id	sid	tid	token	lemma	pos
cana-1749	1	1	communications	communication	NOUN
cana-1749	1	2	on	on	ADP
cana-1749	1	3	applied	apply	VERB
cana-1749	1	4	nonlinear	nonlinear	ADJ
cana-1749	1	5	analysis	analysis	NOUN
cana-1749	1	6	issn	issn	NOUN
cana-1749	1	7	:	:	PUNCT
cana-1749	1	8	1074	1074	NUM
cana-1749	1	9	-	-	PUNCT
cana-1749	1	10	133x	133x	NUM
cana-1749	1	11	vol	vol	NOUN
cana-1749	1	12	32	32	NUM
cana-1749	1	13	no	no	NOUN
cana-1749	1	14	.	.	NOUN
cana-1749	1	15	2	2	NUM
cana-1749	1	16	(	(	PUNCT
cana-1749	1	17	2025	2025	NUM
cana-1749	1	18	)	)	PUNCT
cana-1749	1	19	375	375	NUM
cana-1749	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1749	1	21	on	on	ADP
cana-1749	1	22	slightly	slightly	ADJ
cana-1749	1	23	delta	delta	NOUN
cana-1749	1	24	generalized	generalize	VERB
cana-1749	1	25	pre	pre	ADJ
cana-1749	1	26	-	-	ADJ
cana-1749	1	27	continuous	continuous	ADJ
cana-1749	1	28	functions	function	NOUN
cana-1749	1	29	j.b.toranagatti	j.b.toranagatti	PROPN
cana-1749	1	30	department	department	PROPN
cana-1749	1	31	of	of	ADP
cana-1749	1	32	mathematics	mathematics	PROPN
cana-1749	1	33	,	,	PUNCT
cana-1749	1	34	karnatak	karnatak	PROPN
cana-1749	1	35	university	university	PROPN
cana-1749	1	36	’s	’s	PART
cana-1749	1	37	karnatak	karnatak	PROPN
cana-1749	1	38	science	science	PROPN
cana-1749	1	39	college	college	PROPN
cana-1749	1	40	,	,	PUNCT
cana-1749	1	41	dharwad-580	dharwad-580	NOUN
cana-1749	1	42	001	001	NUM
cana-1749	1	43	,	,	PUNCT
cana-1749	1	44	karnataka	karnataka	PROPN
cana-1749	1	45	state	state	PROPN
cana-1749	1	46	,	,	PUNCT
cana-1749	1	47	india	india	PROPN
cana-1749	1	48	.	.	PUNCT
cana-1749	2	1	e	e	X
cana-1749	2	2	-	-	NOUN
cana-1749	2	3	mail	mail	NOUN
cana-1749	2	4	:	:	PUNCT
cana-1749	2	5	jagadeeshbt2000@gmail.com	jagadeeshbt2000@gmail.com	PROPN
cana-1749	2	6	article	article	PROPN
cana-1749	2	7	history	history	NOUN
cana-1749	2	8	:	:	PUNCT
cana-1749	2	9	received	receive	VERB
cana-1749	2	10	:	:	PUNCT
cana-1749	2	11	02	02	NUM
cana-1749	2	12	-	-	SYM
cana-1749	2	13	08	08	NUM
cana-1749	2	14	-	-	PUNCT
cana-1749	2	15	2024	2024	NUM
cana-1749	2	16	revised	revise	VERB
cana-1749	2	17	:	:	PUNCT
cana-1749	2	18	11	11	NUM
cana-1749	2	19	-	-	SYM
cana-1749	2	20	09	09	NUM
cana-1749	2	21	-	-	PUNCT
cana-1749	2	22	2024	2024	NUM
cana-1749	2	23	accepted	accept	VERB
cana-1749	2	24	:	:	PUNCT
cana-1749	2	25	20	20	NUM
cana-1749	2	26	-	-	SYM
cana-1749	2	27	09	09	NUM
cana-1749	2	28	-	-	PUNCT
cana-1749	2	29	2024	2024	NUM
cana-1749	2	30	abstract	abstract	NOUN
cana-1749	2	31	:	:	PUNCT
cana-1749	2	32	the	the	DET
cana-1749	2	33	purpose	purpose	NOUN
cana-1749	2	34	of	of	ADP
cana-1749	2	35	this	this	DET
cana-1749	2	36	article	article	NOUN
cana-1749	2	37	is	be	AUX
cana-1749	2	38	to	to	PART
cana-1749	2	39	introduce	introduce	VERB
cana-1749	2	40	and	and	CCONJ
cana-1749	2	41	investigate	investigate	VERB
cana-1749	2	42	the	the	DET
cana-1749	2	43	properties	property	NOUN
cana-1749	2	44	of	of	ADP
cana-1749	2	45	slightly	slightly	ADV
cana-1749	2	46	δgpcontinuous	δgpcontinuous	ADJ
cana-1749	2	47	functions	function	NOUN
cana-1749	2	48	.	.	PUNCT
cana-1749	3	1	also	also	ADV
cana-1749	3	2	,	,	PUNCT
cana-1749	3	3	the	the	DET
cana-1749	3	4	relationships	relationship	NOUN
cana-1749	3	5	of	of	ADP
cana-1749	3	6	slightly	slightly	ADV
cana-1749	3	7	δgp	δgp	ADJ
cana-1749	3	8	-	-	PUNCT
cana-1749	3	9	continuous	continuous	ADJ
cana-1749	3	10	functions	function	NOUN
cana-1749	3	11	and	and	CCONJ
cana-1749	3	12	graphs	graph	NOUN
cana-1749	3	13	are	be	AUX
cana-1749	3	14	investigated	investigate	VERB
cana-1749	3	15	.	.	PUNCT
cana-1749	4	1	keywordsslight	keywordsslight	NOUN
cana-1749	4	2	continuity	continuity	NOUN
cana-1749	4	3	,	,	PUNCT
cana-1749	4	4	slight	slight	ADJ
cana-1749	4	5	pre	pre	NOUN
cana-1749	4	6	-	-	NOUN
cana-1749	4	7	continuity	continuity	ADJ
cana-1749	4	8	,	,	PUNCT
cana-1749	4	9	slight	slight	ADJ
cana-1749	4	10	gpr	gpr	PROPN
cana-1749	4	11	-	-	PUNCT
cana-1749	4	12	continuity	continuity	NOUN
cana-1749	4	13	,	,	PUNCT
cana-1749	4	14	slight	slight	ADJ
cana-1749	4	15	δgp	δgp	NOUN
cana-1749	4	16	-	-	PUNCT
cana-1749	4	17	continuity	continuity	NOUN
cana-1749	4	18	.	.	PUNCT
cana-1749	5	1	1.introduction	1.introduction	NUM
cana-1749	5	2	and	and	CCONJ
cana-1749	5	3	preliminaries	preliminary	NOUN
cana-1749	5	4	:	:	PUNCT
cana-1749	5	5	in	in	ADP
cana-1749	5	6	1982,mashhour	1982,mashhour	NUM
cana-1749	5	7	et	et	NOUN
cana-1749	5	8	al	al	PROPN
cana-1749	5	9	.	.	PUNCT
cana-1749	6	1	[	[	X
cana-1749	6	2	9	9	NUM
cana-1749	6	3	]	]	PUNCT
cana-1749	6	4	introduced	introduce	VERB
cana-1749	6	5	preopen	preopen	ADJ
cana-1749	6	6	sets	set	NOUN
cana-1749	6	7	and	and	CCONJ
cana-1749	6	8	pre	pre	NOUN
cana-1749	6	9	-	-	NOUN
cana-1749	6	10	continuity	continuity	NOUN
cana-1749	6	11	in	in	ADP
cana-1749	6	12	topology	topology	NOUN
cana-1749	6	13	.	.	PUNCT
cana-1749	7	1	r.c.jain	r.c.jain	VERB
cana-1749	7	2	[	[	X
cana-1749	7	3	8	8	NUM
cana-1749	7	4	]	]	PUNCT
cana-1749	7	5	presented	present	VERB
cana-1749	7	6	the	the	DET
cana-1749	7	7	idea	idea	NOUN
cana-1749	7	8	of	of	ADP
cana-1749	7	9	slight	slight	ADJ
cana-1749	7	10	continuity	continuity	NOUN
cana-1749	7	11	and	and	CCONJ
cana-1749	7	12	studied	study	VERB
cana-1749	7	13	its	its	PRON
cana-1749	7	14	fundamental	fundamental	ADJ
cana-1749	7	15	characteristics	characteristic	NOUN
cana-1749	7	16	.	.	PUNCT
cana-1749	8	1	balasubramanian	balasubramanian	PROPN
cana-1749	8	2	et	et	PROPN
cana-1749	8	3	al	al	PROPN
cana-1749	8	4	.	.	PROPN
cana-1749	8	5	(	(	PUNCT
cana-1749	8	6	2011	2011	NUM
cana-1749	8	7	)	)	PUNCT
cana-1749	8	8	introduced	introduce	VERB
cana-1749	8	9	the	the	DET
cana-1749	8	10	notion	notion	NOUN
cana-1749	8	11	of	of	ADP
cana-1749	8	12	slight	slight	ADJ
cana-1749	8	13	gpr	gpr	PROPN
cana-1749	8	14	-	-	PUNCT
cana-1749	8	15	continuity[2	continuity[2	PROPN
cana-1749	8	16	]	]	PUNCT
cana-1749	8	17	as	as	ADP
cana-1749	8	18	a	a	DET
cana-1749	8	19	generalization	generalization	NOUN
cana-1749	8	20	of	of	ADP
cana-1749	8	21	slight	slight	ADJ
cana-1749	8	22	precontinuity[2	precontinuity[2	PROPN
cana-1749	8	23	]	]	PUNCT
cana-1749	8	24	.	.	PUNCT
cana-1749	9	1	j.b	j.b	PROPN
cana-1749	9	2	.	.	PUNCT
cana-1749	9	3	toranagatti	toranagatti	PROPN
cana-1749	10	1	[	[	X
cana-1749	10	2	15,16	15,16	NUM
cana-1749	10	3	]	]	PUNCT
cana-1749	10	4	recently	recently	ADV
cana-1749	10	5	introduced	introduce	VERB
cana-1749	10	6	the	the	DET
cana-1749	10	7	concepts	concept	NOUN
cana-1749	10	8	of	of	ADP
cana-1749	10	9	δgp	δgp	NOUN
cana-1749	10	10	-	-	PUNCT
cana-1749	10	11	continuity	continuity	NOUN
cana-1749	10	12	and	and	CCONJ
cana-1749	10	13	contra	contra	PROPN
cana-1749	10	14	δgp	δgp	PROPN
cana-1749	10	15	-	-	PUNCT
cana-1749	10	16	continuity	continuity	NOUN
cana-1749	10	17	in	in	ADP
cana-1749	10	18	topological	topological	ADJ
cana-1749	10	19	spaces	space	NOUN
cana-1749	10	20	.	.	PUNCT
cana-1749	11	1	in	in	ADP
cana-1749	11	2	this	this	DET
cana-1749	11	3	paper	paper	NOUN
cana-1749	11	4	,	,	PUNCT
cana-1749	11	5	a	a	DET
cana-1749	11	6	new	new	ADJ
cana-1749	11	7	strong	strong	ADJ
cana-1749	11	8	form	form	NOUN
cana-1749	11	9	of	of	ADP
cana-1749	11	10	slight	slight	ADJ
cana-1749	11	11	gpr	gpr	PROPN
cana-1749	11	12	-	-	PUNCT
cana-1749	11	13	continuity	continuity	NOUN
cana-1749	11	14	,	,	PUNCT
cana-1749	11	15	called	call	VERB
cana-1749	11	16	slight	slight	ADJ
cana-1749	11	17	δgp	δgp	NOUN
cana-1749	11	18	-	-	PUNCT
cana-1749	11	19	continuity	continuity	NOUN
cana-1749	11	20	,	,	PUNCT
cana-1749	11	21	is	be	AUX
cana-1749	11	22	introduced	introduce	VERB
cana-1749	11	23	.	.	PUNCT
cana-1749	12	1	it	it	PRON
cana-1749	12	2	is	be	AUX
cana-1749	12	3	shown	show	VERB
cana-1749	12	4	that	that	SCONJ
cana-1749	12	5	slight	slight	ADJ
cana-1749	12	6	δgp	δgp	NOUN
cana-1749	12	7	-	-	PUNCT
cana-1749	12	8	continuity	continuity	NOUN
cana-1749	12	9	is	be	AUX
cana-1749	12	10	strictly	strictly	ADV
cana-1749	12	11	weaker	weak	ADJ
cana-1749	12	12	than	than	ADP
cana-1749	12	13	contra	contra	PROPN
cana-1749	12	14	δgp	δgp	PROPN
cana-1749	12	15	-	-	PUNCT
cana-1749	12	16	continuity	continuity	NOUN
cana-1749	12	17	and	and	CCONJ
cana-1749	12	18	δgp	δgp	NOUN
cana-1749	12	19	-	-	PUNCT
cana-1749	12	20	continuity	continuity	NOUN
cana-1749	12	21	.	.	PUNCT
cana-1749	13	1	relationships	relationship	NOUN
cana-1749	13	2	between	between	ADP
cana-1749	13	3	slight	slight	ADJ
cana-1749	13	4	δgp	δgp	NOUN
cana-1749	13	5	-	-	PUNCT
cana-1749	13	6	continuity	continuity	NOUN
cana-1749	13	7	and	and	CCONJ
cana-1749	13	8	graphs	graph	NOUN
cana-1749	13	9	are	be	AUX
cana-1749	13	10	investigated	investigate	VERB
cana-1749	13	11	.	.	PUNCT
cana-1749	14	1	also	also	ADV
cana-1749	14	2	,	,	PUNCT
cana-1749	14	3	additional	additional	ADJ
cana-1749	14	4	properties	property	NOUN
cana-1749	14	5	of	of	ADP
cana-1749	14	6	these	these	DET
cana-1749	14	7	functions	function	NOUN
cana-1749	14	8	are	be	AUX
cana-1749	14	9	investigated	investigate	VERB
cana-1749	14	10	.	.	PUNCT
cana-1749	15	1	throughout	throughout	ADP
cana-1749	15	2	this	this	DET
cana-1749	15	3	paper	paper	NOUN
cana-1749	15	4	,	,	PUNCT
cana-1749	15	5	(	(	PUNCT
cana-1749	15	6	x	x	X
cana-1749	15	7	,	,	PUNCT
cana-1749	15	8	τ	τ	X
cana-1749	15	9	)	)	PUNCT
cana-1749	15	10	and	and	CCONJ
cana-1749	15	11	(	(	PUNCT
cana-1749	15	12	y	y	PROPN
cana-1749	15	13	,	,	PUNCT
cana-1749	15	14	σ	σ	PROPN
cana-1749	15	15	)	)	PUNCT
cana-1749	15	16	(	(	PUNCT
cana-1749	15	17	or	or	CCONJ
cana-1749	15	18	x	x	X
cana-1749	15	19	and	and	CCONJ
cana-1749	15	20	y	y	NOUN
cana-1749	15	21	)	)	PUNCT
cana-1749	15	22	represents	represent	VERB
cana-1749	15	23	a	a	DET
cana-1749	15	24	topological	topological	ADJ
cana-1749	15	25	space	space	NOUN
cana-1749	15	26	on	on	ADP
cana-1749	15	27	which	which	PRON
cana-1749	15	28	no	no	DET
cana-1749	15	29	separation	separation	NOUN
cana-1749	15	30	axioms	axiom	NOUN
cana-1749	15	31	are	be	AUX
cana-1749	15	32	assumed	assume	VERB
cana-1749	15	33	,	,	PUNCT
cana-1749	15	34	unless	unless	SCONJ
cana-1749	15	35	otherwise	otherwise	ADV
cana-1749	15	36	mentioned	mention	VERB
cana-1749	15	37	.	.	PUNCT
cana-1749	16	1	the	the	DET
cana-1749	16	2	closure	closure	NOUN
cana-1749	16	3	and	and	CCONJ
cana-1749	16	4	interior	interior	NOUN
cana-1749	16	5	of	of	ADP
cana-1749	16	6	m	m	NOUN
cana-1749	16	7	x	x	PUNCT
cana-1749	16	8	will	will	AUX
cana-1749	16	9	be	be	AUX
cana-1749	16	10	denoted	denote	VERB
cana-1749	16	11	by	by	ADP
cana-1749	16	12	cl(m	cl(m	PROPN
cana-1749	16	13	)	)	PUNCT
cana-1749	16	14	and	and	CCONJ
cana-1749	16	15	int(m	int(m	PROPN
cana-1749	16	16	)	)	PUNCT
cana-1749	16	17	respectively	respectively	ADV
cana-1749	16	18	.	.	PUNCT
cana-1749	17	1	definition	definition	NOUN
cana-1749	17	2	1.1	1.1	NUM
cana-1749	17	3	.	.	PUNCT
cana-1749	18	1	a	a	DET
cana-1749	18	2	subset	subset	NOUN
cana-1749	18	3	k	k	PROPN
cana-1749	18	4	of	of	ADP
cana-1749	18	5	a	a	DET
cana-1749	18	6	topological	topological	ADJ
cana-1749	18	7	space	space	NOUN
cana-1749	18	8	x	x	PUNCT
cana-1749	18	9	is	be	AUX
cana-1749	18	10	called	call	VERB
cana-1749	18	11	pre	pre	ADJ
cana-1749	18	12	-	-	ADJ
cana-1749	18	13	closed[10](resp	closed[10](resp	ADJ
cana-1749	18	14	.	.	PUNCT
cana-1749	19	1	,regular	,regular	PUNCT
cana-1749	20	1	closed[13	closed[13	PROPN
cana-1749	20	2	]	]	X
cana-1749	20	3	)	)	PUNCT
cana-1749	21	1	if	if	SCONJ
cana-1749	21	2	cl(int(k))⊆k(resp	cl(int(k))⊆k(resp	PROPN
cana-1749	21	3	.	.	PUNCT
cana-1749	21	4	,cl(int(k)=k	,cl(int(k)=k	PUNCT
cana-1749	21	5	.	.	PUNCT
cana-1749	22	1	definition	definition	NOUN
cana-1749	22	2	1.2	1.2	NUM
cana-1749	22	3	a	a	DET
cana-1749	22	4	subset	subset	NOUN
cana-1749	22	5	k	k	PROPN
cana-1749	22	6	of	of	ADP
cana-1749	22	7	a	a	DET
cana-1749	22	8	topological	topological	ADJ
cana-1749	22	9	space	space	NOUN
cana-1749	22	10	x	x	PUNCT
cana-1749	22	11	is	be	AUX
cana-1749	22	12	called	call	VERB
cana-1749	22	13	δ	δ	NOUN
cana-1749	22	14	-	-	NOUN
cana-1749	22	15	closed[18	closed[18	NOUN
cana-1749	22	16	]	]	PUNCT
cana-1749	23	1	if	if	SCONJ
cana-1749	23	2	k	k	PROPN
cana-1749	23	3	=	=	SYM
cana-1749	23	4	clδ(k	clδ(k	PROPN
cana-1749	23	5	)	)	PUNCT
cana-1749	23	6	where	where	SCONJ
cana-1749	23	7	clδ	clδ	PROPN
cana-1749	23	8	(	(	PUNCT
cana-1749	23	9	k	k	NOUN
cana-1749	23	10	)	)	PUNCT
cana-1749	23	11	=	=	SYM
cana-1749	23	12	{	{	PUNCT
cana-1749	23	13	x	x	PUNCT
cana-1749	23	14	∈	∈	PROPN
cana-1749	23	15	x	x	NOUN
cana-1749	23	16	:	:	PUNCT
cana-1749	23	17	int(cl(u))∩k=	int(cl(u))∩k=	NUM
cana-1749	23	18			NOUN
cana-1749	23	19	,	,	PUNCT
cana-1749	23	20	u	u	PROPN
cana-1749	23	21	∈	∈	PROPN
cana-1749	23	22	τ	τ	X
cana-1749	23	23	and	and	CCONJ
cana-1749	23	24	x	x	SYM
cana-1749	23	25	∈	∈	PROPN
cana-1749	23	26	u	u	NOUN
cana-1749	23	27	}	}	PUNCT
cana-1749	23	28	.	.	PUNCT
cana-1749	24	1	definition	definition	NOUN
cana-1749	24	2	1.3	1.3	NUM
cana-1749	24	3	a	a	DET
cana-1749	24	4	subset	subset	NOUN
cana-1749	24	5	k	k	PROPN
cana-1749	24	6	of	of	ADP
cana-1749	24	7	a	a	DET
cana-1749	24	8	topological	topological	ADJ
cana-1749	24	9	space	space	NOUN
cana-1749	24	10	x	x	PUNCT
cana-1749	24	11	is	be	AUX
cana-1749	24	12	called	call	VERB
cana-1749	24	13	δgp	δgp	PROPN
cana-1749	24	14	-	-	PUNCT
cana-1749	24	15	closed[3](resp	closed[3](resp	ADJ
cana-1749	24	16	,	,	PUNCT
cana-1749	24	17	gpr	gpr	PROPN
cana-1749	24	18	-	-	PUNCT
cana-1749	24	19	closed[6	closed[6	NOUN
cana-1749	24	20	]	]	PUNCT
cana-1749	24	21	and	and	CCONJ
cana-1749	24	22	gp	gp	NOUN
cana-1749	24	23	-	-	PUNCT
cana-1749	24	24	closed[9	closed[9	NOUN
cana-1749	24	25	]	]	NOUN
cana-1749	24	26	)	)	PUNCT
cana-1749	24	27	if	if	SCONJ
cana-1749	24	28	pcl(k	pcl(k	NOUN
cana-1749	24	29	)	)	PUNCT
cana-1749	25	1	⊆	⊆	NUM
cana-1749	25	2	u	u	NOUN
cana-1749	25	3	whenever	whenever	SCONJ
cana-1749	25	4	k	k	PROPN
cana-1749	25	5	⊆	⊆	NUM
cana-1749	25	6	u	u	NOUN
cana-1749	25	7	and	and	CCONJ
cana-1749	25	8	u	u	NOUN
cana-1749	25	9	is	be	AUX
cana-1749	25	10	δ	δ	NOUN
cana-1749	25	11	-	-	ADJ
cana-1749	25	12	open	open	ADJ
cana-1749	25	13	(	(	PUNCT
cana-1749	25	14	resp	resp	NOUN
cana-1749	25	15	,	,	PUNCT
cana-1749	25	16	regular	regular	ADJ
cana-1749	25	17	open	open	ADJ
cana-1749	25	18	and	and	CCONJ
cana-1749	25	19	open	open	ADJ
cana-1749	25	20	)	)	PUNCT
cana-1749	25	21	in	in	ADP
cana-1749	25	22	x.	x.	NOUN
cana-1749	25	23	definition	definition	NOUN
cana-1749	25	24	1.4	1.4	NUM
cana-1749	25	25	a	a	DET
cana-1749	25	26	function	function	NOUN
cana-1749	25	27	f	f	NOUN
cana-1749	25	28	:	:	PUNCT
cana-1749	25	29	x→y	x→y	NUM
cana-1749	25	30	from	from	ADP
cana-1749	25	31	a	a	DET
cana-1749	25	32	topological	topological	ADJ
cana-1749	25	33	space	space	NOUN
cana-1749	25	34	x	x	PUNCT
cana-1749	25	35	into	into	ADP
cana-1749	25	36	a	a	DET
cana-1749	25	37	topological	topological	ADJ
cana-1749	25	38	space	space	NOUN
cana-1749	25	39	y	y	PROPN
cana-1749	25	40	is	be	AUX
cana-1749	25	41	called	call	VERB
cana-1749	25	42	,	,	PUNCT
cana-1749	25	43	(	(	PUNCT
cana-1749	25	44	i	i	NOUN
cana-1749	25	45	)	)	PUNCT
cana-1749	25	46	δ	δ	PROPN
cana-1749	25	47	-	-	PUNCT
cana-1749	25	48	irresolute	irresolute	PROPN
cana-1749	25	49	[	[	X
cana-1749	25	50	7	7	NUM
cana-1749	25	51	]	]	X
cana-1749	25	52	if	if	SCONJ
cana-1749	25	53	f-1	f-1	PROPN
cana-1749	25	54	(	(	PUNCT
cana-1749	25	55	m	m	NOUN
cana-1749	25	56	)	)	PUNCT
cana-1749	25	57	is	be	AUX
cana-1749	25	58	δ	δ	PROPN
cana-1749	25	59	-	-	PUNCT
cana-1749	25	60	closed	closed	ADJ
cana-1749	25	61	in	in	ADP
cana-1749	25	62	x	x	PUNCT
cana-1749	25	63	for	for	ADP
cana-1749	25	64	every	every	DET
cana-1749	25	65	δ	δ	PROPN
cana-1749	25	66	-	-	PUNCT
cana-1749	25	67	closed	close	VERB
cana-1749	25	68	set	set	NOUN
cana-1749	25	69	m	m	PROPN
cana-1749	25	70	of	of	ADP
cana-1749	25	71	y.	y.	PROPN
cana-1749	25	72	(	(	PUNCT
cana-1749	25	73	ii	ii	PROPN
cana-1749	25	74	)	)	PUNCT
cana-1749	25	75	slightly	slightly	ADV
cana-1749	25	76	continuous[8	continuous[8	X
cana-1749	25	77	]	]	X
cana-1749	25	78	(	(	PUNCT
cana-1749	25	79	resp	resp	NOUN
cana-1749	25	80	,	,	PUNCT
cana-1749	25	81	slightly	slightly	ADV
cana-1749	25	82	gp	gp	NOUN
cana-1749	25	83	-	-	ADJ
cana-1749	25	84	continuous	continuous	ADJ
cana-1749	25	85	and	and	CCONJ
cana-1749	25	86	slightly	slightly	ADV
cana-1749	25	87	gpr	gpr	PROPN
cana-1749	25	88	-	-	PUNCT
cana-1749	25	89	continuous[2	continuous[2	PROPN
cana-1749	25	90	]	]	PUNCT
cana-1749	25	91	)	)	PUNCT
cana-1749	25	92	if	if	SCONJ
cana-1749	25	93	f-1	f-1	PROPN
cana-1749	25	94	(	(	PUNCT
cana-1749	25	95	m	m	NOUN
cana-1749	25	96	)	)	PUNCT
cana-1749	25	97	is	be	AUX
cana-1749	25	98	closed	close	VERB
cana-1749	25	99	(	(	PUNCT
cana-1749	25	100	resp	resp	NOUN
cana-1749	25	101	.	.	PUNCT
cana-1749	26	1	,gp	,gp	PUNCT
cana-1749	26	2	-	-	PUNCT
cana-1749	26	3	closed	close	VERB
cana-1749	26	4	and	and	CCONJ
cana-1749	26	5	gpr	gpr	PROPN
cana-1749	26	6	-	-	PUNCT
cana-1749	26	7	closed	closed	ADJ
cana-1749	26	8	)	)	PUNCT
cana-1749	26	9	in	in	ADP
cana-1749	26	10	x	x	PUNCT
cana-1749	26	11	for	for	ADP
cana-1749	26	12	every	every	DET
cana-1749	26	13	clopen	clopen	ADJ
cana-1749	26	14	set	set	NOUN
cana-1749	26	15	m	m	PROPN
cana-1749	26	16	of	of	ADP
cana-1749	26	17	y.	y.	PROPN
cana-1749	26	18	mailto:jagadeeshbt2000@gmail.com	mailto:jagadeeshbt2000@gmail.com	PROPN
cana-1749	26	19	communications	communication	NOUN
cana-1749	26	20	on	on	ADP
cana-1749	26	21	applied	apply	VERB
cana-1749	26	22	nonlinear	nonlinear	ADJ
cana-1749	26	23	analysis	analysis	NOUN
cana-1749	26	24	issn	issn	NOUN
cana-1749	26	25	:	:	PUNCT
cana-1749	26	26	1074	1074	NUM
cana-1749	26	27	-	-	PUNCT
cana-1749	26	28	133x	133x	NUM
cana-1749	26	29	vol	vol	NOUN
cana-1749	26	30	32	32	NUM
cana-1749	26	31	no	no	NOUN
cana-1749	26	32	.	.	NOUN
cana-1749	26	33	2	2	NUM
cana-1749	26	34	(	(	PUNCT
cana-1749	26	35	2025	2025	NUM
cana-1749	26	36	)	)	PUNCT
cana-1749	27	1	376	376	NUM
cana-1749	27	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-1749	27	3	(	(	PUNCT
cana-1749	27	4	iii	iii	NOUN
cana-1749	27	5	)	)	PUNCT
cana-1749	27	6	δgp	δgp	PROPN
cana-1749	27	7	-	-	PUNCT
cana-1749	27	8	continuous[15](resp	continuous[15](resp	PROPN
cana-1749	27	9	,	,	PUNCT
cana-1749	27	10	contra	contra	PROPN
cana-1749	27	11	δgp	δgp	PROPN
cana-1749	27	12	-	-	PUNCT
cana-1749	27	13	continuous[16	continuous[16	PROPN
cana-1749	27	14	]	]	PUNCT
cana-1749	27	15	)	)	PUNCT
cana-1749	27	16	f-1	f-1	NOUN
cana-1749	27	17	(	(	PUNCT
cana-1749	27	18	m	m	NOUN
cana-1749	27	19	)	)	PUNCT
cana-1749	27	20	is	be	AUX
cana-1749	27	21	δgp	δgp	NOUN
cana-1749	27	22	-	-	PUNCT
cana-1749	27	23	open](resp	open](resp	ADJ
cana-1749	27	24	,	,	PUNCT
cana-1749	27	25	δgp	δgp	NOUN
cana-1749	27	26	-	-	PUNCT
cana-1749	27	27	closed	closed	ADJ
cana-1749	27	28	)	)	PUNCT
cana-1749	27	29	in	in	ADP
cana-1749	27	30	x	x	PUNCT
cana-1749	27	31	for	for	ADP
cana-1749	27	32	every	every	DET
cana-1749	27	33	open	open	ADJ
cana-1749	27	34	set	set	NOUN
cana-1749	27	35	m	m	PROPN
cana-1749	27	36	of	of	ADP
cana-1749	27	37	y.	y.	PROPN
cana-1749	27	38	(	(	PUNCT
cana-1749	27	39	iv	iv	X
cana-1749	27	40	)	)	PUNCT
cana-1749	27	41	δgp	δgp	NOUN
cana-1749	27	42	-	-	PUNCT
cana-1749	27	43	irresolute	irresolute	NOUN
cana-1749	27	44	[	[	X
cana-1749	27	45	15	15	NUM
cana-1749	27	46	]	]	X
cana-1749	27	47	if	if	SCONJ
cana-1749	27	48	f-1	f-1	PROPN
cana-1749	27	49	(	(	PUNCT
cana-1749	27	50	m	m	NOUN
cana-1749	27	51	)	)	PUNCT
cana-1749	27	52	is	be	AUX
cana-1749	27	53	δgp	δgp	NOUN
cana-1749	27	54	-	-	PUNCT
cana-1749	27	55	closed	closed	ADJ
cana-1749	27	56	in	in	ADP
cana-1749	27	57	x	x	PUNCT
cana-1749	27	58	for	for	ADP
cana-1749	27	59	every	every	DET
cana-1749	27	60	δgp	δgp	NOUN
cana-1749	27	61	-	-	PUNCT
cana-1749	27	62	closed	close	VERB
cana-1749	27	63	set	set	NOUN
cana-1749	27	64	m	m	PROPN
cana-1749	27	65	of	of	ADP
cana-1749	27	66	y.	y.	PROPN
cana-1749	27	67	(	(	PUNCT
cana-1749	27	68	v	v	NOUN
cana-1749	27	69	)	)	PUNCT
cana-1749	27	70	pre	pre	VERB
cana-1749	27	71	δgp	δgp	PROPN
cana-1749	27	72	-	-	PUNCT
cana-1749	27	73	closed[16	closed[16	PROPN
cana-1749	27	74	]	]	X
cana-1749	27	75	if	if	SCONJ
cana-1749	27	76	the	the	DET
cana-1749	27	77	image	image	NOUN
cana-1749	27	78	of	of	ADP
cana-1749	27	79	every	every	DET
cana-1749	27	80	δgp	δgp	NOUN
cana-1749	27	81	-	-	PUNCT
cana-1749	27	82	closed	close	VERB
cana-1749	27	83	set	set	NOUN
cana-1749	27	84	of	of	ADP
cana-1749	27	85	x	x	PUNCT
cana-1749	27	86	is	be	AUX
cana-1749	27	87	δgp	δgp	NOUN
cana-1749	27	88	-	-	PUNCT
cana-1749	27	89	closed	close	VERB
cana-1749	27	90	in	in	ADP
cana-1749	27	91	y.	y.	PROPN
cana-1749	27	92	definition	definition	NOUN
cana-1749	27	93	1.5	1.5	NUM
cana-1749	27	94	a	a	DET
cana-1749	27	95	space	space	NOUN
cana-1749	27	96	x	x	PUNCT
cana-1749	27	97	is	be	AUX
cana-1749	27	98	called	call	VERB
cana-1749	27	99	,	,	PUNCT
cana-1749	27	100	(	(	PUNCT
cana-1749	27	101	i	i	NOUN
cana-1749	27	102	)	)	PUNCT
cana-1749	27	103	locally	locally	ADV
cana-1749	27	104	discrete[11	discrete[11	X
cana-1749	27	105	]	]	X
cana-1749	27	106	if	if	SCONJ
cana-1749	27	107	every	every	DET
cana-1749	27	108	open	open	ADJ
cana-1749	27	109	subset	subset	NOUN
cana-1749	27	110	is	be	AUX
cana-1749	27	111	closed	closed	ADJ
cana-1749	27	112	.	.	PUNCT
cana-1749	28	1	(	(	PUNCT
cana-1749	28	2	ii	ii	NOUN
cana-1749	28	3	)	)	PUNCT
cana-1749	28	4	submaximal[14	submaximal[14	PROPN
cana-1749	28	5	]	]	PUNCT
cana-1749	28	6	if	if	SCONJ
cana-1749	28	7	every	every	DET
cana-1749	28	8	pre	pre	ADJ
cana-1749	28	9	-	-	ADJ
cana-1749	28	10	open	open	ADJ
cana-1749	28	11	set	set	NOUN
cana-1749	28	12	is	be	AUX
cana-1749	28	13	open	open	ADJ
cana-1749	28	14	in	in	ADP
cana-1749	28	15	x.	x.	PROPN
cana-1749	28	16	(	(	PUNCT
cana-1749	28	17	iii	iii	NOUN
cana-1749	28	18	)	)	PUNCT
cana-1749	28	19	δgp	δgp	PROPN
cana-1749	28	20	-	-	PUNCT
cana-1749	28	21	additive[16]if	additive[16]if	NOUN
cana-1749	28	22	δgpc(x	δgpc(x	NOUN
cana-1749	28	23	)	)	PUNCT
cana-1749	28	24	is	be	AUX
cana-1749	28	25	closed	close	VERB
cana-1749	28	26	under	under	ADP
cana-1749	28	27	arbitrary	arbitrary	ADJ
cana-1749	28	28	intersections	intersection	NOUN
cana-1749	28	29	.	.	PUNCT
cana-1749	29	1	(	(	PUNCT
cana-1749	29	2	iv	iv	X
cana-1749	29	3	)	)	PUNCT
cana-1749	29	4	δgpt1/2	δgpt1/2	NOUN
cana-1749	29	5	-	-	PUNCT
cana-1749	29	6	space[15	space[15	NOUN
cana-1749	29	7	]	]	PUNCT
cana-1749	29	8	if	if	SCONJ
cana-1749	29	9	every	every	DET
cana-1749	29	10	δgp	δgp	NOUN
cana-1749	29	11	-	-	PUNCT
cana-1749	29	12	closed	close	VERB
cana-1749	29	13	subset	subset	NOUN
cana-1749	29	14	of	of	ADP
cana-1749	29	15	x	x	PUNCT
cana-1749	29	16	is	be	AUX
cana-1749	29	17	pre	pre	ADJ
cana-1749	29	18	-	-	ADJ
cana-1749	29	19	closed	closed	ADJ
cana-1749	29	20	.	.	PUNCT
cana-1749	30	1	theorem	theorem	VERB
cana-1749	30	2	1.6[16	1.6[16	NUM
cana-1749	30	3	]	]	X
cana-1749	30	4	if	if	SCONJ
cana-1749	30	5	m	m	PROPN
cana-1749	30	6	and	and	CCONJ
cana-1749	30	7	n	n	PRON
cana-1749	30	8	are	be	AUX
cana-1749	30	9	δgp	δgp	NOUN
cana-1749	30	10	-	-	PUNCT
cana-1749	30	11	open	open	ADJ
cana-1749	30	12	subsets	subset	NOUN
cana-1749	30	13	of	of	ADP
cana-1749	30	14	a	a	DET
cana-1749	30	15	submaximal	submaximal	ADJ
cana-1749	30	16	space	space	NOUN
cana-1749	30	17	x	x	NOUN
cana-1749	30	18	,	,	PUNCT
cana-1749	30	19	then	then	ADV
cana-1749	30	20	m	m	PROPN
cana-1749	30	21	n	n	PROPN
cana-1749	30	22	is	be	AUX
cana-1749	30	23	δgp	δgp	NOUN
cana-1749	30	24	-	-	PUNCT
cana-1749	30	25	open	open	ADJ
cana-1749	30	26	in	in	ADP
cana-1749	30	27	x.	x.	NOUN
cana-1749	30	28	2	2	NUM
cana-1749	30	29	.	.	X
cana-1749	30	30	slightly	slightly	ADV
cana-1749	30	31	δgp	δgp	VERB
cana-1749	30	32	-	-	PUNCT
cana-1749	30	33	continuous	continuous	ADJ
cana-1749	30	34	functions	function	NOUN
cana-1749	30	35	definition	definition	NOUN
cana-1749	30	36	2.1	2.1	NUM
cana-1749	30	37	.	.	PUNCT
cana-1749	31	1	a	a	DET
cana-1749	31	2	function	function	NOUN
cana-1749	31	3	f	f	NOUN
cana-1749	31	4	:	:	PUNCT
cana-1749	31	5	x	x	X
cana-1749	31	6	→	→	SYM
cana-1749	31	7	y	y	PROPN
cana-1749	31	8	is	be	AUX
cana-1749	31	9	called	call	VERB
cana-1749	31	10	slightly	slightly	ADV
cana-1749	31	11	δ	δ	NOUN
cana-1749	31	12	-	-	PUNCT
cana-1749	31	13	generalized	generalized	ADJ
cana-1749	31	14	pre	pre	ADJ
cana-1749	31	15	-	-	ADJ
cana-1749	31	16	continuous	continuous	ADJ
cana-1749	31	17	(	(	PUNCT
cana-1749	31	18	briefly	briefly	ADV
cana-1749	31	19	slightly	slightly	ADV
cana-1749	31	20	δgp	δgp	VERB
cana-1749	31	21	-	-	PUNCT
cana-1749	31	22	continuous	continuous	ADJ
cana-1749	31	23	)	)	PUNCT
cana-1749	31	24	if	if	SCONJ
cana-1749	31	25	inverse	inverse	ADJ
cana-1749	31	26	image	image	NOUN
cana-1749	31	27	of	of	ADP
cana-1749	31	28	every	every	DET
cana-1749	31	29	clopen	clopen	ADJ
cana-1749	31	30	subset	subset	NOUN
cana-1749	31	31	of	of	ADP
cana-1749	31	32	y	y	PROPN
cana-1749	31	33	is	be	AUX
cana-1749	31	34	δgp	δgp	NOUN
cana-1749	31	35	-	-	PUNCT
cana-1749	31	36	open	open	ADJ
cana-1749	31	37	in	in	ADP
cana-1749	31	38	x.the	x.the	DET
cana-1749	31	39	proof	proof	NOUN
cana-1749	31	40	of	of	ADP
cana-1749	31	41	the	the	DET
cana-1749	31	42	following	follow	VERB
cana-1749	31	43	theorem	theorem	NOUN
cana-1749	31	44	is	be	AUX
cana-1749	31	45	straightforward	straightforward	ADJ
cana-1749	31	46	and	and	CCONJ
cana-1749	31	47	hence	hence	ADV
cana-1749	31	48	omitted	omit	VERB
cana-1749	31	49	.	.	PUNCT
cana-1749	32	1	theorem	theorem	VERB
cana-1749	32	2	2.2	2.2	NUM
cana-1749	32	3	.	.	PUNCT
cana-1749	33	1	for	for	ADP
cana-1749	33	2	a	a	DET
cana-1749	33	3	function	function	NOUN
cana-1749	33	4	f	f	NOUN
cana-1749	33	5	:	:	PUNCT
cana-1749	33	6	x	x	SYM
cana-1749	33	7	→	→	SYM
cana-1749	33	8	y	y	PROPN
cana-1749	33	9	,	,	PUNCT
cana-1749	33	10	the	the	DET
cana-1749	33	11	following	follow	VERB
cana-1749	33	12	statements	statement	NOUN
cana-1749	33	13	are	be	AUX
cana-1749	33	14	equivalent	equivalent	ADJ
cana-1749	33	15	:	:	PUNCT
cana-1749	33	16	i).f	i).f	NOUN
cana-1749	33	17	is	be	AUX
cana-1749	33	18	slightly	slightly	ADV
cana-1749	33	19	δgp	δgp	ADJ
cana-1749	33	20	-	-	PUNCT
cana-1749	33	21	continuous	continuous	ADJ
cana-1749	33	22	.	.	PUNCT
cana-1749	33	23	ii	ii	PROPN
cana-1749	33	24	)	)	PUNCT
cana-1749	33	25	.	.	PUNCT
cana-1749	34	1	inverse	inverse	ADJ
cana-1749	34	2	image	image	NOUN
cana-1749	34	3	of	of	ADP
cana-1749	34	4	every	every	DET
cana-1749	34	5	clopen	clopen	ADJ
cana-1749	34	6	subset	subset	NOUN
cana-1749	34	7	of	of	ADP
cana-1749	34	8	y	y	PROPN
cana-1749	34	9	is	be	AUX
cana-1749	34	10	δgp	δgp	NOUN
cana-1749	34	11	-	-	PUNCT
cana-1749	34	12	closed	close	VERB
cana-1749	34	13	in	in	ADP
cana-1749	34	14	x.	x.	PROPN
cana-1749	34	15	iii	iii	PROPN
cana-1749	34	16	)	)	PUNCT
cana-1749	34	17	.	.	PUNCT
cana-1749	35	1	inverse	inverse	ADJ
cana-1749	35	2	image	image	NOUN
cana-1749	35	3	of	of	ADP
cana-1749	35	4	every	every	DET
cana-1749	35	5	clopen	clopen	ADJ
cana-1749	35	6	subset	subset	NOUN
cana-1749	35	7	of	of	ADP
cana-1749	35	8	y	y	PROPN
cana-1749	35	9	is	be	AUX
cana-1749	35	10	δgp	δgp	NOUN
cana-1749	35	11	-	-	PUNCT
cana-1749	35	12	clopen	clopen	ADJ
cana-1749	35	13	in	in	ADP
cana-1749	35	14	x.	x.	NOUN
cana-1749	35	15	remark	remark	PROPN
cana-1749	35	16	2.3	2.3	NUM
cana-1749	35	17	.	.	PUNCT
cana-1749	36	1	we	we	PRON
cana-1749	36	2	have	have	VERB
cana-1749	36	3	the	the	DET
cana-1749	36	4	following	follow	VERB
cana-1749	36	5	diagram	diagram	NOUN
cana-1749	36	6	for	for	ADP
cana-1749	36	7	a	a	DET
cana-1749	36	8	function	function	NOUN
cana-1749	36	9	f	f	NOUN
cana-1749	36	10	:	:	PUNCT
cana-1749	36	11	(	(	PUNCT
cana-1749	36	12	x	x	X
cana-1749	36	13	,	,	PUNCT
cana-1749	36	14	τ)→(y	τ)→(y	PROPN
cana-1749	36	15	,	,	PUNCT
cana-1749	36	16	σ	σ	PROPN
cana-1749	36	17	)	)	PUNCT
cana-1749	36	18	:	:	PUNCT
cana-1749	36	19	slight	slight	ADJ
cana-1749	36	20	precontinuity	precontinuity	PROPN
cana-1749	36	21	contra	contra	PROPN
cana-1749	36	22	δgp	δgp	PROPN
cana-1749	36	23	-	-	PUNCT
cana-1749	36	24	continuity	continuity	NOUN
cana-1749	36	25	↓	↓	NOUN
cana-1749	36	26	↓	↓	PROPN
cana-1749	36	27	slight	slight	ADJ
cana-1749	36	28	gp	gp	NOUN
cana-1749	36	29	-	-	NOUN
cana-1749	36	30	continuity	continuity	NOUN
cana-1749	36	31	→	→	SYM
cana-1749	36	32	slight	slight	ADJ
cana-1749	36	33	δgp	δgp	NOUN
cana-1749	36	34	-	-	PUNCT
cana-1749	36	35	continuity	continuity	NOUN
cana-1749	36	36	→	→	PUNCT
cana-1749	36	37	slight	slight	ADJ
cana-1749	36	38	gpr	gpr	PROPN
cana-1749	36	39	-	-	PUNCT
cana-1749	36	40	continuity	continuity	NOUN
cana-1749	36	41	↑	↑	NOUN
cana-1749	36	42	δgp	δgp	NOUN
cana-1749	36	43	-	-	PUNCT
cana-1749	36	44	continuity	continuity	NOUN
cana-1749	36	45	none	none	NOUN
cana-1749	36	46	of	of	ADP
cana-1749	36	47	the	the	DET
cana-1749	36	48	implications	implication	NOUN
cana-1749	36	49	in	in	ADP
cana-1749	36	50	above	above	ADP
cana-1749	36	51	diagram	diagram	NOUN
cana-1749	36	52	is	be	AUX
cana-1749	36	53	reversible	reversible	ADJ
cana-1749	36	54	.	.	PUNCT
cana-1749	37	1	example	example	NOUN
cana-1749	37	2	2.3	2.3	NUM
cana-1749	37	3	.	.	PUNCT
cana-1749	38	1	let	let	VERB
cana-1749	38	2	x	x	PUNCT
cana-1749	39	1	=	=	NOUN
cana-1749	39	2	y	y	NOUN
cana-1749	39	3	=	=	PRON
cana-1749	39	4	{	{	PUNCT
cana-1749	39	5	a	a	PRON
cana-1749	39	6	,	,	PUNCT
cana-1749	39	7	b	b	NOUN
cana-1749	39	8	,	,	PUNCT
cana-1749	39	9	c	c	X
cana-1749	39	10	,	,	PUNCT
cana-1749	39	11	d	d	NOUN
cana-1749	39	12	}	}	PUNCT
cana-1749	39	13	,	,	PUNCT
cana-1749	39	14	τ	τ	X
cana-1749	39	15	=	=	PUNCT
cana-1749	39	16	{	{	PUNCT
cana-1749	39	17	x	x	NOUN
cana-1749	39	18	,	,	PUNCT
cana-1749	39	19			NOUN
cana-1749	39	20	,	,	PUNCT
cana-1749	39	21	{	{	PUNCT
cana-1749	39	22	a},{b},{a	a},{b},{a	ADJ
cana-1749	39	23	,	,	PUNCT
cana-1749	39	24	b},{a	b},{a	PROPN
cana-1749	39	25	,	,	PUNCT
cana-1749	39	26	b	b	NOUN
cana-1749	39	27	,	,	PUNCT
cana-1749	39	28	c	c	NOUN
cana-1749	39	29	}	}	PUNCT
cana-1749	39	30	}	}	PUNCT
cana-1749	39	31	and	and	CCONJ
cana-1749	39	32	σ={y	σ={y	PROPN
cana-1749	39	33	,	,	PUNCT
cana-1749	39	34			NOUN
cana-1749	39	35	,	,	PUNCT
cana-1749	39	36	{	{	PUNCT
cana-1749	39	37	a},{b	a},{b	NOUN
cana-1749	39	38	,	,	PUNCT
cana-1749	39	39	c}{a	c}{a	PROPN
cana-1749	39	40	,	,	PUNCT
cana-1749	39	41	b	b	NOUN
cana-1749	39	42	,	,	PUNCT
cana-1749	39	43	c},{b	c},{b	NOUN
cana-1749	39	44	,	,	PUNCT
cana-1749	39	45	c	c	X
cana-1749	39	46	,	,	PUNCT
cana-1749	39	47	d	d	NOUN
cana-1749	39	48	}	}	PUNCT
cana-1749	39	49	}	}	PUNCT
cana-1749	39	50	.	.	PUNCT
cana-1749	40	1	(	(	PUNCT
cana-1749	40	2	i)define	i)define	NOUN
cana-1749	40	3	f	f	X
cana-1749	40	4	:	:	PUNCT
cana-1749	40	5	(	(	PUNCT
cana-1749	40	6	x	x	X
cana-1749	40	7	,	,	PUNCT
cana-1749	40	8	τ)→(y	τ)→(y	PROPN
cana-1749	40	9	,	,	PUNCT
cana-1749	40	10	σ	σ	PROPN
cana-1749	40	11	)	)	PUNCT
cana-1749	40	12	by	by	ADP
cana-1749	40	13	f(a)=a	f(a)=a	NOUN
cana-1749	40	14	=	=	NOUN
cana-1749	40	15	f(b),f(c)=b	f(b),f(c)=b	PROPN
cana-1749	40	16	and	and	CCONJ
cana-1749	40	17	f(d)=c	f(d)=c	PROPN
cana-1749	40	18	,	,	PUNCT
cana-1749	40	19	then	then	ADV
cana-1749	40	20	f	f	PROPN
cana-1749	40	21	is	be	AUX
cana-1749	40	22	slightly	slightly	ADV
cana-1749	40	23	gpr	gpr	NOUN
cana-1749	40	24	-	-	PUNCT
cana-1749	40	25	continuous	continuous	ADJ
cana-1749	40	26	but	but	CCONJ
cana-1749	40	27	not	not	PART
cana-1749	40	28	slightly	slightly	ADV
cana-1749	40	29	δgp	δgp	VERB
cana-1749	40	30	-	-	PUNCT
cana-1749	40	31	continuous	continuous	ADJ
cana-1749	40	32	since	since	SCONJ
cana-1749	40	33	{	{	PUNCT
cana-1749	40	34	a	a	PRON
cana-1749	40	35	}	}	PUNCT
cana-1749	40	36	is	be	AUX
cana-1749	40	37	clopen	clopen	ADJ
cana-1749	40	38	in	in	ADP
cana-1749	40	39	y	y	PROPN
cana-1749	40	40	but	but	CCONJ
cana-1749	40	41	f-1({a})={a	f-1({a})={a	PROPN
cana-1749	40	42	,	,	PUNCT
cana-1749	40	43	b	b	AUX
cana-1749	40	44	}	}	PUNCT
cana-1749	40	45	is	be	AUX
cana-1749	40	46	not	not	PART
cana-1749	40	47	δgp	δgp	NOUN
cana-1749	40	48	-	-	PUNCT
cana-1749	40	49	closed	closed	ADJ
cana-1749	40	50	in	in	ADP
cana-1749	40	51	x	x	X
cana-1749	40	52	.	.	PUNCT
cana-1749	41	1	communications	communication	NOUN
cana-1749	41	2	on	on	ADP
cana-1749	41	3	applied	apply	VERB
cana-1749	41	4	nonlinear	nonlinear	ADJ
cana-1749	41	5	analysis	analysis	NOUN
cana-1749	41	6	issn	issn	NOUN
cana-1749	41	7	:	:	PUNCT
cana-1749	41	8	1074	1074	NUM
cana-1749	41	9	-	-	PUNCT
cana-1749	41	10	133x	133x	NUM
cana-1749	41	11	vol	vol	NOUN
cana-1749	41	12	32	32	NUM
cana-1749	41	13	no	no	NOUN
cana-1749	41	14	.	.	NOUN
cana-1749	41	15	2	2	NUM
cana-1749	41	16	(	(	PUNCT
cana-1749	41	17	2025	2025	NUM
cana-1749	41	18	)	)	PUNCT
cana-1749	41	19	377	377	NUM
cana-1749	41	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1749	41	21	(	(	PUNCT
cana-1749	41	22	ii	ii	NOUN
cana-1749	41	23	)	)	PUNCT
cana-1749	41	24	define	define	VERB
cana-1749	41	25	g	g	NOUN
cana-1749	41	26	:	:	PUNCT
cana-1749	41	27	(	(	PUNCT
cana-1749	41	28	x	x	X
cana-1749	41	29	,	,	PUNCT
cana-1749	41	30	τ)→(y	τ)→(y	PROPN
cana-1749	41	31	,	,	PUNCT
cana-1749	41	32	σ	σ	PROPN
cana-1749	41	33	)	)	PUNCT
cana-1749	41	34	by	by	ADP
cana-1749	41	35	g(a)=a	g(a)=a	NOUN
cana-1749	41	36	=	=	SYM
cana-1749	41	37	g(c),g(b)=b	g(c),g(b)=b	PROPN
cana-1749	41	38	and	and	CCONJ
cana-1749	41	39	g(d)=c	g(d)=c	NOUN
cana-1749	41	40	,	,	PUNCT
cana-1749	41	41	then	then	ADV
cana-1749	41	42	g	g	PROPN
cana-1749	41	43	is	be	AUX
cana-1749	41	44	slightly	slightly	ADV
cana-1749	41	45	δgp	δgp	ADJ
cana-1749	41	46	-	-	PUNCT
cana-1749	41	47	continuous	continuous	ADJ
cana-1749	41	48	but	but	CCONJ
cana-1749	41	49	not	not	PART
cana-1749	41	50	slightly	slightly	ADV
cana-1749	41	51	gp	gp	VERB
cana-1749	41	52	-	-	ADJ
cana-1749	41	53	continuous	continuous	ADJ
cana-1749	41	54	since	since	SCONJ
cana-1749	41	55	{	{	PUNCT
cana-1749	41	56	a	a	PRON
cana-1749	41	57	}	}	PUNCT
cana-1749	41	58	is	be	AUX
cana-1749	41	59	clopen	clopen	ADJ
cana-1749	41	60	in	in	ADP
cana-1749	41	61	y	y	PROPN
cana-1749	41	62	but	but	CCONJ
cana-1749	41	63	g−1({a})={a	g−1({a})={a	PROPN
cana-1749	41	64	,	,	PUNCT
cana-1749	41	65	c	c	NOUN
cana-1749	41	66	}	}	PUNCT
cana-1749	41	67	is	be	AUX
cana-1749	41	68	not	not	PART
cana-1749	41	69	gp	gp	NOUN
cana-1749	41	70	-	-	VERB
cana-1749	41	71	closed	closed	ADJ
cana-1749	41	72	in	in	ADP
cana-1749	41	73	x	x	PART
cana-1749	41	74	(	(	PUNCT
cana-1749	41	75	iii	iii	NOUN
cana-1749	41	76	)	)	PUNCT
cana-1749	41	77	define	define	VERB
cana-1749	41	78	h	h	NOUN
cana-1749	41	79	:	:	PUNCT
cana-1749	41	80	(	(	PUNCT
cana-1749	41	81	x	x	X
cana-1749	41	82	,	,	PUNCT
cana-1749	41	83	τ)→(y	τ)→(y	PROPN
cana-1749	41	84	,	,	PUNCT
cana-1749	41	85	σ	σ	PROPN
cana-1749	41	86	)	)	PUNCT
cana-1749	41	87	by	by	ADP
cana-1749	41	88	h(a)=a	h(a)=a	NOUN
cana-1749	41	89	=	=	SYM
cana-1749	41	90	h(d),h(b)=d	h(d),h(b)=d	PROPN
cana-1749	41	91	and	and	CCONJ
cana-1749	41	92	h(c)=b	h(c)=b	NOUN
cana-1749	41	93	,	,	PUNCT
cana-1749	41	94	then	then	ADV
cana-1749	41	95	h	h	NOUN
cana-1749	41	96	is	be	AUX
cana-1749	41	97	slightly	slightly	ADV
cana-1749	41	98	δgp	δgp	ADJ
cana-1749	41	99	-	-	PUNCT
cana-1749	41	100	continuous	continuous	ADJ
cana-1749	41	101	but	but	CCONJ
cana-1749	41	102	not	not	PART
cana-1749	41	103	δgp	δgp	VERB
cana-1749	41	104	-	-	PUNCT
cana-1749	41	105	continuous	continuous	ADJ
cana-1749	41	106	since	since	SCONJ
cana-1749	41	107	for	for	ADP
cana-1749	41	108	closed	closed	ADJ
cana-1749	41	109	set	set	NOUN
cana-1749	41	110	{	{	PUNCT
cana-1749	41	111	d	d	NOUN
cana-1749	41	112	}	}	PUNCT
cana-1749	41	113	,	,	PUNCT
cana-1749	41	114	h−1({d})={b	h−1({d})={b	ADV
cana-1749	41	115	}	}	PUNCT
cana-1749	41	116	is	be	AUX
cana-1749	41	117	not	not	PART
cana-1749	41	118	δgp	δgp	NOUN
cana-1749	41	119	-	-	PUNCT
cana-1749	41	120	closed	closed	ADJ
cana-1749	41	121	in	in	ADP
cana-1749	41	122	x	x	X
cana-1749	41	123	.	.	PUNCT
cana-1749	41	124	example	example	NOUN
cana-1749	42	1	2.4	2.4	NUM
cana-1749	42	2	.	.	PUNCT
cana-1749	43	1	let	let	VERB
cana-1749	43	2	r	r	NOUN
cana-1749	43	3	and	and	CCONJ
cana-1749	43	4	q	q	NOUN
cana-1749	43	5	be	be	AUX
cana-1749	43	6	the	the	DET
cana-1749	43	7	real	real	ADJ
cana-1749	43	8	numbers	number	NOUN
cana-1749	43	9	and	and	CCONJ
cana-1749	43	10	rational	rational	ADJ
cana-1749	43	11	numbers	number	NOUN
cana-1749	43	12	,	,	PUNCT
cana-1749	43	13	respectively	respectively	ADV
cana-1749	43	14	.	.	PUNCT
cana-1749	44	1	let	let	VERB
cana-1749	44	2	m={x	m={x	NOUN
cana-1749	44	3			NOUN
cana-1749	44	4	r	r	NOUN
cana-1749	44	5	:x	:x	PROPN
cana-1749	44	6	is	be	AUX
cana-1749	44	7	rational	rational	ADJ
cana-1749	44	8	and	and	CCONJ
cana-1749	44	9	0	0	NUM
cana-1749	44	10	<	<	X
cana-1749	44	11	x<1}.we	x<1}.we	NOUN
cana-1749	44	12	define	define	VERB
cana-1749	44	13	two	two	NUM
cana-1749	44	14	topologies	topology	NOUN
cana-1749	44	15	on	on	ADP
cana-1749	44	16	τ	τ	X
cana-1749	44	17	=	=	NOUN
cana-1749	44	18	{	{	PUNCT
cana-1749	44	19	r	r	NOUN
cana-1749	44	20	,	,	PUNCT
cana-1749	44	21			NOUN
cana-1749	44	22	,	,	PUNCT
cana-1749	44	23	m	m	NOUN
cana-1749	44	24	,	,	PUNCT
cana-1749	44	25	r\m	r\m	X
cana-1749	44	26	}	}	PUNCT
cana-1749	44	27	and	and	CCONJ
cana-1749	44	28	σ	σ	NOUN
cana-1749	44	29	=	=	SYM
cana-1749	44	30	{	{	PUNCT
cana-1749	44	31	r,	r,	PROPN
cana-1749	44	32	,	,	PUNCT
cana-1749	44	33	{	{	PUNCT
cana-1749	44	34	0	0	NUM
cana-1749	44	35	}	}	PUNCT
cana-1749	44	36	}	}	PUNCT
cana-1749	44	37	.	.	PUNCT
cana-1749	45	1	define	define	VERB
cana-1749	45	2	f	f	X
cana-1749	45	3	:	:	PUNCT
cana-1749	45	4	(	(	PUNCT
cana-1749	45	5	r	r	NOUN
cana-1749	45	6	,	,	PUNCT
cana-1749	45	7	τ)→(r	τ)→(r	PROPN
cana-1749	45	8	,	,	PUNCT
cana-1749	45	9	σ	σ	NOUN
cana-1749	45	10	)	)	PUNCT
cana-1749	45	11	by	by	ADP
cana-1749	45	12	f(x)=1	f(x)=1	NOUN
cana-1749	45	13	if	if	SCONJ
cana-1749	45	14	xq	xq	ADJ
cana-1749	45	15	and	and	CCONJ
cana-1749	45	16	f(x)=0	f(x)=0	ADV
cana-1749	45	17	if	if	SCONJ
cana-1749	45	18	x∉q	x∉q	NUM
cana-1749	45	19	.	.	PUNCT
cana-1749	46	1	then	then	ADV
cana-1749	46	2	f	f	PROPN
cana-1749	46	3	is	be	AUX
cana-1749	46	4	slightly	slightly	ADV
cana-1749	46	5	δgp	δgp	ADJ
cana-1749	46	6	-	-	PUNCT
cana-1749	46	7	continuous	continuous	ADJ
cana-1749	46	8	but	but	CCONJ
cana-1749	46	9	not	not	PART
cana-1749	46	10	contra	contra	PROPN
cana-1749	46	11	δgp	δgp	PROPN
cana-1749	46	12	-	-	PUNCT
cana-1749	46	13	continuous	continuous	ADJ
cana-1749	46	14	since	since	SCONJ
cana-1749	46	15	for	for	ADP
cana-1749	46	16	closed	closed	ADJ
cana-1749	46	17	set	set	VERB
cana-1749	46	18	r\{0	r\{0	NOUN
cana-1749	46	19	}	}	PUNCT
cana-1749	46	20	,	,	PUNCT
cana-1749	46	21	f−1(r\{0})=q	f−1(r\{0})=q	PROPN
cana-1749	46	22	is	be	AUX
cana-1749	46	23	not	not	PART
cana-1749	46	24	δgp	δgp	NOUN
cana-1749	46	25	-	-	PUNCT
cana-1749	46	26	closed	closed	ADJ
cana-1749	46	27	in	in	ADP
cana-1749	46	28	(	(	PUNCT
cana-1749	46	29	r	r	NOUN
cana-1749	46	30	,	,	PUNCT
cana-1749	46	31	τ	τ	PROPN
cana-1749	46	32	)	)	PUNCT
cana-1749	46	33	.	.	PUNCT
cana-1749	47	1	theorem	theorem	VERB
cana-1749	47	2	2.5	2.5	NUM
cana-1749	47	3	.	.	PUNCT
cana-1749	48	1	if	if	SCONJ
cana-1749	48	2	x	x	PRON
cana-1749	48	3	is	be	AUX
cana-1749	48	4	locally	locally	ADV
cana-1749	48	5	discrete	discrete	ADJ
cana-1749	48	6	,	,	PUNCT
cana-1749	48	7	then	then	ADV
cana-1749	48	8	the	the	DET
cana-1749	48	9	following	following	NOUN
cana-1749	48	10	are	be	AUX
cana-1749	48	11	equivalent	equivalent	ADJ
cana-1749	48	12	:	:	PUNCT
cana-1749	48	13	(	(	PUNCT
cana-1749	48	14	i	i	NOUN
cana-1749	48	15	)	)	PUNCT
cana-1749	48	16	f	f	NOUN
cana-1749	49	1	:	:	PUNCT
cana-1749	49	2	x	x	X
cana-1749	49	3	→	→	SYM
cana-1749	49	4	y	y	PROPN
cana-1749	49	5	is	be	AUX
cana-1749	49	6	slightly	slightly	ADV
cana-1749	49	7	δgp	δgp	ADJ
cana-1749	49	8	-	-	PUNCT
cana-1749	49	9	continuous	continuous	ADJ
cana-1749	49	10	;	;	PUNCT
cana-1749	49	11	(	(	PUNCT
cana-1749	49	12	ii	ii	NOUN
cana-1749	49	13	)	)	PUNCT
cana-1749	49	14	f	f	NOUN
cana-1749	49	15	:	:	PUNCT
cana-1749	49	16	x	x	X
cana-1749	49	17	→	→	SYM
cana-1749	49	18	y	y	PROPN
cana-1749	49	19	is	be	AUX
cana-1749	49	20	δgp	δgp	ADJ
cana-1749	49	21	-	-	PUNCT
cana-1749	49	22	continuous	continuous	ADJ
cana-1749	49	23	;	;	PUNCT
cana-1749	49	24	(	(	PUNCT
cana-1749	49	25	iii	iii	X
cana-1749	49	26	)	)	PUNCT
cana-1749	49	27	f	f	NOUN
cana-1749	49	28	:	:	PUNCT
cana-1749	49	29	x	x	X
cana-1749	49	30	→	→	SYM
cana-1749	49	31	y	y	PROPN
cana-1749	49	32	is	be	AUX
cana-1749	49	33	contra	contra	PROPN
cana-1749	49	34	δgp	δgp	PROPN
cana-1749	49	35	-	-	PUNCT
cana-1749	49	36	continuous	continuous	ADJ
cana-1749	49	37	.	.	PUNCT
cana-1749	50	1	theorem	theorem	VERB
cana-1749	50	2	2.6	2.6	NUM
cana-1749	50	3	.	.	PUNCT
cana-1749	51	1	if	if	SCONJ
cana-1749	51	2	x	x	PRON
cana-1749	51	3	is	be	AUX
cana-1749	51	4	δgpt1/2	δgpt1/2	NOUN
cana-1749	51	5	-	-	PUNCT
cana-1749	51	6	space	space	NOUN
cana-1749	51	7	,	,	PUNCT
cana-1749	51	8	then	then	ADV
cana-1749	51	9	the	the	DET
cana-1749	51	10	following	following	NOUN
cana-1749	51	11	are	be	AUX
cana-1749	51	12	equivalent	equivalent	ADJ
cana-1749	51	13	:	:	PUNCT
cana-1749	51	14	(	(	PUNCT
cana-1749	51	15	i	i	NOUN
cana-1749	51	16	)	)	PUNCT
cana-1749	51	17	f	f	NOUN
cana-1749	52	1	:	:	PUNCT
cana-1749	52	2	x	x	X
cana-1749	52	3	→	→	SYM
cana-1749	52	4	y	y	PROPN
cana-1749	52	5	is	be	AUX
cana-1749	52	6	slightly	slightly	ADV
cana-1749	52	7	gpr	gpr	NOUN
cana-1749	52	8	-	-	PUNCT
cana-1749	52	9	continuous	continuous	ADJ
cana-1749	52	10	;	;	PUNCT
cana-1749	52	11	(	(	PUNCT
cana-1749	52	12	ii	ii	NOUN
cana-1749	52	13	)	)	PUNCT
cana-1749	52	14	f	f	NOUN
cana-1749	52	15	:	:	PUNCT
cana-1749	52	16	x	x	X
cana-1749	52	17	→	→	SYM
cana-1749	52	18	y	y	PROPN
cana-1749	52	19	is	be	AUX
cana-1749	52	20	slightly	slightly	ADV
cana-1749	52	21	δgp	δgp	ADJ
cana-1749	52	22	-	-	PUNCT
cana-1749	52	23	continuous	continuous	ADJ
cana-1749	52	24	;	;	PUNCT
cana-1749	52	25	(	(	PUNCT
cana-1749	52	26	iii	iii	X
cana-1749	52	27	)	)	PUNCT
cana-1749	52	28	f	f	NOUN
cana-1749	52	29	:	:	PUNCT
cana-1749	52	30	x	x	X
cana-1749	52	31	→	→	SYM
cana-1749	52	32	y	y	PROPN
cana-1749	52	33	is	be	AUX
cana-1749	52	34	slightly	slightly	ADV
cana-1749	52	35	gp	gp	NOUN
cana-1749	52	36	-	-	ADJ
cana-1749	52	37	continuous	continuous	ADJ
cana-1749	52	38	;	;	PUNCT
cana-1749	52	39	(	(	PUNCT
cana-1749	52	40	iv	iv	X
cana-1749	52	41	)	)	PUNCT
cana-1749	52	42	f	f	NOUN
cana-1749	52	43	:	:	PUNCT
cana-1749	52	44	x	x	X
cana-1749	52	45	→	→	SYM
cana-1749	52	46	y	y	PROPN
cana-1749	52	47	is	be	AUX
cana-1749	52	48	slightly	slightly	ADV
cana-1749	52	49	pre	pre	ADJ
cana-1749	52	50	-	-	ADJ
cana-1749	52	51	continuous	continuous	ADJ
cana-1749	52	52	.	.	PUNCT
cana-1749	53	1	theorem	theorem	ADJ
cana-1749	53	2	2.7	2.7	NUM
cana-1749	53	3	let	let	VERB
cana-1749	53	4	f	f	NOUN
cana-1749	53	5	:	:	PUNCT
cana-1749	53	6	(	(	PUNCT
cana-1749	53	7	x	x	X
cana-1749	53	8	,	,	PUNCT
cana-1749	53	9	τ	τ	X
cana-1749	53	10	)	)	PUNCT
cana-1749	53	11	→	→	SYM
cana-1749	53	12	(	(	PUNCT
cana-1749	53	13	y	y	PROPN
cana-1749	53	14	,	,	PUNCT
cana-1749	53	15	σ	σ	PROPN
cana-1749	53	16	)	)	PUNCT
cana-1749	53	17	be	be	VERB
cana-1749	53	18	a	a	DET
cana-1749	53	19	slightly	slightly	ADV
cana-1749	53	20	δgp	δgp	ADJ
cana-1749	53	21	-	-	PUNCT
cana-1749	53	22	continuous	continuous	ADJ
cana-1749	53	23	function	function	NOUN
cana-1749	53	24	,	,	PUNCT
cana-1749	53	25	then	then	ADV
cana-1749	53	26	for	for	ADP
cana-1749	53	27	each	each	DET
cana-1749	53	28	p	p	NOUN
cana-1749	53	29			NOUN
cana-1749	53	30	x	x	PUNCT
cana-1749	53	31	and	and	CCONJ
cana-1749	53	32	each	each	DET
cana-1749	53	33	clopen	clopen	ADJ
cana-1749	53	34	set	set	VERB
cana-1749	53	35	n	n	PRON
cana-1749	53	36	containing	contain	VERB
cana-1749	53	37	f(p	f(p	NOUN
cana-1749	53	38	)	)	PUNCT
cana-1749	53	39	,	,	PUNCT
cana-1749	53	40	there	there	PRON
cana-1749	53	41	exists	exist	VERB
cana-1749	53	42	δgp	δgp	NOUN
cana-1749	53	43	-	-	PUNCT
cana-1749	53	44	open	open	NOUN
cana-1749	53	45	set	set	NOUN
cana-1749	53	46	m	m	AUX
cana-1749	53	47	containing	contain	VERB
cana-1749	53	48	such	such	ADJ
cana-1749	53	49	that	that	SCONJ
cana-1749	53	50	f(m	f(m	PROPN
cana-1749	53	51	)	)	PUNCT
cana-1749	53	52			PROPN
cana-1749	53	53	n.	n.	NOUN
cana-1749	53	54	proof	proof	NOUN
cana-1749	53	55	.	.	PUNCT
cana-1749	54	1	let	let	VERB
cana-1749	54	2	p	p	PRON
cana-1749	54	3			PROPN
cana-1749	54	4	x	x	PUNCT
cana-1749	54	5	and	and	CCONJ
cana-1749	54	6	n	n	CCONJ
cana-1749	54	7	be	be	VERB
cana-1749	54	8	a	a	DET
cana-1749	54	9	clopen	clopen	ADJ
cana-1749	54	10	set	set	NOUN
cana-1749	54	11	such	such	ADJ
cana-1749	54	12	that	that	PRON
cana-1749	54	13	f(n	f(n	PROPN
cana-1749	54	14	)	)	PUNCT
cana-1749	54	15			PROPN
cana-1749	54	16	y.	y.	PROPN
cana-1749	54	17	since	since	SCONJ
cana-1749	54	18	f	f	PROPN
cana-1749	54	19	is	be	AUX
cana-1749	54	20	slightly	slightly	ADV
cana-1749	54	21	δgp	δgp	ADJ
cana-1749	54	22	-	-	PUNCT
cana-1749	54	23	continuous	continuous	ADJ
cana-1749	54	24	f-1	f-1	NOUN
cana-1749	54	25	(	(	PUNCT
cana-1749	54	26	n	n	CCONJ
cana-1749	54	27	)	)	PUNCT
cana-1749	54	28	is	be	AUX
cana-1749	54	29	δgp	δgp	NOUN
cana-1749	54	30	-	-	PUNCT
cana-1749	54	31	open	open	ADJ
cana-1749	54	32	in	in	ADP
cana-1749	54	33	x.	x.	NOUN
cana-1749	54	34	if	if	SCONJ
cana-1749	54	35	we	we	PRON
cana-1749	54	36	put	put	VERB
cana-1749	54	37	m	m	NOUN
cana-1749	54	38	=	=	NOUN
cana-1749	54	39	f-1	f-1	NOUN
cana-1749	54	40	(	(	PUNCT
cana-1749	54	41	n	n	CCONJ
cana-1749	54	42	)	)	PUNCT
cana-1749	54	43	,	,	PUNCT
cana-1749	54	44	we	we	PRON
cana-1749	54	45	have	have	VERB
cana-1749	54	46	p	p	NOUN
cana-1749	54	47			PROPN
cana-1749	54	48	m	m	NOUN
cana-1749	54	49	and	and	CCONJ
cana-1749	54	50	f(m	f(m	PROPN
cana-1749	54	51	)	)	PUNCT
cana-1749	54	52			PROPN
cana-1749	54	53	n.	n.	PROPN
cana-1749	54	54	let	let	VERB
cana-1749	54	55	(	(	PUNCT
cana-1749	54	56	x	x	NOUN
cana-1749	54	57	,	,	PUNCT
cana-1749	54	58	τ	τ	X
cana-1749	54	59	)	)	PUNCT
cana-1749	54	60	be	be	VERB
cana-1749	54	61	a	a	DET
cana-1749	54	62	topological	topological	ADJ
cana-1749	54	63	space	space	NOUN
cana-1749	54	64	.	.	PUNCT
cana-1749	55	1	the	the	DET
cana-1749	55	2	quasi	quasi	NOUN
cana-1749	55	3	-	-	NOUN
cana-1749	55	4	topology	topology	NOUN
cana-1749	55	5	on	on	ADP
cana-1749	55	6	x	x	X
cana-1749	55	7	is	be	AUX
cana-1749	55	8	the	the	DET
cana-1749	55	9	topology	topology	NOUN
cana-1749	55	10	having	have	VERB
cana-1749	55	11	as	as	ADP
cana-1749	55	12	base	base	NOUN
cana-1749	55	13	all	all	DET
cana-1749	55	14	clopen	clopen	ADJ
cana-1749	55	15	subsets	subset	NOUN
cana-1749	55	16	of	of	ADP
cana-1749	55	17	(	(	PUNCT
cana-1749	55	18	x	x	X
cana-1749	55	19	,	,	PUNCT
cana-1749	55	20	τ	τ	PROPN
cana-1749	55	21	)	)	PUNCT
cana-1749	55	22	.	.	PUNCT
cana-1749	56	1	the	the	DET
cana-1749	56	2	open	open	ADJ
cana-1749	56	3	(	(	PUNCT
cana-1749	56	4	resp	resp	NOUN
cana-1749	56	5	.	.	PUNCT
cana-1749	56	6	closed	closed	ADJ
cana-1749	56	7	)	)	PUNCT
cana-1749	56	8	subsets	subset	NOUN
cana-1749	56	9	of	of	ADP
cana-1749	56	10	the	the	DET
cana-1749	56	11	quasitopology	quasitopology	NOUN
cana-1749	56	12	are	be	AUX
cana-1749	56	13	said	say	VERB
cana-1749	56	14	to	to	PART
cana-1749	56	15	be	be	AUX
cana-1749	56	16	quasi	quasi	ADJ
cana-1749	56	17	-	-	ADJ
cana-1749	56	18	open	open	ADJ
cana-1749	56	19	(	(	PUNCT
cana-1749	56	20	resp	resp	NOUN
cana-1749	56	21	.	.	PUNCT
cana-1749	57	1	quasi	quasi	ADJ
cana-1749	57	2	-	-	ADJ
cana-1749	57	3	closed	closed	ADJ
cana-1749	57	4	)	)	PUNCT
cana-1749	57	5	.	.	PUNCT
cana-1749	58	1	a	a	DET
cana-1749	58	2	point	point	NOUN
cana-1749	58	3	x	x	PUNCT
cana-1749	58	4	of	of	ADP
cana-1749	58	5	a	a	DET
cana-1749	58	6	space	space	NOUN
cana-1749	58	7	x	x	PUNCT
cana-1749	58	8	is	be	AUX
cana-1749	58	9	said	say	VERB
cana-1749	58	10	to	to	PART
cana-1749	58	11	be	be	AUX
cana-1749	58	12	quasi	quasi	ADJ
cana-1749	58	13	closure	closure	NOUN
cana-1749	58	14	of	of	ADP
cana-1749	58	15	a	a	DET
cana-1749	58	16	subset	subset	NOUN
cana-1749	58	17	a	a	PRON
cana-1749	58	18	of	of	ADP
cana-1749	58	19	x	x	PRON
cana-1749	58	20	,	,	PUNCT
cana-1749	58	21	denoted	denote	VERB
cana-1749	58	22	by	by	ADP
cana-1749	58	23	clqa	clqa	NOUN
cana-1749	58	24	,	,	PUNCT
cana-1749	58	25	if	if	SCONJ
cana-1749	58	26	a	a	DET
cana-1749	58	27			PUNCT
cana-1749	58	28	u	u	NOUN
cana-1749	58	29	≠	≠	NOUN
cana-1749	58	30			NOUN
cana-1749	58	31	for	for	ADP
cana-1749	58	32	every	every	DET
cana-1749	58	33	clopen	clopen	ADJ
cana-1749	58	34	set	set	NOUN
cana-1749	58	35	u	u	NOUN
cana-1749	58	36	containing	contain	VERB
cana-1749	58	37	x.	x.	NOUN
cana-1749	58	38	a	a	DET
cana-1749	58	39	subset	subset	NOUN
cana-1749	58	40	a	a	PRON
cana-1749	58	41	is	be	AUX
cana-1749	58	42	said	say	VERB
cana-1749	58	43	to	to	PART
cana-1749	58	44	be	be	AUX
cana-1749	58	45	quasi	quasi	NOUN
cana-1749	58	46	closed	close	VERB
cana-1749	58	47	if	if	SCONJ
cana-1749	58	48	and	and	CCONJ
cana-1749	59	1	only	only	ADV
cana-1749	59	2	if	if	SCONJ
cana-1749	59	3	a	a	DET
cana-1749	59	4	=	=	X
cana-1749	59	5	clq	clq	NOUN
cana-1749	59	6	[	[	X
cana-1749	59	7	8	8	NUM
cana-1749	59	8	]	]	PUNCT
cana-1749	59	9	theorem	theorem	VERB
cana-1749	59	10	2.8	2.8	NUM
cana-1749	59	11	the	the	DET
cana-1749	59	12	following	following	NOUN
cana-1749	59	13	are	be	AUX
cana-1749	59	14	equivalent	equivalent	ADJ
cana-1749	59	15	for	for	ADP
cana-1749	59	16	a	a	DET
cana-1749	59	17	function	function	NOUN
cana-1749	59	18	f	f	NOUN
cana-1749	59	19	:	:	PUNCT
cana-1749	59	20	x→y	x→y	NUM
cana-1749	59	21	with	with	ADP
cana-1749	59	22	x	x	PUNCT
cana-1749	59	23	is	be	AUX
cana-1749	59	24	δgp	δgp	NOUN
cana-1749	59	25	-	-	PUNCT
cana-1749	59	26	additive	additive	NOUN
cana-1749	59	27	.	.	PUNCT
cana-1749	60	1	(	(	PUNCT
cana-1749	60	2	i	i	NOUN
cana-1749	60	3	)	)	PUNCT
cana-1749	60	4	f	f	PROPN
cana-1749	60	5	is	be	AUX
cana-1749	60	6	slightly	slightly	ADV
cana-1749	60	7	δgp	δgp	ADJ
cana-1749	60	8	-	-	PUNCT
cana-1749	60	9	continuous	continuous	ADJ
cana-1749	60	10	;	;	PUNCT
cana-1749	60	11	(	(	PUNCT
cana-1749	60	12	ii	ii	NOUN
cana-1749	60	13	)	)	PUNCT
cana-1749	60	14	for	for	ADP
cana-1749	60	15	each	each	DET
cana-1749	60	16	p	p	NOUN
cana-1749	60	17	∈	∈	PROPN
cana-1749	60	18	x	x	X
cana-1749	60	19	and	and	CCONJ
cana-1749	60	20	each	each	DET
cana-1749	60	21	clopen	clopen	ADJ
cana-1749	60	22	set	set	VERB
cana-1749	60	23	n	n	PROPN
cana-1749	60	24	of	of	ADP
cana-1749	60	25	y	y	PROPN
cana-1749	60	26	containing	contain	VERB
cana-1749	60	27	f(p	f(p	PROPN
cana-1749	60	28	)	)	PUNCT
cana-1749	60	29	,	,	PUNCT
cana-1749	60	30	there	there	PRON
cana-1749	60	31	exists	exist	VERB
cana-1749	60	32	an	an	DET
cana-1749	60	33	δgp	δgp	NOUN
cana-1749	60	34	-	-	PUNCT
cana-1749	60	35	open	open	NOUN
cana-1749	60	36	set	set	NOUN
cana-1749	60	37	m	m	NOUN
cana-1749	60	38	in	in	ADP
cana-1749	60	39	x	x	X
cana-1749	60	40	containing	contain	VERB
cana-1749	60	41	p	p	NOUN
cana-1749	60	42	such	such	ADJ
cana-1749	60	43	that	that	SCONJ
cana-1749	60	44	f(m	f(m	PROPN
cana-1749	60	45	)	)	PUNCT
cana-1749	60	46			PROPN
cana-1749	60	47	n	n	CCONJ
cana-1749	60	48	;	;	PUNCT
cana-1749	60	49	communications	communication	NOUN
cana-1749	60	50	on	on	ADP
cana-1749	60	51	applied	apply	VERB
cana-1749	60	52	nonlinear	nonlinear	ADJ
cana-1749	60	53	analysis	analysis	NOUN
cana-1749	60	54	issn	issn	NOUN
cana-1749	60	55	:	:	PUNCT
cana-1749	60	56	1074	1074	NUM
cana-1749	60	57	-	-	PUNCT
cana-1749	60	58	133x	133x	NUM
cana-1749	60	59	vol	vol	NOUN
cana-1749	60	60	32	32	NUM
cana-1749	60	61	no	no	NOUN
cana-1749	60	62	.	.	NOUN
cana-1749	60	63	2	2	NUM
cana-1749	60	64	(	(	PUNCT
cana-1749	60	65	2025	2025	NUM
cana-1749	60	66	)	)	PUNCT
cana-1749	60	67	378	378	NUM
cana-1749	60	68	https://internationalpubls.com	https://internationalpubls.com	X
cana-1749	60	69	(	(	PUNCT
cana-1749	60	70	iii	iii	NOUN
cana-1749	60	71	)	)	PUNCT
cana-1749	60	72	f(δgpcl(a	f(δgpcl(a	NUM
cana-1749	60	73	)	)	PUNCT
cana-1749	60	74	)	)	PUNCT
cana-1749	60	75			PROPN
cana-1749	60	76	clq	clq	CCONJ
cana-1749	60	77	(	(	PUNCT
cana-1749	60	78	f(a	f(a	NOUN
cana-1749	60	79	)	)	PUNCT
cana-1749	60	80	for	for	ADP
cana-1749	60	81	every	every	DET
cana-1749	60	82	subset	subset	NOUN
cana-1749	60	83	a	a	PRON
cana-1749	60	84	of	of	ADP
cana-1749	60	85	x	x	PRON
cana-1749	60	86	;	;	PUNCT
cana-1749	60	87	(	(	PUNCT
cana-1749	60	88	iv	iv	X
cana-1749	60	89	)	)	PUNCT
cana-1749	60	90	δgpcl(f−1(b	δgpcl(f−1(b	NOUN
cana-1749	60	91	)	)	PUNCT
cana-1749	60	92	)	)	PUNCT
cana-1749	61	1			PROPN
cana-1749	61	2	f−1(clq	f−1(clq	PROPN
cana-1749	61	3	(	(	PUNCT
cana-1749	61	4	b	b	NOUN
cana-1749	61	5	)	)	PUNCT
cana-1749	61	6	)	)	PUNCT
cana-1749	61	7	for	for	ADP
cana-1749	61	8	every	every	DET
cana-1749	61	9	subset	subset	NOUN
cana-1749	61	10	b	b	PROPN
cana-1749	61	11	of	of	ADP
cana-1749	61	12	y.	y.	PROPN
cana-1749	61	13	proof	proof	PROPN
cana-1749	61	14	:	:	PUNCT
cana-1749	61	15	(	(	PUNCT
cana-1749	61	16	i)→(ii)it	i)→(ii)it	PROPN
cana-1749	61	17	follows	follow	VERB
cana-1749	61	18	from	from	ADP
cana-1749	61	19	theorem	theorem	ADJ
cana-1749	61	20	2.7	2.7	NUM
cana-1749	61	21	(	(	PUNCT
cana-1749	61	22	ii)→(i	ii)→(i	NOUN
cana-1749	61	23	)	)	PUNCT
cana-1749	61	24	let	let	VERB
cana-1749	61	25	m	m	PROPN
cana-1749	61	26			PROPN
cana-1749	61	27	y	y	PROPN
cana-1749	61	28	be	be	AUX
cana-1749	61	29	a	a	DET
cana-1749	61	30	clopen	clopen	ADJ
cana-1749	61	31	set	set	NOUN
cana-1749	61	32	such	such	ADJ
cana-1749	61	33	that	that	SCONJ
cana-1749	61	34	f(p	f(p	NOUN
cana-1749	61	35	)	)	PUNCT
cana-1749	61	36	∈	∈	PROPN
cana-1749	61	37	m	m	PROPN
cana-1749	61	38	,	,	PUNCT
cana-1749	61	39	then	then	ADV
cana-1749	61	40	p	p	PROPN
cana-1749	61	41	∈	∈	PROPN
cana-1749	61	42	f-1	f-1	NOUN
cana-1749	61	43	(	(	PUNCT
cana-1749	61	44	m	m	NOUN
cana-1749	61	45	)	)	PUNCT
cana-1749	61	46	.	.	PUNCT
cana-1749	62	1	from	from	ADP
cana-1749	62	2	(	(	PUNCT
cana-1749	62	3	ii	ii	NOUN
cana-1749	62	4	)	)	PUNCT
cana-1749	62	5	,	,	PUNCT
cana-1749	62	6	there	there	PRON
cana-1749	62	7	exists	exist	VERB
cana-1749	62	8	a	a	DET
cana-1749	62	9	δgp	δgp	NOUN
cana-1749	62	10	-	-	PUNCT
cana-1749	62	11	open	open	NOUN
cana-1749	62	12	set	set	VERB
cana-1749	62	13	up	up	ADP
cana-1749	62	14	containing	contain	VERB
cana-1749	62	15	p	p	NOUN
cana-1749	62	16	such	such	ADJ
cana-1749	62	17	that	that	DET
cana-1749	62	18	f(up	f(up	NOUN
cana-1749	62	19	)	)	PUNCT
cana-1749	62	20			PROPN
cana-1749	62	21	m	m	PROPN
cana-1749	62	22	,	,	PUNCT
cana-1749	62	23	then	then	ADV
cana-1749	62	24	p∈	p∈	PROPN
cana-1749	62	25	up	up	PROPN
cana-1749	62	26	f-1	f-1	PROPN
cana-1749	62	27	(	(	PUNCT
cana-1749	62	28	m	m	NOUN
cana-1749	62	29	)	)	PUNCT
cana-1749	62	30	.	.	PUNCT
cana-1749	63	1	hence	hence	ADV
cana-1749	63	2	f-1	f-1	PROPN
cana-1749	63	3	(	(	PUNCT
cana-1749	63	4	m	m	NOUN
cana-1749	63	5	)	)	PUNCT
cana-1749	63	6	=	=	VERB
cana-1749	63	7	∪{up	∪{up	PROPN
cana-1749	63	8	:	:	PUNCT
cana-1749	63	9	up⊂	up⊂	PROPN
cana-1749	63	10	f-1	f-1	PROPN
cana-1749	63	11	(	(	PUNCT
cana-1749	63	12	m	m	NOUN
cana-1749	63	13	)	)	PUNCT
cana-1749	63	14	}	}	PUNCT
cana-1749	63	15	is	be	AUX
cana-1749	63	16	δgp	δgp	NOUN
cana-1749	63	17	-	-	PUNCT
cana-1749	63	18	open	open	ADJ
cana-1749	63	19	in	in	ADP
cana-1749	63	20	x	x	X
cana-1749	63	21	.	.	PUNCT
cana-1749	64	1	(	(	PUNCT
cana-1749	64	2	i)→(iii)let	i)→(iii)let	NUM
cana-1749	64	3	a	a	DET
cana-1749	64	4			PROPN
cana-1749	64	5	x.	x.	NOUN
cana-1749	64	6	suppose	suppose	VERB
cana-1749	64	7	y	y	PROPN
cana-1749	64	8	clq	clq	ADP
cana-1749	64	9	(	(	PUNCT
cana-1749	64	10	f(a	f(a	NOUN
cana-1749	64	11	)	)	PUNCT
cana-1749	64	12	)	)	PUNCT
cana-1749	64	13	,	,	PUNCT
cana-1749	64	14	then	then	ADV
cana-1749	64	15	there	there	PRON
cana-1749	64	16	exists	exist	VERB
cana-1749	64	17	a	a	DET
cana-1749	64	18	clopen	clopen	ADJ
cana-1749	64	19	set	set	NOUN
cana-1749	64	20	b	b	NOUN
cana-1749	64	21			PROPN
cana-1749	64	22	y	y	PROPN
cana-1749	64	23	containing	contain	VERB
cana-1749	64	24	y	y	PRON
cana-1749	64	25	such	such	ADJ
cana-1749	64	26	that	that	DET
cana-1749	64	27	f(a	f(a	NOUN
cana-1749	64	28	)	)	PUNCT
cana-1749	64	29	∩	∩	PROPN
cana-1749	65	1	b	b	X
cana-1749	66	1	=	=	NOUN
cana-1749	66	2	.	.	X
cana-1749	66	3	so	so	ADV
cana-1749	66	4	,	,	PUNCT
cana-1749	66	5	we	we	PRON
cana-1749	66	6	have	have	VERB
cana-1749	66	7	,	,	PUNCT
cana-1749	66	8	a	a	DET
cana-1749	66	9	∩	∩	ADJ
cana-1749	66	10	f-1	f-1	NOUN
cana-1749	66	11	(	(	PUNCT
cana-1749	66	12	b)=φ	b)=φ	NOUN
cana-1749	66	13	and	and	CCONJ
cana-1749	66	14	δgp	δgp	PROPN
cana-1749	66	15	-	-	PUNCT
cana-1749	66	16	cl(a	cl(a	NUM
cana-1749	66	17	)	)	PUNCT
cana-1749	66	18	∩	∩	NOUN
cana-1749	66	19	f-1(b)=φ	f-1(b)=φ	NOUN
cana-1749	66	20	which	which	PRON
cana-1749	66	21	implies	imply	VERB
cana-1749	66	22	f(δgpcl(a	f(δgpcl(a	NUM
cana-1749	66	23	)	)	PUNCT
cana-1749	66	24	)	)	PUNCT
cana-1749	66	25	∩	∩	PROPN
cana-1749	66	26	b=	b=	NOUN
cana-1749	66	27	and	and	CCONJ
cana-1749	66	28	hence	hence	ADV
cana-1749	66	29	yf(δgpcl(a	yf(δgpcl(a	NUM
cana-1749	66	30	)	)	PUNCT
cana-1749	66	31	)	)	PUNCT
cana-1749	66	32	.	.	PUNCT
cana-1749	67	1	therefore	therefore	ADV
cana-1749	67	2	f(δgpcl(a	f(δgpcl(a	NUM
cana-1749	67	3	)	)	PUNCT
cana-1749	67	4	)	)	PUNCT
cana-1749	68	1			PROPN
cana-1749	68	2	clq	clq	CCONJ
cana-1749	68	3	(	(	PUNCT
cana-1749	68	4	f(a	f(a	NOUN
cana-1749	68	5	)	)	PUNCT
cana-1749	68	6	)	)	PUNCT
cana-1749	68	7	.	.	PUNCT
cana-1749	69	1	(	(	PUNCT
cana-1749	69	2	iii)→(iv)let	iii)→(iv)let	NUM
cana-1749	69	3	b	b	X
cana-1749	69	4			PROPN
cana-1749	69	5	y	y	PROPN
cana-1749	69	6	,	,	PUNCT
cana-1749	69	7	then	then	ADV
cana-1749	69	8	f-1	f-1	NOUN
cana-1749	69	9	(	(	PUNCT
cana-1749	69	10	b	b	NOUN
cana-1749	69	11	)	)	PUNCT
cana-1749	69	12			PROPN
cana-1749	69	13	x.	x.	NOUN
cana-1749	69	14	by	by	ADP
cana-1749	69	15	(	(	PUNCT
cana-1749	69	16	iii	iii	NOUN
cana-1749	69	17	)	)	PUNCT
cana-1749	69	18	,	,	PUNCT
cana-1749	69	19	f	f	PROPN
cana-1749	69	20	(	(	PUNCT
cana-1749	69	21	δgpcl(f-1(b	δgpcl(f-1(b	PROPN
cana-1749	69	22	)	)	PUNCT
cana-1749	69	23	)	)	PUNCT
cana-1749	70	1			PROPN
cana-1749	70	2	clq	clq	INTJ
cana-1749	70	3	(	(	PUNCT
cana-1749	70	4	f	f	X
cana-1749	70	5	(	(	PUNCT
cana-1749	70	6	f-1	f-1	NOUN
cana-1749	70	7	(	(	PUNCT
cana-1749	70	8	b	b	NOUN
cana-1749	70	9	)	)	PUNCT
cana-1749	70	10	)	)	PUNCT
cana-1749	70	11	)	)	PUNCT
cana-1749	71	1			PROPN
cana-1749	71	2	clq	clq	CCONJ
cana-1749	71	3	(	(	PUNCT
cana-1749	71	4	b	b	NOUN
cana-1749	71	5	)	)	PUNCT
cana-1749	71	6	.	.	PUNCT
cana-1749	72	1	thus	thus	ADV
cana-1749	72	2	δgpcl(f-1	δgpcl(f-1	ADV
cana-1749	72	3	(	(	PUNCT
cana-1749	72	4	b	b	NOUN
cana-1749	72	5	)	)	PUNCT
cana-1749	72	6	)	)	PUNCT
cana-1749	72	7			PROPN
cana-1749	72	8	f-1	f-1	PROPN
cana-1749	72	9	(	(	PUNCT
cana-1749	72	10	clq	clq	X
cana-1749	72	11	(	(	PUNCT
cana-1749	72	12	b	b	NOUN
cana-1749	72	13	)	)	PUNCT
cana-1749	72	14	)	)	PUNCT
cana-1749	72	15	.	.	PUNCT
cana-1749	73	1	(	(	PUNCT
cana-1749	73	2	iv)→(i	iv)→(i	PROPN
cana-1749	73	3	)	)	PUNCT
cana-1749	73	4	let	let	VERB
cana-1749	73	5	m	m	PRON
cana-1749	73	6	be	be	AUX
cana-1749	73	7	any	any	DET
cana-1749	73	8	clopen	clopen	ADJ
cana-1749	73	9	subset	subset	NOUN
cana-1749	73	10	of	of	ADP
cana-1749	73	11	y.	y.	PROPN
cana-1749	73	12	then	then	ADV
cana-1749	73	13	by	by	ADP
cana-1749	73	14	(	(	PUNCT
cana-1749	73	15	iv	iv	NOUN
cana-1749	73	16	)	)	PUNCT
cana-1749	73	17	,	,	PUNCT
cana-1749	73	18	δgpcl(f-1(m	δgpcl(f-1(m	PROPN
cana-1749	73	19	)	)	PUNCT
cana-1749	73	20			PROPN
cana-1749	73	21	f-1	f-1	PROPN
cana-1749	73	22	(	(	PUNCT
cana-1749	73	23	clq	clq	X
cana-1749	73	24	(	(	PUNCT
cana-1749	73	25	m	m	NOUN
cana-1749	73	26	)	)	PUNCT
cana-1749	73	27	)	)	PUNCT
cana-1749	74	1	=	=	SYM
cana-1749	74	2	f-1	f-1	NOUN
cana-1749	74	3	(	(	PUNCT
cana-1749	74	4	m	m	NOUN
cana-1749	74	5	)	)	PUNCT
cana-1749	74	6	and	and	CCONJ
cana-1749	74	7	δgpcl(f-1	δgpcl(f-1	ADV
cana-1749	74	8	(	(	PUNCT
cana-1749	74	9	m	m	NOUN
cana-1749	74	10	)	)	PUNCT
cana-1749	74	11	)	)	PUNCT
cana-1749	75	1	=	=	SYM
cana-1749	75	2	f-1	f-1	NOUN
cana-1749	75	3	(	(	PUNCT
cana-1749	75	4	m	m	NOUN
cana-1749	75	5	)	)	PUNCT
cana-1749	75	6	.	.	PUNCT
cana-1749	76	1	therefore	therefore	ADV
cana-1749	76	2	,	,	PUNCT
cana-1749	76	3	f-1	f-1	PROPN
cana-1749	76	4	(	(	PUNCT
cana-1749	76	5	m	m	NOUN
cana-1749	76	6	)	)	PUNCT
cana-1749	76	7	is	be	AUX
cana-1749	76	8	δgp	δgp	NOUN
cana-1749	76	9	-	-	PUNCT
cana-1749	76	10	closed	close	VERB
cana-1749	76	11	set	set	NOUN
cana-1749	76	12	in	in	ADP
cana-1749	76	13	x.	x.	NOUN
cana-1749	76	14	remark	remark	PROPN
cana-1749	76	15	2.9	2.9	NUM
cana-1749	76	16	the	the	DET
cana-1749	76	17	composition	composition	NOUN
cana-1749	76	18	of	of	ADP
cana-1749	76	19	two	two	NUM
cana-1749	76	20	slightly	slightly	ADV
cana-1749	76	21	δgp	δgp	ADJ
cana-1749	76	22	-	-	PUNCT
cana-1749	76	23	continuous	continuous	ADJ
cana-1749	76	24	functions	function	NOUN
cana-1749	76	25	need	need	AUX
cana-1749	76	26	not	not	PART
cana-1749	76	27	be	be	AUX
cana-1749	76	28	slightly	slightly	ADV
cana-1749	76	29	δgpcontinuous	δgpcontinuous	ADJ
cana-1749	76	30	as	as	SCONJ
cana-1749	76	31	seen	see	VERB
cana-1749	76	32	from	from	ADP
cana-1749	76	33	the	the	DET
cana-1749	76	34	following	following	ADJ
cana-1749	76	35	examples	example	NOUN
cana-1749	76	36	.	.	PUNCT
cana-1749	77	1	example	example	NOUN
cana-1749	77	2	2.10	2.10	NUM
cana-1749	77	3	let	let	VERB
cana-1749	77	4	x	x	NOUN
cana-1749	77	5	=	=	NOUN
cana-1749	77	6	y={a	y={a	PROPN
cana-1749	77	7	,	,	PUNCT
cana-1749	77	8	b	b	NOUN
cana-1749	77	9	,	,	PUNCT
cana-1749	77	10	c	c	X
cana-1749	77	11	,	,	PUNCT
cana-1749	77	12	d},z=={a	d},z=={a	PROPN
cana-1749	77	13	,	,	PUNCT
cana-1749	77	14	b	b	NOUN
cana-1749	77	15	,	,	PUNCT
cana-1749	77	16	c	c	NOUN
cana-1749	77	17	}	}	PUNCT
cana-1749	77	18	,	,	PUNCT
cana-1749	77	19	τ	τ	X
cana-1749	77	20	=	=	PUNCT
cana-1749	77	21	{	{	PUNCT
cana-1749	77	22	x	x	NOUN
cana-1749	77	23	,	,	PUNCT
cana-1749	77	24			NOUN
cana-1749	77	25	,	,	PUNCT
cana-1749	77	26	{	{	PUNCT
cana-1749	77	27	a},{b},{a	a},{b},{a	ADJ
cana-1749	77	28	,	,	PUNCT
cana-1749	77	29	b},{a	b},{a	PROPN
cana-1749	77	30	,	,	PUNCT
cana-1749	77	31	b	b	NOUN
cana-1749	77	32	,	,	PUNCT
cana-1749	77	33	c	c	NOUN
cana-1749	77	34	}	}	PUNCT
cana-1749	77	35	}	}	PUNCT
cana-1749	77	36	and	and	CCONJ
cana-1749	77	37	σ={y	σ={y	PROPN
cana-1749	77	38	,	,	PUNCT
cana-1749	77	39			NOUN
cana-1749	77	40	,	,	PUNCT
cana-1749	77	41	{	{	PUNCT
cana-1749	77	42	a},{b	a},{b	NOUN
cana-1749	77	43	,	,	PUNCT
cana-1749	77	44	c},{a	c},{a	ADJ
cana-1749	77	45	,	,	PUNCT
cana-1749	77	46	b	b	NOUN
cana-1749	77	47	,	,	PUNCT
cana-1749	77	48	c},{b	c},{b	NOUN
cana-1749	77	49	,	,	PUNCT
cana-1749	77	50	c	c	X
cana-1749	77	51	,	,	PUNCT
cana-1749	77	52	d	d	NOUN
cana-1749	77	53	}	}	PUNCT
cana-1749	77	54	}	}	PUNCT
cana-1749	77	55	and	and	CCONJ
cana-1749	77	56	ƞ=={z	ƞ=={z	PROPN
cana-1749	77	57	,	,	PUNCT
cana-1749	77	58			NOUN
cana-1749	77	59	,	,	PUNCT
cana-1749	77	60	{	{	PUNCT
cana-1749	77	61	a},{b	a},{b	NOUN
cana-1749	77	62	,	,	PUNCT
cana-1749	77	63	c	c	NOUN
cana-1749	77	64	}	}	PUNCT
cana-1749	77	65	}	}	PUNCT
cana-1749	77	66	.	.	PUNCT
cana-1749	78	1	define	define	VERB
cana-1749	78	2	f	f	X
cana-1749	78	3	:	:	PUNCT
cana-1749	78	4	(	(	PUNCT
cana-1749	78	5	x	x	X
cana-1749	78	6	,	,	PUNCT
cana-1749	78	7	τ)→(y	τ)→(y	PROPN
cana-1749	78	8	,	,	PUNCT
cana-1749	78	9	σ	σ	PROPN
cana-1749	78	10	)	)	PUNCT
cana-1749	78	11	by	by	ADP
cana-1749	78	12	f(a)=a	f(a)=a	NOUN
cana-1749	78	13	=	=	NOUN
cana-1749	78	14	f(b),f(c)=b	f(b),f(c)=b	PROPN
cana-1749	78	15	and	and	CCONJ
cana-1749	78	16	f(d)=c	f(d)=c	PROPN
cana-1749	78	17	and	and	CCONJ
cana-1749	78	18	g	g	PROPN
cana-1749	78	19	:	:	PUNCT
cana-1749	78	20	(	(	PUNCT
cana-1749	78	21	y	y	PROPN
cana-1749	78	22	,	,	PUNCT
cana-1749	78	23	σ)→(z	σ)→(z	PROPN
cana-1749	78	24	,	,	PUNCT
cana-1749	78	25	ƞ	ƞ	NOUN
cana-1749	78	26	)	)	PUNCT
cana-1749	78	27	by	by	ADP
cana-1749	78	28	g(a)=a	g(a)=a	PROPN
cana-1749	78	29	,	,	PUNCT
cana-1749	78	30	g(b)=b	g(b)=b	PRON
cana-1749	78	31	,	,	PUNCT
cana-1749	78	32	g(c)=c	g(c)=c	PROPN
cana-1749	78	33	=	=	PROPN
cana-1749	78	34	g(d	g(d	PROPN
cana-1749	78	35	)	)	PUNCT
cana-1749	78	36	.	.	PUNCT
cana-1749	79	1	then	then	ADV
cana-1749	79	2	f	f	PROPN
cana-1749	79	3	and	and	CCONJ
cana-1749	79	4	g	g	PROPN
cana-1749	79	5	are	be	AUX
cana-1749	79	6	slightly	slightly	ADV
cana-1749	79	7	δgp	δgp	ADJ
cana-1749	79	8	-	-	PUNCT
cana-1749	79	9	continuous	continuous	ADJ
cana-1749	79	10	but	but	CCONJ
cana-1749	79	11	g⋆f	g⋆f	ADJ
cana-1749	79	12	:	:	PUNCT
cana-1749	79	13	x→z	x→z	NUM
cana-1749	79	14	is	be	AUX
cana-1749	79	15	not	not	PART
cana-1749	79	16	s	s	PART
cana-1749	79	17	l	l	NOUN
cana-1749	80	1	i	i	PRON
cana-1749	81	1	g	g	PROPN
cana-1749	81	2	h	h	NOUN
cana-1749	82	1	t	t	PROPN
cana-1749	82	2	l	l	NOUN
cana-1749	82	3	y	y	PROPN
cana-1749	82	4	δgp	δgp	ADV
cana-1749	82	5	-	-	PUNCT
cana-1749	82	6	continuous	continuous	ADJ
cana-1749	82	7	,	,	PUNCT
cana-1749	82	8	since	since	SCONJ
cana-1749	82	9	for	for	ADP
cana-1749	82	10	the	the	DET
cana-1749	82	11	clopen	clopen	ADJ
cana-1749	82	12	set	set	NOUN
cana-1749	82	13	{	{	PUNCT
cana-1749	82	14	a	a	NOUN
cana-1749	82	15	}	}	PUNCT
cana-1749	82	16	in	in	ADP
cana-1749	82	17	z	z	PROPN
cana-1749	82	18	,	,	PUNCT
cana-1749	82	19	(	(	PUNCT
cana-1749	82	20	g⋆f)−1{a}={a	g⋆f)−1{a}={a	NOUN
cana-1749	82	21	,	,	PUNCT
cana-1749	82	22	b	b	NOUN
cana-1749	82	23	}	}	PUNCT
cana-1749	82	24	is	be	AUX
cana-1749	82	25	not	not	PART
cana-1749	82	26	δgp	δgp	NOUN
cana-1749	82	27	-	-	PUNCT
cana-1749	82	28	closed	close	VERB
cana-1749	82	29	in	in	ADP
cana-1749	82	30	x.	x.	NOUN
cana-1749	82	31	theorem	theorem	VERB
cana-1749	82	32	2.11	2.11	NUM
cana-1749	82	33	for	for	ADP
cana-1749	82	34	any	any	DET
cana-1749	82	35	two	two	NUM
cana-1749	82	36	functions	function	NOUN
cana-1749	82	37	f	f	X
cana-1749	82	38	:	:	PUNCT
cana-1749	82	39	x→y	x→y	NUM
cana-1749	82	40	and	and	CCONJ
cana-1749	82	41	g	g	NOUN
cana-1749	82	42	:	:	PUNCT
cana-1749	82	43	y→z	y→z	NUM
cana-1749	82	44	,	,	PUNCT
cana-1749	82	45	the	the	DET
cana-1749	82	46	following	follow	VERB
cana-1749	82	47	hold	hold	NOUN
cana-1749	82	48	:	:	PUNCT
cana-1749	82	49	(	(	PUNCT
cana-1749	82	50	i)g⋆f	i)g⋆f	VERB
cana-1749	82	51	is	be	AUX
cana-1749	82	52	slightlly	slightlly	ADV
cana-1749	82	53	δgp	δgp	VERB
cana-1749	82	54	-	-	PUNCT
cana-1749	82	55	continuous	continuous	ADJ
cana-1749	82	56	if	if	SCONJ
cana-1749	82	57	f	f	PROPN
cana-1749	82	58	is	be	AUX
cana-1749	82	59	δgp	δgp	NOUN
cana-1749	82	60	-	-	PUNCT
cana-1749	82	61	irresolute	irresolute	ADJ
cana-1749	82	62	and	and	CCONJ
cana-1749	82	63	g	g	NOUN
cana-1749	82	64	is	be	AUX
cana-1749	82	65	slightlly	slightlly	ADV
cana-1749	82	66	δgp	δgp	VERB
cana-1749	82	67	-	-	PUNCT
cana-1749	82	68	continuous	continuous	ADJ
cana-1749	82	69	.	.	PUNCT
cana-1749	83	1	(	(	PUNCT
cana-1749	83	2	ii)g⋆f	ii)g⋆f	NOUN
cana-1749	83	3	is	be	AUX
cana-1749	83	4	slightlly	slightlly	ADV
cana-1749	83	5	δgp	δgp	VERB
cana-1749	83	6	-	-	PUNCT
cana-1749	83	7	continuous	continuous	ADJ
cana-1749	83	8	if	if	SCONJ
cana-1749	83	9	f	f	PROPN
cana-1749	83	10	is	be	AUX
cana-1749	83	11	δgp	δgp	NOUN
cana-1749	83	12	-	-	PUNCT
cana-1749	83	13	irresolute	irresolute	ADJ
cana-1749	83	14	and	and	CCONJ
cana-1749	83	15	g	g	NOUN
cana-1749	83	16	is	be	AUX
cana-1749	83	17	δgp	δgp	ADJ
cana-1749	83	18	-	-	PUNCT
cana-1749	83	19	continuous	continuous	ADJ
cana-1749	83	20	.	.	PUNCT
cana-1749	84	1	(	(	PUNCT
cana-1749	84	2	iii)g⋆f	iii)g⋆f	VERB
cana-1749	84	3	is	be	AUX
cana-1749	84	4	slightlly	slightlly	ADV
cana-1749	84	5	δgp	δgp	ADJ
cana-1749	84	6	-	-	PUNCT
cana-1749	84	7	continuous	continuous	ADJ
cana-1749	84	8	f	f	PROPN
cana-1749	84	9	is	be	AUX
cana-1749	84	10	δgp	δgp	NOUN
cana-1749	84	11	-	-	PUNCT
cana-1749	84	12	irresolute	irresolute	ADJ
cana-1749	84	13	and	and	CCONJ
cana-1749	84	14	g	g	NOUN
cana-1749	84	15	is	be	AUX
cana-1749	84	16	slightlly	slightlly	ADV
cana-1749	84	17	continuous	continuous	ADJ
cana-1749	84	18	.	.	PUNCT
cana-1749	85	1	proof:(i	proof:(i	PROPN
cana-1749	85	2	)	)	PUNCT
cana-1749	85	3	let	let	VERB
cana-1749	85	4	n	n	PRON
cana-1749	85	5	be	be	AUX
cana-1749	85	6	any	any	DET
cana-1749	85	7	clopen	clopen	ADJ
cana-1749	85	8	set	set	NOUN
cana-1749	85	9	in	in	ADP
cana-1749	85	10	z.	z.	PROPN
cana-1749	85	11	then	then	ADV
cana-1749	85	12	g-1(v	g-1(v	PROPN
cana-1749	85	13	)	)	PUNCT
cana-1749	85	14	is	be	AUX
cana-1749	85	15	δgp	δgp	NOUN
cana-1749	85	16	-	-	PUNCT
cana-1749	85	17	open	open	ADJ
cana-1749	85	18	in	in	ADP
cana-1749	85	19	y	y	PROPN
cana-1749	85	20	since	since	SCONJ
cana-1749	85	21	g	g	PROPN
cana-1749	85	22	is	be	AUX
cana-1749	85	23	slightlly	slightlly	ADV
cana-1749	85	24	δgpcontinuous	δgpcontinuous	ADJ
cana-1749	85	25	.	.	PUNCT
cana-1749	86	1	therefore	therefore	ADV
cana-1749	86	2	,	,	PUNCT
cana-1749	86	3	f-1	f-1	NOUN
cana-1749	86	4	[	[	X
cana-1749	86	5	g-1(u)]=(g⋆f	g-1(u)]=(g⋆f	ADJ
cana-1749	86	6	)	)	PUNCT
cana-1749	86	7	-1(u	-1(u	PROPN
cana-1749	86	8	)	)	PUNCT
cana-1749	86	9	is	be	AUX
cana-1749	86	10	δgp	δgp	NOUN
cana-1749	86	11	-	-	PUNCT
cana-1749	86	12	closed	closed	ADJ
cana-1749	86	13	in	in	ADP
cana-1749	86	14	x	x	PUNCT
cana-1749	86	15	because	because	SCONJ
cana-1749	86	16	f	f	PROPN
cana-1749	86	17	is	be	AUX
cana-1749	86	18	contra	contra	PROPN
cana-1749	86	19	δgpcontinuous	δgpcontinuous	ADJ
cana-1749	86	20	.	.	PUNCT
cana-1749	87	1	hence	hence	ADV
cana-1749	87	2	g⋆f	g⋆f	PROPN
cana-1749	87	3	is	be	AUX
cana-1749	87	4	contra	contra	PROPN
cana-1749	87	5	δgp	δgp	PROPN
cana-1749	87	6	-	-	PUNCT
cana-1749	87	7	continuous	continuous	ADJ
cana-1749	87	8	.	.	PUNCT
cana-1749	88	1	the	the	DET
cana-1749	88	2	proofs	proof	NOUN
cana-1749	88	3	of	of	ADP
cana-1749	88	4	(	(	PUNCT
cana-1749	88	5	ii	ii	NOUN
cana-1749	88	6	)	)	PUNCT
cana-1749	88	7	and	and	CCONJ
cana-1749	88	8	(	(	PUNCT
cana-1749	88	9	iii	iii	X
cana-1749	88	10	)	)	PUNCT
cana-1749	88	11	are	be	AUX
cana-1749	88	12	analogous	analogous	ADJ
cana-1749	88	13	to	to	ADP
cana-1749	88	14	(	(	PUNCT
cana-1749	88	15	i	i	NOUN
cana-1749	88	16	)	)	PUNCT
cana-1749	88	17	with	with	ADP
cana-1749	88	18	the	the	DET
cana-1749	88	19	obvious	obvious	ADJ
cana-1749	88	20	changes	change	NOUN
cana-1749	88	21	.	.	PUNCT
cana-1749	89	1	communications	communication	NOUN
cana-1749	89	2	on	on	ADP
cana-1749	89	3	applied	apply	VERB
cana-1749	89	4	nonlinear	nonlinear	ADJ
cana-1749	89	5	analysis	analysis	NOUN
cana-1749	89	6	issn	issn	NOUN
cana-1749	89	7	:	:	PUNCT
cana-1749	89	8	1074	1074	NUM
cana-1749	89	9	-	-	PUNCT
cana-1749	89	10	133x	133x	NUM
cana-1749	89	11	vol	vol	NOUN
cana-1749	89	12	32	32	NUM
cana-1749	89	13	no	no	NOUN
cana-1749	89	14	.	.	NOUN
cana-1749	89	15	2	2	NUM
cana-1749	89	16	(	(	PUNCT
cana-1749	89	17	2025	2025	NUM
cana-1749	89	18	)	)	PUNCT
cana-1749	89	19	379	379	NUM
cana-1749	89	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1749	89	21	theorem	theorem	VERB
cana-1749	89	22	2.12	2.12	NUM
cana-1749	89	23	let	let	VERB
cana-1749	89	24	f	f	X
cana-1749	89	25	:	:	PUNCT
cana-1749	89	26	x→y	x→y	NUM
cana-1749	89	27	be	be	AUX
cana-1749	89	28	δgp	δgp	NOUN
cana-1749	89	29	-	-	PUNCT
cana-1749	89	30	open	open	ADJ
cana-1749	89	31	surjection	surjection	NOUN
cana-1749	89	32	and	and	CCONJ
cana-1749	89	33	g	g	NOUN
cana-1749	89	34	:	:	PUNCT
cana-1749	89	35	y→z	y→z	NUM
cana-1749	89	36	be	be	AUX
cana-1749	89	37	a	a	DET
cana-1749	89	38	function	function	NOUN
cana-1749	89	39	such	such	ADJ
cana-1749	89	40	that	that	DET
cana-1749	89	41	g⋆f	g⋆f	NOUN
cana-1749	89	42	:	:	PUNCT
cana-1749	89	43	x→z	x→z	NUM
cana-1749	89	44	is	be	AUX
cana-1749	89	45	slightly	slightly	ADV
cana-1749	89	46	δgp	δgp	ADJ
cana-1749	89	47	-	-	PUNCT
cana-1749	89	48	continuous	continuous	ADJ
cana-1749	89	49	,	,	PUNCT
cana-1749	89	50	then	then	ADV
cana-1749	89	51	g	g	PROPN
cana-1749	89	52	is	be	AUX
cana-1749	89	53	slightly	slightly	ADV
cana-1749	89	54	δgpcontinuous	δgpcontinuous	ADJ
cana-1749	89	55	.	.	PUNCT
cana-1749	90	1	proof	proof	NOUN
cana-1749	90	2	:	:	PUNCT
cana-1749	90	3	let	let	VERB
cana-1749	90	4	u	u	PRON
cana-1749	90	5	be	be	AUX
cana-1749	90	6	any	any	DET
cana-1749	90	7	c	c	PROPN
cana-1749	90	8	lopen	lopen	NOUN
cana-1749	90	9	set	set	VERB
cana-1749	90	10	in	in	ADP
cana-1749	90	11	z.then	z.then	PROPN
cana-1749	90	12	(	(	PUNCT
cana-1749	90	13	g⋆f	g⋆f	ADJ
cana-1749	90	14	)	)	PUNCT
cana-1749	90	15	-1(u)=	-1(u)=	PROPN
cana-1749	90	16	f-1	f-1	NOUN
cana-1749	90	17	(	(	PUNCT
cana-1749	90	18	g-1(u	g-1(u	NOUN
cana-1749	90	19	)	)	PUNCT
cana-1749	90	20	)	)	PUNCT
cana-1749	90	21	is	be	AUX
cana-1749	90	22	δgp	δgp	NOUN
cana-1749	90	23	-	-	PUNCT
cana-1749	90	24	open	open	ADJ
cana-1749	90	25	in	in	ADP
cana-1749	90	26	x.	x.	NOUN
cana-1749	90	27	since	since	SCONJ
cana-1749	90	28	f	f	PROPN
cana-1749	90	29	is	be	AUX
cana-1749	90	30	a	a	DET
cana-1749	90	31	δgp	δgp	NOUN
cana-1749	90	32	-	-	PUNCT
cana-1749	90	33	open	open	NOUN
cana-1749	90	34	surjection	surjection	NOUN
cana-1749	90	35	,	,	PUNCT
cana-1749	90	36	f(f-1(g-1(u)))=	f(f-1(g-1(u)))=	PROPN
cana-1749	90	37	g-1(u	g-1(u	NOUN
cana-1749	90	38	)	)	PUNCT
cana-1749	90	39	is	be	AUX
cana-1749	90	40	δgpopen	δgpopen	ADJ
cana-1749	90	41	set	set	VERB
cana-1749	90	42	in	in	ADP
cana-1749	90	43	y.	y.	PROPN
cana-1749	90	44	therefore	therefore	ADV
cana-1749	90	45	,	,	PUNCT
cana-1749	90	46	g	g	PROPN
cana-1749	90	47	is	be	AUX
cana-1749	90	48	slightly	slightly	ADV
cana-1749	90	49	δgp	δgp	ADJ
cana-1749	90	50	-	-	PUNCT
cana-1749	90	51	continuous	continuous	ADJ
cana-1749	90	52	.	.	PUNCT
cana-1749	91	1	theorem	theorem	NOUN
cana-1749	91	2	2.13	2.13	NUM
cana-1749	91	3	let	let	VERB
cana-1749	91	4	f	f	NOUN
cana-1749	91	5	:	:	PUNCT
cana-1749	91	6	x	x	X
cana-1749	91	7	→	→	SYM
cana-1749	91	8	y	y	PROPN
cana-1749	91	9	be	be	AUX
cana-1749	91	10	bijective	bijective	ADJ
cana-1749	91	11	,	,	PUNCT
cana-1749	91	12	δ	δ	PROPN
cana-1749	91	13	-	-	PUNCT
cana-1749	91	14	irresolute	irresolute	ADJ
cana-1749	91	15	and	and	CCONJ
cana-1749	91	16	pre	pre	ADJ
cana-1749	91	17	-	-	ADJ
cana-1749	91	18	closed	closed	ADJ
cana-1749	91	19	.	.	PUNCT
cana-1749	92	1	then	then	ADV
cana-1749	92	2	for	for	ADP
cana-1749	92	3	every	every	DET
cana-1749	92	4	δgp	δgp	NOUN
cana-1749	92	5	-	-	PUNCT
cana-1749	92	6	closed	close	VERB
cana-1749	92	7	set	set	NOUN
cana-1749	92	8	m	m	NOUN
cana-1749	92	9	of	of	ADP
cana-1749	92	10	x	x	PROPN
cana-1749	92	11	,	,	PUNCT
cana-1749	92	12	f(m	f(m	PROPN
cana-1749	92	13	)	)	PUNCT
cana-1749	92	14	is	be	AUX
cana-1749	92	15	δgp	δgp	NOUN
cana-1749	92	16	-	-	PUNCT
cana-1749	92	17	closed	closed	ADJ
cana-1749	92	18	in	in	ADP
cana-1749	92	19	y.	y.	PROPN
cana-1749	92	20	proof	proof	NOUN
cana-1749	92	21	.	.	PUNCT
cana-1749	93	1	let	let	VERB
cana-1749	93	2	m	m	PRON
cana-1749	93	3	be	be	AUX
cana-1749	93	4	any	any	DET
cana-1749	93	5	δgp	δgp	NOUN
cana-1749	93	6	-	-	PUNCT
cana-1749	93	7	closed	close	VERB
cana-1749	93	8	set	set	NOUN
cana-1749	93	9	of	of	ADP
cana-1749	93	10	x	x	PROPN
cana-1749	93	11	and	and	CCONJ
cana-1749	93	12	n	n	DET
cana-1749	93	13	a	a	DET
cana-1749	93	14	δ	δ	NOUN
cana-1749	93	15	-	-	ADJ
cana-1749	93	16	open	open	ADJ
cana-1749	93	17	set	set	NOUN
cana-1749	93	18	of	of	ADP
cana-1749	93	19	y	y	PROPN
cana-1749	93	20	containing	contain	VERB
cana-1749	93	21	f(m	f(m	PROPN
cana-1749	93	22	)	)	PUNCT
cana-1749	93	23	.	.	PUNCT
cana-1749	94	1	since	since	SCONJ
cana-1749	94	2	f-1(n	f-1(n	NOUN
cana-1749	94	3	)	)	PUNCT
cana-1749	94	4	is	be	AUX
cana-1749	94	5	a	a	DET
cana-1749	94	6	δ	δ	NOUN
cana-1749	94	7	-	-	ADJ
cana-1749	94	8	open	open	ADJ
cana-1749	94	9	set	set	NOUN
cana-1749	94	10	of	of	ADP
cana-1749	94	11	x	x	PUNCT
cana-1749	94	12	containing	contain	VERB
cana-1749	94	13	m	m	PROPN
cana-1749	94	14	,	,	PUNCT
cana-1749	94	15	pcl(m	pcl(m	PROPN
cana-1749	94	16	)	)	PUNCT
cana-1749	94	17			PROPN
cana-1749	94	18	f-1	f-1	PROPN
cana-1749	94	19	(	(	PUNCT
cana-1749	94	20	n	n	CCONJ
cana-1749	94	21	)	)	PUNCT
cana-1749	94	22	and	and	CCONJ
cana-1749	94	23	hence	hence	ADV
cana-1749	94	24	f(pcl(m	f(pcl(m	PROPN
cana-1749	94	25	)	)	PUNCT
cana-1749	94	26	)	)	PUNCT
cana-1749	94	27			PROPN
cana-1749	94	28	n.	n.	PROPN
cana-1749	94	29	since	since	SCONJ
cana-1749	94	30	f	f	PROPN
cana-1749	94	31	is	be	AUX
cana-1749	94	32	pre	pre	ADJ
cana-1749	94	33	-	-	ADJ
cana-1749	94	34	closed	closed	ADJ
cana-1749	94	35	and	and	CCONJ
cana-1749	94	36	pcl(m	pcl(m	PROPN
cana-1749	94	37	)	)	PUNCT
cana-1749	94	38	is	be	AUX
cana-1749	94	39	pre	pre	ADJ
cana-1749	94	40	-	-	VERB
cana-1749	94	41	closed	closed	ADJ
cana-1749	94	42	in	in	ADP
cana-1749	94	43	x	x	PROPN
cana-1749	94	44	,	,	PUNCT
cana-1749	94	45	f	f	PROPN
cana-1749	94	46	(	(	PUNCT
cana-1749	94	47	pcl(m	pcl(m	PROPN
cana-1749	94	48	)	)	PUNCT
cana-1749	94	49	)	)	PUNCT
cana-1749	95	1	is	be	AUX
cana-1749	95	2	pre	pre	ADJ
cana-1749	95	3	-	-	VERB
cana-1749	95	4	closed	closed	ADJ
cana-1749	95	5	in	in	ADP
cana-1749	95	6	y.	y.	PROPN
cana-1749	95	7	since	since	SCONJ
cana-1749	95	8	pcl	pcl	PROPN
cana-1749	95	9	(	(	PUNCT
cana-1749	95	10	f(m	f(m	PROPN
cana-1749	95	11	)	)	PUNCT
cana-1749	95	12	)	)	PUNCT
cana-1749	96	1			PROPN
cana-1749	96	2	pcl(f(pcl(m	pcl(f(pcl(m	PROPN
cana-1749	96	3	)	)	PUNCT
cana-1749	96	4	)	)	PUNCT
cana-1749	96	5	)	)	PUNCT
cana-1749	97	1			PROPN
cana-1749	97	2	n	n	CCONJ
cana-1749	97	3	,	,	PUNCT
cana-1749	97	4	pcl(f(m	pcl(f(m	NOUN
cana-1749	97	5	)	)	PUNCT
cana-1749	97	6	)	)	PUNCT
cana-1749	98	1			PROPN
cana-1749	98	2	n.	n.	PROPN
cana-1749	98	3	therefore	therefore	ADV
cana-1749	98	4	f(m	f(m	PROPN
cana-1749	98	5	)	)	PUNCT
cana-1749	98	6	is	be	AUX
cana-1749	98	7	δgp	δgp	NOUN
cana-1749	98	8	-	-	PUNCT
cana-1749	98	9	closed	close	VERB
cana-1749	98	10	in	in	ADP
cana-1749	98	11	y.	y.	PROPN
cana-1749	98	12	theorem	theorem	VERB
cana-1749	98	13	2.14	2.14	NUM
cana-1749	98	14	let	let	VERB
cana-1749	98	15	g	g	NOUN
cana-1749	98	16	:	:	PUNCT
cana-1749	98	17	x→x×y	x→x×y	PROPN
cana-1749	98	18	be	be	AUX
cana-1749	98	19	the	the	DET
cana-1749	98	20	graph	graph	NOUN
cana-1749	98	21	function	function	NOUN
cana-1749	98	22	of	of	ADP
cana-1749	98	23	f	f	NOUN
cana-1749	98	24	:	:	PUNCT
cana-1749	98	25	x→y	x→y	NUM
cana-1749	98	26	,	,	PUNCT
cana-1749	98	27	defined	define	VERB
cana-1749	98	28	by	by	ADP
cana-1749	98	29	g(x)=(x	g(x)=(x	PROPN
cana-1749	98	30	,	,	PUNCT
cana-1749	98	31	f(x	f(x	PROPN
cana-1749	98	32	)	)	PUNCT
cana-1749	98	33	)	)	PUNCT
cana-1749	98	34	for	for	ADP
cana-1749	98	35	each	each	DET
cana-1749	98	36	x∈x	x∈x	NOUN
cana-1749	98	37	.	.	PUNCT
cana-1749	99	1	then	then	ADV
cana-1749	99	2	f	f	PROPN
cana-1749	99	3	is	be	AUX
cana-1749	99	4	slightly	slightly	ADV
cana-1749	99	5	δgp	δgp	ADJ
cana-1749	99	6	-	-	PUNCT
cana-1749	99	7	continuous	continuous	ADJ
cana-1749	99	8	,	,	PUNCT
cana-1749	99	9	if	if	SCONJ
cana-1749	99	10	g	g	PROPN
cana-1749	99	11	is	be	AUX
cana-1749	99	12	slightly	slightly	ADV
cana-1749	99	13	δgp	δgp	ADJ
cana-1749	99	14	-	-	PUNCT
cana-1749	99	15	continuous	continuous	ADJ
cana-1749	99	16	.	.	PUNCT
cana-1749	100	1	proof	proof	NOUN
cana-1749	100	2	:	:	PUNCT
cana-1749	100	3	let	let	VERB
cana-1749	100	4	v	v	PART
cana-1749	100	5	be	be	AUX
cana-1749	100	6	any	any	DET
cana-1749	100	7	c	c	NOUN
cana-1749	100	8	l	l	NOUN
cana-1749	100	9	open	open	ADJ
cana-1749	100	10	set	set	VERB
cana-1749	100	11	in	in	ADP
cana-1749	100	12	y	y	PROPN
cana-1749	100	13	,	,	PUNCT
cana-1749	100	14	then	then	ADV
cana-1749	100	15	x×v	x×v	PROPN
cana-1749	100	16	is	be	AUX
cana-1749	100	17	a	a	DET
cana-1749	100	18	clopen	clopen	ADJ
cana-1749	100	19	set	set	NOUN
cana-1749	100	20	in	in	ADP
cana-1749	100	21	x×y	x×y	PROPN
cana-1749	100	22	.	.	PUNCT
cana-1749	101	1	it	it	PRON
cana-1749	101	2	follows	follow	VERB
cana-1749	101	3	that	that	PRON
cana-1749	101	4	f-1	f-1	PROPN
cana-1749	101	5	(	(	PUNCT
cana-1749	101	6	u)=	u)=	NOUN
cana-1749	101	7	g-1(x×u	g-1(x×u	NOUN
cana-1749	101	8	)	)	PUNCT
cana-1749	101	9	is	be	AUX
cana-1749	101	10	δgp	δgp	NOUN
cana-1749	101	11	-	-	PUNCT
cana-1749	101	12	closed	closed	ADJ
cana-1749	101	13	in	in	ADP
cana-1749	101	14	x	x	PUNCT
cana-1749	101	15	since	since	SCONJ
cana-1749	101	16	g	g	PROPN
cana-1749	101	17	is	be	AUX
cana-1749	101	18	slightly	slightly	ADV
cana-1749	101	19	δgp	δgp	ADJ
cana-1749	101	20	-	-	PUNCT
cana-1749	101	21	continuous	continuous	ADJ
cana-1749	101	22	.	.	PUNCT
cana-1749	102	1	hence	hence	ADV
cana-1749	102	2	f	f	PROPN
cana-1749	102	3	is	be	AUX
cana-1749	102	4	slightly	slightly	ADV
cana-1749	102	5	δgp	δgp	ADJ
cana-1749	102	6	-	-	PUNCT
cana-1749	102	7	continuous	continuous	ADJ
cana-1749	102	8	.	.	PUNCT
cana-1749	103	1	definition	definition	NOUN
cana-1749	103	2	2.15	2.15	NUM
cana-1749	103	3	a	a	DET
cana-1749	103	4	space	space	NOUN
cana-1749	103	5	x	x	PUNCT
cana-1749	103	6	is	be	AUX
cana-1749	103	7	called	call	VERB
cana-1749	103	8	,	,	PUNCT
cana-1749	103	9	(	(	PUNCT
cana-1749	103	10	i	i	NOUN
cana-1749	103	11	)	)	PUNCT
cana-1749	103	12	co	co	NOUN
cana-1749	103	13	-	-	NOUN
cana-1749	103	14	t2	t2	ADJ
cana-1749	103	15	[	[	X
cana-1749	103	16	5	5	NUM
cana-1749	103	17	]	]	PUNCT
cana-1749	103	18	(	(	PUNCT
cana-1749	103	19	resp	resp	NOUN
cana-1749	103	20	,	,	PUNCT
cana-1749	103	21	δgp	δgp	NOUN
cana-1749	103	22	-	-	PUNCT
cana-1749	103	23	hausdorff[16])if	hausdorff[16])if	NOUN
cana-1749	103	24	for	for	ADP
cana-1749	103	25	any	any	DET
cana-1749	103	26	pair	pair	NOUN
cana-1749	103	27	of	of	ADP
cana-1749	103	28	distinct	distinct	ADJ
cana-1749	103	29	points	point	NOUN
cana-1749	103	30	x	x	PUNCT
cana-1749	103	31	and	and	CCONJ
cana-1749	103	32	y	y	PROPN
cana-1749	103	33	,	,	PUNCT
cana-1749	103	34	there	there	PRON
cana-1749	103	35	exist	exist	VERB
cana-1749	103	36	disjoint	disjoint	NOUN
cana-1749	103	37	clopen	clopen	ADJ
cana-1749	103	38	(	(	PUNCT
cana-1749	103	39	δgp	δgp	PROPN
cana-1749	103	40	-	-	PUNCT
cana-1749	103	41	open)sets	open)set	NOUN
cana-1749	103	42	g	g	NOUN
cana-1749	103	43	and	and	CCONJ
cana-1749	103	44	h	h	NOUN
cana-1749	104	1	such	such	ADJ
cana-1749	104	2	that	that	SCONJ
cana-1749	104	3	x	x	SYM
cana-1749	104	4	∈	∈	PROPN
cana-1749	104	5	g	g	PROPN
cana-1749	104	6	and	and	CCONJ
cana-1749	104	7	y	y	PROPN
cana-1749	104	8	∈	∈	PROPN
cana-1749	104	9	h.	h.	PROPN
cana-1749	104	10	(	(	PUNCT
cana-1749	104	11	ii	ii	PROPN
cana-1749	104	12	)	)	PUNCT
cana-1749	104	13	co	co	NOUN
cana-1749	104	14	–	–	PUNCT
cana-1749	104	15	normal[5](resp	normal[5](resp	ADJ
cana-1749	104	16	,	,	PUNCT
cana-1749	104	17	strongly	strongly	ADV
cana-1749	104	18	δgp	δgp	VERB
cana-1749	104	19	-	-	PUNCT
cana-1749	104	20	normal	normal	ADJ
cana-1749	104	21	)	)	PUNCT
cana-1749	104	22	if	if	SCONJ
cana-1749	104	23	each	each	DET
cana-1749	104	24	pair	pair	NOUN
cana-1749	104	25	of	of	ADP
cana-1749	104	26	disjoint	disjoint	PROPN
cana-1749	104	27	clopen(resp	clopen(resp	PROPN
cana-1749	104	28	,	,	PUNCT
cana-1749	104	29	δgp	δgp	VERB
cana-1749	104	30	–	–	PUNCT
cana-1749	104	31	closed	closed	ADJ
cana-1749	104	32	)	)	PUNCT
cana-1749	104	33	sets	set	NOUN
cana-1749	104	34	can	can	AUX
cana-1749	104	35	be	be	AUX
cana-1749	104	36	separated	separate	VERB
cana-1749	104	37	by	by	ADP
cana-1749	104	38	disjoint	disjoint	ADJ
cana-1749	104	39	open	open	ADJ
cana-1749	104	40	sets	set	NOUN
cana-1749	104	41	.	.	PUNCT
cana-1749	105	1	(	(	PUNCT
cana-1749	105	2	iii	iii	NOUN
cana-1749	105	3	)	)	PUNCT
cana-1749	105	4	co	co	NOUN
cana-1749	105	5	-	-	NOUN
cana-1749	105	6	regular[5](resp	regular[5](resp	ADJ
cana-1749	105	7	,	,	PUNCT
cana-1749	105	8	strongly	strongly	ADV
cana-1749	105	9	δgp	δgp	VERB
cana-1749	105	10	-	-	PUNCT
cana-1749	105	11	regular	regular	NOUN
cana-1749	105	12	)	)	PUNCT
cana-1749	105	13	if	if	SCONJ
cana-1749	105	14	for	for	ADP
cana-1749	105	15	each	each	DET
cana-1749	105	16	clopen(resp	clopen(resp	PROPN
cana-1749	105	17	,	,	PUNCT
cana-1749	105	18	δgp	δgp	NOUN
cana-1749	105	19	-	-	PUNCT
cana-1749	105	20	closed	closed	ADJ
cana-1749	105	21	)	)	PUNCT
cana-1749	105	22	set	set	NOUN
cana-1749	105	23	b	b	NOUN
cana-1749	105	24	and	and	CCONJ
cana-1749	105	25	each	each	DET
cana-1749	105	26	x	x	PROPN
cana-1749	105	27	∉	∉	PROPN
cana-1749	105	28	b	b	PROPN
cana-1749	105	29	,	,	PUNCT
cana-1749	105	30	there	there	PRON
cana-1749	105	31	exist	exist	VERB
cana-1749	105	32	disjoint	disjoint	ADJ
cana-1749	105	33	open	open	ADJ
cana-1749	105	34	sets	set	NOUN
cana-1749	105	35	m	m	VERB
cana-1749	105	36	and	and	CCONJ
cana-1749	105	37	n	n	CCONJ
cana-1749	105	38	such	such	ADJ
cana-1749	105	39	that	that	DET
cana-1749	105	40	b	b	X
cana-1749	105	41	⊂	⊂	PROPN
cana-1749	105	42	m	m	PROPN
cana-1749	105	43	and	and	CCONJ
cana-1749	105	44	x∈	x∈	PROPN
cana-1749	105	45	m.	m.	NOUN
cana-1749	105	46	(	(	PUNCT
cana-1749	105	47	iv	iv	NOUN
cana-1749	105	48	)	)	PUNCT
cana-1749	105	49	mildly	mildly	ADV
cana-1749	105	50	compact[12	compact[12	PROPN
cana-1749	105	51	]	]	X
cana-1749	105	52	(	(	PUNCT
cana-1749	105	53	δgp	δgp	PROPN
cana-1749	105	54	-	-	PUNCT
cana-1749	105	55	compact[17	compact[17	NOUN
cana-1749	105	56	]	]	X
cana-1749	105	57	)	)	PUNCT
cana-1749	105	58	if	if	SCONJ
cana-1749	105	59	every	every	DET
cana-1749	105	60	clopen(resp	clopen(resp	PROPN
cana-1749	105	61	,	,	PUNCT
cana-1749	105	62	δgpcover	δgpcover	NOUN
cana-1749	105	63	of	of	ADP
cana-1749	105	64	x	x	PUNCT
cana-1749	105	65	has	have	VERB
cana-1749	105	66	a	a	DET
cana-1749	105	67	finite	finite	ADJ
cana-1749	105	68	sucover	sucover	NOUN
cana-1749	105	69	(	(	PUNCT
cana-1749	105	70	v	v	NOUN
cana-1749	105	71	)	)	PUNCT
cana-1749	105	72	δgp	δgp	PROPN
cana-1749	105	73	-	-	PUNCT
cana-1749	105	74	connected[16	connected[16	NOUN
cana-1749	105	75	]	]	PUNCT
cana-1749	106	1	if	if	SCONJ
cana-1749	106	2	x	x	PRON
cana-1749	106	3	is	be	AUX
cana-1749	106	4	not	not	PART
cana-1749	106	5	the	the	DET
cana-1749	106	6	union	union	NOUN
cana-1749	106	7	of	of	ADP
cana-1749	106	8	two	two	NUM
cana-1749	106	9	disjoint	disjoint	NOUN
cana-1749	106	10	nonempty	nonempty	X
cana-1749	106	11	δgp	δgp	NOUN
cana-1749	106	12	-	-	PUNCT
cana-1749	106	13	open	open	ADJ
cana-1749	106	14	sets	set	NOUN
cana-1749	106	15	.	.	PUNCT
cana-1749	107	1	theorem	theorem	VERB
cana-1749	107	2	2.16	2.16	NUM
cana-1749	107	3	if	if	SCONJ
cana-1749	107	4	a	a	DET
cana-1749	107	5	surjective	surjective	ADJ
cana-1749	107	6	function	function	NOUN
cana-1749	107	7	f	f	NOUN
cana-1749	107	8	:	:	PUNCT
cana-1749	107	9	x→y	x→y	NUM
cana-1749	107	10	is	be	AUX
cana-1749	107	11	slightly	slightly	ADV
cana-1749	107	12	δgp	δgp	ADJ
cana-1749	107	13	-	-	PUNCT
cana-1749	107	14	continuous	continuous	ADJ
cana-1749	107	15	and	and	CCONJ
cana-1749	107	16	x	x	NOUN
cana-1749	107	17	is	be	AUX
cana-1749	107	18	δgp	δgp	NOUN
cana-1749	107	19	-	-	PUNCT
cana-1749	107	20	connected	connect	VERB
cana-1749	107	21	space	space	NOUN
cana-1749	107	22	,	,	PUNCT
cana-1749	107	23	then	then	ADV
cana-1749	107	24	y	y	PROPN
cana-1749	107	25	is	be	AUX
cana-1749	107	26	connected	connect	VERB
cana-1749	107	27	.	.	PUNCT
cana-1749	108	1	proof	proof	NOUN
cana-1749	108	2	:	:	PUNCT
cana-1749	108	3	suppose	suppose	VERB
cana-1749	108	4	that	that	SCONJ
cana-1749	108	5	y	y	PROPN
cana-1749	108	6	is	be	AUX
cana-1749	108	7	not	not	PART
cana-1749	108	8	a	a	DET
cana-1749	108	9	connected	connected	ADJ
cana-1749	108	10	space	space	NOUN
cana-1749	108	11	.	.	PUNCT
cana-1749	109	1	then	then	ADV
cana-1749	109	2	there	there	PRON
cana-1749	109	3	exist	exist	VERB
cana-1749	109	4	disjoint	disjoint	ADJ
cana-1749	109	5	open	open	ADJ
cana-1749	109	6	sets	set	NOUN
cana-1749	109	7	u	u	NOUN
cana-1749	109	8	and	and	CCONJ
cana-1749	109	9	v	v	NOUN
cana-1749	109	10	in	in	ADP
cana-1749	109	11	y	y	PRON
cana-1749	109	12	such	such	ADJ
cana-1749	109	13	that	that	SCONJ
cana-1749	109	14	y	y	PROPN
cana-1749	109	15	=	=	NOUN
cana-1749	109	16	u∪v	u∪v	NOUN
cana-1749	109	17	.	.	PUNCT
cana-1749	110	1	therefore	therefore	ADV
cana-1749	110	2	u	u	PROPN
cana-1749	110	3	and	and	CCONJ
cana-1749	110	4	v	v	NOUN
cana-1749	110	5	are	be	AUX
cana-1749	110	6	clopen	clopen	ADJ
cana-1749	110	7	sets	set	NOUN
cana-1749	110	8	in	in	ADP
cana-1749	110	9	y.	y.	PROPN
cana-1749	110	10	since	since	SCONJ
cana-1749	110	11	f	f	PROPN
cana-1749	110	12	is	be	AUX
cana-1749	110	13	slightly	slightly	ADV
cana-1749	110	14	δgpcontinuous	δgpcontinuous	ADJ
cana-1749	110	15	,	,	PUNCT
cana-1749	110	16	f-1(u	f-1(u	NOUN
cana-1749	110	17	)	)	PUNCT
cana-1749	110	18	and	and	CCONJ
cana-1749	110	19	f-1(v	f-1(v	NOUN
cana-1749	110	20	)	)	PUNCT
cana-1749	110	21	are	be	AUX
cana-1749	110	22	δgp	δgp	NOUN
cana-1749	110	23	-	-	PUNCT
cana-1749	110	24	open	open	ADJ
cana-1749	110	25	sets	set	NOUN
cana-1749	110	26	in	in	ADP
cana-1749	110	27	x	x	X
cana-1749	110	28	.	.	PUNCT
cana-1749	111	1	also	also	ADV
cana-1749	111	2	f	f	PROPN
cana-1749	111	3	is	be	AUX
cana-1749	111	4	surjective	surjective	ADJ
cana-1749	111	5	,	,	PUNCT
cana-1749	111	6	f-1(u	f-1(u	NOUN
cana-1749	111	7	)	)	PUNCT
cana-1749	111	8	and	and	CCONJ
cana-1749	111	9	f-1(v	f-1(v	NOUN
cana-1749	111	10	)	)	PUNCT
cana-1749	111	11	communications	communication	NOUN
cana-1749	111	12	on	on	ADP
cana-1749	111	13	applied	apply	VERB
cana-1749	111	14	nonlinear	nonlinear	ADJ
cana-1749	111	15	analysis	analysis	NOUN
cana-1749	111	16	issn	issn	NOUN
cana-1749	111	17	:	:	PUNCT
cana-1749	111	18	1074	1074	NUM
cana-1749	111	19	-	-	PUNCT
cana-1749	111	20	133x	133x	NUM
cana-1749	111	21	vol	vol	NOUN
cana-1749	111	22	32	32	NUM
cana-1749	111	23	no	no	NOUN
cana-1749	111	24	.	.	NOUN
cana-1749	111	25	2	2	NUM
cana-1749	111	26	(	(	PUNCT
cana-1749	111	27	2025	2025	NUM
cana-1749	111	28	)	)	PUNCT
cana-1749	111	29	380	380	NUM
cana-1749	111	30	https://internationalpubls.com	https://internationalpubls.com	X
cana-1749	111	31	are	be	AUX
cana-1749	111	32	non	non	X
cana-1749	111	33	empty	empty	ADJ
cana-1749	111	34	disjoint	disjoint	NOUN
cana-1749	111	35	and	and	CCONJ
cana-1749	111	36	x	x	PUNCT
cana-1749	111	37	=	=	PUNCT
cana-1749	111	38	f-1(u)∪	f-1(u)∪	NOUN
cana-1749	111	39	f-1(v	f-1(v	NOUN
cana-1749	111	40	)	)	PUNCT
cana-1749	111	41	which	which	PRON
cana-1749	111	42	contradicts	contradict	VERB
cana-1749	111	43	the	the	DET
cana-1749	111	44	fact	fact	NOUN
cana-1749	111	45	that	that	SCONJ
cana-1749	111	46	x	x	PRON
cana-1749	111	47	is	be	AUX
cana-1749	111	48	δgpconnected	δgpconnecte	VERB
cana-1749	111	49	space	space	NOUN
cana-1749	111	50	.	.	PUNCT
cana-1749	112	1	hence	hence	ADV
cana-1749	112	2	y	y	PROPN
cana-1749	112	3	is	be	AUX
cana-1749	112	4	connected	connect	VERB
cana-1749	112	5	.	.	PUNCT
cana-1749	113	1	theorem	theorem	VERB
cana-1749	113	2	2.17	2.17	NUM
cana-1749	113	3	if	if	SCONJ
cana-1749	113	4	f	f	X
cana-1749	113	5	:	:	PUNCT
cana-1749	113	6	x	x	X
cana-1749	113	7	→	→	SYM
cana-1749	113	8	y	y	PROPN
cana-1749	113	9	is	be	AUX
cana-1749	113	10	slightly	slightly	ADV
cana-1749	113	11	δgp	δgp	ADJ
cana-1749	113	12	-	-	PUNCT
cana-1749	113	13	continuous	continuous	ADJ
cana-1749	113	14	injection	injection	NOUN
cana-1749	113	15	and	and	CCONJ
cana-1749	113	16	y	y	PROPN
cana-1749	113	17	is	be	AUX
cana-1749	113	18	co	co	NOUN
cana-1749	113	19	-	-	NOUN
cana-1749	113	20	t2	t2	ADJ
cana-1749	113	21	,	,	PUNCT
cana-1749	113	22	then	then	ADV
cana-1749	113	23	x	x	PUNCT
cana-1749	113	24	is	be	AUX
cana-1749	113	25	δgphausdorff	δgphausdorff	ADJ
cana-1749	113	26	proof	proof	NOUN
cana-1749	113	27	.	.	PUNCT
cana-1749	114	1	let	let	VERB
cana-1749	114	2	x1	x1	NUM
cana-1749	114	3	,	,	PUNCT
cana-1749	114	4	x2	x2	PROPN
cana-1749	114	5	,	,	PUNCT
cana-1749	114	6			NOUN
cana-1749	114	7	x	x	PUNCT
cana-1749	114	8	and	and	CCONJ
cana-1749	114	9	x1	x1	ADJ
cana-1749	114	10	≠	≠	ADJ
cana-1749	114	11	x2	x2	PROPN
cana-1749	114	12	.	.	PUNCT
cana-1749	115	1	then	then	ADV
cana-1749	115	2	since	since	SCONJ
cana-1749	115	3	f	f	PROPN
cana-1749	115	4	is	be	AUX
cana-1749	115	5	injective	injective	ADJ
cana-1749	115	6	and	and	CCONJ
cana-1749	115	7	y	y	PROPN
cana-1749	115	8	is	be	AUX
cana-1749	115	9	co	co	NOUN
cana-1749	115	10	-	-	NOUN
cana-1749	115	11	t2	t2	ADJ
cana-1749	115	12	,	,	PUNCT
cana-1749	115	13	f(x1	f(x1	ADJ
cana-1749	115	14	)	)	PUNCT
cana-1749	115	15	≠	≠	PROPN
cana-1749	115	16	f(x2	f(x2	NOUN
cana-1749	115	17	)	)	PUNCT
cana-1749	115	18	and	and	CCONJ
cana-1749	115	19	there	there	PRON
cana-1749	115	20	exist	exist	VERB
cana-1749	115	21	clopen	clopen	ADJ
cana-1749	115	22	subsets	subset	NOUN
cana-1749	115	23	v1	v1	NOUN
cana-1749	115	24	,	,	PUNCT
cana-1749	115	25	v2	v2	PROPN
cana-1749	115	26	of	of	ADP
cana-1749	115	27	y	y	PRON
cana-1749	115	28	such	such	ADJ
cana-1749	115	29	that	that	DET
cana-1749	115	30	f(x1	f(x1	NOUN
cana-1749	115	31	)	)	PUNCT
cana-1749	115	32			NOUN
cana-1749	115	33	v1	v1	NOUN
cana-1749	115	34	and	and	CCONJ
cana-1749	115	35	f(x2	f(x2	NOUN
cana-1749	115	36	)	)	PUNCT
cana-1749	115	37			NOUN
cana-1749	115	38	v2	v2	PROPN
cana-1749	115	39	and	and	CCONJ
cana-1749	115	40	v1	v1	VERB
cana-1749	115	41			PUNCT
cana-1749	115	42	v2	v2	NOUN
cana-1749	115	43	=	=	SYM
cana-1749	115	44	.	.	ADJ
cana-1749	115	45	since	since	SCONJ
cana-1749	115	46	f	f	PROPN
cana-1749	115	47	is	be	AUX
cana-1749	115	48	slightly	slightly	ADV
cana-1749	115	49	δgp	δgp	ADJ
cana-1749	115	50	-	-	PUNCT
cana-1749	115	51	continuous	continuous	ADJ
cana-1749	115	52	,	,	PUNCT
cana-1749	115	53	xi	xi	PROPN
cana-1749	115	54			PROPN
cana-1749	115	55	f-1	f-1	PROPN
cana-1749	115	56	(	(	PUNCT
cana-1749	115	57	vi	vi	NOUN
cana-1749	115	58	)	)	PUNCT
cana-1749	115	59			NOUN
cana-1749	115	60	δgpo	δgpo	NOUN
cana-1749	115	61	(	(	PUNCT
cana-1749	115	62	x	x	NOUN
cana-1749	115	63	)	)	PUNCT
cana-1749	115	64	for	for	ADP
cana-1749	115	65	i	i	PROPN
cana-1749	115	66	=	=	SYM
cana-1749	115	67	1	1	NUM
cana-1749	115	68	,	,	PUNCT
cana-1749	115	69	2	2	NUM
cana-1749	115	70	and	and	CCONJ
cana-1749	115	71	f-1	f-1	NOUN
cana-1749	115	72	(	(	PUNCT
cana-1749	115	73	v1	v1	NOUN
cana-1749	115	74	)	)	PUNCT
cana-1749	115	75			X
cana-1749	115	76	f-1(v2	f-1(v2	NOUN
cana-1749	115	77	)	)	PUNCT
cana-1749	115	78	=	=	PUNCT
cana-1749	115	79	.	.	X
cana-1749	115	80	thus	thus	ADV
cana-1749	115	81	x	x	X
cana-1749	115	82	is	be	AUX
cana-1749	115	83	δgphausdorff	δgphausdorff	NOUN
cana-1749	115	84	.	.	PUNCT
cana-1749	116	1	theorem	theorem	VERB
cana-1749	116	2	2.18	2.18	NUM
cana-1749	116	3	assume	assume	VERB
cana-1749	116	4	that	that	SCONJ
cana-1749	116	5	x	x	PRON
cana-1749	116	6	is	be	AUX
cana-1749	116	7	δgp	δgp	NOUN
cana-1749	116	8	-	-	PUNCT
cana-1749	116	9	additive	additive	NOUN
cana-1749	116	10	.	.	PUNCT
cana-1749	117	1	if	if	SCONJ
cana-1749	117	2	f	f	X
cana-1749	117	3	:	:	PUNCT
cana-1749	117	4	x→y	x→y	NUM
cana-1749	117	5	and	and	CCONJ
cana-1749	117	6	g	g	NOUN
cana-1749	117	7	:	:	PUNCT
cana-1749	117	8	x→y	x→y	NUM
cana-1749	117	9	are	be	AUX
cana-1749	117	10	slightly	slightly	ADV
cana-1749	117	11	δgp	δgp	ADJ
cana-1749	117	12	-	-	PUNCT
cana-1749	117	13	continuous	continuous	ADJ
cana-1749	117	14	,	,	PUNCT
cana-1749	117	15	x	x	PUNCT
cana-1749	117	16	is	be	AUX
cana-1749	117	17	submaximal	submaximal	ADJ
cana-1749	117	18	and	and	CCONJ
cana-1749	117	19	y	y	PROPN
cana-1749	117	20	is	be	AUX
cana-1749	117	21	co	co	NOUN
cana-1749	117	22	-	-	NOUN
cana-1749	117	23	t2	t2	NOUN
cana-1749	117	24	.	.	PUNCT
cana-1749	118	1	then	then	ADV
cana-1749	118	2	f={p	f={p	VERB
cana-1749	118	3	∈	∈	PROPN
cana-1749	118	4	x	x	X
cana-1749	118	5	:	:	PUNCT
cana-1749	118	6	f(p)=g(p	f(p)=g(p	NOUN
cana-1749	118	7	)	)	PUNCT
cana-1749	118	8	}	}	PUNCT
cana-1749	118	9	is	be	AUX
cana-1749	118	10	δgp	δgp	NOUN
cana-1749	118	11	-	-	PUNCT
cana-1749	118	12	closed	close	VERB
cana-1749	118	13	in	in	ADP
cana-1749	118	14	x.	x.	NOUN
cana-1749	118	15	proof	proof	NOUN
cana-1749	118	16	:	:	PUNCT
cana-1749	118	17	let	let	VERB
cana-1749	118	18	p	p	PRON
cana-1749	118	19			ADJ
cana-1749	118	20	f	f	NOUN
cana-1749	118	21	,	,	PUNCT
cana-1749	118	22	then	then	ADV
cana-1749	118	23	f(p	f(p	NUM
cana-1749	118	24	)	)	PUNCT
cana-1749	118	25	≠	≠	PROPN
cana-1749	118	26	g(p	g(p	PROPN
cana-1749	118	27	)	)	PUNCT
cana-1749	118	28	.	.	PUNCT
cana-1749	119	1	since	since	SCONJ
cana-1749	119	2	y	y	PROPN
cana-1749	119	3	is	be	AUX
cana-1749	119	4	co	co	NOUN
cana-1749	119	5	-	-	NOUN
cana-1749	119	6	t2	t2	ADJ
cana-1749	119	7	,	,	PUNCT
cana-1749	119	8	there	there	PRON
cana-1749	119	9	exist	exist	VERB
cana-1749	119	10	clopen	clopen	ADJ
cana-1749	119	11	subsets	subset	NOUN
cana-1749	119	12	m1	m1	PROPN
cana-1749	119	13	and	and	CCONJ
cana-1749	119	14	m2	m2	PROPN
cana-1749	119	15	of	of	ADP
cana-1749	119	16	y	y	PROPN
cana-1749	119	17	such	such	ADJ
cana-1749	119	18	that	that	SCONJ
cana-1749	119	19	f(p)∈	f(p)∈	NOUN
cana-1749	119	20	m1	m1	NOUN
cana-1749	119	21	,	,	PUNCT
cana-1749	119	22	g(p)∈	g(p)∈	PROPN
cana-1749	119	23	m2	m2	PROPN
cana-1749	119	24	and	and	CCONJ
cana-1749	119	25	m1∩	m1∩	NOUN
cana-1749	119	26	m2	m2	PROPN
cana-1749	119	27	=	=	PROPN
cana-1749	119	28	.	.	X
cana-1749	119	29	since	since	SCONJ
cana-1749	119	30	f	f	PROPN
cana-1749	119	31	and	and	CCONJ
cana-1749	119	32	g	g	PROPN
cana-1749	119	33	are	be	AUX
cana-1749	119	34	slightly	slightly	ADV
cana-1749	119	35	δgp	δgp	ADJ
cana-1749	119	36	-	-	PUNCT
cana-1749	119	37	continuous	continuous	ADJ
cana-1749	119	38	,	,	PUNCT
cana-1749	119	39	f-1	f-1	NOUN
cana-1749	119	40	(	(	PUNCT
cana-1749	119	41	m1	m1	PROPN
cana-1749	119	42	)	)	PUNCT
cana-1749	119	43	and	and	CCONJ
cana-1749	119	44	g	g	PROPN
cana-1749	119	45	-1(m2	-1(m2	PUNCT
cana-1749	119	46	)	)	PUNCT
cana-1749	119	47	are	be	AUX
cana-1749	119	48	δgp	δgp	NOUN
cana-1749	119	49	-	-	PUNCT
cana-1749	119	50	open	open	ADJ
cana-1749	119	51	sets	set	NOUN
cana-1749	119	52	in	in	ADP
cana-1749	119	53	x	x	X
cana-1749	119	54	.	.	PUNCT
cana-1749	120	1	let	let	AUX
cana-1749	120	2	m=	m=	X
cana-1749	120	3	f-1	f-1	PROPN
cana-1749	120	4	(	(	PUNCT
cana-1749	120	5	m1	m1	PROPN
cana-1749	120	6	)	)	PUNCT
cana-1749	120	7	and	and	CCONJ
cana-1749	120	8	n	n	PROPN
cana-1749	120	9	=	=	PROPN
cana-1749	120	10	g	g	PROPN
cana-1749	120	11	-1(m2	-1(m2	PUNCT
cana-1749	120	12	)	)	PUNCT
cana-1749	120	13	,	,	PUNCT
cana-1749	120	14	then	then	ADV
cana-1749	120	15	m	m	PROPN
cana-1749	120	16	and	and	CCONJ
cana-1749	120	17	n	n	PROPN
cana-1749	120	18	are	be	AUX
cana-1749	120	19	δgp	δgp	ADJ
cana-1749	120	20	-	-	PUNCT
cana-1749	120	21	open	open	ADJ
cana-1749	120	22	sets	set	NOUN
cana-1749	120	23	containing	contain	VERB
cana-1749	120	24	p.	p.	NOUN
cana-1749	120	25	set	set	VERB
cana-1749	120	26	o	o	NOUN
cana-1749	120	27	=	=	NOUN
cana-1749	120	28	m∩n	m∩n	PROPN
cana-1749	120	29	,	,	PUNCT
cana-1749	120	30	then	then	ADV
cana-1749	120	31	o	o	INTJ
cana-1749	121	1	i	i	PRON
cana-1749	121	2	s	s	VERB
cana-1749	121	3	δgp	δgp	ADV
cana-1749	121	4	-	-	PUNCT
cana-1749	121	5	open	open	ADJ
cana-1749	121	6	set	set	NOUN
cana-1749	121	7	containing	contain	VERB
cana-1749	121	8	p.	p.	NOUN
cana-1749	121	9	hence	hence	ADV
cana-1749	121	10	f(o)∩g(o)=f(m∩n)∩g(m∩n)⊂f(m)∩g(n)=	f(o)∩g(o)=f(m∩n)∩g(m∩n)⊂f(m)∩g(n)=	PROPN
cana-1749	121	11	m1∩	m1∩	NOUN
cana-1749	121	12	m2	m2	PROPN
cana-1749	121	13	=	=	NOUN
cana-1749	121	14			NOUN
cana-1749	121	15	and	and	CCONJ
cana-1749	122	1	so	so	ADV
cana-1749	122	2	o∩f=.	o∩f=.	INTJ
cana-1749	122	3	then	then	ADV
cana-1749	122	4	p	p	PROPN
cana-1749	122	5	∈	∈	PROPN
cana-1749	122	6			PUNCT
cana-1749	122	7	δgpcl(f	δgpcl(f	PROPN
cana-1749	122	8	)	)	PUNCT
cana-1749	122	9	and	and	CCONJ
cana-1749	122	10	hence	hence	ADV
cana-1749	122	11	,	,	PUNCT
cana-1749	122	12	f	f	PROPN
cana-1749	122	13	is	be	AUX
cana-1749	122	14	δgp	δgp	NOUN
cana-1749	122	15	-	-	PUNCT
cana-1749	122	16	closed	close	VERB
cana-1749	122	17	in	in	ADP
cana-1749	122	18	x.	x.	NOUN
cana-1749	122	19	theorem	theorem	VERB
cana-1749	122	20	2.19	2.19	NUM
cana-1749	122	21	if	if	SCONJ
cana-1749	122	22	f	f	X
cana-1749	122	23	:	:	PUNCT
cana-1749	122	24	x→y	x→y	NUM
cana-1749	123	1	be	be	AUX
cana-1749	123	2	slightly	slightly	ADV
cana-1749	123	3	δgp	δgp	ADJ
cana-1749	123	4	-	-	PUNCT
cana-1749	123	5	continuous	continuous	ADJ
cana-1749	123	6	injective	injective	ADJ
cana-1749	123	7	closed	close	VERB
cana-1749	123	8	function	function	NOUN
cana-1749	123	9	from	from	ADP
cana-1749	123	10	a	a	DET
cana-1749	123	11	strongly	strongly	ADV
cana-1749	123	12	δgp	δgp	NOUN
cana-1749	123	13	-	-	PUNCT
cana-1749	123	14	regular	regular	ADJ
cana-1749	123	15	space	space	NOUN
cana-1749	123	16	x	x	X
cana-1749	123	17	onto	onto	ADP
cana-1749	123	18	a	a	DET
cana-1749	123	19	space	space	NOUN
cana-1749	123	20	y	y	NOUN
cana-1749	123	21	,	,	PUNCT
cana-1749	123	22	then	then	ADV
cana-1749	123	23	y	y	PROPN
cana-1749	123	24	is	be	AUX
cana-1749	123	25	co	co	ADJ
cana-1749	123	26	-	-	ADJ
cana-1749	123	27	regular	regular	ADJ
cana-1749	123	28	.	.	PUNCT
cana-1749	124	1	proof	proof	NOUN
cana-1749	124	2	:	:	PUNCT
cana-1749	124	3	let	let	VERB
cana-1749	124	4	m	m	PRON
cana-1749	124	5	be	be	AUX
cana-1749	124	6	a	a	DET
cana-1749	124	7	clopen	clopen	ADJ
cana-1749	124	8	set	set	NOUN
cana-1749	124	9	in	in	ADP
cana-1749	124	10	y	y	PROPN
cana-1749	124	11	and	and	CCONJ
cana-1749	124	12	y	y	PROPN
cana-1749	124	13	∉	∉	PROPN
cana-1749	124	14	m.take	m.take	VERB
cana-1749	124	15	y	y	PROPN
cana-1749	124	16	=	=	PROPN
cana-1749	124	17	f(x).since	f(x).since	PROPN
cana-1749	124	18	f	f	PROPN
cana-1749	124	19	is	be	AUX
cana-1749	124	20	slightly	slightly	ADV
cana-1749	124	21	δgp	δgp	ADJ
cana-1749	124	22	-	-	PUNCT
cana-1749	124	23	continuous	continuous	ADJ
cana-1749	124	24	,	,	PUNCT
cana-1749	124	25	f1(m	f1(m	NOUN
cana-1749	124	26	)	)	PUNCT
cana-1749	124	27	is	be	AUX
cana-1749	124	28	δgp	δgp	NOUN
cana-1749	124	29	-	-	PUNCT
cana-1749	124	30	closed	closed	ADJ
cana-1749	124	31	.	.	PUNCT
cana-1749	125	1	take	take	VERB
cana-1749	125	2	n=	n=	ADJ
cana-1749	125	3	f-1(m	f-1(m	NOUN
cana-1749	125	4	)	)	PUNCT
cana-1749	125	5	.we	.we	PUNCT
cana-1749	126	1	have	have	VERB
cana-1749	126	2	x	x	PROPN
cana-1749	126	3	∉	∉	PROPN
cana-1749	126	4	n.	n.	PROPN
cana-1749	126	5	since	since	SCONJ
cana-1749	126	6	x	x	PRON
cana-1749	126	7	is	be	AUX
cana-1749	126	8	strongly	strongly	ADV
cana-1749	126	9	δgp	δgp	ADJ
cana-1749	126	10	-	-	PUNCT
cana-1749	126	11	regular	regular	ADJ
cana-1749	126	12	,	,	PUNCT
cana-1749	126	13	disjoint	disjoint	NOUN
cana-1749	126	14	o	o	NOUN
cana-1749	126	15	p	p	X
cana-1749	126	16	e	e	NOUN
cana-1749	126	17	n	n	PRON
cana-1749	126	18	sets	set	VERB
cana-1749	126	19	u	u	NOUN
cana-1749	126	20	and	and	CCONJ
cana-1749	126	21	v	v	NOUN
cana-1749	126	22	in	in	ADP
cana-1749	126	23	y	y	PRON
cana-1749	126	24	such	such	ADJ
cana-1749	126	25	that	that	DET
cana-1749	126	26	n⊂	n⊂	PROPN
cana-1749	126	27	u	u	PROPN
cana-1749	126	28	and	and	CCONJ
cana-1749	126	29	x	x	ADP
cana-1749	126	30			NOUN
cana-1749	127	1	v.	v.	CCONJ
cana-1749	127	2	then	then	ADV
cana-1749	127	3	we	we	PRON
cana-1749	127	4	obtain	obtain	VERB
cana-1749	127	5	that	that	SCONJ
cana-1749	127	6	m	m	PROPN
cana-1749	127	7	=	=	NOUN
cana-1749	127	8	f(n)⊂	f(n)⊂	NOUN
cana-1749	127	9	f(u	f(u	PROPN
cana-1749	127	10	)	)	PUNCT
cana-1749	127	11	and	and	CCONJ
cana-1749	127	12	y	y	PROPN
cana-1749	127	13	=	=	PROPN
cana-1749	127	14	f(x	f(x	PROPN
cana-1749	127	15	)	)	PUNCT
cana-1749	127	16			NOUN
cana-1749	127	17	f(v	f(v	NOUN
cana-1749	127	18	)	)	PUNCT
cana-1749	127	19	such	such	ADJ
cana-1749	127	20	that	that	DET
cana-1749	127	21	f(u	f(u	PROPN
cana-1749	127	22	)	)	PUNCT
cana-1749	127	23	and	and	CCONJ
cana-1749	127	24	f(v	f(v	NOUN
cana-1749	127	25	)	)	PUNCT
cana-1749	127	26	are	be	AUX
cana-1749	127	27	disjoint	disjoint	ADJ
cana-1749	127	28	open	open	ADJ
cana-1749	127	29	sets	set	NOUN
cana-1749	127	30	in	in	ADP
cana-1749	127	31	y.	y.	PROPN
cana-1749	127	32	this	this	PRON
cana-1749	127	33	shows	show	VERB
cana-1749	127	34	y	y	PROPN
cana-1749	127	35	is	be	AUX
cana-1749	127	36	co	co	ADJ
cana-1749	127	37	-	-	ADJ
cana-1749	127	38	regular	regular	ADJ
cana-1749	127	39	-	-	PUNCT
cana-1749	127	40	normal	normal	ADJ
cana-1749	127	41	.	.	PUNCT
cana-1749	128	1	theorem	theorem	VERB
cana-1749	128	2	2.20	2.20	NUM
cana-1749	128	3	if	if	SCONJ
cana-1749	128	4	f	f	X
cana-1749	128	5	:	:	PUNCT
cana-1749	128	6	x→y	x→y	NUM
cana-1749	129	1	be	be	AUX
cana-1749	129	2	slightly	slightly	ADV
cana-1749	129	3	δgp	δgp	ADJ
cana-1749	129	4	-	-	PUNCT
cana-1749	129	5	continuous	continuous	ADJ
cana-1749	129	6	injective	injective	ADJ
cana-1749	129	7	open	open	ADJ
cana-1749	129	8	function	function	NOUN
cana-1749	129	9	from	from	ADP
cana-1749	129	10	a	a	DET
cana-1749	129	11	strongly	strongly	ADV
cana-1749	129	12	δgp	δgp	ADJ
cana-1749	129	13	-	-	PUNCT
cana-1749	129	14	normal	normal	ADJ
cana-1749	129	15	space	space	NOUN
cana-1749	129	16	x	x	X
cana-1749	129	17	onto	onto	ADP
cana-1749	129	18	a	a	DET
cana-1749	129	19	space	space	NOUN
cana-1749	129	20	y	y	NOUN
cana-1749	129	21	,	,	PUNCT
cana-1749	129	22	then	then	ADV
cana-1749	129	23	y	y	PROPN
cana-1749	129	24	is	be	AUX
cana-1749	129	25	co	co	ADJ
cana-1749	129	26	-	-	ADJ
cana-1749	129	27	normal	normal	ADJ
cana-1749	129	28	.	.	PUNCT
cana-1749	130	1	proof	proof	NOUN
cana-1749	130	2	:	:	PUNCT
cana-1749	130	3	let	let	VERB
cana-1749	130	4	e	e	NOUN
cana-1749	130	5	and	and	CCONJ
cana-1749	130	6	f	f	PROPN
cana-1749	130	7	be	be	AUX
cana-1749	130	8	disjoint	disjoint	X
cana-1749	130	9	clopen	clopen	ADJ
cana-1749	130	10	subsets	subset	NOUN
cana-1749	130	11	of	of	ADP
cana-1749	130	12	y.	y.	PROPN
cana-1749	130	13	since	since	SCONJ
cana-1749	130	14	f	f	PROPN
cana-1749	130	15	is	be	AUX
cana-1749	130	16	slightly	slightly	ADV
cana-1749	130	17	δgp	δgp	ADJ
cana-1749	130	18	-	-	PUNCT
cana-1749	130	19	continuous	continuous	ADJ
cana-1749	130	20	,	,	PUNCT
cana-1749	130	21	f-1	f-1	NOUN
cana-1749	130	22	(	(	PUNCT
cana-1749	130	23	e	e	NOUN
cana-1749	130	24	)	)	PUNCT
cana-1749	130	25	and	and	CCONJ
cana-1749	130	26	f-1(f	f-1(f	NOUN
cana-1749	130	27	)	)	PUNCT
cana-1749	130	28	are	be	AUX
cana-1749	130	29	disjoint	disjoint	ADJ
cana-1749	130	30	δgp	δgp	NOUN
cana-1749	130	31	-	-	PUNCT
cana-1749	130	32	closed	close	VERB
cana-1749	130	33	sets	set	NOUN
cana-1749	130	34	in	in	ADP
cana-1749	130	35	x.	x.	NOUN
cana-1749	130	36	since	since	SCONJ
cana-1749	130	37	x	x	PRON
cana-1749	130	38	is	be	AUX
cana-1749	130	39	strongly	strongly	ADV
cana-1749	130	40	δgp	δgp	ADJ
cana-1749	130	41	-	-	PUNCT
cana-1749	130	42	normal	normal	ADJ
cana-1749	130	43	,	,	PUNCT
cana-1749	130	44	there	there	PRON
cana-1749	130	45	exist	exist	VERB
cana-1749	130	46	disjoint	disjoint	ADJ
cana-1749	130	47	open	open	ADJ
cana-1749	130	48	sets	set	NOUN
cana-1749	130	49	u	u	NOUN
cana-1749	130	50	and	and	CCONJ
cana-1749	130	51	v	v	ADP
cana-1749	130	52	such	such	ADJ
cana-1749	130	53	that	that	DET
cana-1749	130	54	f-1(e)⊂u	f-1(e)⊂u	NOUN
cana-1749	130	55	and	and	CCONJ
cana-1749	130	56	f-1	f-1	NOUN
cana-1749	130	57	(	(	PUNCT
cana-1749	130	58	f)⊂v	f)⊂v	PROPN
cana-1749	130	59	.	.	PUNCT
cana-1749	131	1	this	this	PRON
cana-1749	131	2	implies	imply	VERB
cana-1749	131	3	e⊂	e⊂	PROPN
cana-1749	131	4	f(u	f(u	PROPN
cana-1749	131	5	)	)	PUNCT
cana-1749	131	6	and	and	CCONJ
cana-1749	131	7	f⊂	f⊂	PROPN
cana-1749	131	8	f(v	f(v	NOUN
cana-1749	131	9	)	)	PUNCT
cana-1749	131	10	and	and	CCONJ
cana-1749	131	11	injectivity	injectivity	NOUN
cana-1749	131	12	and	and	CCONJ
cana-1749	131	13	openness	openness	NOUN
cana-1749	131	14	of	of	ADP
cana-1749	131	15	f	f	PROPN
cana-1749	131	16	implies	imply	VERB
cana-1749	131	17	f(u	f(u	PROPN
cana-1749	131	18	)	)	PUNCT
cana-1749	131	19	and	and	CCONJ
cana-1749	131	20	f(v	f(v	NOUN
cana-1749	131	21	)	)	PUNCT
cana-1749	131	22	are	be	AUX
cana-1749	131	23	disjoint	disjoint	ADJ
cana-1749	131	24	open	open	ADJ
cana-1749	131	25	sets	set	NOUN
cana-1749	131	26	in	in	ADP
cana-1749	131	27	y.	y.	PROPN
cana-1749	131	28	this	this	PRON
cana-1749	131	29	shows	show	VERB
cana-1749	131	30	y	y	PROPN
cana-1749	131	31	is	be	AUX
cana-1749	131	32	co	co	ADJ
cana-1749	131	33	-	-	ADJ
cana-1749	131	34	normal	normal	ADJ
cana-1749	131	35	.	.	PUNCT
cana-1749	132	1	theorem	theorem	VERB
cana-1749	132	2	2.21	2.21	NUM
cana-1749	132	3	if	if	SCONJ
cana-1749	132	4	f	f	X
cana-1749	132	5	:	:	PUNCT
cana-1749	132	6	x	x	X
cana-1749	132	7	→	→	SYM
cana-1749	132	8	y	y	PROPN
cana-1749	132	9	is	be	AUX
cana-1749	132	10	slightly	slightly	ADV
cana-1749	132	11	δgp	δgp	ADJ
cana-1749	132	12	-	-	PUNCT
cana-1749	132	13	continuous	continuous	ADJ
cana-1749	132	14	surjection	surjection	NOUN
cana-1749	132	15	,	,	PUNCT
cana-1749	132	16	and	and	CCONJ
cana-1749	132	17	x	x	X
cana-1749	132	18	is	be	AUX
cana-1749	132	19	δgp	δgp	NOUN
cana-1749	132	20	-	-	PUNCT
cana-1749	132	21	compact	compact	ADJ
cana-1749	132	22	,	,	PUNCT
cana-1749	132	23	then	then	ADV
cana-1749	132	24	y	y	PROPN
cana-1749	132	25	is	be	AUX
cana-1749	132	26	mildly	mildly	ADV
cana-1749	132	27	compact	compact	ADJ
cana-1749	132	28	.	.	PUNCT
cana-1749	133	1	proof	proof	NOUN
cana-1749	133	2	.	.	PUNCT
cana-1749	134	1	let	let	VERB
cana-1749	134	2	{	{	PUNCT
cana-1749	134	3	v	v	ADV
cana-1749	134	4	:	:	PUNCT
cana-1749	134	5	v	v	NUM
cana-1749	134	6			NOUN
cana-1749	134	7	co	co	X
cana-1749	134	8	(	(	PUNCT
cana-1749	134	9	y	y	NOUN
cana-1749	134	10	)	)	PUNCT
cana-1749	134	11	,	,	PUNCT
cana-1749	134	12			PROPN
cana-1749	134	13			PROPN
cana-1749	134	14	i	i	PRON
cana-1749	134	15	}	}	PUNCT
cana-1749	134	16	be	be	VERB
cana-1749	134	17	a	a	DET
cana-1749	134	18	cover	cover	NOUN
cana-1749	134	19	of	of	ADP
cana-1749	134	20	y	y	NOUN
cana-1749	134	21	by	by	ADP
cana-1749	134	22	clopen	clopen	ADJ
cana-1749	134	23	sets	set	NOUN
cana-1749	134	24	.	.	PUNCT
cana-1749	135	1	since	since	SCONJ
cana-1749	135	2	f	f	PROPN
cana-1749	135	3	is	be	AUX
cana-1749	135	4	slightly	slightly	ADV
cana-1749	135	5	communications	communication	NOUN
cana-1749	135	6	on	on	ADP
cana-1749	135	7	applied	apply	VERB
cana-1749	135	8	nonlinear	nonlinear	ADJ
cana-1749	135	9	analysis	analysis	NOUN
cana-1749	135	10	issn	issn	NOUN
cana-1749	135	11	:	:	PUNCT
cana-1749	135	12	1074	1074	NUM
cana-1749	135	13	-	-	PUNCT
cana-1749	135	14	133x	133x	NUM
cana-1749	135	15	vol	vol	NOUN
cana-1749	135	16	32	32	NUM
cana-1749	135	17	no	no	NOUN
cana-1749	135	18	.	.	NOUN
cana-1749	135	19	2	2	NUM
cana-1749	135	20	(	(	PUNCT
cana-1749	135	21	2025	2025	NUM
cana-1749	135	22	)	)	PUNCT
cana-1749	135	23	381	381	NUM
cana-1749	135	24	https://internationalpubls.com	https://internationalpubls.com	X
cana-1749	135	25	δgp	δgp	NOUN
cana-1749	135	26	-	-	PUNCT
cana-1749	135	27	continuous	continuous	ADJ
cana-1749	135	28	,	,	PUNCT
cana-1749	135	29	{	{	PUNCT
cana-1749	135	30	f-1(v	f-1(v	NUM
cana-1749	135	31	)	)	PUNCT
cana-1749	135	32	:	:	PUNCT
cana-1749	135	33			X
cana-1749	135	34			NOUN
cana-1749	135	35	i	i	PRON
cana-1749	135	36	}	}	PUNCT
cana-1749	135	37	be	be	VERB
cana-1749	135	38	δgp	δgp	NOUN
cana-1749	135	39	-	-	PUNCT
cana-1749	135	40	open	open	ADJ
cana-1749	135	41	cover	cover	NOUN
cana-1749	135	42	of	of	ADP
cana-1749	135	43	x	x	PUNCT
cana-1749	136	1	so	so	ADV
cana-1749	136	2	there	there	PRON
cana-1749	136	3	is	be	VERB
cana-1749	136	4	a	a	DET
cana-1749	136	5	finite	finite	NOUN
cana-1749	136	6	subset	subset	NOUN
cana-1749	136	7	i0	i0	PROPN
cana-1749	136	8	of	of	ADP
cana-1749	136	9	i	i	PRON
cana-1749	136	10	such	such	ADJ
cana-1749	136	11	that	that	SCONJ
cana-1749	136	12	x	x	X
cana-1749	136	13	=	=	SYM
cana-1749	136	14			NOUN
cana-1749	136	15	{	{	PUNCT
cana-1749	136	16	f-1	f-1	NOUN
cana-1749	136	17	(	(	PUNCT
cana-1749	136	18	v	v	PROPN
cana-1749	136	19	)	)	PUNCT
cana-1749	136	20	:	:	PUNCT
cana-1749	136	21			X
cana-1749	136	22			PROPN
cana-1749	136	23	i0	i0	PROPN
cana-1749	136	24	}	}	PUNCT
cana-1749	136	25	.	.	PUNCT
cana-1749	137	1	therefore	therefore	ADV
cana-1749	137	2	,	,	PUNCT
cana-1749	137	3	y	y	PROPN
cana-1749	137	4	=	=	SYM
cana-1749	137	5	u	u	PROPN
cana-1749	137	6	{	{	PUNCT
cana-1749	137	7	v	v	ADV
cana-1749	137	8	:	:	PUNCT
cana-1749	137	9			X
cana-1749	137	10			PROPN
cana-1749	137	11	i0	i0	PROPN
cana-1749	137	12	}	}	PUNCT
cana-1749	137	13	as	as	SCONJ
cana-1749	137	14	f	f	PROPN
cana-1749	137	15	is	be	AUX
cana-1749	137	16	surjective	surjective	ADJ
cana-1749	137	17	.	.	PUNCT
cana-1749	138	1	thus	thus	ADV
cana-1749	138	2	y	y	PROPN
cana-1749	138	3	is	be	AUX
cana-1749	138	4	mildly	mildly	ADV
cana-1749	138	5	compact	compact	ADJ
cana-1749	138	6	.	.	PUNCT
cana-1749	139	1	theorem	theorem	VERB
cana-1749	139	2	2.22	2.22	NUM
cana-1749	139	3	if	if	SCONJ
cana-1749	139	4	f	f	X
cana-1749	139	5	:	:	PUNCT
cana-1749	139	6	x	x	X
cana-1749	139	7	→	→	SYM
cana-1749	139	8	y	y	PROPN
cana-1749	139	9	is	be	AUX
cana-1749	139	10	slightly	slightly	ADV
cana-1749	139	11	δgp	δgp	ADJ
cana-1749	139	12	-	-	PUNCT
cana-1749	139	13	continuous	continuous	ADJ
cana-1749	139	14	surjection	surjection	NOUN
cana-1749	139	15	,	,	PUNCT
cana-1749	139	16	and	and	CCONJ
cana-1749	139	17	x	x	X
cana-1749	139	18	is	be	AUX
cana-1749	139	19	δgp	δgp	NOUN
cana-1749	139	20	-	-	PUNCT
cana-1749	139	21	connected	connect	VERB
cana-1749	139	22	,	,	PUNCT
cana-1749	139	23	then	then	ADV
cana-1749	139	24	y	y	PROPN
cana-1749	139	25	is	be	AUX
cana-1749	139	26	connected	connect	VERB
cana-1749	139	27	.	.	PUNCT
cana-1749	140	1	proof	proof	NOUN
cana-1749	140	2	.	.	PUNCT
cana-1749	141	1	assume	assume	VERB
cana-1749	141	2	that	that	SCONJ
cana-1749	141	3	y	y	PROPN
cana-1749	141	4	is	be	AUX
cana-1749	141	5	disconnected	disconnect	VERB
cana-1749	141	6	.	.	PUNCT
cana-1749	142	1	then	then	ADV
cana-1749	142	2	exist	exist	VERB
cana-1749	142	3	disjoint	disjoint	ADJ
cana-1749	142	4	non	non	ADJ
cana-1749	142	5	-	-	ADJ
cana-1749	142	6	empty	empty	ADJ
cana-1749	142	7	clopen	clopen	ADJ
cana-1749	142	8	sets	set	NOUN
cana-1749	142	9	u	u	NOUN
cana-1749	142	10	and	and	CCONJ
cana-1749	142	11	v	v	NOUN
cana-1749	142	12	for	for	ADP
cana-1749	142	13	which	which	PRON
cana-1749	142	14	y	y	PROPN
cana-1749	142	15	=	=	SYM
cana-1749	142	16	u	u	PROPN
cana-1749	142	17			PROPN
cana-1749	142	18	v.	v.	CCONJ
cana-1749	142	19	therefore	therefore	ADV
cana-1749	142	20	,	,	PUNCT
cana-1749	142	21	x	x	SYM
cana-1749	142	22	=	=	PRON
cana-1749	142	23	f-1	f-1	NOUN
cana-1749	142	24	(	(	PUNCT
cana-1749	142	25	u	u	NOUN
cana-1749	142	26	)	)	PUNCT
cana-1749	142	27			NOUN
cana-1749	142	28	f-1	f-1	NOUN
cana-1749	142	29	(	(	PUNCT
cana-1749	142	30	v	v	NOUN
cana-1749	142	31	)	)	PUNCT
cana-1749	142	32	is	be	AUX
cana-1749	142	33	the	the	DET
cana-1749	142	34	union	union	NOUN
cana-1749	142	35	of	of	ADP
cana-1749	142	36	two	two	NUM
cana-1749	142	37	disjoint	disjoint	ADJ
cana-1749	142	38	δgp	δgp	NOUN
cana-1749	142	39	-	-	PUNCT
cana-1749	142	40	open	open	ADJ
cana-1749	142	41	nonempty	nonempty	NOUN
cana-1749	142	42	sets	set	NOUN
cana-1749	142	43	and	and	CCONJ
cana-1749	142	44	hence	hence	ADV
cana-1749	142	45	is	be	AUX
cana-1749	142	46	not	not	PART
cana-1749	142	47	δgp	δgp	NOUN
cana-1749	142	48	-	-	PUNCT
cana-1749	142	49	connected	connect	VERB
cana-1749	142	50	.	.	PUNCT
cana-1749	143	1	definition	definition	NOUN
cana-1749	143	2	2.23	2.23	NUM
cana-1749	143	3	the	the	DET
cana-1749	143	4	graph	graph	NOUN
cana-1749	143	5	g(f	g(f	PROPN
cana-1749	143	6	)	)	PUNCT
cana-1749	143	7	of	of	ADP
cana-1749	143	8	a	a	DET
cana-1749	143	9	function	function	NOUN
cana-1749	143	10	f:(x	f:(x	NOUN
cana-1749	143	11	,	,	PUNCT
cana-1749	143	12	τ	τ	PROPN
cana-1749	143	13	)	)	PUNCT
cana-1749	143	14	→(y	→(y	PROPN
cana-1749	143	15	,	,	PUNCT
cana-1749	143	16	σ	σ	PROPN
cana-1749	143	17	)	)	PUNCT
cana-1749	143	18	is	be	AUX
cana-1749	143	19	said	say	VERB
cana-1749	143	20	to	to	PART
cana-1749	143	21	be	be	AUX
cana-1749	143	22	is	be	AUX
cana-1749	143	23	δgp	δgp	NOUN
cana-1749	143	24	-	-	PUNCT
cana-1749	143	25	co	co	NOUN
cana-1749	143	26	-	-	ADJ
cana-1749	143	27	closed	closed	ADJ
cana-1749	143	28	if	if	SCONJ
cana-1749	143	29	for	for	ADP
cana-1749	143	30	each	each	PRON
cana-1749	143	31	(	(	PUNCT
cana-1749	143	32	x	x	NOUN
cana-1749	143	33	,	,	PUNCT
cana-1749	143	34	y)∈(x×y)-g(f	y)∈(x×y)-g(f	PROPN
cana-1749	143	35	)	)	PUNCT
cana-1749	143	36	there	there	PRON
cana-1749	143	37	exist	exist	VERB
cana-1749	143	38	δgp	δgp	NOUN
cana-1749	143	39	-	-	PUNCT
cana-1749	143	40	open	open	NOUN
cana-1749	143	41	set	set	NOUN
cana-1749	143	42	u	u	NOUN
cana-1749	143	43	in	in	ADP
cana-1749	143	44	x	x	PUNCT
cana-1749	143	45	containing	contain	VERB
cana-1749	143	46	x	x	X
cana-1749	143	47	and	and	CCONJ
cana-1749	143	48	clopen	clopen	ADJ
cana-1749	143	49	set	set	VERB
cana-1749	143	50	v	v	NOUN
cana-1749	143	51	in	in	ADP
cana-1749	143	52	y	y	NOUN
cana-1749	143	53	containing	contain	VERB
cana-1749	143	54	y	y	PRON
cana-1749	143	55	such	such	ADJ
cana-1749	143	56	that	that	PRON
cana-1749	143	57	(	(	PUNCT
cana-1749	143	58	u×v)∩g(f	u×v)∩g(f	NOUN
cana-1749	143	59	)	)	PUNCT
cana-1749	143	60	=	=	PUNCT
cana-1749	143	61	.	.	X
cana-1749	143	62	theorem	theorem	VERB
cana-1749	143	63	2.24	2.24	NUM
cana-1749	143	64	the	the	DET
cana-1749	143	65	graph	graph	NOUN
cana-1749	143	66	g(f	g(f	PROPN
cana-1749	143	67	)	)	PUNCT
cana-1749	143	68	of	of	ADP
cana-1749	143	69	a	a	DET
cana-1749	143	70	function	function	NOUN
cana-1749	143	71	f:(x	f:(x	NOUN
cana-1749	143	72	,	,	PUNCT
cana-1749	143	73	τ	τ	PROPN
cana-1749	143	74	)	)	PUNCT
cana-1749	143	75	→(y	→(y	PROPN
cana-1749	143	76	,	,	PUNCT
cana-1749	143	77	σ	σ	PROPN
cana-1749	143	78	)	)	PUNCT
cana-1749	143	79	is	be	AUX
cana-1749	143	80	δgp	δgp	NOUN
cana-1749	143	81	-	-	PUNCT
cana-1749	143	82	co	co	NOUN
cana-1749	143	83	-	-	VERB
cana-1749	143	84	closed	closed	ADJ
cana-1749	143	85	in	in	ADP
cana-1749	143	86	x×y	x×y	PROPN
cana-1749	144	1	if	if	SCONJ
cana-1749	144	2	and	and	CCONJ
cana-1749	144	3	only	only	ADV
cana-1749	144	4	for	for	ADP
cana-1749	144	5	each	each	DET
cana-1749	144	6	(	(	PUNCT
cana-1749	144	7	x	x	NOUN
cana-1749	144	8	,	,	PUNCT
cana-1749	144	9	y)∈(x×y)-g(f	y)∈(x×y)-g(f	PROPN
cana-1749	144	10	)	)	PUNCT
cana-1749	144	11	there	there	PRON
cana-1749	144	12	exist	exist	VERB
cana-1749	144	13	δgp	δgp	NOUN
cana-1749	144	14	-	-	PUNCT
cana-1749	144	15	open	open	NOUN
cana-1749	144	16	set	set	NOUN
cana-1749	144	17	u	u	NOUN
cana-1749	144	18	in	in	ADP
cana-1749	144	19	x	x	PUNCT
cana-1749	144	20	containing	contain	VERB
cana-1749	144	21	x	x	X
cana-1749	144	22	and	and	CCONJ
cana-1749	144	23	clopen	clopen	ADJ
cana-1749	144	24	set	set	VERB
cana-1749	144	25	v	v	NOUN
cana-1749	144	26	in	in	ADP
cana-1749	144	27	y	y	NOUN
cana-1749	144	28	containing	contain	VERB
cana-1749	144	29	y	y	PRON
cana-1749	144	30	such	such	ADJ
cana-1749	144	31	that	that	DET
cana-1749	144	32	f(u)∩v=.	f(u)∩v=.	NOUN
cana-1749	144	33	theorem	theorem	VERB
cana-1749	144	34	2.25	2.25	NUM
cana-1749	144	35	if	if	SCONJ
cana-1749	144	36	f:(x	f:(x	PROPN
cana-1749	144	37	,	,	PUNCT
cana-1749	144	38	τ	τ	PROPN
cana-1749	144	39	)	)	PUNCT
cana-1749	144	40	→(y	→(y	PROPN
cana-1749	144	41	,	,	PUNCT
cana-1749	144	42	σ	σ	PROPN
cana-1749	144	43	)	)	PUNCT
cana-1749	144	44	is	be	AUX
cana-1749	144	45	slightly	slightly	ADV
cana-1749	144	46	δgp	δgp	ADJ
cana-1749	144	47	-	-	PUNCT
cana-1749	144	48	continuous	continuous	ADJ
cana-1749	144	49	and	and	CCONJ
cana-1749	144	50	y	y	PROPN
cana-1749	144	51	is	be	AUX
cana-1749	144	52	co	co	ADJ
cana-1749	144	53	-	-	ADJ
cana-1749	144	54	hausdorf	hausdorf	ADJ
cana-1749	144	55	,	,	PUNCT
cana-1749	144	56	then	then	ADV
cana-1749	144	57	g(f	g(f	PROPN
cana-1749	144	58	)	)	PUNCT
cana-1749	144	59	is	be	AUX
cana-1749	144	60	is	be	AUX
cana-1749	144	61	δgp	δgp	NOUN
cana-1749	144	62	-	-	PUNCT
cana-1749	144	63	co	co	NOUN
cana-1749	144	64	-	-	VERB
cana-1749	144	65	closed	closed	ADJ
cana-1749	144	66	in	in	ADP
cana-1749	144	67	the	the	DET
cana-1749	144	68	product	product	NOUN
cana-1749	144	69	space	space	NOUN
cana-1749	144	70	x×y	x×y	PROPN
cana-1749	144	71	.	.	PUNCT
cana-1749	145	1	proof	proof	NOUN
cana-1749	145	2	:	:	PUNCT
cana-1749	145	3	let	let	VERB
cana-1749	145	4	(	(	PUNCT
cana-1749	145	5	x	x	X
cana-1749	145	6	,	,	PUNCT
cana-1749	145	7	y)∈(x×y)-g(f	y)∈(x×y)-g(f	PROPN
cana-1749	145	8	)	)	PUNCT
cana-1749	145	9	,	,	PUNCT
cana-1749	145	10	then	then	ADV
cana-1749	145	11	f(x	f(x	PROPN
cana-1749	145	12	)	)	PUNCT
cana-1749	145	13	≠y	≠y	NOUN
cana-1749	145	14	and	and	CCONJ
cana-1749	145	15	there	there	PRON
cana-1749	145	16	exists	exist	VERB
cana-1749	145	17	c	c	NOUN
cana-1749	145	18	l	l	NOUN
cana-1749	145	19	o	o	NOUN
cana-1749	145	20	pen	pen	NOUN
cana-1749	145	21	sets	set	VERB
cana-1749	145	22	u	u	NOUN
cana-1749	145	23	and	and	CCONJ
cana-1749	145	24	v	v	ADP
cana-1749	145	25	such	such	ADJ
cana-1749	145	26	that	that	DET
cana-1749	145	27	f(x)∈u	f(x)∈u	NOUN
cana-1749	145	28	,	,	PUNCT
cana-1749	145	29	y∈v	y∈v	NOUN
cana-1749	145	30	and	and	CCONJ
cana-1749	145	31	u∩v=	u∩v=	NOUN
cana-1749	145	32	.	.	PUNCT
cana-1749	145	33	since	since	SCONJ
cana-1749	145	34	f	f	PROPN
cana-1749	145	35	is	be	AUX
cana-1749	145	36	slightly	slightly	ADV
cana-1749	145	37	δgp	δgp	ADJ
cana-1749	145	38	-	-	PUNCT
cana-1749	145	39	continuous	continuous	ADJ
cana-1749	145	40	,	,	PUNCT
cana-1749	145	41	then	then	ADV
cana-1749	145	42	there	there	PRON
cana-1749	145	43	exists	exist	VERB
cana-1749	145	44	a	a	DET
cana-1749	145	45	δgp	δgp	NOUN
cana-1749	145	46	-	-	PUNCT
cana-1749	145	47	open	open	NOUN
cana-1749	145	48	set	set	NOUN
cana-1749	145	49	g	g	PROPN
cana-1749	145	50	such	such	DET
cana-1749	145	51	that	that	DET
cana-1749	145	52	x∈g	x∈g	NOUN
cana-1749	145	53	and	and	CCONJ
cana-1749	145	54	f(g)⊂	f(g)⊂	PROPN
cana-1749	145	55	u	u	NOUN
cana-1749	145	56	and	and	CCONJ
cana-1749	145	57	hence	hence	ADV
cana-1749	145	58	we	we	PRON
cana-1749	145	59	obtain	obtain	VERB
cana-1749	145	60	f(g)∩v	f(g)∩v	ADJ
cana-1749	145	61	=	=	NOUN
cana-1749	145	62	.this	.this	PRON
cana-1749	145	63	shows	show	VERB
cana-1749	145	64	that	that	SCONJ
cana-1749	145	65	g(f	g(f	NOUN
cana-1749	145	66	)	)	PUNCT
cana-1749	145	67	is	be	AUX
cana-1749	145	68	is	be	AUX
cana-1749	145	69	δgp	δgp	NOUN
cana-1749	145	70	-	-	PUNCT
cana-1749	145	71	co	co	NOUN
cana-1749	145	72	-	-	ADJ
cana-1749	145	73	closed	closed	ADJ
cana-1749	145	74	.	.	PUNCT
cana-1749	146	1	theorem	theorem	VERB
cana-1749	146	2	2.26	2.26	NUM
cana-1749	146	3	if	if	SCONJ
cana-1749	146	4	f:(x	f:(x	PROPN
cana-1749	146	5	,	,	PUNCT
cana-1749	146	6	τ	τ	PROPN
cana-1749	146	7	)	)	PUNCT
cana-1749	146	8	→(y	→(y	PROPN
cana-1749	146	9	,	,	PUNCT
cana-1749	146	10	σ	σ	PROPN
cana-1749	146	11	)	)	PUNCT
cana-1749	146	12	is	be	AUX
cana-1749	146	13	slightly	slightly	ADV
cana-1749	146	14	δgp	δgp	ADJ
cana-1749	146	15	-	-	PUNCT
cana-1749	146	16	continuous	continuous	ADJ
cana-1749	146	17	and	and	CCONJ
cana-1749	146	18	y	y	PROPN
cana-1749	146	19	is	be	AUX
cana-1749	146	20	co	co	ADJ
cana-1749	146	21	-	-	NOUN
cana-1749	146	22	t1	t1	ADJ
cana-1749	146	23	,	,	PUNCT
cana-1749	146	24	then	then	ADV
cana-1749	146	25	g(f	g(f	PROPN
cana-1749	146	26	)	)	PUNCT
cana-1749	146	27	is	be	AUX
cana-1749	146	28	is	be	AUX
cana-1749	146	29	δgp	δgp	NOUN
cana-1749	146	30	-	-	PUNCT
cana-1749	146	31	co	co	NOUN
cana-1749	146	32	-	-	VERB
cana-1749	146	33	closed	closed	ADJ
cana-1749	146	34	in	in	ADP
cana-1749	146	35	the	the	DET
cana-1749	146	36	product	product	NOUN
cana-1749	146	37	space	space	NOUN
cana-1749	146	38	x×y	x×y	PROPN
cana-1749	146	39	.	.	PUNCT
cana-1749	147	1	proof	proof	NOUN
cana-1749	147	2	:	:	PUNCT
cana-1749	147	3	let	let	VERB
cana-1749	147	4	(	(	PUNCT
cana-1749	147	5	x	x	X
cana-1749	147	6	,	,	PUNCT
cana-1749	147	7	y)∈(x×y)-g(f	y)∈(x×y)-g(f	PROPN
cana-1749	147	8	)	)	PUNCT
cana-1749	147	9	,	,	PUNCT
cana-1749	147	10	then	then	ADV
cana-1749	147	11	f(x	f(x	PROPN
cana-1749	147	12	)	)	PUNCT
cana-1749	147	13	≠y	≠y	NOUN
cana-1749	147	14	and	and	CCONJ
cana-1749	147	15	there	there	PRON
cana-1749	147	16	exists	exist	VERB
cana-1749	147	17	a	a	DET
cana-1749	147	18	clopen	clopen	ADJ
cana-1749	147	19	sets	set	NOUN
cana-1749	147	20	m	m	VERB
cana-1749	147	21	o	o	NOUN
cana-1749	147	22	f	f	X
cana-1749	147	23	y	y	PROPN
cana-1749	147	24	such	such	ADJ
cana-1749	147	25	that	that	SCONJ
cana-1749	147	26	f(x)∈	f(x)∈	PROPN
cana-1749	147	27	m	m	VERB
cana-1749	147	28	and	and	CCONJ
cana-1749	147	29	y∉	y∉	PRON
cana-1749	147	30	m	m	NOUN
cana-1749	147	31	and	and	CCONJ
cana-1749	147	32	u∩v=	u∩v=	NOUN
cana-1749	147	33	.	.	PUNCT
cana-1749	147	34	since	since	SCONJ
cana-1749	147	35	f	f	PROPN
cana-1749	147	36	is	be	AUX
cana-1749	147	37	slightly	slightly	ADV
cana-1749	147	38	δgp	δgp	ADJ
cana-1749	147	39	-	-	PUNCT
cana-1749	147	40	continuous	continuous	ADJ
cana-1749	147	41	,	,	PUNCT
cana-1749	147	42	then	then	ADV
cana-1749	147	43	there	there	PRON
cana-1749	147	44	exists	exist	VERB
cana-1749	147	45	a	a	DET
cana-1749	147	46	δgpopen	δgpopen	NOUN
cana-1749	147	47	set	set	VERB
cana-1749	147	48	g	g	PROPN
cana-1749	147	49	such	such	DET
cana-1749	147	50	that	that	DET
cana-1749	147	51	x∈g	x∈g	NOUN
cana-1749	147	52	and	and	CCONJ
cana-1749	147	53	f(g)⊂	f(g)⊂	PROPN
cana-1749	147	54	m	m	PROPN
cana-1749	147	55	and	and	CCONJ
cana-1749	147	56	hence	hence	ADV
cana-1749	147	57	we	we	PRON
cana-1749	147	58	obtain	obtain	VERB
cana-1749	147	59	f(g)∩(y\m	f(g)∩(y\m	NOUN
cana-1749	147	60	)	)	PUNCT
cana-1749	148	1	=	=	SYM
cana-1749	148	2			NOUN
cana-1749	148	3	and	and	CCONJ
cana-1749	148	4	y\m	y\m	NOUN
cana-1749	148	5	is	be	AUX
cana-1749	148	6	clopen	clopen	ADJ
cana-1749	148	7	set	set	NOUN
cana-1749	148	8	containing	contain	VERB
cana-1749	148	9	y.this	y.this	DET
cana-1749	148	10	shows	show	NOUN
cana-1749	148	11	that	that	SCONJ
cana-1749	148	12	g(f	g(f	PROPN
cana-1749	148	13	)	)	PUNCT
cana-1749	148	14	is	be	AUX
cana-1749	148	15	δgp	δgp	NOUN
cana-1749	148	16	-	-	PUNCT
cana-1749	148	17	co	co	NOUN
cana-1749	148	18	-	-	ADJ
cana-1749	148	19	closed	closed	ADJ
cana-1749	148	20	.	.	PUNCT
cana-1749	149	1	references	reference	NOUN
cana-1749	149	2	:	:	PUNCT
cana-1749	150	1	[	[	X
cana-1749	150	2	1	1	NUM
cana-1749	150	3	]	]	PUNCT
cana-1749	150	4	c.w.baker	c.w.baker	NOUN
cana-1749	150	5	,	,	PUNCT
cana-1749	150	6	slightly	slightly	ADV
cana-1749	150	7	precontinuous	precontinuous	ADJ
cana-1749	150	8	functions	function	NOUN
cana-1749	150	9	,	,	PUNCT
cana-1749	150	10	acta	acta	PROPN
cana-1749	150	11	mathematica	mathematica	PROPN
cana-1749	150	12	hungarica	hungarica	PROPN
cana-1749	150	13	,	,	PUNCT
cana-1749	150	14	vol.94	vol.94	NOUN
cana-1749	150	15	,	,	PUNCT
cana-1749	150	16	pp.45	pp.45	NOUN
cana-1749	150	17	-	-	NOUN
cana-1749	150	18	52,2002	52,2002	NOUN
cana-1749	150	19	.	.	PUNCT
cana-1749	151	1	[	[	X
cana-1749	151	2	2	2	NUM
cana-1749	151	3	]	]	PUNCT
cana-1749	151	4	s.balasubramanian	s.balasubramanian	ADJ
cana-1749	151	5	and	and	CCONJ
cana-1749	151	6	m.	m.	NOUN
cana-1749	151	7	lakshmi	lakshmi	PROPN
cana-1749	151	8	sarada	sarada	NOUN
cana-1749	151	9	,	,	PUNCT
cana-1749	151	10	slightly	slightly	ADV
cana-1749	151	11	gpr	gpr	NOUN
cana-1749	151	12	-	-	PUNCT
cana-1749	151	13	continuous	continuous	ADJ
cana-1749	151	14	functions	function	NOUN
cana-1749	151	15	,	,	PUNCT
cana-1749	151	16	scientia	scientia	PROPN
cana-1749	151	17	magna	magna	PROPN
cana-1749	151	18	,	,	PUNCT
cana-1749	151	19	vol.7(3	vol.7(3	PROPN
cana-1749	151	20	)	)	PUNCT
cana-1749	151	21	,	,	PUNCT
cana-1749	151	22	pp	pp	ADP
cana-1749	151	23	.	.	PUNCT
cana-1749	152	1	4652,2011	4652,2011	X
cana-1749	152	2	.	.	PUNCT
cana-1749	153	1	[	[	X
cana-1749	153	2	3	3	X
cana-1749	153	3	]	]	X
cana-1749	153	4	s.s.benchalli	s.s.benchalli	PROPN
cana-1749	153	5	and	and	CCONJ
cana-1749	153	6	j.b.toranagatti	j.b.toranagatti	PROPN
cana-1749	153	7	,	,	PUNCT
cana-1749	153	8	delta	delta	NOUN
cana-1749	153	9	generalized	generalize	VERB
cana-1749	153	10	pre	pre	ADJ
cana-1749	153	11	-	-	ADJ
cana-1749	153	12	closed	closed	ADJ
cana-1749	153	13	sets	set	NOUN
cana-1749	153	14	in	in	ADP
cana-1749	153	15	topological	topological	ADJ
cana-1749	153	16	vspaces	vspace	NOUN
cana-1749	153	17	,	,	PUNCT
cana-1749	153	18	international	international	ADJ
cana-1749	153	19	journal	journal	NOUN
cana-1749	153	20	of	of	ADP
cana-1749	153	21	contemporary	contemporary	PROPN
cana-1749	153	22	mathematical	mathematical	PROPN
cana-1749	153	23	sciences	sciences	PROPN
cana-1749	153	24	,	,	PUNCT
cana-1749	153	25	vol.11,pp.281	vol.11,pp.281	NOUN
cana-1749	153	26	-	-	PUNCT
cana-1749	153	27	292,2016	292,2016	PROPN
cana-1749	153	28	.	.	PUNCT
cana-1749	154	1	communications	communication	NOUN
cana-1749	154	2	on	on	ADP
cana-1749	154	3	applied	apply	VERB
cana-1749	154	4	nonlinear	nonlinear	ADJ
cana-1749	154	5	analysis	analysis	NOUN
cana-1749	154	6	issn	issn	NOUN
cana-1749	154	7	:	:	PUNCT
cana-1749	154	8	1074	1074	NUM
cana-1749	154	9	-	-	PUNCT
cana-1749	154	10	133x	133x	NUM
cana-1749	154	11	vol	vol	NOUN
cana-1749	154	12	32	32	NUM
cana-1749	154	13	no	no	NOUN
cana-1749	154	14	.	.	NOUN
cana-1749	154	15	2	2	NUM
cana-1749	154	16	(	(	PUNCT
cana-1749	154	17	2025	2025	NUM
cana-1749	154	18	)	)	PUNCT
cana-1749	154	19	382	382	NUM
cana-1749	154	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1749	155	1	[	[	X
cana-1749	155	2	4	4	X
cana-1749	155	3	]	]	X
cana-1749	155	4	n.el	n.el	PROPN
cana-1749	155	5	-	-	PUNCT
cana-1749	155	6	deeb	deeb	PROPN
cana-1749	155	7	,	,	PUNCT
cana-1749	155	8	i.a.hasanein	i.a.hasanein	ADJ
cana-1749	155	9	,	,	PUNCT
cana-1749	155	10	a.s.mashhour	a.s.mashhour	ADJ
cana-1749	155	11	and	and	CCONJ
cana-1749	155	12	t.noiri	t.noiri	ADV
cana-1749	155	13	,	,	PUNCT
cana-1749	155	14	on	on	ADP
cana-1749	155	15	p	p	NOUN
cana-1749	155	16	-	-	PUNCT
cana-1749	155	17	regular	regular	ADJ
cana-1749	155	18	spaces	space	NOUN
cana-1749	155	19	,	,	PUNCT
cana-1749	155	20	bulletin	bulletin	NOUN
cana-1749	155	21	mathematique	mathematique	NOUN
cana-1749	155	22	de	de	PROPN
cana-1749	155	23	la	la	PROPN
cana-1749	155	24	societe	societe	PROPN
cana-1749	155	25	des	des	PROPN
cana-1749	155	26	sciences	sciences	PROPN
cana-1749	155	27	mathematiques	mathematiques	PROPN
cana-1749	155	28	de	de	PROPN
cana-1749	155	29	roumanie	roumanie	PROPN
cana-1749	155	30	,	,	PUNCT
cana-1749	155	31	vol.27	vol.27	PROPN
cana-1749	155	32	,	,	PUNCT
cana-1749	155	33	pp.311	pp.311	NOUN
cana-1749	155	34	-	-	PUNCT
cana-1749	155	35	315,1983	315,1983	NOUN
cana-1749	155	36	..	..	PUNCT
cana-1749	156	1	[	[	X
cana-1749	156	2	5	5	X
cana-1749	156	3	]	]	PUNCT
cana-1749	156	4	e.ekici	e.ekici	CCONJ
cana-1749	156	5	,	,	PUNCT
cana-1749	156	6	a	a	DET
cana-1749	156	7	weak	weak	ADJ
cana-1749	156	8	form	form	NOUN
cana-1749	156	9	of	of	ADP
cana-1749	156	10	continuity	continuity	NOUN
cana-1749	156	11	,	,	PUNCT
cana-1749	156	12	act	act	VERB
cana-1749	156	13	et	et	PROPN
cana-1749	156	14	commentationes	commentatione	VERB
cana-1749	156	15	universitatis	universitatis	PROPN
cana-1749	156	16	tartuensis	tartuensis	PROPN
cana-1749	156	17	de	de	X
cana-1749	156	18	mathematica	mathematica	PROPN
cana-1749	156	19	,	,	PUNCT
cana-1749	156	20	vol.9	vol.9	PROPN
cana-1749	156	21	pp.21	pp.21	PROPN
cana-1749	156	22	-	-	PUNCT
cana-1749	156	23	31,2005	31,2005	NUM
cana-1749	156	24	.	.	PUNCT
cana-1749	157	1	[	[	X
cana-1749	157	2	6	6	NUM
cana-1749	157	3	]	]	PUNCT
cana-1749	157	4	y.gnanambal	y.gnanambal	ADJ
cana-1749	157	5	,	,	PUNCT
cana-1749	157	6	on	on	ADP
cana-1749	157	7	generalized	generalized	ADJ
cana-1749	157	8	pre	pre	ADJ
cana-1749	157	9	-	-	ADJ
cana-1749	157	10	regular	regular	ADJ
cana-1749	157	11	closed	closed	ADJ
cana-1749	157	12	sets	set	NOUN
cana-1749	157	13	in	in	ADP
cana-1749	157	14	topological	topological	ADJ
cana-1749	157	15	spaces	space	NOUN
cana-1749	157	16	,	,	PUNCT
cana-1749	157	17	indian	indian	ADJ
cana-1749	157	18	journal	journal	NOUN
cana-1749	157	19	of	of	ADP
cana-1749	157	20	pure	pure	ADJ
cana-1749	157	21	and	and	CCONJ
cana-1749	157	22	applied	applied	ADJ
cana-1749	157	23	mathematics	mathematic	NOUN
cana-1749	157	24	,	,	PUNCT
cana-1749	157	25	vol.28	vol.28	NOUN
cana-1749	157	26	,	,	PUNCT
cana-1749	157	27	pp.351	pp.351	NOUN
cana-1749	157	28	-	-	PUNCT
cana-1749	157	29	360	360	NUM
cana-1749	157	30	,	,	PUNCT
cana-1749	157	31	1997	1997	NUM
cana-1749	157	32	.	.	PUNCT
cana-1749	158	1	[	[	X
cana-1749	158	2	7	7	X
cana-1749	158	3	]	]	X
cana-1749	158	4	r.m	r.m	PROPN
cana-1749	158	5	.	.	PROPN
cana-1749	158	6	latif	latif	PROPN
cana-1749	158	7	,	,	PUNCT
cana-1749	158	8	delta	delta	NOUN
cana-1749	158	9	-	-	PUNCT
cana-1749	158	10	open	open	ADJ
cana-1749	158	11	sets	set	NOUN
cana-1749	158	12	and	and	CCONJ
cana-1749	158	13	delta	delta	NOUN
cana-1749	158	14	-	-	PUNCT
cana-1749	158	15	continuous	continuous	ADJ
cana-1749	158	16	functions	function	NOUN
cana-1749	158	17	,	,	PUNCT
cana-1749	158	18	international	international	ADJ
cana-1749	158	19	journal	journal	NOUN
cana-1749	158	20	of	of	ADP
cana-1749	158	21	pure	pure	ADJ
cana-1749	158	22	mathematics	mathematic	NOUN
cana-1749	158	23	,	,	PUNCT
cana-1749	158	24	vol	vol	NOUN
cana-1749	158	25	.	.	PROPN
cana-1749	158	26	8	8	NUM
cana-1749	158	27	,	,	PUNCT
cana-1749	158	28	pp.122,2021	pp.122,2021	NOUN
cana-1749	158	29	.	.	PUNCT
cana-1749	159	1	[	[	X
cana-1749	159	2	8	8	NUM
cana-1749	159	3	]	]	X
cana-1749	159	4	r.	r.	PROPN
cana-1749	159	5	c.	c.	PROPN
cana-1749	159	6	jain	jain	PROPN
cana-1749	159	7	,	,	PUNCT
cana-1749	159	8	the	the	DET
cana-1749	159	9	role	role	NOUN
cana-1749	159	10	of	of	ADP
cana-1749	159	11	regularly	regularly	ADV
cana-1749	159	12	open	open	ADJ
cana-1749	159	13	sets	set	NOUN
cana-1749	159	14	in	in	ADP
cana-1749	159	15	general	general	ADJ
cana-1749	159	16	topology	topology	NOUN
cana-1749	159	17	,	,	PUNCT
cana-1749	159	18	ph	ph	PROPN
cana-1749	159	19	.	.	PROPN
cana-1749	159	20	d.	d.	PROPN
cana-1749	159	21	thesis	thesis	PROPN
cana-1749	159	22	,	,	PUNCT
cana-1749	159	23	meerut	meerut	PROPN
cana-1749	159	24	university	university	PROPN
cana-1749	159	25	,	,	PUNCT
cana-1749	159	26	institute	institute	NOUN
cana-1749	159	27	of	of	ADP
cana-1749	159	28	advanced	advanced	ADJ
cana-1749	159	29	studies	study	NOUN
cana-1749	159	30	,	,	PUNCT
cana-1749	159	31	meerut	meerut	PROPN
cana-1749	159	32	,	,	PUNCT
cana-1749	159	33	india	india	PROPN
cana-1749	159	34	1980	1980	NUM
cana-1749	159	35	[	[	X
cana-1749	159	36	9	9	NUM
cana-1749	159	37	]	]	X
cana-1749	159	38	h.	h.	NOUN
cana-1749	159	39	maki	maki	PROPN
cana-1749	159	40	,	,	PUNCT
cana-1749	159	41	j.	j.	PROPN
cana-1749	159	42	umehara	umehara	PROPN
cana-1749	159	43	and	and	CCONJ
cana-1749	159	44	t.	t.	PROPN
cana-1749	159	45	noiri	noiri	PROPN
cana-1749	159	46	,	,	PUNCT
cana-1749	159	47	every	every	DET
cana-1749	159	48	topological	topological	ADJ
cana-1749	159	49	space	space	NOUN
cana-1749	159	50	is	be	AUX
cana-1749	159	51	pre	pre	ADJ
cana-1749	159	52	-	-	ADJ
cana-1749	159	53	t1/2,,mem	t1/2,,mem	ADJ
cana-1749	159	54	.	.	PUNCT
cana-1749	160	1	faculty	faculty	NOUN
cana-1749	160	2	of	of	ADP
cana-1749	160	3	science	science	PROPN
cana-1749	160	4	kochi	kochi	PROPN
cana-1749	160	5	university	university	PROPN
cana-1749	160	6	series	series	NOUN
cana-1749	160	7	a	a	DET
cana-1749	160	8	mathematics	mathematic	NOUN
cana-1749	160	9	,	,	PUNCT
cana-1749	160	10	vol.17	vol.17	ADJ
cana-1749	160	11	,	,	PUNCT
cana-1749	160	12	pp.33	pp.33	PROPN
cana-1749	160	13	-	-	PUNCT
cana-1749	160	14	42,1996	42,1996	NOUN
cana-1749	160	15	.	.	PUNCT
cana-1749	161	1	[	[	X
cana-1749	161	2	10	10	NUM
cana-1749	161	3	]	]	PUNCT
cana-1749	161	4	a.	a.	NOUN
cana-1749	161	5	s.	s.	PROPN
cana-1749	161	6	mashhour	mashhour	PROPN
cana-1749	161	7	,	,	PUNCT
cana-1749	161	8	m.	m.	PROPN
cana-1749	161	9	e.	e.	PROPN
cana-1749	161	10	abd	abd	PROPN
cana-1749	162	1	el	el	PROPN
cana-1749	162	2	-	-	PROPN
cana-1749	162	3	monsef	monsef	PROPN
cana-1749	162	4	and	and	CCONJ
cana-1749	162	5	s.	s.	PROPN
cana-1749	162	6	n.	n.	PROPN
cana-1749	162	7	el	el	PROPN
cana-1749	162	8	-	-	PROPN
cana-1749	162	9	deeb	deeb	PROPN
cana-1749	162	10	,	,	PUNCT
cana-1749	162	11	on	on	ADP
cana-1749	162	12	pre	pre	ADJ
cana-1749	162	13	-	-	ADJ
cana-1749	162	14	continuous	continuous	ADJ
cana-1749	162	15	and	and	CCONJ
cana-1749	162	16	weak	weak	ADJ
cana-1749	162	17	pre	pre	ADJ
cana-1749	162	18	continuous	continuous	ADJ
cana-1749	162	19	mappings	mapping	NOUN
cana-1749	162	20	,	,	PUNCT
cana-1749	162	21	proceedings	proceeding	NOUN
cana-1749	162	22	of	of	ADP
cana-1749	162	23	the	the	DET
cana-1749	162	24	mathematical	mathematical	ADJ
cana-1749	162	25	and	and	CCONJ
cana-1749	162	26	physical	physical	ADJ
cana-1749	162	27	society	society	NOUN
cana-1749	162	28	of	of	ADP
cana-1749	162	29	egypt	egypt	PROPN
cana-1749	162	30	,	,	PUNCT
cana-1749	162	31	vol.53	vol.53	NOUN
cana-1749	162	32	,	,	PUNCT
cana-1749	162	33	pp.47	pp.47	NOUN
cana-1749	162	34	-	-	PUNCT
cana-1749	162	35	53,1982	53,1982	NUM
cana-1749	162	36	.	.	PUNCT
cana-1749	163	1	[	[	X
cana-1749	163	2	11	11	NUM
cana-1749	163	3	]	]	SYM
cana-1749	163	4	t.nieminen	t.nieminen	NUM
cana-1749	163	5	,	,	PUNCT
cana-1749	163	6	on	on	ADP
cana-1749	163	7	ultrapseudocompact	ultrapseudocompact	ADJ
cana-1749	163	8	and	and	CCONJ
cana-1749	163	9	related	related	ADJ
cana-1749	163	10	spaces	space	NOUN
cana-1749	163	11	,	,	PUNCT
cana-1749	163	12	annales	annale	VERB
cana-1749	163	13	academiae	academiae	PROPN
cana-1749	163	14	scientiarum	scientiarum	PROPN
cana-1749	163	15	fennicae	fennicae	PROPN
cana-1749	163	16	.	.	PUNCT
cana-1749	164	1	series	series	PROPN
cana-1749	164	2	a	a	DET
cana-1749	164	3	i.	i.	PROPN
cana-1749	164	4	mathematica	mathematica	PROPN
cana-1749	164	5	,	,	PUNCT
cana-1749	164	6	vol.3	vol.3	PROPN
cana-1749	164	7	,	,	PUNCT
cana-1749	164	8	pp	pp	ADJ
cana-1749	164	9	.	.	PUNCT
cana-1749	165	1	185	185	NUM
cana-1749	165	2	-	-	SYM
cana-1749	165	3	205,1977	205,1977	NUM
cana-1749	165	4	.	.	PUNCT
cana-1749	166	1	[	[	X
cana-1749	166	2	12	12	NUM
cana-1749	166	3	]	]	PUNCT
cana-1749	166	4	r.staum	r.staum	NOUN
cana-1749	166	5	,	,	PUNCT
cana-1749	166	6	the	the	DET
cana-1749	166	7	algebra	algebra	NOUN
cana-1749	166	8	of	of	ADP
cana-1749	166	9	bounded	bounded	ADJ
cana-1749	166	10	continuous	continuous	ADJ
cana-1749	166	11	functions	function	NOUN
cana-1749	166	12	into	into	ADP
cana-1749	166	13	a	a	DET
cana-1749	166	14	nonarchimedean	nonarchimedean	ADJ
cana-1749	166	15	field	field	NOUN
cana-1749	166	16	,	,	PUNCT
cana-1749	166	17	pacific	pacific	PROPN
cana-1749	166	18	journal	journal	NOUN
cana-1749	166	19	of	of	ADP
cana-1749	166	20	mathematics	mathematic	NOUN
cana-1749	166	21	,	,	PUNCT
cana-1749	166	22	vol.50	vol.50	NOUN
cana-1749	166	23	,	,	PUNCT
cana-1749	166	24	pp	pp	ADP
cana-1749	166	25	.	.	PUNCT
cana-1749	167	1	169	169	NUM
cana-1749	167	2	-	-	SYM
cana-1749	167	3	185,1974	185,1974	NUM
cana-1749	167	4	.	.	PUNCT
cana-1749	168	1	[	[	X
cana-1749	168	2	13	13	NUM
cana-1749	168	3	]	]	SYM
cana-1749	168	4	m.stone	m.stone	NUM
cana-1749	168	5	,	,	PUNCT
cana-1749	168	6	application	application	NOUN
cana-1749	168	7	of	of	ADP
cana-1749	168	8	the	the	DET
cana-1749	168	9	theory	theory	NOUN
cana-1749	168	10	of	of	ADP
cana-1749	168	11	boolean	boolean	ADJ
cana-1749	168	12	rings	ring	NOUN
cana-1749	168	13	to	to	ADP
cana-1749	168	14	general	general	ADJ
cana-1749	168	15	topology	topology	NOUN
cana-1749	168	16	,	,	PUNCT
cana-1749	168	17	the	the	DET
cana-1749	168	18	transactions	transaction	NOUN
cana-1749	168	19	of	of	ADP
cana-1749	168	20	the	the	DET
cana-1749	168	21	american	american	PROPN
cana-1749	168	22	mathematical	mathematical	PROPN
cana-1749	168	23	society	society	NOUN
cana-1749	168	24	,	,	PUNCT
cana-1749	168	25	vol.41,pp.371	vol.41,pp.371	PROPN
cana-1749	168	26	-	-	PUNCT
cana-1749	168	27	381,1937	381,1937	PROPN
cana-1749	168	28	.	.	PUNCT
cana-1749	169	1	[	[	X
cana-1749	169	2	14	14	NUM
cana-1749	169	3	]	]	PUNCT
cana-1749	169	4	i.l.reilly	i.l.reilly	ADV
cana-1749	169	5	and	and	CCONJ
cana-1749	169	6	m.k.vamanamurthy	m.k.vamanamurthy	ADJ
cana-1749	169	7	,	,	PUNCT
cana-1749	169	8	on	on	ADP
cana-1749	169	9	some	some	DET
cana-1749	169	10	quetions	quetion	NOUN
cana-1749	169	11	concerning	concern	VERB
cana-1749	169	12	preopen	preopen	ADJ
cana-1749	169	13	sets	set	NOUN
cana-1749	169	14	,	,	PUNCT
cana-1749	169	15	kyungpook	kyungpook	PROPN
cana-1749	169	16	mathematical	mathematical	ADJ
cana-1749	169	17	journal	journal	PROPN
cana-1749	169	18	,	,	PUNCT
cana-1749	169	19	vol.30	vol.30	VERB
cana-1749	169	20	,	,	PUNCT
cana-1749	169	21	pp	pp	ADV
cana-1749	169	22	87	87	NUM
cana-1749	169	23	-	-	SYM
cana-1749	169	24	93,1990	93,1990	NUM
cana-1749	169	25	.	.	PUNCT
cana-1749	170	1	[	[	X
cana-1749	170	2	15	15	NUM
cana-1749	170	3	]	]	X
cana-1749	170	4	j.b.toranagatti	j.b.toranagatti	PROPN
cana-1749	170	5	,	,	PUNCT
cana-1749	170	6	delta	delta	NOUN
cana-1749	170	7	generalized	generalize	VERB
cana-1749	170	8	pre	pre	ADJ
cana-1749	170	9	-	-	ADJ
cana-1749	170	10	continuous	continuous	ADJ
cana-1749	170	11	functions	function	NOUN
cana-1749	170	12	in	in	ADP
cana-1749	170	13	topological	topological	ADJ
cana-1749	170	14	spaces	space	NOUN
cana-1749	170	15	,	,	PUNCT
cana-1749	170	16	international	international	ADJ
cana-1749	170	17	journal	journal	NOUN
cana-1749	170	18	of	of	ADP
cana-1749	170	19	pure	pure	ADJ
cana-1749	170	20	and	and	CCONJ
cana-1749	170	21	applied	applied	ADJ
cana-1749	170	22	mathematics	mathematic	NOUN
cana-1749	170	23	,	,	PUNCT
cana-1749	170	24	vol.116	vol.116	NOUN
cana-1749	170	25	,	,	PUNCT
cana-1749	170	26	pp.829	pp.829	NOUN
cana-1749	170	27	-	-	PUNCT
cana-1749	170	28	843,2017	843,2017	NUM
cana-1749	170	29	.	.	PUNCT
cana-1749	171	1	[	[	X
cana-1749	171	2	16	16	NUM
cana-1749	171	3	]	]	X
cana-1749	171	4	j.b.toranagatti	j.b.toranagatti	PROPN
cana-1749	171	5	,	,	PUNCT
cana-1749	171	6	on	on	ADP
cana-1749	171	7	contra	contra	PROPN
cana-1749	171	8	delta	delta	PROPN
cana-1749	171	9	generalized	generalize	VERB
cana-1749	171	10	pre	pre	ADJ
cana-1749	171	11	-	-	ADJ
cana-1749	171	12	continuous	continuous	ADJ
cana-1749	171	13	functions	function	NOUN
cana-1749	171	14	,	,	PUNCT
cana-1749	171	15	international	international	ADJ
cana-1749	171	16	journal	journal	NOUN
cana-1749	171	17	of	of	ADP
cana-1749	171	18	scientific	scientific	ADJ
cana-1749	171	19	research	research	NOUN
cana-1749	171	20	in	in	ADP
cana-1749	171	21	mathematical	mathematical	ADJ
cana-1749	171	22	and	and	CCONJ
cana-1749	171	23	statistical	statistical	ADJ
cana-1749	171	24	sciences	science	NOUN
cana-1749	171	25	,	,	PUNCT
cana-1749	171	26	vol.5(1	vol.5(1	NOUN
cana-1749	171	27	)	)	PUNCT
cana-1749	171	28	,	,	PUNCT
cana-1749	171	29	pp.283288	pp.283288	PROPN
cana-1749	171	30	,	,	PUNCT
cana-1749	171	31	2018	2018	NUM
cana-1749	171	32	.	.	PUNCT
cana-1749	172	1	[	[	X
cana-1749	172	2	17	17	NUM
cana-1749	172	3	]	]	X
cana-1749	172	4	j.b.toranagatti	j.b.toranagatti	PROPN
cana-1749	172	5	,	,	PUNCT
cana-1749	172	6	a	a	DET
cana-1749	172	7	new	new	ADJ
cana-1749	172	8	class	class	NOUN
cana-1749	172	9	of	of	ADP
cana-1749	172	10	continuous	continuous	ADJ
cana-1749	172	11	functions	function	NOUN
cana-1749	172	12	via	via	ADP
cana-1749	172	13	δgp	δgp	PROPN
cana-1749	172	14	–	–	PUNCT
cana-1749	172	15	open	open	ADJ
cana-1749	172	16	sets	set	NOUN
cana-1749	172	17	,	,	PUNCT
cana-1749	172	18	the	the	DET
cana-1749	172	19	aligarh	aligarh	NOUN
cana-1749	172	20	bulletin	bulletin	NOUN
cana-1749	172	21	of	of	ADP
cana-1749	172	22	mathematics	mathematic	NOUN
cana-1749	172	23	,	,	PUNCT
cana-1749	172	24	vol.39(2).pp.103	vol.39(2).pp.103	PROPN
cana-1749	172	25	-	-	PUNCT
cana-1749	172	26	117,2020	117,2020	NUM
cana-1749	172	27	.	.	PUNCT
cana-1749	173	1	[	[	X
cana-1749	173	2	18	18	NUM
cana-1749	173	3	]	]	PUNCT
cana-1749	173	4	n.v.veliko	n.v.veliko	NOUN
cana-1749	173	5	,	,	PUNCT
cana-1749	173	6	h	h	NOUN
cana-1749	173	7	-	-	PUNCT
cana-1749	173	8	closed	closed	ADJ
cana-1749	173	9	topological	topological	ADJ
cana-1749	173	10	spaces	space	NOUN
cana-1749	173	11	,	,	PUNCT
cana-1749	173	12	american	american	PROPN
cana-1749	173	13	mathematical	mathematical	ADJ
cana-1749	173	14	society	society	NOUN
cana-1749	173	15	translations	translation	NOUN
cana-1749	173	16	,	,	PUNCT
cana-1749	173	17	vol.78	vol.78	NOUN
cana-1749	173	18	,	,	PUNCT
cana-1749	173	19	pp.103	pp.103	NOUN
cana-1749	173	20	-	-	ADJ
cana-1749	173	21	118,1968	118,1968	NUM
cana-1749	173	22	.	.	PUNCT