id	sid	tid	token	lemma	pos
ejpam-3352	1	1	eta	eta	INTJ
ejpam-3352	2	1	_	_	PUNCT
ejpam-3352	2	2	1$-paracompactness	1$-paracompactness	NUM
ejpam-3352	2	3	with	with	ADP
ejpam-3352	2	4	respect	respect	NOUN
ejpam-3352	2	5	to	to	ADP
ejpam-3352	2	6	an	an	DET
ejpam-3352	2	7	ideal	ideal	ADJ
ejpam-3352	2	8	european	european	ADJ
ejpam-3352	2	9	journal	journal	NOUN
ejpam-3352	2	10	of	of	ADP
ejpam-3352	2	11	pure	pure	ADJ
ejpam-3352	2	12	and	and	CCONJ
ejpam-3352	2	13	applied	apply	VERB
ejpam-3352	2	14	mathematics	mathematic	NOUN
ejpam-3352	2	15	vol	vol	NOUN
ejpam-3352	2	16	.	.	PROPN
ejpam-3352	3	1	12	12	NUM
ejpam-3352	3	2	,	,	PUNCT
ejpam-3352	3	3	no	no	INTJ
ejpam-3352	3	4	.	.	NOUN
ejpam-3352	3	5	1	1	NUM
ejpam-3352	3	6	,	,	PUNCT
ejpam-3352	3	7	2019	2019	NUM
ejpam-3352	3	8	,	,	PUNCT
ejpam-3352	3	9	135	135	NUM
ejpam-3352	3	10	-	-	SYM
ejpam-3352	3	11	145	145	NUM
ejpam-3352	3	12	issn	issn	PROPN
ejpam-3352	3	13	1307	1307	NUM
ejpam-3352	3	14	-	-	SYM
ejpam-3352	3	15	5543	5543	NUM
ejpam-3352	3	16	–	–	PUNCT
ejpam-3352	3	17	www.ejpam.com	www.ejpam.com	X
ejpam-3352	3	18	published	publish	VERB
ejpam-3352	3	19	by	by	ADP
ejpam-3352	3	20	new	new	PROPN
ejpam-3352	3	21	york	york	PROPN
ejpam-3352	3	22	business	business	PROPN
ejpam-3352	3	23	global	global	ADJ
ejpam-3352	3	24	β1	β1	PROPN
ejpam-3352	3	25	-	-	PUNCT
ejpam-3352	3	26	paracompactness	paracompactness	NOUN
ejpam-3352	3	27	with	with	ADP
ejpam-3352	3	28	respect	respect	NOUN
ejpam-3352	3	29	to	to	ADP
ejpam-3352	3	30	an	an	DET
ejpam-3352	3	31	ideal	ideal	ADJ
ejpam-3352	3	32	abdo	abdo	PROPN
ejpam-3352	3	33	qahis	qahis	PROPN
ejpam-3352	3	34	department	department	PROPN
ejpam-3352	3	35	of	of	ADP
ejpam-3352	3	36	mathematics	mathematic	NOUN
ejpam-3352	3	37	,	,	PUNCT
ejpam-3352	3	38	faculty	faculty	NOUN
ejpam-3352	3	39	of	of	ADP
ejpam-3352	3	40	science	science	NOUN
ejpam-3352	3	41	and	and	CCONJ
ejpam-3352	3	42	arts	art	NOUN
ejpam-3352	3	43	,	,	PUNCT
ejpam-3352	3	44	najran	najran	ADJ
ejpam-3352	3	45	university	university	NOUN
ejpam-3352	3	46	,	,	PUNCT
ejpam-3352	4	1	saudi	saudi	PROPN
ejpam-3352	4	2	arabia	arabia	PROPN
ejpam-3352	4	3	abstract	abstract	NOUN
ejpam-3352	4	4	.	.	PUNCT
ejpam-3352	5	1	the	the	DET
ejpam-3352	5	2	notion	notion	NOUN
ejpam-3352	5	3	of	of	ADP
ejpam-3352	5	4	β1	β1	PROPN
ejpam-3352	5	5	-	-	PUNCT
ejpam-3352	5	6	paracompactness	paracompactness	NOUN
ejpam-3352	5	7	in	in	ADP
ejpam-3352	5	8	topological	topological	ADJ
ejpam-3352	5	9	spaces	space	NOUN
ejpam-3352	5	10	is	be	AUX
ejpam-3352	5	11	introduced	introduce	VERB
ejpam-3352	5	12	and	and	CCONJ
ejpam-3352	5	13	studied	study	VERB
ejpam-3352	5	14	in	in	ADP
ejpam-3352	5	15	[	[	X
ejpam-3352	5	16	1	1	NUM
ejpam-3352	5	17	]	]	PUNCT
ejpam-3352	5	18	.	.	PUNCT
ejpam-3352	6	1	in	in	ADP
ejpam-3352	6	2	this	this	DET
ejpam-3352	6	3	paper	paper	NOUN
ejpam-3352	6	4	,	,	PUNCT
ejpam-3352	6	5	we	we	PRON
ejpam-3352	6	6	introduce	introduce	VERB
ejpam-3352	6	7	and	and	CCONJ
ejpam-3352	6	8	investigate	investigate	VERB
ejpam-3352	6	9	the	the	DET
ejpam-3352	6	10	notion	notion	NOUN
ejpam-3352	6	11	of	of	ADP
ejpam-3352	6	12	β1	β1	NOUN
ejpam-3352	6	13	-	-	PUNCT
ejpam-3352	6	14	paracompact	paracompact	NOUN
ejpam-3352	6	15	spaces	space	NOUN
ejpam-3352	6	16	with	with	ADP
ejpam-3352	6	17	respect	respect	NOUN
ejpam-3352	6	18	to	to	ADP
ejpam-3352	6	19	an	an	DET
ejpam-3352	6	20	ideal	ideal	NOUN
ejpam-3352	6	21	i	i	PRON
ejpam-3352	6	22	which	which	PRON
ejpam-3352	6	23	is	be	AUX
ejpam-3352	6	24	a	a	DET
ejpam-3352	6	25	generalization	generalization	NOUN
ejpam-3352	6	26	of	of	ADP
ejpam-3352	6	27	the	the	DET
ejpam-3352	6	28	notion	notion	NOUN
ejpam-3352	6	29	of	of	ADP
ejpam-3352	6	30	β1	β1	NOUN
ejpam-3352	6	31	-	-	PUNCT
ejpam-3352	6	32	paracompact	paracompact	NOUN
ejpam-3352	6	33	spaces	space	NOUN
ejpam-3352	6	34	.	.	PUNCT
ejpam-3352	7	1	we	we	PRON
ejpam-3352	7	2	study	study	VERB
ejpam-3352	7	3	characterizations	characterization	NOUN
ejpam-3352	7	4	,	,	PUNCT
ejpam-3352	7	5	subsets	subset	NOUN
ejpam-3352	7	6	and	and	CCONJ
ejpam-3352	7	7	subspaces	subspace	NOUN
ejpam-3352	7	8	of	of	ADP
ejpam-3352	7	9	β1i	β1i	ADJ
ejpam-3352	7	10	-	-	ADJ
ejpam-3352	7	11	paracompact	paracompact	ADJ
ejpam-3352	7	12	spaces	space	NOUN
ejpam-3352	7	13	.	.	PUNCT
ejpam-3352	8	1	also	also	ADV
ejpam-3352	8	2	,	,	PUNCT
ejpam-3352	8	3	we	we	PRON
ejpam-3352	8	4	investigate	investigate	VERB
ejpam-3352	8	5	the	the	DET
ejpam-3352	8	6	invariants	invariant	NOUN
ejpam-3352	8	7	of	of	ADP
ejpam-3352	8	8	β1i	β1i	ADJ
ejpam-3352	8	9	-	-	ADJ
ejpam-3352	8	10	paracompact	paracompact	ADJ
ejpam-3352	8	11	spaces	space	NOUN
ejpam-3352	8	12	by	by	ADP
ejpam-3352	8	13	functions	function	NOUN
ejpam-3352	8	14	.	.	PUNCT
ejpam-3352	9	1	2010	2010	NUM
ejpam-3352	9	2	mathematics	mathematic	NOUN
ejpam-3352	9	3	subject	subject	NOUN
ejpam-3352	9	4	classifications	classification	NOUN
ejpam-3352	9	5	:	:	PUNCT
ejpam-3352	9	6	54a05	54a05	NUM
ejpam-3352	9	7	,	,	PUNCT
ejpam-3352	9	8	54a08	54a08	NUM
ejpam-3352	9	9	,	,	PUNCT
ejpam-3352	9	10	54d10	54d10	NUM
ejpam-3352	9	11	key	key	ADJ
ejpam-3352	9	12	words	word	NOUN
ejpam-3352	9	13	and	and	CCONJ
ejpam-3352	9	14	phrases	phrase	NOUN
ejpam-3352	9	15	:	:	PUNCT
ejpam-3352	9	16	ideal	ideal	ADJ
ejpam-3352	9	17	,	,	PUNCT
ejpam-3352	9	18	β	β	ADJ
ejpam-3352	9	19	-	-	ADJ
ejpam-3352	9	20	open	open	ADJ
ejpam-3352	9	21	set	set	NOUN
ejpam-3352	9	22	,	,	PUNCT
ejpam-3352	9	23	β1	β1	NOUN
ejpam-3352	9	24	-	-	PUNCT
ejpam-3352	9	25	paracompact	paracompact	ADJ
ejpam-3352	9	26	,	,	PUNCT
ejpam-3352	9	27	β1	β1	NOUN
ejpam-3352	9	28	-	-	PUNCT
ejpam-3352	9	29	paracompact	paracompact	NOUN
ejpam-3352	9	30	modulo	modulo	NOUN
ejpam-3352	10	1	i	i	PRON
ejpam-3352	10	2	,	,	PUNCT
ejpam-3352	10	3	locally	locally	ADV
ejpam-3352	10	4	finite	finite	ADJ
ejpam-3352	10	5	collection	collection	NOUN
ejpam-3352	10	6	,	,	PUNCT
ejpam-3352	10	7	β	β	NOUN
ejpam-3352	10	8	-	-	PUNCT
ejpam-3352	10	9	irresolute	irresolute	ADJ
ejpam-3352	10	10	,	,	PUNCT
ejpam-3352	10	11	β	β	ADJ
ejpam-3352	10	12	-	-	ADJ
ejpam-3352	10	13	continuous	continuous	ADJ
ejpam-3352	10	14	,	,	PUNCT
ejpam-3352	10	15	strongly	strongly	ADV
ejpam-3352	10	16	β	β	NOUN
ejpam-3352	10	17	-	-	ADJ
ejpam-3352	10	18	continuous	continuous	ADJ
ejpam-3352	10	19	.	.	PUNCT
ejpam-3352	11	1	1	1	X
ejpam-3352	11	2	.	.	X
ejpam-3352	11	3	introduction	introduction	NOUN
ejpam-3352	11	4	and	and	CCONJ
ejpam-3352	11	5	preliminaries	preliminary	NOUN
ejpam-3352	11	6	in	in	ADP
ejpam-3352	11	7	2016	2016	NUM
ejpam-3352	11	8	,	,	PUNCT
ejpam-3352	11	9	heyam	heyam	PROPN
ejpam-3352	11	10	al	al	PROPN
ejpam-3352	11	11	-	-	PUNCT
ejpam-3352	11	12	jarrah	jarrah	PROPN
ejpam-3352	11	13	introduced	introduce	VERB
ejpam-3352	11	14	and	and	CCONJ
ejpam-3352	11	15	studied	study	VERB
ejpam-3352	11	16	the	the	DET
ejpam-3352	11	17	concept	concept	NOUN
ejpam-3352	11	18	of	of	ADP
ejpam-3352	11	19	β1	β1	NOUN
ejpam-3352	11	20	-	-	PUNCT
ejpam-3352	11	21	paracompact	paracompact	NOUN
ejpam-3352	11	22	spaces	space	NOUN
ejpam-3352	11	23	.	.	PUNCT
ejpam-3352	12	1	a	a	DET
ejpam-3352	12	2	space	space	NOUN
ejpam-3352	12	3	(	(	PUNCT
ejpam-3352	12	4	x	x	X
ejpam-3352	12	5	,	,	PUNCT
ejpam-3352	12	6	τ	τ	X
ejpam-3352	12	7	)	)	PUNCT
ejpam-3352	12	8	is	be	AUX
ejpam-3352	12	9	said	say	VERB
ejpam-3352	12	10	to	to	PART
ejpam-3352	12	11	be	be	AUX
ejpam-3352	12	12	β1	β1	NOUN
ejpam-3352	12	13	-	-	PUNCT
ejpam-3352	12	14	paracompact	paracompact	NOUN
ejpam-3352	12	15	space	space	NOUN
ejpam-3352	12	16	[	[	X
ejpam-3352	12	17	1	1	X
ejpam-3352	12	18	]	]	PUNCT
ejpam-3352	12	19	if	if	SCONJ
ejpam-3352	12	20	every	every	DET
ejpam-3352	12	21	β	β	NOUN
ejpam-3352	12	22	-	-	ADJ
ejpam-3352	12	23	open	open	ADJ
ejpam-3352	12	24	cover	cover	NOUN
ejpam-3352	12	25	of	of	ADP
ejpam-3352	12	26	x	x	PUNCT
ejpam-3352	12	27	has	have	VERB
ejpam-3352	12	28	a	a	DET
ejpam-3352	12	29	locally	locally	ADV
ejpam-3352	12	30	finite	finite	ADJ
ejpam-3352	12	31	open	open	ADJ
ejpam-3352	12	32	refinement	refinement	NOUN
ejpam-3352	12	33	.	.	PUNCT
ejpam-3352	13	1	in	in	ADP
ejpam-3352	13	2	this	this	DET
ejpam-3352	13	3	paper	paper	NOUN
ejpam-3352	13	4	,	,	PUNCT
ejpam-3352	13	5	we	we	PRON
ejpam-3352	13	6	introduce	introduce	VERB
ejpam-3352	13	7	a	a	DET
ejpam-3352	13	8	new	new	ADJ
ejpam-3352	13	9	class	class	NOUN
ejpam-3352	13	10	of	of	ADP
ejpam-3352	13	11	spaces	space	NOUN
ejpam-3352	13	12	,	,	PUNCT
ejpam-3352	13	13	called	call	VERB
ejpam-3352	13	14	β1i	β1i	ADJ
ejpam-3352	13	15	-	-	ADJ
ejpam-3352	13	16	paracompact	paracompact	ADJ
ejpam-3352	13	17	spaces	space	NOUN
ejpam-3352	13	18	and	and	CCONJ
ejpam-3352	13	19	investigate	investigate	VERB
ejpam-3352	13	20	their	their	PRON
ejpam-3352	13	21	properties	property	NOUN
ejpam-3352	13	22	and	and	CCONJ
ejpam-3352	13	23	their	their	PRON
ejpam-3352	13	24	relations	relation	NOUN
ejpam-3352	13	25	with	with	ADP
ejpam-3352	13	26	other	other	ADJ
ejpam-3352	13	27	types	type	NOUN
ejpam-3352	13	28	of	of	ADP
ejpam-3352	13	29	spaces	space	NOUN
ejpam-3352	13	30	.	.	PUNCT
ejpam-3352	14	1	the	the	DET
ejpam-3352	14	2	notion	notion	NOUN
ejpam-3352	14	3	of	of	ADP
ejpam-3352	14	4	ideals	ideal	NOUN
ejpam-3352	14	5	in	in	ADP
ejpam-3352	14	6	topological	topological	ADJ
ejpam-3352	14	7	spaces	space	NOUN
ejpam-3352	14	8	was	be	AUX
ejpam-3352	14	9	first	first	ADV
ejpam-3352	14	10	studied	study	VERB
ejpam-3352	14	11	by	by	ADP
ejpam-3352	14	12	kuratowski	kuratowski	NOUN
ejpam-3352	14	13	[	[	X
ejpam-3352	14	14	12	12	NUM
ejpam-3352	14	15	]	]	PUNCT
ejpam-3352	14	16	and	and	CCONJ
ejpam-3352	14	17	vaidyanathaswamy	vaidyanathaswamy	VERB
ejpam-3352	14	18	[	[	X
ejpam-3352	14	19	23	23	NUM
ejpam-3352	14	20	]	]	PUNCT
ejpam-3352	14	21	.	.	PUNCT
ejpam-3352	15	1	an	an	DET
ejpam-3352	15	2	ideal	ideal	NOUN
ejpam-3352	15	3	i	i	PRON
ejpam-3352	15	4	on	on	ADP
ejpam-3352	15	5	a	a	DET
ejpam-3352	15	6	set	set	NOUN
ejpam-3352	15	7	x	x	PUNCT
ejpam-3352	15	8	is	be	AUX
ejpam-3352	15	9	a	a	DET
ejpam-3352	15	10	nonempty	nonempty	ADJ
ejpam-3352	15	11	collection	collection	NOUN
ejpam-3352	15	12	of	of	ADP
ejpam-3352	15	13	subsets	subset	NOUN
ejpam-3352	15	14	of	of	ADP
ejpam-3352	15	15	x	x	PUNCT
ejpam-3352	15	16	which	which	PRON
ejpam-3352	15	17	satisfies	satisfy	VERB
ejpam-3352	15	18	the	the	DET
ejpam-3352	15	19	following	follow	VERB
ejpam-3352	15	20	properties	property	NOUN
ejpam-3352	15	21	:	:	PUNCT
ejpam-3352	15	22	(	(	PUNCT
ejpam-3352	15	23	i	i	NOUN
ejpam-3352	15	24	)	)	PUNCT
ejpam-3352	16	1	a	a	PRON
ejpam-3352	16	2	∈	∈	NOUN
ejpam-3352	17	1	i	i	PRON
ejpam-3352	17	2	and	and	CCONJ
ejpam-3352	17	3	b	b	PROPN
ejpam-3352	17	4	⊂	⊂	PROPN
ejpam-3352	17	5	a	a	PRON
ejpam-3352	17	6	implies	imply	VERB
ejpam-3352	17	7	b	b	X
ejpam-3352	17	8	∈	∈	PROPN
ejpam-3352	18	1	i	i	PRON
ejpam-3352	18	2	;	;	PUNCT
ejpam-3352	18	3	(	(	PUNCT
ejpam-3352	18	4	ii	ii	NOUN
ejpam-3352	18	5	)	)	PUNCT
ejpam-3352	18	6	a	a	PRON
ejpam-3352	18	7	∈	∈	PROPN
ejpam-3352	19	1	i	i	PRON
ejpam-3352	19	2	and	and	CCONJ
ejpam-3352	19	3	b	b	X
ejpam-3352	19	4	∈	∈	PROPN
ejpam-3352	19	5	i	i	PRON
ejpam-3352	19	6	implies	imply	VERB
ejpam-3352	19	7	a	a	DET
ejpam-3352	19	8	∪b	∪b	PUNCT
ejpam-3352	19	9	∈	∈	PROPN
ejpam-3352	19	10	i.	i.	NOUN
ejpam-3352	19	11	given	give	VERB
ejpam-3352	19	12	a	a	DET
ejpam-3352	19	13	topological	topological	ADJ
ejpam-3352	19	14	space	space	NOUN
ejpam-3352	19	15	(	(	PUNCT
ejpam-3352	19	16	x	x	X
ejpam-3352	19	17	,	,	PUNCT
ejpam-3352	19	18	τ	τ	X
ejpam-3352	19	19	)	)	PUNCT
ejpam-3352	19	20	with	with	ADP
ejpam-3352	19	21	an	an	DET
ejpam-3352	19	22	ideal	ideal	ADJ
ejpam-3352	19	23	i	i	PRON
ejpam-3352	19	24	on	on	ADP
ejpam-3352	19	25	x	x	X
ejpam-3352	19	26	and	and	CCONJ
ejpam-3352	19	27	if	if	SCONJ
ejpam-3352	19	28	p(x	p(x	PROPN
ejpam-3352	19	29	)	)	PUNCT
ejpam-3352	19	30	is	be	AUX
ejpam-3352	19	31	the	the	DET
ejpam-3352	19	32	set	set	NOUN
ejpam-3352	19	33	of	of	ADP
ejpam-3352	19	34	all	all	DET
ejpam-3352	19	35	subsets	subset	NOUN
ejpam-3352	19	36	of	of	ADP
ejpam-3352	19	37	x	x	PRON
ejpam-3352	19	38	,	,	PUNCT
ejpam-3352	19	39	a	a	DET
ejpam-3352	19	40	set	set	NOUN
ejpam-3352	19	41	operator	operator	NOUN
ejpam-3352	19	42	(	(	PUNCT
ejpam-3352	19	43	)	)	PUNCT
ejpam-3352	19	44	∗	∗	NOUN
ejpam-3352	19	45	:	:	PUNCT
ejpam-3352	19	46	p(x)→	p(x)→	X
ejpam-3352	19	47	p(x	p(x	NOUN
ejpam-3352	19	48	)	)	PUNCT
ejpam-3352	19	49	,	,	PUNCT
ejpam-3352	19	50	called	call	VERB
ejpam-3352	19	51	a	a	DET
ejpam-3352	19	52	local	local	ADJ
ejpam-3352	19	53	function	function	NOUN
ejpam-3352	19	54	[	[	X
ejpam-3352	19	55	10	10	NUM
ejpam-3352	19	56	]	]	PUNCT
ejpam-3352	19	57	of	of	ADP
ejpam-3352	19	58	a	a	PRON
ejpam-3352	19	59	with	with	ADP
ejpam-3352	19	60	respect	respect	NOUN
ejpam-3352	19	61	to	to	ADP
ejpam-3352	19	62	τ	τ	PROPN
ejpam-3352	20	1	and	and	CCONJ
ejpam-3352	20	2	i	i	PRON
ejpam-3352	20	3	is	be	AUX
ejpam-3352	20	4	defined	define	VERB
ejpam-3352	20	5	as	as	SCONJ
ejpam-3352	20	6	follows	follow	VERB
ejpam-3352	20	7	:	:	PUNCT
ejpam-3352	20	8	for	for	ADP
ejpam-3352	20	9	a	a	DET
ejpam-3352	20	10	⊂	⊂	PROPN
ejpam-3352	20	11	x	x	SYM
ejpam-3352	20	12	,	,	PUNCT
ejpam-3352	20	13	a∗(i	a∗(i	PROPN
ejpam-3352	20	14	,	,	PUNCT
ejpam-3352	20	15	τ	τ	X
ejpam-3352	20	16	)	)	PUNCT
ejpam-3352	20	17	=	=	PRON
ejpam-3352	21	1	{	{	PUNCT
ejpam-3352	21	2	x	x	PUNCT
ejpam-3352	21	3	∈	∈	PROPN
ejpam-3352	21	4	x	x	X
ejpam-3352	21	5	:	:	PUNCT
ejpam-3352	21	6	u	u	NOUN
ejpam-3352	21	7	∩	∩	NOUN
ejpam-3352	21	8	a	a	X
ejpam-3352	21	9	/∈	/∈	PUNCT
ejpam-3352	22	1	i	i	PRON
ejpam-3352	22	2	for	for	ADP
ejpam-3352	22	3	every	every	DET
ejpam-3352	22	4	u	u	PROPN
ejpam-3352	22	5	∈	∈	PROPN
ejpam-3352	22	6	τ(x	τ(x	NOUN
ejpam-3352	22	7	)	)	PUNCT
ejpam-3352	22	8	}	}	PUNCT
ejpam-3352	22	9	where	where	SCONJ
ejpam-3352	22	10	τ(x	τ(x	NOUN
ejpam-3352	22	11	)	)	PUNCT
ejpam-3352	22	12	=	=	PRON
ejpam-3352	22	13	{	{	PUNCT
ejpam-3352	22	14	u	u	X
ejpam-3352	22	15	∈	∈	PROPN
ejpam-3352	22	16	τ	τ	X
ejpam-3352	22	17	:	:	PUNCT
ejpam-3352	22	18	x	x	SYM
ejpam-3352	22	19	∈	∈	PROPN
ejpam-3352	22	20	u	u	NOUN
ejpam-3352	22	21	}	}	PUNCT
ejpam-3352	22	22	.	.	PUNCT
ejpam-3352	23	1	a	a	DET
ejpam-3352	23	2	kuratowski	kuratowski	ADJ
ejpam-3352	23	3	closure	closure	NOUN
ejpam-3352	23	4	operator	operator	NOUN
ejpam-3352	23	5	cl∗	cl∗	PROPN
ejpam-3352	23	6	(	(	PUNCT
ejpam-3352	23	7	)	)	PUNCT
ejpam-3352	23	8	for	for	ADP
ejpam-3352	23	9	a	a	DET
ejpam-3352	23	10	topology	topology	NOUN
ejpam-3352	23	11	τ∗(i	τ∗(i	PROPN
ejpam-3352	23	12	,	,	PUNCT
ejpam-3352	23	13	τ	τ	PROPN
ejpam-3352	23	14	)	)	PUNCT
ejpam-3352	23	15	called	call	VERB
ejpam-3352	23	16	∗-topology	∗-topology	NOUN
ejpam-3352	23	17	finer	fine	ADJ
ejpam-3352	23	18	than	than	SCONJ
ejpam-3352	23	19	τ	τ	PROPN
ejpam-3352	23	20	is	be	AUX
ejpam-3352	23	21	defined	define	VERB
ejpam-3352	23	22	by	by	ADP
ejpam-3352	23	23	cl∗(a	cl∗(a	NOUN
ejpam-3352	23	24	)	)	PUNCT
ejpam-3352	23	25	=	=	PUNCT
ejpam-3352	23	26	a	a	DET
ejpam-3352	23	27	∪	∪	X
ejpam-3352	23	28	a∗(i	a∗(i	PROPN
ejpam-3352	23	29	,	,	PUNCT
ejpam-3352	23	30	τ	τ	X
ejpam-3352	23	31	)	)	PUNCT
ejpam-3352	24	1	[	[	X
ejpam-3352	24	2	10	10	NUM
ejpam-3352	24	3	]	]	PUNCT
ejpam-3352	24	4	and	and	CCONJ
ejpam-3352	24	5	β	β	X
ejpam-3352	24	6	=	=	SYM
ejpam-3352	24	7	{	{	PUNCT
ejpam-3352	24	8	u	u	NOUN
ejpam-3352	24	9	\	\	PROPN
ejpam-3352	25	1	i	i	PRON
ejpam-3352	25	2	:	:	PUNCT
ejpam-3352	25	3	u	u	PROPN
ejpam-3352	25	4	∈	∈	PROPN
ejpam-3352	25	5	τ	τ	PROPN
ejpam-3352	25	6	,	,	PUNCT
ejpam-3352	25	7	i	i	PRON
ejpam-3352	25	8	∈	∈	VERB
ejpam-3352	25	9	i	i	PRON
ejpam-3352	25	10	}	}	PUNCT
ejpam-3352	25	11	is	be	AUX
ejpam-3352	25	12	a	a	DET
ejpam-3352	25	13	basis	basis	NOUN
ejpam-3352	25	14	for	for	ADP
ejpam-3352	25	15	τ∗	τ∗	NOUN
ejpam-3352	26	1	[	[	X
ejpam-3352	26	2	10	10	NUM
ejpam-3352	26	3	]	]	PUNCT
ejpam-3352	26	4	.	.	PUNCT
ejpam-3352	27	1	we	we	PRON
ejpam-3352	27	2	simply	simply	ADV
ejpam-3352	27	3	write	write	VERB
ejpam-3352	27	4	τ∗	τ∗	NOUN
ejpam-3352	27	5	for	for	ADP
ejpam-3352	27	6	τ∗(i	τ∗(i	PROPN
ejpam-3352	27	7	,	,	PUNCT
ejpam-3352	27	8	τ	τ	PROPN
ejpam-3352	27	9	)	)	PUNCT
ejpam-3352	27	10	.	.	PUNCT
ejpam-3352	28	1	if	if	SCONJ
ejpam-3352	28	2	i	i	PRON
ejpam-3352	28	3	is	be	AUX
ejpam-3352	28	4	an	an	DET
ejpam-3352	28	5	ideal	ideal	NOUN
ejpam-3352	28	6	on	on	ADP
ejpam-3352	28	7	x	x	NOUN
ejpam-3352	28	8	,	,	PUNCT
ejpam-3352	28	9	then	then	ADV
ejpam-3352	28	10	(	(	PUNCT
ejpam-3352	28	11	x	x	X
ejpam-3352	28	12	,	,	PUNCT
ejpam-3352	28	13	τ	τ	PROPN
ejpam-3352	28	14	,	,	PUNCT
ejpam-3352	28	15	i	i	PROPN
ejpam-3352	28	16	)	)	PUNCT
ejpam-3352	28	17	is	be	AUX
ejpam-3352	28	18	called	call	VERB
ejpam-3352	28	19	an	an	DET
ejpam-3352	28	20	ideal	ideal	ADJ
ejpam-3352	28	21	space	space	NOUN
ejpam-3352	28	22	.	.	PUNCT
ejpam-3352	29	1	if	if	SCONJ
ejpam-3352	29	2	β	β	NOUN
ejpam-3352	29	3	=	=	SYM
ejpam-3352	29	4	τ∗	τ∗	NOUN
ejpam-3352	29	5	,	,	PUNCT
ejpam-3352	29	6	then	then	ADV
ejpam-3352	29	7	we	we	PRON
ejpam-3352	29	8	say	say	VERB
ejpam-3352	29	9	i	i	PRON
ejpam-3352	29	10	is	be	AUX
ejpam-3352	29	11	τ	τ	PROPN
ejpam-3352	29	12	-simple	-simple	X
ejpam-3352	30	1	[	[	X
ejpam-3352	30	2	10	10	NUM
ejpam-3352	30	3	]	]	PUNCT
ejpam-3352	30	4	.	.	PUNCT
ejpam-3352	31	1	a	a	DET
ejpam-3352	31	2	sufficient	sufficient	ADJ
ejpam-3352	31	3	condition	condition	NOUN
ejpam-3352	31	4	for	for	SCONJ
ejpam-3352	31	5	i	i	PRON
ejpam-3352	31	6	to	to	PART
ejpam-3352	31	7	be	be	AUX
ejpam-3352	31	8	simple	simple	ADJ
ejpam-3352	31	9	is	be	AUX
ejpam-3352	31	10	the	the	DET
ejpam-3352	31	11	following	following	NOUN
ejpam-3352	31	12	:	:	PUNCT
ejpam-3352	31	13	for	for	ADP
ejpam-3352	31	14	a	a	DET
ejpam-3352	31	15	⊂	⊂	PROPN
ejpam-3352	31	16	x	x	X
ejpam-3352	31	17	,	,	PUNCT
ejpam-3352	31	18	if	if	SCONJ
ejpam-3352	31	19	for	for	ADP
ejpam-3352	31	20	doi	doi	NOUN
ejpam-3352	31	21	:	:	PUNCT
ejpam-3352	31	22	https://doi.org/10.29020/nybg.ejpam.v12i1.3552	https://doi.org/10.29020/nybg.ejpam.v12i1.3552	PRON
ejpam-3352	31	23	email	email	NOUN
ejpam-3352	31	24	address	address	NOUN
ejpam-3352	31	25	:	:	PUNCT
ejpam-3352	31	26	cahis82@gmail.com	cahis82@gmail.com	X
ejpam-3352	31	27	(	(	PUNCT
ejpam-3352	31	28	a.	a.	NOUN
ejpam-3352	31	29	qahis	qahis	PROPN
ejpam-3352	31	30	)	)	PUNCT
ejpam-3352	31	31	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3352	32	1	135	135	NUM
ejpam-3352	32	2	c	c	X
ejpam-3352	32	3	©	©	PROPN
ejpam-3352	32	4	2019	2019	NUM
ejpam-3352	32	5	ejpam	ejpam	NOUN
ejpam-3352	32	6	all	all	DET
ejpam-3352	32	7	rights	right	NOUN
ejpam-3352	32	8	reserved	reserve	VERB
ejpam-3352	32	9	.	.	PUNCT
ejpam-3352	33	1	a.	a.	NOUN
ejpam-3352	33	2	qahis	qahis	PROPN
ejpam-3352	33	3	/	/	SYM
ejpam-3352	33	4	eur	eur	PROPN
ejpam-3352	33	5	.	.	PUNCT
ejpam-3352	34	1	j.	j.	PROPN
ejpam-3352	34	2	pure	pure	PROPN
ejpam-3352	34	3	appl	appl	PROPN
ejpam-3352	34	4	.	.	PROPN
ejpam-3352	34	5	math	math	PROPN
ejpam-3352	34	6	,	,	PUNCT
ejpam-3352	34	7	12	12	NUM
ejpam-3352	34	8	(	(	PUNCT
ejpam-3352	34	9	1	1	NUM
ejpam-3352	34	10	)	)	PUNCT
ejpam-3352	34	11	(	(	PUNCT
ejpam-3352	34	12	2019	2019	NUM
ejpam-3352	34	13	)	)	PUNCT
ejpam-3352	34	14	,	,	PUNCT
ejpam-3352	34	15	135	135	NUM
ejpam-3352	34	16	-	-	SYM
ejpam-3352	34	17	145	145	NUM
ejpam-3352	34	18	136	136	NUM
ejpam-3352	34	19	every	every	PRON
ejpam-3352	34	20	a	a	DET
ejpam-3352	34	21	∈	∈	PROPN
ejpam-3352	34	22	a	a	DET
ejpam-3352	34	23	there	there	PRON
ejpam-3352	34	24	exists	exist	VERB
ejpam-3352	34	25	u	u	PROPN
ejpam-3352	34	26	∈	∈	PROPN
ejpam-3352	34	27	τ(a	τ(a	NOUN
ejpam-3352	34	28	)	)	PUNCT
ejpam-3352	34	29	such	such	ADJ
ejpam-3352	34	30	that	that	SCONJ
ejpam-3352	34	31	u	u	PROPN
ejpam-3352	34	32	∩	∩	NOUN
ejpam-3352	34	33	a	a	PRON
ejpam-3352	34	34	∈	∈	X
ejpam-3352	35	1	i	i	PRON
ejpam-3352	35	2	,	,	PUNCT
ejpam-3352	35	3	then	then	ADV
ejpam-3352	35	4	a	a	DET
ejpam-3352	35	5	∈	∈	PROPN
ejpam-3352	35	6	i.	i.	NOUN
ejpam-3352	35	7	if	if	SCONJ
ejpam-3352	35	8	(	(	PUNCT
ejpam-3352	35	9	x	x	X
ejpam-3352	35	10	,	,	PUNCT
ejpam-3352	35	11	τ	τ	PROPN
ejpam-3352	35	12	,	,	PUNCT
ejpam-3352	35	13	i	i	NOUN
ejpam-3352	35	14	)	)	PUNCT
ejpam-3352	35	15	satisfies	satisfy	VERB
ejpam-3352	35	16	this	this	DET
ejpam-3352	35	17	condition	condition	NOUN
ejpam-3352	35	18	,	,	PUNCT
ejpam-3352	35	19	then	then	ADV
ejpam-3352	35	20	τ	τ	PROPN
ejpam-3352	35	21	is	be	AUX
ejpam-3352	35	22	said	say	VERB
ejpam-3352	35	23	to	to	PART
ejpam-3352	35	24	be	be	AUX
ejpam-3352	35	25	compatible	compatible	ADJ
ejpam-3352	35	26	with	with	ADP
ejpam-3352	35	27	respect	respect	NOUN
ejpam-3352	35	28	to	to	ADP
ejpam-3352	35	29	i	i	PRON
ejpam-3352	36	1	[	[	X
ejpam-3352	36	2	10	10	NUM
ejpam-3352	36	3	]	]	PUNCT
ejpam-3352	36	4	or	or	CCONJ
ejpam-3352	36	5	i	i	PRON
ejpam-3352	36	6	is	be	AUX
ejpam-3352	36	7	said	say	VERB
ejpam-3352	36	8	to	to	PART
ejpam-3352	36	9	be	be	AUX
ejpam-3352	36	10	τ	τ	PROPN
ejpam-3352	36	11	-local	-local	PROPN
ejpam-3352	36	12	,	,	PUNCT
ejpam-3352	36	13	denoted	denote	VERB
ejpam-3352	36	14	by	by	ADP
ejpam-3352	36	15	i	i	PRON
ejpam-3352	36	16	∼	∼	VERB
ejpam-3352	36	17	τ	τ	PROPN
ejpam-3352	36	18	.	.	PUNCT
ejpam-3352	37	1	given	give	VERB
ejpam-3352	37	2	an	an	DET
ejpam-3352	37	3	ideal	ideal	ADJ
ejpam-3352	37	4	space	space	NOUN
ejpam-3352	37	5	(	(	PUNCT
ejpam-3352	37	6	x	x	X
ejpam-3352	37	7	,	,	PUNCT
ejpam-3352	37	8	τ	τ	PROPN
ejpam-3352	37	9	,	,	PUNCT
ejpam-3352	37	10	i	i	PROPN
ejpam-3352	37	11	)	)	PUNCT
ejpam-3352	37	12	,	,	PUNCT
ejpam-3352	37	13	we	we	PRON
ejpam-3352	37	14	say	say	VERB
ejpam-3352	37	15	i	i	PRON
ejpam-3352	37	16	is	be	AUX
ejpam-3352	37	17	τ	τ	PROPN
ejpam-3352	37	18	-boundary	-boundary	ADJ
ejpam-3352	38	1	[	[	X
ejpam-3352	38	2	10	10	NUM
ejpam-3352	38	3	]	]	PUNCT
ejpam-3352	38	4	or	or	CCONJ
ejpam-3352	38	5	i	i	NOUN
ejpam-3352	38	6	-	-	PUNCT
ejpam-3352	38	7	codense	codense	NOUN
ejpam-3352	38	8	if	if	SCONJ
ejpam-3352	38	9	i	i	PRON
ejpam-3352	38	10	∩	∩	NOUN
ejpam-3352	38	11	τ	τ	X
ejpam-3352	38	12	=	=	PUNCT
ejpam-3352	38	13	∅.	∅.	VERB
ejpam-3352	38	14	an	an	DET
ejpam-3352	38	15	ideal	ideal	NOUN
ejpam-3352	38	16	i	i	PRON
ejpam-3352	38	17	is	be	AUX
ejpam-3352	38	18	said	say	VERB
ejpam-3352	38	19	to	to	PART
ejpam-3352	38	20	be	be	AUX
ejpam-3352	38	21	weakly	weakly	ADV
ejpam-3352	38	22	τ	τ	X
ejpam-3352	38	23	-local	-local	ADJ
ejpam-3352	39	1	[	[	X
ejpam-3352	39	2	11	11	NUM
ejpam-3352	39	3	]	]	PUNCT
ejpam-3352	39	4	if	if	SCONJ
ejpam-3352	39	5	a∗	a∗	PROPN
ejpam-3352	39	6	=	=	SYM
ejpam-3352	39	7	∅	∅	NOUN
ejpam-3352	39	8	implies	imply	VERB
ejpam-3352	39	9	a	a	DET
ejpam-3352	39	10	∈	∈	PROPN
ejpam-3352	39	11	i.	i.	NOUN
ejpam-3352	39	12	some	some	DET
ejpam-3352	39	13	useful	useful	ADJ
ejpam-3352	39	14	ideals	ideal	NOUN
ejpam-3352	39	15	in	in	ADP
ejpam-3352	39	16	x	x	PUNCT
ejpam-3352	39	17	are	be	AUX
ejpam-3352	39	18	:	:	PUNCT
ejpam-3352	39	19	(	(	PUNCT
ejpam-3352	39	20	i	i	NOUN
ejpam-3352	39	21	)	)	PUNCT
ejpam-3352	39	22	p(a	p(a	PROPN
ejpam-3352	39	23	)	)	PUNCT
ejpam-3352	39	24	,	,	PUNCT
ejpam-3352	39	25	where	where	SCONJ
ejpam-3352	39	26	a	a	DET
ejpam-3352	39	27	⊆	⊆	NUM
ejpam-3352	39	28	x	x	SYM
ejpam-3352	39	29	and	and	CCONJ
ejpam-3352	39	30	(	(	PUNCT
ejpam-3352	39	31	ii	ii	NOUN
ejpam-3352	39	32	)	)	PUNCT
ejpam-3352	39	33	if	if	SCONJ
ejpam-3352	39	34	,	,	PUNCT
ejpam-3352	39	35	the	the	DET
ejpam-3352	39	36	ideal	ideal	NOUN
ejpam-3352	39	37	of	of	ADP
ejpam-3352	39	38	all	all	DET
ejpam-3352	39	39	finite	finite	ADJ
ejpam-3352	39	40	subsets	subset	NOUN
ejpam-3352	39	41	of	of	ADP
ejpam-3352	39	42	x.	x.	NOUN
ejpam-3352	39	43	by	by	ADP
ejpam-3352	39	44	a	a	DET
ejpam-3352	39	45	space	space	NOUN
ejpam-3352	39	46	(	(	PUNCT
ejpam-3352	39	47	x	x	X
ejpam-3352	39	48	,	,	PUNCT
ejpam-3352	39	49	τ	τ	PROPN
ejpam-3352	39	50	)	)	PUNCT
ejpam-3352	39	51	,	,	PUNCT
ejpam-3352	39	52	we	we	PRON
ejpam-3352	39	53	always	always	ADV
ejpam-3352	39	54	mean	mean	VERB
ejpam-3352	39	55	a	a	DET
ejpam-3352	39	56	topological	topological	ADJ
ejpam-3352	39	57	space	space	NOUN
ejpam-3352	39	58	(	(	PUNCT
ejpam-3352	39	59	x	x	X
ejpam-3352	39	60	,	,	PUNCT
ejpam-3352	39	61	τ	τ	X
ejpam-3352	39	62	)	)	PUNCT
ejpam-3352	39	63	with	with	SCONJ
ejpam-3352	39	64	no	no	DET
ejpam-3352	39	65	separation	separation	NOUN
ejpam-3352	39	66	properties	property	NOUN
ejpam-3352	39	67	assumed	assume	VERB
ejpam-3352	39	68	.	.	PUNCT
ejpam-3352	40	1	if	if	SCONJ
ejpam-3352	40	2	a	a	DET
ejpam-3352	40	3	⊆	⊆	NUM
ejpam-3352	40	4	x	x	SYM
ejpam-3352	40	5	,	,	PUNCT
ejpam-3352	40	6	we	we	PRON
ejpam-3352	40	7	denote	denote	VERB
ejpam-3352	40	8	the	the	DET
ejpam-3352	40	9	closure	closure	NOUN
ejpam-3352	40	10	of	of	ADP
ejpam-3352	40	11	a	a	PRON
ejpam-3352	40	12	and	and	CCONJ
ejpam-3352	40	13	the	the	DET
ejpam-3352	40	14	interior	interior	NOUN
ejpam-3352	40	15	of	of	ADP
ejpam-3352	40	16	a	a	PRON
ejpam-3352	40	17	by	by	ADP
ejpam-3352	40	18	cl(a	cl(a	NUM
ejpam-3352	40	19	)	)	PUNCT
ejpam-3352	40	20	and	and	CCONJ
ejpam-3352	40	21	int(a	int(a	PROPN
ejpam-3352	40	22	)	)	PUNCT
ejpam-3352	40	23	,	,	PUNCT
ejpam-3352	40	24	respectively	respectively	ADV
ejpam-3352	40	25	.	.	PUNCT
ejpam-3352	41	1	a	a	DET
ejpam-3352	41	2	subset	subset	NOUN
ejpam-3352	41	3	a	a	PRON
ejpam-3352	41	4	of	of	ADP
ejpam-3352	41	5	(	(	PUNCT
ejpam-3352	41	6	x	x	PROPN
ejpam-3352	41	7	,	,	PUNCT
ejpam-3352	41	8	τ	τ	X
ejpam-3352	41	9	)	)	PUNCT
ejpam-3352	41	10	is	be	AUX
ejpam-3352	41	11	siad	siad	VERB
ejpam-3352	41	12	to	to	PART
ejpam-3352	41	13	be	be	AUX
ejpam-3352	41	14	semi	semi	ADJ
ejpam-3352	41	15	-	-	ADJ
ejpam-3352	41	16	open[13	open[13	ADJ
ejpam-3352	41	17	]	]	X
ejpam-3352	41	18	(	(	PUNCT
ejpam-3352	41	19	resp	resp	NOUN
ejpam-3352	41	20	.	.	PUNCT
ejpam-3352	41	21	,	,	PUNCT
ejpam-3352	41	22	α	α	X
ejpam-3352	41	23	-	-	PUNCT
ejpam-3352	41	24	open[17	open[17	VERB
ejpam-3352	41	25	]	]	X
ejpam-3352	41	26	,	,	PUNCT
ejpam-3352	41	27	regular	regular	ADJ
ejpam-3352	41	28	open	open	ADJ
ejpam-3352	41	29	[	[	X
ejpam-3352	41	30	21	21	NUM
ejpam-3352	41	31	]	]	PUNCT
ejpam-3352	41	32	)	)	PUNCT
ejpam-3352	41	33	if	if	SCONJ
ejpam-3352	41	34	a	a	DET
ejpam-3352	41	35	⊂	⊂	PROPN
ejpam-3352	41	36	cl(int(a	cl(int(a	PROPN
ejpam-3352	41	37	)	)	PUNCT
ejpam-3352	41	38	)	)	PUNCT
ejpam-3352	42	1	,	,	PUNCT
ejpam-3352	42	2	(	(	PUNCT
ejpam-3352	42	3	resp	resp	NOUN
ejpam-3352	42	4	.	.	PROPN
ejpam-3352	42	5	,	,	PUNCT
ejpam-3352	42	6	a	a	DET
ejpam-3352	42	7	⊂	⊂	X
ejpam-3352	42	8	int(cl(int(a	int(cl(int(a	PROPN
ejpam-3352	42	9	)	)	PUNCT
ejpam-3352	42	10	)	)	PUNCT
ejpam-3352	42	11	)	)	PUNCT
ejpam-3352	42	12	,	,	PUNCT
ejpam-3352	42	13	a	a	DET
ejpam-3352	42	14	=	=	X
ejpam-3352	42	15	int(cl(a	int(cl(a	PROPN
ejpam-3352	42	16	)	)	PUNCT
ejpam-3352	42	17	)	)	PUNCT
ejpam-3352	42	18	)	)	PUNCT
ejpam-3352	42	19	.	.	PUNCT
ejpam-3352	43	1	the	the	DET
ejpam-3352	43	2	family	family	NOUN
ejpam-3352	43	3	of	of	ADP
ejpam-3352	43	4	α	α	NOUN
ejpam-3352	43	5	-	-	PUNCT
ejpam-3352	43	6	sets	set	NOUN
ejpam-3352	43	7	of	of	ADP
ejpam-3352	43	8	a	a	DET
ejpam-3352	43	9	space	space	NOUN
ejpam-3352	43	10	(	(	PUNCT
ejpam-3352	43	11	x	x	X
ejpam-3352	43	12	,	,	PUNCT
ejpam-3352	43	13	τ	τ	X
ejpam-3352	43	14	)	)	PUNCT
ejpam-3352	43	15	denoted	denote	VERB
ejpam-3352	43	16	by	by	ADP
ejpam-3352	43	17	τα	τα	NOUN
ejpam-3352	43	18	forms	form	VERB
ejpam-3352	43	19	a	a	DET
ejpam-3352	43	20	topology	topology	NOUN
ejpam-3352	43	21	on	on	ADP
ejpam-3352	43	22	x	x	SYM
ejpam-3352	43	23	finer	fine	ADJ
ejpam-3352	43	24	than	than	ADP
ejpam-3352	43	25	τ	τ	PROPN
ejpam-3352	43	26	[	[	X
ejpam-3352	43	27	17	17	NUM
ejpam-3352	43	28	]	]	PUNCT
ejpam-3352	43	29	.	.	PUNCT
ejpam-3352	44	1	abd	abd	PROPN
ejpam-3352	44	2	el	el	PROPN
ejpam-3352	44	3	-	-	PROPN
ejpam-3352	44	4	monsef	monsef	PROPN
ejpam-3352	44	5	et	et	PROPN
ejpam-3352	44	6	al.[8	al.[8	PROPN
ejpam-3352	44	7	]	]	PUNCT
ejpam-3352	44	8	introduced	introduce	VERB
ejpam-3352	44	9	and	and	CCONJ
ejpam-3352	44	10	studied	study	VERB
ejpam-3352	44	11	the	the	DET
ejpam-3352	44	12	concept	concept	NOUN
ejpam-3352	44	13	of	of	ADP
ejpam-3352	44	14	β	β	ADJ
ejpam-3352	44	15	-	-	ADJ
ejpam-3352	44	16	open	open	ADJ
ejpam-3352	44	17	sets	set	NOUN
ejpam-3352	44	18	in	in	ADP
ejpam-3352	44	19	topological	topological	ADJ
ejpam-3352	44	20	spaces	space	NOUN
ejpam-3352	44	21	.	.	PUNCT
ejpam-3352	45	1	a	a	DET
ejpam-3352	45	2	subset	subset	NOUN
ejpam-3352	45	3	a	a	PRON
ejpam-3352	45	4	of	of	ADP
ejpam-3352	45	5	x	x	PRON
ejpam-3352	45	6	is	be	AUX
ejpam-3352	45	7	called	call	VERB
ejpam-3352	45	8	β	β	NOUN
ejpam-3352	45	9	-	-	VERB
ejpam-3352	45	10	open	open	ADJ
ejpam-3352	45	11	if	if	SCONJ
ejpam-3352	45	12	a	a	DET
ejpam-3352	45	13	⊂	⊂	PROPN
ejpam-3352	45	14	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-3352	45	15	)	)	PUNCT
ejpam-3352	45	16	)	)	PUNCT
ejpam-3352	45	17	)	)	PUNCT
ejpam-3352	45	18	.	.	PUNCT
ejpam-3352	46	1	the	the	DET
ejpam-3352	46	2	complement	complement	NOUN
ejpam-3352	46	3	of	of	ADP
ejpam-3352	46	4	a	a	DET
ejpam-3352	46	5	β	β	X
ejpam-3352	46	6	-	-	ADJ
ejpam-3352	46	7	open	open	ADJ
ejpam-3352	46	8	set	set	NOUN
ejpam-3352	46	9	is	be	AUX
ejpam-3352	46	10	said	say	VERB
ejpam-3352	46	11	to	to	PART
ejpam-3352	46	12	be	be	AUX
ejpam-3352	46	13	β	β	X
ejpam-3352	46	14	-	-	VERB
ejpam-3352	46	15	closed	closed	ADJ
ejpam-3352	46	16	[	[	X
ejpam-3352	46	17	8	8	NUM
ejpam-3352	46	18	]	]	PUNCT
ejpam-3352	46	19	.	.	PUNCT
ejpam-3352	47	1	the	the	DET
ejpam-3352	47	2	family	family	NOUN
ejpam-3352	47	3	of	of	ADP
ejpam-3352	47	4	all	all	DET
ejpam-3352	47	5	β	β	NOUN
ejpam-3352	47	6	-	-	ADJ
ejpam-3352	47	7	open	open	ADJ
ejpam-3352	47	8	(	(	PUNCT
ejpam-3352	47	9	resp	resp	NOUN
ejpam-3352	47	10	.	.	PUNCT
ejpam-3352	47	11	,	,	PUNCT
ejpam-3352	47	12	β	β	X
ejpam-3352	47	13	-	-	ADJ
ejpam-3352	47	14	closed	closed	ADJ
ejpam-3352	47	15	)	)	PUNCT
ejpam-3352	47	16	subsets	subset	NOUN
ejpam-3352	47	17	of	of	ADP
ejpam-3352	47	18	x	x	PROPN
ejpam-3352	47	19	is	be	AUX
ejpam-3352	47	20	denoted	denote	VERB
ejpam-3352	47	21	by	by	ADP
ejpam-3352	47	22	βo(x	βo(x	PROPN
ejpam-3352	47	23	,	,	PUNCT
ejpam-3352	47	24	τ	τ	X
ejpam-3352	47	25	)	)	PUNCT
ejpam-3352	47	26	(	(	PUNCT
ejpam-3352	47	27	resp	resp	NOUN
ejpam-3352	47	28	.	.	PUNCT
ejpam-3352	47	29	,	,	PUNCT
ejpam-3352	47	30	βc(x	βc(x	NUM
ejpam-3352	47	31	,	,	PUNCT
ejpam-3352	47	32	τ	τ	PROPN
ejpam-3352	47	33	)	)	PUNCT
ejpam-3352	47	34	)	)	PUNCT
ejpam-3352	47	35	.	.	PUNCT
ejpam-3352	48	1	the	the	DET
ejpam-3352	48	2	union	union	NOUN
ejpam-3352	48	3	of	of	ADP
ejpam-3352	48	4	all	all	DET
ejpam-3352	48	5	β	β	ADJ
ejpam-3352	48	6	-	-	ADJ
ejpam-3352	48	7	open	open	ADJ
ejpam-3352	48	8	subsets	subset	NOUN
ejpam-3352	48	9	of	of	ADP
ejpam-3352	48	10	x	x	PUNCT
ejpam-3352	48	11	contained	contain	VERB
ejpam-3352	48	12	in	in	ADP
ejpam-3352	48	13	a	a	PRON
ejpam-3352	48	14	is	be	AUX
ejpam-3352	48	15	called	call	VERB
ejpam-3352	48	16	β	β	NOUN
ejpam-3352	48	17	-	-	NOUN
ejpam-3352	48	18	interior	interior	ADJ
ejpam-3352	48	19	of	of	ADP
ejpam-3352	48	20	a	a	PRON
ejpam-3352	48	21	and	and	CCONJ
ejpam-3352	48	22	is	be	AUX
ejpam-3352	48	23	denoted	denote	VERB
ejpam-3352	48	24	by	by	ADP
ejpam-3352	48	25	βint(a	βint(a	NOUN
ejpam-3352	48	26	)	)	PUNCT
ejpam-3352	48	27	and	and	CCONJ
ejpam-3352	48	28	the	the	DET
ejpam-3352	48	29	intersection	intersection	NOUN
ejpam-3352	48	30	of	of	ADP
ejpam-3352	48	31	all	all	DET
ejpam-3352	48	32	β	β	NOUN
ejpam-3352	48	33	-	-	ADJ
ejpam-3352	48	34	closed	closed	ADJ
ejpam-3352	48	35	subsets	subset	NOUN
ejpam-3352	48	36	of	of	ADP
ejpam-3352	48	37	x	x	PUNCT
ejpam-3352	48	38	containing	contain	VERB
ejpam-3352	48	39	a	a	PRON
ejpam-3352	48	40	is	be	AUX
ejpam-3352	48	41	called	call	VERB
ejpam-3352	48	42	the	the	DET
ejpam-3352	48	43	β	β	NOUN
ejpam-3352	48	44	-	-	NOUN
ejpam-3352	48	45	closure	closure	NOUN
ejpam-3352	48	46	of	of	ADP
ejpam-3352	48	47	a	a	PRON
ejpam-3352	48	48	and	and	CCONJ
ejpam-3352	48	49	is	be	AUX
ejpam-3352	48	50	denoted	denote	VERB
ejpam-3352	48	51	by	by	ADP
ejpam-3352	48	52	βcl(a	βcl(a	NOUN
ejpam-3352	48	53	)	)	PUNCT
ejpam-3352	48	54	.	.	PUNCT
ejpam-3352	49	1	a	a	DET
ejpam-3352	49	2	set	set	NOUN
ejpam-3352	49	3	a	a	PRON
ejpam-3352	49	4	is	be	AUX
ejpam-3352	49	5	called	call	VERB
ejpam-3352	49	6	β	β	NOUN
ejpam-3352	49	7	-	-	ADJ
ejpam-3352	49	8	regular	regular	ADJ
ejpam-3352	49	9	[	[	X
ejpam-3352	49	10	19	19	NUM
ejpam-3352	49	11	]	]	X
ejpam-3352	49	12	if	if	SCONJ
ejpam-3352	49	13	it	it	PRON
ejpam-3352	49	14	is	be	AUX
ejpam-3352	49	15	both	both	PRON
ejpam-3352	49	16	β	β	NOUN
ejpam-3352	49	17	-	-	ADJ
ejpam-3352	49	18	open	open	ADJ
ejpam-3352	49	19	and	and	CCONJ
ejpam-3352	49	20	β	β	NOUN
ejpam-3352	49	21	-	-	VERB
ejpam-3352	49	22	closed	closed	ADJ
ejpam-3352	49	23	.	.	PUNCT
ejpam-3352	50	1	a	a	DET
ejpam-3352	50	2	space	space	NOUN
ejpam-3352	50	3	(	(	PUNCT
ejpam-3352	50	4	x	x	X
ejpam-3352	50	5	,	,	PUNCT
ejpam-3352	50	6	τ	τ	X
ejpam-3352	50	7	)	)	PUNCT
ejpam-3352	50	8	is	be	AUX
ejpam-3352	50	9	said	say	VERB
ejpam-3352	50	10	to	to	PART
ejpam-3352	50	11	be	be	AUX
ejpam-3352	50	12	β	β	X
ejpam-3352	50	13	-	-	ADJ
ejpam-3352	50	14	regular	regular	ADJ
ejpam-3352	50	15	[	[	X
ejpam-3352	50	16	14	14	NUM
ejpam-3352	50	17	]	]	X
ejpam-3352	50	18	if	if	SCONJ
ejpam-3352	50	19	for	for	ADP
ejpam-3352	50	20	each	each	DET
ejpam-3352	50	21	β	β	NOUN
ejpam-3352	50	22	-	-	ADJ
ejpam-3352	50	23	open	open	ADJ
ejpam-3352	50	24	set	set	NOUN
ejpam-3352	50	25	u	u	NOUN
ejpam-3352	50	26	and	and	CCONJ
ejpam-3352	50	27	each	each	DET
ejpam-3352	50	28	x	x	SYM
ejpam-3352	50	29	∈	∈	PROPN
ejpam-3352	50	30	u	u	NOUN
ejpam-3352	50	31	,	,	PUNCT
ejpam-3352	50	32	there	there	PRON
ejpam-3352	50	33	exists	exist	VERB
ejpam-3352	50	34	a	a	DET
ejpam-3352	50	35	β	β	NOUN
ejpam-3352	50	36	-	-	ADJ
ejpam-3352	50	37	open	open	ADJ
ejpam-3352	50	38	set	set	VERB
ejpam-3352	50	39	v	v	ADP
ejpam-3352	51	1	such	such	ADJ
ejpam-3352	51	2	that	that	SCONJ
ejpam-3352	51	3	x	x	SYM
ejpam-3352	51	4	∈	∈	NOUN
ejpam-3352	51	5	v	v	ADP
ejpam-3352	51	6	⊂	⊂	PROPN
ejpam-3352	51	7	βcl(v	βcl(v	X
ejpam-3352	51	8	)	)	PUNCT
ejpam-3352	51	9	⊂	⊂	PROPN
ejpam-3352	51	10	u	u	PROPN
ejpam-3352	51	11	.	.	PUNCT
ejpam-3352	52	1	for	for	ADP
ejpam-3352	52	2	any	any	DET
ejpam-3352	52	3	space	space	NOUN
ejpam-3352	52	4	,	,	PUNCT
ejpam-3352	52	5	βo(x	βo(x	PUNCT
ejpam-3352	52	6	,	,	PUNCT
ejpam-3352	52	7	τα	τα	NOUN
ejpam-3352	52	8	)	)	PUNCT
ejpam-3352	52	9	=	=	PUNCT
ejpam-3352	52	10	βo(x	βo(x	X
ejpam-3352	52	11	,	,	PUNCT
ejpam-3352	52	12	τ	τ	X
ejpam-3352	52	13	)	)	PUNCT
ejpam-3352	52	14	[	[	X
ejpam-3352	52	15	2	2	NUM
ejpam-3352	52	16	]	]	PUNCT
ejpam-3352	52	17	.	.	PUNCT
ejpam-3352	53	1	a	a	DET
ejpam-3352	53	2	collection	collection	NOUN
ejpam-3352	53	3	w	w	NOUN
ejpam-3352	53	4	=	=	SYM
ejpam-3352	53	5	{	{	PUNCT
ejpam-3352	53	6	wα	wα	NOUN
ejpam-3352	53	7	:	:	PUNCT
ejpam-3352	53	8	α	α	PROPN
ejpam-3352	53	9	∈	∈	PROPN
ejpam-3352	53	10	∆	∆	PROPN
ejpam-3352	53	11	}	}	PUNCT
ejpam-3352	53	12	of	of	ADP
ejpam-3352	53	13	subsets	subset	NOUN
ejpam-3352	53	14	of	of	ADP
ejpam-3352	53	15	a	a	DET
ejpam-3352	53	16	space	space	NOUN
ejpam-3352	53	17	(	(	PUNCT
ejpam-3352	53	18	x	x	X
ejpam-3352	53	19	,	,	PUNCT
ejpam-3352	53	20	τ	τ	X
ejpam-3352	53	21	)	)	PUNCT
ejpam-3352	53	22	is	be	AUX
ejpam-3352	53	23	said	say	VERB
ejpam-3352	53	24	to	to	PART
ejpam-3352	53	25	be	be	AUX
ejpam-3352	53	26	locally	locally	ADV
ejpam-3352	53	27	finite	finite	ADJ
ejpam-3352	53	28	if	if	SCONJ
ejpam-3352	53	29	for	for	ADP
ejpam-3352	53	30	each	each	DET
ejpam-3352	53	31	x	x	SYM
ejpam-3352	53	32	∈	∈	PROPN
ejpam-3352	53	33	x	x	X
ejpam-3352	53	34	,	,	PUNCT
ejpam-3352	53	35	there	there	PRON
ejpam-3352	53	36	exists	exist	VERB
ejpam-3352	53	37	an	an	DET
ejpam-3352	53	38	open	open	ADJ
ejpam-3352	53	39	set	set	NOUN
ejpam-3352	53	40	u	u	NOUN
ejpam-3352	53	41	containing	contain	VERB
ejpam-3352	53	42	x	x	X
ejpam-3352	53	43	and	and	CCONJ
ejpam-3352	53	44	u	u	NOUN
ejpam-3352	53	45	intersects	intersect	NOUN
ejpam-3352	53	46	at	at	ADV
ejpam-3352	53	47	most	most	ADV
ejpam-3352	53	48	finitely	finitely	ADV
ejpam-3352	53	49	many	many	ADJ
ejpam-3352	53	50	members	member	NOUN
ejpam-3352	53	51	of	of	ADP
ejpam-3352	53	52	w.	w.	PROPN
ejpam-3352	53	53	a	a	DET
ejpam-3352	53	54	subset	subset	VERB
ejpam-3352	53	55	a	a	PRON
ejpam-3352	53	56	of	of	ADP
ejpam-3352	53	57	space	space	NOUN
ejpam-3352	53	58	x	x	PUNCT
ejpam-3352	53	59	is	be	AUX
ejpam-3352	53	60	said	say	VERB
ejpam-3352	53	61	to	to	PART
ejpam-3352	53	62	be	be	AUX
ejpam-3352	53	63	n	n	ADV
ejpam-3352	53	64	-closed	-close	VERB
ejpam-3352	53	65	relative	relative	ADJ
ejpam-3352	53	66	to	to	ADP
ejpam-3352	53	67	x	x	PROPN
ejpam-3352	53	68	(	(	PUNCT
ejpam-3352	53	69	briery	briery	NOUN
ejpam-3352	53	70	,	,	PUNCT
ejpam-3352	53	71	n	n	PROPN
ejpam-3352	53	72	-closed)[7	-closed)[7	PROPN
ejpam-3352	53	73	]	]	PUNCT
ejpam-3352	53	74	if	if	SCONJ
ejpam-3352	53	75	for	for	ADP
ejpam-3352	53	76	every	every	DET
ejpam-3352	53	77	cover	cover	NOUN
ejpam-3352	53	78	{	{	PUNCT
ejpam-3352	53	79	uα	uα	X
ejpam-3352	53	80	:	:	PUNCT
ejpam-3352	53	81	α	α	PROPN
ejpam-3352	53	82	∈	∈	PROPN
ejpam-3352	53	83	∆	∆	PROPN
ejpam-3352	53	84	}	}	PUNCT
ejpam-3352	53	85	of	of	ADP
ejpam-3352	53	86	a	a	PRON
ejpam-3352	53	87	by	by	ADP
ejpam-3352	53	88	open	open	ADJ
ejpam-3352	53	89	subsets	subset	NOUN
ejpam-3352	53	90	of	of	ADP
ejpam-3352	53	91	x	x	PRON
ejpam-3352	53	92	,	,	PUNCT
ejpam-3352	53	93	there	there	PRON
ejpam-3352	53	94	exists	exist	VERB
ejpam-3352	53	95	a	a	DET
ejpam-3352	53	96	finite	finite	NOUN
ejpam-3352	53	97	subfamily	subfamily	ADV
ejpam-3352	53	98	∆0	∆0	NUM
ejpam-3352	53	99	of	of	ADP
ejpam-3352	53	100	∆	∆	PROPN
ejpam-3352	53	101	such	such	ADJ
ejpam-3352	53	102	that	that	SCONJ
ejpam-3352	53	103	a	a	DET
ejpam-3352	53	104	⊂	⊂	PROPN
ejpam-3352	53	105	∪{int(cl(int(uα	∪{int(cl(int(uα	NOUN
ejpam-3352	53	106	)	)	PUNCT
ejpam-3352	53	107	)	)	PUNCT
ejpam-3352	53	108	)	)	PUNCT
ejpam-3352	53	109	:	:	PUNCT
ejpam-3352	54	1	α	α	PROPN
ejpam-3352	54	2	∈	∈	PROPN
ejpam-3352	54	3	∆0	∆0	NUM
ejpam-3352	54	4	}	}	PUNCT
ejpam-3352	54	5	.	.	PUNCT
ejpam-3352	55	1	definition	definition	NOUN
ejpam-3352	55	2	1	1	NUM
ejpam-3352	55	3	.	.	PUNCT
ejpam-3352	56	1	a	a	DET
ejpam-3352	56	2	space	space	NOUN
ejpam-3352	56	3	(	(	PUNCT
ejpam-3352	56	4	x	x	X
ejpam-3352	56	5	,	,	PUNCT
ejpam-3352	56	6	τ	τ	X
ejpam-3352	56	7	)	)	PUNCT
ejpam-3352	56	8	is	be	AUX
ejpam-3352	56	9	said	say	VERB
ejpam-3352	56	10	to	to	PART
ejpam-3352	56	11	be	be	AUX
ejpam-3352	56	12	:	:	PUNCT
ejpam-3352	56	13	(	(	PUNCT
ejpam-3352	56	14	i	i	NOUN
ejpam-3352	56	15	)	)	PUNCT
ejpam-3352	56	16	extremally	extremally	ADV
ejpam-3352	56	17	disconnected	disconnected	ADJ
ejpam-3352	56	18	(	(	PUNCT
ejpam-3352	56	19	briefly	briefly	NOUN
ejpam-3352	56	20	e.d	e.d	PROPN
ejpam-3352	56	21	.	.	PUNCT
ejpam-3352	56	22	)	)	PUNCT
ejpam-3352	57	1	[	[	X
ejpam-3352	57	2	24	24	NUM
ejpam-3352	57	3	]	]	X
ejpam-3352	57	4	if	if	SCONJ
ejpam-3352	57	5	the	the	DET
ejpam-3352	57	6	closure	closure	NOUN
ejpam-3352	57	7	of	of	ADP
ejpam-3352	57	8	every	every	DET
ejpam-3352	57	9	open	open	ADJ
ejpam-3352	57	10	set	set	NOUN
ejpam-3352	57	11	in	in	ADP
ejpam-3352	57	12	(	(	PUNCT
ejpam-3352	57	13	x	x	NOUN
ejpam-3352	57	14	,	,	PUNCT
ejpam-3352	57	15	τ	τ	X
ejpam-3352	57	16	)	)	PUNCT
ejpam-3352	57	17	is	be	AUX
ejpam-3352	57	18	open	open	ADJ
ejpam-3352	57	19	;	;	PUNCT
ejpam-3352	57	20	(	(	PUNCT
ejpam-3352	57	21	ii	ii	NOUN
ejpam-3352	57	22	)	)	PUNCT
ejpam-3352	57	23	submaximal	submaximal	ADJ
ejpam-3352	57	24	[	[	X
ejpam-3352	57	25	5	5	NUM
ejpam-3352	57	26	]	]	PUNCT
ejpam-3352	57	27	if	if	SCONJ
ejpam-3352	57	28	each	each	DET
ejpam-3352	57	29	dense	dense	ADJ
ejpam-3352	57	30	subset	subset	NOUN
ejpam-3352	57	31	of	of	ADP
ejpam-3352	57	32	x	x	PUNCT
ejpam-3352	57	33	is	be	AUX
ejpam-3352	57	34	open	open	ADJ
ejpam-3352	57	35	in	in	ADP
ejpam-3352	57	36	x.	x.	PROPN
ejpam-3352	57	37	lemma	lemma	PROPN
ejpam-3352	57	38	1	1	NUM
ejpam-3352	57	39	.	.	PUNCT
ejpam-3352	58	1	[	[	X
ejpam-3352	58	2	3	3	X
ejpam-3352	58	3	]	]	PUNCT
ejpam-3352	58	4	the	the	DET
ejpam-3352	58	5	union	union	NOUN
ejpam-3352	58	6	of	of	ADP
ejpam-3352	58	7	a	a	DET
ejpam-3352	58	8	finite	finite	ADJ
ejpam-3352	58	9	family	family	NOUN
ejpam-3352	58	10	of	of	ADP
ejpam-3352	58	11	locally	locally	ADV
ejpam-3352	58	12	finite	finite	ADJ
ejpam-3352	58	13	collection	collection	NOUN
ejpam-3352	58	14	of	of	ADP
ejpam-3352	58	15	sets	set	NOUN
ejpam-3352	58	16	in	in	ADP
ejpam-3352	58	17	a	a	DET
ejpam-3352	58	18	space	space	NOUN
ejpam-3352	58	19	is	be	AUX
ejpam-3352	58	20	a	a	DET
ejpam-3352	58	21	locally	locally	ADV
ejpam-3352	58	22	finite	finite	ADJ
ejpam-3352	58	23	family	family	NOUN
ejpam-3352	58	24	of	of	ADP
ejpam-3352	58	25	sets	set	NOUN
ejpam-3352	58	26	.	.	PUNCT
ejpam-3352	59	1	theorem	theorem	NOUN
ejpam-3352	59	2	1	1	NUM
ejpam-3352	59	3	.	.	PUNCT
ejpam-3352	60	1	[	[	X
ejpam-3352	60	2	16	16	NUM
ejpam-3352	60	3	]	]	X
ejpam-3352	60	4	let	let	VERB
ejpam-3352	60	5	(	(	PUNCT
ejpam-3352	60	6	x	x	NOUN
ejpam-3352	60	7	,	,	PUNCT
ejpam-3352	60	8	τ	τ	X
ejpam-3352	60	9	)	)	PUNCT
ejpam-3352	60	10	be	be	VERB
ejpam-3352	60	11	a	a	DET
ejpam-3352	60	12	space	space	NOUN
ejpam-3352	60	13	,	,	PUNCT
ejpam-3352	60	14	a	a	DET
ejpam-3352	60	15	⊂	⊂	X
ejpam-3352	60	16	b	b	X
ejpam-3352	60	17	⊂	⊂	PROPN
ejpam-3352	60	18	x	x	X
ejpam-3352	60	19	and	and	CCONJ
ejpam-3352	60	20	b	b	PROPN
ejpam-3352	60	21	is	be	AUX
ejpam-3352	60	22	β	β	X
ejpam-3352	60	23	-	-	VERB
ejpam-3352	60	24	open	open	ADJ
ejpam-3352	60	25	in	in	ADP
ejpam-3352	60	26	(	(	PUNCT
ejpam-3352	60	27	x	x	NOUN
ejpam-3352	60	28	,	,	PUNCT
ejpam-3352	60	29	τ	τ	PROPN
ejpam-3352	60	30	)	)	PUNCT
ejpam-3352	60	31	.	.	PUNCT
ejpam-3352	61	1	then	then	ADV
ejpam-3352	61	2	a	a	PRON
ejpam-3352	61	3	is	be	AUX
ejpam-3352	61	4	β	β	X
ejpam-3352	61	5	-	-	ADJ
ejpam-3352	61	6	open	open	ADJ
ejpam-3352	61	7	in	in	ADP
ejpam-3352	61	8	(	(	PUNCT
ejpam-3352	61	9	x	x	NOUN
ejpam-3352	61	10	,	,	PUNCT
ejpam-3352	61	11	τ	τ	X
ejpam-3352	61	12	)	)	PUNCT
ejpam-3352	62	1	if	if	SCONJ
ejpam-3352	62	2	and	and	CCONJ
ejpam-3352	62	3	only	only	ADV
ejpam-3352	62	4	if	if	SCONJ
ejpam-3352	62	5	a	a	PRON
ejpam-3352	62	6	is	be	AUX
ejpam-3352	62	7	β	β	X
ejpam-3352	62	8	-	-	ADJ
ejpam-3352	62	9	open	open	ADJ
ejpam-3352	62	10	in	in	ADP
ejpam-3352	62	11	the	the	DET
ejpam-3352	62	12	subspace	subspace	NOUN
ejpam-3352	62	13	(	(	PUNCT
ejpam-3352	62	14	b	b	NOUN
ejpam-3352	62	15	,	,	PUNCT
ejpam-3352	62	16	τb	τb	ADJ
ejpam-3352	62	17	)	)	PUNCT
ejpam-3352	62	18	.	.	PUNCT
ejpam-3352	63	1	theorem	theorem	NOUN
ejpam-3352	63	2	2	2	NUM
ejpam-3352	63	3	.	.	PUNCT
ejpam-3352	64	1	[	[	X
ejpam-3352	64	2	4	4	X
ejpam-3352	64	3	]	]	X
ejpam-3352	64	4	if	if	SCONJ
ejpam-3352	64	5	{	{	PUNCT
ejpam-3352	64	6	uα	uα	X
ejpam-3352	64	7	:	:	PUNCT
ejpam-3352	64	8	α	α	PROPN
ejpam-3352	64	9	∈	∈	PROPN
ejpam-3352	64	10	∆	∆	X
ejpam-3352	64	11	}	}	PUNCT
ejpam-3352	64	12	is	be	AUX
ejpam-3352	64	13	a	a	DET
ejpam-3352	64	14	locally	locally	ADV
ejpam-3352	64	15	finite	finite	ADJ
ejpam-3352	64	16	family	family	NOUN
ejpam-3352	64	17	of	of	ADP
ejpam-3352	64	18	subsets	subset	NOUN
ejpam-3352	64	19	in	in	ADP
ejpam-3352	64	20	a	a	DET
ejpam-3352	64	21	space	space	NOUN
ejpam-3352	64	22	x	x	NOUN
ejpam-3352	64	23	and	and	CCONJ
ejpam-3352	64	24	if	if	SCONJ
ejpam-3352	64	25	vα	vα	X
ejpam-3352	64	26	⊂	⊂	X
ejpam-3352	64	27	uα	uα	PROPN
ejpam-3352	64	28	for	for	ADP
ejpam-3352	64	29	each	each	DET
ejpam-3352	64	30	α	α	NOUN
ejpam-3352	64	31	∈	∈	PROPN
ejpam-3352	64	32	∆	∆	PROPN
ejpam-3352	64	33	,	,	PUNCT
ejpam-3352	64	34	then	then	ADV
ejpam-3352	64	35	the	the	DET
ejpam-3352	64	36	family	family	NOUN
ejpam-3352	64	37	{	{	PUNCT
ejpam-3352	64	38	vα	vα	X
ejpam-3352	64	39	:	:	PUNCT
ejpam-3352	64	40	α	α	PROPN
ejpam-3352	64	41	∈	∈	PROPN
ejpam-3352	64	42	∆	∆	X
ejpam-3352	64	43	}	}	PUNCT
ejpam-3352	64	44	is	be	AUX
ejpam-3352	64	45	a	a	DET
ejpam-3352	64	46	locally	locally	ADV
ejpam-3352	64	47	finite	finite	NOUN
ejpam-3352	64	48	in	in	ADP
ejpam-3352	64	49	x.	x.	PROPN
ejpam-3352	64	50	lemma	lemma	PROPN
ejpam-3352	65	1	2	2	X
ejpam-3352	65	2	.	.	PUNCT
ejpam-3352	66	1	[	[	X
ejpam-3352	66	2	9	9	NUM
ejpam-3352	66	3	]	]	X
ejpam-3352	66	4	if	if	SCONJ
ejpam-3352	66	5	f	f	PROPN
ejpam-3352	66	6	:	:	PUNCT
ejpam-3352	66	7	(	(	PUNCT
ejpam-3352	66	8	x	x	X
ejpam-3352	66	9	,	,	PUNCT
ejpam-3352	66	10	τ	τ	X
ejpam-3352	66	11	)	)	PUNCT
ejpam-3352	66	12	→	→	SYM
ejpam-3352	66	13	(	(	PUNCT
ejpam-3352	66	14	y	y	PROPN
ejpam-3352	66	15	,	,	PUNCT
ejpam-3352	66	16	σ	σ	PROPN
ejpam-3352	66	17	)	)	PUNCT
ejpam-3352	66	18	is	be	AUX
ejpam-3352	66	19	a	a	DET
ejpam-3352	66	20	continuous	continuous	ADJ
ejpam-3352	66	21	surjective	surjective	ADJ
ejpam-3352	66	22	function	function	NOUN
ejpam-3352	66	23	and	and	CCONJ
ejpam-3352	66	24	u	u	NOUN
ejpam-3352	66	25	=	=	PUNCT
ejpam-3352	66	26	{	{	PUNCT
ejpam-3352	66	27	uα	uα	X
ejpam-3352	66	28	:	:	PUNCT
ejpam-3352	66	29	α	α	PROPN
ejpam-3352	66	30	∈	∈	PROPN
ejpam-3352	66	31	∆	∆	X
ejpam-3352	66	32	}	}	PUNCT
ejpam-3352	66	33	is	be	AUX
ejpam-3352	66	34	locally	locally	ADV
ejpam-3352	66	35	finite	finite	ADJ
ejpam-3352	66	36	in	in	ADP
ejpam-3352	66	37	y	y	PROPN
ejpam-3352	66	38	,	,	PUNCT
ejpam-3352	66	39	then	then	ADV
ejpam-3352	66	40	f−1(u	f−1(u	PROPN
ejpam-3352	66	41	)	)	PUNCT
ejpam-3352	67	1	=	=	PRON
ejpam-3352	67	2	{	{	PUNCT
ejpam-3352	67	3	f−1(uα	f−1(uα	PROPN
ejpam-3352	67	4	)	)	PUNCT
ejpam-3352	67	5	:	:	PUNCT
ejpam-3352	67	6	α	α	PROPN
ejpam-3352	67	7	∈	∈	PROPN
ejpam-3352	67	8	∆	∆	X
ejpam-3352	67	9	}	}	PUNCT
ejpam-3352	67	10	is	be	AUX
ejpam-3352	67	11	locally	locally	ADV
ejpam-3352	67	12	finite	finite	ADJ
ejpam-3352	67	13	in	in	ADP
ejpam-3352	67	14	x.	x.	PROPN
ejpam-3352	67	15	lemma	lemma	PROPN
ejpam-3352	67	16	3	3	X
ejpam-3352	67	17	.	.	PUNCT
ejpam-3352	68	1	[	[	X
ejpam-3352	68	2	18	18	NUM
ejpam-3352	68	3	]	]	PUNCT
ejpam-3352	68	4	let	let	VERB
ejpam-3352	68	5	f	f	X
ejpam-3352	68	6	:	:	PUNCT
ejpam-3352	68	7	(	(	PUNCT
ejpam-3352	68	8	x	x	X
ejpam-3352	68	9	,	,	PUNCT
ejpam-3352	68	10	τ	τ	X
ejpam-3352	68	11	)	)	PUNCT
ejpam-3352	68	12	→	→	SYM
ejpam-3352	68	13	(	(	PUNCT
ejpam-3352	68	14	y	y	PROPN
ejpam-3352	68	15	,	,	PUNCT
ejpam-3352	68	16	σ	σ	PROPN
ejpam-3352	68	17	)	)	PUNCT
ejpam-3352	68	18	be	be	AUX
ejpam-3352	68	19	almost	almost	ADV
ejpam-3352	68	20	closed	close	VERB
ejpam-3352	68	21	surjection	surjection	NOUN
ejpam-3352	68	22	with	with	ADP
ejpam-3352	68	23	n	n	ADV
ejpam-3352	68	24	-closed	-closed	ADJ
ejpam-3352	68	25	point	point	NOUN
ejpam-3352	68	26	inverse	inverse	NOUN
ejpam-3352	68	27	.	.	PUNCT
ejpam-3352	69	1	if	if	SCONJ
ejpam-3352	69	2	{	{	PUNCT
ejpam-3352	69	3	uα	uα	X
ejpam-3352	69	4	:	:	PUNCT
ejpam-3352	69	5	α	α	PROPN
ejpam-3352	69	6	∈	∈	PROPN
ejpam-3352	69	7	∆	∆	X
ejpam-3352	69	8	}	}	PUNCT
ejpam-3352	69	9	is	be	AUX
ejpam-3352	69	10	a	a	DET
ejpam-3352	69	11	locally	locally	ADV
ejpam-3352	69	12	finite	finite	ADJ
ejpam-3352	69	13	open	open	ADJ
ejpam-3352	69	14	cover	cover	NOUN
ejpam-3352	69	15	of	of	ADP
ejpam-3352	69	16	x	x	PRON
ejpam-3352	69	17	,	,	PUNCT
ejpam-3352	69	18	then	then	ADV
ejpam-3352	69	19	{	{	PUNCT
ejpam-3352	69	20	f(uα	f(uα	NOUN
ejpam-3352	69	21	)	)	PUNCT
ejpam-3352	69	22	:	:	PUNCT
ejpam-3352	69	23	α	α	PROPN
ejpam-3352	69	24	∈	∈	PROPN
ejpam-3352	69	25	∆	∆	X
ejpam-3352	69	26	}	}	PUNCT
ejpam-3352	69	27	is	be	AUX
ejpam-3352	69	28	a	a	DET
ejpam-3352	69	29	locally	locally	ADV
ejpam-3352	69	30	finite	finite	ADJ
ejpam-3352	69	31	cover	cover	NOUN
ejpam-3352	69	32	of	of	ADP
ejpam-3352	69	33	y	y	PROPN
ejpam-3352	69	34	.	.	PUNCT
ejpam-3352	70	1	a.	a.	NOUN
ejpam-3352	70	2	qahis	qahis	PROPN
ejpam-3352	70	3	/	/	SYM
ejpam-3352	70	4	eur	eur	PROPN
ejpam-3352	70	5	.	.	PUNCT
ejpam-3352	71	1	j.	j.	PROPN
ejpam-3352	71	2	pure	pure	PROPN
ejpam-3352	71	3	appl	appl	PROPN
ejpam-3352	71	4	.	.	PROPN
ejpam-3352	71	5	math	math	PROPN
ejpam-3352	71	6	,	,	PUNCT
ejpam-3352	71	7	12	12	NUM
ejpam-3352	71	8	(	(	PUNCT
ejpam-3352	71	9	1	1	NUM
ejpam-3352	71	10	)	)	PUNCT
ejpam-3352	71	11	(	(	PUNCT
ejpam-3352	71	12	2019	2019	NUM
ejpam-3352	71	13	)	)	PUNCT
ejpam-3352	71	14	,	,	PUNCT
ejpam-3352	71	15	135	135	NUM
ejpam-3352	71	16	-	-	SYM
ejpam-3352	71	17	145	145	NUM
ejpam-3352	71	18	137	137	NUM
ejpam-3352	71	19	lemma	lemma	PROPN
ejpam-3352	71	20	4	4	NUM
ejpam-3352	71	21	.	.	PUNCT
ejpam-3352	72	1	[	[	X
ejpam-3352	72	2	22	22	NUM
ejpam-3352	72	3	]	]	X
ejpam-3352	72	4	i	i	PRON
ejpam-3352	72	5	is	be	AUX
ejpam-3352	72	6	weakly	weakly	ADJ
ejpam-3352	72	7	τ	τ	X
ejpam-3352	72	8	-local	-local	ADJ
ejpam-3352	72	9	implies	imply	VERB
ejpam-3352	72	10	i	i	PRON
ejpam-3352	72	11	is	be	AUX
ejpam-3352	72	12	τ	τ	PROPN
ejpam-3352	72	13	-locally	-locally	ADV
ejpam-3352	72	14	finite	finite	ADJ
ejpam-3352	72	15	.	.	PUNCT
ejpam-3352	73	1	lemma	lemma	PROPN
ejpam-3352	73	2	5	5	NUM
ejpam-3352	73	3	.	.	PUNCT
ejpam-3352	74	1	[	[	X
ejpam-3352	74	2	2	2	X
ejpam-3352	74	3	]	]	X
ejpam-3352	74	4	if	if	SCONJ
ejpam-3352	74	5	v	v	NOUN
ejpam-3352	74	6	is	be	AUX
ejpam-3352	74	7	open	open	ADJ
ejpam-3352	74	8	and	and	CCONJ
ejpam-3352	74	9	a	a	PRON
ejpam-3352	74	10	is	be	AUX
ejpam-3352	74	11	semi	semi	ADJ
ejpam-3352	74	12	-	-	ADJ
ejpam-3352	74	13	preopen	preopen	ADJ
ejpam-3352	74	14	(	(	PUNCT
ejpam-3352	74	15	or	or	CCONJ
ejpam-3352	74	16	β	β	NOUN
ejpam-3352	74	17	-	-	ADJ
ejpam-3352	74	18	open	open	ADJ
ejpam-3352	74	19	)	)	PUNCT
ejpam-3352	74	20	then	then	ADV
ejpam-3352	74	21	v	v	X
ejpam-3352	74	22	∩a	∩a	PROPN
ejpam-3352	74	23	is	be	AUX
ejpam-3352	74	24	semi	semi	ADJ
ejpam-3352	74	25	-	-	ADJ
ejpam-3352	74	26	preopen	preopen	ADJ
ejpam-3352	74	27	(	(	PUNCT
ejpam-3352	74	28	or	or	CCONJ
ejpam-3352	74	29	β	β	NOUN
ejpam-3352	74	30	-	-	ADJ
ejpam-3352	74	31	open	open	ADJ
ejpam-3352	74	32	)	)	PUNCT
ejpam-3352	74	33	.	.	PUNCT
ejpam-3352	75	1	2	2	X
ejpam-3352	75	2	.	.	X
ejpam-3352	75	3	β1i	β1i	ADJ
ejpam-3352	75	4	-	-	ADJ
ejpam-3352	75	5	paracompact	paracompact	ADJ
ejpam-3352	75	6	spaces	space	NOUN
ejpam-3352	75	7	recall	recall	VERB
ejpam-3352	75	8	that	that	SCONJ
ejpam-3352	75	9	an	an	DET
ejpam-3352	75	10	ideal	ideal	ADJ
ejpam-3352	75	11	space	space	NOUN
ejpam-3352	75	12	(	(	PUNCT
ejpam-3352	75	13	x	x	X
ejpam-3352	75	14	,	,	PUNCT
ejpam-3352	75	15	τ	τ	PROPN
ejpam-3352	75	16	,	,	PUNCT
ejpam-3352	75	17	i	i	PROPN
ejpam-3352	75	18	)	)	PUNCT
ejpam-3352	75	19	is	be	AUX
ejpam-3352	75	20	said	say	VERB
ejpam-3352	75	21	to	to	PART
ejpam-3352	75	22	be	be	AUX
ejpam-3352	75	23	i	i	NOUN
ejpam-3352	75	24	-	-	NOUN
ejpam-3352	75	25	paracompact	paracompact	ADJ
ejpam-3352	76	1	[	[	X
ejpam-3352	76	2	22	22	NUM
ejpam-3352	76	3	]	]	PUNCT
ejpam-3352	76	4	(	(	PUNCT
ejpam-3352	76	5	resp	resp	NOUN
ejpam-3352	76	6	.	.	PUNCT
ejpam-3352	76	7	,	,	PUNCT
ejpam-3352	77	1	s1iparacompact	s1iparacompact	PROPN
ejpam-3352	77	2	[	[	X
ejpam-3352	77	3	20	20	NUM
ejpam-3352	77	4	]	]	SYM
ejpam-3352	77	5	)	)	PUNCT
ejpam-3352	77	6	if	if	SCONJ
ejpam-3352	77	7	every	every	DET
ejpam-3352	77	8	open	open	ADJ
ejpam-3352	77	9	(	(	PUNCT
ejpam-3352	77	10	resp	resp	NOUN
ejpam-3352	77	11	.	.	PUNCT
ejpam-3352	77	12	,	,	PUNCT
ejpam-3352	77	13	semi	semi	ADJ
ejpam-3352	77	14	-	-	ADJ
ejpam-3352	77	15	open	open	ADJ
ejpam-3352	77	16	)	)	PUNCT
ejpam-3352	77	17	cover	cover	VERB
ejpam-3352	77	18	u	u	NOUN
ejpam-3352	77	19	of	of	ADP
ejpam-3352	77	20	x	x	PUNCT
ejpam-3352	77	21	has	have	VERB
ejpam-3352	77	22	a	a	DET
ejpam-3352	77	23	locally	locally	ADV
ejpam-3352	77	24	finite	finite	ADJ
ejpam-3352	77	25	open	open	ADJ
ejpam-3352	77	26	refinement	refinement	PROPN
ejpam-3352	77	27	v	v	NOUN
ejpam-3352	77	28	(	(	PUNCT
ejpam-3352	77	29	not	not	PART
ejpam-3352	77	30	necessarily	necessarily	ADV
ejpam-3352	77	31	a	a	DET
ejpam-3352	77	32	cover	cover	NOUN
ejpam-3352	77	33	)	)	PUNCT
ejpam-3352	77	34	such	such	ADJ
ejpam-3352	77	35	that	that	SCONJ
ejpam-3352	77	36	x	x	SYM
ejpam-3352	77	37	\	\	PROPN
ejpam-3352	78	1	∪{v	∪{v	NOUN
ejpam-3352	78	2	:	:	PUNCT
ejpam-3352	78	3	v	v	NUM
ejpam-3352	78	4	∈	∈	PROPN
ejpam-3352	78	5	v	v	NOUN
ejpam-3352	78	6	}	}	PUNCT
ejpam-3352	78	7	∈	∈	PROPN
ejpam-3352	78	8	i.	i.	NOUN
ejpam-3352	78	9	definition	definition	NOUN
ejpam-3352	78	10	2	2	NUM
ejpam-3352	78	11	.	.	PUNCT
ejpam-3352	78	12	an	an	DET
ejpam-3352	78	13	ideal	ideal	ADJ
ejpam-3352	78	14	space	space	NOUN
ejpam-3352	78	15	(	(	PUNCT
ejpam-3352	78	16	x	x	X
ejpam-3352	78	17	,	,	PUNCT
ejpam-3352	78	18	τ	τ	PROPN
ejpam-3352	78	19	,	,	PUNCT
ejpam-3352	78	20	i	i	PROPN
ejpam-3352	78	21	)	)	PUNCT
ejpam-3352	78	22	is	be	AUX
ejpam-3352	78	23	said	say	VERB
ejpam-3352	78	24	to	to	PART
ejpam-3352	78	25	be	be	AUX
ejpam-3352	78	26	β1i	β1i	NOUN
ejpam-3352	78	27	-	-	ADJ
ejpam-3352	78	28	paracompact	paracompact	ADJ
ejpam-3352	78	29	,	,	PUNCT
ejpam-3352	78	30	or	or	CCONJ
ejpam-3352	78	31	β1	β1	NOUN
ejpam-3352	78	32	-	-	PUNCT
ejpam-3352	78	33	paracompact	paracompact	NOUN
ejpam-3352	78	34	modulo	modulo	VERB
ejpam-3352	78	35	an	an	DET
ejpam-3352	78	36	ideal	ideal	NOUN
ejpam-3352	78	37	i	i	PRON
ejpam-3352	78	38	if	if	SCONJ
ejpam-3352	78	39	every	every	DET
ejpam-3352	78	40	β	β	NOUN
ejpam-3352	78	41	-	-	ADJ
ejpam-3352	78	42	open	open	ADJ
ejpam-3352	78	43	cover	cover	NOUN
ejpam-3352	78	44	u	u	NOUN
ejpam-3352	78	45	of	of	ADP
ejpam-3352	78	46	x	x	PUNCT
ejpam-3352	78	47	has	have	VERB
ejpam-3352	78	48	a	a	DET
ejpam-3352	78	49	locally	locally	ADV
ejpam-3352	78	50	finite	finite	ADJ
ejpam-3352	78	51	open	open	ADJ
ejpam-3352	78	52	refinement	refinement	PROPN
ejpam-3352	78	53	v	v	NOUN
ejpam-3352	78	54	(	(	PUNCT
ejpam-3352	78	55	not	not	PART
ejpam-3352	78	56	necessarily	necessarily	ADV
ejpam-3352	78	57	a	a	DET
ejpam-3352	78	58	cover	cover	NOUN
ejpam-3352	78	59	)	)	PUNCT
ejpam-3352	78	60	such	such	ADJ
ejpam-3352	78	61	that	that	SCONJ
ejpam-3352	78	62	x	x	SYM
ejpam-3352	78	63	\	\	PROPN
ejpam-3352	78	64	∪{v	∪{v	NOUN
ejpam-3352	78	65	:	:	PUNCT
ejpam-3352	78	66	v	v	NUM
ejpam-3352	78	67	∈	∈	PROPN
ejpam-3352	78	68	v	v	NOUN
ejpam-3352	78	69	}	}	PUNCT
ejpam-3352	78	70	∈	∈	PROPN
ejpam-3352	78	71	i.	i.	NOUN
ejpam-3352	78	72	a	a	DET
ejpam-3352	78	73	family	family	NOUN
ejpam-3352	78	74	v	v	NOUN
ejpam-3352	78	75	of	of	ADP
ejpam-3352	78	76	subsets	subset	NOUN
ejpam-3352	78	77	of	of	ADP
ejpam-3352	78	78	x	x	SYM
ejpam-3352	78	79	such	such	ADJ
ejpam-3352	78	80	that	that	SCONJ
ejpam-3352	78	81	x	x	SYM
ejpam-3352	78	82	\	\	PROPN
ejpam-3352	79	1	∪{v	∪{v	NOUN
ejpam-3352	79	2	:	:	PUNCT
ejpam-3352	79	3	v	v	NUM
ejpam-3352	79	4	∈	∈	PROPN
ejpam-3352	79	5	v	v	NOUN
ejpam-3352	79	6	}	}	PUNCT
ejpam-3352	79	7	∈	∈	NOUN
ejpam-3352	79	8	i	i	PRON
ejpam-3352	79	9	is	be	AUX
ejpam-3352	79	10	called	call	VERB
ejpam-3352	79	11	an	an	DET
ejpam-3352	79	12	i	i	NOUN
ejpam-3352	79	13	-	-	PUNCT
ejpam-3352	79	14	cover	cover	NOUN
ejpam-3352	79	15	of	of	ADP
ejpam-3352	79	16	x.	x.	NOUN
ejpam-3352	79	17	it	it	PRON
ejpam-3352	79	18	follows	follow	VERB
ejpam-3352	79	19	from	from	ADP
ejpam-3352	79	20	the	the	DET
ejpam-3352	79	21	definitions	definition	NOUN
ejpam-3352	79	22	that	that	PRON
ejpam-3352	79	23	β1	β1	NOUN
ejpam-3352	79	24	-	-	PUNCT
ejpam-3352	79	25	paracompact	paracompact	ADJ
ejpam-3352	79	26	⇒	⇒	NOUN
ejpam-3352	79	27	β1i	β1i	ADJ
ejpam-3352	79	28	-	-	PUNCT
ejpam-3352	79	29	paracompact	paracompact	ADJ
ejpam-3352	79	30	⇒	⇒	NOUN
ejpam-3352	79	31	s1i	s1i	PROPN
ejpam-3352	79	32	-	-	PUNCT
ejpam-3352	79	33	paracompact	paracompact	ADJ
ejpam-3352	79	34	⇒	⇒	NOUN
ejpam-3352	79	35	i	i	PRON
ejpam-3352	79	36	-	-	PUNCT
ejpam-3352	79	37	paracompact	paracompact	VERB
ejpam-3352	80	1	the	the	DET
ejpam-3352	80	2	following	follow	VERB
ejpam-3352	80	3	examples	example	NOUN
ejpam-3352	80	4	show	show	VERB
ejpam-3352	80	5	that	that	SCONJ
ejpam-3352	80	6	the	the	DET
ejpam-3352	80	7	converses	converse	NOUN
ejpam-3352	80	8	of	of	ADP
ejpam-3352	80	9	the	the	DET
ejpam-3352	80	10	above	above	ADJ
ejpam-3352	80	11	implications	implication	NOUN
ejpam-3352	80	12	need	need	AUX
ejpam-3352	80	13	not	not	PART
ejpam-3352	80	14	be	be	AUX
ejpam-3352	80	15	true	true	ADJ
ejpam-3352	80	16	in	in	ADP
ejpam-3352	80	17	general	general	ADJ
ejpam-3352	80	18	.	.	PUNCT
ejpam-3352	81	1	example	example	NOUN
ejpam-3352	82	1	1	1	NUM
ejpam-3352	82	2	.	.	PUNCT
ejpam-3352	82	3	let	let	VERB
ejpam-3352	82	4	x	x	PUNCT
ejpam-3352	82	5	=	=	PUNCT
ejpam-3352	82	6	r	r	NOUN
ejpam-3352	82	7	with	with	ADP
ejpam-3352	82	8	the	the	DET
ejpam-3352	82	9	topology	topology	NOUN
ejpam-3352	82	10	τ	τ	X
ejpam-3352	82	11	=	=	PUNCT
ejpam-3352	82	12	{	{	PUNCT
ejpam-3352	82	13	∅	∅	NOUN
ejpam-3352	82	14	,	,	PUNCT
ejpam-3352	82	15	x	x	X
ejpam-3352	82	16	,	,	PUNCT
ejpam-3352	82	17	{	{	PUNCT
ejpam-3352	82	18	0	0	NUM
ejpam-3352	82	19	}	}	PUNCT
ejpam-3352	82	20	}	}	PUNCT
ejpam-3352	82	21	and	and	CCONJ
ejpam-3352	82	22	i	i	PRON
ejpam-3352	82	23	=	=	PUNCT
ejpam-3352	82	24	if	if	SCONJ
ejpam-3352	82	25	.	.	PUNCT
ejpam-3352	83	1	then	then	ADV
ejpam-3352	83	2	(	(	PUNCT
ejpam-3352	83	3	x	x	X
ejpam-3352	83	4	,	,	PUNCT
ejpam-3352	83	5	τ	τ	X
ejpam-3352	83	6	)	)	PUNCT
ejpam-3352	83	7	is	be	AUX
ejpam-3352	83	8	paracompact	paracompact	ADJ
ejpam-3352	83	9	which	which	PRON
ejpam-3352	83	10	implies	imply	VERB
ejpam-3352	83	11	that	that	SCONJ
ejpam-3352	83	12	(	(	PUNCT
ejpam-3352	83	13	x	x	X
ejpam-3352	83	14	,	,	PUNCT
ejpam-3352	83	15	τ	τ	PROPN
ejpam-3352	83	16	,	,	PUNCT
ejpam-3352	83	17	i	i	PROPN
ejpam-3352	83	18	)	)	PUNCT
ejpam-3352	83	19	is	be	AUX
ejpam-3352	83	20	i	i	NOUN
ejpam-3352	83	21	-	-	PUNCT
ejpam-3352	83	22	paracompact	paracompact	ADJ
ejpam-3352	83	23	.	.	PUNCT
ejpam-3352	84	1	on	on	ADP
ejpam-3352	84	2	the	the	DET
ejpam-3352	84	3	other	other	ADJ
ejpam-3352	84	4	hand	hand	NOUN
ejpam-3352	84	5	(	(	PUNCT
ejpam-3352	84	6	x	x	X
ejpam-3352	84	7	,	,	PUNCT
ejpam-3352	84	8	τ	τ	PROPN
ejpam-3352	84	9	,	,	PUNCT
ejpam-3352	84	10	i	i	PROPN
ejpam-3352	84	11	)	)	PUNCT
ejpam-3352	84	12	is	be	AUX
ejpam-3352	84	13	not	not	PART
ejpam-3352	84	14	β1i	β1i	ADJ
ejpam-3352	84	15	-	-	ADJ
ejpam-3352	84	16	paracompact	paracompact	ADJ
ejpam-3352	84	17	.	.	PUNCT
ejpam-3352	85	1	for	for	ADP
ejpam-3352	85	2	the	the	DET
ejpam-3352	85	3	β	β	NOUN
ejpam-3352	85	4	-	-	ADJ
ejpam-3352	85	5	open	open	ADJ
ejpam-3352	85	6	cover	cover	NOUN
ejpam-3352	85	7	u	u	NOUN
ejpam-3352	85	8	=	=	PUNCT
ejpam-3352	85	9	{	{	PUNCT
ejpam-3352	85	10	{	{	PUNCT
ejpam-3352	85	11	0	0	NUM
ejpam-3352	85	12	,	,	PUNCT
ejpam-3352	85	13	x	x	NOUN
ejpam-3352	85	14	}	}	PUNCT
ejpam-3352	85	15	:	:	PUNCT
ejpam-3352	85	16	x	x	SYM
ejpam-3352	85	17	∈	∈	NOUN
ejpam-3352	85	18	x	x	X
ejpam-3352	85	19	,	,	PUNCT
ejpam-3352	85	20	x	x	PROPN
ejpam-3352	85	21	6=	6=	ADP
ejpam-3352	85	22	0	0	NUM
ejpam-3352	85	23	}	}	PUNCT
ejpam-3352	85	24	,	,	PUNCT
ejpam-3352	85	25	we	we	PRON
ejpam-3352	85	26	can	can	AUX
ejpam-3352	85	27	find	find	VERB
ejpam-3352	85	28	a	a	DET
ejpam-3352	85	29	locally	locally	ADV
ejpam-3352	85	30	finite	finite	ADJ
ejpam-3352	85	31	open	open	ADJ
ejpam-3352	85	32	refinement	refinement	NOUN
ejpam-3352	85	33	v	v	PROPN
ejpam-3352	85	34	=	=	SYM
ejpam-3352	85	35	{	{	PUNCT
ejpam-3352	85	36	0	0	NUM
ejpam-3352	85	37	}	}	PUNCT
ejpam-3352	85	38	of	of	ADP
ejpam-3352	85	39	u	u	PROPN
ejpam-3352	85	40	.	.	PUNCT
ejpam-3352	86	1	but	but	CCONJ
ejpam-3352	86	2	v	v	NOUN
ejpam-3352	86	3	does	do	AUX
ejpam-3352	86	4	not	not	PART
ejpam-3352	86	5	i	i	PRON
ejpam-3352	86	6	-	-	PUNCT
ejpam-3352	86	7	cover	cover	NOUN
ejpam-3352	86	8	of	of	ADP
ejpam-3352	86	9	x.	x.	NOUN
ejpam-3352	86	10	therefore	therefore	ADV
ejpam-3352	86	11	,	,	PUNCT
ejpam-3352	86	12	(	(	PUNCT
ejpam-3352	86	13	x	x	X
ejpam-3352	86	14	,	,	PUNCT
ejpam-3352	86	15	τ	τ	PROPN
ejpam-3352	86	16	,	,	PUNCT
ejpam-3352	86	17	i	i	PROPN
ejpam-3352	86	18	)	)	PUNCT
ejpam-3352	86	19	is	be	AUX
ejpam-3352	86	20	not	not	PART
ejpam-3352	86	21	β1i	β1i	ADJ
ejpam-3352	86	22	-	-	ADJ
ejpam-3352	86	23	paracompact	paracompact	ADJ
ejpam-3352	86	24	.	.	PUNCT
ejpam-3352	86	25	example	example	NOUN
ejpam-3352	87	1	2	2	NUM
ejpam-3352	87	2	.	.	PUNCT
ejpam-3352	88	1	let	let	VERB
ejpam-3352	88	2	x	x	PUNCT
ejpam-3352	88	3	=	=	PRON
ejpam-3352	88	4	{	{	PUNCT
ejpam-3352	88	5	1	1	NUM
ejpam-3352	88	6	,	,	PUNCT
ejpam-3352	88	7	2	2	NUM
ejpam-3352	88	8	,	,	PUNCT
ejpam-3352	88	9	3	3	NUM
ejpam-3352	88	10	,	,	PUNCT
ejpam-3352	88	11	4	4	NUM
ejpam-3352	88	12	}	}	PUNCT
ejpam-3352	88	13	with	with	ADP
ejpam-3352	88	14	the	the	DET
ejpam-3352	88	15	topology	topology	NOUN
ejpam-3352	88	16	τ	τ	X
ejpam-3352	88	17	=	=	PUNCT
ejpam-3352	88	18	{	{	PUNCT
ejpam-3352	88	19	∅	∅	NOUN
ejpam-3352	88	20	,	,	PUNCT
ejpam-3352	88	21	x	x	X
ejpam-3352	88	22	,	,	PUNCT
ejpam-3352	88	23	{	{	PUNCT
ejpam-3352	88	24	1	1	NUM
ejpam-3352	88	25	,	,	PUNCT
ejpam-3352	88	26	2	2	NUM
ejpam-3352	88	27	}	}	PUNCT
ejpam-3352	88	28	,	,	PUNCT
ejpam-3352	88	29	{	{	PUNCT
ejpam-3352	88	30	3	3	NUM
ejpam-3352	88	31	,	,	PUNCT
ejpam-3352	88	32	4	4	NUM
ejpam-3352	88	33	}	}	PUNCT
ejpam-3352	88	34	}	}	PUNCT
ejpam-3352	88	35	and	and	CCONJ
ejpam-3352	88	36	i	i	PRON
ejpam-3352	88	37	=	=	PUNCT
ejpam-3352	89	1	if	if	SCONJ
ejpam-3352	89	2	.	.	PUNCT
ejpam-3352	90	1	then	then	ADV
ejpam-3352	90	2	(	(	PUNCT
ejpam-3352	90	3	x	x	X
ejpam-3352	90	4	,	,	PUNCT
ejpam-3352	90	5	τ	τ	PROPN
ejpam-3352	90	6	,	,	PUNCT
ejpam-3352	90	7	i	i	PROPN
ejpam-3352	90	8	)	)	PUNCT
ejpam-3352	90	9	is	be	AUX
ejpam-3352	90	10	s1i	s1i	NOUN
ejpam-3352	90	11	-	-	PUNCT
ejpam-3352	90	12	paracompact	paracompact	ADJ
ejpam-3352	90	13	,	,	PUNCT
ejpam-3352	90	14	since	since	SCONJ
ejpam-3352	90	15	so(x	so(x	NOUN
ejpam-3352	90	16	,	,	PUNCT
ejpam-3352	90	17	τ	τ	X
ejpam-3352	90	18	)	)	PUNCT
ejpam-3352	90	19	=	=	SYM
ejpam-3352	90	20	τ	τ	PROPN
ejpam-3352	90	21	,	,	PUNCT
ejpam-3352	90	22	but	but	CCONJ
ejpam-3352	90	23	it	it	PRON
ejpam-3352	90	24	is	be	AUX
ejpam-3352	90	25	not	not	PART
ejpam-3352	90	26	β1i	β1i	ADJ
ejpam-3352	90	27	-	-	ADJ
ejpam-3352	90	28	paracompact	paracompact	NOUN
ejpam-3352	90	29	since	since	SCONJ
ejpam-3352	90	30	u	u	NOUN
ejpam-3352	90	31	=	=	X
ejpam-3352	90	32	{	{	PUNCT
ejpam-3352	90	33	{	{	PUNCT
ejpam-3352	90	34	1	1	NUM
ejpam-3352	90	35	}	}	PUNCT
ejpam-3352	90	36	,	,	PUNCT
ejpam-3352	90	37	{	{	PUNCT
ejpam-3352	90	38	2	2	NUM
ejpam-3352	90	39	}	}	PUNCT
ejpam-3352	90	40	,	,	PUNCT
ejpam-3352	90	41	{	{	PUNCT
ejpam-3352	90	42	3	3	NUM
ejpam-3352	90	43	}	}	PUNCT
ejpam-3352	90	44	,	,	PUNCT
ejpam-3352	90	45	{	{	PUNCT
ejpam-3352	90	46	4	4	NUM
ejpam-3352	90	47	}	}	PUNCT
ejpam-3352	90	48	}	}	PUNCT
ejpam-3352	90	49	is	be	AUX
ejpam-3352	90	50	a	a	DET
ejpam-3352	90	51	β	β	NOUN
ejpam-3352	90	52	-	-	ADJ
ejpam-3352	90	53	open	open	ADJ
ejpam-3352	90	54	cover	cover	NOUN
ejpam-3352	90	55	of	of	ADP
ejpam-3352	90	56	x	x	PUNCT
ejpam-3352	90	57	which	which	PRON
ejpam-3352	90	58	admits	admit	VERB
ejpam-3352	90	59	no	no	DET
ejpam-3352	90	60	locally	locally	ADV
ejpam-3352	90	61	finite	finite	ADJ
ejpam-3352	90	62	open	open	ADJ
ejpam-3352	90	63	refinement	refinement	NOUN
ejpam-3352	90	64	.	.	PUNCT
ejpam-3352	91	1	example	example	NOUN
ejpam-3352	92	1	3	3	X
ejpam-3352	92	2	.	.	X
ejpam-3352	92	3	consider	consider	VERB
ejpam-3352	92	4	the	the	DET
ejpam-3352	92	5	ideal	ideal	ADJ
ejpam-3352	92	6	space	space	NOUN
ejpam-3352	92	7	(	(	PUNCT
ejpam-3352	92	8	x	x	X
ejpam-3352	92	9	,	,	PUNCT
ejpam-3352	92	10	τ	τ	PROPN
ejpam-3352	92	11	,	,	PUNCT
ejpam-3352	92	12	i	i	NOUN
ejpam-3352	92	13	)	)	PUNCT
ejpam-3352	92	14	where	where	SCONJ
ejpam-3352	92	15	x	x	X
ejpam-3352	92	16	=	=	PRON
ejpam-3352	92	17	{	{	PUNCT
ejpam-3352	92	18	1	1	NUM
ejpam-3352	92	19	,	,	PUNCT
ejpam-3352	92	20	2	2	NUM
ejpam-3352	92	21	,	,	PUNCT
ejpam-3352	92	22	3	3	NUM
ejpam-3352	92	23	}	}	PUNCT
ejpam-3352	92	24	,	,	PUNCT
ejpam-3352	92	25	τ	τ	X
ejpam-3352	92	26	=	=	PUNCT
ejpam-3352	92	27	{	{	PUNCT
ejpam-3352	92	28	∅	∅	NOUN
ejpam-3352	92	29	,	,	PUNCT
ejpam-3352	92	30	x	x	X
ejpam-3352	92	31	,	,	PUNCT
ejpam-3352	92	32	{	{	PUNCT
ejpam-3352	92	33	1	1	NUM
ejpam-3352	92	34	}	}	PUNCT
ejpam-3352	92	35	}	}	PUNCT
ejpam-3352	92	36	and	and	CCONJ
ejpam-3352	92	37	i	i	PRON
ejpam-3352	92	38	=	=	PUNCT
ejpam-3352	92	39	{	{	PUNCT
ejpam-3352	92	40	a	a	DET
ejpam-3352	92	41	⊂	⊂	X
ejpam-3352	92	42	x	x	X
ejpam-3352	92	43	:	:	PUNCT
ejpam-3352	92	44	1	1	NUM
ejpam-3352	92	45	/∈	/∈	SYM
ejpam-3352	92	46	a	a	PRON
ejpam-3352	92	47	}	}	PUNCT
ejpam-3352	92	48	.	.	PUNCT
ejpam-3352	93	1	then	then	ADV
ejpam-3352	93	2	βo(x	βo(x	PUNCT
ejpam-3352	93	3	,	,	PUNCT
ejpam-3352	93	4	τ	τ	X
ejpam-3352	93	5	)	)	PUNCT
ejpam-3352	93	6	=	=	PRON
ejpam-3352	93	7	{	{	PUNCT
ejpam-3352	93	8	∅	∅	NOUN
ejpam-3352	93	9	,	,	PUNCT
ejpam-3352	93	10	x	x	X
ejpam-3352	93	11	,	,	PUNCT
ejpam-3352	93	12	{	{	PUNCT
ejpam-3352	93	13	1	1	NUM
ejpam-3352	93	14	}	}	PUNCT
ejpam-3352	93	15	,	,	PUNCT
ejpam-3352	93	16	{	{	PUNCT
ejpam-3352	93	17	1	1	NUM
ejpam-3352	93	18	,	,	PUNCT
ejpam-3352	93	19	2	2	NUM
ejpam-3352	93	20	}	}	PUNCT
ejpam-3352	93	21	,	,	PUNCT
ejpam-3352	93	22	{	{	PUNCT
ejpam-3352	93	23	1	1	NUM
ejpam-3352	93	24	,	,	PUNCT
ejpam-3352	93	25	3	3	NUM
ejpam-3352	93	26	}	}	PUNCT
ejpam-3352	93	27	}	}	PUNCT
ejpam-3352	93	28	.	.	PUNCT
ejpam-3352	94	1	therefore	therefore	ADV
ejpam-3352	94	2	(	(	PUNCT
ejpam-3352	94	3	x	x	X
ejpam-3352	94	4	,	,	PUNCT
ejpam-3352	94	5	τ	τ	PROPN
ejpam-3352	94	6	,	,	PUNCT
ejpam-3352	94	7	i	i	PROPN
ejpam-3352	94	8	)	)	PUNCT
ejpam-3352	94	9	is	be	AUX
ejpam-3352	94	10	β1i	β1i	ADJ
ejpam-3352	94	11	-	-	ADJ
ejpam-3352	94	12	paracompact	paracompact	ADJ
ejpam-3352	94	13	space	space	NOUN
ejpam-3352	94	14	.	.	PUNCT
ejpam-3352	95	1	on	on	ADP
ejpam-3352	95	2	the	the	DET
ejpam-3352	95	3	other	other	ADJ
ejpam-3352	95	4	hand	hand	NOUN
ejpam-3352	95	5	,	,	PUNCT
ejpam-3352	95	6	(	(	PUNCT
ejpam-3352	95	7	x	x	X
ejpam-3352	95	8	,	,	PUNCT
ejpam-3352	95	9	τ	τ	X
ejpam-3352	95	10	)	)	PUNCT
ejpam-3352	95	11	is	be	AUX
ejpam-3352	95	12	not	not	PART
ejpam-3352	95	13	β1paracompact	β1paracompact	ADJ
ejpam-3352	95	14	since	since	SCONJ
ejpam-3352	95	15	u	u	NOUN
ejpam-3352	95	16	=	=	X
ejpam-3352	95	17	{	{	PUNCT
ejpam-3352	95	18	{	{	PUNCT
ejpam-3352	95	19	1	1	NUM
ejpam-3352	95	20	,	,	PUNCT
ejpam-3352	95	21	2	2	NUM
ejpam-3352	95	22	}	}	PUNCT
ejpam-3352	95	23	,	,	PUNCT
ejpam-3352	95	24	{	{	PUNCT
ejpam-3352	95	25	1	1	NUM
ejpam-3352	95	26	,	,	PUNCT
ejpam-3352	95	27	3	3	NUM
ejpam-3352	95	28	}	}	PUNCT
ejpam-3352	95	29	}	}	PUNCT
ejpam-3352	95	30	is	be	AUX
ejpam-3352	95	31	a	a	DET
ejpam-3352	95	32	β1	β1	NOUN
ejpam-3352	95	33	-	-	PUNCT
ejpam-3352	95	34	open	open	ADJ
ejpam-3352	95	35	cover	cover	NOUN
ejpam-3352	95	36	of	of	ADP
ejpam-3352	95	37	(	(	PUNCT
ejpam-3352	95	38	x	x	X
ejpam-3352	95	39	,	,	PUNCT
ejpam-3352	95	40	τ	τ	X
ejpam-3352	95	41	)	)	PUNCT
ejpam-3352	95	42	which	which	PRON
ejpam-3352	95	43	admits	admit	VERB
ejpam-3352	95	44	no	no	DET
ejpam-3352	95	45	locally	locally	ADV
ejpam-3352	95	46	finite	finite	ADJ
ejpam-3352	95	47	open	open	ADJ
ejpam-3352	95	48	refinement	refinement	NOUN
ejpam-3352	95	49	.	.	PUNCT
ejpam-3352	96	1	example	example	NOUN
ejpam-3352	97	1	4	4	X
ejpam-3352	97	2	.	.	PUNCT
ejpam-3352	97	3	consider	consider	VERB
ejpam-3352	97	4	the	the	DET
ejpam-3352	97	5	ideal	ideal	ADJ
ejpam-3352	97	6	space	space	NOUN
ejpam-3352	97	7	(	(	PUNCT
ejpam-3352	97	8	x	x	X
ejpam-3352	97	9	,	,	PUNCT
ejpam-3352	97	10	τ	τ	PROPN
ejpam-3352	97	11	,	,	PUNCT
ejpam-3352	97	12	i	i	NOUN
ejpam-3352	97	13	)	)	PUNCT
ejpam-3352	97	14	where	where	SCONJ
ejpam-3352	97	15	x	x	X
ejpam-3352	97	16	=	=	SYM
ejpam-3352	97	17	r	r	NOUN
ejpam-3352	97	18	,	,	PUNCT
ejpam-3352	97	19	the	the	DET
ejpam-3352	97	20	set	set	NOUN
ejpam-3352	97	21	of	of	ADP
ejpam-3352	97	22	all	all	DET
ejpam-3352	97	23	real	real	ADJ
ejpam-3352	97	24	numbers	number	NOUN
ejpam-3352	97	25	,	,	PUNCT
ejpam-3352	97	26	τ	τ	PROPN
ejpam-3352	97	27	=	=	PUNCT
ejpam-3352	97	28	{	{	PUNCT
ejpam-3352	97	29	∅	∅	NOUN
ejpam-3352	97	30	,	,	PUNCT
ejpam-3352	97	31	x	x	X
ejpam-3352	97	32	,	,	PUNCT
ejpam-3352	97	33	{	{	PUNCT
ejpam-3352	97	34	0	0	NUM
ejpam-3352	97	35	}	}	PUNCT
ejpam-3352	97	36	}	}	PUNCT
ejpam-3352	97	37	and	and	CCONJ
ejpam-3352	97	38	i	i	PRON
ejpam-3352	97	39	=	=	PUNCT
ejpam-3352	97	40	{	{	PUNCT
ejpam-3352	97	41	a	a	DET
ejpam-3352	97	42	⊂	⊂	X
ejpam-3352	97	43	x	x	X
ejpam-3352	97	44	:	:	PUNCT
ejpam-3352	97	45	0	0	NUM
ejpam-3352	97	46	/∈	/∈	PUNCT
ejpam-3352	97	47	a	a	PRON
ejpam-3352	97	48	}	}	PUNCT
ejpam-3352	97	49	.	.	PUNCT
ejpam-3352	98	1	then	then	ADV
ejpam-3352	98	2	(	(	PUNCT
ejpam-3352	98	3	x	x	X
ejpam-3352	98	4	,	,	PUNCT
ejpam-3352	98	5	τ	τ	PROPN
ejpam-3352	98	6	,	,	PUNCT
ejpam-3352	98	7	i	i	PROPN
ejpam-3352	98	8	)	)	PUNCT
ejpam-3352	98	9	is	be	AUX
ejpam-3352	98	10	β1i	β1i	ADJ
ejpam-3352	98	11	-	-	ADJ
ejpam-3352	98	12	paracompact	paracompact	ADJ
ejpam-3352	98	13	space	space	NOUN
ejpam-3352	98	14	but	but	CCONJ
ejpam-3352	98	15	(	(	PUNCT
ejpam-3352	98	16	x	x	X
ejpam-3352	98	17	,	,	PUNCT
ejpam-3352	98	18	τ	τ	X
ejpam-3352	98	19	)	)	PUNCT
ejpam-3352	98	20	is	be	AUX
ejpam-3352	98	21	not	not	PART
ejpam-3352	98	22	β1	β1	NOUN
ejpam-3352	98	23	-	-	PUNCT
ejpam-3352	98	24	paracompact	paracompact	ADJ
ejpam-3352	98	25	,	,	PUNCT
ejpam-3352	98	26	since	since	SCONJ
ejpam-3352	98	27	the	the	DET
ejpam-3352	98	28	β	β	NOUN
ejpam-3352	98	29	-	-	ADJ
ejpam-3352	98	30	open	open	ADJ
ejpam-3352	98	31	cover	cover	NOUN
ejpam-3352	98	32	u	u	NOUN
ejpam-3352	98	33	=	=	PUNCT
ejpam-3352	98	34	{	{	PUNCT
ejpam-3352	98	35	{	{	PUNCT
ejpam-3352	98	36	0	0	NUM
ejpam-3352	98	37	,	,	PUNCT
ejpam-3352	98	38	x	x	NOUN
ejpam-3352	98	39	}	}	PUNCT
ejpam-3352	98	40	:	:	PUNCT
ejpam-3352	98	41	x	x	SYM
ejpam-3352	98	42	∈	∈	NOUN
ejpam-3352	98	43	x	x	X
ejpam-3352	98	44	,	,	PUNCT
ejpam-3352	98	45	x	x	PROPN
ejpam-3352	98	46	6=	6=	ADP
ejpam-3352	98	47	0	0	NUM
ejpam-3352	98	48	}	}	PUNCT
ejpam-3352	98	49	admits	admit	VERB
ejpam-3352	98	50	no	no	DET
ejpam-3352	98	51	locally	locally	ADV
ejpam-3352	98	52	finite	finite	ADJ
ejpam-3352	98	53	open	open	ADJ
ejpam-3352	98	54	refinement	refinement	NOUN
ejpam-3352	98	55	.	.	PUNCT
ejpam-3352	99	1	corollary	corollary	ADJ
ejpam-3352	99	2	1	1	NUM
ejpam-3352	99	3	.	.	PUNCT
ejpam-3352	100	1	let	let	VERB
ejpam-3352	100	2	(	(	PUNCT
ejpam-3352	100	3	x	x	NOUN
ejpam-3352	100	4	,	,	PUNCT
ejpam-3352	100	5	τ	τ	X
ejpam-3352	100	6	)	)	PUNCT
ejpam-3352	100	7	be	be	VERB
ejpam-3352	100	8	a	a	DET
ejpam-3352	100	9	space	space	NOUN
ejpam-3352	100	10	with	with	ADP
ejpam-3352	100	11	an	an	DET
ejpam-3352	100	12	ideal	ideal	NOUN
ejpam-3352	100	13	i	i	NOUN
ejpam-3352	100	14	=	=	SYM
ejpam-3352	100	15	{	{	PUNCT
ejpam-3352	100	16	∅	∅	NOUN
ejpam-3352	100	17	}	}	PUNCT
ejpam-3352	100	18	.	.	PUNCT
ejpam-3352	101	1	then	then	ADV
ejpam-3352	101	2	(	(	PUNCT
ejpam-3352	101	3	x	x	X
ejpam-3352	101	4	,	,	PUNCT
ejpam-3352	101	5	τ	τ	X
ejpam-3352	101	6	)	)	PUNCT
ejpam-3352	101	7	is	be	AUX
ejpam-3352	101	8	β1	β1	NOUN
ejpam-3352	101	9	-	-	PUNCT
ejpam-3352	101	10	paracompact	paracompact	NOUN
ejpam-3352	101	11	if	if	SCONJ
ejpam-3352	101	12	and	and	CCONJ
ejpam-3352	101	13	only	only	ADV
ejpam-3352	101	14	if	if	SCONJ
ejpam-3352	101	15	(	(	PUNCT
ejpam-3352	101	16	x	x	NOUN
ejpam-3352	101	17	,	,	PUNCT
ejpam-3352	101	18	τ	τ	PROPN
ejpam-3352	101	19	,	,	PUNCT
ejpam-3352	101	20	i	i	PROPN
ejpam-3352	101	21	)	)	PUNCT
ejpam-3352	101	22	is	be	AUX
ejpam-3352	101	23	β1i	β1i	NOUN
ejpam-3352	101	24	-	-	ADJ
ejpam-3352	101	25	paracompact	paracompact	ADJ
ejpam-3352	101	26	.	.	PUNCT
ejpam-3352	102	1	a.	a.	NOUN
ejpam-3352	102	2	qahis	qahis	PROPN
ejpam-3352	102	3	/	/	SYM
ejpam-3352	102	4	eur	eur	PROPN
ejpam-3352	102	5	.	.	PUNCT
ejpam-3352	103	1	j.	j.	PROPN
ejpam-3352	103	2	pure	pure	PROPN
ejpam-3352	103	3	appl	appl	PROPN
ejpam-3352	103	4	.	.	PROPN
ejpam-3352	103	5	math	math	PROPN
ejpam-3352	103	6	,	,	PUNCT
ejpam-3352	103	7	12	12	NUM
ejpam-3352	103	8	(	(	PUNCT
ejpam-3352	103	9	1	1	NUM
ejpam-3352	103	10	)	)	PUNCT
ejpam-3352	103	11	(	(	PUNCT
ejpam-3352	103	12	2019	2019	NUM
ejpam-3352	103	13	)	)	PUNCT
ejpam-3352	103	14	,	,	PUNCT
ejpam-3352	103	15	135	135	NUM
ejpam-3352	103	16	-	-	SYM
ejpam-3352	103	17	145	145	NUM
ejpam-3352	103	18	138	138	NUM
ejpam-3352	103	19	corollary	corollary	ADJ
ejpam-3352	103	20	2	2	NUM
ejpam-3352	103	21	.	.	PUNCT
ejpam-3352	104	1	for	for	ADP
ejpam-3352	104	2	an	an	DET
ejpam-3352	104	3	e.d	e.d	PROPN
ejpam-3352	104	4	.	.	PROPN
ejpam-3352	104	5	submaximal	submaximal	ADJ
ejpam-3352	104	6	ideal	ideal	ADJ
ejpam-3352	104	7	space	space	NOUN
ejpam-3352	104	8	(	(	PUNCT
ejpam-3352	104	9	x	x	X
ejpam-3352	104	10	,	,	PUNCT
ejpam-3352	104	11	τ	τ	PROPN
ejpam-3352	104	12	,	,	PUNCT
ejpam-3352	104	13	i	i	PROPN
ejpam-3352	104	14	)	)	PUNCT
ejpam-3352	104	15	,	,	PUNCT
ejpam-3352	104	16	the	the	DET
ejpam-3352	104	17	following	follow	VERB
ejpam-3352	104	18	conditions	condition	NOUN
ejpam-3352	104	19	are	be	AUX
ejpam-3352	104	20	equivalent	equivalent	ADJ
ejpam-3352	104	21	:	:	PUNCT
ejpam-3352	104	22	(	(	PUNCT
ejpam-3352	104	23	i	i	NOUN
ejpam-3352	104	24	)	)	PUNCT
ejpam-3352	104	25	(	(	PUNCT
ejpam-3352	104	26	x	x	X
ejpam-3352	104	27	,	,	PUNCT
ejpam-3352	104	28	τ	τ	PROPN
ejpam-3352	104	29	,	,	PUNCT
ejpam-3352	104	30	i	i	PROPN
ejpam-3352	104	31	)	)	PUNCT
ejpam-3352	104	32	is	be	AUX
ejpam-3352	104	33	β1i	β1i	NOUN
ejpam-3352	104	34	-	-	ADJ
ejpam-3352	104	35	paracompact	paracompact	ADJ
ejpam-3352	104	36	;	;	PUNCT
ejpam-3352	104	37	(	(	PUNCT
ejpam-3352	104	38	ii	ii	NOUN
ejpam-3352	104	39	)	)	PUNCT
ejpam-3352	104	40	(	(	PUNCT
ejpam-3352	104	41	x	x	X
ejpam-3352	104	42	,	,	PUNCT
ejpam-3352	104	43	τ	τ	PROPN
ejpam-3352	104	44	,	,	PUNCT
ejpam-3352	104	45	i	i	PROPN
ejpam-3352	104	46	)	)	PUNCT
ejpam-3352	104	47	is	be	AUX
ejpam-3352	104	48	s1i	s1i	NOUN
ejpam-3352	104	49	-	-	PUNCT
ejpam-3352	104	50	paracompact	paracompact	ADJ
ejpam-3352	104	51	;	;	PUNCT
ejpam-3352	104	52	(	(	PUNCT
ejpam-3352	104	53	iii	iii	X
ejpam-3352	104	54	)	)	PUNCT
ejpam-3352	104	55	(	(	PUNCT
ejpam-3352	104	56	x	x	X
ejpam-3352	104	57	,	,	PUNCT
ejpam-3352	104	58	τ	τ	PROPN
ejpam-3352	104	59	,	,	PUNCT
ejpam-3352	104	60	i	i	PROPN
ejpam-3352	104	61	)	)	PUNCT
ejpam-3352	104	62	is	be	AUX
ejpam-3352	104	63	i	i	NOUN
ejpam-3352	104	64	-	-	PUNCT
ejpam-3352	104	65	paracompact	paracompact	ADJ
ejpam-3352	104	66	.	.	PUNCT
ejpam-3352	105	1	proof	proof	NOUN
ejpam-3352	105	2	.	.	PUNCT
ejpam-3352	106	1	this	this	PRON
ejpam-3352	106	2	follows	follow	VERB
ejpam-3352	106	3	directly	directly	ADV
ejpam-3352	106	4	from	from	ADP
ejpam-3352	106	5	the	the	DET
ejpam-3352	106	6	fact	fact	NOUN
ejpam-3352	106	7	that	that	SCONJ
ejpam-3352	106	8	if	if	SCONJ
ejpam-3352	106	9	an	an	DET
ejpam-3352	106	10	ideal	ideal	ADJ
ejpam-3352	106	11	space	space	NOUN
ejpam-3352	106	12	(	(	PUNCT
ejpam-3352	106	13	x	x	X
ejpam-3352	106	14	,	,	PUNCT
ejpam-3352	106	15	τ	τ	PROPN
ejpam-3352	106	16	,	,	PUNCT
ejpam-3352	106	17	i	i	PROPN
ejpam-3352	106	18	)	)	PUNCT
ejpam-3352	106	19	is	be	AUX
ejpam-3352	106	20	an	an	DET
ejpam-3352	106	21	e.d	e.d	PROPN
ejpam-3352	106	22	.	.	PROPN
ejpam-3352	106	23	submaximal	submaximal	ADJ
ejpam-3352	106	24	space	space	NOUN
ejpam-3352	106	25	,	,	PUNCT
ejpam-3352	106	26	then	then	ADV
ejpam-3352	106	27	τ	τ	PROPN
ejpam-3352	106	28	=	=	SYM
ejpam-3352	106	29	so(x	so(x	PROPN
ejpam-3352	106	30	,	,	PUNCT
ejpam-3352	106	31	τ	τ	X
ejpam-3352	106	32	)	)	PUNCT
ejpam-3352	106	33	=	=	SYM
ejpam-3352	106	34	βo(x	βo(x	X
ejpam-3352	106	35	,	,	PUNCT
ejpam-3352	106	36	τ	τ	PROPN
ejpam-3352	106	37	)	)	PUNCT
ejpam-3352	106	38	.	.	PUNCT
ejpam-3352	107	1	proposition	proposition	NOUN
ejpam-3352	107	2	1	1	NUM
ejpam-3352	107	3	.	.	PUNCT
ejpam-3352	108	1	if	if	SCONJ
ejpam-3352	108	2	(	(	PUNCT
ejpam-3352	108	3	x	x	X
ejpam-3352	108	4	,	,	PUNCT
ejpam-3352	108	5	τ	τ	PROPN
ejpam-3352	108	6	,	,	PUNCT
ejpam-3352	108	7	i	i	PROPN
ejpam-3352	108	8	)	)	PUNCT
ejpam-3352	108	9	is	be	AUX
ejpam-3352	108	10	β1i	β1i	NOUN
ejpam-3352	108	11	-	-	ADJ
ejpam-3352	108	12	paracompact	paracompact	ADJ
ejpam-3352	108	13	,	,	PUNCT
ejpam-3352	108	14	then	then	ADV
ejpam-3352	108	15	(	(	PUNCT
ejpam-3352	108	16	x	x	NOUN
ejpam-3352	108	17	,	,	PUNCT
ejpam-3352	108	18	τα	τα	PROPN
ejpam-3352	108	19	,	,	PUNCT
ejpam-3352	108	20	i	i	PROPN
ejpam-3352	108	21	)	)	PUNCT
ejpam-3352	108	22	is	be	AUX
ejpam-3352	108	23	β1i	β1i	NOUN
ejpam-3352	108	24	-	-	ADJ
ejpam-3352	108	25	paracompact	paracompact	ADJ
ejpam-3352	108	26	.	.	PUNCT
ejpam-3352	109	1	proof	proof	NOUN
ejpam-3352	109	2	.	.	PUNCT
ejpam-3352	110	1	suppose	suppose	VERB
ejpam-3352	110	2	(	(	PUNCT
ejpam-3352	110	3	x	x	X
ejpam-3352	110	4	,	,	PUNCT
ejpam-3352	110	5	τ	τ	PROPN
ejpam-3352	110	6	,	,	PUNCT
ejpam-3352	110	7	i	i	PROPN
ejpam-3352	110	8	)	)	PUNCT
ejpam-3352	110	9	is	be	AUX
ejpam-3352	110	10	β1i	β1i	NOUN
ejpam-3352	110	11	-	-	NOUN
ejpam-3352	110	12	paracompact	paracompact	ADJ
ejpam-3352	110	13	.	.	PUNCT
ejpam-3352	111	1	let	let	VERB
ejpam-3352	111	2	u	u	PRON
ejpam-3352	111	3	=	=	PUNCT
ejpam-3352	111	4	{	{	PUNCT
ejpam-3352	111	5	uα	uα	X
ejpam-3352	111	6	:	:	PUNCT
ejpam-3352	111	7	α	α	PROPN
ejpam-3352	111	8	∈	∈	PROPN
ejpam-3352	111	9	∆	∆	PROPN
ejpam-3352	111	10	}	}	PUNCT
ejpam-3352	111	11	be	be	AUX
ejpam-3352	111	12	a	a	DET
ejpam-3352	111	13	β	β	NOUN
ejpam-3352	111	14	-	-	ADJ
ejpam-3352	111	15	open	open	ADJ
ejpam-3352	111	16	cover	cover	NOUN
ejpam-3352	111	17	of	of	ADP
ejpam-3352	111	18	(	(	PUNCT
ejpam-3352	111	19	x	x	NOUN
ejpam-3352	111	20	,	,	PUNCT
ejpam-3352	111	21	τα	τα	PROPN
ejpam-3352	111	22	,	,	PUNCT
ejpam-3352	111	23	i	i	PROPN
ejpam-3352	111	24	)	)	PUNCT
ejpam-3352	111	25	.	.	PUNCT
ejpam-3352	112	1	then	then	ADV
ejpam-3352	112	2	u	u	PRON
ejpam-3352	112	3	is	be	AUX
ejpam-3352	112	4	a	a	DET
ejpam-3352	112	5	β	β	NOUN
ejpam-3352	112	6	-	-	ADJ
ejpam-3352	112	7	open	open	ADJ
ejpam-3352	112	8	cover	cover	NOUN
ejpam-3352	112	9	of	of	ADP
ejpam-3352	112	10	(	(	PUNCT
ejpam-3352	112	11	x	x	X
ejpam-3352	112	12	,	,	PUNCT
ejpam-3352	112	13	τ	τ	PROPN
ejpam-3352	112	14	,	,	PUNCT
ejpam-3352	112	15	i	i	PROPN
ejpam-3352	112	16	)	)	PUNCT
ejpam-3352	112	17	.	.	PUNCT
ejpam-3352	113	1	by	by	ADP
ejpam-3352	113	2	hypothesis	hypothesis	NOUN
ejpam-3352	113	3	,	,	PUNCT
ejpam-3352	113	4	there	there	PRON
ejpam-3352	113	5	exist	exist	VERB
ejpam-3352	113	6	a	a	DET
ejpam-3352	113	7	locally	locally	ADV
ejpam-3352	113	8	finite	finite	ADJ
ejpam-3352	113	9	open	open	ADJ
ejpam-3352	113	10	refinement	refinement	NOUN
ejpam-3352	113	11	v	v	X
ejpam-3352	113	12	=	=	PUNCT
ejpam-3352	113	13	{	{	PUNCT
ejpam-3352	113	14	vλ	vλ	INTJ
ejpam-3352	113	15	:	:	PUNCT
ejpam-3352	113	16	λ	λ	PROPN
ejpam-3352	113	17	∈	∈	PROPN
ejpam-3352	113	18	λ	λ	PROPN
ejpam-3352	113	19	}	}	PUNCT
ejpam-3352	113	20	of	of	ADP
ejpam-3352	113	21	u	u	PRON
ejpam-3352	113	22	such	such	ADJ
ejpam-3352	113	23	that	that	SCONJ
ejpam-3352	113	24	x	x	PUNCT
ejpam-3352	113	25	∪	∪	ADJ
ejpam-3352	113	26	\{vλ	\{vλ	NOUN
ejpam-3352	113	27	:	:	PUNCT
ejpam-3352	113	28	λ	λ	X
ejpam-3352	113	29	∈	∈	PROPN
ejpam-3352	113	30	λ	λ	PROPN
ejpam-3352	113	31	}	}	PUNCT
ejpam-3352	113	32	∈	∈	PROPN
ejpam-3352	113	33	i.	i.	NOUN
ejpam-3352	113	34	since	since	SCONJ
ejpam-3352	113	35	τ	τ	PROPN
ejpam-3352	113	36	⊂	⊂	PROPN
ejpam-3352	113	37	τα	τα	PROPN
ejpam-3352	113	38	,	,	PUNCT
ejpam-3352	113	39	the	the	DET
ejpam-3352	113	40	family	family	NOUN
ejpam-3352	113	41	v	v	NOUN
ejpam-3352	113	42	is	be	AUX
ejpam-3352	113	43	a	a	DET
ejpam-3352	113	44	τα	τα	NOUN
ejpam-3352	113	45	-	-	ADJ
ejpam-3352	113	46	locally	locally	ADV
ejpam-3352	113	47	finite	finite	ADJ
ejpam-3352	113	48	τα	τα	ADJ
ejpam-3352	113	49	-	-	ADJ
ejpam-3352	113	50	open	open	ADJ
ejpam-3352	113	51	refinement	refinement	NOUN
ejpam-3352	113	52	of	of	ADP
ejpam-3352	113	53	u	u	NOUN
ejpam-3352	113	54	and	and	CCONJ
ejpam-3352	113	55	so	so	ADV
ejpam-3352	113	56	(	(	PUNCT
ejpam-3352	113	57	x	x	NOUN
ejpam-3352	113	58	,	,	PUNCT
ejpam-3352	113	59	τα	τα	PROPN
ejpam-3352	113	60	,	,	PUNCT
ejpam-3352	113	61	i	i	PROPN
ejpam-3352	113	62	)	)	PUNCT
ejpam-3352	113	63	is	be	AUX
ejpam-3352	113	64	β1i	β1i	NOUN
ejpam-3352	113	65	-	-	NOUN
ejpam-3352	113	66	paracompact	paracompact	ADJ
ejpam-3352	113	67	.	.	PUNCT
ejpam-3352	114	1	by	by	ADP
ejpam-3352	114	2	replacing	replace	VERB
ejpam-3352	114	3	i	i	PRON
ejpam-3352	114	4	by	by	ADP
ejpam-3352	114	5	{	{	PUNCT
ejpam-3352	114	6	∅	∅	NOUN
ejpam-3352	114	7	}	}	PUNCT
ejpam-3352	114	8	in	in	ADP
ejpam-3352	114	9	proposition	proposition	NOUN
ejpam-3352	114	10	1	1	NUM
ejpam-3352	114	11	,	,	PUNCT
ejpam-3352	114	12	we	we	PRON
ejpam-3352	114	13	have	have	VERB
ejpam-3352	114	14	the	the	DET
ejpam-3352	114	15	following	follow	VERB
ejpam-3352	114	16	corollary	corollary	NOUN
ejpam-3352	114	17	.	.	PUNCT
ejpam-3352	115	1	corollary	corollary	ADJ
ejpam-3352	115	2	3	3	NUM
ejpam-3352	115	3	.	.	PUNCT
ejpam-3352	116	1	[	[	X
ejpam-3352	116	2	1	1	NUM
ejpam-3352	116	3	,	,	PUNCT
ejpam-3352	116	4	theorem	theorem	ADJ
ejpam-3352	116	5	2.8(2	2.8(2	NUM
ejpam-3352	116	6	)	)	PUNCT
ejpam-3352	116	7	]	]	PUNCT
ejpam-3352	117	1	if	if	SCONJ
ejpam-3352	117	2	(	(	PUNCT
ejpam-3352	117	3	x	x	NOUN
ejpam-3352	117	4	,	,	PUNCT
ejpam-3352	117	5	τ	τ	X
ejpam-3352	117	6	)	)	PUNCT
ejpam-3352	117	7	is	be	AUX
ejpam-3352	117	8	β1	β1	NOUN
ejpam-3352	117	9	-	-	PUNCT
ejpam-3352	117	10	paracompact	paracompact	ADJ
ejpam-3352	117	11	,	,	PUNCT
ejpam-3352	117	12	then	then	ADV
ejpam-3352	117	13	(	(	PUNCT
ejpam-3352	117	14	x	x	NOUN
ejpam-3352	117	15	,	,	PUNCT
ejpam-3352	117	16	τα	τα	NOUN
ejpam-3352	117	17	)	)	PUNCT
ejpam-3352	117	18	is	be	AUX
ejpam-3352	117	19	β1	β1	NOUN
ejpam-3352	117	20	-	-	PUNCT
ejpam-3352	117	21	paracompact	paracompact	NOUN
ejpam-3352	117	22	.	.	PUNCT
ejpam-3352	118	1	theorem	theorem	NOUN
ejpam-3352	118	2	3	3	X
ejpam-3352	118	3	.	.	PUNCT
ejpam-3352	119	1	let	let	VERB
ejpam-3352	119	2	(	(	PUNCT
ejpam-3352	119	3	x	x	X
ejpam-3352	119	4	,	,	PUNCT
ejpam-3352	119	5	τ	τ	PROPN
ejpam-3352	119	6	,	,	PUNCT
ejpam-3352	119	7	i	i	PRON
ejpam-3352	119	8	)	)	PUNCT
ejpam-3352	119	9	be	be	VERB
ejpam-3352	119	10	an	an	DET
ejpam-3352	119	11	ideal	ideal	ADJ
ejpam-3352	119	12	space	space	NOUN
ejpam-3352	119	13	.	.	PUNCT
ejpam-3352	120	1	if	if	SCONJ
ejpam-3352	120	2	i	i	PRON
ejpam-3352	120	3	is	be	AUX
ejpam-3352	120	4	codense	codense	NOUN
ejpam-3352	120	5	,	,	PUNCT
ejpam-3352	120	6	(	(	PUNCT
ejpam-3352	120	7	x	x	NOUN
ejpam-3352	120	8	,	,	PUNCT
ejpam-3352	120	9	τ∗	τ∗	PROPN
ejpam-3352	120	10	,	,	PUNCT
ejpam-3352	120	11	i	i	NOUN
ejpam-3352	120	12	)	)	PUNCT
ejpam-3352	120	13	is	be	AUX
ejpam-3352	120	14	β1i	β1i	NOUN
ejpam-3352	120	15	-	-	ADJ
ejpam-3352	120	16	paracompact	paracompact	NOUN
ejpam-3352	121	1	and	and	CCONJ
ejpam-3352	121	2	i	i	PRON
ejpam-3352	121	3	is	be	AUX
ejpam-3352	121	4	τ	τ	PROPN
ejpam-3352	121	5	-simple	-simple	PROPN
ejpam-3352	121	6	,	,	PUNCT
ejpam-3352	121	7	then	then	ADV
ejpam-3352	121	8	every	every	DET
ejpam-3352	121	9	β	β	X
ejpam-3352	121	10	-	-	ADJ
ejpam-3352	121	11	open	open	ADJ
ejpam-3352	121	12	cover	cover	NOUN
ejpam-3352	121	13	of	of	ADP
ejpam-3352	121	14	(	(	PUNCT
ejpam-3352	121	15	x	x	X
ejpam-3352	121	16	,	,	PUNCT
ejpam-3352	121	17	τ	τ	PROPN
ejpam-3352	121	18	,	,	PUNCT
ejpam-3352	121	19	i	i	NOUN
ejpam-3352	121	20	)	)	PUNCT
ejpam-3352	121	21	has	have	AUX
ejpam-3352	121	22	τ	τ	PROPN
ejpam-3352	121	23	-locally	-locally	ADV
ejpam-3352	121	24	finite	finite	VERB
ejpam-3352	121	25	β	β	NOUN
ejpam-3352	121	26	-	-	ADJ
ejpam-3352	121	27	open	open	ADJ
ejpam-3352	121	28	i	i	NOUN
ejpam-3352	121	29	-	-	PUNCT
ejpam-3352	121	30	cover	cover	NOUN
ejpam-3352	121	31	refinement	refinement	NOUN
ejpam-3352	121	32	.	.	PUNCT
ejpam-3352	122	1	proof	proof	NOUN
ejpam-3352	122	2	.	.	PUNCT
ejpam-3352	123	1	let	let	VERB
ejpam-3352	123	2	u	u	PRON
ejpam-3352	123	3	=	=	PUNCT
ejpam-3352	123	4	{	{	PUNCT
ejpam-3352	123	5	uα	uα	X
ejpam-3352	123	6	:	:	PUNCT
ejpam-3352	123	7	α	α	PROPN
ejpam-3352	123	8	∈	∈	PROPN
ejpam-3352	123	9	∆	∆	PROPN
ejpam-3352	123	10	}	}	PUNCT
ejpam-3352	123	11	be	be	AUX
ejpam-3352	123	12	a	a	DET
ejpam-3352	123	13	β	β	NOUN
ejpam-3352	123	14	-	-	ADJ
ejpam-3352	123	15	open	open	ADJ
ejpam-3352	123	16	cover	cover	NOUN
ejpam-3352	123	17	of	of	ADP
ejpam-3352	123	18	x.	x.	NOUN
ejpam-3352	123	19	since	since	SCONJ
ejpam-3352	123	20	,	,	PUNCT
ejpam-3352	123	21	βo(x	βo(x	PUNCT
ejpam-3352	123	22	,	,	PUNCT
ejpam-3352	123	23	τ	τ	PROPN
ejpam-3352	123	24	)	)	PUNCT
ejpam-3352	123	25	⊆	⊆	NUM
ejpam-3352	123	26	βo(x	βo(x	NUM
ejpam-3352	123	27	,	,	PUNCT
ejpam-3352	123	28	τ∗	τ∗	NOUN
ejpam-3352	123	29	)	)	PUNCT
ejpam-3352	123	30	.	.	PUNCT
ejpam-3352	124	1	then	then	ADV
ejpam-3352	124	2	u	u	PRON
ejpam-3352	124	3	is	be	AUX
ejpam-3352	124	4	a	a	DET
ejpam-3352	124	5	β	β	NOUN
ejpam-3352	124	6	-	-	ADJ
ejpam-3352	124	7	open	open	ADJ
ejpam-3352	124	8	cover	cover	NOUN
ejpam-3352	124	9	of	of	ADP
ejpam-3352	124	10	(	(	PUNCT
ejpam-3352	124	11	x	x	NOUN
ejpam-3352	124	12	,	,	PUNCT
ejpam-3352	124	13	τ∗	τ∗	PROPN
ejpam-3352	124	14	,	,	PUNCT
ejpam-3352	124	15	i	i	NOUN
ejpam-3352	124	16	)	)	PUNCT
ejpam-3352	124	17	.	.	PUNCT
ejpam-3352	125	1	by	by	ADP
ejpam-3352	125	2	hypothesis	hypothesis	NOUN
ejpam-3352	125	3	,	,	PUNCT
ejpam-3352	125	4	there	there	PRON
ejpam-3352	125	5	exist	exist	VERB
ejpam-3352	125	6	τ∗-locally	τ∗-locally	ADV
ejpam-3352	125	7	finite	finite	X
ejpam-3352	125	8	τ∗-open	τ∗-open	PUNCT
ejpam-3352	125	9	refinement	refinement	NOUN
ejpam-3352	125	10	v	v	NOUN
ejpam-3352	125	11	=	=	SYM
ejpam-3352	125	12	{	{	PUNCT
ejpam-3352	125	13	vλ	vλ	ADP
ejpam-3352	125	14	\	\	PROPN
ejpam-3352	125	15	iλ	iλ	NOUN
ejpam-3352	125	16	:	:	PUNCT
ejpam-3352	125	17	λ	λ	X
ejpam-3352	125	18	∈	∈	PROPN
ejpam-3352	125	19	λ	λ	PROPN
ejpam-3352	125	20	,	,	PUNCT
ejpam-3352	125	21	vλ	vλ	ADP
ejpam-3352	125	22	∈	∈	PROPN
ejpam-3352	125	23	τ	τ	PROPN
ejpam-3352	125	24	,	,	PUNCT
ejpam-3352	125	25	iλ	iλ	PROPN
ejpam-3352	125	26	∈	∈	PROPN
ejpam-3352	126	1	i	i	X
ejpam-3352	126	2	}	}	PUNCT
ejpam-3352	126	3	of	of	ADP
ejpam-3352	126	4	u	u	PRON
ejpam-3352	126	5	such	such	ADJ
ejpam-3352	126	6	that	that	SCONJ
ejpam-3352	126	7	x	x	SYM
ejpam-3352	126	8	\∪{vλ	\∪{vλ	PROPN
ejpam-3352	126	9	\	\	PROPN
ejpam-3352	126	10	iλ	iλ	NOUN
ejpam-3352	126	11	:	:	PUNCT
ejpam-3352	126	12	λ	λ	X
ejpam-3352	126	13	∈	∈	PROPN
ejpam-3352	126	14	λ	λ	PROPN
ejpam-3352	126	15	}	}	PUNCT
ejpam-3352	126	16	∈	∈	PROPN
ejpam-3352	126	17	i.	i.	NOUN
ejpam-3352	126	18	for	for	ADP
ejpam-3352	126	19	each	each	DET
ejpam-3352	126	20	x	x	SYM
ejpam-3352	126	21	∈	∈	PROPN
ejpam-3352	126	22	x	x	X
ejpam-3352	126	23	,	,	PUNCT
ejpam-3352	126	24	there	there	PRON
ejpam-3352	126	25	exists	exist	VERB
ejpam-3352	126	26	a	a	DET
ejpam-3352	126	27	τ∗-open	τ∗-open	PUNCT
ejpam-3352	126	28	set	set	NOUN
ejpam-3352	126	29	w	w	NOUN
ejpam-3352	126	30	containing	contain	VERB
ejpam-3352	126	31	x	x	PUNCT
ejpam-3352	126	32	such	such	ADJ
ejpam-3352	126	33	that	that	SCONJ
ejpam-3352	126	34	w	w	NOUN
ejpam-3352	126	35	∩	∩	NOUN
ejpam-3352	126	36	(	(	PUNCT
ejpam-3352	126	37	vλ	vλ	ADP
ejpam-3352	126	38	\	\	PROPN
ejpam-3352	126	39	iλ	iλ	NOUN
ejpam-3352	126	40	)	)	PUNCT
ejpam-3352	126	41	=	=	NOUN
ejpam-3352	126	42	∅	∅	NOUN
ejpam-3352	126	43	for	for	ADP
ejpam-3352	126	44	λ	λ	PROPN
ejpam-3352	126	45	6=	6=	SYM
ejpam-3352	126	46	λ1	λ1	ADJ
ejpam-3352	126	47	,	,	PUNCT
ejpam-3352	126	48	λ2	λ2	NOUN
ejpam-3352	126	49	,	,	PUNCT
ejpam-3352	126	50	...	...	PUNCT
ejpam-3352	126	51	,	,	PUNCT
ejpam-3352	126	52	λn	λn	PROPN
ejpam-3352	126	53	.	.	PUNCT
ejpam-3352	127	1	since	since	SCONJ
ejpam-3352	127	2	i	i	PRON
ejpam-3352	127	3	is	be	AUX
ejpam-3352	127	4	τ	τ	PROPN
ejpam-3352	127	5	-simple	-simple	PROPN
ejpam-3352	127	6	,	,	PUNCT
ejpam-3352	127	7	w	w	PROPN
ejpam-3352	127	8	=	=	SYM
ejpam-3352	127	9	u	u	NOUN
ejpam-3352	127	10	\	\	PROPN
ejpam-3352	127	11	i	i	PRON
ejpam-3352	127	12	for	for	ADP
ejpam-3352	127	13	some	some	DET
ejpam-3352	127	14	u	u	NOUN
ejpam-3352	127	15	∈	∈	PROPN
ejpam-3352	127	16	τ	τ	X
ejpam-3352	128	1	and	and	CCONJ
ejpam-3352	128	2	i	i	PROPN
ejpam-3352	128	3	∈	∈	PROPN
ejpam-3352	128	4	i.	i.	NOUN
ejpam-3352	128	5	thus	thus	ADV
ejpam-3352	128	6	,	,	PUNCT
ejpam-3352	128	7	(	(	PUNCT
ejpam-3352	128	8	u	u	NOUN
ejpam-3352	128	9	\	\	PROPN
ejpam-3352	128	10	i	i	PROPN
ejpam-3352	128	11	)	)	PUNCT
ejpam-3352	128	12	∩	∩	NOUN
ejpam-3352	128	13	(	(	PUNCT
ejpam-3352	128	14	vλ	vλ	ADP
ejpam-3352	128	15	\	\	PROPN
ejpam-3352	128	16	iλ	iλ	NOUN
ejpam-3352	128	17	)	)	PUNCT
ejpam-3352	128	18	=	=	NOUN
ejpam-3352	128	19	∅	∅	NOUN
ejpam-3352	128	20	for	for	ADP
ejpam-3352	128	21	λ	λ	PROPN
ejpam-3352	128	22	6=	6=	SYM
ejpam-3352	128	23	λ1	λ1	ADJ
ejpam-3352	128	24	,	,	PUNCT
ejpam-3352	128	25	λ2	λ2	NOUN
ejpam-3352	128	26	,	,	PUNCT
ejpam-3352	128	27	...	...	PUNCT
ejpam-3352	128	28	,	,	PUNCT
ejpam-3352	128	29	λn	λn	X
ejpam-3352	128	30	which	which	PRON
ejpam-3352	128	31	implies	imply	VERB
ejpam-3352	128	32	that	that	SCONJ
ejpam-3352	128	33	(	(	PUNCT
ejpam-3352	128	34	u	u	NOUN
ejpam-3352	128	35	∩	∩	NOUN
ejpam-3352	128	36	vλ	vλ	ADP
ejpam-3352	128	37	)	)	PUNCT
ejpam-3352	128	38	\	\	PUNCT
ejpam-3352	129	1	(	(	PUNCT
ejpam-3352	129	2	i	i	PRON
ejpam-3352	129	3	∪	∪	VERB
ejpam-3352	129	4	iλ	iλ	NOUN
ejpam-3352	129	5	)	)	PUNCT
ejpam-3352	129	6	=	=	NOUN
ejpam-3352	129	7	∅	∅	NOUN
ejpam-3352	129	8	for	for	ADP
ejpam-3352	129	9	λ	λ	PROPN
ejpam-3352	129	10	6=	6=	SYM
ejpam-3352	129	11	λ1	λ1	ADJ
ejpam-3352	129	12	,	,	PUNCT
ejpam-3352	129	13	λ2	λ2	NOUN
ejpam-3352	129	14	,	,	PUNCT
ejpam-3352	129	15	...	...	PUNCT
ejpam-3352	129	16	,	,	PUNCT
ejpam-3352	129	17	λn	λn	PROPN
ejpam-3352	129	18	.	.	PUNCT
ejpam-3352	130	1	since	since	SCONJ
ejpam-3352	130	2	i	i	PRON
ejpam-3352	130	3	is	be	AUX
ejpam-3352	130	4	codense	codense	NOUN
ejpam-3352	130	5	,	,	PUNCT
ejpam-3352	130	6	then	then	ADV
ejpam-3352	130	7	u	u	NOUN
ejpam-3352	130	8	∩	∩	NOUN
ejpam-3352	130	9	vλ	vλ	ADP
ejpam-3352	130	10	=	=	NOUN
ejpam-3352	130	11	∅	∅	NOUN
ejpam-3352	130	12	for	for	ADP
ejpam-3352	130	13	λ	λ	PROPN
ejpam-3352	130	14	6=	6=	SYM
ejpam-3352	130	15	λ1	λ1	ADJ
ejpam-3352	130	16	,	,	PUNCT
ejpam-3352	130	17	λ2	λ2	NOUN
ejpam-3352	130	18	,	,	PUNCT
ejpam-3352	130	19	...	...	PUNCT
ejpam-3352	130	20	,	,	PUNCT
ejpam-3352	130	21	λn	λn	PROPN
ejpam-3352	130	22	.	.	PUNCT
ejpam-3352	131	1	then	then	ADV
ejpam-3352	131	2	u	u	NOUN
ejpam-3352	131	3	∩	∩	NOUN
ejpam-3352	131	4	(	(	PUNCT
ejpam-3352	131	5	vλ	vλ	ADP
ejpam-3352	131	6	∩uα	∩uα	NOUN
ejpam-3352	131	7	)	)	PUNCT
ejpam-3352	131	8	=	=	NOUN
ejpam-3352	131	9	∅	∅	NOUN
ejpam-3352	131	10	for	for	ADP
ejpam-3352	131	11	λ	λ	PROPN
ejpam-3352	131	12	6=	6=	SYM
ejpam-3352	131	13	λ1	λ1	ADJ
ejpam-3352	131	14	,	,	PUNCT
ejpam-3352	131	15	λ2	λ2	NOUN
ejpam-3352	131	16	,	,	PUNCT
ejpam-3352	131	17	...	...	PUNCT
ejpam-3352	131	18	,	,	PUNCT
ejpam-3352	131	19	λn	λn	NOUN
ejpam-3352	131	20	.	.	PUNCT
ejpam-3352	132	1	by	by	ADP
ejpam-3352	132	2	lemma	lemma	PROPN
ejpam-3352	132	3	5	5	NUM
ejpam-3352	132	4	,	,	PUNCT
ejpam-3352	132	5	w	w	NOUN
ejpam-3352	132	6	=	=	PUNCT
ejpam-3352	132	7	{	{	PUNCT
ejpam-3352	132	8	vλ	vλ	INTJ
ejpam-3352	132	9	∩uα	∩uα	NOUN
ejpam-3352	132	10	:	:	PUNCT
ejpam-3352	133	1	λ	λ	X
ejpam-3352	133	2	∈	∈	PROPN
ejpam-3352	133	3	λ	λ	PROPN
ejpam-3352	133	4	}	}	PUNCT
ejpam-3352	133	5	is	be	AUX
ejpam-3352	133	6	τ	τ	PROPN
ejpam-3352	133	7	-locally	-locally	ADV
ejpam-3352	133	8	finite	finite	ADJ
ejpam-3352	133	9	β	β	ADJ
ejpam-3352	133	10	-	-	ADJ
ejpam-3352	133	11	open	open	ADJ
ejpam-3352	133	12	refinement	refinement	NOUN
ejpam-3352	133	13	of	of	ADP
ejpam-3352	133	14	u	u	PROPN
ejpam-3352	133	15	.	.	PUNCT
ejpam-3352	134	1	since	since	SCONJ
ejpam-3352	134	2	v	v	NOUN
ejpam-3352	134	3	refines	refine	VERB
ejpam-3352	134	4	u	u	NOUN
ejpam-3352	134	5	for	for	ADP
ejpam-3352	134	6	every	every	DET
ejpam-3352	134	7	vλ	vλ	ADJ
ejpam-3352	134	8	\	\	PROPN
ejpam-3352	134	9	iλ	iλ	PROPN
ejpam-3352	134	10	∈	∈	PROPN
ejpam-3352	134	11	v	v	NOUN
ejpam-3352	134	12	,	,	PUNCT
ejpam-3352	134	13	there	there	PRON
ejpam-3352	134	14	exists	exist	VERB
ejpam-3352	134	15	uα	uα	PROPN
ejpam-3352	134	16	∈	∈	PROPN
ejpam-3352	134	17	u	u	NOUN
ejpam-3352	134	18	such	such	ADJ
ejpam-3352	134	19	that	that	SCONJ
ejpam-3352	134	20	vλ	vλ	PROPN
ejpam-3352	134	21	\	\	PROPN
ejpam-3352	134	22	iλ	iλ	PROPN
ejpam-3352	134	23	⊂	⊂	PROPN
ejpam-3352	134	24	uα	uα	PROPN
ejpam-3352	134	25	.	.	PUNCT
ejpam-3352	135	1	thus	thus	ADV
ejpam-3352	135	2	,	,	PUNCT
ejpam-3352	135	3	vλ	vλ	ADP
ejpam-3352	135	4	\	\	PROPN
ejpam-3352	135	5	iλ	iλ	PROPN
ejpam-3352	135	6	=	=	SYM
ejpam-3352	135	7	uα	uα	PROPN
ejpam-3352	135	8	∩	∩	NOUN
ejpam-3352	135	9	(	(	PUNCT
ejpam-3352	135	10	vλ	vλ	ADP
ejpam-3352	135	11	\	\	PROPN
ejpam-3352	135	12	iλ	iλ	PROPN
ejpam-3352	135	13	)	)	PUNCT
ejpam-3352	135	14	=	=	SYM
ejpam-3352	135	15	(	(	PUNCT
ejpam-3352	135	16	vλ	vλ	ADP
ejpam-3352	135	17	∩	∩	NOUN
ejpam-3352	135	18	uα	uα	NOUN
ejpam-3352	135	19	)	)	PUNCT
ejpam-3352	135	20	\	\	PROPN
ejpam-3352	136	1	iλ	iλ	PROPN
ejpam-3352	136	2	⊂	⊂	PROPN
ejpam-3352	136	3	vλ	vλ	ADP
ejpam-3352	136	4	∩	∩	NOUN
ejpam-3352	136	5	uα	uα	PROPN
ejpam-3352	136	6	⊂	⊂	PROPN
ejpam-3352	136	7	uα	uα	PROPN
ejpam-3352	136	8	.	.	PUNCT
ejpam-3352	137	1	then	then	ADV
ejpam-3352	137	2	x	x	SYM
ejpam-3352	137	3	\	\	PROPN
ejpam-3352	137	4	∪{vλ	∪{vλ	NUM
ejpam-3352	137	5	∩	∩	X
ejpam-3352	137	6	uα	uα	NOUN
ejpam-3352	137	7	:	:	PUNCT
ejpam-3352	137	8	λ	λ	PROPN
ejpam-3352	137	9	∈	∈	PROPN
ejpam-3352	137	10	λ	λ	PROPN
ejpam-3352	137	11	,	,	PUNCT
ejpam-3352	137	12	α	α	PROPN
ejpam-3352	137	13	∈	∈	NOUN
ejpam-3352	137	14	∆	∆	X
ejpam-3352	137	15	}	}	PUNCT
ejpam-3352	137	16	⊂	⊂	PUNCT
ejpam-3352	137	17	x	x	SYM
ejpam-3352	137	18	\	\	X
ejpam-3352	137	19	∪{vλ	∪{vλ	NUM
ejpam-3352	137	20	\	\	NOUN
ejpam-3352	137	21	iλ	iλ	NOUN
ejpam-3352	137	22	:	:	PUNCT
ejpam-3352	137	23	λ	λ	X
ejpam-3352	137	24	∈	∈	PROPN
ejpam-3352	137	25	λ	λ	PROPN
ejpam-3352	137	26	}	}	PUNCT
ejpam-3352	137	27	∈	∈	PROPN
ejpam-3352	137	28	i	i	PRON
ejpam-3352	137	29	which	which	PRON
ejpam-3352	137	30	implies	imply	VERB
ejpam-3352	137	31	that	that	SCONJ
ejpam-3352	137	32	x	x	SYM
ejpam-3352	137	33	\	\	X
ejpam-3352	137	34	∪{vλ	∪{vλ	NUM
ejpam-3352	137	35	∩	∩	X
ejpam-3352	137	36	uα	uα	NOUN
ejpam-3352	137	37	:	:	PUNCT
ejpam-3352	137	38	λ	λ	PROPN
ejpam-3352	137	39	∈	∈	PROPN
ejpam-3352	137	40	λ	λ	PROPN
ejpam-3352	137	41	,	,	PUNCT
ejpam-3352	137	42	α	α	PROPN
ejpam-3352	137	43	∈	∈	NOUN
ejpam-3352	137	44	∆	∆	X
ejpam-3352	137	45	}	}	PUNCT
ejpam-3352	137	46	∈	∈	PROPN
ejpam-3352	137	47	i.	i.	NOUN
ejpam-3352	137	48	theorem	theorem	VERB
ejpam-3352	137	49	4	4	NUM
ejpam-3352	137	50	.	.	PUNCT
ejpam-3352	138	1	let	let	VERB
ejpam-3352	138	2	(	(	PUNCT
ejpam-3352	138	3	x	x	X
ejpam-3352	138	4	,	,	PUNCT
ejpam-3352	138	5	τ	τ	PROPN
ejpam-3352	138	6	,	,	PUNCT
ejpam-3352	138	7	i	i	PRON
ejpam-3352	138	8	)	)	PUNCT
ejpam-3352	138	9	be	be	VERB
ejpam-3352	138	10	an	an	DET
ejpam-3352	138	11	ideal	ideal	ADJ
ejpam-3352	138	12	space	space	NOUN
ejpam-3352	138	13	.	.	PUNCT
ejpam-3352	139	1	if	if	SCONJ
ejpam-3352	139	2	i	i	PRON
ejpam-3352	139	3	is	be	AUX
ejpam-3352	139	4	weakly	weakly	ADJ
ejpam-3352	139	5	τ	τ	X
ejpam-3352	139	6	-local	-local	ADJ
ejpam-3352	139	7	and	and	CCONJ
ejpam-3352	139	8	(	(	PUNCT
ejpam-3352	139	9	x	x	X
ejpam-3352	139	10	,	,	PUNCT
ejpam-3352	139	11	τ	τ	PROPN
ejpam-3352	139	12	,	,	PUNCT
ejpam-3352	139	13	i	i	PROPN
ejpam-3352	139	14	)	)	PUNCT
ejpam-3352	139	15	is	be	AUX
ejpam-3352	139	16	β1iparacompact	β1iparacompact	PROPN
ejpam-3352	139	17	,	,	PUNCT
ejpam-3352	139	18	then	then	ADV
ejpam-3352	139	19	(	(	PUNCT
ejpam-3352	139	20	x	x	NOUN
ejpam-3352	139	21	,	,	PUNCT
ejpam-3352	139	22	τ∗	τ∗	PROPN
ejpam-3352	139	23	,	,	PUNCT
ejpam-3352	139	24	i	i	NOUN
ejpam-3352	139	25	)	)	PUNCT
ejpam-3352	139	26	is	be	AUX
ejpam-3352	139	27	β1i	β1i	NOUN
ejpam-3352	139	28	-	-	ADJ
ejpam-3352	139	29	paracompact	paracompact	ADJ
ejpam-3352	139	30	.	.	PUNCT
ejpam-3352	140	1	proof	proof	NOUN
ejpam-3352	140	2	.	.	PUNCT
ejpam-3352	141	1	let	let	VERB
ejpam-3352	141	2	u	u	PRON
ejpam-3352	141	3	=	=	X
ejpam-3352	141	4	{	{	PUNCT
ejpam-3352	141	5	uα	uα	PROPN
ejpam-3352	141	6	\	\	PROPN
ejpam-3352	141	7	iα	iα	NOUN
ejpam-3352	141	8	:	:	PUNCT
ejpam-3352	141	9	α	α	PROPN
ejpam-3352	141	10	∈	∈	PROPN
ejpam-3352	141	11	∆	∆	PROPN
ejpam-3352	141	12	,	,	PUNCT
ejpam-3352	141	13	uα	uα	PROPN
ejpam-3352	141	14	∈	∈	PROPN
ejpam-3352	141	15	τ	τ	PROPN
ejpam-3352	141	16	,	,	PUNCT
ejpam-3352	141	17	iα	iα	ADP
ejpam-3352	141	18	∈	∈	PROPN
ejpam-3352	142	1	i	i	PRON
ejpam-3352	142	2	}	}	PUNCT
ejpam-3352	142	3	be	be	VERB
ejpam-3352	142	4	a	a	DET
ejpam-3352	142	5	β	β	NOUN
ejpam-3352	142	6	-	-	ADJ
ejpam-3352	142	7	open	open	ADJ
ejpam-3352	142	8	cover	cover	NOUN
ejpam-3352	142	9	of	of	ADP
ejpam-3352	142	10	(	(	PUNCT
ejpam-3352	142	11	x	x	NOUN
ejpam-3352	142	12	,	,	PUNCT
ejpam-3352	142	13	τ∗	τ∗	PROPN
ejpam-3352	142	14	,	,	PUNCT
ejpam-3352	142	15	i	i	PROPN
ejpam-3352	142	16	)	)	PUNCT
ejpam-3352	142	17	.	.	PUNCT
ejpam-3352	143	1	then	then	ADV
ejpam-3352	143	2	w	w	X
ejpam-3352	143	3	=	=	PRON
ejpam-3352	143	4	{	{	PUNCT
ejpam-3352	143	5	uα	uα	X
ejpam-3352	143	6	:	:	PUNCT
ejpam-3352	143	7	α	α	PROPN
ejpam-3352	143	8	∈	∈	PROPN
ejpam-3352	143	9	∆	∆	X
ejpam-3352	143	10	}	}	PUNCT
ejpam-3352	143	11	is	be	AUX
ejpam-3352	143	12	a	a	DET
ejpam-3352	143	13	β	β	NOUN
ejpam-3352	143	14	-	-	ADJ
ejpam-3352	143	15	open	open	ADJ
ejpam-3352	143	16	cover	cover	NOUN
ejpam-3352	143	17	of	of	ADP
ejpam-3352	143	18	x	x	PUNCT
ejpam-3352	144	1	and	and	CCONJ
ejpam-3352	144	2	so	so	ADV
ejpam-3352	144	3	it	it	PRON
ejpam-3352	144	4	has	have	AUX
ejpam-3352	144	5	locally	locally	ADV
ejpam-3352	144	6	finite	finite	ADJ
ejpam-3352	144	7	open	open	ADJ
ejpam-3352	144	8	refinement	refinement	NOUN
ejpam-3352	144	9	v	v	X
ejpam-3352	144	10	=	=	PUNCT
ejpam-3352	144	11	{	{	PUNCT
ejpam-3352	144	12	vλ	vλ	INTJ
ejpam-3352	144	13	:	:	PUNCT
ejpam-3352	144	14	λ	λ	PROPN
ejpam-3352	144	15	∈	∈	PROPN
ejpam-3352	144	16	λ	λ	NOUN
ejpam-3352	144	17	}	}	PUNCT
ejpam-3352	144	18	such	such	ADJ
ejpam-3352	144	19	that	that	SCONJ
ejpam-3352	144	20	x	x	SYM
ejpam-3352	144	21	\	\	PROPN
ejpam-3352	144	22	∪{vλ	∪{vλ	NUM
ejpam-3352	144	23	:	:	PUNCT
ejpam-3352	144	24	λ	λ	PROPN
ejpam-3352	144	25	∈	∈	PROPN
ejpam-3352	144	26	λ	λ	PROPN
ejpam-3352	144	27	}	}	PUNCT
ejpam-3352	144	28	∈	∈	PROPN
ejpam-3352	144	29	i.	i.	NOUN
ejpam-3352	144	30	now	now	ADV
ejpam-3352	144	31	the	the	DET
ejpam-3352	144	32	family	family	NOUN
ejpam-3352	144	33	{	{	PUNCT
ejpam-3352	144	34	vλ	vλ	PROPN
ejpam-3352	144	35	∩	∩	NOUN
ejpam-3352	144	36	iα	iα	NOUN
ejpam-3352	144	37	:	:	PUNCT
ejpam-3352	144	38	λ	λ	X
ejpam-3352	144	39	∈	∈	PROPN
ejpam-3352	144	40	λ	λ	PROPN
ejpam-3352	144	41	}	}	PUNCT
ejpam-3352	144	42	⊂	⊂	PROPN
ejpam-3352	145	1	i	i	PRON
ejpam-3352	145	2	is	be	AUX
ejpam-3352	145	3	locally	locally	ADV
ejpam-3352	145	4	finite	finite	ADJ
ejpam-3352	145	5	.	.	PUNCT
ejpam-3352	146	1	since	since	ADV
ejpam-3352	146	2	,	,	PUNCT
ejpam-3352	146	3	i	i	PRON
ejpam-3352	146	4	is	be	AUX
ejpam-3352	146	5	weakly	weakly	ADJ
ejpam-3352	146	6	τ	τ	X
ejpam-3352	146	7	-local	-local	PROPN
ejpam-3352	146	8	,	,	PUNCT
ejpam-3352	146	9	∪λ∈λ(vλ	∪λ∈λ(vλ	NOUN
ejpam-3352	146	10	∩	∩	X
ejpam-3352	146	11	iα	iα	NOUN
ejpam-3352	146	12	)	)	PUNCT
ejpam-3352	146	13	∈	∈	PROPN
ejpam-3352	147	1	i	i	PRON
ejpam-3352	147	2	,	,	PUNCT
ejpam-3352	147	3	by	by	ADP
ejpam-3352	147	4	lemma	lemma	PROPN
ejpam-3352	147	5	4	4	NUM
ejpam-3352	147	6	.	.	PUNCT
ejpam-3352	147	7	then	then	ADV
ejpam-3352	147	8	x	x	X
ejpam-3352	147	9	\	\	PROPN
ejpam-3352	147	10	∪λ∈λ(vλ	∪λ∈λ(vλ	X
ejpam-3352	147	11	\	\	PROPN
ejpam-3352	147	12	iα	iα	PROPN
ejpam-3352	147	13	)	)	PUNCT
ejpam-3352	147	14	⊂	⊂	PROPN
ejpam-3352	147	15	(	(	PUNCT
ejpam-3352	147	16	x	x	SYM
ejpam-3352	147	17	\	\	NOUN
ejpam-3352	147	18	⋃	⋃	PUNCT
ejpam-3352	147	19	λ∈λ	λ∈λ	NOUN
ejpam-3352	147	20	vλ)∪	vλ)∪	NOUN
ejpam-3352	147	21	(	(	PUNCT
ejpam-3352	147	22	⋃	⋃	PROPN
ejpam-3352	147	23	λ∈λ	λ∈λ	NOUN
ejpam-3352	147	24	(	(	PUNCT
ejpam-3352	147	25	vλ	vλ	ADP
ejpam-3352	147	26	∩	∩	NOUN
ejpam-3352	147	27	iα	iα	NOUN
ejpam-3352	147	28	)	)	PUNCT
ejpam-3352	147	29	)	)	PUNCT
ejpam-3352	148	1	∈	∈	PROPN
ejpam-3352	148	2	i	i	PRON
ejpam-3352	148	3	which	which	PRON
ejpam-3352	148	4	implies	imply	VERB
ejpam-3352	148	5	a.	a.	NOUN
ejpam-3352	148	6	qahis	qahis	PROPN
ejpam-3352	148	7	/	/	SYM
ejpam-3352	148	8	eur	eur	PROPN
ejpam-3352	148	9	.	.	PUNCT
ejpam-3352	149	1	j.	j.	PROPN
ejpam-3352	149	2	pure	pure	PROPN
ejpam-3352	149	3	appl	appl	PROPN
ejpam-3352	149	4	.	.	PROPN
ejpam-3352	149	5	math	math	PROPN
ejpam-3352	149	6	,	,	PUNCT
ejpam-3352	149	7	12	12	NUM
ejpam-3352	149	8	(	(	PUNCT
ejpam-3352	149	9	1	1	NUM
ejpam-3352	149	10	)	)	PUNCT
ejpam-3352	149	11	(	(	PUNCT
ejpam-3352	149	12	2019	2019	NUM
ejpam-3352	149	13	)	)	PUNCT
ejpam-3352	149	14	,	,	PUNCT
ejpam-3352	149	15	135	135	NUM
ejpam-3352	149	16	-	-	SYM
ejpam-3352	149	17	145	145	NUM
ejpam-3352	149	18	139	139	NUM
ejpam-3352	149	19	x	x	NOUN
ejpam-3352	149	20	\	\	NOUN
ejpam-3352	149	21	∪λ∈λ(vλ	∪λ∈λ(vλ	X
ejpam-3352	149	22	\	\	PROPN
ejpam-3352	149	23	iα	iα	PROPN
ejpam-3352	149	24	)	)	PUNCT
ejpam-3352	149	25	∈	∈	PROPN
ejpam-3352	149	26	i.	i.	NOUN
ejpam-3352	149	27	since	since	SCONJ
ejpam-3352	149	28	v	v	NOUN
ejpam-3352	149	29	is	be	AUX
ejpam-3352	149	30	locally	locally	ADV
ejpam-3352	149	31	finite	finite	ADJ
ejpam-3352	149	32	,	,	PUNCT
ejpam-3352	149	33	v1	v1	NOUN
ejpam-3352	149	34	=	=	SYM
ejpam-3352	149	35	{	{	PUNCT
ejpam-3352	149	36	vλ	vλ	ADP
ejpam-3352	149	37	\	\	NOUN
ejpam-3352	149	38	iα	iα	NOUN
ejpam-3352	149	39	:	:	PUNCT
ejpam-3352	149	40	λ	λ	X
ejpam-3352	149	41	∈	∈	PROPN
ejpam-3352	149	42	λ	λ	PROPN
ejpam-3352	149	43	}	}	PUNCT
ejpam-3352	149	44	is	be	AUX
ejpam-3352	149	45	locally	locally	ADV
ejpam-3352	149	46	finite	finite	ADJ
ejpam-3352	149	47	.	.	PUNCT
ejpam-3352	150	1	since	since	SCONJ
ejpam-3352	150	2	τ∗	τ∗	NOUN
ejpam-3352	150	3	is	be	AUX
ejpam-3352	150	4	finer	fine	ADJ
ejpam-3352	150	5	than	than	ADP
ejpam-3352	150	6	τ	τ	PROPN
ejpam-3352	150	7	,	,	PUNCT
ejpam-3352	150	8	v1	v1	PROPN
ejpam-3352	150	9	is	be	AUX
ejpam-3352	150	10	τ∗-locally	τ∗-locally	ADV
ejpam-3352	150	11	finite	finite	X
ejpam-3352	150	12	τ∗-open	τ∗-open	PUNCT
ejpam-3352	150	13	which	which	PRON
ejpam-3352	150	14	refines	refine	VERB
ejpam-3352	150	15	u	u	PRON
ejpam-3352	150	16	.	.	PUNCT
ejpam-3352	151	1	hence	hence	ADV
ejpam-3352	151	2	(	(	PUNCT
ejpam-3352	151	3	x	x	NOUN
ejpam-3352	151	4	,	,	PUNCT
ejpam-3352	151	5	τ∗	τ∗	PROPN
ejpam-3352	151	6	,	,	PUNCT
ejpam-3352	151	7	i	i	NOUN
ejpam-3352	151	8	)	)	PUNCT
ejpam-3352	151	9	is	be	AUX
ejpam-3352	151	10	β1i	β1i	NOUN
ejpam-3352	151	11	-	-	ADJ
ejpam-3352	151	12	paracompact	paracompact	ADJ
ejpam-3352	151	13	.	.	PUNCT
ejpam-3352	152	1	theorem	theorem	NOUN
ejpam-3352	152	2	5	5	NUM
ejpam-3352	152	3	.	.	PUNCT
ejpam-3352	153	1	let	let	AUX
ejpam-3352	153	2	(	(	PUNCT
ejpam-3352	153	3	x	x	NOUN
ejpam-3352	153	4	,	,	PUNCT
ejpam-3352	153	5	τ	τ	X
ejpam-3352	153	6	)	)	PUNCT
ejpam-3352	153	7	be	be	VERB
ejpam-3352	153	8	a	a	DET
ejpam-3352	153	9	β	β	NOUN
ejpam-3352	153	10	-	-	ADJ
ejpam-3352	153	11	regular	regular	ADJ
ejpam-3352	153	12	space	space	NOUN
ejpam-3352	153	13	.	.	PUNCT
ejpam-3352	154	1	if	if	SCONJ
ejpam-3352	154	2	(	(	PUNCT
ejpam-3352	154	3	x	x	X
ejpam-3352	154	4	,	,	PUNCT
ejpam-3352	154	5	τ	τ	PROPN
ejpam-3352	154	6	,	,	PUNCT
ejpam-3352	154	7	i	i	PROPN
ejpam-3352	154	8	)	)	PUNCT
ejpam-3352	154	9	is	be	AUX
ejpam-3352	154	10	β1i	β1i	NOUN
ejpam-3352	154	11	-	-	ADJ
ejpam-3352	154	12	paracompact	paracompact	ADJ
ejpam-3352	154	13	,	,	PUNCT
ejpam-3352	154	14	then	then	ADV
ejpam-3352	154	15	every	every	DET
ejpam-3352	154	16	β	β	X
ejpam-3352	154	17	-	-	ADJ
ejpam-3352	154	18	open	open	ADJ
ejpam-3352	154	19	cover	cover	NOUN
ejpam-3352	154	20	of	of	ADP
ejpam-3352	154	21	x	x	PUNCT
ejpam-3352	154	22	has	have	VERB
ejpam-3352	154	23	a	a	DET
ejpam-3352	154	24	locally	locally	ADV
ejpam-3352	154	25	finite	finite	ADJ
ejpam-3352	154	26	β	β	NOUN
ejpam-3352	154	27	-	-	ADJ
ejpam-3352	154	28	closed	closed	ADJ
ejpam-3352	154	29	i	i	NOUN
ejpam-3352	154	30	-	-	PUNCT
ejpam-3352	154	31	cover	cover	NOUN
ejpam-3352	154	32	refinement	refinement	NOUN
ejpam-3352	154	33	.	.	PUNCT
ejpam-3352	155	1	proof	proof	NOUN
ejpam-3352	155	2	.	.	PUNCT
ejpam-3352	156	1	let	let	VERB
ejpam-3352	156	2	u	u	PRON
ejpam-3352	156	3	be	be	AUX
ejpam-3352	156	4	a	a	DET
ejpam-3352	156	5	β	β	NOUN
ejpam-3352	156	6	-	-	ADJ
ejpam-3352	156	7	open	open	ADJ
ejpam-3352	156	8	cover	cover	NOUN
ejpam-3352	156	9	of	of	ADP
ejpam-3352	156	10	x.	x.	NOUN
ejpam-3352	156	11	for	for	ADP
ejpam-3352	156	12	each	each	DET
ejpam-3352	156	13	x	x	SYM
ejpam-3352	156	14	∈	∈	PROPN
ejpam-3352	156	15	x	x	NOUN
ejpam-3352	156	16	,	,	PUNCT
ejpam-3352	156	17	let	let	VERB
ejpam-3352	156	18	ux	ux	PROPN
ejpam-3352	156	19	∈	∈	VERB
ejpam-3352	156	20	u	u	PRON
ejpam-3352	157	1	such	such	ADJ
ejpam-3352	157	2	that	that	SCONJ
ejpam-3352	157	3	x	x	SYM
ejpam-3352	157	4	∈	∈	PROPN
ejpam-3352	157	5	ux	ux	PROPN
ejpam-3352	157	6	.	.	PUNCT
ejpam-3352	158	1	since	since	SCONJ
ejpam-3352	158	2	(	(	PUNCT
ejpam-3352	158	3	x	x	X
ejpam-3352	158	4	,	,	PUNCT
ejpam-3352	158	5	τ	τ	X
ejpam-3352	158	6	)	)	PUNCT
ejpam-3352	158	7	is	be	AUX
ejpam-3352	158	8	β	β	NOUN
ejpam-3352	158	9	-	-	ADJ
ejpam-3352	158	10	regular	regular	ADJ
ejpam-3352	158	11	,	,	PUNCT
ejpam-3352	158	12	there	there	PRON
ejpam-3352	158	13	exists	exist	VERB
ejpam-3352	158	14	vx	vx	PROPN
ejpam-3352	158	15	∈	∈	PROPN
ejpam-3352	158	16	βo(x	βo(x	PUNCT
ejpam-3352	158	17	,	,	PUNCT
ejpam-3352	158	18	τ	τ	X
ejpam-3352	158	19	)	)	PUNCT
ejpam-3352	158	20	such	such	ADJ
ejpam-3352	158	21	that	that	SCONJ
ejpam-3352	158	22	x	x	SYM
ejpam-3352	158	23	∈	∈	PROPN
ejpam-3352	158	24	vx	vx	PROPN
ejpam-3352	158	25	⊂	⊂	X
ejpam-3352	158	26	βcl(vx	βcl(vx	PROPN
ejpam-3352	158	27	)	)	PUNCT
ejpam-3352	159	1	⊂	⊂	PROPN
ejpam-3352	159	2	ux	ux	PROPN
ejpam-3352	159	3	.	.	PUNCT
ejpam-3352	160	1	then	then	ADV
ejpam-3352	160	2	the	the	DET
ejpam-3352	160	3	family	family	NOUN
ejpam-3352	160	4	v	v	NOUN
ejpam-3352	160	5	=	=	PUNCT
ejpam-3352	160	6	{	{	PUNCT
ejpam-3352	160	7	vx	vx	X
ejpam-3352	160	8	:	:	PUNCT
ejpam-3352	160	9	x	x	SYM
ejpam-3352	160	10	∈	∈	PROPN
ejpam-3352	160	11	x	x	PRON
ejpam-3352	160	12	}	}	PUNCT
ejpam-3352	160	13	is	be	AUX
ejpam-3352	160	14	a	a	DET
ejpam-3352	160	15	β	β	NOUN
ejpam-3352	160	16	-	-	ADJ
ejpam-3352	160	17	open	open	ADJ
ejpam-3352	160	18	cover	cover	NOUN
ejpam-3352	160	19	refinement	refinement	NOUN
ejpam-3352	160	20	of	of	ADP
ejpam-3352	160	21	u	u	PROPN
ejpam-3352	160	22	.	.	PUNCT
ejpam-3352	161	1	by	by	ADP
ejpam-3352	161	2	hypothesis	hypothesis	NOUN
ejpam-3352	161	3	,	,	PUNCT
ejpam-3352	161	4	there	there	PRON
ejpam-3352	161	5	exist	exist	VERB
ejpam-3352	161	6	a	a	DET
ejpam-3352	161	7	locally	locally	ADV
ejpam-3352	161	8	finite	finite	ADJ
ejpam-3352	161	9	open	open	ADJ
ejpam-3352	161	10	refinement	refinement	NOUN
ejpam-3352	161	11	w	w	PROPN
ejpam-3352	161	12	=	=	PRON
ejpam-3352	161	13	{	{	PUNCT
ejpam-3352	161	14	wα	wα	NOUN
ejpam-3352	161	15	:	:	PUNCT
ejpam-3352	161	16	α	α	PROPN
ejpam-3352	161	17	∈	∈	PROPN
ejpam-3352	161	18	∆	∆	PROPN
ejpam-3352	161	19	}	}	PUNCT
ejpam-3352	161	20	which	which	PRON
ejpam-3352	161	21	refine	refine	VERB
ejpam-3352	161	22	v	v	ADP
ejpam-3352	162	1	such	such	ADJ
ejpam-3352	162	2	that	that	SCONJ
ejpam-3352	162	3	x	x	X
ejpam-3352	162	4	\	\	PROPN
ejpam-3352	162	5	∪{wα	∪{wα	PROPN
ejpam-3352	162	6	:	:	PUNCT
ejpam-3352	162	7	α	α	X
ejpam-3352	162	8	∈	∈	NOUN
ejpam-3352	162	9	∆	∆	X
ejpam-3352	162	10	}	}	PUNCT
ejpam-3352	162	11	∈	∈	PROPN
ejpam-3352	162	12	i.	i.	NOUN
ejpam-3352	162	13	the	the	DET
ejpam-3352	162	14	family	family	NOUN
ejpam-3352	162	15	βcl(w	βcl(w	PROPN
ejpam-3352	162	16	)	)	PUNCT
ejpam-3352	162	17	=	=	SYM
ejpam-3352	162	18	{	{	PUNCT
ejpam-3352	162	19	βcl(wα	βcl(wα	NOUN
ejpam-3352	162	20	)	)	PUNCT
ejpam-3352	162	21	:	:	PUNCT
ejpam-3352	163	1	α	α	PROPN
ejpam-3352	163	2	∈	∈	PROPN
ejpam-3352	163	3	∆	∆	X
ejpam-3352	163	4	}	}	PUNCT
ejpam-3352	163	5	is	be	AUX
ejpam-3352	163	6	locally	locally	ADV
ejpam-3352	163	7	finite	finite	ADJ
ejpam-3352	163	8	for	for	ADP
ejpam-3352	163	9	each	each	DET
ejpam-3352	163	10	α	α	NOUN
ejpam-3352	163	11	∈	∈	PROPN
ejpam-3352	164	1	∆.	∆.	X
ejpam-3352	164	2	now	now	ADV
ejpam-3352	164	3	x	x	SYM
ejpam-3352	164	4	\	\	PROPN
ejpam-3352	164	5	∪{βcl(wα	∪{βcl(wα	NOUN
ejpam-3352	164	6	)	)	PUNCT
ejpam-3352	164	7	:	:	PUNCT
ejpam-3352	165	1	α	α	PROPN
ejpam-3352	165	2	∈	∈	NOUN
ejpam-3352	165	3	∆	∆	X
ejpam-3352	165	4	}	}	PUNCT
ejpam-3352	166	1	⊆	⊆	NUM
ejpam-3352	166	2	x	x	SYM
ejpam-3352	166	3	\	\	X
ejpam-3352	166	4	∪{wα	∪{wα	NUM
ejpam-3352	166	5	:	:	PUNCT
ejpam-3352	166	6	α	α	PROPN
ejpam-3352	166	7	∈	∈	PROPN
ejpam-3352	166	8	∆	∆	PROPN
ejpam-3352	166	9	}	}	PUNCT
ejpam-3352	166	10	implies	imply	VERB
ejpam-3352	166	11	x	x	SYM
ejpam-3352	166	12	\	\	PROPN
ejpam-3352	166	13	∪{βcl(wα	∪{βcl(wα	PROPN
ejpam-3352	166	14	)	)	PUNCT
ejpam-3352	166	15	:	:	PUNCT
ejpam-3352	166	16	α	α	PROPN
ejpam-3352	166	17	∈	∈	NOUN
ejpam-3352	166	18	∆	∆	X
ejpam-3352	166	19	}	}	PUNCT
ejpam-3352	166	20	∈	∈	PROPN
ejpam-3352	166	21	i.	i.	NOUN
ejpam-3352	166	22	hence	hence	ADV
ejpam-3352	166	23	βcl(w	βcl(w	NUM
ejpam-3352	166	24	)	)	PUNCT
ejpam-3352	166	25	is	be	AUX
ejpam-3352	166	26	i	i	NOUN
ejpam-3352	166	27	-	-	PUNCT
ejpam-3352	166	28	cover	cover	NOUN
ejpam-3352	166	29	.	.	PUNCT
ejpam-3352	167	1	let	let	VERB
ejpam-3352	167	2	βcl(wα	βcl(wα	PRON
ejpam-3352	167	3	)	)	PUNCT
ejpam-3352	167	4	∈	∈	PROPN
ejpam-3352	167	5	βcl(w	βcl(w	PROPN
ejpam-3352	167	6	)	)	PUNCT
ejpam-3352	167	7	.	.	PUNCT
ejpam-3352	168	1	since	since	SCONJ
ejpam-3352	168	2	w	w	PROPN
ejpam-3352	168	3	refines	refine	NOUN
ejpam-3352	168	4	v	v	ADP
ejpam-3352	168	5	,	,	PUNCT
ejpam-3352	168	6	there	there	PRON
ejpam-3352	168	7	is	be	VERB
ejpam-3352	168	8	some	some	DET
ejpam-3352	168	9	vx	vx	PROPN
ejpam-3352	168	10	∈	∈	PROPN
ejpam-3352	168	11	v	v	ADP
ejpam-3352	168	12	such	such	ADJ
ejpam-3352	168	13	that	that	DET
ejpam-3352	168	14	wα	wα	NOUN
ejpam-3352	168	15	⊂	⊂	X
ejpam-3352	168	16	vx	vx	PROPN
ejpam-3352	168	17	and	and	CCONJ
ejpam-3352	168	18	so	so	ADV
ejpam-3352	168	19	βcl(wα	βcl(wα	NUM
ejpam-3352	168	20	)	)	PUNCT
ejpam-3352	168	21	⊂	⊂	PROPN
ejpam-3352	168	22	βcl(vx	βcl(vx	NOUN
ejpam-3352	168	23	)	)	PUNCT
ejpam-3352	169	1	⊂	⊂	PROPN
ejpam-3352	169	2	ux	ux	PROPN
ejpam-3352	169	3	implies	imply	VERB
ejpam-3352	169	4	that	that	SCONJ
ejpam-3352	169	5	βcl(wα	βcl(wα	PRON
ejpam-3352	169	6	)	)	PUNCT
ejpam-3352	169	7	⊂	⊂	PROPN
ejpam-3352	169	8	ux	ux	PROPN
ejpam-3352	169	9	.	.	PUNCT
ejpam-3352	169	10	hence	hence	ADV
ejpam-3352	169	11	βcl(w	βcl(w	NUM
ejpam-3352	169	12	)	)	PUNCT
ejpam-3352	169	13	refines	refine	VERB
ejpam-3352	169	14	u	u	PRON
ejpam-3352	169	15	.	.	PUNCT
ejpam-3352	170	1	thus	thus	ADV
ejpam-3352	170	2	,	,	PUNCT
ejpam-3352	170	3	βcl(w	βcl(w	PROPN
ejpam-3352	170	4	)	)	PUNCT
ejpam-3352	170	5	=	=	SYM
ejpam-3352	170	6	{	{	PUNCT
ejpam-3352	170	7	βcl(wα	βcl(wα	NOUN
ejpam-3352	170	8	)	)	PUNCT
ejpam-3352	170	9	:	:	PUNCT
ejpam-3352	171	1	α	α	PROPN
ejpam-3352	171	2	∈	∈	PROPN
ejpam-3352	171	3	∆	∆	X
ejpam-3352	171	4	}	}	PUNCT
ejpam-3352	171	5	is	be	AUX
ejpam-3352	171	6	a	a	DET
ejpam-3352	171	7	locally	locally	ADV
ejpam-3352	171	8	finite	finite	ADJ
ejpam-3352	171	9	β	β	NOUN
ejpam-3352	171	10	-	-	ADJ
ejpam-3352	171	11	closed	closed	ADJ
ejpam-3352	171	12	i	i	NOUN
ejpam-3352	171	13	-	-	PUNCT
ejpam-3352	171	14	cover	cover	NOUN
ejpam-3352	171	15	refinement	refinement	NOUN
ejpam-3352	171	16	of	of	ADP
ejpam-3352	171	17	u	u	PROPN
ejpam-3352	171	18	.	.	PUNCT
ejpam-3352	172	1	if	if	SCONJ
ejpam-3352	172	2	i	i	PRON
ejpam-3352	172	3	=	=	SYM
ejpam-3352	172	4	{	{	PUNCT
ejpam-3352	172	5	∅	∅	NOUN
ejpam-3352	172	6	}	}	PUNCT
ejpam-3352	172	7	in	in	ADP
ejpam-3352	172	8	theorem	theorem	NOUN
ejpam-3352	172	9	5	5	NUM
ejpam-3352	172	10	,	,	PUNCT
ejpam-3352	172	11	then	then	ADV
ejpam-3352	172	12	we	we	PRON
ejpam-3352	172	13	have	have	VERB
ejpam-3352	172	14	the	the	DET
ejpam-3352	172	15	following	follow	VERB
ejpam-3352	172	16	corollary	corollary	NOUN
ejpam-3352	172	17	.	.	PUNCT
ejpam-3352	173	1	corollary	corollary	ADJ
ejpam-3352	173	2	4	4	NUM
ejpam-3352	173	3	.	.	PUNCT
ejpam-3352	174	1	[	[	X
ejpam-3352	174	2	1	1	NUM
ejpam-3352	174	3	,	,	PUNCT
ejpam-3352	174	4	theorem	theorem	VERB
ejpam-3352	174	5	2.12	2.12	NUM
ejpam-3352	174	6	]	]	SYM
ejpam-3352	174	7	let	let	VERB
ejpam-3352	174	8	(	(	PUNCT
ejpam-3352	174	9	x	x	NOUN
ejpam-3352	174	10	,	,	PUNCT
ejpam-3352	174	11	τ	τ	X
ejpam-3352	174	12	)	)	PUNCT
ejpam-3352	174	13	be	be	VERB
ejpam-3352	174	14	a	a	DET
ejpam-3352	174	15	β	β	NOUN
ejpam-3352	174	16	-	-	ADJ
ejpam-3352	174	17	regular	regular	ADJ
ejpam-3352	174	18	space	space	NOUN
ejpam-3352	174	19	..	..	PUNCT
ejpam-3352	175	1	if	if	SCONJ
ejpam-3352	175	2	each	each	DET
ejpam-3352	175	3	β	β	NOUN
ejpam-3352	175	4	-	-	ADJ
ejpam-3352	175	5	open	open	ADJ
ejpam-3352	175	6	cover	cover	NOUN
ejpam-3352	175	7	of	of	ADP
ejpam-3352	175	8	the	the	DET
ejpam-3352	175	9	space	space	NOUN
ejpam-3352	175	10	x	x	PRON
ejpam-3352	175	11	has	have	VERB
ejpam-3352	175	12	a	a	DET
ejpam-3352	175	13	locally	locally	ADV
ejpam-3352	175	14	finite	finite	ADJ
ejpam-3352	175	15	refinement	refinement	NOUN
ejpam-3352	175	16	,	,	PUNCT
ejpam-3352	175	17	then	then	ADV
ejpam-3352	175	18	each	each	DET
ejpam-3352	175	19	β	β	X
ejpam-3352	175	20	-	-	ADJ
ejpam-3352	175	21	open	open	ADJ
ejpam-3352	175	22	cover	cover	NOUN
ejpam-3352	175	23	of	of	ADP
ejpam-3352	175	24	x	x	PUNCT
ejpam-3352	175	25	has	have	VERB
ejpam-3352	175	26	a	a	DET
ejpam-3352	175	27	locally	locally	ADV
ejpam-3352	175	28	finite	finite	ADJ
ejpam-3352	175	29	β	β	ADJ
ejpam-3352	175	30	-	-	ADJ
ejpam-3352	175	31	closed	closed	ADJ
ejpam-3352	175	32	refinement	refinement	NOUN
ejpam-3352	175	33	recall	recall	NOUN
ejpam-3352	175	34	that	that	SCONJ
ejpam-3352	175	35	a	a	DET
ejpam-3352	175	36	function	function	NOUN
ejpam-3352	175	37	f	f	NOUN
ejpam-3352	175	38	:	:	PUNCT
ejpam-3352	175	39	(	(	PUNCT
ejpam-3352	175	40	x	x	X
ejpam-3352	175	41	,	,	PUNCT
ejpam-3352	175	42	τ	τ	X
ejpam-3352	175	43	)	)	PUNCT
ejpam-3352	175	44	→	→	SYM
ejpam-3352	175	45	(	(	PUNCT
ejpam-3352	175	46	y	y	PROPN
ejpam-3352	175	47	,	,	PUNCT
ejpam-3352	175	48	σ	σ	PROPN
ejpam-3352	175	49	)	)	PUNCT
ejpam-3352	175	50	is	be	AUX
ejpam-3352	175	51	said	say	VERB
ejpam-3352	175	52	to	to	PART
ejpam-3352	175	53	be	be	AUX
ejpam-3352	175	54	β	β	X
ejpam-3352	175	55	-	-	ADJ
ejpam-3352	175	56	continuous	continuous	ADJ
ejpam-3352	175	57	[	[	X
ejpam-3352	175	58	8	8	NUM
ejpam-3352	175	59	]	]	PUNCT
ejpam-3352	175	60	(	(	PUNCT
ejpam-3352	175	61	resp	resp	NOUN
ejpam-3352	175	62	.	.	PUNCT
ejpam-3352	175	63	,	,	PUNCT
ejpam-3352	175	64	βirresolute	βirresolute	NOUN
ejpam-3352	176	1	[	[	X
ejpam-3352	176	2	15	15	NUM
ejpam-3352	176	3	]	]	PUNCT
ejpam-3352	176	4	)	)	PUNCT
ejpam-3352	177	1	if	if	SCONJ
ejpam-3352	177	2	f−1(v	f−1(v	PROPN
ejpam-3352	177	3	)	)	PUNCT
ejpam-3352	177	4	∈	∈	PROPN
ejpam-3352	177	5	βo(x	βo(x	PUNCT
ejpam-3352	177	6	,	,	PUNCT
ejpam-3352	177	7	τ	τ	X
ejpam-3352	177	8	)	)	PUNCT
ejpam-3352	177	9	for	for	ADP
ejpam-3352	177	10	each	each	DET
ejpam-3352	177	11	open	open	ADJ
ejpam-3352	177	12	(	(	PUNCT
ejpam-3352	177	13	resp	resp	NOUN
ejpam-3352	177	14	.	.	PUNCT
ejpam-3352	177	15	,	,	PUNCT
ejpam-3352	178	1	β	β	X
ejpam-3352	178	2	-	-	ADJ
ejpam-3352	178	3	open	open	ADJ
ejpam-3352	178	4	)	)	PUNCT
ejpam-3352	178	5	set	set	VERB
ejpam-3352	178	6	v	v	NOUN
ejpam-3352	178	7	in	in	ADP
ejpam-3352	178	8	(	(	PUNCT
ejpam-3352	178	9	y	y	PROPN
ejpam-3352	178	10	,	,	PUNCT
ejpam-3352	178	11	σ	σ	PROPN
ejpam-3352	178	12	)	)	PUNCT
ejpam-3352	178	13	.	.	PUNCT
ejpam-3352	179	1	theorem	theorem	VERB
ejpam-3352	179	2	6	6	NUM
ejpam-3352	179	3	.	.	PUNCT
ejpam-3352	180	1	let	let	VERB
ejpam-3352	180	2	f	f	NOUN
ejpam-3352	180	3	:	:	PUNCT
ejpam-3352	180	4	(	(	PUNCT
ejpam-3352	180	5	x	x	X
ejpam-3352	180	6	,	,	PUNCT
ejpam-3352	180	7	τ	τ	PROPN
ejpam-3352	180	8	,	,	PUNCT
ejpam-3352	180	9	i)→	i)→	PROPN
ejpam-3352	180	10	(	(	PUNCT
ejpam-3352	180	11	y	y	PROPN
ejpam-3352	180	12	,	,	PUNCT
ejpam-3352	180	13	σ	σ	PROPN
ejpam-3352	180	14	)	)	PUNCT
ejpam-3352	180	15	be	be	AUX
ejpam-3352	180	16	an	an	DET
ejpam-3352	180	17	open	open	ADJ
ejpam-3352	180	18	,	,	PUNCT
ejpam-3352	180	19	β	β	NOUN
ejpam-3352	180	20	-	-	NOUN
ejpam-3352	180	21	irresolute	irresolute	ADJ
ejpam-3352	180	22	and	and	CCONJ
ejpam-3352	180	23	almost	almost	ADV
ejpam-3352	180	24	closed	close	VERB
ejpam-3352	180	25	surjective	surjective	ADJ
ejpam-3352	180	26	function	function	NOUN
ejpam-3352	180	27	with	with	ADP
ejpam-3352	180	28	n	n	ADV
ejpam-3352	180	29	-closed	-closed	ADJ
ejpam-3352	180	30	point	point	NOUN
ejpam-3352	180	31	inverse	inverse	NOUN
ejpam-3352	180	32	.	.	PUNCT
ejpam-3352	181	1	if	if	SCONJ
ejpam-3352	181	2	(	(	PUNCT
ejpam-3352	181	3	x	x	X
ejpam-3352	181	4	,	,	PUNCT
ejpam-3352	181	5	τ	τ	PROPN
ejpam-3352	181	6	,	,	PUNCT
ejpam-3352	181	7	i	i	PROPN
ejpam-3352	181	8	)	)	PUNCT
ejpam-3352	181	9	is	be	AUX
ejpam-3352	181	10	β1i	β1i	NOUN
ejpam-3352	181	11	-	-	ADJ
ejpam-3352	181	12	paracompact	paracompact	ADJ
ejpam-3352	181	13	,	,	PUNCT
ejpam-3352	181	14	then	then	ADV
ejpam-3352	181	15	(	(	PUNCT
ejpam-3352	181	16	y	y	PROPN
ejpam-3352	181	17	,	,	PUNCT
ejpam-3352	181	18	σ	σ	PROPN
ejpam-3352	181	19	,	,	PUNCT
ejpam-3352	181	20	f(i	f(i	PROPN
ejpam-3352	181	21	)	)	PUNCT
ejpam-3352	181	22	)	)	PUNCT
ejpam-3352	181	23	is	be	AUX
ejpam-3352	181	24	β1f(i)-paracompact	β1f(i)-paracompact	NUM
ejpam-3352	181	25	.	.	PUNCT
ejpam-3352	182	1	proof	proof	NOUN
ejpam-3352	182	2	.	.	PUNCT
ejpam-3352	183	1	let	let	VERB
ejpam-3352	183	2	u	u	PRON
ejpam-3352	183	3	=	=	PUNCT
ejpam-3352	183	4	{	{	PUNCT
ejpam-3352	183	5	uα	uα	X
ejpam-3352	183	6	:	:	PUNCT
ejpam-3352	183	7	α	α	PROPN
ejpam-3352	183	8	∈	∈	PROPN
ejpam-3352	183	9	∆	∆	PROPN
ejpam-3352	183	10	}	}	PUNCT
ejpam-3352	183	11	be	be	AUX
ejpam-3352	183	12	a	a	DET
ejpam-3352	183	13	β	β	NOUN
ejpam-3352	183	14	-	-	ADJ
ejpam-3352	183	15	open	open	ADJ
ejpam-3352	183	16	cover	cover	NOUN
ejpam-3352	183	17	of	of	ADP
ejpam-3352	183	18	y	y	PROPN
ejpam-3352	183	19	.	.	PUNCT
ejpam-3352	184	1	since	since	SCONJ
ejpam-3352	184	2	f	f	PROPN
ejpam-3352	184	3	is	be	AUX
ejpam-3352	184	4	β	β	NOUN
ejpam-3352	184	5	-	-	NOUN
ejpam-3352	184	6	irresolute	irresolute	ADJ
ejpam-3352	184	7	,	,	PUNCT
ejpam-3352	184	8	u1	u1	NOUN
ejpam-3352	184	9	=	=	SYM
ejpam-3352	184	10	{	{	PUNCT
ejpam-3352	184	11	f−1(uα	f−1(uα	PROPN
ejpam-3352	184	12	)	)	PUNCT
ejpam-3352	184	13	:	:	PUNCT
ejpam-3352	184	14	α	α	PROPN
ejpam-3352	184	15	∈	∈	PROPN
ejpam-3352	184	16	∆	∆	X
ejpam-3352	184	17	}	}	PUNCT
ejpam-3352	184	18	is	be	AUX
ejpam-3352	184	19	a	a	DET
ejpam-3352	184	20	β	β	NOUN
ejpam-3352	184	21	-	-	ADJ
ejpam-3352	184	22	open	open	ADJ
ejpam-3352	184	23	cover	cover	NOUN
ejpam-3352	184	24	of	of	ADP
ejpam-3352	184	25	x.	x.	NOUN
ejpam-3352	184	26	by	by	ADP
ejpam-3352	184	27	hypothesis	hypothesis	NOUN
ejpam-3352	184	28	,	,	PUNCT
ejpam-3352	184	29	there	there	PRON
ejpam-3352	184	30	exists	exist	VERB
ejpam-3352	184	31	a	a	DET
ejpam-3352	184	32	τ	τ	PROPN
ejpam-3352	184	33	-locally	-locally	ADV
ejpam-3352	184	34	finite	finite	ADJ
ejpam-3352	184	35	τ	τ	PROPN
ejpam-3352	184	36	-open	-open	NOUN
ejpam-3352	184	37	refinement	refinement	NOUN
ejpam-3352	184	38	v1	v1	NOUN
ejpam-3352	184	39	=	=	SYM
ejpam-3352	184	40	{	{	PUNCT
ejpam-3352	184	41	vλ	vλ	INTJ
ejpam-3352	184	42	:	:	PUNCT
ejpam-3352	184	43	λ	λ	PROPN
ejpam-3352	184	44	∈	∈	PROPN
ejpam-3352	184	45	λ	λ	PROPN
ejpam-3352	184	46	}	}	PUNCT
ejpam-3352	184	47	of	of	ADP
ejpam-3352	184	48	u1	u1	NOUN
ejpam-3352	184	49	such	such	ADJ
ejpam-3352	184	50	that	that	SCONJ
ejpam-3352	184	51	x	x	SYM
ejpam-3352	184	52	\	\	PROPN
ejpam-3352	184	53	∪{vλ	∪{vλ	NUM
ejpam-3352	184	54	:	:	PUNCT
ejpam-3352	184	55	λ	λ	PROPN
ejpam-3352	184	56	∈	∈	PROPN
ejpam-3352	184	57	λ	λ	PROPN
ejpam-3352	184	58	}	}	PUNCT
ejpam-3352	184	59	∈	∈	PROPN
ejpam-3352	184	60	i.	i.	NOUN
ejpam-3352	184	61	then	then	ADV
ejpam-3352	184	62	f(x\∪{vλ	f(x\∪{vλ	VERB
ejpam-3352	184	63	:	:	PUNCT
ejpam-3352	184	64	λ	λ	X
ejpam-3352	184	65	∈	∈	PROPN
ejpam-3352	184	66	λ	λ	NOUN
ejpam-3352	184	67	}	}	PUNCT
ejpam-3352	184	68	)	)	PUNCT
ejpam-3352	184	69	∈	∈	PROPN
ejpam-3352	184	70	f(i	f(i	PROPN
ejpam-3352	184	71	)	)	PUNCT
ejpam-3352	184	72	.	.	PUNCT
ejpam-3352	185	1	now	now	ADV
ejpam-3352	185	2	,	,	PUNCT
ejpam-3352	185	3	f(x)\∪{f(vλ	f(x)\∪{f(vλ	PROPN
ejpam-3352	185	4	)	)	PUNCT
ejpam-3352	185	5	:	:	PUNCT
ejpam-3352	186	1	λ	λ	X
ejpam-3352	186	2	∈	∈	PROPN
ejpam-3352	186	3	λ	λ	PROPN
ejpam-3352	186	4	}	}	PUNCT
ejpam-3352	186	5	⊆	⊆	NUM
ejpam-3352	186	6	f(x\∪{vλ	f(x\∪{vλ	NOUN
ejpam-3352	186	7	:	:	PUNCT
ejpam-3352	186	8	λ	λ	X
ejpam-3352	186	9	∈	∈	PROPN
ejpam-3352	186	10	λ	λ	NOUN
ejpam-3352	186	11	}	}	PUNCT
ejpam-3352	186	12	)	)	PUNCT
ejpam-3352	186	13	implies	imply	VERB
ejpam-3352	186	14	that	that	SCONJ
ejpam-3352	186	15	f(x	f(x	PROPN
ejpam-3352	186	16	)	)	PUNCT
ejpam-3352	186	17	\	\	NOUN
ejpam-3352	186	18	∪{f(vλ	∪{f(vλ	NUM
ejpam-3352	186	19	)	)	PUNCT
ejpam-3352	186	20	:	:	PUNCT
ejpam-3352	187	1	λ	λ	X
ejpam-3352	187	2	∈	∈	PROPN
ejpam-3352	187	3	λ	λ	PROPN
ejpam-3352	187	4	}	}	PUNCT
ejpam-3352	187	5	∈	∈	PROPN
ejpam-3352	187	6	f(i	f(i	PROPN
ejpam-3352	187	7	)	)	PUNCT
ejpam-3352	187	8	which	which	PRON
ejpam-3352	187	9	implies	imply	VERB
ejpam-3352	187	10	that	that	SCONJ
ejpam-3352	187	11	y	y	PROPN
ejpam-3352	187	12	\	\	PROPN
ejpam-3352	187	13	∪{f(vλ	∪{f(vλ	NUM
ejpam-3352	187	14	)	)	PUNCT
ejpam-3352	187	15	:	:	PUNCT
ejpam-3352	188	1	λ	λ	X
ejpam-3352	188	2	∈	∈	PROPN
ejpam-3352	188	3	λ	λ	PROPN
ejpam-3352	188	4	}	}	PUNCT
ejpam-3352	188	5	∈	∈	PROPN
ejpam-3352	188	6	f(i	f(i	PROPN
ejpam-3352	188	7	)	)	PUNCT
ejpam-3352	188	8	.	.	PUNCT
ejpam-3352	189	1	since	since	SCONJ
ejpam-3352	189	2	f	f	PROPN
ejpam-3352	189	3	is	be	AUX
ejpam-3352	189	4	open	open	ADJ
ejpam-3352	189	5	and	and	CCONJ
ejpam-3352	189	6	v1	v1	NOUN
ejpam-3352	189	7	is	be	AUX
ejpam-3352	189	8	τ	τ	PROPN
ejpam-3352	189	9	-locally	-locally	ADV
ejpam-3352	189	10	finite	finite	ADJ
ejpam-3352	189	11	,	,	PUNCT
ejpam-3352	189	12	v	v	NOUN
ejpam-3352	189	13	=	=	SYM
ejpam-3352	189	14	{	{	PUNCT
ejpam-3352	189	15	f(vλ	f(vλ	NOUN
ejpam-3352	189	16	)	)	PUNCT
ejpam-3352	189	17	:	:	PUNCT
ejpam-3352	190	1	λ	λ	X
ejpam-3352	190	2	∈	∈	PROPN
ejpam-3352	190	3	λ	λ	PROPN
ejpam-3352	190	4	}	}	PUNCT
ejpam-3352	190	5	is	be	AUX
ejpam-3352	190	6	σ	σ	NOUN
ejpam-3352	190	7	-	-	PUNCT
ejpam-3352	190	8	locally	locally	ADV
ejpam-3352	190	9	finite	finite	NOUN
ejpam-3352	190	10	by	by	ADP
ejpam-3352	190	11	lemma	lemma	PROPN
ejpam-3352	190	12	3	3	NUM
ejpam-3352	190	13	.	.	PUNCT
ejpam-3352	191	1	let	let	VERB
ejpam-3352	191	2	f(vλ	f(vλ	NOUN
ejpam-3352	191	3	)	)	PUNCT
ejpam-3352	191	4	∈	∈	NOUN
ejpam-3352	192	1	v.	v.	CCONJ
ejpam-3352	192	2	then	then	ADV
ejpam-3352	192	3	vλ	vλ	INTJ
ejpam-3352	192	4	∈	∈	PROPN
ejpam-3352	192	5	v1	v1	NOUN
ejpam-3352	192	6	.	.	PUNCT
ejpam-3352	193	1	since	since	SCONJ
ejpam-3352	193	2	v1	v1	NOUN
ejpam-3352	193	3	refines	refine	NOUN
ejpam-3352	193	4	u1	u1	NOUN
ejpam-3352	193	5	,	,	PUNCT
ejpam-3352	193	6	there	there	PRON
ejpam-3352	193	7	exists	exist	VERB
ejpam-3352	193	8	f−1(uα	f−1(uα	PROPN
ejpam-3352	193	9	)	)	PUNCT
ejpam-3352	193	10	∈	∈	PROPN
ejpam-3352	193	11	u1	u1	NOUN
ejpam-3352	193	12	such	such	ADJ
ejpam-3352	193	13	that	that	SCONJ
ejpam-3352	193	14	vλ	vλ	ADP
ejpam-3352	193	15	⊂	⊂	PROPN
ejpam-3352	193	16	f−1(uα	f−1(uα	NOUN
ejpam-3352	193	17	)	)	PUNCT
ejpam-3352	193	18	.	.	PUNCT
ejpam-3352	194	1	thus	thus	ADV
ejpam-3352	194	2	f(vλ	f(vλ	NOUN
ejpam-3352	194	3	)	)	PUNCT
ejpam-3352	194	4	⊂	⊂	PROPN
ejpam-3352	194	5	f(f−1(uα	f(f−1(uα	NOUN
ejpam-3352	194	6	)	)	PUNCT
ejpam-3352	194	7	)	)	PUNCT
ejpam-3352	194	8	implies	imply	VERB
ejpam-3352	194	9	that	that	SCONJ
ejpam-3352	194	10	f(vλ	f(vλ	NOUN
ejpam-3352	194	11	)	)	PUNCT
ejpam-3352	195	1	⊂	⊂	PROPN
ejpam-3352	195	2	uα	uα	PROPN
ejpam-3352	195	3	for	for	ADP
ejpam-3352	195	4	some	some	DET
ejpam-3352	195	5	uα	uα	PROPN
ejpam-3352	195	6	∈	∈	PROPN
ejpam-3352	195	7	u	u	NOUN
ejpam-3352	195	8	.	.	PUNCT
ejpam-3352	196	1	hence	hence	ADV
ejpam-3352	196	2	v	v	NOUN
ejpam-3352	196	3	refines	refine	VERB
ejpam-3352	196	4	u	u	PRON
ejpam-3352	196	5	.	.	PUNCT
ejpam-3352	197	1	therefore	therefore	ADV
ejpam-3352	197	2	,	,	PUNCT
ejpam-3352	197	3	(	(	PUNCT
ejpam-3352	197	4	y	y	PROPN
ejpam-3352	197	5	,	,	PUNCT
ejpam-3352	197	6	σ	σ	PROPN
ejpam-3352	197	7	,	,	PUNCT
ejpam-3352	197	8	f(i	f(i	PROPN
ejpam-3352	197	9	)	)	PUNCT
ejpam-3352	197	10	)	)	PUNCT
ejpam-3352	197	11	is	be	AUX
ejpam-3352	197	12	β1f(i)-paracompact	β1f(i)-paracompact	PROPN
ejpam-3352	197	13	.	.	PUNCT
ejpam-3352	198	1	since	since	SCONJ
ejpam-3352	198	2	every	every	DET
ejpam-3352	198	3	compact	compact	ADJ
ejpam-3352	198	4	set	set	NOUN
ejpam-3352	198	5	is	be	AUX
ejpam-3352	198	6	n	n	ADV
ejpam-3352	198	7	-closed	-close	VERB
ejpam-3352	198	8	and	and	CCONJ
ejpam-3352	198	9	every	every	DET
ejpam-3352	198	10	closed	closed	ADJ
ejpam-3352	198	11	map	map	NOUN
ejpam-3352	198	12	is	be	AUX
ejpam-3352	198	13	almost	almost	ADV
ejpam-3352	198	14	closed	closed	ADJ
ejpam-3352	198	15	,	,	PUNCT
ejpam-3352	198	16	we	we	PRON
ejpam-3352	198	17	conclude	conclude	VERB
ejpam-3352	198	18	the	the	DET
ejpam-3352	198	19	following	follow	VERB
ejpam-3352	198	20	corollary	corollary	NOUN
ejpam-3352	198	21	.	.	PUNCT
ejpam-3352	199	1	corollary	corollary	ADJ
ejpam-3352	199	2	5	5	NUM
ejpam-3352	199	3	.	.	PUNCT
ejpam-3352	200	1	let	let	VERB
ejpam-3352	200	2	f	f	NOUN
ejpam-3352	200	3	:	:	PUNCT
ejpam-3352	200	4	(	(	PUNCT
ejpam-3352	200	5	x	x	X
ejpam-3352	200	6	,	,	PUNCT
ejpam-3352	200	7	τ	τ	PROPN
ejpam-3352	200	8	,	,	PUNCT
ejpam-3352	200	9	i)→	i)→	PROPN
ejpam-3352	200	10	(	(	PUNCT
ejpam-3352	200	11	y	y	PROPN
ejpam-3352	200	12	,	,	PUNCT
ejpam-3352	200	13	σ	σ	PROPN
ejpam-3352	200	14	)	)	PUNCT
ejpam-3352	200	15	be	be	AUX
ejpam-3352	200	16	an	an	DET
ejpam-3352	200	17	open	open	ADJ
ejpam-3352	200	18	,	,	PUNCT
ejpam-3352	200	19	β	β	NOUN
ejpam-3352	200	20	-	-	NOUN
ejpam-3352	200	21	irresolute	irresolute	ADJ
ejpam-3352	200	22	,	,	PUNCT
ejpam-3352	200	23	closed	close	VERB
ejpam-3352	200	24	surjective	surjective	ADJ
ejpam-3352	200	25	function	function	NOUN
ejpam-3352	200	26	with	with	ADP
ejpam-3352	200	27	compact	compact	ADJ
ejpam-3352	200	28	point	point	NOUN
ejpam-3352	200	29	inverse	inverse	NOUN
ejpam-3352	200	30	.	.	PUNCT
ejpam-3352	201	1	if	if	SCONJ
ejpam-3352	201	2	(	(	PUNCT
ejpam-3352	201	3	x	x	X
ejpam-3352	201	4	,	,	PUNCT
ejpam-3352	201	5	τ	τ	PROPN
ejpam-3352	201	6	,	,	PUNCT
ejpam-3352	201	7	i	i	PROPN
ejpam-3352	201	8	)	)	PUNCT
ejpam-3352	201	9	is	be	AUX
ejpam-3352	201	10	β1i	β1i	NOUN
ejpam-3352	201	11	-	-	ADJ
ejpam-3352	201	12	paracompact	paracompact	ADJ
ejpam-3352	201	13	,	,	PUNCT
ejpam-3352	201	14	then	then	ADV
ejpam-3352	201	15	(	(	PUNCT
ejpam-3352	201	16	y	y	PROPN
ejpam-3352	201	17	,	,	PUNCT
ejpam-3352	201	18	σ	σ	PROPN
ejpam-3352	201	19	,	,	PUNCT
ejpam-3352	201	20	f(i	f(i	PROPN
ejpam-3352	201	21	)	)	PUNCT
ejpam-3352	201	22	)	)	PUNCT
ejpam-3352	201	23	is	be	AUX
ejpam-3352	201	24	β1f(i)paracompact	β1f(i)paracompact	PRON
ejpam-3352	201	25	.	.	PUNCT
ejpam-3352	202	1	a.	a.	NOUN
ejpam-3352	202	2	qahis	qahis	PROPN
ejpam-3352	202	3	/	/	SYM
ejpam-3352	202	4	eur	eur	PROPN
ejpam-3352	202	5	.	.	PUNCT
ejpam-3352	203	1	j.	j.	PROPN
ejpam-3352	203	2	pure	pure	PROPN
ejpam-3352	203	3	appl	appl	PROPN
ejpam-3352	203	4	.	.	PROPN
ejpam-3352	203	5	math	math	PROPN
ejpam-3352	203	6	,	,	PUNCT
ejpam-3352	203	7	12	12	NUM
ejpam-3352	203	8	(	(	PUNCT
ejpam-3352	203	9	1	1	NUM
ejpam-3352	203	10	)	)	PUNCT
ejpam-3352	203	11	(	(	PUNCT
ejpam-3352	203	12	2019	2019	NUM
ejpam-3352	203	13	)	)	PUNCT
ejpam-3352	203	14	,	,	PUNCT
ejpam-3352	203	15	135	135	NUM
ejpam-3352	203	16	-	-	SYM
ejpam-3352	203	17	145	145	NUM
ejpam-3352	203	18	140	140	NUM
ejpam-3352	203	19	a	a	DET
ejpam-3352	203	20	function	function	NOUN
ejpam-3352	203	21	f	f	NOUN
ejpam-3352	203	22	:	:	PUNCT
ejpam-3352	203	23	(	(	PUNCT
ejpam-3352	203	24	x	x	X
ejpam-3352	203	25	,	,	PUNCT
ejpam-3352	203	26	τ)→	τ)→	PROPN
ejpam-3352	203	27	(	(	PUNCT
ejpam-3352	203	28	y	y	PROPN
ejpam-3352	203	29	,	,	PUNCT
ejpam-3352	203	30	σ	σ	PROPN
ejpam-3352	203	31	)	)	PUNCT
ejpam-3352	203	32	is	be	AUX
ejpam-3352	203	33	said	say	VERB
ejpam-3352	203	34	to	to	PART
ejpam-3352	203	35	be	be	AUX
ejpam-3352	203	36	strongly	strongly	ADV
ejpam-3352	203	37	β	β	ADJ
ejpam-3352	203	38	-	-	ADJ
ejpam-3352	204	1	continuous	continuous	ADJ
ejpam-3352	204	2	[	[	X
ejpam-3352	204	3	1	1	NUM
ejpam-3352	204	4	]	]	PUNCT
ejpam-3352	204	5	if	if	SCONJ
ejpam-3352	204	6	f−1(v	f−1(v	PROPN
ejpam-3352	204	7	)	)	PUNCT
ejpam-3352	204	8	∈	∈	PROPN
ejpam-3352	204	9	τ	τ	PROPN
ejpam-3352	204	10	for	for	ADP
ejpam-3352	204	11	each	each	DET
ejpam-3352	204	12	v	v	NOUN
ejpam-3352	204	13	∈	∈	PROPN
ejpam-3352	204	14	βo(y	βo(y	PUNCT
ejpam-3352	204	15	,	,	PUNCT
ejpam-3352	204	16	σ	σ	PROPN
ejpam-3352	204	17	)	)	PUNCT
ejpam-3352	204	18	.	.	PUNCT
ejpam-3352	205	1	theorem	theorem	ADJ
ejpam-3352	205	2	7	7	NUM
ejpam-3352	205	3	.	.	PUNCT
ejpam-3352	206	1	let	let	VERB
ejpam-3352	206	2	f	f	NOUN
ejpam-3352	206	3	:	:	PUNCT
ejpam-3352	206	4	(	(	PUNCT
ejpam-3352	206	5	x	x	X
ejpam-3352	206	6	,	,	PUNCT
ejpam-3352	206	7	τ	τ	PROPN
ejpam-3352	206	8	,	,	PUNCT
ejpam-3352	206	9	i)→	i)→	PROPN
ejpam-3352	206	10	(	(	PUNCT
ejpam-3352	206	11	y	y	PROPN
ejpam-3352	206	12	,	,	PUNCT
ejpam-3352	206	13	σ	σ	PROPN
ejpam-3352	206	14	)	)	PUNCT
ejpam-3352	206	15	be	be	VERB
ejpam-3352	206	16	an	an	DET
ejpam-3352	206	17	open	open	ADJ
ejpam-3352	206	18	,	,	PUNCT
ejpam-3352	206	19	strongly	strongly	ADV
ejpam-3352	206	20	β	β	NOUN
ejpam-3352	206	21	-	-	ADJ
ejpam-3352	206	22	continuous	continuous	ADJ
ejpam-3352	206	23	,	,	PUNCT
ejpam-3352	206	24	almost	almost	ADV
ejpam-3352	206	25	closed	closed	ADJ
ejpam-3352	206	26	,	,	PUNCT
ejpam-3352	206	27	surjective	surjective	ADJ
ejpam-3352	206	28	function	function	NOUN
ejpam-3352	206	29	with	with	ADP
ejpam-3352	206	30	n	n	CCONJ
ejpam-3352	206	31	-closed	-closed	ADJ
ejpam-3352	206	32	point	point	NOUN
ejpam-3352	206	33	inverse	inverse	NOUN
ejpam-3352	206	34	.	.	PUNCT
ejpam-3352	207	1	if	if	SCONJ
ejpam-3352	207	2	(	(	PUNCT
ejpam-3352	207	3	x	x	X
ejpam-3352	207	4	,	,	PUNCT
ejpam-3352	207	5	τ	τ	PROPN
ejpam-3352	207	6	,	,	PUNCT
ejpam-3352	207	7	i	i	PROPN
ejpam-3352	207	8	)	)	PUNCT
ejpam-3352	207	9	is	be	AUX
ejpam-3352	207	10	i	i	NOUN
ejpam-3352	207	11	-	-	PUNCT
ejpam-3352	207	12	paracompact	paracompact	ADJ
ejpam-3352	207	13	,	,	PUNCT
ejpam-3352	207	14	then	then	ADV
ejpam-3352	207	15	(	(	PUNCT
ejpam-3352	207	16	y	y	PROPN
ejpam-3352	207	17	,	,	PUNCT
ejpam-3352	207	18	σ	σ	PROPN
ejpam-3352	207	19	,	,	PUNCT
ejpam-3352	207	20	f(i	f(i	PROPN
ejpam-3352	207	21	)	)	PUNCT
ejpam-3352	207	22	)	)	PUNCT
ejpam-3352	207	23	is	be	AUX
ejpam-3352	207	24	β1f(i)-paracompact	β1f(i)-paracompact	NUM
ejpam-3352	207	25	.	.	PUNCT
ejpam-3352	208	1	proof	proof	NOUN
ejpam-3352	208	2	.	.	PUNCT
ejpam-3352	209	1	let	let	VERB
ejpam-3352	209	2	u	u	PRON
ejpam-3352	209	3	=	=	PUNCT
ejpam-3352	209	4	{	{	PUNCT
ejpam-3352	209	5	uα	uα	X
ejpam-3352	209	6	:	:	PUNCT
ejpam-3352	209	7	α	α	PROPN
ejpam-3352	209	8	∈	∈	PROPN
ejpam-3352	209	9	∆	∆	PROPN
ejpam-3352	209	10	}	}	PUNCT
ejpam-3352	209	11	be	be	AUX
ejpam-3352	209	12	a	a	DET
ejpam-3352	209	13	β	β	NOUN
ejpam-3352	209	14	-	-	ADJ
ejpam-3352	209	15	open	open	ADJ
ejpam-3352	209	16	cover	cover	NOUN
ejpam-3352	209	17	of	of	ADP
ejpam-3352	209	18	y	y	PROPN
ejpam-3352	209	19	.	.	PUNCT
ejpam-3352	210	1	since	since	SCONJ
ejpam-3352	210	2	f	f	PROPN
ejpam-3352	210	3	is	be	AUX
ejpam-3352	210	4	strongly	strongly	ADV
ejpam-3352	210	5	β	β	ADJ
ejpam-3352	210	6	-	-	ADJ
ejpam-3352	210	7	continuous	continuous	ADJ
ejpam-3352	210	8	,	,	PUNCT
ejpam-3352	210	9	u1	u1	NOUN
ejpam-3352	210	10	=	=	SYM
ejpam-3352	210	11	{	{	PUNCT
ejpam-3352	210	12	f−1(uα	f−1(uα	PROPN
ejpam-3352	210	13	)	)	PUNCT
ejpam-3352	210	14	:	:	PUNCT
ejpam-3352	210	15	α	α	PROPN
ejpam-3352	210	16	∈	∈	PROPN
ejpam-3352	210	17	∆	∆	X
ejpam-3352	210	18	}	}	PUNCT
ejpam-3352	210	19	is	be	AUX
ejpam-3352	210	20	an	an	DET
ejpam-3352	210	21	open	open	ADJ
ejpam-3352	210	22	cover	cover	NOUN
ejpam-3352	210	23	ofx	ofx	NOUN
ejpam-3352	210	24	.	.	PUNCT
ejpam-3352	211	1	by	by	ADP
ejpam-3352	211	2	hypothesis	hypothesis	NOUN
ejpam-3352	211	3	,	,	PUNCT
ejpam-3352	211	4	there	there	PRON
ejpam-3352	211	5	exists	exist	VERB
ejpam-3352	211	6	a	a	DET
ejpam-3352	211	7	τ	τ	PROPN
ejpam-3352	211	8	-locally	-locally	ADV
ejpam-3352	211	9	finite	finite	ADJ
ejpam-3352	211	10	τ	τ	PROPN
ejpam-3352	211	11	-open	-open	NOUN
ejpam-3352	211	12	refinement	refinement	NOUN
ejpam-3352	211	13	v1	v1	NOUN
ejpam-3352	211	14	=	=	SYM
ejpam-3352	211	15	{	{	PUNCT
ejpam-3352	211	16	vλ	vλ	INTJ
ejpam-3352	211	17	:	:	PUNCT
ejpam-3352	211	18	λ	λ	PROPN
ejpam-3352	211	19	∈	∈	PROPN
ejpam-3352	211	20	λ	λ	X
ejpam-3352	211	21	}	}	PUNCT
ejpam-3352	211	22	which	which	PRON
ejpam-3352	211	23	refines	refine	VERB
ejpam-3352	211	24	u1	u1	VERB
ejpam-3352	211	25	such	such	ADJ
ejpam-3352	211	26	that	that	SCONJ
ejpam-3352	211	27	x	x	SYM
ejpam-3352	211	28	\	\	PROPN
ejpam-3352	211	29	∪{vλ	∪{vλ	NUM
ejpam-3352	211	30	:	:	PUNCT
ejpam-3352	211	31	λ	λ	PROPN
ejpam-3352	211	32	∈	∈	PROPN
ejpam-3352	211	33	λ	λ	PROPN
ejpam-3352	211	34	}	}	PUNCT
ejpam-3352	211	35	∈	∈	PROPN
ejpam-3352	211	36	i.	i.	NOUN
ejpam-3352	211	37	then	then	ADV
ejpam-3352	211	38	f(x	f(x	PROPN
ejpam-3352	211	39	\∪{vλ	\∪{vλ	PROPN
ejpam-3352	211	40	:	:	PUNCT
ejpam-3352	211	41	λ	λ	X
ejpam-3352	211	42	∈	∈	PROPN
ejpam-3352	211	43	λ	λ	NOUN
ejpam-3352	211	44	}	}	PUNCT
ejpam-3352	211	45	)	)	PUNCT
ejpam-3352	211	46	∈	∈	PROPN
ejpam-3352	211	47	f(i	f(i	PROPN
ejpam-3352	211	48	)	)	PUNCT
ejpam-3352	211	49	.	.	PUNCT
ejpam-3352	212	1	now	now	ADV
ejpam-3352	212	2	f(x	f(x	PROPN
ejpam-3352	212	3	)	)	PUNCT
ejpam-3352	212	4	\∪{f(vλ	\∪{f(vλ	NOUN
ejpam-3352	212	5	)	)	PUNCT
ejpam-3352	212	6	:	:	PUNCT
ejpam-3352	213	1	λ	λ	X
ejpam-3352	213	2	∈	∈	PROPN
ejpam-3352	213	3	λ	λ	PROPN
ejpam-3352	213	4	}	}	PUNCT
ejpam-3352	213	5	⊆	⊆	NUM
ejpam-3352	213	6	f(x	f(x	PROPN
ejpam-3352	213	7	\∪{vλ	\∪{vλ	PROPN
ejpam-3352	213	8	:	:	PUNCT
ejpam-3352	213	9	λ	λ	X
ejpam-3352	213	10	∈	∈	PROPN
ejpam-3352	213	11	λ	λ	NOUN
ejpam-3352	213	12	}	}	PUNCT
ejpam-3352	213	13	)	)	PUNCT
ejpam-3352	213	14	implies	imply	VERB
ejpam-3352	213	15	that	that	SCONJ
ejpam-3352	213	16	f(x	f(x	PROPN
ejpam-3352	213	17	)	)	PUNCT
ejpam-3352	213	18	\	\	NOUN
ejpam-3352	213	19	∪{f(vλ	∪{f(vλ	NUM
ejpam-3352	213	20	)	)	PUNCT
ejpam-3352	213	21	:	:	PUNCT
ejpam-3352	214	1	λ	λ	X
ejpam-3352	214	2	∈	∈	PROPN
ejpam-3352	214	3	λ	λ	PROPN
ejpam-3352	214	4	}	}	PUNCT
ejpam-3352	214	5	∈	∈	PROPN
ejpam-3352	214	6	f(i	f(i	PROPN
ejpam-3352	214	7	)	)	PUNCT
ejpam-3352	214	8	which	which	PRON
ejpam-3352	214	9	implies	imply	VERB
ejpam-3352	214	10	that	that	SCONJ
ejpam-3352	214	11	y	y	PROPN
ejpam-3352	214	12	\	\	PROPN
ejpam-3352	214	13	∪{f(vλ	∪{f(vλ	NUM
ejpam-3352	214	14	)	)	PUNCT
ejpam-3352	214	15	:	:	PUNCT
ejpam-3352	215	1	λ	λ	X
ejpam-3352	215	2	∈	∈	PROPN
ejpam-3352	215	3	λ	λ	PROPN
ejpam-3352	215	4	}	}	PUNCT
ejpam-3352	215	5	∈	∈	PROPN
ejpam-3352	215	6	f(i	f(i	PROPN
ejpam-3352	215	7	)	)	PUNCT
ejpam-3352	215	8	.	.	PUNCT
ejpam-3352	216	1	since	since	SCONJ
ejpam-3352	216	2	f	f	PROPN
ejpam-3352	216	3	is	be	AUX
ejpam-3352	216	4	open	open	ADJ
ejpam-3352	216	5	and	and	CCONJ
ejpam-3352	216	6	v1	v1	NOUN
ejpam-3352	216	7	is	be	AUX
ejpam-3352	216	8	τ	τ	PROPN
ejpam-3352	216	9	-locally	-locally	ADV
ejpam-3352	216	10	finite	finite	ADJ
ejpam-3352	216	11	,	,	PUNCT
ejpam-3352	216	12	v	v	NOUN
ejpam-3352	216	13	=	=	SYM
ejpam-3352	216	14	{	{	PUNCT
ejpam-3352	216	15	f(vλ	f(vλ	NOUN
ejpam-3352	216	16	)	)	PUNCT
ejpam-3352	216	17	:	:	PUNCT
ejpam-3352	217	1	λ	λ	X
ejpam-3352	217	2	∈	∈	PROPN
ejpam-3352	217	3	λ	λ	PROPN
ejpam-3352	217	4	}	}	PUNCT
ejpam-3352	217	5	is	be	AUX
ejpam-3352	217	6	σ	σ	NOUN
ejpam-3352	217	7	-	-	PUNCT
ejpam-3352	217	8	locally	locally	ADV
ejpam-3352	217	9	finite	finite	NOUN
ejpam-3352	217	10	by	by	ADP
ejpam-3352	217	11	lemma	lemma	PROPN
ejpam-3352	217	12	3	3	NUM
ejpam-3352	217	13	.	.	PUNCT
ejpam-3352	218	1	let	let	VERB
ejpam-3352	218	2	f(vλ	f(vλ	NOUN
ejpam-3352	218	3	)	)	PUNCT
ejpam-3352	218	4	∈	∈	NOUN
ejpam-3352	219	1	v.	v.	CCONJ
ejpam-3352	219	2	then	then	ADV
ejpam-3352	219	3	vλ	vλ	INTJ
ejpam-3352	219	4	∈	∈	PROPN
ejpam-3352	219	5	v1	v1	NOUN
ejpam-3352	219	6	.	.	PUNCT
ejpam-3352	220	1	since	since	SCONJ
ejpam-3352	220	2	v1	v1	NOUN
ejpam-3352	220	3	refines	refine	NOUN
ejpam-3352	220	4	u1	u1	NOUN
ejpam-3352	220	5	,	,	PUNCT
ejpam-3352	220	6	there	there	PRON
ejpam-3352	220	7	exists	exist	VERB
ejpam-3352	220	8	f−1(uα	f−1(uα	PROPN
ejpam-3352	220	9	)	)	PUNCT
ejpam-3352	220	10	∈	∈	PROPN
ejpam-3352	220	11	u1	u1	NOUN
ejpam-3352	220	12	such	such	ADJ
ejpam-3352	220	13	that	that	SCONJ
ejpam-3352	220	14	vλ	vλ	ADP
ejpam-3352	220	15	⊂	⊂	PROPN
ejpam-3352	220	16	f−1(uα	f−1(uα	NOUN
ejpam-3352	220	17	)	)	PUNCT
ejpam-3352	220	18	.	.	PUNCT
ejpam-3352	221	1	thus	thus	ADV
ejpam-3352	221	2	f(vλ	f(vλ	NOUN
ejpam-3352	221	3	)	)	PUNCT
ejpam-3352	221	4	⊂	⊂	PROPN
ejpam-3352	221	5	f(f−1(uα	f(f−1(uα	NOUN
ejpam-3352	221	6	)	)	PUNCT
ejpam-3352	221	7	)	)	PUNCT
ejpam-3352	221	8	implies	imply	VERB
ejpam-3352	221	9	that	that	SCONJ
ejpam-3352	221	10	f(vλ	f(vλ	NOUN
ejpam-3352	221	11	)	)	PUNCT
ejpam-3352	222	1	⊂	⊂	PROPN
ejpam-3352	222	2	uα	uα	PROPN
ejpam-3352	222	3	for	for	ADP
ejpam-3352	222	4	some	some	DET
ejpam-3352	222	5	uα	uα	PROPN
ejpam-3352	222	6	∈	∈	PROPN
ejpam-3352	222	7	u	u	NOUN
ejpam-3352	222	8	.	.	PUNCT
ejpam-3352	223	1	hence	hence	ADV
ejpam-3352	223	2	v	v	NOUN
ejpam-3352	223	3	refines	refine	VERB
ejpam-3352	223	4	u	u	PRON
ejpam-3352	223	5	.	.	PUNCT
ejpam-3352	224	1	therefore	therefore	ADV
ejpam-3352	224	2	,	,	PUNCT
ejpam-3352	224	3	(	(	PUNCT
ejpam-3352	224	4	y	y	PROPN
ejpam-3352	224	5	,	,	PUNCT
ejpam-3352	224	6	σ	σ	PROPN
ejpam-3352	224	7	,	,	PUNCT
ejpam-3352	224	8	f(i	f(i	PROPN
ejpam-3352	224	9	)	)	PUNCT
ejpam-3352	224	10	)	)	PUNCT
ejpam-3352	224	11	is	be	AUX
ejpam-3352	224	12	β1f(i)-paracompact	β1f(i)-paracompact	PROPN
ejpam-3352	224	13	.	.	PUNCT
ejpam-3352	225	1	recall	recall	VERB
ejpam-3352	225	2	that	that	SCONJ
ejpam-3352	225	3	a	a	DET
ejpam-3352	225	4	function	function	NOUN
ejpam-3352	225	5	f	f	NOUN
ejpam-3352	225	6	:	:	PUNCT
ejpam-3352	225	7	(	(	PUNCT
ejpam-3352	225	8	x	x	X
ejpam-3352	225	9	,	,	PUNCT
ejpam-3352	225	10	τ	τ	X
ejpam-3352	225	11	)	)	PUNCT
ejpam-3352	225	12	→	→	SYM
ejpam-3352	225	13	(	(	PUNCT
ejpam-3352	225	14	y	y	PROPN
ejpam-3352	225	15	,	,	PUNCT
ejpam-3352	225	16	σ	σ	PROPN
ejpam-3352	225	17	)	)	PUNCT
ejpam-3352	225	18	is	be	AUX
ejpam-3352	225	19	said	say	VERB
ejpam-3352	225	20	to	to	PART
ejpam-3352	225	21	be	be	AUX
ejpam-3352	225	22	pre	pre	VERB
ejpam-3352	225	23	β	β	X
ejpam-3352	225	24	-	-	PUNCT
ejpam-3352	225	25	open[15	open[15	NOUN
ejpam-3352	225	26	]	]	PUNCT
ejpam-3352	225	27	if	if	SCONJ
ejpam-3352	225	28	for	for	ADP
ejpam-3352	225	29	every	every	DET
ejpam-3352	225	30	β	β	NOUN
ejpam-3352	225	31	-	-	ADJ
ejpam-3352	225	32	open	open	ADJ
ejpam-3352	225	33	set	set	NOUN
ejpam-3352	225	34	v	v	NOUN
ejpam-3352	225	35	of	of	ADP
ejpam-3352	225	36	(	(	PUNCT
ejpam-3352	225	37	x	x	PROPN
ejpam-3352	225	38	,	,	PUNCT
ejpam-3352	225	39	τ	τ	PROPN
ejpam-3352	225	40	)	)	PUNCT
ejpam-3352	225	41	,	,	PUNCT
ejpam-3352	225	42	f(v	f(v	PROPN
ejpam-3352	225	43	)	)	PUNCT
ejpam-3352	225	44	is	be	AUX
ejpam-3352	225	45	β	β	NOUN
ejpam-3352	225	46	-	-	ADJ
ejpam-3352	225	47	open	open	ADJ
ejpam-3352	225	48	in	in	ADP
ejpam-3352	225	49	(	(	PUNCT
ejpam-3352	225	50	y	y	PROPN
ejpam-3352	225	51	,	,	PUNCT
ejpam-3352	225	52	σ	σ	PROPN
ejpam-3352	225	53	)	)	PUNCT
ejpam-3352	225	54	.	.	PUNCT
ejpam-3352	226	1	theorem	theorem	ADJ
ejpam-3352	226	2	8	8	NUM
ejpam-3352	226	3	.	.	PUNCT
ejpam-3352	227	1	let	let	VERB
ejpam-3352	227	2	f	f	NOUN
ejpam-3352	227	3	:	:	PUNCT
ejpam-3352	227	4	(	(	PUNCT
ejpam-3352	227	5	x	x	X
ejpam-3352	227	6	,	,	PUNCT
ejpam-3352	227	7	τ)→	τ)→	PROPN
ejpam-3352	227	8	(	(	PUNCT
ejpam-3352	227	9	y	y	PROPN
ejpam-3352	227	10	,	,	PUNCT
ejpam-3352	227	11	σ	σ	PROPN
ejpam-3352	227	12	,	,	PUNCT
ejpam-3352	227	13	j	j	PROPN
ejpam-3352	227	14	)	)	PUNCT
ejpam-3352	227	15	be	be	AUX
ejpam-3352	227	16	a	a	DET
ejpam-3352	227	17	pre	pre	NOUN
ejpam-3352	227	18	β	β	NOUN
ejpam-3352	227	19	-	-	ADJ
ejpam-3352	227	20	open	open	ADJ
ejpam-3352	227	21	,	,	PUNCT
ejpam-3352	227	22	continuous	continuous	ADJ
ejpam-3352	227	23	,	,	PUNCT
ejpam-3352	227	24	bijective	bijective	ADJ
ejpam-3352	227	25	function	function	NOUN
ejpam-3352	227	26	.	.	PUNCT
ejpam-3352	228	1	if	if	SCONJ
ejpam-3352	228	2	(	(	PUNCT
ejpam-3352	228	3	y	y	PROPN
ejpam-3352	228	4	,	,	PUNCT
ejpam-3352	228	5	σ	σ	PROPN
ejpam-3352	228	6	,	,	PUNCT
ejpam-3352	228	7	j	j	PROPN
ejpam-3352	228	8	)	)	PUNCT
ejpam-3352	228	9	is	be	AUX
ejpam-3352	228	10	a	a	DET
ejpam-3352	228	11	β1j	β1j	PUNCT
ejpam-3352	228	12	-paracompact	-paracompact	NOUN
ejpam-3352	228	13	,	,	PUNCT
ejpam-3352	228	14	then	then	ADV
ejpam-3352	228	15	(	(	PUNCT
ejpam-3352	228	16	x	x	X
ejpam-3352	228	17	,	,	PUNCT
ejpam-3352	228	18	τ	τ	PROPN
ejpam-3352	228	19	,	,	PUNCT
ejpam-3352	228	20	f−1(j	f−1(j	NOUN
ejpam-3352	228	21	)	)	PUNCT
ejpam-3352	228	22	)	)	PUNCT
ejpam-3352	228	23	is	be	AUX
ejpam-3352	228	24	β1f	β1f	X
ejpam-3352	228	25	−1(j	−1(j	NOUN
ejpam-3352	228	26	)	)	PUNCT
ejpam-3352	228	27	-paracompact	-paracompact	NOUN
ejpam-3352	228	28	.	.	PUNCT
ejpam-3352	229	1	proof	proof	NOUN
ejpam-3352	229	2	.	.	PUNCT
ejpam-3352	230	1	let	let	VERB
ejpam-3352	230	2	u	u	PRON
ejpam-3352	230	3	=	=	PUNCT
ejpam-3352	230	4	{	{	PUNCT
ejpam-3352	230	5	uα	uα	X
ejpam-3352	230	6	:	:	PUNCT
ejpam-3352	230	7	α	α	PROPN
ejpam-3352	230	8	∈	∈	PROPN
ejpam-3352	230	9	∆	∆	PROPN
ejpam-3352	230	10	}	}	PUNCT
ejpam-3352	230	11	be	be	AUX
ejpam-3352	230	12	a	a	DET
ejpam-3352	230	13	β	β	NOUN
ejpam-3352	230	14	-	-	ADJ
ejpam-3352	230	15	open	open	ADJ
ejpam-3352	230	16	cover	cover	NOUN
ejpam-3352	230	17	of	of	ADP
ejpam-3352	230	18	x.	x.	NOUN
ejpam-3352	230	19	since	since	SCONJ
ejpam-3352	230	20	f	f	PROPN
ejpam-3352	230	21	is	be	AUX
ejpam-3352	230	22	a	a	DET
ejpam-3352	230	23	pre	pre	NOUN
ejpam-3352	230	24	β	β	NOUN
ejpam-3352	230	25	-	-	ADJ
ejpam-3352	230	26	open	open	ADJ
ejpam-3352	230	27	,	,	PUNCT
ejpam-3352	230	28	f(u	f(u	PROPN
ejpam-3352	230	29	)	)	PUNCT
ejpam-3352	231	1	=	=	PRON
ejpam-3352	231	2	{	{	PUNCT
ejpam-3352	231	3	f(uα	f(uα	NOUN
ejpam-3352	231	4	)	)	PUNCT
ejpam-3352	231	5	:	:	PUNCT
ejpam-3352	231	6	α	α	PROPN
ejpam-3352	231	7	∈	∈	PROPN
ejpam-3352	231	8	∆	∆	X
ejpam-3352	231	9	}	}	PUNCT
ejpam-3352	231	10	is	be	AUX
ejpam-3352	231	11	a	a	DET
ejpam-3352	231	12	β	β	NOUN
ejpam-3352	231	13	-	-	ADJ
ejpam-3352	231	14	open	open	ADJ
ejpam-3352	231	15	cover	cover	NOUN
ejpam-3352	231	16	of	of	ADP
ejpam-3352	231	17	y	y	PROPN
ejpam-3352	231	18	and	and	CCONJ
ejpam-3352	231	19	so	so	ADV
ejpam-3352	231	20	it	it	PRON
ejpam-3352	231	21	has	have	VERB
ejpam-3352	231	22	a	a	DET
ejpam-3352	231	23	σ	σ	PROPN
ejpam-3352	231	24	-	-	PUNCT
ejpam-3352	231	25	locally	locally	ADV
ejpam-3352	231	26	finite	finite	PROPN
ejpam-3352	231	27	σopen	σopen	NOUN
ejpam-3352	231	28	refinement	refinement	NOUN
ejpam-3352	232	1	w	w	PROPN
ejpam-3352	232	2	=	=	PRON
ejpam-3352	232	3	{	{	PUNCT
ejpam-3352	232	4	wλ	wλ	NOUN
ejpam-3352	232	5	:	:	PUNCT
ejpam-3352	232	6	λ	λ	PROPN
ejpam-3352	232	7	∈	∈	PROPN
ejpam-3352	232	8	λ	λ	PROPN
ejpam-3352	232	9	}	}	PUNCT
ejpam-3352	232	10	of	of	ADP
ejpam-3352	232	11	f(u	f(u	PROPN
ejpam-3352	232	12	)	)	PUNCT
ejpam-3352	233	1	such	such	ADJ
ejpam-3352	233	2	that	that	SCONJ
ejpam-3352	233	3	y	y	PROPN
ejpam-3352	233	4	\	\	NOUN
ejpam-3352	233	5	∪{wλ	∪{wλ	NOUN
ejpam-3352	233	6	:	:	PUNCT
ejpam-3352	233	7	λ	λ	PROPN
ejpam-3352	233	8	∈	∈	PROPN
ejpam-3352	233	9	λ	λ	PROPN
ejpam-3352	233	10	}	}	PUNCT
ejpam-3352	233	11	∈	∈	PROPN
ejpam-3352	233	12	j	j	PROPN
ejpam-3352	233	13	.	.	PUNCT
ejpam-3352	234	1	let	let	VERB
ejpam-3352	234	2	y	y	PRON
ejpam-3352	234	3	\	\	NOUN
ejpam-3352	234	4	∪{wλ	∪{wλ	NOUN
ejpam-3352	234	5	:	:	PUNCT
ejpam-3352	235	1	λ	λ	PROPN
ejpam-3352	235	2	∈	∈	PROPN
ejpam-3352	235	3	λ	λ	NOUN
ejpam-3352	235	4	}	}	PUNCT
ejpam-3352	235	5	=	=	PUNCT
ejpam-3352	235	6	j	j	PROPN
ejpam-3352	235	7	∈	∈	PROPN
ejpam-3352	235	8	j	j	PROPN
ejpam-3352	235	9	.	.	PUNCT
ejpam-3352	236	1	this	this	PRON
ejpam-3352	236	2	implies	imply	VERB
ejpam-3352	236	3	y	y	PROPN
ejpam-3352	236	4	=	=	PUNCT
ejpam-3352	236	5	(	(	PUNCT
ejpam-3352	236	6	∪{wλ	∪{wλ	NOUN
ejpam-3352	236	7	:	:	PUNCT
ejpam-3352	236	8	λ	λ	PROPN
ejpam-3352	236	9	∈	∈	PROPN
ejpam-3352	236	10	λ	λ	NOUN
ejpam-3352	236	11	}	}	PUNCT
ejpam-3352	236	12	)	)	PUNCT
ejpam-3352	236	13	∪	∪	ADP
ejpam-3352	236	14	j	j	PROPN
ejpam-3352	236	15	.	.	PUNCT
ejpam-3352	237	1	then	then	ADV
ejpam-3352	237	2	f−1(y	f−1(y	PROPN
ejpam-3352	237	3	)	)	PUNCT
ejpam-3352	238	1	=	=	PUNCT
ejpam-3352	238	2	(	(	PUNCT
ejpam-3352	238	3	∪{f−1(wλ	∪{f−1(wλ	PROPN
ejpam-3352	238	4	)	)	PUNCT
ejpam-3352	238	5	:	:	PUNCT
ejpam-3352	239	1	λ	λ	X
ejpam-3352	239	2	∈	∈	PROPN
ejpam-3352	239	3	λ	λ	NOUN
ejpam-3352	239	4	}	}	PUNCT
ejpam-3352	239	5	)	)	PUNCT
ejpam-3352	239	6	∪	∪	X
ejpam-3352	239	7	f−1(j	f−1(j	PROPN
ejpam-3352	239	8	)	)	PUNCT
ejpam-3352	239	9	which	which	PRON
ejpam-3352	239	10	implies	imply	VERB
ejpam-3352	239	11	x	x	X
ejpam-3352	239	12	=	=	SYM
ejpam-3352	239	13	(	(	PUNCT
ejpam-3352	239	14	∪{f−1(wλ	∪{f−1(wλ	PROPN
ejpam-3352	239	15	)	)	PUNCT
ejpam-3352	239	16	:	:	PUNCT
ejpam-3352	240	1	λ	λ	X
ejpam-3352	240	2	∈	∈	PROPN
ejpam-3352	240	3	λ	λ	NOUN
ejpam-3352	240	4	}	}	PUNCT
ejpam-3352	240	5	)	)	PUNCT
ejpam-3352	240	6	∪	∪	ADP
ejpam-3352	240	7	f−1(j	f−1(j	NOUN
ejpam-3352	240	8	)	)	PUNCT
ejpam-3352	240	9	.	.	PUNCT
ejpam-3352	241	1	it	it	PRON
ejpam-3352	241	2	follows	follow	VERB
ejpam-3352	241	3	that	that	SCONJ
ejpam-3352	241	4	x	x	SYM
ejpam-3352	241	5	\	\	PROPN
ejpam-3352	241	6	∪{f−1(wλ	∪{f−1(wλ	PROPN
ejpam-3352	241	7	)	)	PUNCT
ejpam-3352	241	8	:	:	PUNCT
ejpam-3352	242	1	λ	λ	X
ejpam-3352	242	2	∈	∈	PROPN
ejpam-3352	242	3	λ	λ	PROPN
ejpam-3352	242	4	}	}	PUNCT
ejpam-3352	242	5	∈	∈	PROPN
ejpam-3352	242	6	f−1(j	f−1(j	NOUN
ejpam-3352	242	7	)	)	PUNCT
ejpam-3352	242	8	.	.	PUNCT
ejpam-3352	243	1	since	since	SCONJ
ejpam-3352	243	2	f	f	PROPN
ejpam-3352	243	3	is	be	AUX
ejpam-3352	243	4	continuous	continuous	ADJ
ejpam-3352	243	5	,	,	PUNCT
ejpam-3352	243	6	by	by	ADP
ejpam-3352	243	7	lemma	lemma	PROPN
ejpam-3352	243	8	2	2	NUM
ejpam-3352	243	9	,	,	PUNCT
ejpam-3352	243	10	v	v	NOUN
ejpam-3352	243	11	=	=	SYM
ejpam-3352	243	12	{	{	PUNCT
ejpam-3352	243	13	f−1(wλ	f−1(wλ	NOUN
ejpam-3352	243	14	)	)	PUNCT
ejpam-3352	243	15	:	:	PUNCT
ejpam-3352	243	16	λ	λ	X
ejpam-3352	243	17	∈	∈	PROPN
ejpam-3352	243	18	λ	λ	PROPN
ejpam-3352	243	19	}	}	PUNCT
ejpam-3352	243	20	is	be	AUX
ejpam-3352	243	21	is	be	AUX
ejpam-3352	243	22	τ	τ	PROPN
ejpam-3352	243	23	-open	-open	PROPN
ejpam-3352	243	24	,	,	PUNCT
ejpam-3352	243	25	τ	τ	PROPN
ejpam-3352	243	26	-locally	-locally	ADV
ejpam-3352	243	27	finite	finite	ADJ
ejpam-3352	243	28	.	.	PUNCT
ejpam-3352	244	1	let	let	VERB
ejpam-3352	244	2	f−1(wλ	f−1(wλ	NUM
ejpam-3352	244	3	)	)	PUNCT
ejpam-3352	244	4	∈	∈	PROPN
ejpam-3352	245	1	v.	v.	CCONJ
ejpam-3352	245	2	then	then	ADV
ejpam-3352	245	3	wλ	wλ	PROPN
ejpam-3352	245	4	∈	∈	PROPN
ejpam-3352	245	5	w.	w.	PROPN
ejpam-3352	245	6	since	since	SCONJ
ejpam-3352	245	7	w	w	PROPN
ejpam-3352	245	8	refines	refine	NOUN
ejpam-3352	245	9	f(u	f(u	PROPN
ejpam-3352	245	10	)	)	PUNCT
ejpam-3352	245	11	,	,	PUNCT
ejpam-3352	245	12	there	there	PRON
ejpam-3352	245	13	exists	exist	VERB
ejpam-3352	245	14	f(uα	f(uα	NOUN
ejpam-3352	245	15	)	)	PUNCT
ejpam-3352	245	16	∈	∈	PROPN
ejpam-3352	245	17	f(u	f(u	PROPN
ejpam-3352	245	18	)	)	PUNCT
ejpam-3352	245	19	such	such	ADJ
ejpam-3352	245	20	that	that	SCONJ
ejpam-3352	245	21	wλ	wλ	PROPN
ejpam-3352	245	22	⊂	⊂	PROPN
ejpam-3352	245	23	f(uα	f(uα	NOUN
ejpam-3352	245	24	)	)	PUNCT
ejpam-3352	245	25	.	.	PUNCT
ejpam-3352	246	1	thus	thus	ADV
ejpam-3352	246	2	f−1(wλ	f−1(wλ	X
ejpam-3352	246	3	)	)	PUNCT
ejpam-3352	247	1	⊂	⊂	PROPN
ejpam-3352	247	2	f−1(f(uα	f−1(f(uα	PROPN
ejpam-3352	247	3	)	)	PUNCT
ejpam-3352	247	4	)	)	PUNCT
ejpam-3352	247	5	implies	imply	VERB
ejpam-3352	247	6	that	that	SCONJ
ejpam-3352	247	7	f−1(wλ	f−1(wλ	X
ejpam-3352	247	8	)	)	PUNCT
ejpam-3352	248	1	⊂	⊂	PROPN
ejpam-3352	248	2	uα	uα	PROPN
ejpam-3352	248	3	for	for	ADP
ejpam-3352	248	4	some	some	DET
ejpam-3352	248	5	uα	uα	PROPN
ejpam-3352	248	6	∈	∈	PROPN
ejpam-3352	248	7	u	u	NOUN
ejpam-3352	248	8	.	.	PUNCT
ejpam-3352	249	1	hence	hence	ADV
ejpam-3352	249	2	v	v	NOUN
ejpam-3352	249	3	refines	refine	VERB
ejpam-3352	249	4	u	u	PRON
ejpam-3352	249	5	.	.	PUNCT
ejpam-3352	250	1	therefore	therefore	ADV
ejpam-3352	250	2	,	,	PUNCT
ejpam-3352	250	3	(	(	PUNCT
ejpam-3352	250	4	x	x	X
ejpam-3352	250	5	,	,	PUNCT
ejpam-3352	250	6	τ	τ	PROPN
ejpam-3352	250	7	,	,	PUNCT
ejpam-3352	250	8	f−1(j	f−1(j	NOUN
ejpam-3352	250	9	)	)	PUNCT
ejpam-3352	250	10	)	)	PUNCT
ejpam-3352	250	11	is	be	AUX
ejpam-3352	250	12	β1f	β1f	X
ejpam-3352	250	13	−1(j	−1(j	NOUN
ejpam-3352	250	14	)	)	PUNCT
ejpam-3352	250	15	-paracompact	-paracompact	NOUN
ejpam-3352	250	16	.	.	PUNCT
ejpam-3352	251	1	if	if	SCONJ
ejpam-3352	251	2	i	i	PRON
ejpam-3352	251	3	=	=	SYM
ejpam-3352	251	4	{	{	PUNCT
ejpam-3352	251	5	∅	∅	NOUN
ejpam-3352	251	6	}	}	PUNCT
ejpam-3352	251	7	in	in	ADP
ejpam-3352	251	8	theorem	theorem	NOUN
ejpam-3352	251	9	8	8	NUM
ejpam-3352	251	10	,	,	PUNCT
ejpam-3352	251	11	then	then	ADV
ejpam-3352	251	12	we	we	PRON
ejpam-3352	251	13	have	have	VERB
ejpam-3352	251	14	the	the	DET
ejpam-3352	251	15	following	follow	VERB
ejpam-3352	251	16	corollary	corollary	NOUN
ejpam-3352	251	17	.	.	PUNCT
ejpam-3352	252	1	corollary	corollary	ADJ
ejpam-3352	252	2	6	6	NUM
ejpam-3352	252	3	.	.	PUNCT
ejpam-3352	253	1	let	let	VERB
ejpam-3352	253	2	f	f	NOUN
ejpam-3352	253	3	:	:	PUNCT
ejpam-3352	253	4	(	(	PUNCT
ejpam-3352	253	5	x	x	X
ejpam-3352	253	6	,	,	PUNCT
ejpam-3352	253	7	τ	τ	X
ejpam-3352	253	8	)	)	PUNCT
ejpam-3352	253	9	→	→	SYM
ejpam-3352	253	10	(	(	PUNCT
ejpam-3352	253	11	y	y	PROPN
ejpam-3352	253	12	,	,	PUNCT
ejpam-3352	253	13	σ	σ	PROPN
ejpam-3352	253	14	)	)	PUNCT
ejpam-3352	253	15	be	be	AUX
ejpam-3352	253	16	a	a	DET
ejpam-3352	253	17	pre	pre	NOUN
ejpam-3352	253	18	β	β	NOUN
ejpam-3352	253	19	-	-	ADJ
ejpam-3352	253	20	open	open	ADJ
ejpam-3352	253	21	,	,	PUNCT
ejpam-3352	253	22	continuous	continuous	ADJ
ejpam-3352	253	23	,	,	PUNCT
ejpam-3352	253	24	bijective	bijective	ADJ
ejpam-3352	253	25	function	function	NOUN
ejpam-3352	253	26	.	.	PUNCT
ejpam-3352	254	1	if	if	SCONJ
ejpam-3352	254	2	(	(	PUNCT
ejpam-3352	254	3	y	y	PROPN
ejpam-3352	254	4	,	,	PUNCT
ejpam-3352	254	5	σ	σ	PROPN
ejpam-3352	254	6	)	)	PUNCT
ejpam-3352	254	7	is	be	AUX
ejpam-3352	254	8	β1	β1	NOUN
ejpam-3352	254	9	-	-	PUNCT
ejpam-3352	254	10	paracompact	paracompact	ADJ
ejpam-3352	254	11	,	,	PUNCT
ejpam-3352	254	12	then	then	ADV
ejpam-3352	254	13	(	(	PUNCT
ejpam-3352	254	14	x	x	X
ejpam-3352	254	15	,	,	PUNCT
ejpam-3352	254	16	τ	τ	X
ejpam-3352	254	17	)	)	PUNCT
ejpam-3352	254	18	is	be	AUX
ejpam-3352	254	19	β1	β1	NOUN
ejpam-3352	254	20	-	-	PUNCT
ejpam-3352	254	21	paracompact	paracompact	ADJ
ejpam-3352	254	22	.	.	PUNCT
ejpam-3352	255	1	3	3	X
ejpam-3352	255	2	.	.	X
ejpam-3352	255	3	β1i	β1i	ADJ
ejpam-3352	255	4	-	-	ADJ
ejpam-3352	255	5	paracompact	paracompact	ADJ
ejpam-3352	255	6	subsets	subset	NOUN
ejpam-3352	255	7	in	in	ADP
ejpam-3352	255	8	this	this	DET
ejpam-3352	255	9	section	section	NOUN
ejpam-3352	255	10	,	,	PUNCT
ejpam-3352	255	11	we	we	PRON
ejpam-3352	255	12	define	define	VERB
ejpam-3352	255	13	the	the	DET
ejpam-3352	255	14	subsets	subset	NOUN
ejpam-3352	255	15	and	and	CCONJ
ejpam-3352	255	16	subspaces	subspace	NOUN
ejpam-3352	255	17	of	of	ADP
ejpam-3352	255	18	β1i	β1i	NOUN
ejpam-3352	255	19	-	-	ADJ
ejpam-3352	255	20	paracompact	paracompact	NOUN
ejpam-3352	255	21	and	and	CCONJ
ejpam-3352	255	22	study	study	VERB
ejpam-3352	255	23	some	some	PRON
ejpam-3352	255	24	of	of	ADP
ejpam-3352	255	25	their	their	PRON
ejpam-3352	255	26	properties	property	NOUN
ejpam-3352	255	27	.	.	PUNCT
ejpam-3352	256	1	definition	definition	NOUN
ejpam-3352	256	2	3	3	NUM
ejpam-3352	256	3	.	.	PUNCT
ejpam-3352	257	1	a	a	DET
ejpam-3352	257	2	subset	subset	NOUN
ejpam-3352	257	3	a	a	PRON
ejpam-3352	257	4	of	of	ADP
ejpam-3352	257	5	an	an	DET
ejpam-3352	257	6	ideal	ideal	ADJ
ejpam-3352	257	7	space	space	NOUN
ejpam-3352	257	8	(	(	PUNCT
ejpam-3352	257	9	x	x	X
ejpam-3352	257	10	,	,	PUNCT
ejpam-3352	257	11	τ	τ	PROPN
ejpam-3352	257	12	,	,	PUNCT
ejpam-3352	257	13	i	i	PROPN
ejpam-3352	257	14	)	)	PUNCT
ejpam-3352	257	15	is	be	AUX
ejpam-3352	257	16	said	say	VERB
ejpam-3352	257	17	to	to	PART
ejpam-3352	257	18	be	be	AUX
ejpam-3352	257	19	β1i	β1i	ADJ
ejpam-3352	257	20	-	-	ADJ
ejpam-3352	257	21	paracompact	paracompact	ADJ
ejpam-3352	257	22	relative	relative	NOUN
ejpam-3352	257	23	to	to	ADP
ejpam-3352	257	24	x	x	SYM
ejpam-3352	257	25	(	(	PUNCT
ejpam-3352	257	26	β1i	β1i	ADJ
ejpam-3352	257	27	-	-	ADJ
ejpam-3352	257	28	paracompact	paracompact	ADJ
ejpam-3352	257	29	subset	subset	NOUN
ejpam-3352	257	30	)	)	PUNCT
ejpam-3352	257	31	if	if	SCONJ
ejpam-3352	257	32	each	each	PRON
ejpam-3352	257	33	cover	cover	VERB
ejpam-3352	257	34	u	u	NOUN
ejpam-3352	257	35	of	of	ADP
ejpam-3352	257	36	a	a	PRON
ejpam-3352	257	37	by	by	ADP
ejpam-3352	257	38	β	β	ADJ
ejpam-3352	257	39	-	-	ADJ
ejpam-3352	257	40	open	open	ADJ
ejpam-3352	257	41	sets	set	NOUN
ejpam-3352	257	42	of	of	ADP
ejpam-3352	257	43	x	x	NOUN
ejpam-3352	257	44	,	,	PUNCT
ejpam-3352	257	45	there	there	PRON
ejpam-3352	257	46	exists	exist	VERB
ejpam-3352	257	47	a	a	DET
ejpam-3352	257	48	a.	a.	NOUN
ejpam-3352	257	49	qahis	qahis	PROPN
ejpam-3352	257	50	/	/	SYM
ejpam-3352	257	51	eur	eur	PROPN
ejpam-3352	257	52	.	.	PUNCT
ejpam-3352	258	1	j.	j.	PROPN
ejpam-3352	258	2	pure	pure	PROPN
ejpam-3352	258	3	appl	appl	PROPN
ejpam-3352	258	4	.	.	PROPN
ejpam-3352	258	5	math	math	PROPN
ejpam-3352	258	6	,	,	PUNCT
ejpam-3352	258	7	12	12	NUM
ejpam-3352	258	8	(	(	PUNCT
ejpam-3352	258	9	1	1	NUM
ejpam-3352	258	10	)	)	PUNCT
ejpam-3352	258	11	(	(	PUNCT
ejpam-3352	258	12	2019	2019	NUM
ejpam-3352	258	13	)	)	PUNCT
ejpam-3352	258	14	,	,	PUNCT
ejpam-3352	258	15	135	135	NUM
ejpam-3352	258	16	-	-	SYM
ejpam-3352	258	17	145	145	NUM
ejpam-3352	258	18	141	141	NUM
ejpam-3352	258	19	locally	locally	ADV
ejpam-3352	258	20	finite	finite	VERB
ejpam-3352	258	21	open	open	ADJ
ejpam-3352	258	22	refinement	refinement	PROPN
ejpam-3352	258	23	v	v	NOUN
ejpam-3352	258	24	of	of	ADP
ejpam-3352	258	25	u	u	PRON
ejpam-3352	258	26	such	such	ADJ
ejpam-3352	258	27	that	that	SCONJ
ejpam-3352	258	28	a	a	DET
ejpam-3352	258	29	\	\	NOUN
ejpam-3352	258	30	∪{v	∪{v	NOUN
ejpam-3352	258	31	:	:	PUNCT
ejpam-3352	258	32	v	v	NUM
ejpam-3352	258	33	∈	∈	PROPN
ejpam-3352	258	34	v	v	NOUN
ejpam-3352	258	35	}	}	PUNCT
ejpam-3352	258	36	∈	∈	PROPN
ejpam-3352	258	37	i.	i.	NOUN
ejpam-3352	258	38	a	a	PROPN
ejpam-3352	258	39	is	be	AUX
ejpam-3352	258	40	said	say	VERB
ejpam-3352	258	41	to	to	PART
ejpam-3352	258	42	be	be	AUX
ejpam-3352	258	43	β1aia	β1aia	NOUN
ejpam-3352	258	44	-	-	PUNCT
ejpam-3352	258	45	paracompact	paracompact	ADJ
ejpam-3352	258	46	(	(	PUNCT
ejpam-3352	258	47	β1aia	β1aia	NUM
ejpam-3352	258	48	-	-	PUNCT
ejpam-3352	258	49	paracompact	paracompact	ADJ
ejpam-3352	258	50	subspace	subspace	NOUN
ejpam-3352	258	51	)	)	PUNCT
ejpam-3352	259	1	if	if	SCONJ
ejpam-3352	259	2	(	(	PUNCT
ejpam-3352	259	3	a	a	DET
ejpam-3352	259	4	,	,	PUNCT
ejpam-3352	259	5	τa	τa	PROPN
ejpam-3352	259	6	,	,	PUNCT
ejpam-3352	259	7	ia	ia	PROPN
ejpam-3352	259	8	)	)	PUNCT
ejpam-3352	259	9	is	be	AUX
ejpam-3352	259	10	β1aia	β1aia	NOUN
ejpam-3352	259	11	-	-	PUNCT
ejpam-3352	259	12	paracompact	paracompact	NOUN
ejpam-3352	259	13	as	as	ADP
ejpam-3352	259	14	a	a	DET
ejpam-3352	259	15	subspace	subspace	NOUN
ejpam-3352	259	16	,	,	PUNCT
ejpam-3352	259	17	where	where	SCONJ
ejpam-3352	259	18	τa	τa	ADV
ejpam-3352	259	19	is	be	AUX
ejpam-3352	259	20	the	the	DET
ejpam-3352	259	21	usual	usual	ADJ
ejpam-3352	259	22	subspace	subspace	NOUN
ejpam-3352	259	23	topology	topology	NOUN
ejpam-3352	259	24	and	and	CCONJ
ejpam-3352	259	25	ia	ia	PROPN
ejpam-3352	259	26	=	=	X
ejpam-3352	259	27	{	{	PUNCT
ejpam-3352	259	28	a	a	DET
ejpam-3352	259	29	∩	∩	ADJ
ejpam-3352	260	1	i	i	PRON
ejpam-3352	260	2	:	:	PUNCT
ejpam-3352	260	3	i	i	PRON
ejpam-3352	260	4	∈	∈	VERB
ejpam-3352	260	5	i	i	X
ejpam-3352	260	6	}	}	PUNCT
ejpam-3352	260	7	.	.	PUNCT
ejpam-3352	261	1	a	a	DET
ejpam-3352	261	2	subset	subset	NOUN
ejpam-3352	261	3	a	a	PRON
ejpam-3352	261	4	of	of	ADP
ejpam-3352	261	5	a	a	DET
ejpam-3352	261	6	space	space	NOUN
ejpam-3352	261	7	(	(	PUNCT
ejpam-3352	261	8	x	x	X
ejpam-3352	261	9	,	,	PUNCT
ejpam-3352	261	10	τ	τ	X
ejpam-3352	261	11	)	)	PUNCT
ejpam-3352	261	12	is	be	AUX
ejpam-3352	261	13	said	say	VERB
ejpam-3352	261	14	to	to	PART
ejpam-3352	261	15	be	be	AUX
ejpam-3352	261	16	βg	βg	ADV
ejpam-3352	261	17	-	-	PUNCT
ejpam-3352	261	18	closed	closed	ADJ
ejpam-3352	261	19	[	[	X
ejpam-3352	261	20	6	6	NUM
ejpam-3352	261	21	]	]	PUNCT
ejpam-3352	261	22	if	if	SCONJ
ejpam-3352	261	23	βcl(a	βcl(a	NUM
ejpam-3352	261	24	)	)	PUNCT
ejpam-3352	261	25	⊆	⊆	NUM
ejpam-3352	261	26	u	u	NOUN
ejpam-3352	261	27	whenever	whenever	SCONJ
ejpam-3352	261	28	a	a	DET
ejpam-3352	261	29	⊂	⊂	PROPN
ejpam-3352	261	30	u	u	NOUN
ejpam-3352	261	31	and	and	CCONJ
ejpam-3352	261	32	u	u	NOUN
ejpam-3352	261	33	is	be	AUX
ejpam-3352	261	34	any	any	DET
ejpam-3352	261	35	β	β	NOUN
ejpam-3352	261	36	-	-	ADJ
ejpam-3352	261	37	open	open	ADJ
ejpam-3352	261	38	set	set	NOUN
ejpam-3352	261	39	in	in	ADP
ejpam-3352	261	40	(	(	PUNCT
ejpam-3352	261	41	x	x	NOUN
ejpam-3352	261	42	,	,	PUNCT
ejpam-3352	261	43	τ	τ	PROPN
ejpam-3352	261	44	)	)	PUNCT
ejpam-3352	261	45	.	.	PUNCT
ejpam-3352	261	46	theorem	theorem	VERB
ejpam-3352	261	47	9	9	NUM
ejpam-3352	261	48	.	.	PUNCT
ejpam-3352	262	1	every	every	DET
ejpam-3352	262	2	βg	βg	ADJ
ejpam-3352	262	3	-	-	PUNCT
ejpam-3352	262	4	closed	closed	ADJ
ejpam-3352	262	5	subset	subset	NOUN
ejpam-3352	262	6	of	of	ADP
ejpam-3352	262	7	a	a	DET
ejpam-3352	262	8	β1i	β1i	NOUN
ejpam-3352	262	9	-	-	ADJ
ejpam-3352	262	10	paracompact	paracompact	NOUN
ejpam-3352	262	11	is	be	AUX
ejpam-3352	262	12	β1i	β1i	NOUN
ejpam-3352	262	13	-	-	ADJ
ejpam-3352	262	14	paracompact	paracompact	ADJ
ejpam-3352	262	15	.	.	PUNCT
ejpam-3352	263	1	proof	proof	NOUN
ejpam-3352	263	2	.	.	PUNCT
ejpam-3352	264	1	let	let	VERB
ejpam-3352	264	2	a	a	DET
ejpam-3352	264	3	be	be	AUX
ejpam-3352	264	4	a	a	DET
ejpam-3352	264	5	βg	βg	ADV
ejpam-3352	264	6	-	-	PUNCT
ejpam-3352	264	7	closed	closed	ADJ
ejpam-3352	264	8	subset	subset	NOUN
ejpam-3352	264	9	of	of	ADP
ejpam-3352	264	10	(	(	PUNCT
ejpam-3352	264	11	x	x	PROPN
ejpam-3352	264	12	,	,	PUNCT
ejpam-3352	264	13	τ	τ	PROPN
ejpam-3352	264	14	,	,	PUNCT
ejpam-3352	264	15	i	i	NOUN
ejpam-3352	264	16	)	)	PUNCT
ejpam-3352	264	17	and	and	CCONJ
ejpam-3352	264	18	u	u	NOUN
ejpam-3352	264	19	=	=	PUNCT
ejpam-3352	264	20	{	{	PUNCT
ejpam-3352	264	21	uα	uα	X
ejpam-3352	264	22	:	:	PUNCT
ejpam-3352	264	23	α	α	PROPN
ejpam-3352	264	24	∈	∈	PROPN
ejpam-3352	264	25	∆	∆	PROPN
ejpam-3352	264	26	}	}	PUNCT
ejpam-3352	264	27	be	be	AUX
ejpam-3352	264	28	a	a	DET
ejpam-3352	264	29	cover	cover	NOUN
ejpam-3352	264	30	of	of	ADP
ejpam-3352	264	31	a	a	PRON
ejpam-3352	264	32	by	by	ADP
ejpam-3352	264	33	β	β	ADJ
ejpam-3352	264	34	-	-	ADJ
ejpam-3352	264	35	open	open	ADJ
ejpam-3352	264	36	sets	set	NOUN
ejpam-3352	264	37	of	of	ADP
ejpam-3352	264	38	x.	x.	NOUN
ejpam-3352	264	39	since	since	SCONJ
ejpam-3352	264	40	a	a	DET
ejpam-3352	264	41	⊆	⊆	NUM
ejpam-3352	264	42	∪α∈∆uα	∪α∈∆uα	NOUN
ejpam-3352	264	43	and	and	CCONJ
ejpam-3352	264	44	a	a	PRON
ejpam-3352	264	45	is	be	AUX
ejpam-3352	264	46	a	a	DET
ejpam-3352	264	47	βg	βg	ADV
ejpam-3352	264	48	-	-	PUNCT
ejpam-3352	264	49	closed	closed	ADJ
ejpam-3352	264	50	,	,	PUNCT
ejpam-3352	264	51	we	we	PRON
ejpam-3352	264	52	have	have	VERB
ejpam-3352	264	53	βcl(a	βcl(a	NOUN
ejpam-3352	264	54	)	)	PUNCT
ejpam-3352	264	55	⊆	⊆	NUM
ejpam-3352	264	56	∪α∈∆uα	∪α∈∆uα	NOUN
ejpam-3352	264	57	.	.	PUNCT
ejpam-3352	265	1	then	then	ADV
ejpam-3352	265	2	u1	u1	VERB
ejpam-3352	265	3	=	=	SYM
ejpam-3352	265	4	u	u	NOUN
ejpam-3352	265	5	∪	∪	X
ejpam-3352	265	6	{	{	PUNCT
ejpam-3352	265	7	x	x	SYM
ejpam-3352	265	8	\	\	ADJ
ejpam-3352	265	9	βcl(a	βcl(a	PROPN
ejpam-3352	265	10	)	)	PUNCT
ejpam-3352	265	11	}	}	PUNCT
ejpam-3352	265	12	is	be	AUX
ejpam-3352	265	13	a	a	DET
ejpam-3352	265	14	β	β	NOUN
ejpam-3352	265	15	-	-	ADJ
ejpam-3352	265	16	open	open	ADJ
ejpam-3352	265	17	cover	cover	NOUN
ejpam-3352	265	18	of	of	ADP
ejpam-3352	265	19	x.	x.	NOUN
ejpam-3352	265	20	by	by	ADP
ejpam-3352	265	21	hypothesis	hypothesis	NOUN
ejpam-3352	265	22	,	,	PUNCT
ejpam-3352	265	23	there	there	PRON
ejpam-3352	265	24	exist	exist	VERB
ejpam-3352	265	25	a	a	DET
ejpam-3352	265	26	locally	locally	ADV
ejpam-3352	265	27	finite	finite	ADJ
ejpam-3352	265	28	open	open	ADJ
ejpam-3352	265	29	family	family	NOUN
ejpam-3352	265	30	v	v	NOUN
ejpam-3352	265	31	=	=	PUNCT
ejpam-3352	265	32	{	{	PUNCT
ejpam-3352	265	33	vλ	vλ	INTJ
ejpam-3352	265	34	:	:	PUNCT
ejpam-3352	265	35	λ	λ	PROPN
ejpam-3352	265	36	∈	∈	PROPN
ejpam-3352	265	37	λ	λ	PROPN
ejpam-3352	265	38	}	}	PUNCT
ejpam-3352	265	39	∪	∪	ADJ
ejpam-3352	265	40	{	{	PUNCT
ejpam-3352	265	41	v	v	NOUN
ejpam-3352	265	42	}	}	PUNCT
ejpam-3352	265	43	which	which	PRON
ejpam-3352	265	44	refines	refine	VERB
ejpam-3352	265	45	u1	u1	NOUN
ejpam-3352	265	46	(	(	PUNCT
ejpam-3352	265	47	vλ	vλ	INTJ
ejpam-3352	265	48	⊂	⊂	PROPN
ejpam-3352	265	49	uα	uα	PROPN
ejpam-3352	265	50	for	for	ADP
ejpam-3352	265	51	some	some	DET
ejpam-3352	265	52	α	α	NUM
ejpam-3352	265	53	∈	∈	NOUN
ejpam-3352	265	54	∆	∆	PROPN
ejpam-3352	265	55	and	and	CCONJ
ejpam-3352	265	56	v	v	ADP
ejpam-3352	265	57	⊂	⊂	PROPN
ejpam-3352	265	58	x	x	SYM
ejpam-3352	265	59	\	\	ADJ
ejpam-3352	265	60	βcl(a	βcl(a	PROPN
ejpam-3352	265	61	)	)	PUNCT
ejpam-3352	265	62	)	)	PUNCT
ejpam-3352	266	1	such	such	ADJ
ejpam-3352	266	2	that	that	SCONJ
ejpam-3352	266	3	x	x	SYM
ejpam-3352	266	4	\	\	PROPN
ejpam-3352	266	5	∪[{vλ	∪[{vλ	NOUN
ejpam-3352	266	6	:	:	PUNCT
ejpam-3352	266	7	λ	λ	X
ejpam-3352	266	8	∈	∈	PROPN
ejpam-3352	266	9	λ	λ	PROPN
ejpam-3352	266	10	}	}	PUNCT
ejpam-3352	266	11	∪	∪	ADJ
ejpam-3352	266	12	{	{	PUNCT
ejpam-3352	266	13	v	v	NOUN
ejpam-3352	266	14	}	}	PUNCT
ejpam-3352	266	15	]	]	PUNCT
ejpam-3352	266	16	∈	∈	PROPN
ejpam-3352	266	17	i.	i.	NOUN
ejpam-3352	266	18	then	then	ADV
ejpam-3352	266	19	βcl(a	βcl(a	NUM
ejpam-3352	266	20	)	)	PUNCT
ejpam-3352	266	21	\	\	PROPN
ejpam-3352	267	1	∪{vλ	∪{vλ	NUM
ejpam-3352	267	2	:	:	PUNCT
ejpam-3352	267	3	λ	λ	PROPN
ejpam-3352	267	4	∈	∈	PROPN
ejpam-3352	267	5	λ	λ	NOUN
ejpam-3352	267	6	}	}	PUNCT
ejpam-3352	267	7	=	=	SYM
ejpam-3352	267	8	βcl(a	βcl(a	NUM
ejpam-3352	267	9	)	)	PUNCT
ejpam-3352	267	10	\	\	PUNCT
ejpam-3352	268	1	[	[	X
ejpam-3352	268	2	v	v	ADP
ejpam-3352	268	3	∪	∪	X
ejpam-3352	268	4	(	(	PUNCT
ejpam-3352	268	5	∪{vλ	∪{vλ	NUM
ejpam-3352	268	6	:	:	PUNCT
ejpam-3352	268	7	λ	λ	PROPN
ejpam-3352	268	8	∈	∈	PROPN
ejpam-3352	268	9	λ	λ	NOUN
ejpam-3352	268	10	}	}	PUNCT
ejpam-3352	268	11	)	)	PUNCT
ejpam-3352	268	12	]	]	PUNCT
ejpam-3352	269	1	⊂	⊂	PUNCT
ejpam-3352	269	2	x	x	PUNCT
ejpam-3352	269	3	\	\	PROPN
ejpam-3352	269	4	∪[{vλ	∪[{vλ	NOUN
ejpam-3352	269	5	:	:	PUNCT
ejpam-3352	269	6	λ	λ	X
ejpam-3352	269	7	∈	∈	PROPN
ejpam-3352	269	8	λ	λ	PROPN
ejpam-3352	269	9	}	}	PUNCT
ejpam-3352	269	10	∪	∪	ADJ
ejpam-3352	269	11	{	{	PUNCT
ejpam-3352	269	12	v	v	NOUN
ejpam-3352	269	13	}	}	PUNCT
ejpam-3352	269	14	]	]	PUNCT
ejpam-3352	269	15	∈	∈	PROPN
ejpam-3352	269	16	i.	i.	NOUN
ejpam-3352	269	17	since	since	SCONJ
ejpam-3352	269	18	a	a	DET
ejpam-3352	269	19	\	\	PROPN
ejpam-3352	269	20	∪{vλ	∪{vλ	NUM
ejpam-3352	269	21	:	:	PUNCT
ejpam-3352	269	22	λ	λ	PROPN
ejpam-3352	269	23	∈	∈	PROPN
ejpam-3352	269	24	λ	λ	PROPN
ejpam-3352	269	25	}	}	PUNCT
ejpam-3352	269	26	⊂	⊂	X
ejpam-3352	269	27	βcl(a	βcl(a	PROPN
ejpam-3352	269	28	)	)	PUNCT
ejpam-3352	269	29	\	\	PROPN
ejpam-3352	269	30	∪{vλ	∪{vλ	NUM
ejpam-3352	269	31	:	:	PUNCT
ejpam-3352	269	32	λ	λ	PROPN
ejpam-3352	269	33	∈	∈	PROPN
ejpam-3352	269	34	λ	λ	PROPN
ejpam-3352	269	35	}	}	PUNCT
ejpam-3352	269	36	,	,	PUNCT
ejpam-3352	269	37	a	a	DET
ejpam-3352	269	38	\	\	PROPN
ejpam-3352	269	39	∪{vλ	∪{vλ	NUM
ejpam-3352	269	40	:	:	PUNCT
ejpam-3352	269	41	λ	λ	PROPN
ejpam-3352	269	42	∈	∈	PROPN
ejpam-3352	269	43	λ	λ	PROPN
ejpam-3352	269	44	}	}	PUNCT
ejpam-3352	269	45	∈	∈	PROPN
ejpam-3352	269	46	i	i	PRON
ejpam-3352	269	47	,	,	PUNCT
ejpam-3352	269	48	by	by	ADP
ejpam-3352	269	49	heredity	heredity	NOUN
ejpam-3352	269	50	property	property	NOUN
ejpam-3352	269	51	of	of	ADP
ejpam-3352	269	52	i.	i.	NOUN
ejpam-3352	269	53	since	since	SCONJ
ejpam-3352	269	54	v	v	NUM
ejpam-3352	269	55	=	=	PUNCT
ejpam-3352	269	56	{	{	PUNCT
ejpam-3352	269	57	vλ	vλ	INTJ
ejpam-3352	269	58	:	:	PUNCT
ejpam-3352	269	59	λ	λ	PROPN
ejpam-3352	269	60	∈	∈	PROPN
ejpam-3352	269	61	λ}∪{v	λ}∪{v	NOUN
ejpam-3352	269	62	}	}	PUNCT
ejpam-3352	269	63	is	be	AUX
ejpam-3352	269	64	a	a	DET
ejpam-3352	269	65	locally	locally	ADV
ejpam-3352	269	66	finite	finite	NOUN
ejpam-3352	269	67	,	,	PUNCT
ejpam-3352	269	68	the	the	DET
ejpam-3352	269	69	family	family	NOUN
ejpam-3352	269	70	v1	v1	NOUN
ejpam-3352	269	71	=	=	SYM
ejpam-3352	269	72	{	{	PUNCT
ejpam-3352	269	73	vλ	vλ	INTJ
ejpam-3352	269	74	:	:	PUNCT
ejpam-3352	269	75	λ	λ	PROPN
ejpam-3352	269	76	∈	∈	PROPN
ejpam-3352	269	77	λ	λ	PROPN
ejpam-3352	269	78	}	}	PUNCT
ejpam-3352	269	79	is	be	AUX
ejpam-3352	269	80	locally	locally	ADV
ejpam-3352	269	81	finite	finite	ADJ
ejpam-3352	269	82	.	.	PUNCT
ejpam-3352	270	1	thus	thus	ADV
ejpam-3352	270	2	,	,	PUNCT
ejpam-3352	270	3	the	the	DET
ejpam-3352	270	4	family	family	NOUN
ejpam-3352	270	5	v1	v1	NOUN
ejpam-3352	270	6	is	be	AUX
ejpam-3352	270	7	locally	locally	ADV
ejpam-3352	270	8	finite	finite	ADJ
ejpam-3352	270	9	open	open	ADJ
ejpam-3352	270	10	and	and	CCONJ
ejpam-3352	270	11	v1	v1	VERB
ejpam-3352	270	12	refines	refine	VERB
ejpam-3352	270	13	u	u	NOUN
ejpam-3352	270	14	.	.	PUNCT
ejpam-3352	271	1	therefore	therefore	ADV
ejpam-3352	271	2	,	,	PUNCT
ejpam-3352	271	3	a	a	PRON
ejpam-3352	271	4	is	be	AUX
ejpam-3352	271	5	βi	βi	NOUN
ejpam-3352	271	6	-	-	PUNCT
ejpam-3352	271	7	paracompact	paracompact	ADJ
ejpam-3352	271	8	.	.	PUNCT
ejpam-3352	272	1	theorem	theorem	NOUN
ejpam-3352	272	2	10	10	NUM
ejpam-3352	272	3	.	.	PUNCT
ejpam-3352	273	1	every	every	DET
ejpam-3352	273	2	regular	regular	ADJ
ejpam-3352	273	3	open	open	ADJ
ejpam-3352	273	4	subset	subset	NOUN
ejpam-3352	273	5	of	of	ADP
ejpam-3352	273	6	a	a	DET
ejpam-3352	273	7	β1i	β1i	NOUN
ejpam-3352	273	8	-	-	ADJ
ejpam-3352	273	9	paracompact	paracompact	NOUN
ejpam-3352	273	10	is	be	AUX
ejpam-3352	273	11	β1aia	β1aia	NOUN
ejpam-3352	273	12	-	-	PUNCT
ejpam-3352	273	13	paracompact	paracompact	ADJ
ejpam-3352	273	14	.	.	PUNCT
ejpam-3352	274	1	proof	proof	NOUN
ejpam-3352	274	2	.	.	PUNCT
ejpam-3352	275	1	let	let	VERB
ejpam-3352	275	2	a	a	PRON
ejpam-3352	275	3	be	be	AUX
ejpam-3352	275	4	a	a	DET
ejpam-3352	275	5	regular	regular	ADJ
ejpam-3352	275	6	open	open	NOUN
ejpam-3352	275	7	in	in	ADP
ejpam-3352	275	8	(	(	PUNCT
ejpam-3352	275	9	x	x	NOUN
ejpam-3352	275	10	,	,	PUNCT
ejpam-3352	275	11	τ	τ	X
ejpam-3352	275	12	)	)	PUNCT
ejpam-3352	275	13	and	and	CCONJ
ejpam-3352	275	14	w	w	NOUN
ejpam-3352	275	15	=	=	SYM
ejpam-3352	275	16	{	{	PUNCT
ejpam-3352	275	17	wα	wα	NOUN
ejpam-3352	275	18	:	:	PUNCT
ejpam-3352	276	1	α	α	PROPN
ejpam-3352	276	2	∈	∈	PROPN
ejpam-3352	276	3	∆	∆	PROPN
ejpam-3352	276	4	}	}	PUNCT
ejpam-3352	276	5	be	be	AUX
ejpam-3352	276	6	a	a	DET
ejpam-3352	276	7	β	β	NOUN
ejpam-3352	276	8	-	-	ADJ
ejpam-3352	276	9	open	open	ADJ
ejpam-3352	276	10	cover	cover	NOUN
ejpam-3352	276	11	of	of	ADP
ejpam-3352	276	12	a	a	DET
ejpam-3352	276	13	in	in	ADP
ejpam-3352	276	14	(	(	PUNCT
ejpam-3352	276	15	a	a	DET
ejpam-3352	276	16	,	,	PUNCT
ejpam-3352	276	17	τa	τa	PROPN
ejpam-3352	276	18	,	,	PUNCT
ejpam-3352	276	19	ia	ia	PROPN
ejpam-3352	276	20	)	)	PUNCT
ejpam-3352	276	21	.	.	PUNCT
ejpam-3352	277	1	since	since	SCONJ
ejpam-3352	277	2	a	a	PRON
ejpam-3352	277	3	is	be	AUX
ejpam-3352	277	4	open	open	ADJ
ejpam-3352	277	5	in	in	ADP
ejpam-3352	277	6	(	(	PUNCT
ejpam-3352	277	7	x	x	X
ejpam-3352	277	8	,	,	PUNCT
ejpam-3352	277	9	τ	τ	PROPN
ejpam-3352	277	10	,	,	PUNCT
ejpam-3352	277	11	i	i	PROPN
ejpam-3352	277	12	)	)	PUNCT
ejpam-3352	277	13	,	,	PUNCT
ejpam-3352	277	14	wα	wα	NOUN
ejpam-3352	277	15	is	be	AUX
ejpam-3352	277	16	a	a	DET
ejpam-3352	277	17	β	β	NOUN
ejpam-3352	277	18	-	-	ADJ
ejpam-3352	277	19	open	open	ADJ
ejpam-3352	277	20	set	set	NOUN
ejpam-3352	277	21	in	in	ADP
ejpam-3352	277	22	(	(	PUNCT
ejpam-3352	277	23	x	x	NOUN
ejpam-3352	277	24	,	,	PUNCT
ejpam-3352	277	25	τ	τ	PROPN
ejpam-3352	277	26	,	,	PUNCT
ejpam-3352	277	27	i	i	NOUN
ejpam-3352	277	28	)	)	PUNCT
ejpam-3352	277	29	for	for	ADP
ejpam-3352	277	30	each	each	DET
ejpam-3352	277	31	α	α	NOUN
ejpam-3352	277	32	∈	∈	NOUN
ejpam-3352	277	33	∆	∆	PROPN
ejpam-3352	277	34	,	,	PUNCT
ejpam-3352	277	35	by	by	ADP
ejpam-3352	277	36	theorem	theorem	NOUN
ejpam-3352	277	37	1	1	NUM
ejpam-3352	277	38	.	.	PUNCT
ejpam-3352	278	1	then	then	ADV
ejpam-3352	278	2	u	u	X
ejpam-3352	278	3	=	=	NOUN
ejpam-3352	278	4	{	{	PUNCT
ejpam-3352	278	5	wα	wα	NOUN
ejpam-3352	278	6	:	:	PUNCT
ejpam-3352	278	7	α	α	PROPN
ejpam-3352	278	8	∈	∈	PROPN
ejpam-3352	278	9	∆	∆	X
ejpam-3352	278	10	}	}	PUNCT
ejpam-3352	278	11	∪	∪	X
ejpam-3352	278	12	{	{	PUNCT
ejpam-3352	278	13	x	x	SYM
ejpam-3352	278	14	\	\	PROPN
ejpam-3352	278	15	a	a	PRON
ejpam-3352	278	16	}	}	PUNCT
ejpam-3352	278	17	is	be	AUX
ejpam-3352	278	18	a	a	DET
ejpam-3352	278	19	β	β	NOUN
ejpam-3352	278	20	-	-	ADJ
ejpam-3352	278	21	open	open	ADJ
ejpam-3352	278	22	cover	cover	NOUN
ejpam-3352	278	23	of	of	ADP
ejpam-3352	278	24	the	the	DET
ejpam-3352	278	25	β1iparacompact	β1iparacompact	PROPN
ejpam-3352	278	26	(	(	PUNCT
ejpam-3352	278	27	x	x	X
ejpam-3352	278	28	,	,	PUNCT
ejpam-3352	278	29	τ	τ	PROPN
ejpam-3352	278	30	,	,	PUNCT
ejpam-3352	278	31	i	i	NOUN
ejpam-3352	278	32	)	)	PUNCT
ejpam-3352	279	1	and	and	CCONJ
ejpam-3352	279	2	so	so	ADV
ejpam-3352	279	3	it	it	PRON
ejpam-3352	279	4	has	have	VERB
ejpam-3352	279	5	a	a	DET
ejpam-3352	279	6	locally	locally	ADV
ejpam-3352	279	7	finite	finite	ADJ
ejpam-3352	279	8	open	open	ADJ
ejpam-3352	279	9	refinement	refinement	NOUN
ejpam-3352	279	10	v	v	X
ejpam-3352	279	11	=	=	PUNCT
ejpam-3352	279	12	{	{	PUNCT
ejpam-3352	279	13	vλ	vλ	INTJ
ejpam-3352	279	14	:	:	PUNCT
ejpam-3352	279	15	λ	λ	PROPN
ejpam-3352	279	16	∈	∈	PROPN
ejpam-3352	279	17	λ	λ	NOUN
ejpam-3352	279	18	}	}	PUNCT
ejpam-3352	279	19	such	such	ADJ
ejpam-3352	279	20	that	that	SCONJ
ejpam-3352	279	21	x	x	SYM
ejpam-3352	279	22	\	\	PROPN
ejpam-3352	279	23	∪{vλ	∪{vλ	NUM
ejpam-3352	279	24	:	:	PUNCT
ejpam-3352	279	25	λ	λ	PROPN
ejpam-3352	279	26	∈	∈	PROPN
ejpam-3352	279	27	λ	λ	NOUN
ejpam-3352	279	28	}	}	PUNCT
ejpam-3352	279	29	=	=	PUNCT
ejpam-3352	279	30	i	i	PRON
ejpam-3352	279	31	∈	∈	PROPN
ejpam-3352	279	32	i.	i.	NOUN
ejpam-3352	279	33	then	then	ADV
ejpam-3352	279	34	a	a	DET
ejpam-3352	279	35	⊂	⊂	PROPN
ejpam-3352	279	36	a	a	DET
ejpam-3352	279	37	∩	∩	ADJ
ejpam-3352	279	38	[	[	X
ejpam-3352	279	39	(	(	PUNCT
ejpam-3352	279	40	∪{vλ	∪{vλ	NUM
ejpam-3352	279	41	:	:	PUNCT
ejpam-3352	279	42	λ	λ	PROPN
ejpam-3352	279	43	∈	∈	PROPN
ejpam-3352	279	44	λ	λ	NOUN
ejpam-3352	279	45	}	}	PUNCT
ejpam-3352	279	46	)	)	PUNCT
ejpam-3352	279	47	∪	∪	ADP
ejpam-3352	279	48	i	i	PRON
ejpam-3352	279	49	]	]	X
ejpam-3352	279	50	=	=	SYM
ejpam-3352	279	51	(	(	PUNCT
ejpam-3352	279	52	∪{vλ	∪{vλ	X
ejpam-3352	279	53	∩	∩	NOUN
ejpam-3352	279	54	a	a	X
ejpam-3352	279	55	:	:	PUNCT
ejpam-3352	279	56	λ	λ	X
ejpam-3352	279	57	∈	∈	PROPN
ejpam-3352	279	58	λ	λ	NOUN
ejpam-3352	279	59	}	}	PUNCT
ejpam-3352	279	60	)	)	PUNCT
ejpam-3352	279	61	∪	∪	NOUN
ejpam-3352	279	62	(	(	PUNCT
ejpam-3352	279	63	i	i	PRON
ejpam-3352	279	64	∩a	∩a	PROPN
ejpam-3352	279	65	)	)	PUNCT
ejpam-3352	280	1	=	=	PUNCT
ejpam-3352	280	2	(	(	PUNCT
ejpam-3352	280	3	∪{vλ	∪{vλ	PROPN
ejpam-3352	280	4	∩a	∩a	NOUN
ejpam-3352	280	5	:	:	PUNCT
ejpam-3352	280	6	λ	λ	X
ejpam-3352	280	7	∈	∈	PROPN
ejpam-3352	280	8	λ	λ	NOUN
ejpam-3352	280	9	}	}	PUNCT
ejpam-3352	280	10	)	)	PUNCT
ejpam-3352	280	11	∪	∪	ADP
ejpam-3352	280	12	ia	ia	PROPN
ejpam-3352	280	13	which	which	PRON
ejpam-3352	280	14	implies	imply	VERB
ejpam-3352	280	15	that	that	SCONJ
ejpam-3352	280	16	a	a	DET
ejpam-3352	280	17	\	\	PROPN
ejpam-3352	280	18	∪{vλ	∪{vλ	PROPN
ejpam-3352	280	19	∩a	∩a	NOUN
ejpam-3352	280	20	:	:	PUNCT
ejpam-3352	280	21	λ	λ	X
ejpam-3352	280	22	∈	∈	PROPN
ejpam-3352	280	23	λ	λ	PROPN
ejpam-3352	280	24	}	}	PUNCT
ejpam-3352	280	25	∈	∈	PROPN
ejpam-3352	280	26	ia	ia	PROPN
ejpam-3352	280	27	.	.	PROPN
ejpam-3352	280	28	let	let	VERB
ejpam-3352	280	29	x	x	PUNCT
ejpam-3352	280	30	∈	∈	VERB
ejpam-3352	280	31	a.	a.	NOUN
ejpam-3352	280	32	since	since	SCONJ
ejpam-3352	280	33	v	v	NOUN
ejpam-3352	280	34	is	be	AUX
ejpam-3352	280	35	locally	locally	ADV
ejpam-3352	280	36	finite	finite	ADJ
ejpam-3352	280	37	,	,	PUNCT
ejpam-3352	280	38	there	there	PRON
ejpam-3352	280	39	exists	exist	VERB
ejpam-3352	280	40	v	v	ADP
ejpam-3352	280	41	∈	∈	PROPN
ejpam-3352	280	42	τ(x	τ(x	NOUN
ejpam-3352	280	43	)	)	PUNCT
ejpam-3352	280	44	such	such	ADJ
ejpam-3352	280	45	that	that	SCONJ
ejpam-3352	280	46	vλ	vλ	ADP
ejpam-3352	280	47	∩	∩	NOUN
ejpam-3352	280	48	v	v	NOUN
ejpam-3352	280	49	=	=	NOUN
ejpam-3352	280	50	∅	∅	NOUN
ejpam-3352	280	51	for	for	ADP
ejpam-3352	280	52	λ	λ	PROPN
ejpam-3352	280	53	6=	6=	SYM
ejpam-3352	280	54	λ1	λ1	ADJ
ejpam-3352	280	55	,	,	PUNCT
ejpam-3352	280	56	λ2	λ2	NOUN
ejpam-3352	280	57	,	,	PUNCT
ejpam-3352	280	58	...	...	PUNCT
ejpam-3352	280	59	,	,	PUNCT
ejpam-3352	280	60	λn	λn	PROPN
ejpam-3352	280	61	.	.	PUNCT
ejpam-3352	281	1	then	then	ADV
ejpam-3352	281	2	(	(	PUNCT
ejpam-3352	281	3	vλ∩v	vλ∩v	NOUN
ejpam-3352	281	4	)	)	PUNCT
ejpam-3352	281	5	∩a	∩a	NOUN
ejpam-3352	281	6	=	=	PUNCT
ejpam-3352	281	7	∅	∅	NOUN
ejpam-3352	281	8	for	for	ADP
ejpam-3352	281	9	λ	λ	PROPN
ejpam-3352	281	10	6=	6=	SYM
ejpam-3352	281	11	λ1	λ1	ADJ
ejpam-3352	281	12	,	,	PUNCT
ejpam-3352	281	13	λ2	λ2	NOUN
ejpam-3352	281	14	,	,	PUNCT
ejpam-3352	281	15	...	...	PUNCT
ejpam-3352	281	16	,	,	PUNCT
ejpam-3352	281	17	λn	λn	NOUN
ejpam-3352	281	18	and	and	CCONJ
ejpam-3352	281	19	so	so	ADV
ejpam-3352	281	20	(	(	PUNCT
ejpam-3352	281	21	vλ∩a)∩(v	vλ∩a)∩(v	PROPN
ejpam-3352	281	22	∩a	∩a	PROPN
ejpam-3352	281	23	)	)	PUNCT
ejpam-3352	282	1	=	=	PUNCT
ejpam-3352	282	2	∅.	∅.	VERB
ejpam-3352	282	3	therefore	therefore	ADV
ejpam-3352	282	4	,	,	PUNCT
ejpam-3352	282	5	va	va	PROPN
ejpam-3352	282	6	=	=	PUNCT
ejpam-3352	282	7	{	{	PUNCT
ejpam-3352	282	8	vλ	vλ	ADP
ejpam-3352	282	9	∩	∩	NOUN
ejpam-3352	282	10	a	a	PRON
ejpam-3352	282	11	:	:	PUNCT
ejpam-3352	282	12	λ	λ	X
ejpam-3352	282	13	∈	∈	PROPN
ejpam-3352	282	14	λ	λ	PROPN
ejpam-3352	282	15	}	}	PUNCT
ejpam-3352	282	16	is	be	AUX
ejpam-3352	282	17	τa	τa	NOUN
ejpam-3352	282	18	-	-	PUNCT
ejpam-3352	282	19	locally	locally	ADV
ejpam-3352	282	20	finite	finite	NOUN
ejpam-3352	282	21	.	.	PUNCT
ejpam-3352	283	1	let	let	VERB
ejpam-3352	283	2	vλ	vλ	ADP
ejpam-3352	283	3	∩	∩	NOUN
ejpam-3352	283	4	a	a	DET
ejpam-3352	283	5	∈	∈	PROPN
ejpam-3352	283	6	va	va	NOUN
ejpam-3352	283	7	.	.	PUNCT
ejpam-3352	284	1	since	since	SCONJ
ejpam-3352	284	2	v	v	NOUN
ejpam-3352	284	3	refines	refine	VERB
ejpam-3352	284	4	u	u	NOUN
ejpam-3352	284	5	,	,	PUNCT
ejpam-3352	284	6	there	there	PRON
ejpam-3352	284	7	is	be	VERB
ejpam-3352	284	8	some	some	DET
ejpam-3352	284	9	wα	wα	NOUN
ejpam-3352	284	10	∈	∈	PROPN
ejpam-3352	284	11	u	u	NOUN
ejpam-3352	284	12	such	such	ADJ
ejpam-3352	284	13	that	that	SCONJ
ejpam-3352	284	14	vλ	vλ	PROPN
ejpam-3352	284	15	⊂	⊂	PROPN
ejpam-3352	284	16	wα	wα	NOUN
ejpam-3352	284	17	which	which	PRON
ejpam-3352	284	18	implies	imply	VERB
ejpam-3352	284	19	vλ	vλ	ADP
ejpam-3352	284	20	∩	∩	NOUN
ejpam-3352	284	21	a	a	DET
ejpam-3352	284	22	⊂	⊂	PROPN
ejpam-3352	284	23	wα	wα	NOUN
ejpam-3352	284	24	.	.	PUNCT
ejpam-3352	285	1	therefore	therefore	ADV
ejpam-3352	285	2	,	,	PUNCT
ejpam-3352	285	3	va	va	PROPN
ejpam-3352	285	4	refines	refine	VERB
ejpam-3352	285	5	w.	w.	PROPN
ejpam-3352	285	6	hence	hence	ADV
ejpam-3352	285	7	a	a	PRON
ejpam-3352	285	8	is	be	AUX
ejpam-3352	285	9	β1aia	β1aia	NOUN
ejpam-3352	285	10	-	-	PUNCT
ejpam-3352	285	11	paracompact	paracompact	ADJ
ejpam-3352	285	12	.	.	PUNCT
ejpam-3352	286	1	corollary	corollary	ADJ
ejpam-3352	286	2	7	7	NUM
ejpam-3352	286	3	.	.	PUNCT
ejpam-3352	287	1	every	every	DET
ejpam-3352	287	2	clopen	clopen	ADJ
ejpam-3352	287	3	subset	subset	NOUN
ejpam-3352	287	4	of	of	ADP
ejpam-3352	287	5	a	a	DET
ejpam-3352	287	6	β1i	β1i	NOUN
ejpam-3352	287	7	-	-	ADJ
ejpam-3352	287	8	paracompact	paracompact	NOUN
ejpam-3352	287	9	is	be	AUX
ejpam-3352	287	10	β1aia	β1aia	NOUN
ejpam-3352	287	11	-	-	PUNCT
ejpam-3352	287	12	paracompact	paracompact	NOUN
ejpam-3352	287	13	.	.	PUNCT
ejpam-3352	288	1	if	if	SCONJ
ejpam-3352	288	2	i	i	PRON
ejpam-3352	288	3	=	=	SYM
ejpam-3352	288	4	{	{	PUNCT
ejpam-3352	288	5	∅	∅	NOUN
ejpam-3352	288	6	}	}	PUNCT
ejpam-3352	288	7	in	in	ADP
ejpam-3352	288	8	theorem	theorem	ADJ
ejpam-3352	288	9	9	9	NUM
ejpam-3352	288	10	and	and	CCONJ
ejpam-3352	288	11	theorem	theorem	VERB
ejpam-3352	288	12	10	10	NUM
ejpam-3352	288	13	,	,	PUNCT
ejpam-3352	288	14	then	then	ADV
ejpam-3352	288	15	we	we	PRON
ejpam-3352	288	16	have	have	VERB
ejpam-3352	288	17	the	the	DET
ejpam-3352	288	18	following	follow	VERB
ejpam-3352	288	19	corollary	corollary	NOUN
ejpam-3352	288	20	.	.	PUNCT
ejpam-3352	289	1	corollary	corollary	ADJ
ejpam-3352	289	2	8	8	NUM
ejpam-3352	289	3	.	.	PUNCT
ejpam-3352	290	1	[	[	X
ejpam-3352	290	2	1	1	NUM
ejpam-3352	290	3	,	,	PUNCT
ejpam-3352	290	4	theorem	theorem	VERB
ejpam-3352	290	5	3.5	3.5	NUM
ejpam-3352	290	6	]	]	PUNCT
ejpam-3352	290	7	let	let	VERB
ejpam-3352	290	8	(	(	PUNCT
ejpam-3352	290	9	x	x	NOUN
ejpam-3352	290	10	,	,	PUNCT
ejpam-3352	290	11	τ	τ	X
ejpam-3352	290	12	)	)	PUNCT
ejpam-3352	290	13	be	be	VERB
ejpam-3352	290	14	a	a	DET
ejpam-3352	290	15	β1	β1	NOUN
ejpam-3352	290	16	-	-	PUNCT
ejpam-3352	290	17	paracompact	paracompact	NOUN
ejpam-3352	290	18	space	space	NOUN
ejpam-3352	290	19	.	.	PUNCT
ejpam-3352	291	1	then	then	ADV
ejpam-3352	291	2	:	:	PUNCT
ejpam-3352	291	3	(	(	PUNCT
ejpam-3352	291	4	i	i	NOUN
ejpam-3352	291	5	)	)	PUNCT
ejpam-3352	291	6	if	if	SCONJ
ejpam-3352	291	7	a	a	PRON
ejpam-3352	291	8	is	be	AUX
ejpam-3352	291	9	regular	regular	ADJ
ejpam-3352	291	10	open	open	ADJ
ejpam-3352	291	11	subset	subset	NOUN
ejpam-3352	291	12	of	of	ADP
ejpam-3352	291	13	(	(	PUNCT
ejpam-3352	291	14	x	x	PROPN
ejpam-3352	291	15	,	,	PUNCT
ejpam-3352	291	16	τ	τ	PROPN
ejpam-3352	291	17	)	)	PUNCT
ejpam-3352	291	18	,	,	PUNCT
ejpam-3352	291	19	then	then	ADV
ejpam-3352	291	20	(	(	PUNCT
ejpam-3352	291	21	a	a	DET
ejpam-3352	291	22	,	,	PUNCT
ejpam-3352	291	23	τa	τa	NOUN
ejpam-3352	291	24	)	)	PUNCT
ejpam-3352	291	25	is	be	AUX
ejpam-3352	291	26	β1a	β1a	NOUN
ejpam-3352	291	27	-	-	NOUN
ejpam-3352	291	28	paracompact	paracompact	ADJ
ejpam-3352	291	29	;	;	PUNCT
ejpam-3352	291	30	(	(	PUNCT
ejpam-3352	291	31	ii	ii	NOUN
ejpam-3352	291	32	)	)	PUNCT
ejpam-3352	291	33	if	if	SCONJ
ejpam-3352	291	34	a	a	PRON
ejpam-3352	291	35	is	be	AUX
ejpam-3352	291	36	a	a	DET
ejpam-3352	291	37	βg	βg	ADV
ejpam-3352	291	38	-	-	PUNCT
ejpam-3352	291	39	closed	closed	ADJ
ejpam-3352	291	40	subset	subset	NOUN
ejpam-3352	291	41	of	of	ADP
ejpam-3352	291	42	(	(	PUNCT
ejpam-3352	291	43	x	x	PROPN
ejpam-3352	291	44	,	,	PUNCT
ejpam-3352	291	45	τ	τ	PROPN
ejpam-3352	291	46	)	)	PUNCT
ejpam-3352	291	47	,	,	PUNCT
ejpam-3352	291	48	then	then	ADV
ejpam-3352	291	49	a	a	PRON
ejpam-3352	291	50	is	be	AUX
ejpam-3352	291	51	a	a	DET
ejpam-3352	291	52	β1	β1	NOUN
ejpam-3352	291	53	-	-	PUNCT
ejpam-3352	291	54	paracompact	paracompact	NOUN
ejpam-3352	291	55	.	.	PUNCT
ejpam-3352	292	1	theorem	theorem	NOUN
ejpam-3352	292	2	11	11	NUM
ejpam-3352	292	3	.	.	PUNCT
ejpam-3352	293	1	let	let	VERB
ejpam-3352	293	2	a	a	PRON
ejpam-3352	293	3	and	and	CCONJ
ejpam-3352	293	4	b	b	NOUN
ejpam-3352	293	5	be	be	AUX
ejpam-3352	293	6	subsets	subset	NOUN
ejpam-3352	293	7	of	of	ADP
ejpam-3352	293	8	an	an	DET
ejpam-3352	293	9	ideal	ideal	ADJ
ejpam-3352	293	10	space	space	NOUN
ejpam-3352	293	11	(	(	PUNCT
ejpam-3352	293	12	x	x	X
ejpam-3352	293	13	,	,	PUNCT
ejpam-3352	293	14	τ	τ	PROPN
ejpam-3352	293	15	,	,	PUNCT
ejpam-3352	293	16	i	i	NOUN
ejpam-3352	293	17	)	)	PUNCT
ejpam-3352	293	18	such	such	ADJ
ejpam-3352	293	19	that	that	SCONJ
ejpam-3352	293	20	a	a	DET
ejpam-3352	293	21	⊂	⊂	X
ejpam-3352	293	22	b	b	X
ejpam-3352	293	23	⊂	⊂	PROPN
ejpam-3352	293	24	x.	x.	PROPN
ejpam-3352	294	1	then	then	ADV
ejpam-3352	294	2	the	the	DET
ejpam-3352	294	3	following	follow	VERB
ejpam-3352	294	4	conditions	condition	NOUN
ejpam-3352	294	5	hold	hold	VERB
ejpam-3352	294	6	.	.	PUNCT
ejpam-3352	295	1	(	(	PUNCT
ejpam-3352	295	2	i	i	NOUN
ejpam-3352	295	3	)	)	PUNCT
ejpam-3352	295	4	if	if	SCONJ
ejpam-3352	295	5	a	a	PRON
ejpam-3352	295	6	is	be	AUX
ejpam-3352	295	7	βi	βi	NOUN
ejpam-3352	295	8	-	-	NOUN
ejpam-3352	295	9	paracompact	paracompact	ADJ
ejpam-3352	295	10	and	and	CCONJ
ejpam-3352	295	11	b	b	NOUN
ejpam-3352	295	12	is	be	AUX
ejpam-3352	295	13	β	β	X
ejpam-3352	295	14	-	-	VERB
ejpam-3352	295	15	open	open	ADJ
ejpam-3352	295	16	in	in	ADP
ejpam-3352	295	17	(	(	PUNCT
ejpam-3352	295	18	x	x	NOUN
ejpam-3352	295	19	,	,	PUNCT
ejpam-3352	295	20	τ	τ	PROPN
ejpam-3352	295	21	)	)	PUNCT
ejpam-3352	295	22	,	,	PUNCT
ejpam-3352	295	23	then	then	ADV
ejpam-3352	295	24	a	a	PRON
ejpam-3352	295	25	is	be	AUX
ejpam-3352	295	26	β1bib	β1bib	ADV
ejpam-3352	295	27	-	-	PUNCT
ejpam-3352	295	28	paracompact	paracompact	ADJ
ejpam-3352	295	29	.	.	PUNCT
ejpam-3352	296	1	(	(	PUNCT
ejpam-3352	296	2	ii	ii	NOUN
ejpam-3352	296	3	)	)	PUNCT
ejpam-3352	296	4	if	if	SCONJ
ejpam-3352	296	5	a	a	PRON
ejpam-3352	296	6	is	be	AUX
ejpam-3352	296	7	β1bib	β1bib	NOUN
ejpam-3352	296	8	-	-	PUNCT
ejpam-3352	296	9	paracompact	paracompact	NOUN
ejpam-3352	296	10	and	and	CCONJ
ejpam-3352	296	11	b	b	NOUN
ejpam-3352	296	12	is	be	AUX
ejpam-3352	296	13	open	open	ADJ
ejpam-3352	296	14	in	in	ADP
ejpam-3352	296	15	(	(	PUNCT
ejpam-3352	296	16	x	x	NOUN
ejpam-3352	296	17	,	,	PUNCT
ejpam-3352	296	18	τ	τ	PROPN
ejpam-3352	296	19	)	)	PUNCT
ejpam-3352	296	20	,	,	PUNCT
ejpam-3352	296	21	then	then	ADV
ejpam-3352	296	22	a	a	PRON
ejpam-3352	296	23	is	be	AUX
ejpam-3352	296	24	β1i	β1i	NOUN
ejpam-3352	296	25	-	-	ADJ
ejpam-3352	296	26	paracompact	paracompact	ADJ
ejpam-3352	296	27	.	.	PUNCT
ejpam-3352	297	1	a.	a.	NOUN
ejpam-3352	297	2	qahis	qahis	PROPN
ejpam-3352	297	3	/	/	SYM
ejpam-3352	297	4	eur	eur	PROPN
ejpam-3352	297	5	.	.	PUNCT
ejpam-3352	298	1	j.	j.	PROPN
ejpam-3352	298	2	pure	pure	PROPN
ejpam-3352	298	3	appl	appl	PROPN
ejpam-3352	298	4	.	.	PROPN
ejpam-3352	298	5	math	math	PROPN
ejpam-3352	298	6	,	,	PUNCT
ejpam-3352	298	7	12	12	NUM
ejpam-3352	298	8	(	(	PUNCT
ejpam-3352	298	9	1	1	NUM
ejpam-3352	298	10	)	)	PUNCT
ejpam-3352	298	11	(	(	PUNCT
ejpam-3352	298	12	2019	2019	NUM
ejpam-3352	298	13	)	)	PUNCT
ejpam-3352	298	14	,	,	PUNCT
ejpam-3352	298	15	135	135	NUM
ejpam-3352	298	16	-	-	SYM
ejpam-3352	298	17	145	145	NUM
ejpam-3352	298	18	142	142	NUM
ejpam-3352	298	19	proof	proof	NOUN
ejpam-3352	298	20	.	.	PUNCT
ejpam-3352	299	1	(	(	PUNCT
ejpam-3352	299	2	i	i	NOUN
ejpam-3352	299	3	)	)	PUNCT
ejpam-3352	299	4	let	let	VERB
ejpam-3352	299	5	u	u	PRON
ejpam-3352	299	6	=	=	X
ejpam-3352	299	7	{	{	PUNCT
ejpam-3352	299	8	uα	uα	X
ejpam-3352	299	9	:	:	PUNCT
ejpam-3352	299	10	α	α	PROPN
ejpam-3352	299	11	∈	∈	PROPN
ejpam-3352	299	12	∆	∆	PROPN
ejpam-3352	299	13	}	}	PUNCT
ejpam-3352	299	14	be	be	AUX
ejpam-3352	299	15	a	a	DET
ejpam-3352	299	16	cover	cover	NOUN
ejpam-3352	299	17	of	of	ADP
ejpam-3352	299	18	a	a	DET
ejpam-3352	299	19	such	such	ADJ
ejpam-3352	299	20	that	that	SCONJ
ejpam-3352	299	21	uα	uα	PROPN
ejpam-3352	299	22	∈	∈	PROPN
ejpam-3352	299	23	βo(b	βo(b	NOUN
ejpam-3352	299	24	,	,	PUNCT
ejpam-3352	299	25	τb	τb	ADJ
ejpam-3352	299	26	)	)	PUNCT
ejpam-3352	299	27	.	.	PUNCT
ejpam-3352	300	1	since	since	SCONJ
ejpam-3352	300	2	b	b	PROPN
ejpam-3352	300	3	∈	∈	PROPN
ejpam-3352	300	4	βo(x	βo(x	PUNCT
ejpam-3352	300	5	,	,	PUNCT
ejpam-3352	300	6	τ	τ	PROPN
ejpam-3352	300	7	)	)	PUNCT
ejpam-3352	300	8	,	,	PUNCT
ejpam-3352	300	9	u	u	NOUN
ejpam-3352	300	10	is	be	AUX
ejpam-3352	300	11	a	a	DET
ejpam-3352	300	12	β	β	NOUN
ejpam-3352	300	13	-	-	ADJ
ejpam-3352	300	14	open	open	ADJ
ejpam-3352	300	15	cover	cover	NOUN
ejpam-3352	300	16	of	of	ADP
ejpam-3352	300	17	a	a	DET
ejpam-3352	300	18	in	in	ADP
ejpam-3352	300	19	(	(	PUNCT
ejpam-3352	300	20	x	x	NOUN
ejpam-3352	300	21	,	,	PUNCT
ejpam-3352	300	22	τ	τ	PROPN
ejpam-3352	300	23	)	)	PUNCT
ejpam-3352	300	24	,	,	PUNCT
ejpam-3352	300	25	by	by	ADP
ejpam-3352	300	26	theorem	theorem	NOUN
ejpam-3352	300	27	1	1	NUM
ejpam-3352	300	28	.	.	PUNCT
ejpam-3352	300	29	by	by	ADP
ejpam-3352	300	30	hypothesis	hypothesis	NOUN
ejpam-3352	300	31	,	,	PUNCT
ejpam-3352	300	32	there	there	PRON
ejpam-3352	300	33	exists	exist	VERB
ejpam-3352	300	34	a	a	DET
ejpam-3352	300	35	locally	locally	ADV
ejpam-3352	300	36	finite	finite	ADJ
ejpam-3352	300	37	open	open	ADJ
ejpam-3352	300	38	family	family	NOUN
ejpam-3352	300	39	v	v	NOUN
ejpam-3352	300	40	=	=	PUNCT
ejpam-3352	300	41	{	{	PUNCT
ejpam-3352	300	42	vλ	vλ	INTJ
ejpam-3352	300	43	:	:	PUNCT
ejpam-3352	300	44	λ	λ	PROPN
ejpam-3352	300	45	∈	∈	PROPN
ejpam-3352	300	46	λ	λ	PROPN
ejpam-3352	300	47	}	}	PUNCT
ejpam-3352	300	48	refines	refine	VERB
ejpam-3352	300	49	u	u	PRON
ejpam-3352	300	50	such	such	ADJ
ejpam-3352	300	51	that	that	SCONJ
ejpam-3352	300	52	a	a	DET
ejpam-3352	300	53	\∪{vλ	\∪{vλ	NOUN
ejpam-3352	300	54	:	:	PUNCT
ejpam-3352	301	1	λ	λ	X
ejpam-3352	301	2	∈	∈	PROPN
ejpam-3352	301	3	λ	λ	NOUN
ejpam-3352	301	4	}	}	PUNCT
ejpam-3352	301	5	=	=	PUNCT
ejpam-3352	301	6	i	i	PRON
ejpam-3352	301	7	∈	∈	PROPN
ejpam-3352	301	8	i.	i.	NOUN
ejpam-3352	301	9	then	then	ADV
ejpam-3352	301	10	a	a	DET
ejpam-3352	301	11	⊆	⊆	NUM
ejpam-3352	301	12	(	(	PUNCT
ejpam-3352	301	13	∪{vλ	∪{vλ	NUM
ejpam-3352	301	14	:	:	PUNCT
ejpam-3352	301	15	λ	λ	X
ejpam-3352	301	16	∈	∈	NOUN
ejpam-3352	301	17	λ})∪	λ})∪	PROPN
ejpam-3352	301	18	i	i	PRON
ejpam-3352	301	19	and	and	CCONJ
ejpam-3352	301	20	a	a	DET
ejpam-3352	301	21	=	=	X
ejpam-3352	301	22	a∩b	a∩b	PROPN
ejpam-3352	301	23	⊆	⊆	NUM
ejpam-3352	301	24	[	[	X
ejpam-3352	301	25	∪{vλ	∪{vλ	NUM
ejpam-3352	301	26	:	:	PUNCT
ejpam-3352	301	27	λ	λ	X
ejpam-3352	301	28	∈	∈	PROPN
ejpam-3352	301	29	λ}∪	λ}∪	X
ejpam-3352	302	1	i]∩b	i]∩b	PROPN
ejpam-3352	302	2	=	=	SYM
ejpam-3352	302	3	∪{vλ	∪{vλ	PROPN
ejpam-3352	302	4	∩b	∩b	NOUN
ejpam-3352	302	5	:	:	PUNCT
ejpam-3352	302	6	λ	λ	X
ejpam-3352	302	7	∈	∈	PROPN
ejpam-3352	302	8	λ}∪	λ}∪	X
ejpam-3352	303	1	(	(	PUNCT
ejpam-3352	303	2	i	i	PRON
ejpam-3352	303	3	∩b	∩b	NOUN
ejpam-3352	303	4	)	)	PUNCT
ejpam-3352	303	5	implies	imply	VERB
ejpam-3352	303	6	a\∪{vλ∩b	a\∪{vλ∩b	NOUN
ejpam-3352	303	7	:	:	PUNCT
ejpam-3352	303	8	λ	λ	X
ejpam-3352	303	9	∈	∈	PROPN
ejpam-3352	303	10	λ	λ	PROPN
ejpam-3352	303	11	}	}	PUNCT
ejpam-3352	303	12	∈	∈	PROPN
ejpam-3352	303	13	ib	ib	NOUN
ejpam-3352	303	14	.	.	PUNCT
ejpam-3352	304	1	let	let	VERB
ejpam-3352	304	2	x	x	SYM
ejpam-3352	304	3	∈	∈	PROPN
ejpam-3352	304	4	b.	b.	PROPN
ejpam-3352	305	1	since	since	SCONJ
ejpam-3352	305	2	v	v	NOUN
ejpam-3352	305	3	is	be	AUX
ejpam-3352	305	4	locally	locally	ADV
ejpam-3352	305	5	finite	finite	ADJ
ejpam-3352	305	6	,	,	PUNCT
ejpam-3352	305	7	there	there	PRON
ejpam-3352	305	8	exits	exit	VERB
ejpam-3352	305	9	u	u	NOUN
ejpam-3352	305	10	∈	∈	PROPN
ejpam-3352	305	11	τ(x	τ(x	NOUN
ejpam-3352	305	12	)	)	PUNCT
ejpam-3352	305	13	such	such	ADJ
ejpam-3352	305	14	that	that	SCONJ
ejpam-3352	305	15	u	u	PROPN
ejpam-3352	305	16	∩	∩	NOUN
ejpam-3352	305	17	vλ	vλ	ADP
ejpam-3352	305	18	=	=	NOUN
ejpam-3352	305	19	∅	∅	NOUN
ejpam-3352	305	20	for	for	ADP
ejpam-3352	305	21	λ	λ	PROPN
ejpam-3352	305	22	6=	6=	SYM
ejpam-3352	305	23	λ1	λ1	ADJ
ejpam-3352	305	24	,	,	PUNCT
ejpam-3352	305	25	λ2	λ2	NOUN
ejpam-3352	305	26	,	,	PUNCT
ejpam-3352	305	27	...	...	PUNCT
ejpam-3352	305	28	,	,	PUNCT
ejpam-3352	305	29	λn	λn	NOUN
ejpam-3352	305	30	.	.	PUNCT
ejpam-3352	306	1	this	this	PRON
ejpam-3352	306	2	implies	imply	VERB
ejpam-3352	306	3	(	(	PUNCT
ejpam-3352	306	4	u	u	NOUN
ejpam-3352	306	5	∩	∩	NOUN
ejpam-3352	306	6	vλ	vλ	NOUN
ejpam-3352	306	7	)	)	PUNCT
ejpam-3352	306	8	∩b	∩b	NOUN
ejpam-3352	306	9	=	=	NOUN
ejpam-3352	306	10	∅	∅	NOUN
ejpam-3352	306	11	for	for	ADP
ejpam-3352	306	12	λ	λ	PROPN
ejpam-3352	306	13	6=	6=	SYM
ejpam-3352	306	14	λ1	λ1	ADJ
ejpam-3352	306	15	,	,	PUNCT
ejpam-3352	306	16	λ2	λ2	NOUN
ejpam-3352	306	17	,	,	PUNCT
ejpam-3352	306	18	...	...	PUNCT
ejpam-3352	306	19	,	,	PUNCT
ejpam-3352	306	20	λn	λn	X
ejpam-3352	306	21	which	which	PRON
ejpam-3352	306	22	implies	imply	VERB
ejpam-3352	306	23	(	(	PUNCT
ejpam-3352	306	24	u	u	NOUN
ejpam-3352	306	25	∩b)∩	∩b)∩	PUNCT
ejpam-3352	306	26	(	(	PUNCT
ejpam-3352	306	27	vλ∩b	vλ∩b	NOUN
ejpam-3352	306	28	)	)	PUNCT
ejpam-3352	306	29	=	=	NOUN
ejpam-3352	306	30	∅	∅	NOUN
ejpam-3352	306	31	for	for	ADP
ejpam-3352	306	32	λ	λ	PROPN
ejpam-3352	306	33	6=	6=	SYM
ejpam-3352	306	34	λ1	λ1	ADJ
ejpam-3352	306	35	,	,	PUNCT
ejpam-3352	306	36	λ2	λ2	NOUN
ejpam-3352	306	37	,	,	PUNCT
ejpam-3352	306	38	...	...	PUNCT
ejpam-3352	306	39	,	,	PUNCT
ejpam-3352	306	40	λn	λn	PROPN
ejpam-3352	306	41	.	.	PUNCT
ejpam-3352	307	1	therefore	therefore	ADV
ejpam-3352	307	2	,	,	PUNCT
ejpam-3352	307	3	the	the	DET
ejpam-3352	307	4	family	family	NOUN
ejpam-3352	307	5	vb	vb	NOUN
ejpam-3352	307	6	=	=	PUNCT
ejpam-3352	307	7	{	{	PUNCT
ejpam-3352	307	8	vλ	vλ	INTJ
ejpam-3352	307	9	∩b	∩b	NOUN
ejpam-3352	307	10	:	:	PUNCT
ejpam-3352	307	11	λ	λ	X
ejpam-3352	307	12	∈	∈	PROPN
ejpam-3352	307	13	λ	λ	PROPN
ejpam-3352	307	14	}	}	PUNCT
ejpam-3352	307	15	is	be	AUX
ejpam-3352	307	16	τb	τb	NOUN
ejpam-3352	307	17	-	-	PUNCT
ejpam-3352	307	18	locally	locally	ADV
ejpam-3352	307	19	finite	finite	NOUN
ejpam-3352	307	20	τb	τb	NOUN
ejpam-3352	307	21	-	-	PUNCT
ejpam-3352	307	22	open	open	ADJ
ejpam-3352	307	23	.	.	PUNCT
ejpam-3352	308	1	let	let	VERB
ejpam-3352	308	2	vλ	vλ	ADP
ejpam-3352	308	3	∩b	∩b	NOUN
ejpam-3352	308	4	∈	∈	PROPN
ejpam-3352	308	5	vb	vb	NOUN
ejpam-3352	308	6	.	.	PUNCT
ejpam-3352	309	1	since	since	SCONJ
ejpam-3352	309	2	v	v	NOUN
ejpam-3352	309	3	refines	refine	VERB
ejpam-3352	309	4	u	u	NOUN
ejpam-3352	309	5	there	there	PRON
ejpam-3352	309	6	exits	exit	VERB
ejpam-3352	309	7	uα	uα	PROPN
ejpam-3352	309	8	∈	∈	PROPN
ejpam-3352	309	9	u	u	NOUN
ejpam-3352	309	10	such	such	ADJ
ejpam-3352	309	11	that	that	SCONJ
ejpam-3352	309	12	vλ	vλ	VERB
ejpam-3352	309	13	⊂	⊂	PRON
ejpam-3352	309	14	uα	uα	PROPN
ejpam-3352	310	1	and	and	CCONJ
ejpam-3352	310	2	so	so	ADV
ejpam-3352	310	3	vλ	vλ	INTJ
ejpam-3352	310	4	∩b	∩b	NOUN
ejpam-3352	310	5	⊂	⊂	PROPN
ejpam-3352	310	6	uα	uα	PROPN
ejpam-3352	310	7	.	.	PUNCT
ejpam-3352	311	1	hence	hence	ADV
ejpam-3352	311	2	va	va	PROPN
ejpam-3352	311	3	refines	refine	VERB
ejpam-3352	311	4	u	u	PRON
ejpam-3352	311	5	.	.	PUNCT
ejpam-3352	312	1	therefore	therefore	ADV
ejpam-3352	312	2	a	a	PRON
ejpam-3352	312	3	is	be	AUX
ejpam-3352	312	4	β1bib	β1bib	ADV
ejpam-3352	312	5	-	-	PUNCT
ejpam-3352	312	6	paracompact	paracompact	ADJ
ejpam-3352	312	7	.	.	PUNCT
ejpam-3352	313	1	(	(	PUNCT
ejpam-3352	313	2	ii	ii	NOUN
ejpam-3352	313	3	)	)	PUNCT
ejpam-3352	313	4	let	let	VERB
ejpam-3352	313	5	u	u	PRON
ejpam-3352	313	6	=	=	X
ejpam-3352	313	7	{	{	PUNCT
ejpam-3352	313	8	uα	uα	X
ejpam-3352	313	9	:	:	PUNCT
ejpam-3352	313	10	α	α	PROPN
ejpam-3352	313	11	∈	∈	PROPN
ejpam-3352	313	12	∆	∆	PROPN
ejpam-3352	313	13	}	}	PUNCT
ejpam-3352	313	14	be	be	AUX
ejpam-3352	313	15	a	a	DET
ejpam-3352	313	16	cover	cover	NOUN
ejpam-3352	313	17	of	of	ADP
ejpam-3352	313	18	a	a	PRON
ejpam-3352	313	19	by	by	ADP
ejpam-3352	313	20	β	β	NOUN
ejpam-3352	313	21	-	-	ADJ
ejpam-3352	313	22	open	open	ADJ
ejpam-3352	313	23	subsets	subset	NOUN
ejpam-3352	313	24	of	of	ADP
ejpam-3352	313	25	x.	x.	NOUN
ejpam-3352	313	26	then	then	ADV
ejpam-3352	313	27	the	the	DET
ejpam-3352	313	28	family	family	NOUN
ejpam-3352	313	29	u1	u1	NOUN
ejpam-3352	313	30	=	=	SYM
ejpam-3352	313	31	{	{	PUNCT
ejpam-3352	313	32	b	b	PROPN
ejpam-3352	313	33	∩	∩	ADJ
ejpam-3352	313	34	uα	uα	X
ejpam-3352	313	35	:	:	PUNCT
ejpam-3352	313	36	α	α	PROPN
ejpam-3352	313	37	∈	∈	PROPN
ejpam-3352	313	38	∆	∆	X
ejpam-3352	313	39	}	}	PUNCT
ejpam-3352	313	40	is	be	AUX
ejpam-3352	313	41	a	a	DET
ejpam-3352	313	42	β	β	NOUN
ejpam-3352	313	43	-	-	ADJ
ejpam-3352	313	44	open	open	ADJ
ejpam-3352	313	45	cover	cover	NOUN
ejpam-3352	313	46	of	of	ADP
ejpam-3352	313	47	a	a	DET
ejpam-3352	313	48	in	in	ADP
ejpam-3352	313	49	(	(	PUNCT
ejpam-3352	313	50	b	b	NOUN
ejpam-3352	313	51	,	,	PUNCT
ejpam-3352	313	52	τb	τb	ADJ
ejpam-3352	313	53	,	,	PUNCT
ejpam-3352	313	54	ib	ib	NOUN
ejpam-3352	313	55	)	)	PUNCT
ejpam-3352	313	56	.	.	PUNCT
ejpam-3352	314	1	by	by	ADP
ejpam-3352	314	2	hypothesis	hypothesis	NOUN
ejpam-3352	314	3	,	,	PUNCT
ejpam-3352	314	4	exists	exist	VERB
ejpam-3352	314	5	τblocally	τblocally	ADV
ejpam-3352	314	6	finite	finite	ADJ
ejpam-3352	314	7	τb	τb	ADJ
ejpam-3352	314	8	-	-	PUNCT
ejpam-3352	314	9	open	open	ADJ
ejpam-3352	314	10	family	family	NOUN
ejpam-3352	314	11	v	v	NOUN
ejpam-3352	314	12	=	=	PUNCT
ejpam-3352	314	13	{	{	PUNCT
ejpam-3352	314	14	vλ	vλ	INTJ
ejpam-3352	314	15	:	:	PUNCT
ejpam-3352	314	16	λ	λ	PROPN
ejpam-3352	314	17	∈	∈	PROPN
ejpam-3352	314	18	λ	λ	PROPN
ejpam-3352	314	19	}	}	PUNCT
ejpam-3352	314	20	refines	refine	NOUN
ejpam-3352	314	21	u1	u1	VERB
ejpam-3352	314	22	such	such	ADJ
ejpam-3352	314	23	that	that	DET
ejpam-3352	314	24	a\∪{vλ	a\∪{vλ	NOUN
ejpam-3352	314	25	:	:	PUNCT
ejpam-3352	314	26	λ	λ	X
ejpam-3352	314	27	∈	∈	PROPN
ejpam-3352	314	28	λ	λ	PROPN
ejpam-3352	314	29	}	}	PUNCT
ejpam-3352	314	30	∈	∈	PROPN
ejpam-3352	314	31	ib	ib	NOUN
ejpam-3352	314	32	,	,	PUNCT
ejpam-3352	314	33	where	where	SCONJ
ejpam-3352	314	34	ib	ib	NOUN
ejpam-3352	314	35	=	=	NOUN
ejpam-3352	314	36	i	i	PRON
ejpam-3352	314	37	∩b	∩b	VERB
ejpam-3352	314	38	.	.	PUNCT
ejpam-3352	315	1	it	it	PRON
ejpam-3352	315	2	follows	follow	VERB
ejpam-3352	315	3	that	that	SCONJ
ejpam-3352	315	4	a	a	DET
ejpam-3352	315	5	\	\	PROPN
ejpam-3352	315	6	∪{vλ	∪{vλ	PROPN
ejpam-3352	315	7	:	:	PUNCT
ejpam-3352	315	8	λ	λ	PROPN
ejpam-3352	315	9	∈	∈	PROPN
ejpam-3352	315	10	λ	λ	PROPN
ejpam-3352	315	11	}	}	PUNCT
ejpam-3352	315	12	∈	∈	PROPN
ejpam-3352	315	13	i.	i.	NOUN
ejpam-3352	315	14	since	since	SCONJ
ejpam-3352	315	15	b	b	PROPN
ejpam-3352	315	16	is	be	AUX
ejpam-3352	315	17	open	open	ADJ
ejpam-3352	315	18	in	in	ADP
ejpam-3352	315	19	x.	x.	NOUN
ejpam-3352	315	20	then	then	ADV
ejpam-3352	315	21	by	by	ADP
ejpam-3352	315	22	theorem	theorem	NOUN
ejpam-3352	315	23	1	1	NUM
ejpam-3352	315	24	,	,	PUNCT
ejpam-3352	315	25	v	v	NOUN
ejpam-3352	315	26	is	be	AUX
ejpam-3352	315	27	a	a	DET
ejpam-3352	315	28	locally	locally	ADV
ejpam-3352	315	29	finite	finite	ADJ
ejpam-3352	315	30	open	open	ADJ
ejpam-3352	315	31	refinement	refinement	NOUN
ejpam-3352	315	32	of	of	ADP
ejpam-3352	315	33	u	u	PROPN
ejpam-3352	315	34	.	.	PUNCT
ejpam-3352	316	1	therefore	therefore	ADV
ejpam-3352	316	2	,	,	PUNCT
ejpam-3352	316	3	a	a	PRON
ejpam-3352	316	4	is	be	AUX
ejpam-3352	316	5	β1i	β1i	NOUN
ejpam-3352	316	6	-	-	ADJ
ejpam-3352	316	7	paracompact	paracompact	ADJ
ejpam-3352	316	8	.	.	PUNCT
ejpam-3352	317	1	if	if	SCONJ
ejpam-3352	317	2	i	i	PRON
ejpam-3352	317	3	=	=	SYM
ejpam-3352	317	4	{	{	PUNCT
ejpam-3352	317	5	∅	∅	NOUN
ejpam-3352	317	6	}	}	PUNCT
ejpam-3352	317	7	in	in	ADP
ejpam-3352	317	8	theorem	theorem	NOUN
ejpam-3352	317	9	11	11	NUM
ejpam-3352	317	10	,	,	PUNCT
ejpam-3352	317	11	then	then	ADV
ejpam-3352	317	12	we	we	PRON
ejpam-3352	317	13	have	have	VERB
ejpam-3352	317	14	the	the	DET
ejpam-3352	317	15	following	follow	VERB
ejpam-3352	317	16	corollary	corollary	NOUN
ejpam-3352	317	17	.	.	PUNCT
ejpam-3352	318	1	corollary	corollary	ADJ
ejpam-3352	318	2	9	9	NUM
ejpam-3352	318	3	.	.	PUNCT
ejpam-3352	319	1	[	[	X
ejpam-3352	319	2	1	1	NUM
ejpam-3352	319	3	,	,	PUNCT
ejpam-3352	319	4	theorem	theorem	VERB
ejpam-3352	319	5	3.6	3.6	NUM
ejpam-3352	319	6	]	]	PUNCT
ejpam-3352	319	7	let	let	VERB
ejpam-3352	319	8	a	a	PRON
ejpam-3352	319	9	and	and	CCONJ
ejpam-3352	319	10	b	b	NOUN
ejpam-3352	319	11	be	be	AUX
ejpam-3352	319	12	subsets	subset	NOUN
ejpam-3352	319	13	of	of	ADP
ejpam-3352	319	14	an	an	DET
ejpam-3352	319	15	ideal	ideal	ADJ
ejpam-3352	319	16	space	space	NOUN
ejpam-3352	319	17	(	(	PUNCT
ejpam-3352	319	18	x	x	X
ejpam-3352	319	19	,	,	PUNCT
ejpam-3352	319	20	τ	τ	X
ejpam-3352	319	21	)	)	PUNCT
ejpam-3352	319	22	such	such	ADJ
ejpam-3352	319	23	that	that	SCONJ
ejpam-3352	319	24	a	a	DET
ejpam-3352	319	25	⊂	⊂	X
ejpam-3352	319	26	b	b	X
ejpam-3352	319	27	⊂	⊂	PROPN
ejpam-3352	319	28	x.	x.	PROPN
ejpam-3352	320	1	then	then	ADV
ejpam-3352	320	2	:	:	PUNCT
ejpam-3352	320	3	(	(	PUNCT
ejpam-3352	320	4	i	i	NOUN
ejpam-3352	320	5	)	)	PUNCT
ejpam-3352	320	6	if	if	SCONJ
ejpam-3352	320	7	a	a	PRON
ejpam-3352	320	8	is	be	AUX
ejpam-3352	320	9	β	β	NOUN
ejpam-3352	320	10	-	-	ADJ
ejpam-3352	320	11	paracompact	paracompact	ADJ
ejpam-3352	320	12	and	and	CCONJ
ejpam-3352	320	13	b	b	NOUN
ejpam-3352	320	14	is	be	AUX
ejpam-3352	320	15	β	β	X
ejpam-3352	320	16	-	-	VERB
ejpam-3352	320	17	open	open	ADJ
ejpam-3352	320	18	in	in	ADP
ejpam-3352	320	19	(	(	PUNCT
ejpam-3352	320	20	x	x	NOUN
ejpam-3352	320	21	,	,	PUNCT
ejpam-3352	320	22	τ	τ	PROPN
ejpam-3352	320	23	)	)	PUNCT
ejpam-3352	320	24	,	,	PUNCT
ejpam-3352	320	25	then	then	ADV
ejpam-3352	320	26	a	a	PRON
ejpam-3352	320	27	is	be	AUX
ejpam-3352	320	28	β1b	β1b	NOUN
ejpam-3352	320	29	-	-	PUNCT
ejpam-3352	320	30	paracompact	paracompact	ADJ
ejpam-3352	320	31	.	.	PUNCT
ejpam-3352	321	1	(	(	PUNCT
ejpam-3352	321	2	ii	ii	NOUN
ejpam-3352	321	3	)	)	PUNCT
ejpam-3352	321	4	if	if	SCONJ
ejpam-3352	321	5	a	a	PRON
ejpam-3352	321	6	is	be	AUX
ejpam-3352	321	7	β1b	β1b	NOUN
ejpam-3352	321	8	-	-	PUNCT
ejpam-3352	321	9	paracompact	paracompact	NOUN
ejpam-3352	321	10	and	and	CCONJ
ejpam-3352	321	11	b	b	NOUN
ejpam-3352	321	12	is	be	AUX
ejpam-3352	321	13	open	open	ADJ
ejpam-3352	321	14	in	in	ADP
ejpam-3352	321	15	(	(	PUNCT
ejpam-3352	321	16	x	x	NOUN
ejpam-3352	321	17	,	,	PUNCT
ejpam-3352	321	18	τ	τ	PROPN
ejpam-3352	321	19	)	)	PUNCT
ejpam-3352	321	20	,	,	PUNCT
ejpam-3352	321	21	then	then	ADV
ejpam-3352	321	22	a	a	PRON
ejpam-3352	321	23	is	be	AUX
ejpam-3352	321	24	β1	β1	NOUN
ejpam-3352	321	25	-	-	PUNCT
ejpam-3352	321	26	paracompact	paracompact	NOUN
ejpam-3352	321	27	.	.	PUNCT
ejpam-3352	322	1	theorem	theorem	NOUN
ejpam-3352	322	2	12	12	NUM
ejpam-3352	322	3	.	.	PUNCT
ejpam-3352	323	1	let	let	VERB
ejpam-3352	323	2	a	a	PRON
ejpam-3352	323	3	be	be	AUX
ejpam-3352	323	4	a	a	DET
ejpam-3352	323	5	clopen	clopen	ADJ
ejpam-3352	323	6	subspace	subspace	NOUN
ejpam-3352	323	7	of	of	ADP
ejpam-3352	323	8	an	an	DET
ejpam-3352	323	9	ideal	ideal	ADJ
ejpam-3352	323	10	space	space	NOUN
ejpam-3352	323	11	(	(	PUNCT
ejpam-3352	323	12	x	x	X
ejpam-3352	323	13	,	,	PUNCT
ejpam-3352	323	14	τ	τ	PROPN
ejpam-3352	323	15	,	,	PUNCT
ejpam-3352	323	16	i	i	PROPN
ejpam-3352	323	17	)	)	PUNCT
ejpam-3352	323	18	.	.	PUNCT
ejpam-3352	324	1	then	then	ADV
ejpam-3352	324	2	a	a	DET
ejpam-3352	324	3	a	a	PRON
ejpam-3352	324	4	is	be	AUX
ejpam-3352	324	5	β1aiaparacompact	β1aiaparacompact	ADJ
ejpam-3352	324	6	if	if	SCONJ
ejpam-3352	324	7	and	and	CCONJ
ejpam-3352	324	8	only	only	ADV
ejpam-3352	324	9	if	if	SCONJ
ejpam-3352	324	10	it	it	PRON
ejpam-3352	324	11	is	be	AUX
ejpam-3352	324	12	β1i	β1i	ADJ
ejpam-3352	324	13	-	-	ADJ
ejpam-3352	324	14	paracompact	paracompact	ADJ
ejpam-3352	324	15	.	.	PUNCT
ejpam-3352	325	1	proof	proof	NOUN
ejpam-3352	325	2	.	.	PUNCT
ejpam-3352	326	1	to	to	PART
ejpam-3352	326	2	prove	prove	VERB
ejpam-3352	326	3	necessity	necessity	NOUN
ejpam-3352	326	4	,	,	PUNCT
ejpam-3352	326	5	let	let	VERB
ejpam-3352	326	6	u	u	PRON
ejpam-3352	326	7	=	=	X
ejpam-3352	326	8	{	{	PUNCT
ejpam-3352	326	9	uα	uα	X
ejpam-3352	326	10	:	:	PUNCT
ejpam-3352	326	11	α	α	PROPN
ejpam-3352	326	12	∈	∈	PROPN
ejpam-3352	326	13	∆	∆	PROPN
ejpam-3352	326	14	}	}	PUNCT
ejpam-3352	326	15	be	be	AUX
ejpam-3352	326	16	a	a	DET
ejpam-3352	326	17	cover	cover	NOUN
ejpam-3352	326	18	of	of	ADP
ejpam-3352	326	19	a	a	PRON
ejpam-3352	326	20	by	by	ADP
ejpam-3352	326	21	β	β	NOUN
ejpam-3352	326	22	-	-	ADJ
ejpam-3352	326	23	open	open	ADJ
ejpam-3352	326	24	subsets	subset	NOUN
ejpam-3352	326	25	of	of	ADP
ejpam-3352	326	26	the	the	DET
ejpam-3352	326	27	ideal	ideal	ADJ
ejpam-3352	326	28	subspace	subspace	NOUN
ejpam-3352	326	29	(	(	PUNCT
ejpam-3352	326	30	a	a	DET
ejpam-3352	326	31	,	,	PUNCT
ejpam-3352	326	32	τa	τa	PROPN
ejpam-3352	326	33	,	,	PUNCT
ejpam-3352	326	34	ia	ia	PROPN
ejpam-3352	326	35	)	)	PUNCT
ejpam-3352	326	36	.	.	PUNCT
ejpam-3352	327	1	since	since	SCONJ
ejpam-3352	327	2	a	a	PRON
ejpam-3352	327	3	is	be	AUX
ejpam-3352	327	4	open	open	ADJ
ejpam-3352	327	5	,	,	PUNCT
ejpam-3352	327	6	u	u	NOUN
ejpam-3352	327	7	is	be	AUX
ejpam-3352	327	8	a	a	DET
ejpam-3352	327	9	cover	cover	NOUN
ejpam-3352	327	10	of	of	ADP
ejpam-3352	327	11	a	a	PRON
ejpam-3352	327	12	by	by	ADP
ejpam-3352	327	13	β	β	NOUN
ejpam-3352	327	14	-	-	ADJ
ejpam-3352	327	15	open	open	ADJ
ejpam-3352	327	16	subsets	subset	NOUN
ejpam-3352	327	17	of	of	ADP
ejpam-3352	327	18	x	x	PUNCT
ejpam-3352	328	1	and	and	CCONJ
ejpam-3352	328	2	so	so	ADV
ejpam-3352	328	3	it	it	PRON
ejpam-3352	328	4	has	have	VERB
ejpam-3352	328	5	a	a	DET
ejpam-3352	328	6	locally	locally	ADV
ejpam-3352	328	7	finite	finite	ADJ
ejpam-3352	328	8	open	open	ADJ
ejpam-3352	328	9	refinement	refinement	NOUN
ejpam-3352	328	10	,	,	PUNCT
ejpam-3352	328	11	say	say	VERB
ejpam-3352	328	12	v	v	NOUN
ejpam-3352	328	13	=	=	PUNCT
ejpam-3352	328	14	{	{	PUNCT
ejpam-3352	328	15	vλ	vλ	INTJ
ejpam-3352	328	16	:	:	PUNCT
ejpam-3352	328	17	λ	λ	PROPN
ejpam-3352	328	18	∈	∈	PROPN
ejpam-3352	328	19	λ	λ	NOUN
ejpam-3352	328	20	}	}	PUNCT
ejpam-3352	328	21	such	such	ADJ
ejpam-3352	328	22	that	that	SCONJ
ejpam-3352	328	23	a	a	DET
ejpam-3352	328	24	\	\	PROPN
ejpam-3352	328	25	∪{vλ	∪{vλ	PROPN
ejpam-3352	328	26	:	:	PUNCT
ejpam-3352	328	27	λ	λ	PROPN
ejpam-3352	328	28	∈	∈	PROPN
ejpam-3352	328	29	λ	λ	NOUN
ejpam-3352	328	30	}	}	PUNCT
ejpam-3352	328	31	=	=	PUNCT
ejpam-3352	328	32	i	i	PRON
ejpam-3352	328	33	∈	∈	PROPN
ejpam-3352	328	34	i.	i.	NOUN
ejpam-3352	328	35	then	then	ADV
ejpam-3352	328	36	a	a	DET
ejpam-3352	328	37	⊆	⊆	NUM
ejpam-3352	328	38	(	(	PUNCT
ejpam-3352	328	39	∪{vλ	∪{vλ	NUM
ejpam-3352	328	40	:	:	PUNCT
ejpam-3352	328	41	λ	λ	PROPN
ejpam-3352	328	42	∈	∈	PROPN
ejpam-3352	328	43	λ	λ	NOUN
ejpam-3352	328	44	}	}	PUNCT
ejpam-3352	328	45	)	)	PUNCT
ejpam-3352	328	46	∪	∪	ADP
ejpam-3352	328	47	i.	i.	PROPN
ejpam-3352	328	48	now	now	ADV
ejpam-3352	328	49	a	a	DET
ejpam-3352	328	50	⊆	⊆	NUM
ejpam-3352	328	51	a	a	DET
ejpam-3352	328	52	∩	∩	NOUN
ejpam-3352	328	53	[	[	X
ejpam-3352	328	54	(	(	PUNCT
ejpam-3352	328	55	∪{vλ	∪{vλ	NUM
ejpam-3352	328	56	:	:	PUNCT
ejpam-3352	328	57	λ	λ	PROPN
ejpam-3352	328	58	∈	∈	PROPN
ejpam-3352	328	59	λ	λ	NOUN
ejpam-3352	328	60	}	}	PUNCT
ejpam-3352	328	61	)	)	PUNCT
ejpam-3352	328	62	∪	∪	ADP
ejpam-3352	328	63	i	i	PRON
ejpam-3352	328	64	]	]	X
ejpam-3352	328	65	=	=	SYM
ejpam-3352	328	66	∪{vλ	∪{vλ	X
ejpam-3352	328	67	∩	∩	X
ejpam-3352	328	68	a	a	X
ejpam-3352	328	69	:	:	PUNCT
ejpam-3352	328	70	λ	λ	X
ejpam-3352	328	71	∈	∈	PROPN
ejpam-3352	328	72	λ	λ	PROPN
ejpam-3352	328	73	}	}	PUNCT
ejpam-3352	328	74	∪	∪	NOUN
ejpam-3352	328	75	(	(	PUNCT
ejpam-3352	328	76	a	a	DET
ejpam-3352	328	77	∩	∩	ADJ
ejpam-3352	328	78	i	i	NOUN
ejpam-3352	328	79	)	)	PUNCT
ejpam-3352	328	80	.	.	PUNCT
ejpam-3352	329	1	it	it	PRON
ejpam-3352	329	2	follows	follow	VERB
ejpam-3352	329	3	that	that	SCONJ
ejpam-3352	329	4	a	a	DET
ejpam-3352	329	5	\	\	PROPN
ejpam-3352	329	6	∪{vλ	∪{vλ	PROPN
ejpam-3352	329	7	∩	∩	NOUN
ejpam-3352	329	8	a	a	X
ejpam-3352	329	9	:	:	PUNCT
ejpam-3352	329	10	λ	λ	X
ejpam-3352	329	11	∈	∈	PROPN
ejpam-3352	329	12	λ	λ	PROPN
ejpam-3352	329	13	}	}	PUNCT
ejpam-3352	329	14	∈	∈	PROPN
ejpam-3352	329	15	ia	ia	PROPN
ejpam-3352	329	16	.	.	PROPN
ejpam-3352	329	17	let	let	VERB
ejpam-3352	329	18	x	x	PUNCT
ejpam-3352	329	19	∈	∈	VERB
ejpam-3352	329	20	a.	a.	NOUN
ejpam-3352	329	21	since	since	SCONJ
ejpam-3352	329	22	v	v	NUM
ejpam-3352	329	23	=	=	PUNCT
ejpam-3352	329	24	{	{	PUNCT
ejpam-3352	329	25	vλ	vλ	INTJ
ejpam-3352	329	26	:	:	PUNCT
ejpam-3352	329	27	λ	λ	PROPN
ejpam-3352	329	28	∈	∈	PROPN
ejpam-3352	329	29	λ	λ	PROPN
ejpam-3352	329	30	}	}	PUNCT
ejpam-3352	329	31	is	be	AUX
ejpam-3352	329	32	locally	locally	ADV
ejpam-3352	329	33	finite	finite	ADJ
ejpam-3352	329	34	,	,	PUNCT
ejpam-3352	329	35	there	there	PRON
ejpam-3352	329	36	exists	exist	VERB
ejpam-3352	329	37	w	w	PROPN
ejpam-3352	329	38	∈	∈	NOUN
ejpam-3352	329	39	τ(x	τ(x	NOUN
ejpam-3352	329	40	)	)	PUNCT
ejpam-3352	329	41	such	such	ADJ
ejpam-3352	329	42	that	that	SCONJ
ejpam-3352	329	43	vλ	vλ	ADJ
ejpam-3352	329	44	∩w	∩w	ADJ
ejpam-3352	329	45	=	=	NOUN
ejpam-3352	329	46	∅	∅	NOUN
ejpam-3352	329	47	for	for	ADP
ejpam-3352	329	48	λ	λ	PROPN
ejpam-3352	329	49	6=	6=	SYM
ejpam-3352	329	50	λ1	λ1	ADJ
ejpam-3352	329	51	,	,	PUNCT
ejpam-3352	329	52	λ2	λ2	NOUN
ejpam-3352	329	53	,	,	PUNCT
ejpam-3352	329	54	...	...	PUNCT
ejpam-3352	329	55	,	,	PUNCT
ejpam-3352	329	56	λn	λn	PROPN
ejpam-3352	329	57	.	.	PUNCT
ejpam-3352	330	1	then	then	ADV
ejpam-3352	330	2	(	(	PUNCT
ejpam-3352	330	3	vλ	vλ	ADP
ejpam-3352	330	4	∩w	∩w	ADJ
ejpam-3352	330	5	)	)	PUNCT
ejpam-3352	330	6	∩	∩	NOUN
ejpam-3352	330	7	a	a	DET
ejpam-3352	330	8	=	=	NOUN
ejpam-3352	330	9	∅	∅	NOUN
ejpam-3352	330	10	for	for	ADP
ejpam-3352	330	11	λ	λ	PROPN
ejpam-3352	330	12	6=	6=	SYM
ejpam-3352	330	13	λ1	λ1	ADJ
ejpam-3352	330	14	,	,	PUNCT
ejpam-3352	330	15	λ2	λ2	NOUN
ejpam-3352	330	16	,	,	PUNCT
ejpam-3352	330	17	...	...	PUNCT
ejpam-3352	330	18	,	,	PUNCT
ejpam-3352	330	19	λn	λn	X
ejpam-3352	330	20	which	which	PRON
ejpam-3352	330	21	implies	imply	VERB
ejpam-3352	330	22	(	(	PUNCT
ejpam-3352	330	23	vλ∩a)∩(w	vλ∩a)∩(w	PROPN
ejpam-3352	330	24	∩a	∩a	PROPN
ejpam-3352	330	25	)	)	PUNCT
ejpam-3352	331	1	=	=	NOUN
ejpam-3352	331	2	∅	∅	NOUN
ejpam-3352	331	3	for	for	ADP
ejpam-3352	331	4	λ	λ	PROPN
ejpam-3352	331	5	6=	6=	SYM
ejpam-3352	331	6	λ1	λ1	ADJ
ejpam-3352	331	7	,	,	PUNCT
ejpam-3352	331	8	λ2	λ2	NOUN
ejpam-3352	331	9	,	,	PUNCT
ejpam-3352	331	10	...	...	PUNCT
ejpam-3352	331	11	,	,	PUNCT
ejpam-3352	331	12	λn	λn	NOUN
ejpam-3352	331	13	.	.	PUNCT
ejpam-3352	332	1	thus	thus	ADV
ejpam-3352	332	2	the	the	DET
ejpam-3352	332	3	family	family	NOUN
ejpam-3352	332	4	va	va	NOUN
ejpam-3352	332	5	=	=	PUNCT
ejpam-3352	332	6	{	{	PUNCT
ejpam-3352	332	7	vλ∩a	vλ∩a	NOUN
ejpam-3352	332	8	:	:	PUNCT
ejpam-3352	332	9	λ	λ	PROPN
ejpam-3352	332	10	∈	∈	PROPN
ejpam-3352	332	11	λ	λ	PROPN
ejpam-3352	332	12	}	}	PUNCT
ejpam-3352	332	13	is	be	AUX
ejpam-3352	332	14	τa	τa	NOUN
ejpam-3352	332	15	-	-	PUNCT
ejpam-3352	332	16	locally	locally	ADV
ejpam-3352	332	17	finite	finite	ADJ
ejpam-3352	332	18	τa	τa	ADJ
ejpam-3352	332	19	-	-	PUNCT
ejpam-3352	332	20	open	open	ADJ
ejpam-3352	332	21	refitment	refitment	NOUN
ejpam-3352	332	22	of	of	ADP
ejpam-3352	332	23	u	u	PROPN
ejpam-3352	332	24	.	.	PUNCT
ejpam-3352	333	1	hence	hence	ADV
ejpam-3352	333	2	a	a	PRON
ejpam-3352	333	3	is	be	AUX
ejpam-3352	333	4	β1aia	β1aia	NOUN
ejpam-3352	333	5	-	-	PUNCT
ejpam-3352	333	6	paracompact	paracompact	NOUN
ejpam-3352	333	7	.	.	PUNCT
ejpam-3352	334	1	to	to	PART
ejpam-3352	334	2	prove	prove	VERB
ejpam-3352	334	3	sufficiency	sufficiency	NOUN
ejpam-3352	334	4	,	,	PUNCT
ejpam-3352	334	5	let	let	VERB
ejpam-3352	334	6	u	u	PRON
ejpam-3352	334	7	=	=	X
ejpam-3352	334	8	{	{	PUNCT
ejpam-3352	334	9	uα	uα	X
ejpam-3352	334	10	:	:	PUNCT
ejpam-3352	334	11	α	α	PROPN
ejpam-3352	334	12	∈	∈	PROPN
ejpam-3352	334	13	∆	∆	PROPN
ejpam-3352	334	14	}	}	PUNCT
ejpam-3352	334	15	be	be	AUX
ejpam-3352	334	16	a	a	DET
ejpam-3352	334	17	cover	cover	NOUN
ejpam-3352	334	18	of	of	ADP
ejpam-3352	334	19	a	a	PRON
ejpam-3352	334	20	by	by	ADP
ejpam-3352	334	21	β	β	NOUN
ejpam-3352	334	22	-	-	ADJ
ejpam-3352	334	23	open	open	ADJ
ejpam-3352	334	24	subsets	subset	NOUN
ejpam-3352	334	25	of	of	ADP
ejpam-3352	334	26	an	an	DET
ejpam-3352	334	27	ideal	ideal	ADJ
ejpam-3352	334	28	space	space	NOUN
ejpam-3352	334	29	(	(	PUNCT
ejpam-3352	334	30	x	x	X
ejpam-3352	334	31	,	,	PUNCT
ejpam-3352	334	32	τ	τ	PROPN
ejpam-3352	334	33	,	,	PUNCT
ejpam-3352	334	34	i	i	PROPN
ejpam-3352	334	35	)	)	PUNCT
ejpam-3352	334	36	.	.	PUNCT
ejpam-3352	335	1	then	then	ADV
ejpam-3352	335	2	u1	u1	VERB
ejpam-3352	335	3	=	=	PRON
ejpam-3352	335	4	{	{	PUNCT
ejpam-3352	335	5	a	a	DET
ejpam-3352	335	6	∩	∩	ADJ
ejpam-3352	335	7	uα	uα	NOUN
ejpam-3352	335	8	:	:	PUNCT
ejpam-3352	335	9	α	α	PROPN
ejpam-3352	335	10	∈	∈	PROPN
ejpam-3352	335	11	∆	∆	X
ejpam-3352	335	12	}	}	PUNCT
ejpam-3352	335	13	is	be	AUX
ejpam-3352	335	14	a	a	DET
ejpam-3352	335	15	β	β	NOUN
ejpam-3352	335	16	-	-	ADJ
ejpam-3352	335	17	open	open	ADJ
ejpam-3352	335	18	cover	cover	NOUN
ejpam-3352	335	19	of	of	ADP
ejpam-3352	335	20	the	the	DET
ejpam-3352	335	21	β1aiaparacompact	β1aiaparacompact	ADJ
ejpam-3352	335	22	ideal	ideal	NOUN
ejpam-3352	335	23	subspace	subspace	NOUN
ejpam-3352	335	24	(	(	PUNCT
ejpam-3352	335	25	a	a	DET
ejpam-3352	335	26	,	,	PUNCT
ejpam-3352	335	27	τa	τa	PROPN
ejpam-3352	335	28	,	,	PUNCT
ejpam-3352	335	29	ia	ia	PROPN
ejpam-3352	335	30	)	)	PUNCT
ejpam-3352	335	31	and	and	CCONJ
ejpam-3352	335	32	so	so	ADV
ejpam-3352	335	33	it	it	PRON
ejpam-3352	335	34	has	have	VERB
ejpam-3352	335	35	a	a	DET
ejpam-3352	335	36	τa	τa	NOUN
ejpam-3352	335	37	-	-	PUNCT
ejpam-3352	335	38	locally	locally	ADV
ejpam-3352	335	39	finite	finite	ADJ
ejpam-3352	335	40	τa	τa	ADJ
ejpam-3352	335	41	-	-	PUNCT
ejpam-3352	335	42	open	open	ADJ
ejpam-3352	335	43	refinement	refinement	NOUN
ejpam-3352	335	44	v	v	NOUN
ejpam-3352	335	45	=	=	PUNCT
ejpam-3352	335	46	{	{	PUNCT
ejpam-3352	335	47	vλ	vλ	INTJ
ejpam-3352	335	48	:	:	PUNCT
ejpam-3352	335	49	λ	λ	PROPN
ejpam-3352	335	50	∈	∈	PROPN
ejpam-3352	335	51	λ	λ	NOUN
ejpam-3352	335	52	}	}	PUNCT
ejpam-3352	335	53	such	such	ADJ
ejpam-3352	335	54	that	that	SCONJ
ejpam-3352	335	55	a	a	DET
ejpam-3352	335	56	\	\	PROPN
ejpam-3352	335	57	∪{vλ	∪{vλ	PROPN
ejpam-3352	335	58	:	:	PUNCT
ejpam-3352	335	59	λ	λ	PROPN
ejpam-3352	335	60	∈	∈	PROPN
ejpam-3352	335	61	λ	λ	PROPN
ejpam-3352	335	62	}	}	PUNCT
ejpam-3352	335	63	∈	∈	PROPN
ejpam-3352	335	64	ia	ia	PROPN
ejpam-3352	335	65	.	.	PROPN
ejpam-3352	336	1	then	then	ADV
ejpam-3352	336	2	a	a	DET
ejpam-3352	336	3	\	\	PROPN
ejpam-3352	336	4	∪{vλ	∪{vλ	NUM
ejpam-3352	336	5	:	:	PUNCT
ejpam-3352	336	6	λ	λ	PROPN
ejpam-3352	336	7	∈	∈	PROPN
ejpam-3352	336	8	λ	λ	PROPN
ejpam-3352	336	9	}	}	PUNCT
ejpam-3352	336	10	∈	∈	PROPN
ejpam-3352	336	11	i.	i.	NOUN
ejpam-3352	336	12	but	but	CCONJ
ejpam-3352	336	13	a	a	PRON
ejpam-3352	336	14	is	be	AUX
ejpam-3352	336	15	an	an	DET
ejpam-3352	336	16	open	open	ADJ
ejpam-3352	336	17	set	set	NOUN
ejpam-3352	336	18	in	in	ADP
ejpam-3352	336	19	x	x	NOUN
ejpam-3352	336	20	,	,	PUNCT
ejpam-3352	336	21	so	so	ADV
ejpam-3352	336	22	vλ	vλ	ADV
ejpam-3352	336	23	is	be	AUX
ejpam-3352	336	24	an	an	DET
ejpam-3352	336	25	open	open	ADJ
ejpam-3352	336	26	set	set	NOUN
ejpam-3352	336	27	for	for	ADP
ejpam-3352	336	28	every	every	DET
ejpam-3352	336	29	λ	λ	PROPN
ejpam-3352	336	30	∈	∈	PROPN
ejpam-3352	336	31	λ	λ	PROPN
ejpam-3352	336	32	.	.	PUNCT
ejpam-3352	337	1	now	now	ADV
ejpam-3352	337	2	τa	τa	VERB
ejpam-3352	337	3	⊆	⊆	NUM
ejpam-3352	337	4	τ	τ	X
ejpam-3352	337	5	and	and	CCONJ
ejpam-3352	337	6	x	x	SYM
ejpam-3352	337	7	\	\	PROPN
ejpam-3352	337	8	a	a	PRON
ejpam-3352	337	9	is	be	AUX
ejpam-3352	337	10	an	an	DET
ejpam-3352	337	11	open	open	ADJ
ejpam-3352	337	12	set	set	NOUN
ejpam-3352	337	13	in	in	ADP
ejpam-3352	337	14	x	x	PUNCT
ejpam-3352	337	15	which	which	PRON
ejpam-3352	337	16	intersects	intersect	VERB
ejpam-3352	337	17	no	no	DET
ejpam-3352	337	18	member	member	NOUN
ejpam-3352	337	19	of	of	ADP
ejpam-3352	337	20	v.	v.	ADP
ejpam-3352	337	21	therefore	therefore	ADV
ejpam-3352	337	22	v	v	NOUN
ejpam-3352	337	23	is	be	AUX
ejpam-3352	337	24	locally	locally	ADV
ejpam-3352	337	25	finite	finite	ADJ
ejpam-3352	337	26	and	and	CCONJ
ejpam-3352	337	27	refines	refine	VERB
ejpam-3352	337	28	u	u	PRON
ejpam-3352	337	29	.	.	PUNCT
ejpam-3352	338	1	thus	thus	ADV
ejpam-3352	338	2	a	a	PRON
ejpam-3352	338	3	is	be	AUX
ejpam-3352	338	4	a	a	DET
ejpam-3352	338	5	β1i	β1i	NOUN
ejpam-3352	338	6	-	-	NOUN
ejpam-3352	338	7	paracompact	paracompact	NOUN
ejpam-3352	338	8	.	.	PUNCT
ejpam-3352	339	1	if	if	SCONJ
ejpam-3352	339	2	i	i	PRON
ejpam-3352	339	3	=	=	SYM
ejpam-3352	339	4	{	{	PUNCT
ejpam-3352	339	5	∅	∅	NOUN
ejpam-3352	339	6	}	}	PUNCT
ejpam-3352	339	7	in	in	ADP
ejpam-3352	339	8	theorem	theorem	NOUN
ejpam-3352	339	9	12	12	NUM
ejpam-3352	339	10	,	,	PUNCT
ejpam-3352	339	11	then	then	ADV
ejpam-3352	339	12	we	we	PRON
ejpam-3352	339	13	have	have	VERB
ejpam-3352	339	14	the	the	DET
ejpam-3352	339	15	following	follow	VERB
ejpam-3352	339	16	corollary	corollary	NOUN
ejpam-3352	339	17	.	.	PUNCT
ejpam-3352	340	1	corollary	corollary	ADJ
ejpam-3352	340	2	10	10	NUM
ejpam-3352	340	3	.	.	PUNCT
ejpam-3352	341	1	[	[	X
ejpam-3352	341	2	1	1	NUM
ejpam-3352	341	3	,	,	PUNCT
ejpam-3352	341	4	theorem	theorem	VERB
ejpam-3352	341	5	3.8	3.8	NUM
ejpam-3352	341	6	]	]	PUNCT
ejpam-3352	341	7	let	let	VERB
ejpam-3352	341	8	a	a	PRON
ejpam-3352	341	9	be	be	AUX
ejpam-3352	341	10	a	a	DET
ejpam-3352	341	11	clopen	clopen	ADJ
ejpam-3352	341	12	subspace	subspace	NOUN
ejpam-3352	341	13	of	of	ADP
ejpam-3352	341	14	a	a	DET
ejpam-3352	341	15	space	space	NOUN
ejpam-3352	341	16	(	(	PUNCT
ejpam-3352	341	17	x	x	X
ejpam-3352	341	18	,	,	PUNCT
ejpam-3352	341	19	τ	τ	PROPN
ejpam-3352	341	20	)	)	PUNCT
ejpam-3352	341	21	.	.	PUNCT
ejpam-3352	342	1	then	then	ADV
ejpam-3352	342	2	a	a	PRON
ejpam-3352	342	3	is	be	AUX
ejpam-3352	342	4	a	a	DET
ejpam-3352	342	5	β1a	β1a	NOUN
ejpam-3352	342	6	-	-	NOUN
ejpam-3352	342	7	paracompact	paracompact	NOUN
ejpam-3352	342	8	if	if	SCONJ
ejpam-3352	342	9	and	and	CCONJ
ejpam-3352	342	10	only	only	ADV
ejpam-3352	342	11	if	if	SCONJ
ejpam-3352	342	12	it	it	PRON
ejpam-3352	342	13	is	be	AUX
ejpam-3352	342	14	β1	β1	NOUN
ejpam-3352	342	15	-	-	PUNCT
ejpam-3352	342	16	paracompact	paracompact	NOUN
ejpam-3352	342	17	.	.	PUNCT
ejpam-3352	343	1	a.	a.	NOUN
ejpam-3352	343	2	qahis	qahis	PROPN
ejpam-3352	343	3	/	/	SYM
ejpam-3352	343	4	eur	eur	PROPN
ejpam-3352	343	5	.	.	PUNCT
ejpam-3352	344	1	j.	j.	PROPN
ejpam-3352	344	2	pure	pure	PROPN
ejpam-3352	344	3	appl	appl	PROPN
ejpam-3352	344	4	.	.	PROPN
ejpam-3352	344	5	math	math	PROPN
ejpam-3352	344	6	,	,	PUNCT
ejpam-3352	344	7	12	12	NUM
ejpam-3352	344	8	(	(	PUNCT
ejpam-3352	344	9	1	1	NUM
ejpam-3352	344	10	)	)	PUNCT
ejpam-3352	344	11	(	(	PUNCT
ejpam-3352	344	12	2019	2019	NUM
ejpam-3352	344	13	)	)	PUNCT
ejpam-3352	344	14	,	,	PUNCT
ejpam-3352	344	15	135	135	NUM
ejpam-3352	344	16	-	-	SYM
ejpam-3352	344	17	145	145	NUM
ejpam-3352	344	18	143	143	NUM
ejpam-3352	344	19	theorem	theorem	NOUN
ejpam-3352	344	20	13	13	NUM
ejpam-3352	344	21	.	.	PUNCT
ejpam-3352	345	1	if	if	SCONJ
ejpam-3352	345	2	(	(	PUNCT
ejpam-3352	345	3	x	x	X
ejpam-3352	345	4	,	,	PUNCT
ejpam-3352	345	5	τ	τ	PROPN
ejpam-3352	345	6	,	,	PUNCT
ejpam-3352	345	7	i	i	PROPN
ejpam-3352	345	8	)	)	PUNCT
ejpam-3352	345	9	is	be	AUX
ejpam-3352	345	10	t2	t2	NOUN
ejpam-3352	345	11	space	space	NOUN
ejpam-3352	345	12	and	and	CCONJ
ejpam-3352	345	13	a	a	PRON
ejpam-3352	345	14	is	be	AUX
ejpam-3352	345	15	β1i	β1i	NOUN
ejpam-3352	345	16	-	-	ADJ
ejpam-3352	345	17	paracompact	paracompact	ADJ
ejpam-3352	345	18	relative	relative	NOUN
ejpam-3352	345	19	to	to	ADP
ejpam-3352	345	20	x	x	PRON
ejpam-3352	345	21	,	,	PUNCT
ejpam-3352	345	22	then	then	ADV
ejpam-3352	345	23	a	a	PRON
ejpam-3352	345	24	is	be	AUX
ejpam-3352	345	25	closed	close	VERB
ejpam-3352	345	26	in	in	ADP
ejpam-3352	345	27	(	(	PUNCT
ejpam-3352	345	28	x	x	NOUN
ejpam-3352	345	29	,	,	PUNCT
ejpam-3352	345	30	τ∗	τ∗	NOUN
ejpam-3352	345	31	)	)	PUNCT
ejpam-3352	345	32	.	.	PUNCT
ejpam-3352	346	1	proof	proof	NOUN
ejpam-3352	346	2	.	.	PUNCT
ejpam-3352	347	1	let	let	VERB
ejpam-3352	347	2	x	x	SYM
ejpam-3352	347	3	∈	∈	PROPN
ejpam-3352	347	4	x	x	SYM
ejpam-3352	347	5	\	\	PROPN
ejpam-3352	347	6	a.	a.	NOUN
ejpam-3352	347	7	for	for	ADP
ejpam-3352	347	8	each	each	DET
ejpam-3352	347	9	y	y	PROPN
ejpam-3352	347	10	∈	∈	PROPN
ejpam-3352	347	11	a	a	PRON
ejpam-3352	347	12	,	,	PUNCT
ejpam-3352	347	13	there	there	PRON
ejpam-3352	347	14	exists	exist	VERB
ejpam-3352	347	15	u	u	PROPN
ejpam-3352	347	16	∈	∈	PROPN
ejpam-3352	347	17	τ	τ	X
ejpam-3352	348	1	such	such	ADJ
ejpam-3352	348	2	that	that	SCONJ
ejpam-3352	348	3	y	y	PROPN
ejpam-3352	348	4	∈	∈	PROPN
ejpam-3352	348	5	uy	uy	PROPN
ejpam-3352	348	6	and	and	CCONJ
ejpam-3352	348	7	x	x	PROPN
ejpam-3352	348	8	/∈	/∈	PUNCT
ejpam-3352	348	9	cl(uy	cl(uy	NOUN
ejpam-3352	348	10	)	)	PUNCT
ejpam-3352	348	11	.	.	PUNCT
ejpam-3352	349	1	therefore	therefore	ADV
ejpam-3352	349	2	,	,	PUNCT
ejpam-3352	349	3	the	the	DET
ejpam-3352	349	4	family	family	NOUN
ejpam-3352	349	5	u	u	NOUN
ejpam-3352	349	6	=	=	PUNCT
ejpam-3352	349	7	{	{	PUNCT
ejpam-3352	349	8	uy	uy	NOUN
ejpam-3352	349	9	:	:	PUNCT
ejpam-3352	349	10	y	y	PROPN
ejpam-3352	349	11	∈	∈	PROPN
ejpam-3352	349	12	a	a	PRON
ejpam-3352	349	13	}	}	PUNCT
ejpam-3352	349	14	is	be	AUX
ejpam-3352	349	15	an	an	DET
ejpam-3352	349	16	open	open	ADJ
ejpam-3352	349	17	cover	cover	NOUN
ejpam-3352	349	18	of	of	ADP
ejpam-3352	349	19	a	a	PRON
ejpam-3352	349	20	which	which	PRON
ejpam-3352	349	21	is	be	AUX
ejpam-3352	349	22	β1iparacompact	β1iparacompact	PROPN
ejpam-3352	349	23	relative	relative	ADJ
ejpam-3352	349	24	to	to	ADP
ejpam-3352	349	25	x.	x.	VERB
ejpam-3352	349	26	since	since	SCONJ
ejpam-3352	349	27	u	u	NOUN
ejpam-3352	349	28	is	be	AUX
ejpam-3352	349	29	a	a	DET
ejpam-3352	349	30	β	β	NOUN
ejpam-3352	349	31	-	-	ADJ
ejpam-3352	349	32	open	open	ADJ
ejpam-3352	349	33	cover	cover	NOUN
ejpam-3352	349	34	of	of	ADP
ejpam-3352	349	35	a	a	PRON
ejpam-3352	350	1	and	and	CCONJ
ejpam-3352	350	2	so	so	ADV
ejpam-3352	350	3	it	it	PRON
ejpam-3352	350	4	has	have	VERB
ejpam-3352	350	5	a	a	DET
ejpam-3352	350	6	locally	locally	ADV
ejpam-3352	350	7	finite	finite	ADJ
ejpam-3352	350	8	open	open	ADJ
ejpam-3352	350	9	refinement	refinement	NOUN
ejpam-3352	350	10	v	v	X
ejpam-3352	350	11	=	=	PUNCT
ejpam-3352	350	12	{	{	PUNCT
ejpam-3352	350	13	vλ	vλ	INTJ
ejpam-3352	350	14	:	:	PUNCT
ejpam-3352	350	15	λ	λ	PROPN
ejpam-3352	350	16	∈	∈	PROPN
ejpam-3352	350	17	λ	λ	PROPN
ejpam-3352	350	18	}	}	PUNCT
ejpam-3352	350	19	of	of	ADP
ejpam-3352	350	20	u	u	PRON
ejpam-3352	350	21	such	such	ADJ
ejpam-3352	350	22	that	that	SCONJ
ejpam-3352	350	23	a	a	DET
ejpam-3352	350	24	\	\	PROPN
ejpam-3352	350	25	∪{vλ	∪{vλ	PROPN
ejpam-3352	350	26	:	:	PUNCT
ejpam-3352	350	27	λ	λ	PROPN
ejpam-3352	350	28	∈	∈	PROPN
ejpam-3352	350	29	λ	λ	PROPN
ejpam-3352	350	30	}	}	PUNCT
ejpam-3352	350	31	∈	∈	PROPN
ejpam-3352	350	32	i.	i.	NOUN
ejpam-3352	350	33	now	now	ADV
ejpam-3352	350	34	x	x	X
ejpam-3352	350	35	/∈	/∈	PUNCT
ejpam-3352	350	36	cl(vλ	cl(vλ	PROPN
ejpam-3352	350	37	)	)	PUNCT
ejpam-3352	350	38	for	for	ADP
ejpam-3352	350	39	each	each	DET
ejpam-3352	350	40	λ	λ	PROPN
ejpam-3352	350	41	implies	imply	VERB
ejpam-3352	350	42	that	that	SCONJ
ejpam-3352	350	43	x	x	SYM
ejpam-3352	350	44	/∈	/∈	PUNCT
ejpam-3352	350	45	∪{cl(vλ	∪{cl(vλ	PROPN
ejpam-3352	350	46	)	)	PUNCT
ejpam-3352	350	47	:	:	PUNCT
ejpam-3352	351	1	λ	λ	X
ejpam-3352	351	2	∈	∈	PROPN
ejpam-3352	351	3	λ	λ	NOUN
ejpam-3352	351	4	}	}	PUNCT
ejpam-3352	351	5	.	.	PUNCT
ejpam-3352	352	1	since	since	SCONJ
ejpam-3352	352	2	the	the	DET
ejpam-3352	352	3	locally	locally	ADV
ejpam-3352	352	4	finite	finite	ADJ
ejpam-3352	352	5	family	family	PROPN
ejpam-3352	352	6	v	v	PROPN
ejpam-3352	352	7	is	be	AUX
ejpam-3352	352	8	closurepreserving	closurepreserve	VERB
ejpam-3352	352	9	,	,	PUNCT
ejpam-3352	352	10	x	x	SYM
ejpam-3352	352	11	/∈	/∈	PUNCT
ejpam-3352	352	12	∪{cl(vλ	∪{cl(vλ	PROPN
ejpam-3352	352	13	)	)	PUNCT
ejpam-3352	352	14	:	:	PUNCT
ejpam-3352	353	1	λ	λ	X
ejpam-3352	353	2	∈	∈	NOUN
ejpam-3352	353	3	λ	λ	X
ejpam-3352	353	4	}	}	PUNCT
ejpam-3352	353	5	=	=	SYM
ejpam-3352	353	6	cl(∪{vλ	cl(∪{vλ	NOUN
ejpam-3352	353	7	:	:	PUNCT
ejpam-3352	353	8	λ	λ	PROPN
ejpam-3352	353	9	∈	∈	PROPN
ejpam-3352	353	10	λ	λ	NOUN
ejpam-3352	353	11	}	}	PUNCT
ejpam-3352	353	12	)	)	PUNCT
ejpam-3352	353	13	.	.	PUNCT
ejpam-3352	354	1	let	let	VERB
ejpam-3352	354	2	u	u	PRON
ejpam-3352	354	3	=	=	NOUN
ejpam-3352	354	4	x	x	SYM
ejpam-3352	354	5	\	\	PROPN
ejpam-3352	354	6	cl(∪{vλ	cl(∪{vλ	NOUN
ejpam-3352	354	7	:	:	PUNCT
ejpam-3352	354	8	λ	λ	PROPN
ejpam-3352	354	9	∈	∈	PROPN
ejpam-3352	354	10	λ	λ	NOUN
ejpam-3352	354	11	}	}	PUNCT
ejpam-3352	354	12	)	)	PUNCT
ejpam-3352	354	13	and	and	CCONJ
ejpam-3352	354	14	j	j	PROPN
ejpam-3352	354	15	=	=	PRON
ejpam-3352	354	16	a\cl(∪{vλ	a\cl(∪{vλ	VERB
ejpam-3352	354	17	:	:	PUNCT
ejpam-3352	354	18	λ	λ	PROPN
ejpam-3352	354	19	∈	∈	PROPN
ejpam-3352	354	20	λ	λ	NOUN
ejpam-3352	354	21	}	}	PUNCT
ejpam-3352	354	22	)	)	PUNCT
ejpam-3352	354	23	⊂	⊂	PROPN
ejpam-3352	355	1	a\∪{vλ	a\∪{vλ	ADV
ejpam-3352	355	2	:	:	PUNCT
ejpam-3352	355	3	λ	λ	X
ejpam-3352	355	4	∈	∈	NOUN
ejpam-3352	355	5	λ	λ	NOUN
ejpam-3352	355	6	}	}	PUNCT
ejpam-3352	355	7	=	=	SYM
ejpam-3352	355	8	i1	i1	PROPN
ejpam-3352	355	9	,	,	PUNCT
ejpam-3352	355	10	where	where	SCONJ
ejpam-3352	355	11	i1	i1	PROPN
ejpam-3352	355	12	∈	∈	PROPN
ejpam-3352	355	13	i.	i.	PROPN
ejpam-3352	355	14	then	then	ADV
ejpam-3352	355	15	u	u	PROPN
ejpam-3352	355	16	\j	\j	PROPN
ejpam-3352	355	17	∈	∈	PROPN
ejpam-3352	355	18	τ∗(x	τ∗(x	PROPN
ejpam-3352	355	19	)	)	PUNCT
ejpam-3352	355	20	and	and	CCONJ
ejpam-3352	355	21	(	(	PUNCT
ejpam-3352	355	22	u	u	NOUN
ejpam-3352	355	23	\	\	PROPN
ejpam-3352	355	24	j)∩a	j)∩a	NOUN
ejpam-3352	355	25	=	=	PUNCT
ejpam-3352	355	26	∅	∅	NOUN
ejpam-3352	355	27	which	which	PRON
ejpam-3352	355	28	implies	imply	VERB
ejpam-3352	355	29	x	x	X
ejpam-3352	355	30	/∈	/∈	INTJ
ejpam-3352	355	31	a∗.	a∗.	NOUN
ejpam-3352	355	32	hence	hence	ADV
ejpam-3352	355	33	a∗	a∗	PROPN
ejpam-3352	355	34	⊂	⊂	PROPN
ejpam-3352	355	35	a.	a.	NOUN
ejpam-3352	355	36	this	this	PRON
ejpam-3352	355	37	shows	show	VERB
ejpam-3352	355	38	that	that	SCONJ
ejpam-3352	355	39	a	a	PRON
ejpam-3352	355	40	is	be	AUX
ejpam-3352	355	41	closed	close	VERB
ejpam-3352	355	42	in	in	ADP
ejpam-3352	355	43	(	(	PUNCT
ejpam-3352	355	44	x	x	NOUN
ejpam-3352	355	45	,	,	PUNCT
ejpam-3352	355	46	τ∗	τ∗	NOUN
ejpam-3352	355	47	)	)	PUNCT
ejpam-3352	355	48	.	.	PUNCT
ejpam-3352	356	1	if	if	SCONJ
ejpam-3352	356	2	i	i	PRON
ejpam-3352	356	3	=	=	SYM
ejpam-3352	356	4	{	{	PUNCT
ejpam-3352	356	5	∅	∅	NOUN
ejpam-3352	356	6	}	}	PUNCT
ejpam-3352	356	7	in	in	ADP
ejpam-3352	356	8	theorem	theorem	ADJ
ejpam-3352	356	9	13	13	NUM
ejpam-3352	356	10	,	,	PUNCT
ejpam-3352	356	11	then	then	ADV
ejpam-3352	356	12	we	we	PRON
ejpam-3352	356	13	conclude	conclude	VERB
ejpam-3352	356	14	the	the	DET
ejpam-3352	356	15	following	follow	VERB
ejpam-3352	356	16	corollary	corollary	NOUN
ejpam-3352	356	17	.	.	PUNCT
ejpam-3352	357	1	corollary	corollary	ADJ
ejpam-3352	357	2	11	11	NUM
ejpam-3352	357	3	.	.	PUNCT
ejpam-3352	358	1	let	let	VERB
ejpam-3352	358	2	a	a	PRON
ejpam-3352	358	3	be	be	AUX
ejpam-3352	358	4	a	a	DET
ejpam-3352	358	5	β1	β1	NOUN
ejpam-3352	358	6	-	-	PUNCT
ejpam-3352	358	7	paracompact	paracompact	NOUN
ejpam-3352	358	8	relative	relative	ADJ
ejpam-3352	358	9	subset	subset	NOUN
ejpam-3352	358	10	of	of	ADP
ejpam-3352	358	11	a	a	DET
ejpam-3352	358	12	t2	t2	NOUN
ejpam-3352	358	13	space	space	NOUN
ejpam-3352	358	14	(	(	PUNCT
ejpam-3352	358	15	x	x	X
ejpam-3352	358	16	,	,	PUNCT
ejpam-3352	358	17	τ	τ	PROPN
ejpam-3352	358	18	)	)	PUNCT
ejpam-3352	358	19	.	.	PUNCT
ejpam-3352	359	1	then	then	ADV
ejpam-3352	359	2	a	a	PRON
ejpam-3352	359	3	is	be	AUX
ejpam-3352	359	4	closed	close	VERB
ejpam-3352	359	5	in	in	ADP
ejpam-3352	359	6	(	(	PUNCT
ejpam-3352	359	7	x	x	NOUN
ejpam-3352	359	8	,	,	PUNCT
ejpam-3352	359	9	τ	τ	PROPN
ejpam-3352	359	10	)	)	PUNCT
ejpam-3352	359	11	.	.	PUNCT
ejpam-3352	360	1	theorem	theorem	VERB
ejpam-3352	360	2	14	14	NUM
ejpam-3352	360	3	.	.	PUNCT
ejpam-3352	361	1	in	in	ADP
ejpam-3352	361	2	an	an	DET
ejpam-3352	361	3	ideal	ideal	ADJ
ejpam-3352	361	4	space	space	NOUN
ejpam-3352	361	5	(	(	PUNCT
ejpam-3352	361	6	x	x	X
ejpam-3352	361	7	,	,	PUNCT
ejpam-3352	361	8	τ	τ	PROPN
ejpam-3352	361	9	,	,	PUNCT
ejpam-3352	361	10	i	i	PROPN
ejpam-3352	361	11	)	)	PUNCT
ejpam-3352	361	12	,	,	PUNCT
ejpam-3352	361	13	if	if	SCONJ
ejpam-3352	361	14	a	a	PRON
ejpam-3352	361	15	and	and	CCONJ
ejpam-3352	361	16	b	b	NOUN
ejpam-3352	361	17	are	be	AUX
ejpam-3352	361	18	β1i	β1i	ADJ
ejpam-3352	361	19	-	-	ADJ
ejpam-3352	361	20	paracompact	paracompact	ADJ
ejpam-3352	361	21	,	,	PUNCT
ejpam-3352	361	22	then	then	ADV
ejpam-3352	361	23	a	a	DET
ejpam-3352	361	24	∪	∪	X
ejpam-3352	361	25	b	b	NOUN
ejpam-3352	361	26	is	be	AUX
ejpam-3352	361	27	β1i	β1i	NOUN
ejpam-3352	361	28	-	-	ADJ
ejpam-3352	361	29	paracompact	paracompact	ADJ
ejpam-3352	361	30	.	.	PUNCT
ejpam-3352	362	1	proof	proof	NOUN
ejpam-3352	362	2	.	.	PUNCT
ejpam-3352	363	1	let	let	VERB
ejpam-3352	363	2	u	u	PRON
ejpam-3352	363	3	=	=	PUNCT
ejpam-3352	363	4	{	{	PUNCT
ejpam-3352	363	5	uα	uα	X
ejpam-3352	363	6	:	:	PUNCT
ejpam-3352	363	7	α	α	PROPN
ejpam-3352	363	8	∈	∈	PROPN
ejpam-3352	363	9	∆	∆	PROPN
ejpam-3352	363	10	}	}	PUNCT
ejpam-3352	363	11	be	be	AUX
ejpam-3352	363	12	a	a	DET
ejpam-3352	363	13	cover	cover	NOUN
ejpam-3352	363	14	of	of	ADP
ejpam-3352	363	15	a∪b	a∪b	NOUN
ejpam-3352	363	16	by	by	ADP
ejpam-3352	363	17	β	β	ADJ
ejpam-3352	363	18	-	-	ADJ
ejpam-3352	363	19	open	open	ADJ
ejpam-3352	363	20	sets	set	NOUN
ejpam-3352	363	21	in	in	ADP
ejpam-3352	363	22	x.	x.	NOUN
ejpam-3352	363	23	then	then	ADV
ejpam-3352	363	24	u	u	NOUN
ejpam-3352	363	25	is	be	AUX
ejpam-3352	363	26	a	a	DET
ejpam-3352	363	27	βopen	βopen	ADJ
ejpam-3352	363	28	cover	cover	NOUN
ejpam-3352	363	29	of	of	ADP
ejpam-3352	363	30	a	a	PRON
ejpam-3352	363	31	and	and	CCONJ
ejpam-3352	363	32	b.	b.	PROPN
ejpam-3352	363	33	by	by	ADP
ejpam-3352	363	34	hypothesis	hypothesis	NOUN
ejpam-3352	363	35	,	,	PUNCT
ejpam-3352	363	36	there	there	PRON
ejpam-3352	363	37	exist	exist	VERB
ejpam-3352	363	38	locally	locally	ADV
ejpam-3352	363	39	finite	finite	ADJ
ejpam-3352	363	40	families	family	NOUN
ejpam-3352	363	41	v	v	NOUN
ejpam-3352	363	42	=	=	PUNCT
ejpam-3352	363	43	{	{	PUNCT
ejpam-3352	363	44	vλ	vλ	INTJ
ejpam-3352	363	45	:	:	PUNCT
ejpam-3352	363	46	λ	λ	PROPN
ejpam-3352	363	47	∈	∈	PROPN
ejpam-3352	363	48	λ	λ	NOUN
ejpam-3352	363	49	}	}	PUNCT
ejpam-3352	363	50	of	of	ADP
ejpam-3352	363	51	a	a	PRON
ejpam-3352	363	52	and	and	CCONJ
ejpam-3352	363	53	w	w	NOUN
ejpam-3352	363	54	=	=	PRON
ejpam-3352	363	55	{	{	PUNCT
ejpam-3352	363	56	wγ	wγ	NOUN
ejpam-3352	363	57	:	:	PUNCT
ejpam-3352	363	58	γ	γ	PROPN
ejpam-3352	363	59	∈	∈	PROPN
ejpam-3352	363	60	λ0	λ0	NOUN
ejpam-3352	363	61	}	}	PUNCT
ejpam-3352	363	62	of	of	ADP
ejpam-3352	363	63	b	b	NUM
ejpam-3352	363	64	which	which	PRON
ejpam-3352	363	65	refines	refine	VERB
ejpam-3352	363	66	u	u	PRON
ejpam-3352	363	67	such	such	ADJ
ejpam-3352	363	68	that	that	SCONJ
ejpam-3352	363	69	a	a	DET
ejpam-3352	363	70	\	\	PROPN
ejpam-3352	363	71	∪{vλ	∪{vλ	PROPN
ejpam-3352	363	72	:	:	PUNCT
ejpam-3352	363	73	λ	λ	PROPN
ejpam-3352	363	74	∈	∈	PROPN
ejpam-3352	363	75	λ	λ	PROPN
ejpam-3352	363	76	}	}	PUNCT
ejpam-3352	363	77	∈	∈	PROPN
ejpam-3352	363	78	i	i	PRON
ejpam-3352	363	79	and	and	CCONJ
ejpam-3352	363	80	b	b	PROPN
ejpam-3352	363	81	\	\	PROPN
ejpam-3352	363	82	∪{wγ	∪{wγ	PROPN
ejpam-3352	363	83	:	:	PUNCT
ejpam-3352	363	84	γ	γ	PROPN
ejpam-3352	363	85	∈	∈	PROPN
ejpam-3352	363	86	λ0	λ0	NOUN
ejpam-3352	363	87	}	}	PUNCT
ejpam-3352	363	88	∈	∈	PROPN
ejpam-3352	363	89	i.	i.	NOUN
ejpam-3352	363	90	then	then	ADV
ejpam-3352	363	91	a	a	DET
ejpam-3352	363	92	∪	∪	X
ejpam-3352	363	93	b	b	X
ejpam-3352	363	94	⊂	⊂	PROPN
ejpam-3352	363	95	(	(	PUNCT
ejpam-3352	363	96	∪{vλ	∪{vλ	NUM
ejpam-3352	363	97	:	:	PUNCT
ejpam-3352	363	98	λ	λ	PROPN
ejpam-3352	363	99	∈	∈	PROPN
ejpam-3352	363	100	λ	λ	PROPN
ejpam-3352	363	101	}	}	PUNCT
ejpam-3352	363	102	∪	∪	PROPN
ejpam-3352	363	103	i1	i1	PROPN
ejpam-3352	363	104	)	)	PUNCT
ejpam-3352	363	105	∪	∪	ADP
ejpam-3352	363	106	(	(	PUNCT
ejpam-3352	363	107	∪{wγ	∪{wγ	NUM
ejpam-3352	363	108	:	:	PUNCT
ejpam-3352	363	109	γ	γ	PROPN
ejpam-3352	363	110	∈	∈	PROPN
ejpam-3352	363	111	λ0	λ0	NOUN
ejpam-3352	363	112	}	}	PUNCT
ejpam-3352	363	113	∪	∪	NOUN
ejpam-3352	363	114	i2	i2	NOUN
ejpam-3352	363	115	)	)	PUNCT
ejpam-3352	363	116	,	,	PUNCT
ejpam-3352	363	117	where	where	SCONJ
ejpam-3352	363	118	i1	i1	PROPN
ejpam-3352	363	119	,	,	PUNCT
ejpam-3352	363	120	i2	i2	PROPN
ejpam-3352	363	121	∈	∈	PROPN
ejpam-3352	363	122	i	i	PRON
ejpam-3352	363	123	which	which	PRON
ejpam-3352	363	124	implies	imply	VERB
ejpam-3352	363	125	that	that	SCONJ
ejpam-3352	363	126	a	a	DET
ejpam-3352	363	127	∪	∪	X
ejpam-3352	363	128	b	b	X
ejpam-3352	363	129	⊂	⊂	X
ejpam-3352	363	130	(	(	PUNCT
ejpam-3352	363	131	∪{vλ	∪{vλ	PROPN
ejpam-3352	363	132	∪wγ	∪wγ	NOUN
ejpam-3352	363	133	:	:	PUNCT
ejpam-3352	363	134	λ	λ	X
ejpam-3352	363	135	∈	∈	PROPN
ejpam-3352	363	136	λ	λ	PROPN
ejpam-3352	363	137	,	,	PUNCT
ejpam-3352	363	138	γ	γ	PROPN
ejpam-3352	363	139	∈	∈	PROPN
ejpam-3352	363	140	λ0	λ0	NOUN
ejpam-3352	363	141	}	}	PUNCT
ejpam-3352	363	142	)	)	PUNCT
ejpam-3352	363	143	∪	∪	NOUN
ejpam-3352	363	144	(	(	PUNCT
ejpam-3352	363	145	i1	i1	PROPN
ejpam-3352	363	146	∪	∪	PROPN
ejpam-3352	363	147	i2	i2	PROPN
ejpam-3352	363	148	)	)	PUNCT
ejpam-3352	363	149	.	.	PUNCT
ejpam-3352	364	1	it	it	PRON
ejpam-3352	364	2	follows	follow	VERB
ejpam-3352	364	3	that	that	SCONJ
ejpam-3352	364	4	(	(	PUNCT
ejpam-3352	364	5	a	a	DET
ejpam-3352	364	6	∪	∪	ADJ
ejpam-3352	364	7	b	b	NOUN
ejpam-3352	364	8	)	)	PUNCT
ejpam-3352	364	9	\	\	PROPN
ejpam-3352	364	10	∪{vλ	∪{vλ	PROPN
ejpam-3352	364	11	∪wγ	∪wγ	NOUN
ejpam-3352	364	12	:	:	PUNCT
ejpam-3352	364	13	λ	λ	X
ejpam-3352	364	14	∈	∈	PROPN
ejpam-3352	364	15	λ	λ	PROPN
ejpam-3352	364	16	,	,	PUNCT
ejpam-3352	364	17	γ	γ	PROPN
ejpam-3352	364	18	∈	∈	PROPN
ejpam-3352	364	19	λ0	λ0	NOUN
ejpam-3352	364	20	}	}	PUNCT
ejpam-3352	364	21	∈	∈	PROPN
ejpam-3352	364	22	i.	i.	NOUN
ejpam-3352	364	23	since	since	SCONJ
ejpam-3352	364	24	the	the	DET
ejpam-3352	364	25	families	family	NOUN
ejpam-3352	364	26	v	v	VERB
ejpam-3352	364	27	and	and	CCONJ
ejpam-3352	364	28	w	w	NOUN
ejpam-3352	364	29	are	be	AUX
ejpam-3352	364	30	locally	locally	ADV
ejpam-3352	364	31	finite	finite	ADJ
ejpam-3352	364	32	the	the	DET
ejpam-3352	364	33	family	family	NOUN
ejpam-3352	364	34	v	v	ADP
ejpam-3352	364	35	′	′	NUM
ejpam-3352	365	1	=	=	PUNCT
ejpam-3352	366	1	{	{	PUNCT
ejpam-3352	367	1	vλ	vλ	INTJ
ejpam-3352	367	2	∪wγ	∪wγ	NOUN
ejpam-3352	367	3	:	:	PUNCT
ejpam-3352	367	4	λ	λ	X
ejpam-3352	367	5	∈	∈	PROPN
ejpam-3352	367	6	λ	λ	PROPN
ejpam-3352	367	7	,	,	PUNCT
ejpam-3352	367	8	γ	γ	PROPN
ejpam-3352	367	9	∈	∈	PROPN
ejpam-3352	367	10	λ0	λ0	NOUN
ejpam-3352	367	11	}	}	PUNCT
ejpam-3352	367	12	is	be	AUX
ejpam-3352	367	13	locally	locally	ADV
ejpam-3352	367	14	finite	finite	ADJ
ejpam-3352	367	15	,	,	PUNCT
ejpam-3352	367	16	by	by	ADP
ejpam-3352	367	17	lemma	lemma	PROPN
ejpam-3352	367	18	1	1	NUM
ejpam-3352	367	19	which	which	PRON
ejpam-3352	367	20	refines	refine	VERB
ejpam-3352	367	21	u	u	PRON
ejpam-3352	367	22	.	.	PUNCT
ejpam-3352	368	1	therefore	therefore	ADV
ejpam-3352	368	2	,	,	PUNCT
ejpam-3352	368	3	a	a	DET
ejpam-3352	368	4	∪b	∪b	PRON
ejpam-3352	368	5	is	be	AUX
ejpam-3352	368	6	β1i	β1i	NOUN
ejpam-3352	368	7	-	-	ADJ
ejpam-3352	368	8	paracompact	paracompact	ADJ
ejpam-3352	368	9	.	.	PUNCT
ejpam-3352	369	1	theorem	theorem	NOUN
ejpam-3352	369	2	15	15	NUM
ejpam-3352	369	3	.	.	PUNCT
ejpam-3352	370	1	in	in	ADP
ejpam-3352	370	2	an	an	DET
ejpam-3352	370	3	ideal	ideal	ADJ
ejpam-3352	370	4	space	space	NOUN
ejpam-3352	370	5	(	(	PUNCT
ejpam-3352	370	6	x	x	X
ejpam-3352	370	7	,	,	PUNCT
ejpam-3352	370	8	τ	τ	PROPN
ejpam-3352	370	9	,	,	PUNCT
ejpam-3352	370	10	i	i	PROPN
ejpam-3352	370	11	)	)	PUNCT
ejpam-3352	370	12	,	,	PUNCT
ejpam-3352	370	13	if	if	SCONJ
ejpam-3352	370	14	a	a	PRON
ejpam-3352	370	15	is	be	AUX
ejpam-3352	370	16	β1i	β1i	NOUN
ejpam-3352	370	17	-	-	ADJ
ejpam-3352	370	18	paracompact	paracompact	ADJ
ejpam-3352	370	19	and	and	CCONJ
ejpam-3352	370	20	b	b	NOUN
ejpam-3352	370	21	is	be	AUX
ejpam-3352	370	22	a	a	DET
ejpam-3352	370	23	β	β	NOUN
ejpam-3352	370	24	-	-	ADJ
ejpam-3352	370	25	closed	closed	ADJ
ejpam-3352	370	26	subset	subset	NOUN
ejpam-3352	370	27	of	of	ADP
ejpam-3352	370	28	x	x	PRON
ejpam-3352	370	29	,	,	PUNCT
ejpam-3352	370	30	then	then	ADV
ejpam-3352	370	31	a	a	DET
ejpam-3352	370	32	∩b	∩b	NOUN
ejpam-3352	370	33	is	be	AUX
ejpam-3352	370	34	β1i	β1i	NOUN
ejpam-3352	370	35	-	-	ADJ
ejpam-3352	370	36	paracompact	paracompact	ADJ
ejpam-3352	370	37	.	.	PUNCT
ejpam-3352	371	1	proof	proof	NOUN
ejpam-3352	371	2	.	.	PUNCT
ejpam-3352	372	1	let	let	VERB
ejpam-3352	372	2	u	u	PRON
ejpam-3352	372	3	=	=	PUNCT
ejpam-3352	372	4	{	{	PUNCT
ejpam-3352	372	5	uα	uα	X
ejpam-3352	372	6	:	:	PUNCT
ejpam-3352	372	7	α	α	PROPN
ejpam-3352	372	8	∈	∈	PROPN
ejpam-3352	372	9	∆	∆	PROPN
ejpam-3352	372	10	}	}	PUNCT
ejpam-3352	372	11	be	be	AUX
ejpam-3352	372	12	a	a	DET
ejpam-3352	372	13	cover	cover	NOUN
ejpam-3352	372	14	of	of	ADP
ejpam-3352	372	15	a	a	DET
ejpam-3352	372	16	∩	∩	ADJ
ejpam-3352	372	17	b	b	NOUN
ejpam-3352	372	18	by	by	ADP
ejpam-3352	372	19	β	β	ADJ
ejpam-3352	372	20	-	-	ADJ
ejpam-3352	372	21	open	open	ADJ
ejpam-3352	372	22	subsets	subset	NOUN
ejpam-3352	372	23	of	of	ADP
ejpam-3352	372	24	x.	x.	NOUN
ejpam-3352	372	25	then	then	ADV
ejpam-3352	372	26	ua	ua	PROPN
ejpam-3352	372	27	=	=	SYM
ejpam-3352	372	28	u	u	PROPN
ejpam-3352	372	29	∪{x	∪{x	NOUN
ejpam-3352	372	30	\b	\b	ADJ
ejpam-3352	372	31	}	}	PUNCT
ejpam-3352	372	32	is	be	AUX
ejpam-3352	372	33	a	a	DET
ejpam-3352	372	34	cover	cover	NOUN
ejpam-3352	372	35	of	of	ADP
ejpam-3352	372	36	a	a	PRON
ejpam-3352	372	37	by	by	ADP
ejpam-3352	372	38	β	β	ADJ
ejpam-3352	372	39	-	-	ADJ
ejpam-3352	372	40	open	open	ADJ
ejpam-3352	372	41	sets	set	NOUN
ejpam-3352	372	42	in	in	ADP
ejpam-3352	372	43	x.	x.	NOUN
ejpam-3352	372	44	by	by	ADP
ejpam-3352	372	45	hypothesis	hypothesis	NOUN
ejpam-3352	372	46	,	,	PUNCT
ejpam-3352	372	47	there	there	PRON
ejpam-3352	372	48	exists	exist	VERB
ejpam-3352	372	49	a	a	DET
ejpam-3352	372	50	locally	locally	ADV
ejpam-3352	372	51	finite	finite	ADJ
ejpam-3352	372	52	open	open	ADJ
ejpam-3352	372	53	refinement	refinement	PROPN
ejpam-3352	372	54	va	va	PROPN
ejpam-3352	372	55	=	=	PUNCT
ejpam-3352	372	56	{	{	PUNCT
ejpam-3352	372	57	vλ	vλ	INTJ
ejpam-3352	372	58	:	:	PUNCT
ejpam-3352	372	59	λ	λ	PROPN
ejpam-3352	372	60	∈	∈	PROPN
ejpam-3352	372	61	λ	λ	PROPN
ejpam-3352	372	62	}	}	PUNCT
ejpam-3352	372	63	∪	∪	NOUN
ejpam-3352	372	64	{	{	PUNCT
ejpam-3352	372	65	v	v	NOUN
ejpam-3352	372	66	}	}	PUNCT
ejpam-3352	372	67	of	of	ADP
ejpam-3352	372	68	ua	ua	PROPN
ejpam-3352	372	69	,	,	PUNCT
ejpam-3352	372	70	where	where	SCONJ
ejpam-3352	372	71	vλ	vλ	ADP
ejpam-3352	372	72	⊂	⊂	PRON
ejpam-3352	372	73	uα	uα	PROPN
ejpam-3352	372	74	and	and	CCONJ
ejpam-3352	372	75	v	v	ADP
ejpam-3352	372	76	⊂	⊂	PROPN
ejpam-3352	372	77	x	x	PUNCT
ejpam-3352	373	1	\	\	PROPN
ejpam-3352	373	2	b	b	X
ejpam-3352	373	3	such	such	ADJ
ejpam-3352	373	4	that	that	SCONJ
ejpam-3352	373	5	a	a	DET
ejpam-3352	373	6	\	\	PUNCT
ejpam-3352	374	1	[	[	X
ejpam-3352	374	2	(	(	PUNCT
ejpam-3352	374	3	∪{vλ	∪{vλ	NUM
ejpam-3352	374	4	:	:	PUNCT
ejpam-3352	374	5	λ	λ	PROPN
ejpam-3352	374	6	∈	∈	PROPN
ejpam-3352	374	7	λ	λ	NOUN
ejpam-3352	374	8	}	}	PUNCT
ejpam-3352	374	9	)	)	PUNCT
ejpam-3352	374	10	∪	∪	ADP
ejpam-3352	374	11	{	{	PUNCT
ejpam-3352	374	12	v	v	NOUN
ejpam-3352	374	13	}	}	PUNCT
ejpam-3352	374	14	]	]	PUNCT
ejpam-3352	374	15	∈	∈	PROPN
ejpam-3352	374	16	i.	i.	NOUN
ejpam-3352	374	17	let	let	VERB
ejpam-3352	374	18	a	a	DET
ejpam-3352	374	19	\	\	PUNCT
ejpam-3352	375	1	[	[	X
ejpam-3352	375	2	(	(	PUNCT
ejpam-3352	375	3	∪{vλ	∪{vλ	NUM
ejpam-3352	375	4	:	:	PUNCT
ejpam-3352	375	5	λ	λ	PROPN
ejpam-3352	375	6	∈	∈	PROPN
ejpam-3352	375	7	λ	λ	NOUN
ejpam-3352	375	8	}	}	PUNCT
ejpam-3352	375	9	)	)	PUNCT
ejpam-3352	375	10	∪	∪	ADP
ejpam-3352	375	11	{	{	PUNCT
ejpam-3352	375	12	v	v	NOUN
ejpam-3352	375	13	}	}	PUNCT
ejpam-3352	375	14	]	]	PUNCT
ejpam-3352	375	15	=	=	PUNCT
ejpam-3352	375	16	i.	i.	NOUN
ejpam-3352	375	17	then	then	ADV
ejpam-3352	375	18	i	i	PRON
ejpam-3352	375	19	∩	∩	PROPN
ejpam-3352	375	20	b	b	X
ejpam-3352	375	21	=	=	PUNCT
ejpam-3352	375	22	a	a	PRON
ejpam-3352	375	23	\	\	PUNCT
ejpam-3352	376	1	[	[	X
ejpam-3352	376	2	(	(	PUNCT
ejpam-3352	376	3	∪({vλ	∪({vλ	ADP
ejpam-3352	376	4	:	:	PUNCT
ejpam-3352	376	5	λ	λ	PROPN
ejpam-3352	376	6	∈	∈	PROPN
ejpam-3352	376	7	λ	λ	NOUN
ejpam-3352	376	8	}	}	PUNCT
ejpam-3352	376	9	)	)	PUNCT
ejpam-3352	376	10	∪	∪	ADP
ejpam-3352	376	11	{	{	PUNCT
ejpam-3352	376	12	v	v	NOUN
ejpam-3352	376	13	}	}	PUNCT
ejpam-3352	376	14	]	]	PUNCT
ejpam-3352	376	15	∩	∩	PROPN
ejpam-3352	376	16	b	b	X
ejpam-3352	376	17	=	=	SYM
ejpam-3352	376	18	a	a	DET
ejpam-3352	376	19	∩	∩	NOUN
ejpam-3352	376	20	(	(	PUNCT
ejpam-3352	376	21	x	x	SYM
ejpam-3352	376	22	\	\	PROPN
ejpam-3352	376	23	[	[	X
ejpam-3352	376	24	(	(	PUNCT
ejpam-3352	376	25	∪{vλ	∪{vλ	NUM
ejpam-3352	376	26	:	:	PUNCT
ejpam-3352	376	27	λ	λ	PROPN
ejpam-3352	376	28	∈	∈	PROPN
ejpam-3352	376	29	λ	λ	NOUN
ejpam-3352	376	30	}	}	PUNCT
ejpam-3352	376	31	)	)	PUNCT
ejpam-3352	376	32	∪	∪	ADP
ejpam-3352	376	33	{	{	PUNCT
ejpam-3352	376	34	v	v	NOUN
ejpam-3352	376	35	}	}	PUNCT
ejpam-3352	376	36	]	]	PUNCT
ejpam-3352	376	37	)	)	PUNCT
ejpam-3352	376	38	∩	∩	NOUN
ejpam-3352	376	39	b	b	PROPN
ejpam-3352	376	40	implies	imply	VERB
ejpam-3352	376	41	that	that	SCONJ
ejpam-3352	376	42	i	i	PRON
ejpam-3352	376	43	∩	∩	NOUN
ejpam-3352	376	44	b	b	X
ejpam-3352	376	45	=	=	PUNCT
ejpam-3352	376	46	a	a	DET
ejpam-3352	376	47	∩	∩	NOUN
ejpam-3352	376	48	[	[	X
ejpam-3352	376	49	x	x	X
ejpam-3352	376	50	\	\	PROPN
ejpam-3352	376	51	(	(	PUNCT
ejpam-3352	376	52	∪{vλ	∪{vλ	NUM
ejpam-3352	376	53	:	:	PUNCT
ejpam-3352	376	54	λ	λ	PROPN
ejpam-3352	376	55	∈	∈	PROPN
ejpam-3352	376	56	λ	λ	NOUN
ejpam-3352	376	57	}	}	PUNCT
ejpam-3352	376	58	)	)	PUNCT
ejpam-3352	376	59	∩	∩	NOUN
ejpam-3352	376	60	{	{	PUNCT
ejpam-3352	376	61	x	x	SYM
ejpam-3352	376	62	\	\	PROPN
ejpam-3352	376	63	v	v	ADJ
ejpam-3352	376	64	}	}	PUNCT
ejpam-3352	376	65	∩	∩	ADJ
ejpam-3352	376	66	b	b	NOUN
ejpam-3352	376	67	]	]	PUNCT
ejpam-3352	376	68	.	.	PUNCT
ejpam-3352	377	1	it	it	PRON
ejpam-3352	377	2	follows	follow	VERB
ejpam-3352	377	3	that	that	SCONJ
ejpam-3352	377	4	i	i	PRON
ejpam-3352	377	5	∩	∩	NOUN
ejpam-3352	377	6	b	b	X
ejpam-3352	377	7	=	=	SYM
ejpam-3352	377	8	(	(	PUNCT
ejpam-3352	377	9	a	a	DET
ejpam-3352	377	10	∩	∩	ADJ
ejpam-3352	377	11	b	b	X
ejpam-3352	377	12	)	)	PUNCT
ejpam-3352	377	13	\	\	PROPN
ejpam-3352	378	1	∪{vλ	∪{vλ	NUM
ejpam-3352	378	2	:	:	PUNCT
ejpam-3352	378	3	λ	λ	PROPN
ejpam-3352	378	4	∈	∈	PROPN
ejpam-3352	378	5	λ	λ	PROPN
ejpam-3352	378	6	}	}	PUNCT
ejpam-3352	378	7	∈	∈	PROPN
ejpam-3352	378	8	i.	i.	NOUN
ejpam-3352	378	9	since	since	SCONJ
ejpam-3352	378	10	vλ	vλ	INTJ
ejpam-3352	378	11	⊂	⊂	PRON
ejpam-3352	378	12	vλ	vλ	ADP
ejpam-3352	378	13	∪	∪	PROPN
ejpam-3352	378	14	v	v	NOUN
ejpam-3352	378	15	,	,	PUNCT
ejpam-3352	378	16	v	v	NOUN
ejpam-3352	378	17	=	=	PUNCT
ejpam-3352	378	18	{	{	PUNCT
ejpam-3352	378	19	vλ	vλ	INTJ
ejpam-3352	378	20	:	:	PUNCT
ejpam-3352	378	21	λ	λ	PROPN
ejpam-3352	378	22	∈	∈	PROPN
ejpam-3352	378	23	λ	λ	PROPN
ejpam-3352	378	24	}	}	PUNCT
ejpam-3352	378	25	is	be	AUX
ejpam-3352	378	26	locally	locally	ADV
ejpam-3352	378	27	finite	finite	ADJ
ejpam-3352	378	28	open	open	ADJ
ejpam-3352	378	29	by	by	ADP
ejpam-3352	378	30	theorem	theorem	NOUN
ejpam-3352	378	31	2	2	NUM
ejpam-3352	378	32	which	which	PRON
ejpam-3352	378	33	refines	refine	VERB
ejpam-3352	378	34	u	u	PRON
ejpam-3352	378	35	.	.	PUNCT
ejpam-3352	379	1	hence	hence	ADV
ejpam-3352	379	2	a	a	DET
ejpam-3352	379	3	∩b	∩b	NOUN
ejpam-3352	379	4	is	be	AUX
ejpam-3352	379	5	β1i	β1i	NOUN
ejpam-3352	379	6	-	-	ADJ
ejpam-3352	379	7	paracompact	paracompact	ADJ
ejpam-3352	379	8	.	.	PUNCT
ejpam-3352	380	1	corollary	corollary	ADJ
ejpam-3352	380	2	12	12	NUM
ejpam-3352	380	3	.	.	PUNCT
ejpam-3352	381	1	let	let	VERB
ejpam-3352	381	2	f	f	NOUN
ejpam-3352	381	3	:	:	PUNCT
ejpam-3352	381	4	(	(	PUNCT
ejpam-3352	381	5	x	x	X
ejpam-3352	381	6	,	,	PUNCT
ejpam-3352	381	7	τ)→	τ)→	PROPN
ejpam-3352	381	8	(	(	PUNCT
ejpam-3352	381	9	y	y	PROPN
ejpam-3352	381	10	,	,	PUNCT
ejpam-3352	381	11	σ	σ	PROPN
ejpam-3352	381	12	,	,	PUNCT
ejpam-3352	381	13	j	j	PROPN
ejpam-3352	381	14	)	)	PUNCT
ejpam-3352	381	15	be	be	AUX
ejpam-3352	381	16	a	a	DET
ejpam-3352	381	17	pre	pre	NOUN
ejpam-3352	381	18	β	β	NOUN
ejpam-3352	381	19	-	-	ADJ
ejpam-3352	381	20	open	open	ADJ
ejpam-3352	381	21	,	,	PUNCT
ejpam-3352	381	22	continuous	continuous	ADJ
ejpam-3352	381	23	,	,	PUNCT
ejpam-3352	381	24	bijective	bijective	ADJ
ejpam-3352	381	25	function	function	NOUN
ejpam-3352	381	26	.	.	PUNCT
ejpam-3352	382	1	if	if	SCONJ
ejpam-3352	382	2	a	a	PRON
ejpam-3352	382	3	is	is	ADV
ejpam-3352	382	4	β1j	β1j	PUNCT
ejpam-3352	382	5	-paracompact	-paracompact	ADJ
ejpam-3352	382	6	relative	relative	ADJ
ejpam-3352	382	7	to	to	ADP
ejpam-3352	382	8	y	y	PROPN
ejpam-3352	382	9	,	,	PUNCT
ejpam-3352	382	10	then	then	ADV
ejpam-3352	382	11	f−1(a	f−1(a	PROPN
ejpam-3352	382	12	)	)	PUNCT
ejpam-3352	382	13	is	be	AUX
ejpam-3352	382	14	β1f	β1f	PRON
ejpam-3352	382	15	−1(j	−1(j	NOUN
ejpam-3352	382	16	)	)	PUNCT
ejpam-3352	382	17	-paracompact	-paracompact	NOUN
ejpam-3352	382	18	relative	relative	ADJ
ejpam-3352	382	19	to	to	ADP
ejpam-3352	382	20	x.	x.	NOUN
ejpam-3352	382	21	references	reference	NOUN
ejpam-3352	382	22	144	144	NUM
ejpam-3352	382	23	acknowledgements	acknowledgement	NOUN
ejpam-3352	382	24	the	the	DET
ejpam-3352	382	25	author	author	NOUN
ejpam-3352	382	26	would	would	AUX
ejpam-3352	382	27	like	like	VERB
ejpam-3352	382	28	to	to	PART
ejpam-3352	382	29	thank	thank	VERB
ejpam-3352	382	30	the	the	DET
ejpam-3352	382	31	referees	referee	NOUN
ejpam-3352	382	32	for	for	ADP
ejpam-3352	382	33	their	their	PRON
ejpam-3352	382	34	helpful	helpful	ADJ
ejpam-3352	382	35	suggestions	suggestion	NOUN
ejpam-3352	382	36	.	.	PUNCT
ejpam-3352	383	1	references	reference	NOUN
ejpam-3352	383	2	[	[	X
ejpam-3352	383	3	1	1	NUM
ejpam-3352	383	4	]	]	PUNCT
ejpam-3352	383	5	h.	h.	PROPN
ejpam-3352	383	6	h.	h.	PROPN
ejpam-3352	383	7	aljarrah	aljarrah	PROPN
ejpam-3352	383	8	.	.	PUNCT
ejpam-3352	384	1	β1	β1	NOUN
ejpam-3352	384	2	-	-	PUNCT
ejpam-3352	384	3	paracompact	paracompact	NOUN
ejpam-3352	384	4	spaces	space	NOUN
ejpam-3352	384	5	.	.	PUNCT
ejpam-3352	385	1	j.	j.	PROPN
ejpam-3352	385	2	nonlinear	nonlinear	PROPN
ejpam-3352	385	3	sci	sci	PROPN
ejpam-3352	385	4	.	.	PUNCT
ejpam-3352	385	5	appl	appl	PROPN
ejpam-3352	385	6	,	,	PUNCT
ejpam-3352	385	7	9:1728–1734	9:1728–1734	NUM
ejpam-3352	385	8	,	,	PUNCT
ejpam-3352	385	9	2016	2016	NUM
ejpam-3352	385	10	.	.	PUNCT
ejpam-3352	386	1	[	[	X
ejpam-3352	386	2	2	2	X
ejpam-3352	386	3	]	]	PUNCT
ejpam-3352	386	4	d.	d.	PROPN
ejpam-3352	386	5	andrijević.	andrijević.	PROPN
ejpam-3352	386	6	semi	semi	ADJ
ejpam-3352	386	7	-	-	ADJ
ejpam-3352	386	8	preopen	preopen	ADJ
ejpam-3352	386	9	sets	set	NOUN
ejpam-3352	386	10	.	.	PUNCT
ejpam-3352	387	1	mat	mat	NOUN
ejpam-3352	387	2	.	.	PUNCT
ejpam-3352	387	3	vesn	vesn	PROPN
ejpam-3352	387	4	.	.	PUNCT
ejpam-3352	387	5	,	,	PUNCT
ejpam-3352	387	6	38:24–32	38:24–32	NUM
ejpam-3352	387	7	,	,	PUNCT
ejpam-3352	387	8	1986	1986	NUM
ejpam-3352	387	9	.	.	PUNCT
ejpam-3352	388	1	[	[	X
ejpam-3352	388	2	3	3	NUM
ejpam-3352	388	3	]	]	X
ejpam-3352	388	4	a.	a.	NOUN
ejpam-3352	388	5	v.	v.	ADP
ejpam-3352	388	6	arkhangelski	arkhangelski	PROPN
ejpam-3352	388	7	and	and	CCONJ
ejpam-3352	388	8	v.	v.	ADP
ejpam-3352	388	9	i.	i.	PROPN
ejpam-3352	388	10	ponomarev	ponomarev	PROPN
ejpam-3352	388	11	.	.	PUNCT
ejpam-3352	389	1	fundamentals	fundamental	NOUN
ejpam-3352	389	2	of	of	ADP
ejpam-3352	389	3	general	general	ADJ
ejpam-3352	389	4	topology	topology	NOUN
ejpam-3352	389	5	problems	problem	NOUN
ejpam-3352	389	6	and	and	CCONJ
ejpam-3352	389	7	exercises	exercise	NOUN
ejpam-3352	389	8	.	.	PUNCT
ejpam-3352	390	1	int	int	NOUN
ejpam-3352	390	2	.	.	PUNCT
ejpam-3352	391	1	hindustan	hindustan	PROPN
ejpam-3352	391	2	,	,	PUNCT
ejpam-3352	391	3	india	india	PROPN
ejpam-3352	391	4	,	,	PUNCT
ejpam-3352	391	5	1984	1984	NUM
ejpam-3352	391	6	.	.	PUNCT
ejpam-3352	392	1	[	[	X
ejpam-3352	392	2	4	4	NUM
ejpam-3352	392	3	]	]	X
ejpam-3352	392	4	n.	n.	NOUN
ejpam-3352	392	5	bourbaki	bourbaki	PROPN
ejpam-3352	392	6	.	.	PUNCT
ejpam-3352	393	1	general	general	ADJ
ejpam-3352	393	2	topology	topology	PROPN
ejpam-3352	393	3	.	.	PUNCT
ejpam-3352	394	1	hermann	hermann	PROPN
ejpam-3352	394	2	addison	addison	PROPN
ejpam-3352	394	3	wesley	wesley	PROPN
ejpam-3352	394	4	,	,	PUNCT
ejpam-3352	394	5	massachusets	massachusets	PROPN
ejpam-3352	394	6	,	,	PUNCT
ejpam-3352	394	7	1966	1966	NUM
ejpam-3352	394	8	.	.	PUNCT
ejpam-3352	395	1	[	[	X
ejpam-3352	395	2	5	5	NUM
ejpam-3352	395	3	]	]	X
ejpam-3352	395	4	n.	n.	NOUN
ejpam-3352	395	5	bourbaki	bourbaki	PROPN
ejpam-3352	395	6	.	.	PUNCT
ejpam-3352	396	1	general	general	ADJ
ejpam-3352	396	2	topology	topology	PROPN
ejpam-3352	396	3	part	part	PROPN
ejpam-3352	396	4	i.	i.	PROPN
ejpam-3352	396	5	addison	addison	PROPN
ejpam-3352	396	6	-	-	PUNCT
ejpam-3352	396	7	wesley	wesley	PROPN
ejpam-3352	396	8	,	,	PUNCT
ejpam-3352	396	9	reading	reading	NOUN
ejpam-3352	396	10	,	,	PUNCT
ejpam-3352	396	11	mass	mass	PROPN
ejpam-3352	396	12	,	,	PUNCT
ejpam-3352	396	13	1966	1966	NUM
ejpam-3352	396	14	.	.	PUNCT
ejpam-3352	397	1	[	[	X
ejpam-3352	397	2	6	6	NUM
ejpam-3352	397	3	]	]	PUNCT
ejpam-3352	397	4	m.	m.	NOUN
ejpam-3352	397	5	caldas	caldas	PROPN
ejpam-3352	397	6	and	and	CCONJ
ejpam-3352	397	7	s.	s.	PROPN
ejpam-3352	397	8	jafari	jafari	PROPN
ejpam-3352	397	9	.	.	PUNCT
ejpam-3352	398	1	weak	weak	ADJ
ejpam-3352	398	2	and	and	CCONJ
ejpam-3352	398	3	strong	strong	ADJ
ejpam-3352	398	4	forms	form	NOUN
ejpam-3352	398	5	of	of	ADP
ejpam-3352	398	6	β	β	NOUN
ejpam-3352	398	7	-	-	NOUN
ejpam-3352	398	8	irresolutess	irresolutess	ADJ
ejpam-3352	398	9	.	.	PUNCT
ejpam-3352	399	1	arab	arab	PROPN
ejpam-3352	399	2	.	.	PUNCT
ejpam-3352	400	1	j.	j.	PROPN
ejpam-3352	400	2	sci	sci	PROPN
ejpam-3352	400	3	.	.	PUNCT
ejpam-3352	401	1	eng	eng	PROPN
ejpam-3352	401	2	,	,	PUNCT
ejpam-3352	401	3	31:31–39	31:31–39	PROPN
ejpam-3352	401	4	,	,	PUNCT
ejpam-3352	401	5	2006	2006	NUM
ejpam-3352	401	6	.	.	PUNCT
ejpam-3352	402	1	[	[	X
ejpam-3352	402	2	7	7	X
ejpam-3352	402	3	]	]	X
ejpam-3352	402	4	d.	d.	PROPN
ejpam-3352	402	5	carnahan	carnahan	PROPN
ejpam-3352	402	6	.	.	PUNCT
ejpam-3352	403	1	locally	locally	ADV
ejpam-3352	403	2	nearly	nearly	ADV
ejpam-3352	403	3	-	-	PUNCT
ejpam-3352	403	4	compact	compact	ADJ
ejpam-3352	403	5	spaces	space	NOUN
ejpam-3352	403	6	.	.	PUNCT
ejpam-3352	404	1	boll	boll	NOUN
ejpam-3352	404	2	.	.	PUNCT
ejpam-3352	405	1	un	un	PROPN
ejpam-3352	405	2	.	.	PROPN
ejpam-3352	405	3	mat	mat	PROPN
ejpam-3352	405	4	.	.	PUNCT
ejpam-3352	405	5	ital	ital	PROPN
ejpam-3352	405	6	,	,	PUNCT
ejpam-3352	405	7	6:146–153	6:146–153	PROPN
ejpam-3352	405	8	,	,	PUNCT
ejpam-3352	405	9	1972	1972	NUM
ejpam-3352	405	10	.	.	PUNCT
ejpam-3352	406	1	[	[	X
ejpam-3352	406	2	8	8	NUM
ejpam-3352	406	3	]	]	PUNCT
ejpam-3352	406	4	m.	m.	NOUN
ejpam-3352	406	5	e.	e.	PROPN
ejpam-3352	406	6	abd	abd	PROPN
ejpam-3352	406	7	el	el	PROPN
ejpam-3352	406	8	-	-	PROPN
ejpam-3352	406	9	monsef	monsef	PROPN
ejpam-3352	406	10	,	,	PUNCT
ejpam-3352	406	11	s.	s.	PROPN
ejpam-3352	406	12	n.	n.	PROPN
ejpam-3352	406	13	el	el	PROPN
ejpam-3352	406	14	-	-	PROPN
ejpam-3352	406	15	deeb	deeb	PROPN
ejpam-3352	406	16	,	,	PUNCT
ejpam-3352	406	17	and	and	CCONJ
ejpam-3352	406	18	r.	r.	PROPN
ejpam-3352	406	19	a.	a.	PROPN
ejpam-3352	406	20	mahmoud	mahmoud	PROPN
ejpam-3352	406	21	.	.	PUNCT
ejpam-3352	407	1	β	β	X
ejpam-3352	407	2	-	-	ADJ
ejpam-3352	407	3	open	open	ADJ
ejpam-3352	407	4	sets	set	NOUN
ejpam-3352	407	5	and	and	CCONJ
ejpam-3352	407	6	βcontinuous	βcontinuous	ADJ
ejpam-3352	407	7	mapping	mapping	NOUN
ejpam-3352	407	8	.	.	PUNCT
ejpam-3352	408	1	bull	bull	NOUN
ejpam-3352	408	2	.	.	PUNCT
ejpam-3352	409	1	fac	fac	PROPN
ejpam-3352	409	2	.	.	PROPN
ejpam-3352	409	3	sci.assint	sci.assint	PROPN
ejpam-3352	409	4	univ	univ	PROPN
ejpam-3352	409	5	,	,	PUNCT
ejpam-3352	409	6	12:77–90	12:77–90	NUM
ejpam-3352	409	7	,	,	PUNCT
ejpam-3352	409	8	1983	1983	NUM
ejpam-3352	409	9	.	.	PUNCT
ejpam-3352	410	1	[	[	X
ejpam-3352	410	2	9	9	NUM
ejpam-3352	410	3	]	]	PUNCT
ejpam-3352	410	4	t.	t.	PROPN
ejpam-3352	410	5	r.	r.	PROPN
ejpam-3352	410	6	hamlett	hamlett	PROPN
ejpam-3352	410	7	and	and	CCONJ
ejpam-3352	410	8	d.	d.	PROPN
ejpam-3352	410	9	janković.	janković.	PROPN
ejpam-3352	410	10	on	on	ADP
ejpam-3352	410	11	almost	almost	ADV
ejpam-3352	410	12	paracompact	paracompact	ADJ
ejpam-3352	410	13	and	and	CCONJ
ejpam-3352	410	14	para	para	NOUN
ejpam-3352	410	15	-	-	PUNCT
ejpam-3352	410	16	h	h	NOUN
ejpam-3352	410	17	-	-	PUNCT
ejpam-3352	410	18	closed	closed	ADJ
ejpam-3352	410	19	spaces	space	NOUN
ejpam-3352	410	20	.	.	PUNCT
ejpam-3352	411	1	quest	quest	NOUN
ejpam-3352	411	2	.	.	PUNCT
ejpam-3352	412	1	answers	answer	VERB
ejpam-3352	412	2	gen	gen	PROPN
ejpam-3352	412	3	.	.	PROPN
ejpam-3352	412	4	topology	topology	PROPN
ejpam-3352	412	5	,	,	PUNCT
ejpam-3352	412	6	11:139–143	11:139–143	PROPN
ejpam-3352	412	7	,	,	PUNCT
ejpam-3352	412	8	1993	1993	NUM
ejpam-3352	412	9	.	.	PUNCT
ejpam-3352	413	1	[	[	X
ejpam-3352	413	2	10	10	NUM
ejpam-3352	413	3	]	]	X
ejpam-3352	413	4	d.	d.	PROPN
ejpam-3352	413	5	janković	janković	PROPN
ejpam-3352	413	6	and	and	CCONJ
ejpam-3352	413	7	t.	t.	PROPN
ejpam-3352	413	8	r.	r.	PROPN
ejpam-3352	413	9	hamlett	hamlett	PROPN
ejpam-3352	413	10	.	.	PUNCT
ejpam-3352	414	1	new	new	ADJ
ejpam-3352	414	2	topologies	topology	NOUN
ejpam-3352	414	3	from	from	ADP
ejpam-3352	414	4	old	old	ADJ
ejpam-3352	414	5	via	via	ADP
ejpam-3352	414	6	ideals	ideal	NOUN
ejpam-3352	414	7	.	.	PUNCT
ejpam-3352	415	1	amer	amer	PROPN
ejpam-3352	415	2	.	.	PUNCT
ejpam-3352	415	3	math	math	PROPN
ejpam-3352	415	4	.	.	PUNCT
ejpam-3352	416	1	monthly	monthly	ADJ
ejpam-3352	416	2	,	,	PUNCT
ejpam-3352	416	3	97(4):295–310	97(4):295–310	PROPN
ejpam-3352	416	4	,	,	PUNCT
ejpam-3352	416	5	1990	1990	NUM
ejpam-3352	416	6	.	.	PUNCT
ejpam-3352	417	1	[	[	X
ejpam-3352	417	2	11	11	NUM
ejpam-3352	417	3	]	]	X
ejpam-3352	417	4	d.	d.	PROPN
ejpam-3352	417	5	janković	janković	PROPN
ejpam-3352	417	6	and	and	CCONJ
ejpam-3352	417	7	t.	t.	PROPN
ejpam-3352	417	8	r.	r.	PROPN
ejpam-3352	417	9	hamlett	hamlett	PROPN
ejpam-3352	417	10	.	.	PUNCT
ejpam-3352	418	1	compatible	compatible	ADJ
ejpam-3352	418	2	extensions	extension	NOUN
ejpam-3352	418	3	of	of	ADP
ejpam-3352	418	4	ideals	ideal	NOUN
ejpam-3352	418	5	.	.	PUNCT
ejpam-3352	419	1	boll	boll	NOUN
ejpam-3352	419	2	.	.	PUNCT
ejpam-3352	420	1	u.	u.	PROPN
ejpam-3352	420	2	m.	m.	PROPN
ejpam-3352	421	1	i	i	PRON
ejpam-3352	421	2	,	,	PUNCT
ejpam-3352	421	3	7(6	7(6	PROPN
ejpam-3352	421	4	-	-	PUNCT
ejpam-3352	421	5	b):453–465	b):453–465	NOUN
ejpam-3352	421	6	,	,	PUNCT
ejpam-3352	421	7	1992	1992	NUM
ejpam-3352	421	8	.	.	PUNCT
ejpam-3352	422	1	[	[	X
ejpam-3352	422	2	12	12	NUM
ejpam-3352	422	3	]	]	PUNCT
ejpam-3352	422	4	k.	k.	PROPN
ejpam-3352	422	5	kuratowski	kuratowski	PROPN
ejpam-3352	422	6	.	.	PUNCT
ejpam-3352	423	1	topology	topology	PROPN
ejpam-3352	424	1	i	i	PRON
ejpam-3352	424	2	,	,	PUNCT
ejpam-3352	424	3	warszawa	warszawa	PROPN
ejpam-3352	424	4	.	.	PUNCT
ejpam-3352	424	5	warszawa	warszawa	PROPN
ejpam-3352	424	6	,	,	PUNCT
ejpam-3352	424	7	1933	1933	NUM
ejpam-3352	424	8	.	.	PUNCT
ejpam-3352	425	1	[	[	X
ejpam-3352	425	2	13	13	NUM
ejpam-3352	425	3	]	]	X
ejpam-3352	425	4	n.	n.	PROPN
ejpam-3352	425	5	levine	levine	PROPN
ejpam-3352	425	6	.	.	PUNCT
ejpam-3352	426	1	semi	semi	ADJ
ejpam-3352	426	2	-	-	ADJ
ejpam-3352	426	3	open	open	ADJ
ejpam-3352	426	4	sets	set	NOUN
ejpam-3352	426	5	and	and	CCONJ
ejpam-3352	426	6	semi	semi	ADJ
ejpam-3352	426	7	-	-	NOUN
ejpam-3352	426	8	continuity	continuity	NOUN
ejpam-3352	426	9	in	in	ADP
ejpam-3352	426	10	topological	topological	ADJ
ejpam-3352	426	11	spaces	space	NOUN
ejpam-3352	426	12	.	.	PUNCT
ejpam-3352	427	1	amer	amer	PROPN
ejpam-3352	427	2	.	.	PUNCT
ejpam-3352	427	3	math	math	PROPN
ejpam-3352	427	4	.	.	PUNCT
ejpam-3352	428	1	monthly	monthly	ADJ
ejpam-3352	428	2	,	,	PUNCT
ejpam-3352	428	3	70:36–41	70:36–41	NUM
ejpam-3352	428	4	,	,	PUNCT
ejpam-3352	428	5	1963	1963	NUM
ejpam-3352	428	6	.	.	PUNCT
ejpam-3352	429	1	[	[	X
ejpam-3352	429	2	14	14	NUM
ejpam-3352	429	3	]	]	PUNCT
ejpam-3352	429	4	a.	a.	NOUN
ejpam-3352	429	5	n.	n.	PROPN
ejpam-3352	429	6	geaisa	geaisa	PROPN
ejpam-3352	429	7	m.	m.	PROPN
ejpam-3352	429	8	e.	e.	PROPN
ejpam-3352	429	9	abd	abd	PROPN
ejpam-3352	430	1	el	el	PROPN
ejpam-3352	430	2	-	-	PROPN
ejpam-3352	430	3	monsef	monsef	PROPN
ejpam-3352	430	4	and	and	CCONJ
ejpam-3352	430	5	r.	r.	PROPN
ejpam-3352	430	6	a.	a.	PROPN
ejpam-3352	430	7	mahmoud	mahmoud	PROPN
ejpam-3352	430	8	.	.	PUNCT
ejpam-3352	431	1	β	β	X
ejpam-3352	431	2	-	-	ADJ
ejpam-3352	431	3	regular	regular	ADJ
ejpam-3352	431	4	spaces	space	NOUN
ejpam-3352	431	5	.	.	PUNCT
ejpam-3352	432	1	proc	proc	NOUN
ejpam-3352	432	2	.	.	PUNCT
ejpam-3352	433	1	math	math	NOUN
ejpam-3352	433	2	.	.	PUNCT
ejpam-3352	434	1	phys	phy	NOUN
ejpam-3352	434	2	.	.	PUNCT
ejpam-3352	435	1	soc	soc	PROPN
ejpam-3352	435	2	.	.	PUNCT
ejpam-3352	436	1	egypt	egypt	PROPN
ejpam-3352	436	2	,	,	PUNCT
ejpam-3352	436	3	60:705–717	60:705–717	PROPN
ejpam-3352	436	4	,	,	PUNCT
ejpam-3352	436	5	1985	1985	NUM
ejpam-3352	436	6	.	.	PUNCT
ejpam-3352	437	1	[	[	X
ejpam-3352	437	2	15	15	NUM
ejpam-3352	437	3	]	]	X
ejpam-3352	437	4	r.	r.	PROPN
ejpam-3352	437	5	a.	a.	PROPN
ejpam-3352	437	6	mahmoud	mahmoud	PROPN
ejpam-3352	437	7	and	and	CCONJ
ejpam-3352	437	8	m.	m.	PROPN
ejpam-3352	437	9	e.	e.	PROPN
ejpam-3352	437	10	abd	abd	PROPN
ejpam-3352	437	11	el	el	PROPN
ejpam-3352	437	12	-	-	PROPN
ejpam-3352	437	13	monsef	monsef	ADJ
ejpam-3352	437	14	.	.	PUNCT
ejpam-3352	438	1	β	β	X
ejpam-3352	438	2	-	-	PUNCT
ejpam-3352	438	3	irresolute	irresolute	ADJ
ejpam-3352	438	4	and	and	CCONJ
ejpam-3352	438	5	β	β	NOUN
ejpam-3352	438	6	-	-	ADJ
ejpam-3352	438	7	topological	topological	ADJ
ejpam-3352	438	8	invariant	invariant	ADJ
ejpam-3352	438	9	.	.	PUNCT
ejpam-3352	439	1	proc	proc	PROPN
ejpam-3352	439	2	.	.	PUNCT
ejpam-3352	440	1	pakistan	pakistan	PROPN
ejpam-3352	440	2	acad	acad	PROPN
ejpam-3352	440	3	.	.	PUNCT
ejpam-3352	441	1	sci	sci	PROPN
ejpam-3352	441	2	,	,	PUNCT
ejpam-3352	441	3	27:285–296	27:285–296	NUM
ejpam-3352	441	4	,	,	PUNCT
ejpam-3352	441	5	1990	1990	NUM
ejpam-3352	441	6	.	.	PUNCT
ejpam-3352	442	1	[	[	X
ejpam-3352	442	2	16	16	NUM
ejpam-3352	442	3	]	]	X
ejpam-3352	442	4	g.	g.	PROPN
ejpam-3352	442	5	b.	b.	PROPN
ejpam-3352	442	6	navalagi	navalagi	PROPN
ejpam-3352	442	7	.	.	PUNCT
ejpam-3352	443	1	semi	semi	ADJ
ejpam-3352	443	2	-	-	ADJ
ejpam-3352	443	3	precontinuous	precontinuous	ADJ
ejpam-3352	443	4	functions	function	NOUN
ejpam-3352	443	5	and	and	CCONJ
ejpam-3352	443	6	properties	property	NOUN
ejpam-3352	443	7	of	of	ADP
ejpam-3352	443	8	generalized	generalize	VERB
ejpam-3352	443	9	semipreclosed	semipreclose	VERB
ejpam-3352	443	10	sets	set	NOUN
ejpam-3352	443	11	in	in	ADP
ejpam-3352	443	12	topological	topological	ADJ
ejpam-3352	443	13	spaces	space	NOUN
ejpam-3352	443	14	.	.	PUNCT
ejpam-3352	444	1	int	int	NOUN
ejpam-3352	444	2	.	.	PUNCT
ejpam-3352	445	1	j.	j.	PROPN
ejpam-3352	445	2	math	math	PROPN
ejpam-3352	445	3	.	.	PUNCT
ejpam-3352	446	1	math	math	NOUN
ejpam-3352	446	2	.	.	PUNCT
ejpam-3352	447	1	sci	sci	PROPN
ejpam-3352	447	2	,	,	PUNCT
ejpam-3352	447	3	29:85–98	29:85–98	NUM
ejpam-3352	447	4	,	,	PUNCT
ejpam-3352	447	5	2002	2002	NUM
ejpam-3352	447	6	.	.	PUNCT
ejpam-3352	448	1	[	[	X
ejpam-3352	448	2	17	17	NUM
ejpam-3352	448	3	]	]	X
ejpam-3352	448	4	o.	o.	PROPN
ejpam-3352	448	5	njastad	njastad	PROPN
ejpam-3352	448	6	.	.	PUNCT
ejpam-3352	449	1	on	on	ADP
ejpam-3352	449	2	some	some	DET
ejpam-3352	449	3	classes	class	NOUN
ejpam-3352	449	4	of	of	ADP
ejpam-3352	449	5	nearly	nearly	ADV
ejpam-3352	449	6	open	open	ADJ
ejpam-3352	449	7	sets	set	NOUN
ejpam-3352	449	8	.	.	PUNCT
ejpam-3352	450	1	pac	pac	PROPN
ejpam-3352	450	2	.	.	PUNCT
ejpam-3352	451	1	j.	j.	PROPN
ejpam-3352	451	2	math	math	PROPN
ejpam-3352	451	3	,	,	PUNCT
ejpam-3352	451	4	15:961–970	15:961–970	PROPN
ejpam-3352	451	5	,	,	PUNCT
ejpam-3352	451	6	1965	1965	NUM
ejpam-3352	451	7	.	.	PUNCT
ejpam-3352	452	1	references	reference	NOUN
ejpam-3352	452	2	145	145	NUM
ejpam-3352	452	3	[	[	X
ejpam-3352	452	4	18	18	NUM
ejpam-3352	452	5	]	]	PUNCT
ejpam-3352	452	6	t.	t.	PROPN
ejpam-3352	452	7	noiri	noiri	PROPN
ejpam-3352	452	8	.	.	PUNCT
ejpam-3352	453	1	completely	completely	ADV
ejpam-3352	453	2	continuous	continuous	ADJ
ejpam-3352	453	3	image	image	NOUN
ejpam-3352	453	4	of	of	ADP
ejpam-3352	453	5	nearly	nearly	ADV
ejpam-3352	453	6	paracompact	paracompact	ADJ
ejpam-3352	453	7	space	space	NOUN
ejpam-3352	453	8	.	.	PUNCT
ejpam-3352	454	1	mat	mat	NOUN
ejpam-3352	454	2	.	.	PUNCT
ejpam-3352	454	3	vesn	vesn	PROPN
ejpam-3352	454	4	,	,	PUNCT
ejpam-3352	454	5	29:59–64	29:59–64	PROPN
ejpam-3352	454	6	,	,	PUNCT
ejpam-3352	454	7	1977	1977	NUM
ejpam-3352	454	8	.	.	PUNCT
ejpam-3352	455	1	[	[	X
ejpam-3352	455	2	19	19	NUM
ejpam-3352	455	3	]	]	PUNCT
ejpam-3352	455	4	t.	t.	PROPN
ejpam-3352	455	5	noiri	noiri	PROPN
ejpam-3352	455	6	.	.	PUNCT
ejpam-3352	456	1	weak	weak	ADJ
ejpam-3352	456	2	and	and	CCONJ
ejpam-3352	456	3	strong	strong	ADJ
ejpam-3352	456	4	forms	form	NOUN
ejpam-3352	456	5	of	of	ADP
ejpam-3352	456	6	β	β	NOUN
ejpam-3352	456	7	-	-	PUNCT
ejpam-3352	456	8	irressolute	irressolute	ADJ
ejpam-3352	456	9	functions	function	NOUN
ejpam-3352	456	10	.	.	PUNCT
ejpam-3352	457	1	acta	acta	PROPN
ejpam-3352	457	2	math	math	PROPN
ejpam-3352	457	3	.	.	PUNCT
ejpam-3352	458	1	hungar	hungar	NOUN
ejpam-3352	458	2	,	,	PUNCT
ejpam-3352	458	3	99:315–328	99:315–328	PROPN
ejpam-3352	458	4	,	,	PUNCT
ejpam-3352	458	5	2003	2003	NUM
ejpam-3352	458	6	.	.	PUNCT
ejpam-3352	459	1	[	[	X
ejpam-3352	459	2	20	20	NUM
ejpam-3352	459	3	]	]	X
ejpam-3352	459	4	n.	n.	NOUN
ejpam-3352	459	5	sathiyasundari	sathiyasundari	PROPN
ejpam-3352	459	6	and	and	CCONJ
ejpam-3352	459	7	v.	v.	ADP
ejpam-3352	459	8	renukadevi	renukadevi	NOUN
ejpam-3352	459	9	.	.	PUNCT
ejpam-3352	460	1	s1	s1	NOUN
ejpam-3352	460	2	-	-	PUNCT
ejpam-3352	460	3	paracompactness	paracompactness	NOUN
ejpam-3352	460	4	with	with	ADP
ejpam-3352	460	5	respect	respect	NOUN
ejpam-3352	460	6	to	to	ADP
ejpam-3352	460	7	an	an	DET
ejpam-3352	460	8	ideal	ideal	NOUN
ejpam-3352	460	9	.	.	PUNCT
ejpam-3352	461	1	acta	acta	PROPN
ejpam-3352	461	2	universitatis	universitatis	PROPN
ejpam-3352	461	3	apulensis	apulensis	NOUN
ejpam-3352	461	4	,	,	PUNCT
ejpam-3352	461	5	35:215–227	35:215–227	PROPN
ejpam-3352	461	6	,	,	PUNCT
ejpam-3352	461	7	2013	2013	NUM
ejpam-3352	461	8	.	.	PUNCT
ejpam-3352	462	1	[	[	X
ejpam-3352	462	2	21	21	NUM
ejpam-3352	462	3	]	]	PUNCT
ejpam-3352	462	4	m.	m.	NOUN
ejpam-3352	462	5	h.	h.	PROPN
ejpam-3352	462	6	stone	stone	PROPN
ejpam-3352	462	7	.	.	PUNCT
ejpam-3352	463	1	applications	application	NOUN
ejpam-3352	463	2	of	of	ADP
ejpam-3352	463	3	the	the	DET
ejpam-3352	463	4	theory	theory	NOUN
ejpam-3352	463	5	of	of	ADP
ejpam-3352	463	6	boolean	boolean	ADJ
ejpam-3352	463	7	rings	ring	NOUN
ejpam-3352	463	8	to	to	ADP
ejpam-3352	463	9	general	general	ADJ
ejpam-3352	463	10	topology	topology	NOUN
ejpam-3352	463	11	.	.	PUNCT
ejpam-3352	464	1	trans	trans	PROPN
ejpam-3352	464	2	.	.	PUNCT
ejpam-3352	465	1	amer	amer	PROPN
ejpam-3352	465	2	.	.	PUNCT
ejpam-3352	465	3	math	math	PROPN
ejpam-3352	465	4	.	.	PUNCT
ejpam-3352	466	1	soc	soc	PROPN
ejpam-3352	466	2	,	,	PUNCT
ejpam-3352	466	3	41:375–481	41:375–481	PROPN
ejpam-3352	466	4	,	,	PUNCT
ejpam-3352	466	5	1937	1937	NUM
ejpam-3352	466	6	.	.	PUNCT
ejpam-3352	467	1	[	[	X
ejpam-3352	467	2	22	22	NUM
ejpam-3352	467	3	]	]	X
ejpam-3352	467	4	d.	d.	PROPN
ejpam-3352	467	5	rose	rise	VERB
ejpam-3352	467	6	t.	t.	PROPN
ejpam-3352	467	7	r.	r.	PROPN
ejpam-3352	467	8	hamlett	hamlett	PROPN
ejpam-3352	467	9	and	and	CCONJ
ejpam-3352	467	10	d.	d.	PROPN
ejpam-3352	467	11	janković.	janković.	PROPN
ejpam-3352	467	12	paracompactness	paracompactness	PROPN
ejpam-3352	467	13	with	with	ADP
ejpam-3352	467	14	respect	respect	NOUN
ejpam-3352	467	15	to	to	ADP
ejpam-3352	467	16	an	an	DET
ejpam-3352	467	17	ideal	ideal	NOUN
ejpam-3352	467	18	.	.	PUNCT
ejpam-3352	468	1	internat	internat	PROPN
ejpam-3352	468	2	.	.	PUNCT
ejpam-3352	469	1	j.	j.	PROPN
ejpam-3352	469	2	math	math	PROPN
ejpam-3352	469	3	.	.	PUNCT
ejpam-3352	470	1	math	math	NOUN
ejpam-3352	470	2	.	.	PUNCT
ejpam-3352	471	1	sci	sci	PROPN
ejpam-3352	471	2	,	,	PUNCT
ejpam-3352	471	3	20(3	20(3	NUM
ejpam-3352	471	4	)	)	PUNCT
ejpam-3352	471	5	,	,	PUNCT
ejpam-3352	471	6	pages	page	NOUN
ejpam-3352	471	7	=	=	NOUN
ejpam-3352	471	8	433	433	NUM
ejpam-3352	471	9	-	-	SYM
ejpam-3352	471	10	442	442	NUM
ejpam-3352	471	11	,	,	PUNCT
ejpam-3352	471	12	)	)	PUNCT
ejpam-3352	471	13	,	,	PUNCT
ejpam-3352	471	14	1997	1997	NUM
ejpam-3352	471	15	.	.	PUNCT
ejpam-3352	472	1	[	[	X
ejpam-3352	472	2	23	23	NUM
ejpam-3352	472	3	]	]	X
ejpam-3352	472	4	r.	r.	PROPN
ejpam-3352	472	5	vaidyanathaswamy	vaidyanathaswamy	PROPN
ejpam-3352	472	6	.	.	PUNCT
ejpam-3352	473	1	set	set	VERB
ejpam-3352	473	2	topology	topology	NOUN
ejpam-3352	473	3	.	.	PUNCT
ejpam-3352	474	1	chelsea	chelsea	PROPN
ejpam-3352	474	2	publishing	publishing	PROPN
ejpam-3352	474	3	company	company	NOUN
ejpam-3352	474	4	,	,	PUNCT
ejpam-3352	474	5	new	new	PROPN
ejpam-3352	474	6	york	york	PROPN
ejpam-3352	474	7	,	,	PUNCT
ejpam-3352	474	8	1946	1946	NUM
ejpam-3352	474	9	.	.	PUNCT
ejpam-3352	475	1	[	[	X
ejpam-3352	475	2	24	24	NUM
ejpam-3352	475	3	]	]	X
ejpam-3352	475	4	s.	s.	PROPN
ejpam-3352	475	5	willard	willard	PROPN
ejpam-3352	475	6	.	.	PUNCT
ejpam-3352	475	7	general	general	ADJ
ejpam-3352	475	8	topology	topology	PROPN
ejpam-3352	475	9	.	.	PUNCT
ejpam-3352	476	1	addison	addison	PROPN
ejpam-3352	476	2	-	-	PUNCT
ejpam-3352	476	3	wesley	wesley	PROPN
ejpam-3352	476	4	publishing	publishing	PROPN
ejpam-3352	476	5	company	company	NOUN
ejpam-3352	476	6	,	,	PUNCT
ejpam-3352	476	7	1970	1970	NUM
ejpam-3352	476	8	.	.	PUNCT