id	sid	tid	token	lemma	pos
cana-1058	1	1	communications	communication	NOUN
cana-1058	1	2	on	on	ADP
cana-1058	1	3	applied	apply	VERB
cana-1058	1	4	nonlinear	nonlinear	ADJ
cana-1058	1	5	analysis	analysis	NOUN
cana-1058	1	6	issn	issn	NOUN
cana-1058	1	7	:	:	PUNCT
cana-1058	1	8	1074	1074	NUM
cana-1058	1	9	-	-	PUNCT
cana-1058	1	10	133x	133x	NUM
cana-1058	1	11	vol	vol	NOUN
cana-1058	1	12	31	31	NUM
cana-1058	1	13	no	no	NOUN
cana-1058	1	14	.	.	PUNCT
cana-1058	2	1	5s	5s	NUM
cana-1058	2	2	(	(	PUNCT
cana-1058	2	3	2024	2024	NUM
cana-1058	2	4	)	)	PUNCT
cana-1058	2	5	390	390	NUM
cana-1058	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1058	3	2	exploration	exploration	NOUN
cana-1058	3	3	of	of	ADP
cana-1058	3	4	nαig	nαig	NOUN
cana-1058	3	5	-	-	PUNCT
cana-1058	3	6	closed	close	VERB
cana-1058	3	7	and	and	CCONJ
cana-1058	3	8	nαig	nαig	ADV
cana-1058	3	9	-	-	PUNCT
cana-1058	3	10	open	open	ADJ
cana-1058	3	11	sets	set	NOUN
cana-1058	3	12	in	in	ADP
cana-1058	3	13	nano	nano	NOUN
cana-1058	3	14	ideal	ideal	ADJ
cana-1058	3	15	topological	topological	ADJ
cana-1058	3	16	spaces	space	NOUN
cana-1058	3	17	with	with	ADP
cana-1058	3	18	practical	practical	ADJ
cana-1058	3	19	applications	application	NOUN
cana-1058	3	20	of	of	ADP
cana-1058	3	21	nano	nano	NOUN
cana-1058	3	22	ideal	ideal	ADJ
cana-1058	3	23	topological	topological	ADJ
cana-1058	3	24	spaces	space	NOUN
cana-1058	3	25	m.	m.	NOUN
cana-1058	3	26	mary	mary	PROPN
cana-1058	3	27	jansirani1	jansirani1	PROPN
cana-1058	3	28	,	,	PUNCT
cana-1058	3	29	k.	k.	PROPN
cana-1058	3	30	lakshmanan2	lakshmanan2	PROPN
cana-1058	3	31	,	,	PUNCT
cana-1058	3	32	l.	l.	PROPN
cana-1058	3	33	senthil	senthil	PROPN
cana-1058	3	34	kumar3	kumar3	PROPN
cana-1058	3	35	,	,	PUNCT
cana-1058	3	36	k.	k.	PROPN
cana-1058	3	37	sivaranjani4	sivaranjani4	PROPN
cana-1058	3	38	,	,	PUNCT
cana-1058	3	39	r.kulandaivelu5,s	r.kulandaivelu5,s	PROPN
cana-1058	3	40	.	.	PROPN
cana-1058	3	41	santhiya6	santhiya6	PROPN
cana-1058	3	42	,	,	PUNCT
cana-1058	3	43	a	a	DET
cana-1058	3	44	stephan	stephan	PROPN
cana-1058	3	45	antony	antony	PROPN
cana-1058	3	46	raj7	raj7	PROPN
cana-1058	3	47	,	,	PUNCT
cana-1058	3	48	d.	d.	PROPN
cana-1058	3	49	vinodhini8	vinodhini8	PROPN
cana-1058	3	50	*	*	PROPN
cana-1058	3	51	1	1	X
cana-1058	3	52	.	.	X
cana-1058	3	53	srminstitute	srminstitute	NOUN
cana-1058	3	54	of	of	ADP
cana-1058	3	55	science	science	NOUN
cana-1058	3	56	and	and	CCONJ
cana-1058	3	57	technology	technology	NOUN
cana-1058	3	58	,	,	PUNCT
cana-1058	3	59	trichy621105	trichy621105	PROPN
cana-1058	3	60	,	,	PUNCT
cana-1058	3	61	india	india	PROPN
cana-1058	3	62	2	2	NUM
cana-1058	3	63	.	.	PUNCT
cana-1058	3	64	st	st	PROPN
cana-1058	3	65	.	.	PROPN
cana-1058	3	66	joseph	joseph	PROPN
cana-1058	3	67	university	university	PROPN
cana-1058	3	68	,	,	PUNCT
cana-1058	3	69	chumoukedima	chumoukedima	PROPN
cana-1058	3	70	,	,	PUNCT
cana-1058	3	71	nagaland797	nagaland797	PROPN
cana-1058	3	72	115	115	NUM
cana-1058	3	73	,	,	PUNCT
cana-1058	3	74	india	india	PROPN
cana-1058	3	75	3	3	NUM
cana-1058	3	76	.	.	PUNCT
cana-1058	3	77	dr	dr	PROPN
cana-1058	3	78	.	.	PROPN
cana-1058	3	79	mahalingam	mahalingam	PROPN
cana-1058	3	80	college	college	PROPN
cana-1058	3	81	of	of	ADP
cana-1058	3	82	engineering	engineering	NOUN
cana-1058	3	83	and	and	CCONJ
cana-1058	3	84	technology	technology	NOUN
cana-1058	3	85	,	,	PUNCT
cana-1058	3	86	pollachi	pollachi	PROPN
cana-1058	3	87	,	,	PUNCT
cana-1058	3	88	tamil	tamil	PROPN
cana-1058	3	89	nadu642	nadu642	PROPN
cana-1058	3	90	003,india	003,india	NUM
cana-1058	3	91	.	.	PUNCT
cana-1058	4	1	4	4	X
cana-1058	4	2	.	.	X
cana-1058	4	3	sri	sri	PROPN
cana-1058	4	4	eshwar	eshwar	PROPN
cana-1058	4	5	college	college	PROPN
cana-1058	4	6	of	of	ADP
cana-1058	4	7	engineering	engineering	PROPN
cana-1058	4	8	,	,	PUNCT
cana-1058	4	9	coimbatore	coimbatore	PROPN
cana-1058	4	10	,	,	PUNCT
cana-1058	4	11	india	india	PROPN
cana-1058	4	12	5	5	NUM
cana-1058	4	13	.	.	PUNCT
cana-1058	5	1	dr.n.g.p.instituteoftechnology	dr.n.g.p.instituteoftechnology	NOUN
cana-1058	5	2	,	,	PUNCT
cana-1058	5	3	coimbatore	coimbatore	PROPN
cana-1058	5	4	,	,	PUNCT
cana-1058	5	5	tamilnadu-641	tamilnadu-641	NOUN
cana-1058	5	6	008,india	008,india	PROPN
cana-1058	5	7	.	.	PROPN
cana-1058	5	8	6	6	NUM
cana-1058	5	9	.	.	X
cana-1058	6	1	sri	sri	PROPN
cana-1058	6	2	krishna	krishna	PROPN
cana-1058	6	3	college	college	PROPN
cana-1058	6	4	of	of	ADP
cana-1058	6	5	engineering	engineering	NOUN
cana-1058	6	6	and	and	CCONJ
cana-1058	6	7	technology	technology	NOUN
cana-1058	6	8	,	,	PUNCT
cana-1058	6	9	coimbatore	coimbatore	PROPN
cana-1058	6	10	,	,	PUNCT
cana-1058	6	11	tamil	tamil	PROPN
cana-1058	6	12	nadu641	nadu641	PROPN
cana-1058	6	13	008,india	008,india	PROPN
cana-1058	6	14	.	.	PROPN
cana-1058	7	1	7	7	NUM
cana-1058	7	2	.	.	X
cana-1058	7	3	sns	sns	PROPN
cana-1058	7	4	college	college	PROPN
cana-1058	7	5	of	of	ADP
cana-1058	7	6	engineering	engineering	PROPN
cana-1058	7	7	,	,	PUNCT
cana-1058	7	8	tamilnadu	tamilnadu	PROPN
cana-1058	7	9	,	,	PUNCT
cana-1058	7	10	india	india	PROPN
cana-1058	7	11	8	8	NUM
cana-1058	7	12	.	.	PUNCT
cana-1058	8	1	amrita	amrita	PROPN
cana-1058	8	2	school	school	PROPN
cana-1058	8	3	of	of	ADP
cana-1058	8	4	agricultural	agricultural	ADJ
cana-1058	8	5	sciences	science	NOUN
cana-1058	8	6	,	,	PUNCT
cana-1058	8	7	amrita	amrita	PROPN
cana-1058	8	8	vishwa	vishwa	PROPN
cana-1058	8	9	vidyapeetham	vidyapeetham	PROPN
cana-1058	8	10	university	university	PROPN
cana-1058	8	11	,	,	PUNCT
cana-1058	8	12	coimbatore	coimbatore	PROPN
cana-1058	8	13	,	,	PUNCT
cana-1058	8	14	tamilnadu	tamilnadu	ADJ
cana-1058	8	15	–	–	PUNCT
cana-1058	8	16	642109,india	642109,india	NUM
cana-1058	8	17	.	.	PUNCT
cana-1058	9	1	correspondence	correspondence	NOUN
cana-1058	9	2	:	:	PUNCT
cana-1058	10	1	d.	d.	PROPN
cana-1058	10	2	vinodhini,email:d_vinodhini@cb.amrita.edu	vinodhini,email:d_vinodhini@cb.amrita.edu	PROPN
cana-1058	10	3	article	article	PROPN
cana-1058	10	4	history	history	NOUN
cana-1058	10	5	:	:	PUNCT
cana-1058	10	6	received	receive	VERB
cana-1058	10	7	:	:	PUNCT
cana-1058	10	8	11	11	NUM
cana-1058	10	9	-	-	SYM
cana-1058	10	10	05	05	NUM
cana-1058	10	11	-	-	PUNCT
cana-1058	10	12	2024	2024	NUM
cana-1058	10	13	revised	revise	VERB
cana-1058	10	14	:	:	PUNCT
cana-1058	10	15	20	20	NUM
cana-1058	10	16	-	-	SYM
cana-1058	10	17	06	06	NUM
cana-1058	10	18	-	-	PUNCT
cana-1058	10	19	2024	2024	NUM
cana-1058	10	20	accepted	accept	VERB
cana-1058	10	21	:	:	PUNCT
cana-1058	10	22	07	07	NUM
cana-1058	10	23	-	-	PUNCT
cana-1058	10	24	07	07	NUM
cana-1058	10	25	-	-	PUNCT
cana-1058	10	26	2024	2024	NUM
cana-1058	10	27	abstract	abstract	NOUN
cana-1058	10	28	:	:	PUNCT
cana-1058	10	29	in	in	ADP
cana-1058	10	30	this	this	DET
cana-1058	10	31	study	study	NOUN
cana-1058	10	32	,	,	PUNCT
cana-1058	10	33	we	we	PRON
cana-1058	10	34	explore	explore	VERB
cana-1058	10	35	the	the	DET
cana-1058	10	36	concepts	concept	NOUN
cana-1058	10	37	of	of	ADP
cana-1058	10	38	nαig	nαig	ADV
cana-1058	10	39	-	-	PUNCT
cana-1058	10	40	closed	close	VERB
cana-1058	10	41	sets	set	NOUN
cana-1058	10	42	(	(	PUNCT
cana-1058	10	43	nano	nano	NOUN
cana-1058	10	44	αig	αig	NOUN
cana-1058	10	45	-	-	PUNCT
cana-1058	10	46	closed	close	VERB
cana-1058	10	47	sets	set	NOUN
cana-1058	10	48	)	)	PUNCT
cana-1058	10	49	and	and	CCONJ
cana-1058	10	50	nαig	nαig	ADV
cana-1058	10	51	-	-	PUNCT
cana-1058	10	52	open	open	ADJ
cana-1058	10	53	sets	set	NOUN
cana-1058	10	54	(	(	PUNCT
cana-1058	10	55	nano	nano	NOUN
cana-1058	10	56	αig	αig	NOUN
cana-1058	10	57	-	-	PUNCT
cana-1058	10	58	open	open	ADJ
cana-1058	10	59	sets	set	NOUN
cana-1058	10	60	)	)	PUNCT
cana-1058	10	61	in	in	ADP
cana-1058	10	62	nano	nano	NOUN
cana-1058	10	63	ideal	ideal	ADJ
cana-1058	10	64	topological	topological	ADJ
cana-1058	10	65	spaces	space	NOUN
cana-1058	10	66	.	.	PUNCT
cana-1058	11	1	we	we	PRON
cana-1058	11	2	discuss	discuss	VERB
cana-1058	11	3	their	their	PRON
cana-1058	11	4	relationships	relationship	NOUN
cana-1058	11	5	with	with	ADP
cana-1058	11	6	other	other	ADJ
cana-1058	11	7	forms	form	NOUN
cana-1058	11	8	of	of	ADP
cana-1058	11	9	nano	nano	NOUN
cana-1058	11	10	ideal	ideal	ADJ
cana-1058	11	11	sets	set	NOUN
cana-1058	11	12	and	and	CCONJ
cana-1058	11	13	illustrate	illustrate	VERB
cana-1058	11	14	the	the	DET
cana-1058	11	15	abstract	abstract	ADJ
cana-1058	11	16	concepts	concept	NOUN
cana-1058	11	17	using	use	VERB
cana-1058	11	18	appropriate	appropriate	ADJ
cana-1058	11	19	examples	example	NOUN
cana-1058	11	20	.	.	PUNCT
cana-1058	12	1	this	this	DET
cana-1058	12	2	research	research	NOUN
cana-1058	12	3	delves	delve	VERB
cana-1058	12	4	into	into	ADP
cana-1058	12	5	the	the	DET
cana-1058	12	6	application	application	NOUN
cana-1058	12	7	of	of	ADP
cana-1058	12	8	nαigclosed	nαigclose	VERB
cana-1058	12	9	and	and	CCONJ
cana-1058	12	10	nαig	nαig	ADV
cana-1058	12	11	-	-	PUNCT
cana-1058	12	12	open	open	ADJ
cana-1058	12	13	sets	set	NOUN
cana-1058	12	14	across	across	ADP
cana-1058	12	15	diverse	diverse	ADJ
cana-1058	12	16	domains	domain	NOUN
cana-1058	12	17	,	,	PUNCT
cana-1058	12	18	ranging	range	VERB
cana-1058	12	19	from	from	ADP
cana-1058	12	20	network	network	NOUN
cana-1058	12	21	security	security	NOUN
cana-1058	12	22	to	to	ADP
cana-1058	12	23	healthcare	healthcare	NOUN
cana-1058	12	24	and	and	CCONJ
cana-1058	12	25	environmental	environmental	ADJ
cana-1058	12	26	monitoring	monitoring	NOUN
cana-1058	12	27	.	.	PUNCT
cana-1058	13	1	by	by	ADP
cana-1058	13	2	leveraging	leverage	VERB
cana-1058	13	3	these	these	DET
cana-1058	13	4	mathematical	mathematical	ADJ
cana-1058	13	5	concepts	concept	NOUN
cana-1058	13	6	,	,	PUNCT
cana-1058	13	7	we	we	PRON
cana-1058	13	8	aim	aim	VERB
cana-1058	13	9	to	to	PART
cana-1058	13	10	enhance	enhance	VERB
cana-1058	13	11	anomaly	anomaly	NOUN
cana-1058	13	12	detection	detection	NOUN
cana-1058	13	13	,	,	PUNCT
cana-1058	13	14	optimize	optimize	NOUN
cana-1058	13	15	network	network	NOUN
cana-1058	13	16	operations	operation	NOUN
cana-1058	13	17	,	,	PUNCT
cana-1058	13	18	and	and	CCONJ
cana-1058	13	19	improve	improve	VERB
cana-1058	13	20	decision	decision	NOUN
cana-1058	13	21	-	-	PUNCT
cana-1058	13	22	making	make	VERB
cana-1058	13	23	processes	process	NOUN
cana-1058	13	24	across	across	ADP
cana-1058	13	25	various	various	ADJ
cana-1058	13	26	sectors	sector	NOUN
cana-1058	13	27	.	.	PUNCT
cana-1058	14	1	this	this	DET
cana-1058	14	2	article	article	NOUN
cana-1058	14	3	outlines	outline	VERB
cana-1058	14	4	the	the	DET
cana-1058	14	5	methodologies	methodology	NOUN
cana-1058	14	6	and	and	CCONJ
cana-1058	14	7	potential	potential	ADJ
cana-1058	14	8	benefits	benefit	NOUN
cana-1058	14	9	of	of	ADP
cana-1058	14	10	integrating	integrate	VERB
cana-1058	14	11	nαig	nαig	ADV
cana-1058	14	12	-	-	PUNCT
cana-1058	14	13	closed	close	VERB
cana-1058	14	14	and	and	CCONJ
cana-1058	14	15	nαig	nαig	ADV
cana-1058	14	16	-	-	PUNCT
cana-1058	14	17	open	open	ADJ
cana-1058	14	18	sets	set	NOUN
cana-1058	14	19	in	in	ADP
cana-1058	14	20	different	different	ADJ
cana-1058	14	21	application	application	NOUN
cana-1058	14	22	domains	domain	NOUN
cana-1058	14	23	.	.	PUNCT
cana-1058	15	1	keywords	keyword	NOUN
cana-1058	15	2	:	:	PUNCT
cana-1058	15	3	ideals	ideal	NOUN
cana-1058	15	4	,	,	PUNCT
cana-1058	15	5	nanotopology	nanotopology	NOUN
cana-1058	15	6	,	,	PUNCT
cana-1058	15	7	nαig	nαig	NOUN
cana-1058	15	8	-	-	PUNCT
cana-1058	15	9	closedsets	closedset	NOUN
cana-1058	15	10	,	,	PUNCT
cana-1058	15	11	nαig	nαig	NOUN
cana-1058	15	12	-	-	PUNCT
cana-1058	15	13	opensets	openset	NOUN
cana-1058	15	14	,	,	PUNCT
cana-1058	15	15	nanoαopen	nanoαopen	ADJ
cana-1058	15	16	.	.	PUNCT
cana-1058	16	1	1	1	X
cana-1058	16	2	.	.	X
cana-1058	16	3	introduction	introduction	NOUN
cana-1058	16	4	on	on	ADP
cana-1058	16	5	topological	topological	ADJ
cana-1058	16	6	spaces	space	NOUN
cana-1058	16	7	,	,	PUNCT
cana-1058	16	8	levine	levine	PROPN
cana-1058	17	1	[	[	X
cana-1058	17	2	1	1	NUM
cana-1058	17	3	]	]	PUNCT
cana-1058	17	4	introduced	introduce	VERB
cana-1058	17	5	the	the	DET
cana-1058	17	6	concept	concept	NOUN
cana-1058	17	7	of	of	ADP
cana-1058	17	8	generalized	generalized	ADJ
cana-1058	17	9	closed	close	VERB
cana-1058	17	10	sets	set	NOUN
cana-1058	17	11	during	during	ADP
cana-1058	17	12	1970	1970	NUM
cana-1058	17	13	.	.	PUNCT
cana-1058	18	1	numerous	numerous	ADJ
cana-1058	18	2	results	result	NOUN
cana-1058	18	3	in	in	ADP
cana-1058	18	4	general	general	ADJ
cana-1058	18	5	topology	topology	NOUN
cana-1058	18	6	have	have	AUX
cana-1058	18	7	been	be	AUX
cana-1058	18	8	developed	develop	VERB
cana-1058	18	9	using	use	VERB
cana-1058	18	10	this	this	DET
cana-1058	18	11	concept	concept	NOUN
cana-1058	18	12	.	.	PUNCT
cana-1058	19	1	in	in	ADP
cana-1058	19	2	1991	1991	NUM
cana-1058	19	3	,	,	PUNCT
cana-1058	19	4	balachandran	balachandran	NOUN
cana-1058	19	5	et	et	PROPN
cana-1058	19	6	.	.	PUNCT
cana-1058	20	1	al	al	PROPN
cana-1058	21	1	[	[	X
cana-1058	21	2	2	2	NUM
cana-1058	21	3	]	]	PUNCT
cana-1058	21	4	introduced	introduce	VERB
cana-1058	21	5	and	and	CCONJ
cana-1058	21	6	examined	examine	VERB
cana-1058	21	7	the	the	DET
cana-1058	21	8	notion	notion	NOUN
cana-1058	21	9	of	of	ADP
cana-1058	21	10	generalized	generalized	ADJ
cana-1058	21	11	continuous	continuous	ADJ
cana-1058	21	12	functions	function	NOUN
cana-1058	21	13	in	in	ADP
cana-1058	21	14	topological	topological	ADJ
cana-1058	21	15	spaces	space	NOUN
cana-1058	21	16	.	.	PUNCT
cana-1058	22	1	the	the	DET
cana-1058	22	2	notion	notion	NOUN
cana-1058	22	3	of	of	ADP
cana-1058	22	4	α	α	NOUN
cana-1058	22	5	-	-	ADJ
cana-1058	22	6	open	open	ADJ
cana-1058	22	7	sets	set	NOUN
cana-1058	22	8	was	be	AUX
cana-1058	22	9	introduced	introduce	VERB
cana-1058	22	10	and	and	CCONJ
cana-1058	22	11	investigated	investigate	VERB
cana-1058	22	12	by	by	ADP
cana-1058	22	13	njastad	njastad	NOUN
cana-1058	22	14	[	[	X
cana-1058	22	15	3	3	NUM
cana-1058	22	16	]	]	PUNCT
cana-1058	22	17	.	.	PUNCT
cana-1058	23	1	by	by	ADP
cana-1058	23	2	using	use	VERB
cana-1058	23	3	α	α	NOUN
cana-1058	23	4	-	-	ADJ
cana-1058	23	5	open	open	ADJ
cana-1058	23	6	,	,	PUNCT
cana-1058	23	7	mashbour	mashbour	PROPN
cana-1058	23	8	et	et	PROPN
cana-1058	23	9	al	al	PROPN
cana-1058	23	10	.	.	PUNCT
cana-1058	24	1	[	[	X
cana-1058	24	2	4	4	X
cana-1058	24	3	]	]	PUNCT
cana-1058	24	4	defined	define	VERB
cana-1058	24	5	and	and	CCONJ
cana-1058	24	6	studied	study	VERB
cana-1058	24	7	the	the	DET
cana-1058	24	8	concept	concept	NOUN
cana-1058	24	9	of	of	ADP
cana-1058	24	10	α	α	NOUN
cana-1058	24	11	-	-	PUNCT
cana-1058	24	12	closed	closed	ADJ
cana-1058	24	13	sets	set	NOUN
cana-1058	24	14	,	,	PUNCT
cana-1058	24	15	αclosure	αclosure	NOUN
cana-1058	24	16	of	of	ADP
cana-1058	24	17	a	a	DET
cana-1058	24	18	set	set	NOUN
cana-1058	24	19	,	,	PUNCT
cana-1058	24	20	α	α	NOUN
cana-1058	24	21	-	-	NOUN
cana-1058	24	22	continuity	continuity	NOUN
cana-1058	24	23	.	.	PUNCT
cana-1058	25	1	the	the	DET
cana-1058	25	2	concept	concept	NOUN
cana-1058	25	3	of	of	ADP
cana-1058	25	4	ideal	ideal	ADJ
cana-1058	25	5	topological	topological	ADJ
cana-1058	25	6	space	space	NOUN
cana-1058	25	7	was	be	AUX
cana-1058	25	8	introduced	introduce	VERB
cana-1058	25	9	by	by	ADP
cana-1058	25	10	kuratowski	kuratowski	ADJ
cana-1058	25	11	[	[	X
cana-1058	25	12	5	5	NUM
cana-1058	25	13	]	]	PUNCT
cana-1058	25	14	in	in	ADP
cana-1058	25	15	1966	1966	NUM
cana-1058	25	16	.	.	PUNCT
cana-1058	26	1	he	he	PRON
cana-1058	26	2	also	also	ADV
cana-1058	26	3	defined	define	VERB
cana-1058	26	4	the	the	DET
cana-1058	26	5	local	local	ADJ
cana-1058	26	6	functions	function	NOUN
cana-1058	26	7	in	in	ADP
cana-1058	26	8	ideal	ideal	ADJ
cana-1058	26	9	topological	topological	ADJ
cana-1058	26	10	spaces	space	NOUN
cana-1058	26	11	.	.	PUNCT
cana-1058	27	1	furthermore	furthermore	ADV
cana-1058	27	2	,	,	PUNCT
cana-1058	27	3	during	during	ADP
cana-1058	27	4	the	the	DET
cana-1058	27	5	period	period	NOUN
cana-1058	27	6	1990	1990	NUM
cana-1058	27	7	,	,	PUNCT
cana-1058	27	8	jankovic	jankovic	PROPN
cana-1058	27	9	and	and	CCONJ
cana-1058	27	10	hamlett	hamlett	PROPN
cana-1058	28	1	[	[	X
cana-1058	28	2	6	6	NUM
cana-1058	28	3	]	]	PUNCT
cana-1058	28	4	investigated	investigate	VERB
cana-1058	28	5	the	the	DET
cana-1058	28	6	properties	property	NOUN
cana-1058	28	7	of	of	ADP
cana-1058	28	8	ideal	ideal	ADJ
cana-1058	28	9	topological	topological	ADJ
cana-1058	28	10	spaces	space	NOUN
cana-1058	28	11	.	.	PUNCT
cana-1058	29	1	in	in	ADP
cana-1058	29	2	2014	2014	NUM
cana-1058	29	3	,	,	PUNCT
cana-1058	29	4	αigclosed	αigclose	VERB
cana-1058	29	5	is	be	AUX
cana-1058	29	6	introduced	introduce	VERB
cana-1058	29	7	in	in	ADP
cana-1058	29	8	ideal	ideal	ADJ
cana-1058	29	9	topological	topological	ADJ
cana-1058	29	10	spaces	space	NOUN
cana-1058	29	11	.	.	PUNCT
cana-1058	30	1	the	the	DET
cana-1058	30	2	notion	notion	NOUN
cana-1058	30	3	of	of	ADP
cana-1058	30	4	nano	nano	NOUN
cana-1058	30	5	topology	topology	NOUN
cana-1058	30	6	was	be	AUX
cana-1058	30	7	introduced	introduce	VERB
cana-1058	30	8	by	by	ADP
cana-1058	30	9	lellis	lellis	PROPN
cana-1058	30	10	thivagar	thivagar	NOUN
cana-1058	30	11	[	[	X
cana-1058	30	12	7	7	NUM
cana-1058	30	13	,	,	PUNCT
cana-1058	30	14	8	8	NUM
cana-1058	30	15	,	,	PUNCT
cana-1058	30	16	9	9	NUM
cana-1058	30	17	]	]	PUNCT
cana-1058	30	18	which	which	PRON
cana-1058	30	19	was	be	AUX
cana-1058	30	20	defined	define	VERB
cana-1058	30	21	in	in	ADP
cana-1058	30	22	terms	term	NOUN
cana-1058	30	23	of	of	ADP
cana-1058	30	24	approximations	approximation	NOUN
cana-1058	30	25	and	and	CCONJ
cana-1058	30	26	boundary	boundary	ADJ
cana-1058	30	27	region	region	NOUN
cana-1058	30	28	of	of	ADP
cana-1058	30	29	a	a	DET
cana-1058	30	30	subset	subset	NOUN
cana-1058	30	31	of	of	ADP
cana-1058	30	32	an	an	DET
cana-1058	30	33	universe	universe	NOUN
cana-1058	30	34	using	use	VERB
cana-1058	30	35	an	an	DET
cana-1058	30	36	equivalence	equivalence	NOUN
cana-1058	30	37	relation	relation	NOUN
cana-1058	30	38	on	on	ADP
cana-1058	30	39	it	it	PRON
cana-1058	30	40	.	.	PUNCT
cana-1058	31	1	he	he	PRON
cana-1058	31	2	also	also	ADV
cana-1058	31	3	established	establish	VERB
cana-1058	31	4	and	and	CCONJ
cana-1058	31	5	analyzed	analyze	VERB
cana-1058	31	6	the	the	DET
cana-1058	31	7	nano	nano	NOUN
cana-1058	31	8	forms	form	NOUN
cana-1058	31	9	of	of	ADP
cana-1058	31	10	weakly	weakly	ADJ
cana-1058	31	11	open	open	ADJ
cana-1058	31	12	sets	set	NOUN
cana-1058	31	13	such	such	ADJ
cana-1058	31	14	as	as	ADP
cana-1058	31	15	nanoαopen	nanoαopen	PROPN
cana-1058	31	16	sets	set	NOUN
cana-1058	31	17	,	,	PUNCT
cana-1058	31	18	nano	nano	NOUN
cana-1058	31	19	mailto:d_vinodhini@cb.amrita.edu	mailto:d_vinodhini@cb.amrita.edu	PROPN
cana-1058	31	20	communications	communication	NOUN
cana-1058	31	21	on	on	ADP
cana-1058	31	22	applied	apply	VERB
cana-1058	31	23	nonlinear	nonlinear	ADJ
cana-1058	31	24	analysis	analysis	NOUN
cana-1058	31	25	issn	issn	NOUN
cana-1058	31	26	:	:	PUNCT
cana-1058	31	27	1074	1074	NUM
cana-1058	31	28	-	-	PUNCT
cana-1058	31	29	133x	133x	NUM
cana-1058	31	30	vol	vol	NOUN
cana-1058	31	31	31	31	NUM
cana-1058	31	32	no	no	NOUN
cana-1058	31	33	.	.	PUNCT
cana-1058	32	1	5s	5s	NUM
cana-1058	32	2	(	(	PUNCT
cana-1058	32	3	2024	2024	NUM
cana-1058	32	4	)	)	PUNCT
cana-1058	32	5	391	391	NUM
cana-1058	32	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1058	32	7	semi	semi	ADJ
cana-1058	32	8	-	-	ADJ
cana-1058	32	9	open	open	ADJ
cana-1058	32	10	sets	set	NOUN
cana-1058	32	11	and	and	CCONJ
cana-1058	32	12	nano	nano	ADJ
cana-1058	32	13	pre	pre	ADJ
cana-1058	32	14	-	-	ADJ
cana-1058	32	15	open	open	ADJ
cana-1058	32	16	sets	set	NOUN
cana-1058	32	17	.	.	PUNCT
cana-1058	33	1	in	in	ADP
cana-1058	33	2	2016	2016	NUM
cana-1058	33	3	,	,	PUNCT
cana-1058	33	4	bhuvaneswari	bhuvaneswari	PROPN
cana-1058	33	5	et	et	PROPN
cana-1058	33	6	.	.	PUNCT
cana-1058	34	1	al	al	PROPN
cana-1058	35	1	[	[	X
cana-1058	35	2	10	10	NUM
cana-1058	35	3	]	]	PUNCT
cana-1058	35	4	,	,	PUNCT
cana-1058	35	5	introduced	introduce	VERB
cana-1058	35	6	and	and	CCONJ
cana-1058	35	7	studied	study	VERB
cana-1058	35	8	the	the	DET
cana-1058	35	9	characteristic	characteristic	NOUN
cana-1058	35	10	of	of	ADP
cana-1058	35	11	nano	nano	NOUN
cana-1058	35	12	generalized	generalize	VERB
cana-1058	35	13	closed	closed	ADJ
cana-1058	35	14	sets	set	NOUN
cana-1058	35	15	.	.	PUNCT
cana-1058	36	1	the	the	DET
cana-1058	36	2	structure	structure	NOUN
cana-1058	36	3	of	of	ADP
cana-1058	36	4	this	this	DET
cana-1058	36	5	manuscript	manuscript	NOUN
cana-1058	36	6	is	be	AUX
cana-1058	36	7	as	as	SCONJ
cana-1058	36	8	follows	follow	VERB
cana-1058	36	9	.	.	PUNCT
cana-1058	37	1	in	in	ADP
cana-1058	37	2	section	section	NOUN
cana-1058	37	3	2	2	NUM
cana-1058	37	4	,	,	PUNCT
cana-1058	37	5	we	we	PRON
cana-1058	37	6	recall	recall	VERB
cana-1058	37	7	some	some	DET
cana-1058	37	8	fundamental	fundamental	ADJ
cana-1058	37	9	definitions	definition	NOUN
cana-1058	37	10	and	and	CCONJ
cana-1058	37	11	results	result	NOUN
cana-1058	37	12	which	which	PRON
cana-1058	37	13	are	be	AUX
cana-1058	37	14	useful	useful	ADJ
cana-1058	37	15	to	to	PART
cana-1058	37	16	prove	prove	VERB
cana-1058	37	17	our	our	PRON
cana-1058	37	18	main	main	ADJ
cana-1058	37	19	results	result	NOUN
cana-1058	37	20	.	.	PUNCT
cana-1058	38	1	in	in	ADP
cana-1058	38	2	section	section	NOUN
cana-1058	38	3	3	3	NUM
cana-1058	38	4	,	,	PUNCT
cana-1058	38	5	we	we	PRON
cana-1058	38	6	define	define	VERB
cana-1058	38	7	and	and	CCONJ
cana-1058	38	8	study	study	VERB
cana-1058	38	9	the	the	DET
cana-1058	38	10	notion	notion	NOUN
cana-1058	38	11	of	of	ADP
cana-1058	38	12	nαigclosed	nαigclose	VERB
cana-1058	38	13	sets	set	NOUN
cana-1058	38	14	and	and	CCONJ
cana-1058	38	15	nαigopen	nαigopen	ADJ
cana-1058	38	16	sets	set	NOUN
cana-1058	38	17	in	in	ADP
cana-1058	38	18	nano	nano	NOUN
cana-1058	38	19	ideal	ideal	ADJ
cana-1058	38	20	topological	topological	ADJ
cana-1058	38	21	spaces	space	NOUN
cana-1058	38	22	.	.	PUNCT
cana-1058	39	1	we	we	PRON
cana-1058	39	2	also	also	ADV
cana-1058	39	3	discuss	discuss	VERB
cana-1058	39	4	the	the	DET
cana-1058	39	5	concept	concept	NOUN
cana-1058	39	6	of	of	ADP
cana-1058	39	7	nαig	nαig	ADV
cana-1058	39	8	-	-	PUNCT
cana-1058	39	9	closed	close	VERB
cana-1058	39	10	sets	set	NOUN
cana-1058	39	11	and	and	CCONJ
cana-1058	39	12	discussed	discuss	VERB
cana-1058	39	13	the	the	DET
cana-1058	39	14	relationships	relationship	NOUN
cana-1058	39	15	between	between	ADP
cana-1058	39	16	the	the	DET
cana-1058	39	17	other	other	ADJ
cana-1058	39	18	existing	exist	VERB
cana-1058	39	19	nano	nano	NOUN
cana-1058	39	20	ideal	ideal	ADJ
cana-1058	39	21	sets	set	NOUN
cana-1058	39	22	.	.	PUNCT
cana-1058	40	1	in	in	ADP
cana-1058	40	2	section	section	NOUN
cana-1058	40	3	4	4	NUM
cana-1058	40	4	,	,	PUNCT
cana-1058	40	5	the	the	DET
cana-1058	40	6	integration	integration	NOUN
cana-1058	40	7	of	of	ADP
cana-1058	40	8	nαig	nαig	ADV
cana-1058	40	9	-	-	PUNCT
cana-1058	40	10	closed	close	VERB
cana-1058	40	11	and	and	CCONJ
cana-1058	40	12	nαig	nαig	ADV
cana-1058	40	13	-	-	PUNCT
cana-1058	40	14	open	open	ADJ
cana-1058	40	15	sets	set	NOUN
cana-1058	40	16	in	in	ADP
cana-1058	40	17	nano	nano	NOUN
cana-1058	40	18	ideal	ideal	ADJ
cana-1058	40	19	topological	topological	ADJ
cana-1058	40	20	spaces	space	NOUN
cana-1058	40	21	which	which	PRON
cana-1058	40	22	presents	present	VERB
cana-1058	40	23	a	a	DET
cana-1058	40	24	novel	novel	ADJ
cana-1058	40	25	approach	approach	NOUN
cana-1058	40	26	to	to	PART
cana-1058	40	27	address	address	VERB
cana-1058	40	28	complex	complex	ADJ
cana-1058	40	29	challenges	challenge	NOUN
cana-1058	40	30	across	across	ADP
cana-1058	40	31	multiple	multiple	ADJ
cana-1058	40	32	domains	domain	NOUN
cana-1058	40	33	is	be	AUX
cana-1058	40	34	discussed	discuss	VERB
cana-1058	40	35	.	.	PUNCT
cana-1058	41	1	this	this	DET
cana-1058	41	2	research	research	NOUN
cana-1058	41	3	explores	explore	VERB
cana-1058	41	4	the	the	DET
cana-1058	41	5	diverse	diverse	ADJ
cana-1058	41	6	applications	application	NOUN
cana-1058	41	7	of	of	ADP
cana-1058	41	8	these	these	DET
cana-1058	41	9	mathematical	mathematical	ADJ
cana-1058	41	10	concepts	concept	NOUN
cana-1058	41	11	,	,	PUNCT
cana-1058	41	12	highlighting	highlight	VERB
cana-1058	41	13	their	their	PRON
cana-1058	41	14	potential	potential	NOUN
cana-1058	41	15	to	to	PART
cana-1058	41	16	revolutionize	revolutionize	VERB
cana-1058	41	17	various	various	ADJ
cana-1058	41	18	industries	industry	NOUN
cana-1058	41	19	.	.	PUNCT
cana-1058	42	1	2	2	X
cana-1058	42	2	.	.	X
cana-1058	42	3	premilinaries	premilinarie	NOUN
cana-1058	42	4	throughout	throughout	ADP
cana-1058	42	5	this	this	DET
cana-1058	42	6	study	study	NOUN
cana-1058	42	7	(	(	PUNCT
cana-1058	42	8	u	u	NOUN
cana-1058	42	9	,	,	PUNCT
cana-1058	42	10	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	42	11	)	)	PUNCT
cana-1058	42	12	)	)	PUNCT
cana-1058	42	13	(	(	PUNCT
cana-1058	42	14	or	or	CCONJ
cana-1058	42	15	u	u	NOUN
cana-1058	42	16	)	)	PUNCT
cana-1058	42	17	represent	represent	VERB
cana-1058	42	18	nano	nano	NOUN
cana-1058	42	19	topological	topological	ADJ
cana-1058	42	20	spaces	space	NOUN
cana-1058	42	21	on	on	ADP
cana-1058	42	22	which	which	PRON
cana-1058	42	23	no	no	DET
cana-1058	42	24	separation	separation	NOUN
cana-1058	42	25	axioms	axiom	NOUN
cana-1058	42	26	are	be	AUX
cana-1058	42	27	assumed	assume	VERB
cana-1058	42	28	unless	unless	SCONJ
cana-1058	42	29	otherwise	otherwise	ADV
cana-1058	42	30	mentioned	mention	VERB
cana-1058	42	31	.	.	PUNCT
cana-1058	43	1	for	for	ADP
cana-1058	43	2	a	a	DET
cana-1058	43	3	subset	subset	NOUN
cana-1058	43	4	a	a	PRON
cana-1058	43	5	of	of	ADP
cana-1058	43	6	a	a	DET
cana-1058	43	7	space	space	NOUN
cana-1058	43	8	(	(	PUNCT
cana-1058	43	9	u,𝑟𝑅(𝑥	u,𝑟𝑅(𝑥	PROPN
cana-1058	43	10	)	)	PUNCT
cana-1058	43	11	)	)	PUNCT
cana-1058	43	12	,	,	PUNCT
cana-1058	43	13	ncl(a	ncl(a	PROPN
cana-1058	43	14	)	)	PUNCT
cana-1058	43	15	and	and	CCONJ
cana-1058	43	16	nint(a	nint(a	NOUN
cana-1058	43	17	)	)	PUNCT
cana-1058	43	18	denote	denote	VERB
cana-1058	43	19	the	the	DET
cana-1058	43	20	nano	nano	NOUN
cana-1058	43	21	closure	closure	NOUN
cana-1058	43	22	of	of	ADP
cana-1058	43	23	a	a	PRON
cana-1058	43	24	and	and	CCONJ
cana-1058	43	25	the	the	DET
cana-1058	43	26	nano	nano	ADJ
cana-1058	43	27	interior	interior	NOUN
cana-1058	43	28	of	of	ADP
cana-1058	43	29	a	a	DET
cana-1058	43	30	respectively	respectively	ADV
cana-1058	43	31	.	.	PUNCT
cana-1058	44	1	we	we	PRON
cana-1058	44	2	recall	recall	VERB
cana-1058	44	3	the	the	DET
cana-1058	44	4	following	follow	VERB
cana-1058	44	5	definition	definition	NOUN
cana-1058	44	6	which	which	PRON
cana-1058	44	7	are	be	AUX
cana-1058	44	8	useful	useful	ADJ
cana-1058	44	9	in	in	ADP
cana-1058	44	10	the	the	DET
cana-1058	44	11	sequel	sequel	NOUN
cana-1058	44	12	.	.	PUNCT
cana-1058	45	1	definition	definition	NOUN
cana-1058	45	2	2.1	2.1	NUM
cana-1058	45	3	.	.	PUNCT
cana-1058	46	1	[	[	X
cana-1058	46	2	11	11	NUM
cana-1058	46	3	]	]	PUNCT
cana-1058	46	4	let	let	VERB
cana-1058	46	5	u	u	PRON
cana-1058	46	6	be	be	AUX
cana-1058	46	7	a	a	DET
cana-1058	46	8	non	non	ADJ
cana-1058	46	9	-	-	ADJ
cana-1058	46	10	empty	empty	ADJ
cana-1058	46	11	finite	finite	NOUN
cana-1058	46	12	set	set	NOUN
cana-1058	46	13	of	of	ADP
cana-1058	46	14	objects	object	NOUN
cana-1058	46	15	called	call	VERB
cana-1058	46	16	the	the	DET
cana-1058	46	17	universe	universe	NOUN
cana-1058	46	18	r	r	NOUN
cana-1058	46	19	be	be	VERB
cana-1058	46	20	an	an	DET
cana-1058	46	21	equivalence	equivalence	NOUN
cana-1058	46	22	relation	relation	NOUN
cana-1058	46	23	on	on	ADP
cana-1058	46	24	u	u	NOUN
cana-1058	46	25	named	name	VERB
cana-1058	46	26	as	as	ADP
cana-1058	46	27	the	the	DET
cana-1058	46	28	indiscernibility	indiscernibility	NOUN
cana-1058	46	29	relation	relation	NOUN
cana-1058	46	30	.	.	PUNCT
cana-1058	47	1	elements	element	NOUN
cana-1058	47	2	belonging	belong	VERB
cana-1058	47	3	to	to	ADP
cana-1058	47	4	the	the	DET
cana-1058	47	5	same	same	ADJ
cana-1058	47	6	equivalence	equivalence	NOUN
cana-1058	47	7	class	class	NOUN
cana-1058	47	8	are	be	AUX
cana-1058	47	9	said	say	VERB
cana-1058	47	10	to	to	PART
cana-1058	47	11	be	be	AUX
cana-1058	47	12	indiscernible	indiscernible	ADJ
cana-1058	47	13	with	with	ADP
cana-1058	47	14	one	one	NUM
cana-1058	47	15	another	another	DET
cana-1058	47	16	.	.	PUNCT
cana-1058	48	1	the	the	DET
cana-1058	48	2	pair	pair	NOUN
cana-1058	48	3	(	(	PUNCT
cana-1058	48	4	u	u	NOUN
cana-1058	48	5	,	,	PUNCT
cana-1058	48	6	r	r	NOUN
cana-1058	48	7	)	)	PUNCT
cana-1058	48	8	is	be	AUX
cana-1058	48	9	said	say	VERB
cana-1058	48	10	to	to	PART
cana-1058	48	11	be	be	AUX
cana-1058	48	12	the	the	DET
cana-1058	48	13	approximation	approximation	NOUN
cana-1058	48	14	space	space	NOUN
cana-1058	48	15	.	.	PUNCT
cana-1058	49	1	let	let	VERB
cana-1058	49	2	x	x	SYM
cana-1058	49	3	⊆	⊆	X
cana-1058	49	4	u.	u.	NOUN
cana-1058	49	5	(	(	PUNCT
cana-1058	49	6	1	1	X
cana-1058	49	7	)	)	PUNCT
cana-1058	49	8	the	the	DET
cana-1058	49	9	lower	low	ADJ
cana-1058	49	10	approximation	approximation	NOUN
cana-1058	49	11	of	of	ADP
cana-1058	49	12	x	x	PUNCT
cana-1058	49	13	with	with	ADP
cana-1058	49	14	respect	respect	NOUN
cana-1058	49	15	to	to	ADP
cana-1058	49	16	r	r	NOUN
cana-1058	49	17	is	be	AUX
cana-1058	49	18	the	the	DET
cana-1058	49	19	set	set	NOUN
cana-1058	49	20	of	of	ADP
cana-1058	49	21	all	all	DET
cana-1058	49	22	objects	object	NOUN
cana-1058	49	23	,	,	PUNCT
cana-1058	49	24	which	which	PRON
cana-1058	49	25	can	can	AUX
cana-1058	49	26	be	be	AUX
cana-1058	49	27	for	for	ADP
cana-1058	49	28	certain	certain	ADJ
cana-1058	49	29	classified	classified	ADJ
cana-1058	49	30	as	as	ADP
cana-1058	49	31	x	x	PUNCT
cana-1058	49	32	with	with	ADP
cana-1058	49	33	respect	respect	NOUN
cana-1058	49	34	to	to	ADP
cana-1058	49	35	r	r	NOUN
cana-1058	49	36	and	and	CCONJ
cana-1058	49	37	it	it	PRON
cana-1058	49	38	is	be	AUX
cana-1058	49	39	denoted	denote	VERB
cana-1058	49	40	by	by	ADP
cana-1058	49	41	lr	lr	X
cana-1058	49	42	(	(	PUNCT
cana-1058	49	43	x	x	NOUN
cana-1058	49	44	)	)	PUNCT
cana-1058	49	45	.	.	PUNCT
cana-1058	50	1	that	that	ADV
cana-1058	50	2	is	is	ADV
cana-1058	50	3	,	,	PUNCT
cana-1058	50	4	lr	lr	INTJ
cana-1058	50	5	(	(	PUNCT
cana-1058	50	6	x	x	NOUN
cana-1058	50	7	)	)	PUNCT
cana-1058	50	8	=	=	PRON
cana-1058	50	9	{	{	PUNCT
cana-1058	50	10	𝖴𝑥∈𝑈	𝖴𝑥∈𝑈	PROPN
cana-1058	50	11	{	{	PUNCT
cana-1058	50	12	𝑅(𝑥	𝑅(𝑥	NOUN
cana-1058	50	13	):	):	PUNCT
cana-1058	50	14	𝑅(𝑥	𝑅(𝑥	NOUN
cana-1058	50	15	)	)	PUNCT
cana-1058	50	16	⊆𝑋	⊆𝑋	NOUN
cana-1058	50	17	}	}	PUNCT
cana-1058	50	18	}	}	PUNCT
cana-1058	50	19	,	,	PUNCT
cana-1058	50	20	where	where	SCONJ
cana-1058	50	21	r(x	r(x	NOUN
cana-1058	50	22	)	)	PUNCT
cana-1058	50	23	denotes	denote	VERB
cana-1058	50	24	the	the	DET
cana-1058	50	25	equivalence	equivalence	NOUN
cana-1058	50	26	class	class	NOUN
cana-1058	50	27	determined	determine	VERB
cana-1058	50	28	by	by	ADP
cana-1058	50	29	x.	x.	PROPN
cana-1058	50	30	(	(	PUNCT
cana-1058	50	31	2	2	X
cana-1058	50	32	)	)	PUNCT
cana-1058	50	33	the	the	DET
cana-1058	50	34	upper	upper	ADJ
cana-1058	50	35	approximation	approximation	NOUN
cana-1058	50	36	of	of	ADP
cana-1058	50	37	x	x	PUNCT
cana-1058	50	38	with	with	ADP
cana-1058	50	39	respect	respect	NOUN
cana-1058	50	40	to	to	ADP
cana-1058	50	41	r	r	NOUN
cana-1058	50	42	is	be	AUX
cana-1058	50	43	the	the	DET
cana-1058	50	44	set	set	NOUN
cana-1058	50	45	of	of	ADP
cana-1058	50	46	all	all	DET
cana-1058	50	47	objects	object	NOUN
cana-1058	50	48	,	,	PUNCT
cana-1058	50	49	which	which	PRON
cana-1058	50	50	can	can	AUX
cana-1058	50	51	be	be	AUX
cana-1058	50	52	for	for	ADP
cana-1058	50	53	certain	certain	ADJ
cana-1058	50	54	classified	classified	ADJ
cana-1058	50	55	as	as	ADP
cana-1058	50	56	x	x	PUNCT
cana-1058	50	57	with	with	ADP
cana-1058	50	58	respect	respect	NOUN
cana-1058	50	59	to	to	ADP
cana-1058	50	60	r	r	NOUN
cana-1058	50	61	and	and	CCONJ
cana-1058	50	62	it	it	PRON
cana-1058	50	63	is	be	AUX
cana-1058	50	64	denoted	denote	VERB
cana-1058	50	65	by	by	ADP
cana-1058	50	66	ur	ur	INTJ
cana-1058	50	67	(	(	PUNCT
cana-1058	50	68	x	x	NOUN
cana-1058	50	69	)	)	PUNCT
cana-1058	50	70	.	.	PUNCT
cana-1058	51	1	that	that	ADV
cana-1058	51	2	is	is	ADV
cana-1058	51	3	,	,	PUNCT
cana-1058	51	4	ur(x	ur(x	X
cana-1058	51	5	)	)	PUNCT
cana-1058	51	6	=	=	SYM
cana-1058	51	7	{	{	PUNCT
cana-1058	51	8	𝖴𝑥∈𝑈	𝖴𝑥∈𝑈	PROPN
cana-1058	51	9	{	{	PUNCT
cana-1058	51	10	𝑅(𝑥	𝑅(𝑥	NOUN
cana-1058	51	11	):	):	PUNCT
cana-1058	51	12	𝑅(𝑥	𝑅(𝑥	NOUN
cana-1058	51	13	)	)	PUNCT
cana-1058	51	14	∩	∩	NOUN
cana-1058	51	15	𝑋	𝑋	PROPN
cana-1058	51	16	≠	≠	PROPN
cana-1058	51	17	∅	∅	NOUN
cana-1058	51	18	}	}	PUNCT
cana-1058	51	19	}	}	PUNCT
cana-1058	51	20	(	(	PUNCT
cana-1058	51	21	3	3	X
cana-1058	51	22	)	)	PUNCT
cana-1058	51	23	the	the	DET
cana-1058	51	24	boundary	boundary	ADJ
cana-1058	51	25	region	region	NOUN
cana-1058	51	26	of	of	ADP
cana-1058	51	27	x	x	PUNCT
cana-1058	51	28	with	with	ADP
cana-1058	51	29	respect	respect	NOUN
cana-1058	51	30	to	to	ADP
cana-1058	51	31	r	r	NOUN
cana-1058	51	32	is	be	AUX
cana-1058	51	33	the	the	DET
cana-1058	51	34	set	set	NOUN
cana-1058	51	35	of	of	ADP
cana-1058	51	36	all	all	DET
cana-1058	51	37	objects	object	NOUN
cana-1058	51	38	,	,	PUNCT
cana-1058	51	39	which	which	PRON
cana-1058	51	40	can	can	AUX
cana-1058	51	41	be	be	AUX
cana-1058	51	42	classified	classify	VERB
cana-1058	51	43	as	as	ADP
cana-1058	51	44	neither	neither	CCONJ
cana-1058	51	45	as	as	ADP
cana-1058	51	46	x	x	NUM
cana-1058	51	47	nor	nor	CCONJ
cana-1058	51	48	as	as	ADP
cana-1058	51	49	not	not	PART
cana-1058	51	50	x	x	PUNCT
cana-1058	51	51	with	with	ADP
cana-1058	51	52	respect	respect	NOUN
cana-1058	51	53	to	to	ADP
cana-1058	51	54	r	r	NOUN
cana-1058	51	55	and	and	CCONJ
cana-1058	51	56	it	it	PRON
cana-1058	51	57	is	be	AUX
cana-1058	51	58	denoted	denote	VERB
cana-1058	51	59	by	by	ADP
cana-1058	51	60	br	br	PROPN
cana-1058	51	61	(	(	PUNCT
cana-1058	51	62	x	x	NOUN
cana-1058	51	63	)	)	PUNCT
cana-1058	51	64	.	.	PUNCT
cana-1058	52	1	that	that	PRON
cana-1058	52	2	is	be	AUX
cana-1058	52	3	,	,	PUNCT
cana-1058	52	4	br	br	PROPN
cana-1058	52	5	(	(	PUNCT
cana-1058	52	6	x	x	X
cana-1058	52	7	)	)	PUNCT
cana-1058	52	8	=	=	SYM
cana-1058	52	9	ur(x	ur(x	X
cana-1058	52	10	)	)	PUNCT
cana-1058	52	11	lr	lr	X
cana-1058	52	12	(	(	PUNCT
cana-1058	52	13	x	x	NOUN
cana-1058	52	14	)	)	PUNCT
cana-1058	52	15	definition	definition	NOUN
cana-1058	52	16	2.2	2.2	NUM
cana-1058	52	17	.	.	PUNCT
cana-1058	53	1	let	let	VERB
cana-1058	53	2	u	u	PRON
cana-1058	53	3	be	be	AUX
cana-1058	53	4	the	the	DET
cana-1058	53	5	universe	universe	NOUN
cana-1058	53	6	,	,	PUNCT
cana-1058	53	7	r	r	NOUN
cana-1058	53	8	be	be	VERB
cana-1058	53	9	an	an	DET
cana-1058	53	10	equivalence	equivalence	NOUN
cana-1058	53	11	relation	relation	NOUN
cana-1058	53	12	on	on	ADP
cana-1058	53	13	u	u	NOUN
cana-1058	53	14	and	and	CCONJ
cana-1058	53	15	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	53	16	)	)	PUNCT
cana-1058	53	17	=	=	SYM
cana-1058	53	18	{	{	PUNCT
cana-1058	53	19	𝑈	𝑈	PROPN
cana-1058	53	20	,	,	PUNCT
cana-1058	53	21	∅	∅	NOUN
cana-1058	53	22	,	,	PUNCT
cana-1058	53	23	𝐿𝑅	𝐿𝑅	PROPN
cana-1058	53	24	(	(	PUNCT
cana-1058	53	25	x	x	NOUN
cana-1058	53	26	)	)	PUNCT
cana-1058	53	27	,	,	PUNCT
cana-1058	53	28	ur(x	ur(x	PROPN
cana-1058	53	29	)	)	PUNCT
cana-1058	53	30	,	,	PUNCT
cana-1058	53	31	br(x	br(x	NOUN
cana-1058	53	32	)	)	PUNCT
cana-1058	53	33	}	}	PUNCT
cana-1058	53	34	where	where	SCONJ
cana-1058	53	35	x	x	X
cana-1058	53	36	⊆	⊆	X
cana-1058	53	37	u.	u.	NOUN
cana-1058	53	38	then	then	ADV
cana-1058	53	39	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	53	40	)	)	PUNCT
cana-1058	53	41	satisfies	satisfy	VERB
cana-1058	53	42	the	the	DET
cana-1058	53	43	following	follow	VERB
cana-1058	53	44	axioms	axiom	NOUN
cana-1058	53	45	:	:	PUNCT
cana-1058	53	46	(	(	PUNCT
cana-1058	53	47	1	1	X
cana-1058	53	48	)	)	PUNCT
cana-1058	53	49	u	u	NOUN
cana-1058	53	50	and	and	CCONJ
cana-1058	53	51	∅∈𝑟𝑅(𝑥	∅∈𝑟𝑅(𝑥	NOUN
cana-1058	53	52	)	)	PUNCT
cana-1058	53	53	(	(	PUNCT
cana-1058	53	54	2	2	X
cana-1058	53	55	)	)	PUNCT
cana-1058	53	56	the	the	DET
cana-1058	53	57	union	union	NOUN
cana-1058	53	58	of	of	ADP
cana-1058	53	59	elements	element	NOUN
cana-1058	53	60	of	of	ADP
cana-1058	53	61	any	any	DET
cana-1058	53	62	subcollection	subcollection	NOUN
cana-1058	53	63	of	of	ADP
cana-1058	53	64	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	53	65	)	)	PUNCT
cana-1058	53	66	is	be	AUX
cana-1058	53	67	in	in	ADP
cana-1058	53	68	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	53	69	)	)	PUNCT
cana-1058	53	70	.	.	PUNCT
cana-1058	54	1	(	(	PUNCT
cana-1058	54	2	3	3	X
cana-1058	54	3	)	)	PUNCT
cana-1058	54	4	the	the	DET
cana-1058	54	5	intersection	intersection	NOUN
cana-1058	54	6	of	of	ADP
cana-1058	54	7	the	the	DET
cana-1058	54	8	elements	element	NOUN
cana-1058	54	9	of	of	ADP
cana-1058	54	10	any	any	DET
cana-1058	54	11	finite	finite	ADJ
cana-1058	54	12	subcollection	subcollection	NOUN
cana-1058	54	13	of	of	ADP
cana-1058	54	14	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	54	15	)	)	PUNCT
cana-1058	54	16	is	be	AUX
cana-1058	54	17	in	in	ADP
cana-1058	54	18	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	54	19	)	)	PUNCT
cana-1058	54	20	.	.	PUNCT
cana-1058	55	1	that	that	ADV
cana-1058	55	2	is	is	ADV
cana-1058	55	3	,	,	PUNCT
cana-1058	55	4	𝑟𝑅(𝑥	𝑟𝑅(𝑥	X
cana-1058	55	5	)	)	PUNCT
cana-1058	55	6	forms	form	VERB
cana-1058	55	7	a	a	DET
cana-1058	55	8	topology	topology	NOUN
cana-1058	55	9	on	on	ADP
cana-1058	55	10	u	u	NOUN
cana-1058	55	11	called	call	VERB
cana-1058	55	12	the	the	DET
cana-1058	55	13	nano	nano	NOUN
cana-1058	55	14	topology	topology	NOUN
cana-1058	55	15	on	on	ADP
cana-1058	55	16	u	u	NOUN
cana-1058	55	17	with	with	ADP
cana-1058	55	18	respect	respect	NOUN
cana-1058	55	19	to	to	ADP
cana-1058	55	20	x.	x.	NOUN
cana-1058	55	21	we	we	PRON
cana-1058	55	22	call	call	VERB
cana-1058	55	23	{	{	PUNCT
cana-1058	55	24	u	u	NOUN
cana-1058	55	25	,	,	PUNCT
cana-1058	55	26	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	55	27	)	)	PUNCT
cana-1058	55	28	}	}	PUNCT
cana-1058	55	29	is	be	AUX
cana-1058	55	30	called	call	VERB
cana-1058	55	31	the	the	DET
cana-1058	55	32	nano	nano	NOUN
cana-1058	55	33	topological	topological	ADJ
cana-1058	55	34	space	space	NOUN
cana-1058	55	35	.	.	PUNCT
cana-1058	56	1	the	the	DET
cana-1058	56	2	elements	element	NOUN
cana-1058	56	3	of	of	ADP
cana-1058	56	4	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	56	5	)	)	PUNCT
cana-1058	56	6	are	be	AUX
cana-1058	56	7	called	call	VERB
cana-1058	56	8	as	as	ADP
cana-1058	56	9	nano	nano	NOUN
cana-1058	56	10	-	-	PUNCT
cana-1058	56	11	open	open	ADJ
cana-1058	56	12	sets	set	NOUN
cana-1058	56	13	.	.	PUNCT
cana-1058	57	1	the	the	DET
cana-1058	57	2	complement	complement	NOUN
cana-1058	57	3	of	of	ADP
cana-1058	57	4	the	the	DET
cana-1058	57	5	nano	nano	NOUN
cana-1058	57	6	-	-	PUNCT
cana-1058	57	7	open	open	ADJ
cana-1058	57	8	sets	set	NOUN
cana-1058	57	9	are	be	AUX
cana-1058	57	10	called	call	VERB
cana-1058	57	11	nano	nano	NOUN
cana-1058	57	12	-	-	PUNCT
cana-1058	57	13	closed	close	VERB
cana-1058	57	14	sets	set	NOUN
cana-1058	57	15	.	.	PUNCT
cana-1058	58	1	communications	communication	NOUN
cana-1058	58	2	on	on	ADP
cana-1058	58	3	applied	apply	VERB
cana-1058	58	4	nonlinear	nonlinear	ADJ
cana-1058	58	5	analysis	analysis	NOUN
cana-1058	58	6	issn	issn	NOUN
cana-1058	58	7	:	:	PUNCT
cana-1058	58	8	1074	1074	NUM
cana-1058	58	9	-	-	PUNCT
cana-1058	58	10	133x	133x	NUM
cana-1058	58	11	vol	vol	NOUN
cana-1058	58	12	31	31	NUM
cana-1058	58	13	no	no	NOUN
cana-1058	58	14	.	.	PUNCT
cana-1058	59	1	5s	5s	NUM
cana-1058	59	2	(	(	PUNCT
cana-1058	59	3	2024	2024	NUM
cana-1058	59	4	)	)	PUNCT
cana-1058	59	5	392	392	NUM
cana-1058	59	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1058	59	7	definition	definition	NOUN
cana-1058	59	8	2.3	2.3	NUM
cana-1058	59	9	.	.	PUNCT
cana-1058	60	1	an	an	DET
cana-1058	60	2	ideal	ideal	NOUN
cana-1058	60	3	i	i	PRON
cana-1058	60	4	on	on	ADP
cana-1058	60	5	a	a	DET
cana-1058	60	6	topological	topological	ADJ
cana-1058	60	7	space	space	NOUN
cana-1058	60	8	is	be	AUX
cana-1058	60	9	a	a	DET
cana-1058	60	10	non	non	ADJ
cana-1058	60	11	-	-	ADJ
cana-1058	60	12	empty	empty	ADJ
cana-1058	60	13	collection	collection	NOUN
cana-1058	60	14	of	of	ADP
cana-1058	60	15	subsets	subset	NOUN
cana-1058	60	16	of	of	ADP
cana-1058	60	17	x	x	PUNCT
cana-1058	60	18	which	which	DET
cana-1058	60	19	satisfies	satisfy	VERB
cana-1058	60	20	(	(	PUNCT
cana-1058	60	21	1	1	X
cana-1058	60	22	)	)	PUNCT
cana-1058	60	23	a	a	DET
cana-1058	60	24	𝜖	𝜖	PROPN
cana-1058	60	25	i	i	NOUN
cana-1058	60	26	and	and	CCONJ
cana-1058	60	27	b	b	X
cana-1058	60	28	⊆	⊆	NUM
cana-1058	60	29	a	a	DET
cana-1058	60	30			NOUN
cana-1058	60	31	b	b	PROPN
cana-1058	60	32	𝜖	𝜖	PROPN
cana-1058	60	33	i.	i.	PROPN
cana-1058	60	34	(	(	PUNCT
cana-1058	60	35	2	2	NUM
cana-1058	60	36	)	)	PUNCT
cana-1058	60	37	a	a	DET
cana-1058	60	38	𝜖	𝜖	PROPN
cana-1058	60	39	i	i	PROPN
cana-1058	60	40	and	and	CCONJ
cana-1058	60	41	b	b	PROPN
cana-1058	60	42	𝜖	𝜖	PROPN
cana-1058	60	43	i	i	PRON
cana-1058	60	44			VERB
cana-1058	60	45	a	a	DET
cana-1058	60	46	𝖴	𝖴	PROPN
cana-1058	60	47	b	b	PROPN
cana-1058	60	48	𝜖	𝜖	PROPN
cana-1058	60	49	i.	i.	NOUN
cana-1058	60	50	definition	definition	NOUN
cana-1058	60	51	2.4	2.4	NUM
cana-1058	60	52	[	[	SYM
cana-1058	60	53	13	13	NUM
cana-1058	60	54	]	]	PUNCT
cana-1058	60	55	let	let	VERB
cana-1058	60	56	(	(	PUNCT
cana-1058	60	57	x	x	X
cana-1058	60	58	,	,	PUNCT
cana-1058	60	59	𝑟	𝑟	AUX
cana-1058	60	60	)	)	PUNCT
cana-1058	60	61	be	be	AUX
cana-1058	60	62	a	a	DET
cana-1058	60	63	topological	topological	ADJ
cana-1058	60	64	space	space	NOUN
cana-1058	60	65	and	and	CCONJ
cana-1058	60	66	i	i	PRON
cana-1058	60	67	be	be	VERB
cana-1058	60	68	an	an	DET
cana-1058	60	69	ideal	ideal	NOUN
cana-1058	60	70	on	on	ADP
cana-1058	60	71	x.	x.	PROPN
cana-1058	60	72	a	a	DET
cana-1058	60	73	subset	subset	NOUN
cana-1058	60	74	a	a	PRON
cana-1058	60	75	of	of	ADP
cana-1058	60	76	x	x	SYM
cana-1058	60	77	is	be	AUX
cana-1058	60	78	said	say	VERB
cana-1058	60	79	to	to	PART
cana-1058	60	80	be	be	AUX
cana-1058	60	81	αig	αig	NOUN
cana-1058	60	82	-	-	PUNCT
cana-1058	60	83	closed	closed	ADJ
cana-1058	60	84	if	if	SCONJ
cana-1058	60	85	𝐴∗⊆𝑈	𝐴∗⊆𝑈	ADJ
cana-1058	60	86	whenever	whenever	SCONJ
cana-1058	60	87	a	a	DET
cana-1058	60	88	⊆𝑈	⊆𝑈	NUM
cana-1058	60	89	and	and	CCONJ
cana-1058	60	90	u	u	NOUN
cana-1058	60	91	is	be	AUX
cana-1058	60	92	α	α	NOUN
cana-1058	60	93	-	-	ADJ
cana-1058	60	94	open	open	ADJ
cana-1058	60	95	.	.	PUNCT
cana-1058	61	1	definition	definition	NOUN
cana-1058	61	2	2.5	2.5	NUM
cana-1058	61	3	.	.	PUNCT
cana-1058	62	1	[	[	X
cana-1058	62	2	12	12	NUM
cana-1058	62	3	]	]	PUNCT
cana-1058	62	4	a	a	DET
cana-1058	62	5	nano	nano	ADJ
cana-1058	62	6	topological	topological	ADJ
cana-1058	62	7	space	space	NOUN
cana-1058	62	8	{	{	PUNCT
cana-1058	62	9	u	u	NOUN
cana-1058	62	10	,	,	PUNCT
cana-1058	62	11	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	62	12	)	)	PUNCT
cana-1058	62	13	}	}	PUNCT
cana-1058	62	14	with	with	ADP
cana-1058	62	15	an	an	DET
cana-1058	62	16	ideal	ideal	ADJ
cana-1058	62	17	i	i	PRON
cana-1058	62	18	on	on	ADP
cana-1058	62	19	u	u	NOUN
cana-1058	62	20	is	be	AUX
cana-1058	62	21	called	call	VERB
cana-1058	62	22	a	a	DET
cana-1058	62	23	nano	nano	NOUN
cana-1058	62	24	ideal	ideal	ADJ
cana-1058	62	25	topological	topological	ADJ
cana-1058	62	26	space	space	NOUN
cana-1058	62	27	or	or	CCONJ
cana-1058	62	28	nano	nano	NOUN
cana-1058	62	29	ideal	ideal	ADJ
cana-1058	62	30	space	space	NOUN
cana-1058	62	31	and	and	CCONJ
cana-1058	62	32	denoted	denote	VERB
cana-1058	62	33	as	as	ADP
cana-1058	62	34	(	(	PUNCT
cana-1058	62	35	u	u	NOUN
cana-1058	62	36	,	,	PUNCT
cana-1058	62	37	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	62	38	)	)	PUNCT
cana-1058	62	39	,	,	PUNCT
cana-1058	62	40	i	i	PRON
cana-1058	62	41	)	)	PUNCT
cana-1058	62	42	definition	definition	NOUN
cana-1058	62	43	2.6	2.6	NUM
cana-1058	62	44	.	.	PUNCT
cana-1058	63	1	let	let	VERB
cana-1058	63	2	{	{	PUNCT
cana-1058	63	3	u	u	NOUN
cana-1058	63	4	,	,	PUNCT
cana-1058	63	5	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	63	6	)	)	PUNCT
cana-1058	63	7	,	,	PUNCT
cana-1058	63	8	i	i	PRON
cana-1058	63	9	}	}	PUNCT
cana-1058	63	10	be	be	VERB
cana-1058	63	11	a	a	DET
cana-1058	63	12	nano	nano	NOUN
cana-1058	63	13	ideal	ideal	ADJ
cana-1058	63	14	topological	topological	ADJ
cana-1058	63	15	space	space	NOUN
cana-1058	63	16	.	.	PUNCT
cana-1058	64	1	a	a	DET
cana-1058	64	2	set	set	NOUN
cana-1058	64	3	operator	operator	NOUN
cana-1058	64	4	(	(	PUNCT
cana-1058	64	5	a)*n	a)*n	NOUN
cana-1058	64	6	:	:	PUNCT
cana-1058	64	7	p(u	p(u	NUM
cana-1058	64	8	)	)	PUNCT
cana-1058	64	9	→p(u)is	→p(u)i	VERB
cana-1058	64	10	called	call	VERB
cana-1058	64	11	the	the	DET
cana-1058	64	12	nano	nano	VERB
cana-1058	64	13	local	local	ADJ
cana-1058	64	14	function	function	NOUN
cana-1058	64	15	of	of	ADP
cana-1058	64	16	i	i	PRON
cana-1058	64	17	on	on	ADP
cana-1058	64	18	u	u	NOUN
cana-1058	64	19	with	with	ADP
cana-1058	64	20	respect	respect	NOUN
cana-1058	64	21	to	to	ADP
cana-1058	64	22	i	i	PRON
cana-1058	64	23	on	on	ADP
cana-1058	64	24	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	64	25	)	)	PUNCT
cana-1058	64	26	is	be	AUX
cana-1058	64	27	defined	define	VERB
cana-1058	64	28	as	as	ADP
cana-1058	64	29	(	(	PUNCT
cana-1058	64	30	a)*n	a)*n	X
cana-1058	64	31	=	=	SYM
cana-1058	64	32	{	{	PUNCT
cana-1058	64	33	x	x	X
cana-1058	64	34	𝜖u	𝜖u	PROPN
cana-1058	64	35	:	:	PUNCT
cana-1058	64	36	u	u	NOUN
cana-1058	64	37	∩	∩	NOUN
cana-1058	64	38	a	a	DET
cana-1058	64	39	∉	∉	PROPN
cana-1058	64	40	i	i	X
cana-1058	64	41	;	;	PUNCT
cana-1058	64	42	for	for	ADP
cana-1058	64	43	every	every	DET
cana-1058	64	44	u	u	PROPN
cana-1058	64	45	𝜖𝑟𝑅(𝑥	𝜖𝑟𝑅(𝑥	PROPN
cana-1058	64	46	)	)	PUNCT
cana-1058	64	47	}	}	PUNCT
cana-1058	64	48	and	and	CCONJ
cana-1058	64	49	is	be	AUX
cana-1058	64	50	denoted	denote	VERB
cana-1058	64	51	by	by	ADP
cana-1058	64	52	(	(	PUNCT
cana-1058	64	53	a)*n	a)*n	PROPN
cana-1058	64	54	,	,	PUNCT
cana-1058	64	55	where	where	SCONJ
cana-1058	64	56	nano	nano	NOUN
cana-1058	64	57	closure	closure	NOUN
cana-1058	64	58	operator	operator	NOUN
cana-1058	64	59	is	be	AUX
cana-1058	64	60	defined	define	VERB
cana-1058	64	61	as	as	ADP
cana-1058	64	62	ncl*(a	ncl*(a	NOUN
cana-1058	64	63	)	)	PUNCT
cana-1058	64	64	=	=	NOUN
cana-1058	65	1	a	a	DET
cana-1058	65	2	𝖴(a)*n	𝖴(a)*n	PROPN
cana-1058	65	3	.	.	PUNCT
cana-1058	65	4	result	result	PROPN
cana-1058	65	5	2.7	2.7	NUM
cana-1058	65	6	.	.	PUNCT
cana-1058	66	1	let	let	VERB
cana-1058	66	2	(	(	PUNCT
cana-1058	66	3	u	u	NOUN
cana-1058	66	4	,	,	PUNCT
cana-1058	66	5	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	66	6	)	)	PUNCT
cana-1058	66	7	,	,	PUNCT
cana-1058	66	8	i	i	PRON
cana-1058	66	9	)	)	PUNCT
cana-1058	66	10	be	be	VERB
cana-1058	66	11	a	a	DET
cana-1058	66	12	nano	nano	NOUN
cana-1058	66	13	ideal	ideal	ADJ
cana-1058	66	14	topological	topological	ADJ
cana-1058	66	15	space	space	NOUN
cana-1058	66	16	and	and	CCONJ
cana-1058	66	17	let	let	VERB
cana-1058	66	18	a	a	PRON
cana-1058	66	19	and	and	CCONJ
cana-1058	66	20	b	b	NOUN
cana-1058	66	21	be	be	AUX
cana-1058	66	22	subsets	subset	NOUN
cana-1058	66	23	of	of	ADP
cana-1058	66	24	u	u	NOUN
cana-1058	66	25	,	,	PUNCT
cana-1058	66	26	then	then	ADV
cana-1058	66	27	(	(	PUNCT
cana-1058	66	28	1	1	X
cana-1058	66	29	)	)	PUNCT
cana-1058	66	30	(	(	PUNCT
cana-1058	66	31	∅)∗𝑁	∅)∗𝑁	NOUN
cana-1058	66	32	=	=	NOUN
cana-1058	66	33	∅	∅	NOUN
cana-1058	66	34	(	(	PUNCT
cana-1058	66	35	2	2	NUM
cana-1058	66	36	)	)	PUNCT
cana-1058	66	37	𝐴⊂𝐵	𝐴⊂𝐵	NOUN
cana-1058	66	38	→	→	SYM
cana-1058	66	39	(	(	PUNCT
cana-1058	66	40	𝐴)∗𝑁⊂	𝐴)∗𝑁⊂	X
cana-1058	66	41	(	(	PUNCT
cana-1058	66	42	𝐵)∗𝑁	𝐵)∗𝑁	X
cana-1058	66	43	(	(	PUNCT
cana-1058	66	44	3	3	NUM
cana-1058	66	45	)	)	PUNCT
cana-1058	66	46	for	for	ADP
cana-1058	66	47	another	another	DET
cana-1058	66	48	𝐽⊇𝐼𝑜𝑛𝑈	𝐽⊇𝐼𝑜𝑛𝑈	PROPN
cana-1058	66	49	,	,	PUNCT
cana-1058	66	50	(	(	PUNCT
cana-1058	66	51	𝐴)∗𝑁(𝐽	𝐴)∗𝑁(𝐽	PROPN
cana-1058	66	52	)	)	PUNCT
cana-1058	67	1	⊂	⊂	PROPN
cana-1058	67	2	(	(	PUNCT
cana-1058	67	3	𝐴)∗𝑁(𝐼	𝐴)∗𝑁(𝐼	PROPN
cana-1058	67	4	)	)	PUNCT
cana-1058	67	5	(	(	PUNCT
cana-1058	67	6	4	4	NUM
cana-1058	67	7	)	)	PUNCT
cana-1058	67	8	(	(	PUNCT
cana-1058	67	9	𝐴)∗𝑁⊂𝑁𝑐𝑙∗(𝐴	𝐴)∗𝑁⊂𝑁𝑐𝑙∗(𝐴	NOUN
cana-1058	67	10	)	)	PUNCT
cana-1058	67	11	(	(	PUNCT
cana-1058	67	12	5	5	NUM
cana-1058	67	13	)	)	PUNCT
cana-1058	67	14	(	(	PUNCT
cana-1058	67	15	𝐴)∗𝑁	𝐴)∗𝑁	X
cana-1058	67	16	is	be	AUX
cana-1058	67	17	a	a	DET
cana-1058	67	18	nano	nano	NOUN
cana-1058	67	19	closed	close	VERB
cana-1058	67	20	set	set	NOUN
cana-1058	67	21	.	.	PUNCT
cana-1058	68	1	(	(	PUNCT
cana-1058	68	2	6	6	NUM
cana-1058	68	3	)	)	PUNCT
cana-1058	68	4	(	(	PUNCT
cana-1058	68	5	(	(	PUNCT
cana-1058	68	6	𝐴)∗𝑁)∗𝑁⊂	𝐴)∗𝑁)∗𝑁⊂	NOUN
cana-1058	68	7	(	(	PUNCT
cana-1058	68	8	𝐴)∗𝑁	𝐴)∗𝑁	PUNCT
cana-1058	68	9	(	(	PUNCT
cana-1058	68	10	7	7	NUM
cana-1058	68	11	)	)	PUNCT
cana-1058	68	12	(	(	PUNCT
cana-1058	68	13	𝐴)∗𝑁𝖴	𝐴)∗𝑁𝖴	X
cana-1058	68	14	(	(	PUNCT
cana-1058	68	15	𝐵)∗𝑁	𝐵)∗𝑁	PUNCT
cana-1058	68	16	=	=	SYM
cana-1058	68	17	(	(	PUNCT
cana-1058	68	18	𝐴𝖴𝐵)∗𝑁	𝐴𝖴𝐵)∗𝑁	NOUN
cana-1058	68	19	(	(	PUNCT
cana-1058	68	20	8)	8)	NUM
cana-1058	68	21	(	(	PUNCT
cana-1058	68	22	𝐴	𝐴	PROPN
cana-1058	68	23	∩	∩	NOUN
cana-1058	68	24	𝐵)∗𝑁	𝐵)∗𝑁	PUNCT
cana-1058	68	25	=	=	SYM
cana-1058	68	26	(	(	PUNCT
cana-1058	68	27	𝐴)∗𝑁	𝐴)∗𝑁	X
cana-1058	68	28	∩	∩	NOUN
cana-1058	68	29	(	(	PUNCT
cana-1058	68	30	𝐵)∗𝑁	𝐵)∗𝑁	X
cana-1058	68	31	(	(	PUNCT
cana-1058	68	32	9	9	NUM
cana-1058	68	33	)	)	PUNCT
cana-1058	68	34	for	for	ADP
cana-1058	68	35	every	every	DET
cana-1058	68	36	nano	nano	NOUN
cana-1058	68	37	open	open	ADJ
cana-1058	68	38	set	set	VERB
cana-1058	68	39	v	v	ADP
cana-1058	68	40	,	,	PUNCT
cana-1058	68	41	v∩	v∩	PROPN
cana-1058	68	42	(	(	PUNCT
cana-1058	68	43	𝑉	𝑉	PROPN
cana-1058	68	44	∩	∩	NOUN
cana-1058	68	45	𝐴)∗𝑁⊂	𝐴)∗𝑁⊂	X
cana-1058	68	46	(	(	PUNCT
cana-1058	68	47	𝑉	𝑉	PROPN
cana-1058	68	48	∩	∩	NOUN
cana-1058	68	49	𝐴)∗𝑁	𝐴)∗𝑁	PUNCT
cana-1058	68	50	(	(	PUNCT
cana-1058	68	51	10	10	NUM
cana-1058	68	52	)	)	PUNCT
cana-1058	68	53	for	for	ADP
cana-1058	68	54	i	i	PRON
cana-1058	68	55	∈𝐼	∈𝐼	PROPN
cana-1058	68	56	,	,	PUNCT
cana-1058	68	57	(	(	PUNCT
cana-1058	68	58	𝐴𝖴𝐼)∗𝑁	𝐴𝖴𝐼)∗𝑁	NOUN
cana-1058	68	59	=	=	SYM
cana-1058	68	60	(	(	PUNCT
cana-1058	68	61	𝐴)∗𝑁	𝐴)∗𝑁	X
cana-1058	68	62	=	=	SYM
cana-1058	68	63	(	(	PUNCT
cana-1058	68	64	𝐴	𝐴	PROPN
cana-1058	68	65	−	−	PROPN
cana-1058	68	66	𝐼)∗𝑁	𝐼)∗𝑁	NUM
cana-1058	68	67	result	result	VERB
cana-1058	68	68	2.8	2.8	NUM
cana-1058	68	69	.	.	PUNCT
cana-1058	69	1	let	let	VERB
cana-1058	69	2	{	{	PUNCT
cana-1058	69	3	u	u	NOUN
cana-1058	69	4	,	,	PUNCT
cana-1058	69	5	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	69	6	)	)	PUNCT
cana-1058	69	7	,	,	PUNCT
cana-1058	69	8	i	i	PRON
cana-1058	69	9	}	}	PUNCT
cana-1058	69	10	be	be	VERB
cana-1058	69	11	a	a	DET
cana-1058	69	12	nano	nano	NOUN
cana-1058	69	13	ideal	ideal	ADJ
cana-1058	69	14	topological	topological	ADJ
cana-1058	69	15	space	space	NOUN
cana-1058	69	16	and	and	CCONJ
cana-1058	69	17	a	a	DET
cana-1058	69	18	be	be	AUX
cana-1058	69	19	a	a	DET
cana-1058	69	20	subset	subset	NOUN
cana-1058	69	21	of	of	ADP
cana-1058	69	22	u	u	NOUN
cana-1058	69	23	,	,	PUNCT
cana-1058	69	24	if	if	SCONJ
cana-1058	69	25	a	a	DET
cana-1058	69	26	⊂	⊂	X
cana-1058	69	27	(	(	PUNCT
cana-1058	69	28	𝐴)∗𝑁	𝐴)∗𝑁	PROPN
cana-1058	69	29	,	,	PUNCT
cana-1058	69	30	then	then	ADV
cana-1058	69	31	(	(	PUNCT
cana-1058	69	32	𝐴)∗𝑁	𝐴)∗𝑁	CCONJ
cana-1058	69	33	=	=	SYM
cana-1058	69	34	𝑁𝑐𝑙	𝑁𝑐𝑙	PROPN
cana-1058	69	35	(	(	PUNCT
cana-1058	69	36	𝐴)∗𝑁	𝐴)∗𝑁	VERB
cana-1058	69	37	=	=	SYM
cana-1058	69	38	𝑁𝑐𝑙(𝐴	𝑁𝑐𝑙(𝐴	NOUN
cana-1058	69	39	)	)	PUNCT
cana-1058	69	40	=	=	SYM
cana-1058	69	41	𝑁𝑐𝑙∗(𝐴	𝑁𝑐𝑙∗(𝐴	PROPN
cana-1058	69	42	)	)	PUNCT
cana-1058	69	43	.	.	PUNCT
cana-1058	70	1	definition	definition	NOUN
cana-1058	70	2	2.9	2.9	NUM
cana-1058	70	3	.	.	PUNCT
cana-1058	71	1	let	let	VERB
cana-1058	71	2	{	{	PUNCT
cana-1058	71	3	u	u	NOUN
cana-1058	71	4	,	,	PUNCT
cana-1058	71	5	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	71	6	)	)	PUNCT
cana-1058	71	7	}	}	PUNCT
cana-1058	71	8	be	be	AUX
cana-1058	71	9	a	a	DET
cana-1058	71	10	nano	nano	ADJ
cana-1058	71	11	topological	topological	ADJ
cana-1058	71	12	space	space	NOUN
cana-1058	71	13	and	and	CCONJ
cana-1058	71	14	a	a	DET
cana-1058	71	15	⊆	⊆	NUM
cana-1058	71	16	u.	u.	NOUN
cana-1058	71	17	then	then	ADV
cana-1058	71	18	a	a	PRON
cana-1058	71	19	is	be	AUX
cana-1058	71	20	said	say	VERB
cana-1058	71	21	to	to	PART
cana-1058	71	22	be	be	AUX
cana-1058	71	23	(	(	PUNCT
cana-1058	71	24	1	1	X
cana-1058	71	25	)	)	PUNCT
cana-1058	71	26	nano	nano	NOUN
cana-1058	71	27	semiclosed	semiclose	VERB
cana-1058	71	28	,	,	PUNCT
cana-1058	71	29	if	if	SCONJ
cana-1058	71	30	nint	nint	NOUN
cana-1058	71	31	(	(	PUNCT
cana-1058	71	32	n	n	CCONJ
cana-1058	71	33	cl	cl	NOUN
cana-1058	71	34	(	(	PUNCT
cana-1058	71	35	a	a	NOUN
cana-1058	71	36	)	)	PUNCT
cana-1058	71	37	)	)	PUNCT
cana-1058	72	1	⊆	⊆	NUM
cana-1058	72	2	a.	a.	NOUN
cana-1058	72	3	(	(	PUNCT
cana-1058	72	4	2	2	NUM
cana-1058	72	5	)	)	PUNCT
cana-1058	72	6	ng	ng	NOUN
cana-1058	72	7	-	-	PUNCT
cana-1058	72	8	closed	closed	ADJ
cana-1058	72	9	,	,	PUNCT
cana-1058	72	10	ncl(a	ncl(a	PROPN
cana-1058	72	11	)	)	PUNCT
cana-1058	72	12	⊆g	⊆g	NOUN
cana-1058	72	13	whenever	whenever	SCONJ
cana-1058	72	14	a	a	DET
cana-1058	72	15	⊆	⊆	NUM
cana-1058	72	16	g	g	NOUN
cana-1058	72	17	and	and	CCONJ
cana-1058	72	18	g	g	PROPN
cana-1058	72	19	is	be	AUX
cana-1058	72	20	nano	nano	NOUN
cana-1058	72	21	open	open	ADJ
cana-1058	72	22	.	.	PUNCT
cana-1058	73	1	(	(	PUNCT
cana-1058	73	2	3	3	X
cana-1058	73	3	)	)	PUNCT
cana-1058	73	4	n	n	PRON
cana-1058	73	5	�	�	PROPN
cana-1058	73	6	̂	̂	VERB
cana-1058	73	7	�	�	NOUN
cana-1058	73	8	-closed	-close	VERB
cana-1058	73	9	,	,	PUNCT
cana-1058	73	10	ncl(a	ncl(a	PROPN
cana-1058	73	11	)	)	PUNCT
cana-1058	73	12	⊆g	⊆g	NOUN
cana-1058	73	13	whenever	whenever	SCONJ
cana-1058	73	14	a	a	DET
cana-1058	73	15	⊆	⊆	NUM
cana-1058	73	16	g	g	NOUN
cana-1058	73	17	and	and	CCONJ
cana-1058	73	18	g	g	PROPN
cana-1058	73	19	is	be	AUX
cana-1058	73	20	nano	nano	ADJ
cana-1058	73	21	semi	semi	ADJ
cana-1058	73	22	-	-	ADJ
cana-1058	73	23	open	open	ADJ
cana-1058	73	24	.	.	PUNCT
cana-1058	74	1	(	(	PUNCT
cana-1058	74	2	4	4	X
cana-1058	74	3	)	)	PUNCT
cana-1058	74	4	n𝑔∗-closed	n𝑔∗-close	VERB
cana-1058	74	5	,	,	PUNCT
cana-1058	74	6	ncl(a	ncl(a	PROPN
cana-1058	74	7	)	)	PUNCT
cana-1058	74	8	⊆g	⊆g	NOUN
cana-1058	74	9	whenever	whenever	SCONJ
cana-1058	74	10	a	a	DET
cana-1058	74	11	⊆	⊆	NUM
cana-1058	74	12	g	g	NOUN
cana-1058	74	13	and	and	CCONJ
cana-1058	74	14	g	g	PROPN
cana-1058	74	15	is	be	AUX
cana-1058	74	16	nano	nano	VERB
cana-1058	74	17	g	g	NOUN
cana-1058	74	18	-	-	PUNCT
cana-1058	74	19	open	open	ADJ
cana-1058	74	20	.	.	PUNCT
cana-1058	75	1	(	(	PUNCT
cana-1058	75	2	5	5	X
cana-1058	75	3	)	)	PUNCT
cana-1058	75	4	nano	nano	NOUN
cana-1058	75	5	pre	pre	X
cana-1058	75	6	closed	close	VERB
cana-1058	75	7	if	if	SCONJ
cana-1058	75	8	n	n	PRON
cana-1058	75	9	cl	cl	NOUN
cana-1058	75	10	(	(	PUNCT
cana-1058	75	11	n	n	NUM
cana-1058	75	12	int	int	NOUN
cana-1058	75	13	(	(	PUNCT
cana-1058	75	14	a	a	NOUN
cana-1058	75	15	)	)	PUNCT
cana-1058	75	16	)	)	PUNCT
cana-1058	76	1	⊆	⊆	NUM
cana-1058	76	2	a.	a.	NOUN
cana-1058	76	3	(	(	PUNCT
cana-1058	76	4	6	6	NUM
cana-1058	76	5	)	)	PUNCT
cana-1058	76	6	nano	nano	NOUN
cana-1058	76	7	α	α	NOUN
cana-1058	76	8	-	-	ADJ
cana-1058	76	9	closed	closed	ADJ
cana-1058	76	10	set	set	NOUN
cana-1058	76	11	if	if	SCONJ
cana-1058	76	12	n	n	PRON
cana-1058	76	13	cl	cl	NOUN
cana-1058	76	14	(	(	PUNCT
cana-1058	76	15	n	n	NUM
cana-1058	76	16	int	int	NOUN
cana-1058	76	17	(	(	PUNCT
cana-1058	76	18	n	n	X
cana-1058	76	19	cl(a	cl(a	NUM
cana-1058	76	20	)	)	PUNCT
cana-1058	76	21	)	)	PUNCT
cana-1058	77	1	⊆	⊆	NUM
cana-1058	77	2	a	a	DET
cana-1058	77	3	communications	communication	NOUN
cana-1058	77	4	on	on	ADP
cana-1058	77	5	applied	apply	VERB
cana-1058	77	6	nonlinear	nonlinear	ADJ
cana-1058	77	7	analysis	analysis	NOUN
cana-1058	77	8	issn	issn	NOUN
cana-1058	77	9	:	:	PUNCT
cana-1058	77	10	1074	1074	NUM
cana-1058	77	11	-	-	PUNCT
cana-1058	77	12	133x	133x	NUM
cana-1058	77	13	vol	vol	NOUN
cana-1058	77	14	31	31	NUM
cana-1058	77	15	no	no	NOUN
cana-1058	77	16	.	.	PUNCT
cana-1058	78	1	5s	5s	NUM
cana-1058	78	2	(	(	PUNCT
cana-1058	78	3	2024	2024	NUM
cana-1058	78	4	)	)	PUNCT
cana-1058	78	5	393	393	NUM
cana-1058	78	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1058	78	7	the	the	DET
cana-1058	78	8	complements	complement	NOUN
cana-1058	78	9	of	of	ADP
cana-1058	78	10	the	the	DET
cana-1058	78	11	above	above	ADJ
cana-1058	78	12	mentioned	mention	VERB
cana-1058	78	13	sets	set	NOUN
cana-1058	78	14	are	be	AUX
cana-1058	78	15	called	call	VERB
cana-1058	78	16	their	their	PRON
cana-1058	78	17	respective	respective	ADJ
cana-1058	78	18	open	open	ADJ
cana-1058	78	19	sets	set	NOUN
cana-1058	78	20	.	.	PUNCT
cana-1058	79	1	definition	definition	NOUN
cana-1058	79	2	2.10	2.10	NUM
cana-1058	79	3	.	.	PUNCT
cana-1058	80	1	a	a	DET
cana-1058	80	2	subset	subset	NOUN
cana-1058	80	3	a	a	PRON
cana-1058	80	4	of	of	ADP
cana-1058	80	5	a	a	DET
cana-1058	80	6	nano	nano	ADJ
cana-1058	80	7	ideal	ideal	ADJ
cana-1058	80	8	space	space	NOUN
cana-1058	80	9	.	.	PUNCT
cana-1058	81	1	let	let	VERB
cana-1058	81	2	(	(	PUNCT
cana-1058	81	3	u	u	NOUN
cana-1058	81	4	,	,	PUNCT
cana-1058	81	5	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	81	6	)	)	PUNCT
cana-1058	81	7	,	,	PUNCT
cana-1058	81	8	i	i	PRON
cana-1058	81	9	)	)	PUNCT
cana-1058	81	10	is	be	AUX
cana-1058	81	11	said	say	VERB
cana-1058	81	12	to	to	PART
cana-1058	81	13	be	be	AUX
cana-1058	81	14	(	(	PUNCT
cana-1058	81	15	1	1	NUM
cana-1058	81	16	)	)	PUNCT
cana-1058	81	17	∗𝑁closed	∗𝑁close	VERB
cana-1058	81	18	,	,	PUNCT
cana-1058	81	19	if	if	SCONJ
cana-1058	81	20	(	(	PUNCT
cana-1058	81	21	𝐴)∗𝑁⊆𝐴	𝐴)∗𝑁⊆𝐴	NOUN
cana-1058	81	22	(	(	PUNCT
cana-1058	81	23	2	2	NUM
cana-1058	81	24	)	)	PUNCT
cana-1058	81	25	∗𝑁dense	∗𝑁dense	NOUN
cana-1058	81	26	,	,	PUNCT
cana-1058	81	27	if	if	SCONJ
cana-1058	81	28	a	a	DET
cana-1058	81	29	⊆	⊆	NUM
cana-1058	81	30	(	(	PUNCT
cana-1058	81	31	𝐴)∗𝑁	𝐴)∗𝑁	X
cana-1058	81	32	(	(	PUNCT
cana-1058	81	33	3	3	NUM
cana-1058	81	34	)	)	PUNCT
cana-1058	81	35	n	n	CCONJ
cana-1058	81	36	ig	ig	PROPN
cana-1058	81	37	closed	close	VERB
cana-1058	81	38	,	,	PUNCT
cana-1058	81	39	if	if	SCONJ
cana-1058	81	40	(	(	PUNCT
cana-1058	81	41	𝐴)∗𝑁⊆𝐺	𝐴)∗𝑁⊆𝐺	NOUN
cana-1058	81	42	whenever	whenever	SCONJ
cana-1058	81	43	a	a	PRON
cana-1058	81	44	⊆𝐺	⊆𝐺	PUNCT
cana-1058	81	45	and	and	CCONJ
cana-1058	81	46	g	g	PROPN
cana-1058	81	47	is	be	AUX
cana-1058	81	48	nano	nano	NOUN
cana-1058	81	49	open	open	ADJ
cana-1058	81	50	(	(	PUNCT
cana-1058	81	51	4	4	NUM
cana-1058	81	52	)	)	PUNCT
cana-1058	81	53	n	n	CCONJ
cana-1058	81	54	ig	ig	PROPN
cana-1058	81	55	*	*	NOUN
cana-1058	81	56	closed	closed	ADJ
cana-1058	81	57	,	,	PUNCT
cana-1058	81	58	if	if	SCONJ
cana-1058	81	59	(	(	PUNCT
cana-1058	81	60	𝐴)∗𝑁⊆𝐺	𝐴)∗𝑁⊆𝐺	NOUN
cana-1058	81	61	whenever	whenever	SCONJ
cana-1058	81	62	a	a	PRON
cana-1058	81	63	⊆𝐺	⊆𝐺	PUNCT
cana-1058	81	64	and	and	CCONJ
cana-1058	81	65	g	g	PROPN
cana-1058	81	66	is	be	AUX
cana-1058	81	67	nano	nano	NOUN
cana-1058	81	68	gopen	gopen	NOUN
cana-1058	81	69	3	3	NUM
cana-1058	81	70	.	.	PUNCT
cana-1058	82	1	n	n	PRON
cana-1058	82	2	αigclosed	αigclose	VERB
cana-1058	82	3	sets	set	NOUN
cana-1058	82	4	in	in	ADP
cana-1058	82	5	this	this	DET
cana-1058	82	6	section	section	NOUN
cana-1058	82	7	we	we	PRON
cana-1058	82	8	define	define	VERB
cana-1058	82	9	and	and	CCONJ
cana-1058	82	10	study	study	VERB
cana-1058	82	11	the	the	DET
cana-1058	82	12	notion	notion	NOUN
cana-1058	82	13	of	of	ADP
cana-1058	82	14	nαig	nαig	ADV
cana-1058	82	15	-	-	PUNCT
cana-1058	82	16	closed	close	VERB
cana-1058	82	17	sets	set	NOUN
cana-1058	82	18	and	and	CCONJ
cana-1058	82	19	n	n	DET
cana-1058	82	20	αig	αig	NOUN
cana-1058	82	21	-	-	PUNCT
cana-1058	82	22	open	open	ADJ
cana-1058	82	23	sets	set	NOUN
cana-1058	82	24	in	in	ADP
cana-1058	82	25	nano	nano	NOUN
cana-1058	82	26	ideal	ideal	ADJ
cana-1058	82	27	topological	topological	ADJ
cana-1058	82	28	spaces	space	NOUN
cana-1058	82	29	.	.	PUNCT
cana-1058	83	1	also	also	ADV
cana-1058	83	2	we	we	PRON
cana-1058	83	3	discuss	discuss	VERB
cana-1058	83	4	their	their	PRON
cana-1058	83	5	basic	basic	ADJ
cana-1058	83	6	properties	property	NOUN
cana-1058	83	7	and	and	CCONJ
cana-1058	83	8	study	study	VERB
cana-1058	83	9	the	the	DET
cana-1058	83	10	relationship	relationship	NOUN
cana-1058	83	11	between	between	ADP
cana-1058	83	12	other	other	ADJ
cana-1058	83	13	existing	exist	VERB
cana-1058	83	14	nano	nano	NOUN
cana-1058	83	15	closed	close	VERB
cana-1058	83	16	sets	set	NOUN
cana-1058	83	17	in	in	ADP
cana-1058	83	18	nano	nano	NOUN
cana-1058	83	19	ideal	ideal	ADJ
cana-1058	83	20	topological	topological	ADJ
cana-1058	83	21	spaces	space	NOUN
cana-1058	83	22	.	.	PUNCT
cana-1058	84	1	definition	definition	NOUN
cana-1058	84	2	3.1	3.1	NUM
cana-1058	84	3	.	.	PUNCT
cana-1058	85	1	a	a	DET
cana-1058	85	2	subset	subset	NOUN
cana-1058	85	3	a	a	PRON
cana-1058	85	4	of	of	ADP
cana-1058	85	5	a	a	DET
cana-1058	85	6	nano	nano	NOUN
cana-1058	85	7	ideal	ideal	ADJ
cana-1058	85	8	space	space	NOUN
cana-1058	85	9	(	(	PUNCT
cana-1058	85	10	u	u	NOUN
cana-1058	85	11	,	,	PUNCT
cana-1058	85	12	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	85	13	)	)	PUNCT
cana-1058	85	14	,	,	PUNCT
cana-1058	85	15	i	i	PRON
cana-1058	85	16	)	)	PUNCT
cana-1058	85	17	is	be	AUX
cana-1058	85	18	said	say	VERB
cana-1058	85	19	to	to	PART
cana-1058	85	20	be	be	AUX
cana-1058	85	21	n	n	PRON
cana-1058	85	22	αig	αig	NOUN
cana-1058	85	23	-	-	PUNCT
cana-1058	85	24	closed	closed	ADJ
cana-1058	85	25	if	if	SCONJ
cana-1058	85	26	(	(	PUNCT
cana-1058	85	27	𝐴)∗𝑁⊆𝐺whenever	𝐴)∗𝑁⊆𝐺whenever	NOUN
cana-1058	85	28	a	a	X
cana-1058	85	29	⊆𝐺	⊆𝐺	PUNCT
cana-1058	85	30	and	and	CCONJ
cana-1058	85	31	g	g	PROPN
cana-1058	85	32	is	be	AUX
cana-1058	85	33	nano	nano	VERB
cana-1058	85	34	α	α	NOUN
cana-1058	85	35	open	open	ADJ
cana-1058	85	36	.	.	PUNCT
cana-1058	86	1	example	example	NOUN
cana-1058	86	2	3.2	3.2	NUM
cana-1058	86	3	:	:	PUNCT
cana-1058	86	4	let	let	VERB
cana-1058	86	5	u	u	PRON
cana-1058	86	6	=	=	X
cana-1058	86	7	{	{	PUNCT
cana-1058	86	8	a	a	PRON
cana-1058	86	9	,	,	PUNCT
cana-1058	86	10	b	b	NOUN
cana-1058	86	11	,	,	PUNCT
cana-1058	86	12	c	c	NOUN
cana-1058	86	13	,	,	PUNCT
cana-1058	86	14	d	d	NOUN
cana-1058	86	15	}	}	PUNCT
cana-1058	86	16	,	,	PUNCT
cana-1058	86	17	u	u	NOUN
cana-1058	86	18	/	/	SYM
cana-1058	86	19	r	r	NOUN
cana-1058	86	20	=	=	NOUN
cana-1058	86	21	{	{	PUNCT
cana-1058	86	22	a},{d	a},{d	NOUN
cana-1058	86	23	}	}	PUNCT
cana-1058	86	24	,	,	PUNCT
cana-1058	86	25	{	{	PUNCT
cana-1058	86	26	b	b	X
cana-1058	86	27	,	,	PUNCT
cana-1058	86	28	c	c	NOUN
cana-1058	86	29	}	}	PUNCT
cana-1058	86	30	}	}	PUNCT
cana-1058	86	31	and	and	CCONJ
cana-1058	86	32	x	x	X
cana-1058	86	33	=	=	X
cana-1058	86	34	{	{	PUNCT
cana-1058	86	35	a	a	X
cana-1058	86	36	,	,	PUNCT
cana-1058	86	37	d	d	NOUN
cana-1058	86	38	}	}	PUNCT
cana-1058	86	39	.	.	PUNCT
cana-1058	87	1	let	let	VERB
cana-1058	87	2	the	the	DET
cana-1058	87	3	nano	nano	NOUN
cana-1058	87	4	ideal	ideal	ADJ
cana-1058	87	5	space	space	NOUN
cana-1058	87	6	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	87	7	)	)	PUNCT
cana-1058	87	8	=	=	PRON
cana-1058	87	9	{	{	PUNCT
cana-1058	87	10	u	u	NOUN
cana-1058	87	11	,	,	PUNCT
cana-1058	87	12	∅	∅	NOUN
cana-1058	87	13	,	,	PUNCT
cana-1058	87	14	{	{	PUNCT
cana-1058	87	15	𝑎	𝑎	X
cana-1058	87	16	,	,	PUNCT
cana-1058	87	17	𝑑	𝑑	NOUN
cana-1058	87	18	}	}	PUNCT
cana-1058	87	19	}	}	PUNCT
cana-1058	87	20	with	with	ADP
cana-1058	87	21	a	a	DET
cana-1058	87	22	nano	nano	NOUN
cana-1058	87	23	ideal	ideal	NOUN
cana-1058	87	24	i	i	PRON
cana-1058	87	25	=	=	NOUN
cana-1058	87	26	{	{	PUNCT
cana-1058	87	27	∅	∅	NOUN
cana-1058	87	28	,	,	PUNCT
cana-1058	87	29	{	{	PUNCT
cana-1058	87	30	a	a	X
cana-1058	87	31	}	}	PUNCT
cana-1058	87	32	}	}	PUNCT
cana-1058	87	33	.	.	PUNCT
cana-1058	88	1	then	then	ADV
cana-1058	88	2	n	n	CCONJ
cana-1058	88	3	αig	αig	NOUN
cana-1058	88	4	-	-	PUNCT
cana-1058	88	5	closed	close	VERB
cana-1058	88	6	sets	set	NOUN
cana-1058	88	7	are	be	AUX
cana-1058	88	8	{	{	PUNCT
cana-1058	88	9	u	u	NOUN
cana-1058	88	10	,	,	PUNCT
cana-1058	88	11	∅,{a	∅,{a	ADV
cana-1058	88	12	}	}	PUNCT
cana-1058	88	13	,	,	PUNCT
cana-1058	88	14	{	{	PUNCT
cana-1058	88	15	b	b	X
cana-1058	88	16	,	,	PUNCT
cana-1058	88	17	c	c	NOUN
cana-1058	88	18	}	}	PUNCT
cana-1058	88	19	,	,	PUNCT
cana-1058	88	20	{	{	PUNCT
cana-1058	88	21	b	b	X
cana-1058	88	22	,	,	PUNCT
cana-1058	88	23	d	d	NOUN
cana-1058	88	24	}	}	PUNCT
cana-1058	88	25	,	,	PUNCT
cana-1058	88	26	{	{	PUNCT
cana-1058	88	27	a	a	DET
cana-1058	88	28	,	,	PUNCT
cana-1058	88	29	b	b	NOUN
cana-1058	88	30	,	,	PUNCT
cana-1058	88	31	c	c	NOUN
cana-1058	88	32	}	}	PUNCT
cana-1058	88	33	,	,	PUNCT
cana-1058	88	34	{	{	PUNCT
cana-1058	88	35	b	b	X
cana-1058	88	36	,	,	PUNCT
cana-1058	88	37	c	c	NOUN
cana-1058	88	38	,	,	PUNCT
cana-1058	88	39	d	d	NOUN
cana-1058	88	40	}	}	PUNCT
cana-1058	88	41	}	}	PUNCT
cana-1058	88	42	.	.	PUNCT
cana-1058	89	1	definition	definition	NOUN
cana-1058	89	2	3.3	3.3	NUM
cana-1058	89	3	.	.	PUNCT
cana-1058	90	1	let	let	VERB
cana-1058	90	2	(	(	PUNCT
cana-1058	90	3	u	u	NOUN
cana-1058	90	4	,	,	PUNCT
cana-1058	90	5	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	90	6	)	)	PUNCT
cana-1058	90	7	,	,	PUNCT
cana-1058	90	8	i	i	PRON
cana-1058	90	9	)	)	PUNCT
cana-1058	90	10	be	be	AUX
cana-1058	90	11	a	a	DET
cana-1058	90	12	nano	nano	NOUN
cana-1058	90	13	ideal	ideal	ADJ
cana-1058	90	14	topological	topological	ADJ
cana-1058	90	15	space	space	NOUN
cana-1058	90	16	.	.	PUNCT
cana-1058	91	1	a	a	DET
cana-1058	91	2	subset	subset	NOUN
cana-1058	91	3	a	a	PRON
cana-1058	91	4	of	of	ADP
cana-1058	91	5	x	x	SYM
cana-1058	91	6	is	be	AUX
cana-1058	91	7	said	say	VERB
cana-1058	91	8	to	to	PART
cana-1058	91	9	be	be	AUX
cana-1058	91	10	n	n	PRON
cana-1058	91	11	αig	αig	NOUN
cana-1058	91	12	-	-	PUNCT
cana-1058	91	13	open	open	ADJ
cana-1058	91	14	if	if	SCONJ
cana-1058	91	15	x	x	X
cana-1058	91	16	–	–	PUNCT
cana-1058	91	17	a	a	PRON
cana-1058	91	18	is	be	AUX
cana-1058	91	19	n	n	PRON
cana-1058	91	20	αig	αig	NOUN
cana-1058	91	21	-	-	PUNCT
cana-1058	91	22	closed	closed	ADJ
cana-1058	91	23	.	.	PUNCT
cana-1058	92	1	theorem	theorem	VERB
cana-1058	92	2	3.4	3.4	NUM
cana-1058	92	3	.	.	PUNCT
cana-1058	93	1	if	if	SCONJ
cana-1058	93	2	(	(	PUNCT
cana-1058	93	3	u	u	NOUN
cana-1058	93	4	,	,	PUNCT
cana-1058	93	5	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	93	6	)	)	PUNCT
cana-1058	93	7	,	,	PUNCT
cana-1058	93	8	i	i	PRON
cana-1058	93	9	)	)	PUNCT
cana-1058	93	10	is	be	AUX
cana-1058	93	11	any	any	DET
cana-1058	93	12	nano	nano	ADJ
cana-1058	93	13	ideal	ideal	ADJ
cana-1058	93	14	space	space	NOUN
cana-1058	93	15	,	,	PUNCT
cana-1058	93	16	then	then	ADV
cana-1058	93	17	the	the	DET
cana-1058	93	18	following	follow	VERB
cana-1058	93	19	are	be	AUX
cana-1058	93	20	equivalent	equivalent	ADJ
cana-1058	93	21	(	(	PUNCT
cana-1058	93	22	1	1	X
cana-1058	93	23	)	)	PUNCT
cana-1058	93	24	a	a	PRON
cana-1058	93	25	is	be	AUX
cana-1058	93	26	n	n	PRON
cana-1058	93	27	αig	αig	NOUN
cana-1058	93	28	-	-	PUNCT
cana-1058	93	29	closed	closed	ADJ
cana-1058	93	30	(	(	PUNCT
cana-1058	93	31	2	2	NUM
cana-1058	93	32	)	)	PUNCT
cana-1058	93	33	𝑁𝑐𝑙∗(𝐴	𝑁𝑐𝑙∗(𝐴	PROPN
cana-1058	93	34	)	)	PUNCT
cana-1058	94	1	⊆𝐺	⊆𝐺	PUNCT
cana-1058	94	2	whenever	whenever	SCONJ
cana-1058	94	3	a	a	PRON
cana-1058	94	4	⊆𝐺	⊆𝐺	PUNCT
cana-1058	94	5	and	and	CCONJ
cana-1058	94	6	g	g	PROPN
cana-1058	94	7	is	be	AUX
cana-1058	94	8	nano	nano	NOUN
cana-1058	94	9	α	α	NOUN
cana-1058	94	10	open	open	ADJ
cana-1058	94	11	in	in	ADP
cana-1058	94	12	u	u	NOUN
cana-1058	94	13	(	(	PUNCT
cana-1058	94	14	3	3	NUM
cana-1058	94	15	)	)	PUNCT
cana-1058	94	16	for	for	ADP
cana-1058	94	17	all	all	DET
cana-1058	94	18	x	x	PUNCT
cana-1058	94	19	𝑁𝑐𝑙∗(𝐴	𝑁𝑐𝑙∗(𝐴	PROPN
cana-1058	94	20	)	)	PUNCT
cana-1058	94	21	,	,	PUNCT
cana-1058	94	22	n	n	CCONJ
cana-1058	94	23	αcl({x	αcl({x	NOUN
cana-1058	94	24	}	}	PUNCT
cana-1058	94	25	)	)	PUNCT
cana-1058	95	1			PUNCT
cana-1058	95	2	a	a	DET
cana-1058	95	3	≠	≠	PROPN
cana-1058	95	4			NOUN
cana-1058	95	5	(	(	PUNCT
cana-1058	95	6	4	4	NUM
cana-1058	95	7	)	)	PUNCT
cana-1058	95	8	𝑁𝑐𝑙∗(𝐴	𝑁𝑐𝑙∗(𝐴	PROPN
cana-1058	95	9	)	)	PUNCT
cana-1058	95	10	a	a	PRON
cana-1058	95	11	contains	contain	VERB
cana-1058	95	12	no	no	DET
cana-1058	95	13	non	non	ADJ
cana-1058	95	14	empty	empty	ADJ
cana-1058	95	15	nano	nano	NOUN
cana-1058	95	16	α	α	PROPN
cana-1058	95	17	closed	close	VERB
cana-1058	95	18	set	set	NOUN
cana-1058	95	19	.	.	PUNCT
cana-1058	96	1	(	(	PUNCT
cana-1058	96	2	5	5	NUM
cana-1058	96	3	)	)	PUNCT
cana-1058	96	4	(	(	PUNCT
cana-1058	96	5	𝐴)∗𝑁-acontains	𝐴)∗𝑁-acontain	VERB
cana-1058	96	6	no	no	PRON
cana-1058	96	7	nonempty	nonempty	ADV
cana-1058	96	8	nano	nano	NOUN
cana-1058	96	9	α	α	NUM
cana-1058	96	10	closed	close	VERB
cana-1058	96	11	set	set	NOUN
cana-1058	96	12	.	.	PUNCT
cana-1058	97	1	proof	proof	NOUN
cana-1058	97	2	:	:	PUNCT
cana-1058	97	3	(	(	PUNCT
cana-1058	97	4	1	1	X
cana-1058	97	5	)	)	PUNCT
cana-1058	97	6			NOUN
cana-1058	97	7	(	(	PUNCT
cana-1058	97	8	2	2	NUM
cana-1058	97	9	)	)	PUNCT
cana-1058	97	10	:	:	PUNCT
cana-1058	97	11	if	if	SCONJ
cana-1058	97	12	a	a	PRON
cana-1058	97	13	is	be	AUX
cana-1058	97	14	n	n	PRON
cana-1058	97	15	αig	αig	NOUN
cana-1058	97	16	-	-	PUNCT
cana-1058	97	17	closed	closed	ADJ
cana-1058	97	18	,	,	PUNCT
cana-1058	97	19	then	then	ADV
cana-1058	97	20	(	(	PUNCT
cana-1058	97	21	𝐴)∗𝑁⊆𝐺whenever	𝐴)∗𝑁⊆𝐺whenever	PROPN
cana-1058	97	22	a	a	X
cana-1058	97	23	⊆𝐺	⊆𝐺	PUNCT
cana-1058	97	24	and	and	CCONJ
cana-1058	97	25	g	g	PROPN
cana-1058	97	26	is	be	AUX
cana-1058	97	27	nano	nano	NOUN
cana-1058	97	28	α	α	NOUN
cana-1058	97	29	open	open	ADJ
cana-1058	97	30	in	in	ADP
cana-1058	97	31	x	x	X
cana-1058	97	32	and	and	CCONJ
cana-1058	97	33	so	so	ADV
cana-1058	97	34	𝑁𝑐𝑙∗(𝐴	𝑁𝑐𝑙∗(𝐴	NUM
cana-1058	97	35	)	)	PUNCT
cana-1058	97	36	=	=	PUNCT
cana-1058	98	1	𝐴⋃(𝐴)∗𝑁⊆𝐺	𝐴⋃(𝐴)∗𝑁⊆𝐺	NOUN
cana-1058	98	2	and	and	CCONJ
cana-1058	98	3	g	g	PROPN
cana-1058	98	4	is	be	AUX
cana-1058	98	5	nano	nano	VERB
cana-1058	98	6	α	α	NOUN
cana-1058	98	7	open	open	ADJ
cana-1058	98	8	in	in	ADP
cana-1058	98	9	u.	u.	PROPN
cana-1058	98	10	this	this	PRON
cana-1058	98	11	proves	prove	VERB
cana-1058	98	12	(	(	PUNCT
cana-1058	98	13	2	2	NUM
cana-1058	98	14	)	)	PUNCT
cana-1058	98	15	.	.	PUNCT
cana-1058	99	1	(	(	PUNCT
cana-1058	99	2	2	2	X
cana-1058	99	3	)	)	PUNCT
cana-1058	99	4	(3	(3	PUNCT
cana-1058	99	5	)	)	PUNCT
cana-1058	99	6	:	:	PUNCT
cana-1058	99	7	suppose	suppose	VERB
cana-1058	99	8	x	x	PUNCT
cana-1058	99	9	𝑁𝑐𝑙∗(𝐴	𝑁𝑐𝑙∗(𝐴	PROPN
cana-1058	99	10	)	)	PUNCT
cana-1058	99	11	.	.	PUNCT
cana-1058	100	1	if	if	SCONJ
cana-1058	100	2	n	n	PRON
cana-1058	100	3	αcl({x	αcl({x	NOUN
cana-1058	100	4	}	}	PUNCT
cana-1058	100	5	)	)	PUNCT
cana-1058	100	6	a=,then	a=,then	VERB
cana-1058	100	7	a	a	DET
cana-1058	100	8	⊆	⊆	NUM
cana-1058	100	9	xn	xn	NUM
cana-1058	100	10	αcl({x	αcl({x	NOUN
cana-1058	100	11	}	}	PUNCT
cana-1058	100	12	)	)	PUNCT
cana-1058	100	13	.	.	PUNCT
cana-1058	101	1	by	by	ADP
cana-1058	101	2	(	(	PUNCT
cana-1058	101	3	2	2	NUM
cana-1058	101	4	)	)	PUNCT
cana-1058	101	5	,	,	PUNCT
cana-1058	101	6	𝑁𝑐𝑙∗(𝐴	𝑁𝑐𝑙∗(𝐴	PROPN
cana-1058	101	7	)	)	PUNCT
cana-1058	101	8	⊆	⊆	NUM
cana-1058	101	9	xn	xn	SYM
cana-1058	101	10	αcl({x	αcl({x	NOUN
cana-1058	101	11	}	}	PUNCT
cana-1058	101	12	)	)	PUNCT
cana-1058	101	13	,	,	PUNCT
cana-1058	101	14	which	which	PRON
cana-1058	101	15	is	be	AUX
cana-1058	101	16	a	a	DET
cana-1058	101	17	contradiction	contradiction	NOUN
cana-1058	101	18	to	to	ADP
cana-1058	101	19	x	x	PART
cana-1058	101	20	𝑁𝑐𝑙∗(𝐴	𝑁𝑐𝑙∗(𝐴	PROPN
cana-1058	101	21	)	)	PUNCT
cana-1058	101	22	.	.	PUNCT
cana-1058	102	1	this	this	PRON
cana-1058	102	2	proves	prove	VERB
cana-1058	102	3	(	(	PUNCT
cana-1058	102	4	3	3	NUM
cana-1058	102	5	)	)	PUNCT
cana-1058	102	6	(	(	PUNCT
cana-1058	102	7	3	3	X
cana-1058	102	8	)	)	PUNCT
cana-1058	102	9			NOUN
cana-1058	102	10	(	(	PUNCT
cana-1058	102	11	4	4	NUM
cana-1058	102	12	)	)	PUNCT
cana-1058	102	13	:	:	PUNCT
cana-1058	102	14	suppose	suppose	VERB
cana-1058	102	15	f	f	PROPN
cana-1058	102	16	⊆𝑁𝑐𝑙∗(𝐴	⊆𝑁𝑐𝑙∗(𝐴	NOUN
cana-1058	102	17	)	)	PUNCT
cana-1058	102	18	–	–	PUNCT
cana-1058	102	19	a	a	X
cana-1058	102	20	,	,	PUNCT
cana-1058	102	21	f	f	PROPN
cana-1058	102	22	is	be	AUX
cana-1058	102	23	nano	nano	VERB
cana-1058	102	24	α	α	NOUN
cana-1058	102	25	-	-	ADJ
cana-1058	102	26	closed	closed	ADJ
cana-1058	102	27	and	and	CCONJ
cana-1058	102	28	x	x	SYM
cana-1058	102	29			PROPN
cana-1058	102	30	f.	f.	PROPN
cana-1058	102	31	since	since	SCONJ
cana-1058	102	32	f	f	PROPN
cana-1058	102	33	⊆𝑋	⊆𝑋	PROPN
cana-1058	102	34	−	−	PROPN
cana-1058	102	35	𝐴	𝐴	PROPN
cana-1058	102	36	and	and	CCONJ
cana-1058	102	37	f	f	PROPN
cana-1058	102	38	is	be	AUX
cana-1058	102	39	nano	nano	NOUN
cana-1058	102	40	αclosed	αclose	VERB
cana-1058	102	41	,	,	PUNCT
cana-1058	102	42	then	then	ADV
cana-1058	102	43	a	a	DET
cana-1058	102	44	⊆𝑋	⊆𝑋	PROPN
cana-1058	102	45	−	−	PROPN
cana-1058	102	46	𝐹	𝐹	PROPN
cana-1058	102	47	and	and	CCONJ
cana-1058	102	48	hence	hence	ADV
cana-1058	102	49	n	n	ADV
cana-1058	102	50	αcl({x	αcl({x	NOUN
cana-1058	102	51	}	}	PUNCT
cana-1058	102	52	)	)	PUNCT
cana-1058	102	53			PUNCT
cana-1058	102	54	a	a	X
cana-1058	102	55	=	=	SYM
cana-1058	102	56	.	.	X
cana-1058	102	57	therefore	therefore	ADV
cana-1058	102	58	,	,	PUNCT
cana-1058	102	59	𝑁𝑐𝑙∗(𝐴	𝑁𝑐𝑙∗(𝐴	PROPN
cana-1058	102	60	)	)	PUNCT
cana-1058	102	61	–	–	PUNCT
cana-1058	102	62	a	a	PRON
cana-1058	102	63	contains	contain	VERB
cana-1058	102	64	no	no	DET
cana-1058	102	65	non	non	ADJ
cana-1058	102	66	empty	empty	ADJ
cana-1058	102	67	nano	nano	ADJ
cana-1058	102	68	α	α	NOUN
cana-1058	102	69	-	-	PUNCT
cana-1058	102	70	closed	closed	ADJ
cana-1058	102	71	set	set	NOUN
cana-1058	102	72	.	.	PUNCT
cana-1058	103	1	(	(	PUNCT
cana-1058	103	2	4	4	NUM
cana-1058	103	3	)	)	PUNCT
cana-1058	103	4			NOUN
cana-1058	103	5	(	(	PUNCT
cana-1058	103	6	5	5	NUM
cana-1058	103	7	)	)	PUNCT
cana-1058	103	8	:	:	PUNCT
cana-1058	103	9	since	since	SCONJ
cana-1058	103	10	𝑁𝑐𝑙∗(𝐴	𝑁𝑐𝑙∗(𝐴	PROPN
cana-1058	103	11	)	)	PUNCT
cana-1058	103	12	–	–	PUNCT
cana-1058	103	13	a	a	DET
cana-1058	103	14	=	=	X
cana-1058	103	15	(	(	PUNCT
cana-1058	103	16	a	a	DET
cana-1058	103	17	(𝐴)∗𝑁	(𝐴)∗𝑁	X
cana-1058	103	18	)	)	PUNCT
cana-1058	103	19	–	–	PUNCT
cana-1058	103	20	a	a	X
cana-1058	103	21	=	=	X
cana-1058	103	22	(	(	PUNCT
cana-1058	103	23	a	a	DET
cana-1058	103	24	(𝐴)∗𝑁	(𝐴)∗𝑁	NOUN
cana-1058	103	25	)	)	PUNCT
cana-1058	103	26	∩	∩	NOUN
cana-1058	103	27	𝐴𝑐	𝐴𝑐	PROPN
cana-1058	103	28	=	=	SYM
cana-1058	103	29	(	(	PUNCT
cana-1058	103	30	a∩𝐴𝑐)((𝐴)∗𝑁∩	a∩𝐴𝑐)((𝐴)∗𝑁∩	PROPN
cana-1058	103	31	𝐴𝑐)=(𝐴)∗𝑁∩𝐴𝑐=(𝐴)∗𝑁−𝐴.	𝐴𝑐)=(𝐴)∗𝑁∩𝐴𝑐=(𝐴)∗𝑁−𝐴.	ADV
cana-1058	103	32	therefore	therefore	ADV
cana-1058	103	33	,	,	PUNCT
cana-1058	103	34	(	(	PUNCT
cana-1058	103	35	𝐴)∗𝑁−𝐴contains	𝐴)∗𝑁−𝐴contain	VERB
cana-1058	103	36	no	no	PRON
cana-1058	103	37	nonempty	nonempty	ADV
cana-1058	103	38	nano	nano	NOUN
cana-1058	103	39	α	α	NUM
cana-1058	103	40	closed	close	VERB
cana-1058	103	41	set	set	NOUN
cana-1058	103	42	.	.	PUNCT
cana-1058	104	1	communications	communication	NOUN
cana-1058	104	2	on	on	ADP
cana-1058	104	3	applied	apply	VERB
cana-1058	104	4	nonlinear	nonlinear	ADJ
cana-1058	104	5	analysis	analysis	NOUN
cana-1058	104	6	issn	issn	NOUN
cana-1058	104	7	:	:	PUNCT
cana-1058	104	8	1074	1074	NUM
cana-1058	104	9	-	-	PUNCT
cana-1058	104	10	133x	133x	NUM
cana-1058	104	11	vol	vol	NOUN
cana-1058	104	12	31	31	NUM
cana-1058	104	13	no	no	NOUN
cana-1058	104	14	.	.	PUNCT
cana-1058	105	1	5s	5s	NUM
cana-1058	105	2	(	(	PUNCT
cana-1058	105	3	2024	2024	NUM
cana-1058	105	4	)	)	PUNCT
cana-1058	105	5	394	394	NUM
cana-1058	105	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1058	105	7	theorem	theorem	VERB
cana-1058	105	8	3.5	3.5	NUM
cana-1058	105	9	:	:	PUNCT
cana-1058	106	1	every	every	PRON
cana-1058	106	2	∗𝑁	∗𝑁	ADJ
cana-1058	106	3	closed	closed	ADJ
cana-1058	106	4	set	set	NOUN
cana-1058	106	5	is	be	AUX
cana-1058	106	6	n	n	PRON
cana-1058	106	7	αig	αig	NOUN
cana-1058	106	8	-	-	PUNCT
cana-1058	106	9	closed	closed	ADJ
cana-1058	106	10	but	but	CCONJ
cana-1058	106	11	not	not	PART
cana-1058	106	12	conversely	conversely	ADV
cana-1058	106	13	.	.	PUNCT
cana-1058	107	1	proof	proof	NOUN
cana-1058	107	2	:	:	PUNCT
cana-1058	107	3	let	let	VERB
cana-1058	107	4	a	a	PRON
cana-1058	107	5	be	be	AUX
cana-1058	107	6	a	a	DET
cana-1058	107	7	∗𝑁	∗𝑁	NOUN
cana-1058	107	8	closed	closed	ADJ
cana-1058	107	9	,	,	PUNCT
cana-1058	107	10	then	then	ADV
cana-1058	107	11	(	(	PUNCT
cana-1058	107	12	𝐴)∗𝑁⊆	𝐴)∗𝑁⊆	PROPN
cana-1058	107	13	a.	a.	NOUN
cana-1058	107	14	let	let	VERB
cana-1058	107	15	a	a	DET
cana-1058	107	16	⊆	⊆	NUM
cana-1058	107	17	g	g	NOUN
cana-1058	107	18	and	and	CCONJ
cana-1058	107	19	g	g	PROPN
cana-1058	107	20	is	be	AUX
cana-1058	107	21	nano	nano	NOUN
cana-1058	107	22	α	α	NOUN
cana-1058	107	23	open	open	ADJ
cana-1058	107	24	.	.	PUNCT
cana-1058	108	1	this	this	PRON
cana-1058	108	2	implies	imply	VERB
cana-1058	108	3	(	(	PUNCT
cana-1058	108	4	𝐴)∗𝑁⊆	𝐴)∗𝑁⊆	PROPN
cana-1058	108	5	g.	g.	PROPN
cana-1058	108	6	hence	hence	ADV
cana-1058	108	7	a	a	PRON
cana-1058	108	8	is	be	AUX
cana-1058	108	9	n	n	PRON
cana-1058	108	10	αig	αig	NOUN
cana-1058	108	11	-	-	PUNCT
cana-1058	108	12	closed	closed	ADJ
cana-1058	108	13	.	.	PUNCT
cana-1058	108	14	example	example	NOUN
cana-1058	108	15	3.6	3.6	NUM
cana-1058	108	16	:	:	PUNCT
cana-1058	108	17	let	let	VERB
cana-1058	108	18	u	u	PRON
cana-1058	108	19	=	=	X
cana-1058	108	20	{	{	PUNCT
cana-1058	108	21	a	a	DET
cana-1058	108	22	,	,	PUNCT
cana-1058	108	23	b	b	NOUN
cana-1058	108	24	,	,	PUNCT
cana-1058	108	25	c	c	NOUN
cana-1058	108	26	,	,	PUNCT
cana-1058	108	27	d	d	NOUN
cana-1058	108	28	}	}	PUNCT
cana-1058	108	29	,	,	PUNCT
cana-1058	108	30	u	u	NOUN
cana-1058	108	31	/	/	SYM
cana-1058	108	32	r	r	AUX
cana-1058	108	33	=	=	PUNCT
cana-1058	108	34	{	{	PUNCT
cana-1058	108	35	{	{	PUNCT
cana-1058	108	36	a	a	NOUN
cana-1058	108	37	}	}	PUNCT
cana-1058	108	38	,	,	PUNCT
cana-1058	108	39	{	{	PUNCT
cana-1058	108	40	c	c	NOUN
cana-1058	108	41	}	}	PUNCT
cana-1058	108	42	,	,	PUNCT
cana-1058	108	43	{	{	PUNCT
cana-1058	108	44	b	b	X
cana-1058	108	45	,	,	PUNCT
cana-1058	108	46	d	d	NOUN
cana-1058	108	47	}	}	PUNCT
cana-1058	108	48	}	}	PUNCT
cana-1058	108	49	and	and	CCONJ
cana-1058	108	50	x	x	X
cana-1058	108	51	=	=	X
cana-1058	108	52	{	{	PUNCT
cana-1058	108	53	a	a	DET
cana-1058	108	54	,	,	PUNCT
cana-1058	108	55	b	b	NOUN
cana-1058	108	56	}	}	PUNCT
cana-1058	108	57	.	.	PUNCT
cana-1058	109	1	let	let	VERB
cana-1058	109	2	the	the	DET
cana-1058	109	3	nano	nano	NOUN
cana-1058	109	4	ideal	ideal	ADJ
cana-1058	109	5	space	space	NOUN
cana-1058	109	6	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	109	7	)	)	PUNCT
cana-1058	109	8	=	=	PRON
cana-1058	109	9	{	{	PUNCT
cana-1058	109	10	u	u	NOUN
cana-1058	109	11	,	,	PUNCT
cana-1058	109	12	∅	∅	NOUN
cana-1058	109	13	,	,	PUNCT
cana-1058	109	14	{	{	PUNCT
cana-1058	109	15	a	a	X
cana-1058	109	16	}	}	PUNCT
cana-1058	109	17	,	,	PUNCT
cana-1058	109	18	{	{	PUNCT
cana-1058	109	19	a	a	DET
cana-1058	109	20	,	,	PUNCT
cana-1058	109	21	b	b	NOUN
cana-1058	109	22	,	,	PUNCT
cana-1058	109	23	d	d	NOUN
cana-1058	109	24	}	}	PUNCT
cana-1058	109	25	,	,	PUNCT
cana-1058	109	26	{	{	PUNCT
cana-1058	109	27	b	b	X
cana-1058	109	28	,	,	PUNCT
cana-1058	109	29	d	d	NOUN
cana-1058	109	30	}	}	PUNCT
cana-1058	109	31	}	}	PUNCT
cana-1058	109	32	with	with	ADP
cana-1058	109	33	a	a	DET
cana-1058	109	34	nano	nano	NOUN
cana-1058	109	35	ideal	ideal	NOUN
cana-1058	109	36	i	i	PRON
cana-1058	109	37	=	=	NOUN
cana-1058	109	38	{	{	PUNCT
cana-1058	109	39	∅	∅	NOUN
cana-1058	109	40	,	,	PUNCT
cana-1058	109	41	{	{	PUNCT
cana-1058	109	42	a	a	X
cana-1058	109	43	}	}	PUNCT
cana-1058	109	44	,	,	PUNCT
cana-1058	109	45	{	{	PUNCT
cana-1058	109	46	a	a	PRON
cana-1058	109	47	,	,	PUNCT
cana-1058	109	48	b	b	NOUN
cana-1058	109	49	,	,	PUNCT
cana-1058	109	50	d	d	NOUN
cana-1058	109	51	}	}	PUNCT
cana-1058	109	52	}	}	PUNCT
cana-1058	109	53	.	.	PUNCT
cana-1058	110	1	then	then	ADV
cana-1058	110	2	n	n	NUM
cana-1058	110	3	αigclosed	αigclose	VERB
cana-1058	110	4	sets	set	NOUN
cana-1058	110	5	are	be	AUX
cana-1058	110	6	{	{	PUNCT
cana-1058	110	7	u	u	NOUN
cana-1058	110	8	,	,	PUNCT
cana-1058	110	9	∅	∅	NOUN
cana-1058	110	10	,	,	PUNCT
cana-1058	110	11	{	{	PUNCT
cana-1058	110	12	a	a	X
cana-1058	110	13	}	}	PUNCT
cana-1058	110	14	,	,	PUNCT
cana-1058	110	15	{	{	PUNCT
cana-1058	110	16	c	c	X
cana-1058	110	17	}	}	PUNCT
cana-1058	110	18	,	,	PUNCT
cana-1058	110	19	{	{	PUNCT
cana-1058	110	20	a	a	X
cana-1058	110	21	,	,	PUNCT
cana-1058	110	22	c	c	NOUN
cana-1058	110	23	}	}	PUNCT
cana-1058	110	24	,	,	PUNCT
cana-1058	110	25	{	{	PUNCT
cana-1058	110	26	b	b	X
cana-1058	110	27	,	,	PUNCT
cana-1058	110	28	c	c	NOUN
cana-1058	110	29	}	}	PUNCT
cana-1058	110	30	,	,	PUNCT
cana-1058	110	31	{	{	PUNCT
cana-1058	110	32	c	c	X
cana-1058	110	33	,	,	PUNCT
cana-1058	110	34	d	d	NOUN
cana-1058	110	35	}	}	PUNCT
cana-1058	110	36	,	,	PUNCT
cana-1058	110	37	{	{	PUNCT
cana-1058	110	38	a	a	DET
cana-1058	110	39	,	,	PUNCT
cana-1058	110	40	b	b	NOUN
cana-1058	110	41	,	,	PUNCT
cana-1058	110	42	c	c	NOUN
cana-1058	110	43	}	}	PUNCT
cana-1058	110	44	,	,	PUNCT
cana-1058	110	45	{	{	PUNCT
cana-1058	110	46	a	a	DET
cana-1058	110	47	,	,	PUNCT
cana-1058	110	48	b	b	NOUN
cana-1058	110	49	,	,	PUNCT
cana-1058	110	50	d	d	NOUN
cana-1058	110	51	}	}	PUNCT
cana-1058	110	52	,	,	PUNCT
cana-1058	110	53	{	{	PUNCT
cana-1058	110	54	a	a	PRON
cana-1058	110	55	,	,	PUNCT
cana-1058	110	56	c	c	NOUN
cana-1058	110	57	,	,	PUNCT
cana-1058	110	58	d},{b	d},{b	PROPN
cana-1058	110	59	,	,	PUNCT
cana-1058	110	60	c	c	X
cana-1058	110	61	,	,	PUNCT
cana-1058	110	62	d	d	NOUN
cana-1058	110	63	}	}	PUNCT
cana-1058	110	64	}	}	PUNCT
cana-1058	110	65	and	and	CCONJ
cana-1058	110	66	∗𝑁	∗𝑁	X
cana-1058	110	67	closed	close	VERB
cana-1058	110	68	set	set	NOUN
cana-1058	110	69	are	be	AUX
cana-1058	110	70	{	{	PUNCT
cana-1058	110	71	u	u	NOUN
cana-1058	110	72	,	,	PUNCT
cana-1058	110	73	∅	∅	NOUN
cana-1058	110	74	,	,	PUNCT
cana-1058	110	75	{	{	PUNCT
cana-1058	110	76	a	a	X
cana-1058	110	77	}	}	PUNCT
cana-1058	110	78	,	,	PUNCT
cana-1058	110	79	{	{	PUNCT
cana-1058	110	80	c	c	X
cana-1058	110	81	}	}	PUNCT
cana-1058	110	82	,	,	PUNCT
cana-1058	110	83	{	{	PUNCT
cana-1058	110	84	a	a	X
cana-1058	110	85	,	,	PUNCT
cana-1058	110	86	c	c	NOUN
cana-1058	110	87	}	}	PUNCT
cana-1058	110	88	,	,	PUNCT
cana-1058	110	89	{	{	PUNCT
cana-1058	110	90	b	b	X
cana-1058	110	91	,	,	PUNCT
cana-1058	110	92	c	c	NOUN
cana-1058	110	93	,	,	PUNCT
cana-1058	110	94	d	d	NOUN
cana-1058	110	95	}	}	PUNCT
cana-1058	110	96	}	}	PUNCT
cana-1058	110	97	.	.	PUNCT
cana-1058	111	1	it	it	PRON
cana-1058	111	2	is	be	AUX
cana-1058	111	3	clear	clear	ADJ
cana-1058	111	4	that	that	SCONJ
cana-1058	111	5	{	{	PUNCT
cana-1058	111	6	a	a	DET
cana-1058	111	7	,	,	PUNCT
cana-1058	111	8	b	b	NOUN
cana-1058	111	9	,	,	PUNCT
cana-1058	111	10	c	c	NOUN
cana-1058	111	11	}	}	PUNCT
cana-1058	111	12	is	be	AUX
cana-1058	111	13	n	n	PRON
cana-1058	111	14	αig	αig	NOUN
cana-1058	111	15	-	-	PUNCT
cana-1058	111	16	closed	close	VERB
cana-1058	111	17	set	set	NOUN
cana-1058	111	18	but	but	CCONJ
cana-1058	111	19	it	it	PRON
cana-1058	111	20	is	be	AUX
cana-1058	111	21	not	not	PART
cana-1058	111	22	∗𝑁closed	∗𝑁close	VERB
cana-1058	111	23	.	.	PUNCT
cana-1058	112	1	theorem	theorem	VERB
cana-1058	112	2	3.7	3.7	NUM
cana-1058	112	3	:	:	PUNCT
cana-1058	112	4	every	every	DET
cana-1058	112	5	nig*closed	nig*close	VERB
cana-1058	112	6	set	set	NOUN
cana-1058	112	7	is	be	AUX
cana-1058	112	8	n	n	PRON
cana-1058	112	9	αig	αig	NOUN
cana-1058	112	10	-	-	PUNCT
cana-1058	112	11	closed	closed	ADJ
cana-1058	112	12	.	.	PUNCT
cana-1058	113	1	but	but	CCONJ
cana-1058	113	2	not	not	PART
cana-1058	113	3	conversely	conversely	ADV
cana-1058	113	4	.	.	PUNCT
cana-1058	114	1	proof	proof	NOUN
cana-1058	114	2	:	:	PUNCT
cana-1058	114	3	let	let	VERB
cana-1058	114	4	a	a	DET
cana-1058	114	5	⊆	⊆	NUM
cana-1058	114	6	g	g	NOUN
cana-1058	114	7	and	and	CCONJ
cana-1058	114	8	g	g	PROPN
cana-1058	114	9	is	be	AUX
cana-1058	114	10	nano	nano	VERB
cana-1058	114	11	α	α	NOUN
cana-1058	114	12	open	open	ADJ
cana-1058	114	13	.	.	PUNCT
cana-1058	115	1	clearly	clearly	ADV
cana-1058	115	2	every	every	DET
cana-1058	115	3	nano	nano	NOUN
cana-1058	115	4	α	α	DET
cana-1058	115	5	open	open	ADJ
cana-1058	115	6	set	set	NOUN
cana-1058	115	7	is	be	AUX
cana-1058	115	8	nano	nano	VERB
cana-1058	115	9	semi	semi	ADV
cana-1058	115	10	open	open	ADJ
cana-1058	115	11	.	.	PUNCT
cana-1058	116	1	since	since	SCONJ
cana-1058	116	2	a	a	DET
cana-1058	116	3	is	be	AUX
cana-1058	116	4	nig*closed	nig*close	VERB
cana-1058	116	5	set	set	NOUN
cana-1058	116	6	,	,	PUNCT
cana-1058	116	7	(	(	PUNCT
cana-1058	116	8	a*)n⊆	a*)n⊆	PROPN
cana-1058	116	9	g	g	PROPN
cana-1058	116	10	,	,	PUNCT
cana-1058	116	11	which	which	PRON
cana-1058	116	12	implies	imply	VERB
cana-1058	116	13	that	that	SCONJ
cana-1058	116	14	a	a	PRON
cana-1058	116	15	is	be	AUX
cana-1058	116	16	an	an	DET
cana-1058	116	17	nαig	nαig	ADV
cana-1058	116	18	-	-	PUNCT
cana-1058	116	19	closed	close	VERB
cana-1058	116	20	set	set	NOUN
cana-1058	116	21	.	.	PUNCT
cana-1058	116	22	example	example	NOUN
cana-1058	117	1	3.8	3.8	NUM
cana-1058	117	2	:	:	PUNCT
cana-1058	117	3	let	let	VERB
cana-1058	117	4	u	u	PRON
cana-1058	117	5	=	=	X
cana-1058	117	6	{	{	PUNCT
cana-1058	117	7	a	a	PRON
cana-1058	117	8	,	,	PUNCT
cana-1058	117	9	b	b	NOUN
cana-1058	117	10	,	,	PUNCT
cana-1058	117	11	c	c	NOUN
cana-1058	117	12	,	,	PUNCT
cana-1058	117	13	d	d	NOUN
cana-1058	117	14	}	}	PUNCT
cana-1058	117	15	,	,	PUNCT
cana-1058	117	16	u	u	NOUN
cana-1058	117	17	/	/	SYM
cana-1058	117	18	r	r	NOUN
cana-1058	117	19	=	=	NOUN
cana-1058	117	20	{	{	PUNCT
cana-1058	117	21	a},{d	a},{d	NOUN
cana-1058	117	22	}	}	PUNCT
cana-1058	117	23	,	,	PUNCT
cana-1058	117	24	{	{	PUNCT
cana-1058	117	25	b	b	X
cana-1058	117	26	,	,	PUNCT
cana-1058	117	27	c	c	NOUN
cana-1058	117	28	}	}	PUNCT
cana-1058	117	29	}	}	PUNCT
cana-1058	117	30	and	and	CCONJ
cana-1058	117	31	x	x	X
cana-1058	117	32	=	=	X
cana-1058	117	33	{	{	PUNCT
cana-1058	117	34	a	a	X
cana-1058	117	35	,	,	PUNCT
cana-1058	117	36	d	d	NOUN
cana-1058	117	37	}	}	PUNCT
cana-1058	117	38	.	.	PUNCT
cana-1058	118	1	let	let	VERB
cana-1058	118	2	the	the	DET
cana-1058	118	3	nano	nano	NOUN
cana-1058	118	4	ideal	ideal	ADJ
cana-1058	118	5	space	space	NOUN
cana-1058	118	6	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	118	7	)	)	PUNCT
cana-1058	118	8	=	=	PRON
cana-1058	118	9	{	{	PUNCT
cana-1058	118	10	u	u	NOUN
cana-1058	118	11	,	,	PUNCT
cana-1058	118	12	∅	∅	NOUN
cana-1058	118	13	,	,	PUNCT
cana-1058	118	14	{	{	PUNCT
cana-1058	118	15	𝑎	𝑎	X
cana-1058	118	16	,	,	PUNCT
cana-1058	118	17	𝑑	𝑑	NOUN
cana-1058	118	18	}	}	PUNCT
cana-1058	118	19	}	}	PUNCT
cana-1058	118	20	with	with	ADP
cana-1058	118	21	a	a	DET
cana-1058	118	22	nano	nano	NOUN
cana-1058	118	23	ideal	ideal	NOUN
cana-1058	118	24	i	i	PRON
cana-1058	118	25	=	=	NOUN
cana-1058	118	26	{	{	PUNCT
cana-1058	118	27	∅	∅	NOUN
cana-1058	118	28	,	,	PUNCT
cana-1058	118	29	{	{	PUNCT
cana-1058	118	30	a	a	X
cana-1058	118	31	}	}	PUNCT
cana-1058	118	32	}	}	PUNCT
cana-1058	118	33	.	.	PUNCT
cana-1058	119	1	then	then	ADV
cana-1058	119	2	nαig	nαig	ADV
cana-1058	119	3	-	-	PUNCT
cana-1058	119	4	closed	close	VERB
cana-1058	119	5	sets	set	NOUN
cana-1058	119	6	are	be	AUX
cana-1058	119	7	{	{	PUNCT
cana-1058	119	8	u	u	NOUN
cana-1058	119	9	,	,	PUNCT
cana-1058	119	10	∅	∅	NOUN
cana-1058	119	11	,	,	PUNCT
cana-1058	119	12	{	{	PUNCT
cana-1058	119	13	a	a	X
cana-1058	119	14	}	}	PUNCT
cana-1058	119	15	,	,	PUNCT
cana-1058	119	16	{	{	PUNCT
cana-1058	119	17	b	b	NOUN
cana-1058	119	18	,	,	PUNCT
cana-1058	119	19	c	c	NOUN
cana-1058	119	20	}	}	PUNCT
cana-1058	119	21	,	,	PUNCT
cana-1058	119	22	{	{	PUNCT
cana-1058	119	23	b	b	X
cana-1058	119	24	,	,	PUNCT
cana-1058	119	25	d	d	NOUN
cana-1058	119	26	}	}	PUNCT
cana-1058	119	27	,	,	PUNCT
cana-1058	119	28	{	{	PUNCT
cana-1058	119	29	a	a	DET
cana-1058	119	30	,	,	PUNCT
cana-1058	119	31	b	b	NOUN
cana-1058	119	32	,	,	PUNCT
cana-1058	119	33	c	c	NOUN
cana-1058	119	34	}	}	PUNCT
cana-1058	119	35	,	,	PUNCT
cana-1058	119	36	{	{	PUNCT
cana-1058	119	37	b	b	X
cana-1058	119	38	,	,	PUNCT
cana-1058	119	39	c	c	NOUN
cana-1058	119	40	,	,	PUNCT
cana-1058	119	41	d	d	NOUN
cana-1058	119	42	}	}	PUNCT
cana-1058	119	43	}	}	PUNCT
cana-1058	119	44	and	and	CCONJ
cana-1058	119	45	nig*closed	nig*close	VERB
cana-1058	119	46	sets	set	NOUN
cana-1058	119	47	are	be	AUX
cana-1058	119	48	{	{	PUNCT
cana-1058	119	49	u	u	NOUN
cana-1058	119	50	,	,	PUNCT
cana-1058	119	51	∅	∅	NOUN
cana-1058	119	52	,	,	PUNCT
cana-1058	119	53	{	{	PUNCT
cana-1058	119	54	a	a	X
cana-1058	119	55	}	}	PUNCT
cana-1058	119	56	,	,	PUNCT
cana-1058	119	57	{	{	PUNCT
cana-1058	119	58	b	b	NOUN
cana-1058	119	59	,	,	PUNCT
cana-1058	119	60	c	c	NOUN
cana-1058	119	61	}	}	PUNCT
cana-1058	119	62	,	,	PUNCT
cana-1058	119	63	{	{	PUNCT
cana-1058	119	64	a	a	DET
cana-1058	119	65	,	,	PUNCT
cana-1058	119	66	b	b	NOUN
cana-1058	119	67	,	,	PUNCT
cana-1058	119	68	c	c	NOUN
cana-1058	119	69	}	}	PUNCT
cana-1058	119	70	,	,	PUNCT
cana-1058	119	71	{	{	PUNCT
cana-1058	119	72	b	b	X
cana-1058	119	73	,	,	PUNCT
cana-1058	119	74	c	c	NOUN
cana-1058	119	75	,	,	PUNCT
cana-1058	119	76	d	d	NOUN
cana-1058	119	77	}	}	PUNCT
cana-1058	119	78	}	}	PUNCT
cana-1058	119	79	.	.	PUNCT
cana-1058	120	1	it	it	PRON
cana-1058	120	2	is	be	AUX
cana-1058	120	3	clear	clear	ADJ
cana-1058	120	4	that	that	SCONJ
cana-1058	120	5	{	{	PUNCT
cana-1058	120	6	b	b	X
cana-1058	120	7	,	,	PUNCT
cana-1058	120	8	d	d	NOUN
cana-1058	120	9	}	}	PUNCT
cana-1058	120	10	is	be	AUX
cana-1058	120	11	n	n	PRON
cana-1058	120	12	αig	αig	NOUN
cana-1058	120	13	-	-	PUNCT
cana-1058	120	14	closed	closed	ADJ
cana-1058	120	15	but	but	CCONJ
cana-1058	120	16	it	it	PRON
cana-1058	120	17	is	be	AUX
cana-1058	120	18	not	not	PART
cana-1058	120	19	nig	nig	ADJ
cana-1058	120	20	*	*	PUNCT
cana-1058	120	21	closed	closed	ADJ
cana-1058	120	22	.	.	PUNCT
cana-1058	121	1	theorem	theorem	VERB
cana-1058	121	2	3.9.everynαig	3.9.everynαig	PROPN
cana-1058	121	3	-	-	PUNCT
cana-1058	121	4	closedsetisnigclosed	closedsetisnigclose	VERB
cana-1058	121	5	.	.	PUNCT
cana-1058	122	1	butconverseisnottrue	butconverseisnottrue	NOUN
cana-1058	122	2	.	.	PUNCT
cana-1058	123	1	proof	proof	NOUN
cana-1058	123	2	:	:	PUNCT
cana-1058	123	3	let	let	VERB
cana-1058	123	4	a	a	DET
cana-1058	123	5	⊆	⊆	NUM
cana-1058	123	6	g	g	NOUN
cana-1058	123	7	and	and	CCONJ
cana-1058	123	8	g	g	PROPN
cana-1058	123	9	is	be	AUX
cana-1058	123	10	nano	nano	VERB
cana-1058	123	11	α	α	NOUN
cana-1058	123	12	open	open	ADJ
cana-1058	123	13	.	.	PUNCT
cana-1058	124	1	clearly	clearly	ADV
cana-1058	124	2	every	every	DET
cana-1058	124	3	nano	nano	NOUN
cana-1058	124	4	open	open	ADJ
cana-1058	124	5	set	set	NOUN
cana-1058	124	6	is	be	AUX
cana-1058	124	7	nano	nano	VERB
cana-1058	124	8	α	α	NOUN
cana-1058	124	9	open	open	ADJ
cana-1058	124	10	.	.	PUNCT
cana-1058	125	1	since	since	SCONJ
cana-1058	125	2	a	a	PRON
cana-1058	125	3	is	be	AUX
cana-1058	125	4	n	n	PRON
cana-1058	125	5	αig	αig	NOUN
cana-1058	125	6	-	-	PUNCT
cana-1058	125	7	closed	close	VERB
cana-1058	125	8	set	set	NOUN
cana-1058	125	9	,	,	PUNCT
cana-1058	125	10	(	(	PUNCT
cana-1058	125	11	a*)n⊆	a*)n⊆	PROPN
cana-1058	125	12	g	g	PROPN
cana-1058	125	13	,	,	PUNCT
cana-1058	125	14	which	which	PRON
cana-1058	125	15	implies	imply	VERB
cana-1058	125	16	that	that	SCONJ
cana-1058	125	17	a	a	PRON
cana-1058	125	18	is	be	AUX
cana-1058	125	19	nigclosed	nigclose	VERB
cana-1058	125	20	.	.	PUNCT
cana-1058	126	1	example	example	NOUN
cana-1058	126	2	3.10	3.10	NUM
cana-1058	126	3	.	.	PUNCT
cana-1058	127	1	let	let	VERB
cana-1058	127	2	u	u	PRON
cana-1058	127	3	=	=	X
cana-1058	127	4	{	{	PUNCT
cana-1058	127	5	a	a	PRON
cana-1058	127	6	,	,	PUNCT
cana-1058	127	7	b	b	NOUN
cana-1058	127	8	,	,	PUNCT
cana-1058	127	9	c	c	NOUN
cana-1058	127	10	,	,	PUNCT
cana-1058	127	11	d	d	NOUN
cana-1058	127	12	}	}	PUNCT
cana-1058	127	13	,	,	PUNCT
cana-1058	127	14	u	u	NOUN
cana-1058	127	15	/	/	SYM
cana-1058	127	16	r	r	NOUN
cana-1058	127	17	=	=	NOUN
cana-1058	127	18	{	{	PUNCT
cana-1058	127	19	a},{d	a},{d	NOUN
cana-1058	127	20	}	}	PUNCT
cana-1058	127	21	,	,	PUNCT
cana-1058	127	22	{	{	PUNCT
cana-1058	127	23	b	b	X
cana-1058	127	24	,	,	PUNCT
cana-1058	127	25	c	c	NOUN
cana-1058	127	26	}	}	PUNCT
cana-1058	127	27	}	}	PUNCT
cana-1058	127	28	and	and	CCONJ
cana-1058	127	29	x	x	X
cana-1058	127	30	=	=	X
cana-1058	127	31	{	{	PUNCT
cana-1058	127	32	a	a	X
cana-1058	127	33	,	,	PUNCT
cana-1058	127	34	d	d	NOUN
cana-1058	127	35	}	}	PUNCT
cana-1058	127	36	.	.	PUNCT
cana-1058	128	1	let	let	VERB
cana-1058	128	2	the	the	DET
cana-1058	128	3	nano	nano	NOUN
cana-1058	128	4	ideal	ideal	ADJ
cana-1058	128	5	space	space	NOUN
cana-1058	128	6	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	128	7	)	)	PUNCT
cana-1058	128	8	=	=	PRON
cana-1058	128	9	{	{	PUNCT
cana-1058	128	10	u	u	NOUN
cana-1058	128	11	,	,	PUNCT
cana-1058	128	12	∅	∅	NOUN
cana-1058	128	13	,	,	PUNCT
cana-1058	128	14	{	{	PUNCT
cana-1058	128	15	𝑎	𝑎	X
cana-1058	128	16	,	,	PUNCT
cana-1058	128	17	𝑑	𝑑	NOUN
cana-1058	128	18	}	}	PUNCT
cana-1058	128	19	}	}	PUNCT
cana-1058	128	20	with	with	ADP
cana-1058	128	21	a	a	DET
cana-1058	128	22	nano	nano	NOUN
cana-1058	128	23	ideal	ideal	NOUN
cana-1058	128	24	i	i	PRON
cana-1058	128	25	=	=	NOUN
cana-1058	128	26	{	{	PUNCT
cana-1058	128	27	∅	∅	NOUN
cana-1058	128	28	,	,	PUNCT
cana-1058	128	29	{	{	PUNCT
cana-1058	128	30	a	a	X
cana-1058	128	31	}	}	PUNCT
cana-1058	128	32	}	}	PUNCT
cana-1058	128	33	.	.	PUNCT
cana-1058	129	1	then	then	ADV
cana-1058	129	2	nαig	nαig	ADV
cana-1058	129	3	-	-	PUNCT
cana-1058	129	4	closed	close	VERB
cana-1058	129	5	sets	set	NOUN
cana-1058	129	6	are	be	AUX
cana-1058	129	7	{	{	PUNCT
cana-1058	129	8	u	u	NOUN
cana-1058	129	9	,	,	PUNCT
cana-1058	129	10	∅,{a	∅,{a	ADV
cana-1058	129	11	}	}	PUNCT
cana-1058	129	12	,	,	PUNCT
cana-1058	129	13	{	{	PUNCT
cana-1058	129	14	b	b	X
cana-1058	129	15	,	,	PUNCT
cana-1058	129	16	c	c	NOUN
cana-1058	129	17	}	}	PUNCT
cana-1058	129	18	,	,	PUNCT
cana-1058	129	19	{	{	PUNCT
cana-1058	129	20	b	b	X
cana-1058	129	21	,	,	PUNCT
cana-1058	129	22	d	d	NOUN
cana-1058	129	23	}	}	PUNCT
cana-1058	129	24	,	,	PUNCT
cana-1058	129	25	{	{	PUNCT
cana-1058	129	26	a	a	DET
cana-1058	129	27	,	,	PUNCT
cana-1058	129	28	b	b	NOUN
cana-1058	129	29	,	,	PUNCT
cana-1058	129	30	c	c	NOUN
cana-1058	129	31	}	}	PUNCT
cana-1058	129	32	,	,	PUNCT
cana-1058	129	33	{	{	PUNCT
cana-1058	129	34	b	b	X
cana-1058	129	35	,	,	PUNCT
cana-1058	129	36	c	c	NOUN
cana-1058	129	37	,	,	PUNCT
cana-1058	129	38	d	d	NOUN
cana-1058	129	39	}	}	PUNCT
cana-1058	129	40	}	}	PUNCT
cana-1058	129	41	and	and	CCONJ
cana-1058	129	42	nigclosed	nigclose	VERB
cana-1058	129	43	sets	set	NOUN
cana-1058	129	44	are	be	AUX
cana-1058	129	45	{	{	PUNCT
cana-1058	129	46	u	u	NOUN
cana-1058	129	47	,	,	PUNCT
cana-1058	129	48	∅	∅	NOUN
cana-1058	129	49	,	,	PUNCT
cana-1058	129	50	{	{	PUNCT
cana-1058	129	51	a	a	X
cana-1058	129	52	}	}	PUNCT
cana-1058	129	53	,	,	PUNCT
cana-1058	129	54	{	{	PUNCT
cana-1058	129	55	b	b	NOUN
cana-1058	129	56	}	}	PUNCT
cana-1058	129	57	,	,	PUNCT
cana-1058	129	58	{	{	PUNCT
cana-1058	129	59	c	c	X
cana-1058	129	60	}	}	PUNCT
cana-1058	129	61	,	,	PUNCT
cana-1058	129	62	{	{	PUNCT
cana-1058	129	63	a	a	DET
cana-1058	129	64	,	,	PUNCT
cana-1058	129	65	b	b	NOUN
cana-1058	129	66	}	}	PUNCT
cana-1058	129	67	,	,	PUNCT
cana-1058	129	68	{	{	PUNCT
cana-1058	129	69	a	a	DET
cana-1058	129	70	,	,	PUNCT
cana-1058	129	71	c	c	NOUN
cana-1058	129	72	}	}	PUNCT
cana-1058	129	73	,	,	PUNCT
cana-1058	129	74	{	{	PUNCT
cana-1058	129	75	b	b	X
cana-1058	129	76	,	,	PUNCT
cana-1058	129	77	c	c	NOUN
cana-1058	129	78	}	}	PUNCT
cana-1058	129	79	,	,	PUNCT
cana-1058	129	80	{	{	PUNCT
cana-1058	129	81	b	b	X
cana-1058	129	82	,	,	PUNCT
cana-1058	129	83	d	d	NOUN
cana-1058	129	84	}	}	PUNCT
cana-1058	129	85	,	,	PUNCT
cana-1058	129	86	{	{	PUNCT
cana-1058	129	87	c	c	X
cana-1058	129	88	,	,	PUNCT
cana-1058	129	89	d	d	NOUN
cana-1058	129	90	}	}	PUNCT
cana-1058	129	91	,	,	PUNCT
cana-1058	129	92	{	{	PUNCT
cana-1058	129	93	a	a	DET
cana-1058	129	94	,	,	PUNCT
cana-1058	129	95	b	b	NOUN
cana-1058	129	96	,	,	PUNCT
cana-1058	129	97	c	c	NOUN
cana-1058	129	98	}	}	PUNCT
cana-1058	129	99	,	,	PUNCT
cana-1058	129	100	{	{	PUNCT
cana-1058	129	101	a	a	DET
cana-1058	129	102	,	,	PUNCT
cana-1058	129	103	b	b	NOUN
cana-1058	129	104	,	,	PUNCT
cana-1058	129	105	d	d	NOUN
cana-1058	129	106	}	}	PUNCT
cana-1058	129	107	,	,	PUNCT
cana-1058	129	108	{	{	PUNCT
cana-1058	129	109	a	a	PRON
cana-1058	129	110	,	,	PUNCT
cana-1058	129	111	c	c	NOUN
cana-1058	129	112	,	,	PUNCT
cana-1058	129	113	d	d	NOUN
cana-1058	129	114	}	}	PUNCT
cana-1058	129	115	,	,	PUNCT
cana-1058	129	116	{	{	PUNCT
cana-1058	129	117	b	b	X
cana-1058	129	118	,	,	PUNCT
cana-1058	129	119	c	c	NOUN
cana-1058	129	120	,	,	PUNCT
cana-1058	129	121	d	d	NOUN
cana-1058	129	122	}	}	PUNCT
cana-1058	129	123	}	}	PUNCT
cana-1058	129	124	.	.	PUNCT
cana-1058	130	1	it	it	PRON
cana-1058	130	2	is	be	AUX
cana-1058	130	3	clear	clear	ADJ
cana-1058	130	4	that	that	SCONJ
cana-1058	130	5	{	{	PUNCT
cana-1058	130	6	b	b	X
cana-1058	130	7	}	}	PUNCT
cana-1058	130	8	is	be	AUX
cana-1058	130	9	nigclosed	nigclose	VERB
cana-1058	130	10	but	but	CCONJ
cana-1058	130	11	it	it	PRON
cana-1058	130	12	is	be	AUX
cana-1058	130	13	not	not	PART
cana-1058	130	14	nigclosed	nigclose	VERB
cana-1058	130	15	.	.	PUNCT
cana-1058	131	1	theorem	theorem	NOUN
cana-1058	131	2	3.11	3.11	NUM
cana-1058	131	3	.	.	PUNCT
cana-1058	132	1	let	let	VERB
cana-1058	132	2	(	(	PUNCT
cana-1058	132	3	u	u	NOUN
cana-1058	132	4	,	,	PUNCT
cana-1058	132	5	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	132	6	)	)	PUNCT
cana-1058	132	7	,	,	PUNCT
cana-1058	132	8	i	i	PRON
cana-1058	132	9	)	)	PUNCT
cana-1058	132	10	be	be	AUX
cana-1058	132	11	an	an	DET
cana-1058	132	12	nano	nano	NOUN
cana-1058	132	13	ideal	ideal	ADJ
cana-1058	132	14	space	space	NOUN
cana-1058	132	15	.	.	PUNCT
cana-1058	133	1	for	for	ADP
cana-1058	133	2	every	every	DET
cana-1058	133	3	a	a	DET
cana-1058	133	4			NOUN
cana-1058	133	5	i	i	PRON
cana-1058	133	6	,	,	PUNCT
cana-1058	133	7	a	a	PRON
cana-1058	133	8	is	be	AUX
cana-1058	133	9	n	n	PRON
cana-1058	133	10	αig	αig	NOUN
cana-1058	133	11	-	-	PUNCT
cana-1058	133	12	closed	close	VERB
cana-1058	133	13	set	set	NOUN
cana-1058	133	14	.	.	PUNCT
cana-1058	134	1	proof	proof	NOUN
cana-1058	134	2	:	:	PUNCT
cana-1058	134	3	let	let	VERB
cana-1058	134	4	a	a	DET
cana-1058	134	5	⊆	⊆	NUM
cana-1058	134	6	g	g	NOUN
cana-1058	134	7	and	and	CCONJ
cana-1058	134	8	g	g	PROPN
cana-1058	134	9	is	be	AUX
cana-1058	134	10	nano	nano	VERB
cana-1058	134	11	α	α	NOUN
cana-1058	134	12	open	open	ADJ
cana-1058	134	13	.	.	PUNCT
cana-1058	135	1	since	since	SCONJ
cana-1058	135	2	(	(	PUNCT
cana-1058	135	3	a*)n	a*)n	NOUN
cana-1058	135	4	=	=	NOUN
cana-1058	135	5	∅	∅	NOUN
cana-1058	135	6	for	for	ADP
cana-1058	135	7	every	every	DET
cana-1058	135	8	a	a	DET
cana-1058	135	9			NOUN
cana-1058	135	10	i	i	PRON
cana-1058	135	11	,	,	PUNCT
cana-1058	135	12	then	then	ADV
cana-1058	135	13	(	(	PUNCT
cana-1058	135	14	a*)n⊆	a*)n⊆	PROPN
cana-1058	135	15	a.	a.	NOUN
cana-1058	135	16	this	this	PRON
cana-1058	135	17	implies	imply	VERB
cana-1058	135	18	(	(	PUNCT
cana-1058	135	19	a*)n⊆	a*)n⊆	PROPN
cana-1058	135	20	g.	g.	PROPN
cana-1058	135	21	hence	hence	ADV
cana-1058	135	22	for	for	ADP
cana-1058	135	23	every	every	DET
cana-1058	135	24	a	a	DET
cana-1058	135	25			NOUN
cana-1058	135	26	i	i	PRON
cana-1058	135	27	,	,	PUNCT
cana-1058	135	28	a	a	PRON
cana-1058	135	29	is	be	AUX
cana-1058	135	30	an	an	DET
cana-1058	135	31	n	n	CCONJ
cana-1058	135	32	αig	αig	NOUN
cana-1058	135	33	-	-	PUNCT
cana-1058	135	34	closed	close	VERB
cana-1058	135	35	set	set	NOUN
cana-1058	135	36	.	.	PUNCT
cana-1058	136	1	theorem	theorem	VERB
cana-1058	136	2	3.12	3.12	NUM
cana-1058	136	3	.	.	PUNCT
cana-1058	137	1	if	if	SCONJ
cana-1058	137	2	a	a	PRON
cana-1058	137	3	and	and	CCONJ
cana-1058	137	4	b	b	NOUN
cana-1058	137	5	are	be	AUX
cana-1058	137	6	nαig	nαig	ADV
cana-1058	137	7	-	-	PUNCT
cana-1058	137	8	closed	close	VERB
cana-1058	137	9	sets	set	NOUN
cana-1058	137	10	in	in	ADP
cana-1058	137	11	(	(	PUNCT
cana-1058	137	12	u	u	NOUN
cana-1058	137	13	,	,	PUNCT
cana-1058	137	14	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	137	15	)	)	PUNCT
cana-1058	137	16	,	,	PUNCT
cana-1058	137	17	i	i	NOUN
cana-1058	137	18	)	)	PUNCT
cana-1058	137	19	,	,	PUNCT
cana-1058	137	20	then	then	ADV
cana-1058	137	21	a	a	DET
cana-1058	137	22			NOUN
cana-1058	137	23	b	b	PROPN
cana-1058	137	24	is	be	AUX
cana-1058	137	25	also	also	ADV
cana-1058	137	26	an	an	DET
cana-1058	137	27	n	n	ADV
cana-1058	137	28	αigclosed	αigclose	VERB
cana-1058	137	29	set	set	NOUN
cana-1058	137	30	.	.	PUNCT
cana-1058	138	1	proof	proof	NOUN
cana-1058	138	2	:	:	PUNCT
cana-1058	138	3	let	let	VERB
cana-1058	138	4	a	a	DET
cana-1058	138	5			NOUN
cana-1058	138	6	b	b	PROPN
cana-1058	138	7			PROPN
cana-1058	138	8	gwhere	gwhere	NOUN
cana-1058	138	9	g	g	PROPN
cana-1058	138	10	is	be	AUX
cana-1058	138	11	a	a	DET
cana-1058	138	12	nano	nano	ADJ
cana-1058	138	13	-open	-open	NOUN
cana-1058	138	14	set	set	VERB
cana-1058	138	15	in	in	ADP
cana-1058	138	16	x.	x.	NOUN
cana-1058	138	17	then	then	ADV
cana-1058	138	18	,	,	PUNCT
cana-1058	138	19	a	a	DET
cana-1058	138	20			PROPN
cana-1058	138	21	g	g	PROPN
cana-1058	138	22	and	and	CCONJ
cana-1058	138	23	b	b	PROPN
cana-1058	138	24			PROPN
cana-1058	138	25	g.	g.	PROPN
cana-1058	138	26	by	by	ADP
cana-1058	138	27	hypothesis	hypothesis	NOUN
cana-1058	138	28	,	,	PUNCT
cana-1058	138	29	a	a	PRON
cana-1058	138	30	and	and	CCONJ
cana-1058	138	31	b	b	NOUN
cana-1058	138	32	are	be	AUX
cana-1058	138	33	two	two	NUM
cana-1058	138	34	n	n	CCONJ
cana-1058	138	35	αig	αig	NOUN
cana-1058	138	36	-	-	PUNCT
cana-1058	138	37	closed	close	VERB
cana-1058	138	38	set	set	NOUN
cana-1058	138	39	.	.	PUNCT
cana-1058	139	1	this	this	PRON
cana-1058	139	2	implies	imply	VERB
cana-1058	139	3	(	(	PUNCT
cana-1058	139	4	a*)n	a*)n	INTJ
cana-1058	139	5	g	g	NOUN
cana-1058	139	6	and	and	CCONJ
cana-1058	139	7	(	(	PUNCT
cana-1058	139	8	b*)n	b*)n	NOUN
cana-1058	139	9	g.	g.	PROPN
cana-1058	139	10	hence	hence	ADV
cana-1058	139	11	(	(	PUNCT
cana-1058	139	12	a*)n	a*)n	NOUN
cana-1058	139	13	(	(	PUNCT
cana-1058	139	14	b*)n	b*)n	NOUN
cana-1058	139	15	g.	g.	NOUN
cana-1058	139	16	by	by	ADP
cana-1058	139	17	result	result	NOUN
cana-1058	139	18	2.6	2.6	NUM
cana-1058	139	19	(	(	PUNCT
cana-1058	139	20	vii	vii	PROPN
cana-1058	139	21	)	)	PUNCT
cana-1058	139	22	,	,	PUNCT
cana-1058	139	23	(	(	PUNCT
cana-1058	139	24	(	(	PUNCT
cana-1058	139	25	a	a	DET
cana-1058	139	26			NOUN
cana-1058	139	27	b)*)n	b)*)n	NOUN
cana-1058	139	28	=	=	PUNCT
cana-1058	139	29	(	(	PUNCT
cana-1058	139	30	a*)n	a*)n	NOUN
cana-1058	139	31	(	(	PUNCT
cana-1058	139	32	b*)n	b*)n	NOUN
cana-1058	139	33	g.	g.	PROPN
cana-1058	139	34	therefore	therefore	ADV
cana-1058	139	35	,	,	PUNCT
cana-1058	139	36	a	a	DET
cana-1058	139	37			NOUN
cana-1058	139	38	b	b	PROPN
cana-1058	139	39	is	be	AUX
cana-1058	139	40	an	an	DET
cana-1058	139	41	n	n	CCONJ
cana-1058	139	42	αig	αig	NOUN
cana-1058	139	43	-	-	PUNCT
cana-1058	139	44	closed	close	VERB
cana-1058	139	45	set	set	NOUN
cana-1058	139	46	.	.	PUNCT
cana-1058	140	1	remark	remark	PROPN
cana-1058	140	2	3.13	3.13	NUM
cana-1058	140	3	.	.	PUNCT
cana-1058	141	1	the	the	DET
cana-1058	141	2	intersection	intersection	NOUN
cana-1058	141	3	of	of	ADP
cana-1058	141	4	n	n	DET
cana-1058	141	5	αig	αig	NOUN
cana-1058	141	6	-	-	PUNCT
cana-1058	141	7	closed	close	VERB
cana-1058	141	8	sets	set	NOUN
cana-1058	141	9	in	in	ADP
cana-1058	141	10	(	(	PUNCT
cana-1058	141	11	u	u	NOUN
cana-1058	141	12	,	,	PUNCT
cana-1058	141	13	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	141	14	)	)	PUNCT
cana-1058	141	15	,	,	PUNCT
cana-1058	141	16	i	i	NOUN
cana-1058	141	17	)	)	PUNCT
cana-1058	141	18	need	need	AUX
cana-1058	141	19	not	not	PART
cana-1058	141	20	be	be	AUX
cana-1058	141	21	a	a	DET
cana-1058	141	22	n	n	CCONJ
cana-1058	141	23	αig	αig	NOUN
cana-1058	141	24	-	-	PUNCT
cana-1058	141	25	closed	close	VERB
cana-1058	141	26	set	set	NOUN
cana-1058	141	27	.	.	PUNCT
cana-1058	142	1	this	this	PRON
cana-1058	142	2	can	can	AUX
cana-1058	142	3	be	be	AUX
cana-1058	142	4	proved	prove	VERB
cana-1058	142	5	from	from	ADP
cana-1058	142	6	the	the	DET
cana-1058	142	7	example	example	NOUN
cana-1058	142	8	given	give	VERB
cana-1058	142	9	below	below	ADV
cana-1058	142	10	.	.	PUNCT
cana-1058	143	1	communications	communication	NOUN
cana-1058	143	2	on	on	ADP
cana-1058	143	3	applied	apply	VERB
cana-1058	143	4	nonlinear	nonlinear	ADJ
cana-1058	143	5	analysis	analysis	NOUN
cana-1058	143	6	issn	issn	NOUN
cana-1058	143	7	:	:	PUNCT
cana-1058	143	8	1074	1074	NUM
cana-1058	143	9	-	-	PUNCT
cana-1058	143	10	133x	133x	NUM
cana-1058	143	11	vol	vol	NOUN
cana-1058	143	12	31	31	NUM
cana-1058	143	13	no	no	NOUN
cana-1058	143	14	.	.	PUNCT
cana-1058	144	1	5s	5s	NUM
cana-1058	144	2	(	(	PUNCT
cana-1058	144	3	2024	2024	NUM
cana-1058	144	4	)	)	PUNCT
cana-1058	144	5	395	395	NUM
cana-1058	144	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1058	144	7	example	example	NOUN
cana-1058	144	8	3.14	3.14	NUM
cana-1058	144	9	.	.	PUNCT
cana-1058	145	1	let	let	VERB
cana-1058	145	2	u	u	PRON
cana-1058	145	3	=	=	X
cana-1058	145	4	{	{	PUNCT
cana-1058	145	5	a	a	PRON
cana-1058	145	6	,	,	PUNCT
cana-1058	145	7	b	b	NOUN
cana-1058	145	8	,	,	PUNCT
cana-1058	145	9	c	c	NOUN
cana-1058	145	10	,	,	PUNCT
cana-1058	145	11	d	d	NOUN
cana-1058	145	12	}	}	PUNCT
cana-1058	145	13	,	,	PUNCT
cana-1058	145	14	u	u	NOUN
cana-1058	145	15	/	/	SYM
cana-1058	145	16	r	r	NOUN
cana-1058	145	17	=	=	NOUN
cana-1058	145	18	{	{	PUNCT
cana-1058	145	19	{	{	PUNCT
cana-1058	145	20	a},{d	a},{d	NOUN
cana-1058	145	21	}	}	PUNCT
cana-1058	145	22	,	,	PUNCT
cana-1058	145	23	{	{	PUNCT
cana-1058	145	24	b	b	X
cana-1058	145	25	,	,	PUNCT
cana-1058	145	26	c	c	NOUN
cana-1058	145	27	}	}	PUNCT
cana-1058	145	28	}	}	PUNCT
cana-1058	145	29	and	and	CCONJ
cana-1058	145	30	x	x	X
cana-1058	145	31	=	=	X
cana-1058	145	32	{	{	PUNCT
cana-1058	145	33	a	a	X
cana-1058	145	34	,	,	PUNCT
cana-1058	145	35	d	d	NOUN
cana-1058	145	36	}	}	PUNCT
cana-1058	145	37	.	.	PUNCT
cana-1058	146	1	let	let	VERB
cana-1058	146	2	the	the	DET
cana-1058	146	3	nano	nano	NOUN
cana-1058	146	4	ideal	ideal	ADJ
cana-1058	146	5	space	space	NOUN
cana-1058	146	6	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	146	7	)	)	PUNCT
cana-1058	146	8	=	=	PRON
cana-1058	146	9	{	{	PUNCT
cana-1058	146	10	u	u	NOUN
cana-1058	146	11	,	,	PUNCT
cana-1058	146	12	∅	∅	NOUN
cana-1058	146	13	,	,	PUNCT
cana-1058	146	14	{	{	PUNCT
cana-1058	146	15	𝑎	𝑎	X
cana-1058	146	16	,	,	PUNCT
cana-1058	146	17	𝑑	𝑑	NOUN
cana-1058	146	18	}	}	PUNCT
cana-1058	146	19	}	}	PUNCT
cana-1058	146	20	with	with	ADP
cana-1058	146	21	a	a	DET
cana-1058	146	22	nano	nano	NOUN
cana-1058	146	23	ideal	ideal	NOUN
cana-1058	146	24	i	i	PRON
cana-1058	146	25	=	=	NOUN
cana-1058	146	26	{	{	PUNCT
cana-1058	146	27	∅	∅	NOUN
cana-1058	146	28	,	,	PUNCT
cana-1058	146	29	{	{	PUNCT
cana-1058	146	30	a	a	X
cana-1058	146	31	}	}	PUNCT
cana-1058	146	32	}	}	PUNCT
cana-1058	146	33	.	.	PUNCT
cana-1058	147	1	then	then	ADV
cana-1058	147	2	n	n	CCONJ
cana-1058	147	3	αig	αig	NOUN
cana-1058	147	4	-	-	PUNCT
cana-1058	147	5	closed	close	VERB
cana-1058	147	6	sets	set	NOUN
cana-1058	147	7	are	be	AUX
cana-1058	147	8	{	{	PUNCT
cana-1058	147	9	u	u	NOUN
cana-1058	147	10	,	,	PUNCT
cana-1058	147	11	∅	∅	NOUN
cana-1058	147	12	,	,	PUNCT
cana-1058	147	13	{	{	PUNCT
cana-1058	147	14	a	a	X
cana-1058	147	15	}	}	PUNCT
cana-1058	147	16	,	,	PUNCT
cana-1058	147	17	{	{	PUNCT
cana-1058	147	18	b	b	NOUN
cana-1058	147	19	,	,	PUNCT
cana-1058	147	20	c	c	NOUN
cana-1058	147	21	}	}	PUNCT
cana-1058	147	22	,	,	PUNCT
cana-1058	147	23	{	{	PUNCT
cana-1058	147	24	b	b	X
cana-1058	147	25	,	,	PUNCT
cana-1058	147	26	d	d	NOUN
cana-1058	147	27	}	}	PUNCT
cana-1058	147	28	,	,	PUNCT
cana-1058	147	29	{	{	PUNCT
cana-1058	147	30	a	a	DET
cana-1058	147	31	,	,	PUNCT
cana-1058	147	32	b	b	NOUN
cana-1058	147	33	,	,	PUNCT
cana-1058	147	34	c	c	NOUN
cana-1058	147	35	}	}	PUNCT
cana-1058	147	36	,	,	PUNCT
cana-1058	147	37	{	{	PUNCT
cana-1058	147	38	b	b	X
cana-1058	147	39	,	,	PUNCT
cana-1058	147	40	c	c	NOUN
cana-1058	147	41	,	,	PUNCT
cana-1058	147	42	d	d	NOUN
cana-1058	147	43	}	}	PUNCT
cana-1058	147	44	}	}	PUNCT
cana-1058	147	45	.	.	PUNCT
cana-1058	148	1	if	if	SCONJ
cana-1058	148	2	a	a	PRON
cana-1058	148	3	=	=	X
cana-1058	148	4	{	{	PUNCT
cana-1058	148	5	b	b	NOUN
cana-1058	148	6	,	,	PUNCT
cana-1058	148	7	c	c	NOUN
cana-1058	148	8	}	}	PUNCT
cana-1058	148	9	and	and	CCONJ
cana-1058	148	10	b	b	X
cana-1058	148	11	=	=	SYM
cana-1058	148	12	{	{	PUNCT
cana-1058	148	13	b	b	PROPN
cana-1058	148	14	,	,	PUNCT
cana-1058	148	15	d	d	NOUN
cana-1058	148	16	}	}	PUNCT
cana-1058	148	17	,	,	PUNCT
cana-1058	148	18	then	then	ADV
cana-1058	148	19	their	their	PRON
cana-1058	148	20	intersection	intersection	NOUN
cana-1058	148	21	a	a	X
cana-1058	148	22	b	b	PROPN
cana-1058	148	23	=	=	PRON
cana-1058	148	24	{	{	PUNCT
cana-1058	148	25	b	b	NOUN
cana-1058	148	26	}	}	PUNCT
cana-1058	148	27	is	be	AUX
cana-1058	148	28	not	not	PART
cana-1058	148	29	n	n	ADV
cana-1058	148	30	αig	αig	NOUN
cana-1058	148	31	-	-	PUNCT
cana-1058	148	32	closed	closed	ADJ
cana-1058	148	33	.	.	PUNCT
cana-1058	149	1	theorem	theorem	VERB
cana-1058	149	2	3	3	NUM
cana-1058	149	3	.	.	NOUN
cana-1058	149	4	15	15	NUM
cana-1058	149	5	.	.	PUNCT
cana-1058	150	1	if	if	SCONJ
cana-1058	150	2	(	(	PUNCT
cana-1058	150	3	u	u	NOUN
cana-1058	150	4	,	,	PUNCT
cana-1058	150	5	𝑟𝑅(𝑥	𝑟𝑅(𝑥	NUM
cana-1058	150	6	)	)	PUNCT
cana-1058	150	7	,	,	PUNCT
cana-1058	150	8	i	i	PRON
cana-1058	150	9	)	)	PUNCT
cana-1058	150	10	is	be	AUX
cana-1058	150	11	a	a	DET
cana-1058	150	12	nano	nano	ADJ
cana-1058	150	13	ideal	ideal	ADJ
cana-1058	150	14	space	space	NOUN
cana-1058	150	15	,	,	PUNCT
cana-1058	150	16	then	then	ADV
cana-1058	150	17	(	(	PUNCT
cana-1058	150	18	a*)n	a*)n	PROPN
cana-1058	150	19	is	be	AUX
cana-1058	150	20	always	always	ADV
cana-1058	150	21	a	a	DET
cana-1058	150	22	n	n	CCONJ
cana-1058	150	23	αig	αig	NOUN
cana-1058	150	24	-	-	PUNCT
cana-1058	150	25	closed	close	VERB
cana-1058	150	26	set	set	NOUN
cana-1058	150	27	for	for	ADP
cana-1058	150	28	every	every	DET
cana-1058	150	29	subset	subset	NOUN
cana-1058	150	30	a	a	PRON
cana-1058	150	31	of	of	ADP
cana-1058	150	32	x.	x.	NOUN
cana-1058	150	33	proof	proof	NOUN
cana-1058	150	34	.	.	PUNCT
cana-1058	151	1	let	let	VERB
cana-1058	151	2	(	(	PUNCT
cana-1058	151	3	a*)n	a*)n	INTJ
cana-1058	151	4	g	g	NOUN
cana-1058	151	5	,	,	PUNCT
cana-1058	151	6	where	where	SCONJ
cana-1058	151	7	g	g	PROPN
cana-1058	151	8	is	be	AUX
cana-1058	151	9	nano	nano	NOUN
cana-1058	151	10	α	α	NOUN
cana-1058	151	11	open	open	ADJ
cana-1058	151	12	.	.	PUNCT
cana-1058	152	1	since	since	SCONJ
cana-1058	152	2	,	,	PUNCT
cana-1058	152	3	(	(	PUNCT
cana-1058	152	4	(	(	PUNCT
cana-1058	152	5	a*)n	a*)n	PROPN
cana-1058	152	6	)	)	PUNCT
cana-1058	152	7	*	*	PUNCT
cana-1058	152	8			PROPN
cana-1058	152	9	(	(	PUNCT
cana-1058	152	10	a*)n	a*)n	PROPN
cana-1058	152	11	,	,	PUNCT
cana-1058	152	12	we	we	PRON
cana-1058	152	13	have	have	VERB
cana-1058	152	14	(	(	PUNCT
cana-1058	152	15	(	(	PUNCT
cana-1058	152	16	a*)n	a*)n	PROPN
cana-1058	152	17	)	)	PUNCT
cana-1058	152	18	*	*	PUNCT
cana-1058	153	1			PROPN
cana-1058	153	2	g	g	PROPN
cana-1058	153	3	whenever	whenever	SCONJ
cana-1058	153	4	(	(	PUNCT
cana-1058	153	5	a*)n	a*)n	INTJ
cana-1058	153	6	g	g	NOUN
cana-1058	153	7	and	and	CCONJ
cana-1058	153	8	g	g	PROPN
cana-1058	153	9	is	be	AUX
cana-1058	153	10	nano	nano	VERB
cana-1058	153	11	α	α	NOUN
cana-1058	153	12	open	open	ADJ
cana-1058	153	13	.	.	PUNCT
cana-1058	154	1	hence	hence	ADV
cana-1058	154	2	(	(	PUNCT
cana-1058	154	3	a*)n	a*)n	PROPN
cana-1058	154	4	is	be	AUX
cana-1058	154	5	a	a	PRON
cana-1058	154	6	n	n	CCONJ
cana-1058	154	7	αig	αig	NOUN
cana-1058	154	8	-	-	PUNCT
cana-1058	154	9	closed	close	VERB
cana-1058	154	10	set	set	NOUN
cana-1058	154	11	4	4	NUM
cana-1058	154	12	.	.	PUNCT
cana-1058	155	1	application	application	NOUN
cana-1058	155	2	:	:	PUNCT
cana-1058	155	3	4.1	4.1	NUM
cana-1058	155	4	network	network	NOUN
cana-1058	155	5	anomaly	anomaly	NOUN
cana-1058	155	6	detection	detection	NOUN
cana-1058	155	7	:	:	PUNCT
cana-1058	155	8	the	the	DET
cana-1058	155	9	research	research	NOUN
cana-1058	155	10	would	would	AUX
cana-1058	155	11	enhance	enhance	VERB
cana-1058	155	12	the	the	DET
cana-1058	155	13	potential	potential	NOUN
cana-1058	155	14	of	of	ADP
cana-1058	155	15	anomaly	anomaly	NOUN
cana-1058	155	16	detection	detection	NOUN
cana-1058	155	17	inside	inside	ADP
cana-1058	155	18	computer	computer	NOUN
cana-1058	155	19	networks	network	NOUN
cana-1058	155	20	.	.	PUNCT
cana-1058	156	1	subsequently	subsequently	ADV
cana-1058	156	2	,	,	PUNCT
cana-1058	156	3	this	this	DET
cana-1058	156	4	method	method	NOUN
cana-1058	156	5	empowers	empower	VERB
cana-1058	156	6	the	the	DET
cana-1058	156	7	security	security	NOUN
cana-1058	156	8	system	system	NOUN
cana-1058	156	9	through	through	ADP
cana-1058	156	10	the	the	DET
cana-1058	156	11	early	early	ADJ
cana-1058	156	12	detection	detection	NOUN
cana-1058	156	13	of	of	ADP
cana-1058	156	14	various	various	ADJ
cana-1058	156	15	threats	threat	NOUN
cana-1058	156	16	by	by	ADP
cana-1058	156	17	utilizing	utilize	VERB
cana-1058	156	18	notions	notion	NOUN
cana-1058	156	19	of	of	ADP
cana-1058	156	20	nαig	nαig	ADV
cana-1058	156	21	-	-	PUNCT
cana-1058	156	22	closed	close	VERB
cana-1058	156	23	set	set	NOUN
cana-1058	156	24	and	and	CCONJ
cana-1058	156	25	nαig	nαig	ADV
cana-1058	156	26	-	-	PUNCT
cana-1058	156	27	open	open	NOUN
cana-1058	156	28	set	set	NOUN
cana-1058	156	29	.	.	PUNCT
cana-1058	157	1	thus	thus	ADV
cana-1058	157	2	,	,	PUNCT
cana-1058	157	3	security	security	NOUN
cana-1058	157	4	analyses	analysis	NOUN
cana-1058	157	5	could	could	AUX
cana-1058	157	6	review	review	VERB
cana-1058	157	7	all	all	DET
cana-1058	157	8	network	network	NOUN
cana-1058	157	9	traffic	traffic	NOUN
cana-1058	157	10	,	,	PUNCT
cana-1058	157	11	catching	catch	VERB
cana-1058	157	12	even	even	ADV
cana-1058	157	13	the	the	DET
cana-1058	157	14	smallest	small	ADJ
cana-1058	157	15	irregularities	irregularity	NOUN
cana-1058	157	16	in	in	ADP
cana-1058	157	17	network	network	NOUN
cana-1058	157	18	behavior	behavior	NOUN
cana-1058	157	19	.	.	PUNCT
cana-1058	158	1	4.2	4.2	NUM
cana-1058	158	2	social	social	ADJ
cana-1058	158	3	network	network	NOUN
cana-1058	158	4	analysis	analysis	NOUN
cana-1058	158	5	:	:	PUNCT
cana-1058	158	6	additionally	additionally	ADV
cana-1058	158	7	,	,	PUNCT
cana-1058	158	8	the	the	DET
cana-1058	158	9	research	research	NOUN
cana-1058	158	10	could	could	AUX
cana-1058	158	11	contain	contain	VERB
cana-1058	158	12	applications	application	NOUN
cana-1058	158	13	in	in	ADP
cana-1058	158	14	the	the	DET
cana-1058	158	15	field	field	NOUN
cana-1058	158	16	of	of	ADP
cana-1058	158	17	social	social	ADJ
cana-1058	158	18	network	network	NOUN
cana-1058	158	19	analysis	analysis	NOUN
cana-1058	158	20	.	.	PUNCT
cana-1058	159	1	in	in	ADP
cana-1058	159	2	this	this	DET
cana-1058	159	3	regard	regard	NOUN
cana-1058	159	4	,	,	PUNCT
cana-1058	159	5	nαig	nαig	ADV
cana-1058	159	6	-	-	PUNCT
cana-1058	159	7	closed	close	VERB
cana-1058	159	8	set	set	NOUN
cana-1058	159	9	and	and	CCONJ
cana-1058	159	10	nαig	nαig	ADV
cana-1058	159	11	-	-	PUNCT
cana-1058	159	12	open	open	NOUN
cana-1058	159	13	set	set	NOUN
cana-1058	159	14	demonstrate	demonstrate	VERB
cana-1058	159	15	the	the	DET
cana-1058	159	16	potential	potential	NOUN
cana-1058	159	17	to	to	PART
cana-1058	159	18	identify	identify	VERB
cana-1058	159	19	the	the	DET
cana-1058	159	20	most	most	ADV
cana-1058	159	21	influencing	influence	VERB
cana-1058	159	22	nodes	node	NOUN
cana-1058	159	23	and	and	CCONJ
cana-1058	159	24	communities	community	NOUN
cana-1058	159	25	within	within	ADP
cana-1058	159	26	social	social	ADJ
cana-1058	159	27	networks	network	NOUN
cana-1058	159	28	.	.	PUNCT
cana-1058	160	1	therefore	therefore	ADV
cana-1058	160	2	,	,	PUNCT
cana-1058	160	3	specific	specific	ADJ
cana-1058	160	4	information	information	NOUN
cana-1058	160	5	flow	flow	NOUN
cana-1058	160	6	patterns	pattern	NOUN
cana-1058	160	7	,	,	PUNCT
cana-1058	160	8	social	social	ADJ
cana-1058	160	9	dynamics	dynamic	NOUN
cana-1058	160	10	,	,	PUNCT
cana-1058	160	11	and	and	CCONJ
cana-1058	160	12	community	community	NOUN
cana-1058	160	13	structures	structure	NOUN
cana-1058	160	14	could	could	AUX
cana-1058	160	15	be	be	AUX
cana-1058	160	16	analyzed	analyze	VERB
cana-1058	160	17	and	and	CCONJ
cana-1058	160	18	introduce	introduce	VERB
cana-1058	160	19	some	some	DET
cana-1058	160	20	strategic	strategic	ADJ
cana-1058	160	21	decisions	decision	NOUN
cana-1058	160	22	.	.	PUNCT
cana-1058	161	1	4.3	4.3	NUM
cana-1058	161	2	biological	biological	ADJ
cana-1058	161	3	network	network	NOUN
cana-1058	161	4	analysis	analysis	NOUN
cana-1058	161	5	:	:	PUNCT
cana-1058	161	6	the	the	DET
cana-1058	161	7	use	use	NOUN
cana-1058	161	8	of	of	ADP
cana-1058	161	9	nαig	nαig	ADV
cana-1058	161	10	-	-	PUNCT
cana-1058	161	11	closed	close	VERB
cana-1058	161	12	and	and	CCONJ
cana-1058	161	13	nαig	nαig	ADV
cana-1058	161	14	-	-	PUNCT
cana-1058	161	15	open	open	ADJ
cana-1058	161	16	sets	set	NOUN
cana-1058	161	17	for	for	ADP
cana-1058	161	18	biological	biological	ADJ
cana-1058	161	19	network	network	NOUN
cana-1058	161	20	analysis	analysis	NOUN
cana-1058	161	21	also	also	ADV
cana-1058	161	22	allows	allow	VERB
cana-1058	161	23	understanding	understand	VERB
cana-1058	161	24	some	some	DET
cana-1058	161	25	gene	gene	NOUN
cana-1058	161	26	-	-	PUNCT
cana-1058	161	27	protein	protein	NOUN
cana-1058	161	28	interactions	interaction	NOUN
cana-1058	161	29	and	and	CCONJ
cana-1058	161	30	biological	biological	ADJ
cana-1058	161	31	pathways	pathway	NOUN
cana-1058	161	32	.	.	PUNCT
cana-1058	162	1	the	the	DET
cana-1058	162	2	conducting	conducting	NOUN
cana-1058	162	3	of	of	ADP
cana-1058	162	4	analysis	analysis	NOUN
cana-1058	162	5	of	of	ADP
cana-1058	162	6	complex	complex	ADJ
cana-1058	162	7	biological	biological	ADJ
cana-1058	162	8	systems	system	NOUN
cana-1058	162	9	through	through	ADP
cana-1058	162	10	these	these	DET
cana-1058	162	11	mathematical	mathematical	ADJ
cana-1058	162	12	concepts	concept	NOUN
cana-1058	162	13	will	will	AUX
cana-1058	162	14	help	help	VERB
cana-1058	162	15	determine	determine	VERB
cana-1058	162	16	the	the	DET
cana-1058	162	17	complex	complex	ADJ
cana-1058	162	18	biological	biological	ADJ
cana-1058	162	19	processes	process	NOUN
cana-1058	162	20	and	and	CCONJ
cana-1058	162	21	ways	way	NOUN
cana-1058	162	22	to	to	PART
cana-1058	162	23	affect	affect	VERB
cana-1058	162	24	them	they	PRON
cana-1058	162	25	in	in	ADP
cana-1058	162	26	case	case	NOUN
cana-1058	162	27	of	of	ADP
cana-1058	162	28	drug	drug	NOUN
cana-1058	162	29	development	development	NOUN
cana-1058	162	30	.	.	PUNCT
cana-1058	163	1	4.4	4.4	NUM
cana-1058	163	2	transportation	transportation	NOUN
cana-1058	163	3	network	network	NOUN
cana-1058	163	4	optimization	optimization	NOUN
cana-1058	163	5	:	:	PUNCT
cana-1058	163	6	the	the	DET
cana-1058	163	7	combination	combination	NOUN
cana-1058	163	8	of	of	ADP
cana-1058	163	9	nαig	nαig	ADV
cana-1058	163	10	-	-	PUNCT
cana-1058	163	11	closed	close	VERB
cana-1058	163	12	and	and	CCONJ
cana-1058	163	13	nαig	nαig	ADV
cana-1058	163	14	-	-	PUNCT
cana-1058	163	15	open	open	ADJ
cana-1058	163	16	sets	set	NOUN
cana-1058	163	17	in	in	ADP
cana-1058	163	18	transportation	transportation	NOUN
cana-1058	163	19	network	network	NOUN
cana-1058	163	20	optimization	optimization	NOUN
cana-1058	163	21	implies	imply	VERB
cana-1058	163	22	their	their	PRON
cana-1058	163	23	use	use	NOUN
cana-1058	163	24	for	for	ADP
cana-1058	163	25	planning	planning	NOUN
cana-1058	163	26	routes	route	NOUN
cana-1058	163	27	and	and	CCONJ
cana-1058	163	28	managing	manage	VERB
cana-1058	163	29	traffic	traffic	NOUN
cana-1058	163	30	.	.	PUNCT
cana-1058	164	1	the	the	DET
cana-1058	164	2	optimization	optimization	NOUN
cana-1058	164	3	of	of	ADP
cana-1058	164	4	transportation	transportation	NOUN
cana-1058	164	5	networks	network	NOUN
cana-1058	164	6	through	through	ADP
cana-1058	164	7	these	these	DET
cana-1058	164	8	mathematical	mathematical	ADJ
cana-1058	164	9	concepts	concept	NOUN
cana-1058	164	10	will	will	AUX
cana-1058	164	11	allow	allow	VERB
cana-1058	164	12	improvements	improvement	NOUN
cana-1058	164	13	in	in	ADP
cana-1058	164	14	the	the	DET
cana-1058	164	15	performance	performance	NOUN
cana-1058	164	16	of	of	ADP
cana-1058	164	17	multiple	multiple	ADJ
cana-1058	164	18	systems	system	NOUN
cana-1058	164	19	and	and	CCONJ
cana-1058	164	20	reducing	reduce	VERB
cana-1058	164	21	congestion	congestion	NOUN
cana-1058	164	22	to	to	PART
cana-1058	164	23	achieve	achieve	VERB
cana-1058	164	24	better	well	ADJ
cana-1058	164	25	overall	overall	ADJ
cana-1058	164	26	performance	performance	NOUN
cana-1058	164	27	.	.	PUNCT
cana-1058	165	1	4.5	4.5	NUM
cana-1058	165	2	pattern	pattern	NOUN
cana-1058	165	3	recognition	recognition	NOUN
cana-1058	165	4	:	:	PUNCT
cana-1058	165	5	pattern	pattern	NOUN
cana-1058	165	6	recognition	recognition	NOUN
cana-1058	165	7	uses	use	VERB
cana-1058	165	8	nαig	nαig	ADV
cana-1058	165	9	-	-	PUNCT
cana-1058	165	10	closed	close	VERB
cana-1058	165	11	and	and	CCONJ
cana-1058	165	12	nαig	nαig	ADV
cana-1058	165	13	-	-	PUNCT
cana-1058	165	14	open	open	ADJ
cana-1058	165	15	sets	set	NOUN
cana-1058	165	16	to	to	PART
cana-1058	165	17	improve	improve	VERB
cana-1058	165	18	image	image	NOUN
cana-1058	165	19	processing	processing	NOUN
cana-1058	165	20	,	,	PUNCT
cana-1058	165	21	natural	natural	ADJ
cana-1058	165	22	language	language	NOUN
cana-1058	165	23	processing	processing	NOUN
cana-1058	165	24	,	,	PUNCT
cana-1058	165	25	and	and	CCONJ
cana-1058	165	26	other	other	ADJ
cana-1058	165	27	important	important	ADJ
cana-1058	165	28	machine	machine	NOUN
cana-1058	165	29	learning	learning	NOUN
cana-1058	165	30	tasks	task	NOUN
cana-1058	165	31	.	.	PUNCT
cana-1058	166	1	by	by	ADP
cana-1058	166	2	utilizing	utilize	VERB
cana-1058	166	3	the	the	DET
cana-1058	166	4	defined	define	VERB
cana-1058	166	5	mathematics	mathematic	NOUN
cana-1058	166	6	,	,	PUNCT
cana-1058	166	7	machine	machine	NOUN
cana-1058	166	8	learning	learning	NOUN
cana-1058	166	9	practitioners	practitioner	NOUN
cana-1058	166	10	can	can	AUX
cana-1058	166	11	come	come	VERB
cana-1058	166	12	up	up	ADP
cana-1058	166	13	with	with	ADP
cana-1058	166	14	superior	superior	ADJ
cana-1058	166	15	-	-	PUNCT
cana-1058	166	16	pattern	pattern	NOUN
cana-1058	166	17	-	-	PUNCT
cana-1058	166	18	recognition	recognition	NOUN
cana-1058	166	19	algorithms	algorithm	NOUN
cana-1058	166	20	that	that	PRON
cana-1058	166	21	are	be	AUX
cana-1058	166	22	widely	widely	ADV
cana-1058	166	23	applicable	applicable	ADJ
cana-1058	166	24	.	.	PUNCT
cana-1058	167	1	communications	communication	NOUN
cana-1058	167	2	on	on	ADP
cana-1058	167	3	applied	apply	VERB
cana-1058	167	4	nonlinear	nonlinear	ADJ
cana-1058	167	5	analysis	analysis	NOUN
cana-1058	167	6	issn	issn	NOUN
cana-1058	167	7	:	:	PUNCT
cana-1058	167	8	1074	1074	NUM
cana-1058	167	9	-	-	PUNCT
cana-1058	167	10	133x	133x	NUM
cana-1058	167	11	vol	vol	NOUN
cana-1058	167	12	31	31	NUM
cana-1058	167	13	no	no	NOUN
cana-1058	167	14	.	.	PUNCT
cana-1058	168	1	5s	5s	NUM
cana-1058	168	2	(	(	PUNCT
cana-1058	168	3	2024	2024	NUM
cana-1058	168	4	)	)	PUNCT
cana-1058	168	5	396	396	NUM
cana-1058	168	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-1058	168	7	4.6	4.6	NUM
cana-1058	168	8	iot	iot	NOUN
cana-1058	168	9	security	security	NOUN
cana-1058	168	10	:	:	PUNCT
cana-1058	168	11	nαig	nαig	NOUN
cana-1058	168	12	-	-	PUNCT
cana-1058	168	13	closed	close	VERB
cana-1058	168	14	and	and	CCONJ
cana-1058	168	15	nαig	nαig	ADV
cana-1058	168	16	-	-	PUNCT
cana-1058	168	17	open	open	ADJ
cana-1058	168	18	sets	set	NOUN
cana-1058	168	19	help	help	NOUN
cana-1058	168	20	in	in	ADP
cana-1058	168	21	improving	improve	VERB
cana-1058	168	22	security	security	NOUN
cana-1058	168	23	in	in	ADP
cana-1058	168	24	iot	iot	ADJ
cana-1058	168	25	devices	device	NOUN
cana-1058	168	26	and	and	CCONJ
cana-1058	168	27	networks	network	NOUN
cana-1058	168	28	.	.	PUNCT
cana-1058	169	1	the	the	DET
cana-1058	169	2	mathematics	mathematic	NOUN
cana-1058	169	3	of	of	ADP
cana-1058	169	4	the	the	DET
cana-1058	169	5	two	two	NUM
cana-1058	169	6	important	important	ADJ
cana-1058	169	7	concepts	concept	NOUN
cana-1058	169	8	can	can	AUX
cana-1058	169	9	help	help	VERB
cana-1058	169	10	in	in	ADP
cana-1058	169	11	abnormality	abnormality	NOUN
cana-1058	169	12	detection	detection	NOUN
cana-1058	169	13	and	and	CCONJ
cana-1058	169	14	therefore	therefore	ADV
cana-1058	169	15	providing	provide	VERB
cana-1058	169	16	stake	stake	NOUN
cana-1058	169	17	holders	holder	NOUN
cana-1058	169	18	with	with	ADP
cana-1058	169	19	severe	severe	ADJ
cana-1058	169	20	stability	stability	NOUN
cana-1058	169	21	to	to	PART
cana-1058	169	22	prevent	prevent	VERB
cana-1058	169	23	possible	possible	ADJ
cana-1058	169	24	losses	loss	NOUN
cana-1058	169	25	and	and	CCONJ
cana-1058	169	26	data	datum	NOUN
cana-1058	169	27	compromise	compromise	NOUN
cana-1058	169	28	.	.	PUNCT
cana-1058	170	1	4.7	4.7	NUM
cana-1058	170	2	financial	financial	ADJ
cana-1058	170	3	market	market	NOUN
cana-1058	170	4	analysis	analysis	NOUN
cana-1058	170	5	:	:	PUNCT
cana-1058	170	6	the	the	DET
cana-1058	170	7	integration	integration	NOUN
cana-1058	170	8	of	of	ADP
cana-1058	170	9	nαig	nαig	ADV
cana-1058	170	10	-	-	PUNCT
cana-1058	170	11	closed	close	VERB
cana-1058	170	12	and	and	CCONJ
cana-1058	170	13	nαig	nαig	ADV
cana-1058	170	14	-	-	PUNCT
cana-1058	170	15	open	open	ADJ
cana-1058	170	16	sets	set	NOUN
cana-1058	170	17	in	in	ADP
cana-1058	170	18	financial	financial	ADJ
cana-1058	170	19	market	market	NOUN
cana-1058	170	20	analysis	analysis	NOUN
cana-1058	170	21	enables	enable	VERB
cana-1058	170	22	the	the	DET
cana-1058	170	23	detection	detection	NOUN
cana-1058	170	24	of	of	ADP
cana-1058	170	25	abnormal	abnormal	ADJ
cana-1058	170	26	trading	trading	NOUN
cana-1058	170	27	patterns	pattern	NOUN
cana-1058	170	28	and	and	CCONJ
cana-1058	170	29	assists	assist	NOUN
cana-1058	170	30	in	in	ADP
cana-1058	170	31	risk	risk	NOUN
cana-1058	170	32	management	management	NOUN
cana-1058	170	33	and	and	CCONJ
cana-1058	170	34	fraud	fraud	NOUN
cana-1058	170	35	detection	detection	NOUN
cana-1058	170	36	.	.	PUNCT
cana-1058	171	1	by	by	ADP
cana-1058	171	2	analyzing	analyze	VERB
cana-1058	171	3	financial	financial	ADJ
cana-1058	171	4	market	market	NOUN
cana-1058	171	5	data	datum	NOUN
cana-1058	171	6	using	use	VERB
cana-1058	171	7	these	these	DET
cana-1058	171	8	mathematical	mathematical	ADJ
cana-1058	171	9	concepts	concept	NOUN
cana-1058	171	10	,	,	PUNCT
cana-1058	171	11	stakeholders	stakeholder	NOUN
cana-1058	171	12	can	can	AUX
cana-1058	171	13	make	make	VERB
cana-1058	171	14	informed	informed	ADJ
cana-1058	171	15	decisions	decision	NOUN
cana-1058	171	16	and	and	CCONJ
cana-1058	171	17	mitigate	mitigate	VERB
cana-1058	171	18	financial	financial	ADJ
cana-1058	171	19	risks	risk	NOUN
cana-1058	171	20	.	.	PUNCT
cana-1058	172	1	4.8	4.8	NUM
cana-1058	172	2	environmental	environmental	ADJ
cana-1058	172	3	monitoring	monitoring	NOUN
cana-1058	172	4	:	:	PUNCT
cana-1058	172	5	nαig	nαig	NOUN
cana-1058	172	6	-	-	PUNCT
cana-1058	172	7	closed	close	VERB
cana-1058	172	8	and	and	CCONJ
cana-1058	172	9	nαig	nαig	ADV
cana-1058	172	10	-	-	PUNCT
cana-1058	172	11	open	open	ADJ
cana-1058	172	12	sets	set	NOUN
cana-1058	172	13	play	play	VERB
cana-1058	172	14	a	a	DET
cana-1058	172	15	crucial	crucial	ADJ
cana-1058	172	16	role	role	NOUN
cana-1058	172	17	in	in	ADP
cana-1058	172	18	environmental	environmental	ADJ
cana-1058	172	19	monitoring	monitoring	NOUN
cana-1058	172	20	by	by	ADP
cana-1058	172	21	detecting	detect	VERB
cana-1058	172	22	anomalies	anomaly	NOUN
cana-1058	172	23	in	in	ADP
cana-1058	172	24	ecosystems	ecosystem	NOUN
cana-1058	172	25	and	and	CCONJ
cana-1058	172	26	contributing	contribute	VERB
cana-1058	172	27	to	to	ADP
cana-1058	172	28	conservation	conservation	NOUN
cana-1058	172	29	efforts	effort	NOUN
cana-1058	172	30	.	.	PUNCT
cana-1058	173	1	by	by	ADP
cana-1058	173	2	leveraging	leverage	VERB
cana-1058	173	3	these	these	DET
cana-1058	173	4	mathematical	mathematical	ADJ
cana-1058	173	5	concepts	concept	NOUN
cana-1058	173	6	,	,	PUNCT
cana-1058	173	7	researchers	researcher	NOUN
cana-1058	173	8	can	can	AUX
cana-1058	173	9	monitor	monitor	VERB
cana-1058	173	10	environmental	environmental	ADJ
cana-1058	173	11	data	datum	NOUN
cana-1058	173	12	effectively	effectively	ADV
cana-1058	173	13	and	and	CCONJ
cana-1058	173	14	implement	implement	VERB
cana-1058	173	15	sustainable	sustainable	ADJ
cana-1058	173	16	solutions	solution	NOUN
cana-1058	173	17	to	to	PART
cana-1058	173	18	address	address	VERB
cana-1058	173	19	environmental	environmental	ADJ
cana-1058	173	20	challenges	challenge	NOUN
cana-1058	173	21	.	.	PUNCT
cana-1058	174	1	4.9	4.9	NUM
cana-1058	174	2	healthcare	healthcare	NOUN
cana-1058	174	3	systems	system	NOUN
cana-1058	174	4	analysis	analysis	NOUN
cana-1058	174	5	:	:	PUNCT
cana-1058	174	6	in	in	ADP
cana-1058	174	7	healthcare	healthcare	NOUN
cana-1058	174	8	systems	system	NOUN
cana-1058	174	9	analysis	analysis	NOUN
cana-1058	174	10	,	,	PUNCT
cana-1058	174	11	nαig	nαig	ADV
cana-1058	174	12	-	-	PUNCT
cana-1058	174	13	closed	close	VERB
cana-1058	174	14	and	and	CCONJ
cana-1058	174	15	nαig	nαig	ADV
cana-1058	174	16	-	-	PUNCT
cana-1058	174	17	open	open	ADJ
cana-1058	174	18	sets	set	NOUN
cana-1058	174	19	are	be	AUX
cana-1058	174	20	utilized	utilize	VERB
cana-1058	174	21	for	for	ADP
cana-1058	174	22	analyzing	analyze	VERB
cana-1058	174	23	healthcare	healthcare	NOUN
cana-1058	174	24	data	datum	NOUN
cana-1058	174	25	and	and	CCONJ
cana-1058	174	26	improving	improve	VERB
cana-1058	174	27	patient	patient	ADJ
cana-1058	174	28	outcomes	outcome	NOUN
cana-1058	174	29	.	.	PUNCT
cana-1058	175	1	by	by	ADP
cana-1058	175	2	applying	apply	VERB
cana-1058	175	3	these	these	DET
cana-1058	175	4	mathematical	mathematical	ADJ
cana-1058	175	5	concepts	concept	NOUN
cana-1058	175	6	,	,	PUNCT
cana-1058	175	7	stakeholders	stakeholder	NOUN
cana-1058	175	8	can	can	AUX
cana-1058	175	9	identify	identify	VERB
cana-1058	175	10	irregularities	irregularity	NOUN
cana-1058	175	11	in	in	ADP
cana-1058	175	12	patient	patient	ADJ
cana-1058	175	13	records	record	NOUN
cana-1058	175	14	,	,	PUNCT
cana-1058	175	15	optimize	optimize	VERB
cana-1058	175	16	healthcare	healthcare	NOUN
cana-1058	175	17	delivery	delivery	NOUN
cana-1058	175	18	,	,	PUNCT
cana-1058	175	19	and	and	CCONJ
cana-1058	175	20	enhance	enhance	VERB
cana-1058	175	21	patient	patient	ADJ
cana-1058	175	22	care	care	NOUN
cana-1058	175	23	.	.	PUNCT
cana-1058	176	1	4.10	4.10	NUM
cana-1058	176	2	supply	supply	NOUN
cana-1058	176	3	chain	chain	NOUN
cana-1058	176	4	management	management	NOUN
cana-1058	176	5	:	:	PUNCT
cana-1058	176	6	applying	apply	VERB
cana-1058	176	7	nαig	nαig	NOUN
cana-1058	176	8	-	-	PUNCT
cana-1058	176	9	closed	close	VERB
cana-1058	176	10	and	and	CCONJ
cana-1058	176	11	nαig	nαig	ADV
cana-1058	176	12	-	-	PUNCT
cana-1058	176	13	open	open	ADJ
cana-1058	176	14	sets	set	NOUN
cana-1058	176	15	in	in	ADP
cana-1058	176	16	supply	supply	NOUN
cana-1058	176	17	chain	chain	NOUN
cana-1058	176	18	management	management	NOUN
cana-1058	176	19	enables	enable	VERB
cana-1058	176	20	optimization	optimization	NOUN
cana-1058	176	21	of	of	ADP
cana-1058	176	22	supply	supply	NOUN
cana-1058	176	23	chain	chain	NOUN
cana-1058	176	24	networks	network	NOUN
cana-1058	176	25	and	and	CCONJ
cana-1058	176	26	enhances	enhance	VERB
cana-1058	176	27	resilience	resilience	NOUN
cana-1058	176	28	and	and	CCONJ
cana-1058	176	29	responsiveness	responsiveness	NOUN
cana-1058	176	30	.	.	PUNCT
cana-1058	177	1	by	by	ADP
cana-1058	177	2	leveraging	leverage	VERB
cana-1058	177	3	these	these	DET
cana-1058	177	4	mathematical	mathematical	ADJ
cana-1058	177	5	concepts	concept	NOUN
cana-1058	177	6	,	,	PUNCT
cana-1058	177	7	stakeholders	stakeholder	NOUN
cana-1058	177	8	can	can	AUX
cana-1058	177	9	identify	identify	VERB
cana-1058	177	10	inefficiencies	inefficiency	NOUN
cana-1058	177	11	,	,	PUNCT
cana-1058	177	12	vulnerabilities	vulnerability	NOUN
cana-1058	177	13	,	,	PUNCT
cana-1058	177	14	and	and	CCONJ
cana-1058	177	15	potential	potential	ADJ
cana-1058	177	16	disruptions	disruption	NOUN
cana-1058	177	17	in	in	ADP
cana-1058	177	18	supply	supply	NOUN
cana-1058	177	19	chains	chain	NOUN
cana-1058	177	20	,	,	PUNCT
cana-1058	177	21	leading	lead	VERB
cana-1058	177	22	to	to	ADP
cana-1058	177	23	improved	improve	VERB
cana-1058	177	24	supply	supply	NOUN
cana-1058	177	25	chain	chain	NOUN
cana-1058	177	26	performance	performance	NOUN
cana-1058	177	27	.	.	PUNCT
cana-1058	178	1	conclusion	conclusion	NOUN
cana-1058	178	2	:	:	PUNCT
cana-1058	178	3	the	the	DET
cana-1058	178	4	exploration	exploration	NOUN
cana-1058	178	5	of	of	ADP
cana-1058	178	6	nαig	nαig	ADV
cana-1058	178	7	-	-	PUNCT
cana-1058	178	8	closed	close	VERB
cana-1058	178	9	and	and	CCONJ
cana-1058	178	10	nαig	nαig	ADV
cana-1058	178	11	-	-	PUNCT
cana-1058	178	12	open	open	ADJ
cana-1058	178	13	sets	set	NOUN
cana-1058	178	14	across	across	ADP
cana-1058	178	15	various	various	ADJ
cana-1058	178	16	domains	domain	NOUN
cana-1058	178	17	showcases	showcase	VERB
cana-1058	178	18	their	their	PRON
cana-1058	178	19	versatility	versatility	NOUN
cana-1058	178	20	and	and	CCONJ
cana-1058	178	21	potential	potential	ADJ
cana-1058	178	22	impact	impact	NOUN
cana-1058	178	23	in	in	ADP
cana-1058	178	24	addressing	address	VERB
cana-1058	178	25	complex	complex	ADJ
cana-1058	178	26	challenges	challenge	NOUN
cana-1058	178	27	.	.	PUNCT
cana-1058	179	1	by	by	ADP
cana-1058	179	2	integrating	integrate	VERB
cana-1058	179	3	these	these	DET
cana-1058	179	4	mathematical	mathematical	ADJ
cana-1058	179	5	concepts	concept	NOUN
cana-1058	179	6	into	into	ADP
cana-1058	179	7	diverse	diverse	ADJ
cana-1058	179	8	applications	application	NOUN
cana-1058	179	9	,	,	PUNCT
cana-1058	179	10	researchers	researcher	NOUN
cana-1058	179	11	and	and	CCONJ
cana-1058	179	12	practitioners	practitioner	NOUN
cana-1058	179	13	can	can	AUX
cana-1058	179	14	unlock	unlock	VERB
cana-1058	179	15	new	new	ADJ
cana-1058	179	16	insights	insight	NOUN
cana-1058	179	17	,	,	PUNCT
cana-1058	179	18	optimize	optimize	NOUN
cana-1058	179	19	operations	operation	NOUN
cana-1058	179	20	,	,	PUNCT
cana-1058	179	21	and	and	CCONJ
cana-1058	179	22	drive	drive	VERB
cana-1058	179	23	innovation	innovation	NOUN
cana-1058	179	24	across	across	ADP
cana-1058	179	25	multiple	multiple	ADJ
cana-1058	179	26	industries	industry	NOUN
cana-1058	179	27	.	.	PUNCT
cana-1058	180	1	further	further	ADJ
cana-1058	180	2	research	research	NOUN
cana-1058	180	3	and	and	CCONJ
cana-1058	180	4	experimentation	experimentation	NOUN
cana-1058	180	5	in	in	ADP
cana-1058	180	6	this	this	DET
cana-1058	180	7	area	area	NOUN
cana-1058	180	8	are	be	AUX
cana-1058	180	9	essential	essential	ADJ
cana-1058	180	10	to	to	PART
cana-1058	180	11	fully	fully	ADV
cana-1058	180	12	realize	realize	VERB
cana-1058	180	13	the	the	DET
cana-1058	180	14	potential	potential	NOUN
cana-1058	180	15	of	of	ADP
cana-1058	180	16	nαig	nαig	ADV
cana-1058	180	17	-	-	PUNCT
cana-1058	180	18	closed	close	VERB
cana-1058	180	19	and	and	CCONJ
cana-1058	180	20	nαig	nαig	ADV
cana-1058	180	21	-	-	PUNCT
cana-1058	180	22	open	open	ADJ
cana-1058	180	23	sets	set	NOUN
cana-1058	180	24	in	in	ADP
cana-1058	180	25	multidomain	multidomain	NOUN
cana-1058	180	26	applications	application	NOUN
cana-1058	180	27	.	.	PUNCT
cana-1058	181	1	acknowledgement	acknowledgement	NOUN
cana-1058	181	2	:	:	PUNCT
cana-1058	181	3	we	we	PRON
cana-1058	181	4	would	would	AUX
cana-1058	181	5	like	like	VERB
cana-1058	181	6	to	to	PART
cana-1058	181	7	express	express	VERB
cana-1058	181	8	our	our	PRON
cana-1058	181	9	sincere	sincere	ADJ
cana-1058	181	10	gratitude	gratitude	NOUN
cana-1058	181	11	to	to	ADP
cana-1058	181	12	all	all	DET
cana-1058	181	13	those	those	PRON
cana-1058	181	14	who	who	PRON
cana-1058	181	15	contributed	contribute	VERB
cana-1058	181	16	to	to	ADP
cana-1058	181	17	the	the	DET
cana-1058	181	18	publication	publication	NOUN
cana-1058	181	19	of	of	ADP
cana-1058	181	20	this	this	DET
cana-1058	181	21	paper	paper	NOUN
cana-1058	181	22	.	.	PUNCT
cana-1058	182	1	our	our	PRON
cana-1058	182	2	sincere	sincere	ADJ
cana-1058	182	3	gratitude	gratitude	NOUN
cana-1058	182	4	goes	go	VERB
cana-1058	182	5	out	out	ADP
cana-1058	182	6	to	to	ADP
cana-1058	182	7	amrita	amrita	PROPN
cana-1058	182	8	university	university	PROPN
cana-1058	182	9	and	and	CCONJ
cana-1058	182	10	other	other	ADJ
cana-1058	182	11	organizations	organization	NOUN
cana-1058	182	12	for	for	ADP
cana-1058	182	13	providing	provide	VERB
cana-1058	182	14	us	we	PRON
cana-1058	182	15	with	with	ADP
cana-1058	182	16	resources	resource	NOUN
cana-1058	182	17	and	and	CCONJ
cana-1058	182	18	support	support	NOUN
cana-1058	182	19	.	.	PUNCT
cana-1058	183	1	we	we	PRON
cana-1058	183	2	would	would	AUX
cana-1058	183	3	also	also	ADV
cana-1058	183	4	like	like	VERB
cana-1058	183	5	to	to	PART
cana-1058	183	6	thank	thank	VERB
cana-1058	183	7	our	our	PRON
cana-1058	183	8	colleagues	colleague	NOUN
cana-1058	183	9	for	for	ADP
cana-1058	183	10	their	their	PRON
cana-1058	183	11	feedback	feedback	NOUN
cana-1058	183	12	and	and	CCONJ
cana-1058	183	13	support	support	NOUN
cana-1058	183	14	throughout	throughout	ADP
cana-1058	183	15	the	the	DET
cana-1058	183	16	research	research	NOUN
cana-1058	183	17	process	process	NOUN
cana-1058	183	18	.	.	PUNCT
cana-1058	184	1	without	without	ADP
cana-1058	184	2	their	their	PRON
cana-1058	184	3	support	support	NOUN
cana-1058	184	4	,	,	PUNCT
cana-1058	184	5	we	we	PRON
cana-1058	184	6	would	would	AUX
cana-1058	184	7	not	not	PART
cana-1058	184	8	have	have	AUX
cana-1058	184	9	been	be	AUX
cana-1058	184	10	able	able	ADJ
cana-1058	184	11	to	to	PART
cana-1058	184	12	complete	complete	VERB
cana-1058	184	13	this	this	DET
cana-1058	184	14	study	study	NOUN
cana-1058	184	15	.	.	PUNCT
cana-1058	185	1	communications	communication	NOUN
cana-1058	185	2	on	on	ADP
cana-1058	185	3	applied	apply	VERB
cana-1058	185	4	nonlinear	nonlinear	ADJ
cana-1058	185	5	analysis	analysis	NOUN
cana-1058	185	6	issn	issn	NOUN
cana-1058	185	7	:	:	PUNCT
cana-1058	185	8	1074	1074	NUM
cana-1058	185	9	-	-	PUNCT
cana-1058	185	10	133x	133x	NUM
cana-1058	185	11	vol	vol	NOUN
cana-1058	185	12	31	31	NUM
cana-1058	185	13	no	no	NOUN
cana-1058	185	14	.	.	PUNCT
cana-1058	186	1	5s	5s	NUM
cana-1058	186	2	(	(	PUNCT
cana-1058	186	3	2024	2024	NUM
cana-1058	186	4	)	)	PUNCT
cana-1058	186	5	397	397	NUM
cana-1058	186	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1058	186	7	references	reference	NOUN
cana-1058	186	8	[	[	X
cana-1058	186	9	1	1	NUM
cana-1058	186	10	]	]	PUNCT
cana-1058	186	11	levine	levine	PROPN
cana-1058	186	12	,	,	PUNCT
cana-1058	186	13	n.(1970).generalizedclosedsetsintopology.rendicontidelcircolomatematicodipalermo,19,89	n.(1970).generalizedclosedsetsintopology.rendicontidelcircolomatematicodipalermo,19,89	PROPN
cana-1058	186	14	-	-	SYM
cana-1058	186	15	96	96	NUM
cana-1058	186	16	.	.	PUNCT
cana-1058	187	1	[	[	X
cana-1058	187	2	2	2	NUM
cana-1058	187	3	]	]	PUNCT
cana-1058	187	4	balachandran	balachandran	NOUN
cana-1058	187	5	,	,	PUNCT
cana-1058	187	6	k.	k.	PROPN
cana-1058	187	7	,p.sundaram	,p.sundaram	PUNCT
cana-1058	187	8	andh.maki(1991a).ongeneralizedcontinuousmapsintopologicalspaces.mem.fac.sci.kochiuniv.ser.a	andh.maki(1991a).ongeneralizedcontinuousmapsintopologicalspaces.mem.fac.sci.kochiuniv.ser.a	PROPN
cana-1058	187	9	,	,	PUNCT
cana-1058	187	10	math	math	NOUN
cana-1058	187	11	.	.	PUNCT
cana-1058	188	1	,12,513	,12,513	PROPN
cana-1058	188	2	.	.	PUNCT
cana-1058	189	1	[	[	X
cana-1058	189	2	3	3	NUM
cana-1058	189	3	]	]	X
cana-1058	189	4	njȧstad	njȧstad	NOUN
cana-1058	189	5	,	,	PUNCT
cana-1058	189	6	o.(1965	o.(1965	PROPN
cana-1058	189	7	)	)	PUNCT
cana-1058	189	8	.	.	PUNCT
cana-1058	190	1	onsomeclassesofnearlyopensets	onsomeclassesofnearlyopenset	NOUN
cana-1058	190	2	.	.	PUNCT
cana-1058	191	1	pacificjournalofmathematics	pacificjournalofmathematic	NOUN
cana-1058	191	2	,	,	PUNCT
cana-1058	191	3	15(3),961	15(3),961	NUM
cana-1058	191	4	-	-	SYM
cana-1058	191	5	970	970	NUM
cana-1058	191	6	.	.	PUNCT
cana-1058	192	1	[	[	X
cana-1058	192	2	4	4	X
cana-1058	192	3	]	]	X
cana-1058	192	4	mashhour	mashhour	ADJ
cana-1058	192	5	,	,	PUNCT
cana-1058	192	6	a.	a.	PROPN
cana-1058	192	7	s.	s.	PROPN
cana-1058	192	8	,	,	PUNCT
cana-1058	192	9	hasanein	hasanein	ADV
cana-1058	192	10	,	,	PUNCT
cana-1058	192	11	i.	i.	PROPN
cana-1058	192	12	a.	a.	PROPN
cana-1058	192	13	,	,	PUNCT
cana-1058	192	14	&	&	CCONJ
cana-1058	192	15	el	el	PROPN
cana-1058	192	16	-	-	PROPN
cana-1058	192	17	deeb	deeb	PROPN
cana-1058	192	18	,	,	PUNCT
cana-1058	192	19	s.	s.	PROPN
cana-1058	192	20	n.	n.	PROPN
cana-1058	192	21	(	(	PUNCT
cana-1058	192	22	1983	1983	NUM
cana-1058	192	23	)	)	PUNCT
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