id	sid	tid	token	lemma	pos
ejpam-3420	1	1	european	european	PROPN
ejpam-3420	1	2	journal	journal	PROPN
ejpam-3420	1	3	of	of	ADP
ejpam-3420	1	4	pure	pure	ADJ
ejpam-3420	1	5	and	and	CCONJ
ejpam-3420	1	6	applied	apply	VERB
ejpam-3420	1	7	mathematics	mathematic	NOUN
ejpam-3420	1	8	vol	vol	NOUN
ejpam-3420	1	9	.	.	PROPN
ejpam-3420	2	1	12	12	NUM
ejpam-3420	2	2	,	,	PUNCT
ejpam-3420	2	3	no	no	INTJ
ejpam-3420	2	4	.	.	NOUN
ejpam-3420	2	5	2	2	NUM
ejpam-3420	2	6	,	,	PUNCT
ejpam-3420	2	7	2019	2019	NUM
ejpam-3420	2	8	,	,	PUNCT
ejpam-3420	2	9	533	533	NUM
ejpam-3420	2	10	-	-	SYM
ejpam-3420	2	11	543	543	NUM
ejpam-3420	2	12	issn	issn	PROPN
ejpam-3420	2	13	1307	1307	NUM
ejpam-3420	2	14	-	-	SYM
ejpam-3420	2	15	5543	5543	NUM
ejpam-3420	2	16	–	–	PUNCT
ejpam-3420	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3420	2	18	published	publish	VERB
ejpam-3420	2	19	by	by	ADP
ejpam-3420	2	20	new	new	PROPN
ejpam-3420	2	21	york	york	PROPN
ejpam-3420	2	22	business	business	PROPN
ejpam-3420	2	23	global	global	ADJ
ejpam-3420	2	24	rough	rough	ADJ
ejpam-3420	2	25	spaces	space	NOUN
ejpam-3420	2	26	on	on	ADP
ejpam-3420	2	27	covering	cover	VERB
ejpam-3420	2	28	based	base	VERB
ejpam-3420	2	29	rough	rough	ADJ
ejpam-3420	2	30	sets	set	NOUN
ejpam-3420	2	31	n.	n.	PROPN
ejpam-3420	2	32	alharbi1	alharbi1	PROPN
ejpam-3420	2	33	,	,	PUNCT
ejpam-3420	2	34	h.	h.	PROPN
ejpam-3420	2	35	aydi2,∗	aydi2,∗	PROPN
ejpam-3420	2	36	,	,	PUNCT
ejpam-3420	2	37	c.	c.	NOUN
ejpam-3420	2	38	özel1	özel1	NOUN
ejpam-3420	2	39	1	1	NUM
ejpam-3420	2	40	department	department	NOUN
ejpam-3420	2	41	of	of	ADP
ejpam-3420	2	42	mathematics	mathematic	NOUN
ejpam-3420	2	43	,	,	PUNCT
ejpam-3420	2	44	king	king	PROPN
ejpam-3420	2	45	abdulaziz	abdulaziz	PROPN
ejpam-3420	2	46	university	university	PROPN
ejpam-3420	2	47	,	,	PUNCT
ejpam-3420	2	48	p.o	p.o	PROPN
ejpam-3420	2	49	.	.	PROPN
ejpam-3420	2	50	box	box	PROPN
ejpam-3420	2	51	:	:	PUNCT
ejpam-3420	2	52	80203	80203	NUM
ejpam-3420	2	53	jeddah	jeddah	PROPN
ejpam-3420	2	54	21589	21589	NUM
ejpam-3420	2	55	,	,	PUNCT
ejpam-3420	2	56	saudi	saudi	PROPN
ejpam-3420	2	57	arabia	arabia	PROPN
ejpam-3420	2	58	2	2	NUM
ejpam-3420	2	59	université	université	NOUN
ejpam-3420	2	60	de	de	X
ejpam-3420	2	61	sousse	sousse	PROPN
ejpam-3420	2	62	,	,	PUNCT
ejpam-3420	2	63	institut	institut	PROPN
ejpam-3420	2	64	supérieur	supérieur	PROPN
ejpam-3420	2	65	d’informatique	d’informatique	PROPN
ejpam-3420	2	66	et	et	NOUN
ejpam-3420	2	67	des	des	X
ejpam-3420	2	68	techniques	techniques	X
ejpam-3420	2	69	de	de	X
ejpam-3420	2	70	communication	communication	NOUN
ejpam-3420	2	71	,	,	PUNCT
ejpam-3420	2	72	h.	h.	PROPN
ejpam-3420	2	73	sousse	sousse	PROPN
ejpam-3420	2	74	4000	4000	NUM
ejpam-3420	2	75	,	,	PUNCT
ejpam-3420	2	76	tunisia	tunisia	NOUN
ejpam-3420	2	77	abstract	abstract	NOUN
ejpam-3420	2	78	.	.	PUNCT
ejpam-3420	3	1	in	in	ADP
ejpam-3420	3	2	this	this	DET
ejpam-3420	3	3	paper	paper	NOUN
ejpam-3420	3	4	,	,	PUNCT
ejpam-3420	3	5	we	we	PRON
ejpam-3420	3	6	discuss	discuss	VERB
ejpam-3420	3	7	open	open	ADJ
ejpam-3420	3	8	(	(	PUNCT
ejpam-3420	3	9	closed	closed	ADJ
ejpam-3420	3	10	)	)	PUNCT
ejpam-3420	3	11	sets	set	NOUN
ejpam-3420	3	12	,	,	PUNCT
ejpam-3420	3	13	rough	rough	ADJ
ejpam-3420	3	14	interiors	interior	NOUN
ejpam-3420	3	15	,	,	PUNCT
ejpam-3420	3	16	rough	rough	ADJ
ejpam-3420	3	17	closures	closure	NOUN
ejpam-3420	3	18	,	,	PUNCT
ejpam-3420	3	19	continuous	continuous	ADJ
ejpam-3420	3	20	mappings	mapping	NOUN
ejpam-3420	3	21	,	,	PUNCT
ejpam-3420	3	22	open	open	ADJ
ejpam-3420	3	23	(	(	PUNCT
ejpam-3420	3	24	closed	closed	ADJ
ejpam-3420	3	25	)	)	PUNCT
ejpam-3420	3	26	mappings	mapping	NOUN
ejpam-3420	3	27	and	and	CCONJ
ejpam-3420	3	28	homeomorphism	homeomorphism	NOUN
ejpam-3420	3	29	mappings	mapping	NOUN
ejpam-3420	3	30	of	of	ADP
ejpam-3420	3	31	covering	covering	NOUN
ejpam-3420	3	32	based	base	VERB
ejpam-3420	3	33	rough	rough	ADJ
ejpam-3420	3	34	topology	topology	NOUN
ejpam-3420	3	35	of	of	ADP
ejpam-3420	3	36	akduman	akduman	NOUN
ejpam-3420	3	37	,	,	PUNCT
ejpam-3420	3	38	özcelik	özcelik	PROPN
ejpam-3420	3	39	and	and	CCONJ
ejpam-3420	3	40	özel	özel	PROPN
ejpam-3420	3	41	.	.	PUNCT
ejpam-3420	4	1	we	we	PRON
ejpam-3420	4	2	also	also	ADV
ejpam-3420	4	3	construct	construct	VERB
ejpam-3420	4	4	the	the	DET
ejpam-3420	4	5	nano	nano	NOUN
ejpam-3420	4	6	topology	topology	NOUN
ejpam-3420	4	7	on	on	ADP
ejpam-3420	4	8	a	a	DET
ejpam-3420	4	9	given	give	VERB
ejpam-3420	4	10	covering	covering	NOUN
ejpam-3420	4	11	based	base	VERB
ejpam-3420	4	12	rough	rough	ADJ
ejpam-3420	4	13	set	set	NOUN
ejpam-3420	4	14	.	.	PUNCT
ejpam-3420	5	1	2010	2010	NUM
ejpam-3420	5	2	mathematics	mathematic	NOUN
ejpam-3420	5	3	subject	subject	NOUN
ejpam-3420	5	4	classifications	classification	NOUN
ejpam-3420	5	5	:	:	PUNCT
ejpam-3420	5	6	22a05	22a05	NUM
ejpam-3420	5	7	,	,	PUNCT
ejpam-3420	5	8	54a05	54a05	NUM
ejpam-3420	5	9	,	,	PUNCT
ejpam-3420	5	10	03e25	03e25	NOUN
ejpam-3420	5	11	key	key	ADJ
ejpam-3420	5	12	words	word	NOUN
ejpam-3420	5	13	and	and	CCONJ
ejpam-3420	5	14	phrases	phrase	NOUN
ejpam-3420	5	15	:	:	PUNCT
ejpam-3420	5	16	covering	cover	VERB
ejpam-3420	5	17	based	base	VERB
ejpam-3420	5	18	rough	rough	ADJ
ejpam-3420	5	19	topology	topology	NOUN
ejpam-3420	5	20	,	,	PUNCT
ejpam-3420	5	21	covering	cover	VERB
ejpam-3420	5	22	based	base	VERB
ejpam-3420	5	23	rough	rough	ADJ
ejpam-3420	5	24	sets	set	NOUN
ejpam-3420	5	25	,	,	PUNCT
ejpam-3420	5	26	isomorphisms	isomorphism	NOUN
ejpam-3420	5	27	of	of	ADP
ejpam-3420	5	28	approximations	approximation	NOUN
ejpam-3420	5	29	,	,	PUNCT
ejpam-3420	5	30	rough	rough	ADJ
ejpam-3420	5	31	continuity	continuity	NOUN
ejpam-3420	5	32	this	this	DET
ejpam-3420	5	33	paper	paper	NOUN
ejpam-3420	5	34	is	be	AUX
ejpam-3420	5	35	produced	produce	VERB
ejpam-3420	5	36	from	from	ADP
ejpam-3420	5	37	the	the	DET
ejpam-3420	5	38	phd	phd	NOUN
ejpam-3420	5	39	thesis	thesis	NOUN
ejpam-3420	5	40	of	of	ADP
ejpam-3420	5	41	ms	ms	PROPN
ejpam-3420	5	42	.	.	PROPN
ejpam-3420	5	43	nof	nof	PROPN
ejpam-3420	5	44	alharbi	alharbi	PROPN
ejpam-3420	5	45	registered	register	VERB
ejpam-3420	5	46	in	in	ADP
ejpam-3420	5	47	king	king	PROPN
ejpam-3420	5	48	abdulaziz	abdulaziz	PROPN
ejpam-3420	5	49	university	university	PROPN
ejpam-3420	5	50	.	.	PUNCT
ejpam-3420	6	1	1	1	X
ejpam-3420	6	2	.	.	X
ejpam-3420	6	3	introduction	introduction	NOUN
ejpam-3420	6	4	the	the	DET
ejpam-3420	6	5	rough	rough	ADJ
ejpam-3420	6	6	set	set	NOUN
ejpam-3420	6	7	theory	theory	NOUN
ejpam-3420	6	8	was	be	AUX
ejpam-3420	6	9	introduced	introduce	VERB
ejpam-3420	6	10	by	by	ADP
ejpam-3420	6	11	pawlak	pawlak	NOUN
ejpam-3420	6	12	in	in	ADP
ejpam-3420	6	13	[	[	X
ejpam-3420	6	14	2	2	NUM
ejpam-3420	6	15	]	]	PUNCT
ejpam-3420	6	16	.	.	PUNCT
ejpam-3420	7	1	it	it	PRON
ejpam-3420	7	2	deals	deal	VERB
ejpam-3420	7	3	with	with	ADP
ejpam-3420	7	4	impression	impression	NOUN
ejpam-3420	7	5	,	,	PUNCT
ejpam-3420	7	6	vagueness	vagueness	NOUN
ejpam-3420	7	7	,	,	PUNCT
ejpam-3420	7	8	and	and	CCONJ
ejpam-3420	7	9	uncertainty	uncertainty	NOUN
ejpam-3420	7	10	in	in	ADP
ejpam-3420	7	11	data	data	NOUN
ejpam-3420	7	12	analysis	analysis	NOUN
ejpam-3420	7	13	and	and	CCONJ
ejpam-3420	7	14	information	information	NOUN
ejpam-3420	7	15	systems	system	NOUN
ejpam-3420	7	16	.	.	PUNCT
ejpam-3420	8	1	the	the	DET
ejpam-3420	8	2	rough	rough	ADJ
ejpam-3420	8	3	set	set	NOUN
ejpam-3420	8	4	theory	theory	NOUN
ejpam-3420	8	5	offers	offer	VERB
ejpam-3420	8	6	ways	way	NOUN
ejpam-3420	8	7	to	to	PART
ejpam-3420	8	8	find	find	VERB
ejpam-3420	8	9	the	the	DET
ejpam-3420	8	10	deciding	decide	VERB
ejpam-3420	8	11	factors	factor	NOUN
ejpam-3420	8	12	or	or	CCONJ
ejpam-3420	8	13	core	core	NOUN
ejpam-3420	8	14	from	from	ADP
ejpam-3420	8	15	data	data	PROPN
ejpam-3420	8	16	.	.	PUNCT
ejpam-3420	9	1	the	the	DET
ejpam-3420	9	2	classical	classical	ADJ
ejpam-3420	9	3	rough	rough	ADJ
ejpam-3420	9	4	set	set	NOUN
ejpam-3420	9	5	theory	theory	NOUN
ejpam-3420	9	6	(	(	PUNCT
ejpam-3420	9	7	pawlak	pawlak	ADJ
ejpam-3420	9	8	version	version	NOUN
ejpam-3420	9	9	)	)	PUNCT
ejpam-3420	9	10	is	be	AUX
ejpam-3420	9	11	based	base	VERB
ejpam-3420	9	12	on	on	ADP
ejpam-3420	9	13	the	the	DET
ejpam-3420	9	14	equivalence	equivalence	NOUN
ejpam-3420	9	15	relations	relation	NOUN
ejpam-3420	9	16	.	.	PUNCT
ejpam-3420	10	1	pawlak	pawlak	PROPN
ejpam-3420	10	2	defined	define	VERB
ejpam-3420	10	3	an	an	DET
ejpam-3420	10	4	approximation	approximation	NOUN
ejpam-3420	10	5	space	space	NOUN
ejpam-3420	10	6	as	as	ADP
ejpam-3420	10	7	an	an	DET
ejpam-3420	10	8	ordered	ordered	ADJ
ejpam-3420	10	9	pair	pair	NOUN
ejpam-3420	10	10	<	<	X
ejpam-3420	10	11	u	u	NOUN
ejpam-3420	10	12	,	,	PUNCT
ejpam-3420	10	13	r	r	NOUN
ejpam-3420	10	14	>	>	X
ejpam-3420	10	15	where	where	SCONJ
ejpam-3420	10	16	u	u	NOUN
ejpam-3420	10	17	is	be	AUX
ejpam-3420	10	18	a	a	DET
ejpam-3420	10	19	non	non	ADJ
ejpam-3420	10	20	-	-	ADJ
ejpam-3420	10	21	empty	empty	ADJ
ejpam-3420	10	22	set	set	NOUN
ejpam-3420	10	23	and	and	CCONJ
ejpam-3420	10	24	r	r	NOUN
ejpam-3420	10	25	is	be	AUX
ejpam-3420	10	26	an	an	DET
ejpam-3420	10	27	equivalence	equivalence	NOUN
ejpam-3420	10	28	relation	relation	NOUN
ejpam-3420	10	29	defined	define	VERB
ejpam-3420	10	30	on	on	ADP
ejpam-3420	10	31	u	u	PROPN
ejpam-3420	10	32	.	.	PUNCT
ejpam-3420	11	1	then	then	ADV
ejpam-3420	11	2	for	for	ADP
ejpam-3420	11	3	any	any	DET
ejpam-3420	11	4	x	x	SYM
ejpam-3420	11	5	∈	∈	PROPN
ejpam-3420	11	6	p	p	X
ejpam-3420	11	7	(	(	PUNCT
ejpam-3420	11	8	u	u	NOUN
ejpam-3420	11	9	)	)	PUNCT
ejpam-3420	11	10	,	,	PUNCT
ejpam-3420	11	11	he	he	PRON
ejpam-3420	11	12	presented	present	VERB
ejpam-3420	11	13	definitions	definition	NOUN
ejpam-3420	11	14	for	for	ADP
ejpam-3420	11	15	lower	low	ADJ
ejpam-3420	11	16	and	and	CCONJ
ejpam-3420	11	17	upper	upper	ADJ
ejpam-3420	11	18	approximations	approximation	NOUN
ejpam-3420	11	19	.	.	PUNCT
ejpam-3420	12	1	next	next	ADV
ejpam-3420	12	2	,	,	PUNCT
ejpam-3420	12	3	he	he	PRON
ejpam-3420	12	4	determined	determine	VERB
ejpam-3420	12	5	rough	rough	ADJ
ejpam-3420	12	6	sets	set	NOUN
ejpam-3420	12	7	by	by	ADP
ejpam-3420	12	8	establishing	establish	VERB
ejpam-3420	12	9	rough	rough	ADJ
ejpam-3420	12	10	equivalence	equivalence	NOUN
ejpam-3420	12	11	relation	relation	NOUN
ejpam-3420	12	12	.	.	PUNCT
ejpam-3420	13	1	later	later	ADV
ejpam-3420	13	2	in	in	ADP
ejpam-3420	13	3	1988	1988	NUM
ejpam-3420	13	4	,	,	PUNCT
ejpam-3420	13	5	pomykala	pomykala	NOUN
ejpam-3420	13	6	[	[	X
ejpam-3420	13	7	3	3	NUM
ejpam-3420	13	8	]	]	PUNCT
ejpam-3420	13	9	defined	define	VERB
ejpam-3420	13	10	corresponding	correspond	VERB
ejpam-3420	13	11	operations	operation	NOUN
ejpam-3420	13	12	on	on	ADP
ejpam-3420	13	13	pawlak	pawlak	ADJ
ejpam-3420	13	14	rough	rough	ADJ
ejpam-3420	13	15	sets	set	NOUN
ejpam-3420	13	16	.	.	PUNCT
ejpam-3420	14	1	a	a	DET
ejpam-3420	14	2	year	year	NOUN
ejpam-3420	14	3	later	later	ADV
ejpam-3420	14	4	,	,	PUNCT
ejpam-3420	14	5	bryniarski	bryniarski	VERB
ejpam-3420	14	6	[	[	X
ejpam-3420	14	7	1	1	NUM
ejpam-3420	14	8	]	]	X
ejpam-3420	14	9	extended	extend	VERB
ejpam-3420	14	10	classical	classical	ADJ
ejpam-3420	14	11	rough	rough	ADJ
ejpam-3420	14	12	sets	set	NOUN
ejpam-3420	14	13	given	give	VERB
ejpam-3420	14	14	by	by	ADP
ejpam-3420	14	15	pawlak	pawlak	ADJ
ejpam-3420	14	16	to	to	ADP
ejpam-3420	14	17	covering	cover	VERB
ejpam-3420	14	18	based	base	VERB
ejpam-3420	14	19	rough	rough	ADJ
ejpam-3420	14	20	sets	set	NOUN
ejpam-3420	14	21	.	.	PUNCT
ejpam-3420	15	1	moreover	moreover	ADV
ejpam-3420	15	2	,	,	PUNCT
ejpam-3420	15	3	he	he	PRON
ejpam-3420	15	4	restricted	restrict	VERB
ejpam-3420	15	5	coverings	covering	NOUN
ejpam-3420	15	6	by	by	ADP
ejpam-3420	15	7	some	some	DET
ejpam-3420	15	8	conditions	condition	NOUN
ejpam-3420	15	9	to	to	PART
ejpam-3420	15	10	make	make	VERB
ejpam-3420	15	11	operations	operation	NOUN
ejpam-3420	15	12	of	of	ADP
ejpam-3420	15	13	∗corresponding	∗corresponde	VERB
ejpam-3420	15	14	author	author	NOUN
ejpam-3420	15	15	.	.	PUNCT
ejpam-3420	16	1	doi	doi	NOUN
ejpam-3420	16	2	:	:	PUNCT
ejpam-3420	16	3	https://doi.org/10.29020/nybg.ejpam.v12i2.3420	https://doi.org/10.29020/nybg.ejpam.v12i2.3420	NUM
ejpam-3420	16	4	email	email	NOUN
ejpam-3420	16	5	addresses	address	NOUN
ejpam-3420	16	6	:	:	PUNCT
ejpam-3420	16	7	nof20081900@hotmail.com	nof20081900@hotmail.com	X
ejpam-3420	16	8	(	(	PUNCT
ejpam-3420	16	9	n.	n.	NOUN
ejpam-3420	16	10	alharbi	alharbi	PROPN
ejpam-3420	16	11	)	)	PUNCT
ejpam-3420	16	12	,	,	PUNCT
ejpam-3420	17	1	hassen.aydi@isima.rnu.tn	hassen.aydi@isima.rnu.tn	INTJ
ejpam-3420	17	2	(	(	PUNCT
ejpam-3420	17	3	h.	h.	PROPN
ejpam-3420	17	4	aydi	aydi	VERB
ejpam-3420	17	5	)	)	PUNCT
ejpam-3420	17	6	,	,	PUNCT
ejpam-3420	17	7	cenap.ozel@gmail.com	cenap.ozel@gmail.com	X
ejpam-3420	17	8	(	(	PUNCT
ejpam-3420	17	9	c.	c.	PROPN
ejpam-3420	17	10	özel	özel	PROPN
ejpam-3420	17	11	)	)	PUNCT
ejpam-3420	17	12	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3420	18	1	533	533	NUM
ejpam-3420	18	2	c	c	X
ejpam-3420	18	3	©	©	PROPN
ejpam-3420	18	4	2019	2019	NUM
ejpam-3420	18	5	ejpam	ejpam	NOUN
ejpam-3420	18	6	all	all	DET
ejpam-3420	18	7	rights	right	NOUN
ejpam-3420	18	8	reserved	reserve	VERB
ejpam-3420	18	9	.	.	PUNCT
ejpam-3420	19	1	n.	n.	PROPN
ejpam-3420	19	2	alharbi	alharbi	PROPN
ejpam-3420	19	3	,	,	PUNCT
ejpam-3420	19	4	h.	h.	PROPN
ejpam-3420	19	5	aydi	aydi	PROPN
ejpam-3420	19	6	,	,	PUNCT
ejpam-3420	19	7	c.	c.	PROPN
ejpam-3420	19	8	özel	özel	PROPN
ejpam-3420	19	9	/	/	SYM
ejpam-3420	19	10	eur	eur	PROPN
ejpam-3420	19	11	.	.	PUNCT
ejpam-3420	20	1	j.	j.	PROPN
ejpam-3420	20	2	pure	pure	PROPN
ejpam-3420	20	3	appl	appl	PROPN
ejpam-3420	20	4	.	.	PROPN
ejpam-3420	20	5	math	math	PROPN
ejpam-3420	20	6	,	,	PUNCT
ejpam-3420	20	7	12	12	NUM
ejpam-3420	20	8	(	(	PUNCT
ejpam-3420	20	9	2	2	NUM
ejpam-3420	20	10	)	)	PUNCT
ejpam-3420	20	11	(	(	PUNCT
ejpam-3420	20	12	2019	2019	NUM
ejpam-3420	20	13	)	)	PUNCT
ejpam-3420	20	14	,	,	PUNCT
ejpam-3420	20	15	533	533	NUM
ejpam-3420	20	16	-	-	SYM
ejpam-3420	20	17	543	543	NUM
ejpam-3420	20	18	534	534	NUM
ejpam-3420	20	19	pawlak	pawlak	ADJ
ejpam-3420	20	20	rough	rough	ADJ
ejpam-3420	20	21	sets	set	NOUN
ejpam-3420	20	22	,	,	PUNCT
ejpam-3420	20	23	well	well	INTJ
ejpam-3420	20	24	defined	define	VERB
ejpam-3420	20	25	.	.	PUNCT
ejpam-3420	21	1	•	•	INTJ
ejpam-3420	21	2	let	let	VERB
ejpam-3420	21	3	u	u	PRON
ejpam-3420	21	4	be	be	AUX
ejpam-3420	21	5	a	a	DET
ejpam-3420	21	6	non	non	ADJ
ejpam-3420	21	7	-	-	ADJ
ejpam-3420	21	8	empty	empty	ADJ
ejpam-3420	21	9	set	set	NOUN
ejpam-3420	21	10	and	and	CCONJ
ejpam-3420	21	11	c	c	AUX
ejpam-3420	21	12	be	be	AUX
ejpam-3420	21	13	a	a	DET
ejpam-3420	21	14	family	family	NOUN
ejpam-3420	21	15	of	of	ADP
ejpam-3420	21	16	non	non	ADJ
ejpam-3420	21	17	-	-	ADJ
ejpam-3420	21	18	empty	empty	ADJ
ejpam-3420	21	19	subsets	subset	NOUN
ejpam-3420	21	20	of	of	ADP
ejpam-3420	21	21	u	u	PRON
ejpam-3420	21	22	such	such	ADJ
ejpam-3420	21	23	that	that	SCONJ
ejpam-3420	21	24	∪c	∪c	NUM
ejpam-3420	21	25	=	=	SYM
ejpam-3420	21	26	u	u	NOUN
ejpam-3420	21	27	.	.	PUNCT
ejpam-3420	22	1	then	then	ADV
ejpam-3420	22	2	c	c	PROPN
ejpam-3420	22	3	is	be	AUX
ejpam-3420	22	4	called	call	VERB
ejpam-3420	22	5	a	a	DET
ejpam-3420	22	6	covering	covering	NOUN
ejpam-3420	22	7	of	of	ADP
ejpam-3420	22	8	u	u	PROPN
ejpam-3420	22	9	.	.	PUNCT
ejpam-3420	23	1	obviously	obviously	ADV
ejpam-3420	23	2	,	,	PUNCT
ejpam-3420	23	3	a	a	DET
ejpam-3420	23	4	partition	partition	NOUN
ejpam-3420	23	5	is	be	AUX
ejpam-3420	23	6	indeed	indeed	ADV
ejpam-3420	23	7	a	a	DET
ejpam-3420	23	8	covering	covering	NOUN
ejpam-3420	23	9	of	of	ADP
ejpam-3420	23	10	u	u	NOUN
ejpam-3420	23	11	,	,	PUNCT
ejpam-3420	23	12	so	so	SCONJ
ejpam-3420	23	13	that	that	SCONJ
ejpam-3420	23	14	a	a	DET
ejpam-3420	23	15	covering	covering	NOUN
ejpam-3420	23	16	is	be	AUX
ejpam-3420	23	17	an	an	DET
ejpam-3420	23	18	extension	extension	NOUN
ejpam-3420	23	19	of	of	ADP
ejpam-3420	23	20	a	a	DET
ejpam-3420	23	21	partition	partition	NOUN
ejpam-3420	23	22	.	.	PUNCT
ejpam-3420	24	1	•	•	NUM
ejpam-3420	24	2	bryniarski	bryniarski	NOUN
ejpam-3420	24	3	also	also	ADV
ejpam-3420	24	4	defined	define	VERB
ejpam-3420	24	5	the	the	DET
ejpam-3420	24	6	lower	low	ADJ
ejpam-3420	24	7	and	and	CCONJ
ejpam-3420	24	8	upper	upper	ADJ
ejpam-3420	24	9	approximations	approximation	NOUN
ejpam-3420	24	10	and	and	CCONJ
ejpam-3420	24	11	the	the	DET
ejpam-3420	24	12	boundary	boundary	ADJ
ejpam-3420	24	13	region	region	NOUN
ejpam-3420	24	14	in	in	ADP
ejpam-3420	24	15	a	a	DET
ejpam-3420	24	16	similar	similar	ADJ
ejpam-3420	24	17	way	way	NOUN
ejpam-3420	24	18	as	as	ADP
ejpam-3420	24	19	pawlak	pawlak	ADJ
ejpam-3420	24	20	.	.	PUNCT
ejpam-3420	25	1	but	but	CCONJ
ejpam-3420	25	2	,	,	PUNCT
ejpam-3420	25	3	instead	instead	ADV
ejpam-3420	25	4	of	of	ADP
ejpam-3420	25	5	using	use	VERB
ejpam-3420	25	6	elementary	elementary	ADJ
ejpam-3420	25	7	sets	set	NOUN
ejpam-3420	25	8	(	(	PUNCT
ejpam-3420	25	9	classes	class	NOUN
ejpam-3420	25	10	)	)	PUNCT
ejpam-3420	25	11	,	,	PUNCT
ejpam-3420	25	12	he	he	PRON
ejpam-3420	25	13	used	use	VERB
ejpam-3420	25	14	the	the	DET
ejpam-3420	25	15	elements	element	NOUN
ejpam-3420	25	16	of	of	ADP
ejpam-3420	25	17	coverings	covering	NOUN
ejpam-3420	25	18	.	.	PUNCT
ejpam-3420	26	1	let	let	VERB
ejpam-3420	26	2	x	x	PUNCT
ejpam-3420	26	3	∈	∈	PROPN
ejpam-3420	26	4	p	p	X
ejpam-3420	26	5	(	(	PUNCT
ejpam-3420	26	6	u	u	NOUN
ejpam-3420	26	7	)	)	PUNCT
ejpam-3420	26	8	.	.	PUNCT
ejpam-3420	27	1	the	the	DET
ejpam-3420	27	2	ordered	order	VERB
ejpam-3420	27	3	pair	pair	NOUN
ejpam-3420	27	4	(	(	PUNCT
ejpam-3420	27	5	x	x	NOUN
ejpam-3420	27	6	,	,	PUNCT
ejpam-3420	27	7	x	x	X
ejpam-3420	27	8	)	)	PUNCT
ejpam-3420	27	9	is	be	AUX
ejpam-3420	27	10	then	then	ADV
ejpam-3420	27	11	the	the	DET
ejpam-3420	27	12	covering	covering	NOUN
ejpam-3420	27	13	based	base	VERB
ejpam-3420	27	14	rough	rough	ADJ
ejpam-3420	27	15	set	set	NOUN
ejpam-3420	27	16	of	of	ADP
ejpam-3420	27	17	x.	x.	NOUN
ejpam-3420	27	18	he	he	PRON
ejpam-3420	27	19	also	also	ADV
ejpam-3420	27	20	introduced	introduce	VERB
ejpam-3420	27	21	a	a	DET
ejpam-3420	27	22	restricted	restricted	ADJ
ejpam-3420	27	23	condition	condition	NOUN
ejpam-3420	27	24	for	for	ADP
ejpam-3420	27	25	a	a	DET
ejpam-3420	27	26	covering	covering	NOUN
ejpam-3420	27	27	called	call	VERB
ejpam-3420	27	28	an	an	DET
ejpam-3420	27	29	approximation	approximation	NOUN
ejpam-3420	27	30	condition	condition	NOUN
ejpam-3420	27	31	.	.	PUNCT
ejpam-3420	28	1	since	since	SCONJ
ejpam-3420	28	2	we	we	PRON
ejpam-3420	28	3	extend	extend	VERB
ejpam-3420	28	4	the	the	DET
ejpam-3420	28	5	pawlak	pawlak	ADJ
ejpam-3420	28	6	operations	operation	NOUN
ejpam-3420	28	7	to	to	ADP
ejpam-3420	28	8	the	the	DET
ejpam-3420	28	9	covering	covering	NOUN
ejpam-3420	28	10	based	base	VERB
ejpam-3420	28	11	rough	rough	ADJ
ejpam-3420	28	12	sets	set	NOUN
ejpam-3420	28	13	,	,	PUNCT
ejpam-3420	28	14	these	these	DET
ejpam-3420	28	15	operations	operation	NOUN
ejpam-3420	28	16	may	may	AUX
ejpam-3420	28	17	not	not	PART
ejpam-3420	28	18	be	be	AUX
ejpam-3420	28	19	well	well	ADV
ejpam-3420	28	20	defined	define	VERB
ejpam-3420	28	21	.	.	PUNCT
ejpam-3420	29	1	definition	definition	NOUN
ejpam-3420	29	2	1	1	NUM
ejpam-3420	29	3	.	.	PUNCT
ejpam-3420	30	1	(	(	PUNCT
ejpam-3420	30	2	bryniarski	bryniarski	PROPN
ejpam-3420	30	3	,	,	PUNCT
ejpam-3420	30	4	1989	1989	NUM
ejpam-3420	30	5	)	)	PUNCT
ejpam-3420	30	6	let	let	VERB
ejpam-3420	30	7	u	u	PRON
ejpam-3420	30	8	be	be	AUX
ejpam-3420	30	9	a	a	DET
ejpam-3420	30	10	set	set	NOUN
ejpam-3420	30	11	with	with	ADP
ejpam-3420	30	12	at	at	ADV
ejpam-3420	30	13	least	least	ADV
ejpam-3420	30	14	two	two	NUM
ejpam-3420	30	15	elements	element	NOUN
ejpam-3420	30	16	and	and	CCONJ
ejpam-3420	30	17	c	c	AUX
ejpam-3420	30	18	be	be	AUX
ejpam-3420	30	19	its	its	PRON
ejpam-3420	30	20	covering	covering	NOUN
ejpam-3420	30	21	.	.	PUNCT
ejpam-3420	31	1	•	•	NUM
ejpam-3420	31	2	c	c	NOUN
ejpam-3420	31	3	satisfies	satisfy	VERB
ejpam-3420	31	4	the	the	DET
ejpam-3420	31	5	approximation	approximation	NOUN
ejpam-3420	31	6	condition	condition	NOUN
ejpam-3420	31	7	⇐	⇐	ADJ
ejpam-3420	31	8	⇒	⇒	NOUN
ejpam-3420	31	9	for	for	ADP
ejpam-3420	31	10	all	all	DET
ejpam-3420	31	11	a	a	PRON
ejpam-3420	31	12	,	,	PUNCT
ejpam-3420	31	13	b	b	X
ejpam-3420	31	14	⊂	⊂	PROPN
ejpam-3420	31	15	c	c	PROPN
ejpam-3420	31	16	such	such	ADJ
ejpam-3420	31	17	that	that	SCONJ
ejpam-3420	31	18	a	a	DET
ejpam-3420	31	19	⊂	⊂	PROPN
ejpam-3420	31	20	b	b	NOUN
ejpam-3420	31	21	,	,	PUNCT
ejpam-3420	31	22	there	there	PRON
ejpam-3420	31	23	exists	exist	VERB
ejpam-3420	31	24	x	x	X
ejpam-3420	31	25	⊂	⊂	PROPN
ejpam-3420	31	26	u	u	NOUN
ejpam-3420	31	27	with	with	ADP
ejpam-3420	31	28	a	a	DET
ejpam-3420	31	29	=	=	NOUN
ejpam-3420	31	30	x	x	X
ejpam-3420	31	31	and	and	CCONJ
ejpam-3420	31	32	b	b	X
ejpam-3420	32	1	=	=	SYM
ejpam-3420	32	2	x.	x.	NOUN
ejpam-3420	32	3	•	•	NUM
ejpam-3420	33	1	c	c	NOUN
ejpam-3420	33	2	is	be	AUX
ejpam-3420	33	3	minimal	minimal	ADJ
ejpam-3420	33	4	⇐	⇐	ADJ
ejpam-3420	33	5	⇒	⇒	NOUN
ejpam-3420	33	6	for	for	ADP
ejpam-3420	33	7	all	all	PRON
ejpam-3420	33	8	k	k	PROPN
ejpam-3420	33	9	∈	∈	PROPN
ejpam-3420	33	10	c	c	X
ejpam-3420	33	11	,	,	PUNCT
ejpam-3420	33	12	we	we	PRON
ejpam-3420	33	13	have	have	VERB
ejpam-3420	33	14	∪c	∪c	NOUN
ejpam-3420	33	15	\	\	NOUN
ejpam-3420	33	16	{	{	PUNCT
ejpam-3420	33	17	k	k	NOUN
ejpam-3420	33	18	}	}	PUNCT
ejpam-3420	33	19	6=	6=	NUM
ejpam-3420	33	20	u	u	PROPN
ejpam-3420	33	21	.	.	PUNCT
ejpam-3420	34	1	•	•	NUM
ejpam-3420	34	2	c	c	NOUN
ejpam-3420	34	3	is	be	AUX
ejpam-3420	34	4	representable	representable	ADJ
ejpam-3420	34	5	⇐	⇐	ADJ
ejpam-3420	34	6	⇒	⇒	NOUN
ejpam-3420	34	7	for	for	ADP
ejpam-3420	34	8	all	all	DET
ejpam-3420	34	9	k	k	PROPN
ejpam-3420	34	10	∈	∈	PROPN
ejpam-3420	34	11	c	c	NOUN
ejpam-3420	34	12	there	there	PRON
ejpam-3420	34	13	exists	exist	VERB
ejpam-3420	34	14	x	x	X
ejpam-3420	34	15	∈	∈	PROPN
ejpam-3420	34	16	k	k	NOUN
ejpam-3420	34	17	such	such	ADJ
ejpam-3420	34	18	that	that	PRON
ejpam-3420	34	19	for	for	ADP
ejpam-3420	34	20	all	all	DET
ejpam-3420	34	21	l	l	NOUN
ejpam-3420	34	22	∈	∈	PROPN
ejpam-3420	34	23	c	c	NOUN
ejpam-3420	34	24	and	and	CCONJ
ejpam-3420	34	25	l	l	PROPN
ejpam-3420	35	1	6=	6=	PROPN
ejpam-3420	36	1	k	k	X
ejpam-3420	36	2	,	,	PUNCT
ejpam-3420	36	3	we	we	PRON
ejpam-3420	36	4	have	have	VERB
ejpam-3420	36	5	x	x	X
ejpam-3420	36	6	/∈	/∈	PUNCT
ejpam-3420	36	7	l.	l.	NOUN
ejpam-3420	37	1	this	this	DET
ejpam-3420	37	2	element	element	NOUN
ejpam-3420	37	3	x	x	SYM
ejpam-3420	37	4	∈	∈	PROPN
ejpam-3420	37	5	k	k	PROPN
ejpam-3420	37	6	is	be	AUX
ejpam-3420	37	7	called	call	VERB
ejpam-3420	37	8	a	a	DET
ejpam-3420	37	9	representative	representative	NOUN
ejpam-3420	37	10	of	of	ADP
ejpam-3420	37	11	k.	k.	PROPN
ejpam-3420	37	12	•	•	PROPN
ejpam-3420	38	1	c	c	PROPN
ejpam-3420	38	2	is	be	AUX
ejpam-3420	38	3	a	a	DET
ejpam-3420	38	4	well	well	ADV
ejpam-3420	38	5	representable	representable	ADJ
ejpam-3420	38	6	if	if	SCONJ
ejpam-3420	38	7	every	every	DET
ejpam-3420	38	8	k	k	PROPN
ejpam-3420	38	9	∈	∈	PROPN
ejpam-3420	38	10	c	c	NOUN
ejpam-3420	38	11	has	have	VERB
ejpam-3420	38	12	at	at	ADV
ejpam-3420	38	13	least	least	ADJ
ejpam-3420	38	14	two	two	NUM
ejpam-3420	38	15	distinct	distinct	ADJ
ejpam-3420	38	16	representatives	representative	NOUN
ejpam-3420	38	17	.	.	PUNCT
ejpam-3420	39	1	•	•	NUM
ejpam-3420	39	2	c	c	NOUN
ejpam-3420	39	3	is	be	AUX
ejpam-3420	39	4	called	call	VERB
ejpam-3420	39	5	an	an	DET
ejpam-3420	39	6	exact	exact	ADJ
ejpam-3420	39	7	⇐	⇐	ADJ
ejpam-3420	39	8	⇒	⇒	NOUN
ejpam-3420	39	9	for	for	ADP
ejpam-3420	39	10	all	all	DET
ejpam-3420	39	11	a	a	DET
ejpam-3420	39	12	⊂	⊂	PROPN
ejpam-3420	39	13	c	c	NOUN
ejpam-3420	39	14	,	,	PUNCT
ejpam-3420	39	15	a	a	DET
ejpam-3420	39	16	=	=	X
ejpam-3420	39	17	{	{	PUNCT
ejpam-3420	39	18	k	k	X
ejpam-3420	39	19	∈	∈	PROPN
ejpam-3420	39	20	c	c	PROPN
ejpam-3420	39	21	,	,	PUNCT
ejpam-3420	39	22	k	k	PROPN
ejpam-3420	39	23	⊂	⊂	PROPN
ejpam-3420	39	24	∪a	∪a	NUM
ejpam-3420	39	25	}	}	PUNCT
ejpam-3420	39	26	.	.	PUNCT
ejpam-3420	40	1	theorem	theorem	NOUN
ejpam-3420	40	2	1	1	NUM
ejpam-3420	40	3	.	.	PUNCT
ejpam-3420	41	1	[	[	X
ejpam-3420	41	2	1	1	X
ejpam-3420	41	3	]	]	PUNCT
ejpam-3420	41	4	let	let	VERB
ejpam-3420	41	5	c	c	PRON
ejpam-3420	41	6	be	be	AUX
ejpam-3420	41	7	a	a	DET
ejpam-3420	41	8	covering	covering	NOUN
ejpam-3420	41	9	,	,	PUNCT
ejpam-3420	41	10	then	then	ADV
ejpam-3420	41	11	the	the	DET
ejpam-3420	41	12	following	follow	VERB
ejpam-3420	41	13	conditions	condition	NOUN
ejpam-3420	41	14	are	be	AUX
ejpam-3420	41	15	equivalent	equivalent	ADJ
ejpam-3420	41	16	:	:	PUNCT
ejpam-3420	41	17	(	(	PUNCT
ejpam-3420	41	18	i	i	NOUN
ejpam-3420	41	19	)	)	PUNCT
ejpam-3420	41	20	c	c	PROPN
ejpam-3420	41	21	is	be	AUX
ejpam-3420	41	22	minimal	minimal	ADJ
ejpam-3420	41	23	;	;	PUNCT
ejpam-3420	41	24	(	(	PUNCT
ejpam-3420	41	25	ii	ii	NOUN
ejpam-3420	41	26	)	)	PUNCT
ejpam-3420	41	27	c	c	PROPN
ejpam-3420	41	28	is	be	AUX
ejpam-3420	41	29	representable	representable	ADJ
ejpam-3420	41	30	;	;	PUNCT
ejpam-3420	41	31	(	(	PUNCT
ejpam-3420	41	32	iii	iii	X
ejpam-3420	41	33	)	)	PUNCT
ejpam-3420	41	34	c	c	NOUN
ejpam-3420	41	35	is	be	AUX
ejpam-3420	41	36	exact	exact	ADJ
ejpam-3420	41	37	.	.	PUNCT
ejpam-3420	42	1	theorem	theorem	NOUN
ejpam-3420	42	2	2	2	NUM
ejpam-3420	42	3	.	.	PUNCT
ejpam-3420	43	1	[	[	X
ejpam-3420	43	2	1	1	X
ejpam-3420	43	3	]	]	PUNCT
ejpam-3420	43	4	for	for	ADP
ejpam-3420	43	5	a	a	DET
ejpam-3420	43	6	covering	covering	NOUN
ejpam-3420	43	7	c	c	NOUN
ejpam-3420	43	8	of	of	ADP
ejpam-3420	43	9	the	the	DET
ejpam-3420	43	10	universe	universe	ADJ
ejpam-3420	43	11	u	u	NOUN
ejpam-3420	43	12	,	,	PUNCT
ejpam-3420	43	13	the	the	DET
ejpam-3420	43	14	following	follow	VERB
ejpam-3420	43	15	conditions	condition	NOUN
ejpam-3420	43	16	are	be	AUX
ejpam-3420	43	17	equivalent	equivalent	ADJ
ejpam-3420	43	18	:	:	PUNCT
ejpam-3420	43	19	(	(	PUNCT
ejpam-3420	43	20	i	i	NOUN
ejpam-3420	43	21	)	)	PUNCT
ejpam-3420	43	22	c	c	PROPN
ejpam-3420	43	23	is	be	AUX
ejpam-3420	43	24	well	well	ADV
ejpam-3420	43	25	representable	representable	ADJ
ejpam-3420	43	26	;	;	PUNCT
ejpam-3420	43	27	(	(	PUNCT
ejpam-3420	43	28	ii	ii	NOUN
ejpam-3420	43	29	)	)	PUNCT
ejpam-3420	43	30	c	c	NOUN
ejpam-3420	43	31	satisfies	satisfy	VERB
ejpam-3420	43	32	the	the	DET
ejpam-3420	43	33	approximation	approximation	NOUN
ejpam-3420	43	34	condition	condition	NOUN
ejpam-3420	43	35	.	.	PUNCT
ejpam-3420	44	1	we	we	PRON
ejpam-3420	44	2	can	can	AUX
ejpam-3420	44	3	say	say	VERB
ejpam-3420	44	4	that	that	SCONJ
ejpam-3420	44	5	an	an	DET
ejpam-3420	44	6	ordered	order	VERB
ejpam-3420	44	7	pair	pair	NOUN
ejpam-3420	44	8	(	(	PUNCT
ejpam-3420	44	9	u	u	NOUN
ejpam-3420	44	10	,	,	PUNCT
ejpam-3420	44	11	c	c	NOUN
ejpam-3420	44	12	)	)	PUNCT
ejpam-3420	44	13	is	be	AUX
ejpam-3420	44	14	a	a	DET
ejpam-3420	44	15	covering	covering	NOUN
ejpam-3420	44	16	approximation	approximation	NOUN
ejpam-3420	44	17	space	space	NOUN
ejpam-3420	44	18	if	if	SCONJ
ejpam-3420	44	19	c	c	PROPN
ejpam-3420	44	20	satisfies	satisfy	VERB
ejpam-3420	44	21	one	one	NUM
ejpam-3420	44	22	condition	condition	NOUN
ejpam-3420	44	23	of	of	ADP
ejpam-3420	44	24	theorem	theorem	NOUN
ejpam-3420	44	25	2	2	NUM
ejpam-3420	44	26	.	.	PUNCT
ejpam-3420	45	1	let	let	VERB
ejpam-3420	45	2	(	(	PUNCT
ejpam-3420	45	3	u	u	NOUN
ejpam-3420	45	4	,	,	PUNCT
ejpam-3420	45	5	c	c	NOUN
ejpam-3420	45	6	)	)	PUNCT
ejpam-3420	45	7	be	be	AUX
ejpam-3420	45	8	a	a	DET
ejpam-3420	45	9	covering	covering	NOUN
ejpam-3420	45	10	approximation	approximation	NOUN
ejpam-3420	45	11	space	space	NOUN
ejpam-3420	45	12	and	and	CCONJ
ejpam-3420	45	13	x	x	NOUN
ejpam-3420	45	14	,	,	PUNCT
ejpam-3420	45	15	y	y	PROPN
ejpam-3420	45	16	∈	∈	PROPN
ejpam-3420	45	17	p	p	X
ejpam-3420	45	18	(	(	PUNCT
ejpam-3420	45	19	u	u	NOUN
ejpam-3420	45	20	)	)	PUNCT
ejpam-3420	45	21	.	.	PUNCT
ejpam-3420	46	1	then	then	ADV
ejpam-3420	46	2	the	the	DET
ejpam-3420	46	3	operations	operation	NOUN
ejpam-3420	46	4	∨	∨	NOUN
ejpam-3420	46	5	,	,	PUNCT
ejpam-3420	46	6	∧	∧	NOUN
ejpam-3420	46	7	and	and	CCONJ
ejpam-3420	46	8	complement	complement	VERB
ejpam-3420	46	9	on	on	ADP
ejpam-3420	46	10	the	the	DET
ejpam-3420	46	11	rough	rough	ADJ
ejpam-3420	46	12	sets	set	NOUN
ejpam-3420	46	13	are	be	AUX
ejpam-3420	46	14	defined	define	VERB
ejpam-3420	46	15	in	in	ADP
ejpam-3420	46	16	the	the	DET
ejpam-3420	46	17	following	following	NOUN
ejpam-3420	46	18	:	:	PUNCT
ejpam-3420	46	19	(	(	PUNCT
ejpam-3420	46	20	i	i	NOUN
ejpam-3420	46	21	)	)	PUNCT
ejpam-3420	46	22	x	x	PUNCT
ejpam-3420	47	1	∨	∨	NUM
ejpam-3420	47	2	y	y	PROPN
ejpam-3420	47	3	=	=	SYM
ejpam-3420	47	4	(	(	PUNCT
ejpam-3420	47	5	x	x	PROPN
ejpam-3420	47	6	∨	∨	NUM
ejpam-3420	47	7	y	y	PROPN
ejpam-3420	47	8	,	,	PUNCT
ejpam-3420	47	9	x	x	PROPN
ejpam-3420	47	10	∨	∨	NUM
ejpam-3420	47	11	y	y	PROPN
ejpam-3420	47	12	)	)	PUNCT
ejpam-3420	47	13	;	;	PUNCT
ejpam-3420	47	14	n.	n.	NOUN
ejpam-3420	47	15	alharbi	alharbi	PROPN
ejpam-3420	47	16	,	,	PUNCT
ejpam-3420	47	17	h.	h.	PROPN
ejpam-3420	47	18	aydi	aydi	PROPN
ejpam-3420	47	19	,	,	PUNCT
ejpam-3420	47	20	c.	c.	PROPN
ejpam-3420	47	21	özel	özel	PROPN
ejpam-3420	47	22	/	/	SYM
ejpam-3420	47	23	eur	eur	PROPN
ejpam-3420	47	24	.	.	PUNCT
ejpam-3420	48	1	j.	j.	PROPN
ejpam-3420	48	2	pure	pure	PROPN
ejpam-3420	48	3	appl	appl	PROPN
ejpam-3420	48	4	.	.	PROPN
ejpam-3420	48	5	math	math	PROPN
ejpam-3420	48	6	,	,	PUNCT
ejpam-3420	48	7	12	12	NUM
ejpam-3420	48	8	(	(	PUNCT
ejpam-3420	48	9	2	2	NUM
ejpam-3420	48	10	)	)	PUNCT
ejpam-3420	48	11	(	(	PUNCT
ejpam-3420	48	12	2019	2019	NUM
ejpam-3420	48	13	)	)	PUNCT
ejpam-3420	48	14	,	,	PUNCT
ejpam-3420	48	15	533	533	NUM
ejpam-3420	48	16	-	-	SYM
ejpam-3420	48	17	543	543	NUM
ejpam-3420	48	18	535	535	NUM
ejpam-3420	48	19	(	(	PUNCT
ejpam-3420	48	20	ii	ii	NOUN
ejpam-3420	48	21	)	)	PUNCT
ejpam-3420	48	22	x	x	X
ejpam-3420	49	1	∧	∧	NOUN
ejpam-3420	49	2	y	y	NOUN
ejpam-3420	49	3	=	=	SYM
ejpam-3420	49	4	(	(	PUNCT
ejpam-3420	49	5	x	x	PUNCT
ejpam-3420	49	6	∧	∧	NOUN
ejpam-3420	49	7	y	y	PROPN
ejpam-3420	49	8	,	,	PUNCT
ejpam-3420	49	9	x	x	PROPN
ejpam-3420	49	10	∧	∧	NOUN
ejpam-3420	49	11	y	y	PROPN
ejpam-3420	49	12	)	)	PUNCT
ejpam-3420	49	13	;	;	PUNCT
ejpam-3420	49	14	(	(	PUNCT
ejpam-3420	49	15	iii	iii	X
ejpam-3420	49	16	)	)	PUNCT
ejpam-3420	49	17	xc	xc	X
ejpam-3420	49	18	=	=	PUNCT
ejpam-3420	49	19	(	(	PUNCT
ejpam-3420	49	20	c	c	NOUN
ejpam-3420	49	21	\x	\x	PROPN
ejpam-3420	49	22	,	,	PUNCT
ejpam-3420	49	23	c	c	PROPN
ejpam-3420	49	24	\x	\x	NOUN
ejpam-3420	49	25	)	)	PUNCT
ejpam-3420	49	26	.	.	PUNCT
ejpam-3420	50	1	note	note	VERB
ejpam-3420	50	2	that	that	SCONJ
ejpam-3420	50	3	the	the	DET
ejpam-3420	50	4	union	union	NOUN
ejpam-3420	50	5	and	and	CCONJ
ejpam-3420	50	6	intersection	intersection	NOUN
ejpam-3420	50	7	are	be	AUX
ejpam-3420	50	8	about	about	ADP
ejpam-3420	50	9	classes	class	NOUN
ejpam-3420	50	10	,	,	PUNCT
ejpam-3420	50	11	not	not	PART
ejpam-3420	50	12	elements	element	NOUN
ejpam-3420	50	13	in	in	ADP
ejpam-3420	50	14	the	the	DET
ejpam-3420	50	15	classes	class	NOUN
ejpam-3420	50	16	.	.	PUNCT
ejpam-3420	51	1	a	a	DET
ejpam-3420	51	2	rough	rough	ADJ
ejpam-3420	51	3	set	set	NOUN
ejpam-3420	51	4	x	x	PUNCT
ejpam-3420	51	5	is	be	AUX
ejpam-3420	51	6	exact	exact	ADJ
ejpam-3420	51	7	if	if	SCONJ
ejpam-3420	51	8	x	x	PRON
ejpam-3420	51	9	=	=	PUNCT
ejpam-3420	51	10	x.	x.	NOUN
ejpam-3420	51	11	the	the	DET
ejpam-3420	51	12	family	family	NOUN
ejpam-3420	51	13	of	of	ADP
ejpam-3420	51	14	all	all	DET
ejpam-3420	51	15	rough	rough	ADJ
ejpam-3420	51	16	sets	set	NOUN
ejpam-3420	51	17	of	of	ADP
ejpam-3420	51	18	elements	element	NOUN
ejpam-3420	51	19	belonging	belong	VERB
ejpam-3420	51	20	to	to	ADP
ejpam-3420	51	21	p	p	PROPN
ejpam-3420	51	22	(	(	PUNCT
ejpam-3420	51	23	u	u	NOUN
ejpam-3420	51	24	)	)	PUNCT
ejpam-3420	51	25	is	be	AUX
ejpam-3420	51	26	called	call	VERB
ejpam-3420	51	27	a	a	DET
ejpam-3420	51	28	rough	rough	ADJ
ejpam-3420	51	29	set	set	NOUN
ejpam-3420	51	30	of	of	ADP
ejpam-3420	51	31	the	the	DET
ejpam-3420	51	32	first	first	ADJ
ejpam-3420	51	33	order	order	NOUN
ejpam-3420	51	34	.	.	PUNCT
ejpam-3420	52	1	this	this	DET
ejpam-3420	52	2	family	family	NOUN
ejpam-3420	52	3	is	be	AUX
ejpam-3420	52	4	a	a	DET
ejpam-3420	52	5	distributive	distributive	ADJ
ejpam-3420	52	6	lattice	lattice	NOUN
ejpam-3420	52	7	with	with	ADP
ejpam-3420	52	8	operations	operation	NOUN
ejpam-3420	52	9	∨	∨	ADJ
ejpam-3420	52	10	and	and	CCONJ
ejpam-3420	52	11	∧.	∧.	PROPN
ejpam-3420	52	12	also	also	ADV
ejpam-3420	52	13	,	,	PUNCT
ejpam-3420	52	14	it	it	PRON
ejpam-3420	52	15	satisfies	satisfy	VERB
ejpam-3420	52	16	de	de	PROPN
ejpam-3420	52	17	morgan	morgan	PROPN
ejpam-3420	52	18	laws	law	NOUN
ejpam-3420	52	19	with	with	ADP
ejpam-3420	52	20	operations	operation	NOUN
ejpam-3420	52	21	∨	∨	NOUN
ejpam-3420	52	22	and	and	CCONJ
ejpam-3420	52	23	∧	∧	PROPN
ejpam-3420	52	24	and	and	CCONJ
ejpam-3420	52	25	complement	complement	NOUN
ejpam-3420	52	26	.	.	PUNCT
ejpam-3420	53	1	in	in	ADP
ejpam-3420	53	2	addition	addition	NOUN
ejpam-3420	53	3	,	,	PUNCT
ejpam-3420	53	4	the	the	DET
ejpam-3420	53	5	set	set	NOUN
ejpam-3420	53	6	of	of	ADP
ejpam-3420	53	7	all	all	DET
ejpam-3420	53	8	rough	rough	ADJ
ejpam-3420	53	9	exact	exact	ADJ
ejpam-3420	53	10	sets	set	NOUN
ejpam-3420	53	11	with	with	ADP
ejpam-3420	53	12	operations	operation	NOUN
ejpam-3420	53	13	∨,∧	∨,∧	NOUN
ejpam-3420	53	14	and	and	CCONJ
ejpam-3420	53	15	complement	complement	NOUN
ejpam-3420	53	16	forms	form	NOUN
ejpam-3420	53	17	a	a	DET
ejpam-3420	53	18	boolean	boolean	ADJ
ejpam-3420	53	19	algebra	algebra	NOUN
ejpam-3420	53	20	[	[	X
ejpam-3420	53	21	1	1	NUM
ejpam-3420	53	22	]	]	PUNCT
ejpam-3420	53	23	.	.	PUNCT
ejpam-3420	54	1	proposition	proposition	NOUN
ejpam-3420	54	2	1	1	NUM
ejpam-3420	54	3	.	.	PUNCT
ejpam-3420	55	1	[	[	X
ejpam-3420	55	2	1	1	X
ejpam-3420	55	3	]	]	X
ejpam-3420	55	4	if	if	SCONJ
ejpam-3420	55	5	(	(	PUNCT
ejpam-3420	55	6	u	u	NOUN
ejpam-3420	55	7	,	,	PUNCT
ejpam-3420	55	8	c	c	NOUN
ejpam-3420	55	9	)	)	PUNCT
ejpam-3420	55	10	is	be	AUX
ejpam-3420	55	11	a	a	DET
ejpam-3420	55	12	covering	covering	NOUN
ejpam-3420	55	13	approximation	approximation	NOUN
ejpam-3420	55	14	space	space	NOUN
ejpam-3420	55	15	and	and	CCONJ
ejpam-3420	55	16	a	a	PRON
ejpam-3420	55	17	and	and	CCONJ
ejpam-3420	55	18	b	b	NOUN
ejpam-3420	55	19	are	be	AUX
ejpam-3420	55	20	subsets	subset	NOUN
ejpam-3420	55	21	of	of	ADP
ejpam-3420	55	22	u	u	NOUN
ejpam-3420	55	23	,	,	PUNCT
ejpam-3420	55	24	then	then	ADV
ejpam-3420	55	25	(	(	PUNCT
ejpam-3420	55	26	i	i	NOUN
ejpam-3420	55	27	)	)	PUNCT
ejpam-3420	55	28	a	a	DET
ejpam-3420	55	29	⊆	⊆	NUM
ejpam-3420	55	30	a	a	DET
ejpam-3420	55	31	⊆	⊆	NUM
ejpam-3420	55	32	a	a	PRON
ejpam-3420	55	33	;	;	PUNCT
ejpam-3420	55	34	(	(	PUNCT
ejpam-3420	55	35	ii	ii	NOUN
ejpam-3420	55	36	)	)	PUNCT
ejpam-3420	55	37	∅	∅	NOUN
ejpam-3420	55	38	=	=	NOUN
ejpam-3420	55	39	∅	∅	NOUN
ejpam-3420	55	40	=	=	NOUN
ejpam-3420	55	41	∅	∅	NOUN
ejpam-3420	55	42	;	;	PUNCT
ejpam-3420	55	43	(	(	PUNCT
ejpam-3420	55	44	iii	iii	X
ejpam-3420	55	45	)	)	PUNCT
ejpam-3420	55	46	u	u	NOUN
ejpam-3420	55	47	=	=	SYM
ejpam-3420	55	48	u	u	NOUN
ejpam-3420	55	49	=	=	PROPN
ejpam-3420	55	50	u	u	PROPN
ejpam-3420	55	51	;	;	PUNCT
ejpam-3420	55	52	(	(	PUNCT
ejpam-3420	55	53	iv	iv	X
ejpam-3420	55	54	)	)	PUNCT
ejpam-3420	55	55	a	a	DET
ejpam-3420	55	56	∨b	∨b	NOUN
ejpam-3420	55	57	=	=	PUNCT
ejpam-3420	55	58	a	a	DET
ejpam-3420	55	59	∨b	∨b	NOUN
ejpam-3420	55	60	;	;	PUNCT
ejpam-3420	55	61	(	(	PUNCT
ejpam-3420	55	62	v	v	NOUN
ejpam-3420	55	63	)	)	PUNCT
ejpam-3420	55	64	a	a	DET
ejpam-3420	55	65	∧b	∧b	NOUN
ejpam-3420	55	66	⊆	⊆	NUM
ejpam-3420	55	67	a	a	DET
ejpam-3420	55	68	∧b	∧b	NUM
ejpam-3420	55	69	;	;	PUNCT
ejpam-3420	55	70	(	(	PUNCT
ejpam-3420	55	71	vi	vi	X
ejpam-3420	55	72	)	)	PUNCT
ejpam-3420	55	73	a	a	DET
ejpam-3420	55	74	∧b	∧b	NOUN
ejpam-3420	55	75	=	=	NOUN
ejpam-3420	55	76	a	a	DET
ejpam-3420	55	77	∧b	∧b	NOUN
ejpam-3420	55	78	.	.	PUNCT
ejpam-3420	56	1	(	(	PUNCT
ejpam-3420	56	2	vii	vii	PROPN
ejpam-3420	56	3	)	)	PUNCT
ejpam-3420	56	4	a	a	DET
ejpam-3420	56	5	∨b	∨b	NOUN
ejpam-3420	56	6	⊆	⊆	NUM
ejpam-3420	56	7	a	a	DET
ejpam-3420	56	8	∨b	∨b	NOUN
ejpam-3420	56	9	;	;	PUNCT
ejpam-3420	56	10	(	(	PUNCT
ejpam-3420	56	11	viii	viii	NOUN
ejpam-3420	56	12	)	)	PUNCT
ejpam-3420	56	13	if	if	SCONJ
ejpam-3420	56	14	a	a	DET
ejpam-3420	56	15	⊆	⊆	NUM
ejpam-3420	56	16	b	b	NOUN
ejpam-3420	56	17	,	,	PUNCT
ejpam-3420	56	18	then	then	ADV
ejpam-3420	56	19	a	a	DET
ejpam-3420	56	20	⊆	⊆	NUM
ejpam-3420	56	21	b	b	NOUN
ejpam-3420	56	22	and	and	CCONJ
ejpam-3420	56	23	a	a	DET
ejpam-3420	56	24	⊆	⊆	NUM
ejpam-3420	56	25	b	b	NOUN
ejpam-3420	56	26	;	;	PUNCT
ejpam-3420	56	27	(	(	PUNCT
ejpam-3420	56	28	ix	ix	X
ejpam-3420	56	29	)	)	PUNCT
ejpam-3420	56	30	ac	ac	PROPN
ejpam-3420	56	31	=	=	PUNCT
ejpam-3420	56	32	(	(	PUNCT
ejpam-3420	56	33	a)c	a)c	X
ejpam-3420	56	34	,	,	PUNCT
ejpam-3420	56	35	and	and	CCONJ
ejpam-3420	56	36	ac	ac	PROPN
ejpam-3420	56	37	=	=	PUNCT
ejpam-3420	56	38	(	(	PUNCT
ejpam-3420	56	39	a)c	a)c	X
ejpam-3420	56	40	;	;	PUNCT
ejpam-3420	56	41	(	(	PUNCT
ejpam-3420	56	42	x	x	X
ejpam-3420	56	43	)	)	PUNCT
ejpam-3420	56	44	a	a	PRON
ejpam-3420	56	45	=	=	X
ejpam-3420	56	46	(	(	PUNCT
ejpam-3420	56	47	a	a	NOUN
ejpam-3420	56	48	)	)	PUNCT
ejpam-3420	56	49	=	=	SYM
ejpam-3420	57	1	a	a	NOUN
ejpam-3420	57	2	;	;	PUNCT
ejpam-3420	57	3	(	(	PUNCT
ejpam-3420	57	4	xi	xi	X
ejpam-3420	57	5	)	)	PUNCT
ejpam-3420	57	6	a	a	PRON
ejpam-3420	58	1	=	=	X
ejpam-3420	59	1	(	(	PUNCT
ejpam-3420	59	2	a	a	NOUN
ejpam-3420	59	3	)	)	PUNCT
ejpam-3420	59	4	=	=	SYM
ejpam-3420	59	5	a.	a.	NOUN
ejpam-3420	59	6	definition	definition	NOUN
ejpam-3420	59	7	2	2	NUM
ejpam-3420	59	8	.	.	PUNCT
ejpam-3420	60	1	[	[	X
ejpam-3420	60	2	5	5	NUM
ejpam-3420	60	3	]	]	X
ejpam-3420	60	4	let	let	VERB
ejpam-3420	60	5	(	(	PUNCT
ejpam-3420	60	6	u	u	NOUN
ejpam-3420	60	7	,	,	PUNCT
ejpam-3420	60	8	c	c	NOUN
ejpam-3420	60	9	)	)	PUNCT
ejpam-3420	60	10	be	be	AUX
ejpam-3420	60	11	a	a	DET
ejpam-3420	60	12	covering	covering	NOUN
ejpam-3420	60	13	approximation	approximation	NOUN
ejpam-3420	60	14	space	space	NOUN
ejpam-3420	60	15	and	and	CCONJ
ejpam-3420	60	16	x	x	SYM
ejpam-3420	60	17	=	=	SYM
ejpam-3420	60	18	(	(	PUNCT
ejpam-3420	60	19	x	x	X
ejpam-3420	60	20	,	,	PUNCT
ejpam-3420	60	21	x	x	X
ejpam-3420	60	22	)	)	PUNCT
ejpam-3420	60	23	be	be	AUX
ejpam-3420	60	24	a	a	DET
ejpam-3420	60	25	covering	covering	NOUN
ejpam-3420	60	26	based	base	VERB
ejpam-3420	60	27	rough	rough	ADJ
ejpam-3420	60	28	set	set	NOUN
ejpam-3420	60	29	.	.	PUNCT
ejpam-3420	61	1	the	the	DET
ejpam-3420	61	2	collection	collection	NOUN
ejpam-3420	61	3	τ	τ	X
ejpam-3420	61	4	consisting	consist	VERB
ejpam-3420	61	5	of	of	ADP
ejpam-3420	61	6	rough	rough	ADJ
ejpam-3420	61	7	subsets	subset	NOUN
ejpam-3420	61	8	of	of	ADP
ejpam-3420	61	9	x	x	X
ejpam-3420	61	10	=	=	SYM
ejpam-3420	61	11	(	(	PUNCT
ejpam-3420	61	12	x	x	X
ejpam-3420	61	13	,	,	PUNCT
ejpam-3420	61	14	x	x	X
ejpam-3420	61	15	)	)	PUNCT
ejpam-3420	61	16	is	be	AUX
ejpam-3420	61	17	called	call	VERB
ejpam-3420	61	18	a	a	DET
ejpam-3420	61	19	covering	covering	NOUN
ejpam-3420	61	20	based	base	VERB
ejpam-3420	61	21	rough	rough	ADJ
ejpam-3420	61	22	topology	topology	NOUN
ejpam-3420	61	23	on	on	ADP
ejpam-3420	61	24	x	x	X
ejpam-3420	61	25	=	=	SYM
ejpam-3420	61	26	(	(	PUNCT
ejpam-3420	61	27	x	x	X
ejpam-3420	61	28	,	,	PUNCT
ejpam-3420	61	29	x	x	NOUN
ejpam-3420	61	30	)	)	PUNCT
ejpam-3420	61	31	if	if	SCONJ
ejpam-3420	61	32	the	the	DET
ejpam-3420	61	33	following	follow	VERB
ejpam-3420	61	34	conditions	condition	NOUN
ejpam-3420	61	35	are	be	AUX
ejpam-3420	61	36	satisfied	satisfied	ADJ
ejpam-3420	61	37	:	:	PUNCT
ejpam-3420	61	38	(	(	PUNCT
ejpam-3420	61	39	i	i	NOUN
ejpam-3420	61	40	)	)	PUNCT
ejpam-3420	61	41	∅	∅	NOUN
ejpam-3420	61	42	,	,	PUNCT
ejpam-3420	61	43	x	x	SYM
ejpam-3420	61	44	=	=	SYM
ejpam-3420	61	45	(	(	PUNCT
ejpam-3420	61	46	x	x	X
ejpam-3420	61	47	,	,	PUNCT
ejpam-3420	61	48	x	x	X
ejpam-3420	61	49	)	)	PUNCT
ejpam-3420	61	50	∈	∈	PROPN
ejpam-3420	61	51	τ	τ	X
ejpam-3420	61	52	;	;	PUNCT
ejpam-3420	61	53	(	(	PUNCT
ejpam-3420	61	54	ii	ii	NOUN
ejpam-3420	61	55	)	)	PUNCT
ejpam-3420	61	56	τ	τ	PROPN
ejpam-3420	61	57	is	be	AUX
ejpam-3420	61	58	closed	close	VERB
ejpam-3420	61	59	under	under	ADP
ejpam-3420	61	60	a	a	DET
ejpam-3420	61	61	finite	finite	ADJ
ejpam-3420	61	62	intersection	intersection	NOUN
ejpam-3420	61	63	;	;	PUNCT
ejpam-3420	61	64	(	(	PUNCT
ejpam-3420	61	65	iii	iii	X
ejpam-3420	61	66	)	)	PUNCT
ejpam-3420	61	67	τ	τ	PROPN
ejpam-3420	61	68	is	be	AUX
ejpam-3420	61	69	closed	close	VERB
ejpam-3420	61	70	under	under	ADP
ejpam-3420	61	71	an	an	DET
ejpam-3420	61	72	arbitrary	arbitrary	ADJ
ejpam-3420	61	73	union	union	NOUN
ejpam-3420	61	74	.	.	PUNCT
ejpam-3420	62	1	in	in	ADP
ejpam-3420	62	2	2015	2015	NUM
ejpam-3420	62	3	,	,	PUNCT
ejpam-3420	62	4	akduman	akduman	NOUN
ejpam-3420	62	5	,	,	PUNCT
ejpam-3420	62	6	özcelik	özcelik	PROPN
ejpam-3420	62	7	and	and	CCONJ
ejpam-3420	62	8	özel	özel	ADJ
ejpam-3420	62	9	[	[	X
ejpam-3420	62	10	4	4	X
ejpam-3420	62	11	]	]	PUNCT
ejpam-3420	62	12	constructed	construct	VERB
ejpam-3420	62	13	a	a	DET
ejpam-3420	62	14	topology	topology	NOUN
ejpam-3420	62	15	on	on	ADP
ejpam-3420	62	16	classical	classical	ADJ
ejpam-3420	62	17	rough	rough	ADJ
ejpam-3420	62	18	sets	set	NOUN
ejpam-3420	62	19	.	.	PUNCT
ejpam-3420	63	1	they	they	PRON
ejpam-3420	63	2	called	call	VERB
ejpam-3420	63	3	it	it	PRON
ejpam-3420	63	4	a	a	DET
ejpam-3420	63	5	rough	rough	ADJ
ejpam-3420	63	6	topological	topological	ADJ
ejpam-3420	63	7	space	space	NOUN
ejpam-3420	63	8	.	.	PUNCT
ejpam-3420	64	1	also	also	ADV
ejpam-3420	64	2	,	,	PUNCT
ejpam-3420	64	3	they	they	PRON
ejpam-3420	64	4	defined	define	VERB
ejpam-3420	64	5	a	a	DET
ejpam-3420	64	6	topology	topology	NOUN
ejpam-3420	64	7	on	on	ADP
ejpam-3420	64	8	a	a	DET
ejpam-3420	64	9	given	give	VERB
ejpam-3420	64	10	covering	covering	NOUN
ejpam-3420	64	11	based	base	VERB
ejpam-3420	64	12	rough	rough	ADJ
ejpam-3420	64	13	set	set	NOUN
ejpam-3420	64	14	.	.	PUNCT
ejpam-3420	65	1	n.	n.	PROPN
ejpam-3420	65	2	alharbi	alharbi	PROPN
ejpam-3420	65	3	,	,	PUNCT
ejpam-3420	65	4	h.	h.	PROPN
ejpam-3420	65	5	aydi	aydi	PROPN
ejpam-3420	65	6	,	,	PUNCT
ejpam-3420	65	7	c.	c.	PROPN
ejpam-3420	65	8	özel	özel	PROPN
ejpam-3420	65	9	/	/	SYM
ejpam-3420	65	10	eur	eur	PROPN
ejpam-3420	65	11	.	.	PUNCT
ejpam-3420	66	1	j.	j.	PROPN
ejpam-3420	66	2	pure	pure	PROPN
ejpam-3420	66	3	appl	appl	PROPN
ejpam-3420	66	4	.	.	PROPN
ejpam-3420	66	5	math	math	PROPN
ejpam-3420	66	6	,	,	PUNCT
ejpam-3420	66	7	12	12	NUM
ejpam-3420	66	8	(	(	PUNCT
ejpam-3420	66	9	2	2	NUM
ejpam-3420	66	10	)	)	PUNCT
ejpam-3420	66	11	(	(	PUNCT
ejpam-3420	66	12	2019	2019	NUM
ejpam-3420	66	13	)	)	PUNCT
ejpam-3420	66	14	,	,	PUNCT
ejpam-3420	66	15	533	533	NUM
ejpam-3420	66	16	-	-	SYM
ejpam-3420	66	17	543	543	NUM
ejpam-3420	66	18	536	536	NUM
ejpam-3420	66	19	2	2	NUM
ejpam-3420	66	20	.	.	PUNCT
ejpam-3420	67	1	main	main	ADJ
ejpam-3420	67	2	results	result	NOUN
ejpam-3420	67	3	to	to	PART
ejpam-3420	67	4	define	define	VERB
ejpam-3420	67	5	open	open	ADJ
ejpam-3420	67	6	and	and	CCONJ
ejpam-3420	67	7	closed	closed	ADJ
ejpam-3420	67	8	sets	set	NOUN
ejpam-3420	67	9	on	on	ADP
ejpam-3420	67	10	a	a	DET
ejpam-3420	67	11	covering	covering	NOUN
ejpam-3420	67	12	based	base	VERB
ejpam-3420	67	13	rough	rough	ADJ
ejpam-3420	67	14	topology	topology	NOUN
ejpam-3420	67	15	,	,	PUNCT
ejpam-3420	67	16	we	we	PRON
ejpam-3420	67	17	need	need	VERB
ejpam-3420	67	18	the	the	DET
ejpam-3420	67	19	next	next	ADJ
ejpam-3420	67	20	result	result	NOUN
ejpam-3420	67	21	.	.	PUNCT
ejpam-3420	68	1	theorem	theorem	VERB
ejpam-3420	68	2	3	3	NUM
ejpam-3420	68	3	.	.	PUNCT
ejpam-3420	69	1	[	[	X
ejpam-3420	69	2	1	1	X
ejpam-3420	69	3	]	]	PUNCT
ejpam-3420	69	4	for	for	ADP
ejpam-3420	69	5	every	every	DET
ejpam-3420	69	6	covering	covering	NOUN
ejpam-3420	69	7	approximation	approximation	NOUN
ejpam-3420	69	8	space	space	NOUN
ejpam-3420	69	9	(	(	PUNCT
ejpam-3420	69	10	u	u	NOUN
ejpam-3420	69	11	,	,	PUNCT
ejpam-3420	69	12	c	c	NOUN
ejpam-3420	69	13	)	)	PUNCT
ejpam-3420	69	14	,	,	PUNCT
ejpam-3420	69	15	there	there	PRON
ejpam-3420	69	16	exists	exist	VERB
ejpam-3420	69	17	a	a	DET
ejpam-3420	69	18	one	one	NUM
ejpam-3420	69	19	-	-	PUNCT
ejpam-3420	69	20	to	to	ADP
ejpam-3420	69	21	-	-	PUNCT
ejpam-3420	69	22	one	one	NUM
ejpam-3420	69	23	mapping	mapping	NOUN
ejpam-3420	69	24	of	of	ADP
ejpam-3420	69	25	the	the	DET
ejpam-3420	69	26	set	set	NOUN
ejpam-3420	69	27	of	of	ADP
ejpam-3420	69	28	all	all	DET
ejpam-3420	69	29	pairs	pair	NOUN
ejpam-3420	69	30	(	(	PUNCT
ejpam-3420	69	31	a	a	DET
ejpam-3420	69	32	,	,	PUNCT
ejpam-3420	69	33	b	b	NOUN
ejpam-3420	69	34	)	)	PUNCT
ejpam-3420	69	35	such	such	ADJ
ejpam-3420	69	36	that	that	SCONJ
ejpam-3420	69	37	a	a	DET
ejpam-3420	69	38	⊂	⊂	X
ejpam-3420	69	39	b	b	X
ejpam-3420	69	40	⊂	⊂	PROPN
ejpam-3420	69	41	c	c	PROPN
ejpam-3420	69	42	onto	onto	ADP
ejpam-3420	69	43	the	the	DET
ejpam-3420	69	44	set	set	NOUN
ejpam-3420	69	45	of	of	ADP
ejpam-3420	69	46	all	all	DET
ejpam-3420	69	47	rough	rough	ADJ
ejpam-3420	69	48	sets	set	NOUN
ejpam-3420	69	49	of	of	ADP
ejpam-3420	69	50	the	the	DET
ejpam-3420	69	51	first	first	ADJ
ejpam-3420	69	52	order	order	NOUN
ejpam-3420	69	53	defined	define	VERB
ejpam-3420	69	54	as	as	ADP
ejpam-3420	69	55	(	(	PUNCT
ejpam-3420	69	56	a	a	DET
ejpam-3420	69	57	,	,	PUNCT
ejpam-3420	69	58	b	b	NOUN
ejpam-3420	69	59	)	)	PUNCT
ejpam-3420	70	1	=	=	NOUN
ejpam-3420	70	2	x	x	SYM
ejpam-3420	70	3	⇐	⇐	ADJ
ejpam-3420	70	4	⇒	⇒	NOUN
ejpam-3420	70	5	a	a	DET
ejpam-3420	70	6	=	=	SYM
ejpam-3420	70	7	x	x	X
ejpam-3420	70	8	and	and	CCONJ
ejpam-3420	70	9	b	b	X
ejpam-3420	70	10	=	=	SYM
ejpam-3420	70	11	x	x	NOUN
ejpam-3420	70	12	,	,	PUNCT
ejpam-3420	70	13	where	where	SCONJ
ejpam-3420	70	14	x	x	SYM
ejpam-3420	70	15	⊆	⊆	NUM
ejpam-3420	70	16	u	u	NOUN
ejpam-3420	70	17	.	.	PUNCT
ejpam-3420	71	1	let	let	VERB
ejpam-3420	71	2	y	y	PRON
ejpam-3420	71	3	⊆	⊆	NUM
ejpam-3420	71	4	u	u	NOUN
ejpam-3420	71	5	be	be	VERB
ejpam-3420	71	6	a	a	DET
ejpam-3420	71	7	rough	rough	ADJ
ejpam-3420	71	8	open	open	ADJ
ejpam-3420	71	9	set	set	NOUN
ejpam-3420	71	10	in	in	ADP
ejpam-3420	71	11	a	a	DET
ejpam-3420	71	12	covering	covering	NOUN
ejpam-3420	71	13	based	base	VERB
ejpam-3420	71	14	rough	rough	ADJ
ejpam-3420	71	15	space	space	NOUN
ejpam-3420	71	16	(	(	PUNCT
ejpam-3420	71	17	x	x	X
ejpam-3420	71	18	,	,	PUNCT
ejpam-3420	71	19	τ	τ	PROPN
ejpam-3420	71	20	)	)	PUNCT
ejpam-3420	71	21	,	,	PUNCT
ejpam-3420	71	22	then	then	ADV
ejpam-3420	71	23	there	there	PRON
ejpam-3420	71	24	exist	exist	VERB
ejpam-3420	71	25	a	a	DET
ejpam-3420	71	26	,	,	PUNCT
ejpam-3420	71	27	b	b	X
ejpam-3420	71	28	⊂	⊂	PUNCT
ejpam-3420	71	29	u	u	NOUN
ejpam-3420	71	30	such	such	ADJ
ejpam-3420	71	31	that	that	SCONJ
ejpam-3420	71	32	a	a	DET
ejpam-3420	71	33	⊂	⊂	X
ejpam-3420	71	34	b	b	X
ejpam-3420	71	35	⊂	⊂	PROPN
ejpam-3420	71	36	c	c	PROPN
ejpam-3420	71	37	with	with	ADP
ejpam-3420	71	38	y	y	PROPN
ejpam-3420	71	39	=	=	PUNCT
ejpam-3420	71	40	a	a	PROPN
ejpam-3420	71	41	,	,	PUNCT
ejpam-3420	71	42	y	y	PROPN
ejpam-3420	71	43	=	=	SYM
ejpam-3420	71	44	b	b	PROPN
ejpam-3420	71	45	,	,	PUNCT
ejpam-3420	71	46	a	a	DET
ejpam-3420	71	47	⊆	⊆	NUM
ejpam-3420	71	48	x	x	NOUN
ejpam-3420	71	49	and	and	CCONJ
ejpam-3420	71	50	b	b	NOUN
ejpam-3420	71	51	⊆	⊆	NUM
ejpam-3420	71	52	x.	x.	NOUN
ejpam-3420	71	53	to	to	PART
ejpam-3420	71	54	define	define	VERB
ejpam-3420	71	55	closed	closed	ADJ
ejpam-3420	71	56	sets	set	NOUN
ejpam-3420	71	57	in	in	ADP
ejpam-3420	71	58	x	x	NOUN
ejpam-3420	71	59	,	,	PUNCT
ejpam-3420	71	60	we	we	PRON
ejpam-3420	71	61	define	define	VERB
ejpam-3420	71	62	the	the	DET
ejpam-3420	71	63	rough	rough	ADJ
ejpam-3420	71	64	complement	complement	NOUN
ejpam-3420	71	65	of	of	ADP
ejpam-3420	71	66	any	any	DET
ejpam-3420	71	67	rough	rough	ADJ
ejpam-3420	71	68	open	open	NOUN
ejpam-3420	71	69	set	set	VERB
ejpam-3420	71	70	with	with	ADP
ejpam-3420	71	71	respect	respect	NOUN
ejpam-3420	71	72	to	to	ADP
ejpam-3420	71	73	the	the	DET
ejpam-3420	71	74	rough	rough	ADJ
ejpam-3420	71	75	setx	setx	NOUN
ejpam-3420	71	76	.	.	PUNCT
ejpam-3420	72	1	if	if	SCONJ
ejpam-3420	72	2	y	y	PROPN
ejpam-3420	72	3	=	=	PRON
ejpam-3420	72	4	(	(	PUNCT
ejpam-3420	72	5	y	y	PROPN
ejpam-3420	72	6	,	,	PUNCT
ejpam-3420	72	7	y	y	PROPN
ejpam-3420	72	8	)	)	PUNCT
ejpam-3420	72	9	is	be	AUX
ejpam-3420	72	10	a	a	DET
ejpam-3420	72	11	rough	rough	ADJ
ejpam-3420	72	12	open	open	ADJ
ejpam-3420	72	13	set	set	NOUN
ejpam-3420	72	14	inx	inx	NOUN
ejpam-3420	72	15	,	,	PUNCT
ejpam-3420	72	16	then	then	ADV
ejpam-3420	72	17	y	y	PROPN
ejpam-3420	72	18	c	c	PROPN
ejpam-3420	72	19	=	=	PUNCT
ejpam-3420	72	20	(	(	PUNCT
ejpam-3420	72	21	x\y	x\y	X
ejpam-3420	72	22	,	,	PUNCT
ejpam-3420	72	23	x\y	x\y	PROPN
ejpam-3420	72	24	)	)	PUNCT
ejpam-3420	72	25	is	be	AUX
ejpam-3420	72	26	a	a	DET
ejpam-3420	72	27	closed	closed	ADJ
ejpam-3420	72	28	set	set	NOUN
ejpam-3420	72	29	in	in	ADP
ejpam-3420	72	30	the	the	DET
ejpam-3420	72	31	covering	covering	NOUN
ejpam-3420	72	32	based	base	VERB
ejpam-3420	72	33	rough	rough	ADJ
ejpam-3420	72	34	space	space	NOUN
ejpam-3420	72	35	(	(	PUNCT
ejpam-3420	72	36	x	x	X
ejpam-3420	72	37	,	,	PUNCT
ejpam-3420	72	38	τ	τ	X
ejpam-3420	72	39	)	)	PUNCT
ejpam-3420	72	40	such	such	ADJ
ejpam-3420	72	41	that	that	PRON
ejpam-3420	72	42	x\y	x\y	PROPN
ejpam-3420	73	1	⊆	⊆	NUM
ejpam-3420	73	2	x	x	PUNCT
ejpam-3420	73	3	and	and	CCONJ
ejpam-3420	73	4	x\y	x\y	PROPN
ejpam-3420	74	1	⊆	⊆	NUM
ejpam-3420	74	2	x.	x.	NOUN
ejpam-3420	74	3	similarly	similarly	ADV
ejpam-3420	74	4	as	as	ADP
ejpam-3420	74	5	the	the	DET
ejpam-3420	74	6	classical	classical	ADJ
ejpam-3420	74	7	rough	rough	ADJ
ejpam-3420	74	8	topology	topology	NOUN
ejpam-3420	74	9	,	,	PUNCT
ejpam-3420	74	10	we	we	PRON
ejpam-3420	74	11	can	can	AUX
ejpam-3420	74	12	define	define	VERB
ejpam-3420	74	13	strong	strong	ADJ
ejpam-3420	74	14	and	and	CCONJ
ejpam-3420	74	15	weak	weak	ADJ
ejpam-3420	74	16	elements	element	NOUN
ejpam-3420	74	17	for	for	ADP
ejpam-3420	74	18	any	any	DET
ejpam-3420	74	19	covering	covering	NOUN
ejpam-3420	74	20	based	base	VERB
ejpam-3420	74	21	rough	rough	ADJ
ejpam-3420	74	22	topology	topology	NOUN
ejpam-3420	74	23	.	.	PUNCT
ejpam-3420	75	1	then	then	ADV
ejpam-3420	75	2	we	we	PRON
ejpam-3420	75	3	can	can	AUX
ejpam-3420	75	4	apply	apply	VERB
ejpam-3420	75	5	a	a	DET
ejpam-3420	75	6	neighborhood	neighborhood	NOUN
ejpam-3420	75	7	for	for	ADP
ejpam-3420	75	8	strong	strong	ADJ
ejpam-3420	75	9	(	(	PUNCT
ejpam-3420	75	10	resp	resp	NOUN
ejpam-3420	75	11	.	.	PUNCT
ejpam-3420	75	12	weak	weak	ADJ
ejpam-3420	75	13	)	)	PUNCT
ejpam-3420	75	14	element	element	NOUN
ejpam-3420	75	15	and	and	CCONJ
ejpam-3420	75	16	strong	strong	ADJ
ejpam-3420	75	17	(	(	PUNCT
ejpam-3420	75	18	resp	resp	NOUN
ejpam-3420	75	19	.	.	PUNCT
ejpam-3420	76	1	weak	weak	ADJ
ejpam-3420	76	2	)	)	PUNCT
ejpam-3420	76	3	interior	interior	ADJ
ejpam-3420	76	4	and	and	CCONJ
ejpam-3420	76	5	strong	strong	ADJ
ejpam-3420	76	6	(	(	PUNCT
ejpam-3420	76	7	resp	resp	NOUN
ejpam-3420	76	8	.	.	PUNCT
ejpam-3420	77	1	weak	weak	ADJ
ejpam-3420	77	2	)	)	PUNCT
ejpam-3420	77	3	exterior	exterior	ADJ
ejpam-3420	77	4	point	point	NOUN
ejpam-3420	77	5	in	in	ADP
ejpam-3420	77	6	similar	similar	ADJ
ejpam-3420	77	7	way	way	NOUN
ejpam-3420	77	8	.	.	PUNCT
ejpam-3420	78	1	however	however	ADV
ejpam-3420	78	2	,	,	PUNCT
ejpam-3420	78	3	bryniarski	bryniarski	VERB
ejpam-3420	79	1	[	[	X
ejpam-3420	79	2	1	1	X
ejpam-3420	79	3	]	]	PUNCT
ejpam-3420	79	4	defined	define	VERB
ejpam-3420	79	5	an	an	DET
ejpam-3420	79	6	exact	exact	ADJ
ejpam-3420	79	7	element	element	NOUN
ejpam-3420	79	8	of	of	ADP
ejpam-3420	79	9	a	a	DET
ejpam-3420	79	10	rough	rough	ADJ
ejpam-3420	79	11	set	set	NOUN
ejpam-3420	79	12	as	as	ADP
ejpam-3420	79	13	a	a	DET
ejpam-3420	79	14	nonempty	nonempty	NOUN
ejpam-3420	79	15	subset	subset	NOUN
ejpam-3420	79	16	of	of	ADP
ejpam-3420	79	17	the	the	DET
ejpam-3420	79	18	set	set	NOUN
ejpam-3420	79	19	of	of	ADP
ejpam-3420	79	20	representatives	representative	NOUN
ejpam-3420	79	21	of	of	ADP
ejpam-3420	79	22	some	some	DET
ejpam-3420	79	23	class	class	NOUN
ejpam-3420	79	24	k	k	PROPN
ejpam-3420	79	25	∈	∈	PROPN
ejpam-3420	79	26	c	c	PROPN
ejpam-3420	79	27	or	or	CCONJ
ejpam-3420	79	28	the	the	DET
ejpam-3420	79	29	whole	whole	ADJ
ejpam-3420	79	30	class	class	NOUN
ejpam-3420	79	31	k.	k.	PROPN
ejpam-3420	79	32	definition	definition	NOUN
ejpam-3420	79	33	3	3	NUM
ejpam-3420	79	34	.	.	PUNCT
ejpam-3420	80	1	[	[	X
ejpam-3420	80	2	1	1	X
ejpam-3420	80	3	]	]	PUNCT
ejpam-3420	80	4	let	let	VERB
ejpam-3420	80	5	<	<	X
ejpam-3420	80	6	u	u	PROPN
ejpam-3420	80	7	,	,	PUNCT
ejpam-3420	80	8	c	c	X
ejpam-3420	80	9	>	>	X
ejpam-3420	80	10	be	be	AUX
ejpam-3420	80	11	an	an	DET
ejpam-3420	80	12	approximation	approximation	NOUN
ejpam-3420	80	13	space	space	NOUN
ejpam-3420	80	14	.	.	PUNCT
ejpam-3420	81	1	an	an	DET
ejpam-3420	81	2	element	element	NOUN
ejpam-3420	81	3	of	of	ADP
ejpam-3420	81	4	a	a	DET
ejpam-3420	81	5	rough	rough	ADJ
ejpam-3420	81	6	set	set	NOUN
ejpam-3420	81	7	of	of	ADP
ejpam-3420	81	8	the	the	DET
ejpam-3420	81	9	first	first	ADJ
ejpam-3420	81	10	order	order	NOUN
ejpam-3420	81	11	is	be	AUX
ejpam-3420	81	12	defined	define	VERB
ejpam-3420	81	13	as	as	SCONJ
ejpam-3420	81	14	follows	follow	VERB
ejpam-3420	81	15	:	:	PUNCT
ejpam-3420	81	16	for	for	ADP
ejpam-3420	81	17	all	all	DET
ejpam-3420	81	18	x	x	NOUN
ejpam-3420	81	19	,	,	PUNCT
ejpam-3420	81	20	y	y	PROPN
ejpam-3420	81	21	∈	∈	PROPN
ejpam-3420	81	22	p	p	X
ejpam-3420	81	23	(	(	PUNCT
ejpam-3420	81	24	u	u	NOUN
ejpam-3420	81	25	)	)	PUNCT
ejpam-3420	81	26	,	,	PUNCT
ejpam-3420	81	27	y	y	PROPN
ejpam-3420	81	28	∈	∈	PROPN
ejpam-3420	81	29	xc	xc	PUNCT
ejpam-3420	81	30	⇐	⇐	PROPN
ejpam-3420	81	31	⇒	⇒	PROPN
ejpam-3420	81	32	y	y	PROPN
ejpam-3420	81	33	6=	6=	PROPN
ejpam-3420	81	34	∅	∅	NOUN
ejpam-3420	81	35	,	,	PUNCT
ejpam-3420	81	36	yc	yc	PROPN
ejpam-3420	81	37	⊂	⊂	PROPN
ejpam-3420	81	38	xc	xc	PROPN
ejpam-3420	81	39	and	and	CCONJ
ejpam-3420	81	40	there	there	PRON
ejpam-3420	81	41	exists	exist	VERB
ejpam-3420	81	42	k	k	PROPN
ejpam-3420	81	43	∈	∈	PROPN
ejpam-3420	81	44	c	c	PROPN
ejpam-3420	81	45	such	such	ADJ
ejpam-3420	81	46	that	that	PRON
ejpam-3420	81	47	y	y	PROPN
ejpam-3420	81	48	=	=	PRON
ejpam-3420	81	49	{	{	PUNCT
ejpam-3420	81	50	k	k	NOUN
ejpam-3420	81	51	}	}	PUNCT
ejpam-3420	81	52	.	.	PUNCT
ejpam-3420	82	1	we	we	PRON
ejpam-3420	82	2	are	be	AUX
ejpam-3420	82	3	ready	ready	ADJ
ejpam-3420	82	4	to	to	PART
ejpam-3420	82	5	introduce	introduce	VERB
ejpam-3420	82	6	a	a	DET
ejpam-3420	82	7	rough	rough	ADJ
ejpam-3420	82	8	exact	exact	ADJ
ejpam-3420	82	9	neighborhood	neighborhood	NOUN
ejpam-3420	82	10	of	of	ADP
ejpam-3420	82	11	an	an	DET
ejpam-3420	82	12	exact	exact	ADJ
ejpam-3420	82	13	element	element	NOUN
ejpam-3420	82	14	.	.	PUNCT
ejpam-3420	83	1	let	let	VERB
ejpam-3420	83	2	y	y	PRON
ejpam-3420	83	3	be	be	AUX
ejpam-3420	83	4	an	an	DET
ejpam-3420	83	5	exact	exact	ADJ
ejpam-3420	83	6	element	element	NOUN
ejpam-3420	83	7	in	in	ADP
ejpam-3420	83	8	the	the	DET
ejpam-3420	83	9	covering	covering	NOUN
ejpam-3420	83	10	based	base	VERB
ejpam-3420	83	11	rough	rough	ADJ
ejpam-3420	83	12	space	space	NOUN
ejpam-3420	83	13	x	x	PUNCT
ejpam-3420	83	14	and	and	CCONJ
ejpam-3420	83	15	a	a	DET
ejpam-3420	83	16	be	be	AUX
ejpam-3420	83	17	a	a	DET
ejpam-3420	83	18	covering	covering	NOUN
ejpam-3420	83	19	based	base	VERB
ejpam-3420	83	20	rough	rough	ADJ
ejpam-3420	83	21	open	open	ADJ
ejpam-3420	83	22	set	set	NOUN
ejpam-3420	83	23	in	in	ADP
ejpam-3420	83	24	x.	x.	NOUN
ejpam-3420	83	25	then	then	ADV
ejpam-3420	83	26	a	a	PRON
ejpam-3420	83	27	is	be	AUX
ejpam-3420	83	28	a	a	DET
ejpam-3420	83	29	rough	rough	ADJ
ejpam-3420	83	30	exact	exact	ADJ
ejpam-3420	83	31	neighborhood	neighborhood	NOUN
ejpam-3420	83	32	of	of	ADP
ejpam-3420	83	33	y	y	PRON
ejpam-3420	83	34	if	if	SCONJ
ejpam-3420	83	35	yc	yc	PROPN
ejpam-3420	83	36	⊆	⊆	SYM
ejpam-3420	83	37	a	a	PRON
ejpam-3420	83	38	,	,	PUNCT
ejpam-3420	83	39	that	that	ADV
ejpam-3420	83	40	is	is	ADV
ejpam-3420	83	41	,	,	PUNCT
ejpam-3420	83	42	the	the	DET
ejpam-3420	83	43	exact	exact	ADJ
ejpam-3420	83	44	element	element	NOUN
ejpam-3420	83	45	y	y	PROPN
ejpam-3420	83	46	is	be	AUX
ejpam-3420	83	47	included	include	VERB
ejpam-3420	83	48	in	in	ADP
ejpam-3420	83	49	a	a	DET
ejpam-3420	83	50	roughly	roughly	NOUN
ejpam-3420	83	51	.	.	PUNCT
ejpam-3420	84	1	definition	definition	NOUN
ejpam-3420	84	2	4	4	NUM
ejpam-3420	84	3	.	.	PUNCT
ejpam-3420	85	1	let	let	AUX
ejpam-3420	85	2	(	(	PUNCT
ejpam-3420	85	3	x	x	NOUN
ejpam-3420	85	4	,	,	PUNCT
ejpam-3420	85	5	τ	τ	X
ejpam-3420	85	6	)	)	PUNCT
ejpam-3420	85	7	be	be	VERB
ejpam-3420	85	8	a	a	DET
ejpam-3420	85	9	covering	covering	NOUN
ejpam-3420	85	10	based	base	VERB
ejpam-3420	85	11	rough	rough	ADJ
ejpam-3420	85	12	space	space	NOUN
ejpam-3420	85	13	,	,	PUNCT
ejpam-3420	85	14	where	where	SCONJ
ejpam-3420	85	15	x	x	SYM
ejpam-3420	85	16	⊆	⊆	NUM
ejpam-3420	85	17	u	u	NOUN
ejpam-3420	85	18	.	.	PUNCT
ejpam-3420	86	1	for	for	ADP
ejpam-3420	86	2	a	a	DET
ejpam-3420	86	3	⊆	⊆	NUM
ejpam-3420	86	4	x	x	SYM
ejpam-3420	86	5	,	,	PUNCT
ejpam-3420	86	6	(	(	PUNCT
ejpam-3420	86	7	i	i	NOUN
ejpam-3420	86	8	)	)	PUNCT
ejpam-3420	86	9	the	the	DET
ejpam-3420	86	10	covering	covering	NOUN
ejpam-3420	86	11	based	base	VERB
ejpam-3420	86	12	rough	rough	ADJ
ejpam-3420	86	13	interior	interior	NOUN
ejpam-3420	86	14	of	of	ADP
ejpam-3420	86	15	the	the	DET
ejpam-3420	86	16	set	set	NOUN
ejpam-3420	86	17	a	a	PRON
ejpam-3420	86	18	is	be	AUX
ejpam-3420	86	19	defined	define	VERB
ejpam-3420	86	20	as	as	ADP
ejpam-3420	86	21	the	the	DET
ejpam-3420	86	22	union	union	NOUN
ejpam-3420	86	23	of	of	ADP
ejpam-3420	86	24	all	all	DET
ejpam-3420	86	25	covering	cover	VERB
ejpam-3420	86	26	based	base	VERB
ejpam-3420	86	27	rough	rough	ADJ
ejpam-3420	86	28	open	open	ADJ
ejpam-3420	86	29	subsets	subset	NOUN
ejpam-3420	86	30	contained	contain	VERB
ejpam-3420	86	31	in	in	ADP
ejpam-3420	86	32	a.	a.	NOUN
ejpam-3420	86	33	it	it	PRON
ejpam-3420	86	34	is	be	AUX
ejpam-3420	86	35	denoted	denote	VERB
ejpam-3420	86	36	by	by	ADP
ejpam-3420	86	37	cint(a	cint(a	PROPN
ejpam-3420	86	38	)	)	PUNCT
ejpam-3420	86	39	or	or	CCONJ
ejpam-3420	86	40	a	a	DET
ejpam-3420	86	41	◦	◦	NOUN
ejpam-3420	86	42	;	;	PUNCT
ejpam-3420	86	43	(	(	PUNCT
ejpam-3420	86	44	ii	ii	X
ejpam-3420	86	45	)	)	PUNCT
ejpam-3420	86	46	the	the	DET
ejpam-3420	86	47	covering	covering	NOUN
ejpam-3420	86	48	based	base	VERB
ejpam-3420	86	49	rough	rough	ADJ
ejpam-3420	86	50	closure	closure	NOUN
ejpam-3420	86	51	of	of	ADP
ejpam-3420	86	52	the	the	DET
ejpam-3420	86	53	set	set	NOUN
ejpam-3420	86	54	a	a	PRON
ejpam-3420	86	55	is	be	AUX
ejpam-3420	86	56	defined	define	VERB
ejpam-3420	86	57	as	as	ADP
ejpam-3420	86	58	the	the	DET
ejpam-3420	86	59	intersection	intersection	NOUN
ejpam-3420	86	60	of	of	ADP
ejpam-3420	86	61	all	all	PRON
ejpam-3420	86	62	covering	cover	VERB
ejpam-3420	86	63	based	base	VERB
ejpam-3420	86	64	rough	rough	ADJ
ejpam-3420	86	65	closed	closed	ADJ
ejpam-3420	86	66	subsets	subset	NOUN
ejpam-3420	86	67	containing	contain	VERB
ejpam-3420	86	68	a.	a.	NOUN
ejpam-3420	86	69	it	it	PRON
ejpam-3420	86	70	is	be	AUX
ejpam-3420	86	71	denoted	denote	VERB
ejpam-3420	86	72	by	by	ADP
ejpam-3420	86	73	ccl(a	ccl(a	PROPN
ejpam-3420	86	74	)	)	PUNCT
ejpam-3420	86	75	;	;	PUNCT
ejpam-3420	86	76	(	(	PUNCT
ejpam-3420	86	77	iii	iii	X
ejpam-3420	86	78	)	)	PUNCT
ejpam-3420	86	79	the	the	DET
ejpam-3420	86	80	exact	exact	ADJ
ejpam-3420	86	81	interior	interior	ADJ
ejpam-3420	86	82	point	point	NOUN
ejpam-3420	86	83	of	of	ADP
ejpam-3420	86	84	a	a	DET
ejpam-3420	86	85	subset	subset	NOUN
ejpam-3420	86	86	a	a	PRON
ejpam-3420	86	87	is	be	AUX
ejpam-3420	86	88	an	an	DET
ejpam-3420	86	89	element	element	NOUN
ejpam-3420	86	90	y	y	PROPN
ejpam-3420	86	91	∈	∈	PROPN
ejpam-3420	86	92	xc	xc	X
ejpam-3420	87	1	having	have	VERB
ejpam-3420	87	2	a	a	DET
ejpam-3420	87	3	rough	rough	ADJ
ejpam-3420	87	4	exact	exact	ADJ
ejpam-3420	87	5	neighborhood	neighborhood	NOUN
ejpam-3420	87	6	v	v	ADP
ejpam-3420	87	7	such	such	ADJ
ejpam-3420	87	8	that	that	SCONJ
ejpam-3420	87	9	y	y	PROPN
ejpam-3420	87	10	∈	∈	PROPN
ejpam-3420	87	11	v	v	ADP
ejpam-3420	87	12	⊆	⊆	NUM
ejpam-3420	87	13	a	a	PRON
ejpam-3420	87	14	;	;	PUNCT
ejpam-3420	87	15	(	(	PUNCT
ejpam-3420	87	16	iv	iv	X
ejpam-3420	87	17	)	)	PUNCT
ejpam-3420	87	18	the	the	DET
ejpam-3420	87	19	exact	exact	ADJ
ejpam-3420	87	20	exterior	exterior	ADJ
ejpam-3420	87	21	point	point	NOUN
ejpam-3420	87	22	of	of	ADP
ejpam-3420	87	23	a	a	DET
ejpam-3420	87	24	subset	subset	NOUN
ejpam-3420	87	25	a	a	PRON
ejpam-3420	87	26	is	be	AUX
ejpam-3420	87	27	an	an	DET
ejpam-3420	87	28	element	element	NOUN
ejpam-3420	87	29	y	y	PROPN
ejpam-3420	87	30	∈	∈	PROPN
ejpam-3420	87	31	xc	xc	X
ejpam-3420	88	1	having	have	VERB
ejpam-3420	88	2	a	a	DET
ejpam-3420	88	3	rough	rough	ADJ
ejpam-3420	88	4	exact	exact	ADJ
ejpam-3420	88	5	neighborhood	neighborhood	NOUN
ejpam-3420	88	6	v	v	ADP
ejpam-3420	88	7	such	such	ADJ
ejpam-3420	88	8	that	that	SCONJ
ejpam-3420	88	9	y	y	PROPN
ejpam-3420	88	10	∈	∈	PROPN
ejpam-3420	88	11	v	v	ADP
ejpam-3420	88	12	⊆	⊆	NUM
ejpam-3420	88	13	ac	ac	NOUN
ejpam-3420	88	14	.	.	PUNCT
ejpam-3420	89	1	n.	n.	PROPN
ejpam-3420	89	2	alharbi	alharbi	PROPN
ejpam-3420	89	3	,	,	PUNCT
ejpam-3420	89	4	h.	h.	PROPN
ejpam-3420	89	5	aydi	aydi	PROPN
ejpam-3420	89	6	,	,	PUNCT
ejpam-3420	89	7	c.	c.	PROPN
ejpam-3420	89	8	özel	özel	PROPN
ejpam-3420	89	9	/	/	SYM
ejpam-3420	89	10	eur	eur	PROPN
ejpam-3420	89	11	.	.	PUNCT
ejpam-3420	90	1	j.	j.	PROPN
ejpam-3420	90	2	pure	pure	PROPN
ejpam-3420	90	3	appl	appl	PROPN
ejpam-3420	90	4	.	.	PROPN
ejpam-3420	90	5	math	math	PROPN
ejpam-3420	90	6	,	,	PUNCT
ejpam-3420	90	7	12	12	NUM
ejpam-3420	90	8	(	(	PUNCT
ejpam-3420	90	9	2	2	NUM
ejpam-3420	90	10	)	)	PUNCT
ejpam-3420	90	11	(	(	PUNCT
ejpam-3420	90	12	2019	2019	NUM
ejpam-3420	90	13	)	)	PUNCT
ejpam-3420	90	14	,	,	PUNCT
ejpam-3420	90	15	533	533	NUM
ejpam-3420	90	16	-	-	SYM
ejpam-3420	90	17	543	543	NUM
ejpam-3420	90	18	537	537	NUM
ejpam-3420	90	19	example	example	NOUN
ejpam-3420	90	20	1	1	NUM
ejpam-3420	90	21	.	.	PUNCT
ejpam-3420	91	1	let	let	VERB
ejpam-3420	91	2	u	u	PRON
ejpam-3420	91	3	=	=	X
ejpam-3420	91	4	{	{	PUNCT
ejpam-3420	91	5	a	a	PRON
ejpam-3420	91	6	,	,	PUNCT
ejpam-3420	91	7	b	b	NOUN
ejpam-3420	91	8	,	,	PUNCT
ejpam-3420	91	9	c	c	NOUN
ejpam-3420	91	10	,	,	PUNCT
ejpam-3420	91	11	d	d	NOUN
ejpam-3420	91	12	,	,	PUNCT
ejpam-3420	91	13	e	e	NOUN
ejpam-3420	91	14	}	}	PUNCT
ejpam-3420	91	15	and	and	CCONJ
ejpam-3420	91	16	c	c	X
ejpam-3420	91	17	=	=	PRON
ejpam-3420	91	18	{	{	PUNCT
ejpam-3420	91	19	{	{	PUNCT
ejpam-3420	91	20	a	a	PROPN
ejpam-3420	91	21	,	,	PUNCT
ejpam-3420	91	22	b	b	NOUN
ejpam-3420	91	23	,	,	PUNCT
ejpam-3420	91	24	e	e	NOUN
ejpam-3420	91	25	}	}	PUNCT
ejpam-3420	91	26	,	,	PUNCT
ejpam-3420	91	27	{	{	PUNCT
ejpam-3420	91	28	c	c	X
ejpam-3420	91	29	,	,	PUNCT
ejpam-3420	91	30	d	d	NOUN
ejpam-3420	91	31	,	,	PUNCT
ejpam-3420	91	32	e	e	NOUN
ejpam-3420	91	33	}	}	PUNCT
ejpam-3420	91	34	}	}	PUNCT
ejpam-3420	91	35	.	.	PUNCT
ejpam-3420	92	1	clearly	clearly	ADV
ejpam-3420	92	2	,	,	PUNCT
ejpam-3420	92	3	⋃	⋃	PROPN
ejpam-3420	92	4	c	c	NOUN
ejpam-3420	92	5	=	=	SYM
ejpam-3420	92	6	u	u	PROPN
ejpam-3420	92	7	and	and	CCONJ
ejpam-3420	92	8	the	the	DET
ejpam-3420	92	9	covering	covering	NOUN
ejpam-3420	92	10	c	c	NOUN
ejpam-3420	92	11	is	be	AUX
ejpam-3420	92	12	well	well	ADV
ejpam-3420	92	13	representable	representable	ADJ
ejpam-3420	92	14	,	,	PUNCT
ejpam-3420	92	15	so	so	SCONJ
ejpam-3420	92	16	it	it	PRON
ejpam-3420	92	17	satisfies	satisfy	VERB
ejpam-3420	92	18	an	an	DET
ejpam-3420	92	19	approximation	approximation	NOUN
ejpam-3420	92	20	condition	condition	NOUN
ejpam-3420	92	21	.	.	PUNCT
ejpam-3420	93	1	assume	assume	VERB
ejpam-3420	93	2	that	that	SCONJ
ejpam-3420	93	3	x	x	PRON
ejpam-3420	93	4	=	=	PRON
ejpam-3420	93	5	{	{	PUNCT
ejpam-3420	93	6	a	a	X
ejpam-3420	93	7	,	,	PUNCT
ejpam-3420	93	8	e	e	NOUN
ejpam-3420	93	9	,	,	PUNCT
ejpam-3420	93	10	d	d	PROPN
ejpam-3420	93	11	}	}	PUNCT
ejpam-3420	93	12	,	,	PUNCT
ejpam-3420	93	13	then	then	ADV
ejpam-3420	93	14	x	x	X
ejpam-3420	93	15	=	=	NOUN
ejpam-3420	93	16	∅	∅	NOUN
ejpam-3420	93	17	,	,	PUNCT
ejpam-3420	93	18	and	and	CCONJ
ejpam-3420	93	19	x	x	X
ejpam-3420	93	20	=	=	PRON
ejpam-3420	93	21	{	{	PUNCT
ejpam-3420	93	22	{	{	PUNCT
ejpam-3420	93	23	a	a	PROPN
ejpam-3420	93	24	,	,	PUNCT
ejpam-3420	93	25	b	b	NOUN
ejpam-3420	93	26	,	,	PUNCT
ejpam-3420	93	27	e}{c	e}{c	NUM
ejpam-3420	93	28	,	,	PUNCT
ejpam-3420	93	29	d	d	NOUN
ejpam-3420	93	30	,	,	PUNCT
ejpam-3420	93	31	e	e	NOUN
ejpam-3420	93	32	}	}	PUNCT
ejpam-3420	93	33	}	}	PUNCT
ejpam-3420	93	34	.	.	PUNCT
ejpam-3420	94	1	define	define	VERB
ejpam-3420	94	2	τc	τc	ADV
ejpam-3420	94	3	=	=	PUNCT
ejpam-3420	94	4	{	{	PUNCT
ejpam-3420	94	5	x	x	SYM
ejpam-3420	94	6	=	=	SYM
ejpam-3420	94	7	(	(	PUNCT
ejpam-3420	94	8	x	x	X
ejpam-3420	94	9	,	,	PUNCT
ejpam-3420	94	10	x),=	x),=	PROPN
ejpam-3420	94	11	(	(	PUNCT
ejpam-3420	94	12	∅	∅	NOUN
ejpam-3420	94	13	,	,	PUNCT
ejpam-3420	94	14	∅	∅	NOUN
ejpam-3420	94	15	)	)	PUNCT
ejpam-3420	94	16	,	,	PUNCT
ejpam-3420	94	17	{	{	PUNCT
ejpam-3420	94	18	a	a	X
ejpam-3420	94	19	}	}	PUNCT
ejpam-3420	94	20	,	,	PUNCT
ejpam-3420	94	21	{	{	PUNCT
ejpam-3420	94	22	d	d	NOUN
ejpam-3420	94	23	}	}	PUNCT
ejpam-3420	94	24	}	}	PUNCT
ejpam-3420	94	25	,	,	PUNCT
ejpam-3420	94	26	then	then	ADV
ejpam-3420	94	27	τc	τc	ADV
ejpam-3420	94	28	is	be	AUX
ejpam-3420	94	29	a	a	DET
ejpam-3420	94	30	covering	covering	NOUN
ejpam-3420	94	31	based	base	VERB
ejpam-3420	94	32	rough	rough	ADJ
ejpam-3420	94	33	space	space	NOUN
ejpam-3420	94	34	,	,	PUNCT
ejpam-3420	94	35	where	where	SCONJ
ejpam-3420	94	36	the	the	DET
ejpam-3420	94	37	approximations	approximation	NOUN
ejpam-3420	94	38	of	of	ADP
ejpam-3420	94	39	each	each	DET
ejpam-3420	94	40	rough	rough	ADJ
ejpam-3420	94	41	open	open	ADJ
ejpam-3420	94	42	set	set	NOUN
ejpam-3420	94	43	are	be	AUX
ejpam-3420	94	44	as	as	SCONJ
ejpam-3420	94	45	follows	follow	VERB
ejpam-3420	94	46	:	:	PUNCT
ejpam-3420	94	47	{	{	PUNCT
ejpam-3420	94	48	a	a	PRON
ejpam-3420	94	49	}	}	PUNCT
ejpam-3420	94	50	=	=	SYM
ejpam-3420	94	51	∅	∅	NOUN
ejpam-3420	94	52	,	,	PUNCT
ejpam-3420	94	53	{	{	PUNCT
ejpam-3420	94	54	a	a	NOUN
ejpam-3420	94	55	}	}	PUNCT
ejpam-3420	94	56	=	=	SYM
ejpam-3420	94	57	{	{	PUNCT
ejpam-3420	94	58	{	{	PUNCT
ejpam-3420	94	59	a	a	PROPN
ejpam-3420	94	60	,	,	PUNCT
ejpam-3420	94	61	b	b	NOUN
ejpam-3420	94	62	,	,	PUNCT
ejpam-3420	94	63	e	e	NOUN
ejpam-3420	94	64	}	}	PUNCT
ejpam-3420	94	65	}	}	PUNCT
ejpam-3420	94	66	,	,	PUNCT
ejpam-3420	94	67	and	and	CCONJ
ejpam-3420	94	68	{	{	PUNCT
ejpam-3420	94	69	d	d	NOUN
ejpam-3420	94	70	}	}	PUNCT
ejpam-3420	94	71	=	=	SYM
ejpam-3420	94	72	∅	∅	NOUN
ejpam-3420	94	73	,	,	PUNCT
ejpam-3420	94	74	{	{	PUNCT
ejpam-3420	94	75	d	d	NOUN
ejpam-3420	94	76	}	}	PUNCT
ejpam-3420	94	77	=	=	SYM
ejpam-3420	94	78	{	{	PUNCT
ejpam-3420	94	79	{	{	PUNCT
ejpam-3420	94	80	c	c	NOUN
ejpam-3420	94	81	,	,	PUNCT
ejpam-3420	94	82	d	d	NOUN
ejpam-3420	94	83	,	,	PUNCT
ejpam-3420	94	84	e	e	NOUN
ejpam-3420	94	85	}	}	PUNCT
ejpam-3420	94	86	}	}	PUNCT
ejpam-3420	94	87	.	.	PUNCT
ejpam-3420	95	1	exact	exact	ADJ
ejpam-3420	95	2	elements	element	NOUN
ejpam-3420	95	3	of	of	ADP
ejpam-3420	95	4	this	this	DET
ejpam-3420	95	5	covering	covering	NOUN
ejpam-3420	95	6	based	base	VERB
ejpam-3420	95	7	rough	rough	ADJ
ejpam-3420	95	8	topology	topology	NOUN
ejpam-3420	95	9	are	be	AUX
ejpam-3420	95	10	{	{	PUNCT
ejpam-3420	95	11	a	a	X
ejpam-3420	95	12	}	}	PUNCT
ejpam-3420	95	13	,	,	PUNCT
ejpam-3420	95	14	{	{	PUNCT
ejpam-3420	95	15	d	d	NOUN
ejpam-3420	95	16	}	}	PUNCT
ejpam-3420	95	17	,	,	PUNCT
ejpam-3420	95	18	{	{	PUNCT
ejpam-3420	95	19	e	e	NOUN
ejpam-3420	95	20	}	}	PUNCT
ejpam-3420	95	21	,	,	PUNCT
ejpam-3420	95	22	{	{	PUNCT
ejpam-3420	95	23	a	a	X
ejpam-3420	95	24	,	,	PUNCT
ejpam-3420	95	25	e	e	NOUN
ejpam-3420	95	26	}	}	PUNCT
ejpam-3420	95	27	,	,	PUNCT
ejpam-3420	95	28	{	{	PUNCT
ejpam-3420	95	29	a	a	PRON
ejpam-3420	95	30	,	,	PUNCT
ejpam-3420	95	31	d	d	NOUN
ejpam-3420	95	32	}	}	PUNCT
ejpam-3420	95	33	,	,	PUNCT
ejpam-3420	95	34	{	{	PUNCT
ejpam-3420	95	35	e	e	NOUN
ejpam-3420	95	36	,	,	PUNCT
ejpam-3420	95	37	d	d	NOUN
ejpam-3420	95	38	}	}	PUNCT
ejpam-3420	95	39	,	,	PUNCT
ejpam-3420	95	40	{	{	PUNCT
ejpam-3420	95	41	a	a	DET
ejpam-3420	95	42	,	,	PUNCT
ejpam-3420	95	43	e	e	NOUN
ejpam-3420	95	44	,	,	PUNCT
ejpam-3420	95	45	d	d	NOUN
ejpam-3420	95	46	}	}	PUNCT
ejpam-3420	95	47	,	,	PUNCT
ejpam-3420	95	48	which	which	PRON
ejpam-3420	95	49	satisfy	satisfy	VERB
ejpam-3420	95	50	the	the	DET
ejpam-3420	95	51	condition	condition	NOUN
ejpam-3420	95	52	in	in	ADP
ejpam-3420	95	53	definition	definition	NOUN
ejpam-3420	95	54	3	3	NUM
ejpam-3420	95	55	.	.	PUNCT
ejpam-3420	95	56	proposition	proposition	NOUN
ejpam-3420	95	57	2	2	NUM
ejpam-3420	95	58	.	.	PUNCT
ejpam-3420	96	1	let	let	VERB
ejpam-3420	96	2	(	(	PUNCT
ejpam-3420	96	3	u	u	NOUN
ejpam-3420	96	4	,	,	PUNCT
ejpam-3420	96	5	c	c	NOUN
ejpam-3420	96	6	)	)	PUNCT
ejpam-3420	96	7	be	be	AUX
ejpam-3420	96	8	a	a	DET
ejpam-3420	96	9	covering	covering	NOUN
ejpam-3420	96	10	approximation	approximation	NOUN
ejpam-3420	96	11	space	space	NOUN
ejpam-3420	96	12	and	and	CCONJ
ejpam-3420	96	13	x	x	SYM
ejpam-3420	96	14	⊆	⊆	NUM
ejpam-3420	96	15	u	u	NOUN
ejpam-3420	96	16	be	be	VERB
ejpam-3420	96	17	such	such	ADJ
ejpam-3420	96	18	that	that	SCONJ
ejpam-3420	96	19	(	(	PUNCT
ejpam-3420	96	20	x	x	X
ejpam-3420	96	21	,	,	PUNCT
ejpam-3420	96	22	τ	τ	X
ejpam-3420	96	23	)	)	PUNCT
ejpam-3420	96	24	is	be	AUX
ejpam-3420	96	25	a	a	DET
ejpam-3420	96	26	covering	covering	NOUN
ejpam-3420	96	27	based	base	VERB
ejpam-3420	96	28	rough	rough	ADJ
ejpam-3420	96	29	topology	topology	NOUN
ejpam-3420	96	30	.	.	PUNCT
ejpam-3420	97	1	suppose	suppose	VERB
ejpam-3420	97	2	a	a	PRON
ejpam-3420	97	3	and	and	CCONJ
ejpam-3420	97	4	b	b	NOUN
ejpam-3420	97	5	are	be	AUX
ejpam-3420	97	6	subsets	subset	NOUN
ejpam-3420	97	7	of	of	ADP
ejpam-3420	97	8	x.	x.	PROPN
ejpam-3420	97	9	(	(	PUNCT
ejpam-3420	97	10	i	i	NOUN
ejpam-3420	97	11	)	)	PUNCT
ejpam-3420	98	1	if	if	SCONJ
ejpam-3420	98	2	a	a	DET
ejpam-3420	98	3	⊆	⊆	NUM
ejpam-3420	98	4	b	b	NOUN
ejpam-3420	98	5	,	,	PUNCT
ejpam-3420	98	6	then	then	ADV
ejpam-3420	98	7	a	a	DET
ejpam-3420	98	8	◦	◦	NOUN
ejpam-3420	98	9	⊆	⊆	NUM
ejpam-3420	98	10	b	b	NOUN
ejpam-3420	98	11	◦	◦	NOUN
ejpam-3420	98	12	;	;	PUNCT
ejpam-3420	98	13	(	(	PUNCT
ejpam-3420	98	14	ii	ii	NOUN
ejpam-3420	98	15	)	)	PUNCT
ejpam-3420	98	16	if	if	SCONJ
ejpam-3420	98	17	a	a	DET
ejpam-3420	98	18	⊆	⊆	NUM
ejpam-3420	98	19	b	b	NOUN
ejpam-3420	98	20	,	,	PUNCT
ejpam-3420	98	21	then	then	ADV
ejpam-3420	98	22	ccl(a	ccl(a	X
ejpam-3420	98	23	)	)	PUNCT
ejpam-3420	98	24	⊆	⊆	NUM
ejpam-3420	98	25	ccl(b	ccl(b	PROPN
ejpam-3420	98	26	)	)	PUNCT
ejpam-3420	98	27	;	;	PUNCT
ejpam-3420	98	28	(	(	PUNCT
ejpam-3420	98	29	iii	iii	X
ejpam-3420	98	30	)	)	PUNCT
ejpam-3420	98	31	(	(	PUNCT
ejpam-3420	98	32	a	a	DET
ejpam-3420	98	33	∧b	∧b	NOUN
ejpam-3420	98	34	)	)	PUNCT
ejpam-3420	98	35	◦	◦	NOUN
ejpam-3420	98	36	=	=	SYM
ejpam-3420	98	37	a	a	DET
ejpam-3420	98	38	◦	◦	NOUN
ejpam-3420	98	39	∧b	∧b	NUM
ejpam-3420	98	40	◦	◦	NOUN
ejpam-3420	98	41	;	;	PUNCT
ejpam-3420	98	42	(	(	PUNCT
ejpam-3420	98	43	iv	iv	X
ejpam-3420	98	44	)	)	PUNCT
ejpam-3420	98	45	a	a	DET
ejpam-3420	98	46	◦	◦	NOUN
ejpam-3420	98	47	∨b	∨b	NOUN
ejpam-3420	98	48	◦	◦	NOUN
ejpam-3420	98	49	⊆	⊆	NUM
ejpam-3420	98	50	(	(	PUNCT
ejpam-3420	98	51	a	a	DET
ejpam-3420	98	52	∨b	∨b	NOUN
ejpam-3420	98	53	)	)	PUNCT
ejpam-3420	98	54	◦	◦	NOUN
ejpam-3420	98	55	;	;	PUNCT
ejpam-3420	98	56	(	(	PUNCT
ejpam-3420	98	57	v	v	NOUN
ejpam-3420	98	58	)	)	PUNCT
ejpam-3420	98	59	ccl(a	ccl(a	ADJ
ejpam-3420	98	60	∨b	∨b	NOUN
ejpam-3420	98	61	)	)	PUNCT
ejpam-3420	98	62	=	=	SYM
ejpam-3420	98	63	ccl(a	ccl(a	X
ejpam-3420	98	64	)	)	PUNCT
ejpam-3420	98	65	∨	∨	NUM
ejpam-3420	98	66	ccl(b	ccl(b	PROPN
ejpam-3420	98	67	)	)	PUNCT
ejpam-3420	98	68	;	;	PUNCT
ejpam-3420	98	69	(	(	PUNCT
ejpam-3420	98	70	vi	vi	NOUN
ejpam-3420	98	71	)	)	PUNCT
ejpam-3420	98	72	ccl(a	ccl(a	ADJ
ejpam-3420	98	73	)	)	PUNCT
ejpam-3420	98	74	∧	∧	NOUN
ejpam-3420	98	75	ccl(b	ccl(b	PROPN
ejpam-3420	98	76	)	)	PUNCT
ejpam-3420	98	77	⊆	⊆	NUM
ejpam-3420	98	78	ccl(a	ccl(a	PROPN
ejpam-3420	98	79	∧b	∧b	NUM
ejpam-3420	98	80	)	)	PUNCT
ejpam-3420	98	81	.	.	PUNCT
ejpam-3420	99	1	proof	proof	NOUN
ejpam-3420	99	2	.	.	PUNCT
ejpam-3420	100	1	(	(	PUNCT
ejpam-3420	100	2	i	i	NOUN
ejpam-3420	100	3	)	)	PUNCT
ejpam-3420	100	4	we	we	PRON
ejpam-3420	100	5	know	know	VERB
ejpam-3420	100	6	that	that	SCONJ
ejpam-3420	100	7	a	a	DET
ejpam-3420	100	8	◦	◦	NOUN
ejpam-3420	100	9	⊆	⊆	NUM
ejpam-3420	100	10	a	a	DET
ejpam-3420	100	11	and	and	CCONJ
ejpam-3420	100	12	b	b	NOUN
ejpam-3420	100	13	◦	◦	NOUN
ejpam-3420	100	14	⊆	⊆	NUM
ejpam-3420	100	15	b.	b.	NOUN
ejpam-3420	100	16	but	but	CCONJ
ejpam-3420	100	17	a	a	DET
ejpam-3420	100	18	⊆	⊆	NUM
ejpam-3420	100	19	b	b	NOUN
ejpam-3420	100	20	,	,	PUNCT
ejpam-3420	100	21	so	so	SCONJ
ejpam-3420	100	22	a	a	DET
ejpam-3420	100	23	◦	◦	NOUN
ejpam-3420	100	24	⊆	⊆	NUM
ejpam-3420	100	25	a	a	DET
ejpam-3420	100	26	⊆	⊆	NUM
ejpam-3420	100	27	b	b	NOUN
ejpam-3420	100	28	,	,	PUNCT
ejpam-3420	100	29	a	a	DET
ejpam-3420	100	30	◦	◦	NOUN
ejpam-3420	100	31	⊆	⊆	NUM
ejpam-3420	100	32	a	a	DET
ejpam-3420	100	33	⊆	⊆	NUM
ejpam-3420	100	34	b.	b.	NOUN
ejpam-3420	100	35	thus	thus	ADV
ejpam-3420	100	36	,	,	PUNCT
ejpam-3420	100	37	a	a	DET
ejpam-3420	100	38	◦	◦	NOUN
ejpam-3420	100	39	⊆	⊆	NUM
ejpam-3420	100	40	b.	b.	NOUN
ejpam-3420	100	41	by	by	ADP
ejpam-3420	100	42	definition	definition	NOUN
ejpam-3420	100	43	of	of	ADP
ejpam-3420	100	44	interior	interior	NOUN
ejpam-3420	100	45	of	of	ADP
ejpam-3420	100	46	b	b	NOUN
ejpam-3420	100	47	as	as	ADP
ejpam-3420	100	48	the	the	DET
ejpam-3420	100	49	biggest	big	ADJ
ejpam-3420	100	50	rough	rough	ADJ
ejpam-3420	100	51	open	open	ADJ
ejpam-3420	100	52	set	set	NOUN
ejpam-3420	100	53	contained	contain	VERB
ejpam-3420	100	54	in	in	ADP
ejpam-3420	100	55	b	b	PROPN
ejpam-3420	100	56	,	,	PUNCT
ejpam-3420	100	57	a	a	DET
ejpam-3420	100	58	◦	◦	NOUN
ejpam-3420	100	59	⊆	⊆	NUM
ejpam-3420	100	60	b	b	NOUN
ejpam-3420	100	61	◦	◦	NOUN
ejpam-3420	100	62	.	.	PUNCT
ejpam-3420	101	1	n.	n.	PROPN
ejpam-3420	101	2	alharbi	alharbi	PROPN
ejpam-3420	101	3	,	,	PUNCT
ejpam-3420	101	4	h.	h.	PROPN
ejpam-3420	101	5	aydi	aydi	PROPN
ejpam-3420	101	6	,	,	PUNCT
ejpam-3420	101	7	c.	c.	PROPN
ejpam-3420	101	8	özel	özel	PROPN
ejpam-3420	101	9	/	/	SYM
ejpam-3420	101	10	eur	eur	PROPN
ejpam-3420	101	11	.	.	PUNCT
ejpam-3420	102	1	j.	j.	PROPN
ejpam-3420	102	2	pure	pure	PROPN
ejpam-3420	102	3	appl	appl	PROPN
ejpam-3420	102	4	.	.	PROPN
ejpam-3420	102	5	math	math	PROPN
ejpam-3420	102	6	,	,	PUNCT
ejpam-3420	102	7	12	12	NUM
ejpam-3420	102	8	(	(	PUNCT
ejpam-3420	102	9	2	2	NUM
ejpam-3420	102	10	)	)	PUNCT
ejpam-3420	102	11	(	(	PUNCT
ejpam-3420	102	12	2019	2019	NUM
ejpam-3420	102	13	)	)	PUNCT
ejpam-3420	102	14	,	,	PUNCT
ejpam-3420	102	15	533	533	NUM
ejpam-3420	102	16	-	-	SYM
ejpam-3420	102	17	543	543	NUM
ejpam-3420	102	18	538	538	NUM
ejpam-3420	102	19	(	(	PUNCT
ejpam-3420	102	20	ii	ii	NOUN
ejpam-3420	102	21	)	)	PUNCT
ejpam-3420	103	1	we	we	PRON
ejpam-3420	103	2	have	have	VERB
ejpam-3420	103	3	a	a	DET
ejpam-3420	103	4	⊆	⊆	NUM
ejpam-3420	103	5	ccl(a	ccl(a	NOUN
ejpam-3420	103	6	)	)	PUNCT
ejpam-3420	103	7	and	and	CCONJ
ejpam-3420	103	8	b	b	NOUN
ejpam-3420	103	9	⊆	⊆	NUM
ejpam-3420	103	10	ccl(b	ccl(b	PROPN
ejpam-3420	103	11	)	)	PUNCT
ejpam-3420	103	12	.	.	PUNCT
ejpam-3420	104	1	hence	hence	ADV
ejpam-3420	104	2	a	a	DET
ejpam-3420	104	3	⊆	⊆	NUM
ejpam-3420	104	4	ccl(a	ccl(a	NOUN
ejpam-3420	104	5	)	)	PUNCT
ejpam-3420	104	6	,	,	PUNCT
ejpam-3420	104	7	a	a	DET
ejpam-3420	104	8	⊆	⊆	NUM
ejpam-3420	104	9	ccl(a	ccl(a	NOUN
ejpam-3420	104	10	)	)	PUNCT
ejpam-3420	104	11	,	,	PUNCT
ejpam-3420	104	12	b	b	PROPN
ejpam-3420	104	13	⊆	⊆	NUM
ejpam-3420	104	14	ccl(b	ccl(b	PROPN
ejpam-3420	104	15	)	)	PUNCT
ejpam-3420	104	16	,	,	PUNCT
ejpam-3420	104	17	b	b	X
ejpam-3420	104	18	⊆	⊆	NUM
ejpam-3420	104	19	ccl(b	ccl(b	PROPN
ejpam-3420	104	20	)	)	PUNCT
ejpam-3420	104	21	.	.	PUNCT
ejpam-3420	105	1	as	as	ADP
ejpam-3420	105	2	a	a	DET
ejpam-3420	105	3	⊆	⊆	NUM
ejpam-3420	105	4	b	b	NOUN
ejpam-3420	105	5	,	,	PUNCT
ejpam-3420	105	6	by	by	ADP
ejpam-3420	105	7	definition	definition	NOUN
ejpam-3420	105	8	of	of	ADP
ejpam-3420	105	9	closure	closure	NOUN
ejpam-3420	105	10	of	of	ADP
ejpam-3420	105	11	a	a	PRON
ejpam-3420	105	12	,	,	PUNCT
ejpam-3420	105	13	we	we	PRON
ejpam-3420	105	14	get	get	VERB
ejpam-3420	105	15	ccl(a	ccl(a	NOUN
ejpam-3420	105	16	)	)	PUNCT
ejpam-3420	105	17	⊆	⊆	NUM
ejpam-3420	105	18	ccl(b	ccl(b	PROPN
ejpam-3420	105	19	)	)	PUNCT
ejpam-3420	105	20	,	,	PUNCT
ejpam-3420	105	21	ccl(a	ccl(a	X
ejpam-3420	105	22	)	)	PUNCT
ejpam-3420	105	23	⊆	⊆	NUM
ejpam-3420	105	24	ccl(b	ccl(b	PROPN
ejpam-3420	105	25	)	)	PUNCT
ejpam-3420	105	26	.	.	PUNCT
ejpam-3420	106	1	hence	hence	ADV
ejpam-3420	106	2	ccl(a	ccl(a	NUM
ejpam-3420	106	3	)	)	PUNCT
ejpam-3420	106	4	⊆	⊆	NUM
ejpam-3420	106	5	ccl(b	ccl(b	PROPN
ejpam-3420	106	6	)	)	PUNCT
ejpam-3420	106	7	.	.	PUNCT
ejpam-3420	107	1	rest	rest	NOUN
ejpam-3420	107	2	items	item	NOUN
ejpam-3420	107	3	are	be	AUX
ejpam-3420	107	4	similar	similar	ADJ
ejpam-3420	107	5	to	to	ADP
ejpam-3420	107	6	the	the	DET
ejpam-3420	107	7	classical	classical	ADJ
ejpam-3420	107	8	case	case	NOUN
ejpam-3420	107	9	.	.	PUNCT
ejpam-3420	108	1	subspaces	subspace	VERB
ejpam-3420	108	2	the	the	DET
ejpam-3420	108	3	inclusion	inclusion	NOUN
ejpam-3420	108	4	operation	operation	NOUN
ejpam-3420	108	5	on	on	ADP
ejpam-3420	108	6	covering	cover	VERB
ejpam-3420	108	7	based	base	VERB
ejpam-3420	108	8	rough	rough	ADJ
ejpam-3420	108	9	sets	set	NOUN
ejpam-3420	108	10	x	x	PUNCT
ejpam-3420	108	11	and	and	CCONJ
ejpam-3420	108	12	y	y	PROPN
ejpam-3420	108	13	is	be	AUX
ejpam-3420	108	14	defined	define	VERB
ejpam-3420	108	15	by	by	ADP
ejpam-3420	108	16	y	y	PROPN
ejpam-3420	108	17	⊆c	⊆c	NOUN
ejpam-3420	109	1	x	x	SYM
ejpam-3420	109	2	⇐	⇐	ADJ
ejpam-3420	109	3	⇒	⇒	NOUN
ejpam-3420	109	4	y	y	PROPN
ejpam-3420	109	5	⊆	⊆	NUM
ejpam-3420	109	6	x	x	NOUN
ejpam-3420	109	7	,	,	PUNCT
ejpam-3420	109	8	and	and	CCONJ
ejpam-3420	109	9	y	y	PROPN
ejpam-3420	109	10	⊆	⊆	NUM
ejpam-3420	109	11	x.	x.	NOUN
ejpam-3420	109	12	let	let	VERB
ejpam-3420	109	13	(	(	PUNCT
ejpam-3420	109	14	u	u	NOUN
ejpam-3420	109	15	,	,	PUNCT
ejpam-3420	109	16	c	c	NOUN
ejpam-3420	109	17	)	)	PUNCT
ejpam-3420	109	18	be	be	AUX
ejpam-3420	109	19	a	a	DET
ejpam-3420	109	20	covering	covering	NOUN
ejpam-3420	109	21	approximation	approximation	NOUN
ejpam-3420	109	22	space	space	NOUN
ejpam-3420	109	23	.	.	PUNCT
ejpam-3420	110	1	suppose	suppose	VERB
ejpam-3420	110	2	that	that	SCONJ
ejpam-3420	110	3	y	y	PROPN
ejpam-3420	110	4	and	and	CCONJ
ejpam-3420	110	5	x	x	NOUN
ejpam-3420	110	6	are	be	AUX
ejpam-3420	110	7	elements	element	NOUN
ejpam-3420	110	8	of	of	ADP
ejpam-3420	110	9	the	the	DET
ejpam-3420	110	10	power	power	NOUN
ejpam-3420	110	11	set	set	NOUN
ejpam-3420	110	12	of	of	ADP
ejpam-3420	110	13	u	u	PRON
ejpam-3420	110	14	such	such	ADJ
ejpam-3420	110	15	that	that	SCONJ
ejpam-3420	110	16	y	y	PROPN
ejpam-3420	110	17	⊆c	⊆c	PROPN
ejpam-3420	110	18	x.	x.	PUNCT
ejpam-3420	111	1	our	our	PRON
ejpam-3420	111	2	aim	aim	NOUN
ejpam-3420	111	3	is	be	AUX
ejpam-3420	111	4	to	to	PART
ejpam-3420	111	5	define	define	VERB
ejpam-3420	111	6	topological	topological	ADJ
ejpam-3420	111	7	subspaces	subspace	NOUN
ejpam-3420	111	8	of	of	ADP
ejpam-3420	111	9	a	a	DET
ejpam-3420	111	10	covering	covering	NOUN
ejpam-3420	111	11	based	base	VERB
ejpam-3420	111	12	rough	rough	ADJ
ejpam-3420	111	13	topology	topology	NOUN
ejpam-3420	111	14	.	.	PUNCT
ejpam-3420	112	1	following	follow	VERB
ejpam-3420	112	2	[	[	X
ejpam-3420	112	3	1	1	NUM
ejpam-3420	112	4	]	]	PUNCT
ejpam-3420	112	5	,	,	PUNCT
ejpam-3420	112	6	the	the	DET
ejpam-3420	112	7	set	set	NOUN
ejpam-3420	112	8	of	of	ADP
ejpam-3420	112	9	all	all	DET
ejpam-3420	112	10	rough	rough	ADJ
ejpam-3420	112	11	sets	set	NOUN
ejpam-3420	112	12	of	of	ADP
ejpam-3420	112	13	the	the	DET
ejpam-3420	112	14	first	first	ADJ
ejpam-3420	112	15	order	order	NOUN
ejpam-3420	112	16	with	with	ADP
ejpam-3420	112	17	operations	operation	NOUN
ejpam-3420	112	18	∨	∨	NOUN
ejpam-3420	112	19	and	and	CCONJ
ejpam-3420	112	20	∧	∧	NOUN
ejpam-3420	112	21	is	be	AUX
ejpam-3420	112	22	a	a	DET
ejpam-3420	112	23	distributive	distributive	ADJ
ejpam-3420	112	24	lattice	lattice	NOUN
ejpam-3420	112	25	,	,	PUNCT
ejpam-3420	112	26	where	where	SCONJ
ejpam-3420	112	27	∨i(vi	∨i(vi	PROPN
ejpam-3420	112	28	∧	∧	PROPN
ejpam-3420	112	29	y	y	PROPN
ejpam-3420	112	30	)	)	PUNCT
ejpam-3420	113	1	=	=	SYM
ejpam-3420	113	2	(	(	PUNCT
ejpam-3420	113	3	∨ivi	∨ivi	NOUN
ejpam-3420	113	4	)	)	PUNCT
ejpam-3420	113	5	∧	∧	PROPN
ejpam-3420	113	6	y	y	PROPN
ejpam-3420	113	7	,	,	PUNCT
ejpam-3420	113	8	∧i(vi	∧i(vi	PROPN
ejpam-3420	113	9	∨	∨	PROPN
ejpam-3420	113	10	y	y	PROPN
ejpam-3420	113	11	)	)	PUNCT
ejpam-3420	113	12	=	=	SYM
ejpam-3420	113	13	(	(	PUNCT
ejpam-3420	113	14	∧ivi	∧ivi	NOUN
ejpam-3420	113	15	)	)	PUNCT
ejpam-3420	113	16	∨	∨	NOUN
ejpam-3420	113	17	y.	y.	NOUN
ejpam-3420	113	18	that	that	PRON
ejpam-3420	113	19	is	be	AUX
ejpam-3420	113	20	,	,	PUNCT
ejpam-3420	113	21	vi	vi	PROPN
ejpam-3420	113	22	and	and	CCONJ
ejpam-3420	113	23	y	y	PROPN
ejpam-3420	113	24	are	be	AUX
ejpam-3420	113	25	rough	rough	ADJ
ejpam-3420	113	26	sets	set	NOUN
ejpam-3420	113	27	.	.	PUNCT
ejpam-3420	114	1	then	then	ADV
ejpam-3420	114	2	∨i(vi	∨i(vi	PROPN
ejpam-3420	114	3	∧	∧	PROPN
ejpam-3420	114	4	y	y	PROPN
ejpam-3420	114	5	)	)	PUNCT
ejpam-3420	114	6	=	=	SYM
ejpam-3420	114	7	∨i(vi	∨i(vi	PROPN
ejpam-3420	114	8	)	)	PUNCT
ejpam-3420	114	9	∧	∧	PROPN
ejpam-3420	114	10	y	y	PROPN
ejpam-3420	114	11	,	,	PUNCT
ejpam-3420	114	12	∧i(vi	∧i(vi	PROPN
ejpam-3420	114	13	∨	∨	PROPN
ejpam-3420	114	14	y	y	PROPN
ejpam-3420	114	15	)	)	PUNCT
ejpam-3420	115	1	=	=	SYM
ejpam-3420	115	2	∧i(vi	∧i(vi	NOUN
ejpam-3420	115	3	)	)	PUNCT
ejpam-3420	115	4	∨	∨	NUM
ejpam-3420	115	5	y	y	PROPN
ejpam-3420	115	6	,	,	PUNCT
ejpam-3420	115	7	∨i(vi	∨i(vi	PROPN
ejpam-3420	115	8	∧	∧	PROPN
ejpam-3420	115	9	y	y	PROPN
ejpam-3420	115	10	)	)	PUNCT
ejpam-3420	115	11	=	=	SYM
ejpam-3420	115	12	∨i(vi	∨i(vi	PROPN
ejpam-3420	115	13	)	)	PUNCT
ejpam-3420	115	14	∧	∧	PROPN
ejpam-3420	115	15	y	y	PROPN
ejpam-3420	115	16	,	,	PUNCT
ejpam-3420	115	17	and	and	CCONJ
ejpam-3420	115	18	∧i(vi	∧i(vi	PROPN
ejpam-3420	115	19	∨	∨	PROPN
ejpam-3420	115	20	y	y	PROPN
ejpam-3420	115	21	)	)	PUNCT
ejpam-3420	116	1	=	=	SYM
ejpam-3420	116	2	∧i(vi	∧i(vi	NOUN
ejpam-3420	116	3	)	)	PUNCT
ejpam-3420	116	4	∨	∨	NUM
ejpam-3420	116	5	y	y	PROPN
ejpam-3420	116	6	.	.	PUNCT
ejpam-3420	117	1	therefore	therefore	ADV
ejpam-3420	117	2	,	,	PUNCT
ejpam-3420	117	3	we	we	PRON
ejpam-3420	117	4	can	can	AUX
ejpam-3420	117	5	define	define	VERB
ejpam-3420	117	6	covering	covering	NOUN
ejpam-3420	117	7	based	base	VERB
ejpam-3420	117	8	rough	rough	ADJ
ejpam-3420	117	9	subspaces	subspace	NOUN
ejpam-3420	117	10	.	.	PUNCT
ejpam-3420	118	1	definition	definition	NOUN
ejpam-3420	118	2	5	5	NUM
ejpam-3420	118	3	.	.	PUNCT
ejpam-3420	119	1	suppose	suppose	VERB
ejpam-3420	119	2	(	(	PUNCT
ejpam-3420	119	3	u	u	NOUN
ejpam-3420	119	4	,	,	PUNCT
ejpam-3420	119	5	c	c	NOUN
ejpam-3420	119	6	)	)	PUNCT
ejpam-3420	119	7	be	be	AUX
ejpam-3420	119	8	a	a	DET
ejpam-3420	119	9	covering	covering	NOUN
ejpam-3420	119	10	approximation	approximation	NOUN
ejpam-3420	119	11	space	space	NOUN
ejpam-3420	119	12	.	.	PUNCT
ejpam-3420	120	1	let	let	VERB
ejpam-3420	120	2	(	(	PUNCT
ejpam-3420	120	3	x	x	NOUN
ejpam-3420	120	4	,	,	PUNCT
ejpam-3420	120	5	τ	τ	X
ejpam-3420	120	6	)	)	PUNCT
ejpam-3420	120	7	be	be	VERB
ejpam-3420	120	8	a	a	DET
ejpam-3420	120	9	covering	covering	NOUN
ejpam-3420	120	10	based	base	VERB
ejpam-3420	120	11	rough	rough	ADJ
ejpam-3420	120	12	topology	topology	NOUN
ejpam-3420	120	13	with	with	ADP
ejpam-3420	120	14	the	the	DET
ejpam-3420	120	15	rough	rough	ADJ
ejpam-3420	120	16	set	set	NOUN
ejpam-3420	120	17	y	y	PROPN
ejpam-3420	120	18	⊆	⊆	NUM
ejpam-3420	120	19	x.	x.	NOUN
ejpam-3420	120	20	define	define	VERB
ejpam-3420	120	21	τ	τ	X
ejpam-3420	120	22	′	′	NUM
ejpam-3420	121	1	=	=	PUNCT
ejpam-3420	121	2	{	{	PUNCT
ejpam-3420	121	3	v	v	ADP
ejpam-3420	121	4	∧y	∧y	NOUN
ejpam-3420	121	5	:	:	PUNCT
ejpam-3420	121	6	v	v	NUM
ejpam-3420	121	7	∈	∈	PROPN
ejpam-3420	121	8	τ	τ	X
ejpam-3420	121	9	}	}	PUNCT
ejpam-3420	121	10	.	.	PUNCT
ejpam-3420	122	1	then	then	ADV
ejpam-3420	122	2	(	(	PUNCT
ejpam-3420	122	3	y	y	PROPN
ejpam-3420	122	4	,	,	PUNCT
ejpam-3420	122	5	τ	τ	PROPN
ejpam-3420	122	6	′	′	NUM
ejpam-3420	122	7	)	)	PUNCT
ejpam-3420	122	8	is	be	AUX
ejpam-3420	122	9	a	a	DET
ejpam-3420	122	10	covering	covering	NOUN
ejpam-3420	122	11	based	base	VERB
ejpam-3420	122	12	rough	rough	ADJ
ejpam-3420	122	13	subspace	subspace	NOUN
ejpam-3420	122	14	of	of	ADP
ejpam-3420	122	15	(	(	PUNCT
ejpam-3420	122	16	x	x	PROPN
ejpam-3420	122	17	,	,	PUNCT
ejpam-3420	122	18	τ	τ	PROPN
ejpam-3420	122	19	)	)	PUNCT
ejpam-3420	122	20	.	.	PUNCT
ejpam-3420	123	1	the	the	DET
ejpam-3420	123	2	intersection	intersection	NOUN
ejpam-3420	123	3	of	of	ADP
ejpam-3420	123	4	y	y	PROPN
ejpam-3420	123	5	with	with	ADP
ejpam-3420	123	6	all	all	PRON
ejpam-3420	123	7	covering	cover	VERB
ejpam-3420	123	8	rough	rough	ADJ
ejpam-3420	123	9	open	open	ADJ
ejpam-3420	123	10	sets	set	NOUN
ejpam-3420	123	11	in	in	ADP
ejpam-3420	123	12	τ	τ	PROPN
ejpam-3420	123	13	induces	induce	VERB
ejpam-3420	123	14	a	a	DET
ejpam-3420	123	15	topology	topology	NOUN
ejpam-3420	123	16	space	space	NOUN
ejpam-3420	123	17	on	on	ADP
ejpam-3420	123	18	y	y	PROPN
ejpam-3420	123	19	.	.	PUNCT
ejpam-3420	124	1	n.	n.	PROPN
ejpam-3420	124	2	alharbi	alharbi	PROPN
ejpam-3420	124	3	,	,	PUNCT
ejpam-3420	124	4	h.	h.	PROPN
ejpam-3420	124	5	aydi	aydi	PROPN
ejpam-3420	124	6	,	,	PUNCT
ejpam-3420	124	7	c.	c.	PROPN
ejpam-3420	124	8	özel	özel	PROPN
ejpam-3420	124	9	/	/	SYM
ejpam-3420	124	10	eur	eur	PROPN
ejpam-3420	124	11	.	.	PUNCT
ejpam-3420	125	1	j.	j.	PROPN
ejpam-3420	125	2	pure	pure	PROPN
ejpam-3420	125	3	appl	appl	PROPN
ejpam-3420	125	4	.	.	PROPN
ejpam-3420	125	5	math	math	PROPN
ejpam-3420	125	6	,	,	PUNCT
ejpam-3420	125	7	12	12	NUM
ejpam-3420	125	8	(	(	PUNCT
ejpam-3420	125	9	2	2	NUM
ejpam-3420	125	10	)	)	PUNCT
ejpam-3420	125	11	(	(	PUNCT
ejpam-3420	125	12	2019	2019	NUM
ejpam-3420	125	13	)	)	PUNCT
ejpam-3420	125	14	,	,	PUNCT
ejpam-3420	125	15	533	533	NUM
ejpam-3420	125	16	-	-	SYM
ejpam-3420	125	17	543	543	NUM
ejpam-3420	125	18	539	539	NUM
ejpam-3420	125	19	exact	exact	ADJ
ejpam-3420	125	20	base	base	NOUN
ejpam-3420	125	21	and	and	CCONJ
ejpam-3420	125	22	exact	exact	ADJ
ejpam-3420	125	23	subbase	subbase	NOUN
ejpam-3420	125	24	of	of	ADP
ejpam-3420	125	25	covering	covering	NOUN
ejpam-3420	125	26	based	base	VERB
ejpam-3420	125	27	rough	rough	ADJ
ejpam-3420	125	28	space	space	NOUN
ejpam-3420	125	29	a	a	DET
ejpam-3420	125	30	base	base	NOUN
ejpam-3420	125	31	for	for	ADP
ejpam-3420	125	32	a	a	DET
ejpam-3420	125	33	covering	covering	NOUN
ejpam-3420	125	34	based	base	VERB
ejpam-3420	125	35	rough	rough	ADJ
ejpam-3420	125	36	space	space	NOUN
ejpam-3420	125	37	(	(	PUNCT
ejpam-3420	125	38	x	x	X
ejpam-3420	125	39	,	,	PUNCT
ejpam-3420	125	40	τ	τ	X
ejpam-3420	125	41	)	)	PUNCT
ejpam-3420	125	42	is	be	AUX
ejpam-3420	125	43	a	a	DET
ejpam-3420	125	44	collection	collection	NOUN
ejpam-3420	125	45	of	of	ADP
ejpam-3420	125	46	covering	covering	NOUN
ejpam-3420	125	47	based	base	VERB
ejpam-3420	125	48	rough	rough	ADJ
ejpam-3420	125	49	open	open	ADJ
ejpam-3420	125	50	sets	set	NOUN
ejpam-3420	125	51	,	,	PUNCT
ejpam-3420	125	52	that	that	PRON
ejpam-3420	125	53	are	be	AUX
ejpam-3420	125	54	,	,	PUNCT
ejpam-3420	125	55	subsets	subset	NOUN
ejpam-3420	125	56	of	of	ADP
ejpam-3420	125	57	x	x	PUNCT
ejpam-3420	125	58	satisfying	satisfy	VERB
ejpam-3420	125	59	the	the	DET
ejpam-3420	125	60	following	follow	VERB
ejpam-3420	125	61	fact	fact	NOUN
ejpam-3420	125	62	:	:	PUNCT
ejpam-3420	125	63	for	for	ADP
ejpam-3420	125	64	every	every	DET
ejpam-3420	125	65	rough	rough	ADJ
ejpam-3420	125	66	open	open	ADJ
ejpam-3420	125	67	set	set	NOUN
ejpam-3420	125	68	yc	yc	NOUN
ejpam-3420	125	69	=	=	SYM
ejpam-3420	125	70	(	(	PUNCT
ejpam-3420	125	71	y	y	PROPN
ejpam-3420	125	72	,	,	PUNCT
ejpam-3420	125	73	y	y	PROPN
ejpam-3420	125	74	)	)	PUNCT
ejpam-3420	125	75	and	and	CCONJ
ejpam-3420	125	76	for	for	ADP
ejpam-3420	125	77	every	every	DET
ejpam-3420	125	78	exact	exact	ADJ
ejpam-3420	125	79	element	element	NOUN
ejpam-3420	125	80	y	y	PROPN
ejpam-3420	125	81	∈	∈	PROPN
ejpam-3420	126	1	yc	yc	INTJ
ejpam-3420	126	2	,	,	PUNCT
ejpam-3420	126	3	there	there	PRON
ejpam-3420	126	4	exists	exist	VERB
ejpam-3420	126	5	a	a	DET
ejpam-3420	126	6	basic	basic	ADJ
ejpam-3420	126	7	open	open	ADJ
ejpam-3420	126	8	set	set	NOUN
ejpam-3420	126	9	b	b	NOUN
ejpam-3420	126	10	=	=	SYM
ejpam-3420	126	11	(	(	PUNCT
ejpam-3420	126	12	b	b	PROPN
ejpam-3420	126	13	,	,	PUNCT
ejpam-3420	126	14	b	b	NOUN
ejpam-3420	126	15	)	)	PUNCT
ejpam-3420	126	16	of	of	ADP
ejpam-3420	126	17	that	that	DET
ejpam-3420	126	18	collection	collection	NOUN
ejpam-3420	126	19	such	such	ADJ
ejpam-3420	126	20	that	that	SCONJ
ejpam-3420	126	21	y	y	PROPN
ejpam-3420	126	22	∈	∈	PROPN
ejpam-3420	126	23	b	b	PROPN
ejpam-3420	126	24	⊆	⊆	NUM
ejpam-3420	126	25	yc	yc	PROPN
ejpam-3420	126	26	,	,	PUNCT
ejpam-3420	126	27	i.e.	i.e.	X
ejpam-3420	126	28	,	,	PUNCT
ejpam-3420	126	29	b	b	PROPN
ejpam-3420	126	30	⊆	⊆	NUM
ejpam-3420	126	31	y	y	PROPN
ejpam-3420	126	32	and	and	CCONJ
ejpam-3420	126	33	b	b	PROPN
ejpam-3420	126	34	⊆	⊆	NUM
ejpam-3420	126	35	y	y	NOUN
ejpam-3420	126	36	.	.	PUNCT
ejpam-3420	127	1	now	now	ADV
ejpam-3420	127	2	,	,	PUNCT
ejpam-3420	127	3	we	we	PRON
ejpam-3420	127	4	will	will	AUX
ejpam-3420	127	5	give	give	VERB
ejpam-3420	127	6	a	a	DET
ejpam-3420	127	7	result	result	NOUN
ejpam-3420	127	8	related	relate	VERB
ejpam-3420	127	9	to	to	ADP
ejpam-3420	127	10	a	a	DET
ejpam-3420	127	11	characterization	characterization	NOUN
ejpam-3420	127	12	of	of	ADP
ejpam-3420	127	13	the	the	DET
ejpam-3420	127	14	base	base	NOUN
ejpam-3420	127	15	in	in	ADP
ejpam-3420	127	16	the	the	DET
ejpam-3420	127	17	covering	covering	NOUN
ejpam-3420	127	18	based	base	VERB
ejpam-3420	127	19	rough	rough	ADJ
ejpam-3420	127	20	topology	topology	NOUN
ejpam-3420	127	21	on	on	ADP
ejpam-3420	127	22	rough	rough	ADJ
ejpam-3420	127	23	sets	set	NOUN
ejpam-3420	127	24	.	.	PUNCT
ejpam-3420	128	1	proposition	proposition	NOUN
ejpam-3420	128	2	3	3	NUM
ejpam-3420	128	3	.	.	PUNCT
ejpam-3420	129	1	let	let	VERB
ejpam-3420	129	2	b	b	NOUN
ejpam-3420	129	3	=	=	PRON
ejpam-3420	129	4	{	{	PUNCT
ejpam-3420	129	5	(	(	PUNCT
ejpam-3420	129	6	a	a	NOUN
ejpam-3420	129	7	,	,	PUNCT
ejpam-3420	129	8	b);a	b);a	PROPN
ejpam-3420	129	9	⊆	⊆	NUM
ejpam-3420	129	10	b	b	NOUN
ejpam-3420	129	11	,	,	PUNCT
ejpam-3420	129	12	b	b	PROPN
ejpam-3420	129	13	∈	∈	PROPN
ejpam-3420	129	14	c	c	NOUN
ejpam-3420	129	15	,	,	PUNCT
ejpam-3420	129	16	a	a	PRON
ejpam-3420	129	17	⊆	⊆	NUM
ejpam-3420	129	18	x	x	SYM
ejpam-3420	129	19	∧	∧	PROPN
ejpam-3420	129	20	b	b	PROPN
ejpam-3420	129	21	⊆	⊆	NUM
ejpam-3420	129	22	x	x	SYM
ejpam-3420	129	23	}	}	PUNCT
ejpam-3420	129	24	be	be	AUX
ejpam-3420	129	25	a	a	DET
ejpam-3420	129	26	collection	collection	NOUN
ejpam-3420	129	27	of	of	ADP
ejpam-3420	129	28	covering	covering	NOUN
ejpam-3420	129	29	based	base	VERB
ejpam-3420	129	30	rough	rough	ADJ
ejpam-3420	129	31	open	open	ADJ
ejpam-3420	129	32	subsets	subset	NOUN
ejpam-3420	129	33	of	of	ADP
ejpam-3420	129	34	x.	x.	NOUN
ejpam-3420	129	35	then	then	ADV
ejpam-3420	129	36	b	b	X
ejpam-3420	129	37	satisfies	satisfy	VERB
ejpam-3420	129	38	the	the	DET
ejpam-3420	129	39	following	follow	VERB
ejpam-3420	129	40	two	two	NUM
ejpam-3420	129	41	conditions	condition	NOUN
ejpam-3420	129	42	:	:	PUNCT
ejpam-3420	129	43	(	(	PUNCT
ejpam-3420	129	44	i	i	NOUN
ejpam-3420	129	45	)	)	PUNCT
ejpam-3420	129	46	for	for	ADP
ejpam-3420	129	47	every	every	DET
ejpam-3420	129	48	y	y	PROPN
ejpam-3420	129	49	∈	∈	PROPN
ejpam-3420	129	50	xc	xc	PROPN
ejpam-3420	129	51	,	,	PUNCT
ejpam-3420	129	52	there	there	PRON
ejpam-3420	129	53	exists	exist	VERB
ejpam-3420	129	54	(	(	PUNCT
ejpam-3420	129	55	a	a	PRON
ejpam-3420	129	56	,	,	PUNCT
ejpam-3420	129	57	b	b	NOUN
ejpam-3420	129	58	)	)	PUNCT
ejpam-3420	129	59	∈	∈	PROPN
ejpam-3420	129	60	b	b	NOUN
ejpam-3420	129	61	such	such	ADJ
ejpam-3420	129	62	that	that	SCONJ
ejpam-3420	129	63	y	y	PROPN
ejpam-3420	129	64	∈	∈	PROPN
ejpam-3420	129	65	(	(	PUNCT
ejpam-3420	129	66	a	a	DET
ejpam-3420	129	67	,	,	PUNCT
ejpam-3420	129	68	b	b	NOUN
ejpam-3420	129	69	)	)	PUNCT
ejpam-3420	129	70	;	;	PUNCT
ejpam-3420	129	71	(	(	PUNCT
ejpam-3420	129	72	ii	ii	NOUN
ejpam-3420	129	73	)	)	PUNCT
ejpam-3420	129	74	for	for	ADP
ejpam-3420	129	75	any	any	DET
ejpam-3420	129	76	(	(	PUNCT
ejpam-3420	129	77	a1	a1	NOUN
ejpam-3420	129	78	,	,	PUNCT
ejpam-3420	129	79	b1	b1	NOUN
ejpam-3420	129	80	)	)	PUNCT
ejpam-3420	129	81	,	,	PUNCT
ejpam-3420	129	82	(	(	PUNCT
ejpam-3420	129	83	a2	a2	PROPN
ejpam-3420	129	84	,	,	PUNCT
ejpam-3420	129	85	b2	b2	NOUN
ejpam-3420	129	86	)	)	PUNCT
ejpam-3420	129	87	∈	∈	PROPN
ejpam-3420	129	88	b	b	PROPN
ejpam-3420	129	89	and	and	CCONJ
ejpam-3420	129	90	every	every	DET
ejpam-3420	129	91	element	element	NOUN
ejpam-3420	129	92	y	y	PROPN
ejpam-3420	129	93	∈	∈	PROPN
ejpam-3420	129	94	(	(	PUNCT
ejpam-3420	129	95	(	(	PUNCT
ejpam-3420	129	96	a1	a1	NOUN
ejpam-3420	129	97	,	,	PUNCT
ejpam-3420	129	98	b1	b1	NOUN
ejpam-3420	129	99	)	)	PUNCT
ejpam-3420	129	100	∧	∧	PROPN
ejpam-3420	129	101	(	(	PUNCT
ejpam-3420	129	102	a2	a2	PROPN
ejpam-3420	129	103	,	,	PUNCT
ejpam-3420	129	104	b2	b2	NOUN
ejpam-3420	129	105	)	)	PUNCT
ejpam-3420	129	106	)	)	PUNCT
ejpam-3420	129	107	,	,	PUNCT
ejpam-3420	129	108	there	there	PRON
ejpam-3420	129	109	exists	exist	VERB
ejpam-3420	129	110	(	(	PUNCT
ejpam-3420	129	111	a	a	PRON
ejpam-3420	129	112	,	,	PUNCT
ejpam-3420	129	113	b	b	NOUN
ejpam-3420	129	114	)	)	PUNCT
ejpam-3420	129	115	∈	∈	PROPN
ejpam-3420	129	116	b	b	NOUN
ejpam-3420	129	117	such	such	ADJ
ejpam-3420	129	118	that	that	SCONJ
ejpam-3420	129	119	y	y	PROPN
ejpam-3420	129	120	∈	∈	PROPN
ejpam-3420	129	121	(	(	PUNCT
ejpam-3420	129	122	a	a	DET
ejpam-3420	129	123	,	,	PUNCT
ejpam-3420	129	124	b	b	NOUN
ejpam-3420	129	125	)	)	PUNCT
ejpam-3420	129	126	⊆	⊆	NUM
ejpam-3420	129	127	(	(	PUNCT
ejpam-3420	129	128	(	(	PUNCT
ejpam-3420	129	129	a1	a1	NOUN
ejpam-3420	129	130	,	,	PUNCT
ejpam-3420	129	131	b1	b1	NOUN
ejpam-3420	129	132	)	)	PUNCT
ejpam-3420	129	133	∧	∧	PROPN
ejpam-3420	129	134	(	(	PUNCT
ejpam-3420	129	135	a2	a2	PROPN
ejpam-3420	129	136	,	,	PUNCT
ejpam-3420	129	137	b2	b2	NOUN
ejpam-3420	129	138	)	)	PUNCT
ejpam-3420	129	139	)	)	PUNCT
ejpam-3420	129	140	.	.	PUNCT
ejpam-3420	130	1	therefore	therefore	ADV
ejpam-3420	130	2	b	b	PROPN
ejpam-3420	130	3	is	be	AUX
ejpam-3420	130	4	an	an	DET
ejpam-3420	130	5	exact	exact	ADJ
ejpam-3420	130	6	base	base	NOUN
ejpam-3420	130	7	for	for	ADP
ejpam-3420	130	8	the	the	DET
ejpam-3420	130	9	covering	covering	NOUN
ejpam-3420	130	10	based	base	VERB
ejpam-3420	130	11	rough	rough	ADJ
ejpam-3420	130	12	space	space	NOUN
ejpam-3420	130	13	(	(	PUNCT
ejpam-3420	130	14	x	x	X
ejpam-3420	130	15	,	,	PUNCT
ejpam-3420	130	16	τ	τ	X
ejpam-3420	130	17	)	)	PUNCT
ejpam-3420	130	18	.	.	PUNCT
ejpam-3420	131	1	proof	proof	NOUN
ejpam-3420	131	2	.	.	PUNCT
ejpam-3420	132	1	we	we	PRON
ejpam-3420	132	2	need	need	VERB
ejpam-3420	132	3	the	the	DET
ejpam-3420	132	4	two	two	NUM
ejpam-3420	132	5	following	follow	VERB
ejpam-3420	132	6	conditions	condition	NOUN
ejpam-3420	132	7	:	:	PUNCT
ejpam-3420	132	8	(	(	PUNCT
ejpam-3420	132	9	i	i	NOUN
ejpam-3420	132	10	)	)	PUNCT
ejpam-3420	132	11	let	let	VERB
ejpam-3420	132	12	y	y	PROPN
ejpam-3420	132	13	∈	∈	PROPN
ejpam-3420	132	14	xc	xc	PUNCT
ejpam-3420	133	1	=	=	PRON
ejpam-3420	133	2	⇒	⇒	PROPN
ejpam-3420	133	3	y	y	PROPN
ejpam-3420	133	4	6=	6=	PROPN
ejpam-3420	133	5	∅	∅	NOUN
ejpam-3420	133	6	∧	∧	PROPN
ejpam-3420	133	7	yc	yc	PROPN
ejpam-3420	133	8	⊆	⊆	NUM
ejpam-3420	133	9	xc	xc	PROPN
ejpam-3420	133	10	∧	∧	PROPN
ejpam-3420	133	11	∃k	∃k	PROPN
ejpam-3420	133	12	∈	∈	PROPN
ejpam-3420	133	13	c	c	NOUN
ejpam-3420	133	14	such	such	ADJ
ejpam-3420	133	15	that	that	DET
ejpam-3420	133	16	y	y	PROPN
ejpam-3420	133	17	=	=	PUNCT
ejpam-3420	133	18	k.	k.	PROPN
ejpam-3420	134	1	we	we	PRON
ejpam-3420	134	2	have	have	VERB
ejpam-3420	134	3	y	y	PROPN
ejpam-3420	134	4	⊆	⊆	NUM
ejpam-3420	134	5	y	y	NOUN
ejpam-3420	134	6	=	=	SYM
ejpam-3420	134	7	k	k	PROPN
ejpam-3420	135	1	⊆	⊆	NUM
ejpam-3420	135	2	c.	c.	NOUN
ejpam-3420	135	3	so	so	SCONJ
ejpam-3420	135	4	that	that	SCONJ
ejpam-3420	135	5	yc	yc	PRON
ejpam-3420	135	6	=	=	PUNCT
ejpam-3420	135	7	(	(	PUNCT
ejpam-3420	136	1	y	y	PROPN
ejpam-3420	136	2	,	,	PUNCT
ejpam-3420	136	3	y	y	PROPN
ejpam-3420	136	4	)	)	PUNCT
ejpam-3420	136	5	∈	∈	PROPN
ejpam-3420	136	6	b	b	PROPN
ejpam-3420	136	7	,	,	PUNCT
ejpam-3420	136	8	with	with	ADP
ejpam-3420	136	9	y	y	PROPN
ejpam-3420	136	10	⊆	⊆	NUM
ejpam-3420	136	11	x	x	PUNCT
ejpam-3420	136	12	∧	∧	NOUN
ejpam-3420	136	13	y	y	PROPN
ejpam-3420	136	14	⊆	⊆	NUM
ejpam-3420	136	15	x	x	NOUN
ejpam-3420	136	16	,	,	PUNCT
ejpam-3420	136	17	which	which	PRON
ejpam-3420	136	18	is	be	AUX
ejpam-3420	136	19	the	the	DET
ejpam-3420	136	20	desired	desire	VERB
ejpam-3420	136	21	result	result	NOUN
ejpam-3420	136	22	.	.	PUNCT
ejpam-3420	137	1	(	(	PUNCT
ejpam-3420	137	2	ii	ii	NOUN
ejpam-3420	137	3	)	)	PUNCT
ejpam-3420	137	4	let	let	VERB
ejpam-3420	137	5	y	y	PROPN
ejpam-3420	137	6	∈	∈	PROPN
ejpam-3420	137	7	gc	gc	PROPN
ejpam-3420	137	8	=	=	SYM
ejpam-3420	137	9	(	(	PUNCT
ejpam-3420	137	10	g	g	NOUN
ejpam-3420	137	11	,	,	PUNCT
ejpam-3420	137	12	g	g	NOUN
ejpam-3420	137	13	)	)	PUNCT
ejpam-3420	137	14	=	=	SYM
ejpam-3420	137	15	(	(	PUNCT
ejpam-3420	137	16	(	(	PUNCT
ejpam-3420	137	17	a1	a1	NOUN
ejpam-3420	137	18	,	,	PUNCT
ejpam-3420	137	19	b1	b1	NOUN
ejpam-3420	137	20	)	)	PUNCT
ejpam-3420	138	1	∧	∧	PROPN
ejpam-3420	138	2	(	(	PUNCT
ejpam-3420	138	3	a2	a2	PROPN
ejpam-3420	138	4	,	,	PUNCT
ejpam-3420	138	5	b2	b2	NOUN
ejpam-3420	138	6	)	)	PUNCT
ejpam-3420	138	7	)	)	PUNCT
ejpam-3420	138	8	,	,	PUNCT
ejpam-3420	138	9	where	where	SCONJ
ejpam-3420	138	10	(	(	PUNCT
ejpam-3420	138	11	a1	a1	NOUN
ejpam-3420	138	12	,	,	PUNCT
ejpam-3420	138	13	b1	b1	NOUN
ejpam-3420	138	14	)	)	PUNCT
ejpam-3420	138	15	,	,	PUNCT
ejpam-3420	138	16	(	(	PUNCT
ejpam-3420	138	17	a2	a2	PROPN
ejpam-3420	138	18	,	,	PUNCT
ejpam-3420	138	19	b2	b2	NOUN
ejpam-3420	138	20	)	)	PUNCT
ejpam-3420	138	21	∈	∈	PROPN
ejpam-3420	138	22	b.	b.	PROPN
ejpam-3420	139	1	this	this	PRON
ejpam-3420	139	2	implies	imply	VERB
ejpam-3420	139	3	that	that	SCONJ
ejpam-3420	139	4	y	y	PROPN
ejpam-3420	139	5	6=	6=	PROPN
ejpam-3420	139	6	φ	φ	PROPN
ejpam-3420	139	7	,	,	PUNCT
ejpam-3420	139	8	yc	yc	PROPN
ejpam-3420	139	9	⊆	⊆	NUM
ejpam-3420	139	10	gc	gc	PROPN
ejpam-3420	139	11	and	and	CCONJ
ejpam-3420	139	12	there	there	PRON
ejpam-3420	139	13	exists	exist	VERB
ejpam-3420	139	14	b	b	PROPN
ejpam-3420	139	15	∈	∈	PROPN
ejpam-3420	139	16	c	c	NOUN
ejpam-3420	139	17	such	such	ADJ
ejpam-3420	139	18	that	that	DET
ejpam-3420	139	19	y	y	PROPN
ejpam-3420	139	20	=	=	PROPN
ejpam-3420	139	21	b.	b.	PROPN
ejpam-3420	140	1	then	then	ADV
ejpam-3420	140	2	yc	yc	PROPN
ejpam-3420	140	3	=	=	PUNCT
ejpam-3420	140	4	(	(	PUNCT
ejpam-3420	140	5	y	y	PROPN
ejpam-3420	140	6	,	,	PUNCT
ejpam-3420	140	7	y	y	PROPN
ejpam-3420	140	8	)	)	PUNCT
ejpam-3420	140	9	with	with	ADP
ejpam-3420	140	10	y	y	PROPN
ejpam-3420	140	11	⊆	⊆	NUM
ejpam-3420	140	12	y	y	PROPN
ejpam-3420	140	13	=	=	PROPN
ejpam-3420	140	14	b.	b.	PROPN
ejpam-3420	141	1	hence	hence	ADV
ejpam-3420	141	2	yc	yc	PROPN
ejpam-3420	141	3	∈	∈	PROPN
ejpam-3420	141	4	b	b	PROPN
ejpam-3420	141	5	and	and	CCONJ
ejpam-3420	141	6	y	y	PROPN
ejpam-3420	141	7	∈	∈	PROPN
ejpam-3420	142	1	yc	yc	PROPN
ejpam-3420	142	2	⊆	⊆	NUM
ejpam-3420	142	3	(	(	PUNCT
ejpam-3420	142	4	(	(	PUNCT
ejpam-3420	142	5	a1	a1	NOUN
ejpam-3420	142	6	,	,	PUNCT
ejpam-3420	142	7	b1	b1	NOUN
ejpam-3420	142	8	)	)	PUNCT
ejpam-3420	142	9	∧	∧	PROPN
ejpam-3420	142	10	(	(	PUNCT
ejpam-3420	142	11	a2	a2	PROPN
ejpam-3420	142	12	,	,	PUNCT
ejpam-3420	142	13	b2	b2	NOUN
ejpam-3420	142	14	)	)	PUNCT
ejpam-3420	142	15	)	)	PUNCT
ejpam-3420	142	16	.	.	PUNCT
ejpam-3420	143	1	so	so	ADV
ejpam-3420	143	2	b	b	PROPN
ejpam-3420	143	3	is	be	AUX
ejpam-3420	143	4	an	an	DET
ejpam-3420	143	5	exact	exact	ADJ
ejpam-3420	143	6	base	base	NOUN
ejpam-3420	143	7	for	for	ADP
ejpam-3420	143	8	the	the	DET
ejpam-3420	143	9	covering	covering	NOUN
ejpam-3420	143	10	based	base	VERB
ejpam-3420	143	11	rough	rough	ADJ
ejpam-3420	143	12	topology	topology	NOUN
ejpam-3420	143	13	on	on	ADP
ejpam-3420	143	14	the	the	DET
ejpam-3420	143	15	covering	covering	NOUN
ejpam-3420	143	16	based	base	VERB
ejpam-3420	143	17	rough	rough	ADJ
ejpam-3420	143	18	set	set	NOUN
ejpam-3420	143	19	x.	x.	NOUN
ejpam-3420	144	1	a	a	DET
ejpam-3420	144	2	family	family	NOUN
ejpam-3420	144	3	f	f	PROPN
ejpam-3420	144	4	⊆	⊆	NUM
ejpam-3420	144	5	τ	τ	PROPN
ejpam-3420	144	6	is	be	AUX
ejpam-3420	144	7	called	call	VERB
ejpam-3420	144	8	an	an	DET
ejpam-3420	144	9	exact	exact	ADJ
ejpam-3420	144	10	subbase	subbase	NOUN
ejpam-3420	144	11	for	for	ADP
ejpam-3420	144	12	a	a	DET
ejpam-3420	144	13	topological	topological	ADJ
ejpam-3420	144	14	space	space	NOUN
ejpam-3420	144	15	(	(	PUNCT
ejpam-3420	144	16	x	x	X
ejpam-3420	144	17	,	,	PUNCT
ejpam-3420	144	18	τ	τ	X
ejpam-3420	144	19	)	)	PUNCT
ejpam-3420	144	20	if	if	SCONJ
ejpam-3420	144	21	the	the	DET
ejpam-3420	144	22	family	family	NOUN
ejpam-3420	144	23	of	of	ADP
ejpam-3420	144	24	all	all	DET
ejpam-3420	144	25	finite	finite	ADJ
ejpam-3420	144	26	intersections	intersection	NOUN
ejpam-3420	144	27	of	of	ADP
ejpam-3420	144	28	members	member	NOUN
ejpam-3420	144	29	of	of	ADP
ejpam-3420	144	30	f	f	PROPN
ejpam-3420	144	31	forms	form	VERB
ejpam-3420	144	32	a	a	DET
ejpam-3420	144	33	base	base	NOUN
ejpam-3420	144	34	for	for	ADP
ejpam-3420	144	35	(	(	PUNCT
ejpam-3420	144	36	x	x	X
ejpam-3420	144	37	,	,	PUNCT
ejpam-3420	144	38	τ	τ	PROPN
ejpam-3420	144	39	)	)	PUNCT
ejpam-3420	144	40	.	.	PUNCT
ejpam-3420	145	1	definition	definition	NOUN
ejpam-3420	145	2	6	6	NUM
ejpam-3420	145	3	.	.	PUNCT
ejpam-3420	146	1	let	let	VERB
ejpam-3420	146	2	f	f	PRON
ejpam-3420	146	3	be	be	AUX
ejpam-3420	146	4	a	a	DET
ejpam-3420	146	5	family	family	NOUN
ejpam-3420	146	6	of	of	ADP
ejpam-3420	146	7	covering	cover	VERB
ejpam-3420	146	8	based	base	VERB
ejpam-3420	146	9	rough	rough	ADJ
ejpam-3420	146	10	open	open	ADJ
ejpam-3420	146	11	subsets	subset	NOUN
ejpam-3420	146	12	of	of	ADP
ejpam-3420	146	13	xc	xc	PROPN
ejpam-3420	146	14	.	.	PUNCT
ejpam-3420	147	1	then	then	ADV
ejpam-3420	147	2	f	f	PROPN
ejpam-3420	147	3	is	be	AUX
ejpam-3420	147	4	called	call	VERB
ejpam-3420	147	5	an	an	DET
ejpam-3420	147	6	exact	exact	ADJ
ejpam-3420	147	7	covering	covering	NOUN
ejpam-3420	147	8	based	base	VERB
ejpam-3420	147	9	rough	rough	ADJ
ejpam-3420	147	10	subbase	subbase	NOUN
ejpam-3420	147	11	if	if	SCONJ
ejpam-3420	147	12	the	the	DET
ejpam-3420	147	13	family	family	NOUN
ejpam-3420	147	14	{	{	PUNCT
ejpam-3420	147	15	y1c	y1c	PROPN
ejpam-3420	147	16	∧	∧	PROPN
ejpam-3420	147	17	y2c	y2c	PROPN
ejpam-3420	147	18	∧	∧	PROPN
ejpam-3420	147	19	·	·	PUNCT
ejpam-3420	147	20	·	·	PUNCT
ejpam-3420	147	21	·	·	PUNCT
ejpam-3420	147	22	∧	∧	NOUN
ejpam-3420	147	23	ync	ync	NOUN
ejpam-3420	147	24	:	:	PUNCT
ejpam-3420	147	25	n	n	CCONJ
ejpam-3420	147	26	∈	∈	PROPN
ejpam-3420	147	27	n	n	CCONJ
ejpam-3420	147	28	,	,	PUNCT
ejpam-3420	147	29	yic	yic	PROPN
ejpam-3420	147	30	∈	∈	PROPN
ejpam-3420	147	31	f	f	PROPN
ejpam-3420	147	32	,	,	PUNCT
ejpam-3420	147	33	∀i	∀i	NOUN
ejpam-3420	147	34	∈	∈	NOUN
ejpam-3420	147	35	{	{	PUNCT
ejpam-3420	147	36	1	1	NUM
ejpam-3420	147	37	,	,	PUNCT
ejpam-3420	147	38	2	2	NUM
ejpam-3420	147	39	,	,	PUNCT
ejpam-3420	147	40	.	.	PUNCT
ejpam-3420	147	41	.	.	PUNCT
ejpam-3420	147	42	.	.	PUNCT
ejpam-3420	148	1	,	,	PUNCT
ejpam-3420	148	2	n	n	CCONJ
ejpam-3420	148	3	}	}	PUNCT
ejpam-3420	148	4	}	}	PUNCT
ejpam-3420	148	5	forms	form	VERB
ejpam-3420	148	6	a	a	DET
ejpam-3420	148	7	base	base	NOUN
ejpam-3420	148	8	for	for	ADP
ejpam-3420	148	9	(	(	PUNCT
ejpam-3420	148	10	xc	xc	PROPN
ejpam-3420	148	11	,	,	PUNCT
ejpam-3420	148	12	τ	τ	PROPN
ejpam-3420	148	13	)	)	PUNCT
ejpam-3420	148	14	.	.	PUNCT
ejpam-3420	149	1	remark	remark	PROPN
ejpam-3420	149	2	1	1	NUM
ejpam-3420	149	3	.	.	PUNCT
ejpam-3420	150	1	let	let	VERB
ejpam-3420	150	2	f	f	PRON
ejpam-3420	150	3	be	be	AUX
ejpam-3420	150	4	an	an	DET
ejpam-3420	150	5	exact	exact	ADJ
ejpam-3420	150	6	covering	covering	NOUN
ejpam-3420	150	7	based	base	VERB
ejpam-3420	150	8	rough	rough	ADJ
ejpam-3420	150	9	subbase	subbase	NOUN
ejpam-3420	150	10	.	.	PUNCT
ejpam-3420	151	1	then	then	ADV
ejpam-3420	151	2	it	it	PRON
ejpam-3420	151	3	generates	generate	VERB
ejpam-3420	151	4	a	a	DET
ejpam-3420	151	5	covering	covering	NOUN
ejpam-3420	151	6	based	base	VERB
ejpam-3420	151	7	rough	rough	ADJ
ejpam-3420	151	8	topology	topology	NOUN
ejpam-3420	151	9	on	on	ADP
ejpam-3420	151	10	x.	x.	NOUN
ejpam-3420	151	11	now	now	ADV
ejpam-3420	151	12	,	,	PUNCT
ejpam-3420	151	13	we	we	PRON
ejpam-3420	151	14	introduce	introduce	VERB
ejpam-3420	151	15	separation	separation	NOUN
ejpam-3420	151	16	axioms	axiom	NOUN
ejpam-3420	151	17	with	with	ADP
ejpam-3420	151	18	respect	respect	NOUN
ejpam-3420	151	19	to	to	ADP
ejpam-3420	151	20	exact	exact	ADJ
ejpam-3420	151	21	elements	element	NOUN
ejpam-3420	151	22	for	for	ADP
ejpam-3420	151	23	a	a	DET
ejpam-3420	151	24	covering	covering	NOUN
ejpam-3420	151	25	based	base	VERB
ejpam-3420	151	26	rough	rough	ADJ
ejpam-3420	151	27	space	space	NOUN
ejpam-3420	151	28	(	(	PUNCT
ejpam-3420	151	29	x	x	X
ejpam-3420	151	30	,	,	PUNCT
ejpam-3420	151	31	τ	τ	PROPN
ejpam-3420	151	32	)	)	PUNCT
ejpam-3420	151	33	.	.	PUNCT
ejpam-3420	152	1	the	the	DET
ejpam-3420	152	2	space	space	NOUN
ejpam-3420	152	3	(	(	PUNCT
ejpam-3420	152	4	x	x	X
ejpam-3420	152	5	,	,	PUNCT
ejpam-3420	152	6	τ	τ	X
ejpam-3420	152	7	)	)	PUNCT
ejpam-3420	152	8	is	be	AUX
ejpam-3420	152	9	called	call	VERB
ejpam-3420	152	10	an	an	DET
ejpam-3420	152	11	exact	exact	ADJ
ejpam-3420	152	12	covering	covering	NOUN
ejpam-3420	152	13	based	base	VERB
ejpam-3420	152	14	rough	rough	ADJ
ejpam-3420	152	15	t	t	PROPN
ejpam-3420	152	16	◦	◦	NOUN
ejpam-3420	152	17	-space	-space	NOUN
ejpam-3420	152	18	if	if	SCONJ
ejpam-3420	152	19	for	for	ADP
ejpam-3420	152	20	every	every	DET
ejpam-3420	152	21	exact	exact	ADJ
ejpam-3420	152	22	distinct	distinct	ADJ
ejpam-3420	152	23	elements	element	NOUN
ejpam-3420	152	24	y	y	PROPN
ejpam-3420	152	25	,	,	PUNCT
ejpam-3420	152	26	y	y	PROPN
ejpam-3420	152	27	′	′	NUM
ejpam-3420	152	28	∈	∈	PROPN
ejpam-3420	153	1	x	x	PRON
ejpam-3420	153	2	,	,	PUNCT
ejpam-3420	153	3	there	there	PRON
ejpam-3420	153	4	is	be	VERB
ejpam-3420	153	5	a	a	DET
ejpam-3420	153	6	covering	covering	NOUN
ejpam-3420	153	7	based	base	VERB
ejpam-3420	153	8	rough	rough	ADJ
ejpam-3420	153	9	open	open	ADJ
ejpam-3420	153	10	set	set	VERB
ejpam-3420	153	11	a	a	PRON
ejpam-3420	153	12	of	of	ADP
ejpam-3420	153	13	x	x	SYM
ejpam-3420	153	14	such	such	ADJ
ejpam-3420	153	15	that	that	SCONJ
ejpam-3420	153	16	y	y	PROPN
ejpam-3420	153	17	/∈	/∈	PUNCT
ejpam-3420	154	1	a	a	DET
ejpam-3420	154	2	3	3	NUM
ejpam-3420	154	3	y	y	NOUN
ejpam-3420	154	4	′	′	NOUN
ejpam-3420	154	5	or	or	CCONJ
ejpam-3420	154	6	y	y	PROPN
ejpam-3420	154	7	′	′	NUM
ejpam-3420	154	8	/∈	/∈	PUNCT
ejpam-3420	155	1	a	a	DET
ejpam-3420	155	2	3	3	NUM
ejpam-3420	155	3	y	y	NOUN
ejpam-3420	155	4	.	.	PUNCT
ejpam-3420	156	1	the	the	DET
ejpam-3420	156	2	space	space	NOUN
ejpam-3420	156	3	(	(	PUNCT
ejpam-3420	156	4	x	x	X
ejpam-3420	156	5	,	,	PUNCT
ejpam-3420	156	6	τ	τ	X
ejpam-3420	156	7	)	)	PUNCT
ejpam-3420	156	8	is	be	AUX
ejpam-3420	156	9	called	call	VERB
ejpam-3420	156	10	an	an	DET
ejpam-3420	156	11	exact	exact	ADJ
ejpam-3420	156	12	covering	covering	NOUN
ejpam-3420	156	13	based	base	VERB
ejpam-3420	156	14	rough	rough	ADJ
ejpam-3420	156	15	t1	t1	NOUN
ejpam-3420	156	16	-	-	PUNCT
ejpam-3420	156	17	space	space	NOUN
ejpam-3420	156	18	if	if	SCONJ
ejpam-3420	156	19	for	for	ADP
ejpam-3420	156	20	every	every	DET
ejpam-3420	156	21	exact	exact	ADJ
ejpam-3420	156	22	n.	n.	NOUN
ejpam-3420	156	23	alharbi	alharbi	PROPN
ejpam-3420	156	24	,	,	PUNCT
ejpam-3420	156	25	h.	h.	PROPN
ejpam-3420	156	26	aydi	aydi	PROPN
ejpam-3420	156	27	,	,	PUNCT
ejpam-3420	156	28	c.	c.	PROPN
ejpam-3420	156	29	özel	özel	PROPN
ejpam-3420	156	30	/	/	SYM
ejpam-3420	156	31	eur	eur	PROPN
ejpam-3420	156	32	.	.	PUNCT
ejpam-3420	157	1	j.	j.	PROPN
ejpam-3420	157	2	pure	pure	PROPN
ejpam-3420	157	3	appl	appl	PROPN
ejpam-3420	157	4	.	.	PROPN
ejpam-3420	157	5	math	math	PROPN
ejpam-3420	157	6	,	,	PUNCT
ejpam-3420	157	7	12	12	NUM
ejpam-3420	157	8	(	(	PUNCT
ejpam-3420	157	9	2	2	NUM
ejpam-3420	157	10	)	)	PUNCT
ejpam-3420	157	11	(	(	PUNCT
ejpam-3420	157	12	2019	2019	NUM
ejpam-3420	157	13	)	)	PUNCT
ejpam-3420	157	14	,	,	PUNCT
ejpam-3420	157	15	533	533	NUM
ejpam-3420	157	16	-	-	SYM
ejpam-3420	157	17	543	543	NUM
ejpam-3420	157	18	540	540	NUM
ejpam-3420	157	19	distinct	distinct	ADJ
ejpam-3420	157	20	elements	element	NOUN
ejpam-3420	157	21	y	y	PROPN
ejpam-3420	157	22	,	,	PUNCT
ejpam-3420	157	23	y	y	PROPN
ejpam-3420	157	24	′	′	NUM
ejpam-3420	157	25	∈	∈	PROPN
ejpam-3420	158	1	x	x	PRON
ejpam-3420	158	2	,	,	PUNCT
ejpam-3420	158	3	there	there	PRON
ejpam-3420	158	4	are	be	VERB
ejpam-3420	158	5	two	two	NUM
ejpam-3420	158	6	covering	covering	NOUN
ejpam-3420	158	7	based	base	VERB
ejpam-3420	158	8	rough	rough	ADJ
ejpam-3420	158	9	open	open	ADJ
ejpam-3420	158	10	sets	set	NOUN
ejpam-3420	158	11	a	a	PRON
ejpam-3420	158	12	and	and	CCONJ
ejpam-3420	158	13	b	b	NOUN
ejpam-3420	158	14	of	of	ADP
ejpam-3420	158	15	x	x	SYM
ejpam-3420	158	16	such	such	ADJ
ejpam-3420	158	17	that	that	SCONJ
ejpam-3420	158	18	y	y	PROPN
ejpam-3420	158	19	/∈	/∈	PUNCT
ejpam-3420	159	1	a	a	DET
ejpam-3420	159	2	3	3	NUM
ejpam-3420	159	3	y	y	NOUN
ejpam-3420	159	4	′	′	NOUN
ejpam-3420	160	1	and	and	CCONJ
ejpam-3420	160	2	y	y	PROPN
ejpam-3420	160	3	′	′	NUM
ejpam-3420	160	4	/∈	/∈	PUNCT
ejpam-3420	161	1	b	b	X
ejpam-3420	161	2	3	3	NUM
ejpam-3420	161	3	y	y	NOUN
ejpam-3420	161	4	.	.	PUNCT
ejpam-3420	162	1	the	the	DET
ejpam-3420	162	2	space	space	NOUN
ejpam-3420	162	3	(	(	PUNCT
ejpam-3420	162	4	x	x	X
ejpam-3420	162	5	,	,	PUNCT
ejpam-3420	162	6	τ	τ	X
ejpam-3420	162	7	)	)	PUNCT
ejpam-3420	162	8	is	be	AUX
ejpam-3420	162	9	called	call	VERB
ejpam-3420	162	10	an	an	DET
ejpam-3420	162	11	exact	exact	ADJ
ejpam-3420	162	12	covering	covering	NOUN
ejpam-3420	162	13	based	base	VERB
ejpam-3420	162	14	rough	rough	ADJ
ejpam-3420	162	15	t2	t2	NOUN
ejpam-3420	162	16	-	-	PUNCT
ejpam-3420	162	17	space	space	NOUN
ejpam-3420	162	18	if	if	SCONJ
ejpam-3420	162	19	for	for	ADP
ejpam-3420	162	20	every	every	DET
ejpam-3420	162	21	exact	exact	ADJ
ejpam-3420	162	22	distinct	distinct	ADJ
ejpam-3420	162	23	elements	element	NOUN
ejpam-3420	162	24	y	y	PROPN
ejpam-3420	162	25	,	,	PUNCT
ejpam-3420	162	26	y	y	PROPN
ejpam-3420	162	27	′	′	NUM
ejpam-3420	162	28	∈	∈	PROPN
ejpam-3420	163	1	x	x	PRON
ejpam-3420	163	2	,	,	PUNCT
ejpam-3420	163	3	there	there	PRON
ejpam-3420	163	4	are	be	VERB
ejpam-3420	163	5	two	two	NUM
ejpam-3420	163	6	disjoint	disjoint	NOUN
ejpam-3420	163	7	covering	covering	NOUN
ejpam-3420	163	8	based	base	VERB
ejpam-3420	163	9	rough	rough	ADJ
ejpam-3420	163	10	open	open	ADJ
ejpam-3420	163	11	sets	set	NOUN
ejpam-3420	163	12	a	a	PRON
ejpam-3420	163	13	and	and	CCONJ
ejpam-3420	163	14	b	b	NOUN
ejpam-3420	163	15	of	of	ADP
ejpam-3420	163	16	x	x	SYM
ejpam-3420	163	17	such	such	ADJ
ejpam-3420	163	18	that	that	SCONJ
ejpam-3420	163	19	y	y	PROPN
ejpam-3420	163	20	/∈	/∈	PUNCT
ejpam-3420	164	1	a	a	DET
ejpam-3420	164	2	3	3	NUM
ejpam-3420	164	3	y	y	NOUN
ejpam-3420	164	4	′	′	NOUN
ejpam-3420	165	1	and	and	CCONJ
ejpam-3420	165	2	y	y	PROPN
ejpam-3420	165	3	′	′	NUM
ejpam-3420	165	4	/∈	/∈	PUNCT
ejpam-3420	166	1	b	b	X
ejpam-3420	166	2	3	3	NUM
ejpam-3420	166	3	y	y	PROPN
ejpam-3420	166	4	.	.	PUNCT
ejpam-3420	167	1	note	note	VERB
ejpam-3420	167	2	that	that	SCONJ
ejpam-3420	167	3	,	,	PUNCT
ejpam-3420	167	4	there	there	PRON
ejpam-3420	167	5	is	be	VERB
ejpam-3420	167	6	no	no	DET
ejpam-3420	167	7	exact	exact	ADJ
ejpam-3420	167	8	rough	rough	ADJ
ejpam-3420	167	9	t1	t1	NOUN
ejpam-3420	167	10	-	-	PUNCT
ejpam-3420	167	11	space	space	NOUN
ejpam-3420	167	12	.	.	PUNCT
ejpam-3420	168	1	also	also	ADV
ejpam-3420	168	2	,	,	PUNCT
ejpam-3420	168	3	there	there	PRON
ejpam-3420	168	4	is	be	VERB
ejpam-3420	168	5	no	no	DET
ejpam-3420	168	6	covering	covering	NOUN
ejpam-3420	168	7	based	base	VERB
ejpam-3420	168	8	rough	rough	ADJ
ejpam-3420	168	9	t1	t1	NOUN
ejpam-3420	168	10	-	-	PUNCT
ejpam-3420	168	11	space	space	NOUN
ejpam-3420	168	12	.	.	PUNCT
ejpam-3420	169	1	this	this	PRON
ejpam-3420	169	2	is	be	AUX
ejpam-3420	169	3	true	true	ADJ
ejpam-3420	169	4	only	only	ADV
ejpam-3420	169	5	in	in	ADP
ejpam-3420	169	6	the	the	DET
ejpam-3420	169	7	case	case	NOUN
ejpam-3420	169	8	that	that	SCONJ
ejpam-3420	169	9	the	the	DET
ejpam-3420	169	10	rough	rough	ADJ
ejpam-3420	169	11	set	set	NOUN
ejpam-3420	169	12	x	x	PUNCT
ejpam-3420	169	13	is	be	AUX
ejpam-3420	169	14	a	a	DET
ejpam-3420	169	15	singleton	singleton	NOUN
ejpam-3420	169	16	and	and	CCONJ
ejpam-3420	169	17	this	this	DET
ejpam-3420	169	18	singleton	singleton	NOUN
ejpam-3420	169	19	is	be	AUX
ejpam-3420	169	20	a	a	DET
ejpam-3420	169	21	class	class	NOUN
ejpam-3420	169	22	,	,	PUNCT
ejpam-3420	169	23	since	since	SCONJ
ejpam-3420	169	24	exact	exact	ADJ
ejpam-3420	169	25	elements	element	NOUN
ejpam-3420	169	26	are	be	AUX
ejpam-3420	169	27	subset	subset	VERB
ejpam-3420	169	28	of	of	ADP
ejpam-3420	169	29	each	each	DET
ejpam-3420	169	30	other	other	ADJ
ejpam-3420	169	31	.	.	PUNCT
ejpam-3420	170	1	continuous	continuous	ADJ
ejpam-3420	170	2	,	,	PUNCT
ejpam-3420	170	3	closed	closed	ADJ
ejpam-3420	170	4	,	,	PUNCT
ejpam-3420	170	5	open	open	ADJ
ejpam-3420	170	6	and	and	CCONJ
ejpam-3420	170	7	homeomorphism	homeomorphism	NOUN
ejpam-3420	170	8	mappings	mapping	NOUN
ejpam-3420	170	9	here	here	ADV
ejpam-3420	170	10	,	,	PUNCT
ejpam-3420	170	11	we	we	PRON
ejpam-3420	170	12	will	will	AUX
ejpam-3420	170	13	define	define	VERB
ejpam-3420	170	14	covering	covering	NOUN
ejpam-3420	170	15	based	base	VERB
ejpam-3420	170	16	rough	rough	ADJ
ejpam-3420	170	17	continuous	continuous	ADJ
ejpam-3420	170	18	maps	map	NOUN
ejpam-3420	170	19	between	between	ADP
ejpam-3420	170	20	two	two	NUM
ejpam-3420	170	21	covering	covering	NOUN
ejpam-3420	170	22	based	base	VERB
ejpam-3420	170	23	rough	rough	ADJ
ejpam-3420	170	24	topological	topological	ADJ
ejpam-3420	170	25	spaces	space	NOUN
ejpam-3420	170	26	.	.	PUNCT
ejpam-3420	171	1	definition	definition	NOUN
ejpam-3420	171	2	7	7	NUM
ejpam-3420	171	3	.	.	PUNCT
ejpam-3420	172	1	let	let	VERB
ejpam-3420	172	2	(	(	PUNCT
ejpam-3420	172	3	u	u	NOUN
ejpam-3420	172	4	,	,	PUNCT
ejpam-3420	172	5	c	c	NOUN
ejpam-3420	172	6	)	)	PUNCT
ejpam-3420	172	7	be	be	AUX
ejpam-3420	172	8	a	a	DET
ejpam-3420	172	9	covering	covering	NOUN
ejpam-3420	172	10	based	base	VERB
ejpam-3420	172	11	approximation	approximation	NOUN
ejpam-3420	172	12	space	space	NOUN
ejpam-3420	172	13	and	and	CCONJ
ejpam-3420	172	14	f	f	NOUN
ejpam-3420	172	15	:	:	PUNCT
ejpam-3420	172	16	u	u	X
ejpam-3420	172	17	→	→	SYM
ejpam-3420	172	18	u	u	X
ejpam-3420	172	19	be	be	VERB
ejpam-3420	172	20	a	a	DET
ejpam-3420	172	21	map	map	NOUN
ejpam-3420	172	22	.	.	PUNCT
ejpam-3420	173	1	assume	assume	VERB
ejpam-3420	173	2	that	that	SCONJ
ejpam-3420	173	3	x	x	X
ejpam-3420	173	4	,	,	PUNCT
ejpam-3420	173	5	y	y	PROPN
ejpam-3420	173	6	are	be	AUX
ejpam-3420	173	7	covering	cover	VERB
ejpam-3420	173	8	based	base	VERB
ejpam-3420	173	9	rough	rough	ADJ
ejpam-3420	173	10	topological	topological	ADJ
ejpam-3420	173	11	spaces	space	NOUN
ejpam-3420	173	12	over	over	ADP
ejpam-3420	173	13	u	u	PROPN
ejpam-3420	173	14	.	.	PUNCT
ejpam-3420	174	1	then	then	ADV
ejpam-3420	174	2	the	the	DET
ejpam-3420	174	3	restriction	restriction	NOUN
ejpam-3420	174	4	of	of	ADP
ejpam-3420	174	5	f	f	PROPN
ejpam-3420	174	6	from	from	ADP
ejpam-3420	174	7	x	x	PUNCT
ejpam-3420	174	8	to	to	ADP
ejpam-3420	174	9	y	y	PROPN
ejpam-3420	174	10	is	be	AUX
ejpam-3420	174	11	a	a	DET
ejpam-3420	174	12	covering	covering	NOUN
ejpam-3420	174	13	based	base	VERB
ejpam-3420	174	14	rough	rough	ADJ
ejpam-3420	174	15	continuous	continuous	ADJ
ejpam-3420	174	16	map	map	NOUN
ejpam-3420	174	17	,	,	PUNCT
ejpam-3420	174	18	if	if	SCONJ
ejpam-3420	174	19	the	the	DET
ejpam-3420	174	20	pre	pre	ADJ
ejpam-3420	174	21	image	image	NOUN
ejpam-3420	174	22	of	of	ADP
ejpam-3420	174	23	every	every	DET
ejpam-3420	174	24	v	v	NOUN
ejpam-3420	174	25	⊆	⊆	NUM
ejpam-3420	174	26	y	y	NOUN
ejpam-3420	174	27	,	,	PUNCT
ejpam-3420	174	28	which	which	PRON
ejpam-3420	174	29	is	be	AUX
ejpam-3420	174	30	a	a	DET
ejpam-3420	174	31	covering	covering	NOUN
ejpam-3420	174	32	based	base	VERB
ejpam-3420	174	33	rough	rough	ADJ
ejpam-3420	174	34	open	open	ADJ
ejpam-3420	174	35	(	(	PUNCT
ejpam-3420	174	36	resp	resp	NOUN
ejpam-3420	174	37	.	.	PUNCT
ejpam-3420	175	1	closed	close	VERB
ejpam-3420	175	2	)	)	PUNCT
ejpam-3420	175	3	set	set	VERB
ejpam-3420	175	4	in	in	ADP
ejpam-3420	175	5	y	y	PROPN
ejpam-3420	175	6	,	,	PUNCT
ejpam-3420	175	7	gives	give	VERB
ejpam-3420	175	8	a	a	DET
ejpam-3420	175	9	subset	subset	NOUN
ejpam-3420	175	10	of	of	ADP
ejpam-3420	175	11	x	x	PUNCT
ejpam-3420	175	12	being	be	AUX
ejpam-3420	175	13	a	a	DET
ejpam-3420	175	14	covering	covering	NOUN
ejpam-3420	175	15	based	base	VERB
ejpam-3420	175	16	rough	rough	ADJ
ejpam-3420	175	17	open	open	ADJ
ejpam-3420	175	18	(	(	PUNCT
ejpam-3420	175	19	resp	resp	NOUN
ejpam-3420	175	20	.	.	PUNCT
ejpam-3420	176	1	closed	close	VERB
ejpam-3420	176	2	)	)	PUNCT
ejpam-3420	176	3	in	in	ADP
ejpam-3420	176	4	x.	x.	NOUN
ejpam-3420	176	5	definition	definition	NOUN
ejpam-3420	176	6	8	8	NUM
ejpam-3420	176	7	.	.	PUNCT
ejpam-3420	177	1	let	let	VERB
ejpam-3420	177	2	(	(	PUNCT
ejpam-3420	177	3	u	u	NOUN
ejpam-3420	177	4	,	,	PUNCT
ejpam-3420	177	5	c	c	NOUN
ejpam-3420	177	6	)	)	PUNCT
ejpam-3420	177	7	and	and	CCONJ
ejpam-3420	177	8	(	(	PUNCT
ejpam-3420	177	9	v	v	NOUN
ejpam-3420	177	10	,	,	PUNCT
ejpam-3420	177	11	c	c	NOUN
ejpam-3420	177	12	′	′	NUM
ejpam-3420	177	13	)	)	PUNCT
ejpam-3420	177	14	be	be	AUX
ejpam-3420	177	15	a	a	DET
ejpam-3420	177	16	covering	covering	NOUN
ejpam-3420	177	17	based	base	VERB
ejpam-3420	177	18	approximation	approximation	NOUN
ejpam-3420	177	19	spaces	space	VERB
ejpam-3420	177	20	such	such	ADJ
ejpam-3420	177	21	that	that	SCONJ
ejpam-3420	177	22	x	x	SYM
ejpam-3420	177	23	⊆	⊆	NUM
ejpam-3420	177	24	u	u	NOUN
ejpam-3420	177	25	and	and	CCONJ
ejpam-3420	177	26	y	y	PROPN
ejpam-3420	177	27	⊆	⊆	PROPN
ejpam-3420	177	28	v	v	NOUN
ejpam-3420	177	29	.	.	PUNCT
ejpam-3420	178	1	consider	consider	VERB
ejpam-3420	178	2	the	the	DET
ejpam-3420	178	3	two	two	NUM
ejpam-3420	178	4	rough	rough	ADJ
ejpam-3420	178	5	topological	topological	ADJ
ejpam-3420	178	6	spaces	space	NOUN
ejpam-3420	178	7	(	(	PUNCT
ejpam-3420	178	8	x	x	X
ejpam-3420	178	9	,	,	PUNCT
ejpam-3420	178	10	τ	τ	PROPN
ejpam-3420	178	11	)	)	PUNCT
ejpam-3420	178	12	,	,	PUNCT
ejpam-3420	178	13	(	(	PUNCT
ejpam-3420	178	14	y	y	PROPN
ejpam-3420	178	15	,	,	PUNCT
ejpam-3420	178	16	τ	τ	PROPN
ejpam-3420	178	17	′	′	NUM
ejpam-3420	178	18	)	)	PUNCT
ejpam-3420	178	19	.	.	PUNCT
ejpam-3420	179	1	a	a	DET
ejpam-3420	179	2	mapping	mapping	NOUN
ejpam-3420	179	3	f	f	NOUN
ejpam-3420	179	4	from	from	ADP
ejpam-3420	179	5	x	x	PUNCT
ejpam-3420	179	6	to	to	ADP
ejpam-3420	179	7	y	y	PROPN
ejpam-3420	179	8	is	be	AUX
ejpam-3420	179	9	called	call	VERB
ejpam-3420	179	10	a	a	DET
ejpam-3420	179	11	covering	covering	NOUN
ejpam-3420	179	12	based	base	VERB
ejpam-3420	179	13	rough	rough	ADJ
ejpam-3420	179	14	continuous	continuous	ADJ
ejpam-3420	179	15	if	if	SCONJ
ejpam-3420	179	16	f−1(b	f−1(b	PROPN
ejpam-3420	179	17	)	)	PUNCT
ejpam-3420	179	18	∈	∈	PROPN
ejpam-3420	179	19	τ	τ	PROPN
ejpam-3420	179	20	for	for	ADP
ejpam-3420	179	21	any	any	DET
ejpam-3420	179	22	covering	covering	NOUN
ejpam-3420	179	23	based	base	VERB
ejpam-3420	179	24	rough	rough	ADJ
ejpam-3420	179	25	open	open	ADJ
ejpam-3420	179	26	(	(	PUNCT
ejpam-3420	179	27	resp	resp	NOUN
ejpam-3420	179	28	.	.	PUNCT
ejpam-3420	180	1	closed	close	VERB
ejpam-3420	180	2	)	)	PUNCT
ejpam-3420	180	3	set	set	VERB
ejpam-3420	180	4	b	b	PROPN
ejpam-3420	180	5	⊆	⊆	NUM
ejpam-3420	180	6	y	y	NOUN
ejpam-3420	180	7	.	.	PUNCT
ejpam-3420	181	1	that	that	PRON
ejpam-3420	181	2	is	be	AUX
ejpam-3420	181	3	,	,	PUNCT
ejpam-3420	181	4	if	if	SCONJ
ejpam-3420	181	5	the	the	DET
ejpam-3420	181	6	inverse	inverse	ADJ
ejpam-3420	181	7	image	image	NOUN
ejpam-3420	181	8	of	of	ADP
ejpam-3420	181	9	any	any	DET
ejpam-3420	181	10	subset	subset	NOUN
ejpam-3420	181	11	b	b	PROPN
ejpam-3420	181	12	of	of	ADP
ejpam-3420	181	13	y	y	PROPN
ejpam-3420	181	14	being	be	AUX
ejpam-3420	181	15	covering	cover	VERB
ejpam-3420	181	16	based	base	VERB
ejpam-3420	181	17	rough	rough	ADJ
ejpam-3420	181	18	open	open	ADJ
ejpam-3420	181	19	(	(	PUNCT
ejpam-3420	181	20	resp	resp	NOUN
ejpam-3420	181	21	.	.	PUNCT
ejpam-3420	182	1	closed	closed	ADJ
ejpam-3420	182	2	)	)	PUNCT
ejpam-3420	182	3	subset	subset	NOUN
ejpam-3420	182	4	of	of	ADP
ejpam-3420	182	5	y	y	PROPN
ejpam-3420	182	6	is	be	AUX
ejpam-3420	182	7	a	a	DET
ejpam-3420	182	8	subset	subset	NOUN
ejpam-3420	182	9	of	of	ADP
ejpam-3420	182	10	x	x	SYM
ejpam-3420	182	11	such	such	ADJ
ejpam-3420	182	12	that	that	DET
ejpam-3420	182	13	f−1(b	f−1(b	PROPN
ejpam-3420	182	14	)	)	PUNCT
ejpam-3420	182	15	∈	∈	PROPN
ejpam-3420	183	1	τ	τ	X
ejpam-3420	183	2	.	.	PUNCT
ejpam-3420	184	1	now	now	ADV
ejpam-3420	184	2	,	,	PUNCT
ejpam-3420	184	3	we	we	PRON
ejpam-3420	184	4	define	define	VERB
ejpam-3420	184	5	the	the	DET
ejpam-3420	184	6	concept	concept	NOUN
ejpam-3420	184	7	of	of	ADP
ejpam-3420	184	8	a	a	DET
ejpam-3420	184	9	covering	covering	NOUN
ejpam-3420	184	10	based	base	VERB
ejpam-3420	184	11	rough	rough	ADJ
ejpam-3420	184	12	closed	closed	ADJ
ejpam-3420	184	13	(	(	PUNCT
ejpam-3420	184	14	resp	resp	NOUN
ejpam-3420	184	15	.	.	PUNCT
ejpam-3420	185	1	open	open	ADJ
ejpam-3420	185	2	)	)	PUNCT
ejpam-3420	185	3	map	map	NOUN
ejpam-3420	185	4	.	.	PUNCT
ejpam-3420	186	1	a	a	DET
ejpam-3420	186	2	covering	covering	NOUN
ejpam-3420	186	3	based	base	VERB
ejpam-3420	186	4	rough	rough	ADJ
ejpam-3420	186	5	continuous	continuous	ADJ
ejpam-3420	186	6	map	map	NOUN
ejpam-3420	187	1	f	f	NOUN
ejpam-3420	187	2	:	:	PUNCT
ejpam-3420	187	3	x	x	X
ejpam-3420	187	4	→	→	SYM
ejpam-3420	187	5	y	y	PROPN
ejpam-3420	187	6	is	be	AUX
ejpam-3420	187	7	called	call	VERB
ejpam-3420	187	8	a	a	DET
ejpam-3420	187	9	covering	covering	NOUN
ejpam-3420	187	10	based	base	VERB
ejpam-3420	187	11	rough	rough	ADJ
ejpam-3420	187	12	closed	closed	ADJ
ejpam-3420	187	13	(	(	PUNCT
ejpam-3420	187	14	resp	resp	NOUN
ejpam-3420	187	15	.	.	PUNCT
ejpam-3420	188	1	open	open	ADJ
ejpam-3420	188	2	)	)	PUNCT
ejpam-3420	188	3	,	,	PUNCT
ejpam-3420	188	4	if	if	SCONJ
ejpam-3420	188	5	for	for	ADP
ejpam-3420	188	6	every	every	DET
ejpam-3420	188	7	covering	covering	NOUN
ejpam-3420	188	8	based	base	VERB
ejpam-3420	188	9	rough	rough	ADJ
ejpam-3420	188	10	closed	closed	ADJ
ejpam-3420	188	11	(	(	PUNCT
ejpam-3420	188	12	resp	resp	NOUN
ejpam-3420	188	13	.	.	PUNCT
ejpam-3420	189	1	open	open	ADJ
ejpam-3420	189	2	)	)	PUNCT
ejpam-3420	189	3	set	set	VERB
ejpam-3420	189	4	a	a	DET
ejpam-3420	189	5	⊂	⊂	PROPN
ejpam-3420	189	6	x	x	NOUN
ejpam-3420	189	7	,	,	PUNCT
ejpam-3420	189	8	the	the	DET
ejpam-3420	189	9	image	image	NOUN
ejpam-3420	189	10	f(a	f(a	PROPN
ejpam-3420	189	11	)	)	PUNCT
ejpam-3420	189	12	is	be	AUX
ejpam-3420	189	13	a	a	DET
ejpam-3420	189	14	covering	covering	NOUN
ejpam-3420	189	15	based	base	VERB
ejpam-3420	189	16	rough	rough	ADJ
ejpam-3420	189	17	closed	closed	ADJ
ejpam-3420	189	18	(	(	PUNCT
ejpam-3420	189	19	resp	resp	NOUN
ejpam-3420	189	20	.	.	PUNCT
ejpam-3420	190	1	open	open	ADJ
ejpam-3420	190	2	)	)	PUNCT
ejpam-3420	190	3	in	in	ADP
ejpam-3420	190	4	y	y	PROPN
ejpam-3420	190	5	.	.	PUNCT
ejpam-3420	191	1	a	a	DET
ejpam-3420	191	2	covering	covering	NOUN
ejpam-3420	191	3	based	base	VERB
ejpam-3420	191	4	rough	rough	ADJ
ejpam-3420	191	5	map	map	NOUN
ejpam-3420	191	6	,	,	PUNCT
ejpam-3420	191	7	which	which	PRON
ejpam-3420	191	8	is	be	AUX
ejpam-3420	191	9	both	both	PRON
ejpam-3420	191	10	covering	cover	VERB
ejpam-3420	191	11	based	base	VERB
ejpam-3420	191	12	rough	rough	ADJ
ejpam-3420	191	13	closed	closed	ADJ
ejpam-3420	191	14	and	and	CCONJ
ejpam-3420	191	15	covering	covering	NOUN
ejpam-3420	191	16	based	base	VERB
ejpam-3420	191	17	rough	rough	ADJ
ejpam-3420	191	18	open	open	NOUN
ejpam-3420	191	19	,	,	PUNCT
ejpam-3420	191	20	is	be	AUX
ejpam-3420	191	21	called	call	VERB
ejpam-3420	191	22	a	a	DET
ejpam-3420	191	23	covering	covering	NOUN
ejpam-3420	191	24	based	base	VERB
ejpam-3420	191	25	rough	rough	ADJ
ejpam-3420	191	26	clopen	clopen	ADJ
ejpam-3420	191	27	map	map	NOUN
ejpam-3420	191	28	.	.	PUNCT
ejpam-3420	192	1	a	a	DET
ejpam-3420	192	2	covering	covering	NOUN
ejpam-3420	192	3	based	base	VERB
ejpam-3420	192	4	rough	rough	ADJ
ejpam-3420	192	5	continuous	continuous	ADJ
ejpam-3420	192	6	map	map	NOUN
ejpam-3420	193	1	f	f	NOUN
ejpam-3420	193	2	:	:	PUNCT
ejpam-3420	193	3	x	x	X
ejpam-3420	193	4	→	→	SYM
ejpam-3420	193	5	y	y	PROPN
ejpam-3420	193	6	is	be	AUX
ejpam-3420	193	7	called	call	VERB
ejpam-3420	193	8	a	a	DET
ejpam-3420	193	9	homeomorphism	homeomorphism	NOUN
ejpam-3420	193	10	if	if	SCONJ
ejpam-3420	193	11	f	f	PROPN
ejpam-3420	193	12	maps	map	VERB
ejpam-3420	193	13	x	x	X
ejpam-3420	193	14	onto	onto	ADP
ejpam-3420	193	15	y	y	PROPN
ejpam-3420	193	16	in	in	ADP
ejpam-3420	193	17	a	a	DET
ejpam-3420	193	18	one	one	NUM
ejpam-3420	193	19	-	-	PUNCT
ejpam-3420	193	20	to	to	ADP
ejpam-3420	193	21	-	-	PUNCT
ejpam-3420	193	22	one	one	NUM
ejpam-3420	193	23	way	way	NOUN
ejpam-3420	193	24	and	and	CCONJ
ejpam-3420	193	25	the	the	DET
ejpam-3420	193	26	inverse	inverse	NOUN
ejpam-3420	193	27	map	map	NOUN
ejpam-3420	193	28	f−1	f−1	PROPN
ejpam-3420	193	29	of	of	ADP
ejpam-3420	193	30	y	y	PROPN
ejpam-3420	193	31	to	to	PART
ejpam-3420	193	32	x	x	VERB
ejpam-3420	193	33	is	be	AUX
ejpam-3420	193	34	a	a	DET
ejpam-3420	193	35	covering	covering	NOUN
ejpam-3420	193	36	based	base	VERB
ejpam-3420	193	37	rough	rough	ADJ
ejpam-3420	193	38	continuous	continuous	ADJ
ejpam-3420	193	39	map	map	NOUN
ejpam-3420	193	40	.	.	PUNCT
ejpam-3420	194	1	we	we	PRON
ejpam-3420	194	2	say	say	VERB
ejpam-3420	194	3	that	that	SCONJ
ejpam-3420	194	4	two	two	NUM
ejpam-3420	194	5	covering	covering	NOUN
ejpam-3420	194	6	based	base	VERB
ejpam-3420	194	7	rough	rough	ADJ
ejpam-3420	194	8	topological	topological	ADJ
ejpam-3420	194	9	spaces	space	NOUN
ejpam-3420	194	10	x	x	PUNCT
ejpam-3420	194	11	and	and	CCONJ
ejpam-3420	194	12	y	y	PROPN
ejpam-3420	194	13	are	be	AUX
ejpam-3420	194	14	homeomorphic	homeomorphic	ADJ
ejpam-3420	194	15	if	if	SCONJ
ejpam-3420	194	16	there	there	PRON
ejpam-3420	194	17	exists	exist	VERB
ejpam-3420	194	18	a	a	DET
ejpam-3420	194	19	homeomorphism	homeomorphism	NOUN
ejpam-3420	194	20	of	of	ADP
ejpam-3420	194	21	x	x	PUNCT
ejpam-3420	194	22	onto	onto	ADP
ejpam-3420	194	23	y	y	PROPN
ejpam-3420	194	24	.	.	PUNCT
ejpam-3420	195	1	also	also	ADV
ejpam-3420	195	2	,	,	PUNCT
ejpam-3420	195	3	a	a	DET
ejpam-3420	195	4	covering	covering	NOUN
ejpam-3420	195	5	based	base	VERB
ejpam-3420	195	6	rough	rough	ADJ
ejpam-3420	195	7	homeomorphism	homeomorphism	NOUN
ejpam-3420	195	8	is	be	AUX
ejpam-3420	195	9	always	always	ADV
ejpam-3420	195	10	an	an	DET
ejpam-3420	195	11	isomorphism	isomorphism	NOUN
ejpam-3420	195	12	,	,	PUNCT
ejpam-3420	195	13	but	but	CCONJ
ejpam-3420	195	14	the	the	DET
ejpam-3420	195	15	converse	converse	NOUN
ejpam-3420	195	16	may	may	AUX
ejpam-3420	195	17	be	be	AUX
ejpam-3420	195	18	not	not	PART
ejpam-3420	195	19	true	true	ADJ
ejpam-3420	195	20	.	.	PUNCT
ejpam-3420	196	1	if	if	SCONJ
ejpam-3420	196	2	there	there	PRON
ejpam-3420	196	3	is	be	VERB
ejpam-3420	196	4	a	a	DET
ejpam-3420	196	5	covering	covering	NOUN
ejpam-3420	196	6	based	base	VERB
ejpam-3420	196	7	rough	rough	ADJ
ejpam-3420	196	8	homeomorphism	homeomorphism	NOUN
ejpam-3420	196	9	from	from	ADP
ejpam-3420	196	10	x	x	PUNCT
ejpam-3420	196	11	into	into	ADP
ejpam-3420	196	12	y	y	PROPN
ejpam-3420	196	13	,	,	PUNCT
ejpam-3420	196	14	then	then	ADV
ejpam-3420	196	15	there	there	PRON
ejpam-3420	196	16	is	be	VERB
ejpam-3420	196	17	oneto	oneto	NOUN
ejpam-3420	196	18	-	-	PUNCT
ejpam-3420	196	19	one	one	NUM
ejpam-3420	196	20	corresponding	corresponding	NOUN
ejpam-3420	196	21	between	between	ADP
ejpam-3420	196	22	lower	low	ADJ
ejpam-3420	196	23	(	(	PUNCT
ejpam-3420	196	24	resp	resp	NOUN
ejpam-3420	196	25	.	.	PUNCT
ejpam-3420	197	1	upper	upper	ADJ
ejpam-3420	197	2	)	)	PUNCT
ejpam-3420	197	3	approximations	approximation	NOUN
ejpam-3420	197	4	.	.	PUNCT
ejpam-3420	198	1	then	then	ADV
ejpam-3420	198	2	we	we	PRON
ejpam-3420	198	3	can	can	AUX
ejpam-3420	198	4	say	say	VERB
ejpam-3420	198	5	there	there	PRON
ejpam-3420	198	6	n.	n.	NOUN
ejpam-3420	198	7	alharbi	alharbi	PROPN
ejpam-3420	198	8	,	,	PUNCT
ejpam-3420	198	9	h.	h.	PROPN
ejpam-3420	198	10	aydi	aydi	PROPN
ejpam-3420	198	11	,	,	PUNCT
ejpam-3420	198	12	c.	c.	PROPN
ejpam-3420	198	13	özel	özel	PROPN
ejpam-3420	198	14	/	/	SYM
ejpam-3420	198	15	eur	eur	PROPN
ejpam-3420	198	16	.	.	PUNCT
ejpam-3420	199	1	j.	j.	PROPN
ejpam-3420	199	2	pure	pure	PROPN
ejpam-3420	199	3	appl	appl	PROPN
ejpam-3420	199	4	.	.	PROPN
ejpam-3420	199	5	math	math	PROPN
ejpam-3420	199	6	,	,	PUNCT
ejpam-3420	199	7	12	12	NUM
ejpam-3420	199	8	(	(	PUNCT
ejpam-3420	199	9	2	2	NUM
ejpam-3420	199	10	)	)	PUNCT
ejpam-3420	199	11	(	(	PUNCT
ejpam-3420	199	12	2019	2019	NUM
ejpam-3420	199	13	)	)	PUNCT
ejpam-3420	199	14	,	,	PUNCT
ejpam-3420	199	15	533	533	NUM
ejpam-3420	199	16	-	-	SYM
ejpam-3420	199	17	543	543	NUM
ejpam-3420	199	18	541	541	NUM
ejpam-3420	199	19	exist	exist	VERB
ejpam-3420	199	20	isomorphisms	isomorphisms	PROPN
ejpam-3420	199	21	fl	fl	NOUN
ejpam-3420	199	22	and	and	CCONJ
ejpam-3420	199	23	fu	fu	NOUN
ejpam-3420	199	24	such	such	ADJ
ejpam-3420	199	25	that	that	SCONJ
ejpam-3420	199	26	fl	fl	NOUN
ejpam-3420	199	27	:	:	PUNCT
ejpam-3420	199	28	x	x	X
ejpam-3420	199	29	→	→	SYM
ejpam-3420	199	30	y	y	PROPN
ejpam-3420	199	31	and	and	CCONJ
ejpam-3420	199	32	fu	fu	ADJ
ejpam-3420	199	33	:	:	PUNCT
ejpam-3420	199	34	x	x	X
ejpam-3420	199	35	→	→	SYM
ejpam-3420	199	36	y	y	PROPN
ejpam-3420	199	37	.	.	PUNCT
ejpam-3420	200	1	if	if	SCONJ
ejpam-3420	200	2	there	there	PRON
ejpam-3420	200	3	are	be	VERB
ejpam-3420	200	4	lower	low	ADJ
ejpam-3420	200	5	and	and	CCONJ
ejpam-3420	200	6	upper	upper	ADJ
ejpam-3420	200	7	isomorphisms	isomorphism	NOUN
ejpam-3420	200	8	fl	fl	X
ejpam-3420	200	9	:	:	PUNCT
ejpam-3420	200	10	x	x	X
ejpam-3420	200	11	→	→	SYM
ejpam-3420	200	12	y	y	PROPN
ejpam-3420	200	13	and	and	CCONJ
ejpam-3420	200	14	fu	fu	ADJ
ejpam-3420	200	15	:	:	PUNCT
ejpam-3420	200	16	x	x	X
ejpam-3420	200	17	→	→	SYM
ejpam-3420	200	18	y	y	PROPN
ejpam-3420	200	19	,	,	PUNCT
ejpam-3420	200	20	then	then	ADV
ejpam-3420	200	21	the	the	DET
ejpam-3420	200	22	pair	pair	NOUN
ejpam-3420	200	23	f	f	X
ejpam-3420	200	24	=	=	SYM
ejpam-3420	200	25	(	(	PUNCT
ejpam-3420	200	26	fl	fl	PROPN
ejpam-3420	200	27	,	,	PUNCT
ejpam-3420	200	28	fu	fu	NOUN
ejpam-3420	200	29	)	)	PUNCT
ejpam-3420	200	30	is	be	AUX
ejpam-3420	200	31	not	not	PART
ejpam-3420	200	32	always	always	ADV
ejpam-3420	200	33	a	a	DET
ejpam-3420	200	34	homomorphism	homomorphism	NOUN
ejpam-3420	200	35	from	from	ADP
ejpam-3420	200	36	x	x	PUNCT
ejpam-3420	200	37	to	to	ADP
ejpam-3420	200	38	y	y	PROPN
ejpam-3420	200	39	.	.	PUNCT
ejpam-3420	201	1	the	the	DET
ejpam-3420	201	2	product	product	NOUN
ejpam-3420	201	3	of	of	ADP
ejpam-3420	201	4	covering	cover	VERB
ejpam-3420	201	5	based	base	VERB
ejpam-3420	201	6	rough	rough	ADJ
ejpam-3420	201	7	topologies	topology	NOUN
ejpam-3420	201	8	let	let	VERB
ejpam-3420	201	9	u	u	PRON
ejpam-3420	201	10	and	and	CCONJ
ejpam-3420	201	11	v	v	AUX
ejpam-3420	201	12	be	be	AUX
ejpam-3420	201	13	non	non	ADJ
ejpam-3420	201	14	-	-	ADJ
ejpam-3420	201	15	empty	empty	ADJ
ejpam-3420	201	16	universes	universe	NOUN
ejpam-3420	201	17	and	and	CCONJ
ejpam-3420	201	18	let	let	VERB
ejpam-3420	201	19	c	c	NOUN
ejpam-3420	201	20	and	and	CCONJ
ejpam-3420	201	21	d	d	NOUN
ejpam-3420	201	22	be	be	AUX
ejpam-3420	201	23	a	a	DET
ejpam-3420	201	24	covering	covering	NOUN
ejpam-3420	201	25	of	of	ADP
ejpam-3420	201	26	u	u	NOUN
ejpam-3420	201	27	and	and	CCONJ
ejpam-3420	201	28	v	v	ADP
ejpam-3420	201	29	such	such	ADJ
ejpam-3420	201	30	that	that	SCONJ
ejpam-3420	201	31	c	c	PROPN
ejpam-3420	202	1	and	and	CCONJ
ejpam-3420	202	2	d	d	AUX
ejpam-3420	202	3	satisfy	satisfy	VERB
ejpam-3420	202	4	the	the	DET
ejpam-3420	202	5	approximation	approximation	NOUN
ejpam-3420	202	6	condition	condition	NOUN
ejpam-3420	202	7	.	.	PUNCT
ejpam-3420	203	1	let	let	VERB
ejpam-3420	203	2	(	(	PUNCT
ejpam-3420	203	3	x	x	NOUN
ejpam-3420	203	4	,	,	PUNCT
ejpam-3420	203	5	τc	τc	NOUN
ejpam-3420	203	6	)	)	PUNCT
ejpam-3420	203	7	and	and	CCONJ
ejpam-3420	203	8	(	(	PUNCT
ejpam-3420	203	9	y	y	NOUN
ejpam-3420	203	10	,	,	PUNCT
ejpam-3420	203	11	τd	τd	AUX
ejpam-3420	203	12	)	)	PUNCT
ejpam-3420	203	13	be	be	AUX
ejpam-3420	203	14	covering	cover	VERB
ejpam-3420	203	15	based	base	VERB
ejpam-3420	203	16	rough	rough	ADJ
ejpam-3420	203	17	topologies	topology	NOUN
ejpam-3420	203	18	.	.	PUNCT
ejpam-3420	204	1	consider	consider	VERB
ejpam-3420	204	2	the	the	DET
ejpam-3420	204	3	product	product	NOUN
ejpam-3420	204	4	cartesian	cartesian	ADJ
ejpam-3420	204	5	of	of	ADP
ejpam-3420	204	6	c	c	PROPN
ejpam-3420	204	7	and	and	CCONJ
ejpam-3420	204	8	d	d	PROPN
ejpam-3420	204	9	as	as	SCONJ
ejpam-3420	204	10	follows	follow	VERB
ejpam-3420	204	11	:	:	PUNCT
ejpam-3420	204	12	c	c	NOUN
ejpam-3420	204	13	×d	×d	NOUN
ejpam-3420	204	14	=	=	PUNCT
ejpam-3420	204	15	{	{	PUNCT
ejpam-3420	204	16	l	l	NOUN
ejpam-3420	204	17	×	×	NOUN
ejpam-3420	204	18	k	k	X
ejpam-3420	204	19	:	:	PUNCT
ejpam-3420	205	1	l	l	PUNCT
ejpam-3420	205	2	∈	∈	PROPN
ejpam-3420	205	3	c	c	X
ejpam-3420	205	4	,	,	PUNCT
ejpam-3420	205	5	k	k	PROPN
ejpam-3420	205	6	∈	∈	PROPN
ejpam-3420	205	7	d	d	X
ejpam-3420	205	8	}	}	PUNCT
ejpam-3420	205	9	.	.	PUNCT
ejpam-3420	206	1	let	let	VERB
ejpam-3420	206	2	x	x	PRON
ejpam-3420	206	3	,	,	PUNCT
ejpam-3420	206	4	y	y	PROPN
ejpam-3420	206	5	be	be	VERB
ejpam-3420	206	6	two	two	NUM
ejpam-3420	206	7	representative	representative	ADJ
ejpam-3420	206	8	elements	element	NOUN
ejpam-3420	206	9	of	of	ADP
ejpam-3420	206	10	l	l	NOUN
ejpam-3420	206	11	and	and	CCONJ
ejpam-3420	206	12	x	x	SYM
ejpam-3420	206	13	′	′	NOUN
ejpam-3420	206	14	,	,	PUNCT
ejpam-3420	206	15	y	y	PROPN
ejpam-3420	206	16	′	′	NUM
ejpam-3420	206	17	be	be	VERB
ejpam-3420	206	18	two	two	NUM
ejpam-3420	206	19	representative	representative	ADJ
ejpam-3420	206	20	elements	element	NOUN
ejpam-3420	206	21	of	of	ADP
ejpam-3420	206	22	k.	k.	PROPN
ejpam-3420	206	23	then	then	ADV
ejpam-3420	206	24	(	(	PUNCT
ejpam-3420	206	25	x	x	X
ejpam-3420	206	26	,	,	PUNCT
ejpam-3420	206	27	x	x	NOUN
ejpam-3420	206	28	′	′	NUM
ejpam-3420	206	29	)	)	PUNCT
ejpam-3420	206	30	,	,	PUNCT
ejpam-3420	206	31	(	(	PUNCT
ejpam-3420	206	32	y	y	X
ejpam-3420	206	33	,	,	PUNCT
ejpam-3420	206	34	y	y	PROPN
ejpam-3420	206	35	′	′	NUM
ejpam-3420	206	36	)	)	PUNCT
ejpam-3420	206	37	,	,	PUNCT
ejpam-3420	206	38	(	(	PUNCT
ejpam-3420	206	39	x	x	X
ejpam-3420	206	40	,	,	PUNCT
ejpam-3420	206	41	y	y	PROPN
ejpam-3420	206	42	′	′	NUM
ejpam-3420	206	43	)	)	PUNCT
ejpam-3420	206	44	,	,	PUNCT
ejpam-3420	206	45	(	(	PUNCT
ejpam-3420	206	46	y	y	NOUN
ejpam-3420	206	47	,	,	PUNCT
ejpam-3420	206	48	x	x	NOUN
ejpam-3420	206	49	′	′	NUM
ejpam-3420	206	50	)	)	PUNCT
ejpam-3420	206	51	are	be	AUX
ejpam-3420	206	52	representative	representative	ADJ
ejpam-3420	206	53	elements	element	NOUN
ejpam-3420	206	54	of	of	ADP
ejpam-3420	206	55	l	l	NOUN
ejpam-3420	206	56	×	×	PROPN
ejpam-3420	206	57	k.	k.	NOUN
ejpam-3420	207	1	so	so	SCONJ
ejpam-3420	207	2	that	that	SCONJ
ejpam-3420	207	3	,	,	PUNCT
ejpam-3420	207	4	c	c	PROPN
ejpam-3420	207	5	×d	×d	NOUN
ejpam-3420	207	6	satisfies	satisfy	VERB
ejpam-3420	207	7	the	the	DET
ejpam-3420	207	8	approximation	approximation	NOUN
ejpam-3420	207	9	condition	condition	NOUN
ejpam-3420	207	10	.	.	PUNCT
ejpam-3420	208	1	in	in	ADP
ejpam-3420	208	2	addition	addition	NOUN
ejpam-3420	208	3	,	,	PUNCT
ejpam-3420	208	4	assume	assume	VERB
ejpam-3420	208	5	that	that	SCONJ
ejpam-3420	208	6	for	for	ADP
ejpam-3420	208	7	every	every	DET
ejpam-3420	208	8	ac	ac	PROPN
ejpam-3420	208	9	×ad	×ad	PROPN
ejpam-3420	208	10	,	,	PUNCT
ejpam-3420	208	11	bc	bc	PROPN
ejpam-3420	208	12	×bd	×bd	VERB
ejpam-3420	208	13	⊆	⊆	NUM
ejpam-3420	208	14	c	c	NOUN
ejpam-3420	208	15	×d	×d	NOUN
ejpam-3420	208	16	such	such	ADJ
ejpam-3420	208	17	that	that	SCONJ
ejpam-3420	208	18	ac	ac	PROPN
ejpam-3420	208	19	×ad	×ad	PROPN
ejpam-3420	208	20	⊆	⊆	NUM
ejpam-3420	208	21	bc×bd	bc×bd	PROPN
ejpam-3420	208	22	,	,	PUNCT
ejpam-3420	208	23	there	there	PRON
ejpam-3420	208	24	exists	exist	VERB
ejpam-3420	208	25	a	a	DET
ejpam-3420	208	26	set	set	NOUN
ejpam-3420	208	27	x×y	x×y	NOUN
ejpam-3420	208	28	⊆	⊆	NUM
ejpam-3420	208	29	u×v	u×v	PROPN
ejpam-3420	208	30	with	with	ADP
ejpam-3420	208	31	ac×ad	ac×ad	PROPN
ejpam-3420	208	32	=	=	PUNCT
ejpam-3420	208	33	x	x	SYM
ejpam-3420	208	34	×	×	PROPN
ejpam-3420	208	35	y	y	PROPN
ejpam-3420	208	36	and	and	CCONJ
ejpam-3420	208	37	bc×bd	bc×bd	PROPN
ejpam-3420	208	38	=	=	PUNCT
ejpam-3420	208	39	x	x	SYM
ejpam-3420	208	40	×	×	PROPN
ejpam-3420	208	41	y	y	PROPN
ejpam-3420	208	42	.	.	PUNCT
ejpam-3420	209	1	now	now	ADV
ejpam-3420	209	2	,	,	PUNCT
ejpam-3420	209	3	suppose	suppose	VERB
ejpam-3420	209	4	that	that	SCONJ
ejpam-3420	209	5	x	x	PRON
ejpam-3420	209	6	×	×	NOUN
ejpam-3420	209	7	y	y	PROPN
ejpam-3420	209	8	⊆	⊆	NUM
ejpam-3420	209	9	u	u	NOUN
ejpam-3420	209	10	×	×	PROPN
ejpam-3420	209	11	v	v	NOUN
ejpam-3420	209	12	.	.	PUNCT
ejpam-3420	210	1	then	then	ADV
ejpam-3420	210	2	define	define	VERB
ejpam-3420	210	3	(	(	PUNCT
ejpam-3420	210	4	i	i	NOUN
ejpam-3420	210	5	)	)	PUNCT
ejpam-3420	210	6	x	x	SYM
ejpam-3420	210	7	×	×	NOUN
ejpam-3420	210	8	y	y	NOUN
ejpam-3420	210	9	=	=	PUNCT
ejpam-3420	210	10	{	{	PUNCT
ejpam-3420	210	11	l	l	NOUN
ejpam-3420	210	12	×	×	NOUN
ejpam-3420	210	13	k	k	PROPN
ejpam-3420	210	14	∈	∈	PROPN
ejpam-3420	210	15	c	c	PROPN
ejpam-3420	210	16	×d	×d	NOUN
ejpam-3420	210	17	:	:	PUNCT
ejpam-3420	211	1	l	l	PUNCT
ejpam-3420	211	2	×	×	NOUN
ejpam-3420	211	3	k	k	NOUN
ejpam-3420	211	4	⊆	⊆	NUM
ejpam-3420	211	5	x	x	SYM
ejpam-3420	211	6	×	×	PROPN
ejpam-3420	211	7	y	y	PROPN
ejpam-3420	211	8	}	}	PUNCT
ejpam-3420	211	9	;	;	PUNCT
ejpam-3420	211	10	(	(	PUNCT
ejpam-3420	211	11	ii	ii	NOUN
ejpam-3420	211	12	)	)	PUNCT
ejpam-3420	211	13	x	x	SYM
ejpam-3420	212	1	×	×	NOUN
ejpam-3420	212	2	y	y	NOUN
ejpam-3420	212	3	=	=	PUNCT
ejpam-3420	212	4	{	{	PUNCT
ejpam-3420	212	5	l	l	NOUN
ejpam-3420	212	6	×	×	NOUN
ejpam-3420	212	7	k	k	PROPN
ejpam-3420	212	8	∈	∈	PROPN
ejpam-3420	212	9	x	x	SYM
ejpam-3420	212	10	×	×	NOUN
ejpam-3420	212	11	y	y	NOUN
ejpam-3420	212	12	:	:	PUNCT
ejpam-3420	212	13	l	l	PUNCT
ejpam-3420	212	14	×	×	NOUN
ejpam-3420	212	15	k	k	X
ejpam-3420	212	16	∧x	∧x	PROPN
ejpam-3420	212	17	×	×	PROPN
ejpam-3420	212	18	y	y	PROPN
ejpam-3420	212	19	6=	6=	PROPN
ejpam-3420	212	20	φ	φ	NUM
ejpam-3420	212	21	}	}	PUNCT
ejpam-3420	212	22	;	;	PUNCT
ejpam-3420	212	23	(	(	PUNCT
ejpam-3420	212	24	iii	iii	NOUN
ejpam-3420	212	25	)	)	PUNCT
ejpam-3420	212	26	bn(x	bn(x	PUNCT
ejpam-3420	212	27	×	×	PROPN
ejpam-3420	212	28	y	y	NOUN
ejpam-3420	212	29	)	)	PUNCT
ejpam-3420	212	30	=	=	PUNCT
ejpam-3420	213	1	x	x	SYM
ejpam-3420	213	2	×	×	NOUN
ejpam-3420	213	3	y	y	PROPN
ejpam-3420	213	4	−x	−x	NOUN
ejpam-3420	213	5	×	×	PROPN
ejpam-3420	213	6	y	y	PROPN
ejpam-3420	213	7	.	.	PUNCT
ejpam-3420	214	1	let	let	VERB
ejpam-3420	214	2	a	a	DET
ejpam-3420	214	3	,	,	PUNCT
ejpam-3420	214	4	b	b	PROPN
ejpam-3420	214	5	⊆	⊆	NUM
ejpam-3420	214	6	u	u	NOUN
ejpam-3420	214	7	×	×	NOUN
ejpam-3420	214	8	v	v	NOUN
ejpam-3420	214	9	.	.	PUNCT
ejpam-3420	215	1	we	we	PRON
ejpam-3420	215	2	can	can	AUX
ejpam-3420	215	3	define	define	VERB
ejpam-3420	215	4	the	the	DET
ejpam-3420	215	5	union	union	NOUN
ejpam-3420	215	6	,	,	PUNCT
ejpam-3420	215	7	the	the	DET
ejpam-3420	215	8	intersection	intersection	NOUN
ejpam-3420	215	9	and	and	CCONJ
ejpam-3420	215	10	the	the	DET
ejpam-3420	215	11	complement	complement	NOUN
ejpam-3420	215	12	of	of	ADP
ejpam-3420	215	13	two	two	NUM
ejpam-3420	215	14	sets	set	NOUN
ejpam-3420	215	15	in	in	ADP
ejpam-3420	215	16	u	u	NOUN
ejpam-3420	215	17	×	×	NOUN
ejpam-3420	215	18	v	v	NOUN
ejpam-3420	215	19	as	as	SCONJ
ejpam-3420	215	20	follows	follow	VERB
ejpam-3420	215	21	:	:	PUNCT
ejpam-3420	215	22	(	(	PUNCT
ejpam-3420	215	23	i	i	NOUN
ejpam-3420	215	24	)	)	PUNCT
ejpam-3420	215	25	a	a	DET
ejpam-3420	215	26	∨b	∨b	NOUN
ejpam-3420	215	27	=	=	PUNCT
ejpam-3420	215	28	(	(	PUNCT
ejpam-3420	215	29	a	a	DET
ejpam-3420	215	30	∨b	∨b	PROPN
ejpam-3420	215	31	,	,	PUNCT
ejpam-3420	215	32	a	a	DET
ejpam-3420	215	33	∨b	∨b	NOUN
ejpam-3420	215	34	)	)	PUNCT
ejpam-3420	215	35	;	;	PUNCT
ejpam-3420	215	36	(	(	PUNCT
ejpam-3420	215	37	ii	ii	NOUN
ejpam-3420	215	38	)	)	PUNCT
ejpam-3420	215	39	a	a	DET
ejpam-3420	215	40	∧b	∧b	NOUN
ejpam-3420	215	41	=	=	SYM
ejpam-3420	215	42	(	(	PUNCT
ejpam-3420	215	43	a	a	DET
ejpam-3420	215	44	∧b	∧b	NOUN
ejpam-3420	215	45	,	,	PUNCT
ejpam-3420	215	46	a	a	DET
ejpam-3420	215	47	∧b	∧b	NOUN
ejpam-3420	215	48	)	)	PUNCT
ejpam-3420	215	49	;	;	PUNCT
ejpam-3420	215	50	(	(	PUNCT
ejpam-3420	215	51	iii	iii	X
ejpam-3420	215	52	)	)	PUNCT
ejpam-3420	215	53	a	a	DET
ejpam-3420	215	54	′	′	NOUN
ejpam-3420	216	1	=	=	SYM
ejpam-3420	216	2	(	(	PUNCT
ejpam-3420	216	3	c	c	NOUN
ejpam-3420	216	4	×d	×d	NOUN
ejpam-3420	216	5	\a	\a	ADJ
ejpam-3420	216	6	,	,	PUNCT
ejpam-3420	216	7	c	c	NOUN
ejpam-3420	216	8	×d	×d	NOUN
ejpam-3420	216	9	\a	\a	ADJ
ejpam-3420	216	10	)	)	PUNCT
ejpam-3420	216	11	.	.	PUNCT
ejpam-3420	217	1	definition	definition	NOUN
ejpam-3420	217	2	9	9	NUM
ejpam-3420	217	3	.	.	PUNCT
ejpam-3420	218	1	let	let	VERB
ejpam-3420	218	2	x	x	SYM
ejpam-3420	218	3	⊆	⊆	NUM
ejpam-3420	218	4	u	u	NOUN
ejpam-3420	218	5	and	and	CCONJ
ejpam-3420	218	6	y	y	PROPN
ejpam-3420	218	7	⊆	⊆	PROPN
ejpam-3420	218	8	v	v	ADP
ejpam-3420	218	9	such	such	ADJ
ejpam-3420	218	10	that	that	SCONJ
ejpam-3420	218	11	(	(	PUNCT
ejpam-3420	218	12	x	x	NOUN
ejpam-3420	218	13	,	,	PUNCT
ejpam-3420	218	14	τc	τc	NOUN
ejpam-3420	218	15	)	)	PUNCT
ejpam-3420	218	16	and	and	CCONJ
ejpam-3420	218	17	(	(	PUNCT
ejpam-3420	218	18	y	y	NOUN
ejpam-3420	218	19	,	,	PUNCT
ejpam-3420	218	20	τd	τd	ADJ
ejpam-3420	218	21	)	)	PUNCT
ejpam-3420	218	22	are	be	AUX
ejpam-3420	218	23	covering	cover	VERB
ejpam-3420	218	24	based	base	VERB
ejpam-3420	218	25	rough	rough	ADJ
ejpam-3420	218	26	topological	topological	ADJ
ejpam-3420	218	27	spaces	space	NOUN
ejpam-3420	218	28	on	on	ADP
ejpam-3420	218	29	the	the	DET
ejpam-3420	218	30	coverings	covering	NOUN
ejpam-3420	218	31	c	c	PROPN
ejpam-3420	218	32	and	and	CCONJ
ejpam-3420	218	33	d	d	NOUN
ejpam-3420	218	34	,	,	PUNCT
ejpam-3420	218	35	respectively	respectively	ADV
ejpam-3420	218	36	.	.	PUNCT
ejpam-3420	219	1	then	then	ADV
ejpam-3420	219	2	the	the	DET
ejpam-3420	219	3	set	set	NOUN
ejpam-3420	219	4	{	{	PUNCT
ejpam-3420	219	5	a	a	DET
ejpam-3420	219	6	×	×	PROPN
ejpam-3420	219	7	b	b	NOUN
ejpam-3420	219	8	:	:	PUNCT
ejpam-3420	219	9	a	a	DET
ejpam-3420	219	10	∈	∈	NOUN
ejpam-3420	219	11	τc	τc	X
ejpam-3420	219	12	,	,	PUNCT
ejpam-3420	219	13	b	b	X
ejpam-3420	219	14	∈	∈	PROPN
ejpam-3420	219	15	τd	τd	ADP
ejpam-3420	219	16	}	}	PUNCT
ejpam-3420	219	17	is	be	AUX
ejpam-3420	219	18	a	a	DET
ejpam-3420	219	19	base	base	NOUN
ejpam-3420	219	20	for	for	ADP
ejpam-3420	219	21	the	the	DET
ejpam-3420	219	22	product	product	NOUN
ejpam-3420	219	23	covering	cover	VERB
ejpam-3420	219	24	based	base	VERB
ejpam-3420	219	25	rough	rough	ADJ
ejpam-3420	219	26	topology	topology	NOUN
ejpam-3420	219	27	x	x	NOUN
ejpam-3420	219	28	×	×	PROPN
ejpam-3420	219	29	y	y	PROPN
ejpam-3420	219	30	on	on	ADP
ejpam-3420	219	31	c	c	PROPN
ejpam-3420	219	32	×d	×d	NOUN
ejpam-3420	219	33	.	.	PUNCT
ejpam-3420	220	1	now	now	ADV
ejpam-3420	220	2	,	,	PUNCT
ejpam-3420	220	3	let	let	VERB
ejpam-3420	220	4	w	w	NOUN
ejpam-3420	220	5	be	be	AUX
ejpam-3420	220	6	a	a	DET
ejpam-3420	220	7	covering	covering	NOUN
ejpam-3420	220	8	based	base	VERB
ejpam-3420	220	9	rough	rough	ADJ
ejpam-3420	220	10	open	open	ADJ
ejpam-3420	220	11	set	set	NOUN
ejpam-3420	220	12	in	in	ADP
ejpam-3420	220	13	the	the	DET
ejpam-3420	220	14	product	product	NOUN
ejpam-3420	220	15	topology	topology	NOUN
ejpam-3420	220	16	τx×y	τx×y	NOUN
ejpam-3420	220	17	.	.	PUNCT
ejpam-3420	221	1	consider	consider	VERB
ejpam-3420	221	2	the	the	DET
ejpam-3420	221	3	pair	pair	NOUN
ejpam-3420	221	4	(	(	PUNCT
ejpam-3420	221	5	x1	x1	PROPN
ejpam-3420	221	6	,	,	PUNCT
ejpam-3420	221	7	x2	x2	PROPN
ejpam-3420	221	8	)	)	PUNCT
ejpam-3420	221	9	∈	∈	PROPN
ejpam-3420	221	10	w	w	NOUN
ejpam-3420	221	11	,	,	PUNCT
ejpam-3420	221	12	where	where	SCONJ
ejpam-3420	221	13	x1	x1	X
ejpam-3420	221	14	,	,	PUNCT
ejpam-3420	221	15	x2	x2	PROPN
ejpam-3420	221	16	are	be	AUX
ejpam-3420	221	17	exact	exact	ADJ
ejpam-3420	221	18	elements	element	NOUN
ejpam-3420	221	19	in	in	ADP
ejpam-3420	221	20	x	x	PUNCT
ejpam-3420	221	21	and	and	CCONJ
ejpam-3420	221	22	y	y	PROPN
ejpam-3420	221	23	,	,	PUNCT
ejpam-3420	221	24	respectively	respectively	ADV
ejpam-3420	221	25	.	.	PUNCT
ejpam-3420	222	1	so	so	ADV
ejpam-3420	222	2	there	there	PRON
ejpam-3420	222	3	exist	exist	VERB
ejpam-3420	222	4	wx	wx	PROPN
ejpam-3420	222	5	∈	∈	PROPN
ejpam-3420	222	6	τx	τx	X
ejpam-3420	222	7	and	and	CCONJ
ejpam-3420	222	8	wy	wy	PROPN
ejpam-3420	222	9	∈	∈	PROPN
ejpam-3420	222	10	τy	τy	PRON
ejpam-3420	222	11	such	such	ADJ
ejpam-3420	222	12	that	that	SCONJ
ejpam-3420	222	13	x1	x1	PROPN
ejpam-3420	222	14	∈	∈	PROPN
ejpam-3420	222	15	wx	wx	PROPN
ejpam-3420	222	16	and	and	CCONJ
ejpam-3420	222	17	x2	x2	PROPN
ejpam-3420	222	18	∈	∈	PROPN
ejpam-3420	222	19	wy	wy	PROPN
ejpam-3420	222	20	.	.	PUNCT
ejpam-3420	223	1	this	this	PRON
ejpam-3420	223	2	implies	imply	VERB
ejpam-3420	223	3	that	that	SCONJ
ejpam-3420	223	4	x1	x1	PROPN
ejpam-3420	223	5	6=	6=	PROPN
ejpam-3420	223	6	φ	φ	PROPN
ejpam-3420	223	7	6=	6=	PROPN
ejpam-3420	224	1	x2	x2	PROPN
ejpam-3420	224	2	and	and	CCONJ
ejpam-3420	224	3	x1	x1	PROPN
ejpam-3420	224	4	⊆c	⊆c	NOUN
ejpam-3420	224	5	wc	wc	PROPN
ejpam-3420	224	6	,	,	PUNCT
ejpam-3420	224	7	x2	x2	PROPN
ejpam-3420	224	8	⊆d	⊆d	NOUN
ejpam-3420	224	9	wy	wy	PROPN
ejpam-3420	224	10	.	.	PUNCT
ejpam-3420	225	1	then	then	ADV
ejpam-3420	225	2	∃kc	∃kc	NOUN
ejpam-3420	225	3	∈	∈	PROPN
ejpam-3420	225	4	c	c	X
ejpam-3420	225	5	,	,	PUNCT
ejpam-3420	225	6	kd	kd	PROPN
ejpam-3420	225	7	∈	∈	PROPN
ejpam-3420	225	8	d	d	PROPN
ejpam-3420	225	9	with	with	ADP
ejpam-3420	225	10	x1	x1	PROPN
ejpam-3420	225	11	=	=	SYM
ejpam-3420	225	12	{	{	PUNCT
ejpam-3420	225	13	kc	kc	PROPN
ejpam-3420	225	14	}	}	PUNCT
ejpam-3420	225	15	and	and	CCONJ
ejpam-3420	225	16	x2	x2	PROPN
ejpam-3420	225	17	=	=	PRON
ejpam-3420	225	18	{	{	PUNCT
ejpam-3420	225	19	kd	kd	NOUN
ejpam-3420	225	20	}	}	PUNCT
ejpam-3420	225	21	.	.	PUNCT
ejpam-3420	226	1	hence	hence	ADV
ejpam-3420	226	2	(	(	PUNCT
ejpam-3420	226	3	x1	x1	PROPN
ejpam-3420	226	4	,	,	PUNCT
ejpam-3420	226	5	x2	x2	PROPN
ejpam-3420	226	6	)	)	PUNCT
ejpam-3420	226	7	6=	6=	ADP
ejpam-3420	226	8	∅	∅	NOUN
ejpam-3420	226	9	and	and	CCONJ
ejpam-3420	226	10	x1×x2	x1×x2	PROPN
ejpam-3420	227	1	⊆c×d	⊆c×d	PROPN
ejpam-3420	227	2	w	w	NOUN
ejpam-3420	227	3	.	.	PUNCT
ejpam-3420	228	1	there	there	PRON
ejpam-3420	228	2	exists	exist	VERB
ejpam-3420	228	3	kc	kc	PROPN
ejpam-3420	228	4	×kd	×kd	PROPN
ejpam-3420	228	5	∈	∈	PROPN
ejpam-3420	228	6	c×d	c×d	PROPN
ejpam-3420	228	7	with	with	ADP
ejpam-3420	228	8	w	w	NOUN
ejpam-3420	228	9	=	=	SYM
ejpam-3420	228	10	{	{	PUNCT
ejpam-3420	228	11	kc	kc	PROPN
ejpam-3420	228	12	×kd	×kd	PROPN
ejpam-3420	228	13	}	}	PUNCT
ejpam-3420	228	14	.	.	PUNCT
ejpam-3420	229	1	we	we	PRON
ejpam-3420	229	2	can	can	AUX
ejpam-3420	229	3	then	then	ADV
ejpam-3420	229	4	write	write	VERB
ejpam-3420	229	5	(	(	PUNCT
ejpam-3420	229	6	x1	x1	PROPN
ejpam-3420	229	7	,	,	PUNCT
ejpam-3420	229	8	x2	x2	ADJ
ejpam-3420	229	9	)	)	PUNCT
ejpam-3420	229	10	∈wx	∈wx	PROPN
ejpam-3420	229	11	×wy	×wy	PROPN
ejpam-3420	229	12	⊆w	⊆w	NOUN
ejpam-3420	229	13	,	,	PUNCT
ejpam-3420	229	14	where	where	SCONJ
ejpam-3420	229	15	wx	wx	PROPN
ejpam-3420	229	16	×wy	×wy	PROPN
ejpam-3420	229	17	is	be	AUX
ejpam-3420	229	18	a	a	DET
ejpam-3420	229	19	covering	covering	NOUN
ejpam-3420	229	20	based	base	VERB
ejpam-3420	229	21	rough	rough	ADJ
ejpam-3420	229	22	basic	basic	ADJ
ejpam-3420	229	23	open	open	ADJ
ejpam-3420	229	24	set	set	NOUN
ejpam-3420	229	25	.	.	PUNCT
ejpam-3420	230	1	n.	n.	PROPN
ejpam-3420	230	2	alharbi	alharbi	PROPN
ejpam-3420	230	3	,	,	PUNCT
ejpam-3420	230	4	h.	h.	PROPN
ejpam-3420	230	5	aydi	aydi	PROPN
ejpam-3420	230	6	,	,	PUNCT
ejpam-3420	230	7	c.	c.	PROPN
ejpam-3420	230	8	özel	özel	PROPN
ejpam-3420	230	9	/	/	SYM
ejpam-3420	230	10	eur	eur	PROPN
ejpam-3420	230	11	.	.	PUNCT
ejpam-3420	231	1	j.	j.	PROPN
ejpam-3420	231	2	pure	pure	PROPN
ejpam-3420	231	3	appl	appl	PROPN
ejpam-3420	231	4	.	.	PROPN
ejpam-3420	231	5	math	math	PROPN
ejpam-3420	231	6	,	,	PUNCT
ejpam-3420	231	7	12	12	NUM
ejpam-3420	231	8	(	(	PUNCT
ejpam-3420	231	9	2	2	NUM
ejpam-3420	231	10	)	)	PUNCT
ejpam-3420	231	11	(	(	PUNCT
ejpam-3420	231	12	2019	2019	NUM
ejpam-3420	231	13	)	)	PUNCT
ejpam-3420	231	14	,	,	PUNCT
ejpam-3420	231	15	533	533	NUM
ejpam-3420	231	16	-	-	SYM
ejpam-3420	231	17	543	543	NUM
ejpam-3420	231	18	542	542	NUM
ejpam-3420	231	19	remark	remark	NOUN
ejpam-3420	231	20	2	2	NUM
ejpam-3420	231	21	.	.	PUNCT
ejpam-3420	231	22	consider	consider	VERB
ejpam-3420	231	23	the	the	DET
ejpam-3420	231	24	product	product	NOUN
ejpam-3420	231	25	covering	cover	VERB
ejpam-3420	231	26	based	base	VERB
ejpam-3420	231	27	rough	rough	ADJ
ejpam-3420	231	28	topology	topology	NOUN
ejpam-3420	231	29	x	x	NOUN
ejpam-3420	231	30	×	×	PROPN
ejpam-3420	231	31	y	y	PROPN
ejpam-3420	231	32	on	on	ADP
ejpam-3420	231	33	c	c	PROPN
ejpam-3420	231	34	×	×	PROPN
ejpam-3420	231	35	d.	d.	PROPN
ejpam-3420	231	36	let	let	VERB
ejpam-3420	231	37	a	a	DET
ejpam-3420	231	38	⊂	⊂	PROPN
ejpam-3420	231	39	x	x	X
ejpam-3420	231	40	and	and	CCONJ
ejpam-3420	231	41	b	b	PROPN
ejpam-3420	231	42	⊂	⊂	PROPN
ejpam-3420	231	43	y	y	PROPN
ejpam-3420	231	44	.	.	PUNCT
ejpam-3420	232	1	then	then	ADV
ejpam-3420	232	2	(	(	PUNCT
ejpam-3420	232	3	i	i	NOUN
ejpam-3420	232	4	)	)	PUNCT
ejpam-3420	232	5	cint(a×b	cint(a×b	PROPN
ejpam-3420	232	6	)	)	PUNCT
ejpam-3420	232	7	=	=	NUM
ejpam-3420	232	8	cint(a)×	cint(a)×	NOUN
ejpam-3420	232	9	cint(b	cint(b	NOUN
ejpam-3420	232	10	)	)	PUNCT
ejpam-3420	232	11	;	;	PUNCT
ejpam-3420	232	12	(	(	PUNCT
ejpam-3420	232	13	ii	ii	NOUN
ejpam-3420	232	14	)	)	PUNCT
ejpam-3420	232	15	ccl(a×b	ccl(a×b	PROPN
ejpam-3420	232	16	)	)	PUNCT
ejpam-3420	233	1	=	=	PUNCT
ejpam-3420	234	1	ccl(a)×	ccl(a)×	X
ejpam-3420	234	2	ccl(b	ccl(b	PROPN
ejpam-3420	234	3	)	)	PUNCT
ejpam-3420	234	4	.	.	PUNCT
ejpam-3420	235	1	properties	property	NOUN
ejpam-3420	235	2	of	of	ADP
ejpam-3420	235	3	covering	cover	VERB
ejpam-3420	235	4	based	base	VERB
ejpam-3420	235	5	rough	rough	ADJ
ejpam-3420	235	6	topology	topology	NOUN
ejpam-3420	235	7	let	let	VERB
ejpam-3420	235	8	(	(	PUNCT
ejpam-3420	235	9	u	u	NOUN
ejpam-3420	235	10	,	,	PUNCT
ejpam-3420	235	11	c	c	NOUN
ejpam-3420	235	12	)	)	PUNCT
ejpam-3420	235	13	be	be	AUX
ejpam-3420	235	14	a	a	DET
ejpam-3420	235	15	covering	covering	NOUN
ejpam-3420	235	16	approximation	approximation	NOUN
ejpam-3420	235	17	space	space	NOUN
ejpam-3420	235	18	.	.	PUNCT
ejpam-3420	236	1	suppose	suppose	VERB
ejpam-3420	236	2	that	that	SCONJ
ejpam-3420	236	3	x	x	PROPN
ejpam-3420	236	4	and	and	CCONJ
ejpam-3420	236	5	y	y	PROPN
ejpam-3420	236	6	are	be	AUX
ejpam-3420	236	7	rough	rough	ADJ
ejpam-3420	236	8	subsets	subset	NOUN
ejpam-3420	236	9	of	of	ADP
ejpam-3420	236	10	u	u	PRON
ejpam-3420	236	11	such	such	ADJ
ejpam-3420	236	12	that	that	SCONJ
ejpam-3420	236	13	y	y	PROPN
ejpam-3420	236	14	⊆	⊆	NUM
ejpam-3420	236	15	x.	x.	NOUN
ejpam-3420	236	16	consider	consider	VERB
ejpam-3420	236	17	the	the	DET
ejpam-3420	236	18	covering	covering	NOUN
ejpam-3420	236	19	based	base	VERB
ejpam-3420	236	20	rough	rough	ADJ
ejpam-3420	236	21	topology	topology	NOUN
ejpam-3420	236	22	(	(	PUNCT
ejpam-3420	236	23	x	x	X
ejpam-3420	236	24	,	,	PUNCT
ejpam-3420	236	25	τ	τ	PROPN
ejpam-3420	236	26	)	)	PUNCT
ejpam-3420	236	27	.	.	PUNCT
ejpam-3420	237	1	we	we	PRON
ejpam-3420	237	2	generalize	generalize	VERB
ejpam-3420	237	3	the	the	DET
ejpam-3420	237	4	definition	definition	NOUN
ejpam-3420	237	5	of	of	ADP
ejpam-3420	237	6	density	density	NOUN
ejpam-3420	237	7	from	from	ADP
ejpam-3420	237	8	classical	classical	ADJ
ejpam-3420	237	9	rough	rough	ADJ
ejpam-3420	237	10	topology	topology	NOUN
ejpam-3420	237	11	to	to	ADP
ejpam-3420	237	12	covering	cover	VERB
ejpam-3420	237	13	based	base	VERB
ejpam-3420	237	14	rough	rough	ADJ
ejpam-3420	237	15	topology	topology	NOUN
ejpam-3420	237	16	.	.	PUNCT
ejpam-3420	238	1	definition	definition	NOUN
ejpam-3420	238	2	10	10	NUM
ejpam-3420	238	3	.	.	PUNCT
ejpam-3420	239	1	the	the	DET
ejpam-3420	239	2	rough	rough	ADJ
ejpam-3420	239	3	subset	subset	NOUN
ejpam-3420	239	4	y	y	PROPN
ejpam-3420	239	5	of	of	ADP
ejpam-3420	239	6	x	x	PRON
ejpam-3420	239	7	is	be	AUX
ejpam-3420	239	8	dense	dense	ADJ
ejpam-3420	239	9	in	in	ADP
ejpam-3420	239	10	the	the	DET
ejpam-3420	239	11	covering	covering	NOUN
ejpam-3420	239	12	based	base	VERB
ejpam-3420	239	13	rough	rough	ADJ
ejpam-3420	239	14	topology	topology	NOUN
ejpam-3420	239	15	(	(	PUNCT
ejpam-3420	239	16	x	x	X
ejpam-3420	239	17	,	,	PUNCT
ejpam-3420	239	18	τ	τ	PROPN
ejpam-3420	239	19	)	)	PUNCT
ejpam-3420	239	20	,	,	PUNCT
ejpam-3420	239	21	if	if	SCONJ
ejpam-3420	239	22	the	the	DET
ejpam-3420	239	23	upper	upper	ADJ
ejpam-3420	239	24	approximation	approximation	NOUN
ejpam-3420	239	25	of	of	ADP
ejpam-3420	239	26	y	y	PROPN
ejpam-3420	239	27	is	be	AUX
ejpam-3420	239	28	equal	equal	ADJ
ejpam-3420	239	29	to	to	ADP
ejpam-3420	239	30	the	the	DET
ejpam-3420	239	31	upper	upper	ADJ
ejpam-3420	239	32	approximation	approximation	NOUN
ejpam-3420	239	33	of	of	ADP
ejpam-3420	239	34	x	x	PRON
ejpam-3420	239	35	,	,	PUNCT
ejpam-3420	239	36	i.e.	i.e.	X
ejpam-3420	239	37	,	,	PUNCT
ejpam-3420	239	38	x	x	SYM
ejpam-3420	239	39	=	=	SYM
ejpam-3420	239	40	y	y	PROPN
ejpam-3420	239	41	.	.	PUNCT
ejpam-3420	240	1	generating	generate	VERB
ejpam-3420	240	2	covering	covering	NOUN
ejpam-3420	240	3	based	base	VERB
ejpam-3420	240	4	rough	rough	ADJ
ejpam-3420	240	5	topology	topology	NOUN
ejpam-3420	240	6	let	let	VERB
ejpam-3420	240	7	x	x	PRON
ejpam-3420	240	8	be	be	AUX
ejpam-3420	240	9	a	a	DET
ejpam-3420	240	10	subset	subset	NOUN
ejpam-3420	240	11	of	of	ADP
ejpam-3420	240	12	the	the	DET
ejpam-3420	240	13	universe	universe	ADJ
ejpam-3420	240	14	u	u	NOUN
ejpam-3420	240	15	such	such	ADJ
ejpam-3420	240	16	that	that	SCONJ
ejpam-3420	240	17	the	the	DET
ejpam-3420	240	18	covering	covering	NOUN
ejpam-3420	240	19	based	base	VERB
ejpam-3420	240	20	rough	rough	ADJ
ejpam-3420	240	21	set	set	NOUN
ejpam-3420	240	22	of	of	ADP
ejpam-3420	240	23	x	x	PUNCT
ejpam-3420	240	24	is	be	AUX
ejpam-3420	240	25	xc	xc	PROPN
ejpam-3420	240	26	=	=	PUNCT
ejpam-3420	240	27	(	(	PUNCT
ejpam-3420	240	28	x	x	X
ejpam-3420	240	29	,	,	PUNCT
ejpam-3420	240	30	x	x	NOUN
ejpam-3420	240	31	)	)	PUNCT
ejpam-3420	240	32	.	.	PUNCT
ejpam-3420	241	1	let	let	VERB
ejpam-3420	241	2	o	o	NOUN
ejpam-3420	241	3	be	be	AUX
ejpam-3420	241	4	a	a	DET
ejpam-3420	241	5	family	family	NOUN
ejpam-3420	241	6	of	of	ADP
ejpam-3420	241	7	subsets	subset	NOUN
ejpam-3420	241	8	of	of	ADP
ejpam-3420	241	9	xc	xc	PROPN
ejpam-3420	241	10	that	that	PRON
ejpam-3420	241	11	satisfies	satisfy	VERB
ejpam-3420	241	12	the	the	DET
ejpam-3420	241	13	base	base	NOUN
ejpam-3420	241	14	conditions	condition	NOUN
ejpam-3420	241	15	.	.	PUNCT
ejpam-3420	242	1	definition	definition	NOUN
ejpam-3420	242	2	11	11	NUM
ejpam-3420	242	3	.	.	PUNCT
ejpam-3420	243	1	the	the	DET
ejpam-3420	243	2	set	set	NOUN
ejpam-3420	243	3	τ	τ	PROPN
ejpam-3420	243	4	consisting	consist	VERB
ejpam-3420	243	5	of	of	ADP
ejpam-3420	243	6	all	all	DET
ejpam-3420	243	7	possible	possible	ADJ
ejpam-3420	243	8	unions	union	NOUN
ejpam-3420	243	9	of	of	ADP
ejpam-3420	243	10	members	member	NOUN
ejpam-3420	243	11	of	of	ADP
ejpam-3420	243	12	o	o	PROPN
ejpam-3420	243	13	together	together	ADV
ejpam-3420	243	14	with	with	ADP
ejpam-3420	243	15	∅c	∅c	NOUN
ejpam-3420	243	16	=	=	SYM
ejpam-3420	243	17	(	(	PUNCT
ejpam-3420	243	18	∅	∅	NOUN
ejpam-3420	243	19	,	,	PUNCT
ejpam-3420	243	20	∅	∅	NOUN
ejpam-3420	243	21	)	)	PUNCT
ejpam-3420	243	22	is	be	AUX
ejpam-3420	243	23	called	call	VERB
ejpam-3420	243	24	a	a	DET
ejpam-3420	243	25	generating	generate	VERB
ejpam-3420	243	26	covering	covering	NOUN
ejpam-3420	243	27	based	base	VERB
ejpam-3420	243	28	rough	rough	ADJ
ejpam-3420	243	29	topology	topology	NOUN
ejpam-3420	243	30	and	and	CCONJ
ejpam-3420	243	31	we	we	PRON
ejpam-3420	243	32	write	write	VERB
ejpam-3420	243	33	τ	τ	PROPN
ejpam-3420	243	34	=	=	X
ejpam-3420	243	35	<	<	X
ejpam-3420	243	36	o	o	X
ejpam-3420	243	37	>	>	X
ejpam-3420	243	38	,	,	PUNCT
ejpam-3420	243	39	where	where	SCONJ
ejpam-3420	243	40	o	o	NOUN
ejpam-3420	243	41	is	be	AUX
ejpam-3420	243	42	the	the	DET
ejpam-3420	243	43	base	base	NOUN
ejpam-3420	243	44	of	of	ADP
ejpam-3420	243	45	τ	τ	PROPN
ejpam-3420	243	46	.	.	PUNCT
ejpam-3420	244	1	now	now	ADV
ejpam-3420	244	2	,	,	PUNCT
ejpam-3420	244	3	let	let	VERB
ejpam-3420	244	4	τ1	τ1	VERB
ejpam-3420	244	5	=	=	NOUN
ejpam-3420	244	6	<	<	X
ejpam-3420	244	7	o1	o1	NOUN
ejpam-3420	244	8	>	>	X
ejpam-3420	244	9	and	and	CCONJ
ejpam-3420	244	10	τ2	τ2	PROPN
ejpam-3420	244	11	=	=	PROPN
ejpam-3420	244	12	<	<	X
ejpam-3420	244	13	o2	o2	PROPN
ejpam-3420	244	14	>	>	X
ejpam-3420	244	15	be	be	AUX
ejpam-3420	244	16	two	two	NUM
ejpam-3420	244	17	generating	generating	NOUN
ejpam-3420	244	18	covering	covering	NOUN
ejpam-3420	244	19	based	base	VERB
ejpam-3420	244	20	rough	rough	ADJ
ejpam-3420	244	21	topologies	topology	NOUN
ejpam-3420	244	22	on	on	ADP
ejpam-3420	244	23	the	the	DET
ejpam-3420	244	24	rough	rough	ADJ
ejpam-3420	244	25	set	set	NOUN
ejpam-3420	244	26	xc	xc	PROPN
ejpam-3420	244	27	.	.	PUNCT
ejpam-3420	245	1	then	then	ADV
ejpam-3420	245	2	we	we	PRON
ejpam-3420	245	3	say	say	VERB
ejpam-3420	245	4	that	that	SCONJ
ejpam-3420	245	5	τ1	τ1	NOUN
ejpam-3420	245	6	is	be	AUX
ejpam-3420	245	7	coarser	coarse	ADJ
ejpam-3420	245	8	than	than	SCONJ
ejpam-3420	245	9	τ2	τ2	ADJ
ejpam-3420	245	10	or	or	CCONJ
ejpam-3420	245	11	τ2	τ2	NOUN
ejpam-3420	245	12	is	be	AUX
ejpam-3420	245	13	finer	fine	ADJ
ejpam-3420	245	14	than	than	ADP
ejpam-3420	245	15	τ1	τ1	NOUN
ejpam-3420	245	16	and	and	CCONJ
ejpam-3420	245	17	write	write	VERB
ejpam-3420	245	18	τ1	τ1	ADP
ejpam-3420	245	19	⊆	⊆	NUM
ejpam-3420	245	20	τ2	τ2	NOUN
ejpam-3420	245	21	if	if	SCONJ
ejpam-3420	245	22	and	and	CCONJ
ejpam-3420	245	23	only	only	ADV
ejpam-3420	245	24	if	if	SCONJ
ejpam-3420	245	25	for	for	ADP
ejpam-3420	245	26	every	every	DET
ejpam-3420	245	27	ac	ac	PROPN
ejpam-3420	245	28	∈	∈	PROPN
ejpam-3420	245	29	τ1	τ1	NOUN
ejpam-3420	245	30	and	and	CCONJ
ejpam-3420	245	31	for	for	ADP
ejpam-3420	245	32	every	every	DET
ejpam-3420	245	33	y	y	PROPN
ejpam-3420	245	34	∈	∈	PROPN
ejpam-3420	245	35	ac	ac	PROPN
ejpam-3420	245	36	,	,	PUNCT
ejpam-3420	245	37	there	there	PRON
ejpam-3420	245	38	exists	exist	VERB
ejpam-3420	245	39	b	b	PROPN
ejpam-3420	245	40	∈	∈	PROPN
ejpam-3420	245	41	o2	o2	PROPN
ejpam-3420	245	42	such	such	ADJ
ejpam-3420	245	43	that	that	SCONJ
ejpam-3420	245	44	y	y	PROPN
ejpam-3420	245	45	∈	∈	PROPN
ejpam-3420	245	46	b	b	PROPN
ejpam-3420	245	47	⊆	⊆	NUM
ejpam-3420	245	48	ac	ac	NOUN
ejpam-3420	245	49	.	.	PUNCT
ejpam-3420	246	1	in	in	ADP
ejpam-3420	246	2	addition	addition	NOUN
ejpam-3420	246	3	,	,	PUNCT
ejpam-3420	246	4	we	we	PRON
ejpam-3420	246	5	write	write	VERB
ejpam-3420	246	6	τ1	τ1	NOUN
ejpam-3420	246	7	=	=	SYM
ejpam-3420	246	8	τ2	τ2	NOUN
ejpam-3420	246	9	if	if	SCONJ
ejpam-3420	246	10	and	and	CCONJ
ejpam-3420	246	11	only	only	ADV
ejpam-3420	246	12	if	if	SCONJ
ejpam-3420	246	13	for	for	ADP
ejpam-3420	246	14	every	every	DET
ejpam-3420	246	15	ac	ac	PROPN
ejpam-3420	246	16	∈	∈	PROPN
ejpam-3420	246	17	τ1	τ1	PROPN
ejpam-3420	246	18	,	,	PUNCT
ejpam-3420	246	19	bc	bc	PROPN
ejpam-3420	246	20	∈	∈	PROPN
ejpam-3420	246	21	τ2	τ2	PROPN
ejpam-3420	246	22	,	,	PUNCT
ejpam-3420	246	23	y	y	PROPN
ejpam-3420	246	24	∈	∈	PROPN
ejpam-3420	246	25	ac	ac	PROPN
ejpam-3420	246	26	and	and	CCONJ
ejpam-3420	246	27	y	y	PROPN
ejpam-3420	246	28	′	′	NOUN
ejpam-3420	246	29	∈	∈	PROPN
ejpam-3420	246	30	bc	bc	PROPN
ejpam-3420	246	31	,	,	PUNCT
ejpam-3420	246	32	there	there	PRON
ejpam-3420	246	33	exist	exist	VERB
ejpam-3420	246	34	k	k	PROPN
ejpam-3420	246	35	∈	∈	PROPN
ejpam-3420	246	36	o1	o1	PROPN
ejpam-3420	246	37	and	and	CCONJ
ejpam-3420	246	38	k	k	NOUN
ejpam-3420	246	39	′	′	NUM
ejpam-3420	246	40	∈	∈	PROPN
ejpam-3420	246	41	o2	o2	PROPN
ejpam-3420	246	42	such	such	ADJ
ejpam-3420	246	43	that	that	SCONJ
ejpam-3420	246	44	y	y	PROPN
ejpam-3420	246	45	∈	∈	PROPN
ejpam-3420	247	1	k	k	PROPN
ejpam-3420	247	2	′	′	NUM
ejpam-3420	247	3	⊆	⊆	NUM
ejpam-3420	247	4	ac	ac	PROPN
ejpam-3420	247	5	and	and	CCONJ
ejpam-3420	247	6	y	y	PROPN
ejpam-3420	247	7	′	′	NUM
ejpam-3420	248	1	∈	∈	PROPN
ejpam-3420	248	2	k	k	PROPN
ejpam-3420	249	1	⊆	⊆	NUM
ejpam-3420	249	2	bc	bc	PROPN
ejpam-3420	249	3	.	.	PUNCT
ejpam-3420	250	1	covering	cover	VERB
ejpam-3420	250	2	based	base	VERB
ejpam-3420	250	3	nano	nano	NOUN
ejpam-3420	250	4	space	space	NOUN
ejpam-3420	250	5	in	in	ADP
ejpam-3420	250	6	this	this	DET
ejpam-3420	250	7	section	section	NOUN
ejpam-3420	250	8	,	,	PUNCT
ejpam-3420	250	9	we	we	PRON
ejpam-3420	250	10	extend	extend	VERB
ejpam-3420	250	11	the	the	DET
ejpam-3420	250	12	definition	definition	NOUN
ejpam-3420	250	13	of	of	ADP
ejpam-3420	250	14	classical	classical	ADJ
ejpam-3420	250	15	nano	nano	NOUN
ejpam-3420	250	16	topology	topology	NOUN
ejpam-3420	250	17	to	to	PART
ejpam-3420	250	18	include	include	VERB
ejpam-3420	250	19	a	a	DET
ejpam-3420	250	20	rough	rough	ADJ
ejpam-3420	250	21	set	set	NOUN
ejpam-3420	250	22	x	x	PUNCT
ejpam-3420	250	23	determined	determine	VERB
ejpam-3420	250	24	by	by	ADP
ejpam-3420	250	25	a	a	DET
ejpam-3420	250	26	covering	covering	NOUN
ejpam-3420	250	27	c.	c.	NOUN
ejpam-3420	250	28	let	let	NOUN
ejpam-3420	250	29	(	(	PUNCT
ejpam-3420	250	30	u	u	NOUN
ejpam-3420	250	31	,	,	PUNCT
ejpam-3420	250	32	c	c	NOUN
ejpam-3420	250	33	)	)	PUNCT
ejpam-3420	250	34	be	be	AUX
ejpam-3420	250	35	a	a	DET
ejpam-3420	250	36	covering	covering	NOUN
ejpam-3420	250	37	based	base	VERB
ejpam-3420	250	38	approximation	approximation	NOUN
ejpam-3420	250	39	space	space	NOUN
ejpam-3420	250	40	.	.	PUNCT
ejpam-3420	251	1	similarly	similarly	ADV
ejpam-3420	251	2	,	,	PUNCT
ejpam-3420	251	3	we	we	PRON
ejpam-3420	251	4	define	define	VERB
ejpam-3420	251	5	the	the	DET
ejpam-3420	251	6	covering	covering	NOUN
ejpam-3420	251	7	based	base	VERB
ejpam-3420	251	8	rough	rough	ADJ
ejpam-3420	251	9	set	set	NOUN
ejpam-3420	251	10	x	x	SYM
ejpam-3420	251	11	⊆	⊆	NUM
ejpam-3420	251	12	u	u	NOUN
ejpam-3420	251	13	as	as	ADP
ejpam-3420	251	14	x	x	X
ejpam-3420	251	15	=	=	SYM
ejpam-3420	251	16	(	(	PUNCT
ejpam-3420	251	17	x	x	X
ejpam-3420	251	18	,	,	PUNCT
ejpam-3420	251	19	x	x	NOUN
ejpam-3420	251	20	,	,	PUNCT
ejpam-3420	251	21	bn(x	bn(x	NUM
ejpam-3420	251	22	)	)	PUNCT
ejpam-3420	251	23	)	)	PUNCT
ejpam-3420	251	24	.	.	PUNCT
ejpam-3420	252	1	if	if	SCONJ
ejpam-3420	252	2	y	y	PROPN
ejpam-3420	252	3	⊆	⊆	NUM
ejpam-3420	252	4	x	x	NOUN
ejpam-3420	252	5	,	,	PUNCT
ejpam-3420	252	6	then	then	ADV
ejpam-3420	252	7	y	y	PROPN
ejpam-3420	252	8	⊆	⊆	NUM
ejpam-3420	252	9	x	x	SYM
ejpam-3420	252	10	,	,	PUNCT
ejpam-3420	252	11	y	y	PROPN
ejpam-3420	252	12	⊆	⊆	NUM
ejpam-3420	252	13	x	x	PUNCT
ejpam-3420	252	14	and	and	CCONJ
ejpam-3420	252	15	bn(y	bn(y	NUM
ejpam-3420	252	16	)	)	PUNCT
ejpam-3420	252	17	⊆	⊆	NUM
ejpam-3420	252	18	bn(x	bn(x	NUM
ejpam-3420	252	19	)	)	PUNCT
ejpam-3420	252	20	.	.	PUNCT
ejpam-3420	253	1	now	now	ADV
ejpam-3420	253	2	,	,	PUNCT
ejpam-3420	253	3	we	we	PRON
ejpam-3420	253	4	are	be	AUX
ejpam-3420	253	5	constructing	construct	VERB
ejpam-3420	253	6	a	a	DET
ejpam-3420	253	7	nano	nano	NOUN
ejpam-3420	253	8	topology	topology	NOUN
ejpam-3420	253	9	on	on	ADP
ejpam-3420	253	10	the	the	DET
ejpam-3420	253	11	covering	covering	NOUN
ejpam-3420	253	12	based	base	VERB
ejpam-3420	253	13	rough	rough	ADJ
ejpam-3420	253	14	topology	topology	NOUN
ejpam-3420	253	15	.	.	PUNCT
ejpam-3420	254	1	definition	definition	NOUN
ejpam-3420	254	2	12	12	NUM
ejpam-3420	254	3	.	.	PUNCT
ejpam-3420	255	1	let	let	VERB
ejpam-3420	255	2	(	(	PUNCT
ejpam-3420	255	3	u	u	NOUN
ejpam-3420	255	4	,	,	PUNCT
ejpam-3420	255	5	c	c	NOUN
ejpam-3420	255	6	)	)	PUNCT
ejpam-3420	255	7	be	be	AUX
ejpam-3420	255	8	an	an	DET
ejpam-3420	255	9	approximation	approximation	NOUN
ejpam-3420	255	10	space	space	NOUN
ejpam-3420	255	11	and	and	CCONJ
ejpam-3420	255	12	τ	τ	PROPN
ejpam-3420	255	13	be	be	AUX
ejpam-3420	255	14	the	the	DET
ejpam-3420	255	15	family	family	NOUN
ejpam-3420	255	16	of	of	ADP
ejpam-3420	255	17	all	all	DET
ejpam-3420	255	18	rough	rough	ADJ
ejpam-3420	255	19	subsets	subset	NOUN
ejpam-3420	255	20	of	of	ADP
ejpam-3420	255	21	x	x	X
ejpam-3420	255	22	=	=	SYM
ejpam-3420	255	23	(	(	PUNCT
ejpam-3420	255	24	x	x	X
ejpam-3420	255	25	,	,	PUNCT
ejpam-3420	255	26	x	x	NOUN
ejpam-3420	255	27	,	,	PUNCT
ejpam-3420	255	28	bn(x	bn(x	NUM
ejpam-3420	255	29	)	)	PUNCT
ejpam-3420	255	30	)	)	PUNCT
ejpam-3420	256	1	satisfying	satisfy	VERB
ejpam-3420	256	2	the	the	DET
ejpam-3420	256	3	following	follow	VERB
ejpam-3420	256	4	conditions	condition	NOUN
ejpam-3420	256	5	:	:	PUNCT
ejpam-3420	256	6	(	(	PUNCT
ejpam-3420	256	7	i	i	NOUN
ejpam-3420	256	8	)	)	PUNCT
ejpam-3420	256	9	x	x	NOUN
ejpam-3420	256	10	,	,	PUNCT
ejpam-3420	256	11	∅	∅	NOUN
ejpam-3420	256	12	∈	∈	PROPN
ejpam-3420	256	13	τ	τ	X
ejpam-3420	256	14	;	;	PUNCT
ejpam-3420	256	15	references	reference	NOUN
ejpam-3420	256	16	543	543	NUM
ejpam-3420	256	17	(	(	PUNCT
ejpam-3420	256	18	ii	ii	NOUN
ejpam-3420	256	19	)	)	PUNCT
ejpam-3420	256	20	it	it	PRON
ejpam-3420	256	21	is	be	AUX
ejpam-3420	256	22	closed	close	VERB
ejpam-3420	256	23	under	under	ADP
ejpam-3420	256	24	finite	finite	ADJ
ejpam-3420	256	25	intersection	intersection	NOUN
ejpam-3420	256	26	;	;	PUNCT
ejpam-3420	256	27	(	(	PUNCT
ejpam-3420	256	28	iii	iii	X
ejpam-3420	256	29	)	)	PUNCT
ejpam-3420	256	30	it	it	PRON
ejpam-3420	256	31	is	be	AUX
ejpam-3420	256	32	closed	close	VERB
ejpam-3420	256	33	under	under	ADP
ejpam-3420	256	34	arbitrary	arbitrary	ADJ
ejpam-3420	256	35	union	union	NOUN
ejpam-3420	256	36	.	.	PUNCT
ejpam-3420	257	1	then	then	ADV
ejpam-3420	257	2	τ	τ	PROPN
ejpam-3420	257	3	is	be	AUX
ejpam-3420	257	4	called	call	VERB
ejpam-3420	257	5	a	a	DET
ejpam-3420	257	6	covering	covering	NOUN
ejpam-3420	257	7	based	base	VERB
ejpam-3420	257	8	nano	nano	NOUN
ejpam-3420	257	9	topology	topology	NOUN
ejpam-3420	257	10	on	on	ADP
ejpam-3420	257	11	a	a	DET
ejpam-3420	257	12	given	give	VERB
ejpam-3420	257	13	rough	rough	ADJ
ejpam-3420	257	14	set	set	NOUN
ejpam-3420	257	15	.	.	PUNCT
ejpam-3420	258	1	in	in	ADP
ejpam-3420	258	2	covering	cover	VERB
ejpam-3420	258	3	based	base	VERB
ejpam-3420	258	4	approximation	approximation	NOUN
ejpam-3420	258	5	space	space	NOUN
ejpam-3420	258	6	,	,	PUNCT
ejpam-3420	258	7	we	we	PRON
ejpam-3420	258	8	also	also	ADV
ejpam-3420	258	9	find	find	VERB
ejpam-3420	258	10	that	that	SCONJ
ejpam-3420	258	11	the	the	DET
ejpam-3420	258	12	nano	nano	NOUN
ejpam-3420	258	13	topology	topology	NOUN
ejpam-3420	258	14	coincides	coincide	VERB
ejpam-3420	258	15	with	with	ADP
ejpam-3420	258	16	the	the	DET
ejpam-3420	258	17	rough	rough	ADJ
ejpam-3420	258	18	topology	topology	NOUN
ejpam-3420	258	19	.	.	PUNCT
ejpam-3420	259	1	the	the	DET
ejpam-3420	259	2	reasons	reason	NOUN
ejpam-3420	259	3	are	be	AUX
ejpam-3420	259	4	similar	similar	ADJ
ejpam-3420	259	5	with	with	ADP
ejpam-3420	259	6	the	the	DET
ejpam-3420	259	7	ones	one	NOUN
ejpam-3420	259	8	in	in	ADP
ejpam-3420	259	9	a	a	DET
ejpam-3420	259	10	classical	classical	ADJ
ejpam-3420	259	11	approximation	approximation	NOUN
ejpam-3420	259	12	space	space	NOUN
ejpam-3420	259	13	.	.	PUNCT
ejpam-3420	260	1	acknowledgements	acknowledgement	NOUN
ejpam-3420	260	2	the	the	DET
ejpam-3420	260	3	authors	author	NOUN
ejpam-3420	260	4	wish	wish	VERB
ejpam-3420	260	5	to	to	PART
ejpam-3420	260	6	thank	thank	VERB
ejpam-3420	260	7	the	the	DET
ejpam-3420	260	8	deanship	deanship	NOUN
ejpam-3420	260	9	for	for	ADP
ejpam-3420	260	10	scientific	scientific	ADJ
ejpam-3420	260	11	research	research	NOUN
ejpam-3420	260	12	(	(	PUNCT
ejpam-3420	260	13	dsr	dsr	PROPN
ejpam-3420	260	14	)	)	PUNCT
ejpam-3420	260	15	at	at	ADP
ejpam-3420	260	16	king	king	PROPN
ejpam-3420	260	17	abdulaziz	abdulaziz	PROPN
ejpam-3420	260	18	university	university	PROPN
ejpam-3420	260	19	for	for	ADP
ejpam-3420	260	20	financially	financially	ADV
ejpam-3420	260	21	funding	fund	VERB
ejpam-3420	260	22	this	this	DET
ejpam-3420	260	23	project	project	NOUN
ejpam-3420	260	24	under	under	ADP
ejpam-3420	260	25	grant	grant	NOUN
ejpam-3420	260	26	no	no	INTJ
ejpam-3420	260	27	.	.	PUNCT
ejpam-3420	261	1	kep	kep	NOUN
ejpam-3420	261	2	-	-	PUNCT
ejpam-3420	261	3	phd-2	phd-2	NOUN
ejpam-3420	261	4	-	-	PUNCT
ejpam-3420	261	5	130	130	NUM
ejpam-3420	261	6	-	-	PUNCT
ejpam-3420	261	7	39	39	NUM
ejpam-3420	261	8	.	.	PUNCT
ejpam-3420	262	1	competing	compete	VERB
ejpam-3420	262	2	interests	interest	NOUN
ejpam-3420	262	3	the	the	DET
ejpam-3420	262	4	authors	author	NOUN
ejpam-3420	262	5	declare	declare	VERB
ejpam-3420	262	6	that	that	SCONJ
ejpam-3420	262	7	they	they	PRON
ejpam-3420	262	8	have	have	VERB
ejpam-3420	262	9	no	no	DET
ejpam-3420	262	10	competing	compete	VERB
ejpam-3420	262	11	interests	interest	NOUN
ejpam-3420	262	12	.	.	PUNCT
ejpam-3420	263	1	references	reference	NOUN
ejpam-3420	263	2	[	[	X
ejpam-3420	263	3	1	1	NUM
ejpam-3420	263	4	]	]	PUNCT
ejpam-3420	263	5	e	e	NOUN
ejpam-3420	263	6	brynairski	brynairski	NOUN
ejpam-3420	263	7	.	.	PUNCT
ejpam-3420	264	1	a	a	DET
ejpam-3420	264	2	calculus	calculus	NOUN
ejpam-3420	264	3	of	of	ADP
ejpam-3420	264	4	rough	rough	ADJ
ejpam-3420	264	5	sets	set	NOUN
ejpam-3420	264	6	of	of	ADP
ejpam-3420	264	7	the	the	DET
ejpam-3420	264	8	first	first	ADJ
ejpam-3420	264	9	order	order	NOUN
ejpam-3420	264	10	.	.	PUNCT
ejpam-3420	265	1	bull	bull	NOUN
ejpam-3420	265	2	of	of	ADP
ejpam-3420	265	3	the	the	DET
ejpam-3420	265	4	polish	polish	PROPN
ejpam-3420	265	5	academy	academy	PROPN
ejpam-3420	265	6	sciences	sciences	PROPN
ejpam-3420	265	7	:	:	PUNCT
ejpam-3420	265	8	mathematics	mathematic	NOUN
ejpam-3420	265	9	,	,	PUNCT
ejpam-3420	265	10	37(1	37(1	NUM
ejpam-3420	265	11	-	-	SYM
ejpam-3420	265	12	6):71–78	6):71–78	NUM
ejpam-3420	265	13	,	,	PUNCT
ejpam-3420	265	14	1989	1989	NUM
ejpam-3420	265	15	.	.	PUNCT
ejpam-3420	266	1	[	[	X
ejpam-3420	266	2	2	2	NUM
ejpam-3420	266	3	]	]	PUNCT
ejpam-3420	266	4	z	z	NOUN
ejpam-3420	266	5	pawlak	pawlak	ADJ
ejpam-3420	266	6	.	.	PUNCT
ejpam-3420	267	1	rough	rough	ADJ
ejpam-3420	267	2	sets	set	NOUN
ejpam-3420	267	3	.	.	PUNCT
ejpam-3420	268	1	int	int	NOUN
ejpam-3420	268	2	.	.	PUNCT
ejpam-3420	269	1	j.	j.	PROPN
ejpam-3420	269	2	comput	comput	PROPN
ejpam-3420	269	3	.	.	PUNCT
ejpam-3420	270	1	inform	inform	NOUN
ejpam-3420	270	2	.	.	PUNCT
ejpam-3420	271	1	sci	sci	PROPN
ejpam-3420	271	2	..	..	PROPN
ejpam-3420	271	3	,	,	PUNCT
ejpam-3420	271	4	11(5):341–356	11(5):341–356	NUM
ejpam-3420	271	5	,	,	PUNCT
ejpam-3420	271	6	1982	1982	NUM
ejpam-3420	271	7	.	.	PUNCT
ejpam-3420	272	1	[	[	X
ejpam-3420	272	2	3	3	X
ejpam-3420	272	3	]	]	PUNCT
ejpam-3420	272	4	j	j	NOUN
ejpam-3420	272	5	pomykala	pomykala	NOUN
ejpam-3420	272	6	.	.	PUNCT
ejpam-3420	273	1	the	the	DET
ejpam-3420	273	2	stone	stone	NOUN
ejpam-3420	273	3	algebra	algebra	NOUN
ejpam-3420	273	4	of	of	ADP
ejpam-3420	273	5	rough	rough	ADJ
ejpam-3420	273	6	sets	set	NOUN
ejpam-3420	273	7	.	.	PUNCT
ejpam-3420	274	1	int	int	NOUN
ejpam-3420	274	2	.	.	PUNCT
ejpam-3420	275	1	j.	j.	PROPN
ejpam-3420	275	2	comput	comput	PROPN
ejpam-3420	275	3	.	.	PUNCT
ejpam-3420	276	1	inform	inform	NOUN
ejpam-3420	276	2	.	.	PUNCT
ejpam-3420	277	1	sci	sci	PROPN
ejpam-3420	277	2	.	.	PROPN
ejpam-3420	277	3	,	,	PUNCT
ejpam-3420	277	4	36(78):495–508	36(78):495–508	NUM
ejpam-3420	277	5	,	,	PUNCT
ejpam-3420	277	6	1988	1988	NUM
ejpam-3420	277	7	.	.	PUNCT
ejpam-3420	278	1	[	[	X
ejpam-3420	278	2	4	4	X
ejpam-3420	278	3	]	]	X
ejpam-3420	278	4	a	a	DET
ejpam-3420	278	5	z	z	NOUN
ejpam-3420	278	6	özcelik	özcelik	PROPN
ejpam-3420	278	7	s	s	PART
ejpam-3420	278	8	akduman	akduman	NOUN
ejpam-3420	278	9	and	and	CCONJ
ejpam-3420	278	10	c	c	PROPN
ejpam-3420	278	11	özel	özel	PROPN
ejpam-3420	278	12	.	.	PUNCT
ejpam-3420	279	1	rough	rough	ADJ
ejpam-3420	279	2	topology	topology	NOUN
ejpam-3420	279	3	on	on	ADP
ejpam-3420	279	4	covering	covering	NOUN
ejpam-3420	279	5	-	-	PUNCT
ejpam-3420	279	6	based	base	VERB
ejpam-3420	279	7	rough	rough	ADJ
ejpam-3420	279	8	sets	set	NOUN
ejpam-3420	279	9	.	.	PUNCT
ejpam-3420	280	1	int	int	NOUN
ejpam-3420	280	2	.	.	PUNCT
ejpam-3420	281	1	j.	j.	PROPN
ejpam-3420	281	2	computational	computational	PROPN
ejpam-3420	281	3	systems	systems	PROPN
ejpam-3420	281	4	engineering	engineering	NOUN
ejpam-3420	281	5	,	,	PUNCT
ejpam-3420	281	6	2(2	2(2	NUM
ejpam-3420	281	7	)	)	PUNCT
ejpam-3420	281	8	,	,	PUNCT
ejpam-3420	281	9	2015	2015	NUM
ejpam-3420	281	10	.	.	PUNCT
ejpam-3420	282	1	[	[	X
ejpam-3420	282	2	5	5	NUM
ejpam-3420	282	3	]	]	PUNCT
ejpam-3420	282	4	a	a	DET
ejpam-3420	282	5	z	z	NOUN
ejpam-3420	282	6	özcelik	özcelik	PROPN
ejpam-3420	282	7	s	s	PART
ejpam-3420	282	8	akduman	akduman	NOUN
ejpam-3420	282	9	,	,	PUNCT
ejpam-3420	282	10	e	e	NOUN
ejpam-3420	282	11	zeliha	zeliha	PROPN
ejpam-3420	282	12	and	and	CCONJ
ejpam-3420	282	13	s	s	VERB
ejpam-3420	282	14	narli	narli	ADJ
ejpam-3420	282	15	.	.	PUNCT
ejpam-3420	283	1	rough	rough	ADJ
ejpam-3420	283	2	topology	topology	NOUN
ejpam-3420	283	3	on	on	ADP
ejpam-3420	283	4	covering	cover	VERB
ejpam-3420	283	5	based	base	VERB
ejpam-3420	283	6	rough	rough	ADJ
ejpam-3420	283	7	sets	set	NOUN
ejpam-3420	283	8	.	.	PUNCT
ejpam-3420	284	1	2012	2012	NUM
ejpam-3420	284	2	.	.	PUNCT