id	sid	tid	token	lemma	pos
cana-809	1	1	communications	communication	NOUN
cana-809	1	2	on	on	ADP
cana-809	1	3	applied	apply	VERB
cana-809	1	4	nonlinear	nonlinear	ADJ
cana-809	1	5	analysis	analysis	NOUN
cana-809	1	6	issn	issn	NOUN
cana-809	1	7	:	:	PUNCT
cana-809	1	8	1074	1074	NUM
cana-809	1	9	-	-	PUNCT
cana-809	1	10	133x	133x	NUM
cana-809	1	11	vol	vol	NOUN
cana-809	1	12	31	31	NUM
cana-809	1	13	no	no	NOUN
cana-809	1	14	.	.	PUNCT
cana-809	2	1	2s	2s	NUM
cana-809	2	2	(	(	PUNCT
cana-809	2	3	2024	2024	NUM
cana-809	2	4	)	)	PUNCT
cana-809	2	5	718	718	NUM
cana-809	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-809	2	7	(	(	PUNCT
cana-809	2	8	1	1	NUM
cana-809	2	9	,	,	PUNCT
cana-809	2	10	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	2	11	sets	set	VERB
cana-809	2	12	in	in	ADP
cana-809	2	13	bitopological	bitopological	ADJ
cana-809	2	14	spaces	space	NOUN
cana-809	2	15	p.	p.	PROPN
cana-809	2	16	devi	devi	PROPN
cana-809	3	1	prabha1	prabha1	PROPN
cana-809	3	2	,	,	PUNCT
cana-809	3	3	r.	r.	PROPN
cana-809	3	4	asokan2	asokan2	PROPN
cana-809	4	1	1&2	1&2	PROPN
cana-809	4	2	department	department	PROPN
cana-809	4	3	of	of	ADP
cana-809	4	4	mathematics	mathematics	PROPN
cana-809	4	5	school	school	NOUN
cana-809	4	6	of	of	ADP
cana-809	4	7	mathematics	mathematics	PROPN
cana-809	4	8	madurai	madurai	PROPN
cana-809	4	9	kamaraj	kamaraj	ADJ
cana-809	4	10	university	university	NOUN
cana-809	4	11	madurai-625021	madurai-625021	NOUN
cana-809	4	12	,	,	PUNCT
cana-809	4	13	tamil	tamil	PROPN
cana-809	4	14	nadu	nadu	PROPN
cana-809	4	15	,	,	PUNCT
cana-809	4	16	india	india	PROPN
cana-809	4	17	email	email	NOUN
cana-809	4	18	:	:	PUNCT
cana-809	4	19	1deviprabhaponnusamy@gmail.com	1deviprabhaponnusamy@gmail.com	NUM
cana-809	4	20	,	,	PUNCT
cana-809	4	21	2asokan.maths@mkuniversity.org	2asokan.maths@mkuniversity.org	NUM
cana-809	4	22	article	article	NOUN
cana-809	4	23	history	history	NOUN
cana-809	4	24	:	:	PUNCT
cana-809	4	25	received	receive	VERB
cana-809	4	26	:	:	PUNCT
cana-809	4	27	08	08	NUM
cana-809	4	28	-	-	PUNCT
cana-809	4	29	04	04	NUM
cana-809	4	30	-	-	PUNCT
cana-809	4	31	2024	2024	NUM
cana-809	4	32	revised	revise	VERB
cana-809	4	33	:	:	PUNCT
cana-809	4	34	20	20	NUM
cana-809	4	35	-	-	SYM
cana-809	4	36	05	05	NUM
cana-809	4	37	-	-	PUNCT
cana-809	4	38	2024	2024	NUM
cana-809	4	39	accepted	accept	VERB
cana-809	4	40	:	:	PUNCT
cana-809	4	41	07	07	NUM
cana-809	4	42	-	-	PUNCT
cana-809	4	43	06	06	NUM
cana-809	4	44	-	-	PUNCT
cana-809	4	45	2024	2024	NUM
cana-809	4	46	abstract	abstract	NOUN
cana-809	4	47	this	this	DET
cana-809	4	48	article	article	NOUN
cana-809	4	49	explains	explain	VERB
cana-809	4	50	the	the	DET
cana-809	4	51	concept	concept	NOUN
cana-809	4	52	of	of	ADP
cana-809	4	53	(	(	PUNCT
cana-809	4	54	1	1	NUM
cana-809	4	55	,	,	PUNCT
cana-809	4	56	2)*d	2)*d	NOUN
cana-809	4	57	*	*	NOUN
cana-809	4	58	*	*	PUNCT
cana-809	4	59	semipre	semipre	VERB
cana-809	4	60	-	-	PUNCT
cana-809	4	61	open	open	ADJ
cana-809	4	62	sets	set	NOUN
cana-809	4	63	based	base	VERB
cana-809	4	64	on	on	ADP
cana-809	4	65	the	the	DET
cana-809	4	66	concepts	concept	NOUN
cana-809	4	67	of	of	ADP
cana-809	4	68	semipreopen	semipreopen	ADJ
cana-809	4	69	sets	set	NOUN
cana-809	4	70	and	and	CCONJ
cana-809	4	71	semiprecontinuity	semiprecontinuity	NOUN
cana-809	4	72	in	in	ADP
cana-809	4	73	topological	topological	ADJ
cana-809	4	74	space	space	NOUN
cana-809	4	75	.	.	PUNCT
cana-809	5	1	in	in	ADP
cana-809	5	2	addition	addition	NOUN
cana-809	5	3	to	to	ADP
cana-809	5	4	that	that	SCONJ
cana-809	5	5	the	the	DET
cana-809	5	6	concept	concept	NOUN
cana-809	5	7	of	of	ADP
cana-809	5	8	(	(	PUNCT
cana-809	5	9	1,2)*d**sp	1,2)*d**sp	NUM
cana-809	5	10	generalized	generalize	VERB
cana-809	5	11	continuous	continuous	ADJ
cana-809	5	12	maps	map	NOUN
cana-809	5	13	and	and	CCONJ
cana-809	5	14	generalized	generalized	ADJ
cana-809	5	15	homeomorphisms	homeomorphism	NOUN
cana-809	5	16	are	be	AUX
cana-809	5	17	also	also	ADV
cana-809	5	18	discussed	discuss	VERB
cana-809	5	19	.	.	PUNCT
cana-809	6	1	keywords	keyword	NOUN
cana-809	6	2	:	:	PUNCT
cana-809	6	3	(	(	PUNCT
cana-809	6	4	1	1	NUM
cana-809	6	5	,	,	PUNCT
cana-809	6	6	2)*-open	2)*-open	NUM
cana-809	6	7	map	map	NOUN
cana-809	6	8	,	,	PUNCT
cana-809	6	9	(	(	PUNCT
cana-809	6	10	1	1	NUM
cana-809	6	11	,	,	PUNCT
cana-809	6	12	2)*-d**spos	2)*-d**spos	NUM
cana-809	6	13	,	,	PUNCT
cana-809	6	14	(	(	PUNCT
cana-809	6	15	1	1	NUM
cana-809	6	16	,	,	PUNCT
cana-809	6	17	2)*-d**spcontinuous	2)*-d**spcontinuous	ADJ
cana-809	6	18	map	map	NOUN
cana-809	6	19	.	.	PUNCT
cana-809	7	1	1	1	X
cana-809	7	2	.	.	X
cana-809	7	3	introduction	introduction	NOUN
cana-809	7	4	bhattacharya	bhattacharya	NOUN
cana-809	7	5	and	and	CCONJ
cana-809	7	6	lahiri	lahiri	PROPN
cana-809	8	1	[	[	X
cana-809	8	2	1	1	NUM
cana-809	8	3	]	]	PUNCT
cana-809	8	4	introduced	introduce	VERB
cana-809	8	5	a	a	DET
cana-809	8	6	new	new	ADJ
cana-809	8	7	class	class	NOUN
cana-809	8	8	of	of	ADP
cana-809	8	9	semi	semi	ADJ
cana-809	8	10	generalized	generalized	ADJ
cana-809	8	11	open	open	ADJ
cana-809	8	12	sets	set	NOUN
cana-809	8	13	by	by	ADP
cana-809	8	14	means	mean	NOUN
cana-809	8	15	of	of	ADP
cana-809	8	16	semi	semi	ADJ
cana-809	8	17	open	open	ADJ
cana-809	8	18	sets	set	NOUN
cana-809	8	19	introduced	introduce	VERB
cana-809	8	20	by	by	ADP
cana-809	8	21	levine	levine	PROPN
cana-809	8	22	[	[	X
cana-809	8	23	5	5	NUM
cana-809	8	24	]	]	PUNCT
cana-809	8	25	.	.	PUNCT
cana-809	9	1	in	in	ADP
cana-809	9	2	view	view	NOUN
cana-809	9	3	of	of	ADP
cana-809	9	4	that	that	PRON
cana-809	9	5	we	we	PRON
cana-809	9	6	introduce	introduce	VERB
cana-809	9	7	a	a	DET
cana-809	9	8	new	new	ADJ
cana-809	9	9	class	class	NOUN
cana-809	9	10	of	of	ADP
cana-809	9	11	open	open	ADJ
cana-809	9	12	sets	set	NOUN
cana-809	9	13	namely	namely	ADV
cana-809	9	14	(	(	PUNCT
cana-809	9	15	1	1	NUM
cana-809	9	16	,	,	PUNCT
cana-809	9	17	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	9	18	sets	set	NOUN
cana-809	9	19	and	and	CCONJ
cana-809	9	20	their	their	PRON
cana-809	9	21	properties	property	NOUN
cana-809	9	22	are	be	AUX
cana-809	9	23	also	also	ADV
cana-809	9	24	studied	study	VERB
cana-809	9	25	..	..	PUNCT
cana-809	9	26	also	also	ADV
cana-809	9	27	(	(	PUNCT
cana-809	9	28	1	1	NUM
cana-809	9	29	,	,	PUNCT
cana-809	9	30	2)*-d**spcontinuous	2)*-d**spcontinuous	ADJ
cana-809	9	31	maps	map	NOUN
cana-809	9	32	,	,	PUNCT
cana-809	9	33	irresolute	irresolute	ADJ
cana-809	9	34	maps	map	NOUN
cana-809	9	35	,	,	PUNCT
cana-809	9	36	(	(	PUNCT
cana-809	9	37	1	1	NUM
cana-809	9	38	,	,	PUNCT
cana-809	9	39	2)*-d**spconnected	2)*-d**spconnected	PROPN
cana-809	9	40	sets	set	NOUN
cana-809	9	41	,	,	PUNCT
cana-809	9	42	homeomorphism	homeomorphism	PROPN
cana-809	9	43	are	be	AUX
cana-809	9	44	also	also	ADV
cana-809	9	45	studied	study	VERB
cana-809	9	46	with	with	ADP
cana-809	9	47	their	their	PRON
cana-809	9	48	characterizations	characterization	NOUN
cana-809	9	49	.	.	PUNCT
cana-809	10	1	2	2	X
cana-809	10	2	.	.	X
cana-809	10	3	preliminaries	preliminary	NOUN
cana-809	10	4	entire	entire	ADJ
cana-809	10	5	area	area	NOUN
cana-809	10	6	of	of	ADP
cana-809	10	7	this	this	DET
cana-809	10	8	paper	paper	NOUN
cana-809	10	9	,	,	PUNCT
cana-809	10	10	(	(	PUNCT
cana-809	10	11	x	x	X
cana-809	10	12	,	,	PUNCT
cana-809	10	13	1	1	NUM
cana-809	10	14	,	,	PUNCT
cana-809	10	15	2	2	NUM
cana-809	10	16	)	)	PUNCT
cana-809	10	17	x	x	X
cana-809	10	18	will	will	AUX
cana-809	10	19	denote	denote	VERB
cana-809	10	20	bitopological	bitopological	ADJ
cana-809	10	21	space	space	NOUN
cana-809	10	22	(	(	PUNCT
cana-809	10	23	briefly	briefly	ADV
cana-809	10	24	,	,	PUNCT
cana-809	10	25	btps	btps	NOUN
cana-809	10	26	)	)	PUNCT
cana-809	10	27	.	.	PUNCT
cana-809	11	1	definition	definition	NOUN
cana-809	11	2	2.1	2.1	NUM
cana-809	11	3	:	:	PUNCT
cana-809	11	4	let	let	VERB
cana-809	11	5	h	h	NOUN
cana-809	11	6	be	be	AUX
cana-809	11	7	a	a	DET
cana-809	11	8	subset	subset	NOUN
cana-809	11	9	of	of	ADP
cana-809	11	10	x.	x.	NOUN
cana-809	11	11	then	then	ADV
cana-809	11	12	h	h	PROPN
cana-809	11	13	is	be	AUX
cana-809	11	14	said	say	VERB
cana-809	11	15	to	to	PART
cana-809	11	16	be	be	AUX
cana-809	11	17	1,2	1,2	NOUN
cana-809	11	18	-	-	VERB
cana-809	11	19	open	open	ADJ
cana-809	12	1	[	[	X
cana-809	12	2	7	7	NUM
cana-809	12	3	]	]	X
cana-809	12	4	if	if	SCONJ
cana-809	12	5	h	h	NOUN
cana-809	12	6	=	=	PUNCT
cana-809	12	7	a	a	DET
cana-809	12	8			NOUN
cana-809	12	9	b	b	PROPN
cana-809	12	10	where	where	SCONJ
cana-809	12	11	a	a	DET
cana-809	12	12			NOUN
cana-809	12	13	1	1	NOUN
cana-809	12	14	and	and	CCONJ
cana-809	12	15	b	b	NOUN
cana-809	12	16			NOUN
cana-809	12	17			VERB
cana-809	12	18	the	the	DET
cana-809	12	19	complement	complement	NOUN
cana-809	12	20	of	of	ADP
cana-809	12	21	1,2	1,2	NOUN
cana-809	12	22	-	-	ADJ
cana-809	12	23	open	open	ADJ
cana-809	12	24	set	set	NOUN
cana-809	12	25	is	be	AUX
cana-809	12	26	called	call	VERB
cana-809	12	27	1,2	1,2	ADV
cana-809	12	28	-	-	PUNCT
cana-809	12	29	closed	closed	ADJ
cana-809	12	30	.	.	PUNCT
cana-809	13	1	notice	notice	VERB
cana-809	13	2	that	that	SCONJ
cana-809	13	3	1,2	1,2	ADV
cana-809	13	4	-	-	SYM
cana-809	13	5	open	open	ADJ
cana-809	13	6	sets	set	NOUN
cana-809	13	7	need	need	AUX
cana-809	13	8	not	not	PART
cana-809	13	9	necessarily	necessarily	ADV
cana-809	13	10	form	form	VERB
cana-809	13	11	a	a	DET
cana-809	13	12	topology	topology	NOUN
cana-809	13	13	note	note	NOUN
cana-809	13	14	;	;	PUNCT
cana-809	13	15	1,2	1,2	NUM
cana-809	13	16	-	-	ADJ
cana-809	13	17	open	open	ADJ
cana-809	13	18	sets	set	NOUN
cana-809	13	19	need	need	AUX
cana-809	13	20	not	not	PART
cana-809	13	21	necessarily	necessarily	ADV
cana-809	13	22	form	form	VERB
cana-809	13	23	a	a	DET
cana-809	13	24	topology	topology	NOUN
cana-809	13	25	.	.	PUNCT
cana-809	14	1	definition	definition	NOUN
cana-809	14	2	2.2	2.2	NUM
cana-809	15	1	[	[	X
cana-809	15	2	7	7	NUM
cana-809	15	3	]	]	PUNCT
cana-809	15	4	:	:	PUNCT
cana-809	15	5	let	let	VERB
cana-809	15	6	h	h	NOUN
cana-809	15	7	be	be	AUX
cana-809	15	8	a	a	DET
cana-809	15	9	subset	subset	NOUN
cana-809	15	10	of	of	ADP
cana-809	15	11	a	a	DET
cana-809	15	12	bitopological	bitopological	ADJ
cana-809	15	13	space	space	NOUN
cana-809	15	14	x.	x.	NOUN
cana-809	16	1	then	then	ADV
cana-809	16	2	(	(	PUNCT
cana-809	16	3	i	i	NOUN
cana-809	16	4	)	)	PUNCT
cana-809	16	5	the	the	DET
cana-809	16	6	τ1,2	τ1,2	ADJ
cana-809	16	7	-	-	NOUN
cana-809	16	8	closure	closure	NOUN
cana-809	16	9	of	of	ADP
cana-809	16	10	h	h	NOUN
cana-809	16	11	,	,	PUNCT
cana-809	16	12	denoted	denote	VERB
cana-809	16	13	by	by	ADP
cana-809	16	14	τ1,2	τ1,2	NOUN
cana-809	16	15	-	-	PUNCT
cana-809	16	16	cl(h	cl(h	NUM
cana-809	16	17	)	)	PUNCT
cana-809	16	18	,	,	PUNCT
cana-809	16	19	is	be	AUX
cana-809	16	20	defined	define	VERB
cana-809	16	21	as	as	ADP
cana-809	16	22	{	{	PUNCT
cana-809	16	23	f	f	NOUN
cana-809	16	24	:	:	PUNCT
cana-809	16	25	h	h	PROPN
cana-809	16	26			PROPN
cana-809	16	27	f	f	PROPN
cana-809	16	28	and	and	CCONJ
cana-809	16	29	f	f	PROPN
cana-809	16	30	is	be	AUX
cana-809	16	31	τ1,2	τ1,2	ADJ
cana-809	16	32	-	-	ADJ
cana-809	16	33	closed	closed	ADJ
cana-809	16	34	}	}	PUNCT
cana-809	16	35	(	(	PUNCT
cana-809	16	36	ii	ii	NOUN
cana-809	16	37	)	)	PUNCT
cana-809	16	38	the	the	DET
cana-809	16	39	τ1,2	τ1,2	ADJ
cana-809	16	40	-	-	ADJ
cana-809	16	41	interior	interior	NOUN
cana-809	16	42	of	of	ADP
cana-809	16	43	h	h	NOUN
cana-809	16	44	,	,	PUNCT
cana-809	16	45	denoted	denote	VERB
cana-809	16	46	by	by	ADP
cana-809	16	47	τ1,2	τ1,2	PROPN
cana-809	16	48	-	-	PUNCT
cana-809	16	49	int(h	int(h	ADV
cana-809	16	50	)	)	PUNCT
cana-809	16	51	,	,	PUNCT
cana-809	16	52	is	be	AUX
cana-809	16	53	defined	define	VERB
cana-809	16	54	as	as	ADP
cana-809	16	55	{	{	PUNCT
cana-809	16	56	f	f	NOUN
cana-809	16	57	:	:	PUNCT
cana-809	16	58	f	f	PROPN
cana-809	16	59			PROPN
cana-809	16	60	h	h	PROPN
cana-809	16	61	and	and	CCONJ
cana-809	16	62	f	f	PROPN
cana-809	16	63	is	be	AUX
cana-809	16	64	τ1,2	τ1,2	ADJ
cana-809	16	65	-	-	ADJ
cana-809	16	66	open	open	ADJ
cana-809	16	67	}	}	PUNCT
cana-809	16	68	mailto:1deviprabhaponnusamy@gmail.com	mailto:1deviprabhaponnusamy@gmail.com	NOUN
cana-809	16	69	mailto:2asokan.maths@mkuniversity.org	mailto:2asokan.maths@mkuniversity.org	X
cana-809	16	70	communications	communication	NOUN
cana-809	16	71	on	on	ADP
cana-809	16	72	applied	apply	VERB
cana-809	16	73	nonlinear	nonlinear	ADJ
cana-809	16	74	analysis	analysis	NOUN
cana-809	16	75	issn	issn	NOUN
cana-809	16	76	:	:	PUNCT
cana-809	16	77	1074	1074	NUM
cana-809	16	78	-	-	PUNCT
cana-809	16	79	133x	133x	NUM
cana-809	16	80	vol	vol	NOUN
cana-809	16	81	31	31	NUM
cana-809	16	82	no	no	NOUN
cana-809	16	83	.	.	PUNCT
cana-809	17	1	2s	2s	NUM
cana-809	17	2	(	(	PUNCT
cana-809	17	3	2024	2024	NUM
cana-809	17	4	)	)	PUNCT
cana-809	17	5	719	719	NUM
cana-809	17	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-809	17	7	definition	definition	NOUN
cana-809	17	8	2.3	2.3	NUM
cana-809	17	9	:	:	PUNCT
cana-809	17	10	a	a	DET
cana-809	17	11	subset	subset	ADJ
cana-809	17	12	h	h	NOUN
cana-809	17	13	of	of	ADP
cana-809	17	14	a	a	DET
cana-809	17	15	btps	btps	NOUN
cana-809	17	16	x	x	X
cana-809	17	17	is	be	AUX
cana-809	17	18	called	call	VERB
cana-809	17	19	:	:	PUNCT
cana-809	17	20	(	(	PUNCT
cana-809	17	21	i	i	NOUN
cana-809	17	22	)	)	PUNCT
cana-809	17	23	(	(	PUNCT
cana-809	17	24	1	1	NUM
cana-809	17	25	,	,	PUNCT
cana-809	17	26	2)*-semi	2)*-semi	NUM
cana-809	17	27	-	-	ADJ
cana-809	17	28	open	open	ADJ
cana-809	17	29	set	set	NOUN
cana-809	17	30	[	[	X
cana-809	17	31	8	8	NUM
cana-809	17	32	]	]	X
cana-809	17	33	if	if	SCONJ
cana-809	17	34	h	h	NOUN
cana-809	17	35			PROPN
cana-809	17	36	1,2	1,2	ADV
cana-809	17	37	-	-	PUNCT
cana-809	17	38	cl(1,2	cl(1,2	NOUN
cana-809	17	39	-	-	PUNCT
cana-809	17	40	int(h	int(h	NOUN
cana-809	17	41	)	)	PUNCT
cana-809	17	42	)	)	PUNCT
cana-809	17	43	;	;	PUNCT
cana-809	17	44	(	(	PUNCT
cana-809	17	45	ii	ii	NOUN
cana-809	17	46	)	)	PUNCT
cana-809	17	47	(	(	PUNCT
cana-809	17	48	1	1	NUM
cana-809	17	49	,	,	PUNCT
cana-809	17	50	2)*-preopen	2)*-preopen	PROPN
cana-809	17	51	set	set	VERB
cana-809	17	52	[	[	X
cana-809	17	53	6	6	NUM
cana-809	17	54	]	]	PUNCT
cana-809	17	55	if	if	SCONJ
cana-809	17	56	h	h	NOUN
cana-809	17	57			PROPN
cana-809	17	58	1,2	1,2	NUM
cana-809	17	59	-	-	PUNCT
cana-809	17	60	int(1,2	int(1,2	NOUN
cana-809	17	61	-	-	PUNCT
cana-809	17	62	cl(h	cl(h	NUM
cana-809	17	63	)	)	PUNCT
cana-809	17	64	)	)	PUNCT
cana-809	17	65	;	;	PUNCT
cana-809	17	66	(	(	PUNCT
cana-809	17	67	iii	iii	X
cana-809	17	68	)	)	PUNCT
cana-809	17	69	(	(	PUNCT
cana-809	17	70	1	1	NUM
cana-809	17	71	,	,	PUNCT
cana-809	17	72	2)*--open	2)*--open	NOUN
cana-809	17	73	set	set	NOUN
cana-809	17	74	[	[	X
cana-809	17	75	3	3	X
cana-809	17	76	]	]	PUNCT
cana-809	17	77	if	if	SCONJ
cana-809	17	78	h	h	NOUN
cana-809	17	79			PROPN
cana-809	17	80	1,2	1,2	NUM
cana-809	17	81	-	-	PUNCT
cana-809	17	82	int(1,2	int(1,2	ADJ
cana-809	17	83	-	-	PUNCT
cana-809	17	84	cl(1,2	cl(1,2	NOUN
cana-809	17	85	-	-	PUNCT
cana-809	17	86	int(h	int(h	NOUN
cana-809	17	87	)	)	PUNCT
cana-809	17	88	)	)	PUNCT
cana-809	17	89	)	)	PUNCT
cana-809	17	90	;	;	PUNCT
cana-809	17	91	(	(	PUNCT
cana-809	17	92	iv	iv	X
cana-809	17	93	)	)	PUNCT
cana-809	17	94	regular	regular	ADJ
cana-809	17	95	(	(	PUNCT
cana-809	17	96	1	1	NUM
cana-809	17	97	,	,	PUNCT
cana-809	17	98	2)*-open	2)*-open	NUM
cana-809	17	99	set	set	VERB
cana-809	18	1	[	[	X
cana-809	18	2	6	6	NUM
cana-809	18	3	]	]	PUNCT
cana-809	18	4	if	if	SCONJ
cana-809	18	5	h	h	NOUN
cana-809	18	6	=	=	SYM
cana-809	18	7	1,2	1,2	NUM
cana-809	18	8	-	-	PUNCT
cana-809	18	9	int(1,2	int(1,2	NOUN
cana-809	18	10	-	-	PUNCT
cana-809	18	11	cl(h	cl(h	NUM
cana-809	18	12	)	)	PUNCT
cana-809	18	13	)	)	PUNCT
cana-809	18	14	.	.	PUNCT
cana-809	19	1	(	(	PUNCT
cana-809	19	2	v	v	NOUN
cana-809	19	3	)	)	PUNCT
cana-809	19	4	(	(	PUNCT
cana-809	19	5	1	1	NUM
cana-809	19	6	,	,	PUNCT
cana-809	19	7	2)*-gsp	2)*-gsp	NUM
cana-809	19	8	-	-	ADJ
cana-809	19	9	closed	closed	ADJ
cana-809	19	10	[	[	X
cana-809	19	11	10	10	NUM
cana-809	19	12	,	,	PUNCT
cana-809	19	13	11	11	NUM
cana-809	19	14	]	]	PUNCT
cana-809	19	15	if	if	SCONJ
cana-809	19	16	(	(	PUNCT
cana-809	19	17	1,2)*-cl(a	1,2)*-cl(a	NUM
cana-809	19	18	)	)	PUNCT
cana-809	19	19			PROPN
cana-809	19	20	u	u	PROPN
cana-809	19	21	whenever	whenever	SCONJ
cana-809	19	22	a	a	DET
cana-809	19	23			PROPN
cana-809	19	24	u	u	NOUN
cana-809	19	25	and	and	CCONJ
cana-809	19	26	u	u	NOUN
cana-809	19	27	is	be	AUX
cana-809	19	28	(	(	PUNCT
cana-809	19	29	1	1	NUM
cana-809	19	30	,	,	PUNCT
cana-809	19	31	2)*-open	2)*-open	NUM
cana-809	19	32	.	.	PUNCT
cana-809	20	1	then	then	ADV
cana-809	20	2	complement	complement	VERB
cana-809	20	3	of	of	ADP
cana-809	20	4	(	(	PUNCT
cana-809	20	5	1	1	NUM
cana-809	20	6	,	,	PUNCT
cana-809	20	7	2)*-gsp	2)*-gsp	NUM
cana-809	20	8	-	-	PUNCT
cana-809	20	9	closed	closed	ADJ
cana-809	20	10	set	set	NOUN
cana-809	20	11	is	be	AUX
cana-809	20	12	called	call	VERB
cana-809	20	13	(	(	PUNCT
cana-809	20	14	1	1	NUM
cana-809	20	15	,	,	PUNCT
cana-809	20	16	2)*-gsp	2)*-gsp	ADJ
cana-809	20	17	-	-	ADJ
cana-809	20	18	open	open	ADJ
cana-809	20	19	set	set	NOUN
cana-809	20	20	.	.	PUNCT
cana-809	21	1	the	the	DET
cana-809	21	2	complements	complement	NOUN
cana-809	21	3	of	of	ADP
cana-809	21	4	the	the	DET
cana-809	21	5	above	above	ADV
cana-809	21	6	-	-	PUNCT
cana-809	21	7	mentioned	mention	VERB
cana-809	21	8	open	open	ADJ
cana-809	21	9	sets	set	NOUN
cana-809	21	10	are	be	AUX
cana-809	21	11	called	call	VERB
cana-809	21	12	their	their	PRON
cana-809	21	13	respective	respective	ADJ
cana-809	21	14	closed	closed	ADJ
cana-809	21	15	sets	set	NOUN
cana-809	21	16	.	.	PUNCT
cana-809	22	1	3	3	X
cana-809	22	2	.	.	X
cana-809	22	3	definition	definition	NOUN
cana-809	22	4	2.4	2.4	NUM
cana-809	22	5	:	:	PUNCT
cana-809	23	1	[	[	X
cana-809	23	2	2	2	X
cana-809	23	3	]	]	PUNCT
cana-809	23	4	a	a	DET
cana-809	23	5	subset	subset	ADJ
cana-809	23	6	h	h	NOUN
cana-809	23	7	of	of	ADP
cana-809	23	8	a	a	DET
cana-809	23	9	space	space	NOUN
cana-809	23	10	(	(	PUNCT
cana-809	23	11	x	x	X
cana-809	23	12	,	,	PUNCT
cana-809	23	13	𝜏	𝜏	NOUN
cana-809	23	14	)	)	PUNCT
cana-809	23	15	is	be	AUX
cana-809	23	16	called	call	VERB
cana-809	23	17	-cld	-cld	PROPN
cana-809	23	18	if	if	SCONJ
cana-809	23	19	it	it	PRON
cana-809	23	20	contains	contain	VERB
cana-809	23	21	all	all	DET
cana-809	23	22	its	its	PRON
cana-809	23	23	condensation	condensation	NOUN
cana-809	23	24	points	point	NOUN
cana-809	23	25	.	.	PUNCT
cana-809	24	1	the	the	DET
cana-809	24	2	complement	complement	NOUN
cana-809	24	3	of	of	ADP
cana-809	24	4	-cld	-cld	NOUN
cana-809	24	5	set	set	VERB
cana-809	24	6	is	be	AUX
cana-809	24	7	called	call	VERB
cana-809	24	8	-open	-open	ADJ
cana-809	24	9	.	.	PUNCT
cana-809	25	1	4	4	X
cana-809	25	2	.	.	X
cana-809	25	3	definition	definition	NOUN
cana-809	25	4	2.5	2.5	NUM
cana-809	25	5	.	.	PUNCT
cana-809	26	1	a	a	DET
cana-809	26	2	bijection	bijection	NOUN
cana-809	26	3	f	f	NOUN
cana-809	26	4	:	:	PUNCT
cana-809	26	5	x	x	X
cana-809	26	6	→	→	SYM
cana-809	26	7	y	y	PROPN
cana-809	26	8	is	be	AUX
cana-809	26	9	called	call	VERB
cana-809	26	10	(	(	PUNCT
cana-809	26	11	1	1	NUM
cana-809	26	12	,	,	PUNCT
cana-809	26	13	2)*-homeomorphism	2)*-homeomorphism	NUM
cana-809	26	14	[	[	X
cana-809	26	15	4	4	X
cana-809	26	16	]	]	X
cana-809	26	17	if	if	SCONJ
cana-809	26	18	f	f	PROPN
cana-809	26	19	is	be	AUX
cana-809	26	20	bijection	bijection	ADJ
cana-809	26	21	,	,	PUNCT
cana-809	26	22	(	(	PUNCT
cana-809	26	23	1,2)*continuous	1,2)*continuous	NUM
cana-809	26	24	and	and	CCONJ
cana-809	26	25	(	(	PUNCT
cana-809	26	26	1,2)*-open	1,2)*-open	NUM
cana-809	26	27	.	.	NOUN
cana-809	27	1	5	5	NUM
cana-809	27	2	.	.	X
cana-809	27	3	definition	definition	NOUN
cana-809	27	4	2.6	2.6	NUM
cana-809	27	5	:	:	PUNCT
cana-809	27	6	a	a	DET
cana-809	27	7	subset	subset	NOUN
cana-809	27	8	a	a	PRON
cana-809	27	9	of	of	ADP
cana-809	27	10	x	x	PRON
cana-809	27	11	is	be	AUX
cana-809	27	12	called	call	VERB
cana-809	27	13	(	(	PUNCT
cana-809	27	14	1	1	NUM
cana-809	27	15	,	,	PUNCT
cana-809	27	16	2)*-d*-cld	2)*-d*-cld	PROPN
cana-809	27	17	(	(	PUNCT
cana-809	27	18	briefly	briefly	ADV
cana-809	27	19	,	,	PUNCT
cana-809	27	20	(	(	PUNCT
cana-809	27	21	1,2)*-d*-cld	1,2)*-d*-cld	NUM
cana-809	27	22	)	)	PUNCT
cana-809	27	23	if	if	SCONJ
cana-809	27	24	(	(	PUNCT
cana-809	27	25	1	1	NUM
cana-809	27	26	,	,	PUNCT
cana-809	27	27	2)*scl*(a	2)*scl*(a	NUM
cana-809	27	28	)	)	PUNCT
cana-809	27	29			PROPN
cana-809	27	30	(	(	PUNCT
cana-809	27	31	1	1	NUM
cana-809	27	32	,	,	PUNCT
cana-809	27	33	2)*-int(u	2)*-int(u	NUM
cana-809	27	34	)	)	PUNCT
cana-809	27	35	whenever	whenever	SCONJ
cana-809	27	36	a	a	DET
cana-809	27	37			PROPN
cana-809	27	38	u	u	NOUN
cana-809	27	39	and	and	CCONJ
cana-809	27	40	u	u	NOUN
cana-809	27	41	is	be	AUX
cana-809	27	42	(	(	PUNCT
cana-809	27	43	1	1	NUM
cana-809	27	44	,	,	PUNCT
cana-809	27	45	2)*-ω	2)*-ω	PROPN
cana-809	27	46	-	-	PUNCT
cana-809	27	47	open	open	ADJ
cana-809	27	48	.	.	PUNCT
cana-809	28	1	the	the	DET
cana-809	28	2	complement	complement	NOUN
cana-809	28	3	of	of	ADP
cana-809	28	4	(	(	PUNCT
cana-809	28	5	1	1	NUM
cana-809	28	6	,	,	PUNCT
cana-809	28	7	2)*-d*-cld	2)*-d*-cld	PROPN
cana-809	28	8	set	set	NOUN
cana-809	28	9	is	be	AUX
cana-809	28	10	called	call	VERB
cana-809	28	11	(	(	PUNCT
cana-809	28	12	1	1	NUM
cana-809	28	13	,	,	PUNCT
cana-809	28	14	2)*-d*-open	2)*-d*-open	NUM
cana-809	28	15	.	.	PUNCT
cana-809	29	1	6	6	X
cana-809	29	2	.	.	X
cana-809	29	3	definition	definition	NOUN
cana-809	29	4	2.7	2.7	NUM
cana-809	29	5	:	:	PUNCT
cana-809	29	6	(	(	PUNCT
cana-809	29	7	1	1	NUM
cana-809	29	8	,	,	PUNCT
cana-809	29	9	2)*-d**-closed	2)*-d**-close	VERB
cana-809	29	10	(	(	PUNCT
cana-809	29	11	briefly	briefly	ADV
cana-809	29	12	,	,	PUNCT
cana-809	29	13	(	(	PUNCT
cana-809	29	14	1	1	NUM
cana-809	29	15	,	,	PUNCT
cana-809	29	16	2)*-d**-cld	2)*-d**-cld	NOUN
cana-809	29	17	)	)	PUNCT
cana-809	29	18	if	if	SCONJ
cana-809	29	19	(	(	PUNCT
cana-809	29	20	1	1	NUM
cana-809	29	21	,	,	PUNCT
cana-809	29	22	2)*-spcl(a	2)*-spcl(a	NUM
cana-809	29	23	)	)	PUNCT
cana-809	29	24			PROPN
cana-809	29	25	u	u	PROPN
cana-809	29	26	whenever	whenever	SCONJ
cana-809	29	27	a	a	DET
cana-809	29	28			PROPN
cana-809	29	29	u	u	NOUN
cana-809	29	30	and	and	CCONJ
cana-809	29	31	u	u	NOUN
cana-809	29	32	is	be	AUX
cana-809	29	33	(	(	PUNCT
cana-809	29	34	1,2)*-d*-open	1,2)*-d*-open	NUM
cana-809	29	35	.	.	PUNCT
cana-809	30	1	the	the	DET
cana-809	30	2	complement	complement	NOUN
cana-809	30	3	of	of	ADP
cana-809	30	4	(	(	PUNCT
cana-809	30	5	1	1	NUM
cana-809	30	6	,	,	PUNCT
cana-809	30	7	2)*-d**-closed	2)*-d**-close	VERB
cana-809	30	8	set	set	NOUN
cana-809	30	9	is	be	AUX
cana-809	30	10	called	call	VERB
cana-809	30	11	(	(	PUNCT
cana-809	30	12	1	1	NUM
cana-809	30	13	,	,	PUNCT
cana-809	30	14	2)*-d**-open	2)*-d**-open	NUM
cana-809	30	15	.	.	PUNCT
cana-809	31	1	the	the	DET
cana-809	31	2	class	class	NOUN
cana-809	31	3	of	of	ADP
cana-809	31	4	all	all	PRON
cana-809	31	5	(	(	PUNCT
cana-809	31	6	1	1	NUM
cana-809	31	7	,	,	PUNCT
cana-809	31	8	2)*-d**-cld	2)*-d**-cld	NOUN
cana-809	31	9	in	in	ADP
cana-809	31	10	x	x	PROPN
cana-809	31	11	is	be	AUX
cana-809	31	12	denoted	denote	VERB
cana-809	31	13	by	by	ADP
cana-809	31	14	(	(	PUNCT
cana-809	31	15	1	1	NUM
cana-809	31	16	,	,	PUNCT
cana-809	31	17	2)*-d**-c(x	2)*-d**-c(x	NUM
cana-809	31	18	)	)	PUNCT
cana-809	31	19	.	.	PUNCT
cana-809	32	1	the	the	DET
cana-809	32	2	complements	complement	NOUN
cana-809	32	3	of	of	ADP
cana-809	32	4	the	the	DET
cana-809	32	5	above	above	ADJ
cana-809	32	6	mentioned	mention	VERB
cana-809	32	7	open	open	ADJ
cana-809	32	8	sets	set	NOUN
cana-809	32	9	are	be	AUX
cana-809	32	10	called	call	VERB
cana-809	32	11	their	their	PRON
cana-809	32	12	respective	respective	ADJ
cana-809	32	13	closed	closed	ADJ
cana-809	32	14	sets	set	NOUN
cana-809	32	15	.	.	PUNCT
cana-809	33	1	3	3	NUM
cana-809	33	2	(	(	PUNCT
cana-809	33	3	1	1	NUM
cana-809	33	4	,	,	PUNCT
cana-809	33	5	2)*-d**spopen	2)*-d**spopen	PROPN
cana-809	33	6	sets	set	VERB
cana-809	33	7	definition	definition	NOUN
cana-809	33	8	3.1	3.1	NUM
cana-809	33	9	for	for	ADP
cana-809	33	10	s	s	NOUN
cana-809	33	11	⊆	⊆	NUM
cana-809	33	12	x	x	SYM
cana-809	33	13	,	,	PUNCT
cana-809	33	14	(	(	PUNCT
cana-809	33	15	1	1	NUM
cana-809	33	16	,	,	PUNCT
cana-809	33	17	2)*-spcl**(s	2)*-spcl**(s	NUM
cana-809	33	18	)	)	PUNCT
cana-809	34	1	=	=	SYM
cana-809	34	2	∩{k	∩{k	PROPN
cana-809	34	3	/	/	SYM
cana-809	34	4	s	s	PROPN
cana-809	34	5	⊆	⊆	NUM
cana-809	34	6	k	k	NOUN
cana-809	34	7	,	,	PUNCT
cana-809	34	8	k	k	PROPN
cana-809	34	9	is	be	AUX
cana-809	34	10	(	(	PUNCT
cana-809	34	11	1	1	NUM
cana-809	34	12	,	,	PUNCT
cana-809	34	13	2)*gspclosed	2)*gspclosed	NUM
cana-809	34	14	}	}	PUNCT
cana-809	34	15	.	.	PUNCT
cana-809	35	1	remark	remark	PROPN
cana-809	35	2	3.2	3.2	NUM
cana-809	35	3	(	(	PUNCT
cana-809	35	4	1	1	NUM
cana-809	35	5	,	,	PUNCT
cana-809	35	6	2)*-spcl**(s	2)*-spcl**(s	NUM
cana-809	35	7	)	)	PUNCT
cana-809	36	1	=	=	VERB
cana-809	36	2	kuratowski	kuratowski	ADJ
cana-809	36	3	closure	closure	NOUN
cana-809	36	4	operator	operator	NOUN
cana-809	36	5	on	on	ADP
cana-809	36	6	x	x	X
cana-809	36	7	.	.	PUNCT
cana-809	37	1	definition	definition	NOUN
cana-809	37	2	3.3	3.3	NUM
cana-809	37	3	s	s	NOUN
cana-809	37	4	⊆	⊆	NUM
cana-809	37	5	x	x	SYM
cana-809	37	6	,	,	PUNCT
cana-809	37	7	(	(	PUNCT
cana-809	37	8	1	1	NUM
cana-809	37	9	,	,	PUNCT
cana-809	37	10	2)*-d**spopen	2)*-d**spopen	PROPN
cana-809	37	11	iff	iff	NOUN
cana-809	37	12	there	there	PRON
cana-809	37	13	exists	exist	VERB
cana-809	37	14	an	an	DET
cana-809	37	15	τ1,2os	τ1,2o	NOUN
cana-809	37	16	u	u	NOUN
cana-809	37	17	such	such	ADJ
cana-809	37	18	that	that	SCONJ
cana-809	37	19	u⊆s	u⊆s	PROPN
cana-809	37	20	⊆	⊆	NUM
cana-809	37	21	(	(	PUNCT
cana-809	37	22	1	1	NUM
cana-809	37	23	,	,	PUNCT
cana-809	37	24	2)*spcl**(u	2)*spcl**(u	NUM
cana-809	37	25	)	)	PUNCT
cana-809	37	26	.	.	PUNCT
cana-809	38	1	in	in	ADP
cana-809	38	2	x.	x.	PROPN
cana-809	38	3	example	example	NOUN
cana-809	38	4	3.4	3.4	NUM
cana-809	38	5	let	let	VERB
cana-809	38	6	g	g	NOUN
cana-809	38	7	=	=	PUNCT
cana-809	38	8	{	{	PUNCT
cana-809	38	9	1	1	NUM
cana-809	38	10	,	,	PUNCT
cana-809	38	11	2	2	NUM
cana-809	38	12	,	,	PUNCT
cana-809	38	13	3	3	NUM
cana-809	38	14	}	}	PUNCT
cana-809	38	15	,	,	PUNCT
cana-809	38	16	τ1	τ1	NOUN
cana-809	38	17	=	=	SYM
cana-809	38	18	{	{	PUNCT
cana-809	38	19	g	g	PROPN
cana-809	38	20	,	,	PUNCT
cana-809	38	21	φ	φ	PROPN
cana-809	38	22	,	,	PUNCT
cana-809	38	23	{	{	PUNCT
cana-809	38	24	1	1	NUM
cana-809	38	25	}	}	PUNCT
cana-809	38	26	}	}	PUNCT
cana-809	38	27	and	and	CCONJ
cana-809	38	28	τ2	τ2	NOUN
cana-809	38	29	=	=	SYM
cana-809	38	30	{	{	PUNCT
cana-809	38	31	g	g	PROPN
cana-809	38	32	,	,	PUNCT
cana-809	38	33	φ	φ	PROPN
cana-809	38	34	,	,	PUNCT
cana-809	38	35	{	{	PUNCT
cana-809	38	36	1	1	NUM
cana-809	38	37	}	}	PUNCT
cana-809	38	38	,	,	PUNCT
cana-809	38	39	{	{	PUNCT
cana-809	38	40	1	1	NUM
cana-809	38	41	,	,	PUNCT
cana-809	38	42	2	2	NUM
cana-809	38	43	}	}	PUNCT
cana-809	38	44	}	}	PUNCT
cana-809	38	45	.	.	PUNCT
cana-809	39	1	then	then	ADV
cana-809	39	2	(	(	PUNCT
cana-809	39	3	1	1	NUM
cana-809	39	4	,	,	PUNCT
cana-809	39	5	2)*-d**spos	2)*-d**spos	NUM
cana-809	39	6	of	of	ADP
cana-809	39	7	(	(	PUNCT
cana-809	39	8	x	x	NOUN
cana-809	39	9	,	,	PUNCT
cana-809	39	10	τ1	τ1	NOUN
cana-809	39	11	,	,	PUNCT
cana-809	39	12	τ2	τ2	NOUN
cana-809	39	13	)	)	PUNCT
cana-809	39	14	are	be	AUX
cana-809	39	15	x	x	X
cana-809	39	16	,	,	PUNCT
cana-809	39	17	φ	φ	PROPN
cana-809	39	18	,	,	PUNCT
cana-809	39	19	{	{	PUNCT
cana-809	39	20	1	1	NUM
cana-809	39	21	}	}	PUNCT
cana-809	39	22	,	,	PUNCT
cana-809	39	23	{	{	PUNCT
cana-809	39	24	1,2	1,2	NUM
cana-809	39	25	}	}	PUNCT
cana-809	39	26	and	and	CCONJ
cana-809	39	27	{	{	PUNCT
cana-809	39	28	1,3	1,3	NUM
cana-809	39	29	}	}	PUNCT
cana-809	39	30	.	.	PUNCT
cana-809	40	1	remark	remark	VERB
cana-809	40	2	3.5	3.5	NUM
cana-809	40	3	if	if	SCONJ
cana-809	40	4	c	c	PROPN
cana-809	40	5	⊆	⊆	NUM
cana-809	40	6	x	x	SYM
cana-809	40	7	,	,	PUNCT
cana-809	40	8	d	d	PROPN
cana-809	40	9	⊆	⊆	NUM
cana-809	40	10	x	x	SYM
cana-809	40	11	∋	∋	NOUN
cana-809	40	12	c	c	NOUN
cana-809	40	13	⊆	⊆	NUM
cana-809	40	14	d	d	NOUN
cana-809	40	15	,	,	PUNCT
cana-809	40	16	then	then	ADV
cana-809	40	17	we	we	PRON
cana-809	40	18	have	have	VERB
cana-809	40	19	(	(	PUNCT
cana-809	40	20	1	1	NUM
cana-809	40	21	,	,	PUNCT
cana-809	40	22	2)*-spcl**(c	2)*-spcl**(c	NUM
cana-809	40	23	)	)	PUNCT
cana-809	40	24	⊆	⊆	NUM
cana-809	40	25	(	(	PUNCT
cana-809	40	26	1	1	NUM
cana-809	40	27	,	,	PUNCT
cana-809	40	28	2)*-spcl	2)*-spcl	NOUN
cana-809	40	29	*	*	NOUN
cana-809	40	30	*	*	PUNCT
cana-809	40	31	(	(	PUNCT
cana-809	40	32	d	d	NOUN
cana-809	40	33	)	)	PUNCT
cana-809	40	34	.	.	PUNCT
cana-809	41	1	communications	communication	NOUN
cana-809	41	2	on	on	ADP
cana-809	41	3	applied	apply	VERB
cana-809	41	4	nonlinear	nonlinear	ADJ
cana-809	41	5	analysis	analysis	NOUN
cana-809	41	6	issn	issn	NOUN
cana-809	41	7	:	:	PUNCT
cana-809	41	8	1074	1074	NUM
cana-809	41	9	-	-	PUNCT
cana-809	41	10	133x	133x	NUM
cana-809	41	11	vol	vol	NOUN
cana-809	41	12	31	31	NUM
cana-809	41	13	no	no	NOUN
cana-809	41	14	.	.	PUNCT
cana-809	42	1	2s	2s	NUM
cana-809	42	2	(	(	PUNCT
cana-809	42	3	2024	2024	NUM
cana-809	42	4	)	)	PUNCT
cana-809	42	5	720	720	NUM
cana-809	42	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-809	42	7	theorem	theorem	VERB
cana-809	42	8	3.6	3.6	NUM
cana-809	42	9	for	for	ADP
cana-809	42	10	s	s	NOUN
cana-809	42	11	⊆	⊆	NUM
cana-809	42	12	x.	x.	NOUN
cana-809	42	13	we	we	PRON
cana-809	42	14	have	have	VERB
cana-809	42	15	s	s	VERB
cana-809	42	16	is	be	AUX
cana-809	42	17	(	(	PUNCT
cana-809	42	18	1	1	NUM
cana-809	42	19	,	,	PUNCT
cana-809	42	20	2)*d**spopen	2)*d**spopen	PROPN
cana-809	42	21	iff	iff	PROPN
cana-809	42	22	s	s	PROPN
cana-809	42	23			PROPN
cana-809	42	24	(	(	PUNCT
cana-809	42	25	1	1	NUM
cana-809	42	26	,	,	PUNCT
cana-809	42	27	2)*-spcl**(τ1,2int(s	2)*-spcl**(τ1,2int(s	NUM
cana-809	42	28	)	)	PUNCT
cana-809	42	29	)	)	PUNCT
cana-809	42	30	.	.	PUNCT
cana-809	43	1	proof	proof	NOUN
cana-809	43	2	.	.	PUNCT
cana-809	44	1	assume	assume	VERB
cana-809	44	2	s	s	X
cana-809	44	3	is	be	AUX
cana-809	44	4	(	(	PUNCT
cana-809	44	5	1	1	NUM
cana-809	44	6	,	,	PUNCT
cana-809	44	7	2)*d**spopen	2)*d**spopen	VERB
cana-809	44	8	in	in	ADP
cana-809	44	9	x	x	X
cana-809	44	10	.	.	PUNCT
cana-809	45	1	u	u	PROPN
cana-809	45	2			PROPN
cana-809	45	3	s	s	PROPN
cana-809	45	4	implies	imply	VERB
cana-809	45	5	u	u	NOUN
cana-809	45	6	⊆	⊆	NUM
cana-809	45	7	τ1,2int(s	τ1,2int(s	NUM
cana-809	45	8	)	)	PUNCT
cana-809	45	9	.	.	PUNCT
cana-809	46	1	hence	hence	ADV
cana-809	46	2	from	from	ADP
cana-809	46	3	remark	remark	NOUN
cana-809	46	4	3.5	3.5	NUM
cana-809	46	5	and	and	CCONJ
cana-809	46	6	by	by	ADP
cana-809	46	7	defn	defn	PROPN
cana-809	46	8	3.3	3.3	NUM
cana-809	46	9	,	,	PUNCT
cana-809	46	10	(	(	PUNCT
cana-809	46	11	1	1	NUM
cana-809	46	12	,	,	PUNCT
cana-809	46	13	2)∗-spcl**(u	2)∗-spcl**(u	NUM
cana-809	46	14	)	)	PUNCT
cana-809	46	15			PROPN
cana-809	46	16	(	(	PUNCT
cana-809	46	17	1	1	NUM
cana-809	46	18	,	,	PUNCT
cana-809	46	19	2)*spcl**(τ1,2	2)*spcl**(τ1,2	NOUN
cana-809	46	20	-	-	SYM
cana-809	46	21	int(s	int(s	NUM
cana-809	46	22	)	)	PUNCT
cana-809	46	23	)	)	PUNCT
cana-809	46	24	.	.	PUNCT
cana-809	47	1	so	so	ADV
cana-809	47	2	s	s	VERB
cana-809	47	3	⊆	⊆	NUM
cana-809	47	4	(	(	PUNCT
cana-809	47	5	1,2)*-spcl**(τ1,2	1,2)*-spcl**(τ1,2	NUM
cana-809	47	6	-	-	PUNCT
cana-809	47	7	int(s	int(s	PROPN
cana-809	47	8	)	)	PUNCT
cana-809	47	9	)	)	PUNCT
cana-809	47	10	.	.	PUNCT
cana-809	48	1	to	to	PART
cana-809	48	2	prove	prove	VERB
cana-809	48	3	the	the	DET
cana-809	48	4	converse	converse	NOUN
cana-809	48	5	let	let	VERB
cana-809	48	6	s	s	PRON
cana-809	48	7	⊆	⊆	NUM
cana-809	48	8	(	(	PUNCT
cana-809	48	9	1	1	NUM
cana-809	48	10	,	,	PUNCT
cana-809	48	11	2)*-spcl**(τ1,2int(s	2)*-spcl**(τ1,2int(s	NUM
cana-809	48	12	)	)	PUNCT
cana-809	48	13	)	)	PUNCT
cana-809	48	14	.	.	PUNCT
cana-809	49	1	substitute	substitute	PROPN
cana-809	49	2	u	u	NOUN
cana-809	49	3	=	=	PUNCT
cana-809	49	4	τ1,2	τ1,2	PROPN
cana-809	49	5	-	-	PUNCT
cana-809	49	6	int(s	int(s	PROPN
cana-809	49	7	)	)	PUNCT
cana-809	49	8	.	.	PUNCT
cana-809	50	1	finally	finally	ADV
cana-809	50	2	by	by	ADP
cana-809	50	3	defn	defn	PROPN
cana-809	50	4	3.3	3.3	NUM
cana-809	50	5	.	.	PUNCT
cana-809	51	1	s	s	VERB
cana-809	51	2	in	in	ADP
cana-809	51	3	x	x	SYM
cana-809	51	4	is	be	AUX
cana-809	51	5	(	(	PUNCT
cana-809	51	6	1	1	NUM
cana-809	51	7	,	,	PUNCT
cana-809	51	8	2)*d**spos	2)*d**spos	NUM
cana-809	51	9	.	.	PUNCT
cana-809	52	1	theorem	theorem	VERB
cana-809	52	2	3.7	3.7	NUM
cana-809	52	3	in	in	ADP
cana-809	52	4	bts	bt	NOUN
cana-809	52	5	x	x	SYM
cana-809	52	6	τ1,2	τ1,2	ADJ
cana-809	52	7	-	-	PUNCT
cana-809	52	8	os(r	os(r	NUM
cana-809	52	9	)	)	PUNCT
cana-809	52	10	⇒	⇒	NOUN
cana-809	52	11	(	(	PUNCT
cana-809	52	12	1	1	NUM
cana-809	52	13	,	,	PUNCT
cana-809	52	14	2)*-d**spos(r	2)*-d**spos(r	NUM
cana-809	52	15	)	)	PUNCT
cana-809	52	16	proof	proof	NOUN
cana-809	52	17	.	.	PUNCT
cana-809	53	1	assume	assume	VERB
cana-809	53	2	r	r	NOUN
cana-809	53	3	be	be	AUX
cana-809	53	4	a	a	DET
cana-809	53	5	τ1,2os	τ1,2o	NOUN
cana-809	53	6	in	in	ADP
cana-809	53	7	x	x	PART
cana-809	53	8	let	let	VERB
cana-809	53	9	r	r	NOUN
cana-809	53	10	=	=	PUNCT
cana-809	53	11	τ1,2int(r	τ1,2int(r	NOUN
cana-809	53	12	)	)	PUNCT
cana-809	53	13	⊆	⊆	NUM
cana-809	53	14	(	(	PUNCT
cana-809	53	15	1	1	NUM
cana-809	53	16	,	,	PUNCT
cana-809	53	17	2)*-spcl**(τ1,2int(r	2)*-spcl**(τ1,2int(r	NUM
cana-809	53	18	)	)	PUNCT
cana-809	53	19	)	)	PUNCT
cana-809	53	20	.	.	PUNCT
cana-809	54	1	we	we	PRON
cana-809	54	2	have	have	VERB
cana-809	54	3	s	s	VERB
cana-809	54	4	is	be	AUX
cana-809	54	5	(	(	PUNCT
cana-809	54	6	1	1	NUM
cana-809	54	7	,	,	PUNCT
cana-809	54	8	2)*-d**spos	2)*-d**spos	NUM
cana-809	54	9	in	in	ADP
cana-809	54	10	x	x	PROPN
cana-809	54	11	.	.	PUNCT
cana-809	54	12	example	example	NOUN
cana-809	54	13	3.8	3.8	NUM
cana-809	54	14	reverse	reverse	NOUN
cana-809	54	15	of	of	ADP
cana-809	54	16	theorem	theorem	ADJ
cana-809	54	17	3.7	3.7	NUM
cana-809	54	18	is	be	AUX
cana-809	54	19	proved	prove	VERB
cana-809	54	20	by	by	ADP
cana-809	54	21	this	this	DET
cana-809	54	22	example	example	NOUN
cana-809	54	23	s	s	X
cana-809	54	24	=	=	SYM
cana-809	54	25	{	{	PUNCT
cana-809	54	26	1,3	1,3	NUM
cana-809	54	27	}	}	PUNCT
cana-809	54	28	is	be	AUX
cana-809	54	29	a	a	DET
cana-809	54	30	(	(	PUNCT
cana-809	54	31	1	1	NUM
cana-809	54	32	,	,	PUNCT
cana-809	54	33	2)*-d**sp	2)*-d**sp	NUM
cana-809	54	34	-	-	NOUN
cana-809	54	35	os	os	NOUN
cana-809	54	36	in	in	ADP
cana-809	54	37	x	x	PROPN
cana-809	54	38	but	but	CCONJ
cana-809	54	39	not	not	PART
cana-809	54	40	an	an	DET
cana-809	54	41	τ1,2	τ1,2	ADJ
cana-809	54	42	-	-	NOUN
cana-809	54	43	os	os	NOUN
cana-809	54	44	in	in	ADP
cana-809	54	45	x.	x.	NOUN
cana-809	54	46	definition	definition	NOUN
cana-809	54	47	3.9	3.9	NUM
cana-809	54	48	in	in	ADP
cana-809	54	49	bts	bt	NOUN
cana-809	54	50	x	x	SYM
cana-809	54	51	(	(	PUNCT
cana-809	54	52	1	1	NUM
cana-809	54	53	,	,	PUNCT
cana-809	54	54	2)*-d**spt1/2	2)*-d**spt1/2	NUM
cana-809	54	55	space	space	NOUN
cana-809	54	56	for	for	ADP
cana-809	54	57	every	every	DET
cana-809	54	58	(	(	PUNCT
cana-809	54	59	1	1	NUM
cana-809	54	60	,	,	PUNCT
cana-809	54	61	2)*-d**spos	2)*-d**spos	NUM
cana-809	54	62	is	be	AUX
cana-809	54	63	τ1,2os	τ1,2o	NOUN
cana-809	54	64	remark	remark	VERB
cana-809	54	65	3.10	3.10	NUM
cana-809	54	66	in	in	ADP
cana-809	54	67	(	(	PUNCT
cana-809	54	68	1	1	NUM
cana-809	54	69	,	,	PUNCT
cana-809	54	70	2)*t1/2	2)*t1/2	NUM
cana-809	54	71	space	space	NOUN
cana-809	54	72	,	,	PUNCT
cana-809	54	73	every	every	DET
cana-809	54	74	(	(	PUNCT
cana-809	54	75	1	1	NUM
cana-809	54	76	,	,	PUNCT
cana-809	54	77	2)*spos	2)*spos	NUM
cana-809	54	78	is	be	AUX
cana-809	54	79	(	(	PUNCT
cana-809	54	80	1	1	NUM
cana-809	54	81	,	,	PUNCT
cana-809	54	82	2)*-d**spos	2)*-d**spos	NUM
cana-809	54	83	.	.	PUNCT
cana-809	55	1	remark	remark	NOUN
cana-809	55	2	3.11	3.11	NUM
cana-809	55	3	(	(	PUNCT
cana-809	55	4	1	1	NUM
cana-809	55	5	,	,	PUNCT
cana-809	55	6	2)∗-spcl**(a	2)∗-spcl**(a	NUM
cana-809	55	7	)	)	PUNCT
cana-809	55	8	⊆	⊆	NUM
cana-809	55	9	τ1,2spcl(a	τ1,2spcl(a	NUM
cana-809	55	10	)	)	PUNCT
cana-809	55	11	for	for	ADP
cana-809	55	12	a	a	DET
cana-809	55	13	subset	subset	NOUN
cana-809	55	14	a	a	PRON
cana-809	55	15	in	in	ADP
cana-809	55	16	x.	x.	NOUN
cana-809	55	17	theorem	theorem	VERB
cana-809	55	18	3.12	3.12	NUM
cana-809	55	19	in	in	ADP
cana-809	55	20	bts	bt	NOUN
cana-809	55	21	x.	x.	NOUN
cana-809	55	22	s	s	PART
cana-809	55	23	is	be	AUX
cana-809	55	24	(	(	PUNCT
cana-809	55	25	1	1	NUM
cana-809	55	26	,	,	PUNCT
cana-809	56	1	2)*-d**spos	2)*-d**spos	NUM
cana-809	56	2	⇒	⇒	NOUN
cana-809	56	3	s	s	VERB
cana-809	56	4	i	i	PRON
cana-809	56	5	s	s	X
cana-809	56	6	(	(	PUNCT
cana-809	56	7	1	1	NUM
cana-809	56	8	,	,	PUNCT
cana-809	56	9	2)*-spos	2)*-spos	NUM
cana-809	56	10	.	.	PUNCT
cana-809	57	1	proof	proof	NOUN
cana-809	57	2	in	in	ADP
cana-809	57	3	x	x	X
cana-809	57	4	,	,	PUNCT
cana-809	57	5	suppose	suppose	VERB
cana-809	57	6	s	s	NOUN
cana-809	57	7	is	be	AUX
cana-809	57	8	(	(	PUNCT
cana-809	57	9	1	1	NUM
cana-809	57	10	,	,	PUNCT
cana-809	57	11	2)*d**spos	2)*d**spos	NUM
cana-809	57	12	and	and	CCONJ
cana-809	57	13	by	by	ADP
cana-809	57	14	defn	defn	PROPN
cana-809	57	15	3.3	3.3	NUM
cana-809	57	16	and	and	CCONJ
cana-809	57	17	.	.	PUNCT
cana-809	58	1	by	by	ADP
cana-809	58	2	remark	remark	NOUN
cana-809	58	3	3.11	3.11	NUM
cana-809	58	4	,	,	PUNCT
cana-809	58	5	(	(	PUNCT
cana-809	58	6	1	1	NUM
cana-809	58	7	,	,	PUNCT
cana-809	58	8	2	2	NUM
cana-809	58	9	)	)	PUNCT
cana-809	58	10	*	*	NOUN
cana-809	58	11	spcl**(u	spcl**(u	PROPN
cana-809	58	12	)	)	PUNCT
cana-809	58	13	⊆	⊆	NUM
cana-809	58	14	τ1,2	τ1,2	PROPN
cana-809	58	15	-	-	PUNCT
cana-809	58	16	spcl(u	spcl(u	NOUN
cana-809	58	17	)	)	PUNCT
cana-809	58	18	.	.	PUNCT
cana-809	59	1	hence	hence	ADV
cana-809	59	2	u	u	NOUN
cana-809	59	3	⊆	⊆	NUM
cana-809	59	4	s	s	ADP
cana-809	59	5	⊆	⊆	NUM
cana-809	59	6	τ1,2	τ1,2	PROPN
cana-809	59	7	-	-	PUNCT
cana-809	59	8	spcl(u	spcl(u	NOUN
cana-809	59	9	)	)	PUNCT
cana-809	59	10	⇒	⇒	NOUN
cana-809	59	11	s	s	PART
cana-809	59	12	is	be	AUX
cana-809	59	13	(	(	PUNCT
cana-809	59	14	1,2)*-spos	1,2)*-spos	NUM
cana-809	59	15	.	.	PUNCT
cana-809	59	16	example	example	NOUN
cana-809	59	17	3.13	3.13	NUM
cana-809	59	18	to	to	PART
cana-809	59	19	prove	prove	VERB
cana-809	59	20	the	the	DET
cana-809	59	21	reverse	reverse	NOUN
cana-809	59	22	of	of	ADP
cana-809	59	23	thm	thm	PROPN
cana-809	59	24	3.12	3.12	NUM
cana-809	59	25	is	be	AUX
cana-809	59	26	not	not	PART
cana-809	59	27	true	true	ADJ
cana-809	59	28	assume	assume	NOUN
cana-809	59	29	g	g	NOUN
cana-809	59	30	=	=	SYM
cana-809	59	31	{	{	PUNCT
cana-809	59	32	1	1	NUM
cana-809	59	33	,	,	PUNCT
cana-809	59	34	2	2	NUM
cana-809	59	35	,	,	PUNCT
cana-809	59	36	3	3	NUM
cana-809	59	37	}	}	PUNCT
cana-809	59	38	,	,	PUNCT
cana-809	59	39	t1	t1	NOUN
cana-809	59	40	=	=	PUNCT
cana-809	59	41	{	{	PUNCT
cana-809	59	42	g	g	PROPN
cana-809	59	43	,	,	PUNCT
cana-809	59	44	φ	φ	PROPN
cana-809	59	45	,	,	PUNCT
cana-809	59	46	{	{	PUNCT
cana-809	59	47	2	2	NUM
cana-809	59	48	}	}	PUNCT
cana-809	59	49	}	}	PUNCT
cana-809	59	50	and	and	CCONJ
cana-809	59	51	t2	t2	PROPN
cana-809	59	52	=	=	SYM
cana-809	59	53	{	{	PUNCT
cana-809	59	54	g	g	PROPN
cana-809	59	55	,	,	PUNCT
cana-809	59	56	φ	φ	NUM
cana-809	59	57	}	}	PUNCT
cana-809	59	58	.	.	PUNCT
cana-809	60	1	communications	communication	NOUN
cana-809	60	2	on	on	ADP
cana-809	60	3	applied	apply	VERB
cana-809	60	4	nonlinear	nonlinear	ADJ
cana-809	60	5	analysis	analysis	NOUN
cana-809	60	6	issn	issn	NOUN
cana-809	60	7	:	:	PUNCT
cana-809	60	8	1074	1074	NUM
cana-809	60	9	-	-	PUNCT
cana-809	60	10	133x	133x	NUM
cana-809	60	11	vol	vol	NOUN
cana-809	60	12	31	31	NUM
cana-809	60	13	no	no	NOUN
cana-809	60	14	.	.	PUNCT
cana-809	61	1	2s	2s	NUM
cana-809	61	2	(	(	PUNCT
cana-809	61	3	2024	2024	NUM
cana-809	61	4	)	)	PUNCT
cana-809	61	5	721	721	NUM
cana-809	62	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-809	62	2	then	then	ADV
cana-809	62	3	s	s	VERB
cana-809	62	4	=	=	PUNCT
cana-809	62	5	{	{	PUNCT
cana-809	62	6	2	2	NUM
cana-809	62	7	,	,	PUNCT
cana-809	62	8	3	3	NUM
cana-809	62	9	}	}	PUNCT
cana-809	62	10	is	be	AUX
cana-809	62	11	(	(	PUNCT
cana-809	62	12	1	1	NUM
cana-809	62	13	,	,	PUNCT
cana-809	62	14	2)*spos	2)*spos	NUM
cana-809	62	15	but	but	CCONJ
cana-809	62	16	not	not	PART
cana-809	62	17	a	a	DET
cana-809	62	18	(	(	PUNCT
cana-809	62	19	1	1	NUM
cana-809	62	20	,	,	PUNCT
cana-809	62	21	2)*-d**spos	2)*-d**spos	NUM
cana-809	62	22	.	.	PUNCT
cana-809	63	1	remark	remark	VERB
cana-809	63	2	3.14	3.14	NUM
cana-809	63	3	in	in	ADP
cana-809	63	4	bts	bt	NOUN
cana-809	63	5	x	x	PRON
cana-809	63	6	consider	consider	VERB
cana-809	63	7	c	c	NOUN
cana-809	63	8	⊆	⊆	NUM
cana-809	63	9	x	x	NOUN
cana-809	63	10	,	,	PUNCT
cana-809	63	11	d	d	PROPN
cana-809	63	12	⊆	⊆	NUM
cana-809	63	13	x	x	X
cana-809	63	14	we	we	PRON
cana-809	63	15	have	have	VERB
cana-809	63	16	(	(	PUNCT
cana-809	63	17	1	1	NUM
cana-809	63	18	,	,	PUNCT
cana-809	63	19	2)*-spcl	2)*-spcl	NOUN
cana-809	63	20	*	*	NOUN
cana-809	63	21	*	*	PUNCT
cana-809	63	22	(	(	PUNCT
cana-809	63	23	c∪d	c∪d	NOUN
cana-809	63	24	)	)	PUNCT
cana-809	63	25	=	=	PUNCT
cana-809	64	1	(	(	PUNCT
cana-809	64	2	1	1	NUM
cana-809	64	3	,	,	PUNCT
cana-809	64	4	2)*-spcl**(c)∪	2)*-spcl**(c)∪	ADJ
cana-809	64	5	(	(	PUNCT
cana-809	64	6	1	1	NUM
cana-809	64	7	,	,	PUNCT
cana-809	64	8	2)*-spcl**(d	2)*-spcl**(d	NUM
cana-809	64	9	)	)	PUNCT
cana-809	64	10	.	.	PUNCT
cana-809	65	1	theorem	theorem	VERB
cana-809	65	2	3.15	3.15	NUM
cana-809	65	3	in	in	ADP
cana-809	65	4	bts	bt	NOUN
cana-809	65	5	x	x	PRON
cana-809	65	6	consider	consider	VERB
cana-809	65	7	c	c	NOUN
cana-809	65	8	⊆	⊆	NUM
cana-809	65	9	x	x	NOUN
cana-809	65	10	,	,	PUNCT
cana-809	65	11	d	d	PROPN
cana-809	65	12	⊆	⊆	NUM
cana-809	65	13	x	x	X
cana-809	65	14	we	we	PRON
cana-809	65	15	have	have	VERB
cana-809	65	16	(	(	PUNCT
cana-809	65	17	1	1	NUM
cana-809	65	18	,	,	PUNCT
cana-809	65	19	2)*-d**spos	2)*-d**spos	NUM
cana-809	65	20	⇒	⇒	NOUN
cana-809	65	21	cuniond	cuniond	NOUN
cana-809	65	22	is	be	AUX
cana-809	65	23	also	also	ADV
cana-809	65	24	a	a	DET
cana-809	65	25	(	(	PUNCT
cana-809	65	26	1	1	NUM
cana-809	65	27	,	,	PUNCT
cana-809	65	28	2)*-d**spos	2)*-d**spos	NUM
cana-809	65	29	.	.	PUNCT
cana-809	66	1	proof	proof	NOUN
cana-809	66	2	.	.	PUNCT
cana-809	67	1	in	in	ADP
cana-809	67	2	bts	bt	NOUN
cana-809	67	3	x	x	PRON
cana-809	67	4	suppose	suppose	VERB
cana-809	67	5	c	c	NOUN
cana-809	67	6	and	and	CCONJ
cana-809	67	7	d	d	PROPN
cana-809	67	8	are	be	AUX
cana-809	67	9	(	(	PUNCT
cana-809	67	10	1	1	NUM
cana-809	67	11	,	,	PUNCT
cana-809	67	12	2)*-d**spos	2)*-d**spos	NUM
cana-809	67	13	in	in	ADP
cana-809	67	14	x.	x.	NOUN
cana-809	67	15	by	by	ADP
cana-809	67	16	defn	defn	PROPN
cana-809	67	17	&	&	CCONJ
cana-809	67	18	remark3.14	remark3.14	PROPN
cana-809	67	19	,	,	PUNCT
cana-809	67	20	(	(	PUNCT
cana-809	67	21	1	1	NUM
cana-809	67	22	,	,	PUNCT
cana-809	67	23	2)*-spcl**(s	2)*-spcl**(s	NUM
cana-809	67	24	)	)	PUNCT
cana-809	67	25	⊆	⊆	NUM
cana-809	67	26	(	(	PUNCT
cana-809	67	27	1	1	NUM
cana-809	67	28	,	,	PUNCT
cana-809	67	29	2)*-spcl	2)*-spcl	NOUN
cana-809	67	30	*	*	NOUN
cana-809	67	31	*	*	PUNCT
cana-809	67	32	(	(	PUNCT
cana-809	67	33	t	t	NOUN
cana-809	67	34	)	)	PUNCT
cana-809	67	35	⇒	⇒	NOUN
cana-809	67	36	(	(	PUNCT
cana-809	67	37	1	1	NUM
cana-809	67	38	,	,	PUNCT
cana-809	67	39	2)*-spcl	2)*-spcl	NOUN
cana-809	67	40	*	*	NOUN
cana-809	67	41	*	*	PUNCT
cana-809	67	42	(	(	PUNCT
cana-809	67	43	s	s	X
cana-809	67	44	∪	∪	ADP
cana-809	67	45	t	t	PROPN
cana-809	67	46	)	)	PUNCT
cana-809	67	47	.	.	PUNCT
cana-809	68	1	⇒	⇒	PROPN
cana-809	68	2	s	s	PART
cana-809	68	3	∪	∪	PROPN
cana-809	68	4	t	t	PROPN
cana-809	68	5	is	be	AUX
cana-809	68	6	also	also	ADV
cana-809	68	7	(	(	PUNCT
cana-809	68	8	1	1	NUM
cana-809	68	9	,	,	PUNCT
cana-809	68	10	2)*-d**spos	2)*-d**spos	NUM
cana-809	68	11	.	.	PUNCT
cana-809	69	1	example	example	NOUN
cana-809	69	2	3.16	3.16	NUM
cana-809	69	3	in	in	ADP
cana-809	69	4	bts	bt	NOUN
cana-809	69	5	x	x	PRON
cana-809	69	6	suppose	suppose	VERB
cana-809	69	7	c	c	NOUN
cana-809	69	8	,	,	PUNCT
cana-809	69	9	d	d	X
cana-809	69	10	are	be	AUX
cana-809	69	11	(	(	PUNCT
cana-809	69	12	1	1	NUM
cana-809	69	13	,	,	PUNCT
cana-809	69	14	2)*-d**spos	2)*-d**spos	NUM
cana-809	69	15	⇒	⇒	NOUN
cana-809	69	16	c	c	NOUN
cana-809	69	17	∩	∩	PROPN
cana-809	69	18	d	d	X
cana-809	69	19	may	may	AUX
cana-809	69	20	not	not	PART
cana-809	69	21	(	(	PUNCT
cana-809	69	22	1	1	NUM
cana-809	69	23	,	,	PUNCT
cana-809	69	24	2)*-d**spos	2)*-d**spos	PROPN
cana-809	69	25	.	.	PUNCT
cana-809	70	1	let	let	VERB
cana-809	70	2	x	x	PUNCT
cana-809	70	3	=	=	PUNCT
cana-809	70	4	{	{	PUNCT
cana-809	70	5	1	1	NUM
cana-809	70	6	,	,	PUNCT
cana-809	70	7	2	2	NUM
cana-809	70	8	,	,	PUNCT
cana-809	70	9	3	3	NUM
cana-809	70	10	,	,	PUNCT
cana-809	70	11	4	4	NUM
cana-809	70	12	}	}	PUNCT
cana-809	70	13	,	,	PUNCT
cana-809	70	14	t1	t1	NOUN
cana-809	70	15	=	=	PUNCT
cana-809	70	16	{	{	PUNCT
cana-809	70	17	x	x	X
cana-809	70	18	,	,	PUNCT
cana-809	70	19	φ	φ	PROPN
cana-809	70	20	,	,	PUNCT
cana-809	70	21	{	{	PUNCT
cana-809	70	22	1	1	NUM
cana-809	70	23	,	,	PUNCT
cana-809	70	24	2	2	NUM
cana-809	70	25	}	}	PUNCT
cana-809	70	26	,	,	PUNCT
cana-809	70	27	{	{	PUNCT
cana-809	70	28	1	1	NUM
cana-809	70	29	,	,	PUNCT
cana-809	70	30	2	2	NUM
cana-809	70	31	,	,	PUNCT
cana-809	70	32	3	3	NUM
cana-809	70	33	}	}	PUNCT
cana-809	70	34	,	,	PUNCT
cana-809	70	35	{	{	PUNCT
cana-809	70	36	1	1	NUM
cana-809	70	37	,	,	PUNCT
cana-809	70	38	2	2	NUM
cana-809	70	39	,	,	PUNCT
cana-809	70	40	4	4	NUM
cana-809	70	41	}	}	PUNCT
cana-809	70	42	}	}	PUNCT
cana-809	70	43	,	,	PUNCT
cana-809	70	44	t2	t2	NOUN
cana-809	70	45	=	=	SYM
cana-809	70	46	{	{	PUNCT
cana-809	70	47	x	x	PROPN
cana-809	70	48	,	,	PUNCT
cana-809	70	49	φ	φ	PROPN
cana-809	70	50	,	,	PUNCT
cana-809	70	51	{	{	PUNCT
cana-809	70	52	1	1	NUM
cana-809	70	53	,	,	PUNCT
cana-809	70	54	2	2	NUM
cana-809	70	55	}	}	PUNCT
cana-809	70	56	,	,	PUNCT
cana-809	70	57	{	{	PUNCT
cana-809	70	58	3	3	NUM
cana-809	70	59	,	,	PUNCT
cana-809	70	60	4	4	NUM
cana-809	70	61	}	}	PUNCT
cana-809	70	62	}	}	PUNCT
cana-809	70	63	.	.	PUNCT
cana-809	71	1	then	then	ADV
cana-809	71	2	the	the	DET
cana-809	71	3	set	set	NOUN
cana-809	71	4	a	a	X
cana-809	71	5	=	=	PUNCT
cana-809	71	6	{	{	PUNCT
cana-809	71	7	1,2,3	1,2,3	NOUN
cana-809	71	8	}	}	PUNCT
cana-809	71	9	and	and	CCONJ
cana-809	71	10	b	b	X
cana-809	71	11	=	=	PUNCT
cana-809	71	12	{	{	PUNCT
cana-809	71	13	3,4	3,4	NUM
cana-809	71	14	}	}	PUNCT
cana-809	71	15	are	be	AUX
cana-809	71	16	(	(	PUNCT
cana-809	71	17	1,2)*d**spos	1,2)*d**spos	NUM
cana-809	71	18	in	in	ADP
cana-809	71	19	x	x	X
cana-809	71	20	and	and	CCONJ
cana-809	71	21	a	a	DET
cana-809	71	22	∩	∩	ADJ
cana-809	71	23	b	b	NOUN
cana-809	71	24	=	=	SYM
cana-809	71	25	{	{	PUNCT
cana-809	71	26	3	3	NUM
cana-809	71	27	}	}	PUNCT
cana-809	71	28	is	be	AUX
cana-809	71	29	not	not	PART
cana-809	71	30	a	a	DET
cana-809	71	31	(	(	PUNCT
cana-809	71	32	1,2	1,2	NUM
cana-809	71	33	)	)	PUNCT
cana-809	71	34	*	*	PUNCT
cana-809	71	35	d**spos	d**spos	NOUN
cana-809	71	36	.	.	PUNCT
cana-809	72	1	theorem	theorem	VERB
cana-809	72	2	3.17	3.17	NUM
cana-809	72	3	in	in	ADP
cana-809	72	4	bts	bt	NOUN
cana-809	73	1	x	x	PRON
cana-809	73	2	assume	assume	VERB
cana-809	73	3	b	b	X
cana-809	73	4	be	be	AUX
cana-809	73	5	a	a	DET
cana-809	73	6	(	(	PUNCT
cana-809	73	7	1	1	NUM
cana-809	73	8	,	,	PUNCT
cana-809	73	9	2)*-d**spos	2)*-d**spos	NUM
cana-809	73	10	,	,	PUNCT
cana-809	73	11	b	b	PROPN
cana-809	73	12	⊆	⊆	NUM
cana-809	73	13	c	c	NOUN
cana-809	73	14	and	and	CCONJ
cana-809	73	15	b	b	NOUN
cana-809	74	1	⊆	⊆	NUM
cana-809	74	2	c	c	NOUN
cana-809	74	3	⊆	⊆	NUM
cana-809	74	4	(	(	PUNCT
cana-809	74	5	(	(	PUNCT
cana-809	74	6	1	1	NUM
cana-809	74	7	,	,	PUNCT
cana-809	74	8	2)*spcl**(τ1,2int(a	2)*spcl**(τ1,2int(a	NUM
cana-809	74	9	)	)	PUNCT
cana-809	74	10	)	)	PUNCT
cana-809	74	11	.	.	PUNCT
cana-809	75	1	we	we	PRON
cana-809	75	2	have	have	VERB
cana-809	75	3	c	c	NOUN
cana-809	75	4	is	be	AUX
cana-809	75	5	a	a	DET
cana-809	75	6	(	(	PUNCT
cana-809	75	7	1	1	NUM
cana-809	75	8	,	,	PUNCT
cana-809	75	9	2)*-d**spos	2)*-d**spos	NUM
cana-809	75	10	proof	proof	NOUN
cana-809	75	11	from	from	ADP
cana-809	75	12	statement	statement	NOUN
cana-809	75	13	b	b	PROPN
cana-809	75	14	is	be	AUX
cana-809	75	15	(	(	PUNCT
cana-809	75	16	1	1	NUM
cana-809	75	17	,	,	PUNCT
cana-809	75	18	2)*-d**spos	2)*-d**spos	NUM
cana-809	75	19	and	and	CCONJ
cana-809	75	20	by	by	ADP
cana-809	75	21	thm	thm	PROPN
cana-809	75	22	3.16	3.16	NUM
cana-809	75	23	b	b	NOUN
cana-809	75	24	⊆	⊆	NUM
cana-809	75	25	(	(	PUNCT
cana-809	75	26	1	1	NUM
cana-809	75	27	,	,	PUNCT
cana-809	75	28	2)*-spcl**(τ1,2int(b	2)*-spcl**(τ1,2int(b	NUM
cana-809	75	29	)	)	PUNCT
cana-809	75	30	)	)	PUNCT
cana-809	75	31	,	,	PUNCT
cana-809	75	32	also	also	ADV
cana-809	75	33	b	b	X
cana-809	75	34	⊆	⊆	NUM
cana-809	75	35	c	c	NOUN
cana-809	75	36	communications	communication	NOUN
cana-809	75	37	on	on	ADP
cana-809	75	38	applied	apply	VERB
cana-809	75	39	nonlinear	nonlinear	ADJ
cana-809	75	40	analysis	analysis	NOUN
cana-809	75	41	issn	issn	NOUN
cana-809	75	42	:	:	PUNCT
cana-809	75	43	1074	1074	NUM
cana-809	75	44	-	-	PUNCT
cana-809	75	45	133x	133x	NUM
cana-809	75	46	vol	vol	NOUN
cana-809	75	47	31	31	NUM
cana-809	75	48	no	no	NOUN
cana-809	75	49	.	.	PUNCT
cana-809	76	1	2s	2s	NUM
cana-809	76	2	(	(	PUNCT
cana-809	76	3	2024	2024	NUM
cana-809	76	4	)	)	PUNCT
cana-809	76	5	722	722	NUM
cana-809	76	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-809	76	7	⇒	⇒	PROPN
cana-809	76	8	τ1,2	τ1,2	PROPN
cana-809	76	9	-	-	PUNCT
cana-809	76	10	int(a	int(a	NOUN
cana-809	76	11	)	)	PUNCT
cana-809	76	12	⊆	⊆	NUM
cana-809	76	13	τ1,2int(b	τ1,2int(b	NUM
cana-809	76	14	)	)	PUNCT
cana-809	76	15	.	.	PUNCT
cana-809	77	1	hence	hence	ADV
cana-809	77	2	,	,	PUNCT
cana-809	77	3	(	(	PUNCT
cana-809	77	4	1	1	NUM
cana-809	77	5	,	,	PUNCT
cana-809	77	6	2)*-spcl**(τ1,2int(b	2)*-spcl**(τ1,2int(b	NUM
cana-809	77	7	)	)	PUNCT
cana-809	77	8	)	)	PUNCT
cana-809	78	1	⊆	⊆	NUM
cana-809	78	2	(	(	PUNCT
cana-809	78	3	1	1	NUM
cana-809	78	4	,	,	PUNCT
cana-809	78	5	2)*-spcl**(τ1,2int(c	2)*-spcl**(τ1,2int(c	NUM
cana-809	78	6	)	)	PUNCT
cana-809	78	7	)	)	PUNCT
cana-809	78	8	.	.	PUNCT
cana-809	79	1	we	we	PRON
cana-809	79	2	have	have	VERB
cana-809	79	3	c	c	NOUN
cana-809	79	4	⊆	⊆	NUM
cana-809	79	5	(	(	PUNCT
cana-809	79	6	1	1	NUM
cana-809	79	7	,	,	PUNCT
cana-809	79	8	2)*-spcl**(τ1,2int(b	2)*-spcl**(τ1,2int(b	NUM
cana-809	79	9	)	)	PUNCT
cana-809	79	10	)	)	PUNCT
cana-809	80	1	⊆	⊆	NUM
cana-809	80	2	(	(	PUNCT
cana-809	80	3	1	1	NUM
cana-809	80	4	,	,	PUNCT
cana-809	80	5	2)*-spcl**(τ1,2int(c	2)*-spcl**(τ1,2int(c	NUM
cana-809	80	6	)	)	PUNCT
cana-809	80	7	)	)	PUNCT
cana-809	81	1	it	it	PRON
cana-809	81	2	proves	prove	VERB
cana-809	81	3	c	c	NOUN
cana-809	81	4	is	be	AUX
cana-809	81	5	(	(	PUNCT
cana-809	81	6	1	1	NUM
cana-809	81	7	,	,	PUNCT
cana-809	81	8	2	2	NUM
cana-809	81	9	)	)	PUNCT
cana-809	81	10	*	*	PUNCT
cana-809	82	1	d**spos	d**spos	NOUN
cana-809	82	2	.	.	PUNCT
cana-809	83	1	remark	remark	VERB
cana-809	83	2	3.18	3.18	NUM
cana-809	83	3	a	a	DET
cana-809	83	4	map	map	NOUN
cana-809	83	5	h	h	NOUN
cana-809	83	6	:	:	PUNCT
cana-809	84	1	l	l	PUNCT
cana-809	84	2	→	→	PUNCT
cana-809	84	3	m	m	NOUN
cana-809	84	4	is	be	AUX
cana-809	84	5	(	(	PUNCT
cana-809	84	6	1	1	NUM
cana-809	84	7	,	,	PUNCT
cana-809	84	8	2)*gspcontinuous	2)*gspcontinuous	ADJ
cana-809	84	9	⇒	⇒	NOUN
cana-809	84	10	f((1	f((1	PROPN
cana-809	84	11	,	,	PUNCT
cana-809	84	12	2)*-spcl**(a	2)*-spcl**(a	NUM
cana-809	84	13	)	)	PUNCT
cana-809	84	14	)	)	PUNCT
cana-809	85	1			PROPN
cana-809	85	2	1,2	1,2	NUM
cana-809	85	3	-	-	PUNCT
cana-809	85	4	spcl	spcl	NOUN
cana-809	85	5	(	(	PUNCT
cana-809	85	6	f(a	f(a	NOUN
cana-809	85	7	)	)	PUNCT
cana-809	85	8	)	)	PUNCT
cana-809	85	9	.	.	PUNCT
cana-809	86	1	theorem	theorem	VERB
cana-809	86	2	3.19	3.19	NUM
cana-809	86	3	in	in	ADP
cana-809	86	4	bts	bt	NOUN
cana-809	86	5	x	x	PRON
cana-809	86	6	suppose	suppose	VERB
cana-809	86	7	a	a	DET
cana-809	86	8	map	map	NOUN
cana-809	86	9	h	h	NOUN
cana-809	86	10	:	:	PUNCT
cana-809	86	11	l	l	PUNCT
cana-809	86	12	→	→	AUX
cana-809	86	13	m	m	AUX
cana-809	86	14	be	be	AUX
cana-809	86	15	(	(	PUNCT
cana-809	86	16	1	1	NUM
cana-809	86	17	,	,	PUNCT
cana-809	86	18	2)*gspcontinuous	2)*gspcontinuous	NUM
cana-809	86	19	and	and	CCONJ
cana-809	86	20	(	(	PUNCT
cana-809	86	21	1	1	NUM
cana-809	86	22	,	,	PUNCT
cana-809	86	23	2)*open	2)*open	NUM
cana-809	86	24	⇒	⇒	NOUN
cana-809	86	25	b	b	NOUN
cana-809	86	26	is	be	AUX
cana-809	86	27	(	(	PUNCT
cana-809	86	28	1	1	NUM
cana-809	86	29	,	,	PUNCT
cana-809	86	30	2	2	NUM
cana-809	86	31	)	)	PUNCT
cana-809	86	32	*	*	PUNCT
cana-809	87	1	d**spos	d**spos	NOUN
cana-809	87	2	⇒	⇒	NOUN
cana-809	87	3	f(b	f(b	PROPN
cana-809	87	4	)	)	PUNCT
cana-809	87	5	in	in	ADP
cana-809	87	6	y	y	PROPN
cana-809	87	7	is	be	AUX
cana-809	87	8	(	(	PUNCT
cana-809	87	9	1	1	NUM
cana-809	87	10	,	,	PUNCT
cana-809	87	11	2)*-spos	2)*-spos	NUM
cana-809	87	12	.	.	PUNCT
cana-809	88	1	proof	proof	NOUN
cana-809	88	2	from	from	ADP
cana-809	88	3	statement	statement	NOUN
cana-809	88	4	b	b	PROPN
cana-809	88	5	is	be	AUX
cana-809	88	6	(	(	PUNCT
cana-809	88	7	1	1	NUM
cana-809	88	8	,	,	PUNCT
cana-809	88	9	2)*-d**spos	2)*-d**spos	PROPN
cana-809	88	10	in	in	ADP
cana-809	88	11	l.	l.	PROPN
cana-809	88	12	by	by	ADP
cana-809	88	13	defn	defn	PROPN
cana-809	88	14	3.3	3.3	NUM
cana-809	88	15	and	and	CCONJ
cana-809	88	16	.	.	PUNCT
cana-809	89	1	remark	remark	PROPN
cana-809	89	2	3.18	3.18	NUM
cana-809	89	3	,	,	PUNCT
cana-809	89	4	h((1	h((1	PROPN
cana-809	89	5	,	,	PUNCT
cana-809	89	6	2	2	NUM
cana-809	89	7	)	)	PUNCT
cana-809	89	8	*	*	PUNCT
cana-809	89	9	spcl**(v	spcl**(v	NOUN
cana-809	89	10	)	)	PUNCT
cana-809	89	11	)	)	PUNCT
cana-809	90	1	⊆	⊆	NUM
cana-809	90	2	σ1,2	σ1,2	NUM
cana-809	90	3	-	-	PUNCT
cana-809	90	4	spcl(h(b	spcl(h(b	PROPN
cana-809	90	5	)	)	PUNCT
cana-809	90	6	)	)	PUNCT
cana-809	90	7	.	.	PUNCT
cana-809	91	1	we	we	PRON
cana-809	91	2	have	have	VERB
cana-809	91	3	h(b	h(b	PROPN
cana-809	91	4	)	)	PUNCT
cana-809	91	5	⊆	⊆	NUM
cana-809	91	6	h((1	h((1	PROPN
cana-809	91	7	,	,	PUNCT
cana-809	91	8	2)*spcl**(v	2)*spcl**(v	NUM
cana-809	91	9	)	)	PUNCT
cana-809	91	10	)	)	PUNCT
cana-809	92	1	⊆	⊆	NUM
cana-809	92	2	σ1,2	σ1,2	PROPN
cana-809	92	3	-	-	PUNCT
cana-809	92	4	spcl(h(v	spcl(h(v	NOUN
cana-809	92	5	)	)	PUNCT
cana-809	92	6	)	)	PUNCT
cana-809	92	7	.	.	PUNCT
cana-809	93	1	also	also	ADV
cana-809	93	2	given	give	VERB
cana-809	93	3	h	h	NOUN
cana-809	93	4	is	be	AUX
cana-809	93	5	(	(	PUNCT
cana-809	93	6	1	1	NUM
cana-809	93	7	,	,	PUNCT
cana-809	93	8	2)*open	2)*open	NUM
cana-809	93	9	map	map	NOUN
cana-809	93	10	h(v	h(v	NOUN
cana-809	93	11	)	)	PUNCT
cana-809	93	12	in	in	ADP
cana-809	93	13	m	m	PROPN
cana-809	93	14	is	be	AUX
cana-809	93	15	σ1,2	σ1,2	NOUN
cana-809	93	16	-	-	PUNCT
cana-809	93	17	open	open	ADJ
cana-809	93	18	.	.	PUNCT
cana-809	94	1	it	it	PRON
cana-809	94	2	follows	follow	VERB
cana-809	94	3	that	that	SCONJ
cana-809	94	4	h(b	h(b	PROPN
cana-809	94	5	)	)	PUNCT
cana-809	94	6	in	in	ADP
cana-809	94	7	m	m	PROPN
cana-809	94	8	is	be	AUX
cana-809	94	9	(	(	PUNCT
cana-809	94	10	1	1	NUM
cana-809	94	11	,	,	PUNCT
cana-809	94	12	2	2	NUM
cana-809	94	13	)	)	PUNCT
cana-809	94	14	*	*	PUNCT
cana-809	94	15	spos	spos	PROPN
cana-809	94	16	.	.	PUNCT
cana-809	95	1	theorem	theorem	VERB
cana-809	95	2	3.20	3.20	NUM
cana-809	95	3	in	in	ADP
cana-809	95	4	bts	bt	NOUN
cana-809	95	5	x	x	PRON
cana-809	95	6	consider	consider	VERB
cana-809	95	7	a	a	DET
cana-809	95	8	map	map	NOUN
cana-809	95	9	h	h	NOUN
cana-809	95	10	:	:	PUNCT
cana-809	95	11	l	l	PUNCT
cana-809	96	1	→	→	PUNCT
cana-809	96	2	m	m	AUX
cana-809	96	3	be	be	AUX
cana-809	96	4	a	a	DET
cana-809	96	5	(	(	PUNCT
cana-809	96	6	1	1	NUM
cana-809	96	7	,	,	PUNCT
cana-809	96	8	2)*homeomorphism	2)*homeomorphism	NUM
cana-809	96	9	.	.	PUNCT
cana-809	97	1	if	if	SCONJ
cana-809	97	2	b	b	X
cana-809	97	3	in	in	ADP
cana-809	97	4	l	l	PROPN
cana-809	97	5	is	be	AUX
cana-809	97	6	(	(	PUNCT
cana-809	97	7	1	1	NUM
cana-809	97	8	,	,	PUNCT
cana-809	97	9	2)*-d**spos	2)*-d**spos	NUM
cana-809	97	10	,	,	PUNCT
cana-809	97	11	then	then	ADV
cana-809	97	12	h(b	h(b	PROPN
cana-809	97	13	)	)	PUNCT
cana-809	97	14	is	be	AUX
cana-809	97	15	(	(	PUNCT
cana-809	97	16	1	1	NUM
cana-809	97	17	,	,	PUNCT
cana-809	97	18	2)*d**spos	2)*d**spos	NOUN
cana-809	97	19	in	in	ADP
cana-809	97	20	m.	m.	NOUN
cana-809	97	21	proof	proof	NOUN
cana-809	97	22	b	b	PROPN
cana-809	97	23	is	be	AUX
cana-809	97	24	(	(	PUNCT
cana-809	97	25	1	1	NUM
cana-809	97	26	,	,	PUNCT
cana-809	97	27	2)*-d**spos	2)*-d**spos	PROPN
cana-809	97	28	in	in	ADP
cana-809	97	29	l	l	PROPN
cana-809	97	30	..	..	PUNCT
cana-809	97	31	by	by	ADP
cana-809	97	32	definition	definition	NOUN
cana-809	97	33	3.3	3.3	NUM
cana-809	97	34	h(v	h(v	NOUN
cana-809	97	35	)	)	PUNCT
cana-809	97	36	⊆	⊆	NUM
cana-809	97	37	h(b	h(b	PROPN
cana-809	97	38	)	)	PUNCT
cana-809	97	39	⊆	⊆	NUM
cana-809	97	40	h((1	h((1	PROPN
cana-809	97	41	,	,	PUNCT
cana-809	97	42	2)*-spcl**(v	2)*-spcl**(v	NUM
cana-809	97	43	)	)	PUNCT
cana-809	97	44	)	)	PUNCT
cana-809	97	45	.	.	PUNCT
cana-809	98	1	also	also	ADV
cana-809	98	2	given	give	VERB
cana-809	98	3	h	h	NOUN
cana-809	98	4	is	be	AUX
cana-809	98	5	(	(	PUNCT
cana-809	98	6	1	1	NUM
cana-809	98	7	,	,	PUNCT
cana-809	98	8	2)*homeomorphism	2)*homeomorphism	NUM
cana-809	98	9	we	we	PRON
cana-809	98	10	have	have	VERB
cana-809	98	11	h((1	h((1	PROPN
cana-809	98	12	,	,	PUNCT
cana-809	98	13	2)*-spcl**(v	2)*-spcl**(v	NUM
cana-809	98	14	)	)	PUNCT
cana-809	98	15	)	)	PUNCT
cana-809	99	1	⊆	⊆	NUM
cana-809	99	2	(	(	PUNCT
cana-809	99	3	1	1	NUM
cana-809	99	4	,	,	PUNCT
cana-809	99	5	2)*spcl**h(v	2)*spcl**h(v	NOUN
cana-809	99	6	)	)	PUNCT
cana-809	99	7	)	)	PUNCT
cana-809	99	8	.	.	PUNCT
cana-809	100	1	it	it	PRON
cana-809	100	2	follows	follow	VERB
cana-809	100	3	h(v	h(v	PROPN
cana-809	100	4	)	)	PUNCT
cana-809	100	5	⊆	⊆	NUM
cana-809	100	6	h(b	h(b	PROPN
cana-809	100	7	)	)	PUNCT
cana-809	100	8	⊆(1	⊆(1	NOUN
cana-809	100	9	,	,	PUNCT
cana-809	100	10	2)*spcl	2)*spcl	NUM
cana-809	100	11	*	*	NOUN
cana-809	100	12	*	*	PUNCT
cana-809	100	13	(	(	PUNCT
cana-809	100	14	h(v	h(v	PROPN
cana-809	100	15	)	)	PUNCT
cana-809	100	16	)	)	PUNCT
cana-809	100	17	.	.	PUNCT
cana-809	101	1	so	so	ADV
cana-809	101	2	that	that	SCONJ
cana-809	101	3	h	h	NOUN
cana-809	101	4	(	(	PUNCT
cana-809	101	5	b	b	NOUN
cana-809	101	6	)	)	PUNCT
cana-809	101	7	in	in	ADP
cana-809	101	8	m	m	PROPN
cana-809	101	9	is	be	AUX
cana-809	101	10	(	(	PUNCT
cana-809	101	11	1	1	NUM
cana-809	101	12	,	,	PUNCT
cana-809	101	13	2	2	NUM
cana-809	101	14	)	)	PUNCT
cana-809	101	15	*	*	PUNCT
cana-809	101	16	-d**spos	-d**spos	PROPN
cana-809	101	17	theorem	theorem	VERB
cana-809	101	18	3.21	3.21	NUM
cana-809	101	19	in	in	ADP
cana-809	101	20	bts	bt	NOUN
cana-809	101	21	x	x	PRON
cana-809	101	22	consider	consider	VERB
cana-809	101	23	a	a	DET
cana-809	101	24	map	map	NOUN
cana-809	101	25	h	h	NOUN
cana-809	101	26	:	:	PUNCT
cana-809	101	27	l	l	PUNCT
cana-809	102	1	→	→	PUNCT
cana-809	102	2	m	m	NOUN
cana-809	102	3	if	if	SCONJ
cana-809	102	4	h	h	NOUN
cana-809	102	5	is	be	AUX
cana-809	102	6	(	(	PUNCT
cana-809	102	7	1	1	NUM
cana-809	102	8	,	,	PUNCT
cana-809	102	9	2)*homeomorphism.and	2)*homeomorphism.and	NUM
cana-809	102	10	b	b	NOUN
cana-809	102	11	in	in	ADP
cana-809	102	12	m	m	PROPN
cana-809	102	13	is	be	AUX
cana-809	102	14	(	(	PUNCT
cana-809	102	15	1	1	NUM
cana-809	102	16	,	,	PUNCT
cana-809	102	17	2)*-d**spos	2)*-d**spos	PROPN
cana-809	102	18	then	then	ADV
cana-809	102	19	∃	∃	PROPN
cana-809	102	20	τ1,2	τ1,2	PROPN
cana-809	102	21	-	-	PROPN
cana-809	102	22	os	os	INTJ
cana-809	102	23	such	such	ADJ
cana-809	102	24	that	that	DET
cana-809	102	25	h−1(b	h−1(b	PROPN
cana-809	102	26	)	)	PUNCT
cana-809	102	27	in	in	ADP
cana-809	102	28	m	m	PROPN
cana-809	102	29	is	be	AUX
cana-809	102	30	(	(	PUNCT
cana-809	102	31	1	1	NUM
cana-809	102	32	,	,	PUNCT
cana-809	102	33	2)*d**spos	2)*d**spos	NUM
cana-809	102	34	proof	proof	NOUN
cana-809	102	35	from	from	ADP
cana-809	102	36	statement	statement	NOUN
cana-809	102	37	b	b	PROPN
cana-809	102	38	in	in	ADP
cana-809	102	39	l	l	PROPN
cana-809	102	40	is	be	AUX
cana-809	102	41	(	(	PUNCT
cana-809	102	42	1	1	NUM
cana-809	102	43	,	,	PUNCT
cana-809	102	44	2)*-d**spos	2)*-d**spos	NUM
cana-809	102	45	.	.	PUNCT
cana-809	103	1	by	by	ADP
cana-809	103	2	defn	defn	PROPN
cana-809	103	3	3.3	3.3	NUM
cana-809	103	4	we	we	PRON
cana-809	103	5	have	have	VERB
cana-809	103	6	h−1(v	h−1(v	PROPN
cana-809	103	7	)	)	PUNCT
cana-809	103	8	⊆	⊆	NUM
cana-809	103	9	h−1(b	h−1(b	PROPN
cana-809	103	10	)	)	PUNCT
cana-809	103	11	⊆	⊆	NUM
cana-809	103	12	h−1((1	h−1((1	PROPN
cana-809	103	13	,	,	PUNCT
cana-809	103	14	communications	communication	NOUN
cana-809	103	15	on	on	ADP
cana-809	103	16	applied	apply	VERB
cana-809	103	17	nonlinear	nonlinear	ADJ
cana-809	103	18	analysis	analysis	NOUN
cana-809	103	19	issn	issn	NOUN
cana-809	103	20	:	:	PUNCT
cana-809	103	21	1074	1074	NUM
cana-809	103	22	-	-	PUNCT
cana-809	103	23	133x	133x	NUM
cana-809	103	24	vol	vol	NOUN
cana-809	103	25	31	31	NUM
cana-809	103	26	no	no	NOUN
cana-809	103	27	.	.	PUNCT
cana-809	104	1	2s	2s	NUM
cana-809	104	2	(	(	PUNCT
cana-809	104	3	2024	2024	NUM
cana-809	104	4	)	)	PUNCT
cana-809	104	5	723	723	NUM
cana-809	104	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-809	104	7	2)*-spcl	2)*-spcl	NOUN
cana-809	104	8	*	*	PUNCT
cana-809	104	9	*	*	PUNCT
cana-809	104	10	(	(	PUNCT
cana-809	104	11	v	v	NOUN
cana-809	104	12	)	)	PUNCT
cana-809	104	13	)	)	PUNCT
cana-809	104	14	.	.	PUNCT
cana-809	105	1	and	and	CCONJ
cana-809	105	2	since	since	SCONJ
cana-809	105	3	h	h	NOUN
cana-809	105	4	is	be	AUX
cana-809	105	5	(	(	PUNCT
cana-809	105	6	1	1	NUM
cana-809	105	7	,	,	PUNCT
cana-809	105	8	2)*homeomorphism	2)*homeomorphism	NUM
cana-809	105	9	⇒	⇒	NOUN
cana-809	105	10	h−1((1	h−1((1	PROPN
cana-809	105	11	,	,	PUNCT
cana-809	105	12	2)*-spcl**(v	2)*-spcl**(v	NUM
cana-809	105	13	)	)	PUNCT
cana-809	105	14	)	)	PUNCT
cana-809	106	1	⊆	⊆	NUM
cana-809	106	2	(	(	PUNCT
cana-809	106	3	1	1	NUM
cana-809	106	4	,	,	PUNCT
cana-809	106	5	2)*-spcl**(h−1(v	2)*-spcl**(h−1(v	NUM
cana-809	106	6	)	)	PUNCT
cana-809	106	7	)	)	PUNCT
cana-809	106	8	.	.	PUNCT
cana-809	107	1	hence	hence	ADV
cana-809	107	2	we	we	PRON
cana-809	107	3	have	have	VERB
cana-809	107	4	h−1(v	h−1(v	PROPN
cana-809	107	5	)	)	PUNCT
cana-809	108	1	⊆	⊆	NUM
cana-809	108	2	h−1(b	h−1(b	PROPN
cana-809	108	3	)	)	PUNCT
cana-809	108	4	⊆	⊆	NUM
cana-809	108	5	(	(	PUNCT
cana-809	108	6	1	1	NUM
cana-809	108	7	,	,	PUNCT
cana-809	108	8	2)*-spcl**(h−1(v	2)*-spcl**(h−1(v	NUM
cana-809	108	9	)	)	PUNCT
cana-809	108	10	)	)	PUNCT
cana-809	108	11	thus	thus	ADV
cana-809	108	12	h−1(b	h−1(b	PROPN
cana-809	108	13	)	)	PUNCT
cana-809	108	14	is	be	AUX
cana-809	108	15	(	(	PUNCT
cana-809	108	16	1	1	NUM
cana-809	108	17	,	,	PUNCT
cana-809	108	18	2)*-d**spos	2)*-d**spos	NUM
cana-809	108	19	4	4	NUM
cana-809	108	20	.	.	PUNCT
cana-809	109	1	(	(	PUNCT
cana-809	109	2	1	1	NUM
cana-809	109	3	,	,	PUNCT
cana-809	109	4	2)*-d**spclosed	2)*-d**spclosed	NUM
cana-809	109	5	and	and	CCONJ
cana-809	109	6	(	(	PUNCT
cana-809	109	7	1	1	NUM
cana-809	109	8	,	,	PUNCT
cana-809	109	9	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	109	10	mappings	mapping	NOUN
cana-809	109	11	definition	definition	NOUN
cana-809	109	12	4	4	NUM
cana-809	109	13	.	.	NOUN
cana-809	109	14	1	1	NUM
cana-809	109	15	a	a	DET
cana-809	109	16	map	map	NOUN
cana-809	109	17	h	h	NOUN
cana-809	109	18	:	:	PUNCT
cana-809	109	19	l	l	PUNCT
cana-809	110	1	→	→	PUNCT
cana-809	110	2	m	m	NOUN
cana-809	110	3	is	be	AUX
cana-809	110	4	(	(	PUNCT
cana-809	110	5	1	1	NUM
cana-809	110	6	,	,	PUNCT
cana-809	110	7	2)*-d**spomap	2)*-d**spomap	NUM
cana-809	110	8	if	if	SCONJ
cana-809	110	9	h(v	h(v	PROPN
cana-809	110	10	)	)	PUNCT
cana-809	110	11	in	in	ADP
cana-809	110	12	m	m	PROPN
cana-809	110	13	is	be	AUX
cana-809	110	14	(	(	PUNCT
cana-809	110	15	1	1	NUM
cana-809	110	16	,	,	PUNCT
cana-809	110	17	2)*-d**spos	2)*-d**spos	NUM
cana-809	110	18	∀	∀	NOUN
cana-809	110	19	τ1,2os	τ1,2os	PRON
cana-809	110	20	v	v	NOUN
cana-809	110	21	in	in	ADP
cana-809	110	22	m.	m.	NOUN
cana-809	110	23	theorem	theorem	VERB
cana-809	110	24	4.2	4.2	NUM
cana-809	110	25	a	a	DET
cana-809	110	26	map	map	NOUN
cana-809	110	27	h	h	NOUN
cana-809	110	28	:	:	PUNCT
cana-809	110	29	l	l	X
cana-809	110	30	→m	→m	X
cana-809	110	31	is	be	AUX
cana-809	110	32	(	(	PUNCT
cana-809	110	33	1	1	NUM
cana-809	110	34	,	,	PUNCT
cana-809	110	35	2)*-open	2)*-open	NUM
cana-809	110	36	map	map	VERB
cana-809	110	37	⇒	⇒	NOUN
cana-809	110	38	(	(	PUNCT
cana-809	110	39	1	1	NUM
cana-809	110	40	,	,	PUNCT
cana-809	110	41	2)*d**spopen	2)*d**spopen	ADJ
cana-809	110	42	-map	-map	PUNCT
cana-809	110	43	proof	proof	NOUN
cana-809	110	44	.	.	PUNCT
cana-809	111	1	from	from	ADP
cana-809	111	2	statement	statement	NOUN
cana-809	111	3	h	h	NOUN
cana-809	111	4	:	:	PUNCT
cana-809	111	5	l	l	X
cana-809	111	6	→m	→m	X
cana-809	111	7	is	be	AUX
cana-809	111	8	(	(	PUNCT
cana-809	111	9	1	1	NUM
cana-809	111	10	,	,	PUNCT
cana-809	111	11	2)*-open	2)*-open	NUM
cana-809	111	12	map	map	NOUN
cana-809	111	13	.and	.and	PUNCT
cana-809	112	1	g	g	PROPN
cana-809	112	2	is	be	AUX
cana-809	112	3	τ1,2	τ1,2	ADJ
cana-809	112	4	-	-	NOUN
cana-809	112	5	os	os	NOUN
cana-809	112	6	in	in	ADP
cana-809	112	7	l.	l.	NOUN
cana-809	112	8	we	we	PRON
cana-809	112	9	have	have	VERB
cana-809	112	10	h(g	h(g	NOUN
cana-809	112	11	)	)	PUNCT
cana-809	112	12	in	in	ADP
cana-809	112	13	m	m	PROPN
cana-809	112	14	is	be	AUX
cana-809	112	15	σ1,2open	σ1,2open	X
cana-809	112	16	.	.	PUNCT
cana-809	113	1	from	from	ADP
cana-809	113	2	theorem3.7	theorem3.7	NOUN
cana-809	113	3	,	,	PUNCT
cana-809	113	4	h(g	h(g	NOUN
cana-809	113	5	)	)	PUNCT
cana-809	113	6	is	be	AUX
cana-809	113	7	(	(	PUNCT
cana-809	113	8	1	1	NUM
cana-809	113	9	,	,	PUNCT
cana-809	113	10	2)*-d**spos	2)*-d**spos	NUM
cana-809	113	11	in	in	ADP
cana-809	113	12	m	m	PROPN
cana-809	113	13	..	..	PUNCT
cana-809	113	14	henceforth	henceforth	ADV
cana-809	113	15	h	h	PROPN
cana-809	113	16	is	be	AUX
cana-809	113	17	(	(	PUNCT
cana-809	113	18	1	1	NUM
cana-809	113	19	,	,	PUNCT
cana-809	113	20	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	113	21	.	.	PUNCT
cana-809	114	1	example	example	NOUN
cana-809	114	2	4.3	4.3	NUM
cana-809	114	3	reverse	reverse	NOUN
cana-809	114	4	oftheorem	oftheorem	VERB
cana-809	114	5	4.2	4.2	NUM
cana-809	114	6	is	be	AUX
cana-809	114	7	not	not	PART
cana-809	114	8	true	true	ADJ
cana-809	114	9	.	.	PUNCT
cana-809	115	1	by	by	ADP
cana-809	115	2	this	this	DET
cana-809	115	3	example	example	NOUN
cana-809	115	4	let	let	VERB
cana-809	115	5	l=	l=	ADV
cana-809	115	6	m=	m=	X
cana-809	115	7	{	{	PUNCT
cana-809	115	8	a1	a1	PROPN
cana-809	115	9	,	,	PUNCT
cana-809	115	10	,	,	PUNCT
cana-809	115	11	b1	b1	NOUN
cana-809	115	12	,	,	PUNCT
cana-809	115	13	c1	c1	PROPN
cana-809	115	14	}	}	PUNCT
cana-809	115	15	,	,	PUNCT
cana-809	115	16	τ1	τ1	NOUN
cana-809	115	17	=	=	SYM
cana-809	115	18	{	{	PUNCT
cana-809	115	19	l	l	NOUN
cana-809	115	20	,	,	PUNCT
cana-809	115	21	φ	φ	PROPN
cana-809	115	22	,	,	PUNCT
cana-809	115	23	{	{	PUNCT
cana-809	115	24	a1	a1	NOUN
cana-809	115	25	}	}	PUNCT
cana-809	115	26	}	}	PUNCT
cana-809	115	27	,	,	PUNCT
cana-809	115	28	τ2	τ2	NOUN
cana-809	115	29	=	=	SYM
cana-809	115	30	{	{	PUNCT
cana-809	115	31	l	l	NOUN
cana-809	115	32	,	,	PUNCT
cana-809	115	33	φ	φ	PROPN
cana-809	115	34	,	,	PUNCT
cana-809	115	35	{	{	PUNCT
cana-809	115	36	a1	a1	NOUN
cana-809	115	37	,	,	PUNCT
cana-809	115	38	c1	c1	NOUN
cana-809	115	39	}	}	PUNCT
cana-809	115	40	}	}	PUNCT
cana-809	115	41	.	.	PUNCT
cana-809	116	1	let	let	VERB
cana-809	116	2	σ1	σ1	NOUN
cana-809	116	3	=	=	PUNCT
cana-809	116	4	{	{	PUNCT
cana-809	116	5	l	l	NOUN
cana-809	116	6	,	,	PUNCT
cana-809	116	7	φ	φ	PROPN
cana-809	116	8	,	,	PUNCT
cana-809	116	9	{	{	PUNCT
cana-809	116	10	a1	a1	NOUN
cana-809	116	11	}	}	PUNCT
cana-809	116	12	}	}	PUNCT
cana-809	116	13	,	,	PUNCT
cana-809	116	14	σ2	σ2	NOUN
cana-809	116	15	=	=	SYM
cana-809	116	16	{	{	PUNCT
cana-809	116	17	l	l	NOUN
cana-809	116	18	,	,	PUNCT
cana-809	116	19	φ	φ	PROPN
cana-809	116	20	,	,	PUNCT
cana-809	116	21	{	{	PUNCT
cana-809	116	22	a1	a1	NOUN
cana-809	116	23	}	}	PUNCT
cana-809	116	24	,	,	PUNCT
cana-809	116	25	{	{	PUNCT
cana-809	116	26	a1,b1	a1,b1	NOUN
cana-809	116	27	}	}	PUNCT
cana-809	116	28	}	}	PUNCT
cana-809	116	29	.	.	PUNCT
cana-809	117	1	let	let	VERB
cana-809	117	2	h	h	NOUN
cana-809	117	3	:	:	PUNCT
cana-809	117	4	l	l	PUNCT
cana-809	117	5	→	→	PUNCT
cana-809	117	6	m	m	NOUN
cana-809	117	7	is	be	AUX
cana-809	117	8	an	an	DET
cana-809	117	9	identity	identity	NOUN
cana-809	117	10	map	map	NOUN
cana-809	117	11	.	.	PUNCT
cana-809	118	1	we	we	PRON
cana-809	118	2	have	have	VERB
cana-809	118	3	h	h	NOUN
cana-809	118	4	is	be	AUX
cana-809	118	5	(	(	PUNCT
cana-809	118	6	1	1	NUM
cana-809	118	7	,	,	PUNCT
cana-809	118	8	2)*d**spopen	2)*d**spopen	ADJ
cana-809	119	1	but	but	CCONJ
cana-809	119	2	h	h	NOUN
cana-809	119	3	is	be	AUX
cana-809	119	4	not	not	PART
cana-809	119	5	(	(	PUNCT
cana-809	119	6	1	1	NUM
cana-809	119	7	,	,	PUNCT
cana-809	119	8	2)*-open	2)*-open	NUM
cana-809	119	9	.	.	PUNCT
cana-809	120	1	map	map	NOUN
cana-809	120	2	definition	definition	NOUN
cana-809	120	3	4.4	4.4	NUM
cana-809	120	4	the	the	DET
cana-809	120	5	map	map	NOUN
cana-809	120	6	h	h	NOUN
cana-809	120	7	:	:	PUNCT
cana-809	121	1	l	l	PUNCT
cana-809	121	2	→	→	PUNCT
cana-809	121	3	m	m	NOUN
cana-809	121	4	is	be	AUX
cana-809	121	5	(	(	PUNCT
cana-809	121	6	1	1	NUM
cana-809	121	7	,	,	PUNCT
cana-809	121	8	2)*d**spclosed	2)*d**spclosed	ADJ
cana-809	121	9	map	map	NOUN
cana-809	121	10	if	if	SCONJ
cana-809	121	11	for	for	SCONJ
cana-809	121	12	every	every	DET
cana-809	121	13	τ1,2	τ1,2	PROPN
cana-809	121	14	-	-	PROPN
cana-809	121	15	cs	cs	ADJ
cana-809	121	16	v	v	NOUN
cana-809	121	17	in	in	ADP
cana-809	121	18	l	l	NOUN
cana-809	121	19	,	,	PUNCT
cana-809	121	20	h(v	h(v	PROPN
cana-809	121	21	)	)	PUNCT
cana-809	121	22	in	in	ADP
cana-809	121	23	m	m	PROPN
cana-809	121	24	is	be	AUX
cana-809	121	25	(	(	PUNCT
cana-809	121	26	1	1	NUM
cana-809	121	27	,	,	PUNCT
cana-809	121	28	2)*d**spclosed	2)*d**spclosed	ADJ
cana-809	121	29	remark	remark	NOUN
cana-809	121	30	4.5	4.5	NUM
cana-809	121	31	h	h	NOUN
cana-809	121	32	:	:	PUNCT
cana-809	122	1	l	l	PUNCT
cana-809	122	2	→	→	PUNCT
cana-809	122	3	m	m	NOUN
cana-809	122	4	is	be	AUX
cana-809	122	5	(	(	PUNCT
cana-809	122	6	1	1	NUM
cana-809	122	7	,	,	PUNCT
cana-809	122	8	2)*closed	2)*closed	NUM
cana-809	122	9	⇒	⇒	NOUN
cana-809	122	10	h	h	NOUN
cana-809	122	11	is	be	AUX
cana-809	122	12	(	(	PUNCT
cana-809	122	13	1	1	NUM
cana-809	122	14	,	,	PUNCT
cana-809	122	15	2)*d**spclosed	2)*d**spclosed	ADJ
cana-809	122	16	but	but	CCONJ
cana-809	122	17	conversely	conversely	ADV
cana-809	122	18	not	not	PART
cana-809	122	19	true	true	ADJ
cana-809	122	20	proof	proof	NOUN
cana-809	122	21	.	.	PUNCT
cana-809	123	1	from	from	ADP
cana-809	123	2	theorem	theorem	ADJ
cana-809	123	3	4.2	4.2	NUM
cana-809	123	4	.	.	PUNCT
cana-809	124	1	proof	proof	NOUN
cana-809	124	2	is	be	AUX
cana-809	124	3	clear	clear	ADJ
cana-809	124	4	5	5	NUM
cana-809	124	5	.	.	PUNCT
cana-809	125	1	(	(	PUNCT
cana-809	125	2	1	1	NUM
cana-809	125	3	,	,	PUNCT
cana-809	125	4	2)*-d**spcontinuous	2)*-d**spcontinuous	ADJ
cana-809	125	5	mappings	mapping	NOUN
cana-809	125	6	definition	definition	NOUN
cana-809	125	7	5.1	5.1	NUM
cana-809	125	8	a	a	DET
cana-809	125	9	map	map	NOUN
cana-809	125	10	h	h	NOUN
cana-809	125	11	:	:	PUNCT
cana-809	125	12	k	k	X
cana-809	125	13	→l	→l	PUNCT
cana-809	125	14	is	be	AUX
cana-809	125	15	called	call	VERB
cana-809	125	16	(	(	PUNCT
cana-809	125	17	1	1	NUM
cana-809	125	18	,	,	PUNCT
cana-809	125	19	2)*-d**spcontinuous	2)*-d**spcontinuous	NUM
cana-809	125	20	∀	∀	NOUN
cana-809	125	21	σ1,2	σ1,2	NOUN
cana-809	125	22	-	-	PUNCT
cana-809	125	23	os	os	NOUN
cana-809	125	24	in	in	ADP
cana-809	125	25	l	l	NOUN
cana-809	125	26	its	its	PRON
cana-809	125	27	inverse	inverse	NOUN
cana-809	125	28	image	image	NOUN
cana-809	125	29	of	of	ADP
cana-809	125	30	h	h	NOUN
cana-809	125	31	is	be	AUX
cana-809	125	32	(	(	PUNCT
cana-809	125	33	1	1	NUM
cana-809	125	34	,	,	PUNCT
cana-809	125	35	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	125	36	in	in	ADP
cana-809	125	37	k	k	PROPN
cana-809	125	38	..	..	PUNCT
cana-809	125	39	communications	communication	NOUN
cana-809	125	40	on	on	ADP
cana-809	125	41	applied	apply	VERB
cana-809	125	42	nonlinear	nonlinear	ADJ
cana-809	125	43	analysis	analysis	NOUN
cana-809	125	44	issn	issn	NOUN
cana-809	125	45	:	:	PUNCT
cana-809	125	46	1074	1074	NUM
cana-809	125	47	-	-	PUNCT
cana-809	125	48	133x	133x	NUM
cana-809	125	49	vol	vol	NOUN
cana-809	125	50	31	31	NUM
cana-809	125	51	no	no	NOUN
cana-809	125	52	.	.	PUNCT
cana-809	126	1	2s	2s	NUM
cana-809	126	2	(	(	PUNCT
cana-809	126	3	2024	2024	NUM
cana-809	126	4	)	)	PUNCT
cana-809	126	5	724	724	NUM
cana-809	126	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-809	126	7	theorem	theorem	VERB
cana-809	126	8	5.2	5.2	NUM
cana-809	126	9	a	a	DET
cana-809	126	10	map	map	NOUN
cana-809	126	11	h	h	NOUN
cana-809	126	12	:	:	PUNCT
cana-809	127	1	k	k	X
cana-809	127	2	→l	→l	PUNCT
cana-809	127	3	h	h	NOUN
cana-809	127	4	is	be	AUX
cana-809	127	5	(	(	PUNCT
cana-809	127	6	1	1	NUM
cana-809	127	7	,	,	PUNCT
cana-809	127	8	2)*continuous	2)*continuous	ADJ
cana-809	127	9	⇒	⇒	NOUN
cana-809	127	10	h	h	NOUN
cana-809	127	11	is	be	AUX
cana-809	127	12	(	(	PUNCT
cana-809	127	13	1	1	NUM
cana-809	127	14	,	,	PUNCT
cana-809	127	15	2)*-d**spcontinuous	2)*-d**spcontinuous	PROPN
cana-809	127	16	.	.	PUNCT
cana-809	128	1	proof	proof	NOUN
cana-809	128	2	assume	assume	VERB
cana-809	128	3	r	r	NOUN
cana-809	128	4	as	as	ADP
cana-809	128	5	a	a	DET
cana-809	128	6	σ1,2os	σ1,2o	NOUN
cana-809	128	7	in	in	ADP
cana-809	128	8	l.	l.	PROPN
cana-809	128	9	also	also	ADV
cana-809	128	10	h	h	PROPN
cana-809	128	11	is	be	AUX
cana-809	128	12	(	(	PUNCT
cana-809	128	13	1	1	NUM
cana-809	128	14	,	,	PUNCT
cana-809	128	15	2)*continuous	2)*continuous	NUM
cana-809	128	16	,	,	PUNCT
cana-809	128	17	h−1(r	h−1(r	NOUN
cana-809	128	18	)	)	PUNCT
cana-809	128	19	is	be	AUX
cana-809	128	20	τ1,2open	τ1,2open	ADJ
cana-809	128	21	in	in	ADP
cana-809	128	22	x.	x.	NOUN
cana-809	128	23	from	from	ADP
cana-809	128	24	theorem	theorem	ADJ
cana-809	128	25	3.7	3.7	NUM
cana-809	128	26	,	,	PUNCT
cana-809	128	27	h−1(r	h−1(r	NOUN
cana-809	128	28	)	)	PUNCT
cana-809	128	29	is	be	AUX
cana-809	128	30	(	(	PUNCT
cana-809	128	31	1	1	NUM
cana-809	128	32	,	,	PUNCT
cana-809	128	33	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	128	34	in	in	ADP
cana-809	128	35	x	x	X
cana-809	128	36	.	.	PUNCT
cana-809	129	1	thus	thus	ADV
cana-809	129	2	h	h	NOUN
cana-809	129	3	is	be	AUX
cana-809	129	4	(	(	PUNCT
cana-809	129	5	1	1	NUM
cana-809	129	6	,	,	PUNCT
cana-809	129	7	2)*-d**spcontinuous	2)*-d**spcontinuous	PROPN
cana-809	129	8	.	.	PUNCT
cana-809	129	9	example	example	NOUN
cana-809	129	10	5.3	5.3	NUM
cana-809	129	11	this	this	DET
cana-809	129	12	example	example	NOUN
cana-809	129	13	proves	prove	VERB
cana-809	129	14	reverse	reverse	NOUN
cana-809	129	15	of	of	ADP
cana-809	129	16	thm	thm	PROPN
cana-809	129	17	5.2	5.2	NUM
cana-809	129	18	consider	consider	VERB
cana-809	129	19	a	a	DET
cana-809	129	20	map	map	NOUN
cana-809	129	21	h	h	NOUN
cana-809	129	22	:	:	PUNCT
cana-809	129	23	k	k	X
cana-809	129	24	→l	→l	PUNCT
cana-809	129	25	let	let	VERB
cana-809	129	26	the	the	DET
cana-809	129	27	two	two	NUM
cana-809	129	28	sets	set	NOUN
cana-809	129	29	l	l	NOUN
cana-809	129	30	=	=	PUNCT
cana-809	129	31	k	k	NOUN
cana-809	129	32	=	=	PRON
cana-809	129	33	{	{	PUNCT
cana-809	129	34	a1	a1	NOUN
cana-809	129	35	,	,	PUNCT
cana-809	129	36	a2	a2	PROPN
cana-809	129	37	,	,	PUNCT
cana-809	129	38	a3	a3	NOUN
cana-809	129	39	}	}	PUNCT
cana-809	129	40	,	,	PUNCT
cana-809	129	41	τ1	τ1	NOUN
cana-809	129	42	=	=	SYM
cana-809	129	43	{	{	PUNCT
cana-809	129	44	l	l	NOUN
cana-809	129	45	,	,	PUNCT
cana-809	129	46	φ	φ	PROPN
cana-809	129	47	,	,	PUNCT
cana-809	129	48	{	{	PUNCT
cana-809	129	49	a1	a1	NOUN
cana-809	129	50	}	}	PUNCT
cana-809	129	51	}	}	PUNCT
cana-809	129	52	,	,	PUNCT
cana-809	129	53	τ2	τ2	NOUN
cana-809	129	54	=	=	SYM
cana-809	129	55	{	{	PUNCT
cana-809	129	56	l	l	NOUN
cana-809	129	57	,	,	PUNCT
cana-809	129	58	φ	φ	PROPN
cana-809	129	59	,	,	PUNCT
cana-809	129	60	{	{	PUNCT
cana-809	129	61	a1	a1	NOUN
cana-809	129	62	}	}	PUNCT
cana-809	129	63	,	,	PUNCT
cana-809	129	64	{	{	PUNCT
cana-809	129	65	a1	a1	NOUN
cana-809	129	66	,	,	PUNCT
cana-809	129	67	a2	a2	PROPN
cana-809	129	68	}	}	PUNCT
cana-809	129	69	}	}	PUNCT
cana-809	129	70	,	,	PUNCT
cana-809	129	71	σ1	σ1	NOUN
cana-809	129	72	=	=	SYM
cana-809	129	73	{	{	PUNCT
cana-809	129	74	k	k	PROPN
cana-809	129	75	,	,	PUNCT
cana-809	129	76	φ	φ	PROPN
cana-809	129	77	,	,	PUNCT
cana-809	129	78	{	{	PUNCT
cana-809	129	79	a1	a1	NOUN
cana-809	129	80	}	}	PUNCT
cana-809	129	81	}	}	PUNCT
cana-809	129	82	and	and	CCONJ
cana-809	129	83	σ2	σ2	PROPN
cana-809	129	84	=	=	SYM
cana-809	129	85	{	{	PUNCT
cana-809	129	86	k	k	PROPN
cana-809	129	87	,	,	PUNCT
cana-809	129	88	φ	φ	PROPN
cana-809	129	89	,	,	PUNCT
cana-809	129	90	{	{	PUNCT
cana-809	129	91	a1	a1	NOUN
cana-809	129	92	,	,	PUNCT
cana-809	129	93	a3	a3	NOUN
cana-809	129	94	}	}	PUNCT
cana-809	129	95	}	}	PUNCT
cana-809	129	96	.	.	PUNCT
cana-809	130	1	suppose	suppose	VERB
cana-809	131	1	h	h	NOUN
cana-809	131	2	:	:	PUNCT
cana-809	131	3	k	k	X
cana-809	131	4	→	→	PUNCT
cana-809	131	5	l	l	X
cana-809	131	6	be	be	AUX
cana-809	131	7	the	the	DET
cana-809	131	8	identity	identity	NOUN
cana-809	131	9	map	map	NOUN
cana-809	131	10	we	we	PRON
cana-809	131	11	have	have	VERB
cana-809	131	12	h	h	NOUN
cana-809	131	13	is	be	AUX
cana-809	131	14	(	(	PUNCT
cana-809	131	15	1	1	NUM
cana-809	131	16	,	,	PUNCT
cana-809	131	17	2)*d**spcontinuous	2)*d**spcontinuous	ADJ
cana-809	131	18	but	but	CCONJ
cana-809	131	19	h	h	NOUN
cana-809	131	20	is	be	AUX
cana-809	131	21	not	not	PART
cana-809	131	22	(	(	PUNCT
cana-809	131	23	1	1	NUM
cana-809	131	24	,	,	PUNCT
cana-809	131	25	2)*continuous	2)*continuous	NUM
cana-809	131	26	.	.	PUNCT
cana-809	132	1	remark	remark	NOUN
cana-809	132	2	5.4	5.4	NUM
cana-809	132	3	in	in	ADP
cana-809	132	4	(	(	PUNCT
cana-809	132	5	1	1	NUM
cana-809	132	6	,	,	PUNCT
cana-809	132	7	2)*-d**sp	2)*-d**sp	NUM
cana-809	132	8	-	-	PUNCT
cana-809	132	9	t1/2	t1/2	ADJ
cana-809	132	10	space	space	NOUN
cana-809	132	11	,	,	PUNCT
cana-809	132	12	every	every	DET
cana-809	132	13	(	(	PUNCT
cana-809	132	14	1	1	NUM
cana-809	132	15	,	,	PUNCT
cana-809	132	16	2)*-d**spcontinuous	2)*-d**spcontinuous	ADJ
cana-809	132	17	map	map	NOUN
cana-809	132	18	is	be	AUX
cana-809	132	19	(	(	PUNCT
cana-809	132	20	1	1	NUM
cana-809	132	21	,	,	PUNCT
cana-809	132	22	2)*continuous	2)*continuous	NUM
cana-809	132	23	.	.	PUNCT
cana-809	133	1	theorem	theorem	VERB
cana-809	133	2	5.5	5.5	NUM
cana-809	133	3	h	h	NOUN
cana-809	133	4	:	:	PUNCT
cana-809	133	5	k	k	X
cana-809	133	6	→	→	PUNCT
cana-809	133	7	l	l	NOUN
cana-809	133	8	is	be	AUX
cana-809	133	9	a	a	DET
cana-809	133	10	map	map	NOUN
cana-809	133	11	.	.	PUNCT
cana-809	134	1	we	we	PRON
cana-809	134	2	have	have	VERB
cana-809	134	3	the	the	DET
cana-809	134	4	below	below	ADJ
cana-809	134	5	implications	implication	NOUN
cana-809	134	6	are	be	AUX
cana-809	134	7	true	true	ADJ
cana-809	134	8	.	.	PUNCT
cana-809	135	1	•	•	NUM
cana-809	135	2	1	1	NUM
cana-809	135	3	h	h	NOUN
cana-809	135	4	is	be	AUX
cana-809	135	5	(	(	PUNCT
cana-809	135	6	1	1	NUM
cana-809	135	7	,	,	PUNCT
cana-809	135	8	2)*d**spcontinuous	2)*d**spcontinuous	NUM
cana-809	135	9	.	.	PUNCT
cana-809	136	1	•	•	NUM
cana-809	136	2	2	2	NUM
cana-809	136	3	.	.	X
cana-809	137	1	for	for	ADP
cana-809	137	2	each	each	DET
cana-809	137	3	σ1,2cs	σ1,2cs	PROPN
cana-809	137	4	in	in	ADP
cana-809	137	5	l	l	NOUN
cana-809	137	6	its	its	PRON
cana-809	137	7	inverse	inverse	NOUN
cana-809	137	8	image	image	NOUN
cana-809	137	9	is	be	AUX
cana-809	137	10	(	(	PUNCT
cana-809	137	11	1	1	NUM
cana-809	137	12	,	,	PUNCT
cana-809	137	13	2)*-d**spclosed	2)*-d**spclosed	NUM
cana-809	137	14	in	in	ADP
cana-809	137	15	k	k	PROPN
cana-809	137	16	.	.	PUNCT
cana-809	138	1	proof	proof	NOUN
cana-809	138	2	.	.	PUNCT
cana-809	139	1	(	(	PUNCT
cana-809	139	2	1	1	X
cana-809	139	3	)	)	PUNCT
cana-809	139	4	⇒	⇒	NOUN
cana-809	139	5	(	(	PUNCT
cana-809	139	6	2	2	X
cana-809	139	7	)	)	PUNCT
cana-809	139	8	let	let	VERB
cana-809	139	9	r	r	NOUN
cana-809	139	10	is	be	AUX
cana-809	139	11	σ1,2cs	σ1,2cs	PROPN
cana-809	139	12	in	in	ADP
cana-809	139	13	l.	l.	PROPN
cana-809	139	14	then	then	ADV
cana-809	139	15	l	l	PROPN
cana-809	139	16	r	r	NOUN
cana-809	139	17	is	be	AUX
cana-809	139	18	σ1,2open	σ1,2open	X
cana-809	139	19	in	in	ADP
cana-809	139	20	l.	l.	PROPN
cana-809	139	21	also	also	ADV
cana-809	139	22	h	h	PROPN
cana-809	139	23	is	be	AUX
cana-809	139	24	(	(	PUNCT
cana-809	139	25	1	1	NUM
cana-809	139	26	,	,	PUNCT
cana-809	139	27	2)*-d**spcontinuous	2)*-d**spcontinuous	NUM
cana-809	139	28	,	,	PUNCT
cana-809	139	29	f−1	f−1	PROPN
cana-809	139	30	(	(	PUNCT
cana-809	139	31	l	l	NOUN
cana-809	139	32	r	r	NOUN
cana-809	139	33	)	)	PUNCT
cana-809	139	34	is	be	AUX
cana-809	139	35	(	(	PUNCT
cana-809	139	36	1	1	NUM
cana-809	139	37	,	,	PUNCT
cana-809	139	38	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	139	39	in	in	ADP
cana-809	139	40	k.	k.	PROPN
cana-809	140	1	so	so	SCONJ
cana-809	140	2	that	that	SCONJ
cana-809	140	3	we	we	PRON
cana-809	140	4	have	have	AUX
cana-809	140	5	k	k	PROPN
cana-809	140	6	/	/	SYM
cana-809	140	7	f−1(r	f−1(r	PROPN
cana-809	140	8	)	)	PUNCT
cana-809	140	9	is	be	AUX
cana-809	140	10	(	(	PUNCT
cana-809	140	11	1	1	NUM
cana-809	140	12	,	,	PUNCT
cana-809	140	13	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	140	14	in	in	ADP
cana-809	140	15	k	k	PROPN
cana-809	140	16	⇒	⇒	PROPN
cana-809	140	17	f−1(r	f−1(r	PROPN
cana-809	140	18	)	)	PUNCT
cana-809	141	1	is	be	AUX
cana-809	141	2	(	(	PUNCT
cana-809	141	3	1	1	NUM
cana-809	141	4	,	,	PUNCT
cana-809	141	5	2)*d**spclosed	2)*d**spclose	VERB
cana-809	141	6	in	in	ADP
cana-809	141	7	k.	k.	PROPN
cana-809	141	8	(	(	PUNCT
cana-809	141	9	ii	ii	PROPN
cana-809	141	10	)	)	PUNCT
cana-809	141	11	⇒	⇒	NOUN
cana-809	141	12	(	(	PUNCT
cana-809	141	13	i	i	NOUN
cana-809	141	14	)	)	PUNCT
cana-809	141	15	let	let	VERB
cana-809	141	16	s	s	PRON
cana-809	141	17	is	be	AUX
cana-809	141	18	a	a	DET
cana-809	141	19	σ1,2os	σ1,2o	NOUN
cana-809	141	20	in	in	ADP
cana-809	141	21	l.	l.	PROPN
cana-809	141	22	then	then	ADV
cana-809	141	23	l	l	PROPN
cana-809	141	24	-	-	PUNCT
cana-809	141	25	s	s	X
cana-809	141	26	is	be	AUX
cana-809	141	27	σ1,2open	σ1,2open	X
cana-809	141	28	in	in	ADP
cana-809	141	29	l.	l.	PROPN
cana-809	141	30	⇒	⇒	PROPN
cana-809	141	31	f−1(l	f−1(l	PROPN
cana-809	141	32	\	\	PROPN
cana-809	141	33	s	s	PART
cana-809	141	34	)	)	PUNCT
cana-809	141	35	is	be	AUX
cana-809	141	36	(	(	PUNCT
cana-809	141	37	1	1	NUM
cana-809	141	38	,	,	PUNCT
cana-809	141	39	2	2	NUM
cana-809	141	40	)	)	PUNCT
cana-809	141	41	*	*	PUNCT
cana-809	142	1	d**spclosed	d**spclose	VERB
cana-809	142	2	in	in	ADP
cana-809	142	3	k	k	PROPN
cana-809	142	4	,	,	PUNCT
cana-809	142	5	⇒l	⇒l	PUNCT
cana-809	142	6	\f−1(s	\f−1(	VERB
cana-809	142	7	)	)	PUNCT
cana-809	142	8	is	be	AUX
cana-809	142	9	(	(	PUNCT
cana-809	142	10	1	1	NUM
cana-809	142	11	,	,	PUNCT
cana-809	142	12	2)*-d**spclosed	2)*-d**spclosed	NUM
cana-809	142	13	in	in	ADP
cana-809	142	14	l	l	PROPN
cana-809	142	15	.so	.so	PUNCT
cana-809	142	16	that	that	PRON
cana-809	142	17	h−1(s	h−1(s	PROPN
cana-809	142	18	)	)	PUNCT
cana-809	142	19	is	be	AUX
cana-809	142	20	(	(	PUNCT
cana-809	142	21	1	1	NUM
cana-809	142	22	,	,	PUNCT
cana-809	142	23	2	2	NUM
cana-809	142	24	)	)	PUNCT
cana-809	142	25	*	*	PUNCT
cana-809	142	26	d**spopen	d**spopen	ADJ
cana-809	142	27	in	in	ADP
cana-809	142	28	l.	l.	PROPN
cana-809	142	29	hence	hence	ADV
cana-809	142	30	h	h	PROPN
cana-809	142	31	is	be	AUX
cana-809	142	32	(	(	PUNCT
cana-809	142	33	1	1	NUM
cana-809	142	34	,	,	PUNCT
cana-809	142	35	2)*-d**spcontinous	2)*-d**spcontinous	NUM
cana-809	142	36	.	.	PUNCT
cana-809	142	37	theorem	theorem	VERB
cana-809	142	38	5.6	5.6	NUM
cana-809	142	39	a	a	DET
cana-809	142	40	s	s	NOUN
cana-809	142	41	s	s	X
cana-809	142	42	u	u	NOUN
cana-809	142	43	m	m	NOUN
cana-809	142	44	e	e	NOUN
cana-809	142	45	h	h	NOUN
cana-809	142	46	:	:	PUNCT
cana-809	142	47	k	k	X
cana-809	142	48	→	→	PUNCT
cana-809	142	49	l	l	NOUN
cana-809	142	50	is	be	AUX
cana-809	142	51	a	a	DET
cana-809	142	52	map	map	NOUN
cana-809	142	53	if	if	SCONJ
cana-809	142	54	h	h	NOUN
cana-809	142	55	is	be	AUX
cana-809	142	56	(	(	PUNCT
cana-809	142	57	1	1	NUM
cana-809	142	58	,	,	PUNCT
cana-809	142	59	2)*-d**spcontinuous	2)*-d**spcontinuous	ADJ
cana-809	142	60	map	map	NOUN
cana-809	142	61	,	,	PUNCT
cana-809	142	62	then	then	ADV
cana-809	142	63	h	h	PROPN
cana-809	142	64	(	(	PUNCT
cana-809	142	65	τ1,2	τ1,2	ADJ
cana-809	142	66	-	-	PUNCT
cana-809	142	67	d**spcl(b	d**spcl(b	NOUN
cana-809	142	68	)	)	PUNCT
cana-809	142	69	)	)	PUNCT
cana-809	143	1	⊆	⊆	NUM
cana-809	143	2	σ1,2	σ1,2	NUM
cana-809	143	3	-	-	PUNCT
cana-809	143	4	spcl(h(b	spcl(h(b	PROPN
cana-809	143	5	)	)	PUNCT
cana-809	143	6	)	)	PUNCT
cana-809	143	7	.	.	PUNCT
cana-809	144	1	proof	proof	NOUN
cana-809	144	2	.	.	PUNCT
cana-809	145	1	given	give	VERB
cana-809	145	2	h(b	h(b	PROPN
cana-809	145	3	)	)	PUNCT
cana-809	145	4	⊆	⊆	NUM
cana-809	145	5	σ1,2	σ1,2	PROPN
cana-809	145	6	-	-	PUNCT
cana-809	145	7	spcl(h(b	spcl(h(b	PROPN
cana-809	145	8	)	)	PUNCT
cana-809	145	9	)	)	PUNCT
cana-809	145	10	,	,	PUNCT
cana-809	145	11	⇒	⇒	PROPN
cana-809	145	12	b	b	PROPN
cana-809	145	13	⊆	⊆	NUM
cana-809	145	14	h−1(σ1,2	h−1(σ1,2	NUM
cana-809	145	15	-	-	PUNCT
cana-809	145	16	spcl((b	spcl((b	NOUN
cana-809	145	17	)	)	PUNCT
cana-809	145	18	)	)	PUNCT
cana-809	145	19	.	.	PUNCT
cana-809	146	1	then	then	ADV
cana-809	146	2	σ1,2	σ1,2	PROPN
cana-809	146	3	-	-	PUNCT
cana-809	146	4	spcl(h(b	spcl(h(b	PROPN
cana-809	146	5	)	)	PUNCT
cana-809	146	6	)	)	PUNCT
cana-809	146	7	is	be	AUX
cana-809	146	8	a	a	DET
cana-809	146	9	σ1,2cs	σ1,2cs	PROPN
cana-809	146	10	in	in	ADP
cana-809	146	11	l	l	NOUN
cana-809	146	12	and	and	CCONJ
cana-809	146	13	communications	communication	NOUN
cana-809	146	14	on	on	ADP
cana-809	146	15	applied	apply	VERB
cana-809	146	16	nonlinear	nonlinear	ADJ
cana-809	146	17	analysis	analysis	NOUN
cana-809	146	18	issn	issn	NOUN
cana-809	146	19	:	:	PUNCT
cana-809	146	20	1074	1074	NUM
cana-809	146	21	-	-	PUNCT
cana-809	146	22	133x	133x	NUM
cana-809	146	23	vol	vol	NOUN
cana-809	146	24	31	31	NUM
cana-809	146	25	no	no	NOUN
cana-809	146	26	.	.	PUNCT
cana-809	147	1	2s	2s	NUM
cana-809	147	2	(	(	PUNCT
cana-809	147	3	2024	2024	NUM
cana-809	147	4	)	)	PUNCT
cana-809	147	5	725	725	NUM
cana-809	147	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-809	147	7	h	h	NOUN
cana-809	147	8	is	be	AUX
cana-809	147	9	(	(	PUNCT
cana-809	147	10	1	1	NUM
cana-809	147	11	,	,	PUNCT
cana-809	147	12	2)*-d**spcontinuous	2)*-d**spcontinuous	ADJ
cana-809	147	13	map	map	NOUN
cana-809	147	14	⇒	⇒	NOUN
cana-809	147	15	h−1(σ1,2	h−1(σ1,2	NOUN
cana-809	147	16	-	-	PUNCT
cana-809	147	17	spcl(h(b	spcl(h(b	PROPN
cana-809	147	18	)	)	PUNCT
cana-809	147	19	)	)	PUNCT
cana-809	148	1	is	be	AUX
cana-809	148	2	(	(	PUNCT
cana-809	148	3	1	1	NUM
cana-809	148	4	,	,	PUNCT
cana-809	148	5	2)*-d**spclosed	2)*-d**spclosed	NUM
cana-809	148	6	in	in	ADP
cana-809	148	7	l.	l.	PROPN
cana-809	148	8	hence	hence	ADV
cana-809	148	9	τ1,2	τ1,2	PROPN
cana-809	148	10	-	-	PUNCT
cana-809	148	11	d**spcl(b	d**spcl(b	NOUN
cana-809	148	12	)	)	PUNCT
cana-809	148	13	⊆	⊆	NUM
cana-809	148	14	h−1(σ1,2	h−1(σ1,2	NUM
cana-809	148	15	-	-	PUNCT
cana-809	148	16	scl(f(b).it	scl(f(b).it	NOUN
cana-809	148	17	proves	prove	VERB
cana-809	148	18	h	h	NOUN
cana-809	148	19	(	(	PUNCT
cana-809	148	20	τ1,2	τ1,2	ADJ
cana-809	148	21	-	-	PUNCT
cana-809	148	22	d**spcl	d**spcl	NOUN
cana-809	148	23	(	(	PUNCT
cana-809	148	24	b	b	NOUN
cana-809	148	25	)	)	PUNCT
cana-809	148	26	)	)	PUNCT
cana-809	149	1	⊆	⊆	NUM
cana-809	149	2	σ1,2	σ1,2	NUM
cana-809	149	3	-	-	PUNCT
cana-809	149	4	spcl(h(b	spcl(h(b	PROPN
cana-809	149	5	)	)	PUNCT
cana-809	149	6	)	)	PUNCT
cana-809	149	7	.	.	PUNCT
cana-809	150	1	(	(	PUNCT
cana-809	150	2	1	1	NUM
cana-809	150	3	,	,	PUNCT
cana-809	150	4	2)*-d**spcontinuous	2)*-d**spcontinuous	ADJ
cana-809	150	5	and	and	CCONJ
cana-809	150	6	(	(	PUNCT
cana-809	150	7	1	1	NUM
cana-809	150	8	,	,	PUNCT
cana-809	150	9	2)*-d**spirresolute	2)*-d**spirresolute	NUM
cana-809	150	10	mappings	mapping	NOUN
cana-809	150	11	definition	definition	NOUN
cana-809	150	12	6.1	6.1	NUM
cana-809	150	13	consider	consider	VERB
cana-809	150	14	a	a	DET
cana-809	150	15	map	map	NOUN
cana-809	150	16	h	h	NOUN
cana-809	151	1	:	:	PUNCT
cana-809	151	2	k	k	X
cana-809	151	3	→	→	PUNCT
cana-809	151	4	l	l	NOUN
cana-809	151	5	h	h	NOUN
cana-809	151	6	is	be	AUX
cana-809	151	7	(	(	PUNCT
cana-809	151	8	1	1	NUM
cana-809	151	9	,	,	PUNCT
cana-809	151	10	2)*-d**spirresolute	2)*-d**spirresolute	NUM
cana-809	151	11	if	if	SCONJ
cana-809	151	12	for	for	SCONJ
cana-809	151	13	every	every	DET
cana-809	151	14	(	(	PUNCT
cana-809	151	15	1	1	NUM
cana-809	151	16	,	,	PUNCT
cana-809	151	17	2)*-d**spos	2)*-d**spos	NUM
cana-809	151	18	of	of	ADP
cana-809	151	19	l	l	NOUN
cana-809	151	20	its	its	PRON
cana-809	151	21	inverse	inverse	NOUN
cana-809	151	22	image	image	NOUN
cana-809	151	23	of	of	ADP
cana-809	151	24	h	h	NOUN
cana-809	151	25	is	be	AUX
cana-809	151	26	(	(	PUNCT
cana-809	151	27	1	1	NUM
cana-809	151	28	,	,	PUNCT
cana-809	151	29	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	151	30	in	in	ADP
cana-809	151	31	k	k	PROPN
cana-809	151	32	remark	remark	PROPN
cana-809	151	33	6.2	6.2	NUM
cana-809	151	34	consider	consider	VERB
cana-809	151	35	a	a	DET
cana-809	151	36	map	map	NOUN
cana-809	151	37	h	h	NOUN
cana-809	151	38	:	:	PUNCT
cana-809	151	39	k	k	X
cana-809	151	40	→	→	PUNCT
cana-809	151	41	l	l	NOUN
cana-809	151	42	for	for	ADP
cana-809	151	43	every	every	DET
cana-809	151	44	(	(	PUNCT
cana-809	151	45	1	1	NUM
cana-809	151	46	,	,	PUNCT
cana-809	151	47	2)*-d**spcs	2)*-d**spcs	PROPN
cana-809	151	48	of	of	ADP
cana-809	151	49	l	l	NOUN
cana-809	151	50	by	by	ADP
cana-809	151	51	defn	defn	PROPN
cana-809	151	52	of	of	ADP
cana-809	151	53	6.1	6.1	NUM
cana-809	151	54	(	(	PUNCT
cana-809	151	55	1	1	NUM
cana-809	151	56	,	,	PUNCT
cana-809	151	57	2)*-d**spclosed	2)*-d**spclosed	NUM
cana-809	151	58	in	in	ADP
cana-809	151	59	k	k	PROPN
cana-809	151	60	..	..	PUNCT
cana-809	151	61	theorem	theorem	VERB
cana-809	151	62	6.3	6.3	NUM
cana-809	151	63	consider	consider	VERB
cana-809	151	64	a	a	DET
cana-809	151	65	map	map	NOUN
cana-809	151	66	proof	proof	NOUN
cana-809	151	67	.	.	PUNCT
cana-809	152	1	h	h	NOUN
cana-809	152	2	:	:	PUNCT
cana-809	153	1	k	k	X
cana-809	153	2	→	→	PUNCT
cana-809	153	3	l	l	NOUN
cana-809	153	4	is	be	AUX
cana-809	153	5	(	(	PUNCT
cana-809	153	6	1	1	NUM
cana-809	153	7	,	,	PUNCT
cana-809	153	8	2)*-d**spirresolute	2)*-d**spirresolute	NUM
cana-809	153	9	implies	imply	VERB
cana-809	153	10	h	h	NOUN
cana-809	153	11	is	be	AUX
cana-809	153	12	(	(	PUNCT
cana-809	153	13	1	1	NUM
cana-809	153	14	,	,	PUNCT
cana-809	153	15	2)*-d**spcontinuous	2)*-d**spcontinuous	PROPN
cana-809	153	16	.	.	PUNCT
cana-809	154	1	suppose	suppose	VERB
cana-809	154	2	r	r	NOUN
cana-809	154	3	is	be	AUX
cana-809	154	4	a	a	DET
cana-809	154	5	τ1,2os	τ1,2os	PROPN
cana-809	154	6	in	in	ADP
cana-809	154	7	k.	k.	PROPN
cana-809	155	1	also	also	ADV
cana-809	155	2	h	h	PROPN
cana-809	155	3	is	be	AUX
cana-809	155	4	(	(	PUNCT
cana-809	155	5	1	1	NUM
cana-809	155	6	,	,	PUNCT
cana-809	155	7	2)*-d**spirresolute	2)*-d**spirresolute	NUM
cana-809	155	8	proves	prove	VERB
cana-809	155	9	h−1(r	h−1(r	NOUN
cana-809	155	10	)	)	PUNCT
cana-809	155	11	is	be	AUX
cana-809	155	12	(	(	PUNCT
cana-809	155	13	1	1	NUM
cana-809	155	14	,	,	PUNCT
cana-809	155	15	2	2	NUM
cana-809	155	16	)	)	PUNCT
cana-809	155	17	*	*	PUNCT
cana-809	155	18	d**spopen	d**spopen	ADJ
cana-809	155	19	in	in	ADP
cana-809	155	20	k.	k.	NOUN
cana-809	156	1	thus	thus	ADV
cana-809	156	2	h	h	NOUN
cana-809	156	3	is	be	AUX
cana-809	156	4	(	(	PUNCT
cana-809	156	5	1	1	NUM
cana-809	156	6	,	,	PUNCT
cana-809	156	7	2)*-d**spcontinuous	2)*-d**spcontinuous	PROPN
cana-809	156	8	.	.	PUNCT
cana-809	156	9	example	example	NOUN
cana-809	156	10	6.4	6.4	NUM
cana-809	156	11	reverse	reverse	ADJ
cana-809	156	12	part	part	NOUN
cana-809	156	13	of	of	ADP
cana-809	156	14	the	the	DET
cana-809	156	15	theorem	theorem	NOUN
cana-809	156	16	6.3	6.3	PRON
cana-809	156	17	can	can	AUX
cana-809	156	18	be	be	AUX
cana-809	156	19	proved	prove	VERB
cana-809	156	20	by	by	ADP
cana-809	156	21	the	the	DET
cana-809	156	22	following	following	ADJ
cana-809	156	23	example	example	NOUN
cana-809	156	24	to	to	PART
cana-809	156	25	show	show	VERB
cana-809	156	26	it	it	PRON
cana-809	156	27	is	be	AUX
cana-809	156	28	not	not	PART
cana-809	156	29	true	true	ADJ
cana-809	156	30	let	let	VERB
cana-809	156	31	x	x	SYM
cana-809	156	32	=	=	PUNCT
cana-809	156	33	y	y	PROPN
cana-809	156	34	=	=	PRON
cana-809	156	35	{	{	PUNCT
cana-809	156	36	a1	a1	NOUN
cana-809	156	37	,	,	PUNCT
cana-809	156	38	a2	a2	PROPN
cana-809	156	39	,	,	PUNCT
cana-809	156	40	a3	a3	NOUN
cana-809	156	41	}	}	PUNCT
cana-809	156	42	,	,	PUNCT
cana-809	156	43	τ1	τ1	NOUN
cana-809	156	44	=	=	SYM
cana-809	156	45	{	{	PUNCT
cana-809	156	46	x	x	PROPN
cana-809	156	47	,	,	PUNCT
cana-809	156	48	φ	φ	PROPN
cana-809	156	49	,	,	PUNCT
cana-809	156	50	{	{	PUNCT
cana-809	156	51	a1	a1	NOUN
cana-809	156	52	}	}	PUNCT
cana-809	156	53	,	,	PUNCT
cana-809	156	54	{	{	PUNCT
cana-809	156	55	a2	a2	PROPN
cana-809	156	56	}	}	PUNCT
cana-809	156	57	,	,	PUNCT
cana-809	156	58	{	{	PUNCT
cana-809	156	59	a1,a2	a1,a2	PROPN
cana-809	156	60	}	}	PUNCT
cana-809	156	61	}	}	PUNCT
cana-809	156	62	,	,	PUNCT
cana-809	156	63	τ2	τ2	NOUN
cana-809	156	64	=	=	SYM
cana-809	156	65	{	{	PUNCT
cana-809	156	66	x	x	PROPN
cana-809	156	67	,	,	PUNCT
cana-809	156	68	φ	φ	PROPN
cana-809	156	69	,	,	PUNCT
cana-809	156	70	{	{	PUNCT
cana-809	156	71	a1,a2	a1,a2	PROPN
cana-809	156	72	}	}	PUNCT
cana-809	156	73	}	}	PUNCT
cana-809	156	74	,	,	PUNCT
cana-809	156	75	σ1	σ1	NOUN
cana-809	156	76	=	=	PUNCT
cana-809	156	77	{	{	PUNCT
cana-809	156	78	x	x	PROPN
cana-809	156	79	,	,	PUNCT
cana-809	156	80	φ	φ	PROPN
cana-809	156	81	,	,	PUNCT
cana-809	156	82	{	{	PUNCT
cana-809	156	83	a1	a1	NOUN
cana-809	156	84	}	}	PUNCT
cana-809	156	85	}	}	PUNCT
cana-809	156	86	,	,	PUNCT
cana-809	156	87	σ2	σ2	NOUN
cana-809	156	88	=	=	PUNCT
cana-809	156	89	{	{	PUNCT
cana-809	156	90	x	x	PROPN
cana-809	156	91	,	,	PUNCT
cana-809	156	92	φ	φ	PROPN
cana-809	156	93	,	,	PUNCT
cana-809	156	94	{	{	PUNCT
cana-809	156	95	a1	a1	NOUN
cana-809	156	96	}	}	PUNCT
cana-809	156	97	,	,	PUNCT
cana-809	156	98	{	{	PUNCT
cana-809	156	99	a1,a2	a1,a2	PROPN
cana-809	156	100	}	}	PUNCT
cana-809	156	101	}	}	PUNCT
cana-809	156	102	.	.	PUNCT
cana-809	157	1	let	let	VERB
cana-809	157	2	f	f	NOUN
cana-809	157	3	:	:	PUNCT
cana-809	157	4	x	x	X
cana-809	157	5	→	→	SYM
cana-809	157	6	y	y	PROPN
cana-809	157	7	be	be	AUX
cana-809	157	8	the	the	DET
cana-809	157	9	identity	identity	NOUN
cana-809	157	10	map	map	NOUN
cana-809	157	11	.	.	PUNCT
cana-809	158	1	hence	hence	ADV
cana-809	158	2	f	f	PROPN
cana-809	158	3	is	be	AUX
cana-809	158	4	(	(	PUNCT
cana-809	158	5	1	1	NUM
cana-809	158	6	,	,	PUNCT
cana-809	158	7	2)*d**sp	2)*d**sp	NUM
cana-809	158	8	-	-	ADJ
cana-809	158	9	continuous	continuous	ADJ
cana-809	158	10	but	but	CCONJ
cana-809	158	11	f	f	PROPN
cana-809	158	12	is	be	AUX
cana-809	158	13	not	not	PART
cana-809	158	14	(	(	PUNCT
cana-809	158	15	1	1	NUM
cana-809	158	16	,	,	PUNCT
cana-809	158	17	2)*-d**spirresolute	2)*-d**spirresolute	NUM
cana-809	158	18	.	.	PUNCT
cana-809	158	19	theorem	theorem	VERB
cana-809	158	20	6.5	6.5	NUM
cana-809	158	21	consider	consider	VERB
cana-809	158	22	a	a	DET
cana-809	158	23	map	map	NOUN
cana-809	158	24	h	h	NOUN
cana-809	158	25	:	:	PUNCT
cana-809	158	26	k	k	X
cana-809	158	27	→	→	PUNCT
cana-809	158	28	l	l	NOUN
cana-809	158	29	h	h	NOUN
cana-809	158	30	is	be	AUX
cana-809	158	31	(	(	PUNCT
cana-809	158	32	1	1	NUM
cana-809	158	33	,	,	PUNCT
cana-809	158	34	2)*continuous	2)*continuous	NUM
cana-809	158	35	and	and	CCONJ
cana-809	158	36	l	l	NOUN
cana-809	158	37	is	be	AUX
cana-809	158	38	(	(	PUNCT
cana-809	158	39	1	1	NUM
cana-809	158	40	,	,	PUNCT
cana-809	158	41	2)*-d**sp	2)*-d**sp	NUM
cana-809	158	42	-	-	PUNCT
cana-809	158	43	t1/2	t1/2	NOUN
cana-809	158	44	-	-	PUNCT
cana-809	158	45	space	space	NOUN
cana-809	158	46	implies	imply	VERB
cana-809	158	47	h	h	NOUN
cana-809	158	48	is	be	AUX
cana-809	158	49	(	(	PUNCT
cana-809	158	50	1	1	NUM
cana-809	158	51	,	,	PUNCT
cana-809	158	52	2)*d**spirresolute	2)*d**spirresolute	NOUN
cana-809	158	53	.	.	PUNCT
cana-809	159	1	proof	proof	NOUN
cana-809	159	2	assume	assume	VERB
cana-809	159	3	b	b	X
cana-809	159	4	be	be	AUX
cana-809	159	5	(	(	PUNCT
cana-809	159	6	1	1	NUM
cana-809	159	7	,	,	PUNCT
cana-809	159	8	2)*-d**spos	2)*-d**spos	PROPN
cana-809	159	9	in	in	ADP
cana-809	159	10	l.	l.	PROPN
cana-809	159	11	also	also	ADV
cana-809	159	12	l	l	PROPN
cana-809	159	13	is	be	AUX
cana-809	159	14	(	(	PUNCT
cana-809	159	15	1,2)*-d**sp	1,2)*-d**sp	NUM
cana-809	159	16	-	-	PUNCT
cana-809	159	17	t1/2	t1/2	NOUN
cana-809	159	18	-	-	NOUN
cana-809	159	19	space	space	NOUN
cana-809	159	20	,	,	PUNCT
cana-809	159	21	implies	imply	VERB
cana-809	159	22	b	b	NOUN
cana-809	159	23	is	be	AUX
cana-809	159	24	an	an	DET
cana-809	159	25	σ1,2os	σ1,2o	NOUN
cana-809	159	26	in	in	ADP
cana-809	159	27	l	l	NOUN
cana-809	159	28	and	and	CCONJ
cana-809	159	29	also	also	ADV
cana-809	159	30	h	h	NOUN
cana-809	159	31	is	be	AUX
cana-809	159	32	(	(	PUNCT
cana-809	159	33	1	1	NUM
cana-809	159	34	,	,	PUNCT
cana-809	159	35	2)*continuous	2)*continuous	NOUN
cana-809	159	36	proves	prove	VERB
cana-809	159	37	h−1(b	h−1(b	PROPN
cana-809	159	38	)	)	PUNCT
cana-809	159	39	is	be	AUX
cana-809	159	40	(	(	PUNCT
cana-809	159	41	1	1	NUM
cana-809	159	42	,	,	PUNCT
cana-809	159	43	2)*-d**spos	2)*-d**spos	NUM
cana-809	159	44	in	in	ADP
cana-809	159	45	k.	k.	NOUN
cana-809	159	46	thus	thus	ADV
cana-809	159	47	h	h	NOUN
cana-809	159	48	is	be	AUX
cana-809	159	49	(	(	PUNCT
cana-809	159	50	1	1	NUM
cana-809	159	51	,	,	PUNCT
cana-809	159	52	2)*-d**spirresolute	2)*-d**spirresolute	NUM
cana-809	159	53	.	.	PUNCT
cana-809	160	1	communications	communication	NOUN
cana-809	160	2	on	on	ADP
cana-809	160	3	applied	apply	VERB
cana-809	160	4	nonlinear	nonlinear	ADJ
cana-809	160	5	analysis	analysis	NOUN
cana-809	160	6	issn	issn	NOUN
cana-809	160	7	:	:	PUNCT
cana-809	160	8	1074	1074	NUM
cana-809	160	9	-	-	PUNCT
cana-809	160	10	133x	133x	NUM
cana-809	160	11	vol	vol	NOUN
cana-809	160	12	31	31	NUM
cana-809	160	13	no	no	NOUN
cana-809	160	14	.	.	PUNCT
cana-809	161	1	2s	2s	NUM
cana-809	161	2	(	(	PUNCT
cana-809	161	3	2024	2024	NUM
cana-809	161	4	)	)	PUNCT
cana-809	161	5	726	726	NUM
cana-809	161	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-809	161	7	theorem	theorem	VERB
cana-809	161	8	6.6	6.6	NUM
cana-809	161	9	consider	consider	VERB
cana-809	161	10	a	a	DET
cana-809	161	11	map	map	NOUN
cana-809	161	12	h	h	NOUN
cana-809	161	13	:	:	PUNCT
cana-809	161	14	k	k	X
cana-809	161	15	→	→	PUNCT
cana-809	161	16	l	l	NOUN
cana-809	161	17	h	h	NOUN
cana-809	161	18	is	be	AUX
cana-809	161	19	(	(	PUNCT
cana-809	161	20	1	1	NUM
cana-809	161	21	,	,	PUNCT
cana-809	161	22	2)*-d**spirresolute	2)*-d**spirresolute	NUM
cana-809	161	23	and	and	CCONJ
cana-809	161	24	k	k	NOUN
cana-809	161	25	:	:	PUNCT
cana-809	162	1	l	l	PUNCT
cana-809	162	2	→	→	PUNCT
cana-809	162	3	m	m	AUX
cana-809	162	4	be	be	AUX
cana-809	162	5	an	an	DET
cana-809	162	6	(	(	PUNCT
cana-809	162	7	1	1	NUM
cana-809	162	8	,	,	PUNCT
cana-809	162	9	2)*-d**spirresolute	2)*-d**spirresolute	NUM
cana-809	162	10	maps	map	NOUN
cana-809	162	11	.	.	PUNCT
cana-809	163	1	then	then	ADV
cana-809	163	2	h	h	NOUN
cana-809	164	1	o	o	NOUN
cana-809	164	2	k	k	NOUN
cana-809	164	3	:	:	PUNCT
cana-809	165	1	k	k	X
cana-809	165	2	→	→	PUNCT
cana-809	165	3	m	m	NOUN
cana-809	165	4	is	be	AUX
cana-809	165	5	(	(	PUNCT
cana-809	165	6	1	1	NUM
cana-809	165	7	,	,	PUNCT
cana-809	165	8	2)*d**spirresolute	2)*d**spirresolute	NOUN
cana-809	165	9	.	.	PUNCT
cana-809	166	1	proof	proof	NOUN
cana-809	166	2	.	.	PUNCT
cana-809	167	1	suppose	suppose	VERB
cana-809	167	2	v	v	PART
cana-809	167	3	be	be	AUX
cana-809	167	4	a	a	DET
cana-809	167	5	(	(	PUNCT
cana-809	167	6	1	1	NUM
cana-809	167	7	,	,	PUNCT
cana-809	167	8	2)*-d**spos	2)*-d**spos	NUM
cana-809	167	9	in	in	ADP
cana-809	167	10	m.	m.	NOUN
cana-809	167	11	so	so	ADV
cana-809	167	12	h−1(v	h−1(v	PROPN
cana-809	167	13	)	)	PUNCT
cana-809	167	14	is	be	AUX
cana-809	167	15	(	(	PUNCT
cana-809	167	16	1	1	NUM
cana-809	167	17	,	,	PUNCT
cana-809	167	18	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	167	19	in	in	ADP
cana-809	167	20	l	l	PROPN
cana-809	167	21	implies	imply	VERB
cana-809	167	22	h−1(k−1(v	h−1(k−1(v	PROPN
cana-809	167	23	)	)	PUNCT
cana-809	167	24	)	)	PUNCT
cana-809	168	1	is	be	AUX
cana-809	168	2	(	(	PUNCT
cana-809	168	3	1	1	NUM
cana-809	168	4	,	,	PUNCT
cana-809	168	5	2)*-d**spopen	2)*-d**spopen	PROPN
cana-809	168	6	in	in	ADP
cana-809	168	7	k.	k.	PROPN
cana-809	169	1	thus	thus	ADV
cana-809	169	2	(	(	PUNCT
cana-809	169	3	hok)−1(v	hok)−1(v	X
cana-809	169	4	)	)	PUNCT
cana-809	169	5	is	be	AUX
cana-809	169	6	(	(	PUNCT
cana-809	169	7	1	1	NUM
cana-809	169	8	,	,	PUNCT
cana-809	169	9	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	169	10	in	in	ADP
cana-809	169	11	k.	k.	PROPN
cana-809	169	12	hence	hence	ADV
cana-809	169	13	hok	hok	PROPN
cana-809	169	14	is	be	AUX
cana-809	169	15	(	(	PUNCT
cana-809	169	16	1	1	NUM
cana-809	169	17	,	,	PUNCT
cana-809	169	18	2)*-d**spirresolute	2)*-d**spirresolute	NUM
cana-809	169	19	.	.	NOUN
cana-809	170	1	7	7	NUM
cana-809	170	2	.	.	PUNCT
cana-809	170	3	(	(	PUNCT
cana-809	170	4	1	1	NUM
cana-809	170	5	,	,	PUNCT
cana-809	170	6	2)*-d**spconnected	2)*-d**spconnected	PROPN
cana-809	170	7	sets	set	VERB
cana-809	170	8	definition	definition	NOUN
cana-809	170	9	7.1	7.1	NUM
cana-809	170	10	a	a	DET
cana-809	170	11	space	space	NOUN
cana-809	170	12	x	x	X
cana-809	170	13	is	be	AUX
cana-809	170	14	(	(	PUNCT
cana-809	170	15	1	1	NUM
cana-809	170	16	,	,	PUNCT
cana-809	170	17	2)*-d**spdisconnected	2)*-d**spdisconnecte	VERB
cana-809	170	18	if	if	SCONJ
cana-809	170	19	it	it	PRON
cana-809	170	20	is	be	AUX
cana-809	170	21	the	the	DET
cana-809	170	22	union	union	NOUN
cana-809	170	23	of	of	ADP
cana-809	170	24	two	two	NUM
cana-809	170	25	disjoint	disjoint	ADJ
cana-809	170	26	non	non	ADJ
cana-809	170	27	empty	empty	ADJ
cana-809	170	28	(	(	PUNCT
cana-809	170	29	1	1	NUM
cana-809	170	30	,	,	PUNCT
cana-809	170	31	2	2	NUM
cana-809	170	32	)	)	PUNCT
cana-809	170	33	*	*	PUNCT
cana-809	171	1	d**spos	d**spos	NOUN
cana-809	171	2	otherwise	otherwise	ADV
cana-809	171	3	it	it	PRON
cana-809	171	4	is	be	AUX
cana-809	171	5	said	say	VERB
cana-809	171	6	to	to	PART
cana-809	171	7	be	be	AUX
cana-809	171	8	(	(	PUNCT
cana-809	171	9	1	1	NUM
cana-809	171	10	,	,	PUNCT
cana-809	171	11	2)*-d**spconnected	2)*-d**spconnected	PROPN
cana-809	171	12	theorem	theorem	VERB
cana-809	171	13	7.2	7.2	NUM
cana-809	171	14	in	in	ADP
cana-809	171	15	bts	bt	NOUN
cana-809	171	16	x	x	SYM
cana-809	171	17	,	,	PUNCT
cana-809	171	18	the	the	DET
cana-809	171	19	following	following	ADJ
cana-809	171	20	statements	statement	NOUN
cana-809	171	21	are	be	AUX
cana-809	171	22	true	true	ADJ
cana-809	171	23	.	.	PUNCT
cana-809	172	1	•	•	NOUN
cana-809	172	2	x	x	X
cana-809	172	3	is	be	AUX
cana-809	172	4	(	(	PUNCT
cana-809	172	5	1	1	NUM
cana-809	172	6	,	,	PUNCT
cana-809	172	7	2)*-d**spconnected	2)*-d**spconnected	PROPN
cana-809	172	8	.	.	PROPN
cana-809	173	1	•	•	NUM
cana-809	173	2	φ	φ	PROPN
cana-809	173	3	,	,	PUNCT
cana-809	173	4	x	x	PRON
cana-809	173	5	are	be	AUX
cana-809	173	6	the	the	DET
cana-809	173	7	subsets	subset	NOUN
cana-809	173	8	which	which	PRON
cana-809	173	9	are	be	AUX
cana-809	173	10	both	both	PRON
cana-809	173	11	(	(	PUNCT
cana-809	173	12	1	1	NUM
cana-809	173	13	,	,	PUNCT
cana-809	173	14	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	173	15	and	and	CCONJ
cana-809	173	16	(	(	PUNCT
cana-809	173	17	1	1	NUM
cana-809	173	18	,	,	PUNCT
cana-809	173	19	2)*-d**spclosed	2)*-d**spclosed	NUM
cana-809	173	20	.	.	PUNCT
cana-809	174	1	proof	proof	NOUN
cana-809	174	2	.	.	PUNCT
cana-809	175	1	i	i	PRON
cana-809	175	2	⇒	⇒	VERB
cana-809	175	3	ii	ii	PROPN
cana-809	175	4	let	let	VERB
cana-809	175	5	u	u	PRON
cana-809	175	6	⊆	⊆	NUM
cana-809	175	7	x	x	SYM
cana-809	175	8	which	which	PRON
cana-809	175	9	is	be	AUX
cana-809	175	10	(	(	PUNCT
cana-809	175	11	1	1	NUM
cana-809	175	12	,	,	PUNCT
cana-809	175	13	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	175	14	&	&	CCONJ
cana-809	175	15	(	(	PUNCT
cana-809	175	16	1	1	NUM
cana-809	175	17	,	,	PUNCT
cana-809	175	18	2)*-d**spclosed	2)*-d**spclosed	NUM
cana-809	175	19	.	.	PUNCT
cana-809	176	1	then	then	ADV
cana-809	176	2	x	x	X
cana-809	176	3	/	/	SYM
cana-809	176	4	u	u	NOUN
cana-809	176	5	is	be	AUX
cana-809	176	6	also	also	ADV
cana-809	176	7	(	(	PUNCT
cana-809	176	8	1	1	NUM
cana-809	176	9	,	,	PUNCT
cana-809	176	10	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	176	11	&	&	CCONJ
cana-809	176	12	(	(	PUNCT
cana-809	176	13	1	1	NUM
cana-809	176	14	,	,	PUNCT
cana-809	176	15	2)*-d**spclosed	2)*-d**spclose	VERB
cana-809	176	16	by	by	ADP
cana-809	176	17	defn	defn	PROPN
cana-809	176	18	of	of	ADP
cana-809	176	19	7.1	7.1	NUM
cana-809	176	20	.	.	PUNCT
cana-809	177	1	u	u	NOUN
cana-809	177	2	and	and	CCONJ
cana-809	177	3	x	x	X
cana-809	177	4	/	/	SYM
cana-809	177	5	u	u	PROPN
cana-809	177	6	implies	imply	VERB
cana-809	177	7	either	either	CCONJ
cana-809	177	8	u	u	PROPN
cana-809	177	9	=	=	PROPN
cana-809	177	10	φ	φ	PROPN
cana-809	177	11	or	or	CCONJ
cana-809	177	12	x	x	SYM
cana-809	177	13	/	/	SYM
cana-809	177	14	u	u	X
cana-809	177	15	=	=	PROPN
cana-809	177	16	φ	φ	PROPN
cana-809	177	17	.	.	PUNCT
cana-809	177	18	ii	ii	PROPN
cana-809	177	19	⇒	⇒	NOUN
cana-809	178	1	i	i	PRON
cana-809	178	2	suppose	suppose	VERB
cana-809	178	3	a	a	DET
cana-809	178	4	,	,	PUNCT
cana-809	178	5	b	b	NOUN
cana-809	178	6	in	in	ADP
cana-809	178	7	x	x	SYM
cana-809	178	8	such	such	ADJ
cana-809	178	9	that	that	SCONJ
cana-809	178	10	aub	aub	PROPN
cana-809	179	1	=	=	NOUN
cana-809	179	2	x	x	X
cana-809	179	3	where	where	SCONJ
cana-809	179	4	a	a	DET
cana-809	179	5	,	,	PUNCT
cana-809	179	6	b	b	NOUN
cana-809	179	7	not	not	PART
cana-809	179	8	equal	equal	ADJ
cana-809	179	9	to	to	ADP
cana-809	179	10	empty	empty	ADJ
cana-809	179	11	(	(	PUNCT
cana-809	179	12	1	1	NUM
cana-809	179	13	,	,	PUNCT
cana-809	179	14	2)*-d**spos	2)*-d**spos	PROPN
cana-809	179	15	.	.	PUNCT
cana-809	180	1	so	so	ADV
cana-809	180	2	ax	ax	ADJ
cana-809	180	3	/	/	SYM
cana-809	180	4	b	b	NOUN
cana-809	180	5	is	be	AUX
cana-809	180	6	(	(	PUNCT
cana-809	180	7	1	1	NUM
cana-809	180	8	,	,	PUNCT
cana-809	180	9	2)*-d**spcs	2)*-d**spcs	PROPN
cana-809	180	10	⇒	⇒	VERB
cana-809	180	11	a	a	DET
cana-809	180	12	is(1	is(1	PROPN
cana-809	180	13	,	,	PUNCT
cana-809	180	14	2)*-d**spo⊆	2)*-d**spo⊆	NUM
cana-809	180	15	x	x	PUNCT
cana-809	180	16	and	and	CCONJ
cana-809	180	17	(	(	PUNCT
cana-809	180	18	1	1	NUM
cana-809	180	19	,	,	PUNCT
cana-809	180	20	2	2	NUM
cana-809	180	21	)	)	PUNCT
cana-809	180	22	*	*	PUNCT
cana-809	180	23	d**spcs	d**spc	NOUN
cana-809	180	24	⊆	⊆	NUM
cana-809	180	25	x	x	PUNCT
cana-809	180	26	as	as	SCONJ
cana-809	180	27	we	we	PRON
cana-809	180	28	assumed	assume	VERB
cana-809	180	29	a	a	DET
cana-809	180	30	=	=	SYM
cana-809	180	31	φ	φ	PROPN
cana-809	180	32	or	or	CCONJ
cana-809	180	33	x	x	PRON
cana-809	180	34	proves	prove	VERB
cana-809	180	35	x	x	INTJ
cana-809	180	36	is	be	AUX
cana-809	180	37	(	(	PUNCT
cana-809	180	38	1	1	NUM
cana-809	180	39	,	,	PUNCT
cana-809	180	40	2)*d**spconnected	2)*d**spconnecte	VERB
cana-809	180	41	.	.	PUNCT
cana-809	181	1	theorem	theorem	VERB
cana-809	181	2	6.3	6.3	NUM
cana-809	181	3	suppose	suppose	VERB
cana-809	181	4	a	a	DET
cana-809	181	5	mapping	mapping	NOUN
cana-809	181	6	j	j	NOUN
cana-809	182	1	:	:	PUNCT
cana-809	182	2	k	k	X
cana-809	182	3	→	→	PUNCT
cana-809	182	4	l	l	NOUN
cana-809	182	5	is	be	AUX
cana-809	182	6	i	i	PRON
cana-809	182	7	)	)	PUNCT
cana-809	182	8	(	(	PUNCT
cana-809	182	9	1	1	NUM
cana-809	182	10	,	,	PUNCT
cana-809	182	11	2)*-d**spcontinuous	2)*-d**spcontinuous	PROPN
cana-809	182	12	and	and	CCONJ
cana-809	182	13	onto	onto	ADP
cana-809	182	14	,	,	PUNCT
cana-809	182	15	k	k	PROPN
cana-809	182	16	is	be	AUX
cana-809	182	17	(	(	PUNCT
cana-809	182	18	1	1	NUM
cana-809	182	19	,	,	PUNCT
cana-809	182	20	2)*-d**spconnected	2)*-d**spconnected	PROPN
cana-809	182	21	⇒	⇒	PROPN
cana-809	182	22	l	l	NOUN
cana-809	182	23	is	be	AUX
cana-809	182	24	(	(	PUNCT
cana-809	182	25	1	1	NUM
cana-809	182	26	,	,	PUNCT
cana-809	182	27	2)*connected	2)*connecte	VERB
cana-809	182	28	.	.	PUNCT
cana-809	183	1	communications	communication	NOUN
cana-809	183	2	on	on	ADP
cana-809	183	3	applied	apply	VERB
cana-809	183	4	nonlinear	nonlinear	ADJ
cana-809	183	5	analysis	analysis	NOUN
cana-809	183	6	issn	issn	NOUN
cana-809	183	7	:	:	PUNCT
cana-809	183	8	1074	1074	NUM
cana-809	183	9	-	-	PUNCT
cana-809	183	10	133x	133x	NUM
cana-809	183	11	vol	vol	NOUN
cana-809	183	12	31	31	NUM
cana-809	183	13	no	no	NOUN
cana-809	183	14	.	.	PUNCT
cana-809	184	1	2s	2s	NUM
cana-809	184	2	(	(	PUNCT
cana-809	184	3	2024	2024	NUM
cana-809	184	4	)	)	PUNCT
cana-809	184	5	727	727	NUM
cana-809	184	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-809	184	7	(	(	PUNCT
cana-809	184	8	ii	ii	NOUN
cana-809	184	9	)	)	PUNCT
cana-809	184	10	if	if	SCONJ
cana-809	184	11	j	j	PROPN
cana-809	184	12	:	:	PUNCT
cana-809	184	13	k	k	X
cana-809	184	14	→	→	PUNCT
cana-809	184	15	l	l	NOUN
cana-809	184	16	is	be	AUX
cana-809	184	17	(	(	PUNCT
cana-809	184	18	1	1	NUM
cana-809	184	19	,	,	PUNCT
cana-809	184	20	2)*-d**spirresolute	2)*-d**spirresolute	NUM
cana-809	184	21	surjection	surjection	NOUN
cana-809	184	22	map	map	NOUN
cana-809	184	23	and	and	CCONJ
cana-809	184	24	k	k	X
cana-809	184	25	is	be	AUX
cana-809	184	26	(	(	PUNCT
cana-809	184	27	1	1	NUM
cana-809	184	28	,	,	PUNCT
cana-809	184	29	2)*-d**spconnected	2)*-d**spconnected	PROPN
cana-809	184	30	⇒	⇒	PROPN
cana-809	184	31	l	l	NOUN
cana-809	184	32	is	be	AUX
cana-809	184	33	(	(	PUNCT
cana-809	184	34	1	1	NUM
cana-809	184	35	,	,	PUNCT
cana-809	184	36	2)*-d**spconnected	2)*-d**spconnected	PROPN
cana-809	184	37	.	.	PUNCT
cana-809	185	1	proof	proof	NOUN
cana-809	185	2	.	.	PUNCT
cana-809	186	1	assume	assume	VERB
cana-809	186	2	l	l	NOUN
cana-809	186	3	is	be	AUX
cana-809	186	4	not	not	PART
cana-809	186	5	(	(	PUNCT
cana-809	186	6	1	1	NUM
cana-809	186	7	,	,	PUNCT
cana-809	186	8	2)*connected	2)*connecte	VERB
cana-809	186	9	.	.	PUNCT
cana-809	187	1	we	we	PRON
cana-809	187	2	have	have	VERB
cana-809	187	3	l	l	NOUN
cana-809	187	4	=	=	SYM
cana-809	187	5	c	c	NOUN
cana-809	187	6	∪	∪	NOUN
cana-809	187	7	d	d	NOUN
cana-809	187	8	is	be	AUX
cana-809	187	9	not	not	PART
cana-809	187	10	empty	empty	ADJ
cana-809	187	11	where	where	SCONJ
cana-809	187	12	c	c	NOUN
cana-809	187	13	and	and	CCONJ
cana-809	187	14	d	d	PROPN
cana-809	187	15	are	be	AUX
cana-809	187	16	disjoint	disjoint	ADJ
cana-809	187	17	σ1,2	σ1,2	ADJ
cana-809	187	18	-	-	PUNCT
cana-809	187	19	os	os	NOUN
cana-809	187	20	in	in	ADP
cana-809	187	21	l.	l.	PROPN
cana-809	187	22	also	also	ADV
cana-809	187	23	j	j	PROPN
cana-809	187	24	is	be	AUX
cana-809	187	25	(	(	PUNCT
cana-809	187	26	1	1	NUM
cana-809	187	27	,	,	PUNCT
cana-809	187	28	2)*-d**spcontinuous	2)*-d**spcontinuous	ADJ
cana-809	187	29	,	,	PUNCT
cana-809	187	30	onto	onto	ADP
cana-809	187	31	k	k	PROPN
cana-809	187	32	=	=	SYM
cana-809	187	33	f−1(c	f−1(c	PROPN
cana-809	187	34	)	)	PUNCT
cana-809	187	35	∪	∪	ADP
cana-809	187	36	f−1(d	f−1(d	PROPN
cana-809	187	37	)	)	PUNCT
cana-809	187	38	where	where	SCONJ
cana-809	187	39	f−1(c	f−1(c	PROPN
cana-809	187	40	)	)	PUNCT
cana-809	187	41	and	and	CCONJ
cana-809	187	42	f−1(d	f−1(d	PROPN
cana-809	187	43	)	)	PUNCT
cana-809	187	44	are	be	AUX
cana-809	187	45	disjoint	disjoint	ADJ
cana-809	187	46	but	but	CCONJ
cana-809	187	47	not	not	PART
cana-809	187	48	empty	empty	ADJ
cana-809	187	49	(	(	PUNCT
cana-809	187	50	1	1	NUM
cana-809	187	51	,	,	PUNCT
cana-809	187	52	2)*-d**sposs	2)*-d**sposs	PROPN
cana-809	187	53	contradicts	contradict	VERB
cana-809	187	54	x	x	X
cana-809	187	55	is	be	AUX
cana-809	187	56	(	(	PUNCT
cana-809	187	57	1	1	NUM
cana-809	187	58	,	,	PUNCT
cana-809	187	59	2	2	NUM
cana-809	187	60	)	)	PUNCT
cana-809	187	61	*	*	PUNCT
cana-809	187	62	d**spconnected	d**spconnecte	VERB
cana-809	187	63	.	.	PUNCT
cana-809	188	1	so	so	ADV
cana-809	188	2	that	that	SCONJ
cana-809	188	3	l	l	NOUN
cana-809	188	4	is	be	AUX
cana-809	188	5	(	(	PUNCT
cana-809	188	6	1	1	NUM
cana-809	188	7	,	,	PUNCT
cana-809	188	8	2)*connected	2)*connecte	VERB
cana-809	188	9	.	.	PUNCT
cana-809	189	1	ii	ii	PROPN
cana-809	189	2	proof	proof	NOUN
cana-809	189	3	obvious	obvious	ADJ
cana-809	189	4	from	from	ADP
cana-809	189	5	7.1	7.1	NUM
cana-809	189	6	.	.	PUNCT
cana-809	190	1	8	8	NUM
cana-809	190	2	.	.	PUNCT
cana-809	191	1	(	(	PUNCT
cana-809	191	2	1	1	NUM
cana-809	191	3	,	,	PUNCT
cana-809	191	4	2)*-d**sphomeomorphisms	2)*-d**sphomeomorphisms	NUM
cana-809	191	5	definition	definition	NOUN
cana-809	191	6	8.1	8.1	NUM
cana-809	191	7	f	f	NOUN
cana-809	191	8	:	:	PUNCT
cana-809	191	9	x	x	SYM
cana-809	191	10	→y	→y	PROPN
cana-809	191	11	is	be	AUX
cana-809	191	12	a	a	DET
cana-809	191	13	bijection	bijection	ADJ
cana-809	191	14	map	map	NOUN
cana-809	191	15	called	call	VERB
cana-809	191	16	(	(	PUNCT
cana-809	191	17	1	1	NUM
cana-809	191	18	,	,	PUNCT
cana-809	191	19	2)*-d**sphomeomorphism	2)*-d**sphomeomorphism	NUM
cana-809	191	20	if	if	SCONJ
cana-809	191	21	the	the	DET
cana-809	191	22	mapping	mapping	NOUN
cana-809	191	23	is	be	AUX
cana-809	191	24	(	(	PUNCT
cana-809	191	25	1	1	NUM
cana-809	191	26	,	,	PUNCT
cana-809	191	27	2)*-d**spcontinuous	2)*-d**spcontinuous	ADJ
cana-809	191	28	,	,	PUNCT
cana-809	191	29	(	(	PUNCT
cana-809	191	30	1	1	NUM
cana-809	191	31	,	,	PUNCT
cana-809	191	32	2)*-d**spopen	2)*-d**spopen	PROPN
cana-809	191	33	.	.	PUNCT
cana-809	192	1	remark	remark	VERB
cana-809	192	2	8.2	8.2	NUM
cana-809	192	3	every	every	DET
cana-809	192	4	(	(	PUNCT
cana-809	192	5	1	1	NUM
cana-809	192	6	,	,	PUNCT
cana-809	192	7	2)*homeomorphism	2)*homeomorphism	NUM
cana-809	192	8	is	be	AUX
cana-809	192	9	(	(	PUNCT
cana-809	192	10	1	1	NUM
cana-809	192	11	,	,	PUNCT
cana-809	192	12	2)*d**sphomeomorphism	2)*d**sphomeomorphism	NOUN
cana-809	192	13	but	but	CCONJ
cana-809	192	14	conversely	conversely	ADV
cana-809	192	15	not	not	PART
cana-809	192	16	true	true	ADJ
cana-809	192	17	theorem	theorem	NOUN
cana-809	192	18	8.3	8.3	NUM
cana-809	192	19	consider	consider	VERB
cana-809	192	20	the	the	DET
cana-809	192	21	mapping	mapping	NOUN
cana-809	192	22	h	h	NOUN
cana-809	192	23	:	:	PUNCT
cana-809	192	24	x	x	SYM
cana-809	193	1	→y	→y	PROPN
cana-809	193	2	,	,	PUNCT
cana-809	193	3	is	be	AUX
cana-809	193	4	1	1	NUM
cana-809	193	5	-	-	SYM
cana-809	193	6	1	1	NUM
cana-809	193	7	and	and	CCONJ
cana-809	193	8	onto	onto	ADP
cana-809	193	9	we	we	PRON
cana-809	193	10	have	have	VERB
cana-809	193	11	the	the	DET
cana-809	193	12	following	follow	VERB
cana-809	193	13	statements	statement	NOUN
cana-809	193	14	are	be	AUX
cana-809	193	15	true	true	ADJ
cana-809	193	16	.	.	PUNCT
cana-809	194	1	•	•	PRON
cana-809	194	2	(	(	PUNCT
cana-809	194	3	i)h−1	i)h−1	NOUN
cana-809	194	4	:	:	PUNCT
cana-809	194	5	y	y	PROPN
cana-809	194	6	→	→	PUNCT
cana-809	194	7	x	x	X
cana-809	194	8	is	be	AUX
cana-809	194	9	(	(	PUNCT
cana-809	194	10	1	1	NUM
cana-809	194	11	,	,	PUNCT
cana-809	194	12	2)*d**spcontinuous	2)*d**spcontinuous	NUM
cana-809	194	13	.	.	PUNCT
cana-809	195	1	•	•	NUM
cana-809	195	2	(	(	PUNCT
cana-809	195	3	ii)the	ii)the	DET
cana-809	195	4	mapping	mapping	NOUN
cana-809	195	5	is	be	AUX
cana-809	195	6	(	(	PUNCT
cana-809	195	7	1	1	NUM
cana-809	195	8	,	,	PUNCT
cana-809	195	9	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	195	10	.	.	PUNCT
cana-809	196	1	•	•	X
cana-809	196	2	(	(	PUNCT
cana-809	196	3	iii)the	iii)the	DET
cana-809	196	4	mapping	mapping	NOUN
cana-809	196	5	is	be	AUX
cana-809	196	6	(	(	PUNCT
cana-809	196	7	1	1	NUM
cana-809	196	8	,	,	PUNCT
cana-809	196	9	2)*d**spclosed	2)*d**spclosed	ADJ
cana-809	196	10	.	.	PUNCT
cana-809	197	1	proof	proof	NOUN
cana-809	197	2	.	.	PUNCT
cana-809	198	1	•	•	NUM
cana-809	198	2	(	(	PUNCT
cana-809	198	3	i)⇒	i)⇒	PROPN
cana-809	198	4	(	(	PUNCT
cana-809	198	5	ii	ii	NOUN
cana-809	198	6	)	)	PUNCT
cana-809	198	7	let	let	VERB
cana-809	198	8	k	k	X
cana-809	198	9	be	be	AUX
cana-809	198	10	any	any	DET
cana-809	198	11	τ1,2	τ1,2	ADJ
cana-809	198	12	-	-	NOUN
cana-809	198	13	os	os	NOUN
cana-809	198	14	in	in	ADP
cana-809	198	15	x.	x.	NOUN
cana-809	198	16	since	since	SCONJ
cana-809	198	17	f−1	f−1	PROPN
cana-809	198	18	is	be	AUX
cana-809	198	19	(	(	PUNCT
cana-809	198	20	1	1	NUM
cana-809	198	21	,	,	PUNCT
cana-809	198	22	2)*-d**spcontinuous	2)*-d**spcontinuous	PROPN
cana-809	198	23	,	,	PUNCT
cana-809	198	24	f	f	X
cana-809	198	25	(	(	PUNCT
cana-809	198	26	k	k	NOUN
cana-809	198	27	)	)	PUNCT
cana-809	198	28	in	in	ADP
cana-809	198	29	y	y	PROPN
cana-809	198	30	is	be	AUX
cana-809	198	31	(	(	PUNCT
cana-809	198	32	1	1	NUM
cana-809	198	33	,	,	PUNCT
cana-809	198	34	2)*d**spopen	2)*d**spopen	ADJ
cana-809	198	35	.	.	PUNCT
cana-809	199	1	thus	thus	ADV
cana-809	199	2	the	the	DET
cana-809	199	3	mapping	mapping	NOUN
cana-809	199	4	is	be	AUX
cana-809	199	5	(	(	PUNCT
cana-809	199	6	1	1	NUM
cana-809	199	7	,	,	PUNCT
cana-809	199	8	2	2	NUM
cana-809	199	9	)	)	PUNCT
cana-809	199	10	*	*	PUNCT
cana-809	200	1	d**spopen	d**spopen	ADJ
cana-809	200	2	.	.	PUNCT
cana-809	201	1	•	•	NUM
cana-809	201	2	(	(	PUNCT
cana-809	201	3	ii)implies	ii)implie	NOUN
cana-809	201	4	iii	iii	X
cana-809	201	5	in	in	ADP
cana-809	201	6	x	x	PART
cana-809	201	7	suppose	suppose	VERB
cana-809	201	8	f	f	PROPN
cana-809	201	9	is	be	AUX
cana-809	201	10	τ12cs	τ12cs	PRON
cana-809	201	11	,	,	PUNCT
cana-809	201	12	then	then	ADV
cana-809	201	13	x	x	X
cana-809	201	14	/	/	SYM
cana-809	201	15	f	f	PROPN
cana-809	201	16	is	be	AUX
cana-809	201	17	τ12os	τ12os	PUNCT
cana-809	201	18	and	and	CCONJ
cana-809	201	19	also	also	ADV
cana-809	201	20	the	the	DET
cana-809	201	21	mapping	mapping	NOUN
cana-809	201	22	is	be	AUX
cana-809	201	23	(	(	PUNCT
cana-809	201	24	1	1	NUM
cana-809	201	25	,	,	PUNCT
cana-809	201	26	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	201	27	,	,	PUNCT
cana-809	201	28	f	f	PROPN
cana-809	201	29	(	(	PUNCT
cana-809	201	30	x	x	PROPN
cana-809	201	31	/	/	SYM
cana-809	201	32	f	f	NOUN
cana-809	201	33	)	)	PUNCT
cana-809	201	34	in	in	ADP
cana-809	201	35	y	y	PROPN
cana-809	201	36	is	be	AUX
cana-809	201	37	(	(	PUNCT
cana-809	201	38	1	1	NUM
cana-809	201	39	,	,	PUNCT
cana-809	201	40	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	201	41	in	in	ADP
cana-809	201	42	y	y	PROPN
cana-809	201	43	..	..	PUNCT
cana-809	201	44	but	but	CCONJ
cana-809	201	45	in	in	ADP
cana-809	201	46	y	y	PROPN
cana-809	201	47	,	,	PUNCT
cana-809	201	48	f(x	f(x	PROPN
cana-809	201	49	/	/	SYM
cana-809	201	50	f	f	NOUN
cana-809	201	51	)	)	PUNCT
cana-809	202	1	=	=	SYM
cana-809	202	2	y	y	PROPN
cana-809	202	3	/f(f	/f(f	PROPN
cana-809	202	4	)	)	PUNCT
cana-809	202	5	where	where	SCONJ
cana-809	202	6	f	f	PROPN
cana-809	202	7	(	(	PUNCT
cana-809	202	8	f	f	X
cana-809	202	9	)	)	PUNCT
cana-809	202	10	is	be	AUX
cana-809	202	11	(	(	PUNCT
cana-809	202	12	1	1	NUM
cana-809	202	13	,	,	PUNCT
cana-809	202	14	2)*-d**spclosed	2)*-d**spclosed	NUM
cana-809	202	15	.	.	PUNCT
cana-809	203	1	thus	thus	ADV
cana-809	203	2	the	the	DET
cana-809	203	3	mapping	mapping	NOUN
cana-809	203	4	is	be	AUX
cana-809	203	5	(	(	PUNCT
cana-809	203	6	1	1	NUM
cana-809	203	7	,	,	PUNCT
cana-809	203	8	2)*-d**spclosed	2)*-d**spclosed	NUM
cana-809	203	9	map	map	NOUN
cana-809	203	10	(	(	PUNCT
cana-809	203	11	iii	iii	NOUN
cana-809	203	12	)	)	PUNCT
cana-809	203	13	implies	imply	VERB
cana-809	203	14	(	(	PUNCT
cana-809	203	15	i	i	NOUN
cana-809	203	16	)	)	PUNCT
cana-809	203	17	suppose	suppose	VERB
cana-809	203	18	r	r	NOUN
cana-809	203	19	is	be	AUX
cana-809	203	20	τ1,2	τ1,2	NOUN
cana-809	203	21	-	-	PUNCT
cana-809	203	22	cs	cs	ADJ
cana-809	203	23	in	in	ADP
cana-809	203	24	x	x	SYM
cana-809	203	25	,	,	PUNCT
cana-809	203	26	we	we	PRON
cana-809	203	27	have	have	VERB
cana-809	203	28	f	f	X
cana-809	203	29	(	(	PUNCT
cana-809	203	30	r	r	NOUN
cana-809	203	31	)	)	PUNCT
cana-809	203	32	in	in	ADP
cana-809	203	33	y	y	PROPN
cana-809	203	34	is	be	AUX
cana-809	203	35	(	(	PUNCT
cana-809	203	36	1	1	NUM
cana-809	203	37	,	,	PUNCT
cana-809	203	38	2)*-d**spclosed	2)*-d**spclosed	NUM
cana-809	203	39	.	.	PUNCT
cana-809	204	1	also	also	ADV
cana-809	204	2	the	the	DET
cana-809	204	3	mapping	mapping	NOUN
cana-809	204	4	f	f	X
cana-809	204	5	is	be	AUX
cana-809	204	6	(	(	PUNCT
cana-809	204	7	1	1	NUM
cana-809	204	8	,	,	PUNCT
cana-809	204	9	2)*d**spclosed	2)*d**spclose	VERB
cana-809	204	10	its	its	PRON
cana-809	204	11	inverse	inverse	NOUN
cana-809	204	12	mapping	mapping	NOUN
cana-809	204	13	is	be	AUX
cana-809	204	14	(	(	PUNCT
cana-809	204	15	1	1	NUM
cana-809	204	16	,	,	PUNCT
cana-809	204	17	2)*-d**spcontinuous	2)*-d**spcontinuous	PROPN
cana-809	204	18	.	.	PUNCT
cana-809	205	1	communications	communication	NOUN
cana-809	205	2	on	on	ADP
cana-809	205	3	applied	apply	VERB
cana-809	205	4	nonlinear	nonlinear	ADJ
cana-809	205	5	analysis	analysis	NOUN
cana-809	205	6	issn	issn	NOUN
cana-809	205	7	:	:	PUNCT
cana-809	205	8	1074	1074	NUM
cana-809	205	9	-	-	PUNCT
cana-809	205	10	133x	133x	NUM
cana-809	205	11	vol	vol	NOUN
cana-809	205	12	31	31	NUM
cana-809	205	13	no	no	NOUN
cana-809	205	14	.	.	PUNCT
cana-809	206	1	2s	2s	NUM
cana-809	206	2	(	(	PUNCT
cana-809	206	3	2024	2024	NUM
cana-809	206	4	)	)	PUNCT
cana-809	206	5	728	728	NUM
cana-809	206	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-809	206	7	theorem	theorem	VERB
cana-809	206	8	8.4	8.4	NUM
cana-809	206	9	suppose	suppose	VERB
cana-809	206	10	the	the	DET
cana-809	206	11	mapping	mapping	NOUN
cana-809	206	12	f	f	NOUN
cana-809	206	13	is	be	AUX
cana-809	206	14	1	1	NUM
cana-809	206	15	-	-	SYM
cana-809	206	16	1	1	NUM
cana-809	206	17	,	,	PUNCT
cana-809	206	18	onto	onto	ADP
cana-809	206	19	and	and	CCONJ
cana-809	206	20	(	(	PUNCT
cana-809	206	21	1	1	NUM
cana-809	206	22	,	,	PUNCT
cana-809	206	23	2)*-d**spcontinuous	2)*-d**spcontinuous	PROPN
cana-809	206	24	then	then	ADV
cana-809	206	25	the	the	DET
cana-809	206	26	implications	implication	NOUN
cana-809	206	27	are	be	AUX
cana-809	206	28	true	true	ADJ
cana-809	206	29	.	.	PUNCT
cana-809	207	1	to	to	PART
cana-809	207	2	prove	prove	VERB
cana-809	207	3	the	the	DET
cana-809	207	4	mapping	mapping	NOUN
cana-809	207	5	f	f	NOUN
cana-809	207	6	is	be	AUX
cana-809	207	7	•	•	ADP
cana-809	207	8	i)(1	i)(1	NOUN
cana-809	207	9	,	,	PUNCT
cana-809	207	10	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	207	11	.	.	PUNCT
cana-809	208	1	•	•	NUM
cana-809	208	2	ii)(1	ii)(1	PRON
cana-809	208	3	,	,	PUNCT
cana-809	208	4	2)*-d**sphomeomorphism	2)*-d**sphomeomorphism	NUM
cana-809	208	5	.	.	PUNCT
cana-809	209	1	•	•	NUM
cana-809	209	2	iii)(1	iii)(1	NOUN
cana-809	209	3	,	,	PUNCT
cana-809	209	4	2)*-d**spclosed	2)*-d**spclosed	NUM
cana-809	209	5	proof	proof	NOUN
cana-809	209	6	assume	assume	VERB
cana-809	209	7	i	i	PRON
cana-809	209	8	)	)	PUNCT
cana-809	209	9	f	f	PROPN
cana-809	209	10	is	be	AUX
cana-809	209	11	(	(	PUNCT
cana-809	209	12	1	1	NUM
cana-809	209	13	,	,	PUNCT
cana-809	209	14	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	209	15	also	also	ADV
cana-809	209	16	the	the	DET
cana-809	209	17	mapping	mapping	NOUN
cana-809	209	18	is	be	AUX
cana-809	209	19	1	1	NUM
cana-809	209	20	-	-	SYM
cana-809	209	21	1	1	NUM
cana-809	209	22	and	and	CCONJ
cana-809	209	23	onto	onto	ADP
cana-809	209	24	,	,	PUNCT
cana-809	209	25	(	(	PUNCT
cana-809	209	26	1	1	NUM
cana-809	209	27	,	,	PUNCT
cana-809	209	28	2)*-d**spcontinuous	2)*-d**spcontinuous	ADJ
cana-809	209	29	from	from	ADP
cana-809	209	30	the	the	DET
cana-809	209	31	definition8.1	definition8.1	PROPN
cana-809	209	32	,	,	PUNCT
cana-809	209	33	the	the	DET
cana-809	209	34	mapping	mapping	NOUN
cana-809	209	35	is	be	AUX
cana-809	209	36	(	(	PUNCT
cana-809	209	37	1	1	NUM
cana-809	209	38	,	,	PUNCT
cana-809	209	39	2)*-d**sphomeomorphism	2)*-d**sphomeomorphism	NUM
cana-809	209	40	.	.	PUNCT
cana-809	209	41	(	(	PUNCT
cana-809	209	42	ii	ii	NOUN
cana-809	209	43	)	)	PUNCT
cana-809	209	44	is	be	AUX
cana-809	209	45	proved	prove	VERB
cana-809	209	46	.	.	PUNCT
cana-809	210	1	assume	assume	VERB
cana-809	210	2	(	(	PUNCT
cana-809	210	3	ii	ii	NOUN
cana-809	210	4	)	)	PUNCT
cana-809	210	5	the	the	DET
cana-809	210	6	mapping	mapping	NOUN
cana-809	210	7	is	be	AUX
cana-809	210	8	(	(	PUNCT
cana-809	210	9	1	1	NUM
cana-809	210	10	,	,	PUNCT
cana-809	210	11	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	210	12	,	,	PUNCT
cana-809	210	13	1	1	NUM
cana-809	210	14	-	-	SYM
cana-809	210	15	1	1	NUM
cana-809	210	16	and	and	CCONJ
cana-809	210	17	onto	onto	ADP
cana-809	210	18	it	it	PRON
cana-809	210	19	is	be	AUX
cana-809	210	20	(	(	PUNCT
cana-809	210	21	1	1	NUM
cana-809	210	22	,	,	PUNCT
cana-809	210	23	2	2	NUM
cana-809	210	24	)	)	PUNCT
cana-809	210	25	*	*	PUNCT
cana-809	211	1	d**spclosed	d**spclose	VERB
cana-809	211	2	from	from	ADP
cana-809	211	3	thm	thm	PROPN
cana-809	211	4	7.8	7.8	NUM
cana-809	211	5	.	.	PUNCT
cana-809	212	1	(	(	PUNCT
cana-809	212	2	iii	iii	NOUN
cana-809	212	3	)	)	PUNCT
cana-809	212	4	proved	prove	VERB
cana-809	212	5	assume	assume	VERB
cana-809	212	6	iii	iii	NUM
cana-809	212	7	f	f	NOUN
cana-809	212	8	is	be	AUX
cana-809	212	9	(	(	PUNCT
cana-809	212	10	1	1	NUM
cana-809	212	11	,	,	PUNCT
cana-809	212	12	2)*-d**spclosed	2)*-d**spclosed	NUM
cana-809	212	13	and	and	CCONJ
cana-809	212	14	bijective	bijective	ADJ
cana-809	212	15	.	.	PUNCT
cana-809	213	1	f	f	PROPN
cana-809	213	2	is	be	AUX
cana-809	213	3	(	(	PUNCT
cana-809	213	4	1	1	NUM
cana-809	213	5	,	,	PUNCT
cana-809	213	6	2)*-d**spopen	2)*-d**spopen	NUM
cana-809	213	7	map	map	NOUN
cana-809	213	8	.	.	PUNCT
cana-809	214	1	by	by	ADP
cana-809	214	2	theorem	theorem	NOUN
cana-809	214	3	8.3	8.3	NUM
cana-809	214	4	(	(	PUNCT
cana-809	214	5	i	i	PROPN
cana-809	214	6	)	)	PUNCT
cana-809	214	7	,	,	PUNCT
cana-809	214	8	proved	prove	VERB
cana-809	214	9	references	reference	NOUN
cana-809	214	10	[	[	X
cana-809	214	11	1	1	NUM
cana-809	214	12	]	]	PUNCT
cana-809	214	13	bhattacharya	bhattacharya	NOUN
cana-809	214	14	,	,	PUNCT
cana-809	214	15	p.	p.	NOUN
cana-809	214	16	and	and	CCONJ
cana-809	214	17	lahiri	lahiri	PROPN
cana-809	214	18	,	,	PUNCT
cana-809	214	19	b.	b.	PROPN
cana-809	214	20	k.	k.	PROPN
cana-809	214	21	,	,	PUNCT
cana-809	214	22	semi	semi	ADJ
cana-809	214	23	-	-	ADJ
cana-809	214	24	generalized	generalized	ADJ
cana-809	214	25	closed	closed	ADJ
cana-809	214	26	sets	set	NOUN
cana-809	214	27	in	in	ADP
cana-809	214	28	a	a	DET
cana-809	214	29	topology	topology	NOUN
cana-809	214	30	,	,	PUNCT
cana-809	214	31	indian	indian	PROPN
cana-809	214	32	j.	j.	PROPN
cana-809	214	33	math	math	PROPN
cana-809	214	34	.	.	PUNCT
cana-809	214	35	,	,	PUNCT
cana-809	214	36	1987	1987	NUM
cana-809	214	37	,	,	PUNCT
cana-809	214	38	29(3	29(3	NUM
cana-809	214	39	)	)	PUNCT
cana-809	214	40	,	,	PUNCT
cana-809	214	41	375	375	NUM
cana-809	214	42	.	.	PUNCT
cana-809	215	1	[	[	X
cana-809	215	2	2	2	NUM
cana-809	215	3	]	]	X
cana-809	215	4	hdeib	hdeib	PROPN
cana-809	215	5	,	,	PUNCT
cana-809	215	6	h.z	h.z	PROPN
cana-809	215	7	.	.	PROPN
cana-809	215	8	,	,	PUNCT
cana-809	215	9	-closed	-closed	ADJ
cana-809	215	10	mappings	mapping	NOUN
cana-809	215	11	,	,	PUNCT
cana-809	215	12	rev	rev	PROPN
cana-809	215	13	.	.	PROPN
cana-809	215	14	colomb	colomb	PROPN
cana-809	215	15	.	.	PUNCT
cana-809	216	1	mat	mat	PROPN
cana-809	216	2	.	.	PROPN
cana-809	216	3	,	,	PUNCT
cana-809	216	4	1982	1982	NUM
cana-809	216	5	,	,	PUNCT
cana-809	216	6	16(3	16(3	PROPN
cana-809	216	7	-	-	SYM
cana-809	216	8	4	4	NUM
cana-809	216	9	)	)	PUNCT
cana-809	216	10	65	65	NUM
cana-809	216	11	-	-	SYM
cana-809	216	12	67	67	NUM
cana-809	216	13	.	.	PUNCT
cana-809	217	1	[	[	X
cana-809	217	2	3	3	NUM
cana-809	217	3	]	]	X
cana-809	217	4	lellis	lellis	PROPN
cana-809	217	5	thivagar	thivagar	NOUN
cana-809	217	6	,	,	PUNCT
cana-809	217	7	m.	m.	NOUN
cana-809	217	8	,	,	PUNCT
cana-809	217	9	ravi	ravi	NOUN
cana-809	217	10	,	,	PUNCT
cana-809	217	11	o.	o.	PROPN
cana-809	217	12	and	and	CCONJ
cana-809	217	13	abd	abd	PROPN
cana-809	217	14	el	el	PROPN
cana-809	217	15	-	-	PUNCT
cana-809	217	16	monsef	monsef	ADJ
cana-809	217	17	,	,	PUNCT
cana-809	217	18	m.	m.	PROPN
cana-809	217	19	e.	e.	PROPN
cana-809	217	20	:	:	PUNCT
cana-809	217	21	remarks	remark	NOUN
cana-809	217	22	on	on	ADP
cana-809	217	23	bitopological	bitopological	ADJ
cana-809	217	24	(	(	PUNCT
cana-809	217	25	l,2)*-quotient	l,2)*-quotient	PROPN
cana-809	217	26	mappings	mapping	NOUN
cana-809	217	27	,	,	PUNCT
cana-809	217	28	j.	j.	PROPN
cana-809	217	29	egypt	egypt	PROPN
cana-809	217	30	math	math	PROPN
cana-809	217	31	.	.	PUNCT
cana-809	218	1	soc	soc	PROPN
cana-809	218	2	.	.	PUNCT
cana-809	218	3	,	,	PUNCT
cana-809	218	4	16(1	16(1	X
cana-809	218	5	)	)	PUNCT
cana-809	218	6	(	(	PUNCT
cana-809	218	7	2008	2008	NUM
cana-809	218	8	)	)	PUNCT
cana-809	218	9	,	,	PUNCT
cana-809	218	10	17	17	NUM
cana-809	218	11	-	-	SYM
cana-809	218	12	25	25	NUM
cana-809	218	13	.	.	PUNCT
cana-809	219	1	[	[	X
cana-809	219	2	4	4	NUM
cana-809	219	3	]	]	X
cana-809	219	4	lellis	lellis	PROPN
cana-809	219	5	thivagar	thivagar	NOUN
cana-809	219	6	,	,	PUNCT
cana-809	219	7	m.	m.	NOUN
cana-809	219	8	,	,	PUNCT
cana-809	219	9	ravi	ravi	NOUN
cana-809	219	10	,	,	PUNCT
cana-809	219	11	o.	o.	PROPN
cana-809	219	12	,	,	PUNCT
cana-809	219	13	joseph	joseph	PROPN
cana-809	219	14	israel	israel	PROPN
cana-809	219	15	,	,	PUNCT
cana-809	219	16	m.	m.	NOUN
cana-809	219	17	and	and	CCONJ
cana-809	219	18	kayathri	kayathri	PROPN
cana-809	219	19	,	,	PUNCT
cana-809	219	20	k.	k.	PROPN
cana-809	219	21	decompositions	decomposition	NOUN
cana-809	219	22	of	of	ADP
cana-809	219	23	(	(	PUNCT
cana-809	219	24	1,2)*-rg	1,2)*-rg	NUM
cana-809	219	25	-	-	PUNCT
cana-809	219	26	continuous	continuous	ADJ
cana-809	219	27	maps	map	NOUN
cana-809	219	28	in	in	ADP
cana-809	219	29	bitopological	bitopological	ADJ
cana-809	219	30	spaces	space	NOUN
cana-809	219	31	,	,	PUNCT
cana-809	219	32	2009	2009	NUM
cana-809	219	33	,	,	PUNCT
cana-809	219	34	6(1	6(1	NUM
cana-809	219	35	)	)	PUNCT
cana-809	219	36	,	,	PUNCT
cana-809	219	37	13	13	NUM
cana-809	219	38	-	-	SYM
cana-809	219	39	21	21	NUM
cana-809	219	40	.	.	PUNCT
cana-809	220	1	[	[	X
cana-809	220	2	5	5	NUM
cana-809	220	3	]	]	X
cana-809	220	4	levine	levine	PROPN
cana-809	220	5	,	,	PUNCT
cana-809	220	6	n.	n.	PROPN
cana-809	220	7	,	,	PUNCT
cana-809	220	8	generalized	generalize	VERB
cana-809	220	9	closed	closed	ADJ
cana-809	220	10	sets	set	NOUN
cana-809	220	11	in	in	ADP
cana-809	220	12	topology	topology	NOUN
cana-809	220	13	,	,	PUNCT
cana-809	220	14	rend	rend	VERB
cana-809	220	15	.	.	PUNCT
cana-809	221	1	circ	circ	PROPN
cana-809	221	2	.	.	PUNCT
cana-809	222	1	mat	mat	PROPN
cana-809	222	2	.	.	PUNCT
cana-809	222	3	paleroma	paleroma	PROPN
cana-809	222	4	,	,	PUNCT
cana-809	222	5	1970	1970	NUM
cana-809	222	6	,	,	PUNCT
cana-809	222	7	19	19	NUM
cana-809	222	8	,	,	PUNCT
cana-809	222	9	89	89	NUM
cana-809	222	10	-	-	SYM
cana-809	222	11	96	96	NUM
cana-809	222	12	.	.	PUNCT
cana-809	223	1	[	[	X
cana-809	223	2	6	6	NUM
cana-809	223	3	]	]	X
cana-809	223	4	ravi	ravi	NOUN
cana-809	223	5	,	,	PUNCT
cana-809	223	6	o.	o.	PROPN
cana-809	223	7	,	,	PUNCT
cana-809	223	8	thivagar	thivagar	NOUN
cana-809	223	9	,	,	PUNCT
cana-809	223	10	m.	m.	NOUN
cana-809	223	11	l.	l.	PROPN
cana-809	223	12	and	and	CCONJ
cana-809	223	13	hatir	hatir	PROPN
cana-809	223	14	,	,	PUNCT
cana-809	223	15	e.	e.	NOUN
cana-809	223	16	:	:	PUNCT
cana-809	223	17	decomposition	decomposition	NOUN
cana-809	223	18	of	of	ADP
cana-809	223	19	(	(	PUNCT
cana-809	223	20	l	l	NOUN
cana-809	223	21	,	,	PUNCT
cana-809	223	22	2)*-continuity	2)*-continuity	NUM
cana-809	223	23	and	and	CCONJ
cana-809	223	24	(	(	PUNCT
cana-809	223	25	l,2)*--continuity	l,2)*--continuity	NOUN
cana-809	223	26	,	,	PUNCT
cana-809	223	27	miskolc	miskolc	ADJ
cana-809	223	28	mathematical	mathematical	ADJ
cana-809	223	29	notes	note	NOUN
cana-809	223	30	.	.	PUNCT
cana-809	223	31	,	,	PUNCT
cana-809	223	32	10(2	10(2	NUM
cana-809	223	33	)	)	PUNCT
cana-809	223	34	(	(	PUNCT
cana-809	223	35	2009	2009	NUM
cana-809	223	36	)	)	PUNCT
cana-809	223	37	,	,	PUNCT
cana-809	223	38	163	163	NUM
cana-809	223	39	-	-	SYM
cana-809	223	40	171	171	NUM
cana-809	223	41	.	.	PUNCT
cana-809	224	1	[	[	X
cana-809	224	2	7	7	X
cana-809	224	3	]	]	X
cana-809	224	4	ravi	ravi	NOUN
cana-809	224	5	,	,	PUNCT
cana-809	224	6	o.	o.	PROPN
cana-809	224	7	and	and	CCONJ
cana-809	224	8	lellis	lellis	PROPN
cana-809	224	9	thivagar	thivagar	NOUN
cana-809	224	10	,	,	PUNCT
cana-809	224	11	m.	m.	NOUN
cana-809	224	12	:	:	PUNCT
cana-809	224	13	a	a	DET
cana-809	224	14	bitopological	bitopological	ADJ
cana-809	224	15	(	(	PUNCT
cana-809	224	16	1,2)*-semigeneralized	1,2)*-semigeneralized	NUM
cana-809	224	17	continuous	continuous	ADJ
cana-809	224	18	maps	map	NOUN
cana-809	224	19	,	,	PUNCT
cana-809	224	20	bull	bull	NOUN
cana-809	224	21	.	.	PUNCT
cana-809	225	1	malays	malays	PROPN
cana-809	225	2	.	.	PUNCT
cana-809	226	1	math	math	NOUN
cana-809	226	2	.	.	PUNCT
cana-809	227	1	sci	sci	PROPN
cana-809	227	2	.	.	PROPN
cana-809	227	3	soc	soc	PROPN
cana-809	227	4	.	.	PUNCT
cana-809	227	5	,	,	PUNCT
cana-809	227	6	(	(	PUNCT
cana-809	227	7	2	2	NUM
cana-809	227	8	)	)	PUNCT
cana-809	227	9	,	,	PUNCT
cana-809	227	10	29(1	29(1	NUM
cana-809	227	11	)	)	PUNCT
cana-809	227	12	(	(	PUNCT
cana-809	227	13	2006	2006	NUM
cana-809	227	14	)	)	PUNCT
cana-809	227	15	,	,	PUNCT
cana-809	227	16	79	79	NUM
cana-809	227	17	-	-	SYM
cana-809	227	18	88	88	NUM
cana-809	227	19	.	.	PUNCT
cana-809	228	1	[	[	X
cana-809	228	2	8	8	NUM
cana-809	228	3	]	]	X
cana-809	228	4	ravi	ravi	NOUN
cana-809	228	5	,	,	PUNCT
cana-809	228	6	o.	o.	PROPN
cana-809	228	7	,	,	PUNCT
cana-809	228	8	pious	pious	ADJ
cana-809	228	9	missier	missier	NOUN
cana-809	228	10	,	,	PUNCT
cana-809	228	11	s.	s.	PROPN
cana-809	228	12	and	and	CCONJ
cana-809	228	13	salai	salai	PROPN
cana-809	228	14	parkunan	parkunan	PROPN
cana-809	228	15	,	,	PUNCT
cana-809	228	16	t.	t.	PROPN
cana-809	228	17	:	:	PUNCT
cana-809	228	18	on	on	ADP
cana-809	228	19	bitopological	bitopological	ADJ
cana-809	228	20	(	(	PUNCT
cana-809	228	21	l,2)*-generalized	l,2)*-generalized	ADJ
cana-809	228	22	homeomorphisms	homeomorphism	NOUN
cana-809	228	23	,	,	PUNCT
cana-809	228	24	int	int	PROPN
cana-809	228	25	j.	j.	PROPN
cana-809	228	26	contemp	contemp	PROPN
cana-809	228	27	.	.	PUNCT
cana-809	229	1	math	math	NOUN
cana-809	229	2	.	.	PUNCT
cana-809	230	1	sciences	science	NOUN
cana-809	230	2	.	.	PUNCT
cana-809	230	3	,	,	PUNCT
cana-809	230	4	5(11	5(11	NUM
cana-809	230	5	)	)	PUNCT
cana-809	230	6	(	(	PUNCT
cana-809	230	7	2010	2010	NUM
cana-809	230	8	)	)	PUNCT
cana-809	230	9	,	,	PUNCT
cana-809	230	10	543	543	NUM
cana-809	230	11	-	-	SYM
cana-809	230	12	557	557	NUM
cana-809	230	13	.	.	PUNCT