id	sid	tid	token	lemma	pos
cana-4781	1	1	communications	communication	NOUN
cana-4781	1	2	on	on	ADP
cana-4781	1	3	applied	apply	VERB
cana-4781	1	4	nonlinear	nonlinear	ADJ
cana-4781	1	5	analysis	analysis	NOUN
cana-4781	1	6	issn	issn	NOUN
cana-4781	1	7	:	:	PUNCT
cana-4781	1	8	1074	1074	NUM
cana-4781	1	9	-	-	PUNCT
cana-4781	1	10	133x	133x	NUM
cana-4781	1	11	vol	vol	NOUN
cana-4781	1	12	32	32	NUM
cana-4781	1	13	no	no	NOUN
cana-4781	1	14	.	.	NOUN
cana-4781	1	15	3	3	NUM
cana-4781	1	16	(	(	PUNCT
cana-4781	1	17	2025	2025	NUM
cana-4781	1	18	)	)	PUNCT
cana-4781	1	19	879	879	NUM
cana-4781	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-4781	1	21	compactness	compactness	NOUN
cana-4781	1	22	and	and	CCONJ
cana-4781	1	23	rough	rough	ADJ
cana-4781	1	24	isomorphism	isomorphism	NOUN
cana-4781	1	25	on	on	ADP
cana-4781	1	26	topological	topological	ADJ
cana-4781	1	27	simple	simple	ADJ
cana-4781	1	28	rough	rough	ADJ
cana-4781	1	29	groups	group	NOUN
cana-4781	1	30	p.	p.	NOUN
cana-4781	1	31	tamilarasi	tamilarasi	NOUN
cana-4781	2	1	[	[	X
cana-4781	2	2	1	1	X
cana-4781	2	3	]	]	PUNCT
cana-4781	2	4	and	and	CCONJ
cana-4781	2	5	r.	r.	PROPN
cana-4781	2	6	selvi	selvi	PROPN
cana-4781	3	1	[	[	X
cana-4781	3	2	2	2	X
cana-4781	3	3	]	]	X
cana-4781	3	4	[	[	X
cana-4781	3	5	1	1	NUM
cana-4781	3	6	]	]	PUNCT
cana-4781	3	7	research	research	NOUN
cana-4781	3	8	scholar	scholar	NOUN
cana-4781	3	9	(	(	PUNCT
cana-4781	3	10	reg	reg	NOUN
cana-4781	3	11	.	.	PUNCT
cana-4781	4	1	no	no	DET
cana-4781	4	2	:	:	PUNCT
cana-4781	4	3	22211202092003	22211202092003	NUM
cana-4781	4	4	)	)	PUNCT
cana-4781	4	5	,	,	PUNCT
cana-4781	4	6	department	department	NOUN
cana-4781	4	7	of	of	ADP
cana-4781	4	8	mathematics	mathematic	NOUN
cana-4781	4	9	,	,	PUNCT
cana-4781	4	10	sri	sri	PROPN
cana-4781	4	11	parasakthi	parasakthi	PROPN
cana-4781	4	12	college	college	PROPN
cana-4781	4	13	for	for	ADP
cana-4781	4	14	women	woman	NOUN
cana-4781	4	15	,	,	PUNCT
cana-4781	4	16	courtallam	courtallam	PROPN
cana-4781	4	17	627802	627802	NUM
cana-4781	4	18	,	,	PUNCT
cana-4781	4	19	affiliated	affiliate	VERB
cana-4781	4	20	by	by	ADP
cana-4781	4	21	manonmaniam	manonmaniam	PROPN
cana-4781	4	22	sundaranar	sundaranar	PROPN
cana-4781	4	23	university	university	PROPN
cana-4781	4	24	,	,	PUNCT
cana-4781	4	25	tirunelveli	tirunelveli	NOUN
cana-4781	4	26	627012	627012	NUM
cana-4781	4	27	.	.	PUNCT
cana-4781	5	1	e	e	X
cana-4781	5	2	-	-	NOUN
cana-4781	5	3	mail	mail	NOUN
cana-4781	5	4	:	:	PUNCT
cana-4781	5	5	tamilarasiparamasivan@gmail.com	tamilarasiparamasivan@gmail.com	X
cana-4781	6	1	[	[	X
cana-4781	6	2	2	2	NUM
cana-4781	6	3	]	]	PUNCT
cana-4781	6	4	associate	associate	NOUN
cana-4781	6	5	professor	professor	NOUN
cana-4781	6	6	,	,	PUNCT
cana-4781	6	7	department	department	NOUN
cana-4781	6	8	of	of	ADP
cana-4781	6	9	mathematics	mathematic	NOUN
cana-4781	6	10	,	,	PUNCT
cana-4781	6	11	sri	sri	PROPN
cana-4781	6	12	parasakthi	parasakthi	PROPN
cana-4781	6	13	college	college	PROPN
cana-4781	6	14	for	for	ADP
cana-4781	6	15	women	woman	NOUN
cana-4781	6	16	,	,	PUNCT
cana-4781	6	17	courtallam	courtallam	PROPN
cana-4781	6	18	627802	627802	NUM
cana-4781	6	19	,	,	PUNCT
cana-4781	6	20	affiliated	affiliate	VERB
cana-4781	6	21	by	by	ADP
cana-4781	6	22	manonmaniam	manonmaniam	PROPN
cana-4781	6	23	sundaranar	sundaranar	PROPN
cana-4781	6	24	university	university	PROPN
cana-4781	6	25	,	,	PUNCT
cana-4781	6	26	tirunelveli	tirunelveli	NOUN
cana-4781	6	27	627012	627012	NUM
cana-4781	6	28	.	.	PUNCT
cana-4781	7	1	e	e	X
cana-4781	7	2	-	-	NOUN
cana-4781	7	3	mail	mail	NOUN
cana-4781	7	4	:	:	PUNCT
cana-4781	7	5	r.selvimuthu@gmail.com	r.selvimuthu@gmail.com	X
cana-4781	7	6	article	article	NOUN
cana-4781	7	7	history	history	NOUN
cana-4781	7	8	:	:	PUNCT
cana-4781	7	9	received	receive	VERB
cana-4781	7	10	:	:	PUNCT
cana-4781	7	11	12	12	NUM
cana-4781	7	12	-	-	SYM
cana-4781	7	13	01	01	NUM
cana-4781	7	14	-	-	PUNCT
cana-4781	7	15	2025	2025	NUM
cana-4781	7	16	revised	revise	VERB
cana-4781	7	17	:	:	PUNCT
cana-4781	7	18	15	15	NUM
cana-4781	7	19	-	-	NUM
cana-4781	7	20	02	02	NUM
cana-4781	7	21	-	-	PUNCT
cana-4781	7	22	2025	2025	NUM
cana-4781	7	23	accepted	accept	VERB
cana-4781	7	24	:	:	PUNCT
cana-4781	7	25	01	01	NUM
cana-4781	7	26	-	-	SYM
cana-4781	7	27	03	03	NUM
cana-4781	7	28	-	-	PUNCT
cana-4781	7	29	2025	2025	NUM
cana-4781	7	30	abstract	abstract	NOUN
cana-4781	7	31	:	:	PUNCT
cana-4781	7	32	in	in	ADP
cana-4781	7	33	this	this	DET
cana-4781	7	34	paper	paper	NOUN
cana-4781	7	35	,	,	PUNCT
cana-4781	7	36	we	we	PRON
cana-4781	7	37	study	study	VERB
cana-4781	7	38	about	about	ADP
cana-4781	7	39	the	the	DET
cana-4781	7	40	compactness	compactness	NOUN
cana-4781	7	41	on	on	ADP
cana-4781	7	42	topological	topological	ADJ
cana-4781	7	43	simple	simple	ADJ
cana-4781	7	44	rough	rough	ADJ
cana-4781	7	45	groups	group	NOUN
cana-4781	7	46	.	.	PUNCT
cana-4781	8	1	in	in	ADP
cana-4781	8	2	particular	particular	ADJ
cana-4781	8	3	we	we	PRON
cana-4781	8	4	discuss	discuss	VERB
cana-4781	8	5	the	the	DET
cana-4781	8	6	open	open	ADJ
cana-4781	8	7	mapping	mapping	NOUN
cana-4781	8	8	theorems	theorem	NOUN
cana-4781	8	9	and	and	CCONJ
cana-4781	8	10	rough	rough	ADJ
cana-4781	8	11	isomorphism	isomorphism	NOUN
cana-4781	8	12	theorems	theorem	NOUN
cana-4781	8	13	in	in	ADP
cana-4781	8	14	topological	topological	ADJ
cana-4781	8	15	simple	simple	ADJ
cana-4781	8	16	rough	rough	ADJ
cana-4781	8	17	groups	group	NOUN
cana-4781	8	18	.	.	PUNCT
cana-4781	9	1	also	also	ADV
cana-4781	9	2	,	,	PUNCT
cana-4781	9	3	we	we	PRON
cana-4781	9	4	define	define	VERB
cana-4781	9	5	a	a	DET
cana-4781	9	6	rough	rough	ADJ
cana-4781	9	7	double	double	ADJ
cana-4781	9	8	coset	coset	NOUN
cana-4781	9	9	space	space	NOUN
cana-4781	9	10	and	and	CCONJ
cana-4781	9	11	discuss	discuss	VERB
cana-4781	9	12	their	their	PRON
cana-4781	9	13	role	role	NOUN
cana-4781	9	14	in	in	ADP
cana-4781	9	15	topological	topological	ADJ
cana-4781	9	16	simple	simple	ADJ
cana-4781	9	17	rough	rough	ADJ
cana-4781	9	18	groups	group	NOUN
cana-4781	9	19	.	.	PUNCT
cana-4781	10	1	further	far	ADV
cana-4781	10	2	,	,	PUNCT
cana-4781	10	3	we	we	PRON
cana-4781	10	4	examine	examine	VERB
cana-4781	10	5	the	the	DET
cana-4781	10	6	relationship	relationship	NOUN
cana-4781	10	7	between	between	ADP
cana-4781	10	8	compactness	compactness	NOUN
cana-4781	10	9	and	and	CCONJ
cana-4781	10	10	continuity	continuity	NOUN
cana-4781	10	11	of	of	ADP
cana-4781	10	12	quotient	quotient	NOUN
cana-4781	10	13	maps	map	NOUN
cana-4781	10	14	.	.	PUNCT
cana-4781	11	1	keywords	keyword	NOUN
cana-4781	11	2	:	:	PUNCT
cana-4781	11	3	rough	rough	ADJ
cana-4781	11	4	groups	group	NOUN
cana-4781	11	5	,	,	PUNCT
cana-4781	11	6	rough	rough	ADJ
cana-4781	11	7	subgroups	subgroup	NOUN
cana-4781	11	8	,	,	PUNCT
cana-4781	11	9	topological	topological	ADJ
cana-4781	11	10	simple	simple	ADJ
cana-4781	11	11	rough	rough	ADJ
cana-4781	11	12	groups	group	NOUN
cana-4781	11	13	,	,	PUNCT
cana-4781	11	14	compact	compact	ADJ
cana-4781	11	15	,	,	PUNCT
cana-4781	11	16	topological	topological	ADJ
cana-4781	11	17	rough	rough	ADJ
cana-4781	11	18	group	group	NOUN
cana-4781	11	19	homeomorphism	homeomorphism	NOUN
cana-4781	11	20	,	,	PUNCT
cana-4781	11	21	rough	rough	ADJ
cana-4781	11	22	double	double	ADJ
cana-4781	11	23	coset	coset	NOUN
cana-4781	11	24	spaces	space	NOUN
cana-4781	11	25	,	,	PUNCT
cana-4781	11	26	quotient	quotient	NOUN
cana-4781	11	27	spaces	space	NOUN
cana-4781	11	28	.	.	PUNCT
cana-4781	12	1	2020	2020	NUM
cana-4781	12	2	mathematics	mathematic	NOUN
cana-4781	12	3	subject	subject	ADJ
cana-4781	12	4	classification	classification	NOUN
cana-4781	12	5	:	:	PUNCT
cana-4781	12	6	20e32	20e32	NUM
cana-4781	12	7	,	,	PUNCT
cana-4781	12	8	22c05	22c05	NUM
cana-4781	12	9	,	,	PUNCT
cana-4781	12	10	22d05	22d05	NUM
cana-4781	12	11	,	,	PUNCT
cana-4781	12	12	54d45	54d45	NOUN
cana-4781	12	13	.	.	PUNCT
cana-4781	13	1	1.introduction	1.introduction	NUM
cana-4781	13	2	:	:	PUNCT
cana-4781	13	3	the	the	DET
cana-4781	13	4	rough	rough	ADJ
cana-4781	13	5	set	set	NOUN
cana-4781	13	6	theory	theory	NOUN
cana-4781	13	7	was	be	AUX
cana-4781	13	8	introduced	introduce	VERB
cana-4781	13	9	by	by	ADP
cana-4781	13	10	pawlak	pawlak	ADJ
cana-4781	13	11	[	[	X
cana-4781	13	12	14	14	NUM
cana-4781	13	13	]	]	PUNCT
cana-4781	13	14	in	in	ADP
cana-4781	13	15	1982	1982	NUM
cana-4781	13	16	which	which	PRON
cana-4781	13	17	is	be	AUX
cana-4781	13	18	based	base	VERB
cana-4781	13	19	on	on	ADP
cana-4781	13	20	the	the	DET
cana-4781	13	21	equivalence	equivalence	NOUN
cana-4781	13	22	relations	relation	NOUN
cana-4781	13	23	.	.	PUNCT
cana-4781	14	1	after	after	ADP
cana-4781	14	2	more	more	ADJ
cana-4781	14	3	than	than	ADP
cana-4781	14	4	30	30	NUM
cana-4781	14	5	years	year	NOUN
cana-4781	14	6	of	of	ADP
cana-4781	14	7	research	research	NOUN
cana-4781	14	8	,	,	PUNCT
cana-4781	14	9	the	the	DET
cana-4781	14	10	theory	theory	NOUN
cana-4781	14	11	of	of	ADP
cana-4781	14	12	rough	rough	ADJ
cana-4781	14	13	set	set	NOUN
cana-4781	14	14	has	have	AUX
cana-4781	14	15	been	be	AUX
cana-4781	14	16	continuously	continuously	ADV
cana-4781	14	17	improved	improve	VERB
cana-4781	14	18	and	and	CCONJ
cana-4781	14	19	widely	widely	ADV
cana-4781	14	20	expanded	expand	VERB
cana-4781	14	21	in	in	ADP
cana-4781	14	22	applications	application	NOUN
cana-4781	14	23	.	.	PUNCT
cana-4781	15	1	in	in	ADP
cana-4781	15	2	1994	1994	NUM
cana-4781	15	3	,	,	PUNCT
cana-4781	15	4	biswas	biswas	PROPN
cana-4781	15	5	and	and	CCONJ
cana-4781	15	6	nanda	nanda	ADV
cana-4781	15	7	[	[	X
cana-4781	15	8	3	3	X
cana-4781	15	9	]	]	PUNCT
cana-4781	15	10	introduced	introduce	VERB
cana-4781	15	11	the	the	DET
cana-4781	15	12	notion	notion	NOUN
cana-4781	15	13	of	of	ADP
cana-4781	15	14	rough	rough	ADJ
cana-4781	15	15	groups	group	NOUN
cana-4781	15	16	and	and	CCONJ
cana-4781	15	17	rough	rough	ADJ
cana-4781	15	18	subgroups	subgroup	NOUN
cana-4781	15	19	,	,	PUNCT
cana-4781	15	20	which	which	PRON
cana-4781	15	21	depends	depend	VERB
cana-4781	15	22	on	on	ADP
cana-4781	15	23	the	the	DET
cana-4781	15	24	upper	upper	ADJ
cana-4781	15	25	approximation	approximation	NOUN
cana-4781	15	26	.	.	PUNCT
cana-4781	16	1	then	then	ADV
cana-4781	16	2	,	,	PUNCT
cana-4781	16	3	bagirmaz	bagirmaz	NOUN
cana-4781	16	4	et	et	PROPN
cana-4781	16	5	al	al	PROPN
cana-4781	16	6	.	.	PUNCT
cana-4781	17	1	(	(	PUNCT
cana-4781	17	2	2016	2016	NUM
cana-4781	17	3	)	)	PUNCT
cana-4781	18	1	[	[	X
cana-4781	18	2	13	13	NUM
cana-4781	18	3	]	]	PUNCT
cana-4781	18	4	introduced	introduce	VERB
cana-4781	18	5	the	the	DET
cana-4781	18	6	concept	concept	NOUN
cana-4781	18	7	of	of	ADP
cana-4781	18	8	topological	topological	ADJ
cana-4781	18	9	rough	rough	ADJ
cana-4781	18	10	group	group	NOUN
cana-4781	18	11	and	and	CCONJ
cana-4781	18	12	extended	extend	VERB
cana-4781	18	13	the	the	DET
cana-4781	18	14	notion	notion	NOUN
cana-4781	18	15	of	of	ADP
cana-4781	18	16	a	a	DET
cana-4781	18	17	topological	topological	ADJ
cana-4781	18	18	group	group	NOUN
cana-4781	18	19	to	to	PART
cana-4781	18	20	include	include	VERB
cana-4781	18	21	algebraic	algebraic	ADJ
cana-4781	18	22	structures	structure	NOUN
cana-4781	18	23	of	of	ADP
cana-4781	18	24	rough	rough	ADJ
cana-4781	18	25	groups	group	NOUN
cana-4781	18	26	.	.	PUNCT
cana-4781	19	1	in	in	ADP
cana-4781	19	2	this	this	DET
cana-4781	19	3	paper	paper	NOUN
cana-4781	19	4	,	,	PUNCT
cana-4781	19	5	we	we	PRON
cana-4781	19	6	discussed	discuss	VERB
cana-4781	19	7	compactness	compactness	NOUN
cana-4781	19	8	and	and	CCONJ
cana-4781	19	9	open	open	ADJ
cana-4781	19	10	mapping	mapping	NOUN
cana-4781	19	11	theorems	theorem	NOUN
cana-4781	19	12	in	in	ADP
cana-4781	19	13	topological	topological	ADJ
cana-4781	19	14	simple	simple	ADJ
cana-4781	19	15	rough	rough	ADJ
cana-4781	19	16	groups	group	NOUN
cana-4781	19	17	and	and	CCONJ
cana-4781	19	18	we	we	PRON
cana-4781	19	19	examine	examine	VERB
cana-4781	19	20	the	the	DET
cana-4781	19	21	relationship	relationship	NOUN
cana-4781	19	22	between	between	ADP
cana-4781	19	23	compactness	compactness	NOUN
cana-4781	19	24	and	and	CCONJ
cana-4781	19	25	continuity	continuity	NOUN
cana-4781	19	26	of	of	ADP
cana-4781	19	27	quotient	quotient	NOUN
cana-4781	19	28	maps	map	NOUN
cana-4781	19	29	.	.	PUNCT
cana-4781	20	1	further	far	ADV
cana-4781	20	2	,	,	PUNCT
cana-4781	20	3	we	we	PRON
cana-4781	20	4	analysed	analyse	VERB
cana-4781	20	5	some	some	DET
cana-4781	20	6	results	result	NOUN
cana-4781	20	7	related	relate	VERB
cana-4781	20	8	to	to	ADP
cana-4781	20	9	topological	topological	ADJ
cana-4781	20	10	rough	rough	ADJ
cana-4781	20	11	group	group	NOUN
cana-4781	20	12	homeomorphism	homeomorphism	PROPN
cana-4781	20	13	and	and	CCONJ
cana-4781	20	14	also	also	ADV
cana-4781	20	15	rough	rough	ADJ
cana-4781	20	16	isomorphism	isomorphism	NOUN
cana-4781	20	17	theorems	theorem	NOUN
cana-4781	20	18	are	be	AUX
cana-4781	20	19	discussed	discuss	VERB
cana-4781	20	20	.	.	PUNCT
cana-4781	21	1	finally	finally	ADV
cana-4781	21	2	,	,	PUNCT
cana-4781	21	3	we	we	PRON
cana-4781	21	4	defined	define	VERB
cana-4781	21	5	a	a	DET
cana-4781	21	6	rough	rough	ADJ
cana-4781	21	7	double	double	ADJ
cana-4781	21	8	coset	coset	NOUN
cana-4781	21	9	space	space	NOUN
cana-4781	21	10	and	and	CCONJ
cana-4781	21	11	discuss	discuss	VERB
cana-4781	21	12	their	their	PRON
cana-4781	21	13	role	role	NOUN
cana-4781	21	14	in	in	ADP
cana-4781	21	15	topological	topological	ADJ
cana-4781	21	16	simple	simple	ADJ
cana-4781	21	17	rough	rough	ADJ
cana-4781	21	18	groups	group	NOUN
cana-4781	21	19	.	.	PUNCT
cana-4781	22	1	by	by	ADP
cana-4781	22	2	utilizing	utilize	VERB
cana-4781	22	3	techniques	technique	NOUN
cana-4781	22	4	from	from	ADP
cana-4781	22	5	topology	topology	NOUN
cana-4781	22	6	and	and	CCONJ
cana-4781	22	7	fundamental	fundamental	ADJ
cana-4781	22	8	analysis	analysis	NOUN
cana-4781	22	9	,	,	PUNCT
cana-4781	22	10	we	we	PRON
cana-4781	22	11	establish	establish	VERB
cana-4781	22	12	criteria	criterion	NOUN
cana-4781	22	13	for	for	ADP
cana-4781	22	14	compactness	compactness	NOUN
cana-4781	22	15	in	in	ADP
cana-4781	22	16	rough	rough	ADJ
cana-4781	22	17	double	double	ADJ
cana-4781	22	18	coset	coset	NOUN
cana-4781	22	19	spaces	space	NOUN
cana-4781	22	20	.	.	PUNCT
cana-4781	23	1	2	2	X
cana-4781	23	2	.	.	X
cana-4781	23	3	preliminaries	preliminary	NOUN
cana-4781	23	4	:	:	PUNCT
cana-4781	23	5	definition	definition	NOUN
cana-4781	23	6	2.1	2.1	NUM
cana-4781	23	7	.	.	PUNCT
cana-4781	24	1	[	[	X
cana-4781	24	2	4	4	X
cana-4781	24	3	]	]	PUNCT
cana-4781	24	4	let	let	VERB
cana-4781	24	5	k	k	X
cana-4781	24	6	=	=	SYM
cana-4781	24	7	(	(	PUNCT
cana-4781	24	8	u	u	NOUN
cana-4781	24	9	,	,	PUNCT
cana-4781	24	10	r	r	NOUN
cana-4781	24	11	)	)	PUNCT
cana-4781	24	12	be	be	AUX
cana-4781	24	13	an	an	DET
cana-4781	24	14	approximation	approximation	NOUN
cana-4781	24	15	space	space	NOUN
cana-4781	24	16	and	and	CCONJ
cana-4781	24	17	∗	∗	NOUN
cana-4781	24	18	be	be	VERB
cana-4781	24	19	a	a	DET
cana-4781	24	20	binary	binary	ADJ
cana-4781	24	21	operation	operation	NOUN
cana-4781	24	22	defined	define	VERB
cana-4781	24	23	on	on	ADP
cana-4781	24	24	u.	u.	PROPN
cana-4781	24	25	a	a	PRON
cana-4781	24	26	subset	subset	NOUN
cana-4781	24	27	g	g	NOUN
cana-4781	24	28	of	of	ADP
cana-4781	24	29	universe	universe	NOUN
cana-4781	24	30	u	u	NOUN
cana-4781	24	31	is	be	AUX
cana-4781	24	32	called	call	VERB
cana-4781	24	33	a	a	DET
cana-4781	24	34	rough	rough	ADJ
cana-4781	24	35	group	group	NOUN
cana-4781	24	36	if	if	SCONJ
cana-4781	24	37	the	the	DET
cana-4781	24	38	following	follow	VERB
cana-4781	24	39	properties	property	NOUN
cana-4781	24	40	are	be	AUX
cana-4781	24	41	satisfied	satisfied	ADJ
cana-4781	24	42	:	:	PUNCT
cana-4781	24	43	(	(	PUNCT
cana-4781	24	44	i	i	NOUN
cana-4781	24	45	)	)	PUNCT
cana-4781	24	46	∀	∀	PUNCT
cana-4781	25	1	x	x	X
cana-4781	25	2	,	,	PUNCT
cana-4781	25	3	y	y	PROPN
cana-4781	25	4	∈	∈	PROPN
cana-4781	25	5	g	g	PROPN
cana-4781	25	6	,	,	PUNCT
cana-4781	25	7	x∗y	x∗y	PROPN
cana-4781	25	8	∈	∈	PROPN
cana-4781	25	9	g̅	g̅	PROPN
cana-4781	25	10	;	;	PUNCT
cana-4781	25	11	mailto:tamilarasiparamasivan@gmail.com	mailto:tamilarasiparamasivan@gmail.com	X
cana-4781	25	12	mailto:r.selvimuthu@gmail.com	mailto:r.selvimuthu@gmail.com	X
cana-4781	25	13	communications	communication	NOUN
cana-4781	25	14	on	on	ADP
cana-4781	25	15	applied	apply	VERB
cana-4781	25	16	nonlinear	nonlinear	ADJ
cana-4781	25	17	analysis	analysis	NOUN
cana-4781	25	18	issn	issn	NOUN
cana-4781	25	19	:	:	PUNCT
cana-4781	25	20	1074	1074	NUM
cana-4781	25	21	-	-	PUNCT
cana-4781	25	22	133x	133x	NUM
cana-4781	25	23	vol	vol	NOUN
cana-4781	25	24	32	32	NUM
cana-4781	25	25	no	no	NOUN
cana-4781	25	26	.	.	NOUN
cana-4781	25	27	3	3	NUM
cana-4781	25	28	(	(	PUNCT
cana-4781	25	29	2025	2025	NUM
cana-4781	25	30	)	)	PUNCT
cana-4781	25	31	880	880	NUM
cana-4781	25	32	https://internationalpubls.com	https://internationalpubls.com	X
cana-4781	25	33	(	(	PUNCT
cana-4781	25	34	ii	ii	NOUN
cana-4781	25	35	)	)	PUNCT
cana-4781	25	36	association	association	NOUN
cana-4781	25	37	property	property	NOUN
cana-4781	25	38	holds	hold	VERB
cana-4781	25	39	in	in	ADP
cana-4781	25	40	g̅	g̅	NOUN
cana-4781	25	41	;	;	PUNCT
cana-4781	25	42	(	(	PUNCT
cana-4781	25	43	iii	iii	X
cana-4781	25	44	)	)	PUNCT
cana-4781	25	45	∃	∃	PROPN
cana-4781	25	46	e	e	PROPN
cana-4781	25	47	∈	∈	PROPN
cana-4781	25	48	g̅	g̅	PROPN
cana-4781	26	1	such	such	ADJ
cana-4781	26	2	that	that	SCONJ
cana-4781	26	3	∀	∀	NOUN
cana-4781	26	4	x	x	SYM
cana-4781	26	5	∈	∈	PROPN
cana-4781	26	6	g	g	PROPN
cana-4781	26	7	,	,	PUNCT
cana-4781	26	8	x∗e	x∗e	PROPN
cana-4781	26	9	=	=	PUNCT
cana-4781	26	10	e∗x	e∗x	PUNCT
cana-4781	26	11	=	=	SYM
cana-4781	26	12	x	x	NOUN
cana-4781	26	13	;	;	PUNCT
cana-4781	26	14	e	e	X
cana-4781	26	15	is	be	AUX
cana-4781	26	16	called	call	VERB
cana-4781	26	17	the	the	DET
cana-4781	26	18	rough	rough	ADJ
cana-4781	26	19	identity	identity	NOUN
cana-4781	26	20	element	element	NOUN
cana-4781	26	21	of	of	ADP
cana-4781	26	22	rough	rough	ADJ
cana-4781	26	23	group	group	NOUN
cana-4781	26	24	g	g	NOUN
cana-4781	26	25	;	;	PUNCT
cana-4781	26	26	(	(	PUNCT
cana-4781	26	27	iv	iv	X
cana-4781	26	28	)	)	PUNCT
cana-4781	26	29	∀	∀	X
cana-4781	27	1	x	x	X
cana-4781	27	2	∈	∈	PROPN
cana-4781	27	3	g	g	PROPN
cana-4781	27	4	,	,	PUNCT
cana-4781	27	5	∃	∃	PROPN
cana-4781	27	6	y	y	PROPN
cana-4781	27	7	∈	∈	PROPN
cana-4781	27	8	g	g	PROPN
cana-4781	27	9	such	such	ADJ
cana-4781	27	10	that	that	SCONJ
cana-4781	27	11	x∗y	x∗y	X
cana-4781	27	12	=	=	PUNCT
cana-4781	27	13	y∗x	y∗x	NUM
cana-4781	27	14	=	=	SYM
cana-4781	27	15	e	e	X
cana-4781	27	16	;	;	PUNCT
cana-4781	27	17	y	y	PROPN
cana-4781	27	18	is	be	AUX
cana-4781	27	19	called	call	VERB
cana-4781	27	20	the	the	DET
cana-4781	27	21	rough	rough	ADJ
cana-4781	27	22	inverse	inverse	NOUN
cana-4781	27	23	element	element	NOUN
cana-4781	27	24	of	of	ADP
cana-4781	27	25	x	x	PUNCT
cana-4781	27	26	in	in	ADP
cana-4781	27	27	g	g	NOUN
cana-4781	27	28	;	;	PUNCT
cana-4781	27	29	theorem	theorem	VERB
cana-4781	27	30	2.2	2.2	NUM
cana-4781	27	31	.	.	PUNCT
cana-4781	28	1	[	[	X
cana-4781	28	2	4	4	X
cana-4781	28	3	]	]	X
cana-4781	28	4	a	a	DET
cana-4781	28	5	necessary	necessary	ADJ
cana-4781	28	6	and	and	CCONJ
cana-4781	28	7	sufficient	sufficient	ADJ
cana-4781	28	8	condition	condition	NOUN
cana-4781	28	9	for	for	ADP
cana-4781	28	10	a	a	DET
cana-4781	28	11	subset	subset	ADJ
cana-4781	28	12	h	h	NOUN
cana-4781	28	13	of	of	ADP
cana-4781	28	14	rough	rough	ADJ
cana-4781	28	15	group	group	NOUN
cana-4781	28	16	g	g	PROPN
cana-4781	28	17	to	to	PART
cana-4781	28	18	be	be	AUX
cana-4781	28	19	a	a	DET
cana-4781	28	20	rough	rough	ADJ
cana-4781	28	21	subgroup	subgroup	NOUN
cana-4781	28	22	is	be	AUX
cana-4781	28	23	that	that	SCONJ
cana-4781	28	24	:	:	PUNCT
cana-4781	28	25	(	(	PUNCT
cana-4781	28	26	i	i	NOUN
cana-4781	28	27	)	)	PUNCT
cana-4781	28	28	∀	∀	PUNCT
cana-4781	29	1	x	x	X
cana-4781	29	2	,	,	PUNCT
cana-4781	29	3	y	y	PROPN
cana-4781	29	4	∈	∈	PROPN
cana-4781	29	5	h	h	NOUN
cana-4781	29	6	,	,	PUNCT
cana-4781	29	7	x∗y	x∗y	PROPN
cana-4781	29	8	∈	∈	PROPN
cana-4781	29	9	g̅	g̅	NOUN
cana-4781	29	10	;	;	PUNCT
cana-4781	29	11	(	(	PUNCT
cana-4781	29	12	ii	ii	NOUN
cana-4781	29	13	)	)	PUNCT
cana-4781	29	14	∀	∀	X
cana-4781	30	1	x	x	SYM
cana-4781	30	2	∈	∈	NOUN
cana-4781	30	3	h	h	NOUN
cana-4781	30	4	,	,	PUNCT
cana-4781	30	5	x-1	x-1	PROPN
cana-4781	30	6	∈	∈	PROPN
cana-4781	30	7	h.	h.	PROPN
cana-4781	30	8	definition	definition	NOUN
cana-4781	30	9	2.3	2.3	NUM
cana-4781	30	10	.	.	PUNCT
cana-4781	31	1	[	[	X
cana-4781	31	2	13	13	NUM
cana-4781	31	3	]	]	PUNCT
cana-4781	31	4	a	a	DET
cana-4781	31	5	topological	topological	ADJ
cana-4781	31	6	rough	rough	ADJ
cana-4781	31	7	group	group	NOUN
cana-4781	31	8	is	be	AUX
cana-4781	31	9	a	a	DET
cana-4781	31	10	rough	rough	ADJ
cana-4781	31	11	group	group	NOUN
cana-4781	31	12	(	(	PUNCT
cana-4781	31	13	g	g	NOUN
cana-4781	31	14	,	,	PUNCT
cana-4781	31	15	∗	∗	NOUN
cana-4781	31	16	)	)	PUNCT
cana-4781	31	17	together	together	ADV
cana-4781	31	18	with	with	ADP
cana-4781	31	19	a	a	DET
cana-4781	31	20	topology	topology	NOUN
cana-4781	31	21	t	t	NOUN
cana-4781	31	22	on	on	ADP
cana-4781	31	23	g̅	g̅	NOUN
cana-4781	31	24	satisfying	satisfy	VERB
cana-4781	31	25	the	the	DET
cana-4781	31	26	following	follow	VERB
cana-4781	31	27	two	two	NUM
cana-4781	31	28	properties	property	NOUN
cana-4781	31	29	:	:	PUNCT
cana-4781	31	30	(	(	PUNCT
cana-4781	31	31	i	i	NOUN
cana-4781	31	32	)	)	PUNCT
cana-4781	32	1	the	the	DET
cana-4781	32	2	mapping	mapping	NOUN
cana-4781	33	1	f	f	NOUN
cana-4781	33	2	:	:	PUNCT
cana-4781	33	3	g	g	PROPN
cana-4781	33	4	×	×	PROPN
cana-4781	33	5	g	g	PROPN
cana-4781	33	6	→	→	SYM
cana-4781	33	7	g̅	g̅	NOUN
cana-4781	33	8	defined	define	VERB
cana-4781	33	9	by	by	ADP
cana-4781	33	10	f(x	f(x	PROPN
cana-4781	33	11	,	,	PUNCT
cana-4781	33	12	y	y	NOUN
cana-4781	33	13	)	)	PUNCT
cana-4781	34	1	=	=	PUNCT
cana-4781	34	2	xy	xy	PROPN
cana-4781	34	3	is	be	AUX
cana-4781	34	4	continuous	continuous	ADJ
cana-4781	34	5	with	with	ADP
cana-4781	34	6	respect	respect	NOUN
cana-4781	34	7	to	to	ADP
cana-4781	34	8	product	product	NOUN
cana-4781	34	9	topology	topology	NOUN
cana-4781	34	10	on	on	ADP
cana-4781	34	11	g	g	PROPN
cana-4781	34	12	×	×	PROPN
cana-4781	34	13	g	g	NOUN
cana-4781	34	14	and	and	CCONJ
cana-4781	34	15	the	the	DET
cana-4781	34	16	topology	topology	NOUN
cana-4781	34	17	tg	tg	PROPN
cana-4781	34	18	on	on	ADP
cana-4781	34	19	g	g	PROPN
cana-4781	34	20	induced	induce	VERB
cana-4781	34	21	by	by	ADP
cana-4781	34	22	t	t	PROPN
cana-4781	34	23	,	,	PUNCT
cana-4781	34	24	(	(	PUNCT
cana-4781	34	25	ii	ii	NOUN
cana-4781	34	26	)	)	PUNCT
cana-4781	34	27	the	the	DET
cana-4781	34	28	inverse	inverse	NOUN
cana-4781	34	29	mapping	mapping	NOUN
cana-4781	34	30	g	g	NOUN
cana-4781	34	31	:	:	PUNCT
cana-4781	34	32	g	g	PROPN
cana-4781	34	33	→	→	SYM
cana-4781	34	34	g	g	NOUN
cana-4781	34	35	defined	define	VERB
cana-4781	34	36	by	by	ADP
cana-4781	34	37	g(x	g(x	NOUN
cana-4781	34	38	)	)	PUNCT
cana-4781	35	1	=	=	PUNCT
cana-4781	35	2	x-1	x-1	NOUN
cana-4781	35	3	is	be	AUX
cana-4781	35	4	continuous	continuous	ADJ
cana-4781	35	5	with	with	ADP
cana-4781	35	6	respect	respect	NOUN
cana-4781	35	7	to	to	ADP
cana-4781	35	8	the	the	DET
cana-4781	35	9	topology	topology	NOUN
cana-4781	35	10	tg	tg	PROPN
cana-4781	35	11	on	on	ADP
cana-4781	35	12	g	g	PROPN
cana-4781	35	13	induced	induce	VERB
cana-4781	35	14	by	by	ADP
cana-4781	35	15	t.	t.	ADJ
cana-4781	35	16	definition	definition	NOUN
cana-4781	35	17	2.4	2.4	NUM
cana-4781	35	18	.	.	PUNCT
cana-4781	36	1	[	[	X
cana-4781	36	2	4	4	X
cana-4781	36	3	]	]	X
cana-4781	36	4	let	let	ADJ
cana-4781	36	5	(	(	PUNCT
cana-4781	36	6	𝑈1	𝑈1	PROPN
cana-4781	36	7	,	,	PUNCT
cana-4781	36	8	𝑅1	𝑅1	PROPN
cana-4781	36	9	)	)	PUNCT
cana-4781	36	10	,	,	PUNCT
cana-4781	36	11	(	(	PUNCT
cana-4781	36	12	𝑈2	𝑈2	NOUN
cana-4781	36	13	,	,	PUNCT
cana-4781	36	14	𝑅2	𝑅2	NOUN
cana-4781	36	15	)	)	PUNCT
cana-4781	36	16	be	be	VERB
cana-4781	36	17	two	two	NUM
cana-4781	36	18	approximation	approximation	NOUN
cana-4781	36	19	spaces	space	NOUN
cana-4781	36	20	,	,	PUNCT
cana-4781	36	21	and	and	CCONJ
cana-4781	36	22	∗	∗	NOUN
cana-4781	36	23	,	,	PUNCT
cana-4781	36	24	∗̅	∗̅	PROPN
cana-4781	36	25	be	be	AUX
cana-4781	36	26	binary	binary	ADJ
cana-4781	36	27	operations	operation	NOUN
cana-4781	36	28	over	over	ADP
cana-4781	36	29	universes	universe	NOUN
cana-4781	36	30	𝑈1	𝑈1	PROPN
cana-4781	36	31	,	,	PUNCT
cana-4781	36	32	𝑈2	𝑈2	PROPN
cana-4781	36	33	respectively	respectively	ADV
cana-4781	36	34	.	.	PUNCT
cana-4781	37	1	let	let	VERB
cana-4781	37	2	𝐺1	𝐺1	PROPN
cana-4781	37	3	⊂	⊂	PROPN
cana-4781	37	4	𝑈1	𝑈1	PROPN
cana-4781	37	5	and	and	CCONJ
cana-4781	37	6	𝐺2	𝐺2	NOUN
cana-4781	37	7	⊂	⊂	PROPN
cana-4781	37	8	𝑈2	𝑈2	PROPN
cana-4781	37	9	be	be	AUX
cana-4781	37	10	rough	rough	ADJ
cana-4781	37	11	groups	group	NOUN
cana-4781	37	12	.	.	PUNCT
cana-4781	38	1	𝐺1	𝐺1	PROPN
cana-4781	38	2	,	,	PUNCT
cana-4781	38	3	𝐺2	𝐺2	NOUN
cana-4781	38	4	are	be	AUX
cana-4781	38	5	called	call	VERB
cana-4781	38	6	rough	rough	ADJ
cana-4781	38	7	homomorphism	homomorphism	NOUN
cana-4781	38	8	sets	set	VERB
cana-4781	38	9	if	if	SCONJ
cana-4781	38	10	there	there	PRON
cana-4781	38	11	exists	exist	VERB
cana-4781	38	12	a	a	DET
cana-4781	38	13	surjection	surjection	NOUN
cana-4781	38	14	𝜑	𝜑	PROPN
cana-4781	38	15	:	:	PUNCT
cana-4781	38	16	𝐺1	𝐺1	PROPN
cana-4781	38	17	̅̅	̅̅	PROPN
cana-4781	38	18	̅	̅	PROPN
cana-4781	38	19	→	→	PUNCT
cana-4781	38	20	𝐺2	𝐺2	ADJ
cana-4781	38	21	̅̅	̅̅	PROPN
cana-4781	38	22	̅	̅	NOUN
cana-4781	38	23	such	such	ADJ
cana-4781	38	24	that	that	SCONJ
cana-4781	38	25	∀	∀	NOUN
cana-4781	38	26	𝑥	𝑥	NOUN
cana-4781	38	27	,	,	PUNCT
cana-4781	38	28	𝑦	𝑦	PROPN
cana-4781	38	29	∈	∈	PROPN
cana-4781	38	30	𝐺1	𝐺1	NOUN
cana-4781	38	31	∪	∪	X
cana-4781	38	32	{	{	PUNCT
cana-4781	38	33	𝑒	𝑒	PROPN
cana-4781	38	34	}	}	PUNCT
cana-4781	38	35	,	,	PUNCT
cana-4781	38	36	we	we	PRON
cana-4781	38	37	have	have	VERB
cana-4781	38	38	𝜑(𝑥	𝜑(𝑥	NOUN
cana-4781	38	39	∗	∗	NOUN
cana-4781	38	40	𝑦	𝑦	NOUN
cana-4781	38	41	)	)	PUNCT
cana-4781	38	42	=	=	SYM
cana-4781	38	43	𝜑(𝑥	𝜑(𝑥	X
cana-4781	38	44	)	)	PUNCT
cana-4781	38	45	∗̅	∗̅	ADJ
cana-4781	38	46	𝜑(𝑦	𝜑(𝑦	NOUN
cana-4781	38	47	)	)	PUNCT
cana-4781	38	48	.	.	PUNCT
cana-4781	39	1	if	if	SCONJ
cana-4781	39	2	a	a	DET
cana-4781	39	3	rough	rough	ADJ
cana-4781	39	4	homomorphism	homomorphism	NOUN
cana-4781	39	5	is	be	AUX
cana-4781	39	6	a	a	DET
cana-4781	39	7	bijection	bijection	NOUN
cana-4781	39	8	,	,	PUNCT
cana-4781	39	9	then	then	ADV
cana-4781	39	10	we	we	PRON
cana-4781	39	11	say	say	VERB
cana-4781	39	12	that	that	SCONJ
cana-4781	39	13	𝐺1	𝐺1	PROPN
cana-4781	39	14	and	and	CCONJ
cana-4781	39	15	𝐺2	𝐺2	NOUN
cana-4781	39	16	are	be	AUX
cana-4781	39	17	rough	rough	ADJ
cana-4781	39	18	isomorphism	isomorphism	NOUN
cana-4781	39	19	.	.	PUNCT
cana-4781	40	1	definition	definition	NOUN
cana-4781	40	2	2.5	2.5	NUM
cana-4781	40	3	.	.	PUNCT
cana-4781	41	1	[	[	X
cana-4781	41	2	1	1	X
cana-4781	41	3	]	]	PUNCT
cana-4781	41	4	a	a	DET
cana-4781	41	5	mapping	mapping	NOUN
cana-4781	41	6	𝑓	𝑓	NOUN
cana-4781	41	7	:	:	PUNCT
cana-4781	41	8	𝐺1	𝐺1	PROPN
cana-4781	41	9	̅̅	̅̅	PROPN
cana-4781	41	10	̅	̅	PROPN
cana-4781	41	11	→	→	PUNCT
cana-4781	41	12	𝐺2	𝐺2	ADJ
cana-4781	41	13	̅̅	̅̅	PROPN
cana-4781	41	14	̅	̅	NOUN
cana-4781	41	15	is	be	AUX
cana-4781	41	16	called	call	VERB
cana-4781	41	17	a	a	DET
cana-4781	41	18	topological	topological	ADJ
cana-4781	41	19	rough	rough	ADJ
cana-4781	41	20	group	group	NOUN
cana-4781	41	21	homomorphism	homomorphism	NOUN
cana-4781	41	22	,	,	PUNCT
cana-4781	41	23	if	if	SCONJ
cana-4781	41	24	𝑓	𝑓	PRON
cana-4781	41	25	is	be	AUX
cana-4781	41	26	a	a	DET
cana-4781	41	27	rough	rough	ADJ
cana-4781	41	28	homomorphism	homomorphism	NOUN
cana-4781	41	29	and	and	CCONJ
cana-4781	41	30	continuous	continuous	ADJ
cana-4781	41	31	with	with	ADP
cana-4781	41	32	respect	respect	NOUN
cana-4781	41	33	to	to	ADP
cana-4781	41	34	the	the	DET
cana-4781	41	35	topology	topology	NOUN
cana-4781	41	36	𝜏2	𝜏2	NOUN
cana-4781	41	37	on	on	ADP
cana-4781	41	38	𝐺2	𝐺2	PROPN
cana-4781	41	39	̅̅	̅̅	PROPN
cana-4781	41	40	̅	̅	NOUN
cana-4781	41	41	inducing	induce	VERB
cana-4781	41	42	𝜏𝐺2	𝜏𝐺2	NOUN
cana-4781	41	43	on	on	ADP
cana-4781	41	44	𝐺2	𝐺2	NOUN
cana-4781	41	45	and	and	CCONJ
cana-4781	41	46	a	a	DET
cana-4781	41	47	topology	topology	NOUN
cana-4781	41	48	𝜏1	𝜏1	NOUN
cana-4781	41	49	on	on	ADP
cana-4781	41	50	𝐺1	𝐺1	PROPN
cana-4781	41	51	̅̅	̅̅	PROPN
cana-4781	41	52	̅	̅	NOUN
cana-4781	41	53	inducing	induce	VERB
cana-4781	41	54	𝜏𝐺1	𝜏𝐺1	PROPN
cana-4781	41	55	on	on	ADP
cana-4781	41	56	𝐺1	𝐺1	NOUN
cana-4781	41	57	.	.	PUNCT
cana-4781	42	1	definition	definition	NOUN
cana-4781	42	2	2.6	2.6	NUM
cana-4781	42	3	.	.	PUNCT
cana-4781	43	1	[	[	X
cana-4781	43	2	1	1	X
cana-4781	43	3	]	]	X
cana-4781	43	4	topological	topological	ADJ
cana-4781	43	5	rough	rough	ADJ
cana-4781	43	6	group	group	NOUN
cana-4781	43	7	homomorphism	homomorphism	NOUN
cana-4781	43	8	𝑓	𝑓	X
cana-4781	43	9	:	:	PUNCT
cana-4781	43	10	𝐺1	𝐺1	PROPN
cana-4781	43	11	̅̅	̅̅	PROPN
cana-4781	43	12	̅	̅	PROPN
cana-4781	43	13	→	→	PUNCT
cana-4781	43	14	𝐺2	𝐺2	ADJ
cana-4781	43	15	̅̅	̅̅	PROPN
cana-4781	43	16	̅	̅	NOUN
cana-4781	43	17	is	be	AUX
cana-4781	43	18	called	call	VERB
cana-4781	43	19	a	a	DET
cana-4781	43	20	topological	topological	ADJ
cana-4781	43	21	rough	rough	ADJ
cana-4781	43	22	group	group	NOUN
cana-4781	43	23	homeomorphism	homeomorphism	NOUN
cana-4781	43	24	,	,	PUNCT
cana-4781	43	25	if	if	SCONJ
cana-4781	43	26	there	there	PRON
cana-4781	43	27	exists	exist	VERB
cana-4781	43	28	a	a	DET
cana-4781	43	29	topological	topological	ADJ
cana-4781	43	30	rough	rough	ADJ
cana-4781	43	31	homomorphism	homomorphism	NOUN
cana-4781	43	32	𝑓−1	𝑓−1	PRON
cana-4781	43	33	such	such	ADJ
cana-4781	43	34	that	that	SCONJ
cana-4781	43	35	𝑓−1	𝑓−1	NUM
cana-4781	43	36	∘	∘	NUM
cana-4781	43	37	𝑓	𝑓	PROPN
cana-4781	43	38	=	=	ADJ
cana-4781	43	39	1𝐺1	1𝐺1	NUM
cana-4781	43	40	.	.	PUNCT
cana-4781	44	1	definition	definition	NOUN
cana-4781	44	2	2.7	2.7	NUM
cana-4781	44	3	.	.	PUNCT
cana-4781	45	1	[	[	X
cana-4781	45	2	1	1	X
cana-4781	45	3	]	]	PUNCT
cana-4781	45	4	let	let	VERB
cana-4781	45	5	φ	φ	NOUN
cana-4781	45	6	:	:	PUNCT
cana-4781	45	7	𝐺1	𝐺1	PROPN
cana-4781	45	8	̅̅	̅̅	PROPN
cana-4781	45	9	̅	̅	PROPN
cana-4781	45	10	→	→	PUNCT
cana-4781	45	11	𝐺2	𝐺2	ADJ
cana-4781	45	12	̅̅	̅̅	PROPN
cana-4781	45	13	̅	̅	NOUN
cana-4781	45	14	be	be	VERB
cana-4781	45	15	a	a	DET
cana-4781	45	16	topological	topological	ADJ
cana-4781	45	17	rough	rough	ADJ
cana-4781	45	18	group	group	NOUN
cana-4781	45	19	homomorphism	homomorphism	NOUN
cana-4781	45	20	and	and	CCONJ
cana-4781	45	21	let	let	VERB
cana-4781	45	22	𝑒2	𝑒2	PROPN
cana-4781	45	23	be	be	AUX
cana-4781	45	24	the	the	DET
cana-4781	45	25	rough	rough	ADJ
cana-4781	45	26	identity	identity	NOUN
cana-4781	45	27	element	element	NOUN
cana-4781	45	28	in	in	ADP
cana-4781	45	29	𝐺2	𝐺2	NOUN
cana-4781	45	30	.	.	PUNCT
cana-4781	46	1	then	then	ADV
cana-4781	46	2	ker(φ	ker(φ	X
cana-4781	46	3	)	)	PUNCT
cana-4781	46	4	=	=	PRON
cana-4781	46	5	{	{	PUNCT
cana-4781	46	6	𝑔	𝑔	PART
cana-4781	46	7	∈	∈	PROPN
cana-4781	46	8	𝐺1	𝐺1	NOUN
cana-4781	46	9	∶	∶	NOUN
cana-4781	46	10	φ(𝑔	φ(𝑔	NOUN
cana-4781	46	11	)	)	PUNCT
cana-4781	46	12	=	=	SYM
cana-4781	46	13	𝑒2	𝑒2	PROPN
cana-4781	46	14	}	}	PUNCT
cana-4781	46	15	.	.	PUNCT
cana-4781	47	1	is	be	AUX
cana-4781	47	2	called	call	VERB
cana-4781	47	3	the	the	DET
cana-4781	47	4	rough	rough	ADJ
cana-4781	47	5	kernel	kernel	NOUN
cana-4781	47	6	associated	associate	VERB
cana-4781	47	7	to	to	ADP
cana-4781	47	8	the	the	DET
cana-4781	47	9	map	map	NOUN
cana-4781	47	10	φ	φ	NOUN
cana-4781	47	11	.	.	PUNCT
cana-4781	48	1	definition	definition	NOUN
cana-4781	48	2	2.8	2.8	NUM
cana-4781	48	3	.	.	PUNCT
cana-4781	49	1	[	[	X
cana-4781	49	2	12	12	NUM
cana-4781	49	3	]	]	PUNCT
cana-4781	49	4	let	let	VERB
cana-4781	49	5	g	g	PRON
cana-4781	49	6	be	be	AUX
cana-4781	49	7	a	a	DET
cana-4781	49	8	rough	rough	ADJ
cana-4781	49	9	group	group	NOUN
cana-4781	49	10	such	such	ADJ
cana-4781	49	11	that	that	SCONJ
cana-4781	49	12	g̅	g̅	PROPN
cana-4781	49	13	is	be	AUX
cana-4781	49	14	a	a	DET
cana-4781	49	15	group	group	NOUN
cana-4781	49	16	and	and	CCONJ
cana-4781	49	17	h	h	NOUN
cana-4781	49	18	is	be	AUX
cana-4781	49	19	a	a	DET
cana-4781	49	20	rough	rough	ADJ
cana-4781	49	21	subgroup	subgroup	NOUN
cana-4781	49	22	of	of	ADP
cana-4781	49	23	g.	g.	PROPN
cana-4781	49	24	if	if	SCONJ
cana-4781	49	25	h	h	NOUN
cana-4781	49	26	is	be	AUX
cana-4781	49	27	a	a	DET
cana-4781	49	28	normal	normal	ADJ
cana-4781	49	29	subgroup	subgroup	NOUN
cana-4781	49	30	in	in	ADP
cana-4781	49	31	g̅	g̅	PROPN
cana-4781	49	32	,	,	PUNCT
cana-4781	49	33	then	then	ADV
cana-4781	49	34	g̅	g̅	PROPN
cana-4781	49	35	𝐻⁄	𝐻⁄	PROPN
cana-4781	49	36	is	be	AUX
cana-4781	49	37	a	a	DET
cana-4781	49	38	rough	rough	ADJ
cana-4781	49	39	quotient	quotient	NOUN
cana-4781	49	40	group	group	NOUN
cana-4781	49	41	.	.	PUNCT
cana-4781	50	1	definition	definition	NOUN
cana-4781	50	2	2.9	2.9	NUM
cana-4781	50	3	.	.	PUNCT
cana-4781	51	1	[	[	X
cana-4781	51	2	16	16	NUM
cana-4781	51	3	]	]	X
cana-4781	51	4	a	a	DET
cana-4781	51	5	rough	rough	ADJ
cana-4781	51	6	group	group	NOUN
cana-4781	51	7	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	51	8	is	be	AUX
cana-4781	51	9	called	call	VERB
cana-4781	51	10	a	a	DET
cana-4781	51	11	simple	simple	ADJ
cana-4781	51	12	rough	rough	ADJ
cana-4781	51	13	group	group	NOUN
cana-4781	51	14	if	if	SCONJ
cana-4781	51	15	it	it	PRON
cana-4781	51	16	contains	contain	VERB
cana-4781	51	17	no	no	DET
cana-4781	51	18	proper	proper	ADJ
cana-4781	51	19	nontrivial	nontrivial	ADJ
cana-4781	51	20	rough	rough	ADJ
cana-4781	51	21	normal	normal	ADJ
cana-4781	51	22	subgroups	subgroup	NOUN
cana-4781	51	23	.	.	PUNCT
cana-4781	52	1	that	that	PRON
cana-4781	52	2	is	is	ADV
cana-4781	52	3	,	,	PUNCT
cana-4781	52	4	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	52	5	has	have	VERB
cana-4781	52	6	only	only	ADV
cana-4781	52	7	the	the	DET
cana-4781	52	8	rough	rough	ADJ
cana-4781	52	9	normal	normal	ADJ
cana-4781	52	10	subgroups	subgroup	NOUN
cana-4781	52	11	{	{	PUNCT
cana-4781	52	12	e	e	NOUN
cana-4781	52	13	}	}	PUNCT
cana-4781	52	14	and	and	CCONJ
cana-4781	52	15	𝐺ℜ.	𝐺ℜ.	PUNCT
cana-4781	52	16	communications	communication	NOUN
cana-4781	52	17	on	on	ADP
cana-4781	52	18	applied	apply	VERB
cana-4781	52	19	nonlinear	nonlinear	ADJ
cana-4781	52	20	analysis	analysis	NOUN
cana-4781	52	21	issn	issn	NOUN
cana-4781	52	22	:	:	PUNCT
cana-4781	52	23	1074	1074	NUM
cana-4781	52	24	-	-	PUNCT
cana-4781	52	25	133x	133x	NUM
cana-4781	52	26	vol	vol	NOUN
cana-4781	52	27	32	32	NUM
cana-4781	52	28	no	no	NOUN
cana-4781	52	29	.	.	NOUN
cana-4781	52	30	3	3	NUM
cana-4781	52	31	(	(	PUNCT
cana-4781	52	32	2025	2025	NUM
cana-4781	52	33	)	)	PUNCT
cana-4781	52	34	881	881	NUM
cana-4781	52	35	https://internationalpubls.com	https://internationalpubls.com	X
cana-4781	52	36	definition	definition	NOUN
cana-4781	52	37	2.10	2.10	NUM
cana-4781	52	38	.	.	PUNCT
cana-4781	53	1	[	[	X
cana-4781	53	2	16	16	NUM
cana-4781	53	3	]	]	X
cana-4781	53	4	a	a	DET
cana-4781	53	5	topological	topological	ADJ
cana-4781	53	6	simple	simple	ADJ
cana-4781	53	7	rough	rough	ADJ
cana-4781	53	8	group	group	NOUN
cana-4781	53	9	is	be	AUX
cana-4781	53	10	a	a	DET
cana-4781	53	11	simple	simple	ADJ
cana-4781	53	12	rough	rough	ADJ
cana-4781	53	13	group	group	NOUN
cana-4781	53	14	(	(	PUNCT
cana-4781	53	15	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	53	16	,	,	PUNCT
cana-4781	53	17	∗	∗	NOUN
cana-4781	53	18	)	)	PUNCT
cana-4781	53	19	together	together	ADV
cana-4781	53	20	with	with	ADP
cana-4781	53	21	a	a	DET
cana-4781	53	22	topology	topology	NOUN
cana-4781	53	23	τ̅	τ̅	PUNCT
cana-4781	53	24	on	on	ADP
cana-4781	53	25	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	53	26	̅̅	̅̅	PROPN
cana-4781	53	27	̅̅	̅̅	PROPN
cana-4781	53	28	satisfying	satisfy	VERB
cana-4781	53	29	the	the	DET
cana-4781	53	30	following	follow	VERB
cana-4781	53	31	two	two	NUM
cana-4781	53	32	properties	property	NOUN
cana-4781	53	33	:	:	PUNCT
cana-4781	53	34	(	(	PUNCT
cana-4781	53	35	i	i	NOUN
cana-4781	53	36	)	)	PUNCT
cana-4781	53	37	the	the	DET
cana-4781	53	38	mapping	mapping	NOUN
cana-4781	53	39	f	f	NOUN
cana-4781	53	40	:	:	PUNCT
cana-4781	53	41	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	53	42	×	×	NOUN
cana-4781	53	43	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	53	44	→	→	SYM
cana-4781	53	45	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	53	46	̅̅	̅̅	NOUN
cana-4781	53	47	̅̅	̅̅	PROPN
cana-4781	53	48	defined	define	VERB
cana-4781	53	49	by	by	ADP
cana-4781	53	50	f(x	f(x	PROPN
cana-4781	53	51	,	,	PUNCT
cana-4781	53	52	y	y	PROPN
cana-4781	53	53	)	)	PUNCT
cana-4781	53	54	=	=	SYM
cana-4781	53	55	xy	xy	PROPN
cana-4781	53	56	,	,	PUNCT
cana-4781	53	57	x	x	PRON
cana-4781	53	58	,	,	PUNCT
cana-4781	53	59	y	y	PROPN
cana-4781	53	60	∈	∈	PROPN
cana-4781	53	61	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	53	62	is	be	AUX
cana-4781	53	63	continuous	continuous	ADJ
cana-4781	53	64	with	with	ADP
cana-4781	53	65	respect	respect	NOUN
cana-4781	53	66	to	to	ADP
cana-4781	53	67	the	the	DET
cana-4781	53	68	product	product	NOUN
cana-4781	53	69	topology	topology	NOUN
cana-4781	53	70	on	on	ADP
cana-4781	53	71	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	53	72	×	×	PROPN
cana-4781	53	73	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	53	74	and	and	CCONJ
cana-4781	53	75	the	the	DET
cana-4781	53	76	topology	topology	NOUN
cana-4781	53	77	τ	τ	PROPN
cana-4781	53	78	on	on	ADP
cana-4781	53	79	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	53	80	induced	induce	VERB
cana-4781	53	81	by	by	ADP
cana-4781	53	82	τ̅	τ̅	X
cana-4781	53	83	(	(	PUNCT
cana-4781	53	84	ii	ii	NOUN
cana-4781	53	85	)	)	PUNCT
cana-4781	53	86	the	the	DET
cana-4781	53	87	inverse	inverse	NOUN
cana-4781	53	88	mapping	mapping	NOUN
cana-4781	53	89	g	g	NOUN
cana-4781	53	90	:	:	PUNCT
cana-4781	53	91	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	53	92	→	→	SYM
cana-4781	53	93	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	53	94	defined	define	VERB
cana-4781	53	95	by	by	ADP
cana-4781	53	96	g(x	g(x	NOUN
cana-4781	53	97	)	)	PUNCT
cana-4781	54	1	=	=	SYM
cana-4781	54	2	x-1	x-1	PROPN
cana-4781	54	3	,	,	PUNCT
cana-4781	54	4	x	x	PROPN
cana-4781	54	5	∈	∈	PROPN
cana-4781	54	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	54	7	is	be	AUX
cana-4781	54	8	continuous	continuous	ADJ
cana-4781	54	9	with	with	ADP
cana-4781	54	10	respect	respect	NOUN
cana-4781	54	11	to	to	ADP
cana-4781	54	12	the	the	DET
cana-4781	54	13	topology	topology	NOUN
cana-4781	54	14	τ	τ	PROPN
cana-4781	54	15	on	on	ADP
cana-4781	54	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	54	17	induced	induce	VERB
cana-4781	54	18	by	by	ADP
cana-4781	54	19	τ̅.	τ̅.	NOUN
cana-4781	54	20	proposition	proposition	NOUN
cana-4781	54	21	2.11	2.11	NUM
cana-4781	54	22	.	.	PUNCT
cana-4781	55	1	[	[	X
cana-4781	55	2	16	16	NUM
cana-4781	55	3	]	]	PUNCT
cana-4781	55	4	let	let	VERB
cana-4781	55	5	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	55	6	be	be	AUX
cana-4781	55	7	a	a	DET
cana-4781	55	8	topological	topological	ADJ
cana-4781	55	9	simple	simple	ADJ
cana-4781	55	10	rough	rough	ADJ
cana-4781	55	11	group	group	NOUN
cana-4781	55	12	.	.	PUNCT
cana-4781	56	1	if	if	SCONJ
cana-4781	56	2	u	u	PROPN
cana-4781	56	3	⊆	⊆	NUM
cana-4781	56	4	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	56	5	̅̅	̅̅	PROPN
cana-4781	56	6	̅̅	̅̅	PROPN
cana-4781	56	7	is	be	AUX
cana-4781	56	8	an	an	DET
cana-4781	56	9	open	open	ADJ
cana-4781	56	10	set	set	NOUN
cana-4781	56	11	with	with	ADP
cana-4781	56	12	e	e	PROPN
cana-4781	56	13	∈	∈	PROPN
cana-4781	56	14	u	u	NOUN
cana-4781	56	15	,	,	PUNCT
cana-4781	56	16	then	then	ADV
cana-4781	56	17	there	there	PRON
cana-4781	56	18	exists	exist	VERB
cana-4781	56	19	a	a	DET
cana-4781	56	20	symmetric	symmetric	ADJ
cana-4781	56	21	open	open	NOUN
cana-4781	56	22	set	set	VERB
cana-4781	56	23	v	v	NOUN
cana-4781	56	24	of	of	ADP
cana-4781	56	25	e	e	PROPN
cana-4781	56	26	in	in	ADP
cana-4781	56	27	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	56	28	such	such	ADJ
cana-4781	56	29	that	that	PRON
cana-4781	56	30	vv	vv	PROPN
cana-4781	56	31	⊆	⊆	NUM
cana-4781	56	32	u.	u.	PROPN
cana-4781	56	33	lemma	lemma	PROPN
cana-4781	56	34	2.12	2.12	NUM
cana-4781	56	35	.	.	PUNCT
cana-4781	57	1	[	[	X
cana-4781	57	2	17	17	NUM
cana-4781	57	3	]	]	PUNCT
cana-4781	57	4	let	let	VERB
cana-4781	57	5	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	57	6	be	be	AUX
cana-4781	57	7	a	a	DET
cana-4781	57	8	topological	topological	ADJ
cana-4781	57	9	simple	simple	ADJ
cana-4781	57	10	rough	rough	ADJ
cana-4781	57	11	group	group	NOUN
cana-4781	57	12	such	such	ADJ
cana-4781	57	13	that	that	SCONJ
cana-4781	57	14	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	57	15	is	be	AUX
cana-4781	57	16	open	open	ADJ
cana-4781	57	17	in	in	ADP
cana-4781	57	18	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	57	19	̅̅	̅̅	NOUN
cana-4781	57	20	̅̅	̅̅	PROPN
cana-4781	57	21	and	and	CCONJ
cana-4781	57	22	𝑊	𝑊	PROPN
cana-4781	57	23	be	be	VERB
cana-4781	57	24	a	a	DET
cana-4781	57	25	neighbourhood	neighbourhood	NOUN
cana-4781	57	26	of	of	ADP
cana-4781	57	27	𝑒	𝑒	PROPN
cana-4781	57	28	in	in	ADP
cana-4781	57	29	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	57	30	̅̅	̅̅	PROPN
cana-4781	57	31	̅̅	̅̅	PROPN
cana-4781	57	32	.	.	PUNCT
cana-4781	58	1	then	then	ADV
cana-4781	58	2	there	there	PRON
cana-4781	58	3	is	be	VERB
cana-4781	58	4	an	an	DET
cana-4781	58	5	open	open	ADJ
cana-4781	58	6	set	set	NOUN
cana-4781	58	7	𝑈	𝑈	PROPN
cana-4781	58	8	of	of	ADP
cana-4781	58	9	𝑒	𝑒	PROPN
cana-4781	58	10	in	in	ADP
cana-4781	58	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	58	12	such	such	ADJ
cana-4781	58	13	that	that	SCONJ
cana-4781	58	14	𝑈	𝑈	PROPN
cana-4781	58	15	⊆	⊆	NUM
cana-4781	58	16	𝑈𝑛	𝑈𝑛	PROPN
cana-4781	58	17	⊆	⊆	NUM
cana-4781	58	18	𝑊	𝑊	PROPN
cana-4781	58	19	,	,	PUNCT
cana-4781	58	20	for	for	ADP
cana-4781	58	21	every	every	DET
cana-4781	58	22	𝑛	𝑛	PRON
cana-4781	58	23	∈	∈	PROPN
cana-4781	58	24	ℕ	ℕ	PROPN
cana-4781	58	25	−	−	PROPN
cana-4781	58	26	{	{	PUNCT
cana-4781	58	27	0	0	NUM
cana-4781	58	28	}	}	PUNCT
cana-4781	58	29	.	.	PUNCT
cana-4781	59	1	lemma	lemma	PROPN
cana-4781	59	2	2.13	2.13	NUM
cana-4781	59	3	.	.	PUNCT
cana-4781	60	1	[	[	X
cana-4781	60	2	7	7	X
cana-4781	60	3	]	]	X
cana-4781	60	4	let	let	VERB
cana-4781	60	5	y	y	PRON
cana-4781	60	6	be	be	AUX
cana-4781	60	7	a	a	DET
cana-4781	60	8	subspace	subspace	NOUN
cana-4781	60	9	of	of	ADP
cana-4781	60	10	x.	x.	NOUN
cana-4781	60	11	if	if	SCONJ
cana-4781	60	12	u	u	NOUN
cana-4781	60	13	is	be	AUX
cana-4781	60	14	open	open	ADJ
cana-4781	60	15	in	in	ADP
cana-4781	60	16	y	y	PROPN
cana-4781	60	17	and	and	CCONJ
cana-4781	60	18	y	y	PROPN
cana-4781	60	19	is	be	AUX
cana-4781	60	20	open	open	ADJ
cana-4781	60	21	in	in	ADP
cana-4781	60	22	x	x	NOUN
cana-4781	60	23	,	,	PUNCT
cana-4781	60	24	then	then	ADV
cana-4781	60	25	u	u	NOUN
cana-4781	60	26	is	be	AUX
cana-4781	60	27	open	open	ADJ
cana-4781	60	28	in	in	ADP
cana-4781	60	29	x.	x.	NOUN
cana-4781	60	30	theorem	theorem	VERB
cana-4781	60	31	2.14	2.14	NUM
cana-4781	60	32	.	.	PUNCT
cana-4781	61	1	[	[	X
cana-4781	61	2	5	5	NUM
cana-4781	61	3	]	]	PUNCT
cana-4781	61	4	every	every	DET
cana-4781	61	5	locally	locally	ADV
cana-4781	61	6	compact	compact	ADJ
cana-4781	61	7	subspace	subspace	NOUN
cana-4781	61	8	m	m	VERB
cana-4781	61	9	of	of	ADP
cana-4781	61	10	a	a	DET
cana-4781	61	11	hausdorff	hausdorff	NOUN
cana-4781	61	12	space	space	NOUN
cana-4781	61	13	x	x	PRON
cana-4781	61	14	is	be	AUX
cana-4781	61	15	an	an	DET
cana-4781	61	16	open	open	ADJ
cana-4781	61	17	subset	subset	NOUN
cana-4781	61	18	of	of	ADP
cana-4781	61	19	the	the	DET
cana-4781	61	20	closure	closure	NOUN
cana-4781	61	21	�	�	NOUN
cana-4781	61	22	̅	̅	NOUN
cana-4781	61	23	�	�	NOUN
cana-4781	61	24	of	of	ADP
cana-4781	61	25	the	the	DET
cana-4781	61	26	set	set	NOUN
cana-4781	61	27	m	m	PROPN
cana-4781	61	28	in	in	ADP
cana-4781	61	29	the	the	DET
cana-4781	61	30	space	space	NOUN
cana-4781	61	31	x.	x.	NOUN
cana-4781	61	32	remark	remark	VERB
cana-4781	61	33	2.15	2.15	NUM
cana-4781	61	34	.	.	PUNCT
cana-4781	62	1	[	[	X
cana-4781	62	2	16	16	NUM
cana-4781	62	3	]	]	PUNCT
cana-4781	62	4	the	the	DET
cana-4781	62	5	topological	topological	ADJ
cana-4781	62	6	closure	closure	NOUN
cana-4781	62	7	of	of	ADP
cana-4781	62	8	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	62	9	,	,	PUNCT
cana-4781	62	10	𝑐𝑙(𝐻ℜ	𝑐𝑙(𝐻ℜ	NOUN
cana-4781	62	11	)	)	PUNCT
cana-4781	62	12	,	,	PUNCT
cana-4781	62	13	in	in	ADP
cana-4781	62	14	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	62	15	̅̅	̅̅	PROPN
cana-4781	62	16	̅̅	̅̅	PROPN
cana-4781	62	17	is	be	AUX
cana-4781	62	18	a	a	DET
cana-4781	62	19	topological	topological	ADJ
cana-4781	62	20	rough	rough	ADJ
cana-4781	62	21	subgroup	subgroup	NOUN
cana-4781	62	22	in	in	ADP
cana-4781	62	23	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	62	24	̅̅	̅̅	PROPN
cana-4781	62	25	̅̅	̅̅	PROPN
cana-4781	62	26	.	.	PUNCT
cana-4781	63	1	definition	definition	NOUN
cana-4781	63	2	2.16	2.16	NUM
cana-4781	63	3	.	.	PUNCT
cana-4781	64	1	[	[	X
cana-4781	64	2	16	16	NUM
cana-4781	64	3	]	]	PUNCT
cana-4781	64	4	a	a	DET
cana-4781	64	5	continuous	continuous	ADJ
cana-4781	64	6	mapping	mapping	NOUN
cana-4781	64	7	𝑓	𝑓	X
cana-4781	64	8	:	:	PUNCT
cana-4781	64	9	𝑋	𝑋	PROPN
cana-4781	64	10	→	→	SYM
cana-4781	64	11	𝑌	𝑌	PROPN
cana-4781	64	12	is	be	AUX
cana-4781	64	13	perfect	perfect	ADJ
cana-4781	64	14	if	if	SCONJ
cana-4781	64	15	x	x	PRON
cana-4781	64	16	is	be	AUX
cana-4781	64	17	a	a	DET
cana-4781	64	18	hausdorff	hausdorff	NOUN
cana-4781	64	19	space	space	NOUN
cana-4781	64	20	,	,	PUNCT
cana-4781	64	21	𝑓	𝑓	PRON
cana-4781	64	22	is	be	AUX
cana-4781	64	23	a	a	DET
cana-4781	64	24	closed	closed	ADJ
cana-4781	64	25	mapping	mapping	NOUN
cana-4781	64	26	and	and	CCONJ
cana-4781	64	27	all	all	DET
cana-4781	64	28	fibers	fiber	NOUN
cana-4781	64	29	𝑓−1(𝑦	𝑓−1(𝑦	PROPN
cana-4781	64	30	)	)	PUNCT
cana-4781	64	31	are	be	AUX
cana-4781	64	32	compact	compact	ADJ
cana-4781	64	33	subsets	subset	NOUN
cana-4781	64	34	of	of	ADP
cana-4781	64	35	x.	x.	NOUN
cana-4781	64	36	result	result	NOUN
cana-4781	64	37	2.17	2.17	NUM
cana-4781	64	38	.	.	PUNCT
cana-4781	65	1	[	[	X
cana-4781	65	2	5	5	NUM
cana-4781	65	3	]	]	PUNCT
cana-4781	65	4	a	a	DET
cana-4781	65	5	𝑇1space	𝑇1space	NOUN
cana-4781	65	6	𝑋	𝑋	NOUN
cana-4781	65	7	is	be	AUX
cana-4781	65	8	a	a	DET
cana-4781	65	9	regular	regular	ADJ
cana-4781	65	10	space	space	NOUN
cana-4781	65	11	if	if	SCONJ
cana-4781	65	12	and	and	CCONJ
cana-4781	65	13	only	only	ADV
cana-4781	65	14	if	if	SCONJ
cana-4781	65	15	for	for	ADP
cana-4781	65	16	every	every	DET
cana-4781	65	17	𝑥	𝑥	DET
cana-4781	65	18	∈	∈	PROPN
cana-4781	65	19	𝑋	𝑋	NOUN
cana-4781	65	20	and	and	CCONJ
cana-4781	65	21	every	every	DET
cana-4781	65	22	neighbourhood	neighbourhood	NOUN
cana-4781	65	23	𝑉	𝑉	PROPN
cana-4781	65	24	of	of	ADP
cana-4781	65	25	𝑥	𝑥	PRON
cana-4781	65	26	there	there	PRON
cana-4781	65	27	exists	exist	VERB
cana-4781	65	28	a	a	DET
cana-4781	65	29	neighbourhood	neighbourhood	NOUN
cana-4781	65	30	𝑈	𝑈	NOUN
cana-4781	65	31	of	of	ADP
cana-4781	65	32	𝑥	𝑥	PRON
cana-4781	65	33	such	such	ADJ
cana-4781	65	34	that	that	SCONJ
cana-4781	65	35	𝑉	𝑉	PROPN
cana-4781	65	36	⊆	⊆	NUM
cana-4781	65	37	𝑈.	𝑈.	PROPN
cana-4781	65	38	theorem	theorem	VERB
cana-4781	65	39	2.18	2.18	NUM
cana-4781	65	40	.	.	PUNCT
cana-4781	66	1	[	[	X
cana-4781	66	2	5	5	NUM
cana-4781	66	3	]	]	PUNCT
cana-4781	66	4	a	a	DET
cana-4781	66	5	continuous	continuous	ADJ
cana-4781	66	6	mapping	mapping	NOUN
cana-4781	66	7	𝑓	𝑓	X
cana-4781	66	8	:	:	PUNCT
cana-4781	66	9	𝑋	𝑋	PROPN
cana-4781	66	10	→	→	SYM
cana-4781	66	11	𝑌	𝑌	PROPN
cana-4781	66	12	is	be	AUX
cana-4781	66	13	closed	close	VERB
cana-4781	66	14	if	if	SCONJ
cana-4781	66	15	and	and	CCONJ
cana-4781	66	16	only	only	ADV
cana-4781	66	17	if	if	SCONJ
cana-4781	66	18	for	for	ADP
cana-4781	66	19	every	every	DET
cana-4781	66	20	point	point	NOUN
cana-4781	66	21	𝑦	𝑦	NOUN
cana-4781	66	22	∈	∈	NOUN
cana-4781	66	23	𝑌	𝑌	PROPN
cana-4781	66	24	and	and	CCONJ
cana-4781	66	25	every	every	PRON
cana-4781	66	26	open	open	ADJ
cana-4781	66	27	set	set	NOUN
cana-4781	66	28	𝑈	𝑈	PROPN
cana-4781	66	29	⊂	⊂	PROPN
cana-4781	66	30	𝑋	𝑋	PROPN
cana-4781	66	31	which	which	PRON
cana-4781	66	32	contains	contain	VERB
cana-4781	66	33	𝑓−1(𝑦	𝑓−1(𝑦	PROPN
cana-4781	66	34	)	)	PUNCT
cana-4781	66	35	,	,	PUNCT
cana-4781	66	36	there	there	PRON
cana-4781	66	37	exists	exist	VERB
cana-4781	66	38	a	a	DET
cana-4781	66	39	neighbourhood	neighbourhood	NOUN
cana-4781	66	40	v	v	NOUN
cana-4781	66	41	of	of	ADP
cana-4781	66	42	the	the	DET
cana-4781	66	43	point	point	NOUN
cana-4781	66	44	y	y	PROPN
cana-4781	66	45	in	in	ADP
cana-4781	66	46	y	y	PRON
cana-4781	66	47	such	such	ADJ
cana-4781	66	48	that	that	DET
cana-4781	66	49	𝑓−1(𝑉	𝑓−1(𝑉	NOUN
cana-4781	66	50	)	)	PUNCT
cana-4781	67	1	⊂	⊂	PROPN
cana-4781	67	2	𝑈.	𝑈.	PROPN
cana-4781	67	3	throughout	throughout	ADP
cana-4781	67	4	this	this	DET
cana-4781	67	5	paper	paper	NOUN
cana-4781	67	6	,	,	PUNCT
cana-4781	67	7	we	we	PRON
cana-4781	67	8	consider	consider	VERB
cana-4781	67	9	𝑋	𝑋	NOUN
cana-4781	67	10	be	be	AUX
cana-4781	67	11	the	the	DET
cana-4781	67	12	universal	universal	ADJ
cana-4781	67	13	set	set	NOUN
cana-4781	67	14	,	,	PUNCT
cana-4781	67	15	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	67	16	be	be	AUX
cana-4781	67	17	a	a	DET
cana-4781	67	18	simple	simple	ADJ
cana-4781	67	19	rough	rough	ADJ
cana-4781	67	20	group	group	NOUN
cana-4781	67	21	with	with	ADP
cana-4781	67	22	identity	identity	NOUN
cana-4781	67	23	𝑒	𝑒	NOUN
cana-4781	67	24	and	and	CCONJ
cana-4781	67	25	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	67	26	̅̅	̅̅	PROPN
cana-4781	67	27	̅̅	̅̅	PROPN
cana-4781	67	28	be	be	AUX
cana-4781	67	29	the	the	DET
cana-4781	67	30	upper	upper	ADJ
cana-4781	67	31	rough	rough	ADJ
cana-4781	67	32	approximation	approximation	NOUN
cana-4781	67	33	of	of	ADP
cana-4781	67	34	𝐺ℜ.	𝐺ℜ.	PUNCT
cana-4781	67	35	also	also	ADV
cana-4781	67	36	,	,	PUNCT
cana-4781	67	37	the	the	DET
cana-4781	67	38	corresponding	corresponding	ADJ
cana-4781	67	39	topologies	topology	NOUN
cana-4781	67	40	are	be	AUX
cana-4781	67	41	denoted	denote	VERB
cana-4781	67	42	by	by	ADP
cana-4781	67	43	τ̅	τ̅	X
cana-4781	67	44	for	for	ADP
cana-4781	67	45	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	67	46	̅̅	̅̅	PROPN
cana-4781	67	47	̅̅	̅̅	PROPN
cana-4781	67	48	and	and	CCONJ
cana-4781	67	49	τ	τ	PROPN
cana-4781	67	50	for	for	ADP
cana-4781	67	51	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	67	52	induced	induce	VERB
cana-4781	67	53	from	from	ADP
cana-4781	67	54	τ̅.	τ̅.	NOUN
cana-4781	67	55	3	3	NUM
cana-4781	67	56	.	.	PUNCT
cana-4781	67	57	compactness	compactness	NOUN
cana-4781	67	58	:	:	PUNCT
cana-4781	67	59	theorem	theorem	VERB
cana-4781	67	60	3.1	3.1	NUM
cana-4781	67	61	.	.	PUNCT
cana-4781	68	1	let	let	VERB
cana-4781	68	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	68	3	be	be	AUX
cana-4781	68	4	a	a	DET
cana-4781	68	5	topological	topological	ADJ
cana-4781	68	6	simple	simple	ADJ
cana-4781	68	7	rough	rough	ADJ
cana-4781	68	8	group	group	NOUN
cana-4781	68	9	such	such	ADJ
cana-4781	68	10	that	that	SCONJ
cana-4781	68	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	68	12	is	be	AUX
cana-4781	68	13	open	open	ADJ
cana-4781	68	14	in	in	ADP
cana-4781	68	15	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	68	16	̅̅	̅̅	PROPN
cana-4781	68	17	̅̅	̅̅	PROPN
cana-4781	68	18	and	and	CCONJ
cana-4781	68	19	𝐴	𝐴	PROPN
cana-4781	68	20	be	be	VERB
cana-4781	68	21	a	a	DET
cana-4781	68	22	compact	compact	ADJ
cana-4781	68	23	subset	subset	NOUN
cana-4781	68	24	of	of	ADP
cana-4781	68	25	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	68	26	̅̅	̅̅	PROPN
cana-4781	68	27	̅̅	̅̅	PROPN
cana-4781	68	28	.	.	PUNCT
cana-4781	69	1	if	if	SCONJ
cana-4781	69	2	𝑀	𝑀	PROPN
cana-4781	69	3	is	be	AUX
cana-4781	69	4	a	a	DET
cana-4781	69	5	closed	closed	ADJ
cana-4781	69	6	subset	subset	NOUN
cana-4781	69	7	of	of	ADP
cana-4781	69	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	69	9	̅̅	̅̅	PROPN
cana-4781	69	10	̅̅	̅̅	PROPN
cana-4781	69	11	with	with	ADP
cana-4781	69	12	𝐴	𝐴	PROPN
cana-4781	69	13	∩	∩	ADJ
cana-4781	69	14	𝑀	𝑀	NOUN
cana-4781	69	15	=	=	NOUN
cana-4781	69	16	∅	∅	NOUN
cana-4781	69	17	,	,	PUNCT
cana-4781	69	18	then	then	ADV
cana-4781	69	19	there	there	PRON
cana-4781	69	20	is	be	VERB
cana-4781	69	21	an	an	DET
cana-4781	69	22	open	open	ADJ
cana-4781	69	23	neighbourhood	neighbourhood	NOUN
cana-4781	69	24	𝑉	𝑉	PROPN
cana-4781	69	25	of	of	ADP
cana-4781	69	26	e	e	PROPN
cana-4781	69	27	in	in	ADP
cana-4781	69	28	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	69	29	such	such	ADJ
cana-4781	69	30	that	that	SCONJ
cana-4781	69	31	𝐴𝑉	𝐴𝑉	PROPN
cana-4781	69	32	∩	∩	NOUN
cana-4781	69	33	𝑀	𝑀	NOUN
cana-4781	69	34	=	=	NOUN
cana-4781	69	35	∅	∅	NOUN
cana-4781	69	36	and	and	CCONJ
cana-4781	69	37	𝑉𝐴	𝑉𝐴	PROPN
cana-4781	69	38	∩	∩	NOUN
cana-4781	69	39	𝑀	𝑀	NOUN
cana-4781	69	40	=	=	PUNCT
cana-4781	69	41	∅.	∅.	ADP
cana-4781	69	42	proof	proof	NOUN
cana-4781	69	43	:	:	PUNCT
cana-4781	69	44	since	since	SCONJ
cana-4781	69	45	the	the	DET
cana-4781	69	46	map	map	NOUN
cana-4781	69	47	𝐿𝑔	𝐿𝑔	NOUN
cana-4781	69	48	:	:	PUNCT
cana-4781	69	49	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	69	50	→	→	SYM
cana-4781	69	51	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	69	52	̅̅	̅̅	PROPN
cana-4781	69	53	̅̅	̅̅	PROPN
cana-4781	69	54	is	be	AUX
cana-4781	69	55	continuous	continuous	ADJ
cana-4781	69	56	and	and	CCONJ
cana-4781	69	57	𝑀	𝑀	PROPN
cana-4781	69	58	is	be	AUX
cana-4781	69	59	closed	closed	ADJ
cana-4781	69	60	,	,	PUNCT
cana-4781	69	61	there	there	PRON
cana-4781	69	62	exists	exist	VERB
cana-4781	69	63	an	an	DET
cana-4781	69	64	open	open	ADJ
cana-4781	69	65	neighbourhood	neighbourhood	NOUN
cana-4781	69	66	𝑈𝑎	𝑈𝑎	NOUN
cana-4781	69	67	of	of	ADP
cana-4781	69	68	𝑒	𝑒	PROPN
cana-4781	69	69	in	in	ADP
cana-4781	69	70	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	69	71	̅̅	̅̅	NOUN
cana-4781	69	72	̅̅	̅̅	NOUN
cana-4781	69	73	such	such	ADJ
cana-4781	69	74	that	that	SCONJ
cana-4781	69	75	𝑎𝑈𝑎	𝑎𝑈𝑎	NOUN
cana-4781	69	76	∩	∩	ADJ
cana-4781	69	77	𝑀	𝑀	NOUN
cana-4781	69	78	=	=	NOUN
cana-4781	69	79	∅	∅	NOUN
cana-4781	69	80	,	,	PUNCT
cana-4781	69	81	for	for	ADP
cana-4781	69	82	all	all	DET
cana-4781	69	83	𝑎	𝑎	DET
cana-4781	69	84	∈	∈	NOUN
cana-4781	69	85	𝐴.	𝐴.	NOUN
cana-4781	69	86	by	by	ADP
cana-4781	69	87	proposition	proposition	NOUN
cana-4781	69	88	2.9	2.9	NUM
cana-4781	69	89	,	,	PUNCT
cana-4781	69	90	there	there	PRON
cana-4781	69	91	is	be	VERB
cana-4781	69	92	a	a	DET
cana-4781	69	93	symmetric	symmetric	ADJ
cana-4781	69	94	open	open	ADJ
cana-4781	69	95	set	set	NOUN
cana-4781	69	96	𝑉𝑎	𝑉𝑎	PROPN
cana-4781	69	97	of	of	ADP
cana-4781	69	98	e	e	PROPN
cana-4781	69	99	in	in	ADP
cana-4781	69	100	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	69	101	such	such	ADJ
cana-4781	69	102	that	that	SCONJ
cana-4781	69	103	𝑉𝑎𝑉𝑎	𝑉𝑎𝑉𝑎	PROPN
cana-4781	69	104	⊆	⊆	NUM
cana-4781	69	105	𝑈𝑎.	𝑈𝑎.	PROPN
cana-4781	69	106	since	since	SCONJ
cana-4781	69	107	𝐴	𝐴	PROPN
cana-4781	69	108	is	be	AUX
cana-4781	69	109	compact	compact	ADJ
cana-4781	69	110	,	,	PUNCT
cana-4781	69	111	there	there	PRON
cana-4781	69	112	exists	exist	VERB
cana-4781	69	113	an	an	DET
cana-4781	69	114	open	open	ADJ
cana-4781	69	115	cover	cover	NOUN
cana-4781	69	116	⋃	⋃	PUNCT
cana-4781	69	117	𝑎𝑎∈𝐴	𝑎𝑎∈𝐴	PROPN
cana-4781	69	118	𝑉𝑎	𝑉𝑎	PROPN
cana-4781	69	119	such	such	ADJ
cana-4781	69	120	that	that	SCONJ
cana-4781	69	121	𝐴	𝐴	PROPN
cana-4781	69	122	⊆	⊆	NUM
cana-4781	69	123	⋃	⋃	PROPN
cana-4781	69	124	𝑎𝑎∈𝐴	𝑎𝑎∈𝐴	PROPN
cana-4781	69	125	𝑉𝑎.	𝑉𝑎.	PROPN
cana-4781	69	126	let	let	VERB
cana-4781	69	127	𝑉	𝑉	PROPN
cana-4781	69	128	=	=	SYM
cana-4781	69	129	⋂	⋂	PROPN
cana-4781	69	130	𝑉𝑎𝑎∈𝐴	𝑉𝑎𝑎∈𝐴	PROPN
cana-4781	69	131	.	.	PUNCT
cana-4781	70	1	suppose	suppose	VERB
cana-4781	70	2	there	there	PRON
cana-4781	70	3	exists	exist	VERB
cana-4781	70	4	an	an	DET
cana-4781	70	5	arbitrary	arbitrary	ADJ
cana-4781	70	6	element	element	NOUN
cana-4781	70	7	𝑏	𝑏	PROPN
cana-4781	70	8	∈	∈	PROPN
cana-4781	70	9	𝐴	𝐴	PROPN
cana-4781	70	10	,	,	PUNCT
cana-4781	70	11	then	then	ADV
cana-4781	70	12	𝑏	𝑏	PROPN
cana-4781	70	13	∈	∈	PROPN
cana-4781	70	14	𝑎𝑉𝑎.	𝑎𝑉𝑎.	PROPN
cana-4781	70	15	now	now	ADV
cana-4781	70	16	,	,	PUNCT
cana-4781	70	17	𝑏𝑉	𝑏𝑉	ADJ
cana-4781	70	18	⊆	⊆	NUM
cana-4781	70	19	𝑏𝑉𝑎	𝑏𝑉𝑎	NOUN
cana-4781	70	20	⊆	⊆	NUM
cana-4781	70	21	𝑎𝑉𝑎𝑉𝑎	𝑎𝑉𝑎𝑉𝑎	NOUN
cana-4781	70	22	⊆	⊆	NUM
cana-4781	70	23	𝑎𝑈𝑎	𝑎𝑈𝑎	NOUN
cana-4781	70	24	which	which	PRON
cana-4781	70	25	implies	imply	VERB
cana-4781	70	26	𝑏𝑉	𝑏𝑉	ADJ
cana-4781	70	27	∩	∩	ADJ
cana-4781	70	28	𝑀	𝑀	NOUN
cana-4781	70	29	=	=	PUNCT
cana-4781	70	30	∅.	∅.	VERB
cana-4781	70	31	therefore	therefore	ADV
cana-4781	70	32	,	,	PUNCT
cana-4781	70	33	𝐴𝑉	𝐴𝑉	PROPN
cana-4781	70	34	∩	∩	X
cana-4781	70	35	𝑀	𝑀	NOUN
cana-4781	70	36	=	=	PUNCT
cana-4781	70	37	∅.	∅.	VERB
cana-4781	70	38	similarly	similarly	ADV
cana-4781	70	39	,	,	PUNCT
cana-4781	70	40	we	we	PRON
cana-4781	70	41	prove	prove	VERB
cana-4781	70	42	𝑉𝐴	𝑉𝐴	PROPN
cana-4781	70	43	∩	∩	NOUN
cana-4781	70	44	𝑀	𝑀	NOUN
cana-4781	70	45	=	=	PUNCT
cana-4781	70	46	∅.	∅.	NOUN
cana-4781	70	47	communications	communication	NOUN
cana-4781	70	48	on	on	ADP
cana-4781	70	49	applied	apply	VERB
cana-4781	70	50	nonlinear	nonlinear	ADJ
cana-4781	70	51	analysis	analysis	NOUN
cana-4781	70	52	issn	issn	NOUN
cana-4781	70	53	:	:	PUNCT
cana-4781	70	54	1074	1074	NUM
cana-4781	70	55	-	-	PUNCT
cana-4781	70	56	133x	133x	NUM
cana-4781	70	57	vol	vol	NOUN
cana-4781	70	58	32	32	NUM
cana-4781	70	59	no	no	NOUN
cana-4781	70	60	.	.	NOUN
cana-4781	70	61	3	3	NUM
cana-4781	70	62	(	(	PUNCT
cana-4781	70	63	2025	2025	NUM
cana-4781	70	64	)	)	PUNCT
cana-4781	70	65	882	882	NUM
cana-4781	70	66	https://internationalpubls.com	https://internationalpubls.com	X
cana-4781	70	67	theorem	theorem	ADJ
cana-4781	70	68	3.2	3.2	NUM
cana-4781	70	69	.	.	PUNCT
cana-4781	71	1	(	(	PUNCT
cana-4781	71	2	second	second	ADJ
cana-4781	71	3	closure	closure	NOUN
cana-4781	71	4	lemma	lemma	PROPN
cana-4781	71	5	)	)	PUNCT
cana-4781	71	6	let	let	VERB
cana-4781	71	7	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	71	8	be	be	AUX
cana-4781	71	9	a	a	DET
cana-4781	71	10	topological	topological	ADJ
cana-4781	71	11	simple	simple	ADJ
cana-4781	71	12	rough	rough	ADJ
cana-4781	71	13	group	group	NOUN
cana-4781	71	14	such	such	ADJ
cana-4781	71	15	that	that	SCONJ
cana-4781	71	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	71	17	is	be	AUX
cana-4781	71	18	open	open	ADJ
cana-4781	71	19	in	in	ADP
cana-4781	71	20	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	71	21	̅̅	̅̅	PROPN
cana-4781	71	22	̅̅	̅̅	PROPN
cana-4781	71	23	.	.	PUNCT
cana-4781	72	1	suppose	suppose	VERB
cana-4781	72	2	𝐴	𝐴	PROPN
cana-4781	72	3	is	be	AUX
cana-4781	72	4	a	a	DET
cana-4781	72	5	compact	compact	ADJ
cana-4781	72	6	subset	subset	NOUN
cana-4781	72	7	of	of	ADP
cana-4781	72	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	72	9	̅̅	̅̅	PROPN
cana-4781	72	10	̅̅	̅̅	PROPN
cana-4781	72	11	and	and	CCONJ
cana-4781	72	12	𝑀	𝑀	PROPN
cana-4781	72	13	is	be	AUX
cana-4781	72	14	a	a	DET
cana-4781	72	15	closed	closed	ADJ
cana-4781	72	16	subset	subset	NOUN
cana-4781	72	17	of	of	ADP
cana-4781	72	18	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	72	19	̅̅	̅̅	PROPN
cana-4781	72	20	̅̅	̅̅	PROPN
cana-4781	72	21	.	.	PUNCT
cana-4781	73	1	then	then	ADV
cana-4781	73	2	𝐴𝑀	𝐴𝑀	PROPN
cana-4781	73	3	and	and	CCONJ
cana-4781	73	4	𝑀𝐴	𝑀𝐴	PROPN
cana-4781	73	5	are	be	AUX
cana-4781	73	6	closed	close	VERB
cana-4781	73	7	sets	set	NOUN
cana-4781	73	8	in	in	ADP
cana-4781	73	9	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	73	10	̅̅	̅̅	PROPN
cana-4781	73	11	̅̅	̅̅	PROPN
cana-4781	73	12	.	.	PUNCT
cana-4781	74	1	proof	proof	NOUN
cana-4781	74	2	:	:	PUNCT
cana-4781	74	3	let	let	VERB
cana-4781	74	4	𝑎	𝑎	NOUN
cana-4781	74	5	∉	∉	PROPN
cana-4781	74	6	𝐴𝑀.	𝐴𝑀.	X
cana-4781	74	7	then	then	ADV
cana-4781	74	8	𝐴−1𝑎	𝐴−1𝑎	PROPN
cana-4781	74	9	∩	∩	ADJ
cana-4781	74	10	𝑀	𝑀	NOUN
cana-4781	74	11	=	=	PUNCT
cana-4781	74	12	∅.	∅.	NOUN
cana-4781	74	13	since	since	SCONJ
cana-4781	74	14	𝐴	𝐴	PROPN
cana-4781	74	15	is	be	AUX
cana-4781	74	16	a	a	DET
cana-4781	74	17	compact	compact	ADJ
cana-4781	74	18	subset	subset	NOUN
cana-4781	74	19	,	,	PUNCT
cana-4781	74	20	𝐴−1𝑎	𝐴−1𝑎	PROPN
cana-4781	74	21	is	be	AUX
cana-4781	74	22	compact	compact	ADJ
cana-4781	74	23	.	.	PUNCT
cana-4781	75	1	therefore	therefore	ADV
cana-4781	75	2	,	,	PUNCT
cana-4781	75	3	by	by	ADP
cana-4781	75	4	theorem	theorem	NOUN
cana-4781	75	5	3.1	3.1	NUM
cana-4781	75	6	,	,	PUNCT
cana-4781	75	7	there	there	PRON
cana-4781	75	8	is	be	VERB
cana-4781	75	9	an	an	DET
cana-4781	75	10	open	open	ADJ
cana-4781	75	11	neighbourhood	neighbourhood	NOUN
cana-4781	75	12	𝑉	𝑉	PROPN
cana-4781	75	13	of	of	ADP
cana-4781	75	14	e	e	PROPN
cana-4781	75	15	in	in	ADP
cana-4781	75	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	75	17	such	such	ADJ
cana-4781	75	18	that	that	DET
cana-4781	75	19	𝐴−1𝑎𝑉	𝐴−1𝑎𝑉	PROPN
cana-4781	75	20	∩	∩	PROPN
cana-4781	75	21	𝑀	𝑀	PROPN
cana-4781	75	22	=	=	PUNCT
cana-4781	75	23	∅	∅	NOUN
cana-4781	75	24	which	which	PRON
cana-4781	75	25	implies	imply	VERB
cana-4781	75	26	𝑎𝑉	𝑎𝑉	NOUN
cana-4781	75	27	∩	∩	ADJ
cana-4781	75	28	𝐴𝑀	𝐴𝑀	NOUN
cana-4781	75	29	=	=	NOUN
cana-4781	75	30	∅	∅	NOUN
cana-4781	75	31	and	and	CCONJ
cana-4781	75	32	𝑎𝑉	𝑎𝑉	PROPN
cana-4781	75	33	is	be	AUX
cana-4781	75	34	an	an	DET
cana-4781	75	35	open	open	ADJ
cana-4781	75	36	neighbourhood	neighbourhood	NOUN
cana-4781	75	37	of	of	ADP
cana-4781	75	38	𝑎	𝑎	NOUN
cana-4781	75	39	in	in	ADP
cana-4781	75	40	the	the	DET
cana-4781	75	41	complement	complement	NOUN
cana-4781	75	42	of	of	ADP
cana-4781	75	43	𝐴𝑀.	𝐴𝑀.	NOUN
cana-4781	75	44	hence	hence	ADV
cana-4781	75	45	𝐴𝑀	𝐴𝑀	PROPN
cana-4781	75	46	is	be	AUX
cana-4781	75	47	a	a	DET
cana-4781	75	48	closed	closed	ADJ
cana-4781	75	49	subset	subset	NOUN
cana-4781	75	50	in	in	ADP
cana-4781	75	51	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	75	52	̅̅	̅̅	PROPN
cana-4781	75	53	̅̅	̅̅	PROPN
cana-4781	75	54	.	.	PUNCT
cana-4781	76	1	similarly	similarly	ADV
cana-4781	76	2	,	,	PUNCT
cana-4781	76	3	𝑀𝐴	𝑀𝐴	PROPN
cana-4781	76	4	is	be	AUX
cana-4781	76	5	a	a	DET
cana-4781	76	6	closed	closed	ADJ
cana-4781	76	7	subset	subset	NOUN
cana-4781	76	8	in	in	ADP
cana-4781	76	9	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	76	10	̅̅	̅̅	PROPN
cana-4781	76	11	̅̅	̅̅	PROPN
cana-4781	76	12	.	.	PUNCT
cana-4781	77	1	theorem	theorem	VERB
cana-4781	77	2	3.3	3.3	NUM
cana-4781	77	3	.	.	PUNCT
cana-4781	78	1	let	let	VERB
cana-4781	78	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	78	3	be	be	AUX
cana-4781	78	4	a	a	DET
cana-4781	78	5	topological	topological	ADJ
cana-4781	78	6	simple	simple	ADJ
cana-4781	78	7	rough	rough	ADJ
cana-4781	78	8	group	group	NOUN
cana-4781	78	9	such	such	ADJ
cana-4781	78	10	that	that	SCONJ
cana-4781	78	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	78	12	is	be	AUX
cana-4781	78	13	open	open	ADJ
cana-4781	78	14	in	in	ADP
cana-4781	78	15	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	78	16	̅̅	̅̅	PROPN
cana-4781	78	17	̅̅	̅̅	PROPN
cana-4781	78	18	.	.	PUNCT
cana-4781	79	1	suppose	suppose	VERB
cana-4781	79	2	𝐴	𝐴	PROPN
cana-4781	79	3	is	be	AUX
cana-4781	79	4	a	a	DET
cana-4781	79	5	compact	compact	ADJ
cana-4781	79	6	subset	subset	NOUN
cana-4781	79	7	of	of	ADP
cana-4781	79	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	79	9	̅̅	̅̅	PROPN
cana-4781	79	10	̅̅	̅̅	PROPN
cana-4781	79	11	.	.	PUNCT
cana-4781	80	1	then	then	ADV
cana-4781	80	2	there	there	PRON
cana-4781	80	3	exists	exist	VERB
cana-4781	80	4	an	an	DET
cana-4781	80	5	identity	identity	NOUN
cana-4781	80	6	neighbourhood	neighbourhood	NOUN
cana-4781	80	7	𝑉	𝑉	PROPN
cana-4781	80	8	⊆	⊆	NUM
cana-4781	80	9	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	80	10	̅̅	̅̅	NOUN
cana-4781	80	11	̅̅	̅̅	NOUN
cana-4781	80	12	such	such	ADJ
cana-4781	80	13	that	that	DET
cana-4781	80	14	𝑎𝑉𝑎−1	𝑎𝑉𝑎−1	ADJ
cana-4781	80	15	⊆	⊆	PROPN
cana-4781	80	16	𝑊	𝑊	PROPN
cana-4781	80	17	,	,	PUNCT
cana-4781	80	18	for	for	ADP
cana-4781	80	19	every	every	DET
cana-4781	80	20	open	open	ADJ
cana-4781	80	21	neighbourhood	neighbourhood	NOUN
cana-4781	80	22	𝑊	𝑊	NOUN
cana-4781	80	23	of	of	ADP
cana-4781	80	24	𝑒	𝑒	PROPN
cana-4781	80	25	in	in	ADP
cana-4781	80	26	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	80	27	̅̅	̅̅	NOUN
cana-4781	80	28	̅̅	̅̅	PROPN
cana-4781	80	29	and	and	CCONJ
cana-4781	80	30	𝑎	𝑎	PROPN
cana-4781	80	31	∈	∈	NOUN
cana-4781	80	32	𝐴.	𝐴.	NOUN
cana-4781	80	33	proof	proof	NOUN
cana-4781	80	34	:	:	PUNCT
cana-4781	80	35	let	let	VERB
cana-4781	80	36	𝑊	𝑊	PRON
cana-4781	80	37	be	be	AUX
cana-4781	80	38	a	a	DET
cana-4781	80	39	neighbourhood	neighbourhood	NOUN
cana-4781	80	40	of	of	ADP
cana-4781	80	41	𝑒	𝑒	PROPN
cana-4781	80	42	in	in	ADP
cana-4781	80	43	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	80	44	̅̅	̅̅	PROPN
cana-4781	80	45	̅̅	̅̅	PROPN
cana-4781	80	46	.	.	PUNCT
cana-4781	81	1	from	from	ADP
cana-4781	81	2	lemma	lemma	PROPN
cana-4781	81	3	2.10	2.10	NUM
cana-4781	81	4	,	,	PUNCT
cana-4781	81	5	there	there	PRON
cana-4781	81	6	is	be	VERB
cana-4781	81	7	an	an	DET
cana-4781	81	8	open	open	ADJ
cana-4781	81	9	set	set	NOUN
cana-4781	81	10	𝑈	𝑈	PROPN
cana-4781	81	11	of	of	ADP
cana-4781	81	12	𝑒	𝑒	PROPN
cana-4781	81	13	in	in	ADP
cana-4781	81	14	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	81	15	such	such	ADJ
cana-4781	81	16	that	that	SCONJ
cana-4781	81	17	𝑈	𝑈	PROPN
cana-4781	81	18	⊆	⊆	NUM
cana-4781	81	19	𝑈𝑛	𝑈𝑛	PROPN
cana-4781	81	20	⊆	⊆	NUM
cana-4781	81	21	𝑊	𝑊	PROPN
cana-4781	81	22	,	,	PUNCT
cana-4781	81	23	for	for	ADP
cana-4781	81	24	every	every	DET
cana-4781	81	25	𝑛	𝑛	PRON
cana-4781	81	26	∈	∈	PROPN
cana-4781	81	27	ℕ	ℕ	PROPN
cana-4781	81	28	−	−	PROPN
cana-4781	81	29	{	{	PUNCT
cana-4781	81	30	0	0	NUM
cana-4781	81	31	}	}	PUNCT
cana-4781	81	32	.	.	PUNCT
cana-4781	82	1	since	since	SCONJ
cana-4781	82	2	𝐴	𝐴	PROPN
cana-4781	82	3	is	be	AUX
cana-4781	82	4	compact	compact	ADJ
cana-4781	82	5	,	,	PUNCT
cana-4781	82	6	there	there	PRON
cana-4781	82	7	exists	exist	VERB
cana-4781	82	8	an	an	DET
cana-4781	82	9	open	open	ADJ
cana-4781	82	10	cover	cover	NOUN
cana-4781	82	11	𝐴	𝐴	PROPN
cana-4781	82	12	⊆	⊆	NUM
cana-4781	82	13	𝑈𝑀	𝑈𝑀	PROPN
cana-4781	82	14	such	such	ADJ
cana-4781	82	15	that	that	SCONJ
cana-4781	82	16	𝑀	𝑀	PROPN
cana-4781	82	17	is	be	AUX
cana-4781	82	18	a	a	DET
cana-4781	82	19	finite	finite	NOUN
cana-4781	82	20	subset	subset	NOUN
cana-4781	82	21	of	of	ADP
cana-4781	82	22	𝐴.	𝐴.	PROPN
cana-4781	82	23	consider	consider	VERB
cana-4781	82	24	𝑉	𝑉	PROPN
cana-4781	82	25	=	=	SYM
cana-4781	82	26	⋂	⋂	PROPN
cana-4781	82	27	𝑥−1𝑈𝑥𝑥∈𝑀	𝑥−1𝑈𝑥𝑥∈𝑀	PROPN
cana-4781	82	28	.	.	PUNCT
cana-4781	83	1	then	then	ADV
cana-4781	83	2	e	e	PROPN
cana-4781	83	3	∈	∈	PROPN
cana-4781	83	4	v	v	NOUN
cana-4781	83	5	is	be	AUX
cana-4781	83	6	open	open	ADJ
cana-4781	83	7	in	in	ADP
cana-4781	83	8	𝐺ℜ.	𝐺ℜ.	PUNCT
cana-4781	83	9	also	also	ADV
cana-4781	83	10	,	,	PUNCT
cana-4781	83	11	by	by	ADP
cana-4781	83	12	theorem	theorem	NOUN
cana-4781	83	13	2.11	2.11	NUM
cana-4781	83	14	,	,	PUNCT
cana-4781	83	15	𝑉	𝑉	PROPN
cana-4781	83	16	is	be	AUX
cana-4781	83	17	open	open	ADJ
cana-4781	83	18	in	in	ADP
cana-4781	83	19	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	83	20	̅̅	̅̅	PROPN
cana-4781	83	21	̅̅	̅̅	PROPN
cana-4781	83	22	.	.	PUNCT
cana-4781	84	1	now	now	ADV
cana-4781	84	2	we	we	PRON
cana-4781	84	3	choose	choose	VERB
cana-4781	84	4	an	an	DET
cana-4781	84	5	element	element	NOUN
cana-4781	84	6	𝑎	𝑎	PROPN
cana-4781	84	7	∈	∈	PROPN
cana-4781	84	8	𝐴	𝐴	NOUN
cana-4781	84	9	such	such	ADJ
cana-4781	84	10	that	that	SCONJ
cana-4781	84	11	𝑎	𝑎	PROPN
cana-4781	84	12	=	=	SYM
cana-4781	84	13	𝑢𝑥	𝑢𝑥	NOUN
cana-4781	84	14	,	,	PUNCT
cana-4781	84	15	for	for	ADP
cana-4781	84	16	some	some	DET
cana-4781	84	17	𝑢	𝑢	PRON
cana-4781	84	18	∈	∈	PROPN
cana-4781	84	19	𝑈	𝑈	PROPN
cana-4781	84	20	and	and	CCONJ
cana-4781	84	21	𝑥	𝑥	PRON
cana-4781	84	22	∈	∈	NOUN
cana-4781	84	23	𝑀.	𝑀.	PROPN
cana-4781	84	24	hence	hence	ADV
cana-4781	84	25	,	,	PUNCT
cana-4781	84	26	𝑎𝑉𝑎−1	𝑎𝑉𝑎−1	ADJ
cana-4781	84	27	=	=	PUNCT
cana-4781	84	28	𝑢𝑥𝑉𝑥−1𝑢−1	𝑢𝑥𝑉𝑥−1𝑢−1	PROPN
cana-4781	84	29	⊆	⊆	NUM
cana-4781	84	30	𝑢𝑈𝑢−1	𝑢𝑈𝑢−1	ADV
cana-4781	84	31	⊆	⊆	NUM
cana-4781	84	32	𝑈3	𝑈3	NOUN
cana-4781	84	33	⊆	⊆	NUM
cana-4781	84	34	𝑊	𝑊	PROPN
cana-4781	84	35	,	,	PUNCT
cana-4781	84	36	for	for	ADP
cana-4781	84	37	any	any	DET
cana-4781	84	38	𝑎	𝑎	PROPN
cana-4781	84	39	∈	∈	PROPN
cana-4781	84	40	𝐴.	𝐴.	NOUN
cana-4781	84	41	theorem	theorem	NOUN
cana-4781	84	42	3.4	3.4	NUM
cana-4781	84	43	.	.	PUNCT
cana-4781	85	1	let	let	VERB
cana-4781	85	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	85	3	be	be	AUX
cana-4781	85	4	a	a	DET
cana-4781	85	5	topological	topological	ADJ
cana-4781	85	6	simple	simple	ADJ
cana-4781	85	7	rough	rough	ADJ
cana-4781	85	8	group	group	NOUN
cana-4781	85	9	such	such	ADJ
cana-4781	85	10	that	that	SCONJ
cana-4781	85	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	85	12	̅̅	̅̅	PROPN
cana-4781	85	13	̅̅	̅̅	PROPN
cana-4781	85	14	is	be	AUX
cana-4781	85	15	a	a	DET
cana-4781	85	16	group	group	NOUN
cana-4781	85	17	and	and	CCONJ
cana-4781	85	18	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	85	19	is	be	AUX
cana-4781	85	20	a	a	DET
cana-4781	85	21	subgroup	subgroup	NOUN
cana-4781	85	22	of	of	ADP
cana-4781	85	23	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	85	24	̅̅	̅̅	PROPN
cana-4781	85	25	̅̅	̅̅	PROPN
cana-4781	85	26	.	.	PUNCT
cana-4781	86	1	if	if	SCONJ
cana-4781	86	2	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	86	3	is	be	AUX
cana-4781	86	4	open	open	ADJ
cana-4781	86	5	in	in	ADP
cana-4781	86	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	86	7	̅̅	̅̅	PROPN
cana-4781	86	8	̅̅	̅̅	PROPN
cana-4781	86	9	,	,	PUNCT
cana-4781	86	10	then	then	ADV
cana-4781	86	11	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	86	12	is	be	AUX
cana-4781	86	13	closed	close	VERB
cana-4781	86	14	in	in	ADP
cana-4781	86	15	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	86	16	̅̅	̅̅	PROPN
cana-4781	86	17	̅̅	̅̅	PROPN
cana-4781	86	18	.	.	PUNCT
cana-4781	87	1	proof	proof	NOUN
cana-4781	87	2	:	:	PUNCT
cana-4781	87	3	the	the	DET
cana-4781	87	4	rough	rough	ADJ
cana-4781	87	5	quotient	quotient	NOUN
cana-4781	87	6	space	space	NOUN
cana-4781	87	7	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	87	8	̅̅	̅̅	PROPN
cana-4781	87	9	̅̅	̅̅	PROPN
cana-4781	87	10	𝐻ℜ⁄	𝐻ℜ⁄	PROPN
cana-4781	87	11	=	=	PUNCT
cana-4781	87	12	{	{	PUNCT
cana-4781	87	13	𝑎𝐻ℜ	𝑎𝐻ℜ	NOUN
cana-4781	87	14	∶	∶	VERB
cana-4781	87	15	𝑎	𝑎	PROPN
cana-4781	87	16	∈	∈	PROPN
cana-4781	87	17	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	87	18	̅̅	̅̅	NOUN
cana-4781	87	19	̅̅	̅̅	PROPN
cana-4781	87	20	}	}	PUNCT
cana-4781	87	21	.	.	PUNCT
cana-4781	88	1	it	it	PRON
cana-4781	88	2	is	be	AUX
cana-4781	88	3	a	a	DET
cana-4781	88	4	disjoint	disjoint	ADJ
cana-4781	88	5	open	open	ADJ
cana-4781	88	6	cover	cover	NOUN
cana-4781	88	7	of	of	ADP
cana-4781	88	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	88	9	̅̅	̅̅	PROPN
cana-4781	88	10	̅̅	̅̅	PROPN
cana-4781	88	11	.	.	PUNCT
cana-4781	89	1	since	since	SCONJ
cana-4781	89	2	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	89	3	is	be	AUX
cana-4781	89	4	open	open	ADJ
cana-4781	89	5	,	,	PUNCT
cana-4781	89	6	𝑎𝐻ℜ	𝑎𝐻ℜ	PRON
cana-4781	89	7	is	be	AUX
cana-4781	89	8	also	also	ADV
cana-4781	89	9	open	open	ADJ
cana-4781	89	10	.	.	PUNCT
cana-4781	90	1	therefore	therefore	ADV
cana-4781	90	2	,	,	PUNCT
cana-4781	90	3	the	the	DET
cana-4781	90	4	complement	complement	NOUN
cana-4781	90	5	of	of	ADP
cana-4781	90	6	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	90	7	=	=	SYM
cana-4781	90	8	⋃	⋃	PROPN
cana-4781	90	9	𝑎𝐻ℜ𝑎∉𝐻ℜ	𝑎𝐻ℜ𝑎∉𝐻ℜ	PROPN
cana-4781	90	10	is	be	AUX
cana-4781	90	11	open	open	ADJ
cana-4781	90	12	in	in	ADP
cana-4781	90	13	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	90	14	̅̅	̅̅	PROPN
cana-4781	90	15	̅̅	̅̅	PROPN
cana-4781	90	16	.	.	PUNCT
cana-4781	91	1	hence	hence	ADV
cana-4781	91	2	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	91	3	is	be	AUX
cana-4781	91	4	closed	close	VERB
cana-4781	91	5	in	in	ADP
cana-4781	91	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	91	7	̅̅	̅̅	PROPN
cana-4781	91	8	̅̅	̅̅	PROPN
cana-4781	91	9	.	.	PUNCT
cana-4781	92	1	theorem	theorem	VERB
cana-4781	92	2	3.5	3.5	NUM
cana-4781	92	3	.	.	PUNCT
cana-4781	93	1	let	let	VERB
cana-4781	93	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	93	3	be	be	AUX
cana-4781	93	4	a	a	DET
cana-4781	93	5	topological	topological	ADJ
cana-4781	93	6	simple	simple	ADJ
cana-4781	93	7	rough	rough	ADJ
cana-4781	93	8	group	group	NOUN
cana-4781	93	9	such	such	ADJ
cana-4781	93	10	that	that	SCONJ
cana-4781	93	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	93	12	̅̅	̅̅	PROPN
cana-4781	93	13	̅̅	̅̅	PROPN
cana-4781	93	14	is	be	AUX
cana-4781	93	15	a	a	DET
cana-4781	93	16	group	group	NOUN
cana-4781	93	17	and	and	CCONJ
cana-4781	93	18	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	93	19	is	be	AUX
cana-4781	93	20	open	open	ADJ
cana-4781	93	21	in	in	ADP
cana-4781	93	22	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	93	23	̅̅	̅̅	NOUN
cana-4781	93	24	̅̅	̅̅	PROPN
cana-4781	93	25	.	.	PUNCT
cana-4781	94	1	if	if	SCONJ
cana-4781	94	2	𝐴	𝐴	PROPN
cana-4781	94	3	is	be	AUX
cana-4781	94	4	a	a	DET
cana-4781	94	5	compact	compact	ADJ
cana-4781	94	6	open	open	ADJ
cana-4781	94	7	neighbourhood	neighbourhood	NOUN
cana-4781	94	8	of	of	ADP
cana-4781	94	9	𝑒	𝑒	PROPN
cana-4781	94	10	in	in	ADP
cana-4781	94	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	94	12	̅̅	̅̅	NOUN
cana-4781	94	13	̅̅	̅̅	PROPN
cana-4781	94	14	,	,	PUNCT
cana-4781	94	15	there	there	PRON
cana-4781	94	16	is	be	VERB
cana-4781	94	17	a	a	DET
cana-4781	94	18	compact	compact	ADJ
cana-4781	94	19	subgroup	subgroup	NOUN
cana-4781	94	20	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	94	21	of	of	ADP
cana-4781	94	22	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	94	23	̅̅	̅̅	PROPN
cana-4781	94	24	̅̅	̅̅	PROPN
cana-4781	94	25	such	such	ADJ
cana-4781	94	26	that	that	SCONJ
cana-4781	94	27	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	94	28	⊆	⊆	NUM
cana-4781	94	29	𝐴.	𝐴.	PROPN
cana-4781	94	30	proof	proof	NOUN
cana-4781	94	31	:	:	PUNCT
cana-4781	94	32	since	since	SCONJ
cana-4781	94	33	𝐴	𝐴	PROPN
cana-4781	94	34	⊆	⊆	NUM
cana-4781	94	35	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	94	36	̅̅	̅̅	PROPN
cana-4781	94	37	̅̅	̅̅	PROPN
cana-4781	94	38	is	be	AUX
cana-4781	94	39	an	an	DET
cana-4781	94	40	open	open	ADJ
cana-4781	94	41	neighbourhood	neighbourhood	NOUN
cana-4781	94	42	of	of	ADP
cana-4781	94	43	𝑒	𝑒	PROPN
cana-4781	94	44	,	,	PUNCT
cana-4781	94	45	there	there	PRON
cana-4781	94	46	exists	exist	VERB
cana-4781	94	47	a	a	DET
cana-4781	94	48	symmetric	symmetric	ADJ
cana-4781	94	49	open	open	ADJ
cana-4781	94	50	neighbourhood	neighbourhood	NOUN
cana-4781	94	51	𝑉	𝑉	PROPN
cana-4781	94	52	of	of	ADP
cana-4781	94	53	𝑒	𝑒	PROPN
cana-4781	94	54	in	in	ADP
cana-4781	94	55	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	94	56	such	such	ADJ
cana-4781	94	57	that	that	SCONJ
cana-4781	94	58	𝑉𝑉	𝑉𝑉	PROPN
cana-4781	94	59	⊆	⊆	NUM
cana-4781	94	60	𝐴	𝐴	PROPN
cana-4781	94	61	and	and	CCONJ
cana-4781	94	62	by	by	ADP
cana-4781	94	63	lemma	lemma	PROPN
cana-4781	94	64	2.10	2.10	NUM
cana-4781	94	65	,	,	PUNCT
cana-4781	94	66	𝑉	𝑉	PROPN
cana-4781	94	67	⊆	⊆	NUM
cana-4781	94	68	𝑉𝑛	𝑉𝑛	PROPN
cana-4781	94	69	⊆	⊆	NUM
cana-4781	94	70	𝐴	𝐴	PROPN
cana-4781	94	71	,	,	PUNCT
cana-4781	94	72	for	for	ADP
cana-4781	94	73	every	every	DET
cana-4781	94	74	𝑛	𝑛	PRON
cana-4781	94	75	∈	∈	PROPN
cana-4781	94	76	ℕ	ℕ	PROPN
cana-4781	94	77	−	−	PROPN
cana-4781	94	78	{	{	PUNCT
cana-4781	94	79	0	0	NUM
cana-4781	94	80	}	}	PUNCT
cana-4781	94	81	.	.	PUNCT
cana-4781	95	1	now	now	ADV
cana-4781	95	2	consider	consider	VERB
cana-4781	95	3	𝐻ℜ	𝐻ℜ	NOUN
cana-4781	95	4	=	=	SYM
cana-4781	95	5	⋃	⋃	ADP
cana-4781	95	6	𝑉𝑛	𝑉𝑛	NOUN
cana-4781	95	7	𝑛∈ℕ−{0	𝑛∈ℕ−{0	NOUN
cana-4781	95	8	}	}	PUNCT
cana-4781	95	9	.	.	PUNCT
cana-4781	96	1	then	then	ADV
cana-4781	96	2	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	96	3	is	be	AUX
cana-4781	96	4	open	open	ADJ
cana-4781	96	5	in	in	ADP
cana-4781	96	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	96	7	and	and	CCONJ
cana-4781	96	8	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	96	9	⊆	⊆	NUM
cana-4781	96	10	𝐴.	𝐴.	NOUN
cana-4781	96	11	since	since	SCONJ
cana-4781	96	12	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	96	13	is	be	AUX
cana-4781	96	14	open	open	ADJ
cana-4781	96	15	in	in	ADP
cana-4781	96	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	96	17	̅̅	̅̅	PROPN
cana-4781	96	18	̅̅	̅̅	PROPN
cana-4781	96	19	,	,	PUNCT
cana-4781	96	20	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	96	21	is	be	AUX
cana-4781	96	22	open	open	ADJ
cana-4781	96	23	in	in	ADP
cana-4781	96	24	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	96	25	̅̅	̅̅	PROPN
cana-4781	96	26	̅̅	̅̅	PROPN
cana-4781	96	27	.	.	PUNCT
cana-4781	97	1	let	let	VERB
cana-4781	97	2	us	we	PRON
cana-4781	97	3	prove	prove	VERB
cana-4781	97	4	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	97	5	is	be	AUX
cana-4781	97	6	a	a	DET
cana-4781	97	7	subgroup	subgroup	NOUN
cana-4781	97	8	of	of	ADP
cana-4781	97	9	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	97	10	̅̅	̅̅	PROPN
cana-4781	97	11	̅̅	̅̅	PROPN
cana-4781	97	12	.	.	PUNCT
cana-4781	98	1	let	let	VERB
cana-4781	98	2	𝑎	𝑎	X
cana-4781	98	3	,	,	PUNCT
cana-4781	98	4	𝑏	𝑏	PROPN
cana-4781	98	5	∈	∈	NOUN
cana-4781	98	6	𝐻ℜ.	𝐻ℜ.	NOUN
cana-4781	98	7	then	then	ADV
cana-4781	98	8	𝑎	𝑎	X
cana-4781	98	9	∈	∈	PROPN
cana-4781	98	10	𝑉𝑛	𝑉𝑛	PROPN
cana-4781	98	11	and	and	CCONJ
cana-4781	98	12	𝑏	𝑏	PRON
cana-4781	98	13	∈	∈	PROPN
cana-4781	98	14	𝑉𝑚	𝑉𝑚	PROPN
cana-4781	98	15	,	,	PUNCT
cana-4781	98	16	for	for	ADP
cana-4781	98	17	some	some	DET
cana-4781	98	18	𝑛	𝑛	NOUN
cana-4781	98	19	,	,	PUNCT
cana-4781	98	20	𝑚	𝑚	PROPN
cana-4781	98	21	∈	∈	PROPN
cana-4781	98	22	ℕ	ℕ	PROPN
cana-4781	98	23	−	−	PROPN
cana-4781	98	24	{	{	PUNCT
cana-4781	98	25	0	0	NUM
cana-4781	98	26	}	}	PUNCT
cana-4781	98	27	which	which	PRON
cana-4781	98	28	implies	imply	VERB
cana-4781	98	29	𝑎𝑏	𝑎𝑏	PROPN
cana-4781	98	30	∈	∈	PROPN
cana-4781	98	31	𝑉𝑛+𝑚	𝑉𝑛+𝑚	PROPN
cana-4781	98	32	∈	∈	NOUN
cana-4781	98	33	𝐻ℜ.	𝐻ℜ.	X
cana-4781	98	34	also	also	ADV
cana-4781	98	35	,	,	PUNCT
cana-4781	98	36	𝑎	𝑎	PROPN
cana-4781	98	37	∈	∈	NOUN
cana-4781	99	1	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	99	2	implies	imply	VERB
cana-4781	99	3	𝑎	𝑎	PRON
cana-4781	99	4	∈	∈	PROPN
cana-4781	99	5	𝑉𝑛	𝑉𝑛	PROPN
cana-4781	99	6	and	and	CCONJ
cana-4781	99	7	𝑎−1	𝑎−1	PROPN
cana-4781	99	8	∈	∈	PROPN
cana-4781	99	9	(	(	PUNCT
cana-4781	99	10	𝑉𝑛)−1	𝑉𝑛)−1	NOUN
cana-4781	99	11	=	=	SYM
cana-4781	99	12	(	(	PUNCT
cana-4781	99	13	𝑉−1)𝑛	𝑉−1)𝑛	PRON
cana-4781	99	14	=	=	PUNCT
cana-4781	99	15	𝑉𝑛	𝑉𝑛	PRON
cana-4781	99	16	∈	∈	NOUN
cana-4781	99	17	𝐻ℜ.	𝐻ℜ.	X
cana-4781	99	18	therefore	therefore	ADV
cana-4781	99	19	,	,	PUNCT
cana-4781	99	20	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	99	21	is	be	AUX
cana-4781	99	22	a	a	DET
cana-4781	99	23	subgroup	subgroup	NOUN
cana-4781	99	24	of	of	ADP
cana-4781	99	25	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	99	26	̅̅	̅̅	PROPN
cana-4781	99	27	̅̅	̅̅	PROPN
cana-4781	99	28	.	.	PUNCT
cana-4781	100	1	applying	apply	VERB
cana-4781	100	2	theorem	theorem	NOUN
cana-4781	100	3	3.4	3.4	NUM
cana-4781	100	4	,	,	PUNCT
cana-4781	100	5	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	100	6	is	be	AUX
cana-4781	100	7	closed	close	VERB
cana-4781	100	8	in	in	ADP
cana-4781	100	9	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	100	10	̅̅	̅̅	PROPN
cana-4781	100	11	̅̅	̅̅	PROPN
cana-4781	100	12	.	.	PUNCT
cana-4781	101	1	hence	hence	ADV
cana-4781	101	2	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	101	3	is	be	AUX
cana-4781	101	4	a	a	DET
cana-4781	101	5	compact	compact	ADJ
cana-4781	101	6	subgroup	subgroup	NOUN
cana-4781	101	7	of	of	ADP
cana-4781	101	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	101	9	̅̅	̅̅	PROPN
cana-4781	101	10	̅̅	̅̅	PROPN
cana-4781	101	11	.	.	PUNCT
cana-4781	102	1	proposition	proposition	NOUN
cana-4781	102	2	3.6	3.6	NUM
cana-4781	102	3	.	.	PUNCT
cana-4781	103	1	let	let	VERB
cana-4781	103	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	103	3	be	be	AUX
cana-4781	103	4	a	a	DET
cana-4781	103	5	topological	topological	ADJ
cana-4781	103	6	simple	simple	ADJ
cana-4781	103	7	rough	rough	ADJ
cana-4781	103	8	group	group	NOUN
cana-4781	103	9	such	such	ADJ
cana-4781	103	10	that	that	SCONJ
cana-4781	103	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	103	12	̅̅	̅̅	PROPN
cana-4781	103	13	̅̅	̅̅	PROPN
cana-4781	103	14	is	be	AUX
cana-4781	103	15	a	a	DET
cana-4781	103	16	group	group	NOUN
cana-4781	103	17	and	and	CCONJ
cana-4781	103	18	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	103	19	be	be	AUX
cana-4781	103	20	a	a	DET
cana-4781	103	21	locally	locally	ADV
cana-4781	103	22	compact	compact	ADJ
cana-4781	103	23	subgroup	subgroup	NOUN
cana-4781	103	24	of	of	ADP
cana-4781	103	25	a	a	DET
cana-4781	103	26	hausdorff	hausdorff	NOUN
cana-4781	103	27	topological	topological	PROPN
cana-4781	103	28	group	group	NOUN
cana-4781	103	29	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	103	30	̅̅	̅̅	PROPN
cana-4781	103	31	̅̅	̅̅	PROPN
cana-4781	103	32	.	.	PUNCT
cana-4781	104	1	then	then	ADV
cana-4781	104	2	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	104	3	is	be	AUX
cana-4781	104	4	closed	close	VERB
cana-4781	104	5	in	in	ADP
cana-4781	104	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	104	7	̅̅	̅̅	PROPN
cana-4781	104	8	̅̅	̅̅	PROPN
cana-4781	104	9	.	.	PUNCT
cana-4781	105	1	proof	proof	NOUN
cana-4781	105	2	:	:	PUNCT
cana-4781	105	3	since	since	SCONJ
cana-4781	105	4	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	105	5	be	be	AUX
cana-4781	105	6	a	a	DET
cana-4781	105	7	locally	locally	ADV
cana-4781	105	8	compact	compact	ADJ
cana-4781	105	9	subgroup	subgroup	NOUN
cana-4781	105	10	of	of	ADP
cana-4781	105	11	a	a	DET
cana-4781	105	12	hausdorff	hausdorff	NOUN
cana-4781	105	13	topological	topological	PROPN
cana-4781	105	14	group	group	NOUN
cana-4781	105	15	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	105	16	̅̅	̅̅	PROPN
cana-4781	105	17	̅̅	̅̅	PROPN
cana-4781	105	18	,	,	PUNCT
cana-4781	105	19	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	105	20	is	be	AUX
cana-4781	105	21	open	open	ADJ
cana-4781	105	22	in	in	ADP
cana-4781	105	23	𝑐𝑙(𝐻ℜ	𝑐𝑙(𝐻ℜ	NOUN
cana-4781	105	24	)	)	PUNCT
cana-4781	105	25	,	,	PUNCT
cana-4781	105	26	closure	closure	NOUN
cana-4781	105	27	of	of	ADP
cana-4781	105	28	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	105	29	and	and	CCONJ
cana-4781	105	30	𝑐𝑙(𝐻ℜ	𝑐𝑙(𝐻ℜ	NOUN
cana-4781	105	31	)	)	PUNCT
cana-4781	105	32	is	be	AUX
cana-4781	105	33	a	a	DET
cana-4781	105	34	topological	topological	ADJ
cana-4781	105	35	rough	rough	ADJ
cana-4781	105	36	subgroup	subgroup	NOUN
cana-4781	105	37	in	in	ADP
cana-4781	105	38	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	105	39	̅̅	̅̅	PROPN
cana-4781	105	40	̅̅	̅̅	PROPN
cana-4781	105	41	.	.	PUNCT
cana-4781	106	1	therefore	therefore	ADV
cana-4781	106	2	,	,	PUNCT
cana-4781	106	3	by	by	ADP
cana-4781	106	4	theorem	theorem	NOUN
cana-4781	106	5	3.4	3.4	NUM
cana-4781	106	6	,	,	PUNCT
cana-4781	106	7	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	106	8	is	be	AUX
cana-4781	106	9	closed	close	VERB
cana-4781	106	10	in	in	ADP
cana-4781	106	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	106	12	̅̅	̅̅	PROPN
cana-4781	106	13	̅̅	̅̅	PROPN
cana-4781	106	14	.	.	PUNCT
cana-4781	107	1	theorem	theorem	VERB
cana-4781	107	2	3.7	3.7	NUM
cana-4781	107	3	.	.	PUNCT
cana-4781	108	1	let	let	VERB
cana-4781	108	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	108	3	be	be	AUX
cana-4781	108	4	a	a	DET
cana-4781	108	5	topological	topological	ADJ
cana-4781	108	6	simple	simple	ADJ
cana-4781	108	7	rough	rough	ADJ
cana-4781	108	8	group	group	NOUN
cana-4781	108	9	such	such	ADJ
cana-4781	108	10	that	that	SCONJ
cana-4781	108	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	108	12	̅̅	̅̅	PROPN
cana-4781	108	13	̅̅	̅̅	PROPN
cana-4781	108	14	is	be	AUX
cana-4781	108	15	a	a	DET
cana-4781	108	16	hausdorff	hausdorff	NOUN
cana-4781	108	17	topological	topological	ADJ
cana-4781	108	18	group	group	NOUN
cana-4781	108	19	and	and	CCONJ
cana-4781	108	20	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	108	21	is	be	AUX
cana-4781	108	22	a	a	DET
cana-4781	108	23	compact	compact	ADJ
cana-4781	108	24	subgroup	subgroup	NOUN
cana-4781	108	25	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	108	26	̅̅	̅̅	PROPN
cana-4781	108	27	̅̅	̅̅	PROPN
cana-4781	108	28	.	.	PUNCT
cana-4781	109	1	then	then	ADV
cana-4781	109	2	the	the	DET
cana-4781	109	3	rough	rough	ADJ
cana-4781	109	4	quotient	quotient	NOUN
cana-4781	109	5	mapping	map	VERB
cana-4781	109	6	𝜑	𝜑	NOUN
cana-4781	109	7	:	:	PUNCT
cana-4781	109	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	109	9	̅̅	̅̅	PROPN
cana-4781	109	10	̅̅	̅̅	PROPN
cana-4781	109	11	→	→	PUNCT
cana-4781	109	12	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	109	13	̅̅	̅̅	PROPN
cana-4781	109	14	̅̅	̅̅	PROPN
cana-4781	109	15	𝐻ℜ⁄	𝐻ℜ⁄	PROPN
cana-4781	109	16	is	be	AUX
cana-4781	109	17	perfect	perfect	ADJ
cana-4781	109	18	.	.	PUNCT
cana-4781	110	1	communications	communication	NOUN
cana-4781	110	2	on	on	ADP
cana-4781	110	3	applied	apply	VERB
cana-4781	110	4	nonlinear	nonlinear	ADJ
cana-4781	110	5	analysis	analysis	NOUN
cana-4781	110	6	issn	issn	NOUN
cana-4781	110	7	:	:	PUNCT
cana-4781	110	8	1074	1074	NUM
cana-4781	110	9	-	-	PUNCT
cana-4781	110	10	133x	133x	NUM
cana-4781	110	11	vol	vol	NOUN
cana-4781	110	12	32	32	NUM
cana-4781	110	13	no	no	NOUN
cana-4781	110	14	.	.	NOUN
cana-4781	110	15	3	3	NUM
cana-4781	110	16	(	(	PUNCT
cana-4781	110	17	2025	2025	NUM
cana-4781	110	18	)	)	PUNCT
cana-4781	110	19	883	883	NUM
cana-4781	110	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-4781	110	21	proof	proof	NOUN
cana-4781	110	22	:	:	PUNCT
cana-4781	110	23	let	let	VERB
cana-4781	110	24	𝑀	𝑀	PRON
cana-4781	110	25	be	be	AUX
cana-4781	110	26	a	a	DET
cana-4781	110	27	closed	closed	ADJ
cana-4781	110	28	subset	subset	NOUN
cana-4781	110	29	of	of	ADP
cana-4781	110	30	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	110	31	̅̅	̅̅	PROPN
cana-4781	110	32	̅̅	̅̅	PROPN
cana-4781	110	33	.	.	PUNCT
cana-4781	111	1	then	then	ADV
cana-4781	111	2	by	by	ADP
cana-4781	111	3	the	the	DET
cana-4781	111	4	second	second	ADJ
cana-4781	111	5	closure	closure	NOUN
cana-4781	111	6	lemma	lemma	PROPN
cana-4781	111	7	,	,	PUNCT
cana-4781	111	8	𝑀𝐻ℜ	𝑀𝐻ℜ	PROPN
cana-4781	111	9	is	be	AUX
cana-4781	111	10	closed	close	VERB
cana-4781	111	11	in	in	ADP
cana-4781	111	12	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	111	13	̅̅	̅̅	PROPN
cana-4781	111	14	̅̅	̅̅	PROPN
cana-4781	111	15	.	.	PUNCT
cana-4781	112	1	that	that	PRON
cana-4781	112	2	is	is	AUX
cana-4781	112	3	,	,	PUNCT
cana-4781	112	4	𝜑(𝑀	𝜑(𝑀	VERB
cana-4781	112	5	)	)	PUNCT
cana-4781	113	1	is	be	AUX
cana-4781	113	2	closed	close	VERB
cana-4781	113	3	in	in	ADP
cana-4781	113	4	rough	rough	ADJ
cana-4781	113	5	quotient	quotient	NOUN
cana-4781	113	6	space	space	NOUN
cana-4781	113	7	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	113	8	̅̅	̅̅	PROPN
cana-4781	113	9	̅̅	̅̅	PROPN
cana-4781	113	10	𝐻ℜ⁄	𝐻ℜ⁄	PROPN
cana-4781	113	11	.	.	PUNCT
cana-4781	114	1	therefore	therefore	ADV
cana-4781	114	2	,	,	PUNCT
cana-4781	114	3	the	the	DET
cana-4781	114	4	rough	rough	ADJ
cana-4781	114	5	quotient	quotient	NOUN
cana-4781	114	6	mapping	mapping	NOUN
cana-4781	114	7	𝜑	𝜑	NOUN
cana-4781	114	8	is	be	AUX
cana-4781	114	9	closed	closed	ADJ
cana-4781	114	10	.	.	PUNCT
cana-4781	115	1	let	let	VERB
cana-4781	115	2	𝑏	𝑏	PRON
cana-4781	115	3	∈	∈	PROPN
cana-4781	115	4	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	115	5	̅̅	̅̅	NOUN
cana-4781	115	6	̅̅	̅̅	PROPN
cana-4781	115	7	𝐻ℜ⁄	𝐻ℜ⁄	PROPN
cana-4781	115	8	and	and	CCONJ
cana-4781	115	9	𝑎	𝑎	PROPN
cana-4781	115	10	∈	∈	PROPN
cana-4781	115	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	115	12	̅̅	̅̅	NOUN
cana-4781	115	13	̅̅	̅̅	PROPN
cana-4781	115	14	such	such	ADJ
cana-4781	115	15	that	that	DET
cana-4781	115	16	𝜑(𝑎	𝜑(𝑎	NOUN
cana-4781	115	17	)	)	PUNCT
cana-4781	115	18	=	=	VERB
cana-4781	116	1	𝑏.	𝑏.	ADV
cana-4781	116	2	then	then	ADV
cana-4781	116	3	𝜑−1(𝑏	𝜑−1(𝑏	NUM
cana-4781	116	4	)	)	PUNCT
cana-4781	116	5	=	=	PUNCT
cana-4781	117	1	𝑎𝐻ℜ	𝑎𝐻ℜ	NOUN
cana-4781	117	2	,	,	PUNCT
cana-4781	117	3	is	be	AUX
cana-4781	117	4	a	a	DET
cana-4781	117	5	compact	compact	ADJ
cana-4781	117	6	subset	subset	NOUN
cana-4781	117	7	of	of	ADP
cana-4781	117	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	117	9	̅̅	̅̅	PROPN
cana-4781	117	10	̅̅	̅̅	PROPN
cana-4781	117	11	.	.	PUNCT
cana-4781	118	1	hence	hence	ADV
cana-4781	118	2	,	,	PUNCT
cana-4781	118	3	by	by	ADP
cana-4781	118	4	the	the	DET
cana-4781	118	5	definition	definition	NOUN
cana-4781	118	6	of	of	ADP
cana-4781	118	7	2.14	2.14	NUM
cana-4781	118	8	,	,	PUNCT
cana-4781	118	9	the	the	DET
cana-4781	118	10	rough	rough	ADJ
cana-4781	118	11	quotient	quotient	NOUN
cana-4781	118	12	mapping	mapping	NOUN
cana-4781	118	13	𝜑	𝜑	NOUN
cana-4781	118	14	is	be	AUX
cana-4781	118	15	perfect	perfect	ADJ
cana-4781	118	16	.	.	PUNCT
cana-4781	119	1	theorem	theorem	VERB
cana-4781	119	2	3.8	3.8	NUM
cana-4781	119	3	.	.	PUNCT
cana-4781	120	1	suppose	suppose	VERB
cana-4781	120	2	𝑀	𝑀	PROPN
cana-4781	120	3	is	be	AUX
cana-4781	120	4	a	a	DET
cana-4781	120	5	compact	compact	ADJ
cana-4781	120	6	subset	subset	NOUN
cana-4781	120	7	of	of	ADP
cana-4781	120	8	a	a	DET
cana-4781	120	9	topological	topological	ADJ
cana-4781	120	10	simple	simple	ADJ
cana-4781	120	11	rough	rough	ADJ
cana-4781	120	12	group	group	NOUN
cana-4781	120	13	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	120	14	such	such	ADJ
cana-4781	120	15	that	that	SCONJ
cana-4781	120	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	120	17	̅̅	̅̅	PROPN
cana-4781	120	18	̅̅	̅̅	PROPN
cana-4781	120	19	is	be	AUX
cana-4781	120	20	a	a	DET
cana-4781	120	21	group	group	NOUN
cana-4781	120	22	.	.	PUNCT
cana-4781	121	1	then	then	ADV
cana-4781	121	2	there	there	PRON
cana-4781	121	3	exists	exist	VERB
cana-4781	121	4	a	a	DET
cana-4781	121	5	smallest	small	ADJ
cana-4781	121	6	rough	rough	ADJ
cana-4781	121	7	subgroup	subgroup	NOUN
cana-4781	121	8	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	121	9	in	in	ADP
cana-4781	121	10	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	121	11	̅̅	̅̅	PROPN
cana-4781	121	12	̅̅	̅̅	PROPN
cana-4781	121	13	containing	contain	VERB
cana-4781	121	14	m	m	VERB
cana-4781	121	15	such	such	ADJ
cana-4781	121	16	that	that	SCONJ
cana-4781	121	17	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	121	18	is	be	AUX
cana-4781	121	19	𝜎compact	𝜎compact	NOUN
cana-4781	121	20	.	.	PUNCT
cana-4781	122	1	proof	proof	NOUN
cana-4781	122	2	:	:	PUNCT
cana-4781	122	3	let	let	VERB
cana-4781	122	4	𝐴	𝐴	PROPN
cana-4781	122	5	=	=	SYM
cana-4781	122	6	𝑀	𝑀	PROPN
cana-4781	122	7	∪	∪	NOUN
cana-4781	122	8	{	{	PUNCT
cana-4781	122	9	e	e	NOUN
cana-4781	122	10	}	}	PUNCT
cana-4781	122	11	∪	∪	ADJ
cana-4781	122	12	𝑀−1	𝑀−1	PROPN
cana-4781	122	13	.	.	PUNCT
cana-4781	123	1	since	since	SCONJ
cana-4781	123	2	𝑀	𝑀	PROPN
cana-4781	123	3	is	be	AUX
cana-4781	123	4	a	a	DET
cana-4781	123	5	compact	compact	ADJ
cana-4781	123	6	subset	subset	NOUN
cana-4781	123	7	of	of	ADP
cana-4781	123	8	a	a	DET
cana-4781	123	9	topological	topological	ADJ
cana-4781	123	10	simple	simple	ADJ
cana-4781	123	11	rough	rough	ADJ
cana-4781	123	12	group	group	NOUN
cana-4781	123	13	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	123	14	,	,	PUNCT
cana-4781	123	15	𝐴	𝐴	PROPN
cana-4781	123	16	is	be	AUX
cana-4781	123	17	compact	compact	ADJ
cana-4781	123	18	in	in	ADP
cana-4781	123	19	𝐺ℜ.	𝐺ℜ.	PUNCT
cana-4781	123	20	now	now	ADV
cana-4781	123	21	define	define	VERB
cana-4781	123	22	the	the	DET
cana-4781	123	23	multiplication	multiplication	NOUN
cana-4781	123	24	mapping	mapping	NOUN
cana-4781	123	25	𝑓𝑖	𝑓𝑖	NOUN
cana-4781	123	26	:	:	PUNCT
cana-4781	123	27	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	123	28	𝑖	𝑖	PROPN
cana-4781	123	29	→	→	SYM
cana-4781	123	30	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	123	31	̅̅	̅̅	NOUN
cana-4781	123	32	̅̅	̅̅	PROPN
cana-4781	123	33	by	by	ADP
cana-4781	123	34	𝑓𝑖(𝑥1	𝑓𝑖(𝑥1	ADJ
cana-4781	123	35	,	,	PUNCT
cana-4781	123	36	𝑥2	𝑥2	NOUN
cana-4781	123	37	,	,	PUNCT
cana-4781	123	38	…	…	PUNCT
cana-4781	123	39	,	,	PUNCT
cana-4781	123	40	𝑥𝑖	𝑥𝑖	X
cana-4781	123	41	)	)	PUNCT
cana-4781	123	42	=	=	SYM
cana-4781	123	43	𝑥1𝑥2	𝑥1𝑥2	PROPN
cana-4781	123	44	…	…	SYM
cana-4781	123	45	𝑥𝑖	𝑥𝑖	PROPN
cana-4781	123	46	,	,	PUNCT
cana-4781	123	47	for	for	ADP
cana-4781	123	48	𝑥1	𝑥1	NOUN
cana-4781	123	49	,	,	PUNCT
cana-4781	123	50	𝑥2	𝑥2	NOUN
cana-4781	123	51	,	,	PUNCT
cana-4781	123	52	…	…	PUNCT
cana-4781	123	53	,	,	PUNCT
cana-4781	123	54	𝑥𝑖	𝑥𝑖	PROPN
cana-4781	123	55	∈	∈	PROPN
cana-4781	123	56	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	123	57	and	and	CCONJ
cana-4781	123	58	for	for	ADP
cana-4781	123	59	every	every	DET
cana-4781	123	60	𝑖	𝑖	PROPN
cana-4781	123	61	∈	∈	PROPN
cana-4781	123	62	ℕ.	ℕ.	PROPN
cana-4781	123	63	since	since	SCONJ
cana-4781	123	64	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	123	65	is	be	AUX
cana-4781	123	66	a	a	DET
cana-4781	123	67	topological	topological	ADJ
cana-4781	123	68	simple	simple	ADJ
cana-4781	123	69	rough	rough	ADJ
cana-4781	123	70	group	group	NOUN
cana-4781	123	71	and	and	CCONJ
cana-4781	123	72	continuous	continuous	ADJ
cana-4781	123	73	image	image	NOUN
cana-4781	123	74	of	of	ADP
cana-4781	123	75	compact	compact	ADJ
cana-4781	123	76	set	set	NOUN
cana-4781	123	77	is	be	AUX
cana-4781	123	78	compact	compact	ADJ
cana-4781	123	79	,	,	PUNCT
cana-4781	123	80	the	the	DET
cana-4781	123	81	mappings	mapping	NOUN
cana-4781	123	82	𝑓𝑖	𝑓𝑖	VERB
cana-4781	123	83	are	be	AUX
cana-4781	123	84	continuous	continuous	ADJ
cana-4781	123	85	which	which	PRON
cana-4781	123	86	implies	imply	VERB
cana-4781	123	87	𝑓𝑖(𝐴𝑖	𝑓𝑖(𝐴𝑖	PROPN
cana-4781	123	88	)	)	PUNCT
cana-4781	123	89	is	be	AUX
cana-4781	123	90	compact	compact	ADJ
cana-4781	123	91	in	in	ADP
cana-4781	123	92	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	123	93	̅̅	̅̅	PROPN
cana-4781	123	94	̅̅	̅̅	PROPN
cana-4781	123	95	,	,	PUNCT
cana-4781	123	96	for	for	ADP
cana-4781	123	97	every	every	DET
cana-4781	123	98	𝑖	𝑖	PROPN
cana-4781	123	99	∈	∈	PROPN
cana-4781	123	100	ℕ.	ℕ.	PROPN
cana-4781	123	101	therefore	therefore	ADV
cana-4781	123	102	,	,	PUNCT
cana-4781	123	103	the	the	DET
cana-4781	123	104	rough	rough	ADJ
cana-4781	123	105	subgroup	subgroup	NOUN
cana-4781	123	106	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	123	107	=	=	SYM
cana-4781	123	108	⋃	⋃	PROPN
cana-4781	123	109	𝑓𝑖(𝐴𝑖	𝑓𝑖(𝐴𝑖	PROPN
cana-4781	123	110	)	)	PUNCT
cana-4781	123	111	𝑛	𝑛	PRON
cana-4781	123	112	𝑖=1	𝑖=1	PROPN
cana-4781	123	113	is	be	AUX
cana-4781	123	114	generated	generate	VERB
cana-4781	123	115	by	by	ADP
cana-4781	123	116	𝑀.	𝑀.	PROPN
cana-4781	123	117	hence	hence	ADV
cana-4781	123	118	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	123	119	is	be	AUX
cana-4781	123	120	𝜎-compact	𝜎-compact	PROPN
cana-4781	123	121	.	.	PUNCT
cana-4781	124	1	theorem	theorem	VERB
cana-4781	124	2	3.9	3.9	NUM
cana-4781	124	3	.	.	PUNCT
cana-4781	125	1	(	(	PUNCT
cana-4781	125	2	open	open	ADJ
cana-4781	125	3	mapping	mapping	NOUN
cana-4781	125	4	theorem	theorem	VERB
cana-4781	125	5	i	i	PRON
cana-4781	125	6	)	)	PUNCT
cana-4781	125	7	let	let	VERB
cana-4781	125	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	125	9	and	and	CCONJ
cana-4781	125	10	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	125	11	be	be	AUX
cana-4781	125	12	topological	topological	ADJ
cana-4781	125	13	simple	simple	ADJ
cana-4781	125	14	rough	rough	ADJ
cana-4781	125	15	groups	group	NOUN
cana-4781	125	16	such	such	ADJ
cana-4781	125	17	that	that	SCONJ
cana-4781	125	18	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	125	19	and	and	CCONJ
cana-4781	125	20	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	125	21	are	be	AUX
cana-4781	125	22	open	open	ADJ
cana-4781	125	23	in	in	ADP
cana-4781	125	24	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	125	25	̅̅	̅̅	PROPN
cana-4781	125	26	̅̅	̅̅	PROPN
cana-4781	125	27	and	and	CCONJ
cana-4781	125	28	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	125	29	̅̅	̅̅	PROPN
cana-4781	125	30	̅̅	̅̅	PROPN
cana-4781	125	31	.	.	PUNCT
cana-4781	126	1	let	let	VERB
cana-4781	126	2	𝜋	𝜋	NOUN
cana-4781	126	3	:	:	PUNCT
cana-4781	126	4	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	126	5	̅̅	̅̅	PROPN
cana-4781	126	6	̅̅	̅̅	PROPN
cana-4781	126	7	→	→	PUNCT
cana-4781	126	8	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	126	9	̅̅	̅̅	PROPN
cana-4781	126	10	̅̅	̅̅	NOUN
cana-4781	126	11	be	be	AUX
cana-4781	126	12	a	a	DET
cana-4781	126	13	surjective	surjective	ADJ
cana-4781	126	14	mapping	mapping	NOUN
cana-4781	126	15	topological	topological	ADJ
cana-4781	126	16	rough	rough	ADJ
cana-4781	126	17	group	group	NOUN
cana-4781	126	18	homomorphism	homomorphism	NOUN
cana-4781	126	19	.	.	PUNCT
cana-4781	127	1	if	if	SCONJ
cana-4781	127	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	127	3	̅̅	̅̅	PROPN
cana-4781	127	4	̅̅	̅̅	PROPN
cana-4781	127	5	is	be	AUX
cana-4781	127	6	a	a	DET
cana-4781	127	7	compact	compact	ADJ
cana-4781	127	8	space	space	NOUN
cana-4781	127	9	and	and	CCONJ
cana-4781	127	10	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	127	11	̅̅	̅̅	PROPN
cana-4781	127	12	̅̅	̅̅	PROPN
cana-4781	127	13	is	be	AUX
cana-4781	127	14	a	a	DET
cana-4781	127	15	hausdorff	hausdorff	NOUN
cana-4781	127	16	space	space	NOUN
cana-4781	127	17	,	,	PUNCT
cana-4781	127	18	then	then	ADV
cana-4781	127	19	the	the	DET
cana-4781	127	20	mapping	mapping	NOUN
cana-4781	127	21	𝜋	𝜋	NOUN
cana-4781	127	22	is	be	AUX
cana-4781	127	23	open	open	ADJ
cana-4781	127	24	.	.	PUNCT
cana-4781	128	1	proof	proof	NOUN
cana-4781	128	2	:	:	PUNCT
cana-4781	128	3	since	since	SCONJ
cana-4781	128	4	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	128	5	̅̅	̅̅	PROPN
cana-4781	128	6	̅̅	̅̅	PROPN
cana-4781	128	7	is	be	AUX
cana-4781	128	8	a	a	DET
cana-4781	128	9	compact	compact	ADJ
cana-4781	128	10	space	space	NOUN
cana-4781	128	11	and	and	CCONJ
cana-4781	128	12	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	128	13	̅̅	̅̅	PROPN
cana-4781	128	14	̅̅	̅̅	PROPN
cana-4781	128	15	is	be	AUX
cana-4781	128	16	a	a	DET
cana-4781	128	17	hausdorff	hausdorff	NOUN
cana-4781	128	18	space	space	NOUN
cana-4781	128	19	,	,	PUNCT
cana-4781	128	20	𝜋	𝜋	NOUN
cana-4781	128	21	is	be	AUX
cana-4781	128	22	closed	closed	ADJ
cana-4781	128	23	.	.	PUNCT
cana-4781	129	1	also	also	ADV
cana-4781	129	2	,	,	PUNCT
cana-4781	129	3	by	by	ADP
cana-4781	129	4	the	the	DET
cana-4781	129	5	continuity	continuity	NOUN
cana-4781	129	6	of	of	ADP
cana-4781	129	7	𝜋	𝜋	NOUN
cana-4781	129	8	,	,	PUNCT
cana-4781	129	9	the	the	DET
cana-4781	129	10	mapping	mapping	NOUN
cana-4781	129	11	𝜋	𝜋	NOUN
cana-4781	129	12	is	be	AUX
cana-4781	129	13	a	a	DET
cana-4781	129	14	quotient	quotient	NOUN
cana-4781	129	15	map	map	NOUN
cana-4781	129	16	that	that	PRON
cana-4781	129	17	means	mean	VERB
cana-4781	129	18	a	a	DET
cana-4781	129	19	subset	subset	NOUN
cana-4781	130	1	𝑈	𝑈	PROPN
cana-4781	130	2	⊆	⊆	NUM
cana-4781	130	3	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	130	4	̅̅	̅̅	PROPN
cana-4781	130	5	̅̅	̅̅	NOUN
cana-4781	130	6	is	be	AUX
cana-4781	130	7	open	open	ADJ
cana-4781	130	8	if	if	SCONJ
cana-4781	130	9	and	and	CCONJ
cana-4781	130	10	only	only	ADV
cana-4781	130	11	if	if	SCONJ
cana-4781	130	12	𝜋−1(𝑈	𝜋−1(𝑈	NOUN
cana-4781	130	13	)	)	PUNCT
cana-4781	130	14	is	be	AUX
cana-4781	130	15	open	open	ADJ
cana-4781	130	16	in	in	ADP
cana-4781	130	17	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	130	18	̅̅	̅̅	PROPN
cana-4781	130	19	̅̅	̅̅	PROPN
cana-4781	130	20	.	.	PUNCT
cana-4781	131	1	let	let	VERB
cana-4781	131	2	us	we	PRON
cana-4781	131	3	prove	prove	VERB
cana-4781	131	4	𝜋	𝜋	NOUN
cana-4781	131	5	is	be	AUX
cana-4781	131	6	an	an	DET
cana-4781	131	7	open	open	ADJ
cana-4781	131	8	mapping	mapping	NOUN
cana-4781	131	9	.	.	PUNCT
cana-4781	132	1	let	let	VERB
cana-4781	132	2	𝑉	𝑉	PRON
cana-4781	132	3	be	be	AUX
cana-4781	132	4	an	an	DET
cana-4781	132	5	open	open	ADJ
cana-4781	132	6	set	set	NOUN
cana-4781	132	7	in	in	ADP
cana-4781	132	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	132	9	̅̅	̅̅	PROPN
cana-4781	132	10	̅̅	̅̅	PROPN
cana-4781	132	11	.	.	PUNCT
cana-4781	133	1	then	then	ADV
cana-4781	133	2	𝜋−1(𝜋(𝑉	𝜋−1(𝜋(𝑉	NOUN
cana-4781	133	3	)	)	PUNCT
cana-4781	133	4	)	)	PUNCT
cana-4781	134	1	=	=	SYM
cana-4781	134	2	𝒦𝜋𝑉	𝒦𝜋𝑉	PROPN
cana-4781	134	3	is	be	AUX
cana-4781	134	4	open	open	ADJ
cana-4781	134	5	in	in	ADP
cana-4781	134	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	134	7	̅̅	̅̅	NOUN
cana-4781	134	8	̅̅	̅̅	PROPN
cana-4781	134	9	,	,	PUNCT
cana-4781	134	10	where	where	SCONJ
cana-4781	134	11	𝒦𝜋	𝒦𝜋	PROPN
cana-4781	134	12	is	be	AUX
cana-4781	134	13	the	the	DET
cana-4781	134	14	rough	rough	ADJ
cana-4781	134	15	kernel	kernel	NOUN
cana-4781	134	16	of	of	ADP
cana-4781	134	17	𝜋.	𝜋.	ADV
cana-4781	134	18	now	now	ADV
cana-4781	134	19	consider	consider	VERB
cana-4781	134	20	𝑈	𝑈	PROPN
cana-4781	134	21	=	=	PUNCT
cana-4781	134	22	𝜋(𝑉	𝜋(𝑉	PROPN
cana-4781	134	23	)	)	PUNCT
cana-4781	134	24	that	that	PRON
cana-4781	134	25	implies	imply	VERB
cana-4781	134	26	𝜋−1(𝑈	𝜋−1(𝑈	NOUN
cana-4781	134	27	)	)	PUNCT
cana-4781	134	28	is	be	AUX
cana-4781	134	29	open	open	ADJ
cana-4781	134	30	in	in	ADP
cana-4781	134	31	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	134	32	̅̅	̅̅	PROPN
cana-4781	134	33	̅̅	̅̅	PROPN
cana-4781	134	34	.	.	PUNCT
cana-4781	135	1	since	since	SCONJ
cana-4781	135	2	𝜋	𝜋	NOUN
cana-4781	135	3	is	be	AUX
cana-4781	135	4	a	a	DET
cana-4781	135	5	quotient	quotient	NOUN
cana-4781	135	6	map	map	NOUN
cana-4781	135	7	,	,	PUNCT
cana-4781	135	8	𝑈	𝑈	PROPN
cana-4781	135	9	=	=	PUNCT
cana-4781	135	10	𝜋(𝑉	𝜋(𝑉	PROPN
cana-4781	135	11	)	)	PUNCT
cana-4781	135	12	is	be	AUX
cana-4781	135	13	open	open	ADJ
cana-4781	135	14	.	.	PUNCT
cana-4781	136	1	hence	hence	ADV
cana-4781	136	2	the	the	DET
cana-4781	136	3	mapping	mapping	NOUN
cana-4781	136	4	𝜋	𝜋	NOUN
cana-4781	136	5	is	be	AUX
cana-4781	136	6	open	open	ADJ
cana-4781	136	7	.	.	PUNCT
cana-4781	137	1	proposition	proposition	NOUN
cana-4781	137	2	3.10	3.10	NUM
cana-4781	137	3	.	.	PUNCT
cana-4781	138	1	let	let	VERB
cana-4781	138	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	138	3	be	be	AUX
cana-4781	138	4	a	a	DET
cana-4781	138	5	topological	topological	ADJ
cana-4781	138	6	simple	simple	ADJ
cana-4781	138	7	rough	rough	ADJ
cana-4781	138	8	group	group	NOUN
cana-4781	138	9	and	and	CCONJ
cana-4781	138	10	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	138	11	be	be	AUX
cana-4781	138	12	an	an	DET
cana-4781	138	13	open	open	ADJ
cana-4781	138	14	set	set	NOUN
cana-4781	138	15	in	in	ADP
cana-4781	138	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	138	17	̅̅	̅̅	PROPN
cana-4781	138	18	̅̅	̅̅	PROPN
cana-4781	138	19	.	.	PUNCT
cana-4781	139	1	suppose	suppose	VERB
cana-4781	139	2	the	the	DET
cana-4781	139	3	subset	subset	NOUN
cana-4781	139	4	𝐴	𝐴	PROPN
cana-4781	139	5	⊆	⊆	NUM
cana-4781	139	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	139	7	and	and	CCONJ
cana-4781	139	8	𝑖𝑛𝑡(𝑥𝐴	𝑖𝑛𝑡(𝑥𝐴	NOUN
cana-4781	139	9	∩	∩	ADJ
cana-4781	139	10	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	139	11	)	)	PUNCT
cana-4781	139	12	≠	≠	PROPN
cana-4781	139	13	∅.	∅.	VERB
cana-4781	139	14	then	then	ADV
cana-4781	139	15	𝑖𝑛𝑡(𝐴	𝑖𝑛𝑡(𝐴	NUM
cana-4781	139	16	)	)	PUNCT
cana-4781	139	17	≠	≠	PROPN
cana-4781	139	18	∅	∅	NOUN
cana-4781	139	19	,	,	PUNCT
cana-4781	139	20	where	where	SCONJ
cana-4781	139	21	𝑖𝑛𝑡	𝑖𝑛𝑡	PROPN
cana-4781	139	22	means	mean	VERB
cana-4781	139	23	interior	interior	ADJ
cana-4781	139	24	of	of	ADP
cana-4781	139	25	the	the	DET
cana-4781	139	26	set	set	NOUN
cana-4781	139	27	.	.	PUNCT
cana-4781	140	1	proof	proof	NOUN
cana-4781	140	2	:	:	PUNCT
cana-4781	140	3	consider	consider	VERB
cana-4781	140	4	an	an	DET
cana-4781	140	5	arbitrary	arbitrary	ADJ
cana-4781	140	6	element	element	NOUN
cana-4781	140	7	𝑏	𝑏	PRON
cana-4781	140	8	∈	∈	PROPN
cana-4781	140	9	𝑖𝑛𝑡(𝑥𝐴	𝑖𝑛𝑡(𝑥𝐴	NOUN
cana-4781	140	10	∩	∩	ADJ
cana-4781	140	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	140	12	)	)	PUNCT
cana-4781	140	13	.	.	PUNCT
cana-4781	141	1	then	then	ADV
cana-4781	141	2	there	there	PRON
cana-4781	141	3	is	be	VERB
cana-4781	141	4	a	a	DET
cana-4781	141	5	point	point	NOUN
cana-4781	141	6	𝑎	𝑎	PRON
cana-4781	141	7	∈	∈	NOUN
cana-4781	141	8	𝐴	𝐴	NOUN
cana-4781	141	9	such	such	ADJ
cana-4781	141	10	that	that	SCONJ
cana-4781	141	11	𝑏	𝑏	NOUN
cana-4781	141	12	=	=	PUNCT
cana-4781	141	13	𝑥𝑎.	𝑥𝑎.	NOUN
cana-4781	141	14	since	since	SCONJ
cana-4781	141	15	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	141	16	is	be	AUX
cana-4781	141	17	a	a	DET
cana-4781	141	18	topological	topological	ADJ
cana-4781	141	19	simple	simple	ADJ
cana-4781	141	20	rough	rough	ADJ
cana-4781	141	21	group	group	NOUN
cana-4781	141	22	and	and	CCONJ
cana-4781	141	23	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	141	24	is	be	AUX
cana-4781	141	25	an	an	DET
cana-4781	141	26	open	open	ADJ
cana-4781	141	27	set	set	NOUN
cana-4781	141	28	in	in	ADP
cana-4781	141	29	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	141	30	̅̅	̅̅	PROPN
cana-4781	141	31	̅̅	̅̅	PROPN
cana-4781	141	32	,	,	PUNCT
cana-4781	141	33	there	there	PRON
cana-4781	141	34	is	be	VERB
cana-4781	141	35	a	a	DET
cana-4781	141	36	neighbourhood	neighbourhood	NOUN
cana-4781	141	37	𝑈	𝑈	NOUN
cana-4781	141	38	of	of	ADP
cana-4781	141	39	𝑎	𝑎	NOUN
cana-4781	141	40	in	in	ADP
cana-4781	141	41	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	141	42	such	such	ADJ
cana-4781	141	43	that	that	SCONJ
cana-4781	141	44	𝑥𝑈	𝑥𝑈	NOUN
cana-4781	141	45	⊆	⊆	NUM
cana-4781	141	46	𝑖𝑛𝑡(𝑥𝐴	𝑖𝑛𝑡(𝑥𝐴	NOUN
cana-4781	141	47	∩	∩	ADJ
cana-4781	141	48	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	141	49	)	)	PUNCT
cana-4781	141	50	which	which	PRON
cana-4781	141	51	implies	imply	VERB
cana-4781	141	52	𝑥𝑈	𝑥𝑈	NOUN
cana-4781	141	53	⊆	⊆	NUM
cana-4781	141	54	𝑥𝐴	𝑥𝐴	NOUN
cana-4781	141	55	that	that	PRON
cana-4781	141	56	is	be	AUX
cana-4781	141	57	,	,	PUNCT
cana-4781	141	58	𝑈	𝑈	PROPN
cana-4781	141	59	⊆	⊆	NUM
cana-4781	141	60	𝐴.	𝐴.	NOUN
cana-4781	141	61	hence	hence	ADV
cana-4781	141	62	,	,	PUNCT
cana-4781	141	63	𝑖𝑛𝑡(𝐴	𝑖𝑛𝑡(𝐴	NUM
cana-4781	141	64	)	)	PUNCT
cana-4781	141	65	≠	≠	PROPN
cana-4781	141	66	∅.	∅.	VERB
cana-4781	141	67	theorem	theorem	VERB
cana-4781	141	68	3.11	3.11	NUM
cana-4781	141	69	.	.	PUNCT
cana-4781	142	1	(	(	PUNCT
cana-4781	142	2	open	open	ADJ
cana-4781	142	3	mapping	mapping	NOUN
cana-4781	142	4	theorem	theorem	ADJ
cana-4781	142	5	ii	ii	NOUN
cana-4781	142	6	)	)	PUNCT
cana-4781	142	7	let	let	VERB
cana-4781	142	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	142	9	and	and	CCONJ
cana-4781	142	10	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	142	11	be	be	AUX
cana-4781	142	12	two	two	NUM
cana-4781	142	13	locally	locally	ADV
cana-4781	142	14	compact	compact	ADJ
cana-4781	142	15	hausdorff	hausdorff	NOUN
cana-4781	142	16	topological	topological	ADJ
cana-4781	142	17	simple	simple	ADJ
cana-4781	142	18	rough	rough	ADJ
cana-4781	142	19	groups	group	NOUN
cana-4781	142	20	such	such	ADJ
cana-4781	142	21	that	that	SCONJ
cana-4781	142	22	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	142	23	and	and	CCONJ
cana-4781	142	24	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	142	25	are	be	AUX
cana-4781	142	26	open	open	ADJ
cana-4781	142	27	in	in	ADP
cana-4781	142	28	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	142	29	̅̅	̅̅	PROPN
cana-4781	142	30	̅̅	̅̅	PROPN
cana-4781	142	31	and	and	CCONJ
cana-4781	142	32	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	142	33	̅̅	̅̅	PROPN
cana-4781	142	34	̅̅	̅̅	PROPN
cana-4781	142	35	.	.	PUNCT
cana-4781	143	1	suppose	suppose	VERB
cana-4781	143	2	a	a	DET
cana-4781	143	3	surjective	surjective	ADJ
cana-4781	143	4	mapping	mapping	NOUN
cana-4781	143	5	𝜋	𝜋	NOUN
cana-4781	143	6	:	:	PUNCT
cana-4781	143	7	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	143	8	→	→	SYM
cana-4781	143	9	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	143	10	is	be	AUX
cana-4781	143	11	a	a	DET
cana-4781	143	12	continuous	continuous	ADJ
cana-4781	143	13	rough	rough	ADJ
cana-4781	143	14	homomorphism	homomorphism	NOUN
cana-4781	143	15	and	and	CCONJ
cana-4781	143	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	143	17	is	be	AUX
cana-4781	143	18	a	a	DET
cana-4781	143	19	𝜎-compact	𝜎-compact	NOUN
cana-4781	143	20	space	space	NOUN
cana-4781	143	21	.	.	PUNCT
cana-4781	144	1	then	then	ADV
cana-4781	144	2	𝜋	𝜋	PRON
cana-4781	144	3	is	be	AUX
cana-4781	144	4	an	an	DET
cana-4781	144	5	open	open	ADJ
cana-4781	144	6	mapping	mapping	NOUN
cana-4781	144	7	.	.	PUNCT
cana-4781	145	1	proof	proof	NOUN
cana-4781	145	2	:	:	PUNCT
cana-4781	145	3	let	let	VERB
cana-4781	145	4	𝑈	𝑈	PROPN
cana-4781	145	5	⊆	⊆	NUM
cana-4781	145	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	145	7	be	be	AUX
cana-4781	145	8	a	a	DET
cana-4781	145	9	symmetric	symmetric	ADJ
cana-4781	145	10	identity	identity	NOUN
cana-4781	145	11	neighbourhood	neighbourhood	NOUN
cana-4781	145	12	.	.	PUNCT
cana-4781	146	1	since	since	SCONJ
cana-4781	146	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	146	3	is	be	AUX
cana-4781	146	4	locally	locally	ADV
cana-4781	146	5	compact	compact	ADJ
cana-4781	146	6	,	,	PUNCT
cana-4781	146	7	there	there	PRON
cana-4781	146	8	exists	exist	VERB
cana-4781	146	9	a	a	DET
cana-4781	146	10	symmetric	symmetric	ADJ
cana-4781	146	11	open	open	ADJ
cana-4781	146	12	neighbourhood	neighbourhood	NOUN
cana-4781	146	13	𝑁	𝑁	PROPN
cana-4781	146	14	of	of	ADP
cana-4781	146	15	e	e	PROPN
cana-4781	146	16	in	in	ADP
cana-4781	146	17	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	146	18	such	such	ADJ
cana-4781	146	19	that	that	SCONJ
cana-4781	146	20	𝑐𝑙(𝑁	𝑐𝑙(𝑁	NOUN
cana-4781	146	21	)	)	PUNCT
cana-4781	146	22	is	be	AUX
cana-4781	146	23	compact	compact	ADJ
cana-4781	146	24	and	and	CCONJ
cana-4781	146	25	𝑐𝑙(𝑁)𝑐𝑙(𝑁	𝑐𝑙(𝑁)𝑐𝑙(𝑁	NOUN
cana-4781	146	26	)	)	PUNCT
cana-4781	146	27	⊆	⊆	X
cana-4781	146	28	𝑈	𝑈	PROPN
cana-4781	146	29	,	,	PUNCT
cana-4781	146	30	where	where	SCONJ
cana-4781	146	31	𝑐𝑙(𝑁	𝑐𝑙(𝑁	X
cana-4781	146	32	)	)	PUNCT
cana-4781	146	33	is	be	AUX
cana-4781	146	34	the	the	DET
cana-4781	146	35	closure	closure	NOUN
cana-4781	146	36	of	of	ADP
cana-4781	146	37	𝑁.	𝑁.	PROPN
cana-4781	146	38	since	since	SCONJ
cana-4781	146	39	𝑁	𝑁	PROPN
cana-4781	146	40	is	be	AUX
cana-4781	146	41	open	open	ADJ
cana-4781	146	42	,	,	PUNCT
cana-4781	146	43	𝑥𝑁	𝑥𝑁	NOUN
cana-4781	146	44	is	be	AUX
cana-4781	146	45	open	open	ADJ
cana-4781	146	46	,	,	PUNCT
cana-4781	146	47	for	for	ADP
cana-4781	146	48	every	every	DET
cana-4781	146	49	𝑥	𝑥	PROPN
cana-4781	146	50	∈	∈	ADJ
cana-4781	146	51	𝐺ℜ.	𝐺ℜ.	PUNCT
cana-4781	147	1	so	so	ADV
cana-4781	147	2	,	,	PUNCT
cana-4781	147	3	⋃	⋃	ADP
cana-4781	147	4	𝑥𝑁𝑥∈𝐺ℜ	𝑥𝑁𝑥∈𝐺ℜ	PRON
cana-4781	147	5	covers	cover	VERB
cana-4781	147	6	𝐺ℜ.	𝐺ℜ.	PUNCT
cana-4781	147	7	therefore	therefore	ADV
cana-4781	147	8	,	,	PUNCT
cana-4781	147	9	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	147	10	=	=	SYM
cana-4781	147	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	147	12	∩	∩	NOUN
cana-4781	147	13	⋃	⋃	PROPN
cana-4781	147	14	𝑥𝑁𝑥∈𝐺ℜ	𝑥𝑁𝑥∈𝐺ℜ	PRON
cana-4781	147	15	.	.	PUNCT
cana-4781	148	1	since	since	SCONJ
cana-4781	148	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	148	3	is	be	AUX
cana-4781	148	4	a	a	DET
cana-4781	148	5	𝜎-compact	𝜎-compact	ADJ
cana-4781	148	6	space	space	NOUN
cana-4781	148	7	,	,	PUNCT
cana-4781	148	8	there	there	PRON
cana-4781	148	9	exists	exist	VERB
cana-4781	148	10	a	a	DET
cana-4781	148	11	countable	countable	ADJ
cana-4781	148	12	set	set	NOUN
cana-4781	148	13	{	{	PUNCT
cana-4781	148	14	𝑥𝑖}𝑖∈ℕ	𝑥𝑖}𝑖∈ℕ	PROPN
cana-4781	148	15	,	,	PUNCT
cana-4781	148	16	such	such	ADJ
cana-4781	148	17	that	that	DET
cana-4781	148	18	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	148	19	=	=	SYM
cana-4781	148	20	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	148	21	∩	∩	NOUN
cana-4781	148	22	⋃	⋃	PROPN
cana-4781	148	23	𝑥𝑖𝑁𝑖∈ℕ	𝑥𝑖𝑁𝑖∈ℕ	NOUN
cana-4781	148	24	.	.	PUNCT
cana-4781	149	1	since	since	SCONJ
cana-4781	149	2	𝑐𝑙(𝑁	𝑐𝑙(𝑁	NUM
cana-4781	149	3	)	)	PUNCT
cana-4781	149	4	is	be	AUX
cana-4781	149	5	compact	compact	ADJ
cana-4781	149	6	and	and	CCONJ
cana-4781	149	7	𝜋	𝜋	NOUN
cana-4781	149	8	is	be	AUX
cana-4781	149	9	continuous	continuous	ADJ
cana-4781	149	10	,	,	PUNCT
cana-4781	149	11	𝜋(𝐺ℜ	𝜋(𝐺ℜ	PROPN
cana-4781	149	12	)	)	PUNCT
cana-4781	149	13	=	=	SYM
cana-4781	149	14	𝜋(𝐺ℜ	𝜋(𝐺ℜ	ADJ
cana-4781	149	15	)	)	PUNCT
cana-4781	149	16	∩	∩	NOUN
cana-4781	149	17	⋃	⋃	NOUN
cana-4781	149	18	𝜋(𝑥𝑖𝑐𝑙(𝑁𝑖∈ℕ	𝜋(𝑥𝑖𝑐𝑙(𝑁𝑖∈ℕ	NUM
cana-4781	149	19	)	)	PUNCT
cana-4781	149	20	)	)	PUNCT
cana-4781	149	21	,	,	PUNCT
cana-4781	149	22	for	for	ADP
cana-4781	149	23	every	every	DET
cana-4781	149	24	𝑥𝑖	𝑥𝑖	PROPN
cana-4781	149	25	∈	∈	PROPN
cana-4781	149	26	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	149	27	implies	imply	VERB
cana-4781	149	28	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	149	29	=	=	SYM
cana-4781	149	30	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	149	31	∩	∩	NOUN
cana-4781	149	32	⋃	⋃	NOUN
cana-4781	149	33	𝜋(𝑥𝑖	𝜋(𝑥𝑖	NUM
cana-4781	149	34	)	)	PUNCT
cana-4781	149	35	𝜋(𝑐𝑙(𝑁𝑖∈ℕ	𝜋(𝑐𝑙(𝑁𝑖∈ℕ	NUM
cana-4781	149	36	)	)	PUNCT
cana-4781	149	37	)	)	PUNCT
cana-4781	150	1	=	=	PUNCT
cana-4781	150	2	𝐻ℜ	𝐻ℜ	SCONJ
cana-4781	150	3	∩	∩	ADJ
cana-4781	150	4	communications	communication	NOUN
cana-4781	150	5	on	on	ADP
cana-4781	150	6	applied	apply	VERB
cana-4781	150	7	nonlinear	nonlinear	ADJ
cana-4781	150	8	analysis	analysis	NOUN
cana-4781	150	9	issn	issn	NOUN
cana-4781	150	10	:	:	PUNCT
cana-4781	150	11	1074	1074	NUM
cana-4781	150	12	-	-	PUNCT
cana-4781	150	13	133x	133x	NUM
cana-4781	150	14	vol	vol	NOUN
cana-4781	150	15	32	32	NUM
cana-4781	150	16	no	no	NOUN
cana-4781	150	17	.	.	NOUN
cana-4781	150	18	3	3	NUM
cana-4781	150	19	(	(	PUNCT
cana-4781	150	20	2025	2025	NUM
cana-4781	150	21	)	)	PUNCT
cana-4781	150	22	884	884	NUM
cana-4781	151	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-4781	151	2	⋃	⋃	ADP
cana-4781	151	3	𝑦𝑖	𝑦𝑖	NOUN
cana-4781	151	4	𝜋(𝑐𝑙(𝑁𝑖∈ℕ	𝜋(𝑐𝑙(𝑁𝑖∈ℕ	NUM
cana-4781	151	5	)	)	PUNCT
cana-4781	151	6	)	)	PUNCT
cana-4781	151	7	,	,	PUNCT
cana-4781	151	8	where	where	SCONJ
cana-4781	151	9	𝑦𝑖	𝑦𝑖	PROPN
cana-4781	151	10	=	=	PUNCT
cana-4781	151	11	𝜋(𝑥𝑖	𝜋(𝑥𝑖	PROPN
cana-4781	151	12	)	)	PUNCT
cana-4781	151	13	∈	∈	NOUN
cana-4781	151	14	𝐻ℜ.	𝐻ℜ.	X
cana-4781	151	15	thus	thus	ADV
cana-4781	151	16	,	,	PUNCT
cana-4781	151	17	𝑦𝑖	𝑦𝑖	PROPN
cana-4781	151	18	𝜋(𝑐𝑙(𝑁	𝜋(𝑐𝑙(𝑁	NOUN
cana-4781	151	19	)	)	PUNCT
cana-4781	151	20	)	)	PUNCT
cana-4781	151	21	is	be	AUX
cana-4781	151	22	closed	close	VERB
cana-4781	151	23	in	in	ADP
cana-4781	151	24	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	151	25	̅̅	̅̅	NOUN
cana-4781	151	26	̅̅	̅̅	NOUN
cana-4781	151	27	,	,	PUNCT
cana-4781	151	28	for	for	ADP
cana-4781	151	29	every	every	DET
cana-4781	151	30	𝑖	𝑖	PROPN
cana-4781	151	31	∈	∈	PROPN
cana-4781	151	32	ℕ.	ℕ.	PROPN
cana-4781	151	33	therefore	therefore	ADV
cana-4781	151	34	,	,	PUNCT
cana-4781	151	35	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	151	36	∩	∩	NOUN
cana-4781	151	37	𝑦𝑖	𝑦𝑖	ADP
cana-4781	151	38	𝜋(𝑐𝑙(𝑁	𝜋(𝑐𝑙(𝑁	NOUN
cana-4781	151	39	)	)	PUNCT
cana-4781	151	40	)	)	PUNCT
cana-4781	151	41	is	be	AUX
cana-4781	151	42	closed	close	VERB
cana-4781	151	43	in	in	ADP
cana-4781	151	44	𝐻ℜ.	𝐻ℜ.	NOUN
cana-4781	151	45	since	since	SCONJ
cana-4781	151	46	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	151	47	is	be	AUX
cana-4781	151	48	locally	locally	ADV
cana-4781	151	49	compact	compact	ADJ
cana-4781	151	50	,	,	PUNCT
cana-4781	151	51	𝑖𝑛𝑡(𝐻ℜ	𝑖𝑛𝑡(𝐻ℜ	NOUN
cana-4781	151	52	∩	∩	NOUN
cana-4781	151	53	𝑦𝑖	𝑦𝑖	ADP
cana-4781	151	54	𝜋(𝑐𝑙(𝑁	𝜋(𝑐𝑙(𝑁	NOUN
cana-4781	151	55	)	)	PUNCT
cana-4781	151	56	)	)	PUNCT
cana-4781	151	57	)	)	PUNCT
cana-4781	152	1	≠	≠	PROPN
cana-4781	152	2	∅	∅	NOUN
cana-4781	152	3	,	,	PUNCT
cana-4781	152	4	for	for	ADP
cana-4781	152	5	every	every	DET
cana-4781	152	6	𝑖	𝑖	SYM
cana-4781	152	7	∈	∈	PROPN
cana-4781	152	8	ℕ.	ℕ.	PROPN
cana-4781	152	9	by	by	ADP
cana-4781	152	10	proposition	proposition	NOUN
cana-4781	152	11	3.10	3.10	NUM
cana-4781	152	12	,	,	PUNCT
cana-4781	152	13	we	we	PRON
cana-4781	152	14	get	get	VERB
cana-4781	152	15	𝑖𝑛𝑡(𝜋(𝑐𝑙(𝑁	𝑖𝑛𝑡(𝜋(𝑐𝑙(𝑁	NOUN
cana-4781	152	16	)	)	PUNCT
cana-4781	152	17	)	)	PUNCT
cana-4781	152	18	)	)	PUNCT
cana-4781	153	1	≠	≠	PROPN
cana-4781	153	2	∅.	∅.	VERB
cana-4781	153	3	then	then	ADV
cana-4781	153	4	there	there	PRON
cana-4781	153	5	exists	exist	VERB
cana-4781	153	6	an	an	DET
cana-4781	153	7	open	open	ADJ
cana-4781	153	8	set	set	NOUN
cana-4781	153	9	𝑉	𝑉	PROPN
cana-4781	153	10	⊆	⊆	NUM
cana-4781	153	11	𝐻ℜ	𝐻ℜ	ADP
cana-4781	153	12	such	such	ADJ
cana-4781	153	13	that	that	SCONJ
cana-4781	153	14	𝑉	𝑉	PROPN
cana-4781	153	15	⊆	⊆	NUM
cana-4781	153	16	𝜋(𝑐𝑙(𝑁	𝜋(𝑐𝑙(𝑁	NUM
cana-4781	153	17	)	)	PUNCT
cana-4781	153	18	)	)	PUNCT
cana-4781	153	19	.	.	PUNCT
cana-4781	154	1	let	let	VERB
cana-4781	154	2	𝑣	𝑣	PRON
cana-4781	154	3	∈	∈	PROPN
cana-4781	154	4	𝑉.	𝑉.	PROPN
cana-4781	154	5	then	then	ADV
cana-4781	154	6	there	there	PRON
cana-4781	154	7	exists	exist	VERB
cana-4781	154	8	a	a	DET
cana-4781	154	9	point	point	NOUN
cana-4781	154	10	𝑛	𝑛	PRON
cana-4781	154	11	∈	∈	PROPN
cana-4781	154	12	𝑐𝑙(𝑁	𝑐𝑙(𝑁	NOUN
cana-4781	154	13	)	)	PUNCT
cana-4781	154	14	such	such	ADJ
cana-4781	154	15	that	that	SCONJ
cana-4781	154	16	𝜋(𝑛	𝜋(𝑛	X
cana-4781	154	17	)	)	PUNCT
cana-4781	154	18	=	=	VERB
cana-4781	154	19	𝑣.	𝑣.	NOUN
cana-4781	154	20	therefore	therefore	ADV
cana-4781	154	21	,	,	PUNCT
cana-4781	154	22	𝑒′	𝑒′	PROPN
cana-4781	154	23	∈	∈	PROPN
cana-4781	154	24	𝑣−1𝑉	𝑣−1𝑉	VERB
cana-4781	154	25	⊆	⊆	NUM
cana-4781	154	26	𝑣−1𝜋(𝑐𝑙(𝑁	𝑣−1𝜋(𝑐𝑙(𝑁	NOUN
cana-4781	154	27	)	)	PUNCT
cana-4781	154	28	)	)	PUNCT
cana-4781	155	1	=	=	SYM
cana-4781	155	2	𝜋(𝑛−1)𝜋(𝑐𝑙(𝑁	𝜋(𝑛−1)𝜋(𝑐𝑙(𝑁	NUM
cana-4781	155	3	)	)	PUNCT
cana-4781	155	4	)	)	PUNCT
cana-4781	156	1	⊆	⊆	NUM
cana-4781	156	2	𝜋(𝑛−1𝑐𝑙(𝑁	𝜋(𝑛−1𝑐𝑙(𝑁	NOUN
cana-4781	156	3	)	)	PUNCT
cana-4781	156	4	)	)	PUNCT
cana-4781	156	5	⊆	⊆	NUM
cana-4781	156	6	𝜋(𝑐𝑙(𝑁)𝑐𝑙(𝑁	𝜋(𝑐𝑙(𝑁)𝑐𝑙(𝑁	NOUN
cana-4781	156	7	)	)	PUNCT
cana-4781	156	8	)	)	PUNCT
cana-4781	156	9	⊆	⊆	NUM
cana-4781	156	10	𝜋(𝑈	𝜋(𝑈	NOUN
cana-4781	156	11	)	)	PUNCT
cana-4781	156	12	.	.	PUNCT
cana-4781	157	1	hence	hence	ADV
cana-4781	157	2	𝜋	𝜋	PRON
cana-4781	157	3	is	be	AUX
cana-4781	157	4	an	an	DET
cana-4781	157	5	open	open	ADJ
cana-4781	157	6	mapping	mapping	NOUN
cana-4781	157	7	.	.	PUNCT
cana-4781	158	1	4	4	X
cana-4781	158	2	.	.	X
cana-4781	158	3	topological	topological	ADJ
cana-4781	158	4	rough	rough	ADJ
cana-4781	158	5	group	group	NOUN
cana-4781	158	6	homomorphism	homomorphism	NOUN
cana-4781	158	7	:	:	PUNCT
cana-4781	158	8	proposition	proposition	NOUN
cana-4781	158	9	4.1	4.1	NUM
cana-4781	158	10	.	.	PUNCT
cana-4781	159	1	let	let	VERB
cana-4781	159	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	159	3	and	and	CCONJ
cana-4781	159	4	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	159	5	be	be	AUX
cana-4781	159	6	topological	topological	ADJ
cana-4781	159	7	simple	simple	ADJ
cana-4781	159	8	rough	rough	ADJ
cana-4781	159	9	groups	group	NOUN
cana-4781	159	10	such	such	ADJ
cana-4781	159	11	that	that	SCONJ
cana-4781	159	12	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	159	13	and	and	CCONJ
cana-4781	159	14	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	159	15	are	be	AUX
cana-4781	159	16	open	open	ADJ
cana-4781	159	17	in	in	ADP
cana-4781	159	18	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	159	19	̅̅	̅̅	PROPN
cana-4781	159	20	̅̅	̅̅	PROPN
cana-4781	159	21	and	and	CCONJ
cana-4781	159	22	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	159	23	̅̅	̅̅	PROPN
cana-4781	159	24	̅̅	̅̅	PROPN
cana-4781	159	25	.	.	PUNCT
cana-4781	160	1	let	let	VERB
cana-4781	160	2	the	the	DET
cana-4781	160	3	map	map	VERB
cana-4781	160	4	𝑓	𝑓	NOUN
cana-4781	160	5	:	:	PUNCT
cana-4781	160	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	160	7	̅̅	̅̅	PROPN
cana-4781	160	8	̅̅	̅̅	PROPN
cana-4781	160	9	→	→	PUNCT
cana-4781	160	10	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	160	11	̅̅	̅̅	PROPN
cana-4781	160	12	̅̅	̅̅	NOUN
cana-4781	160	13	be	be	AUX
cana-4781	160	14	a	a	DET
cana-4781	160	15	continuous	continuous	ADJ
cana-4781	160	16	topological	topological	ADJ
cana-4781	160	17	rough	rough	ADJ
cana-4781	160	18	group	group	NOUN
cana-4781	160	19	homomorphism	homomorphism	NOUN
cana-4781	160	20	.	.	PUNCT
cana-4781	161	1	suppose	suppose	VERB
cana-4781	161	2	for	for	SCONJ
cana-4781	161	3	every	every	DET
cana-4781	161	4	open	open	ADJ
cana-4781	161	5	neighbourhood	neighbourhood	NOUN
cana-4781	161	6	𝑁	𝑁	PROPN
cana-4781	161	7	of	of	ADP
cana-4781	161	8	𝑒	𝑒	PROPN
cana-4781	161	9	in	in	ADP
cana-4781	161	10	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	161	11	̅̅	̅̅	PROPN
cana-4781	161	12	̅̅	̅̅	PROPN
cana-4781	161	13	,	,	PUNCT
cana-4781	161	14	𝑓(𝑁	𝑓(𝑁	PROPN
cana-4781	161	15	)	)	PUNCT
cana-4781	161	16	has	have	VERB
cana-4781	161	17	a	a	DET
cana-4781	161	18	non	non	ADJ
cana-4781	161	19	-	-	ADJ
cana-4781	161	20	empty	empty	ADJ
cana-4781	161	21	open	open	ADJ
cana-4781	161	22	set	set	NOUN
cana-4781	161	23	in	in	ADP
cana-4781	161	24	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	161	25	̅̅	̅̅	NOUN
cana-4781	161	26	̅̅	̅̅	PROPN
cana-4781	161	27	.	.	PUNCT
cana-4781	162	1	then	then	ADV
cana-4781	162	2	𝑓	𝑓	PRON
cana-4781	162	3	is	be	AUX
cana-4781	162	4	an	an	DET
cana-4781	162	5	open	open	ADJ
cana-4781	162	6	mapping	mapping	NOUN
cana-4781	162	7	.	.	PUNCT
cana-4781	163	1	proof	proof	NOUN
cana-4781	163	2	:	:	PUNCT
cana-4781	163	3	let	let	VERB
cana-4781	163	4	𝑈	𝑈	PROPN
cana-4781	163	5	be	be	AUX
cana-4781	163	6	an	an	DET
cana-4781	163	7	open	open	ADJ
cana-4781	163	8	neighbourhood	neighbourhood	NOUN
cana-4781	163	9	of	of	ADP
cana-4781	163	10	𝑒	𝑒	PROPN
cana-4781	163	11	in	in	ADP
cana-4781	163	12	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	163	13	̅̅	̅̅	NOUN
cana-4781	163	14	̅̅	̅̅	NOUN
cana-4781	163	15	such	such	ADJ
cana-4781	163	16	that	that	SCONJ
cana-4781	163	17	𝑈−1𝑈	𝑈−1𝑈	VERB
cana-4781	163	18	⊆	⊆	NUM
cana-4781	163	19	𝑁.	𝑁.	PROPN
cana-4781	163	20	but	but	CCONJ
cana-4781	163	21	by	by	ADP
cana-4781	163	22	the	the	DET
cana-4781	163	23	hypothesis	hypothesis	NOUN
cana-4781	163	24	,	,	PUNCT
cana-4781	163	25	𝑓(𝑈	𝑓(𝑈	NUM
cana-4781	163	26	)	)	PUNCT
cana-4781	163	27	has	have	VERB
cana-4781	163	28	a	a	DET
cana-4781	163	29	non	non	ADJ
cana-4781	163	30	-	-	ADJ
cana-4781	163	31	empty	empty	ADJ
cana-4781	163	32	open	open	ADJ
cana-4781	163	33	set	set	NOUN
cana-4781	163	34	in	in	ADP
cana-4781	163	35	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	163	36	̅̅	̅̅	PROPN
cana-4781	163	37	̅̅	̅̅	PROPN
cana-4781	163	38	.	.	PUNCT
cana-4781	164	1	consider	consider	VERB
cana-4781	164	2	that	that	PRON
cana-4781	164	3	open	open	ADJ
cana-4781	164	4	set	set	VERB
cana-4781	164	5	𝑉	𝑉	PROPN
cana-4781	164	6	in	in	ADP
cana-4781	164	7	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	164	8	̅̅	̅̅	NOUN
cana-4781	164	9	̅̅	̅̅	PROPN
cana-4781	164	10	.	.	PUNCT
cana-4781	165	1	so	so	ADV
cana-4781	165	2	,	,	PUNCT
cana-4781	165	3	𝑉−1𝑉	𝑉−1𝑉	PROPN
cana-4781	165	4	is	be	AUX
cana-4781	165	5	an	an	DET
cana-4781	165	6	identity	identity	NOUN
cana-4781	165	7	neighbourhood	neighbourhood	NOUN
cana-4781	165	8	.	.	PUNCT
cana-4781	166	1	therefore	therefore	ADV
cana-4781	166	2	,	,	PUNCT
cana-4781	166	3	𝑉−1𝑉	𝑉−1𝑉	PROPN
cana-4781	166	4	⊆	⊆	NUM
cana-4781	166	5	𝑓(𝑈)−1𝑓(𝑈	𝑓(𝑈)−1𝑓(𝑈	NOUN
cana-4781	166	6	)	)	PUNCT
cana-4781	166	7	=	=	SYM
cana-4781	167	1	𝑓(𝑈−1𝑈	𝑓(𝑈−1𝑈	X
cana-4781	167	2	)	)	PUNCT
cana-4781	167	3	⊆	⊆	NUM
cana-4781	167	4	𝑓(𝑁	𝑓(𝑁	NOUN
cana-4781	167	5	)	)	PUNCT
cana-4781	167	6	which	which	PRON
cana-4781	167	7	implies	imply	VERB
cana-4781	167	8	𝑓(𝑁	𝑓(𝑁	NOUN
cana-4781	167	9	)	)	PUNCT
cana-4781	167	10	has	have	VERB
cana-4781	167	11	an	an	DET
cana-4781	167	12	identity	identity	NOUN
cana-4781	167	13	in	in	ADP
cana-4781	167	14	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	167	15	̅̅	̅̅	NOUN
cana-4781	167	16	̅̅	̅̅	PROPN
cana-4781	167	17	.	.	PUNCT
cana-4781	168	1	let	let	VERB
cana-4781	168	2	𝑏	𝑏	PRON
cana-4781	168	3	∈	∈	PROPN
cana-4781	168	4	𝑓(𝑁	𝑓(𝑁	NOUN
cana-4781	168	5	)	)	PUNCT
cana-4781	168	6	.	.	PUNCT
cana-4781	169	1	since	since	SCONJ
cana-4781	169	2	𝑓	𝑓	PRON
cana-4781	169	3	is	be	AUX
cana-4781	169	4	a	a	DET
cana-4781	169	5	continuous	continuous	ADJ
cana-4781	169	6	topological	topological	ADJ
cana-4781	169	7	rough	rough	ADJ
cana-4781	169	8	group	group	NOUN
cana-4781	169	9	homomorphism	homomorphism	NOUN
cana-4781	169	10	,	,	PUNCT
cana-4781	169	11	there	there	PRON
cana-4781	169	12	exists	exist	VERB
cana-4781	169	13	an	an	DET
cana-4781	169	14	arbitrary	arbitrary	ADJ
cana-4781	169	15	element	element	NOUN
cana-4781	169	16	𝑎	𝑎	PROPN
cana-4781	169	17	∈	∈	NOUN
cana-4781	169	18	n	n	CCONJ
cana-4781	169	19	such	such	ADJ
cana-4781	169	20	that	that	DET
cana-4781	169	21	𝑓(𝑎	𝑓(𝑎	NOUN
cana-4781	169	22	)	)	PUNCT
cana-4781	169	23	=	=	SYM
cana-4781	169	24	𝑏	𝑏	NOUN
cana-4781	169	25	and	and	CCONJ
cana-4781	169	26	𝑎𝑈	𝑎𝑈	VERB
cana-4781	169	27	⊆	⊆	NUM
cana-4781	169	28	𝑁.	𝑁.	PROPN
cana-4781	169	29	also	also	ADV
cana-4781	169	30	,	,	PUNCT
cana-4781	169	31	𝑉	𝑉	PROPN
cana-4781	169	32	⊆	⊆	NUM
cana-4781	169	33	𝑓(𝑈	𝑓(𝑈	NUM
cana-4781	169	34	)	)	PUNCT
cana-4781	169	35	and	and	CCONJ
cana-4781	169	36	𝑏𝑉	𝑏𝑉	ADJ
cana-4781	169	37	is	be	AUX
cana-4781	169	38	an	an	DET
cana-4781	169	39	open	open	ADJ
cana-4781	169	40	neighbourhood	neighbourhood	NOUN
cana-4781	169	41	in	in	ADP
cana-4781	169	42	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	169	43	̅̅	̅̅	NOUN
cana-4781	169	44	̅̅	̅̅	PROPN
cana-4781	169	45	.	.	PUNCT
cana-4781	170	1	then	then	ADV
cana-4781	170	2	𝑏𝑉	𝑏𝑉	PROPN
cana-4781	170	3	⊆	⊆	NUM
cana-4781	170	4	𝑓(𝑎𝑈	𝑓(𝑎𝑈	PROPN
cana-4781	170	5	)	)	PUNCT
cana-4781	170	6	⊆	⊆	NUM
cana-4781	170	7	𝑓(𝑁	𝑓(𝑁	NOUN
cana-4781	170	8	)	)	PUNCT
cana-4781	170	9	.	.	PUNCT
cana-4781	171	1	hence	hence	ADV
cana-4781	171	2	the	the	DET
cana-4781	171	3	map	map	NOUN
cana-4781	171	4	𝑓	𝑓	NOUN
cana-4781	171	5	is	be	AUX
cana-4781	171	6	open	open	ADJ
cana-4781	171	7	.	.	PUNCT
cana-4781	172	1	proposition	proposition	NOUN
cana-4781	172	2	4.2	4.2	NUM
cana-4781	172	3	.	.	PUNCT
cana-4781	173	1	(	(	PUNCT
cana-4781	173	2	i	i	NOUN
cana-4781	173	3	)	)	PUNCT
cana-4781	173	4	let	let	VERB
cana-4781	173	5	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	173	6	,	,	PUNCT
cana-4781	173	7	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	173	8	and	and	CCONJ
cana-4781	173	9	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	173	10	be	be	VERB
cana-4781	173	11	simple	simple	ADJ
cana-4781	173	12	rough	rough	ADJ
cana-4781	173	13	groups	group	NOUN
cana-4781	173	14	.	.	PUNCT
cana-4781	174	1	let	let	VERB
cana-4781	174	2	𝑓	𝑓	X
cana-4781	174	3	:	:	PUNCT
cana-4781	174	4	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	174	5	̅̅	̅̅	PROPN
cana-4781	174	6	̅̅	̅̅	PROPN
cana-4781	174	7	→	→	PUNCT
cana-4781	175	1	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	175	2	̅̅	̅̅	PROPN
cana-4781	175	3	̅̅	̅̅	PROPN
cana-4781	175	4	and	and	CCONJ
cana-4781	175	5	𝑔	𝑔	ADJ
cana-4781	175	6	:	:	PUNCT
cana-4781	175	7	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	175	8	̅̅	̅̅	PROPN
cana-4781	175	9	̅̅	̅̅	PROPN
cana-4781	175	10	→	→	SYM
cana-4781	175	11	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	175	12	̅̅	̅̅	NOUN
cana-4781	175	13	̅̅	̅̅	NOUN
cana-4781	175	14	be	be	AUX
cana-4781	175	15	rough	rough	ADJ
cana-4781	175	16	group	group	NOUN
cana-4781	175	17	homomorphisms	homomorphism	NOUN
cana-4781	175	18	,	,	PUNCT
cana-4781	175	19	where	where	SCONJ
cana-4781	175	20	𝑔(𝐺ℜ	𝑔(𝐺ℜ	PROPN
cana-4781	175	21	̅̅	̅̅	NOUN
cana-4781	175	22	̅̅	̅̅	PROPN
cana-4781	175	23	)	)	PUNCT
cana-4781	176	1	=	=	SYM
cana-4781	176	2	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	176	3	̅̅	̅̅	NOUN
cana-4781	176	4	̅̅	̅̅	PROPN
cana-4781	176	5	and	and	CCONJ
cana-4781	176	6	𝒦𝑓	𝒦𝑓	PROPN
cana-4781	176	7	⊆	⊆	NUM
cana-4781	176	8	𝒦𝑔	𝒦𝑔	PROPN
cana-4781	176	9	,	,	PUNCT
cana-4781	176	10	where	where	SCONJ
cana-4781	176	11	𝒦𝑓	𝒦𝑓	PROPN
cana-4781	176	12	and	and	CCONJ
cana-4781	176	13	𝒦𝑔	𝒦𝑔	PROPN
cana-4781	176	14	represent	represent	VERB
cana-4781	176	15	the	the	DET
cana-4781	176	16	rough	rough	ADJ
cana-4781	176	17	kernel	kernel	NOUN
cana-4781	176	18	of	of	ADP
cana-4781	176	19	𝑓	𝑓	PROPN
cana-4781	176	20	and	and	CCONJ
cana-4781	176	21	𝑔.	𝑔.	NOUN
cana-4781	176	22	then	then	ADV
cana-4781	176	23	ℎ	ℎ	PROPN
cana-4781	176	24	:	:	PUNCT
cana-4781	176	25	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	176	26	̅̅	̅̅	NOUN
cana-4781	176	27	̅̅	̅̅	PROPN
cana-4781	176	28	→	→	PUNCT
cana-4781	176	29	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	176	30	̅̅	̅̅	PROPN
cana-4781	176	31	̅̅	̅̅	NOUN
cana-4781	176	32	is	be	AUX
cana-4781	176	33	a	a	DET
cana-4781	176	34	rough	rough	ADJ
cana-4781	176	35	group	group	NOUN
cana-4781	176	36	homomorphism	homomorphism	NOUN
cana-4781	176	37	such	such	ADJ
cana-4781	176	38	that	that	SCONJ
cana-4781	176	39	𝑓	𝑓	PRON
cana-4781	176	40	=	=	SYM
cana-4781	176	41	ℎ	ℎ	PROPN
cana-4781	176	42	∘	∘	PROPN
cana-4781	176	43	𝑔.	𝑔.	PROPN
cana-4781	176	44	(	(	PUNCT
cana-4781	176	45	ii	ii	NOUN
cana-4781	176	46	)	)	PUNCT
cana-4781	176	47	let	let	VERB
cana-4781	176	48	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	176	49	,	,	PUNCT
cana-4781	176	50	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	176	51	and	and	CCONJ
cana-4781	176	52	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	176	53	be	be	AUX
cana-4781	176	54	topological	topological	ADJ
cana-4781	176	55	simple	simple	ADJ
cana-4781	176	56	rough	rough	ADJ
cana-4781	176	57	groups	group	NOUN
cana-4781	176	58	such	such	ADJ
cana-4781	176	59	that	that	SCONJ
cana-4781	176	60	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	176	61	,	,	PUNCT
cana-4781	176	62	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	176	63	and	and	CCONJ
cana-4781	176	64	𝐾ℜ	𝐾ℜ	NOUN
cana-4781	176	65	are	be	AUX
cana-4781	176	66	open	open	ADJ
cana-4781	176	67	in	in	ADP
cana-4781	176	68	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	176	69	̅̅	̅̅	PROPN
cana-4781	176	70	̅̅	̅̅	PROPN
cana-4781	176	71	,	,	PUNCT
cana-4781	176	72	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	176	73	̅̅	̅̅	PROPN
cana-4781	176	74	̅̅	̅̅	PROPN
cana-4781	176	75	and	and	CCONJ
cana-4781	176	76	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	176	77	̅̅	̅̅	PROPN
cana-4781	176	78	̅̅	̅̅	PROPN
cana-4781	176	79	.	.	PUNCT
cana-4781	177	1	suppose	suppose	VERB
cana-4781	177	2	𝑔−1(𝑈	𝑔−1(𝑈	X
cana-4781	177	3	)	)	PUNCT
cana-4781	177	4	⊆	⊆	NUM
cana-4781	177	5	𝑓−1(𝑉	𝑓−1(𝑉	NOUN
cana-4781	177	6	)	)	PUNCT
cana-4781	177	7	,	,	PUNCT
cana-4781	177	8	for	for	ADP
cana-4781	177	9	every	every	DET
cana-4781	177	10	identity	identity	NOUN
cana-4781	177	11	neighbourhood	neighbourhood	NOUN
cana-4781	177	12	𝑈	𝑈	PROPN
cana-4781	177	13	in	in	ADP
cana-4781	177	14	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	177	15	̅̅	̅̅	NOUN
cana-4781	177	16	̅̅	̅̅	NOUN
cana-4781	177	17	,	,	PUNCT
cana-4781	177	18	there	there	PRON
cana-4781	177	19	exists	exist	VERB
cana-4781	177	20	an	an	DET
cana-4781	177	21	identity	identity	NOUN
cana-4781	177	22	neighbourhood	neighbourhood	NOUN
cana-4781	177	23	𝑉	𝑉	PROPN
cana-4781	177	24	in	in	ADP
cana-4781	177	25	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	177	26	̅̅	̅̅	NOUN
cana-4781	177	27	̅̅	̅̅	NOUN
cana-4781	177	28	,	,	PUNCT
cana-4781	177	29	then	then	ADV
cana-4781	177	30	ℎ	ℎ	PROPN
cana-4781	177	31	:	:	PUNCT
cana-4781	177	32	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	177	33	̅̅	̅̅	NOUN
cana-4781	177	34	̅̅	̅̅	PROPN
cana-4781	177	35	→	→	PUNCT
cana-4781	177	36	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	177	37	̅̅	̅̅	PROPN
cana-4781	177	38	̅̅	̅̅	NOUN
cana-4781	177	39	is	be	AUX
cana-4781	177	40	continuous	continuous	ADJ
cana-4781	177	41	.	.	PUNCT
cana-4781	178	1	proof	proof	NOUN
cana-4781	178	2	:	:	PUNCT
cana-4781	178	3	(	(	PUNCT
cana-4781	178	4	i	i	NOUN
cana-4781	178	5	)	)	PUNCT
cana-4781	178	6	since	since	SCONJ
cana-4781	178	7	𝑓	𝑓	PRON
cana-4781	178	8	and	and	CCONJ
cana-4781	178	9	𝑔	𝑔	PROPN
cana-4781	178	10	are	be	AUX
cana-4781	178	11	rough	rough	ADJ
cana-4781	178	12	group	group	NOUN
cana-4781	178	13	homomorphism	homomorphism	NOUN
cana-4781	178	14	and	and	CCONJ
cana-4781	178	15	𝑓	𝑓	PRON
cana-4781	178	16	=	=	NOUN
cana-4781	178	17	ℎ	ℎ	PROPN
cana-4781	178	18	∘	∘	PROPN
cana-4781	178	19	𝑔	𝑔	PROPN
cana-4781	178	20	,	,	PUNCT
cana-4781	178	21	ℎ	ℎ	PROPN
cana-4781	178	22	is	be	AUX
cana-4781	178	23	a	a	DET
cana-4781	178	24	rough	rough	ADJ
cana-4781	178	25	group	group	NOUN
cana-4781	178	26	homomorphism	homomorphism	NOUN
cana-4781	178	27	.	.	PUNCT
cana-4781	179	1	(	(	PUNCT
cana-4781	179	2	ii	ii	NOUN
cana-4781	179	3	)	)	PUNCT
cana-4781	179	4	let	let	VERB
cana-4781	179	5	𝑈	𝑈	PROPN
cana-4781	179	6	be	be	AUX
cana-4781	179	7	an	an	DET
cana-4781	179	8	identity	identity	NOUN
cana-4781	179	9	neighbourhood	neighbourhood	NOUN
cana-4781	179	10	in	in	ADP
cana-4781	179	11	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	179	12	̅̅	̅̅	NOUN
cana-4781	179	13	̅̅	̅̅	PROPN
cana-4781	179	14	.	.	PUNCT
cana-4781	180	1	by	by	ADP
cana-4781	180	2	the	the	DET
cana-4781	180	3	hypothesis	hypothesis	NOUN
cana-4781	180	4	,	,	PUNCT
cana-4781	180	5	there	there	PRON
cana-4781	180	6	exists	exist	VERB
cana-4781	180	7	an	an	DET
cana-4781	180	8	identity	identity	NOUN
cana-4781	180	9	neighbourhood	neighbourhood	NOUN
cana-4781	180	10	𝑉	𝑉	PROPN
cana-4781	180	11	in	in	ADP
cana-4781	180	12	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	180	13	̅̅	̅̅	NOUN
cana-4781	180	14	̅̅	̅̅	NOUN
cana-4781	180	15	such	such	ADJ
cana-4781	180	16	that	that	DET
cana-4781	180	17	𝑔−1(𝑉	𝑔−1(𝑉	NOUN
cana-4781	180	18	)	)	PUNCT
cana-4781	180	19	⊆	⊆	NUM
cana-4781	180	20	𝑓−1(𝑈	𝑓−1(𝑈	NUM
cana-4781	180	21	)	)	PUNCT
cana-4781	180	22	.	.	PUNCT
cana-4781	181	1	consider	consider	VERB
cana-4781	181	2	𝑁	𝑁	PROPN
cana-4781	181	3	=	=	PUNCT
cana-4781	181	4	𝑔−1(𝑉	𝑔−1(𝑉	NOUN
cana-4781	181	5	)	)	PUNCT
cana-4781	181	6	.	.	PUNCT
cana-4781	182	1	therefore	therefore	ADV
cana-4781	182	2	,	,	PUNCT
cana-4781	182	3	𝑓(𝑁	𝑓(𝑁	NOUN
cana-4781	182	4	)	)	PUNCT
cana-4781	182	5	⊆	⊆	NUM
cana-4781	182	6	𝑈	𝑈	PROPN
cana-4781	182	7	and	and	CCONJ
cana-4781	182	8	ℎ(𝑉	ℎ(𝑉	PROPN
cana-4781	182	9	)	)	PUNCT
cana-4781	182	10	=	=	SYM
cana-4781	182	11	𝑓(𝑁	𝑓(𝑁	X
cana-4781	182	12	)	)	PUNCT
cana-4781	182	13	which	which	PRON
cana-4781	182	14	implies	imply	VERB
cana-4781	182	15	ℎ(𝑉	ℎ(𝑉	PROPN
cana-4781	182	16	)	)	PUNCT
cana-4781	183	1	⊆	⊆	X
cana-4781	183	2	𝑈.	𝑈.	NOUN
cana-4781	183	3	hence	hence	ADV
cana-4781	183	4	ℎ	ℎ	PROPN
cana-4781	183	5	is	be	AUX
cana-4781	183	6	continuous	continuous	ADJ
cana-4781	183	7	at	at	ADP
cana-4781	183	8	𝑒	𝑒	PROPN
cana-4781	183	9	in	in	ADP
cana-4781	183	10	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	183	11	̅̅	̅̅	NOUN
cana-4781	183	12	̅̅	̅̅	PROPN
cana-4781	183	13	,	,	PUNCT
cana-4781	183	14	which	which	PRON
cana-4781	183	15	implies	imply	VERB
cana-4781	183	16	ℎ	ℎ	PROPN
cana-4781	183	17	is	be	AUX
cana-4781	183	18	continuous	continuous	ADJ
cana-4781	183	19	.	.	PUNCT
cana-4781	184	1	corollary	corollary	ADJ
cana-4781	184	2	4.3	4.3	NUM
cana-4781	184	3	.	.	PUNCT
cana-4781	185	1	let	let	VERB
cana-4781	185	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	185	3	,	,	PUNCT
cana-4781	185	4	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	185	5	and	and	CCONJ
cana-4781	185	6	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	185	7	be	be	AUX
cana-4781	185	8	topological	topological	ADJ
cana-4781	185	9	simple	simple	ADJ
cana-4781	185	10	rough	rough	ADJ
cana-4781	185	11	groups	group	NOUN
cana-4781	185	12	,	,	PUNCT
cana-4781	185	13	where	where	SCONJ
cana-4781	185	14	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	185	15	,	,	PUNCT
cana-4781	185	16	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	185	17	and	and	CCONJ
cana-4781	185	18	𝐾ℜ	𝐾ℜ	NOUN
cana-4781	185	19	are	be	AUX
cana-4781	185	20	open	open	ADJ
cana-4781	185	21	in	in	ADP
cana-4781	185	22	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	185	23	̅̅	̅̅	PROPN
cana-4781	185	24	̅̅	̅̅	PROPN
cana-4781	185	25	,	,	PUNCT
cana-4781	185	26	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	185	27	̅̅	̅̅	PROPN
cana-4781	185	28	̅̅	̅̅	PROPN
cana-4781	185	29	and	and	CCONJ
cana-4781	185	30	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	185	31	̅̅	̅̅	PROPN
cana-4781	185	32	̅̅	̅̅	PROPN
cana-4781	185	33	.	.	PUNCT
cana-4781	186	1	let	let	VERB
cana-4781	186	2	𝑓	𝑓	X
cana-4781	186	3	:	:	PUNCT
cana-4781	186	4	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	186	5	̅̅	̅̅	PROPN
cana-4781	186	6	̅̅	̅̅	PROPN
cana-4781	186	7	→	→	PUNCT
cana-4781	187	1	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	187	2	̅̅	̅̅	PROPN
cana-4781	187	3	̅̅	̅̅	PROPN
cana-4781	187	4	and	and	CCONJ
cana-4781	187	5	𝑔	𝑔	ADJ
cana-4781	187	6	:	:	PUNCT
cana-4781	187	7	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	187	8	̅̅	̅̅	PROPN
cana-4781	187	9	̅̅	̅̅	PROPN
cana-4781	187	10	→	→	SYM
cana-4781	187	11	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	187	12	̅̅	̅̅	NOUN
cana-4781	187	13	̅̅	̅̅	NOUN
cana-4781	187	14	be	be	AUX
cana-4781	187	15	continuous	continuous	ADJ
cana-4781	187	16	rough	rough	ADJ
cana-4781	187	17	group	group	NOUN
cana-4781	187	18	homomorphisms	homomorphism	NOUN
cana-4781	187	19	such	such	ADJ
cana-4781	187	20	that	that	DET
cana-4781	187	21	𝑔(𝐺ℜ	𝑔(𝐺ℜ	PROPN
cana-4781	187	22	̅̅	̅̅	NOUN
cana-4781	187	23	̅̅	̅̅	PROPN
cana-4781	187	24	)	)	PUNCT
cana-4781	188	1	=	=	SYM
cana-4781	188	2	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	188	3	̅̅	̅̅	NOUN
cana-4781	188	4	̅̅	̅̅	PROPN
cana-4781	188	5	and	and	CCONJ
cana-4781	188	6	𝒦𝑓	𝒦𝑓	PROPN
cana-4781	188	7	⊆	⊆	NUM
cana-4781	188	8	𝒦𝑔	𝒦𝑔	PROPN
cana-4781	188	9	,	,	PUNCT
cana-4781	188	10	where	where	SCONJ
cana-4781	188	11	𝒦𝑓	𝒦𝑓	PROPN
cana-4781	188	12	and	and	CCONJ
cana-4781	188	13	𝒦𝑔	𝒦𝑔	PROPN
cana-4781	188	14	represent	represent	VERB
cana-4781	188	15	the	the	DET
cana-4781	188	16	rough	rough	ADJ
cana-4781	188	17	kernel	kernel	NOUN
cana-4781	188	18	of	of	ADP
cana-4781	188	19	𝑓	𝑓	PROPN
cana-4781	188	20	and	and	CCONJ
cana-4781	188	21	𝑔.	𝑔.	NOUN
cana-4781	188	22	suppose	suppose	VERB
cana-4781	188	23	𝑔	𝑔	NOUN
cana-4781	188	24	is	be	AUX
cana-4781	188	25	open	open	ADJ
cana-4781	188	26	,	,	PUNCT
cana-4781	188	27	then	then	ADV
cana-4781	188	28	there	there	PRON
cana-4781	188	29	exists	exist	VERB
cana-4781	188	30	a	a	DET
cana-4781	188	31	continuous	continuous	ADJ
cana-4781	188	32	rough	rough	ADJ
cana-4781	188	33	group	group	NOUN
cana-4781	188	34	homomorphism	homomorphism	NOUN
cana-4781	188	35	ℎ	ℎ	PROPN
cana-4781	188	36	:	:	PUNCT
cana-4781	188	37	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	188	38	̅̅	̅̅	NOUN
cana-4781	188	39	̅̅	̅̅	PROPN
cana-4781	188	40	→	→	PUNCT
cana-4781	188	41	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	188	42	̅̅	̅̅	NOUN
cana-4781	188	43	̅̅	̅̅	NOUN
cana-4781	188	44	such	such	ADJ
cana-4781	188	45	that	that	SCONJ
cana-4781	188	46	𝑓	𝑓	DET
cana-4781	188	47	=	=	SYM
cana-4781	188	48	ℎ	ℎ	PROPN
cana-4781	188	49	∘	∘	PROPN
cana-4781	188	50	𝑔.	𝑔.	PROPN
cana-4781	188	51	communications	communication	NOUN
cana-4781	188	52	on	on	ADP
cana-4781	188	53	applied	apply	VERB
cana-4781	188	54	nonlinear	nonlinear	ADJ
cana-4781	188	55	analysis	analysis	NOUN
cana-4781	188	56	issn	issn	NOUN
cana-4781	188	57	:	:	PUNCT
cana-4781	188	58	1074	1074	NUM
cana-4781	188	59	-	-	PUNCT
cana-4781	188	60	133x	133x	NUM
cana-4781	188	61	vol	vol	NOUN
cana-4781	188	62	32	32	NUM
cana-4781	188	63	no	no	NOUN
cana-4781	188	64	.	.	NOUN
cana-4781	188	65	3	3	NUM
cana-4781	188	66	(	(	PUNCT
cana-4781	188	67	2025	2025	NUM
cana-4781	188	68	)	)	PUNCT
cana-4781	188	69	885	885	NUM
cana-4781	188	70	https://internationalpubls.com	https://internationalpubls.com	X
cana-4781	188	71	proof	proof	NOUN
cana-4781	188	72	:	:	PUNCT
cana-4781	188	73	by	by	ADP
cana-4781	188	74	proposition	proposition	NOUN
cana-4781	188	75	4.2	4.2	NUM
cana-4781	188	76	(	(	PUNCT
cana-4781	188	77	i	i	NOUN
cana-4781	188	78	)	)	PUNCT
cana-4781	188	79	,	,	PUNCT
cana-4781	188	80	there	there	PRON
cana-4781	188	81	exists	exist	VERB
cana-4781	188	82	a	a	DET
cana-4781	188	83	rough	rough	ADJ
cana-4781	188	84	group	group	NOUN
cana-4781	188	85	homomorphism	homomorphism	NOUN
cana-4781	188	86	ℎ	ℎ	PROPN
cana-4781	188	87	:	:	PUNCT
cana-4781	188	88	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	188	89	̅̅	̅̅	NOUN
cana-4781	188	90	̅̅	̅̅	PROPN
cana-4781	188	91	→	→	PUNCT
cana-4781	188	92	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	188	93	̅̅	̅̅	NOUN
cana-4781	188	94	̅̅	̅̅	NOUN
cana-4781	188	95	such	such	ADJ
cana-4781	188	96	that	that	SCONJ
cana-4781	188	97	𝑓	𝑓	DET
cana-4781	188	98	=	=	SYM
cana-4781	188	99	ℎ	ℎ	PROPN
cana-4781	188	100	∘	∘	PROPN
cana-4781	188	101	𝑔.	𝑔.	PROPN
cana-4781	188	102	let	let	VERB
cana-4781	188	103	𝑈	𝑈	PROPN
cana-4781	188	104	be	be	AUX
cana-4781	188	105	an	an	DET
cana-4781	188	106	open	open	ADJ
cana-4781	188	107	set	set	NOUN
cana-4781	188	108	in	in	ADP
cana-4781	188	109	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	188	110	̅̅	̅̅	NOUN
cana-4781	188	111	̅̅	̅̅	PROPN
cana-4781	188	112	.	.	PUNCT
cana-4781	189	1	then	then	ADV
cana-4781	189	2	ℎ−1(𝑈	ℎ−1(𝑈	NUM
cana-4781	189	3	)	)	PUNCT
cana-4781	189	4	=	=	SYM
cana-4781	189	5	𝑔(𝑓−1(𝑈	𝑔(𝑓−1(𝑈	PROPN
cana-4781	189	6	)	)	PUNCT
cana-4781	189	7	)	)	PUNCT
cana-4781	189	8	.	.	PUNCT
cana-4781	190	1	since	since	SCONJ
cana-4781	190	2	𝑓	𝑓	PRON
cana-4781	190	3	is	be	AUX
cana-4781	190	4	continuous	continuous	ADJ
cana-4781	190	5	and	and	CCONJ
cana-4781	190	6	𝑔	𝑔	PROPN
cana-4781	190	7	is	be	AUX
cana-4781	190	8	open	open	ADJ
cana-4781	190	9	,	,	PUNCT
cana-4781	190	10	ℎ−1(𝑈	ℎ−1(𝑈	NOUN
cana-4781	190	11	)	)	PUNCT
cana-4781	190	12	is	be	AUX
cana-4781	190	13	open	open	ADJ
cana-4781	190	14	.	.	PUNCT
cana-4781	191	1	therefore	therefore	ADV
cana-4781	191	2	,	,	PUNCT
cana-4781	191	3	ℎ	ℎ	PROPN
cana-4781	191	4	is	be	AUX
cana-4781	191	5	continuous	continuous	ADJ
cana-4781	191	6	and	and	CCONJ
cana-4781	191	7	hence	hence	ADV
cana-4781	191	8	ℎ	ℎ	PROPN
cana-4781	191	9	is	be	AUX
cana-4781	191	10	continuous	continuous	ADJ
cana-4781	191	11	rough	rough	ADJ
cana-4781	191	12	group	group	NOUN
cana-4781	191	13	homomorphism	homomorphism	NOUN
cana-4781	191	14	.	.	PUNCT
cana-4781	192	1	proposition	proposition	NOUN
cana-4781	192	2	4.4	4.4	NUM
cana-4781	192	3	.	.	PUNCT
cana-4781	193	1	let	let	VERB
cana-4781	193	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	193	3	and	and	CCONJ
cana-4781	193	4	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	193	5	be	be	AUX
cana-4781	193	6	topological	topological	ADJ
cana-4781	193	7	simple	simple	ADJ
cana-4781	193	8	rough	rough	ADJ
cana-4781	193	9	groups	group	NOUN
cana-4781	193	10	such	such	ADJ
cana-4781	193	11	that	that	SCONJ
cana-4781	193	12	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	193	13	̅̅	̅̅	PROPN
cana-4781	193	14	̅̅	̅̅	PROPN
cana-4781	193	15	and	and	CCONJ
cana-4781	193	16	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	193	17	̅̅	̅̅	PROPN
cana-4781	193	18	̅̅	̅̅	NOUN
cana-4781	193	19	are	be	AUX
cana-4781	193	20	groups	group	NOUN
cana-4781	193	21	.	.	PUNCT
cana-4781	194	1	let	let	VERB
cana-4781	194	2	ℒ	ℒ	NOUN
cana-4781	194	3	be	be	AUX
cana-4781	194	4	a	a	DET
cana-4781	194	5	subgroup	subgroup	NOUN
cana-4781	194	6	of	of	ADP
cana-4781	194	7	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	194	8	and	and	CCONJ
cana-4781	194	9	normal	normal	ADJ
cana-4781	194	10	subgroup	subgroup	NOUN
cana-4781	194	11	of	of	ADP
cana-4781	194	12	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	194	13	̅̅	̅̅	PROPN
cana-4781	194	14	̅̅	̅̅	PROPN
cana-4781	194	15	.	.	PUNCT
cana-4781	195	1	let	let	VERB
cana-4781	195	2	ℳ	ℳ	PROPN
cana-4781	195	3	=	=	SYM
cana-4781	195	4	𝜌(ℒ	𝜌(ℒ	PROPN
cana-4781	195	5	)	)	PUNCT
cana-4781	195	6	be	be	AUX
cana-4781	195	7	a	a	DET
cana-4781	195	8	subgroup	subgroup	NOUN
cana-4781	195	9	of	of	ADP
cana-4781	195	10	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	195	11	and	and	CCONJ
cana-4781	195	12	normal	normal	ADJ
cana-4781	195	13	subgroup	subgroup	NOUN
cana-4781	195	14	of	of	ADP
cana-4781	195	15	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	195	16	̅̅	̅̅	PROPN
cana-4781	195	17	̅̅	̅̅	PROPN
cana-4781	195	18	.	.	PUNCT
cana-4781	196	1	suppose	suppose	VERB
cana-4781	196	2	𝜌	𝜌	ADP
cana-4781	196	3	:	:	PUNCT
cana-4781	196	4	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	196	5	̅̅	̅̅	PROPN
cana-4781	196	6	̅̅	̅̅	PROPN
cana-4781	196	7	→	→	PUNCT
cana-4781	196	8	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	196	9	̅̅	̅̅	PROPN
cana-4781	196	10	̅̅	̅̅	NOUN
cana-4781	196	11	is	be	AUX
cana-4781	196	12	a	a	DET
cana-4781	196	13	rough	rough	ADJ
cana-4781	196	14	homeomorphism	homeomorphism	NOUN
cana-4781	196	15	.	.	PUNCT
cana-4781	197	1	then	then	ADV
cana-4781	197	2	the	the	DET
cana-4781	197	3	quotient	quotient	NOUN
cana-4781	197	4	map	map	NOUN
cana-4781	197	5	𝛾	𝛾	NOUN
cana-4781	197	6	:	:	PUNCT
cana-4781	197	7	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	197	8	̅̅	̅̅	PROPN
cana-4781	197	9	̅̅	̅̅	PROPN
cana-4781	197	10	ℒ⁄	ℒ⁄	PROPN
cana-4781	197	11	→	→	SYM
cana-4781	197	12	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	197	13	̅̅	̅̅	PROPN
cana-4781	197	14	̅̅	̅̅	PROPN
cana-4781	197	15	ℳ⁄	ℳ⁄	PROPN
cana-4781	197	16	is	be	AUX
cana-4781	197	17	a	a	DET
cana-4781	197	18	topological	topological	ADJ
cana-4781	197	19	rough	rough	ADJ
cana-4781	197	20	group	group	NOUN
cana-4781	197	21	homeomorphism	homeomorphism	NOUN
cana-4781	197	22	which	which	PRON
cana-4781	197	23	is	be	AUX
cana-4781	197	24	defined	define	VERB
cana-4781	197	25	by	by	ADP
cana-4781	197	26	𝛾(𝑎ℒ	𝛾(𝑎ℒ	PROPN
cana-4781	197	27	)	)	PUNCT
cana-4781	198	1	=	=	SYM
cana-4781	198	2	𝑏ℳ	𝑏ℳ	PROPN
cana-4781	198	3	,	,	PUNCT
cana-4781	198	4	for	for	ADP
cana-4781	198	5	𝑎	𝑎	PROPN
cana-4781	198	6	∈	∈	PROPN
cana-4781	198	7	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	198	8	̅̅	̅̅	NOUN
cana-4781	198	9	̅̅	̅̅	PROPN
cana-4781	198	10	and	and	CCONJ
cana-4781	198	11	𝑏	𝑏	NOUN
cana-4781	198	12	=	=	SYM
cana-4781	198	13	𝜌(𝑎	𝜌(𝑎	PROPN
cana-4781	198	14	)	)	PUNCT
cana-4781	198	15	.	.	PUNCT
cana-4781	199	1	proof	proof	NOUN
cana-4781	199	2	:	:	PUNCT
cana-4781	199	3	consider	consider	VERB
cana-4781	199	4	the	the	DET
cana-4781	199	5	quotient	quotient	NOUN
cana-4781	199	6	maps	map	NOUN
cana-4781	199	7	𝜇	𝜇	ADP
cana-4781	199	8	:	:	PUNCT
cana-4781	199	9	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	199	10	̅̅	̅̅	PROPN
cana-4781	199	11	̅̅	̅̅	PROPN
cana-4781	199	12	→	→	PUNCT
cana-4781	199	13	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	199	14	̅̅	̅̅	NOUN
cana-4781	199	15	̅̅	̅̅	PROPN
cana-4781	199	16	ℒ⁄	ℒ⁄	PROPN
cana-4781	199	17	and	and	CCONJ
cana-4781	199	18	𝜇′	𝜇′	NUM
cana-4781	199	19	:	:	PUNCT
cana-4781	199	20	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	199	21	̅̅	̅̅	PROPN
cana-4781	199	22	̅̅	̅̅	PROPN
cana-4781	199	23	→	→	PUNCT
cana-4781	199	24	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	199	25	̅̅	̅̅	PROPN
cana-4781	199	26	̅̅	̅̅	PROPN
cana-4781	199	27	ℳ⁄	ℳ⁄	PROPN
cana-4781	199	28	.	.	PUNCT
cana-4781	200	1	then	then	ADV
cana-4781	200	2	𝜇′	𝜇′	ADV
cana-4781	200	3	∘	∘	VERB
cana-4781	200	4	𝜌	𝜌	X
cana-4781	200	5	=	=	SYM
cana-4781	200	6	𝛾	𝛾	ADP
cana-4781	200	7	∘	∘	NOUN
cana-4781	200	8	𝜇.	𝜇.	NOUN
cana-4781	200	9	since	since	SCONJ
cana-4781	200	10	the	the	DET
cana-4781	200	11	maps	map	NOUN
cana-4781	200	12	𝜇′	𝜇′	NOUN
cana-4781	200	13	,	,	PUNCT
cana-4781	200	14	𝜌	𝜌	X
cana-4781	200	15	and	and	CCONJ
cana-4781	200	16	𝜇	𝜇	X
cana-4781	200	17	are	be	AUX
cana-4781	200	18	open	open	ADJ
cana-4781	200	19	continuous	continuous	ADJ
cana-4781	200	20	rough	rough	ADJ
cana-4781	200	21	homomorphisms	homomorphism	NOUN
cana-4781	200	22	,	,	PUNCT
cana-4781	200	23	the	the	DET
cana-4781	200	24	map	map	NOUN
cana-4781	200	25	𝛾	𝛾	NOUN
cana-4781	200	26	is	be	AUX
cana-4781	200	27	an	an	DET
cana-4781	200	28	open	open	ADJ
cana-4781	200	29	continuous	continuous	ADJ
cana-4781	200	30	homomorphism	homomorphism	NOUN
cana-4781	200	31	.	.	PUNCT
cana-4781	201	1	let	let	VERB
cana-4781	201	2	𝑎ℒ	𝑎ℒ	PROPN
cana-4781	201	3	∈	∈	PROPN
cana-4781	201	4	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	201	5	̅̅	̅̅	NOUN
cana-4781	201	6	̅̅	̅̅	PROPN
cana-4781	201	7	ℒ⁄	ℒ⁄	PROPN
cana-4781	201	8	and	and	CCONJ
cana-4781	201	9	𝛾(𝑎ℒ	𝛾(𝑎ℒ	PROPN
cana-4781	201	10	)	)	PUNCT
cana-4781	202	1	=	=	SYM
cana-4781	202	2	ℳ.	ℳ.	PROPN
cana-4781	202	3	then	then	ADV
cana-4781	202	4	𝜇′(𝑏	𝜇′(𝑏	PROPN
cana-4781	202	5	)	)	PUNCT
cana-4781	202	6	=	=	SYM
cana-4781	202	7	ℳ	ℳ	PROPN
cana-4781	202	8	and	and	CCONJ
cana-4781	202	9	𝑏ℳ	𝑏ℳ	PROPN
cana-4781	202	10	=	=	SYM
cana-4781	202	11	ℳ	ℳ	PROPN
cana-4781	202	12	which	which	PRON
cana-4781	202	13	implies	imply	VERB
cana-4781	202	14	𝑏	𝑏	PROPN
cana-4781	202	15	=	=	SYM
cana-4781	202	16	𝜌(𝑎	𝜌(𝑎	PROPN
cana-4781	202	17	)	)	PUNCT
cana-4781	202	18	∈	∈	PROPN
cana-4781	202	19	ℳ.	ℳ.	PROPN
cana-4781	202	20	since	since	SCONJ
cana-4781	202	21	𝜌	𝜌	NOUN
cana-4781	202	22	is	be	AUX
cana-4781	202	23	a	a	DET
cana-4781	202	24	rough	rough	ADJ
cana-4781	202	25	homeomorphism	homeomorphism	NOUN
cana-4781	202	26	and	and	CCONJ
cana-4781	202	27	ℳ	ℳ	PROPN
cana-4781	202	28	=	=	SYM
cana-4781	202	29	𝜌(ℒ	𝜌(ℒ	PROPN
cana-4781	202	30	)	)	PUNCT
cana-4781	202	31	,	,	PUNCT
cana-4781	202	32	𝜌(𝑎	𝜌(𝑎	NUM
cana-4781	202	33	)	)	PUNCT
cana-4781	202	34	=	=	SYM
cana-4781	202	35	𝜌(𝑐	𝜌(𝑐	NOUN
cana-4781	202	36	)	)	PUNCT
cana-4781	202	37	,	,	PUNCT
cana-4781	202	38	for	for	ADP
cana-4781	202	39	some	some	DET
cana-4781	202	40	𝑐	𝑐	PROPN
cana-4781	202	41	∈	∈	PROPN
cana-4781	202	42	ℒ	ℒ	PROPN
cana-4781	202	43	which	which	PRON
cana-4781	202	44	implies	imply	VERB
cana-4781	202	45	𝑎	𝑎	PROPN
cana-4781	202	46	=	=	X
cana-4781	202	47	𝑐.	𝑐.	NOUN
cana-4781	202	48	therefore	therefore	ADV
cana-4781	202	49	,	,	PUNCT
cana-4781	202	50	𝑎	𝑎	PROPN
cana-4781	202	51	∈	∈	NOUN
cana-4781	202	52	ℒ	ℒ	PROPN
cana-4781	202	53	implies	imply	VERB
cana-4781	202	54	the	the	DET
cana-4781	202	55	kernel	kernel	NOUN
cana-4781	202	56	of	of	ADP
cana-4781	202	57	𝛾	𝛾	PROPN
cana-4781	202	58	is	be	AUX
cana-4781	202	59	ℒ	ℒ	NOUN
cana-4781	202	60	that	that	PRON
cana-4781	202	61	mean	mean	VERB
cana-4781	202	62	the	the	DET
cana-4781	202	63	quotient	quotient	NOUN
cana-4781	202	64	map	map	NOUN
cana-4781	202	65	𝛾	𝛾	NOUN
cana-4781	202	66	is	be	AUX
cana-4781	202	67	injective	injective	ADJ
cana-4781	202	68	.	.	PUNCT
cana-4781	203	1	hence	hence	ADV
cana-4781	203	2	the	the	DET
cana-4781	203	3	quotient	quotient	NOUN
cana-4781	203	4	map	map	NOUN
cana-4781	203	5	𝛾	𝛾	NOUN
cana-4781	203	6	is	be	AUX
cana-4781	203	7	a	a	DET
cana-4781	203	8	topological	topological	ADJ
cana-4781	203	9	rough	rough	ADJ
cana-4781	203	10	group	group	NOUN
cana-4781	203	11	homeomorphism	homeomorphism	NOUN
cana-4781	203	12	.	.	PUNCT
cana-4781	204	1	5	5	NUM
cana-4781	204	2	.	.	X
cana-4781	204	3	rough	rough	ADJ
cana-4781	204	4	isomorphism	isomorphism	NOUN
cana-4781	204	5	:	:	PUNCT
cana-4781	204	6	theorem	theorem	VERB
cana-4781	204	7	5.1	5.1	NUM
cana-4781	204	8	.	.	PUNCT
cana-4781	205	1	(	(	PUNCT
cana-4781	205	2	rough	rough	ADJ
cana-4781	205	3	isomorphism	isomorphism	NOUN
cana-4781	205	4	theorem	theorem	VERB
cana-4781	205	5	i	i	PRON
cana-4781	205	6	)	)	PUNCT
cana-4781	205	7	let	let	VERB
cana-4781	205	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	205	9	and	and	CCONJ
cana-4781	205	10	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	205	11	be	be	AUX
cana-4781	205	12	topological	topological	ADJ
cana-4781	205	13	simple	simple	ADJ
cana-4781	205	14	rough	rough	ADJ
cana-4781	205	15	groups	group	NOUN
cana-4781	205	16	such	such	ADJ
cana-4781	205	17	that	that	SCONJ
cana-4781	205	18	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	205	19	̅̅	̅̅	PROPN
cana-4781	205	20	̅̅	̅̅	PROPN
cana-4781	205	21	is	be	AUX
cana-4781	205	22	a	a	DET
cana-4781	205	23	group	group	NOUN
cana-4781	205	24	.	.	PUNCT
cana-4781	206	1	let	let	VERB
cana-4781	206	2	𝜌	𝜌	X
cana-4781	206	3	:	:	PUNCT
cana-4781	206	4	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	206	5	̅̅	̅̅	PROPN
cana-4781	206	6	̅̅	̅̅	PROPN
cana-4781	206	7	→	→	PUNCT
cana-4781	206	8	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	206	9	̅̅	̅̅	PROPN
cana-4781	206	10	̅̅	̅̅	NOUN
cana-4781	206	11	be	be	AUX
cana-4781	206	12	a	a	DET
cana-4781	206	13	topological	topological	ADJ
cana-4781	206	14	rough	rough	ADJ
cana-4781	206	15	group	group	NOUN
cana-4781	206	16	homomorphism	homomorphism	NOUN
cana-4781	206	17	and	and	CCONJ
cana-4781	206	18	𝒦𝜌	𝒦𝜌	PROPN
cana-4781	206	19	be	be	AUX
cana-4781	206	20	the	the	DET
cana-4781	206	21	rough	rough	ADJ
cana-4781	206	22	kernel	kernel	NOUN
cana-4781	206	23	of	of	ADP
cana-4781	206	24	𝜌.	𝜌.	NOUN
cana-4781	206	25	then	then	ADV
cana-4781	206	26	the	the	DET
cana-4781	206	27	map	map	NOUN
cana-4781	206	28	𝜑	𝜑	NOUN
cana-4781	206	29	:	:	PUNCT
cana-4781	206	30	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	206	31	̅̅	̅̅	NOUN
cana-4781	206	32	̅̅	̅̅	PROPN
cana-4781	207	1	𝒦𝜌⁄	𝒦𝜌⁄	PROPN
cana-4781	207	2	→	→	PUNCT
cana-4781	207	3	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	207	4	̅̅	̅̅	PROPN
cana-4781	207	5	̅̅	̅̅	NOUN
cana-4781	207	6	is	be	AUX
cana-4781	207	7	a	a	DET
cana-4781	207	8	continuous	continuous	ADJ
cana-4781	207	9	rough	rough	ADJ
cana-4781	207	10	isomorphism	isomorphism	NOUN
cana-4781	207	11	which	which	PRON
cana-4781	207	12	is	be	AUX
cana-4781	207	13	defined	define	VERB
cana-4781	207	14	by	by	ADP
cana-4781	207	15	φ(𝑎𝒦𝜌	φ(𝑎𝒦𝜌	NOUN
cana-4781	207	16	)	)	PUNCT
cana-4781	207	17	=	=	PUNCT
cana-4781	208	1	𝜌(𝑎	𝜌(𝑎	PROPN
cana-4781	208	2	)	)	PUNCT
cana-4781	208	3	,	,	PUNCT
cana-4781	208	4	for	for	ADP
cana-4781	208	5	every	every	DET
cana-4781	208	6	𝑎	𝑎	PROPN
cana-4781	208	7	∈	∈	PROPN
cana-4781	208	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	208	9	̅̅	̅̅	NOUN
cana-4781	208	10	̅̅	̅̅	PROPN
cana-4781	208	11	.	.	PUNCT
cana-4781	209	1	also	also	ADV
cana-4781	209	2	,	,	PUNCT
cana-4781	209	3	if	if	SCONJ
cana-4781	209	4	𝜌	𝜌	PRON
cana-4781	209	5	is	be	AUX
cana-4781	209	6	open	open	ADJ
cana-4781	209	7	,	,	PUNCT
cana-4781	209	8	the	the	DET
cana-4781	209	9	map	map	NOUN
cana-4781	209	10	𝜑	𝜑	NOUN
cana-4781	209	11	is	be	AUX
cana-4781	209	12	a	a	DET
cana-4781	209	13	rough	rough	ADJ
cana-4781	209	14	homeomorphism	homeomorphism	NOUN
cana-4781	209	15	.	.	PUNCT
cana-4781	210	1	proof	proof	NOUN
cana-4781	210	2	:	:	PUNCT
cana-4781	210	3	let	let	VERB
cana-4781	210	4	𝜇	𝜇	X
cana-4781	210	5	:	:	PUNCT
cana-4781	210	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	210	7	̅̅	̅̅	PROPN
cana-4781	210	8	̅̅	̅̅	PROPN
cana-4781	210	9	→	→	PUNCT
cana-4781	210	10	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	210	11	̅̅	̅̅	NOUN
cana-4781	210	12	̅̅	̅̅	PROPN
cana-4781	210	13	𝒦𝜌⁄	𝒦𝜌⁄	PROPN
cana-4781	210	14	be	be	AUX
cana-4781	210	15	a	a	DET
cana-4781	210	16	quotient	quotient	NOUN
cana-4781	210	17	map	map	NOUN
cana-4781	210	18	.	.	PUNCT
cana-4781	211	1	then	then	ADV
cana-4781	211	2	𝜌	𝜌	X
cana-4781	211	3	=	=	SYM
cana-4781	211	4	φ	φ	PROPN
cana-4781	211	5	∘	∘	PROPN
cana-4781	211	6	𝜇.	𝜇.	PROPN
cana-4781	212	1	(	(	PUNCT
cana-4781	212	2	i	i	NOUN
cana-4781	212	3	)	)	PUNCT
cana-4781	212	4	𝜑	𝜑	PROPN
cana-4781	212	5	is	be	AUX
cana-4781	212	6	injective	injective	ADJ
cana-4781	212	7	:	:	PUNCT
cana-4781	212	8	since	since	SCONJ
cana-4781	212	9	𝜌	𝜌	PRON
cana-4781	212	10	is	be	AUX
cana-4781	212	11	a	a	DET
cana-4781	212	12	topological	topological	ADJ
cana-4781	212	13	rough	rough	ADJ
cana-4781	212	14	group	group	NOUN
cana-4781	212	15	homomorphism	homomorphism	NOUN
cana-4781	212	16	and	and	CCONJ
cana-4781	212	17	𝒦𝜌	𝒦𝜌	PROPN
cana-4781	212	18	be	be	AUX
cana-4781	212	19	the	the	DET
cana-4781	212	20	rough	rough	ADJ
cana-4781	212	21	kernel	kernel	NOUN
cana-4781	212	22	of	of	ADP
cana-4781	212	23	𝜌	𝜌	ADP
cana-4781	212	24	,	,	PUNCT
cana-4781	212	25	for	for	ADP
cana-4781	212	26	some	some	DET
cana-4781	212	27	𝑎	𝑎	NOUN
cana-4781	212	28	,	,	PUNCT
cana-4781	212	29	𝑏	𝑏	PROPN
cana-4781	212	30	∈	∈	PROPN
cana-4781	212	31	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	212	32	̅̅	̅̅	PROPN
cana-4781	212	33	̅̅	̅̅	PROPN
cana-4781	212	34	,	,	PUNCT
cana-4781	212	35	𝑎𝒦𝜌	𝑎𝒦𝜌	NOUN
cana-4781	212	36	=	=	NOUN
cana-4781	212	37	𝑏𝒦𝜌	𝑏𝒦𝜌	NOUN
cana-4781	212	38	which	which	PRON
cana-4781	212	39	implies	imply	VERB
cana-4781	212	40	𝑎𝑏−1	𝑎𝑏−1	NOUN
cana-4781	212	41	∈	∈	PROPN
cana-4781	212	42	𝒦𝜌.	𝒦𝜌.	PROPN
cana-4781	212	43	then	then	ADV
cana-4781	212	44	𝜌(𝑎)𝜌(𝑏)−1	𝜌(𝑎)𝜌(𝑏)−1	NUM
cana-4781	212	45	=	=	SYM
cana-4781	212	46	𝜌(𝑎𝑏−1	𝜌(𝑎𝑏−1	NUM
cana-4781	212	47	)	)	PUNCT
cana-4781	212	48	=	=	SYM
cana-4781	212	49	𝑒′	𝑒′	NOUN
cana-4781	212	50	,	,	PUNCT
cana-4781	212	51	where	where	SCONJ
cana-4781	212	52	𝑒′	𝑒′	NOUN
cana-4781	212	53	is	be	AUX
cana-4781	212	54	the	the	DET
cana-4781	212	55	identity	identity	NOUN
cana-4781	212	56	in	in	ADP
cana-4781	212	57	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	212	58	̅̅	̅̅	NOUN
cana-4781	212	59	̅̅	̅̅	PROPN
cana-4781	212	60	.	.	PUNCT
cana-4781	213	1	therefore	therefore	ADV
cana-4781	213	2	,	,	PUNCT
cana-4781	213	3	𝜌(𝑎	𝜌(𝑎	PROPN
cana-4781	213	4	)	)	PUNCT
cana-4781	213	5	=	=	SYM
cana-4781	213	6	𝜌(𝑏	𝜌(𝑏	NUM
cana-4781	213	7	)	)	PUNCT
cana-4781	213	8	implies	imply	VERB
cana-4781	213	9	that	that	SCONJ
cana-4781	213	10	𝜑	𝜑	PROPN
cana-4781	213	11	is	be	AUX
cana-4781	213	12	one	one	NUM
cana-4781	213	13	-	-	PUNCT
cana-4781	213	14	one	one	NUM
cana-4781	213	15	.	.	PUNCT
cana-4781	214	1	(	(	PUNCT
cana-4781	214	2	ii	ii	NOUN
cana-4781	214	3	)	)	PUNCT
cana-4781	214	4	𝜑	𝜑	PROPN
cana-4781	214	5	is	be	AUX
cana-4781	214	6	surjective	surjective	ADJ
cana-4781	214	7	:	:	PUNCT
cana-4781	214	8	let	let	VERB
cana-4781	214	9	𝑥	𝑥	X
cana-4781	214	10	∈	∈	PROPN
cana-4781	214	11	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	214	12	̅̅	̅̅	NOUN
cana-4781	214	13	̅̅	̅̅	PROPN
cana-4781	214	14	.	.	PUNCT
cana-4781	215	1	then	then	ADV
cana-4781	215	2	there	there	PRON
cana-4781	215	3	exists	exist	VERB
cana-4781	215	4	an	an	DET
cana-4781	215	5	element	element	NOUN
cana-4781	215	6	𝑎	𝑎	PROPN
cana-4781	215	7	∈	∈	PROPN
cana-4781	215	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	215	9	̅̅	̅̅	NOUN
cana-4781	215	10	̅̅	̅̅	PROPN
cana-4781	215	11	such	such	ADJ
cana-4781	215	12	that	that	SCONJ
cana-4781	215	13	𝜌(𝑎	𝜌(𝑎	PROPN
cana-4781	215	14	)	)	PUNCT
cana-4781	215	15	=	=	PUNCT
cana-4781	216	1	𝑥.	𝑥.	VERB
cana-4781	216	2	since	since	SCONJ
cana-4781	216	3	φ(𝑎𝒦𝜌	φ(𝑎𝒦𝜌	NUM
cana-4781	216	4	)	)	PUNCT
cana-4781	216	5	=	=	SYM
cana-4781	217	1	𝜌(𝑎	𝜌(𝑎	PROPN
cana-4781	217	2	)	)	PUNCT
cana-4781	217	3	,	,	PUNCT
cana-4781	217	4	for	for	ADP
cana-4781	217	5	every	every	DET
cana-4781	217	6	element	element	NOUN
cana-4781	217	7	of	of	ADP
cana-4781	217	8	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	217	9	̅̅	̅̅	PROPN
cana-4781	217	10	̅̅	̅̅	NOUN
cana-4781	217	11	is	be	AUX
cana-4781	217	12	in	in	ADP
cana-4781	217	13	the	the	DET
cana-4781	217	14	image	image	NOUN
cana-4781	217	15	of	of	ADP
cana-4781	217	16	𝜑.	𝜑.	NOUN
cana-4781	217	17	hence	hence	ADV
cana-4781	217	18	𝜑	𝜑	PROPN
cana-4781	217	19	is	be	AUX
cana-4781	217	20	onto	onto	ADP
cana-4781	217	21	.	.	PUNCT
cana-4781	218	1	(	(	PUNCT
cana-4781	218	2	iii	iii	X
cana-4781	218	3	)	)	PUNCT
cana-4781	218	4	𝜑	𝜑	NOUN
cana-4781	218	5	is	be	AUX
cana-4781	218	6	rough	rough	ADJ
cana-4781	218	7	homomorphism	homomorphism	NOUN
cana-4781	218	8	:	:	PUNCT
cana-4781	218	9	let	let	VERB
cana-4781	218	10	𝑎	𝑎	X
cana-4781	218	11	,	,	PUNCT
cana-4781	218	12	𝑏	𝑏	PROPN
cana-4781	218	13	∈	∈	PROPN
cana-4781	218	14	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	218	15	̅̅	̅̅	PROPN
cana-4781	218	16	̅̅	̅̅	PROPN
cana-4781	218	17	.	.	PUNCT
cana-4781	219	1	then	then	ADV
cana-4781	219	2	𝑎𝒦𝜌	𝑎𝒦𝜌	NUM
cana-4781	219	3	,	,	PUNCT
cana-4781	219	4	𝑏𝒦𝜌	𝑏𝒦𝜌	NOUN
cana-4781	219	5	∈	∈	PROPN
cana-4781	219	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	219	7	̅̅	̅̅	NOUN
cana-4781	219	8	̅̅	̅̅	PROPN
cana-4781	219	9	𝒦𝜌⁄	𝒦𝜌⁄	INTJ
cana-4781	219	10	.	.	PUNCT
cana-4781	220	1	since	since	SCONJ
cana-4781	220	2	𝜌	𝜌	PRON
cana-4781	220	3	is	be	AUX
cana-4781	220	4	a	a	DET
cana-4781	220	5	topological	topological	ADJ
cana-4781	220	6	rough	rough	ADJ
cana-4781	220	7	group	group	NOUN
cana-4781	220	8	homomorphism	homomorphism	NOUN
cana-4781	220	9	,	,	PUNCT
cana-4781	220	10	φ(𝑎𝑏𝒦𝜌	φ(𝑎𝑏𝒦𝜌	NOUN
cana-4781	220	11	)	)	PUNCT
cana-4781	220	12	=	=	SYM
cana-4781	220	13	𝜌(𝑎𝑏	𝜌(𝑎𝑏	X
cana-4781	220	14	)	)	PUNCT
cana-4781	220	15	=	=	SYM
cana-4781	220	16	𝜌(𝑎)𝜌(𝑏	𝜌(𝑎)𝜌(𝑏	X
cana-4781	220	17	)	)	PUNCT
cana-4781	220	18	=	=	SYM
cana-4781	220	19	φ(𝑎𝒦𝜌)φ(𝑏𝒦𝜌	φ(𝑎𝒦𝜌)φ(𝑏𝒦𝜌	ADJ
cana-4781	220	20	)	)	PUNCT
cana-4781	220	21	.	.	PUNCT
cana-4781	221	1	therefore	therefore	ADV
cana-4781	221	2	,	,	PUNCT
cana-4781	221	3	𝜑	𝜑	PROPN
cana-4781	221	4	is	be	AUX
cana-4781	221	5	a	a	DET
cana-4781	221	6	rough	rough	ADJ
cana-4781	221	7	homomorphism	homomorphism	NOUN
cana-4781	221	8	.	.	PUNCT
cana-4781	222	1	communications	communication	NOUN
cana-4781	222	2	on	on	ADP
cana-4781	222	3	applied	apply	VERB
cana-4781	222	4	nonlinear	nonlinear	ADJ
cana-4781	222	5	analysis	analysis	NOUN
cana-4781	222	6	issn	issn	NOUN
cana-4781	222	7	:	:	PUNCT
cana-4781	222	8	1074	1074	NUM
cana-4781	222	9	-	-	PUNCT
cana-4781	222	10	133x	133x	NUM
cana-4781	222	11	vol	vol	NOUN
cana-4781	222	12	32	32	NUM
cana-4781	222	13	no	no	NOUN
cana-4781	222	14	.	.	NOUN
cana-4781	222	15	3	3	NUM
cana-4781	222	16	(	(	PUNCT
cana-4781	222	17	2025	2025	NUM
cana-4781	222	18	)	)	PUNCT
cana-4781	222	19	886	886	NUM
cana-4781	222	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-4781	222	21	(	(	PUNCT
cana-4781	222	22	iv	iv	X
cana-4781	222	23	)	)	PUNCT
cana-4781	222	24	𝜑	𝜑	NOUN
cana-4781	222	25	is	be	AUX
cana-4781	222	26	continuous	continuous	ADJ
cana-4781	222	27	:	:	PUNCT
cana-4781	222	28	since	since	SCONJ
cana-4781	222	29	𝜌	𝜌	NOUN
cana-4781	222	30	is	be	AUX
cana-4781	222	31	continuous	continuous	ADJ
cana-4781	222	32	and	and	CCONJ
cana-4781	222	33	𝜇	𝜇	ADV
cana-4781	222	34	is	be	AUX
cana-4781	222	35	a	a	DET
cana-4781	222	36	quotient	quotient	NOUN
cana-4781	222	37	map	map	NOUN
cana-4781	222	38	,	,	PUNCT
cana-4781	222	39	φ	φ	NOUN
cana-4781	222	40	=	=	SYM
cana-4781	222	41	𝜌	𝜌	ADP
cana-4781	222	42	∘	∘	PROPN
cana-4781	222	43	𝜇−1	𝜇−1	PROPN
cana-4781	222	44	is	be	AUX
cana-4781	222	45	continuous	continuous	ADJ
cana-4781	222	46	.	.	PUNCT
cana-4781	223	1	hence	hence	ADV
cana-4781	223	2	the	the	DET
cana-4781	223	3	map	map	NOUN
cana-4781	223	4	𝜑	𝜑	NOUN
cana-4781	223	5	:	:	PUNCT
cana-4781	223	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	223	7	̅̅	̅̅	NOUN
cana-4781	223	8	̅̅	̅̅	PROPN
cana-4781	224	1	𝒦𝜌⁄	𝒦𝜌⁄	PROPN
cana-4781	224	2	→	→	PUNCT
cana-4781	224	3	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	224	4	̅̅	̅̅	PROPN
cana-4781	224	5	̅̅	̅̅	NOUN
cana-4781	224	6	is	be	AUX
cana-4781	224	7	a	a	DET
cana-4781	224	8	continuous	continuous	ADJ
cana-4781	224	9	rough	rough	ADJ
cana-4781	224	10	isomorphism	isomorphism	NOUN
cana-4781	224	11	.	.	PUNCT
cana-4781	225	1	(	(	PUNCT
cana-4781	225	2	v	v	NOUN
cana-4781	225	3	)	)	PUNCT
cana-4781	225	4	𝜑	𝜑	NOUN
cana-4781	225	5	is	be	AUX
cana-4781	225	6	a	a	DET
cana-4781	225	7	rough	rough	ADJ
cana-4781	225	8	homeomorphism	homeomorphism	NOUN
cana-4781	225	9	if	if	SCONJ
cana-4781	225	10	𝜌	𝜌	ADV
cana-4781	225	11	is	be	AUX
cana-4781	225	12	open	open	ADJ
cana-4781	225	13	:	:	PUNCT
cana-4781	225	14	let	let	VERB
cana-4781	225	15	𝑈	𝑈	PROPN
cana-4781	225	16	be	be	AUX
cana-4781	225	17	an	an	DET
cana-4781	225	18	open	open	ADJ
cana-4781	225	19	neighbourhood	neighbourhood	NOUN
cana-4781	225	20	in	in	ADP
cana-4781	225	21	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	225	22	̅̅	̅̅	PROPN
cana-4781	225	23	̅̅	̅̅	PROPN
cana-4781	225	24	.	.	PUNCT
cana-4781	226	1	since	since	SCONJ
cana-4781	226	2	𝜌	𝜌	NOUN
cana-4781	226	3	is	be	AUX
cana-4781	226	4	open	open	ADJ
cana-4781	226	5	,	,	PUNCT
cana-4781	226	6	𝜌(𝑈	𝜌(𝑈	NUM
cana-4781	226	7	)	)	PUNCT
cana-4781	226	8	is	be	AUX
cana-4781	226	9	open	open	ADJ
cana-4781	226	10	.	.	PUNCT
cana-4781	227	1	then	then	ADV
cana-4781	227	2	φ(u	φ(u	NOUN
cana-4781	227	3	)	)	PUNCT
cana-4781	227	4	=	=	SYM
cana-4781	227	5	𝜌(𝜇−1(𝑈	𝜌(𝜇−1(𝑈	NOUN
cana-4781	227	6	)	)	PUNCT
cana-4781	227	7	)	)	PUNCT
cana-4781	227	8	is	be	AUX
cana-4781	227	9	open	open	ADJ
cana-4781	227	10	.	.	PUNCT
cana-4781	228	1	therefore	therefore	ADV
cana-4781	228	2	,	,	PUNCT
cana-4781	228	3	φ−1	φ−1	PROPN
cana-4781	228	4	is	be	AUX
cana-4781	228	5	continuous	continuous	ADJ
cana-4781	228	6	.	.	PUNCT
cana-4781	229	1	hence	hence	ADV
cana-4781	229	2	𝜑	𝜑	PROPN
cana-4781	229	3	is	be	AUX
cana-4781	229	4	a	a	DET
cana-4781	229	5	rough	rough	ADJ
cana-4781	229	6	homeomorphism	homeomorphism	NOUN
cana-4781	229	7	.	.	PUNCT
cana-4781	230	1	theorem	theorem	VERB
cana-4781	230	2	5.2	5.2	NUM
cana-4781	230	3	.	.	PUNCT
cana-4781	231	1	(	(	PUNCT
cana-4781	231	2	rough	rough	ADJ
cana-4781	231	3	isomorphism	isomorphism	PROPN
cana-4781	231	4	theorem	theorem	PROPN
cana-4781	231	5	ii	ii	PROPN
cana-4781	231	6	)	)	PUNCT
cana-4781	231	7	let	let	VERB
cana-4781	231	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	231	9	and	and	CCONJ
cana-4781	231	10	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	231	11	be	be	AUX
cana-4781	231	12	topological	topological	ADJ
cana-4781	231	13	simple	simple	ADJ
cana-4781	231	14	rough	rough	ADJ
cana-4781	231	15	groups	group	NOUN
cana-4781	231	16	.	.	PUNCT
cana-4781	232	1	let	let	VERB
cana-4781	232	2	𝜌	𝜌	X
cana-4781	232	3	:	:	PUNCT
cana-4781	232	4	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	232	5	̅̅	̅̅	PROPN
cana-4781	232	6	̅̅	̅̅	PROPN
cana-4781	232	7	→	→	PUNCT
cana-4781	232	8	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	232	9	̅̅	̅̅	PROPN
cana-4781	232	10	̅̅	̅̅	NOUN
cana-4781	232	11	be	be	AUX
cana-4781	232	12	a	a	DET
cana-4781	232	13	topological	topological	ADJ
cana-4781	232	14	rough	rough	ADJ
cana-4781	232	15	group	group	NOUN
cana-4781	232	16	homomorphism	homomorphism	NOUN
cana-4781	232	17	such	such	ADJ
cana-4781	232	18	that	that	SCONJ
cana-4781	232	19	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	232	20	̅̅	̅̅	PROPN
cana-4781	232	21	̅̅	̅̅	PROPN
cana-4781	232	22	and	and	CCONJ
cana-4781	232	23	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	232	24	̅̅	̅̅	PROPN
cana-4781	232	25	̅̅	̅̅	PROPN
cana-4781	232	26	are	be	AUX
cana-4781	232	27	groups	group	NOUN
cana-4781	232	28	and	and	CCONJ
cana-4781	232	29	ℒ	ℒ	NOUN
cana-4781	232	30	be	be	VERB
cana-4781	232	31	a	a	DET
cana-4781	232	32	normal	normal	ADJ
cana-4781	232	33	subgroup	subgroup	NOUN
cana-4781	232	34	of	of	ADP
cana-4781	232	35	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	232	36	̅̅	̅̅	PROPN
cana-4781	232	37	̅̅	̅̅	PROPN
cana-4781	232	38	.	.	PUNCT
cana-4781	233	1	define	define	VERB
cana-4781	233	2	ℳ	ℳ	PROPN
cana-4781	233	3	=	=	PUNCT
cana-4781	233	4	𝜌−1(ℒ	𝜌−1(ℒ	NUM
cana-4781	233	5	)	)	PUNCT
cana-4781	233	6	and	and	CCONJ
cana-4781	233	7	𝒦𝜌	𝒦𝜌	PROPN
cana-4781	233	8	be	be	AUX
cana-4781	233	9	the	the	DET
cana-4781	233	10	rough	rough	ADJ
cana-4781	233	11	kernel	kernel	NOUN
cana-4781	233	12	of	of	ADP
cana-4781	233	13	𝜌.	𝜌.	NOUN
cana-4781	233	14	then	then	ADV
cana-4781	233	15	the	the	DET
cana-4781	233	16	map	map	NOUN
cana-4781	233	17	𝜑	𝜑	X
cana-4781	233	18	:	:	PUNCT
cana-4781	233	19	(	(	PUNCT
cana-4781	233	20	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	233	21	̅̅	̅̅	PROPN
cana-4781	233	22	̅̅	̅̅	PROPN
cana-4781	233	23	𝒦𝜌⁄	𝒦𝜌⁄	PROPN
cana-4781	233	24	)	)	PUNCT
cana-4781	234	1	/(ℳ	/(ℳ	PUNCT
cana-4781	235	1	𝒦𝜌⁄	𝒦𝜌⁄	NOUN
cana-4781	235	2	)	)	PUNCT
cana-4781	236	1	→	→	PUNCT
cana-4781	236	2	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	236	3	̅̅	̅̅	PROPN
cana-4781	236	4	̅̅	̅̅	PROPN
cana-4781	236	5	ℒ⁄	ℒ⁄	PROPN
cana-4781	236	6	is	be	AUX
cana-4781	236	7	a	a	DET
cana-4781	236	8	topological	topological	ADJ
cana-4781	236	9	rough	rough	ADJ
cana-4781	236	10	group	group	NOUN
cana-4781	236	11	homeomorphism	homeomorphism	NOUN
cana-4781	236	12	.	.	PUNCT
cana-4781	237	1	proof	proof	NOUN
cana-4781	237	2	:	:	PUNCT
cana-4781	237	3	consider	consider	VERB
cana-4781	237	4	the	the	DET
cana-4781	237	5	rough	rough	ADJ
cana-4781	237	6	quotient	quotient	NOUN
cana-4781	237	7	map	map	NOUN
cana-4781	237	8	𝜇	𝜇	ADP
cana-4781	237	9	:	:	PUNCT
cana-4781	237	10	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	237	11	̅̅	̅̅	PROPN
cana-4781	237	12	̅̅	̅̅	PROPN
cana-4781	237	13	→	→	PUNCT
cana-4781	238	1	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	238	2	̅̅	̅̅	NOUN
cana-4781	238	3	̅̅	̅̅	NOUN
cana-4781	238	4	ℒ⁄	ℒ⁄	PROPN
cana-4781	238	5	which	which	PRON
cana-4781	238	6	is	be	AUX
cana-4781	238	7	an	an	DET
cana-4781	238	8	open	open	ADJ
cana-4781	238	9	continuous	continuous	ADJ
cana-4781	238	10	rough	rough	ADJ
cana-4781	238	11	homomorphism	homomorphism	NOUN
cana-4781	238	12	.	.	PUNCT
cana-4781	239	1	then	then	ADV
cana-4781	239	2	𝜇	𝜇	ADP
cana-4781	239	3	∘	∘	ADP
cana-4781	239	4	𝜌	𝜌	ADP
cana-4781	239	5	∶	∶	NOUN
cana-4781	239	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	239	7	̅̅	̅̅	PROPN
cana-4781	239	8	̅̅	̅̅	PROPN
cana-4781	239	9	→	→	PUNCT
cana-4781	239	10	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	239	11	̅̅	̅̅	PROPN
cana-4781	239	12	̅̅	̅̅	PROPN
cana-4781	239	13	ℒ⁄	ℒ⁄	PROPN
cana-4781	239	14	is	be	AUX
cana-4781	239	15	also	also	ADV
cana-4781	239	16	a	a	DET
cana-4781	239	17	continuous	continuous	ADJ
cana-4781	239	18	open	open	ADJ
cana-4781	239	19	rough	rough	ADJ
cana-4781	239	20	homomorphism	homomorphism	NOUN
cana-4781	239	21	.	.	PUNCT
cana-4781	240	1	let	let	VERB
cana-4781	240	2	𝜌′	𝜌′	NOUN
cana-4781	240	3	=	=	PUNCT
cana-4781	240	4	𝜇	𝜇	ADP
cana-4781	240	5	∘	∘	NOUN
cana-4781	240	6	𝜌.	𝜌.	NOUN
cana-4781	240	7	then	then	ADV
cana-4781	240	8	the	the	DET
cana-4781	240	9	rough	rough	ADJ
cana-4781	240	10	kernel	kernel	NOUN
cana-4781	240	11	of	of	ADP
cana-4781	240	12	𝜌′	𝜌′	PROPN
cana-4781	240	13	,	,	PUNCT
cana-4781	240	14	𝒦𝜌′	𝒦𝜌′	NOUN
cana-4781	240	15	=	=	SYM
cana-4781	240	16	{	{	PUNCT
cana-4781	240	17	𝑎	𝑎	PROPN
cana-4781	240	18	∈	∈	PROPN
cana-4781	240	19	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	240	20	̅̅	̅̅	NOUN
cana-4781	240	21	̅̅	̅̅	PROPN
cana-4781	240	22	∶	∶	PROPN
cana-4781	240	23	𝜌′(𝑎	𝜌′(𝑎	PUNCT
cana-4781	240	24	)	)	PUNCT
cana-4781	240	25	=	=	SYM
cana-4781	240	26	ℒ	ℒ	NOUN
cana-4781	240	27	}	}	PUNCT
cana-4781	240	28	.	.	PUNCT
cana-4781	241	1	since	since	SCONJ
cana-4781	241	2	𝜌′(𝑎	𝜌′(𝑎	PRON
cana-4781	241	3	)	)	PUNCT
cana-4781	241	4	=	=	SYM
cana-4781	241	5	𝜇(𝜌(𝑎	𝜇(𝜌(𝑎	NUM
cana-4781	241	6	)	)	PUNCT
cana-4781	241	7	)	)	PUNCT
cana-4781	242	1	=	=	SYM
cana-4781	242	2	𝜌(𝑎)ℒ	𝜌(𝑎)ℒ	NUM
cana-4781	242	3	,	,	PUNCT
cana-4781	242	4	𝒦𝜌′	𝒦𝜌′	NOUN
cana-4781	242	5	=	=	PUNCT
cana-4781	242	6	{	{	PUNCT
cana-4781	242	7	𝑎	𝑎	PROPN
cana-4781	242	8	∈	∈	PROPN
cana-4781	242	9	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	242	10	̅̅	̅̅	NOUN
cana-4781	242	11	̅̅	̅̅	PROPN
cana-4781	242	12	∶	∶	PROPN
cana-4781	242	13	𝜌(𝑎	𝜌(𝑎	PROPN
cana-4781	242	14	)	)	PUNCT
cana-4781	242	15	∈	∈	PROPN
cana-4781	242	16	ℒ	ℒ	PROPN
cana-4781	242	17	}	}	PUNCT
cana-4781	242	18	.	.	PUNCT
cana-4781	243	1	but	but	CCONJ
cana-4781	243	2	ℳ	ℳ	PROPN
cana-4781	243	3	=	=	SYM
cana-4781	243	4	𝜌−1(ℒ	𝜌−1(ℒ	NUM
cana-4781	243	5	)	)	PUNCT
cana-4781	243	6	=	=	PRON
cana-4781	243	7	{	{	PUNCT
cana-4781	243	8	𝑎	𝑎	PROPN
cana-4781	243	9	∈	∈	PROPN
cana-4781	243	10	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	243	11	̅̅	̅̅	NOUN
cana-4781	243	12	̅̅	̅̅	PROPN
cana-4781	243	13	∶	∶	PROPN
cana-4781	243	14	𝜌(𝑎	𝜌(𝑎	PROPN
cana-4781	243	15	)	)	PUNCT
cana-4781	243	16	∈	∈	PROPN
cana-4781	243	17	ℒ	ℒ	PROPN
cana-4781	243	18	}	}	PUNCT
cana-4781	243	19	.	.	PUNCT
cana-4781	244	1	therefore	therefore	ADV
cana-4781	244	2	,	,	PUNCT
cana-4781	244	3	𝒦𝜌′	𝒦𝜌′	PROPN
cana-4781	244	4	=	=	PUNCT
cana-4781	244	5	ℳ.	ℳ.	PROPN
cana-4781	244	6	by	by	ADP
cana-4781	244	7	theorem	theorem	VERB
cana-4781	244	8	5.1	5.1	NUM
cana-4781	244	9	,	,	PUNCT
cana-4781	244	10	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	244	11	̅̅	̅̅	PROPN
cana-4781	244	12	̅̅	̅̅	PROPN
cana-4781	244	13	ℳ⁄	ℳ⁄	PROPN
cana-4781	244	14	is	be	AUX
cana-4781	244	15	topological	topological	ADJ
cana-4781	244	16	rough	rough	ADJ
cana-4781	244	17	group	group	NOUN
cana-4781	244	18	homeomorphism	homeomorphism	PROPN
cana-4781	244	19	to	to	ADP
cana-4781	244	20	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	244	21	̅̅	̅̅	PROPN
cana-4781	244	22	̅̅	̅̅	NOUN
cana-4781	244	23	ℒ⁄	ℒ⁄	PROPN
cana-4781	244	24	.	.	PUNCT
cana-4781	245	1	in	in	ADP
cana-4781	245	2	the	the	DET
cana-4781	245	3	similar	similar	ADJ
cana-4781	245	4	way	way	NOUN
cana-4781	245	5	,	,	PUNCT
cana-4781	245	6	define	define	VERB
cana-4781	245	7	the	the	DET
cana-4781	245	8	map	map	NOUN
cana-4781	245	9	𝜑	𝜑	NOUN
cana-4781	245	10	:	:	PUNCT
cana-4781	245	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	245	12	̅̅	̅̅	NOUN
cana-4781	245	13	̅̅	̅̅	PROPN
cana-4781	246	1	𝒦𝜌⁄	𝒦𝜌⁄	PROPN
cana-4781	246	2	→	→	PUNCT
cana-4781	246	3	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	246	4	̅̅	̅̅	NOUN
cana-4781	246	5	̅̅	̅̅	NOUN
cana-4781	246	6	ℒ⁄	ℒ⁄	VERB
cana-4781	246	7	by	by	ADP
cana-4781	246	8	𝜑(𝑎𝒦𝜌	𝜑(𝑎𝒦𝜌	NOUN
cana-4781	246	9	)	)	PUNCT
cana-4781	246	10	=	=	PUNCT
cana-4781	247	1	𝜌(𝑎)ℒ.	𝜌(𝑎)ℒ.	ADP
cana-4781	247	2	then	then	ADV
cana-4781	247	3	the	the	DET
cana-4781	247	4	rough	rough	ADJ
cana-4781	247	5	kernel	kernel	NOUN
cana-4781	247	6	of	of	ADP
cana-4781	247	7	𝜑	𝜑	PRON
cana-4781	247	8	,	,	PUNCT
cana-4781	247	9	𝒦𝜑	𝒦𝜑	NOUN
cana-4781	247	10	=	=	PUNCT
cana-4781	247	11	{	{	PUNCT
cana-4781	247	12	𝑎𝒦𝜌	𝑎𝒦𝜌	NOUN
cana-4781	247	13	∈	∈	PROPN
cana-4781	247	14	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	247	15	̅̅	̅̅	NOUN
cana-4781	247	16	̅̅	̅̅	PROPN
cana-4781	248	1	𝒦𝜌⁄	𝒦𝜌⁄	PROPN
cana-4781	248	2	∶	∶	PROPN
cana-4781	248	3	𝜑(𝑎𝒦𝜌	𝜑(𝑎𝒦𝜌	PUNCT
cana-4781	248	4	)	)	PUNCT
cana-4781	249	1	=	=	PUNCT
cana-4781	249	2	ℒ	ℒ	X
cana-4781	249	3	}	}	PUNCT
cana-4781	249	4	which	which	PRON
cana-4781	249	5	implies	imply	VERB
cana-4781	249	6	𝒦𝜑	𝒦𝜑	PROPN
cana-4781	249	7	=	=	PRON
cana-4781	249	8	{	{	PUNCT
cana-4781	249	9	𝑎𝒦𝜌	𝑎𝒦𝜌	NOUN
cana-4781	249	10	∈	∈	PROPN
cana-4781	249	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	249	12	̅̅	̅̅	NOUN
cana-4781	249	13	̅̅	̅̅	PROPN
cana-4781	249	14	𝒦𝜌⁄	𝒦𝜌⁄	PROPN
cana-4781	249	15	∶	∶	PROPN
cana-4781	249	16	𝜌(𝑎	𝜌(𝑎	PROPN
cana-4781	249	17	)	)	PUNCT
cana-4781	249	18	∈	∈	PROPN
cana-4781	249	19	ℒ	ℒ	PROPN
cana-4781	249	20	}	}	PUNCT
cana-4781	249	21	.	.	PUNCT
cana-4781	250	1	but	but	CCONJ
cana-4781	250	2	ℳ	ℳ	PROPN
cana-4781	250	3	=	=	SYM
cana-4781	250	4	𝜌−1(ℒ	𝜌−1(ℒ	NUM
cana-4781	250	5	)	)	PUNCT
cana-4781	250	6	=	=	PRON
cana-4781	250	7	{	{	PUNCT
cana-4781	250	8	𝑎	𝑎	PROPN
cana-4781	250	9	∈	∈	PROPN
cana-4781	250	10	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	250	11	̅̅	̅̅	NOUN
cana-4781	250	12	̅̅	̅̅	PROPN
cana-4781	250	13	∶	∶	PROPN
cana-4781	250	14	𝜌(𝑎	𝜌(𝑎	PROPN
cana-4781	250	15	)	)	PUNCT
cana-4781	250	16	∈	∈	PROPN
cana-4781	250	17	ℒ	ℒ	PROPN
cana-4781	250	18	}	}	PUNCT
cana-4781	250	19	.	.	PUNCT
cana-4781	251	1	that	that	PRON
cana-4781	251	2	is	be	AUX
cana-4781	251	3	,	,	PUNCT
cana-4781	251	4	the	the	DET
cana-4781	251	5	element	element	NOUN
cana-4781	251	6	𝑎	𝑎	PROPN
cana-4781	251	7	∈	∈	PROPN
cana-4781	251	8	ℳ	ℳ	NOUN
cana-4781	251	9	mapped	map	VERB
cana-4781	251	10	to	to	ADP
cana-4781	251	11	the	the	DET
cana-4781	251	12	identity	identity	NOUN
cana-4781	251	13	ℒ	ℒ	NOUN
cana-4781	251	14	in	in	ADP
cana-4781	251	15	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	251	16	̅̅	̅̅	NOUN
cana-4781	251	17	̅̅	̅̅	NOUN
cana-4781	251	18	ℒ⁄	ℒ⁄	PROPN
cana-4781	251	19	.	.	PUNCT
cana-4781	252	1	therefore	therefore	ADV
cana-4781	252	2	,	,	PUNCT
cana-4781	252	3	the	the	DET
cana-4781	252	4	element	element	NOUN
cana-4781	252	5	𝑎𝒦𝜌	𝑎𝒦𝜌	NOUN
cana-4781	252	6	∈	∈	PROPN
cana-4781	252	7	ℳ	ℳ	NOUN
cana-4781	252	8	𝒦𝜌⁄	𝒦𝜌⁄	PROPN
cana-4781	252	9	mapped	map	VERB
cana-4781	252	10	to	to	ADP
cana-4781	252	11	the	the	DET
cana-4781	252	12	identity	identity	NOUN
cana-4781	252	13	ℒ	ℒ	NOUN
cana-4781	252	14	in	in	ADP
cana-4781	252	15	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	252	16	̅̅	̅̅	NOUN
cana-4781	252	17	̅̅	̅̅	NOUN
cana-4781	252	18	ℒ⁄	ℒ⁄	PROPN
cana-4781	252	19	which	which	PRON
cana-4781	252	20	implies	imply	VERB
cana-4781	252	21	the	the	DET
cana-4781	252	22	rough	rough	ADJ
cana-4781	252	23	kernel	kernel	NOUN
cana-4781	252	24	of	of	ADP
cana-4781	252	25	𝜑	𝜑	PRON
cana-4781	252	26	,	,	PUNCT
cana-4781	252	27	𝒦𝜑	𝒦𝜑	PROPN
cana-4781	252	28	=	=	SYM
cana-4781	252	29	ℳ	ℳ	PROPN
cana-4781	252	30	𝒦𝜌⁄	𝒦𝜌⁄	PROPN
cana-4781	252	31	.	.	PUNCT
cana-4781	253	1	by	by	ADP
cana-4781	253	2	theorem	theorem	NOUN
cana-4781	253	3	5.1	5.1	NUM
cana-4781	253	4	,	,	PUNCT
cana-4781	253	5	(	(	PUNCT
cana-4781	253	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	253	7	̅̅	̅̅	PROPN
cana-4781	253	8	̅̅	̅̅	PROPN
cana-4781	253	9	𝒦𝜌⁄	𝒦𝜌⁄	PROPN
cana-4781	253	10	)	)	PUNCT
cana-4781	253	11	/(ℳ	/(ℳ	PUNCT
cana-4781	254	1	𝒦𝜌⁄	𝒦𝜌⁄	NOUN
cana-4781	254	2	)	)	PUNCT
cana-4781	254	3	is	be	AUX
cana-4781	254	4	topological	topological	ADJ
cana-4781	254	5	rough	rough	ADJ
cana-4781	254	6	group	group	NOUN
cana-4781	254	7	homeomorphism	homeomorphism	PROPN
cana-4781	254	8	to	to	ADP
cana-4781	254	9	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	254	10	̅̅	̅̅	PROPN
cana-4781	254	11	̅̅	̅̅	NOUN
cana-4781	254	12	ℒ⁄	ℒ⁄	PROPN
cana-4781	254	13	.	.	PUNCT
cana-4781	255	1	theorem	theorem	VERB
cana-4781	255	2	5.3	5.3	NUM
cana-4781	255	3	.	.	PUNCT
cana-4781	256	1	(	(	PUNCT
cana-4781	256	2	rough	rough	ADJ
cana-4781	256	3	isomorphism	isomorphism	NOUN
cana-4781	256	4	theorem	theorem	VERB
cana-4781	256	5	iii	iii	NOUN
cana-4781	256	6	)	)	PUNCT
cana-4781	256	7	let	let	VERB
cana-4781	256	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	256	9	be	be	AUX
cana-4781	256	10	a	a	DET
cana-4781	256	11	topological	topological	ADJ
cana-4781	256	12	simple	simple	ADJ
cana-4781	256	13	rough	rough	ADJ
cana-4781	256	14	group	group	NOUN
cana-4781	256	15	and	and	CCONJ
cana-4781	256	16	let	let	VERB
cana-4781	256	17	ℒ	ℒ	NOUN
cana-4781	256	18	be	be	AUX
cana-4781	256	19	a	a	DET
cana-4781	256	20	normal	normal	ADJ
cana-4781	256	21	rough	rough	ADJ
cana-4781	256	22	subgroup	subgroup	NOUN
cana-4781	256	23	of	of	ADP
cana-4781	256	24	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	256	25	̅̅	̅̅	PROPN
cana-4781	256	26	̅̅	̅̅	PROPN
cana-4781	256	27	.	.	PUNCT
cana-4781	257	1	for	for	ADP
cana-4781	257	2	any	any	DET
cana-4781	257	3	topological	topological	ADJ
cana-4781	257	4	rough	rough	ADJ
cana-4781	257	5	subgroup	subgroup	NOUN
cana-4781	257	6	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	257	7	of	of	ADP
cana-4781	257	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	257	9	,	,	PUNCT
cana-4781	257	10	if	if	SCONJ
cana-4781	257	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	257	12	̅̅	̅̅	PROPN
cana-4781	257	13	̅̅	̅̅	PROPN
cana-4781	257	14	,	,	PUNCT
cana-4781	257	15	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	257	16	̅̅	̅̅	PROPN
cana-4781	257	17	̅̅	̅̅	PROPN
cana-4781	257	18	and	and	CCONJ
cana-4781	257	19	ℒ̅	ℒ̅	PROPN
cana-4781	257	20	are	be	AUX
cana-4781	257	21	groups	group	NOUN
cana-4781	257	22	,	,	PUNCT
cana-4781	257	23	where	where	SCONJ
cana-4781	257	24	ℒ̅	ℒ̅	PROPN
cana-4781	257	25	is	be	AUX
cana-4781	257	26	the	the	DET
cana-4781	257	27	upper	upper	ADJ
cana-4781	257	28	approximation	approximation	NOUN
cana-4781	257	29	of	of	ADP
cana-4781	257	30	ℒ	ℒ	NOUN
cana-4781	257	31	,	,	PUNCT
cana-4781	257	32	then	then	ADV
cana-4781	257	33	the	the	DET
cana-4781	257	34	rough	rough	ADJ
cana-4781	257	35	quotient	quotient	NOUN
cana-4781	257	36	map	map	NOUN
cana-4781	257	37	𝛾	𝛾	NOUN
cana-4781	257	38	:	:	PUNCT
cana-4781	257	39	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	257	40	̅̅	̅̅	PROPN
cana-4781	257	41	̅̅	̅̅	NOUN
cana-4781	257	42	ℒ	ℒ	PROPN
cana-4781	257	43	ℒ⁄	ℒ⁄	ADJ
cana-4781	257	44	→	→	SYM
cana-4781	257	45	𝜑(𝐻ℜ	𝜑(𝐻ℜ	NOUN
cana-4781	257	46	̅̅	̅̅	NOUN
cana-4781	257	47	̅̅	̅̅	PROPN
cana-4781	257	48	)	)	PUNCT
cana-4781	257	49	is	be	AUX
cana-4781	257	50	a	a	DET
cana-4781	257	51	topological	topological	ADJ
cana-4781	257	52	rough	rough	ADJ
cana-4781	257	53	group	group	NOUN
cana-4781	257	54	homeomorphism	homeomorphism	NOUN
cana-4781	257	55	,	,	PUNCT
cana-4781	257	56	where	where	SCONJ
cana-4781	257	57	𝜑	𝜑	NOUN
cana-4781	257	58	:	:	PUNCT
cana-4781	257	59	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	257	60	̅̅	̅̅	PROPN
cana-4781	257	61	̅̅	̅̅	PROPN
cana-4781	257	62	→	→	PUNCT
cana-4781	257	63	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	257	64	̅̅	̅̅	PROPN
cana-4781	257	65	̅̅	̅̅	PROPN
cana-4781	257	66	ℒ⁄	ℒ⁄	PROPN
cana-4781	257	67	is	be	AUX
cana-4781	257	68	a	a	DET
cana-4781	257	69	rough	rough	ADJ
cana-4781	257	70	quotient	quotient	NOUN
cana-4781	257	71	map	map	NOUN
cana-4781	257	72	and	and	CCONJ
cana-4781	257	73	𝜑(𝐻ℜ	𝜑(𝐻ℜ	NOUN
cana-4781	257	74	̅̅	̅̅	PROPN
cana-4781	257	75	̅̅	̅̅	PROPN
cana-4781	257	76	)	)	PUNCT
cana-4781	257	77	is	be	AUX
cana-4781	257	78	a	a	DET
cana-4781	257	79	subgroup	subgroup	NOUN
cana-4781	257	80	of	of	ADP
cana-4781	257	81	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	257	82	̅̅	̅̅	PROPN
cana-4781	257	83	̅̅	̅̅	PROPN
cana-4781	257	84	ℒ⁄	ℒ⁄	PROPN
cana-4781	257	85	.	.	PUNCT
cana-4781	258	1	proof	proof	NOUN
cana-4781	258	2	:	:	PUNCT
cana-4781	258	3	by	by	ADP
cana-4781	258	4	our	our	PRON
cana-4781	258	5	assumption	assumption	NOUN
cana-4781	258	6	,	,	PUNCT
cana-4781	258	7	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	258	8	̅̅	̅̅	NOUN
cana-4781	258	9	̅̅	̅̅	NOUN
cana-4781	258	10	ℒ	ℒ	PROPN
cana-4781	258	11	=	=	PUNCT
cana-4781	258	12	𝜑−1(𝜑(𝐻ℜ	𝜑−1(𝜑(𝐻ℜ	PROPN
cana-4781	258	13	̅̅	̅̅	NOUN
cana-4781	258	14	̅̅	̅̅	PROPN
cana-4781	258	15	)	)	PUNCT
cana-4781	258	16	)	)	PUNCT
cana-4781	258	17	.	.	PUNCT
cana-4781	259	1	let	let	VERB
cana-4781	259	2	𝜇	𝜇	VERB
cana-4781	259	3	:	:	PUNCT
cana-4781	259	4	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	259	5	̅̅	̅̅	PROPN
cana-4781	259	6	̅̅	̅̅	NOUN
cana-4781	259	7	ℒ	ℒ	PROPN
cana-4781	259	8	→	→	SYM
cana-4781	259	9	𝜑(𝐻ℜ	𝜑(𝐻ℜ	PROPN
cana-4781	259	10	̅̅	̅̅	NOUN
cana-4781	259	11	̅̅	̅̅	PROPN
cana-4781	259	12	)	)	PUNCT
cana-4781	259	13	defined	define	VERB
cana-4781	259	14	by	by	ADP
cana-4781	259	15	𝜇(𝑎ℒ	𝜇(𝑎ℒ	PROPN
cana-4781	259	16	)	)	PUNCT
cana-4781	259	17	=	=	SYM
cana-4781	259	18	𝜑(𝑎	𝜑(𝑎	NOUN
cana-4781	259	19	)	)	PUNCT
cana-4781	259	20	.	.	PUNCT
cana-4781	260	1	since	since	SCONJ
cana-4781	260	2	𝜑	𝜑	PROPN
cana-4781	260	3	is	be	AUX
cana-4781	260	4	homomorphism	homomorphism	NOUN
cana-4781	260	5	,	,	PUNCT
cana-4781	260	6	the	the	DET
cana-4781	260	7	rough	rough	ADJ
cana-4781	260	8	quotient	quotient	NOUN
cana-4781	260	9	map	map	NOUN
cana-4781	260	10	𝜇	𝜇	ADP
cana-4781	260	11	is	be	AUX
cana-4781	260	12	homomorphism	homomorphism	NOUN
cana-4781	260	13	.	.	PUNCT
cana-4781	261	1	then	then	ADV
cana-4781	261	2	the	the	DET
cana-4781	261	3	rough	rough	ADJ
cana-4781	261	4	kernel	kernel	NOUN
cana-4781	261	5	of	of	ADP
cana-4781	261	6	𝜇	𝜇	ADP
cana-4781	261	7	,	,	PUNCT
cana-4781	261	8	𝒦𝜇	𝒦𝜇	PROPN
cana-4781	261	9	=	=	X
cana-4781	261	10	{	{	PUNCT
cana-4781	261	11	𝑎	𝑎	NOUN
cana-4781	261	12	∈	∈	PRON
cana-4781	261	13	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	261	14	̅̅	̅̅	NOUN
cana-4781	261	15	̅̅	̅̅	NOUN
cana-4781	261	16	ℒ	ℒ	PROPN
cana-4781	261	17	∶	∶	PROPN
cana-4781	261	18	𝜇(𝑎ℒ	𝜇(𝑎ℒ	PROPN
cana-4781	261	19	)	)	PUNCT
cana-4781	262	1	=	=	SYM
cana-4781	262	2	𝑒	𝑒	X
cana-4781	262	3	,	,	PUNCT
cana-4781	262	4	is	be	AUX
cana-4781	262	5	the	the	DET
cana-4781	262	6	identity	identity	NOUN
cana-4781	262	7	of	of	ADP
cana-4781	262	8	𝜑(𝐻ℜ	𝜑(𝐻ℜ	PROPN
cana-4781	262	9	̅̅	̅̅	NOUN
cana-4781	262	10	̅̅	̅̅	PROPN
cana-4781	262	11	)	)	PUNCT
cana-4781	262	12	}	}	PUNCT
cana-4781	262	13	.	.	PUNCT
cana-4781	263	1	but	but	CCONJ
cana-4781	263	2	the	the	DET
cana-4781	263	3	identity	identity	NOUN
cana-4781	263	4	of	of	ADP
cana-4781	263	5	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	263	6	̅̅	̅̅	PROPN
cana-4781	263	7	̅̅	̅̅	PROPN
cana-4781	263	8	ℒ⁄	ℒ⁄	PROPN
cana-4781	263	9	is	be	AUX
cana-4781	263	10	ℒ	ℒ	PROPN
cana-4781	263	11	and	and	CCONJ
cana-4781	263	12	𝜑(𝐻ℜ	𝜑(𝐻ℜ	PROPN
cana-4781	263	13	̅̅	̅̅	PROPN
cana-4781	263	14	̅̅	̅̅	PROPN
cana-4781	263	15	)	)	PUNCT
cana-4781	263	16	is	be	AUX
cana-4781	263	17	a	a	DET
cana-4781	263	18	subgroup	subgroup	NOUN
cana-4781	263	19	of	of	ADP
cana-4781	263	20	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	263	21	̅̅	̅̅	PROPN
cana-4781	263	22	̅̅	̅̅	PROPN
cana-4781	263	23	ℒ⁄	ℒ⁄	PROPN
cana-4781	263	24	.	.	PUNCT
cana-4781	264	1	so	so	ADV
cana-4781	264	2	,	,	PUNCT
cana-4781	264	3	𝒦𝜇	𝒦𝜇	PROPN
cana-4781	264	4	=	=	SYM
cana-4781	264	5	{	{	PUNCT
cana-4781	264	6	𝑎ℒ	𝑎ℒ	PROPN
cana-4781	264	7	∈	∈	PROPN
cana-4781	264	8	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	264	9	̅̅	̅̅	NOUN
cana-4781	264	10	̅̅	̅̅	NOUN
cana-4781	264	11	ℒ	ℒ	PROPN
cana-4781	264	12	∶	∶	PROPN
cana-4781	264	13	𝜑(𝑎	𝜑(𝑎	NOUN
cana-4781	264	14	)	)	PUNCT
cana-4781	264	15	=	=	SYM
cana-4781	264	16	ℒ	ℒ	VERB
cana-4781	264	17	}	}	PUNCT
cana-4781	264	18	=	=	SYM
cana-4781	264	19	ℒ.	ℒ.	PROPN
cana-4781	264	20	therefore	therefore	ADV
cana-4781	264	21	,	,	PUNCT
cana-4781	264	22	by	by	ADP
cana-4781	264	23	the	the	DET
cana-4781	264	24	theorem	theorem	NOUN
cana-4781	264	25	5.1	5.1	NUM
cana-4781	264	26	,	,	PUNCT
cana-4781	264	27	the	the	DET
cana-4781	264	28	rough	rough	ADJ
cana-4781	264	29	quotient	quotient	NOUN
cana-4781	264	30	map	map	NOUN
cana-4781	264	31	𝛾	𝛾	NOUN
cana-4781	264	32	:	:	PUNCT
cana-4781	264	33	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	264	34	̅̅	̅̅	PROPN
cana-4781	264	35	̅̅	̅̅	NOUN
cana-4781	264	36	ℒ	ℒ	PROPN
cana-4781	264	37	ℒ⁄	ℒ⁄	ADJ
cana-4781	264	38	→	→	SYM
cana-4781	264	39	𝜑(𝐻ℜ	𝜑(𝐻ℜ	NOUN
cana-4781	264	40	̅̅	̅̅	NOUN
cana-4781	264	41	̅̅	̅̅	PROPN
cana-4781	264	42	)	)	PUNCT
cana-4781	264	43	is	be	AUX
cana-4781	264	44	a	a	DET
cana-4781	264	45	topological	topological	ADJ
cana-4781	264	46	rough	rough	ADJ
cana-4781	264	47	group	group	NOUN
cana-4781	264	48	homeomorphism	homeomorphism	NOUN
cana-4781	264	49	.	.	PUNCT
cana-4781	265	1	6	6	NUM
cana-4781	265	2	.	.	X
cana-4781	265	3	rough	rough	ADJ
cana-4781	265	4	double	double	ADJ
cana-4781	265	5	coset	coset	NOUN
cana-4781	265	6	spaces	space	NOUN
cana-4781	265	7	:	:	PUNCT
cana-4781	265	8	definition	definition	NOUN
cana-4781	265	9	6.1	6.1	NUM
cana-4781	265	10	.	.	PUNCT
cana-4781	266	1	let	let	VERB
cana-4781	266	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	266	3	be	be	AUX
cana-4781	266	4	a	a	DET
cana-4781	266	5	rough	rough	ADJ
cana-4781	266	6	group	group	NOUN
cana-4781	266	7	such	such	ADJ
cana-4781	266	8	that	that	SCONJ
cana-4781	266	9	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	266	10	̅̅	̅̅	PROPN
cana-4781	266	11	̅̅	̅̅	PROPN
cana-4781	266	12	is	be	AUX
cana-4781	266	13	a	a	DET
cana-4781	266	14	group	group	NOUN
cana-4781	266	15	and	and	CCONJ
cana-4781	266	16	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	266	17	,	,	PUNCT
cana-4781	266	18	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	266	19	be	be	VERB
cana-4781	266	20	rough	rough	ADJ
cana-4781	266	21	subgroups	subgroup	NOUN
cana-4781	266	22	in	in	ADP
cana-4781	266	23	𝐺ℜ.	𝐺ℜ.	NOUN
cana-4781	266	24	if	if	SCONJ
cana-4781	266	25	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	266	26	,	,	PUNCT
cana-4781	266	27	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	266	28	be	be	AUX
cana-4781	266	29	subgroups	subgroup	NOUN
cana-4781	266	30	in	in	ADP
cana-4781	266	31	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	266	32	̅̅	̅̅	PROPN
cana-4781	266	33	̅̅	̅̅	PROPN
cana-4781	266	34	,	,	PUNCT
cana-4781	266	35	then	then	ADV
cana-4781	266	36	𝐾ℜ\𝐺ℜ	𝐾ℜ\𝐺ℜ	PROPN
cana-4781	266	37	̅̅	̅̅	PROPN
cana-4781	266	38	̅̅	̅̅	PROPN
cana-4781	266	39	/𝐻ℜ	/𝐻ℜ	PROPN
cana-4781	266	40	=	=	SYM
cana-4781	267	1	{	{	PUNCT
cana-4781	267	2	𝐾ℜ𝑥𝐻ℜ	𝐾ℜ𝑥𝐻ℜ	NUM
cana-4781	267	3	∶	∶	NOUN
cana-4781	267	4	𝑥	𝑥	X
cana-4781	267	5	∈	∈	PROPN
cana-4781	267	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	267	7	̅̅	̅̅	NOUN
cana-4781	267	8	̅̅	̅̅	PROPN
cana-4781	267	9	}	}	PUNCT
cana-4781	267	10	is	be	AUX
cana-4781	267	11	a	a	DET
cana-4781	267	12	rough	rough	ADJ
cana-4781	267	13	double	double	ADJ
cana-4781	267	14	coset	coset	NOUN
cana-4781	267	15	space	space	NOUN
cana-4781	267	16	(	(	PUNCT
cana-4781	267	17	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	267	18	)	)	PUNCT
cana-4781	267	19	.	.	PUNCT
cana-4781	268	1	also	also	ADV
cana-4781	268	2	,	,	PUNCT
cana-4781	268	3	for	for	ADP
cana-4781	268	4	any	any	DET
cana-4781	268	5	𝑥	𝑥	PROPN
cana-4781	268	6	,	,	PUNCT
cana-4781	268	7	𝑦	𝑦	PROPN
cana-4781	268	8	∈	∈	PROPN
cana-4781	268	9	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	268	10	̅̅	̅̅	NOUN
cana-4781	268	11	̅̅	̅̅	PROPN
cana-4781	268	12	,	,	PUNCT
cana-4781	268	13	either	either	CCONJ
cana-4781	268	14	𝐾ℜ𝑥𝐻ℜ	𝐾ℜ𝑥𝐻ℜ	X
cana-4781	268	15	=	=	SYM
cana-4781	268	16	𝐾ℜ𝑦𝐻ℜ	𝐾ℜ𝑦𝐻ℜ	PROPN
cana-4781	268	17	or	or	CCONJ
cana-4781	268	18	𝐾ℜ𝑥𝐻ℜ	𝐾ℜ𝑥𝐻ℜ	VERB
cana-4781	268	19	∩	∩	X
cana-4781	268	20	𝐾ℜ𝑦𝐻ℜ	𝐾ℜ𝑦𝐻ℜ	X
cana-4781	268	21	=	=	PUNCT
cana-4781	268	22	∅.	∅.	NOUN
cana-4781	268	23	communications	communication	NOUN
cana-4781	268	24	on	on	ADP
cana-4781	268	25	applied	apply	VERB
cana-4781	268	26	nonlinear	nonlinear	ADJ
cana-4781	268	27	analysis	analysis	NOUN
cana-4781	268	28	issn	issn	NOUN
cana-4781	268	29	:	:	PUNCT
cana-4781	268	30	1074	1074	NUM
cana-4781	268	31	-	-	PUNCT
cana-4781	268	32	133x	133x	NUM
cana-4781	268	33	vol	vol	NOUN
cana-4781	268	34	32	32	NUM
cana-4781	268	35	no	no	NOUN
cana-4781	268	36	.	.	NOUN
cana-4781	268	37	3	3	NUM
cana-4781	268	38	(	(	PUNCT
cana-4781	268	39	2025	2025	NUM
cana-4781	268	40	)	)	PUNCT
cana-4781	268	41	887	887	NUM
cana-4781	268	42	https://internationalpubls.com	https://internationalpubls.com	X
cana-4781	268	43	lemma	lemma	PROPN
cana-4781	268	44	6.2	6.2	NUM
cana-4781	268	45	.	.	PUNCT
cana-4781	269	1	let	let	VERB
cana-4781	269	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	269	3	be	be	AUX
cana-4781	269	4	a	a	DET
cana-4781	269	5	rough	rough	ADJ
cana-4781	269	6	group	group	NOUN
cana-4781	269	7	such	such	ADJ
cana-4781	269	8	that	that	SCONJ
cana-4781	269	9	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	269	10	̅̅	̅̅	PROPN
cana-4781	269	11	̅̅	̅̅	PROPN
cana-4781	269	12	is	be	AUX
cana-4781	269	13	a	a	DET
cana-4781	269	14	group	group	NOUN
cana-4781	269	15	and	and	CCONJ
cana-4781	269	16	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	269	17	,	,	PUNCT
cana-4781	269	18	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	269	19	be	be	VERB
cana-4781	269	20	rough	rough	ADJ
cana-4781	269	21	subgroups	subgroup	NOUN
cana-4781	269	22	in	in	ADP
cana-4781	269	23	𝐺ℜ.	𝐺ℜ.	NOUN
cana-4781	269	24	if	if	SCONJ
cana-4781	269	25	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	269	26	,	,	PUNCT
cana-4781	269	27	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	269	28	be	be	AUX
cana-4781	269	29	subgroups	subgroup	NOUN
cana-4781	269	30	in	in	ADP
cana-4781	269	31	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	269	32	̅̅	̅̅	PROPN
cana-4781	269	33	̅̅	̅̅	PROPN
cana-4781	269	34	,	,	PUNCT
cana-4781	269	35	then	then	ADV
cana-4781	269	36	all	all	DET
cana-4781	269	37	the	the	DET
cana-4781	269	38	double	double	ADJ
cana-4781	269	39	cosets	coset	NOUN
cana-4781	269	40	form	form	VERB
cana-4781	269	41	a	a	DET
cana-4781	269	42	partition	partition	NOUN
cana-4781	269	43	of	of	ADP
cana-4781	269	44	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	269	45	̅̅	̅̅	PROPN
cana-4781	269	46	̅̅	̅̅	PROPN
cana-4781	269	47	.	.	PUNCT
cana-4781	270	1	proof	proof	NOUN
cana-4781	270	2	:	:	PUNCT
cana-4781	270	3	let	let	VERB
cana-4781	270	4	𝑥	𝑥	PRON
cana-4781	270	5	,	,	PUNCT
cana-4781	270	6	𝑦	𝑦	PRON
cana-4781	270	7	∈	∈	PROPN
cana-4781	270	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	270	9	̅̅	̅̅	PROPN
cana-4781	270	10	̅̅	̅̅	PROPN
cana-4781	270	11	.	.	PUNCT
cana-4781	271	1	then	then	ADV
cana-4781	271	2	the	the	DET
cana-4781	271	3	corresponding	correspond	VERB
cana-4781	271	4	cosets	coset	NOUN
cana-4781	271	5	are	be	AUX
cana-4781	271	6	𝐾ℜ𝑥𝐻ℜ	𝐾ℜ𝑥𝐻ℜ	NOUN
cana-4781	271	7	,	,	PUNCT
cana-4781	271	8	𝐾ℜ𝑦𝐻ℜ.	𝐾ℜ𝑦𝐻ℜ.	PROPN
cana-4781	271	9	these	these	PRON
cana-4781	271	10	are	be	AUX
cana-4781	271	11	either	either	CCONJ
cana-4781	271	12	disjoint	disjoint	NOUN
cana-4781	271	13	or	or	CCONJ
cana-4781	271	14	coincide	coincide	NOUN
cana-4781	271	15	.	.	PUNCT
cana-4781	272	1	consider	consider	VERB
cana-4781	272	2	an	an	DET
cana-4781	272	3	arbitrary	arbitrary	ADJ
cana-4781	272	4	element	element	NOUN
cana-4781	272	5	𝑧	𝑧	PRON
cana-4781	272	6	∈	∈	PROPN
cana-4781	272	7	𝐾ℜ𝑥𝐻ℜ	𝐾ℜ𝑥𝐻ℜ	NUM
cana-4781	272	8	∩	∩	PROPN
cana-4781	272	9	𝐾ℜ𝑦𝐻ℜ	𝐾ℜ𝑦𝐻ℜ	PROPN
cana-4781	272	10	,	,	PUNCT
cana-4781	272	11	that	that	ADV
cana-4781	272	12	is	is	ADV
cana-4781	272	13	,	,	PUNCT
cana-4781	272	14	𝑧	𝑧	PROPN
cana-4781	272	15	=	=	PUNCT
cana-4781	272	16	𝑘1𝑥ℎ1	𝑘1𝑥ℎ1	ADJ
cana-4781	272	17	=	=	SYM
cana-4781	272	18	𝑘2𝑦ℎ2	𝑘2𝑦ℎ2	PROPN
cana-4781	272	19	,	,	PUNCT
cana-4781	272	20	for	for	ADP
cana-4781	272	21	some	some	DET
cana-4781	272	22	𝑘1	𝑘1	NOUN
cana-4781	272	23	,	,	PUNCT
cana-4781	272	24	𝑘2	𝑘2	PROPN
cana-4781	272	25	∈	∈	PROPN
cana-4781	272	26	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	272	27	and	and	CCONJ
cana-4781	272	28	ℎ1	ℎ1	PROPN
cana-4781	272	29	,	,	PUNCT
cana-4781	272	30	ℎ2	ℎ2	ADJ
cana-4781	272	31	∈	∈	PROPN
cana-4781	272	32	𝐻ℜ.	𝐻ℜ.	X
cana-4781	272	33	therefore	therefore	ADV
cana-4781	272	34	,	,	PUNCT
cana-4781	272	35	𝑥	𝑥	PROPN
cana-4781	272	36	∈	∈	PROPN
cana-4781	272	37	𝐾ℜ𝑦𝐻ℜ	𝐾ℜ𝑦𝐻ℜ	PROPN
cana-4781	272	38	which	which	PRON
cana-4781	272	39	implies	imply	VERB
cana-4781	272	40	𝐾ℜ𝑥𝐻ℜ	𝐾ℜ𝑥𝐻ℜ	NUM
cana-4781	272	41	⊆	⊆	NUM
cana-4781	272	42	𝐾ℜ𝑦𝐻ℜ.	𝐾ℜ𝑦𝐻ℜ.	PROPN
cana-4781	272	43	similarly	similarly	ADV
cana-4781	272	44	,	,	PUNCT
cana-4781	272	45	we	we	PRON
cana-4781	272	46	can	can	AUX
cana-4781	272	47	prove	prove	VERB
cana-4781	272	48	𝐾ℜ𝑦𝐻ℜ	𝐾ℜ𝑦𝐻ℜ	PRON
cana-4781	272	49	⊆	⊆	NUM
cana-4781	272	50	𝐾ℜ𝑥𝐻ℜ.	𝐾ℜ𝑥𝐻ℜ.	NOUN
cana-4781	272	51	hence	hence	ADV
cana-4781	272	52	,	,	PUNCT
cana-4781	272	53	𝐾ℜ𝑥𝐻ℜ	𝐾ℜ𝑥𝐻ℜ	PUNCT
cana-4781	272	54	=	=	SYM
cana-4781	272	55	𝐾ℜ𝑦𝐻ℜ.	𝐾ℜ𝑦𝐻ℜ.	PROPN
cana-4781	272	56	lemma	lemma	PROPN
cana-4781	272	57	6.3	6.3	NUM
cana-4781	272	58	.	.	PUNCT
cana-4781	273	1	let	let	VERB
cana-4781	273	2	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	273	3	and	and	CCONJ
cana-4781	273	4	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	273	5	be	be	AUX
cana-4781	273	6	rough	rough	ADJ
cana-4781	273	7	subgroups	subgroup	NOUN
cana-4781	273	8	of	of	ADP
cana-4781	273	9	a	a	DET
cana-4781	273	10	topological	topological	ADJ
cana-4781	273	11	simple	simple	ADJ
cana-4781	273	12	rough	rough	ADJ
cana-4781	273	13	group	group	NOUN
cana-4781	273	14	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	273	15	such	such	ADJ
cana-4781	273	16	that	that	SCONJ
cana-4781	273	17	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	273	18	̅̅	̅̅	PROPN
cana-4781	273	19	̅̅	̅̅	PROPN
cana-4781	273	20	is	be	AUX
cana-4781	273	21	a	a	DET
cana-4781	273	22	group	group	NOUN
cana-4781	273	23	and	and	CCONJ
cana-4781	273	24	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	273	25	,	,	PUNCT
cana-4781	273	26	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	273	27	are	be	AUX
cana-4781	273	28	subgroups	subgroup	NOUN
cana-4781	273	29	of	of	ADP
cana-4781	273	30	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	273	31	̅̅	̅̅	PROPN
cana-4781	273	32	̅̅	̅̅	PROPN
cana-4781	273	33	.	.	PUNCT
cana-4781	274	1	if	if	SCONJ
cana-4781	274	2	𝐴	𝐴	PROPN
cana-4781	274	3	is	be	AUX
cana-4781	274	4	a	a	DET
cana-4781	274	5	compact	compact	ADJ
cana-4781	274	6	subset	subset	NOUN
cana-4781	274	7	in	in	ADP
cana-4781	274	8	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	274	9	and	and	CCONJ
cana-4781	274	10	the	the	DET
cana-4781	274	11	rough	rough	ADJ
cana-4781	274	12	quotient	quotient	NOUN
cana-4781	274	13	map	map	NOUN
cana-4781	274	14	𝜑	𝜑	NOUN
cana-4781	274	15	:	:	PUNCT
cana-4781	274	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	274	17	̅̅	̅̅	PROPN
cana-4781	274	18	̅̅	̅̅	PROPN
cana-4781	274	19	→	→	PUNCT
cana-4781	274	20	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	274	21	is	be	AUX
cana-4781	274	22	defined	define	VERB
cana-4781	274	23	by	by	ADP
cana-4781	274	24	𝜑(𝑥	𝜑(𝑥	NOUN
cana-4781	274	25	)	)	PUNCT
cana-4781	274	26	=	=	PUNCT
cana-4781	275	1	𝐾ℜ𝑥𝐻ℜ	𝐾ℜ𝑥𝐻ℜ	PROPN
cana-4781	275	2	,	,	PUNCT
cana-4781	275	3	for	for	ADP
cana-4781	275	4	𝑥	𝑥	PROPN
cana-4781	275	5	∈	∈	PROPN
cana-4781	275	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	275	7	̅̅	̅̅	PROPN
cana-4781	275	8	̅̅	̅̅	PROPN
cana-4781	275	9	,	,	PUNCT
cana-4781	275	10	then	then	ADV
cana-4781	275	11	there	there	PRON
cana-4781	275	12	exists	exist	VERB
cana-4781	275	13	a	a	DET
cana-4781	275	14	compact	compact	ADJ
cana-4781	275	15	subset	subset	NOUN
cana-4781	275	16	𝐵	𝐵	PROPN
cana-4781	275	17	in	in	ADP
cana-4781	275	18	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	275	19	̅̅	̅̅	NOUN
cana-4781	275	20	̅̅	̅̅	PROPN
cana-4781	275	21	such	such	ADJ
cana-4781	275	22	that	that	PRON
cana-4781	275	23	𝜑(𝐵	𝜑(𝐵	NUM
cana-4781	275	24	)	)	PUNCT
cana-4781	275	25	=	=	SYM
cana-4781	276	1	𝐴.	𝐴.	NOUN
cana-4781	276	2	proof	proof	NOUN
cana-4781	276	3	:	:	PUNCT
cana-4781	276	4	let	let	VERB
cana-4781	276	5	𝑈	𝑈	PROPN
cana-4781	276	6	be	be	AUX
cana-4781	276	7	an	an	DET
cana-4781	276	8	identity	identity	NOUN
cana-4781	276	9	neighbourhood	neighbourhood	NOUN
cana-4781	276	10	in	in	ADP
cana-4781	276	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	276	12	̅̅	̅̅	PROPN
cana-4781	276	13	̅̅	̅̅	PROPN
cana-4781	276	14	which	which	PRON
cana-4781	276	15	has	have	VERB
cana-4781	276	16	a	a	DET
cana-4781	276	17	compact	compact	ADJ
cana-4781	276	18	closure	closure	NOUN
cana-4781	276	19	.	.	PUNCT
cana-4781	277	1	since	since	SCONJ
cana-4781	277	2	𝐴	𝐴	PROPN
cana-4781	277	3	is	be	AUX
cana-4781	277	4	compact	compact	ADJ
cana-4781	277	5	,	,	PUNCT
cana-4781	277	6	there	there	PRON
cana-4781	277	7	exists	exist	VERB
cana-4781	277	8	a	a	DET
cana-4781	277	9	cover	cover	NOUN
cana-4781	277	10	⋃	⋃	PUNCT
cana-4781	277	11	𝜑(𝑥𝑖𝑈)𝑛	𝜑(𝑥𝑖𝑈)𝑛	PROPN
cana-4781	277	12	𝑖=1	𝑖=1	PROPN
cana-4781	277	13	,	,	PUNCT
cana-4781	277	14	where	where	SCONJ
cana-4781	277	15	𝑥1	𝑥1	NOUN
cana-4781	277	16	,	,	PUNCT
cana-4781	277	17	𝑥2	𝑥2	NOUN
cana-4781	277	18	,	,	PUNCT
cana-4781	277	19	…	…	PUNCT
cana-4781	277	20	𝑥𝑛	𝑥𝑛	PROPN
cana-4781	277	21	∈	∈	PROPN
cana-4781	277	22	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	277	23	̅̅	̅̅	PROPN
cana-4781	277	24	̅̅	̅̅	PROPN
cana-4781	277	25	.	.	PUNCT
cana-4781	278	1	that	that	PRON
cana-4781	278	2	is	be	AUX
cana-4781	278	3	,	,	PUNCT
cana-4781	278	4	𝐴	𝐴	PROPN
cana-4781	278	5	⊆	⊆	NUM
cana-4781	278	6	⋃	⋃	PUNCT
cana-4781	278	7	𝜑(𝑥𝑖𝑈)𝑛	𝜑(𝑥𝑖𝑈)𝑛	PROPN
cana-4781	278	8	𝑖=1	𝑖=1	PUNCT
cana-4781	278	9	.	.	PUNCT
cana-4781	279	1	now	now	ADV
cana-4781	279	2	consider	consider	VERB
cana-4781	279	3	𝐵	𝐵	NOUN
cana-4781	279	4	=	=	PUNCT
cana-4781	279	5	𝜑−1(𝐴	𝜑−1(𝐴	PROPN
cana-4781	279	6	)	)	PUNCT
cana-4781	279	7	∩	∩	NOUN
cana-4781	279	8	⋃	⋃	PROPN
cana-4781	279	9	𝑥𝑖𝑐𝑙(𝑈)𝑛	𝑥𝑖𝑐𝑙(𝑈)𝑛	PROPN
cana-4781	279	10	𝑖=1	𝑖=1	PROPN
cana-4781	279	11	,	,	PUNCT
cana-4781	279	12	𝑐𝑙(𝑈	𝑐𝑙(𝑈	NOUN
cana-4781	279	13	)	)	PUNCT
cana-4781	279	14	means	mean	VERB
cana-4781	279	15	closure	closure	NOUN
cana-4781	279	16	of	of	ADP
cana-4781	279	17	𝑈.	𝑈.	PROPN
cana-4781	279	18	since	since	SCONJ
cana-4781	279	19	𝑐𝑙(𝑈	𝑐𝑙(𝑈	NOUN
cana-4781	279	20	)	)	PUNCT
cana-4781	279	21	is	be	AUX
cana-4781	279	22	compact	compact	ADJ
cana-4781	279	23	,	,	PUNCT
cana-4781	279	24	𝜑−1(𝐴	𝜑−1(𝐴	PRON
cana-4781	279	25	)	)	PUNCT
cana-4781	279	26	lies	lie	VERB
cana-4781	279	27	in	in	ADP
cana-4781	279	28	the	the	DET
cana-4781	279	29	compact	compact	ADJ
cana-4781	279	30	set	set	NOUN
cana-4781	279	31	⋃	⋃	PUNCT
cana-4781	279	32	𝑥𝑖𝑐𝑙(𝑈)𝑛	𝑥𝑖𝑐𝑙(𝑈)𝑛	NOUN
cana-4781	279	33	𝑖=1	𝑖=1	PROPN
cana-4781	279	34	.	.	PUNCT
cana-4781	280	1	therefore	therefore	ADV
cana-4781	280	2	,	,	PUNCT
cana-4781	280	3	𝐵	𝐵	NOUN
cana-4781	280	4	is	be	AUX
cana-4781	280	5	compact	compact	ADJ
cana-4781	280	6	and	and	CCONJ
cana-4781	280	7	𝜑(𝐵	𝜑(𝐵	NUM
cana-4781	280	8	)	)	PUNCT
cana-4781	280	9	=	=	SYM
cana-4781	280	10	𝜑(𝜑−1(𝐴	𝜑(𝜑−1(𝐴	NOUN
cana-4781	280	11	)	)	PUNCT
cana-4781	280	12	∩	∩	NOUN
cana-4781	280	13	⋃	⋃	PROPN
cana-4781	280	14	𝑥𝑖𝑐𝑙(𝑈)𝑛	𝑥𝑖𝑐𝑙(𝑈)𝑛	NOUN
cana-4781	280	15	𝑖=1	𝑖=1	PUNCT
cana-4781	280	16	)	)	PUNCT
cana-4781	281	1	=	=	SYM
cana-4781	281	2	𝐴	𝐴	PROPN
cana-4781	281	3	∩	∩	NOUN
cana-4781	281	4	⋃	⋃	PUNCT
cana-4781	281	5	𝜑(𝑥𝑖𝑐𝑙(𝑈))𝑛	𝜑(𝑥𝑖𝑐𝑙(𝑈))𝑛	PROPN
cana-4781	281	6	𝑖=1	𝑖=1	PROPN
cana-4781	281	7	=	=	SYM
cana-4781	281	8	𝐴.	𝐴.	PROPN
cana-4781	281	9	lemma	lemma	PROPN
cana-4781	281	10	6.4	6.4	NUM
cana-4781	281	11	.	.	PUNCT
cana-4781	282	1	let	let	VERB
cana-4781	282	2	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	282	3	be	be	AUX
cana-4781	282	4	a	a	DET
cana-4781	282	5	topological	topological	ADJ
cana-4781	282	6	simple	simple	ADJ
cana-4781	282	7	rough	rough	ADJ
cana-4781	282	8	group	group	NOUN
cana-4781	282	9	such	such	ADJ
cana-4781	282	10	that	that	SCONJ
cana-4781	282	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	282	12	̅̅	̅̅	PROPN
cana-4781	282	13	̅̅	̅̅	PROPN
cana-4781	282	14	is	be	AUX
cana-4781	282	15	a	a	DET
cana-4781	282	16	group	group	NOUN
cana-4781	282	17	and	and	CCONJ
cana-4781	282	18	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	282	19	is	be	AUX
cana-4781	282	20	open	open	ADJ
cana-4781	282	21	in	in	ADP
cana-4781	282	22	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	282	23	̅̅	̅̅	PROPN
cana-4781	282	24	̅̅	̅̅	PROPN
cana-4781	282	25	.	.	PUNCT
cana-4781	283	1	suppose	suppose	VERB
cana-4781	283	2	𝐾ℜ	𝐾ℜ	NOUN
cana-4781	283	3	and	and	CCONJ
cana-4781	283	4	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	283	5	are	be	AUX
cana-4781	283	6	closed	close	VERB
cana-4781	283	7	in	in	ADP
cana-4781	283	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	283	9	̅̅	̅̅	PROPN
cana-4781	283	10	̅̅	̅̅	PROPN
cana-4781	283	11	and	and	CCONJ
cana-4781	283	12	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	283	13	is	be	AUX
cana-4781	283	14	compact	compact	ADJ
cana-4781	283	15	in	in	ADP
cana-4781	283	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	283	17	̅̅	̅̅	PROPN
cana-4781	283	18	̅̅	̅̅	PROPN
cana-4781	283	19	.	.	PUNCT
cana-4781	284	1	then	then	ADV
cana-4781	284	2	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	284	3	is	be	AUX
cana-4781	284	4	a	a	DET
cana-4781	284	5	closed	closed	ADJ
cana-4781	284	6	set	set	NOUN
cana-4781	284	7	in	in	ADP
cana-4781	284	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	284	9	̅̅	̅̅	PROPN
cana-4781	284	10	̅̅	̅̅	PROPN
cana-4781	284	11	.	.	PUNCT
cana-4781	285	1	proof	proof	NOUN
cana-4781	285	2	:	:	PUNCT
cana-4781	285	3	since	since	SCONJ
cana-4781	285	4	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	285	5	is	be	AUX
cana-4781	285	6	closed	close	VERB
cana-4781	285	7	in	in	ADP
cana-4781	285	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	285	9	̅̅	̅̅	PROPN
cana-4781	285	10	̅̅	̅̅	PROPN
cana-4781	285	11	,	,	PUNCT
cana-4781	285	12	𝑥𝐻ℜ	𝑥𝐻ℜ	PROPN
cana-4781	285	13	is	be	AUX
cana-4781	285	14	closed	close	VERB
cana-4781	285	15	in	in	ADP
cana-4781	285	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	285	17	̅̅	̅̅	PROPN
cana-4781	285	18	̅̅	̅̅	PROPN
cana-4781	285	19	.	.	PUNCT
cana-4781	286	1	then	then	ADV
cana-4781	286	2	using	use	VERB
cana-4781	286	3	theorem	theorem	NOUN
cana-4781	286	4	3.2	3.2	NUM
cana-4781	286	5	,	,	PUNCT
cana-4781	286	6	𝐾ℜ𝑥𝐻ℜ	𝐾ℜ𝑥𝐻ℜ	X
cana-4781	286	7	is	be	AUX
cana-4781	286	8	closed	close	VERB
cana-4781	286	9	in	in	ADP
cana-4781	286	10	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	286	11	̅̅	̅̅	NOUN
cana-4781	286	12	̅̅	̅̅	NOUN
cana-4781	286	13	that	that	PRON
cana-4781	286	14	is	be	AUX
cana-4781	286	15	,	,	PUNCT
cana-4781	286	16	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	286	17	is	be	AUX
cana-4781	286	18	a	a	DET
cana-4781	286	19	closed	closed	ADJ
cana-4781	286	20	set	set	NOUN
cana-4781	286	21	in	in	ADP
cana-4781	286	22	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	286	23	̅̅	̅̅	PROPN
cana-4781	286	24	̅̅	̅̅	PROPN
cana-4781	286	25	.	.	PUNCT
cana-4781	287	1	proposition	proposition	NOUN
cana-4781	287	2	6.5	6.5	NUM
cana-4781	287	3	.	.	PUNCT
cana-4781	288	1	let	let	VERB
cana-4781	288	2	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	288	3	and	and	CCONJ
cana-4781	288	4	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	288	5	be	be	AUX
cana-4781	288	6	rough	rough	ADJ
cana-4781	288	7	subgroups	subgroup	NOUN
cana-4781	288	8	of	of	ADP
cana-4781	288	9	a	a	DET
cana-4781	288	10	topological	topological	ADJ
cana-4781	288	11	simple	simple	ADJ
cana-4781	288	12	rough	rough	ADJ
cana-4781	288	13	group	group	NOUN
cana-4781	288	14	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	288	15	such	such	ADJ
cana-4781	288	16	that	that	SCONJ
cana-4781	288	17	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	288	18	̅̅	̅̅	PROPN
cana-4781	288	19	̅̅	̅̅	PROPN
cana-4781	288	20	is	be	AUX
cana-4781	288	21	a	a	DET
cana-4781	288	22	group	group	NOUN
cana-4781	288	23	and	and	CCONJ
cana-4781	288	24	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	288	25	,	,	PUNCT
cana-4781	288	26	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	288	27	are	be	AUX
cana-4781	288	28	subgroups	subgroup	NOUN
cana-4781	288	29	of	of	ADP
cana-4781	288	30	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	288	31	̅̅	̅̅	PROPN
cana-4781	288	32	̅̅	̅̅	PROPN
cana-4781	288	33	.	.	PUNCT
cana-4781	289	1	if	if	SCONJ
cana-4781	289	2	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	289	3	is	be	AUX
cana-4781	289	4	a	a	DET
cana-4781	289	5	compact	compact	ADJ
cana-4781	289	6	subset	subset	NOUN
cana-4781	289	7	in	in	ADP
cana-4781	289	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	289	9	̅̅	̅̅	PROPN
cana-4781	289	10	̅̅	̅̅	PROPN
cana-4781	289	11	and	and	CCONJ
cana-4781	289	12	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	289	13	is	be	AUX
cana-4781	289	14	a	a	DET
cana-4781	289	15	closed	closed	ADJ
cana-4781	289	16	subset	subset	NOUN
cana-4781	289	17	in	in	ADP
cana-4781	289	18	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	289	19	̅̅	̅̅	PROPN
cana-4781	289	20	̅̅	̅̅	PROPN
cana-4781	289	21	,	,	PUNCT
cana-4781	289	22	then	then	ADV
cana-4781	289	23	the	the	DET
cana-4781	289	24	rough	rough	ADJ
cana-4781	289	25	quotient	quotient	NOUN
cana-4781	289	26	map	map	NOUN
cana-4781	289	27	𝜑	𝜑	NOUN
cana-4781	289	28	:	:	PUNCT
cana-4781	289	29	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	289	30	̅̅	̅̅	PROPN
cana-4781	289	31	̅̅	̅̅	PROPN
cana-4781	289	32	→	→	PUNCT
cana-4781	289	33	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	289	34	is	be	AUX
cana-4781	289	35	open	open	ADJ
cana-4781	289	36	.	.	PUNCT
cana-4781	290	1	proof	proof	NOUN
cana-4781	290	2	:	:	PUNCT
cana-4781	290	3	let	let	VERB
cana-4781	290	4	𝑈	𝑈	PROPN
cana-4781	290	5	be	be	AUX
cana-4781	290	6	an	an	DET
cana-4781	290	7	open	open	ADJ
cana-4781	290	8	set	set	NOUN
cana-4781	290	9	in	in	ADP
cana-4781	290	10	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	290	11	̅̅	̅̅	PROPN
cana-4781	290	12	̅̅	̅̅	PROPN
cana-4781	290	13	.	.	PUNCT
cana-4781	291	1	then	then	ADV
cana-4781	291	2	𝜑−1𝜑(𝑈	𝜑−1𝜑(𝑈	ADV
cana-4781	291	3	)	)	PUNCT
cana-4781	291	4	=	=	SYM
cana-4781	292	1	𝐾ℜ𝑈𝐻ℜ	𝐾ℜ𝑈𝐻ℜ	PROPN
cana-4781	292	2	is	be	AUX
cana-4781	292	3	open	open	ADJ
cana-4781	292	4	in	in	ADP
cana-4781	292	5	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	292	6	̅̅	̅̅	PROPN
cana-4781	292	7	̅̅	̅̅	PROPN
cana-4781	292	8	which	which	PRON
cana-4781	292	9	implies	imply	VERB
cana-4781	292	10	𝜑(𝑈	𝜑(𝑈	ADV
cana-4781	292	11	)	)	PUNCT
cana-4781	292	12	is	be	AUX
cana-4781	292	13	open	open	ADJ
cana-4781	292	14	.	.	PUNCT
cana-4781	293	1	therefore	therefore	ADV
cana-4781	293	2	,	,	PUNCT
cana-4781	293	3	the	the	DET
cana-4781	293	4	rough	rough	ADJ
cana-4781	293	5	quotient	quotient	NOUN
cana-4781	293	6	map	map	NOUN
cana-4781	293	7	𝜑	𝜑	NOUN
cana-4781	293	8	is	be	AUX
cana-4781	293	9	open	open	ADJ
cana-4781	293	10	.	.	PUNCT
cana-4781	294	1	proposition	proposition	NOUN
cana-4781	294	2	6.6	6.6	NUM
cana-4781	294	3	.	.	PUNCT
cana-4781	295	1	let	let	VERB
cana-4781	295	2	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	295	3	and	and	CCONJ
cana-4781	295	4	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	295	5	be	be	AUX
cana-4781	295	6	rough	rough	ADJ
cana-4781	295	7	subgroups	subgroup	NOUN
cana-4781	295	8	of	of	ADP
cana-4781	295	9	a	a	DET
cana-4781	295	10	topological	topological	ADJ
cana-4781	295	11	simple	simple	ADJ
cana-4781	295	12	rough	rough	ADJ
cana-4781	295	13	group	group	NOUN
cana-4781	295	14	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	295	15	such	such	ADJ
cana-4781	295	16	that	that	SCONJ
cana-4781	295	17	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	295	18	̅̅	̅̅	PROPN
cana-4781	295	19	̅̅	̅̅	PROPN
cana-4781	295	20	is	be	AUX
cana-4781	295	21	a	a	DET
cana-4781	295	22	group	group	NOUN
cana-4781	295	23	and	and	CCONJ
cana-4781	295	24	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	295	25	,	,	PUNCT
cana-4781	295	26	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	295	27	are	be	AUX
cana-4781	295	28	subgroups	subgroup	NOUN
cana-4781	295	29	of	of	ADP
cana-4781	295	30	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	295	31	̅̅	̅̅	PROPN
cana-4781	295	32	̅̅	̅̅	PROPN
cana-4781	295	33	.	.	PUNCT
cana-4781	296	1	then	then	ADV
cana-4781	296	2	the	the	DET
cana-4781	296	3	rough	rough	ADJ
cana-4781	296	4	double	double	ADJ
cana-4781	296	5	coset	coset	NOUN
cana-4781	296	6	space	space	NOUN
cana-4781	296	7	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	296	8	is	be	AUX
cana-4781	296	9	regular	regular	ADJ
cana-4781	296	10	.	.	PUNCT
cana-4781	297	1	proof	proof	NOUN
cana-4781	297	2	:	:	PUNCT
cana-4781	297	3	consider	consider	VERB
cana-4781	297	4	an	an	DET
cana-4781	297	5	arbitrary	arbitrary	ADJ
cana-4781	297	6	point	point	NOUN
cana-4781	297	7	𝑐	𝑐	PROPN
cana-4781	297	8	∈	∈	PROPN
cana-4781	297	9	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	297	10	and	and	CCONJ
cana-4781	297	11	𝜑(𝑥	𝜑(𝑥	NOUN
cana-4781	297	12	)	)	PUNCT
cana-4781	297	13	=	=	SYM
cana-4781	297	14	𝑐	𝑐	NOUN
cana-4781	297	15	,	,	PUNCT
cana-4781	297	16	for	for	ADP
cana-4781	297	17	some	some	DET
cana-4781	297	18	𝑥	𝑥	PRON
cana-4781	297	19	∈	∈	PROPN
cana-4781	297	20	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	297	21	̅̅	̅̅	PROPN
cana-4781	297	22	̅̅	̅̅	PROPN
cana-4781	297	23	.	.	PUNCT
cana-4781	298	1	then	then	ADV
cana-4781	298	2	𝜑−1(𝑐	𝜑−1(𝑐	PROPN
cana-4781	298	3	)	)	PUNCT
cana-4781	298	4	=	=	PUNCT
cana-4781	298	5	𝐾ℜ𝑥𝐻ℜ.	𝐾ℜ𝑥𝐻ℜ.	NOUN
cana-4781	298	6	from	from	ADP
cana-4781	298	7	theorem	theorem	ADJ
cana-4781	298	8	4.2	4.2	NUM
cana-4781	298	9	,	,	PUNCT
cana-4781	298	10	𝜑−1(𝑐	𝜑−1(𝑐	PROPN
cana-4781	298	11	)	)	PUNCT
cana-4781	298	12	is	be	AUX
cana-4781	298	13	closed	close	VERB
cana-4781	298	14	which	which	PRON
cana-4781	298	15	implies	imply	VERB
cana-4781	298	16	{	{	PUNCT
cana-4781	298	17	𝑐	𝑐	X
cana-4781	298	18	}	}	PUNCT
cana-4781	298	19	is	be	AUX
cana-4781	298	20	closed	close	VERB
cana-4781	298	21	in	in	ADP
cana-4781	298	22	𝒟ℭ.	𝒟ℭ.	X
cana-4781	298	23	therefore	therefore	ADV
cana-4781	298	24	,	,	PUNCT
cana-4781	298	25	for	for	ADP
cana-4781	298	26	any	any	DET
cana-4781	298	27	𝑐	𝑐	PROPN
cana-4781	298	28	∈	∈	PROPN
cana-4781	298	29	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	298	30	,	,	PUNCT
cana-4781	298	31	the	the	DET
cana-4781	298	32	rough	rough	ADJ
cana-4781	298	33	double	double	ADJ
cana-4781	298	34	coset	coset	NOUN
cana-4781	298	35	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	298	36	is	be	AUX
cana-4781	298	37	a	a	DET
cana-4781	298	38	𝑇1space	𝑇1space	NOUN
cana-4781	298	39	.	.	PUNCT
cana-4781	299	1	let	let	VERB
cana-4781	299	2	𝑈	𝑈	NOUN
cana-4781	299	3	be	be	AUX
cana-4781	299	4	an	an	DET
cana-4781	299	5	open	open	ADJ
cana-4781	299	6	neighbourhood	neighbourhood	NOUN
cana-4781	299	7	of	of	ADP
cana-4781	299	8	𝑐	𝑐	PROPN
cana-4781	299	9	in	in	ADP
cana-4781	299	10	𝒟ℭ.	𝒟ℭ.	NOUN
cana-4781	299	11	then	then	ADV
cana-4781	299	12	there	there	PRON
cana-4781	299	13	exist	exist	VERB
cana-4781	299	14	identity	identity	NOUN
cana-4781	299	15	neighbourhoods	neighbourhood	NOUN
cana-4781	299	16	𝑉	𝑉	PROPN
cana-4781	299	17	and	and	CCONJ
cana-4781	299	18	w	w	NOUN
cana-4781	299	19	in	in	ADP
cana-4781	299	20	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	299	21	̅̅	̅̅	NOUN
cana-4781	299	22	̅̅	̅̅	PROPN
cana-4781	299	23	such	such	ADJ
cana-4781	299	24	that	that	DET
cana-4781	299	25	𝜑(𝑉𝑥	𝜑(𝑉𝑥	NOUN
cana-4781	299	26	)	)	PUNCT
cana-4781	299	27	⊆	⊆	PROPN
cana-4781	299	28	𝑈	𝑈	PROPN
cana-4781	299	29	and	and	CCONJ
cana-4781	299	30	ww	ww	PROPN
cana-4781	299	31	⊆	⊆	NUM
cana-4781	299	32	v.	v.	ADP
cana-4781	299	33	also	also	ADV
cana-4781	299	34	,	,	PUNCT
cana-4781	299	35	by	by	ADP
cana-4781	299	36	theorem	theorem	NOUN
cana-4781	299	37	3.3	3.3	NUM
cana-4781	299	38	,	,	PUNCT
cana-4781	299	39	there	there	PRON
cana-4781	299	40	exists	exist	VERB
cana-4781	299	41	a	a	DET
cana-4781	299	42	symmetric	symmetric	ADJ
cana-4781	299	43	identity	identity	NOUN
cana-4781	299	44	neighbourhood	neighbourhood	NOUN
cana-4781	299	45	𝑁	𝑁	PROPN
cana-4781	299	46	⊆	⊆	NUM
cana-4781	299	47	𝑊	𝑊	PROPN
cana-4781	299	48	in	in	ADP
cana-4781	299	49	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	299	50	̅̅	̅̅	NOUN
cana-4781	299	51	̅̅	̅̅	NOUN
cana-4781	299	52	such	such	ADJ
cana-4781	299	53	that	that	DET
cana-4781	299	54	𝑥𝑁𝑥−1	𝑥𝑁𝑥−1	NOUN
cana-4781	299	55	⊆	⊆	NUM
cana-4781	299	56	𝑊	𝑊	PROPN
cana-4781	299	57	,	,	PUNCT
cana-4781	299	58	for	for	ADP
cana-4781	299	59	every	every	DET
cana-4781	299	60	𝑥	𝑥	PROPN
cana-4781	299	61	∈	∈	PROPN
cana-4781	299	62	𝐾ℜ	𝐾ℜ	NOUN
cana-4781	299	63	which	which	PRON
cana-4781	299	64	implies	imply	VERB
cana-4781	299	65	𝑁𝐾ℜ	𝑁𝐾ℜ	PROPN
cana-4781	299	66	⊆	⊆	NUM
cana-4781	299	67	𝐾ℜ𝑊.	𝐾ℜ𝑊.	PUNCT
cana-4781	299	68	since	since	SCONJ
cana-4781	299	69	𝜑	𝜑	NOUN
cana-4781	299	70	is	be	AUX
cana-4781	299	71	an	an	DET
cana-4781	299	72	open	open	ADJ
cana-4781	299	73	mapping	mapping	NOUN
cana-4781	299	74	,	,	PUNCT
cana-4781	299	75	𝜑(𝑁𝑥	𝜑(𝑁𝑥	ADJ
cana-4781	299	76	)	)	PUNCT
cana-4781	299	77	is	be	AUX
cana-4781	299	78	an	an	DET
cana-4781	299	79	open	open	ADJ
cana-4781	299	80	neighbourhood	neighbourhood	NOUN
cana-4781	299	81	of	of	ADP
cana-4781	299	82	𝜑(𝑥	𝜑(𝑥	NOUN
cana-4781	299	83	)	)	PUNCT
cana-4781	299	84	in	in	ADP
cana-4781	299	85	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	299	86	and	and	CCONJ
cana-4781	299	87	𝜑(𝑁𝑥	𝜑(𝑁𝑥	ADJ
cana-4781	299	88	)	)	PUNCT
cana-4781	299	89	⊆	⊆	NUM
cana-4781	299	90	𝑈.	𝑈.	PROPN
cana-4781	299	91	now	now	ADV
cana-4781	299	92	let	let	VERB
cana-4781	299	93	us	we	PRON
cana-4781	299	94	prove	prove	VERB
cana-4781	299	95	the	the	DET
cana-4781	299	96	closure	closure	NOUN
cana-4781	299	97	of	of	ADP
cana-4781	299	98	𝜑(𝑁𝑥	𝜑(𝑁𝑥	ADJ
cana-4781	299	99	)	)	PUNCT
cana-4781	299	100	is	be	AUX
cana-4781	299	101	contained	contain	VERB
cana-4781	299	102	in	in	ADP
cana-4781	299	103	𝑈.	𝑈.	PROPN
cana-4781	299	104	let	let	VERB
cana-4781	299	105	𝜑(𝑁𝑦	𝜑(𝑁𝑦	X
cana-4781	299	106	)	)	PUNCT
cana-4781	299	107	be	be	AUX
cana-4781	299	108	an	an	DET
cana-4781	299	109	open	open	ADJ
cana-4781	299	110	neighbourhood	neighbourhood	NOUN
cana-4781	299	111	of	of	ADP
cana-4781	299	112	𝑦	𝑦	NOUN
cana-4781	299	113	in	in	ADP
cana-4781	299	114	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	299	115	̅̅	̅̅	PROPN
cana-4781	299	116	̅̅	̅̅	PROPN
cana-4781	299	117	and	and	CCONJ
cana-4781	299	118	𝑦	𝑦	NOUN
cana-4781	299	119	be	be	AUX
cana-4781	299	120	an	an	DET
cana-4781	299	121	accumulation	accumulation	NOUN
cana-4781	299	122	point	point	NOUN
cana-4781	299	123	of	of	ADP
cana-4781	299	124	𝜑(𝑁𝑥).that	𝜑(𝑁𝑥).that	PROPN
cana-4781	299	125	is	be	AUX
cana-4781	299	126	,	,	PUNCT
cana-4781	299	127	𝑦	𝑦	NOUN
cana-4781	299	128	in	in	ADP
cana-4781	299	129	closure	closure	NOUN
cana-4781	299	130	of	of	ADP
cana-4781	299	131	𝜑(𝑁𝑥	𝜑(𝑁𝑥	ADJ
cana-4781	299	132	)	)	PUNCT
cana-4781	299	133	.	.	PUNCT
cana-4781	300	1	then	then	ADV
cana-4781	300	2	𝜑(𝑁𝑦	𝜑(𝑁𝑦	PRON
cana-4781	300	3	)	)	PUNCT
cana-4781	300	4	∩	∩	NOUN
cana-4781	300	5	𝜑(𝑁𝑥	𝜑(𝑁𝑥	ADJ
cana-4781	300	6	)	)	PUNCT
cana-4781	300	7	≠	≠	PROPN
cana-4781	300	8	∅	∅	NOUN
cana-4781	300	9	implies	imply	VERB
cana-4781	300	10	𝑁𝑦	𝑁𝑦	PROPN
cana-4781	300	11	∩	∩	NOUN
cana-4781	300	12	𝐾ℜ𝑁𝑥𝐻ℜ	𝐾ℜ𝑁𝑥𝐻ℜ	VERB
cana-4781	300	13	≠	≠	PROPN
cana-4781	300	14	∅.	∅.	VERB
cana-4781	300	15	therefore	therefore	ADV
cana-4781	300	16	,	,	PUNCT
cana-4781	300	17	𝑦	𝑦	NOUN
cana-4781	300	18	∈	∈	NOUN
cana-4781	300	19	𝑁𝐾ℜ𝑁𝑥𝐻ℜ	𝑁𝐾ℜ𝑁𝑥𝐻ℜ	ADJ
cana-4781	300	20	⊆	⊆	NUM
cana-4781	300	21	𝐾ℜ𝑊𝑁𝑥𝐻ℜ	𝐾ℜ𝑊𝑁𝑥𝐻ℜ	NOUN
cana-4781	300	22	⊆	⊆	NUM
cana-4781	300	23	𝐾ℜ𝑊𝑊𝑥𝐻ℜ	𝐾ℜ𝑊𝑊𝑥𝐻ℜ	NOUN
cana-4781	300	24	⊆	⊆	NUM
cana-4781	300	25	𝐾ℜ𝑉𝑥𝐻ℜ	𝐾ℜ𝑉𝑥𝐻ℜ	NOUN
cana-4781	300	26	=	=	SYM
cana-4781	300	27	𝜑(𝑉𝑥	𝜑(𝑉𝑥	NOUN
cana-4781	300	28	)	)	PUNCT
cana-4781	300	29	⊆	⊆	NUM
cana-4781	300	30	𝑈.	𝑈.	PROPN
cana-4781	300	31	so	so	ADV
cana-4781	300	32	,	,	PUNCT
cana-4781	300	33	closure	closure	NOUN
cana-4781	300	34	of	of	ADP
cana-4781	300	35	𝜑(𝑁𝑥	𝜑(𝑁𝑥	ADJ
cana-4781	300	36	)	)	PUNCT
cana-4781	300	37	is	be	AUX
cana-4781	300	38	contained	contain	VERB
cana-4781	300	39	in	in	ADP
cana-4781	300	40	𝑈.	𝑈.	PROPN
cana-4781	300	41	hence	hence	ADV
cana-4781	300	42	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	300	43	is	be	AUX
cana-4781	300	44	a	a	DET
cana-4781	300	45	regular	regular	ADJ
cana-4781	300	46	space	space	NOUN
cana-4781	300	47	.	.	PUNCT
cana-4781	301	1	communications	communication	NOUN
cana-4781	301	2	on	on	ADP
cana-4781	301	3	applied	apply	VERB
cana-4781	301	4	nonlinear	nonlinear	ADJ
cana-4781	301	5	analysis	analysis	NOUN
cana-4781	301	6	issn	issn	NOUN
cana-4781	301	7	:	:	PUNCT
cana-4781	301	8	1074	1074	NUM
cana-4781	301	9	-	-	PUNCT
cana-4781	301	10	133x	133x	NUM
cana-4781	301	11	vol	vol	NOUN
cana-4781	301	12	32	32	NUM
cana-4781	301	13	no	no	NOUN
cana-4781	301	14	.	.	NOUN
cana-4781	301	15	3	3	NUM
cana-4781	301	16	(	(	PUNCT
cana-4781	301	17	2025	2025	NUM
cana-4781	301	18	)	)	PUNCT
cana-4781	301	19	888	888	NUM
cana-4781	301	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-4781	301	21	proposition	proposition	NOUN
cana-4781	301	22	6.7	6.7	NUM
cana-4781	301	23	.	.	PUNCT
cana-4781	302	1	let	let	VERB
cana-4781	302	2	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	302	3	and	and	CCONJ
cana-4781	302	4	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	302	5	be	be	AUX
cana-4781	302	6	rough	rough	ADJ
cana-4781	302	7	subgroups	subgroup	NOUN
cana-4781	302	8	of	of	ADP
cana-4781	302	9	a	a	DET
cana-4781	302	10	topological	topological	ADJ
cana-4781	302	11	simple	simple	ADJ
cana-4781	302	12	rough	rough	ADJ
cana-4781	302	13	group	group	NOUN
cana-4781	302	14	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	302	15	such	such	ADJ
cana-4781	302	16	that	that	SCONJ
cana-4781	302	17	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	302	18	̅̅	̅̅	PROPN
cana-4781	302	19	̅̅	̅̅	PROPN
cana-4781	302	20	is	be	AUX
cana-4781	302	21	a	a	DET
cana-4781	302	22	hausdorff	hausdorff	NOUN
cana-4781	302	23	topological	topological	ADJ
cana-4781	302	24	group	group	NOUN
cana-4781	302	25	and	and	CCONJ
cana-4781	302	26	𝐻ℜ	𝐻ℜ	PROPN
cana-4781	302	27	,	,	PUNCT
cana-4781	302	28	𝐾ℜ	𝐾ℜ	PROPN
cana-4781	302	29	are	be	AUX
cana-4781	302	30	subgroups	subgroup	NOUN
cana-4781	302	31	of	of	ADP
cana-4781	302	32	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	302	33	̅̅	̅̅	PROPN
cana-4781	302	34	̅̅	̅̅	PROPN
cana-4781	302	35	.	.	PUNCT
cana-4781	303	1	then	then	ADV
cana-4781	303	2	the	the	DET
cana-4781	303	3	mapping	mapping	NOUN
cana-4781	303	4	𝜌	𝜌	ADP
cana-4781	303	5	:	:	PUNCT
cana-4781	303	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	303	7	̅̅	̅̅	PROPN
cana-4781	303	8	̅̅	̅̅	PROPN
cana-4781	303	9	/𝐻ℜ	/𝐻ℜ	PROPN
cana-4781	303	10	→	→	SYM
cana-4781	303	11	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	303	12	defined	define	VERB
cana-4781	303	13	by	by	ADP
cana-4781	303	14	ρ(𝑥𝐻ℜ	ρ(𝑥𝐻ℜ	NOUN
cana-4781	303	15	)	)	PUNCT
cana-4781	303	16	=	=	SYM
cana-4781	304	1	𝐾ℜ𝑥𝐻ℜ	𝐾ℜ𝑥𝐻ℜ	PROPN
cana-4781	304	2	,	,	PUNCT
cana-4781	304	3	for	for	ADP
cana-4781	304	4	every	every	DET
cana-4781	304	5	𝑥	𝑥	PROPN
cana-4781	304	6	∈	∈	PROPN
cana-4781	304	7	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	304	8	̅̅	̅̅	PROPN
cana-4781	304	9	̅̅	̅̅	PROPN
cana-4781	304	10	is	be	AUX
cana-4781	304	11	open	open	ADJ
cana-4781	304	12	and	and	CCONJ
cana-4781	304	13	perfect	perfect	ADJ
cana-4781	304	14	.	.	PUNCT
cana-4781	305	1	proof	proof	NOUN
cana-4781	305	2	:	:	PUNCT
cana-4781	305	3	let	let	VERB
cana-4781	305	4	the	the	DET
cana-4781	305	5	mappings	mapping	NOUN
cana-4781	305	6	𝜑	𝜑	NOUN
cana-4781	305	7	:	:	PUNCT
cana-4781	305	8	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	305	9	̅̅	̅̅	PROPN
cana-4781	305	10	̅̅	̅̅	PROPN
cana-4781	305	11	→	→	SYM
cana-4781	305	12	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	305	13	and	and	CCONJ
cana-4781	305	14	𝜑∗	𝜑∗	PROPN
cana-4781	305	15	:	:	PUNCT
cana-4781	305	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	305	17	̅̅	̅̅	PROPN
cana-4781	305	18	̅̅	̅̅	PROPN
cana-4781	305	19	→	→	PUNCT
cana-4781	305	20	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	305	21	̅̅	̅̅	PROPN
cana-4781	305	22	̅̅	̅̅	PROPN
cana-4781	305	23	𝐻ℜ⁄	𝐻ℜ⁄	NOUN
cana-4781	305	24	defined	define	VERB
cana-4781	305	25	by	by	ADP
cana-4781	305	26	𝜑(𝑥	𝜑(𝑥	NOUN
cana-4781	305	27	)	)	PUNCT
cana-4781	305	28	=	=	PUNCT
cana-4781	305	29	𝐾ℜ𝑥𝐻ℜ	𝐾ℜ𝑥𝐻ℜ	X
cana-4781	305	30	and	and	CCONJ
cana-4781	305	31	𝜑∗(𝑥	𝜑∗(𝑥	PROPN
cana-4781	305	32	)	)	PUNCT
cana-4781	305	33	=	=	PUNCT
cana-4781	306	1	𝑥𝐻ℜ	𝑥𝐻ℜ	PROPN
cana-4781	306	2	,	,	PUNCT
cana-4781	306	3	for	for	ADP
cana-4781	306	4	𝑥	𝑥	PROPN
cana-4781	306	5	∈	∈	PROPN
cana-4781	306	6	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	306	7	̅̅	̅̅	PROPN
cana-4781	306	8	̅̅	̅̅	PROPN
cana-4781	306	9	.	.	PUNCT
cana-4781	307	1	then	then	ADV
cana-4781	307	2	𝜑	𝜑	X
cana-4781	307	3	=	=	PUNCT
cana-4781	307	4	𝜌	𝜌	ADP
cana-4781	307	5	∘	∘	X
cana-4781	307	6	𝜑∗.	𝜑∗.	NOUN
cana-4781	307	7	since	since	SCONJ
cana-4781	307	8	𝜑	𝜑	PROPN
cana-4781	307	9	and	and	CCONJ
cana-4781	307	10	𝜑∗	𝜑∗	NOUN
cana-4781	307	11	are	be	AUX
cana-4781	307	12	continuous	continuous	ADJ
cana-4781	307	13	and	and	CCONJ
cana-4781	307	14	open	open	ADJ
cana-4781	307	15	,	,	PUNCT
cana-4781	307	16	𝜌	𝜌	X
cana-4781	307	17	is	be	AUX
cana-4781	307	18	an	an	DET
cana-4781	307	19	open	open	ADJ
cana-4781	307	20	mapping	mapping	NOUN
cana-4781	307	21	.	.	PUNCT
cana-4781	308	1	let	let	VERB
cana-4781	308	2	𝑐	𝑐	PROPN
cana-4781	308	3	∈	∈	VERB
cana-4781	308	4	𝒟ℭ	𝒟ℭ	PROPN
cana-4781	308	5	such	such	ADJ
cana-4781	308	6	that	that	SCONJ
cana-4781	308	7	𝜑(𝑥	𝜑(𝑥	NOUN
cana-4781	308	8	)	)	PUNCT
cana-4781	308	9	=	=	SYM
cana-4781	308	10	𝑐	𝑐	NOUN
cana-4781	308	11	,	,	PUNCT
cana-4781	308	12	for	for	ADP
cana-4781	308	13	some	some	DET
cana-4781	308	14	𝑥	𝑥	PRON
cana-4781	308	15	∈	∈	PROPN
cana-4781	308	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	308	17	̅̅	̅̅	PROPN
cana-4781	308	18	̅̅	̅̅	PROPN
cana-4781	308	19	.	.	PUNCT
cana-4781	309	1	then	then	ADV
cana-4781	309	2	𝜑−1(𝑐	𝜑−1(𝑐	PROPN
cana-4781	309	3	)	)	PUNCT
cana-4781	309	4	=	=	PUNCT
cana-4781	310	1	𝐾ℜ𝑥𝐻ℜ	𝐾ℜ𝑥𝐻ℜ	X
cana-4781	310	2	and	and	CCONJ
cana-4781	310	3	the	the	DET
cana-4781	310	4	preimage	preimage	PROPN
cana-4781	310	5	𝜌−1(𝑐	𝜌−1(𝑐	PROPN
cana-4781	310	6	)	)	PUNCT
cana-4781	310	7	=	=	SYM
cana-4781	310	8	𝜑∗(𝐾ℜ𝑥𝐻ℜ	𝜑∗(𝐾ℜ𝑥𝐻ℜ	X
cana-4781	310	9	)	)	PUNCT
cana-4781	310	10	=	=	SYM
cana-4781	310	11	𝜑∗(𝐾ℜ𝑥	𝜑∗(𝐾ℜ𝑥	NOUN
cana-4781	310	12	)	)	PUNCT
cana-4781	310	13	,	,	PUNCT
cana-4781	310	14	𝐾ℜ𝑥	𝐾ℜ𝑥	VERB
cana-4781	310	15	⊆	⊆	NUM
cana-4781	310	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	310	17	̅̅	̅̅	NOUN
cana-4781	310	18	̅̅	̅̅	PROPN
cana-4781	310	19	.	.	PUNCT
cana-4781	311	1	since	since	SCONJ
cana-4781	311	2	continuous	continuous	ADJ
cana-4781	311	3	image	image	NOUN
cana-4781	311	4	of	of	ADP
cana-4781	311	5	a	a	DET
cana-4781	311	6	compact	compact	ADJ
cana-4781	311	7	set	set	NOUN
cana-4781	311	8	is	be	AUX
cana-4781	311	9	compact	compact	ADJ
cana-4781	311	10	,	,	PUNCT
cana-4781	311	11	the	the	DET
cana-4781	311	12	set	set	NOUN
cana-4781	311	13	of	of	ADP
cana-4781	311	14	all	all	DET
cana-4781	311	15	preimages	preimage	NOUN
cana-4781	311	16	,	,	PUNCT
cana-4781	311	17	𝜌−1(𝑐	𝜌−1(𝑐	PROPN
cana-4781	311	18	)	)	PUNCT
cana-4781	311	19	is	be	AUX
cana-4781	311	20	compact	compact	ADJ
cana-4781	311	21	in	in	ADP
cana-4781	311	22	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	311	23	̅̅	̅̅	PROPN
cana-4781	311	24	̅̅	̅̅	PROPN
cana-4781	311	25	/𝐻ℜ	/𝐻ℜ	PROPN
cana-4781	311	26	,	,	PUNCT
cana-4781	311	27	for	for	SCONJ
cana-4781	311	28	every	every	DET
cana-4781	311	29	𝑐	𝑐	PROPN
cana-4781	311	30	∈	∈	PROPN
cana-4781	311	31	𝒟ℭ.	𝒟ℭ.	PUNCT
cana-4781	311	32	now	now	ADV
cana-4781	311	33	let	let	VERB
cana-4781	311	34	us	we	PRON
cana-4781	311	35	prove	prove	VERB
cana-4781	311	36	𝜌	𝜌	PART
cana-4781	311	37	is	be	AUX
cana-4781	311	38	closed	closed	ADJ
cana-4781	311	39	,	,	PUNCT
cana-4781	311	40	using	use	VERB
cana-4781	311	41	the	the	DET
cana-4781	311	42	theorem	theorem	NOUN
cana-4781	311	43	2.16	2.16	NUM
cana-4781	311	44	.	.	PUNCT
cana-4781	312	1	let	let	VERB
cana-4781	312	2	𝑁	𝑁	PROPN
cana-4781	312	3	be	be	AUX
cana-4781	312	4	an	an	DET
cana-4781	312	5	open	open	ADJ
cana-4781	312	6	neighbourhood	neighbourhood	NOUN
cana-4781	312	7	of	of	ADP
cana-4781	312	8	𝜑∗(𝐾ℜ𝑥	𝜑∗(𝐾ℜ𝑥	NOUN
cana-4781	312	9	)	)	PUNCT
cana-4781	312	10	in	in	ADP
cana-4781	312	11	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	312	12	̅̅	̅̅	PROPN
cana-4781	312	13	̅̅	̅̅	PROPN
cana-4781	312	14	/𝐻ℜ.	/𝐻ℜ.	PUNCT
cana-4781	313	1	then	then	ADV
cana-4781	313	2	,	,	PUNCT
cana-4781	313	3	𝐾ℜ𝑥	𝐾ℜ𝑥	VERB
cana-4781	313	4	⊆	⊆	NUM
cana-4781	313	5	𝜑∗−1(𝑁	𝜑∗−1(𝑁	NOUN
cana-4781	313	6	)	)	PUNCT
cana-4781	313	7	which	which	PRON
cana-4781	313	8	implies	imply	VERB
cana-4781	313	9	there	there	PRON
cana-4781	313	10	exists	exist	VERB
cana-4781	313	11	an	an	DET
cana-4781	313	12	open	open	ADJ
cana-4781	313	13	neighbourhood	neighbourhood	NOUN
cana-4781	313	14	𝑈	𝑈	PROPN
cana-4781	313	15	⊆	⊆	NUM
cana-4781	313	16	𝐺ℜ	𝐺ℜ	PROPN
cana-4781	313	17	̅̅	̅̅	NOUN
cana-4781	313	18	̅̅	̅̅	NOUN
cana-4781	313	19	such	such	ADJ
cana-4781	313	20	that	that	SCONJ
cana-4781	313	21	𝐾ℜ𝑥𝑈	𝐾ℜ𝑥𝑈	PROPN
cana-4781	313	22	⊆	⊆	NUM
cana-4781	313	23	𝜑∗−1(𝑁	𝜑∗−1(𝑁	NOUN
cana-4781	313	24	)	)	PUNCT
cana-4781	313	25	that	that	PRON
cana-4781	313	26	is	be	AUX
cana-4781	313	27	,	,	PUNCT
cana-4781	313	28	𝜑∗(𝐾ℜ𝑥𝑈	𝜑∗(𝐾ℜ𝑥𝑈	NOUN
cana-4781	313	29	)	)	PUNCT
cana-4781	313	30	⊆	⊆	X
cana-4781	313	31	𝑁.	𝑁.	PROPN
cana-4781	313	32	let	let	VERB
cana-4781	313	33	𝑉	𝑉	PROPN
cana-4781	313	34	=	=	SYM
cana-4781	313	35	𝜑(𝑥𝑈	𝜑(𝑥𝑈	PROPN
cana-4781	313	36	)	)	PUNCT
cana-4781	313	37	be	be	VERB
cana-4781	313	38	an	an	DET
cana-4781	313	39	open	open	ADJ
cana-4781	313	40	neighbourhood	neighbourhood	NOUN
cana-4781	313	41	of	of	ADP
cana-4781	313	42	𝑐	𝑐	PROPN
cana-4781	313	43	∈	∈	PROPN
cana-4781	313	44	𝒟ℭ.	𝒟ℭ.	PUNCT
cana-4781	313	45	therefore	therefore	ADV
cana-4781	313	46	,	,	PUNCT
cana-4781	313	47	𝜌−1(𝑉	𝜌−1(𝑉	NOUN
cana-4781	313	48	)	)	PUNCT
cana-4781	313	49	=	=	SYM
cana-4781	313	50	𝜌−1(𝜑(𝑥𝑈	𝜌−1(𝜑(𝑥𝑈	PROPN
cana-4781	313	51	)	)	PUNCT
cana-4781	313	52	)	)	PUNCT
cana-4781	314	1	=	=	SYM
cana-4781	314	2	𝜑∗(𝜑−1(𝜑(𝑥𝑈	𝜑∗(𝜑−1(𝜑(𝑥𝑈	NOUN
cana-4781	314	3	)	)	PUNCT
cana-4781	314	4	)	)	PUNCT
cana-4781	314	5	)	)	PUNCT
cana-4781	315	1	=	=	PUNCT
cana-4781	315	2	𝜑∗(𝐾ℜ𝑥𝑈𝐻ℜ	𝜑∗(𝐾ℜ𝑥𝑈𝐻ℜ	PROPN
cana-4781	315	3	)	)	PUNCT
cana-4781	315	4	=	=	SYM
cana-4781	315	5	𝜑∗(𝐾ℜ𝑥𝑈	𝜑∗(𝐾ℜ𝑥𝑈	NOUN
cana-4781	315	6	)	)	PUNCT
cana-4781	316	1	⊆	⊆	X
cana-4781	316	2	𝑁.	𝑁.	PROPN
cana-4781	316	3	hence	hence	ADV
cana-4781	316	4	,	,	PUNCT
cana-4781	316	5	𝜌	𝜌	X
cana-4781	316	6	is	be	AUX
cana-4781	316	7	closed	close	VERB
cana-4781	316	8	which	which	PRON
cana-4781	316	9	implies	imply	VERB
cana-4781	316	10	𝜌	𝜌	PART
cana-4781	316	11	is	be	AUX
cana-4781	316	12	a	a	DET
cana-4781	316	13	perfect	perfect	ADJ
cana-4781	316	14	mapping	mapping	NOUN
cana-4781	316	15	.	.	PUNCT
cana-4781	317	1	references	reference	NOUN
cana-4781	317	2	:	:	PUNCT
cana-4781	318	1	[	[	X
cana-4781	318	2	1	1	X
cana-4781	318	3	]	]	X
cana-4781	318	4	alaa	alaa	PROPN
cana-4781	318	5	altassan	altassan	PROPN
cana-4781	318	6	,	,	PUNCT
cana-4781	318	7	nof	nof	PROPN
cana-4781	318	8	alharbi	alharbi	PROPN
cana-4781	318	9	,	,	PUNCT
cana-4781	318	10	hassen	hassen	PROPN
cana-4781	318	11	aydi	aydi	ADV
cana-4781	318	12	,	,	PUNCT
cana-4781	318	13	cenap	cenap	VERB
cana-4781	318	14	ozel	ozel	ADJ
cana-4781	318	15	,	,	PUNCT
cana-4781	318	16	rough	rough	ADJ
cana-4781	318	17	action	action	NOUN
cana-4781	318	18	on	on	ADP
cana-4781	318	19	topological	topological	ADJ
cana-4781	318	20	rough	rough	ADJ
cana-4781	318	21	groups	group	NOUN
cana-4781	318	22	,	,	PUNCT
cana-4781	318	23	appl	appl	PROPN
cana-4781	318	24	.	.	PUNCT
cana-4781	319	1	gen	gen	PROPN
cana-4781	319	2	.	.	PROPN
cana-4781	319	3	topol	topol	PROPN
cana-4781	319	4	.	.	PUNCT
cana-4781	320	1	21	21	NUM
cana-4781	320	2	,	,	PUNCT
cana-4781	320	3	no	no	INTJ
cana-4781	320	4	.	.	PUNCT
cana-4781	320	5	2(2020	2(2020	NUM
cana-4781	320	6	)	)	PUNCT
cana-4781	320	7	,	,	PUNCT
cana-4781	320	8	295	295	NUM
cana-4781	320	9	-	-	SYM
cana-4781	320	10	304	304	NUM
cana-4781	320	11	.	.	PUNCT
cana-4781	321	1	[	[	X
cana-4781	321	2	2	2	NUM
cana-4781	321	3	]	]	PUNCT
cana-4781	321	4	arhangel	arhangel	NOUN
cana-4781	321	5	skii	skii	PROPN
cana-4781	321	6	av	av	PROPN
cana-4781	321	7	,	,	PUNCT
cana-4781	321	8	tkachenko	tkachenko	PROPN
cana-4781	321	9	m	m	PROPN
cana-4781	321	10	,	,	PUNCT
cana-4781	321	11	topological	topological	ADJ
cana-4781	321	12	groups	group	NOUN
cana-4781	321	13	and	and	CCONJ
cana-4781	321	14	related	related	ADJ
cana-4781	321	15	structures	structure	NOUN
cana-4781	321	16	,	,	PUNCT
cana-4781	321	17	atlantis	atlantis	PROPN
cana-4781	321	18	press	press	PROPN
cana-4781	321	19	and	and	CCONJ
cana-4781	321	20	world	world	PROPN
cana-4781	321	21	sci	sci	PROPN
cana-4781	321	22	,	,	PUNCT
cana-4781	321	23	paris	paris	PROPN
cana-4781	321	24	(	(	PUNCT
cana-4781	321	25	2008	2008	NUM
cana-4781	321	26	)	)	PUNCT
cana-4781	321	27	.	.	PUNCT
cana-4781	322	1	[	[	X
cana-4781	322	2	3	3	X
cana-4781	322	3	]	]	X
cana-4781	322	4	biswas	biswas	NOUN
cana-4781	322	5	.	.	PUNCT
cana-4781	323	1	r	r	NOUN
cana-4781	323	2	,	,	PUNCT
cana-4781	323	3	nanda	nanda	ADJ
cana-4781	323	4	.	.	PUNCT
cana-4781	324	1	s	s	X
cana-4781	324	2	,	,	PUNCT
cana-4781	324	3	rough	rough	ADJ
cana-4781	324	4	groups	group	NOUN
cana-4781	324	5	and	and	CCONJ
cana-4781	324	6	rough	rough	ADJ
cana-4781	324	7	subgroups	subgroup	NOUN
cana-4781	324	8	,	,	PUNCT
cana-4781	324	9	bull	bull	NOUN
cana-4781	324	10	.	.	PUNCT
cana-4781	325	1	pol	pol	PROPN
cana-4781	325	2	.	.	PUNCT
cana-4781	326	1	ac	ac	PROPN
cana-4781	326	2	.	.	PROPN
cana-4781	326	3	math	math	PROPN
cana-4781	326	4	.	.	PUNCT
cana-4781	326	5	,	,	PUNCT
cana-4781	326	6	42(1994	42(1994	NUM
cana-4781	326	7	)	)	PUNCT
cana-4781	326	8	251254	251254	NUM
cana-4781	326	9	.	.	PUNCT
cana-4781	327	1	[	[	X
cana-4781	327	2	4	4	NUM
cana-4781	327	3	]	]	X
cana-4781	327	4	duoqian	duoqian	ADJ
cana-4781	327	5	miao	miao	PROPN
cana-4781	327	6	,	,	PUNCT
cana-4781	327	7	suqing	suqe	VERB
cana-4781	327	8	han	han	PROPN
cana-4781	327	9	,	,	PUNCT
cana-4781	327	10	daoguo	daoguo	PROPN
cana-4781	327	11	li	li	PROPN
cana-4781	327	12	,	,	PUNCT
cana-4781	327	13	and	and	CCONJ
cana-4781	327	14	lijun	lijun	PROPN
cana-4781	327	15	sun	sun	NOUN
cana-4781	327	16	,	,	PUNCT
cana-4781	327	17	rough	rough	ADJ
cana-4781	327	18	groups	group	NOUN
cana-4781	327	19	,	,	PUNCT
cana-4781	327	20	rough	rough	ADJ
cana-4781	327	21	subgroup	subgroup	NOUN
cana-4781	327	22	and	and	CCONJ
cana-4781	327	23	their	their	PRON
cana-4781	327	24	properties	property	NOUN
cana-4781	327	25	,	,	PUNCT
cana-4781	327	26	rough	rough	ADJ
cana-4781	327	27	sets	set	NOUN
cana-4781	327	28	,	,	PUNCT
cana-4781	327	29	fuzzy	fuzzy	ADJ
cana-4781	327	30	sets	set	NOUN
cana-4781	327	31	,	,	PUNCT
cana-4781	327	32	data	datum	NOUN
cana-4781	327	33	mining	mining	NOUN
cana-4781	327	34	,	,	PUNCT
cana-4781	327	35	and	and	CCONJ
cana-4781	327	36	granular	granular	ADJ
cana-4781	327	37	computing	computing	NOUN
cana-4781	327	38	.	.	PUNCT
cana-4781	328	1	2005	2005	NUM
cana-4781	328	2	,	,	PUNCT
cana-4781	328	3	pp	pp	ADJ
cana-4781	328	4	.	.	PUNCT
cana-4781	328	5	104113	104113	NUM
cana-4781	328	6	.	.	PUNCT
cana-4781	329	1	[	[	X
cana-4781	329	2	5	5	NUM
cana-4781	329	3	]	]	X
cana-4781	329	4	engelking	engelke	VERB
cana-4781	329	5	.	.	PUNCT
cana-4781	330	1	r	r	NOUN
cana-4781	330	2	,	,	PUNCT
cana-4781	330	3	general	general	ADJ
cana-4781	330	4	topology	topology	NOUN
cana-4781	330	5	(	(	PUNCT
cana-4781	330	6	revised	revise	VERB
cana-4781	330	7	and	and	CCONJ
cana-4781	330	8	completed	complete	VERB
cana-4781	330	9	)	)	PUNCT
cana-4781	330	10	,	,	PUNCT
cana-4781	330	11	heldermann	heldermann	PROPN
cana-4781	330	12	verlag	verlag	PROPN
cana-4781	330	13	,	,	PUNCT
cana-4781	330	14	berlin	berlin	PROPN
cana-4781	330	15	,	,	PUNCT
cana-4781	330	16	1989	1989	NUM
cana-4781	330	17	.	.	PUNCT
cana-4781	331	1	[	[	X
cana-4781	331	2	6	6	NUM
cana-4781	331	3	]	]	PUNCT
cana-4781	331	4	fucai	fucai	PROPN
cana-4781	331	5	lin	lin	PROPN
cana-4781	331	6	,	,	PUNCT
cana-4781	331	7	qianqian	qianqian	PROPN
cana-4781	331	8	sun	sun	PROPN
cana-4781	331	9	,	,	PUNCT
cana-4781	331	10	yujin	yujin	PROPN
cana-4781	331	11	lin	lin	PROPN
cana-4781	331	12	,	,	PUNCT
cana-4781	331	13	jinjin	jinjin	PROPN
cana-4781	331	14	li	li	PROPN
cana-4781	331	15	,	,	PUNCT
cana-4781	331	16	some	some	DET
cana-4781	331	17	topological	topological	ADJ
cana-4781	331	18	properties	property	NOUN
cana-4781	331	19	of	of	ADP
cana-4781	331	20	topological	topological	ADJ
cana-4781	331	21	rough	rough	ADJ
cana-4781	331	22	groups	group	NOUN
cana-4781	331	23	,	,	PUNCT
cana-4781	331	24	soft	soft	ADJ
cana-4781	331	25	computing	computing	NOUN
cana-4781	331	26	(	(	PUNCT
cana-4781	331	27	2021	2021	NUM
cana-4781	331	28	)	)	PUNCT
cana-4781	331	29	25	25	NUM
cana-4781	331	30	:	:	SYM
cana-4781	331	31	3441	3441	NUM
cana-4781	331	32	3453	3453	NUM
cana-4781	331	33	.	.	PUNCT
cana-4781	332	1	[	[	X
cana-4781	332	2	7	7	X
cana-4781	332	3	]	]	X
cana-4781	332	4	james	james	PROPN
cana-4781	332	5	r.	r.	PROPN
cana-4781	332	6	munkres	munkres	PROPN
cana-4781	332	7	,	,	PUNCT
cana-4781	332	8	topology	topology	NOUN
cana-4781	332	9	,	,	PUNCT
cana-4781	332	10	updated	update	VERB
cana-4781	332	11	second	second	ADJ
cana-4781	332	12	edition	edition	NOUN
cana-4781	332	13	,	,	PUNCT
cana-4781	332	14	pearson	pearson	PROPN
cana-4781	332	15	india	india	PROPN
cana-4781	332	16	education	education	PROPN
cana-4781	332	17	services	services	PROPN
cana-4781	332	18	pvt	pvt	PROPN
cana-4781	332	19	.	.	PROPN
cana-4781	332	20	ltd	ltd	PROPN
cana-4781	332	21	,	,	PUNCT
cana-4781	332	22	chennai	chennai	PROPN
cana-4781	332	23	.	.	PUNCT
cana-4781	333	1	[	[	X
cana-4781	333	2	8	8	NUM
cana-4781	333	3	]	]	X
cana-4781	333	4	joseph	joseph	PROPN
cana-4781	333	5	a.	a.	PROPN
cana-4781	333	6	gallian	gallian	PROPN
cana-4781	333	7	,	,	PUNCT
cana-4781	333	8	contemporary	contemporary	ADJ
cana-4781	333	9	abstract	abstract	ADJ
cana-4781	333	10	algebra	algebra	NOUN
cana-4781	333	11	,	,	PUNCT
cana-4781	333	12	ninth	ninth	ADJ
cana-4781	333	13	edition	edition	NOUN
cana-4781	333	14	,	,	PUNCT
cana-4781	333	15	cengage	cengage	PROPN
cana-4781	333	16	learning	learn	VERB
cana-4781	333	17	india	india	PROPN
cana-4781	333	18	private	private	PROPN
cana-4781	333	19	limited	limited	ADJ
cana-4781	333	20	,	,	PUNCT
cana-4781	333	21	delhi	delhi	ADJ
cana-4781	333	22	.	.	PUNCT
cana-4781	334	1	[	[	X
cana-4781	334	2	9	9	NUM
cana-4781	334	3	]	]	PUNCT
cana-4781	334	4	laurens	lauren	NOUN
cana-4781	334	5	diels	diel	NOUN
cana-4781	334	6	and	and	CCONJ
cana-4781	334	7	philip	philip	PROPN
cana-4781	334	8	a.	a.	PROPN
cana-4781	334	9	dowerk	dowerk	PROPN
cana-4781	334	10	,	,	PUNCT
cana-4781	334	11	invarient	invarient	NOUN
cana-4781	334	12	automatic	automatic	ADJ
cana-4781	334	13	continuity	continuity	NOUN
cana-4781	334	14	for	for	ADP
cana-4781	334	15	compact	compact	ADJ
cana-4781	334	16	connected	connect	VERB
cana-4781	334	17	simple	simple	ADJ
cana-4781	334	18	lie	lie	NOUN
cana-4781	334	19	groups	group	NOUN
cana-4781	334	20	,	,	PUNCT
cana-4781	334	21	topology	topology	NOUN
cana-4781	334	22	and	and	CCONJ
cana-4781	334	23	its	its	PRON
cana-4781	334	24	applications	application	NOUN
cana-4781	334	25	,	,	PUNCT
cana-4781	334	26	266	266	NUM
cana-4781	334	27	(	(	PUNCT
cana-4781	334	28	2019	2019	NUM
cana-4781	334	29	)	)	PUNCT
cana-4781	334	30	,	,	PUNCT
cana-4781	334	31	106858	106858	NUM
cana-4781	334	32	.	.	PUNCT
cana-4781	335	1	communications	communication	NOUN
cana-4781	335	2	on	on	ADP
cana-4781	335	3	applied	apply	VERB
cana-4781	335	4	nonlinear	nonlinear	ADJ
cana-4781	335	5	analysis	analysis	NOUN
cana-4781	335	6	issn	issn	NOUN
cana-4781	335	7	:	:	PUNCT
cana-4781	335	8	1074	1074	NUM
cana-4781	335	9	-	-	PUNCT
cana-4781	335	10	133x	133x	NUM
cana-4781	335	11	vol	vol	NOUN
cana-4781	335	12	32	32	NUM
cana-4781	335	13	no	no	NOUN
cana-4781	335	14	.	.	NOUN
cana-4781	335	15	3	3	NUM
cana-4781	335	16	(	(	PUNCT
cana-4781	335	17	2025	2025	NUM
cana-4781	335	18	)	)	PUNCT
cana-4781	336	1	889	889	NUM
cana-4781	336	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-4781	337	1	[	[	X
cana-4781	337	2	10	10	NUM
cana-4781	337	3	]	]	X
cana-4781	337	4	mangesh	mangesh	PROPN
cana-4781	337	5	g.	g.	PROPN
cana-4781	337	6	murdeshwar	murdeshwar	PROPN
cana-4781	337	7	,	,	PUNCT
cana-4781	337	8	general	general	ADJ
cana-4781	337	9	topology	topology	NOUN
cana-4781	337	10	,	,	PUNCT
cana-4781	337	11	wiley	wiley	PROPN
cana-4781	337	12	eastern	eastern	PROPN
cana-4781	337	13	limited	limited	PROPN
cana-4781	337	14	,	,	PUNCT
cana-4781	337	15	india	india	PROPN
cana-4781	337	16	,	,	PUNCT
cana-4781	337	17	1983	1983	NUM
cana-4781	337	18	.	.	PUNCT
cana-4781	338	1	[	[	X
cana-4781	338	2	11	11	NUM
cana-4781	338	3	]	]	X
cana-4781	338	4	mikhail	mikhail	PROPN
cana-4781	338	5	tkaccnko	tkaccnko	NOUN
cana-4781	338	6	,	,	PUNCT
cana-4781	338	7	introduction	introduction	NOUN
cana-4781	338	8	to	to	ADP
cana-4781	338	9	topological	topological	ADJ
cana-4781	338	10	groups	group	NOUN
cana-4781	338	11	,	,	PUNCT
cana-4781	338	12	topology	topology	NOUN
cana-4781	338	13	and	and	CCONJ
cana-4781	338	14	its	its	PRON
cana-4781	338	15	application	application	NOUN
cana-4781	338	16	,	,	PUNCT
cana-4781	338	17	86(1998	86(1998	NUM
cana-4781	338	18	)	)	PUNCT
cana-4781	338	19	,	,	PUNCT
cana-4781	338	20	179	179	NUM
cana-4781	338	21	-	-	SYM
cana-4781	338	22	231	231	NUM
cana-4781	338	23	.	.	PUNCT
cana-4781	339	1	[	[	X
cana-4781	339	2	12	12	NUM
cana-4781	339	3	]	]	PUNCT
cana-4781	339	4	nof	nof	PROPN
cana-4781	339	5	alharbi	alharbi	PROPN
cana-4781	339	6	,	,	PUNCT
cana-4781	339	7	alaa	alaa	PROPN
cana-4781	339	8	altassan	altassan	PROPN
cana-4781	339	9	,	,	PUNCT
cana-4781	339	10	hassen	hassen	PROPN
cana-4781	339	11	aydi	aydi	ADJ
cana-4781	339	12	and	and	CCONJ
cana-4781	339	13	cenap	cenap	VERB
cana-4781	339	14	�	�	PROPN
cana-4781	339	15	̈	̈	X
cana-4781	339	16	�	�	PROPN
cana-4781	339	17	zel	zel	PROPN
cana-4781	339	18	,	,	PUNCT
cana-4781	339	19	rough	rough	ADJ
cana-4781	339	20	quotient	quotient	NOUN
cana-4781	339	21	in	in	ADP
cana-4781	339	22	topological	topological	ADJ
cana-4781	339	23	rough	rough	ADJ
cana-4781	339	24	sets	set	NOUN
cana-4781	339	25	,	,	PUNCT
cana-4781	339	26	open	open	ADJ
cana-4781	339	27	mathematics	mathematic	NOUN
cana-4781	339	28	,	,	PUNCT
cana-4781	339	29	2019	2019	NUM
cana-4781	339	30	;	;	PUNCT
cana-4781	339	31	17	17	NUM
cana-4781	339	32	:	:	SYM
cana-4781	339	33	1750	1750	NUM
cana-4781	339	34	-	-	SYM
cana-4781	339	35	1755	1755	NUM
cana-4781	339	36	.	.	PUNCT
cana-4781	340	1	[	[	X
cana-4781	340	2	13	13	NUM
cana-4781	340	3	]	]	X
cana-4781	340	4	nurettin	nurettin	ADJ
cana-4781	340	5	bagirmaz	bagirmaz	NOUN
cana-4781	340	6	,	,	PUNCT
cana-4781	340	7	ilhan	ilhan	PROPN
cana-4781	340	8	icen	icen	PROPN
cana-4781	340	9	and	and	CCONJ
cana-4781	340	10	abdullah	abdullah	PROPN
cana-4781	340	11	f.	f.	PROPN
cana-4781	340	12	ozcan	ozcan	PROPN
cana-4781	340	13	,	,	PUNCT
cana-4781	340	14	topological	topological	ADJ
cana-4781	340	15	rough	rough	ADJ
cana-4781	340	16	groups	group	NOUN
cana-4781	340	17	,	,	PUNCT
cana-4781	340	18	topol	topol	NOUN
cana-4781	340	19	.	.	PUNCT
cana-4781	341	1	algebra	algebra	PROPN
cana-4781	341	2	appl	appl	NOUN
cana-4781	341	3	.	.	PROPN
cana-4781	341	4	2016	2016	NUM
cana-4781	341	5	;	;	PUNCT
cana-4781	341	6	4	4	NUM
cana-4781	341	7	:	:	SYM
cana-4781	341	8	31	31	NUM
cana-4781	341	9	-	-	SYM
cana-4781	341	10	38	38	NUM
cana-4781	341	11	.	.	PUNCT
cana-4781	342	1	[	[	X
cana-4781	342	2	14	14	NUM
cana-4781	342	3	]	]	X
cana-4781	342	4	pawlak	pawlak	ADJ
cana-4781	342	5	.	.	PUNCT
cana-4781	343	1	z	z	X
cana-4781	343	2	,	,	PUNCT
cana-4781	343	3	rough	rough	ADJ
cana-4781	343	4	sets	set	NOUN
cana-4781	343	5	,	,	PUNCT
cana-4781	343	6	international	international	ADJ
cana-4781	343	7	journal	journal	NOUN
cana-4781	343	8	of	of	ADP
cana-4781	343	9	computer	computer	NOUN
cana-4781	343	10	and	and	CCONJ
cana-4781	343	11	information	information	NOUN
cana-4781	343	12	sciences	science	NOUN
cana-4781	343	13	,	,	PUNCT
cana-4781	343	14	vol	vol	NOUN
cana-4781	343	15	.	.	PROPN
cana-4781	343	16	11	11	NUM
cana-4781	343	17	,	,	PUNCT
cana-4781	343	18	no	no	INTJ
cana-4781	343	19	.	.	NOUN
cana-4781	343	20	5	5	NUM
cana-4781	343	21	,	,	PUNCT
cana-4781	343	22	1982	1982	NUM
cana-4781	343	23	,	,	PUNCT
cana-4781	343	24	341	341	NUM
cana-4781	343	25	-	-	SYM
cana-4781	343	26	356	356	NUM
cana-4781	343	27	.	.	PUNCT
cana-4781	344	1	[	[	X
cana-4781	344	2	15	15	NUM
cana-4781	344	3	]	]	X
cana-4781	344	4	pi	pi	PROPN
cana-4781	344	5	-	-	PUNCT
cana-4781	344	6	yu	yu	PROPN
cana-4781	344	7	li	li	PROPN
cana-4781	344	8	,	,	PUNCT
cana-4781	344	9	wen	wen	PROPN
cana-4781	344	10	-	-	PUNCT
cana-4781	344	11	li	li	PROPN
cana-4781	344	12	liu	liu	PROPN
cana-4781	344	13	,	,	PUNCT
cana-4781	344	14	lei	lei	PROPN
cana-4781	344	15	mou	mou	PROPN
cana-4781	344	16	,	,	PUNCT
cana-4781	344	17	zhi	zhi	PROPN
cana-4781	344	18	-	-	PUNCT
cana-4781	344	19	fang	fang	X
cana-4781	344	20	guo	guo	PROPN
cana-4781	344	21	,	,	PUNCT
cana-4781	344	22	on	on	ADP
cana-4781	344	23	separation	separation	NOUN
cana-4781	344	24	axioms	axiom	NOUN
cana-4781	344	25	of	of	ADP
cana-4781	344	26	topological	topological	ADJ
cana-4781	344	27	rough	rough	ADJ
cana-4781	344	28	groups	group	NOUN
cana-4781	344	29	,	,	PUNCT
cana-4781	344	30	foundation	foundation	NOUN
cana-4781	344	31	,	,	PUNCT
cana-4781	344	32	algebraic	algebraic	ADJ
cana-4781	344	33	,	,	PUNCT
cana-4781	344	34	and	and	CCONJ
cana-4781	344	35	analytical	analytical	ADJ
cana-4781	344	36	methods	method	NOUN
cana-4781	344	37	in	in	ADP
cana-4781	344	38	soft	soft	ADJ
cana-4781	344	39	computing	computing	NOUN
cana-4781	344	40	,	,	PUNCT
cana-4781	344	41	volume	volume	NOUN
cana-4781	344	42	27	27	NUM
cana-4781	344	43	,	,	PUNCT
cana-4781	344	44	pages	page	NOUN
cana-4781	344	45	57	57	NUM
cana-4781	344	46	-	-	SYM
cana-4781	344	47	61	61	NUM
cana-4781	344	48	(	(	PUNCT
cana-4781	344	49	2023	2023	NUM
cana-4781	344	50	)	)	PUNCT
cana-4781	344	51	.	.	PUNCT
cana-4781	345	1	[	[	X
cana-4781	345	2	16	16	NUM
cana-4781	345	3	]	]	PUNCT
cana-4781	345	4	tamilarasi	tamilarasi	NOUN
cana-4781	345	5	.	.	PUNCT
cana-4781	346	1	p	p	X
cana-4781	346	2	,	,	PUNCT
cana-4781	346	3	selvi	selvi	PROPN
cana-4781	346	4	.	.	PUNCT
cana-4781	347	1	r	r	X
cana-4781	347	2	,	,	PUNCT
cana-4781	347	3	topological	topological	ADJ
cana-4781	347	4	simple	simple	ADJ
cana-4781	347	5	rough	rough	ADJ
cana-4781	347	6	groups	group	NOUN
cana-4781	347	7	,	,	PUNCT
cana-4781	347	8	journal	journal	NOUN
cana-4781	347	9	of	of	ADP
cana-4781	347	10	the	the	DET
cana-4781	347	11	indian	indian	PROPN
cana-4781	347	12	academy	academy	PROPN
cana-4781	347	13	of	of	ADP
cana-4781	347	14	mathematics	mathematics	PROPN
cana-4781	347	15	(	(	PUNCT
cana-4781	347	16	accepted	accept	VERB
cana-4781	347	17	)	)	PUNCT
cana-4781	347	18	.	.	PUNCT
cana-4781	348	1	[	[	X
cana-4781	348	2	17	17	NUM
cana-4781	348	3	]	]	PUNCT
cana-4781	348	4	tamilarasi	tamilarasi	NOUN
cana-4781	348	5	.	.	PUNCT
cana-4781	349	1	p	p	X
cana-4781	349	2	,	,	PUNCT
cana-4781	349	3	selvi	selvi	PROPN
cana-4781	349	4	.	.	PUNCT
cana-4781	350	1	r	r	NOUN
cana-4781	350	2	,	,	PUNCT
cana-4781	350	3	separation	separation	NOUN
cana-4781	350	4	axioms	axiom	NOUN
cana-4781	350	5	on	on	ADP
cana-4781	350	6	topological	topological	ADJ
cana-4781	350	7	simple	simple	ADJ
cana-4781	350	8	rough	rough	ADJ
cana-4781	350	9	groups	group	NOUN
cana-4781	350	10	,	,	PUNCT
cana-4781	350	11	proceeding	proceed	VERB
cana-4781	350	12	of	of	ADP
cana-4781	350	13	the	the	DET
cana-4781	350	14	international	international	ADJ
cana-4781	350	15	conference	conference	NOUN
cana-4781	350	16	on	on	ADP
cana-4781	350	17	recent	recent	ADJ
cana-4781	350	18	trends	trend	NOUN
cana-4781	350	19	in	in	ADP
cana-4781	350	20	applied	applied	ADJ
cana-4781	350	21	mathematics	mathematic	NOUN
cana-4781	350	22	and	and	CCONJ
cana-4781	350	23	computer	computer	NOUN
cana-4781	350	24	science	science	NOUN
cana-4781	350	25	,	,	PUNCT
cana-4781	350	26	virudhunagar	virudhunagar	ADJ
cana-4781	350	27	hindu	hindu	NOUN
cana-4781	350	28	nadars	nadar	NOUN
cana-4781	350	29	’	'	PUNCT
cana-4781	350	30	senthikumara	senthikumara	PROPN
cana-4781	350	31	nadar	nadar	PROPN
cana-4781	350	32	college	college	PROPN
cana-4781	350	33	,	,	PUNCT
cana-4781	350	34	2025	2025	NUM
cana-4781	350	35	,	,	PUNCT
cana-4781	350	36	p.no	p.no	NOUN
cana-4781	350	37	:	:	PUNCT
cana-4781	350	38	82	82	NUM
cana-4781	350	39	-	-	SYM
cana-4781	350	40	89	89	NUM
cana-4781	350	41	.	.	PUNCT