id	sid	tid	token	lemma	pos
ejpam-4248	1	1	european	european	PROPN
ejpam-4248	1	2	journal	journal	PROPN
ejpam-4248	1	3	of	of	ADP
ejpam-4248	1	4	pure	pure	ADJ
ejpam-4248	1	5	and	and	CCONJ
ejpam-4248	1	6	applied	apply	VERB
ejpam-4248	1	7	mathematics	mathematic	NOUN
ejpam-4248	1	8	vol	vol	NOUN
ejpam-4248	1	9	.	.	PROPN
ejpam-4248	2	1	15	15	NUM
ejpam-4248	2	2	,	,	PUNCT
ejpam-4248	2	3	no	no	INTJ
ejpam-4248	2	4	.	.	NOUN
ejpam-4248	2	5	1	1	NUM
ejpam-4248	2	6	,	,	PUNCT
ejpam-4248	2	7	2022	2022	NUM
ejpam-4248	2	8	,	,	PUNCT
ejpam-4248	2	9	199	199	NUM
ejpam-4248	2	10	-	-	SYM
ejpam-4248	2	11	206	206	NUM
ejpam-4248	2	12	issn	issn	PROPN
ejpam-4248	2	13	1307	1307	NUM
ejpam-4248	2	14	-	-	SYM
ejpam-4248	2	15	5543	5543	NUM
ejpam-4248	2	16	–	–	PUNCT
ejpam-4248	2	17	ejpam.com	ejpam.com	X
ejpam-4248	2	18	published	publish	VERB
ejpam-4248	2	19	by	by	ADP
ejpam-4248	2	20	new	new	PROPN
ejpam-4248	2	21	york	york	PROPN
ejpam-4248	2	22	business	business	PROPN
ejpam-4248	2	23	global	global	ADJ
ejpam-4248	2	24	weakly	weakly	ADJ
ejpam-4248	2	25	covering	covering	NOUN
ejpam-4248	2	26	spaces	space	NOUN
ejpam-4248	2	27	in	in	ADP
ejpam-4248	2	28	coc	coc	NOUN
ejpam-4248	2	29	-	-	PUNCT
ejpam-4248	2	30	open	open	ADJ
ejpam-4248	2	31	sets	set	VERB
ejpam-4248	2	32	fuad	fuad	PROPN
ejpam-4248	2	33	a.	a.	PROPN
ejpam-4248	2	34	abushaheen1,∗	abushaheen1,∗	PROPN
ejpam-4248	2	35	,	,	PUNCT
ejpam-4248	2	36	fadi	fadi	NOUN
ejpam-4248	2	37	alrimawi2	alrimawi2	NOUN
ejpam-4248	3	1	1	1	NUM
ejpam-4248	3	2	basic	basic	ADJ
ejpam-4248	3	3	science	science	NOUN
ejpam-4248	3	4	department	department	NOUN
ejpam-4248	3	5	,	,	PUNCT
ejpam-4248	3	6	middle	middle	PROPN
ejpam-4248	3	7	east	east	PROPN
ejpam-4248	3	8	university	university	PROPN
ejpam-4248	3	9	,	,	PUNCT
ejpam-4248	3	10	amman	amman	PROPN
ejpam-4248	3	11	,	,	PUNCT
ejpam-4248	3	12	jordan	jordan	PROPN
ejpam-4248	3	13	2	2	NUM
ejpam-4248	3	14	department	department	NOUN
ejpam-4248	3	15	of	of	ADP
ejpam-4248	3	16	basic	basic	ADJ
ejpam-4248	3	17	sciences	sciences	PROPN
ejpam-4248	3	18	,	,	PUNCT
ejpam-4248	3	19	al	al	PROPN
ejpam-4248	3	20	-	-	PUNCT
ejpam-4248	3	21	ahliyya	ahliyya	PROPN
ejpam-4248	3	22	amman	amman	PROPN
ejpam-4248	3	23	university	university	PROPN
ejpam-4248	3	24	,	,	PUNCT
ejpam-4248	3	25	amman	amman	PROPN
ejpam-4248	3	26	,	,	PUNCT
ejpam-4248	3	27	jordan	jordan	PROPN
ejpam-4248	3	28	abstract	abstract	PROPN
ejpam-4248	3	29	.	.	PUNCT
ejpam-4248	4	1	in	in	ADP
ejpam-4248	4	2	this	this	DET
ejpam-4248	4	3	paper	paper	NOUN
ejpam-4248	4	4	,	,	PUNCT
ejpam-4248	4	5	we	we	PRON
ejpam-4248	4	6	define	define	VERB
ejpam-4248	4	7	several	several	ADJ
ejpam-4248	4	8	types	type	NOUN
ejpam-4248	4	9	of	of	ADP
ejpam-4248	4	10	weakly	weakly	ADJ
ejpam-4248	4	11	covering	covering	NOUN
ejpam-4248	4	12	spaces	space	NOUN
ejpam-4248	4	13	concerning	concern	VERB
ejpam-4248	4	14	coc	coc	NOUN
ejpam-4248	4	15	-	-	PUNCT
ejpam-4248	4	16	open	open	ADJ
ejpam-4248	4	17	sets	set	NOUN
ejpam-4248	4	18	,	,	PUNCT
ejpam-4248	4	19	we	we	PRON
ejpam-4248	4	20	study	study	VERB
ejpam-4248	4	21	some	some	DET
ejpam-4248	4	22	implications	implication	NOUN
ejpam-4248	4	23	between	between	ADP
ejpam-4248	4	24	these	these	DET
ejpam-4248	4	25	weakly	weakly	ADJ
ejpam-4248	4	26	covering	covering	NOUN
ejpam-4248	4	27	spaces	space	NOUN
ejpam-4248	4	28	.	.	PUNCT
ejpam-4248	5	1	also	also	ADV
ejpam-4248	5	2	,	,	PUNCT
ejpam-4248	5	3	we	we	PRON
ejpam-4248	5	4	give	give	VERB
ejpam-4248	5	5	several	several	ADJ
ejpam-4248	5	6	product	product	NOUN
ejpam-4248	5	7	theorems	theorem	NOUN
ejpam-4248	5	8	under	under	ADP
ejpam-4248	5	9	coc	coc	NOUN
ejpam-4248	5	10	-	-	PUNCT
ejpam-4248	5	11	almost	almost	ADV
ejpam-4248	5	12	open	open	ADJ
ejpam-4248	5	13	function	function	NOUN
ejpam-4248	5	14	.	.	PUNCT
ejpam-4248	6	1	2020	2020	NUM
ejpam-4248	6	2	mathematics	mathematic	NOUN
ejpam-4248	6	3	subject	subject	NOUN
ejpam-4248	6	4	classifications	classification	NOUN
ejpam-4248	6	5	:	:	PUNCT
ejpam-4248	6	6	54b05	54b05	NUM
ejpam-4248	6	7	,	,	PUNCT
ejpam-4248	6	8	54b10	54b10	NUM
ejpam-4248	6	9	,	,	PUNCT
ejpam-4248	6	10	54d20	54d20	NUM
ejpam-4248	6	11	key	key	ADJ
ejpam-4248	6	12	words	word	NOUN
ejpam-4248	6	13	and	and	CCONJ
ejpam-4248	6	14	phrases	phrase	NOUN
ejpam-4248	6	15	:	:	PUNCT
ejpam-4248	6	16	coc	coc	ADJ
ejpam-4248	6	17	-	-	PUNCT
ejpam-4248	6	18	weakly	weakly	ADJ
ejpam-4248	6	19	-	-	PUNCT
ejpam-4248	6	20	compact	compact	ADJ
ejpam-4248	6	21	,	,	PUNCT
ejpam-4248	6	22	coc	coc	NOUN
ejpam-4248	6	23	-	-	PUNCT
ejpam-4248	6	24	almost	almost	ADV
ejpam-4248	6	25	-	-	PUNCT
ejpam-4248	6	26	compact	compact	ADJ
ejpam-4248	6	27	,	,	PUNCT
ejpam-4248	6	28	coc	coc	NOUN
ejpam-4248	6	29	-	-	PUNCT
ejpam-4248	6	30	nearly	nearly	ADV
ejpam-4248	6	31	-	-	PUNCT
ejpam-4248	6	32	compact	compact	ADJ
ejpam-4248	6	33	,	,	PUNCT
ejpam-4248	6	34	cocr	cocr	ADJ
ejpam-4248	6	35	-	-	PUNCT
ejpam-4248	6	36	compact	compact	ADJ
ejpam-4248	6	37	1	1	NUM
ejpam-4248	6	38	.	.	PUNCT
ejpam-4248	7	1	introduction	introduction	NOUN
ejpam-4248	7	2	and	and	CCONJ
ejpam-4248	7	3	preliminaries	preliminary	NOUN
ejpam-4248	7	4	it	it	PRON
ejpam-4248	7	5	is	be	AUX
ejpam-4248	7	6	known	know	VERB
ejpam-4248	7	7	that	that	SCONJ
ejpam-4248	7	8	covering	cover	VERB
ejpam-4248	7	9	spaces	space	NOUN
ejpam-4248	7	10	play	play	VERB
ejpam-4248	7	11	very	very	ADV
ejpam-4248	7	12	important	important	ADJ
ejpam-4248	7	13	role	role	NOUN
ejpam-4248	7	14	in	in	ADP
ejpam-4248	7	15	topology	topology	NOUN
ejpam-4248	7	16	.	.	PUNCT
ejpam-4248	8	1	after	after	ADP
ejpam-4248	8	2	the	the	DET
ejpam-4248	8	3	concept	concept	NOUN
ejpam-4248	8	4	of	of	ADP
ejpam-4248	8	5	compactness	compactness	NOUN
ejpam-4248	8	6	defined	define	VERB
ejpam-4248	8	7	,	,	PUNCT
ejpam-4248	8	8	many	many	ADJ
ejpam-4248	8	9	authors	author	NOUN
ejpam-4248	8	10	studied	study	VERB
ejpam-4248	8	11	the	the	DET
ejpam-4248	8	12	relations	relation	NOUN
ejpam-4248	8	13	between	between	ADP
ejpam-4248	8	14	this	this	DET
ejpam-4248	8	15	notion	notion	NOUN
ejpam-4248	8	16	and	and	CCONJ
ejpam-4248	8	17	other	other	ADJ
ejpam-4248	8	18	topological	topological	ADJ
ejpam-4248	8	19	and	and	CCONJ
ejpam-4248	8	20	analytical	analytical	ADJ
ejpam-4248	8	21	concepts	concept	NOUN
ejpam-4248	8	22	.	.	PUNCT
ejpam-4248	9	1	moreover	moreover	ADV
ejpam-4248	9	2	the	the	DET
ejpam-4248	9	3	topologists	topologist	NOUN
ejpam-4248	9	4	gave	give	VERB
ejpam-4248	9	5	many	many	ADJ
ejpam-4248	9	6	different	different	ADJ
ejpam-4248	9	7	generalizations	generalization	NOUN
ejpam-4248	9	8	of	of	ADP
ejpam-4248	9	9	compactness	compactness	NOUN
ejpam-4248	9	10	stands	stand	VERB
ejpam-4248	9	11	on	on	ADP
ejpam-4248	9	12	the	the	DET
ejpam-4248	9	13	types	type	NOUN
ejpam-4248	9	14	of	of	ADP
ejpam-4248	9	15	covers	cover	NOUN
ejpam-4248	9	16	,	,	PUNCT
ejpam-4248	9	17	open	open	ADJ
ejpam-4248	9	18	sets	set	NOUN
ejpam-4248	9	19	and	and	CCONJ
ejpam-4248	9	20	subcovers	subcover	NOUN
ejpam-4248	9	21	.	.	PUNCT
ejpam-4248	10	1	the	the	DET
ejpam-4248	10	2	discussion	discussion	NOUN
ejpam-4248	10	3	about	about	ADP
ejpam-4248	10	4	these	these	DET
ejpam-4248	10	5	covering	cover	VERB
ejpam-4248	10	6	spaces	space	NOUN
ejpam-4248	10	7	is	be	AUX
ejpam-4248	10	8	still	still	ADV
ejpam-4248	10	9	a	a	DET
ejpam-4248	10	10	rich	rich	ADJ
ejpam-4248	10	11	area	area	NOUN
ejpam-4248	10	12	to	to	PART
ejpam-4248	10	13	study	study	VERB
ejpam-4248	10	14	in	in	ADP
ejpam-4248	10	15	topology	topology	NOUN
ejpam-4248	10	16	.	.	PUNCT
ejpam-4248	11	1	also	also	ADV
ejpam-4248	11	2	,	,	PUNCT
ejpam-4248	11	3	the	the	DET
ejpam-4248	11	4	idea	idea	NOUN
ejpam-4248	11	5	of	of	ADP
ejpam-4248	11	6	giving	give	VERB
ejpam-4248	11	7	weaker	weak	ADJ
ejpam-4248	11	8	and	and	CCONJ
ejpam-4248	11	9	stronger	strong	ADJ
ejpam-4248	11	10	forms	form	NOUN
ejpam-4248	11	11	of	of	ADP
ejpam-4248	11	12	open	open	ADJ
ejpam-4248	11	13	sets	set	NOUN
ejpam-4248	11	14	is	be	AUX
ejpam-4248	11	15	also	also	ADV
ejpam-4248	11	16	a	a	DET
ejpam-4248	11	17	hot	hot	ADJ
ejpam-4248	11	18	topic	topic	NOUN
ejpam-4248	11	19	in	in	ADP
ejpam-4248	11	20	research	research	NOUN
ejpam-4248	11	21	.	.	PUNCT
ejpam-4248	12	1	for	for	ADP
ejpam-4248	12	2	example	example	NOUN
ejpam-4248	12	3	semi	semi	ADJ
ejpam-4248	12	4	-	-	ADJ
ejpam-4248	12	5	open	open	ADJ
ejpam-4248	12	6	set	set	NOUN
ejpam-4248	12	7	[	[	X
ejpam-4248	12	8	9	9	NUM
ejpam-4248	12	9	]	]	PUNCT
ejpam-4248	12	10	,	,	PUNCT
ejpam-4248	12	11	pre	pre	ADJ
ejpam-4248	12	12	-	-	ADJ
ejpam-4248	12	13	open	open	ADJ
ejpam-4248	12	14	[	[	X
ejpam-4248	12	15	10	10	NUM
ejpam-4248	12	16	]	]	PUNCT
ejpam-4248	12	17	,	,	PUNCT
ejpam-4248	12	18	no	no	INTJ
ejpam-4248	12	19	where	where	SCONJ
ejpam-4248	12	20	dense	dense	ADJ
ejpam-4248	12	21	[	[	X
ejpam-4248	12	22	12	12	NUM
ejpam-4248	12	23	]	]	PUNCT
ejpam-4248	12	24	and	and	CCONJ
ejpam-4248	12	25	many	many	ADJ
ejpam-4248	12	26	more	more	ADJ
ejpam-4248	12	27	.	.	PUNCT
ejpam-4248	13	1	in	in	ADP
ejpam-4248	13	2	[	[	X
ejpam-4248	13	3	8	8	NUM
ejpam-4248	13	4	]	]	PUNCT
ejpam-4248	13	5	,	,	PUNCT
ejpam-4248	13	6	a	a	DET
ejpam-4248	13	7	regular	regular	ADJ
ejpam-4248	13	8	-	-	PUNCT
ejpam-4248	13	9	open	open	ADJ
ejpam-4248	13	10	set	set	NOUN
ejpam-4248	13	11	is	be	AUX
ejpam-4248	13	12	a	a	DET
ejpam-4248	13	13	set	set	NOUN
ejpam-4248	13	14	that	that	PRON
ejpam-4248	13	15	equals	equal	VERB
ejpam-4248	13	16	to	to	ADP
ejpam-4248	13	17	interior	interior	NOUN
ejpam-4248	13	18	of	of	ADP
ejpam-4248	13	19	its	its	PRON
ejpam-4248	13	20	closure	closure	NOUN
ejpam-4248	13	21	or	or	CCONJ
ejpam-4248	13	22	equivalently	equivalently	ADV
ejpam-4248	13	23	,	,	PUNCT
ejpam-4248	13	24	a	a	DET
ejpam-4248	13	25	subset	subset	NOUN
ejpam-4248	13	26	a	a	PRON
ejpam-4248	13	27	of	of	ADP
ejpam-4248	13	28	a	a	DET
ejpam-4248	13	29	topological	topological	ADJ
ejpam-4248	13	30	space	space	NOUN
ejpam-4248	13	31	x	x	PUNCT
ejpam-4248	13	32	is	be	AUX
ejpam-4248	13	33	regular	regular	ADJ
ejpam-4248	13	34	open	open	ADJ
ejpam-4248	13	35	if	if	SCONJ
ejpam-4248	13	36	int(a	int(a	PROPN
ejpam-4248	13	37	)	)	PUNCT
ejpam-4248	13	38	=	=	NOUN
ejpam-4248	14	1	a	a	PRON
ejpam-4248	14	2	where	where	SCONJ
ejpam-4248	14	3	the	the	DET
ejpam-4248	14	4	closure	closure	NOUN
ejpam-4248	14	5	of	of	ADP
ejpam-4248	14	6	a	a	PRON
ejpam-4248	14	7	and	and	CCONJ
ejpam-4248	14	8	the	the	DET
ejpam-4248	14	9	interior	interior	NOUN
ejpam-4248	14	10	of	of	ADP
ejpam-4248	14	11	a	a	PRON
ejpam-4248	14	12	are	be	AUX
ejpam-4248	14	13	denoted	denote	VERB
ejpam-4248	14	14	by	by	ADP
ejpam-4248	14	15	a	a	PRON
ejpam-4248	14	16	and	and	CCONJ
ejpam-4248	14	17	int(a	int(a	PROPN
ejpam-4248	14	18	)	)	PUNCT
ejpam-4248	14	19	,	,	PUNCT
ejpam-4248	14	20	respectively	respectively	ADV
ejpam-4248	14	21	.	.	PUNCT
ejpam-4248	15	1	the	the	DET
ejpam-4248	15	2	complement	complement	NOUN
ejpam-4248	15	3	of	of	ADP
ejpam-4248	15	4	a	a	DET
ejpam-4248	15	5	regular	regular	ADJ
ejpam-4248	15	6	open	open	NOUN
ejpam-4248	15	7	said	say	VERB
ejpam-4248	15	8	is	be	AUX
ejpam-4248	15	9	said	say	VERB
ejpam-4248	15	10	to	to	PART
ejpam-4248	15	11	be	be	AUX
ejpam-4248	15	12	regular	regular	ADV
ejpam-4248	15	13	closed	closed	ADJ
ejpam-4248	15	14	,	,	PUNCT
ejpam-4248	15	15	or	or	CCONJ
ejpam-4248	15	16	a	a	DET
ejpam-4248	15	17	=	=	SYM
ejpam-4248	15	18	int(a	int(a	NOUN
ejpam-4248	15	19	)	)	PUNCT
ejpam-4248	15	20	.	.	PUNCT
ejpam-4248	16	1	it	it	PRON
ejpam-4248	16	2	is	be	AUX
ejpam-4248	16	3	clearly	clearly	ADV
ejpam-4248	16	4	that	that	SCONJ
ejpam-4248	16	5	regularly	regularly	ADV
ejpam-4248	16	6	open	open	ADJ
ejpam-4248	16	7	is	be	AUX
ejpam-4248	16	8	an	an	DET
ejpam-4248	16	9	open	open	ADJ
ejpam-4248	16	10	set	set	NOUN
ejpam-4248	16	11	.	.	PUNCT
ejpam-4248	17	1	for	for	ADP
ejpam-4248	17	2	further	far	ADV
ejpam-4248	17	3	studied	study	VERB
ejpam-4248	17	4	in	in	ADP
ejpam-4248	17	5	regular	regular	ADJ
ejpam-4248	17	6	open	open	ADJ
ejpam-4248	17	7	spaces	space	NOUN
ejpam-4248	17	8	in	in	ADP
ejpam-4248	17	9	covering	cover	VERB
ejpam-4248	17	10	spaces	space	NOUN
ejpam-4248	17	11	and	and	CCONJ
ejpam-4248	17	12	separation	separation	NOUN
ejpam-4248	17	13	axioms	axiom	NOUN
ejpam-4248	17	14	,	,	PUNCT
ejpam-4248	17	15	see	see	VERB
ejpam-4248	17	16	[	[	X
ejpam-4248	17	17	1–5	1–5	X
ejpam-4248	17	18	]	]	X
ejpam-4248	17	19	.	.	PUNCT
ejpam-4248	18	1	in	in	ADP
ejpam-4248	18	2	2012	2012	NUM
ejpam-4248	18	3	,	,	PUNCT
ejpam-4248	18	4	al	al	PROPN
ejpam-4248	18	5	-	-	PUNCT
ejpam-4248	18	6	ghour	ghour	PROPN
ejpam-4248	18	7	and	and	CCONJ
ejpam-4248	18	8	samarah	samarah	NOUN
ejpam-4248	19	1	[	[	X
ejpam-4248	19	2	7	7	X
ejpam-4248	19	3	]	]	X
ejpam-4248	19	4	defined	define	VERB
ejpam-4248	19	5	a	a	DET
ejpam-4248	19	6	new	new	ADJ
ejpam-4248	19	7	type	type	NOUN
ejpam-4248	19	8	of	of	ADP
ejpam-4248	19	9	open	open	ADJ
ejpam-4248	19	10	sets	set	NOUN
ejpam-4248	19	11	called	call	VERB
ejpam-4248	19	12	coc	coc	NOUN
ejpam-4248	19	13	-	-	PUNCT
ejpam-4248	19	14	compact	compact	ADJ
ejpam-4248	19	15	,	,	PUNCT
ejpam-4248	19	16	a	a	DET
ejpam-4248	19	17	subset	subset	NOUN
ejpam-4248	19	18	a	a	PRON
ejpam-4248	19	19	of	of	ADP
ejpam-4248	19	20	a	a	DET
ejpam-4248	19	21	topological	topological	ADJ
ejpam-4248	19	22	space	space	NOUN
ejpam-4248	19	23	x	x	PUNCT
ejpam-4248	19	24	is	be	AUX
ejpam-4248	19	25	called	call	VERB
ejpam-4248	19	26	coc	coc	ADJ
ejpam-4248	19	27	-	-	PUNCT
ejpam-4248	19	28	open	open	NOUN
ejpam-4248	19	29	set	set	NOUN
ejpam-4248	19	30	if	if	SCONJ
ejpam-4248	19	31	a	a	PRON
ejpam-4248	19	32	is	be	AUX
ejpam-4248	19	33	a	a	DET
ejpam-4248	19	34	union	union	NOUN
ejpam-4248	19	35	of	of	ADP
ejpam-4248	19	36	sets	set	NOUN
ejpam-4248	19	37	of	of	ADP
ejpam-4248	19	38	the	the	DET
ejpam-4248	19	39	form	form	NOUN
ejpam-4248	19	40	v	v	ADP
ejpam-4248	19	41	−c	−c	NOUN
ejpam-4248	19	42	,	,	PUNCT
ejpam-4248	19	43	where	where	SCONJ
ejpam-4248	19	44	v	v	NOUN
ejpam-4248	19	45	is	be	AUX
ejpam-4248	19	46	open	open	ADJ
ejpam-4248	19	47	set	set	VERB
ejpam-4248	19	48	and	and	CCONJ
ejpam-4248	19	49	c	c	NOUN
ejpam-4248	19	50	is	be	AUX
ejpam-4248	19	51	a	a	DET
ejpam-4248	19	52	compact	compact	ADJ
ejpam-4248	19	53	subset	subset	NOUN
ejpam-4248	19	54	of	of	ADP
ejpam-4248	19	55	x.	x.	NOUN
ejpam-4248	19	56	also	also	ADV
ejpam-4248	19	57	,	,	PUNCT
ejpam-4248	19	58	they	they	PRON
ejpam-4248	19	59	proved	prove	VERB
ejpam-4248	19	60	that	that	SCONJ
ejpam-4248	19	61	the	the	DET
ejpam-4248	19	62	family	family	NOUN
ejpam-4248	19	63	of	of	ADP
ejpam-4248	19	64	all	all	DET
ejpam-4248	19	65	coc	coc	ADJ
ejpam-4248	19	66	-	-	PUNCT
ejpam-4248	19	67	open	open	ADJ
ejpam-4248	19	68	sets	set	NOUN
ejpam-4248	19	69	of	of	ADP
ejpam-4248	19	70	a	a	DET
ejpam-4248	19	71	topological	topological	ADJ
ejpam-4248	19	72	space	space	NOUN
ejpam-4248	19	73	(	(	PUNCT
ejpam-4248	19	74	x	x	X
ejpam-4248	19	75	,	,	PUNCT
ejpam-4248	19	76	τ	τ	X
ejpam-4248	19	77	)	)	PUNCT
ejpam-4248	19	78	forms	form	VERB
ejpam-4248	19	79	a	a	DET
ejpam-4248	19	80	topology	topology	NOUN
ejpam-4248	19	81	on	on	ADP
ejpam-4248	19	82	x	x	SYM
ejpam-4248	19	83	finer	fine	ADJ
ejpam-4248	19	84	than	than	ADP
ejpam-4248	19	85	τ	τ	PROPN
ejpam-4248	19	86	.	.	PUNCT
ejpam-4248	20	1	for	for	ADP
ejpam-4248	20	2	more	more	ADJ
ejpam-4248	20	3	results	result	NOUN
ejpam-4248	20	4	about	about	ADP
ejpam-4248	20	5	coc	coc	NOUN
ejpam-4248	20	6	-	-	PUNCT
ejpam-4248	20	7	open	open	ADJ
ejpam-4248	20	8	subsets	subset	NOUN
ejpam-4248	20	9	,	,	PUNCT
ejpam-4248	20	10	see	see	VERB
ejpam-4248	20	11	[	[	X
ejpam-4248	20	12	11	11	NUM
ejpam-4248	20	13	]	]	PUNCT
ejpam-4248	20	14	.	.	PUNCT
ejpam-4248	21	1	∗corresponding	∗corresponde	VERB
ejpam-4248	21	2	author	author	NOUN
ejpam-4248	21	3	.	.	PUNCT
ejpam-4248	22	1	doi	doi	NOUN
ejpam-4248	22	2	:	:	PUNCT
ejpam-4248	22	3	https://doi.org/10.29020/nybg.ejpam.v15i1.4248	https://doi.org/10.29020/nybg.ejpam.v15i1.4248	NUM
ejpam-4248	22	4	email	email	NOUN
ejpam-4248	22	5	addresses	address	NOUN
ejpam-4248	22	6	:	:	PUNCT
ejpam-4248	22	7	fshaheen@meu.edu.jo	fshaheen@meu.edu.jo	ADJ
ejpam-4248	22	8	(	(	PUNCT
ejpam-4248	22	9	f.a	f.a	PROPN
ejpam-4248	22	10	.	.	PROPN
ejpam-4248	22	11	abushaheen	abushaheen	PROPN
ejpam-4248	22	12	)	)	PUNCT
ejpam-4248	22	13	,	,	PUNCT
ejpam-4248	22	14	f.rimawi@ammanu.edu.jo	f.rimawi@ammanu.edu.jo	NOUN
ejpam-4248	22	15	(	(	PUNCT
ejpam-4248	22	16	f.	f.	PROPN
ejpam-4248	22	17	alrimawi	alrimawi	PROPN
ejpam-4248	22	18	)	)	PUNCT
ejpam-4248	22	19	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-4248	23	1	199	199	NUM
ejpam-4248	23	2	©	©	PROPN
ejpam-4248	23	3	2022	2022	NUM
ejpam-4248	23	4	ejpam	ejpam	VERB
ejpam-4248	23	5	all	all	DET
ejpam-4248	23	6	rights	right	NOUN
ejpam-4248	23	7	reserved	reserve	VERB
ejpam-4248	23	8	.	.	PUNCT
ejpam-4248	24	1	f.a	f.a	PROPN
ejpam-4248	24	2	.	.	PROPN
ejpam-4248	24	3	abushaheen	abushaheen	PROPN
ejpam-4248	24	4	,	,	PUNCT
ejpam-4248	24	5	f.	f.	PROPN
ejpam-4248	24	6	alrimawi	alrimawi	PROPN
ejpam-4248	24	7	/	/	SYM
ejpam-4248	24	8	eur	eur	PROPN
ejpam-4248	24	9	.	.	PUNCT
ejpam-4248	25	1	j.	j.	PROPN
ejpam-4248	25	2	pure	pure	PROPN
ejpam-4248	25	3	appl	appl	PROPN
ejpam-4248	25	4	.	.	PROPN
ejpam-4248	25	5	math	math	PROPN
ejpam-4248	25	6	,	,	PUNCT
ejpam-4248	25	7	15	15	NUM
ejpam-4248	25	8	(	(	PUNCT
ejpam-4248	25	9	1	1	NUM
ejpam-4248	25	10	)	)	PUNCT
ejpam-4248	25	11	(	(	PUNCT
ejpam-4248	25	12	2022	2022	NUM
ejpam-4248	25	13	)	)	PUNCT
ejpam-4248	25	14	,	,	PUNCT
ejpam-4248	25	15	199	199	NUM
ejpam-4248	25	16	-	-	SYM
ejpam-4248	25	17	206	206	NUM
ejpam-4248	25	18	200	200	NUM
ejpam-4248	25	19	in	in	ADP
ejpam-4248	25	20	this	this	DET
ejpam-4248	25	21	paper	paper	NOUN
ejpam-4248	25	22	,	,	PUNCT
ejpam-4248	25	23	no	no	DET
ejpam-4248	25	24	separation	separation	NOUN
ejpam-4248	25	25	axioms	axiom	VERB
ejpam-4248	25	26	to	to	PART
ejpam-4248	25	27	be	be	AUX
ejpam-4248	25	28	assumed	assume	VERB
ejpam-4248	25	29	.	.	PUNCT
ejpam-4248	26	1	a	a	DET
ejpam-4248	26	2	space	space	NOUN
ejpam-4248	26	3	x	x	PUNCT
ejpam-4248	26	4	means	mean	VERB
ejpam-4248	26	5	nonempty	nonempty	ADJ
ejpam-4248	26	6	sets	set	NOUN
ejpam-4248	26	7	with	with	ADP
ejpam-4248	26	8	topology	topology	NOUN
ejpam-4248	26	9	.	.	PUNCT
ejpam-4248	27	1	ω0	ω0	PROPN
ejpam-4248	27	2	denoted	denote	VERB
ejpam-4248	27	3	the	the	DET
ejpam-4248	27	4	cardinal	cardinal	ADJ
ejpam-4248	27	5	number	number	NOUN
ejpam-4248	27	6	of	of	ADP
ejpam-4248	27	7	n	n	CCONJ
ejpam-4248	27	8	,	,	PUNCT
ejpam-4248	27	9	for	for	ADP
ejpam-4248	27	10	a	a	DET
ejpam-4248	27	11	subset	subset	NOUN
ejpam-4248	27	12	a	a	PRON
ejpam-4248	27	13	of	of	ADP
ejpam-4248	27	14	x	x	PRON
ejpam-4248	27	15	,	,	PUNCT
ejpam-4248	27	16	a	a	DET
ejpam-4248	27	17	coc	coc	NOUN
ejpam-4248	27	18	and	and	CCONJ
ejpam-4248	27	19	intcoc(a	intcoc(a	NOUN
ejpam-4248	27	20	)	)	PUNCT
ejpam-4248	27	21	will	will	AUX
ejpam-4248	27	22	denote	denote	VERB
ejpam-4248	27	23	the	the	DET
ejpam-4248	27	24	closure	closure	NOUN
ejpam-4248	27	25	of	of	ADP
ejpam-4248	27	26	a	a	PRON
ejpam-4248	27	27	and	and	CCONJ
ejpam-4248	27	28	the	the	DET
ejpam-4248	27	29	interior	interior	NOUN
ejpam-4248	27	30	of	of	ADP
ejpam-4248	27	31	a	a	PRON
ejpam-4248	27	32	in	in	ADP
ejpam-4248	27	33	τk	τk	ADP
ejpam-4248	27	34	,	,	PUNCT
ejpam-4248	27	35	respectively	respectively	ADV
ejpam-4248	27	36	.	.	PUNCT
ejpam-4248	28	1	for	for	ADP
ejpam-4248	28	2	notation	notation	NOUN
ejpam-4248	28	3	not	not	PART
ejpam-4248	28	4	mention	mention	VERB
ejpam-4248	28	5	here	here	ADV
ejpam-4248	28	6	,	,	PUNCT
ejpam-4248	28	7	we	we	PRON
ejpam-4248	28	8	refer	refer	VERB
ejpam-4248	28	9	the	the	DET
ejpam-4248	28	10	reader	reader	NOUN
ejpam-4248	28	11	to	to	ADP
ejpam-4248	28	12	[	[	PUNCT
ejpam-4248	28	13	6	6	NUM
ejpam-4248	28	14	]	]	PUNCT
ejpam-4248	28	15	.	.	PUNCT
ejpam-4248	29	1	the	the	DET
ejpam-4248	29	2	following	follow	VERB
ejpam-4248	29	3	definitions	definition	NOUN
ejpam-4248	29	4	are	be	AUX
ejpam-4248	29	5	important	important	ADJ
ejpam-4248	29	6	for	for	ADP
ejpam-4248	29	7	our	our	PRON
ejpam-4248	29	8	paper	paper	NOUN
ejpam-4248	29	9	.	.	PUNCT
ejpam-4248	30	1	definition	definition	NOUN
ejpam-4248	30	2	1	1	NUM
ejpam-4248	30	3	.	.	PUNCT
ejpam-4248	31	1	[	[	X
ejpam-4248	31	2	7	7	X
ejpam-4248	31	3	]	]	X
ejpam-4248	31	4	a	a	DET
ejpam-4248	31	5	subset	subset	NOUN
ejpam-4248	31	6	a	a	PRON
ejpam-4248	31	7	of	of	ADP
ejpam-4248	31	8	a	a	DET
ejpam-4248	31	9	topological	topological	ADJ
ejpam-4248	31	10	space	space	NOUN
ejpam-4248	31	11	x	x	PRON
ejpam-4248	31	12	is	be	AUX
ejpam-4248	31	13	called	call	VERB
ejpam-4248	31	14	co	co	ADJ
ejpam-4248	31	15	-	-	ADJ
ejpam-4248	31	16	compact	compact	ADJ
ejpam-4248	31	17	open	open	ADJ
ejpam-4248	31	18	set	set	NOUN
ejpam-4248	31	19	(	(	PUNCT
ejpam-4248	31	20	notation	notation	NOUN
ejpam-4248	31	21	:	:	PUNCT
ejpam-4248	31	22	coc	coc	NOUN
ejpam-4248	31	23	-	-	PUNCT
ejpam-4248	31	24	open	open	ADJ
ejpam-4248	31	25	)	)	PUNCT
ejpam-4248	31	26	if	if	SCONJ
ejpam-4248	31	27	for	for	ADP
ejpam-4248	31	28	every	every	DET
ejpam-4248	31	29	x	x	PROPN
ejpam-4248	31	30	∈	∈	PROPN
ejpam-4248	31	31	a	a	PRON
ejpam-4248	31	32	,	,	PUNCT
ejpam-4248	31	33	there	there	PRON
ejpam-4248	31	34	exists	exist	VERB
ejpam-4248	31	35	an	an	DET
ejpam-4248	31	36	open	open	ADJ
ejpam-4248	31	37	set	set	NOUN
ejpam-4248	31	38	u	u	NOUN
ejpam-4248	31	39	⊆	⊆	NUM
ejpam-4248	31	40	x	x	PUNCT
ejpam-4248	31	41	and	and	CCONJ
ejpam-4248	31	42	a	a	DET
ejpam-4248	31	43	compact	compact	ADJ
ejpam-4248	31	44	subset	subset	NOUN
ejpam-4248	31	45	k	k	PROPN
ejpam-4248	31	46	of	of	ADP
ejpam-4248	31	47	x	x	INTJ
ejpam-4248	31	48	such	such	ADJ
ejpam-4248	31	49	that	that	SCONJ
ejpam-4248	31	50	x	x	SYM
ejpam-4248	31	51	∈	∈	NOUN
ejpam-4248	31	52	u	u	NOUN
ejpam-4248	31	53	−k	−k	NOUN
ejpam-4248	31	54	⊆	⊆	NUM
ejpam-4248	31	55	a.	a.	NOUN
ejpam-4248	31	56	the	the	DET
ejpam-4248	31	57	complement	complement	NOUN
ejpam-4248	31	58	of	of	ADP
ejpam-4248	31	59	a	a	DET
ejpam-4248	31	60	coc	coc	NOUN
ejpam-4248	31	61	-	-	PUNCT
ejpam-4248	31	62	open	open	ADJ
ejpam-4248	31	63	subset	subset	NOUN
ejpam-4248	31	64	is	be	AUX
ejpam-4248	31	65	called	call	VERB
ejpam-4248	31	66	coc	coc	NOUN
ejpam-4248	31	67	-	-	PUNCT
ejpam-4248	31	68	closed	closed	ADJ
ejpam-4248	31	69	.	.	PUNCT
ejpam-4248	32	1	the	the	DET
ejpam-4248	32	2	family	family	NOUN
ejpam-4248	32	3	of	of	ADP
ejpam-4248	32	4	all	all	DET
ejpam-4248	32	5	coc	coc	ADJ
ejpam-4248	32	6	-	-	PUNCT
ejpam-4248	32	7	open	open	ADJ
ejpam-4248	32	8	subsets	subset	NOUN
ejpam-4248	32	9	of	of	ADP
ejpam-4248	32	10	a	a	DET
ejpam-4248	32	11	topological	topological	ADJ
ejpam-4248	32	12	space	space	NOUN
ejpam-4248	32	13	(	(	PUNCT
ejpam-4248	32	14	x	x	X
ejpam-4248	32	15	,	,	PUNCT
ejpam-4248	32	16	τ	τ	X
ejpam-4248	32	17	)	)	PUNCT
ejpam-4248	32	18	will	will	AUX
ejpam-4248	32	19	be	be	AUX
ejpam-4248	32	20	denoted	denote	VERB
ejpam-4248	32	21	by	by	ADP
ejpam-4248	32	22	τk	τk	ADP
ejpam-4248	32	23	.	.	PUNCT
ejpam-4248	32	24	definition	definition	NOUN
ejpam-4248	32	25	2	2	NUM
ejpam-4248	32	26	.	.	PUNCT
ejpam-4248	33	1	[	[	X
ejpam-4248	33	2	11	11	NUM
ejpam-4248	33	3	]	]	PUNCT
ejpam-4248	33	4	a	a	DET
ejpam-4248	33	5	topological	topological	ADJ
ejpam-4248	33	6	space	space	NOUN
ejpam-4248	33	7	x	x	PUNCT
ejpam-4248	33	8	is	be	AUX
ejpam-4248	33	9	called	call	VERB
ejpam-4248	33	10	coc	coc	PROPN
ejpam-4248	33	11	-	-	PUNCT
ejpam-4248	33	12	t2	t2	NOUN
ejpam-4248	33	13	-	-	PUNCT
ejpam-4248	33	14	space	space	NOUN
ejpam-4248	33	15	if	if	SCONJ
ejpam-4248	33	16	and	and	CCONJ
ejpam-4248	33	17	only	only	ADV
ejpam-4248	33	18	if	if	SCONJ
ejpam-4248	33	19	for	for	ADP
ejpam-4248	33	20	all	all	DET
ejpam-4248	33	21	x	x	NOUN
ejpam-4248	33	22	,	,	PUNCT
ejpam-4248	33	23	y	y	PROPN
ejpam-4248	33	24	∈	∈	PROPN
ejpam-4248	33	25	x	x	PUNCT
ejpam-4248	33	26	with	with	ADP
ejpam-4248	33	27	x	x	X
ejpam-4248	33	28	̸=	̸=	PROPN
ejpam-4248	33	29	y	y	PROPN
ejpam-4248	33	30	∈	∈	PROPN
ejpam-4248	34	1	x	x	PRON
ejpam-4248	34	2	,	,	PUNCT
ejpam-4248	34	3	there	there	PRON
ejpam-4248	34	4	exist	exist	VERB
ejpam-4248	34	5	u	u	NOUN
ejpam-4248	34	6	,	,	PUNCT
ejpam-4248	34	7	v	v	PROPN
ejpam-4248	34	8	∈	∈	NOUN
ejpam-4248	34	9	τk	τk	ADP
ejpam-4248	34	10	such	such	ADJ
ejpam-4248	34	11	that	that	SCONJ
ejpam-4248	34	12	x	x	SYM
ejpam-4248	34	13	∈	∈	PROPN
ejpam-4248	34	14	u	u	NOUN
ejpam-4248	34	15	,	,	PUNCT
ejpam-4248	34	16	y	y	PROPN
ejpam-4248	34	17	∈	∈	PROPN
ejpam-4248	34	18	v	v	NOUN
ejpam-4248	34	19	with	with	ADP
ejpam-4248	34	20	u	u	NOUN
ejpam-4248	34	21	∩	∩	NOUN
ejpam-4248	34	22	v	v	NOUN
ejpam-4248	34	23	=	=	SYM
ejpam-4248	34	24	ϕ.	ϕ.	ADJ
ejpam-4248	34	25	definition	definition	NOUN
ejpam-4248	34	26	3	3	X
ejpam-4248	34	27	.	.	PUNCT
ejpam-4248	35	1	a	a	DET
ejpam-4248	35	2	subset	subset	NOUN
ejpam-4248	35	3	a	a	PRON
ejpam-4248	35	4	of	of	ADP
ejpam-4248	35	5	a	a	DET
ejpam-4248	35	6	topological	topological	ADJ
ejpam-4248	35	7	space	space	NOUN
ejpam-4248	35	8	x	x	PUNCT
ejpam-4248	35	9	is	be	AUX
ejpam-4248	35	10	coc	coc	ADJ
ejpam-4248	35	11	-	-	PUNCT
ejpam-4248	35	12	regular	regular	ADV
ejpam-4248	35	13	open	open	ADJ
ejpam-4248	35	14	if	if	SCONJ
ejpam-4248	35	15	intcoc(a	intcoc(a	NOUN
ejpam-4248	35	16	coc	coc	NOUN
ejpam-4248	35	17	)	)	PUNCT
ejpam-4248	36	1	=	=	SYM
ejpam-4248	36	2	a	a	PRON
ejpam-4248	36	3	,	,	PUNCT
ejpam-4248	36	4	the	the	DET
ejpam-4248	36	5	complement	complement	NOUN
ejpam-4248	36	6	of	of	ADP
ejpam-4248	36	7	a	a	DET
ejpam-4248	36	8	coc	coc	NOUN
ejpam-4248	36	9	-	-	PUNCT
ejpam-4248	36	10	regular	regular	ADJ
ejpam-4248	36	11	open	open	NOUN
ejpam-4248	36	12	is	be	AUX
ejpam-4248	36	13	said	say	VERB
ejpam-4248	36	14	to	to	PART
ejpam-4248	36	15	be	be	AUX
ejpam-4248	36	16	coc	coc	ADJ
ejpam-4248	36	17	-	-	PUNCT
ejpam-4248	36	18	regular	regular	ADJ
ejpam-4248	36	19	closed	close	VERB
ejpam-4248	36	20	,	,	PUNCT
ejpam-4248	36	21	or	or	CCONJ
ejpam-4248	36	22	a	a	DET
ejpam-4248	36	23	=	=	PUNCT
ejpam-4248	36	24	intcoc(a	intcoc(a	NOUN
ejpam-4248	36	25	)	)	PUNCT
ejpam-4248	36	26	coc	coc	NOUN
ejpam-4248	36	27	.	.	PUNCT
ejpam-4248	37	1	definition	definition	NOUN
ejpam-4248	37	2	4	4	NUM
ejpam-4248	37	3	.	.	PUNCT
ejpam-4248	38	1	[	[	X
ejpam-4248	38	2	7	7	X
ejpam-4248	38	3	]	]	X
ejpam-4248	38	4	a	a	DET
ejpam-4248	38	5	family	family	NOUN
ejpam-4248	38	6	u	u	NOUN
ejpam-4248	38	7	⊆	⊆	NUM
ejpam-4248	38	8	x	x	X
ejpam-4248	38	9	is	be	AUX
ejpam-4248	38	10	coc	coc	ADJ
ejpam-4248	38	11	-	-	PUNCT
ejpam-4248	38	12	open	open	ADJ
ejpam-4248	38	13	cover	cover	NOUN
ejpam-4248	38	14	of	of	ADP
ejpam-4248	38	15	x	x	PRON
ejpam-4248	38	16	if	if	SCONJ
ejpam-4248	38	17	u	u	PRON
ejpam-4248	38	18	covers	cover	VERB
ejpam-4248	38	19	x	x	PUNCT
ejpam-4248	38	20	and	and	CCONJ
ejpam-4248	38	21	u	u	NOUN
ejpam-4248	38	22	is	be	AUX
ejpam-4248	38	23	a	a	DET
ejpam-4248	38	24	subfamily	subfamily	NOUN
ejpam-4248	38	25	of	of	ADP
ejpam-4248	38	26	τk	τk	ADP
ejpam-4248	38	27	.	.	PUNCT
ejpam-4248	38	28	definition	definition	NOUN
ejpam-4248	38	29	5	5	NUM
ejpam-4248	38	30	.	.	PUNCT
ejpam-4248	39	1	a	a	DET
ejpam-4248	39	2	coc	coc	NOUN
ejpam-4248	39	3	-	-	PUNCT
ejpam-4248	39	4	open	open	ADJ
ejpam-4248	39	5	cover	cover	NOUN
ejpam-4248	39	6	u	u	NOUN
ejpam-4248	39	7	=	=	PUNCT
ejpam-4248	39	8	{	{	PUNCT
ejpam-4248	39	9	uα|α	uα|α	PROPN
ejpam-4248	39	10	∈	∈	PROPN
ejpam-4248	39	11	∆	∆	NOUN
ejpam-4248	39	12	}	}	PUNCT
ejpam-4248	39	13	of	of	ADP
ejpam-4248	39	14	x	x	SYM
ejpam-4248	39	15	is	be	AUX
ejpam-4248	39	16	called	call	VERB
ejpam-4248	39	17	coc	coc	ADJ
ejpam-4248	39	18	-	-	PUNCT
ejpam-4248	39	19	regular	regular	ADJ
ejpam-4248	39	20	cover	cover	NOUN
ejpam-4248	39	21	,	,	PUNCT
ejpam-4248	39	22	if	if	SCONJ
ejpam-4248	39	23	for	for	ADP
ejpam-4248	39	24	each	each	DET
ejpam-4248	39	25	α	α	PRON
ejpam-4248	39	26	∈	∈	NOUN
ejpam-4248	39	27	∆	∆	NOUN
ejpam-4248	39	28	there	there	PRON
ejpam-4248	39	29	exists	exist	VERB
ejpam-4248	39	30	a	a	DET
ejpam-4248	39	31	coc	coc	ADJ
ejpam-4248	39	32	-	-	PUNCT
ejpam-4248	39	33	regular	regular	ADJ
ejpam-4248	39	34	closed	close	VERB
ejpam-4248	39	35	set	set	NOUN
ejpam-4248	39	36	fα	fα	ADP
ejpam-4248	39	37	such	such	ADJ
ejpam-4248	39	38	that	that	DET
ejpam-4248	39	39	fα	fα	ADP
ejpam-4248	39	40	⊆	⊆	NUM
ejpam-4248	39	41	uα	uα	NOUN
ejpam-4248	39	42	and	and	CCONJ
ejpam-4248	39	43	x	x	PUNCT
ejpam-4248	39	44	=	=	NOUN
ejpam-4248	39	45	⋃	⋃	NOUN
ejpam-4248	39	46	{	{	PUNCT
ejpam-4248	39	47	intcoc(fα)|α	intcoc(fα)|α	NOUN
ejpam-4248	39	48	∈	∈	PROPN
ejpam-4248	39	49	∆	∆	X
ejpam-4248	39	50	}	}	PUNCT
ejpam-4248	39	51	.	.	PUNCT
ejpam-4248	40	1	definition	definition	NOUN
ejpam-4248	40	2	6	6	NUM
ejpam-4248	40	3	.	.	PUNCT
ejpam-4248	41	1	[	[	X
ejpam-4248	41	2	7	7	X
ejpam-4248	41	3	]	]	PUNCT
ejpam-4248	41	4	a	a	DET
ejpam-4248	41	5	space	space	NOUN
ejpam-4248	41	6	(	(	PUNCT
ejpam-4248	41	7	x	x	X
ejpam-4248	41	8	,	,	PUNCT
ejpam-4248	41	9	τ	τ	X
ejpam-4248	41	10	)	)	PUNCT
ejpam-4248	41	11	is	be	AUX
ejpam-4248	41	12	coc	coc	ADJ
ejpam-4248	41	13	-	-	ADJ
ejpam-4248	41	14	compact	compact	ADJ
ejpam-4248	41	15	if	if	SCONJ
ejpam-4248	41	16	every	every	DET
ejpam-4248	41	17	coc	coc	NOUN
ejpam-4248	41	18	-	-	PUNCT
ejpam-4248	41	19	open	open	ADJ
ejpam-4248	41	20	cover	cover	NOUN
ejpam-4248	41	21	has	have	VERB
ejpam-4248	41	22	a	a	DET
ejpam-4248	41	23	finite	finite	ADJ
ejpam-4248	41	24	subcover	subcover	PROPN
ejpam-4248	41	25	.	.	PUNCT
ejpam-4248	42	1	definition	definition	NOUN
ejpam-4248	42	2	7	7	NUM
ejpam-4248	42	3	.	.	PUNCT
ejpam-4248	43	1	a	a	DET
ejpam-4248	43	2	space	space	NOUN
ejpam-4248	43	3	(	(	PUNCT
ejpam-4248	43	4	x	x	X
ejpam-4248	43	5	,	,	PUNCT
ejpam-4248	43	6	τ	τ	X
ejpam-4248	43	7	)	)	PUNCT
ejpam-4248	43	8	is	be	AUX
ejpam-4248	43	9	coc	coc	ADJ
ejpam-4248	43	10	-	-	PUNCT
ejpam-4248	43	11	almost	almost	ADV
ejpam-4248	43	12	-	-	PUNCT
ejpam-4248	43	13	compact	compact	ADJ
ejpam-4248	43	14	if	if	SCONJ
ejpam-4248	43	15	every	every	DET
ejpam-4248	43	16	coc	coc	NOUN
ejpam-4248	43	17	-	-	PUNCT
ejpam-4248	43	18	open	open	ADJ
ejpam-4248	43	19	cover	cover	NOUN
ejpam-4248	43	20	of	of	ADP
ejpam-4248	43	21	x	x	PUNCT
ejpam-4248	43	22	has	have	VERB
ejpam-4248	43	23	a	a	DET
ejpam-4248	43	24	finite	finite	ADJ
ejpam-4248	43	25	collection	collection	NOUN
ejpam-4248	43	26	such	such	ADJ
ejpam-4248	43	27	that	that	SCONJ
ejpam-4248	43	28	the	the	DET
ejpam-4248	43	29	coc	coc	NOUN
ejpam-4248	43	30	-	-	PUNCT
ejpam-4248	43	31	closure	closure	NOUN
ejpam-4248	43	32	of	of	ADP
ejpam-4248	43	33	the	the	DET
ejpam-4248	43	34	union	union	NOUN
ejpam-4248	43	35	is	be	AUX
ejpam-4248	43	36	x.	x.	NOUN
ejpam-4248	43	37	definition	definition	NOUN
ejpam-4248	43	38	8	8	NUM
ejpam-4248	43	39	.	.	PUNCT
ejpam-4248	44	1	a	a	DET
ejpam-4248	44	2	space	space	NOUN
ejpam-4248	44	3	(	(	PUNCT
ejpam-4248	44	4	x	x	X
ejpam-4248	44	5	,	,	PUNCT
ejpam-4248	44	6	τ	τ	X
ejpam-4248	44	7	)	)	PUNCT
ejpam-4248	44	8	is	be	AUX
ejpam-4248	44	9	coc	coc	ADJ
ejpam-4248	44	10	-	-	PUNCT
ejpam-4248	44	11	weakly	weakly	ADJ
ejpam-4248	44	12	-	-	PUNCT
ejpam-4248	44	13	compact	compact	ADJ
ejpam-4248	44	14	if	if	SCONJ
ejpam-4248	44	15	every	every	DET
ejpam-4248	44	16	coc	coc	NOUN
ejpam-4248	44	17	-	-	PUNCT
ejpam-4248	44	18	regular	regular	ADJ
ejpam-4248	44	19	cover	cover	NOUN
ejpam-4248	44	20	of	of	ADP
ejpam-4248	44	21	x	x	PUNCT
ejpam-4248	44	22	has	have	VERB
ejpam-4248	44	23	a	a	DET
ejpam-4248	44	24	finite	finite	ADJ
ejpam-4248	44	25	collection	collection	NOUN
ejpam-4248	44	26	such	such	ADJ
ejpam-4248	44	27	that	that	SCONJ
ejpam-4248	44	28	the	the	DET
ejpam-4248	44	29	coc	coc	NOUN
ejpam-4248	44	30	-	-	PUNCT
ejpam-4248	44	31	closure	closure	NOUN
ejpam-4248	44	32	of	of	ADP
ejpam-4248	44	33	the	the	DET
ejpam-4248	44	34	union	union	NOUN
ejpam-4248	44	35	is	be	AUX
ejpam-4248	44	36	x.	x.	NOUN
ejpam-4248	44	37	definition	definition	NOUN
ejpam-4248	44	38	9	9	NUM
ejpam-4248	44	39	.	.	PUNCT
ejpam-4248	45	1	a	a	DET
ejpam-4248	45	2	space	space	NOUN
ejpam-4248	45	3	(	(	PUNCT
ejpam-4248	45	4	x	x	X
ejpam-4248	45	5	,	,	PUNCT
ejpam-4248	45	6	τ	τ	X
ejpam-4248	45	7	)	)	PUNCT
ejpam-4248	45	8	is	be	AUX
ejpam-4248	45	9	coc	coc	ADJ
ejpam-4248	45	10	-	-	PUNCT
ejpam-4248	45	11	nearly	nearly	ADV
ejpam-4248	45	12	-	-	PUNCT
ejpam-4248	45	13	compact	compact	ADJ
ejpam-4248	45	14	if	if	SCONJ
ejpam-4248	45	15	every	every	DET
ejpam-4248	45	16	coc	coc	NOUN
ejpam-4248	45	17	-	-	PUNCT
ejpam-4248	45	18	open	open	ADJ
ejpam-4248	45	19	cover	cover	NOUN
ejpam-4248	45	20	of	of	ADP
ejpam-4248	45	21	x	x	PUNCT
ejpam-4248	45	22	by	by	ADP
ejpam-4248	45	23	cocregular	cocregular	ADJ
ejpam-4248	45	24	open	open	ADJ
ejpam-4248	45	25	sets	set	NOUN
ejpam-4248	45	26	has	have	VERB
ejpam-4248	45	27	a	a	DET
ejpam-4248	45	28	finite	finite	ADJ
ejpam-4248	45	29	subcover	subcover	PROPN
ejpam-4248	45	30	.	.	PUNCT
ejpam-4248	46	1	clearly	clearly	ADV
ejpam-4248	46	2	the	the	DET
ejpam-4248	46	3	following	follow	VERB
ejpam-4248	46	4	implications	implication	NOUN
ejpam-4248	46	5	are	be	AUX
ejpam-4248	46	6	true	true	ADJ
ejpam-4248	46	7	for	for	ADP
ejpam-4248	46	8	a	a	DET
ejpam-4248	46	9	topological	topological	ADJ
ejpam-4248	46	10	space	space	NOUN
ejpam-4248	46	11	x	x	NOUN
ejpam-4248	46	12	:	:	PUNCT
ejpam-4248	46	13	coc	coc	ADJ
ejpam-4248	46	14	-	-	PUNCT
ejpam-4248	46	15	compact	compact	ADJ
ejpam-4248	46	16	⇒	⇒	NOUN
ejpam-4248	46	17	coc	coc	PROPN
ejpam-4248	46	18	-	-	PUNCT
ejpam-4248	46	19	nearly	nearly	ADV
ejpam-4248	46	20	compact	compact	ADJ
ejpam-4248	46	21	⇒	⇒	NOUN
ejpam-4248	46	22	coc	coc	VERB
ejpam-4248	46	23	-	-	PUNCT
ejpam-4248	46	24	almost	almost	ADV
ejpam-4248	46	25	compact	compact	ADJ
ejpam-4248	46	26	⇒	⇒	NOUN
ejpam-4248	46	27	coc	coc	ADJ
ejpam-4248	46	28	-	-	PUNCT
ejpam-4248	46	29	weakly	weakly	ADJ
ejpam-4248	46	30	compact	compact	ADJ
ejpam-4248	46	31	.	.	PUNCT
ejpam-4248	47	1	the	the	DET
ejpam-4248	47	2	following	follow	VERB
ejpam-4248	47	3	definition	definition	NOUN
ejpam-4248	47	4	gives	give	VERB
ejpam-4248	47	5	a	a	DET
ejpam-4248	47	6	coc	coc	NOUN
ejpam-4248	47	7	-	-	PUNCT
ejpam-4248	47	8	covering	cover	VERB
ejpam-4248	47	9	space	space	NOUN
ejpam-4248	47	10	between	between	ADP
ejpam-4248	47	11	coc	coc	NOUN
ejpam-4248	47	12	-	-	PUNCT
ejpam-4248	47	13	almost	almost	ADV
ejpam-4248	47	14	-	-	PUNCT
ejpam-4248	47	15	compact	compact	ADJ
ejpam-4248	47	16	and	and	CCONJ
ejpam-4248	47	17	coc	coc	ADJ
ejpam-4248	47	18	-	-	PUNCT
ejpam-4248	47	19	weakly	weakly	ADJ
ejpam-4248	47	20	-	-	PUNCT
ejpam-4248	47	21	compact	compact	ADJ
ejpam-4248	47	22	spaces	space	NOUN
ejpam-4248	47	23	.	.	PUNCT
ejpam-4248	48	1	definition	definition	NOUN
ejpam-4248	48	2	10	10	NUM
ejpam-4248	48	3	.	.	PUNCT
ejpam-4248	49	1	a	a	DET
ejpam-4248	49	2	space	space	NOUN
ejpam-4248	49	3	x	x	PUNCT
ejpam-4248	49	4	is	be	AUX
ejpam-4248	49	5	called	call	VERB
ejpam-4248	49	6	coc	coc	PROPN
ejpam-4248	49	7	-	-	PUNCT
ejpam-4248	49	8	r	r	NOUN
ejpam-4248	49	9	-	-	PUNCT
ejpam-4248	49	10	compact	compact	ADJ
ejpam-4248	49	11	if	if	SCONJ
ejpam-4248	49	12	every	every	DET
ejpam-4248	49	13	coc	coc	NOUN
ejpam-4248	49	14	-	-	PUNCT
ejpam-4248	49	15	regular	regular	ADJ
ejpam-4248	49	16	cover	cover	NOUN
ejpam-4248	49	17	of	of	ADP
ejpam-4248	49	18	x	x	PUNCT
ejpam-4248	49	19	has	have	VERB
ejpam-4248	49	20	a	a	DET
ejpam-4248	49	21	finite	finite	ADJ
ejpam-4248	49	22	subcover	subcover	PROPN
ejpam-4248	49	23	.	.	PUNCT
ejpam-4248	50	1	it	it	PRON
ejpam-4248	50	2	is	be	AUX
ejpam-4248	50	3	obvious	obvious	ADJ
ejpam-4248	50	4	that	that	SCONJ
ejpam-4248	50	5	coc	coc	NOUN
ejpam-4248	50	6	-	-	PUNCT
ejpam-4248	50	7	r	r	NOUN
ejpam-4248	50	8	-	-	PUNCT
ejpam-4248	50	9	compactness	compactness	NOUN
ejpam-4248	50	10	implies	imply	VERB
ejpam-4248	50	11	coc	coc	ADJ
ejpam-4248	50	12	-	-	PUNCT
ejpam-4248	50	13	weak	weak	ADJ
ejpam-4248	50	14	-	-	PUNCT
ejpam-4248	50	15	compactness	compactness	NOUN
ejpam-4248	50	16	.	.	PUNCT
ejpam-4248	51	1	theorem	theorem	NOUN
ejpam-4248	51	2	1	1	NUM
ejpam-4248	51	3	.	.	PUNCT
ejpam-4248	52	1	a	a	DET
ejpam-4248	52	2	coc	coc	NOUN
ejpam-4248	52	3	-	-	PUNCT
ejpam-4248	52	4	almost	almost	ADV
ejpam-4248	52	5	-	-	PUNCT
ejpam-4248	52	6	compact	compact	ADJ
ejpam-4248	52	7	space	space	NOUN
ejpam-4248	52	8	x	x	PUNCT
ejpam-4248	52	9	is	be	AUX
ejpam-4248	52	10	coc	coc	ADJ
ejpam-4248	52	11	-	-	PUNCT
ejpam-4248	52	12	r	r	NOUN
ejpam-4248	52	13	-	-	PUNCT
ejpam-4248	52	14	compact	compact	ADJ
ejpam-4248	52	15	-	-	PUNCT
ejpam-4248	52	16	space	space	NOUN
ejpam-4248	52	17	.	.	PUNCT
ejpam-4248	53	1	proof	proof	NOUN
ejpam-4248	53	2	.	.	PUNCT
ejpam-4248	54	1	let	let	VERB
ejpam-4248	54	2	u	u	PRON
ejpam-4248	54	3	=	=	X
ejpam-4248	54	4	{	{	PUNCT
ejpam-4248	54	5	uα|α	uα|α	PROPN
ejpam-4248	54	6	∈	∈	PROPN
ejpam-4248	54	7	∆	∆	PROPN
ejpam-4248	54	8	}	}	PUNCT
ejpam-4248	54	9	be	be	AUX
ejpam-4248	54	10	a	a	DET
ejpam-4248	54	11	coc	coc	ADJ
ejpam-4248	54	12	-	-	PUNCT
ejpam-4248	54	13	regular	regular	ADJ
ejpam-4248	54	14	cover	cover	NOUN
ejpam-4248	54	15	of	of	ADP
ejpam-4248	54	16	x	x	PUNCT
ejpam-4248	54	17	with	with	ADP
ejpam-4248	54	18	fα	fα	ADP
ejpam-4248	54	19	⊆	⊆	NUM
ejpam-4248	54	20	uα	uα	NOUN
ejpam-4248	54	21	and	and	CCONJ
ejpam-4248	54	22	x	x	PUNCT
ejpam-4248	55	1	=	=	VERB
ejpam-4248	55	2	⋃	⋃	X
ejpam-4248	55	3	{	{	PUNCT
ejpam-4248	55	4	intcoc(fα)|α	intcoc(fα)|α	NOUN
ejpam-4248	55	5	∈	∈	PROPN
ejpam-4248	55	6	∆	∆	PROPN
ejpam-4248	55	7	}	}	PUNCT
ejpam-4248	55	8	,	,	PUNCT
ejpam-4248	55	9	but	but	CCONJ
ejpam-4248	55	10	x	x	X
ejpam-4248	55	11	is	be	AUX
ejpam-4248	55	12	coc	coc	ADJ
ejpam-4248	55	13	-	-	PUNCT
ejpam-4248	55	14	almost	almost	ADV
ejpam-4248	55	15	-	-	PUNCT
ejpam-4248	55	16	compact	compact	ADJ
ejpam-4248	55	17	,	,	PUNCT
ejpam-4248	55	18	so	so	CCONJ
ejpam-4248	55	19	there	there	PRON
ejpam-4248	55	20	exists	exist	VERB
ejpam-4248	55	21	a	a	DET
ejpam-4248	55	22	finite	finite	NOUN
ejpam-4248	55	23	subset	subset	VERB
ejpam-4248	55	24	∆0	∆0	NUM
ejpam-4248	55	25	of	of	ADP
ejpam-4248	55	26	∆	∆	PROPN
ejpam-4248	55	27	such	such	ADJ
ejpam-4248	55	28	that	that	SCONJ
ejpam-4248	55	29	x	x	X
ejpam-4248	55	30	=	=	SYM
ejpam-4248	55	31	⋃	⋃	NOUN
ejpam-4248	55	32	{	{	PUNCT
ejpam-4248	55	33	intcoc(fα	intcoc(fα	NOUN
ejpam-4248	55	34	)	)	PUNCT
ejpam-4248	55	35	coc|α	coc|α	NOUN
ejpam-4248	55	36	∈	∈	PROPN
ejpam-4248	55	37	∆0	∆0	PROPN
ejpam-4248	55	38	}	}	PUNCT
ejpam-4248	55	39	,	,	PUNCT
ejpam-4248	55	40	so	so	CCONJ
ejpam-4248	55	41	x	x	X
ejpam-4248	55	42	=	=	SYM
ejpam-4248	55	43	⋃	⋃	X
ejpam-4248	55	44	{	{	PUNCT
ejpam-4248	55	45	uα|α	uα|α	PROPN
ejpam-4248	55	46	∈	∈	PROPN
ejpam-4248	55	47	∆0	∆0	PROPN
ejpam-4248	55	48	}	}	PUNCT
ejpam-4248	55	49	=	=	SYM
ejpam-4248	55	50	⋃	⋃	X
ejpam-4248	55	51	{	{	PUNCT
ejpam-4248	55	52	uα|α	uα|α	PROPN
ejpam-4248	55	53	∈	∈	PROPN
ejpam-4248	55	54	∆0	∆0	PRON
ejpam-4248	55	55	}	}	PUNCT
ejpam-4248	55	56	coc	coc	NOUN
ejpam-4248	55	57	,	,	PUNCT
ejpam-4248	55	58	hence	hence	ADV
ejpam-4248	55	59	x	x	VERB
ejpam-4248	55	60	is	be	AUX
ejpam-4248	55	61	coc	coc	ADJ
ejpam-4248	55	62	-	-	PUNCT
ejpam-4248	55	63	r	r	NOUN
ejpam-4248	55	64	-	-	PUNCT
ejpam-4248	55	65	compact	compact	ADJ
ejpam-4248	55	66	-	-	PUNCT
ejpam-4248	55	67	space	space	NOUN
ejpam-4248	55	68	.	.	PUNCT
ejpam-4248	56	1	f.a	f.a	PROPN
ejpam-4248	56	2	.	.	PROPN
ejpam-4248	56	3	abushaheen	abushaheen	PROPN
ejpam-4248	56	4	,	,	PUNCT
ejpam-4248	56	5	f.	f.	PROPN
ejpam-4248	56	6	alrimawi	alrimawi	PROPN
ejpam-4248	56	7	/	/	SYM
ejpam-4248	56	8	eur	eur	PROPN
ejpam-4248	56	9	.	.	PUNCT
ejpam-4248	57	1	j.	j.	PROPN
ejpam-4248	57	2	pure	pure	PROPN
ejpam-4248	57	3	appl	appl	PROPN
ejpam-4248	57	4	.	.	PROPN
ejpam-4248	57	5	math	math	PROPN
ejpam-4248	57	6	,	,	PUNCT
ejpam-4248	57	7	15	15	NUM
ejpam-4248	57	8	(	(	PUNCT
ejpam-4248	57	9	1	1	NUM
ejpam-4248	57	10	)	)	PUNCT
ejpam-4248	57	11	(	(	PUNCT
ejpam-4248	57	12	2022	2022	NUM
ejpam-4248	57	13	)	)	PUNCT
ejpam-4248	57	14	,	,	PUNCT
ejpam-4248	57	15	199	199	NUM
ejpam-4248	57	16	-	-	SYM
ejpam-4248	57	17	206	206	NUM
ejpam-4248	57	18	201	201	NUM
ejpam-4248	57	19	2	2	NUM
ejpam-4248	57	20	.	.	PUNCT
ejpam-4248	57	21	main	main	ADJ
ejpam-4248	57	22	results	result	NOUN
ejpam-4248	57	23	in	in	ADP
ejpam-4248	57	24	the	the	DET
ejpam-4248	57	25	beginning	beginning	NOUN
ejpam-4248	57	26	of	of	ADP
ejpam-4248	57	27	this	this	DET
ejpam-4248	57	28	section	section	NOUN
ejpam-4248	57	29	,	,	PUNCT
ejpam-4248	57	30	we	we	PRON
ejpam-4248	57	31	explore	explore	VERB
ejpam-4248	57	32	some	some	DET
ejpam-4248	57	33	results	result	NOUN
ejpam-4248	57	34	concerning	concern	VERB
ejpam-4248	57	35	subspaces	subspace	NOUN
ejpam-4248	57	36	via	via	ADP
ejpam-4248	57	37	coc	coc	NOUN
ejpam-4248	57	38	-	-	PUNCT
ejpam-4248	57	39	weakly	weakly	ADJ
ejpam-4248	57	40	covering	covering	NOUN
ejpam-4248	57	41	spaces	space	NOUN
ejpam-4248	57	42	.	.	PUNCT
ejpam-4248	58	1	definition	definition	NOUN
ejpam-4248	58	2	11	11	NUM
ejpam-4248	58	3	.	.	PUNCT
ejpam-4248	59	1	a	a	DET
ejpam-4248	59	2	subset	subset	NOUN
ejpam-4248	59	3	s	s	NOUN
ejpam-4248	59	4	of	of	ADP
ejpam-4248	59	5	a	a	DET
ejpam-4248	59	6	space	space	NOUN
ejpam-4248	59	7	x	x	PUNCT
ejpam-4248	59	8	is	be	AUX
ejpam-4248	59	9	said	say	VERB
ejpam-4248	59	10	to	to	PART
ejpam-4248	59	11	be	be	AUX
ejpam-4248	59	12	coc	coc	ADJ
ejpam-4248	59	13	-	-	PUNCT
ejpam-4248	59	14	weakly	weakly	ADJ
ejpam-4248	59	15	-	-	PUNCT
ejpam-4248	59	16	compact	compact	ADJ
ejpam-4248	59	17	subset	subset	NOUN
ejpam-4248	59	18	if	if	SCONJ
ejpam-4248	59	19	s	s	NOUN
ejpam-4248	59	20	is	be	AUX
ejpam-4248	59	21	coc	coc	ADJ
ejpam-4248	59	22	-	-	PUNCT
ejpam-4248	59	23	weakly	weakly	ADJ
ejpam-4248	59	24	-	-	PUNCT
ejpam-4248	59	25	compact	compact	ADJ
ejpam-4248	59	26	subspace	subspace	NOUN
ejpam-4248	59	27	of	of	ADP
ejpam-4248	59	28	x.	x.	NOUN
ejpam-4248	59	29	definition	definition	NOUN
ejpam-4248	59	30	12	12	NUM
ejpam-4248	59	31	.	.	PUNCT
ejpam-4248	60	1	a	a	DET
ejpam-4248	60	2	subset	subset	NOUN
ejpam-4248	60	3	s	s	NOUN
ejpam-4248	60	4	of	of	ADP
ejpam-4248	60	5	a	a	DET
ejpam-4248	60	6	space	space	NOUN
ejpam-4248	60	7	x	x	PUNCT
ejpam-4248	60	8	is	be	AUX
ejpam-4248	60	9	said	say	VERB
ejpam-4248	60	10	to	to	PART
ejpam-4248	60	11	be	be	AUX
ejpam-4248	60	12	coc	coc	ADJ
ejpam-4248	60	13	-	-	PUNCT
ejpam-4248	60	14	weakly	weakly	ADJ
ejpam-4248	60	15	-	-	PUNCT
ejpam-4248	60	16	compact	compact	ADJ
ejpam-4248	60	17	relative	relative	NOUN
ejpam-4248	60	18	to	to	ADP
ejpam-4248	60	19	x	x	SYM
ejpam-4248	60	20	,	,	PUNCT
ejpam-4248	60	21	if	if	SCONJ
ejpam-4248	60	22	for	for	ADP
ejpam-4248	60	23	each	each	DET
ejpam-4248	60	24	cover	cover	NOUN
ejpam-4248	60	25	{	{	PUNCT
ejpam-4248	60	26	vα|α	vα|α	NOUN
ejpam-4248	60	27	∈	∈	PROPN
ejpam-4248	60	28	∆	∆	X
ejpam-4248	60	29	}	}	PUNCT
ejpam-4248	60	30	of	of	ADP
ejpam-4248	60	31	s	s	PRON
ejpam-4248	60	32	by	by	ADP
ejpam-4248	60	33	coc	coc	NOUN
ejpam-4248	60	34	-	-	PUNCT
ejpam-4248	60	35	open	open	ADJ
ejpam-4248	60	36	sets	set	NOUN
ejpam-4248	60	37	of	of	ADP
ejpam-4248	60	38	x	x	PUNCT
ejpam-4248	60	39	satisfying	satisfy	VERB
ejpam-4248	60	40	the	the	DET
ejpam-4248	60	41	following	follow	VERB
ejpam-4248	60	42	condition	condition	NOUN
ejpam-4248	60	43	(	(	PUNCT
ejpam-4248	60	44	∗∗	∗∗	PROPN
ejpam-4248	60	45	)	)	PUNCT
ejpam-4248	60	46	,	,	PUNCT
ejpam-4248	60	47	there	there	PRON
ejpam-4248	60	48	exists	exist	VERB
ejpam-4248	60	49	a	a	DET
ejpam-4248	60	50	set	set	NOUN
ejpam-4248	60	51	∆0	∆0	NOUN
ejpam-4248	60	52	⊆	⊆	NUM
ejpam-4248	60	53	∆	∆	PROPN
ejpam-4248	60	54	such	such	ADJ
ejpam-4248	60	55	that	that	PRON
ejpam-4248	60	56	s	s	VERB
ejpam-4248	60	57	⊆	⊆	NUM
ejpam-4248	60	58	⋃	⋃	X
ejpam-4248	60	59	{	{	PUNCT
ejpam-4248	60	60	vα	vα	ADP
ejpam-4248	60	61	coc|α	coc|α	NOUN
ejpam-4248	60	62	∈	∈	PROPN
ejpam-4248	60	63	∆0	∆0	NOUN
ejpam-4248	60	64	⊆	⊆	NUM
ejpam-4248	60	65	∆	∆	PROPN
ejpam-4248	60	66	,	,	PUNCT
ejpam-4248	60	67	|∆0|	|∆0|	X
ejpam-4248	60	68	<	<	X
ejpam-4248	60	69	ω0	ω0	PROPN
ejpam-4248	60	70	}	}	PUNCT
ejpam-4248	60	71	,	,	PUNCT
ejpam-4248	60	72	where	where	SCONJ
ejpam-4248	60	73	the	the	DET
ejpam-4248	60	74	condition	condition	NOUN
ejpam-4248	60	75	(	(	PUNCT
ejpam-4248	60	76	∗∗	∗∗	NOUN
ejpam-4248	60	77	)	)	PUNCT
ejpam-4248	60	78	is	be	AUX
ejpam-4248	60	79	:	:	PUNCT
ejpam-4248	60	80	”	"	PUNCT
ejpam-4248	60	81	for	for	ADP
ejpam-4248	60	82	each	each	DET
ejpam-4248	60	83	α	α	PROPN
ejpam-4248	60	84	∈	∈	PROPN
ejpam-4248	60	85	∆	∆	PROPN
ejpam-4248	60	86	,	,	PUNCT
ejpam-4248	60	87	there	there	PRON
ejpam-4248	60	88	exists	exist	VERB
ejpam-4248	60	89	a	a	DET
ejpam-4248	60	90	nonempty	nonempty	ADJ
ejpam-4248	60	91	coc	coc	ADJ
ejpam-4248	60	92	-	-	PUNCT
ejpam-4248	60	93	regular	regular	ADJ
ejpam-4248	60	94	closed	close	VERB
ejpam-4248	60	95	set	set	NOUN
ejpam-4248	60	96	fα	fα	ADP
ejpam-4248	60	97	such	such	ADJ
ejpam-4248	60	98	that	that	DET
ejpam-4248	60	99	fα	fα	ADP
ejpam-4248	60	100	⊆	⊆	NUM
ejpam-4248	60	101	vα	vα	NOUN
ejpam-4248	60	102	and	and	CCONJ
ejpam-4248	60	103	s	s	VERB
ejpam-4248	60	104	⊆	⊆	NUM
ejpam-4248	60	105	⋃	⋃	NUM
ejpam-4248	60	106	{	{	PUNCT
ejpam-4248	60	107	intcoc(fα)|α	intcoc(fα)|α	NOUN
ejpam-4248	60	108	∈	∈	PROPN
ejpam-4248	60	109	∆	∆	PROPN
ejpam-4248	60	110	}	}	PUNCT
ejpam-4248	60	111	”	"	PUNCT
ejpam-4248	60	112	(	(	PUNCT
ejpam-4248	60	113	∗∗	∗∗	NOUN
ejpam-4248	60	114	)	)	PUNCT
ejpam-4248	60	115	.	.	PUNCT
ejpam-4248	61	1	theorem	theorem	NOUN
ejpam-4248	61	2	2	2	NUM
ejpam-4248	61	3	.	.	PUNCT
ejpam-4248	62	1	if	if	SCONJ
ejpam-4248	62	2	a	a	PRON
ejpam-4248	62	3	is	be	AUX
ejpam-4248	62	4	a	a	DET
ejpam-4248	62	5	coc	coc	NOUN
ejpam-4248	62	6	-	-	PUNCT
ejpam-4248	62	7	weakly	weakly	ADJ
ejpam-4248	62	8	-	-	PUNCT
ejpam-4248	62	9	compact	compact	ADJ
ejpam-4248	62	10	subspace	subspace	NOUN
ejpam-4248	62	11	of	of	ADP
ejpam-4248	62	12	a	a	DET
ejpam-4248	62	13	space	space	NOUN
ejpam-4248	62	14	x	x	NOUN
ejpam-4248	62	15	,	,	PUNCT
ejpam-4248	62	16	then	then	ADV
ejpam-4248	62	17	a	a	PRON
ejpam-4248	62	18	is	be	AUX
ejpam-4248	62	19	coc	coc	ADJ
ejpam-4248	62	20	-	-	PUNCT
ejpam-4248	62	21	weaklycompact	weaklycompact	NOUN
ejpam-4248	62	22	relative	relative	ADJ
ejpam-4248	62	23	to	to	ADP
ejpam-4248	62	24	x.	x.	NOUN
ejpam-4248	62	25	proof	proof	NOUN
ejpam-4248	62	26	.	.	PUNCT
ejpam-4248	63	1	let	let	AUX
ejpam-4248	63	2	{	{	PUNCT
ejpam-4248	63	3	vα|α	vα|α	NOUN
ejpam-4248	63	4	∈	∈	PROPN
ejpam-4248	63	5	∆	∆	X
ejpam-4248	63	6	}	}	PUNCT
ejpam-4248	63	7	be	be	AUX
ejpam-4248	63	8	a	a	DET
ejpam-4248	63	9	cover	cover	NOUN
ejpam-4248	63	10	of	of	ADP
ejpam-4248	63	11	a	a	PRON
ejpam-4248	63	12	by	by	ADP
ejpam-4248	63	13	coc	coc	ADJ
ejpam-4248	63	14	-	-	PUNCT
ejpam-4248	63	15	open	open	ADJ
ejpam-4248	63	16	sets	set	NOUN
ejpam-4248	63	17	of	of	ADP
ejpam-4248	63	18	x	x	PUNCT
ejpam-4248	63	19	satisfying	satisfy	VERB
ejpam-4248	63	20	the	the	DET
ejpam-4248	63	21	condition	condition	NOUN
ejpam-4248	63	22	(	(	PUNCT
ejpam-4248	63	23	∗∗	∗∗	NOUN
ejpam-4248	63	24	)	)	PUNCT
ejpam-4248	63	25	,	,	PUNCT
ejpam-4248	63	26	for	for	ADP
ejpam-4248	63	27	each	each	DET
ejpam-4248	63	28	α	α	PROPN
ejpam-4248	63	29	∈	∈	NOUN
ejpam-4248	63	30	∆	∆	NOUN
ejpam-4248	63	31	there	there	PRON
ejpam-4248	63	32	exists	exist	VERB
ejpam-4248	63	33	a	a	DET
ejpam-4248	63	34	non	non	X
ejpam-4248	63	35	empty	empty	ADJ
ejpam-4248	63	36	coc	coc	NOUN
ejpam-4248	63	37	-	-	PUNCT
ejpam-4248	63	38	regular	regular	ADJ
ejpam-4248	63	39	closed	close	VERB
ejpam-4248	63	40	set	set	NOUN
ejpam-4248	63	41	fα	fα	ADP
ejpam-4248	63	42	such	such	ADJ
ejpam-4248	63	43	that	that	DET
ejpam-4248	63	44	fα	fα	ADP
ejpam-4248	63	45	⊆	⊆	NUM
ejpam-4248	63	46	vα	vα	NOUN
ejpam-4248	63	47	and	and	CCONJ
ejpam-4248	63	48	a	a	DET
ejpam-4248	63	49	⊆	⊆	NUM
ejpam-4248	63	50	⋃	⋃	NUM
ejpam-4248	63	51	{	{	PUNCT
ejpam-4248	63	52	intcoc(fα)|α	intcoc(fα)|α	NOUN
ejpam-4248	63	53	∈	∈	PROPN
ejpam-4248	63	54	∆	∆	PROPN
ejpam-4248	63	55	}	}	PUNCT
ejpam-4248	63	56	.	.	PUNCT
ejpam-4248	64	1	for	for	ADP
ejpam-4248	64	2	each	each	DET
ejpam-4248	64	3	α	α	PROPN
ejpam-4248	64	4	∈	∈	PROPN
ejpam-4248	64	5	∆	∆	X
ejpam-4248	64	6	,	,	PUNCT
ejpam-4248	64	7	intcoc(fα	intcoc(fα	NOUN
ejpam-4248	64	8	)	)	PUNCT
ejpam-4248	64	9	∩a	∩a	PROPN
ejpam-4248	64	10	and	and	CCONJ
ejpam-4248	64	11	vα	vα	INTJ
ejpam-4248	64	12	∩a	∩a	PROPN
ejpam-4248	64	13	are	be	AUX
ejpam-4248	64	14	coc	coc	ADJ
ejpam-4248	64	15	-	-	PUNCT
ejpam-4248	64	16	open	open	ADJ
ejpam-4248	64	17	subsets	subset	NOUN
ejpam-4248	64	18	in	in	ADP
ejpam-4248	64	19	a	a	DET
ejpam-4248	64	20	and	and	CCONJ
ejpam-4248	64	21	fα	fα	ADP
ejpam-4248	64	22	∩	∩	NOUN
ejpam-4248	65	1	a	a	PRON
ejpam-4248	65	2	is	be	AUX
ejpam-4248	65	3	coc	coc	NOUN
ejpam-4248	65	4	-	-	PUNCT
ejpam-4248	65	5	closed	closed	ADJ
ejpam-4248	65	6	in	in	ADP
ejpam-4248	65	7	a	a	PRON
ejpam-4248	65	8	,	,	PUNCT
ejpam-4248	65	9	so	so	SCONJ
ejpam-4248	65	10	the	the	DET
ejpam-4248	65	11	family	family	NOUN
ejpam-4248	65	12	{	{	PUNCT
ejpam-4248	65	13	vα	vα	ADP
ejpam-4248	65	14	∩	∩	NOUN
ejpam-4248	65	15	a|α	a|α	NOUN
ejpam-4248	65	16	∈	∈	PROPN
ejpam-4248	65	17	∆	∆	PROPN
ejpam-4248	65	18	}	}	PUNCT
ejpam-4248	65	19	is	be	AUX
ejpam-4248	65	20	coc	coc	ADJ
ejpam-4248	65	21	-	-	PUNCT
ejpam-4248	65	22	open	open	ADJ
ejpam-4248	65	23	cover	cover	NOUN
ejpam-4248	65	24	of	of	ADP
ejpam-4248	65	25	a	a	PRON
ejpam-4248	65	26	,	,	PUNCT
ejpam-4248	65	27	therefore	therefore	ADV
ejpam-4248	65	28	for	for	ADP
ejpam-4248	65	29	each	each	DET
ejpam-4248	65	30	α	α	NOUN
ejpam-4248	65	31	∈	∈	PROPN
ejpam-4248	65	32	∆	∆	PROPN
ejpam-4248	65	33	,	,	PUNCT
ejpam-4248	65	34	we	we	PRON
ejpam-4248	65	35	have	have	VERB
ejpam-4248	65	36	intcoc(fα	intcoc(fα	NOUN
ejpam-4248	65	37	)	)	PUNCT
ejpam-4248	65	38	∩a	∩a	PROPN
ejpam-4248	65	39	⊆	⊆	NUM
ejpam-4248	65	40	intcoc(fα	intcoc(fα	NOUN
ejpam-4248	65	41	)	)	PUNCT
ejpam-4248	66	1	∩a	∩a	PROPN
ejpam-4248	66	2	coc(a	coc(a	PROPN
ejpam-4248	66	3	)	)	PUNCT
ejpam-4248	66	4	⊆	⊆	NUM
ejpam-4248	66	5	fα	fα	ADP
ejpam-4248	66	6	∩a	∩a	PROPN
ejpam-4248	66	7	⊆	⊆	NUM
ejpam-4248	66	8	vα	vα	ADP
ejpam-4248	66	9	∩a	∩a	PROPN
ejpam-4248	66	10	⊆	⊆	NUM
ejpam-4248	66	11	a	a	PRON
ejpam-4248	66	12	,	,	PUNCT
ejpam-4248	66	13	moreover	moreover	ADV
ejpam-4248	66	14	a	a	DET
ejpam-4248	66	15	=	=	PUNCT
ejpam-4248	66	16	⋃	⋃	NOUN
ejpam-4248	66	17	{	{	PUNCT
ejpam-4248	66	18	intcoc(fα	intcoc(fα	NOUN
ejpam-4248	66	19	)	)	PUNCT
ejpam-4248	66	20	∩a|α	∩a|α	NOUN
ejpam-4248	66	21	∈	∈	PROPN
ejpam-4248	66	22	∆	∆	X
ejpam-4248	66	23	}	}	PUNCT
ejpam-4248	66	24	and	and	CCONJ
ejpam-4248	66	25	intcoc(fα	intcoc(fα	NOUN
ejpam-4248	66	26	)	)	PUNCT
ejpam-4248	66	27	∩a	∩a	PROPN
ejpam-4248	66	28	⊆	⊆	NUM
ejpam-4248	66	29	intcoc(a	intcoc(a	NOUN
ejpam-4248	66	30	)	)	PUNCT
ejpam-4248	66	31	(	(	PUNCT
ejpam-4248	66	32	intcoc(fα	intcoc(fα	NOUN
ejpam-4248	66	33	∩a	∩a	PROPN
ejpam-4248	66	34	)	)	PUNCT
ejpam-4248	67	1	coc(a	coc(a	PROPN
ejpam-4248	67	2	)	)	PUNCT
ejpam-4248	67	3	.	.	PUNCT
ejpam-4248	68	1	now	now	ADV
ejpam-4248	68	2	,	,	PUNCT
ejpam-4248	68	3	the	the	DET
ejpam-4248	68	4	set	set	NOUN
ejpam-4248	68	5	intcoc(fα	intcoc(fα	NOUN
ejpam-4248	68	6	∩a	∩a	PROPN
ejpam-4248	68	7	)	)	PUNCT
ejpam-4248	69	1	coc(a	coc(a	X
ejpam-4248	69	2	)	)	PUNCT
ejpam-4248	69	3	is	be	AUX
ejpam-4248	69	4	coc	coc	ADJ
ejpam-4248	69	5	-	-	PUNCT
ejpam-4248	69	6	regular	regular	NOUN
ejpam-4248	69	7	closed	close	VERB
ejpam-4248	69	8	in	in	ADP
ejpam-4248	69	9	a	a	PRON
ejpam-4248	69	10	and	and	CCONJ
ejpam-4248	69	11	{	{	PUNCT
ejpam-4248	69	12	vα	vα	ADP
ejpam-4248	69	13	∩	∩	NOUN
ejpam-4248	69	14	a|α	a|α	NOUN
ejpam-4248	69	15	∈	∈	PROPN
ejpam-4248	69	16	∆	∆	X
ejpam-4248	69	17	}	}	PUNCT
ejpam-4248	69	18	is	be	AUX
ejpam-4248	69	19	a	a	DET
ejpam-4248	69	20	coc	coc	ADJ
ejpam-4248	69	21	-	-	PUNCT
ejpam-4248	69	22	regular	regular	ADJ
ejpam-4248	69	23	cover	cover	NOUN
ejpam-4248	69	24	of	of	ADP
ejpam-4248	69	25	a	a	PRON
ejpam-4248	69	26	,	,	PUNCT
ejpam-4248	69	27	so	so	SCONJ
ejpam-4248	69	28	there	there	PRON
ejpam-4248	69	29	exists	exist	VERB
ejpam-4248	69	30	a	a	DET
ejpam-4248	69	31	finite	finite	NOUN
ejpam-4248	69	32	subset	subset	VERB
ejpam-4248	69	33	∆0	∆0	NUM
ejpam-4248	69	34	⊆	⊆	NUM
ejpam-4248	69	35	∆	∆	PROPN
ejpam-4248	69	36	such	such	ADJ
ejpam-4248	69	37	that	that	SCONJ
ejpam-4248	69	38	a	a	DET
ejpam-4248	69	39	=	=	X
ejpam-4248	69	40	⋃	⋃	X
ejpam-4248	69	41	{	{	PUNCT
ejpam-4248	69	42	vα	vα	ADP
ejpam-4248	69	43	∩a	∩a	PROPN
ejpam-4248	69	44	coc(a)|α	coc(a)|α	VERB
ejpam-4248	69	45	∈	∈	PROPN
ejpam-4248	69	46	∆	∆	X
ejpam-4248	69	47	}	}	PUNCT
ejpam-4248	69	48	,	,	PUNCT
ejpam-4248	69	49	but	but	CCONJ
ejpam-4248	69	50	vα	vα	INTJ
ejpam-4248	69	51	∩a	∩a	PROPN
ejpam-4248	69	52	coc(a	coc(a	PROPN
ejpam-4248	69	53	)	)	PUNCT
ejpam-4248	69	54	⊆	⊆	NUM
ejpam-4248	69	55	vα	vα	ADP
ejpam-4248	69	56	coc(a	coc(a	NOUN
ejpam-4248	69	57	)	)	PUNCT
ejpam-4248	69	58	for	for	ADP
ejpam-4248	69	59	each	each	DET
ejpam-4248	69	60	α	α	PROPN
ejpam-4248	69	61	∈	∈	PROPN
ejpam-4248	69	62	∆0	∆0	NOUN
ejpam-4248	69	63	,	,	PUNCT
ejpam-4248	69	64	therefore	therefore	ADV
ejpam-4248	69	65	a	a	DET
ejpam-4248	69	66	⊆	⊆	NUM
ejpam-4248	69	67	⋃	⋃	NOUN
ejpam-4248	69	68	{	{	PUNCT
ejpam-4248	69	69	vα	vα	ADP
ejpam-4248	69	70	coc|α	coc|α	NOUN
ejpam-4248	69	71	∈	∈	PROPN
ejpam-4248	69	72	∆0	∆0	PROPN
ejpam-4248	69	73	}	}	PUNCT
ejpam-4248	69	74	,	,	PUNCT
ejpam-4248	69	75	hence	hence	ADV
ejpam-4248	69	76	the	the	DET
ejpam-4248	69	77	result	result	NOUN
ejpam-4248	69	78	.	.	PUNCT
ejpam-4248	70	1	corollary	corollary	ADJ
ejpam-4248	70	2	1	1	NUM
ejpam-4248	70	3	.	.	PUNCT
ejpam-4248	71	1	if	if	SCONJ
ejpam-4248	71	2	a	a	PRON
ejpam-4248	71	3	is	be	AUX
ejpam-4248	71	4	coc	coc	ADJ
ejpam-4248	71	5	-	-	ADJ
ejpam-4248	71	6	compact	compact	ADJ
ejpam-4248	71	7	(	(	PUNCT
ejpam-4248	71	8	coc	coc	NOUN
ejpam-4248	71	9	-	-	PUNCT
ejpam-4248	71	10	almost	almost	ADV
ejpam-4248	71	11	-	-	PUNCT
ejpam-4248	71	12	compact	compact	ADJ
ejpam-4248	71	13	,	,	PUNCT
ejpam-4248	71	14	coc	coc	NOUN
ejpam-4248	71	15	-	-	PUNCT
ejpam-4248	71	16	nearly	nearly	ADV
ejpam-4248	71	17	-	-	PUNCT
ejpam-4248	71	18	compact	compact	ADJ
ejpam-4248	71	19	)	)	PUNCT
ejpam-4248	71	20	subspace	subspace	NOUN
ejpam-4248	71	21	of	of	ADP
ejpam-4248	71	22	x	x	PRON
ejpam-4248	71	23	,	,	PUNCT
ejpam-4248	71	24	then	then	ADV
ejpam-4248	71	25	a	a	PRON
ejpam-4248	71	26	is	be	AUX
ejpam-4248	71	27	coc	coc	ADJ
ejpam-4248	71	28	-	-	PUNCT
ejpam-4248	71	29	weakly	weakly	ADJ
ejpam-4248	71	30	-	-	PUNCT
ejpam-4248	71	31	compact	compact	ADJ
ejpam-4248	71	32	relative	relative	NOUN
ejpam-4248	71	33	to	to	ADP
ejpam-4248	71	34	x.	x.	NOUN
ejpam-4248	71	35	theorem	theorem	NOUN
ejpam-4248	71	36	3	3	X
ejpam-4248	71	37	.	.	PUNCT
ejpam-4248	72	1	if	if	SCONJ
ejpam-4248	72	2	every	every	DET
ejpam-4248	72	3	coc	coc	ADJ
ejpam-4248	72	4	-	-	PUNCT
ejpam-4248	72	5	regular	regular	ADJ
ejpam-4248	72	6	closed	closed	ADJ
ejpam-4248	72	7	subset	subset	NOUN
ejpam-4248	72	8	of	of	ADP
ejpam-4248	72	9	a	a	DET
ejpam-4248	72	10	space	space	NOUN
ejpam-4248	72	11	x	x	PUNCT
ejpam-4248	72	12	is	be	AUX
ejpam-4248	72	13	coc	coc	ADJ
ejpam-4248	72	14	-	-	PUNCT
ejpam-4248	72	15	weakly	weakly	ADJ
ejpam-4248	72	16	compact	compact	ADJ
ejpam-4248	72	17	relative	relative	NOUN
ejpam-4248	72	18	to	to	ADP
ejpam-4248	72	19	x	x	PRON
ejpam-4248	72	20	,	,	PUNCT
ejpam-4248	72	21	then	then	ADV
ejpam-4248	72	22	x	x	PUNCT
ejpam-4248	72	23	is	be	AUX
ejpam-4248	72	24	coc	coc	ADJ
ejpam-4248	72	25	-	-	PUNCT
ejpam-4248	72	26	weakly	weakly	ADV
ejpam-4248	72	27	compact	compact	ADJ
ejpam-4248	72	28	.	.	PUNCT
ejpam-4248	73	1	proof	proof	NOUN
ejpam-4248	73	2	.	.	PUNCT
ejpam-4248	74	1	let	let	AUX
ejpam-4248	74	2	{	{	PUNCT
ejpam-4248	74	3	vα|α	vα|α	NOUN
ejpam-4248	74	4	∈	∈	PROPN
ejpam-4248	74	5	∆	∆	X
ejpam-4248	74	6	}	}	PUNCT
ejpam-4248	74	7	be	be	AUX
ejpam-4248	74	8	a	a	DET
ejpam-4248	74	9	coc	coc	ADJ
ejpam-4248	74	10	-	-	PUNCT
ejpam-4248	74	11	regular	regular	ADJ
ejpam-4248	74	12	cover	cover	NOUN
ejpam-4248	74	13	of	of	ADP
ejpam-4248	74	14	x	x	PRON
ejpam-4248	74	15	,	,	PUNCT
ejpam-4248	74	16	for	for	ADP
ejpam-4248	74	17	each	each	DET
ejpam-4248	74	18	α	α	PROPN
ejpam-4248	74	19	∈	∈	PROPN
ejpam-4248	74	20	∆	∆	PROPN
ejpam-4248	74	21	,	,	PUNCT
ejpam-4248	74	22	there	there	PRON
ejpam-4248	74	23	exists	exist	VERB
ejpam-4248	74	24	nonempty	nonempty	ADJ
ejpam-4248	74	25	coc	coc	ADJ
ejpam-4248	74	26	-	-	PUNCT
ejpam-4248	74	27	regular	regular	ADJ
ejpam-4248	74	28	set	set	NOUN
ejpam-4248	74	29	such	such	ADJ
ejpam-4248	74	30	that	that	DET
ejpam-4248	74	31	fα	fα	ADP
ejpam-4248	74	32	⊆	⊆	NUM
ejpam-4248	74	33	vα	vα	NOUN
ejpam-4248	74	34	and	and	CCONJ
ejpam-4248	74	35	x	x	SYM
ejpam-4248	74	36	=	=	SYM
ejpam-4248	74	37	⋃	⋃	X
ejpam-4248	74	38	{	{	PUNCT
ejpam-4248	74	39	intcoc(fα)|α	intcoc(fα)|α	NOUN
ejpam-4248	74	40	∈	∈	PROPN
ejpam-4248	74	41	∆	∆	PROPN
ejpam-4248	74	42	}	}	PUNCT
ejpam-4248	74	43	.	.	PUNCT
ejpam-4248	75	1	for	for	ADP
ejpam-4248	75	2	α0	α0	ADJ
ejpam-4248	75	3	∈	∈	PROPN
ejpam-4248	75	4	∆	∆	X
ejpam-4248	75	5	let	let	VERB
ejpam-4248	75	6	k	k	PROPN
ejpam-4248	75	7	=	=	PUNCT
ejpam-4248	75	8	x	x	SYM
ejpam-4248	75	9	−	−	PROPN
ejpam-4248	75	10	intcoc(fα0	intcoc(fα0	NOUN
ejpam-4248	75	11	)	)	PUNCT
ejpam-4248	75	12	,	,	PUNCT
ejpam-4248	75	13	then	then	ADV
ejpam-4248	75	14	k	k	PROPN
ejpam-4248	75	15	is	be	AUX
ejpam-4248	75	16	a	a	DET
ejpam-4248	75	17	coc	coc	ADJ
ejpam-4248	75	18	-	-	PUNCT
ejpam-4248	75	19	regular	regular	ADJ
ejpam-4248	75	20	closed	closed	ADJ
ejpam-4248	75	21	and	and	CCONJ
ejpam-4248	75	22	k	k	NOUN
ejpam-4248	75	23	⊆	⊆	NUM
ejpam-4248	75	24	⋃	⋃	NUM
ejpam-4248	75	25	{	{	PUNCT
ejpam-4248	75	26	intcoc(fα)|α	intcoc(fα)|α	NOUN
ejpam-4248	75	27	∈	∈	PROPN
ejpam-4248	75	28	f.a	f.a	PROPN
ejpam-4248	75	29	.	.	PROPN
ejpam-4248	75	30	abushaheen	abushaheen	PROPN
ejpam-4248	75	31	,	,	PUNCT
ejpam-4248	75	32	f.	f.	PROPN
ejpam-4248	75	33	alrimawi	alrimawi	PROPN
ejpam-4248	75	34	/	/	SYM
ejpam-4248	75	35	eur	eur	PROPN
ejpam-4248	75	36	.	.	PUNCT
ejpam-4248	76	1	j.	j.	PROPN
ejpam-4248	76	2	pure	pure	PROPN
ejpam-4248	76	3	appl	appl	PROPN
ejpam-4248	76	4	.	.	PROPN
ejpam-4248	76	5	math	math	PROPN
ejpam-4248	76	6	,	,	PUNCT
ejpam-4248	76	7	15	15	NUM
ejpam-4248	76	8	(	(	PUNCT
ejpam-4248	76	9	1	1	NUM
ejpam-4248	76	10	)	)	PUNCT
ejpam-4248	76	11	(	(	PUNCT
ejpam-4248	76	12	2022	2022	NUM
ejpam-4248	76	13	)	)	PUNCT
ejpam-4248	76	14	,	,	PUNCT
ejpam-4248	76	15	199	199	NUM
ejpam-4248	76	16	-	-	SYM
ejpam-4248	76	17	206	206	NUM
ejpam-4248	76	18	202	202	NUM
ejpam-4248	76	19	∆−	∆−	NOUN
ejpam-4248	76	20	{	{	PUNCT
ejpam-4248	76	21	α0	α0	ADJ
ejpam-4248	76	22	}	}	PUNCT
ejpam-4248	76	23	}	}	PUNCT
ejpam-4248	76	24	,	,	PUNCT
ejpam-4248	76	25	so	so	ADV
ejpam-4248	76	26	v∗	v∗	PROPN
ejpam-4248	76	27	=	=	NOUN
ejpam-4248	76	28	{	{	PUNCT
ejpam-4248	76	29	vα|α	vα|α	NOUN
ejpam-4248	76	30	∈	∈	PROPN
ejpam-4248	76	31	∆−	∆−	NOUN
ejpam-4248	76	32	{	{	PUNCT
ejpam-4248	76	33	α0	α0	ADJ
ejpam-4248	76	34	}	}	PUNCT
ejpam-4248	76	35	}	}	PUNCT
ejpam-4248	76	36	is	be	AUX
ejpam-4248	76	37	a	a	DET
ejpam-4248	76	38	cover	cover	NOUN
ejpam-4248	76	39	of	of	ADP
ejpam-4248	76	40	k	k	PROPN
ejpam-4248	76	41	by	by	ADP
ejpam-4248	76	42	coc	coc	ADJ
ejpam-4248	76	43	-	-	PUNCT
ejpam-4248	76	44	open	open	ADJ
ejpam-4248	76	45	sets	set	NOUN
ejpam-4248	76	46	of	of	ADP
ejpam-4248	76	47	x	x	PUNCT
ejpam-4248	76	48	satisfying	satisfy	VERB
ejpam-4248	76	49	condition	condition	NOUN
ejpam-4248	76	50	(	(	PUNCT
ejpam-4248	76	51	*	*	PUNCT
ejpam-4248	76	52	*	*	PUNCT
ejpam-4248	76	53	)	)	PUNCT
ejpam-4248	76	54	and	and	CCONJ
ejpam-4248	76	55	hence	hence	ADV
ejpam-4248	76	56	for	for	SCONJ
ejpam-4248	76	57	some	some	DET
ejpam-4248	76	58	finite	finite	NOUN
ejpam-4248	76	59	set	set	VERB
ejpam-4248	76	60	∆0	∆0	NUM
ejpam-4248	76	61	⊆	⊆	NUM
ejpam-4248	76	62	∆	∆	PROPN
ejpam-4248	76	63	we	we	PRON
ejpam-4248	76	64	have	have	VERB
ejpam-4248	76	65	k	k	PROPN
ejpam-4248	76	66	⊆	⊆	NUM
ejpam-4248	76	67	⋃	⋃	X
ejpam-4248	76	68	{	{	PUNCT
ejpam-4248	76	69	vα	vα	ADP
ejpam-4248	76	70	coc|α	coc|α	NOUN
ejpam-4248	76	71	∈	∈	PROPN
ejpam-4248	76	72	∆0	∆0	PROPN
ejpam-4248	76	73	}	}	PUNCT
ejpam-4248	76	74	,	,	PUNCT
ejpam-4248	76	75	thus	thus	ADV
ejpam-4248	76	76	x	x	X
ejpam-4248	76	77	=	=	SYM
ejpam-4248	76	78	k	k	PROPN
ejpam-4248	76	79	∪	∪	PROPN
ejpam-4248	76	80	intcoc(fα0	intcoc(fα0	NOUN
ejpam-4248	76	81	)	)	PUNCT
ejpam-4248	76	82	⊆	⊆	NUM
ejpam-4248	76	83	⋃	⋃	X
ejpam-4248	76	84	{	{	PUNCT
ejpam-4248	76	85	vα	vα	ADP
ejpam-4248	76	86	coc|α	coc|α	NOUN
ejpam-4248	76	87	∈	∈	PROPN
ejpam-4248	76	88	∆0	∆0	PROPN
ejpam-4248	76	89	∪	∪	X
ejpam-4248	76	90	{	{	PUNCT
ejpam-4248	76	91	α0	α0	ADJ
ejpam-4248	76	92	}	}	PUNCT
ejpam-4248	76	93	}	}	PUNCT
ejpam-4248	76	94	,	,	PUNCT
ejpam-4248	76	95	so	so	CCONJ
ejpam-4248	76	96	x	x	PUNCT
ejpam-4248	76	97	is	be	AUX
ejpam-4248	76	98	coc	coc	ADJ
ejpam-4248	76	99	-	-	PUNCT
ejpam-4248	76	100	weakly	weakly	ADJ
ejpam-4248	76	101	compact	compact	ADJ
ejpam-4248	76	102	.	.	PUNCT
ejpam-4248	77	1	corollary	corollary	ADJ
ejpam-4248	77	2	2	2	NUM
ejpam-4248	77	3	.	.	PUNCT
ejpam-4248	78	1	if	if	SCONJ
ejpam-4248	78	2	every	every	DET
ejpam-4248	78	3	proper	proper	ADJ
ejpam-4248	78	4	coc	coc	ADJ
ejpam-4248	78	5	-	-	PUNCT
ejpam-4248	78	6	regular	regular	ADJ
ejpam-4248	78	7	closed	closed	ADJ
ejpam-4248	78	8	subset	subset	NOUN
ejpam-4248	78	9	of	of	ADP
ejpam-4248	78	10	a	a	DET
ejpam-4248	78	11	space	space	NOUN
ejpam-4248	78	12	x	x	PUNCT
ejpam-4248	78	13	is	be	AUX
ejpam-4248	78	14	coc	coc	ADJ
ejpam-4248	78	15	-	-	PUNCT
ejpam-4248	78	16	weakly	weakly	ADJ
ejpam-4248	78	17	compact	compact	ADJ
ejpam-4248	78	18	,	,	PUNCT
ejpam-4248	78	19	then	then	ADV
ejpam-4248	78	20	x	x	PUNCT
ejpam-4248	78	21	is	be	AUX
ejpam-4248	78	22	coc	coc	ADJ
ejpam-4248	78	23	-	-	PUNCT
ejpam-4248	78	24	weakly	weakly	ADV
ejpam-4248	78	25	compact	compact	ADJ
ejpam-4248	78	26	.	.	PUNCT
ejpam-4248	79	1	proof	proof	NOUN
ejpam-4248	79	2	.	.	PUNCT
ejpam-4248	80	1	clearly	clearly	ADV
ejpam-4248	80	2	.	.	PUNCT
ejpam-4248	81	1	corollary	corollary	ADJ
ejpam-4248	81	2	3	3	X
ejpam-4248	81	3	.	.	PUNCT
ejpam-4248	82	1	if	if	SCONJ
ejpam-4248	82	2	every	every	DET
ejpam-4248	82	3	proper	proper	ADJ
ejpam-4248	82	4	coc	coc	ADJ
ejpam-4248	82	5	-	-	PUNCT
ejpam-4248	82	6	regular	regular	ADJ
ejpam-4248	82	7	closed	closed	ADJ
ejpam-4248	82	8	subset	subset	NOUN
ejpam-4248	82	9	of	of	ADP
ejpam-4248	82	10	a	a	DET
ejpam-4248	82	11	space	space	NOUN
ejpam-4248	82	12	x	x	PUNCT
ejpam-4248	82	13	is	be	AUX
ejpam-4248	82	14	coc	coc	ADJ
ejpam-4248	82	15	-	-	ADJ
ejpam-4248	82	16	compact	compact	ADJ
ejpam-4248	82	17	(	(	PUNCT
ejpam-4248	82	18	cocalmost	cocalmost	NOUN
ejpam-4248	82	19	-	-	PUNCT
ejpam-4248	82	20	compact	compact	ADJ
ejpam-4248	82	21	,	,	PUNCT
ejpam-4248	82	22	coc	coc	NOUN
ejpam-4248	82	23	-	-	PUNCT
ejpam-4248	82	24	nearly	nearly	ADV
ejpam-4248	82	25	-	-	PUNCT
ejpam-4248	82	26	compact	compact	ADJ
ejpam-4248	82	27	)	)	PUNCT
ejpam-4248	82	28	subspace	subspace	NOUN
ejpam-4248	82	29	,	,	PUNCT
ejpam-4248	82	30	then	then	ADV
ejpam-4248	82	31	x	x	PUNCT
ejpam-4248	82	32	is	be	AUX
ejpam-4248	82	33	coc	coc	ADJ
ejpam-4248	82	34	-	-	PUNCT
ejpam-4248	82	35	weakly	weakly	ADJ
ejpam-4248	82	36	compact	compact	ADJ
ejpam-4248	82	37	.	.	PUNCT
ejpam-4248	83	1	theorem	theorem	VERB
ejpam-4248	83	2	4	4	NUM
ejpam-4248	83	3	.	.	PUNCT
ejpam-4248	84	1	if	if	SCONJ
ejpam-4248	84	2	a	a	DET
ejpam-4248	84	3	subset	subset	NOUN
ejpam-4248	84	4	a	a	PRON
ejpam-4248	84	5	of	of	ADP
ejpam-4248	84	6	a	a	DET
ejpam-4248	84	7	coc	coc	NOUN
ejpam-4248	84	8	-	-	PUNCT
ejpam-4248	84	9	weakly	weakly	ADJ
ejpam-4248	84	10	-	-	PUNCT
ejpam-4248	84	11	compact	compact	ADJ
ejpam-4248	84	12	space	space	NOUN
ejpam-4248	84	13	x	x	PUNCT
ejpam-4248	84	14	is	be	AUX
ejpam-4248	84	15	coc	coc	NOUN
ejpam-4248	84	16	-	-	PUNCT
ejpam-4248	84	17	clopen	clopen	ADJ
ejpam-4248	84	18	,	,	PUNCT
ejpam-4248	84	19	then	then	ADV
ejpam-4248	84	20	a	a	PRON
ejpam-4248	84	21	is	be	AUX
ejpam-4248	84	22	coc	coc	ADJ
ejpam-4248	84	23	-	-	PUNCT
ejpam-4248	84	24	weakly	weakly	ADJ
ejpam-4248	84	25	-	-	PUNCT
ejpam-4248	84	26	compact	compact	ADJ
ejpam-4248	84	27	relative	relative	NOUN
ejpam-4248	84	28	to	to	ADP
ejpam-4248	84	29	x.	x.	NOUN
ejpam-4248	84	30	proof	proof	NOUN
ejpam-4248	84	31	.	.	PUNCT
ejpam-4248	85	1	let	let	VERB
ejpam-4248	85	2	ua	ua	PROPN
ejpam-4248	85	3	=	=	PRON
ejpam-4248	85	4	{	{	PUNCT
ejpam-4248	85	5	uα|α	uα|α	PROPN
ejpam-4248	85	6	∈	∈	PROPN
ejpam-4248	85	7	∆	∆	PROPN
ejpam-4248	85	8	}	}	PUNCT
ejpam-4248	85	9	be	be	AUX
ejpam-4248	85	10	a	a	DET
ejpam-4248	85	11	cover	cover	NOUN
ejpam-4248	85	12	of	of	ADP
ejpam-4248	85	13	a	a	PRON
ejpam-4248	85	14	by	by	ADP
ejpam-4248	85	15	coc	coc	ADJ
ejpam-4248	85	16	-	-	PUNCT
ejpam-4248	85	17	open	open	ADJ
ejpam-4248	85	18	sets	set	NOUN
ejpam-4248	85	19	of	of	ADP
ejpam-4248	85	20	x	x	PUNCT
ejpam-4248	85	21	satisfying	satisfy	VERB
ejpam-4248	85	22	condition	condition	NOUN
ejpam-4248	85	23	(	(	PUNCT
ejpam-4248	85	24	*	*	NOUN
ejpam-4248	85	25	*	*	PUNCT
ejpam-4248	85	26	)	)	PUNCT
ejpam-4248	85	27	,	,	PUNCT
ejpam-4248	85	28	hence	hence	ADV
ejpam-4248	85	29	the	the	DET
ejpam-4248	85	30	family	family	NOUN
ejpam-4248	85	31	{	{	PUNCT
ejpam-4248	85	32	uα|α	uα|α	PROPN
ejpam-4248	85	33	∈	∈	PROPN
ejpam-4248	85	34	∆	∆	X
ejpam-4248	85	35	}	}	PUNCT
ejpam-4248	85	36	∪	∪	VERB
ejpam-4248	85	37	{	{	PUNCT
ejpam-4248	85	38	x	x	PART
ejpam-4248	85	39	−a	−a	NOUN
ejpam-4248	85	40	}	}	PUNCT
ejpam-4248	85	41	is	be	AUX
ejpam-4248	85	42	a	a	DET
ejpam-4248	85	43	coc	coc	ADJ
ejpam-4248	85	44	-	-	PUNCT
ejpam-4248	85	45	regular	regular	ADJ
ejpam-4248	85	46	cover	cover	NOUN
ejpam-4248	85	47	of	of	ADP
ejpam-4248	85	48	x	x	NOUN
ejpam-4248	85	49	,	,	PUNCT
ejpam-4248	85	50	but	but	CCONJ
ejpam-4248	85	51	x	x	X
ejpam-4248	85	52	is	be	AUX
ejpam-4248	85	53	coc	coc	ADJ
ejpam-4248	85	54	-	-	PUNCT
ejpam-4248	85	55	weakly	weakly	ADJ
ejpam-4248	85	56	-	-	PUNCT
ejpam-4248	85	57	compact	compact	ADJ
ejpam-4248	85	58	,	,	PUNCT
ejpam-4248	85	59	so	so	CCONJ
ejpam-4248	85	60	there	there	PRON
ejpam-4248	85	61	exists	exist	VERB
ejpam-4248	85	62	a	a	DET
ejpam-4248	85	63	finite	finite	NOUN
ejpam-4248	85	64	set	set	VERB
ejpam-4248	85	65	∆0	∆0	ADP
ejpam-4248	85	66	⊆	⊆	NUM
ejpam-4248	85	67	∆	∆	PROPN
ejpam-4248	85	68	such	such	ADJ
ejpam-4248	85	69	that	that	SCONJ
ejpam-4248	85	70	x	x	NOUN
ejpam-4248	85	71	∪	∪	X
ejpam-4248	85	72	(	(	PUNCT
ejpam-4248	85	73	⋃	⋃	ADP
ejpam-4248	85	74	α∈∆0	α∈∆0	NUM
ejpam-4248	85	75	vα	vα	ADP
ejpam-4248	85	76	coc	coc	NOUN
ejpam-4248	85	77	)	)	PUNCT
ejpam-4248	85	78	∪	∪	NOUN
ejpam-4248	85	79	(	(	PUNCT
ejpam-4248	85	80	x	x	SYM
ejpam-4248	85	81	−a	−a	ADP
ejpam-4248	85	82	coc	coc	PROPN
ejpam-4248	85	83	)	)	PUNCT
ejpam-4248	85	84	=	=	PRON
ejpam-4248	85	85	(	(	PUNCT
ejpam-4248	85	86	⋃	⋃	NOUN
ejpam-4248	85	87	α∈∆0	α∈∆0	NOUN
ejpam-4248	85	88	vα	vα	ADP
ejpam-4248	85	89	coc	coc	NOUN
ejpam-4248	85	90	)	)	PUNCT
ejpam-4248	85	91	∪	∪	NOUN
ejpam-4248	85	92	(	(	PUNCT
ejpam-4248	85	93	x	x	SYM
ejpam-4248	85	94	−a	−a	NOUN
ejpam-4248	85	95	)	)	PUNCT
ejpam-4248	85	96	,	,	PUNCT
ejpam-4248	85	97	therefore	therefore	ADV
ejpam-4248	85	98	a	a	DET
ejpam-4248	85	99	⊆	⊆	NUM
ejpam-4248	85	100	⋃	⋃	NOUN
ejpam-4248	85	101	α∈∆0	α∈∆0	NUM
ejpam-4248	85	102	vα	vα	ADP
ejpam-4248	85	103	coc	coc	NOUN
ejpam-4248	85	104	,	,	PUNCT
ejpam-4248	85	105	therefore	therefore	ADV
ejpam-4248	85	106	a	a	PRON
ejpam-4248	85	107	is	be	AUX
ejpam-4248	85	108	coc	coc	ADJ
ejpam-4248	85	109	-	-	PUNCT
ejpam-4248	85	110	weakly	weakly	ADJ
ejpam-4248	85	111	-	-	PUNCT
ejpam-4248	85	112	compact	compact	ADJ
ejpam-4248	85	113	relative	relative	NOUN
ejpam-4248	85	114	to	to	AUX
ejpam-4248	85	115	x.	x.	VERB
ejpam-4248	85	116	the	the	DET
ejpam-4248	85	117	next	next	ADJ
ejpam-4248	85	118	definition	definition	NOUN
ejpam-4248	85	119	makes	make	VERB
ejpam-4248	85	120	our	our	PRON
ejpam-4248	85	121	coc	coc	NOUN
ejpam-4248	85	122	-	-	PUNCT
ejpam-4248	85	123	weakly	weakly	ADJ
ejpam-4248	85	124	covering	covering	NOUN
ejpam-4248	85	125	spaces	space	NOUN
ejpam-4248	85	126	equivalent	equivalent	ADJ
ejpam-4248	85	127	.	.	PUNCT
ejpam-4248	86	1	definition	definition	NOUN
ejpam-4248	86	2	13	13	NUM
ejpam-4248	86	3	.	.	PUNCT
ejpam-4248	87	1	a	a	DET
ejpam-4248	87	2	space	space	NOUN
ejpam-4248	87	3	x	x	PUNCT
ejpam-4248	87	4	is	be	AUX
ejpam-4248	87	5	said	say	VERB
ejpam-4248	87	6	to	to	PART
ejpam-4248	87	7	be	be	AUX
ejpam-4248	87	8	coc	coc	VERB
ejpam-4248	87	9	-	-	PUNCT
ejpam-4248	87	10	almost	almost	ADV
ejpam-4248	87	11	regular	regular	ADJ
ejpam-4248	87	12	if	if	SCONJ
ejpam-4248	87	13	for	for	ADP
ejpam-4248	87	14	each	each	DET
ejpam-4248	87	15	coc	coc	NOUN
ejpam-4248	87	16	-	-	PUNCT
ejpam-4248	87	17	regular	regular	ADJ
ejpam-4248	87	18	closed	closed	ADJ
ejpam-4248	87	19	f	f	NOUN
ejpam-4248	87	20	and	and	CCONJ
ejpam-4248	87	21	x	x	SYM
ejpam-4248	87	22	∈	∈	PROPN
ejpam-4248	87	23	x	x	X
ejpam-4248	87	24	−	−	PROPN
ejpam-4248	87	25	f	f	NOUN
ejpam-4248	87	26	,	,	PUNCT
ejpam-4248	87	27	there	there	PRON
ejpam-4248	87	28	exist	exist	VERB
ejpam-4248	87	29	disjoint	disjoint	ADJ
ejpam-4248	87	30	coc	coc	ADJ
ejpam-4248	87	31	-	-	PUNCT
ejpam-4248	87	32	open	open	ADJ
ejpam-4248	87	33	sets	set	NOUN
ejpam-4248	87	34	u	u	NOUN
ejpam-4248	87	35	,	,	PUNCT
ejpam-4248	87	36	v	v	ADP
ejpam-4248	87	37	such	such	ADJ
ejpam-4248	87	38	that	that	SCONJ
ejpam-4248	87	39	f	f	PROPN
ejpam-4248	87	40	⊆	⊆	NUM
ejpam-4248	87	41	u	u	NOUN
ejpam-4248	87	42	and	and	CCONJ
ejpam-4248	87	43	x	x	SYM
ejpam-4248	87	44	∈	∈	PROPN
ejpam-4248	87	45	v	v	NOUN
ejpam-4248	87	46	.	.	PUNCT
ejpam-4248	88	1	theorem	theorem	NOUN
ejpam-4248	88	2	5	5	NUM
ejpam-4248	88	3	.	.	PUNCT
ejpam-4248	89	1	if	if	SCONJ
ejpam-4248	89	2	a	a	DET
ejpam-4248	89	3	space	space	NOUN
ejpam-4248	89	4	x	x	PUNCT
ejpam-4248	89	5	is	be	AUX
ejpam-4248	89	6	coc	coc	ADJ
ejpam-4248	89	7	-	-	PUNCT
ejpam-4248	89	8	weakly	weakly	ADJ
ejpam-4248	89	9	-	-	PUNCT
ejpam-4248	89	10	compact	compact	ADJ
ejpam-4248	89	11	and	and	CCONJ
ejpam-4248	89	12	coc	coc	NOUN
ejpam-4248	89	13	-	-	PUNCT
ejpam-4248	89	14	almost	almost	ADV
ejpam-4248	89	15	-	-	PUNCT
ejpam-4248	89	16	regular	regular	ADJ
ejpam-4248	89	17	,	,	PUNCT
ejpam-4248	89	18	then	then	ADV
ejpam-4248	89	19	x	x	PUNCT
ejpam-4248	89	20	is	be	AUX
ejpam-4248	89	21	cocnearly	cocnearly	ADV
ejpam-4248	89	22	-	-	PUNCT
ejpam-4248	89	23	compact	compact	ADJ
ejpam-4248	89	24	.	.	PUNCT
ejpam-4248	90	1	proof	proof	NOUN
ejpam-4248	90	2	.	.	PUNCT
ejpam-4248	91	1	let	let	VERB
ejpam-4248	91	2	u	u	PRON
ejpam-4248	91	3	=	=	X
ejpam-4248	91	4	{	{	PUNCT
ejpam-4248	91	5	uα|α	uα|α	PROPN
ejpam-4248	91	6	∈	∈	PROPN
ejpam-4248	91	7	∆	∆	PROPN
ejpam-4248	91	8	}	}	PUNCT
ejpam-4248	91	9	be	be	AUX
ejpam-4248	91	10	coc	coc	ADJ
ejpam-4248	91	11	-	-	PUNCT
ejpam-4248	91	12	regular	regular	ADJ
ejpam-4248	91	13	open	open	ADJ
ejpam-4248	91	14	cover	cover	NOUN
ejpam-4248	91	15	of	of	ADP
ejpam-4248	91	16	x	x	PRON
ejpam-4248	91	17	,	,	PUNCT
ejpam-4248	91	18	so	so	ADV
ejpam-4248	91	19	for	for	ADP
ejpam-4248	91	20	each	each	DET
ejpam-4248	91	21	x	x	SYM
ejpam-4248	91	22	∈	∈	PROPN
ejpam-4248	91	23	x	x	NOUN
ejpam-4248	91	24	,	,	PUNCT
ejpam-4248	91	25	there	there	PRON
ejpam-4248	91	26	exists	exist	VERB
ejpam-4248	91	27	α(x	α(x	NOUN
ejpam-4248	91	28	)	)	PUNCT
ejpam-4248	91	29	∈	∈	PROPN
ejpam-4248	91	30	∆	∆	PROPN
ejpam-4248	91	31	such	such	ADJ
ejpam-4248	91	32	that	that	SCONJ
ejpam-4248	91	33	x	x	SYM
ejpam-4248	91	34	∈	∈	NOUN
ejpam-4248	91	35	uα(x	uα(x	NOUN
ejpam-4248	91	36	)	)	PUNCT
ejpam-4248	91	37	,	,	PUNCT
ejpam-4248	91	38	but	but	CCONJ
ejpam-4248	91	39	x	x	X
ejpam-4248	91	40	is	be	AUX
ejpam-4248	91	41	coc	coc	ADJ
ejpam-4248	91	42	-	-	PUNCT
ejpam-4248	91	43	almost	almost	ADV
ejpam-4248	91	44	regular	regular	ADJ
ejpam-4248	91	45	space	space	NOUN
ejpam-4248	91	46	,	,	PUNCT
ejpam-4248	91	47	so	so	SCONJ
ejpam-4248	91	48	there	there	PRON
ejpam-4248	91	49	exist	exist	VERB
ejpam-4248	91	50	coc	coc	ADJ
ejpam-4248	91	51	-	-	PUNCT
ejpam-4248	91	52	regular	regular	ADJ
ejpam-4248	91	53	open	open	ADJ
ejpam-4248	91	54	sets	set	NOUN
ejpam-4248	91	55	vα(x	vα(x	NOUN
ejpam-4248	91	56	)	)	PUNCT
ejpam-4248	91	57	and	and	CCONJ
ejpam-4248	91	58	wα(x	wα(x	NOUN
ejpam-4248	91	59	)	)	PUNCT
ejpam-4248	91	60	such	such	ADJ
ejpam-4248	91	61	that	that	SCONJ
ejpam-4248	91	62	x	x	SYM
ejpam-4248	91	63	∈	∈	PROPN
ejpam-4248	91	64	vα(x	vα(x	NOUN
ejpam-4248	91	65	)	)	PUNCT
ejpam-4248	91	66	⊆	⊆	NUM
ejpam-4248	91	67	vα(x	vα(x	NOUN
ejpam-4248	91	68	)	)	PUNCT
ejpam-4248	91	69	coc	coc	NOUN
ejpam-4248	91	70	⊆	⊆	NUM
ejpam-4248	91	71	wα(x	wα(x	NUM
ejpam-4248	91	72	)	)	PUNCT
ejpam-4248	91	73	⊆	⊆	NUM
ejpam-4248	91	74	wα(x	wα(x	NUM
ejpam-4248	91	75	)	)	PUNCT
ejpam-4248	91	76	coc	coc	VERB
ejpam-4248	91	77	⊆	⊆	NUM
ejpam-4248	91	78	uα(x	uα(x	NOUN
ejpam-4248	91	79	)	)	PUNCT
ejpam-4248	91	80	,	,	PUNCT
ejpam-4248	91	81	and	and	CCONJ
ejpam-4248	91	82	clearly	clearly	ADV
ejpam-4248	91	83	the	the	DET
ejpam-4248	91	84	family	family	NOUN
ejpam-4248	91	85	{	{	PUNCT
ejpam-4248	91	86	wα(x)|α(x	wα(x)|α(x	PROPN
ejpam-4248	91	87	)	)	PUNCT
ejpam-4248	91	88	∈	∈	PROPN
ejpam-4248	91	89	∆	∆	PROPN
ejpam-4248	91	90	}	}	PUNCT
ejpam-4248	91	91	is	be	AUX
ejpam-4248	91	92	coc	coc	ADJ
ejpam-4248	91	93	-	-	PUNCT
ejpam-4248	91	94	regular	regular	ADJ
ejpam-4248	91	95	cover	cover	NOUN
ejpam-4248	91	96	of	of	ADP
ejpam-4248	91	97	x	x	PRON
ejpam-4248	91	98	,	,	PUNCT
ejpam-4248	91	99	so	so	CCONJ
ejpam-4248	91	100	there	there	PRON
ejpam-4248	91	101	exists	exist	VERB
ejpam-4248	91	102	a	a	DET
ejpam-4248	91	103	finite	finite	ADJ
ejpam-4248	91	104	elements	element	NOUN
ejpam-4248	91	105	x1	x1	NUM
ejpam-4248	91	106	,	,	PUNCT
ejpam-4248	91	107	x2	x2	PROPN
ejpam-4248	91	108	,	,	PUNCT
ejpam-4248	91	109	...	...	PUNCT
ejpam-4248	91	110	,	,	PUNCT
ejpam-4248	91	111	xn	xn	PUNCT
ejpam-4248	91	112	∈	∈	PROPN
ejpam-4248	91	113	x	x	PUNCT
ejpam-4248	91	114	such	such	ADJ
ejpam-4248	91	115	that	that	SCONJ
ejpam-4248	91	116	x	x	NOUN
ejpam-4248	91	117	=	=	PRON
ejpam-4248	91	118	n⋃	n⋃	PROPN
ejpam-4248	91	119	i=1	i=1	PROPN
ejpam-4248	91	120	wα(xi	wα(xi	X
ejpam-4248	91	121	)	)	PUNCT
ejpam-4248	91	122	coc	coc	NOUN
ejpam-4248	91	123	⊆	⊆	NUM
ejpam-4248	91	124	n⋃	n⋃	NOUN
ejpam-4248	91	125	i=1	i=1	PROPN
ejpam-4248	91	126	uα(xi	uα(xi	PROPN
ejpam-4248	91	127	)	)	PUNCT
ejpam-4248	91	128	,	,	PUNCT
ejpam-4248	91	129	therefore	therefore	ADV
ejpam-4248	91	130	x	x	X
ejpam-4248	91	131	is	be	AUX
ejpam-4248	91	132	coc	coc	ADJ
ejpam-4248	91	133	-	-	PUNCT
ejpam-4248	91	134	nearly	nearly	ADV
ejpam-4248	91	135	-	-	PUNCT
ejpam-4248	91	136	compact	compact	ADJ
ejpam-4248	91	137	space	space	NOUN
ejpam-4248	91	138	.	.	PUNCT
ejpam-4248	92	1	f.a	f.a	PROPN
ejpam-4248	92	2	.	.	PROPN
ejpam-4248	92	3	abushaheen	abushaheen	PROPN
ejpam-4248	92	4	,	,	PUNCT
ejpam-4248	92	5	f.	f.	PROPN
ejpam-4248	92	6	alrimawi	alrimawi	PROPN
ejpam-4248	92	7	/	/	SYM
ejpam-4248	92	8	eur	eur	PROPN
ejpam-4248	92	9	.	.	PUNCT
ejpam-4248	93	1	j.	j.	PROPN
ejpam-4248	93	2	pure	pure	PROPN
ejpam-4248	93	3	appl	appl	PROPN
ejpam-4248	93	4	.	.	PROPN
ejpam-4248	93	5	math	math	PROPN
ejpam-4248	93	6	,	,	PUNCT
ejpam-4248	93	7	15	15	NUM
ejpam-4248	93	8	(	(	PUNCT
ejpam-4248	93	9	1	1	NUM
ejpam-4248	93	10	)	)	PUNCT
ejpam-4248	93	11	(	(	PUNCT
ejpam-4248	93	12	2022	2022	NUM
ejpam-4248	93	13	)	)	PUNCT
ejpam-4248	93	14	,	,	PUNCT
ejpam-4248	93	15	199	199	NUM
ejpam-4248	93	16	-	-	SYM
ejpam-4248	93	17	206	206	NUM
ejpam-4248	93	18	203	203	NUM
ejpam-4248	93	19	corollary	corollary	ADJ
ejpam-4248	93	20	4	4	NUM
ejpam-4248	93	21	.	.	PUNCT
ejpam-4248	94	1	a	a	DET
ejpam-4248	94	2	coc	coc	NOUN
ejpam-4248	94	3	-	-	PUNCT
ejpam-4248	94	4	almost	almost	ADV
ejpam-4248	94	5	-	-	PUNCT
ejpam-4248	94	6	regular	regular	ADJ
ejpam-4248	94	7	space	space	NOUN
ejpam-4248	94	8	is	be	AUX
ejpam-4248	94	9	coc	coc	ADJ
ejpam-4248	94	10	-	-	PUNCT
ejpam-4248	94	11	weakly	weakly	ADJ
ejpam-4248	94	12	-	-	PUNCT
ejpam-4248	94	13	compact	compact	ADJ
ejpam-4248	94	14	if	if	SCONJ
ejpam-4248	95	1	and	and	CCONJ
ejpam-4248	95	2	only	only	ADV
ejpam-4248	95	3	if	if	SCONJ
ejpam-4248	95	4	it	it	PRON
ejpam-4248	95	5	is	be	AUX
ejpam-4248	95	6	cocnearly	cocnearly	ADV
ejpam-4248	95	7	-	-	PUNCT
ejpam-4248	95	8	compact	compact	ADJ
ejpam-4248	95	9	.	.	PUNCT
ejpam-4248	96	1	proof	proof	NOUN
ejpam-4248	96	2	.	.	PUNCT
ejpam-4248	97	1	obvious	obvious	ADJ
ejpam-4248	97	2	.	.	PUNCT
ejpam-4248	98	1	corollary	corollary	ADJ
ejpam-4248	98	2	5	5	NUM
ejpam-4248	98	3	.	.	PUNCT
ejpam-4248	99	1	a	a	DET
ejpam-4248	99	2	coc	coc	PROPN
ejpam-4248	99	3	-	-	PUNCT
ejpam-4248	99	4	t2	t2	NOUN
ejpam-4248	99	5	-	-	PUNCT
ejpam-4248	99	6	space	space	NOUN
ejpam-4248	99	7	is	be	AUX
ejpam-4248	99	8	coc	coc	ADJ
ejpam-4248	99	9	-	-	PUNCT
ejpam-4248	99	10	nearly	nearly	ADV
ejpam-4248	99	11	-	-	PUNCT
ejpam-4248	99	12	compact	compact	ADJ
ejpam-4248	99	13	if	if	SCONJ
ejpam-4248	100	1	and	and	CCONJ
ejpam-4248	100	2	only	only	ADV
ejpam-4248	100	3	if	if	SCONJ
ejpam-4248	100	4	it	it	PRON
ejpam-4248	100	5	is	be	AUX
ejpam-4248	100	6	coc	coc	ADJ
ejpam-4248	100	7	-	-	PUNCT
ejpam-4248	100	8	weakly	weakly	ADJ
ejpam-4248	100	9	compact	compact	ADJ
ejpam-4248	100	10	and	and	CCONJ
ejpam-4248	100	11	coc	coc	NOUN
ejpam-4248	100	12	-	-	PUNCT
ejpam-4248	100	13	almost	almost	ADV
ejpam-4248	100	14	regular	regular	ADJ
ejpam-4248	100	15	.	.	PUNCT
ejpam-4248	101	1	proof	proof	NOUN
ejpam-4248	101	2	.	.	PUNCT
ejpam-4248	102	1	(	(	PUNCT
ejpam-4248	102	2	⇐	⇐	NOUN
ejpam-4248	102	3	)	)	PUNCT
ejpam-4248	102	4	clearly	clearly	ADV
ejpam-4248	102	5	.	.	PUNCT
ejpam-4248	103	1	(	(	PUNCT
ejpam-4248	103	2	⇒	⇒	NOUN
ejpam-4248	103	3	)	)	PUNCT
ejpam-4248	103	4	enough	enough	ADV
ejpam-4248	103	5	to	to	PART
ejpam-4248	103	6	show	show	VERB
ejpam-4248	103	7	that	that	SCONJ
ejpam-4248	103	8	x	x	PRON
ejpam-4248	103	9	is	be	AUX
ejpam-4248	103	10	coc	coc	ADJ
ejpam-4248	103	11	-	-	PUNCT
ejpam-4248	103	12	almost	almost	ADV
ejpam-4248	103	13	regular	regular	ADJ
ejpam-4248	103	14	.	.	PUNCT
ejpam-4248	104	1	let	let	VERB
ejpam-4248	104	2	a	a	PRON
ejpam-4248	104	3	be	be	AUX
ejpam-4248	104	4	a	a	DET
ejpam-4248	104	5	coc	coc	ADJ
ejpam-4248	104	6	-	-	PUNCT
ejpam-4248	104	7	regular	regular	ADJ
ejpam-4248	104	8	closed	close	VERB
ejpam-4248	104	9	set	set	NOUN
ejpam-4248	104	10	and	and	CCONJ
ejpam-4248	104	11	x	x	NOUN
ejpam-4248	104	12	−	−	PROPN
ejpam-4248	104	13	a	a	X
ejpam-4248	104	14	,	,	PUNCT
ejpam-4248	104	15	assume	assume	VERB
ejpam-4248	104	16	y	y	PROPN
ejpam-4248	104	17	∈	∈	PROPN
ejpam-4248	104	18	a.	a.	NOUN
ejpam-4248	104	19	since	since	SCONJ
ejpam-4248	104	20	x	x	PROPN
ejpam-4248	104	21	is	be	AUX
ejpam-4248	104	22	coc	coc	ADJ
ejpam-4248	104	23	-	-	PUNCT
ejpam-4248	104	24	t2	t2	NOUN
ejpam-4248	104	25	-	-	PUNCT
ejpam-4248	104	26	space	space	NOUN
ejpam-4248	104	27	,	,	PUNCT
ejpam-4248	104	28	there	there	PRON
ejpam-4248	104	29	exists	exist	VERB
ejpam-4248	104	30	a	a	DET
ejpam-4248	104	31	coc	coc	NOUN
ejpam-4248	104	32	-	-	PUNCT
ejpam-4248	104	33	open	open	ADJ
ejpam-4248	104	34	set	set	NOUN
ejpam-4248	104	35	vy	vy	NOUN
ejpam-4248	104	36	contains	contain	VERB
ejpam-4248	104	37	y	y	PROPN
ejpam-4248	104	38	with	with	ADP
ejpam-4248	104	39	x	x	PROPN
ejpam-4248	104	40	/∈	/∈	PROPN
ejpam-4248	104	41	vy	vy	PROPN
ejpam-4248	104	42	coc	coc	PROPN
ejpam-4248	104	43	.	.	PUNCT
ejpam-4248	105	1	the	the	DET
ejpam-4248	105	2	family	family	NOUN
ejpam-4248	105	3	v	v	NOUN
ejpam-4248	105	4	=	=	PUNCT
ejpam-4248	105	5	{	{	PUNCT
ejpam-4248	105	6	vy|y	vy|y	NOUN
ejpam-4248	105	7	∈	∈	PROPN
ejpam-4248	105	8	a	a	DET
ejpam-4248	105	9	}	}	PUNCT
ejpam-4248	105	10	∪	∪	NOUN
ejpam-4248	105	11	{	{	PUNCT
ejpam-4248	105	12	x	x	NOUN
ejpam-4248	105	13	−	−	PROPN
ejpam-4248	105	14	a	a	PRON
ejpam-4248	105	15	}	}	PUNCT
ejpam-4248	105	16	is	be	AUX
ejpam-4248	105	17	coc	coc	ADJ
ejpam-4248	105	18	-	-	PUNCT
ejpam-4248	105	19	regular	regular	ADJ
ejpam-4248	105	20	open	open	ADJ
ejpam-4248	105	21	cover	cover	NOUN
ejpam-4248	105	22	of	of	ADP
ejpam-4248	105	23	x	x	PRON
ejpam-4248	105	24	,	,	PUNCT
ejpam-4248	105	25	so	so	CCONJ
ejpam-4248	105	26	there	there	PRON
ejpam-4248	105	27	exists	exist	VERB
ejpam-4248	105	28	a	a	DET
ejpam-4248	105	29	finite	finite	PROPN
ejpam-4248	105	30	subcover	subcover	PROPN
ejpam-4248	105	31	v∗	v∗	PROPN
ejpam-4248	105	32	of	of	ADP
ejpam-4248	105	33	x	x	PRON
ejpam-4248	105	34	,	,	PUNCT
ejpam-4248	105	35	let	let	VERB
ejpam-4248	105	36	v	v	NOUN
ejpam-4248	105	37	=	=	SYM
ejpam-4248	105	38	⋃	⋃	X
ejpam-4248	105	39	{	{	PUNCT
ejpam-4248	105	40	vα	vα	ADP
ejpam-4248	105	41	∈	∈	PROPN
ejpam-4248	105	42	v∗|vα	v∗|vα	NUM
ejpam-4248	105	43	∩	∩	NOUN
ejpam-4248	105	44	a	a	DET
ejpam-4248	105	45	̸=	̸=	PROPN
ejpam-4248	105	46	ϕ	ϕ	PROPN
ejpam-4248	105	47	}	}	PUNCT
ejpam-4248	105	48	,	,	PUNCT
ejpam-4248	105	49	then	then	ADV
ejpam-4248	105	50	v	v	NOUN
ejpam-4248	105	51	is	be	AUX
ejpam-4248	105	52	coc	coc	ADJ
ejpam-4248	105	53	-	-	PUNCT
ejpam-4248	105	54	open	open	ADJ
ejpam-4248	105	55	subset	subset	NOUN
ejpam-4248	105	56	of	of	ADP
ejpam-4248	105	57	x	x	X
ejpam-4248	105	58	and	and	CCONJ
ejpam-4248	105	59	a	a	DET
ejpam-4248	105	60	⊆	⊆	NUM
ejpam-4248	105	61	v	v	NOUN
ejpam-4248	105	62	,	,	PUNCT
ejpam-4248	105	63	therefore	therefore	ADV
ejpam-4248	105	64	for	for	ADP
ejpam-4248	105	65	each	each	DET
ejpam-4248	105	66	vα	vα	ADP
ejpam-4248	105	67	∈	∈	PROPN
ejpam-4248	105	68	v∗	v∗	NOUN
ejpam-4248	105	69	we	we	PRON
ejpam-4248	105	70	have	have	VERB
ejpam-4248	105	71	vα	vα	INTJ
ejpam-4248	105	72	⊆	⊆	NUM
ejpam-4248	105	73	vy	vy	NOUN
ejpam-4248	105	74	and	and	CCONJ
ejpam-4248	105	75	x	x	PROPN
ejpam-4248	105	76	/∈	/∈	PROPN
ejpam-4248	105	77	v	v	NUM
ejpam-4248	105	78	coc	coc	NOUN
ejpam-4248	105	79	,	,	PUNCT
ejpam-4248	105	80	hence	hence	ADV
ejpam-4248	105	81	x	x	VERB
ejpam-4248	105	82	is	be	AUX
ejpam-4248	105	83	coc	coc	ADJ
ejpam-4248	105	84	-	-	PUNCT
ejpam-4248	105	85	almost	almost	ADV
ejpam-4248	105	86	-	-	PUNCT
ejpam-4248	105	87	regular	regular	ADJ
ejpam-4248	105	88	.	.	PUNCT
ejpam-4248	106	1	corollary	corollary	ADJ
ejpam-4248	106	2	6	6	NUM
ejpam-4248	106	3	.	.	PUNCT
ejpam-4248	107	1	the	the	DET
ejpam-4248	107	2	following	follow	VERB
ejpam-4248	107	3	are	be	AUX
ejpam-4248	107	4	equivalent	equivalent	ADJ
ejpam-4248	107	5	for	for	ADP
ejpam-4248	107	6	a	a	DET
ejpam-4248	107	7	coc	coc	NOUN
ejpam-4248	107	8	-	-	PUNCT
ejpam-4248	107	9	almost	almost	ADV
ejpam-4248	107	10	-	-	PUNCT
ejpam-4248	107	11	regular	regular	ADJ
ejpam-4248	107	12	space	space	NOUN
ejpam-4248	107	13	:	:	PUNCT
ejpam-4248	107	14	(	(	PUNCT
ejpam-4248	107	15	i	i	NOUN
ejpam-4248	107	16	)	)	PUNCT
ejpam-4248	107	17	x	x	X
ejpam-4248	107	18	is	be	AUX
ejpam-4248	107	19	coc	coc	ADJ
ejpam-4248	107	20	-	-	PUNCT
ejpam-4248	107	21	nearly	nearly	ADV
ejpam-4248	107	22	-	-	PUNCT
ejpam-4248	107	23	compact	compact	ADJ
ejpam-4248	107	24	,	,	PUNCT
ejpam-4248	107	25	(	(	PUNCT
ejpam-4248	107	26	ii	ii	NOUN
ejpam-4248	107	27	)	)	PUNCT
ejpam-4248	107	28	x	x	X
ejpam-4248	107	29	is	be	AUX
ejpam-4248	107	30	coc	coc	ADJ
ejpam-4248	107	31	-	-	PUNCT
ejpam-4248	107	32	almost	almost	ADV
ejpam-4248	107	33	-	-	PUNCT
ejpam-4248	107	34	compact	compact	ADJ
ejpam-4248	107	35	,	,	PUNCT
ejpam-4248	107	36	(	(	PUNCT
ejpam-4248	107	37	iii	iii	X
ejpam-4248	107	38	)	)	PUNCT
ejpam-4248	107	39	x	x	X
ejpam-4248	107	40	is	be	AUX
ejpam-4248	107	41	coc	coc	ADJ
ejpam-4248	107	42	-	-	PUNCT
ejpam-4248	107	43	r	r	NOUN
ejpam-4248	107	44	-	-	PUNCT
ejpam-4248	107	45	compact	compact	ADJ
ejpam-4248	107	46	,	,	PUNCT
ejpam-4248	107	47	(	(	PUNCT
ejpam-4248	107	48	iv	iv	X
ejpam-4248	107	49	)	)	PUNCT
ejpam-4248	107	50	x	x	X
ejpam-4248	107	51	is	be	AUX
ejpam-4248	107	52	coc	coc	ADJ
ejpam-4248	107	53	-	-	PUNCT
ejpam-4248	107	54	weakly	weakly	ADJ
ejpam-4248	107	55	-	-	PUNCT
ejpam-4248	107	56	compact	compact	ADJ
ejpam-4248	107	57	.	.	PUNCT
ejpam-4248	108	1	proof	proof	NOUN
ejpam-4248	108	2	.	.	PUNCT
ejpam-4248	109	1	the	the	DET
ejpam-4248	109	2	prove	prove	NOUN
ejpam-4248	109	3	comes	come	VERB
ejpam-4248	109	4	from	from	ADP
ejpam-4248	109	5	theorems	theorem	NOUN
ejpam-4248	109	6	5	5	NUM
ejpam-4248	109	7	and	and	CCONJ
ejpam-4248	109	8	1	1	NUM
ejpam-4248	109	9	.	.	PUNCT
ejpam-4248	109	10	corollary	corollary	ADJ
ejpam-4248	109	11	7	7	NUM
ejpam-4248	109	12	.	.	PUNCT
ejpam-4248	110	1	let	let	AUX
ejpam-4248	110	2	(	(	PUNCT
ejpam-4248	110	3	x	x	NOUN
ejpam-4248	110	4	,	,	PUNCT
ejpam-4248	110	5	τk	τk	ADV
ejpam-4248	110	6	)	)	PUNCT
ejpam-4248	110	7	be	be	AUX
ejpam-4248	110	8	a	a	DET
ejpam-4248	110	9	t3	t3	NOUN
ejpam-4248	110	10	-	-	PUNCT
ejpam-4248	110	11	space	space	NOUN
ejpam-4248	110	12	and	and	CCONJ
ejpam-4248	110	13	x	x	NOUN
ejpam-4248	110	14	is	be	AUX
ejpam-4248	110	15	coc	coc	ADJ
ejpam-4248	110	16	-	-	PUNCT
ejpam-4248	110	17	weakly	weakly	ADJ
ejpam-4248	110	18	-	-	PUNCT
ejpam-4248	110	19	compact	compact	ADJ
ejpam-4248	110	20	,	,	PUNCT
ejpam-4248	110	21	then	then	ADV
ejpam-4248	110	22	x	x	PUNCT
ejpam-4248	110	23	is	be	AUX
ejpam-4248	110	24	coccompact	coccompact	ADJ
ejpam-4248	110	25	space	space	NOUN
ejpam-4248	110	26	.	.	PUNCT
ejpam-4248	111	1	proof	proof	NOUN
ejpam-4248	111	2	.	.	PUNCT
ejpam-4248	112	1	the	the	DET
ejpam-4248	112	2	result	result	NOUN
ejpam-4248	112	3	comes	come	VERB
ejpam-4248	112	4	from	from	ADP
ejpam-4248	112	5	theorem	theorem	ADJ
ejpam-4248	112	6	5	5	NUM
ejpam-4248	112	7	and	and	CCONJ
ejpam-4248	112	8	the	the	DET
ejpam-4248	112	9	fact	fact	NOUN
ejpam-4248	112	10	that	that	SCONJ
ejpam-4248	112	11	every	every	DET
ejpam-4248	112	12	t3	t3	NOUN
ejpam-4248	112	13	-	-	PUNCT
ejpam-4248	112	14	space	space	NOUN
ejpam-4248	112	15	in	in	ADP
ejpam-4248	112	16	(	(	PUNCT
ejpam-4248	112	17	x	x	NOUN
ejpam-4248	112	18	,	,	PUNCT
ejpam-4248	112	19	τk	τk	ADV
ejpam-4248	112	20	)	)	PUNCT
ejpam-4248	112	21	is	be	AUX
ejpam-4248	112	22	coc	coc	ADJ
ejpam-4248	112	23	-	-	PUNCT
ejpam-4248	112	24	almost	almost	ADV
ejpam-4248	112	25	regular	regular	ADJ
ejpam-4248	112	26	.	.	PUNCT
ejpam-4248	113	1	definition	definition	NOUN
ejpam-4248	113	2	14	14	NUM
ejpam-4248	113	3	.	.	PUNCT
ejpam-4248	114	1	a	a	DET
ejpam-4248	114	2	function	function	NOUN
ejpam-4248	114	3	f	f	NOUN
ejpam-4248	114	4	:	:	PUNCT
ejpam-4248	114	5	x	x	X
ejpam-4248	114	6	→	→	SYM
ejpam-4248	114	7	y	y	PROPN
ejpam-4248	114	8	is	be	AUX
ejpam-4248	114	9	said	say	VERB
ejpam-4248	114	10	to	to	PART
ejpam-4248	114	11	be	be	AUX
ejpam-4248	114	12	coc	coc	VERB
ejpam-4248	114	13	-	-	PUNCT
ejpam-4248	114	14	almost	almost	ADV
ejpam-4248	114	15	open	open	ADJ
ejpam-4248	114	16	function	function	NOUN
ejpam-4248	114	17	if	if	SCONJ
ejpam-4248	114	18	f(u	f(u	PROPN
ejpam-4248	114	19	)	)	PUNCT
ejpam-4248	114	20	⊆	⊆	NUM
ejpam-4248	114	21	intcoc	intcoc	PROPN
ejpam-4248	114	22	(	(	PUNCT
ejpam-4248	114	23	f(u	f(u	PROPN
ejpam-4248	114	24	)	)	PUNCT
ejpam-4248	114	25	coc	coc	PROPN
ejpam-4248	114	26	)	)	PUNCT
ejpam-4248	114	27	for	for	ADP
ejpam-4248	114	28	every	every	DET
ejpam-4248	114	29	coc	coc	NOUN
ejpam-4248	114	30	-	-	PUNCT
ejpam-4248	114	31	open	open	ADJ
ejpam-4248	114	32	set	set	NOUN
ejpam-4248	114	33	u	u	NOUN
ejpam-4248	114	34	of	of	ADP
ejpam-4248	114	35	x.	x.	PROPN
ejpam-4248	114	36	lemma	lemma	PROPN
ejpam-4248	115	1	1	1	NUM
ejpam-4248	115	2	.	.	PUNCT
ejpam-4248	116	1	a	a	DET
ejpam-4248	116	2	function	function	NOUN
ejpam-4248	116	3	f	f	NOUN
ejpam-4248	116	4	:	:	PUNCT
ejpam-4248	116	5	x	x	X
ejpam-4248	116	6	→	→	SYM
ejpam-4248	116	7	y	y	PROPN
ejpam-4248	116	8	is	be	AUX
ejpam-4248	116	9	coc	coc	ADJ
ejpam-4248	116	10	-	-	PUNCT
ejpam-4248	116	11	almost	almost	ADV
ejpam-4248	116	12	open	open	ADJ
ejpam-4248	116	13	if	if	SCONJ
ejpam-4248	116	14	and	and	CCONJ
ejpam-4248	116	15	only	only	ADV
ejpam-4248	116	16	if	if	SCONJ
ejpam-4248	116	17	f−1	f−1	PROPN
ejpam-4248	116	18	(	(	PUNCT
ejpam-4248	116	19	v	v	X
ejpam-4248	116	20	coc	coc	NOUN
ejpam-4248	116	21	)	)	PUNCT
ejpam-4248	116	22	⊆	⊆	NUM
ejpam-4248	116	23	f−1	f−1	PROPN
ejpam-4248	116	24	(	(	PUNCT
ejpam-4248	116	25	v	v	NOUN
ejpam-4248	116	26	)	)	PUNCT
ejpam-4248	116	27	coc	coc	NOUN
ejpam-4248	116	28	for	for	ADP
ejpam-4248	116	29	each	each	DET
ejpam-4248	116	30	coc	coc	NOUN
ejpam-4248	116	31	-	-	PUNCT
ejpam-4248	116	32	open	open	NOUN
ejpam-4248	116	33	subset	subset	NOUN
ejpam-4248	116	34	v	v	NOUN
ejpam-4248	116	35	of	of	ADP
ejpam-4248	116	36	y	y	PROPN
ejpam-4248	116	37	.	.	PUNCT
ejpam-4248	117	1	proof	proof	NOUN
ejpam-4248	117	2	.	.	PUNCT
ejpam-4248	118	1	(	(	PUNCT
ejpam-4248	118	2	⇒	⇒	NOUN
ejpam-4248	118	3	):	):	PUNCT
ejpam-4248	118	4	let	let	VERB
ejpam-4248	118	5	v	v	PART
ejpam-4248	118	6	be	be	AUX
ejpam-4248	118	7	a	a	DET
ejpam-4248	118	8	coc	coc	NOUN
ejpam-4248	118	9	-	-	PUNCT
ejpam-4248	118	10	open	open	ADJ
ejpam-4248	118	11	subset	subset	NOUN
ejpam-4248	118	12	of	of	ADP
ejpam-4248	118	13	y	y	PROPN
ejpam-4248	118	14	with	with	ADP
ejpam-4248	118	15	x	x	PROPN
ejpam-4248	118	16	∈	∈	PROPN
ejpam-4248	118	17	f−1	f−1	PROPN
ejpam-4248	118	18	(	(	PUNCT
ejpam-4248	118	19	v	v	X
ejpam-4248	118	20	coc	coc	NOUN
ejpam-4248	118	21	)	)	PUNCT
ejpam-4248	118	22	and	and	CCONJ
ejpam-4248	118	23	u	u	NOUN
ejpam-4248	118	24	is	be	AUX
ejpam-4248	118	25	a	a	DET
ejpam-4248	118	26	coc	coc	ADJ
ejpam-4248	118	27	-	-	PUNCT
ejpam-4248	118	28	open	open	ADJ
ejpam-4248	118	29	subset	subset	ADJ
ejpam-4248	118	30	ofx	ofx	PROPN
ejpam-4248	118	31	contains	contain	VERB
ejpam-4248	118	32	x	x	PRON
ejpam-4248	118	33	,	,	PUNCT
ejpam-4248	118	34	then	then	ADV
ejpam-4248	118	35	f(x	f(x	PROPN
ejpam-4248	118	36	)	)	PUNCT
ejpam-4248	118	37	∈	∈	PROPN
ejpam-4248	118	38	f(u)∩v	f(u)∩v	NOUN
ejpam-4248	118	39	coc	coc	VERB
ejpam-4248	118	40	⊆	⊆	NUM
ejpam-4248	118	41	intcoc	intcoc	PROPN
ejpam-4248	118	42	(	(	PUNCT
ejpam-4248	118	43	u	u	NOUN
ejpam-4248	118	44	coc)∩v	coc)∩v	X
ejpam-4248	118	45	coc	coc	NOUN
ejpam-4248	118	46	,	,	PUNCT
ejpam-4248	118	47	so	so	ADV
ejpam-4248	118	48	v	v	ADP
ejpam-4248	118	49	∩intcoc	∩intcoc	PROPN
ejpam-4248	118	50	(	(	PUNCT
ejpam-4248	118	51	u	u	NOUN
ejpam-4248	118	52	coc	coc	PROPN
ejpam-4248	118	53	)	)	PUNCT
ejpam-4248	118	54	̸=	̸=	PROPN
ejpam-4248	118	55	ϕ	ϕ	PROPN
ejpam-4248	118	56	and	and	CCONJ
ejpam-4248	118	57	u	u	PROPN
ejpam-4248	118	58	∩	∩	ADJ
ejpam-4248	118	59	f−1(v	f−1(v	NOUN
ejpam-4248	118	60	)	)	PUNCT
ejpam-4248	119	1	̸=	̸=	PROPN
ejpam-4248	119	2	ϕ	ϕ	NOUN
ejpam-4248	119	3	,	,	PUNCT
ejpam-4248	119	4	therefore	therefore	ADV
ejpam-4248	119	5	x	x	PROPN
ejpam-4248	119	6	∈	∈	PROPN
ejpam-4248	119	7	f−1	f−1	PROPN
ejpam-4248	119	8	(	(	PUNCT
ejpam-4248	119	9	v	v	NOUN
ejpam-4248	119	10	)	)	PUNCT
ejpam-4248	119	11	coc	coc	NOUN
ejpam-4248	119	12	.	.	PUNCT
ejpam-4248	120	1	(	(	PUNCT
ejpam-4248	120	2	⇐	⇐	ADJ
ejpam-4248	120	3	):	):	PUNCT
ejpam-4248	120	4	if	if	SCONJ
ejpam-4248	120	5	f	f	PROPN
ejpam-4248	120	6	is	be	AUX
ejpam-4248	120	7	not	not	PART
ejpam-4248	120	8	coc	coc	ADJ
ejpam-4248	120	9	-	-	PUNCT
ejpam-4248	120	10	almost	almost	ADV
ejpam-4248	120	11	open	open	ADJ
ejpam-4248	120	12	function	function	NOUN
ejpam-4248	120	13	,	,	PUNCT
ejpam-4248	120	14	so	so	CCONJ
ejpam-4248	120	15	there	there	PRON
ejpam-4248	120	16	is	be	VERB
ejpam-4248	120	17	a	a	DET
ejpam-4248	120	18	coc	coc	NOUN
ejpam-4248	120	19	-	-	PUNCT
ejpam-4248	120	20	open	open	ADJ
ejpam-4248	120	21	set	set	NOUN
ejpam-4248	120	22	u	u	NOUN
ejpam-4248	120	23	with	with	ADP
ejpam-4248	120	24	f(u	f(u	PROPN
ejpam-4248	120	25	)	)	PUNCT
ejpam-4248	121	1	⊈	⊈	PROPN
ejpam-4248	121	2	intcocf	intcocf	NOUN
ejpam-4248	121	3	(	(	PUNCT
ejpam-4248	121	4	u	u	NOUN
ejpam-4248	121	5	coc	coc	PROPN
ejpam-4248	121	6	)	)	PUNCT
ejpam-4248	121	7	.	.	PUNCT
ejpam-4248	122	1	let	let	VERB
ejpam-4248	122	2	v	v	NOUN
ejpam-4248	122	3	=	=	SYM
ejpam-4248	122	4	y−f(u	y−f(u	ADJ
ejpam-4248	122	5	)	)	PUNCT
ejpam-4248	122	6	coc	coc	NOUN
ejpam-4248	122	7	,	,	PUNCT
ejpam-4248	122	8	then	then	ADV
ejpam-4248	122	9	v	v	ADP
ejpam-4248	122	10	∩f(u	∩f(u	ADJ
ejpam-4248	122	11	)	)	PUNCT
ejpam-4248	123	1	=	=	SYM
ejpam-4248	123	2	ϕ	ϕ	NOUN
ejpam-4248	123	3	,	,	PUNCT
ejpam-4248	123	4	but	but	CCONJ
ejpam-4248	123	5	v	v	ADP
ejpam-4248	123	6	⊆	⊆	NUM
ejpam-4248	123	7	y−f(u	y−f(u	NOUN
ejpam-4248	123	8	)	)	PUNCT
ejpam-4248	123	9	,	,	PUNCT
ejpam-4248	123	10	so	so	ADV
ejpam-4248	123	11	v	v	ADP
ejpam-4248	123	12	coc∩f(u	coc∩f(u	ADJ
ejpam-4248	123	13	)	)	PUNCT
ejpam-4248	124	1	̸=	̸=	PROPN
ejpam-4248	124	2	ϕ	ϕ	NOUN
ejpam-4248	124	3	,	,	PUNCT
ejpam-4248	124	4	so	so	SCONJ
ejpam-4248	124	5	u	u	NOUN
ejpam-4248	124	6	∩f−1	∩f−1	PUNCT
ejpam-4248	124	7	(	(	PUNCT
ejpam-4248	124	8	v	v	X
ejpam-4248	124	9	coc	coc	NOUN
ejpam-4248	124	10	)	)	PUNCT
ejpam-4248	124	11	̸=	̸=	PROPN
ejpam-4248	124	12	ϕ	ϕ	NOUN
ejpam-4248	124	13	and	and	CCONJ
ejpam-4248	124	14	u	u	PRON
ejpam-4248	124	15	∩f−1	∩f−1	SYM
ejpam-4248	124	16	(	(	PUNCT
ejpam-4248	124	17	v	v	NOUN
ejpam-4248	124	18	)	)	PUNCT
ejpam-4248	124	19	coc	coc	NOUN
ejpam-4248	124	20	̸=	̸=	PROPN
ejpam-4248	124	21	ϕ	ϕ	NOUN
ejpam-4248	124	22	,	,	PUNCT
ejpam-4248	124	23	hence	hence	ADV
ejpam-4248	124	24	u	u	NOUN
ejpam-4248	124	25	∩f−1(v	∩f−1(v	PUNCT
ejpam-4248	124	26	)	)	PUNCT
ejpam-4248	124	27	̸=	̸=	PROPN
ejpam-4248	124	28	ϕ	ϕ	NOUN
ejpam-4248	124	29	which	which	PRON
ejpam-4248	124	30	contradicts	contradict	VERB
ejpam-4248	124	31	the	the	DET
ejpam-4248	124	32	fact	fact	NOUN
ejpam-4248	124	33	that	that	SCONJ
ejpam-4248	124	34	f(u	f(u	PROPN
ejpam-4248	124	35	)	)	PUNCT
ejpam-4248	124	36	∩	∩	PROPN
ejpam-4248	124	37	v	v	ADP
ejpam-4248	124	38	=	=	SYM
ejpam-4248	124	39	ϕ	ϕ	NOUN
ejpam-4248	124	40	,	,	PUNCT
ejpam-4248	124	41	hence	hence	ADV
ejpam-4248	124	42	the	the	DET
ejpam-4248	124	43	result	result	NOUN
ejpam-4248	124	44	.	.	PUNCT
ejpam-4248	125	1	f.a	f.a	PROPN
ejpam-4248	125	2	.	.	PROPN
ejpam-4248	125	3	abushaheen	abushaheen	PROPN
ejpam-4248	125	4	,	,	PUNCT
ejpam-4248	125	5	f.	f.	PROPN
ejpam-4248	125	6	alrimawi	alrimawi	PROPN
ejpam-4248	125	7	/	/	SYM
ejpam-4248	125	8	eur	eur	PROPN
ejpam-4248	125	9	.	.	PUNCT
ejpam-4248	126	1	j.	j.	PROPN
ejpam-4248	126	2	pure	pure	PROPN
ejpam-4248	126	3	appl	appl	PROPN
ejpam-4248	126	4	.	.	PROPN
ejpam-4248	126	5	math	math	PROPN
ejpam-4248	126	6	,	,	PUNCT
ejpam-4248	126	7	15	15	NUM
ejpam-4248	126	8	(	(	PUNCT
ejpam-4248	126	9	1	1	NUM
ejpam-4248	126	10	)	)	PUNCT
ejpam-4248	126	11	(	(	PUNCT
ejpam-4248	126	12	2022	2022	NUM
ejpam-4248	126	13	)	)	PUNCT
ejpam-4248	126	14	,	,	PUNCT
ejpam-4248	126	15	199	199	NUM
ejpam-4248	126	16	-	-	SYM
ejpam-4248	126	17	206	206	NUM
ejpam-4248	126	18	204	204	NUM
ejpam-4248	126	19	definition	definition	NOUN
ejpam-4248	126	20	15	15	NUM
ejpam-4248	126	21	.	.	PUNCT
ejpam-4248	127	1	a	a	DET
ejpam-4248	127	2	function	function	NOUN
ejpam-4248	127	3	f	f	NOUN
ejpam-4248	127	4	:	:	PUNCT
ejpam-4248	127	5	x	x	X
ejpam-4248	127	6	→	→	SYM
ejpam-4248	127	7	y	y	PROPN
ejpam-4248	127	8	is	be	AUX
ejpam-4248	127	9	called	call	VERB
ejpam-4248	127	10	coc	coc	ADJ
ejpam-4248	127	11	-	-	PUNCT
ejpam-4248	127	12	perfect	perfect	ADJ
ejpam-4248	127	13	function	function	NOUN
ejpam-4248	127	14	if	if	SCONJ
ejpam-4248	127	15	it	it	PRON
ejpam-4248	127	16	is	be	AUX
ejpam-4248	127	17	surjective	surjective	ADJ
ejpam-4248	127	18	,	,	PUNCT
ejpam-4248	127	19	coc	coc	NOUN
ejpam-4248	127	20	-	-	PUNCT
ejpam-4248	127	21	closed	closed	ADJ
ejpam-4248	127	22	and	and	CCONJ
ejpam-4248	127	23	f−1(y	f−1(y	PROPN
ejpam-4248	127	24	)	)	PUNCT
ejpam-4248	127	25	coc	coc	NOUN
ejpam-4248	127	26	-	-	PUNCT
ejpam-4248	127	27	compact	compact	ADJ
ejpam-4248	127	28	.	.	PUNCT
ejpam-4248	128	1	theorem	theorem	ADJ
ejpam-4248	128	2	6	6	NUM
ejpam-4248	128	3	.	.	PUNCT
ejpam-4248	129	1	let	let	VERB
ejpam-4248	129	2	f	f	NOUN
ejpam-4248	129	3	:	:	PUNCT
ejpam-4248	129	4	x	x	X
ejpam-4248	129	5	→	→	SYM
ejpam-4248	129	6	y	y	X
ejpam-4248	129	7	be	be	AUX
ejpam-4248	129	8	a	a	DET
ejpam-4248	129	9	coc	coc	NOUN
ejpam-4248	129	10	-	-	PUNCT
ejpam-4248	129	11	almost	almost	ADV
ejpam-4248	129	12	-	-	PUNCT
ejpam-4248	129	13	open	open	ADJ
ejpam-4248	129	14	coc	coc	ADJ
ejpam-4248	129	15	-	-	PUNCT
ejpam-4248	129	16	perfect	perfect	ADJ
ejpam-4248	129	17	function	function	NOUN
ejpam-4248	129	18	.	.	PUNCT
ejpam-4248	130	1	if	if	SCONJ
ejpam-4248	130	2	k	k	PROPN
ejpam-4248	130	3	is	be	AUX
ejpam-4248	130	4	a	a	DET
ejpam-4248	130	5	cocweakly	cocweakly	ADV
ejpam-4248	130	6	-	-	PUNCT
ejpam-4248	130	7	compact	compact	ADJ
ejpam-4248	130	8	relative	relative	NOUN
ejpam-4248	130	9	to	to	ADP
ejpam-4248	130	10	y	y	PROPN
ejpam-4248	130	11	,	,	PUNCT
ejpam-4248	130	12	then	then	ADV
ejpam-4248	130	13	f−1	f−1	PROPN
ejpam-4248	130	14	(	(	PUNCT
ejpam-4248	130	15	k	k	PROPN
ejpam-4248	130	16	)	)	PUNCT
ejpam-4248	130	17	is	be	AUX
ejpam-4248	130	18	coc	coc	ADJ
ejpam-4248	130	19	-	-	PUNCT
ejpam-4248	130	20	weakly	weakly	ADJ
ejpam-4248	130	21	-	-	PUNCT
ejpam-4248	130	22	compact	compact	ADJ
ejpam-4248	130	23	relative	relative	NOUN
ejpam-4248	130	24	to	to	ADP
ejpam-4248	130	25	x.	x.	NOUN
ejpam-4248	130	26	proof	proof	NOUN
ejpam-4248	130	27	.	.	PUNCT
ejpam-4248	131	1	let	let	VERB
ejpam-4248	131	2	k	k	PRON
ejpam-4248	131	3	be	be	AUX
ejpam-4248	131	4	coc	coc	ADJ
ejpam-4248	131	5	-	-	PUNCT
ejpam-4248	131	6	weakly	weakly	ADJ
ejpam-4248	131	7	-	-	PUNCT
ejpam-4248	131	8	compact	compact	ADJ
ejpam-4248	131	9	relative	relative	NOUN
ejpam-4248	131	10	to	to	ADP
ejpam-4248	131	11	y	y	PROPN
ejpam-4248	131	12	and	and	CCONJ
ejpam-4248	131	13	{	{	PUNCT
ejpam-4248	131	14	uα|α	uα|α	PROPN
ejpam-4248	131	15	∈	∈	PROPN
ejpam-4248	131	16	∆	∆	PROPN
ejpam-4248	131	17	}	}	PUNCT
ejpam-4248	131	18	any	any	DET
ejpam-4248	131	19	cover	cover	NOUN
ejpam-4248	131	20	of	of	ADP
ejpam-4248	131	21	f−1(k	f−1(k	PROPN
ejpam-4248	131	22	)	)	PUNCT
ejpam-4248	131	23	by	by	ADP
ejpam-4248	131	24	coc	coc	ADJ
ejpam-4248	131	25	-	-	PUNCT
ejpam-4248	131	26	open	open	ADJ
ejpam-4248	131	27	sets	set	NOUN
ejpam-4248	131	28	of	of	ADP
ejpam-4248	131	29	x	x	SYM
ejpam-4248	131	30	satisfies	satisfie	NOUN
ejpam-4248	131	31	condition	condition	NOUN
ejpam-4248	131	32	(	(	PUNCT
ejpam-4248	131	33	∗∗	∗∗	NOUN
ejpam-4248	131	34	)	)	PUNCT
ejpam-4248	131	35	.	.	PUNCT
ejpam-4248	132	1	for	for	ADP
ejpam-4248	132	2	each	each	DET
ejpam-4248	132	3	α	α	PROPN
ejpam-4248	132	4	∈	∈	NOUN
ejpam-4248	132	5	∆	∆	NOUN
ejpam-4248	132	6	there	there	PRON
ejpam-4248	132	7	exists	exist	VERB
ejpam-4248	132	8	a	a	DET
ejpam-4248	132	9	coc	coc	ADJ
ejpam-4248	132	10	-	-	PUNCT
ejpam-4248	132	11	regular	regular	ADJ
ejpam-4248	132	12	closed	close	VERB
ejpam-4248	132	13	set	set	VERB
ejpam-4248	132	14	fα	fα	NOUN
ejpam-4248	132	15	of	of	ADP
ejpam-4248	132	16	x	x	SYM
ejpam-4248	132	17	such	such	ADJ
ejpam-4248	132	18	that	that	DET
ejpam-4248	132	19	fα	fα	ADP
ejpam-4248	132	20	⊆	⊆	NUM
ejpam-4248	132	21	uα	uα	NOUN
ejpam-4248	132	22	and	and	CCONJ
ejpam-4248	132	23	f−1(k	f−1(k	PROPN
ejpam-4248	132	24	)	)	PUNCT
ejpam-4248	132	25	⊆	⊆	NUM
ejpam-4248	132	26	⋃	⋃	X
ejpam-4248	132	27	{	{	PUNCT
ejpam-4248	132	28	intcoc	intcoc	NOUN
ejpam-4248	132	29	(	(	PUNCT
ejpam-4248	132	30	fα	fα	NOUN
ejpam-4248	132	31	)	)	PUNCT
ejpam-4248	132	32	|α	|α	NOUN
ejpam-4248	132	33	∈	∈	NOUN
ejpam-4248	132	34	∆	∆	X
ejpam-4248	132	35	}	}	PUNCT
ejpam-4248	132	36	.	.	PUNCT
ejpam-4248	133	1	now	now	ADV
ejpam-4248	133	2	for	for	ADP
ejpam-4248	133	3	each	each	DET
ejpam-4248	133	4	y	y	PROPN
ejpam-4248	133	5	∈	∈	PROPN
ejpam-4248	133	6	y	y	PROPN
ejpam-4248	133	7	,	,	PUNCT
ejpam-4248	133	8	we	we	PRON
ejpam-4248	133	9	have	have	VERB
ejpam-4248	133	10	f−1(y	f−1(y	PROPN
ejpam-4248	133	11	)	)	PUNCT
ejpam-4248	134	1	⊆	⊆	NUM
ejpam-4248	134	2	⋃	⋃	X
ejpam-4248	134	3	{	{	PUNCT
ejpam-4248	134	4	intcoc	intcoc	NOUN
ejpam-4248	134	5	(	(	PUNCT
ejpam-4248	134	6	fα	fα	NOUN
ejpam-4248	134	7	)	)	PUNCT
ejpam-4248	134	8	|α	|α	NOUN
ejpam-4248	134	9	∈	∈	PROPN
ejpam-4248	134	10	∆(y	∆(y	NOUN
ejpam-4248	134	11	)	)	PUNCT
ejpam-4248	134	12	,	,	PUNCT
ejpam-4248	134	13	|∆(y)|	|∆(y)|	X
ejpam-4248	134	14	<	<	X
ejpam-4248	134	15	ω0	ω0	PROPN
ejpam-4248	134	16	}	}	PUNCT
ejpam-4248	134	17	and	and	CCONJ
ejpam-4248	134	18	f−1(y	f−1(y	PROPN
ejpam-4248	134	19	)	)	PUNCT
ejpam-4248	135	1	⊆	⊆	NUM
ejpam-4248	135	2	⋃	⋃	NOUN
ejpam-4248	135	3	α∈∆(y	α∈∆(y	NUM
ejpam-4248	135	4	)	)	PUNCT
ejpam-4248	135	5	intcoc	intcoc	PROPN
ejpam-4248	135	6	(	(	PUNCT
ejpam-4248	135	7	fα	fα	ADP
ejpam-4248	135	8	)	)	PUNCT
ejpam-4248	135	9	⊆	⊆	NUM
ejpam-4248	135	10	⋃	⋃	NOUN
ejpam-4248	135	11	α∈∆(y	α∈∆(y	NOUN
ejpam-4248	135	12	)	)	PUNCT
ejpam-4248	135	13	uα	uα	PROPN
ejpam-4248	135	14	.	.	PUNCT
ejpam-4248	135	15	define	define	VERB
ejpam-4248	135	16	gy	gy	PROPN
ejpam-4248	135	17	=	=	SYM
ejpam-4248	135	18	y	y	PROPN
ejpam-4248	136	1	−	−	PROPN
ejpam-4248	137	1	f	f	PROPN
ejpam-4248	138	1	(	(	PUNCT
ejpam-4248	138	2	x	x	X
ejpam-4248	138	3	−	−	ADP
ejpam-4248	138	4	⋃	⋃	NOUN
ejpam-4248	138	5	{	{	PUNCT
ejpam-4248	138	6	intcoc	intcoc	NOUN
ejpam-4248	138	7	(	(	PUNCT
ejpam-4248	138	8	fα	fα	NOUN
ejpam-4248	138	9	)	)	PUNCT
ejpam-4248	138	10	|α	|α	NOUN
ejpam-4248	138	11	∈	∈	PROPN
ejpam-4248	138	12	∆(y	∆(y	NOUN
ejpam-4248	138	13	)	)	PUNCT
ejpam-4248	138	14	,	,	PUNCT
ejpam-4248	138	15	|∆(y)|	|∆(y)|	X
ejpam-4248	138	16	<	<	X
ejpam-4248	138	17	ω0	ω0	PROPN
ejpam-4248	138	18	}	}	PUNCT
ejpam-4248	138	19	)	)	PUNCT
ejpam-4248	138	20	,	,	PUNCT
ejpam-4248	139	1	hy	hy	NOUN
ejpam-4248	139	2	=	=	SYM
ejpam-4248	139	3	y	y	PROPN
ejpam-4248	139	4	−	−	PROPN
ejpam-4248	140	1	intcoc	intcoc	PROPN
ejpam-4248	140	2	(	(	PUNCT
ejpam-4248	140	3	f	f	X
ejpam-4248	140	4	(	(	PUNCT
ejpam-4248	140	5	x	x	X
ejpam-4248	140	6	−	−	ADP
ejpam-4248	140	7	⋃	⋃	NOUN
ejpam-4248	140	8	{	{	PUNCT
ejpam-4248	140	9	fα|α	fα|α	PROPN
ejpam-4248	140	10	∈	∈	PROPN
ejpam-4248	140	11	∆(y	∆(y	NOUN
ejpam-4248	140	12	)	)	PUNCT
ejpam-4248	140	13	,	,	PUNCT
ejpam-4248	140	14	|∆(y)|	|∆(y)|	X
ejpam-4248	140	15	<	<	X
ejpam-4248	140	16	ω0	ω0	PROPN
ejpam-4248	140	17	}	}	PUNCT
ejpam-4248	140	18	)	)	PUNCT
ejpam-4248	140	19	coc	coc	NOUN
ejpam-4248	140	20	,	,	PUNCT
ejpam-4248	140	21	and	and	CCONJ
ejpam-4248	140	22	vy	vy	X
ejpam-4248	140	23	=	=	NOUN
ejpam-4248	140	24	v	v	ADP
ejpam-4248	140	25	−	−	PROPN
ejpam-4248	140	26	f	f	PROPN
ejpam-4248	140	27	(	(	PUNCT
ejpam-4248	140	28	x	x	X
ejpam-4248	140	29	−	−	PROPN
ejpam-4248	140	30	⋃	⋃	NOUN
ejpam-4248	140	31	{	{	PUNCT
ejpam-4248	140	32	uα|α	uα|α	PROPN
ejpam-4248	140	33	∈	∈	PROPN
ejpam-4248	140	34	∆(y	∆(y	NOUN
ejpam-4248	140	35	)	)	PUNCT
ejpam-4248	140	36	,	,	PUNCT
ejpam-4248	140	37	|∆(y)|	|∆(y)|	X
ejpam-4248	140	38	<	<	X
ejpam-4248	140	39	ω0	ω0	PROPN
ejpam-4248	140	40	}	}	PUNCT
ejpam-4248	140	41	)	)	PUNCT
ejpam-4248	140	42	,	,	PUNCT
ejpam-4248	140	43	clearly	clearly	ADV
ejpam-4248	140	44	gy	gy	VERB
ejpam-4248	140	45	and	and	CCONJ
ejpam-4248	140	46	vy	vy	NOUN
ejpam-4248	140	47	are	be	AUX
ejpam-4248	140	48	coc	coc	ADJ
ejpam-4248	140	49	-	-	PUNCT
ejpam-4248	140	50	open	open	ADJ
ejpam-4248	140	51	subsets	subset	NOUN
ejpam-4248	140	52	in	in	ADP
ejpam-4248	140	53	y	y	PROPN
ejpam-4248	140	54	,	,	PUNCT
ejpam-4248	140	55	hy	hy	PROPN
ejpam-4248	140	56	is	be	AUX
ejpam-4248	140	57	coc	coc	ADJ
ejpam-4248	140	58	-	-	PUNCT
ejpam-4248	140	59	regular	regular	NOUN
ejpam-4248	140	60	closed	close	VERB
ejpam-4248	140	61	with	with	ADP
ejpam-4248	140	62	y	y	PROPN
ejpam-4248	140	63	∈	∈	PROPN
ejpam-4248	140	64	gy	gy	VERB
ejpam-4248	140	65	⊆	⊆	NUM
ejpam-4248	140	66	hy	hy	NOUN
ejpam-4248	140	67	⊆	⊆	NUM
ejpam-4248	140	68	vy	vy	NOUN
ejpam-4248	140	69	,	,	PUNCT
ejpam-4248	140	70	therefore	therefore	ADV
ejpam-4248	140	71	the	the	DET
ejpam-4248	140	72	family	family	NOUN
ejpam-4248	140	73	v	v	NOUN
ejpam-4248	140	74	=	=	PUNCT
ejpam-4248	140	75	{	{	PUNCT
ejpam-4248	140	76	vy|y	vy|y	NOUN
ejpam-4248	140	77	∈	∈	PROPN
ejpam-4248	141	1	k	k	X
ejpam-4248	141	2	}	}	PUNCT
ejpam-4248	141	3	is	be	AUX
ejpam-4248	141	4	a	a	DET
ejpam-4248	141	5	cover	cover	NOUN
ejpam-4248	141	6	of	of	ADP
ejpam-4248	141	7	k	k	PROPN
ejpam-4248	141	8	by	by	ADP
ejpam-4248	141	9	coc	coc	ADJ
ejpam-4248	141	10	-	-	PUNCT
ejpam-4248	141	11	open	open	ADJ
ejpam-4248	141	12	subsets	subset	NOUN
ejpam-4248	141	13	of	of	ADP
ejpam-4248	141	14	y	y	PROPN
ejpam-4248	141	15	satisfies	satisfy	VERB
ejpam-4248	141	16	condition	condition	NOUN
ejpam-4248	141	17	(	(	PUNCT
ejpam-4248	141	18	∗∗	∗∗	NOUN
ejpam-4248	141	19	)	)	PUNCT
ejpam-4248	141	20	,	,	PUNCT
ejpam-4248	141	21	but	but	CCONJ
ejpam-4248	141	22	k	k	PROPN
ejpam-4248	141	23	is	be	AUX
ejpam-4248	141	24	a	a	DET
ejpam-4248	141	25	coc	coc	NOUN
ejpam-4248	141	26	-	-	PUNCT
ejpam-4248	141	27	weakly	weakly	ADJ
ejpam-4248	141	28	-	-	PUNCT
ejpam-4248	141	29	compact	compact	ADJ
ejpam-4248	141	30	relative	relative	NOUN
ejpam-4248	141	31	to	to	ADP
ejpam-4248	141	32	y	y	PROPN
ejpam-4248	141	33	,	,	PUNCT
ejpam-4248	141	34	so	so	ADV
ejpam-4248	141	35	there	there	PRON
ejpam-4248	141	36	exists	exist	VERB
ejpam-4248	141	37	set	set	NOUN
ejpam-4248	141	38	y1	y1	NOUN
ejpam-4248	141	39	,	,	PUNCT
ejpam-4248	141	40	y2	y2	PROPN
ejpam-4248	141	41	,	,	PUNCT
ejpam-4248	141	42	...	...	PUNCT
ejpam-4248	141	43	,	,	PUNCT
ejpam-4248	141	44	yn	yn	PROPN
ejpam-4248	141	45	∈	∈	PROPN
ejpam-4248	141	46	y	y	PROPN
ejpam-4248	141	47	of	of	ADP
ejpam-4248	141	48	such	such	ADJ
ejpam-4248	141	49	that	that	SCONJ
ejpam-4248	141	50	k	k	PROPN
ejpam-4248	141	51	⊆	⊆	NUM
ejpam-4248	141	52	⋃	⋃	NOUN
ejpam-4248	141	53	{	{	PUNCT
ejpam-4248	141	54	vyi	vyi	NOUN
ejpam-4248	141	55	coc|i	coc|i	NOUN
ejpam-4248	141	56	=	=	SYM
ejpam-4248	141	57	1	1	NUM
ejpam-4248	141	58	,	,	PUNCT
ejpam-4248	141	59	2	2	NUM
ejpam-4248	141	60	,	,	PUNCT
ejpam-4248	141	61	...	...	PUNCT
ejpam-4248	141	62	,	,	PUNCT
ejpam-4248	141	63	n	n	CCONJ
ejpam-4248	141	64	}	}	PUNCT
ejpam-4248	141	65	,	,	PUNCT
ejpam-4248	141	66	but	but	CCONJ
ejpam-4248	141	67	f−1	f−1	PROPN
ejpam-4248	141	68	(	(	PUNCT
ejpam-4248	141	69	vy	vy	X
ejpam-4248	141	70	coc	coc	PROPN
ejpam-4248	141	71	)	)	PUNCT
ejpam-4248	141	72	⊆	⊆	NUM
ejpam-4248	141	73	f−1	f−1	PROPN
ejpam-4248	141	74	(	(	PUNCT
ejpam-4248	141	75	vy	vy	INTJ
ejpam-4248	141	76	)	)	PUNCT
ejpam-4248	141	77	coc	coc	NOUN
ejpam-4248	141	78	for	for	ADP
ejpam-4248	141	79	all	all	DET
ejpam-4248	141	80	y	y	PROPN
ejpam-4248	141	81	∈	∈	PROPN
ejpam-4248	141	82	y	y	PROPN
ejpam-4248	141	83	,	,	PUNCT
ejpam-4248	142	1	so	so	ADV
ejpam-4248	142	2	f−1	f−1	PROPN
ejpam-4248	142	3	(	(	PUNCT
ejpam-4248	142	4	vy	vy	INTJ
ejpam-4248	142	5	)	)	PUNCT
ejpam-4248	142	6	⊆	⊆	NUM
ejpam-4248	142	7	⋃	⋃	NOUN
ejpam-4248	142	8	{	{	PUNCT
ejpam-4248	142	9	uα|α	uα|α	PROPN
ejpam-4248	142	10	∈	∈	PROPN
ejpam-4248	142	11	∆(y	∆(y	NOUN
ejpam-4248	142	12	)	)	PUNCT
ejpam-4248	142	13	}	}	PUNCT
ejpam-4248	142	14	,	,	PUNCT
ejpam-4248	142	15	therefore	therefore	ADV
ejpam-4248	142	16	f−1	f−1	PROPN
ejpam-4248	142	17	(	(	PUNCT
ejpam-4248	142	18	k	k	PROPN
ejpam-4248	142	19	)	)	PUNCT
ejpam-4248	142	20	⊆	⊆	X
ejpam-4248	142	21	{	{	PUNCT
ejpam-4248	142	22	uα	uα	PROPN
ejpam-4248	142	23	coc|α	coc|α	NOUN
ejpam-4248	142	24	∈	∈	PROPN
ejpam-4248	142	25	∆(yi	∆(yi	ADV
ejpam-4248	142	26	)	)	PUNCT
ejpam-4248	142	27	,	,	PUNCT
ejpam-4248	142	28	i	i	PRON
ejpam-4248	142	29	=	=	NOUN
ejpam-4248	142	30	1	1	NUM
ejpam-4248	142	31	,	,	PUNCT
ejpam-4248	142	32	2	2	NUM
ejpam-4248	142	33	,	,	PUNCT
ejpam-4248	142	34	...	...	PUNCT
ejpam-4248	142	35	,	,	PUNCT
ejpam-4248	142	36	n	n	CCONJ
ejpam-4248	142	37	}	}	PUNCT
ejpam-4248	142	38	,	,	PUNCT
ejpam-4248	142	39	hence	hence	ADV
ejpam-4248	142	40	f−1	f−1	PROPN
ejpam-4248	142	41	(	(	PUNCT
ejpam-4248	142	42	k	k	PROPN
ejpam-4248	142	43	)	)	PUNCT
ejpam-4248	142	44	is	be	AUX
ejpam-4248	142	45	coc	coc	ADJ
ejpam-4248	142	46	-	-	PUNCT
ejpam-4248	142	47	weakly	weakly	ADJ
ejpam-4248	142	48	-	-	PUNCT
ejpam-4248	142	49	compact	compact	ADJ
ejpam-4248	142	50	relative	relative	NOUN
ejpam-4248	142	51	to	to	ADP
ejpam-4248	142	52	x.	x.	NOUN
ejpam-4248	142	53	corollary	corollary	ADJ
ejpam-4248	142	54	8	8	NUM
ejpam-4248	142	55	.	.	PUNCT
ejpam-4248	143	1	the	the	DET
ejpam-4248	143	2	coc	coc	PROPN
ejpam-4248	143	3	-	-	PUNCT
ejpam-4248	143	4	open	open	ADJ
ejpam-4248	143	5	coc	coc	ADJ
ejpam-4248	143	6	-	-	PUNCT
ejpam-4248	143	7	perfect	perfect	ADJ
ejpam-4248	143	8	function	function	NOUN
ejpam-4248	143	9	of	of	ADP
ejpam-4248	143	10	a	a	DET
ejpam-4248	143	11	coc	coc	NOUN
ejpam-4248	143	12	-	-	PUNCT
ejpam-4248	143	13	weakly	weakly	ADJ
ejpam-4248	143	14	-	-	PUNCT
ejpam-4248	143	15	compact	compact	ADJ
ejpam-4248	143	16	space	space	NOUN
ejpam-4248	143	17	is	be	AUX
ejpam-4248	143	18	cocweakly	cocweakly	ADV
ejpam-4248	143	19	-	-	PUNCT
ejpam-4248	143	20	compact	compact	ADJ
ejpam-4248	143	21	.	.	PUNCT
ejpam-4248	144	1	theorem	theorem	VERB
ejpam-4248	144	2	7	7	NUM
ejpam-4248	144	3	.	.	PUNCT
ejpam-4248	145	1	if	if	SCONJ
ejpam-4248	145	2	f	f	PROPN
ejpam-4248	145	3	:	:	PUNCT
ejpam-4248	145	4	x	x	X
ejpam-4248	145	5	→	→	SYM
ejpam-4248	145	6	y	y	PROPN
ejpam-4248	145	7	is	be	AUX
ejpam-4248	145	8	a	a	DET
ejpam-4248	145	9	coc	coc	ADJ
ejpam-4248	145	10	-	-	ADJ
ejpam-4248	145	11	continuous	continuous	ADJ
ejpam-4248	145	12	surjective	surjective	ADJ
ejpam-4248	145	13	function	function	NOUN
ejpam-4248	145	14	and	and	CCONJ
ejpam-4248	145	15	x	x	NOUN
ejpam-4248	145	16	is	be	AUX
ejpam-4248	145	17	coc	coc	NOUN
ejpam-4248	145	18	-	-	PUNCT
ejpam-4248	145	19	rcompact	rcompact	NOUN
ejpam-4248	145	20	,	,	PUNCT
ejpam-4248	145	21	then	then	ADV
ejpam-4248	145	22	y	y	PROPN
ejpam-4248	145	23	is	be	AUX
ejpam-4248	145	24	coc	coc	ADJ
ejpam-4248	145	25	-	-	PUNCT
ejpam-4248	145	26	r	r	NOUN
ejpam-4248	145	27	-	-	PUNCT
ejpam-4248	145	28	compact	compact	ADJ
ejpam-4248	145	29	.	.	PUNCT
ejpam-4248	146	1	f.a	f.a	PROPN
ejpam-4248	146	2	.	.	PROPN
ejpam-4248	146	3	abushaheen	abushaheen	PROPN
ejpam-4248	146	4	,	,	PUNCT
ejpam-4248	146	5	f.	f.	PROPN
ejpam-4248	146	6	alrimawi	alrimawi	PROPN
ejpam-4248	146	7	/	/	SYM
ejpam-4248	146	8	eur	eur	PROPN
ejpam-4248	146	9	.	.	PUNCT
ejpam-4248	147	1	j.	j.	PROPN
ejpam-4248	147	2	pure	pure	PROPN
ejpam-4248	147	3	appl	appl	PROPN
ejpam-4248	147	4	.	.	PROPN
ejpam-4248	147	5	math	math	PROPN
ejpam-4248	147	6	,	,	PUNCT
ejpam-4248	147	7	15	15	NUM
ejpam-4248	147	8	(	(	PUNCT
ejpam-4248	147	9	1	1	NUM
ejpam-4248	147	10	)	)	PUNCT
ejpam-4248	147	11	(	(	PUNCT
ejpam-4248	147	12	2022	2022	NUM
ejpam-4248	147	13	)	)	PUNCT
ejpam-4248	147	14	,	,	PUNCT
ejpam-4248	147	15	199	199	NUM
ejpam-4248	147	16	-	-	SYM
ejpam-4248	147	17	206	206	NUM
ejpam-4248	147	18	205	205	NUM
ejpam-4248	147	19	proof	proof	NOUN
ejpam-4248	147	20	.	.	PUNCT
ejpam-4248	148	1	let	let	VERB
ejpam-4248	148	2	u	u	PRON
ejpam-4248	148	3	=	=	X
ejpam-4248	148	4	{	{	PUNCT
ejpam-4248	148	5	uα|α	uα|α	PROPN
ejpam-4248	148	6	∈	∈	PROPN
ejpam-4248	148	7	∆	∆	PROPN
ejpam-4248	148	8	}	}	PUNCT
ejpam-4248	148	9	be	be	AUX
ejpam-4248	148	10	a	a	DET
ejpam-4248	148	11	coc	coc	ADJ
ejpam-4248	148	12	-	-	PUNCT
ejpam-4248	148	13	regular	regular	ADJ
ejpam-4248	148	14	cover	cover	NOUN
ejpam-4248	148	15	of	of	ADP
ejpam-4248	148	16	y	y	PROPN
ejpam-4248	148	17	.	.	PUNCT
ejpam-4248	149	1	for	for	ADP
ejpam-4248	149	2	each	each	DET
ejpam-4248	149	3	α	α	PROPN
ejpam-4248	149	4	∈	∈	PROPN
ejpam-4248	149	5	∆	∆	PROPN
ejpam-4248	149	6	,	,	PUNCT
ejpam-4248	149	7	there	there	PRON
ejpam-4248	149	8	exists	exist	VERB
ejpam-4248	149	9	a	a	DET
ejpam-4248	149	10	coc	coc	NOUN
ejpam-4248	149	11	-	-	PUNCT
ejpam-4248	149	12	closed	closed	ADJ
ejpam-4248	149	13	subset	subset	NOUN
ejpam-4248	149	14	fα	fα	ADP
ejpam-4248	149	15	⊆	⊆	NUM
ejpam-4248	149	16	uα	uα	PROPN
ejpam-4248	149	17	and	and	CCONJ
ejpam-4248	149	18	y	y	PROPN
ejpam-4248	149	19	=	=	PUNCT
ejpam-4248	149	20	⋃	⋃	NOUN
ejpam-4248	149	21	{	{	PUNCT
ejpam-4248	149	22	intcoc(fα	intcoc(fα	NOUN
ejpam-4248	149	23	)	)	PUNCT
ejpam-4248	149	24	}	}	PUNCT
ejpam-4248	149	25	.	.	PUNCT
ejpam-4248	150	1	now	now	ADV
ejpam-4248	150	2	f−1	f−1	PROPN
ejpam-4248	150	3	(	(	PUNCT
ejpam-4248	150	4	u	u	NOUN
ejpam-4248	150	5	)	)	PUNCT
ejpam-4248	150	6	=	=	SYM
ejpam-4248	150	7	{	{	PUNCT
ejpam-4248	150	8	f−1(uα)|α	f−1(uα)|α	NOUN
ejpam-4248	150	9	∈	∈	PROPN
ejpam-4248	150	10	∆	∆	X
ejpam-4248	150	11	}	}	PUNCT
ejpam-4248	150	12	,	,	PUNCT
ejpam-4248	150	13	f−1(fα	f−1(fα	NOUN
ejpam-4248	150	14	)	)	PUNCT
ejpam-4248	150	15	⊆	⊆	NUM
ejpam-4248	150	16	f−1(uα	f−1(uα	NOUN
ejpam-4248	150	17	)	)	PUNCT
ejpam-4248	150	18	and	and	CCONJ
ejpam-4248	150	19	x	x	X
ejpam-4248	150	20	=	=	PUNCT
ejpam-4248	150	21	f−1(y	f−1(y	PROPN
ejpam-4248	150	22	)	)	PUNCT
ejpam-4248	151	1	=	=	SYM
ejpam-4248	151	2	f−1	f−1	PROPN
ejpam-4248	151	3	(	(	PUNCT
ejpam-4248	151	4	⋃	⋃	NOUN
ejpam-4248	151	5	intcoc(fα	intcoc(fα	NOUN
ejpam-4248	151	6	)	)	PUNCT
ejpam-4248	151	7	)	)	PUNCT
ejpam-4248	152	1	⊆	⊆	NUM
ejpam-4248	152	2	⋃	⋃	PROPN
ejpam-4248	152	3	f−1	f−1	PROPN
ejpam-4248	152	4	(	(	PUNCT
ejpam-4248	152	5	uα	uα	NOUN
ejpam-4248	152	6	)	)	PUNCT
ejpam-4248	152	7	=	=	SYM
ejpam-4248	153	1	x	x	X
ejpam-4248	153	2	,	,	PUNCT
ejpam-4248	153	3	clearly	clearly	ADV
ejpam-4248	153	4	f−1	f−1	PROPN
ejpam-4248	153	5	(	(	PUNCT
ejpam-4248	153	6	u	u	NOUN
ejpam-4248	153	7	)	)	PUNCT
ejpam-4248	153	8	is	be	AUX
ejpam-4248	153	9	coc	coc	ADJ
ejpam-4248	153	10	-	-	PUNCT
ejpam-4248	153	11	regular	regular	ADJ
ejpam-4248	153	12	cover	cover	NOUN
ejpam-4248	153	13	of	of	ADP
ejpam-4248	153	14	x	x	PRON
ejpam-4248	153	15	,	,	PUNCT
ejpam-4248	153	16	so	so	CCONJ
ejpam-4248	153	17	there	there	PRON
ejpam-4248	153	18	exists	exist	VERB
ejpam-4248	153	19	a	a	DET
ejpam-4248	153	20	finite	finite	ADJ
ejpam-4248	153	21	subcover	subcover	NOUN
ejpam-4248	153	22	such	such	ADJ
ejpam-4248	153	23	that	that	SCONJ
ejpam-4248	153	24	x	x	X
ejpam-4248	153	25	=	=	SYM
ejpam-4248	153	26	⋃	⋃	NOUN
ejpam-4248	153	27	{	{	PUNCT
ejpam-4248	153	28	f−1(uα)|α	f−1(uα)|α	NOUN
ejpam-4248	153	29	∈	∈	PROPN
ejpam-4248	153	30	∆0	∆0	NUM
ejpam-4248	153	31	⊆	⊆	NUM
ejpam-4248	153	32	∆	∆	PROPN
ejpam-4248	153	33	,	,	PUNCT
ejpam-4248	153	34	|∆0|	|∆0|	X
ejpam-4248	153	35	<	<	X
ejpam-4248	153	36	ω0	ω0	PROPN
ejpam-4248	153	37	}	}	PUNCT
ejpam-4248	153	38	,	,	PUNCT
ejpam-4248	153	39	hence	hence	ADV
ejpam-4248	153	40	the	the	DET
ejpam-4248	153	41	result	result	NOUN
ejpam-4248	153	42	.	.	PUNCT
ejpam-4248	154	1	theorem	theorem	ADJ
ejpam-4248	154	2	8	8	NUM
ejpam-4248	154	3	.	.	PUNCT
ejpam-4248	155	1	let	let	VERB
ejpam-4248	155	2	f	f	NOUN
ejpam-4248	155	3	:	:	PUNCT
ejpam-4248	155	4	x	x	X
ejpam-4248	155	5	→	→	SYM
ejpam-4248	155	6	y	y	X
ejpam-4248	155	7	be	be	AUX
ejpam-4248	155	8	a	a	DET
ejpam-4248	155	9	coc	coc	NOUN
ejpam-4248	155	10	-	-	PUNCT
ejpam-4248	155	11	almost	almost	ADV
ejpam-4248	155	12	coc	coc	ADJ
ejpam-4248	155	13	-	-	PUNCT
ejpam-4248	155	14	open	open	ADJ
ejpam-4248	155	15	and	and	CCONJ
ejpam-4248	155	16	coc	coc	ADJ
ejpam-4248	155	17	-	-	PUNCT
ejpam-4248	155	18	perfect	perfect	ADJ
ejpam-4248	155	19	function	function	NOUN
ejpam-4248	155	20	.	.	PUNCT
ejpam-4248	156	1	if	if	SCONJ
ejpam-4248	156	2	y	y	PROPN
ejpam-4248	156	3	is	be	AUX
ejpam-4248	156	4	coc	coc	ADJ
ejpam-4248	156	5	-	-	PUNCT
ejpam-4248	156	6	r	r	NOUN
ejpam-4248	156	7	-	-	PUNCT
ejpam-4248	156	8	compact	compact	ADJ
ejpam-4248	156	9	space	space	NOUN
ejpam-4248	156	10	,	,	PUNCT
ejpam-4248	156	11	then	then	ADV
ejpam-4248	156	12	x	x	PUNCT
ejpam-4248	156	13	is	be	AUX
ejpam-4248	156	14	so	so	ADV
ejpam-4248	156	15	.	.	PUNCT
ejpam-4248	157	1	proof	proof	NOUN
ejpam-4248	157	2	.	.	PUNCT
ejpam-4248	158	1	let	let	VERB
ejpam-4248	158	2	u	u	PRON
ejpam-4248	158	3	=	=	X
ejpam-4248	158	4	{	{	PUNCT
ejpam-4248	158	5	uα|α	uα|α	PROPN
ejpam-4248	158	6	∈	∈	PROPN
ejpam-4248	158	7	∆	∆	PROPN
ejpam-4248	158	8	}	}	PUNCT
ejpam-4248	158	9	be	be	AUX
ejpam-4248	158	10	a	a	DET
ejpam-4248	158	11	coc	coc	ADJ
ejpam-4248	158	12	-	-	PUNCT
ejpam-4248	158	13	regular	regular	ADJ
ejpam-4248	158	14	cover	cover	NOUN
ejpam-4248	158	15	of	of	ADP
ejpam-4248	158	16	x	x	NOUN
ejpam-4248	158	17	,	,	PUNCT
ejpam-4248	158	18	so	so	ADV
ejpam-4248	158	19	for	for	ADP
ejpam-4248	158	20	each	each	DET
ejpam-4248	158	21	α	α	NOUN
ejpam-4248	158	22	∈	∈	PROPN
ejpam-4248	158	23	∆	∆	PROPN
ejpam-4248	158	24	,	,	PUNCT
ejpam-4248	158	25	there	there	PRON
ejpam-4248	158	26	exists	exist	VERB
ejpam-4248	158	27	a	a	DET
ejpam-4248	158	28	coc	coc	NOUN
ejpam-4248	158	29	-	-	PUNCT
ejpam-4248	158	30	closed	closed	ADJ
ejpam-4248	158	31	subset	subset	NOUN
ejpam-4248	158	32	fα	fα	ADP
ejpam-4248	158	33	⊆	⊆	NUM
ejpam-4248	158	34	uα	uα	NOUN
ejpam-4248	158	35	and	and	CCONJ
ejpam-4248	158	36	x	x	NOUN
ejpam-4248	158	37	=	=	SYM
ejpam-4248	158	38	⋃	⋃	NOUN
ejpam-4248	158	39	{	{	PUNCT
ejpam-4248	158	40	intcoc(fα	intcoc(fα	NOUN
ejpam-4248	158	41	)	)	PUNCT
ejpam-4248	158	42	}	}	PUNCT
ejpam-4248	158	43	,	,	PUNCT
ejpam-4248	158	44	but	but	CCONJ
ejpam-4248	158	45	f	f	PROPN
ejpam-4248	158	46	is	be	AUX
ejpam-4248	158	47	coc	coc	ADJ
ejpam-4248	158	48	-	-	PUNCT
ejpam-4248	158	49	perfect	perfect	ADJ
ejpam-4248	158	50	function	function	NOUN
ejpam-4248	158	51	,	,	PUNCT
ejpam-4248	158	52	so	so	SCONJ
ejpam-4248	158	53	for	for	ADP
ejpam-4248	158	54	each	each	DET
ejpam-4248	158	55	y	y	PROPN
ejpam-4248	158	56	∈	∈	PROPN
ejpam-4248	158	57	y	y	PROPN
ejpam-4248	158	58	,	,	PUNCT
ejpam-4248	158	59	there	there	PRON
ejpam-4248	158	60	is	be	VERB
ejpam-4248	158	61	a	a	DET
ejpam-4248	158	62	finite	finite	NOUN
ejpam-4248	158	63	set	set	VERB
ejpam-4248	158	64	∆0	∆0	ADP
ejpam-4248	158	65	⊆	⊆	NUM
ejpam-4248	158	66	∆	∆	PROPN
ejpam-4248	158	67	such	such	ADJ
ejpam-4248	159	1	that	that	SCONJ
ejpam-4248	159	2	f−1(y	f−1(y	PROPN
ejpam-4248	159	3	)	)	PUNCT
ejpam-4248	159	4	⊆	⊆	NUM
ejpam-4248	159	5	⋃	⋃	ADP
ejpam-4248	159	6	α∈∆0	α∈∆0	NUM
ejpam-4248	159	7	intcoc(fα	intcoc(fα	NOUN
ejpam-4248	159	8	)	)	PUNCT
ejpam-4248	159	9	⊆	⊆	NUM
ejpam-4248	159	10	⋃	⋃	NOUN
ejpam-4248	159	11	α∈∆0	α∈∆0	NOUN
ejpam-4248	159	12	fα	fα	ADP
ejpam-4248	159	13	⊆	⊆	NUM
ejpam-4248	159	14	⋃	⋃	NOUN
ejpam-4248	159	15	α∈∆0	α∈∆0	X
ejpam-4248	159	16	uα	uα	PROPN
ejpam-4248	159	17	.	.	PUNCT
ejpam-4248	160	1	define	define	VERB
ejpam-4248	160	2	ky	ky	PROPN
ejpam-4248	160	3	=	=	PUNCT
ejpam-4248	160	4	y	y	PROPN
ejpam-4248	161	1	−	−	PROPN
ejpam-4248	161	2	f	f	PROPN
ejpam-4248	161	3	(	(	PUNCT
ejpam-4248	161	4	x	x	X
ejpam-4248	161	5	−	−	ADP
ejpam-4248	161	6	⋃	⋃	NOUN
ejpam-4248	161	7	{	{	PUNCT
ejpam-4248	161	8	intcoc	intcoc	NOUN
ejpam-4248	161	9	(	(	PUNCT
ejpam-4248	161	10	fα	fα	NOUN
ejpam-4248	161	11	)	)	PUNCT
ejpam-4248	161	12	|α	|α	NOUN
ejpam-4248	161	13	∈	∈	PROPN
ejpam-4248	161	14	∆0	∆0	NOUN
ejpam-4248	161	15	}	}	PUNCT
ejpam-4248	161	16	)	)	PUNCT
ejpam-4248	161	17	,	,	PUNCT
ejpam-4248	161	18	jy	jy	PROPN
ejpam-4248	161	19	=	=	SYM
ejpam-4248	161	20	y	y	PROPN
ejpam-4248	162	1	−	−	PROPN
ejpam-4248	162	2	f	f	PROPN
ejpam-4248	162	3	(	(	PUNCT
ejpam-4248	162	4	x	x	X
ejpam-4248	162	5	−	−	X
ejpam-4248	162	6	⋃	⋃	NOUN
ejpam-4248	162	7	{	{	PUNCT
ejpam-4248	162	8	intcoc(fα)|α	intcoc(fα)|α	NOUN
ejpam-4248	162	9	∈	∈	PROPN
ejpam-4248	162	10	∆0	∆0	PRON
ejpam-4248	162	11	}	}	PUNCT
ejpam-4248	162	12	coc	coc	PROPN
ejpam-4248	162	13	)	)	PUNCT
ejpam-4248	162	14	,	,	PUNCT
ejpam-4248	162	15	and	and	CCONJ
ejpam-4248	162	16	ly	ly	X
ejpam-4248	163	1	=	=	SYM
ejpam-4248	163	2	y	y	PROPN
ejpam-4248	163	3	−	−	PROPN
ejpam-4248	163	4	f	f	PROPN
ejpam-4248	163	5	(	(	PUNCT
ejpam-4248	163	6	x	x	X
ejpam-4248	163	7	−	−	PROPN
ejpam-4248	163	8	⋃	⋃	NOUN
ejpam-4248	163	9	{	{	PUNCT
ejpam-4248	163	10	uα|α	uα|α	PROPN
ejpam-4248	163	11	∈	∈	PROPN
ejpam-4248	163	12	∆0	∆0	PRON
ejpam-4248	163	13	}	}	PUNCT
ejpam-4248	163	14	)	)	PUNCT
ejpam-4248	163	15	.	.	PUNCT
ejpam-4248	164	1	clearlyky	clearlyky	VERB
ejpam-4248	164	2	⊆	⊆	NUM
ejpam-4248	164	3	jy	jy	PROPN
ejpam-4248	164	4	⊆	⊆	NUM
ejpam-4248	164	5	ly	ly	NOUN
ejpam-4248	164	6	,	,	PUNCT
ejpam-4248	164	7	alsoky	alsoky	NOUN
ejpam-4248	164	8	and	and	CCONJ
ejpam-4248	164	9	ly	ly	NOUN
ejpam-4248	164	10	are	be	AUX
ejpam-4248	164	11	coc	coc	ADJ
ejpam-4248	164	12	-	-	PUNCT
ejpam-4248	164	13	open	open	ADJ
ejpam-4248	164	14	subsets	subset	NOUN
ejpam-4248	164	15	of	of	ADP
ejpam-4248	164	16	y	y	PROPN
ejpam-4248	164	17	and	and	CCONJ
ejpam-4248	164	18	jy	jy	PROPN
ejpam-4248	164	19	is	be	AUX
ejpam-4248	164	20	coc	coc	ADJ
ejpam-4248	164	21	-	-	PUNCT
ejpam-4248	164	22	closed	close	VERB
ejpam-4248	164	23	subset	subset	NOUN
ejpam-4248	164	24	in	in	ADP
ejpam-4248	164	25	y	y	PROPN
ejpam-4248	164	26	,	,	PUNCT
ejpam-4248	164	27	so	so	CCONJ
ejpam-4248	164	28	for	for	ADP
ejpam-4248	164	29	each	each	DET
ejpam-4248	164	30	y	y	PROPN
ejpam-4248	164	31	∈	∈	PROPN
ejpam-4248	164	32	y	y	PROPN
ejpam-4248	164	33	,	,	PUNCT
ejpam-4248	164	34	there	there	PRON
ejpam-4248	164	35	exists	exist	VERB
ejpam-4248	164	36	α	α	PROPN
ejpam-4248	164	37	∈	∈	PROPN
ejpam-4248	164	38	∆0	∆0	NUM
ejpam-4248	164	39	such	such	ADJ
ejpam-4248	164	40	that	that	SCONJ
ejpam-4248	164	41	y	y	PROPN
ejpam-4248	164	42	∈	∈	PROPN
ejpam-4248	164	43	ky	ky	PROPN
ejpam-4248	164	44	,	,	PUNCT
ejpam-4248	164	45	therefore	therefore	ADV
ejpam-4248	164	46	{	{	PUNCT
ejpam-4248	164	47	ky|y	ky|y	PROPN
ejpam-4248	164	48	∈	∈	PROPN
ejpam-4248	164	49	y	y	PROPN
ejpam-4248	164	50	}	}	PUNCT
ejpam-4248	164	51	is	be	AUX
ejpam-4248	164	52	a	a	DET
ejpam-4248	164	53	coc	coc	ADJ
ejpam-4248	164	54	-	-	PUNCT
ejpam-4248	164	55	regular	regular	ADJ
ejpam-4248	164	56	cover	cover	NOUN
ejpam-4248	164	57	of	of	ADP
ejpam-4248	164	58	y	y	PROPN
ejpam-4248	164	59	,	,	PUNCT
ejpam-4248	164	60	but	but	CCONJ
ejpam-4248	164	61	y	y	PROPN
ejpam-4248	164	62	is	be	AUX
ejpam-4248	164	63	coc	coc	ADJ
ejpam-4248	164	64	-	-	PUNCT
ejpam-4248	164	65	r	r	NOUN
ejpam-4248	164	66	-	-	PUNCT
ejpam-4248	164	67	compact	compact	ADJ
ejpam-4248	164	68	,	,	PUNCT
ejpam-4248	164	69	so	so	ADV
ejpam-4248	164	70	y	y	PROPN
ejpam-4248	164	71	=	=	SYM
ejpam-4248	164	72	⋃	⋃	NOUN
ejpam-4248	164	73	{	{	PUNCT
ejpam-4248	164	74	lyi	lyi	NOUN
ejpam-4248	164	75	|i	|i	NOUN
ejpam-4248	164	76	=	=	NOUN
ejpam-4248	164	77	1	1	NUM
ejpam-4248	164	78	,	,	PUNCT
ejpam-4248	164	79	2	2	NUM
ejpam-4248	164	80	,	,	PUNCT
ejpam-4248	164	81	...	...	PUNCT
ejpam-4248	164	82	,	,	PUNCT
ejpam-4248	164	83	n	n	CCONJ
ejpam-4248	164	84	}	}	PUNCT
ejpam-4248	164	85	,	,	PUNCT
ejpam-4248	164	86	moreover	moreover	ADV
ejpam-4248	164	87	x	x	X
ejpam-4248	164	88	=	=	SYM
ejpam-4248	164	89	f−1	f−1	PROPN
ejpam-4248	164	90	(	(	PUNCT
ejpam-4248	164	91	y	y	NOUN
ejpam-4248	164	92	)	)	PUNCT
ejpam-4248	165	1	=	=	SYM
ejpam-4248	165	2	f−1	f−1	PROPN
ejpam-4248	165	3	(	(	PUNCT
ejpam-4248	165	4	{	{	PUNCT
ejpam-4248	165	5	lyi	lyi	PROPN
ejpam-4248	165	6	|i	|i	NOUN
ejpam-4248	165	7	=	=	NOUN
ejpam-4248	165	8	1	1	NUM
ejpam-4248	165	9	,	,	PUNCT
ejpam-4248	165	10	2	2	NUM
ejpam-4248	165	11	,	,	PUNCT
ejpam-4248	165	12	...	...	PUNCT
ejpam-4248	165	13	,	,	PUNCT
ejpam-4248	165	14	n	n	CCONJ
ejpam-4248	165	15	}	}	PUNCT
ejpam-4248	165	16	)	)	PUNCT
ejpam-4248	166	1	⊆	⊆	NUM
ejpam-4248	166	2	⋃	⋃	NOUN
ejpam-4248	166	3	{	{	PUNCT
ejpam-4248	166	4	uα|α	uα|α	PROPN
ejpam-4248	166	5	∈	∈	PROPN
ejpam-4248	166	6	∆0	∆0	PROPN
ejpam-4248	166	7	}	}	PUNCT
ejpam-4248	166	8	,	,	PUNCT
ejpam-4248	166	9	this	this	PRON
ejpam-4248	166	10	complete	complete	VERB
ejpam-4248	166	11	the	the	DET
ejpam-4248	166	12	proof	proof	NOUN
ejpam-4248	166	13	.	.	PUNCT
ejpam-4248	167	1	corollary	corollary	ADJ
ejpam-4248	167	2	9	9	NUM
ejpam-4248	167	3	.	.	PUNCT
ejpam-4248	168	1	let	let	VERB
ejpam-4248	168	2	f	f	NOUN
ejpam-4248	168	3	:	:	PUNCT
ejpam-4248	168	4	x	x	X
ejpam-4248	168	5	→	→	SYM
ejpam-4248	168	6	y	y	X
ejpam-4248	168	7	be	be	AUX
ejpam-4248	168	8	a	a	DET
ejpam-4248	168	9	coc	coc	ADJ
ejpam-4248	168	10	-	-	PUNCT
ejpam-4248	168	11	open	open	ADJ
ejpam-4248	168	12	coc	coc	ADJ
ejpam-4248	168	13	-	-	PUNCT
ejpam-4248	168	14	perfect	perfect	ADJ
ejpam-4248	168	15	coc	coc	NOUN
ejpam-4248	168	16	-	-	ADJ
ejpam-4248	168	17	continuous	continuous	ADJ
ejpam-4248	168	18	,	,	PUNCT
ejpam-4248	168	19	then	then	ADV
ejpam-4248	168	20	x	x	PUNCT
ejpam-4248	168	21	is	be	AUX
ejpam-4248	168	22	coc	coc	ADJ
ejpam-4248	168	23	-	-	PUNCT
ejpam-4248	168	24	rcompact	rcompact	ADJ
ejpam-4248	168	25	space	space	NOUN
ejpam-4248	168	26	if	if	SCONJ
ejpam-4248	169	1	and	and	CCONJ
ejpam-4248	169	2	only	only	ADV
ejpam-4248	169	3	if	if	SCONJ
ejpam-4248	169	4	y	y	PROPN
ejpam-4248	169	5	is	be	AUX
ejpam-4248	169	6	coc	coc	ADJ
ejpam-4248	169	7	-	-	PUNCT
ejpam-4248	169	8	r	r	NOUN
ejpam-4248	169	9	-	-	PUNCT
ejpam-4248	169	10	compact	compact	ADJ
ejpam-4248	169	11	space	space	NOUN
ejpam-4248	169	12	.	.	PUNCT
ejpam-4248	170	1	corollary	corollary	ADJ
ejpam-4248	170	2	10	10	NUM
ejpam-4248	170	3	.	.	PUNCT
ejpam-4248	171	1	if	if	SCONJ
ejpam-4248	171	2	x	x	PRON
ejpam-4248	171	3	is	be	AUX
ejpam-4248	171	4	coc	coc	ADJ
ejpam-4248	171	5	-	-	PUNCT
ejpam-4248	171	6	weakly	weakly	ADJ
ejpam-4248	171	7	-	-	PUNCT
ejpam-4248	171	8	compact	compact	ADJ
ejpam-4248	171	9	space	space	NOUN
ejpam-4248	171	10	and	and	CCONJ
ejpam-4248	171	11	y	y	PROPN
ejpam-4248	171	12	is	be	AUX
ejpam-4248	171	13	coc	coc	ADJ
ejpam-4248	171	14	-	-	ADJ
ejpam-4248	171	15	compact	compact	ADJ
ejpam-4248	171	16	space	space	NOUN
ejpam-4248	171	17	,	,	PUNCT
ejpam-4248	171	18	then	then	ADV
ejpam-4248	171	19	x×y	x×y	PUNCT
ejpam-4248	171	20	is	be	AUX
ejpam-4248	171	21	coc	coc	ADJ
ejpam-4248	171	22	-	-	PUNCT
ejpam-4248	171	23	weakly	weakly	ADJ
ejpam-4248	171	24	-	-	PUNCT
ejpam-4248	171	25	compact	compact	ADJ
ejpam-4248	171	26	.	.	PUNCT
ejpam-4248	172	1	proof	proof	NOUN
ejpam-4248	172	2	.	.	PUNCT
ejpam-4248	173	1	notice	notice	VERB
ejpam-4248	173	2	that	that	SCONJ
ejpam-4248	173	3	the	the	DET
ejpam-4248	173	4	projection	projection	NOUN
ejpam-4248	173	5	px	px	X
ejpam-4248	173	6	:	:	PUNCT
ejpam-4248	173	7	x	x	SYM
ejpam-4248	173	8	×	×	NOUN
ejpam-4248	173	9	y	y	PROPN
ejpam-4248	173	10	→	→	PUNCT
ejpam-4248	173	11	x	x	X
ejpam-4248	173	12	is	be	AUX
ejpam-4248	173	13	coc	coc	NOUN
ejpam-4248	173	14	-	-	PUNCT
ejpam-4248	173	15	closed	close	VERB
ejpam-4248	173	16	and	and	CCONJ
ejpam-4248	173	17	the	the	DET
ejpam-4248	173	18	result	result	NOUN
ejpam-4248	173	19	comes	come	VERB
ejpam-4248	173	20	from	from	ADP
ejpam-4248	173	21	theorem	theorem	ADJ
ejpam-4248	173	22	6	6	NUM
ejpam-4248	173	23	.	.	PUNCT
ejpam-4248	173	24	corollary	corollary	ADJ
ejpam-4248	173	25	11	11	NUM
ejpam-4248	173	26	.	.	PUNCT
ejpam-4248	174	1	for	for	ADP
ejpam-4248	174	2	a	a	DET
ejpam-4248	174	3	coc	coc	NOUN
ejpam-4248	174	4	-	-	PUNCT
ejpam-4248	174	5	almost	almost	ADV
ejpam-4248	174	6	-	-	PUNCT
ejpam-4248	174	7	regular	regular	ADJ
ejpam-4248	174	8	spaces	space	NOUN
ejpam-4248	174	9	x	x	PUNCT
ejpam-4248	174	10	and	and	CCONJ
ejpam-4248	174	11	y	y	PROPN
ejpam-4248	174	12	,	,	PUNCT
ejpam-4248	174	13	if	if	SCONJ
ejpam-4248	174	14	x	x	PRON
ejpam-4248	174	15	is	be	AUX
ejpam-4248	174	16	coc	coc	ADJ
ejpam-4248	174	17	-	-	ADJ
ejpam-4248	174	18	compact	compact	ADJ
ejpam-4248	174	19	,	,	PUNCT
ejpam-4248	174	20	then	then	ADV
ejpam-4248	174	21	x×y	x×y	PUNCT
ejpam-4248	174	22	is	be	AUX
ejpam-4248	174	23	coc	coc	ADJ
ejpam-4248	174	24	-	-	PUNCT
ejpam-4248	174	25	nearly	nearly	ADV
ejpam-4248	174	26	(	(	PUNCT
ejpam-4248	174	27	coc	coc	NOUN
ejpam-4248	174	28	-	-	PUNCT
ejpam-4248	174	29	weakly	weakly	ADJ
ejpam-4248	174	30	-	-	PUNCT
ejpam-4248	174	31	compact	compact	ADJ
ejpam-4248	174	32	,	,	PUNCT
ejpam-4248	174	33	coc	coc	NOUN
ejpam-4248	174	34	-	-	PUNCT
ejpam-4248	174	35	r	r	NOUN
ejpam-4248	174	36	-	-	PUNCT
ejpam-4248	174	37	compact	compact	ADJ
ejpam-4248	174	38	)	)	PUNCT
ejpam-4248	174	39	.	.	PUNCT
ejpam-4248	175	1	corollary	corollary	ADJ
ejpam-4248	175	2	12	12	NUM
ejpam-4248	175	3	.	.	PUNCT
ejpam-4248	176	1	if	if	SCONJ
ejpam-4248	176	2	y	y	PROPN
ejpam-4248	176	3	is	be	AUX
ejpam-4248	176	4	coc	coc	ADJ
ejpam-4248	176	5	-	-	ADJ
ejpam-4248	176	6	compact	compact	ADJ
ejpam-4248	176	7	space	space	NOUN
ejpam-4248	176	8	,	,	PUNCT
ejpam-4248	176	9	then	then	ADV
ejpam-4248	176	10	x	x	SYM
ejpam-4248	176	11	×	×	PROPN
ejpam-4248	176	12	y	y	PROPN
ejpam-4248	176	13	is	be	AUX
ejpam-4248	176	14	coc	coc	ADJ
ejpam-4248	176	15	-	-	PUNCT
ejpam-4248	176	16	almost	almost	ADV
ejpam-4248	176	17	-	-	PUNCT
ejpam-4248	176	18	compact	compact	ADJ
ejpam-4248	176	19	(	(	PUNCT
ejpam-4248	176	20	coc	coc	NOUN
ejpam-4248	176	21	-	-	PUNCT
ejpam-4248	176	22	nearlycompact	nearlycompact	ADJ
ejpam-4248	176	23	,	,	PUNCT
ejpam-4248	176	24	coc	coc	NOUN
ejpam-4248	176	25	-	-	PUNCT
ejpam-4248	176	26	weakly	weakly	ADJ
ejpam-4248	176	27	-	-	PUNCT
ejpam-4248	176	28	compact	compact	ADJ
ejpam-4248	176	29	)	)	PUNCT
ejpam-4248	176	30	if	if	SCONJ
ejpam-4248	176	31	and	and	CCONJ
ejpam-4248	176	32	only	only	ADV
ejpam-4248	176	33	if	if	SCONJ
ejpam-4248	176	34	x	x	PRON
ejpam-4248	176	35	is	be	AUX
ejpam-4248	176	36	coc	coc	ADJ
ejpam-4248	176	37	-	-	PUNCT
ejpam-4248	176	38	r	r	NOUN
ejpam-4248	176	39	-	-	PUNCT
ejpam-4248	176	40	compact	compact	ADJ
ejpam-4248	176	41	space	space	NOUN
ejpam-4248	176	42	.	.	PUNCT
ejpam-4248	177	1	references	reference	NOUN
ejpam-4248	177	2	206	206	NUM
ejpam-4248	177	3	acknowledgements	acknowledgement	NOUN
ejpam-4248	177	4	the	the	DET
ejpam-4248	177	5	authors	author	NOUN
ejpam-4248	177	6	are	be	AUX
ejpam-4248	177	7	grateful	grateful	ADJ
ejpam-4248	177	8	to	to	ADP
ejpam-4248	177	9	the	the	DET
ejpam-4248	177	10	middle	middle	PROPN
ejpam-4248	177	11	east	east	PROPN
ejpam-4248	177	12	university	university	PROPN
ejpam-4248	177	13	,	,	PUNCT
ejpam-4248	177	14	amman	amman	PROPN
ejpam-4248	177	15	,	,	PUNCT
ejpam-4248	177	16	jordan	jordan	PROPN
ejpam-4248	177	17	for	for	ADP
ejpam-4248	177	18	the	the	DET
ejpam-4248	177	19	financial	financial	ADJ
ejpam-4248	177	20	support	support	NOUN
ejpam-4248	177	21	granted	grant	VERB
ejpam-4248	177	22	to	to	PART
ejpam-4248	177	23	cover	cover	VERB
ejpam-4248	177	24	the	the	DET
ejpam-4248	177	25	publication	publication	NOUN
ejpam-4248	177	26	fee	fee	NOUN
ejpam-4248	177	27	of	of	ADP
ejpam-4248	177	28	this	this	DET
ejpam-4248	177	29	research	research	NOUN
ejpam-4248	177	30	article	article	NOUN
ejpam-4248	177	31	.	.	PUNCT
ejpam-4248	178	1	references	reference	NOUN
ejpam-4248	178	2	[	[	X
ejpam-4248	178	3	1	1	NUM
ejpam-4248	178	4	]	]	PUNCT
ejpam-4248	178	5	z	z	NOUN
ejpam-4248	178	6	altawalbeh	altawalbeh	NOUN
ejpam-4248	178	7	.	.	PUNCT
ejpam-4248	179	1	more	more	ADV
ejpam-4248	179	2	on	on	ADP
ejpam-4248	179	3	almost	almost	ADV
ejpam-4248	179	4	countably	countably	ADV
ejpam-4248	179	5	compact	compact	ADJ
ejpam-4248	179	6	spaces	space	NOUN
ejpam-4248	179	7	.	.	PUNCT
ejpam-4248	180	1	italian	italian	ADJ
ejpam-4248	180	2	journal	journal	NOUN
ejpam-4248	180	3	of	of	ADP
ejpam-4248	180	4	pure	pure	ADJ
ejpam-4248	180	5	and	and	CCONJ
ejpam-4248	180	6	applied	applied	ADJ
ejpam-4248	180	7	mathematics	mathematic	NOUN
ejpam-4248	180	8	,	,	PUNCT
ejpam-4248	180	9	43:177–184	43:177–184	PROPN
ejpam-4248	180	10	,	,	PUNCT
ejpam-4248	180	11	2020	2020	NUM
ejpam-4248	180	12	.	.	PUNCT
ejpam-4248	181	1	[	[	X
ejpam-4248	181	2	2	2	NUM
ejpam-4248	181	3	]	]	PUNCT
ejpam-4248	181	4	z	z	NOUN
ejpam-4248	181	5	altawallbeh	altawallbeh	NOUN
ejpam-4248	181	6	and	and	CCONJ
ejpam-4248	181	7	a	a	DET
ejpam-4248	181	8	al	al	PROPN
ejpam-4248	181	9	-	-	PUNCT
ejpam-4248	181	10	momany	momany	NOUN
ejpam-4248	181	11	.	.	PUNCT
ejpam-4248	182	1	nearly	nearly	ADV
ejpam-4248	182	2	countably	countably	ADV
ejpam-4248	182	3	compact	compact	ADJ
ejpam-4248	182	4	spaces	space	NOUN
ejpam-4248	182	5	.	.	PUNCT
ejpam-4248	183	1	international	international	ADJ
ejpam-4248	183	2	electronic	electronic	ADJ
ejpam-4248	183	3	journal	journal	NOUN
ejpam-4248	183	4	of	of	ADP
ejpam-4248	183	5	pure	pure	ADJ
ejpam-4248	183	6	and	and	CCONJ
ejpam-4248	183	7	applied	applied	ADJ
ejpam-4248	183	8	mathematics	mathematic	NOUN
ejpam-4248	183	9	,	,	PUNCT
ejpam-4248	183	10	8(4):59–65	8(4):59–65	NUM
ejpam-4248	183	11	,	,	PUNCT
ejpam-4248	183	12	2014	2014	NUM
ejpam-4248	183	13	.	.	PUNCT
ejpam-4248	184	1	[	[	X
ejpam-4248	184	2	3	3	NUM
ejpam-4248	184	3	]	]	X
ejpam-4248	184	4	f	f	PROPN
ejpam-4248	184	5	cammaroto	cammaroto	NOUN
ejpam-4248	184	6	.	.	PUNCT
ejpam-4248	185	1	a	a	DET
ejpam-4248	185	2	note	note	NOUN
ejpam-4248	185	3	on	on	ADP
ejpam-4248	185	4	weakly	weakly	ADJ
ejpam-4248	185	5	-	-	PUNCT
ejpam-4248	185	6	compact	compact	ADJ
ejpam-4248	185	7	spaces	space	NOUN
ejpam-4248	185	8	.	.	PUNCT
ejpam-4248	186	1	indian	indian	PROPN
ejpam-4248	186	2	j.	j.	PROPN
ejpam-4248	186	3	pure	pure	PROPN
ejpam-4248	186	4	appl	appl	PROPN
ejpam-4248	186	5	.	.	PUNCT
ejpam-4248	186	6	math	math	PROPN
ejpam-4248	186	7	,	,	PUNCT
ejpam-4248	186	8	16(12):1472–1477	16(12):1472–1477	NUM
ejpam-4248	186	9	,	,	PUNCT
ejpam-4248	186	10	1985	1985	NUM
ejpam-4248	186	11	.	.	PUNCT
ejpam-4248	187	1	[	[	X
ejpam-4248	187	2	4	4	NUM
ejpam-4248	187	3	]	]	SYM
ejpam-4248	187	4	f	f	PROPN
ejpam-4248	187	5	cammaroto	cammaroto	NOUN
ejpam-4248	187	6	and	and	CCONJ
ejpam-4248	187	7	g	g	PROPN
ejpam-4248	187	8	lo	lo	PROPN
ejpam-4248	187	9	faro	faro	NOUN
ejpam-4248	187	10	.	.	PUNCT
ejpam-4248	188	1	weakly	weakly	ADJ
ejpam-4248	188	2	compact	compact	ADJ
ejpam-4248	188	3	spaces	space	NOUN
ejpam-4248	188	4	.	.	PUNCT
ejpam-4248	189	1	riv	riv	PROPN
ejpam-4248	189	2	.	.	PROPN
ejpam-4248	189	3	mat	mat	PROPN
ejpam-4248	189	4	.	.	PROPN
ejpam-4248	189	5	univ	univ	PROPN
ejpam-4248	189	6	.	.	PUNCT
ejpam-4248	190	1	parma	parma	PROPN
ejpam-4248	190	2	,	,	PUNCT
ejpam-4248	190	3	7(4):383–395	7(4):383–395	PROPN
ejpam-4248	190	4	,	,	PUNCT
ejpam-4248	190	5	1981	1981	NUM
ejpam-4248	190	6	.	.	PUNCT
ejpam-4248	191	1	[	[	X
ejpam-4248	191	2	5	5	NUM
ejpam-4248	191	3	]	]	SYM
ejpam-4248	191	4	f	f	PROPN
ejpam-4248	191	5	cammaroto	cammaroto	NOUN
ejpam-4248	191	6	and	and	CCONJ
ejpam-4248	191	7	t	t	PROPN
ejpam-4248	191	8	noiri	noiri	PROPN
ejpam-4248	191	9	.	.	PUNCT
ejpam-4248	192	1	on	on	ADP
ejpam-4248	192	2	r	r	NOUN
ejpam-4248	192	3	-	-	ADJ
ejpam-4248	192	4	compact	compact	ADJ
ejpam-4248	192	5	spaces	space	NOUN
ejpam-4248	192	6	.	.	PUNCT
ejpam-4248	193	1	mat	mat	NOUN
ejpam-4248	193	2	.	.	PUNCT
ejpam-4248	193	3	bec	bec	PROPN
ejpam-4248	193	4	.	.	PROPN
ejpam-4248	193	5	,	,	PUNCT
ejpam-4248	193	6	41:141–147	41:141–147	PROPN
ejpam-4248	193	7	,	,	PUNCT
ejpam-4248	193	8	1989	1989	NUM
ejpam-4248	193	9	.	.	PUNCT
ejpam-4248	194	1	[	[	X
ejpam-4248	194	2	6	6	NUM
ejpam-4248	194	3	]	]	X
ejpam-4248	194	4	r	r	NOUN
ejpam-4248	194	5	engelking	engelking	NOUN
ejpam-4248	194	6	.	.	PUNCT
ejpam-4248	195	1	general	general	ADJ
ejpam-4248	195	2	topology	topology	PROPN
ejpam-4248	195	3	.	.	PUNCT
ejpam-4248	196	1	revised	revise	VERB
ejpam-4248	196	2	and	and	CCONJ
ejpam-4248	196	3	completed	complete	VERB
ejpam-4248	196	4	edition	edition	NOUN
ejpam-4248	196	5	.	.	PUNCT
ejpam-4248	197	1	heldermann	heldermann	PROPN
ejpam-4248	197	2	verlag	verlag	PROPN
ejpam-4248	197	3	,	,	PUNCT
ejpam-4248	197	4	berlin	berlin	PROPN
ejpam-4248	197	5	,	,	PUNCT
ejpam-4248	197	6	1989	1989	NUM
ejpam-4248	197	7	.	.	PUNCT
ejpam-4248	198	1	[	[	X
ejpam-4248	198	2	7	7	X
ejpam-4248	198	3	]	]	X
ejpam-4248	198	4	al	al	PROPN
ejpam-4248	198	5	ghour	ghour	PROPN
ejpam-4248	198	6	,	,	PUNCT
ejpam-4248	198	7	s	s	X
ejpam-4248	198	8	,	,	PUNCT
ejpam-4248	198	9	and	and	CCONJ
ejpam-4248	198	10	s	s	VERB
ejpam-4248	198	11	samarah	samarah	NOUN
ejpam-4248	198	12	.	.	PUNCT
ejpam-4248	199	1	cocompact	cocompact	PROPN
ejpam-4248	199	2	open	open	ADJ
ejpam-4248	199	3	sets	set	NOUN
ejpam-4248	199	4	and	and	CCONJ
ejpam-4248	199	5	continuity	continuity	NOUN
ejpam-4248	199	6	.	.	PUNCT
ejpam-4248	200	1	in	in	ADP
ejpam-4248	200	2	abstarct	abstarct	PROPN
ejpam-4248	200	3	and	and	CCONJ
ejpam-4248	200	4	applied	apply	VERB
ejpam-4248	200	5	analysis	analysis	NOUN
ejpam-4248	200	6	,	,	PUNCT
ejpam-4248	200	7	p548612	p548612	NOUN
ejpam-4248	200	8	,	,	PUNCT
ejpam-4248	200	9	2012	2012	NUM
ejpam-4248	200	10	.	.	PUNCT
ejpam-4248	201	1	[	[	X
ejpam-4248	201	2	8	8	NUM
ejpam-4248	201	3	]	]	X
ejpam-4248	201	4	c	c	PROPN
ejpam-4248	201	5	kuratowski	kuratowski	PROPN
ejpam-4248	201	6	.	.	PUNCT
ejpam-4248	202	1	topologie	topologie	NOUN
ejpam-4248	203	1	i	i	PRON
ejpam-4248	203	2	.4th	.4th	PUNCT
ejpam-4248	203	3	edition	edition	PROPN
ejpam-4248	203	4	,	,	PUNCT
ejpam-4248	203	5	in	in	ADP
ejpam-4248	203	6	french	french	PROPN
ejpam-4248	203	7	.	.	PUNCT
ejpam-4248	204	1	hanfner	hanfner	ADJ
ejpam-4248	204	2	,	,	PUNCT
ejpam-4248	204	3	new	new	PROPN
ejpam-4248	204	4	york	york	PROPN
ejpam-4248	204	5	,	,	PUNCT
ejpam-4248	204	6	1958	1958	NUM
ejpam-4248	204	7	.	.	PUNCT
ejpam-4248	205	1	[	[	X
ejpam-4248	205	2	9	9	NUM
ejpam-4248	205	3	]	]	PUNCT
ejpam-4248	205	4	n	n	DET
ejpam-4248	205	5	levine	levine	PROPN
ejpam-4248	205	6	.	.	PUNCT
ejpam-4248	206	1	semi	semi	ADJ
ejpam-4248	206	2	-	-	ADJ
ejpam-4248	206	3	open	open	ADJ
ejpam-4248	206	4	sets	set	NOUN
ejpam-4248	206	5	and	and	CCONJ
ejpam-4248	206	6	semicontinuity	semicontinuity	NOUN
ejpam-4248	206	7	in	in	ADP
ejpam-4248	206	8	topological	topological	ADJ
ejpam-4248	206	9	spaces	space	NOUN
ejpam-4248	206	10	.	.	PUNCT
ejpam-4248	207	1	amer	amer	PROPN
ejpam-4248	207	2	.	.	PUNCT
ejpam-4248	207	3	math	math	PROPN
ejpam-4248	207	4	.	.	PUNCT
ejpam-4248	207	5	,	,	PUNCT
ejpam-4248	208	1	70:36–41	70:36–41	NUM
ejpam-4248	208	2	,	,	PUNCT
ejpam-4248	208	3	1963	1963	NUM
ejpam-4248	208	4	.	.	PUNCT
ejpam-4248	209	1	[	[	X
ejpam-4248	209	2	10	10	NUM
ejpam-4248	209	3	]	]	X
ejpam-4248	209	4	a	a	DET
ejpam-4248	209	5	mashhour	mashhour	NOUN
ejpam-4248	209	6	,	,	PUNCT
ejpam-4248	209	7	m	m	VERB
ejpam-4248	209	8	abd	abd	PROPN
ejpam-4248	209	9	el	el	PROPN
ejpam-4248	209	10	-	-	NOUN
ejpam-4248	209	11	monsef	monsef	ADJ
ejpam-4248	209	12	,	,	PUNCT
ejpam-4248	209	13	and	and	CCONJ
ejpam-4248	209	14	s	s	PROPN
ejpam-4248	209	15	el	el	PROPN
ejpam-4248	209	16	deeb	deeb	PROPN
ejpam-4248	209	17	.	.	PUNCT
ejpam-4248	210	1	on	on	ADP
ejpam-4248	210	2	precontinuous	precontinuous	ADJ
ejpam-4248	210	3	and	and	CCONJ
ejpam-4248	210	4	weak	weak	ADJ
ejpam-4248	210	5	precontinuous	precontinuous	NOUN
ejpam-4248	210	6	.	.	PUNCT
ejpam-4248	211	1	proc	proc	PROPN
ejpam-4248	211	2	.	.	PUNCT
ejpam-4248	212	1	math	math	NOUN
ejpam-4248	212	2	.	.	PUNCT
ejpam-4248	213	1	phs	phs	PROPN
ejpam-4248	213	2	.	.	PROPN
ejpam-4248	213	3	soc	soc	PROPN
ejpam-4248	213	4	.	.	PUNCT
ejpam-4248	214	1	egypt	egypt	PROPN
ejpam-4248	214	2	,	,	PUNCT
ejpam-4248	214	3	53:47–53	53:47–53	NUM
ejpam-4248	214	4	,	,	PUNCT
ejpam-4248	214	5	1982	1982	NUM
ejpam-4248	214	6	.	.	PUNCT
ejpam-4248	215	1	[	[	X
ejpam-4248	215	2	11	11	NUM
ejpam-4248	215	3	]	]	X
ejpam-4248	215	4	al	al	PROPN
ejpam-4248	215	5	ghour	ghour	PROPN
ejpam-4248	215	6	s	s	PART
ejpam-4248	215	7	and	and	CCONJ
ejpam-4248	215	8	e	e	NOUN
ejpam-4248	215	9	maghrabi	maghrabi	NOUN
ejpam-4248	215	10	.	.	PUNCT
ejpam-4248	216	1	co	co	ADJ
ejpam-4248	216	2	-	-	ADJ
ejpam-4248	216	3	compact	compact	ADJ
ejpam-4248	216	4	separation	separation	NOUN
ejpam-4248	216	5	axoims	axoim	NOUN
ejpam-4248	216	6	and	and	CCONJ
ejpam-4248	216	7	slight	slight	ADJ
ejpam-4248	216	8	co	co	NOUN
ejpam-4248	216	9	-	-	NOUN
ejpam-4248	216	10	continuity	continuity	NOUN
ejpam-4248	216	11	.	.	PUNCT
ejpam-4248	217	1	symmetry	symmetry	NOUN
ejpam-4248	217	2	,	,	PUNCT
ejpam-4248	217	3	12	12	NUM
ejpam-4248	217	4	,	,	PUNCT
ejpam-4248	217	5	2020	2020	NUM
ejpam-4248	217	6	.	.	PUNCT
ejpam-4248	218	1	[	[	X
ejpam-4248	218	2	12	12	NUM
ejpam-4248	218	3	]	]	X
ejpam-4248	218	4	m	m	NOUN
ejpam-4248	218	5	singal	singal	ADJ
ejpam-4248	218	6	and	and	CCONJ
ejpam-4248	218	7	s	s	VERB
ejpam-4248	218	8	arya	arya	NOUN
ejpam-4248	218	9	.	.	PUNCT
ejpam-4248	219	1	on	on	ADP
ejpam-4248	219	2	almost	almost	ADV
ejpam-4248	219	3	-	-	PUNCT
ejpam-4248	219	4	regular	regular	ADJ
ejpam-4248	219	5	spaces	space	NOUN
ejpam-4248	219	6	.	.	PUNCT
ejpam-4248	220	1	glasnik	glasnik	PROPN
ejpam-4248	220	2	mat	mat	PROPN
ejpam-4248	220	3	.	.	PROPN
ejpam-4248	220	4	ser	ser	PROPN
ejpam-4248	220	5	3	3	NUM
ejpam-4248	220	6	,	,	PUNCT
ejpam-4248	220	7	4:89–99	4:89–99	NOUN
ejpam-4248	220	8	,	,	PUNCT
ejpam-4248	220	9	1969	1969	NUM
ejpam-4248	220	10	.	.	PUNCT