id	sid	tid	token	lemma	pos
ejpam-4767	1	1	european	european	PROPN
ejpam-4767	1	2	journal	journal	PROPN
ejpam-4767	1	3	of	of	ADP
ejpam-4767	1	4	pure	pure	ADJ
ejpam-4767	1	5	and	and	CCONJ
ejpam-4767	1	6	applied	apply	VERB
ejpam-4767	1	7	mathematics	mathematic	NOUN
ejpam-4767	1	8	vol	vol	NOUN
ejpam-4767	1	9	.	.	PUNCT
ejpam-4767	2	1	16	16	NUM
ejpam-4767	2	2	,	,	PUNCT
ejpam-4767	2	3	no	no	INTJ
ejpam-4767	2	4	.	.	NOUN
ejpam-4767	2	5	2	2	NUM
ejpam-4767	2	6	,	,	PUNCT
ejpam-4767	2	7	2023	2023	NUM
ejpam-4767	2	8	,	,	PUNCT
ejpam-4767	2	9	1228	1228	NUM
ejpam-4767	2	10	-	-	SYM
ejpam-4767	2	11	1235	1235	NUM
ejpam-4767	2	12	issn	issn	PROPN
ejpam-4767	2	13	1307	1307	NUM
ejpam-4767	2	14	-	-	SYM
ejpam-4767	2	15	5543	5543	NUM
ejpam-4767	2	16	–	–	PUNCT
ejpam-4767	3	1	ejpam.com	ejpam.com	X
ejpam-4767	3	2	published	publish	VERB
ejpam-4767	3	3	by	by	ADP
ejpam-4767	3	4	new	new	PROPN
ejpam-4767	3	5	york	york	PROPN
ejpam-4767	3	6	business	business	PROPN
ejpam-4767	3	7	global	global	ADJ
ejpam-4767	3	8	locally	locally	ADV
ejpam-4767	3	9	compact	compact	ADJ
ejpam-4767	3	10	spaces	space	NOUN
ejpam-4767	3	11	with	with	ADP
ejpam-4767	3	12	defects	defect	NOUN
ejpam-4767	3	13	mohmmad	mohmmad	PROPN
ejpam-4767	3	14	zailai	zailai	PROPN
ejpam-4767	3	15	department	department	PROPN
ejpam-4767	3	16	of	of	ADP
ejpam-4767	3	17	mathematics	mathematic	NOUN
ejpam-4767	3	18	,	,	PUNCT
ejpam-4767	3	19	faculty	faculty	NOUN
ejpam-4767	3	20	of	of	ADP
ejpam-4767	3	21	science	science	NOUN
ejpam-4767	3	22	,	,	PUNCT
ejpam-4767	3	23	king	king	PROPN
ejpam-4767	3	24	abdulaziz	abdulaziz	PROPN
ejpam-4767	3	25	university	university	PROPN
ejpam-4767	3	26	,	,	PUNCT
ejpam-4767	3	27	p.o.box	p.o.box	PROPN
ejpam-4767	3	28	80200	80200	NUM
ejpam-4767	3	29	jeddah	jeddah	PROPN
ejpam-4767	3	30	21589	21589	NUM
ejpam-4767	3	31	,	,	PUNCT
ejpam-4767	3	32	saudi	saudi	PROPN
ejpam-4767	3	33	arabia	arabia	PROPN
ejpam-4767	3	34	abstract	abstract	NOUN
ejpam-4767	3	35	.	.	PUNCT
ejpam-4767	4	1	we	we	PRON
ejpam-4767	4	2	call	call	VERB
ejpam-4767	4	3	a	a	DET
ejpam-4767	4	4	topological	topological	ADJ
ejpam-4767	4	5	space	space	NOUN
ejpam-4767	4	6	x	x	PUNCT
ejpam-4767	4	7	a	a	DET
ejpam-4767	4	8	locally	locally	ADV
ejpam-4767	4	9	compact	compact	ADJ
ejpam-4767	4	10	space	space	NOUN
ejpam-4767	4	11	with	with	ADP
ejpam-4767	4	12	defects	defect	NOUN
ejpam-4767	4	13	if	if	SCONJ
ejpam-4767	4	14	all	all	DET
ejpam-4767	4	15	points	point	VERB
ejpam-4767	4	16	in	in	ADP
ejpam-4767	4	17	x	x	PUNCT
ejpam-4767	4	18	possess	possess	VERB
ejpam-4767	4	19	compact	compact	ADJ
ejpam-4767	4	20	neighborhoods	neighborhood	NOUN
ejpam-4767	4	21	except	except	SCONJ
ejpam-4767	4	22	for	for	ADP
ejpam-4767	4	23	some	some	DET
ejpam-4767	4	24	points	point	NOUN
ejpam-4767	4	25	.	.	PUNCT
ejpam-4767	5	1	we	we	PRON
ejpam-4767	5	2	investigate	investigate	VERB
ejpam-4767	5	3	this	this	DET
ejpam-4767	5	4	weaker	weak	ADJ
ejpam-4767	5	5	version	version	NOUN
ejpam-4767	5	6	of	of	ADP
ejpam-4767	5	7	local	local	ADJ
ejpam-4767	5	8	compactness	compactness	NOUN
ejpam-4767	5	9	.	.	PUNCT
ejpam-4767	6	1	we	we	PRON
ejpam-4767	6	2	show	show	VERB
ejpam-4767	6	3	that	that	SCONJ
ejpam-4767	6	4	for	for	ADP
ejpam-4767	6	5	x	x	PROPN
ejpam-4767	6	6	∈	∈	PROPN
ejpam-4767	6	7	x•	x•	NOUN
ejpam-4767	6	8	if	if	SCONJ
ejpam-4767	6	9	the	the	DET
ejpam-4767	6	10	partition	partition	NOUN
ejpam-4767	6	11	of	of	ADP
ejpam-4767	6	12	singletons	singleton	NOUN
ejpam-4767	6	13	of	of	ADP
ejpam-4767	6	14	x\(x•	x\(x•	NOUN
ejpam-4767	6	15	∪	∪	ADJ
ejpam-4767	6	16	(	(	PUNCT
ejpam-4767	6	17	u\u	u\u	NOUN
ejpam-4767	6	18	)	)	PUNCT
ejpam-4767	6	19	)	)	PUNCT
ejpam-4767	6	20	is	be	AUX
ejpam-4767	6	21	locally	locally	ADV
ejpam-4767	6	22	finite	finite	ADJ
ejpam-4767	6	23	,	,	PUNCT
ejpam-4767	6	24	where	where	SCONJ
ejpam-4767	6	25	u	u	NOUN
ejpam-4767	6	26	̸=	̸=	PROPN
ejpam-4767	6	27	x	x	NUM
ejpam-4767	6	28	is	be	AUX
ejpam-4767	6	29	an	an	DET
ejpam-4767	6	30	open	open	ADJ
ejpam-4767	6	31	neighborhood	neighborhood	NOUN
ejpam-4767	6	32	of	of	ADP
ejpam-4767	6	33	x	x	NOUN
ejpam-4767	6	34	,	,	PUNCT
ejpam-4767	6	35	then	then	ADV
ejpam-4767	6	36	x	x	PUNCT
ejpam-4767	6	37	is	be	AUX
ejpam-4767	6	38	a	a	DET
ejpam-4767	6	39	tychonoff	tychonoff	NOUN
ejpam-4767	6	40	space	space	NOUN
ejpam-4767	6	41	.	.	PUNCT
ejpam-4767	7	1	let	let	VERB
ejpam-4767	7	2	x	x	PRON
ejpam-4767	7	3	be	be	AUX
ejpam-4767	7	4	a	a	DET
ejpam-4767	7	5	t1c	t1c	PRON
ejpam-4767	7	6	locally	locally	ADV
ejpam-4767	7	7	compact	compact	ADJ
ejpam-4767	7	8	space	space	NOUN
ejpam-4767	7	9	with	with	ADP
ejpam-4767	7	10	defects	defect	NOUN
ejpam-4767	7	11	such	such	ADJ
ejpam-4767	7	12	that	that	SCONJ
ejpam-4767	7	13	each	each	DET
ejpam-4767	7	14	x	x	SYM
ejpam-4767	7	15	∈	∈	PROPN
ejpam-4767	7	16	x•	x•	NOUN
ejpam-4767	7	17	has	have	VERB
ejpam-4767	7	18	an	an	DET
ejpam-4767	7	19	open	open	ADJ
ejpam-4767	7	20	neighborhood	neighborhood	NOUN
ejpam-4767	7	21	u	u	NOUN
ejpam-4767	7	22	such	such	ADJ
ejpam-4767	7	23	that	that	SCONJ
ejpam-4767	7	24	u	u	PROPN
ejpam-4767	7	25	is	be	AUX
ejpam-4767	7	26	a	a	DET
ejpam-4767	7	27	union	union	NOUN
ejpam-4767	7	28	of	of	ADP
ejpam-4767	7	29	pairwise	pairwise	PROPN
ejpam-4767	7	30	disjoint	disjoint	NOUN
ejpam-4767	7	31	compact	compact	ADJ
ejpam-4767	7	32	subsets	subset	NOUN
ejpam-4767	7	33	⋃	⋃	PUNCT
ejpam-4767	7	34	s∈s	s∈s	NOUN
ejpam-4767	7	35	fs	fs	NOUN
ejpam-4767	7	36	.	.	PUNCT
ejpam-4767	8	1	then	then	ADV
ejpam-4767	8	2	,	,	PUNCT
ejpam-4767	8	3	we	we	PRON
ejpam-4767	8	4	show	show	VERB
ejpam-4767	8	5	that	that	SCONJ
ejpam-4767	8	6	if	if	SCONJ
ejpam-4767	8	7	the	the	DET
ejpam-4767	8	8	family	family	NOUN
ejpam-4767	8	9	{	{	PUNCT
ejpam-4767	8	10	fs}s∈s	fs}s∈s	X
ejpam-4767	8	11	is	be	AUX
ejpam-4767	8	12	locally	locally	ADV
ejpam-4767	8	13	finite	finite	ADJ
ejpam-4767	8	14	except	except	SCONJ
ejpam-4767	8	15	for	for	ADP
ejpam-4767	8	16	a	a	DET
ejpam-4767	8	17	finite	finite	ADJ
ejpam-4767	8	18	number	number	NOUN
ejpam-4767	8	19	of	of	ADP
ejpam-4767	8	20	points	point	NOUN
ejpam-4767	8	21	,	,	PUNCT
ejpam-4767	8	22	then	then	ADV
ejpam-4767	8	23	x	x	PUNCT
ejpam-4767	8	24	is	be	AUX
ejpam-4767	8	25	a	a	DET
ejpam-4767	8	26	tychonoff	tychonoff	NOUN
ejpam-4767	8	27	space	space	NOUN
ejpam-4767	8	28	.	.	PUNCT
ejpam-4767	9	1	2020	2020	NUM
ejpam-4767	9	2	mathematics	mathematic	NOUN
ejpam-4767	9	3	subject	subject	NOUN
ejpam-4767	9	4	classifications	classification	NOUN
ejpam-4767	9	5	:	:	PUNCT
ejpam-4767	9	6	54d45	54d45	NUM
ejpam-4767	9	7	key	key	ADJ
ejpam-4767	9	8	words	word	NOUN
ejpam-4767	9	9	and	and	CCONJ
ejpam-4767	9	10	phrases	phrase	NOUN
ejpam-4767	9	11	:	:	PUNCT
ejpam-4767	9	12	compact	compact	ADJ
ejpam-4767	9	13	,	,	PUNCT
ejpam-4767	9	14	locally	locally	ADV
ejpam-4767	9	15	compact	compact	ADJ
ejpam-4767	9	16	,	,	PUNCT
ejpam-4767	9	17	defects	defect	NOUN
ejpam-4767	9	18	,	,	PUNCT
ejpam-4767	9	19	tychonoff	tychonoff	NOUN
ejpam-4767	9	20	1	1	NUM
ejpam-4767	9	21	.	.	PUNCT
ejpam-4767	9	22	introduction	introduction	NOUN
ejpam-4767	9	23	a	a	DET
ejpam-4767	9	24	t1	t1	NOUN
ejpam-4767	9	25	space	space	NOUN
ejpam-4767	9	26	x	x	PUNCT
ejpam-4767	9	27	is	be	AUX
ejpam-4767	9	28	said	say	VERB
ejpam-4767	9	29	to	to	PART
ejpam-4767	9	30	be	be	AUX
ejpam-4767	9	31	locally	locally	ADV
ejpam-4767	9	32	compact	compact	ADJ
ejpam-4767	9	33	if	if	SCONJ
ejpam-4767	9	34	every	every	DET
ejpam-4767	9	35	point	point	NOUN
ejpam-4767	9	36	x	x	X
ejpam-4767	9	37	∈	∈	NOUN
ejpam-4767	9	38	x	x	PRON
ejpam-4767	9	39	possesses	possess	VERB
ejpam-4767	9	40	a	a	DET
ejpam-4767	9	41	compact	compact	ADJ
ejpam-4767	9	42	neighborhood	neighborhood	NOUN
ejpam-4767	9	43	,	,	PUNCT
ejpam-4767	9	44	i.e	i.e	PRON
ejpam-4767	9	45	,	,	PUNCT
ejpam-4767	9	46	an	an	DET
ejpam-4767	9	47	open	open	ADJ
ejpam-4767	9	48	neighborhood	neighborhood	NOUN
ejpam-4767	9	49	such	such	ADJ
ejpam-4767	9	50	that	that	SCONJ
ejpam-4767	9	51	its	its	PRON
ejpam-4767	9	52	closure	closure	NOUN
ejpam-4767	9	53	is	be	AUX
ejpam-4767	9	54	a	a	DET
ejpam-4767	9	55	compact	compact	ADJ
ejpam-4767	9	56	subspace	subspace	NOUN
ejpam-4767	9	57	.	.	PUNCT
ejpam-4767	10	1	in	in	ADP
ejpam-4767	10	2	this	this	DET
ejpam-4767	10	3	paper	paper	NOUN
ejpam-4767	10	4	we	we	PRON
ejpam-4767	10	5	introduce	introduce	VERB
ejpam-4767	10	6	a	a	DET
ejpam-4767	10	7	weaker	weak	ADJ
ejpam-4767	10	8	version	version	NOUN
ejpam-4767	10	9	of	of	ADP
ejpam-4767	10	10	local	local	ADJ
ejpam-4767	10	11	compactness	compactness	NOUN
ejpam-4767	10	12	,	,	PUNCT
ejpam-4767	10	13	which	which	PRON
ejpam-4767	10	14	we	we	PRON
ejpam-4767	10	15	call	call	VERB
ejpam-4767	10	16	local	local	ADJ
ejpam-4767	10	17	compactness	compactness	NOUN
ejpam-4767	10	18	with	with	ADP
ejpam-4767	10	19	defects	defect	NOUN
ejpam-4767	10	20	.	.	PUNCT
ejpam-4767	11	1	a	a	DET
ejpam-4767	11	2	t1	t1	NOUN
ejpam-4767	11	3	space	space	NOUN
ejpam-4767	11	4	x	x	PUNCT
ejpam-4767	11	5	is	be	AUX
ejpam-4767	11	6	locally	locally	ADV
ejpam-4767	11	7	compact	compact	ADJ
ejpam-4767	11	8	with	with	ADP
ejpam-4767	11	9	defects	defect	NOUN
ejpam-4767	11	10	if	if	SCONJ
ejpam-4767	11	11	each	each	DET
ejpam-4767	11	12	point	point	NOUN
ejpam-4767	11	13	of	of	ADP
ejpam-4767	11	14	the	the	DET
ejpam-4767	11	15	space	space	NOUN
ejpam-4767	11	16	has	have	VERB
ejpam-4767	11	17	a	a	DET
ejpam-4767	11	18	compact	compact	ADJ
ejpam-4767	11	19	neighborhood	neighborhood	NOUN
ejpam-4767	11	20	except	except	SCONJ
ejpam-4767	11	21	for	for	ADP
ejpam-4767	11	22	some	some	DET
ejpam-4767	11	23	points	point	NOUN
ejpam-4767	11	24	.	.	PUNCT
ejpam-4767	12	1	we	we	PRON
ejpam-4767	12	2	denote	denote	VERB
ejpam-4767	12	3	by	by	ADP
ejpam-4767	12	4	x•	x•	NOUN
ejpam-4767	12	5	the	the	DET
ejpam-4767	12	6	set	set	NOUN
ejpam-4767	12	7	of	of	ADP
ejpam-4767	12	8	points	point	NOUN
ejpam-4767	12	9	of	of	ADP
ejpam-4767	12	10	x	x	PUNCT
ejpam-4767	12	11	which	which	PRON
ejpam-4767	12	12	do	do	AUX
ejpam-4767	12	13	not	not	PART
ejpam-4767	12	14	have	have	VERB
ejpam-4767	12	15	compact	compact	ADJ
ejpam-4767	12	16	neighborhoods	neighborhood	NOUN
ejpam-4767	12	17	.	.	PUNCT
ejpam-4767	13	1	points	point	NOUN
ejpam-4767	13	2	of	of	ADP
ejpam-4767	13	3	x•	x•	NOUN
ejpam-4767	13	4	are	be	AUX
ejpam-4767	13	5	called	call	VERB
ejpam-4767	13	6	defects	defect	NOUN
ejpam-4767	13	7	.	.	PUNCT
ejpam-4767	14	1	a	a	DET
ejpam-4767	14	2	space	space	NOUN
ejpam-4767	14	3	x	x	PUNCT
ejpam-4767	14	4	is	be	AUX
ejpam-4767	14	5	said	say	VERB
ejpam-4767	14	6	to	to	PART
ejpam-4767	14	7	be	be	AUX
ejpam-4767	14	8	scattered	scatter	VERB
ejpam-4767	14	9	if	if	SCONJ
ejpam-4767	14	10	it	it	PRON
ejpam-4767	14	11	contains	contain	VERB
ejpam-4767	14	12	no	no	DET
ejpam-4767	14	13	non	non	ADJ
ejpam-4767	14	14	-	-	ADJ
ejpam-4767	14	15	empty	empty	ADJ
ejpam-4767	14	16	subset	subset	NOUN
ejpam-4767	14	17	which	which	PRON
ejpam-4767	14	18	is	be	AUX
ejpam-4767	14	19	dense	dense	ADJ
ejpam-4767	14	20	-	-	PUNCT
ejpam-4767	14	21	initself	initself	PRON
ejpam-4767	14	22	.	.	PUNCT
ejpam-4767	15	1	it	it	PRON
ejpam-4767	15	2	is	be	AUX
ejpam-4767	15	3	proved	prove	VERB
ejpam-4767	15	4	in	in	ADP
ejpam-4767	15	5	[	[	X
ejpam-4767	15	6	4	4	X
ejpam-4767	15	7	]	]	PUNCT
ejpam-4767	15	8	that	that	SCONJ
ejpam-4767	15	9	for	for	ADP
ejpam-4767	15	10	a	a	DET
ejpam-4767	15	11	tychonoff	tychonoff	NOUN
ejpam-4767	15	12	space	space	NOUN
ejpam-4767	15	13	x	x	PUNCT
ejpam-4767	15	14	the	the	DET
ejpam-4767	15	15	set	set	NOUN
ejpam-4767	15	16	x•	x•	NOUN
ejpam-4767	15	17	is	be	AUX
ejpam-4767	15	18	closed	closed	ADJ
ejpam-4767	15	19	.	.	PUNCT
ejpam-4767	16	1	we	we	PRON
ejpam-4767	16	2	extend	extend	VERB
ejpam-4767	16	3	this	this	DET
ejpam-4767	16	4	result	result	NOUN
ejpam-4767	16	5	and	and	CCONJ
ejpam-4767	16	6	show	show	VERB
ejpam-4767	16	7	that	that	SCONJ
ejpam-4767	16	8	for	for	ADP
ejpam-4767	16	9	any	any	DET
ejpam-4767	16	10	space	space	NOUN
ejpam-4767	16	11	,	,	PUNCT
ejpam-4767	16	12	the	the	DET
ejpam-4767	16	13	set	set	NOUN
ejpam-4767	16	14	of	of	ADP
ejpam-4767	16	15	defects	defect	NOUN
ejpam-4767	16	16	is	be	AUX
ejpam-4767	16	17	a	a	DET
ejpam-4767	16	18	closed	closed	ADJ
ejpam-4767	16	19	subset	subset	NOUN
ejpam-4767	16	20	.	.	PUNCT
ejpam-4767	17	1	we	we	PRON
ejpam-4767	17	2	use	use	VERB
ejpam-4767	17	3	that	that	DET
ejpam-4767	17	4	result	result	NOUN
ejpam-4767	17	5	to	to	PART
ejpam-4767	17	6	show	show	VERB
ejpam-4767	17	7	that	that	SCONJ
ejpam-4767	17	8	for	for	ADP
ejpam-4767	17	9	any	any	DET
ejpam-4767	17	10	t1	t1	NOUN
ejpam-4767	17	11	topological	topological	ADJ
ejpam-4767	17	12	space	space	NOUN
ejpam-4767	17	13	,	,	PUNCT
ejpam-4767	17	14	if	if	SCONJ
ejpam-4767	17	15	the	the	DET
ejpam-4767	17	16	set	set	NOUN
ejpam-4767	17	17	of	of	ADP
ejpam-4767	17	18	defects	defect	NOUN
ejpam-4767	17	19	is	be	AUX
ejpam-4767	17	20	not	not	PART
ejpam-4767	17	21	empty	empty	ADJ
ejpam-4767	17	22	then	then	ADV
ejpam-4767	17	23	the	the	DET
ejpam-4767	17	24	space	space	NOUN
ejpam-4767	17	25	is	be	AUX
ejpam-4767	17	26	not	not	PART
ejpam-4767	17	27	scattered	scatter	VERB
ejpam-4767	17	28	.	.	PUNCT
ejpam-4767	18	1	all	all	DET
ejpam-4767	18	2	compact	compact	ADJ
ejpam-4767	18	3	subspaces	subspace	NOUN
ejpam-4767	18	4	in	in	ADP
ejpam-4767	18	5	this	this	DET
ejpam-4767	18	6	paper	paper	NOUN
ejpam-4767	18	7	are	be	AUX
ejpam-4767	18	8	assumed	assume	VERB
ejpam-4767	18	9	to	to	PART
ejpam-4767	18	10	be	be	AUX
ejpam-4767	18	11	closed	close	VERB
ejpam-4767	18	12	and	and	CCONJ
ejpam-4767	18	13	t2	t2	NOUN
ejpam-4767	18	14	.	.	PUNCT
ejpam-4767	19	1	we	we	PRON
ejpam-4767	19	2	denote	denote	VERB
ejpam-4767	19	3	by	by	ADP
ejpam-4767	19	4	t1c	t1c	PRON
ejpam-4767	19	5	a	a	DET
ejpam-4767	19	6	t1	t1	NOUN
ejpam-4767	19	7	space	space	NOUN
ejpam-4767	19	8	such	such	ADJ
ejpam-4767	19	9	that	that	SCONJ
ejpam-4767	19	10	each	each	DET
ejpam-4767	19	11	compact	compact	ADJ
ejpam-4767	19	12	subspace	subspace	NOUN
ejpam-4767	19	13	is	be	AUX
ejpam-4767	19	14	closed	close	VERB
ejpam-4767	19	15	.	.	PUNCT
ejpam-4767	20	1	this	this	PRON
ejpam-4767	20	2	is	be	AUX
ejpam-4767	20	3	a	a	DET
ejpam-4767	20	4	space	space	NOUN
ejpam-4767	20	5	which	which	PRON
ejpam-4767	20	6	lies	lie	VERB
ejpam-4767	20	7	between	between	ADP
ejpam-4767	20	8	t1	t1	NOUN
ejpam-4767	20	9	and	and	CCONJ
ejpam-4767	20	10	t2	t2	NOUN
ejpam-4767	20	11	spaces	space	NOUN
ejpam-4767	20	12	.	.	PUNCT
ejpam-4767	21	1	ℵ0	ℵ0	PROPN
ejpam-4767	21	2	stands	stand	VERB
ejpam-4767	21	3	for	for	ADP
ejpam-4767	21	4	a	a	DET
ejpam-4767	21	5	cardinality	cardinality	NOUN
ejpam-4767	21	6	of	of	ADP
ejpam-4767	21	7	a	a	DET
ejpam-4767	21	8	countable	countable	ADJ
ejpam-4767	21	9	set	set	NOUN
ejpam-4767	21	10	.	.	PUNCT
ejpam-4767	22	1	n	n	PROPN
ejpam-4767	22	2	stands	stand	VERB
ejpam-4767	22	3	for	for	ADP
ejpam-4767	22	4	the	the	DET
ejpam-4767	22	5	set	set	NOUN
ejpam-4767	22	6	of	of	ADP
ejpam-4767	22	7	all	all	DET
ejpam-4767	22	8	natural	natural	ADJ
ejpam-4767	22	9	numbers	number	NOUN
ejpam-4767	22	10	.	.	PUNCT
ejpam-4767	23	1	by	by	ADP
ejpam-4767	23	2	a	a	DET
ejpam-4767	23	3	t3	t3	PROPN
ejpam-4767	23	4	1	1	NUM
ejpam-4767	23	5	2	2	NUM
ejpam-4767	23	6	space	space	NOUN
ejpam-4767	23	7	we	we	PRON
ejpam-4767	23	8	mean	mean	VERB
ejpam-4767	23	9	a	a	DET
ejpam-4767	23	10	tychonoff	tychonoff	NOUN
ejpam-4767	23	11	space	space	NOUN
ejpam-4767	23	12	.	.	PUNCT
ejpam-4767	24	1	k	k	PROPN
ejpam-4767	24	2	stands	stand	VERB
ejpam-4767	24	3	for	for	ADP
ejpam-4767	24	4	the	the	DET
ejpam-4767	24	5	sorgenfrey	sorgenfrey	PROPN
ejpam-4767	24	6	line	line	NOUN
ejpam-4767	24	7	,	,	PUNCT
ejpam-4767	24	8	i.e.	i.e.	X
ejpam-4767	24	9	,	,	PUNCT
ejpam-4767	24	10	the	the	DET
ejpam-4767	24	11	space	space	NOUN
ejpam-4767	24	12	generated	generate	VERB
ejpam-4767	24	13	by	by	ADP
ejpam-4767	24	14	the	the	DET
ejpam-4767	24	15	base	base	NOUN
ejpam-4767	24	16	b	b	PROPN
ejpam-4767	24	17	=	=	PUNCT
ejpam-4767	24	18	{	{	PUNCT
ejpam-4767	24	19	[	[	X
ejpam-4767	24	20	x	x	X
ejpam-4767	24	21	,	,	PUNCT
ejpam-4767	24	22	y	y	NOUN
ejpam-4767	24	23	)	)	PUNCT
ejpam-4767	24	24	}	}	PUNCT
ejpam-4767	24	25	,	,	PUNCT
ejpam-4767	24	26	where	where	SCONJ
ejpam-4767	24	27	x	x	X
ejpam-4767	24	28	,	,	PUNCT
ejpam-4767	24	29	y	y	PROPN
ejpam-4767	24	30	are	be	AUX
ejpam-4767	24	31	real	real	ADJ
ejpam-4767	24	32	numbers	number	NOUN
ejpam-4767	24	33	such	such	ADJ
ejpam-4767	24	34	that	that	SCONJ
ejpam-4767	24	35	x	x	PRON
ejpam-4767	24	36	<	<	X
ejpam-4767	24	37	y	y	PROPN
ejpam-4767	24	38	,	,	PUNCT
ejpam-4767	24	39	and	and	CCONJ
ejpam-4767	24	40	y	y	PROPN
ejpam-4767	24	41	is	be	AUX
ejpam-4767	24	42	a	a	DET
ejpam-4767	24	43	rational	rational	ADJ
ejpam-4767	24	44	number	number	NOUN
ejpam-4767	24	45	.	.	PUNCT
ejpam-4767	25	1	for	for	ADP
ejpam-4767	25	2	more	more	ADJ
ejpam-4767	25	3	details	detail	NOUN
ejpam-4767	25	4	about	about	ADP
ejpam-4767	25	5	locally	locally	ADV
ejpam-4767	25	6	compact	compact	ADJ
ejpam-4767	25	7	spaces	space	NOUN
ejpam-4767	25	8	,	,	PUNCT
ejpam-4767	25	9	see	see	VERB
ejpam-4767	25	10	[	[	X
ejpam-4767	25	11	1	1	NUM
ejpam-4767	25	12	]	]	PUNCT
ejpam-4767	25	13	.	.	PUNCT
ejpam-4767	26	1	more	more	ADJ
ejpam-4767	26	2	details	detail	NOUN
ejpam-4767	26	3	about	about	ADP
ejpam-4767	26	4	points	point	NOUN
ejpam-4767	26	5	that	that	PRON
ejpam-4767	26	6	do	do	AUX
ejpam-4767	26	7	not	not	PART
ejpam-4767	26	8	have	have	VERB
ejpam-4767	26	9	compact	compact	ADJ
ejpam-4767	26	10	neighborhoods	neighborhood	NOUN
ejpam-4767	26	11	,	,	PUNCT
ejpam-4767	26	12	which	which	PRON
ejpam-4767	26	13	we	we	PRON
ejpam-4767	26	14	call	call	VERB
ejpam-4767	26	15	defects	defect	NOUN
ejpam-4767	26	16	,	,	PUNCT
ejpam-4767	26	17	can	can	AUX
ejpam-4767	26	18	be	be	AUX
ejpam-4767	26	19	found	find	VERB
ejpam-4767	26	20	in	in	ADP
ejpam-4767	26	21	[	[	X
ejpam-4767	26	22	3	3	NUM
ejpam-4767	26	23	]	]	PUNCT
ejpam-4767	26	24	,	,	PUNCT
ejpam-4767	26	25	[	[	X
ejpam-4767	26	26	4	4	X
ejpam-4767	26	27	]	]	PUNCT
ejpam-4767	26	28	and	and	CCONJ
ejpam-4767	26	29	[	[	X
ejpam-4767	26	30	5	5	NUM
ejpam-4767	26	31	]	]	PUNCT
ejpam-4767	26	32	.	.	PUNCT
ejpam-4767	27	1	doi	doi	NOUN
ejpam-4767	27	2	:	:	PUNCT
ejpam-4767	27	3	https://doi.org/10.29020/nybg.ejpam.v16i2.4767	https://doi.org/10.29020/nybg.ejpam.v16i2.4767	DET
ejpam-4767	27	4	email	email	NOUN
ejpam-4767	27	5	address	address	NOUN
ejpam-4767	27	6	:	:	PUNCT
ejpam-4767	27	7	mzailai@kau.edu.sa	mzailai@kau.edu.sa	PROPN
ejpam-4767	27	8	(	(	PUNCT
ejpam-4767	27	9	m.	m.	PROPN
ejpam-4767	27	10	zailai	zailai	PROPN
ejpam-4767	27	11	)	)	PUNCT
ejpam-4767	27	12	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4767	27	13	1228	1228	NUM
ejpam-4767	28	1	©	©	PROPN
ejpam-4767	28	2	2023	2023	NUM
ejpam-4767	28	3	ejpam	ejpam	NOUN
ejpam-4767	28	4	all	all	DET
ejpam-4767	28	5	rights	right	NOUN
ejpam-4767	28	6	reserved	reserve	VERB
ejpam-4767	28	7	.	.	PUNCT
ejpam-4767	29	1	m.	m.	NOUN
ejpam-4767	29	2	zailai	zailai	PROPN
ejpam-4767	29	3	/	/	SYM
ejpam-4767	29	4	eur	eur	PROPN
ejpam-4767	29	5	.	.	PUNCT
ejpam-4767	30	1	j.	j.	PROPN
ejpam-4767	30	2	pure	pure	PROPN
ejpam-4767	30	3	appl	appl	PROPN
ejpam-4767	30	4	.	.	PROPN
ejpam-4767	30	5	math	math	PROPN
ejpam-4767	30	6	,	,	PUNCT
ejpam-4767	30	7	16	16	NUM
ejpam-4767	30	8	(	(	PUNCT
ejpam-4767	30	9	2	2	NUM
ejpam-4767	30	10	)	)	PUNCT
ejpam-4767	30	11	(	(	PUNCT
ejpam-4767	30	12	2023	2023	NUM
ejpam-4767	30	13	)	)	PUNCT
ejpam-4767	30	14	,	,	PUNCT
ejpam-4767	30	15	1228	1228	NUM
ejpam-4767	30	16	-	-	SYM
ejpam-4767	30	17	1235	1235	NUM
ejpam-4767	30	18	1229	1229	NUM
ejpam-4767	30	19	2	2	NUM
ejpam-4767	30	20	.	.	PUNCT
ejpam-4767	30	21	local	local	ADJ
ejpam-4767	30	22	compactness	compactness	NOUN
ejpam-4767	30	23	with	with	ADP
ejpam-4767	30	24	defects	defect	NOUN
ejpam-4767	30	25	definition	definition	NOUN
ejpam-4767	30	26	2.1	2.1	NUM
ejpam-4767	30	27	.	.	PUNCT
ejpam-4767	31	1	let	let	VERB
ejpam-4767	31	2	x	x	PRON
ejpam-4767	31	3	be	be	AUX
ejpam-4767	31	4	a	a	DET
ejpam-4767	31	5	topological	topological	ADJ
ejpam-4767	31	6	space	space	NOUN
ejpam-4767	31	7	and	and	CCONJ
ejpam-4767	31	8	let	let	VERB
ejpam-4767	31	9	f	f	PRON
ejpam-4767	31	10	be	be	AUX
ejpam-4767	31	11	any	any	DET
ejpam-4767	31	12	property	property	NOUN
ejpam-4767	31	13	.	.	PUNCT
ejpam-4767	32	1	if	if	SCONJ
ejpam-4767	32	2	y	y	PROPN
ejpam-4767	32	3	⊂	⊂	PROPN
ejpam-4767	32	4	x	x	X
ejpam-4767	32	5	is	be	AUX
ejpam-4767	32	6	the	the	DET
ejpam-4767	32	7	set	set	NOUN
ejpam-4767	32	8	of	of	ADP
ejpam-4767	32	9	points	point	NOUN
ejpam-4767	32	10	which	which	PRON
ejpam-4767	32	11	do	do	AUX
ejpam-4767	32	12	not	not	PART
ejpam-4767	32	13	satisfy	satisfy	VERB
ejpam-4767	32	14	the	the	DET
ejpam-4767	32	15	property	property	NOUN
ejpam-4767	32	16	f	f	NOUN
ejpam-4767	32	17	,	,	PUNCT
ejpam-4767	32	18	i.e.	i.e.	X
ejpam-4767	32	19	,	,	PUNCT
ejpam-4767	32	20	only	only	ADV
ejpam-4767	32	21	x\y	x\y	PROPN
ejpam-4767	32	22	has	have	VERB
ejpam-4767	32	23	the	the	DET
ejpam-4767	32	24	property	property	NOUN
ejpam-4767	33	1	f	f	NOUN
ejpam-4767	33	2	then	then	ADV
ejpam-4767	33	3	we	we	PRON
ejpam-4767	33	4	call	call	VERB
ejpam-4767	33	5	the	the	DET
ejpam-4767	33	6	topological	topological	ADJ
ejpam-4767	33	7	space	space	NOUN
ejpam-4767	33	8	x	x	PUNCT
ejpam-4767	33	9	a	a	DET
ejpam-4767	33	10	space	space	NOUN
ejpam-4767	33	11	with	with	ADP
ejpam-4767	33	12	defects	defect	NOUN
ejpam-4767	33	13	of	of	ADP
ejpam-4767	33	14	type	type	NOUN
ejpam-4767	33	15	f	f	PROPN
ejpam-4767	33	16	.	.	PUNCT
ejpam-4767	34	1	remark	remark	PROPN
ejpam-4767	34	2	1	1	NUM
ejpam-4767	34	3	.	.	PUNCT
ejpam-4767	35	1	this	this	DET
ejpam-4767	35	2	paper	paper	NOUN
ejpam-4767	35	3	concerns	concern	NOUN
ejpam-4767	35	4	about	about	ADP
ejpam-4767	35	5	defect	defect	NOUN
ejpam-4767	35	6	of	of	ADP
ejpam-4767	35	7	type	type	NOUN
ejpam-4767	35	8	local	local	ADJ
ejpam-4767	35	9	compactness	compactness	NOUN
ejpam-4767	35	10	.	.	PUNCT
ejpam-4767	36	1	throughout	throughout	ADP
ejpam-4767	36	2	this	this	DET
ejpam-4767	36	3	paper	paper	NOUN
ejpam-4767	36	4	we	we	PRON
ejpam-4767	36	5	write	write	VERB
ejpam-4767	36	6	l.c.w.d	l.c.w.d	PROPN
ejpam-4767	36	7	.	.	PUNCT
ejpam-4767	37	1	for	for	ADP
ejpam-4767	37	2	a	a	DET
ejpam-4767	37	3	space	space	NOUN
ejpam-4767	37	4	with	with	ADP
ejpam-4767	37	5	defects	defect	NOUN
ejpam-4767	37	6	of	of	ADP
ejpam-4767	37	7	type	type	NOUN
ejpam-4767	37	8	local	local	ADJ
ejpam-4767	37	9	compactness	compactness	NOUN
ejpam-4767	37	10	.	.	PUNCT
ejpam-4767	38	1	definition	definition	NOUN
ejpam-4767	38	2	2.2	2.2	NUM
ejpam-4767	38	3	.	.	PUNCT
ejpam-4767	39	1	a	a	DET
ejpam-4767	39	2	t1	t1	NOUN
ejpam-4767	39	3	space	space	NOUN
ejpam-4767	39	4	x	x	PUNCT
ejpam-4767	39	5	is	be	AUX
ejpam-4767	39	6	a	a	DET
ejpam-4767	39	7	space	space	NOUN
ejpam-4767	39	8	with	with	ADP
ejpam-4767	39	9	defects	defect	NOUN
ejpam-4767	39	10	of	of	ADP
ejpam-4767	39	11	type	type	NOUN
ejpam-4767	39	12	local	local	ADJ
ejpam-4767	39	13	compactness	compactness	NOUN
ejpam-4767	39	14	,	,	PUNCT
ejpam-4767	39	15	l.c.w.d	l.c.w.d	PROPN
ejpam-4767	39	16	.	.	PROPN
ejpam-4767	39	17	,	,	PUNCT
ejpam-4767	39	18	if	if	SCONJ
ejpam-4767	39	19	all	all	DET
ejpam-4767	39	20	points	point	NOUN
ejpam-4767	39	21	have	have	VERB
ejpam-4767	39	22	compact	compact	ADJ
ejpam-4767	39	23	neighborhoods	neighborhood	NOUN
ejpam-4767	39	24	except	except	SCONJ
ejpam-4767	39	25	for	for	ADP
ejpam-4767	39	26	some	some	DET
ejpam-4767	39	27	points	point	NOUN
ejpam-4767	39	28	.	.	PUNCT
ejpam-4767	40	1	we	we	PRON
ejpam-4767	40	2	denote	denote	VERB
ejpam-4767	40	3	by	by	ADP
ejpam-4767	40	4	x•	x•	NOUN
ejpam-4767	40	5	the	the	DET
ejpam-4767	40	6	set	set	NOUN
ejpam-4767	40	7	of	of	ADP
ejpam-4767	40	8	points	point	NOUN
ejpam-4767	40	9	which	which	PRON
ejpam-4767	40	10	do	do	AUX
ejpam-4767	40	11	not	not	PART
ejpam-4767	40	12	possess	possess	VERB
ejpam-4767	40	13	compact	compact	ADJ
ejpam-4767	40	14	neighborhoods	neighborhood	NOUN
ejpam-4767	40	15	.	.	PUNCT
ejpam-4767	41	1	remark	remark	NOUN
ejpam-4767	41	2	2	2	NUM
ejpam-4767	41	3	.	.	PUNCT
ejpam-4767	42	1	it	it	PRON
ejpam-4767	42	2	is	be	AUX
ejpam-4767	42	3	clear	clear	ADJ
ejpam-4767	42	4	that	that	SCONJ
ejpam-4767	42	5	if	if	SCONJ
ejpam-4767	42	6	x•	x•	PROPN
ejpam-4767	42	7	=	=	SYM
ejpam-4767	42	8	ϕ	ϕ	PROPN
ejpam-4767	42	9	then	then	ADV
ejpam-4767	42	10	x	x	INTJ
ejpam-4767	42	11	is	be	AUX
ejpam-4767	42	12	a	a	DET
ejpam-4767	42	13	locally	locally	ADV
ejpam-4767	42	14	compact	compact	ADJ
ejpam-4767	42	15	space	space	NOUN
ejpam-4767	42	16	.	.	PUNCT
ejpam-4767	43	1	we	we	PRON
ejpam-4767	43	2	always	always	ADV
ejpam-4767	43	3	assume	assume	VERB
ejpam-4767	43	4	that	that	SCONJ
ejpam-4767	43	5	x•	x•	PROPN
ejpam-4767	43	6	̸=	̸=	PROPN
ejpam-4767	43	7	ϕ	ϕ	PROPN
ejpam-4767	43	8	unless	unless	SCONJ
ejpam-4767	43	9	stated	state	VERB
ejpam-4767	43	10	otherwise	otherwise	ADV
ejpam-4767	43	11	.	.	PUNCT
ejpam-4767	44	1	example	example	NOUN
ejpam-4767	45	1	1	1	NUM
ejpam-4767	45	2	.	.	PUNCT
ejpam-4767	46	1	[	[	X
ejpam-4767	46	2	6	6	NUM
ejpam-4767	46	3	,	,	PUNCT
ejpam-4767	46	4	118	118	NUM
ejpam-4767	46	5	page	page	NOUN
ejpam-4767	46	6	137	137	NUM
ejpam-4767	46	7	]	]	PUNCT
ejpam-4767	46	8	denote	denote	NOUN
ejpam-4767	46	9	t	t	PROPN
ejpam-4767	46	10	the	the	DET
ejpam-4767	46	11	graph	graph	NOUN
ejpam-4767	46	12	of	of	ADP
ejpam-4767	46	13	the	the	DET
ejpam-4767	46	14	function	function	NOUN
ejpam-4767	46	15	h(t	h(t	PROPN
ejpam-4767	46	16	)	)	PUNCT
ejpam-4767	46	17	=	=	SYM
ejpam-4767	46	18	sin(1	sin(1	NOUN
ejpam-4767	46	19	/	/	SYM
ejpam-4767	46	20	t	t	NOUN
ejpam-4767	46	21	)	)	PUNCT
ejpam-4767	46	22	where	where	SCONJ
ejpam-4767	46	23	0	0	NUM
ejpam-4767	46	24	<	<	X
ejpam-4767	46	25	t	t	X
ejpam-4767	46	26	≤	≤	NUM
ejpam-4767	46	27	1	1	NUM
ejpam-4767	46	28	,	,	PUNCT
ejpam-4767	46	29	as	as	ADP
ejpam-4767	46	30	a	a	DET
ejpam-4767	46	31	subset	subset	NOUN
ejpam-4767	46	32	of	of	ADP
ejpam-4767	46	33	the	the	DET
ejpam-4767	46	34	euclidean	euclidean	ADJ
ejpam-4767	46	35	space	space	NOUN
ejpam-4767	46	36	r2	r2	PROPN
ejpam-4767	46	37	with	with	ADP
ejpam-4767	46	38	the	the	DET
ejpam-4767	46	39	relative	relative	ADJ
ejpam-4767	46	40	topology	topology	NOUN
ejpam-4767	46	41	.	.	PUNCT
ejpam-4767	47	1	the	the	DET
ejpam-4767	47	2	set	set	NOUN
ejpam-4767	47	3	t	t	PROPN
ejpam-4767	47	4	⋆	⋆	NOUN
ejpam-4767	47	5	=	=	SYM
ejpam-4767	47	6	{	{	PUNCT
ejpam-4767	47	7	(	(	PUNCT
ejpam-4767	47	8	0	0	NUM
ejpam-4767	47	9	,	,	PUNCT
ejpam-4767	47	10	0	0	NUM
ejpam-4767	47	11	)	)	PUNCT
ejpam-4767	47	12	}	}	PUNCT
ejpam-4767	47	13	∪	∪	ADP
ejpam-4767	47	14	t	t	PROPN
ejpam-4767	47	15	is	be	AUX
ejpam-4767	47	16	not	not	PART
ejpam-4767	47	17	locally	locally	ADV
ejpam-4767	47	18	compact	compact	ADJ
ejpam-4767	47	19	since	since	SCONJ
ejpam-4767	47	20	the	the	DET
ejpam-4767	47	21	point	point	NOUN
ejpam-4767	47	22	(	(	PUNCT
ejpam-4767	47	23	0	0	NUM
ejpam-4767	47	24	,	,	PUNCT
ejpam-4767	47	25	0	0	NUM
ejpam-4767	47	26	)	)	PUNCT
ejpam-4767	47	27	has	have	VERB
ejpam-4767	47	28	no	no	DET
ejpam-4767	47	29	compact	compact	ADJ
ejpam-4767	47	30	neighborhood	neighborhood	NOUN
ejpam-4767	47	31	.	.	PUNCT
ejpam-4767	48	1	therefore	therefore	ADV
ejpam-4767	48	2	,	,	PUNCT
ejpam-4767	48	3	the	the	DET
ejpam-4767	48	4	topological	topological	ADJ
ejpam-4767	48	5	space	space	NOUN
ejpam-4767	48	6	t	t	PROPN
ejpam-4767	48	7	⋆	⋆	VERB
ejpam-4767	48	8	is	be	AUX
ejpam-4767	48	9	a	a	DET
ejpam-4767	48	10	locally	locally	ADV
ejpam-4767	48	11	compact	compact	ADJ
ejpam-4767	48	12	space	space	NOUN
ejpam-4767	48	13	with	with	ADP
ejpam-4767	48	14	defect	defect	NOUN
ejpam-4767	48	15	of	of	ADP
ejpam-4767	48	16	type	type	NOUN
ejpam-4767	48	17	local	local	ADJ
ejpam-4767	48	18	compactness	compactness	NOUN
ejpam-4767	48	19	.	.	PUNCT
ejpam-4767	49	1	proposition	proposition	NOUN
ejpam-4767	49	2	2.3	2.3	NUM
ejpam-4767	49	3	.	.	PUNCT
ejpam-4767	49	4	suppose	suppose	VERB
ejpam-4767	49	5	that	that	SCONJ
ejpam-4767	49	6	the	the	DET
ejpam-4767	49	7	topological	topological	ADJ
ejpam-4767	49	8	spaces	space	NOUN
ejpam-4767	49	9	x	x	PUNCT
ejpam-4767	49	10	and	and	CCONJ
ejpam-4767	49	11	y	y	PROPN
ejpam-4767	49	12	are	be	AUX
ejpam-4767	49	13	l.c.w.d	l.c.w.d	PROPN
ejpam-4767	49	14	.	.	PUNCT
ejpam-4767	50	1	if	if	SCONJ
ejpam-4767	50	2	x	x	PRON
ejpam-4767	50	3	is	be	AUX
ejpam-4767	50	4	homeomorphic	homeomorphic	ADJ
ejpam-4767	50	5	to	to	ADP
ejpam-4767	50	6	y	y	PROPN
ejpam-4767	50	7	then	then	ADV
ejpam-4767	50	8	x•	x•	PROPN
ejpam-4767	50	9	and	and	CCONJ
ejpam-4767	50	10	y	y	PROPN
ejpam-4767	50	11	•	•	VERB
ejpam-4767	50	12	have	have	VERB
ejpam-4767	50	13	same	same	ADJ
ejpam-4767	50	14	cardinality	cardinality	NOUN
ejpam-4767	50	15	,	,	PUNCT
ejpam-4767	50	16	i.e.	i.e.	X
ejpam-4767	50	17	,	,	PUNCT
ejpam-4767	50	18	the	the	DET
ejpam-4767	50	19	number	number	NOUN
ejpam-4767	50	20	of	of	ADP
ejpam-4767	50	21	defect	defect	ADJ
ejpam-4767	50	22	points	point	NOUN
ejpam-4767	50	23	is	be	AUX
ejpam-4767	50	24	a	a	DET
ejpam-4767	50	25	topological	topological	ADJ
ejpam-4767	50	26	invariant	invariant	ADJ
ejpam-4767	50	27	.	.	PUNCT
ejpam-4767	51	1	proof	proof	NOUN
ejpam-4767	51	2	.	.	PUNCT
ejpam-4767	52	1	it	it	PRON
ejpam-4767	52	2	is	be	AUX
ejpam-4767	52	3	enough	enough	ADJ
ejpam-4767	52	4	to	to	PART
ejpam-4767	52	5	check	check	VERB
ejpam-4767	52	6	that	that	SCONJ
ejpam-4767	52	7	the	the	DET
ejpam-4767	52	8	homeomorphic	homeomorphic	ADJ
ejpam-4767	52	9	image	image	NOUN
ejpam-4767	52	10	of	of	ADP
ejpam-4767	52	11	any	any	DET
ejpam-4767	52	12	point	point	NOUN
ejpam-4767	52	13	in	in	ADP
ejpam-4767	52	14	x•	x•	PROPN
ejpam-4767	52	15	lies	lie	NOUN
ejpam-4767	52	16	in	in	ADP
ejpam-4767	52	17	y	y	PROPN
ejpam-4767	52	18	•.	•.	NOUN
ejpam-4767	52	19	suppose	suppose	VERB
ejpam-4767	52	20	that	that	SCONJ
ejpam-4767	52	21	f	f	X
ejpam-4767	52	22	:	:	PUNCT
ejpam-4767	52	23	x	x	X
ejpam-4767	52	24	→	→	SYM
ejpam-4767	52	25	y	y	PROPN
ejpam-4767	52	26	is	be	AUX
ejpam-4767	52	27	a	a	DET
ejpam-4767	52	28	homeomorphism	homeomorphism	NOUN
ejpam-4767	52	29	.	.	PUNCT
ejpam-4767	53	1	take	take	VERB
ejpam-4767	53	2	any	any	DET
ejpam-4767	53	3	point	point	NOUN
ejpam-4767	53	4	x	x	X
ejpam-4767	53	5	∈	∈	PROPN
ejpam-4767	53	6	x•.	x•.	PROPN
ejpam-4767	54	1	if	if	SCONJ
ejpam-4767	54	2	f(x	f(x	PROPN
ejpam-4767	54	3	)	)	PUNCT
ejpam-4767	55	1	/∈	/∈	PUNCT
ejpam-4767	56	1	y	y	NOUN
ejpam-4767	56	2	•	•	ADV
ejpam-4767	56	3	then	then	ADV
ejpam-4767	56	4	there	there	PRON
ejpam-4767	56	5	is	be	VERB
ejpam-4767	56	6	a	a	DET
ejpam-4767	56	7	neighborhood	neighborhood	NOUN
ejpam-4767	56	8	u	u	NOUN
ejpam-4767	56	9	of	of	ADP
ejpam-4767	56	10	y	y	PROPN
ejpam-4767	56	11	=	=	SYM
ejpam-4767	56	12	f(x	f(x	PROPN
ejpam-4767	56	13	)	)	PUNCT
ejpam-4767	56	14	such	such	ADJ
ejpam-4767	56	15	that	that	SCONJ
ejpam-4767	56	16	the	the	DET
ejpam-4767	56	17	closure	closure	NOUN
ejpam-4767	56	18	u	u	NOUN
ejpam-4767	56	19	is	be	AUX
ejpam-4767	56	20	a	a	DET
ejpam-4767	56	21	compact	compact	ADJ
ejpam-4767	56	22	subspace	subspace	NOUN
ejpam-4767	56	23	.	.	PUNCT
ejpam-4767	57	1	now	now	ADV
ejpam-4767	57	2	x	x	X
ejpam-4767	57	3	∈	∈	PROPN
ejpam-4767	57	4	f−1(u	f−1(u	PROPN
ejpam-4767	57	5	)	)	PUNCT
ejpam-4767	57	6	and	and	CCONJ
ejpam-4767	57	7	also	also	ADV
ejpam-4767	57	8	have	have	VERB
ejpam-4767	57	9	f−1(u	f−1(u	NOUN
ejpam-4767	57	10	)	)	PUNCT
ejpam-4767	57	11	=	=	SYM
ejpam-4767	57	12	f−1(u	f−1(u	PROPN
ejpam-4767	57	13	)	)	PUNCT
ejpam-4767	57	14	which	which	PRON
ejpam-4767	57	15	is	be	AUX
ejpam-4767	57	16	compact	compact	ADJ
ejpam-4767	57	17	.	.	PUNCT
ejpam-4767	58	1	therefore	therefore	ADV
ejpam-4767	58	2	,	,	PUNCT
ejpam-4767	58	3	f−1(u	f−1(u	PROPN
ejpam-4767	58	4	)	)	PUNCT
ejpam-4767	58	5	is	be	AUX
ejpam-4767	58	6	a	a	DET
ejpam-4767	58	7	compact	compact	ADJ
ejpam-4767	58	8	neighborhood	neighborhood	NOUN
ejpam-4767	58	9	of	of	ADP
ejpam-4767	58	10	the	the	DET
ejpam-4767	58	11	point	point	NOUN
ejpam-4767	58	12	x	x	PUNCT
ejpam-4767	58	13	which	which	PRON
ejpam-4767	58	14	is	be	AUX
ejpam-4767	58	15	a	a	DET
ejpam-4767	58	16	contradiction	contradiction	NOUN
ejpam-4767	58	17	as	as	SCONJ
ejpam-4767	58	18	x	x	PRON
ejpam-4767	58	19	is	be	AUX
ejpam-4767	58	20	a	a	DET
ejpam-4767	58	21	defect	defect	NOUN
ejpam-4767	58	22	point	point	NOUN
ejpam-4767	58	23	.	.	PUNCT
ejpam-4767	59	1	lemma	lemma	PROPN
ejpam-4767	59	2	2.4	2.4	NUM
ejpam-4767	59	3	.	.	PUNCT
ejpam-4767	60	1	let	let	VERB
ejpam-4767	60	2	x	x	PRON
ejpam-4767	60	3	be	be	AUX
ejpam-4767	60	4	l.c.w.d	l.c.w.d	PROPN
ejpam-4767	60	5	then	then	ADV
ejpam-4767	60	6	x•	x•	PROPN
ejpam-4767	60	7	is	be	AUX
ejpam-4767	60	8	a	a	DET
ejpam-4767	60	9	closed	closed	ADJ
ejpam-4767	60	10	subset	subset	NOUN
ejpam-4767	60	11	of	of	ADP
ejpam-4767	60	12	x.	x.	NOUN
ejpam-4767	60	13	proof	proof	NOUN
ejpam-4767	60	14	.	.	PUNCT
ejpam-4767	61	1	it	it	PRON
ejpam-4767	61	2	is	be	AUX
ejpam-4767	61	3	sufficient	sufficient	ADJ
ejpam-4767	61	4	to	to	ADP
ejpam-4767	61	5	that	that	SCONJ
ejpam-4767	62	1	the	the	DET
ejpam-4767	62	2	complement	complement	NOUN
ejpam-4767	62	3	x\x•	x\x•	VERB
ejpam-4767	62	4	is	be	AUX
ejpam-4767	62	5	open	open	ADJ
ejpam-4767	62	6	.	.	PUNCT
ejpam-4767	63	1	if	if	SCONJ
ejpam-4767	63	2	x•	x•	PROPN
ejpam-4767	63	3	=	=	SYM
ejpam-4767	63	4	ϕ	ϕ	PROPN
ejpam-4767	63	5	,	,	PUNCT
ejpam-4767	63	6	then	then	ADV
ejpam-4767	63	7	x\x•	x\x•	PUNCT
ejpam-4767	64	1	=	=	PUNCT
ejpam-4767	64	2	x	x	PUNCT
ejpam-4767	64	3	is	be	AUX
ejpam-4767	64	4	closed	closed	ADJ
ejpam-4767	64	5	.	.	PUNCT
ejpam-4767	65	1	if	if	SCONJ
ejpam-4767	65	2	x•	x•	NOUN
ejpam-4767	65	3	=	=	SYM
ejpam-4767	65	4	x	x	NOUN
ejpam-4767	65	5	,	,	PUNCT
ejpam-4767	65	6	then	then	ADV
ejpam-4767	65	7	x\x•	x\x•	PUNCT
ejpam-4767	66	1	=	=	PUNCT
ejpam-4767	66	2	ϕ	ϕ	PROPN
ejpam-4767	66	3	is	be	AUX
ejpam-4767	66	4	closed	closed	ADJ
ejpam-4767	66	5	.	.	PUNCT
ejpam-4767	67	1	now	now	ADV
ejpam-4767	67	2	suppose	suppose	VERB
ejpam-4767	67	3	x•	x•	PROPN
ejpam-4767	67	4	̸=	̸=	PROPN
ejpam-4767	67	5	x	x	X
ejpam-4767	67	6	and	and	CCONJ
ejpam-4767	67	7	x•	x•	PROPN
ejpam-4767	67	8	̸=	̸=	PROPN
ejpam-4767	67	9	ϕ	ϕ	NOUN
ejpam-4767	67	10	,	,	PUNCT
ejpam-4767	67	11	take	take	VERB
ejpam-4767	67	12	any	any	DET
ejpam-4767	67	13	arbitrary	arbitrary	ADJ
ejpam-4767	67	14	point	point	NOUN
ejpam-4767	67	15	x	x	X
ejpam-4767	67	16	∈	∈	NOUN
ejpam-4767	67	17	x\x•.	x\x•.	PROPN
ejpam-4767	67	18	assume	assume	VERB
ejpam-4767	67	19	that	that	SCONJ
ejpam-4767	67	20	for	for	ADP
ejpam-4767	67	21	any	any	DET
ejpam-4767	67	22	neighborhood	neighborhood	NOUN
ejpam-4767	67	23	of	of	ADP
ejpam-4767	67	24	ux	ux	NOUN
ejpam-4767	67	25	of	of	ADP
ejpam-4767	67	26	x	x	PRON
ejpam-4767	67	27	we	we	PRON
ejpam-4767	67	28	have	have	VERB
ejpam-4767	67	29	that	that	PRON
ejpam-4767	67	30	ux	ux	PROPN
ejpam-4767	67	31	∩x•	∩x•	PRON
ejpam-4767	67	32	=	=	PROPN
ejpam-4767	67	33	ϕ.	ϕ.	PROPN
ejpam-4767	67	34	since	since	SCONJ
ejpam-4767	67	35	x	x	PROPN
ejpam-4767	67	36	∈	∈	PROPN
ejpam-4767	67	37	x\x•	x\x•	X
ejpam-4767	67	38	,	,	PUNCT
ejpam-4767	67	39	then	then	ADV
ejpam-4767	67	40	there	there	PRON
ejpam-4767	67	41	is	be	VERB
ejpam-4767	67	42	a	a	DET
ejpam-4767	67	43	compact	compact	ADJ
ejpam-4767	67	44	neighborhood	neighborhood	NOUN
ejpam-4767	67	45	ux	ux	ADP
ejpam-4767	67	46	of	of	ADP
ejpam-4767	67	47	x.	x.	PROPN
ejpam-4767	67	48	let	let	VERB
ejpam-4767	67	49	y	y	PROPN
ejpam-4767	67	50	∈	∈	PROPN
ejpam-4767	67	51	x•	x•	PROPN
ejpam-4767	67	52	,	,	PUNCT
ejpam-4767	67	53	then	then	ADV
ejpam-4767	67	54	y	y	PROPN
ejpam-4767	67	55	does	do	AUX
ejpam-4767	67	56	not	not	PART
ejpam-4767	67	57	belong	belong	VERB
ejpam-4767	67	58	to	to	ADP
ejpam-4767	67	59	ux	ux	PROPN
ejpam-4767	67	60	.	.	PUNCT
ejpam-4767	68	1	suppose	suppose	VERB
ejpam-4767	68	2	otherwise	otherwise	ADV
ejpam-4767	68	3	,	,	PUNCT
ejpam-4767	68	4	i.e.	i.e.	X
ejpam-4767	68	5	,	,	PUNCT
ejpam-4767	68	6	let	let	VERB
ejpam-4767	68	7	y	y	PROPN
ejpam-4767	68	8	∈	∈	PROPN
ejpam-4767	68	9	ux	ux	PROPN
ejpam-4767	68	10	.	.	PUNCT
ejpam-4767	69	1	then	then	ADV
ejpam-4767	69	2	y	y	PROPN
ejpam-4767	69	3	∈	∈	PROPN
ejpam-4767	69	4	ux	ux	PROPN
ejpam-4767	69	5	.	.	PUNCT
ejpam-4767	70	1	however	however	ADV
ejpam-4767	70	2	,	,	PUNCT
ejpam-4767	70	3	that	that	PRON
ejpam-4767	70	4	means	mean	VERB
ejpam-4767	70	5	ux	ux	PROPN
ejpam-4767	70	6	is	be	AUX
ejpam-4767	70	7	a	a	DET
ejpam-4767	70	8	compact	compact	ADJ
ejpam-4767	70	9	neighborhood	neighborhood	NOUN
ejpam-4767	70	10	of	of	ADP
ejpam-4767	70	11	y.	y.	PROPN
ejpam-4767	70	12	this	this	PRON
ejpam-4767	70	13	is	be	AUX
ejpam-4767	70	14	a	a	DET
ejpam-4767	70	15	contradiction	contradiction	NOUN
ejpam-4767	70	16	as	as	SCONJ
ejpam-4767	70	17	y	y	PROPN
ejpam-4767	70	18	is	be	AUX
ejpam-4767	70	19	a	a	DET
ejpam-4767	70	20	defect	defect	NOUN
ejpam-4767	70	21	.	.	PUNCT
ejpam-4767	71	1	proposition	proposition	NOUN
ejpam-4767	71	2	2.5	2.5	NUM
ejpam-4767	71	3	.	.	PUNCT
ejpam-4767	72	1	if	if	SCONJ
ejpam-4767	72	2	x	x	PRON
ejpam-4767	72	3	is	be	AUX
ejpam-4767	72	4	locally	locally	ADV
ejpam-4767	72	5	compact	compact	ADJ
ejpam-4767	72	6	with	with	ADP
ejpam-4767	72	7	defects	defect	NOUN
ejpam-4767	72	8	,	,	PUNCT
ejpam-4767	72	9	then	then	ADV
ejpam-4767	72	10	x•	x•	PROPN
ejpam-4767	72	11	is	be	AUX
ejpam-4767	72	12	dense	dense	ADJ
ejpam-4767	72	13	-	-	PUNCT
ejpam-4767	72	14	in	in	ADP
ejpam-4767	72	15	-	-	PUNCT
ejpam-4767	72	16	itself	itself	PRON
ejpam-4767	72	17	.	.	PUNCT
ejpam-4767	73	1	proof	proof	NOUN
ejpam-4767	73	2	.	.	PUNCT
ejpam-4767	74	1	if	if	SCONJ
ejpam-4767	74	2	x•	x•	PROPN
ejpam-4767	74	3	=	=	SYM
ejpam-4767	74	4	ϕ	ϕ	PROPN
ejpam-4767	74	5	then	then	ADV
ejpam-4767	74	6	it	it	PRON
ejpam-4767	74	7	is	be	AUX
ejpam-4767	74	8	clearly	clearly	ADV
ejpam-4767	74	9	dense	dense	ADJ
ejpam-4767	74	10	-	-	PUNCT
ejpam-4767	74	11	in	in	ADP
ejpam-4767	74	12	-	-	PUNCT
ejpam-4767	74	13	itself	itself	PRON
ejpam-4767	74	14	.	.	PUNCT
ejpam-4767	75	1	assume	assume	VERB
ejpam-4767	75	2	that	that	SCONJ
ejpam-4767	75	3	x•	x•	PROPN
ejpam-4767	75	4	̸=	̸=	PROPN
ejpam-4767	75	5	ϕ.	ϕ.	NOUN
ejpam-4767	75	6	take	take	VERB
ejpam-4767	75	7	any	any	DET
ejpam-4767	75	8	point	point	NOUN
ejpam-4767	75	9	x	x	X
ejpam-4767	75	10	∈	∈	NOUN
ejpam-4767	75	11	x\x•	x\x•	X
ejpam-4767	75	12	,	,	PUNCT
ejpam-4767	75	13	then	then	ADV
ejpam-4767	75	14	x	x	PUNCT
ejpam-4767	75	15	is	be	AUX
ejpam-4767	75	16	not	not	PART
ejpam-4767	75	17	an	an	DET
ejpam-4767	75	18	accumulation	accumulation	NOUN
ejpam-4767	75	19	point	point	NOUN
ejpam-4767	75	20	of	of	ADP
ejpam-4767	75	21	x•.	x•.	PROPN
ejpam-4767	75	22	now	now	ADV
ejpam-4767	75	23	,	,	PUNCT
ejpam-4767	75	24	let	let	VERB
ejpam-4767	75	25	x	x	X
ejpam-4767	75	26	∈	∈	PRON
ejpam-4767	75	27	x•	x•	NOUN
ejpam-4767	75	28	then	then	ADV
ejpam-4767	75	29	the	the	DET
ejpam-4767	75	30	closure	closure	NOUN
ejpam-4767	75	31	x•\{x	x•\{x	PROPN
ejpam-4767	75	32	}	}	PUNCT
ejpam-4767	75	33	is	be	AUX
ejpam-4767	75	34	the	the	DET
ejpam-4767	75	35	set	set	NOUN
ejpam-4767	75	36	x•.	x•.	PROPN
ejpam-4767	75	37	therefore	therefore	ADV
ejpam-4767	75	38	,	,	PUNCT
ejpam-4767	75	39	the	the	DET
ejpam-4767	75	40	set	set	NOUN
ejpam-4767	75	41	x•	x•	NOUN
ejpam-4767	75	42	contains	contain	VERB
ejpam-4767	75	43	all	all	PRON
ejpam-4767	75	44	of	of	ADP
ejpam-4767	75	45	its	its	PRON
ejpam-4767	75	46	accumulation	accumulation	NOUN
ejpam-4767	75	47	points	point	NOUN
ejpam-4767	75	48	.	.	PUNCT
ejpam-4767	76	1	hence	hence	ADV
ejpam-4767	76	2	,	,	PUNCT
ejpam-4767	76	3	x•	x•	PROPN
ejpam-4767	76	4	is	be	AUX
ejpam-4767	76	5	dense	dense	ADJ
ejpam-4767	76	6	-	-	PUNCT
ejpam-4767	76	7	in	in	ADP
ejpam-4767	76	8	-	-	PUNCT
ejpam-4767	76	9	itself	itself	PRON
ejpam-4767	76	10	.	.	PUNCT
ejpam-4767	77	1	m.	m.	PROPN
ejpam-4767	77	2	zailai	zailai	PROPN
ejpam-4767	77	3	/	/	SYM
ejpam-4767	77	4	eur	eur	PROPN
ejpam-4767	77	5	.	.	PUNCT
ejpam-4767	78	1	j.	j.	PROPN
ejpam-4767	78	2	pure	pure	PROPN
ejpam-4767	78	3	appl	appl	PROPN
ejpam-4767	78	4	.	.	PROPN
ejpam-4767	78	5	math	math	PROPN
ejpam-4767	78	6	,	,	PUNCT
ejpam-4767	78	7	16	16	NUM
ejpam-4767	78	8	(	(	PUNCT
ejpam-4767	78	9	2	2	NUM
ejpam-4767	78	10	)	)	PUNCT
ejpam-4767	78	11	(	(	PUNCT
ejpam-4767	78	12	2023	2023	NUM
ejpam-4767	78	13	)	)	PUNCT
ejpam-4767	78	14	,	,	PUNCT
ejpam-4767	78	15	1228	1228	NUM
ejpam-4767	78	16	-	-	SYM
ejpam-4767	78	17	1235	1235	NUM
ejpam-4767	78	18	1230	1230	NUM
ejpam-4767	78	19	corollary	corollary	NOUN
ejpam-4767	78	20	2.6	2.6	NUM
ejpam-4767	78	21	.	.	PUNCT
ejpam-4767	79	1	let	let	VERB
ejpam-4767	79	2	x	x	PRON
ejpam-4767	79	3	be	be	AUX
ejpam-4767	79	4	l.c.w.d	l.c.w.d	PRON
ejpam-4767	79	5	such	such	ADJ
ejpam-4767	79	6	that	that	SCONJ
ejpam-4767	79	7	x•	x•	PROPN
ejpam-4767	79	8	̸=	̸=	PROPN
ejpam-4767	79	9	ϕ	ϕ	PROPN
ejpam-4767	79	10	then	then	ADV
ejpam-4767	79	11	x	x	X
ejpam-4767	79	12	is	be	AUX
ejpam-4767	79	13	not	not	PART
ejpam-4767	79	14	a	a	DET
ejpam-4767	79	15	scattered	scatter	VERB
ejpam-4767	79	16	space	space	NOUN
ejpam-4767	79	17	.	.	PUNCT
ejpam-4767	80	1	lemma	lemma	PROPN
ejpam-4767	80	2	2.7	2.7	NUM
ejpam-4767	80	3	.	.	PUNCT
ejpam-4767	81	1	[	[	X
ejpam-4767	81	2	2	2	NUM
ejpam-4767	81	3	,	,	PUNCT
ejpam-4767	81	4	page	page	NOUN
ejpam-4767	81	5	17	17	NUM
ejpam-4767	81	6	]	]	PUNCT
ejpam-4767	81	7	suppose	suppose	VERB
ejpam-4767	81	8	that	that	SCONJ
ejpam-4767	81	9	the	the	DET
ejpam-4767	81	10	family	family	NOUN
ejpam-4767	81	11	{	{	PUNCT
ejpam-4767	81	12	ws}s∈s	ws}s∈s	PROPN
ejpam-4767	81	13	is	be	AUX
ejpam-4767	81	14	locally	locally	ADV
ejpam-4767	81	15	finite	finite	ADJ
ejpam-4767	81	16	,	,	PUNCT
ejpam-4767	81	17	then	then	ADV
ejpam-4767	81	18	the	the	DET
ejpam-4767	81	19	following	following	NOUN
ejpam-4767	81	20	⋃	⋃	PROPN
ejpam-4767	81	21	s∈s	s∈s	NOUN
ejpam-4767	81	22	ws	ws	NOUN
ejpam-4767	81	23	=	=	PUNCT
ejpam-4767	81	24	⋃	⋃	ADP
ejpam-4767	81	25	s∈s	s∈s	NOUN
ejpam-4767	81	26	ws	ws	NOUN
ejpam-4767	81	27	is	be	AUX
ejpam-4767	81	28	always	always	ADV
ejpam-4767	81	29	true	true	ADJ
ejpam-4767	81	30	.	.	PUNCT
ejpam-4767	82	1	lemma	lemma	PROPN
ejpam-4767	82	2	2.8	2.8	NUM
ejpam-4767	82	3	.	.	PUNCT
ejpam-4767	83	1	let	let	VERB
ejpam-4767	83	2	x	x	PRON
ejpam-4767	83	3	be	be	AUX
ejpam-4767	83	4	a	a	DET
ejpam-4767	83	5	t1	t1	NOUN
ejpam-4767	83	6	space	space	NOUN
ejpam-4767	83	7	and	and	CCONJ
ejpam-4767	83	8	suppose	suppose	VERB
ejpam-4767	83	9	space	space	NOUN
ejpam-4767	83	10	that	that	PRON
ejpam-4767	83	11	the	the	DET
ejpam-4767	83	12	family	family	NOUN
ejpam-4767	83	13	{	{	PUNCT
ejpam-4767	83	14	ws}s∈s	ws}s∈s	PROPN
ejpam-4767	83	15	is	be	AUX
ejpam-4767	83	16	locally	locally	ADV
ejpam-4767	83	17	finite	finite	ADJ
ejpam-4767	83	18	except	except	SCONJ
ejpam-4767	83	19	at	at	ADP
ejpam-4767	83	20	a	a	DET
ejpam-4767	83	21	finite	finite	ADJ
ejpam-4767	83	22	number	number	NOUN
ejpam-4767	83	23	of	of	ADP
ejpam-4767	83	24	points	point	NOUN
ejpam-4767	83	25	,	,	PUNCT
ejpam-4767	83	26	say	say	VERB
ejpam-4767	83	27	a1	a1	NOUN
ejpam-4767	83	28	,	,	PUNCT
ejpam-4767	83	29	a2	a2	PROPN
ejpam-4767	83	30	,	,	PUNCT
ejpam-4767	83	31	...	...	PUNCT
ejpam-4767	83	32	,	,	PUNCT
ejpam-4767	83	33	an	an	X
ejpam-4767	83	34	,	,	PUNCT
ejpam-4767	83	35	then	then	ADV
ejpam-4767	83	36	we	we	PRON
ejpam-4767	83	37	have	have	VERB
ejpam-4767	83	38	⋃	⋃	ADV
ejpam-4767	83	39	s∈s	s∈s	NOUN
ejpam-4767	83	40	ws	ws	NOUN
ejpam-4767	83	41	n⋃	n⋃	VERB
ejpam-4767	83	42	i=1	i=1	PRON
ejpam-4767	83	43	{	{	PUNCT
ejpam-4767	83	44	ai	ai	PROPN
ejpam-4767	83	45	}	}	PUNCT
ejpam-4767	83	46	=	=	SYM
ejpam-4767	83	47	⋃	⋃	ADP
ejpam-4767	83	48	s∈s	s∈s	NOUN
ejpam-4767	83	49	ws	ws	NOUN
ejpam-4767	83	50	n⋃	n⋃	VERB
ejpam-4767	83	51	i=1	i=1	PRON
ejpam-4767	83	52	{	{	PUNCT
ejpam-4767	83	53	ai	ai	NOUN
ejpam-4767	83	54	}	}	PUNCT
ejpam-4767	83	55	.	.	PUNCT
ejpam-4767	84	1	proof	proof	NOUN
ejpam-4767	84	2	.	.	PUNCT
ejpam-4767	85	1	it	it	PRON
ejpam-4767	85	2	is	be	AUX
ejpam-4767	85	3	clearly	clearly	ADV
ejpam-4767	85	4	that	that	SCONJ
ejpam-4767	85	5	⋃	⋃	PUNCT
ejpam-4767	85	6	s∈s	s∈s	NOUN
ejpam-4767	85	7	ws	ws	NOUN
ejpam-4767	85	8	n⋃	n⋃	VERB
ejpam-4767	85	9	i=1	i=1	PRON
ejpam-4767	85	10	{	{	PUNCT
ejpam-4767	85	11	ai	ai	PROPN
ejpam-4767	85	12	}	}	PUNCT
ejpam-4767	85	13	⊂	⊂	PROPN
ejpam-4767	85	14	⋃	⋃	ADV
ejpam-4767	85	15	s∈s	s∈s	NOUN
ejpam-4767	85	16	ws	ws	NOUN
ejpam-4767	85	17	n⋃	n⋃	VERB
ejpam-4767	85	18	i=1	i=1	PRON
ejpam-4767	85	19	{	{	PUNCT
ejpam-4767	85	20	ai	ai	NOUN
ejpam-4767	85	21	}	}	PUNCT
ejpam-4767	85	22	.	.	PUNCT
ejpam-4767	86	1	now	now	ADV
ejpam-4767	86	2	suppose	suppose	VERB
ejpam-4767	86	3	that	that	SCONJ
ejpam-4767	86	4	x	x	PUNCT
ejpam-4767	86	5	∈	∈	PROPN
ejpam-4767	86	6	⋃	⋃	PUNCT
ejpam-4767	86	7	s∈s	s∈s	NOUN
ejpam-4767	86	8	ws	ws	NOUN
ejpam-4767	86	9	∪	∪	X
ejpam-4767	86	10	{	{	PUNCT
ejpam-4767	86	11	a1	a1	PROPN
ejpam-4767	86	12	}	}	PUNCT
ejpam-4767	86	13	...	...	PUNCT
ejpam-4767	86	14	∪	∪	X
ejpam-4767	86	15	{	{	PUNCT
ejpam-4767	86	16	an	an	PRON
ejpam-4767	86	17	}	}	PUNCT
ejpam-4767	86	18	.	.	PUNCT
ejpam-4767	87	1	let	let	VERB
ejpam-4767	87	2	x	x	PRON
ejpam-4767	87	3	has	have	VERB
ejpam-4767	87	4	an	an	DET
ejpam-4767	87	5	open	open	ADJ
ejpam-4767	87	6	neighborhood	neighborhood	NOUN
ejpam-4767	87	7	u	u	NOUN
ejpam-4767	87	8	which	which	PRON
ejpam-4767	87	9	intersects	intersect	VERB
ejpam-4767	87	10	finitely	finitely	ADV
ejpam-4767	87	11	many	many	ADJ
ejpam-4767	87	12	members	member	NOUN
ejpam-4767	87	13	of	of	ADP
ejpam-4767	87	14	the	the	DET
ejpam-4767	87	15	family	family	NOUN
ejpam-4767	87	16	{	{	PUNCT
ejpam-4767	87	17	fs	fs	PROPN
ejpam-4767	87	18	,	,	PUNCT
ejpam-4767	87	19	{	{	PUNCT
ejpam-4767	87	20	a1	a1	NOUN
ejpam-4767	87	21	}	}	PUNCT
ejpam-4767	87	22	,	,	PUNCT
ejpam-4767	87	23	{	{	PUNCT
ejpam-4767	87	24	a2	a2	NOUN
ejpam-4767	87	25	}	}	PUNCT
ejpam-4767	87	26	,	,	PUNCT
ejpam-4767	87	27	...	...	PUNCT
ejpam-4767	87	28	,	,	PUNCT
ejpam-4767	87	29	{	{	PUNCT
ejpam-4767	87	30	an}}s∈s	an}}s∈s	X
ejpam-4767	87	31	.	.	PUNCT
ejpam-4767	88	1	let	let	VERB
ejpam-4767	88	2	s1	s1	PROPN
ejpam-4767	88	3	=	=	PUNCT
ejpam-4767	88	4	{	{	PUNCT
ejpam-4767	88	5	s	s	NOUN
ejpam-4767	88	6	∈	∈	NOUN
ejpam-4767	88	7	s	s	PART
ejpam-4767	88	8	:	:	PUNCT
ejpam-4767	88	9	u	u	NOUN
ejpam-4767	88	10	∩ws	∩ws	PROPN
ejpam-4767	88	11	}	}	PUNCT
ejpam-4767	88	12	be	be	VERB
ejpam-4767	88	13	the	the	DET
ejpam-4767	88	14	finite	finite	NOUN
ejpam-4767	88	15	indexing	indexing	NOUN
ejpam-4767	88	16	set	set	NOUN
ejpam-4767	88	17	.	.	PUNCT
ejpam-4767	89	1	it	it	PRON
ejpam-4767	89	2	is	be	AUX
ejpam-4767	89	3	clear	clear	ADJ
ejpam-4767	89	4	that	that	SCONJ
ejpam-4767	89	5	x	x	PUNCT
ejpam-4767	89	6	̸∈	̸∈	PROPN
ejpam-4767	89	7	⋃	⋃	PROPN
ejpam-4767	89	8	s∈s\s1	s∈s\s1	ADJ
ejpam-4767	89	9	ws	ws	NOUN
ejpam-4767	89	10	.	.	PUNCT
ejpam-4767	89	11	note	note	VERB
ejpam-4767	89	12	that	that	SCONJ
ejpam-4767	89	13	x	x	PUNCT
ejpam-4767	89	14	∈	∈	PROPN
ejpam-4767	89	15	⋃	⋃	VERB
ejpam-4767	89	16	s∈s1	s∈s1	ADJ
ejpam-4767	89	17	ws	ws	NOUN
ejpam-4767	89	18	n⋃	n⋃	VERB
ejpam-4767	89	19	i=1	i=1	PRON
ejpam-4767	89	20	{	{	PUNCT
ejpam-4767	89	21	ai	ai	VERB
ejpam-4767	89	22	}	}	PUNCT
ejpam-4767	89	23	⋃	⋃	NOUN
ejpam-4767	89	24	s	s	PART
ejpam-4767	89	25	̸∈s\s1	̸∈s\s1	X
ejpam-4767	89	26	ws	ws	NOUN
ejpam-4767	89	27	.	.	PUNCT
ejpam-4767	90	1	then	then	ADV
ejpam-4767	90	2	,	,	PUNCT
ejpam-4767	90	3	we	we	PRON
ejpam-4767	90	4	get	get	VERB
ejpam-4767	90	5	x	x	PUNCT
ejpam-4767	90	6	∈	∈	PROPN
ejpam-4767	90	7	⋃	⋃	ADP
ejpam-4767	90	8	s∈s1	s∈s1	ADJ
ejpam-4767	90	9	ws	ws	PROPN
ejpam-4767	90	10	⋃n	⋃n	PROPN
ejpam-4767	90	11	i=1{ai	i=1{ai	PROPN
ejpam-4767	90	12	}	}	PUNCT
ejpam-4767	90	13	=	=	SYM
ejpam-4767	90	14	⋃	⋃	ADP
ejpam-4767	90	15	s∈s1	s∈s1	ADJ
ejpam-4767	90	16	ws	ws	PROPN
ejpam-4767	90	17	⋃n	⋃n	PROPN
ejpam-4767	90	18	i=1	i=1	PROPN
ejpam-4767	90	19	{	{	PUNCT
ejpam-4767	90	20	ai	ai	NOUN
ejpam-4767	90	21	}	}	PUNCT
ejpam-4767	90	22	.	.	PUNCT
ejpam-4767	91	1	therefore	therefore	ADV
ejpam-4767	91	2	,	,	PUNCT
ejpam-4767	91	3	x	x	PUNCT
ejpam-4767	91	4	∈	∈	PROPN
ejpam-4767	91	5	⋃	⋃	PUNCT
ejpam-4767	91	6	s∈s	s∈s	NOUN
ejpam-4767	91	7	ws	ws	NOUN
ejpam-4767	91	8	∪	∪	X
ejpam-4767	91	9	{	{	PUNCT
ejpam-4767	91	10	a1	a1	PROPN
ejpam-4767	91	11	}	}	PUNCT
ejpam-4767	91	12	∪	∪	ADJ
ejpam-4767	91	13	{	{	PUNCT
ejpam-4767	91	14	a2	a2	NOUN
ejpam-4767	91	15	}	}	PUNCT
ejpam-4767	91	16	∪	∪	ADJ
ejpam-4767	91	17	...	...	PUNCT
ejpam-4767	91	18	∪	∪	X
ejpam-4767	91	19	{	{	PUNCT
ejpam-4767	91	20	an	an	PRON
ejpam-4767	91	21	}	}	PUNCT
ejpam-4767	91	22	.	.	PUNCT
ejpam-4767	91	23	assume	assume	VERB
ejpam-4767	91	24	that	that	SCONJ
ejpam-4767	91	25	x	x	PRON
ejpam-4767	91	26	does	do	AUX
ejpam-4767	91	27	not	not	PART
ejpam-4767	91	28	have	have	VERB
ejpam-4767	91	29	a	a	DET
ejpam-4767	91	30	locally	locally	ADV
ejpam-4767	91	31	finite	finite	ADJ
ejpam-4767	91	32	neighborhood	neighborhood	NOUN
ejpam-4767	91	33	,	,	PUNCT
ejpam-4767	91	34	then	then	ADV
ejpam-4767	91	35	x	x	X
ejpam-4767	91	36	=	=	PRON
ejpam-4767	91	37	ai	ai	VERB
ejpam-4767	91	38	for	for	ADP
ejpam-4767	91	39	some	some	DET
ejpam-4767	91	40	i.	i.	NOUN
ejpam-4767	91	41	therefore	therefore	ADV
ejpam-4767	91	42	,	,	PUNCT
ejpam-4767	91	43	it	it	PRON
ejpam-4767	91	44	is	be	AUX
ejpam-4767	91	45	easy	easy	ADJ
ejpam-4767	91	46	to	to	PART
ejpam-4767	91	47	see	see	VERB
ejpam-4767	91	48	that	that	PRON
ejpam-4767	91	49	x	x	PUNCT
ejpam-4767	91	50	=	=	PUNCT
ejpam-4767	92	1	ai	ai	VERB
ejpam-4767	92	2	∈	∈	PROPN
ejpam-4767	92	3	⋃	⋃	PUNCT
ejpam-4767	92	4	s∈s	s∈s	NOUN
ejpam-4767	92	5	ws	ws	NOUN
ejpam-4767	92	6	∪	∪	X
ejpam-4767	92	7	{	{	PUNCT
ejpam-4767	92	8	a1	a1	PROPN
ejpam-4767	92	9	}	}	PUNCT
ejpam-4767	92	10	∪	∪	ADJ
ejpam-4767	92	11	{	{	PUNCT
ejpam-4767	92	12	a2	a2	NOUN
ejpam-4767	92	13	}	}	PUNCT
ejpam-4767	92	14	∪	∪	ADJ
ejpam-4767	92	15	...	...	PUNCT
ejpam-4767	92	16	∪	∪	X
ejpam-4767	92	17	{	{	PUNCT
ejpam-4767	92	18	an	an	PRON
ejpam-4767	92	19	}	}	PUNCT
ejpam-4767	92	20	.	.	PUNCT
ejpam-4767	93	1	hence	hence	ADV
ejpam-4767	93	2	,	,	PUNCT
ejpam-4767	93	3	⋃	⋃	SCONJ
ejpam-4767	93	4	s∈s	s∈s	NOUN
ejpam-4767	93	5	ws	ws	NOUN
ejpam-4767	93	6	n⋃	n⋃	VERB
ejpam-4767	93	7	i=1	i=1	PRON
ejpam-4767	93	8	{	{	PUNCT
ejpam-4767	93	9	ai	ai	PROPN
ejpam-4767	93	10	}	}	PUNCT
ejpam-4767	93	11	=	=	SYM
ejpam-4767	93	12	⋃	⋃	ADP
ejpam-4767	93	13	s∈s	s∈s	NOUN
ejpam-4767	93	14	ws	ws	NOUN
ejpam-4767	93	15	n⋃	n⋃	VERB
ejpam-4767	93	16	i=1	i=1	PRON
ejpam-4767	93	17	{	{	PUNCT
ejpam-4767	93	18	ai	ai	NOUN
ejpam-4767	93	19	}	}	PUNCT
ejpam-4767	93	20	.	.	PUNCT
ejpam-4767	94	1	remark	remark	PROPN
ejpam-4767	94	2	3	3	NUM
ejpam-4767	94	3	.	.	PUNCT
ejpam-4767	95	1	l.c.w.d	l.c.w.d	PROPN
ejpam-4767	95	2	.	.	PUNCT
ejpam-4767	96	1	spaces	space	NOUN
ejpam-4767	96	2	do	do	AUX
ejpam-4767	96	3	not	not	PART
ejpam-4767	96	4	have	have	VERB
ejpam-4767	96	5	to	to	PART
ejpam-4767	96	6	be	be	AUX
ejpam-4767	96	7	normal	normal	ADJ
ejpam-4767	96	8	.	.	PUNCT
ejpam-4767	97	1	consider	consider	VERB
ejpam-4767	97	2	product	product	NOUN
ejpam-4767	97	3	of	of	ADP
ejpam-4767	97	4	sorgenfrey	sorgenfrey	PROPN
ejpam-4767	97	5	line	line	NOUN
ejpam-4767	97	6	,	,	PUNCT
ejpam-4767	97	7	x	x	X
ejpam-4767	97	8	=	=	SYM
ejpam-4767	97	9	k	k	X
ejpam-4767	97	10	×k	×k	PROPN
ejpam-4767	97	11	,	,	PUNCT
ejpam-4767	97	12	which	which	PRON
ejpam-4767	97	13	is	be	AUX
ejpam-4767	97	14	locally	locally	ADV
ejpam-4767	97	15	compact	compact	ADJ
ejpam-4767	97	16	i.e.	i.e.	X
ejpam-4767	97	17	x•	x•	X
ejpam-4767	98	1	=	=	SYM
ejpam-4767	98	2	ϕ	ϕ	PROPN
ejpam-4767	99	1	but	but	CCONJ
ejpam-4767	99	2	it	it	PRON
ejpam-4767	99	3	is	be	AUX
ejpam-4767	99	4	not	not	PART
ejpam-4767	99	5	normal	normal	ADJ
ejpam-4767	99	6	.	.	PUNCT
ejpam-4767	100	1	m.	m.	PROPN
ejpam-4767	100	2	zailai	zailai	PROPN
ejpam-4767	100	3	/	/	SYM
ejpam-4767	100	4	eur	eur	PROPN
ejpam-4767	100	5	.	.	PUNCT
ejpam-4767	101	1	j.	j.	PROPN
ejpam-4767	101	2	pure	pure	PROPN
ejpam-4767	101	3	appl	appl	PROPN
ejpam-4767	101	4	.	.	PROPN
ejpam-4767	101	5	math	math	PROPN
ejpam-4767	101	6	,	,	PUNCT
ejpam-4767	101	7	16	16	NUM
ejpam-4767	101	8	(	(	PUNCT
ejpam-4767	101	9	2	2	NUM
ejpam-4767	101	10	)	)	PUNCT
ejpam-4767	101	11	(	(	PUNCT
ejpam-4767	101	12	2023	2023	NUM
ejpam-4767	101	13	)	)	PUNCT
ejpam-4767	101	14	,	,	PUNCT
ejpam-4767	101	15	1228	1228	NUM
ejpam-4767	101	16	-	-	SYM
ejpam-4767	101	17	1235	1235	NUM
ejpam-4767	101	18	1231	1231	NUM
ejpam-4767	101	19	definition	definition	NOUN
ejpam-4767	101	20	2.9	2.9	NUM
ejpam-4767	101	21	.	.	PUNCT
ejpam-4767	102	1	[	[	X
ejpam-4767	102	2	2	2	NUM
ejpam-4767	102	3	,	,	PUNCT
ejpam-4767	102	4	page	page	NOUN
ejpam-4767	102	5	71	71	NUM
ejpam-4767	102	6	]	]	PUNCT
ejpam-4767	102	7	let	let	VERB
ejpam-4767	102	8	{	{	PUNCT
ejpam-4767	102	9	bi}i∈i	bi}i∈i	PRON
ejpam-4767	102	10	be	be	AUX
ejpam-4767	102	11	a	a	DET
ejpam-4767	102	12	cover	cover	NOUN
ejpam-4767	102	13	of	of	ADP
ejpam-4767	102	14	the	the	DET
ejpam-4767	102	15	space	space	NOUN
ejpam-4767	102	16	x.	x.	NOUN
ejpam-4767	102	17	consider	consider	VERB
ejpam-4767	102	18	any	any	DET
ejpam-4767	102	19	family	family	NOUN
ejpam-4767	102	20	of	of	ADP
ejpam-4767	102	21	continuous	continuous	ADJ
ejpam-4767	102	22	maps	map	NOUN
ejpam-4767	102	23	{	{	PUNCT
ejpam-4767	102	24	gi}i∈i	gi}i∈i	INTJ
ejpam-4767	102	25	,	,	PUNCT
ejpam-4767	102	26	where	where	SCONJ
ejpam-4767	102	27	gi	gi	X
ejpam-4767	102	28	:	:	PUNCT
ejpam-4767	102	29	bi	bi	PROPN
ejpam-4767	102	30	→	→	PUNCT
ejpam-4767	102	31	y.	y.	NOUN
ejpam-4767	102	32	the	the	DET
ejpam-4767	102	33	maps	map	NOUN
ejpam-4767	102	34	gi	gi	INTJ
ejpam-4767	102	35	are	be	AUX
ejpam-4767	102	36	said	say	VERB
ejpam-4767	102	37	to	to	PART
ejpam-4767	102	38	be	be	AUX
ejpam-4767	102	39	compatible	compatible	ADJ
ejpam-4767	102	40	if	if	SCONJ
ejpam-4767	102	41	for	for	ADP
ejpam-4767	102	42	every	every	DET
ejpam-4767	102	43	i1	i1	PROPN
ejpam-4767	102	44	,	,	PUNCT
ejpam-4767	102	45	i2	i2	PROPN
ejpam-4767	102	46	of	of	ADP
ejpam-4767	102	47	i	i	PRON
ejpam-4767	102	48	we	we	PRON
ejpam-4767	102	49	have	have	VERB
ejpam-4767	102	50	gi1	gi1	PROPN
ejpam-4767	102	51	|bi1	|bi1	PROPN
ejpam-4767	103	1	⋂	⋂	PROPN
ejpam-4767	103	2	bi2	bi2	PROPN
ejpam-4767	103	3	=	=	PROPN
ejpam-4767	103	4	gi2	gi2	PROPN
ejpam-4767	103	5	|bi1	|bi1	PROPN
ejpam-4767	103	6	⋂	⋂	PROPN
ejpam-4767	103	7	bi2	bi2	PROPN
ejpam-4767	103	8	.	.	PUNCT
ejpam-4767	104	1	the	the	DET
ejpam-4767	104	2	combination	combination	NOUN
ejpam-4767	104	3	is	be	AUX
ejpam-4767	104	4	defined	define	VERB
ejpam-4767	104	5	as	as	ADP
ejpam-4767	104	6	g	g	NOUN
ejpam-4767	104	7	=	=	SYM
ejpam-4767	104	8	⋃	⋃	PROPN
ejpam-4767	104	9	i∈i	i∈i	ADJ
ejpam-4767	104	10	gi	gi	INTJ
ejpam-4767	104	11	:	:	PUNCT
ejpam-4767	104	12	x	x	X
ejpam-4767	104	13	→	→	SYM
ejpam-4767	104	14	y.	y.	NOUN
ejpam-4767	104	15	remark	remark	NOUN
ejpam-4767	104	16	4	4	NUM
ejpam-4767	104	17	.	.	PUNCT
ejpam-4767	105	1	the	the	DET
ejpam-4767	105	2	following	follow	VERB
ejpam-4767	105	3	two	two	NUM
ejpam-4767	105	4	lemmas	lemma	NOUN
ejpam-4767	105	5	and	and	CCONJ
ejpam-4767	105	6	theorem	theorem	NOUN
ejpam-4767	105	7	are	be	AUX
ejpam-4767	105	8	used	use	VERB
ejpam-4767	105	9	in	in	ADP
ejpam-4767	105	10	the	the	DET
ejpam-4767	105	11	proofs	proof	NOUN
ejpam-4767	105	12	of	of	ADP
ejpam-4767	105	13	theorem	theorem	ADJ
ejpam-4767	105	14	2.13	2.13	NUM
ejpam-4767	105	15	and	and	CCONJ
ejpam-4767	105	16	theorem	theorem	VERB
ejpam-4767	105	17	2.15	2.15	NUM
ejpam-4767	105	18	.	.	PUNCT
ejpam-4767	106	1	lemma	lemma	PROPN
ejpam-4767	106	2	2.10	2.10	NUM
ejpam-4767	106	3	.	.	PUNCT
ejpam-4767	107	1	[	[	X
ejpam-4767	107	2	2	2	NUM
ejpam-4767	107	3	,	,	PUNCT
ejpam-4767	107	4	page	page	NOUN
ejpam-4767	107	5	17	17	NUM
ejpam-4767	107	6	]	]	PUNCT
ejpam-4767	107	7	if	if	SCONJ
ejpam-4767	107	8	{	{	PUNCT
ejpam-4767	107	9	si}i∈i	si}i∈i	VERB
ejpam-4767	107	10	is	be	AUX
ejpam-4767	107	11	a	a	DET
ejpam-4767	107	12	locally	locally	ADV
ejpam-4767	107	13	finite	finite	ADJ
ejpam-4767	107	14	closed	close	VERB
ejpam-4767	107	15	cover	cover	NOUN
ejpam-4767	107	16	of	of	ADP
ejpam-4767	107	17	x	x	PUNCT
ejpam-4767	107	18	and	and	CCONJ
ejpam-4767	107	19	{	{	PUNCT
ejpam-4767	107	20	gi}i∈i	gi}i∈i	INTJ
ejpam-4767	107	21	is	be	AUX
ejpam-4767	107	22	a	a	DET
ejpam-4767	107	23	family	family	NOUN
ejpam-4767	107	24	of	of	ADP
ejpam-4767	107	25	compatible	compatible	ADJ
ejpam-4767	107	26	maps	map	NOUN
ejpam-4767	107	27	,	,	PUNCT
ejpam-4767	107	28	where	where	SCONJ
ejpam-4767	107	29	gi	gi	ADV
ejpam-4767	107	30	:	:	PUNCT
ejpam-4767	107	31	si	si	PROPN
ejpam-4767	107	32	→	→	SYM
ejpam-4767	107	33	y	y	PROPN
ejpam-4767	107	34	.	.	PUNCT
ejpam-4767	108	1	then	then	ADV
ejpam-4767	108	2	the	the	DET
ejpam-4767	108	3	combination	combination	NOUN
ejpam-4767	108	4	is	be	AUX
ejpam-4767	108	5	continuous	continuous	ADJ
ejpam-4767	108	6	.	.	PUNCT
ejpam-4767	109	1	lemma	lemma	PROPN
ejpam-4767	109	2	2.11	2.11	NUM
ejpam-4767	109	3	.	.	PUNCT
ejpam-4767	110	1	let	let	VERB
ejpam-4767	110	2	w	w	VERB
ejpam-4767	110	3	=	=	VERB
ejpam-4767	110	4	{	{	PUNCT
ejpam-4767	110	5	bs}s∈s	bs}s∈s	INTJ
ejpam-4767	110	6	∪	∪	X
ejpam-4767	110	7	{	{	PUNCT
ejpam-4767	110	8	{	{	PUNCT
ejpam-4767	110	9	a1	a1	NOUN
ejpam-4767	110	10	}	}	PUNCT
ejpam-4767	110	11	,	,	PUNCT
ejpam-4767	110	12	{	{	PUNCT
ejpam-4767	110	13	a2	a2	NOUN
ejpam-4767	110	14	}	}	PUNCT
ejpam-4767	110	15	,	,	PUNCT
ejpam-4767	110	16	...	...	PUNCT
ejpam-4767	110	17	{	{	PUNCT
ejpam-4767	110	18	an	an	X
ejpam-4767	110	19	}	}	PUNCT
ejpam-4767	110	20	}	}	PUNCT
ejpam-4767	110	21	be	be	AUX
ejpam-4767	110	22	a	a	DET
ejpam-4767	110	23	closed	closed	ADJ
ejpam-4767	110	24	cover	cover	NOUN
ejpam-4767	110	25	of	of	ADP
ejpam-4767	110	26	x	x	SYM
ejpam-4767	110	27	such	such	ADJ
ejpam-4767	110	28	that	that	SCONJ
ejpam-4767	110	29	the	the	DET
ejpam-4767	110	30	family	family	NOUN
ejpam-4767	110	31	w	w	NOUN
ejpam-4767	110	32	is	be	AUX
ejpam-4767	110	33	locally	locally	ADV
ejpam-4767	110	34	finite	finite	ADJ
ejpam-4767	110	35	except	except	SCONJ
ejpam-4767	110	36	for	for	ADP
ejpam-4767	110	37	a1	a1	NOUN
ejpam-4767	110	38	,	,	PUNCT
ejpam-4767	110	39	a2	a2	PROPN
ejpam-4767	110	40	,	,	PUNCT
ejpam-4767	110	41	...	...	PUNCT
ejpam-4767	110	42	,	,	PUNCT
ejpam-4767	110	43	an	an	X
ejpam-4767	110	44	.	.	PUNCT
ejpam-4767	111	1	let	let	VERB
ejpam-4767	111	2	{	{	PUNCT
ejpam-4767	111	3	gs}s∈s	gs}s∈s	AUX
ejpam-4767	111	4	be	be	AUX
ejpam-4767	111	5	a	a	DET
ejpam-4767	111	6	family	family	NOUN
ejpam-4767	111	7	of	of	ADP
ejpam-4767	111	8	compatible	compatible	ADJ
ejpam-4767	111	9	maps	map	NOUN
ejpam-4767	111	10	,	,	PUNCT
ejpam-4767	111	11	where	where	SCONJ
ejpam-4767	111	12	gs	gs	INTJ
ejpam-4767	111	13	:	:	PUNCT
ejpam-4767	111	14	bs	bs	PROPN
ejpam-4767	111	15	→	→	PUNCT
ejpam-4767	111	16	y	y	PROPN
ejpam-4767	111	17	such	such	ADJ
ejpam-4767	111	18	that	that	SCONJ
ejpam-4767	111	19	all	all	DET
ejpam-4767	111	20	members	member	NOUN
ejpam-4767	111	21	of	of	ADP
ejpam-4767	111	22	the	the	DET
ejpam-4767	111	23	family	family	NOUN
ejpam-4767	111	24	are	be	AUX
ejpam-4767	111	25	constant	constant	ADJ
ejpam-4767	111	26	of	of	ADP
ejpam-4767	111	27	the	the	DET
ejpam-4767	111	28	form	form	NOUN
ejpam-4767	111	29	gs(bs	gs(bs	NOUN
ejpam-4767	111	30	)	)	PUNCT
ejpam-4767	112	1	=	=	SYM
ejpam-4767	112	2	k	k	PROPN
ejpam-4767	112	3	except	except	SCONJ
ejpam-4767	112	4	for	for	ADP
ejpam-4767	112	5	a	a	DET
ejpam-4767	112	6	finite	finite	ADJ
ejpam-4767	112	7	number	number	NOUN
ejpam-4767	112	8	of	of	ADP
ejpam-4767	112	9	members	member	NOUN
ejpam-4767	112	10	.	.	PUNCT
ejpam-4767	113	1	then	then	ADV
ejpam-4767	113	2	,	,	PUNCT
ejpam-4767	113	3	the	the	DET
ejpam-4767	113	4	combination	combination	NOUN
ejpam-4767	113	5	g	g	NOUN
ejpam-4767	113	6	=	=	SYM
ejpam-4767	113	7	⋃	⋃	ADP
ejpam-4767	113	8	s∈s	s∈s	NOUN
ejpam-4767	113	9	gs	gs	PROPN
ejpam-4767	113	10	n⋃	n⋃	VERB
ejpam-4767	113	11	i=1	i=1	PROPN
ejpam-4767	113	12	fi	fi	NOUN
ejpam-4767	113	13	:	:	PUNCT
ejpam-4767	113	14	x	x	X
ejpam-4767	113	15	→	→	SYM
ejpam-4767	113	16	y	y	PROPN
ejpam-4767	113	17	is	be	AUX
ejpam-4767	113	18	continuous	continuous	ADJ
ejpam-4767	113	19	,	,	PUNCT
ejpam-4767	113	20	where	where	SCONJ
ejpam-4767	113	21	fi	fi	NOUN
ejpam-4767	113	22	:	:	PUNCT
ejpam-4767	113	23	{	{	PUNCT
ejpam-4767	113	24	ai	ai	VERB
ejpam-4767	113	25	}	}	PUNCT
ejpam-4767	113	26	→	→	SYM
ejpam-4767	113	27	y	y	PROPN
ejpam-4767	113	28	is	be	AUX
ejpam-4767	113	29	defined	define	VERB
ejpam-4767	113	30	as	as	ADP
ejpam-4767	113	31	fi(ai	fi(ai	PROPN
ejpam-4767	113	32	)	)	PUNCT
ejpam-4767	113	33	=	=	SYM
ejpam-4767	113	34	k	k	PROPN
ejpam-4767	113	35	for	for	ADP
ejpam-4767	113	36	i	i	PRON
ejpam-4767	113	37	=	=	NOUN
ejpam-4767	113	38	1	1	NUM
ejpam-4767	113	39	,	,	PUNCT
ejpam-4767	113	40	2	2	NUM
ejpam-4767	113	41	,	,	PUNCT
ejpam-4767	113	42	...	...	PUNCT
ejpam-4767	113	43	,	,	PUNCT
ejpam-4767	113	44	n.	n.	NOUN
ejpam-4767	113	45	proof	proof	NOUN
ejpam-4767	113	46	.	.	PUNCT
ejpam-4767	114	1	let	let	VERB
ejpam-4767	114	2	f	f	PRON
ejpam-4767	114	3	be	be	AUX
ejpam-4767	114	4	a	a	DET
ejpam-4767	114	5	closed	closed	ADJ
ejpam-4767	114	6	subset	subset	NOUN
ejpam-4767	114	7	of	of	ADP
ejpam-4767	114	8	y.	y.	NOUN
ejpam-4767	114	9	we	we	PRON
ejpam-4767	114	10	want	want	VERB
ejpam-4767	114	11	to	to	PART
ejpam-4767	114	12	show	show	VERB
ejpam-4767	114	13	that	that	SCONJ
ejpam-4767	114	14	the	the	DET
ejpam-4767	114	15	the	the	DET
ejpam-4767	114	16	inverse	inverse	ADJ
ejpam-4767	114	17	image	image	NOUN
ejpam-4767	114	18	of	of	ADP
ejpam-4767	114	19	f	f	PROPN
ejpam-4767	114	20	is	be	AUX
ejpam-4767	114	21	closed	close	VERB
ejpam-4767	114	22	in	in	ADP
ejpam-4767	114	23	x	x	PUNCT
ejpam-4767	114	24	under	under	ADP
ejpam-4767	114	25	that	that	DET
ejpam-4767	114	26	map	map	NOUN
ejpam-4767	114	27	g.	g.	PROPN
ejpam-4767	114	28	assume	assume	VERB
ejpam-4767	114	29	that	that	SCONJ
ejpam-4767	114	30	k	k	PROPN
ejpam-4767	114	31	̸∈	̸∈	PROPN
ejpam-4767	114	32	f	f	PROPN
ejpam-4767	114	33	,	,	PUNCT
ejpam-4767	114	34	then	then	ADV
ejpam-4767	114	35	g−1(f	g−1(f	PROPN
ejpam-4767	114	36	)	)	PUNCT
ejpam-4767	115	1	=	=	PUNCT
ejpam-4767	115	2	⋃	⋃	ADP
ejpam-4767	115	3	s∈s1	s∈s1	NOUN
ejpam-4767	115	4	gs(f	gs(f	NUM
ejpam-4767	115	5	)	)	PUNCT
ejpam-4767	115	6	for	for	ADP
ejpam-4767	115	7	a	a	DET
ejpam-4767	115	8	finite	finite	NOUN
ejpam-4767	115	9	subset	subset	NOUN
ejpam-4767	115	10	s1	s1	PROPN
ejpam-4767	115	11	⊂	⊂	PROPN
ejpam-4767	115	12	s.	s.	PROPN
ejpam-4767	115	13	therefore	therefore	ADV
ejpam-4767	115	14	,	,	PUNCT
ejpam-4767	115	15	g−1(f	g−1(f	PROPN
ejpam-4767	115	16	)	)	PUNCT
ejpam-4767	115	17	is	be	AUX
ejpam-4767	115	18	a	a	DET
ejpam-4767	115	19	finite	finite	ADJ
ejpam-4767	115	20	union	union	NOUN
ejpam-4767	115	21	of	of	ADP
ejpam-4767	115	22	closed	closed	ADJ
ejpam-4767	115	23	subsets	subset	NOUN
ejpam-4767	115	24	,	,	PUNCT
ejpam-4767	115	25	i.e.	i.e.	X
ejpam-4767	115	26	,	,	PUNCT
ejpam-4767	115	27	g−1	g−1	PROPN
ejpam-4767	115	28	is	be	AUX
ejpam-4767	115	29	closed	closed	ADJ
ejpam-4767	115	30	.	.	PUNCT
ejpam-4767	116	1	now	now	ADV
ejpam-4767	116	2	,	,	PUNCT
ejpam-4767	116	3	let	let	VERB
ejpam-4767	116	4	us	we	PRON
ejpam-4767	116	5	assume	assume	VERB
ejpam-4767	116	6	that	that	SCONJ
ejpam-4767	116	7	k	k	PROPN
ejpam-4767	116	8	∈	∈	PROPN
ejpam-4767	116	9	f.	f.	NOUN
ejpam-4767	116	10	we	we	PRON
ejpam-4767	116	11	have	have	VERB
ejpam-4767	116	12	g−1(f	g−1(f	PROPN
ejpam-4767	116	13	)	)	PUNCT
ejpam-4767	117	1	=	=	SYM
ejpam-4767	117	2	⋃	⋃	NOUN
ejpam-4767	117	3	s∈s\s1	s∈s\s1	X
ejpam-4767	117	4	g−1	g−1	PROPN
ejpam-4767	117	5	s	s	X
ejpam-4767	117	6	(	(	PUNCT
ejpam-4767	117	7	f	f	PROPN
ejpam-4767	117	8	)	)	PUNCT
ejpam-4767	117	9	n⋃	n⋃	VERB
ejpam-4767	117	10	i=1	i=1	PROPN
ejpam-4767	118	1	f−1	f−1	INTJ
ejpam-4767	118	2	i	i	PRON
ejpam-4767	118	3	(	(	PUNCT
ejpam-4767	118	4	f	f	PROPN
ejpam-4767	118	5	)	)	PUNCT
ejpam-4767	119	1	=	=	SYM
ejpam-4767	119	2	⋃	⋃	NOUN
ejpam-4767	119	3	s∈s\s1	s∈s\s1	X
ejpam-4767	119	4	g−1	g−1	PROPN
ejpam-4767	119	5	s	s	X
ejpam-4767	119	6	(	(	PUNCT
ejpam-4767	119	7	f	f	PROPN
ejpam-4767	119	8	)	)	PUNCT
ejpam-4767	119	9	n⋃	n⋃	PROPN
ejpam-4767	119	10	i=1	i=1	PRON
ejpam-4767	119	11	{	{	PUNCT
ejpam-4767	119	12	ai	ai	PROPN
ejpam-4767	119	13	}	}	PUNCT
ejpam-4767	119	14	therefore	therefore	ADV
ejpam-4767	119	15	,	,	PUNCT
ejpam-4767	119	16	by	by	ADP
ejpam-4767	119	17	using	use	VERB
ejpam-4767	119	18	lemma	lemma	PROPN
ejpam-4767	119	19	2.8	2.8	NUM
ejpam-4767	119	20	we	we	PRON
ejpam-4767	119	21	have	have	VERB
ejpam-4767	119	22	that	that	DET
ejpam-4767	119	23	g−1(f	g−1(f	PROPN
ejpam-4767	119	24	)	)	PUNCT
ejpam-4767	119	25	is	be	AUX
ejpam-4767	119	26	a	a	DET
ejpam-4767	119	27	closed	closed	ADJ
ejpam-4767	119	28	subset	subset	NOUN
ejpam-4767	119	29	of	of	ADP
ejpam-4767	119	30	x.	x.	NOUN
ejpam-4767	119	31	hence	hence	ADV
ejpam-4767	119	32	,	,	PUNCT
ejpam-4767	119	33	g	g	PROPN
ejpam-4767	119	34	is	be	AUX
ejpam-4767	119	35	continuous	continuous	ADJ
ejpam-4767	119	36	.	.	PUNCT
ejpam-4767	120	1	theorem	theorem	VERB
ejpam-4767	120	2	2.12	2.12	NUM
ejpam-4767	120	3	.	.	PUNCT
ejpam-4767	121	1	[	[	X
ejpam-4767	121	2	2	2	NUM
ejpam-4767	121	3	,	,	PUNCT
ejpam-4767	121	4	page	page	NOUN
ejpam-4767	121	5	148	148	NUM
ejpam-4767	121	6	]	]	PUNCT
ejpam-4767	121	7	let	let	VERB
ejpam-4767	121	8	x	x	PRON
ejpam-4767	121	9	be	be	AUX
ejpam-4767	121	10	a	a	DET
ejpam-4767	121	11	t1	t1	NOUN
ejpam-4767	121	12	space	space	NOUN
ejpam-4767	121	13	such	such	ADJ
ejpam-4767	121	14	that	that	SCONJ
ejpam-4767	121	15	any	any	DET
ejpam-4767	121	16	x	x	SYM
ejpam-4767	121	17	∈	∈	PROPN
ejpam-4767	121	18	x	x	PRON
ejpam-4767	121	19	possesses	possess	VERB
ejpam-4767	121	20	a	a	DET
ejpam-4767	121	21	compact	compact	ADJ
ejpam-4767	121	22	neighborhood	neighborhood	NOUN
ejpam-4767	121	23	.	.	PUNCT
ejpam-4767	122	1	then	then	ADV
ejpam-4767	122	2	for	for	ADP
ejpam-4767	122	3	any	any	DET
ejpam-4767	122	4	closed	closed	ADJ
ejpam-4767	122	5	subset	subset	NOUN
ejpam-4767	122	6	f	f	PROPN
ejpam-4767	122	7	⊂	⊂	PROPN
ejpam-4767	122	8	x	x	PUNCT
ejpam-4767	122	9	such	such	ADJ
ejpam-4767	122	10	that	that	SCONJ
ejpam-4767	122	11	x	x	SYM
ejpam-4767	122	12	̸∈	̸∈	PROPN
ejpam-4767	122	13	f	f	PROPN
ejpam-4767	122	14	there	there	PRON
ejpam-4767	122	15	exists	exist	VERB
ejpam-4767	122	16	a	a	DET
ejpam-4767	122	17	continuous	continuous	ADJ
ejpam-4767	122	18	f	f	NOUN
ejpam-4767	122	19	:	:	PUNCT
ejpam-4767	122	20	x	x	X
ejpam-4767	122	21	→	→	PUNCT
ejpam-4767	122	22	i	i	PRON
ejpam-4767	122	23	where	where	SCONJ
ejpam-4767	122	24	f(x	f(x	PROPN
ejpam-4767	122	25	)	)	PUNCT
ejpam-4767	122	26	=	=	SYM
ejpam-4767	122	27	0	0	NUM
ejpam-4767	122	28	and	and	CCONJ
ejpam-4767	122	29	f(f	f(f	PROPN
ejpam-4767	122	30	)	)	PUNCT
ejpam-4767	123	1	⊂	⊂	PROPN
ejpam-4767	123	2	{	{	PUNCT
ejpam-4767	123	3	1	1	NUM
ejpam-4767	123	4	}	}	PUNCT
ejpam-4767	123	5	.	.	PUNCT
ejpam-4767	124	1	theorem	theorem	NOUN
ejpam-4767	124	2	2.13	2.13	NUM
ejpam-4767	124	3	.	.	PUNCT
ejpam-4767	125	1	let	let	VERB
ejpam-4767	125	2	x•	x•	NOUN
ejpam-4767	125	3	be	be	AUX
ejpam-4767	125	4	a	a	DET
ejpam-4767	125	5	compact	compact	ADJ
ejpam-4767	125	6	subset	subset	NOUN
ejpam-4767	125	7	of	of	ADP
ejpam-4767	125	8	x.	x.	PROPN
ejpam-4767	125	9	suppose	suppose	VERB
ejpam-4767	125	10	that	that	SCONJ
ejpam-4767	125	11	each	each	DET
ejpam-4767	125	12	point	point	NOUN
ejpam-4767	125	13	x	x	X
ejpam-4767	125	14	∈	∈	NOUN
ejpam-4767	125	15	x•	x•	NOUN
ejpam-4767	125	16	has	have	VERB
ejpam-4767	125	17	an	an	DET
ejpam-4767	125	18	open	open	ADJ
ejpam-4767	125	19	neighborhood	neighborhood	NOUN
ejpam-4767	125	20	u	u	NOUN
ejpam-4767	125	21	̸=	̸=	PROPN
ejpam-4767	125	22	x	x	SYM
ejpam-4767	125	23	such	such	ADJ
ejpam-4767	125	24	that	that	SCONJ
ejpam-4767	125	25	the	the	DET
ejpam-4767	125	26	partition	partition	NOUN
ejpam-4767	125	27	of	of	ADP
ejpam-4767	125	28	singletons	singleton	NOUN
ejpam-4767	125	29	of	of	ADP
ejpam-4767	125	30	the	the	DET
ejpam-4767	125	31	complement	complement	NOUN
ejpam-4767	125	32	of	of	ADP
ejpam-4767	125	33	x•	x•	PROPN
ejpam-4767	125	34	∪	∪	X
ejpam-4767	125	35	(	(	PUNCT
ejpam-4767	125	36	u\u	u\u	NOUN
ejpam-4767	125	37	)	)	PUNCT
ejpam-4767	125	38	is	be	AUX
ejpam-4767	125	39	locally	locally	ADV
ejpam-4767	125	40	finite	finite	ADJ
ejpam-4767	125	41	,	,	PUNCT
ejpam-4767	125	42	then	then	ADV
ejpam-4767	125	43	x	x	PUNCT
ejpam-4767	125	44	is	be	AUX
ejpam-4767	125	45	t3	t3	PROPN
ejpam-4767	125	46	1	1	NUM
ejpam-4767	125	47	2	2	NUM
ejpam-4767	125	48	.	.	PUNCT
ejpam-4767	126	1	m.	m.	PROPN
ejpam-4767	126	2	zailai	zailai	PROPN
ejpam-4767	126	3	/	/	SYM
ejpam-4767	126	4	eur	eur	PROPN
ejpam-4767	126	5	.	.	PUNCT
ejpam-4767	127	1	j.	j.	PROPN
ejpam-4767	127	2	pure	pure	PROPN
ejpam-4767	127	3	appl	appl	PROPN
ejpam-4767	127	4	.	.	PROPN
ejpam-4767	127	5	math	math	PROPN
ejpam-4767	127	6	,	,	PUNCT
ejpam-4767	127	7	16	16	NUM
ejpam-4767	127	8	(	(	PUNCT
ejpam-4767	127	9	2	2	NUM
ejpam-4767	127	10	)	)	PUNCT
ejpam-4767	127	11	(	(	PUNCT
ejpam-4767	127	12	2023	2023	NUM
ejpam-4767	127	13	)	)	PUNCT
ejpam-4767	127	14	,	,	PUNCT
ejpam-4767	127	15	1228	1228	NUM
ejpam-4767	127	16	-	-	SYM
ejpam-4767	127	17	1235	1235	NUM
ejpam-4767	127	18	1232	1232	NUM
ejpam-4767	127	19	proof	proof	NOUN
ejpam-4767	127	20	.	.	PUNCT
ejpam-4767	128	1	let	let	VERB
ejpam-4767	128	2	x	x	SYM
ejpam-4767	128	3	∈	∈	PRON
ejpam-4767	128	4	x•	x•	NOUN
ejpam-4767	128	5	and	and	CCONJ
ejpam-4767	128	6	take	take	VERB
ejpam-4767	128	7	any	any	DET
ejpam-4767	128	8	closed	closed	ADJ
ejpam-4767	128	9	subset	subset	NOUN
ejpam-4767	128	10	f	f	PROPN
ejpam-4767	128	11	of	of	ADP
ejpam-4767	128	12	x	x	INTJ
ejpam-4767	128	13	such	such	ADJ
ejpam-4767	128	14	that	that	SCONJ
ejpam-4767	128	15	x	x	PRON
ejpam-4767	128	16	̸∈	̸∈	PROPN
ejpam-4767	128	17	f.	f.	PROPN
ejpam-4767	128	18	let	let	VERB
ejpam-4767	128	19	u	u	PRON
ejpam-4767	128	20	̸=	̸=	PROPN
ejpam-4767	128	21	x	x	PART
ejpam-4767	128	22	be	be	AUX
ejpam-4767	128	23	an	an	DET
ejpam-4767	128	24	open	open	ADJ
ejpam-4767	128	25	neighborhood	neighborhood	NOUN
ejpam-4767	128	26	of	of	ADP
ejpam-4767	128	27	x	x	PUNCT
ejpam-4767	128	28	which	which	PRON
ejpam-4767	128	29	satisfies	satisfy	VERB
ejpam-4767	128	30	assumption	assumption	NOUN
ejpam-4767	128	31	above	above	ADV
ejpam-4767	128	32	.	.	PUNCT
ejpam-4767	129	1	define	define	VERB
ejpam-4767	129	2	f0	f0	PROPN
ejpam-4767	129	3	=	=	SYM
ejpam-4767	129	4	(	(	PUNCT
ejpam-4767	129	5	(	(	PUNCT
ejpam-4767	129	6	u\u	u\u	NOUN
ejpam-4767	129	7	)	)	PUNCT
ejpam-4767	129	8	∪	∪	NOUN
ejpam-4767	129	9	(	(	PUNCT
ejpam-4767	129	10	u	u	NOUN
ejpam-4767	129	11	∩	∩	NOUN
ejpam-4767	129	12	f	f	PROPN
ejpam-4767	129	13	)	)	PUNCT
ejpam-4767	129	14	)	)	PUNCT
ejpam-4767	130	1	∩x•	∩x•	PROPN
ejpam-4767	130	2	,	,	PUNCT
ejpam-4767	130	3	which	which	PRON
ejpam-4767	130	4	is	be	AUX
ejpam-4767	130	5	a	a	DET
ejpam-4767	130	6	closed	closed	ADJ
ejpam-4767	130	7	subset	subset	NOUN
ejpam-4767	130	8	of	of	ADP
ejpam-4767	130	9	the	the	DET
ejpam-4767	130	10	closed	closed	ADJ
ejpam-4767	130	11	subspace	subspace	NOUN
ejpam-4767	130	12	x•	x•	NOUN
ejpam-4767	130	13	such	such	ADJ
ejpam-4767	130	14	that	that	SCONJ
ejpam-4767	130	15	x	x	PRON
ejpam-4767	130	16	̸∈	̸∈	PROPN
ejpam-4767	130	17	f0	f0	PROPN
ejpam-4767	130	18	.	.	PUNCT
ejpam-4767	131	1	therefore	therefore	ADV
ejpam-4767	131	2	,	,	PUNCT
ejpam-4767	131	3	there	there	PRON
ejpam-4767	131	4	exists	exist	VERB
ejpam-4767	131	5	a	a	DET
ejpam-4767	131	6	map	map	NOUN
ejpam-4767	132	1	f	f	NOUN
ejpam-4767	132	2	:	:	PUNCT
ejpam-4767	132	3	x•	x•	PROPN
ejpam-4767	132	4	→	→	PUNCT
ejpam-4767	132	5	i	i	PRON
ejpam-4767	132	6	such	such	ADJ
ejpam-4767	132	7	that	that	SCONJ
ejpam-4767	132	8	f(x	f(x	NOUN
ejpam-4767	132	9	)	)	PUNCT
ejpam-4767	133	1	=	=	SYM
ejpam-4767	133	2	0	0	NUM
ejpam-4767	133	3	and	and	CCONJ
ejpam-4767	133	4	f(f0	f(f0	NOUN
ejpam-4767	133	5	)	)	PUNCT
ejpam-4767	134	1	⊂	⊂	PROPN
ejpam-4767	134	2	{	{	PUNCT
ejpam-4767	134	3	1	1	NUM
ejpam-4767	134	4	}	}	PUNCT
ejpam-4767	134	5	.	.	PUNCT
ejpam-4767	135	1	let	let	VERB
ejpam-4767	135	2	g	g	NOUN
ejpam-4767	135	3	:	:	PUNCT
ejpam-4767	135	4	u\u	u\u	PROPN
ejpam-4767	135	5	→	→	SYM
ejpam-4767	135	6	i	i	PRON
ejpam-4767	135	7	be	be	VERB
ejpam-4767	135	8	a	a	DET
ejpam-4767	135	9	constant	constant	ADJ
ejpam-4767	135	10	map	map	NOUN
ejpam-4767	135	11	which	which	PRON
ejpam-4767	135	12	is	be	AUX
ejpam-4767	135	13	defined	define	VERB
ejpam-4767	135	14	as	as	ADP
ejpam-4767	135	15	g(y	g(y	NOUN
ejpam-4767	135	16	)	)	PUNCT
ejpam-4767	135	17	=	=	SYM
ejpam-4767	135	18	1	1	NUM
ejpam-4767	135	19	for	for	ADP
ejpam-4767	135	20	any	any	DET
ejpam-4767	135	21	y	y	PROPN
ejpam-4767	135	22	∈	∈	PROPN
ejpam-4767	135	23	u\u	u\u	PROPN
ejpam-4767	135	24	.	.	PUNCT
ejpam-4767	135	25	let	let	VERB
ejpam-4767	135	26	us	we	PRON
ejpam-4767	135	27	define	define	VERB
ejpam-4767	135	28	also	also	ADV
ejpam-4767	135	29	the	the	DET
ejpam-4767	135	30	following	follow	VERB
ejpam-4767	135	31	maps	map	NOUN
ejpam-4767	135	32	fs∈s	fs∈s	NOUN
ejpam-4767	135	33	:	:	PUNCT
ejpam-4767	135	34	{	{	PUNCT
ejpam-4767	135	35	as	as	ADP
ejpam-4767	135	36	}	}	PUNCT
ejpam-4767	135	37	→	→	SYM
ejpam-4767	135	38	i	i	NOUN
ejpam-4767	135	39	;	;	PUNCT
ejpam-4767	135	40	as	as	ADP
ejpam-4767	135	41	7→	7→	NUM
ejpam-4767	135	42	1	1	NUM
ejpam-4767	135	43	.	.	PUNCT
ejpam-4767	136	1	now	now	ADV
ejpam-4767	136	2	,	,	PUNCT
ejpam-4767	136	3	the	the	DET
ejpam-4767	136	4	combination	combination	NOUN
ejpam-4767	136	5	h	h	NOUN
ejpam-4767	136	6	=	=	SYM
ejpam-4767	136	7	f	f	PROPN
ejpam-4767	136	8	∪	∪	ADP
ejpam-4767	136	9	g	g	NOUN
ejpam-4767	136	10	⋃	⋃	PROPN
ejpam-4767	136	11	s∈s	s∈s	NOUN
ejpam-4767	136	12	fs	fs	ADP
ejpam-4767	136	13	:	:	PUNCT
ejpam-4767	136	14	x	x	X
ejpam-4767	136	15	→	→	SYM
ejpam-4767	136	16	i	i	PRON
ejpam-4767	136	17	is	be	AUX
ejpam-4767	136	18	continuous	continuous	ADJ
ejpam-4767	136	19	such	such	ADJ
ejpam-4767	136	20	that	that	SCONJ
ejpam-4767	136	21	h(x	h(x	PROPN
ejpam-4767	136	22	)	)	PUNCT
ejpam-4767	137	1	=	=	SYM
ejpam-4767	137	2	0	0	NUM
ejpam-4767	137	3	and	and	CCONJ
ejpam-4767	137	4	h(f	h(f	PROPN
ejpam-4767	137	5	)	)	PUNCT
ejpam-4767	138	1	⊂	⊂	PRON
ejpam-4767	138	2	{	{	PUNCT
ejpam-4767	138	3	1	1	NUM
ejpam-4767	138	4	}	}	PUNCT
ejpam-4767	138	5	.	.	PUNCT
ejpam-4767	139	1	proposition	proposition	NOUN
ejpam-4767	139	2	2.14	2.14	NUM
ejpam-4767	139	3	.	.	PUNCT
ejpam-4767	140	1	let	let	VERB
ejpam-4767	140	2	x	x	PRON
ejpam-4767	140	3	be	be	AUX
ejpam-4767	140	4	a	a	DET
ejpam-4767	140	5	second	second	ADJ
ejpam-4767	140	6	countable	countable	ADJ
ejpam-4767	140	7	space	space	NOUN
ejpam-4767	140	8	such	such	ADJ
ejpam-4767	140	9	that	that	SCONJ
ejpam-4767	140	10	x•	x•	PROPN
ejpam-4767	140	11	is	be	AUX
ejpam-4767	140	12	a	a	DET
ejpam-4767	140	13	discrete	discrete	ADJ
ejpam-4767	140	14	subspace	subspace	NOUN
ejpam-4767	140	15	.	.	PUNCT
ejpam-4767	141	1	if	if	SCONJ
ejpam-4767	141	2	x•	x•	NOUN
ejpam-4767	141	3	is	be	AUX
ejpam-4767	141	4	compact	compact	ADJ
ejpam-4767	141	5	such	such	ADJ
ejpam-4767	141	6	that	that	SCONJ
ejpam-4767	141	7	each	each	PRON
ejpam-4767	141	8	of	of	ADP
ejpam-4767	141	9	its	its	PRON
ejpam-4767	141	10	points	point	NOUN
ejpam-4767	141	11	satisfies	satisfy	VERB
ejpam-4767	141	12	the	the	DET
ejpam-4767	141	13	assumption	assumption	NOUN
ejpam-4767	141	14	in	in	ADP
ejpam-4767	141	15	theorem	theorem	NOUN
ejpam-4767	141	16	2.13	2.13	NUM
ejpam-4767	141	17	,	,	PUNCT
ejpam-4767	141	18	then	then	ADV
ejpam-4767	141	19	x•	x•	PROPN
ejpam-4767	141	20	is	be	AUX
ejpam-4767	141	21	of	of	ADP
ejpam-4767	141	22	cardinality	cardinality	NOUN
ejpam-4767	141	23	ℵ0	ℵ0	PROPN
ejpam-4767	141	24	.	.	PUNCT
ejpam-4767	142	1	proof	proof	NOUN
ejpam-4767	142	2	.	.	PUNCT
ejpam-4767	143	1	first	first	ADV
ejpam-4767	143	2	from	from	ADP
ejpam-4767	143	3	theorem	theorem	ADJ
ejpam-4767	143	4	2.13	2.13	NUM
ejpam-4767	143	5	we	we	PRON
ejpam-4767	143	6	have	have	VERB
ejpam-4767	143	7	that	that	PRON
ejpam-4767	143	8	x	x	PRON
ejpam-4767	143	9	is	be	AUX
ejpam-4767	143	10	a	a	DET
ejpam-4767	143	11	t3	t3	PROPN
ejpam-4767	143	12	1	1	NUM
ejpam-4767	143	13	2	2	NUM
ejpam-4767	143	14	space	space	NOUN
ejpam-4767	143	15	which	which	PRON
ejpam-4767	143	16	tells	tell	VERB
ejpam-4767	143	17	us	we	PRON
ejpam-4767	143	18	that	that	SCONJ
ejpam-4767	143	19	x	x	PRON
ejpam-4767	143	20	is	be	AUX
ejpam-4767	143	21	a	a	DET
ejpam-4767	143	22	regular	regular	ADJ
ejpam-4767	143	23	space	space	NOUN
ejpam-4767	143	24	.	.	PUNCT
ejpam-4767	144	1	since	since	SCONJ
ejpam-4767	144	2	every	every	DET
ejpam-4767	144	3	second	second	ADJ
ejpam-4767	144	4	countable	countable	ADJ
ejpam-4767	144	5	regular	regular	ADJ
ejpam-4767	144	6	space	space	NOUN
ejpam-4767	144	7	is	be	AUX
ejpam-4767	144	8	metrizable	metrizable	ADJ
ejpam-4767	144	9	,	,	PUNCT
ejpam-4767	144	10	then	then	ADV
ejpam-4767	144	11	x	x	PUNCT
ejpam-4767	144	12	is	be	AUX
ejpam-4767	144	13	a	a	DET
ejpam-4767	144	14	metrizable	metrizable	ADJ
ejpam-4767	144	15	space	space	NOUN
ejpam-4767	144	16	.	.	PUNCT
ejpam-4767	145	1	separability	separability	NOUN
ejpam-4767	145	2	and	and	CCONJ
ejpam-4767	145	3	second	second	ADJ
ejpam-4767	145	4	countability	countability	NOUN
ejpam-4767	145	5	are	be	AUX
ejpam-4767	145	6	equivalent	equivalent	ADJ
ejpam-4767	145	7	in	in	ADP
ejpam-4767	145	8	metrizable	metrizable	ADJ
ejpam-4767	145	9	spaces	space	NOUN
ejpam-4767	145	10	.	.	PUNCT
ejpam-4767	146	1	hence	hence	ADV
ejpam-4767	146	2	,	,	PUNCT
ejpam-4767	146	3	x	x	X
ejpam-4767	146	4	is	be	AUX
ejpam-4767	146	5	a	a	DET
ejpam-4767	146	6	separable	separable	ADJ
ejpam-4767	146	7	space	space	NOUN
ejpam-4767	146	8	.	.	PUNCT
ejpam-4767	147	1	however	however	ADV
ejpam-4767	147	2	,	,	PUNCT
ejpam-4767	147	3	we	we	PRON
ejpam-4767	147	4	know	know	VERB
ejpam-4767	147	5	that	that	SCONJ
ejpam-4767	147	6	every	every	DET
ejpam-4767	147	7	closed	close	VERB
ejpam-4767	147	8	discrete	discrete	ADJ
ejpam-4767	147	9	subspace	subspace	NOUN
ejpam-4767	147	10	of	of	ADP
ejpam-4767	147	11	a	a	DET
ejpam-4767	147	12	separable	separable	ADJ
ejpam-4767	147	13	normal	normal	ADJ
ejpam-4767	147	14	space	space	NOUN
ejpam-4767	147	15	has	have	VERB
ejpam-4767	147	16	cardinality	cardinality	NOUN
ejpam-4767	147	17	≤	≤	PROPN
ejpam-4767	147	18	ℵ0	ℵ0	PROPN
ejpam-4767	147	19	.	.	PUNCT
ejpam-4767	148	1	theorem	theorem	VERB
ejpam-4767	148	2	2.15	2.15	NUM
ejpam-4767	148	3	.	.	PUNCT
ejpam-4767	149	1	let	let	VERB
ejpam-4767	149	2	x	x	PRON
ejpam-4767	149	3	be	be	AUX
ejpam-4767	149	4	a	a	DET
ejpam-4767	149	5	t1c	t1c	PRON
ejpam-4767	149	6	l.c.w.d	l.c.w.d	PROPN
ejpam-4767	149	7	space	space	NOUN
ejpam-4767	149	8	such	such	ADJ
ejpam-4767	149	9	that	that	PRON
ejpam-4767	149	10	for	for	ADP
ejpam-4767	149	11	each	each	DET
ejpam-4767	149	12	point	point	NOUN
ejpam-4767	149	13	x	x	X
ejpam-4767	149	14	∈	∈	PRON
ejpam-4767	149	15	x•	x•	NOUN
ejpam-4767	149	16	there	there	PRON
ejpam-4767	149	17	exists	exist	VERB
ejpam-4767	149	18	an	an	DET
ejpam-4767	149	19	open	open	ADJ
ejpam-4767	149	20	neighborhood	neighborhood	NOUN
ejpam-4767	149	21	u	u	NOUN
ejpam-4767	149	22	of	of	ADP
ejpam-4767	149	23	x	x	SYM
ejpam-4767	149	24	such	such	ADJ
ejpam-4767	149	25	that	that	SCONJ
ejpam-4767	149	26	the	the	DET
ejpam-4767	149	27	closure	closure	NOUN
ejpam-4767	149	28	u	u	NOUN
ejpam-4767	149	29	=	=	PUNCT
ejpam-4767	149	30	⋃	⋃	ADP
ejpam-4767	149	31	s∈s	s∈s	NOUN
ejpam-4767	149	32	fs	fs	NOUN
ejpam-4767	149	33	is	be	AUX
ejpam-4767	149	34	a	a	DET
ejpam-4767	149	35	union	union	NOUN
ejpam-4767	149	36	of	of	ADP
ejpam-4767	149	37	compact	compact	ADJ
ejpam-4767	149	38	subsets	subset	NOUN
ejpam-4767	149	39	.	.	PUNCT
ejpam-4767	150	1	if	if	SCONJ
ejpam-4767	150	2	the	the	DET
ejpam-4767	150	3	family	family	NOUN
ejpam-4767	150	4	{	{	PUNCT
ejpam-4767	150	5	fs}s∈s	fs}s∈s	X
ejpam-4767	150	6	is	be	AUX
ejpam-4767	150	7	pairwise	pairwise	NOUN
ejpam-4767	150	8	disjoint	disjoint	NOUN
ejpam-4767	150	9	and	and	CCONJ
ejpam-4767	150	10	locally	locally	ADV
ejpam-4767	150	11	finite	finite	NOUN
ejpam-4767	150	12	except	except	SCONJ
ejpam-4767	150	13	for	for	ADP
ejpam-4767	150	14	a	a	DET
ejpam-4767	150	15	finite	finite	ADJ
ejpam-4767	150	16	number	number	NOUN
ejpam-4767	150	17	of	of	ADP
ejpam-4767	150	18	points	point	NOUN
ejpam-4767	150	19	,	,	PUNCT
ejpam-4767	150	20	then	then	ADV
ejpam-4767	150	21	x	x	PUNCT
ejpam-4767	150	22	is	be	AUX
ejpam-4767	150	23	t3	t3	PROPN
ejpam-4767	150	24	1	1	NUM
ejpam-4767	150	25	2	2	NUM
ejpam-4767	150	26	.	.	PUNCT
ejpam-4767	151	1	proof	proof	NOUN
ejpam-4767	151	2	.	.	PUNCT
ejpam-4767	152	1	let	let	VERB
ejpam-4767	152	2	x	x	PRON
ejpam-4767	152	3	be	be	AUX
ejpam-4767	152	4	a	a	DET
ejpam-4767	152	5	defect	defect	NOUN
ejpam-4767	152	6	,	,	PUNCT
ejpam-4767	152	7	i.e.	i.e.	X
ejpam-4767	152	8	,	,	PUNCT
ejpam-4767	152	9	x	x	SYM
ejpam-4767	152	10	∈	∈	PROPN
ejpam-4767	152	11	x•.	x•.	NOUN
ejpam-4767	152	12	let	let	VERB
ejpam-4767	152	13	f	f	PRON
ejpam-4767	152	14	be	be	AUX
ejpam-4767	152	15	closed	close	VERB
ejpam-4767	152	16	such	such	ADJ
ejpam-4767	152	17	that	that	SCONJ
ejpam-4767	152	18	x	x	PRON
ejpam-4767	152	19	̸∈	̸∈	PROPN
ejpam-4767	152	20	f.	f.	PROPN
ejpam-4767	152	21	take	take	VERB
ejpam-4767	152	22	an	an	DET
ejpam-4767	152	23	open	open	ADJ
ejpam-4767	152	24	neighborhood	neighborhood	NOUN
ejpam-4767	152	25	u	u	NOUN
ejpam-4767	152	26	of	of	ADP
ejpam-4767	152	27	x	x	SYM
ejpam-4767	152	28	such	such	ADJ
ejpam-4767	152	29	that	that	SCONJ
ejpam-4767	152	30	the	the	DET
ejpam-4767	152	31	closure	closure	NOUN
ejpam-4767	152	32	u	u	NOUN
ejpam-4767	152	33	=	=	PUNCT
ejpam-4767	152	34	⋃	⋃	ADP
ejpam-4767	152	35	s∈s	s∈s	NOUN
ejpam-4767	152	36	fs	fs	NOUN
ejpam-4767	152	37	is	be	AUX
ejpam-4767	152	38	a	a	DET
ejpam-4767	152	39	union	union	NOUN
ejpam-4767	152	40	pairwise	pairwise	NOUN
ejpam-4767	152	41	disjoint	disjoint	NOUN
ejpam-4767	152	42	compact	compact	ADJ
ejpam-4767	152	43	subsets	subset	NOUN
ejpam-4767	152	44	,	,	PUNCT
ejpam-4767	152	45	where	where	SCONJ
ejpam-4767	152	46	w	w	NOUN
ejpam-4767	152	47	=	=	PRON
ejpam-4767	152	48	{	{	PUNCT
ejpam-4767	152	49	fs}s∈s	fs}s∈s	X
ejpam-4767	152	50	is	be	AUX
ejpam-4767	152	51	locally	locally	ADV
ejpam-4767	152	52	finite	finite	ADJ
ejpam-4767	152	53	except	except	SCONJ
ejpam-4767	152	54	at	at	ADP
ejpam-4767	152	55	a1	a1	NOUN
ejpam-4767	152	56	,	,	PUNCT
ejpam-4767	152	57	a2	a2	PROPN
ejpam-4767	152	58	,	,	PUNCT
ejpam-4767	152	59	...	...	PUNCT
ejpam-4767	152	60	,	,	PUNCT
ejpam-4767	152	61	an	an	X
ejpam-4767	152	62	.	.	PUNCT
ejpam-4767	152	63	note	note	VERB
ejpam-4767	152	64	that	that	SCONJ
ejpam-4767	152	65	x	x	PUNCT
ejpam-4767	152	66	belongs	belong	VERB
ejpam-4767	152	67	to	to	ADP
ejpam-4767	152	68	only	only	ADV
ejpam-4767	152	69	one	one	NUM
ejpam-4767	152	70	member	member	NOUN
ejpam-4767	152	71	of	of	ADP
ejpam-4767	152	72	the	the	DET
ejpam-4767	152	73	family	family	NOUN
ejpam-4767	152	74	w	w	PROPN
ejpam-4767	152	75	,	,	PUNCT
ejpam-4767	152	76	say	say	VERB
ejpam-4767	152	77	fsk	fsk	PROPN
ejpam-4767	152	78	for	for	ADP
ejpam-4767	152	79	some	some	DET
ejpam-4767	152	80	sk	sk	PROPN
ejpam-4767	152	81	∈	∈	PROPN
ejpam-4767	152	82	s.	s.	PROPN
ejpam-4767	152	83	define	define	VERB
ejpam-4767	152	84	f0	f0	PROPN
ejpam-4767	152	85	=	=	SYM
ejpam-4767	152	86	(	(	PUNCT
ejpam-4767	152	87	(	(	PUNCT
ejpam-4767	152	88	u\u	u\u	NOUN
ejpam-4767	152	89	)	)	PUNCT
ejpam-4767	152	90	∪	∪	NOUN
ejpam-4767	152	91	(	(	PUNCT
ejpam-4767	152	92	u	u	NOUN
ejpam-4767	152	93	∩	∩	NOUN
ejpam-4767	152	94	f	f	PROPN
ejpam-4767	152	95	)	)	PUNCT
ejpam-4767	152	96	)	)	PUNCT
ejpam-4767	153	1	∩	∩	PROPN
ejpam-4767	153	2	fsk	fsk	PROPN
ejpam-4767	153	3	which	which	PRON
ejpam-4767	153	4	is	be	AUX
ejpam-4767	153	5	a	a	DET
ejpam-4767	153	6	closed	closed	ADJ
ejpam-4767	153	7	subset	subset	NOUN
ejpam-4767	153	8	of	of	ADP
ejpam-4767	153	9	the	the	DET
ejpam-4767	153	10	subspace	subspace	NOUN
ejpam-4767	153	11	fsk	fsk	PROPN
ejpam-4767	154	1	and	and	CCONJ
ejpam-4767	154	2	we	we	PRON
ejpam-4767	154	3	have	have	VERB
ejpam-4767	154	4	that	that	PRON
ejpam-4767	154	5	x	x	PROPN
ejpam-4767	154	6	̸∈	̸∈	PROPN
ejpam-4767	154	7	f0	f0	PROPN
ejpam-4767	154	8	.	.	PUNCT
ejpam-4767	155	1	therefore	therefore	ADV
ejpam-4767	155	2	,	,	PUNCT
ejpam-4767	155	3	there	there	PRON
ejpam-4767	155	4	is	be	VERB
ejpam-4767	155	5	a	a	DET
ejpam-4767	155	6	map	map	NOUN
ejpam-4767	155	7	fsk	fsk	PROPN
ejpam-4767	155	8	:	:	PUNCT
ejpam-4767	155	9	fsk	fsk	PROPN
ejpam-4767	155	10	→	→	PUNCT
ejpam-4767	155	11	i	i	PRON
ejpam-4767	155	12	such	such	ADJ
ejpam-4767	155	13	that	that	PRON
ejpam-4767	155	14	fsk(x	fsk(x	PROPN
ejpam-4767	155	15	)	)	PUNCT
ejpam-4767	155	16	=	=	SYM
ejpam-4767	155	17	0	0	NUM
ejpam-4767	155	18	and	and	CCONJ
ejpam-4767	155	19	fsk(f0	fsk(f0	ADJ
ejpam-4767	155	20	)	)	PUNCT
ejpam-4767	155	21	⊂	⊂	PROPN
ejpam-4767	155	22	{	{	PUNCT
ejpam-4767	155	23	1	1	NUM
ejpam-4767	155	24	}	}	PUNCT
ejpam-4767	155	25	.	.	PUNCT
ejpam-4767	156	1	define	define	VERB
ejpam-4767	156	2	also	also	ADV
ejpam-4767	156	3	following	follow	VERB
ejpam-4767	156	4	constant	constant	ADJ
ejpam-4767	156	5	maps	map	NOUN
ejpam-4767	156	6	fs	f	NOUN
ejpam-4767	156	7	:	:	PUNCT
ejpam-4767	156	8	fs	fs	PROPN
ejpam-4767	156	9	→	→	SYM
ejpam-4767	156	10	i	i	PROPN
ejpam-4767	156	11	,	,	PUNCT
ejpam-4767	156	12	y	y	PROPN
ejpam-4767	156	13	7→	7→	PROPN
ejpam-4767	156	14	1	1	NUM
ejpam-4767	156	15	;	;	PUNCT
ejpam-4767	156	16	for	for	ADP
ejpam-4767	156	17	s	s	NOUN
ejpam-4767	156	18	̸=	̸=	PROPN
ejpam-4767	156	19	sk	sk	VERB
ejpam-4767	156	20	g	g	NOUN
ejpam-4767	156	21	:	:	PUNCT
ejpam-4767	156	22	x\u	x\u	X
ejpam-4767	156	23	→	→	SYM
ejpam-4767	156	24	i	i	PROPN
ejpam-4767	156	25	,	,	PUNCT
ejpam-4767	156	26	y	y	PROPN
ejpam-4767	156	27	7→	7→	PROPN
ejpam-4767	156	28	1	1	NUM
ejpam-4767	156	29	.	.	PUNCT
ejpam-4767	156	30	m.	m.	PROPN
ejpam-4767	156	31	zailai	zailai	PROPN
ejpam-4767	156	32	/	/	SYM
ejpam-4767	156	33	eur	eur	PROPN
ejpam-4767	156	34	.	.	PUNCT
ejpam-4767	157	1	j.	j.	PROPN
ejpam-4767	157	2	pure	pure	PROPN
ejpam-4767	157	3	appl	appl	PROPN
ejpam-4767	157	4	.	.	PROPN
ejpam-4767	157	5	math	math	PROPN
ejpam-4767	157	6	,	,	PUNCT
ejpam-4767	157	7	16	16	NUM
ejpam-4767	157	8	(	(	PUNCT
ejpam-4767	157	9	2	2	NUM
ejpam-4767	157	10	)	)	PUNCT
ejpam-4767	157	11	(	(	PUNCT
ejpam-4767	157	12	2023	2023	NUM
ejpam-4767	157	13	)	)	PUNCT
ejpam-4767	157	14	,	,	PUNCT
ejpam-4767	157	15	1228	1228	NUM
ejpam-4767	157	16	-	-	SYM
ejpam-4767	157	17	1235	1235	NUM
ejpam-4767	157	18	1233	1233	NUM
ejpam-4767	157	19	now	now	ADV
ejpam-4767	157	20	,	,	PUNCT
ejpam-4767	157	21	suppose	suppose	VERB
ejpam-4767	157	22	that	that	SCONJ
ejpam-4767	157	23	one	one	NUM
ejpam-4767	157	24	of	of	ADP
ejpam-4767	157	25	the	the	DET
ejpam-4767	157	26	a′is	a′is	NOUN
ejpam-4767	157	27	is	be	AUX
ejpam-4767	157	28	x	x	X
ejpam-4767	157	29	,	,	PUNCT
ejpam-4767	157	30	say	say	VERB
ejpam-4767	157	31	am	be	AUX
ejpam-4767	157	32	=	=	ADJ
ejpam-4767	157	33	x.	x.	NOUN
ejpam-4767	157	34	then	then	ADV
ejpam-4767	157	35	,	,	PUNCT
ejpam-4767	157	36	we	we	PRON
ejpam-4767	157	37	define	define	VERB
ejpam-4767	157	38	the	the	DET
ejpam-4767	157	39	following	follow	VERB
ejpam-4767	157	40	maps	map	NOUN
ejpam-4767	157	41	gi	gi	INTJ
ejpam-4767	157	42	:	:	PUNCT
ejpam-4767	157	43	{	{	PUNCT
ejpam-4767	157	44	ai	ai	VERB
ejpam-4767	157	45	}	}	PUNCT
ejpam-4767	157	46	→	→	SYM
ejpam-4767	157	47	i	i	PROPN
ejpam-4767	157	48	,	,	PUNCT
ejpam-4767	157	49	ai	ai	VERB
ejpam-4767	157	50	7→	7→	ADV
ejpam-4767	157	51	1	1	NUM
ejpam-4767	157	52	for	for	ADP
ejpam-4767	157	53	i	i	PRON
ejpam-4767	157	54	=	=	NOUN
ejpam-4767	157	55	1	1	NUM
ejpam-4767	157	56	,	,	PUNCT
ejpam-4767	157	57	...	...	PUNCT
ejpam-4767	157	58	,	,	PUNCT
ejpam-4767	157	59	n	n	CCONJ
ejpam-4767	157	60	and	and	CCONJ
ejpam-4767	157	61	i	i	PRON
ejpam-4767	157	62	̸=	̸=	PROPN
ejpam-4767	157	63	m	m	VERB
ejpam-4767	157	64	gm	gm	PROPN
ejpam-4767	157	65	:	:	PUNCT
ejpam-4767	157	66	{	{	PUNCT
ejpam-4767	157	67	am	be	AUX
ejpam-4767	157	68	}	}	PUNCT
ejpam-4767	157	69	→	→	SYM
ejpam-4767	157	70	i	i	PROPN
ejpam-4767	157	71	,	,	PUNCT
ejpam-4767	157	72	am	be	AUX
ejpam-4767	157	73	7→	7→	NUM
ejpam-4767	157	74	0	0	NUM
ejpam-4767	157	75	.	.	PUNCT
ejpam-4767	158	1	if	if	SCONJ
ejpam-4767	158	2	all	all	DET
ejpam-4767	158	3	a′is	a′is	NOUN
ejpam-4767	158	4	are	be	AUX
ejpam-4767	158	5	distinct	distinct	ADJ
ejpam-4767	158	6	from	from	ADP
ejpam-4767	158	7	x	x	PRON
ejpam-4767	158	8	,	,	PUNCT
ejpam-4767	158	9	then	then	ADV
ejpam-4767	158	10	we	we	PRON
ejpam-4767	158	11	define	define	VERB
ejpam-4767	158	12	:	:	PUNCT
ejpam-4767	158	13	gi	gi	INTJ
ejpam-4767	158	14	:	:	PUNCT
ejpam-4767	158	15	{	{	PUNCT
ejpam-4767	158	16	ai	ai	VERB
ejpam-4767	158	17	}	}	PUNCT
ejpam-4767	158	18	→	→	SYM
ejpam-4767	158	19	i	i	PROPN
ejpam-4767	158	20	,	,	PUNCT
ejpam-4767	158	21	ai	ai	VERB
ejpam-4767	158	22	7→	7→	ADV
ejpam-4767	158	23	1	1	NUM
ejpam-4767	158	24	for	for	ADP
ejpam-4767	158	25	i	i	PRON
ejpam-4767	158	26	=	=	NOUN
ejpam-4767	158	27	1	1	NUM
ejpam-4767	158	28	,	,	PUNCT
ejpam-4767	158	29	...	...	PUNCT
ejpam-4767	158	30	,	,	PUNCT
ejpam-4767	158	31	n	n	CCONJ
ejpam-4767	158	32	first	first	ADV
ejpam-4767	158	33	,	,	PUNCT
ejpam-4767	158	34	we	we	PRON
ejpam-4767	158	35	need	need	VERB
ejpam-4767	158	36	to	to	PART
ejpam-4767	158	37	check	check	VERB
ejpam-4767	158	38	that	that	SCONJ
ejpam-4767	158	39	the	the	DET
ejpam-4767	158	40	map	map	NOUN
ejpam-4767	158	41	h	h	NOUN
ejpam-4767	158	42	=	=	PUNCT
ejpam-4767	158	43	⋃	⋃	ADP
ejpam-4767	158	44	s∈s	s∈s	NOUN
ejpam-4767	158	45	fs	fs	ADP
ejpam-4767	158	46	n⋃	n⋃	PROPN
ejpam-4767	158	47	i=1	i=1	PROPN
ejpam-4767	158	48	gi	gi	PROPN
ejpam-4767	158	49	∪	∪	ADP
ejpam-4767	158	50	g	g	PROPN
ejpam-4767	158	51	:	:	PUNCT
ejpam-4767	158	52	x	x	X
ejpam-4767	158	53	→	→	SYM
ejpam-4767	158	54	i	i	PRON
ejpam-4767	158	55	is	be	AUX
ejpam-4767	158	56	continuous	continuous	ADJ
ejpam-4767	158	57	.	.	PUNCT
ejpam-4767	159	1	let	let	VERB
ejpam-4767	159	2	c	c	NOUN
ejpam-4767	159	3	⊂	⊂	PRON
ejpam-4767	159	4	i	i	PRON
ejpam-4767	159	5	be	be	AUX
ejpam-4767	159	6	closed	close	VERB
ejpam-4767	159	7	.	.	PUNCT
ejpam-4767	160	1	if	if	SCONJ
ejpam-4767	160	2	1	1	NUM
ejpam-4767	160	3	̸∈	̸∈	PROPN
ejpam-4767	160	4	c	c	PROPN
ejpam-4767	160	5	,	,	PUNCT
ejpam-4767	160	6	then	then	ADV
ejpam-4767	160	7	h−1(c	h−1(c	NOUN
ejpam-4767	160	8	)	)	PUNCT
ejpam-4767	161	1	=	=	SYM
ejpam-4767	161	2	f−1	f−1	PROPN
ejpam-4767	161	3	sk	sk	INTJ
ejpam-4767	161	4	(	(	PUNCT
ejpam-4767	161	5	c	c	NOUN
ejpam-4767	161	6	)	)	PUNCT
ejpam-4767	161	7	is	be	AUX
ejpam-4767	161	8	closed	close	VERB
ejpam-4767	161	9	or	or	CCONJ
ejpam-4767	161	10	h−1(c	h−1(c	NOUN
ejpam-4767	161	11	)	)	PUNCT
ejpam-4767	162	1	=	=	SYM
ejpam-4767	162	2	f−1	f−1	PROPN
ejpam-4767	162	3	sk	sk	INTJ
ejpam-4767	162	4	(	(	PUNCT
ejpam-4767	162	5	c	c	NOUN
ejpam-4767	162	6	)	)	PUNCT
ejpam-4767	162	7	∪	∪	NOUN
ejpam-4767	162	8	g−1	g−1	PROPN
ejpam-4767	162	9	m	m	PROPN
ejpam-4767	162	10	(	(	PUNCT
ejpam-4767	162	11	c	c	NOUN
ejpam-4767	162	12	)	)	PUNCT
ejpam-4767	162	13	which	which	PRON
ejpam-4767	162	14	is	be	AUX
ejpam-4767	162	15	also	also	ADV
ejpam-4767	162	16	closed	closed	ADJ
ejpam-4767	162	17	.	.	PUNCT
ejpam-4767	163	1	assume	assume	VERB
ejpam-4767	163	2	that	that	SCONJ
ejpam-4767	163	3	1	1	NUM
ejpam-4767	163	4	∈	∈	NOUN
ejpam-4767	163	5	c	c	NOUN
ejpam-4767	163	6	,	,	PUNCT
ejpam-4767	163	7	then	then	ADV
ejpam-4767	163	8	h−1(c	h−1(c	NOUN
ejpam-4767	163	9	)	)	PUNCT
ejpam-4767	164	1	=	=	NOUN
ejpam-4767	164	2	⋃	⋃	NOUN
ejpam-4767	164	3	s∈s	s∈s	NOUN
ejpam-4767	164	4	f−1	f−1	PROPN
ejpam-4767	164	5	s	s	PART
ejpam-4767	164	6	(	(	PUNCT
ejpam-4767	164	7	c	c	NOUN
ejpam-4767	164	8	)	)	PUNCT
ejpam-4767	164	9	⋃n	⋃n	PROPN
ejpam-4767	164	10	i=1	i=1	ADP
ejpam-4767	164	11	g	g	PROPN
ejpam-4767	164	12	−1	−1	NOUN
ejpam-4767	164	13	i	i	PRON
ejpam-4767	164	14	(	(	PUNCT
ejpam-4767	164	15	c	c	NOUN
ejpam-4767	164	16	)	)	PUNCT
ejpam-4767	164	17	⋃	⋃	NOUN
ejpam-4767	164	18	g−1(c	g−1(c	PROPN
ejpam-4767	164	19	)	)	PUNCT
ejpam-4767	164	20	which	which	PRON
ejpam-4767	164	21	is	be	AUX
ejpam-4767	164	22	clearly	clearly	ADV
ejpam-4767	164	23	closed	close	VERB
ejpam-4767	164	24	by	by	ADP
ejpam-4767	164	25	using	use	VERB
ejpam-4767	164	26	lemma	lemma	PROPN
ejpam-4767	164	27	2.8	2.8	NUM
ejpam-4767	164	28	.	.	PUNCT
ejpam-4767	165	1	hence	hence	ADV
ejpam-4767	165	2	,	,	PUNCT
ejpam-4767	165	3	h	h	NOUN
ejpam-4767	165	4	is	be	AUX
ejpam-4767	165	5	continuous	continuous	ADJ
ejpam-4767	165	6	.	.	PUNCT
ejpam-4767	166	1	it	it	PRON
ejpam-4767	166	2	clear	clear	ADJ
ejpam-4767	166	3	that	that	SCONJ
ejpam-4767	166	4	h(x	h(x	PROPN
ejpam-4767	166	5	)	)	PUNCT
ejpam-4767	166	6	=	=	PUNCT
ejpam-4767	167	1	0	0	X
ejpam-4767	167	2	.	.	PUNCT
ejpam-4767	168	1	now	now	ADV
ejpam-4767	168	2	,	,	PUNCT
ejpam-4767	168	3	take	take	VERB
ejpam-4767	168	4	y	y	PROPN
ejpam-4767	168	5	∈	∈	PROPN
ejpam-4767	168	6	f.	f.	PROPN
ejpam-4767	169	1	if	if	SCONJ
ejpam-4767	169	2	y	y	PROPN
ejpam-4767	169	3	∈	∈	PROPN
ejpam-4767	169	4	f0	f0	PROPN
ejpam-4767	169	5	,	,	PUNCT
ejpam-4767	169	6	then	then	ADV
ejpam-4767	169	7	we	we	PRON
ejpam-4767	169	8	have	have	VERB
ejpam-4767	169	9	h(y	h(y	NOUN
ejpam-4767	169	10	)	)	PUNCT
ejpam-4767	170	1	=	=	SYM
ejpam-4767	170	2	1	1	X
ejpam-4767	170	3	.	.	X
ejpam-4767	170	4	assume	assume	VERB
ejpam-4767	170	5	that	that	SCONJ
ejpam-4767	170	6	y	y	PROPN
ejpam-4767	170	7	̸∈	̸∈	PROPN
ejpam-4767	170	8	f0	f0	PROPN
ejpam-4767	170	9	,	,	PUNCT
ejpam-4767	170	10	then	then	ADV
ejpam-4767	170	11	we	we	PRON
ejpam-4767	170	12	have	have	VERB
ejpam-4767	170	13	two	two	NUM
ejpam-4767	170	14	cases	case	NOUN
ejpam-4767	170	15	:	:	PUNCT
ejpam-4767	170	16	•	•	NUM
ejpam-4767	170	17	case	case	NOUN
ejpam-4767	170	18	1	1	NUM
ejpam-4767	170	19	:	:	PUNCT
ejpam-4767	170	20	y	y	PROPN
ejpam-4767	170	21	̸∈	̸∈	PROPN
ejpam-4767	170	22	fsk	fsk	PROPN
ejpam-4767	170	23	,	,	PUNCT
ejpam-4767	170	24	then	then	ADV
ejpam-4767	170	25	it	it	PRON
ejpam-4767	170	26	is	be	AUX
ejpam-4767	170	27	easy	easy	ADJ
ejpam-4767	170	28	to	to	PART
ejpam-4767	170	29	see	see	VERB
ejpam-4767	170	30	that	that	SCONJ
ejpam-4767	170	31	h(y	h(y	ADV
ejpam-4767	170	32	)	)	PUNCT
ejpam-4767	171	1	=	=	SYM
ejpam-4767	171	2	1	1	NUM
ejpam-4767	171	3	,	,	PUNCT
ejpam-4767	171	4	•	•	NUM
ejpam-4767	171	5	case	case	NOUN
ejpam-4767	171	6	2	2	NUM
ejpam-4767	171	7	:	:	PUNCT
ejpam-4767	171	8	y	y	PROPN
ejpam-4767	171	9	∈	∈	PROPN
ejpam-4767	171	10	fsk	fsk	PROPN
ejpam-4767	171	11	,	,	PUNCT
ejpam-4767	171	12	and	and	CCONJ
ejpam-4767	171	13	y	y	PROPN
ejpam-4767	171	14	̸∈	̸∈	PROPN
ejpam-4767	171	15	u	u	PROPN
ejpam-4767	171	16	which	which	PRON
ejpam-4767	171	17	can	can	AUX
ejpam-4767	171	18	not	not	PART
ejpam-4767	171	19	happen	happen	VERB
ejpam-4767	171	20	as	as	SCONJ
ejpam-4767	171	21	fsk	fsk	PROPN
ejpam-4767	171	22	⊂	⊂	PROPN
ejpam-4767	171	23	u.	u.	PROPN
ejpam-4767	172	1	then	then	ADV
ejpam-4767	172	2	,	,	PUNCT
ejpam-4767	172	3	we	we	PRON
ejpam-4767	172	4	conclude	conclude	VERB
ejpam-4767	172	5	that	that	SCONJ
ejpam-4767	172	6	if	if	SCONJ
ejpam-4767	172	7	y	y	PROPN
ejpam-4767	172	8	∈	∈	PROPN
ejpam-4767	172	9	f	f	PROPN
ejpam-4767	172	10	and	and	CCONJ
ejpam-4767	172	11	y	y	PROPN
ejpam-4767	172	12	̸∈	̸∈	PROPN
ejpam-4767	172	13	fsk	fsk	PROPN
ejpam-4767	172	14	.	.	PUNCT
ejpam-4767	173	1	therefore	therefore	ADV
ejpam-4767	173	2	,	,	PUNCT
ejpam-4767	173	3	h(y	h(y	ADV
ejpam-4767	173	4	)	)	PUNCT
ejpam-4767	173	5	=	=	SYM
ejpam-4767	173	6	1	1	X
ejpam-4767	173	7	.	.	PUNCT
ejpam-4767	174	1	hence	hence	ADV
ejpam-4767	174	2	,	,	PUNCT
ejpam-4767	174	3	x	x	X
ejpam-4767	174	4	is	be	AUX
ejpam-4767	174	5	a	a	DET
ejpam-4767	174	6	t3	t3	PROPN
ejpam-4767	174	7	1	1	NUM
ejpam-4767	174	8	2	2	NUM
ejpam-4767	174	9	space	space	NOUN
ejpam-4767	174	10	.	.	PUNCT
ejpam-4767	175	1	proposition	proposition	NOUN
ejpam-4767	175	2	2.16	2.16	NUM
ejpam-4767	175	3	.	.	PUNCT
ejpam-4767	176	1	let	let	VERB
ejpam-4767	176	2	{	{	PUNCT
ejpam-4767	176	3	xs}s∈s	xs}s∈s	PRON
ejpam-4767	176	4	be	be	AUX
ejpam-4767	176	5	a	a	DET
ejpam-4767	176	6	collection	collection	NOUN
ejpam-4767	176	7	of	of	ADP
ejpam-4767	176	8	pairwise	pairwise	NOUN
ejpam-4767	176	9	disjoint	disjoint	NOUN
ejpam-4767	176	10	l.c.w.d	l.c.w.d	PROPN
ejpam-4767	176	11	topological	topological	ADJ
ejpam-4767	176	12	spaces	space	NOUN
ejpam-4767	176	13	.	.	PUNCT
ejpam-4767	177	1	if	if	SCONJ
ejpam-4767	177	2	each	each	DET
ejpam-4767	177	3	point	point	NOUN
ejpam-4767	177	4	xs	xs	PROPN
ejpam-4767	177	5	∈	∈	PROPN
ejpam-4767	177	6	xs	xs	PROPN
ejpam-4767	177	7	has	have	VERB
ejpam-4767	177	8	an	an	DET
ejpam-4767	177	9	open	open	ADJ
ejpam-4767	177	10	neighborhood	neighborhood	NOUN
ejpam-4767	177	11	u	u	NOUN
ejpam-4767	177	12	such	such	ADJ
ejpam-4767	177	13	that	that	SCONJ
ejpam-4767	177	14	its	its	PRON
ejpam-4767	177	15	closure	closure	NOUN
ejpam-4767	177	16	is	be	AUX
ejpam-4767	177	17	a	a	DET
ejpam-4767	177	18	union	union	NOUN
ejpam-4767	177	19	of	of	ADP
ejpam-4767	177	20	pairwise	pairwise	PROPN
ejpam-4767	177	21	disjoint	disjoint	NOUN
ejpam-4767	177	22	compact	compact	ADJ
ejpam-4767	177	23	subsets	subset	NOUN
ejpam-4767	177	24	,	,	PUNCT
ejpam-4767	177	25	then	then	ADV
ejpam-4767	177	26	so	so	ADV
ejpam-4767	177	27	does	do	AUX
ejpam-4767	177	28	each	each	DET
ejpam-4767	177	29	point	point	VERB
ejpam-4767	177	30	x	x	PUNCT
ejpam-4767	177	31	∈	∈	NOUN
ejpam-4767	177	32	x	x	X
ejpam-4767	177	33	=	=	SYM
ejpam-4767	177	34	⊕s∈sxs	⊕s∈sxs	PROPN
ejpam-4767	177	35	.	.	PUNCT
ejpam-4767	177	36	example	example	NOUN
ejpam-4767	178	1	2	2	NUM
ejpam-4767	178	2	.	.	PUNCT
ejpam-4767	178	3	(	(	PUNCT
ejpam-4767	178	4	modified	modify	VERB
ejpam-4767	178	5	arens	arens	PROPN
ejpam-4767	178	6	-	-	PUNCT
ejpam-4767	178	7	fort	fort	NOUN
ejpam-4767	178	8	space	space	NOUN
ejpam-4767	178	9	):	):	PUNCT
ejpam-4767	178	10	here	here	ADV
ejpam-4767	178	11	we	we	PRON
ejpam-4767	178	12	modify	modify	VERB
ejpam-4767	178	13	the	the	DET
ejpam-4767	178	14	arens	arens	PROPN
ejpam-4767	178	15	-	-	PUNCT
ejpam-4767	178	16	fort	fort	NOUN
ejpam-4767	178	17	space	space	NOUN
ejpam-4767	178	18	.	.	PUNCT
ejpam-4767	179	1	let	let	VERB
ejpam-4767	179	2	(	(	PUNCT
ejpam-4767	179	3	a	a	PRON
ejpam-4767	179	4	,	,	PUNCT
ejpam-4767	179	5	τ	τ	X
ejpam-4767	179	6	)	)	PUNCT
ejpam-4767	179	7	be	be	VERB
ejpam-4767	179	8	the	the	DET
ejpam-4767	179	9	set	set	NOUN
ejpam-4767	179	10	of	of	ADP
ejpam-4767	179	11	all	all	DET
ejpam-4767	179	12	ordered	order	VERB
ejpam-4767	179	13	pairs	pair	NOUN
ejpam-4767	179	14	of	of	ADP
ejpam-4767	179	15	n×n	n×n	PROPN
ejpam-4767	179	16	.	.	PUNCT
ejpam-4767	180	1	we	we	PRON
ejpam-4767	180	2	declare	declare	VERB
ejpam-4767	180	3	that	that	SCONJ
ejpam-4767	180	4	all	all	DET
ejpam-4767	180	5	the	the	DET
ejpam-4767	180	6	singletons	singleton	NOUN
ejpam-4767	180	7	of	of	ADP
ejpam-4767	180	8	this	this	DET
ejpam-4767	180	9	set	set	NOUN
ejpam-4767	180	10	are	be	AUX
ejpam-4767	180	11	open	open	ADJ
ejpam-4767	180	12	sets	set	NOUN
ejpam-4767	180	13	except	except	SCONJ
ejpam-4767	180	14	the	the	DET
ejpam-4767	180	15	points	point	NOUN
ejpam-4767	180	16	(	(	PUNCT
ejpam-4767	180	17	0	0	NUM
ejpam-4767	180	18	,	,	PUNCT
ejpam-4767	180	19	0	0	NUM
ejpam-4767	180	20	)	)	PUNCT
ejpam-4767	180	21	,	,	PUNCT
ejpam-4767	180	22	(	(	PUNCT
ejpam-4767	180	23	1	1	NUM
ejpam-4767	180	24	,	,	PUNCT
ejpam-4767	180	25	0	0	NUM
ejpam-4767	180	26	)	)	PUNCT
ejpam-4767	180	27	,	,	PUNCT
ejpam-4767	180	28	...	...	PUNCT
ejpam-4767	180	29	,	,	PUNCT
ejpam-4767	180	30	(	(	PUNCT
ejpam-4767	180	31	n	n	CCONJ
ejpam-4767	180	32	,	,	PUNCT
ejpam-4767	180	33	0	0	NUM
ejpam-4767	180	34	)	)	PUNCT
ejpam-4767	180	35	for	for	ADP
ejpam-4767	180	36	some	some	DET
ejpam-4767	180	37	positive	positive	ADJ
ejpam-4767	180	38	integer	integer	NOUN
ejpam-4767	180	39	n.	n.	NOUN
ejpam-4767	180	40	let	let	VERB
ejpam-4767	180	41	us	we	PRON
ejpam-4767	180	42	define	define	VERB
ejpam-4767	180	43	open	open	ADJ
ejpam-4767	180	44	neighborhoods	neighborhood	NOUN
ejpam-4767	180	45	of	of	ADP
ejpam-4767	180	46	each	each	DET
ejpam-4767	180	47	point	point	NOUN
ejpam-4767	180	48	of	of	ADP
ejpam-4767	180	49	{	{	PUNCT
ejpam-4767	180	50	(	(	PUNCT
ejpam-4767	180	51	0	0	NUM
ejpam-4767	180	52	,	,	PUNCT
ejpam-4767	180	53	0	0	NUM
ejpam-4767	180	54	)	)	PUNCT
ejpam-4767	180	55	,	,	PUNCT
ejpam-4767	180	56	(	(	PUNCT
ejpam-4767	180	57	1	1	NUM
ejpam-4767	180	58	,	,	PUNCT
ejpam-4767	180	59	0	0	NUM
ejpam-4767	180	60	)	)	PUNCT
ejpam-4767	180	61	,	,	PUNCT
ejpam-4767	180	62	...	...	PUNCT
ejpam-4767	180	63	,	,	PUNCT
ejpam-4767	180	64	(	(	PUNCT
ejpam-4767	180	65	n	n	CCONJ
ejpam-4767	180	66	,	,	PUNCT
ejpam-4767	180	67	0	0	NUM
ejpam-4767	180	68	)	)	PUNCT
ejpam-4767	180	69	}	}	PUNCT
ejpam-4767	180	70	as	as	ADP
ejpam-4767	180	71	any	any	DET
ejpam-4767	180	72	set	set	NOUN
ejpam-4767	180	73	u	u	PRON
ejpam-4767	180	74	such	such	ADJ
ejpam-4767	180	75	that	that	SCONJ
ejpam-4767	180	76	{	{	PUNCT
ejpam-4767	180	77	(	(	PUNCT
ejpam-4767	180	78	0	0	NUM
ejpam-4767	180	79	,	,	PUNCT
ejpam-4767	180	80	0	0	NUM
ejpam-4767	180	81	)	)	PUNCT
ejpam-4767	180	82	,	,	PUNCT
ejpam-4767	180	83	(	(	PUNCT
ejpam-4767	180	84	1	1	NUM
ejpam-4767	180	85	,	,	PUNCT
ejpam-4767	180	86	0	0	NUM
ejpam-4767	180	87	)	)	PUNCT
ejpam-4767	180	88	,	,	PUNCT
ejpam-4767	180	89	...	...	PUNCT
ejpam-4767	180	90	,	,	PUNCT
ejpam-4767	180	91	(	(	PUNCT
ejpam-4767	180	92	n	n	CCONJ
ejpam-4767	180	93	,	,	PUNCT
ejpam-4767	180	94	0	0	NUM
ejpam-4767	180	95	)	)	PUNCT
ejpam-4767	180	96	}	}	PUNCT
ejpam-4767	180	97	⊂	⊂	PROPN
ejpam-4767	180	98	u	u	PROPN
ejpam-4767	180	99	,	,	PUNCT
ejpam-4767	180	100	and	and	CCONJ
ejpam-4767	180	101	all	all	PRON
ejpam-4767	180	102	but	but	SCONJ
ejpam-4767	180	103	a	a	DET
ejpam-4767	180	104	finite	finite	ADJ
ejpam-4767	180	105	number	number	NOUN
ejpam-4767	180	106	of	of	ADP
ejpam-4767	180	107	points	point	NOUN
ejpam-4767	180	108	of	of	ADP
ejpam-4767	180	109	each	each	PRON
ejpam-4767	180	110	but	but	CCONJ
ejpam-4767	180	111	a	a	DET
ejpam-4767	180	112	finite	finite	ADJ
ejpam-4767	180	113	number	number	NOUN
ejpam-4767	180	114	of	of	ADP
ejpam-4767	180	115	the	the	DET
ejpam-4767	180	116	sets	set	NOUN
ejpam-4767	180	117	td	td	NOUN
ejpam-4767	180	118	=	=	PUNCT
ejpam-4767	180	119	{	{	PUNCT
ejpam-4767	180	120	(	(	PUNCT
ejpam-4767	180	121	l	l	NOUN
ejpam-4767	180	122	,	,	PUNCT
ejpam-4767	180	123	d	d	NOUN
ejpam-4767	180	124	)	)	PUNCT
ejpam-4767	180	125	:	:	PUNCT
ejpam-4767	180	126	l	l	NOUN
ejpam-4767	180	127	is	be	AUX
ejpam-4767	180	128	fixed	fix	VERB
ejpam-4767	180	129	and	and	CCONJ
ejpam-4767	180	130	d	d	ADP
ejpam-4767	180	131	∈	∈	PROPN
ejpam-4767	180	132	n	n	CCONJ
ejpam-4767	180	133	}	}	PUNCT
ejpam-4767	180	134	.	.	PUNCT
ejpam-4767	181	1	note	note	VERB
ejpam-4767	181	2	that	that	SCONJ
ejpam-4767	181	3	this	this	DET
ejpam-4767	181	4	space	space	NOUN
ejpam-4767	181	5	is	be	AUX
ejpam-4767	181	6	not	not	PART
ejpam-4767	181	7	locally	locally	ADV
ejpam-4767	181	8	compact	compact	ADJ
ejpam-4767	181	9	as	as	ADP
ejpam-4767	181	10	the	the	DET
ejpam-4767	181	11	points	point	NOUN
ejpam-4767	181	12	(	(	PUNCT
ejpam-4767	181	13	0	0	NUM
ejpam-4767	181	14	,	,	PUNCT
ejpam-4767	181	15	0	0	NUM
ejpam-4767	181	16	)	)	PUNCT
ejpam-4767	181	17	,	,	PUNCT
ejpam-4767	181	18	(	(	PUNCT
ejpam-4767	181	19	1	1	NUM
ejpam-4767	181	20	,	,	PUNCT
ejpam-4767	181	21	0	0	NUM
ejpam-4767	181	22	)	)	PUNCT
ejpam-4767	181	23	,	,	PUNCT
ejpam-4767	181	24	...	...	PUNCT
ejpam-4767	181	25	,	,	PUNCT
ejpam-4767	181	26	(	(	PUNCT
ejpam-4767	181	27	n	n	CCONJ
ejpam-4767	181	28	,	,	PUNCT
ejpam-4767	181	29	0	0	NUM
ejpam-4767	181	30	)	)	PUNCT
ejpam-4767	181	31	do	do	AUX
ejpam-4767	181	32	not	not	PART
ejpam-4767	181	33	possess	possess	VERB
ejpam-4767	181	34	compact	compact	ADJ
ejpam-4767	181	35	neighborhoods	neighborhood	NOUN
ejpam-4767	181	36	.	.	PUNCT
ejpam-4767	182	1	let	let	VERB
ejpam-4767	182	2	us	we	PRON
ejpam-4767	182	3	check	check	VERB
ejpam-4767	182	4	that	that	SCONJ
ejpam-4767	182	5	the	the	DET
ejpam-4767	182	6	point	point	NOUN
ejpam-4767	182	7	(	(	PUNCT
ejpam-4767	182	8	0	0	NUM
ejpam-4767	182	9	,	,	PUNCT
ejpam-4767	182	10	0	0	NUM
ejpam-4767	182	11	)	)	PUNCT
ejpam-4767	182	12	does	do	AUX
ejpam-4767	182	13	not	not	PART
ejpam-4767	182	14	have	have	VERB
ejpam-4767	182	15	a	a	DET
ejpam-4767	182	16	compact	compact	ADJ
ejpam-4767	182	17	neighborhood	neighborhood	NOUN
ejpam-4767	182	18	and	and	CCONJ
ejpam-4767	182	19	all	all	DET
ejpam-4767	182	20	other	other	ADJ
ejpam-4767	182	21	points	point	NOUN
ejpam-4767	182	22	can	can	AUX
ejpam-4767	182	23	be	be	AUX
ejpam-4767	182	24	verified	verify	VERB
ejpam-4767	182	25	analogously	analogously	ADV
ejpam-4767	182	26	.	.	PUNCT
ejpam-4767	183	1	let	let	VERB
ejpam-4767	183	2	u	u	PRON
ejpam-4767	183	3	be	be	AUX
ejpam-4767	183	4	an	an	DET
ejpam-4767	183	5	open	open	ADJ
ejpam-4767	183	6	neighborhood	neighborhood	NOUN
ejpam-4767	183	7	of	of	ADP
ejpam-4767	183	8	(	(	PUNCT
ejpam-4767	183	9	0	0	NUM
ejpam-4767	183	10	,	,	PUNCT
ejpam-4767	183	11	0	0	NUM
ejpam-4767	183	12	)	)	PUNCT
ejpam-4767	183	13	.	.	PUNCT
ejpam-4767	184	1	consider	consider	VERB
ejpam-4767	184	2	the	the	DET
ejpam-4767	184	3	following	follow	VERB
ejpam-4767	184	4	u	u	NOUN
ejpam-4767	184	5	=	=	PUNCT
ejpam-4767	184	6	{	{	PUNCT
ejpam-4767	184	7	{	{	PUNCT
ejpam-4767	184	8	as}s∈s	as}s∈s	NOUN
ejpam-4767	184	9	,	,	PUNCT
ejpam-4767	184	10	v	v	NOUN
ejpam-4767	184	11	}	}	PUNCT
ejpam-4767	184	12	such	such	ADJ
ejpam-4767	184	13	that	that	SCONJ
ejpam-4767	184	14	each	each	PRON
ejpam-4767	184	15	as	as	ADP
ejpam-4767	184	16	is	be	AUX
ejpam-4767	184	17	distinct	distinct	ADJ
ejpam-4767	184	18	from	from	ADP
ejpam-4767	184	19	all	all	DET
ejpam-4767	184	20	the	the	DET
ejpam-4767	184	21	points	point	NOUN
ejpam-4767	184	22	(	(	PUNCT
ejpam-4767	184	23	0	0	NUM
ejpam-4767	184	24	,	,	PUNCT
ejpam-4767	184	25	0	0	NUM
ejpam-4767	184	26	)	)	PUNCT
ejpam-4767	184	27	,	,	PUNCT
ejpam-4767	184	28	(	(	PUNCT
ejpam-4767	184	29	1	1	NUM
ejpam-4767	184	30	,	,	PUNCT
ejpam-4767	184	31	0	0	NUM
ejpam-4767	184	32	)	)	PUNCT
ejpam-4767	184	33	,	,	PUNCT
ejpam-4767	184	34	...	...	PUNCT
ejpam-4767	184	35	,	,	PUNCT
ejpam-4767	184	36	(	(	PUNCT
ejpam-4767	184	37	n	n	CCONJ
ejpam-4767	184	38	,	,	PUNCT
ejpam-4767	184	39	0	0	NUM
ejpam-4767	184	40	)	)	PUNCT
ejpam-4767	184	41	,	,	PUNCT
ejpam-4767	184	42	and	and	CCONJ
ejpam-4767	184	43	v	v	NOUN
ejpam-4767	184	44	is	be	AUX
ejpam-4767	184	45	an	an	DET
ejpam-4767	184	46	open	open	ADJ
ejpam-4767	184	47	neighborhood	neighborhood	NOUN
ejpam-4767	184	48	of	of	ADP
ejpam-4767	184	49	(	(	PUNCT
ejpam-4767	184	50	0	0	NUM
ejpam-4767	184	51	,	,	PUNCT
ejpam-4767	184	52	0	0	NUM
ejpam-4767	184	53	)	)	PUNCT
ejpam-4767	184	54	which	which	PRON
ejpam-4767	184	55	is	be	AUX
ejpam-4767	184	56	distinct	distinct	ADJ
ejpam-4767	184	57	from	from	ADP
ejpam-4767	184	58	u	u	NOUN
ejpam-4767	184	59	in	in	ADP
ejpam-4767	184	60	the	the	DET
ejpam-4767	184	61	following	follow	VERB
ejpam-4767	184	62	sense	sense	NOUN
ejpam-4767	184	63	.	.	PUNCT
ejpam-4767	185	1	if	if	SCONJ
ejpam-4767	185	2	d	d	NOUN
ejpam-4767	185	3	=	=	PRON
ejpam-4767	185	4	{	{	PUNCT
ejpam-4767	185	5	(	(	PUNCT
ejpam-4767	185	6	l	l	NOUN
ejpam-4767	185	7	,	,	PUNCT
ejpam-4767	185	8	d	d	NOUN
ejpam-4767	185	9	)	)	PUNCT
ejpam-4767	185	10	:	:	PUNCT
ejpam-4767	185	11	l	l	NOUN
ejpam-4767	185	12	is	be	AUX
ejpam-4767	185	13	fixed	fix	VERB
ejpam-4767	185	14	and	and	CCONJ
ejpam-4767	185	15	l	l	NOUN
ejpam-4767	185	16	̸=	̸=	PROPN
ejpam-4767	185	17	0	0	NUM
ejpam-4767	185	18	}	}	PUNCT
ejpam-4767	185	19	⊂	⊂	PROPN
ejpam-4767	185	20	u	u	PROPN
ejpam-4767	185	21	,	,	PUNCT
ejpam-4767	185	22	then	then	ADV
ejpam-4767	185	23	we	we	PRON
ejpam-4767	185	24	require	require	VERB
ejpam-4767	185	25	that	that	SCONJ
ejpam-4767	186	1	d	d	X
ejpam-4767	186	2	̸⊂	̸⊂	ADV
ejpam-4767	186	3	v.	v.	CCONJ
ejpam-4767	186	4	now	now	ADV
ejpam-4767	186	5	,	,	PUNCT
ejpam-4767	186	6	u	u	NOUN
ejpam-4767	186	7	is	be	AUX
ejpam-4767	186	8	an	an	DET
ejpam-4767	186	9	open	open	ADJ
ejpam-4767	186	10	cover	cover	NOUN
ejpam-4767	186	11	of	of	ADP
ejpam-4767	186	12	the	the	DET
ejpam-4767	186	13	closure	closure	NOUN
ejpam-4767	186	14	u	u	NOUN
ejpam-4767	186	15	which	which	PRON
ejpam-4767	186	16	does	do	AUX
ejpam-4767	186	17	not	not	PART
ejpam-4767	186	18	have	have	VERB
ejpam-4767	186	19	a	a	DET
ejpam-4767	186	20	finite	finite	ADJ
ejpam-4767	186	21	open	open	PROPN
ejpam-4767	186	22	subcover	subcover	PROPN
ejpam-4767	186	23	.	.	PUNCT
ejpam-4767	187	1	for	for	ADP
ejpam-4767	187	2	any	any	DET
ejpam-4767	187	3	point	point	NOUN
ejpam-4767	187	4	x	x	X
ejpam-4767	187	5	∈	∈	NOUN
ejpam-4767	187	6	x•	x•	NOUN
ejpam-4767	187	7	one	one	NOUN
ejpam-4767	187	8	can	can	AUX
ejpam-4767	187	9	take	take	VERB
ejpam-4767	187	10	x	x	PUNCT
ejpam-4767	187	11	as	as	ADP
ejpam-4767	187	12	a	a	DET
ejpam-4767	187	13	neighborhood	neighborhood	NOUN
ejpam-4767	187	14	.	.	PUNCT
ejpam-4767	188	1	now	now	ADV
ejpam-4767	188	2	,	,	PUNCT
ejpam-4767	188	3	x	x	PRON
ejpam-4767	188	4	can	can	AUX
ejpam-4767	188	5	be	be	AUX
ejpam-4767	188	6	written	write	VERB
ejpam-4767	188	7	references	reference	NOUN
ejpam-4767	188	8	1234	1234	NUM
ejpam-4767	188	9	as	as	ADP
ejpam-4767	188	10	a	a	DET
ejpam-4767	188	11	union	union	NOUN
ejpam-4767	188	12	of	of	ADP
ejpam-4767	188	13	singletons	singleton	NOUN
ejpam-4767	188	14	.	.	PUNCT
ejpam-4767	189	1	clearly	clearly	ADV
ejpam-4767	189	2	,	,	PUNCT
ejpam-4767	189	3	each	each	DET
ejpam-4767	189	4	one	one	NUM
ejpam-4767	189	5	-	-	PUNCT
ejpam-4767	189	6	point	point	NOUN
ejpam-4767	189	7	set	set	VERB
ejpam-4767	189	8	in	in	ADP
ejpam-4767	189	9	x	x	PUNCT
ejpam-4767	189	10	is	be	AUX
ejpam-4767	189	11	compact	compact	ADJ
ejpam-4767	189	12	.	.	PUNCT
ejpam-4767	190	1	also	also	ADV
ejpam-4767	190	2	,	,	PUNCT
ejpam-4767	190	3	note	note	VERB
ejpam-4767	190	4	that	that	SCONJ
ejpam-4767	190	5	the	the	DET
ejpam-4767	190	6	partition	partition	NOUN
ejpam-4767	190	7	of	of	ADP
ejpam-4767	190	8	singletons	singleton	NOUN
ejpam-4767	190	9	is	be	AUX
ejpam-4767	190	10	locally	locally	ADV
ejpam-4767	190	11	finite	finite	ADJ
ejpam-4767	190	12	except	except	SCONJ
ejpam-4767	190	13	for	for	ADP
ejpam-4767	190	14	a	a	DET
ejpam-4767	190	15	finite	finite	ADJ
ejpam-4767	190	16	number	number	NOUN
ejpam-4767	190	17	of	of	ADP
ejpam-4767	190	18	points	point	NOUN
ejpam-4767	190	19	.	.	PUNCT
ejpam-4767	191	1	therefore	therefore	ADV
ejpam-4767	191	2	,	,	PUNCT
ejpam-4767	191	3	by	by	ADP
ejpam-4767	191	4	using	use	VERB
ejpam-4767	191	5	theorem	theorem	ADJ
ejpam-4767	191	6	2.15	2.15	NUM
ejpam-4767	191	7	we	we	PRON
ejpam-4767	191	8	see	see	VERB
ejpam-4767	191	9	that	that	SCONJ
ejpam-4767	191	10	x	x	PRON
ejpam-4767	191	11	is	be	AUX
ejpam-4767	191	12	a	a	DET
ejpam-4767	191	13	tychonoff	tychonoff	NOUN
ejpam-4767	191	14	space	space	NOUN
ejpam-4767	191	15	.	.	PUNCT
ejpam-4767	192	1	we	we	PRON
ejpam-4767	192	2	can	can	AUX
ejpam-4767	192	3	also	also	ADV
ejpam-4767	192	4	apply	apply	VERB
ejpam-4767	192	5	theorem	theorem	ADJ
ejpam-4767	192	6	2.13	2.13	NUM
ejpam-4767	192	7	to	to	PART
ejpam-4767	192	8	see	see	VERB
ejpam-4767	192	9	that	that	SCONJ
ejpam-4767	192	10	this	this	DET
ejpam-4767	192	11	space	space	NOUN
ejpam-4767	192	12	is	be	AUX
ejpam-4767	192	13	a	a	DET
ejpam-4767	192	14	tychonoff	tychonoff	NOUN
ejpam-4767	192	15	space	space	NOUN
ejpam-4767	192	16	.	.	PUNCT
ejpam-4767	193	1	namely	namely	ADV
ejpam-4767	193	2	,	,	PUNCT
ejpam-4767	193	3	x•	x•	PROPN
ejpam-4767	193	4	is	be	AUX
ejpam-4767	193	5	finite	finite	ADJ
ejpam-4767	193	6	,	,	PUNCT
ejpam-4767	193	7	then	then	ADV
ejpam-4767	193	8	is	be	AUX
ejpam-4767	193	9	compact	compact	ADJ
ejpam-4767	193	10	.	.	PUNCT
ejpam-4767	194	1	observe	observe	VERB
ejpam-4767	194	2	that	that	DET
ejpam-4767	194	3	partition	partition	NOUN
ejpam-4767	194	4	of	of	ADP
ejpam-4767	194	5	singletons	singleton	NOUN
ejpam-4767	194	6	of	of	ADP
ejpam-4767	194	7	x\(x•	x\(x•	NOUN
ejpam-4767	194	8	∪	∪	ADJ
ejpam-4767	194	9	(	(	PUNCT
ejpam-4767	194	10	x\x	x\x	X
ejpam-4767	194	11	)	)	PUNCT
ejpam-4767	194	12	is	be	AUX
ejpam-4767	194	13	locally	locally	ADV
ejpam-4767	194	14	finite	finite	ADJ
ejpam-4767	194	15	.	.	PUNCT
ejpam-4767	195	1	proposition	proposition	NOUN
ejpam-4767	195	2	2.17	2.17	NUM
ejpam-4767	195	3	.	.	PUNCT
ejpam-4767	196	1	let	let	AUX
ejpam-4767	196	2	{	{	PUNCT
ejpam-4767	196	3	x1	x1	ADJ
ejpam-4767	196	4	,	,	PUNCT
ejpam-4767	196	5	x2	x2	PROPN
ejpam-4767	196	6	,	,	PUNCT
ejpam-4767	196	7	...	...	PUNCT
ejpam-4767	196	8	,	,	PUNCT
ejpam-4767	196	9	xn	xn	PRON
ejpam-4767	196	10	}	}	PUNCT
ejpam-4767	196	11	be	be	AUX
ejpam-4767	196	12	a	a	DET
ejpam-4767	196	13	collection	collection	NOUN
ejpam-4767	196	14	of	of	ADP
ejpam-4767	196	15	l.c.w.d	l.c.w.d	PROPN
ejpam-4767	196	16	.	.	PUNCT
ejpam-4767	197	1	topological	topological	ADJ
ejpam-4767	197	2	spaces	space	NOUN
ejpam-4767	197	3	.	.	PUNCT
ejpam-4767	198	1	suppose	suppose	VERB
ejpam-4767	198	2	that	that	SCONJ
ejpam-4767	198	3	for	for	ADP
ejpam-4767	198	4	each	each	DET
ejpam-4767	198	5	topological	topological	ADJ
ejpam-4767	198	6	space	space	NOUN
ejpam-4767	198	7	xi	xi	INTJ
ejpam-4767	198	8	we	we	PRON
ejpam-4767	198	9	have	have	VERB
ejpam-4767	198	10	x•	x•	NOUN
ejpam-4767	198	11	̸=	̸=	PROPN
ejpam-4767	198	12	ϕ	ϕ	PROPN
ejpam-4767	198	13	̸=	̸=	PROPN
ejpam-4767	198	14	xi	xi	PROPN
ejpam-4767	198	15	.	.	PUNCT
ejpam-4767	199	1	then	then	ADV
ejpam-4767	199	2	for	for	ADP
ejpam-4767	199	3	x	x	SYM
ejpam-4767	199	4	=	=	PROPN
ejpam-4767	199	5	∏n	∏n	PROPN
ejpam-4767	199	6	i=1xi	i=1xi	PROPN
ejpam-4767	199	7	,	,	PUNCT
ejpam-4767	199	8	we	we	PRON
ejpam-4767	199	9	have	have	VERB
ejpam-4767	199	10	x•	x•	NOUN
ejpam-4767	199	11	̸=	̸=	PROPN
ejpam-4767	199	12	ϕ	ϕ	PROPN
ejpam-4767	199	13	̸=	̸=	PROPN
ejpam-4767	199	14	x.	x.	NOUN
ejpam-4767	199	15	proof	proof	NOUN
ejpam-4767	199	16	.	.	PUNCT
ejpam-4767	200	1	it	it	PRON
ejpam-4767	200	2	is	be	AUX
ejpam-4767	200	3	straightforward	straightforward	ADJ
ejpam-4767	200	4	.	.	PUNCT
ejpam-4767	201	1	proposition	proposition	NOUN
ejpam-4767	201	2	2.18	2.18	NUM
ejpam-4767	201	3	.	.	PUNCT
ejpam-4767	202	1	let	let	VERB
ejpam-4767	202	2	each	each	PRON
ejpam-4767	202	3	of	of	ADP
ejpam-4767	202	4	{	{	PUNCT
ejpam-4767	202	5	xs}s∈s	xs}s∈s	PRON
ejpam-4767	202	6	be	be	AUX
ejpam-4767	202	7	a	a	DET
ejpam-4767	202	8	collection	collection	NOUN
ejpam-4767	202	9	of	of	ADP
ejpam-4767	202	10	l.c.w.d	l.c.w.d	PROPN
ejpam-4767	202	11	spaces	space	VERB
ejpam-4767	202	12	such	such	DET
ejpam-4767	202	13	each	each	DET
ejpam-4767	202	14	point	point	NOUN
ejpam-4767	202	15	of	of	ADP
ejpam-4767	202	16	xs	xs	PROPN
ejpam-4767	202	17	has	have	VERB
ejpam-4767	202	18	an	an	DET
ejpam-4767	202	19	open	open	ADJ
ejpam-4767	202	20	neighborhood	neighborhood	NOUN
ejpam-4767	202	21	with	with	ADP
ejpam-4767	202	22	closure	closure	NOUN
ejpam-4767	202	23	being	be	AUX
ejpam-4767	202	24	a	a	DET
ejpam-4767	202	25	union	union	NOUN
ejpam-4767	202	26	of	of	ADP
ejpam-4767	202	27	pairwise	pairwise	PROPN
ejpam-4767	202	28	disjoint	disjoint	NOUN
ejpam-4767	202	29	compact	compact	ADJ
ejpam-4767	202	30	subsets	subset	NOUN
ejpam-4767	202	31	,	,	PUNCT
ejpam-4767	202	32	then	then	ADV
ejpam-4767	202	33	so	so	ADV
ejpam-4767	202	34	does	do	AUX
ejpam-4767	202	35	each	each	DET
ejpam-4767	202	36	point	point	NOUN
ejpam-4767	202	37	of	of	ADP
ejpam-4767	202	38	the	the	DET
ejpam-4767	202	39	cartesian	cartesian	ADJ
ejpam-4767	202	40	product	product	NOUN
ejpam-4767	202	41	∏	∏	PROPN
ejpam-4767	202	42	s∈s	s∈s	NOUN
ejpam-4767	202	43	xs	xs	PROPN
ejpam-4767	202	44	.	.	PUNCT
ejpam-4767	203	1	proof	proof	NOUN
ejpam-4767	203	2	.	.	PUNCT
ejpam-4767	204	1	it	it	PRON
ejpam-4767	204	2	is	be	AUX
ejpam-4767	204	3	straightforward	straightforward	ADJ
ejpam-4767	204	4	.	.	PUNCT
ejpam-4767	205	1	proposition	proposition	NOUN
ejpam-4767	205	2	2.19	2.19	NUM
ejpam-4767	205	3	.	.	PUNCT
ejpam-4767	206	1	let	let	VERB
ejpam-4767	206	2	x	x	PRON
ejpam-4767	206	3	be	be	AUX
ejpam-4767	206	4	l.c.w.d	l.c.w.d	PROPN
ejpam-4767	206	5	such	such	ADJ
ejpam-4767	206	6	that	that	SCONJ
ejpam-4767	206	7	any	any	DET
ejpam-4767	206	8	x	x	SYM
ejpam-4767	206	9	∈	∈	NOUN
ejpam-4767	206	10	x•	x•	NOUN
ejpam-4767	206	11	has	have	VERB
ejpam-4767	206	12	a	a	DET
ejpam-4767	206	13	σ	σ	ADJ
ejpam-4767	206	14	-	-	ADJ
ejpam-4767	206	15	compact	compact	ADJ
ejpam-4767	206	16	neighborhood	neighborhood	NOUN
ejpam-4767	206	17	.	.	PUNCT
ejpam-4767	207	1	then	then	ADV
ejpam-4767	207	2	for	for	ADP
ejpam-4767	207	3	any	any	DET
ejpam-4767	207	4	closed	closed	ADJ
ejpam-4767	207	5	subspace	subspace	NOUN
ejpam-4767	207	6	f	f	PROPN
ejpam-4767	207	7	⊂	⊂	PROPN
ejpam-4767	207	8	x	x	PROPN
ejpam-4767	207	9	,	,	PUNCT
ejpam-4767	207	10	x	x	SYM
ejpam-4767	207	11	∈	∈	NOUN
ejpam-4767	207	12	x•	x•	NOUN
ejpam-4767	207	13	∩	∩	PROPN
ejpam-4767	207	14	f	f	PROPN
ejpam-4767	207	15	has	have	VERB
ejpam-4767	207	16	a	a	DET
ejpam-4767	207	17	σ	σ	ADJ
ejpam-4767	207	18	-	-	ADJ
ejpam-4767	207	19	compact	compact	ADJ
ejpam-4767	207	20	neighborhood	neighborhood	NOUN
ejpam-4767	207	21	of	of	ADP
ejpam-4767	207	22	the	the	DET
ejpam-4767	207	23	subspace	subspace	PROPN
ejpam-4767	207	24	f	f	PROPN
ejpam-4767	207	25	,	,	PUNCT
ejpam-4767	207	26	i.e.	i.e.	X
ejpam-4767	207	27	,	,	PUNCT
ejpam-4767	207	28	this	this	DET
ejpam-4767	207	29	space	space	NOUN
ejpam-4767	207	30	is	be	AUX
ejpam-4767	207	31	hereditarily	hereditarily	ADV
ejpam-4767	207	32	with	with	ADP
ejpam-4767	207	33	respect	respect	NOUN
ejpam-4767	207	34	to	to	ADP
ejpam-4767	207	35	closed	closed	ADJ
ejpam-4767	207	36	subspaces	subspace	NOUN
ejpam-4767	207	37	.	.	PUNCT
ejpam-4767	208	1	proof	proof	NOUN
ejpam-4767	208	2	.	.	PUNCT
ejpam-4767	209	1	take	take	VERB
ejpam-4767	209	2	any	any	DET
ejpam-4767	209	3	x	x	SYM
ejpam-4767	209	4	∈	∈	PROPN
ejpam-4767	209	5	x•	x•	NOUN
ejpam-4767	209	6	∩	∩	PROPN
ejpam-4767	209	7	f	f	PROPN
ejpam-4767	209	8	,	,	PUNCT
ejpam-4767	209	9	then	then	ADV
ejpam-4767	209	10	there	there	PRON
ejpam-4767	209	11	is	be	VERB
ejpam-4767	209	12	an	an	DET
ejpam-4767	209	13	open	open	ADJ
ejpam-4767	209	14	neighborhood	neighborhood	NOUN
ejpam-4767	209	15	u	u	NOUN
ejpam-4767	209	16	⊂	⊂	X
ejpam-4767	209	17	x	x	X
ejpam-4767	209	18	of	of	ADP
ejpam-4767	209	19	x	x	SYM
ejpam-4767	209	20	such	such	ADJ
ejpam-4767	209	21	that	that	DET
ejpam-4767	209	22	u	u	NOUN
ejpam-4767	209	23	=	=	PUNCT
ejpam-4767	209	24	⋃∞	⋃∞	X
ejpam-4767	209	25	s=1	s=1	NOUN
ejpam-4767	209	26	fs	fs	INTJ
ejpam-4767	209	27	where	where	SCONJ
ejpam-4767	209	28	each	each	DET
ejpam-4767	209	29	fs	f	NOUN
ejpam-4767	209	30	is	be	AUX
ejpam-4767	209	31	compact	compact	ADJ
ejpam-4767	209	32	as	as	ADP
ejpam-4767	209	33	a	a	DET
ejpam-4767	209	34	subset	subset	NOUN
ejpam-4767	209	35	of	of	ADP
ejpam-4767	209	36	x.	x.	PROPN
ejpam-4767	209	37	observe	observe	VERB
ejpam-4767	209	38	that	that	SCONJ
ejpam-4767	209	39	u	u	PROPN
ejpam-4767	209	40	∩	∩	NOUN
ejpam-4767	209	41	f	f	PROPN
ejpam-4767	209	42	is	be	AUX
ejpam-4767	209	43	an	an	DET
ejpam-4767	209	44	open	open	NOUN
ejpam-4767	209	45	of	of	ADP
ejpam-4767	209	46	x	x	PUNCT
ejpam-4767	209	47	in	in	ADP
ejpam-4767	209	48	f	f	PROPN
ejpam-4767	209	49	such	such	ADJ
ejpam-4767	209	50	that	that	SCONJ
ejpam-4767	209	51	its	its	PRON
ejpam-4767	209	52	closure	closure	NOUN
ejpam-4767	209	53	,	,	PUNCT
ejpam-4767	209	54	(	(	PUNCT
ejpam-4767	209	55	u	u	NOUN
ejpam-4767	209	56	∩	∩	X
ejpam-4767	209	57	f	f	PROPN
ejpam-4767	209	58	)	)	PUNCT
ejpam-4767	209	59	∩	∩	PROPN
ejpam-4767	209	60	f	f	X
ejpam-4767	209	61	,	,	PUNCT
ejpam-4767	209	62	in	in	ADP
ejpam-4767	209	63	f	f	PROPN
ejpam-4767	209	64	σ	σ	PROPN
ejpam-4767	209	65	-	-	PROPN
ejpam-4767	209	66	compact	compact	ADJ
ejpam-4767	209	67	.	.	PUNCT
ejpam-4767	210	1	3	3	X
ejpam-4767	210	2	.	.	X
ejpam-4767	210	3	conclusion	conclusion	NOUN
ejpam-4767	210	4	a	a	DET
ejpam-4767	210	5	well	well	ADV
ejpam-4767	210	6	-	-	PUNCT
ejpam-4767	210	7	known	know	VERB
ejpam-4767	210	8	result	result	NOUN
ejpam-4767	210	9	in	in	ADP
ejpam-4767	210	10	general	general	ADJ
ejpam-4767	210	11	topology	topology	NOUN
ejpam-4767	210	12	states	state	VERB
ejpam-4767	210	13	that	that	SCONJ
ejpam-4767	210	14	any	any	DET
ejpam-4767	210	15	locally	locally	ADV
ejpam-4767	210	16	compact	compact	ADJ
ejpam-4767	210	17	space	space	NOUN
ejpam-4767	210	18	is	be	AUX
ejpam-4767	210	19	a	a	DET
ejpam-4767	210	20	tychonoff	tychonoff	NOUN
ejpam-4767	210	21	space	space	NOUN
ejpam-4767	210	22	.	.	PUNCT
ejpam-4767	211	1	in	in	ADP
ejpam-4767	211	2	this	this	DET
ejpam-4767	211	3	paper	paper	NOUN
ejpam-4767	211	4	we	we	PRON
ejpam-4767	211	5	investigate	investigate	VERB
ejpam-4767	211	6	a	a	DET
ejpam-4767	211	7	weaker	weak	ADJ
ejpam-4767	211	8	version	version	NOUN
ejpam-4767	211	9	of	of	ADP
ejpam-4767	211	10	local	local	ADJ
ejpam-4767	211	11	compactness	compactness	NOUN
ejpam-4767	211	12	.	.	PUNCT
ejpam-4767	212	1	instead	instead	ADV
ejpam-4767	212	2	of	of	ADP
ejpam-4767	212	3	assuming	assume	VERB
ejpam-4767	212	4	that	that	SCONJ
ejpam-4767	212	5	all	all	DET
ejpam-4767	212	6	points	point	NOUN
ejpam-4767	212	7	in	in	ADP
ejpam-4767	212	8	a	a	DET
ejpam-4767	212	9	space	space	NOUN
ejpam-4767	212	10	have	have	VERB
ejpam-4767	212	11	compact	compact	ADJ
ejpam-4767	212	12	neighborhoods	neighborhood	NOUN
ejpam-4767	212	13	,	,	PUNCT
ejpam-4767	212	14	we	we	PRON
ejpam-4767	212	15	allow	allow	VERB
ejpam-4767	212	16	a	a	DET
ejpam-4767	212	17	possibility	possibility	NOUN
ejpam-4767	212	18	of	of	ADP
ejpam-4767	212	19	having	have	VERB
ejpam-4767	212	20	some	some	DET
ejpam-4767	212	21	points	point	NOUN
ejpam-4767	212	22	which	which	PRON
ejpam-4767	212	23	do	do	AUX
ejpam-4767	212	24	not	not	PART
ejpam-4767	212	25	possess	possess	VERB
ejpam-4767	212	26	compact	compact	ADJ
ejpam-4767	212	27	neighborhoods	neighborhood	NOUN
ejpam-4767	212	28	.	.	PUNCT
ejpam-4767	213	1	we	we	PRON
ejpam-4767	213	2	denote	denote	VERB
ejpam-4767	213	3	by	by	ADP
ejpam-4767	213	4	x•	x•	PROPN
ejpam-4767	213	5	a	a	DET
ejpam-4767	213	6	set	set	NOUN
ejpam-4767	213	7	of	of	ADP
ejpam-4767	213	8	points	point	NOUN
ejpam-4767	213	9	which	which	PRON
ejpam-4767	213	10	do	do	AUX
ejpam-4767	213	11	not	not	PART
ejpam-4767	213	12	have	have	VERB
ejpam-4767	213	13	open	open	ADJ
ejpam-4767	213	14	neighborhoods	neighborhood	NOUN
ejpam-4767	213	15	with	with	ADP
ejpam-4767	213	16	compact	compact	ADJ
ejpam-4767	213	17	closures	closure	NOUN
ejpam-4767	213	18	.	.	PUNCT
ejpam-4767	214	1	one	one	NUM
ejpam-4767	214	2	of	of	ADP
ejpam-4767	214	3	the	the	DET
ejpam-4767	214	4	results	result	NOUN
ejpam-4767	214	5	we	we	PRON
ejpam-4767	214	6	obtain	obtain	VERB
ejpam-4767	214	7	is	be	AUX
ejpam-4767	214	8	that	that	SCONJ
ejpam-4767	214	9	by	by	ADP
ejpam-4767	214	10	requiring	require	VERB
ejpam-4767	214	11	the	the	DET
ejpam-4767	214	12	set	set	NOUN
ejpam-4767	214	13	x•	x•	NOUN
ejpam-4767	214	14	to	to	PART
ejpam-4767	214	15	be	be	AUX
ejpam-4767	214	16	compact	compact	ADJ
ejpam-4767	214	17	,	,	PUNCT
ejpam-4767	214	18	we	we	PRON
ejpam-4767	214	19	show	show	VERB
ejpam-4767	214	20	that	that	SCONJ
ejpam-4767	214	21	if	if	SCONJ
ejpam-4767	214	22	each	each	DET
ejpam-4767	214	23	point	point	NOUN
ejpam-4767	214	24	x	x	X
ejpam-4767	214	25	∈	∈	NOUN
ejpam-4767	214	26	x•	x•	NOUN
ejpam-4767	214	27	has	have	VERB
ejpam-4767	214	28	an	an	DET
ejpam-4767	214	29	open	open	ADJ
ejpam-4767	214	30	neighborhood	neighborhood	NOUN
ejpam-4767	214	31	u	u	NOUN
ejpam-4767	214	32	̸=	̸=	PROPN
ejpam-4767	214	33	x	x	SYM
ejpam-4767	214	34	such	such	ADJ
ejpam-4767	214	35	that	that	SCONJ
ejpam-4767	214	36	the	the	DET
ejpam-4767	214	37	partition	partition	NOUN
ejpam-4767	214	38	of	of	ADP
ejpam-4767	214	39	singletons	singleton	NOUN
ejpam-4767	214	40	of	of	ADP
ejpam-4767	214	41	the	the	DET
ejpam-4767	214	42	complement	complement	NOUN
ejpam-4767	214	43	of	of	ADP
ejpam-4767	214	44	x•	x•	PROPN
ejpam-4767	214	45	∪	∪	X
ejpam-4767	214	46	(	(	PUNCT
ejpam-4767	214	47	u\u	u\u	NOUN
ejpam-4767	214	48	)	)	PUNCT
ejpam-4767	214	49	is	be	AUX
ejpam-4767	214	50	locally	locally	ADV
ejpam-4767	214	51	finite	finite	ADJ
ejpam-4767	214	52	,	,	PUNCT
ejpam-4767	214	53	then	then	ADV
ejpam-4767	214	54	the	the	DET
ejpam-4767	214	55	space	space	NOUN
ejpam-4767	214	56	is	be	AUX
ejpam-4767	214	57	a	a	DET
ejpam-4767	214	58	tychonoff	tychonoff	NOUN
ejpam-4767	214	59	space	space	NOUN
ejpam-4767	214	60	.	.	PUNCT
ejpam-4767	215	1	the	the	DET
ejpam-4767	215	2	following	follow	VERB
ejpam-4767	215	3	questions	question	NOUN
ejpam-4767	215	4	are	be	AUX
ejpam-4767	215	5	still	still	ADV
ejpam-4767	215	6	not	not	PART
ejpam-4767	215	7	answered	answer	VERB
ejpam-4767	215	8	.	.	PUNCT
ejpam-4767	216	1	could	could	AUX
ejpam-4767	216	2	we	we	PRON
ejpam-4767	216	3	assume	assume	VERB
ejpam-4767	216	4	that	that	SCONJ
ejpam-4767	216	5	x•	x•	PROPN
ejpam-4767	216	6	is	be	AUX
ejpam-4767	216	7	locally	locally	ADV
ejpam-4767	216	8	compact	compact	ADJ
ejpam-4767	216	9	instead	instead	ADV
ejpam-4767	216	10	of	of	ADP
ejpam-4767	216	11	being	be	AUX
ejpam-4767	216	12	compact	compact	ADJ
ejpam-4767	216	13	in	in	ADP
ejpam-4767	216	14	theorem	theorem	NOUN
ejpam-4767	216	15	2.13	2.13	NUM
ejpam-4767	216	16	?	?	PUNCT
ejpam-4767	217	1	could	could	AUX
ejpam-4767	217	2	we	we	PRON
ejpam-4767	217	3	drop	drop	VERB
ejpam-4767	217	4	the	the	DET
ejpam-4767	217	5	requirement	requirement	NOUN
ejpam-4767	217	6	of	of	ADP
ejpam-4767	217	7	compact	compact	ADJ
ejpam-4767	217	8	subsets	subset	NOUN
ejpam-4767	217	9	need	need	VERB
ejpam-4767	217	10	to	to	PART
ejpam-4767	217	11	be	be	AUX
ejpam-4767	217	12	closed	close	VERB
ejpam-4767	217	13	in	in	ADP
ejpam-4767	217	14	theorem	theorem	NOUN
ejpam-4767	217	15	2.15	2.15	NUM
ejpam-4767	217	16	?	?	PUNCT
ejpam-4767	218	1	references	reference	NOUN
ejpam-4767	218	2	[	[	X
ejpam-4767	218	3	1	1	NUM
ejpam-4767	218	4	]	]	PUNCT
ejpam-4767	218	5	j	j	PROPN
ejpam-4767	218	6	dugundji	dugundji	PROPN
ejpam-4767	218	7	.	.	PUNCT
ejpam-4767	219	1	topology	topology	PROPN
ejpam-4767	219	2	.	.	PUNCT
ejpam-4767	220	1	allyn	allyn	PROPN
ejpam-4767	220	2	and	and	CCONJ
ejpam-4767	220	3	bacon	bacon	PROPN
ejpam-4767	220	4	,	,	PUNCT
ejpam-4767	220	5	inc	inc	PROPN
ejpam-4767	220	6	,	,	PUNCT
ejpam-4767	220	7	boston	boston	PROPN
ejpam-4767	220	8	,	,	PUNCT
ejpam-4767	220	9	1972	1972	NUM
ejpam-4767	220	10	.	.	PUNCT
ejpam-4767	221	1	[	[	X
ejpam-4767	221	2	2	2	NUM
ejpam-4767	221	3	]	]	X
ejpam-4767	221	4	r	r	NOUN
ejpam-4767	221	5	engelking	engelking	NOUN
ejpam-4767	221	6	.	.	PUNCT
ejpam-4767	222	1	general	general	ADJ
ejpam-4767	222	2	topology	topology	PROPN
ejpam-4767	222	3	.	.	PUNCT
ejpam-4767	223	1	heldermann	heldermann	PROPN
ejpam-4767	223	2	verlag	verlag	PROPN
ejpam-4767	223	3	,	,	PUNCT
ejpam-4767	223	4	berlin	berlin	PROPN
ejpam-4767	223	5	,	,	PUNCT
ejpam-4767	223	6	germany	germany	PROPN
ejpam-4767	223	7	,	,	PUNCT
ejpam-4767	223	8	1989	1989	NUM
ejpam-4767	223	9	.	.	PUNCT
ejpam-4767	224	1	references	reference	NOUN
ejpam-4767	224	2	1235	1235	NUM
ejpam-4767	225	1	[	[	X
ejpam-4767	225	2	3	3	NUM
ejpam-4767	225	3	]	]	X
ejpam-4767	225	4	j	j	PROPN
ejpam-4767	225	5	hatzenbuhler	hatzenbuhler	NOUN
ejpam-4767	225	6	and	and	CCONJ
ejpam-4767	225	7	d	d	PROPN
ejpam-4767	225	8	mattson	mattson	NOUN
ejpam-4767	225	9	.	.	PUNCT
ejpam-4767	226	1	on	on	ADP
ejpam-4767	226	2	hausdorff	hausdorff	PROPN
ejpam-4767	226	3	compactifications	compactification	NOUN
ejpam-4767	226	4	of	of	ADP
ejpam-4767	226	5	non	non	ADJ
ejpam-4767	226	6	-	-	ADJ
ejpam-4767	226	7	locally	locally	ADV
ejpam-4767	226	8	compact	compact	ADJ
ejpam-4767	226	9	spaces	space	NOUN
ejpam-4767	226	10	.	.	PUNCT
ejpam-4767	227	1	international	international	ADJ
ejpam-4767	227	2	journal	journal	PROPN
ejpam-4767	227	3	of	of	ADP
ejpam-4767	227	4	mathematics	mathematics	PROPN
ejpam-4767	227	5	and	and	CCONJ
ejpam-4767	227	6	mathematical	mathematical	ADJ
ejpam-4767	227	7	sciences	science	NOUN
ejpam-4767	227	8	,	,	PUNCT
ejpam-4767	227	9	2(3):481–486	2(3):481–486	NUM
ejpam-4767	227	10	,	,	PUNCT
ejpam-4767	227	11	1979	1979	NUM
ejpam-4767	227	12	.	.	PUNCT
ejpam-4767	228	1	[	[	X
ejpam-4767	228	2	4	4	X
ejpam-4767	228	3	]	]	PUNCT
ejpam-4767	228	4	m	m	VERB
ejpam-4767	228	5	rayburn	rayburn	NOUN
ejpam-4767	228	6	.	.	PUNCT
ejpam-4767	229	1	on	on	ADP
ejpam-4767	229	2	hausdorff	hausdorff	PROPN
ejpam-4767	229	3	compactifications	compactification	NOUN
ejpam-4767	229	4	.	.	PUNCT
ejpam-4767	230	1	pacific	pacific	PROPN
ejpam-4767	230	2	journal	journal	PROPN
ejpam-4767	230	3	of	of	ADP
ejpam-4767	230	4	mathematics	mathematic	NOUN
ejpam-4767	230	5	,	,	PUNCT
ejpam-4767	230	6	44(2):707–714	44(2):707–714	PROPN
ejpam-4767	230	7	,	,	PUNCT
ejpam-4767	230	8	1973	1973	NUM
ejpam-4767	230	9	.	.	PUNCT
ejpam-4767	231	1	[	[	X
ejpam-4767	231	2	5	5	NUM
ejpam-4767	231	3	]	]	PUNCT
ejpam-4767	231	4	m	m	VERB
ejpam-4767	231	5	rayburn	rayburn	NOUN
ejpam-4767	231	6	.	.	PUNCT
ejpam-4767	232	1	compactifications	compactification	NOUN
ejpam-4767	232	2	with	with	ADP
ejpam-4767	232	3	almost	almost	ADV
ejpam-4767	232	4	locally	locally	ADV
ejpam-4767	232	5	compact	compact	ADJ
ejpam-4767	232	6	outgrowth	outgrowth	NOUN
ejpam-4767	232	7	.	.	PUNCT
ejpam-4767	233	1	proceedings	proceeding	NOUN
ejpam-4767	233	2	of	of	ADP
ejpam-4767	233	3	the	the	DET
ejpam-4767	233	4	american	american	PROPN
ejpam-4767	233	5	mathematical	mathematical	PROPN
ejpam-4767	233	6	society	society	NOUN
ejpam-4767	233	7	,	,	PUNCT
ejpam-4767	233	8	106(1):223–229	106(1):223–229	NUM
ejpam-4767	233	9	,	,	PUNCT
ejpam-4767	233	10	1989	1989	NUM
ejpam-4767	233	11	.	.	PUNCT
ejpam-4767	234	1	[	[	X
ejpam-4767	234	2	6	6	NUM
ejpam-4767	234	3	]	]	PUNCT
ejpam-4767	234	4	l	l	NOUN
ejpam-4767	234	5	steen	steen	PROPN
ejpam-4767	234	6	and	and	CCONJ
ejpam-4767	234	7	j	j	PROPN
ejpam-4767	234	8	seebach	seebach	NOUN
ejpam-4767	234	9	.	.	PUNCT
ejpam-4767	235	1	counterexamples	counterexample	NOUN
ejpam-4767	235	2	in	in	ADP
ejpam-4767	235	3	topology	topology	NOUN
ejpam-4767	235	4	.	.	PUNCT
ejpam-4767	236	1	dover	dover	PROPN
ejpam-4767	236	2	publ	publ	PROPN
ejpam-4767	236	3	.	.	PUNCT
ejpam-4767	236	4	,	,	PUNCT
ejpam-4767	236	5	new	new	PROPN
ejpam-4767	236	6	york	york	PROPN
ejpam-4767	236	7	,	,	PUNCT
ejpam-4767	236	8	1995	1995	NUM
ejpam-4767	236	9	.	.	PUNCT