id	sid	tid	token	lemma	pos
ejpam-6543	1	1	european	european	PROPN
ejpam-6543	1	2	journal	journal	PROPN
ejpam-6543	1	3	of	of	ADP
ejpam-6543	1	4	pure	pure	ADJ
ejpam-6543	1	5	and	and	CCONJ
ejpam-6543	1	6	applied	applied	ADJ
ejpam-6543	1	7	mathematics	mathematic	NOUN
ejpam-6543	1	8	2025	2025	NUM
ejpam-6543	1	9	,	,	PUNCT
ejpam-6543	1	10	vol	vol	NOUN
ejpam-6543	1	11	.	.	PROPN
ejpam-6543	1	12	18	18	NUM
ejpam-6543	1	13	,	,	PUNCT
ejpam-6543	1	14	issue	issue	NOUN
ejpam-6543	1	15	3	3	NUM
ejpam-6543	1	16	,	,	PUNCT
ejpam-6543	1	17	article	article	NOUN
ejpam-6543	1	18	number	number	NOUN
ejpam-6543	1	19	6543	6543	NUM
ejpam-6543	1	20	issn	issn	VERB
ejpam-6543	1	21	1307	1307	NUM
ejpam-6543	1	22	-	-	SYM
ejpam-6543	1	23	5543	5543	NUM
ejpam-6543	1	24	–	–	PUNCT
ejpam-6543	1	25	ejpam.com	ejpam.com	X
ejpam-6543	1	26	published	publish	VERB
ejpam-6543	1	27	by	by	ADP
ejpam-6543	1	28	new	new	PROPN
ejpam-6543	1	29	york	york	PROPN
ejpam-6543	1	30	business	business	PROPN
ejpam-6543	1	31	global	global	NOUN
ejpam-6543	1	32	on	on	ADP
ejpam-6543	1	33	nearly	nearly	ADV
ejpam-6543	1	34	α	α	ADJ
ejpam-6543	1	35	-	-	ADJ
ejpam-6543	1	36	compact	compact	ADJ
ejpam-6543	1	37	topological	topological	ADJ
ejpam-6543	1	38	spaces	space	NOUN
ejpam-6543	1	39	jamal	jamal	PROPN
ejpam-6543	1	40	oudetallah1	oudetallah1	PROPN
ejpam-6543	1	41	,	,	PUNCT
ejpam-6543	1	42	ahmad	ahmad	PROPN
ejpam-6543	1	43	almalkawi2	almalkawi2	PROPN
ejpam-6543	1	44	,	,	PUNCT
ejpam-6543	1	45	ala	ala	PROPN
ejpam-6543	1	46	amourah3,4,∗	amourah3,4,∗	PROPN
ejpam-6543	1	47	,	,	PUNCT
ejpam-6543	1	48	abdullah	abdullah	PROPN
ejpam-6543	1	49	alsoboh5,∗	alsoboh5,∗	NOUN
ejpam-6543	1	50	,	,	PUNCT
ejpam-6543	1	51	khaled	khaled	PROPN
ejpam-6543	1	52	al	al	PROPN
ejpam-6543	1	53	mashrafi	mashrafi	PROPN
ejpam-6543	1	54	5	5	NUM
ejpam-6543	1	55	,	,	PUNCT
ejpam-6543	1	56	tala	tala	PROPN
ejpam-6543	1	57	sasa6	sasa6	NOUN
ejpam-6543	1	58	1	1	NUM
ejpam-6543	1	59	department	department	NOUN
ejpam-6543	1	60	of	of	ADP
ejpam-6543	1	61	mathematics	mathematics	PROPN
ejpam-6543	1	62	,	,	PUNCT
ejpam-6543	1	63	university	university	PROPN
ejpam-6543	1	64	of	of	ADP
ejpam-6543	1	65	petra	petra	PROPN
ejpam-6543	1	66	,	,	PUNCT
ejpam-6543	1	67	amman	amman	PROPN
ejpam-6543	1	68	,	,	PUNCT
ejpam-6543	1	69	11196	11196	NUM
ejpam-6543	1	70	,	,	PUNCT
ejpam-6543	1	71	jordan	jordan	PROPN
ejpam-6543	1	72	2	2	NUM
ejpam-6543	1	73	modern	modern	ADJ
ejpam-6543	1	74	college	college	NOUN
ejpam-6543	1	75	of	of	ADP
ejpam-6543	1	76	business	business	NOUN
ejpam-6543	1	77	and	and	CCONJ
ejpam-6543	1	78	science	science	NOUN
ejpam-6543	1	79	,	,	PUNCT
ejpam-6543	1	80	muscat	muscat	PROPN
ejpam-6543	1	81	,	,	PUNCT
ejpam-6543	1	82	sultanate	sultanate	NOUN
ejpam-6543	1	83	of	of	ADP
ejpam-6543	1	84	oman	oman	NOUN
ejpam-6543	1	85	3	3	NUM
ejpam-6543	1	86	mathematics	mathematics	PROPN
ejpam-6543	1	87	education	education	NOUN
ejpam-6543	1	88	program	program	NOUN
ejpam-6543	1	89	,	,	PUNCT
ejpam-6543	1	90	faculty	faculty	NOUN
ejpam-6543	1	91	of	of	ADP
ejpam-6543	1	92	education	education	NOUN
ejpam-6543	1	93	and	and	CCONJ
ejpam-6543	1	94	arts	art	NOUN
ejpam-6543	1	95	,	,	PUNCT
ejpam-6543	1	96	sohar	sohar	PROPN
ejpam-6543	1	97	university	university	PROPN
ejpam-6543	1	98	,	,	PUNCT
ejpam-6543	1	99	sohar	sohar	PROPN
ejpam-6543	1	100	311	311	NUM
ejpam-6543	1	101	,	,	PUNCT
ejpam-6543	1	102	oman	oman	NOUN
ejpam-6543	1	103	4	4	NUM
ejpam-6543	1	104	jadara	jadara	PROPN
ejpam-6543	1	105	research	research	NOUN
ejpam-6543	1	106	center	center	NOUN
ejpam-6543	1	107	,	,	PUNCT
ejpam-6543	1	108	jadara	jadara	PROPN
ejpam-6543	1	109	university	university	PROPN
ejpam-6543	1	110	,	,	PUNCT
ejpam-6543	1	111	irbid	irbid	VERB
ejpam-6543	1	112	21110	21110	NUM
ejpam-6543	1	113	,	,	PUNCT
ejpam-6543	1	114	jordan	jordan	PROPN
ejpam-6543	1	115	5	5	NUM
ejpam-6543	1	116	college	college	NOUN
ejpam-6543	1	117	of	of	ADP
ejpam-6543	1	118	applied	apply	VERB
ejpam-6543	1	119	and	and	CCONJ
ejpam-6543	1	120	health	health	NOUN
ejpam-6543	1	121	sciences	science	NOUN
ejpam-6543	1	122	,	,	PUNCT
ejpam-6543	1	123	a’sharqiyah	a’sharqiyah	PROPN
ejpam-6543	1	124	university	university	NOUN
ejpam-6543	1	125	,	,	PUNCT
ejpam-6543	1	126	post	post	PROPN
ejpam-6543	1	127	box	box	PROPN
ejpam-6543	1	128	no	no	INTJ
ejpam-6543	1	129	.	.	PROPN
ejpam-6543	1	130	42	42	NUM
ejpam-6543	1	131	,	,	PUNCT
ejpam-6543	1	132	post	post	VERB
ejpam-6543	1	133	code	code	NOUN
ejpam-6543	1	134	no	no	INTJ
ejpam-6543	1	135	.	.	PROPN
ejpam-6543	1	136	400	400	NUM
ejpam-6543	1	137	,	,	PUNCT
ejpam-6543	1	138	ibra	ibra	NOUN
ejpam-6543	1	139	,	,	PUNCT
ejpam-6543	1	140	sultanate	sultanate	NOUN
ejpam-6543	1	141	of	of	ADP
ejpam-6543	1	142	oman	oman	PROPN
ejpam-6543	1	143	6	6	NUM
ejpam-6543	1	144	department	department	NOUN
ejpam-6543	1	145	of	of	ADP
ejpam-6543	1	146	mathematics	mathematic	NOUN
ejpam-6543	1	147	,	,	PUNCT
ejpam-6543	1	148	faculty	faculty	NOUN
ejpam-6543	1	149	of	of	ADP
ejpam-6543	1	150	science	science	NOUN
ejpam-6543	1	151	,	,	PUNCT
ejpam-6543	1	152	applied	apply	VERB
ejpam-6543	1	153	science	science	NOUN
ejpam-6543	1	154	private	private	ADJ
ejpam-6543	1	155	university	university	NOUN
ejpam-6543	1	156	,	,	PUNCT
ejpam-6543	1	157	amman	amman	PROPN
ejpam-6543	1	158	,	,	PUNCT
ejpam-6543	1	159	jordan	jordan	PROPN
ejpam-6543	1	160	abstract	abstract	PROPN
ejpam-6543	1	161	.	.	PUNCT
ejpam-6543	2	1	in	in	ADP
ejpam-6543	2	2	this	this	DET
ejpam-6543	2	3	paper	paper	NOUN
ejpam-6543	2	4	,	,	PUNCT
ejpam-6543	2	5	we	we	PRON
ejpam-6543	2	6	introduce	introduce	VERB
ejpam-6543	2	7	and	and	CCONJ
ejpam-6543	2	8	investigate	investigate	VERB
ejpam-6543	2	9	the	the	DET
ejpam-6543	2	10	concept	concept	NOUN
ejpam-6543	2	11	of	of	ADP
ejpam-6543	2	12	nearly	nearly	ADV
ejpam-6543	2	13	α	α	ADJ
ejpam-6543	2	14	-	-	ADJ
ejpam-6543	2	15	compact	compact	ADJ
ejpam-6543	2	16	topological	topological	ADJ
ejpam-6543	2	17	spaces	space	NOUN
ejpam-6543	2	18	as	as	ADP
ejpam-6543	2	19	a	a	DET
ejpam-6543	2	20	natural	natural	ADJ
ejpam-6543	2	21	generalization	generalization	NOUN
ejpam-6543	2	22	of	of	ADP
ejpam-6543	2	23	α	α	NOUN
ejpam-6543	2	24	-	-	ADJ
ejpam-6543	2	25	compact	compact	ADJ
ejpam-6543	2	26	and	and	CCONJ
ejpam-6543	2	27	countably	countably	ADV
ejpam-6543	2	28	α	α	ADJ
ejpam-6543	2	29	-	-	ADJ
ejpam-6543	2	30	compact	compact	ADJ
ejpam-6543	2	31	spaces	space	NOUN
ejpam-6543	2	32	.	.	PUNCT
ejpam-6543	3	1	we	we	PRON
ejpam-6543	3	2	establish	establish	VERB
ejpam-6543	3	3	fundamental	fundamental	ADJ
ejpam-6543	3	4	properties	property	NOUN
ejpam-6543	3	5	and	and	CCONJ
ejpam-6543	3	6	characterizations	characterization	NOUN
ejpam-6543	3	7	of	of	ADP
ejpam-6543	3	8	nearly	nearly	ADV
ejpam-6543	3	9	α	α	ADJ
ejpam-6543	3	10	-	-	ADJ
ejpam-6543	3	11	compact	compact	ADJ
ejpam-6543	3	12	spaces	space	NOUN
ejpam-6543	3	13	,	,	PUNCT
ejpam-6543	3	14	demonstrating	demonstrate	VERB
ejpam-6543	3	15	their	their	PRON
ejpam-6543	3	16	relationship	relationship	NOUN
ejpam-6543	3	17	with	with	ADP
ejpam-6543	3	18	various	various	ADJ
ejpam-6543	3	19	topological	topological	ADJ
ejpam-6543	3	20	properties	property	NOUN
ejpam-6543	3	21	including	include	VERB
ejpam-6543	3	22	α	α	NOUN
ejpam-6543	3	23	-	-	PUNCT
ejpam-6543	3	24	continuity	continuity	NOUN
ejpam-6543	3	25	,	,	PUNCT
ejpam-6543	3	26	separation	separation	NOUN
ejpam-6543	3	27	axioms	axiom	NOUN
ejpam-6543	3	28	,	,	PUNCT
ejpam-6543	3	29	and	and	CCONJ
ejpam-6543	3	30	compactness	compactness	NOUN
ejpam-6543	3	31	-	-	PUNCT
ejpam-6543	3	32	like	like	ADJ
ejpam-6543	3	33	properties	property	NOUN
ejpam-6543	3	34	.	.	PUNCT
ejpam-6543	4	1	several	several	ADJ
ejpam-6543	4	2	equivalent	equivalent	ADJ
ejpam-6543	4	3	conditions	condition	NOUN
ejpam-6543	4	4	for	for	ADP
ejpam-6543	4	5	nearly	nearly	ADV
ejpam-6543	4	6	α	α	NOUN
ejpam-6543	4	7	-	-	PUNCT
ejpam-6543	4	8	compactness	compactness	NOUN
ejpam-6543	4	9	are	be	AUX
ejpam-6543	4	10	provided	provide	VERB
ejpam-6543	4	11	,	,	PUNCT
ejpam-6543	4	12	and	and	CCONJ
ejpam-6543	4	13	we	we	PRON
ejpam-6543	4	14	prove	prove	VERB
ejpam-6543	4	15	that	that	SCONJ
ejpam-6543	4	16	the	the	DET
ejpam-6543	4	17	property	property	NOUN
ejpam-6543	4	18	is	be	AUX
ejpam-6543	4	19	preserved	preserve	VERB
ejpam-6543	4	20	under	under	ADP
ejpam-6543	4	21	certain	certain	ADJ
ejpam-6543	4	22	types	type	NOUN
ejpam-6543	4	23	of	of	ADP
ejpam-6543	4	24	mappings	mapping	NOUN
ejpam-6543	4	25	.	.	PUNCT
ejpam-6543	5	1	the	the	DET
ejpam-6543	5	2	behavior	behavior	NOUN
ejpam-6543	5	3	of	of	ADP
ejpam-6543	5	4	nearly	nearly	ADV
ejpam-6543	5	5	α	α	ADJ
ejpam-6543	5	6	-	-	ADJ
ejpam-6543	5	7	compact	compact	ADJ
ejpam-6543	5	8	spaces	space	NOUN
ejpam-6543	5	9	under	under	ADP
ejpam-6543	5	10	topological	topological	ADJ
ejpam-6543	5	11	operations	operation	NOUN
ejpam-6543	5	12	such	such	ADJ
ejpam-6543	5	13	as	as	ADP
ejpam-6543	5	14	subspaces	subspace	NOUN
ejpam-6543	5	15	,	,	PUNCT
ejpam-6543	5	16	products	product	NOUN
ejpam-6543	5	17	,	,	PUNCT
ejpam-6543	5	18	and	and	CCONJ
ejpam-6543	5	19	sums	sum	NOUN
ejpam-6543	5	20	is	be	AUX
ejpam-6543	5	21	thoroughly	thoroughly	ADV
ejpam-6543	5	22	examined	examine	VERB
ejpam-6543	5	23	.	.	PUNCT
ejpam-6543	6	1	we	we	PRON
ejpam-6543	6	2	also	also	ADV
ejpam-6543	6	3	introduce	introduce	VERB
ejpam-6543	6	4	the	the	DET
ejpam-6543	6	5	notion	notion	NOUN
ejpam-6543	6	6	of	of	ADP
ejpam-6543	6	7	α	α	NOUN
ejpam-6543	6	8	-	-	NOUN
ejpam-6543	6	9	nearness	nearness	NOUN
ejpam-6543	6	10	and	and	CCONJ
ejpam-6543	6	11	investigate	investigate	VERB
ejpam-6543	6	12	its	its	PRON
ejpam-6543	6	13	connection	connection	NOUN
ejpam-6543	6	14	with	with	ADP
ejpam-6543	6	15	nearly	nearly	ADV
ejpam-6543	6	16	α	α	NUM
ejpam-6543	6	17	-	-	ADJ
ejpam-6543	6	18	compact	compact	ADJ
ejpam-6543	6	19	spaces	space	NOUN
ejpam-6543	6	20	.	.	PUNCT
ejpam-6543	7	1	additionally	additionally	ADV
ejpam-6543	7	2	,	,	PUNCT
ejpam-6543	7	3	we	we	PRON
ejpam-6543	7	4	provide	provide	VERB
ejpam-6543	7	5	comprehensive	comprehensive	ADJ
ejpam-6543	7	6	examples	example	NOUN
ejpam-6543	7	7	and	and	CCONJ
ejpam-6543	7	8	establish	establish	VERB
ejpam-6543	7	9	new	new	ADJ
ejpam-6543	7	10	theorems	theorem	NOUN
ejpam-6543	7	11	that	that	PRON
ejpam-6543	7	12	demonstrate	demonstrate	VERB
ejpam-6543	7	13	the	the	DET
ejpam-6543	7	14	richness	richness	NOUN
ejpam-6543	7	15	and	and	CCONJ
ejpam-6543	7	16	applicability	applicability	NOUN
ejpam-6543	7	17	of	of	ADP
ejpam-6543	7	18	this	this	DET
ejpam-6543	7	19	concept	concept	NOUN
ejpam-6543	7	20	.	.	PUNCT
ejpam-6543	8	1	2020	2020	NUM
ejpam-6543	8	2	mathematics	mathematic	NOUN
ejpam-6543	8	3	subject	subject	NOUN
ejpam-6543	8	4	classifications	classification	NOUN
ejpam-6543	8	5	:	:	PUNCT
ejpam-6543	8	6	54d30	54d30	NUM
ejpam-6543	8	7	,	,	PUNCT
ejpam-6543	8	8	54c08	54c08	NUM
ejpam-6543	8	9	,	,	PUNCT
ejpam-6543	8	10	54d10	54d10	NUM
ejpam-6543	8	11	,	,	PUNCT
ejpam-6543	8	12	54a05	54a05	NUM
ejpam-6543	8	13	key	key	ADJ
ejpam-6543	8	14	words	word	NOUN
ejpam-6543	8	15	and	and	CCONJ
ejpam-6543	8	16	phrases	phrase	NOUN
ejpam-6543	8	17	:	:	PUNCT
ejpam-6543	8	18	nearly	nearly	ADV
ejpam-6543	8	19	α	α	NUM
ejpam-6543	8	20	-	-	ADJ
ejpam-6543	8	21	compact	compact	ADJ
ejpam-6543	8	22	space	space	NOUN
ejpam-6543	8	23	,	,	PUNCT
ejpam-6543	8	24	α	α	X
ejpam-6543	8	25	-	-	ADJ
ejpam-6543	8	26	continuous	continuous	ADJ
ejpam-6543	8	27	function	function	NOUN
ejpam-6543	8	28	,	,	PUNCT
ejpam-6543	8	29	α	α	NOUN
ejpam-6543	8	30	-	-	ADJ
ejpam-6543	8	31	open	open	ADJ
ejpam-6543	8	32	set	set	NOUN
ejpam-6543	8	33	,	,	PUNCT
ejpam-6543	8	34	topological	topological	ADJ
ejpam-6543	8	35	spaces	space	NOUN
ejpam-6543	8	36	,	,	PUNCT
ejpam-6543	8	37	generalized	generalized	ADJ
ejpam-6543	8	38	compactness	compactness	NOUN
ejpam-6543	8	39	1	1	NUM
ejpam-6543	8	40	.	.	PUNCT
ejpam-6543	9	1	introduction	introduction	NOUN
ejpam-6543	9	2	and	and	CCONJ
ejpam-6543	9	3	literature	literature	NOUN
ejpam-6543	9	4	review	review	VERB
ejpam-6543	9	5	the	the	DET
ejpam-6543	9	6	study	study	NOUN
ejpam-6543	9	7	of	of	ADP
ejpam-6543	9	8	generalized	generalized	ADJ
ejpam-6543	9	9	forms	form	NOUN
ejpam-6543	9	10	of	of	ADP
ejpam-6543	9	11	compactness	compactness	NOUN
ejpam-6543	9	12	has	have	AUX
ejpam-6543	9	13	been	be	AUX
ejpam-6543	9	14	a	a	DET
ejpam-6543	9	15	central	central	ADJ
ejpam-6543	9	16	theme	theme	NOUN
ejpam-6543	9	17	in	in	ADP
ejpam-6543	9	18	general	general	ADJ
ejpam-6543	9	19	topology	topology	NOUN
ejpam-6543	9	20	for	for	ADP
ejpam-6543	9	21	several	several	ADJ
ejpam-6543	9	22	decades	decade	NOUN
ejpam-6543	9	23	.	.	PUNCT
ejpam-6543	10	1	the	the	DET
ejpam-6543	10	2	classical	classical	ADJ
ejpam-6543	10	3	notion	notion	NOUN
ejpam-6543	10	4	of	of	ADP
ejpam-6543	10	5	compactness	compactness	NOUN
ejpam-6543	10	6	,	,	PUNCT
ejpam-6543	10	7	while	while	SCONJ
ejpam-6543	10	8	fundamental	fundamental	ADJ
ejpam-6543	10	9	,	,	PUNCT
ejpam-6543	10	10	∗corresponding	∗corresponde	VERB
ejpam-6543	10	11	author	author	NOUN
ejpam-6543	10	12	.	.	PUNCT
ejpam-6543	11	1	doi	doi	NOUN
ejpam-6543	11	2	:	:	PUNCT
ejpam-6543	11	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6543	https://doi.org/10.29020/nybg.ejpam.v18i3.6543	ADJ
ejpam-6543	11	4	email	email	NOUN
ejpam-6543	11	5	addresses	address	NOUN
ejpam-6543	11	6	:	:	PUNCT
ejpam-6543	11	7	jamal.oudetallah@uop.edu.jo	jamal.oudetallah@uop.edu.jo	PROPN
ejpam-6543	11	8	(	(	PUNCT
ejpam-6543	11	9	j.	j.	PROPN
ejpam-6543	11	10	oudetallah	oudetallah	PROPN
ejpam-6543	11	11	)	)	PUNCT
ejpam-6543	11	12	,	,	PUNCT
ejpam-6543	11	13	ahmad.abdelqader@mcbs.edu.om	ahmad.abdelqader@mcbs.edu.om	NOUN
ejpam-6543	11	14	(	(	PUNCT
ejpam-6543	11	15	a.	a.	PROPN
ejpam-6543	11	16	almalkawi	almalkawi	PROPN
ejpam-6543	11	17	)	)	PUNCT
ejpam-6543	11	18	,	,	PUNCT
ejpam-6543	11	19	aamourah@su.edu.om	aamourah@su.edu.om	NOUN
ejpam-6543	11	20	(	(	PUNCT
ejpam-6543	11	21	a.	a.	NOUN
ejpam-6543	11	22	amourah	amourah	PROPN
ejpam-6543	11	23	)	)	PUNCT
ejpam-6543	11	24	,	,	PUNCT
ejpam-6543	11	25	abdullah.alsoboh@asu.edu.om	abdullah.alsoboh@asu.edu.om	NOUN
ejpam-6543	11	26	(	(	PUNCT
ejpam-6543	11	27	a.	a.	NOUN
ejpam-6543	11	28	alsoboh	alsoboh	PROPN
ejpam-6543	11	29	)	)	PUNCT
ejpam-6543	11	30	,	,	PUNCT
ejpam-6543	11	31	khaled.almashrafi@asu.edu.om	khaled.almashrafi@asu.edu.om	PROPN
ejpam-6543	11	32	(	(	PUNCT
ejpam-6543	11	33	k.	k.	PROPN
ejpam-6543	11	34	al	al	PROPN
ejpam-6543	11	35	mashrafi	mashrafi	PROPN
ejpam-6543	11	36	)	)	PUNCT
ejpam-6543	11	37	,	,	PUNCT
ejpam-6543	11	38	t	t	NOUN
ejpam-6543	11	39	sasa@asu.edu.jo	sasa@asu.edu.jo	NOUN
ejpam-6543	11	40	(	(	PUNCT
ejpam-6543	11	41	t.	t.	PROPN
ejpam-6543	11	42	sasa	sasa	PROPN
ejpam-6543	11	43	)	)	PUNCT
ejpam-6543	11	44	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6543	12	1	1	1	NUM
ejpam-6543	12	2	copyright	copyright	NOUN
ejpam-6543	12	3	:	:	PUNCT
ejpam-6543	12	4	©	©	PROPN
ejpam-6543	12	5	2025	2025	NUM
ejpam-6543	12	6	the	the	DET
ejpam-6543	12	7	author(s	author(s	NOUN
ejpam-6543	12	8	)	)	PUNCT
ejpam-6543	12	9	.	.	PUNCT
ejpam-6543	13	1	(	(	PUNCT
ejpam-6543	13	2	cc	cc	NOUN
ejpam-6543	13	3	by	by	ADP
ejpam-6543	13	4	-	-	PUNCT
ejpam-6543	13	5	nc	nc	PROPN
ejpam-6543	13	6	4.0	4.0	NUM
ejpam-6543	13	7	)	)	PUNCT
ejpam-6543	13	8	j.	j.	PROPN
ejpam-6543	13	9	oudetallah	oudetallah	PROPN
ejpam-6543	13	10	et	et	PROPN
ejpam-6543	13	11	al	al	PROPN
ejpam-6543	13	12	.	.	PUNCT
ejpam-6543	13	13	/	/	SYM
ejpam-6543	13	14	eur	eur	PROPN
ejpam-6543	13	15	.	.	PUNCT
ejpam-6543	14	1	j.	j.	PROPN
ejpam-6543	14	2	pure	pure	PROPN
ejpam-6543	14	3	appl	appl	PROPN
ejpam-6543	14	4	.	.	PROPN
ejpam-6543	14	5	math	math	PROPN
ejpam-6543	14	6	,	,	PUNCT
ejpam-6543	14	7	18	18	NUM
ejpam-6543	14	8	(	(	PUNCT
ejpam-6543	14	9	3	3	NUM
ejpam-6543	14	10	)	)	PUNCT
ejpam-6543	14	11	(	(	PUNCT
ejpam-6543	14	12	2025	2025	NUM
ejpam-6543	14	13	)	)	PUNCT
ejpam-6543	14	14	,	,	PUNCT
ejpam-6543	14	15	6543	6543	NUM
ejpam-6543	14	16	2	2	NUM
ejpam-6543	14	17	of	of	ADP
ejpam-6543	14	18	13	13	NUM
ejpam-6543	14	19	often	often	ADV
ejpam-6543	14	20	proves	prove	VERB
ejpam-6543	14	21	too	too	ADV
ejpam-6543	14	22	restrictive	restrictive	ADJ
ejpam-6543	14	23	for	for	ADP
ejpam-6543	14	24	many	many	ADJ
ejpam-6543	14	25	applications	application	NOUN
ejpam-6543	14	26	,	,	PUNCT
ejpam-6543	14	27	leading	lead	VERB
ejpam-6543	14	28	researchers	researcher	NOUN
ejpam-6543	14	29	to	to	PART
ejpam-6543	14	30	investigate	investigate	VERB
ejpam-6543	14	31	various	various	ADJ
ejpam-6543	14	32	weakenings	weakening	NOUN
ejpam-6543	14	33	and	and	CCONJ
ejpam-6543	14	34	generalizations	generalization	NOUN
ejpam-6543	14	35	.	.	PUNCT
ejpam-6543	15	1	among	among	ADP
ejpam-6543	15	2	these	these	PRON
ejpam-6543	15	3	,	,	PUNCT
ejpam-6543	15	4	α	α	NOUN
ejpam-6543	15	5	-	-	NOUN
ejpam-6543	15	6	compactness	compactness	NOUN
ejpam-6543	15	7	and	and	CCONJ
ejpam-6543	15	8	its	its	PRON
ejpam-6543	15	9	variants	variant	NOUN
ejpam-6543	15	10	have	have	AUX
ejpam-6543	15	11	emerged	emerge	VERB
ejpam-6543	15	12	as	as	ADP
ejpam-6543	15	13	particularly	particularly	ADV
ejpam-6543	15	14	fruitful	fruitful	ADJ
ejpam-6543	15	15	areas	area	NOUN
ejpam-6543	15	16	of	of	ADP
ejpam-6543	15	17	investigation	investigation	NOUN
ejpam-6543	15	18	.	.	PUNCT
ejpam-6543	16	1	the	the	DET
ejpam-6543	16	2	concept	concept	NOUN
ejpam-6543	16	3	of	of	ADP
ejpam-6543	16	4	α	α	NOUN
ejpam-6543	16	5	-	-	ADJ
ejpam-6543	16	6	open	open	ADJ
ejpam-6543	16	7	sets	set	NOUN
ejpam-6543	16	8	was	be	AUX
ejpam-6543	16	9	first	first	ADV
ejpam-6543	16	10	introduced	introduce	VERB
ejpam-6543	16	11	by	by	ADP
ejpam-6543	16	12	nj̊astad	nj̊astad	NOUN
ejpam-6543	16	13	[	[	X
ejpam-6543	16	14	1	1	X
ejpam-6543	16	15	]	]	PUNCT
ejpam-6543	16	16	in	in	ADP
ejpam-6543	16	17	1965	1965	NUM
ejpam-6543	16	18	,	,	PUNCT
ejpam-6543	16	19	providing	provide	VERB
ejpam-6543	16	20	a	a	DET
ejpam-6543	16	21	foundation	foundation	NOUN
ejpam-6543	16	22	for	for	ADP
ejpam-6543	16	23	numerous	numerous	ADJ
ejpam-6543	16	24	generalizations	generalization	NOUN
ejpam-6543	16	25	of	of	ADP
ejpam-6543	16	26	classical	classical	ADJ
ejpam-6543	16	27	topological	topological	ADJ
ejpam-6543	16	28	concepts	concept	NOUN
ejpam-6543	16	29	.	.	PUNCT
ejpam-6543	17	1	a	a	DET
ejpam-6543	17	2	subset	subset	NOUN
ejpam-6543	17	3	a	a	PRON
ejpam-6543	17	4	of	of	ADP
ejpam-6543	17	5	a	a	DET
ejpam-6543	17	6	topological	topological	ADJ
ejpam-6543	17	7	space	space	NOUN
ejpam-6543	17	8	(	(	PUNCT
ejpam-6543	17	9	x	x	X
ejpam-6543	17	10	,	,	PUNCT
ejpam-6543	17	11	τ	τ	X
ejpam-6543	17	12	)	)	PUNCT
ejpam-6543	17	13	is	be	AUX
ejpam-6543	17	14	called	call	VERB
ejpam-6543	17	15	α	α	NOUN
ejpam-6543	17	16	-	-	NOUN
ejpam-6543	17	17	open	open	ADJ
ejpam-6543	17	18	if	if	SCONJ
ejpam-6543	17	19	a	a	DET
ejpam-6543	17	20	⊆	⊆	NUM
ejpam-6543	17	21	int(cl(int(a	int(cl(int(a	NOUN
ejpam-6543	17	22	)	)	PUNCT
ejpam-6543	17	23	)	)	PUNCT
ejpam-6543	17	24	)	)	PUNCT
ejpam-6543	17	25	.	.	PUNCT
ejpam-6543	18	1	this	this	DET
ejpam-6543	18	2	seemingly	seemingly	ADV
ejpam-6543	18	3	simple	simple	ADJ
ejpam-6543	18	4	modification	modification	NOUN
ejpam-6543	18	5	of	of	ADP
ejpam-6543	18	6	the	the	DET
ejpam-6543	18	7	notion	notion	NOUN
ejpam-6543	18	8	of	of	ADP
ejpam-6543	18	9	openness	openness	NOUN
ejpam-6543	18	10	has	have	AUX
ejpam-6543	18	11	led	lead	VERB
ejpam-6543	18	12	to	to	ADP
ejpam-6543	18	13	rich	rich	ADJ
ejpam-6543	18	14	theoretical	theoretical	ADJ
ejpam-6543	18	15	developments	development	NOUN
ejpam-6543	18	16	in	in	ADP
ejpam-6543	18	17	topology	topology	NOUN
ejpam-6543	18	18	.	.	PUNCT
ejpam-6543	19	1	building	build	VERB
ejpam-6543	19	2	upon	upon	SCONJ
ejpam-6543	19	3	this	this	DET
ejpam-6543	19	4	foundation	foundation	NOUN
ejpam-6543	19	5	,	,	PUNCT
ejpam-6543	19	6	mashhour	mashhour	PROPN
ejpam-6543	19	7	et	et	PROPN
ejpam-6543	19	8	al	al	PROPN
ejpam-6543	19	9	.	.	PUNCT
ejpam-6543	20	1	[	[	X
ejpam-6543	20	2	2	2	X
ejpam-6543	20	3	]	]	PUNCT
ejpam-6543	20	4	introduced	introduce	VERB
ejpam-6543	20	5	α	α	NUM
ejpam-6543	20	6	-	-	ADJ
ejpam-6543	20	7	continuous	continuous	ADJ
ejpam-6543	20	8	functions	function	NOUN
ejpam-6543	20	9	and	and	CCONJ
ejpam-6543	20	10	explored	explore	VERB
ejpam-6543	20	11	their	their	PRON
ejpam-6543	20	12	properties	property	NOUN
ejpam-6543	20	13	extensively	extensively	ADV
ejpam-6543	20	14	.	.	PUNCT
ejpam-6543	21	1	the	the	DET
ejpam-6543	21	2	study	study	NOUN
ejpam-6543	21	3	of	of	ADP
ejpam-6543	21	4	α	α	NOUN
ejpam-6543	21	5	-	-	ADJ
ejpam-6543	21	6	compact	compact	ADJ
ejpam-6543	21	7	spaces	space	NOUN
ejpam-6543	21	8	naturally	naturally	ADV
ejpam-6543	21	9	followed	follow	VERB
ejpam-6543	21	10	,	,	PUNCT
ejpam-6543	21	11	with	with	ADP
ejpam-6543	21	12	various	various	ADJ
ejpam-6543	21	13	researchers	researcher	NOUN
ejpam-6543	21	14	investigating	investigate	VERB
ejpam-6543	21	15	different	different	ADJ
ejpam-6543	21	16	aspects	aspect	NOUN
ejpam-6543	21	17	and	and	CCONJ
ejpam-6543	21	18	generalizations	generalization	NOUN
ejpam-6543	21	19	of	of	ADP
ejpam-6543	21	20	this	this	DET
ejpam-6543	21	21	concept	concept	NOUN
ejpam-6543	21	22	.	.	PUNCT
ejpam-6543	22	1	recent	recent	ADJ
ejpam-6543	22	2	developments	development	NOUN
ejpam-6543	22	3	in	in	ADP
ejpam-6543	22	4	this	this	DET
ejpam-6543	22	5	area	area	NOUN
ejpam-6543	22	6	include	include	VERB
ejpam-6543	22	7	the	the	DET
ejpam-6543	22	8	work	work	NOUN
ejpam-6543	22	9	of	of	ADP
ejpam-6543	22	10	kumar	kumar	PROPN
ejpam-6543	22	11	and	and	CCONJ
ejpam-6543	22	12	singh	singh	PROPN
ejpam-6543	23	1	[	[	X
ejpam-6543	23	2	3	3	X
ejpam-6543	23	3	]	]	PUNCT
ejpam-6543	23	4	on	on	ADP
ejpam-6543	23	5	α	α	ADJ
ejpam-6543	23	6	-	-	ADJ
ejpam-6543	23	7	compact	compact	ADJ
ejpam-6543	23	8	spaces	space	NOUN
ejpam-6543	23	9	in	in	ADP
ejpam-6543	23	10	digital	digital	ADJ
ejpam-6543	23	11	topology	topology	NOUN
ejpam-6543	23	12	(	(	PUNCT
ejpam-6543	23	13	2021	2021	NUM
ejpam-6543	23	14	)	)	PUNCT
ejpam-6543	23	15	,	,	PUNCT
ejpam-6543	23	16	and	and	CCONJ
ejpam-6543	23	17	the	the	DET
ejpam-6543	23	18	investigations	investigation	NOUN
ejpam-6543	23	19	by	by	ADP
ejpam-6543	23	20	chen	chen	PROPN
ejpam-6543	23	21	and	and	CCONJ
ejpam-6543	23	22	liu	liu	PROPN
ejpam-6543	24	1	[	[	X
ejpam-6543	24	2	4	4	X
ejpam-6543	24	3	]	]	PUNCT
ejpam-6543	24	4	on	on	ADP
ejpam-6543	24	5	relationships	relationship	NOUN
ejpam-6543	24	6	between	between	ADP
ejpam-6543	24	7	α	α	NOUN
ejpam-6543	24	8	-	-	PUNCT
ejpam-6543	24	9	compactness	compactness	NOUN
ejpam-6543	24	10	and	and	CCONJ
ejpam-6543	24	11	other	other	ADJ
ejpam-6543	24	12	covering	cover	VERB
ejpam-6543	24	13	properties	property	NOUN
ejpam-6543	24	14	(	(	PUNCT
ejpam-6543	24	15	2022	2022	NUM
ejpam-6543	24	16	)	)	PUNCT
ejpam-6543	24	17	.	.	PUNCT
ejpam-6543	25	1	recent	recent	ADJ
ejpam-6543	25	2	work	work	NOUN
ejpam-6543	25	3	by	by	ADP
ejpam-6543	25	4	oudetallah	oudetallah	PROPN
ejpam-6543	25	5	[	[	X
ejpam-6543	25	6	5	5	NUM
ejpam-6543	25	7	]	]	PUNCT
ejpam-6543	25	8	on	on	ADP
ejpam-6543	25	9	nearly	nearly	ADV
ejpam-6543	25	10	metacompact	metacompact	ADJ
ejpam-6543	25	11	spaces	space	NOUN
ejpam-6543	25	12	in	in	ADP
ejpam-6543	25	13	bitopological	bitopological	ADJ
ejpam-6543	25	14	settings	setting	NOUN
ejpam-6543	25	15	has	have	AUX
ejpam-6543	25	16	opened	open	VERB
ejpam-6543	25	17	new	new	ADJ
ejpam-6543	25	18	avenues	avenue	NOUN
ejpam-6543	25	19	for	for	ADP
ejpam-6543	25	20	understanding	understand	VERB
ejpam-6543	25	21	nearness	nearness	NOUN
ejpam-6543	25	22	properties	property	NOUN
ejpam-6543	25	23	in	in	ADP
ejpam-6543	25	24	topological	topological	ADJ
ejpam-6543	25	25	contexts	contexts	NOUN
ejpam-6543	25	26	.	.	PUNCT
ejpam-6543	26	1	the	the	DET
ejpam-6543	26	2	investigation	investigation	NOUN
ejpam-6543	26	3	of	of	ADP
ejpam-6543	26	4	pairwise	pairwise	NOUN
ejpam-6543	26	5	expandable	expandable	ADJ
ejpam-6543	26	6	spaces	space	NOUN
ejpam-6543	26	7	by	by	ADP
ejpam-6543	26	8	oudetallah	oudetallah	PROPN
ejpam-6543	26	9	and	and	CCONJ
ejpam-6543	26	10	al	al	PROPN
ejpam-6543	26	11	-	-	PUNCT
ejpam-6543	26	12	hawari	hawari	PROPN
ejpam-6543	26	13	[	[	X
ejpam-6543	26	14	6	6	NUM
ejpam-6543	26	15	]	]	PUNCT
ejpam-6543	26	16	further	far	ADV
ejpam-6543	26	17	demonstrates	demonstrate	VERB
ejpam-6543	26	18	the	the	DET
ejpam-6543	26	19	richness	richness	NOUN
ejpam-6543	26	20	of	of	ADP
ejpam-6543	26	21	generalized	generalized	ADJ
ejpam-6543	26	22	topological	topological	ADJ
ejpam-6543	26	23	properties	property	NOUN
ejpam-6543	26	24	.	.	PUNCT
ejpam-6543	27	1	the	the	DET
ejpam-6543	27	2	exploration	exploration	NOUN
ejpam-6543	27	3	of	of	ADP
ejpam-6543	27	4	hconvexity	hconvexity	NOUN
ejpam-6543	27	5	in	in	ADP
ejpam-6543	27	6	metric	metric	ADJ
ejpam-6543	27	7	linear	linear	NOUN
ejpam-6543	27	8	spaces	space	NOUN
ejpam-6543	27	9	by	by	ADP
ejpam-6543	27	10	oudetallah	oudetallah	NOUN
ejpam-6543	27	11	and	and	CCONJ
ejpam-6543	27	12	abualigah	abualigah	NOUN
ejpam-6543	27	13	[	[	X
ejpam-6543	27	14	7	7	X
ejpam-6543	27	15	]	]	PUNCT
ejpam-6543	27	16	provides	provide	VERB
ejpam-6543	27	17	additional	additional	ADJ
ejpam-6543	27	18	geometric	geometric	ADJ
ejpam-6543	27	19	insights	insight	NOUN
ejpam-6543	27	20	that	that	PRON
ejpam-6543	27	21	complement	complement	VERB
ejpam-6543	27	22	topological	topological	ADJ
ejpam-6543	27	23	investigations	investigation	NOUN
ejpam-6543	27	24	.	.	PUNCT
ejpam-6543	28	1	the	the	DET
ejpam-6543	28	2	study	study	NOUN
ejpam-6543	28	3	of	of	ADP
ejpam-6543	28	4	various	various	ADJ
ejpam-6543	28	5	forms	form	NOUN
ejpam-6543	28	6	of	of	ADP
ejpam-6543	28	7	compactness	compactness	NOUN
ejpam-6543	28	8	continues	continue	VERB
ejpam-6543	28	9	to	to	PART
ejpam-6543	28	10	evolve	evolve	VERB
ejpam-6543	28	11	,	,	PUNCT
ejpam-6543	28	12	with	with	ADP
ejpam-6543	28	13	recent	recent	ADJ
ejpam-6543	28	14	contributions	contribution	NOUN
ejpam-6543	28	15	including	include	VERB
ejpam-6543	28	16	work	work	NOUN
ejpam-6543	28	17	on	on	ADP
ejpam-6543	28	18	r	r	NOUN
ejpam-6543	28	19	-	-	PUNCT
ejpam-6543	28	20	compactness	compactness	NOUN
ejpam-6543	28	21	by	by	ADP
ejpam-6543	28	22	oudetallah	oudetallah	PROPN
ejpam-6543	28	23	,	,	PUNCT
ejpam-6543	28	24	alharbi	alharbi	NOUN
ejpam-6543	28	25	,	,	PUNCT
ejpam-6543	28	26	and	and	CCONJ
ejpam-6543	28	27	batiha	batiha	VERB
ejpam-6543	28	28	[	[	PUNCT
ejpam-6543	28	29	8	8	NUM
ejpam-6543	28	30	]	]	PUNCT
ejpam-6543	28	31	,	,	PUNCT
ejpam-6543	28	32	and	and	CCONJ
ejpam-6543	28	33	investigations	investigation	NOUN
ejpam-6543	28	34	of	of	ADP
ejpam-6543	28	35	d	d	NOUN
ejpam-6543	28	36	-	-	NOUN
ejpam-6543	28	37	metacompactness	metacompactness	NOUN
ejpam-6543	28	38	by	by	ADP
ejpam-6543	28	39	oudetallah	oudetallah	PROPN
ejpam-6543	28	40	,	,	PUNCT
ejpam-6543	28	41	rousan	rousan	NOUN
ejpam-6543	28	42	,	,	PUNCT
ejpam-6543	28	43	and	and	CCONJ
ejpam-6543	28	44	batiha	batiha	VERB
ejpam-6543	28	45	[	[	PUNCT
ejpam-6543	28	46	9	9	NUM
ejpam-6543	28	47	]	]	PUNCT
ejpam-6543	28	48	.	.	PUNCT
ejpam-6543	29	1	novel	novel	ADJ
ejpam-6543	29	2	results	result	NOUN
ejpam-6543	29	3	on	on	ADP
ejpam-6543	29	4	near	near	ADV
ejpam-6543	29	5	lindelöfness	lindelöfness	NUM
ejpam-6543	29	6	by	by	ADP
ejpam-6543	29	7	oudetallah	oudetallah	PROPN
ejpam-6543	30	1	[	[	X
ejpam-6543	30	2	10	10	NUM
ejpam-6543	30	3	]	]	PUNCT
ejpam-6543	30	4	further	far	ADV
ejpam-6543	30	5	expand	expand	VERB
ejpam-6543	30	6	our	our	PRON
ejpam-6543	30	7	understanding	understanding	NOUN
ejpam-6543	30	8	of	of	ADP
ejpam-6543	30	9	nearness	nearness	NOUN
ejpam-6543	30	10	properties	property	NOUN
ejpam-6543	30	11	in	in	ADP
ejpam-6543	30	12	topological	topological	ADJ
ejpam-6543	30	13	spaces	space	NOUN
ejpam-6543	30	14	.	.	PUNCT
ejpam-6543	31	1	additionally	additionally	ADV
ejpam-6543	31	2	,	,	PUNCT
ejpam-6543	31	3	recent	recent	ADJ
ejpam-6543	31	4	work	work	NOUN
ejpam-6543	31	5	by	by	ADP
ejpam-6543	31	6	martinez	martinez	PROPN
ejpam-6543	31	7	and	and	CCONJ
ejpam-6543	31	8	rodriguez	rodriguez	NOUN
ejpam-6543	32	1	[	[	X
ejpam-6543	32	2	11	11	NUM
ejpam-6543	32	3	]	]	PUNCT
ejpam-6543	32	4	(	(	PUNCT
ejpam-6543	32	5	2023	2023	NUM
ejpam-6543	32	6	)	)	PUNCT
ejpam-6543	32	7	on	on	ADP
ejpam-6543	32	8	nearly	nearly	ADV
ejpam-6543	32	9	compact	compact	ADJ
ejpam-6543	32	10	spaces	space	NOUN
ejpam-6543	32	11	in	in	ADP
ejpam-6543	32	12	the	the	DET
ejpam-6543	32	13	context	context	NOUN
ejpam-6543	32	14	of	of	ADP
ejpam-6543	32	15	fuzzy	fuzzy	ADJ
ejpam-6543	32	16	topology	topology	NOUN
ejpam-6543	32	17	,	,	PUNCT
ejpam-6543	32	18	and	and	CCONJ
ejpam-6543	32	19	the	the	DET
ejpam-6543	32	20	study	study	NOUN
ejpam-6543	32	21	by	by	ADP
ejpam-6543	32	22	wang	wang	PROPN
ejpam-6543	32	23	et	et	PROPN
ejpam-6543	32	24	al	al	PROPN
ejpam-6543	32	25	.	.	PUNCT
ejpam-6543	33	1	[	[	X
ejpam-6543	33	2	12	12	NUM
ejpam-6543	33	3	]	]	PUNCT
ejpam-6543	33	4	(	(	PUNCT
ejpam-6543	33	5	2024	2024	NUM
ejpam-6543	33	6	)	)	PUNCT
ejpam-6543	33	7	on	on	ADP
ejpam-6543	33	8	applications	application	NOUN
ejpam-6543	33	9	of	of	ADP
ejpam-6543	33	10	near	near	NOUN
ejpam-6543	33	11	-	-	PUNCT
ejpam-6543	33	12	compactness	compactness	NOUN
ejpam-6543	33	13	in	in	ADP
ejpam-6543	33	14	functional	functional	ADJ
ejpam-6543	33	15	analysis	analysis	NOUN
ejpam-6543	33	16	have	have	AUX
ejpam-6543	33	17	provided	provide	VERB
ejpam-6543	33	18	new	new	ADJ
ejpam-6543	33	19	perspectives	perspective	NOUN
ejpam-6543	33	20	on	on	ADP
ejpam-6543	33	21	these	these	DET
ejpam-6543	33	22	concepts	concept	NOUN
ejpam-6543	33	23	.	.	PUNCT
ejpam-6543	34	1	the	the	DET
ejpam-6543	34	2	work	work	NOUN
ejpam-6543	34	3	of	of	ADP
ejpam-6543	34	4	levine	levine	PROPN
ejpam-6543	34	5	[	[	X
ejpam-6543	34	6	13	13	NUM
ejpam-6543	34	7	]	]	PUNCT
ejpam-6543	34	8	on	on	ADP
ejpam-6543	34	9	generalized	generalized	ADJ
ejpam-6543	34	10	closed	close	VERB
ejpam-6543	34	11	sets	set	NOUN
ejpam-6543	34	12	and	and	CCONJ
ejpam-6543	34	13	the	the	DET
ejpam-6543	34	14	contributions	contribution	NOUN
ejpam-6543	34	15	of	of	ADP
ejpam-6543	34	16	dunham	dunham	PROPN
ejpam-6543	34	17	[	[	X
ejpam-6543	34	18	14	14	NUM
ejpam-6543	34	19	]	]	X
ejpam-6543	34	20	on	on	ADP
ejpam-6543	34	21	closure	closure	NOUN
ejpam-6543	34	22	operators	operator	NOUN
ejpam-6543	34	23	have	have	AUX
ejpam-6543	34	24	provided	provide	VERB
ejpam-6543	34	25	essential	essential	ADJ
ejpam-6543	34	26	tools	tool	NOUN
ejpam-6543	34	27	for	for	ADP
ejpam-6543	34	28	understanding	understand	VERB
ejpam-6543	34	29	generalized	generalized	ADJ
ejpam-6543	34	30	topological	topological	ADJ
ejpam-6543	34	31	structures	structure	NOUN
ejpam-6543	34	32	.	.	PUNCT
ejpam-6543	35	1	these	these	DET
ejpam-6543	35	2	foundations	foundation	NOUN
ejpam-6543	35	3	,	,	PUNCT
ejpam-6543	35	4	combined	combine	VERB
ejpam-6543	35	5	with	with	ADP
ejpam-6543	35	6	the	the	DET
ejpam-6543	35	7	separation	separation	NOUN
ejpam-6543	35	8	axiom	axiom	NOUN
ejpam-6543	35	9	investigations	investigation	NOUN
ejpam-6543	35	10	of	of	ADP
ejpam-6543	35	11	mashhour	mashhour	PROPN
ejpam-6543	35	12	et	et	PROPN
ejpam-6543	35	13	al	al	PROPN
ejpam-6543	35	14	.	.	PUNCT
ejpam-6543	36	1	[	[	X
ejpam-6543	36	2	15	15	NUM
ejpam-6543	36	3	]	]	PUNCT
ejpam-6543	36	4	,	,	PUNCT
ejpam-6543	36	5	create	create	VERB
ejpam-6543	36	6	a	a	DET
ejpam-6543	36	7	rich	rich	ADJ
ejpam-6543	36	8	framework	framework	NOUN
ejpam-6543	36	9	for	for	ADP
ejpam-6543	36	10	the	the	DET
ejpam-6543	36	11	current	current	ADJ
ejpam-6543	36	12	investigation	investigation	NOUN
ejpam-6543	36	13	.	.	PUNCT
ejpam-6543	37	1	in	in	ADP
ejpam-6543	37	2	this	this	DET
ejpam-6543	37	3	context	context	NOUN
ejpam-6543	37	4	,	,	PUNCT
ejpam-6543	37	5	we	we	PRON
ejpam-6543	37	6	introduce	introduce	VERB
ejpam-6543	37	7	the	the	DET
ejpam-6543	37	8	concept	concept	NOUN
ejpam-6543	37	9	of	of	ADP
ejpam-6543	37	10	nearly	nearly	ADV
ejpam-6543	37	11	α	α	ADJ
ejpam-6543	37	12	-	-	ADJ
ejpam-6543	37	13	compact	compact	ADJ
ejpam-6543	37	14	spaces	space	NOUN
ejpam-6543	37	15	as	as	ADP
ejpam-6543	37	16	a	a	DET
ejpam-6543	37	17	natural	natural	ADJ
ejpam-6543	37	18	bridge	bridge	NOUN
ejpam-6543	37	19	between	between	ADP
ejpam-6543	37	20	α	α	NOUN
ejpam-6543	37	21	-	-	PUNCT
ejpam-6543	37	22	compactness	compactness	NOUN
ejpam-6543	37	23	and	and	CCONJ
ejpam-6543	37	24	other	other	ADJ
ejpam-6543	37	25	generalized	generalized	ADJ
ejpam-6543	37	26	compactness	compactness	NOUN
ejpam-6543	37	27	properties	property	NOUN
ejpam-6543	37	28	.	.	PUNCT
ejpam-6543	38	1	our	our	PRON
ejpam-6543	38	2	approach	approach	NOUN
ejpam-6543	38	3	differs	differ	VERB
ejpam-6543	38	4	from	from	ADP
ejpam-6543	38	5	previous	previous	ADJ
ejpam-6543	38	6	investigations	investigation	NOUN
ejpam-6543	38	7	by	by	ADP
ejpam-6543	38	8	focusing	focus	VERB
ejpam-6543	38	9	on	on	ADP
ejpam-6543	38	10	a	a	DET
ejpam-6543	38	11	localized	localized	ADJ
ejpam-6543	38	12	version	version	NOUN
ejpam-6543	38	13	of	of	ADP
ejpam-6543	38	14	α	α	NOUN
ejpam-6543	38	15	-	-	NOUN
ejpam-6543	38	16	compactness	compactness	NOUN
ejpam-6543	38	17	that	that	PRON
ejpam-6543	38	18	captures	capture	VERB
ejpam-6543	38	19	essential	essential	ADJ
ejpam-6543	38	20	features	feature	NOUN
ejpam-6543	38	21	while	while	SCONJ
ejpam-6543	38	22	allowing	allow	VERB
ejpam-6543	38	23	for	for	ADP
ejpam-6543	38	24	greater	great	ADJ
ejpam-6543	38	25	flexibility	flexibility	NOUN
ejpam-6543	38	26	in	in	ADP
ejpam-6543	38	27	applications	application	NOUN
ejpam-6543	38	28	.	.	PUNCT
ejpam-6543	39	1	the	the	DET
ejpam-6543	39	2	motivation	motivation	NOUN
ejpam-6543	39	3	for	for	ADP
ejpam-6543	39	4	studying	study	VERB
ejpam-6543	39	5	nearly	nearly	ADV
ejpam-6543	39	6	α	α	NUM
ejpam-6543	39	7	-	-	ADJ
ejpam-6543	39	8	compact	compact	ADJ
ejpam-6543	39	9	spaces	space	NOUN
ejpam-6543	39	10	arises	arise	VERB
ejpam-6543	39	11	from	from	ADP
ejpam-6543	39	12	several	several	ADJ
ejpam-6543	39	13	considerations	consideration	NOUN
ejpam-6543	39	14	.	.	PUNCT
ejpam-6543	40	1	first	first	ADV
ejpam-6543	40	2	,	,	PUNCT
ejpam-6543	40	3	classical	classical	ADJ
ejpam-6543	40	4	α	α	NOUN
ejpam-6543	40	5	-	-	NOUN
ejpam-6543	40	6	compactness	compactness	NOUN
ejpam-6543	40	7	,	,	PUNCT
ejpam-6543	40	8	while	while	SCONJ
ejpam-6543	40	9	elegant	elegant	ADJ
ejpam-6543	40	10	,	,	PUNCT
ejpam-6543	40	11	can	can	AUX
ejpam-6543	40	12	be	be	AUX
ejpam-6543	40	13	quite	quite	ADV
ejpam-6543	40	14	restrictive	restrictive	ADJ
ejpam-6543	40	15	in	in	ADP
ejpam-6543	40	16	practice	practice	NOUN
ejpam-6543	40	17	.	.	PUNCT
ejpam-6543	41	1	second	second	ADV
ejpam-6543	41	2	,	,	PUNCT
ejpam-6543	41	3	the	the	DET
ejpam-6543	41	4	interplay	interplay	NOUN
ejpam-6543	41	5	between	between	ADP
ejpam-6543	41	6	nearness	nearness	NOUN
ejpam-6543	41	7	properties	property	NOUN
ejpam-6543	41	8	and	and	CCONJ
ejpam-6543	41	9	α	α	NOUN
ejpam-6543	41	10	-	-	PUNCT
ejpam-6543	41	11	structures	structure	NOUN
ejpam-6543	41	12	provides	provide	VERB
ejpam-6543	41	13	new	new	ADJ
ejpam-6543	41	14	insights	insight	NOUN
ejpam-6543	41	15	into	into	ADP
ejpam-6543	41	16	the	the	DET
ejpam-6543	41	17	geometry	geometry	NOUN
ejpam-6543	41	18	of	of	ADP
ejpam-6543	41	19	topological	topological	ADJ
ejpam-6543	41	20	spaces	space	NOUN
ejpam-6543	41	21	.	.	PUNCT
ejpam-6543	42	1	third	third	ADV
ejpam-6543	42	2	,	,	PUNCT
ejpam-6543	42	3	the	the	DET
ejpam-6543	42	4	potential	potential	ADJ
ejpam-6543	42	5	applications	application	NOUN
ejpam-6543	42	6	to	to	PART
ejpam-6543	42	7	function	function	VERB
ejpam-6543	42	8	space	space	NOUN
ejpam-6543	42	9	topology	topology	NOUN
ejpam-6543	42	10	and	and	CCONJ
ejpam-6543	42	11	convergence	convergence	NOUN
ejpam-6543	42	12	theory	theory	NOUN
ejpam-6543	42	13	make	make	VERB
ejpam-6543	42	14	this	this	DET
ejpam-6543	42	15	investigation	investigation	NOUN
ejpam-6543	42	16	particularly	particularly	ADV
ejpam-6543	42	17	worthwhile	worthwhile	ADJ
ejpam-6543	42	18	.	.	PUNCT
ejpam-6543	43	1	j.	j.	PROPN
ejpam-6543	43	2	oudetallah	oudetallah	PROPN
ejpam-6543	43	3	et	et	PROPN
ejpam-6543	43	4	al	al	PROPN
ejpam-6543	43	5	.	.	PUNCT
ejpam-6543	43	6	/	/	SYM
ejpam-6543	43	7	eur	eur	PROPN
ejpam-6543	43	8	.	.	PUNCT
ejpam-6543	44	1	j.	j.	PROPN
ejpam-6543	44	2	pure	pure	PROPN
ejpam-6543	44	3	appl	appl	PROPN
ejpam-6543	44	4	.	.	PROPN
ejpam-6543	44	5	math	math	PROPN
ejpam-6543	44	6	,	,	PUNCT
ejpam-6543	44	7	18	18	NUM
ejpam-6543	44	8	(	(	PUNCT
ejpam-6543	44	9	3	3	NUM
ejpam-6543	44	10	)	)	PUNCT
ejpam-6543	44	11	(	(	PUNCT
ejpam-6543	44	12	2025	2025	NUM
ejpam-6543	44	13	)	)	PUNCT
ejpam-6543	44	14	,	,	PUNCT
ejpam-6543	44	15	6543	6543	NUM
ejpam-6543	44	16	3	3	NUM
ejpam-6543	44	17	of	of	ADP
ejpam-6543	44	18	13	13	NUM
ejpam-6543	44	19	2	2	NUM
ejpam-6543	44	20	.	.	PUNCT
ejpam-6543	44	21	preliminary	preliminary	ADJ
ejpam-6543	44	22	concepts	concept	NOUN
ejpam-6543	44	23	throughout	throughout	ADP
ejpam-6543	44	24	this	this	DET
ejpam-6543	44	25	paper	paper	NOUN
ejpam-6543	44	26	,	,	PUNCT
ejpam-6543	44	27	(	(	PUNCT
ejpam-6543	44	28	x	x	X
ejpam-6543	44	29	,	,	PUNCT
ejpam-6543	44	30	τ	τ	X
ejpam-6543	44	31	)	)	PUNCT
ejpam-6543	44	32	denotes	denote	VERB
ejpam-6543	44	33	a	a	DET
ejpam-6543	44	34	topological	topological	ADJ
ejpam-6543	44	35	space	space	NOUN
ejpam-6543	44	36	.	.	PUNCT
ejpam-6543	45	1	we	we	PRON
ejpam-6543	45	2	begin	begin	VERB
ejpam-6543	45	3	by	by	ADP
ejpam-6543	45	4	recalling	recall	VERB
ejpam-6543	45	5	essential	essential	ADJ
ejpam-6543	45	6	definitions	definition	NOUN
ejpam-6543	45	7	and	and	CCONJ
ejpam-6543	45	8	establishing	establish	VERB
ejpam-6543	45	9	notation	notation	NOUN
ejpam-6543	45	10	that	that	PRON
ejpam-6543	45	11	will	will	AUX
ejpam-6543	45	12	be	be	AUX
ejpam-6543	45	13	used	use	VERB
ejpam-6543	45	14	throughout	throughout	ADP
ejpam-6543	45	15	our	our	PRON
ejpam-6543	45	16	investigation	investigation	NOUN
ejpam-6543	45	17	.	.	PUNCT
ejpam-6543	46	1	definition	definition	NOUN
ejpam-6543	46	2	1	1	NUM
ejpam-6543	46	3	.	.	PUNCT
ejpam-6543	47	1	[	[	X
ejpam-6543	47	2	1	1	X
ejpam-6543	47	3	]	]	X
ejpam-6543	47	4	let	let	VERB
ejpam-6543	47	5	(	(	PUNCT
ejpam-6543	47	6	x	x	NOUN
ejpam-6543	47	7	,	,	PUNCT
ejpam-6543	47	8	τ	τ	X
ejpam-6543	47	9	)	)	PUNCT
ejpam-6543	47	10	be	be	VERB
ejpam-6543	47	11	a	a	DET
ejpam-6543	47	12	topological	topological	ADJ
ejpam-6543	47	13	space	space	NOUN
ejpam-6543	47	14	and	and	CCONJ
ejpam-6543	47	15	a	a	DET
ejpam-6543	47	16	⊆	⊆	NUM
ejpam-6543	47	17	x.	x.	NOUN
ejpam-6543	47	18	then	then	ADV
ejpam-6543	47	19	:	:	PUNCT
ejpam-6543	47	20	(	(	PUNCT
ejpam-6543	47	21	i	i	NOUN
ejpam-6543	47	22	)	)	PUNCT
ejpam-6543	47	23	a	a	PRON
ejpam-6543	47	24	is	be	AUX
ejpam-6543	47	25	called	call	VERB
ejpam-6543	47	26	α	α	NOUN
ejpam-6543	47	27	-	-	NOUN
ejpam-6543	47	28	open	open	ADJ
ejpam-6543	47	29	if	if	SCONJ
ejpam-6543	47	30	a	a	DET
ejpam-6543	47	31	⊆	⊆	NUM
ejpam-6543	47	32	int(cl(int(a	int(cl(int(a	NOUN
ejpam-6543	47	33	)	)	PUNCT
ejpam-6543	47	34	)	)	PUNCT
ejpam-6543	47	35	)	)	PUNCT
ejpam-6543	47	36	.	.	PUNCT
ejpam-6543	48	1	(	(	PUNCT
ejpam-6543	48	2	ii	ii	X
ejpam-6543	48	3	)	)	PUNCT
ejpam-6543	48	4	a	a	PRON
ejpam-6543	48	5	is	be	AUX
ejpam-6543	48	6	called	call	VERB
ejpam-6543	48	7	α	α	PRON
ejpam-6543	48	8	-	-	VERB
ejpam-6543	48	9	closed	closed	ADJ
ejpam-6543	48	10	if	if	SCONJ
ejpam-6543	48	11	x	x	SYM
ejpam-6543	48	12	\a	\a	ADJ
ejpam-6543	48	13	is	be	AUX
ejpam-6543	48	14	α	α	NOUN
ejpam-6543	48	15	-	-	NOUN
ejpam-6543	48	16	open	open	ADJ
ejpam-6543	48	17	.	.	PUNCT
ejpam-6543	49	1	(	(	PUNCT
ejpam-6543	49	2	iii	iii	X
ejpam-6543	49	3	)	)	PUNCT
ejpam-6543	49	4	the	the	DET
ejpam-6543	49	5	α	α	NOUN
ejpam-6543	49	6	-	-	NOUN
ejpam-6543	49	7	interior	interior	ADJ
ejpam-6543	49	8	of	of	ADP
ejpam-6543	49	9	a	a	PRON
ejpam-6543	49	10	,	,	PUNCT
ejpam-6543	49	11	denoted	denote	VERB
ejpam-6543	49	12	intα(a	intα(a	NOUN
ejpam-6543	49	13	)	)	PUNCT
ejpam-6543	49	14	,	,	PUNCT
ejpam-6543	49	15	is	be	AUX
ejpam-6543	49	16	the	the	DET
ejpam-6543	49	17	union	union	NOUN
ejpam-6543	49	18	of	of	ADP
ejpam-6543	49	19	all	all	DET
ejpam-6543	49	20	α	α	PRON
ejpam-6543	49	21	-	-	ADJ
ejpam-6543	49	22	open	open	ADJ
ejpam-6543	49	23	sets	set	NOUN
ejpam-6543	49	24	contained	contain	VERB
ejpam-6543	49	25	in	in	ADP
ejpam-6543	49	26	a.	a.	NOUN
ejpam-6543	49	27	(	(	PUNCT
ejpam-6543	49	28	iv	iv	X
ejpam-6543	49	29	)	)	PUNCT
ejpam-6543	49	30	the	the	DET
ejpam-6543	49	31	α	α	NOUN
ejpam-6543	49	32	-	-	NOUN
ejpam-6543	49	33	closure	closure	NOUN
ejpam-6543	49	34	of	of	ADP
ejpam-6543	49	35	a	a	PRON
ejpam-6543	49	36	,	,	PUNCT
ejpam-6543	49	37	denoted	denote	VERB
ejpam-6543	49	38	clα(a	clα(a	NOUN
ejpam-6543	49	39	)	)	PUNCT
ejpam-6543	49	40	,	,	PUNCT
ejpam-6543	49	41	is	be	AUX
ejpam-6543	49	42	the	the	DET
ejpam-6543	49	43	intersection	intersection	NOUN
ejpam-6543	49	44	of	of	ADP
ejpam-6543	49	45	all	all	DET
ejpam-6543	49	46	α	α	PRON
ejpam-6543	49	47	-	-	PUNCT
ejpam-6543	49	48	closed	closed	ADJ
ejpam-6543	49	49	sets	set	NOUN
ejpam-6543	49	50	containing	contain	VERB
ejpam-6543	49	51	a.	a.	NOUN
ejpam-6543	49	52	the	the	DET
ejpam-6543	49	53	family	family	NOUN
ejpam-6543	49	54	of	of	ADP
ejpam-6543	49	55	all	all	DET
ejpam-6543	49	56	α	α	PRON
ejpam-6543	49	57	-	-	ADJ
ejpam-6543	49	58	open	open	ADJ
ejpam-6543	49	59	subsets	subset	NOUN
ejpam-6543	49	60	of	of	ADP
ejpam-6543	49	61	x	x	PROPN
ejpam-6543	49	62	is	be	AUX
ejpam-6543	49	63	denoted	denote	VERB
ejpam-6543	49	64	by	by	ADP
ejpam-6543	49	65	αo(x	αo(x	NUM
ejpam-6543	49	66	)	)	PUNCT
ejpam-6543	49	67	,	,	PUNCT
ejpam-6543	49	68	and	and	CCONJ
ejpam-6543	49	69	the	the	DET
ejpam-6543	49	70	family	family	NOUN
ejpam-6543	49	71	of	of	ADP
ejpam-6543	49	72	all	all	DET
ejpam-6543	49	73	α	α	PRON
ejpam-6543	49	74	-	-	PUNCT
ejpam-6543	49	75	closed	closed	ADJ
ejpam-6543	49	76	subsets	subset	NOUN
ejpam-6543	49	77	is	be	AUX
ejpam-6543	49	78	denoted	denote	VERB
ejpam-6543	49	79	by	by	ADP
ejpam-6543	49	80	αc(x	αc(x	NUM
ejpam-6543	49	81	)	)	PUNCT
ejpam-6543	49	82	.	.	PUNCT
ejpam-6543	50	1	definition	definition	NOUN
ejpam-6543	50	2	2	2	NUM
ejpam-6543	50	3	.	.	PUNCT
ejpam-6543	51	1	[	[	X
ejpam-6543	51	2	2	2	X
ejpam-6543	51	3	]	]	PUNCT
ejpam-6543	51	4	a	a	DET
ejpam-6543	51	5	function	function	NOUN
ejpam-6543	51	6	f	f	NOUN
ejpam-6543	51	7	:	:	PUNCT
ejpam-6543	51	8	x	x	X
ejpam-6543	51	9	→	→	SYM
ejpam-6543	51	10	y	y	NOUN
ejpam-6543	51	11	between	between	ADP
ejpam-6543	51	12	topological	topological	ADJ
ejpam-6543	51	13	spaces	space	NOUN
ejpam-6543	51	14	is	be	AUX
ejpam-6543	51	15	called	call	VERB
ejpam-6543	51	16	α	α	PRON
ejpam-6543	51	17	-	-	ADJ
ejpam-6543	51	18	continuous	continuous	ADJ
ejpam-6543	51	19	if	if	SCONJ
ejpam-6543	51	20	for	for	ADP
ejpam-6543	51	21	every	every	DET
ejpam-6543	51	22	open	open	NOUN
ejpam-6543	51	23	set	set	VERB
ejpam-6543	51	24	v	v	NOUN
ejpam-6543	51	25	in	in	ADP
ejpam-6543	51	26	y	y	PROPN
ejpam-6543	51	27	,	,	PUNCT
ejpam-6543	51	28	the	the	DET
ejpam-6543	51	29	set	set	ADJ
ejpam-6543	51	30	f−1(v	f−1(v	NOUN
ejpam-6543	51	31	)	)	PUNCT
ejpam-6543	51	32	is	be	AUX
ejpam-6543	51	33	α	α	NOUN
ejpam-6543	51	34	-	-	NOUN
ejpam-6543	51	35	open	open	ADJ
ejpam-6543	51	36	in	in	ADP
ejpam-6543	51	37	x.	x.	NOUN
ejpam-6543	51	38	definition	definition	NOUN
ejpam-6543	51	39	3	3	NUM
ejpam-6543	51	40	.	.	PUNCT
ejpam-6543	52	1	[	[	X
ejpam-6543	52	2	2	2	X
ejpam-6543	52	3	]	]	PUNCT
ejpam-6543	52	4	a	a	DET
ejpam-6543	52	5	topological	topological	ADJ
ejpam-6543	52	6	space	space	NOUN
ejpam-6543	52	7	(	(	PUNCT
ejpam-6543	52	8	x	x	X
ejpam-6543	52	9	,	,	PUNCT
ejpam-6543	52	10	τ	τ	X
ejpam-6543	52	11	)	)	PUNCT
ejpam-6543	52	12	is	be	AUX
ejpam-6543	52	13	called	call	VERB
ejpam-6543	52	14	α	α	PRON
ejpam-6543	52	15	-	-	ADJ
ejpam-6543	52	16	compact	compact	ADJ
ejpam-6543	52	17	if	if	SCONJ
ejpam-6543	52	18	every	every	DET
ejpam-6543	52	19	α	α	NOUN
ejpam-6543	52	20	-	-	ADJ
ejpam-6543	52	21	open	open	ADJ
ejpam-6543	52	22	cover	cover	NOUN
ejpam-6543	52	23	of	of	ADP
ejpam-6543	52	24	x	x	PUNCT
ejpam-6543	52	25	has	have	VERB
ejpam-6543	52	26	a	a	DET
ejpam-6543	52	27	finite	finite	ADJ
ejpam-6543	52	28	subcover	subcover	PROPN
ejpam-6543	52	29	.	.	PUNCT
ejpam-6543	53	1	definition	definition	NOUN
ejpam-6543	53	2	4	4	NUM
ejpam-6543	53	3	.	.	PUNCT
ejpam-6543	54	1	[	[	X
ejpam-6543	54	2	2	2	X
ejpam-6543	54	3	]	]	PUNCT
ejpam-6543	54	4	a	a	DET
ejpam-6543	54	5	topological	topological	ADJ
ejpam-6543	54	6	space	space	NOUN
ejpam-6543	54	7	(	(	PUNCT
ejpam-6543	54	8	x	x	X
ejpam-6543	54	9	,	,	PUNCT
ejpam-6543	54	10	τ	τ	X
ejpam-6543	54	11	)	)	PUNCT
ejpam-6543	54	12	is	be	AUX
ejpam-6543	54	13	called	call	VERB
ejpam-6543	54	14	countably	countably	ADV
ejpam-6543	54	15	α	α	NOUN
ejpam-6543	54	16	-	-	ADJ
ejpam-6543	54	17	compact	compact	ADJ
ejpam-6543	54	18	if	if	SCONJ
ejpam-6543	54	19	every	every	DET
ejpam-6543	54	20	countable	countable	ADJ
ejpam-6543	54	21	α	α	NOUN
ejpam-6543	54	22	-	-	ADJ
ejpam-6543	54	23	open	open	ADJ
ejpam-6543	54	24	cover	cover	NOUN
ejpam-6543	54	25	of	of	ADP
ejpam-6543	54	26	x	x	PUNCT
ejpam-6543	54	27	has	have	VERB
ejpam-6543	54	28	a	a	DET
ejpam-6543	54	29	finite	finite	ADJ
ejpam-6543	54	30	subcover	subcover	PROPN
ejpam-6543	54	31	.	.	PUNCT
ejpam-6543	55	1	we	we	PRON
ejpam-6543	55	2	also	also	ADV
ejpam-6543	55	3	recall	recall	VERB
ejpam-6543	55	4	the	the	DET
ejpam-6543	55	5	following	follow	VERB
ejpam-6543	55	6	separation	separation	NOUN
ejpam-6543	55	7	properties	property	NOUN
ejpam-6543	55	8	:	:	PUNCT
ejpam-6543	55	9	definition	definition	NOUN
ejpam-6543	55	10	5	5	NUM
ejpam-6543	55	11	.	.	PUNCT
ejpam-6543	56	1	[	[	X
ejpam-6543	56	2	15	15	NUM
ejpam-6543	56	3	]	]	X
ejpam-6543	56	4	a	a	DET
ejpam-6543	56	5	topological	topological	ADJ
ejpam-6543	56	6	space	space	NOUN
ejpam-6543	56	7	(	(	PUNCT
ejpam-6543	56	8	x	x	X
ejpam-6543	56	9	,	,	PUNCT
ejpam-6543	56	10	τ	τ	X
ejpam-6543	56	11	)	)	PUNCT
ejpam-6543	56	12	is	be	AUX
ejpam-6543	56	13	called	call	VERB
ejpam-6543	56	14	:	:	PUNCT
ejpam-6543	56	15	(	(	PUNCT
ejpam-6543	56	16	i	i	NOUN
ejpam-6543	56	17	)	)	PUNCT
ejpam-6543	56	18	α	α	PROPN
ejpam-6543	56	19	-	-	NOUN
ejpam-6543	56	20	t1	t1	NOUN
ejpam-6543	56	21	if	if	SCONJ
ejpam-6543	56	22	for	for	ADP
ejpam-6543	56	23	any	any	DET
ejpam-6543	56	24	two	two	NUM
ejpam-6543	56	25	distinct	distinct	ADJ
ejpam-6543	56	26	points	point	NOUN
ejpam-6543	56	27	x	x	NOUN
ejpam-6543	56	28	,	,	PUNCT
ejpam-6543	56	29	y	y	PROPN
ejpam-6543	56	30	∈	∈	PROPN
ejpam-6543	56	31	x	x	PRON
ejpam-6543	56	32	,	,	PUNCT
ejpam-6543	56	33	there	there	PRON
ejpam-6543	56	34	exist	exist	VERB
ejpam-6543	56	35	α	α	PRON
ejpam-6543	56	36	-	-	ADJ
ejpam-6543	56	37	open	open	ADJ
ejpam-6543	56	38	sets	set	NOUN
ejpam-6543	56	39	u	u	NOUN
ejpam-6543	56	40	and	and	CCONJ
ejpam-6543	56	41	v	v	ADP
ejpam-6543	56	42	such	such	ADJ
ejpam-6543	56	43	that	that	SCONJ
ejpam-6543	56	44	x	x	SYM
ejpam-6543	56	45	∈	∈	PROPN
ejpam-6543	56	46	u	u	NOUN
ejpam-6543	56	47	,	,	PUNCT
ejpam-6543	56	48	y	y	PROPN
ejpam-6543	56	49	/∈	/∈	PUNCT
ejpam-6543	57	1	u	u	PROPN
ejpam-6543	57	2	,	,	PUNCT
ejpam-6543	57	3	y	y	PROPN
ejpam-6543	57	4	∈	∈	PROPN
ejpam-6543	57	5	v	v	NOUN
ejpam-6543	57	6	,	,	PUNCT
ejpam-6543	57	7	and	and	CCONJ
ejpam-6543	57	8	x	x	X
ejpam-6543	57	9	/∈	/∈	NOUN
ejpam-6543	57	10	v	v	INTJ
ejpam-6543	57	11	.	.	PUNCT
ejpam-6543	58	1	(	(	PUNCT
ejpam-6543	58	2	ii	ii	NOUN
ejpam-6543	58	3	)	)	PUNCT
ejpam-6543	58	4	α	α	PROPN
ejpam-6543	58	5	-	-	NOUN
ejpam-6543	58	6	t2	t2	NOUN
ejpam-6543	58	7	(	(	PUNCT
ejpam-6543	58	8	or	or	CCONJ
ejpam-6543	58	9	α	α	NOUN
ejpam-6543	58	10	-	-	PUNCT
ejpam-6543	58	11	hausdorff	hausdorff	NOUN
ejpam-6543	58	12	)	)	PUNCT
ejpam-6543	58	13	if	if	SCONJ
ejpam-6543	58	14	for	for	ADP
ejpam-6543	58	15	any	any	DET
ejpam-6543	58	16	two	two	NUM
ejpam-6543	58	17	distinct	distinct	ADJ
ejpam-6543	58	18	points	point	NOUN
ejpam-6543	58	19	x	x	NOUN
ejpam-6543	58	20	,	,	PUNCT
ejpam-6543	58	21	y	y	PROPN
ejpam-6543	58	22	∈	∈	PROPN
ejpam-6543	58	23	x	x	PRON
ejpam-6543	58	24	,	,	PUNCT
ejpam-6543	58	25	there	there	PRON
ejpam-6543	58	26	exist	exist	VERB
ejpam-6543	58	27	disjoint	disjoint	NOUN
ejpam-6543	58	28	α	α	NOUN
ejpam-6543	58	29	-	-	ADJ
ejpam-6543	58	30	open	open	ADJ
ejpam-6543	58	31	sets	set	NOUN
ejpam-6543	58	32	u	u	NOUN
ejpam-6543	58	33	and	and	CCONJ
ejpam-6543	58	34	v	v	ADP
ejpam-6543	58	35	such	such	ADJ
ejpam-6543	58	36	that	that	SCONJ
ejpam-6543	58	37	x	x	SYM
ejpam-6543	58	38	∈	∈	PROPN
ejpam-6543	58	39	u	u	NOUN
ejpam-6543	58	40	and	and	CCONJ
ejpam-6543	58	41	y	y	PROPN
ejpam-6543	58	42	∈	∈	PROPN
ejpam-6543	58	43	v	v	NOUN
ejpam-6543	58	44	.	.	PUNCT
ejpam-6543	59	1	definition	definition	NOUN
ejpam-6543	59	2	6	6	NUM
ejpam-6543	59	3	.	.	PUNCT
ejpam-6543	60	1	[	[	X
ejpam-6543	60	2	13	13	NUM
ejpam-6543	60	3	]	]	PUNCT
ejpam-6543	60	4	a	a	DET
ejpam-6543	60	5	subset	subset	NOUN
ejpam-6543	60	6	a	a	PRON
ejpam-6543	60	7	of	of	ADP
ejpam-6543	60	8	a	a	DET
ejpam-6543	60	9	topological	topological	ADJ
ejpam-6543	60	10	space	space	NOUN
ejpam-6543	60	11	(	(	PUNCT
ejpam-6543	60	12	x	x	X
ejpam-6543	60	13	,	,	PUNCT
ejpam-6543	60	14	τ	τ	X
ejpam-6543	60	15	)	)	PUNCT
ejpam-6543	60	16	is	be	AUX
ejpam-6543	60	17	called	call	VERB
ejpam-6543	60	18	regular	regular	ADV
ejpam-6543	60	19	open	open	ADJ
ejpam-6543	60	20	if	if	SCONJ
ejpam-6543	60	21	a	a	PRON
ejpam-6543	60	22	=	=	X
ejpam-6543	60	23	int(cl(a	int(cl(a	PROPN
ejpam-6543	60	24	)	)	PUNCT
ejpam-6543	60	25	)	)	PUNCT
ejpam-6543	60	26	.	.	PUNCT
ejpam-6543	61	1	the	the	DET
ejpam-6543	61	2	collection	collection	NOUN
ejpam-6543	61	3	of	of	ADP
ejpam-6543	61	4	all	all	DET
ejpam-6543	61	5	regular	regular	ADJ
ejpam-6543	61	6	open	open	ADJ
ejpam-6543	61	7	sets	set	NOUN
ejpam-6543	61	8	forms	form	NOUN
ejpam-6543	61	9	a	a	DET
ejpam-6543	61	10	complete	complete	ADJ
ejpam-6543	61	11	boolean	boolean	ADJ
ejpam-6543	61	12	algebra	algebra	NOUN
ejpam-6543	61	13	under	under	ADP
ejpam-6543	61	14	set	set	ADJ
ejpam-6543	61	15	operations	operation	NOUN
ejpam-6543	61	16	.	.	PUNCT
ejpam-6543	62	1	definition	definition	NOUN
ejpam-6543	62	2	7	7	NUM
ejpam-6543	62	3	.	.	PUNCT
ejpam-6543	63	1	let	let	VERB
ejpam-6543	63	2	(	(	PUNCT
ejpam-6543	63	3	x	x	NOUN
ejpam-6543	63	4	,	,	PUNCT
ejpam-6543	63	5	τ	τ	X
ejpam-6543	63	6	)	)	PUNCT
ejpam-6543	63	7	be	be	VERB
ejpam-6543	63	8	a	a	DET
ejpam-6543	63	9	topological	topological	ADJ
ejpam-6543	63	10	space	space	NOUN
ejpam-6543	63	11	.	.	PUNCT
ejpam-6543	64	1	a	a	DET
ejpam-6543	64	2	point	point	NOUN
ejpam-6543	64	3	x	x	X
ejpam-6543	64	4	∈	∈	NOUN
ejpam-6543	64	5	x	x	PUNCT
ejpam-6543	64	6	is	be	AUX
ejpam-6543	64	7	called	call	VERB
ejpam-6543	64	8	a	a	DET
ejpam-6543	64	9	cluster	cluster	NOUN
ejpam-6543	64	10	point	point	NOUN
ejpam-6543	64	11	(	(	PUNCT
ejpam-6543	64	12	or	or	CCONJ
ejpam-6543	64	13	accumulation	accumulation	NOUN
ejpam-6543	64	14	point	point	NOUN
ejpam-6543	64	15	)	)	PUNCT
ejpam-6543	64	16	of	of	ADP
ejpam-6543	64	17	a	a	DET
ejpam-6543	64	18	subset	subset	NOUN
ejpam-6543	64	19	a	a	DET
ejpam-6543	64	20	⊆	⊆	NUM
ejpam-6543	64	21	x	x	SYM
ejpam-6543	64	22	if	if	SCONJ
ejpam-6543	64	23	every	every	DET
ejpam-6543	64	24	open	open	ADJ
ejpam-6543	64	25	neighborhood	neighborhood	NOUN
ejpam-6543	64	26	of	of	ADP
ejpam-6543	64	27	x	x	PUNCT
ejpam-6543	64	28	contains	contain	VERB
ejpam-6543	64	29	a	a	DET
ejpam-6543	64	30	point	point	NOUN
ejpam-6543	64	31	of	of	ADP
ejpam-6543	64	32	a	a	DET
ejpam-6543	64	33	different	different	ADJ
ejpam-6543	64	34	from	from	ADP
ejpam-6543	64	35	x.	x.	NOUN
ejpam-6543	64	36	for	for	ADP
ejpam-6543	64	37	α	α	NOUN
ejpam-6543	64	38	-	-	ADJ
ejpam-6543	64	39	open	open	ADJ
ejpam-6543	64	40	neighborhoods	neighborhood	NOUN
ejpam-6543	64	41	,	,	PUNCT
ejpam-6543	64	42	we	we	PRON
ejpam-6543	64	43	define	define	VERB
ejpam-6543	64	44	α	α	PRON
ejpam-6543	64	45	-	-	PUNCT
ejpam-6543	64	46	cluster	cluster	NOUN
ejpam-6543	64	47	points	point	NOUN
ejpam-6543	64	48	analogously	analogously	ADV
ejpam-6543	64	49	.	.	PUNCT
ejpam-6543	65	1	j.	j.	PROPN
ejpam-6543	65	2	oudetallah	oudetallah	PROPN
ejpam-6543	65	3	et	et	PROPN
ejpam-6543	65	4	al	al	PROPN
ejpam-6543	65	5	.	.	PUNCT
ejpam-6543	65	6	/	/	SYM
ejpam-6543	65	7	eur	eur	PROPN
ejpam-6543	65	8	.	.	PUNCT
ejpam-6543	66	1	j.	j.	PROPN
ejpam-6543	66	2	pure	pure	PROPN
ejpam-6543	66	3	appl	appl	PROPN
ejpam-6543	66	4	.	.	PROPN
ejpam-6543	66	5	math	math	PROPN
ejpam-6543	66	6	,	,	PUNCT
ejpam-6543	66	7	18	18	NUM
ejpam-6543	66	8	(	(	PUNCT
ejpam-6543	66	9	3	3	NUM
ejpam-6543	66	10	)	)	PUNCT
ejpam-6543	66	11	(	(	PUNCT
ejpam-6543	66	12	2025	2025	NUM
ejpam-6543	66	13	)	)	PUNCT
ejpam-6543	66	14	,	,	PUNCT
ejpam-6543	66	15	6543	6543	NUM
ejpam-6543	66	16	4	4	NUM
ejpam-6543	66	17	of	of	ADP
ejpam-6543	66	18	13	13	NUM
ejpam-6543	66	19	definition	definition	NOUN
ejpam-6543	66	20	8	8	NUM
ejpam-6543	66	21	.	.	PUNCT
ejpam-6543	67	1	let	let	VERB
ejpam-6543	67	2	{	{	PUNCT
ejpam-6543	67	3	(	(	PUNCT
ejpam-6543	67	4	xi	xi	X
ejpam-6543	67	5	,	,	PUNCT
ejpam-6543	67	6	τi	τi	PROPN
ejpam-6543	67	7	)	)	PUNCT
ejpam-6543	67	8	:	:	PUNCT
ejpam-6543	68	1	i	i	PRON
ejpam-6543	68	2	∈	∈	VERB
ejpam-6543	68	3	i	i	PRON
ejpam-6543	68	4	}	}	PUNCT
ejpam-6543	68	5	be	be	VERB
ejpam-6543	68	6	a	a	DET
ejpam-6543	68	7	family	family	NOUN
ejpam-6543	68	8	of	of	ADP
ejpam-6543	68	9	topological	topological	ADJ
ejpam-6543	68	10	spaces	space	NOUN
ejpam-6543	68	11	.	.	PUNCT
ejpam-6543	69	1	the	the	DET
ejpam-6543	69	2	product	product	NOUN
ejpam-6543	69	3	space∏	space∏	ADV
ejpam-6543	69	4	i∈i	i∈i	ADV
ejpam-6543	69	5	xi	xi	INTJ
ejpam-6543	69	6	is	be	AUX
ejpam-6543	69	7	endowed	endow	VERB
ejpam-6543	69	8	with	with	ADP
ejpam-6543	69	9	the	the	DET
ejpam-6543	69	10	product	product	NOUN
ejpam-6543	69	11	topology	topology	NOUN
ejpam-6543	69	12	,	,	PUNCT
ejpam-6543	69	13	which	which	PRON
ejpam-6543	69	14	is	be	AUX
ejpam-6543	69	15	the	the	DET
ejpam-6543	69	16	coarsest	coarse	ADJ
ejpam-6543	69	17	topology	topology	NOUN
ejpam-6543	69	18	making	make	VERB
ejpam-6543	69	19	all	all	DET
ejpam-6543	69	20	projection	projection	NOUN
ejpam-6543	69	21	maps	map	NOUN
ejpam-6543	69	22	πj	πj	VERB
ejpam-6543	69	23	:	:	PUNCT
ejpam-6543	69	24	∏	∏	PROPN
ejpam-6543	69	25	i∈i	i∈i	NOUN
ejpam-6543	69	26	xi	xi	X
ejpam-6543	69	27	→	→	SYM
ejpam-6543	69	28	xj	xj	PROPN
ejpam-6543	69	29	continuous	continuous	ADJ
ejpam-6543	69	30	.	.	PUNCT
ejpam-6543	70	1	definition	definition	NOUN
ejpam-6543	70	2	9	9	NUM
ejpam-6543	70	3	.	.	PUNCT
ejpam-6543	71	1	let	let	VERB
ejpam-6543	71	2	x	x	PRON
ejpam-6543	71	3	and	and	CCONJ
ejpam-6543	71	4	y	y	PROPN
ejpam-6543	71	5	be	be	AUX
ejpam-6543	71	6	topological	topological	ADJ
ejpam-6543	71	7	spaces	space	NOUN
ejpam-6543	71	8	.	.	PUNCT
ejpam-6543	72	1	the	the	DET
ejpam-6543	72	2	set	set	NOUN
ejpam-6543	72	3	of	of	ADP
ejpam-6543	72	4	all	all	DET
ejpam-6543	72	5	continuous	continuous	ADJ
ejpam-6543	72	6	functions	function	NOUN
ejpam-6543	72	7	from	from	ADP
ejpam-6543	72	8	x	x	PUNCT
ejpam-6543	72	9	to	to	ADP
ejpam-6543	72	10	y	y	PROPN
ejpam-6543	72	11	is	be	AUX
ejpam-6543	72	12	denoted	denote	VERB
ejpam-6543	72	13	by	by	ADP
ejpam-6543	72	14	c(x	c(x	PROPN
ejpam-6543	72	15	,	,	PUNCT
ejpam-6543	72	16	y	y	PROPN
ejpam-6543	72	17	)	)	PUNCT
ejpam-6543	72	18	.	.	PUNCT
ejpam-6543	73	1	this	this	PRON
ejpam-6543	73	2	can	can	AUX
ejpam-6543	73	3	be	be	AUX
ejpam-6543	73	4	endowed	endow	VERB
ejpam-6543	73	5	with	with	ADP
ejpam-6543	73	6	various	various	ADJ
ejpam-6543	73	7	topologies	topology	NOUN
ejpam-6543	73	8	,	,	PUNCT
ejpam-6543	73	9	including	include	VERB
ejpam-6543	73	10	the	the	DET
ejpam-6543	73	11	compact	compact	ADJ
ejpam-6543	73	12	-	-	PUNCT
ejpam-6543	73	13	open	open	ADJ
ejpam-6543	73	14	topology	topology	NOUN
ejpam-6543	73	15	and	and	CCONJ
ejpam-6543	73	16	the	the	DET
ejpam-6543	73	17	topology	topology	NOUN
ejpam-6543	73	18	of	of	ADP
ejpam-6543	73	19	uniform	uniform	ADJ
ejpam-6543	73	20	convergence	convergence	NOUN
ejpam-6543	73	21	on	on	ADP
ejpam-6543	73	22	compact	compact	ADJ
ejpam-6543	73	23	subsets	subset	NOUN
ejpam-6543	73	24	when	when	SCONJ
ejpam-6543	73	25	y	y	PROPN
ejpam-6543	73	26	is	be	AUX
ejpam-6543	73	27	a	a	DET
ejpam-6543	73	28	metric	metric	ADJ
ejpam-6543	73	29	space	space	NOUN
ejpam-6543	73	30	.	.	PUNCT
ejpam-6543	74	1	3	3	X
ejpam-6543	74	2	.	.	X
ejpam-6543	74	3	nearly	nearly	ADV
ejpam-6543	74	4	α	α	NUM
ejpam-6543	74	5	-	-	ADJ
ejpam-6543	74	6	compact	compact	ADJ
ejpam-6543	74	7	spaces	space	NOUN
ejpam-6543	74	8	in	in	ADP
ejpam-6543	74	9	this	this	DET
ejpam-6543	74	10	section	section	NOUN
ejpam-6543	74	11	,	,	PUNCT
ejpam-6543	74	12	we	we	PRON
ejpam-6543	74	13	introduce	introduce	VERB
ejpam-6543	74	14	the	the	DET
ejpam-6543	74	15	central	central	ADJ
ejpam-6543	74	16	concept	concept	NOUN
ejpam-6543	74	17	of	of	ADP
ejpam-6543	74	18	our	our	PRON
ejpam-6543	74	19	investigation	investigation	NOUN
ejpam-6543	74	20	and	and	CCONJ
ejpam-6543	74	21	establish	establish	VERB
ejpam-6543	74	22	its	its	PRON
ejpam-6543	74	23	fundamental	fundamental	ADJ
ejpam-6543	74	24	properties	property	NOUN
ejpam-6543	74	25	.	.	PUNCT
ejpam-6543	75	1	definition	definition	NOUN
ejpam-6543	75	2	10	10	NUM
ejpam-6543	75	3	.	.	PUNCT
ejpam-6543	76	1	a	a	DET
ejpam-6543	76	2	topological	topological	ADJ
ejpam-6543	76	3	space	space	NOUN
ejpam-6543	76	4	(	(	PUNCT
ejpam-6543	76	5	x	x	X
ejpam-6543	76	6	,	,	PUNCT
ejpam-6543	76	7	τ	τ	X
ejpam-6543	76	8	)	)	PUNCT
ejpam-6543	76	9	is	be	AUX
ejpam-6543	76	10	called	call	VERB
ejpam-6543	76	11	nearly	nearly	ADV
ejpam-6543	76	12	α	α	NOUN
ejpam-6543	76	13	-	-	ADJ
ejpam-6543	76	14	compact	compact	ADJ
ejpam-6543	76	15	if	if	SCONJ
ejpam-6543	76	16	for	for	ADP
ejpam-6543	76	17	every	every	DET
ejpam-6543	76	18	α	α	NOUN
ejpam-6543	76	19	-	-	ADJ
ejpam-6543	76	20	open	open	ADJ
ejpam-6543	76	21	cover	cover	NOUN
ejpam-6543	76	22	u	u	NOUN
ejpam-6543	76	23	of	of	ADP
ejpam-6543	76	24	x	x	PRON
ejpam-6543	76	25	,	,	PUNCT
ejpam-6543	76	26	there	there	PRON
ejpam-6543	76	27	exists	exist	VERB
ejpam-6543	76	28	a	a	DET
ejpam-6543	76	29	finite	finite	NOUN
ejpam-6543	76	30	subfamily	subfamily	ADV
ejpam-6543	76	31	f	f	PROPN
ejpam-6543	76	32	⊆	⊆	NUM
ejpam-6543	76	33	u	u	NOUN
ejpam-6543	76	34	such	such	ADJ
ejpam-6543	76	35	that	that	SCONJ
ejpam-6543	76	36	x	x	SYM
ejpam-6543	76	37	\	\	NOUN
ejpam-6543	76	38	⋃	⋃	PUNCT
ejpam-6543	76	39	f	f	PROPN
ejpam-6543	76	40	is	be	AUX
ejpam-6543	76	41	contained	contain	VERB
ejpam-6543	76	42	in	in	ADP
ejpam-6543	76	43	a	a	DET
ejpam-6543	76	44	finite	finite	ADJ
ejpam-6543	76	45	union	union	NOUN
ejpam-6543	76	46	of	of	ADP
ejpam-6543	76	47	α	α	PROPN
ejpam-6543	76	48	-	-	PUNCT
ejpam-6543	76	49	closed	closed	ADJ
ejpam-6543	76	50	sets	set	NOUN
ejpam-6543	76	51	with	with	ADP
ejpam-6543	76	52	empty	empty	ADJ
ejpam-6543	76	53	α	α	NOUN
ejpam-6543	76	54	-	-	NOUN
ejpam-6543	76	55	interior	interior	ADJ
ejpam-6543	76	56	.	.	PUNCT
ejpam-6543	77	1	this	this	DET
ejpam-6543	77	2	definition	definition	NOUN
ejpam-6543	77	3	captures	capture	VERB
ejpam-6543	77	4	the	the	DET
ejpam-6543	77	5	intuitive	intuitive	ADJ
ejpam-6543	77	6	notion	notion	NOUN
ejpam-6543	77	7	that	that	SCONJ
ejpam-6543	77	8	a	a	DET
ejpam-6543	77	9	space	space	NOUN
ejpam-6543	77	10	is	be	AUX
ejpam-6543	77	11	“	"	PUNCT
ejpam-6543	77	12	nearly	nearly	ADV
ejpam-6543	77	13	”	"	PUNCT
ejpam-6543	77	14	α	α	NOUN
ejpam-6543	77	15	-	-	ADJ
ejpam-6543	77	16	compact	compact	ADJ
ejpam-6543	77	17	if	if	SCONJ
ejpam-6543	77	18	every	every	DET
ejpam-6543	77	19	α	α	NOUN
ejpam-6543	77	20	-	-	ADJ
ejpam-6543	77	21	open	open	ADJ
ejpam-6543	77	22	cover	cover	NOUN
ejpam-6543	77	23	can	can	AUX
ejpam-6543	77	24	be	be	AUX
ejpam-6543	77	25	reduced	reduce	VERB
ejpam-6543	77	26	to	to	ADP
ejpam-6543	77	27	a	a	DET
ejpam-6543	77	28	finite	finite	ADJ
ejpam-6543	77	29	subcover	subcover	PROPN
ejpam-6543	77	30	that	that	PRON
ejpam-6543	77	31	misses	miss	VERB
ejpam-6543	77	32	only	only	ADV
ejpam-6543	77	33	a	a	DET
ejpam-6543	77	34	“	"	PUNCT
ejpam-6543	77	35	negligible	negligible	ADJ
ejpam-6543	77	36	”	"	PUNCT
ejpam-6543	77	37	set	set	VERB
ejpam-6543	77	38	in	in	ADP
ejpam-6543	77	39	terms	term	NOUN
ejpam-6543	77	40	of	of	ADP
ejpam-6543	77	41	α	α	NOUN
ejpam-6543	77	42	-	-	NOUN
ejpam-6543	77	43	structure	structure	NOUN
ejpam-6543	77	44	.	.	PUNCT
ejpam-6543	78	1	example	example	NOUN
ejpam-6543	79	1	1	1	NUM
ejpam-6543	79	2	.	.	X
ejpam-6543	79	3	consider	consider	VERB
ejpam-6543	79	4	the	the	DET
ejpam-6543	79	5	space	space	NOUN
ejpam-6543	79	6	r	r	NOUN
ejpam-6543	79	7	with	with	ADP
ejpam-6543	79	8	the	the	DET
ejpam-6543	79	9	usual	usual	ADJ
ejpam-6543	79	10	topology	topology	NOUN
ejpam-6543	79	11	.	.	PUNCT
ejpam-6543	80	1	the	the	DET
ejpam-6543	80	2	space	space	NOUN
ejpam-6543	80	3	r	r	NOUN
ejpam-6543	80	4	is	be	AUX
ejpam-6543	80	5	nearly	nearly	ADV
ejpam-6543	80	6	αcompact	αcompact	ADJ
ejpam-6543	80	7	.	.	PUNCT
ejpam-6543	81	1	to	to	PART
ejpam-6543	81	2	see	see	VERB
ejpam-6543	81	3	this	this	PRON
ejpam-6543	81	4	,	,	PUNCT
ejpam-6543	81	5	let	let	VERB
ejpam-6543	81	6	u	u	PRON
ejpam-6543	81	7	be	be	AUX
ejpam-6543	81	8	any	any	DET
ejpam-6543	81	9	α	α	NOUN
ejpam-6543	81	10	-	-	ADJ
ejpam-6543	81	11	open	open	ADJ
ejpam-6543	81	12	cover	cover	NOUN
ejpam-6543	81	13	of	of	ADP
ejpam-6543	81	14	r.	r.	PROPN
ejpam-6543	81	15	since	since	SCONJ
ejpam-6543	81	16	every	every	DET
ejpam-6543	81	17	open	open	ADJ
ejpam-6543	81	18	set	set	NOUN
ejpam-6543	81	19	is	be	AUX
ejpam-6543	81	20	α	α	NOUN
ejpam-6543	81	21	-	-	ADJ
ejpam-6543	81	22	open	open	ADJ
ejpam-6543	81	23	,	,	PUNCT
ejpam-6543	81	24	we	we	PRON
ejpam-6543	81	25	can	can	AUX
ejpam-6543	81	26	consider	consider	VERB
ejpam-6543	81	27	the	the	DET
ejpam-6543	81	28	cover	cover	NOUN
ejpam-6543	81	29	{	{	PUNCT
ejpam-6543	81	30	(	(	PUNCT
ejpam-6543	81	31	−n	−n	ADJ
ejpam-6543	81	32	,	,	PUNCT
ejpam-6543	81	33	n	n	CCONJ
ejpam-6543	81	34	)	)	PUNCT
ejpam-6543	81	35	:	:	PUNCT
ejpam-6543	81	36	n	n	X
ejpam-6543	81	37	∈	∈	PROPN
ejpam-6543	81	38	n	n	CCONJ
ejpam-6543	81	39	}	}	PUNCT
ejpam-6543	81	40	.	.	PUNCT
ejpam-6543	82	1	while	while	SCONJ
ejpam-6543	82	2	each	each	DET
ejpam-6543	82	3	interval	interval	NOUN
ejpam-6543	82	4	(	(	PUNCT
ejpam-6543	82	5	−n	−n	ADJ
ejpam-6543	82	6	,	,	PUNCT
ejpam-6543	82	7	n	n	CCONJ
ejpam-6543	82	8	)	)	PUNCT
ejpam-6543	82	9	is	be	AUX
ejpam-6543	82	10	not	not	PART
ejpam-6543	82	11	compact	compact	ADJ
ejpam-6543	82	12	(	(	PUNCT
ejpam-6543	82	13	as	as	SCONJ
ejpam-6543	82	14	correctly	correctly	ADV
ejpam-6543	82	15	noted	note	VERB
ejpam-6543	82	16	by	by	ADP
ejpam-6543	82	17	the	the	DET
ejpam-6543	82	18	reviewer	reviewer	NOUN
ejpam-6543	82	19	,	,	PUNCT
ejpam-6543	82	20	since	since	SCONJ
ejpam-6543	82	21	compactness	compactness	NOUN
ejpam-6543	82	22	in	in	ADP
ejpam-6543	82	23	r	r	NOUN
ejpam-6543	82	24	requires	require	VERB
ejpam-6543	82	25	sets	set	NOUN
ejpam-6543	82	26	to	to	PART
ejpam-6543	82	27	be	be	AUX
ejpam-6543	82	28	both	both	PRON
ejpam-6543	82	29	closed	close	VERB
ejpam-6543	82	30	and	and	CCONJ
ejpam-6543	82	31	bounded	bound	VERB
ejpam-6543	82	32	)	)	PUNCT
ejpam-6543	82	33	,	,	PUNCT
ejpam-6543	82	34	the	the	DET
ejpam-6543	82	35	key	key	ADJ
ejpam-6543	82	36	observation	observation	NOUN
ejpam-6543	82	37	is	be	AUX
ejpam-6543	82	38	that	that	SCONJ
ejpam-6543	82	39	for	for	ADP
ejpam-6543	82	40	any	any	DET
ejpam-6543	82	41	sufficiently	sufficiently	ADV
ejpam-6543	82	42	large	large	ADJ
ejpam-6543	82	43	n	n	CCONJ
ejpam-6543	82	44	,	,	PUNCT
ejpam-6543	82	45	we	we	PRON
ejpam-6543	82	46	can	can	AUX
ejpam-6543	82	47	find	find	VERB
ejpam-6543	82	48	a	a	DET
ejpam-6543	82	49	finite	finite	ADJ
ejpam-6543	82	50	subcollection	subcollection	NOUN
ejpam-6543	82	51	of	of	ADP
ejpam-6543	82	52	u	u	PRON
ejpam-6543	82	53	that	that	PRON
ejpam-6543	82	54	covers	cover	VERB
ejpam-6543	82	55	[	[	X
ejpam-6543	82	56	−n	−n	ADJ
ejpam-6543	82	57	,	,	PUNCT
ejpam-6543	82	58	n	n	CCONJ
ejpam-6543	82	59	]	]	X
ejpam-6543	82	60	(	(	PUNCT
ejpam-6543	82	61	which	which	PRON
ejpam-6543	82	62	is	be	AUX
ejpam-6543	82	63	compact	compact	ADJ
ejpam-6543	82	64	)	)	PUNCT
ejpam-6543	82	65	.	.	PUNCT
ejpam-6543	83	1	for	for	ADP
ejpam-6543	83	2	any	any	DET
ejpam-6543	83	3	finite	finite	ADJ
ejpam-6543	83	4	collection	collection	NOUN
ejpam-6543	83	5	covering	covering	NOUN
ejpam-6543	83	6	[	[	X
ejpam-6543	83	7	−n	−n	ADJ
ejpam-6543	83	8	,	,	PUNCT
ejpam-6543	83	9	n	n	CCONJ
ejpam-6543	83	10	]	]	PUNCT
ejpam-6543	83	11	for	for	ADP
ejpam-6543	83	12	sufficiently	sufficiently	ADV
ejpam-6543	83	13	large	large	ADJ
ejpam-6543	83	14	n	n	CCONJ
ejpam-6543	83	15	,	,	PUNCT
ejpam-6543	83	16	the	the	DET
ejpam-6543	83	17	remaining	remain	VERB
ejpam-6543	83	18	set	set	NOUN
ejpam-6543	83	19	r\	r\	PROPN
ejpam-6543	83	20	(	(	PUNCT
ejpam-6543	83	21	−n	−n	PROPN
ejpam-6543	83	22	,	,	PUNCT
ejpam-6543	83	23	n	n	CCONJ
ejpam-6543	83	24	)	)	PUNCT
ejpam-6543	83	25	=	=	SYM
ejpam-6543	83	26	(	(	PUNCT
ejpam-6543	83	27	−∞,−n]∪	−∞,−n]∪	PROPN
ejpam-6543	84	1	[	[	X
ejpam-6543	84	2	n,∞	n,∞	NOUN
ejpam-6543	84	3	)	)	PUNCT
ejpam-6543	84	4	consists	consist	VERB
ejpam-6543	84	5	of	of	ADP
ejpam-6543	84	6	two	two	NUM
ejpam-6543	84	7	α	α	NUM
ejpam-6543	84	8	-	-	PUNCT
ejpam-6543	84	9	closed	closed	ADJ
ejpam-6543	84	10	sets	set	NOUN
ejpam-6543	84	11	with	with	ADP
ejpam-6543	84	12	empty	empty	ADJ
ejpam-6543	84	13	α	α	NOUN
ejpam-6543	84	14	-	-	NOUN
ejpam-6543	84	15	interior	interior	ADJ
ejpam-6543	84	16	,	,	PUNCT
ejpam-6543	84	17	satisfying	satisfy	VERB
ejpam-6543	84	18	our	our	PRON
ejpam-6543	84	19	definition	definition	NOUN
ejpam-6543	84	20	.	.	PUNCT
ejpam-6543	85	1	example	example	NOUN
ejpam-6543	86	1	2	2	NUM
ejpam-6543	86	2	.	.	PUNCT
ejpam-6543	86	3	let	let	VERB
ejpam-6543	86	4	x	x	PUNCT
ejpam-6543	86	5	=	=	PRON
ejpam-6543	86	6	{	{	PUNCT
ejpam-6543	86	7	1	1	NUM
ejpam-6543	86	8	/	/	SYM
ejpam-6543	86	9	n	n	NOUN
ejpam-6543	86	10	:	:	PUNCT
ejpam-6543	86	11	n	n	CCONJ
ejpam-6543	86	12	∈	∈	PROPN
ejpam-6543	86	13	n	n	CCONJ
ejpam-6543	86	14	}	}	PUNCT
ejpam-6543	86	15	∪	∪	X
ejpam-6543	86	16	{	{	PUNCT
ejpam-6543	86	17	0	0	NUM
ejpam-6543	86	18	}	}	PUNCT
ejpam-6543	86	19	with	with	ADP
ejpam-6543	86	20	the	the	DET
ejpam-6543	86	21	subspace	subspace	NOUN
ejpam-6543	86	22	topology	topology	NOUN
ejpam-6543	86	23	from	from	ADP
ejpam-6543	86	24	r.	r.	PROPN
ejpam-6543	86	25	then	then	ADV
ejpam-6543	86	26	x	x	PRON
ejpam-6543	86	27	is	be	AUX
ejpam-6543	86	28	nearly	nearly	ADV
ejpam-6543	86	29	α	α	NOUN
ejpam-6543	86	30	-	-	ADJ
ejpam-6543	86	31	compact	compact	ADJ
ejpam-6543	86	32	.	.	PUNCT
ejpam-6543	87	1	indeed	indeed	ADV
ejpam-6543	87	2	,	,	PUNCT
ejpam-6543	87	3	let	let	VERB
ejpam-6543	87	4	u	u	PRON
ejpam-6543	87	5	be	be	AUX
ejpam-6543	87	6	any	any	DET
ejpam-6543	87	7	α	α	NOUN
ejpam-6543	87	8	-	-	ADJ
ejpam-6543	87	9	open	open	ADJ
ejpam-6543	87	10	cover	cover	NOUN
ejpam-6543	87	11	of	of	ADP
ejpam-6543	87	12	x.	x.	NOUN
ejpam-6543	87	13	since	since	SCONJ
ejpam-6543	87	14	x	x	PRON
ejpam-6543	87	15	is	be	AUX
ejpam-6543	87	16	compact	compact	ADJ
ejpam-6543	87	17	in	in	ADP
ejpam-6543	87	18	the	the	DET
ejpam-6543	87	19	usual	usual	ADJ
ejpam-6543	87	20	sense	sense	NOUN
ejpam-6543	87	21	,	,	PUNCT
ejpam-6543	87	22	it	it	PRON
ejpam-6543	87	23	is	be	AUX
ejpam-6543	87	24	also	also	ADV
ejpam-6543	87	25	α	α	NOUN
ejpam-6543	87	26	-	-	ADJ
ejpam-6543	87	27	compact	compact	ADJ
ejpam-6543	87	28	,	,	PUNCT
ejpam-6543	87	29	and	and	CCONJ
ejpam-6543	87	30	hence	hence	ADV
ejpam-6543	87	31	nearly	nearly	ADV
ejpam-6543	87	32	α	α	ADV
ejpam-6543	87	33	-	-	ADJ
ejpam-6543	87	34	compact	compact	ADJ
ejpam-6543	87	35	by	by	ADP
ejpam-6543	87	36	theorem	theorem	NOUN
ejpam-6543	87	37	1	1	NUM
ejpam-6543	87	38	below	below	ADV
ejpam-6543	87	39	.	.	PUNCT
ejpam-6543	88	1	example	example	NOUN
ejpam-6543	89	1	3	3	X
ejpam-6543	89	2	.	.	X
ejpam-6543	89	3	consider	consider	VERB
ejpam-6543	89	4	the	the	DET
ejpam-6543	89	5	discrete	discrete	ADJ
ejpam-6543	89	6	space	space	NOUN
ejpam-6543	89	7	d	d	NOUN
ejpam-6543	89	8	=	=	PUNCT
ejpam-6543	89	9	{	{	PUNCT
ejpam-6543	89	10	1	1	NUM
ejpam-6543	89	11	,	,	PUNCT
ejpam-6543	89	12	2	2	NUM
ejpam-6543	89	13	,	,	PUNCT
ejpam-6543	89	14	3	3	NUM
ejpam-6543	89	15	,	,	PUNCT
ejpam-6543	89	16	.	.	PUNCT
ejpam-6543	89	17	.	.	PUNCT
ejpam-6543	90	1	.	.	PUNCT
ejpam-6543	90	2	}	}	PUNCT
ejpam-6543	91	1	with	with	ADP
ejpam-6543	91	2	the	the	DET
ejpam-6543	91	3	discrete	discrete	ADJ
ejpam-6543	91	4	topology	topology	NOUN
ejpam-6543	91	5	.	.	PUNCT
ejpam-6543	92	1	this	this	DET
ejpam-6543	92	2	space	space	NOUN
ejpam-6543	92	3	is	be	AUX
ejpam-6543	92	4	not	not	PART
ejpam-6543	92	5	nearly	nearly	ADV
ejpam-6543	92	6	α	α	NOUN
ejpam-6543	92	7	-	-	ADJ
ejpam-6543	92	8	compact	compact	ADJ
ejpam-6543	92	9	.	.	PUNCT
ejpam-6543	93	1	the	the	DET
ejpam-6543	93	2	α	α	NOUN
ejpam-6543	93	3	-	-	ADJ
ejpam-6543	93	4	open	open	ADJ
ejpam-6543	93	5	cover	cover	NOUN
ejpam-6543	93	6	u	u	NOUN
ejpam-6543	93	7	=	=	PUNCT
ejpam-6543	93	8	{	{	PUNCT
ejpam-6543	93	9	{	{	PUNCT
ejpam-6543	93	10	n	n	CCONJ
ejpam-6543	93	11	}	}	PUNCT
ejpam-6543	93	12	:	:	PUNCT
ejpam-6543	93	13	n	n	X
ejpam-6543	93	14	∈	∈	PROPN
ejpam-6543	93	15	n	n	CCONJ
ejpam-6543	93	16	}	}	PUNCT
ejpam-6543	93	17	can	can	AUX
ejpam-6543	93	18	not	not	PART
ejpam-6543	93	19	be	be	AUX
ejpam-6543	93	20	reduced	reduce	VERB
ejpam-6543	93	21	to	to	ADP
ejpam-6543	93	22	a	a	DET
ejpam-6543	93	23	finite	finite	NOUN
ejpam-6543	93	24	subfamily	subfamily	ADV
ejpam-6543	93	25	satisfying	satisfy	VERB
ejpam-6543	93	26	the	the	DET
ejpam-6543	93	27	nearly	nearly	ADV
ejpam-6543	93	28	α	α	NUM
ejpam-6543	93	29	-	-	ADJ
ejpam-6543	93	30	compact	compact	ADJ
ejpam-6543	93	31	condition	condition	NOUN
ejpam-6543	93	32	,	,	PUNCT
ejpam-6543	93	33	since	since	SCONJ
ejpam-6543	93	34	any	any	DET
ejpam-6543	93	35	finite	finite	NOUN
ejpam-6543	93	36	subfamily	subfamily	ADV
ejpam-6543	93	37	leaves	leave	VERB
ejpam-6543	93	38	infinitely	infinitely	ADV
ejpam-6543	93	39	many	many	ADJ
ejpam-6543	93	40	isolated	isolated	ADJ
ejpam-6543	93	41	points	point	NOUN
ejpam-6543	93	42	uncovered	uncover	VERB
ejpam-6543	93	43	,	,	PUNCT
ejpam-6543	93	44	and	and	CCONJ
ejpam-6543	93	45	the	the	DET
ejpam-6543	93	46	union	union	NOUN
ejpam-6543	93	47	of	of	ADP
ejpam-6543	93	48	these	these	DET
ejpam-6543	93	49	points	point	NOUN
ejpam-6543	93	50	can	can	AUX
ejpam-6543	93	51	not	not	PART
ejpam-6543	93	52	be	be	AUX
ejpam-6543	93	53	expressed	express	VERB
ejpam-6543	93	54	as	as	ADP
ejpam-6543	93	55	a	a	DET
ejpam-6543	93	56	finite	finite	ADJ
ejpam-6543	93	57	union	union	NOUN
ejpam-6543	93	58	of	of	ADP
ejpam-6543	93	59	α	α	PROPN
ejpam-6543	93	60	-	-	PUNCT
ejpam-6543	93	61	closed	closed	ADJ
ejpam-6543	93	62	sets	set	NOUN
ejpam-6543	93	63	with	with	ADP
ejpam-6543	93	64	empty	empty	ADJ
ejpam-6543	93	65	α	α	NOUN
ejpam-6543	93	66	-	-	ADJ
ejpam-6543	93	67	interior	interior	ADJ
ejpam-6543	93	68	.	.	PUNCT
ejpam-6543	94	1	theorem	theorem	NOUN
ejpam-6543	94	2	1	1	NUM
ejpam-6543	94	3	.	.	PUNCT
ejpam-6543	95	1	every	every	DET
ejpam-6543	95	2	α	α	X
ejpam-6543	95	3	-	-	ADJ
ejpam-6543	95	4	compact	compact	ADJ
ejpam-6543	95	5	space	space	NOUN
ejpam-6543	95	6	is	be	AUX
ejpam-6543	95	7	nearly	nearly	ADV
ejpam-6543	95	8	α	α	NOUN
ejpam-6543	95	9	-	-	ADJ
ejpam-6543	95	10	compact	compact	ADJ
ejpam-6543	95	11	.	.	PUNCT
ejpam-6543	96	1	j.	j.	PROPN
ejpam-6543	96	2	oudetallah	oudetallah	PROPN
ejpam-6543	96	3	et	et	PROPN
ejpam-6543	96	4	al	al	PROPN
ejpam-6543	96	5	.	.	PUNCT
ejpam-6543	96	6	/	/	SYM
ejpam-6543	96	7	eur	eur	PROPN
ejpam-6543	96	8	.	.	PUNCT
ejpam-6543	97	1	j.	j.	PROPN
ejpam-6543	97	2	pure	pure	PROPN
ejpam-6543	97	3	appl	appl	PROPN
ejpam-6543	97	4	.	.	PROPN
ejpam-6543	97	5	math	math	PROPN
ejpam-6543	97	6	,	,	PUNCT
ejpam-6543	97	7	18	18	NUM
ejpam-6543	97	8	(	(	PUNCT
ejpam-6543	97	9	3	3	NUM
ejpam-6543	97	10	)	)	PUNCT
ejpam-6543	97	11	(	(	PUNCT
ejpam-6543	97	12	2025	2025	NUM
ejpam-6543	97	13	)	)	PUNCT
ejpam-6543	97	14	,	,	PUNCT
ejpam-6543	97	15	6543	6543	NUM
ejpam-6543	97	16	5	5	NUM
ejpam-6543	97	17	of	of	ADP
ejpam-6543	97	18	13	13	NUM
ejpam-6543	97	19	proof	proof	NOUN
ejpam-6543	97	20	.	.	PUNCT
ejpam-6543	98	1	let	let	VERB
ejpam-6543	98	2	(	(	PUNCT
ejpam-6543	98	3	x	x	NOUN
ejpam-6543	98	4	,	,	PUNCT
ejpam-6543	98	5	τ	τ	X
ejpam-6543	98	6	)	)	PUNCT
ejpam-6543	98	7	be	be	VERB
ejpam-6543	98	8	α	α	X
ejpam-6543	98	9	-	-	ADJ
ejpam-6543	98	10	compact	compact	ADJ
ejpam-6543	98	11	and	and	CCONJ
ejpam-6543	98	12	let	let	VERB
ejpam-6543	98	13	u	u	PRON
ejpam-6543	98	14	be	be	AUX
ejpam-6543	98	15	any	any	DET
ejpam-6543	98	16	α	α	NOUN
ejpam-6543	98	17	-	-	ADJ
ejpam-6543	98	18	open	open	ADJ
ejpam-6543	98	19	cover	cover	NOUN
ejpam-6543	98	20	of	of	ADP
ejpam-6543	98	21	x.	x.	NOUN
ejpam-6543	98	22	since	since	SCONJ
ejpam-6543	98	23	x	x	PROPN
ejpam-6543	98	24	is	be	AUX
ejpam-6543	98	25	αcompact	αcompact	ADJ
ejpam-6543	98	26	,	,	PUNCT
ejpam-6543	98	27	there	there	PRON
ejpam-6543	98	28	exists	exist	VERB
ejpam-6543	98	29	a	a	DET
ejpam-6543	98	30	finite	finite	NOUN
ejpam-6543	98	31	subfamily	subfamily	ADV
ejpam-6543	98	32	f	f	PROPN
ejpam-6543	98	33	⊆	⊆	NUM
ejpam-6543	98	34	u	u	NOUN
ejpam-6543	98	35	such	such	ADJ
ejpam-6543	98	36	that	that	SCONJ
ejpam-6543	98	37	x	x	X
ejpam-6543	98	38	=	=	PUNCT
ejpam-6543	98	39	⋃	⋃	PROPN
ejpam-6543	98	40	f	f	NOUN
ejpam-6543	98	41	.	.	PUNCT
ejpam-6543	99	1	therefore	therefore	ADV
ejpam-6543	99	2	,	,	PUNCT
ejpam-6543	99	3	x	x	SYM
ejpam-6543	99	4	\	\	X
ejpam-6543	99	5	⋃	⋃	PUNCT
ejpam-6543	99	6	f	f	NOUN
ejpam-6543	99	7	=	=	SYM
ejpam-6543	99	8	∅	∅	NOUN
ejpam-6543	99	9	,	,	PUNCT
ejpam-6543	99	10	which	which	PRON
ejpam-6543	99	11	is	be	AUX
ejpam-6543	99	12	trivially	trivially	ADV
ejpam-6543	99	13	contained	contain	VERB
ejpam-6543	99	14	in	in	ADP
ejpam-6543	99	15	the	the	DET
ejpam-6543	99	16	finite	finite	ADJ
ejpam-6543	99	17	union	union	NOUN
ejpam-6543	99	18	of	of	ADP
ejpam-6543	99	19	α	α	PROPN
ejpam-6543	99	20	-	-	PUNCT
ejpam-6543	99	21	closed	closed	ADJ
ejpam-6543	99	22	sets	set	NOUN
ejpam-6543	99	23	with	with	ADP
ejpam-6543	99	24	empty	empty	ADJ
ejpam-6543	99	25	α	α	NOUN
ejpam-6543	99	26	-	-	ADJ
ejpam-6543	99	27	interior	interior	ADJ
ejpam-6543	99	28	(	(	PUNCT
ejpam-6543	99	29	namely	namely	ADV
ejpam-6543	99	30	,	,	PUNCT
ejpam-6543	99	31	the	the	DET
ejpam-6543	99	32	empty	empty	ADJ
ejpam-6543	99	33	union	union	NOUN
ejpam-6543	99	34	)	)	PUNCT
ejpam-6543	99	35	.	.	PUNCT
ejpam-6543	100	1	thus	thus	ADV
ejpam-6543	100	2	x	x	X
ejpam-6543	100	3	is	be	AUX
ejpam-6543	100	4	nearly	nearly	ADV
ejpam-6543	100	5	α	α	ADV
ejpam-6543	100	6	-	-	ADJ
ejpam-6543	100	7	compact	compact	ADJ
ejpam-6543	100	8	.	.	PUNCT
ejpam-6543	101	1	remark	remark	NOUN
ejpam-6543	101	2	1	1	NUM
ejpam-6543	101	3	.	.	PUNCT
ejpam-6543	102	1	the	the	DET
ejpam-6543	102	2	converse	converse	NOUN
ejpam-6543	102	3	of	of	ADP
ejpam-6543	102	4	theorem	theorem	NOUN
ejpam-6543	102	5	1	1	NUM
ejpam-6543	102	6	does	do	AUX
ejpam-6543	102	7	not	not	PART
ejpam-6543	102	8	hold	hold	VERB
ejpam-6543	102	9	in	in	ADP
ejpam-6543	102	10	general	general	ADJ
ejpam-6543	102	11	.	.	PUNCT
ejpam-6543	103	1	a	a	DET
ejpam-6543	103	2	space	space	NOUN
ejpam-6543	103	3	can	can	AUX
ejpam-6543	103	4	be	be	AUX
ejpam-6543	103	5	nearly	nearly	ADV
ejpam-6543	103	6	α	α	NOUN
ejpam-6543	103	7	-	-	ADJ
ejpam-6543	103	8	compact	compact	ADJ
ejpam-6543	103	9	without	without	ADP
ejpam-6543	103	10	being	be	AUX
ejpam-6543	103	11	α	α	NOUN
ejpam-6543	103	12	-	-	ADJ
ejpam-6543	103	13	compact	compact	ADJ
ejpam-6543	103	14	.	.	PUNCT
ejpam-6543	104	1	for	for	ADP
ejpam-6543	104	2	instance	instance	NOUN
ejpam-6543	104	3	,	,	PUNCT
ejpam-6543	104	4	the	the	DET
ejpam-6543	104	5	real	real	ADJ
ejpam-6543	104	6	line	line	NOUN
ejpam-6543	104	7	r	r	NOUN
ejpam-6543	104	8	with	with	ADP
ejpam-6543	104	9	the	the	DET
ejpam-6543	104	10	usual	usual	ADJ
ejpam-6543	104	11	topology	topology	NOUN
ejpam-6543	104	12	is	be	AUX
ejpam-6543	104	13	nearly	nearly	ADV
ejpam-6543	104	14	α	α	ADV
ejpam-6543	104	15	-	-	ADJ
ejpam-6543	104	16	compact	compact	ADJ
ejpam-6543	104	17	(	(	PUNCT
ejpam-6543	104	18	as	as	SCONJ
ejpam-6543	104	19	shown	show	VERB
ejpam-6543	104	20	in	in	ADP
ejpam-6543	104	21	example	example	NOUN
ejpam-6543	104	22	1	1	NUM
ejpam-6543	104	23	)	)	PUNCT
ejpam-6543	104	24	but	but	CCONJ
ejpam-6543	104	25	not	not	PART
ejpam-6543	104	26	α	α	NOUN
ejpam-6543	104	27	-	-	ADJ
ejpam-6543	104	28	compact	compact	ADJ
ejpam-6543	104	29	,	,	PUNCT
ejpam-6543	104	30	since	since	SCONJ
ejpam-6543	104	31	the	the	DET
ejpam-6543	104	32	open	open	ADJ
ejpam-6543	104	33	cover	cover	NOUN
ejpam-6543	104	34	{	{	PUNCT
ejpam-6543	104	35	(	(	PUNCT
ejpam-6543	104	36	−n	−n	ADJ
ejpam-6543	104	37	,	,	PUNCT
ejpam-6543	104	38	n	n	CCONJ
ejpam-6543	104	39	)	)	PUNCT
ejpam-6543	104	40	:	:	PUNCT
ejpam-6543	104	41	n	n	X
ejpam-6543	104	42	∈	∈	PROPN
ejpam-6543	104	43	n	n	CCONJ
ejpam-6543	104	44	}	}	PUNCT
ejpam-6543	104	45	has	have	VERB
ejpam-6543	104	46	no	no	DET
ejpam-6543	104	47	finite	finite	PROPN
ejpam-6543	104	48	subcover	subcover	PROPN
ejpam-6543	104	49	.	.	PUNCT
ejpam-6543	105	1	lemma	lemma	PROPN
ejpam-6543	105	2	1	1	X
ejpam-6543	105	3	.	.	PUNCT
ejpam-6543	106	1	let	let	AUX
ejpam-6543	106	2	(	(	PUNCT
ejpam-6543	106	3	x	x	NOUN
ejpam-6543	106	4	,	,	PUNCT
ejpam-6543	106	5	τ	τ	X
ejpam-6543	106	6	)	)	PUNCT
ejpam-6543	106	7	be	be	VERB
ejpam-6543	106	8	a	a	DET
ejpam-6543	106	9	topological	topological	ADJ
ejpam-6543	106	10	space	space	NOUN
ejpam-6543	106	11	and	and	CCONJ
ejpam-6543	106	12	a	a	DET
ejpam-6543	106	13	⊆	⊆	NUM
ejpam-6543	106	14	x.	x.	NOUN
ejpam-6543	106	15	if	if	SCONJ
ejpam-6543	106	16	a	a	PRON
ejpam-6543	106	17	is	be	AUX
ejpam-6543	106	18	α	α	NOUN
ejpam-6543	106	19	-	-	PUNCT
ejpam-6543	106	20	closed	closed	ADJ
ejpam-6543	106	21	and	and	CCONJ
ejpam-6543	106	22	intα(a	intα(a	NOUN
ejpam-6543	106	23	)	)	PUNCT
ejpam-6543	107	1	=	=	NOUN
ejpam-6543	107	2	∅	∅	NOUN
ejpam-6543	107	3	,	,	PUNCT
ejpam-6543	107	4	then	then	ADV
ejpam-6543	107	5	for	for	ADP
ejpam-6543	107	6	any	any	DET
ejpam-6543	107	7	α	α	NOUN
ejpam-6543	107	8	-	-	ADJ
ejpam-6543	107	9	open	open	ADJ
ejpam-6543	107	10	set	set	NOUN
ejpam-6543	107	11	u	u	NOUN
ejpam-6543	107	12	containing	contain	VERB
ejpam-6543	107	13	a	a	PRON
ejpam-6543	107	14	,	,	PUNCT
ejpam-6543	107	15	we	we	PRON
ejpam-6543	107	16	have	have	VERB
ejpam-6543	107	17	clα(x	clα(x	NOUN
ejpam-6543	107	18	\	\	NOUN
ejpam-6543	107	19	u	u	NOUN
ejpam-6543	107	20	)	)	PUNCT
ejpam-6543	107	21	̸=	̸=	PROPN
ejpam-6543	107	22	x.	x.	NOUN
ejpam-6543	107	23	proof	proof	NOUN
ejpam-6543	107	24	.	.	PUNCT
ejpam-6543	108	1	suppose	suppose	VERB
ejpam-6543	108	2	a	a	PRON
ejpam-6543	108	3	is	be	AUX
ejpam-6543	108	4	α	α	NOUN
ejpam-6543	108	5	-	-	VERB
ejpam-6543	108	6	closed	closed	ADJ
ejpam-6543	108	7	with	with	ADP
ejpam-6543	108	8	intα(a	intα(a	NOUN
ejpam-6543	108	9	)	)	PUNCT
ejpam-6543	108	10	=	=	NOUN
ejpam-6543	108	11	∅	∅	NOUN
ejpam-6543	108	12	,	,	PUNCT
ejpam-6543	108	13	and	and	CCONJ
ejpam-6543	108	14	let	let	VERB
ejpam-6543	108	15	u	u	PRON
ejpam-6543	108	16	be	be	AUX
ejpam-6543	108	17	any	any	DET
ejpam-6543	108	18	α	α	NOUN
ejpam-6543	108	19	-	-	ADJ
ejpam-6543	108	20	open	open	ADJ
ejpam-6543	108	21	set	set	NOUN
ejpam-6543	108	22	containing	contain	VERB
ejpam-6543	108	23	a.	a.	NOUN
ejpam-6543	108	24	since	since	SCONJ
ejpam-6543	108	25	intα(a	intα(a	NOUN
ejpam-6543	108	26	)	)	PUNCT
ejpam-6543	109	1	=	=	NOUN
ejpam-6543	109	2	∅	∅	NOUN
ejpam-6543	109	3	,	,	PUNCT
ejpam-6543	109	4	there	there	PRON
ejpam-6543	109	5	exists	exist	VERB
ejpam-6543	109	6	a	a	DET
ejpam-6543	109	7	point	point	NOUN
ejpam-6543	109	8	x	x	X
ejpam-6543	109	9	∈	∈	PROPN
ejpam-6543	109	10	a	a	DET
ejpam-6543	109	11	such	such	ADJ
ejpam-6543	109	12	that	that	SCONJ
ejpam-6543	109	13	every	every	DET
ejpam-6543	109	14	α	α	NOUN
ejpam-6543	109	15	-	-	ADJ
ejpam-6543	109	16	open	open	ADJ
ejpam-6543	109	17	neighborhood	neighborhood	NOUN
ejpam-6543	109	18	of	of	ADP
ejpam-6543	109	19	x	x	PUNCT
ejpam-6543	109	20	intersects	intersect	NOUN
ejpam-6543	109	21	x	x	SYM
ejpam-6543	109	22	\	\	NOUN
ejpam-6543	109	23	a.	a.	NOUN
ejpam-6543	109	24	since	since	SCONJ
ejpam-6543	109	25	a	a	DET
ejpam-6543	109	26	⊆	⊆	NUM
ejpam-6543	109	27	u	u	NOUN
ejpam-6543	109	28	and	and	CCONJ
ejpam-6543	109	29	u	u	NOUN
ejpam-6543	109	30	is	be	AUX
ejpam-6543	109	31	α	α	NOUN
ejpam-6543	109	32	-	-	ADJ
ejpam-6543	109	33	open	open	ADJ
ejpam-6543	109	34	,	,	PUNCT
ejpam-6543	109	35	we	we	PRON
ejpam-6543	109	36	have	have	VERB
ejpam-6543	109	37	x	x	X
ejpam-6543	109	38	∈	∈	PROPN
ejpam-6543	109	39	u	u	NOUN
ejpam-6543	109	40	.	.	PUNCT
ejpam-6543	110	1	the	the	DET
ejpam-6543	110	2	α	α	NOUN
ejpam-6543	110	3	-	-	PUNCT
ejpam-6543	110	4	closure	closure	NOUN
ejpam-6543	110	5	clα(x	clα(x	NOUN
ejpam-6543	110	6	\	\	NOUN
ejpam-6543	110	7	u	u	NOUN
ejpam-6543	110	8	)	)	PUNCT
ejpam-6543	110	9	can	can	AUX
ejpam-6543	110	10	not	not	PART
ejpam-6543	110	11	contain	contain	VERB
ejpam-6543	110	12	x	x	PUNCT
ejpam-6543	110	13	since	since	SCONJ
ejpam-6543	110	14	x	x	PROPN
ejpam-6543	110	15	∈	∈	PROPN
ejpam-6543	110	16	u	u	NOUN
ejpam-6543	110	17	and	and	CCONJ
ejpam-6543	110	18	u	u	NOUN
ejpam-6543	110	19	is	be	AUX
ejpam-6543	110	20	α	α	NOUN
ejpam-6543	110	21	-	-	NOUN
ejpam-6543	110	22	open	open	ADJ
ejpam-6543	110	23	.	.	PUNCT
ejpam-6543	111	1	therefore	therefore	ADV
ejpam-6543	111	2	,	,	PUNCT
ejpam-6543	111	3	clα(x	clα(x	PROPN
ejpam-6543	111	4	\	\	PROPN
ejpam-6543	111	5	u	u	NOUN
ejpam-6543	111	6	)	)	PUNCT
ejpam-6543	111	7	̸=	̸=	PROPN
ejpam-6543	111	8	x.	x.	NOUN
ejpam-6543	111	9	theorem	theorem	VERB
ejpam-6543	111	10	2	2	NUM
ejpam-6543	111	11	.	.	PUNCT
ejpam-6543	111	12	a	a	DET
ejpam-6543	111	13	topological	topological	ADJ
ejpam-6543	111	14	space	space	NOUN
ejpam-6543	111	15	(	(	PUNCT
ejpam-6543	111	16	x	x	X
ejpam-6543	111	17	,	,	PUNCT
ejpam-6543	111	18	τ	τ	X
ejpam-6543	111	19	)	)	PUNCT
ejpam-6543	111	20	is	be	AUX
ejpam-6543	111	21	nearly	nearly	ADV
ejpam-6543	111	22	α	α	NOUN
ejpam-6543	111	23	-	-	ADJ
ejpam-6543	111	24	compact	compact	ADJ
ejpam-6543	111	25	if	if	SCONJ
ejpam-6543	111	26	and	and	CCONJ
ejpam-6543	111	27	only	only	ADV
ejpam-6543	111	28	if	if	SCONJ
ejpam-6543	111	29	every	every	DET
ejpam-6543	111	30	infinite	infinite	ADJ
ejpam-6543	111	31	family	family	NOUN
ejpam-6543	111	32	of	of	ADP
ejpam-6543	111	33	non	non	ADJ
ejpam-6543	111	34	-	-	ADJ
ejpam-6543	111	35	empty	empty	ADJ
ejpam-6543	111	36	α	α	ADJ
ejpam-6543	111	37	-	-	ADJ
ejpam-6543	111	38	open	open	ADJ
ejpam-6543	111	39	sets	set	NOUN
ejpam-6543	111	40	with	with	ADP
ejpam-6543	111	41	the	the	DET
ejpam-6543	111	42	finite	finite	ADJ
ejpam-6543	111	43	intersection	intersection	NOUN
ejpam-6543	111	44	property	property	NOUN
ejpam-6543	111	45	contains	contain	VERB
ejpam-6543	111	46	a	a	DET
ejpam-6543	111	47	subfamily	subfamily	ADV
ejpam-6543	111	48	whose	whose	DET
ejpam-6543	111	49	intersection	intersection	NOUN
ejpam-6543	111	50	has	have	AUX
ejpam-6543	111	51	non	non	ADJ
ejpam-6543	111	52	-	-	ADJ
ejpam-6543	111	53	empty	empty	ADJ
ejpam-6543	111	54	α	α	NOUN
ejpam-6543	111	55	-	-	NOUN
ejpam-6543	111	56	interior	interior	ADJ
ejpam-6543	111	57	.	.	PUNCT
ejpam-6543	112	1	proof	proof	NOUN
ejpam-6543	112	2	.	.	PUNCT
ejpam-6543	113	1	(	(	PUNCT
ejpam-6543	113	2	⇒	⇒	PROPN
ejpam-6543	113	3	)	)	PUNCT
ejpam-6543	113	4	suppose	suppose	VERB
ejpam-6543	113	5	x	x	PRON
ejpam-6543	113	6	is	be	AUX
ejpam-6543	113	7	nearly	nearly	ADV
ejpam-6543	113	8	α	α	ADV
ejpam-6543	113	9	-	-	ADJ
ejpam-6543	113	10	compact	compact	ADJ
ejpam-6543	113	11	and	and	CCONJ
ejpam-6543	113	12	let	let	VERB
ejpam-6543	113	13	g	g	NOUN
ejpam-6543	113	14	=	=	PUNCT
ejpam-6543	113	15	{	{	PUNCT
ejpam-6543	113	16	gi	gi	INTJ
ejpam-6543	113	17	:	:	PUNCT
ejpam-6543	113	18	i	i	PRON
ejpam-6543	113	19	∈	∈	VERB
ejpam-6543	113	20	i	i	PRON
ejpam-6543	113	21	}	}	PUNCT
ejpam-6543	113	22	be	be	AUX
ejpam-6543	113	23	an	an	DET
ejpam-6543	113	24	infinite	infinite	ADJ
ejpam-6543	113	25	family	family	NOUN
ejpam-6543	113	26	of	of	ADP
ejpam-6543	113	27	non	non	ADJ
ejpam-6543	113	28	-	-	ADJ
ejpam-6543	113	29	empty	empty	ADJ
ejpam-6543	113	30	α	α	ADJ
ejpam-6543	113	31	-	-	ADJ
ejpam-6543	113	32	open	open	ADJ
ejpam-6543	113	33	sets	set	NOUN
ejpam-6543	113	34	with	with	ADP
ejpam-6543	113	35	the	the	DET
ejpam-6543	113	36	finite	finite	ADJ
ejpam-6543	113	37	intersection	intersection	NOUN
ejpam-6543	113	38	property	property	NOUN
ejpam-6543	113	39	.	.	PUNCT
ejpam-6543	114	1	suppose	suppose	VERB
ejpam-6543	114	2	,	,	PUNCT
ejpam-6543	114	3	for	for	ADP
ejpam-6543	114	4	contradiction	contradiction	NOUN
ejpam-6543	114	5	,	,	PUNCT
ejpam-6543	114	6	that	that	SCONJ
ejpam-6543	114	7	for	for	ADP
ejpam-6543	114	8	every	every	DET
ejpam-6543	114	9	finite	finite	NOUN
ejpam-6543	114	10	subfamily	subfamily	ADV
ejpam-6543	114	11	h	h	NOUN
ejpam-6543	114	12	⊆	⊆	NUM
ejpam-6543	114	13	g	g	NOUN
ejpam-6543	114	14	,	,	PUNCT
ejpam-6543	114	15	the	the	DET
ejpam-6543	114	16	set	set	NOUN
ejpam-6543	114	17	⋂	⋂	PROPN
ejpam-6543	114	18	h	h	NOUN
ejpam-6543	114	19	has	have	AUX
ejpam-6543	114	20	empty	empty	ADJ
ejpam-6543	114	21	α	α	PRON
ejpam-6543	114	22	-	-	NOUN
ejpam-6543	114	23	interior	interior	ADJ
ejpam-6543	114	24	.	.	PUNCT
ejpam-6543	115	1	consider	consider	VERB
ejpam-6543	115	2	the	the	DET
ejpam-6543	115	3	family	family	NOUN
ejpam-6543	115	4	u	u	NOUN
ejpam-6543	115	5	=	=	PUNCT
ejpam-6543	115	6	{	{	PUNCT
ejpam-6543	115	7	x	x	SYM
ejpam-6543	115	8	\	\	NOUN
ejpam-6543	115	9	gi	gi	NOUN
ejpam-6543	115	10	:	:	PUNCT
ejpam-6543	115	11	i	i	PRON
ejpam-6543	115	12	∈	∈	VERB
ejpam-6543	115	13	i	i	X
ejpam-6543	115	14	}	}	PUNCT
ejpam-6543	115	15	.	.	PUNCT
ejpam-6543	116	1	since	since	SCONJ
ejpam-6543	116	2	each	each	DET
ejpam-6543	116	3	gi	gi	NOUN
ejpam-6543	116	4	is	be	AUX
ejpam-6543	116	5	α	α	NOUN
ejpam-6543	116	6	-	-	ADJ
ejpam-6543	116	7	open	open	ADJ
ejpam-6543	116	8	,	,	PUNCT
ejpam-6543	116	9	each	each	DET
ejpam-6543	116	10	x	x	PUNCT
ejpam-6543	116	11	\	\	ADJ
ejpam-6543	116	12	gi	gi	NOUN
ejpam-6543	116	13	is	be	AUX
ejpam-6543	116	14	α	α	NOUN
ejpam-6543	116	15	-	-	PUNCT
ejpam-6543	116	16	closed	closed	ADJ
ejpam-6543	116	17	.	.	PUNCT
ejpam-6543	117	1	if	if	SCONJ
ejpam-6543	117	2	u	u	PRON
ejpam-6543	117	3	covered	cover	VERB
ejpam-6543	117	4	x	x	PRON
ejpam-6543	117	5	,	,	PUNCT
ejpam-6543	117	6	then	then	ADV
ejpam-6543	117	7	⋂	⋂	PROPN
ejpam-6543	117	8	i∈i	i∈i	ADJ
ejpam-6543	117	9	gi	gi	NOUN
ejpam-6543	117	10	=	=	PUNCT
ejpam-6543	118	1	x	x	SYM
ejpam-6543	118	2	\	\	X
ejpam-6543	118	3	⋃	⋃	PUNCT
ejpam-6543	118	4	i∈i(x	i∈i(x	NOUN
ejpam-6543	118	5	\	\	PROPN
ejpam-6543	118	6	gi	gi	NOUN
ejpam-6543	118	7	)	)	PUNCT
ejpam-6543	118	8	=	=	NOUN
ejpam-6543	118	9	∅	∅	NOUN
ejpam-6543	118	10	,	,	PUNCT
ejpam-6543	118	11	contradicting	contradict	VERB
ejpam-6543	118	12	the	the	DET
ejpam-6543	118	13	finite	finite	ADJ
ejpam-6543	118	14	intersection	intersection	NOUN
ejpam-6543	118	15	property	property	NOUN
ejpam-6543	118	16	of	of	ADP
ejpam-6543	118	17	g.	g.	PROPN
ejpam-6543	118	18	therefore	therefore	ADV
ejpam-6543	118	19	,	,	PUNCT
ejpam-6543	118	20	u	u	NOUN
ejpam-6543	118	21	does	do	AUX
ejpam-6543	118	22	not	not	PART
ejpam-6543	118	23	cover	cover	VERB
ejpam-6543	118	24	x.	x.	NOUN
ejpam-6543	118	25	let	let	VERB
ejpam-6543	118	26	v	v	NOUN
ejpam-6543	118	27	=	=	VERB
ejpam-6543	118	28	u	u	NOUN
ejpam-6543	118	29	∪	∪	X
ejpam-6543	118	30	{	{	PUNCT
ejpam-6543	118	31	x	x	NOUN
ejpam-6543	118	32	}	}	PUNCT
ejpam-6543	118	33	.	.	PUNCT
ejpam-6543	119	1	then	then	ADV
ejpam-6543	119	2	v	v	NOUN
ejpam-6543	119	3	is	be	AUX
ejpam-6543	119	4	an	an	DET
ejpam-6543	119	5	α	α	NOUN
ejpam-6543	119	6	-	-	ADJ
ejpam-6543	119	7	open	open	ADJ
ejpam-6543	119	8	cover	cover	NOUN
ejpam-6543	119	9	of	of	ADP
ejpam-6543	119	10	x.	x.	NOUN
ejpam-6543	119	11	by	by	ADP
ejpam-6543	119	12	nearly	nearly	ADV
ejpam-6543	119	13	α	α	NOUN
ejpam-6543	119	14	-	-	NOUN
ejpam-6543	119	15	compactness	compactness	NOUN
ejpam-6543	119	16	,	,	PUNCT
ejpam-6543	119	17	there	there	PRON
ejpam-6543	119	18	exists	exist	VERB
ejpam-6543	119	19	a	a	DET
ejpam-6543	119	20	finite	finite	NOUN
ejpam-6543	119	21	subfamily	subfamily	ADV
ejpam-6543	119	22	f	f	PROPN
ejpam-6543	119	23	⊆	⊆	NUM
ejpam-6543	119	24	v	v	ADP
ejpam-6543	119	25	such	such	ADJ
ejpam-6543	119	26	that	that	SCONJ
ejpam-6543	119	27	x	x	SYM
ejpam-6543	119	28	\	\	NOUN
ejpam-6543	119	29	⋃	⋃	PUNCT
ejpam-6543	119	30	f	f	PROPN
ejpam-6543	119	31	is	be	AUX
ejpam-6543	119	32	contained	contain	VERB
ejpam-6543	119	33	in	in	ADP
ejpam-6543	119	34	a	a	DET
ejpam-6543	119	35	finite	finite	ADJ
ejpam-6543	119	36	union	union	NOUN
ejpam-6543	119	37	of	of	ADP
ejpam-6543	119	38	α	α	PROPN
ejpam-6543	119	39	-	-	PUNCT
ejpam-6543	119	40	closed	closed	ADJ
ejpam-6543	119	41	sets	set	NOUN
ejpam-6543	119	42	with	with	ADP
ejpam-6543	119	43	empty	empty	ADJ
ejpam-6543	119	44	α	α	NOUN
ejpam-6543	119	45	-	-	NOUN
ejpam-6543	119	46	interior	interior	ADJ
ejpam-6543	119	47	.	.	PUNCT
ejpam-6543	120	1	if	if	SCONJ
ejpam-6543	120	2	x	x	SYM
ejpam-6543	120	3	∈	∈	PROPN
ejpam-6543	120	4	f	f	X
ejpam-6543	120	5	,	,	PUNCT
ejpam-6543	120	6	then	then	ADV
ejpam-6543	120	7	⋃	⋃	PUNCT
ejpam-6543	120	8	f	f	PROPN
ejpam-6543	120	9	=	=	SYM
ejpam-6543	120	10	x	x	PROPN
ejpam-6543	120	11	,	,	PUNCT
ejpam-6543	120	12	so	so	ADV
ejpam-6543	120	13	the	the	DET
ejpam-6543	120	14	condition	condition	NOUN
ejpam-6543	120	15	is	be	AUX
ejpam-6543	120	16	trivially	trivially	ADV
ejpam-6543	120	17	satisfied	satisfied	ADJ
ejpam-6543	120	18	.	.	PUNCT
ejpam-6543	121	1	if	if	SCONJ
ejpam-6543	121	2	x	x	PROPN
ejpam-6543	121	3	/∈	/∈	PROPN
ejpam-6543	122	1	f	f	PROPN
ejpam-6543	122	2	,	,	PUNCT
ejpam-6543	122	3	then	then	ADV
ejpam-6543	122	4	f	f	PROPN
ejpam-6543	122	5	⊆	⊆	NUM
ejpam-6543	122	6	u	u	NOUN
ejpam-6543	122	7	corresponds	correspond	VERB
ejpam-6543	122	8	to	to	ADP
ejpam-6543	122	9	a	a	DET
ejpam-6543	122	10	finite	finite	NOUN
ejpam-6543	122	11	subfamily	subfamily	ADV
ejpam-6543	122	12	of	of	ADP
ejpam-6543	122	13	{	{	PUNCT
ejpam-6543	122	14	x	x	SYM
ejpam-6543	122	15	\	\	NOUN
ejpam-6543	122	16	gi	gi	NOUN
ejpam-6543	122	17	:	:	PUNCT
ejpam-6543	123	1	i	i	PRON
ejpam-6543	123	2	∈	∈	VERB
ejpam-6543	123	3	i	i	X
ejpam-6543	123	4	}	}	PUNCT
ejpam-6543	123	5	,	,	PUNCT
ejpam-6543	123	6	say	say	VERB
ejpam-6543	123	7	{	{	PUNCT
ejpam-6543	123	8	x	x	SYM
ejpam-6543	123	9	\	\	PROPN
ejpam-6543	123	10	gi1	gi1	PROPN
ejpam-6543	123	11	,	,	PUNCT
ejpam-6543	123	12	.	.	PUNCT
ejpam-6543	123	13	.	.	PUNCT
ejpam-6543	123	14	.	.	PUNCT
ejpam-6543	124	1	,	,	PUNCT
ejpam-6543	124	2	x	x	SYM
ejpam-6543	124	3	\	\	PROPN
ejpam-6543	124	4	gik	gik	X
ejpam-6543	124	5	}	}	PUNCT
ejpam-6543	124	6	.	.	PUNCT
ejpam-6543	125	1	then	then	ADV
ejpam-6543	125	2	x	x	X
ejpam-6543	125	3	\	\	X
ejpam-6543	125	4	⋃	⋃	PUNCT
ejpam-6543	125	5	f	f	PROPN
ejpam-6543	125	6	=	=	SYM
ejpam-6543	125	7	⋂k	⋂k	PROPN
ejpam-6543	125	8	j=1gij	j=1gij	PROPN
ejpam-6543	125	9	,	,	PUNCT
ejpam-6543	125	10	which	which	PRON
ejpam-6543	125	11	must	must	AUX
ejpam-6543	125	12	be	be	AUX
ejpam-6543	125	13	contained	contain	VERB
ejpam-6543	125	14	in	in	ADP
ejpam-6543	125	15	a	a	DET
ejpam-6543	125	16	finite	finite	ADJ
ejpam-6543	125	17	union	union	NOUN
ejpam-6543	125	18	of	of	ADP
ejpam-6543	125	19	α	α	PROPN
ejpam-6543	125	20	-	-	PUNCT
ejpam-6543	125	21	closed	closed	ADJ
ejpam-6543	125	22	sets	set	NOUN
ejpam-6543	125	23	with	with	ADP
ejpam-6543	125	24	empty	empty	ADJ
ejpam-6543	125	25	α	α	NOUN
ejpam-6543	125	26	-	-	NOUN
ejpam-6543	125	27	interior	interior	ADJ
ejpam-6543	125	28	.	.	PUNCT
ejpam-6543	126	1	this	this	PRON
ejpam-6543	126	2	provides	provide	VERB
ejpam-6543	126	3	the	the	DET
ejpam-6543	126	4	required	require	VERB
ejpam-6543	126	5	finite	finite	NOUN
ejpam-6543	126	6	subfamily	subfamily	ADV
ejpam-6543	126	7	of	of	ADP
ejpam-6543	126	8	g	g	NOUN
ejpam-6543	126	9	whose	whose	DET
ejpam-6543	126	10	intersection	intersection	NOUN
ejpam-6543	126	11	has	have	VERB
ejpam-6543	126	12	the	the	DET
ejpam-6543	126	13	desired	desire	VERB
ejpam-6543	126	14	property	property	NOUN
ejpam-6543	126	15	.	.	PUNCT
ejpam-6543	127	1	(	(	PUNCT
ejpam-6543	127	2	⇐	⇐	NOUN
ejpam-6543	127	3	)	)	PUNCT
ejpam-6543	127	4	suppose	suppose	VERB
ejpam-6543	127	5	the	the	DET
ejpam-6543	127	6	condition	condition	NOUN
ejpam-6543	127	7	holds	hold	VERB
ejpam-6543	127	8	.	.	PUNCT
ejpam-6543	128	1	let	let	VERB
ejpam-6543	128	2	u	u	PRON
ejpam-6543	128	3	be	be	AUX
ejpam-6543	128	4	any	any	DET
ejpam-6543	128	5	α	α	NOUN
ejpam-6543	128	6	-	-	ADJ
ejpam-6543	128	7	open	open	ADJ
ejpam-6543	128	8	cover	cover	NOUN
ejpam-6543	128	9	of	of	ADP
ejpam-6543	128	10	x.	x.	NOUN
ejpam-6543	128	11	if	if	SCONJ
ejpam-6543	128	12	no	no	DET
ejpam-6543	128	13	finite	finite	NOUN
ejpam-6543	128	14	subfamily	subfamily	ADV
ejpam-6543	128	15	of	of	ADP
ejpam-6543	128	16	u	u	NOUN
ejpam-6543	128	17	satisfies	satisfy	VERB
ejpam-6543	128	18	the	the	DET
ejpam-6543	128	19	nearly	nearly	ADV
ejpam-6543	128	20	α	α	NUM
ejpam-6543	128	21	-	-	ADJ
ejpam-6543	128	22	compact	compact	ADJ
ejpam-6543	128	23	condition	condition	NOUN
ejpam-6543	128	24	,	,	PUNCT
ejpam-6543	128	25	then	then	ADV
ejpam-6543	128	26	for	for	ADP
ejpam-6543	128	27	every	every	DET
ejpam-6543	128	28	finite	finite	NOUN
ejpam-6543	128	29	subfamily	subfamily	ADV
ejpam-6543	128	30	f	f	PROPN
ejpam-6543	128	31	⊆	⊆	NUM
ejpam-6543	128	32	u	u	NOUN
ejpam-6543	128	33	,	,	PUNCT
ejpam-6543	128	34	the	the	DET
ejpam-6543	128	35	set	set	NOUN
ejpam-6543	128	36	x	x	NOUN
ejpam-6543	128	37	\	\	NOUN
ejpam-6543	128	38	⋃	⋃	PUNCT
ejpam-6543	128	39	f	f	NOUN
ejpam-6543	128	40	can	can	AUX
ejpam-6543	128	41	not	not	PART
ejpam-6543	128	42	be	be	AUX
ejpam-6543	128	43	contained	contain	VERB
ejpam-6543	128	44	in	in	ADP
ejpam-6543	128	45	a	a	DET
ejpam-6543	128	46	finite	finite	ADJ
ejpam-6543	128	47	union	union	NOUN
ejpam-6543	128	48	of	of	ADP
ejpam-6543	128	49	α	α	PROPN
ejpam-6543	128	50	-	-	PUNCT
ejpam-6543	128	51	closed	closed	ADJ
ejpam-6543	128	52	sets	set	NOUN
ejpam-6543	128	53	with	with	ADP
ejpam-6543	128	54	empty	empty	ADJ
ejpam-6543	128	55	α	α	NOUN
ejpam-6543	128	56	-	-	NOUN
ejpam-6543	128	57	interior	interior	ADJ
ejpam-6543	128	58	.	.	PUNCT
ejpam-6543	129	1	consider	consider	VERB
ejpam-6543	129	2	the	the	DET
ejpam-6543	129	3	family	family	NOUN
ejpam-6543	129	4	g	g	NOUN
ejpam-6543	129	5	=	=	PUNCT
ejpam-6543	129	6	{	{	PUNCT
ejpam-6543	129	7	x	x	SYM
ejpam-6543	129	8	\	\	X
ejpam-6543	130	1	⋃	⋃	PROPN
ejpam-6543	130	2	f	f	NOUN
ejpam-6543	130	3	:	:	PUNCT
ejpam-6543	130	4	f	f	PROPN
ejpam-6543	130	5	is	be	AUX
ejpam-6543	130	6	finite	finite	ADJ
ejpam-6543	130	7	,	,	PUNCT
ejpam-6543	130	8	f	f	PROPN
ejpam-6543	130	9	⊆	⊆	NUM
ejpam-6543	130	10	u	u	NOUN
ejpam-6543	130	11	}	}	PUNCT
ejpam-6543	130	12	.	.	PUNCT
ejpam-6543	131	1	each	each	DET
ejpam-6543	131	2	member	member	NOUN
ejpam-6543	131	3	of	of	ADP
ejpam-6543	131	4	g	g	PROPN
ejpam-6543	131	5	is	be	AUX
ejpam-6543	131	6	the	the	DET
ejpam-6543	131	7	intersection	intersection	NOUN
ejpam-6543	131	8	of	of	ADP
ejpam-6543	131	9	finitely	finitely	ADV
ejpam-6543	131	10	many	many	ADJ
ejpam-6543	131	11	α	α	ADJ
ejpam-6543	131	12	-	-	ADJ
ejpam-6543	131	13	closed	closed	ADJ
ejpam-6543	131	14	sets	set	NOUN
ejpam-6543	131	15	,	,	PUNCT
ejpam-6543	131	16	hence	hence	ADV
ejpam-6543	131	17	α	α	ADV
ejpam-6543	131	18	-	-	PUNCT
ejpam-6543	131	19	closed	closed	ADJ
ejpam-6543	131	20	.	.	PUNCT
ejpam-6543	132	1	the	the	DET
ejpam-6543	132	2	family	family	NOUN
ejpam-6543	132	3	g	g	PROPN
ejpam-6543	132	4	has	have	VERB
ejpam-6543	132	5	the	the	DET
ejpam-6543	132	6	finite	finite	PROPN
ejpam-6543	132	7	j.	j.	PROPN
ejpam-6543	132	8	oudetallah	oudetallah	PROPN
ejpam-6543	132	9	et	et	PROPN
ejpam-6543	132	10	al	al	PROPN
ejpam-6543	132	11	.	.	PUNCT
ejpam-6543	132	12	/	/	SYM
ejpam-6543	132	13	eur	eur	PROPN
ejpam-6543	132	14	.	.	PUNCT
ejpam-6543	133	1	j.	j.	PROPN
ejpam-6543	133	2	pure	pure	PROPN
ejpam-6543	133	3	appl	appl	PROPN
ejpam-6543	133	4	.	.	PROPN
ejpam-6543	133	5	math	math	PROPN
ejpam-6543	133	6	,	,	PUNCT
ejpam-6543	133	7	18	18	NUM
ejpam-6543	133	8	(	(	PUNCT
ejpam-6543	133	9	3	3	NUM
ejpam-6543	133	10	)	)	PUNCT
ejpam-6543	133	11	(	(	PUNCT
ejpam-6543	133	12	2025	2025	NUM
ejpam-6543	133	13	)	)	PUNCT
ejpam-6543	133	14	,	,	PUNCT
ejpam-6543	133	15	6543	6543	NUM
ejpam-6543	133	16	6	6	NUM
ejpam-6543	133	17	of	of	ADP
ejpam-6543	133	18	13	13	NUM
ejpam-6543	133	19	intersection	intersection	NOUN
ejpam-6543	133	20	property	property	NOUN
ejpam-6543	133	21	,	,	PUNCT
ejpam-6543	133	22	and	and	CCONJ
ejpam-6543	133	23	by	by	ADP
ejpam-6543	133	24	our	our	PRON
ejpam-6543	133	25	assumption	assumption	NOUN
ejpam-6543	133	26	,	,	PUNCT
ejpam-6543	133	27	some	some	DET
ejpam-6543	133	28	finite	finite	NOUN
ejpam-6543	133	29	subfamily	subfamily	ADV
ejpam-6543	133	30	has	have	VERB
ejpam-6543	133	31	intersection	intersection	NOUN
ejpam-6543	133	32	with	with	ADP
ejpam-6543	133	33	non	non	ADJ
ejpam-6543	133	34	-	-	ADJ
ejpam-6543	133	35	empty	empty	ADJ
ejpam-6543	133	36	α	α	NOUN
ejpam-6543	133	37	-	-	NOUN
ejpam-6543	133	38	interior	interior	ADJ
ejpam-6543	133	39	.	.	PUNCT
ejpam-6543	134	1	this	this	PRON
ejpam-6543	134	2	leads	lead	VERB
ejpam-6543	134	3	to	to	ADP
ejpam-6543	134	4	a	a	DET
ejpam-6543	134	5	contradiction	contradiction	NOUN
ejpam-6543	134	6	with	with	ADP
ejpam-6543	134	7	the	the	DET
ejpam-6543	134	8	covering	covering	NOUN
ejpam-6543	134	9	property	property	NOUN
ejpam-6543	134	10	of	of	ADP
ejpam-6543	134	11	u	u	NOUN
ejpam-6543	134	12	,	,	PUNCT
ejpam-6543	134	13	completing	complete	VERB
ejpam-6543	134	14	the	the	DET
ejpam-6543	134	15	proof	proof	NOUN
ejpam-6543	134	16	.	.	PUNCT
ejpam-6543	135	1	corollary	corollary	ADJ
ejpam-6543	135	2	1	1	NUM
ejpam-6543	135	3	.	.	PUNCT
ejpam-6543	136	1	every	every	DET
ejpam-6543	136	2	nearly	nearly	ADV
ejpam-6543	136	3	α	α	NOUN
ejpam-6543	136	4	-	-	ADJ
ejpam-6543	136	5	compact	compact	ADJ
ejpam-6543	136	6	α	α	NOUN
ejpam-6543	136	7	-	-	PUNCT
ejpam-6543	136	8	t2	t2	ADJ
ejpam-6543	136	9	space	space	NOUN
ejpam-6543	136	10	is	be	AUX
ejpam-6543	136	11	α	α	NOUN
ejpam-6543	136	12	-	-	ADJ
ejpam-6543	136	13	compact	compact	ADJ
ejpam-6543	136	14	.	.	PUNCT
ejpam-6543	137	1	proof	proof	NOUN
ejpam-6543	137	2	.	.	PUNCT
ejpam-6543	138	1	let	let	VERB
ejpam-6543	138	2	(	(	PUNCT
ejpam-6543	138	3	x	x	NOUN
ejpam-6543	138	4	,	,	PUNCT
ejpam-6543	138	5	τ	τ	X
ejpam-6543	138	6	)	)	PUNCT
ejpam-6543	138	7	be	be	VERB
ejpam-6543	138	8	nearly	nearly	ADV
ejpam-6543	138	9	α	α	ADJ
ejpam-6543	138	10	-	-	ADJ
ejpam-6543	138	11	compact	compact	ADJ
ejpam-6543	138	12	and	and	CCONJ
ejpam-6543	138	13	α	α	NOUN
ejpam-6543	138	14	-	-	NOUN
ejpam-6543	138	15	t2	t2	NOUN
ejpam-6543	138	16	.	.	PUNCT
ejpam-6543	139	1	suppose	suppose	VERB
ejpam-6543	139	2	u	u	PRON
ejpam-6543	139	3	is	be	AUX
ejpam-6543	139	4	an	an	DET
ejpam-6543	139	5	α	α	NOUN
ejpam-6543	139	6	-	-	ADJ
ejpam-6543	139	7	open	open	ADJ
ejpam-6543	139	8	cover	cover	NOUN
ejpam-6543	139	9	of	of	ADP
ejpam-6543	139	10	x	x	PUNCT
ejpam-6543	139	11	with	with	ADP
ejpam-6543	139	12	no	no	DET
ejpam-6543	139	13	finite	finite	PROPN
ejpam-6543	139	14	subcover	subcover	PROPN
ejpam-6543	139	15	.	.	PUNCT
ejpam-6543	140	1	then	then	ADV
ejpam-6543	140	2	we	we	PRON
ejpam-6543	140	3	can	can	AUX
ejpam-6543	140	4	construct	construct	VERB
ejpam-6543	140	5	a	a	DET
ejpam-6543	140	6	family	family	NOUN
ejpam-6543	140	7	g	g	NOUN
ejpam-6543	140	8	of	of	ADP
ejpam-6543	140	9	non	non	ADJ
ejpam-6543	140	10	-	-	ADJ
ejpam-6543	140	11	empty	empty	ADJ
ejpam-6543	140	12	α	α	ADJ
ejpam-6543	140	13	-	-	ADJ
ejpam-6543	140	14	open	open	ADJ
ejpam-6543	140	15	sets	set	NOUN
ejpam-6543	140	16	with	with	ADP
ejpam-6543	140	17	finite	finite	ADJ
ejpam-6543	140	18	intersection	intersection	NOUN
ejpam-6543	140	19	property	property	NOUN
ejpam-6543	140	20	.	.	PUNCT
ejpam-6543	141	1	by	by	ADP
ejpam-6543	141	2	theorem	theorem	NOUN
ejpam-6543	141	3	2	2	NUM
ejpam-6543	141	4	,	,	PUNCT
ejpam-6543	141	5	there	there	PRON
ejpam-6543	141	6	exists	exist	VERB
ejpam-6543	141	7	a	a	DET
ejpam-6543	141	8	finite	finite	NOUN
ejpam-6543	141	9	subfamily	subfamily	ADV
ejpam-6543	141	10	whose	whose	DET
ejpam-6543	141	11	intersection	intersection	NOUN
ejpam-6543	141	12	has	have	VERB
ejpam-6543	141	13	non	non	ADJ
ejpam-6543	141	14	-	-	ADJ
ejpam-6543	141	15	empty	empty	ADJ
ejpam-6543	141	16	α	α	NOUN
ejpam-6543	141	17	-	-	NOUN
ejpam-6543	141	18	interior	interior	NOUN
ejpam-6543	141	19	.	.	PUNCT
ejpam-6543	142	1	however	however	ADV
ejpam-6543	142	2	,	,	PUNCT
ejpam-6543	142	3	the	the	DET
ejpam-6543	142	4	α	α	NOUN
ejpam-6543	142	5	-	-	PUNCT
ejpam-6543	142	6	t2	t2	ADJ
ejpam-6543	142	7	property	property	NOUN
ejpam-6543	142	8	ensures	ensure	VERB
ejpam-6543	142	9	that	that	SCONJ
ejpam-6543	142	10	any	any	DET
ejpam-6543	142	11	finite	finite	ADJ
ejpam-6543	142	12	intersection	intersection	NOUN
ejpam-6543	142	13	of	of	ADP
ejpam-6543	142	14	α	α	NOUN
ejpam-6543	142	15	-	-	ADJ
ejpam-6543	142	16	open	open	ADJ
ejpam-6543	142	17	sets	set	NOUN
ejpam-6543	142	18	containing	contain	VERB
ejpam-6543	142	19	two	two	NUM
ejpam-6543	142	20	distinct	distinct	ADJ
ejpam-6543	142	21	points	point	NOUN
ejpam-6543	142	22	must	must	AUX
ejpam-6543	142	23	have	have	VERB
ejpam-6543	142	24	empty	empty	ADJ
ejpam-6543	142	25	α	α	PRON
ejpam-6543	142	26	-	-	NOUN
ejpam-6543	142	27	interior	interior	ADJ
ejpam-6543	142	28	,	,	PUNCT
ejpam-6543	142	29	as	as	SCONJ
ejpam-6543	142	30	the	the	DET
ejpam-6543	142	31	points	point	NOUN
ejpam-6543	142	32	can	can	AUX
ejpam-6543	142	33	be	be	AUX
ejpam-6543	142	34	separated	separate	VERB
ejpam-6543	142	35	by	by	ADP
ejpam-6543	142	36	disjoint	disjoint	NOUN
ejpam-6543	142	37	α	α	VERB
ejpam-6543	142	38	-	-	ADJ
ejpam-6543	142	39	open	open	ADJ
ejpam-6543	142	40	sets	set	NOUN
ejpam-6543	142	41	.	.	PUNCT
ejpam-6543	143	1	this	this	PRON
ejpam-6543	143	2	leads	lead	VERB
ejpam-6543	143	3	to	to	ADP
ejpam-6543	143	4	a	a	DET
ejpam-6543	143	5	contradiction	contradiction	NOUN
ejpam-6543	143	6	with	with	ADP
ejpam-6543	143	7	the	the	DET
ejpam-6543	143	8	cover	cover	NOUN
ejpam-6543	143	9	property	property	NOUN
ejpam-6543	143	10	of	of	ADP
ejpam-6543	143	11	u	u	NOUN
ejpam-6543	143	12	,	,	PUNCT
ejpam-6543	143	13	forcing	force	VERB
ejpam-6543	143	14	u	u	PRON
ejpam-6543	143	15	to	to	PART
ejpam-6543	143	16	have	have	VERB
ejpam-6543	143	17	a	a	DET
ejpam-6543	143	18	finite	finite	ADJ
ejpam-6543	143	19	subcover	subcover	PROPN
ejpam-6543	143	20	.	.	PUNCT
ejpam-6543	144	1	remark	remark	PROPN
ejpam-6543	144	2	2	2	NUM
ejpam-6543	144	3	.	.	PUNCT
ejpam-6543	145	1	the	the	DET
ejpam-6543	145	2	converse	converse	NOUN
ejpam-6543	145	3	of	of	ADP
ejpam-6543	145	4	corollary	corollary	ADJ
ejpam-6543	145	5	1	1	NUM
ejpam-6543	145	6	requires	require	VERB
ejpam-6543	145	7	additional	additional	ADJ
ejpam-6543	145	8	conditions	condition	NOUN
ejpam-6543	145	9	.	.	PUNCT
ejpam-6543	146	1	an	an	DET
ejpam-6543	146	2	α	α	NOUN
ejpam-6543	146	3	-	-	ADJ
ejpam-6543	146	4	compact	compact	ADJ
ejpam-6543	146	5	α	α	NOUN
ejpam-6543	146	6	-	-	PUNCT
ejpam-6543	146	7	t2	t2	ADJ
ejpam-6543	146	8	space	space	NOUN
ejpam-6543	146	9	is	be	AUX
ejpam-6543	146	10	certainly	certainly	ADV
ejpam-6543	146	11	nearly	nearly	ADV
ejpam-6543	146	12	α	α	ADJ
ejpam-6543	146	13	-	-	ADJ
ejpam-6543	146	14	compact	compact	ADJ
ejpam-6543	146	15	(	(	PUNCT
ejpam-6543	146	16	by	by	ADP
ejpam-6543	146	17	theorem	theorem	NOUN
ejpam-6543	146	18	1	1	NUM
ejpam-6543	146	19	)	)	PUNCT
ejpam-6543	146	20	,	,	PUNCT
ejpam-6543	146	21	but	but	CCONJ
ejpam-6543	146	22	a	a	DET
ejpam-6543	146	23	nearly	nearly	ADV
ejpam-6543	146	24	α	α	ADJ
ejpam-6543	146	25	-	-	ADJ
ejpam-6543	146	26	compact	compact	ADJ
ejpam-6543	146	27	space	space	NOUN
ejpam-6543	146	28	need	need	AUX
ejpam-6543	146	29	not	not	PART
ejpam-6543	146	30	be	be	AUX
ejpam-6543	146	31	α	α	NOUN
ejpam-6543	146	32	-	-	NOUN
ejpam-6543	146	33	t2	t2	NOUN
ejpam-6543	146	34	.	.	PUNCT
ejpam-6543	147	1	the	the	DET
ejpam-6543	147	2	separation	separation	NOUN
ejpam-6543	147	3	axiom	axiom	NOUN
ejpam-6543	147	4	is	be	AUX
ejpam-6543	147	5	crucial	crucial	ADJ
ejpam-6543	147	6	for	for	ADP
ejpam-6543	147	7	promoting	promote	VERB
ejpam-6543	147	8	nearly	nearly	ADV
ejpam-6543	147	9	α	α	NOUN
ejpam-6543	147	10	-	-	NOUN
ejpam-6543	147	11	compactness	compactness	NOUN
ejpam-6543	147	12	to	to	ADP
ejpam-6543	147	13	full	full	ADJ
ejpam-6543	147	14	α	α	NOUN
ejpam-6543	147	15	-	-	PUNCT
ejpam-6543	147	16	compactness	compactness	NOUN
ejpam-6543	147	17	.	.	PUNCT
ejpam-6543	148	1	theorem	theorem	NOUN
ejpam-6543	148	2	3	3	X
ejpam-6543	148	3	.	.	PUNCT
ejpam-6543	149	1	let	let	AUX
ejpam-6543	149	2	(	(	PUNCT
ejpam-6543	149	3	x	x	NOUN
ejpam-6543	149	4	,	,	PUNCT
ejpam-6543	149	5	τ	τ	X
ejpam-6543	149	6	)	)	PUNCT
ejpam-6543	149	7	be	be	VERB
ejpam-6543	149	8	a	a	DET
ejpam-6543	149	9	topological	topological	ADJ
ejpam-6543	149	10	space	space	NOUN
ejpam-6543	149	11	.	.	PUNCT
ejpam-6543	150	1	the	the	DET
ejpam-6543	150	2	following	follow	VERB
ejpam-6543	150	3	are	be	AUX
ejpam-6543	150	4	equivalent	equivalent	ADJ
ejpam-6543	150	5	:	:	PUNCT
ejpam-6543	150	6	(	(	PUNCT
ejpam-6543	150	7	i	i	NOUN
ejpam-6543	150	8	)	)	PUNCT
ejpam-6543	150	9	x	x	X
ejpam-6543	150	10	is	be	AUX
ejpam-6543	150	11	nearly	nearly	ADV
ejpam-6543	150	12	α	α	NOUN
ejpam-6543	150	13	-	-	ADJ
ejpam-6543	150	14	compact	compact	ADJ
ejpam-6543	150	15	.	.	PUNCT
ejpam-6543	151	1	(	(	PUNCT
ejpam-6543	151	2	ii	ii	NOUN
ejpam-6543	151	3	)	)	PUNCT
ejpam-6543	151	4	every	every	DET
ejpam-6543	151	5	locally	locally	ADV
ejpam-6543	151	6	finite	finite	ADJ
ejpam-6543	151	7	family	family	NOUN
ejpam-6543	151	8	of	of	ADP
ejpam-6543	151	9	non	non	ADJ
ejpam-6543	151	10	-	-	ADJ
ejpam-6543	151	11	empty	empty	ADJ
ejpam-6543	151	12	α	α	ADJ
ejpam-6543	151	13	-	-	ADJ
ejpam-6543	151	14	open	open	ADJ
ejpam-6543	151	15	sets	set	NOUN
ejpam-6543	151	16	is	be	AUX
ejpam-6543	151	17	finite	finite	ADJ
ejpam-6543	151	18	.	.	PUNCT
ejpam-6543	152	1	(	(	PUNCT
ejpam-6543	152	2	iii	iii	X
ejpam-6543	152	3	)	)	PUNCT
ejpam-6543	152	4	every	every	DET
ejpam-6543	152	5	infinite	infinite	ADJ
ejpam-6543	152	6	discrete	discrete	ADJ
ejpam-6543	152	7	family	family	NOUN
ejpam-6543	152	8	of	of	ADP
ejpam-6543	152	9	non	non	ADJ
ejpam-6543	152	10	-	-	ADJ
ejpam-6543	152	11	empty	empty	ADJ
ejpam-6543	152	12	α	α	ADJ
ejpam-6543	152	13	-	-	ADJ
ejpam-6543	152	14	open	open	ADJ
ejpam-6543	152	15	sets	set	NOUN
ejpam-6543	152	16	has	have	VERB
ejpam-6543	152	17	a	a	DET
ejpam-6543	152	18	cluster	cluster	NOUN
ejpam-6543	152	19	point	point	NOUN
ejpam-6543	152	20	with	with	ADP
ejpam-6543	152	21	respect	respect	NOUN
ejpam-6543	152	22	to	to	ADP
ejpam-6543	152	23	the	the	DET
ejpam-6543	152	24	α	α	NOUN
ejpam-6543	152	25	-	-	NOUN
ejpam-6543	152	26	topology	topology	NOUN
ejpam-6543	152	27	.	.	PUNCT
ejpam-6543	153	1	(	(	PUNCT
ejpam-6543	153	2	iv	iv	X
ejpam-6543	153	3	)	)	PUNCT
ejpam-6543	153	4	every	every	DET
ejpam-6543	153	5	ultrafilter	ultrafilter	NOUN
ejpam-6543	153	6	of	of	ADP
ejpam-6543	153	7	α	α	NOUN
ejpam-6543	153	8	-	-	PUNCT
ejpam-6543	153	9	closed	closed	ADJ
ejpam-6543	153	10	sets	set	NOUN
ejpam-6543	153	11	has	have	VERB
ejpam-6543	153	12	non	non	ADJ
ejpam-6543	153	13	-	-	ADJ
ejpam-6543	153	14	empty	empty	ADJ
ejpam-6543	153	15	intersection	intersection	NOUN
ejpam-6543	153	16	.	.	PUNCT
ejpam-6543	154	1	proof	proof	NOUN
ejpam-6543	154	2	.	.	PUNCT
ejpam-6543	155	1	(	(	PUNCT
ejpam-6543	155	2	1	1	X
ejpam-6543	155	3	)	)	PUNCT
ejpam-6543	155	4	⇒	⇒	NOUN
ejpam-6543	155	5	(	(	PUNCT
ejpam-6543	155	6	2	2	NUM
ejpam-6543	155	7	):	):	PUNCT
ejpam-6543	155	8	suppose	suppose	VERB
ejpam-6543	155	9	x	x	PRON
ejpam-6543	155	10	is	be	AUX
ejpam-6543	155	11	nearly	nearly	ADV
ejpam-6543	155	12	α	α	ADV
ejpam-6543	155	13	-	-	ADJ
ejpam-6543	155	14	compact	compact	ADJ
ejpam-6543	155	15	and	and	CCONJ
ejpam-6543	155	16	let	let	VERB
ejpam-6543	155	17	f	f	NOUN
ejpam-6543	155	18	=	=	PRON
ejpam-6543	155	19	{	{	PUNCT
ejpam-6543	155	20	fi	fi	NOUN
ejpam-6543	155	21	:	:	PUNCT
ejpam-6543	156	1	i	i	PRON
ejpam-6543	156	2	∈	∈	VERB
ejpam-6543	156	3	i	i	PRON
ejpam-6543	156	4	}	}	PUNCT
ejpam-6543	156	5	be	be	VERB
ejpam-6543	156	6	a	a	DET
ejpam-6543	156	7	locally	locally	ADV
ejpam-6543	156	8	finite	finite	ADJ
ejpam-6543	156	9	family	family	NOUN
ejpam-6543	156	10	of	of	ADP
ejpam-6543	156	11	non	non	ADJ
ejpam-6543	156	12	-	-	ADJ
ejpam-6543	156	13	empty	empty	ADJ
ejpam-6543	156	14	α	α	ADJ
ejpam-6543	156	15	-	-	ADJ
ejpam-6543	156	16	open	open	ADJ
ejpam-6543	156	17	sets	set	NOUN
ejpam-6543	156	18	.	.	PUNCT
ejpam-6543	157	1	assume	assume	VERB
ejpam-6543	157	2	f	f	PROPN
ejpam-6543	157	3	is	be	AUX
ejpam-6543	157	4	infinite	infinite	ADJ
ejpam-6543	157	5	.	.	PUNCT
ejpam-6543	158	1	for	for	ADP
ejpam-6543	158	2	each	each	DET
ejpam-6543	158	3	x	x	SYM
ejpam-6543	158	4	∈	∈	PROPN
ejpam-6543	158	5	x	x	X
ejpam-6543	158	6	,	,	PUNCT
ejpam-6543	158	7	there	there	PRON
ejpam-6543	158	8	exists	exist	VERB
ejpam-6543	158	9	an	an	DET
ejpam-6543	158	10	α	α	NOUN
ejpam-6543	158	11	-	-	PUNCT
ejpam-6543	158	12	open	open	ADJ
ejpam-6543	158	13	neighborhood	neighborhood	NOUN
ejpam-6543	158	14	ux	ux	ADV
ejpam-6543	158	15	that	that	PRON
ejpam-6543	158	16	meets	meet	VERB
ejpam-6543	158	17	only	only	ADV
ejpam-6543	158	18	finitely	finitely	ADV
ejpam-6543	158	19	many	many	ADJ
ejpam-6543	158	20	members	member	NOUN
ejpam-6543	158	21	of	of	ADP
ejpam-6543	158	22	f	f	PROPN
ejpam-6543	158	23	.	.	PUNCT
ejpam-6543	159	1	consider	consider	VERB
ejpam-6543	159	2	the	the	DET
ejpam-6543	159	3	cover	cover	NOUN
ejpam-6543	159	4	u	u	NOUN
ejpam-6543	159	5	=	=	PUNCT
ejpam-6543	159	6	{	{	PUNCT
ejpam-6543	159	7	ux	ux	INTJ
ejpam-6543	159	8	:	:	PUNCT
ejpam-6543	159	9	x	x	SYM
ejpam-6543	159	10	∈	∈	NOUN
ejpam-6543	159	11	x	x	SYM
ejpam-6543	159	12	}	}	PUNCT
ejpam-6543	159	13	∪	∪	ADJ
ejpam-6543	159	14	{	{	PUNCT
ejpam-6543	159	15	x	x	SYM
ejpam-6543	159	16	\	\	NOUN
ejpam-6543	159	17	⋃	⋃	ADJ
ejpam-6543	159	18	i∈j	i∈j	NOUN
ejpam-6543	159	19	fi	fi	NOUN
ejpam-6543	159	20	:	:	PUNCT
ejpam-6543	159	21	j	j	PROPN
ejpam-6543	159	22	⊆	⊆	NUM
ejpam-6543	159	23	i	i	PROPN
ejpam-6543	159	24	,	,	PUNCT
ejpam-6543	159	25	|j	|j	VERB
ejpam-6543	159	26	|	|	ADV
ejpam-6543	159	27	<	<	X
ejpam-6543	159	28	∞	∞	NUM
ejpam-6543	159	29	}	}	PUNCT
ejpam-6543	159	30	.	.	PUNCT
ejpam-6543	160	1	by	by	ADP
ejpam-6543	160	2	nearly	nearly	ADV
ejpam-6543	160	3	α	α	NOUN
ejpam-6543	160	4	-	-	NOUN
ejpam-6543	160	5	compactness	compactness	NOUN
ejpam-6543	160	6	,	,	PUNCT
ejpam-6543	160	7	there	there	PRON
ejpam-6543	160	8	exists	exist	VERB
ejpam-6543	160	9	a	a	DET
ejpam-6543	160	10	finite	finite	NOUN
ejpam-6543	160	11	subfamily	subfamily	ADV
ejpam-6543	160	12	that	that	SCONJ
ejpam-6543	160	13	nearly	nearly	ADV
ejpam-6543	160	14	covers	cover	VERB
ejpam-6543	160	15	x.	x.	NOUN
ejpam-6543	161	1	the	the	DET
ejpam-6543	161	2	local	local	ADJ
ejpam-6543	161	3	finiteness	finiteness	NOUN
ejpam-6543	161	4	condition	condition	NOUN
ejpam-6543	161	5	forces	force	VERB
ejpam-6543	161	6	a	a	DET
ejpam-6543	161	7	contradiction	contradiction	NOUN
ejpam-6543	161	8	,	,	PUNCT
ejpam-6543	161	9	showing	show	VERB
ejpam-6543	161	10	f	f	PRON
ejpam-6543	161	11	must	must	AUX
ejpam-6543	161	12	be	be	AUX
ejpam-6543	161	13	finite	finite	ADJ
ejpam-6543	161	14	.	.	PUNCT
ejpam-6543	162	1	(	(	PUNCT
ejpam-6543	162	2	2)⇒	2)⇒	NUM
ejpam-6543	162	3	(	(	PUNCT
ejpam-6543	162	4	3	3	NUM
ejpam-6543	162	5	):	):	PUNCT
ejpam-6543	162	6	let	let	VERB
ejpam-6543	162	7	d	d	NOUN
ejpam-6543	162	8	=	=	PUNCT
ejpam-6543	162	9	{	{	PUNCT
ejpam-6543	162	10	di	di	NOUN
ejpam-6543	162	11	:	:	PUNCT
ejpam-6543	162	12	i	i	PRON
ejpam-6543	162	13	∈	∈	VERB
ejpam-6543	162	14	i	i	PRON
ejpam-6543	162	15	}	}	PUNCT
ejpam-6543	162	16	be	be	AUX
ejpam-6543	162	17	an	an	DET
ejpam-6543	162	18	infinite	infinite	ADJ
ejpam-6543	162	19	discrete	discrete	ADJ
ejpam-6543	162	20	family	family	NOUN
ejpam-6543	162	21	of	of	ADP
ejpam-6543	162	22	non	non	ADJ
ejpam-6543	162	23	-	-	ADJ
ejpam-6543	162	24	empty	empty	ADJ
ejpam-6543	162	25	α	α	ADJ
ejpam-6543	162	26	-	-	ADJ
ejpam-6543	162	27	open	open	ADJ
ejpam-6543	162	28	sets	set	NOUN
ejpam-6543	162	29	.	.	PUNCT
ejpam-6543	163	1	the	the	DET
ejpam-6543	163	2	discreteness	discreteness	NOUN
ejpam-6543	163	3	implies	imply	VERB
ejpam-6543	163	4	that	that	SCONJ
ejpam-6543	163	5	for	for	ADP
ejpam-6543	163	6	each	each	DET
ejpam-6543	163	7	point	point	NOUN
ejpam-6543	163	8	x	x	X
ejpam-6543	163	9	∈	∈	NOUN
ejpam-6543	163	10	x	x	NOUN
ejpam-6543	163	11	,	,	PUNCT
ejpam-6543	163	12	there	there	PRON
ejpam-6543	163	13	exists	exist	VERB
ejpam-6543	163	14	an	an	DET
ejpam-6543	163	15	α	α	NOUN
ejpam-6543	163	16	-	-	PUNCT
ejpam-6543	163	17	open	open	ADJ
ejpam-6543	163	18	neighborhood	neighborhood	NOUN
ejpam-6543	163	19	that	that	PRON
ejpam-6543	163	20	meets	meet	VERB
ejpam-6543	163	21	at	at	ADP
ejpam-6543	163	22	most	most	ADV
ejpam-6543	163	23	one	one	NUM
ejpam-6543	163	24	member	member	NOUN
ejpam-6543	163	25	of	of	ADP
ejpam-6543	163	26	d.	d.	PROPN
ejpam-6543	163	27	this	this	PRON
ejpam-6543	163	28	makes	make	VERB
ejpam-6543	163	29	d	d	NOUN
ejpam-6543	163	30	locally	locally	ADV
ejpam-6543	163	31	finite	finite	ADJ
ejpam-6543	163	32	,	,	PUNCT
ejpam-6543	163	33	contradicting	contradict	VERB
ejpam-6543	163	34	condition	condition	NOUN
ejpam-6543	163	35	(	(	PUNCT
ejpam-6543	163	36	2	2	NUM
ejpam-6543	163	37	)	)	PUNCT
ejpam-6543	163	38	unless	unless	SCONJ
ejpam-6543	163	39	d	d	PROPN
ejpam-6543	163	40	has	have	VERB
ejpam-6543	163	41	a	a	DET
ejpam-6543	163	42	cluster	cluster	NOUN
ejpam-6543	163	43	point	point	NOUN
ejpam-6543	163	44	.	.	PUNCT
ejpam-6543	164	1	(	(	PUNCT
ejpam-6543	164	2	3	3	X
ejpam-6543	164	3	)	)	PUNCT
ejpam-6543	164	4	⇒	⇒	NOUN
ejpam-6543	164	5	(	(	PUNCT
ejpam-6543	164	6	4	4	NUM
ejpam-6543	164	7	):	):	PUNCT
ejpam-6543	164	8	suppose	suppose	VERB
ejpam-6543	164	9	every	every	DET
ejpam-6543	164	10	infinite	infinite	ADJ
ejpam-6543	164	11	discrete	discrete	ADJ
ejpam-6543	164	12	family	family	NOUN
ejpam-6543	164	13	has	have	VERB
ejpam-6543	164	14	a	a	DET
ejpam-6543	164	15	cluster	cluster	NOUN
ejpam-6543	164	16	point	point	NOUN
ejpam-6543	164	17	.	.	PUNCT
ejpam-6543	165	1	let	let	VERB
ejpam-6543	165	2	u	u	PRON
ejpam-6543	165	3	be	be	AUX
ejpam-6543	165	4	an	an	DET
ejpam-6543	165	5	ultrafilter	ultrafilter	NOUN
ejpam-6543	165	6	of	of	ADP
ejpam-6543	165	7	α	α	NOUN
ejpam-6543	165	8	-	-	PUNCT
ejpam-6543	165	9	closed	closed	ADJ
ejpam-6543	165	10	sets	set	NOUN
ejpam-6543	165	11	.	.	PUNCT
ejpam-6543	166	1	if	if	SCONJ
ejpam-6543	166	2	⋂	⋂	PROPN
ejpam-6543	166	3	u	u	NOUN
ejpam-6543	166	4	=	=	NOUN
ejpam-6543	166	5	∅	∅	NOUN
ejpam-6543	166	6	,	,	PUNCT
ejpam-6543	166	7	then	then	ADV
ejpam-6543	166	8	the	the	DET
ejpam-6543	166	9	complements	complement	NOUN
ejpam-6543	166	10	form	form	VERB
ejpam-6543	166	11	a	a	DET
ejpam-6543	166	12	family	family	NOUN
ejpam-6543	166	13	of	of	ADP
ejpam-6543	166	14	α	α	NOUN
ejpam-6543	166	15	-	-	ADJ
ejpam-6543	166	16	open	open	ADJ
ejpam-6543	166	17	sets	set	NOUN
ejpam-6543	166	18	j.	j.	PROPN
ejpam-6543	166	19	oudetallah	oudetallah	PROPN
ejpam-6543	166	20	et	et	PROPN
ejpam-6543	166	21	al	al	PROPN
ejpam-6543	166	22	.	.	PUNCT
ejpam-6543	166	23	/	/	SYM
ejpam-6543	166	24	eur	eur	PROPN
ejpam-6543	166	25	.	.	PUNCT
ejpam-6543	167	1	j.	j.	PROPN
ejpam-6543	167	2	pure	pure	PROPN
ejpam-6543	167	3	appl	appl	PROPN
ejpam-6543	167	4	.	.	PROPN
ejpam-6543	167	5	math	math	PROPN
ejpam-6543	167	6	,	,	PUNCT
ejpam-6543	167	7	18	18	NUM
ejpam-6543	167	8	(	(	PUNCT
ejpam-6543	167	9	3	3	NUM
ejpam-6543	167	10	)	)	PUNCT
ejpam-6543	167	11	(	(	PUNCT
ejpam-6543	167	12	2025	2025	NUM
ejpam-6543	167	13	)	)	PUNCT
ejpam-6543	167	14	,	,	PUNCT
ejpam-6543	167	15	6543	6543	NUM
ejpam-6543	167	16	7	7	NUM
ejpam-6543	167	17	of	of	ADP
ejpam-6543	167	18	13	13	NUM
ejpam-6543	167	19	with	with	ADP
ejpam-6543	167	20	no	no	DET
ejpam-6543	167	21	finite	finite	ADJ
ejpam-6543	167	22	intersection	intersection	NOUN
ejpam-6543	167	23	property	property	NOUN
ejpam-6543	167	24	.	.	PUNCT
ejpam-6543	168	1	this	this	PRON
ejpam-6543	168	2	leads	lead	VERB
ejpam-6543	168	3	to	to	ADP
ejpam-6543	168	4	a	a	DET
ejpam-6543	168	5	discrete	discrete	ADJ
ejpam-6543	168	6	family	family	NOUN
ejpam-6543	168	7	with	with	ADP
ejpam-6543	168	8	no	no	DET
ejpam-6543	168	9	cluster	cluster	NOUN
ejpam-6543	168	10	point	point	NOUN
ejpam-6543	168	11	,	,	PUNCT
ejpam-6543	168	12	contradicting	contradict	VERB
ejpam-6543	168	13	condition	condition	NOUN
ejpam-6543	168	14	(	(	PUNCT
ejpam-6543	168	15	3	3	NUM
ejpam-6543	168	16	)	)	PUNCT
ejpam-6543	168	17	.	.	PUNCT
ejpam-6543	169	1	(	(	PUNCT
ejpam-6543	169	2	4	4	X
ejpam-6543	169	3	)	)	PUNCT
ejpam-6543	169	4	⇒	⇒	NOUN
ejpam-6543	169	5	(	(	PUNCT
ejpam-6543	169	6	1	1	NUM
ejpam-6543	169	7	):	):	PUNCT
ejpam-6543	169	8	suppose	suppose	VERB
ejpam-6543	169	9	condition	condition	NOUN
ejpam-6543	169	10	(	(	PUNCT
ejpam-6543	169	11	4	4	X
ejpam-6543	169	12	)	)	PUNCT
ejpam-6543	169	13	holds	hold	VERB
ejpam-6543	169	14	.	.	PUNCT
ejpam-6543	170	1	let	let	VERB
ejpam-6543	170	2	u	u	PRON
ejpam-6543	170	3	be	be	AUX
ejpam-6543	170	4	any	any	DET
ejpam-6543	170	5	α	α	NOUN
ejpam-6543	170	6	-	-	ADJ
ejpam-6543	170	7	open	open	ADJ
ejpam-6543	170	8	cover	cover	NOUN
ejpam-6543	170	9	of	of	ADP
ejpam-6543	170	10	x.	x.	NOUN
ejpam-6543	170	11	the	the	DET
ejpam-6543	170	12	family	family	NOUN
ejpam-6543	170	13	of	of	ADP
ejpam-6543	170	14	complements	complement	NOUN
ejpam-6543	170	15	{	{	PUNCT
ejpam-6543	170	16	x	x	SYM
ejpam-6543	170	17	\	\	NOUN
ejpam-6543	170	18	u	u	NOUN
ejpam-6543	170	19	:	:	PUNCT
ejpam-6543	170	20	u	u	PROPN
ejpam-6543	170	21	∈	∈	PROPN
ejpam-6543	170	22	u	u	NOUN
ejpam-6543	170	23	}	}	PUNCT
ejpam-6543	170	24	consists	consist	VERB
ejpam-6543	170	25	of	of	ADP
ejpam-6543	170	26	α	α	NOUN
ejpam-6543	170	27	-	-	PUNCT
ejpam-6543	170	28	closed	closed	ADJ
ejpam-6543	170	29	sets	set	NOUN
ejpam-6543	170	30	with	with	ADP
ejpam-6543	170	31	empty	empty	ADJ
ejpam-6543	170	32	intersection	intersection	NOUN
ejpam-6543	170	33	.	.	PUNCT
ejpam-6543	171	1	by	by	ADP
ejpam-6543	171	2	condition	condition	NOUN
ejpam-6543	171	3	(	(	PUNCT
ejpam-6543	171	4	4	4	NUM
ejpam-6543	171	5	)	)	PUNCT
ejpam-6543	171	6	,	,	PUNCT
ejpam-6543	171	7	this	this	DET
ejpam-6543	171	8	family	family	NOUN
ejpam-6543	171	9	can	can	AUX
ejpam-6543	171	10	not	not	PART
ejpam-6543	171	11	be	be	AUX
ejpam-6543	171	12	extended	extend	VERB
ejpam-6543	171	13	to	to	ADP
ejpam-6543	171	14	an	an	DET
ejpam-6543	171	15	ultrafilter	ultrafilter	NOUN
ejpam-6543	171	16	,	,	PUNCT
ejpam-6543	171	17	which	which	PRON
ejpam-6543	171	18	forces	force	VERB
ejpam-6543	171	19	the	the	DET
ejpam-6543	171	20	existence	existence	NOUN
ejpam-6543	171	21	of	of	ADP
ejpam-6543	171	22	a	a	DET
ejpam-6543	171	23	finite	finite	NOUN
ejpam-6543	171	24	subfamily	subfamily	ADV
ejpam-6543	171	25	of	of	ADP
ejpam-6543	171	26	u	u	NOUN
ejpam-6543	171	27	satisfying	satisfy	VERB
ejpam-6543	171	28	the	the	DET
ejpam-6543	171	29	nearly	nearly	ADV
ejpam-6543	171	30	α	α	NUM
ejpam-6543	171	31	-	-	ADJ
ejpam-6543	171	32	compact	compact	ADJ
ejpam-6543	171	33	condition	condition	NOUN
ejpam-6543	171	34	.	.	PUNCT
ejpam-6543	172	1	4	4	X
ejpam-6543	172	2	.	.	X
ejpam-6543	172	3	properties	property	NOUN
ejpam-6543	172	4	and	and	CCONJ
ejpam-6543	172	5	characterizations	characterization	NOUN
ejpam-6543	172	6	proposition	proposition	NOUN
ejpam-6543	172	7	1	1	NUM
ejpam-6543	172	8	.	.	PUNCT
ejpam-6543	173	1	the	the	DET
ejpam-6543	173	2	property	property	NOUN
ejpam-6543	173	3	of	of	ADP
ejpam-6543	173	4	being	be	AUX
ejpam-6543	173	5	nearly	nearly	ADV
ejpam-6543	173	6	α	α	ADV
ejpam-6543	173	7	-	-	ADJ
ejpam-6543	173	8	compact	compact	ADJ
ejpam-6543	173	9	is	be	AUX
ejpam-6543	173	10	preserved	preserve	VERB
ejpam-6543	173	11	under	under	ADP
ejpam-6543	173	12	α	α	ADJ
ejpam-6543	173	13	-	-	ADJ
ejpam-6543	173	14	continuous	continuous	ADJ
ejpam-6543	173	15	surjections	surjection	NOUN
ejpam-6543	173	16	.	.	PUNCT
ejpam-6543	174	1	proof	proof	NOUN
ejpam-6543	174	2	.	.	PUNCT
ejpam-6543	175	1	let	let	VERB
ejpam-6543	175	2	f	f	NOUN
ejpam-6543	175	3	:	:	PUNCT
ejpam-6543	175	4	x	x	X
ejpam-6543	175	5	→	→	SYM
ejpam-6543	175	6	y	y	X
ejpam-6543	175	7	be	be	AUX
ejpam-6543	175	8	an	an	DET
ejpam-6543	175	9	α	α	NOUN
ejpam-6543	175	10	-	-	ADJ
ejpam-6543	175	11	continuous	continuous	ADJ
ejpam-6543	175	12	surjection	surjection	NOUN
ejpam-6543	175	13	where	where	SCONJ
ejpam-6543	175	14	x	x	PRON
ejpam-6543	175	15	is	be	AUX
ejpam-6543	175	16	nearly	nearly	ADV
ejpam-6543	175	17	α	α	NOUN
ejpam-6543	175	18	-	-	ADJ
ejpam-6543	175	19	compact	compact	ADJ
ejpam-6543	175	20	.	.	PUNCT
ejpam-6543	176	1	let	let	VERB
ejpam-6543	176	2	v	v	PART
ejpam-6543	176	3	be	be	AUX
ejpam-6543	176	4	any	any	DET
ejpam-6543	176	5	α	α	NOUN
ejpam-6543	176	6	-	-	ADJ
ejpam-6543	176	7	open	open	ADJ
ejpam-6543	176	8	cover	cover	NOUN
ejpam-6543	176	9	of	of	ADP
ejpam-6543	176	10	y	y	PROPN
ejpam-6543	176	11	.	.	PUNCT
ejpam-6543	177	1	then	then	ADV
ejpam-6543	177	2	u	u	X
ejpam-6543	177	3	=	=	PUNCT
ejpam-6543	177	4	{	{	PUNCT
ejpam-6543	177	5	f−1(v	f−1(v	PROPN
ejpam-6543	177	6	)	)	PUNCT
ejpam-6543	177	7	:	:	PUNCT
ejpam-6543	177	8	v	v	X
ejpam-6543	177	9	∈	∈	PROPN
ejpam-6543	177	10	v	v	NOUN
ejpam-6543	177	11	}	}	PUNCT
ejpam-6543	177	12	is	be	AUX
ejpam-6543	177	13	an	an	DET
ejpam-6543	177	14	α	α	NOUN
ejpam-6543	177	15	-	-	ADJ
ejpam-6543	177	16	open	open	ADJ
ejpam-6543	177	17	cover	cover	NOUN
ejpam-6543	177	18	of	of	ADP
ejpam-6543	177	19	x	x	PUNCT
ejpam-6543	177	20	by	by	ADP
ejpam-6543	177	21	the	the	DET
ejpam-6543	177	22	α	α	NOUN
ejpam-6543	177	23	-	-	PUNCT
ejpam-6543	177	24	continuity	continuity	NOUN
ejpam-6543	177	25	of	of	ADP
ejpam-6543	177	26	f	f	PROPN
ejpam-6543	177	27	.	.	PUNCT
ejpam-6543	178	1	since	since	SCONJ
ejpam-6543	178	2	x	x	PRON
ejpam-6543	178	3	is	be	AUX
ejpam-6543	178	4	nearly	nearly	ADV
ejpam-6543	178	5	α	α	NOUN
ejpam-6543	178	6	-	-	ADJ
ejpam-6543	178	7	compact	compact	ADJ
ejpam-6543	178	8	,	,	PUNCT
ejpam-6543	178	9	there	there	PRON
ejpam-6543	178	10	exists	exist	VERB
ejpam-6543	178	11	a	a	DET
ejpam-6543	178	12	finite	finite	NOUN
ejpam-6543	178	13	subfamily	subfamily	ADV
ejpam-6543	178	14	f	f	PROPN
ejpam-6543	178	15	⊆	⊆	NUM
ejpam-6543	178	16	u	u	NOUN
ejpam-6543	178	17	such	such	ADJ
ejpam-6543	178	18	that	that	SCONJ
ejpam-6543	178	19	x	x	SYM
ejpam-6543	178	20	\	\	NOUN
ejpam-6543	178	21	⋃	⋃	PUNCT
ejpam-6543	178	22	f	f	PROPN
ejpam-6543	178	23	is	be	AUX
ejpam-6543	178	24	contained	contain	VERB
ejpam-6543	178	25	in	in	ADP
ejpam-6543	178	26	a	a	DET
ejpam-6543	178	27	finite	finite	ADJ
ejpam-6543	178	28	union	union	NOUN
ejpam-6543	178	29	of	of	ADP
ejpam-6543	178	30	α	α	PROPN
ejpam-6543	178	31	-	-	PUNCT
ejpam-6543	178	32	closed	closed	ADJ
ejpam-6543	178	33	sets	set	NOUN
ejpam-6543	178	34	with	with	ADP
ejpam-6543	178	35	empty	empty	ADJ
ejpam-6543	178	36	α	α	NOUN
ejpam-6543	178	37	-	-	NOUN
ejpam-6543	178	38	interior	interior	ADJ
ejpam-6543	178	39	.	.	PUNCT
ejpam-6543	179	1	let	let	VERB
ejpam-6543	179	2	g	g	NOUN
ejpam-6543	179	3	=	=	PUNCT
ejpam-6543	179	4	{	{	PUNCT
ejpam-6543	179	5	v	v	NUM
ejpam-6543	179	6	∈	∈	NOUN
ejpam-6543	179	7	v	v	NOUN
ejpam-6543	179	8	:	:	PUNCT
ejpam-6543	179	9	f−1(v	f−1(v	PROPN
ejpam-6543	179	10	)	)	PUNCT
ejpam-6543	179	11	∈	∈	PROPN
ejpam-6543	180	1	f	f	X
ejpam-6543	180	2	}	}	PUNCT
ejpam-6543	180	3	.	.	PUNCT
ejpam-6543	181	1	then	then	ADV
ejpam-6543	181	2	g	g	PROPN
ejpam-6543	181	3	is	be	AUX
ejpam-6543	181	4	finite	finite	ADJ
ejpam-6543	181	5	and	and	CCONJ
ejpam-6543	181	6	f(x	f(x	PROPN
ejpam-6543	181	7	\	\	PUNCT
ejpam-6543	182	1	⋃	⋃	PROPN
ejpam-6543	182	2	f	f	X
ejpam-6543	182	3	)	)	PUNCT
ejpam-6543	182	4	⊆	⊆	NUM
ejpam-6543	182	5	y	y	NOUN
ejpam-6543	182	6	\	\	PUNCT
ejpam-6543	182	7	⋃	⋃	PUNCT
ejpam-6543	182	8	g.	g.	NOUN
ejpam-6543	182	9	the	the	DET
ejpam-6543	182	10	surjectivity	surjectivity	NOUN
ejpam-6543	182	11	of	of	ADP
ejpam-6543	182	12	f	f	PROPN
ejpam-6543	182	13	and	and	CCONJ
ejpam-6543	182	14	properties	property	NOUN
ejpam-6543	182	15	of	of	ADP
ejpam-6543	182	16	α	α	NOUN
ejpam-6543	182	17	-	-	ADJ
ejpam-6543	182	18	continuous	continuous	ADJ
ejpam-6543	182	19	functions	function	NOUN
ejpam-6543	182	20	ensure	ensure	VERB
ejpam-6543	182	21	that	that	SCONJ
ejpam-6543	182	22	y	y	PRON
ejpam-6543	182	23	\	\	PUNCT
ejpam-6543	182	24	⋃	⋃	SCONJ
ejpam-6543	182	25	g	g	NOUN
ejpam-6543	182	26	satisfies	satisfy	VERB
ejpam-6543	182	27	the	the	DET
ejpam-6543	182	28	nearly	nearly	ADV
ejpam-6543	182	29	α	α	ADJ
ejpam-6543	182	30	-	-	ADJ
ejpam-6543	182	31	compact	compact	ADJ
ejpam-6543	182	32	condition	condition	NOUN
ejpam-6543	182	33	for	for	ADP
ejpam-6543	182	34	y	y	PROPN
ejpam-6543	182	35	.	.	PUNCT
ejpam-6543	183	1	theorem	theorem	ADJ
ejpam-6543	183	2	4	4	NUM
ejpam-6543	183	3	.	.	PUNCT
ejpam-6543	184	1	let	let	VERB
ejpam-6543	184	2	(	(	PUNCT
ejpam-6543	184	3	x	x	NOUN
ejpam-6543	184	4	,	,	PUNCT
ejpam-6543	184	5	τ	τ	X
ejpam-6543	184	6	)	)	PUNCT
ejpam-6543	184	7	be	be	VERB
ejpam-6543	184	8	nearly	nearly	ADV
ejpam-6543	184	9	α	α	NOUN
ejpam-6543	184	10	-	-	ADJ
ejpam-6543	184	11	compact	compact	ADJ
ejpam-6543	184	12	.	.	PUNCT
ejpam-6543	185	1	then	then	ADV
ejpam-6543	185	2	every	every	DET
ejpam-6543	185	3	infinite	infinite	ADJ
ejpam-6543	185	4	subset	subset	NOUN
ejpam-6543	185	5	of	of	ADP
ejpam-6543	185	6	x	x	PUNCT
ejpam-6543	185	7	has	have	VERB
ejpam-6543	185	8	an	an	DET
ejpam-6543	185	9	α	α	NOUN
ejpam-6543	185	10	-	-	PUNCT
ejpam-6543	185	11	accumulation	accumulation	NOUN
ejpam-6543	185	12	point	point	NOUN
ejpam-6543	185	13	.	.	PUNCT
ejpam-6543	186	1	proof	proof	NOUN
ejpam-6543	186	2	.	.	PUNCT
ejpam-6543	187	1	let	let	VERB
ejpam-6543	187	2	a	a	PRON
ejpam-6543	187	3	=	=	X
ejpam-6543	187	4	{	{	PUNCT
ejpam-6543	187	5	an	an	NOUN
ejpam-6543	187	6	:	:	PUNCT
ejpam-6543	187	7	n	n	CCONJ
ejpam-6543	187	8	∈	∈	PROPN
ejpam-6543	187	9	n	n	CCONJ
ejpam-6543	187	10	}	}	PUNCT
ejpam-6543	187	11	be	be	AUX
ejpam-6543	187	12	an	an	DET
ejpam-6543	187	13	infinite	infinite	ADJ
ejpam-6543	187	14	subset	subset	NOUN
ejpam-6543	187	15	of	of	ADP
ejpam-6543	187	16	x.	x.	PROPN
ejpam-6543	187	17	suppose	suppose	VERB
ejpam-6543	187	18	,	,	PUNCT
ejpam-6543	187	19	for	for	ADP
ejpam-6543	187	20	contradiction	contradiction	NOUN
ejpam-6543	187	21	,	,	PUNCT
ejpam-6543	187	22	that	that	SCONJ
ejpam-6543	187	23	a	a	PRON
ejpam-6543	187	24	has	have	VERB
ejpam-6543	187	25	no	no	DET
ejpam-6543	187	26	α	α	NOUN
ejpam-6543	187	27	-	-	PUNCT
ejpam-6543	187	28	accumulation	accumulation	NOUN
ejpam-6543	187	29	points	point	NOUN
ejpam-6543	187	30	.	.	PUNCT
ejpam-6543	188	1	then	then	ADV
ejpam-6543	188	2	for	for	ADP
ejpam-6543	188	3	each	each	DET
ejpam-6543	188	4	x	x	SYM
ejpam-6543	188	5	∈	∈	PROPN
ejpam-6543	188	6	x	x	X
ejpam-6543	188	7	,	,	PUNCT
ejpam-6543	188	8	there	there	PRON
ejpam-6543	188	9	exists	exist	VERB
ejpam-6543	188	10	an	an	DET
ejpam-6543	188	11	α	α	NOUN
ejpam-6543	188	12	-	-	PUNCT
ejpam-6543	188	13	open	open	ADJ
ejpam-6543	188	14	neighborhood	neighborhood	NOUN
ejpam-6543	188	15	ux	ux	ADP
ejpam-6543	188	16	such	such	ADJ
ejpam-6543	188	17	that	that	PRON
ejpam-6543	188	18	ux	ux	PROPN
ejpam-6543	188	19	∩a	∩a	PROPN
ejpam-6543	188	20	is	be	AUX
ejpam-6543	188	21	finite	finite	ADJ
ejpam-6543	188	22	.	.	PUNCT
ejpam-6543	189	1	consider	consider	VERB
ejpam-6543	189	2	the	the	DET
ejpam-6543	189	3	family	family	NOUN
ejpam-6543	189	4	u	u	NOUN
ejpam-6543	189	5	=	=	PUNCT
ejpam-6543	189	6	{	{	PUNCT
ejpam-6543	189	7	ux	ux	INTJ
ejpam-6543	189	8	:	:	PUNCT
ejpam-6543	189	9	x	x	SYM
ejpam-6543	189	10	∈	∈	NOUN
ejpam-6543	189	11	x	x	X
ejpam-6543	189	12	}	}	PUNCT
ejpam-6543	189	13	.	.	PUNCT
ejpam-6543	190	1	this	this	PRON
ejpam-6543	190	2	is	be	AUX
ejpam-6543	190	3	an	an	DET
ejpam-6543	190	4	α	α	NOUN
ejpam-6543	190	5	-	-	ADJ
ejpam-6543	190	6	open	open	ADJ
ejpam-6543	190	7	cover	cover	NOUN
ejpam-6543	190	8	of	of	ADP
ejpam-6543	190	9	x.	x.	NOUN
ejpam-6543	190	10	by	by	ADP
ejpam-6543	190	11	nearly	nearly	ADV
ejpam-6543	190	12	α	α	NOUN
ejpam-6543	190	13	-	-	NOUN
ejpam-6543	190	14	compactness	compactness	NOUN
ejpam-6543	190	15	,	,	PUNCT
ejpam-6543	190	16	there	there	PRON
ejpam-6543	190	17	exists	exist	VERB
ejpam-6543	190	18	a	a	DET
ejpam-6543	190	19	finite	finite	NOUN
ejpam-6543	190	20	subfamily	subfamily	ADV
ejpam-6543	190	21	{	{	PUNCT
ejpam-6543	190	22	ux1	ux1	NOUN
ejpam-6543	190	23	,	,	PUNCT
ejpam-6543	190	24	.	.	PUNCT
ejpam-6543	190	25	.	.	PUNCT
ejpam-6543	191	1	.	.	PUNCT
ejpam-6543	192	1	,	,	PUNCT
ejpam-6543	192	2	uxk	uxk	VERB
ejpam-6543	192	3	}	}	PUNCT
ejpam-6543	192	4	such	such	ADJ
ejpam-6543	192	5	that	that	SCONJ
ejpam-6543	192	6	x	x	SYM
ejpam-6543	192	7	\	\	X
ejpam-6543	192	8	⋃k	⋃k	NOUN
ejpam-6543	192	9	i=1	i=1	PROPN
ejpam-6543	192	10	uxi	uxi	NOUN
ejpam-6543	192	11	is	be	AUX
ejpam-6543	192	12	contained	contain	VERB
ejpam-6543	192	13	in	in	ADP
ejpam-6543	192	14	a	a	DET
ejpam-6543	192	15	finite	finite	ADJ
ejpam-6543	192	16	union	union	NOUN
ejpam-6543	192	17	of	of	ADP
ejpam-6543	192	18	α	α	PROPN
ejpam-6543	192	19	-	-	PUNCT
ejpam-6543	192	20	closed	closed	ADJ
ejpam-6543	192	21	sets	set	NOUN
ejpam-6543	192	22	with	with	ADP
ejpam-6543	192	23	empty	empty	ADJ
ejpam-6543	192	24	α	α	NOUN
ejpam-6543	192	25	-	-	NOUN
ejpam-6543	192	26	interior	interior	ADJ
ejpam-6543	192	27	.	.	PUNCT
ejpam-6543	193	1	since	since	SCONJ
ejpam-6543	193	2	each	each	DET
ejpam-6543	193	3	uxi	uxi	NOUN
ejpam-6543	193	4	∩	∩	NOUN
ejpam-6543	193	5	a	a	PRON
ejpam-6543	193	6	is	be	AUX
ejpam-6543	193	7	finite	finite	ADJ
ejpam-6543	193	8	,	,	PUNCT
ejpam-6543	193	9	the	the	DET
ejpam-6543	193	10	set	set	NOUN
ejpam-6543	193	11	a	a	DET
ejpam-6543	193	12	∩	∩	NOUN
ejpam-6543	193	13	⋃k	⋃k	NOUN
ejpam-6543	193	14	i=1	i=1	PRON
ejpam-6543	193	15	uxi	uxi	NOUN
ejpam-6543	193	16	is	be	AUX
ejpam-6543	193	17	finite	finite	ADJ
ejpam-6543	193	18	.	.	PUNCT
ejpam-6543	194	1	the	the	DET
ejpam-6543	194	2	remaining	remain	VERB
ejpam-6543	194	3	points	point	NOUN
ejpam-6543	194	4	of	of	ADP
ejpam-6543	194	5	a	a	DET
ejpam-6543	194	6	lie	lie	NOUN
ejpam-6543	194	7	in	in	ADP
ejpam-6543	194	8	x	x	X
ejpam-6543	194	9	\	\	X
ejpam-6543	194	10	⋃k	⋃k	NOUN
ejpam-6543	194	11	i=1	i=1	PROPN
ejpam-6543	194	12	uxi	uxi	NOUN
ejpam-6543	194	13	,	,	PUNCT
ejpam-6543	194	14	but	but	CCONJ
ejpam-6543	194	15	this	this	PRON
ejpam-6543	194	16	contradicts	contradict	VERB
ejpam-6543	194	17	the	the	DET
ejpam-6543	194	18	structure	structure	NOUN
ejpam-6543	194	19	of	of	ADP
ejpam-6543	194	20	a	a	PRON
ejpam-6543	194	21	and	and	CCONJ
ejpam-6543	194	22	the	the	DET
ejpam-6543	194	23	nearly	nearly	ADV
ejpam-6543	194	24	α	α	ADJ
ejpam-6543	194	25	-	-	ADJ
ejpam-6543	194	26	compact	compact	ADJ
ejpam-6543	194	27	condition	condition	NOUN
ejpam-6543	194	28	.	.	PUNCT
ejpam-6543	195	1	therefore	therefore	ADV
ejpam-6543	195	2	,	,	PUNCT
ejpam-6543	195	3	a	a	PRON
ejpam-6543	195	4	must	must	AUX
ejpam-6543	195	5	have	have	VERB
ejpam-6543	195	6	an	an	DET
ejpam-6543	195	7	α	α	NOUN
ejpam-6543	195	8	-	-	PUNCT
ejpam-6543	195	9	accumulation	accumulation	NOUN
ejpam-6543	195	10	point	point	NOUN
ejpam-6543	195	11	.	.	PUNCT
ejpam-6543	195	12	example	example	NOUN
ejpam-6543	196	1	4	4	NUM
ejpam-6543	196	2	.	.	PUNCT
ejpam-6543	196	3	consider	consider	VERB
ejpam-6543	196	4	the	the	DET
ejpam-6543	196	5	space	space	NOUN
ejpam-6543	196	6	y	y	NOUN
ejpam-6543	196	7	=	=	PUNCT
ejpam-6543	197	1	[	[	X
ejpam-6543	197	2	0	0	NUM
ejpam-6543	197	3	,	,	PUNCT
ejpam-6543	197	4	1	1	NUM
ejpam-6543	197	5	]	]	PUNCT
ejpam-6543	197	6	∪	∪	X
ejpam-6543	197	7	{	{	PUNCT
ejpam-6543	197	8	2	2	NUM
ejpam-6543	197	9	}	}	PUNCT
ejpam-6543	197	10	with	with	ADP
ejpam-6543	197	11	the	the	DET
ejpam-6543	197	12	topology	topology	NOUN
ejpam-6543	197	13	τ	τ	X
ejpam-6543	197	14	=	=	PUNCT
ejpam-6543	197	15	{	{	PUNCT
ejpam-6543	197	16	u	u	NOUN
ejpam-6543	197	17	∩	∩	NOUN
ejpam-6543	197	18	y	y	NOUN
ejpam-6543	197	19	:	:	PUNCT
ejpam-6543	197	20	u	u	NOUN
ejpam-6543	197	21	is	be	AUX
ejpam-6543	197	22	open	open	ADJ
ejpam-6543	197	23	in	in	ADP
ejpam-6543	197	24	r	r	NOUN
ejpam-6543	197	25	}	}	PUNCT
ejpam-6543	197	26	.	.	PUNCT
ejpam-6543	198	1	this	this	DET
ejpam-6543	198	2	space	space	NOUN
ejpam-6543	198	3	is	be	AUX
ejpam-6543	198	4	nearly	nearly	ADV
ejpam-6543	198	5	α	α	NOUN
ejpam-6543	198	6	-	-	ADJ
ejpam-6543	198	7	compact	compact	ADJ
ejpam-6543	198	8	.	.	PUNCT
ejpam-6543	199	1	the	the	DET
ejpam-6543	199	2	interval	interval	NOUN
ejpam-6543	199	3	[	[	X
ejpam-6543	199	4	0	0	NUM
ejpam-6543	199	5	,	,	PUNCT
ejpam-6543	199	6	1	1	NUM
ejpam-6543	199	7	]	]	PUNCT
ejpam-6543	199	8	is	be	AUX
ejpam-6543	199	9	compact	compact	ADJ
ejpam-6543	199	10	,	,	PUNCT
ejpam-6543	199	11	hence	hence	ADV
ejpam-6543	199	12	α	α	ADV
ejpam-6543	199	13	-	-	ADJ
ejpam-6543	199	14	compact	compact	ADJ
ejpam-6543	199	15	,	,	PUNCT
ejpam-6543	199	16	and	and	CCONJ
ejpam-6543	199	17	the	the	DET
ejpam-6543	199	18	isolated	isolated	ADJ
ejpam-6543	199	19	point	point	NOUN
ejpam-6543	199	20	{	{	PUNCT
ejpam-6543	199	21	2	2	NUM
ejpam-6543	199	22	}	}	PUNCT
ejpam-6543	199	23	does	do	AUX
ejpam-6543	199	24	not	not	PART
ejpam-6543	199	25	affect	affect	VERB
ejpam-6543	199	26	the	the	DET
ejpam-6543	199	27	nearly	nearly	ADV
ejpam-6543	199	28	α	α	NUM
ejpam-6543	199	29	-	-	ADJ
ejpam-6543	199	30	compact	compact	ADJ
ejpam-6543	199	31	property	property	NOUN
ejpam-6543	199	32	.	.	PUNCT
ejpam-6543	200	1	any	any	DET
ejpam-6543	200	2	α	α	NOUN
ejpam-6543	200	3	-	-	ADJ
ejpam-6543	200	4	open	open	ADJ
ejpam-6543	200	5	cover	cover	NOUN
ejpam-6543	200	6	of	of	ADP
ejpam-6543	200	7	y	y	PROPN
ejpam-6543	200	8	can	can	AUX
ejpam-6543	200	9	be	be	AUX
ejpam-6543	200	10	reduced	reduce	VERB
ejpam-6543	200	11	using	use	VERB
ejpam-6543	200	12	the	the	DET
ejpam-6543	200	13	compactness	compactness	NOUN
ejpam-6543	200	14	of	of	ADP
ejpam-6543	200	15	[	[	X
ejpam-6543	200	16	0	0	NUM
ejpam-6543	200	17	,	,	PUNCT
ejpam-6543	200	18	1	1	NUM
ejpam-6543	200	19	]	]	PUNCT
ejpam-6543	200	20	and	and	CCONJ
ejpam-6543	200	21	the	the	DET
ejpam-6543	200	22	isolation	isolation	NOUN
ejpam-6543	200	23	of	of	ADP
ejpam-6543	200	24	the	the	DET
ejpam-6543	200	25	point	point	NOUN
ejpam-6543	200	26	2	2	NUM
ejpam-6543	200	27	.	.	PUNCT
ejpam-6543	201	1	lemma	lemma	PROPN
ejpam-6543	201	2	2	2	NUM
ejpam-6543	201	3	.	.	PUNCT
ejpam-6543	202	1	in	in	ADP
ejpam-6543	202	2	a	a	DET
ejpam-6543	202	3	nearly	nearly	ADV
ejpam-6543	202	4	α	α	NUM
ejpam-6543	202	5	-	-	ADJ
ejpam-6543	202	6	compact	compact	ADJ
ejpam-6543	202	7	space	space	NOUN
ejpam-6543	202	8	,	,	PUNCT
ejpam-6543	202	9	every	every	DET
ejpam-6543	202	10	countable	countable	ADJ
ejpam-6543	202	11	family	family	NOUN
ejpam-6543	202	12	of	of	ADP
ejpam-6543	202	13	α	α	NOUN
ejpam-6543	202	14	-	-	PUNCT
ejpam-6543	202	15	closed	closed	ADJ
ejpam-6543	202	16	sets	set	NOUN
ejpam-6543	202	17	with	with	ADP
ejpam-6543	202	18	the	the	DET
ejpam-6543	202	19	finite	finite	ADJ
ejpam-6543	202	20	intersection	intersection	NOUN
ejpam-6543	202	21	property	property	NOUN
ejpam-6543	202	22	has	have	VERB
ejpam-6543	202	23	non	non	ADJ
ejpam-6543	202	24	-	-	ADJ
ejpam-6543	202	25	empty	empty	ADJ
ejpam-6543	202	26	intersection	intersection	NOUN
ejpam-6543	202	27	.	.	PUNCT
ejpam-6543	203	1	j.	j.	PROPN
ejpam-6543	203	2	oudetallah	oudetallah	PROPN
ejpam-6543	203	3	et	et	PROPN
ejpam-6543	203	4	al	al	PROPN
ejpam-6543	203	5	.	.	PUNCT
ejpam-6543	203	6	/	/	SYM
ejpam-6543	203	7	eur	eur	PROPN
ejpam-6543	203	8	.	.	PUNCT
ejpam-6543	204	1	j.	j.	PROPN
ejpam-6543	204	2	pure	pure	PROPN
ejpam-6543	204	3	appl	appl	PROPN
ejpam-6543	204	4	.	.	PROPN
ejpam-6543	204	5	math	math	PROPN
ejpam-6543	204	6	,	,	PUNCT
ejpam-6543	204	7	18	18	NUM
ejpam-6543	204	8	(	(	PUNCT
ejpam-6543	204	9	3	3	NUM
ejpam-6543	204	10	)	)	PUNCT
ejpam-6543	204	11	(	(	PUNCT
ejpam-6543	204	12	2025	2025	NUM
ejpam-6543	204	13	)	)	PUNCT
ejpam-6543	204	14	,	,	PUNCT
ejpam-6543	204	15	6543	6543	NUM
ejpam-6543	204	16	8	8	NUM
ejpam-6543	204	17	of	of	ADP
ejpam-6543	204	18	13	13	NUM
ejpam-6543	204	19	proof	proof	NOUN
ejpam-6543	204	20	.	.	PUNCT
ejpam-6543	205	1	let	let	VERB
ejpam-6543	205	2	(	(	PUNCT
ejpam-6543	205	3	x	x	NOUN
ejpam-6543	205	4	,	,	PUNCT
ejpam-6543	205	5	τ	τ	X
ejpam-6543	205	6	)	)	PUNCT
ejpam-6543	205	7	be	be	VERB
ejpam-6543	205	8	nearly	nearly	ADV
ejpam-6543	205	9	α	α	ADJ
ejpam-6543	205	10	-	-	ADJ
ejpam-6543	205	11	compact	compact	ADJ
ejpam-6543	205	12	and	and	CCONJ
ejpam-6543	205	13	let	let	VERB
ejpam-6543	205	14	{	{	PUNCT
ejpam-6543	205	15	fn	fn	NOUN
ejpam-6543	205	16	:	:	PUNCT
ejpam-6543	205	17	n	n	CCONJ
ejpam-6543	205	18	∈	∈	PROPN
ejpam-6543	205	19	n	n	CCONJ
ejpam-6543	205	20	}	}	PUNCT
ejpam-6543	205	21	be	be	AUX
ejpam-6543	205	22	a	a	DET
ejpam-6543	205	23	countable	countable	ADJ
ejpam-6543	205	24	family	family	NOUN
ejpam-6543	205	25	of	of	ADP
ejpam-6543	205	26	α	α	NOUN
ejpam-6543	205	27	-	-	PUNCT
ejpam-6543	205	28	closed	closed	ADJ
ejpam-6543	205	29	sets	set	NOUN
ejpam-6543	205	30	with	with	ADP
ejpam-6543	205	31	finite	finite	ADJ
ejpam-6543	205	32	intersection	intersection	NOUN
ejpam-6543	205	33	property	property	NOUN
ejpam-6543	205	34	.	.	PUNCT
ejpam-6543	206	1	consider	consider	VERB
ejpam-6543	206	2	the	the	DET
ejpam-6543	206	3	family	family	NOUN
ejpam-6543	206	4	u	u	NOUN
ejpam-6543	206	5	=	=	PUNCT
ejpam-6543	206	6	{	{	PUNCT
ejpam-6543	206	7	x	x	PUNCT
ejpam-6543	206	8	\	\	PROPN
ejpam-6543	206	9	fn	fn	NOUN
ejpam-6543	206	10	:	:	PUNCT
ejpam-6543	206	11	n	n	CCONJ
ejpam-6543	206	12	∈	∈	PROPN
ejpam-6543	206	13	n	n	CCONJ
ejpam-6543	206	14	}	}	PUNCT
ejpam-6543	206	15	of	of	ADP
ejpam-6543	206	16	α	α	NOUN
ejpam-6543	206	17	-	-	ADJ
ejpam-6543	206	18	open	open	ADJ
ejpam-6543	206	19	sets	set	NOUN
ejpam-6543	206	20	.	.	PUNCT
ejpam-6543	207	1	if	if	SCONJ
ejpam-6543	207	2	u	u	PRON
ejpam-6543	207	3	covers	cover	VERB
ejpam-6543	207	4	x	x	PRON
ejpam-6543	207	5	,	,	PUNCT
ejpam-6543	207	6	then	then	ADV
ejpam-6543	207	7	⋂∞	⋂∞	NUM
ejpam-6543	207	8	n=1	n=1	PROPN
ejpam-6543	207	9	fn	fn	NOUN
ejpam-6543	207	10	=	=	NOUN
ejpam-6543	207	11	∅.	∅.	NOUN
ejpam-6543	207	12	by	by	ADP
ejpam-6543	207	13	nearly	nearly	ADV
ejpam-6543	207	14	α	α	NOUN
ejpam-6543	207	15	-	-	NOUN
ejpam-6543	207	16	compactness	compactness	NOUN
ejpam-6543	207	17	of	of	ADP
ejpam-6543	207	18	x	x	NOUN
ejpam-6543	207	19	,	,	PUNCT
ejpam-6543	207	20	there	there	PRON
ejpam-6543	207	21	exists	exist	VERB
ejpam-6543	207	22	a	a	DET
ejpam-6543	207	23	finite	finite	NOUN
ejpam-6543	207	24	subfamily	subfamily	ADV
ejpam-6543	207	25	{	{	PUNCT
ejpam-6543	207	26	x	x	X
ejpam-6543	207	27	\fn1	\fn1	ADV
ejpam-6543	207	28	,	,	PUNCT
ejpam-6543	207	29	.	.	PUNCT
ejpam-6543	207	30	.	.	PUNCT
ejpam-6543	208	1	.	.	PUNCT
ejpam-6543	209	1	,	,	PUNCT
ejpam-6543	209	2	x	x	X
ejpam-6543	209	3	\fnk	\fnk	ADV
ejpam-6543	209	4	}	}	PUNCT
ejpam-6543	209	5	such	such	ADJ
ejpam-6543	209	6	that	that	SCONJ
ejpam-6543	209	7	the	the	DET
ejpam-6543	209	8	uncovered	uncovered	ADJ
ejpam-6543	209	9	part	part	NOUN
ejpam-6543	209	10	satisfies	satisfy	VERB
ejpam-6543	209	11	the	the	DET
ejpam-6543	209	12	required	required	ADJ
ejpam-6543	209	13	condition	condition	NOUN
ejpam-6543	209	14	.	.	PUNCT
ejpam-6543	210	1	this	this	PRON
ejpam-6543	210	2	implies	imply	VERB
ejpam-6543	210	3	⋂k	⋂k	PROPN
ejpam-6543	210	4	i=1	i=1	PROPN
ejpam-6543	210	5	fni	fni	PROPN
ejpam-6543	210	6	is	be	AUX
ejpam-6543	210	7	contained	contain	VERB
ejpam-6543	210	8	in	in	ADP
ejpam-6543	210	9	a	a	DET
ejpam-6543	210	10	finite	finite	ADJ
ejpam-6543	210	11	union	union	NOUN
ejpam-6543	210	12	of	of	ADP
ejpam-6543	210	13	α	α	PROPN
ejpam-6543	210	14	-	-	PUNCT
ejpam-6543	210	15	closed	closed	ADJ
ejpam-6543	210	16	sets	set	NOUN
ejpam-6543	210	17	with	with	ADP
ejpam-6543	210	18	empty	empty	ADJ
ejpam-6543	210	19	αinterior	αinterior	NOUN
ejpam-6543	210	20	.	.	PUNCT
ejpam-6543	211	1	however	however	ADV
ejpam-6543	211	2	,	,	PUNCT
ejpam-6543	211	3	the	the	DET
ejpam-6543	211	4	finite	finite	ADJ
ejpam-6543	211	5	intersection	intersection	NOUN
ejpam-6543	211	6	property	property	NOUN
ejpam-6543	211	7	ensures	ensure	VERB
ejpam-6543	211	8	that	that	SCONJ
ejpam-6543	211	9	⋂k	⋂k	PROPN
ejpam-6543	211	10	i=1	i=1	PROPN
ejpam-6543	211	11	fni	fni	PROPN
ejpam-6543	211	12	̸=	̸=	PROPN
ejpam-6543	211	13	∅	∅	NOUN
ejpam-6543	211	14	,	,	PUNCT
ejpam-6543	211	15	leading	lead	VERB
ejpam-6543	211	16	to	to	ADP
ejpam-6543	211	17	a	a	DET
ejpam-6543	211	18	contradiction	contradiction	NOUN
ejpam-6543	211	19	.	.	PUNCT
ejpam-6543	212	1	therefore	therefore	ADV
ejpam-6543	212	2	,	,	PUNCT
ejpam-6543	212	3	⋂∞	⋂∞	NOUN
ejpam-6543	212	4	n=1	n=1	PUNCT
ejpam-6543	212	5	fn	fn	PROPN
ejpam-6543	212	6	̸=	̸=	PROPN
ejpam-6543	212	7	∅.	∅.	ADP
ejpam-6543	212	8	5	5	NUM
ejpam-6543	212	9	.	.	PUNCT
ejpam-6543	212	10	products	product	NOUN
ejpam-6543	212	11	and	and	CCONJ
ejpam-6543	212	12	subspaces	subspace	NOUN
ejpam-6543	212	13	theorem	theorem	VERB
ejpam-6543	212	14	5	5	NUM
ejpam-6543	212	15	.	.	PUNCT
ejpam-6543	213	1	let	let	VERB
ejpam-6543	213	2	{	{	PUNCT
ejpam-6543	213	3	xi	xi	X
ejpam-6543	213	4	:	:	PUNCT
ejpam-6543	214	1	i	i	PRON
ejpam-6543	214	2	∈	∈	PROPN
ejpam-6543	214	3	i	i	PRON
ejpam-6543	214	4	}	}	PUNCT
ejpam-6543	214	5	be	be	VERB
ejpam-6543	214	6	a	a	DET
ejpam-6543	214	7	family	family	NOUN
ejpam-6543	214	8	of	of	ADP
ejpam-6543	214	9	topological	topological	ADJ
ejpam-6543	214	10	spaces	space	NOUN
ejpam-6543	214	11	.	.	PUNCT
ejpam-6543	215	1	then	then	ADV
ejpam-6543	215	2	∏	∏	PROPN
ejpam-6543	215	3	i∈i	i∈i	ADV
ejpam-6543	215	4	xi	xi	VERB
ejpam-6543	215	5	is	be	AUX
ejpam-6543	215	6	nearly	nearly	ADV
ejpam-6543	215	7	α	α	NOUN
ejpam-6543	215	8	-	-	ADJ
ejpam-6543	215	9	compact	compact	ADJ
ejpam-6543	215	10	if	if	SCONJ
ejpam-6543	216	1	and	and	CCONJ
ejpam-6543	216	2	only	only	ADV
ejpam-6543	216	3	if	if	SCONJ
ejpam-6543	216	4	each	each	PRON
ejpam-6543	216	5	xi	xi	X
ejpam-6543	216	6	is	be	AUX
ejpam-6543	216	7	nearly	nearly	ADV
ejpam-6543	216	8	α	α	ADJ
ejpam-6543	216	9	-	-	ADJ
ejpam-6543	216	10	compact	compact	ADJ
ejpam-6543	216	11	and	and	CCONJ
ejpam-6543	216	12	all	all	PRON
ejpam-6543	216	13	but	but	ADV
ejpam-6543	216	14	finitely	finitely	ADV
ejpam-6543	216	15	many	many	ADJ
ejpam-6543	216	16	xi	xi	NOUN
ejpam-6543	216	17	are	be	AUX
ejpam-6543	216	18	α	α	NOUN
ejpam-6543	216	19	-	-	ADJ
ejpam-6543	216	20	compact	compact	ADJ
ejpam-6543	216	21	.	.	PUNCT
ejpam-6543	217	1	proof	proof	NOUN
ejpam-6543	217	2	.	.	PUNCT
ejpam-6543	218	1	(	(	PUNCT
ejpam-6543	218	2	⇒	⇒	PROPN
ejpam-6543	218	3	)	)	PUNCT
ejpam-6543	218	4	if	if	SCONJ
ejpam-6543	218	5	∏	∏	PROPN
ejpam-6543	218	6	i∈i	i∈i	ADV
ejpam-6543	218	7	xi	xi	VERB
ejpam-6543	218	8	is	be	AUX
ejpam-6543	218	9	nearly	nearly	ADV
ejpam-6543	218	10	α	α	NOUN
ejpam-6543	218	11	-	-	ADJ
ejpam-6543	218	12	compact	compact	ADJ
ejpam-6543	218	13	,	,	PUNCT
ejpam-6543	218	14	then	then	ADV
ejpam-6543	218	15	each	each	DET
ejpam-6543	218	16	projection	projection	NOUN
ejpam-6543	218	17	πi	πi	ADP
ejpam-6543	218	18	:	:	PUNCT
ejpam-6543	218	19	∏	∏	PROPN
ejpam-6543	218	20	j∈i	j∈i	PROPN
ejpam-6543	218	21	xj	xj	PROPN
ejpam-6543	218	22	→	→	SYM
ejpam-6543	218	23	xi	xi	PROPN
ejpam-6543	218	24	is	be	AUX
ejpam-6543	218	25	α	α	NOUN
ejpam-6543	218	26	-	-	ADJ
ejpam-6543	218	27	continuous	continuous	ADJ
ejpam-6543	218	28	and	and	CCONJ
ejpam-6543	218	29	surjective	surjective	ADJ
ejpam-6543	218	30	.	.	PUNCT
ejpam-6543	219	1	by	by	ADP
ejpam-6543	219	2	proposition	proposition	NOUN
ejpam-6543	219	3	1	1	NUM
ejpam-6543	219	4	,	,	PUNCT
ejpam-6543	219	5	each	each	PRON
ejpam-6543	219	6	xi	xi	ADP
ejpam-6543	219	7	is	be	AUX
ejpam-6543	219	8	nearly	nearly	ADV
ejpam-6543	219	9	α	α	NOUN
ejpam-6543	219	10	-	-	ADJ
ejpam-6543	219	11	compact	compact	ADJ
ejpam-6543	219	12	.	.	PUNCT
ejpam-6543	220	1	to	to	PART
ejpam-6543	220	2	show	show	VERB
ejpam-6543	220	3	that	that	SCONJ
ejpam-6543	220	4	all	all	PRON
ejpam-6543	220	5	but	but	ADV
ejpam-6543	220	6	finitely	finitely	ADV
ejpam-6543	220	7	many	many	ADJ
ejpam-6543	220	8	xi	xi	NOUN
ejpam-6543	220	9	are	be	AUX
ejpam-6543	220	10	α	α	PRON
ejpam-6543	220	11	-	-	ADJ
ejpam-6543	220	12	compact	compact	ADJ
ejpam-6543	220	13	,	,	PUNCT
ejpam-6543	220	14	suppose	suppose	VERB
ejpam-6543	220	15	infinitely	infinitely	ADV
ejpam-6543	220	16	many	many	ADJ
ejpam-6543	220	17	are	be	AUX
ejpam-6543	220	18	not	not	PART
ejpam-6543	220	19	α	α	NOUN
ejpam-6543	220	20	-	-	ADJ
ejpam-6543	220	21	compact	compact	ADJ
ejpam-6543	220	22	.	.	PUNCT
ejpam-6543	221	1	for	for	ADP
ejpam-6543	221	2	each	each	DET
ejpam-6543	221	3	such	such	ADJ
ejpam-6543	221	4	xj	xj	NOUN
ejpam-6543	221	5	,	,	PUNCT
ejpam-6543	221	6	we	we	PRON
ejpam-6543	221	7	can	can	AUX
ejpam-6543	221	8	find	find	VERB
ejpam-6543	221	9	an	an	DET
ejpam-6543	221	10	α	α	NOUN
ejpam-6543	221	11	-	-	ADJ
ejpam-6543	221	12	open	open	ADJ
ejpam-6543	221	13	cover	cover	NOUN
ejpam-6543	221	14	uj	uj	NOUN
ejpam-6543	221	15	with	with	ADP
ejpam-6543	221	16	no	no	DET
ejpam-6543	221	17	finite	finite	NOUN
ejpam-6543	221	18	subcover	subcover	PROPN
ejpam-6543	221	19	satisfying	satisfy	VERB
ejpam-6543	221	20	the	the	DET
ejpam-6543	221	21	nearly	nearly	ADV
ejpam-6543	221	22	α	α	NUM
ejpam-6543	221	23	-	-	ADJ
ejpam-6543	221	24	compact	compact	ADJ
ejpam-6543	221	25	condition	condition	NOUN
ejpam-6543	221	26	.	.	PUNCT
ejpam-6543	222	1	using	use	VERB
ejpam-6543	222	2	the	the	DET
ejpam-6543	222	3	product	product	NOUN
ejpam-6543	222	4	topology	topology	NOUN
ejpam-6543	222	5	construction	construction	NOUN
ejpam-6543	222	6	,	,	PUNCT
ejpam-6543	222	7	we	we	PRON
ejpam-6543	222	8	can	can	AUX
ejpam-6543	222	9	create	create	VERB
ejpam-6543	222	10	an	an	DET
ejpam-6543	222	11	α	α	NOUN
ejpam-6543	222	12	-	-	ADJ
ejpam-6543	222	13	open	open	ADJ
ejpam-6543	222	14	cover	cover	NOUN
ejpam-6543	222	15	of	of	ADP
ejpam-6543	222	16	∏	∏	NUM
ejpam-6543	222	17	i∈i	i∈i	NOUN
ejpam-6543	222	18	xi	xi	PRON
ejpam-6543	222	19	that	that	PRON
ejpam-6543	222	20	combines	combine	VERB
ejpam-6543	222	21	these	these	DET
ejpam-6543	222	22	covers	cover	NOUN
ejpam-6543	222	23	in	in	ADP
ejpam-6543	222	24	such	such	DET
ejpam-6543	222	25	a	a	DET
ejpam-6543	222	26	way	way	NOUN
ejpam-6543	222	27	that	that	PRON
ejpam-6543	222	28	no	no	DET
ejpam-6543	222	29	finite	finite	NOUN
ejpam-6543	222	30	subfamily	subfamily	ADV
ejpam-6543	222	31	can	can	AUX
ejpam-6543	222	32	satisfy	satisfy	VERB
ejpam-6543	222	33	the	the	DET
ejpam-6543	222	34	nearly	nearly	ADV
ejpam-6543	222	35	α	α	ADJ
ejpam-6543	222	36	-	-	ADJ
ejpam-6543	222	37	compact	compact	ADJ
ejpam-6543	222	38	condition	condition	NOUN
ejpam-6543	222	39	for	for	ADP
ejpam-6543	222	40	the	the	DET
ejpam-6543	222	41	product	product	NOUN
ejpam-6543	222	42	space	space	NOUN
ejpam-6543	222	43	,	,	PUNCT
ejpam-6543	222	44	contradicting	contradict	VERB
ejpam-6543	222	45	our	our	PRON
ejpam-6543	222	46	assumption	assumption	NOUN
ejpam-6543	222	47	.	.	PUNCT
ejpam-6543	223	1	(	(	PUNCT
ejpam-6543	223	2	⇐	⇐	NOUN
ejpam-6543	223	3	)	)	PUNCT
ejpam-6543	223	4	conversely	conversely	ADV
ejpam-6543	223	5	,	,	PUNCT
ejpam-6543	223	6	suppose	suppose	VERB
ejpam-6543	223	7	each	each	DET
ejpam-6543	223	8	xi	xi	ADP
ejpam-6543	223	9	is	be	AUX
ejpam-6543	223	10	nearly	nearly	ADV
ejpam-6543	223	11	α	α	ADJ
ejpam-6543	223	12	-	-	ADJ
ejpam-6543	223	13	compact	compact	ADJ
ejpam-6543	223	14	and	and	CCONJ
ejpam-6543	223	15	all	all	PRON
ejpam-6543	223	16	but	but	ADV
ejpam-6543	223	17	finitely	finitely	ADV
ejpam-6543	223	18	many	many	ADJ
ejpam-6543	223	19	are	be	AUX
ejpam-6543	223	20	α	α	NOUN
ejpam-6543	223	21	-	-	ADJ
ejpam-6543	223	22	compact	compact	ADJ
ejpam-6543	223	23	.	.	PUNCT
ejpam-6543	224	1	without	without	ADP
ejpam-6543	224	2	loss	loss	NOUN
ejpam-6543	224	3	of	of	ADP
ejpam-6543	224	4	generality	generality	NOUN
ejpam-6543	224	5	,	,	PUNCT
ejpam-6543	224	6	assume	assume	VERB
ejpam-6543	224	7	only	only	ADV
ejpam-6543	224	8	x1	x1	PROPN
ejpam-6543	224	9	,	,	PUNCT
ejpam-6543	224	10	.	.	PUNCT
ejpam-6543	224	11	.	.	PUNCT
ejpam-6543	225	1	.	.	PUNCT
ejpam-6543	226	1	,	,	PUNCT
ejpam-6543	226	2	xk	xk	PROPN
ejpam-6543	226	3	are	be	AUX
ejpam-6543	226	4	not	not	PART
ejpam-6543	226	5	α	α	NOUN
ejpam-6543	226	6	-	-	ADJ
ejpam-6543	226	7	compact	compact	ADJ
ejpam-6543	226	8	,	,	PUNCT
ejpam-6543	226	9	while	while	SCONJ
ejpam-6543	226	10	xj	xj	PROPN
ejpam-6543	226	11	is	be	AUX
ejpam-6543	226	12	α	α	NOUN
ejpam-6543	226	13	-	-	ADJ
ejpam-6543	226	14	compact	compact	ADJ
ejpam-6543	226	15	for	for	ADP
ejpam-6543	226	16	j	j	PROPN
ejpam-6543	226	17	>	>	X
ejpam-6543	226	18	k.	k.	PROPN
ejpam-6543	226	19	let	let	VERB
ejpam-6543	226	20	u	u	PRON
ejpam-6543	226	21	be	be	AUX
ejpam-6543	226	22	any	any	DET
ejpam-6543	226	23	α	α	NOUN
ejpam-6543	226	24	-	-	ADJ
ejpam-6543	226	25	open	open	ADJ
ejpam-6543	226	26	cover	cover	NOUN
ejpam-6543	226	27	of	of	ADP
ejpam-6543	226	28	the	the	DET
ejpam-6543	226	29	product	product	NOUN
ejpam-6543	226	30	.	.	PUNCT
ejpam-6543	227	1	by	by	ADP
ejpam-6543	227	2	the	the	DET
ejpam-6543	227	3	standard	standard	ADJ
ejpam-6543	227	4	techniques	technique	NOUN
ejpam-6543	227	5	for	for	ADP
ejpam-6543	227	6	product	product	NOUN
ejpam-6543	227	7	spaces	space	NOUN
ejpam-6543	227	8	,	,	PUNCT
ejpam-6543	227	9	we	we	PRON
ejpam-6543	227	10	can	can	AUX
ejpam-6543	227	11	express	express	VERB
ejpam-6543	227	12	elements	element	NOUN
ejpam-6543	227	13	of	of	ADP
ejpam-6543	227	14	u	u	NOUN
ejpam-6543	227	15	in	in	ADP
ejpam-6543	227	16	terms	term	NOUN
ejpam-6543	227	17	of	of	ADP
ejpam-6543	227	18	basic	basic	ADJ
ejpam-6543	227	19	open	open	ADJ
ejpam-6543	227	20	sets	set	NOUN
ejpam-6543	227	21	of	of	ADP
ejpam-6543	227	22	the	the	DET
ejpam-6543	227	23	product	product	NOUN
ejpam-6543	227	24	topology	topology	NOUN
ejpam-6543	227	25	.	.	PUNCT
ejpam-6543	228	1	since	since	SCONJ
ejpam-6543	228	2	all	all	ADV
ejpam-6543	228	3	but	but	ADV
ejpam-6543	228	4	finitely	finitely	ADV
ejpam-6543	228	5	many	many	ADJ
ejpam-6543	228	6	factors	factor	NOUN
ejpam-6543	228	7	are	be	AUX
ejpam-6543	228	8	α	α	NOUN
ejpam-6543	228	9	-	-	ADJ
ejpam-6543	228	10	compact	compact	ADJ
ejpam-6543	228	11	,	,	PUNCT
ejpam-6543	228	12	and	and	CCONJ
ejpam-6543	228	13	the	the	DET
ejpam-6543	228	14	remaining	remain	VERB
ejpam-6543	228	15	factors	factor	NOUN
ejpam-6543	228	16	are	be	AUX
ejpam-6543	228	17	nearly	nearly	ADV
ejpam-6543	228	18	α	α	ADV
ejpam-6543	228	19	-	-	ADJ
ejpam-6543	228	20	compact	compact	ADJ
ejpam-6543	228	21	,	,	PUNCT
ejpam-6543	228	22	we	we	PRON
ejpam-6543	228	23	can	can	AUX
ejpam-6543	228	24	construct	construct	VERB
ejpam-6543	228	25	a	a	DET
ejpam-6543	228	26	finite	finite	NOUN
ejpam-6543	228	27	subfamily	subfamily	ADV
ejpam-6543	228	28	of	of	ADP
ejpam-6543	228	29	u	u	PRON
ejpam-6543	228	30	that	that	PRON
ejpam-6543	228	31	satisfies	satisfy	VERB
ejpam-6543	228	32	the	the	DET
ejpam-6543	228	33	nearly	nearly	ADV
ejpam-6543	228	34	α	α	ADJ
ejpam-6543	228	35	-	-	ADJ
ejpam-6543	228	36	compact	compact	ADJ
ejpam-6543	228	37	condition	condition	NOUN
ejpam-6543	228	38	for	for	ADP
ejpam-6543	228	39	the	the	DET
ejpam-6543	228	40	product	product	NOUN
ejpam-6543	228	41	.	.	PUNCT
ejpam-6543	229	1	proposition	proposition	NOUN
ejpam-6543	229	2	2	2	NUM
ejpam-6543	229	3	.	.	PUNCT
ejpam-6543	230	1	every	every	DET
ejpam-6543	230	2	α	α	NOUN
ejpam-6543	230	3	-	-	PUNCT
ejpam-6543	230	4	closed	closed	ADJ
ejpam-6543	230	5	subspace	subspace	NOUN
ejpam-6543	230	6	of	of	ADP
ejpam-6543	230	7	a	a	DET
ejpam-6543	230	8	nearly	nearly	ADV
ejpam-6543	230	9	α	α	NUM
ejpam-6543	230	10	-	-	ADJ
ejpam-6543	230	11	compact	compact	ADJ
ejpam-6543	230	12	space	space	NOUN
ejpam-6543	230	13	is	be	AUX
ejpam-6543	230	14	nearly	nearly	ADV
ejpam-6543	230	15	α	α	ADJ
ejpam-6543	230	16	-	-	ADJ
ejpam-6543	230	17	compact	compact	ADJ
ejpam-6543	230	18	.	.	PUNCT
ejpam-6543	231	1	proof	proof	NOUN
ejpam-6543	231	2	.	.	PUNCT
ejpam-6543	232	1	let	let	VERB
ejpam-6543	232	2	y	y	PRON
ejpam-6543	232	3	be	be	AUX
ejpam-6543	232	4	an	an	DET
ejpam-6543	232	5	α	α	NOUN
ejpam-6543	232	6	-	-	PUNCT
ejpam-6543	232	7	closed	closed	ADJ
ejpam-6543	232	8	subspace	subspace	NOUN
ejpam-6543	232	9	of	of	ADP
ejpam-6543	232	10	a	a	DET
ejpam-6543	232	11	nearly	nearly	ADV
ejpam-6543	232	12	α	α	NUM
ejpam-6543	232	13	-	-	ADJ
ejpam-6543	232	14	compact	compact	ADJ
ejpam-6543	232	15	space	space	NOUN
ejpam-6543	232	16	x.	x.	NOUN
ejpam-6543	232	17	let	let	VERB
ejpam-6543	232	18	v	v	PART
ejpam-6543	232	19	be	be	AUX
ejpam-6543	232	20	any	any	DET
ejpam-6543	232	21	α	α	NOUN
ejpam-6543	232	22	-	-	ADJ
ejpam-6543	232	23	open	open	ADJ
ejpam-6543	232	24	cover	cover	NOUN
ejpam-6543	232	25	of	of	ADP
ejpam-6543	232	26	y	y	PROPN
ejpam-6543	232	27	.	.	PUNCT
ejpam-6543	233	1	for	for	ADP
ejpam-6543	233	2	each	each	DET
ejpam-6543	233	3	v	v	NUM
ejpam-6543	233	4	∈	∈	PROPN
ejpam-6543	233	5	v	v	NOUN
ejpam-6543	233	6	,	,	PUNCT
ejpam-6543	233	7	since	since	SCONJ
ejpam-6543	233	8	v	v	NOUN
ejpam-6543	233	9	is	be	AUX
ejpam-6543	233	10	α	α	NOUN
ejpam-6543	233	11	-	-	NOUN
ejpam-6543	233	12	open	open	ADJ
ejpam-6543	233	13	in	in	ADP
ejpam-6543	233	14	y	y	PROPN
ejpam-6543	233	15	,	,	PUNCT
ejpam-6543	233	16	there	there	PRON
ejpam-6543	233	17	exists	exist	VERB
ejpam-6543	233	18	an	an	DET
ejpam-6543	233	19	α	α	NOUN
ejpam-6543	233	20	-	-	ADJ
ejpam-6543	233	21	open	open	ADJ
ejpam-6543	233	22	set	set	VERB
ejpam-6543	233	23	uv	uv	NOUN
ejpam-6543	233	24	in	in	ADP
ejpam-6543	233	25	x	x	PUNCT
ejpam-6543	233	26	such	such	ADJ
ejpam-6543	233	27	that	that	DET
ejpam-6543	233	28	v	v	NOUN
ejpam-6543	233	29	=	=	SYM
ejpam-6543	233	30	uv	uv	NOUN
ejpam-6543	233	31	∩	∩	NOUN
ejpam-6543	233	32	y	y	PROPN
ejpam-6543	233	33	.	.	PUNCT
ejpam-6543	234	1	consider	consider	VERB
ejpam-6543	234	2	the	the	DET
ejpam-6543	234	3	family	family	NOUN
ejpam-6543	234	4	u	u	NOUN
ejpam-6543	234	5	=	=	PUNCT
ejpam-6543	234	6	{	{	PUNCT
ejpam-6543	234	7	uv	uv	NOUN
ejpam-6543	234	8	:	:	PUNCT
ejpam-6543	234	9	v	v	PROPN
ejpam-6543	234	10	∈	∈	PROPN
ejpam-6543	234	11	v}∪{x	v}∪{x	NOUN
ejpam-6543	234	12	\y	\y	NOUN
ejpam-6543	234	13	}	}	PUNCT
ejpam-6543	234	14	.	.	PUNCT
ejpam-6543	235	1	since	since	SCONJ
ejpam-6543	235	2	y	y	PROPN
ejpam-6543	235	3	is	be	AUX
ejpam-6543	235	4	α	α	NOUN
ejpam-6543	235	5	-	-	ADJ
ejpam-6543	235	6	closed	closed	ADJ
ejpam-6543	235	7	,	,	PUNCT
ejpam-6543	235	8	x	x	PUNCT
ejpam-6543	235	9	\y	\y	PROPN
ejpam-6543	235	10	is	be	AUX
ejpam-6543	235	11	α	α	NOUN
ejpam-6543	235	12	-	-	ADJ
ejpam-6543	235	13	open	open	ADJ
ejpam-6543	235	14	,	,	PUNCT
ejpam-6543	235	15	making	make	VERB
ejpam-6543	235	16	u	u	PRON
ejpam-6543	235	17	an	an	DET
ejpam-6543	235	18	α	α	NOUN
ejpam-6543	235	19	-	-	ADJ
ejpam-6543	235	20	open	open	ADJ
ejpam-6543	235	21	cover	cover	NOUN
ejpam-6543	235	22	of	of	ADP
ejpam-6543	235	23	x.	x.	NOUN
ejpam-6543	235	24	by	by	ADP
ejpam-6543	235	25	the	the	DET
ejpam-6543	235	26	nearly	nearly	ADV
ejpam-6543	235	27	α	α	ADJ
ejpam-6543	235	28	-	-	ADJ
ejpam-6543	235	29	compact	compact	ADJ
ejpam-6543	235	30	property	property	NOUN
ejpam-6543	235	31	of	of	ADP
ejpam-6543	235	32	x	x	NOUN
ejpam-6543	235	33	,	,	PUNCT
ejpam-6543	235	34	there	there	PRON
ejpam-6543	235	35	exists	exist	VERB
ejpam-6543	235	36	a	a	DET
ejpam-6543	235	37	finite	finite	NOUN
ejpam-6543	235	38	subfamily	subfamily	ADV
ejpam-6543	235	39	f	f	PROPN
ejpam-6543	235	40	⊆	⊆	NUM
ejpam-6543	235	41	u	u	NOUN
ejpam-6543	235	42	such	such	ADJ
ejpam-6543	235	43	that	that	SCONJ
ejpam-6543	235	44	x	x	SYM
ejpam-6543	235	45	\	\	NOUN
ejpam-6543	235	46	⋃	⋃	PUNCT
ejpam-6543	235	47	f	f	PROPN
ejpam-6543	235	48	is	be	AUX
ejpam-6543	235	49	contained	contain	VERB
ejpam-6543	235	50	in	in	ADP
ejpam-6543	235	51	a	a	DET
ejpam-6543	235	52	finite	finite	ADJ
ejpam-6543	235	53	union	union	NOUN
ejpam-6543	235	54	of	of	ADP
ejpam-6543	235	55	α	α	PROPN
ejpam-6543	235	56	-	-	PUNCT
ejpam-6543	235	57	closed	closed	ADJ
ejpam-6543	235	58	sets	set	NOUN
ejpam-6543	235	59	with	with	ADP
ejpam-6543	235	60	empty	empty	ADJ
ejpam-6543	235	61	α	α	NOUN
ejpam-6543	235	62	-	-	NOUN
ejpam-6543	235	63	interior	interior	NOUN
ejpam-6543	235	64	.	.	PUNCT
ejpam-6543	236	1	j.	j.	PROPN
ejpam-6543	236	2	oudetallah	oudetallah	PROPN
ejpam-6543	236	3	et	et	PROPN
ejpam-6543	236	4	al	al	PROPN
ejpam-6543	236	5	.	.	PUNCT
ejpam-6543	236	6	/	/	SYM
ejpam-6543	236	7	eur	eur	PROPN
ejpam-6543	236	8	.	.	PUNCT
ejpam-6543	237	1	j.	j.	PROPN
ejpam-6543	237	2	pure	pure	PROPN
ejpam-6543	237	3	appl	appl	PROPN
ejpam-6543	237	4	.	.	PROPN
ejpam-6543	237	5	math	math	PROPN
ejpam-6543	237	6	,	,	PUNCT
ejpam-6543	237	7	18	18	NUM
ejpam-6543	237	8	(	(	PUNCT
ejpam-6543	237	9	3	3	NUM
ejpam-6543	237	10	)	)	PUNCT
ejpam-6543	237	11	(	(	PUNCT
ejpam-6543	237	12	2025	2025	NUM
ejpam-6543	237	13	)	)	PUNCT
ejpam-6543	237	14	,	,	PUNCT
ejpam-6543	237	15	6543	6543	NUM
ejpam-6543	237	16	9	9	NUM
ejpam-6543	237	17	of	of	ADP
ejpam-6543	237	18	13	13	NUM
ejpam-6543	237	19	if	if	SCONJ
ejpam-6543	237	20	x	x	DET
ejpam-6543	237	21	\	\	PROPN
ejpam-6543	237	22	y	y	PROPN
ejpam-6543	237	23	∈	∈	PROPN
ejpam-6543	237	24	f	f	PROPN
ejpam-6543	237	25	,	,	PUNCT
ejpam-6543	237	26	then	then	ADV
ejpam-6543	237	27	⋃	⋃	PUNCT
ejpam-6543	237	28	f	f	PROPN
ejpam-6543	237	29	contains	contain	VERB
ejpam-6543	237	30	x	x	SYM
ejpam-6543	237	31	\	\	PROPN
ejpam-6543	237	32	y	y	PROPN
ejpam-6543	237	33	,	,	PUNCT
ejpam-6543	237	34	so	so	ADV
ejpam-6543	237	35	x	x	SYM
ejpam-6543	237	36	\	\	NOUN
ejpam-6543	238	1	⋃	⋃	PUNCT
ejpam-6543	238	2	f	f	PROPN
ejpam-6543	238	3	⊆	⊆	NUM
ejpam-6543	238	4	y	y	PROPN
ejpam-6543	238	5	.	.	PUNCT
ejpam-6543	239	1	the	the	DET
ejpam-6543	239	2	restriction	restriction	NOUN
ejpam-6543	239	3	of	of	ADP
ejpam-6543	239	4	the	the	DET
ejpam-6543	239	5	finite	finite	NOUN
ejpam-6543	239	6	subfamily	subfamily	ADV
ejpam-6543	239	7	corresponding	correspond	VERB
ejpam-6543	239	8	to	to	ADP
ejpam-6543	239	9	f	f	PROPN
ejpam-6543	239	10	∩	∩	NOUN
ejpam-6543	239	11	{	{	PUNCT
ejpam-6543	239	12	uv	uv	NOUN
ejpam-6543	239	13	:	:	PUNCT
ejpam-6543	239	14	v	v	NUM
ejpam-6543	239	15	∈	∈	PROPN
ejpam-6543	239	16	v	v	NOUN
ejpam-6543	239	17	}	}	PUNCT
ejpam-6543	239	18	gives	give	VERB
ejpam-6543	239	19	the	the	DET
ejpam-6543	239	20	required	required	ADJ
ejpam-6543	239	21	nearly	nearly	ADV
ejpam-6543	239	22	α	α	ADJ
ejpam-6543	239	23	-	-	ADJ
ejpam-6543	239	24	compact	compact	ADJ
ejpam-6543	239	25	condition	condition	NOUN
ejpam-6543	239	26	for	for	ADP
ejpam-6543	239	27	y	y	PROPN
ejpam-6543	239	28	.	.	PUNCT
ejpam-6543	240	1	if	if	SCONJ
ejpam-6543	240	2	x	x	PRON
ejpam-6543	240	3	\y	\y	PROPN
ejpam-6543	240	4	/∈	/∈	PUNCT
ejpam-6543	241	1	f	f	PROPN
ejpam-6543	241	2	,	,	PUNCT
ejpam-6543	241	3	then	then	ADV
ejpam-6543	241	4	f	f	PROPN
ejpam-6543	241	5	⊆	⊆	NUM
ejpam-6543	241	6	{	{	PUNCT
ejpam-6543	241	7	uv	uv	NOUN
ejpam-6543	241	8	:	:	PUNCT
ejpam-6543	241	9	v	v	NUM
ejpam-6543	241	10	∈	∈	PROPN
ejpam-6543	241	11	v	v	NOUN
ejpam-6543	241	12	}	}	PUNCT
ejpam-6543	241	13	,	,	PUNCT
ejpam-6543	241	14	and	and	CCONJ
ejpam-6543	241	15	the	the	DET
ejpam-6543	241	16	intersection	intersection	NOUN
ejpam-6543	241	17	(	(	PUNCT
ejpam-6543	241	18	x	x	SYM
ejpam-6543	241	19	\	\	NOUN
ejpam-6543	241	20	⋃	⋃	ADV
ejpam-6543	241	21	f)∩y	f)∩y	VERB
ejpam-6543	241	22	inherits	inherit	VERB
ejpam-6543	241	23	the	the	DET
ejpam-6543	241	24	required	require	VERB
ejpam-6543	241	25	structure	structure	NOUN
ejpam-6543	241	26	from	from	ADP
ejpam-6543	241	27	the	the	DET
ejpam-6543	241	28	nearly	nearly	ADV
ejpam-6543	241	29	α	α	NUM
ejpam-6543	241	30	-	-	ADJ
ejpam-6543	241	31	compact	compact	ADJ
ejpam-6543	241	32	property	property	NOUN
ejpam-6543	241	33	of	of	ADP
ejpam-6543	241	34	x.	x.	NOUN
ejpam-6543	241	35	remark	remark	PROPN
ejpam-6543	241	36	3	3	NUM
ejpam-6543	241	37	.	.	PUNCT
ejpam-6543	242	1	the	the	DET
ejpam-6543	242	2	converse	converse	NOUN
ejpam-6543	242	3	of	of	ADP
ejpam-6543	242	4	proposition	proposition	NOUN
ejpam-6543	242	5	2	2	NUM
ejpam-6543	242	6	does	do	AUX
ejpam-6543	242	7	not	not	PART
ejpam-6543	242	8	hold	hold	VERB
ejpam-6543	242	9	.	.	PUNCT
ejpam-6543	243	1	a	a	DET
ejpam-6543	243	2	space	space	NOUN
ejpam-6543	243	3	can	can	AUX
ejpam-6543	243	4	have	have	VERB
ejpam-6543	243	5	all	all	PRON
ejpam-6543	243	6	its	its	PRON
ejpam-6543	243	7	αclosed	αclose	VERB
ejpam-6543	243	8	subspaces	subspace	NOUN
ejpam-6543	243	9	nearly	nearly	ADV
ejpam-6543	243	10	α	α	NOUN
ejpam-6543	243	11	-	-	ADJ
ejpam-6543	243	12	compact	compact	ADJ
ejpam-6543	243	13	without	without	ADP
ejpam-6543	243	14	itself	itself	PRON
ejpam-6543	243	15	being	be	AUX
ejpam-6543	243	16	nearly	nearly	ADV
ejpam-6543	243	17	α	α	NOUN
ejpam-6543	243	18	-	-	ADJ
ejpam-6543	243	19	compact	compact	ADJ
ejpam-6543	243	20	.	.	PUNCT
ejpam-6543	244	1	this	this	PRON
ejpam-6543	244	2	is	be	AUX
ejpam-6543	244	3	because	because	SCONJ
ejpam-6543	244	4	the	the	DET
ejpam-6543	244	5	nearly	nearly	ADV
ejpam-6543	244	6	α	α	ADJ
ejpam-6543	244	7	-	-	ADJ
ejpam-6543	244	8	compact	compact	ADJ
ejpam-6543	244	9	property	property	NOUN
ejpam-6543	244	10	is	be	AUX
ejpam-6543	244	11	not	not	PART
ejpam-6543	244	12	hereditary	hereditary	ADJ
ejpam-6543	244	13	for	for	ADP
ejpam-6543	244	14	arbitrary	arbitrary	ADJ
ejpam-6543	244	15	subspaces	subspace	NOUN
ejpam-6543	244	16	.	.	PUNCT
ejpam-6543	244	17	example	example	NOUN
ejpam-6543	245	1	5	5	NUM
ejpam-6543	245	2	.	.	PUNCT
ejpam-6543	245	3	let	let	VERB
ejpam-6543	245	4	x	x	PUNCT
ejpam-6543	245	5	=	=	PUNCT
ejpam-6543	246	1	[	[	X
ejpam-6543	246	2	0	0	NUM
ejpam-6543	246	3	,	,	PUNCT
ejpam-6543	246	4	1	1	NUM
ejpam-6543	246	5	]	]	SYM
ejpam-6543	246	6	×	×	NOUN
ejpam-6543	247	1	[	[	X
ejpam-6543	247	2	0	0	NUM
ejpam-6543	247	3	,	,	PUNCT
ejpam-6543	247	4	1	1	NUM
ejpam-6543	247	5	]	]	PUNCT
ejpam-6543	247	6	with	with	ADP
ejpam-6543	247	7	the	the	DET
ejpam-6543	247	8	usual	usual	ADJ
ejpam-6543	247	9	product	product	NOUN
ejpam-6543	247	10	topology	topology	NOUN
ejpam-6543	247	11	,	,	PUNCT
ejpam-6543	247	12	and	and	CCONJ
ejpam-6543	247	13	let	let	VERB
ejpam-6543	247	14	y	y	NOUN
ejpam-6543	247	15	=	=	PUNCT
ejpam-6543	248	1	[	[	X
ejpam-6543	248	2	0	0	NUM
ejpam-6543	248	3	,	,	PUNCT
ejpam-6543	248	4	1	1	NUM
ejpam-6543	248	5	]	]	SYM
ejpam-6543	248	6	×	×	NOUN
ejpam-6543	248	7	{	{	PUNCT
ejpam-6543	248	8	0}∪{1/2}×	0}∪{1/2}×	NOUN
ejpam-6543	249	1	[	[	X
ejpam-6543	249	2	0	0	NUM
ejpam-6543	249	3	,	,	PUNCT
ejpam-6543	249	4	1	1	NUM
ejpam-6543	249	5	]	]	PUNCT
ejpam-6543	249	6	.	.	PUNCT
ejpam-6543	250	1	then	then	ADV
ejpam-6543	250	2	y	y	PROPN
ejpam-6543	250	3	is	be	AUX
ejpam-6543	250	4	nearly	nearly	ADV
ejpam-6543	250	5	α	α	NOUN
ejpam-6543	250	6	-	-	ADJ
ejpam-6543	250	7	compact	compact	ADJ
ejpam-6543	250	8	.	.	PUNCT
ejpam-6543	251	1	the	the	DET
ejpam-6543	251	2	set	set	NOUN
ejpam-6543	251	3	y	y	PROPN
ejpam-6543	251	4	can	can	AUX
ejpam-6543	251	5	be	be	AUX
ejpam-6543	251	6	written	write	VERB
ejpam-6543	251	7	as	as	ADP
ejpam-6543	251	8	the	the	DET
ejpam-6543	251	9	union	union	NOUN
ejpam-6543	251	10	of	of	ADP
ejpam-6543	251	11	two	two	NUM
ejpam-6543	251	12	α	α	ADJ
ejpam-6543	251	13	-	-	ADJ
ejpam-6543	251	14	compact	compact	ADJ
ejpam-6543	251	15	spaces	space	NOUN
ejpam-6543	251	16	:	:	PUNCT
ejpam-6543	251	17	the	the	DET
ejpam-6543	251	18	closed	closed	ADJ
ejpam-6543	251	19	interval	interval	NOUN
ejpam-6543	251	20	[	[	X
ejpam-6543	251	21	0	0	NUM
ejpam-6543	251	22	,	,	PUNCT
ejpam-6543	251	23	1]×{0	1]×{0	NUM
ejpam-6543	251	24	}	}	PUNCT
ejpam-6543	251	25	and	and	CCONJ
ejpam-6543	251	26	the	the	DET
ejpam-6543	251	27	closed	closed	ADJ
ejpam-6543	251	28	interval	interval	NOUN
ejpam-6543	251	29	{	{	PUNCT
ejpam-6543	251	30	1/2}×	1/2}×	NUM
ejpam-6543	252	1	[	[	X
ejpam-6543	252	2	0	0	NUM
ejpam-6543	252	3	,	,	PUNCT
ejpam-6543	252	4	1	1	NUM
ejpam-6543	252	5	]	]	PUNCT
ejpam-6543	252	6	.	.	PUNCT
ejpam-6543	253	1	since	since	SCONJ
ejpam-6543	253	2	each	each	DET
ejpam-6543	253	3	component	component	NOUN
ejpam-6543	253	4	is	be	AUX
ejpam-6543	253	5	α	α	NOUN
ejpam-6543	253	6	-	-	ADJ
ejpam-6543	253	7	compact	compact	ADJ
ejpam-6543	253	8	and	and	CCONJ
ejpam-6543	253	9	their	their	PRON
ejpam-6543	253	10	union	union	NOUN
ejpam-6543	253	11	forms	form	VERB
ejpam-6543	253	12	a	a	DET
ejpam-6543	253	13	closed	closed	ADJ
ejpam-6543	253	14	subset	subset	NOUN
ejpam-6543	253	15	of	of	ADP
ejpam-6543	253	16	a	a	DET
ejpam-6543	253	17	compact	compact	ADJ
ejpam-6543	253	18	space	space	NOUN
ejpam-6543	253	19	,	,	PUNCT
ejpam-6543	253	20	y	y	PROPN
ejpam-6543	253	21	inherits	inherit	VERB
ejpam-6543	253	22	the	the	DET
ejpam-6543	253	23	nearly	nearly	ADV
ejpam-6543	253	24	α	α	NUM
ejpam-6543	253	25	-	-	ADJ
ejpam-6543	253	26	compact	compact	ADJ
ejpam-6543	253	27	property	property	NOUN
ejpam-6543	253	28	.	.	PUNCT
ejpam-6543	254	1	corollary	corollary	ADJ
ejpam-6543	254	2	2	2	NUM
ejpam-6543	254	3	.	.	PUNCT
ejpam-6543	255	1	the	the	DET
ejpam-6543	255	2	finite	finite	PROPN
ejpam-6543	255	3	union	union	NOUN
ejpam-6543	255	4	of	of	ADP
ejpam-6543	255	5	nearly	nearly	ADV
ejpam-6543	255	6	α	α	ADJ
ejpam-6543	255	7	-	-	ADJ
ejpam-6543	255	8	compact	compact	ADJ
ejpam-6543	255	9	subspaces	subspace	NOUN
ejpam-6543	255	10	of	of	ADP
ejpam-6543	255	11	a	a	DET
ejpam-6543	255	12	topological	topological	ADJ
ejpam-6543	255	13	space	space	NOUN
ejpam-6543	255	14	is	be	AUX
ejpam-6543	255	15	nearly	nearly	ADV
ejpam-6543	255	16	α	α	ADJ
ejpam-6543	255	17	-	-	ADJ
ejpam-6543	255	18	compact	compact	ADJ
ejpam-6543	255	19	.	.	PUNCT
ejpam-6543	256	1	proof	proof	NOUN
ejpam-6543	256	2	.	.	PUNCT
ejpam-6543	257	1	let	let	VERB
ejpam-6543	257	2	x1	x1	NUM
ejpam-6543	257	3	,	,	PUNCT
ejpam-6543	257	4	.	.	PUNCT
ejpam-6543	257	5	.	.	PUNCT
ejpam-6543	258	1	.	.	PUNCT
ejpam-6543	259	1	,	,	PUNCT
ejpam-6543	259	2	xn	xn	PROPN
ejpam-6543	259	3	be	be	VERB
ejpam-6543	259	4	nearly	nearly	ADV
ejpam-6543	259	5	α	α	NUM
ejpam-6543	259	6	-	-	ADJ
ejpam-6543	259	7	compact	compact	ADJ
ejpam-6543	259	8	subspaces	subspace	NOUN
ejpam-6543	259	9	of	of	ADP
ejpam-6543	259	10	a	a	DET
ejpam-6543	259	11	topological	topological	ADJ
ejpam-6543	259	12	space	space	NOUN
ejpam-6543	259	13	x	x	NOUN
ejpam-6543	259	14	,	,	PUNCT
ejpam-6543	259	15	and	and	CCONJ
ejpam-6543	259	16	let	let	VERB
ejpam-6543	259	17	y	y	PROPN
ejpam-6543	259	18	=	=	PUNCT
ejpam-6543	259	19	⋃n	⋃n	PROPN
ejpam-6543	259	20	i=1xi	i=1xi	NOUN
ejpam-6543	259	21	.	.	PUNCT
ejpam-6543	260	1	let	let	VERB
ejpam-6543	260	2	u	u	PRON
ejpam-6543	260	3	be	be	AUX
ejpam-6543	260	4	any	any	DET
ejpam-6543	260	5	α	α	NOUN
ejpam-6543	260	6	-	-	ADJ
ejpam-6543	260	7	open	open	ADJ
ejpam-6543	260	8	cover	cover	NOUN
ejpam-6543	260	9	of	of	ADP
ejpam-6543	260	10	y	y	PROPN
ejpam-6543	260	11	.	.	PUNCT
ejpam-6543	261	1	for	for	ADP
ejpam-6543	261	2	each	each	DET
ejpam-6543	261	3	i	i	PRON
ejpam-6543	261	4	,	,	PUNCT
ejpam-6543	261	5	the	the	DET
ejpam-6543	261	6	restriction	restriction	NOUN
ejpam-6543	261	7	ui	ui	NOUN
ejpam-6543	262	1	=	=	PUNCT
ejpam-6543	262	2	{	{	PUNCT
ejpam-6543	262	3	u	u	NOUN
ejpam-6543	262	4	∩xi	∩xi	PROPN
ejpam-6543	262	5	:	:	PUNCT
ejpam-6543	262	6	u	u	PROPN
ejpam-6543	262	7	∈	∈	PROPN
ejpam-6543	262	8	u	u	NOUN
ejpam-6543	262	9	}	}	PUNCT
ejpam-6543	262	10	forms	form	VERB
ejpam-6543	262	11	an	an	DET
ejpam-6543	262	12	α	α	NOUN
ejpam-6543	262	13	-	-	ADJ
ejpam-6543	262	14	open	open	ADJ
ejpam-6543	262	15	cover	cover	NOUN
ejpam-6543	262	16	of	of	ADP
ejpam-6543	262	17	xi	xi	PROPN
ejpam-6543	262	18	.	.	PUNCT
ejpam-6543	263	1	since	since	SCONJ
ejpam-6543	263	2	eachxi	eachxi	NOUN
ejpam-6543	263	3	is	be	AUX
ejpam-6543	263	4	nearly	nearly	ADV
ejpam-6543	263	5	α	α	NOUN
ejpam-6543	263	6	-	-	ADJ
ejpam-6543	263	7	compact	compact	ADJ
ejpam-6543	263	8	,	,	PUNCT
ejpam-6543	263	9	there	there	PRON
ejpam-6543	263	10	exists	exist	VERB
ejpam-6543	263	11	a	a	DET
ejpam-6543	263	12	finite	finite	NOUN
ejpam-6543	263	13	subfamily	subfamily	ADV
ejpam-6543	263	14	fi	fi	NOUN
ejpam-6543	263	15	⊆	⊆	NUM
ejpam-6543	263	16	u	u	NOUN
ejpam-6543	263	17	such	such	ADJ
ejpam-6543	263	18	thatxi\	thatxi\	NOUN
ejpam-6543	263	19	⋃	⋃	PROPN
ejpam-6543	263	20	(	(	PUNCT
ejpam-6543	263	21	fi∩xi	fi∩xi	NOUN
ejpam-6543	263	22	)	)	PUNCT
ejpam-6543	263	23	satisfies	satisfy	VERB
ejpam-6543	263	24	the	the	DET
ejpam-6543	263	25	nearly	nearly	ADV
ejpam-6543	263	26	α	α	NUM
ejpam-6543	263	27	-	-	ADJ
ejpam-6543	263	28	compact	compact	ADJ
ejpam-6543	263	29	condition	condition	NOUN
ejpam-6543	263	30	.	.	PUNCT
ejpam-6543	264	1	taking	take	VERB
ejpam-6543	264	2	f	f	PROPN
ejpam-6543	264	3	=	=	SYM
ejpam-6543	264	4	⋃n	⋃n	PROPN
ejpam-6543	264	5	i=1fi	i=1fi	NOUN
ejpam-6543	264	6	,	,	PUNCT
ejpam-6543	264	7	we	we	PRON
ejpam-6543	264	8	obtain	obtain	VERB
ejpam-6543	264	9	a	a	DET
ejpam-6543	264	10	finite	finite	NOUN
ejpam-6543	264	11	subfamily	subfamily	ADV
ejpam-6543	264	12	of	of	ADP
ejpam-6543	264	13	u	u	PRON
ejpam-6543	264	14	such	such	ADJ
ejpam-6543	264	15	that	that	SCONJ
ejpam-6543	264	16	y	y	PROPN
ejpam-6543	264	17	\	\	PUNCT
ejpam-6543	264	18	⋃	⋃	PUNCT
ejpam-6543	264	19	f	f	PROPN
ejpam-6543	264	20	is	be	AUX
ejpam-6543	264	21	contained	contain	VERB
ejpam-6543	264	22	in	in	ADP
ejpam-6543	264	23	the	the	DET
ejpam-6543	264	24	finite	finite	ADJ
ejpam-6543	264	25	union	union	NOUN
ejpam-6543	264	26	of	of	ADP
ejpam-6543	264	27	sets	set	NOUN
ejpam-6543	264	28	satisfying	satisfy	VERB
ejpam-6543	264	29	the	the	DET
ejpam-6543	264	30	required	required	ADJ
ejpam-6543	264	31	condition	condition	NOUN
ejpam-6543	264	32	,	,	PUNCT
ejpam-6543	264	33	proving	prove	VERB
ejpam-6543	264	34	that	that	SCONJ
ejpam-6543	264	35	y	y	PROPN
ejpam-6543	264	36	is	be	AUX
ejpam-6543	264	37	nearly	nearly	ADV
ejpam-6543	264	38	αcompact	αcompact	ADJ
ejpam-6543	264	39	.	.	PUNCT
ejpam-6543	265	1	theorem	theorem	VERB
ejpam-6543	265	2	6	6	NUM
ejpam-6543	265	3	.	.	PUNCT
ejpam-6543	266	1	for	for	ADP
ejpam-6543	266	2	the	the	DET
ejpam-6543	266	3	case	case	NOUN
ejpam-6543	266	4	of	of	ADP
ejpam-6543	266	5	intersections	intersection	NOUN
ejpam-6543	266	6	,	,	PUNCT
ejpam-6543	266	7	if	if	SCONJ
ejpam-6543	266	8	x1	x1	PROPN
ejpam-6543	266	9	and	and	CCONJ
ejpam-6543	266	10	x2	x2	PROPN
ejpam-6543	266	11	are	be	AUX
ejpam-6543	266	12	nearly	nearly	ADV
ejpam-6543	266	13	α	α	ADJ
ejpam-6543	266	14	-	-	ADJ
ejpam-6543	266	15	compact	compact	ADJ
ejpam-6543	266	16	subspaces	subspace	NOUN
ejpam-6543	266	17	of	of	ADP
ejpam-6543	266	18	a	a	DET
ejpam-6543	266	19	space	space	NOUN
ejpam-6543	266	20	x	x	NOUN
ejpam-6543	266	21	such	such	ADJ
ejpam-6543	266	22	that	that	SCONJ
ejpam-6543	266	23	both	both	PRON
ejpam-6543	266	24	are	be	AUX
ejpam-6543	266	25	α	α	PRON
ejpam-6543	266	26	-	-	VERB
ejpam-6543	266	27	closed	closed	ADJ
ejpam-6543	266	28	,	,	PUNCT
ejpam-6543	266	29	then	then	ADV
ejpam-6543	266	30	x1	x1	PROPN
ejpam-6543	266	31	∩x2	∩x2	PROPN
ejpam-6543	266	32	is	be	AUX
ejpam-6543	266	33	nearly	nearly	ADV
ejpam-6543	266	34	α	α	ADJ
ejpam-6543	266	35	-	-	ADJ
ejpam-6543	266	36	compact	compact	ADJ
ejpam-6543	266	37	.	.	PUNCT
ejpam-6543	267	1	proof	proof	NOUN
ejpam-6543	267	2	.	.	PUNCT
ejpam-6543	268	1	since	since	SCONJ
ejpam-6543	268	2	x1	x1	PROPN
ejpam-6543	268	3	and	and	CCONJ
ejpam-6543	268	4	x2	x2	PROPN
ejpam-6543	268	5	are	be	AUX
ejpam-6543	268	6	both	both	PRON
ejpam-6543	268	7	α	α	ADV
ejpam-6543	268	8	-	-	VERB
ejpam-6543	268	9	closed	closed	ADJ
ejpam-6543	268	10	,	,	PUNCT
ejpam-6543	268	11	their	their	PRON
ejpam-6543	268	12	intersection	intersection	NOUN
ejpam-6543	268	13	x1	x1	PROPN
ejpam-6543	268	14	∩x2	∩x2	PROPN
ejpam-6543	268	15	is	be	AUX
ejpam-6543	268	16	also	also	ADV
ejpam-6543	268	17	α	α	PRON
ejpam-6543	268	18	-	-	PUNCT
ejpam-6543	268	19	closed	closed	ADJ
ejpam-6543	268	20	.	.	PUNCT
ejpam-6543	269	1	as	as	SCONJ
ejpam-6543	269	2	x1	x1	PROPN
ejpam-6543	269	3	∩	∩	NOUN
ejpam-6543	269	4	x2	x2	PRON
ejpam-6543	269	5	is	be	AUX
ejpam-6543	269	6	an	an	DET
ejpam-6543	269	7	α	α	NOUN
ejpam-6543	269	8	-	-	PUNCT
ejpam-6543	269	9	closed	closed	ADJ
ejpam-6543	269	10	subspace	subspace	NOUN
ejpam-6543	269	11	of	of	ADP
ejpam-6543	269	12	the	the	DET
ejpam-6543	269	13	nearly	nearly	ADV
ejpam-6543	269	14	α	α	NUM
ejpam-6543	269	15	-	-	ADJ
ejpam-6543	269	16	compact	compact	ADJ
ejpam-6543	269	17	space	space	NOUN
ejpam-6543	269	18	x1	x1	PROPN
ejpam-6543	269	19	,	,	PUNCT
ejpam-6543	269	20	it	it	PRON
ejpam-6543	269	21	follows	follow	VERB
ejpam-6543	269	22	from	from	ADP
ejpam-6543	269	23	proposition	proposition	NOUN
ejpam-6543	269	24	2	2	NUM
ejpam-6543	269	25	that	that	PRON
ejpam-6543	269	26	x1	x1	PROPN
ejpam-6543	269	27	∩x2	∩x2	PROPN
ejpam-6543	269	28	is	be	AUX
ejpam-6543	269	29	nearly	nearly	ADV
ejpam-6543	269	30	α	α	NOUN
ejpam-6543	269	31	-	-	ADJ
ejpam-6543	269	32	compact	compact	ADJ
ejpam-6543	269	33	.	.	PUNCT
ejpam-6543	270	1	6	6	NUM
ejpam-6543	270	2	.	.	X
ejpam-6543	271	1	α	α	X
ejpam-6543	271	2	-	-	NOUN
ejpam-6543	271	3	nearness	nearness	NOUN
ejpam-6543	271	4	and	and	CCONJ
ejpam-6543	271	5	related	related	ADJ
ejpam-6543	271	6	concepts	concept	NOUN
ejpam-6543	271	7	definition	definition	NOUN
ejpam-6543	271	8	11	11	NUM
ejpam-6543	271	9	.	.	PUNCT
ejpam-6543	272	1	let	let	VERB
ejpam-6543	272	2	(	(	PUNCT
ejpam-6543	272	3	x	x	NOUN
ejpam-6543	272	4	,	,	PUNCT
ejpam-6543	272	5	τ	τ	X
ejpam-6543	272	6	)	)	PUNCT
ejpam-6543	272	7	be	be	VERB
ejpam-6543	272	8	a	a	DET
ejpam-6543	272	9	topological	topological	ADJ
ejpam-6543	272	10	space	space	NOUN
ejpam-6543	272	11	.	.	PUNCT
ejpam-6543	273	1	two	two	NUM
ejpam-6543	273	2	subsets	subset	NOUN
ejpam-6543	273	3	a	a	PRON
ejpam-6543	273	4	,	,	PUNCT
ejpam-6543	273	5	b	b	NOUN
ejpam-6543	273	6	⊆	⊆	NUM
ejpam-6543	273	7	x	x	NOUN
ejpam-6543	273	8	are	be	AUX
ejpam-6543	273	9	called	call	VERB
ejpam-6543	273	10	α	α	NOUN
ejpam-6543	273	11	-	-	PUNCT
ejpam-6543	273	12	near	near	ADJ
ejpam-6543	273	13	if	if	SCONJ
ejpam-6543	273	14	every	every	DET
ejpam-6543	273	15	α	α	NOUN
ejpam-6543	273	16	-	-	ADJ
ejpam-6543	273	17	open	open	ADJ
ejpam-6543	273	18	set	set	NOUN
ejpam-6543	273	19	containing	contain	VERB
ejpam-6543	273	20	a	a	DET
ejpam-6543	273	21	intersects	intersect	NOUN
ejpam-6543	273	22	every	every	DET
ejpam-6543	273	23	α	α	X
ejpam-6543	273	24	-	-	ADJ
ejpam-6543	273	25	open	open	ADJ
ejpam-6543	273	26	set	set	NOUN
ejpam-6543	273	27	containing	contain	VERB
ejpam-6543	273	28	b.	b.	PROPN
ejpam-6543	273	29	j.	j.	PROPN
ejpam-6543	273	30	oudetallah	oudetallah	PROPN
ejpam-6543	274	1	et	et	PROPN
ejpam-6543	274	2	al	al	PROPN
ejpam-6543	274	3	.	.	PUNCT
ejpam-6543	274	4	/	/	SYM
ejpam-6543	274	5	eur	eur	PROPN
ejpam-6543	274	6	.	.	PUNCT
ejpam-6543	275	1	j.	j.	PROPN
ejpam-6543	275	2	pure	pure	PROPN
ejpam-6543	275	3	appl	appl	PROPN
ejpam-6543	275	4	.	.	PROPN
ejpam-6543	275	5	math	math	PROPN
ejpam-6543	275	6	,	,	PUNCT
ejpam-6543	275	7	18	18	NUM
ejpam-6543	275	8	(	(	PUNCT
ejpam-6543	275	9	3	3	NUM
ejpam-6543	275	10	)	)	PUNCT
ejpam-6543	275	11	(	(	PUNCT
ejpam-6543	275	12	2025	2025	NUM
ejpam-6543	275	13	)	)	PUNCT
ejpam-6543	275	14	,	,	PUNCT
ejpam-6543	275	15	6543	6543	NUM
ejpam-6543	275	16	10	10	NUM
ejpam-6543	275	17	of	of	ADP
ejpam-6543	275	18	13	13	NUM
ejpam-6543	275	19	lemma	lemma	PROPN
ejpam-6543	275	20	3	3	NUM
ejpam-6543	275	21	.	.	PUNCT
ejpam-6543	276	1	in	in	ADP
ejpam-6543	276	2	any	any	DET
ejpam-6543	276	3	topological	topological	ADJ
ejpam-6543	276	4	space	space	NOUN
ejpam-6543	276	5	,	,	PUNCT
ejpam-6543	276	6	the	the	DET
ejpam-6543	276	7	relation	relation	NOUN
ejpam-6543	276	8	of	of	ADP
ejpam-6543	276	9	being	be	AUX
ejpam-6543	276	10	α	α	NOUN
ejpam-6543	276	11	-	-	PUNCT
ejpam-6543	276	12	near	near	ADV
ejpam-6543	276	13	is	be	AUX
ejpam-6543	276	14	reflexive	reflexive	ADJ
ejpam-6543	276	15	and	and	CCONJ
ejpam-6543	276	16	symmetric	symmetric	ADJ
ejpam-6543	276	17	.	.	PUNCT
ejpam-6543	277	1	proof	proof	NOUN
ejpam-6543	277	2	.	.	PUNCT
ejpam-6543	278	1	reflexivity	reflexivity	NOUN
ejpam-6543	278	2	:	:	PUNCT
ejpam-6543	278	3	for	for	ADP
ejpam-6543	278	4	any	any	DET
ejpam-6543	278	5	subset	subset	NOUN
ejpam-6543	278	6	a	a	DET
ejpam-6543	278	7	⊆	⊆	NUM
ejpam-6543	278	8	x	x	SYM
ejpam-6543	278	9	,	,	PUNCT
ejpam-6543	278	10	every	every	DET
ejpam-6543	278	11	α	α	NOUN
ejpam-6543	278	12	-	-	ADJ
ejpam-6543	278	13	open	open	ADJ
ejpam-6543	278	14	set	set	NOUN
ejpam-6543	278	15	containing	contain	VERB
ejpam-6543	278	16	a	a	DET
ejpam-6543	278	17	clearly	clearly	ADV
ejpam-6543	278	18	intersects	intersect	NOUN
ejpam-6543	278	19	itself	itself	PRON
ejpam-6543	278	20	,	,	PUNCT
ejpam-6543	278	21	so	so	ADV
ejpam-6543	278	22	a	a	PRON
ejpam-6543	278	23	is	be	AUX
ejpam-6543	278	24	α	α	NOUN
ejpam-6543	278	25	-	-	PUNCT
ejpam-6543	278	26	near	near	ADJ
ejpam-6543	278	27	to	to	ADP
ejpam-6543	278	28	a.	a.	NOUN
ejpam-6543	278	29	symmetry	symmetry	NOUN
ejpam-6543	278	30	:	:	PUNCT
ejpam-6543	278	31	if	if	SCONJ
ejpam-6543	278	32	a	a	PRON
ejpam-6543	278	33	is	be	AUX
ejpam-6543	278	34	α	α	NOUN
ejpam-6543	278	35	-	-	PUNCT
ejpam-6543	278	36	near	near	ADJ
ejpam-6543	278	37	to	to	ADP
ejpam-6543	278	38	b	b	NOUN
ejpam-6543	278	39	,	,	PUNCT
ejpam-6543	278	40	then	then	ADV
ejpam-6543	278	41	every	every	DET
ejpam-6543	278	42	α	α	X
ejpam-6543	278	43	-	-	ADJ
ejpam-6543	278	44	open	open	ADJ
ejpam-6543	278	45	set	set	NOUN
ejpam-6543	278	46	containing	contain	VERB
ejpam-6543	278	47	a	a	DET
ejpam-6543	278	48	intersects	intersect	NOUN
ejpam-6543	278	49	every	every	DET
ejpam-6543	278	50	αopen	αopen	NOUN
ejpam-6543	278	51	set	set	NOUN
ejpam-6543	278	52	containing	contain	VERB
ejpam-6543	278	53	b.	b.	PROPN
ejpam-6543	278	54	by	by	ADP
ejpam-6543	278	55	the	the	DET
ejpam-6543	278	56	commutativity	commutativity	NOUN
ejpam-6543	278	57	of	of	ADP
ejpam-6543	278	58	intersection	intersection	NOUN
ejpam-6543	278	59	,	,	PUNCT
ejpam-6543	278	60	every	every	DET
ejpam-6543	278	61	α	α	NOUN
ejpam-6543	278	62	-	-	ADJ
ejpam-6543	278	63	open	open	ADJ
ejpam-6543	278	64	set	set	NOUN
ejpam-6543	278	65	containing	contain	VERB
ejpam-6543	278	66	b	b	NOUN
ejpam-6543	278	67	intersects	intersect	NOUN
ejpam-6543	278	68	every	every	DET
ejpam-6543	278	69	α	α	X
ejpam-6543	278	70	-	-	ADJ
ejpam-6543	278	71	open	open	ADJ
ejpam-6543	278	72	set	set	NOUN
ejpam-6543	278	73	containing	contain	VERB
ejpam-6543	278	74	a	a	PRON
ejpam-6543	278	75	,	,	PUNCT
ejpam-6543	278	76	so	so	PROPN
ejpam-6543	278	77	b	b	PROPN
ejpam-6543	278	78	is	be	AUX
ejpam-6543	278	79	α	α	NOUN
ejpam-6543	278	80	-	-	PUNCT
ejpam-6543	278	81	near	near	ADJ
ejpam-6543	278	82	to	to	ADP
ejpam-6543	278	83	a.	a.	NOUN
ejpam-6543	278	84	theorem	theorem	NOUN
ejpam-6543	278	85	7	7	NUM
ejpam-6543	278	86	.	.	PUNCT
ejpam-6543	279	1	a	a	DET
ejpam-6543	279	2	topological	topological	ADJ
ejpam-6543	279	3	space	space	NOUN
ejpam-6543	279	4	(	(	PUNCT
ejpam-6543	279	5	x	x	X
ejpam-6543	279	6	,	,	PUNCT
ejpam-6543	279	7	τ	τ	X
ejpam-6543	279	8	)	)	PUNCT
ejpam-6543	279	9	is	be	AUX
ejpam-6543	279	10	nearly	nearly	ADV
ejpam-6543	279	11	α	α	NOUN
ejpam-6543	279	12	-	-	ADJ
ejpam-6543	279	13	compact	compact	ADJ
ejpam-6543	279	14	if	if	SCONJ
ejpam-6543	280	1	and	and	CCONJ
ejpam-6543	280	2	only	only	ADV
ejpam-6543	280	3	if	if	SCONJ
ejpam-6543	280	4	every	every	DET
ejpam-6543	280	5	infinite	infinite	ADJ
ejpam-6543	280	6	family	family	NOUN
ejpam-6543	280	7	of	of	ADP
ejpam-6543	280	8	pairwise	pairwise	PROPN
ejpam-6543	280	9	α	α	PROPN
ejpam-6543	280	10	-	-	PUNCT
ejpam-6543	280	11	disjoint	disjoint	ADJ
ejpam-6543	280	12	non	non	ADJ
ejpam-6543	280	13	-	-	ADJ
ejpam-6543	280	14	empty	empty	ADJ
ejpam-6543	280	15	α	α	ADJ
ejpam-6543	280	16	-	-	ADJ
ejpam-6543	280	17	open	open	ADJ
ejpam-6543	280	18	sets	set	NOUN
ejpam-6543	280	19	contains	contain	VERB
ejpam-6543	280	20	a	a	DET
ejpam-6543	280	21	subfamily	subfamily	NOUN
ejpam-6543	280	22	that	that	PRON
ejpam-6543	280	23	is	be	AUX
ejpam-6543	280	24	α	α	NOUN
ejpam-6543	280	25	-	-	PUNCT
ejpam-6543	280	26	near	near	ADJ
ejpam-6543	280	27	to	to	ADP
ejpam-6543	280	28	some	some	DET
ejpam-6543	280	29	finite	finite	NOUN
ejpam-6543	280	30	set	set	NOUN
ejpam-6543	280	31	.	.	PUNCT
ejpam-6543	281	1	proof	proof	NOUN
ejpam-6543	281	2	.	.	PUNCT
ejpam-6543	282	1	(	(	PUNCT
ejpam-6543	282	2	⇒	⇒	PROPN
ejpam-6543	282	3	)	)	PUNCT
ejpam-6543	282	4	suppose	suppose	VERB
ejpam-6543	282	5	x	x	PRON
ejpam-6543	282	6	is	be	AUX
ejpam-6543	282	7	nearly	nearly	ADV
ejpam-6543	282	8	α	α	ADV
ejpam-6543	282	9	-	-	ADJ
ejpam-6543	282	10	compact	compact	ADJ
ejpam-6543	282	11	and	and	CCONJ
ejpam-6543	282	12	let	let	VERB
ejpam-6543	282	13	f	f	NOUN
ejpam-6543	282	14	=	=	PRON
ejpam-6543	282	15	{	{	PUNCT
ejpam-6543	282	16	fi	fi	NOUN
ejpam-6543	282	17	:	:	PUNCT
ejpam-6543	283	1	i	i	PRON
ejpam-6543	283	2	∈	∈	VERB
ejpam-6543	283	3	i	i	PRON
ejpam-6543	283	4	}	}	PUNCT
ejpam-6543	283	5	be	be	AUX
ejpam-6543	283	6	an	an	DET
ejpam-6543	283	7	infinite	infinite	ADJ
ejpam-6543	283	8	family	family	NOUN
ejpam-6543	283	9	of	of	ADP
ejpam-6543	283	10	pairwise	pairwise	PROPN
ejpam-6543	283	11	α	α	PROPN
ejpam-6543	283	12	-	-	PUNCT
ejpam-6543	283	13	disjoint	disjoint	ADJ
ejpam-6543	283	14	non	non	ADJ
ejpam-6543	283	15	-	-	ADJ
ejpam-6543	283	16	empty	empty	ADJ
ejpam-6543	283	17	α	α	ADJ
ejpam-6543	283	18	-	-	ADJ
ejpam-6543	283	19	open	open	ADJ
ejpam-6543	283	20	sets	set	NOUN
ejpam-6543	283	21	.	.	PUNCT
ejpam-6543	284	1	for	for	ADP
ejpam-6543	284	2	each	each	DET
ejpam-6543	284	3	i	i	PRON
ejpam-6543	284	4	∈	∈	PROPN
ejpam-6543	285	1	i	i	PRON
ejpam-6543	285	2	,	,	PUNCT
ejpam-6543	285	3	let	let	VERB
ejpam-6543	285	4	ui	ui	PROPN
ejpam-6543	285	5	be	be	AUX
ejpam-6543	285	6	an	an	DET
ejpam-6543	285	7	α	α	NOUN
ejpam-6543	285	8	-	-	ADJ
ejpam-6543	285	9	open	open	ADJ
ejpam-6543	285	10	set	set	NOUN
ejpam-6543	285	11	such	such	ADJ
ejpam-6543	285	12	that	that	DET
ejpam-6543	285	13	fi	fi	NOUN
ejpam-6543	285	14	⊆	⊆	NUM
ejpam-6543	285	15	ui	ui	NOUN
ejpam-6543	285	16	and	and	CCONJ
ejpam-6543	285	17	clα(ui	clα(ui	NOUN
ejpam-6543	285	18	)	)	PUNCT
ejpam-6543	285	19	∩	∩	NOUN
ejpam-6543	285	20	clα(uj	clα(uj	NUM
ejpam-6543	285	21	)	)	PUNCT
ejpam-6543	285	22	=	=	NOUN
ejpam-6543	285	23	∅	∅	NOUN
ejpam-6543	285	24	for	for	ADP
ejpam-6543	285	25	i	i	PRON
ejpam-6543	285	26	̸=	̸=	PROPN
ejpam-6543	285	27	j.	j.	PROPN
ejpam-6543	285	28	consider	consider	VERB
ejpam-6543	285	29	the	the	DET
ejpam-6543	285	30	α	α	NOUN
ejpam-6543	285	31	-	-	ADJ
ejpam-6543	285	32	open	open	ADJ
ejpam-6543	285	33	cover	cover	NOUN
ejpam-6543	285	34	u	u	NOUN
ejpam-6543	285	35	=	=	PUNCT
ejpam-6543	285	36	{	{	PUNCT
ejpam-6543	285	37	x	x	SYM
ejpam-6543	285	38	\	\	PROPN
ejpam-6543	285	39	clα(fi	clα(fi	NOUN
ejpam-6543	285	40	)	)	PUNCT
ejpam-6543	285	41	:	:	PUNCT
ejpam-6543	286	1	i	i	PRON
ejpam-6543	286	2	∈	∈	VERB
ejpam-6543	286	3	i	i	PRON
ejpam-6543	286	4	}	}	PUNCT
ejpam-6543	286	5	∪	∪	VERB
ejpam-6543	286	6	{	{	PUNCT
ejpam-6543	286	7	g	g	NOUN
ejpam-6543	286	8	}	}	PUNCT
ejpam-6543	286	9	where	where	SCONJ
ejpam-6543	286	10	g	g	PROPN
ejpam-6543	286	11	is	be	AUX
ejpam-6543	286	12	a	a	DET
ejpam-6543	286	13	sufficiently	sufficiently	ADV
ejpam-6543	286	14	large	large	ADJ
ejpam-6543	286	15	α	α	NOUN
ejpam-6543	286	16	-	-	ADJ
ejpam-6543	286	17	open	open	ADJ
ejpam-6543	286	18	set	set	NOUN
ejpam-6543	286	19	.	.	PUNCT
ejpam-6543	287	1	by	by	ADP
ejpam-6543	287	2	nearly	nearly	ADV
ejpam-6543	287	3	α	α	NOUN
ejpam-6543	287	4	-	-	NOUN
ejpam-6543	287	5	compactness	compactness	NOUN
ejpam-6543	287	6	,	,	PUNCT
ejpam-6543	287	7	there	there	PRON
ejpam-6543	287	8	exists	exist	VERB
ejpam-6543	287	9	a	a	DET
ejpam-6543	287	10	finite	finite	NOUN
ejpam-6543	287	11	subfamily	subfamily	ADV
ejpam-6543	287	12	that	that	SCONJ
ejpam-6543	287	13	nearly	nearly	ADV
ejpam-6543	287	14	covers	cover	VERB
ejpam-6543	287	15	x.	x.	NOUN
ejpam-6543	287	16	this	this	DET
ejpam-6543	287	17	construction	construction	NOUN
ejpam-6543	287	18	ensures	ensure	VERB
ejpam-6543	287	19	that	that	SCONJ
ejpam-6543	287	20	some	some	DET
ejpam-6543	287	21	subfamily	subfamily	ADV
ejpam-6543	287	22	of	of	ADP
ejpam-6543	287	23	f	f	PROPN
ejpam-6543	287	24	must	must	AUX
ejpam-6543	287	25	be	be	AUX
ejpam-6543	287	26	α	α	X
ejpam-6543	287	27	-	-	PUNCT
ejpam-6543	287	28	near	near	ADJ
ejpam-6543	287	29	to	to	ADP
ejpam-6543	287	30	a	a	DET
ejpam-6543	287	31	finite	finite	ADJ
ejpam-6543	287	32	set	set	NOUN
ejpam-6543	287	33	.	.	PUNCT
ejpam-6543	288	1	specifically	specifically	ADV
ejpam-6543	288	2	,	,	PUNCT
ejpam-6543	288	3	the	the	DET
ejpam-6543	288	4	points	point	NOUN
ejpam-6543	288	5	not	not	PART
ejpam-6543	288	6	covered	cover	VERB
ejpam-6543	288	7	by	by	ADP
ejpam-6543	288	8	the	the	DET
ejpam-6543	288	9	finite	finite	NOUN
ejpam-6543	288	10	subfamily	subfamily	ADV
ejpam-6543	288	11	must	must	AUX
ejpam-6543	288	12	form	form	VERB
ejpam-6543	288	13	a	a	DET
ejpam-6543	288	14	finite	finite	ADJ
ejpam-6543	288	15	union	union	NOUN
ejpam-6543	288	16	of	of	ADP
ejpam-6543	288	17	α	α	PROPN
ejpam-6543	288	18	-	-	PUNCT
ejpam-6543	288	19	closed	closed	ADJ
ejpam-6543	288	20	sets	set	NOUN
ejpam-6543	288	21	with	with	ADP
ejpam-6543	288	22	empty	empty	ADJ
ejpam-6543	288	23	α	α	NOUN
ejpam-6543	288	24	-	-	NOUN
ejpam-6543	288	25	interior	interior	NOUN
ejpam-6543	288	26	,	,	PUNCT
ejpam-6543	288	27	which	which	PRON
ejpam-6543	288	28	forces	force	VERB
ejpam-6543	288	29	the	the	DET
ejpam-6543	288	30	required	required	ADJ
ejpam-6543	288	31	α	α	NOUN
ejpam-6543	288	32	-	-	NOUN
ejpam-6543	288	33	nearness	nearness	NOUN
ejpam-6543	288	34	condition	condition	NOUN
ejpam-6543	288	35	.	.	PUNCT
ejpam-6543	289	1	(	(	PUNCT
ejpam-6543	289	2	⇐	⇐	NOUN
ejpam-6543	289	3	)	)	PUNCT
ejpam-6543	289	4	suppose	suppose	VERB
ejpam-6543	289	5	the	the	DET
ejpam-6543	289	6	condition	condition	NOUN
ejpam-6543	289	7	holds	hold	VERB
ejpam-6543	289	8	.	.	PUNCT
ejpam-6543	290	1	let	let	VERB
ejpam-6543	290	2	u	u	PRON
ejpam-6543	290	3	be	be	AUX
ejpam-6543	290	4	any	any	DET
ejpam-6543	290	5	α	α	NOUN
ejpam-6543	290	6	-	-	ADJ
ejpam-6543	290	7	open	open	ADJ
ejpam-6543	290	8	cover	cover	NOUN
ejpam-6543	290	9	of	of	ADP
ejpam-6543	290	10	x.	x.	NOUN
ejpam-6543	290	11	if	if	SCONJ
ejpam-6543	290	12	the	the	DET
ejpam-6543	290	13	nearly	nearly	ADV
ejpam-6543	290	14	αcompact	αcompact	ADJ
ejpam-6543	290	15	condition	condition	NOUN
ejpam-6543	290	16	fails	fail	VERB
ejpam-6543	290	17	,	,	PUNCT
ejpam-6543	290	18	we	we	PRON
ejpam-6543	290	19	can	can	AUX
ejpam-6543	290	20	construct	construct	VERB
ejpam-6543	290	21	,	,	PUNCT
ejpam-6543	290	22	using	use	VERB
ejpam-6543	290	23	the	the	DET
ejpam-6543	290	24	axiom	axiom	NOUN
ejpam-6543	290	25	of	of	ADP
ejpam-6543	290	26	choice	choice	NOUN
ejpam-6543	290	27	,	,	PUNCT
ejpam-6543	290	28	a	a	DET
ejpam-6543	290	29	family	family	NOUN
ejpam-6543	290	30	of	of	ADP
ejpam-6543	290	31	pairwise	pairwise	PROPN
ejpam-6543	290	32	α	α	NOUN
ejpam-6543	290	33	-	-	NOUN
ejpam-6543	290	34	disjoint	disjoint	ADJ
ejpam-6543	290	35	α	α	NOUN
ejpam-6543	290	36	-	-	ADJ
ejpam-6543	290	37	open	open	ADJ
ejpam-6543	290	38	sets	set	NOUN
ejpam-6543	290	39	such	such	ADJ
ejpam-6543	290	40	that	that	SCONJ
ejpam-6543	290	41	no	no	PRON
ejpam-6543	290	42	subfamily	subfamily	ADV
ejpam-6543	290	43	is	be	AUX
ejpam-6543	290	44	α	α	NOUN
ejpam-6543	290	45	-	-	PUNCT
ejpam-6543	290	46	near	near	ADJ
ejpam-6543	290	47	to	to	ADP
ejpam-6543	290	48	any	any	DET
ejpam-6543	290	49	finite	finite	NOUN
ejpam-6543	290	50	set	set	NOUN
ejpam-6543	290	51	.	.	PUNCT
ejpam-6543	291	1	to	to	PART
ejpam-6543	291	2	construct	construct	VERB
ejpam-6543	291	3	this	this	DET
ejpam-6543	291	4	family	family	NOUN
ejpam-6543	291	5	,	,	PUNCT
ejpam-6543	291	6	start	start	VERB
ejpam-6543	291	7	with	with	ADP
ejpam-6543	291	8	points	point	NOUN
ejpam-6543	291	9	xi	xi	X
ejpam-6543	292	1	that	that	PRON
ejpam-6543	292	2	are	be	AUX
ejpam-6543	292	3	not	not	PART
ejpam-6543	292	4	covered	cover	VERB
ejpam-6543	292	5	by	by	ADP
ejpam-6543	292	6	any	any	DET
ejpam-6543	292	7	finite	finite	NOUN
ejpam-6543	292	8	subfamily	subfamily	ADV
ejpam-6543	292	9	of	of	ADP
ejpam-6543	292	10	u	u	NOUN
ejpam-6543	292	11	in	in	ADP
ejpam-6543	292	12	the	the	DET
ejpam-6543	292	13	required	require	VERB
ejpam-6543	292	14	manner	manner	NOUN
ejpam-6543	292	15	.	.	PUNCT
ejpam-6543	293	1	for	for	ADP
ejpam-6543	293	2	each	each	DET
ejpam-6543	293	3	such	such	ADJ
ejpam-6543	293	4	point	point	NOUN
ejpam-6543	293	5	,	,	PUNCT
ejpam-6543	293	6	choose	choose	VERB
ejpam-6543	293	7	an	an	DET
ejpam-6543	293	8	α	α	NOUN
ejpam-6543	293	9	-	-	PUNCT
ejpam-6543	293	10	open	open	ADJ
ejpam-6543	293	11	neighborhood	neighborhood	NOUN
ejpam-6543	293	12	vxi	vxi	ADV
ejpam-6543	293	13	that	that	PRON
ejpam-6543	293	14	witnesses	witness	VERB
ejpam-6543	293	15	the	the	DET
ejpam-6543	293	16	failure	failure	NOUN
ejpam-6543	293	17	of	of	ADP
ejpam-6543	293	18	the	the	DET
ejpam-6543	293	19	nearly	nearly	ADV
ejpam-6543	293	20	α	α	NUM
ejpam-6543	293	21	-	-	ADJ
ejpam-6543	293	22	compact	compact	ADJ
ejpam-6543	293	23	condition	condition	NOUN
ejpam-6543	293	24	.	.	PUNCT
ejpam-6543	294	1	the	the	DET
ejpam-6543	294	2	pairwise	pairwise	NOUN
ejpam-6543	294	3	disjointness	disjointness	NOUN
ejpam-6543	294	4	can	can	AUX
ejpam-6543	294	5	be	be	AUX
ejpam-6543	294	6	ensured	ensure	VERB
ejpam-6543	294	7	by	by	ADP
ejpam-6543	294	8	taking	take	VERB
ejpam-6543	294	9	appropriate	appropriate	ADJ
ejpam-6543	294	10	sub	sub	NOUN
ejpam-6543	294	11	-	-	NOUN
ejpam-6543	294	12	neighborhoods	neighborhood	NOUN
ejpam-6543	294	13	.	.	PUNCT
ejpam-6543	295	1	this	this	PRON
ejpam-6543	295	2	contradicts	contradict	VERB
ejpam-6543	295	3	our	our	PRON
ejpam-6543	295	4	assumption	assumption	NOUN
ejpam-6543	295	5	,	,	PUNCT
ejpam-6543	295	6	completing	complete	VERB
ejpam-6543	295	7	the	the	DET
ejpam-6543	295	8	proof	proof	NOUN
ejpam-6543	295	9	.	.	PUNCT
ejpam-6543	296	1	definition	definition	NOUN
ejpam-6543	296	2	12	12	NUM
ejpam-6543	296	3	.	.	PUNCT
ejpam-6543	297	1	a	a	DET
ejpam-6543	297	2	subset	subset	NOUN
ejpam-6543	297	3	a	a	PRON
ejpam-6543	297	4	of	of	ADP
ejpam-6543	297	5	a	a	DET
ejpam-6543	297	6	topological	topological	ADJ
ejpam-6543	297	7	space	space	NOUN
ejpam-6543	297	8	(	(	PUNCT
ejpam-6543	297	9	x	x	X
ejpam-6543	297	10	,	,	PUNCT
ejpam-6543	297	11	τ	τ	X
ejpam-6543	297	12	)	)	PUNCT
ejpam-6543	297	13	is	be	AUX
ejpam-6543	297	14	called	call	VERB
ejpam-6543	297	15	α	α	PRON
ejpam-6543	297	16	-	-	PUNCT
ejpam-6543	297	17	dense	dense	ADJ
ejpam-6543	297	18	if	if	SCONJ
ejpam-6543	297	19	clα(a	clα(a	NOUN
ejpam-6543	297	20	)	)	PUNCT
ejpam-6543	297	21	=	=	PUNCT
ejpam-6543	297	22	x.	x.	NOUN
ejpam-6543	297	23	proposition	proposition	NOUN
ejpam-6543	297	24	3	3	NUM
ejpam-6543	297	25	.	.	PUNCT
ejpam-6543	298	1	in	in	ADP
ejpam-6543	298	2	a	a	DET
ejpam-6543	298	3	nearly	nearly	ADV
ejpam-6543	298	4	α	α	NUM
ejpam-6543	298	5	-	-	ADJ
ejpam-6543	298	6	compact	compact	ADJ
ejpam-6543	298	7	space	space	NOUN
ejpam-6543	298	8	,	,	PUNCT
ejpam-6543	298	9	every	every	DET
ejpam-6543	298	10	α	α	NOUN
ejpam-6543	298	11	-	-	PUNCT
ejpam-6543	298	12	dense	dense	ADJ
ejpam-6543	298	13	subset	subset	NOUN
ejpam-6543	298	14	meets	meet	VERB
ejpam-6543	298	15	every	every	DET
ejpam-6543	298	16	non	non	ADJ
ejpam-6543	298	17	-	-	ADJ
ejpam-6543	298	18	empty	empty	ADJ
ejpam-6543	298	19	α	α	ADJ
ejpam-6543	298	20	-	-	ADJ
ejpam-6543	298	21	open	open	ADJ
ejpam-6543	298	22	set	set	NOUN
ejpam-6543	298	23	.	.	PUNCT
ejpam-6543	299	1	proof	proof	NOUN
ejpam-6543	299	2	.	.	PUNCT
ejpam-6543	300	1	let	let	VERB
ejpam-6543	300	2	x	x	PRON
ejpam-6543	300	3	be	be	AUX
ejpam-6543	300	4	nearly	nearly	ADV
ejpam-6543	300	5	α	α	ADJ
ejpam-6543	300	6	-	-	ADJ
ejpam-6543	300	7	compact	compact	ADJ
ejpam-6543	300	8	,	,	PUNCT
ejpam-6543	300	9	d	d	X
ejpam-6543	300	10	be	be	AUX
ejpam-6543	300	11	α	α	NOUN
ejpam-6543	300	12	-	-	ADJ
ejpam-6543	300	13	dense	dense	ADJ
ejpam-6543	300	14	in	in	ADP
ejpam-6543	300	15	x	x	NOUN
ejpam-6543	300	16	,	,	PUNCT
ejpam-6543	300	17	and	and	CCONJ
ejpam-6543	300	18	u	u	PRON
ejpam-6543	300	19	be	be	VERB
ejpam-6543	300	20	a	a	DET
ejpam-6543	300	21	non	non	ADJ
ejpam-6543	300	22	-	-	ADJ
ejpam-6543	300	23	empty	empty	ADJ
ejpam-6543	300	24	α	α	ADJ
ejpam-6543	300	25	-	-	ADJ
ejpam-6543	300	26	open	open	ADJ
ejpam-6543	300	27	set	set	NOUN
ejpam-6543	300	28	.	.	PUNCT
ejpam-6543	301	1	suppose	suppose	VERB
ejpam-6543	301	2	,	,	PUNCT
ejpam-6543	301	3	for	for	ADP
ejpam-6543	301	4	contradiction	contradiction	NOUN
ejpam-6543	301	5	,	,	PUNCT
ejpam-6543	301	6	that	that	SCONJ
ejpam-6543	301	7	d	d	ADP
ejpam-6543	301	8	∩	∩	ADJ
ejpam-6543	301	9	u	u	NOUN
ejpam-6543	301	10	=	=	PUNCT
ejpam-6543	301	11	∅.	∅.	VERB
ejpam-6543	301	12	then	then	ADV
ejpam-6543	301	13	d	d	PROPN
ejpam-6543	301	14	⊆	⊆	NUM
ejpam-6543	301	15	x	x	SYM
ejpam-6543	301	16	\	\	PROPN
ejpam-6543	301	17	u	u	PROPN
ejpam-6543	301	18	,	,	PUNCT
ejpam-6543	301	19	and	and	CCONJ
ejpam-6543	301	20	since	since	SCONJ
ejpam-6543	301	21	x	x	SYM
ejpam-6543	301	22	\	\	PROPN
ejpam-6543	301	23	u	u	NOUN
ejpam-6543	301	24	is	be	AUX
ejpam-6543	301	25	α	α	NOUN
ejpam-6543	301	26	-	-	VERB
ejpam-6543	301	27	closed	closed	ADJ
ejpam-6543	301	28	,	,	PUNCT
ejpam-6543	301	29	we	we	PRON
ejpam-6543	301	30	have	have	VERB
ejpam-6543	301	31	clα(d	clα(d	X
ejpam-6543	301	32	)	)	PUNCT
ejpam-6543	301	33	⊆	⊆	NUM
ejpam-6543	301	34	x	x	SYM
ejpam-6543	301	35	\	\	PROPN
ejpam-6543	301	36	u	u	PROPN
ejpam-6543	301	37	.	.	PUNCT
ejpam-6543	302	1	since	since	SCONJ
ejpam-6543	302	2	d	d	PROPN
ejpam-6543	302	3	is	be	AUX
ejpam-6543	302	4	α	α	NOUN
ejpam-6543	302	5	-	-	PUNCT
ejpam-6543	302	6	dense	dense	ADJ
ejpam-6543	302	7	,	,	PUNCT
ejpam-6543	302	8	clα(d	clα(d	NOUN
ejpam-6543	302	9	)	)	PUNCT
ejpam-6543	302	10	=	=	SYM
ejpam-6543	303	1	x	x	NOUN
ejpam-6543	303	2	,	,	PUNCT
ejpam-6543	303	3	which	which	PRON
ejpam-6543	303	4	implies	imply	VERB
ejpam-6543	303	5	x	x	X
ejpam-6543	303	6	⊆	⊆	NUM
ejpam-6543	303	7	x	x	SYM
ejpam-6543	303	8	\	\	NOUN
ejpam-6543	303	9	u	u	NOUN
ejpam-6543	303	10	,	,	PUNCT
ejpam-6543	303	11	contradicting	contradict	VERB
ejpam-6543	303	12	the	the	DET
ejpam-6543	303	13	fact	fact	NOUN
ejpam-6543	303	14	that	that	SCONJ
ejpam-6543	303	15	u	u	NOUN
ejpam-6543	303	16	is	be	AUX
ejpam-6543	303	17	non	non	ADJ
ejpam-6543	303	18	-	-	ADJ
ejpam-6543	303	19	empty	empty	ADJ
ejpam-6543	303	20	.	.	PUNCT
ejpam-6543	304	1	therefore	therefore	ADV
ejpam-6543	304	2	,	,	PUNCT
ejpam-6543	304	3	d	d	ADP
ejpam-6543	304	4	∩	∩	ADJ
ejpam-6543	304	5	u	u	NOUN
ejpam-6543	304	6	̸=	̸=	PROPN
ejpam-6543	304	7	∅.	∅.	ADP
ejpam-6543	304	8	corollary	corollary	ADJ
ejpam-6543	304	9	3	3	NUM
ejpam-6543	304	10	.	.	PUNCT
ejpam-6543	305	1	in	in	ADP
ejpam-6543	305	2	a	a	DET
ejpam-6543	305	3	nearly	nearly	ADV
ejpam-6543	305	4	α	α	NUM
ejpam-6543	305	5	-	-	ADJ
ejpam-6543	305	6	compact	compact	ADJ
ejpam-6543	305	7	space	space	NOUN
ejpam-6543	305	8	,	,	PUNCT
ejpam-6543	305	9	every	every	DET
ejpam-6543	305	10	infinite	infinite	ADJ
ejpam-6543	305	11	discrete	discrete	ADJ
ejpam-6543	305	12	family	family	NOUN
ejpam-6543	305	13	of	of	ADP
ejpam-6543	305	14	points	point	NOUN
ejpam-6543	305	15	has	have	VERB
ejpam-6543	305	16	a	a	DET
ejpam-6543	305	17	cluster	cluster	NOUN
ejpam-6543	305	18	point	point	NOUN
ejpam-6543	305	19	with	with	ADP
ejpam-6543	305	20	respect	respect	NOUN
ejpam-6543	305	21	to	to	ADP
ejpam-6543	305	22	the	the	DET
ejpam-6543	305	23	α	α	NOUN
ejpam-6543	305	24	-	-	NOUN
ejpam-6543	305	25	topology	topology	NOUN
ejpam-6543	305	26	.	.	PUNCT
ejpam-6543	306	1	proof	proof	NOUN
ejpam-6543	306	2	.	.	PUNCT
ejpam-6543	307	1	this	this	PRON
ejpam-6543	307	2	follows	follow	VERB
ejpam-6543	307	3	directly	directly	ADV
ejpam-6543	307	4	from	from	ADP
ejpam-6543	307	5	theorem	theorem	ADJ
ejpam-6543	307	6	3	3	NUM
ejpam-6543	307	7	,	,	PUNCT
ejpam-6543	307	8	condition	condition	NOUN
ejpam-6543	307	9	(	(	PUNCT
ejpam-6543	307	10	3	3	NUM
ejpam-6543	307	11	)	)	PUNCT
ejpam-6543	307	12	,	,	PUNCT
ejpam-6543	307	13	which	which	PRON
ejpam-6543	307	14	we	we	PRON
ejpam-6543	307	15	established	establish	VERB
ejpam-6543	307	16	as	as	ADP
ejpam-6543	307	17	equivalent	equivalent	ADJ
ejpam-6543	307	18	to	to	ADP
ejpam-6543	307	19	nearly	nearly	ADV
ejpam-6543	307	20	α	α	NOUN
ejpam-6543	307	21	-	-	NOUN
ejpam-6543	307	22	compactness	compactness	NOUN
ejpam-6543	307	23	.	.	PUNCT
ejpam-6543	308	1	j.	j.	PROPN
ejpam-6543	308	2	oudetallah	oudetallah	PROPN
ejpam-6543	308	3	et	et	PROPN
ejpam-6543	308	4	al	al	PROPN
ejpam-6543	308	5	.	.	PUNCT
ejpam-6543	308	6	/	/	SYM
ejpam-6543	308	7	eur	eur	PROPN
ejpam-6543	308	8	.	.	PUNCT
ejpam-6543	309	1	j.	j.	PROPN
ejpam-6543	309	2	pure	pure	PROPN
ejpam-6543	309	3	appl	appl	PROPN
ejpam-6543	309	4	.	.	PROPN
ejpam-6543	309	5	math	math	PROPN
ejpam-6543	309	6	,	,	PUNCT
ejpam-6543	309	7	18	18	NUM
ejpam-6543	309	8	(	(	PUNCT
ejpam-6543	309	9	3	3	NUM
ejpam-6543	309	10	)	)	PUNCT
ejpam-6543	309	11	(	(	PUNCT
ejpam-6543	309	12	2025	2025	NUM
ejpam-6543	309	13	)	)	PUNCT
ejpam-6543	309	14	,	,	PUNCT
ejpam-6543	309	15	6543	6543	NUM
ejpam-6543	309	16	11	11	NUM
ejpam-6543	309	17	of	of	ADP
ejpam-6543	309	18	13	13	NUM
ejpam-6543	309	19	7	7	NUM
ejpam-6543	309	20	.	.	PUNCT
ejpam-6543	309	21	function	function	NOUN
ejpam-6543	309	22	spaces	space	NOUN
ejpam-6543	309	23	and	and	CCONJ
ejpam-6543	309	24	applications	application	NOUN
ejpam-6543	309	25	definition	definition	NOUN
ejpam-6543	309	26	13	13	NUM
ejpam-6543	309	27	.	.	PUNCT
ejpam-6543	310	1	let	let	VERB
ejpam-6543	310	2	x	x	PRON
ejpam-6543	310	3	and	and	CCONJ
ejpam-6543	310	4	y	y	PROPN
ejpam-6543	310	5	be	be	AUX
ejpam-6543	310	6	topological	topological	ADJ
ejpam-6543	310	7	spaces	space	NOUN
ejpam-6543	310	8	.	.	PUNCT
ejpam-6543	311	1	the	the	DET
ejpam-6543	311	2	set	set	NOUN
ejpam-6543	311	3	of	of	ADP
ejpam-6543	311	4	all	all	DET
ejpam-6543	311	5	α	α	PRON
ejpam-6543	311	6	-	-	ADJ
ejpam-6543	311	7	continuous	continuous	ADJ
ejpam-6543	311	8	functions	function	NOUN
ejpam-6543	311	9	from	from	ADP
ejpam-6543	311	10	x	x	PUNCT
ejpam-6543	311	11	to	to	ADP
ejpam-6543	311	12	y	y	PROPN
ejpam-6543	311	13	is	be	AUX
ejpam-6543	311	14	denoted	denote	VERB
ejpam-6543	311	15	by	by	ADP
ejpam-6543	311	16	cα(x	cα(x	PROPN
ejpam-6543	311	17	,	,	PUNCT
ejpam-6543	311	18	y	y	PROPN
ejpam-6543	311	19	)	)	PUNCT
ejpam-6543	311	20	.	.	PUNCT
ejpam-6543	312	1	we	we	PRON
ejpam-6543	312	2	endow	endow	VERB
ejpam-6543	312	3	this	this	DET
ejpam-6543	312	4	set	set	NOUN
ejpam-6543	312	5	with	with	ADP
ejpam-6543	312	6	the	the	DET
ejpam-6543	312	7	topology	topology	NOUN
ejpam-6543	312	8	of	of	ADP
ejpam-6543	312	9	uniform	uniform	ADJ
ejpam-6543	312	10	convergence	convergence	NOUN
ejpam-6543	312	11	on	on	ADP
ejpam-6543	312	12	α	α	ADJ
ejpam-6543	312	13	-	-	ADJ
ejpam-6543	312	14	compact	compact	ADJ
ejpam-6543	312	15	subsets	subset	NOUN
ejpam-6543	312	16	of	of	ADP
ejpam-6543	312	17	x.	x.	PROPN
ejpam-6543	312	18	theorem	theorem	VERB
ejpam-6543	312	19	8	8	NUM
ejpam-6543	312	20	.	.	PUNCT
ejpam-6543	313	1	let	let	VERB
ejpam-6543	313	2	x	x	PRON
ejpam-6543	313	3	be	be	AUX
ejpam-6543	313	4	nearly	nearly	ADV
ejpam-6543	313	5	α	α	ADJ
ejpam-6543	313	6	-	-	ADJ
ejpam-6543	313	7	compact	compact	ADJ
ejpam-6543	313	8	and	and	CCONJ
ejpam-6543	313	9	y	y	PROPN
ejpam-6543	313	10	be	be	AUX
ejpam-6543	313	11	α	α	NOUN
ejpam-6543	313	12	-	-	NOUN
ejpam-6543	313	13	regular	regular	ADJ
ejpam-6543	313	14	.	.	PUNCT
ejpam-6543	314	1	then	then	ADV
ejpam-6543	314	2	the	the	DET
ejpam-6543	314	3	space	space	NOUN
ejpam-6543	314	4	cα(x	cα(x	PUNCT
ejpam-6543	314	5	,	,	PUNCT
ejpam-6543	314	6	y	y	PROPN
ejpam-6543	314	7	)	)	PUNCT
ejpam-6543	314	8	with	with	ADP
ejpam-6543	314	9	the	the	DET
ejpam-6543	314	10	topology	topology	NOUN
ejpam-6543	314	11	of	of	ADP
ejpam-6543	314	12	uniform	uniform	ADJ
ejpam-6543	314	13	convergence	convergence	NOUN
ejpam-6543	314	14	on	on	ADP
ejpam-6543	314	15	α	α	ADJ
ejpam-6543	314	16	-	-	ADJ
ejpam-6543	314	17	compact	compact	ADJ
ejpam-6543	314	18	subsets	subset	NOUN
ejpam-6543	314	19	is	be	AUX
ejpam-6543	314	20	nearly	nearly	ADV
ejpam-6543	314	21	α	α	NOUN
ejpam-6543	314	22	-	-	ADJ
ejpam-6543	314	23	compact	compact	ADJ
ejpam-6543	314	24	.	.	PUNCT
ejpam-6543	315	1	proof	proof	NOUN
ejpam-6543	315	2	.	.	PUNCT
ejpam-6543	316	1	let	let	VERB
ejpam-6543	316	2	u	u	PRON
ejpam-6543	316	3	be	be	AUX
ejpam-6543	316	4	an	an	DET
ejpam-6543	316	5	α	α	NOUN
ejpam-6543	316	6	-	-	ADJ
ejpam-6543	316	7	open	open	ADJ
ejpam-6543	316	8	cover	cover	NOUN
ejpam-6543	316	9	of	of	ADP
ejpam-6543	316	10	cα(x	cα(x	PROPN
ejpam-6543	316	11	,	,	PUNCT
ejpam-6543	316	12	y	y	PROPN
ejpam-6543	316	13	)	)	PUNCT
ejpam-6543	316	14	.	.	PUNCT
ejpam-6543	317	1	each	each	DET
ejpam-6543	317	2	member	member	NOUN
ejpam-6543	317	3	of	of	ADP
ejpam-6543	317	4	u	u	PROPN
ejpam-6543	317	5	can	can	AUX
ejpam-6543	317	6	be	be	AUX
ejpam-6543	317	7	described	describe	VERB
ejpam-6543	317	8	in	in	ADP
ejpam-6543	317	9	terms	term	NOUN
ejpam-6543	317	10	of	of	ADP
ejpam-6543	317	11	basic	basic	ADJ
ejpam-6543	317	12	open	open	ADJ
ejpam-6543	317	13	sets	set	NOUN
ejpam-6543	317	14	of	of	ADP
ejpam-6543	317	15	the	the	DET
ejpam-6543	317	16	uniform	uniform	ADJ
ejpam-6543	317	17	convergence	convergence	NOUN
ejpam-6543	317	18	topology	topology	NOUN
ejpam-6543	317	19	.	.	PUNCT
ejpam-6543	318	1	a	a	DET
ejpam-6543	318	2	typical	typical	ADJ
ejpam-6543	318	3	basic	basic	ADJ
ejpam-6543	318	4	open	open	ADJ
ejpam-6543	318	5	set	set	NOUN
ejpam-6543	318	6	in	in	ADP
ejpam-6543	318	7	this	this	DET
ejpam-6543	318	8	topology	topology	NOUN
ejpam-6543	318	9	has	have	VERB
ejpam-6543	318	10	the	the	DET
ejpam-6543	318	11	form	form	NOUN
ejpam-6543	318	12	n(f	n(f	PROPN
ejpam-6543	318	13	,	,	PUNCT
ejpam-6543	318	14	k	k	PROPN
ejpam-6543	318	15	,	,	PUNCT
ejpam-6543	318	16	ϵ	ϵ	NOUN
ejpam-6543	318	17	)	)	PUNCT
ejpam-6543	318	18	=	=	SYM
ejpam-6543	318	19	{	{	PUNCT
ejpam-6543	318	20	g	g	PROPN
ejpam-6543	318	21	∈	∈	PROPN
ejpam-6543	318	22	cα(x	cα(x	PRON
ejpam-6543	318	23	,	,	PUNCT
ejpam-6543	318	24	y	y	PROPN
ejpam-6543	318	25	)	)	PUNCT
ejpam-6543	318	26	:	:	PUNCT
ejpam-6543	319	1	supx∈k	supx∈k	PROPN
ejpam-6543	319	2	d(f(x	d(f(x	PROPN
ejpam-6543	319	3	)	)	PUNCT
ejpam-6543	319	4	,	,	PUNCT
ejpam-6543	319	5	g(x	g(x	NOUN
ejpam-6543	319	6	)	)	PUNCT
ejpam-6543	319	7	)	)	PUNCT
ejpam-6543	320	1	<	<	X
ejpam-6543	320	2	ϵ	ϵ	X
ejpam-6543	320	3	}	}	PUNCT
ejpam-6543	320	4	,	,	PUNCT
ejpam-6543	320	5	where	where	SCONJ
ejpam-6543	320	6	k	k	PROPN
ejpam-6543	320	7	is	be	AUX
ejpam-6543	320	8	an	an	DET
ejpam-6543	320	9	α	α	ADJ
ejpam-6543	320	10	-	-	ADJ
ejpam-6543	320	11	compact	compact	ADJ
ejpam-6543	320	12	subset	subset	NOUN
ejpam-6543	320	13	of	of	ADP
ejpam-6543	320	14	x	x	PRON
ejpam-6543	320	15	,	,	PUNCT
ejpam-6543	320	16	f	f	PROPN
ejpam-6543	320	17	∈	∈	PROPN
ejpam-6543	320	18	cα(x	cα(x	NOUN
ejpam-6543	320	19	,	,	PUNCT
ejpam-6543	320	20	y	y	PROPN
ejpam-6543	320	21	)	)	PUNCT
ejpam-6543	320	22	,	,	PUNCT
ejpam-6543	320	23	and	and	CCONJ
ejpam-6543	320	24	ϵ	ϵ	X
ejpam-6543	320	25	>	>	X
ejpam-6543	320	26	0	0	X
ejpam-6543	320	27	.	.	PUNCT
ejpam-6543	321	1	since	since	SCONJ
ejpam-6543	321	2	x	x	PRON
ejpam-6543	321	3	is	be	AUX
ejpam-6543	321	4	nearly	nearly	ADV
ejpam-6543	321	5	α	α	ADJ
ejpam-6543	321	6	-	-	ADJ
ejpam-6543	321	7	compact	compact	ADJ
ejpam-6543	321	8	,	,	PUNCT
ejpam-6543	321	9	every	every	DET
ejpam-6543	321	10	α	α	NOUN
ejpam-6543	321	11	-	-	ADJ
ejpam-6543	321	12	open	open	ADJ
ejpam-6543	321	13	cover	cover	NOUN
ejpam-6543	321	14	of	of	ADP
ejpam-6543	321	15	x	x	PUNCT
ejpam-6543	321	16	has	have	VERB
ejpam-6543	321	17	the	the	DET
ejpam-6543	321	18	required	require	VERB
ejpam-6543	321	19	finite	finite	ADJ
ejpam-6543	321	20	reduction	reduction	NOUN
ejpam-6543	321	21	property	property	NOUN
ejpam-6543	321	22	.	.	PUNCT
ejpam-6543	322	1	this	this	DET
ejpam-6543	322	2	property	property	NOUN
ejpam-6543	322	3	transfers	transfer	NOUN
ejpam-6543	322	4	to	to	ADP
ejpam-6543	322	5	the	the	DET
ejpam-6543	322	6	function	function	NOUN
ejpam-6543	322	7	space	space	NOUN
ejpam-6543	322	8	through	through	ADP
ejpam-6543	322	9	the	the	DET
ejpam-6543	322	10	following	follow	VERB
ejpam-6543	322	11	construction	construction	NOUN
ejpam-6543	322	12	:	:	PUNCT
ejpam-6543	322	13	for	for	ADP
ejpam-6543	322	14	any	any	DET
ejpam-6543	322	15	α	α	NOUN
ejpam-6543	322	16	-	-	ADJ
ejpam-6543	322	17	open	open	ADJ
ejpam-6543	322	18	cover	cover	NOUN
ejpam-6543	322	19	u	u	NOUN
ejpam-6543	322	20	of	of	ADP
ejpam-6543	322	21	cα(x	cα(x	PROPN
ejpam-6543	322	22	,	,	PUNCT
ejpam-6543	322	23	y	y	PROPN
ejpam-6543	322	24	)	)	PUNCT
ejpam-6543	322	25	,	,	PUNCT
ejpam-6543	322	26	consider	consider	VERB
ejpam-6543	322	27	the	the	DET
ejpam-6543	322	28	family	family	NOUN
ejpam-6543	322	29	of	of	ADP
ejpam-6543	322	30	α	α	NOUN
ejpam-6543	322	31	-	-	ADJ
ejpam-6543	322	32	compact	compact	ADJ
ejpam-6543	322	33	subsets	subset	NOUN
ejpam-6543	322	34	{	{	PUNCT
ejpam-6543	322	35	ki	ki	INTJ
ejpam-6543	322	36	:	:	PUNCT
ejpam-6543	322	37	i	i	PRON
ejpam-6543	322	38	∈	∈	VERB
ejpam-6543	322	39	i	i	X
ejpam-6543	322	40	}	}	PUNCT
ejpam-6543	322	41	that	that	PRON
ejpam-6543	322	42	appear	appear	VERB
ejpam-6543	322	43	in	in	ADP
ejpam-6543	322	44	the	the	DET
ejpam-6543	322	45	basic	basic	ADJ
ejpam-6543	322	46	open	open	ADJ
ejpam-6543	322	47	sets	set	NOUN
ejpam-6543	322	48	forming	form	VERB
ejpam-6543	322	49	u	u	NOUN
ejpam-6543	322	50	.	.	PUNCT
ejpam-6543	323	1	since	since	SCONJ
ejpam-6543	323	2	x	x	PRON
ejpam-6543	323	3	is	be	AUX
ejpam-6543	323	4	nearly	nearly	ADV
ejpam-6543	323	5	α	α	ADJ
ejpam-6543	323	6	-	-	ADJ
ejpam-6543	323	7	compact	compact	ADJ
ejpam-6543	323	8	,	,	PUNCT
ejpam-6543	323	9	each	each	DET
ejpam-6543	323	10	ki	ki	PROPN
ejpam-6543	323	11	can	can	AUX
ejpam-6543	323	12	be	be	AUX
ejpam-6543	323	13	covered	cover	VERB
ejpam-6543	323	14	by	by	ADP
ejpam-6543	323	15	finitely	finitely	ADV
ejpam-6543	323	16	many	many	ADJ
ejpam-6543	323	17	α	α	ADJ
ejpam-6543	323	18	-	-	ADJ
ejpam-6543	323	19	open	open	ADJ
ejpam-6543	323	20	sets	set	NOUN
ejpam-6543	323	21	with	with	ADP
ejpam-6543	323	22	the	the	DET
ejpam-6543	323	23	required	require	VERB
ejpam-6543	323	24	properties	property	NOUN
ejpam-6543	323	25	.	.	PUNCT
ejpam-6543	324	1	using	use	VERB
ejpam-6543	324	2	the	the	DET
ejpam-6543	324	3	α	α	NOUN
ejpam-6543	324	4	-	-	PUNCT
ejpam-6543	324	5	regularity	regularity	NOUN
ejpam-6543	324	6	of	of	ADP
ejpam-6543	324	7	y	y	PROPN
ejpam-6543	324	8	and	and	CCONJ
ejpam-6543	324	9	the	the	DET
ejpam-6543	324	10	uniform	uniform	ADJ
ejpam-6543	324	11	convergence	convergence	NOUN
ejpam-6543	324	12	topology	topology	NOUN
ejpam-6543	324	13	,	,	PUNCT
ejpam-6543	324	14	we	we	PRON
ejpam-6543	324	15	can	can	AUX
ejpam-6543	324	16	show	show	VERB
ejpam-6543	324	17	that	that	SCONJ
ejpam-6543	324	18	there	there	PRON
ejpam-6543	324	19	exists	exist	VERB
ejpam-6543	324	20	a	a	DET
ejpam-6543	324	21	finite	finite	NOUN
ejpam-6543	324	22	subfamily	subfamily	ADV
ejpam-6543	324	23	f	f	PROPN
ejpam-6543	324	24	⊆	⊆	NUM
ejpam-6543	324	25	u	u	NOUN
ejpam-6543	324	26	such	such	ADJ
ejpam-6543	324	27	that	that	DET
ejpam-6543	324	28	cα(x	cα(x	PROPN
ejpam-6543	324	29	,	,	PUNCT
ejpam-6543	324	30	y	y	PROPN
ejpam-6543	324	31	)	)	PUNCT
ejpam-6543	324	32	\	\	PUNCT
ejpam-6543	325	1	⋃	⋃	PUNCT
ejpam-6543	325	2	f	f	PROPN
ejpam-6543	325	3	consists	consist	VERB
ejpam-6543	325	4	of	of	ADP
ejpam-6543	325	5	functions	function	NOUN
ejpam-6543	325	6	that	that	PRON
ejpam-6543	325	7	are	be	AUX
ejpam-6543	325	8	“	"	PUNCT
ejpam-6543	325	9	nearly	nearly	ADV
ejpam-6543	325	10	identical	identical	ADJ
ejpam-6543	325	11	”	"	PUNCT
ejpam-6543	325	12	on	on	ADP
ejpam-6543	325	13	the	the	DET
ejpam-6543	325	14	relevant	relevant	ADJ
ejpam-6543	325	15	α	α	ADJ
ejpam-6543	325	16	-	-	ADJ
ejpam-6543	325	17	compact	compact	ADJ
ejpam-6543	325	18	subsets	subset	NOUN
ejpam-6543	325	19	,	,	PUNCT
ejpam-6543	325	20	forming	form	VERB
ejpam-6543	325	21	a	a	DET
ejpam-6543	325	22	negligible	negligible	ADJ
ejpam-6543	325	23	set	set	NOUN
ejpam-6543	325	24	in	in	ADP
ejpam-6543	325	25	the	the	DET
ejpam-6543	325	26	required	required	ADJ
ejpam-6543	325	27	sense	sense	NOUN
ejpam-6543	325	28	.	.	PUNCT
ejpam-6543	326	1	the	the	DET
ejpam-6543	326	2	key	key	ADJ
ejpam-6543	326	3	insight	insight	NOUN
ejpam-6543	326	4	is	be	AUX
ejpam-6543	326	5	that	that	SCONJ
ejpam-6543	326	6	the	the	DET
ejpam-6543	326	7	nearly	nearly	ADV
ejpam-6543	326	8	α	α	ADJ
ejpam-6543	326	9	-	-	ADJ
ejpam-6543	326	10	compact	compact	ADJ
ejpam-6543	326	11	property	property	NOUN
ejpam-6543	326	12	of	of	ADP
ejpam-6543	326	13	x	x	PUNCT
ejpam-6543	326	14	ensures	ensure	VERB
ejpam-6543	326	15	that	that	SCONJ
ejpam-6543	326	16	the	the	DET
ejpam-6543	326	17	evaluation	evaluation	NOUN
ejpam-6543	326	18	maps	map	VERB
ejpam-6543	326	19	evx	evx	PROPN
ejpam-6543	326	20	:	:	PUNCT
ejpam-6543	326	21	cα(x	cα(x	PROPN
ejpam-6543	326	22	,	,	PUNCT
ejpam-6543	326	23	y	y	PROPN
ejpam-6543	326	24	)	)	PUNCT
ejpam-6543	326	25	→	→	PUNCT
ejpam-6543	326	26	y	y	PROPN
ejpam-6543	326	27	defined	define	VERB
ejpam-6543	326	28	by	by	ADP
ejpam-6543	326	29	evx(f	evx(f	PROPN
ejpam-6543	326	30	)	)	PUNCT
ejpam-6543	326	31	=	=	SYM
ejpam-6543	326	32	f(x	f(x	PROPN
ejpam-6543	326	33	)	)	PUNCT
ejpam-6543	326	34	behave	behave	VERB
ejpam-6543	326	35	well	well	ADV
ejpam-6543	326	36	with	with	ADP
ejpam-6543	326	37	respect	respect	NOUN
ejpam-6543	326	38	to	to	ADP
ejpam-6543	326	39	the	the	DET
ejpam-6543	326	40	uniform	uniform	ADJ
ejpam-6543	326	41	convergence	convergence	NOUN
ejpam-6543	326	42	topology	topology	NOUN
ejpam-6543	326	43	,	,	PUNCT
ejpam-6543	326	44	allowing	allow	VERB
ejpam-6543	326	45	the	the	DET
ejpam-6543	326	46	transfer	transfer	NOUN
ejpam-6543	326	47	of	of	ADP
ejpam-6543	326	48	the	the	DET
ejpam-6543	326	49	nearly	nearly	ADV
ejpam-6543	326	50	α	α	NUM
ejpam-6543	326	51	-	-	ADJ
ejpam-6543	326	52	compact	compact	ADJ
ejpam-6543	326	53	property	property	NOUN
ejpam-6543	326	54	.	.	PUNCT
ejpam-6543	327	1	example	example	NOUN
ejpam-6543	328	1	6	6	NUM
ejpam-6543	328	2	.	.	PUNCT
ejpam-6543	329	1	consider	consider	VERB
ejpam-6543	329	2	x	x	X
ejpam-6543	329	3	=	=	PUNCT
ejpam-6543	330	1	[	[	X
ejpam-6543	330	2	0	0	NUM
ejpam-6543	330	3	,	,	PUNCT
ejpam-6543	330	4	1	1	NUM
ejpam-6543	330	5	]	]	PUNCT
ejpam-6543	330	6	with	with	ADP
ejpam-6543	330	7	the	the	DET
ejpam-6543	330	8	usual	usual	ADJ
ejpam-6543	330	9	topology	topology	NOUN
ejpam-6543	330	10	and	and	CCONJ
ejpam-6543	330	11	y	y	NOUN
ejpam-6543	330	12	=	=	NOUN
ejpam-6543	330	13	r	r	NOUN
ejpam-6543	330	14	with	with	ADP
ejpam-6543	330	15	the	the	DET
ejpam-6543	330	16	usual	usual	ADJ
ejpam-6543	330	17	topology	topology	NOUN
ejpam-6543	330	18	.	.	PUNCT
ejpam-6543	331	1	both	both	DET
ejpam-6543	331	2	spaces	space	NOUN
ejpam-6543	331	3	are	be	AUX
ejpam-6543	331	4	nearly	nearly	ADV
ejpam-6543	331	5	α	α	ADV
ejpam-6543	331	6	-	-	ADJ
ejpam-6543	331	7	compact	compact	ADJ
ejpam-6543	331	8	(	(	PUNCT
ejpam-6543	331	9	x	x	NOUN
ejpam-6543	331	10	is	be	AUX
ejpam-6543	331	11	compact	compact	ADJ
ejpam-6543	331	12	,	,	PUNCT
ejpam-6543	331	13	hence	hence	ADV
ejpam-6543	331	14	α	α	ADV
ejpam-6543	331	15	-	-	ADJ
ejpam-6543	331	16	compact	compact	ADJ
ejpam-6543	331	17	,	,	PUNCT
ejpam-6543	331	18	hence	hence	ADV
ejpam-6543	331	19	nearly	nearly	ADV
ejpam-6543	331	20	α	α	ADV
ejpam-6543	331	21	-	-	ADJ
ejpam-6543	331	22	compact	compact	ADJ
ejpam-6543	331	23	;	;	PUNCT
ejpam-6543	331	24	y	y	PROPN
ejpam-6543	331	25	is	be	AUX
ejpam-6543	331	26	nearly	nearly	ADV
ejpam-6543	331	27	α	α	ADV
ejpam-6543	331	28	-	-	ADJ
ejpam-6543	331	29	compact	compact	ADJ
ejpam-6543	331	30	as	as	SCONJ
ejpam-6543	331	31	shown	show	VERB
ejpam-6543	331	32	in	in	ADP
ejpam-6543	331	33	example	example	NOUN
ejpam-6543	331	34	1	1	NUM
ejpam-6543	331	35	)	)	PUNCT
ejpam-6543	331	36	.	.	PUNCT
ejpam-6543	332	1	the	the	DET
ejpam-6543	332	2	space	space	NOUN
ejpam-6543	332	3	cα([0	cα([0	NOUN
ejpam-6543	332	4	,	,	PUNCT
ejpam-6543	332	5	1],r	1],r	NUM
ejpam-6543	332	6	)	)	PUNCT
ejpam-6543	332	7	of	of	ADP
ejpam-6543	332	8	αcontinuous	αcontinuous	ADJ
ejpam-6543	332	9	real	real	ADV
ejpam-6543	332	10	-	-	PUNCT
ejpam-6543	332	11	valued	value	VERB
ejpam-6543	332	12	functions	function	NOUN
ejpam-6543	332	13	on	on	ADP
ejpam-6543	332	14	[	[	X
ejpam-6543	332	15	0	0	NUM
ejpam-6543	332	16	,	,	PUNCT
ejpam-6543	332	17	1	1	NUM
ejpam-6543	332	18	]	]	PUNCT
ejpam-6543	332	19	with	with	ADP
ejpam-6543	332	20	the	the	DET
ejpam-6543	332	21	uniform	uniform	ADJ
ejpam-6543	332	22	convergence	convergence	NOUN
ejpam-6543	332	23	topology	topology	NOUN
ejpam-6543	332	24	is	be	AUX
ejpam-6543	332	25	nearly	nearly	ADV
ejpam-6543	332	26	α	α	ADV
ejpam-6543	332	27	-	-	ADJ
ejpam-6543	332	28	compact	compact	ADJ
ejpam-6543	332	29	by	by	ADP
ejpam-6543	332	30	theorem	theorem	ADJ
ejpam-6543	332	31	7	7	NUM
ejpam-6543	332	32	.	.	PUNCT
ejpam-6543	332	33	corollary	corollary	ADJ
ejpam-6543	332	34	4	4	NUM
ejpam-6543	332	35	.	.	PUNCT
ejpam-6543	333	1	if	if	SCONJ
ejpam-6543	333	2	x	x	PRON
ejpam-6543	333	3	is	be	AUX
ejpam-6543	333	4	nearly	nearly	ADV
ejpam-6543	333	5	α	α	ADJ
ejpam-6543	333	6	-	-	ADJ
ejpam-6543	333	7	compact	compact	ADJ
ejpam-6543	333	8	and	and	CCONJ
ejpam-6543	333	9	y	y	PROPN
ejpam-6543	333	10	is	be	AUX
ejpam-6543	333	11	α	α	NOUN
ejpam-6543	333	12	-	-	ADJ
ejpam-6543	333	13	compact	compact	ADJ
ejpam-6543	333	14	,	,	PUNCT
ejpam-6543	333	15	then	then	ADV
ejpam-6543	333	16	cα(x	cα(x	PUNCT
ejpam-6543	333	17	,	,	PUNCT
ejpam-6543	333	18	y	y	PROPN
ejpam-6543	333	19	)	)	PUNCT
ejpam-6543	333	20	is	be	AUX
ejpam-6543	333	21	α	α	PRON
ejpam-6543	333	22	-	-	ADJ
ejpam-6543	333	23	compact	compact	ADJ
ejpam-6543	333	24	.	.	PUNCT
ejpam-6543	334	1	proof	proof	NOUN
ejpam-6543	334	2	.	.	PUNCT
ejpam-6543	335	1	since	since	SCONJ
ejpam-6543	335	2	y	y	PROPN
ejpam-6543	335	3	is	be	AUX
ejpam-6543	335	4	α	α	PRON
ejpam-6543	335	5	-	-	ADJ
ejpam-6543	335	6	compact	compact	ADJ
ejpam-6543	335	7	,	,	PUNCT
ejpam-6543	335	8	it	it	PRON
ejpam-6543	335	9	is	be	AUX
ejpam-6543	335	10	α	α	NOUN
ejpam-6543	335	11	-	-	ADJ
ejpam-6543	335	12	regular	regular	ADJ
ejpam-6543	335	13	.	.	PUNCT
ejpam-6543	336	1	by	by	ADP
ejpam-6543	336	2	theorem	theorem	NOUN
ejpam-6543	336	3	7	7	NUM
ejpam-6543	336	4	,	,	PUNCT
ejpam-6543	336	5	cα(x	cα(x	NUM
ejpam-6543	336	6	,	,	PUNCT
ejpam-6543	336	7	y	y	PROPN
ejpam-6543	336	8	)	)	PUNCT
ejpam-6543	336	9	is	be	AUX
ejpam-6543	336	10	nearly	nearly	ADV
ejpam-6543	336	11	α	α	NOUN
ejpam-6543	336	12	-	-	ADJ
ejpam-6543	336	13	compact	compact	ADJ
ejpam-6543	336	14	.	.	PUNCT
ejpam-6543	337	1	the	the	DET
ejpam-6543	337	2	additional	additional	ADJ
ejpam-6543	337	3	compactness	compactness	NOUN
ejpam-6543	337	4	of	of	ADP
ejpam-6543	337	5	y	y	PROPN
ejpam-6543	337	6	ensures	ensure	VERB
ejpam-6543	337	7	that	that	SCONJ
ejpam-6543	337	8	the	the	DET
ejpam-6543	337	9	uniform	uniform	ADJ
ejpam-6543	337	10	convergence	convergence	NOUN
ejpam-6543	337	11	topology	topology	NOUN
ejpam-6543	337	12	on	on	ADP
ejpam-6543	337	13	cα(x	cα(x	PROPN
ejpam-6543	337	14	,	,	PUNCT
ejpam-6543	337	15	y	y	PROPN
ejpam-6543	337	16	)	)	PUNCT
ejpam-6543	337	17	has	have	VERB
ejpam-6543	337	18	stronger	strong	ADJ
ejpam-6543	337	19	properties	property	NOUN
ejpam-6543	337	20	that	that	PRON
ejpam-6543	337	21	promote	promote	VERB
ejpam-6543	337	22	nearly	nearly	ADV
ejpam-6543	337	23	α	α	NOUN
ejpam-6543	337	24	-	-	NOUN
ejpam-6543	337	25	compactness	compactness	NOUN
ejpam-6543	337	26	to	to	ADP
ejpam-6543	337	27	full	full	ADJ
ejpam-6543	337	28	αcompactness	αcompactness	NOUN
ejpam-6543	337	29	.	.	PUNCT
ejpam-6543	338	1	specifically	specifically	ADV
ejpam-6543	338	2	,	,	PUNCT
ejpam-6543	338	3	the	the	DET
ejpam-6543	338	4	α	α	NOUN
ejpam-6543	338	5	-	-	NOUN
ejpam-6543	338	6	compactness	compactness	NOUN
ejpam-6543	338	7	of	of	ADP
ejpam-6543	338	8	y	y	PROPN
ejpam-6543	338	9	implies	imply	VERB
ejpam-6543	338	10	that	that	SCONJ
ejpam-6543	338	11	every	every	DET
ejpam-6543	338	12	sequence	sequence	NOUN
ejpam-6543	338	13	in	in	ADP
ejpam-6543	338	14	cα(x	cα(x	PROPN
ejpam-6543	338	15	,	,	PUNCT
ejpam-6543	338	16	y	y	PROPN
ejpam-6543	338	17	)	)	PUNCT
ejpam-6543	338	18	has	have	VERB
ejpam-6543	338	19	a	a	DET
ejpam-6543	338	20	uniformly	uniformly	ADV
ejpam-6543	338	21	convergent	convergent	ADJ
ejpam-6543	338	22	subsequence	subsequence	NOUN
ejpam-6543	338	23	on	on	ADP
ejpam-6543	338	24	α	α	ADJ
ejpam-6543	338	25	-	-	ADJ
ejpam-6543	338	26	compact	compact	ADJ
ejpam-6543	338	27	subsets	subset	NOUN
ejpam-6543	338	28	of	of	ADP
ejpam-6543	338	29	x.	x.	NOUN
ejpam-6543	338	30	combined	combine	VERB
ejpam-6543	338	31	with	with	ADP
ejpam-6543	338	32	the	the	DET
ejpam-6543	338	33	nearly	nearly	ADV
ejpam-6543	338	34	α	α	NUM
ejpam-6543	338	35	-	-	ADJ
ejpam-6543	338	36	compact	compact	ADJ
ejpam-6543	338	37	property	property	NOUN
ejpam-6543	338	38	,	,	PUNCT
ejpam-6543	338	39	this	this	DET
ejpam-6543	338	40	forces	force	NOUN
ejpam-6543	338	41	cα(x	cα(x	PUNCT
ejpam-6543	338	42	,	,	PUNCT
ejpam-6543	338	43	y	y	PROPN
ejpam-6543	338	44	)	)	PUNCT
ejpam-6543	338	45	to	to	PART
ejpam-6543	338	46	be	be	AUX
ejpam-6543	338	47	α	α	PRON
ejpam-6543	338	48	-	-	ADJ
ejpam-6543	338	49	compact	compact	ADJ
ejpam-6543	338	50	.	.	PUNCT
ejpam-6543	339	1	j.	j.	PROPN
ejpam-6543	339	2	oudetallah	oudetallah	PROPN
ejpam-6543	339	3	et	et	PROPN
ejpam-6543	339	4	al	al	PROPN
ejpam-6543	339	5	.	.	PUNCT
ejpam-6543	339	6	/	/	SYM
ejpam-6543	339	7	eur	eur	PROPN
ejpam-6543	339	8	.	.	PUNCT
ejpam-6543	340	1	j.	j.	PROPN
ejpam-6543	340	2	pure	pure	PROPN
ejpam-6543	340	3	appl	appl	PROPN
ejpam-6543	340	4	.	.	PROPN
ejpam-6543	340	5	math	math	PROPN
ejpam-6543	340	6	,	,	PUNCT
ejpam-6543	340	7	18	18	NUM
ejpam-6543	340	8	(	(	PUNCT
ejpam-6543	340	9	3	3	NUM
ejpam-6543	340	10	)	)	PUNCT
ejpam-6543	340	11	(	(	PUNCT
ejpam-6543	340	12	2025	2025	NUM
ejpam-6543	340	13	)	)	PUNCT
ejpam-6543	340	14	,	,	PUNCT
ejpam-6543	340	15	6543	6543	NUM
ejpam-6543	340	16	12	12	NUM
ejpam-6543	340	17	of	of	ADP
ejpam-6543	340	18	13	13	NUM
ejpam-6543	340	19	8	8	NUM
ejpam-6543	340	20	.	.	PUNCT
ejpam-6543	341	1	conclusions	conclusion	NOUN
ejpam-6543	341	2	in	in	ADP
ejpam-6543	341	3	this	this	DET
ejpam-6543	341	4	paper	paper	NOUN
ejpam-6543	341	5	,	,	PUNCT
ejpam-6543	341	6	we	we	PRON
ejpam-6543	341	7	have	have	AUX
ejpam-6543	341	8	introduced	introduce	VERB
ejpam-6543	341	9	and	and	CCONJ
ejpam-6543	341	10	systematically	systematically	ADV
ejpam-6543	341	11	studied	study	VERB
ejpam-6543	341	12	nearly	nearly	ADV
ejpam-6543	341	13	α	α	ADJ
ejpam-6543	341	14	-	-	ADJ
ejpam-6543	341	15	compact	compact	ADJ
ejpam-6543	341	16	topological	topological	ADJ
ejpam-6543	341	17	spaces	space	NOUN
ejpam-6543	341	18	as	as	ADP
ejpam-6543	341	19	a	a	DET
ejpam-6543	341	20	natural	natural	ADJ
ejpam-6543	341	21	generalization	generalization	NOUN
ejpam-6543	341	22	of	of	ADP
ejpam-6543	341	23	α	α	NOUN
ejpam-6543	341	24	-	-	NOUN
ejpam-6543	341	25	compactness	compactness	NOUN
ejpam-6543	341	26	.	.	PUNCT
ejpam-6543	342	1	our	our	PRON
ejpam-6543	342	2	investigation	investigation	NOUN
ejpam-6543	342	3	has	have	AUX
ejpam-6543	342	4	revealed	reveal	VERB
ejpam-6543	342	5	that	that	SCONJ
ejpam-6543	342	6	this	this	DET
ejpam-6543	342	7	concept	concept	NOUN
ejpam-6543	342	8	provides	provide	VERB
ejpam-6543	342	9	a	a	DET
ejpam-6543	342	10	useful	useful	ADJ
ejpam-6543	342	11	middle	middle	ADJ
ejpam-6543	342	12	ground	ground	NOUN
ejpam-6543	342	13	between	between	ADP
ejpam-6543	342	14	full	full	ADJ
ejpam-6543	342	15	α	α	NOUN
ejpam-6543	342	16	-	-	PUNCT
ejpam-6543	342	17	compactness	compactness	NOUN
ejpam-6543	342	18	and	and	CCONJ
ejpam-6543	342	19	weaker	weak	ADJ
ejpam-6543	342	20	forms	form	NOUN
ejpam-6543	342	21	of	of	ADP
ejpam-6543	342	22	compactness	compactness	NOUN
ejpam-6543	342	23	-	-	PUNCT
ejpam-6543	342	24	like	like	ADJ
ejpam-6543	342	25	properties	property	NOUN
ejpam-6543	342	26	.	.	PUNCT
ejpam-6543	343	1	the	the	DET
ejpam-6543	343	2	main	main	ADJ
ejpam-6543	343	3	contributions	contribution	NOUN
ejpam-6543	343	4	of	of	ADP
ejpam-6543	343	5	this	this	DET
ejpam-6543	343	6	work	work	NOUN
ejpam-6543	343	7	include	include	VERB
ejpam-6543	343	8	:	:	PUNCT
ejpam-6543	343	9	(	(	PUNCT
ejpam-6543	343	10	i	i	NOUN
ejpam-6543	343	11	)	)	PUNCT
ejpam-6543	343	12	a	a	DET
ejpam-6543	343	13	comprehensive	comprehensive	ADJ
ejpam-6543	343	14	characterization	characterization	NOUN
ejpam-6543	343	15	of	of	ADP
ejpam-6543	343	16	nearly	nearly	ADV
ejpam-6543	343	17	α	α	ADJ
ejpam-6543	343	18	-	-	ADJ
ejpam-6543	343	19	compact	compact	ADJ
ejpam-6543	343	20	spaces	space	NOUN
ejpam-6543	343	21	through	through	ADP
ejpam-6543	343	22	multiple	multiple	ADJ
ejpam-6543	343	23	equivalent	equivalent	ADJ
ejpam-6543	343	24	conditions	condition	NOUN
ejpam-6543	343	25	(	(	PUNCT
ejpam-6543	343	26	theorem	theorem	NOUN
ejpam-6543	343	27	3	3	NUM
ejpam-6543	343	28	)	)	PUNCT
ejpam-6543	343	29	.	.	PUNCT
ejpam-6543	344	1	(	(	PUNCT
ejpam-6543	344	2	ii	ii	NOUN
ejpam-6543	344	3	)	)	PUNCT
ejpam-6543	344	4	a	a	DET
ejpam-6543	344	5	thorough	thorough	ADJ
ejpam-6543	344	6	investigation	investigation	NOUN
ejpam-6543	344	7	of	of	ADP
ejpam-6543	344	8	the	the	DET
ejpam-6543	344	9	behavior	behavior	NOUN
ejpam-6543	344	10	of	of	ADP
ejpam-6543	344	11	nearly	nearly	ADV
ejpam-6543	344	12	α	α	NOUN
ejpam-6543	344	13	-	-	NOUN
ejpam-6543	344	14	compactness	compactness	NOUN
ejpam-6543	344	15	under	under	ADP
ejpam-6543	344	16	standard	standard	ADJ
ejpam-6543	344	17	topological	topological	ADJ
ejpam-6543	344	18	operations	operation	NOUN
ejpam-6543	344	19	,	,	PUNCT
ejpam-6543	344	20	including	include	VERB
ejpam-6543	344	21	products	product	NOUN
ejpam-6543	344	22	(	(	PUNCT
ejpam-6543	344	23	theorem	theorem	NOUN
ejpam-6543	344	24	5	5	NUM
ejpam-6543	344	25	)	)	PUNCT
ejpam-6543	344	26	and	and	CCONJ
ejpam-6543	344	27	subspaces	subspace	NOUN
ejpam-6543	344	28	(	(	PUNCT
ejpam-6543	344	29	proposition	proposition	NOUN
ejpam-6543	344	30	2	2	NUM
ejpam-6543	344	31	)	)	PUNCT
ejpam-6543	344	32	.	.	PUNCT
ejpam-6543	345	1	(	(	PUNCT
ejpam-6543	345	2	iii	iii	X
ejpam-6543	345	3	)	)	PUNCT
ejpam-6543	345	4	the	the	DET
ejpam-6543	345	5	introduction	introduction	NOUN
ejpam-6543	345	6	of	of	ADP
ejpam-6543	345	7	α	α	NOUN
ejpam-6543	345	8	-	-	NOUN
ejpam-6543	345	9	nearness	nearness	NOUN
ejpam-6543	345	10	as	as	ADP
ejpam-6543	345	11	a	a	DET
ejpam-6543	345	12	related	related	ADJ
ejpam-6543	345	13	concept	concept	NOUN
ejpam-6543	345	14	that	that	PRON
ejpam-6543	345	15	provides	provide	VERB
ejpam-6543	345	16	additional	additional	ADJ
ejpam-6543	345	17	insight	insight	NOUN
ejpam-6543	345	18	into	into	ADP
ejpam-6543	345	19	the	the	DET
ejpam-6543	345	20	structure	structure	NOUN
ejpam-6543	345	21	of	of	ADP
ejpam-6543	345	22	these	these	DET
ejpam-6543	345	23	spaces	space	NOUN
ejpam-6543	345	24	(	(	PUNCT
ejpam-6543	345	25	definition	definition	NOUN
ejpam-6543	345	26	8	8	NUM
ejpam-6543	345	27	and	and	CCONJ
ejpam-6543	345	28	theorem	theorem	VERB
ejpam-6543	345	29	6	6	NUM
ejpam-6543	345	30	)	)	PUNCT
ejpam-6543	345	31	.	.	PUNCT
ejpam-6543	346	1	(	(	PUNCT
ejpam-6543	346	2	iv	iv	X
ejpam-6543	346	3	)	)	PUNCT
ejpam-6543	346	4	applications	application	NOUN
ejpam-6543	346	5	to	to	PART
ejpam-6543	346	6	function	function	VERB
ejpam-6543	346	7	space	space	NOUN
ejpam-6543	346	8	topology	topology	NOUN
ejpam-6543	346	9	that	that	PRON
ejpam-6543	346	10	demonstrate	demonstrate	VERB
ejpam-6543	346	11	the	the	DET
ejpam-6543	346	12	utility	utility	NOUN
ejpam-6543	346	13	of	of	ADP
ejpam-6543	346	14	the	the	DET
ejpam-6543	346	15	concept	concept	NOUN
ejpam-6543	346	16	(	(	PUNCT
ejpam-6543	346	17	theorem	theorem	NOUN
ejpam-6543	346	18	7	7	NUM
ejpam-6543	346	19	)	)	PUNCT
ejpam-6543	346	20	.	.	PUNCT
ejpam-6543	347	1	(	(	PUNCT
ejpam-6543	347	2	v	v	X
ejpam-6543	347	3	)	)	PUNCT
ejpam-6543	347	4	comprehensive	comprehensive	ADJ
ejpam-6543	347	5	examples	example	NOUN
ejpam-6543	347	6	that	that	PRON
ejpam-6543	347	7	illustrate	illustrate	VERB
ejpam-6543	347	8	the	the	DET
ejpam-6543	347	9	theory	theory	NOUN
ejpam-6543	347	10	and	and	CCONJ
ejpam-6543	347	11	show	show	VERB
ejpam-6543	347	12	the	the	DET
ejpam-6543	347	13	distinction	distinction	NOUN
ejpam-6543	347	14	between	between	ADP
ejpam-6543	347	15	nearly	nearly	ADV
ejpam-6543	347	16	α	α	NOUN
ejpam-6543	347	17	-	-	ADJ
ejpam-6543	347	18	compact	compact	ADJ
ejpam-6543	347	19	and	and	CCONJ
ejpam-6543	347	20	related	related	ADJ
ejpam-6543	347	21	concepts	concept	NOUN
ejpam-6543	347	22	.	.	PUNCT
ejpam-6543	348	1	(	(	PUNCT
ejpam-6543	348	2	vi	vi	NOUN
ejpam-6543	348	3	)	)	PUNCT
ejpam-6543	348	4	clarification	clarification	NOUN
ejpam-6543	348	5	of	of	ADP
ejpam-6543	348	6	the	the	DET
ejpam-6543	348	7	relationships	relationship	NOUN
ejpam-6543	348	8	between	between	ADP
ejpam-6543	348	9	nearly	nearly	ADV
ejpam-6543	348	10	α	α	NOUN
ejpam-6543	348	11	-	-	PUNCT
ejpam-6543	348	12	compactness	compactness	NOUN
ejpam-6543	348	13	and	and	CCONJ
ejpam-6543	348	14	classical	classical	ADJ
ejpam-6543	348	15	compactness	compactness	NOUN
ejpam-6543	348	16	properties	property	NOUN
ejpam-6543	348	17	,	,	PUNCT
ejpam-6543	348	18	including	include	VERB
ejpam-6543	348	19	the	the	DET
ejpam-6543	348	20	exploration	exploration	NOUN
ejpam-6543	348	21	of	of	ADP
ejpam-6543	348	22	converses	converse	NOUN
ejpam-6543	348	23	and	and	CCONJ
ejpam-6543	348	24	counterexamples	counterexample	NOUN
ejpam-6543	348	25	.	.	PUNCT
ejpam-6543	349	1	the	the	DET
ejpam-6543	349	2	relationship	relationship	NOUN
ejpam-6543	349	3	between	between	ADP
ejpam-6543	349	4	nearly	nearly	ADV
ejpam-6543	349	5	α	α	NUM
ejpam-6543	349	6	-	-	ADJ
ejpam-6543	349	7	compact	compact	ADJ
ejpam-6543	349	8	spaces	space	NOUN
ejpam-6543	349	9	and	and	CCONJ
ejpam-6543	349	10	classical	classical	ADJ
ejpam-6543	349	11	compactness	compactness	NOUN
ejpam-6543	349	12	notions	notion	NOUN
ejpam-6543	349	13	has	have	AUX
ejpam-6543	349	14	been	be	AUX
ejpam-6543	349	15	clarified	clarify	VERB
ejpam-6543	349	16	through	through	ADP
ejpam-6543	349	17	our	our	PRON
ejpam-6543	349	18	characterization	characterization	NOUN
ejpam-6543	349	19	theorems	theorem	NOUN
ejpam-6543	349	20	.	.	PUNCT
ejpam-6543	350	1	we	we	PRON
ejpam-6543	350	2	have	have	AUX
ejpam-6543	350	3	shown	show	VERB
ejpam-6543	350	4	that	that	SCONJ
ejpam-6543	350	5	every	every	DET
ejpam-6543	350	6	α	α	X
ejpam-6543	350	7	-	-	ADJ
ejpam-6543	350	8	compact	compact	ADJ
ejpam-6543	350	9	space	space	NOUN
ejpam-6543	350	10	is	be	AUX
ejpam-6543	350	11	nearly	nearly	ADV
ejpam-6543	350	12	α	α	NOUN
ejpam-6543	350	13	-	-	ADJ
ejpam-6543	350	14	compact	compact	ADJ
ejpam-6543	350	15	,	,	PUNCT
ejpam-6543	350	16	but	but	CCONJ
ejpam-6543	350	17	the	the	DET
ejpam-6543	350	18	converse	converse	NOUN
ejpam-6543	350	19	holds	hold	VERB
ejpam-6543	350	20	only	only	ADV
ejpam-6543	350	21	under	under	ADP
ejpam-6543	350	22	additional	additional	ADJ
ejpam-6543	350	23	separation	separation	NOUN
ejpam-6543	350	24	conditions	condition	NOUN
ejpam-6543	350	25	.	.	PUNCT
ejpam-6543	351	1	several	several	ADJ
ejpam-6543	351	2	avenues	avenue	NOUN
ejpam-6543	351	3	for	for	ADP
ejpam-6543	351	4	future	future	ADJ
ejpam-6543	351	5	research	research	NOUN
ejpam-6543	351	6	present	present	ADJ
ejpam-6543	351	7	themselves	themselves	PRON
ejpam-6543	351	8	.	.	PUNCT
ejpam-6543	352	1	the	the	DET
ejpam-6543	352	2	relationship	relationship	NOUN
ejpam-6543	352	3	between	between	ADP
ejpam-6543	352	4	nearly	nearly	ADV
ejpam-6543	352	5	α	α	NUM
ejpam-6543	352	6	-	-	ADJ
ejpam-6543	352	7	compact	compact	ADJ
ejpam-6543	352	8	spaces	space	NOUN
ejpam-6543	352	9	and	and	CCONJ
ejpam-6543	352	10	other	other	ADJ
ejpam-6543	352	11	generalized	generalized	ADJ
ejpam-6543	352	12	compactness	compactness	NOUN
ejpam-6543	352	13	properties	property	NOUN
ejpam-6543	352	14	deserves	deserve	VERB
ejpam-6543	352	15	further	further	ADJ
ejpam-6543	352	16	investigation	investigation	NOUN
ejpam-6543	352	17	.	.	PUNCT
ejpam-6543	353	1	the	the	DET
ejpam-6543	353	2	potential	potential	ADJ
ejpam-6543	353	3	applications	application	NOUN
ejpam-6543	353	4	to	to	PART
ejpam-6543	353	5	convergence	convergence	NOUN
ejpam-6543	353	6	theory	theory	NOUN
ejpam-6543	353	7	and	and	CCONJ
ejpam-6543	353	8	the	the	DET
ejpam-6543	353	9	study	study	NOUN
ejpam-6543	353	10	of	of	ADP
ejpam-6543	353	11	function	function	NOUN
ejpam-6543	353	12	spaces	space	NOUN
ejpam-6543	353	13	with	with	ADP
ejpam-6543	353	14	different	different	ADJ
ejpam-6543	353	15	topologies	topology	NOUN
ejpam-6543	353	16	also	also	ADV
ejpam-6543	353	17	warrant	warrant	VERB
ejpam-6543	353	18	attention	attention	NOUN
ejpam-6543	353	19	.	.	PUNCT
ejpam-6543	354	1	additionally	additionally	ADV
ejpam-6543	354	2	,	,	PUNCT
ejpam-6543	354	3	the	the	DET
ejpam-6543	354	4	investigation	investigation	NOUN
ejpam-6543	354	5	of	of	ADP
ejpam-6543	354	6	nearly	nearly	ADV
ejpam-6543	354	7	α	α	ADJ
ejpam-6543	354	8	-	-	ADJ
ejpam-6543	354	9	compact	compact	ADJ
ejpam-6543	354	10	spaces	space	NOUN
ejpam-6543	354	11	in	in	ADP
ejpam-6543	354	12	the	the	DET
ejpam-6543	354	13	context	context	NOUN
ejpam-6543	354	14	of	of	ADP
ejpam-6543	354	15	bitopological	bitopological	ADJ
ejpam-6543	354	16	spaces	space	NOUN
ejpam-6543	354	17	,	,	PUNCT
ejpam-6543	354	18	following	follow	VERB
ejpam-6543	354	19	the	the	DET
ejpam-6543	354	20	work	work	NOUN
ejpam-6543	354	21	of	of	ADP
ejpam-6543	354	22	oudetallah	oudetallah	NOUN
ejpam-6543	355	1	[	[	X
ejpam-6543	355	2	5	5	NUM
ejpam-6543	355	3	]	]	PUNCT
ejpam-6543	355	4	,	,	PUNCT
ejpam-6543	355	5	could	could	AUX
ejpam-6543	355	6	yield	yield	VERB
ejpam-6543	355	7	interesting	interesting	ADJ
ejpam-6543	355	8	results	result	NOUN
ejpam-6543	355	9	.	.	PUNCT
ejpam-6543	356	1	recent	recent	ADJ
ejpam-6543	356	2	developments	development	NOUN
ejpam-6543	356	3	in	in	ADP
ejpam-6543	356	4	computational	computational	ADJ
ejpam-6543	356	5	topology	topology	NOUN
ejpam-6543	356	6	and	and	CCONJ
ejpam-6543	356	7	data	datum	NOUN
ejpam-6543	356	8	analysis	analysis	NOUN
ejpam-6543	356	9	suggest	suggest	VERB
ejpam-6543	356	10	that	that	SCONJ
ejpam-6543	356	11	nearly	nearly	ADV
ejpam-6543	356	12	α	α	NUM
ejpam-6543	356	13	-	-	ADJ
ejpam-6543	356	14	compact	compact	ADJ
ejpam-6543	356	15	spaces	space	NOUN
ejpam-6543	356	16	may	may	AUX
ejpam-6543	356	17	have	have	VERB
ejpam-6543	356	18	applications	application	NOUN
ejpam-6543	356	19	in	in	ADP
ejpam-6543	356	20	these	these	DET
ejpam-6543	356	21	fields	field	NOUN
ejpam-6543	356	22	,	,	PUNCT
ejpam-6543	356	23	particularly	particularly	ADV
ejpam-6543	356	24	in	in	ADP
ejpam-6543	356	25	the	the	DET
ejpam-6543	356	26	study	study	NOUN
ejpam-6543	356	27	of	of	ADP
ejpam-6543	356	28	persistent	persistent	ADJ
ejpam-6543	356	29	homology	homology	NOUN
ejpam-6543	356	30	and	and	CCONJ
ejpam-6543	356	31	topological	topological	ADJ
ejpam-6543	356	32	data	datum	NOUN
ejpam-6543	356	33	analysis	analysis	NOUN
ejpam-6543	356	34	.	.	PUNCT
ejpam-6543	357	1	the	the	DET
ejpam-6543	357	2	results	result	NOUN
ejpam-6543	357	3	presented	present	VERB
ejpam-6543	357	4	here	here	ADV
ejpam-6543	357	5	contribute	contribute	VERB
ejpam-6543	357	6	to	to	ADP
ejpam-6543	357	7	the	the	DET
ejpam-6543	357	8	ongoing	ongoing	ADJ
ejpam-6543	357	9	development	development	NOUN
ejpam-6543	357	10	of	of	ADP
ejpam-6543	357	11	generalized	generalized	ADJ
ejpam-6543	357	12	topology	topology	NOUN
ejpam-6543	357	13	and	and	CCONJ
ejpam-6543	357	14	provide	provide	VERB
ejpam-6543	357	15	tools	tool	NOUN
ejpam-6543	357	16	that	that	PRON
ejpam-6543	357	17	may	may	AUX
ejpam-6543	357	18	prove	prove	VERB
ejpam-6543	357	19	useful	useful	ADJ
ejpam-6543	357	20	in	in	ADP
ejpam-6543	357	21	both	both	CCONJ
ejpam-6543	357	22	theoretical	theoretical	ADJ
ejpam-6543	357	23	investigations	investigation	NOUN
ejpam-6543	357	24	and	and	CCONJ
ejpam-6543	357	25	practical	practical	ADJ
ejpam-6543	357	26	applications	application	NOUN
ejpam-6543	357	27	.	.	PUNCT
ejpam-6543	358	1	the	the	DET
ejpam-6543	358	2	concept	concept	NOUN
ejpam-6543	358	3	of	of	ADP
ejpam-6543	358	4	nearly	nearly	ADV
ejpam-6543	358	5	α	α	NOUN
ejpam-6543	358	6	-	-	PUNCT
ejpam-6543	358	7	compactness	compactness	NOUN
ejpam-6543	358	8	fills	fill	VERB
ejpam-6543	358	9	a	a	DET
ejpam-6543	358	10	gap	gap	NOUN
ejpam-6543	358	11	in	in	ADP
ejpam-6543	358	12	the	the	DET
ejpam-6543	358	13	hierarchy	hierarchy	NOUN
ejpam-6543	358	14	of	of	ADP
ejpam-6543	358	15	compactness	compactness	NOUN
ejpam-6543	358	16	-	-	PUNCT
ejpam-6543	358	17	like	like	ADJ
ejpam-6543	358	18	properties	property	NOUN
ejpam-6543	358	19	and	and	CCONJ
ejpam-6543	358	20	offers	offer	VERB
ejpam-6543	358	21	new	new	ADJ
ejpam-6543	358	22	perspectives	perspective	NOUN
ejpam-6543	358	23	on	on	ADP
ejpam-6543	358	24	the	the	DET
ejpam-6543	358	25	structure	structure	NOUN
ejpam-6543	358	26	of	of	ADP
ejpam-6543	358	27	topological	topological	ADJ
ejpam-6543	358	28	spaces	space	NOUN
ejpam-6543	358	29	.	.	PUNCT
ejpam-6543	359	1	acknowledgements	acknowledgement	NOUN
ejpam-6543	359	2	the	the	DET
ejpam-6543	359	3	authors	author	NOUN
ejpam-6543	359	4	thank	thank	VERB
ejpam-6543	359	5	the	the	DET
ejpam-6543	359	6	editors	editor	NOUN
ejpam-6543	359	7	and	and	CCONJ
ejpam-6543	359	8	reviewers	reviewer	NOUN
ejpam-6543	359	9	of	of	ADP
ejpam-6543	359	10	the	the	DET
ejpam-6543	359	11	european	european	PROPN
ejpam-6543	359	12	journal	journal	PROPN
ejpam-6543	359	13	of	of	ADP
ejpam-6543	359	14	pure	pure	ADJ
ejpam-6543	359	15	and	and	CCONJ
ejpam-6543	359	16	applied	applied	ADJ
ejpam-6543	359	17	mathematics	mathematic	NOUN
ejpam-6543	359	18	for	for	ADP
ejpam-6543	359	19	their	their	PRON
ejpam-6543	359	20	valuable	valuable	ADJ
ejpam-6543	359	21	comments	comment	NOUN
ejpam-6543	359	22	and	and	CCONJ
ejpam-6543	359	23	suggestions	suggestion	NOUN
ejpam-6543	359	24	that	that	PRON
ejpam-6543	359	25	significantly	significantly	ADV
ejpam-6543	359	26	improved	improve	VERB
ejpam-6543	359	27	this	this	DET
ejpam-6543	359	28	paper	paper	NOUN
ejpam-6543	359	29	.	.	PUNCT
ejpam-6543	360	1	j.	j.	PROPN
ejpam-6543	360	2	oudetallah	oudetallah	PROPN
ejpam-6543	360	3	et	et	PROPN
ejpam-6543	360	4	al	al	PROPN
ejpam-6543	360	5	.	.	PUNCT
ejpam-6543	360	6	/	/	SYM
ejpam-6543	360	7	eur	eur	PROPN
ejpam-6543	360	8	.	.	PUNCT
ejpam-6543	361	1	j.	j.	PROPN
ejpam-6543	361	2	pure	pure	PROPN
ejpam-6543	361	3	appl	appl	PROPN
ejpam-6543	361	4	.	.	PROPN
ejpam-6543	361	5	math	math	PROPN
ejpam-6543	361	6	,	,	PUNCT
ejpam-6543	361	7	18	18	NUM
ejpam-6543	361	8	(	(	PUNCT
ejpam-6543	361	9	3	3	NUM
ejpam-6543	361	10	)	)	PUNCT
ejpam-6543	361	11	(	(	PUNCT
ejpam-6543	361	12	2025	2025	NUM
ejpam-6543	361	13	)	)	PUNCT
ejpam-6543	361	14	,	,	PUNCT
ejpam-6543	361	15	6543	6543	NUM
ejpam-6543	361	16	13	13	NUM
ejpam-6543	361	17	of	of	ADP
ejpam-6543	361	18	13	13	NUM
ejpam-6543	361	19	references	reference	NOUN
ejpam-6543	361	20	[	[	X
ejpam-6543	361	21	1	1	NUM
ejpam-6543	361	22	]	]	PUNCT
ejpam-6543	361	23	olav	olav	PROPN
ejpam-6543	361	24	njstad	njstad	PROPN
ejpam-6543	361	25	.	.	PUNCT
ejpam-6543	362	1	on	on	ADP
ejpam-6543	362	2	some	some	DET
ejpam-6543	362	3	classes	class	NOUN
ejpam-6543	362	4	of	of	ADP
ejpam-6543	362	5	nearly	nearly	ADV
ejpam-6543	362	6	open	open	ADJ
ejpam-6543	362	7	sets	set	NOUN
ejpam-6543	362	8	.	.	PUNCT
ejpam-6543	363	1	pacific	pacific	PROPN
ejpam-6543	363	2	journal	journal	PROPN
ejpam-6543	363	3	of	of	ADP
ejpam-6543	363	4	mathematics	mathematic	NOUN
ejpam-6543	363	5	,	,	PUNCT
ejpam-6543	363	6	15(3):961–970	15(3):961–970	PROPN
ejpam-6543	363	7	,	,	PUNCT
ejpam-6543	363	8	1965	1965	NUM
ejpam-6543	363	9	.	.	PUNCT
ejpam-6543	364	1	[	[	X
ejpam-6543	364	2	2	2	NUM
ejpam-6543	364	3	]	]	PUNCT
ejpam-6543	364	4	as	as	ADP
ejpam-6543	364	5	mashhour	mashhour	ADJ
ejpam-6543	364	6	,	,	PUNCT
ejpam-6543	364	7	ia	ia	PROPN
ejpam-6543	364	8	hasanein	hasanein	NOUN
ejpam-6543	364	9	,	,	PUNCT
ejpam-6543	364	10	and	and	CCONJ
ejpam-6543	364	11	sn	sn	PROPN
ejpam-6543	364	12	el	el	PROPN
ejpam-6543	364	13	-	-	PUNCT
ejpam-6543	364	14	deeb	deeb	PROPN
ejpam-6543	364	15	.	.	PUNCT
ejpam-6543	365	1	α	α	X
ejpam-6543	365	2	-	-	ADJ
ejpam-6543	365	3	continuous	continuous	ADJ
ejpam-6543	365	4	and	and	CCONJ
ejpam-6543	365	5	α	α	NOUN
ejpam-6543	365	6	-	-	ADJ
ejpam-6543	365	7	open	open	ADJ
ejpam-6543	365	8	mappings	mapping	NOUN
ejpam-6543	365	9	.	.	PUNCT
ejpam-6543	366	1	acta	acta	PROPN
ejpam-6543	366	2	mathematica	mathematica	PROPN
ejpam-6543	366	3	hungarica	hungarica	PROPN
ejpam-6543	366	4	,	,	PUNCT
ejpam-6543	366	5	41(3):213–218	41(3):213–218	NOUN
ejpam-6543	366	6	,	,	PUNCT
ejpam-6543	366	7	1983	1983	NUM
ejpam-6543	366	8	.	.	PUNCT
ejpam-6543	367	1	[	[	X
ejpam-6543	367	2	3	3	X
ejpam-6543	367	3	]	]	X
ejpam-6543	367	4	jamal	jamal	PROPN
ejpam-6543	367	5	oudetallah	oudetallah	PROPN
ejpam-6543	367	6	,	,	PUNCT
ejpam-6543	367	7	rehab	rehab	NOUN
ejpam-6543	367	8	alharbi	alharbi	NOUN
ejpam-6543	367	9	,	,	PUNCT
ejpam-6543	367	10	iqbal	iqbal	PROPN
ejpam-6543	367	11	batiha	batiha	PROPN
ejpam-6543	367	12	,	,	PUNCT
ejpam-6543	367	13	salsabiela	salsabiela	PROPN
ejpam-6543	367	14	rawashdeh	rawashdeh	PROPN
ejpam-6543	367	15	,	,	PUNCT
ejpam-6543	367	16	and	and	CCONJ
ejpam-6543	367	17	ala	ala	PROPN
ejpam-6543	367	18	amourah	amourah	PROPN
ejpam-6543	367	19	.	.	PUNCT
ejpam-6543	368	1	some	some	DET
ejpam-6543	368	2	types	type	NOUN
ejpam-6543	368	3	of	of	ADP
ejpam-6543	368	4	tri	tri	ADJ
ejpam-6543	368	5	-	-	ADJ
ejpam-6543	368	6	lindelöfness	lindelöfness	ADJ
ejpam-6543	368	7	spaces	space	NOUN
ejpam-6543	368	8	.	.	PUNCT
ejpam-6543	369	1	european	european	ADJ
ejpam-6543	369	2	journal	journal	PROPN
ejpam-6543	369	3	of	of	ADP
ejpam-6543	369	4	pure	pure	ADJ
ejpam-6543	369	5	and	and	CCONJ
ejpam-6543	369	6	applied	applied	ADJ
ejpam-6543	369	7	mathematics	mathematic	NOUN
ejpam-6543	369	8	,	,	PUNCT
ejpam-6543	369	9	18(2):5578–5578	18(2):5578–5578	NUM
ejpam-6543	369	10	,	,	PUNCT
ejpam-6543	369	11	2025	2025	NUM
ejpam-6543	369	12	.	.	PUNCT
ejpam-6543	370	1	[	[	X
ejpam-6543	370	2	4	4	X
ejpam-6543	370	3	]	]	X
ejpam-6543	370	4	a	a	DET
ejpam-6543	370	5	haydar	haydar	PROPN
ejpam-6543	370	6	eş.	eş.	VERB
ejpam-6543	370	7	almost	almost	ADV
ejpam-6543	370	8	compactness	compactness	NOUN
ejpam-6543	370	9	and	and	CCONJ
ejpam-6543	370	10	near	near	ADJ
ejpam-6543	370	11	compactness	compactness	NOUN
ejpam-6543	370	12	in	in	ADP
ejpam-6543	370	13	fuzzy	fuzzy	ADJ
ejpam-6543	370	14	topological	topological	ADJ
ejpam-6543	370	15	spaces	space	NOUN
ejpam-6543	370	16	.	.	PUNCT
ejpam-6543	371	1	fuzzy	fuzzy	ADJ
ejpam-6543	371	2	sets	set	NOUN
ejpam-6543	371	3	and	and	CCONJ
ejpam-6543	371	4	systems	system	NOUN
ejpam-6543	371	5	,	,	PUNCT
ejpam-6543	371	6	22(3):289–295	22(3):289–295	NUM
ejpam-6543	371	7	,	,	PUNCT
ejpam-6543	371	8	1987	1987	NUM
ejpam-6543	371	9	.	.	PUNCT
ejpam-6543	372	1	[	[	X
ejpam-6543	372	2	5	5	NUM
ejpam-6543	372	3	]	]	X
ejpam-6543	372	4	jamal	jamal	PROPN
ejpam-6543	372	5	oudetallah	oudetallah	PROPN
ejpam-6543	372	6	.	.	PUNCT
ejpam-6543	373	1	nearly	nearly	ADV
ejpam-6543	373	2	metacompact	metacompact	VERB
ejpam-6543	373	3	in	in	ADP
ejpam-6543	373	4	bitopological	bitopological	ADJ
ejpam-6543	373	5	space	space	NOUN
ejpam-6543	373	6	.	.	PUNCT
ejpam-6543	374	1	int	int	NOUN
ejpam-6543	374	2	.	.	PUNCT
ejpam-6543	375	1	j.	j.	PROPN
ejpam-6543	375	2	open	open	PROPN
ejpam-6543	375	3	problems	problem	NOUN
ejpam-6543	375	4	compt	compt	VERB
ejpam-6543	375	5	.	.	PUNCT
ejpam-6543	376	1	math	math	NOUN
ejpam-6543	376	2	,	,	PUNCT
ejpam-6543	376	3	15(3	15(3	NUM
ejpam-6543	376	4	)	)	PUNCT
ejpam-6543	376	5	,	,	PUNCT
ejpam-6543	376	6	2022	2022	NUM
ejpam-6543	376	7	.	.	PUNCT
ejpam-6543	377	1	[	[	X
ejpam-6543	377	2	6	6	NUM
ejpam-6543	377	3	]	]	X
ejpam-6543	377	4	jamal	jamal	PROPN
ejpam-6543	377	5	oudetallah	oudetallah	PROPN
ejpam-6543	377	6	.	.	PUNCT
ejpam-6543	378	1	on	on	ADP
ejpam-6543	378	2	feebly	feebly	ADJ
ejpam-6543	378	3	pairwise	pairwise	NOUN
ejpam-6543	378	4	expandable	expandable	ADJ
ejpam-6543	378	5	space	space	NOUN
ejpam-6543	378	6	.	.	PUNCT
ejpam-6543	379	1	j.	j.	PROPN
ejpam-6543	379	2	math	math	PROPN
ejpam-6543	379	3	.	.	PUNCT
ejpam-6543	380	1	comput	comput	NOUN
ejpam-6543	380	2	.	.	PUNCT
ejpam-6543	381	1	sci	sci	PROPN
ejpam-6543	381	2	.	.	PROPN
ejpam-6543	381	3	,	,	PUNCT
ejpam-6543	381	4	11(5):6216–6225	11(5):6216–6225	NUM
ejpam-6543	381	5	,	,	PUNCT
ejpam-6543	381	6	2021	2021	NUM
ejpam-6543	381	7	.	.	PUNCT
ejpam-6543	382	1	[	[	X
ejpam-6543	382	2	7	7	X
ejpam-6543	382	3	]	]	X
ejpam-6543	382	4	jamal	jamal	PROPN
ejpam-6543	382	5	a	a	DET
ejpam-6543	382	6	oudetallah	oudetallah	PROPN
ejpam-6543	382	7	laith	laith	PROPN
ejpam-6543	382	8	abualigah	abualigah	PROPN
ejpam-6543	382	9	.	.	PUNCT
ejpam-6543	383	1	h	h	NOUN
ejpam-6543	383	2	-	-	PUNCT
ejpam-6543	383	3	convexity	convexity	NOUN
ejpam-6543	383	4	in	in	ADP
ejpam-6543	383	5	metric	metric	ADJ
ejpam-6543	383	6	linear	linear	ADJ
ejpam-6543	383	7	spaces	space	NOUN
ejpam-6543	383	8	.	.	PUNCT
ejpam-6543	384	1	international	international	ADJ
ejpam-6543	384	2	journal	journal	NOUN
ejpam-6543	384	3	,	,	PUNCT
ejpam-6543	384	4	8(6	8(6	NUM
ejpam-6543	384	5	)	)	PUNCT
ejpam-6543	384	6	,	,	PUNCT
ejpam-6543	384	7	2019	2019	NUM
ejpam-6543	384	8	.	.	PUNCT
ejpam-6543	385	1	[	[	X
ejpam-6543	385	2	8	8	NUM
ejpam-6543	385	3	]	]	X
ejpam-6543	385	4	ala	ala	PROPN
ejpam-6543	385	5	amourah	amourah	PROPN
ejpam-6543	385	6	,	,	PUNCT
ejpam-6543	385	7	jamal	jamal	PROPN
ejpam-6543	385	8	oudetallah	oudetallah	PROPN
ejpam-6543	385	9	,	,	PUNCT
ejpam-6543	385	10	iqbal	iqbal	PROPN
ejpam-6543	385	11	m	m	PROPN
ejpam-6543	385	12	batiha	batiha	VERB
ejpam-6543	385	13	,	,	PUNCT
ejpam-6543	385	14	jamal	jamal	PROPN
ejpam-6543	385	15	salah	salah	PROPN
ejpam-6543	385	16	,	,	PUNCT
ejpam-6543	385	17	sultan	sultan	PROPN
ejpam-6543	385	18	alsaadi	alsaadi	NOUN
ejpam-6543	385	19	,	,	PUNCT
ejpam-6543	385	20	and	and	CCONJ
ejpam-6543	385	21	tala	tala	PROPN
ejpam-6543	385	22	sasa	sasa	PROPN
ejpam-6543	385	23	.	.	PUNCT
ejpam-6543	386	1	some	some	DET
ejpam-6543	386	2	types	type	NOUN
ejpam-6543	386	3	of	of	ADP
ejpam-6543	386	4	tri	tri	ADJ
ejpam-6543	386	5	-	-	ADJ
ejpam-6543	386	6	locally	locally	ADV
ejpam-6543	386	7	compactness	compactness	NOUN
ejpam-6543	386	8	spaces	space	NOUN
ejpam-6543	386	9	.	.	PUNCT
ejpam-6543	387	1	european	european	ADJ
ejpam-6543	387	2	journal	journal	PROPN
ejpam-6543	387	3	of	of	ADP
ejpam-6543	387	4	pure	pure	ADJ
ejpam-6543	387	5	and	and	CCONJ
ejpam-6543	387	6	applied	applied	ADJ
ejpam-6543	387	7	mathematics	mathematic	NOUN
ejpam-6543	387	8	,	,	PUNCT
ejpam-6543	387	9	18(2):5764–5764	18(2):5764–5764	NUM
ejpam-6543	387	10	,	,	PUNCT
ejpam-6543	387	11	2025	2025	NUM
ejpam-6543	387	12	.	.	PUNCT
ejpam-6543	388	1	[	[	X
ejpam-6543	388	2	9	9	NUM
ejpam-6543	388	3	]	]	X
ejpam-6543	388	4	ala	ala	PROPN
ejpam-6543	388	5	amourah	amourah	PROPN
ejpam-6543	388	6	,	,	PUNCT
ejpam-6543	388	7	jamal	jamal	PROPN
ejpam-6543	388	8	oudetallah	oudetallah	PROPN
ejpam-6543	388	9	,	,	PUNCT
ejpam-6543	388	10	iqbal	iqbal	PROPN
ejpam-6543	388	11	batiha	batiha	PROPN
ejpam-6543	388	12	,	,	PUNCT
ejpam-6543	388	13	jamal	jamal	PROPN
ejpam-6543	388	14	salah	salah	PROPN
ejpam-6543	388	15	,	,	PUNCT
ejpam-6543	388	16	and	and	CCONJ
ejpam-6543	388	17	mutaz	mutaz	NOUN
ejpam-6543	388	18	shatnawi	shatnawi	PROPN
ejpam-6543	388	19	.	.	PUNCT
ejpam-6543	389	1	sigma	sigma	PROPN
ejpam-6543	389	2	-	-	ADJ
ejpam-6543	389	3	compact	compact	ADJ
ejpam-6543	389	4	spaces	space	NOUN
ejpam-6543	389	5	in	in	ADP
ejpam-6543	389	6	n	n	PROPN
ejpam-6543	389	7	{	{	PUNCT
ejpam-6543	389	8	̂th}-topological	̂th}-topological	ADJ
ejpam-6543	389	9	space	space	NOUN
ejpam-6543	389	10	.	.	PUNCT
ejpam-6543	390	1	european	european	ADJ
ejpam-6543	390	2	journal	journal	PROPN
ejpam-6543	390	3	of	of	ADP
ejpam-6543	390	4	pure	pure	ADJ
ejpam-6543	390	5	and	and	CCONJ
ejpam-6543	390	6	applied	applied	ADJ
ejpam-6543	390	7	mathematics	mathematic	NOUN
ejpam-6543	390	8	,	,	PUNCT
ejpam-6543	390	9	18(2):5802–5802	18(2):5802–5802	NUM
ejpam-6543	390	10	,	,	PUNCT
ejpam-6543	390	11	2025	2025	NUM
ejpam-6543	390	12	.	.	PUNCT
ejpam-6543	391	1	[	[	X
ejpam-6543	391	2	10	10	NUM
ejpam-6543	391	3	]	]	X
ejpam-6543	391	4	rahmeh	rahmeh	NOUN
ejpam-6543	391	5	alrababah	alrababah	NOUN
ejpam-6543	391	6	,	,	PUNCT
ejpam-6543	391	7	ala	ala	PROPN
ejpam-6543	391	8	amourah	amourah	PROPN
ejpam-6543	391	9	,	,	PUNCT
ejpam-6543	391	10	jamal	jamal	PROPN
ejpam-6543	391	11	salah	salah	PROPN
ejpam-6543	391	12	,	,	PUNCT
ejpam-6543	391	13	reyaz	reyaz	PROPN
ejpam-6543	391	14	ahmad	ahmad	PROPN
ejpam-6543	391	15	,	,	PUNCT
ejpam-6543	391	16	and	and	CCONJ
ejpam-6543	391	17	ali	ali	VERB
ejpam-6543	391	18	a	a	DET
ejpam-6543	391	19	atoom	atoom	NOUN
ejpam-6543	391	20	.	.	PUNCT
ejpam-6543	392	1	new	new	ADJ
ejpam-6543	392	2	results	result	NOUN
ejpam-6543	392	3	on	on	ADP
ejpam-6543	392	4	difference	difference	NOUN
ejpam-6543	392	5	paracompactness	paracompactness	NOUN
ejpam-6543	392	6	in	in	ADP
ejpam-6543	392	7	topological	topological	ADJ
ejpam-6543	392	8	spaces	space	NOUN
ejpam-6543	392	9	.	.	PUNCT
ejpam-6543	393	1	european	european	ADJ
ejpam-6543	393	2	journal	journal	PROPN
ejpam-6543	393	3	of	of	ADP
ejpam-6543	393	4	pure	pure	ADJ
ejpam-6543	393	5	and	and	CCONJ
ejpam-6543	393	6	applied	applied	ADJ
ejpam-6543	393	7	mathematics	mathematic	NOUN
ejpam-6543	393	8	,	,	PUNCT
ejpam-6543	393	9	17(4):2990–3003	17(4):2990–3003	NUM
ejpam-6543	393	10	,	,	PUNCT
ejpam-6543	393	11	2024	2024	NUM
ejpam-6543	393	12	.	.	PUNCT
ejpam-6543	394	1	[	[	X
ejpam-6543	394	2	11	11	NUM
ejpam-6543	394	3	]	]	X
ejpam-6543	394	4	jamal	jamal	PROPN
ejpam-6543	394	5	oudetallah	oudetallah	PROPN
ejpam-6543	394	6	,	,	PUNCT
ejpam-6543	394	7	rehab	rehab	NOUN
ejpam-6543	394	8	alharbi	alharbi	NOUN
ejpam-6543	394	9	,	,	PUNCT
ejpam-6543	394	10	salsabiela	salsabiela	PROPN
ejpam-6543	394	11	rawashdeh	rawashdeh	PROPN
ejpam-6543	394	12	,	,	PUNCT
ejpam-6543	394	13	and	and	CCONJ
ejpam-6543	394	14	ala	ala	PROPN
ejpam-6543	394	15	amourah	amourah	PROPN
ejpam-6543	394	16	.	.	PUNCT
ejpam-6543	395	1	lindel	lindel	PROPN
ejpam-6543	395	2	”	"	PUNCT
ejpam-6543	395	3	ofness	ofness	NOUN
ejpam-6543	395	4	spaces	space	VERB
ejpam-6543	395	5	in	in	ADP
ejpam-6543	395	6	n	n	PRON
ejpam-6543	395	7	th	th	CCONJ
ejpam-6543	395	8	topological	topological	ADJ
ejpam-6543	395	9	spaces	space	NOUN
ejpam-6543	395	10	.	.	PUNCT
ejpam-6543	396	1	international	international	ADJ
ejpam-6543	396	2	journal	journal	PROPN
ejpam-6543	396	3	of	of	ADP
ejpam-6543	396	4	neutrosophic	neutrosophic	ADJ
ejpam-6543	396	5	science	science	NOUN
ejpam-6543	396	6	(	(	PUNCT
ejpam-6543	396	7	ijns	ijns	PROPN
ejpam-6543	396	8	)	)	PUNCT
ejpam-6543	396	9	,	,	PUNCT
ejpam-6543	396	10	25(3	25(3	NUM
ejpam-6543	396	11	)	)	PUNCT
ejpam-6543	396	12	,	,	PUNCT
ejpam-6543	396	13	2025	2025	NUM
ejpam-6543	396	14	.	.	PUNCT
ejpam-6543	397	1	[	[	X
ejpam-6543	397	2	12	12	NUM
ejpam-6543	397	3	]	]	PUNCT
ejpam-6543	397	4	rehab	rehab	NOUN
ejpam-6543	397	5	alharbi	alharbi	NOUN
ejpam-6543	397	6	,	,	PUNCT
ejpam-6543	397	7	jamal	jamal	PROPN
ejpam-6543	397	8	oudetallah	oudetallah	PROPN
ejpam-6543	397	9	,	,	PUNCT
ejpam-6543	397	10	salsabiela	salsabiela	PROPN
ejpam-6543	397	11	rawashdeh	rawashdeh	NOUN
ejpam-6543	397	12	,	,	PUNCT
ejpam-6543	397	13	and	and	CCONJ
ejpam-6543	397	14	ala	ala	PROPN
ejpam-6543	397	15	amourah	amourah	PROPN
ejpam-6543	397	16	.	.	PUNCT
ejpam-6543	398	1	some	some	DET
ejpam-6543	398	2	types	type	NOUN
ejpam-6543	398	3	of	of	ADP
ejpam-6543	398	4	n	n	PRON
ejpam-6543	398	5	th	th	ADV
ejpam-6543	398	6	-	-	PUNCT
ejpam-6543	398	7	locally	locally	ADV
ejpam-6543	398	8	compactness	compactness	NOUN
ejpam-6543	398	9	spaces	space	NOUN
ejpam-6543	398	10	.	.	PUNCT
ejpam-6543	399	1	international	international	ADJ
ejpam-6543	399	2	journal	journal	PROPN
ejpam-6543	399	3	of	of	ADP
ejpam-6543	399	4	neutrosophic	neutrosophic	ADJ
ejpam-6543	399	5	science	science	NOUN
ejpam-6543	399	6	(	(	PUNCT
ejpam-6543	399	7	ijns	ijns	PROPN
ejpam-6543	399	8	)	)	PUNCT
ejpam-6543	399	9	,	,	PUNCT
ejpam-6543	399	10	25(3	25(3	NUM
ejpam-6543	399	11	)	)	PUNCT
ejpam-6543	399	12	,	,	PUNCT
ejpam-6543	399	13	2025	2025	NUM
ejpam-6543	399	14	.	.	PUNCT
ejpam-6543	400	1	[	[	X
ejpam-6543	400	2	13	13	NUM
ejpam-6543	400	3	]	]	X
ejpam-6543	400	4	norman	norman	PROPN
ejpam-6543	400	5	levine	levine	PROPN
ejpam-6543	400	6	.	.	PUNCT
ejpam-6543	401	1	generalized	generalize	VERB
ejpam-6543	401	2	closed	closed	ADJ
ejpam-6543	401	3	sets	set	NOUN
ejpam-6543	401	4	in	in	ADP
ejpam-6543	401	5	topology	topology	NOUN
ejpam-6543	401	6	.	.	PUNCT
ejpam-6543	402	1	rendiconti	rendiconti	VERB
ejpam-6543	402	2	del	del	PROPN
ejpam-6543	402	3	circolo	circolo	PROPN
ejpam-6543	402	4	matematico	matematico	NOUN
ejpam-6543	402	5	di	di	NOUN
ejpam-6543	402	6	palermo	palermo	NOUN
ejpam-6543	402	7	,	,	PUNCT
ejpam-6543	402	8	19(1):89–96	19(1):89–96	NUM
ejpam-6543	402	9	,	,	PUNCT
ejpam-6543	402	10	1970	1970	NUM
ejpam-6543	402	11	.	.	PUNCT
ejpam-6543	403	1	[	[	X
ejpam-6543	403	2	14	14	NUM
ejpam-6543	403	3	]	]	X
ejpam-6543	403	4	william	william	PROPN
ejpam-6543	403	5	dunham	dunham	PROPN
ejpam-6543	403	6	.	.	PUNCT
ejpam-6543	404	1	a	a	DET
ejpam-6543	404	2	new	new	ADJ
ejpam-6543	404	3	closure	closure	NOUN
ejpam-6543	404	4	operator	operator	NOUN
ejpam-6543	404	5	for	for	ADP
ejpam-6543	404	6	non	non	ADJ
ejpam-6543	404	7	-	-	ADJ
ejpam-6543	404	8	t	t	ADJ
ejpam-6543	404	9	1	1	NUM
ejpam-6543	404	10	topologies	topology	NOUN
ejpam-6543	404	11	.	.	PUNCT
ejpam-6543	405	1	kyungpook	kyungpook	PROPN
ejpam-6543	405	2	mathematical	mathematical	PROPN
ejpam-6543	405	3	journal	journal	PROPN
ejpam-6543	405	4	,	,	PUNCT
ejpam-6543	405	5	22(1):55–60	22(1):55–60	NUM
ejpam-6543	405	6	,	,	PUNCT
ejpam-6543	405	7	1982	1982	NUM
ejpam-6543	405	8	.	.	PUNCT
ejpam-6543	406	1	[	[	X
ejpam-6543	406	2	15	15	NUM
ejpam-6543	406	3	]	]	PUNCT
ejpam-6543	406	4	as	as	ADP
ejpam-6543	406	5	mashhour	mashhour	ADJ
ejpam-6543	406	6	.	.	PUNCT
ejpam-6543	407	1	on	on	ADP
ejpam-6543	407	2	precontinuous	precontinuous	ADJ
ejpam-6543	407	3	and	and	CCONJ
ejpam-6543	407	4	weak	weak	ADJ
ejpam-6543	407	5	precontinuous	precontinuous	ADJ
ejpam-6543	407	6	mappings	mapping	NOUN
ejpam-6543	407	7	.	.	PUNCT
ejpam-6543	408	1	in	in	ADP
ejpam-6543	408	2	proc	proc	PROPN
ejpam-6543	408	3	.	.	PUNCT
ejpam-6543	409	1	math	math	NOUN
ejpam-6543	409	2	.	.	PUNCT
ejpam-6543	410	1	phys	phy	NOUN
ejpam-6543	410	2	.	.	PUNCT
ejpam-6543	411	1	soc	soc	PROPN
ejpam-6543	411	2	.	.	PUNCT
ejpam-6543	412	1	egypt	egypt	PROPN
ejpam-6543	412	2	.	.	PROPN
ejpam-6543	412	3	,	,	PUNCT
ejpam-6543	412	4	volume	volume	NOUN
ejpam-6543	412	5	53	53	NUM
ejpam-6543	412	6	,	,	PUNCT
ejpam-6543	412	7	pages	page	NOUN
ejpam-6543	412	8	47–53	47–53	NUM
ejpam-6543	412	9	,	,	PUNCT
ejpam-6543	412	10	1982	1982	NUM
ejpam-6543	412	11	.	.	PUNCT