-5570	149	8	that	that	SCONJ
ejpam-5570	149	9	g	g	PROPN
ejpam-5570	149	10	∩	∩	PROPN
ejpam-5570	149	11	δ	δ	PROPN
ejpam-5570	149	12	-	-	PUNCT
ejpam-5570	149	13	βcli(vλ	βcli(vλ	NOUN
ejpam-5570	149	14	)	)	PUNCT
ejpam-5570	149	15	=	=	NOUN
ejpam-5570	149	16	∅	∅	NOUN
ejpam-5570	149	17	for	for	ADP
ejpam-5570	149	18	every	every	DET
ejpam-5570	149	19	λ	λ	PROPN
ejpam-5570	149	20	̸=	̸=	PROPN
ejpam-5570	149	21	λ1	λ1	PROPN
ejpam-5570	149	22	,	,	PUNCT
ejpam-5570	149	23	λ2	λ2	NOUN
ejpam-5570	149	24	,	,	PUNCT
ejpam-5570	149	25	.	.	PUNCT
ejpam-5570	149	26	.	.	PUNCT
ejpam-5570	150	1	.	.	PUNCT
ejpam-5570	151	1	,	,	PUNCT
ejpam-5570	151	2	λn	λn	NOUN
ejpam-5570	151	3	.	.	PUNCT
ejpam-5570	152	1	thus	thus	ADV
ejpam-5570	152	2	,	,	PUNCT
ejpam-5570	152	3	{	{	PUNCT
ejpam-5570	152	4	δ	δ	NOUN
ejpam-5570	152	5	-	-	PUNCT
ejpam-5570	152	6	βcli(vλ	βcli(vλ	PROPN
ejpam-5570	152	7	)	)	PUNCT
ejpam-5570	152	8	:	:	PUNCT
ejpam-5570	153	1	λ	λ	X
ejpam-5570	153	2	∈	∈	PROPN
ejpam-5570	153	3	λ	λ	PROPN
ejpam-5570	153	4	}	}	PUNCT
ejpam-5570	153	5	is	be	AUX
ejpam-5570	153	6	δ	δ	PROPN
ejpam-5570	153	7	-	-	PUNCT
ejpam-5570	153	8	βi	βi	ADV
ejpam-5570	153	9	-	-	PUNCT
ejpam-5570	153	10	locally	locally	ADV
ejpam-5570	153	11	finite	finite	NOUN
ejpam-5570	153	12	.	.	PUNCT
ejpam-5570	154	1	if	if	SCONJ
ejpam-5570	154	2	{	{	PUNCT
ejpam-5570	154	3	δ	δ	NOUN
ejpam-5570	154	4	-	-	PUNCT
ejpam-5570	154	5	βcli(vλ	βcli(vλ	PROPN
ejpam-5570	154	6	)	)	PUNCT
ejpam-5570	154	7	:	:	PUNCT
ejpam-5570	154	8	λ	λ	X
ejpam-5570	154	9	∈	∈	PROPN
ejpam-5570	154	10	λ	λ	PROPN
ejpam-5570	154	11	}	}	PUNCT
ejpam-5570	154	12	is	be	AUX
ejpam-5570	154	13	δ	δ	PROPN
ejpam-5570	154	14	-	-	PUNCT
ejpam-5570	154	15	βi	βi	ADV
ejpam-5570	154	16	-	-	PUNCT
ejpam-5570	154	17	locally	locally	ADV
ejpam-5570	154	18	finite	finite	NOUN
ejpam-5570	154	19	,	,	PUNCT
ejpam-5570	154	20	then	then	ADV
ejpam-5570	154	21	v	v	NOUN
ejpam-5570	154	22	is	be	AUX
ejpam-5570	154	23	δ	δ	PROPN
ejpam-5570	154	24	-	-	PUNCT
ejpam-5570	154	25	βi	βi	ADV
ejpam-5570	154	26	-	-	PUNCT
ejpam-5570	154	27	locally	locally	ADV
ejpam-5570	154	28	finite	finite	NOUN
ejpam-5570	154	29	,	,	PUNCT
ejpam-5570	154	30	according	accord	VERB
ejpam-5570	154	31	to	to	ADP
ejpam-5570	154	32	(	(	PUNCT
ejpam-5570	154	33	i	i	NOUN
ejpam-5570	154	34	)	)	PUNCT
ejpam-5570	154	35	.	.	PUNCT
ejpam-5570	155	1	therefore	therefore	ADV
ejpam-5570	155	2	,	,	PUNCT
ejpam-5570	155	3	(	(	PUNCT
ejpam-5570	155	4	ii	ii	NOUN
ejpam-5570	155	5	)	)	PUNCT
ejpam-5570	155	6	has	have	AUX
ejpam-5570	155	7	been	be	AUX
ejpam-5570	155	8	demonstrated	demonstrate	VERB
ejpam-5570	155	9	.	.	PUNCT
ejpam-5570	156	1	theorem	theorem	ADJ
ejpam-5570	156	2	5	5	NUM
ejpam-5570	156	3	.	.	PUNCT
ejpam-5570	157	1	if	if	SCONJ
ejpam-5570	157	2	(	(	PUNCT
ejpam-5570	157	3	x	x	X
ejpam-5570	157	4	,	,	PUNCT
ejpam-5570	157	5	τ	τ	PROPN
ejpam-5570	157	6	,	,	PUNCT
ejpam-5570	157	7	i	i	PROPN
ejpam-5570	157	8	)	)	PUNCT
ejpam-5570	157	9	is	be	AUX
ejpam-5570	157	10	δ	δ	PROPN
ejpam-5570	157	11	-	-	PUNCT
ejpam-5570	157	12	βi	βi	ADV
ejpam-5570	157	13	-	-	PUNCT
ejpam-5570	157	14	paracompact	paracompact	NOUN
ejpam-5570	157	15	and	and	CCONJ
ejpam-5570	157	16	j	j	PROPN
ejpam-5570	157	17	is	be	AUX
ejpam-5570	157	18	an	an	DET
ejpam-5570	157	19	ideal	ideal	NOUN
ejpam-5570	157	20	on	on	ADP
ejpam-5570	157	21	x	x	PUNCT
ejpam-5570	157	22	with	with	ADP
ejpam-5570	157	23	i	i	PRON
ejpam-5570	157	24	⊂	⊂	PROPN
ejpam-5570	157	25	j	j	PROPN
ejpam-5570	157	26	,	,	PUNCT
ejpam-5570	157	27	then	then	ADV
ejpam-5570	157	28	(	(	PUNCT
ejpam-5570	157	29	x	x	X
ejpam-5570	157	30	,	,	PUNCT
ejpam-5570	157	31	τ	τ	PROPN
ejpam-5570	157	32	,	,	PUNCT
ejpam-5570	157	33	j	j	PROPN
ejpam-5570	157	34	)	)	PUNCT
ejpam-5570	157	35	is	be	AUX
ejpam-5570	157	36	δ	δ	PROPN
ejpam-5570	157	37	-	-	PUNCT
ejpam-5570	157	38	βj	βj	PUNCT
ejpam-5570	157	39	-paracompact	-paracompact	ADJ
ejpam-5570	157	40	.	.	PUNCT
ejpam-5570	158	1	proof	proof	NOUN
ejpam-5570	158	2	.	.	PUNCT
ejpam-5570	159	1	let	let	VERB
ejpam-5570	159	2	(	(	PUNCT
ejpam-5570	159	3	x	x	X
ejpam-5570	159	4	,	,	PUNCT
ejpam-5570	159	5	τ	τ	PROPN
ejpam-5570	159	6	,	,	PUNCT
ejpam-5570	159	7	i	i	PRON
ejpam-5570	159	8	)	)	PUNCT
ejpam-5570	159	9	be	be	VERB
ejpam-5570	159	10	δ	δ	PROPN
ejpam-5570	159	11	-	-	PUNCT
ejpam-5570	159	12	βi	βi	ADV
ejpam-5570	159	13	-	-	PUNCT
ejpam-5570	159	14	paracompact	paracompact	NOUN
ejpam-5570	159	15	and	and	CCONJ
ejpam-5570	159	16	i	i	PRON
ejpam-5570	159	17	⊂	⊂	PROPN
ejpam-5570	159	18	j	j	PROPN
ejpam-5570	159	19	,	,	PUNCT
ejpam-5570	159	20	and	and	CCONJ
ejpam-5570	159	21	let	let	VERB
ejpam-5570	159	22	u	u	PRON
ejpam-5570	159	23	=	=	X
ejpam-5570	159	24	{	{	PUNCT
ejpam-5570	159	25	uα	uα	X
ejpam-5570	159	26	:	:	PUNCT
ejpam-5570	159	27	α	α	PROPN
ejpam-5570	159	28	∈	∈	PROPN
ejpam-5570	159	29	λ	λ	NOUN
ejpam-5570	159	30	}	}	PUNCT
ejpam-5570	159	31	be	be	VERB
ejpam-5570	159	32	an	an	DET
ejpam-5570	159	33	open	open	ADJ
ejpam-5570	159	34	cover	cover	NOUN
ejpam-5570	159	35	ofx	ofx	NOUN
ejpam-5570	159	36	.	.	PUNCT
ejpam-5570	160	1	since	since	SCONJ
ejpam-5570	160	2	(	(	PUNCT
ejpam-5570	160	3	x	x	X
ejpam-5570	160	4	,	,	PUNCT
ejpam-5570	160	5	τ	τ	PROPN
ejpam-5570	160	6	,	,	PUNCT
ejpam-5570	160	7	i	i	PROPN
ejpam-5570	160	8	)	)	PUNCT
ejpam-5570	160	9	is	be	AUX
ejpam-5570	160	10	δ	δ	PROPN
ejpam-5570	160	11	-	-	PUNCT
ejpam-5570	160	12	βi	βi	ADV
ejpam-5570	160	13	-	-	PUNCT
ejpam-5570	160	14	paracompact	paracompact	ADJ
ejpam-5570	160	15	,	,	PUNCT
ejpam-5570	160	16	u	u	NOUN
ejpam-5570	160	17	has	have	VERB
ejpam-5570	160	18	a	a	DET
ejpam-5570	160	19	δ	δ	PROPN
ejpam-5570	160	20	-	-	PUNCT
ejpam-5570	160	21	βi	βi	ADV
ejpam-5570	160	22	-	-	PUNCT
ejpam-5570	160	23	locally	locally	ADV
ejpam-5570	160	24	finite	finite	PROPN
ejpam-5570	160	25	refinement	refinement	PROPN
ejpam-5570	160	26	v	v	PROPN
ejpam-5570	160	27	of	of	ADP
ejpam-5570	160	28	δ	δ	PROPN
ejpam-5570	160	29	-	-	PUNCT
ejpam-5570	160	30	βi	βi	ADV
ejpam-5570	160	31	-	-	PUNCT
ejpam-5570	160	32	open	open	ADJ
ejpam-5570	160	33	sets	set	VERB
ejpam-5570	160	34	such	such	ADJ
ejpam-5570	160	35	that	that	SCONJ
ejpam-5570	160	36	x	x	SYM
ejpam-5570	160	37	−∪{v	−∪{v	NOUN
ejpam-5570	160	38	:	:	PUNCT
ejpam-5570	160	39	v	v	NUM
ejpam-5570	160	40	∈	∈	PROPN
ejpam-5570	160	41	v	v	NOUN
ejpam-5570	160	42	}	}	PUNCT
ejpam-5570	160	43	∈	∈	PROPN
ejpam-5570	160	44	i.	i.	NOUN
ejpam-5570	160	45	as	as	ADP
ejpam-5570	160	46	i	i	PROPN
ejpam-5570	160	47	⊂	⊂	PROPN
ejpam-5570	160	48	j	j	PROPN
ejpam-5570	160	49	,	,	PUNCT
ejpam-5570	160	50	x	x	PUNCT
ejpam-5570	160	51	−∪{v	−∪{v	NOUN
ejpam-5570	160	52	:	:	PUNCT
ejpam-5570	160	53	v	v	NUM
ejpam-5570	160	54	∈	∈	PROPN
ejpam-5570	160	55	v	v	NOUN
ejpam-5570	160	56	}	}	PUNCT
ejpam-5570	160	57	∈	∈	PROPN
ejpam-5570	160	58	j	j	PROPN
ejpam-5570	160	59	.	.	PUNCT
ejpam-5570	161	1	consequently	consequently	ADV
ejpam-5570	161	2	,	,	PUNCT
ejpam-5570	161	3	(	(	PUNCT
ejpam-5570	161	4	x	x	X
ejpam-5570	161	5	,	,	PUNCT
ejpam-5570	161	6	τ	τ	PROPN
ejpam-5570	161	7	,	,	PUNCT
ejpam-5570	161	8	j	j	PROPN
ejpam-5570	161	9	)	)	PUNCT
ejpam-5570	161	10	is	be	AUX
ejpam-5570	161	11	δ	δ	PROPN
ejpam-5570	161	12	-	-	PUNCT
ejpam-5570	161	13	βj	βj	PUNCT
ejpam-5570	161	14	-paracompact	-paracompact	NOUN
ejpam-5570	161	15	.	.	PUNCT
ejpam-5570	162	1	c.	c.	PROPN
ejpam-5570	162	2	boonpok	boonpok	PROPN
ejpam-5570	162	3	,	,	PUNCT
ejpam-5570	162	4	a.	a.	PROPN
ejpam-5570	162	5	sama	sama	PROPN
ejpam-5570	162	6	-	-	PUNCT
ejpam-5570	162	7	ae	ae	PROPN
ejpam-5570	162	8	,	,	PUNCT
ejpam-5570	162	9	p.	p.	NOUN
ejpam-5570	162	10	raktaow	raktaow	PROPN
ejpam-5570	162	11	/	/	SYM
ejpam-5570	162	12	eur	eur	PROPN
ejpam-5570	162	13	.	.	PUNCT
ejpam-5570	163	1	j.	j.	PROPN
ejpam-5570	163	2	pure	pure	PROPN
ejpam-5570	163	3	appl	appl	PROPN
ejpam-5570	163	4	.	.	PROPN
ejpam-5570	163	5	math	math	PROPN
ejpam-5570	163	6	,	,	PUNCT
ejpam-5570	163	7	18	18	NUM
ejpam-5570	163	8	(	(	PUNCT
ejpam-5570	163	9	1	1	NUM
ejpam-5570	163	10	)	)	PUNCT
ejpam-5570	163	11	(	(	PUNCT
ejpam-5570	163	12	2025	2025	NUM
ejpam-5570	163	13	)	)	PUNCT
ejpam-5570	163	14	,	,	PUNCT
ejpam-5570	163	15	5570	5570	NUM
ejpam-5570	163	16	6	6	NUM
ejpam-5570	163	17	of	of	ADP
ejpam-5570	163	18	12	12	NUM
ejpam-5570	163	19	lemma	lemma	PROPN
ejpam-5570	163	20	5	5	NUM
ejpam-5570	163	21	.	.	PUNCT
ejpam-5570	164	1	if	if	SCONJ
ejpam-5570	164	2	an	an	DET
ejpam-5570	164	3	open	open	ADJ
ejpam-5570	164	4	cover	cover	NOUN
ejpam-5570	164	5	u	u	NOUN
ejpam-5570	164	6	=	=	PUNCT
ejpam-5570	164	7	{	{	PUNCT
ejpam-5570	164	8	uλ	uλ	X
ejpam-5570	164	9	:	:	PUNCT
ejpam-5570	164	10	λ	λ	X
ejpam-5570	164	11	∈	∈	PROPN
ejpam-5570	164	12	λ	λ	NOUN
ejpam-5570	164	13	}	}	PUNCT
ejpam-5570	164	14	of	of	ADP
ejpam-5570	164	15	an	an	DET
ejpam-5570	164	16	ideal	ideal	ADJ
ejpam-5570	164	17	topological	topological	ADJ
ejpam-5570	164	18	space	space	NOUN
ejpam-5570	164	19	(	(	PUNCT
ejpam-5570	164	20	x	x	X
ejpam-5570	164	21	,	,	PUNCT
ejpam-5570	164	22	τ	τ	PROPN
ejpam-5570	164	23	,	,	PUNCT
ejpam-5570	164	24	i	i	NOUN
ejpam-5570	164	25	)	)	PUNCT
ejpam-5570	164	26	has	have	VERB
ejpam-5570	164	27	a	a	DET
ejpam-5570	164	28	δ	δ	PROPN
ejpam-5570	164	29	-	-	PUNCT
ejpam-5570	164	30	βi	βi	ADV
ejpam-5570	164	31	-	-	PUNCT
ejpam-5570	164	32	locally	locally	ADV
ejpam-5570	164	33	finite	finite	NOUN
ejpam-5570	164	34	of	of	ADP
ejpam-5570	164	35	δ	δ	PROPN
ejpam-5570	164	36	-	-	PUNCT
ejpam-5570	164	37	βi	βi	ADV
ejpam-5570	164	38	-	-	PUNCT
ejpam-5570	164	39	open	open	ADJ
ejpam-5570	164	40	refinement	refinement	NOUN
ejpam-5570	164	41	v	v	ADP
ejpam-5570	164	42	such	such	ADJ
ejpam-5570	164	43	that	that	SCONJ
ejpam-5570	164	44	x	x	SYM
ejpam-5570	164	45	−∪{v	−∪{v	NOUN
ejpam-5570	164	46	:	:	PUNCT
ejpam-5570	164	47	v	v	NUM
ejpam-5570	164	48	∈	∈	PROPN
ejpam-5570	164	49	v	v	NOUN
ejpam-5570	164	50	}	}	PUNCT
ejpam-5570	164	51	∈	∈	PROPN
ejpam-5570	164	52	i	i	PRON
ejpam-5570	164	53	,	,	PUNCT
ejpam-5570	164	54	then	then	ADV
ejpam-5570	164	55	there	there	PRON
ejpam-5570	164	56	exists	exist	VERB
ejpam-5570	164	57	a	a	DET
ejpam-5570	164	58	precise	precise	ADJ
ejpam-5570	164	59	δ	δ	PROPN
ejpam-5570	164	60	-	-	PUNCT
ejpam-5570	164	61	βi	βi	ADV
ejpam-5570	164	62	-	-	PUNCT
ejpam-5570	164	63	locally	locally	ADV
ejpam-5570	164	64	finite	finite	PROPN
ejpam-5570	164	65	δ	δ	PROPN
ejpam-5570	164	66	-	-	ADJ
ejpam-5570	164	67	βi	βi	ADV
ejpam-5570	164	68	-	-	PUNCT
ejpam-5570	164	69	open	open	ADJ
ejpam-5570	164	70	refinement	refinement	NOUN
ejpam-5570	164	71	h	h	NOUN
ejpam-5570	164	72	=	=	PRON
ejpam-5570	164	73	{	{	PUNCT
ejpam-5570	164	74	hλ	hλ	X
ejpam-5570	164	75	:	:	PUNCT
ejpam-5570	164	76	λ	λ	PROPN
ejpam-5570	164	77	∈	∈	PROPN
ejpam-5570	164	78	λ	λ	PROPN
ejpam-5570	164	79	}	}	PUNCT
ejpam-5570	164	80	of	of	ADP
ejpam-5570	164	81	u	u	PRON
ejpam-5570	164	82	such	such	ADJ
ejpam-5570	164	83	that	that	SCONJ
ejpam-5570	164	84	x	x	PROPN
ejpam-5570	164	85	−	−	PROPN
ejpam-5570	164	86	∪{hλ	∪{hλ	PROPN
ejpam-5570	164	87	:	:	PUNCT
ejpam-5570	164	88	λ	λ	PROPN
ejpam-5570	164	89	∈	∈	PROPN
ejpam-5570	164	90	λ	λ	PROPN
ejpam-5570	164	91	}	}	PUNCT
ejpam-5570	164	92	∈	∈	PROPN
ejpam-5570	164	93	i.	i.	NOUN
ejpam-5570	164	94	proof	proof	NOUN
ejpam-5570	164	95	.	.	PUNCT
ejpam-5570	165	1	the	the	DET
ejpam-5570	165	2	proof	proof	NOUN
ejpam-5570	165	3	is	be	AUX
ejpam-5570	165	4	comparable	comparable	ADJ
ejpam-5570	165	5	to	to	ADP
ejpam-5570	165	6	that	that	PRON
ejpam-5570	165	7	of	of	ADP
ejpam-5570	165	8	lemma	lemma	PROPN
ejpam-5570	165	9	1.3	1.3	NUM
ejpam-5570	165	10	in	in	ADP
ejpam-5570	165	11	[	[	X
ejpam-5570	165	12	15	15	NUM
ejpam-5570	165	13	]	]	PUNCT
ejpam-5570	165	14	.	.	PUNCT
ejpam-5570	166	1	definition	definition	NOUN
ejpam-5570	166	2	6	6	NUM
ejpam-5570	166	3	.	.	PUNCT
ejpam-5570	167	1	an	an	DET
ejpam-5570	167	2	ideal	ideal	ADJ
ejpam-5570	167	3	topological	topological	ADJ
ejpam-5570	167	4	space	space	NOUN
ejpam-5570	167	5	(	(	PUNCT
ejpam-5570	167	6	x	x	X
ejpam-5570	167	7	,	,	PUNCT
ejpam-5570	167	8	τ	τ	PROPN
ejpam-5570	167	9	,	,	PUNCT
ejpam-5570	167	10	i	i	PROPN
ejpam-5570	167	11	)	)	PUNCT
ejpam-5570	167	12	is	be	AUX
ejpam-5570	167	13	δ	δ	PROPN
ejpam-5570	167	14	-	-	PUNCT
ejpam-5570	167	15	βi	βi	ADV
ejpam-5570	167	16	-	-	ADJ
ejpam-5570	167	17	regular	regular	ADJ
ejpam-5570	167	18	if	if	SCONJ
ejpam-5570	167	19	for	for	ADP
ejpam-5570	167	20	any	any	DET
ejpam-5570	167	21	closed	closed	ADJ
ejpam-5570	167	22	subset	subset	NOUN
ejpam-5570	167	23	f	f	PROPN
ejpam-5570	167	24	of	of	ADP
ejpam-5570	167	25	x	x	X
ejpam-5570	167	26	and	and	CCONJ
ejpam-5570	167	27	x	x	PUNCT
ejpam-5570	167	28	̸∈	̸∈	PROPN
ejpam-5570	167	29	f	f	PROPN
ejpam-5570	167	30	,	,	PUNCT
ejpam-5570	167	31	there	there	PRON
ejpam-5570	167	32	exist	exist	VERB
ejpam-5570	167	33	disjoint	disjoint	PROPN
ejpam-5570	167	34	δ	δ	PROPN
ejpam-5570	167	35	-	-	PUNCT
ejpam-5570	167	36	βi	βi	ADV
ejpam-5570	167	37	-	-	PUNCT
ejpam-5570	167	38	open	open	ADJ
ejpam-5570	167	39	sets	set	VERB
ejpam-5570	167	40	u	u	NOUN
ejpam-5570	167	41	and	and	CCONJ
ejpam-5570	167	42	v	v	ADP
ejpam-5570	167	43	such	such	ADJ
ejpam-5570	167	44	that	that	SCONJ
ejpam-5570	167	45	x	x	SYM
ejpam-5570	167	46	∈	∈	PROPN
ejpam-5570	167	47	u	u	NOUN
ejpam-5570	167	48	and	and	CCONJ
ejpam-5570	167	49	f	f	PROPN
ejpam-5570	167	50	−	−	PROPN
ejpam-5570	167	51	v	v	PROPN
ejpam-5570	167	52	∈	∈	PROPN
ejpam-5570	167	53	i.	i.	NOUN
ejpam-5570	167	54	theorem	theorem	VERB
ejpam-5570	167	55	6	6	NUM
ejpam-5570	167	56	.	.	PUNCT
ejpam-5570	168	1	let	let	VERB
ejpam-5570	168	2	(	(	PUNCT
ejpam-5570	168	3	x	x	X
ejpam-5570	168	4	,	,	PUNCT
ejpam-5570	168	5	τ	τ	PROPN
ejpam-5570	168	6	,	,	PUNCT
ejpam-5570	168	7	i	i	PRON
ejpam-5570	168	8	)	)	PUNCT
ejpam-5570	168	9	be	be	VERB
ejpam-5570	168	10	an	an	DET
ejpam-5570	168	11	ideal	ideal	ADJ
ejpam-5570	168	12	topological	topological	ADJ
ejpam-5570	168	13	space	space	NOUN
ejpam-5570	168	14	.	.	PUNCT
ejpam-5570	169	1	assume	assume	VERB
ejpam-5570	169	2	that	that	SCONJ
ejpam-5570	169	3	the	the	DET
ejpam-5570	169	4	subsequent	subsequent	ADJ
ejpam-5570	169	5	assertions	assertion	NOUN
ejpam-5570	169	6	are	be	AUX
ejpam-5570	169	7	true	true	ADJ
ejpam-5570	169	8	:	:	PUNCT
ejpam-5570	169	9	(	(	PUNCT
ejpam-5570	169	10	i	i	NOUN
ejpam-5570	169	11	)	)	PUNCT
ejpam-5570	169	12	x	x	X
ejpam-5570	169	13	is	be	AUX
ejpam-5570	169	14	δ	δ	PROPN
ejpam-5570	169	15	-	-	PUNCT
ejpam-5570	169	16	βi	βi	ADV
ejpam-5570	169	17	-	-	PUNCT
ejpam-5570	169	18	paracompact	paracompact	ADJ
ejpam-5570	169	19	;	;	PUNCT
ejpam-5570	169	20	(	(	PUNCT
ejpam-5570	169	21	ii	ii	NOUN
ejpam-5570	169	22	)	)	PUNCT
ejpam-5570	169	23	x	x	X
ejpam-5570	169	24	is	be	AUX
ejpam-5570	169	25	hausdorff	hausdorff	NOUN
ejpam-5570	169	26	;	;	PUNCT
ejpam-5570	169	27	(	(	PUNCT
ejpam-5570	169	28	iii	iii	X
ejpam-5570	169	29	)	)	PUNCT
ejpam-5570	169	30	the	the	DET
ejpam-5570	169	31	arbitrary	arbitrary	ADJ
ejpam-5570	169	32	union	union	NOUN
ejpam-5570	169	33	of	of	ADP
ejpam-5570	169	34	δ	δ	PROPN
ejpam-5570	169	35	-	-	PUNCT
ejpam-5570	169	36	βi	βi	ADV
ejpam-5570	169	37	-	-	PUNCT
ejpam-5570	169	38	closed	closed	ADJ
ejpam-5570	169	39	;	;	PUNCT
ejpam-5570	169	40	sets	set	NOUN
ejpam-5570	169	41	remains	remain	VERB
ejpam-5570	169	42	δ	δ	PROPN
ejpam-5570	169	43	-	-	PUNCT
ejpam-5570	169	44	βi	βi	ADV
ejpam-5570	169	45	-	-	PUNCT
ejpam-5570	169	46	closed	closed	ADJ
ejpam-5570	169	47	.	.	PUNCT
ejpam-5570	170	1	then	then	ADV
ejpam-5570	170	2	(	(	PUNCT
ejpam-5570	170	3	x	x	X
ejpam-5570	170	4	,	,	PUNCT
ejpam-5570	170	5	τ	τ	PROPN
ejpam-5570	170	6	,	,	PUNCT
ejpam-5570	170	7	i	i	PROPN
ejpam-5570	170	8	)	)	PUNCT
ejpam-5570	170	9	is	be	AUX
ejpam-5570	170	10	δ	δ	PROPN
ejpam-5570	170	11	-	-	PUNCT
ejpam-5570	170	12	βi	βi	ADV
ejpam-5570	170	13	-	-	PUNCT
ejpam-5570	170	14	regular	regular	ADJ
ejpam-5570	170	15	.	.	PUNCT
ejpam-5570	171	1	proof	proof	NOUN
ejpam-5570	171	2	.	.	PUNCT
ejpam-5570	172	1	let	let	VERB
ejpam-5570	172	2	f	f	PRON
ejpam-5570	172	3	be	be	AUX
ejpam-5570	172	4	a	a	DET
ejpam-5570	172	5	closed	closed	ADJ
ejpam-5570	172	6	subset	subset	NOUN
ejpam-5570	172	7	of	of	ADP
ejpam-5570	172	8	x	x	X
ejpam-5570	172	9	,	,	PUNCT
ejpam-5570	172	10	and	and	CCONJ
ejpam-5570	172	11	let	let	VERB
ejpam-5570	172	12	x	x	PROPN
ejpam-5570	172	13	̸∈	̸∈	PROPN
ejpam-5570	172	14	f	f	PROPN
ejpam-5570	172	15	.	.	PUNCT
ejpam-5570	173	1	utilizing	utilize	VERB
ejpam-5570	173	2	(	(	PUNCT
ejpam-5570	173	3	ii	ii	NOUN
ejpam-5570	173	4	)	)	PUNCT
ejpam-5570	173	5	,	,	PUNCT
ejpam-5570	173	6	there	there	PRON
ejpam-5570	173	7	exist	exist	VERB
ejpam-5570	173	8	disjoint	disjoint	ADJ
ejpam-5570	173	9	open	open	ADJ
ejpam-5570	173	10	sets	set	NOUN
ejpam-5570	173	11	vx	vx	PUNCT
ejpam-5570	173	12	and	and	CCONJ
ejpam-5570	173	13	ox	ox	NOUN
ejpam-5570	173	14	that	that	PRON
ejpam-5570	173	15	include	include	VERB
ejpam-5570	173	16	x	x	PUNCT
ejpam-5570	173	17	and	and	CCONJ
ejpam-5570	173	18	y	y	PROPN
ejpam-5570	173	19	,	,	PUNCT
ejpam-5570	173	20	respectively	respectively	ADV
ejpam-5570	173	21	,	,	PUNCT
ejpam-5570	173	22	for	for	ADP
ejpam-5570	173	23	any	any	DET
ejpam-5570	173	24	point	point	NOUN
ejpam-5570	173	25	y	y	PROPN
ejpam-5570	173	26	∈	∈	PROPN
ejpam-5570	174	1	f	f	X
ejpam-5570	174	2	.	.	PUNCT
ejpam-5570	175	1	it	it	PRON
ejpam-5570	175	2	suggests	suggest	VERB
ejpam-5570	175	3	that	that	SCONJ
ejpam-5570	175	4	y	y	PROPN
ejpam-5570	175	5	̸∈	̸∈	PROPN
ejpam-5570	175	6	cl(vx	cl(vx	PROPN
ejpam-5570	175	7	)	)	PUNCT
ejpam-5570	175	8	.	.	PUNCT
ejpam-5570	176	1	now	now	ADV
ejpam-5570	176	2	we	we	PRON
ejpam-5570	176	3	have	have	VERB
ejpam-5570	176	4	that	that	DET
ejpam-5570	176	5	u	u	NOUN
ejpam-5570	176	6	=	=	PUNCT
ejpam-5570	176	7	{	{	PUNCT
ejpam-5570	176	8	ox	ox	NOUN
ejpam-5570	176	9	:	:	PUNCT
ejpam-5570	176	10	x	x	PUNCT
ejpam-5570	176	11	∈	∈	PROPN
ejpam-5570	176	12	f	f	X
ejpam-5570	176	13	}	}	PUNCT
ejpam-5570	176	14	∪	∪	NOUN
ejpam-5570	176	15	{	{	PUNCT
ejpam-5570	176	16	x	x	NOUN
ejpam-5570	176	17	−	−	PROPN
ejpam-5570	176	18	f	f	X
ejpam-5570	176	19	}	}	PUNCT
ejpam-5570	176	20	is	be	AUX
ejpam-5570	176	21	an	an	DET
ejpam-5570	176	22	open	open	ADJ
ejpam-5570	176	23	cover	cover	NOUN
ejpam-5570	176	24	of	of	ADP
ejpam-5570	176	25	x.	x.	NOUN
ejpam-5570	176	26	by	by	ADP
ejpam-5570	176	27	(	(	PUNCT
ejpam-5570	176	28	i	i	NOUN
ejpam-5570	176	29	)	)	PUNCT
ejpam-5570	176	30	,	,	PUNCT
ejpam-5570	176	31	there	there	PRON
ejpam-5570	176	32	exists	exist	VERB
ejpam-5570	176	33	a	a	DET
ejpam-5570	176	34	δ	δ	PROPN
ejpam-5570	176	35	-	-	PUNCT
ejpam-5570	176	36	βi	βi	ADV
ejpam-5570	176	37	-	-	PUNCT
ejpam-5570	176	38	locally	locally	ADV
ejpam-5570	176	39	finite	finite	PROPN
ejpam-5570	176	40	δ	δ	PROPN
ejpam-5570	176	41	-	-	ADJ
ejpam-5570	176	42	βi	βi	ADV
ejpam-5570	176	43	-	-	PUNCT
ejpam-5570	176	44	open	open	ADJ
ejpam-5570	176	45	refinement	refinement	NOUN
ejpam-5570	176	46	h	h	NOUN
ejpam-5570	176	47	=	=	PUNCT
ejpam-5570	176	48	{	{	PUNCT
ejpam-5570	176	49	hx	hx	NOUN
ejpam-5570	176	50	:	:	PUNCT
ejpam-5570	176	51	x	x	PUNCT
ejpam-5570	176	52	∈	∈	PROPN
ejpam-5570	176	53	f	f	X
ejpam-5570	176	54	}	}	PUNCT
ejpam-5570	176	55	∪	∪	X
ejpam-5570	176	56	{	{	PUNCT
ejpam-5570	176	57	w	w	NOUN
ejpam-5570	176	58	}	}	PUNCT
ejpam-5570	176	59	such	such	ADJ
ejpam-5570	176	60	that	that	SCONJ
ejpam-5570	176	61	hx	hx	PROPN
ejpam-5570	176	62	⊂	⊂	PROPN
ejpam-5570	176	63	ox	ox	PROPN
ejpam-5570	176	64	for	for	ADP
ejpam-5570	176	65	each	each	DET
ejpam-5570	176	66	x	x	NOUN
ejpam-5570	176	67	,	,	PUNCT
ejpam-5570	176	68	w	w	PROPN
ejpam-5570	176	69	⊂	⊂	PROPN
ejpam-5570	176	70	x	x	PUNCT
ejpam-5570	177	1	−	−	PROPN
ejpam-5570	177	2	f	f	X
ejpam-5570	177	3	,	,	PUNCT
ejpam-5570	177	4	and	and	CCONJ
ejpam-5570	177	5	x	x	X
ejpam-5570	177	6	−	−	PROPN
ejpam-5570	177	7	(	(	PUNCT
ejpam-5570	177	8	∪{hx	∪{hx	PROPN
ejpam-5570	177	9	:	:	PUNCT
ejpam-5570	177	10	x	x	SYM
ejpam-5570	177	11	∈	∈	PROPN
ejpam-5570	177	12	f	f	X
ejpam-5570	177	13	}	}	PUNCT
ejpam-5570	177	14	∪	∪	X
ejpam-5570	177	15	{	{	PUNCT
ejpam-5570	177	16	w	w	NOUN
ejpam-5570	177	17	}	}	PUNCT
ejpam-5570	177	18	)	)	PUNCT
ejpam-5570	177	19	∈	∈	PROPN
ejpam-5570	177	20	i.	i.	NOUN
ejpam-5570	177	21	assume	assume	VERB
ejpam-5570	177	22	that	that	SCONJ
ejpam-5570	177	23	v	v	NOUN
ejpam-5570	177	24	=	=	PRON
ejpam-5570	177	25	∪{hx	∪{hx	PROPN
ejpam-5570	177	26	:	:	PUNCT
ejpam-5570	177	27	x	x	SYM
ejpam-5570	178	1	∈	∈	PROPN
ejpam-5570	178	2	f	f	X
ejpam-5570	178	3	}	}	PUNCT
ejpam-5570	178	4	and	and	CCONJ
ejpam-5570	178	5	u	u	X
ejpam-5570	178	6	=	=	NOUN
ejpam-5570	178	7	x	x	SYM
ejpam-5570	178	8	−	−	PROPN
ejpam-5570	178	9	∪{δ	∪{δ	PROPN
ejpam-5570	178	10	-	-	NOUN
ejpam-5570	178	11	βcli(hx	βcli(hx	NOUN
ejpam-5570	178	12	)	)	PUNCT
ejpam-5570	178	13	:	:	PUNCT
ejpam-5570	179	1	x	x	PUNCT
ejpam-5570	179	2	∈	∈	NOUN
ejpam-5570	179	3	f	f	X
ejpam-5570	179	4	}	}	PUNCT
ejpam-5570	179	5	.	.	PUNCT
ejpam-5570	180	1	using	use	VERB
ejpam-5570	180	2	(	(	PUNCT
ejpam-5570	180	3	iii	iii	NOUN
ejpam-5570	180	4	)	)	PUNCT
ejpam-5570	180	5	,	,	PUNCT
ejpam-5570	180	6	u	u	NOUN
ejpam-5570	180	7	and	and	CCONJ
ejpam-5570	180	8	v	v	NOUN
ejpam-5570	180	9	,	,	PUNCT
ejpam-5570	180	10	therefore	therefore	ADV
ejpam-5570	180	11	,	,	PUNCT
ejpam-5570	180	12	are	be	AUX
ejpam-5570	180	13	disjoint	disjoint	PROPN
ejpam-5570	180	14	δ	δ	PROPN
ejpam-5570	180	15	-	-	PUNCT
ejpam-5570	180	16	βi	βi	ADV
ejpam-5570	180	17	-	-	PUNCT
ejpam-5570	180	18	open	open	ADJ
ejpam-5570	180	19	sets	set	NOUN
ejpam-5570	180	20	in	in	ADP
ejpam-5570	180	21	which	which	PRON
ejpam-5570	180	22	x	x	X
ejpam-5570	180	23	∈	∈	PROPN
ejpam-5570	180	24	u	u	NOUN
ejpam-5570	180	25	and	and	CCONJ
ejpam-5570	180	26	f	f	PROPN
ejpam-5570	180	27	−	−	PROPN
ejpam-5570	180	28	v	v	ADP
ejpam-5570	180	29	⊂	⊂	PROPN
ejpam-5570	180	30	x	x	PUNCT
ejpam-5570	181	1	−	−	PROPN
ejpam-5570	181	2	f	f	PROPN
ejpam-5570	181	3	∈	∈	PROPN
ejpam-5570	181	4	i.	i.	NOUN
ejpam-5570	181	5	consequently	consequently	ADV
ejpam-5570	181	6	,	,	PUNCT
ejpam-5570	181	7	(	(	PUNCT
ejpam-5570	181	8	x	x	X
ejpam-5570	181	9	,	,	PUNCT
ejpam-5570	181	10	τ	τ	PROPN
ejpam-5570	181	11	,	,	PUNCT
ejpam-5570	181	12	i	i	PROPN
ejpam-5570	181	13	)	)	PUNCT
ejpam-5570	181	14	is	be	AUX
ejpam-5570	181	15	δ	δ	PROPN
ejpam-5570	181	16	-	-	PUNCT
ejpam-5570	181	17	βi	βi	ADV
ejpam-5570	181	18	-	-	PUNCT
ejpam-5570	181	19	regular	regular	ADJ
ejpam-5570	181	20	.	.	PUNCT
ejpam-5570	182	1	theorem	theorem	VERB
ejpam-5570	182	2	7	7	NUM
ejpam-5570	182	3	.	.	PUNCT
ejpam-5570	183	1	let	let	VERB
ejpam-5570	183	2	(	(	PUNCT
ejpam-5570	183	3	x	x	X
ejpam-5570	183	4	,	,	PUNCT
ejpam-5570	183	5	τ	τ	PROPN
ejpam-5570	183	6	,	,	PUNCT
ejpam-5570	183	7	i	i	PRON
ejpam-5570	183	8	)	)	PUNCT
ejpam-5570	183	9	be	be	VERB
ejpam-5570	183	10	an	an	DET
ejpam-5570	183	11	ideal	ideal	ADJ
ejpam-5570	183	12	topological	topological	ADJ
ejpam-5570	183	13	space	space	NOUN
ejpam-5570	183	14	.	.	PUNCT
ejpam-5570	184	1	suppose	suppose	VERB
ejpam-5570	184	2	that	that	SCONJ
ejpam-5570	184	3	the	the	DET
ejpam-5570	184	4	following	follow	VERB
ejpam-5570	184	5	statements	statement	NOUN
ejpam-5570	184	6	hold	hold	VERB
ejpam-5570	184	7	:	:	PUNCT
ejpam-5570	184	8	(	(	PUNCT
ejpam-5570	184	9	i	i	NOUN
ejpam-5570	184	10	)	)	PUNCT
ejpam-5570	184	11	x	x	X
ejpam-5570	184	12	is	be	AUX
ejpam-5570	184	13	δ	δ	PROPN
ejpam-5570	184	14	-	-	PUNCT
ejpam-5570	184	15	βi	βi	ADV
ejpam-5570	184	16	-	-	PUNCT
ejpam-5570	184	17	paracompact	paracompact	ADJ
ejpam-5570	184	18	;	;	PUNCT
ejpam-5570	184	19	(	(	PUNCT
ejpam-5570	184	20	ii	ii	NOUN
ejpam-5570	184	21	)	)	PUNCT
ejpam-5570	184	22	x	x	X
ejpam-5570	184	23	is	be	AUX
ejpam-5570	184	24	hausdorff	hausdorff	NOUN
ejpam-5570	184	25	;	;	PUNCT
ejpam-5570	184	26	(	(	PUNCT
ejpam-5570	184	27	iii	iii	X
ejpam-5570	184	28	)	)	PUNCT
ejpam-5570	184	29	δ	δ	NOUN
ejpam-5570	184	30	-	-	PUNCT
ejpam-5570	184	31	βcli(∪{vλ	βcli(∪{vλ	PUNCT
ejpam-5570	184	32	:	:	PUNCT
ejpam-5570	184	33	λ	λ	PROPN
ejpam-5570	184	34	∈	∈	PROPN
ejpam-5570	184	35	λ	λ	NOUN
ejpam-5570	184	36	}	}	PUNCT
ejpam-5570	184	37	)	)	PUNCT
ejpam-5570	184	38	=	=	PUNCT
ejpam-5570	185	1	∪{δ	∪{δ	NOUN
ejpam-5570	185	2	-	-	PUNCT
ejpam-5570	185	3	βcli(vλ	βcli(vλ	NOUN
ejpam-5570	185	4	)	)	PUNCT
ejpam-5570	185	5	:	:	PUNCT
ejpam-5570	186	1	λ	λ	X
ejpam-5570	186	2	∈	∈	PROPN
ejpam-5570	186	3	λ	λ	NOUN
ejpam-5570	186	4	}	}	PUNCT
ejpam-5570	186	5	,	,	PUNCT
ejpam-5570	186	6	for	for	SCONJ
ejpam-5570	186	7	any	any	DET
ejpam-5570	186	8	δ	δ	PROPN
ejpam-5570	186	9	-	-	PUNCT
ejpam-5570	186	10	βi	βi	ADV
ejpam-5570	186	11	-	-	PUNCT
ejpam-5570	186	12	locally	locally	ADV
ejpam-5570	186	13	finite	finite	ADJ
ejpam-5570	186	14	collection	collection	NOUN
ejpam-5570	186	15	v	v	NOUN
ejpam-5570	186	16	=	=	PUNCT
ejpam-5570	186	17	{	{	PUNCT
ejpam-5570	186	18	vλ	vλ	INTJ
ejpam-5570	186	19	:	:	PUNCT
ejpam-5570	186	20	λ	λ	PROPN
ejpam-5570	186	21	∈	∈	PROPN
ejpam-5570	186	22	λ	λ	PROPN
ejpam-5570	186	23	}	}	PUNCT
ejpam-5570	186	24	of	of	ADP
ejpam-5570	186	25	x.	x.	NOUN
ejpam-5570	186	26	then	then	ADV
ejpam-5570	186	27	(	(	PUNCT
ejpam-5570	186	28	x	x	X
ejpam-5570	186	29	,	,	PUNCT
ejpam-5570	186	30	τ	τ	PROPN
ejpam-5570	186	31	,	,	PUNCT
ejpam-5570	186	32	i	i	PROPN
ejpam-5570	186	33	)	)	PUNCT
ejpam-5570	186	34	is	be	AUX
ejpam-5570	186	35	δ	δ	PROPN
ejpam-5570	186	36	-	-	PUNCT
ejpam-5570	186	37	βi	βi	ADV
ejpam-5570	186	38	-	-	PUNCT
ejpam-5570	186	39	regular	regular	ADJ
ejpam-5570	186	40	.	.	PUNCT
ejpam-5570	187	1	proof	proof	NOUN
ejpam-5570	187	2	.	.	PUNCT
ejpam-5570	188	1	let	let	VERB
ejpam-5570	188	2	f	f	PRON
ejpam-5570	188	3	be	be	AUX
ejpam-5570	188	4	a	a	DET
ejpam-5570	188	5	closed	closed	ADJ
ejpam-5570	188	6	set	set	NOUN
ejpam-5570	188	7	and	and	CCONJ
ejpam-5570	188	8	x	x	PART
ejpam-5570	188	9	̸∈	̸∈	PROPN
ejpam-5570	188	10	f	f	PROPN
ejpam-5570	188	11	.	.	PUNCT
ejpam-5570	189	1	using	use	VERB
ejpam-5570	189	2	(	(	PUNCT
ejpam-5570	189	3	ii	ii	NOUN
ejpam-5570	189	4	)	)	PUNCT
ejpam-5570	189	5	,	,	PUNCT
ejpam-5570	189	6	for	for	ADP
ejpam-5570	189	7	any	any	DET
ejpam-5570	189	8	y	y	PROPN
ejpam-5570	189	9	∈	∈	PROPN
ejpam-5570	189	10	f	f	PROPN
ejpam-5570	189	11	,	,	PUNCT
ejpam-5570	189	12	there	there	PRON
ejpam-5570	189	13	exists	exist	VERB
ejpam-5570	189	14	an	an	DET
ejpam-5570	189	15	open	open	ADJ
ejpam-5570	189	16	set	set	NOUN
ejpam-5570	189	17	gy	gy	NOUN
ejpam-5570	189	18	containing	contain	VERB
ejpam-5570	189	19	y	y	PRON
ejpam-5570	189	20	such	such	ADJ
ejpam-5570	189	21	that	that	SCONJ
ejpam-5570	189	22	x	x	PROPN
ejpam-5570	189	23	̸∈	̸∈	PROPN
ejpam-5570	189	24	cl(gy	cl(gy	PROPN
ejpam-5570	189	25	)	)	PUNCT
ejpam-5570	189	26	.	.	PUNCT
ejpam-5570	190	1	then	then	ADV
ejpam-5570	190	2	,	,	PUNCT
ejpam-5570	190	3	g	g	PROPN
ejpam-5570	190	4	=	=	PUNCT
ejpam-5570	190	5	{	{	PUNCT
ejpam-5570	190	6	gy	gy	INTJ
ejpam-5570	190	7	:	:	PUNCT
ejpam-5570	190	8	y	y	PROPN
ejpam-5570	190	9	∈	∈	PROPN
ejpam-5570	190	10	f	f	X
ejpam-5570	190	11	}	}	PUNCT
ejpam-5570	190	12	∪	∪	NOUN
ejpam-5570	190	13	{	{	PUNCT
ejpam-5570	190	14	x	x	NOUN
ejpam-5570	190	15	−	−	PROPN
ejpam-5570	190	16	f	f	X
ejpam-5570	190	17	}	}	PUNCT
ejpam-5570	190	18	is	be	AUX
ejpam-5570	190	19	an	an	DET
ejpam-5570	190	20	open	open	ADJ
ejpam-5570	190	21	cover	cover	NOUN
ejpam-5570	190	22	of	of	ADP
ejpam-5570	190	23	x.	x.	NOUN
ejpam-5570	190	24	by	by	ADP
ejpam-5570	190	25	(	(	PUNCT
ejpam-5570	190	26	i	i	NOUN
ejpam-5570	190	27	)	)	PUNCT
ejpam-5570	190	28	and	and	CCONJ
ejpam-5570	190	29	lemma	lemma	PROPN
ejpam-5570	190	30	5	5	NUM
ejpam-5570	190	31	,	,	PUNCT
ejpam-5570	190	32	g	g	PROPN
ejpam-5570	190	33	has	have	VERB
ejpam-5570	190	34	a	a	DET
ejpam-5570	190	35	precise	precise	ADJ
ejpam-5570	190	36	δ	δ	NOUN
ejpam-5570	190	37	-	-	PUNCT
ejpam-5570	190	38	βi	βi	ADV
ejpam-5570	190	39	-	-	PUNCT
ejpam-5570	190	40	locally	locally	ADV
ejpam-5570	190	41	finite	finite	PROPN
ejpam-5570	190	42	δ	δ	PROPN
ejpam-5570	190	43	-	-	ADJ
ejpam-5570	190	44	βi	βi	ADV
ejpam-5570	190	45	-	-	PUNCT
ejpam-5570	190	46	open	open	ADJ
ejpam-5570	190	47	refinement	refinement	NOUN
ejpam-5570	190	48	w	w	PROPN
ejpam-5570	190	49	=	=	PUNCT
ejpam-5570	190	50	{	{	PUNCT
ejpam-5570	190	51	wy	wy	PROPN
ejpam-5570	190	52	:	:	PUNCT
ejpam-5570	190	53	y	y	PROPN
ejpam-5570	190	54	∈	∈	PROPN
ejpam-5570	190	55	f	f	X
ejpam-5570	190	56	}	}	PUNCT
ejpam-5570	190	57	∪	∪	ADJ
ejpam-5570	190	58	{	{	PUNCT
ejpam-5570	190	59	g	g	NOUN
ejpam-5570	190	60	}	}	PUNCT
ejpam-5570	190	61	such	such	ADJ
ejpam-5570	190	62	that	that	SCONJ
ejpam-5570	190	63	wy	wy	PROPN
ejpam-5570	190	64	⊂	⊂	PROPN
ejpam-5570	190	65	gy	gy	VERB
ejpam-5570	190	66	for	for	ADP
ejpam-5570	190	67	each	each	DET
ejpam-5570	190	68	y	y	PROPN
ejpam-5570	190	69	∈	∈	PROPN
ejpam-5570	190	70	f	f	PROPN
ejpam-5570	190	71	,	,	PUNCT
ejpam-5570	190	72	g	g	PROPN
ejpam-5570	190	73	⊂	⊂	PROPN
ejpam-5570	190	74	x	x	PUNCT
ejpam-5570	191	1	−	−	PROPN
ejpam-5570	191	2	f	f	PROPN
ejpam-5570	191	3	c.	c.	PROPN
ejpam-5570	191	4	boonpok	boonpok	PROPN
ejpam-5570	191	5	,	,	PUNCT
ejpam-5570	191	6	a.	a.	PROPN
ejpam-5570	191	7	sama	sama	PROPN
ejpam-5570	191	8	-	-	PUNCT
ejpam-5570	191	9	ae	ae	PROPN
ejpam-5570	191	10	,	,	PUNCT
ejpam-5570	191	11	p.	p.	NOUN
ejpam-5570	191	12	raktaow	raktaow	PROPN
ejpam-5570	191	13	/	/	SYM
ejpam-5570	191	14	eur	eur	PROPN
ejpam-5570	191	15	.	.	PUNCT
ejpam-5570	192	1	j.	j.	PROPN
ejpam-5570	192	2	pure	pure	PROPN
ejpam-5570	192	3	appl	appl	PROPN
ejpam-5570	192	4	.	.	PROPN
ejpam-5570	192	5	math	math	PROPN
ejpam-5570	192	6	,	,	PUNCT
ejpam-5570	192	7	18	18	NUM
ejpam-5570	192	8	(	(	PUNCT
ejpam-5570	192	9	1	1	NUM
ejpam-5570	192	10	)	)	PUNCT
ejpam-5570	192	11	(	(	PUNCT
ejpam-5570	192	12	2025	2025	NUM
ejpam-5570	192	13	)	)	PUNCT
ejpam-5570	192	14	,	,	PUNCT
ejpam-5570	192	15	5570	5570	NUM
ejpam-5570	192	16	7	7	NUM
ejpam-5570	192	17	of	of	ADP
ejpam-5570	192	18	12	12	NUM
ejpam-5570	192	19	and	and	CCONJ
ejpam-5570	192	20	x	x	SYM
ejpam-5570	192	21	−	−	PROPN
ejpam-5570	192	22	(	(	PUNCT
ejpam-5570	192	23	∪{wy	∪{wy	PROPN
ejpam-5570	192	24	:	:	PUNCT
ejpam-5570	192	25	y	y	PROPN
ejpam-5570	192	26	∈	∈	PROPN
ejpam-5570	192	27	f	f	X
ejpam-5570	192	28	}	}	PUNCT
ejpam-5570	192	29	∪	∪	ADJ
ejpam-5570	192	30	{	{	PUNCT
ejpam-5570	192	31	g	g	NOUN
ejpam-5570	192	32	}	}	PUNCT
ejpam-5570	192	33	)	)	PUNCT
ejpam-5570	192	34	∈	∈	PROPN
ejpam-5570	192	35	i.	i.	NOUN
ejpam-5570	192	36	as	as	ADP
ejpam-5570	192	37	f	f	PROPN
ejpam-5570	192	38	−	−	PROPN
ejpam-5570	192	39	∪{wy	∪{wy	PROPN
ejpam-5570	192	40	:	:	PUNCT
ejpam-5570	193	1	y	y	PROPN
ejpam-5570	193	2	∈	∈	PROPN
ejpam-5570	193	3	f	f	X
ejpam-5570	193	4	}	}	PUNCT
ejpam-5570	193	5	=	=	SYM
ejpam-5570	193	6	f	f	X
ejpam-5570	193	7	−	−	PROPN
ejpam-5570	193	8	(	(	PUNCT
ejpam-5570	193	9	∪{wy	∪{wy	PROPN
ejpam-5570	193	10	:	:	PUNCT
ejpam-5570	193	11	y	y	PROPN
ejpam-5570	193	12	∈	∈	PROPN
ejpam-5570	193	13	f	f	X
ejpam-5570	193	14	}	}	PUNCT
ejpam-5570	193	15	∪	∪	ADJ
ejpam-5570	193	16	{	{	PUNCT
ejpam-5570	193	17	g	g	NOUN
ejpam-5570	193	18	}	}	PUNCT
ejpam-5570	193	19	)	)	PUNCT
ejpam-5570	194	1	⊂	⊂	PROPN
ejpam-5570	194	2	x	x	X
ejpam-5570	195	1	−	−	PROPN
ejpam-5570	195	2	(	(	PUNCT
ejpam-5570	195	3	∪{wy	∪{wy	PROPN
ejpam-5570	195	4	:	:	PUNCT
ejpam-5570	195	5	y	y	PROPN
ejpam-5570	195	6	∈	∈	PROPN
ejpam-5570	195	7	f	f	X
ejpam-5570	195	8	}	}	PUNCT
ejpam-5570	195	9	∪	∪	ADJ
ejpam-5570	195	10	{	{	PUNCT
ejpam-5570	195	11	g	g	NOUN
ejpam-5570	195	12	}	}	PUNCT
ejpam-5570	195	13	)	)	PUNCT
ejpam-5570	195	14	,	,	PUNCT
ejpam-5570	195	15	we	we	PRON
ejpam-5570	195	16	get	get	VERB
ejpam-5570	195	17	that	that	PRON
ejpam-5570	195	18	f	f	PROPN
ejpam-5570	196	1	−∪{wy	−∪{wy	NOUN
ejpam-5570	196	2	:	:	PUNCT
ejpam-5570	196	3	y	y	PROPN
ejpam-5570	196	4	∈	∈	PROPN
ejpam-5570	196	5	f	f	X
ejpam-5570	196	6	}	}	PUNCT
ejpam-5570	196	7	∈	∈	PROPN
ejpam-5570	196	8	i.	i.	NOUN
ejpam-5570	196	9	it	it	PRON
ejpam-5570	196	10	follows	follow	VERB
ejpam-5570	196	11	that	that	SCONJ
ejpam-5570	196	12	v	v	NOUN
ejpam-5570	196	13	=	=	SYM
ejpam-5570	196	14	∪{wy	∪{wy	PROPN
ejpam-5570	196	15	:	:	PUNCT
ejpam-5570	196	16	y	y	PROPN
ejpam-5570	196	17	∈	∈	PROPN
ejpam-5570	196	18	f	f	AUX
ejpam-5570	196	19	}	}	PUNCT
ejpam-5570	196	20	is	be	AUX
ejpam-5570	196	21	a	a	DET
ejpam-5570	196	22	δ	δ	PROPN
ejpam-5570	196	23	-	-	PUNCT
ejpam-5570	196	24	βi	βi	ADV
ejpam-5570	196	25	-	-	PUNCT
ejpam-5570	196	26	open	open	NOUN
ejpam-5570	196	27	set	set	NOUN
ejpam-5570	196	28	in	in	ADP
ejpam-5570	196	29	x	x	PUNCT
ejpam-5570	196	30	and	and	CCONJ
ejpam-5570	196	31	f	f	PROPN
ejpam-5570	196	32	−	−	PROPN
ejpam-5570	196	33	v	v	PROPN
ejpam-5570	196	34	∈	∈	PROPN
ejpam-5570	196	35	i.	i.	NOUN
ejpam-5570	196	36	given	give	VERB
ejpam-5570	196	37	that	that	SCONJ
ejpam-5570	196	38	x	x	PROPN
ejpam-5570	196	39	̸∈	̸∈	PROPN
ejpam-5570	196	40	cl(gy	cl(gy	PROPN
ejpam-5570	196	41	)	)	PUNCT
ejpam-5570	196	42	,	,	PUNCT
ejpam-5570	196	43	it	it	PRON
ejpam-5570	196	44	follows	follow	VERB
ejpam-5570	196	45	that	that	SCONJ
ejpam-5570	196	46	x	x	PROPN
ejpam-5570	196	47	̸∈	̸∈	PROPN
ejpam-5570	196	48	cl(wy	cl(wy	PROPN
ejpam-5570	196	49	)	)	PUNCT
ejpam-5570	196	50	,	,	PUNCT
ejpam-5570	196	51	and	and	CCONJ
ejpam-5570	196	52	consequently	consequently	ADV
ejpam-5570	196	53	,	,	PUNCT
ejpam-5570	196	54	x	x	PROPN
ejpam-5570	196	55	̸∈	̸∈	PROPN
ejpam-5570	196	56	δ	δ	PROPN
ejpam-5570	196	57	-	-	PROPN
ejpam-5570	196	58	βcli(wy	βcli(wy	PROPN
ejpam-5570	196	59	)	)	PUNCT
ejpam-5570	196	60	.	.	PUNCT
ejpam-5570	197	1	by	by	ADP
ejpam-5570	197	2	(	(	PUNCT
ejpam-5570	197	3	iii	iii	NOUN
ejpam-5570	197	4	)	)	PUNCT
ejpam-5570	197	5	,	,	PUNCT
ejpam-5570	197	6	the	the	DET
ejpam-5570	197	7	fact	fact	NOUN
ejpam-5570	197	8	that	that	SCONJ
ejpam-5570	197	9	w	w	NOUN
ejpam-5570	197	10	is	be	AUX
ejpam-5570	197	11	δ	δ	PROPN
ejpam-5570	197	12	-	-	PUNCT
ejpam-5570	197	13	βi	βi	PRON
ejpam-5570	197	14	-	-	PUNCT
ejpam-5570	197	15	locally	locally	ADV
ejpam-5570	197	16	finite	finite	NOUN
ejpam-5570	197	17	suggests	suggest	VERB
ejpam-5570	197	18	that	that	SCONJ
ejpam-5570	197	19	δ	δ	PROPN
ejpam-5570	197	20	-	-	PUNCT
ejpam-5570	197	21	βcli(∪{wy	βcli(∪{wy	PUNCT
ejpam-5570	197	22	:	:	PUNCT
ejpam-5570	197	23	y	y	PROPN
ejpam-5570	197	24	∈	∈	PROPN
ejpam-5570	197	25	f	f	X
ejpam-5570	197	26	}	}	PUNCT
ejpam-5570	197	27	)	)	PUNCT
ejpam-5570	197	28	=	=	PUNCT
ejpam-5570	198	1	∪{δ	∪{δ	NOUN
ejpam-5570	198	2	-	-	NOUN
ejpam-5570	198	3	βcli(wy	βcli(wy	PROPN
ejpam-5570	198	4	)	)	PUNCT
ejpam-5570	198	5	:	:	PUNCT
ejpam-5570	199	1	y	y	PROPN
ejpam-5570	199	2	∈	∈	PROPN
ejpam-5570	199	3	f	f	X
ejpam-5570	199	4	}	}	PUNCT
ejpam-5570	199	5	.	.	PUNCT
ejpam-5570	200	1	we	we	PRON
ejpam-5570	200	2	now	now	ADV
ejpam-5570	200	3	obtain	obtain	VERB
ejpam-5570	200	4	u	u	NOUN
ejpam-5570	200	5	∩v	∩v	NOUN
ejpam-5570	200	6	=	=	NOUN
ejpam-5570	200	7	∅	∅	NOUN
ejpam-5570	200	8	such	such	ADJ
ejpam-5570	200	9	that	that	SCONJ
ejpam-5570	200	10	x	x	SYM
ejpam-5570	200	11	∈	∈	PROPN
ejpam-5570	200	12	u	u	NOUN
ejpam-5570	200	13	for	for	ADP
ejpam-5570	200	14	a	a	DET
ejpam-5570	200	15	δ	δ	PROPN
ejpam-5570	200	16	-	-	PUNCT
ejpam-5570	200	17	βi	βi	ADV
ejpam-5570	200	18	-	-	PUNCT
ejpam-5570	200	19	open	open	ADJ
ejpam-5570	200	20	set	set	NOUN
ejpam-5570	200	21	u	u	NOUN
ejpam-5570	200	22	=	=	PROPN
ejpam-5570	200	23	x−δ	x−δ	PROPN
ejpam-5570	200	24	-	-	PUNCT
ejpam-5570	200	25	βcli(v	βcli(v	PROPN
ejpam-5570	200	26	)	)	PUNCT
ejpam-5570	200	27	.	.	PUNCT
ejpam-5570	201	1	therefore	therefore	ADV
ejpam-5570	201	2	,	,	PUNCT
ejpam-5570	201	3	(	(	PUNCT
ejpam-5570	201	4	x	x	X
ejpam-5570	201	5	,	,	PUNCT
ejpam-5570	201	6	τ	τ	PROPN
ejpam-5570	201	7	,	,	PUNCT
ejpam-5570	201	8	i	i	PROPN
ejpam-5570	201	9	)	)	PUNCT
ejpam-5570	201	10	is	be	AUX
ejpam-5570	201	11	δ	δ	PROPN
ejpam-5570	201	12	-	-	PUNCT
ejpam-5570	201	13	βi	βi	ADV
ejpam-5570	201	14	-	-	PUNCT
ejpam-5570	201	15	regular	regular	ADJ
ejpam-5570	201	16	.	.	PUNCT
ejpam-5570	202	1	theorem	theorem	VERB
ejpam-5570	202	2	8	8	NUM
ejpam-5570	202	3	.	.	PUNCT
ejpam-5570	203	1	if	if	SCONJ
ejpam-5570	203	2	an	an	DET
ejpam-5570	203	3	ideal	ideal	ADJ
ejpam-5570	203	4	topological	topological	ADJ
ejpam-5570	203	5	space	space	NOUN
ejpam-5570	203	6	(	(	PUNCT
ejpam-5570	203	7	x	x	X
ejpam-5570	203	8	,	,	PUNCT
ejpam-5570	203	9	τ	τ	PROPN
ejpam-5570	203	10	,	,	PUNCT
ejpam-5570	203	11	i	i	PROPN
ejpam-5570	203	12	)	)	PUNCT
ejpam-5570	203	13	is	be	AUX
ejpam-5570	203	14	δ	δ	PROPN
ejpam-5570	203	15	-	-	PUNCT
ejpam-5570	203	16	βi	βi	ADV
ejpam-5570	203	17	-	-	PUNCT
ejpam-5570	203	18	paracompact	paracompact	NOUN
ejpam-5570	203	19	and	and	CCONJ
ejpam-5570	203	20	regular	regular	ADJ
ejpam-5570	203	21	,	,	PUNCT
ejpam-5570	203	22	then	then	ADV
ejpam-5570	203	23	every	every	DET
ejpam-5570	203	24	open	open	ADJ
ejpam-5570	203	25	cover	cover	NOUN
ejpam-5570	203	26	of	of	ADP
ejpam-5570	203	27	x	x	PUNCT
ejpam-5570	203	28	has	have	VERB
ejpam-5570	203	29	a	a	DET
ejpam-5570	203	30	δ	δ	PROPN
ejpam-5570	203	31	-	-	PUNCT
ejpam-5570	203	32	βi	βi	ADV
ejpam-5570	203	33	-	-	PUNCT
ejpam-5570	203	34	locally	locally	ADV
ejpam-5570	203	35	finite	finite	NOUN
ejpam-5570	203	36	i	i	NOUN
ejpam-5570	203	37	-	-	PUNCT
ejpam-5570	203	38	cover	cover	NOUN
ejpam-5570	203	39	refinement	refinement	NOUN
ejpam-5570	203	40	of	of	ADP
ejpam-5570	203	41	closed	closed	ADJ
ejpam-5570	203	42	sets	set	NOUN
ejpam-5570	203	43	.	.	PUNCT
ejpam-5570	204	1	proof	proof	NOUN
ejpam-5570	204	2	.	.	PUNCT
ejpam-5570	205	1	let	let	VERB
ejpam-5570	205	2	u	u	PRON
ejpam-5570	205	3	be	be	AUX
ejpam-5570	205	4	an	an	DET
ejpam-5570	205	5	open	open	ADJ
ejpam-5570	205	6	cover	cover	NOUN
ejpam-5570	205	7	of	of	ADP
ejpam-5570	205	8	x.	x.	NOUN
ejpam-5570	205	9	by	by	ADP
ejpam-5570	205	10	regularity	regularity	NOUN
ejpam-5570	205	11	of	of	ADP
ejpam-5570	205	12	x	x	PRON
ejpam-5570	205	13	,	,	PUNCT
ejpam-5570	205	14	for	for	ADP
ejpam-5570	205	15	each	each	DET
ejpam-5570	205	16	x	x	SYM
ejpam-5570	205	17	∈	∈	PROPN
ejpam-5570	205	18	x	x	X
ejpam-5570	205	19	and	and	CCONJ
ejpam-5570	205	20	ux	ux	PROPN
ejpam-5570	205	21	∈	∈	PROPN
ejpam-5570	205	22	u	u	NOUN
ejpam-5570	205	23	containing	contain	VERB
ejpam-5570	205	24	x	x	PRON
ejpam-5570	205	25	,	,	PUNCT
ejpam-5570	205	26	there	there	PRON
ejpam-5570	205	27	exists	exist	VERB
ejpam-5570	205	28	an	an	DET
ejpam-5570	205	29	open	open	ADJ
ejpam-5570	205	30	set	set	NOUN
ejpam-5570	205	31	gx	gx	PROPN
ejpam-5570	205	32	of	of	ADP
ejpam-5570	205	33	x	x	INTJ
ejpam-5570	205	34	such	such	ADJ
ejpam-5570	205	35	that	that	DET
ejpam-5570	205	36	cl(gx	cl(gx	NOUN
ejpam-5570	205	37	)	)	PUNCT
ejpam-5570	206	1	⊂	⊂	PROPN
ejpam-5570	206	2	ux	ux	PROPN
ejpam-5570	206	3	.	.	PUNCT
ejpam-5570	206	4	thus	thus	ADV
ejpam-5570	206	5	u1	u1	NOUN
ejpam-5570	206	6	=	=	SYM
ejpam-5570	206	7	{	{	PUNCT
ejpam-5570	206	8	gx	gx	PROPN
ejpam-5570	206	9	:	:	PUNCT
ejpam-5570	206	10	x	x	SYM
ejpam-5570	206	11	∈	∈	PROPN
ejpam-5570	206	12	x	x	PRON
ejpam-5570	206	13	}	}	PUNCT
ejpam-5570	206	14	is	be	AUX
ejpam-5570	206	15	an	an	DET
ejpam-5570	206	16	open	open	ADJ
ejpam-5570	206	17	cover	cover	NOUN
ejpam-5570	206	18	of	of	ADP
ejpam-5570	206	19	x.	x.	NOUN
ejpam-5570	206	20	since	since	SCONJ
ejpam-5570	206	21	x	x	PROPN
ejpam-5570	206	22	is	be	AUX
ejpam-5570	206	23	δ	δ	PROPN
ejpam-5570	206	24	-	-	PUNCT
ejpam-5570	206	25	βi	βi	ADV
ejpam-5570	206	26	-	-	PUNCT
ejpam-5570	206	27	paracompact	paracompact	ADJ
ejpam-5570	206	28	,	,	PUNCT
ejpam-5570	206	29	u1	u1	NOUN
ejpam-5570	206	30	has	have	VERB
ejpam-5570	206	31	a	a	DET
ejpam-5570	206	32	δ	δ	PROPN
ejpam-5570	206	33	-	-	PUNCT
ejpam-5570	206	34	βi	βi	ADV
ejpam-5570	206	35	-	-	PUNCT
ejpam-5570	206	36	locally	locally	ADV
ejpam-5570	206	37	finite	finite	VERB
ejpam-5570	206	38	refinement	refinement	NOUN
ejpam-5570	206	39	v1	v1	PROPN
ejpam-5570	206	40	=	=	SYM
ejpam-5570	206	41	{	{	PUNCT
ejpam-5570	206	42	vλ	vλ	INTJ
ejpam-5570	206	43	:	:	PUNCT
ejpam-5570	206	44	λ	λ	PROPN
ejpam-5570	206	45	∈	∈	PROPN
ejpam-5570	206	46	λ	λ	PROPN
ejpam-5570	206	47	}	}	PUNCT
ejpam-5570	206	48	of	of	ADP
ejpam-5570	206	49	δ	δ	PROPN
ejpam-5570	206	50	-	-	PUNCT
ejpam-5570	206	51	βi	βi	ADV
ejpam-5570	206	52	-	-	PUNCT
ejpam-5570	206	53	open	open	ADJ
ejpam-5570	206	54	sets	set	VERB
ejpam-5570	206	55	such	such	ADJ
ejpam-5570	206	56	that	that	SCONJ
ejpam-5570	206	57	x	x	X
ejpam-5570	206	58	−	−	NOUN
ejpam-5570	206	59	∪{vλ	∪{vλ	NUM
ejpam-5570	206	60	:	:	PUNCT
ejpam-5570	206	61	λ	λ	PROPN
ejpam-5570	206	62	∈	∈	PROPN
ejpam-5570	206	63	λ	λ	PROPN
ejpam-5570	206	64	}	}	PUNCT
ejpam-5570	206	65	∈	∈	PROPN
ejpam-5570	206	66	i.	i.	NOUN
ejpam-5570	206	67	as	as	ADP
ejpam-5570	206	68	vλ	vλ	ADP
ejpam-5570	206	69	⊂	⊂	PROPN
ejpam-5570	206	70	δ	δ	PROPN
ejpam-5570	206	71	-	-	PUNCT
ejpam-5570	206	72	βcli(vλ	βcli(vλ	PROPN
ejpam-5570	206	73	)	)	PUNCT
ejpam-5570	206	74	and	and	CCONJ
ejpam-5570	206	75	i	i	PRON
ejpam-5570	206	76	is	be	AUX
ejpam-5570	206	77	an	an	DET
ejpam-5570	206	78	ideal	ideal	NOUN
ejpam-5570	206	79	,	,	PUNCT
ejpam-5570	206	80	x	x	PRON
ejpam-5570	206	81	−	−	PROPN
ejpam-5570	206	82	∪{δ	∪{δ	NOUN
ejpam-5570	206	83	-	-	PUNCT
ejpam-5570	206	84	βcli(vλ	βcli(vλ	NOUN
ejpam-5570	206	85	)	)	PUNCT
ejpam-5570	206	86	:	:	PUNCT
ejpam-5570	207	1	λ	λ	X
ejpam-5570	207	2	∈	∈	PROPN
ejpam-5570	207	3	λ	λ	PROPN
ejpam-5570	207	4	}	}	PUNCT
ejpam-5570	207	5	∈	∈	PROPN
ejpam-5570	207	6	i.	i.	NOUN
ejpam-5570	207	7	by	by	ADP
ejpam-5570	207	8	theorem	theorem	NOUN
ejpam-5570	207	9	4	4	NUM
ejpam-5570	207	10	,	,	PUNCT
ejpam-5570	207	11	v	v	NOUN
ejpam-5570	207	12	=	=	SYM
ejpam-5570	207	13	{	{	PUNCT
ejpam-5570	207	14	δ	δ	NOUN
ejpam-5570	207	15	-	-	PUNCT
ejpam-5570	207	16	βcli(vλ	βcli(vλ	PROPN
ejpam-5570	207	17	)	)	PUNCT
ejpam-5570	207	18	:	:	PUNCT
ejpam-5570	207	19	vλ	vλ	ADP
ejpam-5570	207	20	∈	∈	PROPN
ejpam-5570	207	21	v1	v1	NOUN
ejpam-5570	207	22	}	}	PUNCT
ejpam-5570	207	23	is	be	AUX
ejpam-5570	207	24	δ	δ	PROPN
ejpam-5570	207	25	-	-	PUNCT
ejpam-5570	207	26	βi	βi	ADV
ejpam-5570	207	27	-	-	PUNCT
ejpam-5570	207	28	locally	locally	ADV
ejpam-5570	207	29	finite	finite	NOUN
ejpam-5570	207	30	.	.	PUNCT
ejpam-5570	208	1	because	because	SCONJ
ejpam-5570	208	2	v1	v1	NOUN
ejpam-5570	208	3	refines	refine	VERB
ejpam-5570	208	4	u1	u1	NOUN
ejpam-5570	208	5	,	,	PUNCT
ejpam-5570	208	6	for	for	ADP
ejpam-5570	208	7	every	every	DET
ejpam-5570	208	8	λ	λ	PROPN
ejpam-5570	208	9	∈	∈	PROPN
ejpam-5570	208	10	λ	λ	PROPN
ejpam-5570	208	11	,	,	PUNCT
ejpam-5570	208	12	there	there	PRON
ejpam-5570	208	13	is	be	VERB
ejpam-5570	208	14	some	some	DET
ejpam-5570	208	15	gx	gx	PROPN
ejpam-5570	208	16	∈	∈	PROPN
ejpam-5570	208	17	u1	u1	NOUN
ejpam-5570	208	18	such	such	ADJ
ejpam-5570	208	19	that	that	SCONJ
ejpam-5570	208	20	vλ	vλ	PROPN
ejpam-5570	208	21	⊂	⊂	PROPN
ejpam-5570	208	22	gx	gx	PROPN
ejpam-5570	208	23	.	.	PUNCT
ejpam-5570	208	24	then	then	ADV
ejpam-5570	208	25	δ	δ	PROPN
ejpam-5570	208	26	-	-	PUNCT
ejpam-5570	208	27	βcli(vλ	βcli(vλ	PROPN
ejpam-5570	208	28	)	)	PUNCT
ejpam-5570	208	29	⊂	⊂	PROPN
ejpam-5570	208	30	cl(vλ	cl(vλ	PROPN
ejpam-5570	208	31	)	)	PUNCT
ejpam-5570	208	32	⊂	⊂	PROPN
ejpam-5570	208	33	cl(gx	cl(gx	PROPN
ejpam-5570	208	34	)	)	PUNCT
ejpam-5570	208	35	,	,	PUNCT
ejpam-5570	208	36	and	and	CCONJ
ejpam-5570	208	37	hence	hence	ADV
ejpam-5570	208	38	δ	δ	PROPN
ejpam-5570	208	39	-	-	PUNCT
ejpam-5570	208	40	βcli(vλ	βcli(vλ	PROPN
ejpam-5570	208	41	)	)	PUNCT
ejpam-5570	208	42	⊂	⊂	PROPN
ejpam-5570	208	43	ux	ux	PROPN
ejpam-5570	208	44	.	.	PUNCT
ejpam-5570	209	1	therefore	therefore	ADV
ejpam-5570	209	2	v	v	X
ejpam-5570	209	3	=	=	PUNCT
ejpam-5570	209	4	{	{	PUNCT
ejpam-5570	209	5	δ	δ	NOUN
ejpam-5570	209	6	-	-	PUNCT
ejpam-5570	209	7	βcli(vλ	βcli(vλ	PROPN
ejpam-5570	209	8	)	)	PUNCT
ejpam-5570	209	9	:	:	PUNCT
ejpam-5570	209	10	vλ	vλ	ADP
ejpam-5570	209	11	∈	∈	PROPN
ejpam-5570	209	12	v1	v1	NOUN
ejpam-5570	209	13	}	}	PUNCT
ejpam-5570	209	14	refines	refine	VERB
ejpam-5570	209	15	u	u	NOUN
ejpam-5570	209	16	,	,	PUNCT
ejpam-5570	209	17	and	and	CCONJ
ejpam-5570	209	18	hence	hence	ADV
ejpam-5570	209	19	v	v	NOUN
ejpam-5570	209	20	is	be	AUX
ejpam-5570	209	21	a	a	DET
ejpam-5570	209	22	δ	δ	PROPN
ejpam-5570	209	23	-	-	PUNCT
ejpam-5570	209	24	βi	βi	ADV
ejpam-5570	209	25	-	-	PUNCT
ejpam-5570	209	26	locally	locally	ADV
ejpam-5570	209	27	finite	finite	NOUN
ejpam-5570	209	28	i	i	NOUN
ejpam-5570	209	29	-	-	PUNCT
ejpam-5570	209	30	cover	cover	NOUN
ejpam-5570	209	31	refinement	refinement	NOUN
ejpam-5570	209	32	of	of	ADP
ejpam-5570	209	33	closed	closed	ADJ
ejpam-5570	209	34	sets	set	NOUN
ejpam-5570	209	35	.	.	PUNCT
ejpam-5570	210	1	theorem	theorem	NOUN
ejpam-5570	210	2	9	9	NUM
ejpam-5570	210	3	.	.	PUNCT
ejpam-5570	211	1	let	let	VERB
ejpam-5570	211	2	(	(	PUNCT
ejpam-5570	211	3	x	x	X
ejpam-5570	211	4	,	,	PUNCT
ejpam-5570	211	5	τ	τ	PROPN
ejpam-5570	211	6	,	,	PUNCT
ejpam-5570	211	7	i	i	PRON
ejpam-5570	211	8	)	)	PUNCT
ejpam-5570	211	9	be	be	VERB
ejpam-5570	211	10	an	an	DET
ejpam-5570	211	11	ideal	ideal	ADJ
ejpam-5570	211	12	topological	topological	ADJ
ejpam-5570	211	13	space	space	NOUN
ejpam-5570	211	14	.	.	PUNCT
ejpam-5570	212	1	the	the	DET
ejpam-5570	212	2	following	follow	VERB
ejpam-5570	212	3	statements	statement	NOUN
ejpam-5570	212	4	are	be	AUX
ejpam-5570	212	5	equivalent	equivalent	ADJ
ejpam-5570	212	6	:	:	PUNCT
ejpam-5570	212	7	(	(	PUNCT
ejpam-5570	212	8	i	i	NOUN
ejpam-5570	212	9	)	)	PUNCT
ejpam-5570	212	10	for	for	ADP
ejpam-5570	212	11	every	every	DET
ejpam-5570	212	12	closed	close	VERB
ejpam-5570	212	13	subset	subset	NOUN
ejpam-5570	212	14	f	f	PROPN
ejpam-5570	212	15	of	of	ADP
ejpam-5570	212	16	x	x	PUNCT
ejpam-5570	212	17	and	and	CCONJ
ejpam-5570	212	18	every	every	DET
ejpam-5570	212	19	x	x	X
ejpam-5570	212	20	̸∈	̸∈	PROPN
ejpam-5570	212	21	f	f	PROPN
ejpam-5570	212	22	,	,	PUNCT
ejpam-5570	212	23	there	there	PRON
ejpam-5570	212	24	exist	exist	VERB
ejpam-5570	212	25	disjoint	disjoint	PROPN
ejpam-5570	212	26	δ	δ	PROPN
ejpam-5570	212	27	-	-	PUNCT
ejpam-5570	212	28	βi	βi	ADV
ejpam-5570	212	29	-	-	PUNCT
ejpam-5570	212	30	open	open	ADJ
ejpam-5570	212	31	sets	set	VERB
ejpam-5570	212	32	u	u	NOUN
ejpam-5570	212	33	and	and	CCONJ
ejpam-5570	212	34	v	v	ADP
ejpam-5570	212	35	such	such	ADJ
ejpam-5570	212	36	that	that	SCONJ
ejpam-5570	212	37	x	x	SYM
ejpam-5570	212	38	∈	∈	PROPN
ejpam-5570	212	39	u	u	NOUN
ejpam-5570	212	40	and	and	CCONJ
ejpam-5570	212	41	f	f	PROPN
ejpam-5570	212	42	−	−	PROPN
ejpam-5570	212	43	v	v	PROPN
ejpam-5570	212	44	∈	∈	PROPN
ejpam-5570	212	45	i.	i.	NOUN
ejpam-5570	212	46	(	(	PUNCT
ejpam-5570	212	47	ii	ii	PROPN
ejpam-5570	212	48	)	)	PUNCT
ejpam-5570	212	49	for	for	ADP
ejpam-5570	212	50	every	every	DET
ejpam-5570	212	51	open	open	NOUN
ejpam-5570	212	52	subset	subset	NOUN
ejpam-5570	212	53	g	g	NOUN
ejpam-5570	212	54	of	of	ADP
ejpam-5570	212	55	x	x	PUNCT
ejpam-5570	212	56	and	and	CCONJ
ejpam-5570	212	57	every	every	DET
ejpam-5570	212	58	x	x	PROPN
ejpam-5570	212	59	∈	∈	PROPN
ejpam-5570	212	60	g	g	NOUN
ejpam-5570	212	61	,	,	PUNCT
ejpam-5570	212	62	there	there	PRON
ejpam-5570	212	63	exists	exist	VERB
ejpam-5570	212	64	a	a	DET
ejpam-5570	212	65	δ	δ	PROPN
ejpam-5570	212	66	-	-	PUNCT
ejpam-5570	212	67	βi	βi	ADV
ejpam-5570	212	68	-	-	PUNCT
ejpam-5570	212	69	open	open	NOUN
ejpam-5570	212	70	set	set	VERB
ejpam-5570	212	71	u	u	PRON
ejpam-5570	212	72	such	such	ADJ
ejpam-5570	212	73	that	that	SCONJ
ejpam-5570	212	74	x	x	SYM
ejpam-5570	212	75	∈	∈	PROPN
ejpam-5570	212	76	u	u	NOUN
ejpam-5570	212	77	and	and	CCONJ
ejpam-5570	212	78	δ	δ	PROPN
ejpam-5570	212	79	-	-	PUNCT
ejpam-5570	212	80	βcli(u)−g	βcli(u)−g	PROPN
ejpam-5570	212	81	∈	∈	PROPN
ejpam-5570	212	82	i.	i.	NOUN
ejpam-5570	212	83	proof	proof	NOUN
ejpam-5570	212	84	.	.	PUNCT
ejpam-5570	213	1	(	(	PUNCT
ejpam-5570	213	2	i	i	NOUN
ejpam-5570	213	3	)	)	PUNCT
ejpam-5570	213	4	⇒	⇒	PROPN
ejpam-5570	213	5	(	(	PUNCT
ejpam-5570	213	6	ii	ii	NOUN
ejpam-5570	213	7	)	)	PUNCT
ejpam-5570	213	8	letg	letg	NOUN
ejpam-5570	213	9	be	be	AUX
ejpam-5570	213	10	open	open	ADJ
ejpam-5570	213	11	and	and	CCONJ
ejpam-5570	213	12	x	x	SYM
ejpam-5570	213	13	∈	∈	PROPN
ejpam-5570	213	14	g.	g.	NOUN
ejpam-5570	213	15	then	then	ADV
ejpam-5570	213	16	,	,	PUNCT
ejpam-5570	213	17	x−g	x−g	PROPN
ejpam-5570	213	18	remains	remain	VERB
ejpam-5570	213	19	closed	closed	ADJ
ejpam-5570	213	20	,	,	PUNCT
ejpam-5570	213	21	and	and	CCONJ
ejpam-5570	213	22	x	x	PRON
ejpam-5570	213	23	does	do	AUX
ejpam-5570	213	24	not	not	PART
ejpam-5570	213	25	belong	belong	VERB
ejpam-5570	213	26	to	to	ADP
ejpam-5570	213	27	x	x	PART
ejpam-5570	213	28	−g	−g	NOUN
ejpam-5570	213	29	.	.	PUNCT
ejpam-5570	214	1	by	by	ADP
ejpam-5570	214	2	(	(	PUNCT
ejpam-5570	214	3	i	i	NOUN
ejpam-5570	214	4	)	)	PUNCT
ejpam-5570	214	5	,	,	PUNCT
ejpam-5570	214	6	there	there	PRON
ejpam-5570	214	7	exist	exist	VERB
ejpam-5570	214	8	disjoint	disjoint	PROPN
ejpam-5570	214	9	δ	δ	PROPN
ejpam-5570	214	10	-	-	PUNCT
ejpam-5570	214	11	βi	βi	ADV
ejpam-5570	214	12	-	-	PUNCT
ejpam-5570	214	13	open	open	ADJ
ejpam-5570	214	14	sets	set	VERB
ejpam-5570	214	15	u	u	NOUN
ejpam-5570	214	16	and	and	CCONJ
ejpam-5570	214	17	v	v	ADP
ejpam-5570	214	18	such	such	ADJ
ejpam-5570	214	19	that	that	SCONJ
ejpam-5570	214	20	x	x	SYM
ejpam-5570	214	21	∈	∈	PROPN
ejpam-5570	214	22	u	u	NOUN
ejpam-5570	214	23	and	and	CCONJ
ejpam-5570	214	24	(	(	PUNCT
ejpam-5570	214	25	x	x	PART
ejpam-5570	214	26	−g)−	−g)−	PROPN
ejpam-5570	214	27	v	v	ADP
ejpam-5570	214	28	∈	∈	PROPN
ejpam-5570	214	29	i.	i.	NOUN
ejpam-5570	214	30	as	as	ADP
ejpam-5570	214	31	u	u	PROPN
ejpam-5570	214	32	and	and	CCONJ
ejpam-5570	214	33	v	v	NOUN
ejpam-5570	214	34	are	be	AUX
ejpam-5570	214	35	disjoint	disjoint	ADJ
ejpam-5570	214	36	,	,	PUNCT
ejpam-5570	214	37	by	by	ADP
ejpam-5570	214	38	theorem	theorem	NOUN
ejpam-5570	214	39	3	3	NUM
ejpam-5570	214	40	,	,	PUNCT
ejpam-5570	214	41	we	we	PRON
ejpam-5570	214	42	have	have	VERB
ejpam-5570	214	43	δ	δ	NOUN
ejpam-5570	214	44	-	-	PUNCT
ejpam-5570	214	45	βcli(u	βcli(u	NOUN
ejpam-5570	214	46	)	)	PUNCT
ejpam-5570	215	1	⊂	⊂	PROPN
ejpam-5570	215	2	x	x	PUNCT
ejpam-5570	216	1	−	−	NOUN
ejpam-5570	216	2	v	v	NOUN
ejpam-5570	216	3	.	.	PUNCT
ejpam-5570	217	1	that	that	PRON
ejpam-5570	217	2	implies	imply	VERB
ejpam-5570	217	3	δ	δ	PROPN
ejpam-5570	217	4	-	-	PUNCT
ejpam-5570	217	5	βcli(u	βcli(u	NOUN
ejpam-5570	217	6	)	)	PUNCT
ejpam-5570	217	7	∩	∩	NOUN
ejpam-5570	217	8	(	(	PUNCT
ejpam-5570	217	9	x	x	SYM
ejpam-5570	217	10	−	−	PROPN
ejpam-5570	217	11	g	g	NOUN
ejpam-5570	217	12	)	)	PUNCT
ejpam-5570	218	1	⊂	⊂	PROPN
ejpam-5570	218	2	(	(	PUNCT
ejpam-5570	218	3	x	x	X
ejpam-5570	218	4	−	−	PROPN
ejpam-5570	218	5	g	g	NOUN
ejpam-5570	218	6	)	)	PUNCT
ejpam-5570	218	7	−	−	PROPN
ejpam-5570	218	8	v	v	NOUN
ejpam-5570	218	9	.	.	PUNCT
ejpam-5570	219	1	hence	hence	ADV
ejpam-5570	219	2	δ	δ	PROPN
ejpam-5570	219	3	-	-	PUNCT
ejpam-5570	219	4	βcli(u	βcli(u	NOUN
ejpam-5570	219	5	)	)	PUNCT
ejpam-5570	219	6	∩	∩	NOUN
ejpam-5570	219	7	(	(	PUNCT
ejpam-5570	219	8	x	x	SYM
ejpam-5570	219	9	−	−	PROPN
ejpam-5570	219	10	g	g	NOUN
ejpam-5570	219	11	)	)	PUNCT
ejpam-5570	219	12	=	=	SYM
ejpam-5570	219	13	δ	δ	PROPN
ejpam-5570	219	14	-	-	PUNCT
ejpam-5570	219	15	βcli(u)−g	βcli(u)−g	PROPN
ejpam-5570	219	16	∈	∈	PROPN
ejpam-5570	219	17	i.	i.	NOUN
ejpam-5570	219	18	(	(	PUNCT
ejpam-5570	219	19	ii	ii	NOUN
ejpam-5570	219	20	)	)	PUNCT
ejpam-5570	219	21	⇒	⇒	NOUN
ejpam-5570	219	22	(	(	PUNCT
ejpam-5570	219	23	i	i	NOUN
ejpam-5570	219	24	)	)	PUNCT
ejpam-5570	219	25	let	let	VERB
ejpam-5570	219	26	f	f	PRON
ejpam-5570	219	27	be	be	AUX
ejpam-5570	219	28	closed	close	VERB
ejpam-5570	219	29	and	and	CCONJ
ejpam-5570	219	30	x	x	PUNCT
ejpam-5570	219	31	̸∈	̸∈	PROPN
ejpam-5570	219	32	f	f	PROPN
ejpam-5570	219	33	.	.	PUNCT
ejpam-5570	220	1	it	it	PRON
ejpam-5570	220	2	implies	imply	VERB
ejpam-5570	220	3	that	that	SCONJ
ejpam-5570	220	4	x	x	PUNCT
ejpam-5570	221	1	−	−	NOUN
ejpam-5570	221	2	f	f	PROPN
ejpam-5570	221	3	is	be	AUX
ejpam-5570	221	4	open	open	ADJ
ejpam-5570	221	5	and	and	CCONJ
ejpam-5570	221	6	x	x	SYM
ejpam-5570	221	7	∈	∈	NOUN
ejpam-5570	221	8	x	x	X
ejpam-5570	222	1	−	−	PROPN
ejpam-5570	222	2	f	f	X
ejpam-5570	222	3	.	.	PUNCT
ejpam-5570	223	1	by	by	ADP
ejpam-5570	223	2	(	(	PUNCT
ejpam-5570	223	3	ii	ii	NOUN
ejpam-5570	223	4	)	)	PUNCT
ejpam-5570	223	5	,	,	PUNCT
ejpam-5570	223	6	there	there	PRON
ejpam-5570	223	7	exists	exist	VERB
ejpam-5570	223	8	a	a	DET
ejpam-5570	223	9	δ	δ	PROPN
ejpam-5570	223	10	-	-	PUNCT
ejpam-5570	223	11	βi	βi	ADV
ejpam-5570	223	12	-	-	PUNCT
ejpam-5570	223	13	open	open	NOUN
ejpam-5570	223	14	set	set	VERB
ejpam-5570	223	15	u	u	PRON
ejpam-5570	223	16	such	such	ADJ
ejpam-5570	223	17	that	that	SCONJ
ejpam-5570	223	18	x	x	SYM
ejpam-5570	223	19	∈	∈	PROPN
ejpam-5570	223	20	u	u	NOUN
ejpam-5570	223	21	and	and	CCONJ
ejpam-5570	223	22	δ	δ	PROPN
ejpam-5570	223	23	-	-	PUNCT
ejpam-5570	223	24	βcli(u	βcli(u	NOUN
ejpam-5570	223	25	)	)	PUNCT
ejpam-5570	223	26	−	−	PROPN
ejpam-5570	224	1	(	(	PUNCT
ejpam-5570	224	2	x	x	SYM
ejpam-5570	224	3	−	−	PROPN
ejpam-5570	224	4	f	f	X
ejpam-5570	224	5	)	)	PUNCT
ejpam-5570	224	6	∈	∈	PROPN
ejpam-5570	224	7	i.	i.	NOUN
ejpam-5570	224	8	hence	hence	ADV
ejpam-5570	224	9	,	,	PUNCT
ejpam-5570	224	10	v	v	NOUN
ejpam-5570	224	11	=	=	SYM
ejpam-5570	224	12	x	x	PUNCT
ejpam-5570	224	13	−	−	PROPN
ejpam-5570	224	14	δ	δ	PROPN
ejpam-5570	224	15	-	-	PUNCT
ejpam-5570	224	16	βcli(u	βcli(u	NOUN
ejpam-5570	224	17	)	)	PUNCT
ejpam-5570	224	18	is	be	AUX
ejpam-5570	224	19	a	a	DET
ejpam-5570	224	20	δ	δ	PROPN
ejpam-5570	224	21	-	-	PUNCT
ejpam-5570	224	22	βi	βi	ADV
ejpam-5570	224	23	-	-	PUNCT
ejpam-5570	224	24	open	open	ADJ
ejpam-5570	224	25	set	set	NOUN
ejpam-5570	224	26	,	,	PUNCT
ejpam-5570	224	27	and	and	CCONJ
ejpam-5570	224	28	u	u	NOUN
ejpam-5570	224	29	∩	∩	NOUN
ejpam-5570	224	30	v	v	NOUN
ejpam-5570	224	31	=	=	PUNCT
ejpam-5570	224	32	∅.	∅.	NOUN
ejpam-5570	224	33	that	that	PRON
ejpam-5570	224	34	is	be	AUX
ejpam-5570	224	35	f	f	PROPN
ejpam-5570	224	36	−	−	PROPN
ejpam-5570	224	37	v	v	NOUN
ejpam-5570	224	38	=	=	SYM
ejpam-5570	224	39	f	f	X
ejpam-5570	225	1	−	−	PROPN
ejpam-5570	225	2	(	(	PUNCT
ejpam-5570	225	3	x	x	NOUN
ejpam-5570	225	4	−	−	PROPN
ejpam-5570	225	5	δ	δ	PROPN
ejpam-5570	225	6	-	-	PUNCT
ejpam-5570	225	7	βcli(u	βcli(u	NOUN
ejpam-5570	225	8	)	)	PUNCT
ejpam-5570	225	9	)	)	PUNCT
ejpam-5570	226	1	=	=	PUNCT
ejpam-5570	226	2	δ	δ	PROPN
ejpam-5570	226	3	-	-	PUNCT
ejpam-5570	226	4	βcli(u)−	βcli(u)−	ADJ
ejpam-5570	226	5	(	(	PUNCT
ejpam-5570	226	6	x	x	SYM
ejpam-5570	226	7	−	−	PROPN
ejpam-5570	226	8	f	f	X
ejpam-5570	226	9	)	)	PUNCT
ejpam-5570	226	10	∈	∈	PROPN
ejpam-5570	226	11	i.	i.	NOUN
ejpam-5570	226	12	4	4	NUM
ejpam-5570	226	13	.	.	PUNCT
ejpam-5570	226	14	δ	δ	PROPN
ejpam-5570	226	15	-	-	PUNCT
ejpam-5570	226	16	βi	βi	PROPN
ejpam-5570	226	17	-	-	PUNCT
ejpam-5570	226	18	paracompactness	paracompactness	NOUN
ejpam-5570	226	19	of	of	ADP
ejpam-5570	226	20	subsets	subset	NOUN
ejpam-5570	226	21	in	in	ADP
ejpam-5570	226	22	the	the	DET
ejpam-5570	226	23	preceding	precede	VERB
ejpam-5570	226	24	section	section	NOUN
ejpam-5570	226	25	,	,	PUNCT
ejpam-5570	226	26	we	we	PRON
ejpam-5570	226	27	presented	present	VERB
ejpam-5570	226	28	the	the	DET
ejpam-5570	226	29	notion	notion	NOUN
ejpam-5570	226	30	of	of	ADP
ejpam-5570	226	31	δ	δ	PROPN
ejpam-5570	226	32	-	-	PUNCT
ejpam-5570	226	33	βi	βi	PROPN
ejpam-5570	226	34	-	-	NOUN
ejpam-5570	226	35	paracompactness	paracompactness	NOUN
ejpam-5570	226	36	for	for	ADP
ejpam-5570	226	37	subsets	subset	NOUN
ejpam-5570	226	38	of	of	ADP
ejpam-5570	226	39	an	an	DET
ejpam-5570	226	40	ideal	ideal	ADJ
ejpam-5570	226	41	topological	topological	ADJ
ejpam-5570	226	42	space	space	NOUN
ejpam-5570	226	43	.	.	PUNCT
ejpam-5570	227	1	prior	prior	ADV
ejpam-5570	227	2	to	to	ADP
ejpam-5570	227	3	delineating	delineate	VERB
ejpam-5570	227	4	the	the	DET
ejpam-5570	227	5	characterizations	characterization	NOUN
ejpam-5570	227	6	,	,	PUNCT
ejpam-5570	227	7	we	we	PRON
ejpam-5570	227	8	note	note	VERB
ejpam-5570	227	9	that	that	SCONJ
ejpam-5570	227	10	the	the	DET
ejpam-5570	227	11	c.	c.	PROPN
ejpam-5570	227	12	boonpok	boonpok	PROPN
ejpam-5570	227	13	,	,	PUNCT
ejpam-5570	227	14	a.	a.	PROPN
ejpam-5570	227	15	sama	sama	PROPN
ejpam-5570	227	16	-	-	PUNCT
ejpam-5570	227	17	ae	ae	PROPN
ejpam-5570	227	18	,	,	PUNCT
ejpam-5570	227	19	p.	p.	NOUN
ejpam-5570	227	20	raktaow	raktaow	PROPN
ejpam-5570	227	21	/	/	SYM
ejpam-5570	227	22	eur	eur	PROPN
ejpam-5570	227	23	.	.	PUNCT
ejpam-5570	228	1	j.	j.	PROPN
ejpam-5570	228	2	pure	pure	PROPN
ejpam-5570	228	3	appl	appl	PROPN
ejpam-5570	228	4	.	.	PROPN
ejpam-5570	228	5	math	math	PROPN
ejpam-5570	228	6	,	,	PUNCT
ejpam-5570	228	7	18	18	NUM
ejpam-5570	228	8	(	(	PUNCT
ejpam-5570	228	9	1	1	NUM
ejpam-5570	228	10	)	)	PUNCT
ejpam-5570	228	11	(	(	PUNCT
ejpam-5570	228	12	2025	2025	NUM
ejpam-5570	228	13	)	)	PUNCT
ejpam-5570	228	14	,	,	PUNCT
ejpam-5570	228	15	5570	5570	NUM
ejpam-5570	228	16	8	8	NUM
ejpam-5570	228	17	of	of	ADP
ejpam-5570	228	18	12	12	NUM
ejpam-5570	228	19	union	union	NOUN
ejpam-5570	228	20	of	of	ADP
ejpam-5570	228	21	a	a	DET
ejpam-5570	228	22	finite	finite	ADJ
ejpam-5570	228	23	family	family	NOUN
ejpam-5570	228	24	of	of	ADP
ejpam-5570	228	25	δ	δ	PROPN
ejpam-5570	228	26	-	-	PUNCT
ejpam-5570	228	27	βi	βi	ADV
ejpam-5570	228	28	-	-	PUNCT
ejpam-5570	228	29	locally	locally	ADV
ejpam-5570	228	30	finite	finite	ADJ
ejpam-5570	228	31	collections	collection	NOUN
ejpam-5570	228	32	of	of	ADP
ejpam-5570	228	33	sets	set	NOUN
ejpam-5570	228	34	inside	inside	ADP
ejpam-5570	228	35	an	an	DET
ejpam-5570	228	36	ideal	ideal	ADJ
ejpam-5570	228	37	topological	topological	ADJ
ejpam-5570	228	38	space	space	NOUN
ejpam-5570	228	39	remains	remain	VERB
ejpam-5570	228	40	δ	δ	PROPN
ejpam-5570	228	41	-	-	PUNCT
ejpam-5570	228	42	βi	βi	ADV
ejpam-5570	228	43	-	-	PUNCT
ejpam-5570	228	44	locally	locally	ADV
ejpam-5570	228	45	finite	finite	X
ejpam-5570	228	46	.	.	PUNCT
ejpam-5570	229	1	theorem	theorem	ADJ
ejpam-5570	229	2	10	10	NUM
ejpam-5570	229	3	.	.	PUNCT
ejpam-5570	230	1	let	let	VERB
ejpam-5570	230	2	a	a	PRON
ejpam-5570	230	3	and	and	CCONJ
ejpam-5570	230	4	b	b	NOUN
ejpam-5570	230	5	be	be	AUX
ejpam-5570	230	6	subsets	subset	NOUN
ejpam-5570	230	7	of	of	ADP
ejpam-5570	230	8	an	an	DET
ejpam-5570	230	9	ideal	ideal	ADJ
ejpam-5570	230	10	topological	topological	ADJ
ejpam-5570	230	11	space	space	NOUN
ejpam-5570	230	12	(	(	PUNCT
ejpam-5570	230	13	x	x	X
ejpam-5570	230	14	,	,	PUNCT
ejpam-5570	230	15	τ	τ	PROPN
ejpam-5570	230	16	,	,	PUNCT
ejpam-5570	230	17	i	i	PROPN
ejpam-5570	230	18	)	)	PUNCT
ejpam-5570	230	19	.	.	PUNCT
ejpam-5570	231	1	if	if	SCONJ
ejpam-5570	231	2	a	a	PRON
ejpam-5570	231	3	and	and	CCONJ
ejpam-5570	231	4	b	b	NOUN
ejpam-5570	231	5	are	be	AUX
ejpam-5570	231	6	δ	δ	PROPN
ejpam-5570	231	7	-	-	PUNCT
ejpam-5570	231	8	βi	βi	ADV
ejpam-5570	231	9	-	-	NOUN
ejpam-5570	231	10	paracompact	paracompact	NOUN
ejpam-5570	231	11	in	in	ADP
ejpam-5570	231	12	x	x	PRON
ejpam-5570	231	13	,	,	PUNCT
ejpam-5570	231	14	then	then	ADV
ejpam-5570	231	15	a	a	DET
ejpam-5570	231	16	∪b	∪b	PRON
ejpam-5570	231	17	is	be	AUX
ejpam-5570	231	18	also	also	ADV
ejpam-5570	231	19	δ	δ	PROPN
ejpam-5570	231	20	-	-	PUNCT
ejpam-5570	231	21	βi	βi	ADV
ejpam-5570	231	22	-	-	PUNCT
ejpam-5570	231	23	paracompact	paracompact	ADJ
ejpam-5570	231	24	.	.	PUNCT
ejpam-5570	232	1	proof	proof	NOUN
ejpam-5570	232	2	.	.	PUNCT
ejpam-5570	233	1	let	let	VERB
ejpam-5570	233	2	u	u	PRON
ejpam-5570	233	3	=	=	PUNCT
ejpam-5570	233	4	{	{	PUNCT
ejpam-5570	233	5	uλ	uλ	X
ejpam-5570	233	6	:	:	PUNCT
ejpam-5570	233	7	λ	λ	X
ejpam-5570	233	8	∈	∈	PROPN
ejpam-5570	233	9	λ	λ	PROPN
ejpam-5570	233	10	}	}	PUNCT
ejpam-5570	233	11	be	be	VERB
ejpam-5570	233	12	an	an	DET
ejpam-5570	233	13	open	open	ADJ
ejpam-5570	233	14	cover	cover	NOUN
ejpam-5570	233	15	of	of	ADP
ejpam-5570	233	16	a	a	DET
ejpam-5570	233	17	∪	∪	X
ejpam-5570	233	18	b.	b.	NOUN
ejpam-5570	233	19	then	then	ADV
ejpam-5570	233	20	u	u	NOUN
ejpam-5570	233	21	is	be	AUX
ejpam-5570	233	22	an	an	DET
ejpam-5570	233	23	open	open	ADJ
ejpam-5570	233	24	cover	cover	NOUN
ejpam-5570	233	25	of	of	ADP
ejpam-5570	233	26	a	a	PRON
ejpam-5570	233	27	and	and	CCONJ
ejpam-5570	233	28	b.	b.	PROPN
ejpam-5570	233	29	hence	hence	ADV
ejpam-5570	233	30	there	there	PRON
ejpam-5570	233	31	are	be	VERB
ejpam-5570	233	32	δ	δ	PROPN
ejpam-5570	233	33	-	-	PUNCT
ejpam-5570	233	34	βi	βi	ADV
ejpam-5570	233	35	-	-	PUNCT
ejpam-5570	233	36	locally	locally	ADV
ejpam-5570	233	37	finite	finite	PROPN
ejpam-5570	233	38	δ	δ	PROPN
ejpam-5570	233	39	-	-	ADJ
ejpam-5570	233	40	βi	βi	ADV
ejpam-5570	233	41	-	-	PUNCT
ejpam-5570	233	42	open	open	ADJ
ejpam-5570	233	43	families	family	NOUN
ejpam-5570	233	44	v	v	NOUN
ejpam-5570	233	45	=	=	SYM
ejpam-5570	233	46	{	{	PUNCT
ejpam-5570	233	47	vα	vα	X
ejpam-5570	233	48	:	:	PUNCT
ejpam-5570	233	49	α	α	PROPN
ejpam-5570	233	50	∈	∈	PROPN
ejpam-5570	233	51	λ1	λ1	PROPN
ejpam-5570	233	52	}	}	PUNCT
ejpam-5570	233	53	of	of	ADP
ejpam-5570	233	54	a	a	PRON
ejpam-5570	233	55	and	and	CCONJ
ejpam-5570	233	56	w	w	NOUN
ejpam-5570	233	57	=	=	PUNCT
ejpam-5570	233	58	{	{	PUNCT
ejpam-5570	233	59	wµ	wµ	NOUN
ejpam-5570	233	60	:	:	PUNCT
ejpam-5570	233	61	µ	µ	X
ejpam-5570	233	62	∈	∈	PROPN
ejpam-5570	233	63	λ2	λ2	PROPN
ejpam-5570	233	64	}	}	PUNCT
ejpam-5570	233	65	of	of	ADP
ejpam-5570	233	66	b	b	NUM
ejpam-5570	233	67	which	which	PRON
ejpam-5570	233	68	refine	refine	VERB
ejpam-5570	233	69	u	u	PRON
ejpam-5570	233	70	such	such	ADJ
ejpam-5570	233	71	that	that	SCONJ
ejpam-5570	233	72	a	a	DET
ejpam-5570	233	73	−	−	NOUN
ejpam-5570	233	74	∪{vα	∪{vα	NUM
ejpam-5570	233	75	:	:	PUNCT
ejpam-5570	233	76	α	α	PROPN
ejpam-5570	233	77	∈	∈	PROPN
ejpam-5570	233	78	λ1	λ1	PROPN
ejpam-5570	233	79	}	}	PUNCT
ejpam-5570	233	80	∈	∈	PROPN
ejpam-5570	233	81	i	i	PROPN
ejpam-5570	233	82	and	and	CCONJ
ejpam-5570	233	83	b	b	NOUN
ejpam-5570	234	1	−	−	X
ejpam-5570	234	2	∪{wµ	∪{wµ	NOUN
ejpam-5570	234	3	:	:	PUNCT
ejpam-5570	234	4	µ	µ	PROPN
ejpam-5570	234	5	∈	∈	PROPN
ejpam-5570	234	6	λ2	λ2	PROPN
ejpam-5570	234	7	}	}	PUNCT
ejpam-5570	234	8	∈	∈	PROPN
ejpam-5570	234	9	i.	i.	NOUN
ejpam-5570	234	10	it	it	PRON
ejpam-5570	234	11	implies	imply	VERB
ejpam-5570	234	12	that	that	SCONJ
ejpam-5570	234	13	a	a	DET
ejpam-5570	234	14	−	−	NOUN
ejpam-5570	234	15	∪{vα	∪{vα	NUM
ejpam-5570	234	16	:	:	PUNCT
ejpam-5570	234	17	α	α	PROPN
ejpam-5570	234	18	∈	∈	PROPN
ejpam-5570	234	19	λ1	λ1	PROPN
ejpam-5570	234	20	}	}	PUNCT
ejpam-5570	234	21	=	=	SYM
ejpam-5570	234	22	i1	i1	PROPN
ejpam-5570	234	23	and	and	CCONJ
ejpam-5570	234	24	b	b	NOUN
ejpam-5570	234	25	−	−	X
ejpam-5570	234	26	∪{wµ	∪{wµ	NOUN
ejpam-5570	234	27	:	:	PUNCT
ejpam-5570	234	28	µ	µ	PROPN
ejpam-5570	234	29	∈	∈	PROPN
ejpam-5570	234	30	λ2	λ2	NOUN
ejpam-5570	234	31	}	}	PUNCT
ejpam-5570	234	32	=	=	SYM
ejpam-5570	234	33	i2	i2	PROPN
ejpam-5570	234	34	,	,	PUNCT
ejpam-5570	234	35	where	where	SCONJ
ejpam-5570	234	36	i1	i1	PROPN
ejpam-5570	234	37	,	,	PUNCT
ejpam-5570	234	38	i2	i2	PROPN
ejpam-5570	234	39	∈	∈	PROPN
ejpam-5570	234	40	i.	i.	NOUN
ejpam-5570	234	41	therefore	therefore	ADV
ejpam-5570	234	42	,	,	PUNCT
ejpam-5570	234	43	a∪b	a∪b	NOUN
ejpam-5570	234	44	=	=	X
ejpam-5570	234	45	∪{vα	∪{vα	NOUN
ejpam-5570	234	46	:	:	PUNCT
ejpam-5570	234	47	α	α	PROPN
ejpam-5570	234	48	∈	∈	PROPN
ejpam-5570	234	49	λ1}∪{wµ	λ1}∪{wµ	NOUN
ejpam-5570	234	50	:	:	PUNCT
ejpam-5570	234	51	µ	µ	X
ejpam-5570	234	52	∈	∈	ADJ
ejpam-5570	234	53	λ2}∪	λ2}∪	NUM
ejpam-5570	234	54	(	(	PUNCT
ejpam-5570	234	55	i1∪i2	i1∪i2	NOUN
ejpam-5570	234	56	)	)	PUNCT
ejpam-5570	234	57	.	.	PUNCT
ejpam-5570	235	1	it	it	PRON
ejpam-5570	235	2	follows	follow	VERB
ejpam-5570	235	3	that	that	SCONJ
ejpam-5570	235	4	a	a	DET
ejpam-5570	235	5	∪	∪	NOUN
ejpam-5570	235	6	b	b	NOUN
ejpam-5570	235	7	−	−	NOUN
ejpam-5570	235	8	∪{vα	∪{vα	NOUN
ejpam-5570	235	9	∪	∪	NOUN
ejpam-5570	235	10	wµ	wµ	ADP
ejpam-5570	235	11	:	:	PUNCT
ejpam-5570	235	12	α	α	PROPN
ejpam-5570	235	13	∈	∈	PROPN
ejpam-5570	235	14	λ1	λ1	PROPN
ejpam-5570	235	15	,	,	PUNCT
ejpam-5570	235	16	µ	µ	PROPN
ejpam-5570	235	17	∈	∈	PROPN
ejpam-5570	235	18	λ2	λ2	PROPN
ejpam-5570	235	19	}	}	PUNCT
ejpam-5570	235	20	∈	∈	PROPN
ejpam-5570	235	21	i.	i.	NOUN
ejpam-5570	235	22	we	we	PRON
ejpam-5570	235	23	see	see	VERB
ejpam-5570	235	24	that	that	SCONJ
ejpam-5570	235	25	the	the	DET
ejpam-5570	235	26	collection	collection	NOUN
ejpam-5570	235	27	h	h	NOUN
ejpam-5570	235	28	=	=	PRON
ejpam-5570	235	29	{	{	PUNCT
ejpam-5570	235	30	vα	vα	X
ejpam-5570	235	31	∪wµ	∪wµ	PROPN
ejpam-5570	235	32	:	:	PUNCT
ejpam-5570	235	33	α	α	PROPN
ejpam-5570	235	34	∈	∈	PROPN
ejpam-5570	235	35	λ1	λ1	PROPN
ejpam-5570	235	36	,	,	PUNCT
ejpam-5570	235	37	µ	µ	PROPN
ejpam-5570	235	38	∈	∈	PROPN
ejpam-5570	235	39	λ2	λ2	NOUN
ejpam-5570	235	40	}	}	PUNCT
ejpam-5570	235	41	of	of	ADP
ejpam-5570	235	42	δ	δ	PROPN
ejpam-5570	235	43	-	-	PUNCT
ejpam-5570	235	44	βi	βi	PRON
ejpam-5570	235	45	-	-	PUNCT
ejpam-5570	235	46	open	open	ADJ
ejpam-5570	235	47	sets	set	NOUN
ejpam-5570	235	48	is	be	AUX
ejpam-5570	235	49	a	a	DET
ejpam-5570	235	50	δ	δ	PROPN
ejpam-5570	235	51	-	-	PUNCT
ejpam-5570	235	52	βi	βi	ADV
ejpam-5570	235	53	-	-	PUNCT
ejpam-5570	235	54	locally	locally	ADV
ejpam-5570	235	55	finite	finite	NOUN
ejpam-5570	235	56	and	and	CCONJ
ejpam-5570	235	57	h	h	NOUN
ejpam-5570	235	58	refines	refine	VERB
ejpam-5570	235	59	u	u	PRON
ejpam-5570	235	60	.	.	PUNCT
ejpam-5570	236	1	consequently	consequently	ADV
ejpam-5570	236	2	,	,	PUNCT
ejpam-5570	236	3	a	a	DET
ejpam-5570	236	4	∪b	∪b	PRON
ejpam-5570	236	5	is	be	AUX
ejpam-5570	236	6	δ	δ	PROPN
ejpam-5570	236	7	-	-	PUNCT
ejpam-5570	236	8	βi	βi	ADV
ejpam-5570	236	9	-	-	PUNCT
ejpam-5570	236	10	paracompact	paracompact	NOUN
ejpam-5570	236	11	.	.	PUNCT
ejpam-5570	237	1	theorem	theorem	NOUN
ejpam-5570	237	2	11	11	NUM
ejpam-5570	237	3	.	.	PUNCT
ejpam-5570	238	1	let	let	VERB
ejpam-5570	238	2	(	(	PUNCT
ejpam-5570	238	3	x	x	X
ejpam-5570	238	4	,	,	PUNCT
ejpam-5570	238	5	τ	τ	PROPN
ejpam-5570	238	6	,	,	PUNCT
ejpam-5570	238	7	i	i	PRON
ejpam-5570	238	8	)	)	PUNCT
ejpam-5570	238	9	be	be	VERB
ejpam-5570	238	10	an	an	DET
ejpam-5570	238	11	ideal	ideal	ADJ
ejpam-5570	238	12	topological	topological	ADJ
ejpam-5570	238	13	space	space	NOUN
ejpam-5570	238	14	.	.	PUNCT
ejpam-5570	239	1	if	if	SCONJ
ejpam-5570	239	2	a	a	PRON
ejpam-5570	239	3	is	be	AUX
ejpam-5570	239	4	δ	δ	PROPN
ejpam-5570	239	5	-	-	PUNCT
ejpam-5570	239	6	βi	βi	ADV
ejpam-5570	239	7	-	-	PUNCT
ejpam-5570	239	8	paracompact	paracompact	NOUN
ejpam-5570	239	9	and	and	CCONJ
ejpam-5570	239	10	b	b	NOUN
ejpam-5570	239	11	is	be	AUX
ejpam-5570	239	12	closed	close	VERB
ejpam-5570	239	13	in	in	ADP
ejpam-5570	239	14	x	x	NOUN
ejpam-5570	239	15	,	,	PUNCT
ejpam-5570	239	16	then	then	ADV
ejpam-5570	239	17	a	a	DET
ejpam-5570	239	18	∩b	∩b	NOUN
ejpam-5570	239	19	is	be	AUX
ejpam-5570	239	20	δ	δ	PROPN
ejpam-5570	239	21	-	-	PUNCT
ejpam-5570	239	22	βi	βi	ADV
ejpam-5570	239	23	-	-	PUNCT
ejpam-5570	239	24	paracompact	paracompact	ADJ
ejpam-5570	239	25	.	.	PUNCT
ejpam-5570	240	1	proof	proof	NOUN
ejpam-5570	240	2	.	.	PUNCT
ejpam-5570	241	1	let	let	VERB
ejpam-5570	241	2	u	u	PRON
ejpam-5570	241	3	=	=	PUNCT
ejpam-5570	241	4	{	{	PUNCT
ejpam-5570	241	5	uλ	uλ	X
ejpam-5570	241	6	:	:	PUNCT
ejpam-5570	241	7	λ	λ	X
ejpam-5570	241	8	∈	∈	PROPN
ejpam-5570	241	9	λ	λ	PROPN
ejpam-5570	241	10	}	}	PUNCT
ejpam-5570	241	11	be	be	VERB
ejpam-5570	241	12	an	an	DET
ejpam-5570	241	13	open	open	ADJ
ejpam-5570	241	14	cover	cover	NOUN
ejpam-5570	241	15	of	of	ADP
ejpam-5570	241	16	a	a	DET
ejpam-5570	241	17	∩	∩	ADJ
ejpam-5570	241	18	b.	b.	NOUN
ejpam-5570	241	19	as	as	SCONJ
ejpam-5570	241	20	x	x	X
ejpam-5570	241	21	−	−	PROPN
ejpam-5570	241	22	b	b	NOUN
ejpam-5570	241	23	is	be	AUX
ejpam-5570	241	24	open	open	ADJ
ejpam-5570	241	25	in	in	ADP
ejpam-5570	241	26	x	x	NOUN
ejpam-5570	241	27	,	,	PUNCT
ejpam-5570	241	28	u1	u1	NOUN
ejpam-5570	241	29	=	=	SYM
ejpam-5570	241	30	{	{	PUNCT
ejpam-5570	241	31	uλ	uλ	X
ejpam-5570	241	32	:	:	PUNCT
ejpam-5570	241	33	λ	λ	X
ejpam-5570	241	34	∈	∈	PROPN
ejpam-5570	241	35	λ	λ	PROPN
ejpam-5570	241	36	}	}	PUNCT
ejpam-5570	241	37	∪	∪	NOUN
ejpam-5570	241	38	{	{	PUNCT
ejpam-5570	241	39	x	x	NOUN
ejpam-5570	241	40	−	−	PROPN
ejpam-5570	241	41	b	b	X
ejpam-5570	241	42	}	}	PUNCT
ejpam-5570	241	43	is	be	AUX
ejpam-5570	241	44	an	an	DET
ejpam-5570	241	45	open	open	ADJ
ejpam-5570	241	46	cover	cover	NOUN
ejpam-5570	241	47	of	of	ADP
ejpam-5570	241	48	a.	a.	NOUN
ejpam-5570	241	49	by	by	ADP
ejpam-5570	241	50	assumption	assumption	NOUN
ejpam-5570	241	51	and	and	CCONJ
ejpam-5570	241	52	lemma	lemma	PROPN
ejpam-5570	241	53	5	5	NUM
ejpam-5570	241	54	,	,	PUNCT
ejpam-5570	241	55	u1	u1	NOUN
ejpam-5570	241	56	has	have	VERB
ejpam-5570	241	57	a	a	DET
ejpam-5570	241	58	δ	δ	PROPN
ejpam-5570	241	59	-	-	PUNCT
ejpam-5570	241	60	βi	βi	ADV
ejpam-5570	241	61	-	-	PUNCT
ejpam-5570	241	62	locally	locally	ADV
ejpam-5570	241	63	finite	finite	VERB
ejpam-5570	241	64	precise	precise	PROPN
ejpam-5570	241	65	δ	δ	PROPN
ejpam-5570	241	66	-	-	PUNCT
ejpam-5570	241	67	βi	βi	ADV
ejpam-5570	241	68	-	-	PUNCT
ejpam-5570	241	69	open	open	ADJ
ejpam-5570	241	70	refinement	refinement	NOUN
ejpam-5570	241	71	v	v	NOUN
ejpam-5570	241	72	=	=	PUNCT
ejpam-5570	241	73	{	{	PUNCT
ejpam-5570	241	74	vλ	vλ	INTJ
ejpam-5570	241	75	:	:	PUNCT
ejpam-5570	241	76	λ	λ	PROPN
ejpam-5570	241	77	∈	∈	PROPN
ejpam-5570	241	78	λ	λ	PROPN
ejpam-5570	241	79	}	}	PUNCT
ejpam-5570	241	80	∪	∪	ADJ
ejpam-5570	241	81	{	{	PUNCT
ejpam-5570	241	82	v	v	NOUN
ejpam-5570	241	83	}	}	PUNCT
ejpam-5570	241	84	such	such	ADJ
ejpam-5570	241	85	that	that	SCONJ
ejpam-5570	241	86	vλ	vλ	ADP
ejpam-5570	241	87	⊂	⊂	PROPN
ejpam-5570	241	88	uλ	uλ	NOUN
ejpam-5570	241	89	for	for	ADP
ejpam-5570	241	90	every	every	DET
ejpam-5570	241	91	λ	λ	PROPN
ejpam-5570	241	92	∈	∈	PROPN
ejpam-5570	241	93	λ	λ	PROPN
ejpam-5570	241	94	,	,	PUNCT
ejpam-5570	241	95	and	and	CCONJ
ejpam-5570	241	96	v	v	ADP
ejpam-5570	241	97	⊂	⊂	PROPN
ejpam-5570	241	98	x	x	PUNCT
ejpam-5570	241	99	−b	−b	ADP
ejpam-5570	241	100	such	such	ADJ
ejpam-5570	241	101	that	that	DET
ejpam-5570	241	102	a−∪({vλ	a−∪({vλ	NOUN
ejpam-5570	241	103	:	:	PUNCT
ejpam-5570	242	1	λ	λ	X
ejpam-5570	242	2	∈	∈	PROPN
ejpam-5570	242	3	λ}∪	λ}∪	X
ejpam-5570	242	4	{	{	PUNCT
ejpam-5570	242	5	v	v	NOUN
ejpam-5570	242	6	}	}	PUNCT
ejpam-5570	242	7	)	)	PUNCT
ejpam-5570	242	8	∈	∈	PROPN
ejpam-5570	242	9	i.	i.	NOUN
ejpam-5570	242	10	since	since	SCONJ
ejpam-5570	242	11	a∩b−∪{vλ	a∩b−∪{vλ	PROPN
ejpam-5570	242	12	:	:	PUNCT
ejpam-5570	242	13	λ	λ	PROPN
ejpam-5570	242	14	∈	∈	PROPN
ejpam-5570	242	15	λ	λ	X
ejpam-5570	242	16	}	}	PUNCT
ejpam-5570	242	17	=	=	PRON
ejpam-5570	242	18	a∩b−∪({vλ	a∩b−∪({vλ	NOUN
ejpam-5570	242	19	:	:	PUNCT
ejpam-5570	242	20	λ	λ	PROPN
ejpam-5570	242	21	∈	∈	PROPN
ejpam-5570	242	22	λ}∪{v	λ}∪{v	NOUN
ejpam-5570	242	23	}	}	PUNCT
ejpam-5570	242	24	)	)	PUNCT
ejpam-5570	243	1	⊂	⊂	PROPN
ejpam-5570	243	2	a−∪({vλ	a−∪({vλ	NOUN
ejpam-5570	243	3	:	:	PUNCT
ejpam-5570	243	4	λ	λ	PROPN
ejpam-5570	243	5	∈	∈	PROPN
ejpam-5570	243	6	λ}∪{v	λ}∪{v	NOUN
ejpam-5570	243	7	}	}	PUNCT
ejpam-5570	243	8	)	)	PUNCT
ejpam-5570	243	9	,	,	PUNCT
ejpam-5570	243	10	we	we	PRON
ejpam-5570	243	11	have	have	VERB
ejpam-5570	243	12	that	that	DET
ejpam-5570	243	13	a∩b−∪{vλ	a∩b−∪{vλ	NOUN
ejpam-5570	243	14	:	:	PUNCT
ejpam-5570	243	15	λ	λ	PROPN
ejpam-5570	243	16	∈	∈	PROPN
ejpam-5570	243	17	λ	λ	PROPN
ejpam-5570	243	18	}	}	PUNCT
ejpam-5570	243	19	∈	∈	PROPN
ejpam-5570	243	20	i.	i.	NOUN
ejpam-5570	243	21	it	it	PRON
ejpam-5570	243	22	is	be	AUX
ejpam-5570	243	23	obvious	obvious	ADJ
ejpam-5570	243	24	that	that	SCONJ
ejpam-5570	243	25	the	the	DET
ejpam-5570	243	26	collection	collection	NOUN
ejpam-5570	243	27	v1	v1	NOUN
ejpam-5570	243	28	=	=	SYM
ejpam-5570	243	29	{	{	PUNCT
ejpam-5570	243	30	vλ	vλ	INTJ
ejpam-5570	243	31	:	:	PUNCT
ejpam-5570	243	32	λ	λ	PROPN
ejpam-5570	243	33	∈	∈	PROPN
ejpam-5570	243	34	λ	λ	PROPN
ejpam-5570	243	35	}	}	PUNCT
ejpam-5570	243	36	of	of	ADP
ejpam-5570	243	37	δ	δ	PROPN
ejpam-5570	243	38	-	-	PUNCT
ejpam-5570	243	39	βi	βi	PRON
ejpam-5570	243	40	-	-	PUNCT
ejpam-5570	243	41	open	open	ADJ
ejpam-5570	243	42	sets	set	NOUN
ejpam-5570	243	43	is	be	AUX
ejpam-5570	243	44	a	a	DET
ejpam-5570	243	45	δ	δ	PROPN
ejpam-5570	243	46	-	-	PUNCT
ejpam-5570	243	47	βi	βi	ADV
ejpam-5570	243	48	-	-	PUNCT
ejpam-5570	243	49	locally	locally	ADV
ejpam-5570	243	50	finite	finite	NOUN
ejpam-5570	243	51	and	and	CCONJ
ejpam-5570	243	52	refines	refine	VERB
ejpam-5570	243	53	u	u	PRON
ejpam-5570	243	54	.	.	PUNCT
ejpam-5570	244	1	a∩b	a∩b	PROPN
ejpam-5570	244	2	is	be	AUX
ejpam-5570	244	3	therefore	therefore	ADV
ejpam-5570	244	4	δ	δ	PROPN
ejpam-5570	244	5	-	-	PUNCT
ejpam-5570	244	6	βi	βi	ADV
ejpam-5570	244	7	-	-	PUNCT
ejpam-5570	244	8	paracompact	paracompact	ADJ
ejpam-5570	244	9	.	.	PUNCT
ejpam-5570	245	1	corollary	corollary	ADJ
ejpam-5570	245	2	1	1	NUM
ejpam-5570	245	3	.	.	PUNCT
ejpam-5570	246	1	if	if	SCONJ
ejpam-5570	246	2	a	a	PRON
ejpam-5570	246	3	is	be	AUX
ejpam-5570	246	4	a	a	DET
ejpam-5570	246	5	closed	closed	ADJ
ejpam-5570	246	6	subset	subset	NOUN
ejpam-5570	246	7	of	of	ADP
ejpam-5570	246	8	ideal	ideal	ADJ
ejpam-5570	246	9	topological	topological	ADJ
ejpam-5570	246	10	space	space	NOUN
ejpam-5570	246	11	(	(	PUNCT
ejpam-5570	246	12	x	x	X
ejpam-5570	246	13	,	,	PUNCT
ejpam-5570	246	14	τ	τ	PROPN
ejpam-5570	246	15	,	,	PUNCT
ejpam-5570	246	16	i	i	PROPN
ejpam-5570	246	17	)	)	PUNCT
ejpam-5570	246	18	which	which	PRON
ejpam-5570	246	19	is	be	AUX
ejpam-5570	246	20	δ	δ	PROPN
ejpam-5570	246	21	-	-	NOUN
ejpam-5570	246	22	βiparacompact	βiparacompact	ADJ
ejpam-5570	246	23	,	,	PUNCT
ejpam-5570	246	24	then	then	ADV
ejpam-5570	246	25	a	a	PRON
ejpam-5570	246	26	is	be	AUX
ejpam-5570	246	27	δ	δ	PROPN
ejpam-5570	246	28	-	-	PUNCT
ejpam-5570	246	29	βi	βi	ADV
ejpam-5570	246	30	-	-	PUNCT
ejpam-5570	246	31	paracompact	paracompact	ADJ
ejpam-5570	246	32	.	.	PUNCT
ejpam-5570	247	1	corollary	corollary	ADJ
ejpam-5570	247	2	2	2	NUM
ejpam-5570	247	3	.	.	PUNCT
ejpam-5570	248	1	if	if	SCONJ
ejpam-5570	248	2	a	a	PRON
ejpam-5570	248	3	is	be	AUX
ejpam-5570	248	4	δ	δ	PROPN
ejpam-5570	248	5	-	-	PUNCT
ejpam-5570	248	6	βi	βi	ADV
ejpam-5570	248	7	-	-	NOUN
ejpam-5570	248	8	paracompact	paracompact	NOUN
ejpam-5570	248	9	in	in	ADP
ejpam-5570	248	10	x	x	PROPN
ejpam-5570	248	11	and	and	CCONJ
ejpam-5570	248	12	b	b	PROPN
ejpam-5570	248	13	is	be	AUX
ejpam-5570	248	14	an	an	DET
ejpam-5570	248	15	open	open	ADJ
ejpam-5570	248	16	contained	contain	VERB
ejpam-5570	248	17	a	a	PRON
ejpam-5570	248	18	,	,	PUNCT
ejpam-5570	248	19	then	then	ADV
ejpam-5570	248	20	a−b	a−b	PROPN
ejpam-5570	248	21	is	be	AUX
ejpam-5570	248	22	δ	δ	PROPN
ejpam-5570	248	23	-	-	PUNCT
ejpam-5570	248	24	βi	βi	ADV
ejpam-5570	248	25	-	-	PUNCT
ejpam-5570	248	26	paracompact	paracompact	ADJ
ejpam-5570	248	27	.	.	PUNCT
ejpam-5570	249	1	lemma	lemma	PROPN
ejpam-5570	249	2	6	6	NUM
ejpam-5570	249	3	.	.	PUNCT
ejpam-5570	250	1	[	[	X
ejpam-5570	250	2	9	9	NUM
ejpam-5570	250	3	]	]	PUNCT
ejpam-5570	250	4	let	let	VERB
ejpam-5570	250	5	i	i	PRON
ejpam-5570	250	6	be	be	AUX
ejpam-5570	250	7	an	an	DET
ejpam-5570	250	8	ideal	ideal	NOUN
ejpam-5570	250	9	on	on	ADP
ejpam-5570	250	10	a	a	DET
ejpam-5570	250	11	topological	topological	ADJ
ejpam-5570	250	12	space	space	NOUN
ejpam-5570	250	13	x.	x.	NOUN
ejpam-5570	251	1	if	if	SCONJ
ejpam-5570	251	2	y	y	PROPN
ejpam-5570	251	3	is	be	AUX
ejpam-5570	251	4	a	a	DET
ejpam-5570	251	5	subset	subset	NOUN
ejpam-5570	251	6	of	of	ADP
ejpam-5570	251	7	x	x	PRON
ejpam-5570	251	8	,	,	PUNCT
ejpam-5570	251	9	then	then	ADV
ejpam-5570	251	10	iy	iy	PROPN
ejpam-5570	251	11	=	=	PUNCT
ejpam-5570	251	12	{	{	PUNCT
ejpam-5570	251	13	i	i	PROPN
ejpam-5570	251	14	∩	∩	NOUN
ejpam-5570	251	15	y	y	NOUN
ejpam-5570	251	16	:	:	PUNCT
ejpam-5570	251	17	i	i	PRON
ejpam-5570	251	18	∈	∈	VERB
ejpam-5570	251	19	i	i	PRON
ejpam-5570	251	20	}	}	PUNCT
ejpam-5570	251	21	is	be	AUX
ejpam-5570	251	22	an	an	DET
ejpam-5570	251	23	ideal	ideal	NOUN
ejpam-5570	251	24	on	on	ADP
ejpam-5570	251	25	y	y	PROPN
ejpam-5570	251	26	.	.	PUNCT
ejpam-5570	252	1	theorem	theorem	PROPN
ejpam-5570	252	2	12	12	NUM
ejpam-5570	252	3	.	.	PUNCT
ejpam-5570	253	1	let	let	VERB
ejpam-5570	253	2	a	a	PRON
ejpam-5570	253	3	and	and	CCONJ
ejpam-5570	253	4	b	b	NOUN
ejpam-5570	253	5	be	be	AUX
ejpam-5570	253	6	subsets	subset	NOUN
ejpam-5570	253	7	of	of	ADP
ejpam-5570	253	8	an	an	DET
ejpam-5570	253	9	ideal	ideal	ADJ
ejpam-5570	253	10	topological	topological	ADJ
ejpam-5570	253	11	space	space	NOUN
ejpam-5570	253	12	(	(	PUNCT
ejpam-5570	253	13	x	x	X
ejpam-5570	253	14	,	,	PUNCT
ejpam-5570	253	15	τ	τ	PROPN
ejpam-5570	253	16	,	,	PUNCT
ejpam-5570	253	17	i	i	NOUN
ejpam-5570	253	18	)	)	PUNCT
ejpam-5570	253	19	such	such	ADJ
ejpam-5570	253	20	that	that	SCONJ
ejpam-5570	253	21	a	a	DET
ejpam-5570	253	22	⊂	⊂	PROPN
ejpam-5570	253	23	b.	b.	PROPN
ejpam-5570	254	1	if	if	SCONJ
ejpam-5570	254	2	a	a	PRON
ejpam-5570	254	3	is	be	AUX
ejpam-5570	254	4	δ	δ	NOUN
ejpam-5570	254	5	-	-	PUNCT
ejpam-5570	254	6	βib	βib	ADJ
ejpam-5570	254	7	-	-	PUNCT
ejpam-5570	254	8	paracompact	paracompact	NOUN
ejpam-5570	254	9	in	in	ADP
ejpam-5570	254	10	b	b	PROPN
ejpam-5570	254	11	and	and	CCONJ
ejpam-5570	254	12	b	b	PROPN
ejpam-5570	254	13	is	be	AUX
ejpam-5570	254	14	δ	δ	PROPN
ejpam-5570	254	15	-	-	PUNCT
ejpam-5570	254	16	βi	βi	ADV
ejpam-5570	254	17	-	-	PUNCT
ejpam-5570	254	18	open	open	ADJ
ejpam-5570	254	19	in	in	ADP
ejpam-5570	254	20	x	x	NOUN
ejpam-5570	254	21	,	,	PUNCT
ejpam-5570	254	22	then	then	ADV
ejpam-5570	254	23	a	a	PRON
ejpam-5570	254	24	is	be	AUX
ejpam-5570	254	25	δ	δ	NOUN
ejpam-5570	254	26	-	-	NOUN
ejpam-5570	254	27	βiparacompact	βiparacompact	NOUN
ejpam-5570	254	28	in	in	ADP
ejpam-5570	254	29	x.	x.	NOUN
ejpam-5570	254	30	proof	proof	NOUN
ejpam-5570	254	31	.	.	PUNCT
ejpam-5570	255	1	let	let	VERB
ejpam-5570	255	2	u	u	PRON
ejpam-5570	255	3	=	=	PUNCT
ejpam-5570	255	4	{	{	PUNCT
ejpam-5570	255	5	uλ	uλ	X
ejpam-5570	255	6	:	:	PUNCT
ejpam-5570	255	7	λ	λ	X
ejpam-5570	255	8	∈	∈	PROPN
ejpam-5570	255	9	λ	λ	PROPN
ejpam-5570	255	10	}	}	PUNCT
ejpam-5570	255	11	be	be	VERB
ejpam-5570	255	12	an	an	DET
ejpam-5570	255	13	open	open	ADJ
ejpam-5570	255	14	cover	cover	NOUN
ejpam-5570	255	15	of	of	ADP
ejpam-5570	255	16	a	a	PRON
ejpam-5570	255	17	in	in	ADP
ejpam-5570	255	18	x.	x.	NOUN
ejpam-5570	255	19	then	then	ADV
ejpam-5570	255	20	ua	ua	PROPN
ejpam-5570	255	21	=	=	PROPN
ejpam-5570	255	22	{	{	PUNCT
ejpam-5570	255	23	uλ∩b	uλ∩b	NOUN
ejpam-5570	255	24	:	:	PUNCT
ejpam-5570	255	25	λ	λ	X
ejpam-5570	255	26	∈	∈	PROPN
ejpam-5570	255	27	λ	λ	PROPN
ejpam-5570	255	28	}	}	PUNCT
ejpam-5570	255	29	is	be	AUX
ejpam-5570	255	30	an	an	DET
ejpam-5570	255	31	open	open	ADJ
ejpam-5570	255	32	cover	cover	NOUN
ejpam-5570	255	33	of	of	ADP
ejpam-5570	255	34	a	a	PRON
ejpam-5570	255	35	in	in	ADP
ejpam-5570	255	36	b.	b.	PROPN
ejpam-5570	255	37	as	as	ADP
ejpam-5570	255	38	a	a	DET
ejpam-5570	255	39	is	be	AUX
ejpam-5570	255	40	δ	δ	NOUN
ejpam-5570	255	41	-	-	PUNCT
ejpam-5570	255	42	βib	βib	ADJ
ejpam-5570	255	43	-paracompact	-paracompact	NOUN
ejpam-5570	255	44	in	in	ADP
ejpam-5570	255	45	b	b	PROPN
ejpam-5570	255	46	,	,	PUNCT
ejpam-5570	255	47	ua	ua	PROPN
ejpam-5570	255	48	has	have	VERB
ejpam-5570	255	49	a	a	DET
ejpam-5570	255	50	δ	δ	NOUN
ejpam-5570	255	51	-	-	PUNCT
ejpam-5570	255	52	βib	βib	ADJ
ejpam-5570	255	53	-locally	-locally	ADV
ejpam-5570	255	54	finite	finite	VERB
ejpam-5570	255	55	precise	precise	ADJ
ejpam-5570	255	56	δ	δ	PROPN
ejpam-5570	255	57	-	-	PUNCT
ejpam-5570	255	58	βib	βib	ADJ
ejpam-5570	255	59	-open	-open	PROPN
ejpam-5570	255	60	refinement	refinement	NOUN
ejpam-5570	255	61	va	va	PROPN
ejpam-5570	255	62	=	=	PUNCT
ejpam-5570	255	63	{	{	PUNCT
ejpam-5570	255	64	vλ∩b	vλ∩b	NOUN
ejpam-5570	255	65	:	:	PUNCT
ejpam-5570	255	66	λ	λ	X
ejpam-5570	255	67	∈	∈	PROPN
ejpam-5570	255	68	λ	λ	NOUN
ejpam-5570	255	69	}	}	PUNCT
ejpam-5570	255	70	such	such	ADJ
ejpam-5570	255	71	that	that	SCONJ
ejpam-5570	255	72	vλ	vλ	ADP
ejpam-5570	255	73	⊂	⊂	PRON
ejpam-5570	255	74	uλ	uλ	PRON
ejpam-5570	255	75	for	for	ADP
ejpam-5570	255	76	all	all	DET
ejpam-5570	255	77	λ	λ	PROPN
ejpam-5570	255	78	∈	∈	PROPN
ejpam-5570	255	79	λ	λ	PROPN
ejpam-5570	255	80	,	,	PUNCT
ejpam-5570	255	81	and	and	CCONJ
ejpam-5570	256	1	a−	a−	PROPN
ejpam-5570	256	2	∪{vλ	∪{vλ	PROPN
ejpam-5570	256	3	∩b	∩b	NOUN
ejpam-5570	256	4	:	:	PUNCT
ejpam-5570	256	5	λ	λ	X
ejpam-5570	256	6	∈	∈	PROPN
ejpam-5570	256	7	λ	λ	PROPN
ejpam-5570	256	8	}	}	PUNCT
ejpam-5570	256	9	∈	∈	PROPN
ejpam-5570	256	10	ib	ib	NOUN
ejpam-5570	256	11	.	.	PUNCT
ejpam-5570	257	1	since	since	SCONJ
ejpam-5570	257	2	vλ	vλ	ADV
ejpam-5570	257	3	is	be	AUX
ejpam-5570	257	4	δ	δ	PROPN
ejpam-5570	257	5	-	-	PUNCT
ejpam-5570	257	6	βi	βi	ADV
ejpam-5570	257	7	-	-	PUNCT
ejpam-5570	257	8	open	open	ADJ
ejpam-5570	257	9	in	in	ADP
ejpam-5570	257	10	x	x	X
ejpam-5570	257	11	,	,	PUNCT
ejpam-5570	257	12	the	the	DET
ejpam-5570	257	13	collection	collection	NOUN
ejpam-5570	257	14	v	v	NOUN
ejpam-5570	257	15	=	=	SYM
ejpam-5570	257	16	{	{	PUNCT
ejpam-5570	257	17	vλ	vλ	INTJ
ejpam-5570	257	18	:	:	PUNCT
ejpam-5570	257	19	λ	λ	PROPN
ejpam-5570	257	20	∈	∈	PROPN
ejpam-5570	257	21	λ	λ	PROPN
ejpam-5570	257	22	}	}	PUNCT
ejpam-5570	257	23	of	of	ADP
ejpam-5570	257	24	δ	δ	PROPN
ejpam-5570	257	25	-	-	PUNCT
ejpam-5570	257	26	βi	βi	ADV
ejpam-5570	257	27	-	-	PUNCT
ejpam-5570	257	28	open	open	ADJ
ejpam-5570	257	29	sets	set	NOUN
ejpam-5570	257	30	in	in	ADP
ejpam-5570	257	31	x	x	PROPN
ejpam-5570	257	32	is	be	AUX
ejpam-5570	257	33	δ	δ	PROPN
ejpam-5570	257	34	-	-	PUNCT
ejpam-5570	257	35	βi	βi	ADV
ejpam-5570	257	36	-	-	PUNCT
ejpam-5570	257	37	locally	locally	ADV
ejpam-5570	257	38	finite	finite	NOUN
ejpam-5570	257	39	that	that	PRON
ejpam-5570	257	40	refines	refine	VERB
ejpam-5570	257	41	u	u	PRON
ejpam-5570	257	42	,	,	PUNCT
ejpam-5570	257	43	and	and	CCONJ
ejpam-5570	257	44	a	a	DET
ejpam-5570	257	45	−	−	PROPN
ejpam-5570	257	46	∪{vλ	∪{vλ	NUM
ejpam-5570	257	47	:	:	PUNCT
ejpam-5570	257	48	λ	λ	PROPN
ejpam-5570	257	49	∈	∈	PROPN
ejpam-5570	257	50	λ	λ	PROPN
ejpam-5570	257	51	}	}	PUNCT
ejpam-5570	257	52	⊂	⊂	PROPN
ejpam-5570	257	53	a−	a−	PROPN
ejpam-5570	257	54	∪{vλ	∪{vλ	PROPN
ejpam-5570	257	55	∩b	∩b	NOUN
ejpam-5570	257	56	:	:	PUNCT
ejpam-5570	257	57	λ	λ	X
ejpam-5570	257	58	∈	∈	PROPN
ejpam-5570	257	59	λ	λ	PROPN
ejpam-5570	257	60	}	}	PUNCT
ejpam-5570	257	61	∈	∈	PROPN
ejpam-5570	257	62	ib	ib	NOUN
ejpam-5570	257	63	⊂	⊂	PROPN
ejpam-5570	257	64	i.	i.	PROPN
ejpam-5570	257	65	subsequently	subsequently	ADV
ejpam-5570	257	66	,	,	PUNCT
ejpam-5570	257	67	a	a	PRON
ejpam-5570	257	68	is	be	AUX
ejpam-5570	257	69	δ	δ	PROPN
ejpam-5570	257	70	-	-	PUNCT
ejpam-5570	257	71	βi	βi	ADV
ejpam-5570	257	72	-	-	NOUN
ejpam-5570	257	73	paracompact	paracompact	NOUN
ejpam-5570	257	74	in	in	ADP
ejpam-5570	257	75	x.	x.	PROPN
ejpam-5570	257	76	c.	c.	PROPN
ejpam-5570	257	77	boonpok	boonpok	PROPN
ejpam-5570	257	78	,	,	PUNCT
ejpam-5570	257	79	a.	a.	PROPN
ejpam-5570	257	80	sama	sama	PROPN
ejpam-5570	257	81	-	-	PUNCT
ejpam-5570	257	82	ae	ae	PROPN
ejpam-5570	257	83	,	,	PUNCT
ejpam-5570	257	84	p.	p.	NOUN
ejpam-5570	257	85	raktaow	raktaow	PROPN
ejpam-5570	257	86	/	/	SYM
ejpam-5570	257	87	eur	eur	PROPN
ejpam-5570	257	88	.	.	PUNCT
ejpam-5570	258	1	j.	j.	PROPN
ejpam-5570	258	2	pure	pure	PROPN
ejpam-5570	258	3	appl	appl	PROPN
ejpam-5570	258	4	.	.	PROPN
ejpam-5570	258	5	math	math	PROPN
ejpam-5570	258	6	,	,	PUNCT
ejpam-5570	258	7	18	18	NUM
ejpam-5570	258	8	(	(	PUNCT
ejpam-5570	258	9	1	1	NUM
ejpam-5570	258	10	)	)	PUNCT
ejpam-5570	258	11	(	(	PUNCT
ejpam-5570	258	12	2025	2025	NUM
ejpam-5570	258	13	)	)	PUNCT
ejpam-5570	258	14	,	,	PUNCT
ejpam-5570	258	15	5570	5570	NUM
ejpam-5570	258	16	9	9	NUM
ejpam-5570	258	17	of	of	ADP
ejpam-5570	258	18	12	12	NUM
ejpam-5570	258	19	5	5	NUM
ejpam-5570	258	20	.	.	PUNCT
ejpam-5570	258	21	preserving	preserve	VERB
ejpam-5570	258	22	paracompactness	paracompactness	NOUN
ejpam-5570	258	23	this	this	DET
ejpam-5570	258	24	section	section	NOUN
ejpam-5570	258	25	will	will	AUX
ejpam-5570	258	26	illustrate	illustrate	VERB
ejpam-5570	258	27	the	the	DET
ejpam-5570	258	28	preservation	preservation	NOUN
ejpam-5570	258	29	of	of	ADP
ejpam-5570	258	30	δ	δ	PROPN
ejpam-5570	258	31	-	-	PUNCT
ejpam-5570	258	32	βj	βj	NOUN
ejpam-5570	258	33	-paracompactness	-paracompactness	ADJ
ejpam-5570	258	34	under	under	ADP
ejpam-5570	258	35	specific	specific	ADJ
ejpam-5570	258	36	situations	situation	NOUN
ejpam-5570	258	37	.	.	PUNCT
ejpam-5570	259	1	we	we	PRON
ejpam-5570	259	2	start	start	VERB
ejpam-5570	259	3	by	by	ADP
ejpam-5570	259	4	delineating	delineate	VERB
ejpam-5570	259	5	the	the	DET
ejpam-5570	259	6	subsequent	subsequent	ADJ
ejpam-5570	259	7	definitions	definition	NOUN
ejpam-5570	259	8	.	.	PUNCT
ejpam-5570	260	1	definition	definition	NOUN
ejpam-5570	260	2	7	7	NUM
ejpam-5570	260	3	.	.	PUNCT
ejpam-5570	261	1	let	let	VERB
ejpam-5570	261	2	(	(	PUNCT
ejpam-5570	261	3	x	x	X
ejpam-5570	261	4	,	,	PUNCT
ejpam-5570	261	5	τ	τ	PROPN
ejpam-5570	261	6	,	,	PUNCT
ejpam-5570	261	7	i	i	PROPN
ejpam-5570	261	8	)	)	PUNCT
ejpam-5570	261	9	,	,	PUNCT
ejpam-5570	261	10	and	and	CCONJ
ejpam-5570	261	11	(	(	PUNCT
ejpam-5570	261	12	y	y	PROPN
ejpam-5570	261	13	,	,	PUNCT
ejpam-5570	261	14	τ	τ	PROPN
ejpam-5570	261	15	′,j	′,j	NOUN
ejpam-5570	261	16	)	)	PUNCT
ejpam-5570	261	17	be	be	AUX
ejpam-5570	261	18	ideal	ideal	ADJ
ejpam-5570	261	19	topological	topological	ADJ
ejpam-5570	261	20	spaces	space	NOUN
ejpam-5570	261	21	,	,	PUNCT
ejpam-5570	261	22	and	and	CCONJ
ejpam-5570	261	23	f	f	X
ejpam-5570	261	24	:	:	PUNCT
ejpam-5570	261	25	x	x	X
ejpam-5570	261	26	→	→	SYM
ejpam-5570	261	27	y	y	X
ejpam-5570	261	28	be	be	AUX
ejpam-5570	261	29	a	a	DET
ejpam-5570	261	30	function	function	NOUN
ejpam-5570	261	31	.	.	PUNCT
ejpam-5570	262	1	(	(	PUNCT
ejpam-5570	262	2	i	i	NOUN
ejpam-5570	262	3	)	)	PUNCT
ejpam-5570	262	4	f	f	PROPN
ejpam-5570	262	5	is	be	AUX
ejpam-5570	262	6	called	call	VERB
ejpam-5570	262	7	δ	δ	PROPN
ejpam-5570	262	8	-	-	PUNCT
ejpam-5570	262	9	βi	βi	ADV
ejpam-5570	262	10	-	-	PUNCT
ejpam-5570	262	11	open	open	ADJ
ejpam-5570	262	12	if	if	SCONJ
ejpam-5570	262	13	f(g	f(g	NOUN
ejpam-5570	262	14	)	)	PUNCT
ejpam-5570	262	15	is	be	AUX
ejpam-5570	262	16	a	a	DET
ejpam-5570	262	17	δ	δ	PROPN
ejpam-5570	262	18	-	-	PUNCT
ejpam-5570	262	19	βj	βj	PUNCT
ejpam-5570	262	20	-open	-open	NOUN
ejpam-5570	262	21	set	set	VERB
ejpam-5570	262	22	in	in	ADP
ejpam-5570	262	23	y	y	PROPN
ejpam-5570	262	24	for	for	ADP
ejpam-5570	262	25	every	every	DET
ejpam-5570	262	26	δ	δ	PROPN
ejpam-5570	262	27	-	-	PUNCT
ejpam-5570	262	28	βi	βi	ADV
ejpam-5570	262	29	-	-	PUNCT
ejpam-5570	262	30	open	open	NOUN
ejpam-5570	262	31	set	set	VERB
ejpam-5570	262	32	g	g	NOUN
ejpam-5570	262	33	in	in	ADP
ejpam-5570	262	34	x.	x.	PROPN
ejpam-5570	262	35	(	(	PUNCT
ejpam-5570	262	36	ii	ii	PROPN
ejpam-5570	262	37	)	)	PUNCT
ejpam-5570	262	38	f	f	PROPN
ejpam-5570	262	39	is	be	AUX
ejpam-5570	262	40	called	call	VERB
ejpam-5570	262	41	δ	δ	PROPN
ejpam-5570	262	42	-	-	PUNCT
ejpam-5570	262	43	βi	βi	ADV
ejpam-5570	262	44	-	-	PUNCT
ejpam-5570	262	45	closed	closed	ADJ
ejpam-5570	262	46	if	if	SCONJ
ejpam-5570	262	47	f(f	f(f	PROPN
ejpam-5570	262	48	)	)	PUNCT
ejpam-5570	262	49	is	be	AUX
ejpam-5570	262	50	a	a	DET
ejpam-5570	262	51	δ	δ	PROPN
ejpam-5570	262	52	-	-	PUNCT
ejpam-5570	262	53	βj	βj	PUNCT
ejpam-5570	262	54	-closed	-close	VERB
ejpam-5570	262	55	set	set	NOUN
ejpam-5570	262	56	in	in	ADP
ejpam-5570	262	57	y	y	PROPN
ejpam-5570	262	58	for	for	ADP
ejpam-5570	262	59	every	every	DET
ejpam-5570	262	60	δ	δ	PROPN
ejpam-5570	262	61	-	-	PUNCT
ejpam-5570	262	62	βi	βi	ADV
ejpam-5570	262	63	-	-	PUNCT
ejpam-5570	262	64	closed	close	VERB
ejpam-5570	262	65	set	set	VERB
ejpam-5570	262	66	f	f	PROPN
ejpam-5570	262	67	in	in	ADP
ejpam-5570	262	68	x.	x.	PROPN
ejpam-5570	262	69	(	(	PUNCT
ejpam-5570	262	70	iii	iii	X
ejpam-5570	262	71	)	)	PUNCT
ejpam-5570	262	72	f	f	PROPN
ejpam-5570	262	73	is	be	AUX
ejpam-5570	262	74	called	call	VERB
ejpam-5570	262	75	δ	δ	PROPN
ejpam-5570	262	76	-	-	PUNCT
ejpam-5570	262	77	βi	βi	ADV
ejpam-5570	262	78	-	-	PUNCT
ejpam-5570	262	79	irresolute	irresolute	ADJ
ejpam-5570	262	80	if	if	SCONJ
ejpam-5570	262	81	f−1(v	f−1(v	PROPN
ejpam-5570	262	82	)	)	PUNCT
ejpam-5570	262	83	is	be	AUX
ejpam-5570	262	84	a	a	DET
ejpam-5570	262	85	δ	δ	PROPN
ejpam-5570	262	86	-	-	PUNCT
ejpam-5570	262	87	βi	βi	ADV
ejpam-5570	262	88	-	-	PUNCT
ejpam-5570	262	89	open	open	NOUN
ejpam-5570	262	90	set	set	NOUN
ejpam-5570	262	91	in	in	ADP
ejpam-5570	262	92	x	x	PUNCT
ejpam-5570	262	93	for	for	SCONJ
ejpam-5570	262	94	every	every	DET
ejpam-5570	262	95	δ	δ	PROPN
ejpam-5570	262	96	-	-	PUNCT
ejpam-5570	262	97	βj	βj	PUNCT
ejpam-5570	262	98	-open	-open	NOUN
ejpam-5570	262	99	set	set	VERB
ejpam-5570	262	100	v	v	NOUN
ejpam-5570	262	101	in	in	ADP
ejpam-5570	262	102	y	y	PROPN
ejpam-5570	262	103	.	.	PUNCT
ejpam-5570	263	1	definition	definition	NOUN
ejpam-5570	263	2	8	8	NUM
ejpam-5570	263	3	.	.	PUNCT
ejpam-5570	264	1	an	an	DET
ejpam-5570	264	2	ideal	ideal	ADJ
ejpam-5570	264	3	topological	topological	ADJ
ejpam-5570	264	4	space	space	NOUN
ejpam-5570	264	5	(	(	PUNCT
ejpam-5570	264	6	x	x	X
ejpam-5570	264	7	,	,	PUNCT
ejpam-5570	264	8	τ	τ	PROPN
ejpam-5570	264	9	,	,	PUNCT
ejpam-5570	264	10	i	i	PROPN
ejpam-5570	264	11	)	)	PUNCT
ejpam-5570	264	12	is	be	AUX
ejpam-5570	264	13	said	say	VERB
ejpam-5570	264	14	to	to	PART
ejpam-5570	264	15	be	be	AUX
ejpam-5570	264	16	δ	δ	PROPN
ejpam-5570	264	17	-	-	PUNCT
ejpam-5570	264	18	βi	βi	ADV
ejpam-5570	264	19	-	-	ADJ
ejpam-5570	264	20	compact	compact	ADJ
ejpam-5570	264	21	if	if	SCONJ
ejpam-5570	264	22	every	every	DET
ejpam-5570	264	23	cover	cover	NOUN
ejpam-5570	264	24	v	v	NOUN
ejpam-5570	264	25	of	of	ADP
ejpam-5570	264	26	δ	δ	PROPN
ejpam-5570	264	27	-	-	PUNCT
ejpam-5570	264	28	βi	βi	ADV
ejpam-5570	264	29	-	-	PUNCT
ejpam-5570	264	30	open	open	ADJ
ejpam-5570	264	31	subsets	subset	NOUN
ejpam-5570	264	32	of	of	ADP
ejpam-5570	264	33	x	x	PUNCT
ejpam-5570	264	34	has	have	VERB
ejpam-5570	264	35	v1	v1	NOUN
ejpam-5570	264	36	,	,	PUNCT
ejpam-5570	264	37	v2	v2	PROPN
ejpam-5570	264	38	,	,	PUNCT
ejpam-5570	264	39	...	...	PUNCT
ejpam-5570	264	40	,	,	PUNCT
ejpam-5570	264	41	vn	vn	PROPN
ejpam-5570	264	42	∈	∈	PROPN
ejpam-5570	264	43	v	v	ADP
ejpam-5570	264	44	such	such	ADJ
ejpam-5570	264	45	that	that	SCONJ
ejpam-5570	264	46	x	x	SYM
ejpam-5570	264	47	⊂	⊂	PROPN
ejpam-5570	264	48	v1	v1	VERB
ejpam-5570	264	49	∪	∪	ADJ
ejpam-5570	264	50	v2	v2	PROPN
ejpam-5570	264	51	∪	∪	X
ejpam-5570	264	52	·	·	PUNCT
ejpam-5570	264	53	·	·	PUNCT
ejpam-5570	264	54	·	·	PUNCT
ejpam-5570	264	55	∪	∪	ADP
ejpam-5570	264	56	vn	vn	PROPN
ejpam-5570	264	57	note	note	VERB
ejpam-5570	264	58	that	that	SCONJ
ejpam-5570	264	59	f−1(j	f−1(j	NOUN
ejpam-5570	264	60	)	)	PUNCT
ejpam-5570	264	61	is	be	AUX
ejpam-5570	264	62	an	an	DET
ejpam-5570	264	63	ideal	ideal	NOUN
ejpam-5570	264	64	on	on	ADP
ejpam-5570	264	65	x	x	SYM
ejpam-5570	264	66	if	if	SCONJ
ejpam-5570	264	67	f	f	X
ejpam-5570	264	68	:	:	PUNCT
ejpam-5570	264	69	x	x	X
ejpam-5570	264	70	→	→	SYM
ejpam-5570	264	71	y	y	PROPN
ejpam-5570	264	72	is	be	AUX
ejpam-5570	264	73	a	a	DET
ejpam-5570	264	74	function	function	NOUN
ejpam-5570	264	75	,	,	PUNCT
ejpam-5570	264	76	(	(	PUNCT
ejpam-5570	264	77	x	x	X
ejpam-5570	264	78	,	,	PUNCT
ejpam-5570	264	79	τ	τ	X
ejpam-5570	264	80	)	)	PUNCT
ejpam-5570	264	81	is	be	AUX
ejpam-5570	264	82	a	a	DET
ejpam-5570	264	83	topological	topological	ADJ
ejpam-5570	264	84	space	space	NOUN
ejpam-5570	264	85	,	,	PUNCT
ejpam-5570	264	86	and	and	CCONJ
ejpam-5570	264	87	(	(	PUNCT
ejpam-5570	264	88	y	y	PROPN
ejpam-5570	264	89	,	,	PUNCT
ejpam-5570	264	90	τ	τ	PROPN
ejpam-5570	264	91	′	′	NUM
ejpam-5570	264	92	)	)	PUNCT
ejpam-5570	264	93	is	be	AUX
ejpam-5570	264	94	a	a	DET
ejpam-5570	264	95	topological	topological	ADJ
ejpam-5570	264	96	space	space	NOUN
ejpam-5570	264	97	with	with	ADP
ejpam-5570	264	98	an	an	DET
ejpam-5570	264	99	ideal	ideal	ADJ
ejpam-5570	264	100	j	j	PROPN
ejpam-5570	264	101	.	.	PUNCT
ejpam-5570	265	1	furthermore	furthermore	ADV
ejpam-5570	265	2	,	,	PUNCT
ejpam-5570	265	3	given	give	VERB
ejpam-5570	265	4	that	that	SCONJ
ejpam-5570	265	5	f	f	PROPN
ejpam-5570	265	6	is	be	AUX
ejpam-5570	265	7	surjective	surjective	ADJ
ejpam-5570	265	8	and	and	CCONJ
ejpam-5570	265	9	x	x	PRON
ejpam-5570	265	10	possesses	possess	VERB
ejpam-5570	265	11	an	an	DET
ejpam-5570	265	12	ideal	ideal	NOUN
ejpam-5570	265	13	i	i	PRON
ejpam-5570	265	14	,	,	PUNCT
ejpam-5570	265	15	f(i	f(i	PROPN
ejpam-5570	265	16	)	)	PUNCT
ejpam-5570	265	17	is	be	AUX
ejpam-5570	265	18	an	an	DET
ejpam-5570	265	19	ideal	ideal	NOUN
ejpam-5570	265	20	on	on	ADP
ejpam-5570	265	21	y	y	PROPN
ejpam-5570	265	22	.	.	PUNCT
ejpam-5570	266	1	before	before	ADP
ejpam-5570	266	2	establishing	establish	VERB
ejpam-5570	266	3	theorem	theorem	VERB
ejpam-5570	266	4	13	13	NUM
ejpam-5570	266	5	,	,	PUNCT
ejpam-5570	266	6	we	we	PRON
ejpam-5570	266	7	will	will	AUX
ejpam-5570	266	8	first	first	ADV
ejpam-5570	266	9	present	present	VERB
ejpam-5570	266	10	the	the	DET
ejpam-5570	266	11	following	follow	VERB
ejpam-5570	266	12	lemma	lemma	PROPN
ejpam-5570	266	13	.	.	PUNCT
ejpam-5570	267	1	lemma	lemma	PROPN
ejpam-5570	267	2	7	7	X
ejpam-5570	267	3	.	.	PUNCT
ejpam-5570	268	1	let	let	VERB
ejpam-5570	268	2	(	(	PUNCT
ejpam-5570	268	3	x	x	X
ejpam-5570	268	4	,	,	PUNCT
ejpam-5570	268	5	τ	τ	PROPN
ejpam-5570	268	6	,	,	PUNCT
ejpam-5570	268	7	i	i	PROPN
ejpam-5570	268	8	)	)	PUNCT
ejpam-5570	268	9	and	and	CCONJ
ejpam-5570	268	10	(	(	PUNCT
ejpam-5570	268	11	y	y	PROPN
ejpam-5570	268	12	,	,	PUNCT
ejpam-5570	268	13	τ	τ	PROPN
ejpam-5570	268	14	′,j	′,j	NOUN
ejpam-5570	268	15	)	)	PUNCT
ejpam-5570	268	16	be	be	AUX
ejpam-5570	268	17	ideal	ideal	ADJ
ejpam-5570	268	18	topological	topological	ADJ
ejpam-5570	268	19	spaces	space	NOUN
ejpam-5570	268	20	,	,	PUNCT
ejpam-5570	268	21	and	and	CCONJ
ejpam-5570	268	22	f	f	X
ejpam-5570	268	23	:	:	PUNCT
ejpam-5570	268	24	x	x	X
ejpam-5570	268	25	→	→	SYM
ejpam-5570	268	26	y	y	PROPN
ejpam-5570	268	27	be	be	AUX
ejpam-5570	268	28	surjective	surjective	ADJ
ejpam-5570	268	29	.	.	PUNCT
ejpam-5570	269	1	then	then	ADV
ejpam-5570	269	2	f	f	PROPN
ejpam-5570	269	3	is	be	AUX
ejpam-5570	269	4	δ	δ	PROPN
ejpam-5570	269	5	-	-	PUNCT
ejpam-5570	269	6	βi	βi	ADV
ejpam-5570	269	7	-	-	PUNCT
ejpam-5570	269	8	closed	closed	ADJ
ejpam-5570	269	9	if	if	SCONJ
ejpam-5570	269	10	and	and	CCONJ
ejpam-5570	269	11	only	only	ADV
ejpam-5570	269	12	if	if	SCONJ
ejpam-5570	269	13	for	for	ADP
ejpam-5570	269	14	every	every	DET
ejpam-5570	269	15	y	y	PROPN
ejpam-5570	269	16	∈	∈	PROPN
ejpam-5570	269	17	y	y	PROPN
ejpam-5570	269	18	and	and	CCONJ
ejpam-5570	269	19	a	a	DET
ejpam-5570	269	20	δ	δ	PROPN
ejpam-5570	269	21	-	-	PUNCT
ejpam-5570	269	22	βi	βi	ADV
ejpam-5570	269	23	-	-	PUNCT
ejpam-5570	269	24	open	open	ADJ
ejpam-5570	269	25	set	set	VERB
ejpam-5570	269	26	u	u	NOUN
ejpam-5570	269	27	in	in	ADP
ejpam-5570	269	28	x	x	SYM
ejpam-5570	269	29	containing	contain	VERB
ejpam-5570	269	30	{	{	PUNCT
ejpam-5570	269	31	f−1(y	f−1(y	PROPN
ejpam-5570	269	32	)	)	PUNCT
ejpam-5570	269	33	}	}	PUNCT
ejpam-5570	269	34	,	,	PUNCT
ejpam-5570	269	35	there	there	PRON
ejpam-5570	269	36	exists	exist	VERB
ejpam-5570	269	37	a	a	DET
ejpam-5570	269	38	δ	δ	PROPN
ejpam-5570	269	39	-	-	PUNCT
ejpam-5570	269	40	βj	βj	PUNCT
ejpam-5570	269	41	-open	-open	NOUN
ejpam-5570	269	42	set	set	VERB
ejpam-5570	269	43	v	v	NOUN
ejpam-5570	269	44	containing	contain	VERB
ejpam-5570	269	45	y	y	PRON
ejpam-5570	269	46	such	such	ADJ
ejpam-5570	269	47	that	that	DET
ejpam-5570	269	48	f−1(v	f−1(v	PROPN
ejpam-5570	269	49	)	)	PUNCT
ejpam-5570	270	1	⊂	⊂	PROPN
ejpam-5570	270	2	u	u	PROPN
ejpam-5570	270	3	.	.	PUNCT
ejpam-5570	271	1	proof	proof	NOUN
ejpam-5570	271	2	.	.	PUNCT
ejpam-5570	272	1	let	let	VERB
ejpam-5570	272	2	y	y	PROPN
ejpam-5570	272	3	∈	∈	PROPN
ejpam-5570	272	4	y	y	PROPN
ejpam-5570	272	5	and	and	CCONJ
ejpam-5570	272	6	u	u	PRON
ejpam-5570	272	7	be	be	VERB
ejpam-5570	272	8	a	a	DET
ejpam-5570	272	9	δ	δ	PROPN
ejpam-5570	272	10	-	-	PUNCT
ejpam-5570	272	11	βi	βi	ADV
ejpam-5570	272	12	-	-	PUNCT
ejpam-5570	272	13	open	open	NOUN
ejpam-5570	272	14	set	set	NOUN
ejpam-5570	272	15	in	in	ADP
ejpam-5570	272	16	x	x	INTJ
ejpam-5570	272	17	such	such	ADJ
ejpam-5570	272	18	that	that	SCONJ
ejpam-5570	272	19	{	{	PUNCT
ejpam-5570	272	20	f−1(y	f−1(y	PROPN
ejpam-5570	272	21	)	)	PUNCT
ejpam-5570	272	22	}	}	PUNCT
ejpam-5570	273	1	⊂	⊂	PROPN
ejpam-5570	273	2	u	u	NOUN
ejpam-5570	273	3	.	.	PUNCT
ejpam-5570	274	1	we	we	PRON
ejpam-5570	274	2	have	have	VERB
ejpam-5570	274	3	that	that	PRON
ejpam-5570	274	4	v	v	NOUN
ejpam-5570	274	5	=	=	SYM
ejpam-5570	274	6	y	y	PROPN
ejpam-5570	274	7	−	−	PROPN
ejpam-5570	275	1	f(x	f(x	PROPN
ejpam-5570	275	2	−	−	PROPN
ejpam-5570	275	3	u	u	NOUN
ejpam-5570	275	4	)	)	PUNCT
ejpam-5570	275	5	is	be	AUX
ejpam-5570	275	6	a	a	DET
ejpam-5570	275	7	δ	δ	PROPN
ejpam-5570	275	8	-	-	PUNCT
ejpam-5570	275	9	βj	βj	AUX
ejpam-5570	275	10	-closed	-close	VERB
ejpam-5570	275	11	set	set	VERB
ejpam-5570	275	12	such	such	ADJ
ejpam-5570	275	13	that	that	SCONJ
ejpam-5570	275	14	y	y	PROPN
ejpam-5570	275	15	∈	∈	PROPN
ejpam-5570	275	16	v	v	NOUN
ejpam-5570	275	17	and	and	CCONJ
ejpam-5570	275	18	f−1(v	f−1(v	PROPN
ejpam-5570	275	19	)	)	PUNCT
ejpam-5570	276	1	⊂	⊂	PROPN
ejpam-5570	276	2	u	u	PROPN
ejpam-5570	276	3	.	.	PUNCT
ejpam-5570	277	1	subsequently	subsequently	ADV
ejpam-5570	277	2	,	,	PUNCT
ejpam-5570	277	3	the	the	DET
ejpam-5570	277	4	necessity	necessity	NOUN
ejpam-5570	277	5	is	be	AUX
ejpam-5570	277	6	verified	verify	VERB
ejpam-5570	277	7	.	.	PUNCT
ejpam-5570	278	1	we	we	PRON
ejpam-5570	278	2	next	next	ADV
ejpam-5570	278	3	demonstrate	demonstrate	VERB
ejpam-5570	278	4	sufficiency	sufficiency	PROPN
ejpam-5570	278	5	.	.	PUNCT
ejpam-5570	279	1	let	let	VERB
ejpam-5570	279	2	f	f	PRON
ejpam-5570	279	3	be	be	AUX
ejpam-5570	279	4	a	a	DET
ejpam-5570	279	5	δ	δ	NOUN
ejpam-5570	279	6	-	-	PUNCT
ejpam-5570	279	7	βiclosed	βiclose	VERB
ejpam-5570	279	8	subset	subset	NOUN
ejpam-5570	279	9	in	in	ADP
ejpam-5570	279	10	x	x	PUNCT
ejpam-5570	279	11	and	and	CCONJ
ejpam-5570	279	12	y	y	PROPN
ejpam-5570	279	13	∈	∈	PROPN
ejpam-5570	280	1	y	y	PROPN
ejpam-5570	280	2	−	−	PROPN
ejpam-5570	280	3	f(f	f(f	PROPN
ejpam-5570	280	4	)	)	PUNCT
ejpam-5570	280	5	.	.	PUNCT
ejpam-5570	281	1	thus	thus	ADV
ejpam-5570	281	2	,	,	PUNCT
ejpam-5570	281	3	{	{	PUNCT
ejpam-5570	281	4	f−1(y	f−1(y	PROPN
ejpam-5570	281	5	)	)	PUNCT
ejpam-5570	281	6	}	}	PUNCT
ejpam-5570	282	1	⊂	⊂	X
ejpam-5570	282	2	x	x	X
ejpam-5570	283	1	−f	−f	NOUN
ejpam-5570	283	2	.	.	PUNCT
ejpam-5570	284	1	by	by	ADP
ejpam-5570	284	2	hypothesis	hypothesis	NOUN
ejpam-5570	284	3	,	,	PUNCT
ejpam-5570	284	4	there	there	PRON
ejpam-5570	284	5	is	be	VERB
ejpam-5570	284	6	a	a	DET
ejpam-5570	284	7	δ	δ	PROPN
ejpam-5570	284	8	-	-	PUNCT
ejpam-5570	284	9	βj	βj	PUNCT
ejpam-5570	284	10	-open	-open	ADJ
ejpam-5570	284	11	set	set	NOUN
ejpam-5570	284	12	vy	vy	ADP
ejpam-5570	284	13	such	such	ADJ
ejpam-5570	284	14	that	that	SCONJ
ejpam-5570	284	15	f−1(vy	f−1(vy	PROPN
ejpam-5570	284	16	)	)	PUNCT
ejpam-5570	284	17	⊂	⊂	PROPN
ejpam-5570	284	18	x	x	X
ejpam-5570	285	1	−	−	PROPN
ejpam-5570	285	2	f	f	X
ejpam-5570	285	3	,	,	PUNCT
ejpam-5570	285	4	and	and	CCONJ
ejpam-5570	285	5	hence	hence	ADV
ejpam-5570	285	6	,	,	PUNCT
ejpam-5570	285	7	y	y	PROPN
ejpam-5570	285	8	∈	∈	PROPN
ejpam-5570	285	9	vy	vy	NOUN
ejpam-5570	285	10	⊂	⊂	PROPN
ejpam-5570	285	11	y	y	PROPN
ejpam-5570	285	12	−	−	PROPN
ejpam-5570	285	13	f(f	f(f	PROPN
ejpam-5570	285	14	)	)	PUNCT
ejpam-5570	285	15	.	.	PUNCT
ejpam-5570	286	1	therefore	therefore	ADV
ejpam-5570	286	2	y	y	PROPN
ejpam-5570	286	3	−	−	PROPN
ejpam-5570	286	4	f(f	f(f	PROPN
ejpam-5570	286	5	)	)	PUNCT
ejpam-5570	287	1	=	=	PUNCT
ejpam-5570	288	1	∪{vy	∪{vy	PROPN
ejpam-5570	288	2	:	:	PUNCT
ejpam-5570	288	3	y	y	PROPN
ejpam-5570	288	4	∈	∈	PROPN
ejpam-5570	288	5	y	y	PROPN
ejpam-5570	288	6	}	}	PUNCT
ejpam-5570	288	7	is	be	AUX
ejpam-5570	288	8	a	a	DET
ejpam-5570	288	9	δ	δ	PROPN
ejpam-5570	288	10	-	-	PUNCT
ejpam-5570	288	11	βj	βj	PUNCT
ejpam-5570	288	12	-open	-open	NOUN
ejpam-5570	288	13	set	set	VERB
ejpam-5570	288	14	in	in	ADP
ejpam-5570	288	15	y	y	PROPN
ejpam-5570	288	16	.	.	PUNCT
ejpam-5570	289	1	consequently	consequently	ADV
ejpam-5570	289	2	,	,	PUNCT
ejpam-5570	289	3	f(f	f(f	PROPN
ejpam-5570	289	4	)	)	PUNCT
ejpam-5570	289	5	is	be	AUX
ejpam-5570	289	6	a	a	DET
ejpam-5570	289	7	δ	δ	PROPN
ejpam-5570	289	8	-	-	PUNCT
ejpam-5570	289	9	βj	βj	PUNCT
ejpam-5570	289	10	-closed	-close	VERB
ejpam-5570	289	11	set	set	NOUN
ejpam-5570	289	12	.	.	PUNCT
ejpam-5570	290	1	theorem	theorem	VERB
ejpam-5570	290	2	13	13	NUM
ejpam-5570	290	3	.	.	PUNCT
ejpam-5570	291	1	let	let	AUX
ejpam-5570	291	2	(	(	PUNCT
ejpam-5570	291	3	x	x	X
ejpam-5570	291	4	,	,	PUNCT
ejpam-5570	291	5	τ	τ	PROPN
ejpam-5570	291	6	,	,	PUNCT
ejpam-5570	291	7	i	i	PROPN
ejpam-5570	291	8	)	)	PUNCT
ejpam-5570	291	9	and	and	CCONJ
ejpam-5570	291	10	(	(	PUNCT
ejpam-5570	291	11	y	y	PROPN
ejpam-5570	291	12	,	,	PUNCT
ejpam-5570	291	13	τ	τ	PROPN
ejpam-5570	291	14	′,j	′,j	NOUN
ejpam-5570	291	15	)	)	PUNCT
ejpam-5570	291	16	be	be	AUX
ejpam-5570	291	17	ideal	ideal	ADJ
ejpam-5570	291	18	topological	topological	ADJ
ejpam-5570	291	19	spaces	space	NOUN
ejpam-5570	291	20	,	,	PUNCT
ejpam-5570	291	21	and	and	CCONJ
ejpam-5570	291	22	f	f	X
ejpam-5570	291	23	:	:	PUNCT
ejpam-5570	291	24	x	x	X
ejpam-5570	291	25	→	→	SYM
ejpam-5570	291	26	y	y	PROPN
ejpam-5570	291	27	be	be	AUX
ejpam-5570	291	28	continuous	continuous	ADJ
ejpam-5570	291	29	,	,	PUNCT
ejpam-5570	291	30	δ	δ	PROPN
ejpam-5570	291	31	-	-	PUNCT
ejpam-5570	291	32	βi	βi	ADV
ejpam-5570	291	33	-	-	PUNCT
ejpam-5570	291	34	open	open	ADJ
ejpam-5570	291	35	,	,	PUNCT
ejpam-5570	291	36	δ	δ	PROPN
ejpam-5570	291	37	-	-	PUNCT
ejpam-5570	291	38	βi	βi	ADV
ejpam-5570	291	39	-	-	PUNCT
ejpam-5570	291	40	closed	closed	ADJ
ejpam-5570	291	41	,	,	PUNCT
ejpam-5570	291	42	surjective	surjective	ADJ
ejpam-5570	291	43	with	with	ADP
ejpam-5570	291	44	{	{	PUNCT
ejpam-5570	291	45	f−1(y	f−1(y	PROPN
ejpam-5570	291	46	)	)	PUNCT
ejpam-5570	291	47	}	}	PUNCT
ejpam-5570	291	48	being	be	AUX
ejpam-5570	291	49	δ	δ	PROPN
ejpam-5570	291	50	-	-	PUNCT
ejpam-5570	291	51	βi	βi	ADV
ejpam-5570	291	52	-	-	NOUN
ejpam-5570	291	53	compact	compact	ADJ
ejpam-5570	291	54	for	for	ADP
ejpam-5570	291	55	every	every	DET
ejpam-5570	291	56	y	y	PROPN
ejpam-5570	291	57	∈	∈	PROPN
ejpam-5570	291	58	y	y	PROPN
ejpam-5570	291	59	and	and	CCONJ
ejpam-5570	291	60	f(i	f(i	NUM
ejpam-5570	291	61	)	)	PUNCT
ejpam-5570	292	1	⊂	⊂	PROPN
ejpam-5570	292	2	j	j	PROPN
ejpam-5570	292	3	.	.	PUNCT
ejpam-5570	293	1	if	if	SCONJ
ejpam-5570	293	2	x	x	PRON
ejpam-5570	293	3	is	be	AUX
ejpam-5570	293	4	δ	δ	PROPN
ejpam-5570	293	5	-	-	PUNCT
ejpam-5570	293	6	βi	βi	ADV
ejpam-5570	293	7	-	-	PUNCT
ejpam-5570	293	8	paracompact	paracompact	ADJ
ejpam-5570	293	9	,	,	PUNCT
ejpam-5570	293	10	then	then	ADV
ejpam-5570	293	11	y	y	PROPN
ejpam-5570	293	12	is	be	AUX
ejpam-5570	293	13	δ	δ	PROPN
ejpam-5570	293	14	-	-	PUNCT
ejpam-5570	293	15	βj	βj	PUNCT
ejpam-5570	293	16	-paracompact	-paracompact	ADJ
ejpam-5570	293	17	.	.	PUNCT
ejpam-5570	294	1	proof	proof	NOUN
ejpam-5570	294	2	.	.	PUNCT
ejpam-5570	295	1	let	let	VERB
ejpam-5570	295	2	u	u	PRON
ejpam-5570	295	3	=	=	PUNCT
ejpam-5570	295	4	{	{	PUNCT
ejpam-5570	295	5	uλ	uλ	X
ejpam-5570	295	6	:	:	PUNCT
ejpam-5570	295	7	λ	λ	X
ejpam-5570	295	8	∈	∈	PROPN
ejpam-5570	295	9	λ	λ	PROPN
ejpam-5570	295	10	}	}	PUNCT
ejpam-5570	295	11	be	be	VERB
ejpam-5570	295	12	an	an	DET
ejpam-5570	295	13	open	open	ADJ
ejpam-5570	295	14	cover	cover	NOUN
ejpam-5570	295	15	of	of	ADP
ejpam-5570	295	16	y	y	PROPN
ejpam-5570	295	17	.	.	PUNCT
ejpam-5570	296	1	as	as	SCONJ
ejpam-5570	296	2	f	f	PROPN
ejpam-5570	296	3	is	be	AUX
ejpam-5570	296	4	continuous	continuous	ADJ
ejpam-5570	296	5	,	,	PUNCT
ejpam-5570	296	6	h	h	NOUN
ejpam-5570	296	7	=	=	PRON
ejpam-5570	296	8	{	{	PUNCT
ejpam-5570	296	9	f−1(uλ	f−1(uλ	PROPN
ejpam-5570	296	10	)	)	PUNCT
ejpam-5570	296	11	:	:	PUNCT
ejpam-5570	297	1	λ	λ	X
ejpam-5570	297	2	∈	∈	PROPN
ejpam-5570	297	3	λ	λ	PROPN
ejpam-5570	297	4	}	}	PUNCT
ejpam-5570	297	5	is	be	AUX
ejpam-5570	297	6	an	an	DET
ejpam-5570	297	7	open	open	ADJ
ejpam-5570	297	8	cover	cover	NOUN
ejpam-5570	297	9	of	of	ADP
ejpam-5570	297	10	x.	x.	NOUN
ejpam-5570	297	11	given	give	VERB
ejpam-5570	297	12	that	that	SCONJ
ejpam-5570	297	13	x	x	PRON
ejpam-5570	297	14	is	be	AUX
ejpam-5570	297	15	δ	δ	PROPN
ejpam-5570	297	16	-	-	PUNCT
ejpam-5570	297	17	βi	βi	ADV
ejpam-5570	297	18	-	-	PUNCT
ejpam-5570	297	19	paracompact	paracompact	ADJ
ejpam-5570	297	20	,	,	PUNCT
ejpam-5570	297	21	h	h	NOUN
ejpam-5570	297	22	has	have	VERB
ejpam-5570	297	23	a	a	DET
ejpam-5570	297	24	precise	precise	ADJ
ejpam-5570	297	25	δ	δ	NOUN
ejpam-5570	297	26	-	-	PUNCT
ejpam-5570	297	27	βi	βi	ADV
ejpam-5570	297	28	-	-	PUNCT
ejpam-5570	297	29	locally	locally	ADV
ejpam-5570	297	30	finite	finite	PROPN
ejpam-5570	297	31	refinement	refinement	NOUN
ejpam-5570	297	32	v	v	ADP
ejpam-5570	297	33	=	=	PUNCT
ejpam-5570	297	34	{	{	PUNCT
ejpam-5570	297	35	vλ	vλ	INTJ
ejpam-5570	297	36	:	:	PUNCT
ejpam-5570	297	37	λ	λ	PROPN
ejpam-5570	297	38	∈	∈	PROPN
ejpam-5570	297	39	λ	λ	PROPN
ejpam-5570	297	40	}	}	PUNCT
ejpam-5570	297	41	of	of	ADP
ejpam-5570	297	42	δ	δ	PROPN
ejpam-5570	297	43	-	-	PUNCT
ejpam-5570	297	44	βi	βi	ADV
ejpam-5570	297	45	-	-	PUNCT
ejpam-5570	297	46	open	open	ADJ
ejpam-5570	297	47	subsets	subset	NOUN
ejpam-5570	297	48	such	such	ADJ
ejpam-5570	297	49	that	that	SCONJ
ejpam-5570	297	50	x	x	X
ejpam-5570	298	1	−	−	NOUN
ejpam-5570	298	2	∪{vλ	∪{vλ	NUM
ejpam-5570	298	3	:	:	PUNCT
ejpam-5570	298	4	λ	λ	PROPN
ejpam-5570	298	5	∈	∈	PROPN
ejpam-5570	298	6	λ	λ	PROPN
ejpam-5570	298	7	}	}	PUNCT
ejpam-5570	298	8	∈	∈	PROPN
ejpam-5570	298	9	i.	i.	NOUN
ejpam-5570	298	10	since	since	SCONJ
ejpam-5570	298	11	f	f	PROPN
ejpam-5570	298	12	is	be	AUX
ejpam-5570	298	13	δ	δ	PROPN
ejpam-5570	298	14	-	-	PUNCT
ejpam-5570	298	15	βi	βi	ADV
ejpam-5570	298	16	-	-	PUNCT
ejpam-5570	298	17	open	open	ADJ
ejpam-5570	298	18	,	,	PUNCT
ejpam-5570	298	19	f(v	f(v	NOUN
ejpam-5570	298	20	)	)	PUNCT
ejpam-5570	298	21	=	=	SYM
ejpam-5570	298	22	{	{	PUNCT
ejpam-5570	298	23	f(vλ	f(vλ	NOUN
ejpam-5570	298	24	)	)	PUNCT
ejpam-5570	298	25	:	:	PUNCT
ejpam-5570	298	26	λ	λ	X
ejpam-5570	298	27	∈	∈	PROPN
ejpam-5570	298	28	λ	λ	PROPN
ejpam-5570	298	29	}	}	PUNCT
ejpam-5570	298	30	is	be	AUX
ejpam-5570	298	31	a	a	DET
ejpam-5570	298	32	precise	precise	ADJ
ejpam-5570	298	33	δ	δ	PROPN
ejpam-5570	298	34	-	-	PUNCT
ejpam-5570	298	35	βj	βj	PUNCT
ejpam-5570	298	36	-open	-open	ADJ
ejpam-5570	298	37	refinement	refinement	NOUN
ejpam-5570	298	38	of	of	ADP
ejpam-5570	298	39	u	u	PROPN
ejpam-5570	298	40	,	,	PUNCT
ejpam-5570	298	41	and	and	CCONJ
ejpam-5570	298	42	y	y	PROPN
ejpam-5570	298	43	−	−	PROPN
ejpam-5570	298	44	∪{f(vλ	∪{f(vλ	NUM
ejpam-5570	298	45	)	)	PUNCT
ejpam-5570	298	46	:	:	PUNCT
ejpam-5570	299	1	λ	λ	X
ejpam-5570	299	2	∈	∈	PROPN
ejpam-5570	299	3	λ	λ	PROPN
ejpam-5570	299	4	}	}	PUNCT
ejpam-5570	299	5	∈	∈	PROPN
ejpam-5570	299	6	j	j	PROPN
ejpam-5570	299	7	.	.	PUNCT
ejpam-5570	300	1	next	next	ADV
ejpam-5570	300	2	,	,	PUNCT
ejpam-5570	300	3	we	we	PRON
ejpam-5570	300	4	shall	shall	AUX
ejpam-5570	300	5	verify	verify	VERB
ejpam-5570	300	6	that	that	DET
ejpam-5570	300	7	f(v	f(v	NOUN
ejpam-5570	300	8	)	)	PUNCT
ejpam-5570	300	9	is	be	AUX
ejpam-5570	300	10	δ	δ	PROPN
ejpam-5570	300	11	-	-	PUNCT
ejpam-5570	300	12	βj	βj	PUNCT
ejpam-5570	300	13	-locally	-locally	ADV
ejpam-5570	300	14	finite	finite	ADJ
ejpam-5570	300	15	.	.	PUNCT
ejpam-5570	301	1	let	let	VERB
ejpam-5570	301	2	y	y	PROPN
ejpam-5570	301	3	∈	∈	PROPN
ejpam-5570	301	4	y	y	PROPN
ejpam-5570	301	5	.	.	PUNCT
ejpam-5570	302	1	as	as	SCONJ
ejpam-5570	302	2	v	v	NOUN
ejpam-5570	302	3	is	be	AUX
ejpam-5570	302	4	δ	δ	PROPN
ejpam-5570	302	5	-	-	PUNCT
ejpam-5570	302	6	βi	βi	ADV
ejpam-5570	302	7	-	-	PUNCT
ejpam-5570	302	8	locally	locally	ADV
ejpam-5570	302	9	finite	finite	NOUN
ejpam-5570	302	10	,	,	PUNCT
ejpam-5570	302	11	for	for	ADP
ejpam-5570	302	12	x	x	PROPN
ejpam-5570	302	13	∈	∈	PROPN
ejpam-5570	302	14	{	{	PUNCT
ejpam-5570	302	15	f−1(y	f−1(y	PROPN
ejpam-5570	302	16	)	)	PUNCT
ejpam-5570	302	17	}	}	PUNCT
ejpam-5570	302	18	,	,	PUNCT
ejpam-5570	302	19	there	there	ADV
ejpam-5570	302	20	c.	c.	PROPN
ejpam-5570	302	21	boonpok	boonpok	PROPN
ejpam-5570	302	22	,	,	PUNCT
ejpam-5570	302	23	a.	a.	PROPN
ejpam-5570	302	24	sama	sama	PROPN
ejpam-5570	302	25	-	-	PUNCT
ejpam-5570	302	26	ae	ae	PROPN
ejpam-5570	302	27	,	,	PUNCT
ejpam-5570	302	28	p.	p.	NOUN
ejpam-5570	302	29	raktaow	raktaow	PROPN
ejpam-5570	302	30	/	/	SYM
ejpam-5570	302	31	eur	eur	PROPN
ejpam-5570	302	32	.	.	PUNCT
ejpam-5570	303	1	j.	j.	PROPN
ejpam-5570	303	2	pure	pure	PROPN
ejpam-5570	303	3	appl	appl	PROPN
ejpam-5570	303	4	.	.	PROPN
ejpam-5570	303	5	math	math	PROPN
ejpam-5570	303	6	,	,	PUNCT
ejpam-5570	303	7	18	18	NUM
ejpam-5570	303	8	(	(	PUNCT
ejpam-5570	303	9	1	1	NUM
ejpam-5570	303	10	)	)	PUNCT
ejpam-5570	303	11	(	(	PUNCT
ejpam-5570	303	12	2025	2025	NUM
ejpam-5570	303	13	)	)	PUNCT
ejpam-5570	303	14	,	,	PUNCT
ejpam-5570	303	15	5570	5570	NUM
ejpam-5570	303	16	10	10	NUM
ejpam-5570	303	17	of	of	ADP
ejpam-5570	303	18	12	12	NUM
ejpam-5570	303	19	exists	exist	VERB
ejpam-5570	303	20	a	a	DET
ejpam-5570	303	21	δ	δ	PROPN
ejpam-5570	303	22	-	-	PUNCT
ejpam-5570	303	23	βi	βi	ADV
ejpam-5570	303	24	-	-	PUNCT
ejpam-5570	303	25	open	open	NOUN
ejpam-5570	303	26	set	set	VERB
ejpam-5570	303	27	gx	gx	PROPN
ejpam-5570	303	28	containing	contain	VERB
ejpam-5570	303	29	x	x	PUNCT
ejpam-5570	303	30	such	such	ADJ
ejpam-5570	303	31	that	that	SCONJ
ejpam-5570	303	32	gx	gx	PROPN
ejpam-5570	303	33	intersects	intersect	NOUN
ejpam-5570	303	34	at	at	ADV
ejpam-5570	303	35	most	most	ADV
ejpam-5570	303	36	finitely	finitely	ADV
ejpam-5570	303	37	many	many	ADJ
ejpam-5570	303	38	members	member	NOUN
ejpam-5570	303	39	of	of	ADP
ejpam-5570	303	40	v.	v.	ADV
ejpam-5570	303	41	because	because	SCONJ
ejpam-5570	303	42	{	{	PUNCT
ejpam-5570	303	43	f−1(y	f−1(y	PROPN
ejpam-5570	303	44	)	)	PUNCT
ejpam-5570	303	45	}	}	PUNCT
ejpam-5570	303	46	is	be	AUX
ejpam-5570	303	47	δ	δ	PROPN
ejpam-5570	303	48	-	-	PUNCT
ejpam-5570	303	49	βi	βi	ADV
ejpam-5570	303	50	-	-	ADJ
ejpam-5570	303	51	compact	compact	ADJ
ejpam-5570	303	52	and	and	CCONJ
ejpam-5570	303	53	{	{	PUNCT
ejpam-5570	303	54	gx	gx	PROPN
ejpam-5570	303	55	:	:	PUNCT
ejpam-5570	303	56	f(x	f(x	PROPN
ejpam-5570	303	57	)	)	PUNCT
ejpam-5570	304	1	=	=	SYM
ejpam-5570	304	2	y	y	X
ejpam-5570	304	3	}	}	PUNCT
ejpam-5570	304	4	is	be	AUX
ejpam-5570	304	5	a	a	DET
ejpam-5570	304	6	δ	δ	PROPN
ejpam-5570	304	7	-	-	PUNCT
ejpam-5570	304	8	βi	βi	ADV
ejpam-5570	304	9	-	-	PUNCT
ejpam-5570	304	10	open	open	ADJ
ejpam-5570	304	11	cover	cover	NOUN
ejpam-5570	304	12	of	of	ADP
ejpam-5570	304	13	{	{	PUNCT
ejpam-5570	304	14	f−1(y	f−1(y	PROPN
ejpam-5570	304	15	)	)	PUNCT
ejpam-5570	304	16	}	}	PUNCT
ejpam-5570	304	17	,	,	PUNCT
ejpam-5570	304	18	there	there	PRON
ejpam-5570	304	19	exists	exist	VERB
ejpam-5570	304	20	a	a	DET
ejpam-5570	304	21	finite	finite	ADJ
ejpam-5570	304	22	subcollection	subcollection	NOUN
ejpam-5570	304	23	hy	hy	PROPN
ejpam-5570	304	24	,	,	PUNCT
ejpam-5570	304	25	such	such	ADJ
ejpam-5570	304	26	that	that	SCONJ
ejpam-5570	304	27	{	{	PUNCT
ejpam-5570	304	28	f−1(y	f−1(y	PROPN
ejpam-5570	304	29	)	)	PUNCT
ejpam-5570	304	30	}	}	PUNCT
ejpam-5570	304	31	⊂	⊂	PROPN
ejpam-5570	304	32	∪hy	∪hy	NOUN
ejpam-5570	304	33	,	,	PUNCT
ejpam-5570	304	34	and	and	CCONJ
ejpam-5570	304	35	∪hy	∪hy	NOUN
ejpam-5570	304	36	intersects	intersect	NOUN
ejpam-5570	304	37	at	at	ADP
ejpam-5570	304	38	most	most	ADV
ejpam-5570	304	39	finitely	finitely	ADV
ejpam-5570	304	40	many	many	ADJ
ejpam-5570	304	41	members	member	NOUN
ejpam-5570	304	42	of	of	ADP
ejpam-5570	304	43	v.	v.	PROPN
ejpam-5570	304	44	as	as	SCONJ
ejpam-5570	304	45	f	f	PROPN
ejpam-5570	304	46	is	be	AUX
ejpam-5570	304	47	δ	δ	PROPN
ejpam-5570	304	48	-	-	PUNCT
ejpam-5570	304	49	βi	βi	ADV
ejpam-5570	304	50	-	-	PUNCT
ejpam-5570	304	51	closed	closed	ADJ
ejpam-5570	304	52	,	,	PUNCT
ejpam-5570	304	53	using	use	VERB
ejpam-5570	304	54	lemma	lemma	PROPN
ejpam-5570	304	55	7	7	NUM
ejpam-5570	304	56	,	,	PUNCT
ejpam-5570	304	57	there	there	PRON
ejpam-5570	304	58	exists	exist	VERB
ejpam-5570	304	59	a	a	DET
ejpam-5570	304	60	δ	δ	PROPN
ejpam-5570	304	61	-	-	PUNCT
ejpam-5570	304	62	βj	βj	PUNCT
ejpam-5570	304	63	-open	-open	NOUN
ejpam-5570	304	64	set	set	VERB
ejpam-5570	304	65	wy	wy	PROPN
ejpam-5570	304	66	containing	contain	VERB
ejpam-5570	304	67	y	y	PRON
ejpam-5570	304	68	such	such	ADJ
ejpam-5570	304	69	that	that	SCONJ
ejpam-5570	304	70	f−1(wy	f−1(wy	PROPN
ejpam-5570	304	71	)	)	PUNCT
ejpam-5570	304	72	⊂	⊂	PROPN
ejpam-5570	304	73	∪hy	∪hy	NOUN
ejpam-5570	304	74	.	.	PUNCT
ejpam-5570	305	1	hence	hence	ADV
ejpam-5570	305	2	,	,	PUNCT
ejpam-5570	305	3	f	f	PROPN
ejpam-5570	305	4	−1(wy	−1(wy	NOUN
ejpam-5570	305	5	)	)	PUNCT
ejpam-5570	305	6	intersects	intersect	NOUN
ejpam-5570	305	7	at	at	ADP
ejpam-5570	305	8	most	most	ADV
ejpam-5570	305	9	finitely	finitely	ADV
ejpam-5570	305	10	many	many	ADJ
ejpam-5570	305	11	members	member	NOUN
ejpam-5570	305	12	of	of	ADP
ejpam-5570	305	13	v.	v.	ADP
ejpam-5570	305	14	this	this	PRON
ejpam-5570	305	15	implies	imply	VERB
ejpam-5570	305	16	that	that	SCONJ
ejpam-5570	305	17	wy	wy	PROPN
ejpam-5570	305	18	intersects	intersect	NOUN
ejpam-5570	305	19	at	at	ADV
ejpam-5570	305	20	most	most	ADV
ejpam-5570	305	21	finitely	finitely	ADV
ejpam-5570	305	22	many	many	ADJ
ejpam-5570	305	23	members	member	NOUN
ejpam-5570	305	24	of	of	ADP
ejpam-5570	305	25	f(v	f(v	NOUN
ejpam-5570	305	26	)	)	PUNCT
ejpam-5570	305	27	.	.	PUNCT
ejpam-5570	306	1	hence	hence	ADV
ejpam-5570	306	2	,	,	PUNCT
ejpam-5570	306	3	f(v	f(v	PROPN
ejpam-5570	306	4	)	)	PUNCT
ejpam-5570	306	5	is	be	AUX
ejpam-5570	306	6	a	a	DET
ejpam-5570	306	7	δ	δ	PROPN
ejpam-5570	306	8	-	-	PUNCT
ejpam-5570	306	9	βj	βj	PUNCT
ejpam-5570	306	10	-locally	-locally	ADV
ejpam-5570	306	11	finite	finite	ADJ
ejpam-5570	306	12	in	in	ADP
ejpam-5570	306	13	y	y	PROPN
ejpam-5570	306	14	.	.	PUNCT
ejpam-5570	307	1	therefore	therefore	ADV
ejpam-5570	307	2	,	,	PUNCT
ejpam-5570	307	3	(	(	PUNCT
ejpam-5570	307	4	y	y	PROPN
ejpam-5570	307	5	,	,	PUNCT
ejpam-5570	307	6	τ	τ	PROPN
ejpam-5570	307	7	′,j	′,j	NOUN
ejpam-5570	307	8	)	)	PUNCT
ejpam-5570	307	9	is	be	AUX
ejpam-5570	307	10	δ	δ	PROPN
ejpam-5570	307	11	-	-	PUNCT
ejpam-5570	307	12	βj	βj	PUNCT
ejpam-5570	307	13	-paracompact	-paracompact	ADJ
ejpam-5570	307	14	.	.	PUNCT
ejpam-5570	308	1	theorem	theorem	NOUN
ejpam-5570	308	2	14	14	NUM
ejpam-5570	308	3	.	.	PUNCT
ejpam-5570	309	1	let	let	VERB
ejpam-5570	309	2	(	(	PUNCT
ejpam-5570	309	3	x	x	X
ejpam-5570	309	4	,	,	PUNCT
ejpam-5570	309	5	τ	τ	PROPN
ejpam-5570	309	6	,	,	PUNCT
ejpam-5570	309	7	i	i	PRON
ejpam-5570	309	8	)	)	PUNCT
ejpam-5570	309	9	be	be	VERB
ejpam-5570	309	10	an	an	DET
ejpam-5570	309	11	ideal	ideal	ADJ
ejpam-5570	309	12	topological	topological	ADJ
ejpam-5570	309	13	space	space	NOUN
ejpam-5570	309	14	and	and	CCONJ
ejpam-5570	309	15	(	(	PUNCT
ejpam-5570	309	16	y	y	PROPN
ejpam-5570	309	17	,	,	PUNCT
ejpam-5570	309	18	τ	τ	PROPN
ejpam-5570	309	19	′	′	NUM
ejpam-5570	309	20	)	)	PUNCT
ejpam-5570	309	21	be	be	AUX
ejpam-5570	309	22	a	a	DET
ejpam-5570	309	23	topological	topological	ADJ
ejpam-5570	309	24	space	space	NOUN
ejpam-5570	309	25	.	.	PUNCT
ejpam-5570	310	1	let	let	VERB
ejpam-5570	310	2	f	f	NOUN
ejpam-5570	310	3	:	:	PUNCT
ejpam-5570	310	4	x	x	X
ejpam-5570	310	5	→	→	SYM
ejpam-5570	310	6	y	y	PROPN
ejpam-5570	310	7	be	be	AUX
ejpam-5570	310	8	δ	δ	PROPN
ejpam-5570	310	9	-	-	PUNCT
ejpam-5570	310	10	βi	βi	ADV
ejpam-5570	310	11	-	-	PUNCT
ejpam-5570	310	12	irresolute	irresolute	ADJ
ejpam-5570	310	13	,	,	PUNCT
ejpam-5570	310	14	continuous	continuous	ADJ
ejpam-5570	310	15	,	,	PUNCT
ejpam-5570	310	16	δ	δ	PROPN
ejpam-5570	310	17	-	-	PUNCT
ejpam-5570	310	18	βi	βi	ADV
ejpam-5570	310	19	-	-	PUNCT
ejpam-5570	310	20	open	open	ADJ
ejpam-5570	310	21	,	,	PUNCT
ejpam-5570	310	22	surjective	surjective	ADJ
ejpam-5570	310	23	,	,	PUNCT
ejpam-5570	310	24	and	and	CCONJ
ejpam-5570	310	25	f(v	f(v	NOUN
ejpam-5570	310	26	)	)	PUNCT
ejpam-5570	310	27	be	be	VERB
ejpam-5570	310	28	a	a	DET
ejpam-5570	310	29	δ	δ	NOUN
ejpam-5570	310	30	-	-	PUNCT
ejpam-5570	310	31	βf(i)locally	βf(i)locally	ADV
ejpam-5570	310	32	finite	finite	NOUN
ejpam-5570	310	33	in	in	ADP
ejpam-5570	310	34	y	y	PROPN
ejpam-5570	310	35	for	for	ADP
ejpam-5570	310	36	every	every	DET
ejpam-5570	310	37	δ	δ	PROPN
ejpam-5570	310	38	-	-	PUNCT
ejpam-5570	310	39	βi	βi	ADV
ejpam-5570	310	40	-	-	PUNCT
ejpam-5570	310	41	locally	locally	ADV
ejpam-5570	310	42	finite	finite	VERB
ejpam-5570	310	43	v	v	NOUN
ejpam-5570	310	44	in	in	ADP
ejpam-5570	310	45	x.	x.	NOUN
ejpam-5570	310	46	if	if	SCONJ
ejpam-5570	310	47	(	(	PUNCT
ejpam-5570	310	48	x	x	NOUN
ejpam-5570	310	49	,	,	PUNCT
ejpam-5570	310	50	τ	τ	PROPN
ejpam-5570	310	51	,	,	PUNCT
ejpam-5570	310	52	i	i	PROPN
ejpam-5570	310	53	)	)	PUNCT
ejpam-5570	310	54	is	be	AUX
ejpam-5570	310	55	δ	δ	PROPN
ejpam-5570	310	56	-	-	PUNCT
ejpam-5570	310	57	βi	βi	ADV
ejpam-5570	310	58	-	-	PUNCT
ejpam-5570	310	59	paracompact	paracompact	ADJ
ejpam-5570	310	60	,	,	PUNCT
ejpam-5570	310	61	then	then	ADV
ejpam-5570	310	62	(	(	PUNCT
ejpam-5570	310	63	y	y	PROPN
ejpam-5570	310	64	,	,	PUNCT
ejpam-5570	310	65	τ	τ	PROPN
ejpam-5570	310	66	′	′	NUM
ejpam-5570	310	67	,	,	PUNCT
ejpam-5570	310	68	f(i	f(i	PROPN
ejpam-5570	310	69	)	)	PUNCT
ejpam-5570	310	70	)	)	PUNCT
ejpam-5570	310	71	is	be	AUX
ejpam-5570	310	72	δ	δ	PROPN
ejpam-5570	310	73	-	-	PUNCT
ejpam-5570	310	74	βf(i)-paracompact	βf(i)-paracompact	PROPN
ejpam-5570	310	75	.	.	PUNCT
ejpam-5570	311	1	proof	proof	NOUN
ejpam-5570	311	2	.	.	PUNCT
ejpam-5570	312	1	let	let	VERB
ejpam-5570	312	2	u	u	PRON
ejpam-5570	312	3	=	=	PUNCT
ejpam-5570	312	4	{	{	PUNCT
ejpam-5570	312	5	uλ	uλ	X
ejpam-5570	312	6	:	:	PUNCT
ejpam-5570	312	7	λ	λ	X
ejpam-5570	312	8	∈	∈	PROPN
ejpam-5570	312	9	λ	λ	PROPN
ejpam-5570	312	10	}	}	PUNCT
ejpam-5570	312	11	represent	represent	VERB
ejpam-5570	312	12	an	an	DET
ejpam-5570	312	13	open	open	ADJ
ejpam-5570	312	14	cover	cover	NOUN
ejpam-5570	312	15	of	of	ADP
ejpam-5570	312	16	y	y	PROPN
ejpam-5570	312	17	.	.	PUNCT
ejpam-5570	313	1	consequently	consequently	ADV
ejpam-5570	313	2	,	,	PUNCT
ejpam-5570	313	3	we	we	PRON
ejpam-5570	313	4	have	have	VERB
ejpam-5570	313	5	that	that	DET
ejpam-5570	313	6	h	h	NOUN
ejpam-5570	313	7	=	=	PRON
ejpam-5570	313	8	{	{	PUNCT
ejpam-5570	313	9	f−1(uλ	f−1(uλ	PROPN
ejpam-5570	313	10	)	)	PUNCT
ejpam-5570	313	11	:	:	PUNCT
ejpam-5570	314	1	λ	λ	X
ejpam-5570	314	2	∈	∈	PROPN
ejpam-5570	314	3	λ	λ	PROPN
ejpam-5570	314	4	}	}	PUNCT
ejpam-5570	314	5	creates	create	VERB
ejpam-5570	314	6	an	an	DET
ejpam-5570	314	7	open	open	ADJ
ejpam-5570	314	8	cover	cover	NOUN
ejpam-5570	314	9	of	of	ADP
ejpam-5570	314	10	x.	x.	NOUN
ejpam-5570	314	11	since	since	SCONJ
ejpam-5570	314	12	x	x	PROPN
ejpam-5570	314	13	is	be	AUX
ejpam-5570	314	14	δ	δ	PROPN
ejpam-5570	314	15	-	-	PUNCT
ejpam-5570	314	16	βi	βi	ADV
ejpam-5570	314	17	-	-	PUNCT
ejpam-5570	314	18	paracompact	paracompact	ADJ
ejpam-5570	314	19	,	,	PUNCT
ejpam-5570	314	20	h	h	NOUN
ejpam-5570	314	21	has	have	VERB
ejpam-5570	314	22	a	a	DET
ejpam-5570	314	23	δ	δ	PROPN
ejpam-5570	314	24	-	-	PUNCT
ejpam-5570	314	25	βi	βi	ADV
ejpam-5570	314	26	-	-	PUNCT
ejpam-5570	314	27	locally	locally	ADV
ejpam-5570	314	28	finite	finite	VERB
ejpam-5570	314	29	precise	precise	PROPN
ejpam-5570	314	30	δ	δ	PROPN
ejpam-5570	314	31	-	-	PUNCT
ejpam-5570	314	32	βi	βi	ADV
ejpam-5570	314	33	-	-	PUNCT
ejpam-5570	314	34	open	open	ADJ
ejpam-5570	314	35	refinement	refinement	NOUN
ejpam-5570	314	36	v	v	NOUN
ejpam-5570	314	37	=	=	PUNCT
ejpam-5570	314	38	{	{	PUNCT
ejpam-5570	314	39	vλ	vλ	INTJ
ejpam-5570	314	40	:	:	PUNCT
ejpam-5570	314	41	λ	λ	PROPN
ejpam-5570	314	42	∈	∈	PROPN
ejpam-5570	314	43	λ	λ	NOUN
ejpam-5570	314	44	}	}	PUNCT
ejpam-5570	314	45	such	such	ADJ
ejpam-5570	314	46	that	that	SCONJ
ejpam-5570	314	47	x	x	X
ejpam-5570	315	1	−	−	NOUN
ejpam-5570	315	2	∪{vλ	∪{vλ	NUM
ejpam-5570	315	3	:	:	PUNCT
ejpam-5570	315	4	λ	λ	PROPN
ejpam-5570	315	5	∈	∈	PROPN
ejpam-5570	315	6	λ	λ	PROPN
ejpam-5570	315	7	}	}	PUNCT
ejpam-5570	315	8	∈	∈	PROPN
ejpam-5570	315	9	i.	i.	NOUN
ejpam-5570	315	10	as	as	ADP
ejpam-5570	315	11	y	y	PROPN
ejpam-5570	315	12	−	−	PROPN
ejpam-5570	315	13	∪{f(vλ	∪{f(vλ	NUM
ejpam-5570	315	14	)	)	PUNCT
ejpam-5570	315	15	:	:	PUNCT
ejpam-5570	316	1	λ	λ	X
ejpam-5570	316	2	∈	∈	PROPN
ejpam-5570	316	3	λ	λ	NOUN
ejpam-5570	316	4	}	}	PUNCT
ejpam-5570	316	5	)	)	PUNCT
ejpam-5570	317	1	⊂	⊂	PROPN
ejpam-5570	317	2	f(x	f(x	PROPN
ejpam-5570	317	3	−	−	PROPN
ejpam-5570	317	4	∪{vλ	∪{vλ	NUM
ejpam-5570	317	5	:	:	PUNCT
ejpam-5570	317	6	λ	λ	PROPN
ejpam-5570	317	7	∈	∈	PROPN
ejpam-5570	317	8	λ	λ	NOUN
ejpam-5570	317	9	}	}	PUNCT
ejpam-5570	317	10	)	)	PUNCT
ejpam-5570	317	11	and	and	CCONJ
ejpam-5570	317	12	f(x	f(x	PROPN
ejpam-5570	317	13	−	−	PROPN
ejpam-5570	317	14	∪{vλ	∪{vλ	PROPN
ejpam-5570	317	15	:	:	PUNCT
ejpam-5570	317	16	λ	λ	PROPN
ejpam-5570	317	17	∈	∈	PROPN
ejpam-5570	317	18	λ	λ	NOUN
ejpam-5570	317	19	}	}	PUNCT
ejpam-5570	317	20	)	)	PUNCT
ejpam-5570	317	21	∈	∈	PROPN
ejpam-5570	318	1	f(i	f(i	PROPN
ejpam-5570	318	2	)	)	PUNCT
ejpam-5570	319	1	,	,	PUNCT
ejpam-5570	319	2	we	we	PRON
ejpam-5570	319	3	have	have	VERB
ejpam-5570	319	4	y	y	PROPN
ejpam-5570	319	5	−	−	PROPN
ejpam-5570	319	6	∪{f(vλ	∪{f(vλ	NUM
ejpam-5570	319	7	)	)	PUNCT
ejpam-5570	319	8	:	:	PUNCT
ejpam-5570	320	1	λ	λ	X
ejpam-5570	320	2	∈	∈	PROPN
ejpam-5570	320	3	λ	λ	NOUN
ejpam-5570	320	4	}	}	PUNCT
ejpam-5570	320	5	)	)	PUNCT
ejpam-5570	320	6	∈	∈	PROPN
ejpam-5570	320	7	f(i	f(i	PROPN
ejpam-5570	320	8	)	)	PUNCT
ejpam-5570	320	9	.	.	PUNCT
ejpam-5570	321	1	given	give	VERB
ejpam-5570	321	2	that	that	SCONJ
ejpam-5570	321	3	f	f	PROPN
ejpam-5570	321	4	is	be	AUX
ejpam-5570	321	5	surjective	surjective	ADJ
ejpam-5570	321	6	,	,	PUNCT
ejpam-5570	321	7	f(i	f(i	NUM
ejpam-5570	321	8	)	)	PUNCT
ejpam-5570	321	9	is	be	AUX
ejpam-5570	321	10	an	an	DET
ejpam-5570	321	11	ideal	ideal	NOUN
ejpam-5570	321	12	of	of	ADP
ejpam-5570	321	13	y	y	PROPN
ejpam-5570	321	14	.	.	PUNCT
ejpam-5570	322	1	by	by	ADP
ejpam-5570	322	2	assumption	assumption	NOUN
ejpam-5570	322	3	,	,	PUNCT
ejpam-5570	322	4	we	we	PRON
ejpam-5570	322	5	have	have	VERB
ejpam-5570	322	6	that	that	DET
ejpam-5570	322	7	f(v	f(v	NOUN
ejpam-5570	322	8	)	)	PUNCT
ejpam-5570	322	9	=	=	SYM
ejpam-5570	322	10	{	{	PUNCT
ejpam-5570	322	11	f(vλ	f(vλ	NOUN
ejpam-5570	322	12	)	)	PUNCT
ejpam-5570	322	13	:	:	PUNCT
ejpam-5570	323	1	λ	λ	X
ejpam-5570	323	2	∈	∈	PROPN
ejpam-5570	323	3	λ	λ	PROPN
ejpam-5570	323	4	}	}	PUNCT
ejpam-5570	323	5	is	be	AUX
ejpam-5570	323	6	a	a	DET
ejpam-5570	323	7	precise	precise	ADJ
ejpam-5570	323	8	δ	δ	NOUN
ejpam-5570	323	9	-	-	PUNCT
ejpam-5570	323	10	βf(i)-locally	βf(i)-locally	ADV
ejpam-5570	323	11	finite	finite	NOUN
ejpam-5570	323	12	of	of	ADP
ejpam-5570	323	13	δ	δ	PROPN
ejpam-5570	323	14	-	-	PUNCT
ejpam-5570	323	15	βf(i)-open	βf(i)-open	ADJ
ejpam-5570	323	16	subsets	subset	NOUN
ejpam-5570	323	17	in	in	ADP
ejpam-5570	323	18	y	y	PROPN
ejpam-5570	323	19	.	.	PUNCT
ejpam-5570	324	1	next	next	ADV
ejpam-5570	324	2	,	,	PUNCT
ejpam-5570	324	3	we	we	PRON
ejpam-5570	324	4	will	will	AUX
ejpam-5570	324	5	confirm	confirm	VERB
ejpam-5570	324	6	that	that	DET
ejpam-5570	324	7	f(v	f(v	NOUN
ejpam-5570	324	8	)	)	PUNCT
ejpam-5570	324	9	refines	refine	VERB
ejpam-5570	324	10	u	u	PRON
ejpam-5570	324	11	.	.	PUNCT
ejpam-5570	325	1	let	let	VERB
ejpam-5570	325	2	f(vλ	f(vλ	NOUN
ejpam-5570	325	3	)	)	PUNCT
ejpam-5570	325	4	∈	∈	PROPN
ejpam-5570	325	5	f(v	f(v	NOUN
ejpam-5570	325	6	)	)	PUNCT
ejpam-5570	325	7	.	.	PUNCT
ejpam-5570	326	1	thus	thus	ADV
ejpam-5570	326	2	,	,	PUNCT
ejpam-5570	326	3	vλ	vλ	INTJ
ejpam-5570	326	4	⊂	⊂	PROPN
ejpam-5570	326	5	f−1(uλ	f−1(uλ	PROPN
ejpam-5570	326	6	)	)	PUNCT
ejpam-5570	326	7	for	for	ADP
ejpam-5570	326	8	some	some	DET
ejpam-5570	326	9	uλ	uλ	DET
ejpam-5570	326	10	∈	∈	PROPN
ejpam-5570	326	11	h	h	NOUN
ejpam-5570	326	12	,	,	PUNCT
ejpam-5570	326	13	as	as	SCONJ
ejpam-5570	326	14	v	v	NOUN
ejpam-5570	326	15	refines	refine	VERB
ejpam-5570	326	16	h.	h.	NOUN
ejpam-5570	326	17	this	this	PRON
ejpam-5570	326	18	indicates	indicate	VERB
ejpam-5570	326	19	that	that	SCONJ
ejpam-5570	326	20	f(vλ	f(vλ	NOUN
ejpam-5570	326	21	)	)	PUNCT
ejpam-5570	326	22	⊂	⊂	PROPN
ejpam-5570	326	23	f(f−1(uλ	f(f−1(uλ	PROPN
ejpam-5570	326	24	)	)	PUNCT
ejpam-5570	326	25	)	)	PUNCT
ejpam-5570	327	1	⊂	⊂	PROPN
ejpam-5570	327	2	uλ	uλ	PROPN
ejpam-5570	327	3	.	.	PUNCT
ejpam-5570	328	1	subsequently	subsequently	ADV
ejpam-5570	328	2	,	,	PUNCT
ejpam-5570	328	3	(	(	PUNCT
ejpam-5570	328	4	y	y	PROPN
ejpam-5570	328	5	,	,	PUNCT
ejpam-5570	328	6	τ	τ	PROPN
ejpam-5570	328	7	′	′	NUM
ejpam-5570	328	8	,	,	PUNCT
ejpam-5570	328	9	f(i	f(i	PROPN
ejpam-5570	328	10	)	)	PUNCT
ejpam-5570	328	11	)	)	PUNCT
ejpam-5570	328	12	is	be	AUX
ejpam-5570	328	13	δ	δ	NOUN
ejpam-5570	328	14	-	-	NOUN
ejpam-5570	328	15	βf(i)paracompact	βf(i)paracompact	NOUN
ejpam-5570	328	16	.	.	PUNCT
ejpam-5570	329	1	theorem	theorem	VERB
ejpam-5570	329	2	15	15	NUM
ejpam-5570	329	3	.	.	PUNCT
ejpam-5570	330	1	let	let	AUX
ejpam-5570	330	2	(	(	PUNCT
ejpam-5570	330	3	x	x	NOUN
ejpam-5570	330	4	,	,	PUNCT
ejpam-5570	330	5	τ	τ	X
ejpam-5570	330	6	)	)	PUNCT
ejpam-5570	330	7	be	be	VERB
ejpam-5570	330	8	a	a	DET
ejpam-5570	330	9	topological	topological	ADJ
ejpam-5570	330	10	space	space	NOUN
ejpam-5570	330	11	and	and	CCONJ
ejpam-5570	330	12	(	(	PUNCT
ejpam-5570	330	13	y	y	PROPN
ejpam-5570	330	14	,	,	PUNCT
ejpam-5570	330	15	τ	τ	PROPN
ejpam-5570	330	16	′,j	′,j	NOUN
ejpam-5570	330	17	)	)	PUNCT
ejpam-5570	330	18	be	be	AUX
ejpam-5570	330	19	an	an	DET
ejpam-5570	330	20	ideal	ideal	ADJ
ejpam-5570	330	21	topological	topological	ADJ
ejpam-5570	330	22	space	space	NOUN
ejpam-5570	330	23	.	.	PUNCT
ejpam-5570	331	1	let	let	VERB
ejpam-5570	331	2	f	f	NOUN
ejpam-5570	331	3	:	:	PUNCT
ejpam-5570	331	4	x	x	X
ejpam-5570	331	5	→	→	SYM
ejpam-5570	331	6	y	y	PROPN
ejpam-5570	331	7	be	be	AUX
ejpam-5570	331	8	open	open	ADJ
ejpam-5570	331	9	,	,	PUNCT
ejpam-5570	331	10	δ	δ	NOUN
ejpam-5570	331	11	-	-	PUNCT
ejpam-5570	331	12	βf−1(j	βf−1(j	ADJ
ejpam-5570	331	13	)	)	PUNCT
ejpam-5570	331	14	-irresolute	-irresolute	ADJ
ejpam-5570	331	15	,	,	PUNCT
ejpam-5570	331	16	and	and	CCONJ
ejpam-5570	331	17	bijective	bijective	ADJ
ejpam-5570	331	18	.	.	PUNCT
ejpam-5570	332	1	if	if	SCONJ
ejpam-5570	332	2	(	(	PUNCT
ejpam-5570	332	3	y	y	PROPN
ejpam-5570	332	4	,	,	PUNCT
ejpam-5570	332	5	τ	τ	PROPN
ejpam-5570	332	6	′,j	′,j	NOUN
ejpam-5570	332	7	)	)	PUNCT
ejpam-5570	332	8	is	be	AUX
ejpam-5570	332	9	δ	δ	PROPN
ejpam-5570	332	10	-	-	PUNCT
ejpam-5570	332	11	βj	βj	NOUN
ejpam-5570	332	12	paracompact	paracompact	ADJ
ejpam-5570	332	13	,	,	PUNCT
ejpam-5570	332	14	then	then	ADV
ejpam-5570	332	15	(	(	PUNCT
ejpam-5570	332	16	x	x	X
ejpam-5570	332	17	,	,	PUNCT
ejpam-5570	332	18	τ	τ	PROPN
ejpam-5570	332	19	,	,	PUNCT
ejpam-5570	332	20	f−1(j	f−1(j	NOUN
ejpam-5570	332	21	)	)	PUNCT
ejpam-5570	332	22	)	)	PUNCT
ejpam-5570	332	23	is	be	AUX
ejpam-5570	332	24	δ	δ	NOUN
ejpam-5570	332	25	-	-	PUNCT
ejpam-5570	332	26	βf−1(j	βf−1(j	ADJ
ejpam-5570	332	27	)	)	PUNCT
ejpam-5570	332	28	-paracompact	-paracompact	ADJ
ejpam-5570	332	29	.	.	PUNCT
ejpam-5570	333	1	proof	proof	NOUN
ejpam-5570	333	2	.	.	PUNCT
ejpam-5570	334	1	let	let	VERB
ejpam-5570	334	2	u	u	PRON
ejpam-5570	334	3	=	=	PUNCT
ejpam-5570	334	4	{	{	PUNCT
ejpam-5570	334	5	uλ	uλ	X
ejpam-5570	334	6	:	:	PUNCT
ejpam-5570	334	7	λ	λ	X
ejpam-5570	334	8	∈	∈	PROPN
ejpam-5570	334	9	λ	λ	PROPN
ejpam-5570	334	10	}	}	PUNCT
ejpam-5570	334	11	be	be	VERB
ejpam-5570	334	12	an	an	DET
ejpam-5570	334	13	open	open	ADJ
ejpam-5570	334	14	cover	cover	NOUN
ejpam-5570	334	15	of	of	ADP
ejpam-5570	334	16	x.	x.	NOUN
ejpam-5570	334	17	as	as	SCONJ
ejpam-5570	334	18	f	f	PROPN
ejpam-5570	334	19	is	be	AUX
ejpam-5570	334	20	open	open	ADJ
ejpam-5570	334	21	,	,	PUNCT
ejpam-5570	334	22	f(u	f(u	PROPN
ejpam-5570	334	23	)	)	PUNCT
ejpam-5570	334	24	=	=	PRON
ejpam-5570	334	25	{	{	PUNCT
ejpam-5570	334	26	f(uλ	f(uλ	PROPN
ejpam-5570	334	27	)	)	PUNCT
ejpam-5570	334	28	:	:	PUNCT
ejpam-5570	335	1	λ	λ	X
ejpam-5570	335	2	∈	∈	PROPN
ejpam-5570	335	3	λ	λ	PROPN
ejpam-5570	335	4	}	}	PUNCT
ejpam-5570	335	5	is	be	AUX
ejpam-5570	335	6	an	an	DET
ejpam-5570	335	7	open	open	ADJ
ejpam-5570	335	8	cover	cover	NOUN
ejpam-5570	335	9	of	of	ADP
ejpam-5570	335	10	y	y	PROPN
ejpam-5570	335	11	.	.	PUNCT
ejpam-5570	336	1	by	by	ADP
ejpam-5570	336	2	hypothesis	hypothesis	NOUN
ejpam-5570	336	3	,	,	PUNCT
ejpam-5570	336	4	f(u	f(u	PROPN
ejpam-5570	336	5	)	)	PUNCT
ejpam-5570	336	6	has	have	VERB
ejpam-5570	336	7	a	a	DET
ejpam-5570	336	8	δ	δ	PROPN
ejpam-5570	336	9	-	-	PUNCT
ejpam-5570	336	10	βj	βj	PRON
ejpam-5570	336	11	-locally	-locally	ADV
ejpam-5570	336	12	finite	finite	VERB
ejpam-5570	336	13	precise	precise	ADJ
ejpam-5570	336	14	δ	δ	PROPN
ejpam-5570	336	15	-	-	PUNCT
ejpam-5570	336	16	βj	βj	PUNCT
ejpam-5570	336	17	-open	-open	ADJ
ejpam-5570	336	18	refinement	refinement	NOUN
ejpam-5570	336	19	h	h	NOUN
ejpam-5570	337	1	=	=	PUNCT
ejpam-5570	337	2	{	{	PUNCT
ejpam-5570	337	3	vλ	vλ	INTJ
ejpam-5570	337	4	:	:	PUNCT
ejpam-5570	337	5	λ	λ	PROPN
ejpam-5570	337	6	∈	∈	PROPN
ejpam-5570	337	7	λ	λ	NOUN
ejpam-5570	337	8	}	}	PUNCT
ejpam-5570	337	9	such	such	ADJ
ejpam-5570	337	10	that	that	SCONJ
ejpam-5570	337	11	y	y	PROPN
ejpam-5570	337	12	−∪{vλ	−∪{vλ	PROPN
ejpam-5570	337	13	:	:	PUNCT
ejpam-5570	337	14	λ	λ	PROPN
ejpam-5570	337	15	∈	∈	PROPN
ejpam-5570	337	16	λ	λ	PROPN
ejpam-5570	337	17	}	}	PUNCT
ejpam-5570	337	18	∈	∈	PROPN
ejpam-5570	337	19	j	j	PROPN
ejpam-5570	337	20	.	.	PUNCT
ejpam-5570	338	1	it	it	PRON
ejpam-5570	338	2	implies	imply	VERB
ejpam-5570	338	3	that	that	SCONJ
ejpam-5570	338	4	y	y	PROPN
ejpam-5570	338	5	−∪{vλ	−∪{vλ	PROPN
ejpam-5570	338	6	:	:	PUNCT
ejpam-5570	338	7	λ	λ	PROPN
ejpam-5570	338	8	∈	∈	PROPN
ejpam-5570	338	9	λ	λ	NOUN
ejpam-5570	338	10	}	}	PUNCT
ejpam-5570	338	11	=	=	SYM
ejpam-5570	338	12	j	j	PROPN
ejpam-5570	338	13	for	for	ADP
ejpam-5570	338	14	some	some	DET
ejpam-5570	338	15	j	j	PROPN
ejpam-5570	338	16	∈	∈	PROPN
ejpam-5570	338	17	j	j	PROPN
ejpam-5570	338	18	,	,	PUNCT
ejpam-5570	338	19	which	which	PRON
ejpam-5570	338	20	follows	follow	VERB
ejpam-5570	338	21	that	that	SCONJ
ejpam-5570	338	22	f−1(y	f−1(y	PROPN
ejpam-5570	338	23	)	)	PUNCT
ejpam-5570	338	24	−	−	PROPN
ejpam-5570	338	25	∪{f−1(vλ	∪{f−1(vλ	NOUN
ejpam-5570	338	26	)	)	PUNCT
ejpam-5570	338	27	:	:	PUNCT
ejpam-5570	339	1	λ	λ	X
ejpam-5570	339	2	∈	∈	NOUN
ejpam-5570	339	3	λ	λ	NOUN
ejpam-5570	339	4	}	}	PUNCT
ejpam-5570	339	5	=	=	SYM
ejpam-5570	339	6	f−1(j	f−1(j	NOUN
ejpam-5570	339	7	)	)	PUNCT
ejpam-5570	339	8	.	.	PUNCT
ejpam-5570	340	1	then	then	ADV
ejpam-5570	340	2	,	,	PUNCT
ejpam-5570	340	3	x	x	X
ejpam-5570	340	4	−	−	ADP
ejpam-5570	340	5	∪{f−1(vλ	∪{f−1(vλ	NOUN
ejpam-5570	340	6	)	)	PUNCT
ejpam-5570	340	7	:	:	PUNCT
ejpam-5570	341	1	λ	λ	X
ejpam-5570	341	2	∈	∈	PROPN
ejpam-5570	341	3	λ	λ	PROPN
ejpam-5570	341	4	}	}	PUNCT
ejpam-5570	341	5	∈	∈	PROPN
ejpam-5570	341	6	f−1(j	f−1(j	NOUN
ejpam-5570	341	7	)	)	PUNCT
ejpam-5570	341	8	.	.	PUNCT
ejpam-5570	342	1	as	as	SCONJ
ejpam-5570	342	2	f	f	PROPN
ejpam-5570	342	3	is	be	AUX
ejpam-5570	342	4	δ	δ	NOUN
ejpam-5570	342	5	-	-	PUNCT
ejpam-5570	342	6	βf−1(j	βf−1(j	ADJ
ejpam-5570	342	7	)	)	PUNCT
ejpam-5570	342	8	-irresolute	-irresolute	ADJ
ejpam-5570	342	9	,	,	PUNCT
ejpam-5570	342	10	v	v	NOUN
ejpam-5570	342	11	=	=	SYM
ejpam-5570	342	12	{	{	PUNCT
ejpam-5570	342	13	f−1(vλ	f−1(vλ	X
ejpam-5570	342	14	)	)	PUNCT
ejpam-5570	343	1	:	:	PUNCT
ejpam-5570	343	2	λ	λ	X
ejpam-5570	343	3	∈	∈	PROPN
ejpam-5570	343	4	λ	λ	PROPN
ejpam-5570	343	5	}	}	PUNCT
ejpam-5570	343	6	is	be	AUX
ejpam-5570	343	7	a	a	DET
ejpam-5570	343	8	δ	δ	NOUN
ejpam-5570	343	9	-	-	PUNCT
ejpam-5570	343	10	βf−1(j	βf−1(j	NOUN
ejpam-5570	343	11	)	)	PUNCT
ejpam-5570	343	12	-locally	-locally	ADV
ejpam-5570	343	13	finite	finite	ADJ
ejpam-5570	343	14	δ	δ	NOUN
ejpam-5570	343	15	-	-	PUNCT
ejpam-5570	343	16	βf−1(j	βf−1(j	ADJ
ejpam-5570	343	17	)	)	PUNCT
ejpam-5570	343	18	-open	-open	ADJ
ejpam-5570	343	19	collection	collection	NOUN
ejpam-5570	343	20	.	.	PUNCT
ejpam-5570	344	1	the	the	DET
ejpam-5570	344	2	refinement	refinement	NOUN
ejpam-5570	344	3	of	of	ADP
ejpam-5570	344	4	u	u	NOUN
ejpam-5570	344	5	by	by	ADP
ejpam-5570	344	6	v	v	NUM
ejpam-5570	344	7	will	will	AUX
ejpam-5570	344	8	be	be	AUX
ejpam-5570	344	9	asserted	assert	VERB
ejpam-5570	344	10	.	.	PUNCT
ejpam-5570	345	1	let	let	VERB
ejpam-5570	345	2	f−1(vλ	f−1(vλ	PRON
ejpam-5570	345	3	)	)	PUNCT
ejpam-5570	345	4	∈	∈	PROPN
ejpam-5570	345	5	v.	v.	ADP
ejpam-5570	345	6	hence	hence	ADV
ejpam-5570	345	7	,	,	PUNCT
ejpam-5570	345	8	vλ	vλ	INTJ
ejpam-5570	345	9	∈	∈	PROPN
ejpam-5570	345	10	h	h	NOUN
ejpam-5570	345	11	,	,	PUNCT
ejpam-5570	345	12	and	and	CCONJ
ejpam-5570	345	13	there	there	PRON
ejpam-5570	345	14	exists	exist	VERB
ejpam-5570	345	15	uλ	uλ	ADP
ejpam-5570	345	16	∈	∈	PROPN
ejpam-5570	345	17	u	u	NOUN
ejpam-5570	345	18	such	such	ADJ
ejpam-5570	345	19	that	that	SCONJ
ejpam-5570	345	20	vλ	vλ	PROPN
ejpam-5570	345	21	⊂	⊂	PRON
ejpam-5570	345	22	f(uλ	f(uλ	PROPN
ejpam-5570	345	23	)	)	PUNCT
ejpam-5570	345	24	as	as	SCONJ
ejpam-5570	345	25	h	h	NOUN
ejpam-5570	345	26	refines	refine	VERB
ejpam-5570	345	27	f(u	f(u	PROPN
ejpam-5570	345	28	)	)	PUNCT
ejpam-5570	345	29	.	.	PUNCT
ejpam-5570	346	1	therefore	therefore	ADV
ejpam-5570	346	2	f−1(vλ	f−1(vλ	X
ejpam-5570	346	3	)	)	PUNCT
ejpam-5570	346	4	⊂	⊂	PROPN
ejpam-5570	346	5	f−1(f(uλ	f−1(f(uλ	NOUN
ejpam-5570	346	6	)	)	PUNCT
ejpam-5570	346	7	)	)	PUNCT
ejpam-5570	347	1	=	=	PUNCT
ejpam-5570	347	2	uλ	uλ	ADP
ejpam-5570	347	3	∈	∈	PROPN
ejpam-5570	347	4	u	u	PROPN
ejpam-5570	347	5	.	.	PUNCT
ejpam-5570	348	1	accordingly	accordingly	ADV
ejpam-5570	348	2	,	,	PUNCT
ejpam-5570	348	3	x	x	X
ejpam-5570	348	4	is	be	AUX
ejpam-5570	348	5	δ	δ	NOUN
ejpam-5570	348	6	-	-	PUNCT
ejpam-5570	348	7	βf−1(j	βf−1(j	ADJ
ejpam-5570	348	8	)	)	PUNCT
ejpam-5570	348	9	-paracompact	-paracompact	NOUN
ejpam-5570	348	10	.	.	PUNCT
ejpam-5570	349	1	theorem	theorem	VERB
ejpam-5570	349	2	16	16	NUM
ejpam-5570	349	3	.	.	PUNCT
ejpam-5570	350	1	let	let	VERB
ejpam-5570	350	2	(	(	PUNCT
ejpam-5570	350	3	x	x	X
ejpam-5570	350	4	,	,	PUNCT
ejpam-5570	350	5	τ	τ	PROPN
ejpam-5570	350	6	,	,	PUNCT
ejpam-5570	350	7	i	i	PROPN
ejpam-5570	350	8	)	)	PUNCT
ejpam-5570	350	9	and	and	CCONJ
ejpam-5570	350	10	(	(	PUNCT
ejpam-5570	350	11	y	y	PROPN
ejpam-5570	350	12	,	,	PUNCT
ejpam-5570	350	13	τ	τ	PROPN
ejpam-5570	350	14	′,j	′,j	NOUN
ejpam-5570	350	15	)	)	PUNCT
ejpam-5570	350	16	be	be	AUX
ejpam-5570	350	17	ideal	ideal	ADJ
ejpam-5570	350	18	topological	topological	ADJ
ejpam-5570	350	19	spaces	space	NOUN
ejpam-5570	350	20	,	,	PUNCT
ejpam-5570	350	21	and	and	CCONJ
ejpam-5570	350	22	let	let	VERB
ejpam-5570	350	23	f	f	X
ejpam-5570	350	24	:	:	PUNCT
ejpam-5570	350	25	(	(	PUNCT
ejpam-5570	350	26	x	x	X
ejpam-5570	350	27	,	,	PUNCT
ejpam-5570	350	28	τ	τ	PROPN
ejpam-5570	350	29	,	,	PUNCT
ejpam-5570	350	30	i	i	NOUN
ejpam-5570	350	31	)	)	PUNCT
ejpam-5570	350	32	→	→	SYM
ejpam-5570	350	33	(	(	PUNCT
ejpam-5570	350	34	y	y	PROPN
ejpam-5570	350	35	,	,	PUNCT
ejpam-5570	350	36	τ	τ	PROPN
ejpam-5570	350	37	′,j	′,j	NOUN
ejpam-5570	350	38	)	)	PUNCT
ejpam-5570	350	39	be	be	AUX
ejpam-5570	350	40	open	open	ADJ
ejpam-5570	350	41	,	,	PUNCT
ejpam-5570	350	42	δ	δ	PROPN
ejpam-5570	350	43	-	-	PUNCT
ejpam-5570	350	44	βi	βi	PRON
ejpam-5570	350	45	-	-	PUNCT
ejpam-5570	350	46	irresolute	irresolute	ADJ
ejpam-5570	350	47	,	,	PUNCT
ejpam-5570	350	48	bijective	bijective	ADJ
ejpam-5570	350	49	,	,	PUNCT
ejpam-5570	350	50	and	and	CCONJ
ejpam-5570	350	51	f(i	f(i	NUM
ejpam-5570	350	52	)	)	PUNCT
ejpam-5570	351	1	=	=	SYM
ejpam-5570	351	2	j	j	PROPN
ejpam-5570	351	3	.	.	PUNCT
ejpam-5570	352	1	if	if	SCONJ
ejpam-5570	352	2	a	a	DET
ejpam-5570	352	3	⊂	⊂	PROPN
ejpam-5570	352	4	y	y	PROPN
ejpam-5570	352	5	is	be	AUX
ejpam-5570	352	6	δ	δ	PROPN
ejpam-5570	352	7	-	-	PUNCT
ejpam-5570	352	8	βj	βj	NOUN
ejpam-5570	352	9	-paracompact	-paracompact	PROPN
ejpam-5570	352	10	in	in	ADP
ejpam-5570	352	11	y	y	PROPN
ejpam-5570	352	12	,	,	PUNCT
ejpam-5570	352	13	then	then	ADV
ejpam-5570	352	14	f−1(a	f−1(a	PROPN
ejpam-5570	352	15	)	)	PUNCT
ejpam-5570	353	1	⊂	⊂	PROPN
ejpam-5570	353	2	x	x	X
ejpam-5570	353	3	is	be	AUX
ejpam-5570	353	4	δ	δ	PROPN
ejpam-5570	353	5	-	-	PUNCT
ejpam-5570	353	6	βi	βi	ADV
ejpam-5570	353	7	-	-	PUNCT
ejpam-5570	353	8	paracompact	paracompact	ADJ
ejpam-5570	353	9	.	.	PUNCT
ejpam-5570	354	1	proof	proof	NOUN
ejpam-5570	354	2	.	.	PUNCT
ejpam-5570	355	1	let	let	VERB
ejpam-5570	355	2	u	u	PRON
ejpam-5570	355	3	=	=	PUNCT
ejpam-5570	355	4	{	{	PUNCT
ejpam-5570	355	5	uλ	uλ	X
ejpam-5570	355	6	:	:	PUNCT
ejpam-5570	355	7	λ	λ	X
ejpam-5570	355	8	∈	∈	PROPN
ejpam-5570	355	9	λ	λ	PROPN
ejpam-5570	355	10	}	}	PUNCT
ejpam-5570	355	11	be	be	VERB
ejpam-5570	355	12	an	an	DET
ejpam-5570	355	13	open	open	ADJ
ejpam-5570	355	14	cover	cover	NOUN
ejpam-5570	355	15	of	of	ADP
ejpam-5570	355	16	f−1(a	f−1(a	NOUN
ejpam-5570	355	17	)	)	PUNCT
ejpam-5570	355	18	.	.	PUNCT
ejpam-5570	356	1	given	give	VERB
ejpam-5570	356	2	that	that	SCONJ
ejpam-5570	356	3	f	f	PROPN
ejpam-5570	356	4	is	be	AUX
ejpam-5570	356	5	an	an	DET
ejpam-5570	356	6	open	open	ADJ
ejpam-5570	356	7	mapping	mapping	NOUN
ejpam-5570	356	8	,	,	PUNCT
ejpam-5570	356	9	f(u	f(u	PROPN
ejpam-5570	356	10	)	)	PUNCT
ejpam-5570	356	11	=	=	PRON
ejpam-5570	356	12	{	{	PUNCT
ejpam-5570	356	13	f(uλ	f(uλ	PROPN
ejpam-5570	356	14	)	)	PUNCT
ejpam-5570	356	15	:	:	PUNCT
ejpam-5570	356	16	λ	λ	X
ejpam-5570	356	17	∈	∈	PROPN
ejpam-5570	356	18	λ	λ	PROPN
ejpam-5570	356	19	}	}	PUNCT
ejpam-5570	356	20	forms	form	VERB
ejpam-5570	356	21	an	an	DET
ejpam-5570	356	22	open	open	ADJ
ejpam-5570	356	23	cover	cover	NOUN
ejpam-5570	356	24	of	of	ADP
ejpam-5570	356	25	a.	a.	NOUN
ejpam-5570	356	26	by	by	ADP
ejpam-5570	356	27	hypothesis	hypothesis	NOUN
ejpam-5570	356	28	,	,	PUNCT
ejpam-5570	356	29	f(u	f(u	PROPN
ejpam-5570	356	30	)	)	PUNCT
ejpam-5570	356	31	c.	c.	PROPN
ejpam-5570	356	32	boonpok	boonpok	PROPN
ejpam-5570	356	33	,	,	PUNCT
ejpam-5570	356	34	a.	a.	PROPN
ejpam-5570	356	35	sama	sama	PROPN
ejpam-5570	356	36	-	-	PUNCT
ejpam-5570	356	37	ae	ae	PROPN
ejpam-5570	356	38	,	,	PUNCT
ejpam-5570	356	39	p.	p.	NOUN
ejpam-5570	356	40	raktaow	raktaow	PROPN
ejpam-5570	356	41	/	/	SYM
ejpam-5570	356	42	eur	eur	PROPN
ejpam-5570	356	43	.	.	PUNCT
ejpam-5570	357	1	j.	j.	PROPN
ejpam-5570	357	2	pure	pure	PROPN
ejpam-5570	357	3	appl	appl	PROPN
ejpam-5570	357	4	.	.	PROPN
ejpam-5570	357	5	math	math	PROPN
ejpam-5570	357	6	,	,	PUNCT
ejpam-5570	357	7	18	18	NUM
ejpam-5570	357	8	(	(	PUNCT
ejpam-5570	357	9	1	1	NUM
ejpam-5570	357	10	)	)	PUNCT
ejpam-5570	357	11	(	(	PUNCT
ejpam-5570	357	12	2025	2025	NUM
ejpam-5570	357	13	)	)	PUNCT
ejpam-5570	357	14	,	,	PUNCT
ejpam-5570	357	15	5570	5570	NUM
ejpam-5570	357	16	11	11	NUM
ejpam-5570	357	17	of	of	ADP
ejpam-5570	357	18	12	12	NUM
ejpam-5570	357	19	possesses	possesse	NOUN
ejpam-5570	357	20	a	a	DET
ejpam-5570	357	21	δ	δ	PROPN
ejpam-5570	357	22	-	-	PUNCT
ejpam-5570	357	23	βj	βj	PRON
ejpam-5570	357	24	-locally	-locally	ADV
ejpam-5570	357	25	finite	finite	VERB
ejpam-5570	357	26	precise	precise	ADJ
ejpam-5570	357	27	δ	δ	PROPN
ejpam-5570	357	28	-	-	PUNCT
ejpam-5570	357	29	βj	βj	PUNCT
ejpam-5570	357	30	-open	-open	ADJ
ejpam-5570	357	31	refinement	refinement	NOUN
ejpam-5570	357	32	h	h	NOUN
ejpam-5570	358	1	=	=	PUNCT
ejpam-5570	358	2	{	{	PUNCT
ejpam-5570	358	3	vλ	vλ	INTJ
ejpam-5570	358	4	:	:	PUNCT
ejpam-5570	358	5	λ	λ	PROPN
ejpam-5570	358	6	∈	∈	PROPN
ejpam-5570	358	7	λ	λ	NOUN
ejpam-5570	358	8	}	}	PUNCT
ejpam-5570	358	9	such	such	ADJ
ejpam-5570	358	10	that	that	SCONJ
ejpam-5570	358	11	a	a	DET
ejpam-5570	358	12	−	−	PROPN
ejpam-5570	358	13	∪{vλ	∪{vλ	NUM
ejpam-5570	358	14	:	:	PUNCT
ejpam-5570	358	15	λ	λ	PROPN
ejpam-5570	358	16	∈	∈	PROPN
ejpam-5570	358	17	λ	λ	PROPN
ejpam-5570	358	18	}	}	PUNCT
ejpam-5570	358	19	∈	∈	PROPN
ejpam-5570	358	20	j	j	PROPN
ejpam-5570	358	21	.	.	PUNCT
ejpam-5570	359	1	then	then	ADV
ejpam-5570	359	2	,	,	PUNCT
ejpam-5570	359	3	f−1(a	f−1(a	PROPN
ejpam-5570	359	4	)	)	PUNCT
ejpam-5570	360	1	−	−	ADP
ejpam-5570	360	2	∪{f−1(vλ	∪{f−1(vλ	NOUN
ejpam-5570	360	3	)	)	PUNCT
ejpam-5570	360	4	:	:	PUNCT
ejpam-5570	361	1	λ	λ	X
ejpam-5570	361	2	∈	∈	PROPN
ejpam-5570	361	3	λ	λ	PROPN
ejpam-5570	361	4	}	}	PUNCT
ejpam-5570	361	5	∈	∈	PROPN
ejpam-5570	361	6	f−1(j	f−1(j	NOUN
ejpam-5570	361	7	)	)	PUNCT
ejpam-5570	362	1	=	=	SYM
ejpam-5570	362	2	i.	i.	NOUN
ejpam-5570	362	3	as	as	SCONJ
ejpam-5570	362	4	f	f	PROPN
ejpam-5570	362	5	is	be	AUX
ejpam-5570	362	6	δ	δ	PROPN
ejpam-5570	362	7	-	-	PUNCT
ejpam-5570	362	8	βi	βi	ADV
ejpam-5570	362	9	-	-	PUNCT
ejpam-5570	362	10	irresolute	irresolute	ADJ
ejpam-5570	362	11	,	,	PUNCT
ejpam-5570	362	12	v	v	NOUN
ejpam-5570	362	13	=	=	SYM
ejpam-5570	362	14	{	{	PUNCT
ejpam-5570	362	15	f−1(vλ	f−1(vλ	X
ejpam-5570	362	16	)	)	PUNCT
ejpam-5570	362	17	:	:	PUNCT
ejpam-5570	363	1	λ	λ	X
ejpam-5570	363	2	∈	∈	PROPN
ejpam-5570	363	3	λ	λ	PROPN
ejpam-5570	363	4	}	}	PUNCT
ejpam-5570	363	5	is	be	AUX
ejpam-5570	363	6	a	a	DET
ejpam-5570	363	7	δ	δ	PROPN
ejpam-5570	363	8	-	-	PUNCT
ejpam-5570	363	9	βi	βi	ADV
ejpam-5570	363	10	-	-	PUNCT
ejpam-5570	363	11	locally	locally	ADV
ejpam-5570	363	12	finite	finite	PROPN
ejpam-5570	363	13	δ	δ	PROPN
ejpam-5570	363	14	-	-	ADJ
ejpam-5570	363	15	βi	βi	ADV
ejpam-5570	363	16	-	-	PUNCT
ejpam-5570	363	17	open	open	ADJ
ejpam-5570	363	18	collection	collection	NOUN
ejpam-5570	363	19	.	.	PUNCT
ejpam-5570	364	1	let	let	VERB
ejpam-5570	364	2	f−1(vλ	f−1(vλ	PRON
ejpam-5570	364	3	)	)	PUNCT
ejpam-5570	364	4	∈	∈	PROPN
ejpam-5570	365	1	v.	v.	CCONJ
ejpam-5570	365	2	there	there	PRON
ejpam-5570	365	3	exists	exist	VERB
ejpam-5570	365	4	f(uλ	f(uλ	PROPN
ejpam-5570	365	5	)	)	PUNCT
ejpam-5570	365	6	∈	∈	PROPN
ejpam-5570	365	7	f(u	f(u	PROPN
ejpam-5570	365	8	)	)	PUNCT
ejpam-5570	365	9	such	such	ADJ
ejpam-5570	365	10	that	that	SCONJ
ejpam-5570	365	11	vλ	vλ	ADP
ejpam-5570	365	12	⊂	⊂	PRON
ejpam-5570	365	13	f(uλ	f(uλ	PROPN
ejpam-5570	365	14	)	)	PUNCT
ejpam-5570	365	15	,	,	PUNCT
ejpam-5570	365	16	since	since	SCONJ
ejpam-5570	365	17	h	h	NOUN
ejpam-5570	365	18	refines	refine	VERB
ejpam-5570	365	19	f(u	f(u	PROPN
ejpam-5570	365	20	)	)	PUNCT
ejpam-5570	365	21	.	.	PUNCT
ejpam-5570	366	1	it	it	PRON
ejpam-5570	366	2	follows	follow	VERB
ejpam-5570	366	3	that	that	SCONJ
ejpam-5570	366	4	f−1(vλ	f−1(vλ	X
ejpam-5570	366	5	)	)	PUNCT
ejpam-5570	366	6	⊂	⊂	PROPN
ejpam-5570	366	7	f−1(f(uλ	f−1(f(uλ	NOUN
ejpam-5570	366	8	)	)	PUNCT
ejpam-5570	366	9	)	)	PUNCT
ejpam-5570	367	1	=	=	SYM
ejpam-5570	367	2	uλ	uλ	NOUN
ejpam-5570	367	3	,	,	PUNCT
ejpam-5570	367	4	thereby	thereby	ADV
ejpam-5570	367	5	indicating	indicate	VERB
ejpam-5570	367	6	that	that	SCONJ
ejpam-5570	367	7	h	h	NOUN
ejpam-5570	367	8	refines	refine	VERB
ejpam-5570	367	9	u	u	PRON
ejpam-5570	367	10	.	.	PUNCT
ejpam-5570	368	1	therefore	therefore	ADV
ejpam-5570	368	2	f−1(a	f−1(a	PROPN
ejpam-5570	368	3	)	)	PUNCT
ejpam-5570	368	4	is	be	AUX
ejpam-5570	368	5	δ	δ	PROPN
ejpam-5570	368	6	-	-	PUNCT
ejpam-5570	368	7	βi	βi	ADV
ejpam-5570	368	8	-	-	NOUN
ejpam-5570	368	9	paracompact	paracompact	NOUN
ejpam-5570	368	10	in	in	ADP
ejpam-5570	368	11	x.	x.	PROPN
ejpam-5570	368	12	6	6	NUM
ejpam-5570	368	13	.	.	PUNCT
ejpam-5570	368	14	conclusion	conclusion	NOUN
ejpam-5570	368	15	this	this	DET
ejpam-5570	368	16	paper	paper	NOUN
ejpam-5570	368	17	looks	look	VERB
ejpam-5570	368	18	at	at	ADP
ejpam-5570	368	19	different	different	ADJ
ejpam-5570	368	20	ways	way	NOUN
ejpam-5570	368	21	to	to	PART
ejpam-5570	368	22	describe	describe	VERB
ejpam-5570	368	23	the	the	DET
ejpam-5570	368	24	δ	δ	PROPN
ejpam-5570	368	25	-	-	PUNCT
ejpam-5570	368	26	βi	βi	PROPN
ejpam-5570	368	27	-	-	PUNCT
ejpam-5570	368	28	paracompactness	paracompactness	NOUN
ejpam-5570	368	29	of	of	ADP
ejpam-5570	368	30	an	an	DET
ejpam-5570	368	31	ideal	ideal	ADJ
ejpam-5570	368	32	topological	topological	ADJ
ejpam-5570	368	33	space	space	NOUN
ejpam-5570	368	34	as	as	ADP
ejpam-5570	368	35	a	a	DET
ejpam-5570	368	36	weaker	weak	ADJ
ejpam-5570	368	37	form	form	NOUN
ejpam-5570	368	38	of	of	ADP
ejpam-5570	368	39	β	β	NOUN
ejpam-5570	368	40	-	-	NOUN
ejpam-5570	368	41	paracompactness	paracompactness	NOUN
ejpam-5570	368	42	compared	compare	VERB
ejpam-5570	368	43	to	to	ADP
ejpam-5570	368	44	an	an	DET
ejpam-5570	368	45	ideal	ideal	NOUN
ejpam-5570	368	46	i	i	PRON
ejpam-5570	368	47	(	(	PUNCT
ejpam-5570	368	48	or	or	CCONJ
ejpam-5570	368	49	i	i	NOUN
ejpam-5570	368	50	-	-	PUNCT
ejpam-5570	368	51	βparacompactness	βparacompactness	NOUN
ejpam-5570	368	52	)	)	PUNCT
ejpam-5570	368	53	.	.	PUNCT
ejpam-5570	369	1	we	we	PRON
ejpam-5570	369	2	found	find	VERB
ejpam-5570	369	3	that	that	SCONJ
ejpam-5570	369	4	every	every	DET
ejpam-5570	369	5	i	i	PROPN
ejpam-5570	369	6	-	-	PUNCT
ejpam-5570	369	7	β	β	NOUN
ejpam-5570	369	8	-	-	ADJ
ejpam-5570	369	9	paracompact	paracompact	ADJ
ejpam-5570	369	10	space	space	NOUN
ejpam-5570	369	11	is	be	AUX
ejpam-5570	369	12	a	a	DET
ejpam-5570	369	13	δ	δ	PROPN
ejpam-5570	369	14	-	-	PUNCT
ejpam-5570	369	15	βi	βi	PRON
ejpam-5570	369	16	-	-	PUNCT
ejpam-5570	369	17	paracompact	paracompact	ADJ
ejpam-5570	369	18	space	space	NOUN
ejpam-5570	369	19	,	,	PUNCT
ejpam-5570	369	20	and	and	CCONJ
ejpam-5570	369	21	every	every	DET
ejpam-5570	369	22	hausdorff	hausdorff	PROPN
ejpam-5570	369	23	δ	δ	PROPN
ejpam-5570	369	24	-	-	PUNCT
ejpam-5570	369	25	βi	βi	ADV
ejpam-5570	369	26	-	-	PUNCT
ejpam-5570	369	27	paracompact	paracompact	ADJ
ejpam-5570	369	28	space	space	NOUN
ejpam-5570	369	29	under	under	ADP
ejpam-5570	369	30	some	some	DET
ejpam-5570	369	31	conditions	condition	NOUN
ejpam-5570	369	32	is	be	AUX
ejpam-5570	369	33	δ	δ	PROPN
ejpam-5570	369	34	-	-	PUNCT
ejpam-5570	369	35	βi	βi	ADV
ejpam-5570	369	36	-	-	NOUN
ejpam-5570	369	37	regular	regular	ADJ
ejpam-5570	369	38	.	.	PUNCT
ejpam-5570	370	1	the	the	DET
ejpam-5570	370	2	union	union	NOUN
ejpam-5570	370	3	of	of	ADP
ejpam-5570	370	4	two	two	NUM
ejpam-5570	370	5	δ	δ	PROPN
ejpam-5570	370	6	-	-	PUNCT
ejpam-5570	370	7	βi	βi	PRON
ejpam-5570	370	8	-	-	PUNCT
ejpam-5570	370	9	paracompact	paracompact	ADJ
ejpam-5570	370	10	subsets	subset	NOUN
ejpam-5570	370	11	is	be	AUX
ejpam-5570	370	12	a	a	DET
ejpam-5570	370	13	δ	δ	PROPN
ejpam-5570	370	14	-	-	PUNCT
ejpam-5570	370	15	βi	βi	PRON
ejpam-5570	370	16	-	-	PUNCT
ejpam-5570	370	17	paracompact	paracompact	NOUN
ejpam-5570	370	18	subset	subset	NOUN
ejpam-5570	370	19	,	,	PUNCT
ejpam-5570	370	20	and	and	CCONJ
ejpam-5570	370	21	the	the	DET
ejpam-5570	370	22	intersection	intersection	NOUN
ejpam-5570	370	23	of	of	ADP
ejpam-5570	370	24	a	a	DET
ejpam-5570	370	25	δ	δ	PROPN
ejpam-5570	370	26	-	-	PUNCT
ejpam-5570	370	27	βi	βi	ADV
ejpam-5570	370	28	-	-	PUNCT
ejpam-5570	370	29	paracompact	paracompact	NOUN
ejpam-5570	370	30	subset	subset	NOUN
ejpam-5570	370	31	and	and	CCONJ
ejpam-5570	370	32	a	a	DET
ejpam-5570	370	33	δ	δ	PROPN
ejpam-5570	370	34	-	-	PUNCT
ejpam-5570	370	35	βi	βi	ADV
ejpam-5570	370	36	-	-	PUNCT
ejpam-5570	370	37	closed	close	VERB
ejpam-5570	370	38	set	set	NOUN
ejpam-5570	370	39	is	be	AUX
ejpam-5570	370	40	δ	δ	PROPN
ejpam-5570	370	41	-	-	PUNCT
ejpam-5570	370	42	βi	βi	ADV
ejpam-5570	370	43	-	-	PUNCT
ejpam-5570	370	44	paracompact	paracompact	ADJ
ejpam-5570	370	45	.	.	PUNCT
ejpam-5570	371	1	in	in	ADP
ejpam-5570	371	2	addition	addition	NOUN
ejpam-5570	371	3	,	,	PUNCT
ejpam-5570	371	4	we	we	PRON
ejpam-5570	371	5	illustrate	illustrate	VERB
ejpam-5570	371	6	that	that	SCONJ
ejpam-5570	371	7	δ	δ	PROPN
ejpam-5570	371	8	-	-	PUNCT
ejpam-5570	371	9	βi	βi	PROPN
ejpam-5570	371	10	-	-	NOUN
ejpam-5570	371	11	paracompactness	paracompactness	NOUN
ejpam-5570	371	12	is	be	AUX
ejpam-5570	371	13	preserved	preserve	VERB
ejpam-5570	371	14	under	under	ADP
ejpam-5570	371	15	certain	certain	ADJ
ejpam-5570	371	16	conditions	condition	NOUN
ejpam-5570	371	17	.	.	PUNCT
ejpam-5570	372	1	if	if	SCONJ
ejpam-5570	372	2	f	f	PROPN
ejpam-5570	372	3	:	:	PUNCT
ejpam-5570	372	4	x	x	X
ejpam-5570	372	5	→	→	SYM
ejpam-5570	372	6	y	y	PROPN
ejpam-5570	372	7	is	be	AUX
ejpam-5570	372	8	δ	δ	PROPN
ejpam-5570	372	9	-	-	PUNCT
ejpam-5570	372	10	βi	βi	ADV
ejpam-5570	372	11	-	-	PUNCT
ejpam-5570	372	12	irresolute	irresolute	ADJ
ejpam-5570	372	13	,	,	PUNCT
ejpam-5570	372	14	continuous	continuous	ADJ
ejpam-5570	372	15	,	,	PUNCT
ejpam-5570	372	16	δ	δ	PROPN
ejpam-5570	372	17	-	-	PUNCT
ejpam-5570	372	18	βi	βi	ADV
ejpam-5570	372	19	-	-	PUNCT
ejpam-5570	372	20	open	open	ADJ
ejpam-5570	372	21	,	,	PUNCT
ejpam-5570	372	22	and	and	CCONJ
ejpam-5570	372	23	surjective	surjective	ADJ
ejpam-5570	372	24	,	,	PUNCT
ejpam-5570	372	25	and	and	CCONJ
ejpam-5570	372	26	x	x	X
ejpam-5570	372	27	is	be	AUX
ejpam-5570	372	28	δ	δ	PROPN
ejpam-5570	372	29	-	-	PUNCT
ejpam-5570	372	30	βi	βi	ADV
ejpam-5570	372	31	-	-	PUNCT
ejpam-5570	372	32	paracompact	paracompact	ADJ
ejpam-5570	372	33	,	,	PUNCT
ejpam-5570	372	34	then	then	ADV
ejpam-5570	372	35	y	y	PROPN
ejpam-5570	372	36	is	be	AUX
ejpam-5570	372	37	δ	δ	PROPN
ejpam-5570	372	38	-	-	PUNCT
ejpam-5570	372	39	βf(i)-paracompact	βf(i)-paracompact	PROPN
ejpam-5570	372	40	.	.	PUNCT
ejpam-5570	373	1	additionally	additionally	ADV
ejpam-5570	373	2	,	,	PUNCT
ejpam-5570	373	3	provided	provide	VERB
ejpam-5570	373	4	that	that	SCONJ
ejpam-5570	373	5	f	f	NOUN
ejpam-5570	373	6	:	:	PUNCT
ejpam-5570	373	7	x	x	X
ejpam-5570	373	8	→	→	SYM
ejpam-5570	373	9	y	y	PROPN
ejpam-5570	373	10	is	be	AUX
ejpam-5570	373	11	open	open	ADJ
ejpam-5570	373	12	,	,	PUNCT
ejpam-5570	373	13	δ	δ	NOUN
ejpam-5570	373	14	-	-	NOUN
ejpam-5570	373	15	βiirresolute	βiirresolute	NOUN
ejpam-5570	373	16	,	,	PUNCT
ejpam-5570	373	17	bijective	bijective	ADJ
ejpam-5570	373	18	,	,	PUNCT
ejpam-5570	373	19	and	and	CCONJ
ejpam-5570	373	20	y	y	PROPN
ejpam-5570	373	21	is	be	AUX
ejpam-5570	373	22	δ	δ	PROPN
ejpam-5570	373	23	-	-	PUNCT
ejpam-5570	373	24	βf(i)-paracompact	βf(i)-paracompact	PROPN
ejpam-5570	373	25	,	,	PUNCT
ejpam-5570	373	26	then	then	ADV
ejpam-5570	373	27	x	x	PUNCT
ejpam-5570	373	28	is	be	AUX
ejpam-5570	373	29	δ	δ	PROPN
ejpam-5570	373	30	-	-	PUNCT
ejpam-5570	373	31	βi	βi	ADV
ejpam-5570	373	32	-	-	PUNCT
ejpam-5570	373	33	paracompact	paracompact	NOUN
ejpam-5570	373	34	.	.	PUNCT
ejpam-5570	374	1	acknowledgements	acknowledgement	NOUN
ejpam-5570	374	2	we	we	PRON
ejpam-5570	374	3	sincerely	sincerely	ADV
ejpam-5570	374	4	appreciate	appreciate	VERB
ejpam-5570	374	5	all	all	DET
ejpam-5570	374	6	those	those	PRON
ejpam-5570	374	7	who	who	PRON
ejpam-5570	374	8	contributed	contribute	VERB
ejpam-5570	374	9	to	to	ADP
ejpam-5570	374	10	our	our	PRON
ejpam-5570	374	11	research	research	NOUN
ejpam-5570	374	12	.	.	PUNCT
ejpam-5570	375	1	their	their	PRON
ejpam-5570	375	2	direction	direction	NOUN
ejpam-5570	375	3	,	,	PUNCT
ejpam-5570	375	4	collaboration	collaboration	NOUN
ejpam-5570	375	5	,	,	PUNCT
ejpam-5570	375	6	and	and	CCONJ
ejpam-5570	375	7	assistance	assistance	NOUN
ejpam-5570	375	8	have	have	AUX
ejpam-5570	375	9	been	be	AUX
ejpam-5570	375	10	essential	essential	ADJ
ejpam-5570	375	11	to	to	ADP
ejpam-5570	375	12	the	the	DET
ejpam-5570	375	13	success	success	NOUN
ejpam-5570	375	14	of	of	ADP
ejpam-5570	375	15	this	this	DET
ejpam-5570	375	16	project	project	NOUN
ejpam-5570	375	17	.	.	PUNCT
ejpam-5570	376	1	this	this	DET
ejpam-5570	376	2	research	research	NOUN
ejpam-5570	376	3	was	be	AUX
ejpam-5570	376	4	supported	support	VERB
ejpam-5570	376	5	by	by	ADP
ejpam-5570	376	6	the	the	DET
ejpam-5570	376	7	national	national	ADJ
ejpam-5570	376	8	science	science	NOUN
ejpam-5570	376	9	,	,	PUNCT
ejpam-5570	376	10	research	research	NOUN
ejpam-5570	376	11	,	,	PUNCT
ejpam-5570	376	12	and	and	CCONJ
ejpam-5570	376	13	innovation	innovation	NOUN
ejpam-5570	376	14	fund	fund	NOUN
ejpam-5570	376	15	(	(	PUNCT
ejpam-5570	376	16	nsrf	nsrf	NOUN
ejpam-5570	376	17	)	)	PUNCT
ejpam-5570	376	18	and	and	CCONJ
ejpam-5570	376	19	prince	prince	NOUN
ejpam-5570	376	20	of	of	ADP
ejpam-5570	376	21	songkla	songkla	PROPN
ejpam-5570	376	22	university	university	PROPN
ejpam-5570	376	23	(	(	PUNCT
ejpam-5570	376	24	ref	ref	NOUN
ejpam-5570	376	25	.	.	PUNCT
ejpam-5570	377	1	no	no	INTJ
ejpam-5570	377	2	.	.	PUNCT
ejpam-5570	377	3	sat6701343s	sat6701343s	PROPN
ejpam-5570	377	4	)	)	PUNCT
ejpam-5570	377	5	.	.	PUNCT
ejpam-5570	378	1	references	reference	NOUN
ejpam-5570	378	2	[	[	X
ejpam-5570	378	3	1	1	NUM
ejpam-5570	378	4	]	]	PUNCT
ejpam-5570	378	5	k.	k.	PROPN
ejpam-5570	378	6	al	al	PROPN
ejpam-5570	378	7	-	-	PROPN
ejpam-5570	378	8	zoubi	zoubi	PROPN
ejpam-5570	378	9	.	.	PUNCT
ejpam-5570	379	1	s	s	X
ejpam-5570	379	2	-	-	PUNCT
ejpam-5570	379	3	paracompact	paracompact	ADJ
ejpam-5570	379	4	spaces	space	NOUN
ejpam-5570	379	5	.	.	PUNCT
ejpam-5570	380	1	acta	acta	PROPN
ejpam-5570	380	2	mathematica	mathematica	PROPN
ejpam-5570	380	3	hungarica	hungarica	PROPN
ejpam-5570	380	4	,	,	PUNCT
ejpam-5570	380	5	110:203–212	110:203–212	NUM
ejpam-5570	380	6	,	,	PUNCT
ejpam-5570	380	7	2006	2006	NUM
ejpam-5570	380	8	.	.	PUNCT
ejpam-5570	381	1	[	[	X
ejpam-5570	381	2	2	2	X
ejpam-5570	381	3	]	]	PUNCT
ejpam-5570	381	4	k.	k.	PROPN
ejpam-5570	381	5	al	al	PROPN
ejpam-5570	381	6	-	-	PROPN
ejpam-5570	381	7	zoubi	zoubi	PROPN
ejpam-5570	381	8	and	and	CCONJ
ejpam-5570	381	9	s.	s.	PROPN
ejpam-5570	381	10	al	al	PROPN
ejpam-5570	381	11	-	-	PROPN
ejpam-5570	381	12	ghour	ghour	PROPN
ejpam-5570	381	13	.	.	PUNCT
ejpam-5570	382	1	on	on	ADP
ejpam-5570	382	2	p3	p3	NOUN
ejpam-5570	382	3	-	-	PUNCT
ejpam-5570	382	4	paracompact	paracompact	ADJ
ejpam-5570	382	5	spaces	space	NOUN
ejpam-5570	382	6	.	.	PUNCT
ejpam-5570	383	1	international	international	ADJ
ejpam-5570	383	2	journal	journal	PROPN
ejpam-5570	383	3	of	of	ADP
ejpam-5570	383	4	mathematics	mathematics	PROPN
ejpam-5570	383	5	and	and	CCONJ
ejpam-5570	383	6	mathematical	mathematical	ADJ
ejpam-5570	383	7	sciences	science	NOUN
ejpam-5570	383	8	,	,	PUNCT
ejpam-5570	383	9	2007:1–16	2007:1–16	NUM
ejpam-5570	383	10	,	,	PUNCT
ejpam-5570	383	11	2007	2007	NUM
ejpam-5570	383	12	.	.	PUNCT
ejpam-5570	384	1	[	[	X
ejpam-5570	384	2	3	3	NUM
ejpam-5570	384	3	]	]	X
ejpam-5570	384	4	i.	i.	PROPN
ejpam-5570	384	5	demir	demir	PROPN
ejpam-5570	384	6	and	and	CCONJ
ejpam-5570	384	7	o.b	o.b	PROPN
ejpam-5570	384	8	.	.	PROPN
ejpam-5570	384	9	ozbakir	ozbakir	PROPN
ejpam-5570	384	10	.	.	PUNCT
ejpam-5570	385	1	on	on	ADP
ejpam-5570	385	2	β	β	ADJ
ejpam-5570	385	3	-	-	ADJ
ejpam-5570	385	4	paracompact	paracompact	ADJ
ejpam-5570	385	5	spaces	space	NOUN
ejpam-5570	385	6	.	.	PUNCT
ejpam-5570	386	1	filomat	filomat	NOUN
ejpam-5570	386	2	,	,	PUNCT
ejpam-5570	386	3	27(6):971–976	27(6):971–976	PROPN
ejpam-5570	386	4	,	,	PUNCT
ejpam-5570	386	5	2013	2013	NUM
ejpam-5570	386	6	.	.	PUNCT
ejpam-5570	387	1	[	[	X
ejpam-5570	387	2	4	4	X
ejpam-5570	387	3	]	]	PUNCT
ejpam-5570	387	4	j.	j.	PROPN
ejpam-5570	387	5	dieudonné.	dieudonné.	PROPN
ejpam-5570	387	6	une	une	PROPN
ejpam-5570	387	7	généralisation	généralisation	PROPN
ejpam-5570	387	8	des	des	PROPN
ejpam-5570	387	9	espaces	espace	NOUN
ejpam-5570	387	10	compacts	compact	NOUN
ejpam-5570	387	11	.	.	PUNCT
ejpam-5570	388	1	journal	journal	PROPN
ejpam-5570	388	2	de	de	PROPN
ejpam-5570	388	3	mathématiques	mathématiques	PROPN
ejpam-5570	388	4	pures	pure	NOUN
ejpam-5570	388	5	et	et	NOUN
ejpam-5570	388	6	appliquées	appliquée	NOUN
ejpam-5570	388	7	,	,	PUNCT
ejpam-5570	388	8	23(9):65–76	23(9):65–76	NUM
ejpam-5570	388	9	,	,	PUNCT
ejpam-5570	388	10	1944	1944	NUM
ejpam-5570	388	11	.	.	PUNCT
ejpam-5570	389	1	[	[	X
ejpam-5570	389	2	5	5	X
ejpam-5570	389	3	]	]	PUNCT
ejpam-5570	389	4	j.	j.	PROPN
ejpam-5570	389	5	dontchev	dontchev	PROPN
ejpam-5570	389	6	,	,	PUNCT
ejpam-5570	389	7	m.	m.	NOUN
ejpam-5570	389	8	ganster	ganster	NOUN
ejpam-5570	389	9	,	,	PUNCT
ejpam-5570	389	10	and	and	CCONJ
ejpam-5570	389	11	t.	t.	PROPN
ejpam-5570	389	12	noiri	noiri	PROPN
ejpam-5570	389	13	.	.	PUNCT
ejpam-5570	390	1	unified	unified	ADJ
ejpam-5570	390	2	operation	operation	NOUN
ejpam-5570	390	3	approach	approach	NOUN
ejpam-5570	390	4	of	of	ADP
ejpam-5570	390	5	generalized	generalized	ADJ
ejpam-5570	390	6	closed	close	VERB
ejpam-5570	390	7	sets	set	NOUN
ejpam-5570	390	8	via	via	ADP
ejpam-5570	390	9	topological	topological	ADJ
ejpam-5570	390	10	ideals	ideal	NOUN
ejpam-5570	390	11	.	.	PUNCT
ejpam-5570	391	1	mathematica	mathematica	PROPN
ejpam-5570	391	2	japonica	japonica	PROPN
ejpam-5570	391	3	,	,	PUNCT
ejpam-5570	391	4	49:395–402	49:395–402	PROPN
ejpam-5570	391	5	,	,	PUNCT
ejpam-5570	391	6	1999	1999	NUM
ejpam-5570	391	7	.	.	PUNCT
ejpam-5570	392	1	[	[	X
ejpam-5570	392	2	6	6	NUM
ejpam-5570	392	3	]	]	PUNCT
ejpam-5570	392	4	j.	j.	PROPN
ejpam-5570	392	5	dugundji	dugundji	PROPN
ejpam-5570	392	6	.	.	PUNCT
ejpam-5570	392	7	topology	topology	PROPN
ejpam-5570	392	8	.	.	PUNCT
ejpam-5570	393	1	allyn	allyn	PROPN
ejpam-5570	393	2	and	and	CCONJ
ejpam-5570	393	3	bacon	bacon	PROPN
ejpam-5570	393	4	,	,	PUNCT
ejpam-5570	393	5	boston	boston	PROPN
ejpam-5570	393	6	,	,	PUNCT
ejpam-5570	393	7	1966	1966	NUM
ejpam-5570	393	8	.	.	PUNCT
ejpam-5570	394	1	[	[	X
ejpam-5570	394	2	7	7	X
ejpam-5570	394	3	]	]	PUNCT
ejpam-5570	394	4	m.	m.	NOUN
ejpam-5570	394	5	e.	e.	PROPN
ejpam-5570	394	6	abd	abd	PROPN
ejpam-5570	395	1	el	el	PROPN
ejpam-5570	395	2	-	-	PROPN
ejpam-5570	395	3	monsef	monsef	PROPN
ejpam-5570	395	4	,	,	PUNCT
ejpam-5570	395	5	e.	e.	PROPN
ejpam-5570	395	6	f.	f.	PROPN
ejpam-5570	395	7	lashien	lashien	PROPN
ejpam-5570	395	8	,	,	PUNCT
ejpam-5570	395	9	and	and	CCONJ
ejpam-5570	395	10	a.	a.	NOUN
ejpam-5570	395	11	a.	a.	NOUN
ejpam-5570	395	12	nasef	nasef	PROPN
ejpam-5570	395	13	.	.	PUNCT
ejpam-5570	396	1	on	on	ADP
ejpam-5570	396	2	i	i	NOUN
ejpam-5570	396	3	-	-	PUNCT
ejpam-5570	396	4	open	open	ADJ
ejpam-5570	396	5	sets	set	NOUN
ejpam-5570	396	6	and	and	CCONJ
ejpam-5570	396	7	icontinuous	icontinuous	ADJ
ejpam-5570	396	8	functions	function	NOUN
ejpam-5570	396	9	.	.	PUNCT
ejpam-5570	397	1	kyungpook	kyungpook	PROPN
ejpam-5570	397	2	mathematical	mathematical	PROPN
ejpam-5570	397	3	journal	journal	PROPN
ejpam-5570	397	4	,	,	PUNCT
ejpam-5570	397	5	32(2):21–30	32(2):21–30	NUM
ejpam-5570	397	6	,	,	PUNCT
ejpam-5570	397	7	1992	1992	NUM
ejpam-5570	397	8	.	.	PUNCT
ejpam-5570	398	1	[	[	X
ejpam-5570	398	2	8	8	NUM
ejpam-5570	398	3	]	]	PUNCT
ejpam-5570	398	4	m.	m.	NOUN
ejpam-5570	398	5	e.	e.	PROPN
ejpam-5570	398	6	abd	abd	PROPN
ejpam-5570	398	7	el	el	PROPN
ejpam-5570	398	8	-	-	PROPN
ejpam-5570	398	9	monsef	monsef	PROPN
ejpam-5570	398	10	,	,	PUNCT
ejpam-5570	398	11	e.	e.	PROPN
ejpam-5570	398	12	f.	f.	PROPN
ejpam-5570	398	13	lashien	lashien	PROPN
ejpam-5570	398	14	,	,	PUNCT
ejpam-5570	398	15	and	and	CCONJ
ejpam-5570	398	16	a.	a.	NOUN
ejpam-5570	398	17	a.	a.	NOUN
ejpam-5570	398	18	nasef	nasef	PROPN
ejpam-5570	398	19	.	.	PUNCT
ejpam-5570	399	1	some	some	DET
ejpam-5570	399	2	topological	topological	ADJ
ejpam-5570	399	3	operators	operator	NOUN
ejpam-5570	399	4	via	via	ADP
ejpam-5570	399	5	ideals	ideal	NOUN
ejpam-5570	399	6	.	.	PUNCT
ejpam-5570	400	1	kyungpook	kyungpook	PROPN
ejpam-5570	400	2	mathematical	mathematical	PROPN
ejpam-5570	400	3	journal	journal	PROPN
ejpam-5570	400	4	,	,	PUNCT
ejpam-5570	400	5	32(2):273284	32(2):273284	NUM
ejpam-5570	400	6	,	,	PUNCT
ejpam-5570	400	7	1992	1992	NUM
ejpam-5570	400	8	.	.	PUNCT
ejpam-5570	401	1	c.	c.	PROPN
ejpam-5570	401	2	boonpok	boonpok	PROPN
ejpam-5570	401	3	,	,	PUNCT
ejpam-5570	401	4	a.	a.	PROPN
ejpam-5570	401	5	sama	sama	PROPN
ejpam-5570	401	6	-	-	PUNCT
ejpam-5570	401	7	ae	ae	PROPN
ejpam-5570	401	8	,	,	PUNCT
ejpam-5570	401	9	p.	p.	NOUN
ejpam-5570	401	10	raktaow	raktaow	PROPN
ejpam-5570	401	11	/	/	SYM
ejpam-5570	401	12	eur	eur	PROPN
ejpam-5570	401	13	.	.	PUNCT
ejpam-5570	402	1	j.	j.	PROPN
ejpam-5570	402	2	pure	pure	PROPN
ejpam-5570	402	3	appl	appl	PROPN
ejpam-5570	402	4	.	.	PROPN
ejpam-5570	402	5	math	math	PROPN
ejpam-5570	402	6	,	,	PUNCT
ejpam-5570	402	7	18	18	NUM
ejpam-5570	402	8	(	(	PUNCT
ejpam-5570	402	9	1	1	NUM
ejpam-5570	402	10	)	)	PUNCT
ejpam-5570	402	11	(	(	PUNCT
ejpam-5570	402	12	2025	2025	NUM
ejpam-5570	402	13	)	)	PUNCT
ejpam-5570	402	14	,	,	PUNCT
ejpam-5570	402	15	5570	5570	NUM
ejpam-5570	402	16	12	12	NUM
ejpam-5570	402	17	of	of	ADP
ejpam-5570	402	18	12	12	NUM
ejpam-5570	402	19	[	[	SYM
ejpam-5570	402	20	9	9	NUM
ejpam-5570	402	21	]	]	PUNCT
ejpam-5570	402	22	t.	t.	PROPN
ejpam-5570	402	23	r.	r.	PROPN
ejpam-5570	402	24	hamlett	hamlett	PROPN
ejpam-5570	402	25	,	,	PUNCT
ejpam-5570	402	26	d.	d.	PROPN
ejpam-5570	402	27	rose	rise	VERB
ejpam-5570	402	28	,	,	PUNCT
ejpam-5570	402	29	and	and	CCONJ
ejpam-5570	402	30	d.	d.	PROPN
ejpam-5570	402	31	jankovic	jankovic	PROPN
ejpam-5570	402	32	.	.	PUNCT
ejpam-5570	403	1	paracompactness	paracompactness	PROPN
ejpam-5570	403	2	with	with	ADP
ejpam-5570	403	3	respect	respect	NOUN
ejpam-5570	403	4	to	to	ADP
ejpam-5570	403	5	an	an	DET
ejpam-5570	403	6	ideal	ideal	NOUN
ejpam-5570	403	7	.	.	PUNCT
ejpam-5570	404	1	international	international	ADJ
ejpam-5570	404	2	journal	journal	PROPN
ejpam-5570	404	3	of	of	ADP
ejpam-5570	404	4	mathematics	mathematics	PROPN
ejpam-5570	404	5	and	and	CCONJ
ejpam-5570	404	6	mathematical	mathematical	ADJ
ejpam-5570	404	7	sciences	science	NOUN
ejpam-5570	404	8	,	,	PUNCT
ejpam-5570	404	9	20:433–442	20:433–442	NUM
ejpam-5570	404	10	,	,	PUNCT
ejpam-5570	404	11	1997	1997	NUM
ejpam-5570	404	12	.	.	PUNCT
ejpam-5570	405	1	[	[	X
ejpam-5570	405	2	10	10	NUM
ejpam-5570	405	3	]	]	X
ejpam-5570	405	4	e.	e.	PROPN
ejpam-5570	405	5	hatir	hatir	PROPN
ejpam-5570	405	6	.	.	PUNCT
ejpam-5570	406	1	on	on	ADP
ejpam-5570	406	2	decompositions	decomposition	NOUN
ejpam-5570	406	3	of	of	ADP
ejpam-5570	406	4	continuity	continuity	NOUN
ejpam-5570	406	5	and	and	CCONJ
ejpam-5570	406	6	complete	complete	ADJ
ejpam-5570	406	7	continuity	continuity	NOUN
ejpam-5570	406	8	in	in	ADP
ejpam-5570	406	9	ideal	ideal	ADJ
ejpam-5570	406	10	topological	topological	ADJ
ejpam-5570	406	11	spaces	space	NOUN
ejpam-5570	406	12	.	.	PUNCT
ejpam-5570	407	1	european	european	ADJ
ejpam-5570	407	2	journal	journal	PROPN
ejpam-5570	407	3	of	of	ADP
ejpam-5570	407	4	pure	pure	ADJ
ejpam-5570	407	5	and	and	CCONJ
ejpam-5570	407	6	applied	applied	ADJ
ejpam-5570	407	7	mathematics	mathematic	NOUN
ejpam-5570	407	8	,	,	PUNCT
ejpam-5570	407	9	6(3):352–362	6(3):352–362	NUM
ejpam-5570	407	10	,	,	PUNCT
ejpam-5570	407	11	2013	2013	NUM
ejpam-5570	407	12	.	.	PUNCT
ejpam-5570	408	1	[	[	X
ejpam-5570	408	2	11	11	NUM
ejpam-5570	408	3	]	]	X
ejpam-5570	408	4	d.	d.	PROPN
ejpam-5570	408	5	jankovic	jankovic	PROPN
ejpam-5570	408	6	and	and	CCONJ
ejpam-5570	408	7	t.	t.	PROPN
ejpam-5570	408	8	r.	r.	PROPN
ejpam-5570	408	9	hamlett	hamlett	PROPN
ejpam-5570	408	10	.	.	PUNCT
ejpam-5570	409	1	new	new	ADJ
ejpam-5570	409	2	topologies	topology	NOUN
ejpam-5570	409	3	from	from	ADP
ejpam-5570	409	4	old	old	ADJ
ejpam-5570	409	5	via	via	ADP
ejpam-5570	409	6	ideals	ideal	NOUN
ejpam-5570	409	7	.	.	PUNCT
ejpam-5570	410	1	the	the	DET
ejpam-5570	410	2	american	american	PROPN
ejpam-5570	410	3	mathematical	mathematical	PROPN
ejpam-5570	410	4	monthly	monthly	PROPN
ejpam-5570	410	5	,	,	PUNCT
ejpam-5570	410	6	97(4):295–310	97(4):295–310	PROPN
ejpam-5570	410	7	,	,	PUNCT
ejpam-5570	410	8	1990	1990	NUM
ejpam-5570	410	9	.	.	PUNCT
ejpam-5570	411	1	[	[	X
ejpam-5570	411	2	12	12	NUM
ejpam-5570	411	3	]	]	PUNCT
ejpam-5570	411	4	m.	m.	NOUN
ejpam-5570	411	5	khan	khan	PROPN
ejpam-5570	411	6	and	and	CCONJ
ejpam-5570	411	7	t.	t.	PROPN
ejpam-5570	411	8	noiri	noiri	PROPN
ejpam-5570	411	9	.	.	PUNCT
ejpam-5570	412	1	semi	semi	ADJ
ejpam-5570	412	2	-	-	ADJ
ejpam-5570	412	3	local	local	ADJ
ejpam-5570	412	4	functions	function	NOUN
ejpam-5570	412	5	in	in	ADP
ejpam-5570	412	6	ideal	ideal	ADJ
ejpam-5570	412	7	topological	topological	ADJ
ejpam-5570	412	8	spaces	space	NOUN
ejpam-5570	412	9	.	.	PUNCT
ejpam-5570	413	1	journal	journal	NOUN
ejpam-5570	413	2	of	of	ADP
ejpam-5570	413	3	advanced	advanced	ADJ
ejpam-5570	413	4	research	research	NOUN
ejpam-5570	413	5	in	in	ADP
ejpam-5570	413	6	pure	pure	ADJ
ejpam-5570	413	7	mathematics	mathematic	NOUN
ejpam-5570	413	8	,	,	PUNCT
ejpam-5570	413	9	2(1):36–42	2(1):36–42	NUM
ejpam-5570	413	10	,	,	PUNCT
ejpam-5570	413	11	2010	2010	NUM
ejpam-5570	413	12	.	.	PUNCT
ejpam-5570	414	1	[	[	X
ejpam-5570	414	2	13	13	NUM
ejpam-5570	414	3	]	]	PUNCT
ejpam-5570	414	4	k.	k.	PROPN
ejpam-5570	414	5	kuratowski	kuratowski	PROPN
ejpam-5570	414	6	.	.	PUNCT
ejpam-5570	415	1	topology	topology	PROPN
ejpam-5570	415	2	i.	i.	PROPN
ejpam-5570	415	3	panstwowe	panstwowe	PROPN
ejpam-5570	415	4	wydawnictwo	wydawnictwo	PROPN
ejpam-5570	415	5	naukowe	naukowe	PROPN
ejpam-5570	415	6	,	,	PUNCT
ejpam-5570	415	7	warszawa	warszawa	PROPN
ejpam-5570	415	8	,	,	PUNCT
ejpam-5570	415	9	1933	1933	NUM
ejpam-5570	415	10	.	.	PUNCT
ejpam-5570	416	1	[	[	X
ejpam-5570	416	2	14	14	NUM
ejpam-5570	416	3	]	]	PUNCT
ejpam-5570	416	4	p.	p.	PROPN
ejpam-5570	416	5	y.	y.	PROPN
ejpam-5570	416	6	li	li	PROPN
ejpam-5570	416	7	and	and	CCONJ
ejpam-5570	416	8	y.	y.	PROPN
ejpam-5570	416	9	k.	k.	PROPN
ejpam-5570	416	10	song	song	PROPN
ejpam-5570	416	11	.	.	PUNCT
ejpam-5570	417	1	some	some	DET
ejpam-5570	417	2	remarks	remark	NOUN
ejpam-5570	417	3	on	on	ADP
ejpam-5570	417	4	s	s	NOUN
ejpam-5570	417	5	-	-	PUNCT
ejpam-5570	417	6	paracompact	paracompact	ADJ
ejpam-5570	417	7	spaces	space	NOUN
ejpam-5570	417	8	.	.	PUNCT
ejpam-5570	418	1	acta	acta	PROPN
ejpam-5570	418	2	mathematica	mathematica	PROPN
ejpam-5570	418	3	hungarica	hungarica	PROPN
ejpam-5570	418	4	,	,	PUNCT
ejpam-5570	418	5	118:345–355	118:345–355	NUM
ejpam-5570	418	6	,	,	PUNCT
ejpam-5570	418	7	2008	2008	NUM
ejpam-5570	418	8	.	.	PUNCT
ejpam-5570	419	1	[	[	X
ejpam-5570	419	2	15	15	NUM
ejpam-5570	419	3	]	]	X
ejpam-5570	419	4	j.	j.	PROPN
ejpam-5570	419	5	sanabria	sanabria	PROPN
ejpam-5570	419	6	,	,	PUNCT
ejpam-5570	419	7	e.	e.	PROPN
ejpam-5570	419	8	rosas	rosas	PROPN
ejpam-5570	419	9	,	,	PUNCT
ejpam-5570	419	10	c.	c.	PROPN
ejpam-5570	419	11	carpintero	carpintero	PROPN
ejpam-5570	419	12	,	,	PUNCT
ejpam-5570	419	13	m.	m.	NOUN
ejpam-5570	419	14	salas	salas	PROPN
ejpam-5570	419	15	-	-	PUNCT
ejpam-5570	419	16	brown	brown	PROPN
ejpam-5570	419	17	,	,	PUNCT
ejpam-5570	419	18	and	and	CCONJ
ejpam-5570	419	19	o.	o.	PROPN
ejpam-5570	419	20	garćıa	garćıa	PROPN
ejpam-5570	419	21	.	.	PROPN
ejpam-5570	419	22	sparacompactness	sparacompactness	PROPN
ejpam-5570	419	23	in	in	ADP
ejpam-5570	419	24	ideal	ideal	ADJ
ejpam-5570	419	25	topological	topological	ADJ
ejpam-5570	419	26	spaces	space	NOUN
ejpam-5570	419	27	.	.	PUNCT
ejpam-5570	420	1	matematički	matematički	PROPN
ejpam-5570	420	2	vesnik	vesnik	PROPN
ejpam-5570	420	3	,	,	PUNCT
ejpam-5570	420	4	68(3):192–203	68(3):192–203	NOUN
ejpam-5570	420	5	,	,	PUNCT
ejpam-5570	420	6	2016	2016	NUM
ejpam-5570	420	7	.	.	PUNCT
ejpam-5570	421	1	[	[	X
ejpam-5570	421	2	16	16	NUM
ejpam-5570	421	3	]	]	X
ejpam-5570	421	4	n.	n.	NOUN
ejpam-5570	421	5	sathiyasundari	sathiyasundari	PROPN
ejpam-5570	421	6	and	and	CCONJ
ejpam-5570	421	7	v.	v.	ADP
ejpam-5570	421	8	renukadevi	renukadevi	NOUN
ejpam-5570	421	9	.	.	PUNCT
ejpam-5570	422	1	paracompactness	paracompactness	NOUN
ejpam-5570	422	2	with	with	ADP
ejpam-5570	422	3	respect	respect	NOUN
ejpam-5570	422	4	to	to	ADP
ejpam-5570	422	5	an	an	DET
ejpam-5570	422	6	ideal	ideal	NOUN
ejpam-5570	422	7	.	.	PUNCT
ejpam-5570	423	1	filomat	filomat	NOUN
ejpam-5570	423	2	,	,	PUNCT
ejpam-5570	423	3	20(2):333–339	20(2):333–339	NOUN
ejpam-5570	423	4	,	,	PUNCT
ejpam-5570	423	5	2013	2013	NUM
ejpam-5570	423	6	.	.	PUNCT
ejpam-5570	424	1	[	[	X
ejpam-5570	424	2	17	17	NUM
ejpam-5570	424	3	]	]	X
ejpam-5570	424	4	e.	e.	PROPN
ejpam-5570	424	5	d.	d.	PROPN
ejpam-5570	424	6	yildirim	yildirim	PROPN
ejpam-5570	424	7	,	,	PUNCT
ejpam-5570	424	8	o.	o.	PROPN
ejpam-5570	424	9	b.	b.	PROPN
ejpam-5570	424	10	ozbakir	ozbakir	PROPN
ejpam-5570	424	11	,	,	PUNCT
ejpam-5570	424	12	and	and	CCONJ
ejpam-5570	424	13	a.	a.	PROPN
ejpam-5570	424	14	c.	c.	PROPN
ejpam-5570	424	15	guler	guler	NOUN
ejpam-5570	424	16	.	.	PUNCT
ejpam-5570	425	1	β	β	X
ejpam-5570	425	2	-	-	PUNCT
ejpam-5570	425	3	paracompactness	paracompactness	NOUN
ejpam-5570	425	4	in	in	ADP
ejpam-5570	425	5	ideal	ideal	ADJ
ejpam-5570	425	6	topological	topological	ADJ
ejpam-5570	425	7	space	space	NOUN
ejpam-5570	425	8	.	.	PUNCT
ejpam-5570	426	1	european	european	ADJ
ejpam-5570	426	2	journal	journal	PROPN
ejpam-5570	426	3	of	of	ADP
ejpam-5570	426	4	pure	pure	ADJ
ejpam-5570	426	5	and	and	CCONJ
ejpam-5570	426	6	applied	applied	ADJ
ejpam-5570	426	7	mathematics	mathematic	NOUN
ejpam-5570	426	8	,	,	PUNCT
ejpam-5570	426	9	12(2):270–278	12(2):270–278	PROPN
ejpam-5570	426	10	,	,	PUNCT
ejpam-5570	426	11	2019	2019	NUM
ejpam-5570	426	12	.	.	PUNCT
ejpam-5570	427	1	[	[	X
ejpam-5570	427	2	18	18	NUM
ejpam-5570	427	3	]	]	X
ejpam-5570	427	4	s.	s.	PROPN
ejpam-5570	427	5	yuksel	yuksel	PROPN
ejpam-5570	427	6	,	,	PUNCT
ejpam-5570	427	7	a.	a.	PROPN
ejpam-5570	427	8	acikgoz	acikgoz	PROPN
ejpam-5570	427	9	,	,	PUNCT
ejpam-5570	427	10	and	and	CCONJ
ejpam-5570	427	11	t.	t.	PROPN
ejpam-5570	427	12	noiri	noiri	PROPN
ejpam-5570	427	13	.	.	PUNCT
ejpam-5570	428	1	on	on	ADP
ejpam-5570	428	2	δ	δ	PROPN
ejpam-5570	428	3	-	-	PUNCT
ejpam-5570	428	4	i	i	NOUN
ejpam-5570	428	5	-	-	PUNCT
ejpam-5570	428	6	continuous	continuous	ADJ
ejpam-5570	428	7	functions	function	NOUN
ejpam-5570	428	8	.	.	PUNCT
ejpam-5570	429	1	acta	acta	PROPN
ejpam-5570	429	2	mathematica	mathematica	PROPN
ejpam-5570	429	3	hungarica	hungarica	PROPN
ejpam-5570	429	4	,	,	PUNCT
ejpam-5570	429	5	29:39–51	29:39–51	NUM
ejpam-5570	429	6	,	,	PUNCT
ejpam-5570	429	7	2005	2005	NUM
ejpam-5570	429	8	.	.	PUNCT
ejpam-5570	430	1	[	[	X
ejpam-5570	430	2	19	19	NUM
ejpam-5570	430	3	]	]	PUNCT
ejpam-5570	430	4	m.	m.	NOUN
ejpam-5570	430	5	i.	i.	PROPN
ejpam-5570	430	6	zahid	zahid	PROPN
ejpam-5570	430	7	.	.	PUNCT
ejpam-5570	430	8	para	para	PROPN
ejpam-5570	430	9	h	h	NOUN
ejpam-5570	430	10	-	-	PUNCT
ejpam-5570	430	11	closed	closed	ADJ
ejpam-5570	430	12	spaces	space	NOUN
ejpam-5570	430	13	,	,	PUNCT
ejpam-5570	430	14	locally	locally	ADV
ejpam-5570	430	15	para	para	ADJ
ejpam-5570	430	16	h	h	NOUN
ejpam-5570	430	17	-	-	PUNCT
ejpam-5570	430	18	closed	closed	ADJ
ejpam-5570	430	19	spaces	space	NOUN
ejpam-5570	430	20	and	and	CCONJ
ejpam-5570	430	21	their	their	PRON
ejpam-5570	430	22	minimal	minimal	ADJ
ejpam-5570	430	23	topologies	topology	NOUN
ejpam-5570	430	24	.	.	PUNCT
ejpam-5570	431	1	phd	phd	NOUN
ejpam-5570	431	2	thesis	thesis	PROPN
ejpam-5570	431	3	,	,	PUNCT
ejpam-5570	431	4	university	university	PROPN
ejpam-5570	431	5	of	of	ADP
ejpam-5570	431	6	pittsburgh	pittsburgh	PROPN
ejpam-5570	431	7	,	,	PUNCT
ejpam-5570	431	8	1981	1981	NUM
ejpam-5570	431	9	.	.	PUNCT