id	sid	tid	token	lemma	pos
cana-1480	1	1	communications	communication	NOUN
cana-1480	1	2	on	on	ADP
cana-1480	1	3	applied	apply	VERB
cana-1480	1	4	nonlinear	nonlinear	ADJ
cana-1480	1	5	analysis	analysis	NOUN
cana-1480	1	6	issn	issn	NOUN
cana-1480	1	7	:	:	PUNCT
cana-1480	1	8	1074	1074	NUM
cana-1480	1	9	-	-	PUNCT
cana-1480	1	10	133x	133x	NUM
cana-1480	1	11	vol	vol	NOUN
cana-1480	1	12	31	31	NUM
cana-1480	1	13	no	no	NOUN
cana-1480	1	14	.	.	PUNCT
cana-1480	2	1	8s	8s	PROPN
cana-1480	2	2	(	(	PUNCT
cana-1480	2	3	2024	2024	NUM
cana-1480	2	4	)	)	PUNCT
cana-1480	2	5	253	253	NUM
cana-1480	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1480	2	7	nano	nano	NOUN
cana-1480	2	8	c	c	PROPN
cana-1480	2	9	τ	τ	PROPN
cana-1480	2	10	&	&	CCONJ
cana-1480	2	11	nano*gα	nano*gα	PROPN
cana-1480	2	12	-compactness	-compactness	NOUN
cana-1480	2	13	in	in	ADP
cana-1480	2	14	nts	nts	PROPN
cana-1480	2	15	k.	k.	PROPN
cana-1480	2	16	baby1	baby1	PROPN
cana-1480	2	17	,	,	PUNCT
cana-1480	2	18	h.	h.	PROPN
cana-1480	2	19	aaminumariyam2	aaminumariyam2	PROPN
cana-1480	2	20	,	,	PUNCT
cana-1480	2	21	a.	a.	NOUN
cana-1480	2	22	singaravelan3	singaravelan3	PROPN
cana-1480	3	1	1,3assistant	1,3assistant	NUM
cana-1480	3	2	professor	professor	NOUN
cana-1480	3	3	,	,	PUNCT
cana-1480	3	4	2research	2research	NUM
cana-1480	3	5	scholar	scholar	NOUN
cana-1480	3	6	department	department	NOUN
cana-1480	3	7	of	of	ADP
cana-1480	3	8	mathematics	mathematic	NOUN
cana-1480	3	9	,	,	PUNCT
cana-1480	3	10	kongunadu	kongunadu	ADJ
cana-1480	3	11	arts	art	NOUN
cana-1480	3	12	and	and	CCONJ
cana-1480	3	13	science	science	NOUN
cana-1480	3	14	college(autonomous	college(autonomous	ADJ
cana-1480	3	15	)	)	PUNCT
cana-1480	3	16	coimbatore	coimbatore	NOUN
cana-1480	3	17	–	–	PUNCT
cana-1480	3	18	641	641	NUM
cana-1480	3	19	029	029	NUM
cana-1480	3	20	,	,	PUNCT
cana-1480	3	21	tamilnadu	tamilnadu	NOUN
cana-1480	3	22	,	,	PUNCT
cana-1480	3	23	india	india	PROPN
cana-1480	3	24	.	.	PUNCT
cana-1480	4	1	1email	1email	NUM
cana-1480	4	2	:	:	PUNCT
cana-1480	5	1	babymanoharan31@gmail.com	babymanoharan31@gmail.com	NOUN
cana-1480	5	2	,	,	PUNCT
cana-1480	5	3	2email	2email	NUM
cana-1480	5	4	:	:	PUNCT
cana-1480	5	5	aaminumariyam3@gmail.com	aaminumariyam3@gmail.com	X
cana-1480	5	6	3email	3email	NUM
cana-1480	5	7	:	:	PUNCT
cana-1480	5	8	siveanand@gmail.com	siveanand@gmail.com	X
cana-1480	5	9	article	article	NOUN
cana-1480	5	10	history	history	NOUN
cana-1480	5	11	:	:	PUNCT
cana-1480	5	12	received	receive	VERB
cana-1480	5	13	:	:	PUNCT
cana-1480	5	14	28	28	NUM
cana-1480	5	15	-	-	PUNCT
cana-1480	5	16	04	04	NUM
cana-1480	5	17	-	-	PUNCT
cana-1480	5	18	2024	2024	NUM
cana-1480	5	19	revised	revise	VERB
cana-1480	5	20	:	:	PUNCT
cana-1480	5	21	17	17	NUM
cana-1480	5	22	-	-	SYM
cana-1480	5	23	06	06	NUM
cana-1480	5	24	-	-	PUNCT
cana-1480	5	25	2024	2024	NUM
cana-1480	5	26	accepted	accept	VERB
cana-1480	5	27	:	:	PUNCT
cana-1480	5	28	30	30	NUM
cana-1480	5	29	-	-	SYM
cana-1480	5	30	06	06	NUM
cana-1480	5	31	-	-	PUNCT
cana-1480	5	32	2024	2024	NUM
cana-1480	5	33	abstract	abstract	NOUN
cana-1480	5	34	:	:	PUNCT
cana-1480	5	35	the	the	DET
cana-1480	5	36	purpose	purpose	NOUN
cana-1480	5	37	of	of	ADP
cana-1480	5	38	this	this	DET
cana-1480	5	39	paper	paper	NOUN
cana-1480	5	40	is	be	AUX
cana-1480	5	41	to	to	PART
cana-1480	5	42	introduce	introduce	VERB
cana-1480	5	43	and	and	CCONJ
cana-1480	5	44	study	study	VERB
cana-1480	5	45	the	the	DET
cana-1480	5	46	concept	concept	NOUN
cana-1480	5	47	of	of	ADP
cana-1480	5	48	c	c	NOUN
cana-1480	5	49	τ	τ	NOUN
cana-1480	5	50	-compactness	-compactness	NOUN
cana-1480	5	51	in	in	ADP
cana-1480	5	52	nts	nt	NOUN
cana-1480	5	53	and	and	CCONJ
cana-1480	5	54	entrenched	entrench	VERB
cana-1480	5	55	few	few	ADJ
cana-1480	5	56	of	of	ADP
cana-1480	5	57	their	their	PRON
cana-1480	5	58	accompanying	accompany	VERB
cana-1480	5	59	features	feature	NOUN
cana-1480	5	60	.	.	PUNCT
cana-1480	6	1	ferther	ferther	ADV
cana-1480	6	2	we	we	PRON
cana-1480	6	3	investigate	investigate	VERB
cana-1480	6	4	the	the	DET
cana-1480	6	5	nano*gα	nano*gα	ADJ
cana-1480	6	6	compact	compact	ADJ
cana-1480	6	7	and	and	CCONJ
cana-1480	6	8	connectedness	connectedness	NOUN
cana-1480	6	9	in	in	ADP
cana-1480	6	10	nts	nt	NOUN
cana-1480	6	11	.	.	PUNCT
cana-1480	7	1	keywords	keyword	NOUN
cana-1480	7	2	:	:	PUNCT
cana-1480	7	3	nano*gα	nano*gα	PROPN
cana-1480	7	4	-closed	-close	VERB
cana-1480	7	5	set	set	NOUN
cana-1480	7	6	,	,	PUNCT
cana-1480	7	7	nano*gα	nano*gα	ADJ
cana-1480	7	8	continuous	continuous	ADJ
cana-1480	7	9	function	function	NOUN
cana-1480	7	10	,	,	PUNCT
cana-1480	7	11	nano	nano	NOUN
cana-1480	7	12	c	c	PROPN
cana-1480	7	13	τ	τ	PROPN
cana-1480	7	14	-compact	-compact	PROPN
cana-1480	7	15	,	,	PUNCT
cana-1480	7	16	nano*gα	nano*gα	PROPN
cana-1480	7	17	compact	compact	ADJ
cana-1480	7	18	,	,	PUNCT
cana-1480	7	19	nano*gα	nano*gα	PROPN
cana-1480	7	20	connected	connect	VERB
cana-1480	7	21	.	.	PUNCT
cana-1480	8	1	1	1	NUM
cana-1480	8	2	introduction	introduction	NOUN
cana-1480	8	3	connectedness	connectedness	NOUN
cana-1480	8	4	and	and	CCONJ
cana-1480	8	5	disconnectedness	disconnectedness	NOUN
cana-1480	8	6	in	in	ADP
cana-1480	8	7	topology	topology	NOUN
cana-1480	8	8	is	be	AUX
cana-1480	8	9	introduced	introduce	VERB
cana-1480	8	10	by	by	ADP
cana-1480	8	11	a.v.arhangelskii	a.v.arhangelskii	PROPN
cana-1480	8	12	and	and	CCONJ
cana-1480	8	13	r.wiegandt	r.wiegandt	PRON
cana-1480	9	1	[	[	X
cana-1480	9	2	7].ctness	7].ctness	NUM
cana-1480	9	3	in	in	ADP
cana-1480	9	4	general	general	ADJ
cana-1480	9	5	is	be	AUX
cana-1480	9	6	an	an	DET
cana-1480	9	7	essential	essential	ADJ
cana-1480	9	8	part	part	NOUN
cana-1480	9	9	of	of	ADP
cana-1480	9	10	the	the	DET
cana-1480	9	11	topological	topological	ADJ
cana-1480	9	12	space	space	NOUN
cana-1480	9	13	with	with	ADP
cana-1480	9	14	regard	regard	NOUN
cana-1480	9	15	to	to	ADP
cana-1480	9	16	the	the	DET
cana-1480	9	17	property	property	NOUN
cana-1480	9	18	of	of	ADP
cana-1480	9	19	closed	closed	ADJ
cana-1480	9	20	and	and	CCONJ
cana-1480	9	21	bounded	bound	VERB
cana-1480	9	22	subsets.the	subsets.the	DET
cana-1480	9	23	idea	idea	NOUN
cana-1480	9	24	of	of	ADP
cana-1480	9	25	compacttness	compacttness	NOUN
cana-1480	9	26	and	and	CCONJ
cana-1480	9	27	connectedness	connectedness	NOUN
cana-1480	9	28	are	be	AUX
cana-1480	9	29	beneficial	beneficial	ADJ
cana-1480	9	30	for	for	ADP
cana-1480	9	31	the	the	DET
cana-1480	9	32	basis	basis	NOUN
cana-1480	9	33	ideas	idea	NOUN
cana-1480	9	34	of	of	ADP
cana-1480	9	35	general	general	ADJ
cana-1480	9	36	topology	topology	NOUN
cana-1480	9	37	as	as	ADV
cana-1480	9	38	well	well	ADV
cana-1480	9	39	as	as	ADP
cana-1480	9	40	for	for	ADP
cana-1480	9	41	advanced	advanced	ADJ
cana-1480	9	42	branches	branch	NOUN
cana-1480	9	43	of	of	ADP
cana-1480	9	44	mathematics	mathematic	NOUN
cana-1480	9	45	.	.	PUNCT
cana-1480	10	1	m.	m.	PROPN
cana-1480	10	2	vigneshwaran	vigneshwaran	PROPN
cana-1480	10	3	and	and	CCONJ
cana-1480	10	4	r.	r.	PROPN
cana-1480	10	5	devi	devi	PROPN
cana-1480	11	1	[	[	X
cana-1480	11	2	3	3	NUM
cana-1480	11	3	]	]	PUNCT
cana-1480	11	4	introduced	introduce	VERB
cana-1480	11	5	the	the	DET
cana-1480	11	6	concepts	concept	NOUN
cana-1480	11	7	of	of	ADP
cana-1480	11	8	*	*	PUNCT
cana-1480	11	9	gα	gα	ADP
cana-1480	11	10	-	-	PUNCT
cana-1480	11	11	closed	close	VERB
cana-1480	11	12	sets	set	NOUN
cana-1480	11	13	in	in	ADP
cana-1480	11	14	topological	topological	PROPN
cana-1480	11	15	spaces.in	spaces.in	PROPN
cana-1480	11	16	1970,levine	1970,levine	NUM
cana-1480	12	1	[	[	X
cana-1480	12	2	6	6	NUM
cana-1480	12	3	]	]	PUNCT
cana-1480	12	4	introduced	introduce	VERB
cana-1480	12	5	the	the	DET
cana-1480	12	6	concept	concept	NOUN
cana-1480	12	7	of	of	ADP
cana-1480	12	8	generalized	generalized	ADJ
cana-1480	12	9	closed	close	VERB
cana-1480	12	10	sets	set	NOUN
cana-1480	12	11	as	as	ADP
cana-1480	12	12	a	a	DET
cana-1480	12	13	generalization	generalization	NOUN
cana-1480	12	14	of	of	ADP
cana-1480	12	15	closed	closed	ADJ
cana-1480	12	16	sets	set	NOUN
cana-1480	12	17	in	in	ADP
cana-1480	12	18	topological	topological	ADJ
cana-1480	12	19	spaces	space	NOUN
cana-1480	12	20	.	.	PUNCT
cana-1480	13	1	lellis	lellis	PROPN
cana-1480	13	2	thivagar	thivagar	NOUN
cana-1480	13	3	[	[	X
cana-1480	13	4	4]and	4]and	PROPN
cana-1480	13	5	carmel	carmel	PROPN
cana-1480	13	6	richard	richard	PROPN
cana-1480	13	7	introduced	introduce	VERB
cana-1480	13	8	the	the	DET
cana-1480	13	9	concept	concept	NOUN
cana-1480	13	10	of	of	ADP
cana-1480	13	11	nano	nano	NOUN
cana-1480	13	12	topology	topology	NOUN
cana-1480	13	13	,	,	PUNCT
cana-1480	13	14	which	which	PRON
cana-1480	13	15	was	be	AUX
cana-1480	13	16	defined	define	VERB
cana-1480	13	17	in	in	ADP
cana-1480	13	18	terms	term	NOUN
cana-1480	13	19	of	of	ADP
cana-1480	13	20	approximations	approximation	NOUN
cana-1480	13	21	and	and	CCONJ
cana-1480	13	22	boundry	boundry	ADJ
cana-1480	13	23	region	region	NOUN
cana-1480	13	24	of	of	ADP
cana-1480	13	25	a	a	DET
cana-1480	13	26	universe	universe	NOUN
cana-1480	13	27	using	use	VERB
cana-1480	13	28	a	a	DET
cana-1480	13	29	equivalence	equivalence	NOUN
cana-1480	13	30	relation	relation	NOUN
cana-1480	13	31	on	on	ADP
cana-1480	13	32	it.he	it.he	NOUN
cana-1480	13	33	also	also	ADV
cana-1480	13	34	introduced	introduce	VERB
cana-1480	13	35	nano	nano	ADJ
cana-1480	13	36	continuous	continuous	ADJ
cana-1480	13	37	functions	function	NOUN
cana-1480	13	38	,	,	PUNCT
cana-1480	13	39	nano	nano	NOUN
cana-1480	13	40	open	open	ADJ
cana-1480	13	41	mappings	mapping	NOUN
cana-1480	13	42	,	,	PUNCT
cana-1480	13	43	nano	nano	NOUN
cana-1480	13	44	closed	close	VERB
cana-1480	13	45	mappings	mapping	NOUN
cana-1480	13	46	and	and	CCONJ
cana-1480	13	47	nano	nano	NOUN
cana-1480	13	48	homeomorphisms	homeomorphism	NOUN
cana-1480	13	49	in	in	ADP
cana-1480	13	50	nts.s.krishnaprakash	nts.s.krishnaprakash	NOUN
cana-1480	13	51	et.al	et.al	ADJ
cana-1480	14	1	[	[	X
cana-1480	14	2	8	8	NUM
cana-1480	14	3	]	]	PUNCT
cana-1480	14	4	innovative	innovative	ADJ
cana-1480	14	5	some	some	DET
cana-1480	14	6	concept	concept	NOUN
cana-1480	14	7	of	of	ADP
cana-1480	14	8	nano	nano	ADJ
cana-1480	14	9	compact	compact	ADJ
cana-1480	14	10	space	space	NOUN
cana-1480	14	11	and	and	CCONJ
cana-1480	14	12	nano	nano	NOUN
cana-1480	14	13	connected	connect	VERB
cana-1480	14	14	in	in	ADP
cana-1480	14	15	nano	nano	VERB
cana-1480	14	16	topology.the	topology.the	DET
cana-1480	14	17	intension	intension	NOUN
cana-1480	14	18	of	of	ADP
cana-1480	14	19	this	this	DET
cana-1480	14	20	paper	paper	NOUN
cana-1480	14	21	is	be	AUX
cana-1480	14	22	to	to	PART
cana-1480	14	23	establish	establish	VERB
cana-1480	14	24	the	the	DET
cana-1480	14	25	conception	conception	NOUN
cana-1480	14	26	of	of	ADP
cana-1480	14	27	c	c	PROPN
cana-1480	14	28	τ	τ	PROPN
cana-1480	14	29	compact	compact	ADV
cana-1480	14	30	set	set	VERB
cana-1480	14	31	and	and	CCONJ
cana-1480	14	32	find	find	VERB
cana-1480	14	33	few	few	ADJ
cana-1480	14	34	of	of	ADP
cana-1480	14	35	their	their	PRON
cana-1480	14	36	features	feature	NOUN
cana-1480	14	37	.	.	PUNCT
cana-1480	15	1	it	it	PRON
cana-1480	15	2	also	also	ADV
cana-1480	15	3	established	establish	VERB
cana-1480	15	4	the	the	DET
cana-1480	15	5	conception	conception	NOUN
cana-1480	15	6	of	of	ADP
cana-1480	15	7	n	n	PROPN
cana-1480	15	8	ano	ano	PROPN
cana-1480	15	9	-compact	-compact	PROPN
cana-1480	15	10	and	and	CCONJ
cana-1480	15	11	n	n	CCONJ
cana-1480	15	12	ano*gα	ano*gα	PROPN
cana-1480	15	13	-connected	-connected	PROPN
cana-1480	15	14	.	.	PUNCT
cana-1480	16	1	the	the	DET
cana-1480	16	2	current	current	ADJ
cana-1480	16	3	study	study	NOUN
cana-1480	16	4	is	be	AUX
cana-1480	16	5	about	about	ADV
cana-1480	16	6	few	few	ADJ
cana-1480	16	7	of	of	ADP
cana-1480	16	8	associated	associated	ADJ
cana-1480	16	9	theorems	theorem	NOUN
cana-1480	16	10	and	and	CCONJ
cana-1480	16	11	results	result	NOUN
cana-1480	16	12	.	.	PUNCT
cana-1480	17	1	2	2	X
cana-1480	17	2	.	.	X
cana-1480	17	3	preliminaries	preliminary	NOUN
cana-1480	17	4	in	in	ADP
cana-1480	17	5	this	this	DET
cana-1480	17	6	section	section	NOUN
cana-1480	17	7	,	,	PUNCT
cana-1480	17	8	we	we	PRON
cana-1480	17	9	recall	recall	VERB
cana-1480	17	10	some	some	DET
cana-1480	17	11	basic	basic	ADJ
cana-1480	17	12	definitions	definition	NOUN
cana-1480	17	13	and	and	CCONJ
cana-1480	17	14	results	result	NOUN
cana-1480	17	15	in	in	ADP
cana-1480	17	16	nano	nano	ADJ
cana-1480	17	17	topological	topological	ADJ
cana-1480	17	18	spaces	space	NOUN
cana-1480	17	19	.	.	PUNCT
cana-1480	18	1	definition	definition	NOUN
cana-1480	18	2	2.1.[4	2.1.[4	NUM
cana-1480	18	3	]	]	PUNCT
cana-1480	18	4	let	let	VERB
cana-1480	18	5	𝑈	𝑈	PROPN
cana-1480	18	6	be	be	AUX
cana-1480	18	7	a	a	DET
cana-1480	18	8	non	non	ADJ
cana-1480	18	9	-	-	ADJ
cana-1480	18	10	empty	empty	ADJ
cana-1480	18	11	finite	finite	NOUN
cana-1480	18	12	set	set	NOUN
cana-1480	18	13	of	of	ADP
cana-1480	18	14	objects	object	NOUN
cana-1480	18	15	called	call	VERB
cana-1480	18	16	the	the	DET
cana-1480	18	17	universe	universe	NOUN
cana-1480	18	18	and	and	CCONJ
cana-1480	18	19	𝑅	𝑅	PROPN
cana-1480	18	20	be	be	AUX
cana-1480	18	21	an	an	DET
cana-1480	18	22	equivalence	equivalence	NOUN
cana-1480	18	23	relation	relation	NOUN
cana-1480	18	24	on	on	ADP
cana-1480	18	25	𝑈	𝑈	PROPN
cana-1480	18	26	named	name	VERB
cana-1480	18	27	as	as	ADP
cana-1480	18	28	in	in	ADP
cana-1480	18	29	discernibility	discernibility	NOUN
cana-1480	18	30	relation	relation	NOUN
cana-1480	18	31	.	.	PUNCT
cana-1480	19	1	then	then	ADV
cana-1480	19	2	𝑈	𝑈	PROPN
cana-1480	19	3	is	be	AUX
cana-1480	19	4	divided	divide	VERB
cana-1480	19	5	into	into	ADP
cana-1480	19	6	equivalence	equivalence	NOUN
cana-1480	19	7	classes	class	NOUN
cana-1480	19	8	.	.	PUNCT
cana-1480	20	1	elements	element	NOUN
cana-1480	20	2	belonging	belong	VERB
cana-1480	20	3	to	to	ADP
cana-1480	20	4	the	the	DET
cana-1480	20	5	same	same	ADJ
cana-1480	20	6	equivalence	equivalence	NOUN
cana-1480	20	7	class	class	NOUN
cana-1480	20	8	are	be	AUX
cana-1480	20	9	said	say	VERB
cana-1480	20	10	to	to	PART
cana-1480	20	11	be	be	AUX
cana-1480	20	12	indiscernible	indiscernible	ADJ
cana-1480	20	13	with	with	ADP
cana-1480	20	14	one	one	NUM
cana-1480	20	15	another	another	DET
cana-1480	20	16	.	.	PUNCT
cana-1480	21	1	the	the	DET
cana-1480	21	2	pair	pair	NOUN
cana-1480	21	3	(	(	PUNCT
cana-1480	21	4	𝑈	𝑈	PROPN
cana-1480	21	5	,	,	PUNCT
cana-1480	21	6	𝑅	𝑅	PROPN
cana-1480	21	7	)	)	PUNCT
cana-1480	21	8	is	be	AUX
cana-1480	21	9	said	say	VERB
cana-1480	21	10	to	to	PART
cana-1480	21	11	be	be	AUX
cana-1480	21	12	the	the	DET
cana-1480	21	13	approximation	approximation	NOUN
cana-1480	21	14	space	space	NOUN
cana-1480	21	15	.	.	PUNCT
cana-1480	22	1	let	let	VERB
cana-1480	22	2	𝑋	𝑋	PROPN
cana-1480	22	3	⊆	⊆	NUM
cana-1480	22	4	𝑈.	𝑈.	PROPN
cana-1480	22	5	then	then	ADV
cana-1480	22	6	,	,	PUNCT
cana-1480	22	7	communications	communication	NOUN
cana-1480	22	8	on	on	ADP
cana-1480	22	9	applied	apply	VERB
cana-1480	22	10	nonlinear	nonlinear	ADJ
cana-1480	22	11	analysis	analysis	NOUN
cana-1480	22	12	issn	issn	NOUN
cana-1480	22	13	:	:	PUNCT
cana-1480	22	14	1074	1074	NUM
cana-1480	22	15	-	-	PUNCT
cana-1480	22	16	133x	133x	NUM
cana-1480	22	17	vol	vol	NOUN
cana-1480	22	18	31	31	NUM
cana-1480	22	19	no	no	NOUN
cana-1480	22	20	.	.	PUNCT
cana-1480	23	1	8s	8s	PROPN
cana-1480	23	2	(	(	PUNCT
cana-1480	23	3	2024	2024	NUM
cana-1480	23	4	)	)	PUNCT
cana-1480	23	5	254	254	NUM
cana-1480	23	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1480	23	7	•	•	ADP
cana-1480	23	8	the	the	DET
cana-1480	23	9	lower	low	ADJ
cana-1480	23	10	approximation	approximation	NOUN
cana-1480	23	11	of	of	ADP
cana-1480	23	12	𝑋	𝑋	PROPN
cana-1480	23	13	with	with	ADP
cana-1480	23	14	respect	respect	NOUN
cana-1480	23	15	to	to	ADP
cana-1480	23	16	𝑅	𝑅	PROPN
cana-1480	23	17	is	be	AUX
cana-1480	23	18	the	the	DET
cana-1480	23	19	set	set	NOUN
cana-1480	23	20	of	of	ADP
cana-1480	23	21	all	all	DET
cana-1480	23	22	objects	object	NOUN
cana-1480	23	23	which	which	PRON
cana-1480	23	24	can	can	AUX
cana-1480	23	25	be	be	AUX
cana-1480	23	26	for	for	ADP
cana-1480	23	27	certain	certain	ADJ
cana-1480	23	28	classified	classified	ADJ
cana-1480	23	29	as	as	ADP
cana-1480	23	30	𝑋	𝑋	NOUN
cana-1480	23	31	with	with	ADP
cana-1480	23	32	respect	respect	NOUN
cana-1480	23	33	to	to	ADP
cana-1480	23	34	𝑅	𝑅	PROPN
cana-1480	23	35	and	and	CCONJ
cana-1480	23	36	is	be	AUX
cana-1480	23	37	denoted	denote	VERB
cana-1480	23	38	by	by	ADP
cana-1480	23	39	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
cana-1480	23	40	)	)	PUNCT
cana-1480	23	41	.	.	PUNCT
cana-1480	24	1	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
cana-1480	24	2	)	)	PUNCT
cana-1480	25	1	=	=	SYM
cana-1480	25	2	𝑈{𝑅(𝑋	𝑈{𝑅(𝑋	PROPN
cana-1480	25	3	):	):	PUNCT
cana-1480	25	4	𝑅(𝑋	𝑅(𝑋	NOUN
cana-1480	25	5	)	)	PUNCT
cana-1480	25	6	⊆	⊆	NUM
cana-1480	25	7	x	x	NOUN
cana-1480	25	8	,	,	PUNCT
cana-1480	25	9	x	x	SYM
cana-1480	25	10	∈	∈	PROPN
cana-1480	25	11	𝑈	𝑈	PROPN
cana-1480	25	12	}	}	PUNCT
cana-1480	25	13	•	•	NOUN
cana-1480	25	14	the	the	DET
cana-1480	25	15	upper	upper	ADJ
cana-1480	25	16	approximation	approximation	NOUN
cana-1480	25	17	of	of	ADP
cana-1480	25	18	𝑋	𝑋	PROPN
cana-1480	25	19	with	with	ADP
cana-1480	25	20	respect	respect	NOUN
cana-1480	25	21	to	to	ADP
cana-1480	25	22	𝑅	𝑅	PROPN
cana-1480	25	23	is	be	AUX
cana-1480	25	24	the	the	DET
cana-1480	25	25	set	set	NOUN
cana-1480	25	26	of	of	ADP
cana-1480	25	27	all	all	DET
cana-1480	25	28	objects	object	NOUN
cana-1480	25	29	which	which	PRON
cana-1480	25	30	can	can	AUX
cana-1480	25	31	be	be	AUX
cana-1480	25	32	possibly	possibly	ADV
cana-1480	25	33	classified	classify	VERB
cana-1480	25	34	as	as	ADP
cana-1480	25	35	𝑋	𝑋	NOUN
cana-1480	25	36	with	with	ADP
cana-1480	25	37	respect	respect	NOUN
cana-1480	25	38	to	to	ADP
cana-1480	25	39	𝑅	𝑅	PROPN
cana-1480	25	40	and	and	CCONJ
cana-1480	25	41	is	be	AUX
cana-1480	25	42	denoted	denote	VERB
cana-1480	25	43	by	by	ADP
cana-1480	25	44	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
cana-1480	25	45	)	)	PUNCT
cana-1480	25	46	.	.	PUNCT
cana-1480	26	1	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
cana-1480	26	2	)	)	PUNCT
cana-1480	26	3	=	=	SYM
cana-1480	26	4	𝑈{𝑅(𝑋	𝑈{𝑅(𝑋	PROPN
cana-1480	26	5	):	):	PUNCT
cana-1480	26	6	𝑅(𝑋	𝑅(𝑋	NOUN
cana-1480	26	7	)	)	PUNCT
cana-1480	26	8	∩	∩	NOUN
cana-1480	26	9	𝑋	𝑋	PROPN
cana-1480	26	10	≠	≠	PROPN
cana-1480	26	11	𝜙	𝜙	NOUN
cana-1480	26	12	,	,	PUNCT
cana-1480	26	13	x	x	SYM
cana-1480	26	14	∈	∈	PROPN
cana-1480	26	15	u	u	NOUN
cana-1480	26	16	}	}	PUNCT
cana-1480	26	17	.	.	PUNCT
cana-1480	27	1	•	•	NUM
cana-1480	27	2	the	the	DET
cana-1480	27	3	boundary	boundary	ADJ
cana-1480	27	4	region	region	NOUN
cana-1480	27	5	of	of	ADP
cana-1480	27	6	x	x	PUNCT
cana-1480	27	7	with	with	ADP
cana-1480	27	8	respect	respect	NOUN
cana-1480	27	9	to	to	ADP
cana-1480	27	10	r	r	NOUN
cana-1480	27	11	is	be	AUX
cana-1480	27	12	the	the	DET
cana-1480	27	13	set	set	NOUN
cana-1480	27	14	of	of	ADP
cana-1480	27	15	all	all	DET
cana-1480	27	16	objects	object	NOUN
cana-1480	27	17	which	which	PRON
cana-1480	27	18	can	can	AUX
cana-1480	27	19	be	be	AUX
cana-1480	27	20	classified	classify	VERB
cana-1480	27	21	neither	neither	CCONJ
cana-1480	27	22	as	as	ADP
cana-1480	27	23	𝑋	𝑋	NOUN
cana-1480	27	24	nor	nor	CCONJ
cana-1480	27	25	as	as	ADV
cana-1480	27	26	not	not	PART
cana-1480	27	27	−𝑋	−𝑋	VERB
cana-1480	27	28	with	with	ADP
cana-1480	27	29	respect	respect	NOUN
cana-1480	27	30	to	to	ADP
cana-1480	27	31	r	r	NOUN
cana-1480	27	32	and	and	CCONJ
cana-1480	27	33	it	it	PRON
cana-1480	27	34	is	be	AUX
cana-1480	27	35	denoted	denote	VERB
cana-1480	27	36	by	by	ADP
cana-1480	27	37	𝐵𝑅(𝑋	𝐵𝑅(𝑋	NOUN
cana-1480	27	38	)	)	PUNCT
cana-1480	28	1	=	=	SYM
cana-1480	28	2	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
cana-1480	28	3	)	)	PUNCT
cana-1480	28	4	−	−	PROPN
cana-1480	28	5	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
cana-1480	28	6	)	)	PUNCT
cana-1480	28	7	.	.	PUNCT
cana-1480	29	1	definition	definition	NOUN
cana-1480	29	2	2.2.[4	2.2.[4	NUM
cana-1480	29	3	]	]	PUNCT
cana-1480	29	4	let	let	VERB
cana-1480	29	5	u	u	PRON
cana-1480	29	6	be	be	AUX
cana-1480	29	7	the	the	DET
cana-1480	29	8	universe	universe	NOUN
cana-1480	29	9	,	,	PUNCT
cana-1480	29	10	r	r	NOUN
cana-1480	29	11	be	be	VERB
cana-1480	29	12	an	an	DET
cana-1480	29	13	equivalence	equivalence	NOUN
cana-1480	29	14	relation	relation	NOUN
cana-1480	29	15	on	on	ADP
cana-1480	29	16	u	u	NOUN
cana-1480	29	17	and	and	CCONJ
cana-1480	29	18	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	29	19	)	)	PUNCT
cana-1480	30	1	=	=	PRON
cana-1480	30	2	{	{	PUNCT
cana-1480	30	3	𝑈	𝑈	PROPN
cana-1480	30	4	,	,	PUNCT
cana-1480	30	5	𝜙	𝜙	NOUN
cana-1480	30	6	,	,	PUNCT
cana-1480	30	7	𝐿𝑅(𝑋	𝐿𝑅(𝑋	NOUN
cana-1480	30	8	)	)	PUNCT
cana-1480	30	9	,	,	PUNCT
cana-1480	30	10	𝑈𝑅(𝑋	𝑈𝑅(𝑋	NOUN
cana-1480	30	11	)	)	PUNCT
cana-1480	30	12	,	,	PUNCT
cana-1480	30	13	𝐵𝑅(𝑋)}where	𝐵𝑅(𝑋)}where	ADP
cana-1480	30	14	𝑋	𝑋	PROPN
cana-1480	30	15	⊆	⊆	NUM
cana-1480	30	16	𝑈.	𝑈.	PROPN
cana-1480	30	17	then	then	ADV
cana-1480	30	18	it	it	PRON
cana-1480	30	19	satisfies	satisfy	VERB
cana-1480	30	20	the	the	DET
cana-1480	30	21	following	follow	VERB
cana-1480	30	22	axioms	axiom	NOUN
cana-1480	30	23	:	:	PUNCT
cana-1480	30	24	1	1	X
cana-1480	30	25	.	.	X
cana-1480	30	26	𝑈	𝑈	NOUN
cana-1480	30	27	and	and	CCONJ
cana-1480	30	28	𝜙	𝜙	PROPN
cana-1480	30	29	belongs	belong	VERB
cana-1480	30	30	to	to	ADP
cana-1480	30	31	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	30	32	)	)	PUNCT
cana-1480	30	33	2	2	NUM
cana-1480	30	34	.	.	X
cana-1480	31	1	the	the	DET
cana-1480	31	2	union	union	NOUN
cana-1480	31	3	of	of	ADP
cana-1480	31	4	the	the	DET
cana-1480	31	5	elements	element	NOUN
cana-1480	31	6	of	of	ADP
cana-1480	31	7	any	any	DET
cana-1480	31	8	sub	sub	NOUN
cana-1480	31	9	-	-	NOUN
cana-1480	31	10	collection	collection	NOUN
cana-1480	31	11	of	of	ADP
cana-1480	31	12	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	31	13	)	)	PUNCT
cana-1480	31	14	is	be	AUX
cana-1480	31	15	in	in	ADP
cana-1480	31	16	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	31	17	)	)	PUNCT
cana-1480	31	18	.	.	PUNCT
cana-1480	32	1	3	3	X
cana-1480	32	2	.	.	X
cana-1480	32	3	the	the	DET
cana-1480	32	4	intersection	intersection	NOUN
cana-1480	32	5	of	of	ADP
cana-1480	32	6	the	the	DET
cana-1480	32	7	elements	element	NOUN
cana-1480	32	8	of	of	ADP
cana-1480	32	9	any	any	DET
cana-1480	32	10	finite	finite	ADJ
cana-1480	32	11	sub	sub	NOUN
cana-1480	32	12	collection	collection	NOUN
cana-1480	32	13	of	of	ADP
cana-1480	32	14	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	32	15	)	)	PUNCT
cana-1480	32	16	is	be	AUX
cana-1480	32	17	in	in	ADP
cana-1480	32	18	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	32	19	)	)	PUNCT
cana-1480	32	20	.	.	PUNCT
cana-1480	33	1	then	then	ADV
cana-1480	33	2	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	33	3	)	)	PUNCT
cana-1480	33	4	is	be	AUX
cana-1480	33	5	a	a	DET
cana-1480	33	6	topology	topology	NOUN
cana-1480	33	7	on	on	ADP
cana-1480	33	8	u	u	NOUN
cana-1480	33	9	called	call	VERB
cana-1480	33	10	the	the	DET
cana-1480	33	11	nano	nano	NOUN
cana-1480	33	12	topology	topology	NOUN
cana-1480	33	13	on	on	ADP
cana-1480	33	14	u	u	NOUN
cana-1480	33	15	with	with	ADP
cana-1480	33	16	respect	respect	NOUN
cana-1480	33	17	to	to	ADP
cana-1480	33	18	𝑋.	𝑋.	PROPN
cana-1480	33	19	(	(	PUNCT
cana-1480	33	20	𝑈	𝑈	PROPN
cana-1480	33	21	,	,	PUNCT
cana-1480	33	22	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	33	23	)	)	PUNCT
cana-1480	33	24	)	)	PUNCT
cana-1480	33	25	is	be	AUX
cana-1480	33	26	called	call	VERB
cana-1480	33	27	the	the	DET
cana-1480	33	28	nano	nano	NOUN
cana-1480	33	29	topological	topological	ADJ
cana-1480	33	30	space	space	NOUN
cana-1480	33	31	.	.	PUNCT
cana-1480	34	1	elements	element	NOUN
cana-1480	34	2	of	of	ADP
cana-1480	34	3	the	the	DET
cana-1480	34	4	nano	nano	NOUN
cana-1480	34	5	topology	topology	NOUN
cana-1480	34	6	are	be	AUX
cana-1480	34	7	known	know	VERB
cana-1480	34	8	as	as	ADP
cana-1480	34	9	nano	nano	NOUN
cana-1480	34	10	sets	set	NOUN
cana-1480	34	11	in	in	ADP
cana-1480	34	12	u.	u.	NOUN
cana-1480	34	13	elements	element	NOUN
cana-1480	34	14	of	of	ADP
cana-1480	34	15	[	[	X
cana-1480	34	16	𝜏𝑅(𝑋)]𝐶	𝜏𝑅(𝑋)]𝐶	NOUN
cana-1480	34	17	are	be	AUX
cana-1480	34	18	called	call	VERB
cana-1480	34	19	nano	nano	NOUN
cana-1480	34	20	closed	close	VERB
cana-1480	34	21	sets	set	NOUN
cana-1480	34	22	with	with	ADP
cana-1480	34	23	[	[	AUX
cana-1480	34	24	𝜏𝑅(𝑋)]𝐶	𝜏𝑅(𝑋)]𝐶	NOUN
cana-1480	34	25	being	be	AUX
cana-1480	34	26	called	call	VERB
cana-1480	34	27	dual	dual	ADJ
cana-1480	34	28	nano	nano	NOUN
cana-1480	34	29	topology	topology	NOUN
cana-1480	34	30	of	of	ADP
cana-1480	34	31	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	34	32	)	)	PUNCT
cana-1480	34	33	.	.	PUNCT
cana-1480	35	1	definition	definition	NOUN
cana-1480	35	2	2.3.[9	2.3.[9	NUM
cana-1480	35	3	]	]	PUNCT
cana-1480	35	4	a	a	DET
cana-1480	35	5	subset	subset	ADJ
cana-1480	35	6	𝐴	𝐴	PROPN
cana-1480	35	7	of	of	ADP
cana-1480	35	8	(	(	PUNCT
cana-1480	35	9	𝑈	𝑈	PROPN
cana-1480	35	10	,	,	PUNCT
cana-1480	35	11	𝜏𝑅(𝑋))is	𝜏𝑅(𝑋))is	PROPN
cana-1480	35	12	called	call	VERB
cana-1480	35	13	nano*gα	nano*gα	PROPN
cana-1480	35	14	-	-	PUNCT
cana-1480	35	15	closed	closed	ADJ
cana-1480	35	16	set	set	NOUN
cana-1480	35	17	if	if	SCONJ
cana-1480	35	18	𝑁𝑐𝑙(𝐴	𝑁𝑐𝑙(𝐴	NOUN
cana-1480	35	19	)	)	PUNCT
cana-1480	36	1	⊆	⊆	X
cana-1480	36	2	𝑉	𝑉	PROPN
cana-1480	36	3	whenever	whenever	SCONJ
cana-1480	36	4	𝐴	𝐴	PROPN
cana-1480	36	5	⊆	⊆	NUM
cana-1480	36	6	𝑉	𝑉	PROPN
cana-1480	36	7	and	and	CCONJ
cana-1480	36	8	v	v	NOUN
cana-1480	36	9	is	be	AUX
cana-1480	36	10	𝑁gα	𝑁gα	PROPN
cana-1480	36	11	open	open	ADJ
cana-1480	36	12	in	in	ADP
cana-1480	36	13	(	(	PUNCT
cana-1480	36	14	𝑈	𝑈	PROPN
cana-1480	36	15	,	,	PUNCT
cana-1480	36	16	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	36	17	)	)	PUNCT
cana-1480	36	18	)	)	PUNCT
cana-1480	36	19	.	.	PUNCT
cana-1480	37	1	definition	definition	NOUN
cana-1480	37	2	2.4.[10	2.4.[10	NUM
cana-1480	37	3	]	]	PUNCT
cana-1480	37	4	a	a	DET
cana-1480	37	5	function	function	NOUN
cana-1480	37	6	𝑓	𝑓	NOUN
cana-1480	37	7	:	:	PUNCT
cana-1480	37	8	(	(	PUNCT
cana-1480	37	9	𝑈	𝑈	PROPN
cana-1480	37	10	,	,	PUNCT
cana-1480	37	11	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	37	12	)	)	PUNCT
cana-1480	37	13	)	)	PUNCT
cana-1480	38	1	→	→	PUNCT
cana-1480	38	2	(	(	PUNCT
cana-1480	38	3	𝑉	𝑉	PROPN
cana-1480	38	4	,	,	PUNCT
cana-1480	38	5	𝜎𝑅(𝑌	𝜎𝑅(𝑌	NOUN
cana-1480	38	6	)	)	PUNCT
cana-1480	38	7	)	)	PUNCT
cana-1480	38	8	is	be	AUX
cana-1480	38	9	said	say	VERB
cana-1480	38	10	to	to	PART
cana-1480	38	11	be	be	AUX
cana-1480	38	12	nano	nano	NOUN
cana-1480	38	13	star	star	NOUN
cana-1480	38	14	generalized	generalize	VERB
cana-1480	38	15	α	α	PRON
cana-1480	38	16	continuous	continuous	ADJ
cana-1480	38	17	(	(	PUNCT
cana-1480	38	18	briefly	briefly	ADV
cana-1480	38	19	nano*gα	nano*gα	ADJ
cana-1480	38	20	-	-	ADJ
cana-1480	38	21	continuous	continuous	ADJ
cana-1480	38	22	)	)	PUNCT
cana-1480	38	23	,	,	PUNCT
cana-1480	38	24	if	if	SCONJ
cana-1480	38	25	the	the	DET
cana-1480	38	26	inverse	inverse	ADJ
cana-1480	38	27	image	image	NOUN
cana-1480	38	28	of	of	ADP
cana-1480	38	29	every	every	DET
cana-1480	38	30	nano	nano	NOUN
cana-1480	38	31	closed	close	VERB
cana-1480	38	32	set	set	NOUN
cana-1480	38	33	is	be	AUX
cana-1480	38	34	(	(	PUNCT
cana-1480	38	35	𝑉	𝑉	PROPN
cana-1480	38	36	,	,	PUNCT
cana-1480	38	37	𝜎𝑅(𝑌	𝜎𝑅(𝑌	NOUN
cana-1480	38	38	)	)	PUNCT
cana-1480	38	39	)	)	PUNCT
cana-1480	38	40	is	be	AUX
cana-1480	38	41	nano*gα	nano*gα	PROPN
cana-1480	38	42	closed	close	VERB
cana-1480	38	43	set	set	VERB
cana-1480	38	44	in	in	ADP
cana-1480	38	45	(	(	PUNCT
cana-1480	38	46	𝑈	𝑈	PROPN
cana-1480	38	47	,	,	PUNCT
cana-1480	38	48	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	38	49	)	)	PUNCT
cana-1480	38	50	)	)	PUNCT
cana-1480	38	51	.	.	PUNCT
cana-1480	39	1	3	3	NUM
cana-1480	39	2	.nano*gα	.nano*gα	ADP
cana-1480	39	3	compact	compact	ADJ
cana-1480	39	4	space	space	NOUN
cana-1480	39	5	definition	definition	NOUN
cana-1480	39	6	3.1	3.1	NUM
cana-1480	39	7	.	.	PUNCT
cana-1480	40	1	a	a	DET
cana-1480	40	2	collection	collection	NOUN
cana-1480	40	3	{	{	PUNCT
cana-1480	40	4	𝐴𝑖	𝐴𝑖	INTJ
cana-1480	40	5	:	:	PUNCT
cana-1480	40	6	i	i	PRON
cana-1480	40	7	∈	∈	VERB
cana-1480	40	8	i	i	PRON
cana-1480	40	9	}	}	PUNCT
cana-1480	40	10	of	of	ADP
cana-1480	40	11	nano*gα	nano*gα	PROPN
cana-1480	40	12	-open	-open	ADJ
cana-1480	40	13	sets	set	NOUN
cana-1480	40	14	in	in	ADP
cana-1480	40	15	nts	nt	NOUN
cana-1480	40	16	(	(	PUNCT
cana-1480	40	17	𝑈	𝑈	PROPN
cana-1480	40	18	,	,	PUNCT
cana-1480	40	19	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	40	20	)	)	PUNCT
cana-1480	40	21	)	)	PUNCT
cana-1480	40	22	is	be	AUX
cana-1480	40	23	called	call	VERB
cana-1480	40	24	nano∗	nano∗	PROPN
cana-1480	40	25	gα	gα	NOUN
cana-1480	40	26	-	-	PUNCT
cana-1480	40	27	open	open	ADJ
cana-1480	40	28	cover	cover	NOUN
cana-1480	40	29	of	of	ADP
cana-1480	40	30	a	a	DET
cana-1480	40	31	subset	subset	NOUN
cana-1480	40	32	a	a	DET
cana-1480	40	33	in	in	ADP
cana-1480	40	34	(	(	PUNCT
cana-1480	40	35	𝑈	𝑈	PROPN
cana-1480	40	36	,	,	PUNCT
cana-1480	40	37	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	40	38	)	)	PUNCT
cana-1480	40	39	)	)	PUNCT
cana-1480	41	1	if	if	SCONJ
cana-1480	41	2	a	a	DET
cana-1480	41	3	⊆∑	⊆∑	NOUN
cana-1480	41	4	(	(	PUNCT
cana-1480	41	5	ai).𝑖⊆1	ai).𝑖⊆1	PROPN
cana-1480	41	6	definition	definition	NOUN
cana-1480	41	7	3.2	3.2	NUM
cana-1480	41	8	.	.	PUNCT
cana-1480	42	1	a	a	DET
cana-1480	42	2	subset	subset	NOUN
cana-1480	42	3	a	a	PRON
cana-1480	42	4	of	of	ADP
cana-1480	42	5	nts	nt	NOUN
cana-1480	42	6	(	(	PUNCT
cana-1480	42	7	𝑈	𝑈	PROPN
cana-1480	42	8	,	,	PUNCT
cana-1480	42	9	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	42	10	)	)	PUNCT
cana-1480	42	11	)	)	PUNCT
cana-1480	42	12	is	be	AUX
cana-1480	42	13	called	call	VERB
cana-1480	42	14	nano*gα	nano*gα	PROPN
cana-1480	42	15	-ct	-ct	NOUN
cana-1480	42	16	relative	relative	ADJ
cana-1480	42	17	to	to	ADP
cana-1480	42	18	u	u	PRON
cana-1480	42	19	if	if	SCONJ
cana-1480	42	20	for	for	SCONJ
cana-1480	42	21	every	every	DET
cana-1480	42	22	nano∗	nano∗	NOUN
cana-1480	42	23	gα	gα	ADP
cana-1480	42	24	-open	-open	ADJ
cana-1480	42	25	cover	cover	NOUN
cana-1480	42	26	of	of	ADP
cana-1480	42	27	u	u	NOUN
cana-1480	42	28	has	have	VERB
cana-1480	42	29	finite	finite	PROPN
cana-1480	42	30	subcover	subcover	PROPN
cana-1480	42	31	.	.	PUNCT
cana-1480	43	1	definition	definition	NOUN
cana-1480	43	2	3.3	3.3	NUM
cana-1480	43	3	.	.	PUNCT
cana-1480	44	1	a	a	DET
cana-1480	44	2	nts	nt	NOUN
cana-1480	44	3	(	(	PUNCT
cana-1480	44	4	𝑈	𝑈	PROPN
cana-1480	44	5	,	,	PUNCT
cana-1480	44	6	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	44	7	)	)	PUNCT
cana-1480	44	8	)	)	PUNCT
cana-1480	44	9	is	be	AUX
cana-1480	44	10	called	call	VERB
cana-1480	44	11	nano*gα	nano*gα	PROPN
cana-1480	44	12	-ct	-ct	NOUN
cana-1480	44	13	if	if	SCONJ
cana-1480	44	14	every	every	DET
cana-1480	44	15	n	n	NOUN
cana-1480	44	16	ano∗	ano∗	X
cana-1480	44	17	gα	gα	NOUN
cana-1480	44	18	-	-	PUNCT
cana-1480	44	19	open	open	ADJ
cana-1480	44	20	cover	cover	NOUN
cana-1480	44	21	of	of	ADP
cana-1480	44	22	u	u	NOUN
cana-1480	44	23	has	have	VERB
cana-1480	44	24	finite	finite	PROPN
cana-1480	44	25	subcover	subcover	PROPN
cana-1480	44	26	.	.	PUNCT
cana-1480	45	1	definition	definition	NOUN
cana-1480	45	2	3.4	3.4	NUM
cana-1480	45	3	.	.	PUNCT
cana-1480	46	1	a	a	DET
cana-1480	46	2	subset	subset	NOUN
cana-1480	46	3	a	a	PRON
cana-1480	46	4	of	of	ADP
cana-1480	46	5	a	a	DET
cana-1480	46	6	nts	nt	NOUN
cana-1480	46	7	(	(	PUNCT
cana-1480	46	8	𝑈	𝑈	PROPN
cana-1480	46	9	,	,	PUNCT
cana-1480	46	10	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	46	11	)	)	PUNCT
cana-1480	46	12	)	)	PUNCT
cana-1480	46	13	is	be	AUX
cana-1480	46	14	called	call	VERB
cana-1480	46	15	nano*gα	nano*gα	PROPN
cana-1480	46	16	ct	ct	PROPN
cana-1480	46	17	if	if	SCONJ
cana-1480	46	18	a	a	PRON
cana-1480	46	19	is	be	AUX
cana-1480	46	20	nano*gα	nano*gα	PROPN
cana-1480	46	21	-ct	-ct	NOUN
cana-1480	46	22	of	of	ADP
cana-1480	46	23	the	the	DET
cana-1480	46	24	subspace	subspace	NOUN
cana-1480	46	25	of	of	ADP
cana-1480	46	26	(	(	PUNCT
cana-1480	46	27	𝑈	𝑈	PROPN
cana-1480	46	28	,	,	PUNCT
cana-1480	46	29	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	46	30	)	)	PUNCT
cana-1480	46	31	)	)	PUNCT
cana-1480	46	32	.	.	PUNCT
cana-1480	47	1	theorem	theorem	VERB
cana-1480	47	2	3.5	3.5	NUM
cana-1480	47	3	.	.	PUNCT
cana-1480	48	1	a	a	DET
cana-1480	48	2	n	n	NUM
cana-1480	48	3	ano∗	ano∗	ADP
cana-1480	48	4	gα	gα	NOUN
cana-1480	48	5	-	-	PUNCT
cana-1480	48	6	closed	closed	ADJ
cana-1480	48	7	subset	subset	NOUN
cana-1480	48	8	of	of	ADP
cana-1480	48	9	n	n	NUM
cana-1480	48	10	ano∗	ano∗	PROPN
cana-1480	48	11	gα	gα	PROPN
cana-1480	48	12	-	-	PUNCT
cana-1480	48	13	ct	ct	NOUN
cana-1480	48	14	space	space	NOUN
cana-1480	48	15	is	be	AUX
cana-1480	48	16	nano*gα	nano*gα	PROPN
cana-1480	48	17	-ct	-ct	NOUN
cana-1480	48	18	relative	relative	ADJ
cana-1480	48	19	to	to	ADP
cana-1480	48	20	(	(	PUNCT
cana-1480	48	21	𝑈	𝑈	PROPN
cana-1480	48	22	,	,	PUNCT
cana-1480	48	23	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	48	24	)	)	PUNCT
cana-1480	48	25	)	)	PUNCT
cana-1480	48	26	.	.	PUNCT
cana-1480	49	1	proof	proof	NOUN
cana-1480	49	2	.	.	PUNCT
cana-1480	50	1	let	let	VERB
cana-1480	50	2	a	a	DET
cana-1480	50	3	be	be	AUX
cana-1480	50	4	a	a	DET
cana-1480	50	5	nano*gα	nano*gα	PROPN
cana-1480	50	6	-closed	-close	VERB
cana-1480	50	7	subset	subset	NOUN
cana-1480	50	8	of	of	ADP
cana-1480	50	9	a	a	DET
cana-1480	50	10	nts	nts	INTJ
cana-1480	50	11	u	u	NOUN
cana-1480	50	12	.then	.then	VERB
cana-1480	50	13	u	u	NOUN
cana-1480	50	14	−	−	PROPN
cana-1480	50	15	a	a	PRON
cana-1480	50	16	is	be	AUX
cana-1480	50	17	nano*gα	nano*gα	PROPN
cana-1480	50	18	-open	-open	NOUN
cana-1480	50	19	in	in	ADP
cana-1480	50	20	u	u	NOUN
cana-1480	50	21	.	.	PUNCT
cana-1480	51	1	let	let	VERB
cana-1480	51	2	h	h	NOUN
cana-1480	51	3	=	=	PRON
cana-1480	51	4	{	{	PUNCT
cana-1480	51	5	𝐴𝑖	𝐴𝑖	INTJ
cana-1480	51	6	:	:	PUNCT
cana-1480	51	7	i	i	PRON
cana-1480	51	8	∈	∈	VERB
cana-1480	51	9	i	i	PRON
cana-1480	51	10	}	}	PUNCT
cana-1480	51	11	be	be	VERB
cana-1480	51	12	a	a	DET
cana-1480	51	13	nano*gα	nano*gα	ADJ
cana-1480	51	14	-open	-open	ADJ
cana-1480	51	15	cover	cover	NOUN
cana-1480	51	16	of	of	ADP
cana-1480	51	17	a	a	DET
cana-1480	51	18	by	by	ADP
cana-1480	51	19	n	n	CCONJ
cana-1480	51	20	ano∗	ano∗	PROPN
cana-1480	51	21	gα	gα	NOUN
cana-1480	51	22	-	-	PUNCT
cana-1480	51	23	open	open	ADJ
cana-1480	51	24	subsets	subset	NOUN
cana-1480	51	25	in	in	ADP
cana-1480	51	26	u	u	PROPN
cana-1480	51	27	.	.	PUNCT
cana-1480	52	1	then	then	ADV
cana-1480	52	2	h	h	PROPN
cana-1480	52	3	∪	∪	NOUN
cana-1480	52	4	{	{	PUNCT
cana-1480	52	5	u	u	NOUN
cana-1480	52	6	−	−	PROPN
cana-1480	52	7	a	a	PRON
cana-1480	52	8	}	}	PUNCT
cana-1480	52	9	is	be	AUX
cana-1480	52	10	communications	communication	NOUN
cana-1480	52	11	on	on	ADP
cana-1480	52	12	applied	apply	VERB
cana-1480	52	13	nonlinear	nonlinear	ADJ
cana-1480	52	14	analysis	analysis	NOUN
cana-1480	52	15	issn	issn	NOUN
cana-1480	52	16	:	:	PUNCT
cana-1480	52	17	1074	1074	NUM
cana-1480	52	18	-	-	PUNCT
cana-1480	52	19	133x	133x	NUM
cana-1480	52	20	vol	vol	NOUN
cana-1480	52	21	31	31	NUM
cana-1480	52	22	no	no	NOUN
cana-1480	52	23	.	.	PUNCT
cana-1480	53	1	8s	8s	PROPN
cana-1480	53	2	(	(	PUNCT
cana-1480	53	3	2024	2024	NUM
cana-1480	53	4	)	)	PUNCT
cana-1480	53	5	255	255	NUM
cana-1480	53	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1480	53	7	a	a	DET
cana-1480	53	8	nano*gα	nano*gα	PROPN
cana-1480	53	9	-open	-open	ADJ
cana-1480	53	10	cover	cover	NOUN
cana-1480	53	11	of	of	ADP
cana-1480	53	12	u	u	NOUN
cana-1480	53	13	.	.	PUNCT
cana-1480	54	1	since	since	SCONJ
cana-1480	54	2	u	u	NOUN
cana-1480	54	3	is	be	AUX
cana-1480	54	4	nano*gα	nano*gα	PROPN
cana-1480	54	5	-ct	-ct	NOUN
cana-1480	54	6	,	,	PUNCT
cana-1480	54	7	then	then	ADV
cana-1480	54	8	there	there	PRON
cana-1480	54	9	exists	exist	VERB
cana-1480	54	10	a	a	DET
cana-1480	54	11	finite	finite	ADJ
cana-1480	54	12	subcover	subcover	PROPN
cana-1480	54	13	say	say	VERB
cana-1480	54	14	{	{	PUNCT
cana-1480	54	15	𝐴1	𝐴1	PROPN
cana-1480	54	16	,	,	PUNCT
cana-1480	54	17	𝐴2	𝐴2	PROPN
cana-1480	54	18	,	,	PUNCT
cana-1480	54	19	…	…	PUNCT
cana-1480	54	20	𝐴𝑛	𝐴𝑛	NOUN
cana-1480	54	21	}	}	PUNCT
cana-1480	54	22	is	be	AUX
cana-1480	54	23	finite	finite	ADJ
cana-1480	54	24	n	n	CCONJ
cana-1480	54	25	ano∗	ano∗	PROPN
cana-1480	54	26	gα	gα	NOUN
cana-1480	54	27	-	-	PUNCT
cana-1480	54	28	open	open	ADJ
cana-1480	54	29	cover	cover	NOUN
cana-1480	54	30	of	of	ADP
cana-1480	54	31	a	a	PRON
cana-1480	54	32	.	.	PUNCT
cana-1480	55	1	hence	hence	ADV
cana-1480	55	2	a	a	PRON
cana-1480	55	3	is	be	AUX
cana-1480	55	4	nano*gα	nano*gα	PROPN
cana-1480	55	5	-ct	-ct	NOUN
cana-1480	55	6	relative	relative	ADJ
cana-1480	55	7	to	to	ADP
cana-1480	55	8	u.	u.	PROPN
cana-1480	55	9	theorem	theorem	PROPN
cana-1480	55	10	3.6	3.6	NUM
cana-1480	55	11	.	.	PUNCT
cana-1480	56	1	let	let	VERB
cana-1480	56	2	𝑓	𝑓	X
cana-1480	56	3	:	:	PUNCT
cana-1480	56	4	(	(	PUNCT
cana-1480	56	5	𝑈	𝑈	PROPN
cana-1480	56	6	,	,	PUNCT
cana-1480	56	7	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	56	8	)	)	PUNCT
cana-1480	56	9	)	)	PUNCT
cana-1480	57	1	→	→	PUNCT
cana-1480	57	2	(	(	PUNCT
cana-1480	57	3	𝑉	𝑉	PROPN
cana-1480	57	4	,	,	PUNCT
cana-1480	57	5	𝜎𝑅(𝑌	𝜎𝑅(𝑌	NOUN
cana-1480	57	6	)	)	PUNCT
cana-1480	57	7	)	)	PUNCT
cana-1480	57	8	be	be	AUX
cana-1480	57	9	surjective	surjective	ADJ
cana-1480	57	10	,	,	PUNCT
cana-1480	57	11	n	n	PRON
cana-1480	57	12	ano∗	ano∗	NOUN
cana-1480	57	13	gα	gα	NOUN
cana-1480	57	14	-	-	PUNCT
cana-1480	57	15	continuous	continuous	ADJ
cana-1480	57	16	function	function	NOUN
cana-1480	57	17	.	.	PUNCT
cana-1480	58	1	if	if	SCONJ
cana-1480	58	2	u	u	NOUN
cana-1480	58	3	is	be	AUX
cana-1480	58	4	n	n	ADP
cana-1480	58	5	∗	∗	NOUN
cana-1480	58	6	gα	gα	NOUN
cana-1480	58	7	-	-	PUNCT
cana-1480	58	8	ct	ct	PROPN
cana-1480	58	9	,	,	PUNCT
cana-1480	58	10	then	then	ADV
cana-1480	58	11	v	v	NOUN
cana-1480	58	12	is	be	AUX
cana-1480	58	13	nano	nano	NOUN
cana-1480	58	14	ct	ct	NOUN
cana-1480	58	15	.	.	PUNCT
cana-1480	59	1	proof	proof	NOUN
cana-1480	59	2	.	.	PUNCT
cana-1480	60	1	let	let	VERB
cana-1480	60	2	{	{	PUNCT
cana-1480	60	3	𝐴𝑖	𝐴𝑖	INTJ
cana-1480	60	4	:	:	PUNCT
cana-1480	60	5	i	i	PRON
cana-1480	60	6	∈	∈	VERB
cana-1480	60	7	i	i	PRON
cana-1480	60	8	}	}	PUNCT
cana-1480	60	9	be	be	AUX
cana-1480	60	10	nano	nano	VERB
cana-1480	60	11	open	open	ADJ
cana-1480	60	12	cover	cover	NOUN
cana-1480	60	13	of	of	ADP
cana-1480	60	14	v	v	NOUN
cana-1480	60	15	.	.	PUNCT
cana-1480	61	1	since	since	SCONJ
cana-1480	61	2	f	f	PROPN
cana-1480	61	3	is	be	AUX
cana-1480	61	4	nano*gα	nano*gα	PROPN
cana-1480	61	5	-continuous	-continuous	ADJ
cana-1480	61	6	function	function	NOUN
cana-1480	61	7	,	,	PUNCT
cana-1480	61	8	then	then	ADV
cana-1480	61	9	{	{	PUNCT
cana-1480	61	10	𝑓−1(𝐴𝑖	𝑓−1(𝐴𝑖	PROPN
cana-1480	61	11	)	)	PUNCT
cana-1480	61	12	:	:	PUNCT
cana-1480	62	1	i	i	PRON
cana-1480	62	2	∈	∈	VERB
cana-1480	62	3	i	i	PRON
cana-1480	62	4	}	}	PUNCT
cana-1480	62	5	is	be	AUX
cana-1480	62	6	nano*gα	nano*gα	PROPN
cana-1480	62	7	-open	-open	ADJ
cana-1480	62	8	cover	cover	NOUN
cana-1480	62	9	of	of	ADP
cana-1480	62	10	u	u	NOUN
cana-1480	62	11	.	.	PUNCT
cana-1480	63	1	since	since	SCONJ
cana-1480	63	2	u	u	NOUN
cana-1480	63	3	is	be	AUX
cana-1480	63	4	nano*gα	nano*gα	PROPN
cana-1480	63	5	-ct	-ct	NOUN
cana-1480	63	6	,	,	PUNCT
cana-1480	63	7	{	{	PUNCT
cana-1480	63	8	𝑓−1(𝐴𝑖	𝑓−1(𝐴𝑖	NOUN
cana-1480	63	9	)	)	PUNCT
cana-1480	64	1	:	:	PUNCT
cana-1480	64	2	i	i	PRON
cana-1480	64	3	∈	∈	VERB
cana-1480	64	4	i	i	PRON
cana-1480	64	5	}	}	PUNCT
cana-1480	64	6	contains	contain	VERB
cana-1480	64	7	a	a	DET
cana-1480	64	8	finite	finite	PROPN
cana-1480	64	9	subcover	subcover	PROPN
cana-1480	64	10	say	say	VERB
cana-1480	64	11	{	{	PUNCT
cana-1480	64	12	𝑓−1(𝐴𝑖	𝑓−1(𝐴𝑖	PROPN
cana-1480	64	13	)	)	PUNCT
cana-1480	64	14	:	:	PUNCT
cana-1480	65	1	i	i	PRON
cana-1480	65	2	∈	∈	VERB
cana-1480	65	3	i	i	PRON
cana-1480	65	4	}	}	PUNCT
cana-1480	65	5	since	since	SCONJ
cana-1480	65	6	f	f	PROPN
cana-1480	65	7	is	be	AUX
cana-1480	65	8	surjective	surjective	ADJ
cana-1480	65	9	,	,	PUNCT
cana-1480	65	10	then	then	ADV
cana-1480	65	11	{	{	PUNCT
cana-1480	65	12	𝐴1	𝐴1	PROPN
cana-1480	65	13	,	,	PUNCT
cana-1480	65	14	𝐴2	𝐴2	PROPN
cana-1480	65	15	,	,	PUNCT
cana-1480	65	16	…	…	PUNCT
cana-1480	65	17	𝐴𝑛	𝐴𝑛	NOUN
cana-1480	65	18	}	}	PUNCT
cana-1480	65	19	is	be	AUX
cana-1480	65	20	finite	finite	PROPN
cana-1480	65	21	subcover	subcover	PROPN
cana-1480	65	22	of	of	ADP
cana-1480	65	23	{	{	PUNCT
cana-1480	65	24	𝐴𝑖	𝐴𝑖	INTJ
cana-1480	65	25	:	:	PUNCT
cana-1480	65	26	i	i	PRON
cana-1480	65	27	∈	∈	VERB
cana-1480	66	1	i	i	X
cana-1480	66	2	}	}	PUNCT
cana-1480	66	3	,	,	PUNCT
cana-1480	66	4	for	for	ADP
cana-1480	66	5	v	v	NOUN
cana-1480	66	6	.	.	PUNCT
cana-1480	67	1	therefote	therefote	ADJ
cana-1480	67	2	v	v	NOUN
cana-1480	67	3	is	be	AUX
cana-1480	67	4	nano	nano	ADJ
cana-1480	67	5	ct	ct	PROPN
cana-1480	67	6	.	.	PUNCT
cana-1480	68	1	theorem	theorem	VERB
cana-1480	68	2	3.7	3.7	NUM
cana-1480	68	3	.	.	PUNCT
cana-1480	69	1	every	every	DET
cana-1480	69	2	nano*gα	nano*gα	PROPN
cana-1480	69	3	-compact	-compact	NOUN
cana-1480	69	4	space	space	NOUN
cana-1480	69	5	is	be	AUX
cana-1480	69	6	nano	nano	NOUN
cana-1480	69	7	compact	compact	ADJ
cana-1480	69	8	.	.	PUNCT
cana-1480	70	1	proof	proof	NOUN
cana-1480	70	2	.	.	PUNCT
cana-1480	71	1	let	let	VERB
cana-1480	71	2	u	u	PRON
cana-1480	71	3	be	be	AUX
cana-1480	71	4	nano*gα	nano*gα	PROPN
cana-1480	71	5	-ct	-ct	NOUN
cana-1480	71	6	.	.	PUNCT
cana-1480	72	1	let	let	VERB
cana-1480	72	2	{	{	PUNCT
cana-1480	72	3	𝐴𝑗	𝐴𝑗	ADV
cana-1480	72	4	:	:	PUNCT
cana-1480	72	5	j	j	PROPN
cana-1480	72	6	∈	∈	PROPN
cana-1480	72	7	j	j	PROPN
cana-1480	72	8	}	}	PUNCT
cana-1480	72	9	is	be	AUX
cana-1480	72	10	a	a	DET
cana-1480	72	11	nano*gα	nano*gα	ADJ
cana-1480	72	12	-open	-open	ADJ
cana-1480	72	13	cover	cover	NOUN
cana-1480	72	14	of	of	ADP
cana-1480	72	15	u	u	NOUN
cana-1480	72	16	.	.	PUNCT
cana-1480	73	1	since	since	SCONJ
cana-1480	73	2	every	every	DET
cana-1480	73	3	nano	nano	NOUN
cana-1480	73	4	open	open	ADJ
cana-1480	73	5	set	set	NOUN
cana-1480	73	6	is	be	AUX
cana-1480	73	7	nano*gα	nano*gα	PROPN
cana-1480	73	8	-open	-open	NOUN
cana-1480	73	9	.	.	PUNCT
cana-1480	74	1	since	since	SCONJ
cana-1480	74	2	u	u	NOUN
cana-1480	74	3	is	be	AUX
cana-1480	74	4	nano*gα	nano*gα	PROPN
cana-1480	74	5	-ct	-ct	NOUN
cana-1480	74	6	,	,	PUNCT
cana-1480	74	7	then	then	ADV
cana-1480	74	8	nano*gα	nano*gα	PROPN
cana-1480	74	9	-open	-open	ADJ
cana-1480	74	10	cover	cover	NOUN
cana-1480	74	11	{	{	PUNCT
cana-1480	74	12	𝐴𝑗	𝐴𝑗	NOUN
cana-1480	74	13	:	:	PUNCT
cana-1480	75	1	j	j	PROPN
cana-1480	75	2	∈	∈	PROPN
cana-1480	75	3	j	j	PROPN
cana-1480	75	4	}	}	PUNCT
cana-1480	75	5	of	of	ADP
cana-1480	75	6	u	u	NOUN
cana-1480	75	7	has	have	VERB
cana-1480	75	8	a	a	DET
cana-1480	75	9	finite	finite	ADJ
cana-1480	75	10	subcover	subcover	PROPN
cana-1480	75	11	,	,	PUNCT
cana-1480	75	12	say	say	VERB
cana-1480	75	13	{	{	PUNCT
cana-1480	75	14	𝐴𝑗	𝐴𝑗	ADV
cana-1480	75	15	:	:	PUNCT
cana-1480	75	16	j	j	PROPN
cana-1480	75	17	∈	∈	PROPN
cana-1480	75	18	j	j	PROPN
cana-1480	75	19	}	}	PUNCT
cana-1480	75	20	for	for	ADP
cana-1480	75	21	u	u	PRON
cana-1480	75	22	.	.	PUNCT
cana-1480	76	1	hence	hence	ADV
cana-1480	76	2	u	u	PROPN
cana-1480	76	3	is	be	AUX
cana-1480	76	4	nano	nano	ADJ
cana-1480	76	5	ct	ct	PROPN
cana-1480	76	6	.	.	PUNCT
cana-1480	76	7	theorem	theorem	VERB
cana-1480	76	8	3.8	3.8	NUM
cana-1480	76	9	.	.	PUNCT
cana-1480	77	1	if	if	SCONJ
cana-1480	77	2	a	a	DET
cana-1480	77	3	function	function	NOUN
cana-1480	77	4	𝑓	𝑓	NOUN
cana-1480	77	5	:	:	PUNCT
cana-1480	77	6	(	(	PUNCT
cana-1480	77	7	𝑈	𝑈	PROPN
cana-1480	77	8	,	,	PUNCT
cana-1480	77	9	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	77	10	)	)	PUNCT
cana-1480	77	11	)	)	PUNCT
cana-1480	77	12	→	→	PUNCT
cana-1480	77	13	(	(	PUNCT
cana-1480	77	14	𝑉	𝑉	PROPN
cana-1480	77	15	,	,	PUNCT
cana-1480	77	16	𝜎𝑅(𝑌	𝜎𝑅(𝑌	NOUN
cana-1480	77	17	)	)	PUNCT
cana-1480	77	18	)	)	PUNCT
cana-1480	77	19	is	be	AUX
cana-1480	77	20	nano*gα	nano*gα	PROPN
cana-1480	77	21	-irresolute	-irresolute	ADJ
cana-1480	77	22	and	and	CCONJ
cana-1480	77	23	a	a	DET
cana-1480	77	24	subset	subset	NOUN
cana-1480	77	25	a	a	PRON
cana-1480	77	26	of	of	ADP
cana-1480	77	27	u	u	NOUN
cana-1480	77	28	is	be	AUX
cana-1480	77	29	nano*gα	nano*gα	PROPN
cana-1480	77	30	ct	ct	NUM
cana-1480	77	31	relative	relative	ADJ
cana-1480	77	32	to	to	ADP
cana-1480	77	33	u	u	PROPN
cana-1480	77	34	,	,	PUNCT
cana-1480	77	35	then	then	ADV
cana-1480	77	36	the	the	DET
cana-1480	77	37	image	image	NOUN
cana-1480	77	38	f(a	f(a	PROPN
cana-1480	77	39	)	)	PUNCT
cana-1480	77	40	is	be	AUX
cana-1480	77	41	nano*gα	nano*gα	PROPN
cana-1480	77	42	-ct	-ct	NOUN
cana-1480	77	43	relative	relative	ADJ
cana-1480	77	44	to	to	ADP
cana-1480	77	45	v	v	NOUN
cana-1480	77	46	.	.	PUNCT
cana-1480	78	1	proof	proof	NOUN
cana-1480	78	2	.	.	PUNCT
cana-1480	79	1	let	let	VERB
cana-1480	79	2	{	{	PUNCT
cana-1480	79	3	𝐴𝑖	𝐴𝑖	INTJ
cana-1480	79	4	:	:	PUNCT
cana-1480	79	5	i	i	PRON
cana-1480	79	6	∈	∈	VERB
cana-1480	79	7	i	i	PRON
cana-1480	79	8	}	}	PUNCT
cana-1480	79	9	be	be	VERB
cana-1480	79	10	any	any	DET
cana-1480	79	11	collection	collection	NOUN
cana-1480	79	12	of	of	ADP
cana-1480	79	13	nano*gα	nano*gα	ADJ
cana-1480	79	14	-open	-open	ADJ
cana-1480	79	15	sets	set	NOUN
cana-1480	79	16	in	in	ADP
cana-1480	79	17	v	v	ADP
cana-1480	79	18	such	such	ADJ
cana-1480	79	19	that	that	DET
cana-1480	79	20	𝑓(𝐴	𝑓(𝐴	NOUN
cana-1480	79	21	)	)	PUNCT
cana-1480	80	1	=	=	SYM
cana-1480	80	2	⋃	⋃	NOUN
cana-1480	80	3	{	{	PUNCT
cana-1480	80	4	𝑓−1(𝐴𝑖).𝑖∈𝐼	𝑓−1(𝐴𝑖).𝑖∈𝐼	ADJ
cana-1480	80	5	then	then	ADV
cana-1480	80	6	a	a	DET
cana-1480	80	7	⊆	⊆	NUM
cana-1480	80	8	⋃	⋃	NOUN
cana-1480	80	9	{	{	PUNCT
cana-1480	80	10	𝑓−1(𝐴𝑖).𝑖∈𝐼	𝑓−1(𝐴𝑖).𝑖∈𝐼	ADJ
cana-1480	80	11	where	where	SCONJ
cana-1480	80	12	{	{	PUNCT
cana-1480	80	13	𝑓−1(𝐴𝑖	𝑓−1(𝐴𝑖	NUM
cana-1480	80	14	)	)	PUNCT
cana-1480	80	15	:	:	PUNCT
cana-1480	81	1	i	i	PRON
cana-1480	81	2	∈	∈	VERB
cana-1480	81	3	i	i	PRON
cana-1480	81	4	}	}	PUNCT
cana-1480	81	5	is	be	AUX
cana-1480	81	6	n	n	NUM
cana-1480	81	7	ano∗	ano∗	X
cana-1480	81	8	gα	gα	NOUN
cana-1480	81	9	-	-	PUNCT
cana-1480	81	10	open	open	ADJ
cana-1480	81	11	sets	set	NOUN
cana-1480	81	12	in	in	ADP
cana-1480	81	13	u	u	NOUN
cana-1480	81	14	.since	.since	NOUN
cana-1480	81	15	a	a	PRON
cana-1480	81	16	is	be	AUX
cana-1480	81	17	nano*gα	nano*gα	PROPN
cana-1480	81	18	ct	ct	PROPN
cana-1480	81	19	relative	relative	ADJ
cana-1480	81	20	to	to	ADP
cana-1480	81	21	u	u	NOUN
cana-1480	81	22	,	,	PUNCT
cana-1480	81	23	there	there	PRON
cana-1480	81	24	is	be	VERB
cana-1480	81	25	a	a	DET
cana-1480	81	26	finite	finite	ADJ
cana-1480	81	27	subcollection	subcollection	NOUN
cana-1480	81	28	{	{	PUNCT
cana-1480	81	29	𝐴1	𝐴1	PROPN
cana-1480	81	30	,	,	PUNCT
cana-1480	81	31	𝐴2	𝐴2	PROPN
cana-1480	81	32	,	,	PUNCT
cana-1480	81	33	…	…	PUNCT
cana-1480	82	1	𝐴𝑛	𝐴𝑛	NOUN
cana-1480	82	2	}	}	PUNCT
cana-1480	82	3	such	such	ADJ
cana-1480	82	4	that	that	SCONJ
cana-1480	82	5	𝐴	𝐴	PROPN
cana-1480	82	6	⊆	⊆	NUM
cana-1480	82	7	⋃	⋃	NOUN
cana-1480	82	8	{	{	PUNCT
cana-1480	82	9	𝑓−1(𝐴𝑖).𝑖∈𝐼	𝑓−1(𝐴𝑖).𝑖∈𝐼	ADJ
cana-1480	82	10	.	.	PUNCT
cana-1480	83	1	therefore	therefore	ADV
cana-1480	83	2	𝑓(𝐴	𝑓(𝐴	NOUN
cana-1480	83	3	)	)	PUNCT
cana-1480	84	1	=	=	SYM
cana-1480	84	2	⋃	⋃	NOUN
cana-1480	84	3	{	{	PUNCT
cana-1480	84	4	𝑓−1(𝐴𝑖).𝑖∈𝐼	𝑓−1(𝐴𝑖).𝑖∈𝐼	ADJ
cana-1480	84	5	hence	hence	ADJ
cana-1480	84	6	𝑓(𝐴	𝑓(𝐴	NOUN
cana-1480	84	7	)	)	PUNCT
cana-1480	84	8	is	be	AUX
cana-1480	84	9	nano*gα	nano*gα	PROPN
cana-1480	84	10	ct	ct	NUM
cana-1480	84	11	relative	relative	ADJ
cana-1480	84	12	to	to	ADP
cana-1480	84	13	v	v	NUM
cana-1480	84	14	.	.	PUNCT
cana-1480	85	1	4	4	NUM
cana-1480	85	2	.nano	.nano	NOUN
cana-1480	85	3	c	c	NOUN
cana-1480	85	4	τ	τ	PROPN
cana-1480	85	5	-compact	-compact	PROPN
cana-1480	85	6	space	space	NOUN
cana-1480	85	7	definition	definition	NOUN
cana-1480	85	8	4.1	4.1	NUM
cana-1480	85	9	.	.	PUNCT
cana-1480	86	1	a	a	DET
cana-1480	86	2	subset	subset	NOUN
cana-1480	86	3	a	a	PRON
cana-1480	86	4	of	of	ADP
cana-1480	86	5	a	a	DET
cana-1480	86	6	nts	nt	NOUN
cana-1480	86	7	(	(	PUNCT
cana-1480	86	8	𝑈	𝑈	PROPN
cana-1480	86	9	,	,	PUNCT
cana-1480	86	10	𝜏𝑅(𝑋))is	𝜏𝑅(𝑋))is	PROPN
cana-1480	86	11	called	call	VERB
cana-1480	86	12	a	a	DET
cana-1480	86	13	nanoc	nanoc	NOUN
cana-1480	86	14	τ	τ	X
cana-1480	86	15	-set	-set	VERB
cana-1480	86	16	if	if	SCONJ
cana-1480	86	17	there	there	PRON
cana-1480	86	18	are	be	VERB
cana-1480	86	19	two	two	NUM
cana-1480	86	20	sets	set	NOUN
cana-1480	86	21	𝐺	𝐺	PROPN
cana-1480	86	22	,	,	PUNCT
cana-1480	87	1	𝐹	𝐹	PROPN
cana-1480	87	2	∈	∈	NOUN
cana-1480	88	1	𝑈such	𝑈such	ADP
cana-1480	88	2	that	that	DET
cana-1480	88	3	𝐺	𝐺	PROPN
cana-1480	88	4	≠	≠	PROPN
cana-1480	88	5	𝑈	𝑈	PROPN
cana-1480	88	6	and	and	CCONJ
cana-1480	88	7	𝐴	𝐴	PROPN
cana-1480	88	8	≠	≠	PROPN
cana-1480	88	9	𝐺	𝐺	NOUN
cana-1480	88	10	−	−	PROPN
cana-1480	88	11	𝐹.	𝐹.	PROPN
cana-1480	88	12	definition	definition	NOUN
cana-1480	88	13	4.2	4.2	NUM
cana-1480	88	14	.	.	PUNCT
cana-1480	89	1	a	a	DET
cana-1480	89	2	collection	collection	NOUN
cana-1480	89	3	r	r	NOUN
cana-1480	89	4	of	of	ADP
cana-1480	89	5	subset	subset	NOUN
cana-1480	89	6	of	of	ADP
cana-1480	89	7	nano	nano	NOUN
cana-1480	89	8	generalized	generalize	VERB
cana-1480	89	9	nts	nt	NOUN
cana-1480	89	10	(	(	PUNCT
cana-1480	89	11	𝑈	𝑈	PROPN
cana-1480	89	12	,	,	PUNCT
cana-1480	89	13	𝜏𝑅(𝑋))is	𝜏𝑅(𝑋))is	PROPN
cana-1480	89	14	said	say	VERB
cana-1480	89	15	to	to	PART
cana-1480	89	16	be	be	AUX
cana-1480	89	17	a	a	DET
cana-1480	89	18	cover	cover	NOUN
cana-1480	89	19	of	of	ADP
cana-1480	89	20	u	u	PRON
cana-1480	89	21	if	if	SCONJ
cana-1480	89	22	the	the	DET
cana-1480	89	23	union	union	NOUN
cana-1480	89	24	of	of	ADP
cana-1480	89	25	the	the	DET
cana-1480	89	26	elements	element	NOUN
cana-1480	89	27	r	r	NOUN
cana-1480	89	28	is	be	AUX
cana-1480	89	29	equal	equal	ADJ
cana-1480	89	30	to	to	ADP
cana-1480	89	31	u	u	PRON
cana-1480	89	32	.	.	PUNCT
cana-1480	90	1	it	it	PRON
cana-1480	90	2	is	be	AUX
cana-1480	90	3	called	call	VERB
cana-1480	90	4	a	a	DET
cana-1480	90	5	nanoc	nanoc	NOUN
cana-1480	90	6	τ	τ	PROPN
cana-1480	90	7	-cover	-cover	NOUN
cana-1480	90	8	of	of	ADP
cana-1480	90	9	u	u	PRON
cana-1480	90	10	if	if	SCONJ
cana-1480	90	11	its	its	PRON
cana-1480	90	12	elements	element	NOUN
cana-1480	90	13	are	be	AUX
cana-1480	90	14	nanoc	nanoc	NOUN
cana-1480	90	15	τ	τ	NOUN
cana-1480	90	16	-subsets	-subset	NOUN
cana-1480	90	17	of	of	ADP
cana-1480	90	18	u	u	NOUN
cana-1480	90	19	.	.	PUNCT
cana-1480	91	1	the	the	DET
cana-1480	91	2	nano	nano	NOUN
cana-1480	91	3	generalized	generalize	VERB
cana-1480	91	4	nts	nt	NOUN
cana-1480	91	5	(	(	PUNCT
cana-1480	91	6	𝑈	𝑈	PROPN
cana-1480	91	7	,	,	PUNCT
cana-1480	91	8	𝜏𝑅(𝑋))is	𝜏𝑅(𝑋))is	PROPN
cana-1480	91	9	called	call	VERB
cana-1480	91	10	nanoc	nanoc	NOUN
cana-1480	91	11	τ	τ	PROPN
cana-1480	91	12	-ct	-ct	NOUN
cana-1480	91	13	if	if	SCONJ
cana-1480	91	14	every	every	DET
cana-1480	91	15	nanoc	nanoc	NOUN
cana-1480	91	16	τ	τ	PROPN
cana-1480	91	17	-cover	-cover	NOUN
cana-1480	91	18	of	of	ADP
cana-1480	91	19	u	u	PROPN
cana-1480	91	20	has	have	VERB
cana-1480	91	21	finite	finite	PROPN
cana-1480	91	22	subcover	subcover	PROPN
cana-1480	91	23	.	.	PUNCT
cana-1480	92	1	definition	definition	NOUN
cana-1480	92	2	4.3	4.3	NUM
cana-1480	92	3	.	.	PUNCT
cana-1480	93	1	a	a	DET
cana-1480	93	2	space	space	NOUN
cana-1480	93	3	(	(	PUNCT
cana-1480	93	4	𝑈	𝑈	PROPN
cana-1480	93	5	,	,	PUNCT
cana-1480	93	6	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	93	7	)	)	PUNCT
cana-1480	93	8	)	)	PUNCT
cana-1480	93	9	is	be	AUX
cana-1480	93	10	called	call	VERB
cana-1480	93	11	nano	nano	ADJ
cana-1480	93	12	𝑇2	𝑇2	NOUN
cana-1480	93	13	space	space	NOUN
cana-1480	93	14	if	if	SCONJ
cana-1480	93	15	for	for	ADP
cana-1480	93	16	any	any	DET
cana-1480	93	17	pair	pair	NOUN
cana-1480	93	18	of	of	ADP
cana-1480	93	19	distinct	distinct	ADJ
cana-1480	93	20	points	point	NOUN
cana-1480	93	21	𝛼1	𝛼1	NOUN
cana-1480	93	22	,	,	PUNCT
cana-1480	93	23	𝛼2	𝛼2	PROPN
cana-1480	93	24	of	of	ADP
cana-1480	93	25	u	u	PRON
cana-1480	93	26	there	there	PRON
cana-1480	93	27	exists	exist	VERB
cana-1480	93	28	disjoint	disjoint	NOUN
cana-1480	93	29	nanoc	nanoc	NOUN
cana-1480	93	30	τ	τ	X
cana-1480	93	31	-set	-set	NOUN
cana-1480	93	32	g	g	NOUN
cana-1480	93	33	and	and	CCONJ
cana-1480	93	34	h	h	NOUN
cana-1480	93	35	of	of	ADP
cana-1480	93	36	u	u	PRON
cana-1480	93	37	containing	contain	VERB
cana-1480	93	38	𝛼1	𝛼1	NOUN
cana-1480	93	39	,	,	PUNCT
cana-1480	93	40	𝛼2respectively	𝛼2respectively	ADV
cana-1480	93	41	.	.	PUNCT
cana-1480	93	42	theorem	theorem	VERB
cana-1480	93	43	4.4	4.4	NUM
cana-1480	93	44	.	.	PUNCT
cana-1480	94	1	if	if	SCONJ
cana-1480	94	2	(	(	PUNCT
cana-1480	94	3	𝑈	𝑈	PROPN
cana-1480	94	4	,	,	PUNCT
cana-1480	94	5	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	94	6	)	)	PUNCT
cana-1480	94	7	)	)	PUNCT
cana-1480	94	8	is	be	AUX
cana-1480	94	9	finite	finite	VERB
cana-1480	94	10	nano	nano	NOUN
cana-1480	94	11	generalized	generalize	VERB
cana-1480	94	12	nts	nt	NOUN
cana-1480	94	13	.	.	PUNCT
cana-1480	95	1	then	then	ADV
cana-1480	95	2	u	u	NOUN
cana-1480	95	3	is	be	AUX
cana-1480	95	4	nanoc	nanoc	NOUN
cana-1480	95	5	τ	τ	X
cana-1480	95	6	-ct	-ct	NOUN
cana-1480	95	7	.	.	PUNCT
cana-1480	96	1	proof	proof	NOUN
cana-1480	96	2	.	.	PUNCT
cana-1480	97	1	let	let	VERB
cana-1480	97	2	{	{	PUNCT
cana-1480	97	3	𝐴𝑖	𝐴𝑖	INTJ
cana-1480	97	4	:	:	PUNCT
cana-1480	98	1	i	i	PRON
cana-1480	98	2	∈	∈	PROPN
cana-1480	98	3	u	u	NOUN
cana-1480	98	4	}	}	PUNCT
cana-1480	98	5	be	be	AUX
cana-1480	98	6	a	a	DET
cana-1480	98	7	nano	nano	NOUN
cana-1480	98	8	cover	cover	NOUN
cana-1480	98	9	of	of	ADP
cana-1480	98	10	u	u	NOUN
cana-1480	98	11	.let	.let	PUNCT
cana-1480	99	1	r	r	NOUN
cana-1480	99	2	be	be	AUX
cana-1480	99	3	a	a	DET
cana-1480	99	4	nanoc	nanoc	NOUN
cana-1480	99	5	τ	τ	X
cana-1480	99	6	-covering	-covering	NOUN
cana-1480	99	7	of	of	ADP
cana-1480	99	8	u	u	PROPN
cana-1480	99	9	.	.	PUNCT
cana-1480	100	1	then	then	ADV
cana-1480	100	2	the	the	DET
cana-1480	100	3	element	element	NOUN
cana-1480	100	4	in	in	ADP
cana-1480	100	5	u	u	PROPN
cana-1480	100	6	belongs	belong	VERB
cana-1480	100	7	to	to	ADP
cana-1480	100	8	one	one	NUM
cana-1480	100	9	of	of	ADP
cana-1480	100	10	the	the	DET
cana-1480	100	11	members	member	NOUN
cana-1480	100	12	of	of	ADP
cana-1480	100	13	r	r	NOUN
cana-1480	100	14	say	say	VERB
cana-1480	100	15	{	{	PUNCT
cana-1480	100	16	𝐴1	𝐴1	PROPN
cana-1480	100	17	,	,	PUNCT
cana-1480	100	18	𝐴2	𝐴2	PROPN
cana-1480	100	19	,	,	PUNCT
cana-1480	100	20	…	…	PUNCT
cana-1480	100	21	𝐴𝑛	𝐴𝑛	NOUN
cana-1480	100	22	}	}	PUNCT
cana-1480	100	23	∈	∈	PROPN
cana-1480	100	24	h	h	NOUN
cana-1480	100	25	.	.	PUNCT
cana-1480	101	1	where	where	SCONJ
cana-1480	101	2	every	every	DET
cana-1480	101	3	{	{	PUNCT
cana-1480	101	4	𝐺𝑖	𝐺𝑖	PROPN
cana-1480	101	5	:	:	PUNCT
cana-1480	101	6	i	i	PRON
cana-1480	101	7	∈	∈	VERB
cana-1480	101	8	r	r	X
cana-1480	101	9	}	}	PUNCT
cana-1480	101	10	,	,	PUNCT
cana-1480	101	11	𝐺	𝐺	PROPN
cana-1480	101	12	≠	≠	PROPN
cana-1480	101	13	𝑈	𝑈	PROPN
cana-1480	101	14	,	,	PUNCT
cana-1480	101	15	i	i	PRON
cana-1480	101	16	=	=	NOUN
cana-1480	101	17	1	1	NUM
cana-1480	101	18	,	,	PUNCT
cana-1480	101	19	2	2	NUM
cana-1480	101	20	,	,	PUNCT
cana-1480	101	21	..	..	PUNCT
cana-1480	101	22	n.	n.	NOUN
cana-1480	101	23	since	since	SCONJ
cana-1480	101	24	each	each	DET
cana-1480	101	25	g	g	NOUN
cana-1480	101	26	is	be	AUX
cana-1480	101	27	nanoc	nanoc	NOUN
cana-1480	101	28	τ	τ	X
cana-1480	101	29	-set	-set	PUNCT
cana-1480	101	30	the	the	DET
cana-1480	101	31	collection	collection	NOUN
cana-1480	101	32	{	{	PUNCT
cana-1480	101	33	𝐴1	𝐴1	PROPN
cana-1480	101	34	,	,	PUNCT
cana-1480	101	35	𝐴2	𝐴2	PROPN
cana-1480	101	36	,	,	PUNCT
cana-1480	101	37	…	…	PUNCT
cana-1480	101	38	𝐴𝑛	𝐴𝑛	NOUN
cana-1480	101	39	}	}	PUNCT
cana-1480	101	40	is	be	AUX
cana-1480	101	41	finite	finite	ADJ
cana-1480	101	42	subcollection	subcollection	NOUN
cana-1480	101	43	of	of	ADP
cana-1480	101	44	nanoc	nanoc	NOUN
cana-1480	101	45	τ	τ	X
cana-1480	101	46	-set	-set	ADJ
cana-1480	101	47	which	which	PRON
cana-1480	101	48	covers	cover	VERB
cana-1480	101	49	u	u	PRON
cana-1480	101	50	.	.	PUNCT
cana-1480	102	1	hence	hence	ADV
cana-1480	102	2	u	u	NOUN
cana-1480	102	3	is	be	AUX
cana-1480	102	4	nanoc	nanoc	NOUN
cana-1480	102	5	τ	τ	X
cana-1480	102	6	–	–	PUNCT
cana-1480	102	7	ct	ct	PROPN
cana-1480	102	8	.	.	PUNCT
cana-1480	102	9	theorem	theorem	VERB
cana-1480	102	10	4.5	4.5	NUM
cana-1480	102	11	.	.	PUNCT
cana-1480	103	1	let	let	VERB
cana-1480	103	2	a	a	DET
cana-1480	103	3	be	be	AUX
cana-1480	103	4	nanoc	nanoc	NOUN
cana-1480	104	1	τ	τ	X
cana-1480	104	2	-ct	-ct	NOUN
cana-1480	104	3	subsets	subset	NOUN
cana-1480	104	4	of	of	ADP
cana-1480	104	5	nano	nano	NOUN
cana-1480	104	6	𝑇2	𝑇2	NOUN
cana-1480	104	7	space	space	NOUN
cana-1480	104	8	in	in	ADP
cana-1480	104	9	(	(	PUNCT
cana-1480	104	10	𝑈	𝑈	PROPN
cana-1480	104	11	,	,	PUNCT
cana-1480	104	12	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	104	13	)	)	PUNCT
cana-1480	104	14	)	)	PUNCT
cana-1480	104	15	and	and	CCONJ
cana-1480	104	16	α	α	PRON
cana-1480	104	17	∈	∈	NOUN
cana-1480	104	18	u	u	NOUN
cana-1480	104	19	is	be	AUX
cana-1480	104	20	not	not	PART
cana-1480	104	21	in	in	ADP
cana-1480	104	22	a	a	PRON
cana-1480	104	23	,	,	PUNCT
cana-1480	104	24	then	then	ADV
cana-1480	104	25	there	there	PRON
cana-1480	104	26	is	be	VERB
cana-1480	104	27	a	a	DET
cana-1480	104	28	nanoc	nanoc	NOUN
cana-1480	104	29	τ	τ	NOUN
cana-1480	104	30	-set	-set	NOUN
cana-1480	104	31	g	g	ADP
cana-1480	104	32	such	such	ADJ
cana-1480	104	33	that	that	SCONJ
cana-1480	104	34	a	a	DET
cana-1480	104	35	⊂	⊂	PROPN
cana-1480	104	36	g	g	PROPN
cana-1480	104	37	.	.	PUNCT
cana-1480	105	1	communications	communication	NOUN
cana-1480	105	2	on	on	ADP
cana-1480	105	3	applied	apply	VERB
cana-1480	105	4	nonlinear	nonlinear	ADJ
cana-1480	105	5	analysis	analysis	NOUN
cana-1480	105	6	issn	issn	NOUN
cana-1480	105	7	:	:	PUNCT
cana-1480	105	8	1074	1074	NUM
cana-1480	105	9	-	-	PUNCT
cana-1480	105	10	133x	133x	NUM
cana-1480	105	11	vol	vol	NOUN
cana-1480	105	12	31	31	NUM
cana-1480	105	13	no	no	NOUN
cana-1480	105	14	.	.	PUNCT
cana-1480	106	1	8s	8s	PROPN
cana-1480	106	2	(	(	PUNCT
cana-1480	106	3	2024	2024	NUM
cana-1480	106	4	)	)	PUNCT
cana-1480	106	5	256	256	NUM
cana-1480	106	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1480	106	7	proof	proof	NOUN
cana-1480	106	8	.	.	PUNCT
cana-1480	107	1	let	let	VERB
cana-1480	107	2	a	a	DET
cana-1480	107	3	be	be	AUX
cana-1480	107	4	nanoc	nanoc	NOUN
cana-1480	107	5	τ	τ	X
cana-1480	107	6	-ct	-ct	NOUN
cana-1480	107	7	subsets	subset	NOUN
cana-1480	107	8	of	of	ADP
cana-1480	107	9	nano	nano	NOUN
cana-1480	107	10	𝑇2space	𝑇2space	NOUN
cana-1480	107	11	in	in	ADP
cana-1480	107	12	(	(	PUNCT
cana-1480	107	13	𝑈	𝑈	PROPN
cana-1480	107	14	,	,	PUNCT
cana-1480	107	15	𝜏𝑅(𝑋)).since	𝜏𝑅(𝑋)).since	ADP
cana-1480	107	16	(	(	PUNCT
cana-1480	107	17	𝑈	𝑈	PROPN
cana-1480	107	18	,	,	PUNCT
cana-1480	107	19	𝜏𝑅(𝑋))is	𝜏𝑅(𝑋))is	ADJ
cana-1480	107	20	nanoc	nanoc	NOUN
cana-1480	107	21	τ	τ	X
cana-1480	107	22	-set	-set	X
cana-1480	107	23	,	,	PUNCT
cana-1480	107	24	for	for	ADP
cana-1480	107	25	each	each	DET
cana-1480	107	26	β	β	X
cana-1480	107	27	∈	∈	PROPN
cana-1480	107	28	u	u	NOUN
cana-1480	107	29	,	,	PUNCT
cana-1480	107	30	there	there	PRON
cana-1480	107	31	exists	exist	VERB
cana-1480	107	32	nanoc	nanoc	NOUN
cana-1480	107	33	τ	τ	X
cana-1480	107	34	-set	-set	PUNCT
cana-1480	108	1	𝐴𝛼	𝐴𝛼	PROPN
cana-1480	108	2	∈	∈	PROPN
cana-1480	108	3	α	α	NOUN
cana-1480	108	4	and	and	CCONJ
cana-1480	108	5	𝐴𝛽	𝐴𝛽	PROPN
cana-1480	108	6	∈	∈	PROPN
cana-1480	108	7	β	β	X
cana-1480	108	8	then	then	ADV
cana-1480	109	1	𝐴𝛼⋂𝐴𝛽	𝐴𝛼⋂𝐴𝛽	PROPN
cana-1480	109	2	=	=	PROPN
cana-1480	109	3	𝐴𝜓are	𝐴𝜓are	PROPN
cana-1480	109	4	nanoc	nanoc	NOUN
cana-1480	109	5	τ	τ	X
cana-1480	109	6	-set	-set	PUNCT
cana-1480	109	7	.	.	PUNCT
cana-1480	110	1	the	the	DET
cana-1480	110	2	collection	collection	NOUN
cana-1480	110	3	{	{	PUNCT
cana-1480	110	4	𝐴𝛽	𝐴𝛽	NOUN
cana-1480	110	5	:	:	PUNCT
cana-1480	110	6	β	β	NOUN
cana-1480	110	7	∈	∈	PART
cana-1480	110	8	u	u	NOUN
cana-1480	110	9	}	}	PUNCT
cana-1480	110	10	is	be	AUX
cana-1480	110	11	nanoc	nanoc	NOUN
cana-1480	110	12	τ	τ	NOUN
cana-1480	110	13	-covering	-covering	NOUN
cana-1480	110	14	of	of	ADP
cana-1480	110	15	u	u	PROPN
cana-1480	110	16	.	.	PUNCT
cana-1480	111	1	there	there	PRON
cana-1480	111	2	exist	exist	VERB
cana-1480	111	3	is	be	AUX
cana-1480	111	4	a	a	DET
cana-1480	111	5	finite	finite	ADJ
cana-1480	111	6	subcollection	subcollection	NOUN
cana-1480	111	7	{	{	PUNCT
cana-1480	111	8	𝐴𝛼1	𝐴𝛼1	NOUN
cana-1480	111	9	,	,	PUNCT
cana-1480	111	10	𝐴𝛼2	𝐴𝛼2	NOUN
cana-1480	111	11	,	,	PUNCT
cana-1480	111	12	.	.	PUNCT
cana-1480	111	13	.	.	PUNCT
cana-1480	112	1	𝐴𝛼𝑛	𝐴𝛼𝑛	PROPN
cana-1480	112	2	}	}	PUNCT
cana-1480	112	3	∈	∈	PROPN
cana-1480	112	4	𝐴	𝐴	PROPN
cana-1480	112	5	is	be	AUX
cana-1480	112	6	a	a	DET
cana-1480	112	7	nanocτ	nanocτ	ADJ
cana-1480	112	8	covering	covering	NOUN
cana-1480	112	9	of	of	ADP
cana-1480	112	10	u	u	PROPN
cana-1480	112	11	.	.	PUNCT
cana-1480	113	1	thus	thus	ADV
cana-1480	113	2	𝐴	𝐴	PROPN
cana-1480	113	3	⊆	⊆	NUM
cana-1480	113	4	⋃	⋃	ADP
cana-1480	113	5	𝐴𝛼𝑖	𝐴𝛼𝑖	PROPN
cana-1480	113	6	=	=	NOUN
cana-1480	113	7	𝑛	𝑛	PRON
cana-1480	113	8	𝑖=1	𝑖=1	PROPN
cana-1480	113	9	⋃	⋃	PROPN
cana-1480	113	10	𝐴𝛽𝑖	𝐴𝛽𝑖	PROPN
cana-1480	113	11	−	−	PROPN
cana-1480	114	1	𝐺𝛽𝑖	𝐺𝛽𝑖	PROPN
cana-1480	114	2	⊂	⊂	PROPN
cana-1480	114	3	⋃	⋃	PROPN
cana-1480	114	4	𝐴𝛽𝑖	𝐴𝛽𝑖	PROPN
cana-1480	114	5	𝑛	𝑛	ADJ
cana-1480	114	6	𝑖=1	𝑖=1	PUNCT
cana-1480	114	7	𝑛	𝑛	PRON
cana-1480	114	8	𝑖=1	𝑖=1	PROPN
cana-1480	114	9	since	since	SCONJ
cana-1480	114	10	𝐴𝛽𝑖	𝐴𝛽𝑖	PROPN
cana-1480	114	11	is	be	AUX
cana-1480	114	12	nanoc	nanoc	NOUN
cana-1480	114	13	τ	τ	X
cana-1480	114	14	-open	-open	NOUN
cana-1480	114	15	.	.	PUNCT
cana-1480	115	1	theorem	theorem	VERB
cana-1480	115	2	4.6	4.6	NUM
cana-1480	115	3	.	.	PUNCT
cana-1480	116	1	let	let	VERB
cana-1480	116	2	(	(	PUNCT
cana-1480	116	3	𝑈	𝑈	PROPN
cana-1480	116	4	,	,	PUNCT
cana-1480	116	5	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	116	6	)	)	PUNCT
cana-1480	116	7	)	)	PUNCT
cana-1480	117	1	be	be	AUX
cana-1480	117	2	strong	strong	ADJ
cana-1480	117	3	nano	nano	ADJ
cana-1480	117	4	generalized	generalize	VERB
cana-1480	117	5	nts	nt	NOUN
cana-1480	117	6	.	.	PUNCT
cana-1480	118	1	then	then	ADV
cana-1480	118	2	finite	finite	VERB
cana-1480	118	3	union	union	NOUN
cana-1480	118	4	of	of	ADP
cana-1480	118	5	nanoc	nanoc	NOUN
cana-1480	118	6	τ	τ	X
cana-1480	118	7	-ct	-ct	AUX
cana-1480	118	8	set	set	VERB
cana-1480	118	9	.	.	PUNCT
cana-1480	119	1	proof	proof	NOUN
cana-1480	119	2	.	.	PUNCT
cana-1480	120	1	assume	assume	VERB
cana-1480	120	2	that	that	SCONJ
cana-1480	120	3	g	g	PROPN
cana-1480	120	4	⊆	⊆	NUM
cana-1480	120	5	u	u	NOUN
cana-1480	120	6	and	and	CCONJ
cana-1480	120	7	f	f	PROPN
cana-1480	120	8	⊆	⊆	NUM
cana-1480	120	9	u	u	NOUN
cana-1480	120	10	are	be	AUX
cana-1480	120	11	any	any	DET
cana-1480	120	12	nanoc	nanoc	NOUN
cana-1480	120	13	τ	τ	X
cana-1480	120	14	-ct	-ct	NOUN
cana-1480	120	15	subset	subset	NOUN
cana-1480	120	16	of	of	ADP
cana-1480	120	17	u	u	PROPN
cana-1480	120	18	.	.	PUNCT
cana-1480	121	1	let	let	VERB
cana-1480	121	2	r	r	NOUN
cana-1480	121	3	be	be	AUX
cana-1480	121	4	nanoc	nanoc	NOUN
cana-1480	121	5	τ	τ	NOUN
cana-1480	121	6	a	a	DET
cana-1480	121	7	cover	cover	NOUN
cana-1480	121	8	of	of	ADP
cana-1480	121	9	g∪f	g∪f	NOUN
cana-1480	121	10	.	.	PUNCT
cana-1480	122	1	then	then	ADV
cana-1480	122	2	r	r	NOUN
cana-1480	122	3	will	will	AUX
cana-1480	122	4	also	also	ADV
cana-1480	122	5	nano	nano	VERB
cana-1480	122	6	cτ	cτ	ADP
cana-1480	122	7	cover	cover	NOUN
cana-1480	122	8	of	of	ADP
cana-1480	122	9	both	both	CCONJ
cana-1480	122	10	g	g	PROPN
cana-1480	122	11	and	and	CCONJ
cana-1480	122	12	f.	f.	PROPN
cana-1480	122	13	so	so	ADV
cana-1480	122	14	by	by	ADP
cana-1480	122	15	hypothesis	hypothesis	NOUN
cana-1480	122	16	,	,	PUNCT
cana-1480	122	17	there	there	PRON
cana-1480	122	18	exist	exist	VERB
cana-1480	122	19	a	a	DET
cana-1480	122	20	finite	finite	ADJ
cana-1480	122	21	subcollection	subcollection	NOUN
cana-1480	122	22	of	of	ADP
cana-1480	122	23	r	r	NOUN
cana-1480	122	24	of	of	ADP
cana-1480	122	25	nanoc	nanoc	NOUN
cana-1480	122	26	τ	τ	X
cana-1480	122	27	-set	-set	PUNCT
cana-1480	122	28	say	say	VERB
cana-1480	122	29	{	{	PUNCT
cana-1480	122	30	𝐺1	𝐺1	NOUN
cana-1480	122	31	,	,	PUNCT
cana-1480	122	32	𝐺2	𝐺2	ADJ
cana-1480	122	33	,	,	PUNCT
cana-1480	122	34	…	…	PUNCT
cana-1480	122	35	𝐺𝑛	𝐺𝑛	ADJ
cana-1480	122	36	}	}	PUNCT
cana-1480	122	37	and	and	CCONJ
cana-1480	122	38	{	{	PUNCT
cana-1480	122	39	𝐹1	𝐹1	PROPN
cana-1480	122	40	,	,	PUNCT
cana-1480	122	41	𝐹2	𝐹2	NOUN
cana-1480	122	42	,	,	PUNCT
cana-1480	122	43	…	…	PUNCT
cana-1480	122	44	𝐹𝑛	𝐹𝑛	AUX
cana-1480	122	45	}	}	PUNCT
cana-1480	122	46	covering	cover	VERB
cana-1480	122	47	g	g	NOUN
cana-1480	122	48	and	and	CCONJ
cana-1480	122	49	f	f	PROPN
cana-1480	122	50	respectively	respectively	ADV
cana-1480	122	51	,	,	PUNCT
cana-1480	122	52	where	where	SCONJ
cana-1480	122	53	g	g	NOUN
cana-1480	122	54	=	=	PUNCT
cana-1480	122	55	a	a	DET
cana-1480	122	56	−	−	PROPN
cana-1480	122	57	b	b	PROPN
cana-1480	122	58	,	,	PUNCT
cana-1480	122	59	a	a	DET
cana-1480	122	60	≠	≠	PROPN
cana-1480	122	61	u	u	NOUN
cana-1480	122	62	and	and	CCONJ
cana-1480	122	63	a	a	PRON
cana-1480	122	64	and	and	CCONJ
cana-1480	122	65	b	b	NOUN
cana-1480	122	66	are	be	AUX
cana-1480	122	67	nano	nano	VERB
cana-1480	122	68	open.clearly	open.clearly	ADV
cana-1480	122	69	the	the	DET
cana-1480	122	70	collection	collection	NOUN
cana-1480	122	71	{	{	PUNCT
cana-1480	122	72	𝐺1	𝐺1	NOUN
cana-1480	122	73	,	,	PUNCT
cana-1480	122	74	𝐺2	𝐺2	ADJ
cana-1480	122	75	,	,	PUNCT
cana-1480	122	76	…	…	PUNCT
cana-1480	122	77	𝐺𝑛	𝐺𝑛	PROPN
cana-1480	122	78	,	,	PUNCT
cana-1480	122	79	𝐹1	𝐹1	PROPN
cana-1480	122	80	,	,	PUNCT
cana-1480	122	81	𝐹2	𝐹2	NOUN
cana-1480	122	82	,	,	PUNCT
cana-1480	122	83	…	…	PUNCT
cana-1480	122	84	𝐹𝑛	𝐹𝑛	PRON
cana-1480	122	85	}	}	PUNCT
cana-1480	122	86	is	be	AUX
cana-1480	122	87	a	a	DET
cana-1480	122	88	finite	finite	ADJ
cana-1480	122	89	subcollection	subcollection	NOUN
cana-1480	122	90	of	of	ADP
cana-1480	122	91	r	r	NOUN
cana-1480	122	92	of	of	ADP
cana-1480	122	93	nanoc	nanoc	NOUN
cana-1480	122	94	τ	τ	PROPN
cana-1480	122	95	-sets	-set	NOUN
cana-1480	122	96	covering	cover	VERB
cana-1480	122	97	g∪f	g∪f	NOUN
cana-1480	122	98	.	.	PUNCT
cana-1480	123	1	by	by	ADP
cana-1480	123	2	induction	induction	NOUN
cana-1480	123	3	,	,	PUNCT
cana-1480	123	4	every	every	DET
cana-1480	123	5	finite	finite	PROPN
cana-1480	123	6	union	union	NOUN
cana-1480	123	7	of	of	ADP
cana-1480	123	8	nanoc	nanoc	NOUN
cana-1480	123	9	τ	τ	PROPN
cana-1480	123	10	-compaact	-compaact	ADJ
cana-1480	123	11	sets	set	NOUN
cana-1480	123	12	is	be	AUX
cana-1480	123	13	nanoc	nanoc	NOUN
cana-1480	123	14	τ	τ	NOUN
cana-1480	123	15	-compaact	-compaact	ADJ
cana-1480	123	16	.	.	PUNCT
cana-1480	124	1	theorem	theorem	NOUN
cana-1480	124	2	4.7	4.7	NUM
cana-1480	124	3	.	.	PUNCT
cana-1480	125	1	let	let	VERB
cana-1480	125	2	(	(	PUNCT
cana-1480	125	3	𝑈	𝑈	PROPN
cana-1480	125	4	,	,	PUNCT
cana-1480	125	5	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	125	6	)	)	PUNCT
cana-1480	125	7	)	)	PUNCT
cana-1480	126	1	strong	strong	ADJ
cana-1480	126	2	nano	nano	NOUN
cana-1480	126	3	generalized	generalize	VERB
cana-1480	126	4	nts	nt	NOUN
cana-1480	126	5	.	.	PUNCT
cana-1480	127	1	if	if	SCONJ
cana-1480	127	2	r	r	NOUN
cana-1480	127	3	is	be	AUX
cana-1480	127	4	a	a	DET
cana-1480	127	5	collection	collection	NOUN
cana-1480	127	6	of	of	ADP
cana-1480	127	7	all	all	DET
cana-1480	127	8	nano	nano	NOUN
cana-1480	127	9	open	open	ADJ
cana-1480	127	10	set	set	NOUN
cana-1480	127	11	then	then	ADV
cana-1480	127	12	the	the	DET
cana-1480	127	13	non	non	ADJ
cana-1480	127	14	-	-	ADJ
cana-1480	127	15	empty	empty	ADJ
cana-1480	127	16	subset	subset	NOUN
cana-1480	127	17	of	of	ADP
cana-1480	127	18	a	a	DET
cana-1480	127	19	nanoc	nanoc	NOUN
cana-1480	127	20	τ	τ	NOUN
cana-1480	127	21	space	space	NOUN
cana-1480	127	22	is	be	AUX
cana-1480	127	23	nanoc	nanoc	NOUN
cana-1480	127	24	τ	τ	X
cana-1480	127	25	-ct	-ct	NOUN
cana-1480	127	26	.	.	PUNCT
cana-1480	128	1	proof	proof	NOUN
cana-1480	128	2	.	.	PUNCT
cana-1480	129	1	a	a	DET
cana-1480	129	2	(	(	PUNCT
cana-1480	129	3	𝑈	𝑈	PROPN
cana-1480	129	4	,	,	PUNCT
cana-1480	129	5	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	129	6	)	)	PUNCT
cana-1480	129	7	)	)	PUNCT
cana-1480	129	8	is	be	AUX
cana-1480	129	9	nano	nano	NOUN
cana-1480	129	10	generalized	generalize	VERB
cana-1480	129	11	nts	nt	NOUN
cana-1480	129	12	and	and	CCONJ
cana-1480	129	13	u	u	PRON
cana-1480	129	14	be	be	AUX
cana-1480	129	15	nano	nano	NOUN
cana-1480	129	16	cτ	cτ	ADP
cana-1480	129	17	-ct	-ct	ADP
cana-1480	129	18	space.let	space.let	NOUN
cana-1480	129	19	g	g	NOUN
cana-1480	129	20	be	be	AUX
cana-1480	129	21	non	non	X
cana-1480	129	22	empty	empty	ADJ
cana-1480	129	23	nano	nano	NOUN
cana-1480	129	24	subset	subset	NOUN
cana-1480	129	25	of	of	ADP
cana-1480	129	26	u	u	PROPN
cana-1480	129	27	.	.	PUNCT
cana-1480	130	1	by	by	ADP
cana-1480	130	2	hypothesis	hypothesis	NOUN
cana-1480	130	3	there	there	PRON
cana-1480	130	4	exist	exist	VERB
cana-1480	130	5	two	two	NUM
cana-1480	130	6	nano	nano	NOUN
cana-1480	130	7	open	open	ADJ
cana-1480	130	8	p	p	NOUN
cana-1480	130	9	and	and	CCONJ
cana-1480	130	10	q	q	NOUN
cana-1480	130	11	,	,	PUNCT
cana-1480	130	12	p	p	X
cana-1480	130	13	6=	6=	ADP
cana-1480	130	14	q	q	NOUN
cana-1480	130	15	such	such	ADJ
cana-1480	130	16	that	that	PRON
cana-1480	130	17	g	g	NOUN
cana-1480	131	1	=	=	PUNCT
cana-1480	131	2	p	p	X
cana-1480	131	3	−	−	PROPN
cana-1480	131	4	q.	q.	NOUN
cana-1480	131	5	u	u	NOUN
cana-1480	131	6	−	−	PROPN
cana-1480	131	7	g	g	PROPN
cana-1480	131	8	=	=	SYM
cana-1480	131	9	u	u	NOUN
cana-1480	131	10	−	−	PROPN
cana-1480	131	11	(	(	PUNCT
cana-1480	131	12	p	p	NOUN
cana-1480	131	13	−	−	PROPN
cana-1480	131	14	q	q	NOUN
cana-1480	131	15	)	)	PUNCT
cana-1480	131	16	which	which	PRON
cana-1480	131	17	implies	imply	VERB
cana-1480	131	18	u	u	PRON
cana-1480	131	19	−	−	PROPN
cana-1480	131	20	g	g	NOUN
cana-1480	131	21	is	be	AUX
cana-1480	131	22	nanoc	nanoc	NOUN
cana-1480	131	23	τ	τ	X
cana-1480	131	24	-set	-set	ADJ
cana-1480	131	25	.	.	PUNCT
cana-1480	132	1	consider	consider	VERB
cana-1480	132	2	the	the	DET
cana-1480	132	3	collection	collection	NOUN
cana-1480	132	4	r	r	NOUN
cana-1480	132	5	=	=	NOUN
cana-1480	132	6	ai	ai	VERB
cana-1480	132	7	:	:	PUNCT
cana-1480	132	8	i	i	PRON
cana-1480	132	9	∈	∈	PROPN
cana-1480	132	10	u	u	NOUN
cana-1480	132	11	are	be	AUX
cana-1480	132	12	nano	nano	ADJ
cana-1480	132	13	open	open	ADJ
cana-1480	132	14	sets	set	NOUN
cana-1480	132	15	be	be	VERB
cana-1480	132	16	a	a	DET
cana-1480	132	17	nanoc	nanoc	NOUN
cana-1480	132	18	τ	τ	NOUN
cana-1480	132	19	-cover	-cover	NOUN
cana-1480	132	20	of	of	ADP
cana-1480	132	21	g	g	PROPN
cana-1480	132	22	.	.	PUNCT
cana-1480	133	1	it	it	PRON
cana-1480	133	2	is	be	AUX
cana-1480	133	3	given	give	VERB
cana-1480	133	4	that	that	SCONJ
cana-1480	133	5	u	u	NOUN
cana-1480	133	6	is	be	AUX
cana-1480	133	7	nanoc	nanoc	NOUN
cana-1480	133	8	τ	τ	X
cana-1480	133	9	-ct	-ct	NOUN
cana-1480	133	10	,	,	PUNCT
cana-1480	133	11	then	then	ADV
cana-1480	133	12	there	there	PRON
cana-1480	133	13	exist	exist	VERB
cana-1480	133	14	a	a	DET
cana-1480	133	15	collection	collection	NOUN
cana-1480	133	16	r	r	NOUN
cana-1480	133	17	of	of	ADP
cana-1480	133	18	nanocτ	nanocτ	NOUN
cana-1480	133	19	-covering	-covering	PROPN
cana-1480	133	20	u	u	PROPN
cana-1480	133	21	.	.	PUNCT
cana-1480	134	1	which	which	PRON
cana-1480	134	2	can	can	AUX
cana-1480	134	3	be	be	AUX
cana-1480	134	4	either	either	CCONJ
cana-1480	134	5	{	{	PUNCT
cana-1480	134	6	𝐴𝛼1	𝐴𝛼1	ADJ
cana-1480	134	7	,	,	PUNCT
cana-1480	134	8	𝐴𝛼2	𝐴𝛼2	NOUN
cana-1480	134	9	,	,	PUNCT
cana-1480	134	10	.	.	PUNCT
cana-1480	134	11	.	.	PUNCT
cana-1480	135	1	𝐴𝛼𝑛	𝐴𝛼𝑛	PROPN
cana-1480	135	2	}	}	PUNCT
cana-1480	135	3	or	or	CCONJ
cana-1480	135	4	{	{	PUNCT
cana-1480	135	5	𝐴𝛼1	𝐴𝛼1	ADJ
cana-1480	135	6	,	,	PUNCT
cana-1480	135	7	𝐴𝛼2	𝐴𝛼2	NOUN
cana-1480	135	8	,	,	PUNCT
cana-1480	135	9	.	.	PUNCT
cana-1480	135	10	.	.	PUNCT
cana-1480	136	1	𝐴𝛼𝑛	𝐴𝛼𝑛	INTJ
cana-1480	136	2	,	,	PUNCT
cana-1480	136	3	u	u	NOUN
cana-1480	136	4	−	−	PROPN
cana-1480	136	5	g	g	PROPN
cana-1480	136	6	}	}	PUNCT
cana-1480	136	7	.	.	PUNCT
cana-1480	137	1	since	since	SCONJ
cana-1480	137	2	𝐴	𝐴	PROPN
cana-1480	137	3	⊆	⊆	NUM
cana-1480	137	4	⋃	⋃	ADP
cana-1480	137	5	𝐴𝛼𝑖	𝐴𝛼𝑖	PROPN
cana-1480	137	6	=	=	NOUN
cana-1480	137	7	𝑛	𝑛	PRON
cana-1480	137	8	𝑖=1	𝑖=1	PUNCT
cana-1480	137	9	u	u	NOUN
cana-1480	137	10	and	and	CCONJ
cana-1480	137	11	g	g	PROPN
cana-1480	137	12	⊆	⊆	NUM
cana-1480	137	13	u	u	NOUN
cana-1480	137	14	,	,	PUNCT
cana-1480	137	15	g	g	NOUN
cana-1480	137	16	=	=	PUNCT
cana-1480	137	17	⋃	⋃	PROPN
cana-1480	137	18	𝐴𝛼𝑖	𝐴𝛼𝑖	NOUN
cana-1480	137	19	𝑛	𝑛	PRON
cana-1480	137	20	𝑖=1	𝑖=1	PROPN
cana-1480	137	21	then	then	ADV
cana-1480	137	22	the	the	DET
cana-1480	137	23	collection	collection	NOUN
cana-1480	137	24	aαi	aαi	ADP
cana-1480	137	25	i=1,2	i=1,2	NOUN
cana-1480	137	26	...	...	PUNCT
cana-1480	137	27	n	n	NUM
cana-1480	137	28	of	of	ADP
cana-1480	137	29	nanoc	nanoc	NOUN
cana-1480	137	30	τ	τ	PROPN
cana-1480	137	31	-sets	-set	NOUN
cana-1480	137	32	is	be	AUX
cana-1480	137	33	finite	finite	ADJ
cana-1480	137	34	subcollection	subcollection	NOUN
cana-1480	137	35	of	of	ADP
cana-1480	137	36	r	r	NOUN
cana-1480	137	37	covering	cover	VERB
cana-1480	137	38	g	g	NOUN
cana-1480	137	39	.	.	PUNCT
cana-1480	138	1	hence	hence	ADV
cana-1480	138	2	g	g	PROPN
cana-1480	138	3	is	be	AUX
cana-1480	138	4	nanoc	nanoc	NOUN
cana-1480	138	5	τ	τ	NOUN
cana-1480	138	6	-compact	-compact	NOUN
cana-1480	138	7	.	.	PUNCT
cana-1480	139	1	5	5	NUM
cana-1480	139	2	nano*gα	nano*gα	PROPN
cana-1480	139	3	-connected	-connected	ADJ
cana-1480	139	4	definition	definition	NOUN
cana-1480	139	5	5.1	5.1	NUM
cana-1480	139	6	.	.	PUNCT
cana-1480	140	1	a	a	DET
cana-1480	140	2	nts	nt	NOUN
cana-1480	140	3	(	(	PUNCT
cana-1480	140	4	𝑈	𝑈	PROPN
cana-1480	140	5	,	,	PUNCT
cana-1480	140	6	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	140	7	)	)	PUNCT
cana-1480	140	8	)	)	PUNCT
cana-1480	140	9	is	be	AUX
cana-1480	140	10	said	say	VERB
cana-1480	140	11	to	to	PART
cana-1480	140	12	be	be	AUX
cana-1480	140	13	is	be	AUX
cana-1480	140	14	nano*gα	nano*gα	PROPN
cana-1480	140	15	-connected	-connected	ADJ
cana-1480	140	16	if	if	SCONJ
cana-1480	140	17	u	u	NOUN
cana-1480	140	18	can	can	AUX
cana-1480	140	19	not	not	PART
cana-1480	140	20	be	be	AUX
cana-1480	140	21	written	write	VERB
cana-1480	140	22	as	as	ADP
cana-1480	140	23	a	a	DET
cana-1480	140	24	union	union	NOUN
cana-1480	140	25	of	of	ADP
cana-1480	140	26	two	two	NUM
cana-1480	140	27	disjoint	disjoint	NOUN
cana-1480	140	28	nonempty	nonempty	NOUN
cana-1480	140	29	is	be	AUX
cana-1480	140	30	nano*gα	nano*gα	PROPN
cana-1480	140	31	-open	-open	ADJ
cana-1480	140	32	sets	set	NOUN
cana-1480	140	33	.	.	PUNCT
cana-1480	141	1	definition	definition	NOUN
cana-1480	141	2	5.2	5.2	NUM
cana-1480	141	3	.	.	PUNCT
cana-1480	142	1	a	a	DET
cana-1480	142	2	subset	subset	NOUN
cana-1480	142	3	g	g	NOUN
cana-1480	142	4	of	of	ADP
cana-1480	142	5	a	a	DET
cana-1480	142	6	nts	nt	NOUN
cana-1480	142	7	(	(	PUNCT
cana-1480	142	8	𝑈	𝑈	PROPN
cana-1480	142	9	,	,	PUNCT
cana-1480	142	10	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	142	11	)	)	PUNCT
cana-1480	142	12	)	)	PUNCT
cana-1480	142	13	is	be	AUX
cana-1480	142	14	said	say	VERB
cana-1480	142	15	to	to	PART
cana-1480	142	16	be	be	AUX
cana-1480	142	17	is	be	AUX
cana-1480	142	18	nano*gα	nano*gα	PROPN
cana-1480	142	19	is	be	AUX
cana-1480	142	20	said	say	VERB
cana-1480	142	21	to	to	PART
cana-1480	142	22	be	be	AUX
cana-1480	142	23	nano*gα	nano*gα	PROPN
cana-1480	142	24	connected	connect	VERB
cana-1480	142	25	set	set	VERB
cana-1480	142	26	in	in	ADP
cana-1480	142	27	u	u	NOUN
cana-1480	142	28	if	if	SCONJ
cana-1480	142	29	g	g	PROPN
cana-1480	142	30	can	can	AUX
cana-1480	142	31	not	not	PART
cana-1480	142	32	be	be	AUX
cana-1480	142	33	expressed	express	VERB
cana-1480	142	34	as	as	ADP
cana-1480	142	35	the	the	DET
cana-1480	142	36	union	union	NOUN
cana-1480	142	37	of	of	ADP
cana-1480	142	38	two	two	NUM
cana-1480	142	39	disjoint	disjoint	NOUN
cana-1480	142	40	nonempty	nonempty	ADJ
cana-1480	142	41	nano*gα	nano*gα	PROPN
cana-1480	142	42	open	open	ADJ
cana-1480	142	43	sets	set	NOUN
cana-1480	142	44	in	in	ADP
cana-1480	142	45	(	(	PUNCT
cana-1480	142	46	𝑈	𝑈	PROPN
cana-1480	142	47	,	,	PUNCT
cana-1480	142	48	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	142	49	)	)	PUNCT
cana-1480	142	50	)	)	PUNCT
cana-1480	142	51	.	.	PUNCT
cana-1480	143	1	theorem	theorem	VERB
cana-1480	143	2	5.3	5.3	NUM
cana-1480	143	3	.	.	PUNCT
cana-1480	144	1	for	for	ADP
cana-1480	144	2	a	a	DET
cana-1480	144	3	nts	nt	NOUN
cana-1480	144	4	(	(	PUNCT
cana-1480	144	5	𝑈	𝑈	PROPN
cana-1480	144	6	,	,	PUNCT
cana-1480	144	7	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	144	8	)	)	PUNCT
cana-1480	144	9	)	)	PUNCT
cana-1480	145	1	the	the	DET
cana-1480	145	2	following	follow	VERB
cana-1480	145	3	statements	statement	NOUN
cana-1480	145	4	are	be	AUX
cana-1480	145	5	equivalent	equivalent	ADJ
cana-1480	145	6	.	.	PUNCT
cana-1480	146	1	(	(	PUNCT
cana-1480	146	2	i)u	i)u	NOUN
cana-1480	146	3	is	be	AUX
cana-1480	146	4	nano*gα	nano*gα	PROPN
cana-1480	146	5	-connected	-connected	PROPN
cana-1480	146	6	.	.	PUNCT
cana-1480	147	1	(	(	PUNCT
cana-1480	147	2	ii	ii	NOUN
cana-1480	147	3	)	)	PUNCT
cana-1480	147	4	the	the	DET
cana-1480	147	5	only	only	ADJ
cana-1480	147	6	subsets	subset	NOUN
cana-1480	147	7	of	of	ADP
cana-1480	147	8	u	u	PRON
cana-1480	147	9	which	which	PRON
cana-1480	147	10	are	be	AUX
cana-1480	147	11	both	both	PRON
cana-1480	147	12	nano*gα	nano*gα	PROPN
cana-1480	147	13	-open	-open	PROPN
cana-1480	147	14	and	and	CCONJ
cana-1480	147	15	n	n	NOUN
cana-1480	147	16	ano∗	ano∗	PROPN
cana-1480	147	17	gα	gα	NOUN
cana-1480	147	18	-	-	PUNCT
cana-1480	147	19	closed	closed	ADJ
cana-1480	147	20	are	be	AUX
cana-1480	147	21	the	the	DET
cana-1480	147	22	empty	empty	ADJ
cana-1480	147	23	set	set	NOUN
cana-1480	147	24	ϕ	ϕ	NOUN
cana-1480	147	25	and	and	CCONJ
cana-1480	147	26	u	u	NOUN
cana-1480	147	27	.	.	PUNCT
cana-1480	148	1	(	(	PUNCT
cana-1480	148	2	iii	iii	X
cana-1480	148	3	)	)	PUNCT
cana-1480	148	4	each	each	DET
cana-1480	148	5	nano*gα	nano*gα	PROPN
cana-1480	148	6	-continuous	-continuous	ADJ
cana-1480	148	7	function	function	NOUN
cana-1480	148	8	of	of	ADP
cana-1480	148	9	u	u	NOUN
cana-1480	148	10	into	into	ADP
cana-1480	148	11	a	a	DET
cana-1480	148	12	discrete	discrete	ADJ
cana-1480	148	13	space	space	NOUN
cana-1480	148	14	v	v	NOUN
cana-1480	148	15	with	with	ADP
cana-1480	148	16	atleast	atleast	ADJ
cana-1480	148	17	two	two	NUM
cana-1480	148	18	points	point	NOUN
cana-1480	148	19	is	be	AUX
cana-1480	148	20	a	a	DET
cana-1480	148	21	constant	constant	ADJ
cana-1480	148	22	function	function	NOUN
cana-1480	148	23	.	.	PUNCT
cana-1480	149	1	proof	proof	NOUN
cana-1480	149	2	.	.	PUNCT
cana-1480	150	1	(	(	PUNCT
cana-1480	150	2	i	i	NOUN
cana-1480	150	3	)	)	PUNCT
cana-1480	150	4	⇒	⇒	PROPN
cana-1480	150	5	(	(	PUNCT
cana-1480	150	6	ii	ii	NOUN
cana-1480	150	7	)	)	PUNCT
cana-1480	150	8	let	let	VERB
cana-1480	150	9	u	u	PRON
cana-1480	150	10	be	be	AUX
cana-1480	150	11	a	a	DET
cana-1480	150	12	nano*gα	nano*gα	ADJ
cana-1480	150	13	-connected	-connected	ADJ
cana-1480	150	14	space	space	NOUN
cana-1480	150	15	.	.	PUNCT
cana-1480	151	1	let	let	VERB
cana-1480	151	2	a	a	DET
cana-1480	151	3	be	be	AUX
cana-1480	151	4	nano*gα	nano*gα	PROPN
cana-1480	151	5	-open	-open	ADJ
cana-1480	151	6	and	and	CCONJ
cana-1480	151	7	nano*gα	nano*gα	ADJ
cana-1480	151	8	closed	close	VERB
cana-1480	151	9	subset	subset	NOUN
cana-1480	151	10	of	of	ADP
cana-1480	151	11	u	u	PROPN
cana-1480	151	12	.	.	PUNCT
cana-1480	152	1	then	then	ADV
cana-1480	152	2	u	u	NOUN
cana-1480	152	3	−	−	PROPN
cana-1480	153	1	a	a	PRON
cana-1480	153	2	is	be	AUX
cana-1480	153	3	both	both	PRON
cana-1480	153	4	nano*gα	nano*gα	PROPN
cana-1480	153	5	-open	-open	PROPN
cana-1480	153	6	and	and	CCONJ
cana-1480	153	7	nano*gα	nano*gα	PROPN
cana-1480	153	8	-closed	-close	VERB
cana-1480	153	9	in	in	ADP
cana-1480	153	10	u	u	PROPN
cana-1480	153	11	.	.	PUNCT
cana-1480	154	1	that	that	PRON
cana-1480	154	2	implies	imply	VERB
cana-1480	154	3	,	,	PUNCT
cana-1480	154	4	u	u	NOUN
cana-1480	154	5	communications	communication	NOUN
cana-1480	154	6	on	on	ADP
cana-1480	154	7	applied	apply	VERB
cana-1480	154	8	nonlinear	nonlinear	ADJ
cana-1480	154	9	analysis	analysis	NOUN
cana-1480	154	10	issn	issn	NOUN
cana-1480	154	11	:	:	PUNCT
cana-1480	154	12	1074	1074	NUM
cana-1480	154	13	-	-	PUNCT
cana-1480	154	14	133x	133x	NUM
cana-1480	154	15	vol	vol	NOUN
cana-1480	154	16	31	31	NUM
cana-1480	154	17	no	no	NOUN
cana-1480	154	18	.	.	PUNCT
cana-1480	155	1	8s	8s	PROPN
cana-1480	155	2	(	(	PUNCT
cana-1480	155	3	2024	2024	NUM
cana-1480	155	4	)	)	PUNCT
cana-1480	155	5	257	257	NUM
cana-1480	155	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1480	155	7	is	be	AUX
cana-1480	155	8	the	the	DET
cana-1480	155	9	union	union	NOUN
cana-1480	155	10	of	of	ADP
cana-1480	155	11	disjoint	disjoint	ADJ
cana-1480	155	12	nano*gα	nano*gα	PROPN
cana-1480	155	13	-open	-open	PROPN
cana-1480	155	14	sets	set	NOUN
cana-1480	155	15	a	a	PRON
cana-1480	155	16	and	and	CCONJ
cana-1480	155	17	u	u	PRON
cana-1480	155	18	⊆	⊆	NUM
cana-1480	155	19	a	a	PRON
cana-1480	155	20	.	.	PUNCT
cana-1480	156	1	since	since	SCONJ
cana-1480	156	2	u	u	NOUN
cana-1480	156	3	is	be	AUX
cana-1480	156	4	nano*gα	nano*gα	PROPN
cana-1480	156	5	-connected	-connected	ADJ
cana-1480	156	6	either	either	CCONJ
cana-1480	156	7	a	a	DET
cana-1480	156	8	=	=	X
cana-1480	156	9	ϕ	ϕ	NOUN
cana-1480	156	10	or	or	CCONJ
cana-1480	156	11	u	u	NOUN
cana-1480	156	12	−	−	PROPN
cana-1480	156	13	a	a	DET
cana-1480	156	14	=	=	PUNCT
cana-1480	156	15	ϕ.	ϕ.	NOUN
cana-1480	156	16	this	this	PRON
cana-1480	156	17	is	be	AUX
cana-1480	156	18	,	,	PUNCT
cana-1480	156	19	a	a	DET
cana-1480	156	20	=	=	X
cana-1480	156	21	ϕ	ϕ	NOUN
cana-1480	156	22	or	or	CCONJ
cana-1480	156	23	a	a	DET
cana-1480	156	24	=	=	SYM
cana-1480	156	25	u	u	NOUN
cana-1480	156	26	.	.	PUNCT
cana-1480	157	1	(	(	PUNCT
cana-1480	157	2	ii	ii	NOUN
cana-1480	157	3	)	)	PUNCT
cana-1480	157	4	⇒	⇒	NOUN
cana-1480	157	5	(	(	PUNCT
cana-1480	157	6	i	i	NOUN
cana-1480	157	7	)	)	PUNCT
cana-1480	157	8	suppose	suppose	VERB
cana-1480	157	9	that	that	SCONJ
cana-1480	157	10	u	u	NOUN
cana-1480	157	11	=	=	NOUN
cana-1480	157	12	a	a	DET
cana-1480	157	13	∪	∪	X
cana-1480	157	14	b	b	NOUN
cana-1480	157	15	,	,	PUNCT
cana-1480	157	16	where	where	SCONJ
cana-1480	157	17	a	a	PRON
cana-1480	157	18	and	and	CCONJ
cana-1480	157	19	b	b	NOUN
cana-1480	157	20	are	be	AUX
cana-1480	157	21	disjoint	disjoint	X
cana-1480	157	22	non	non	X
cana-1480	157	23	empty	empty	ADJ
cana-1480	157	24	nano*gα	nano*gα	ADJ
cana-1480	157	25	open	open	ADJ
cana-1480	157	26	subsets	subset	NOUN
cana-1480	157	27	of	of	ADP
cana-1480	157	28	u	u	PROPN
cana-1480	157	29	.	.	PUNCT
cana-1480	158	1	then	then	ADV
cana-1480	158	2	a	a	PRON
cana-1480	158	3	and	and	CCONJ
cana-1480	158	4	b	b	NOUN
cana-1480	158	5	are	be	AUX
cana-1480	158	6	proper	proper	ADJ
cana-1480	158	7	subsets	subset	NOUN
cana-1480	158	8	of	of	ADP
cana-1480	158	9	u	u	NOUN
cana-1480	158	10	.since	.since	NOUN
cana-1480	158	11	a	a	DET
cana-1480	158	12	=	=	X
cana-1480	158	13	u	u	NOUN
cana-1480	158	14	−	−	PROPN
cana-1480	158	15	b	b	PROPN
cana-1480	158	16	,	,	PUNCT
cana-1480	158	17	a	a	PRON
cana-1480	158	18	is	be	AUX
cana-1480	158	19	nano*gα	nano*gα	PROPN
cana-1480	158	20	-closed	-closed	ADJ
cana-1480	158	21	subset	subset	NOUN
cana-1480	158	22	of	of	ADP
cana-1480	158	23	u	u	PROPN
cana-1480	158	24	,	,	PUNCT
cana-1480	158	25	then	then	ADV
cana-1480	158	26	a	a	PRON
cana-1480	158	27	is	be	AUX
cana-1480	158	28	both	both	PRON
cana-1480	158	29	nano*gα	nano*gα	PROPN
cana-1480	158	30	-open	-open	ADJ
cana-1480	158	31	and	and	CCONJ
cana-1480	158	32	nano*gα	nano*gα	PROPN
cana-1480	158	33	-closed	-close	VERB
cana-1480	158	34	subset	subset	NOUN
cana-1480	158	35	of	of	ADP
cana-1480	158	36	u	u	PROPN
cana-1480	158	37	.therefore	.therefore	ADV
cana-1480	158	38	,	,	PUNCT
cana-1480	158	39	a	a	DET
cana-1480	158	40	=	=	X
cana-1480	158	41	ϕ	ϕ	NOUN
cana-1480	158	42	and	and	CCONJ
cana-1480	158	43	a	a	DET
cana-1480	158	44	=	=	X
cana-1480	158	45	u	u	PROPN
cana-1480	158	46	,	,	PUNCT
cana-1480	158	47	which	which	PRON
cana-1480	158	48	is	be	AUX
cana-1480	158	49	contradiction	contradiction	NOUN
cana-1480	158	50	.	.	PUNCT
cana-1480	159	1	thus	thus	ADV
cana-1480	159	2	u	u	NOUN
cana-1480	159	3	is	be	AUX
cana-1480	159	4	nano*gα	nano*gα	PROPN
cana-1480	159	5	.	.	PUNCT
cana-1480	160	1	(	(	PUNCT
cana-1480	160	2	ii	ii	NOUN
cana-1480	160	3	)	)	PUNCT
cana-1480	160	4	⇒	⇒	NOUN
cana-1480	160	5	(	(	PUNCT
cana-1480	160	6	iii	iii	X
cana-1480	160	7	)	)	PUNCT
cana-1480	160	8	let	let	VERB
cana-1480	160	9	𝑓	𝑓	X
cana-1480	160	10	:	:	PUNCT
cana-1480	160	11	(	(	PUNCT
cana-1480	160	12	𝑈	𝑈	PROPN
cana-1480	160	13	,	,	PUNCT
cana-1480	160	14	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	160	15	)	)	PUNCT
cana-1480	160	16	)	)	PUNCT
cana-1480	161	1	→	→	PUNCT
cana-1480	161	2	(	(	PUNCT
cana-1480	161	3	𝑉	𝑉	PROPN
cana-1480	161	4	,	,	PUNCT
cana-1480	161	5	𝜎𝑅(𝑌	𝜎𝑅(𝑌	NOUN
cana-1480	161	6	)	)	PUNCT
cana-1480	161	7	)	)	PUNCT
cana-1480	161	8	be	be	AUX
cana-1480	161	9	nano*gα	nano*gα	PROPN
cana-1480	161	10	-continuous	-continuous	ADJ
cana-1480	161	11	,	,	PUNCT
cana-1480	161	12	where	where	SCONJ
cana-1480	161	13	v	v	NOUN
cana-1480	161	14	is	be	AUX
cana-1480	161	15	discrete	discrete	ADJ
cana-1480	161	16	space	space	NOUN
cana-1480	161	17	with	with	ADP
cana-1480	161	18	atleast	atleast	ADJ
cana-1480	161	19	two	two	NUM
cana-1480	161	20	points	point	NOUN
cana-1480	161	21	.	.	PUNCT
cana-1480	162	1	then	then	ADV
cana-1480	162	2	u	u	PRON
cana-1480	162	3	is	be	AUX
cana-1480	162	4	covered	cover	VERB
cana-1480	162	5	by	by	ADP
cana-1480	162	6	nano*gα	nano*gα	PROPN
cana-1480	162	7	-open	-open	PROPN
cana-1480	162	8	and	and	CCONJ
cana-1480	162	9	nano*gα	nano*gα	PROPN
cana-1480	162	10	closed	closed	ADJ
cana-1480	162	11	covering	cover	VERB
cana-1480	162	12	{	{	PUNCT
cana-1480	162	13	f	f	NOUN
cana-1480	162	14	−1	−1	NOUN
cana-1480	162	15	(	(	PUNCT
cana-1480	162	16	y	y	NOUN
cana-1480	162	17	)	)	PUNCT
cana-1480	162	18	:	:	PUNCT
cana-1480	162	19	y	y	PROPN
cana-1480	162	20	∈	∈	PROPN
cana-1480	162	21	v	v	ADP
cana-1480	162	22	}	}	PUNCT
cana-1480	162	23	.	.	PUNCT
cana-1480	163	1	by	by	ADP
cana-1480	163	2	part	part	NOUN
cana-1480	163	3	(	(	PUNCT
cana-1480	163	4	i	i	NOUN
cana-1480	163	5	)	)	PUNCT
cana-1480	163	6	,	,	PUNCT
cana-1480	163	7	f(y	f(y	NOUN
cana-1480	163	8	)	)	PUNCT
cana-1480	163	9	=	=	SYM
cana-1480	163	10	ϕ	ϕ	NOUN
cana-1480	163	11	or	or	CCONJ
cana-1480	163	12	u	u	NOUN
cana-1480	163	13	,	,	PUNCT
cana-1480	163	14	each	each	DET
cana-1480	163	15	y	y	PROPN
cana-1480	163	16	∈	∈	PROPN
cana-1480	163	17	v.	v.	ADP
cana-1480	163	18	if	if	SCONJ
cana-1480	163	19	𝑓−1(𝑦	𝑓−1(𝑦	PROPN
cana-1480	163	20	)	)	PUNCT
cana-1480	164	1	=	=	SYM
cana-1480	164	2	ϕ	ϕ	NOUN
cana-1480	164	3	,	,	PUNCT
cana-1480	164	4	for	for	ADP
cana-1480	164	5	all	all	DET
cana-1480	164	6	y	y	PROPN
cana-1480	164	7	∈	∈	PROPN
cana-1480	164	8	v	v	ADP
cana-1480	164	9	if	if	SCONJ
cana-1480	164	10	f	f	PROPN
cana-1480	164	11	fails	fail	VERB
cana-1480	164	12	to	to	PART
cana-1480	164	13	be	be	AUX
cana-1480	164	14	a	a	DET
cana-1480	164	15	function	function	NOUN
cana-1480	164	16	.	.	PUNCT
cana-1480	165	1	therefore	therefore	ADV
cana-1480	165	2	there	there	PRON
cana-1480	165	3	exists	exist	VERB
cana-1480	165	4	atleast	atleast	ADV
cana-1480	165	5	one	one	NUM
cana-1480	165	6	point	point	NOUN
cana-1480	165	7	say	say	VERB
cana-1480	165	8	𝑦1	𝑦1	PROPN
cana-1480	165	9	∈	∈	PROPN
cana-1480	165	10	v	v	NOUN
cana-1480	165	11	,	,	PUNCT
cana-1480	165	12	such	such	ADJ
cana-1480	165	13	that	that	SCONJ
cana-1480	165	14	𝑓−1(𝑦1	𝑓−1(𝑦1	PROPN
cana-1480	165	15	)	)	PUNCT
cana-1480	165	16	≠	≠	PROPN
cana-1480	165	17	and	and	CCONJ
cana-1480	165	18	hence	hence	ADV
cana-1480	165	19	𝑓−1(𝑦1	𝑓−1(𝑦1	PROPN
cana-1480	165	20	)	)	PUNCT
cana-1480	166	1	=	=	SYM
cana-1480	166	2	u	u	PROPN
cana-1480	166	3	,	,	PUNCT
cana-1480	166	4	which	which	PRON
cana-1480	166	5	shows	show	VERB
cana-1480	166	6	that	that	SCONJ
cana-1480	166	7	f	f	PROPN
cana-1480	166	8	is	be	AUX
cana-1480	166	9	a	a	DET
cana-1480	166	10	constant	constant	ADJ
cana-1480	166	11	function	function	NOUN
cana-1480	166	12	.	.	PUNCT
cana-1480	167	1	(	(	PUNCT
cana-1480	167	2	iii	iii	X
cana-1480	167	3	)	)	PUNCT
cana-1480	167	4	⇒	⇒	NOUN
cana-1480	167	5	(	(	PUNCT
cana-1480	167	6	ii	ii	NOUN
cana-1480	167	7	)	)	PUNCT
cana-1480	167	8	let	let	VERB
cana-1480	167	9	g	g	NOUN
cana-1480	167	10	be	be	AUX
cana-1480	167	11	both	both	CCONJ
cana-1480	167	12	nano*gα	nano*gα	PROPN
cana-1480	167	13	-closed	-close	VERB
cana-1480	167	14	in	in	ADP
cana-1480	167	15	u	u	PROPN
cana-1480	167	16	.	.	PUNCT
cana-1480	167	17	suppose	suppose	VERB
cana-1480	167	18	that	that	SCONJ
cana-1480	167	19	g	g	PROPN
cana-1480	167	20	≠ϕ.	≠ϕ.	PROPN
cana-1480	167	21	let	let	VERB
cana-1480	167	22	v	v	PART
cana-1480	167	23	be	be	AUX
cana-1480	167	24	a	a	DET
cana-1480	167	25	discreate	discreate	ADJ
cana-1480	167	26	space	space	NOUN
cana-1480	167	27	with	with	ADP
cana-1480	167	28	atleast	atleast	ADJ
cana-1480	167	29	two	two	NUM
cana-1480	167	30	points	point	NOUN
cana-1480	167	31	,	,	PUNCT
cana-1480	167	32	fix	fix	VERB
cana-1480	167	33	𝑦1	𝑦1	NOUN
cana-1480	167	34	and	and	CCONJ
cana-1480	167	35	𝑦0	𝑦0	NOUN
cana-1480	167	36	in	in	ADP
cana-1480	167	37	v	v	NOUN
cana-1480	167	38	and	and	CCONJ
cana-1480	167	39	𝑦0	𝑦0	NOUN
cana-1480	167	40	.	.	PUNCT
cana-1480	168	1	define	define	VERB
cana-1480	168	2	𝑓	𝑓	PRON
cana-1480	168	3	:	:	PUNCT
cana-1480	168	4	(	(	PUNCT
cana-1480	168	5	𝑈	𝑈	PROPN
cana-1480	168	6	,	,	PUNCT
cana-1480	168	7	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	168	8	)	)	PUNCT
cana-1480	168	9	)	)	PUNCT
cana-1480	169	1	→	→	PUNCT
cana-1480	169	2	(	(	PUNCT
cana-1480	169	3	𝑉	𝑉	PROPN
cana-1480	169	4	,	,	PUNCT
cana-1480	169	5	𝜎𝑅(𝑌))by	𝜎𝑅(𝑌))by	PROPN
cana-1480	169	6	f(x	f(x	PROPN
cana-1480	169	7	)	)	PUNCT
cana-1480	169	8	=	=	PRON
cana-1480	169	9	{	{	PUNCT
cana-1480	169	10	𝑦0	𝑦0	NOUN
cana-1480	169	11	}	}	PUNCT
cana-1480	169	12	,	,	PUNCT
cana-1480	169	13	for	for	ADP
cana-1480	169	14	x	x	PROPN
cana-1480	169	15	∈	∈	PROPN
cana-1480	169	16	g	g	PROPN
cana-1480	169	17	and	and	CCONJ
cana-1480	169	18	f(x	f(x	PROPN
cana-1480	169	19	)	)	PUNCT
cana-1480	169	20	=	=	PRON
cana-1480	169	21	{	{	PUNCT
cana-1480	169	22	𝑦0	𝑦0	NOUN
cana-1480	169	23	}	}	PUNCT
cana-1480	169	24	,	,	PUNCT
cana-1480	169	25	for	for	SCONJ
cana-1480	169	26	x	x	PROPN
cana-1480	169	27	∉	∉	PROPN
cana-1480	169	28	g.let	g.let	PROPN
cana-1480	169	29	f	f	PROPN
cana-1480	169	30	be	be	AUX
cana-1480	169	31	a	a	DET
cana-1480	169	32	nano	nano	NOUN
cana-1480	169	33	open	open	NOUN
cana-1480	169	34	set	set	NOUN
cana-1480	169	35	in	in	ADP
cana-1480	169	36	v	v	NUM
cana-1480	169	37	.if	.if	PUNCT
cana-1480	170	1	f	f	PROPN
cana-1480	170	2	contains	contain	VERB
cana-1480	170	3	alone	alone	ADV
cana-1480	170	4	,	,	PUNCT
cana-1480	170	5	then	then	ADV
cana-1480	170	6	𝑓−1	𝑓−1	NUM
cana-1480	170	7	(	(	PUNCT
cana-1480	170	8	f	f	X
cana-1480	170	9	)	)	PUNCT
cana-1480	170	10	=	=	SYM
cana-1480	170	11	g	g	PROPN
cana-1480	170	12	.if	.if	PUNCT
cana-1480	171	1	f	f	PROPN
cana-1480	171	2	contains	contain	VERB
cana-1480	171	3	both	both	DET
cana-1480	171	4	𝑦0	𝑦0	NOUN
cana-1480	171	5	and	and	CCONJ
cana-1480	171	6	𝑦0	𝑦0	NOUN
cana-1480	171	7	,	,	PUNCT
cana-1480	171	8	then	then	ADV
cana-1480	171	9	𝑓−1	𝑓−1	NUM
cana-1480	171	10	(	(	PUNCT
cana-1480	171	11	f	f	X
cana-1480	171	12	)	)	PUNCT
cana-1480	171	13	=	=	VERB
cana-1480	172	1	u.	u.	NOUN
cana-1480	172	2	otherwise	otherwise	ADV
cana-1480	172	3	𝑓−1	𝑓−1	NUM
cana-1480	172	4	(	(	PUNCT
cana-1480	172	5	f	f	X
cana-1480	172	6	)	)	PUNCT
cana-1480	172	7	=	=	SYM
cana-1480	172	8	ϕ	ϕ	PROPN
cana-1480	172	9	.	.	PUNCT
cana-1480	173	1	in	in	ADP
cana-1480	173	2	all	all	DET
cana-1480	173	3	case	case	NOUN
cana-1480	173	4	𝑓−1	𝑓−1	NUM
cana-1480	173	5	(	(	PUNCT
cana-1480	173	6	f	f	X
cana-1480	173	7	)	)	PUNCT
cana-1480	173	8	is	be	AUX
cana-1480	173	9	n	n	NUM
cana-1480	173	10	ano∗	ano∗	X
cana-1480	173	11	gα	gα	NOUN
cana-1480	173	12	-	-	PUNCT
cana-1480	173	13	open	open	ADJ
cana-1480	173	14	in	in	ADP
cana-1480	173	15	u	u	PROPN
cana-1480	173	16	.	.	PUNCT
cana-1480	174	1	therefore	therefore	ADV
cana-1480	174	2	f	f	PROPN
cana-1480	174	3	is	be	AUX
cana-1480	174	4	nano*gα	nano*gα	PROPN
cana-1480	174	5	-continuous	-continuous	ADJ
cana-1480	174	6	function	function	NOUN
cana-1480	174	7	.	.	PUNCT
cana-1480	175	1	then	then	ADV
cana-1480	175	2	by	by	ADP
cana-1480	175	3	assumption	assumption	NOUN
cana-1480	175	4	f	f	PROPN
cana-1480	175	5	is	be	AUX
cana-1480	175	6	a	a	DET
cana-1480	175	7	constant	constant	ADJ
cana-1480	175	8	function	function	NOUN
cana-1480	175	9	.	.	PUNCT
cana-1480	176	1	therefore	therefore	ADV
cana-1480	176	2	f(x	f(x	PROPN
cana-1480	176	3	)	)	PUNCT
cana-1480	177	1	=	=	SYM
cana-1480	177	2	𝑦0	𝑦0	NOUN
cana-1480	177	3	or	or	CCONJ
cana-1480	177	4	f(x	f(x	PROPN
cana-1480	177	5	)	)	PUNCT
cana-1480	178	1	=	=	SYM
cana-1480	178	2	𝑦1	𝑦1	PROPN
cana-1480	178	3	,	,	PUNCT
cana-1480	178	4	for	for	ADP
cana-1480	178	5	all	all	DET
cana-1480	178	6	x	x	PUNCT
cana-1480	178	7	in	in	ADP
cana-1480	178	8	u	u	PROPN
cana-1480	178	9	.if	.if	PUNCT
cana-1480	178	10	f(x	f(x	PROPN
cana-1480	178	11	)	)	PUNCT
cana-1480	178	12	=	=	SYM
cana-1480	178	13	𝑦0	𝑦0	NOUN
cana-1480	178	14	,	,	PUNCT
cana-1480	178	15	for	for	ADP
cana-1480	178	16	all	all	DET
cana-1480	178	17	x	x	PUNCT
cana-1480	178	18	in	in	ADP
cana-1480	178	19	u	u	NOUN
cana-1480	178	20	,	,	PUNCT
cana-1480	178	21	then	then	ADV
cana-1480	178	22	g	g	PROPN
cana-1480	178	23	=	=	SYM
cana-1480	178	24	u	u	PROPN
cana-1480	178	25	.if	.if	PROPN
cana-1480	178	26	f(x	f(x	PROPN
cana-1480	178	27	)	)	PUNCT
cana-1480	178	28	=	=	PUNCT
cana-1480	179	1	𝑦1,for	𝑦1,for	ADP
cana-1480	179	2	all	all	DET
cana-1480	179	3	x	x	NOUN
cana-1480	179	4	in	in	ADP
cana-1480	179	5	u	u	NOUN
cana-1480	179	6	,	,	PUNCT
cana-1480	179	7	then	then	ADV
cana-1480	179	8	g	g	PROPN
cana-1480	179	9	=	=	PROPN
cana-1480	179	10	ϕ.	ϕ.	PROPN
cana-1480	179	11	theorem	theorem	VERB
cana-1480	179	12	5.4	5.4	NUM
cana-1480	179	13	.	.	PUNCT
cana-1480	180	1	if	if	SCONJ
cana-1480	180	2	a	a	DET
cana-1480	180	3	space	space	NOUN
cana-1480	180	4	u	u	NOUN
cana-1480	180	5	is	be	AUX
cana-1480	180	6	nano*gα	nano*gα	PROPN
cana-1480	180	7	-connected	-connected	ADJ
cana-1480	180	8	space	space	NOUN
cana-1480	180	9	,	,	PUNCT
cana-1480	180	10	then	then	ADV
cana-1480	180	11	it	it	PRON
cana-1480	180	12	is	be	AUX
cana-1480	180	13	nano	nano	NOUN
cana-1480	180	14	connected	connect	VERB
cana-1480	180	15	.	.	PUNCT
cana-1480	181	1	proof	proof	NOUN
cana-1480	181	2	.	.	PUNCT
cana-1480	182	1	let	let	VERB
cana-1480	182	2	u	u	PRON
cana-1480	182	3	be	be	AUX
cana-1480	182	4	a	a	DET
cana-1480	182	5	nano*gα	nano*gα	ADJ
cana-1480	182	6	-connected	-connected	ADJ
cana-1480	182	7	space	space	NOUN
cana-1480	182	8	.	.	PUNCT
cana-1480	183	1	suppose	suppose	VERB
cana-1480	183	2	that	that	SCONJ
cana-1480	183	3	u	u	PROPN
cana-1480	183	4	is	be	AUX
cana-1480	183	5	not	not	PART
cana-1480	183	6	nano	nano	NOUN
cana-1480	183	7	connected	connect	VERB
cana-1480	183	8	then	then	ADV
cana-1480	183	9	u	u	X
cana-1480	183	10	=	=	NOUN
cana-1480	183	11	a	a	DET
cana-1480	183	12	∪	∪	X
cana-1480	183	13	b	b	NOUN
cana-1480	183	14	,	,	PUNCT
cana-1480	183	15	where	where	SCONJ
cana-1480	183	16	a	a	PRON
cana-1480	183	17	and	and	CCONJ
cana-1480	183	18	b	b	NOUN
cana-1480	183	19	are	be	AUX
cana-1480	183	20	disjoint	disjoint	VERB
cana-1480	183	21	non	non	ADJ
cana-1480	183	22	empty	empty	ADJ
cana-1480	183	23	nano	nano	NOUN
cana-1480	183	24	open	open	ADJ
cana-1480	183	25	sets	set	NOUN
cana-1480	183	26	in	in	ADP
cana-1480	183	27	u	u	PROPN
cana-1480	183	28	.	.	PUNCT
cana-1480	184	1	since	since	SCONJ
cana-1480	184	2	every	every	DET
cana-1480	184	3	nano	nano	NOUN
cana-1480	184	4	open	open	ADJ
cana-1480	184	5	set	set	NOUN
cana-1480	184	6	is	be	AUX
cana-1480	184	7	n	n	NUM
cana-1480	184	8	ano∗	ano∗	NOUN
cana-1480	184	9	gαopen	gαopen	NOUN
cana-1480	184	10	,	,	PUNCT
cana-1480	184	11	a	a	PRON
cana-1480	184	12	and	and	CCONJ
cana-1480	184	13	b	b	NOUN
cana-1480	184	14	are	be	AUX
cana-1480	184	15	disjoint	disjoint	X
cana-1480	184	16	non	non	X
cana-1480	184	17	empty	empty	ADJ
cana-1480	184	18	.	.	PUNCT
cana-1480	185	1	nano*gα	nano*gα	ADJ
cana-1480	185	2	open	open	ADJ
cana-1480	185	3	sets	set	NOUN
cana-1480	185	4	in	in	ADP
cana-1480	185	5	u	u	PROPN
cana-1480	185	6	.	.	PUNCT
cana-1480	186	1	this	this	PRON
cana-1480	186	2	contradicts	contradict	VERB
cana-1480	186	3	the	the	DET
cana-1480	186	4	fact	fact	NOUN
cana-1480	186	5	that	that	SCONJ
cana-1480	186	6	u	u	NOUN
cana-1480	186	7	is	be	AUX
cana-1480	186	8	nano*gα	nano*gα	PROPN
cana-1480	186	9	-connected	-connected	PROPN
cana-1480	186	10	.	.	PUNCT
cana-1480	187	1	hence	hence	ADV
cana-1480	187	2	u	u	PROPN
cana-1480	187	3	is	be	AUX
cana-1480	187	4	nano	nano	NOUN
cana-1480	187	5	connected	connect	VERB
cana-1480	187	6	.	.	PUNCT
cana-1480	188	1	example	example	NOUN
cana-1480	188	2	5.5	5.5	NUM
cana-1480	188	3	.	.	PUNCT
cana-1480	189	1	let	let	VERB
cana-1480	189	2	u	u	PRON
cana-1480	189	3	=	=	X
cana-1480	189	4	{	{	PUNCT
cana-1480	189	5	a	a	PRON
cana-1480	189	6	,	,	PUNCT
cana-1480	189	7	b	b	NOUN
cana-1480	189	8	,	,	PUNCT
cana-1480	189	9	c	c	NOUN
cana-1480	189	10	,	,	PUNCT
cana-1480	189	11	d	d	NOUN
cana-1480	189	12	}	}	PUNCT
cana-1480	189	13	and	and	CCONJ
cana-1480	189	14	u	u	NOUN
cana-1480	189	15	/r={a	/r={a	PUNCT
cana-1480	189	16	}	}	PUNCT
cana-1480	189	17	,	,	PUNCT
cana-1480	189	18	{	{	PUNCT
cana-1480	189	19	b	b	X
cana-1480	189	20	}	}	PUNCT
cana-1480	189	21	,	,	PUNCT
cana-1480	189	22	{	{	PUNCT
cana-1480	189	23	c	c	X
cana-1480	189	24	,	,	PUNCT
cana-1480	189	25	d	d	NOUN
cana-1480	189	26	}	}	PUNCT
cana-1480	189	27	and	and	CCONJ
cana-1480	189	28	x	x	SYM
cana-1480	189	29	=	=	X
cana-1480	189	30	{	{	PUNCT
cana-1480	189	31	a	a	X
cana-1480	189	32	,	,	PUNCT
cana-1480	189	33	d	d	NOUN
cana-1480	189	34	}	}	PUNCT
cana-1480	189	35	.	.	PUNCT
cana-1480	190	1	then	then	ADV
cana-1480	190	2	𝜏𝑅(𝑋)={u	𝜏𝑅(𝑋)={u	PROPN
cana-1480	190	3	,	,	PUNCT
cana-1480	190	4	ϕ	ϕ	PROPN
cana-1480	190	5	,	,	PUNCT
cana-1480	190	6	{	{	PUNCT
cana-1480	190	7	a	a	X
cana-1480	190	8	}	}	PUNCT
cana-1480	190	9	,	,	PUNCT
cana-1480	190	10	{	{	PUNCT
cana-1480	190	11	c	c	X
cana-1480	190	12	,	,	PUNCT
cana-1480	190	13	d	d	NOUN
cana-1480	190	14	}	}	PUNCT
cana-1480	190	15	,	,	PUNCT
cana-1480	190	16	{	{	PUNCT
cana-1480	190	17	a	a	PRON
cana-1480	190	18	,	,	PUNCT
cana-1480	190	19	c	c	NOUN
cana-1480	190	20	,	,	PUNCT
cana-1480	190	21	d	d	NOUN
cana-1480	190	22	}	}	PUNCT
cana-1480	190	23	}	}	PUNCT
cana-1480	190	24	.	.	PUNCT
cana-1480	191	1	then	then	ADV
cana-1480	191	2	nano*gα	nano*gα	PROPN
cana-1480	191	3	=	=	SYM
cana-1480	191	4	{	{	PUNCT
cana-1480	191	5	u	u	NOUN
cana-1480	191	6	,	,	PUNCT
cana-1480	191	7	ϕ	ϕ	PROPN
cana-1480	191	8	,	,	PUNCT
cana-1480	191	9	{	{	PUNCT
cana-1480	191	10	a	a	X
cana-1480	191	11	}	}	PUNCT
cana-1480	191	12	,	,	PUNCT
cana-1480	191	13	{	{	PUNCT
cana-1480	191	14	c	c	X
cana-1480	191	15	}	}	PUNCT
cana-1480	191	16	,	,	PUNCT
cana-1480	191	17	{	{	PUNCT
cana-1480	191	18	d	d	X
cana-1480	191	19	}	}	PUNCT
cana-1480	191	20	,	,	PUNCT
cana-1480	191	21	{	{	PUNCT
cana-1480	191	22	a	a	PRON
cana-1480	191	23	,	,	PUNCT
cana-1480	191	24	b	b	NOUN
cana-1480	191	25	}	}	PUNCT
cana-1480	191	26	,	,	PUNCT
cana-1480	191	27	{	{	PUNCT
cana-1480	191	28	a	a	PRON
cana-1480	191	29	,	,	PUNCT
cana-1480	191	30	d	d	NOUN
cana-1480	191	31	}	}	PUNCT
cana-1480	191	32	,	,	PUNCT
cana-1480	191	33	{	{	PUNCT
cana-1480	191	34	c	c	X
cana-1480	191	35	,	,	PUNCT
cana-1480	191	36	d	d	NOUN
cana-1480	191	37	}	}	PUNCT
cana-1480	191	38	,	,	PUNCT
cana-1480	191	39	{	{	PUNCT
cana-1480	191	40	a	a	DET
cana-1480	191	41	,	,	PUNCT
cana-1480	191	42	b	b	NOUN
cana-1480	191	43	,	,	PUNCT
cana-1480	191	44	c	c	NOUN
cana-1480	191	45	}	}	PUNCT
cana-1480	191	46	,	,	PUNCT
cana-1480	191	47	{	{	PUNCT
cana-1480	191	48	a	a	DET
cana-1480	191	49	,	,	PUNCT
cana-1480	191	50	b	b	NOUN
cana-1480	191	51	,	,	PUNCT
cana-1480	191	52	d	d	NOUN
cana-1480	191	53	}	}	PUNCT
cana-1480	191	54	,	,	PUNCT
cana-1480	191	55	{	{	PUNCT
cana-1480	191	56	a	a	PRON
cana-1480	191	57	,	,	PUNCT
cana-1480	191	58	c	c	NOUN
cana-1480	191	59	,	,	PUNCT
cana-1480	191	60	d	d	NOUN
cana-1480	191	61	}	}	PUNCT
cana-1480	191	62	,	,	PUNCT
cana-1480	191	63	{	{	PUNCT
cana-1480	191	64	b	b	X
cana-1480	191	65	,	,	PUNCT
cana-1480	191	66	c	c	NOUN
cana-1480	191	67	,	,	PUNCT
cana-1480	191	68	d	d	NOUN
cana-1480	191	69	}	}	PUNCT
cana-1480	191	70	}	}	PUNCT
cana-1480	191	71	.	.	PUNCT
cana-1480	192	1	here	here	ADV
cana-1480	192	2	u	u	NOUN
cana-1480	192	3	is	be	AUX
cana-1480	192	4	nano	nano	NOUN
cana-1480	192	5	connected	connect	VERB
cana-1480	192	6	but	but	CCONJ
cana-1480	192	7	not	not	PART
cana-1480	192	8	nano*gα	nano*gα	PROPN
cana-1480	192	9	-connected	-connected	ADJ
cana-1480	192	10	because	because	SCONJ
cana-1480	192	11	u	u	PRON
cana-1480	192	12	can	can	AUX
cana-1480	192	13	be	be	AUX
cana-1480	192	14	written	write	VERB
cana-1480	192	15	as	as	ADP
cana-1480	192	16	union	union	NOUN
cana-1480	192	17	of	of	ADP
cana-1480	192	18	two	two	NUM
cana-1480	192	19	disjoint	disjoint	ADJ
cana-1480	192	20	non	non	ADJ
cana-1480	192	21	-	-	ADJ
cana-1480	192	22	empty	empty	ADJ
cana-1480	192	23	nano*gα	nano*gα	ADJ
cana-1480	192	24	-open	-open	ADJ
cana-1480	192	25	sets	set	NOUN
cana-1480	192	26	{	{	PUNCT
cana-1480	192	27	d	d	NOUN
cana-1480	192	28	}	}	PUNCT
cana-1480	192	29	∪	∪	ADJ
cana-1480	192	30	{	{	PUNCT
cana-1480	192	31	a	a	PRON
cana-1480	192	32	,	,	PUNCT
cana-1480	192	33	b	b	NOUN
cana-1480	192	34	,	,	PUNCT
cana-1480	192	35	c	c	NOUN
cana-1480	192	36	}	}	PUNCT
cana-1480	192	37	.	.	PUNCT
cana-1480	193	1	theorem	theorem	VERB
cana-1480	193	2	5.6	5.6	NUM
cana-1480	193	3	.	.	PUNCT
cana-1480	194	1	if	if	SCONJ
cana-1480	194	2	𝑓	𝑓	ADV
cana-1480	194	3	:	:	PUNCT
cana-1480	194	4	(	(	PUNCT
cana-1480	194	5	𝑈	𝑈	PROPN
cana-1480	194	6	,	,	PUNCT
cana-1480	194	7	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	194	8	)	)	PUNCT
cana-1480	194	9	)	)	PUNCT
cana-1480	195	1	→	→	PUNCT
cana-1480	195	2	(	(	PUNCT
cana-1480	195	3	𝑉	𝑉	PROPN
cana-1480	195	4	,	,	PUNCT
cana-1480	195	5	𝜎𝑅(𝑌))is	𝜎𝑅(𝑌))i	VERB
cana-1480	195	6	nano*gα	nano*gα	PROPN
cana-1480	195	7	-irresolute	-irresolute	PROPN
cana-1480	195	8	surjection	surjection	NOUN
cana-1480	195	9	and	and	CCONJ
cana-1480	195	10	u	u	NOUN
cana-1480	195	11	is	be	AUX
cana-1480	195	12	nano*gα	nano*gα	PROPN
cana-1480	195	13	connected	connect	VERB
cana-1480	195	14	,	,	PUNCT
cana-1480	195	15	then	then	ADV
cana-1480	195	16	v	v	NOUN
cana-1480	195	17	is	be	AUX
cana-1480	195	18	nano*gα	nano*gα	PROPN
cana-1480	195	19	-connected	-connected	ADJ
cana-1480	195	20	.	.	PUNCT
cana-1480	196	1	proof	proof	NOUN
cana-1480	196	2	.	.	PUNCT
cana-1480	197	1	assume	assume	VERB
cana-1480	197	2	that	that	SCONJ
cana-1480	197	3	v	v	NOUN
cana-1480	197	4	is	be	AUX
cana-1480	197	5	not	not	PART
cana-1480	197	6	nano*gα	nano*gα	PROPN
cana-1480	197	7	-connected	-connected	PROPN
cana-1480	197	8	.	.	PUNCT
cana-1480	198	1	then	then	ADV
cana-1480	198	2	there	there	PRON
cana-1480	198	3	disjoint	disjoint	VERB
cana-1480	198	4	non	non	ADJ
cana-1480	198	5	empty	empty	ADJ
cana-1480	198	6	nano*gαopen	nano*gαopen	NOUN
cana-1480	198	7	sets	set	NOUN
cana-1480	198	8	a	a	PRON
cana-1480	198	9	and	and	CCONJ
cana-1480	198	10	b	b	NOUN
cana-1480	198	11	in	in	ADP
cana-1480	198	12	v	v	NUM
cana-1480	198	13	such	such	DET
cana-1480	198	14	that	that	PRON
cana-1480	198	15	v	v	NOUN
cana-1480	198	16	=	=	PUNCT
cana-1480	198	17	a	a	DET
cana-1480	198	18	∪	∪	X
cana-1480	198	19	b.	b.	NOUN
cana-1480	198	20	since	since	SCONJ
cana-1480	198	21	f	f	PROPN
cana-1480	198	22	is	be	AUX
cana-1480	198	23	nano*gα	nano*gα	PROPN
cana-1480	198	24	-irresolute	-irresolute	NOUN
cana-1480	198	25	,	,	PUNCT
cana-1480	198	26	𝑓−1	𝑓−1	NUM
cana-1480	198	27	(	(	PUNCT
cana-1480	198	28	a	a	NOUN
cana-1480	198	29	)	)	PUNCT
cana-1480	198	30	and	and	CCONJ
cana-1480	198	31	𝑓−1	𝑓−1	NUM
cana-1480	198	32	(	(	PUNCT
cana-1480	198	33	b	b	NOUN
cana-1480	198	34	)	)	PUNCT
cana-1480	198	35	are	be	AUX
cana-1480	198	36	v	v	ADP
cana-1480	198	37	nano*gα	nano*gα	ADJ
cana-1480	198	38	-open	-open	NOUN
cana-1480	198	39	sets	set	NOUN
cana-1480	198	40	in	in	ADP
cana-1480	198	41	u	u	NOUN
cana-1480	198	42	.	.	PUNCT
cana-1480	199	1	as	as	SCONJ
cana-1480	199	2	f	f	PROPN
cana-1480	199	3	is	be	AUX
cana-1480	199	4	a	a	DET
cana-1480	199	5	surjective	surjective	ADJ
cana-1480	199	6	function	function	NOUN
cana-1480	199	7	,	,	PUNCT
cana-1480	199	8	𝑓−1	𝑓−1	NUM
cana-1480	199	9	(	(	PUNCT
cana-1480	199	10	a	a	NOUN
cana-1480	199	11	)	)	PUNCT
cana-1480	199	12	≠	≠	PROPN
cana-1480	199	13	ϕ	ϕ	NOUN
cana-1480	199	14	and	and	CCONJ
cana-1480	199	15	𝑓−1	𝑓−1	NUM
cana-1480	199	16	(	(	PUNCT
cana-1480	199	17	b	b	NOUN
cana-1480	199	18	)	)	PUNCT
cana-1480	199	19	≠	≠	PROPN
cana-1480	199	20	ϕ	ϕ	PROPN
cana-1480	199	21	,	,	PUNCT
cana-1480	199	22	where	where	SCONJ
cana-1480	199	23	u	u	NOUN
cana-1480	199	24	=	=	NOUN
cana-1480	199	25	𝑓−1	𝑓−1	PROPN
cana-1480	199	26	(	(	PUNCT
cana-1480	199	27	v	v	NOUN
cana-1480	199	28	)	)	PUNCT
cana-1480	199	29	=	=	SYM
cana-1480	200	1	𝑓−1	𝑓−1	PROPN
cana-1480	200	2	(	(	PUNCT
cana-1480	200	3	a	a	DET
cana-1480	200	4	∪	∪	ADJ
cana-1480	200	5	b	b	NOUN
cana-1480	200	6	)	)	PUNCT
cana-1480	200	7	=	=	SYM
cana-1480	201	1	𝑓−1	𝑓−1	PROPN
cana-1480	201	2	(	(	PUNCT
cana-1480	201	3	a)−1	a)−1	PROPN
cana-1480	201	4	(	(	PUNCT
cana-1480	201	5	b	b	NOUN
cana-1480	201	6	)	)	PUNCT
cana-1480	201	7	which	which	PRON
cana-1480	201	8	is	be	AUX
cana-1480	201	9	a	a	DET
cana-1480	201	10	contradiction	contradiction	NOUN
cana-1480	201	11	.	.	PUNCT
cana-1480	202	1	this	this	PRON
cana-1480	202	2	shows	show	VERB
cana-1480	202	3	that	that	SCONJ
cana-1480	202	4	v	v	NOUN
cana-1480	202	5	is	be	AUX
cana-1480	202	6	nano*gα	nano*gα	PROPN
cana-1480	202	7	connected	connect	VERB
cana-1480	202	8	.	.	PUNCT
cana-1480	203	1	theorem	theorem	VERB
cana-1480	203	2	5.7	5.7	NUM
cana-1480	203	3	.	.	PUNCT
cana-1480	204	1	if	if	SCONJ
cana-1480	204	2	g	g	PROPN
cana-1480	204	3	is	be	AUX
cana-1480	204	4	a	a	DET
cana-1480	204	5	nano	nano	NOUN
cana-1480	204	6	-	-	ADJ
cana-1480	204	7	compact	compact	ADJ
cana-1480	204	8	of	of	ADP
cana-1480	204	9	a	a	DET
cana-1480	204	10	nano*gα	nano*gα	PROPN
cana-1480	204	11	-connected	-connected	ADJ
cana-1480	204	12	space	space	NOUN
cana-1480	204	13	(	(	PUNCT
cana-1480	204	14	𝑈	𝑈	PROPN
cana-1480	204	15	,	,	PUNCT
cana-1480	204	16	𝜏𝑅(𝑋	𝜏𝑅(𝑋	NOUN
cana-1480	204	17	)	)	PUNCT
cana-1480	204	18	)	)	PUNCT
cana-1480	204	19	onto	onto	ADP
cana-1480	204	20	an	an	DET
cana-1480	204	21	arbitrary	arbitrary	ADJ
cana-1480	204	22	nts	nt	NOUN
cana-1480	204	23	(	(	PUNCT
cana-1480	204	24	𝑉	𝑉	PROPN
cana-1480	204	25	,	,	PUNCT
cana-1480	204	26	𝜎𝑅(𝑌	𝜎𝑅(𝑌	NOUN
cana-1480	204	27	)	)	PUNCT
cana-1480	204	28	)	)	PUNCT
cana-1480	204	29	,	,	PUNCT
cana-1480	204	30	then	then	ADV
cana-1480	204	31	(	(	PUNCT
cana-1480	204	32	𝑉	𝑉	PROPN
cana-1480	204	33	,	,	PUNCT
cana-1480	204	34	𝜎𝑅(𝑌))is	𝜎𝑅(𝑌))i	VERB
cana-1480	204	35	nano*gα	nano*gα	PROPN
cana-1480	204	36	-connected	-connected	PROPN
cana-1480	204	37	.	.	PUNCT
cana-1480	205	1	communications	communication	NOUN
cana-1480	205	2	on	on	ADP
cana-1480	205	3	applied	apply	VERB
cana-1480	205	4	nonlinear	nonlinear	ADJ
cana-1480	205	5	analysis	analysis	NOUN
cana-1480	205	6	issn	issn	NOUN
cana-1480	205	7	:	:	PUNCT
cana-1480	205	8	1074	1074	NUM
cana-1480	205	9	-	-	PUNCT
cana-1480	205	10	133x	133x	NUM
cana-1480	205	11	vol	vol	NOUN
cana-1480	205	12	31	31	NUM
cana-1480	205	13	no	no	NOUN
cana-1480	205	14	.	.	PUNCT
cana-1480	206	1	8s	8s	PROPN
cana-1480	206	2	(	(	PUNCT
cana-1480	206	3	2024	2024	NUM
cana-1480	206	4	)	)	PUNCT
cana-1480	206	5	258	258	NUM
cana-1480	206	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1480	206	7	proof	proof	NOUN
cana-1480	206	8	.	.	PUNCT
cana-1480	207	1	let	let	VERB
cana-1480	207	2	(	(	PUNCT
cana-1480	207	3	𝑉	𝑉	PROPN
cana-1480	207	4	,	,	PUNCT
cana-1480	207	5	𝜎𝑅(𝑌))be	𝜎𝑅(𝑌))be	ADJ
cana-1480	207	6	a	a	DET
cana-1480	207	7	nano*gα	nano*gα	PROPN
cana-1480	207	8	-connected	-connected	PROPN
cana-1480	207	9	.	.	PUNCT
cana-1480	208	1	then	then	ADV
cana-1480	208	2	there	there	PRON
cana-1480	208	3	exists	exist	VERB
cana-1480	208	4	a	a	DET
cana-1480	208	5	non	non	ADJ
cana-1480	208	6	-	-	ADJ
cana-1480	208	7	empty	empty	ADJ
cana-1480	208	8	proper	proper	ADJ
cana-1480	208	9	subset	subset	NOUN
cana-1480	208	10	g	g	PROPN
cana-1480	208	11	of	of	ADP
cana-1480	208	12	(	(	PUNCT
cana-1480	208	13	𝑉	𝑉	PROPN
cana-1480	208	14	,	,	PUNCT
cana-1480	208	15	𝜎𝑅(𝑌))which	𝜎𝑅(𝑌))which	PROPN
cana-1480	208	16	is	be	AUX
cana-1480	208	17	both	both	PRON
cana-1480	208	18	nano*gα	nano*gα	PROPN
cana-1480	208	19	-open	-open	PROPN
cana-1480	208	20	and	and	CCONJ
cana-1480	208	21	nano*gα	nano*gα	PROPN
cana-1480	208	22	-closed	-close	VERB
cana-1480	208	23	in	in	ADP
cana-1480	208	24	(	(	PUNCT
cana-1480	208	25	𝑉	𝑉	PROPN
cana-1480	208	26	,	,	PUNCT
cana-1480	208	27	𝜎𝑅(𝑌))since	𝜎𝑅(𝑌))since	PROPN
cana-1480	208	28	f	f	PROPN
cana-1480	208	29	is	be	AUX
cana-1480	208	30	nano*gα	nano*gα	PROPN
cana-1480	208	31	continuous	continuous	ADJ
cana-1480	208	32	and	and	CCONJ
cana-1480	208	33	onto	onto	ADP
cana-1480	208	34	(	(	PUNCT
cana-1480	208	35	𝑉	𝑉	PROPN
cana-1480	208	36	,	,	PUNCT
cana-1480	208	37	𝜎𝑅(𝑌))f1(g	𝜎𝑅(𝑌))f1(g	NUM
cana-1480	208	38	)	)	PUNCT
cana-1480	208	39	is	be	AUX
cana-1480	208	40	a	a	DET
cana-1480	208	41	non	non	ADJ
cana-1480	208	42	-	-	ADJ
cana-1480	208	43	empty	empty	ADJ
cana-1480	208	44	proper	proper	ADJ
cana-1480	208	45	subset	subset	NOUN
cana-1480	208	46	of	of	ADP
cana-1480	208	47	(	(	PUNCT
cana-1480	208	48	𝑈	𝑈	PROPN
cana-1480	208	49	,	,	PUNCT
cana-1480	208	50	𝜏𝑅(𝑋))which	𝜏𝑅(𝑋))which	PROPN
cana-1480	208	51	is	be	AUX
cana-1480	208	52	both	both	PRON
cana-1480	208	53	nano*gα	nano*gα	PROPN
cana-1480	208	54	-open	-open	PROPN
cana-1480	208	55	and	and	CCONJ
cana-1480	208	56	nano*gα	nano*gα	PROPN
cana-1480	208	57	-closed	-close	VERB
cana-1480	208	58	in	in	ADP
cana-1480	208	59	(	(	PUNCT
cana-1480	208	60	𝑈	𝑈	PROPN
cana-1480	208	61	,	,	PUNCT
cana-1480	208	62	𝜏𝑅(𝑋))and	𝜏𝑅(𝑋))and	X
cana-1480	209	1	therefore	therefore	ADV
cana-1480	209	2	(	(	PUNCT
cana-1480	209	3	𝑈	𝑈	PROPN
cana-1480	209	4	,	,	PUNCT
cana-1480	209	5	𝜏𝑅(𝑋	𝜏𝑅(𝑋	PROPN
cana-1480	209	6	)	)	PUNCT
cana-1480	209	7	)	)	PUNCT
cana-1480	209	8	is	be	AUX
cana-1480	209	9	disconnected	disconnect	VERB
cana-1480	209	10	which	which	PRON
cana-1480	209	11	is	be	AUX
cana-1480	209	12	a	a	DET
cana-1480	209	13	contradiction	contradiction	NOUN
cana-1480	209	14	.	.	PUNCT
cana-1480	210	1	hence	hence	ADV
cana-1480	210	2	(	(	PUNCT
cana-1480	210	3	𝑉	𝑉	PROPN
cana-1480	210	4	,	,	PUNCT
cana-1480	210	5	𝜎𝑅(𝑌))must	𝜎𝑅(𝑌))must	PROPN
cana-1480	210	6	be	be	AUX
cana-1480	210	7	connected	connect	VERB
cana-1480	210	8	.	.	PUNCT
cana-1480	211	1	theorem	theorem	VERB
cana-1480	211	2	5.8	5.8	NUM
cana-1480	211	3	.	.	PUNCT
cana-1480	212	1	a	a	DET
cana-1480	212	2	space	space	NOUN
cana-1480	212	3	u	u	NOUN
cana-1480	212	4	is	be	AUX
cana-1480	212	5	nano*gα	nano*gα	PROPN
cana-1480	212	6	–	–	PUNCT
cana-1480	212	7	disconnected	disconnected	ADJ
cana-1480	212	8	if	if	SCONJ
cana-1480	212	9	and	and	CCONJ
cana-1480	212	10	only	only	ADV
cana-1480	212	11	if	if	SCONJ
cana-1480	212	12	there	there	PRON
cana-1480	212	13	exists	exist	VERB
cana-1480	212	14	a	a	DET
cana-1480	212	15	non	non	ADJ
cana-1480	212	16	-	-	ADJ
cana-1480	212	17	empty	empty	ADJ
cana-1480	212	18	proper	proper	ADJ
cana-1480	212	19	subset	subset	NOUN
cana-1480	212	20	of	of	ADP
cana-1480	212	21	u	u	PRON
cana-1480	212	22	which	which	PRON
cana-1480	212	23	is	be	AUX
cana-1480	212	24	both	both	PRON
cana-1480	212	25	nano*gα	nano*gα	PROPN
cana-1480	212	26	-open	-open	PROPN
cana-1480	212	27	and	and	CCONJ
cana-1480	212	28	nano*gα	nano*gα	PROPN
cana-1480	212	29	-closed	-close	VERB
cana-1480	212	30	in	in	ADP
cana-1480	212	31	u	u	NOUN
cana-1480	212	32	.	.	PUNCT
cana-1480	213	1	proof	proof	NOUN
cana-1480	213	2	.	.	PUNCT
cana-1480	214	1	let	let	VERB
cana-1480	214	2	g	g	PRON
cana-1480	214	3	be	be	AUX
cana-1480	214	4	a	a	DET
cana-1480	214	5	non	non	ADJ
cana-1480	214	6	-	-	ADJ
cana-1480	214	7	empty	empty	ADJ
cana-1480	214	8	proper	proper	ADJ
cana-1480	214	9	subset	subset	NOUN
cana-1480	214	10	of	of	ADP
cana-1480	214	11	u	u	PRON
cana-1480	214	12	which	which	PRON
cana-1480	214	13	is	be	AUX
cana-1480	214	14	both	both	PRON
cana-1480	214	15	nano*gα	nano*gα	ADJ
cana-1480	214	16	–	–	PUNCT
cana-1480	214	17	open	open	ADJ
cana-1480	214	18	and	and	CCONJ
cana-1480	214	19	nano*gαclosed	nano*gαclosed	ADJ
cana-1480	214	20	.	.	PUNCT
cana-1480	215	1	we	we	PRON
cana-1480	215	2	have	have	VERB
cana-1480	215	3	to	to	PART
cana-1480	215	4	prove	prove	VERB
cana-1480	215	5	that	that	SCONJ
cana-1480	215	6	u	u	NOUN
cana-1480	215	7	is	be	AUX
cana-1480	215	8	nano*gα	nano*gα	PROPN
cana-1480	215	9	-disconnected	-disconnected	ADJ
cana-1480	215	10	.	.	PUNCT
cana-1480	216	1	let	let	VERB
cana-1480	216	2	f	f	NOUN
cana-1480	216	3	=	=	SYM
cana-1480	216	4	u	u	PROPN
cana-1480	216	5	−	−	PROPN
cana-1480	216	6	g	g	PROPN
cana-1480	216	7	.	.	PUNCT
cana-1480	217	1	then	then	ADV
cana-1480	217	2	f	f	PROPN
cana-1480	217	3	is	be	AUX
cana-1480	217	4	a	a	DET
cana-1480	217	5	non	non	ADJ
cana-1480	217	6	-	-	ADJ
cana-1480	217	7	empty	empty	ADJ
cana-1480	217	8	set	set	NOUN
cana-1480	217	9	and	and	CCONJ
cana-1480	217	10	g	g	NOUN
cana-1480	217	11	∪	∪	NOUN
cana-1480	217	12	f	f	PROPN
cana-1480	217	13	=	=	SYM
cana-1480	217	14	u	u	PROPN
cana-1480	217	15	and	and	CCONJ
cana-1480	217	16	g	g	NOUN
cana-1480	217	17	∩	∩	ADJ
cana-1480	217	18	f	f	NOUN
cana-1480	217	19	=	=	PUNCT
cana-1480	217	20	ϕ.since	ϕ.since	NOUN
cana-1480	217	21	g	g	NOUN
cana-1480	217	22	is	be	AUX
cana-1480	217	23	both	both	CCONJ
cana-1480	217	24	nano*gα	nano*gα	PROPN
cana-1480	217	25	-open	-open	PROPN
cana-1480	217	26	and	and	CCONJ
cana-1480	217	27	n	n	NOUN
cana-1480	217	28	ano∗	ano∗	NOUN
cana-1480	217	29	gα	gα	NOUN
cana-1480	217	30	-	-	PUNCT
cana-1480	217	31	closed	closed	ADJ
cana-1480	217	32	,	,	PUNCT
cana-1480	217	33	f	f	PROPN
cana-1480	217	34	is	be	AUX
cana-1480	217	35	both	both	CCONJ
cana-1480	217	36	nano*gα	nano*gα	PROPN
cana-1480	217	37	open	open	ADJ
cana-1480	217	38	and	and	CCONJ
cana-1480	217	39	nano*gα	nano*gα	PROPN
cana-1480	217	40	-closed	-closed	PROPN
cana-1480	217	41	.	.	PUNCT
cana-1480	218	1	thus	thus	ADV
cana-1480	218	2	u	u	PRON
cana-1480	218	3	can	can	AUX
cana-1480	218	4	be	be	AUX
cana-1480	218	5	written	write	VERB
cana-1480	218	6	as	as	ADP
cana-1480	218	7	the	the	DET
cana-1480	218	8	union	union	NOUN
cana-1480	218	9	of	of	ADP
cana-1480	218	10	two	two	NUM
cana-1480	218	11	disjoint	disjoint	ADJ
cana-1480	218	12	non	non	ADJ
cana-1480	218	13	-	-	ADJ
cana-1480	218	14	empty	empty	ADJ
cana-1480	218	15	nano*gα	nano*gα	ADJ
cana-1480	218	16	open	open	ADJ
cana-1480	218	17	sets	set	NOUN
cana-1480	218	18	.	.	PUNCT
cana-1480	219	1	hence	hence	ADV
cana-1480	219	2	u	u	PROPN
cana-1480	219	3	is	be	AUX
cana-1480	219	4	nano*gα	nano*gα	PROPN
cana-1480	219	5	disconnected	disconnected	ADJ
cana-1480	219	6	.	.	PUNCT
cana-1480	220	1	conversely	conversely	ADV
cana-1480	220	2	,	,	PUNCT
cana-1480	220	3	let	let	VERB
cana-1480	220	4	u	u	PRON
cana-1480	220	5	be	be	AUX
cana-1480	220	6	nano*gα	nano*gα	PROPN
cana-1480	220	7	-disconnected	-disconnected	ADJ
cana-1480	220	8	.	.	PUNCT
cana-1480	221	1	then	then	ADV
cana-1480	221	2	there	there	PRON
cana-1480	221	3	exist	exist	VERB
cana-1480	221	4	non	non	ADJ
cana-1480	221	5	-	-	ADJ
cana-1480	221	6	empty	empty	ADJ
cana-1480	221	7	nano*gα	nano*gα	ADJ
cana-1480	221	8	-open	-open	ADJ
cana-1480	221	9	subsets	subset	NOUN
cana-1480	221	10	g	g	NOUN
cana-1480	221	11	and	and	CCONJ
cana-1480	221	12	f	f	PROPN
cana-1480	221	13	such	such	ADJ
cana-1480	221	14	that	that	SCONJ
cana-1480	221	15	x	x	X
cana-1480	221	16	=	=	PUNCT
cana-1480	221	17	g	g	PROPN
cana-1480	221	18	∪	∪	PROPN
cana-1480	221	19	f	f	PROPN
cana-1480	221	20	.	.	PUNCT
cana-1480	222	1	then	then	ADV
cana-1480	222	2	f	f	PROPN
cana-1480	222	3	=	=	SYM
cana-1480	222	4	u	u	PROPN
cana-1480	222	5	−	−	PROPN
cana-1480	222	6	g	g	PROPN
cana-1480	222	7	and	and	CCONJ
cana-1480	222	8	g	g	NOUN
cana-1480	222	9	=	=	SYM
cana-1480	222	10	u	u	PROPN
cana-1480	223	1	−	−	PROPN
cana-1480	223	2	f	f	NOUN
cana-1480	223	3	,	,	PUNCT
cana-1480	223	4	which	which	PRON
cana-1480	223	5	are	be	AUX
cana-1480	223	6	nano*gα	nano*gα	PROPN
cana-1480	223	7	-closed	-closed	ADJ
cana-1480	223	8	in	in	ADP
cana-1480	223	9	u	u	PROPN
cana-1480	223	10	.	.	PUNCT
cana-1480	224	1	hence	hence	ADV
cana-1480	224	2	gand	gand	PROPN
cana-1480	224	3	f	f	PROPN
cana-1480	224	4	are	be	AUX
cana-1480	224	5	both	both	PRON
cana-1480	224	6	nano*gα	nano*gα	PROPN
cana-1480	224	7	-open	-open	PROPN
cana-1480	224	8	and	and	CCONJ
cana-1480	224	9	nano*gα	nano*gα	PROPN
cana-1480	224	10	closed	close	VERB
cana-1480	224	11	in	in	ADP
cana-1480	224	12	u.	u.	NOUN
cana-1480	224	13	theorem	theorem	PROPN
cana-1480	224	14	5.9	5.9	NUM
cana-1480	224	15	.	.	PUNCT
cana-1480	225	1	let	let	VERB
cana-1480	225	2	(	(	PUNCT
cana-1480	225	3	𝑈	𝑈	PROPN
cana-1480	225	4	,	,	PUNCT
cana-1480	225	5	𝜏𝑅(𝑋))be	𝜏𝑅(𝑋))be	NOUN
cana-1480	225	6	a	a	DET
cana-1480	225	7	nts	nt	NOUN
cana-1480	225	8	and	and	CCONJ
cana-1480	225	9	let	let	VERB
cana-1480	225	10	a	a	PRON
cana-1480	225	11	be	be	AUX
cana-1480	225	12	a	a	DET
cana-1480	225	13	subset	subset	NOUN
cana-1480	225	14	of	of	ADP
cana-1480	225	15	u	u	PROPN
cana-1480	225	16	.	.	PUNCT
cana-1480	226	1	then	then	ADV
cana-1480	226	2	a	a	PRON
cana-1480	226	3	is	be	AUX
cana-1480	226	4	nano	nano	NOUN
cana-1480	226	5	disconnected	disconnect	VERB
cana-1480	226	6	if	if	SCONJ
cana-1480	226	7	and	and	CCONJ
cana-1480	226	8	only	only	ADV
cana-1480	226	9	if	if	SCONJ
cana-1480	226	10	there	there	PRON
cana-1480	226	11	exist	exist	VERB
cana-1480	226	12	non	non	ADJ
cana-1480	226	13	-	-	ADJ
cana-1480	226	14	empty	empty	ADJ
cana-1480	226	15	sets	set	NOUN
cana-1480	226	16	g	g	PROPN
cana-1480	226	17	and	and	CCONJ
cana-1480	226	18	f	f	PROPN
cana-1480	226	19	both	both	CCONJ
cana-1480	226	20	nano*gα	nano*gα	PROPN
cana-1480	226	21	–	–	PUNCT
cana-1480	226	22	open	open	ADJ
cana-1480	226	23	in	in	ADP
cana-1480	226	24	u	u	PRON
cana-1480	226	25	such	such	ADJ
cana-1480	226	26	that	that	SCONJ
cana-1480	226	27	g	g	PROPN
cana-1480	226	28	∩	∩	NOUN
cana-1480	226	29	a	a	DET
cana-1480	226	30	≠	≠	PROPN
cana-1480	226	31	ϕ	ϕ	PROPN
cana-1480	226	32	,	,	PUNCT
cana-1480	226	33	f	f	PROPN
cana-1480	226	34	∩	∩	PROPN
cana-1480	226	35	a	a	DET
cana-1480	226	36	≠	≠	PROPN
cana-1480	226	37	ϕ	ϕ	NOUN
cana-1480	226	38	,	,	PUNCT
cana-1480	226	39	a	a	DET
cana-1480	226	40	⊆	⊆	NUM
cana-1480	226	41	g	g	NOUN
cana-1480	226	42	∪	∪	ADJ
cana-1480	226	43	f	f	PROPN
cana-1480	226	44	and	and	CCONJ
cana-1480	226	45	g	g	PROPN
cana-1480	226	46	∩	∩	NOUN
cana-1480	226	47	f	f	PROPN
cana-1480	226	48	⊆	⊆	NUM
cana-1480	226	49	u	u	NOUN
cana-1480	226	50	−	−	PROPN
cana-1480	226	51	a	a	DET
cana-1480	226	52	.	.	PUNCT
cana-1480	227	1	proof	proof	NOUN
cana-1480	227	2	.	.	PUNCT
cana-1480	228	1	a	a	PRON
cana-1480	228	2	is	be	AUX
cana-1480	228	3	nano*gα	nano*gα	PROPN
cana-1480	228	4	-disconnected	-disconnected	ADJ
cana-1480	228	5	if	if	SCONJ
cana-1480	228	6	and	and	CCONJ
cana-1480	228	7	only	only	ADV
cana-1480	228	8	if	if	SCONJ
cana-1480	228	9	there	there	PRON
cana-1480	228	10	exist	exist	VERB
cana-1480	228	11	non	non	ADJ
cana-1480	228	12	empty	empty	ADJ
cana-1480	228	13	sets	set	NOUN
cana-1480	228	14	g	g	PROPN
cana-1480	228	15	and	and	CCONJ
cana-1480	228	16	f	f	PROPN
cana-1480	228	17	both	both	CCONJ
cana-1480	228	18	nano*gα	nano*gα	PROPN
cana-1480	228	19	open	open	ADJ
cana-1480	228	20	in	in	ADP
cana-1480	228	21	u	u	PRON
cana-1480	228	22	such	such	ADJ
cana-1480	228	23	that	that	SCONJ
cana-1480	228	24	g	g	PROPN
cana-1480	228	25	∩	∩	NOUN
cana-1480	228	26	a	a	DET
cana-1480	228	27	≠ϕ	≠ϕ	NOUN
cana-1480	228	28	,	,	PUNCT
cana-1480	228	29	f	f	PROPN
cana-1480	228	30	∩	∩	PROPN
cana-1480	228	31	a	a	DET
cana-1480	228	32	≠ϕ	≠ϕ	PROPN
cana-1480	228	33	,	,	PUNCT
cana-1480	228	34	(	(	PUNCT
cana-1480	228	35	g	g	PROPN
cana-1480	228	36	∩	∩	PROPN
cana-1480	228	37	a	a	PRON
cana-1480	228	38	)	)	PUNCT
cana-1480	228	39	t	t	PROPN
cana-1480	228	40	(	(	PUNCT
cana-1480	228	41	f	f	PROPN
cana-1480	228	42	∩	∩	PROPN
cana-1480	228	43	a	a	X
cana-1480	228	44	)	)	PUNCT
cana-1480	228	45	=	=	SYM
cana-1480	228	46	a	a	PRON
cana-1480	228	47	.	.	PUNCT
cana-1480	229	1	now	now	ADV
cana-1480	229	2	(	(	PUNCT
cana-1480	229	3	g	g	PROPN
cana-1480	229	4	∩	∩	ADJ
cana-1480	229	5	a	a	PRON
cana-1480	229	6	)	)	PUNCT
cana-1480	229	7	∩	∩	NOUN
cana-1480	229	8	(	(	PUNCT
cana-1480	229	9	f	f	PROPN
cana-1480	229	10	∩	∩	PROPN
cana-1480	229	11	a	a	X
cana-1480	229	12	)	)	PUNCT
cana-1480	229	13	=	=	SYM
cana-1480	229	14	ϕ	ϕ	NOUN
cana-1480	230	1	if	if	SCONJ
cana-1480	230	2	and	and	CCONJ
cana-1480	230	3	only	only	ADV
cana-1480	230	4	if	if	SCONJ
cana-1480	230	5	(	(	PUNCT
cana-1480	230	6	g	g	PROPN
cana-1480	230	7	∩	∩	NOUN
cana-1480	230	8	a	a	PRON
cana-1480	230	9	)	)	PUNCT
cana-1480	230	10	∩	∩	NOUN
cana-1480	230	11	a	a	X
cana-1480	230	12	)	)	PUNCT
cana-1480	230	13	=	=	SYM
cana-1480	230	14	ϕ	ϕ	NOUN
cana-1480	231	1	if	if	SCONJ
cana-1480	232	1	and	and	CCONJ
cana-1480	232	2	only	only	ADV
cana-1480	232	3	if	if	SCONJ
cana-1480	232	4	g	g	PROPN
cana-1480	232	5	∩f	∩f	PROPN
cana-1480	232	6	⊆	⊆	NUM
cana-1480	232	7	u	u	NOUN
cana-1480	232	8	−	−	PROPN
cana-1480	232	9	a	a	PRON
cana-1480	232	10	and	and	CCONJ
cana-1480	232	11	(	(	PUNCT
cana-1480	232	12	g	g	PROPN
cana-1480	232	13	∩	∩	NOUN
cana-1480	232	14	a	a	PRON
cana-1480	232	15	)	)	PUNCT
cana-1480	232	16	∪	∪	NOUN
cana-1480	232	17	(	(	PUNCT
cana-1480	232	18	f	f	PROPN
cana-1480	232	19	∩	∩	NOUN
cana-1480	232	20	a	a	X
cana-1480	232	21	)	)	PUNCT
cana-1480	232	22	=	=	PUNCT
cana-1480	233	1	a	a	DET
cana-1480	233	2	if	if	NOUN
cana-1480	233	3	and	and	CCONJ
cana-1480	233	4	only	only	ADV
cana-1480	233	5	if	if	SCONJ
cana-1480	233	6	(	(	PUNCT
cana-1480	233	7	g	g	NOUN
cana-1480	233	8	∪	∪	ADJ
cana-1480	233	9	)	)	PUNCT
cana-1480	233	10	∩	∩	NOUN
cana-1480	233	11	a	a	PRON
cana-1480	233	12	=	=	SYM
cana-1480	233	13	a	a	DET
cana-1480	233	14	if	if	NOUN
cana-1480	234	1	and	and	CCONJ
cana-1480	234	2	only	only	ADV
cana-1480	234	3	if	if	SCONJ
cana-1480	234	4	a	a	DET
cana-1480	234	5	⊆	⊆	NUM
cana-1480	234	6	g	g	NOUN
cana-1480	234	7	∪	∪	PROPN
cana-1480	234	8	f	f	PROPN
cana-1480	234	9	.	.	PUNCT
cana-1480	235	1	references	reference	NOUN
cana-1480	235	2	[	[	X
cana-1480	235	3	1	1	NUM
cana-1480	235	4	]	]	PUNCT
cana-1480	235	5	k.bhuvaneswari	k.bhuvaneswari	PROPN
cana-1480	235	6	and	and	CCONJ
cana-1480	235	7	k.m.gnanapriya	k.m.gnanapriya	PROPN
cana-1480	235	8	,	,	PUNCT
cana-1480	235	9	nano	nano	NOUN
cana-1480	235	10	generalized	generalize	VERB
cana-1480	235	11	closed	closed	ADJ
cana-1480	235	12	sets	set	NOUN
cana-1480	235	13	in	in	ADP
cana-1480	235	14	nts	nt	NOUN
cana-1480	235	15	,	,	PUNCT
cana-1480	235	16	international	international	ADJ
cana-1480	235	17	journal	journal	NOUN
cana-1480	235	18	of	of	ADP
cana-1480	235	19	scientific	scientific	ADJ
cana-1480	235	20	and	and	CCONJ
cana-1480	235	21	research	research	NOUN
cana-1480	235	22	publication,4(5)(2014	publication,4(5)(2014	PROPN
cana-1480	235	23	)	)	PUNCT
cana-1480	235	24	,	,	PUNCT
cana-1480	235	25	1	1	NUM
cana-1480	235	26	−	−	NOUN
cana-1480	235	27	3	3	NUM
cana-1480	235	28	.	.	PUNCT
cana-1480	236	1	[	[	X
cana-1480	236	2	2	2	NUM
cana-1480	236	3	]	]	PUNCT
cana-1480	236	4	k.bhuvaneswari	k.bhuvaneswari	PROPN
cana-1480	236	5	and	and	CCONJ
cana-1480	236	6	k.m.gnanapriya	k.m.gnanapriya	PROPN
cana-1480	236	7	,	,	PUNCT
cana-1480	236	8	on	on	ADP
cana-1480	236	9	nano	nano	NOUN
cana-1480	236	10	generalized	generalize	VERB
cana-1480	236	11	continuous	continuous	ADJ
cana-1480	236	12	function	function	NOUN
cana-1480	236	13	in	in	ADP
cana-1480	236	14	nts	nt	NOUN
cana-1480	236	15	,	,	PUNCT
cana-1480	236	16	international	international	ADJ
cana-1480	236	17	journal	journal	NOUN
cana-1480	236	18	of	of	ADP
cana-1480	236	19	mathematics	mathematics	PROPN
cana-1480	236	20	and	and	CCONJ
cana-1480	236	21	statistics	statistic	NOUN
cana-1480	236	22	invention	invention	NOUN
cana-1480	236	23	,	,	PUNCT
cana-1480	236	24	1(1)(2013),31	1(1)(2013),31	NUM
cana-1480	236	25	-	-	SYM
cana-1480	236	26	37	37	NUM
cana-1480	236	27	.	.	PUNCT
cana-1480	237	1	[	[	X
cana-1480	237	2	3	3	X
cana-1480	237	3	]	]	SYM
cana-1480	237	4	m.vigneshwaran	m.vigneshwaran	NOUN
cana-1480	237	5	and	and	CCONJ
cana-1480	237	6	r.devi	r.devi	ADJ
cana-1480	237	7	,	,	PUNCT
cana-1480	237	8	on	on	ADP
cana-1480	237	9	gao	gao	PROPN
cana-1480	237	10	-	-	PUNCT
cana-1480	237	11	kernel	kernel	PROPN
cana-1480	237	12	in	in	ADP
cana-1480	237	13	the	the	DET
cana-1480	237	14	digital	digital	ADJ
cana-1480	237	15	plane	plane	NOUN
cana-1480	237	16	,	,	PUNCT
cana-1480	237	17	international	international	ADJ
cana-1480	237	18	journal	journal	NOUN
cana-1480	237	19	of	of	ADP
cana-1480	237	20	mathematical	mathematical	ADJ
cana-1480	237	21	archive3(6	archive3(6	PROPN
cana-1480	237	22	)	)	PUNCT
cana-1480	237	23	,	,	PUNCT
cana-1480	237	24	2012	2012	NUM
cana-1480	237	25	[	[	X
cana-1480	237	26	4	4	X
cana-1480	237	27	]	]	PUNCT
cana-1480	237	28	m.lellis	m.lelli	VERB
cana-1480	237	29	thivagar	thivagar	NOUN
cana-1480	237	30	and	and	CCONJ
cana-1480	237	31	carmel	carmel	PROPN
cana-1480	237	32	richard	richard	PROPN
cana-1480	237	33	,	,	PUNCT
cana-1480	237	34	on	on	ADP
cana-1480	237	35	nano	nano	NOUN
cana-1480	237	36	forms	form	NOUN
cana-1480	237	37	of	of	ADP
cana-1480	237	38	weakly	weakly	ADJ
cana-1480	237	39	open	open	ADJ
cana-1480	237	40	sets	set	NOUN
cana-1480	237	41	,	,	PUNCT
cana-1480	237	42	international	international	ADJ
cana-1480	237	43	journal	journal	NOUN
cana-1480	237	44	of	of	ADP
cana-1480	237	45	mathematics	mathematics	PROPN
cana-1480	237	46	and	and	CCONJ
cana-1480	237	47	statistics	statistic	NOUN
cana-1480	237	48	invention	invention	NOUN
cana-1480	237	49	,	,	PUNCT
cana-1480	237	50	volume	volume	NOUN
cana-1480	237	51	1,august	1,august	NUM
cana-1480	237	52	2013,pp.3137	2013,pp.3137	NUM
cana-1480	237	53	.	.	PUNCT
cana-1480	238	1	[	[	X
cana-1480	238	2	5	5	X
cana-1480	238	3	]	]	PUNCT
cana-1480	238	4	k.bhuvaneswari	k.bhuvaneswari	NOUN
cana-1480	238	5	and	and	CCONJ
cana-1480	238	6	a.ezhilarasi	a.ezhilarasi	NOUN
cana-1480	238	7	,	,	PUNCT
cana-1480	238	8	on	on	ADP
cana-1480	238	9	nano	nano	NOUN
cana-1480	238	10	semi	semi	ADJ
cana-1480	238	11	-	-	ADJ
cana-1480	238	12	generalized	generalized	ADJ
cana-1480	238	13	and	and	CCONJ
cana-1480	238	14	nano	nano	NOUN
cana-1480	238	15	generalizedsemiclosed	generalizedsemiclose	VERB
cana-1480	238	16	sets	set	NOUN
cana-1480	238	17	in	in	ADP
cana-1480	238	18	nts	nt	NOUN
cana-1480	238	19	,	,	PUNCT
cana-1480	238	20	international	international	ADJ
cana-1480	238	21	journal	journal	NOUN
cana-1480	238	22	of	of	ADP
cana-1480	238	23	mathemaatics	mathemaatic	NOUN
cana-1480	238	24	and	and	CCONJ
cana-1480	238	25	computeer	computeer	NOUN
cana-1480	238	26	applications	application	NOUN
cana-1480	238	27	research(ijmcar	research(ijmcar	PROPN
cana-1480	238	28	)	)	PUNCT
cana-1480	238	29	,	,	PUNCT
cana-1480	238	30	vol.4	vol.4	PROPN
cana-1480	238	31	,	,	PUNCT
cana-1480	238	32	issue3,jun	issue3,jun	NOUN
cana-1480	238	33	2014,117	2014,117	NUM
cana-1480	238	34	-	-	SYM
cana-1480	238	35	124	124	NUM
cana-1480	238	36	.	.	PUNCT
cana-1480	238	37	9	9	NUM
cana-1480	239	1	[	[	SYM
cana-1480	239	2	6	6	NUM
cana-1480	239	3	]	]	PUNCT
cana-1480	239	4	n.levine	n.levine	PRON
cana-1480	239	5	,	,	PUNCT
cana-1480	239	6	generalized	generalize	VERB
cana-1480	239	7	closed	closed	ADJ
cana-1480	239	8	sets	set	NOUN
cana-1480	239	9	in	in	ADP
cana-1480	239	10	topology	topology	NOUN
cana-1480	239	11	,	,	PUNCT
cana-1480	239	12	rend.cire.math.palermo	rend.cire.math.palermo	NOUN
cana-1480	239	13	,	,	PUNCT
cana-1480	239	14	(	(	PUNCT
cana-1480	239	15	1963	1963	NUM
cana-1480	239	16	)	)	PUNCT
cana-1480	239	17	,	,	PUNCT
cana-1480	239	18	19(2),86	19(2),86	NUM
cana-1480	239	19	-	-	SYM
cana-1480	239	20	96	96	NUM
cana-1480	239	21	.	.	PUNCT
cana-1480	240	1	[	[	X
cana-1480	240	2	7	7	NUM
cana-1480	240	3	]	]	X
cana-1480	240	4	a.v.arhangelskii	a.v.arhangelskii	PROPN
cana-1480	240	5	and	and	CCONJ
cana-1480	240	6	r.	r.	PROPN
cana-1480	240	7	wiegandt	wiegandt	PROPN
cana-1480	240	8	,	,	PUNCT
cana-1480	240	9	connectedness	connectedness	NOUN
cana-1480	240	10	and	and	CCONJ
cana-1480	240	11	disconnectedness	disconnectedness	NOUN
cana-1480	240	12	in	in	ADP
cana-1480	240	13	topology	topology	NOUN
cana-1480	240	14	,	,	PUNCT
cana-1480	240	15	top.app.5(1975	top.app.5(1975	NUM
cana-1480	240	16	)	)	PUNCT
cana-1480	240	17	.	.	PUNCT
cana-1480	241	1	[	[	X
cana-1480	241	2	8	8	NUM
cana-1480	241	3	]	]	PUNCT
cana-1480	241	4	s.krishnaprakash	s.krishnaprakash	NOUN
cana-1480	241	5	,	,	PUNCT
cana-1480	241	6	r.ramesh	r.ramesh	NOUN
cana-1480	241	7	and	and	CCONJ
cana-1480	241	8	r.suresh	r.suresh	NOUN
cana-1480	241	9	,	,	PUNCT
cana-1480	241	10	”	"	PUNCT
cana-1480	241	11	nano	nano	NOUN
cana-1480	241	12	compactness	compactness	NOUN
cana-1480	241	13	and	and	CCONJ
cana-1480	241	14	nano	nano	NOUN
cana-1480	241	15	connectedness	connectedness	NOUN
cana-1480	241	16	in	in	ADP
cana-1480	241	17	nts	nt	NOUN
cana-1480	241	18	”	"	PUNCT
cana-1480	241	19	,	,	PUNCT
cana-1480	241	20	internal	internal	ADJ
cana-1480	241	21	journal	journal	NOUN
cana-1480	241	22	of	of	ADP
cana-1480	241	23	pure	pure	ADJ
cana-1480	241	24	and	and	CCONJ
cana-1480	241	25	applied	applied	ADJ
cana-1480	241	26	mathematics	mathematic	NOUN
cana-1480	241	27	,	,	PUNCT
cana-1480	241	28	volume	volume	NOUN
cana-1480	241	29	119,n0.13	119,n0.13	NUM
cana-1480	241	30	(	(	PUNCT
cana-1480	241	31	2018),107	2018),107	NUM
cana-1480	241	32	-	-	SYM
cana-1480	241	33	115	115	NUM
cana-1480	241	34	.	.	PUNCT
cana-1480	242	1	[	[	X
cana-1480	242	2	9	9	NUM
cana-1480	242	3	]	]	SYM
cana-1480	242	4	k.baby	k.baby	NOUN
cana-1480	242	5	and	and	CCONJ
cana-1480	242	6	h.aaminumariyam	h.aaminumariyam	PROPN
cana-1480	242	7	”	"	PUNCT
cana-1480	242	8	on	on	ADP
cana-1480	242	9	nano	nano	NOUN
cana-1480	242	10	star	star	NOUN
cana-1480	242	11	generalized	generalize	VERB
cana-1480	242	12	alpha	alpha	NOUN
cana-1480	242	13	closed	close	VERB
cana-1480	242	14	set	set	VERB
cana-1480	242	15	and	and	CCONJ
cana-1480	242	16	nano	nano	NOUN
cana-1480	242	17	star	star	NOUN
cana-1480	242	18	generalized	generalize	VERB
cana-1480	242	19	alpha	alpha	NOUN
cana-1480	242	20	continuous	continuous	ADJ
cana-1480	242	21	function	function	NOUN
cana-1480	242	22	in	in	ADP
cana-1480	242	23	nts	nt	NOUN
cana-1480	242	24	”	"	PUNCT
cana-1480	242	25	,	,	PUNCT
cana-1480	242	26	indian	indian	ADJ
cana-1480	242	27	journal	journal	NOUN
cana-1480	242	28	of	of	ADP
cana-1480	242	29	natural	natural	ADJ
cana-1480	242	30	science	science	NOUN
cana-1480	242	31	,	,	PUNCT
cana-1480	242	32	vol.15	vol.15	NOUN
cana-1480	242	33	/	/	SYM
cana-1480	242	34	issue	issue	NOUN
cana-1480	242	35	84	84	NUM
cana-1480	242	36	/	/	SYM
cana-1480	242	37	jun/2024	jun/2024	NOUN
cana-1480	242	38	.	.	PUNCT
cana-1480	243	1	[	[	X
cana-1480	243	2	10	10	NUM
cana-1480	243	3	]	]	X
cana-1480	243	4	qays	qays	PROPN
cana-1480	243	5	hatem	hatem	PROPN
cana-1480	243	6	imran	imran	PROPN
cana-1480	243	7	,	,	PUNCT
cana-1480	243	8	murtadha	murtadha	VERB
cana-1480	243	9	m.abdulkadhim	m.abdulkadhim	PRON
cana-1480	243	10	and	and	CCONJ
cana-1480	243	11	mustafa	mustafa	PROPN
cana-1480	243	12	h.hadi	h.hadi	PROPN
cana-1480	243	13	,	,	PUNCT
cana-1480	243	14	nano	nano	NOUN
cana-1480	243	15	generalized	generalize	VERB
cana-1480	243	16	alpha	alpha	NOUN
cana-1480	243	17	closed	close	VERB
cana-1480	243	18	sets	set	NOUN
cana-1480	243	19	in	in	ADP
cana-1480	243	20	nts	nt	NOUN
cana-1480	243	21	,	,	PUNCT
cana-1480	243	22	gen.math.notes	gen.math.notes	PROPN
cana-1480	243	23	,	,	PUNCT
cana-1480	243	24	vol.34	vol.34	PROPN
cana-1480	243	25	,	,	PUNCT
cana-1480	243	26	no.2	no.2	PROPN
cana-1480	243	27	,	,	PUNCT
cana-1480	243	28	june	june	PROPN
cana-1480	243	29	2016,pp.39	2016,pp.39	NUM
cana-1480	243	30	-	-	SYM
cana-1480	243	31	51	51	NUM
cana-1480	243	32	.	.	PUNCT
cana-1480	244	1	[	[	X
cana-1480	244	2	11	11	NUM
cana-1480	244	3	]	]	PUNCT
cana-1480	244	4	r.x.shen	r.x.shen	ADV
cana-1480	244	5	,	,	PUNCT
cana-1480	244	6	a	a	DET
cana-1480	244	7	note	note	NOUN
cana-1480	244	8	on	on	ADP
cana-1480	244	9	generalized	generalized	ADJ
cana-1480	244	10	connectedness	connectedness	NOUN
cana-1480	244	11	,	,	PUNCT
cana-1480	244	12	acta	acta	PROPN
cana-1480	244	13	math	math	PROPN
cana-1480	244	14	hungar	hungar	NOUN
cana-1480	244	15	,	,	PUNCT
cana-1480	244	16	122(3)(2009),231	122(3)(2009),231	NUM
cana-1480	244	17	-	-	SYM
cana-1480	244	18	235	235	NUM
cana-1480	244	19	.	.	PUNCT
cana-1480	245	1	[	[	X
cana-1480	245	2	12	12	NUM
cana-1480	245	3	]	]	PUNCT
cana-1480	245	4	p.subbulakshmi	p.subbulakshmi	NOUN
cana-1480	245	5	and	and	CCONJ
cana-1480	245	6	n.r.santhi	n.r.santhi	NOUN
cana-1480	245	7	maheswari,”some	maheswari,”some	DET
cana-1480	245	8	new	new	ADJ
cana-1480	245	9	form	form	NOUN
cana-1480	245	10	of	of	ADP
cana-1480	245	11	nano	nano	NOUN
cana-1480	245	12	connectedness	connectedness	NOUN
cana-1480	245	13	and	and	CCONJ
cana-1480	245	14	nano	nano	NOUN
cana-1480	245	15	compactness	compactness	NOUN
cana-1480	245	16	in	in	ADP
cana-1480	245	17	nts”,international	nts”,international	ADJ
cana-1480	245	18	journal	journal	NOUN
cana-1480	245	19	of	of	ADP
cana-1480	245	20	creative	creative	ADJ
cana-1480	245	21	research	research	NOUN
cana-1480	245	22	thoughts(ijcrt),issn:2320	thoughts(ijcrt),issn:2320	PROPN
cana-1480	245	23	-	-	PUNCT
cana-1480	245	24	2882	2882	NUM
cana-1480	245	25	.	.	PUNCT