id	sid	tid	token	lemma	pos
cana-1907	1	1	communications	communication	NOUN
cana-1907	1	2	on	on	ADP
cana-1907	1	3	applied	apply	VERB
cana-1907	1	4	nonlinear	nonlinear	ADJ
cana-1907	1	5	analysis	analysis	NOUN
cana-1907	1	6	issn	issn	NOUN
cana-1907	1	7	:	:	PUNCT
cana-1907	1	8	1074	1074	NUM
cana-1907	1	9	-	-	PUNCT
cana-1907	1	10	133x	133x	NUM
cana-1907	1	11	vol	vol	NOUN
cana-1907	1	12	32	32	NUM
cana-1907	1	13	no	no	NOUN
cana-1907	1	14	.	.	NOUN
cana-1907	1	15	2	2	NUM
cana-1907	1	16	(	(	PUNCT
cana-1907	1	17	2025	2025	NUM
cana-1907	1	18	)	)	PUNCT
cana-1907	1	19	52	52	NUM
cana-1907	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1907	1	21	stability	stability	NOUN
cana-1907	1	22	,	,	PUNCT
cana-1907	1	23	and	and	CCONJ
cana-1907	1	24	almost	almost	ADV
cana-1907	1	25	sensitivity	sensitivity	NOUN
cana-1907	1	26	of	of	ADP
cana-1907	1	27	induced	induced	ADJ
cana-1907	1	28	maps	map	NOUN
cana-1907	1	29	amalraj	amalraj	PROPN
cana-1907	1	30	.	.	PUNCT
cana-1907	2	1	p1	p1	PROPN
cana-1907	2	2	*	*	PROPN
cana-1907	2	3	,	,	PUNCT
cana-1907	2	4	p.b.vinod	p.b.vinod	VERB
cana-1907	2	5	kumar2	kumar2	X
cana-1907	2	6	1*research	1*research	NUM
cana-1907	2	7	scholar	scholar	NOUN
cana-1907	2	8	,	,	PUNCT
cana-1907	2	9	apj	apj	PROPN
cana-1907	2	10	abdul	abdul	PROPN
cana-1907	2	11	kalam	kalam	PROPN
cana-1907	2	12	technological	technological	PROPN
cana-1907	2	13	university	university	NOUN
cana-1907	2	14	,	,	PUNCT
cana-1907	2	15	(	(	PUNCT
cana-1907	2	16	department	department	NOUN
cana-1907	2	17	of	of	ADP
cana-1907	2	18	mathematics	mathematics	PROPN
cana-1907	2	19	,	,	PUNCT
cana-1907	2	20	sanatana	sanatana	PROPN
cana-1907	2	21	dharma	dharma	PROPN
cana-1907	2	22	college	college	PROPN
cana-1907	2	23	,	,	PUNCT
cana-1907	2	24	alappuzha	alappuzha	PROPN
cana-1907	2	25	,	,	PUNCT
cana-1907	2	26	kerala	kerala	PROPN
cana-1907	2	27	,	,	PUNCT
cana-1907	2	28	india	india	PROPN
cana-1907	2	29	)	)	PUNCT
cana-1907	2	30	email	email	NOUN
cana-1907	2	31	address	address	NOUN
cana-1907	2	32	:	:	PUNCT
cana-1907	2	33	amalrp2929@gmail.com	amalrp2929@gmail.com	X
cana-1907	3	1	2research	2research	NUM
cana-1907	3	2	supervisor	supervisor	NOUN
cana-1907	3	3	,	,	PUNCT
cana-1907	3	4	apj	apj	PROPN
cana-1907	3	5	abdul	abdul	PROPN
cana-1907	3	6	kalam	kalam	PROPN
cana-1907	3	7	technological	technological	PROPN
cana-1907	3	8	university	university	NOUN
cana-1907	3	9	,	,	PUNCT
cana-1907	3	10	(	(	PUNCT
cana-1907	3	11	department	department	NOUN
cana-1907	3	12	of	of	ADP
cana-1907	3	13	mathematics	mathematic	NOUN
cana-1907	3	14	,	,	PUNCT
cana-1907	3	15	rajagiri	rajagiri	NOUN
cana-1907	3	16	school	school	NOUN
cana-1907	3	17	of	of	ADP
cana-1907	3	18	engineering	engineering	NOUN
cana-1907	3	19	and	and	CCONJ
cana-1907	3	20	technology	technology	NOUN
cana-1907	3	21	,	,	PUNCT
cana-1907	3	22	cochin	cochin	PROPN
cana-1907	3	23	,	,	PUNCT
cana-1907	3	24	kerala	kerala	PROPN
cana-1907	3	25	,	,	PUNCT
cana-1907	3	26	india	india	PROPN
cana-1907	3	27	)	)	PUNCT
cana-1907	3	28	,	,	PUNCT
cana-1907	3	29	email	email	NOUN
cana-1907	3	30	address	address	NOUN
cana-1907	3	31	:	:	PUNCT
cana-1907	3	32	vinodkumar.rajagiri@gmail.com	vinodkumar.rajagiri@gmail.com	X
cana-1907	3	33	article	article	NOUN
cana-1907	3	34	history	history	NOUN
cana-1907	3	35	:	:	PUNCT
cana-1907	3	36	received	receive	VERB
cana-1907	3	37	:	:	PUNCT
cana-1907	3	38	22	22	NUM
cana-1907	3	39	-	-	SYM
cana-1907	3	40	07	07	NUM
cana-1907	3	41	-	-	PUNCT
cana-1907	3	42	2024	2024	NUM
cana-1907	3	43	revised	revise	VERB
cana-1907	3	44	:	:	PUNCT
cana-1907	3	45	09	09	NUM
cana-1907	3	46	-	-	SYM
cana-1907	3	47	09	09	NUM
cana-1907	3	48	-	-	PUNCT
cana-1907	3	49	2024	2024	NUM
cana-1907	3	50	accepted	accept	VERB
cana-1907	3	51	:	:	PUNCT
cana-1907	3	52	29	29	NUM
cana-1907	3	53	-	-	SYM
cana-1907	3	54	09	09	NUM
cana-1907	3	55	-	-	PUNCT
cana-1907	3	56	2024	2024	NUM
cana-1907	3	57	abstract	abstract	NOUN
cana-1907	3	58	:	:	PUNCT
cana-1907	3	59	suppose	suppose	VERB
cana-1907	3	60	that	that	SCONJ
cana-1907	3	61	x	x	PRON
cana-1907	3	62	is	be	AUX
cana-1907	3	63	a	a	DET
cana-1907	3	64	compact	compact	ADJ
cana-1907	3	65	hausdorff	hausdorff	NOUN
cana-1907	3	66	space	space	NOUN
cana-1907	3	67	and	and	CCONJ
cana-1907	3	68	f	f	NOUN
cana-1907	3	69	:	:	PUNCT
cana-1907	3	70	x	x	X
cana-1907	3	71	→	→	PUNCT
cana-1907	3	72	x	x	X
cana-1907	3	73	is	be	AUX
cana-1907	3	74	continuous	continuous	ADJ
cana-1907	3	75	.	.	PUNCT
cana-1907	4	1	we	we	PRON
cana-1907	4	2	consider	consider	VERB
cana-1907	4	3	the	the	DET
cana-1907	4	4	space	space	NOUN
cana-1907	4	5	k(x	k(x	PROPN
cana-1907	4	6	)	)	PUNCT
cana-1907	4	7	,	,	PUNCT
cana-1907	4	8	the	the	DET
cana-1907	4	9	space	space	NOUN
cana-1907	4	10	of	of	ADP
cana-1907	4	11	all	all	DET
cana-1907	4	12	compact	compact	ADJ
cana-1907	4	13	subsets	subset	NOUN
cana-1907	4	14	of	of	ADP
cana-1907	4	15	x	x	PUNCT
cana-1907	4	16	with	with	ADP
cana-1907	4	17	hausdorff	hausdorff	PROPN
cana-1907	4	18	metric	metric	PROPN
cana-1907	4	19	h.	h.	PROPN
cana-1907	4	20	let	let	VERB
cana-1907	4	21	f˜	f˜	VERB
cana-1907	4	22	:	:	PUNCT
cana-1907	4	23	k(x	k(x	PROPN
cana-1907	4	24	)	)	PUNCT
cana-1907	5	1	→	→	SYM
cana-1907	5	2	k(x	k(x	PROPN
cana-1907	5	3	)	)	PUNCT
cana-1907	5	4	defined	define	VERB
cana-1907	5	5	by	by	ADP
cana-1907	5	6	f	f	PROPN
cana-1907	5	7	̃(k)=f(k	̃(k)=f(k	X
cana-1907	5	8	)	)	PUNCT
cana-1907	5	9	.	.	PUNCT
cana-1907	6	1	we	we	PRON
cana-1907	6	2	discuss	discuss	VERB
cana-1907	6	3	some	some	DET
cana-1907	6	4	interconnections	interconnection	NOUN
cana-1907	6	5	between	between	ADP
cana-1907	6	6	the	the	DET
cana-1907	6	7	orbit	orbit	NOUN
cana-1907	6	8	of	of	ADP
cana-1907	6	9	f	f	PROPN
cana-1907	6	10	̃and	̃and	PROPN
cana-1907	6	11	the	the	DET
cana-1907	6	12	orbit	orbit	NOUN
cana-1907	6	13	of	of	ADP
cana-1907	6	14	f.	f.	PROPN
cana-1907	6	15	by	by	ADP
cana-1907	6	16	assuming	assume	VERB
cana-1907	6	17	the	the	DET
cana-1907	6	18	transitivity	transitivity	NOUN
cana-1907	6	19	of	of	ADP
cana-1907	6	20	f	f	PROPN
cana-1907	6	21	̃	̃	PROPN
cana-1907	6	22	,	,	PUNCT
cana-1907	6	23	we	we	PRON
cana-1907	6	24	conclude	conclude	VERB
cana-1907	6	25	that	that	SCONJ
cana-1907	6	26	x	x	PUNCT
cana-1907	6	27	contains	contain	VERB
cana-1907	6	28	a	a	DET
cana-1907	6	29	cantor	cantor	NOUN
cana-1907	6	30	set	set	NOUN
cana-1907	6	31	c	c	PROPN
cana-1907	6	32	with	with	ADP
cana-1907	6	33	(	(	PUNCT
cana-1907	6	34	orb(f	orb(f	PROPN
cana-1907	6	35	̃,c	̃,c	NOUN
cana-1907	6	36	)	)	PUNCT
cana-1907	6	37	)	)	PUNCT
cana-1907	6	38	̅	̅	NOUN
cana-1907	6	39	=	=	SYM
cana-1907	6	40	k(x	k(x	PROPN
cana-1907	6	41	)	)	PUNCT
cana-1907	6	42	.	.	PUNCT
cana-1907	7	1	that	that	PRON
cana-1907	7	2	is	be	AUX
cana-1907	7	3	,	,	PUNCT
cana-1907	7	4	orbit	orbit	NOUN
cana-1907	7	5	of	of	ADP
cana-1907	7	6	a	a	DET
cana-1907	7	7	nowhere	nowhere	ADV
cana-1907	7	8	dense	dense	ADJ
cana-1907	7	9	set	set	NOUN
cana-1907	7	10	is	be	AUX
cana-1907	7	11	dense	dense	ADJ
cana-1907	7	12	in	in	ADP
cana-1907	7	13	the	the	DET
cana-1907	7	14	hyperspace	hyperspace	NOUN
cana-1907	7	15	.	.	PUNCT
cana-1907	8	1	along	along	ADP
cana-1907	8	2	with	with	ADP
cana-1907	8	3	this	this	PRON
cana-1907	8	4	we	we	PRON
cana-1907	8	5	introduce	introduce	VERB
cana-1907	8	6	”	"	PUNCT
cana-1907	8	7	almost	almost	ADV
cana-1907	8	8	sensitivity	sensitivity	NOUN
cana-1907	8	9	”	"	PUNCT
cana-1907	8	10	and	and	CCONJ
cana-1907	8	11	”	"	PUNCT
cana-1907	8	12	stability	stability	NOUN
cana-1907	8	13	”	"	PUNCT
cana-1907	8	14	in	in	ADP
cana-1907	8	15	k(x	k(x	PROPN
cana-1907	8	16	)	)	PUNCT
cana-1907	8	17	.	.	PUNCT
cana-1907	9	1	we	we	PRON
cana-1907	9	2	prove	prove	VERB
cana-1907	9	3	that	that	SCONJ
cana-1907	9	4	f	f	PROPN
cana-1907	9	5	is	be	AUX
cana-1907	9	6	stable	stable	ADJ
cana-1907	9	7	in	in	ADP
cana-1907	9	8	x	x	SYM
cana-1907	9	9	if	if	SCONJ
cana-1907	10	1	and	and	CCONJ
cana-1907	10	2	only	only	ADV
cana-1907	10	3	if	if	SCONJ
cana-1907	10	4	f	f	PROPN
cana-1907	10	5	̃	̃	PROPN
cana-1907	10	6	is	be	AUX
cana-1907	10	7	stable	stable	ADJ
cana-1907	10	8	in	in	ADP
cana-1907	10	9	k(x	k(x	PROPN
cana-1907	10	10	)	)	PUNCT
cana-1907	10	11	.	.	PUNCT
cana-1907	11	1	again	again	ADV
cana-1907	11	2	we	we	PRON
cana-1907	11	3	prove	prove	VERB
cana-1907	11	4	that	that	SCONJ
cana-1907	11	5	transitive	transitive	ADJ
cana-1907	11	6	maps	map	NOUN
cana-1907	11	7	are	be	AUX
cana-1907	11	8	always	always	ADV
cana-1907	11	9	’	'	PUNCT
cana-1907	11	10	almost	almost	ADV
cana-1907	11	11	sensitive	sensitive	ADJ
cana-1907	11	12	’	'	PUNCT
cana-1907	11	13	in	in	ADP
cana-1907	11	14	k(x	k(x	PROPN
cana-1907	11	15	)	)	PUNCT
cana-1907	11	16	and	and	CCONJ
cana-1907	11	17	hence	hence	ADV
cana-1907	11	18	the	the	DET
cana-1907	11	19	base	base	NOUN
cana-1907	11	20	map	map	NOUN
cana-1907	11	21	is	be	AUX
cana-1907	11	22	’	'	PUNCT
cana-1907	11	23	almost	almost	ADV
cana-1907	11	24	sensitive	sensitive	ADJ
cana-1907	11	25	’	'	PUNCT
cana-1907	11	26	in	in	ADP
cana-1907	11	27	x.	x.	NOUN
cana-1907	11	28	keywords	keyword	NOUN
cana-1907	11	29	:	:	PUNCT
cana-1907	12	1	maps	map	NOUN
cana-1907	12	2	,	,	PUNCT
cana-1907	12	3	continuous	continuous	ADJ
cana-1907	12	4	map	map	NOUN
cana-1907	12	5	,	,	PUNCT
cana-1907	12	6	compact	compact	ADJ
cana-1907	12	7	metric	metric	ADJ
cana-1907	12	8	space	space	NOUN
cana-1907	12	9	.	.	PUNCT
cana-1907	13	1	1	1	X
cana-1907	13	2	.	.	X
cana-1907	13	3	introduction	introduction	NOUN
cana-1907	13	4	normally	normally	ADV
cana-1907	13	5	,	,	PUNCT
cana-1907	13	6	for	for	ADP
cana-1907	13	7	studying	study	VERB
cana-1907	13	8	the	the	DET
cana-1907	13	9	dynamics	dynamic	NOUN
cana-1907	13	10	of	of	ADP
cana-1907	13	11	the	the	DET
cana-1907	13	12	hyperspace	hyperspace	NOUN
cana-1907	13	13	of	of	ADP
cana-1907	13	14	a	a	DET
cana-1907	13	15	compact	compact	ADJ
cana-1907	13	16	metric	metric	ADJ
cana-1907	13	17	space	space	NOUN
cana-1907	13	18	,	,	PUNCT
cana-1907	13	19	we	we	PRON
cana-1907	13	20	consider	consider	VERB
cana-1907	13	21	a	a	DET
cana-1907	13	22	dynamical	dynamical	ADJ
cana-1907	13	23	system	system	NOUN
cana-1907	13	24	defined	define	VERB
cana-1907	13	25	by	by	ADP
cana-1907	13	26	a	a	DET
cana-1907	13	27	continuous	continuous	ADJ
cana-1907	13	28	map	map	NOUN
cana-1907	14	1	f	f	NOUN
cana-1907	14	2	:	:	PUNCT
cana-1907	14	3	x	x	X
cana-1907	14	4	→	→	SYM
cana-1907	14	5	x	x	X
cana-1907	14	6	,	,	PUNCT
cana-1907	14	7	describing	describe	VERB
cana-1907	14	8	the	the	DET
cana-1907	14	9	dynamics	dynamic	NOUN
cana-1907	14	10	of	of	ADP
cana-1907	14	11	points	point	NOUN
cana-1907	14	12	in	in	ADP
cana-1907	14	13	the	the	DET
cana-1907	14	14	base	base	NOUN
cana-1907	14	15	space	space	NOUN
cana-1907	14	16	x	x	PUNCT
cana-1907	15	1	and	and	CCONJ
cana-1907	15	2	then	then	ADV
cana-1907	15	3	we	we	PRON
cana-1907	15	4	study	study	VERB
cana-1907	15	5	the	the	DET
cana-1907	15	6	induced	induced	ADJ
cana-1907	15	7	map	map	NOUN
cana-1907	15	8	𝑓	𝑓	PRON
cana-1907	15	9	:	:	PUNCT
cana-1907	15	10	k(x	k(x	PROPN
cana-1907	15	11	)	)	PUNCT
cana-1907	15	12	→	→	SYM
cana-1907	15	13	k(x	k(x	PROPN
cana-1907	15	14	)	)	PUNCT
cana-1907	15	15	defined	define	VERB
cana-1907	15	16	by	by	ADP
cana-1907	15	17	𝑓(𝐾	𝑓(𝐾	ADJ
cana-1907	15	18	)	)	PUNCT
cana-1907	15	19	=	=	SYM
cana-1907	15	20	𝑓(𝐾	𝑓(𝐾	NOUN
cana-1907	15	21	)	)	PUNCT
cana-1907	15	22	for	for	ADP
cana-1907	15	23	a	a	DET
cana-1907	15	24	compact	compact	ADJ
cana-1907	15	25	set	set	NOUN
cana-1907	15	26	k	k	PROPN
cana-1907	15	27	⊆	⊆	NUM
cana-1907	15	28	x	x	PUNCT
cana-1907	15	29	as	as	ADP
cana-1907	15	30	a	a	DET
cana-1907	15	31	form	form	NOUN
cana-1907	15	32	of	of	ADP
cana-1907	15	33	collective	collective	ADJ
cana-1907	15	34	dynamics	dynamic	NOUN
cana-1907	15	35	.	.	PUNCT
cana-1907	16	1	in	in	ADP
cana-1907	16	2	this	this	DET
cana-1907	16	3	context	context	NOUN
cana-1907	16	4	,	,	PUNCT
cana-1907	16	5	a	a	DET
cana-1907	16	6	very	very	ADV
cana-1907	16	7	natural	natural	ADJ
cana-1907	16	8	question	question	NOUN
cana-1907	16	9	arises	arise	VERB
cana-1907	16	10	:	:	PUNCT
cana-1907	16	11	what	what	PRON
cana-1907	16	12	is	be	AUX
cana-1907	16	13	the	the	DET
cana-1907	16	14	connection	connection	NOUN
cana-1907	16	15	between	between	ADP
cana-1907	16	16	dynamical	dynamical	ADJ
cana-1907	16	17	properties	property	NOUN
cana-1907	16	18	of	of	ADP
cana-1907	16	19	the	the	DET
cana-1907	16	20	base	base	NOUN
cana-1907	16	21	map	map	NOUN
cana-1907	16	22	f	f	PROPN
cana-1907	16	23	and	and	CCONJ
cana-1907	16	24	the	the	DET
cana-1907	16	25	induced	induced	ADJ
cana-1907	16	26	map	map	NOUN
cana-1907	16	27	𝑓	𝑓	PRON
cana-1907	16	28	?	?	PUNCT
cana-1907	17	1	during	during	ADP
cana-1907	17	2	the	the	DET
cana-1907	17	3	past	past	ADJ
cana-1907	17	4	years	year	NOUN
cana-1907	17	5	this	this	DET
cana-1907	17	6	question	question	NOUN
cana-1907	17	7	has	have	AUX
cana-1907	17	8	attracted	attract	VERB
cana-1907	17	9	many	many	ADJ
cana-1907	17	10	researchers	researcher	NOUN
cana-1907	17	11	.	.	PUNCT
cana-1907	18	1	(	(	PUNCT
cana-1907	18	2	see[4],[5],[6],[7],[8	see[4],[5],[6],[7],[8	NOUN
cana-1907	18	3	]	]	X
cana-1907	18	4	,	,	PUNCT
cana-1907	18	5	[	[	X
cana-1907	18	6	9],[10	9],[10	X
cana-1907	18	7	]	]	PUNCT
cana-1907	18	8	and	and	CCONJ
cana-1907	18	9	[	[	X
cana-1907	18	10	11	11	NUM
cana-1907	18	11	]	]	PUNCT
cana-1907	18	12	)	)	PUNCT
cana-1907	18	13	in	in	ADP
cana-1907	18	14	this	this	DET
cana-1907	18	15	paper	paper	NOUN
cana-1907	18	16	,	,	PUNCT
cana-1907	18	17	we	we	PRON
cana-1907	18	18	prove	prove	VERB
cana-1907	18	19	that	that	SCONJ
cana-1907	18	20	there	there	PRON
cana-1907	18	21	exists	exist	VERB
cana-1907	18	22	a	a	DET
cana-1907	18	23	cantor	cantor	NOUN
cana-1907	18	24	set	set	NOUN
cana-1907	18	25	c	c	PROPN
cana-1907	18	26	in	in	ADP
cana-1907	18	27	x	x	NOUN
cana-1907	18	28	,	,	PUNCT
cana-1907	18	29	nowhere	nowhere	ADV
cana-1907	18	30	dense	dense	ADJ
cana-1907	18	31	in	in	ADP
cana-1907	18	32	x	x	SYM
cana-1907	18	33	but	but	CCONJ
cana-1907	18	34	its	its	PRON
cana-1907	18	35	orbit	orbit	NOUN
cana-1907	18	36	is	be	AUX
cana-1907	18	37	dense	dense	ADJ
cana-1907	18	38	in	in	ADP
cana-1907	18	39	k(x	k(x	PROPN
cana-1907	18	40	)	)	PUNCT
cana-1907	18	41	.	.	PUNCT
cana-1907	19	1	at	at	ADP
cana-1907	19	2	the	the	DET
cana-1907	19	3	same	same	ADJ
cana-1907	19	4	time	time	NOUN
cana-1907	19	5	we	we	PRON
cana-1907	19	6	establish	establish	VERB
cana-1907	19	7	the	the	DET
cana-1907	19	8	fact	fact	NOUN
cana-1907	19	9	that	that	SCONJ
cana-1907	19	10	the	the	DET
cana-1907	19	11	set	set	NOUN
cana-1907	19	12	𝐷	𝐷	NOUN
cana-1907	19	13	=	=	PUNCT
cana-1907	19	14	{	{	PUNCT
cana-1907	19	15	𝑥	𝑥	PRON
cana-1907	19	16	∈	∈	PROPN
cana-1907	19	17	𝑋	𝑋	PROPN
cana-1907	19	18	,	,	PUNCT
cana-1907	19	19	𝑜𝑟𝑏(𝑓	𝑜𝑟𝑏(𝑓	PROPN
cana-1907	19	20	,	,	PUNCT
cana-1907	19	21	𝑥)̅̅	𝑥)̅̅	PROPN
cana-1907	19	22	̅̅	̅̅	PROPN
cana-1907	19	23	̅̅	̅̅	PROPN
cana-1907	19	24	̅̅	̅̅	PROPN
cana-1907	19	25	̅̅	̅̅	PROPN
cana-1907	19	26	̅̅	̅̅	PROPN
cana-1907	19	27	=	=	PROPN
cana-1907	19	28	𝑋	𝑋	PROPN
cana-1907	19	29	}	}	PUNCT
cana-1907	19	30	,	,	PUNCT
cana-1907	19	31	𝐷	𝐷	PROPN
cana-1907	19	32	≠	≠	PROPN
cana-1907	19	33	𝑋	𝑋	NOUN
cana-1907	19	34	is	be	AUX
cana-1907	19	35	dense	dense	ADJ
cana-1907	19	36	in	in	ADP
cana-1907	19	37	x	x	PUNCT
cana-1907	20	1	and	and	CCONJ
cana-1907	20	2	so	so	ADV
cana-1907	20	3	it	it	PRON
cana-1907	20	4	is	be	AUX
cana-1907	20	5	not	not	PART
cana-1907	20	6	closed	closed	ADJ
cana-1907	20	7	and	and	CCONJ
cana-1907	20	8	there	there	ADV
cana-1907	20	9	fore	fore	NOUN
cana-1907	20	10	𝐷	𝐷	PROPN
cana-1907	20	11	≠	≠	PROPN
cana-1907	20	12	𝐾(𝑋	𝐾(𝑋	NOUN
cana-1907	20	13	)	)	PUNCT
cana-1907	20	14	.	.	PUNCT
cana-1907	21	1	we	we	PRON
cana-1907	21	2	also	also	ADV
cana-1907	21	3	prove	prove	VERB
cana-1907	21	4	that	that	SCONJ
cana-1907	21	5	f	f	PROPN
cana-1907	21	6	is	be	AUX
cana-1907	21	7	’	'	PUNCT
cana-1907	21	8	stable	stable	ADJ
cana-1907	21	9	’	'	PUNCT
cana-1907	21	10	if	if	SCONJ
cana-1907	22	1	and	and	CCONJ
cana-1907	22	2	only	only	ADV
cana-1907	22	3	if	if	SCONJ
cana-1907	22	4	𝑓	𝑓	PRON
cana-1907	22	5	is	be	AUX
cana-1907	22	6	stable	stable	ADJ
cana-1907	22	7	and	and	CCONJ
cana-1907	22	8	𝑓	𝑓	PRON
cana-1907	22	9	is	be	AUX
cana-1907	22	10	’	'	PUNCT
cana-1907	22	11	almost	almost	ADV
cana-1907	22	12	sensitive	sensitive	ADJ
cana-1907	22	13	’	'	PUNCT
cana-1907	22	14	if	if	SCONJ
cana-1907	22	15	𝑓	𝑓	PRON
cana-1907	22	16	is	be	AUX
cana-1907	22	17	transitive	transitive	ADJ
cana-1907	22	18	and	and	CCONJ
cana-1907	22	19	this	this	PRON
cana-1907	22	20	implies	imply	VERB
cana-1907	22	21	the	the	DET
cana-1907	22	22	’	'	PUNCT
cana-1907	22	23	almost	almost	ADV
cana-1907	22	24	sensitivity	sensitivity	NOUN
cana-1907	22	25	’	'	PUNCT
cana-1907	22	26	of	of	ADP
cana-1907	22	27	f.	f.	PROPN
cana-1907	22	28	through	through	ADP
cana-1907	22	29	out	out	ADP
cana-1907	22	30	this	this	DET
cana-1907	22	31	paper	paper	NOUN
cana-1907	22	32	,	,	PUNCT
cana-1907	22	33	x	x	PRON
cana-1907	22	34	denotes	denote	VERB
cana-1907	22	35	a	a	DET
cana-1907	22	36	hausdorff	hausdorff	NOUN
cana-1907	22	37	compact	compact	ADJ
cana-1907	22	38	metric	metric	ADJ
cana-1907	22	39	space	space	NOUN
cana-1907	22	40	without	without	ADP
cana-1907	22	41	isolated	isolated	ADJ
cana-1907	22	42	points	point	NOUN
cana-1907	22	43	with	with	ADP
cana-1907	22	44	metric	metric	ADJ
cana-1907	22	45	d	d	NOUN
cana-1907	22	46	and	and	CCONJ
cana-1907	22	47	f	f	NOUN
cana-1907	22	48	:	:	PUNCT
cana-1907	22	49	x	x	X
cana-1907	22	50	→	→	PUNCT
cana-1907	22	51	x	x	PUNCT
cana-1907	22	52	is	be	AUX
cana-1907	22	53	continuous	continuous	ADJ
cana-1907	22	54	.	.	PUNCT
cana-1907	23	1	k(x	k(x	PROPN
cana-1907	23	2	)	)	PUNCT
cana-1907	23	3	denotes	denote	VERB
cana-1907	23	4	the	the	DET
cana-1907	23	5	space	space	NOUN
cana-1907	23	6	of	of	ADP
cana-1907	23	7	all	all	DET
cana-1907	23	8	compact	compact	ADJ
cana-1907	23	9	subsets	subset	NOUN
cana-1907	23	10	of	of	ADP
cana-1907	23	11	x	x	PUNCT
cana-1907	23	12	with	with	ADP
cana-1907	23	13	hausdorff	hausdorff	PROPN
cana-1907	23	14	metric	metric	ADJ
cana-1907	23	15	h	h	NOUN
cana-1907	23	16	induced	induce	VERB
cana-1907	23	17	by	by	ADP
cana-1907	23	18	d	d	PROPN
cana-1907	23	19	.	.	PUNCT
cana-1907	24	1	let𝑓	let𝑓	PROPN
cana-1907	24	2	:	:	PUNCT
cana-1907	24	3	k(x	k(x	PROPN
cana-1907	24	4	)	)	PUNCT
cana-1907	24	5	→	→	SYM
cana-1907	24	6	k(x	k(x	PROPN
cana-1907	24	7	)	)	PUNCT
cana-1907	24	8	defined	define	VERB
cana-1907	24	9	by	by	ADP
cana-1907	24	10	𝑓(𝐾	𝑓(𝐾	ADJ
cana-1907	24	11	)	)	PUNCT
cana-1907	24	12	=	=	SYM
cana-1907	24	13	𝑓(𝐾	𝑓(𝐾	NOUN
cana-1907	24	14	)	)	PUNCT
cana-1907	24	15	.	.	PUNCT
cana-1907	25	1	2	2	X
cana-1907	25	2	.	.	X
cana-1907	25	3	hyperspace	hyperspace	NOUN
cana-1907	25	4	and	and	CCONJ
cana-1907	25	5	induced	induce	VERB
cana-1907	25	6	map	map	NOUN
cana-1907	25	7	1.in	1.in	NUM
cana-1907	26	1	this	this	DET
cana-1907	26	2	section	section	NOUN
cana-1907	26	3	we	we	PRON
cana-1907	26	4	have	have	AUX
cana-1907	26	5	study	study	VERB
cana-1907	26	6	some	some	DET
cana-1907	26	7	properties	property	NOUN
cana-1907	26	8	of	of	ADP
cana-1907	26	9	the	the	DET
cana-1907	26	10	induced	induced	ADJ
cana-1907	26	11	map	map	NOUN
cana-1907	26	12	𝑓	𝑓	PRON
cana-1907	26	13	and	and	CCONJ
cana-1907	26	14	the	the	DET
cana-1907	26	15	base	base	NOUN
cana-1907	26	16	map	map	NOUN
cana-1907	26	17	f.	f.	PROPN
cana-1907	26	18	by	by	ADP
cana-1907	26	19	assuming	assume	VERB
cana-1907	26	20	the	the	DET
cana-1907	26	21	transitivity	transitivity	NOUN
cana-1907	26	22	of	of	ADP
cana-1907	26	23	𝑓	𝑓	PRON
cana-1907	26	24	,	,	PUNCT
cana-1907	26	25	we	we	PRON
cana-1907	26	26	show	show	VERB
cana-1907	26	27	that	that	SCONJ
cana-1907	26	28	the	the	DET
cana-1907	26	29	base	base	NOUN
cana-1907	26	30	space	space	NOUN
cana-1907	26	31	x	x	PRON
cana-1907	26	32	contains	contain	VERB
cana-1907	26	33	a	a	DET
cana-1907	26	34	cantor	cantor	NOUN
cana-1907	26	35	set	set	NOUN
cana-1907	26	36	c	c	PROPN
cana-1907	26	37	with	with	ADP
cana-1907	26	38	its	its	PRON
cana-1907	26	39	orbit	orbit	NOUN
cana-1907	26	40	is	be	AUX
cana-1907	26	41	dense	dense	ADJ
cana-1907	26	42	in	in	ADP
cana-1907	26	43	k(x	k(x	PROPN
cana-1907	26	44	)	)	PUNCT
cana-1907	26	45	.	.	PUNCT
cana-1907	27	1	along	along	ADP
cana-1907	27	2	with	with	ADP
cana-1907	27	3	this	this	PRON
cana-1907	27	4	we	we	PRON
cana-1907	27	5	also	also	ADV
cana-1907	27	6	study	study	VERB
cana-1907	27	7	the	the	DET
cana-1907	27	8	concept	concept	NOUN
cana-1907	27	9	of	of	ADP
cana-1907	27	10	’	'	PUNCT
cana-1907	27	11	almost	almost	ADV
cana-1907	27	12	sensitivity	sensitivity	NOUN
cana-1907	27	13	’	'	PUNCT
cana-1907	27	14	in	in	ADP
cana-1907	27	15	x	x	X
cana-1907	27	16	and	and	CCONJ
cana-1907	27	17	k(x	k(x	PROPN
cana-1907	27	18	)	)	PUNCT
cana-1907	27	19	.	.	PUNCT
cana-1907	28	1	let	let	VERB
cana-1907	28	2	us	we	PRON
cana-1907	28	3	use	use	VERB
cana-1907	28	4	the	the	DET
cana-1907	28	5	symbol	symbol	NOUN
cana-1907	28	6	𝜙	𝜙	NOUN
cana-1907	28	7	for	for	ADP
cana-1907	28	8	the	the	DET
cana-1907	28	9	induced	induced	ADJ
cana-1907	28	10	map	map	NOUN
cana-1907	28	11	𝑓	𝑓	PRON
cana-1907	28	12	for	for	ADP
cana-1907	28	13	the	the	DET
cana-1907	28	14	sake	sake	NOUN
cana-1907	28	15	of	of	ADP
cana-1907	28	16	convenience	convenience	NOUN
cana-1907	28	17	.	.	PUNCT
cana-1907	29	1	communications	communication	NOUN
cana-1907	29	2	on	on	ADP
cana-1907	29	3	applied	apply	VERB
cana-1907	29	4	nonlinear	nonlinear	ADJ
cana-1907	29	5	analysis	analysis	NOUN
cana-1907	29	6	issn	issn	NOUN
cana-1907	29	7	:	:	PUNCT
cana-1907	29	8	1074	1074	NUM
cana-1907	29	9	-	-	PUNCT
cana-1907	29	10	133x	133x	NUM
cana-1907	29	11	vol	vol	NOUN
cana-1907	29	12	32	32	NUM
cana-1907	29	13	no	no	NOUN
cana-1907	29	14	.	.	NOUN
cana-1907	29	15	2	2	NUM
cana-1907	29	16	(	(	PUNCT
cana-1907	29	17	2025	2025	NUM
cana-1907	29	18	)	)	PUNCT
cana-1907	30	1	53	53	NUM
cana-1907	30	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-1907	30	3	definition	definition	NOUN
cana-1907	30	4	2.1	2.1	NUM
cana-1907	30	5	.	.	PUNCT
cana-1907	31	1	if	if	SCONJ
cana-1907	31	2	a	a	PRON
cana-1907	31	3	=	=	X
cana-1907	31	4	{	{	PUNCT
cana-1907	31	5	aµ}µ∈ℸ	aµ}µ∈ℸ	PROPN
cana-1907	31	6	is	be	AUX
cana-1907	31	7	a	a	DET
cana-1907	31	8	collection	collection	NOUN
cana-1907	31	9	of	of	ADP
cana-1907	31	10	non	non	ADJ
cana-1907	31	11	-	-	ADJ
cana-1907	31	12	empty	empty	ADJ
cana-1907	31	13	subsections	subsection	NOUN
cana-1907	31	14	of	of	ADP
cana-1907	31	15	x	x	NOUN
cana-1907	31	16	,	,	PUNCT
cana-1907	31	17	then	then	ADV
cana-1907	31	18	mesh(a	mesh(a	NOUN
cana-1907	31	19	)	)	PUNCT
cana-1907	32	1	=	=	PUNCT
cana-1907	32	2	sup{diam(aµ),µ	sup{diam(aµ),µ	PROPN
cana-1907	32	3	∈	∈	PROPN
cana-1907	32	4	ℸ	ℸ	X
cana-1907	32	5	}	}	PUNCT
cana-1907	32	6	.	.	PUNCT
cana-1907	33	1	in	in	ADP
cana-1907	33	2	[	[	X
cana-1907	33	3	2	2	X
cana-1907	33	4	]	]	PUNCT
cana-1907	33	5	it	it	PRON
cana-1907	33	6	is	be	AUX
cana-1907	33	7	proved	prove	VERB
cana-1907	33	8	that	that	SCONJ
cana-1907	33	9	(	(	PUNCT
cana-1907	33	10	k(x),h	k(x),h	NOUN
cana-1907	33	11	)	)	PUNCT
cana-1907	33	12	is	be	AUX
cana-1907	33	13	compact	compact	ADJ
cana-1907	33	14	.	.	PUNCT
cana-1907	34	1	lemma	lemma	PROPN
cana-1907	34	2	2.1	2.1	NUM
cana-1907	34	3	if	if	SCONJ
cana-1907	34	4	𝑜𝑟𝑏(𝜙	𝑜𝑟𝑏(𝜙	PROPN
cana-1907	34	5	,	,	PUNCT
cana-1907	34	6	𝐾	𝐾	PROPN
cana-1907	34	7	)	)	PUNCT
cana-1907	34	8	=	=	SYM
cana-1907	34	9	𝑋	𝑋	PROPN
cana-1907	34	10	then	then	ADV
cana-1907	34	11	for	for	ADP
cana-1907	34	12	individually	individually	ADV
cana-1907	34	13	a	a	DET
cana-1907	34	14	∈	∈	ADJ
cana-1907	34	15	orb(ϕ,k	orb(ϕ,k	NOUN
cana-1907	34	16	)	)	PUNCT
cana-1907	34	17	and	and	CCONJ
cana-1907	34	18	for	for	ADP
cana-1907	34	19	all	all	DET
cana-1907	34	20	x	x	SYM
cana-1907	34	21	∈	∈	PROPN
cana-1907	34	22	a	a	DET
cana-1907	34	23	,	,	PUNCT
cana-1907	34	24	𝑜𝑟𝑏(𝑓	𝑜𝑟𝑏(𝑓	PROPN
cana-1907	34	25	,	,	PUNCT
cana-1907	34	26	𝑥	𝑥	NOUN
cana-1907	34	27	)	)	PUNCT
cana-1907	34	28	=	=	SYM
cana-1907	35	1	𝑋	𝑋	NOUN
cana-1907	35	2	proof	proof	NOUN
cana-1907	35	3	.	.	PUNCT
cana-1907	36	1	let	let	VERB
cana-1907	36	2	n	n	PRON
cana-1907	36	3	∈	∈	PROPN
cana-1907	36	4	ℕ	ℕ	PROPN
cana-1907	36	5	and	and	CCONJ
cana-1907	36	6	x	x	SYM
cana-1907	36	7	∈	∈	PROPN
cana-1907	36	8	f	f	PROPN
cana-1907	36	9	n(k	n(k	PROPN
cana-1907	36	10	)	)	PUNCT
cana-1907	36	11	.	.	PUNCT
cana-1907	37	1	we	we	PRON
cana-1907	37	2	have	have	VERB
cana-1907	37	3	to	to	PART
cana-1907	37	4	demonstrate	demonstrate	VERB
cana-1907	37	5	that	that	SCONJ
cana-1907	37	6	𝑜𝑟𝑏(𝑓	𝑜𝑟𝑏(𝑓	NOUN
cana-1907	37	7	,	,	PUNCT
cana-1907	37	8	𝑥	𝑥	NOUN
cana-1907	37	9	)	)	PUNCT
cana-1907	37	10	=	=	SYM
cana-1907	37	11	𝑋.	𝑋.	PROPN
cana-1907	37	12	take	take	VERB
cana-1907	37	13	y	y	PROPN
cana-1907	37	14	∈	∈	PROPN
cana-1907	37	15	x	x	X
cana-1907	37	16	and	and	CCONJ
cana-1907	37	17	ϵ	ϵ	X
cana-1907	37	18	>	>	X
cana-1907	37	19	0	0	X
cana-1907	37	20	.	.	PUNCT
cana-1907	38	1	since	since	SCONJ
cana-1907	38	2	𝑜𝑟𝑏(𝜙	𝑜𝑟𝑏(𝜙	PROPN
cana-1907	38	3	,	,	PUNCT
cana-1907	38	4	𝐾	𝐾	PROPN
cana-1907	38	5	)	)	PUNCT
cana-1907	38	6	=	=	SYM
cana-1907	39	1	𝑋	𝑋	NOUN
cana-1907	39	2	we	we	PRON
cana-1907	39	3	have	have	VERB
cana-1907	39	4	𝑜𝑟𝑏(𝜙	𝑜𝑟𝑏(𝜙	PROPN
cana-1907	39	5	,	,	PUNCT
cana-1907	39	6	𝑓𝑛(𝐾	𝑓𝑛(𝐾	PROPN
cana-1907	39	7	)	)	PUNCT
cana-1907	39	8	)	)	PUNCT
cana-1907	40	1	=	=	SYM
cana-1907	40	2	𝐾(𝑋	𝐾(𝑋	NOUN
cana-1907	40	3	)	)	PUNCT
cana-1907	40	4	let	let	VERB
cana-1907	40	5	{	{	PUNCT
cana-1907	40	6	𝑦	𝑦	NOUN
cana-1907	40	7	}	}	PUNCT
cana-1907	40	8	∈	∈	PROPN
cana-1907	40	9	𝑜𝑟𝑏(𝜙	𝑜𝑟𝑏(𝜙	PROPN
cana-1907	40	10	,	,	PUNCT
cana-1907	40	11	𝑓𝑛(𝐾	𝑓𝑛(𝐾	PROPN
cana-1907	40	12	)	)	PUNCT
cana-1907	40	13	)	)	PUNCT
cana-1907	40	14	,	,	PUNCT
cana-1907	40	15	then	then	ADV
cana-1907	40	16	there	there	PRON
cana-1907	40	17	exist	exist	VERB
cana-1907	40	18	j	j	PROPN
cana-1907	40	19	∈	∈	PROPN
cana-1907	40	20	ℕ	ℕ	PROPN
cana-1907	40	21	such	such	ADJ
cana-1907	40	22	that	that	SCONJ
cana-1907	40	23	𝐻	𝐻	PROPN
cana-1907	40	24	(	(	PUNCT
cana-1907	40	25	𝑓𝑗(𝑓𝑛(𝐾	𝑓𝑗(𝑓𝑛(𝐾	PROPN
cana-1907	40	26	)	)	PUNCT
cana-1907	40	27	,	,	PUNCT
cana-1907	40	28	{	{	PUNCT
cana-1907	40	29	𝑦	𝑦	NOUN
cana-1907	40	30	}	}	PUNCT
cana-1907	40	31	)	)	PUNCT
cana-1907	40	32	)	)	PUNCT
cana-1907	40	33	<	<	X
cana-1907	40	34	𝜖.	𝜖.	NOUN
cana-1907	40	35	hence	hence	ADV
cana-1907	40	36	𝑓𝑗(𝑥	𝑓𝑗(𝑥	PUNCT
cana-1907	40	37	)	)	PUNCT
cana-1907	40	38	∈	∈	PROPN
cana-1907	40	39	𝑓𝑗(𝑓𝑛(𝐾	𝑓𝑗(𝑓𝑛(𝐾	PROPN
cana-1907	40	40	)	)	PUNCT
cana-1907	40	41	)	)	PUNCT
cana-1907	41	1	⊆	⊆	NUM
cana-1907	41	2	𝐵𝜖(𝑦	𝐵𝜖(𝑦	NOUN
cana-1907	41	3	)	)	PUNCT
cana-1907	41	4	so	so	ADV
cana-1907	41	5	𝐵𝜖(𝑦	𝐵𝜖(𝑦	NOUN
cana-1907	41	6	)	)	PUNCT
cana-1907	41	7	∩	∩	NOUN
cana-1907	41	8	,	,	PUNCT
cana-1907	41	9	𝑜𝑟𝑏(𝑓	𝑜𝑟𝑏(𝑓	PROPN
cana-1907	41	10	,	,	PUNCT
cana-1907	41	11	𝑥	𝑥	NOUN
cana-1907	41	12	)	)	PUNCT
cana-1907	41	13	≠	≠	PROPN
cana-1907	41	14	∅	∅	NOUN
cana-1907	41	15	there	there	ADV
cana-1907	41	16	fore	fore	NOUN
cana-1907	41	17	,	,	PUNCT
cana-1907	41	18	𝑜𝑟𝑏(𝑓	𝑜𝑟𝑏(𝑓	PROPN
cana-1907	41	19	,	,	PUNCT
cana-1907	41	20	𝑥	𝑥	NOUN
cana-1907	41	21	)	)	PUNCT
cana-1907	41	22	=	=	SYM
cana-1907	41	23	𝑋.	𝑋.	PROPN
cana-1907	41	24	lemma	lemma	PROPN
cana-1907	41	25	2.2	2.2	NUM
cana-1907	41	26	.	.	PUNCT
cana-1907	42	1	if	if	SCONJ
cana-1907	42	2	x	x	PRON
cana-1907	42	3	is	be	AUX
cana-1907	42	4	weekly	weekly	ADJ
cana-1907	42	5	mixing	mixing	NOUN
cana-1907	42	6	,	,	PUNCT
cana-1907	42	7	then	then	ADV
cana-1907	42	8	any	any	DET
cana-1907	42	9	power	power	NOUN
cana-1907	42	10	x	x	X
cana-1907	42	11	×	×	NOUN
cana-1907	42	12	x	x	SYM
cana-1907	42	13	×	×	PROPN
cana-1907	42	14	·	·	PUNCT
cana-1907	42	15	·	·	PUNCT
cana-1907	42	16	·	·	SYM
cana-1907	42	17	×	×	NOUN
cana-1907	42	18	x	x	PUNCT
cana-1907	42	19	is	be	AUX
cana-1907	42	20	ergodic	ergodic	ADJ
cana-1907	42	21	.	.	PUNCT
cana-1907	43	1	proof	proof	NOUN
cana-1907	43	2	.	.	PUNCT
cana-1907	44	1	see[1	see[1	PROPN
cana-1907	44	2	]	]	PUNCT
cana-1907	44	3	lemma	lemma	PROPN
cana-1907	44	4	2.3	2.3	NUM
cana-1907	44	5	.	.	PUNCT
cana-1907	45	1	let	let	VERB
cana-1907	45	2	ϕ	ϕ	NOUN
cana-1907	45	3	:	:	PUNCT
cana-1907	45	4	k(x	k(x	PROPN
cana-1907	45	5	)	)	PUNCT
cana-1907	45	6	→	→	SYM
cana-1907	45	7	k(x	k(x	PROPN
cana-1907	45	8	)	)	PUNCT
cana-1907	45	9	be	be	AUX
cana-1907	45	10	transitive	transitive	ADJ
cana-1907	45	11	.	.	PUNCT
cana-1907	46	1	then	then	ADV
cana-1907	46	2	there	there	PRON
cana-1907	46	3	exist	exist	VERB
cana-1907	46	4	a	a	DET
cana-1907	46	5	cantor	cantor	NOUN
cana-1907	46	6	set	set	NOUN
cana-1907	46	7	c	c	NOUN
cana-1907	46	8	⊆	⊆	NUM
cana-1907	46	9	x	x	SYM
cana-1907	46	10	such	such	ADJ
cana-1907	46	11	that	that	SCONJ
cana-1907	46	12	𝑜𝑟𝑏(𝜙	𝑜𝑟𝑏(𝜙	PROPN
cana-1907	46	13	,	,	PUNCT
cana-1907	46	14	𝐶	𝐶	PROPN
cana-1907	46	15	)	)	PUNCT
cana-1907	46	16	=	=	SYM
cana-1907	46	17	𝐾(𝑋	𝐾(𝑋	NOUN
cana-1907	46	18	)	)	PUNCT
cana-1907	46	19	.	.	PUNCT
cana-1907	47	1	proof	proof	NOUN
cana-1907	47	2	.	.	PUNCT
cana-1907	48	1	let	let	VERB
cana-1907	48	2	δ₀	δ₀	PROPN
cana-1907	48	3	represent	represent	VERB
cana-1907	48	4	the	the	DET
cana-1907	48	5	diameter	diameter	NOUN
cana-1907	48	6	of	of	ADP
cana-1907	48	7	the	the	DET
cana-1907	48	8	set	set	NOUN
cana-1907	48	9	x.	x.	NOUN
cana-1907	48	10	for	for	ADP
cana-1907	48	11	every	every	DET
cana-1907	48	12	positive	positive	ADJ
cana-1907	48	13	integer	integer	NOUN
cana-1907	48	14	n	n	CCONJ
cana-1907	48	15	,	,	PUNCT
cana-1907	48	16	we	we	PRON
cana-1907	48	17	define	define	VERB
cana-1907	48	18	a	a	DET
cana-1907	48	19	series	series	NOUN
cana-1907	48	20	of	of	ADP
cana-1907	48	21	finite	finite	PROPN
cana-1907	48	22	open	open	ADJ
cana-1907	48	23	covers	cover	NOUN
cana-1907	48	24	of	of	ADP
cana-1907	48	25	x	x	PRON
cana-1907	48	26	,	,	PUNCT
cana-1907	48	27	denoted	denote	VERB
cana-1907	48	28	by	by	ADP
cana-1907	48	29	𝑉	𝑉	PROPN
cana-1907	48	30	�	�	PROPN
cana-1907	48	31	̃	̃	PROPN
cana-1907	48	32	�	�	NOUN
cana-1907	48	33	=	=	SYM
cana-1907	48	34	{	{	PUNCT
cana-1907	48	35	𝑉𝑛,1	𝑉𝑛,1	PROPN
cana-1907	48	36	,	,	PUNCT
cana-1907	48	37	𝑉𝑛,2	𝑉𝑛,2	NOUN
cana-1907	48	38	,	,	PUNCT
cana-1907	48	39	…	…	PUNCT
cana-1907	48	40	.	.	PUNCT
cana-1907	48	41	,	,	PUNCT
cana-1907	48	42	𝑉𝑛,𝑡𝑛	𝑉𝑛,𝑡𝑛	PROPN
cana-1907	48	43	}	}	PUNCT
cana-1907	48	44	where	where	SCONJ
cana-1907	48	45	each	each	DET
cana-1907	48	46	𝑉𝑛,𝑖	𝑉𝑛,𝑖	PROPN
cana-1907	48	47	is	be	AUX
cana-1907	48	48	a	a	DET
cana-1907	48	49	nonempty	nonempty	NOUN
cana-1907	48	50	subset	subset	NOUN
cana-1907	48	51	.	.	PUNCT
cana-1907	49	1	these	these	DET
cana-1907	49	2	covers	cover	NOUN
cana-1907	49	3	are	be	AUX
cana-1907	49	4	constructed	construct	VERB
cana-1907	49	5	such	such	ADJ
cana-1907	49	6	that	that	SCONJ
cana-1907	49	7	the	the	DET
cana-1907	49	8	size	size	NOUN
cana-1907	49	9	of	of	ADP
cana-1907	49	10	each	each	DET
cana-1907	49	11	cover	cover	NOUN
cana-1907	49	12	is	be	AUX
cana-1907	49	13	smaller	small	ADJ
cana-1907	49	14	than	than	ADP
cana-1907	49	15	δₙ.	δₙ.	NOUN
cana-1907	49	16	in	in	ADP
cana-1907	49	17	essence	essence	NOUN
cana-1907	49	18	,	,	PUNCT
cana-1907	49	19	this	this	DET
cana-1907	49	20	arrangement	arrangement	NOUN
cana-1907	49	21	ensures	ensure	VERB
cana-1907	49	22	that	that	SCONJ
cana-1907	49	23	each	each	DET
cana-1907	49	24	element	element	NOUN
cana-1907	49	25	of	of	ADP
cana-1907	49	26	the	the	DET
cana-1907	49	27	set	set	NOUN
cana-1907	49	28	x	x	PUNCT
cana-1907	49	29	is	be	AUX
cana-1907	49	30	contained	contain	VERB
cana-1907	49	31	within	within	ADP
cana-1907	49	32	at	at	ADV
cana-1907	49	33	least	least	ADV
cana-1907	49	34	one	one	NUM
cana-1907	49	35	open	open	ADJ
cana-1907	49	36	set	set	NOUN
cana-1907	49	37	in	in	ADP
cana-1907	49	38	the	the	DET
cana-1907	49	39	cover	cover	NOUN
cana-1907	49	40	,	,	PUNCT
cana-1907	49	41	and	and	CCONJ
cana-1907	49	42	as	as	ADP
cana-1907	49	43	n	n	PRON
cana-1907	49	44	increases	increase	NOUN
cana-1907	49	45	,	,	PUNCT
cana-1907	49	46	the	the	DET
cana-1907	49	47	covers	cover	NOUN
cana-1907	49	48	become	become	VERB
cana-1907	49	49	increasingly	increasingly	ADV
cana-1907	49	50	finer	fine	ADJ
cana-1907	49	51	,	,	PUNCT
cana-1907	49	52	converging	converge	VERB
cana-1907	49	53	towards	towards	ADP
cana-1907	49	54	the	the	DET
cana-1907	49	55	set	set	NOUN
cana-1907	49	56	's	's	PART
cana-1907	49	57	diameter	diameter	NOUN
cana-1907	49	58	.	.	PUNCT
cana-1907	50	1	step	step	NOUN
cana-1907	50	2	1	1	NUM
cana-1907	50	3	let	let	VERB
cana-1907	50	4	us	we	PRON
cana-1907	50	5	assume	assume	VERB
cana-1907	50	6	w0	w0	PROPN
cana-1907	50	7	and	and	CCONJ
cana-1907	50	8	w1	w1	NOUN
cana-1907	50	9	to	to	PART
cana-1907	50	10	be	be	AUX
cana-1907	50	11	two	two	NUM
cana-1907	50	12	non	non	ADJ
cana-1907	50	13	-	-	ADJ
cana-1907	50	14	empty	empty	ADJ
cana-1907	50	15	disjoint	disjoint	ADJ
cana-1907	50	16	open	open	ADJ
cana-1907	50	17	sets	set	NOUN
cana-1907	50	18	in	in	ADP
cana-1907	50	19	x	x	PUNCT
cana-1907	50	20	and	and	CCONJ
cana-1907	50	21	mesh({w0∩w1	mesh({w0∩w1	PROPN
cana-1907	50	22	}	}	PUNCT
cana-1907	50	23	<	<	X
cana-1907	50	24	δ1	δ1	NOUN
cana-1907	50	25	.	.	PUNCT
cana-1907	51	1	let	let	VERB
cana-1907	51	2	𝜆1	𝜆1	VERB
cana-1907	51	3	=	=	PUNCT
cana-1907	51	4	{	{	PUNCT
cana-1907	51	5	1,2	1,2	NUM
cana-1907	51	6	,	,	PUNCT
cana-1907	51	7	…	…	PUNCT
cana-1907	51	8	.	.	PUNCT
cana-1907	52	1	,	,	PUNCT
cana-1907	52	2	𝑡1	𝑡1	NOUN
cana-1907	52	3	}	}	PUNCT
cana-1907	52	4	×	×	NOUN
cana-1907	52	5	{	{	PUNCT
cana-1907	52	6	1,2	1,2	NUM
cana-1907	52	7	,	,	PUNCT
cana-1907	52	8	…	…	PUNCT
cana-1907	52	9	.	.	PUNCT
cana-1907	53	1	,	,	PUNCT
cana-1907	53	2	𝑡1	𝑡1	NOUN
cana-1907	53	3	}	}	PUNCT
cana-1907	53	4	=	=	SYM
cana-1907	53	5	{	{	PUNCT
cana-1907	53	6	(	(	PUNCT
cana-1907	53	7	𝑎	𝑎	X
cana-1907	53	8	,	,	PUNCT
cana-1907	53	9	𝑏)|𝑎	𝑏)|𝑎	ADJ
cana-1907	53	10	,	,	PUNCT
cana-1907	53	11	𝑏	𝑏	PROPN
cana-1907	53	12	∈	∈	PROPN
cana-1907	53	13	{	{	PUNCT
cana-1907	53	14	1,2	1,2	NUM
cana-1907	53	15	,	,	PUNCT
cana-1907	53	16	…	…	PUNCT
cana-1907	53	17	.	.	PUNCT
cana-1907	54	1	,	,	PUNCT
cana-1907	54	2	𝑡1	𝑡1	NOUN
cana-1907	54	3	}	}	PUNCT
cana-1907	54	4	}	}	PUNCT
cana-1907	54	5	.	.	PUNCT
cana-1907	55	1	let	let	VERB
cana-1907	55	2	us	we	PRON
cana-1907	55	3	take	take	VERB
cana-1907	55	4	into	into	ADP
cana-1907	55	5	consideration	consideration	NOUN
cana-1907	55	6	the	the	DET
cana-1907	55	7	following	follow	VERB
cana-1907	55	8	𝑡1	𝑡1	NOUN
cana-1907	55	9	2	2	NUM
cana-1907	55	10	+	+	SYM
cana-1907	55	11	1	1	NUM
cana-1907	55	12	collection	collection	NOUN
cana-1907	55	13	of	of	ADP
cana-1907	55	14	open	open	ADJ
cana-1907	55	15	sets	set	NOUN
cana-1907	55	16	(	(	PUNCT
cana-1907	55	17	w0,w1	w0,w1	PROPN
cana-1907	55	18	)	)	PUNCT
cana-1907	55	19	and	and	CCONJ
cana-1907	55	20	{	{	PUNCT
cana-1907	55	21	(	(	PUNCT
cana-1907	55	22	v1,a	v1,a	PROPN
cana-1907	55	23	,	,	PUNCT
cana-1907	55	24	v1,b	v1,b	PROPN
cana-1907	55	25	)	)	PUNCT
cana-1907	55	26	:	:	PUNCT
cana-1907	55	27	(	(	PUNCT
cana-1907	55	28	a	a	PRON
cana-1907	55	29	,	,	PUNCT
cana-1907	55	30	b	b	NOUN
cana-1907	55	31	)	)	PUNCT
cana-1907	55	32	∈	∈	PROPN
cana-1907	55	33	λ1	λ1	PROPN
cana-1907	55	34	}	}	PUNCT
cana-1907	55	35	then	then	ADV
cana-1907	55	36	,	,	PUNCT
cana-1907	55	37	by	by	ADP
cana-1907	55	38	lemma	lemma	PROPN
cana-1907	55	39	1.4	1.4	NUM
cana-1907	56	1	[	[	NOUN
cana-1907	56	2	see	see	VERB
cana-1907	56	3	1	1	NUM
cana-1907	56	4	]	]	PUNCT
cana-1907	56	5	,	,	PUNCT
cana-1907	56	6	two	two	NUM
cana-1907	56	7	closed	closed	ADJ
cana-1907	56	8	subsets	subset	NOUN
cana-1907	56	9	of	of	ADP
cana-1907	56	10	x	x	PROPN
cana-1907	56	11	,	,	PUNCT
cana-1907	56	12	𝐶0	𝐶0	NOUN
cana-1907	56	13	and	and	CCONJ
cana-1907	56	14	𝐶1	𝐶1	PRON
cana-1907	56	15	,	,	PUNCT
cana-1907	56	16	having	have	VERB
cana-1907	56	17	the	the	DET
cana-1907	56	18	following	follow	VERB
cana-1907	56	19	characteristics	characteristic	NOUN
cana-1907	56	20	,	,	PUNCT
cana-1907	56	21	exist	exist	VERB
cana-1907	56	22	:	:	PUNCT
cana-1907	56	23	•	•	ADP
cana-1907	56	24	each	each	DET
cana-1907	56	25	int(ci	int(ci	NOUN
cana-1907	56	26	)	)	PUNCT
cana-1907	56	27	is	be	AUX
cana-1907	56	28	nonempty	nonempty	ADJ
cana-1907	56	29	and	and	CCONJ
cana-1907	56	30	which	which	PRON
cana-1907	56	31	is	be	AUX
cana-1907	56	32	contained	contain	VERB
cana-1907	56	33	in	in	ADP
cana-1907	56	34	wi	wi	PROPN
cana-1907	56	35	for	for	ADP
cana-1907	56	36	i	i	PROPN
cana-1907	56	37	=	=	NOUN
cana-1907	56	38	0,1	0,1	NUM
cana-1907	56	39	.	.	PUNCT
cana-1907	57	1	hence	hence	ADV
cana-1907	57	2	c0	c0	PROPN
cana-1907	57	3	and	and	CCONJ
cana-1907	57	4	c1	c1	PROPN
cana-1907	57	5	are	be	AUX
cana-1907	57	6	disjoint	disjoint	ADJ
cana-1907	57	7	.	.	PUNCT
cana-1907	58	1	•	•	NUM
cana-1907	58	2	for	for	ADP
cana-1907	58	3	each	each	DET
cana-1907	58	4	a	a	DET
cana-1907	58	5	∈	∈	PROPN
cana-1907	58	6	⟨c0,c1⟩	⟨c0,c1⟩	NOUN
cana-1907	59	1	=	=	PUNCT
cana-1907	59	2	{	{	PUNCT
cana-1907	59	3	𝐵	𝐵	NOUN
cana-1907	59	4	∈	∈	PROPN
cana-1907	59	5	𝐾(𝑋	𝐾(𝑋	NOUN
cana-1907	59	6	):	):	PUNCT
cana-1907	59	7	𝐵	𝐵	PROPN
cana-1907	59	8	⊂	⊂	PROPN
cana-1907	59	9	(	(	PUNCT
cana-1907	59	10	𝐶0	𝐶0	PROPN
cana-1907	59	11	∪	∪	ADP
cana-1907	59	12	𝐶1)𝑎𝑛𝑑	𝐶1)𝑎𝑛𝑑	PROPN
cana-1907	59	13	𝐵	𝐵	PROPN
cana-1907	59	14	∩	∩	NOUN
cana-1907	59	15	𝐶𝑖	𝐶𝑖	PROPN
cana-1907	59	16	≠	≠	PROPN
cana-1907	59	17	∅	∅	NOUN
cana-1907	59	18	}	}	PUNCT
cana-1907	59	19	and	and	CCONJ
cana-1907	59	20	for	for	ADP
cana-1907	59	21	each	each	DET
cana-1907	59	22	(	(	PUNCT
cana-1907	59	23	a	a	PRON
cana-1907	59	24	,	,	PUNCT
cana-1907	59	25	b	b	NOUN
cana-1907	59	26	)	)	PUNCT
cana-1907	59	27	∈	∈	PROPN
cana-1907	59	28	λ1,there	λ1,there	VERB
cana-1907	59	29	exist	exist	VERB
cana-1907	59	30	n	n	DET
cana-1907	59	31	∈	∈	PROPN
cana-1907	59	32	n	n	PRON
cana-1907	59	33	such	such	ADJ
cana-1907	59	34	that	that	SCONJ
cana-1907	59	35	fn(a	fn(a	PUNCT
cana-1907	59	36	)	)	PUNCT
cana-1907	59	37	∈	∈	PROPN
cana-1907	59	38	⟨u1,a	⟨u1,a	PROPN
cana-1907	59	39	,	,	PUNCT
cana-1907	59	40	u1,b⟩.	u1,b⟩.	ADJ
cana-1907	59	41	communications	communication	NOUN
cana-1907	59	42	on	on	ADP
cana-1907	59	43	applied	apply	VERB
cana-1907	59	44	nonlinear	nonlinear	ADJ
cana-1907	59	45	analysis	analysis	NOUN
cana-1907	59	46	issn	issn	NOUN
cana-1907	59	47	:	:	PUNCT
cana-1907	59	48	1074	1074	NUM
cana-1907	59	49	-	-	PUNCT
cana-1907	59	50	133x	133x	NUM
cana-1907	59	51	vol	vol	NOUN
cana-1907	59	52	32	32	NUM
cana-1907	59	53	no	no	NOUN
cana-1907	59	54	.	.	NOUN
cana-1907	59	55	2	2	NUM
cana-1907	59	56	(	(	PUNCT
cana-1907	59	57	2025	2025	NUM
cana-1907	59	58	)	)	PUNCT
cana-1907	59	59	54	54	NUM
cana-1907	59	60	https://internationalpubls.com	https://internationalpubls.com	X
cana-1907	59	61	•	•	VERB
cana-1907	59	62	also	also	ADV
cana-1907	59	63	,	,	PUNCT
cana-1907	59	64	fn(a	fn(a	X
cana-1907	59	65	∩	∩	PROPN
cana-1907	59	66	c0	c0	NOUN
cana-1907	59	67	)	)	PUNCT
cana-1907	59	68	⊂	⊂	PROPN
cana-1907	60	1	u1,a	u1,a	PROPN
cana-1907	60	2	and	and	CCONJ
cana-1907	60	3	fn(a	fn(a	NOUN
cana-1907	60	4	∩	∩	PROPN
cana-1907	60	5	c1	c1	PROPN
cana-1907	60	6	)	)	PUNCT
cana-1907	61	1	⊂	⊂	PROPN
cana-1907	62	1	u1,b	u1,b	PROPN
cana-1907	62	2	.	.	PUNCT
cana-1907	63	1	let	let	VERB
cana-1907	63	2	ℂ1=	ℂ1=	VERB
cana-1907	63	3	⟨c0,c1⟩.	⟨c0,c1⟩.	PROPN
cana-1907	63	4	then	then	ADV
cana-1907	63	5	diam(ℂ1	diam(ℂ1	PUNCT
cana-1907	63	6	)	)	PUNCT
cana-1907	64	1	<	<	X
cana-1907	64	2	δ1	δ1	NOUN
cana-1907	64	3	for	for	ADP
cana-1907	64	4	each	each	DET
cana-1907	64	5	a	a	DET
cana-1907	64	6	∈	∈	PROPN
cana-1907	64	7	ℂ1	ℂ1	NOUN
cana-1907	64	8	,	,	PUNCT
cana-1907	64	9	orb(ϕ,a	orb(ϕ,a	PROPN
cana-1907	64	10	)	)	PUNCT
cana-1907	64	11	is	be	AUX
cana-1907	64	12	δ1	δ1	NOUN
cana-1907	64	13	-	-	PUNCT
cana-1907	64	14	close	close	ADJ
cana-1907	64	15	to	to	ADP
cana-1907	64	16	f2(x),where	f2(x),where	PROPN
cana-1907	64	17	f2(x	f2(x	NUM
cana-1907	64	18	)	)	PUNCT
cana-1907	64	19	=	=	PRON
cana-1907	64	20	{	{	PUNCT
cana-1907	64	21	a	a	DET
cana-1907	64	22	∈	∈	PROPN
cana-1907	64	23	k(x)\|a|	k(x)\|a|	NOUN
cana-1907	64	24	≤	≤	ADJ
cana-1907	64	25	2	2	NUM
cana-1907	64	26	}	}	PUNCT
cana-1907	64	27	.	.	PUNCT
cana-1907	65	1	for	for	ADP
cana-1907	65	2	,	,	PUNCT
cana-1907	65	3	given	give	VERB
cana-1907	65	4	any	any	DET
cana-1907	65	5	{	{	PUNCT
cana-1907	65	6	p	p	X
cana-1907	65	7	,	,	PUNCT
cana-1907	65	8	q	q	ADJ
cana-1907	65	9	}	}	PUNCT
cana-1907	65	10	∈	∈	PROPN
cana-1907	65	11	f2(x	f2(x	PROPN
cana-1907	65	12	)	)	PUNCT
cana-1907	65	13	,	,	PUNCT
cana-1907	65	14	there	there	PRON
cana-1907	65	15	exist	exist	VERB
cana-1907	65	16	(	(	PUNCT
cana-1907	65	17	a	a	PRON
cana-1907	65	18	,	,	PUNCT
cana-1907	65	19	b	b	NOUN
cana-1907	65	20	)	)	PUNCT
cana-1907	65	21	∈	∈	NOUN
cana-1907	65	22	λ1	λ1	NOUN
cana-1907	65	23	such	such	ADJ
cana-1907	65	24	that	that	SCONJ
cana-1907	65	25	p	p	PROPN
cana-1907	65	26	∈	∈	PROPN
cana-1907	65	27	u1,a	u1,a	PROPN
cana-1907	65	28	and	and	CCONJ
cana-1907	65	29	q	q	NOUN
cana-1907	65	30	∈	∈	NOUN
cana-1907	65	31	𝑈1,𝑏.	𝑈1,𝑏.	VERB
cana-1907	65	32	since	since	SCONJ
cana-1907	65	33	there	there	PRON
cana-1907	65	34	exist	exist	VERB
cana-1907	65	35	n	n	ADV
cana-1907	65	36	so	so	SCONJ
cana-1907	65	37	that	that	SCONJ
cana-1907	65	38	fn(a	fn(a	PUNCT
cana-1907	65	39	)	)	PUNCT
cana-1907	65	40	∈	∈	PROPN
cana-1907	65	41	⟨u1,a	⟨u1,a	PROPN
cana-1907	65	42	,	,	PUNCT
cana-1907	65	43	u1,b⟩.	u1,b⟩.	VERB
cana-1907	65	44	we	we	PRON
cana-1907	65	45	conclude	conclude	VERB
cana-1907	65	46	that	that	DET
cana-1907	65	47	h({p	h({p	PROPN
cana-1907	65	48	,	,	PUNCT
cana-1907	65	49	q},fn(a	q},fn(a	PROPN
cana-1907	65	50	)	)	PUNCT
cana-1907	65	51	)	)	PUNCT
cana-1907	66	1	<	<	X
cana-1907	66	2	δ1	δ1	NOUN
cana-1907	66	3	step	step	NOUN
cana-1907	66	4	2	2	NUM
cana-1907	66	5	let	let	VERB
cana-1907	66	6	w0,0,w1,0,w0,1.w1,1	w0,0,w1,0,w0,1.w1,1	AUX
cana-1907	66	7	are	be	AUX
cana-1907	66	8	4	4	NUM
cana-1907	66	9	non	non	ADJ
cana-1907	66	10	-	-	ADJ
cana-1907	66	11	empty	empty	ADJ
cana-1907	66	12	open	open	ADJ
cana-1907	66	13	subsets	subset	NOUN
cana-1907	66	14	of	of	ADP
cana-1907	66	15	x	x	PUNCT
cana-1907	66	16	with	with	ADP
cana-1907	66	17	𝑊0,0	𝑊0,0	NOUN
cana-1907	66	18	∩	∩	NOUN
cana-1907	66	19	𝑊1,0	𝑊1,0	PART
cana-1907	66	20	=	=	SYM
cana-1907	66	21	∅	∅	NOUN
cana-1907	66	22	and	and	CCONJ
cana-1907	66	23	𝑊0,1	𝑊0,1	NOUN
cana-1907	66	24	∩	∩	ADJ
cana-1907	66	25	𝑊1,1	𝑊1,1	NOUN
cana-1907	66	26	=	=	NOUN
cana-1907	66	27	∅	∅	NOUN
cana-1907	66	28	𝑊0,0	𝑊0,0	NOUN
cana-1907	66	29	∪	∪	VERB
cana-1907	66	30	𝑊1,0	𝑊1,0	PROPN
cana-1907	66	31	⊂	⊂	PROPN
cana-1907	66	32	𝐶0	𝐶0	PROPN
cana-1907	66	33	and	and	CCONJ
cana-1907	66	34	𝑊0,1	𝑊0,1	PROPN
cana-1907	66	35	∪	∪	ADP
cana-1907	66	36	𝑊1,1	𝑊1,1	PROPN
cana-1907	66	37	⊂	⊂	ADJ
cana-1907	66	38	𝐶1	𝐶1	X
cana-1907	66	39	and	and	CCONJ
cana-1907	66	40	𝑚𝑒𝑠ℎ({𝑊0,0	𝑚𝑒𝑠ℎ({𝑊0,0	NOUN
cana-1907	66	41	,	,	PUNCT
cana-1907	66	42	𝑊1,0	𝑊1,0	NOUN
cana-1907	66	43	,	,	PUNCT
cana-1907	66	44	𝑊0,1	𝑊0,1	NOUN
cana-1907	66	45	,	,	PUNCT
cana-1907	66	46	𝑊1,1	𝑊1,1	NOUN
cana-1907	66	47	}	}	PUNCT
cana-1907	66	48	)	)	PUNCT
cana-1907	66	49	<	<	X
cana-1907	66	50	𝛿2	𝛿2	PROPN
cana-1907	66	51	.	.	PUNCT
cana-1907	67	1	let	let	VERB
cana-1907	67	2	λ2	λ2	NOUN
cana-1907	67	3	=	=	PUNCT
cana-1907	67	4	{	{	PUNCT
cana-1907	67	5	1,2,	1,2,	NUM
cana-1907	67	6	...	...	PUNCT
cana-1907	67	7	t2	t2	NOUN
cana-1907	67	8	}	}	PUNCT
cana-1907	67	9	4	4	NUM
cana-1907	67	10	=	=	SYM
cana-1907	67	11	{	{	PUNCT
cana-1907	67	12	(	(	PUNCT
cana-1907	67	13	a1,a2,a3,a4)\ai	a1,a2,a3,a4)\ai	X
cana-1907	67	14	∈	∈	PROPN
cana-1907	67	15	{	{	PUNCT
cana-1907	67	16	1,2,	1,2,	NUM
cana-1907	67	17	...	...	PUNCT
cana-1907	67	18	t2	t2	NOUN
cana-1907	67	19	}	}	PUNCT
cana-1907	67	20	}	}	PUNCT
cana-1907	67	21	.	.	PUNCT
cana-1907	68	1	consider	consider	VERB
cana-1907	68	2	the	the	DET
cana-1907	68	3	following	follow	VERB
cana-1907	68	4	𝑡2	𝑡2	NOUN
cana-1907	68	5	4	4	NUM
cana-1907	68	6	+	+	SYM
cana-1907	68	7	1	1	NUM
cana-1907	68	8	collection	collection	NOUN
cana-1907	68	9	of	of	ADP
cana-1907	68	10	open	open	ADJ
cana-1907	68	11	sets	set	NOUN
cana-1907	68	12	(	(	PUNCT
cana-1907	68	13	𝑊0,0	𝑊0,0	NOUN
cana-1907	68	14	,	,	PUNCT
cana-1907	68	15	𝑊1,0	𝑊1,0	NOUN
cana-1907	68	16	,	,	PUNCT
cana-1907	68	17	𝑊0,1	𝑊0,1	NOUN
cana-1907	68	18	,	,	PUNCT
cana-1907	68	19	𝑊1,1),{(𝑈2	𝑊1,1),{(𝑈2	NOUN
cana-1907	68	20	,	,	PUNCT
cana-1907	68	21	𝑎1	𝑎1	PROPN
cana-1907	68	22	,	,	PUNCT
cana-1907	68	23	𝑈2	𝑈2	PROPN
cana-1907	68	24	,	,	PUNCT
cana-1907	68	25	𝑎2	𝑎2	PROPN
cana-1907	68	26	,	,	PUNCT
cana-1907	68	27	𝑈2	𝑈2	PROPN
cana-1907	68	28	,	,	PUNCT
cana-1907	68	29	𝑎3	𝑎3	PROPN
cana-1907	68	30	,	,	PUNCT
cana-1907	68	31	𝑈2	𝑈2	PROPN
cana-1907	68	32	,	,	PUNCT
cana-1907	68	33	𝑎4	𝑎4	PROPN
cana-1907	68	34	)	)	PUNCT
cana-1907	68	35	}	}	PUNCT
cana-1907	68	36	.	.	PUNCT
cana-1907	69	1	given	give	VERB
cana-1907	69	2	four	four	NUM
cana-1907	69	3	non	non	ADJ
cana-1907	69	4	-	-	ADJ
cana-1907	69	5	empty	empty	ADJ
cana-1907	69	6	open	open	ADJ
cana-1907	69	7	subsets	subset	NOUN
cana-1907	69	8	of	of	ADP
cana-1907	69	9	x	x	NOUN
cana-1907	69	10	,	,	PUNCT
cana-1907	69	11	𝑊0,0	𝑊0,0	NOUN
cana-1907	69	12	,	,	PUNCT
cana-1907	69	13	𝑊1,0	𝑊1,0	NOUN
cana-1907	69	14	,	,	PUNCT
cana-1907	69	15	𝑊0,1	𝑊0,1	PROPN
cana-1907	69	16	,	,	PUNCT
cana-1907	69	17	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1907	69	18	𝑊1,1	𝑊1,1	PROPN
cana-1907	69	19	,	,	PUNCT
cana-1907	69	20	where	where	SCONJ
cana-1907	69	21	𝑊0,0	𝑊0,0	NOUN
cana-1907	69	22	and	and	CCONJ
cana-1907	69	23	𝑊1,0	𝑊1,0	NOUN
cana-1907	69	24	are	be	AUX
cana-1907	69	25	disjoint	disjoint	ADJ
cana-1907	69	26	,	,	PUNCT
cana-1907	69	27	as	as	SCONJ
cana-1907	69	28	are	be	AUX
cana-1907	69	29	𝑊0,1	𝑊0,1	NOUN
cana-1907	69	30	and	and	CCONJ
cana-1907	69	31	𝑊1,1	𝑊1,1	PROPN
cana-1907	69	32	.	.	PUNCT
cana-1907	70	1	each	each	DET
cana-1907	70	2	pair	pair	NOUN
cana-1907	70	3	of	of	ADP
cana-1907	70	4	subsets	subset	NOUN
cana-1907	70	5	is	be	AUX
cana-1907	70	6	contained	contain	VERB
cana-1907	70	7	within	within	ADP
cana-1907	70	8	different	different	ADJ
cana-1907	70	9	closed	closed	ADJ
cana-1907	70	10	sets	set	NOUN
cana-1907	70	11	,	,	PUNCT
cana-1907	70	12	𝐶0	𝐶0	NOUN
cana-1907	70	13	and	and	CCONJ
cana-1907	70	14	𝐶1	𝐶1	NUM
cana-1907	70	15	,	,	PUNCT
cana-1907	70	16	respectively	respectively	ADV
cana-1907	70	17	.	.	PUNCT
cana-1907	71	1	the	the	DET
cana-1907	71	2	mesh	mesh	NOUN
cana-1907	71	3	of	of	ADP
cana-1907	71	4	these	these	DET
cana-1907	71	5	subsets	subset	NOUN
cana-1907	71	6	is	be	AUX
cana-1907	71	7	less	less	ADJ
cana-1907	71	8	than	than	ADP
cana-1907	71	9	𝛿2	𝛿2	PROPN
cana-1907	71	10	.	.	PUNCT
cana-1907	72	1	then	then	ADV
cana-1907	72	2	,	,	PUNCT
cana-1907	72	3	𝜆2	𝜆2	PROPN
cana-1907	72	4	consists	consist	VERB
cana-1907	72	5	of	of	ADP
cana-1907	72	6	all	all	DET
cana-1907	72	7	possible	possible	ADJ
cana-1907	72	8	combinations	combination	NOUN
cana-1907	72	9	of	of	ADP
cana-1907	72	10	indices	index	NOUN
cana-1907	72	11	from	from	ADP
cana-1907	72	12	1	1	NUM
cana-1907	72	13	to	to	ADP
cana-1907	72	14	𝑡2	𝑡2	PROPN
cana-1907	72	15	.	.	PUNCT
cana-1907	73	1	a	a	DET
cana-1907	73	2	collection	collection	NOUN
cana-1907	73	3	of	of	ADP
cana-1907	73	4	open	open	ADJ
cana-1907	73	5	sets	set	NOUN
cana-1907	73	6	is	be	AUX
cana-1907	73	7	formed	form	VERB
cana-1907	73	8	from	from	ADP
cana-1907	73	9	the	the	DET
cana-1907	73	10	given	give	VERB
cana-1907	73	11	subsets	subset	NOUN
cana-1907	73	12	and	and	CCONJ
cana-1907	73	13	𝜆2	𝜆2	NOUN
cana-1907	73	14	.	.	PUNCT
cana-1907	74	1	by	by	ADP
cana-1907	74	2	the	the	DET
cana-1907	74	3	same	same	ADJ
cana-1907	74	4	result	result	NOUN
cana-1907	74	5	of	of	ADP
cana-1907	74	6	lemma	lemma	PROPN
cana-1907	74	7	1.2[see	1.2[see	NUM
cana-1907	74	8	1	1	NUM
cana-1907	74	9	]	]	PUNCT
cana-1907	74	10	there	there	PRON
cana-1907	74	11	exist	exist	VERB
cana-1907	74	12	4	4	NUM
cana-1907	74	13	closed	closed	ADJ
cana-1907	74	14	sets	set	NOUN
cana-1907	74	15	c0,0	c0,0	NOUN
cana-1907	74	16	,	,	PUNCT
cana-1907	74	17	c0,1,c1,0	c0,1,c1,0	X
cana-1907	74	18	,	,	PUNCT
cana-1907	74	19	c1,1	c1,1	NOUN
cana-1907	74	20	with	with	ADP
cana-1907	74	21	the	the	DET
cana-1907	74	22	following	follow	VERB
cana-1907	74	23	properties	property	NOUN
cana-1907	74	24	:	:	PUNCT
cana-1907	75	1	1	1	X
cana-1907	75	2	.	.	X
cana-1907	75	3	each	each	DET
cana-1907	75	4	int(ci	int(ci	PROPN
cana-1907	75	5	,	,	PUNCT
cana-1907	75	6	j	j	NOUN
cana-1907	75	7	)	)	PUNCT
cana-1907	75	8	is	be	AUX
cana-1907	75	9	nonempty	nonempty	ADJ
cana-1907	75	10	and	and	CCONJ
cana-1907	75	11	contained	contain	VERB
cana-1907	75	12	in	in	ADP
cana-1907	75	13	wi	wi	PROPN
cana-1907	75	14	,	,	PUNCT
cana-1907	75	15	j	j	PROPN
cana-1907	75	16	for	for	ADP
cana-1907	75	17	{	{	PUNCT
cana-1907	75	18	i	i	PROPN
cana-1907	75	19	,	,	PUNCT
cana-1907	75	20	j	j	PROPN
cana-1907	75	21	}	}	PUNCT
cana-1907	75	22	∈	∈	PROPN
cana-1907	75	23	{	{	PUNCT
cana-1907	75	24	0,1	0,1	NOUN
cana-1907	75	25	}	}	PUNCT
cana-1907	75	26	×	×	NOUN
cana-1907	75	27	{	{	PUNCT
cana-1907	75	28	0,1	0,1	NOUN
cana-1907	75	29	}	}	SYM
cana-1907	75	30	2	2	NUM
cana-1907	75	31	.	.	X
cana-1907	76	1	for	for	ADP
cana-1907	76	2	each	each	PRON
cana-1907	76	3	a	a	DET
cana-1907	76	4	∈	∈	NOUN
cana-1907	76	5	⟨c0,0,c0,1,c1,0,c1,1⟩	⟨c0,0,c0,1,c1,0,c1,1⟩	NOUN
cana-1907	76	6	and	and	CCONJ
cana-1907	76	7	for	for	ADP
cana-1907	76	8	each	each	DET
cana-1907	76	9	(	(	PUNCT
cana-1907	76	10	a1,a2,a3,a4	a1,a2,a3,a4	PROPN
cana-1907	76	11	)	)	PUNCT
cana-1907	76	12	∈	∈	PROPN
cana-1907	76	13	λ2	λ2	NOUN
cana-1907	76	14	there	there	ADV
cana-1907	76	15	exist	exist	VERB
cana-1907	76	16	𝑛	𝑛	DET
cana-1907	76	17	∈	∈	PROPN
cana-1907	76	18	𝑁such	𝑁such	PROPN
cana-1907	76	19	that	that	PRON
cana-1907	76	20	fn(a	fn(a	PUNCT
cana-1907	76	21	)	)	PUNCT
cana-1907	76	22	∈	∈	NOUN
cana-1907	76	23	⟨u2,a1,u2,a2,u2,a3,u2,a4⟩	⟨u2,a1,u2,a2,u2,a3,u2,a4⟩	NOUN
cana-1907	76	24	,	,	PUNCT
cana-1907	76	25	subsets	subset	NOUN
cana-1907	76	26	of	of	ADP
cana-1907	76	27	ℂ2	ℂ2	NOUN
cana-1907	76	28	namely	namely	ADV
cana-1907	76	29	ℂ2	ℂ2	X
cana-1907	76	30	𝑖	𝑖	PUNCT
cana-1907	76	31	with	with	ADP
cana-1907	76	32	𝑓𝑛(𝐴	𝑓𝑛(𝐴	PROPN
cana-1907	76	33	∩	∩	PROPN
cana-1907	76	34	ℂ2	ℂ2	NOUN
cana-1907	76	35	𝑖	𝑖	SYM
cana-1907	76	36	)	)	PUNCT
cana-1907	77	1	⊂	⊂	PROPN
cana-1907	77	2	𝑈2	𝑈2	PROPN
cana-1907	77	3	,	,	PUNCT
cana-1907	77	4	𝑎𝑖	𝑎𝑖	ADP
cana-1907	77	5	for	for	ADP
cana-1907	77	6	1	1	NUM
cana-1907	77	7	≤	≤	NUM
cana-1907	77	8	𝑖	𝑖	SYM
cana-1907	77	9	≤	≤	NOUN
cana-1907	77	10	4	4	NUM
cana-1907	77	11	,	,	PUNCT
cana-1907	77	12	where	where	SCONJ
cana-1907	77	13	ℂ2	ℂ2	NOUN
cana-1907	77	14	=	=	SYM
cana-1907	77	15	⟨𝐶0,0	⟨𝐶0,0	NOUN
cana-1907	77	16	,	,	PUNCT
cana-1907	77	17	𝐶0,1	𝐶0,1	ADV
cana-1907	77	18	,	,	PUNCT
cana-1907	77	19	𝐶1,0	𝐶1,0	PROPN
cana-1907	77	20	,	,	PUNCT
cana-1907	77	21	𝐶1,1⟩.	𝐶1,1⟩.	PROPN
cana-1907	77	22	note	note	NOUN
cana-1907	77	23	that	that	SCONJ
cana-1907	77	24	diam(ℂ2	diam(ℂ2	PUNCT
cana-1907	77	25	)	)	PUNCT
cana-1907	77	26	<	<	X
cana-1907	77	27	δ2	δ2	VERB
cana-1907	77	28	and	and	CCONJ
cana-1907	77	29	ℂ2⊂	ℂ2⊂	NOUN
cana-1907	77	30	ℂ1	ℂ1	PROPN
cana-1907	77	31	let	let	VERB
cana-1907	77	32	a	a	DET
cana-1907	77	33	∈	∈	NOUN
cana-1907	77	34	ℂ2	ℂ2	NOUN
cana-1907	77	35	and	and	CCONJ
cana-1907	77	36	{	{	PUNCT
cana-1907	77	37	p1,p2,p3,p4	p1,p2,p3,p4	PROPN
cana-1907	77	38	}	}	PUNCT
cana-1907	77	39	∈	∈	NOUN
cana-1907	77	40	f4(x	f4(x	NUM
cana-1907	77	41	)	)	PUNCT
cana-1907	77	42	,	,	PUNCT
cana-1907	77	43	then	then	ADV
cana-1907	77	44	there	there	PRON
cana-1907	77	45	exist	exist	VERB
cana-1907	77	46	(	(	PUNCT
cana-1907	77	47	a1,a2,a3,a4	a1,a2,a3,a4	PROPN
cana-1907	77	48	)	)	PUNCT
cana-1907	77	49	∈	∈	PROPN
cana-1907	77	50	λ2	λ2	NOUN
cana-1907	77	51	such	such	ADJ
cana-1907	77	52	that	that	DET
cana-1907	77	53	pi	pi	PROPN
cana-1907	77	54	∈	∈	PROPN
cana-1907	77	55	u2,ai	u2,ai	NOUN
cana-1907	77	56	since	since	SCONJ
cana-1907	77	57	there	there	PRON
cana-1907	77	58	exist	exist	VERB
cana-1907	77	59	n	n	ADV
cana-1907	77	60	so	so	SCONJ
cana-1907	77	61	that	that	PRON
cana-1907	77	62	fn(a	fn(a	PUNCT
cana-1907	77	63	)	)	PUNCT
cana-1907	78	1	∈	∈	PROPN
cana-1907	78	2	⟨u2,a1,u2,a2,u2,a3,u2,a4⟩.	⟨u2,a1,u2,a2,u2,a3,u2,a4⟩.	NOUN
cana-1907	78	3	we	we	PRON
cana-1907	78	4	conclude	conclude	VERB
cana-1907	78	5	that	that	DET
cana-1907	78	6	h({p1,a2,a3,a4},fn(a	h({p1,a2,a3,a4},fn(a	NOUN
cana-1907	78	7	)	)	PUNCT
cana-1907	78	8	)	)	PUNCT
cana-1907	78	9	<	<	X
cana-1907	78	10	δ2	δ2	VERB
cana-1907	78	11	.	.	PUNCT
cana-1907	79	1	so	so	ADV
cana-1907	79	2	for	for	SCONJ
cana-1907	79	3	every	every	DET
cana-1907	79	4	a	a	DET
cana-1907	79	5	∈ℂ2	∈ℂ2	NOUN
cana-1907	79	6	,	,	PUNCT
cana-1907	79	7	orb(ϕ	orb(ϕ	PROPN
cana-1907	79	8	,	,	PUNCT
cana-1907	79	9	a	a	PRON
cana-1907	79	10	)	)	PUNCT
cana-1907	79	11	is	be	AUX
cana-1907	79	12	δ2	δ2	VERB
cana-1907	79	13	-	-	PUNCT
cana-1907	79	14	close	close	NOUN
cana-1907	79	15	to	to	ADP
cana-1907	79	16	f4(x	f4(x	NUM
cana-1907	79	17	)	)	PUNCT
cana-1907	79	18	.	.	PUNCT
cana-1907	80	1	step	step	NOUN
cana-1907	80	2	3	3	NUM
cana-1907	80	3	let	let	VERB
cana-1907	80	4	's	us	PRON
cana-1907	80	5	say	say	VERB
cana-1907	80	6	that	that	SCONJ
cana-1907	80	7	ℂ𝑟	ℂ𝑟	PROPN
cana-1907	80	8	has	have	AUX
cana-1907	80	9	previously	previously	ADV
cana-1907	80	10	been	be	AUX
cana-1907	80	11	defined	define	VERB
cana-1907	80	12	and	and	CCONJ
cana-1907	80	13	has	have	VERB
cana-1907	80	14	the	the	DET
cana-1907	80	15	following	follow	VERB
cana-1907	80	16	attributes	attribute	NOUN
cana-1907	80	17	:	:	PUNCT
cana-1907	80	18	ℂ𝑟	ℂ𝑟	PROPN
cana-1907	80	19	is	be	AUX
cana-1907	80	20	defined	define	VERB
cana-1907	80	21	as	as	ADP
cana-1907	80	22	a	a	DET
cana-1907	80	23	collection	collection	NOUN
cana-1907	80	24	of	of	ADP
cana-1907	80	25	closed	closed	ADJ
cana-1907	80	26	sets	set	NOUN
cana-1907	80	27	of	of	ADP
cana-1907	80	28	x	x	PRON
cana-1907	80	29	,	,	PUNCT
cana-1907	80	30	represented	represent	VERB
cana-1907	80	31	as	as	ADP
cana-1907	80	32	⟨c0,0,0,	⟨c0,0,0,	NOUN
cana-1907	80	33	...	...	PUNCT
cana-1907	80	34	,	,	PUNCT
cana-1907	80	35	...	...	PUNCT
cana-1907	81	1	,c1,1,	,c1,1,	PROPN
cana-1907	81	2	...	...	PUNCT
cana-1907	81	3	⟩	⟩	NOUN
cana-1907	81	4	,	,	PUNCT
cana-1907	81	5	where	where	SCONJ
cana-1907	81	6	each	each	DET
cana-1907	81	7	closed	closed	ADJ
cana-1907	81	8	set	set	NOUN
cana-1907	81	9	is	be	AUX
cana-1907	81	10	indexed	index	VERB
cana-1907	81	11	by	by	ADP
cana-1907	81	12	a	a	DET
cana-1907	81	13	binary	binary	ADJ
cana-1907	81	14	sequence	sequence	NOUN
cana-1907	81	15	:	:	PUNCT
cana-1907	81	16	{	{	PUNCT
cana-1907	81	17	cj1,j2,	cj1,j2,	NOUN
cana-1907	81	18	...	...	PUNCT
cana-1907	81	19	,jr\(j1,j2,	,jr\(j1,j2,	NUM
cana-1907	81	20	...	...	PUNCT
cana-1907	81	21	,jr	,jr	PUNCT
cana-1907	81	22	)	)	PUNCT
cana-1907	81	23	∈	∈	PROPN
cana-1907	81	24	{	{	PUNCT
cana-1907	81	25	0,1}r	0,1}r	NOUN
cana-1907	81	26	}	}	PUNCT
cana-1907	81	27	.	.	PUNCT
cana-1907	82	1	this	this	DET
cana-1907	82	2	indexing	indexing	NOUN
cana-1907	82	3	scheme	scheme	NOUN
cana-1907	82	4	corresponds	correspond	VERB
cana-1907	82	5	to	to	ADP
cana-1907	82	6	2r	2r	NUM
cana-1907	82	7	possible	possible	ADJ
cana-1907	82	8	combinations	combination	NOUN
cana-1907	82	9	,	,	PUNCT
cana-1907	82	10	denoting	denote	VERB
cana-1907	82	11	the	the	DET
cana-1907	82	12	presence	presence	NOUN
cana-1907	82	13	or	or	CCONJ
cana-1907	82	14	absence	absence	NOUN
cana-1907	82	15	of	of	ADP
cana-1907	82	16	each	each	DET
cana-1907	82	17	closed	close	VERB
cana-1907	82	18	set	set	VERB
cana-1907	82	19	in	in	ADP
cana-1907	82	20	ℂ𝑟	ℂ𝑟	PROPN
cana-1907	82	21	.	.	PUNCT
cana-1907	83	1	each	each	DET
cana-1907	83	2	combination	combination	NOUN
cana-1907	83	3	delineates	delineate	VERB
cana-1907	83	4	the	the	DET
cana-1907	83	5	composition	composition	NOUN
cana-1907	83	6	of	of	ADP
cana-1907	83	7	ℂ𝑟	ℂ𝑟	PROPN
cana-1907	83	8	and	and	CCONJ
cana-1907	83	9	its	its	PRON
cana-1907	83	10	constituent	constituent	NOUN
cana-1907	83	11	closed	close	VERB
cana-1907	83	12	sets	set	NOUN
cana-1907	83	13	.	.	PUNCT
cana-1907	84	1	•	•	NUM
cana-1907	84	2	(	(	PUNCT
cana-1907	84	3	𝑗1	𝑗1	PROPN
cana-1907	84	4	,	,	PUNCT
cana-1907	84	5	𝑗2	𝑗2	PROPN
cana-1907	84	6	,	,	PUNCT
cana-1907	84	7	…	…	PUNCT
cana-1907	84	8	,	,	PUNCT
cana-1907	84	9	𝑗𝑟	𝑗𝑟	INTJ
cana-1907	84	10	)	)	PUNCT
cana-1907	84	11	∈	∈	PROPN
cana-1907	84	12	{	{	PUNCT
cana-1907	84	13	0,1}𝑟	0,1}𝑟	NUM
cana-1907	84	14	and	and	CCONJ
cana-1907	84	15	𝑖𝑛𝑡(𝐶𝑗1𝑗2	𝑖𝑛𝑡(𝐶𝑗1𝑗2	NUM
cana-1907	84	16	…	…	SYM
cana-1907	84	17	.𝑗𝑟	.𝑗𝑟	X
cana-1907	84	18	)	)	PUNCT
cana-1907	84	19	is	be	AUX
cana-1907	84	20	non	non	PROPN
cana-1907	84	21	empty,𝐶𝑗1𝑗2	empty,𝐶𝑗1𝑗2	PROPN
cana-1907	84	22	…	…	SYM
cana-1907	84	23	.𝑗𝑟	.𝑗𝑟	PUNCT
cana-1907	85	1	⊂	⊂	PROPN
cana-1907	85	2	𝐶𝑗2	𝐶𝑗2	VERB
cana-1907	85	3	…	…	PUNCT
cana-1907	85	4	.𝑗𝑟	.𝑗𝑟	PROPN
cana-1907	85	5	and	and	CCONJ
cana-1907	85	6	𝑑𝑖𝑎𝑚(𝐶𝑗1𝑗2	𝑑𝑖𝑎𝑚(𝐶𝑗1𝑗2	PROPN
cana-1907	85	7	…	…	SYM
cana-1907	85	8	.𝑗𝑟	.𝑗𝑟	X
cana-1907	85	9	)	)	PUNCT
cana-1907	86	1	<	<	X
cana-1907	86	2	𝛿𝑟	𝛿𝑟	INTJ
cana-1907	86	3	•	•	NUM
cana-1907	86	4	diam(ℂ𝑟	diam(ℂ𝑟	NOUN
cana-1907	86	5	)	)	PUNCT
cana-1907	86	6	is	be	AUX
cana-1907	86	7	less	less	ADJ
cana-1907	86	8	than	than	ADP
cana-1907	86	9	δr	δr	NOUN
cana-1907	86	10	and	and	CCONJ
cana-1907	86	11	ℂ𝑟	ℂ𝑟	PROPN
cana-1907	86	12	contained	contain	VERB
cana-1907	86	13	in	in	ADP
cana-1907	86	14	ℂ𝑟−1	ℂ𝑟−1	PROPN
cana-1907	86	15	communications	communication	NOUN
cana-1907	86	16	on	on	ADP
cana-1907	86	17	applied	apply	VERB
cana-1907	86	18	nonlinear	nonlinear	ADJ
cana-1907	86	19	analysis	analysis	NOUN
cana-1907	86	20	issn	issn	NOUN
cana-1907	86	21	:	:	PUNCT
cana-1907	86	22	1074	1074	NUM
cana-1907	86	23	-	-	PUNCT
cana-1907	86	24	133x	133x	NUM
cana-1907	86	25	vol	vol	NOUN
cana-1907	86	26	32	32	NUM
cana-1907	86	27	no	no	NOUN
cana-1907	86	28	.	.	NOUN
cana-1907	86	29	2	2	NUM
cana-1907	86	30	(	(	PUNCT
cana-1907	86	31	2025	2025	NUM
cana-1907	86	32	)	)	PUNCT
cana-1907	86	33	55	55	NUM
cana-1907	86	34	https://internationalpubls.com	https://internationalpubls.com	X
cana-1907	86	35	•	•	NOUN
cana-1907	86	36	for	for	ADP
cana-1907	86	37	each	each	DET
cana-1907	86	38	pair	pair	NOUN
cana-1907	86	39	(	(	PUNCT
cana-1907	86	40	𝑗1	𝑗1	PROPN
cana-1907	86	41	,	,	PUNCT
cana-1907	86	42	𝑗2	𝑗2	PROPN
cana-1907	86	43	,	,	PUNCT
cana-1907	86	44	…	…	PUNCT
cana-1907	86	45	𝑗𝑟	𝑗𝑟	X
cana-1907	86	46	)	)	PUNCT
cana-1907	86	47	≠	≠	PROPN
cana-1907	86	48	(	(	PUNCT
cana-1907	86	49	𝑙1	𝑙1	PROPN
cana-1907	86	50	,	,	PUNCT
cana-1907	86	51	𝑙2	𝑙2	PROPN
cana-1907	86	52	,	,	PUNCT
cana-1907	86	53	…	…	PUNCT
cana-1907	86	54	.	.	PUNCT
cana-1907	87	1	,	,	PUNCT
cana-1907	87	2	𝑙𝑟	𝑙𝑟	ADP
cana-1907	87	3	)	)	PUNCT
cana-1907	87	4	in	in	ADP
cana-1907	87	5	{	{	PUNCT
cana-1907	87	6	0,1}r	0,1}r	NUM
cana-1907	87	7	;	;	PUNCT
cana-1907	87	8	𝐶𝑗1𝑗2	𝐶𝑗1𝑗2	ADV
cana-1907	87	9	…	…	PUNCT
cana-1907	87	10	.𝑗𝑟	.𝑗𝑟	PROPN
cana-1907	87	11	and	and	CCONJ
cana-1907	87	12	𝐶𝑙1𝑙2	𝐶𝑙1𝑙2	NOUN
cana-1907	87	13	…	…	PUNCT
cana-1907	87	14	.𝑙𝑟	.𝑙𝑟	PUNCT
cana-1907	87	15	are	be	AUX
cana-1907	87	16	disjoint	disjoint	ADJ
cana-1907	87	17	.	.	PUNCT
cana-1907	88	1	•	•	NUM
cana-1907	88	2	for	for	ADP
cana-1907	88	3	each	each	DET
cana-1907	88	4	a	a	DET
cana-1907	88	5	∈	∈	NOUN
cana-1907	88	6	ℂ𝑟	ℂ𝑟	PROPN
cana-1907	88	7	and	and	CCONJ
cana-1907	88	8	each	each	PRON
cana-1907	88	9	(	(	PUNCT
cana-1907	88	10	𝑡1	𝑡1	PROPN
cana-1907	88	11	,	,	PUNCT
cana-1907	88	12	𝑡2	𝑡2	PROPN
cana-1907	88	13	,	,	PUNCT
cana-1907	88	14	…	…	PUNCT
cana-1907	88	15	,	,	PUNCT
cana-1907	88	16	𝑡2𝑟	𝑡2𝑟	NUM
cana-1907	88	17	)	)	PUNCT
cana-1907	88	18	in	in	ADP
cana-1907	88	19	λr	λr	NOUN
cana-1907	88	20	=	=	SYM
cana-1907	88	21	{	{	PUNCT
cana-1907	88	22	𝑡1	𝑡1	PROPN
cana-1907	88	23	,	,	PUNCT
cana-1907	88	24	𝑡2	𝑡2	PROPN
cana-1907	88	25	,	,	PUNCT
cana-1907	88	26	…	…	PUNCT
cana-1907	89	1	𝑡𝑟}2𝑟	𝑡𝑟}2𝑟	NOUN
cana-1907	89	2	there	there	ADV
cana-1907	89	3	exist	exist	VERB
cana-1907	89	4	𝑛	𝑛	DET
cana-1907	89	5	∈	∈	PROPN
cana-1907	89	6	𝑁	𝑁	PROPN
cana-1907	89	7	,	,	PUNCT
cana-1907	89	8	𝑓𝑛(𝐴	𝑓𝑛(𝐴	PROPN
cana-1907	89	9	)	)	PUNCT
cana-1907	89	10	∈	∈	PROPN
cana-1907	90	1	〈	〈	PROPN
cana-1907	90	2	𝑈𝑟	𝑈𝑟	PROPN
cana-1907	90	3	,	,	PUNCT
cana-1907	90	4	𝑡1	𝑡1	PROPN
cana-1907	90	5	,	,	PUNCT
cana-1907	90	6	𝑈𝑟	𝑈𝑟	PROPN
cana-1907	90	7	,	,	PUNCT
cana-1907	90	8	𝑡2	𝑡2	PROPN
cana-1907	90	9	,	,	PUNCT
cana-1907	90	10	…	…	PUNCT
cana-1907	90	11	.	.	PUNCT
cana-1907	90	12	,	,	PUNCT
cana-1907	91	1	𝑈𝑟	𝑈𝑟	PROPN
cana-1907	91	2	,	,	PUNCT
cana-1907	91	3	𝑡2𝑟	𝑡2𝑟	ADJ
cana-1907	91	4	〉	〉	NOUN
cana-1907	91	5	,	,	PUNCT
cana-1907	91	6	subsets	subset	NOUN
cana-1907	91	7	of	of	ADP
cana-1907	91	8	ℂ𝑟	ℂ𝑟	PROPN
cana-1907	91	9	namely	namely	ADV
cana-1907	91	10	ℂ𝑟	ℂ𝑟	PROPN
cana-1907	91	11	𝑖	𝑖	PROPN
cana-1907	91	12	with	with	ADP
cana-1907	91	13	𝑓𝑛(𝐴	𝑓𝑛(𝐴	PROPN
cana-1907	91	14	∩	∩	NOUN
cana-1907	91	15	ℂ𝑟	ℂ𝑟	PROPN
cana-1907	91	16	𝑖	𝑖	SYM
cana-1907	91	17	)	)	PUNCT
cana-1907	91	18	⊂	⊂	PROPN
cana-1907	92	1	𝑈𝑟	𝑈𝑟	PROPN
cana-1907	92	2	,	,	PUNCT
cana-1907	92	3	𝑡𝑖	𝑡𝑖	NOUN
cana-1907	92	4	,	,	PUNCT
cana-1907	92	5	for	for	ADP
cana-1907	92	6	1	1	NUM
cana-1907	92	7	≤	≤	NUM
cana-1907	92	8	𝑖	𝑖	SYM
cana-1907	92	9	≤	≤	NOUN
cana-1907	92	10	2𝑟.	2𝑟.	NUM
cana-1907	92	11	then	then	ADV
cana-1907	92	12	for	for	ADP
cana-1907	92	13	if	if	SCONJ
cana-1907	92	14	𝐴	𝐴	PROPN
cana-1907	92	15	∈	∈	PROPN
cana-1907	92	16	ℂ𝑟	ℂ𝑟	PROPN
cana-1907	92	17	,	,	PUNCT
cana-1907	92	18	then	then	ADV
cana-1907	92	19	𝑜𝑟𝑏(𝜙	𝑜𝑟𝑏(𝜙	PROPN
cana-1907	92	20	,	,	PUNCT
cana-1907	92	21	𝐴	𝐴	PROPN
cana-1907	92	22	)	)	PUNCT
cana-1907	92	23	is	be	AUX
cana-1907	92	24	𝛿𝑟	𝛿𝑟	ADV
cana-1907	92	25	close	close	ADJ
cana-1907	92	26	to	to	ADP
cana-1907	92	27	𝐹2𝑟(x	𝐹2𝑟(x	NUM
cana-1907	92	28	)	)	PUNCT
cana-1907	92	29	.	.	PUNCT
cana-1907	93	1	similarly	similarly	ADV
cana-1907	93	2	,	,	PUNCT
cana-1907	93	3	for	for	ADP
cana-1907	93	4	ℂ𝑟+1	ℂ𝑟+1	NOUN
cana-1907	93	5	.	.	PUNCT
cana-1907	93	6	step	step	NOUN
cana-1907	93	7	4	4	NUM
cana-1907	93	8	so	so	SCONJ
cana-1907	93	9	we	we	PRON
cana-1907	93	10	get	get	VERB
cana-1907	93	11	a	a	DET
cana-1907	93	12	declining	decline	VERB
cana-1907	93	13	order	order	NOUN
cana-1907	93	14	of	of	ADP
cana-1907	93	15	compact	compact	ADJ
cana-1907	93	16	subsets	subset	NOUN
cana-1907	93	17	of	of	ADP
cana-1907	93	18	𝑋	𝑋	PROPN
cana-1907	93	19	,	,	PUNCT
cana-1907	93	20	{	{	PUNCT
cana-1907	93	21	ℂ𝑟}1	ℂ𝑟}1	NOUN
cana-1907	93	22	∞	∞	NOUN
cana-1907	93	23	let	let	VERB
cana-1907	93	24	{	{	PUNCT
cana-1907	93	25	𝐶𝑙1𝑙2	𝐶𝑙1𝑙2	NOUN
cana-1907	93	26	…	…	PUNCT
cana-1907	93	27	.𝑙𝑟:(𝑙1	.𝑙𝑟:(𝑙1	PROPN
cana-1907	93	28	,	,	PUNCT
cana-1907	93	29	𝑙2	𝑙2	PROPN
cana-1907	93	30	,	,	PUNCT
cana-1907	93	31	…	…	PUNCT
cana-1907	93	32	,	,	PUNCT
cana-1907	93	33	𝑙𝑟	𝑙𝑟	ADP
cana-1907	93	34	)	)	PUNCT
cana-1907	93	35	∈	∈	PROPN
cana-1907	93	36	{	{	PUNCT
cana-1907	93	37	0,1}𝑟	0,1}𝑟	NOUN
cana-1907	93	38	}	}	PUNCT
cana-1907	93	39	be	be	VERB
cana-1907	93	40	the	the	DET
cana-1907	93	41	2𝑟	2𝑟	NUM
cana-1907	93	42	compact	compact	ADJ
cana-1907	93	43	subset	subset	NOUN
cana-1907	93	44	of	of	ADP
cana-1907	93	45	x	x	PRON
cana-1907	93	46	that	that	PRON
cana-1907	93	47	define	define	VERB
cana-1907	93	48	ℂ𝑟	ℂ𝑟	PROPN
cana-1907	93	49	.	.	PUNCT
cana-1907	94	1	∞	∞	PROPN
cana-1907	94	2	let	let	VERB
cana-1907	94	3	𝐶	𝐶	PROPN
cana-1907	94	4	=	=	SYM
cana-1907	94	5	⋂	⋂	PROPN
cana-1907	94	6	(	(	PUNCT
cana-1907	94	7	∪	∪	X
cana-1907	94	8	{	{	PUNCT
cana-1907	94	9	𝐶𝑙1,𝑙2,	𝐶𝑙1,𝑙2,	NOUN
cana-1907	94	10	…	…	PUNCT
cana-1907	94	11	𝑙𝑟	𝑙𝑟	NUM
cana-1907	94	12	,	,	PUNCT
cana-1907	94	13	(	(	PUNCT
cana-1907	94	14	𝑙1	𝑙1	PROPN
cana-1907	94	15	,	,	PUNCT
cana-1907	94	16	𝑙2	𝑙2	PROPN
cana-1907	94	17	,	,	PUNCT
cana-1907	94	18	…	…	PUNCT
cana-1907	94	19	,	,	PUNCT
cana-1907	94	20	𝑙𝑟	𝑙𝑟	ADP
cana-1907	94	21	)	)	PUNCT
cana-1907	94	22	∈	∈	PROPN
cana-1907	94	23	{	{	PUNCT
cana-1907	94	24	0,1}𝑟})∞	0,1}𝑟})∞	PROPN
cana-1907	94	25	𝑟=1	𝑟=1	PROPN
cana-1907	94	26	.	.	PUNCT
cana-1907	95	1	then	then	ADV
cana-1907	95	2	c	c	PROPN
cana-1907	95	3	is	be	AUX
cana-1907	95	4	a	a	DET
cana-1907	95	5	cantor	cantor	NOUN
cana-1907	95	6	set	set	VERB
cana-1907	95	7	in	in	ADP
cana-1907	95	8	x	x	X
cana-1907	95	9	.	.	PUNCT
cana-1907	96	1	for	for	ADP
cana-1907	96	2	all	all	DET
cana-1907	96	3	𝑟	𝑟	NOUN
cana-1907	96	4	,	,	PUNCT
cana-1907	96	5	𝐶	𝐶	PROPN
cana-1907	96	6	∈	∈	PROPN
cana-1907	96	7	ℂ𝑟	ℂ𝑟	PROPN
cana-1907	96	8	,	,	PUNCT
cana-1907	96	9	so	so	ADV
cana-1907	96	10	𝑜𝑟𝑏(𝜙	𝑜𝑟𝑏(𝜙	PROPN
cana-1907	96	11	,	,	PUNCT
cana-1907	96	12	𝐶	𝐶	PROPN
cana-1907	96	13	)	)	PUNCT
cana-1907	96	14	=	=	SYM
cana-1907	96	15	𝐾(𝑋	𝐾(𝑋	NOUN
cana-1907	96	16	)	)	PUNCT
cana-1907	96	17	.	.	PUNCT
cana-1907	97	1	theorem	theorem	VERB
cana-1907	97	2	2.1	2.1	NUM
cana-1907	97	3	.	.	PUNCT
cana-1907	98	1	there	there	PRON
cana-1907	98	2	exist	exist	VERB
cana-1907	98	3	a	a	DET
cana-1907	98	4	cantor	cantor	NOUN
cana-1907	98	5	set	set	NOUN
cana-1907	98	6	c	c	NOUN
cana-1907	98	7	⊆	⊆	NUM
cana-1907	98	8	x	x	SYM
cana-1907	98	9	such	such	ADJ
cana-1907	98	10	that	that	DET
cana-1907	98	11	𝑜𝑟𝑏(𝑓	𝑜𝑟𝑏(𝑓	PROPN
cana-1907	98	12	,	,	PUNCT
cana-1907	98	13	𝑥)̅̅	𝑥)̅̅	PROPN
cana-1907	98	14	̅̅	̅̅	PROPN
cana-1907	98	15	̅̅	̅̅	PROPN
cana-1907	98	16	̅̅	̅̅	PROPN
cana-1907	98	17	̅̅	̅̅	PROPN
cana-1907	98	18	̅̅	̅̅	PROPN
cana-1907	98	19	=	=	PROPN
cana-1907	98	20	𝑋	𝑋	PROPN
cana-1907	98	21	for	for	ADP
cana-1907	98	22	every	every	DET
cana-1907	98	23	x	x	PROPN
cana-1907	98	24	∈	∈	PROPN
cana-1907	98	25	𝑓𝑛(𝐶	𝑓𝑛(𝐶	PROPN
cana-1907	98	26	)	)	PUNCT
cana-1907	98	27	for	for	ADP
cana-1907	98	28	all	all	DET
cana-1907	98	29	n	n	PRON
cana-1907	98	30	𝑛	𝑛	PRON
cana-1907	98	31	∈ℕ	∈ℕ	NOUN
cana-1907	98	32	.	.	PUNCT
cana-1907	99	1	proof	proof	NOUN
cana-1907	99	2	.	.	PUNCT
cana-1907	100	1	clear	clear	ADJ
cana-1907	100	2	from	from	ADP
cana-1907	100	3	lemma	lemma	PROPN
cana-1907	100	4	2.3	2.3	NUM
cana-1907	100	5	theorem	theorem	NOUN
cana-1907	100	6	2.2	2.2	NUM
cana-1907	100	7	.	.	PUNCT
cana-1907	101	1	let	let	VERB
cana-1907	101	2	𝑓	𝑓	PRON
cana-1907	101	3	:	:	PUNCT
cana-1907	101	4	k(x	k(x	PROPN
cana-1907	101	5	)	)	PUNCT
cana-1907	101	6	→	→	SYM
cana-1907	101	7	k(x	k(x	PROPN
cana-1907	101	8	)	)	PUNCT
cana-1907	101	9	be	be	AUX
cana-1907	101	10	transitive	transitive	ADJ
cana-1907	101	11	.	.	PUNCT
cana-1907	102	1	then	then	ADV
cana-1907	102	2	there	there	PRON
cana-1907	102	3	exist	exist	VERB
cana-1907	102	4	a	a	DET
cana-1907	102	5	cantor	cantor	NOUN
cana-1907	102	6	set	set	NOUN
cana-1907	102	7	c	c	NOUN
cana-1907	102	8	⊆	⊆	NUM
cana-1907	102	9	x	x	SYM
cana-1907	102	10	such	such	ADJ
cana-1907	102	11	that	that	DET
cana-1907	102	12	𝑜𝑟𝑏(𝑓	𝑜𝑟𝑏(𝑓	PROPN
cana-1907	102	13	,	,	PUNCT
cana-1907	102	14	𝑥)̅̅	𝑥)̅̅	PROPN
cana-1907	102	15	̅̅	̅̅	PROPN
cana-1907	102	16	̅̅	̅̅	PROPN
cana-1907	102	17	̅̅	̅̅	PROPN
cana-1907	102	18	̅̅	̅̅	PROPN
cana-1907	102	19	̅̅	̅̅	PROPN
cana-1907	102	20	=	=	PROPN
cana-1907	102	21	𝑋	𝑋	PROPN
cana-1907	102	22	for	for	ADP
cana-1907	102	23	every	every	DET
cana-1907	102	24	x	x	SYM
cana-1907	102	25	∈	∈	PROPN
cana-1907	102	26	c	c	NOUN
cana-1907	102	27	proof	proof	NOUN
cana-1907	102	28	.	.	PUNCT
cana-1907	103	1	clear	clear	ADJ
cana-1907	103	2	from	from	ADP
cana-1907	103	3	lemma	lemma	PROPN
cana-1907	103	4	2.1	2.1	NUM
cana-1907	103	5	and	and	CCONJ
cana-1907	103	6	theorem	theorem	VERB
cana-1907	103	7	2.2	2.2	NUM
cana-1907	103	8	□	□	PUNCT
cana-1907	103	9	theorem	theorem	ADJ
cana-1907	103	10	2.3	2.3	NUM
cana-1907	103	11	.	.	PUNCT
cana-1907	104	1	let	let	VERB
cana-1907	104	2	𝐷	𝐷	NOUN
cana-1907	104	3	=	=	PUNCT
cana-1907	104	4	{	{	PUNCT
cana-1907	104	5	𝑥	𝑥	PRON
cana-1907	104	6	∈	∈	PROPN
cana-1907	104	7	𝑋	𝑋	PROPN
cana-1907	104	8	,	,	PUNCT
cana-1907	104	9	𝑜𝑟𝑏(𝑓	𝑜𝑟𝑏(𝑓	PROPN
cana-1907	104	10	,	,	PUNCT
cana-1907	104	11	𝑥)̅̅	𝑥)̅̅	PROPN
cana-1907	104	12	̅̅	̅̅	PROPN
cana-1907	104	13	̅̅	̅̅	PROPN
cana-1907	104	14	̅̅	̅̅	PROPN
cana-1907	104	15	̅̅	̅̅	PROPN
cana-1907	104	16	̅̅	̅̅	PROPN
cana-1907	104	17	=	=	PROPN
cana-1907	104	18	𝑋	𝑋	PROPN
cana-1907	104	19	}	}	PUNCT
cana-1907	104	20	and	and	CCONJ
cana-1907	104	21	𝑓	𝑓	PRON
cana-1907	104	22	:	:	PUNCT
cana-1907	104	23	k(x	k(x	PROPN
cana-1907	104	24	)	)	PUNCT
cana-1907	104	25	→	→	SYM
cana-1907	104	26	k(x	k(x	PROPN
cana-1907	104	27	)	)	PUNCT
cana-1907	104	28	is	be	AUX
cana-1907	104	29	transitive	transitive	ADJ
cana-1907	104	30	.	.	PUNCT
cana-1907	105	1	then	then	ADV
cana-1907	105	2	c	c	PROPN
cana-1907	105	3	⊆	⊆	NUM
cana-1907	105	4	d	d	NOUN
cana-1907	105	5	,	,	PUNCT
cana-1907	105	6	that	that	PRON
cana-1907	105	7	is	is	ADV
cana-1907	105	8	d	d	X
cana-1907	105	9	nonempty	nonempty	X
cana-1907	105	10	.	.	PUNCT
cana-1907	106	1	also	also	ADV
cana-1907	106	2	d	d	PRON
cana-1907	106	3	is	be	AUX
cana-1907	106	4	a	a	DET
cana-1907	106	5	dense	dense	ADJ
cana-1907	106	6	subset	subset	NOUN
cana-1907	106	7	of	of	ADP
cana-1907	106	8	x	x	PUNCT
cana-1907	106	9	fully	fully	ADV
cana-1907	106	10	invariant	invariant	ADJ
cana-1907	106	11	under	under	ADP
cana-1907	106	12	f	f	PROPN
cana-1907	106	13	,	,	PUNCT
cana-1907	106	14	that	that	PRON
cana-1907	106	15	is	is	ADV
cana-1907	106	16	f(d	f(d	PROPN
cana-1907	106	17	)	)	PUNCT
cana-1907	106	18	⊆	⊆	NUM
cana-1907	106	19	d	d	NOUN
cana-1907	106	20	proof	proof	NOUN
cana-1907	106	21	.	.	PUNCT
cana-1907	107	1	from	from	ADP
cana-1907	107	2	theorem	theorem	ADJ
cana-1907	107	3	2.3	2.3	NUM
cana-1907	107	4	,	,	PUNCT
cana-1907	107	5	we	we	PRON
cana-1907	107	6	can	can	AUX
cana-1907	107	7	find	find	VERB
cana-1907	107	8	a	a	DET
cana-1907	107	9	cantor	cantor	NOUN
cana-1907	107	10	set	set	NOUN
cana-1907	107	11	c	c	PROPN
cana-1907	107	12	in	in	ADP
cana-1907	107	13	x	x	PUNCT
cana-1907	107	14	with	with	ADP
cana-1907	107	15	c	c	PROPN
cana-1907	107	16	⊆	⊆	NUM
cana-1907	107	17	d.	d.	NOUN
cana-1907	107	18	that	that	PRON
cana-1907	107	19	is	be	AUX
cana-1907	107	20	d	d	NOUN
cana-1907	107	21	is	be	AUX
cana-1907	107	22	nonempty	nonempty	X
cana-1907	107	23	.	.	PUNCT
cana-1907	108	1	for	for	ADP
cana-1907	108	2	the	the	DET
cana-1907	108	3	proof	proof	NOUN
cana-1907	108	4	of	of	ADP
cana-1907	108	5	the	the	DET
cana-1907	108	6	remaining	remain	VERB
cana-1907	108	7	part	part	NOUN
cana-1907	108	8	,	,	PUNCT
cana-1907	108	9	see[3	see[3	X
cana-1907	108	10	]	]	PUNCT
cana-1907	108	11	and[4	and[4	NUM
cana-1907	108	12	]	]	X
cana-1907	108	13	□	□	PUNCT
cana-1907	108	14	theorem	theorem	VERB
cana-1907	108	15	2.4	2.4	NUM
cana-1907	108	16	.	.	PUNCT
cana-1907	109	1	let	let	VERB
cana-1907	109	2	ϕ	ϕ	NOUN
cana-1907	109	3	:	:	PUNCT
cana-1907	109	4	k(x	k(x	PROPN
cana-1907	109	5	)	)	PUNCT
cana-1907	109	6	→	→	SYM
cana-1907	109	7	k(x	k(x	PROPN
cana-1907	109	8	)	)	PUNCT
cana-1907	109	9	is	be	AUX
cana-1907	109	10	transitive	transitive	ADJ
cana-1907	109	11	and	and	CCONJ
cana-1907	109	12	𝐷	𝐷	PROPN
cana-1907	109	13	≠	≠	PROPN
cana-1907	109	14	𝑋.	𝑋.	NOUN
cana-1907	109	15	then	then	ADV
cana-1907	109	16	𝐷	𝐷	PROPN
cana-1907	109	17	∉	∉	PROPN
cana-1907	109	18	𝐾(𝑋	𝐾(𝑋	NOUN
cana-1907	109	19	)	)	PUNCT
cana-1907	109	20	.	.	PUNCT
cana-1907	110	1	proof	proof	NOUN
cana-1907	110	2	.	.	PUNCT
cana-1907	111	1	from	from	ADP
cana-1907	111	2	theorem	theorem	ADJ
cana-1907	111	3	2.4	2.4	NUM
cana-1907	111	4	,	,	PUNCT
cana-1907	111	5	d	d	NOUN
cana-1907	111	6	is	be	AUX
cana-1907	111	7	dense	dense	ADJ
cana-1907	111	8	in	in	ADP
cana-1907	111	9	x.	x.	NOUN
cana-1907	112	1	so	so	ADV
cana-1907	112	2	d	d	NOUN
cana-1907	112	3	is	be	AUX
cana-1907	112	4	not	not	PART
cana-1907	112	5	closed	closed	ADJ
cana-1907	113	1	and	and	CCONJ
cana-1907	113	2	so	so	ADV
cana-1907	113	3	it	it	PRON
cana-1907	113	4	is	be	AUX
cana-1907	113	5	not	not	PART
cana-1907	113	6	compact	compact	ADJ
cana-1907	113	7	.	.	PUNCT
cana-1907	114	1	□	□	PUNCT
cana-1907	114	2	in	in	ADP
cana-1907	114	3	the	the	DET
cana-1907	114	4	theorem	theorem	NOUN
cana-1907	114	5	we	we	PRON
cana-1907	114	6	have	have	VERB
cana-1907	114	7	a	a	DET
cana-1907	114	8	nowhere	nowhere	ADV
cana-1907	114	9	dense	dense	ADJ
cana-1907	114	10	set	set	NOUN
cana-1907	114	11	c	c	NOUN
cana-1907	114	12	in	in	ADP
cana-1907	114	13	x	x	PUNCT
cana-1907	114	14	with	with	ADP
cana-1907	114	15	its	its	PRON
cana-1907	114	16	orbit	orbit	NOUN
cana-1907	114	17	orb(ϕ	orb(ϕ	PROPN
cana-1907	114	18	,	,	PUNCT
cana-1907	114	19	c	c	X
cana-1907	114	20	)	)	PUNCT
cana-1907	114	21	is	be	AUX
cana-1907	114	22	dense	dense	ADJ
cana-1907	114	23	in	in	ADP
cana-1907	114	24	k(x	k(x	PROPN
cana-1907	114	25	)	)	PUNCT
cana-1907	114	26	.	.	PUNCT
cana-1907	115	1	but	but	CCONJ
cana-1907	115	2	at	at	ADP
cana-1907	115	3	the	the	DET
cana-1907	115	4	same	same	ADJ
cana-1907	115	5	time	time	NOUN
cana-1907	115	6	we	we	PRON
cana-1907	115	7	have	have	VERB
cana-1907	115	8	a	a	DET
cana-1907	115	9	dense	dense	ADJ
cana-1907	115	10	set	set	NOUN
cana-1907	115	11	d	d	NOUN
cana-1907	115	12	in	in	ADP
cana-1907	115	13	x	x	PUNCT
cana-1907	115	14	with	with	ADP
cana-1907	115	15	even	even	ADV
cana-1907	115	16	𝐷	𝐷	PROPN
cana-1907	115	17	∉	∉	ADJ
cana-1907	115	18	𝐾(𝑋	𝐾(𝑋	NOUN
cana-1907	115	19	)	)	PUNCT
cana-1907	115	20	.	.	PUNCT
cana-1907	116	1	theorem	theorem	VERB
cana-1907	116	2	2.5	2.5	NUM
cana-1907	116	3	.	.	PUNCT
cana-1907	117	1	let	let	AUX
cana-1907	117	2	f˜	f˜	VERB
cana-1907	117	3	:	:	PUNCT
cana-1907	117	4	k(x	k(x	PROPN
cana-1907	117	5	)	)	PUNCT
cana-1907	117	6	→	→	SYM
cana-1907	117	7	k(x	k(x	PROPN
cana-1907	117	8	)	)	PUNCT
cana-1907	117	9	is	be	AUX
cana-1907	117	10	transitive	transitive	ADJ
cana-1907	117	11	and	and	CCONJ
cana-1907	117	12	let	let	VERB
cana-1907	117	13	c	c	PRON
cana-1907	117	14	be	be	AUX
cana-1907	117	15	the	the	DET
cana-1907	117	16	cantor	cantor	NOUN
cana-1907	117	17	set	set	VERB
cana-1907	117	18	in	in	ADP
cana-1907	117	19	x.	x.	NOUN
cana-1907	117	20	then	then	ADV
cana-1907	117	21	for	for	ADP
cana-1907	117	22	any	any	DET
cana-1907	117	23	u	u	NOUN
cana-1907	117	24	⊆	⊆	NUM
cana-1907	117	25	x	x	PUNCT
cana-1907	117	26	and	and	CCONJ
cana-1907	117	27	for	for	ADP
cana-1907	117	28	any	any	DET
cana-1907	117	29	integer	integer	NOUN
cana-1907	117	30	m	m	NOUN
cana-1907	117	31	,	,	PUNCT
cana-1907	117	32	lim	lim	PROPN
cana-1907	117	33	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-1907	117	34	𝑛→∞	𝑛→∞	NUM
cana-1907	117	35	𝐻	𝐻	PROPN
cana-1907	117	36	(	(	PUNCT
cana-1907	117	37	𝑓𝑛(𝐶	𝑓𝑛(𝐶	PROPN
cana-1907	117	38	)	)	PUNCT
cana-1907	117	39	,	,	PUNCT
cana-1907	117	40	𝑓𝑛+𝑚(𝐶	𝑓𝑛+𝑚(𝐶	PROPN
cana-1907	117	41	)	)	PUNCT
cana-1907	117	42	)	)	PUNCT
cana-1907	117	43	≥	≥	NOUN
cana-1907	117	44	𝐻(𝑈	𝐻(𝑈	NOUN
cana-1907	117	45	,	,	PUNCT
cana-1907	117	46	𝑓𝑚(𝑈	𝑓𝑚(𝑈	PROPN
cana-1907	117	47	)	)	PUNCT
cana-1907	117	48	)	)	PUNCT
cana-1907	117	49	proof	proof	NOUN
cana-1907	117	50	.	.	PUNCT
cana-1907	118	1	let	let	VERB
cana-1907	118	2	u	u	PRON
cana-1907	118	3	and	and	CCONJ
cana-1907	118	4	m	m	AUX
cana-1907	118	5	be	be	AUX
cana-1907	118	6	given	give	VERB
cana-1907	118	7	and	and	CCONJ
cana-1907	118	8	let	let	VERB
cana-1907	118	9	be	be	AUX
cana-1907	118	10	the	the	DET
cana-1907	118	11	sequence	sequence	NOUN
cana-1907	118	12	.	.	PUNCT
cana-1907	119	1	since	since	SCONJ
cana-1907	119	2	f˜	f˜	PROPN
cana-1907	119	3	is	be	AUX
cana-1907	119	4	continuous	continuous	ADJ
cana-1907	119	5	and	and	CCONJ
cana-1907	119	6	𝑜𝑟𝑏(𝑓	𝑜𝑟𝑏(𝑓	ADJ
cana-1907	119	7	,	,	PUNCT
cana-1907	119	8	𝐶)̅̅	𝐶)̅̅	ADJ
cana-1907	119	9	̅̅	̅̅	PROPN
cana-1907	119	10	̅̅	̅̅	PROPN
cana-1907	119	11	̅̅	̅̅	PROPN
cana-1907	119	12	̅̅	̅̅	PROPN
cana-1907	119	13	̅̅	̅̅	PROPN
cana-1907	119	14	=	=	SYM
cana-1907	119	15	𝐾(𝑋	𝐾(𝑋	NOUN
cana-1907	119	16	)	)	PUNCT
cana-1907	119	17	there	there	PRON
cana-1907	119	18	exist	exist	VERB
cana-1907	119	19	for	for	ADP
cana-1907	119	20	every	every	DET
cana-1907	119	21	1	1	NUM
cana-1907	119	22	2𝑛	2𝑛	NUM
cana-1907	119	23	,	,	PUNCT
cana-1907	119	24	a	a	DET
cana-1907	119	25	positive	positive	ADJ
cana-1907	119	26	integer	integer	NOUN
cana-1907	119	27	𝑠𝑛	𝑠𝑛	NOUN
cana-1907	119	28	such	such	ADJ
cana-1907	119	29	that	that	SCONJ
cana-1907	119	30	𝑓𝑠𝑛(𝐶	𝑓𝑠𝑛(𝐶	NUM
cana-1907	119	31	)	)	PUNCT
cana-1907	119	32	so	so	ADV
cana-1907	119	33	close	close	ADJ
cana-1907	119	34	as	as	ADP
cana-1907	119	35	to	to	ADP
cana-1907	119	36	u	u	PRON
cana-1907	119	37	such	such	ADJ
cana-1907	119	38	that	that	SCONJ
cana-1907	119	39	𝐻(𝑓𝑠𝑛(𝐶	𝐻(𝑓𝑠𝑛(𝐶	PROPN
cana-1907	119	40	)	)	PUNCT
cana-1907	119	41	,	,	PUNCT
cana-1907	119	42	𝑈	𝑈	PROPN
cana-1907	119	43	)	)	PUNCT
cana-1907	119	44	<	<	X
cana-1907	119	45	1	1	NUM
cana-1907	119	46	2𝑛+1	2𝑛+1	NUM
cana-1907	119	47	and	and	CCONJ
cana-1907	119	48	𝐻(𝑓𝑚	𝐻(𝑓𝑚	X
cana-1907	119	49	(	(	PUNCT
cana-1907	119	50	𝑓𝑠𝑛(𝐶	𝑓𝑠𝑛(𝐶	NUM
cana-1907	119	51	)	)	PUNCT
cana-1907	119	52	,	,	PUNCT
cana-1907	119	53	𝑓𝑚(𝑈	𝑓𝑚(𝑈	PROPN
cana-1907	119	54	)	)	PUNCT
cana-1907	119	55	)	)	PUNCT
cana-1907	119	56	<	<	X
cana-1907	119	57	1	1	NUM
cana-1907	119	58	2𝑛+1	2𝑛+1	NUM
cana-1907	119	59	.	.	PUNCT
cana-1907	120	1	communications	communication	NOUN
cana-1907	120	2	on	on	ADP
cana-1907	120	3	applied	apply	VERB
cana-1907	120	4	nonlinear	nonlinear	ADJ
cana-1907	120	5	analysis	analysis	NOUN
cana-1907	120	6	issn	issn	NOUN
cana-1907	120	7	:	:	PUNCT
cana-1907	120	8	1074	1074	NUM
cana-1907	120	9	-	-	PUNCT
cana-1907	120	10	133x	133x	NUM
cana-1907	120	11	vol	vol	NOUN
cana-1907	120	12	32	32	NUM
cana-1907	120	13	no	no	NOUN
cana-1907	120	14	.	.	NOUN
cana-1907	120	15	2	2	NUM
cana-1907	120	16	(	(	PUNCT
cana-1907	120	17	2025	2025	NUM
cana-1907	120	18	)	)	PUNCT
cana-1907	120	19	56	56	NUM
cana-1907	120	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1907	120	21	this	this	PRON
cana-1907	120	22	implies	imply	VERB
cana-1907	120	23	𝐻	𝐻	PROPN
cana-1907	120	24	(	(	PUNCT
cana-1907	120	25	𝑓𝑠𝑛(𝐶	𝑓𝑠𝑛(𝐶	NUM
cana-1907	120	26	)	)	PUNCT
cana-1907	120	27	,	,	PUNCT
cana-1907	120	28	𝑓𝑠𝑛+𝑚(𝐶	𝑓𝑠𝑛+𝑚(𝐶	NOUN
cana-1907	120	29	)	)	PUNCT
cana-1907	120	30	)	)	PUNCT
cana-1907	120	31	>	>	PUNCT
cana-1907	121	1	𝐻	𝐻	PROPN
cana-1907	121	2	(	(	PUNCT
cana-1907	121	3	𝑈	𝑈	PROPN
cana-1907	121	4	,	,	PUNCT
cana-1907	121	5	𝑓𝑚(𝑈	𝑓𝑚(𝑈	PROPN
cana-1907	121	6	)	)	PUNCT
cana-1907	121	7	)	)	PUNCT
cana-1907	122	1	−	−	PROPN
cana-1907	122	2	1	1	NUM
cana-1907	122	3	2𝑛	2𝑛	PROPN
cana-1907	122	4	[	[	X
cana-1907	122	5	by	by	ADP
cana-1907	122	6	using	use	VERB
cana-1907	122	7	triangle	triangle	NOUN
cana-1907	122	8	inequality	inequality	NOUN
cana-1907	122	9	]	]	PUNCT
cana-1907	122	10	hence	hence	ADV
cana-1907	122	11	the	the	DET
cana-1907	122	12	result	result	NOUN
cana-1907	122	13	.	.	PUNCT
cana-1907	123	1	definition	definition	NOUN
cana-1907	123	2	2.3	2.3	NUM
cana-1907	123	3	.	.	PUNCT
cana-1907	124	1	let	let	VERB
cana-1907	124	2	x	x	PRON
cana-1907	124	3	be	be	AUX
cana-1907	124	4	a	a	DET
cana-1907	124	5	metric	metric	ADJ
cana-1907	124	6	space	space	NOUN
cana-1907	124	7	with	with	ADP
cana-1907	124	8	metric	metric	ADJ
cana-1907	124	9	d	d	NOUN
cana-1907	124	10	and	and	CCONJ
cana-1907	124	11	f	f	NOUN
cana-1907	124	12	:	:	PUNCT
cana-1907	124	13	x	x	X
cana-1907	124	14	→	→	PUNCT
cana-1907	124	15	x	x	PUNCT
cana-1907	124	16	be	be	AUX
cana-1907	124	17	a	a	DET
cana-1907	124	18	continuous	continuous	ADJ
cana-1907	124	19	map	map	NOUN
cana-1907	124	20	.	.	PUNCT
cana-1907	125	1	f	f	PROPN
cana-1907	125	2	is	be	AUX
cana-1907	125	3	said	say	VERB
cana-1907	125	4	to	to	PART
cana-1907	125	5	be	be	AUX
cana-1907	125	6	almost	almost	ADV
cana-1907	125	7	sensitive	sensitive	ADJ
cana-1907	125	8	if	if	SCONJ
cana-1907	125	9	we	we	PRON
cana-1907	125	10	can	can	AUX
cana-1907	125	11	find	find	VERB
cana-1907	125	12	an	an	DET
cana-1907	125	13	x	x	SYM
cana-1907	125	14	∈	∈	PROPN
cana-1907	125	15	x	x	X
cana-1907	125	16	and	and	CCONJ
cana-1907	125	17	𝑚	𝑚	PROPN
cana-1907	125	18	∈	∈	PROPN
cana-1907	125	19	ℕ	ℕ	PROPN
cana-1907	125	20	such	such	ADJ
cana-1907	125	21	that	that	PRON
cana-1907	125	22	for	for	ADP
cana-1907	125	23	any	any	DET
cana-1907	125	24	lim	lim	NOUN
cana-1907	125	25	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-1907	125	26	𝑛→∞	𝑛→∞	NUM
cana-1907	125	27	𝑑(𝑓𝑛(𝑥	𝑑(𝑓𝑛(𝑥	NUM
cana-1907	125	28	)	)	PUNCT
cana-1907	125	29	,	,	PUNCT
cana-1907	125	30	𝑓𝑛+𝑚(𝑦	𝑓𝑛+𝑚(𝑦	PROPN
cana-1907	125	31	)	)	PUNCT
cana-1907	125	32	)	)	PUNCT
cana-1907	125	33	≥	≥	NOUN
cana-1907	125	34	𝑑(𝑦	𝑑(𝑦	NOUN
cana-1907	125	35	,	,	PUNCT
cana-1907	125	36	𝑓𝑚(𝑦	𝑓𝑚(𝑦	NUM
cana-1907	125	37	)	)	PUNCT
cana-1907	125	38	)	)	PUNCT
cana-1907	125	39	.	.	PUNCT
cana-1907	126	1	theorem	theorem	VERB
cana-1907	126	2	2.6	2.6	NUM
cana-1907	126	3	.	.	PUNCT
cana-1907	127	1	in	in	ADP
cana-1907	127	2	k(x	k(x	PROPN
cana-1907	127	3	)	)	PUNCT
cana-1907	127	4	,	,	PUNCT
cana-1907	127	5	transitive	transitive	ADJ
cana-1907	127	6	maps	map	NOUN
cana-1907	127	7	are	be	AUX
cana-1907	127	8	almost	almost	ADV
cana-1907	127	9	sensitive	sensitive	ADJ
cana-1907	127	10	.	.	PUNCT
cana-1907	128	1	proof	proof	NOUN
cana-1907	128	2	.	.	PUNCT
cana-1907	129	1	clear	clear	ADJ
cana-1907	129	2	from	from	ADP
cana-1907	129	3	lemma	lemma	PROPN
cana-1907	129	4	2.3	2.3	NUM
cana-1907	129	5	and	and	CCONJ
cana-1907	129	6	theorem	theorem	VERB
cana-1907	129	7	2.6	2.6	NUM
cana-1907	129	8	□	□	PUNCT
cana-1907	129	9	theorem	theorem	ADJ
cana-1907	129	10	2.7	2.7	NUM
cana-1907	129	11	.	.	PUNCT
cana-1907	130	1	𝑓is	𝑓is	PRON
cana-1907	130	2	almost	almost	ADV
cana-1907	130	3	sensitive	sensitive	ADJ
cana-1907	130	4	in	in	ADP
cana-1907	130	5	k(x	k(x	PROPN
cana-1907	130	6	)	)	PUNCT
cana-1907	130	7	implies	imply	VERB
cana-1907	130	8	f	f	PROPN
cana-1907	130	9	is	be	AUX
cana-1907	130	10	almost	almost	ADV
cana-1907	130	11	sensitive	sensitive	ADJ
cana-1907	130	12	in	in	ADP
cana-1907	130	13	x	x	NOUN
cana-1907	130	14	proof	proof	NOUN
cana-1907	130	15	.	.	PUNCT
cana-1907	131	1	clear	clear	ADJ
cana-1907	131	2	from	from	ADP
cana-1907	131	3	theorem	theorem	ADJ
cana-1907	131	4	2.7	2.7	NUM
cana-1907	131	5	3.stability	3.stability	NUM
cana-1907	131	6	of	of	ADP
cana-1907	131	7	induced	induced	ADJ
cana-1907	131	8	maps	map	NOUN
cana-1907	131	9	in	in	ADP
cana-1907	131	10	this	this	DET
cana-1907	131	11	section	section	NOUN
cana-1907	131	12	we	we	PRON
cana-1907	131	13	study	study	VERB
cana-1907	131	14	the	the	DET
cana-1907	131	15	stability	stability	NOUN
cana-1907	131	16	of	of	ADP
cana-1907	131	17	the	the	DET
cana-1907	131	18	induced	induced	ADJ
cana-1907	131	19	map	map	NOUN
cana-1907	131	20	𝑓	𝑓	PRON
cana-1907	131	21	:	:	PUNCT
cana-1907	131	22	k(x	k(x	PROPN
cana-1907	131	23	)	)	PUNCT
cana-1907	131	24	→	→	SYM
cana-1907	131	25	k(x	k(x	PROPN
cana-1907	131	26	)	)	PUNCT
cana-1907	131	27	and	and	CCONJ
cana-1907	131	28	its	its	PRON
cana-1907	131	29	connection	connection	NOUN
cana-1907	131	30	with	with	ADP
cana-1907	131	31	the	the	DET
cana-1907	131	32	stability	stability	NOUN
cana-1907	131	33	of	of	ADP
cana-1907	131	34	the	the	DET
cana-1907	131	35	continuous	continuous	ADJ
cana-1907	131	36	map	map	NOUN
cana-1907	132	1	f	f	X
cana-1907	132	2	:	:	PUNCT
cana-1907	132	3	x	x	X
cana-1907	132	4	→	→	PUNCT
cana-1907	132	5	x.	x.	NOUN
cana-1907	132	6	in	in	ADP
cana-1907	132	7	this	this	DET
cana-1907	132	8	section	section	NOUN
cana-1907	132	9	c(x	c(x	NOUN
cana-1907	132	10	)	)	PUNCT
cana-1907	132	11	denotes	denote	VERB
cana-1907	132	12	the	the	DET
cana-1907	132	13	space	space	NOUN
cana-1907	132	14	of	of	ADP
cana-1907	132	15	all	all	DET
cana-1907	132	16	continuous	continuous	ADJ
cana-1907	132	17	functions	function	NOUN
cana-1907	132	18	on	on	ADP
cana-1907	132	19	x	x	PUNCT
cana-1907	132	20	and	and	CCONJ
cana-1907	132	21	c(k(x	c(k(x	PROPN
cana-1907	132	22	)	)	PUNCT
cana-1907	132	23	)	)	PUNCT
cana-1907	133	1	denotes	denote	VERB
cana-1907	133	2	the	the	DET
cana-1907	133	3	space	space	NOUN
cana-1907	133	4	of	of	ADP
cana-1907	133	5	all	all	DET
cana-1907	133	6	continuous	continuous	ADJ
cana-1907	133	7	functions	function	NOUN
cana-1907	133	8	on	on	ADP
cana-1907	133	9	k(x	k(x	PROPN
cana-1907	133	10	)	)	PUNCT
cana-1907	133	11	.	.	PUNCT
cana-1907	134	1	definition	definition	NOUN
cana-1907	134	2	3.1	3.1	NUM
cana-1907	134	3	.	.	PUNCT
cana-1907	135	1	a	a	DET
cana-1907	135	2	point	point	NOUN
cana-1907	135	3	x	x	X
cana-1907	135	4	∈	∈	NOUN
cana-1907	135	5	x	x	PUNCT
cana-1907	135	6	is	be	AUX
cana-1907	135	7	said	say	VERB
cana-1907	135	8	to	to	PART
cana-1907	135	9	be	be	AUX
cana-1907	135	10	stable	stable	ADJ
cana-1907	135	11	if	if	SCONJ
cana-1907	135	12	for	for	ADP
cana-1907	135	13	all	all	PRON
cana-1907	135	14	ϵ	ϵ	X
cana-1907	135	15	>	>	X
cana-1907	135	16	0	0	PUNCT
cana-1907	136	1	there	there	PRON
cana-1907	136	2	is	be	VERB
cana-1907	136	3	a	a	DET
cana-1907	136	4	δ	δ	PROPN
cana-1907	136	5	>	>	X
cana-1907	136	6	0	0	NUM
cana-1907	136	7	such	such	ADJ
cana-1907	136	8	that	that	SCONJ
cana-1907	136	9	if	if	SCONJ
cana-1907	136	10	d(y	d(y	PROPN
cana-1907	136	11	,	,	PUNCT
cana-1907	136	12	x	x	X
cana-1907	136	13	)	)	PUNCT
cana-1907	136	14	<	<	X
cana-1907	136	15	δ	δ	PROPN
cana-1907	136	16	then	then	ADV
cana-1907	136	17	d(f	d(f	VERB
cana-1907	136	18	n(y),f	n(y),f	VERB
cana-1907	136	19	n(x	n(x	NOUN
cana-1907	136	20	)	)	PUNCT
cana-1907	136	21	)	)	PUNCT
cana-1907	136	22	<	<	X
cana-1907	137	1	ϵ	ϵ	X
cana-1907	137	2	for	for	ADP
cana-1907	137	3	every	every	DET
cana-1907	137	4	n.	n.	NOUN
cana-1907	137	5	a	a	DET
cana-1907	137	6	point	point	NOUN
cana-1907	137	7	x	x	VERB
cana-1907	137	8	is	be	AUX
cana-1907	137	9	said	say	VERB
cana-1907	137	10	to	to	PART
cana-1907	137	11	be	be	AUX
cana-1907	137	12	unstable	unstable	ADJ
cana-1907	137	13	if	if	SCONJ
cana-1907	137	14	it	it	PRON
cana-1907	137	15	is	be	AUX
cana-1907	137	16	stable	stable	ADJ
cana-1907	137	17	for	for	ADP
cana-1907	137	18	𝑓−1	𝑓−1	NUM
cana-1907	137	19	.	.	PROPN
cana-1907	137	20	definition	definition	NOUN
cana-1907	137	21	3.2	3.2	NUM
cana-1907	137	22	.	.	PUNCT
cana-1907	138	1	𝑓∈	𝑓∈	ADJ
cana-1907	138	2	k(x	k(x	PROPN
cana-1907	138	3	)	)	PUNCT
cana-1907	138	4	is	be	AUX
cana-1907	138	5	stable	stable	ADJ
cana-1907	138	6	if	if	SCONJ
cana-1907	138	7	given	give	VERB
cana-1907	138	8	ϵ	ϵ	PROPN
cana-1907	138	9	>	>	X
cana-1907	138	10	0	0	NUM
cana-1907	138	11	,	,	PUNCT
cana-1907	138	12	there	there	PRON
cana-1907	138	13	exists	exist	VERB
cana-1907	138	14	a	a	DET
cana-1907	138	15	δ	δ	PROPN
cana-1907	138	16	>	>	X
cana-1907	138	17	0	0	NUM
cana-1907	138	18	such	such	ADJ
cana-1907	138	19	that	that	PRON
cana-1907	138	20	for	for	ADP
cana-1907	138	21	each	each	DET
cana-1907	138	22	�	�	PROPN
cana-1907	138	23	̃	̃	PROPN
cana-1907	138	24	�	�	PROPN
cana-1907	138	25	∈	∈	NOUN
cana-1907	138	26	c(k(x	c(k(x	NOUN
cana-1907	138	27	)	)	PUNCT
cana-1907	138	28	)	)	PUNCT
cana-1907	138	29	with	with	ADP
cana-1907	138	30	dh(𝑓	dh(𝑓	NOUN
cana-1907	138	31	,	,	PUNCT
cana-1907	138	32	�	�	PROPN
cana-1907	138	33	̃	̃	PROPN
cana-1907	138	34	�	�	PROPN
cana-1907	138	35	)	)	PUNCT
cana-1907	138	36	<	<	X
cana-1907	138	37	δ	δ	PROPN
cana-1907	138	38	,	,	PUNCT
cana-1907	138	39	there	there	PRON
cana-1907	138	40	exist	exist	VERB
cana-1907	138	41	a	a	DET
cana-1907	138	42	continuous	continuous	ADJ
cana-1907	138	43	map	map	NOUN
cana-1907	138	44	h	h	PROPN
cana-1907	138	45	∈	∈	PROPN
cana-1907	138	46	c(x	c(x	NOUN
cana-1907	138	47	)	)	PUNCT
cana-1907	138	48	such	such	ADJ
cana-1907	138	49	that	that	SCONJ
cana-1907	138	50	𝑓	𝑓	DET
cana-1907	138	51	∘	∘	PROPN
cana-1907	138	52	�	�	PROPN
cana-1907	138	53	̃	̃	PROPN
cana-1907	138	54	�	�	PROPN
cana-1907	138	55	=	=	SYM
cana-1907	138	56	�	�	PROPN
cana-1907	138	57	̃	̃	PROPN
cana-1907	138	58	�	�	PROPN
cana-1907	138	59	∘	∘	NOUN
cana-1907	138	60	𝑓	𝑓	PROPN
cana-1907	138	61	and	and	CCONJ
cana-1907	138	62	dh(ℎ̃	dh(ℎ̃	PROPN
cana-1907	138	63	,	,	PUNCT
cana-1907	138	64	𝑖̃	𝑖̃	PROPN
cana-1907	138	65	)	)	PUNCT
cana-1907	138	66	<	<	X
cana-1907	139	1	ϵ	ϵ	X
cana-1907	139	2	where	where	SCONJ
cana-1907	139	3	𝑖̃	𝑖̃	PROPN
cana-1907	139	4	:	:	PUNCT
cana-1907	139	5	k(x	k(x	PROPN
cana-1907	139	6	)	)	PUNCT
cana-1907	139	7	→	→	SYM
cana-1907	139	8	k(x	k(x	PROPN
cana-1907	139	9	)	)	PUNCT
cana-1907	139	10	is	be	AUX
cana-1907	139	11	the	the	DET
cana-1907	139	12	identity	identity	NOUN
cana-1907	139	13	map	map	NOUN
cana-1907	139	14	and	and	CCONJ
cana-1907	139	15	dh(𝑓	dh(𝑓	NOUN
cana-1907	139	16	,	,	PUNCT
cana-1907	139	17	�	�	PROPN
cana-1907	139	18	̃	̃	PROPN
cana-1907	139	19	�	�	NOUN
cana-1907	139	20	)=sup	)=sup	PROPN
cana-1907	139	21	{	{	PUNCT
cana-1907	139	22	𝑑𝐻	𝑑𝐻	ADJ
cana-1907	139	23	(	(	PUNCT
cana-1907	139	24	𝑓(𝐴	𝑓(𝐴	NOUN
cana-1907	139	25	)	)	PUNCT
cana-1907	139	26	,	,	PUNCT
cana-1907	139	27	�	�	PROPN
cana-1907	139	28	̃	̃	PROPN
cana-1907	139	29	�	�	PROPN
cana-1907	139	30	(𝐴	(𝐴	NOUN
cana-1907	139	31	)	)	PUNCT
cana-1907	139	32	)	)	PUNCT
cana-1907	139	33	,	,	PUNCT
cana-1907	139	34	𝐴	𝐴	PROPN
cana-1907	139	35	∈	∈	PROPN
cana-1907	139	36	𝐾(𝑋	𝐾(𝑋	NOUN
cana-1907	139	37	)	)	PUNCT
cana-1907	139	38	}	}	PUNCT
cana-1907	139	39	.	.	PUNCT
cana-1907	140	1	theorem	theorem	VERB
cana-1907	140	2	3.1	3.1	NUM
cana-1907	140	3	.	.	PUNCT
cana-1907	141	1	f	f	X
cana-1907	141	2	:	:	PUNCT
cana-1907	142	1	x	x	X
cana-1907	142	2	→	→	PUNCT
cana-1907	142	3	x	x	PUNCT
cana-1907	142	4	is	be	AUX
cana-1907	142	5	stable	stable	ADJ
cana-1907	142	6	in	in	ADP
cana-1907	142	7	c(x	c(x	NOUN
cana-1907	142	8	)	)	PUNCT
cana-1907	143	1	if	if	SCONJ
cana-1907	143	2	and	and	CCONJ
cana-1907	143	3	only	only	ADV
cana-1907	143	4	if	if	SCONJ
cana-1907	143	5	𝑓	𝑓	PRON
cana-1907	143	6	:	:	PUNCT
cana-1907	143	7	k(x	k(x	PROPN
cana-1907	143	8	)	)	PUNCT
cana-1907	143	9	→	→	SYM
cana-1907	143	10	k(x	k(x	PROPN
cana-1907	143	11	)	)	PUNCT
cana-1907	143	12	is	be	AUX
cana-1907	143	13	stable	stable	ADJ
cana-1907	143	14	in	in	ADP
cana-1907	143	15	c(k(x	c(k(x	PROPN
cana-1907	143	16	)	)	PUNCT
cana-1907	143	17	)	)	PUNCT
cana-1907	143	18	.	.	PUNCT
cana-1907	144	1	proof	proof	NOUN
cana-1907	144	2	.	.	PUNCT
cana-1907	145	1	to	to	PART
cana-1907	145	2	prove	prove	VERB
cana-1907	145	3	the	the	DET
cana-1907	145	4	theorem	theorem	NOUN
cana-1907	145	5	,	,	PUNCT
cana-1907	145	6	first	first	ADV
cana-1907	145	7	we	we	PRON
cana-1907	145	8	prove	prove	VERB
cana-1907	145	9	the	the	DET
cana-1907	145	10	following	follow	VERB
cana-1907	145	11	lemmas	lemmas	PROPN
cana-1907	145	12	lemma	lemma	PROPN
cana-1907	145	13	3.1	3.1	NUM
cana-1907	145	14	.	.	PUNCT
cana-1907	146	1	the	the	DET
cana-1907	146	2	map	map	NOUN
cana-1907	146	3	𝜓	𝜓	X
cana-1907	146	4	:	:	PUNCT
cana-1907	146	5	c(x	c(x	NOUN
cana-1907	146	6	)	)	PUNCT
cana-1907	146	7	→	→	SYM
cana-1907	146	8	c(k(x	c(k(x	PROPN
cana-1907	146	9	)	)	PUNCT
cana-1907	146	10	)	)	PUNCT
cana-1907	146	11	given	give	VERB
cana-1907	146	12	by	by	ADP
cana-1907	146	13	𝜓(𝑓	𝜓(𝑓	ADJ
cana-1907	146	14	)	)	PUNCT
cana-1907	146	15	=	=	SYM
cana-1907	146	16	𝑓is	𝑓is	PROPN
cana-1907	146	17	an	an	DET
cana-1907	146	18	embedding	embedding	NOUN
cana-1907	146	19	of	of	ADP
cana-1907	146	20	c(x	c(x	NOUN
cana-1907	146	21	)	)	PUNCT
cana-1907	146	22	in	in	ADP
cana-1907	146	23	to	to	ADP
cana-1907	146	24	k(c(x	k(c(x	NOUN
cana-1907	146	25	)	)	PUNCT
cana-1907	146	26	)	)	PUNCT
cana-1907	146	27	proof	proof	NOUN
cana-1907	146	28	.	.	PUNCT
cana-1907	147	1	𝜓	𝜓	NOUN
cana-1907	147	2	is	be	AUX
cana-1907	147	3	well	well	ADV
cana-1907	147	4	defined	define	VERB
cana-1907	147	5	since	since	SCONJ
cana-1907	147	6	the	the	DET
cana-1907	147	7	induced	induced	ADJ
cana-1907	147	8	map	map	NOUN
cana-1907	147	9	𝑓	𝑓	PRON
cana-1907	147	10	given	give	VERB
cana-1907	147	11	by	by	ADP
cana-1907	147	12	a	a	DET
cana-1907	147	13	continuous	continuous	ADJ
cana-1907	147	14	map	map	NOUN
cana-1907	147	15	is	be	AUX
cana-1907	147	16	continuous	continuous	ADJ
cana-1907	147	17	.	.	PUNCT
cana-1907	148	1	next	next	ADV
cana-1907	148	2	we	we	PRON
cana-1907	148	3	have	have	VERB
cana-1907	148	4	to	to	PART
cana-1907	148	5	show	show	VERB
cana-1907	148	6	that	that	SCONJ
cana-1907	148	7	𝜓	𝜓	PROPN
cana-1907	148	8	is	be	AUX
cana-1907	148	9	injective	injective	ADJ
cana-1907	148	10	.	.	PUNCT
cana-1907	149	1	suppose	suppose	VERB
cana-1907	149	2	𝜓(𝑓	𝜓(𝑓	ADJ
cana-1907	149	3	)	)	PUNCT
cana-1907	149	4	=	=	SYM
cana-1907	149	5	𝜓(𝑔	𝜓(𝑔	NOUN
cana-1907	149	6	)	)	PUNCT
cana-1907	149	7	.	.	PUNCT
cana-1907	150	1	let	let	VERB
cana-1907	150	2	a	a	DET
cana-1907	150	3	=	=	SYM
cana-1907	150	4	{	{	PUNCT
cana-1907	150	5	x	x	NOUN
cana-1907	150	6	}	}	PUNCT
cana-1907	150	7	for	for	ADP
cana-1907	150	8	each	each	DET
cana-1907	150	9	x	x	SYM
cana-1907	150	10	∈	∈	PROPN
cana-1907	150	11	x	x	X
cana-1907	150	12	we	we	PRON
cana-1907	150	13	have,{f(x	have,{f(x	PROPN
cana-1907	150	14	)	)	PUNCT
cana-1907	150	15	}	}	PUNCT
cana-1907	150	16	=	=	SYM
cana-1907	150	17	𝑓(a	𝑓(a	NUM
cana-1907	150	18	)	)	PUNCT
cana-1907	150	19	=	=	SYM
cana-1907	150	20	�	�	PROPN
cana-1907	150	21	̃	̃	PROPN
cana-1907	150	22	�	�	NOUN
cana-1907	150	23	(a	(a	NOUN
cana-1907	150	24	)	)	PUNCT
cana-1907	150	25	=	=	SYM
cana-1907	150	26	{	{	PUNCT
cana-1907	150	27	g(x	g(x	NOUN
cana-1907	150	28	)	)	PUNCT
cana-1907	150	29	}	}	PUNCT
cana-1907	150	30	for	for	ADP
cana-1907	150	31	each	each	DET
cana-1907	150	32	x	x	SYM
cana-1907	150	33	∈	∈	PROPN
cana-1907	150	34	x.	x.	NOUN
cana-1907	150	35	then	then	ADV
cana-1907	150	36	,	,	PUNCT
cana-1907	150	37	𝜓	𝜓	PROPN
cana-1907	150	38	is	be	AUX
cana-1907	150	39	injective	injective	ADJ
cana-1907	150	40	.	.	PUNCT
cana-1907	151	1	next	next	ADV
cana-1907	151	2	we	we	PRON
cana-1907	151	3	have	have	VERB
cana-1907	151	4	to	to	PART
cana-1907	151	5	show	show	VERB
cana-1907	151	6	that	that	SCONJ
cana-1907	151	7	𝜓	𝜓	PROPN
cana-1907	151	8	and	and	CCONJ
cana-1907	151	9	𝜓−1	𝜓−1	PRON
cana-1907	151	10	are	be	AUX
cana-1907	151	11	continuous	continuous	ADJ
cana-1907	151	12	.	.	PUNCT
cana-1907	152	1	let	let	VERB
cana-1907	152	2	{	{	PUNCT
cana-1907	152	3	𝑓𝑛	𝑓𝑛	AUX
cana-1907	152	4	}	}	PUNCT
cana-1907	152	5	be	be	AUX
cana-1907	152	6	a	a	DET
cana-1907	152	7	sequence	sequence	NOUN
cana-1907	152	8	in	in	ADP
cana-1907	152	9	sequence	sequence	NOUN
cana-1907	152	10	in	in	ADP
cana-1907	152	11	c(x	c(x	NOUN
cana-1907	152	12	)	)	PUNCT
cana-1907	152	13	which	which	PRON
cana-1907	152	14	converges	converge	VERB
cana-1907	152	15	to	to	ADP
cana-1907	152	16	f	f	PROPN
cana-1907	152	17	.	.	PUNCT
cana-1907	153	1	then	then	ADV
cana-1907	153	2	{	{	PUNCT
cana-1907	153	3	𝑓𝑛	𝑓𝑛	NOUN
cana-1907	153	4	}	}	PUNCT
cana-1907	153	5	converges	converge	NOUN
cana-1907	153	6	to	to	ADP
cana-1907	153	7	𝑓	𝑓	DET
cana-1907	153	8	̃in	̃in	NOUN
cana-1907	153	9	c(k(x	c(k(x	NOUN
cana-1907	153	10	)	)	PUNCT
cana-1907	153	11	)	)	PUNCT
cana-1907	153	12	assume	assume	VERB
cana-1907	153	13	that	that	SCONJ
cana-1907	153	14	for	for	ADP
cana-1907	153	15	any	any	DET
cana-1907	153	16	ϵ	ϵ	X
cana-1907	153	17	>	>	X
cana-1907	153	18	0	0	PUNCT
cana-1907	153	19	and	and	CCONJ
cana-1907	153	20	each	each	DET
cana-1907	153	21	a	a	DET
cana-1907	153	22	∈	∈	PROPN
cana-1907	153	23	k(x	k(x	PROPN
cana-1907	153	24	)	)	PUNCT
cana-1907	153	25	,	,	PUNCT
cana-1907	153	26	there	there	PRON
cana-1907	153	27	exist	exist	VERB
cana-1907	153	28	an	an	DET
cana-1907	153	29	n	n	ADV
cana-1907	153	30	∈	∈	NOUN
cana-1907	153	31	ℤ+	ℤ+	PUNCT
cana-1907	153	32	such	such	ADJ
cana-1907	153	33	that	that	SCONJ
cana-1907	153	34	d(𝑓𝑛	d(𝑓𝑛	NOUN
cana-1907	153	35	,	,	PUNCT
cana-1907	153	36	f	f	X
cana-1907	153	37	)	)	PUNCT
cana-1907	153	38	<	<	X
cana-1907	153	39	ϵ	ϵ	X
cana-1907	153	40	for	for	ADP
cana-1907	153	41	every	every	DET
cana-1907	153	42	n	n	NOUN
cana-1907	153	43	⩾	⩾	PROPN
cana-1907	153	44	n.	n.	NOUN
cana-1907	153	45	communications	communication	NOUN
cana-1907	153	46	on	on	ADP
cana-1907	153	47	applied	apply	VERB
cana-1907	153	48	nonlinear	nonlinear	ADJ
cana-1907	153	49	analysis	analysis	NOUN
cana-1907	153	50	issn	issn	NOUN
cana-1907	153	51	:	:	PUNCT
cana-1907	153	52	1074	1074	NUM
cana-1907	153	53	-	-	PUNCT
cana-1907	153	54	133x	133x	NUM
cana-1907	153	55	vol	vol	NOUN
cana-1907	153	56	32	32	NUM
cana-1907	153	57	no	no	NOUN
cana-1907	153	58	.	.	NOUN
cana-1907	153	59	2	2	NUM
cana-1907	153	60	(	(	PUNCT
cana-1907	153	61	2025	2025	NUM
cana-1907	153	62	)	)	PUNCT
cana-1907	153	63	57	57	NUM
cana-1907	153	64	https://internationalpubls.com	https://internationalpubls.com	X
cana-1907	153	65	for	for	ADP
cana-1907	153	66	all	all	DET
cana-1907	153	67	a	a	DET
cana-1907	153	68	∈	∈	PROPN
cana-1907	153	69	a	a	PRON
cana-1907	153	70	and	and	CCONJ
cana-1907	153	71	n	n	CCONJ
cana-1907	153	72	⩾	⩾	PROPN
cana-1907	153	73	n	n	CCONJ
cana-1907	153	74	,	,	PUNCT
cana-1907	153	75	we	we	PRON
cana-1907	153	76	have	have	VERB
cana-1907	153	77	d(f	d(f	NOUN
cana-1907	153	78	n(a),f(a	n(a),f(a	NUM
cana-1907	153	79	)	)	PUNCT
cana-1907	153	80	)	)	PUNCT
cana-1907	153	81	<	<	X
cana-1907	154	1	ϵ	ϵ	X
cana-1907	155	1	so	so	ADV
cana-1907	155	2	,	,	PUNCT
cana-1907	155	3	f	f	PROPN
cana-1907	155	4	n(a	n(a	PROPN
cana-1907	155	5	)	)	PUNCT
cana-1907	156	1	⊆	⊆	NUM
cana-1907	156	2	b(fn(a),ϵ),n	b(fn(a),ϵ),n	PROPN
cana-1907	156	3	⩾	⩾	VERB
cana-1907	156	4	n	n	CCONJ
cana-1907	156	5	for	for	ADP
cana-1907	156	6	all	all	DET
cana-1907	156	7	a	a	DET
cana-1907	156	8	∈	∈	PROPN
cana-1907	156	9	k(x	k(x	PROPN
cana-1907	156	10	)	)	PUNCT
cana-1907	156	11	,	,	PUNCT
cana-1907	156	12	we	we	PRON
cana-1907	156	13	get	get	VERB
cana-1907	156	14	f(a	f(a	NOUN
cana-1907	156	15	)	)	PUNCT
cana-1907	156	16	⊆	⊆	NUM
cana-1907	156	17	b(fn	b(fn	PROPN
cana-1907	156	18	(	(	PUNCT
cana-1907	156	19	a),ϵ	a),ϵ	ADV
cana-1907	156	20	)	)	PUNCT
cana-1907	156	21	for	for	ADP
cana-1907	156	22	every	every	DET
cana-1907	156	23	n	n	NOUN
cana-1907	156	24	⩾	⩾	NOUN
cana-1907	157	1	n	n	CCONJ
cana-1907	157	2	then	then	ADV
cana-1907	157	3	there	there	PRON
cana-1907	157	4	exist	exist	VERB
cana-1907	157	5	an	an	DET
cana-1907	157	6	n	n	ADV
cana-1907	157	7	∈	∈	NOUN
cana-1907	157	8	ℤ+	ℤ+	PUNCT
cana-1907	157	9	such	such	ADJ
cana-1907	157	10	that	that	DET
cana-1907	157	11	𝑑(𝑓𝑛(𝐴	𝑑(𝑓𝑛(𝐴	NOUN
cana-1907	157	12	)	)	PUNCT
cana-1907	157	13	,	,	PUNCT
cana-1907	157	14	𝑓(𝐴	𝑓(𝐴	PROPN
cana-1907	157	15	)	)	PUNCT
cana-1907	157	16	)	)	PUNCT
cana-1907	158	1	<	<	X
cana-1907	158	2	ϵ	ϵ	X
cana-1907	158	3	for	for	ADP
cana-1907	158	4	every	every	DET
cana-1907	158	5	n	n	NOUN
cana-1907	158	6	⩾	⩾	PROPN
cana-1907	158	7	n	n	NOUN
cana-1907	158	8	and	and	CCONJ
cana-1907	158	9	each	each	DET
cana-1907	158	10	a	a	DET
cana-1907	158	11	∈	∈	PROPN
cana-1907	158	12	k(x	k(x	PROPN
cana-1907	158	13	)	)	PUNCT
cana-1907	158	14	.	.	PUNCT
cana-1907	159	1	similarly	similarly	ADV
cana-1907	159	2	we	we	PRON
cana-1907	159	3	show	show	VERB
cana-1907	159	4	that	that	SCONJ
cana-1907	159	5	if	if	SCONJ
cana-1907	159	6	𝑓	𝑓	PROPN
cana-1907	159	7	�	�	PROPN
cana-1907	159	8	̃	̃	PROPN
cana-1907	159	9	�	�	PROPN
cana-1907	159	10	→	→	SYM
cana-1907	159	11	𝑓	𝑓	PROPN
cana-1907	159	12	in	in	ADP
cana-1907	159	13	c(k(x	c(k(x	PROPN
cana-1907	159	14	)	)	PUNCT
cana-1907	159	15	)	)	PUNCT
cana-1907	159	16	then	then	ADV
cana-1907	159	17	𝑓𝑛	𝑓𝑛	ADP
cana-1907	159	18	→	→	SYM
cana-1907	159	19	𝑓	𝑓	X
cana-1907	159	20	in	in	ADP
cana-1907	159	21	c(x	c(x	NOUN
cana-1907	159	22	)	)	PUNCT
cana-1907	159	23	.	.	PUNCT
cana-1907	160	1	there	there	ADV
cana-1907	160	2	fore	fore	NOUN
cana-1907	160	3	,	,	PUNCT
cana-1907	160	4	𝜓	𝜓	PROPN
cana-1907	160	5	is	be	AUX
cana-1907	160	6	an	an	DET
cana-1907	160	7	embedding	embedding	NOUN
cana-1907	160	8	of	of	ADP
cana-1907	160	9	c(x	c(x	NOUN
cana-1907	160	10	)	)	PUNCT
cana-1907	160	11	in	in	ADP
cana-1907	160	12	to	to	ADP
cana-1907	160	13	k(c(x	k(c(x	NOUN
cana-1907	160	14	)	)	PUNCT
cana-1907	160	15	)	)	PUNCT
cana-1907	160	16	.	.	PUNCT
cana-1907	161	1	]	]	PUNCT
cana-1907	161	2	□	□	PUNCT
cana-1907	161	3	lemma	lemma	PROPN
cana-1907	161	4	3.2	3.2	NUM
cana-1907	161	5	.	.	PUNCT
cana-1907	162	1	𝜓(𝐶(𝑋))is	𝜓(𝐶(𝑋))i	NOUN
cana-1907	162	2	closed	close	VERB
cana-1907	162	3	in	in	ADP
cana-1907	162	4	c(k(x	c(k(x	PROPN
cana-1907	162	5	)	)	PUNCT
cana-1907	162	6	)	)	PUNCT
cana-1907	162	7	proof	proof	NOUN
cana-1907	162	8	.	.	PUNCT
cana-1907	163	1	we	we	PRON
cana-1907	163	2	prove	prove	VERB
cana-1907	163	3	that	that	SCONJ
cana-1907	163	4	there	there	PRON
cana-1907	163	5	exist	exist	VERB
cana-1907	163	6	a	a	DET
cana-1907	163	7	sequence	sequence	NOUN
cana-1907	163	8	{	{	PUNCT
cana-1907	163	9	𝑓	𝑓	PROPN
cana-1907	163	10	�	�	PROPN
cana-1907	163	11	̃	̃	PROPN
cana-1907	163	12	�	�	PROPN
cana-1907	163	13	}	}	PUNCT
cana-1907	163	14	in	in	ADP
cana-1907	163	15	𝜓(𝐶(𝑋))which	𝜓(𝐶(𝑋))which	NOUN
cana-1907	163	16	converges	converge	VERB
cana-1907	163	17	to	to	ADP
cana-1907	163	18	f	f	PROPN
cana-1907	163	19	such	such	ADJ
cana-1907	163	20	that	that	SCONJ
cana-1907	163	21	there	there	PRON
cana-1907	163	22	is	be	VERB
cana-1907	163	23	f	f	PROPN
cana-1907	163	24	∈	∈	PROPN
cana-1907	163	25	c(x	c(x	NOUN
cana-1907	163	26	)	)	PUNCT
cana-1907	163	27	satisfies	satisfy	VERB
cana-1907	163	28	𝑓	𝑓	PROPN
cana-1907	163	29	=	=	SYM
cana-1907	163	30	𝐹.	𝐹.	PROPN
cana-1907	163	31	consider	consider	VERB
cana-1907	163	32	𝜓−1(𝑓𝑛	𝜓−1(𝑓𝑛	NOUN
cana-1907	163	33	)	)	PUNCT
cana-1907	163	34	=	=	SYM
cana-1907	163	35	𝑓𝑛.	𝑓𝑛.	NOUN
cana-1907	163	36	for	for	ADP
cana-1907	163	37	every	every	DET
cana-1907	163	38	ϵ	ϵ	PROPN
cana-1907	163	39	>	>	X
cana-1907	163	40	0	0	NUM
cana-1907	163	41	,	,	PUNCT
cana-1907	163	42	take	take	VERB
cana-1907	163	43	n	n	ADV
cana-1907	163	44	>	>	X
cana-1907	163	45	0	0	NUM
cana-1907	164	1	such	such	ADJ
cana-1907	164	2	that	that	SCONJ
cana-1907	164	3	𝑑(𝑓𝑛	𝑑(𝑓𝑛	NOUN
cana-1907	164	4	,	,	PUNCT
cana-1907	164	5	𝑓𝑚	𝑓𝑚	PROPN
cana-1907	164	6	)	)	PUNCT
cana-1907	164	7	<	<	X
cana-1907	164	8	ϵ	ϵ	X
cana-1907	164	9	,	,	PUNCT
cana-1907	164	10	for	for	ADP
cana-1907	164	11	every	every	DET
cana-1907	164	12	n	n	CCONJ
cana-1907	164	13	,	,	PUNCT
cana-1907	164	14	m	m	VERB
cana-1907	164	15	⩾	⩾	ADJ
cana-1907	164	16	n.	n.	NOUN
cana-1907	164	17	this	this	PRON
cana-1907	164	18	means	mean	VERB
cana-1907	164	19	that	that	SCONJ
cana-1907	164	20	any	any	DET
cana-1907	164	21	compact	compact	NOUN
cana-1907	164	22	set	set	VERB
cana-1907	164	23	a	a	DET
cana-1907	164	24	satisfies	satisfie	NOUN
cana-1907	164	25	𝑑(𝑓𝑛(𝐴	𝑑(𝑓𝑛(𝐴	NOUN
cana-1907	164	26	)	)	PUNCT
cana-1907	164	27	,	,	PUNCT
cana-1907	164	28	𝑓𝑚(𝐴	𝑓𝑚(𝐴	NOUN
cana-1907	164	29	)	)	PUNCT
cana-1907	164	30	)	)	PUNCT
cana-1907	165	1	<	<	X
cana-1907	165	2	ϵ	ϵ	X
cana-1907	165	3	for	for	ADP
cana-1907	165	4	every	every	DET
cana-1907	165	5	n	n	CCONJ
cana-1907	165	6	,	,	PUNCT
cana-1907	165	7	m	m	VERB
cana-1907	165	8	⩾	⩾	ADJ
cana-1907	165	9	n.	n.	NOUN
cana-1907	165	10	for	for	ADP
cana-1907	165	11	a	a	DET
cana-1907	165	12	∈	∈	PROPN
cana-1907	165	13	k(x	k(x	PROPN
cana-1907	165	14	)	)	PUNCT
cana-1907	165	15	,	,	PUNCT
cana-1907	165	16	choose	choose	VERB
cana-1907	165	17	ax	ax	NOUN
cana-1907	165	18	=	=	PUNCT
cana-1907	165	19	{	{	PUNCT
cana-1907	165	20	x	x	NOUN
cana-1907	165	21	}	}	PUNCT
cana-1907	165	22	for	for	ADP
cana-1907	165	23	every	every	DET
cana-1907	165	24	x	x	SYM
cana-1907	165	25	∈	∈	PROPN
cana-1907	165	26	x.	x.	NOUN
cana-1907	165	27	then	then	ADV
cana-1907	165	28	d(fn	d(fn	PROPN
cana-1907	165	29	,	,	PUNCT
cana-1907	165	30	fm	fm	PROPN
cana-1907	165	31	)	)	PUNCT
cana-1907	165	32	<	<	X
cana-1907	165	33	ϵ	ϵ	X
cana-1907	165	34	for	for	ADP
cana-1907	165	35	every	every	DET
cana-1907	165	36	n	n	CCONJ
cana-1907	165	37	,	,	PUNCT
cana-1907	165	38	m	m	VERB
cana-1907	165	39	⩾	⩾	ADJ
cana-1907	165	40	n	n	ADJ
cana-1907	165	41	and	and	CCONJ
cana-1907	165	42	all	all	DET
cana-1907	165	43	x	x	SYM
cana-1907	165	44	∈	∈	PROPN
cana-1907	165	45	x.	x.	NOUN
cana-1907	165	46	since	since	SCONJ
cana-1907	165	47	{	{	PUNCT
cana-1907	165	48	fn	fn	NOUN
cana-1907	165	49	}	}	PUNCT
cana-1907	165	50	is	be	AUX
cana-1907	165	51	a	a	DET
cana-1907	165	52	cauchy	cauchy	NOUN
cana-1907	165	53	in	in	ADP
cana-1907	165	54	c(x	c(x	NOUN
cana-1907	165	55	)	)	PUNCT
cana-1907	165	56	,	,	PUNCT
cana-1907	165	57	which	which	PRON
cana-1907	165	58	is	be	AUX
cana-1907	165	59	a	a	DET
cana-1907	165	60	complete	complete	ADJ
cana-1907	165	61	metric	metric	ADJ
cana-1907	165	62	space	space	NOUN
cana-1907	165	63	,	,	PUNCT
cana-1907	165	64	there	there	PRON
cana-1907	165	65	exist	exist	VERB
cana-1907	165	66	f	f	PROPN
cana-1907	165	67	∈	∈	PROPN
cana-1907	165	68	c(x	c(x	NOUN
cana-1907	165	69	)	)	PUNCT
cana-1907	165	70	such	such	ADJ
cana-1907	165	71	that	that	SCONJ
cana-1907	165	72	{	{	PUNCT
cana-1907	165	73	fn	fn	NOUN
cana-1907	165	74	}	}	PUNCT
cana-1907	165	75	converges	converge	NOUN
cana-1907	165	76	to	to	ADP
cana-1907	165	77	f.	f.	PROPN
cana-1907	165	78	since	since	SCONJ
cana-1907	165	79	𝜓	𝜓	PROPN
cana-1907	165	80	is	be	AUX
cana-1907	165	81	continuous	continuous	ADJ
cana-1907	165	82	{	{	PUNCT
cana-1907	165	83	𝜓(𝑓𝑛	𝜓(𝑓𝑛	NOUN
cana-1907	165	84	)	)	PUNCT
cana-1907	165	85	}	}	PUNCT
cana-1907	165	86	=	=	SYM
cana-1907	165	87	{	{	PUNCT
cana-1907	165	88	𝑓𝑛}since	𝑓𝑛}since	NOUN
cana-1907	165	89	the	the	DET
cana-1907	165	90	limit	limit	NOUN
cana-1907	165	91	is	be	AUX
cana-1907	165	92	unique	unique	ADJ
cana-1907	165	93	,	,	PUNCT
cana-1907	166	1	f	f	PROPN
cana-1907	166	2	=	=	PUNCT
cana-1907	166	3	𝑓	𝑓	ADV
cana-1907	166	4	there	there	ADV
cana-1907	166	5	fore	fore	NOUN
cana-1907	166	6	f	f	PROPN
cana-1907	166	7	∈	∈	PROPN
cana-1907	166	8	𝜓(c(x	𝜓(c(x	PROPN
cana-1907	166	9	)	)	PUNCT
cana-1907	166	10	)	)	PUNCT
cana-1907	166	11	.	.	PUNCT
cana-1907	167	1	there	there	ADV
cana-1907	167	2	fore	fore	PROPN
cana-1907	167	3	𝜓(c(x	𝜓(c(x	PROPN
cana-1907	167	4	)	)	PUNCT
cana-1907	167	5	)	)	PUNCT
cana-1907	167	6	is	be	AUX
cana-1907	167	7	closed	close	VERB
cana-1907	167	8	in	in	ADP
cana-1907	167	9	c(k(x	c(k(x	PROPN
cana-1907	167	10	)	)	PUNCT
cana-1907	167	11	)	)	PUNCT
cana-1907	167	12	.	.	PUNCT
cana-1907	168	1	proof	proof	NOUN
cana-1907	168	2	of	of	ADP
cana-1907	168	3	the	the	DET
cana-1907	168	4	theorem	theorem	ADJ
cana-1907	168	5	3.1	3.1	NUM
cana-1907	168	6	suppose	suppose	VERB
cana-1907	168	7	f	f	X
cana-1907	168	8	:	:	PUNCT
cana-1907	168	9	x	x	X
cana-1907	168	10	→	→	PUNCT
cana-1907	168	11	x	x	PUNCT
cana-1907	168	12	is	be	AUX
cana-1907	168	13	stable	stable	ADJ
cana-1907	168	14	and	and	CCONJ
cana-1907	168	15	ϵ	ϵ	X
cana-1907	168	16	>	>	X
cana-1907	168	17	0	0	NUM
cana-1907	168	18	is	be	AUX
cana-1907	168	19	given	give	VERB
cana-1907	168	20	.then	.then	PUNCT
cana-1907	168	21	there	there	PRON
cana-1907	168	22	are	be	VERB
cana-1907	168	23	δ	δ	PROPN
cana-1907	168	24	>	>	X
cana-1907	168	25	0	0	PUNCT
cana-1907	169	1	and	and	CCONJ
cana-1907	169	2	h	h	PROPN
cana-1907	169	3	∈	∈	PROPN
cana-1907	169	4	c(x	c(x	NOUN
cana-1907	169	5	)	)	PUNCT
cana-1907	169	6	as	as	SCONJ
cana-1907	169	7	is	be	AUX
cana-1907	169	8	the	the	DET
cana-1907	169	9	definition	definition	NOUN
cana-1907	169	10	of	of	ADP
cana-1907	169	11	stability	stability	NOUN
cana-1907	169	12	for	for	ADP
cana-1907	169	13	f.	f.	PROPN
cana-1907	169	14	since	since	SCONJ
cana-1907	169	15	𝜓−1	𝜓−1	PROPN
cana-1907	169	16	is	be	AUX
cana-1907	169	17	continuous	continuous	ADJ
cana-1907	169	18	at	at	ADP
cana-1907	169	19	𝑓	𝑓	DET
cana-1907	169	20	∈	∈	ADJ
cana-1907	169	21	𝜓(c(x)),then	𝜓(c(x)),then	NOUN
cana-1907	169	22	for	for	ADP
cana-1907	169	23	δ	δ	PROPN
cana-1907	169	24	>	>	X
cana-1907	169	25	0	0	PROPN
cana-1907	169	26	,	,	PUNCT
cana-1907	169	27	there	there	PRON
cana-1907	169	28	exist	exist	VERB
cana-1907	169	29	a	a	DET
cana-1907	169	30	δ1	δ1	NOUN
cana-1907	169	31	>	>	X
cana-1907	169	32	0	0	NUM
cana-1907	170	1	such	such	ADJ
cana-1907	170	2	that	that	SCONJ
cana-1907	170	3	if	if	SCONJ
cana-1907	170	4	�	�	PROPN
cana-1907	170	5	̃	̃	PROPN
cana-1907	170	6	�	�	PROPN
cana-1907	170	7	∈	∈	PROPN
cana-1907	170	8	𝜓(c(x	𝜓(c(x	NOUN
cana-1907	170	9	)	)	PUNCT
cana-1907	170	10	)	)	PUNCT
cana-1907	170	11	with	with	ADP
cana-1907	170	12	𝑑(𝑓	𝑑(𝑓	PROPN
cana-1907	170	13	,	,	PUNCT
cana-1907	170	14	�	�	PROPN
cana-1907	170	15	̃	̃	PROPN
cana-1907	170	16	�	�	PROPN
cana-1907	170	17	)	)	PUNCT
cana-1907	170	18	<	<	X
cana-1907	170	19	δ	δ	PROPN
cana-1907	170	20	,	,	PUNCT
cana-1907	170	21	then	then	ADV
cana-1907	170	22	we	we	PRON
cana-1907	170	23	have	have	VERB
cana-1907	170	24	,	,	PUNCT
cana-1907	170	25	d(𝜓−1(𝑓),𝜓−1(	d(𝜓−1(𝑓),𝜓−1(	PROPN
cana-1907	170	26	�	�	PROPN
cana-1907	170	27	̃	̃	NOUN
cana-1907	170	28	�	�	NOUN
cana-1907	170	29	)	)	PUNCT
cana-1907	170	30	=	=	SYM
cana-1907	171	1	d(f	d(f	NOUN
cana-1907	171	2	,	,	PUNCT
cana-1907	171	3	g	g	NOUN
cana-1907	171	4	)	)	PUNCT
cana-1907	171	5	<	<	X
cana-1907	171	6	δ	δ	PROPN
cana-1907	171	7	.	.	PUNCT
cana-1907	172	1	from	from	ADP
cana-1907	172	2	f	f	PROPN
cana-1907	172	3	◦	◦	NOUN
cana-1907	172	4	h	h	NOUN
cana-1907	173	1	=	=	NOUN
cana-1907	173	2	h	h	PROPN
cana-1907	174	1	◦	◦	NOUN
cana-1907	174	2	g	g	NOUN
cana-1907	174	3	,	,	PUNCT
cana-1907	174	4	we	we	PRON
cana-1907	174	5	have	have	VERB
cana-1907	174	6	𝜓(f	𝜓(f	NOUN
cana-1907	174	7	◦	◦	NOUN
cana-1907	174	8	h)(a	h)(a	NOUN
cana-1907	174	9	)	)	PUNCT
cana-1907	175	1	=	=	PUNCT
cana-1907	176	1	(	(	PUNCT
cana-1907	176	2	𝑓	𝑓	DET
cana-1907	176	3	∘	∘	PROPN
cana-1907	176	4	�	�	PROPN
cana-1907	176	5	̃	̃	PROPN
cana-1907	176	6	�	�	NOUN
cana-1907	176	7	)(a	)(a	NOUN
cana-1907	176	8	)	)	PUNCT
cana-1907	177	1	=	=	PUNCT
cana-1907	177	2	(	(	PUNCT
cana-1907	177	3	f	f	X
cana-1907	177	4	◦	◦	NOUN
cana-1907	177	5	h)(a	h)(a	NOUN
cana-1907	177	6	)	)	PUNCT
cana-1907	178	1	=	=	SYM
cana-1907	178	2	𝑓(h(a	𝑓(h(a	NOUN
cana-1907	178	3	)	)	PUNCT
cana-1907	178	4	)	)	PUNCT
cana-1907	179	1	=	=	VERB
cana-1907	179	2	𝑓(ℎ̃(a	𝑓(ℎ̃(a	NOUN
cana-1907	179	3	)	)	PUNCT
cana-1907	179	4	)	)	PUNCT
cana-1907	180	1	=	=	PRON
cana-1907	180	2	(	(	PUNCT
cana-1907	180	3	𝑓	𝑓	DET
cana-1907	180	4	∘	∘	PROPN
cana-1907	180	5	�	�	PROPN
cana-1907	180	6	̃	̃	PROPN
cana-1907	180	7	�	�	PROPN
cana-1907	180	8	)(a),a	)(a),a	PART
cana-1907	180	9	∈	∈	PROPN
cana-1907	180	10	k(x	k(x	PROPN
cana-1907	180	11	)	)	PUNCT
cana-1907	180	12	and	and	CCONJ
cana-1907	180	13	𝜓(h	𝜓(h	PROPN
cana-1907	180	14	◦	◦	PROPN
cana-1907	180	15	g)(a	g)(a	NOUN
cana-1907	180	16	)	)	PUNCT
cana-1907	180	17	=	=	SYM
cana-1907	180	18	(	(	PUNCT
cana-1907	180	19	ℎ̃	ℎ̃	PROPN
cana-1907	180	20	◦	◦	PROPN
cana-1907	180	21	�	�	PROPN
cana-1907	180	22	̃	̃	PROPN
cana-1907	180	23	�	�	PROPN
cana-1907	180	24	)(a),a	)(a),a	PART
cana-1907	180	25	∈	∈	PROPN
cana-1907	180	26	k(x	k(x	PROPN
cana-1907	180	27	)	)	PUNCT
cana-1907	180	28	thus	thus	ADV
cana-1907	180	29	𝑓	𝑓	DET
cana-1907	180	30	∘	∘	ADJ
cana-1907	180	31	ℎ̃	ℎ̃	PROPN
cana-1907	180	32	=	=	SYM
cana-1907	180	33	ℎ̃	ℎ̃	PROPN
cana-1907	180	34	∘	∘	PROPN
cana-1907	180	35	�	�	PROPN
cana-1907	180	36	̃	̃	PROPN
cana-1907	180	37	�	�	PROPN
cana-1907	180	38	.	.	PUNCT
cana-1907	181	1	let	let	VERB
cana-1907	181	2	a	a	DET
cana-1907	181	3	∈	∈	NOUN
cana-1907	181	4	k(x).from	k(x).from	VERB
cana-1907	181	5	the	the	DET
cana-1907	181	6	continuity	continuity	NOUN
cana-1907	181	7	of	of	ADP
cana-1907	181	8	h	h	NOUN
cana-1907	181	9	and	and	CCONJ
cana-1907	181	10	compactness	compactness	NOUN
cana-1907	181	11	of	of	ADP
cana-1907	181	12	a	a	PRON
cana-1907	181	13	,	,	PUNCT
cana-1907	181	14	we	we	PRON
cana-1907	181	15	can	can	AUX
cana-1907	181	16	see	see	VERB
cana-1907	181	17	that	that	DET
cana-1907	181	18	ℎ(𝐴	ℎ(𝐴	NOUN
cana-1907	181	19	)	)	PUNCT
cana-1907	181	20	⊆	⊆	NUM
cana-1907	181	21	⋃	⋃	NOUN
cana-1907	181	22	𝐵(𝑎	𝐵(𝑎	NUM
cana-1907	181	23	,	,	PUNCT
cana-1907	181	24	𝜖	𝜖	NOUN
cana-1907	181	25	)	)	PUNCT
cana-1907	181	26	𝑎∈𝐴	𝑎∈𝐴	NOUN
cana-1907	181	27	and	and	CCONJ
cana-1907	181	28	communications	communication	NOUN
cana-1907	181	29	on	on	ADP
cana-1907	181	30	applied	apply	VERB
cana-1907	181	31	nonlinear	nonlinear	ADJ
cana-1907	181	32	analysis	analysis	NOUN
cana-1907	181	33	issn	issn	NOUN
cana-1907	181	34	:	:	PUNCT
cana-1907	181	35	1074	1074	NUM
cana-1907	181	36	-	-	PUNCT
cana-1907	181	37	133x	133x	NUM
cana-1907	181	38	vol	vol	NOUN
cana-1907	181	39	32	32	NUM
cana-1907	181	40	no	no	NOUN
cana-1907	181	41	.	.	NOUN
cana-1907	181	42	2	2	NUM
cana-1907	181	43	(	(	PUNCT
cana-1907	181	44	2025	2025	NUM
cana-1907	181	45	)	)	PUNCT
cana-1907	182	1	58	58	NUM
cana-1907	182	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-1907	182	3	𝐴	𝐴	PROPN
cana-1907	182	4	⊆	⊆	NUM
cana-1907	182	5	⋃	⋃	ADP
cana-1907	182	6	𝐵(ℎ(𝑎	𝐵(ℎ(𝑎	PROPN
cana-1907	182	7	)	)	PUNCT
cana-1907	182	8	,	,	PUNCT
cana-1907	182	9	𝜖	𝜖	X
cana-1907	182	10	)	)	PUNCT
cana-1907	182	11	𝑎∈𝐴	𝑎∈𝐴	NOUN
cana-1907	182	12	if	if	SCONJ
cana-1907	182	13	and	and	CCONJ
cana-1907	182	14	only	only	ADV
cana-1907	182	15	if	if	SCONJ
cana-1907	182	16	ℎ(𝐴	ℎ(𝐴	PROPN
cana-1907	182	17	)	)	PUNCT
cana-1907	182	18	⊆	⊆	NUM
cana-1907	182	19	𝐵(𝐴	𝐵(𝐴	NOUN
cana-1907	182	20	,	,	PUNCT
cana-1907	182	21	𝜖	𝜖	NOUN
cana-1907	182	22	)	)	PUNCT
cana-1907	182	23	and	and	CCONJ
cana-1907	182	24	𝐴	𝐴	PROPN
cana-1907	182	25	⊆	⊆	NUM
cana-1907	182	26	𝐵(ℎ(𝐴	𝐵(ℎ(𝐴	NOUN
cana-1907	182	27	)	)	PUNCT
cana-1907	182	28	,	,	PUNCT
cana-1907	182	29	𝜖	𝜖	X
cana-1907	182	30	)	)	PUNCT
cana-1907	182	31	that	that	PRON
cana-1907	182	32	is	be	AUX
cana-1907	182	33	,	,	PUNCT
cana-1907	182	34	h(a	h(a	PROPN
cana-1907	182	35	)	)	PUNCT
cana-1907	182	36	⊆	⊆	NUM
cana-1907	182	37	b(a,ϵ	b(a,ϵ	NOUN
cana-1907	182	38	)	)	PUNCT
cana-1907	182	39	and	and	CCONJ
cana-1907	182	40	a	a	DET
cana-1907	182	41	⊆	⊆	NUM
cana-1907	182	42	b(h(a),ϵ	b(h(a),ϵ	NOUN
cana-1907	182	43	)	)	PUNCT
cana-1907	182	44	for	for	ADP
cana-1907	182	45	any	any	DET
cana-1907	182	46	a	a	DET
cana-1907	182	47	∈	∈	PROPN
cana-1907	182	48	k(x	k(x	PROPN
cana-1907	182	49	)	)	PUNCT
cana-1907	182	50	.	.	PUNCT
cana-1907	183	1	this	this	PRON
cana-1907	183	2	implies	imply	VERB
cana-1907	183	3	𝑑(ℎ̃	𝑑(ℎ̃	PROPN
cana-1907	183	4	,	,	PUNCT
cana-1907	183	5	𝑖̃	𝑖̃	PROPN
cana-1907	183	6	)	)	PUNCT
cana-1907	183	7	<	<	X
cana-1907	184	1	ϵ	ϵ	X
cana-1907	184	2	there	there	PRON
cana-1907	184	3	fore	fore	NOUN
cana-1907	184	4	,	,	PUNCT
cana-1907	184	5	𝑓	𝑓	PRON
cana-1907	184	6	is	be	AUX
cana-1907	184	7	stable	stable	ADJ
cana-1907	184	8	in	in	ADP
cana-1907	184	9	𝜓(c(x	𝜓(c(x	NOUN
cana-1907	184	10	)	)	PUNCT
cana-1907	184	11	)	)	PUNCT
cana-1907	184	12	.	.	PUNCT
cana-1907	185	1	conversely	conversely	ADV
cana-1907	185	2	,	,	PUNCT
cana-1907	185	3	suppose	suppose	VERB
cana-1907	185	4	𝑓	𝑓	DET
cana-1907	185	5	∈	∈	PROPN
cana-1907	185	6	𝜓(𝐶(𝑋	𝜓(𝐶(𝑋	NOUN
cana-1907	185	7	)	)	PUNCT
cana-1907	185	8	)	)	PUNCT
cana-1907	185	9	is	be	AUX
cana-1907	185	10	stable	stable	ADJ
cana-1907	185	11	.	.	PUNCT
cana-1907	186	1	for	for	ADP
cana-1907	186	2	any	any	DET
cana-1907	186	3	ϵ	ϵ	X
cana-1907	186	4	>	>	X
cana-1907	186	5	0	0	NUM
cana-1907	186	6	,	,	PUNCT
cana-1907	186	7	there	there	PRON
cana-1907	186	8	exist	exist	VERB
cana-1907	186	9	a	a	DET
cana-1907	186	10	δ	δ	PROPN
cana-1907	186	11	>	>	X
cana-1907	186	12	0	0	PUNCT
cana-1907	187	1	and	and	CCONJ
cana-1907	187	2	ℎ̃	ℎ̃	PROPN
cana-1907	187	3	∈	∈	PROPN
cana-1907	187	4	𝜓(𝐶(𝑋	𝜓(𝐶(𝑋	NOUN
cana-1907	187	5	)	)	PUNCT
cana-1907	187	6	)	)	PUNCT
cana-1907	188	1	satisfying	satisfy	VERB
cana-1907	188	2	topological	topological	ADJ
cana-1907	188	3	stability	stability	NOUN
cana-1907	188	4	of	of	ADP
cana-1907	188	5	𝑓.	𝑓.	NOUN
cana-1907	188	6	since	since	SCONJ
cana-1907	188	7	𝜓	𝜓	PROPN
cana-1907	188	8	is	be	AUX
cana-1907	188	9	continuous	continuous	ADJ
cana-1907	188	10	at	at	ADP
cana-1907	188	11	f	f	PROPN
cana-1907	188	12	∈	∈	PROPN
cana-1907	188	13	c(x	c(x	NOUN
cana-1907	188	14	)	)	PUNCT
cana-1907	188	15	,	,	PUNCT
cana-1907	188	16	there	there	PRON
cana-1907	188	17	is	be	VERB
cana-1907	188	18	a	a	DET
cana-1907	188	19	δ2	δ2	VERB
cana-1907	188	20	>	>	X
cana-1907	188	21	0	0	NUM
cana-1907	188	22	such	such	ADJ
cana-1907	188	23	that	that	PRON
cana-1907	188	24	for	for	ADP
cana-1907	188	25	any	any	DET
cana-1907	188	26	g	g	PROPN
cana-1907	188	27	∈	∈	PROPN
cana-1907	188	28	c(x	c(x	NOUN
cana-1907	188	29	)	)	PUNCT
cana-1907	188	30	with	with	ADP
cana-1907	188	31	d(f	d(f	NOUN
cana-1907	188	32	,	,	PUNCT
cana-1907	188	33	g	g	NOUN
cana-1907	188	34	)	)	PUNCT
cana-1907	188	35	<	<	X
cana-1907	188	36	δ2,we	δ2,we	PROPN
cana-1907	188	37	have	have	VERB
cana-1907	188	38	𝑑(𝜓(𝑓	𝑑(𝜓(𝑓	ADJ
cana-1907	188	39	)	)	PUNCT
cana-1907	188	40	,	,	PUNCT
cana-1907	188	41	𝜓(𝑔	𝜓(𝑔	NUM
cana-1907	188	42	)	)	PUNCT
cana-1907	188	43	)	)	PUNCT
cana-1907	189	1	=	=	SYM
cana-1907	189	2	𝑑(𝑓	𝑑(𝑓	PROPN
cana-1907	189	3	,	,	PUNCT
cana-1907	189	4	�	�	PROPN
cana-1907	189	5	̃	̃	PROPN
cana-1907	189	6	�	�	NOUN
cana-1907	189	7	)	)	PUNCT
cana-1907	189	8	<	<	X
cana-1907	190	1	δ	δ	PROPN
cana-1907	190	2	take	take	VERB
cana-1907	190	3	a	a	DET
cana-1907	190	4	=	=	PUNCT
cana-1907	190	5	{	{	PUNCT
cana-1907	190	6	x	x	NOUN
cana-1907	190	7	}	}	PUNCT
cana-1907	190	8	for	for	ADP
cana-1907	190	9	each	each	DET
cana-1907	190	10	x	x	SYM
cana-1907	190	11	∈	∈	PROPN
cana-1907	190	12	x	x	NOUN
cana-1907	190	13	,	,	PUNCT
cana-1907	190	14	then	then	ADV
cana-1907	190	15	𝑓	𝑓	DET
cana-1907	190	16	∘	∘	PROPN
cana-1907	190	17	�	�	PROPN
cana-1907	190	18	̃	̃	PROPN
cana-1907	190	19	�	�	PROPN
cana-1907	190	20	=	=	SYM
cana-1907	190	21	�	�	PROPN
cana-1907	190	22	̃	̃	PROPN
cana-1907	190	23	�	�	NOUN
cana-1907	190	24	∘	∘	VERB
cana-1907	190	25	𝑓	𝑓	ADP
cana-1907	190	26	that	that	PRON
cana-1907	190	27	(	(	PUNCT
cana-1907	190	28	𝑓	𝑓	DET
cana-1907	190	29	∘	∘	PROPN
cana-1907	190	30	�	�	PROPN
cana-1907	190	31	̃	̃	PROPN
cana-1907	190	32	�	�	PROPN
cana-1907	190	33	)	)	PUNCT
cana-1907	190	34	(	(	PUNCT
cana-1907	190	35	a	a	X
cana-1907	190	36	)	)	PUNCT
cana-1907	190	37	=	=	SYM
cana-1907	190	38	{	{	PUNCT
cana-1907	190	39	f(h(x	f(h(x	PROPN
cana-1907	190	40	)	)	PUNCT
cana-1907	190	41	)	)	PUNCT
cana-1907	190	42	}	}	PUNCT
cana-1907	191	1	=	=	PRON
cana-1907	191	2	{	{	PUNCT
cana-1907	191	3	h(g(x	h(g(x	NOUN
cana-1907	191	4	)	)	PUNCT
cana-1907	191	5	)	)	PUNCT
cana-1907	191	6	}	}	PUNCT
cana-1907	191	7	=(	=(	PROPN
cana-1907	191	8	�	�	PROPN
cana-1907	191	9	̃	̃	PROPN
cana-1907	191	10	�	�	PROPN
cana-1907	191	11	∘	∘	NOUN
cana-1907	191	12	𝑓	𝑓	PROPN
cana-1907	191	13	)	)	PUNCT
cana-1907	191	14	(	(	PUNCT
cana-1907	191	15	a	a	X
cana-1907	191	16	)	)	PUNCT
cana-1907	191	17	,	,	PUNCT
cana-1907	191	18	x	x	PUNCT
cana-1907	191	19	∈	∈	PROPN
cana-1907	191	20	x	x	ADP
cana-1907	191	21	this	this	PRON
cana-1907	191	22	implies	imply	VERB
cana-1907	191	23	f	f	X
cana-1907	191	24	◦	◦	NOUN
cana-1907	191	25	h	h	NOUN
cana-1907	192	1	=	=	NOUN
cana-1907	192	2	h	h	PROPN
cana-1907	192	3	◦	◦	NOUN
cana-1907	192	4	g.	g.	NOUN
cana-1907	192	5	from	from	ADP
cana-1907	192	6	𝑑(ℎ̃	𝑑(ℎ̃	PROPN
cana-1907	192	7	,	,	PUNCT
cana-1907	192	8	𝑖̃)<𝜖	𝑖̃)<𝜖	PROPN
cana-1907	192	9	,	,	PUNCT
cana-1907	192	10	we	we	PRON
cana-1907	192	11	have	have	VERB
cana-1907	192	12	𝑓(𝐴	𝑓(𝐴	NOUN
cana-1907	192	13	)	)	PUNCT
cana-1907	192	14	⊆	⊆	NUM
cana-1907	192	15	𝐵(𝐴	𝐵(𝐴	NOUN
cana-1907	192	16	,	,	PUNCT
cana-1907	192	17	𝜖	𝜖	NOUN
cana-1907	192	18	)	)	PUNCT
cana-1907	192	19	and	and	CCONJ
cana-1907	192	20	𝐴	𝐴	PROPN
cana-1907	192	21	⊆	⊆	NUM
cana-1907	192	22	𝐵(ℎ̃(𝐴	𝐵(ℎ̃(𝐴	NOUN
cana-1907	192	23	)	)	PUNCT
cana-1907	192	24	,	,	PUNCT
cana-1907	192	25	𝜖	𝜖	PROPN
cana-1907	192	26	)	)	PUNCT
cana-1907	192	27	for	for	ADP
cana-1907	192	28	𝐴	𝐴	PROPN
cana-1907	192	29	∈	∈	PROPN
cana-1907	192	30	𝐾(𝑋	𝐾(𝑋	NOUN
cana-1907	192	31	)	)	PUNCT
cana-1907	192	32	.	.	PUNCT
cana-1907	193	1	taking	take	VERB
cana-1907	193	2	a	a	DET
cana-1907	193	3	=	=	X
cana-1907	193	4	{	{	PUNCT
cana-1907	193	5	a	a	NOUN
cana-1907	193	6	}	}	PUNCT
cana-1907	193	7	,	,	PUNCT
cana-1907	193	8	for	for	ADP
cana-1907	193	9	each	each	DET
cana-1907	193	10	a	a	DET
cana-1907	193	11	∈	∈	PROPN
cana-1907	193	12	x	x	NOUN
cana-1907	193	13	,	,	PUNCT
cana-1907	193	14	implies	imply	VERB
cana-1907	193	15	{	{	PUNCT
cana-1907	193	16	h(a	h(a	PROPN
cana-1907	193	17	)	)	PUNCT
cana-1907	193	18	}	}	PUNCT
cana-1907	193	19	⊆	⊆	NUM
cana-1907	193	20	b(a,ϵ	b(a,ϵ	NOUN
cana-1907	193	21	)	)	PUNCT
cana-1907	193	22	and	and	CCONJ
cana-1907	193	23	{	{	PUNCT
cana-1907	193	24	a	a	PRON
cana-1907	193	25	}	}	PUNCT
cana-1907	193	26	⊆	⊆	NUM
cana-1907	193	27	b(h(a),ϵ	b(h(a),ϵ	NOUN
cana-1907	193	28	)	)	PUNCT
cana-1907	193	29	since	since	SCONJ
cana-1907	193	30	a	a	DET
cana-1907	193	31	∈	∈	PROPN
cana-1907	193	32	x	x	PUNCT
cana-1907	193	33	is	be	AUX
cana-1907	193	34	arbitrary	arbitrary	ADJ
cana-1907	193	35	,	,	PUNCT
cana-1907	193	36	we	we	PRON
cana-1907	193	37	obtain	obtain	VERB
cana-1907	193	38	d(h	d(h	PROPN
cana-1907	193	39	,	,	PUNCT
cana-1907	193	40	i	i	NOUN
cana-1907	193	41	)	)	PUNCT
cana-1907	193	42	<	<	X
cana-1907	194	1	ϵ	ϵ	X
cana-1907	194	2	hence	hence	ADV
cana-1907	194	3	f	f	PROPN
cana-1907	194	4	∈	∈	PROPN
cana-1907	194	5	c(x	c(x	NOUN
cana-1907	194	6	)	)	PUNCT
cana-1907	194	7	is	be	AUX
cana-1907	194	8	stable	stable	ADJ
cana-1907	194	9	.	.	PUNCT
cana-1907	195	1	□	□	PUNCT
cana-1907	195	2	the	the	DET
cana-1907	195	3	authors	author	NOUN
cana-1907	195	4	would	would	AUX
cana-1907	195	5	like	like	VERB
cana-1907	195	6	to	to	PART
cana-1907	195	7	express	express	VERB
cana-1907	195	8	their	their	PRON
cana-1907	195	9	sincere	sincere	ADJ
cana-1907	195	10	gratitude	gratitude	NOUN
cana-1907	195	11	to	to	ADP
cana-1907	195	12	apj	apj	PROPN
cana-1907	195	13	abdul	abdul	PROPN
cana-1907	195	14	kalam	kalam	PROPN
cana-1907	195	15	technological	technological	PROPN
cana-1907	195	16	university	university	NOUN
cana-1907	195	17	,	,	PUNCT
cana-1907	195	18	kerala	kerala	PROPN
cana-1907	195	19	,	,	PUNCT
cana-1907	195	20	india	india	PROPN
cana-1907	195	21	;	;	PUNCT
cana-1907	195	22	the	the	DET
cana-1907	195	23	management	management	NOUN
cana-1907	195	24	and	and	CCONJ
cana-1907	195	25	staff	staff	NOUN
cana-1907	195	26	of	of	ADP
cana-1907	195	27	sanatana	sanatana	PROPN
cana-1907	195	28	dharma	dharma	PROPN
cana-1907	195	29	college	college	PROPN
cana-1907	195	30	,	,	PUNCT
cana-1907	195	31	alappuzha	alappuzha	PROPN
cana-1907	195	32	,	,	PUNCT
cana-1907	195	33	kerala	kerala	PROPN
cana-1907	195	34	,	,	PUNCT
cana-1907	195	35	india	india	PROPN
cana-1907	195	36	and	and	CCONJ
cana-1907	195	37	the	the	DET
cana-1907	195	38	management	management	NOUN
cana-1907	195	39	and	and	CCONJ
cana-1907	195	40	staff	staff	NOUN
cana-1907	195	41	of	of	ADP
cana-1907	195	42	rajagiri	rajagiri	NOUN
cana-1907	195	43	school	school	NOUN
cana-1907	195	44	of	of	ADP
cana-1907	195	45	engineering	engineering	NOUN
cana-1907	195	46	and	and	CCONJ
cana-1907	195	47	technology	technology	NOUN
cana-1907	195	48	,	,	PUNCT
cana-1907	195	49	kerala	kerala	PROPN
cana-1907	195	50	,	,	PUNCT
cana-1907	195	51	india	india	PROPN
cana-1907	195	52	for	for	ADP
cana-1907	195	53	their	their	PRON
cana-1907	195	54	support	support	NOUN
cana-1907	195	55	.	.	PUNCT
cana-1907	196	1	references	reference	NOUN
cana-1907	196	2	[	[	X
cana-1907	196	3	1	1	X
cana-1907	196	4	]	]	X
cana-1907	196	5	harry	harry	PROPN
cana-1907	196	6	frustenberg	frustenberg	PROPN
cana-1907	196	7	,	,	PUNCT
cana-1907	196	8	disjointness	disjointness	NOUN
cana-1907	196	9	in	in	ADP
cana-1907	196	10	ergodic	ergodic	ADJ
cana-1907	196	11	theory	theory	NOUN
cana-1907	196	12	,	,	PUNCT
cana-1907	196	13	minimal	minimal	ADJ
cana-1907	196	14	sets	set	NOUN
cana-1907	196	15	,	,	PUNCT
cana-1907	196	16	and	and	CCONJ
cana-1907	196	17	a	a	DET
cana-1907	196	18	problem	problem	NOUN
cana-1907	196	19	in	in	ADP
cana-1907	196	20	diophantine	diophantine	NOUN
cana-1907	196	21	approximation	approximation	NOUN
cana-1907	196	22	,	,	PUNCT
cana-1907	196	23	mathematical	mathematical	ADJ
cana-1907	196	24	system	system	NOUN
cana-1907	196	25	theory	theory	NOUN
cana-1907	196	26	,	,	PUNCT
cana-1907	196	27	1	1	NUM
cana-1907	196	28	(	(	PUNCT
cana-1907	196	29	1967	1967	NUM
cana-1907	196	30	)	)	PUNCT
cana-1907	196	31	,	,	PUNCT
cana-1907	196	32	1	1	NUM
cana-1907	196	33	-	-	SYM
cana-1907	196	34	49	49	NUM
cana-1907	196	35	.	.	PUNCT
cana-1907	197	1	[	[	X
cana-1907	197	2	2	2	NUM
cana-1907	197	3	]	]	PUNCT
cana-1907	197	4	jack	jack	NOUN
cana-1907	197	5	t.goodykoontz	t.goodykoontz	PROPN
cana-1907	197	6	,	,	PUNCT
cana-1907	197	7	jr.and	jr.and	PROPN
cana-1907	197	8	choon	choon	PROPN
cana-1907	197	9	jai	jai	PROPN
cana-1907	197	10	rhee	rhee	PROPN
cana-1907	197	11	local	local	ADJ
cana-1907	197	12	properties	property	NOUN
cana-1907	197	13	of	of	ADP
cana-1907	197	14	hyperspaces	hyperspace	NOUN
cana-1907	197	15	,	,	PUNCT
cana-1907	197	16	topology	topology	NOUN
cana-1907	197	17	proceedings	proceeding	NOUN
cana-1907	197	18	,	,	PUNCT
cana-1907	197	19	23(1998),183	23(1998),183	NUM
cana-1907	197	20	–	–	PUNCT
cana-1907	197	21	200	200	NUM
cana-1907	197	22	..	..	PUNCT
cana-1907	198	1	[	[	X
cana-1907	198	2	3	3	X
cana-1907	198	3	]	]	X
cana-1907	198	4	paul	paul	PROPN
cana-1907	198	5	s.bourdon	s.bourdon	PROPN
cana-1907	198	6	,	,	PUNCT
cana-1907	198	7	second	second	ADJ
cana-1907	198	8	iterate	iterate	NOUN
cana-1907	198	9	of	of	ADP
cana-1907	198	10	a	a	DET
cana-1907	198	11	map	map	NOUN
cana-1907	198	12	with	with	ADP
cana-1907	198	13	dense	dense	ADJ
cana-1907	198	14	orbit	orbit	NOUN
cana-1907	198	15	,	,	PUNCT
cana-1907	198	16	proceedings	proceeding	NOUN
cana-1907	198	17	of	of	ADP
cana-1907	198	18	american	american	PROPN
cana-1907	198	19	mathematical	mathematical	PROPN
cana-1907	198	20	society	society	NOUN
cana-1907	198	21	,	,	PUNCT
cana-1907	198	22	124,(1996),1577	124,(1996),1577	NUM
cana-1907	198	23	-	-	PUNCT
cana-1907	198	24	1581	1581	NUM
cana-1907	198	25	.	.	PUNCT
cana-1907	199	1	communications	communication	NOUN
cana-1907	199	2	on	on	ADP
cana-1907	199	3	applied	apply	VERB
cana-1907	199	4	nonlinear	nonlinear	ADJ
cana-1907	199	5	analysis	analysis	NOUN
cana-1907	199	6	issn	issn	NOUN
cana-1907	199	7	:	:	PUNCT
cana-1907	199	8	1074	1074	NUM
cana-1907	199	9	-	-	PUNCT
cana-1907	199	10	133x	133x	NUM
cana-1907	199	11	vol	vol	NOUN
cana-1907	199	12	32	32	NUM
cana-1907	199	13	no	no	NOUN
cana-1907	199	14	.	.	NOUN
cana-1907	199	15	2	2	NUM
cana-1907	199	16	(	(	PUNCT
cana-1907	199	17	2025	2025	NUM
cana-1907	199	18	)	)	PUNCT
cana-1907	199	19	59	59	NUM
cana-1907	199	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1907	200	1	[	[	X
cana-1907	200	2	4	4	X
cana-1907	200	3	]	]	PUNCT
cana-1907	200	4	amalraj.p	amalraj.p	NOUN
cana-1907	200	5	and	and	CCONJ
cana-1907	200	6	p.b.vinod	p.b.vinod	VERB
cana-1907	200	7	kumar	kumar	PROPN
cana-1907	200	8	,	,	PUNCT
cana-1907	200	9	nth	nth	PROPN
cana-1907	200	10	iterate	iterate	NOUN
cana-1907	200	11	of	of	ADP
cana-1907	200	12	a	a	DET
cana-1907	200	13	map	map	NOUN
cana-1907	200	14	with	with	ADP
cana-1907	200	15	dense	dense	ADJ
cana-1907	200	16	orbit	orbit	NOUN
cana-1907	200	17	,	,	PUNCT
cana-1907	200	18	topological	topological	ADJ
cana-1907	200	19	dynamics	dynamic	NOUN
cana-1907	200	20	and	and	CCONJ
cana-1907	200	21	topological	topological	ADJ
cana-1907	200	22	data	datum	NOUN
cana-1907	200	23	analysis	analysis	NOUN
cana-1907	200	24	,	,	PUNCT
cana-1907	200	25	springer	springer	NOUN
cana-1907	200	26	,	,	PUNCT
cana-1907	200	27	(	(	PUNCT
cana-1907	200	28	2018),177	2018),177	NUM
cana-1907	200	29	-	-	SYM
cana-1907	200	30	180	180	NUM
cana-1907	200	31	.	.	PUNCT
cana-1907	201	1	[	[	X
cana-1907	201	2	5	5	NUM
cana-1907	201	3	]	]	X
cana-1907	201	4	hiroshi	hiroshi	PROPN
cana-1907	201	5	hosokava	hosokava	PROPN
cana-1907	201	6	,	,	PUNCT
cana-1907	201	7	induced	induce	VERB
cana-1907	201	8	mappings	mapping	NOUN
cana-1907	201	9	on	on	ADP
cana-1907	201	10	hyperspace	hyperspace	NOUN
cana-1907	201	11	,	,	PUNCT
cana-1907	201	12	tsukuba	tsukuba	PROPN
cana-1907	201	13	j.math	j.math	PROPN
cana-1907	201	14	,	,	PUNCT
cana-1907	201	15	21,1,(1997),239	21,1,(1997),239	NUM
cana-1907	201	16	-	-	PUNCT
cana-1907	201	17	250	250	NUM
cana-1907	201	18	.	.	PUNCT
cana-1907	202	1	[	[	X
cana-1907	202	2	6	6	NUM
cana-1907	202	3	]	]	PUNCT
cana-1907	202	4	amalraj.p	amalraj.p	NOUN
cana-1907	202	5	and	and	CCONJ
cana-1907	202	6	p.b	p.b	PROPN
cana-1907	202	7	.	.	PROPN
cana-1907	202	8	vinod	vinod	PROPN
cana-1907	202	9	kumar	kumar	PROPN
cana-1907	202	10	,	,	PUNCT
cana-1907	202	11	some	some	DET
cana-1907	202	12	properties	property	NOUN
cana-1907	202	13	of	of	ADP
cana-1907	202	14	the	the	DET
cana-1907	202	15	cantor	cantor	NOUN
cana-1907	202	16	set	set	VERB
cana-1907	202	17	in	in	ADP
cana-1907	202	18	the	the	DET
cana-1907	202	19	hyperspace	hyperspace	NOUN
cana-1907	202	20	k(x	k(x	PROPN
cana-1907	202	21	)	)	PUNCT
cana-1907	202	22	,	,	PUNCT
cana-1907	202	23	nanotechnology	nanotechnology	NOUN
cana-1907	202	24	perceptions,20	perceptions,20	ADP
cana-1907	202	25	no	no	DET
cana-1907	202	26	s8(2024),1267	s8(2024),1267	NOUN
cana-1907	202	27	-	-	PUNCT
cana-1907	202	28	1274	1274	NUM
cana-1907	202	29	.	.	PUNCT