id	sid	tid	token	lemma	pos
cana-1932	1	1	communications	communication	NOUN
cana-1932	1	2	on	on	ADP
cana-1932	1	3	applied	apply	VERB
cana-1932	1	4	nonlinear	nonlinear	ADJ
cana-1932	1	5	analysis	analysis	NOUN
cana-1932	1	6	issn	issn	NOUN
cana-1932	1	7	:	:	PUNCT
cana-1932	1	8	1074	1074	NUM
cana-1932	1	9	-	-	PUNCT
cana-1932	1	10	133x	133x	NUM
cana-1932	1	11	vol	vol	NOUN
cana-1932	1	12	32	32	NUM
cana-1932	1	13	no	no	NOUN
cana-1932	1	14	.	.	NOUN
cana-1932	1	15	3	3	NUM
cana-1932	1	16	(	(	PUNCT
cana-1932	1	17	2025	2025	NUM
cana-1932	1	18	)	)	PUNCT
cana-1932	1	19	153	153	NUM
cana-1932	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1932	1	21	fuzzy	fuzzy	ADJ
cana-1932	1	22	double	double	PROPN
cana-1932	1	23	s	s	PART
cana-1932	1	24	–	–	PUNCT
cana-1932	1	25	hausdorff	hausdorff	NOUN
cana-1932	1	26	spaces	space	NOUN
cana-1932	1	27	sowmya.r#1	sowmya.r#1	NOUN
cana-1932	1	28	,	,	PUNCT
cana-1932	1	29	dr.j.srikiruthika#2	dr.j.srikiruthika#2	PROPN
cana-1932	1	30	#	#	SYM
cana-1932	1	31	1	1	NUM
cana-1932	1	32	research	research	NOUN
cana-1932	1	33	scholar	scholar	NOUN
cana-1932	1	34	,	,	PUNCT
cana-1932	1	35	department	department	NOUN
cana-1932	1	36	of	of	ADP
cana-1932	1	37	mathematics	mathematic	NOUN
cana-1932	1	38	,	,	PUNCT
cana-1932	1	39	avinashilingam	avinashilingam	PROPN
cana-1932	1	40	institute	institute	PROPN
cana-1932	1	41	for	for	ADP
cana-1932	1	42	home	home	NOUN
cana-1932	1	43	science	science	NOUN
cana-1932	1	44	and	and	CCONJ
cana-1932	1	45	higher	high	ADJ
cana-1932	1	46	education	education	NOUN
cana-1932	1	47	for	for	ADP
cana-1932	1	48	women	woman	NOUN
cana-1932	1	49	,	,	PUNCT
cana-1932	1	50	coimbatore	coimbatore	PROPN
cana-1932	1	51	,	,	PUNCT
cana-1932	1	52	india	india	PROPN
cana-1932	1	53	.	.	PUNCT
cana-1932	2	1	email	email	NOUN
cana-1932	3	1	i	i	PROPN
cana-1932	3	2	d	d	PROPN
cana-1932	3	3	:	:	PUNCT
cana-1932	3	4	18phmap001@avinuty.ac.in	18phmap001@avinuty.ac.in	NUM
cana-1932	3	5	#	#	SYM
cana-1932	3	6	2	2	NUM
cana-1932	3	7	assistant	assistant	NOUN
cana-1932	3	8	professor	professor	NOUN
cana-1932	3	9	of	of	ADP
cana-1932	3	10	mathematics	mathematics	PROPN
cana-1932	3	11	,	,	PUNCT
cana-1932	3	12	department	department	NOUN
cana-1932	3	13	of	of	ADP
cana-1932	3	14	science	science	NOUN
cana-1932	3	15	and	and	CCONJ
cana-1932	3	16	humanities	humanity	NOUN
cana-1932	3	17	,	,	PUNCT
cana-1932	3	18	school	school	NOUN
cana-1932	3	19	of	of	ADP
cana-1932	3	20	engineering	engineering	NOUN
cana-1932	3	21	,	,	PUNCT
cana-1932	3	22	avinashilingam	avinashilingam	PROPN
cana-1932	3	23	institute	institute	PROPN
cana-1932	3	24	for	for	ADP
cana-1932	3	25	home	home	NOUN
cana-1932	3	26	science	science	NOUN
cana-1932	3	27	and	and	CCONJ
cana-1932	3	28	higher	high	ADJ
cana-1932	3	29	education	education	NOUN
cana-1932	3	30	for	for	ADP
cana-1932	3	31	women	woman	NOUN
cana-1932	3	32	,	,	PUNCT
cana-1932	3	33	coimbatore	coimbatore	PROPN
cana-1932	3	34	,	,	PUNCT
cana-1932	3	35	india	india	PROPN
cana-1932	3	36	.	.	PUNCT
cana-1932	4	1	email	email	NOUN
cana-1932	5	1	i	i	PROPN
cana-1932	5	2	d	d	PROPN
cana-1932	5	3	:	:	PUNCT
cana-1932	5	4	kiruthika_sh@avinuty.ac.in	kiruthika_sh@avinuty.ac.in	PROPN
cana-1932	5	5	article	article	NOUN
cana-1932	5	6	history	history	NOUN
cana-1932	5	7	:	:	PUNCT
cana-1932	5	8	received	receive	VERB
cana-1932	5	9	:	:	PUNCT
cana-1932	5	10	25	25	NUM
cana-1932	5	11	-	-	PUNCT
cana-1932	5	12	07	07	NUM
cana-1932	5	13	-	-	PUNCT
cana-1932	5	14	2024	2024	NUM
cana-1932	5	15	revised	revise	VERB
cana-1932	5	16	:	:	PUNCT
cana-1932	5	17	12	12	NUM
cana-1932	5	18	-	-	SYM
cana-1932	5	19	09	09	NUM
cana-1932	5	20	-	-	PUNCT
cana-1932	5	21	2024	2024	NUM
cana-1932	5	22	accepted	accept	VERB
cana-1932	5	23	:	:	PUNCT
cana-1932	5	24	29	29	NUM
cana-1932	5	25	-	-	SYM
cana-1932	5	26	09	09	NUM
cana-1932	5	27	-	-	PUNCT
cana-1932	5	28	2024	2024	NUM
cana-1932	5	29	abstract	abstract	NOUN
cana-1932	5	30	:	:	PUNCT
cana-1932	5	31	this	this	DET
cana-1932	5	32	study	study	NOUN
cana-1932	5	33	introduces	introduce	NOUN
cana-1932	5	34	and	and	CCONJ
cana-1932	5	35	explores	explore	VERB
cana-1932	5	36	the	the	DET
cana-1932	5	37	concept	concept	NOUN
cana-1932	5	38	of	of	ADP
cana-1932	5	39	fuzzy	fuzzy	ADJ
cana-1932	5	40	double	double	ADJ
cana-1932	5	41	s	s	NOUN
cana-1932	5	42	-	-	PUNCT
cana-1932	5	43	hausdorff	hausdorff	NOUN
cana-1932	5	44	spaces	space	NOUN
cana-1932	5	45	within	within	ADP
cana-1932	5	46	the	the	DET
cana-1932	5	47	field	field	NOUN
cana-1932	5	48	of	of	ADP
cana-1932	5	49	fuzzy	fuzzy	ADJ
cana-1932	5	50	topology	topology	NOUN
cana-1932	5	51	.	.	PUNCT
cana-1932	6	1	fuzzy	fuzzy	ADJ
cana-1932	6	2	double	double	ADJ
cana-1932	6	3	s	s	NOUN
cana-1932	6	4	-	-	PUNCT
cana-1932	6	5	hausdorff	hausdorff	ADJ
cana-1932	6	6	spaces	space	NOUN
cana-1932	6	7	extend	extend	VERB
cana-1932	6	8	the	the	DET
cana-1932	6	9	classical	classical	ADJ
cana-1932	6	10	hausdorff	hausdorff	NOUN
cana-1932	6	11	condition	condition	NOUN
cana-1932	6	12	by	by	ADP
cana-1932	6	13	incorporating	incorporate	VERB
cana-1932	6	14	two	two	NUM
cana-1932	6	15	distinct	distinct	ADJ
cana-1932	6	16	fuzzy	fuzzy	ADJ
cana-1932	6	17	topologies	topology	NOUN
cana-1932	6	18	on	on	ADP
cana-1932	6	19	a	a	DET
cana-1932	6	20	single	single	ADJ
cana-1932	6	21	set	set	NOUN
cana-1932	6	22	,	,	PUNCT
cana-1932	6	23	allowing	allow	VERB
cana-1932	6	24	for	for	ADP
cana-1932	6	25	a	a	DET
cana-1932	6	26	more	more	ADV
cana-1932	6	27	nuanced	nuanced	ADJ
cana-1932	6	28	approach	approach	NOUN
cana-1932	6	29	to	to	ADP
cana-1932	6	30	separation	separation	NOUN
cana-1932	6	31	where	where	SCONJ
cana-1932	6	32	uncertainty	uncertainty	NOUN
cana-1932	6	33	and	and	CCONJ
cana-1932	6	34	gradation	gradation	NOUN
cana-1932	6	35	are	be	AUX
cana-1932	6	36	involved	involve	VERB
cana-1932	6	37	.	.	PUNCT
cana-1932	7	1	the	the	DET
cana-1932	7	2	research	research	NOUN
cana-1932	7	3	develops	develop	VERB
cana-1932	7	4	formal	formal	ADJ
cana-1932	7	5	definitions	definition	NOUN
cana-1932	7	6	and	and	CCONJ
cana-1932	7	7	characterizations	characterization	NOUN
cana-1932	7	8	of	of	ADP
cana-1932	7	9	these	these	DET
cana-1932	7	10	spaces	space	NOUN
cana-1932	7	11	,	,	PUNCT
cana-1932	7	12	establishing	establish	VERB
cana-1932	7	13	key	key	ADJ
cana-1932	7	14	theorems	theorem	NOUN
cana-1932	7	15	that	that	PRON
cana-1932	7	16	outline	outline	VERB
cana-1932	7	17	their	their	PRON
cana-1932	7	18	properties	property	NOUN
cana-1932	7	19	and	and	CCONJ
cana-1932	7	20	behavior	behavior	NOUN
cana-1932	7	21	.	.	PUNCT
cana-1932	8	1	in	in	ADP
cana-1932	8	2	this	this	DET
cana-1932	8	3	article	article	NOUN
cana-1932	8	4	,	,	PUNCT
cana-1932	8	5	a	a	DET
cana-1932	8	6	meaning	meaning	NOUN
cana-1932	8	7	of	of	ADP
cana-1932	8	8	fds	fds	NOUN
cana-1932	8	9	-t2	-t2	PROPN
cana-1932	8	10	remains	remains	AUX
cana-1932	8	11	presented	present	VERB
cana-1932	8	12	through	through	ADP
cana-1932	8	13	covering	cover	VERB
cana-1932	8	14	the	the	DET
cana-1932	8	15	description	description	NOUN
cana-1932	8	16	of	of	ADP
cana-1932	8	17	st2	st2	NOUN
cana-1932	8	18	introduce	introduce	NOUN
cana-1932	8	19	by	by	ADP
cana-1932	8	20	srivastava	srivastava	PROPN
cana-1932	9	1	[	[	X
cana-1932	9	2	3	3	NUM
cana-1932	9	3	]	]	PUNCT
cana-1932	9	4	and	and	CCONJ
cana-1932	9	5	obtain	obtain	VERB
cana-1932	9	6	result	result	NOUN
cana-1932	9	7	analogues	analogue	NOUN
cana-1932	9	8	towards	towards	ADP
cana-1932	9	9	the	the	DET
cana-1932	9	10	result	result	NOUN
cana-1932	9	11	of	of	ADP
cana-1932	9	12	st2	st2	NOUN
cana-1932	9	13	.	.	PUNCT
cana-1932	10	1	keywords	keyword	NOUN
cana-1932	10	2	:	:	PUNCT
cana-1932	10	3	fds	fds	NOUN
cana-1932	10	4	,	,	PUNCT
cana-1932	10	5	fdt	fdt	NOUN
cana-1932	10	6	,	,	PUNCT
cana-1932	10	7	fdts	fdts	ADJ
cana-1932	10	8	,	,	PUNCT
cana-1932	10	9	fds	fds	NOUN
cana-1932	10	10	-	-	PUNCT
cana-1932	10	11	t2	t2	NOUN
cana-1932	10	12	.	.	PUNCT
cana-1932	11	1	1	1	X
cana-1932	11	2	.	.	X
cana-1932	11	3	introduction	introduction	NOUN
cana-1932	11	4	the	the	DET
cana-1932	11	5	concept	concept	NOUN
cana-1932	11	6	of	of	ADP
cana-1932	11	7	fuzzy	fuzzy	ADJ
cana-1932	11	8	set	set	NOUN
cana-1932	11	9	theory	theory	NOUN
cana-1932	11	10	,	,	PUNCT
cana-1932	11	11	introduced	introduce	VERB
cana-1932	11	12	by	by	ADP
cana-1932	11	13	zadeh	zadeh	PROPN
cana-1932	11	14	in	in	ADP
cana-1932	11	15	1965	1965	NUM
cana-1932	11	16	,	,	PUNCT
cana-1932	11	17	has	have	AUX
cana-1932	11	18	revolutionized	revolutionize	VERB
cana-1932	11	19	various	various	ADJ
cana-1932	11	20	fields	field	NOUN
cana-1932	11	21	by	by	ADP
cana-1932	11	22	allowing	allow	VERB
cana-1932	11	23	the	the	DET
cana-1932	11	24	representation	representation	NOUN
cana-1932	11	25	of	of	ADP
cana-1932	11	26	uncertainty	uncertainty	NOUN
cana-1932	11	27	and	and	CCONJ
cana-1932	11	28	imprecision	imprecision	NOUN
cana-1932	11	29	.	.	PUNCT
cana-1932	12	1	in	in	ADP
cana-1932	12	2	topology	topology	NOUN
cana-1932	12	3	,	,	PUNCT
cana-1932	12	4	this	this	PRON
cana-1932	12	5	has	have	AUX
cana-1932	12	6	led	lead	VERB
cana-1932	12	7	to	to	ADP
cana-1932	12	8	the	the	DET
cana-1932	12	9	development	development	NOUN
cana-1932	12	10	of	of	ADP
cana-1932	12	11	fuzzy	fuzzy	ADJ
cana-1932	12	12	topological	topological	ADJ
cana-1932	12	13	spaces	space	NOUN
cana-1932	12	14	,	,	PUNCT
cana-1932	12	15	which	which	PRON
cana-1932	12	16	extend	extend	VERB
cana-1932	12	17	classical	classical	ADJ
cana-1932	12	18	topological	topological	ADJ
cana-1932	12	19	concepts	concept	NOUN
cana-1932	12	20	to	to	PART
cana-1932	12	21	accommodate	accommodate	VERB
cana-1932	12	22	the	the	DET
cana-1932	12	23	vagueness	vagueness	NOUN
cana-1932	12	24	inherent	inherent	ADJ
cana-1932	12	25	in	in	ADP
cana-1932	12	26	many	many	ADJ
cana-1932	12	27	real	real	ADJ
cana-1932	12	28	-	-	PUNCT
cana-1932	12	29	world	world	NOUN
cana-1932	12	30	scenarios	scenario	NOUN
cana-1932	12	31	.	.	PUNCT
cana-1932	13	1	among	among	ADP
cana-1932	13	2	these	these	PRON
cana-1932	13	3	,	,	PUNCT
cana-1932	13	4	the	the	DET
cana-1932	13	5	notion	notion	NOUN
cana-1932	13	6	of	of	ADP
cana-1932	13	7	fuzzy	fuzzy	ADJ
cana-1932	13	8	s	s	NOUN
cana-1932	13	9	-	-	PUNCT
cana-1932	13	10	hausdorff	hausdorff	NOUN
cana-1932	13	11	spaces	space	NOUN
cana-1932	13	12	,	,	PUNCT
cana-1932	13	13	which	which	PRON
cana-1932	13	14	generalizes	generalize	VERB
cana-1932	13	15	the	the	DET
cana-1932	13	16	classical	classical	ADJ
cana-1932	13	17	hausdorff	hausdorff	NOUN
cana-1932	13	18	condition	condition	NOUN
cana-1932	13	19	,	,	PUNCT
cana-1932	13	20	plays	play	VERB
cana-1932	13	21	a	a	DET
cana-1932	13	22	crucial	crucial	ADJ
cana-1932	13	23	role	role	NOUN
cana-1932	13	24	in	in	ADP
cana-1932	13	25	understanding	understand	VERB
cana-1932	13	26	separation	separation	NOUN
cana-1932	13	27	axioms	axiom	NOUN
cana-1932	13	28	within	within	ADP
cana-1932	13	29	a	a	DET
cana-1932	13	30	fuzzy	fuzzy	ADJ
cana-1932	13	31	context	context	NOUN
cana-1932	13	32	.	.	PUNCT
cana-1932	14	1	the	the	DET
cana-1932	14	2	study	study	NOUN
cana-1932	14	3	of	of	ADP
cana-1932	14	4	fuzzy	fuzzy	ADJ
cana-1932	14	5	topological	topological	ADJ
cana-1932	14	6	spaces	space	NOUN
cana-1932	14	7	has	have	AUX
cana-1932	14	8	been	be	AUX
cana-1932	14	9	further	far	ADV
cana-1932	14	10	enriched	enrich	VERB
cana-1932	14	11	by	by	ADP
cana-1932	14	12	the	the	DET
cana-1932	14	13	introduction	introduction	NOUN
cana-1932	14	14	of	of	ADP
cana-1932	14	15	the	the	DET
cana-1932	14	16	double	double	ADJ
cana-1932	14	17	shausdorff	shausdorff	NOUN
cana-1932	14	18	space	space	NOUN
cana-1932	14	19	,	,	PUNCT
cana-1932	14	20	a	a	DET
cana-1932	14	21	variant	variant	NOUN
cana-1932	14	22	that	that	PRON
cana-1932	14	23	incorporates	incorporate	VERB
cana-1932	14	24	two	two	NUM
cana-1932	14	25	distinct	distinct	ADJ
cana-1932	14	26	hausdorff	hausdorff	NOUN
cana-1932	14	27	-	-	PUNCT
cana-1932	14	28	type	type	NOUN
cana-1932	14	29	conditions	condition	NOUN
cana-1932	14	30	into	into	ADP
cana-1932	14	31	the	the	DET
cana-1932	14	32	fuzzy	fuzzy	ADJ
cana-1932	14	33	framework	framework	NOUN
cana-1932	14	34	.	.	PUNCT
cana-1932	15	1	this	this	DET
cana-1932	15	2	development	development	NOUN
cana-1932	15	3	is	be	AUX
cana-1932	15	4	essential	essential	ADJ
cana-1932	15	5	for	for	ADP
cana-1932	15	6	analyzing	analyze	VERB
cana-1932	15	7	complex	complex	ADJ
cana-1932	15	8	topological	topological	ADJ
cana-1932	15	9	structures	structure	NOUN
cana-1932	15	10	where	where	SCONJ
cana-1932	15	11	traditional	traditional	ADJ
cana-1932	15	12	binary	binary	ADJ
cana-1932	15	13	logic	logic	NOUN
cana-1932	15	14	falls	fall	VERB
cana-1932	15	15	short	short	ADJ
cana-1932	15	16	.	.	PUNCT
cana-1932	16	1	this	this	DET
cana-1932	16	2	article	article	NOUN
cana-1932	16	3	explores	explore	VERB
cana-1932	16	4	the	the	DET
cana-1932	16	5	fuzzy	fuzzy	ADJ
cana-1932	16	6	double	double	ADJ
cana-1932	16	7	s	s	NOUN
cana-1932	16	8	-	-	PUNCT
cana-1932	16	9	hausdorff	hausdorff	NOUN
cana-1932	16	10	space	space	NOUN
cana-1932	16	11	,	,	PUNCT
cana-1932	16	12	focusing	focus	VERB
cana-1932	16	13	on	on	ADP
cana-1932	16	14	its	its	PRON
cana-1932	16	15	fundamental	fundamental	ADJ
cana-1932	16	16	properties	property	NOUN
cana-1932	16	17	and	and	CCONJ
cana-1932	16	18	implications	implication	NOUN
cana-1932	16	19	in	in	ADP
cana-1932	16	20	broader	broad	ADJ
cana-1932	16	21	topological	topological	ADJ
cana-1932	16	22	theory	theory	NOUN
cana-1932	16	23	.	.	PUNCT
cana-1932	17	1	we	we	PRON
cana-1932	17	2	begin	begin	VERB
cana-1932	17	3	by	by	ADP
cana-1932	17	4	revisiting	revisit	VERB
cana-1932	17	5	the	the	DET
cana-1932	17	6	basic	basic	ADJ
cana-1932	17	7	definitions	definition	NOUN
cana-1932	17	8	and	and	CCONJ
cana-1932	17	9	concepts	concept	NOUN
cana-1932	17	10	related	relate	VERB
cana-1932	17	11	to	to	ADP
cana-1932	17	12	fuzzy	fuzzy	ADJ
cana-1932	17	13	topologies	topology	NOUN
cana-1932	17	14	,	,	PUNCT
cana-1932	17	15	followed	follow	VERB
cana-1932	17	16	by	by	ADP
cana-1932	17	17	an	an	DET
cana-1932	17	18	in	in	ADP
cana-1932	17	19	-	-	PUNCT
cana-1932	17	20	depth	depth	NOUN
cana-1932	17	21	examination	examination	NOUN
cana-1932	17	22	of	of	ADP
cana-1932	17	23	double	double	ADJ
cana-1932	17	24	s	s	NOUN
cana-1932	17	25	-	-	PUNCT
cana-1932	17	26	hausdorff	hausdorff	NOUN
cana-1932	17	27	spaces	space	NOUN
cana-1932	17	28	.	.	PUNCT
cana-1932	18	1	by	by	ADP
cana-1932	18	2	establishing	establish	VERB
cana-1932	18	3	key	key	ADJ
cana-1932	18	4	theorems	theorem	NOUN
cana-1932	18	5	and	and	CCONJ
cana-1932	18	6	proving	prove	VERB
cana-1932	18	7	their	their	PRON
cana-1932	18	8	significance	significance	NOUN
cana-1932	18	9	,	,	PUNCT
cana-1932	18	10	we	we	PRON
cana-1932	18	11	aim	aim	VERB
cana-1932	18	12	to	to	PART
cana-1932	18	13	provide	provide	VERB
cana-1932	18	14	a	a	DET
cana-1932	18	15	comprehensive	comprehensive	ADJ
cana-1932	18	16	understanding	understanding	NOUN
cana-1932	18	17	of	of	ADP
cana-1932	18	18	this	this	DET
cana-1932	18	19	advanced	advanced	ADJ
cana-1932	18	20	topic	topic	NOUN
cana-1932	18	21	in	in	ADP
cana-1932	18	22	fuzzy	fuzzy	ADJ
cana-1932	18	23	topology	topology	NOUN
cana-1932	18	24	.	.	PUNCT
cana-1932	19	1	objectives	objective	NOUN
cana-1932	19	2	:	:	PUNCT
cana-1932	19	3	•	•	NOUN
cana-1932	19	4	to	to	PART
cana-1932	19	5	formally	formally	ADV
cana-1932	19	6	define	define	VERB
cana-1932	19	7	the	the	DET
cana-1932	19	8	concept	concept	NOUN
cana-1932	19	9	of	of	ADP
cana-1932	19	10	fuzzy	fuzzy	ADJ
cana-1932	19	11	double	double	ADJ
cana-1932	19	12	s	s	NOUN
cana-1932	19	13	-	-	PUNCT
cana-1932	19	14	hausdorff	hausdorff	NOUN
cana-1932	19	15	spaces	space	NOUN
cana-1932	19	16	and	and	CCONJ
cana-1932	19	17	provide	provide	VERB
cana-1932	19	18	clear	clear	ADJ
cana-1932	19	19	characterizations	characterization	NOUN
cana-1932	19	20	that	that	PRON
cana-1932	19	21	distinguish	distinguish	VERB
cana-1932	19	22	them	they	PRON
cana-1932	19	23	from	from	ADP
cana-1932	19	24	other	other	ADJ
cana-1932	19	25	fuzzy	fuzzy	ADJ
cana-1932	19	26	topological	topological	ADJ
cana-1932	19	27	spaces	space	NOUN
cana-1932	19	28	.	.	PUNCT
cana-1932	20	1	•	•	X
cana-1932	20	2	to	to	PART
cana-1932	20	3	investigate	investigate	VERB
cana-1932	20	4	the	the	DET
cana-1932	20	5	fundamental	fundamental	ADJ
cana-1932	20	6	properties	property	NOUN
cana-1932	20	7	of	of	ADP
cana-1932	20	8	fuzzy	fuzzy	ADJ
cana-1932	20	9	double	double	ADJ
cana-1932	20	10	s	s	NOUN
cana-1932	20	11	-	-	PUNCT
cana-1932	20	12	hausdorff	hausdorff	NOUN
cana-1932	20	13	spaces	space	NOUN
cana-1932	20	14	,	,	PUNCT
cana-1932	20	15	including	include	VERB
cana-1932	20	16	their	their	PRON
cana-1932	20	17	relationships	relationship	NOUN
cana-1932	20	18	with	with	ADP
cana-1932	20	19	classical	classical	ADJ
cana-1932	20	20	hausdorff	hausdorff	NOUN
cana-1932	20	21	spaces	space	NOUN
cana-1932	20	22	and	and	CCONJ
cana-1932	20	23	other	other	ADJ
cana-1932	20	24	fuzzy	fuzzy	ADJ
cana-1932	20	25	separation	separation	NOUN
cana-1932	20	26	axioms	axiom	NOUN
cana-1932	20	27	.	.	PUNCT
cana-1932	21	1	•	•	ADP
cana-1932	21	2	to	to	PART
cana-1932	21	3	develop	develop	VERB
cana-1932	21	4	and	and	CCONJ
cana-1932	21	5	prove	prove	VERB
cana-1932	21	6	key	key	ADJ
cana-1932	21	7	theorems	theorem	NOUN
cana-1932	21	8	related	relate	VERB
cana-1932	21	9	to	to	ADP
cana-1932	21	10	fuzzy	fuzzy	ADJ
cana-1932	21	11	double	double	ADJ
cana-1932	21	12	s	s	NOUN
cana-1932	21	13	-	-	PUNCT
cana-1932	21	14	hausdorff	hausdorff	NOUN
cana-1932	21	15	spaces	space	NOUN
cana-1932	21	16	,	,	PUNCT
cana-1932	21	17	contributing	contribute	VERB
cana-1932	21	18	to	to	ADP
cana-1932	21	19	the	the	DET
cana-1932	21	20	theoretical	theoretical	ADJ
cana-1932	21	21	foundation	foundation	NOUN
cana-1932	21	22	of	of	ADP
cana-1932	21	23	fuzzy	fuzzy	ADJ
cana-1932	21	24	topology	topology	NOUN
cana-1932	21	25	.	.	PUNCT
cana-1932	22	1	communications	communication	NOUN
cana-1932	22	2	on	on	ADP
cana-1932	22	3	applied	apply	VERB
cana-1932	22	4	nonlinear	nonlinear	ADJ
cana-1932	22	5	analysis	analysis	NOUN
cana-1932	22	6	issn	issn	NOUN
cana-1932	22	7	:	:	PUNCT
cana-1932	22	8	1074	1074	NUM
cana-1932	22	9	-	-	PUNCT
cana-1932	22	10	133x	133x	NUM
cana-1932	22	11	vol	vol	NOUN
cana-1932	22	12	32	32	NUM
cana-1932	22	13	no	no	NOUN
cana-1932	22	14	.	.	NOUN
cana-1932	22	15	3	3	NUM
cana-1932	22	16	(	(	PUNCT
cana-1932	22	17	2025	2025	NUM
cana-1932	22	18	)	)	PUNCT
cana-1932	22	19	154	154	NUM
cana-1932	22	20	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-1932	22	21	methods	method	NOUN
cana-1932	22	22	:	:	PUNCT
cana-1932	22	23	this	this	DET
cana-1932	22	24	study	study	NOUN
cana-1932	22	25	on	on	ADP
cana-1932	22	26	fuzzy	fuzzy	ADJ
cana-1932	22	27	double	double	ADJ
cana-1932	22	28	s	s	NOUN
cana-1932	22	29	-	-	PUNCT
cana-1932	22	30	hausdorff	hausdorff	NOUN
cana-1932	22	31	spaces	space	NOUN
cana-1932	22	32	employs	employ	VERB
cana-1932	22	33	a	a	DET
cana-1932	22	34	combination	combination	NOUN
cana-1932	22	35	of	of	ADP
cana-1932	22	36	theoretical	theoretical	ADJ
cana-1932	22	37	analysis	analysis	NOUN
cana-1932	22	38	,	,	PUNCT
cana-1932	22	39	formal	formal	ADJ
cana-1932	22	40	proofs	proof	NOUN
cana-1932	22	41	,	,	PUNCT
cana-1932	22	42	and	and	CCONJ
cana-1932	22	43	comparative	comparative	ADJ
cana-1932	22	44	evaluations	evaluation	NOUN
cana-1932	22	45	.	.	PUNCT
cana-1932	23	1	the	the	DET
cana-1932	23	2	methods	method	NOUN
cana-1932	23	3	are	be	AUX
cana-1932	23	4	structured	structure	VERB
cana-1932	23	5	to	to	PART
cana-1932	23	6	ensure	ensure	VERB
cana-1932	23	7	a	a	DET
cana-1932	23	8	comprehensive	comprehensive	ADJ
cana-1932	23	9	understanding	understanding	NOUN
cana-1932	23	10	of	of	ADP
cana-1932	23	11	the	the	DET
cana-1932	23	12	subject	subject	NOUN
cana-1932	23	13	while	while	SCONJ
cana-1932	23	14	contributing	contribute	VERB
cana-1932	23	15	novel	novel	ADJ
cana-1932	23	16	insights	insight	NOUN
cana-1932	23	17	to	to	ADP
cana-1932	23	18	the	the	DET
cana-1932	23	19	field	field	NOUN
cana-1932	23	20	of	of	ADP
cana-1932	23	21	fuzzy	fuzzy	ADJ
cana-1932	23	22	topology	topology	NOUN
cana-1932	23	23	.	.	PUNCT
cana-1932	24	1	1	1	X
cana-1932	24	2	.	.	X
cana-1932	24	3	literature	literature	NOUN
cana-1932	24	4	review	review	PROPN
cana-1932	24	5	and	and	CCONJ
cana-1932	24	6	conceptual	conceptual	ADJ
cana-1932	24	7	framework	framework	NOUN
cana-1932	24	8	development	development	NOUN
cana-1932	24	9	o	o	PROPN
cana-1932	24	10	a	a	DET
cana-1932	24	11	detailed	detailed	ADJ
cana-1932	24	12	review	review	NOUN
cana-1932	24	13	of	of	ADP
cana-1932	24	14	existing	exist	VERB
cana-1932	24	15	literature	literature	NOUN
cana-1932	24	16	was	be	AUX
cana-1932	24	17	conducted	conduct	VERB
cana-1932	24	18	to	to	PART
cana-1932	24	19	establish	establish	VERB
cana-1932	24	20	the	the	DET
cana-1932	24	21	foundational	foundational	ADJ
cana-1932	24	22	concepts	concept	NOUN
cana-1932	24	23	related	relate	VERB
cana-1932	24	24	to	to	ADP
cana-1932	24	25	fuzzy	fuzzy	ADJ
cana-1932	24	26	sets	set	NOUN
cana-1932	24	27	,	,	PUNCT
cana-1932	24	28	fuzzy	fuzzy	ADJ
cana-1932	24	29	topologies	topology	NOUN
cana-1932	24	30	,	,	PUNCT
cana-1932	24	31	and	and	CCONJ
cana-1932	24	32	fuzzy	fuzzy	ADJ
cana-1932	24	33	separation	separation	NOUN
cana-1932	24	34	axioms	axiom	NOUN
cana-1932	24	35	.	.	PUNCT
cana-1932	25	1	this	this	DET
cana-1932	25	2	review	review	NOUN
cana-1932	25	3	helped	help	VERB
cana-1932	25	4	to	to	PART
cana-1932	25	5	identify	identify	VERB
cana-1932	25	6	gaps	gap	NOUN
cana-1932	25	7	in	in	ADP
cana-1932	25	8	current	current	ADJ
cana-1932	25	9	knowledge	knowledge	NOUN
cana-1932	25	10	and	and	CCONJ
cana-1932	25	11	guided	guide	VERB
cana-1932	25	12	the	the	DET
cana-1932	25	13	formulation	formulation	NOUN
cana-1932	25	14	of	of	ADP
cana-1932	25	15	the	the	DET
cana-1932	25	16	research	research	NOUN
cana-1932	25	17	objectives	objective	NOUN
cana-1932	25	18	.	.	PUNCT
cana-1932	26	1	o	o	X
cana-1932	26	2	based	base	VERB
cana-1932	26	3	on	on	ADP
cana-1932	26	4	the	the	DET
cana-1932	26	5	literature	literature	PROPN
cana-1932	26	6	review	review	PROPN
cana-1932	26	7	,	,	PUNCT
cana-1932	26	8	a	a	DET
cana-1932	26	9	conceptual	conceptual	ADJ
cana-1932	26	10	framework	framework	NOUN
cana-1932	26	11	for	for	ADP
cana-1932	26	12	fuzzy	fuzzy	ADJ
cana-1932	26	13	double	double	ADJ
cana-1932	26	14	s	s	NOUN
cana-1932	26	15	-	-	PUNCT
cana-1932	26	16	hausdorff	hausdorff	ADJ
cana-1932	26	17	spaces	space	NOUN
cana-1932	26	18	was	be	AUX
cana-1932	26	19	developed	develop	VERB
cana-1932	26	20	.	.	PUNCT
cana-1932	27	1	this	this	DET
cana-1932	27	2	framework	framework	NOUN
cana-1932	27	3	includes	include	VERB
cana-1932	27	4	the	the	DET
cana-1932	27	5	definition	definition	NOUN
cana-1932	27	6	of	of	ADP
cana-1932	27	7	fuzzy	fuzzy	ADJ
cana-1932	27	8	double	double	ADJ
cana-1932	27	9	s	s	NOUN
cana-1932	27	10	-	-	PUNCT
cana-1932	27	11	hausdorff	hausdorff	NOUN
cana-1932	27	12	spaces	space	NOUN
cana-1932	27	13	,	,	PUNCT
cana-1932	27	14	the	the	DET
cana-1932	27	15	identification	identification	NOUN
cana-1932	27	16	of	of	ADP
cana-1932	27	17	key	key	ADJ
cana-1932	27	18	properties	property	NOUN
cana-1932	27	19	,	,	PUNCT
cana-1932	27	20	and	and	CCONJ
cana-1932	27	21	the	the	DET
cana-1932	27	22	establishment	establishment	NOUN
cana-1932	27	23	of	of	ADP
cana-1932	27	24	their	their	PRON
cana-1932	27	25	relationships	relationship	NOUN
cana-1932	27	26	with	with	ADP
cana-1932	27	27	existing	exist	VERB
cana-1932	27	28	fuzzy	fuzzy	ADJ
cana-1932	27	29	topological	topological	ADJ
cana-1932	27	30	spaces	space	NOUN
cana-1932	27	31	.	.	PUNCT
cana-1932	28	1	2	2	X
cana-1932	28	2	.	.	X
cana-1932	28	3	formal	formal	ADJ
cana-1932	28	4	definitions	definition	NOUN
cana-1932	28	5	and	and	CCONJ
cana-1932	28	6	characterizations	characterization	NOUN
cana-1932	28	7	o	o	NOUN
cana-1932	28	8	the	the	DET
cana-1932	28	9	study	study	NOUN
cana-1932	28	10	begins	begin	VERB
cana-1932	28	11	by	by	ADP
cana-1932	28	12	formally	formally	ADV
cana-1932	28	13	defining	define	VERB
cana-1932	28	14	fuzzy	fuzzy	ADJ
cana-1932	28	15	double	double	ADJ
cana-1932	28	16	s	s	NOUN
cana-1932	28	17	-	-	PUNCT
cana-1932	28	18	hausdorff	hausdorff	NOUN
cana-1932	28	19	spaces	space	NOUN
cana-1932	28	20	within	within	ADP
cana-1932	28	21	the	the	DET
cana-1932	28	22	context	context	NOUN
cana-1932	28	23	of	of	ADP
cana-1932	28	24	fuzzy	fuzzy	ADJ
cana-1932	28	25	topology	topology	NOUN
cana-1932	28	26	.	.	PUNCT
cana-1932	29	1	definitions	definition	NOUN
cana-1932	29	2	are	be	AUX
cana-1932	29	3	constructed	construct	VERB
cana-1932	29	4	to	to	PART
cana-1932	29	5	extend	extend	VERB
cana-1932	29	6	the	the	DET
cana-1932	29	7	classical	classical	ADJ
cana-1932	29	8	hausdorff	hausdorff	NOUN
cana-1932	29	9	condition	condition	NOUN
cana-1932	29	10	into	into	ADP
cana-1932	29	11	a	a	DET
cana-1932	29	12	fuzzy	fuzzy	ADJ
cana-1932	29	13	framework	framework	NOUN
cana-1932	29	14	,	,	PUNCT
cana-1932	29	15	taking	take	VERB
cana-1932	29	16	into	into	ADP
cana-1932	29	17	account	account	NOUN
cana-1932	29	18	the	the	DET
cana-1932	29	19	dual	dual	ADJ
cana-1932	29	20	nature	nature	NOUN
cana-1932	29	21	of	of	ADP
cana-1932	29	22	the	the	DET
cana-1932	29	23	topologies	topology	NOUN
cana-1932	29	24	involved	involve	VERB
cana-1932	29	25	.	.	PUNCT
cana-1932	30	1	o	o	X
cana-1932	30	2	key	key	ADJ
cana-1932	30	3	characterizations	characterization	NOUN
cana-1932	30	4	of	of	ADP
cana-1932	30	5	fuzzy	fuzzy	ADJ
cana-1932	30	6	double	double	ADJ
cana-1932	30	7	s	s	NOUN
cana-1932	30	8	-	-	PUNCT
cana-1932	30	9	hausdorff	hausdorff	ADJ
cana-1932	30	10	spaces	space	NOUN
cana-1932	30	11	are	be	AUX
cana-1932	30	12	provided	provide	VERB
cana-1932	30	13	,	,	PUNCT
cana-1932	30	14	highlighting	highlight	VERB
cana-1932	30	15	their	their	PRON
cana-1932	30	16	unique	unique	ADJ
cana-1932	30	17	properties	property	NOUN
cana-1932	30	18	and	and	CCONJ
cana-1932	30	19	distinguishing	distinguish	VERB
cana-1932	30	20	them	they	PRON
cana-1932	30	21	from	from	ADP
cana-1932	30	22	other	other	ADJ
cana-1932	30	23	related	relate	VERB
cana-1932	30	24	concepts	concept	NOUN
cana-1932	30	25	such	such	ADJ
cana-1932	30	26	as	as	ADP
cana-1932	30	27	fuzzy	fuzzy	ADJ
cana-1932	30	28	s	s	NOUN
cana-1932	30	29	-	-	PUNCT
cana-1932	30	30	hausdorff	hausdorff	NOUN
cana-1932	30	31	spaces	space	NOUN
cana-1932	30	32	and	and	CCONJ
cana-1932	30	33	fuzzy	fuzzy	ADJ
cana-1932	30	34	hausdorff	hausdorff	NOUN
cana-1932	30	35	spaces	space	NOUN
cana-1932	30	36	.	.	PUNCT
cana-1932	31	1	3	3	X
cana-1932	31	2	.	.	X
cana-1932	31	3	theorem	theorem	VERB
cana-1932	31	4	development	development	NOUN
cana-1932	31	5	and	and	CCONJ
cana-1932	31	6	proofs	proof	VERB
cana-1932	31	7	o	o	NOUN
cana-1932	31	8	a	a	DET
cana-1932	31	9	series	series	NOUN
cana-1932	31	10	of	of	ADP
cana-1932	31	11	theorems	theorem	NOUN
cana-1932	31	12	are	be	AUX
cana-1932	31	13	developed	develop	VERB
cana-1932	31	14	to	to	PART
cana-1932	31	15	establish	establish	VERB
cana-1932	31	16	the	the	DET
cana-1932	31	17	theoretical	theoretical	ADJ
cana-1932	31	18	underpinnings	underpinning	NOUN
cana-1932	31	19	of	of	ADP
cana-1932	31	20	fuzzy	fuzzy	ADJ
cana-1932	31	21	double	double	ADJ
cana-1932	31	22	s	s	NOUN
cana-1932	31	23	-	-	PUNCT
cana-1932	31	24	hausdorff	hausdorff	NOUN
cana-1932	31	25	spaces	space	NOUN
cana-1932	31	26	.	.	PUNCT
cana-1932	32	1	these	these	DET
cana-1932	32	2	theorems	theorem	NOUN
cana-1932	32	3	are	be	AUX
cana-1932	32	4	carefully	carefully	ADV
cana-1932	32	5	constructed	construct	VERB
cana-1932	32	6	to	to	PART
cana-1932	32	7	explore	explore	VERB
cana-1932	32	8	the	the	DET
cana-1932	32	9	implications	implication	NOUN
cana-1932	32	10	of	of	ADP
cana-1932	32	11	the	the	DET
cana-1932	32	12	fuzzy	fuzzy	ADJ
cana-1932	32	13	double	double	ADJ
cana-1932	32	14	s	s	NOUN
cana-1932	32	15	-	-	PUNCT
cana-1932	32	16	hausdorff	hausdorff	NOUN
cana-1932	32	17	condition	condition	NOUN
cana-1932	32	18	in	in	ADP
cana-1932	32	19	various	various	ADJ
cana-1932	32	20	topological	topological	ADJ
cana-1932	32	21	scenarios	scenario	NOUN
cana-1932	32	22	.	.	PUNCT
cana-1932	33	1	o	o	NOUN
cana-1932	33	2	each	each	DET
cana-1932	33	3	theorem	theorem	NOUN
cana-1932	33	4	is	be	AUX
cana-1932	33	5	accompanied	accompany	VERB
cana-1932	33	6	by	by	ADP
cana-1932	33	7	a	a	DET
cana-1932	33	8	formal	formal	ADJ
cana-1932	33	9	proof	proof	NOUN
cana-1932	33	10	,	,	PUNCT
cana-1932	33	11	which	which	PRON
cana-1932	33	12	rigorously	rigorously	ADV
cana-1932	33	13	demonstrates	demonstrate	VERB
cana-1932	33	14	the	the	DET
cana-1932	33	15	validity	validity	NOUN
cana-1932	33	16	of	of	ADP
cana-1932	33	17	the	the	DET
cana-1932	33	18	proposed	propose	VERB
cana-1932	33	19	concepts	concept	NOUN
cana-1932	33	20	.	.	PUNCT
cana-1932	34	1	the	the	DET
cana-1932	34	2	proofs	proof	NOUN
cana-1932	34	3	rely	rely	VERB
cana-1932	34	4	on	on	ADP
cana-1932	34	5	established	establish	VERB
cana-1932	34	6	techniques	technique	NOUN
cana-1932	34	7	in	in	ADP
cana-1932	34	8	fuzzy	fuzzy	ADJ
cana-1932	34	9	topology	topology	NOUN
cana-1932	34	10	,	,	PUNCT
cana-1932	34	11	including	include	VERB
cana-1932	34	12	set	set	VERB
cana-1932	34	13	-	-	PUNCT
cana-1932	34	14	theoretic	theoretic	NOUN
cana-1932	34	15	approaches	approach	NOUN
cana-1932	34	16	and	and	CCONJ
cana-1932	34	17	logical	logical	ADJ
cana-1932	34	18	reasoning	reasoning	NOUN
cana-1932	34	19	within	within	ADP
cana-1932	34	20	the	the	DET
cana-1932	34	21	fuzzy	fuzzy	ADJ
cana-1932	34	22	context	context	NOUN
cana-1932	34	23	.	.	PUNCT
cana-1932	35	1	2.preliminary	2.preliminary	NUM
cana-1932	35	2	descriptions	description	NOUN
cana-1932	35	3	definition:2.1	definition:2.1	NOUN
cana-1932	35	4	assume	assume	VERB
cana-1932	35	5	,	,	PUNCT
cana-1932	35	6	𝑋	𝑋	PROPN
cana-1932	35	7	remain	remain	VERB
cana-1932	35	8	a	a	DET
cana-1932	35	9	non	non	ADJ
cana-1932	35	10	-	-	ADJ
cana-1932	35	11	empty	empty	ADJ
cana-1932	35	12	set	set	NOUN
cana-1932	35	13	.	.	PUNCT
cana-1932	36	1	a	a	DET
cana-1932	36	2	fds	fds	NOUN
cana-1932	36	3	f	f	PROPN
cana-1932	36	4	remains	remain	VERB
cana-1932	36	5	an	an	DET
cana-1932	36	6	ordered	order	VERB
cana-1932	36	7	pair	pair	NOUN
cana-1932	36	8	(	(	PUNCT
cana-1932	36	9	f1	f1	NOUN
cana-1932	36	10	,	,	PUNCT
cana-1932	36	11	f2	f2	PROPN
cana-1932	36	12	)	)	PUNCT
cana-1932	36	13	∈	∈	PROPN
cana-1932	36	14	ix	ix	ADP
cana-1932	36	15	×	×	NOUN
cana-1932	36	16	ix	ix	ADP
cana-1932	36	17	suchthat	suchthat	PROPN
cana-1932	36	18	f1	f1	PROPN
cana-1932	36	19	≤	≤	PUNCT
cana-1932	36	20	f2	f2	PROPN
cana-1932	36	21	,	,	PUNCT
cana-1932	36	22	where	where	SCONJ
cana-1932	36	23	i	i	PRON
cana-1932	36	24	=	=	PUNCT
cana-1932	37	1	[	[	X
cana-1932	37	2	0,1	0,1	NUM
cana-1932	37	3	]	]	PUNCT
cana-1932	37	4	the	the	DET
cana-1932	37	5	family	family	NOUN
cana-1932	37	6	of	of	ADP
cana-1932	37	7	fds	fds	NOUN
cana-1932	37	8	is	be	AUX
cana-1932	37	9	denoted	denote	VERB
cana-1932	37	10	by	by	ADP
cana-1932	37	11	fd	fd	PROPN
cana-1932	37	12	(	(	PUNCT
cana-1932	37	13	x	x	X
cana-1932	37	14	)	)	PUNCT
cana-1932	37	15	definition	definition	NOUN
cana-1932	37	16	:	:	PUNCT
cana-1932	37	17	2.2	2.2	NUM
cana-1932	37	18	a	a	DET
cana-1932	37	19	fuzzy	fuzzy	ADJ
cana-1932	37	20	topological	topological	ADJ
cana-1932	37	21	space	space	NOUN
cana-1932	37	22	(	(	PUNCT
cana-1932	37	23	x	x	X
cana-1932	37	24	,	,	PUNCT
cana-1932	37	25	τ	τ	X
cana-1932	37	26	)	)	PUNCT
cana-1932	37	27	is	be	AUX
cana-1932	37	28	supposed	suppose	VERB
cana-1932	37	29	towards	towards	PART
cana-1932	37	30	be	be	AUX
cana-1932	37	31	fuzzy	fuzzy	ADJ
cana-1932	37	32	st2	st2	NOUN
cana-1932	37	33	,	,	PUNCT
cana-1932	37	34	if	if	SCONJ
cana-1932	37	35	for	for	ADP
cana-1932	37	36	any	any	DET
cana-1932	37	37	twosome	twosome	NOUN
cana-1932	37	38	off	off	ADP
cana-1932	37	39	different	different	ADJ
cana-1932	37	40	fuzzy	fuzzy	ADJ
cana-1932	37	41	points	point	NOUN
cana-1932	37	42	xt	xt	PROPN
cana-1932	37	43	,	,	PUNCT
cana-1932	37	44	yu	yu	PROPN
cana-1932	37	45	in	in	ADP
cana-1932	37	46	x	x	PROPN
cana-1932	37	47	,	,	PUNCT
cana-1932	37	48	∃	∃	PROPN
cana-1932	37	49	f	f	PROPN
cana-1932	37	50	,	,	PUNCT
cana-1932	37	51	g	g	PROPN
cana-1932	37	52	∈	∈	PROPN
cana-1932	37	53	τ	τ	X
cana-1932	38	1	suchthat	suchthat	VERB
cana-1932	38	2	xt	xt	PROPN
cana-1932	39	1	∈	∈	PROPN
cana-1932	39	2	f	f	PROPN
cana-1932	39	3	,	,	PUNCT
cana-1932	39	4	y	y	PROPN
cana-1932	39	5	u	u	NOUN
cana-1932	39	6	∈	∈	PROPN
cana-1932	39	7	g	g	NOUN
cana-1932	39	8	and	and	CCONJ
cana-1932	39	9	f	f	NOUN
cana-1932	39	10	g	g	PROPN
cana-1932	39	11	=	=	PUNCT
cana-1932	39	12	𝟎.	𝟎.	NOUN
cana-1932	39	13	communications	communication	NOUN
cana-1932	39	14	on	on	ADP
cana-1932	39	15	applied	apply	VERB
cana-1932	39	16	nonlinear	nonlinear	ADJ
cana-1932	39	17	analysis	analysis	NOUN
cana-1932	39	18	issn	issn	NOUN
cana-1932	39	19	:	:	PUNCT
cana-1932	39	20	1074	1074	NUM
cana-1932	39	21	-	-	PUNCT
cana-1932	39	22	133x	133x	NUM
cana-1932	39	23	vol	vol	NOUN
cana-1932	39	24	32	32	NUM
cana-1932	39	25	no	no	NOUN
cana-1932	39	26	.	.	NOUN
cana-1932	39	27	3	3	NUM
cana-1932	39	28	(	(	PUNCT
cana-1932	39	29	2025	2025	NUM
cana-1932	39	30	)	)	PUNCT
cana-1932	39	31	155	155	NUM
cana-1932	39	32	https://internationalpubls.com	https://internationalpubls.com	X
cana-1932	39	33	3	3	X
cana-1932	39	34	.	.	NOUN
cana-1932	39	35	fuzzy	fuzzy	ADJ
cana-1932	39	36	double	double	ADJ
cana-1932	39	37	st2	st2	NOUN
cana-1932	39	38	definition	definition	NOUN
cana-1932	39	39	:	:	PUNCT
cana-1932	39	40	3.1	3.1	NUM
cana-1932	39	41	a	a	DET
cana-1932	39	42	fdts	fdts	ADJ
cana-1932	39	43	(	(	PUNCT
cana-1932	39	44	x	x	SYM
cana-1932	39	45	,	,	PUNCT
cana-1932	39	46	τ	τ	PROPN
cana-1932	39	47	)	)	PUNCT
cana-1932	39	48	is	be	AUX
cana-1932	39	49	supposed	suppose	VERB
cana-1932	39	50	towards	towards	PART
cana-1932	39	51	remain	remain	VERB
cana-1932	39	52	fds	fds	NOUN
cana-1932	39	53	-	-	PUNCT
cana-1932	39	54	t2	t2	NOUN
cana-1932	39	55	,	,	PUNCT
cana-1932	39	56	meant	mean	VERB
cana-1932	39	57	for	for	SCONJ
cana-1932	39	58	all	all	DET
cana-1932	39	59	two	two	NUM
cana-1932	39	60	some	some	PRON
cana-1932	39	61	of	of	ADP
cana-1932	39	62	different	different	ADJ
cana-1932	39	63	fd	fd	PROPN
cana-1932	39	64	points	point	NOUN
cana-1932	39	65	x	x	INTJ
cana-1932	39	66	(	(	PUNCT
cana-1932	39	67	r	r	NOUN
cana-1932	39	68	,	,	PUNCT
cana-1932	39	69	s	s	PART
cana-1932	39	70	)	)	PUNCT
cana-1932	39	71	,	,	PUNCT
cana-1932	39	72	y	y	PROPN
cana-1932	39	73	(	(	PUNCT
cana-1932	39	74	u	u	PROPN
cana-1932	39	75	,	,	PUNCT
cana-1932	39	76	v	v	NOUN
cana-1932	39	77	)	)	PUNCT
cana-1932	39	78	in	in	ADP
cana-1932	39	79	𝑋	𝑋	PROPN
cana-1932	39	80	,	,	PUNCT
cana-1932	39	81	near	near	ADP
cana-1932	39	82	occurs	occur	VERB
cana-1932	39	83	binary	binary	ADJ
cana-1932	39	84	fuzzy	fuzzy	ADJ
cana-1932	39	85	double	double	ADJ
cana-1932	39	86	open	open	ADJ
cana-1932	39	87	setsf	setsf	NOUN
cana-1932	39	88	=	=	SYM
cana-1932	39	89	(	(	PUNCT
cana-1932	39	90	f1	f1	NOUN
cana-1932	39	91	,	,	PUNCT
cana-1932	39	92	f2	f2	PROPN
cana-1932	39	93	)	)	PUNCT
cana-1932	39	94	,	,	PUNCT
cana-1932	39	95	g	g	PROPN
cana-1932	39	96	=	=	SYM
cana-1932	39	97	(	(	PUNCT
cana-1932	39	98	g1	g1	PROPN
cana-1932	39	99	,	,	PUNCT
cana-1932	39	100	g2	g2	PROPN
cana-1932	39	101	)	)	PUNCT
cana-1932	39	102	∈	∈	PROPN
cana-1932	40	1	τ	τ	X
cana-1932	40	2	suchthat	suchthat	VERB
cana-1932	40	3	x	x	X
cana-1932	40	4	(	(	PUNCT
cana-1932	40	5	r	r	NOUN
cana-1932	40	6	,	,	PUNCT
cana-1932	40	7	s	s	PART
cana-1932	40	8	)	)	PUNCT
cana-1932	40	9	∈	∈	PROPN
cana-1932	41	1	f	f	X
cana-1932	41	2	,	,	PUNCT
cana-1932	41	3	y	y	PROPN
cana-1932	41	4	(	(	PUNCT
cana-1932	41	5	u	u	NOUN
cana-1932	41	6	,	,	PUNCT
cana-1932	41	7	v	v	NOUN
cana-1932	41	8	)	)	PUNCT
cana-1932	41	9	∈	∈	PROPN
cana-1932	41	10	g	g	NOUN
cana-1932	41	11	that	that	PRON
cana-1932	41	12	is	be	AUX
cana-1932	41	13	f1(x	f1(x	PROPN
cana-1932	41	14	)	)	PUNCT
cana-1932	41	15	≥	≥	NOUN
cana-1932	41	16	r	r	NOUN
cana-1932	41	17	,	,	PUNCT
cana-1932	41	18	f2(x	f2(x	PROPN
cana-1932	41	19	)	)	PUNCT
cana-1932	41	20	≥	≥	NUM
cana-1932	41	21	s	s	NOUN
cana-1932	41	22	,	,	PUNCT
cana-1932	41	23	g	g	PROPN
cana-1932	41	24	1	1	NUM
cana-1932	41	25	(	(	PUNCT
cana-1932	41	26	y	y	NOUN
cana-1932	41	27	)	)	PUNCT
cana-1932	41	28	≥	≥	NOUN
cana-1932	41	29	u	u	NOUN
cana-1932	41	30	,	,	PUNCT
cana-1932	41	31	g	g	PROPN
cana-1932	41	32	2	2	NUM
cana-1932	41	33	(	(	PUNCT
cana-1932	41	34	y	y	NOUN
cana-1932	41	35	)	)	PUNCT
cana-1932	41	36	≥	≥	NOUN
cana-1932	41	37	v	v	NOUN
cana-1932	41	38	and	and	CCONJ
cana-1932	41	39	f	f	PROPN
cana-1932	41	40	∩	∩	NOUN
cana-1932	41	41	g	g	PROPN
cana-1932	41	42	=	=	SYM
cana-1932	41	43	0	0	PROPN
cana-1932	41	44	.	.	PUNCT
cana-1932	42	1	definition	definition	NOUN
cana-1932	42	2	:	:	PUNCT
cana-1932	42	3	3.2	3.2	NUM
cana-1932	42	4	assume	assume	VERB
cana-1932	42	5	,	,	PUNCT
cana-1932	42	6	(	(	PUNCT
cana-1932	42	7	x	x	X
cana-1932	42	8	,	,	PUNCT
cana-1932	42	9	τ	τ	PROPN
cana-1932	42	10	)	)	PUNCT
cana-1932	42	11	be	be	AUX
cana-1932	42	12	a	a	DET
cana-1932	42	13	fdts	fdts	NOUN
cana-1932	42	14	.	.	PUNCT
cana-1932	43	1	assume	assume	VERB
cana-1932	43	2	,	,	PUNCT
cana-1932	43	3	x1	x1	PROPN
cana-1932	43	4	⊆	⊆	NUM
cana-1932	43	5	x.	x.	NOUN
cana-1932	43	6	assume	assume	PROPN
cana-1932	43	7	,	,	PUNCT
cana-1932	43	8	f	f	PROPN
cana-1932	43	9	=	=	PUNCT
cana-1932	43	10	(	(	PUNCT
cana-1932	43	11	f1	f1	PROPN
cana-1932	43	12	,	,	PUNCT
cana-1932	43	13	f2	f2	PROPN
cana-1932	43	14	)	)	PUNCT
cana-1932	43	15	∈	∈	PROPN
cana-1932	44	1	τ	τ	PROPN
cana-1932	44	2	.	.	PUNCT
cana-1932	45	1	describe	describe	NOUN
cana-1932	45	2	,	,	PUNCT
cana-1932	45	3	f	f	PROPN
cana-1932	45	4	/x1	/x1	NOUN
cana-1932	46	1	=	=	PUNCT
cana-1932	46	2	(	(	PUNCT
cana-1932	46	3	f1/	f1/	VERB
cana-1932	46	4	x1	x1	PROPN
cana-1932	46	5	,	,	PUNCT
cana-1932	46	6	f2/	f2/	VERB
cana-1932	46	7	x1	x1	NOUN
cana-1932	46	8	)	)	PUNCT
cana-1932	46	9	suchthat	suchthat	PROPN
cana-1932	46	10	(	(	PUNCT
cana-1932	46	11	f1	f1	PROPN
cana-1932	46	12	/	/	SYM
cana-1932	46	13	x1	x1	PROPN
cana-1932	46	14	)	)	PUNCT
cana-1932	47	1	(	(	PUNCT
cana-1932	47	2	z	z	NOUN
cana-1932	47	3	)	)	PUNCT
cana-1932	47	4	=	=	SYM
cana-1932	47	5	f1	f1	NOUN
cana-1932	47	6	(	(	PUNCT
cana-1932	47	7	z	z	NOUN
cana-1932	47	8	)	)	PUNCT
cana-1932	47	9	,	,	PUNCT
cana-1932	47	10	(	(	PUNCT
cana-1932	47	11	f2	f2	X
cana-1932	47	12	/	/	SYM
cana-1932	47	13	x1	x1	PROPN
cana-1932	47	14	)	)	PUNCT
cana-1932	47	15	(	(	PUNCT
cana-1932	47	16	z	z	NOUN
cana-1932	47	17	)	)	PUNCT
cana-1932	47	18	=	=	SYM
cana-1932	48	1	f2	f2	PROPN
cana-1932	48	2	(	(	PUNCT
cana-1932	48	3	z	z	NOUN
cana-1932	48	4	)	)	PUNCT
cana-1932	48	5	for	for	ADP
cana-1932	48	6	all	all	DET
cana-1932	48	7	z	z	NOUN
cana-1932	48	8	∈	∈	NOUN
cana-1932	48	9	x	x	SYM
cana-1932	48	10	1	1	NUM
cana-1932	48	11	define	define	NOUN
cana-1932	48	12	(	(	PUNCT
cana-1932	48	13	τ	τ	PROPN
cana-1932	48	14	/	/	SYM
cana-1932	48	15	x1	x1	PROPN
cana-1932	48	16	)	)	PUNCT
cana-1932	49	1	=	=	PRON
cana-1932	49	2	{	{	PUNCT
cana-1932	49	3	(	(	PUNCT
cana-1932	49	4	f	f	X
cana-1932	49	5	/	/	SYM
cana-1932	49	6	x1	x1	PROPN
cana-1932	49	7	)	)	PUNCT
cana-1932	49	8	|f	|f	PROPN
cana-1932	50	1	∈	∈	PROPN
cana-1932	50	2	τ	τ	PROPN
cana-1932	50	3	}	}	PUNCT
cana-1932	50	4	then	then	ADV
cana-1932	50	5	(	(	PUNCT
cana-1932	50	6	τ	τ	X
cana-1932	50	7	/	/	SYM
cana-1932	50	8	x1	x1	PROPN
cana-1932	50	9	)	)	PUNCT
cana-1932	50	10	is	be	AUX
cana-1932	50	11	called	call	VERB
cana-1932	50	12	fd	fd	PROPN
cana-1932	50	13	subspace	subspace	NOUN
cana-1932	50	14	topology	topology	NOUN
cana-1932	50	15	on	on	ADP
cana-1932	50	16	x1,then(x1	x1,then(x1	PROPN
cana-1932	50	17	,	,	PUNCT
cana-1932	50	18	τ	τ	PROPN
cana-1932	50	19	/	/	SYM
cana-1932	50	20	x1	x1	PROPN
cana-1932	50	21	)	)	PUNCT
cana-1932	50	22	is	be	AUX
cana-1932	50	23	said	say	VERB
cana-1932	50	24	to	to	ADP
cana-1932	50	25	befd	befd	VERB
cana-1932	50	26	subspace	subspace	NOUN
cana-1932	50	27	of	of	ADP
cana-1932	50	28	(	(	PUNCT
cana-1932	50	29	𝐗	𝐗	PROPN
cana-1932	50	30	,	,	PUNCT
cana-1932	50	31	𝛕	𝛕	X
cana-1932	50	32	)	)	PUNCT
cana-1932	50	33	proposition	proposition	NOUN
cana-1932	50	34	:	:	PUNCT
cana-1932	50	35	3.3	3.3	NUM
cana-1932	50	36	sub	sub	NOUN
cana-1932	50	37	-	-	NOUN
cana-1932	50	38	space	space	NOUN
cana-1932	50	39	of	of	ADP
cana-1932	50	40	fuzzy	fuzzy	ADJ
cana-1932	50	41	double	double	ADJ
cana-1932	50	42	s	s	NOUN
cana-1932	50	43	-	-	NOUN
cana-1932	50	44	t2space	t2space	NOUN
cana-1932	50	45	is	be	AUX
cana-1932	50	46	fuzzy	fuzzy	ADJ
cana-1932	50	47	double	double	ADJ
cana-1932	50	48	s	s	NOUN
cana-1932	50	49	-	-	NOUN
cana-1932	50	50	t2	t2	NOUN
cana-1932	50	51	.	.	PUNCT
cana-1932	51	1	proof	proof	NOUN
cana-1932	51	2	:	:	PUNCT
cana-1932	51	3	assume,(𝑋	assume,(𝑋	INTJ
cana-1932	51	4	,	,	PUNCT
cana-1932	51	5	τ)remain	τ)remain	PUNCT
cana-1932	51	6	a	a	DET
cana-1932	51	7	fuzzy	fuzzy	ADJ
cana-1932	51	8	double	double	ADJ
cana-1932	51	9	s	s	NOUN
cana-1932	51	10	-	-	PUNCT
cana-1932	51	11	t2space.assume	t2space.assume	PROPN
cana-1932	51	12	,	,	PUNCT
cana-1932	51	13	x1remain	x1remain	VERB
cana-1932	51	14	a	a	DET
cana-1932	51	15	sub	sub	NOUN
cana-1932	51	16	-	-	NOUN
cana-1932	51	17	space	space	NOUN
cana-1932	51	18	of	of	ADP
cana-1932	51	19	𝑋.	𝑋.	PROPN
cana-1932	51	20	to	to	ADP
cana-1932	51	21	-	-	PUNCT
cana-1932	51	22	show:(x1	show:(x1	PROPN
cana-1932	51	23	,	,	PUNCT
cana-1932	51	24	(	(	PUNCT
cana-1932	51	25	τ	τ	X
cana-1932	51	26	/x1	/x1	NUM
cana-1932	51	27	)	)	PUNCT
cana-1932	51	28	)	)	PUNCT
cana-1932	51	29	is	be	AUX
cana-1932	51	30	a	a	DET
cana-1932	51	31	fuzzy	fuzzy	ADJ
cana-1932	51	32	double	double	ADJ
cana-1932	51	33	s	s	NOUN
cana-1932	51	34	-	-	NOUN
cana-1932	51	35	t2space	t2space	NOUN
cana-1932	51	36	consider	consider	NOUN
cana-1932	51	37	,	,	PUNCT
cana-1932	51	38	y	y	PROPN
cana-1932	51	39	(	(	PUNCT
cana-1932	51	40	r	r	PROPN
cana-1932	51	41	,	,	PUNCT
cana-1932	51	42	s	s	PART
cana-1932	51	43	)	)	PUNCT
cana-1932	51	44	,	,	PUNCT
cana-1932	51	45	z(u	z(u	PROPN
cana-1932	51	46	,	,	PUNCT
cana-1932	51	47	v	v	NOUN
cana-1932	51	48	)	)	PUNCT
cana-1932	52	1	ϵ	ϵ	NOUN
cana-1932	52	2	x1	x1	NUM
cana-1932	52	3	such	such	ADJ
cana-1932	52	4	that	that	SCONJ
cana-1932	52	5	y	y	PROPN
cana-1932	52	6	(	(	PUNCT
cana-1932	52	7	r	r	PROPN
cana-1932	52	8	,	,	PUNCT
cana-1932	52	9	s	s	NOUN
cana-1932	52	10	)	)	PUNCT
cana-1932	52	11	≠	≠	PROPN
cana-1932	52	12	z(u	z(u	PROPN
cana-1932	52	13	,	,	PUNCT
cana-1932	52	14	v)then	v)then	ADJ
cana-1932	52	15	y	y	PROPN
cana-1932	52	16	(	(	PUNCT
cana-1932	52	17	r	r	NOUN
cana-1932	52	18	,	,	PUNCT
cana-1932	52	19	s	s	PART
cana-1932	52	20	)	)	PUNCT
cana-1932	52	21	,	,	PUNCT
cana-1932	52	22	z(u	z(u	PROPN
cana-1932	52	23	,	,	PUNCT
cana-1932	52	24	v	v	NOUN
cana-1932	52	25	)	)	PUNCT
cana-1932	52	26	∈	∈	PROPN
cana-1932	52	27	x	x	NOUN
cana-1932	52	28	,	,	PUNCT
cana-1932	52	29	there	there	PRON
cana-1932	52	30	exists	exist	VERB
cana-1932	52	31	two	two	NUM
cana-1932	52	32	fuzzy	fuzzy	ADJ
cana-1932	52	33	double	double	ADJ
cana-1932	52	34	open	open	ADJ
cana-1932	52	35	sets	set	NOUN
cana-1932	52	36	.	.	PUNCT
cana-1932	53	1	f	f	X
cana-1932	54	1	=	=	PRON
cana-1932	54	2	(	(	PUNCT
cana-1932	54	3	f1	f1	PROPN
cana-1932	54	4	,	,	PUNCT
cana-1932	54	5	f2	f2	PROPN
cana-1932	54	6	)	)	PUNCT
cana-1932	54	7	,	,	PUNCT
cana-1932	54	8	g	g	NOUN
cana-1932	54	9	=	=	PUNCT
cana-1932	54	10	(	(	PUNCT
cana-1932	54	11	g	g	PROPN
cana-1932	54	12	1	1	NUM
cana-1932	54	13	,	,	PUNCT
cana-1932	54	14	g	g	PROPN
cana-1932	54	15	2	2	NUM
cana-1932	54	16	)	)	PUNCT
cana-1932	54	17	∈	∈	PROPN
cana-1932	54	18	τ	τ	X
cana-1932	54	19	suchthat	suchthat	PROPN
cana-1932	54	20	f1	f1	PROPN
cana-1932	54	21	(	(	PUNCT
cana-1932	54	22	x	x	SYM
cana-1932	54	23	)	)	PUNCT
cana-1932	54	24	≥	≥	PROPN
cana-1932	54	25	r	r	NOUN
cana-1932	54	26	,	,	PUNCT
cana-1932	54	27	f2	f2	PROPN
cana-1932	54	28	(	(	PUNCT
cana-1932	54	29	x	x	SYM
cana-1932	54	30	)	)	PUNCT
cana-1932	54	31	≥	≥	NUM
cana-1932	54	32	s	s	PROPN
cana-1932	54	33	,	,	PUNCT
cana-1932	54	34	g	g	PROPN
cana-1932	54	35	1	1	NUM
cana-1932	54	36	(	(	PUNCT
cana-1932	54	37	y	y	NOUN
cana-1932	54	38	)	)	PUNCT
cana-1932	54	39	≥	≥	NOUN
cana-1932	54	40	u	u	NOUN
cana-1932	54	41	,	,	PUNCT
cana-1932	54	42	g	g	PROPN
cana-1932	54	43	2	2	NUM
cana-1932	54	44	(	(	PUNCT
cana-1932	54	45	y	y	NOUN
cana-1932	54	46	)	)	PUNCT
cana-1932	54	47	≥	≥	NOUN
cana-1932	54	48	v	v	ADP
cana-1932	54	49	also	also	ADV
cana-1932	54	50	f	f	PROPN
cana-1932	54	51	⋂	⋂	PROPN
cana-1932	54	52	g	g	PROPN
cana-1932	54	53	=	=	SYM
cana-1932	54	54	0	0	NUM
cana-1932	54	55	subsequently	subsequently	ADV
cana-1932	54	56	,	,	PUNCT
cana-1932	54	57	x1	x1	PROPN
cana-1932	54	58	is	be	AUX
cana-1932	54	59	a	a	DET
cana-1932	54	60	sub	sub	NOUN
cana-1932	54	61	-	-	NOUN
cana-1932	54	62	space	space	NOUN
cana-1932	54	63	of	of	ADP
cana-1932	54	64	𝑋,𝑓	𝑋,𝑓	X
cana-1932	54	65	/	/	SYM
cana-1932	54	66	x1	x1	PROPN
cana-1932	54	67	,	,	PUNCT
cana-1932	54	68	𝑔	𝑔	PROPN
cana-1932	54	69	/	/	SYM
cana-1932	54	70	x1	x1	PROPN
cana-1932	54	71	∈	∈	PROPN
cana-1932	54	72	𝜏	𝜏	X
cana-1932	54	73	/	/	SYM
cana-1932	54	74	x1	x1	PROPN
cana-1932	54	75	,	,	PUNCT
cana-1932	54	76	where	where	SCONJ
cana-1932	54	77	𝑓	𝑓	X
cana-1932	54	78	/	/	SYM
cana-1932	54	79	x1	x1	PROPN
cana-1932	54	80	=	=	SYM
cana-1932	54	81	(	(	PUNCT
cana-1932	54	82	𝑓1	𝑓1	PROPN
cana-1932	54	83	/	/	SYM
cana-1932	54	84	x1	x1	PROPN
cana-1932	54	85	,	,	PUNCT
cana-1932	54	86	𝑓2	𝑓2	PROPN
cana-1932	54	87	/	/	SYM
cana-1932	54	88	x1	x1	PROPN
cana-1932	54	89	)	)	PUNCT
cana-1932	54	90	,	,	PUNCT
cana-1932	54	91	𝑔	𝑔	PROPN
cana-1932	54	92	/	/	SYM
cana-1932	54	93	x1	x1	PROPN
cana-1932	54	94	=	=	PUNCT
cana-1932	54	95	(	(	PUNCT
cana-1932	54	96	𝑔1/	𝑔1/	X
cana-1932	54	97	x1	x1	PROPN
cana-1932	54	98	,	,	PUNCT
cana-1932	54	99	𝑔2/	𝑔2/	PROPN
cana-1932	54	100	x1	x1	PROPN
cana-1932	54	101	)	)	PUNCT
cana-1932	55	1	∴	∴	PROPN
cana-1932	55	2	(	(	PUNCT
cana-1932	55	3	f1	f1	PROPN
cana-1932	55	4	/	/	SYM
cana-1932	55	5	x	x	SYM
cana-1932	55	6	1	1	NUM
cana-1932	55	7	)	)	PUNCT
cana-1932	55	8	(	(	PUNCT
cana-1932	55	9	y	y	NOUN
cana-1932	55	10	)	)	PUNCT
cana-1932	55	11	=	=	SYM
cana-1932	55	12	f1	f1	NOUN
cana-1932	55	13	(	(	PUNCT
cana-1932	55	14	y	y	PROPN
cana-1932	55	15	)	)	PUNCT
cana-1932	55	16	≥	≥	PROPN
cana-1932	55	17	r	r	NOUN
cana-1932	55	18	(	(	PUNCT
cana-1932	55	19	f2	f2	PROPN
cana-1932	55	20	/	/	SYM
cana-1932	55	21	x	x	NOUN
cana-1932	55	22	1	1	NUM
cana-1932	55	23	)	)	PUNCT
cana-1932	55	24	(	(	PUNCT
cana-1932	55	25	y	y	PROPN
cana-1932	55	26	)	)	PUNCT
cana-1932	56	1	=	=	PUNCT
cana-1932	56	2	f	f	PROPN
cana-1932	56	3	2	2	NUM
cana-1932	56	4	(	(	PUNCT
cana-1932	56	5	y	y	PROPN
cana-1932	56	6	)	)	PUNCT
cana-1932	56	7	≥	≥	NOUN
cana-1932	56	8	s	s	X
cana-1932	56	9	(	(	PUNCT
cana-1932	56	10	g	g	PROPN
cana-1932	56	11	1	1	NUM
cana-1932	56	12	/	/	SYM
cana-1932	56	13	x1)(z	x1)(z	PROPN
cana-1932	56	14	)	)	PUNCT
cana-1932	57	1	=	=	SYM
cana-1932	57	2	g	g	NOUN
cana-1932	57	3	1	1	NUM
cana-1932	57	4	(	(	PUNCT
cana-1932	57	5	z	z	NOUN
cana-1932	57	6	)	)	PUNCT
cana-1932	57	7	≥	≥	NOUN
cana-1932	57	8	u	u	NOUN
cana-1932	57	9	(	(	PUNCT
cana-1932	57	10	g	g	PROPN
cana-1932	57	11	2	2	NUM
cana-1932	57	12	/	/	SYM
cana-1932	57	13	x1)(z	x1)(z	PROPN
cana-1932	57	14	)	)	PUNCT
cana-1932	58	1	=	=	SYM
cana-1932	58	2	g	g	ADP
cana-1932	58	3	2	2	NUM
cana-1932	58	4	(	(	PUNCT
cana-1932	58	5	z	z	NOUN
cana-1932	58	6	)	)	PUNCT
cana-1932	58	7	≥	≥	NOUN
cana-1932	58	8	v	v	NOUN
cana-1932	58	9	consider	consider	VERB
cana-1932	58	10	,	,	PUNCT
cana-1932	58	11	(	(	PUNCT
cana-1932	58	12	f	f	PROPN
cana-1932	58	13	/	/	SYM
cana-1932	58	14	x1	x1	PROPN
cana-1932	58	15	)	)	PUNCT
cana-1932	58	16	⋂	⋂	PROPN
cana-1932	58	17	(	(	PUNCT
cana-1932	58	18	g	g	PROPN
cana-1932	58	19	/	/	SYM
cana-1932	58	20	x1	x1	PROPN
cana-1932	58	21	)	)	PUNCT
cana-1932	59	1	=	=	PUNCT
cana-1932	60	1	(	(	PUNCT
cana-1932	60	2	(	(	PUNCT
cana-1932	60	3	f	f	NOUN
cana-1932	60	4	1	1	NUM
cana-1932	60	5	/	/	SYM
cana-1932	60	6	x1	x1	PROPN
cana-1932	60	7	)	)	PUNCT
cana-1932	60	8	∧	∧	PROPN
cana-1932	60	9	(	(	PUNCT
cana-1932	60	10	g	g	PROPN
cana-1932	60	11	1	1	NUM
cana-1932	60	12	/	/	SYM
cana-1932	60	13	x1	x1	PROPN
cana-1932	60	14	)	)	PUNCT
cana-1932	60	15	,	,	PUNCT
cana-1932	60	16	(	(	PUNCT
cana-1932	60	17	f	f	PROPN
cana-1932	60	18	2	2	NUM
cana-1932	60	19	/	/	SYM
cana-1932	60	20	x1	x1	PROPN
cana-1932	60	21	)	)	PUNCT
cana-1932	60	22	∧	∧	PROPN
cana-1932	60	23	(	(	PUNCT
cana-1932	60	24	g	g	PROPN
cana-1932	60	25	2	2	NUM
cana-1932	60	26	/	/	SYM
cana-1932	60	27	x1	x1	NUM
cana-1932	60	28	)	)	PUNCT
cana-1932	60	29	)	)	PUNCT
cana-1932	60	30	communications	communication	NOUN
cana-1932	60	31	on	on	ADP
cana-1932	60	32	applied	apply	VERB
cana-1932	60	33	nonlinear	nonlinear	ADJ
cana-1932	60	34	analysis	analysis	NOUN
cana-1932	60	35	issn	issn	NOUN
cana-1932	60	36	:	:	PUNCT
cana-1932	60	37	1074	1074	NUM
cana-1932	60	38	-	-	PUNCT
cana-1932	60	39	133x	133x	NUM
cana-1932	60	40	vol	vol	NOUN
cana-1932	60	41	32	32	NUM
cana-1932	60	42	no	no	NOUN
cana-1932	60	43	.	.	NOUN
cana-1932	60	44	3	3	NUM
cana-1932	60	45	(	(	PUNCT
cana-1932	60	46	2025	2025	NUM
cana-1932	60	47	)	)	PUNCT
cana-1932	60	48	156	156	NUM
cana-1932	60	49	https://internationalpubls.com	https://internationalpubls.com	X
cana-1932	60	50	(	(	PUNCT
cana-1932	60	51	(	(	PUNCT
cana-1932	60	52	f1/	f1/	NOUN
cana-1932	60	53	x	x	SYM
cana-1932	60	54	1	1	X
cana-1932	60	55	)	)	PUNCT
cana-1932	60	56	∧	∧	PROPN
cana-1932	60	57	(	(	PUNCT
cana-1932	60	58	g	g	PROPN
cana-1932	60	59	1	1	NUM
cana-1932	60	60	/	/	SYM
cana-1932	60	61	x	x	NOUN
cana-1932	60	62	1	1	NUM
cana-1932	60	63	)	)	PUNCT
cana-1932	60	64	)	)	PUNCT
cana-1932	61	1	(	(	PUNCT
cana-1932	61	2	y	y	NOUN
cana-1932	61	3	)	)	PUNCT
cana-1932	61	4	=	=	PRON
cana-1932	61	5	(	(	PUNCT
cana-1932	61	6	f1	f1	PROPN
cana-1932	61	7	/	/	SYM
cana-1932	61	8	x	x	SYM
cana-1932	61	9	1	1	NUM
cana-1932	61	10	)	)	PUNCT
cana-1932	61	11	(	(	PUNCT
cana-1932	61	12	y	y	X
cana-1932	61	13	)	)	PUNCT
cana-1932	61	14	∧	∧	PROPN
cana-1932	61	15	(	(	PUNCT
cana-1932	61	16	g	g	PROPN
cana-1932	61	17	1	1	NUM
cana-1932	61	18	/	/	SYM
cana-1932	61	19	x	x	NOUN
cana-1932	61	20	1	1	NUM
cana-1932	61	21	)	)	PUNCT
cana-1932	61	22	(	(	PUNCT
cana-1932	61	23	y	y	NOUN
cana-1932	61	24	)	)	PUNCT
cana-1932	61	25	,	,	PUNCT
cana-1932	61	26	for	for	ADP
cana-1932	61	27	ally	ally	NOUN
cana-1932	61	28	(	(	PUNCT
cana-1932	61	29	r	r	NOUN
cana-1932	61	30	,	,	PUNCT
cana-1932	61	31	s	s	PART
cana-1932	61	32	)	)	PUNCT
cana-1932	61	33	∈	∈	NOUN
cana-1932	61	34	x1	x1	NOUN
cana-1932	62	1	⊆	⊆	NUM
cana-1932	62	2	x	x	X
cana-1932	62	3	=	=	SYM
cana-1932	62	4	f1(y)∧g	f1(y)∧g	NOUN
cana-1932	62	5	1	1	NUM
cana-1932	62	6	(	(	PUNCT
cana-1932	62	7	y	y	PROPN
cana-1932	62	8	)	)	PUNCT
cana-1932	62	9	,	,	PUNCT
cana-1932	62	10	∀	∀	X
cana-1932	62	11	y	y	PROPN
cana-1932	62	12	(	(	PUNCT
cana-1932	62	13	r	r	NOUN
cana-1932	62	14	,	,	PUNCT
cana-1932	62	15	s	s	PART
cana-1932	62	16	)	)	PUNCT
cana-1932	62	17	∈	∈	NOUN
cana-1932	62	18	x1	x1	NOUN
cana-1932	63	1	⊆	⊆	NUM
cana-1932	63	2	x	x	SYM
cana-1932	63	3	=(	=(	PROPN
cana-1932	63	4	f1∧g	f1∧g	PROPN
cana-1932	63	5	1	1	NUM
cana-1932	63	6	)	)	PUNCT
cana-1932	63	7	(	(	PUNCT
cana-1932	63	8	y	y	PROPN
cana-1932	63	9	)	)	PUNCT
cana-1932	63	10	,	,	PUNCT
cana-1932	63	11	∀	∀	X
cana-1932	63	12	y	y	PROPN
cana-1932	63	13	(	(	PUNCT
cana-1932	63	14	r	r	NOUN
cana-1932	63	15	,	,	PUNCT
cana-1932	63	16	s	s	PART
cana-1932	63	17	)	)	PUNCT
cana-1932	63	18	∈	∈	NOUN
cana-1932	63	19	x1	x1	NOUN
cana-1932	63	20	⊆	⊆	NUM
cana-1932	63	21	𝑋	𝑋	PROPN
cana-1932	63	22	=	=	SYM
cana-1932	63	23	01	01	PROPN
cana-1932	63	24	(	(	PUNCT
cana-1932	63	25	y	y	PROPN
cana-1932	63	26	)	)	PUNCT
cana-1932	63	27	,	,	PUNCT
cana-1932	63	28	∀	∀	X
cana-1932	63	29	y	y	PROPN
cana-1932	63	30	(	(	PUNCT
cana-1932	63	31	r	r	NOUN
cana-1932	63	32	,	,	PUNCT
cana-1932	63	33	s	s	PART
cana-1932	63	34	)	)	PUNCT
cana-1932	63	35	∈	∈	NOUN
cana-1932	64	1	x1	x1	NOUN
cana-1932	65	1	⊆	⊆	NUM
cana-1932	65	2	x	x	SYM
cana-1932	65	3	(	(	PUNCT
cana-1932	65	4	f1	f1	PROPN
cana-1932	65	5	/	/	SYM
cana-1932	65	6	x1	x1	PROPN
cana-1932	65	7	)	)	PUNCT
cana-1932	65	8	∧	∧	PROPN
cana-1932	65	9	(	(	PUNCT
cana-1932	65	10	g	g	PROPN
cana-1932	65	11	1	1	NUM
cana-1932	65	12	/	/	SYM
cana-1932	65	13	x1	x1	PROPN
cana-1932	65	14	)	)	PUNCT
cana-1932	66	1	=	=	SYM
cana-1932	66	2	01	01	NUM
cana-1932	66	3	(	(	PUNCT
cana-1932	66	4	(	(	PUNCT
cana-1932	66	5	f	f	PROPN
cana-1932	66	6	2	2	NUM
cana-1932	66	7	/	/	SYM
cana-1932	66	8	x	x	NOUN
cana-1932	66	9	1	1	X
cana-1932	66	10	)	)	PUNCT
cana-1932	66	11	∧	∧	NOUN
cana-1932	66	12	(	(	PUNCT
cana-1932	66	13	g	g	PROPN
cana-1932	66	14	2	2	NUM
cana-1932	66	15	/	/	SYM
cana-1932	66	16	x	x	NOUN
cana-1932	66	17	1	1	NUM
cana-1932	66	18	)	)	PUNCT
cana-1932	66	19	)	)	PUNCT
cana-1932	66	20	(	(	PUNCT
cana-1932	66	21	y	y	NOUN
cana-1932	66	22	)	)	PUNCT
cana-1932	66	23	=	=	PUNCT
cana-1932	67	1	(	(	PUNCT
cana-1932	67	2	f	f	PROPN
cana-1932	67	3	2	2	NUM
cana-1932	67	4	/	/	SYM
cana-1932	67	5	x	x	NOUN
cana-1932	67	6	1	1	NUM
cana-1932	67	7	)	)	PUNCT
cana-1932	67	8	(	(	PUNCT
cana-1932	67	9	y	y	NOUN
cana-1932	67	10	)	)	PUNCT
cana-1932	67	11	∧	∧	PROPN
cana-1932	67	12	(	(	PUNCT
cana-1932	67	13	g	g	PROPN
cana-1932	67	14	2	2	NUM
cana-1932	67	15	/	/	SYM
cana-1932	67	16	x	x	NOUN
cana-1932	67	17	1	1	NUM
cana-1932	67	18	)	)	PUNCT
cana-1932	67	19	(	(	PUNCT
cana-1932	67	20	y	y	PROPN
cana-1932	67	21	)	)	PUNCT
cana-1932	67	22	,	,	PUNCT
cana-1932	67	23	for	for	ADP
cana-1932	67	24	ally	ally	NOUN
cana-1932	67	25	(	(	PUNCT
cana-1932	67	26	r	r	NOUN
cana-1932	67	27	,	,	PUNCT
cana-1932	67	28	s	s	PART
cana-1932	67	29	)	)	PUNCT
cana-1932	67	30	∈	∈	NOUN
cana-1932	67	31	x1	x1	NOUN
cana-1932	68	1	⊆	⊆	NUM
cana-1932	68	2	x	x	SYM
cana-1932	68	3	=	=	SYM
cana-1932	68	4	f2(y	f2(y	PROPN
cana-1932	68	5	)	)	PUNCT
cana-1932	68	6	g	g	NOUN
cana-1932	68	7	2	2	NUM
cana-1932	68	8	(	(	PUNCT
cana-1932	68	9	y	y	NOUN
cana-1932	68	10	)	)	PUNCT
cana-1932	68	11	,	,	PUNCT
cana-1932	68	12	for	for	ADP
cana-1932	68	13	ally	ally	NOUN
cana-1932	68	14	(	(	PUNCT
cana-1932	68	15	r	r	NOUN
cana-1932	68	16	,	,	PUNCT
cana-1932	68	17	s	s	PART
cana-1932	68	18	)	)	PUNCT
cana-1932	68	19	∈	∈	NOUN
cana-1932	68	20	x1	x1	NOUN
cana-1932	69	1	⊆	⊆	NUM
cana-1932	69	2	x	x	SYM
cana-1932	69	3	=	=	SYM
cana-1932	69	4	(	(	PUNCT
cana-1932	69	5	f2	f2	PROPN
cana-1932	69	6	∧	∧	PROPN
cana-1932	69	7	g	g	PROPN
cana-1932	69	8	2	2	NUM
cana-1932	69	9	)	)	PUNCT
cana-1932	69	10	(	(	PUNCT
cana-1932	69	11	y	y	NOUN
cana-1932	69	12	)	)	PUNCT
cana-1932	69	13	,	,	PUNCT
cana-1932	69	14	for	for	ADP
cana-1932	69	15	ally	ally	NOUN
cana-1932	69	16	(	(	PUNCT
cana-1932	69	17	r	r	NOUN
cana-1932	69	18	,	,	PUNCT
cana-1932	69	19	s	s	PART
cana-1932	69	20	)	)	PUNCT
cana-1932	69	21	∈	∈	NOUN
cana-1932	69	22	x1	x1	NOUN
cana-1932	70	1	⊆	⊆	NUM
cana-1932	70	2	x	x	SYM
cana-1932	70	3	=	=	PUNCT
cana-1932	70	4	02(y	02(y	NUM
cana-1932	70	5	)	)	PUNCT
cana-1932	70	6	,	,	PUNCT
cana-1932	70	7	for	for	ADP
cana-1932	70	8	ally	ally	NOUN
cana-1932	70	9	(	(	PUNCT
cana-1932	70	10	r	r	NOUN
cana-1932	70	11	,	,	PUNCT
cana-1932	70	12	s	s	PART
cana-1932	70	13	)	)	PUNCT
cana-1932	70	14	∈	∈	NOUN
cana-1932	70	15	x1	x1	NOUN
cana-1932	71	1	⊆	⊆	NUM
cana-1932	71	2	x	x	SYM
cana-1932	71	3	=(	=(	NOUN
cana-1932	71	4	f2	f2	PROPN
cana-1932	71	5	/	/	SYM
cana-1932	71	6	x1	x1	PROPN
cana-1932	71	7	)	)	PUNCT
cana-1932	71	8	∧	∧	PROPN
cana-1932	71	9	(	(	PUNCT
cana-1932	71	10	g	g	PROPN
cana-1932	71	11	2	2	NUM
cana-1932	71	12	/	/	SYM
cana-1932	71	13	x1	x1	PROPN
cana-1932	71	14	)	)	PUNCT
cana-1932	72	1	=	=	SYM
cana-1932	72	2	02	02	NUM
cana-1932	72	3	⇒	⇒	NOUN
cana-1932	72	4	(	(	PUNCT
cana-1932	72	5	f	f	PROPN
cana-1932	72	6	/	/	SYM
cana-1932	72	7	x1)⋂	x1)⋂	PROPN
cana-1932	72	8	(	(	PUNCT
cana-1932	72	9	g	g	PROPN
cana-1932	72	10	/	/	SYM
cana-1932	72	11	x1	x1	PROPN
cana-1932	72	12	)	)	PUNCT
cana-1932	72	13	=(	=(	PROPN
cana-1932	72	14	01	01	NUM
cana-1932	72	15	,	,	PUNCT
cana-1932	72	16	02)=	02)=	NOUN
cana-1932	72	17	0	0	NUM
cana-1932	72	18	∴sub	∴sub	NUM
cana-1932	72	19	-	-	NOUN
cana-1932	72	20	space	space	NOUN
cana-1932	72	21	of	of	ADP
cana-1932	72	22	fds	fds	NOUN
cana-1932	72	23	-	-	PUNCT
cana-1932	72	24	t2space	t2space	NOUN
cana-1932	72	25	is	be	AUX
cana-1932	72	26	afds	afds	ADJ
cana-1932	72	27	-	-	PUNCT
cana-1932	72	28	t2	t2	NOUN
cana-1932	72	29	.	.	PUNCT
cana-1932	73	1	definition	definition	NOUN
cana-1932	73	2	:	:	PUNCT
cana-1932	73	3	3.4	3.4	NUM
cana-1932	73	4	let	let	VERB
cana-1932	73	5	f	f	PROPN
cana-1932	73	6	=	=	SYM
cana-1932	73	7	(	(	PUNCT
cana-1932	73	8	f1	f1	PROPN
cana-1932	73	9	,	,	PUNCT
cana-1932	73	10	f2	f2	PROPN
cana-1932	73	11	)	)	PUNCT
cana-1932	73	12	and	and	CCONJ
cana-1932	73	13	g	g	NOUN
cana-1932	73	14	=	=	PUNCT
cana-1932	73	15	(	(	PUNCT
cana-1932	73	16	g	g	PROPN
cana-1932	73	17	1	1	NUM
cana-1932	73	18	,	,	PUNCT
cana-1932	73	19	g	g	PROPN
cana-1932	73	20	2	2	NUM
cana-1932	73	21	)	)	PUNCT
cana-1932	73	22	be	be	AUX
cana-1932	73	23	binary	binary	ADJ
cana-1932	73	24	fds	fds	NOUN
cana-1932	73	25	on	on	ADP
cana-1932	73	26	x	x	PROPN
cana-1932	73	27	&	&	CCONJ
cana-1932	73	28	y	y	PROPN
cana-1932	73	29	correspondingly	correspondingly	ADV
cana-1932	73	30	.	.	PUNCT
cana-1932	74	1	then	then	ADV
cana-1932	74	2	cartesian	cartesian	ADJ
cana-1932	74	3	product	product	NOUN
cana-1932	74	4	of	of	ADP
cana-1932	74	5	f	f	PROPN
cana-1932	74	6	and	and	CCONJ
cana-1932	74	7	g	g	PROPN
cana-1932	74	8	is	be	AUX
cana-1932	74	9	a	a	DET
cana-1932	74	10	fds	fds	NOUN
cana-1932	74	11	in	in	ADP
cana-1932	74	12	x	x	PROPN
cana-1932	74	13	xy	xy	PROPN
cana-1932	74	14	definite	definite	ADJ
cana-1932	74	15	as	as	SCONJ
cana-1932	74	16	(	(	PUNCT
cana-1932	74	17	f	f	PROPN
cana-1932	74	18	∗	∗	PROPN
cana-1932	74	19	g)and	g)and	PROPN
cana-1932	74	20	is	be	AUX
cana-1932	74	21	definite	definite	ADJ
cana-1932	74	22	as	as	ADP
cana-1932	74	23	f	f	PROPN
cana-1932	74	24	∗	∗	NOUN
cana-1932	74	25	g	g	NOUN
cana-1932	74	26	=	=	PUNCT
cana-1932	74	27	(	(	PUNCT
cana-1932	74	28	f1	f1	PROPN
cana-1932	74	29	∗	∗	NOUN
cana-1932	74	30	g	g	PROPN
cana-1932	74	31	1	1	NUM
cana-1932	74	32	,	,	PUNCT
cana-1932	74	33	f2	f2	PROPN
cana-1932	74	34	∗	∗	VERB
cana-1932	74	35	g	g	PROPN
cana-1932	74	36	2	2	NUM
cana-1932	74	37	)	)	PUNCT
cana-1932	75	1	where	where	SCONJ
cana-1932	75	2	,	,	PUNCT
cana-1932	75	3	(	(	PUNCT
cana-1932	75	4	f	f	NOUN
cana-1932	75	5	1	1	NUM
cana-1932	75	6	∗	∗	NOUN
cana-1932	75	7	g	g	NOUN
cana-1932	75	8	1	1	NUM
cana-1932	75	9	)	)	PUNCT
cana-1932	75	10	(	(	PUNCT
cana-1932	75	11	x	x	X
cana-1932	75	12	,	,	PUNCT
cana-1932	75	13	y	y	NOUN
cana-1932	75	14	)	)	PUNCT
cana-1932	75	15	=	=	SYM
cana-1932	75	16	min	min	NOUN
cana-1932	75	17	{	{	PUNCT
cana-1932	75	18	f1	f1	NOUN
cana-1932	75	19	(	(	PUNCT
cana-1932	75	20	x	x	PROPN
cana-1932	75	21	)	)	PUNCT
cana-1932	75	22	,	,	PUNCT
cana-1932	75	23	g	g	PROPN
cana-1932	75	24	1	1	NUM
cana-1932	75	25	(	(	PUNCT
cana-1932	75	26	y	y	PROPN
cana-1932	75	27	)	)	PUNCT
cana-1932	75	28	}	}	PUNCT
cana-1932	75	29	and	and	CCONJ
cana-1932	75	30	(	(	PUNCT
cana-1932	75	31	f	f	PROPN
cana-1932	75	32	2	2	NUM
cana-1932	75	33	∗	∗	NOUN
cana-1932	75	34	g	g	NOUN
cana-1932	75	35	2	2	NUM
cana-1932	75	36	)	)	PUNCT
cana-1932	75	37	(	(	PUNCT
cana-1932	75	38	x	x	X
cana-1932	75	39	,	,	PUNCT
cana-1932	75	40	y	y	NOUN
cana-1932	75	41	)	)	PUNCT
cana-1932	75	42	=	=	SYM
cana-1932	75	43	min	min	X
cana-1932	75	44	{	{	PUNCT
cana-1932	75	45	f2	f2	PROPN
cana-1932	75	46	(	(	PUNCT
cana-1932	75	47	x	x	PROPN
cana-1932	75	48	)	)	PUNCT
cana-1932	75	49	,	,	PUNCT
cana-1932	75	50	g	g	PROPN
cana-1932	75	51	2	2	NUM
cana-1932	75	52	(	(	PUNCT
cana-1932	75	53	y	y	PROPN
cana-1932	75	54	)	)	PUNCT
cana-1932	75	55	}	}	PUNCT
cana-1932	75	56	,	,	PUNCT
cana-1932	75	57	for	for	ADP
cana-1932	75	58	every	every	DET
cana-1932	75	59	(	(	PUNCT
cana-1932	75	60	x	x	NOUN
cana-1932	75	61	,	,	PUNCT
cana-1932	75	62	y	y	NOUN
cana-1932	75	63	)	)	PUNCT
cana-1932	75	64	∈	∈	PROPN
cana-1932	75	65	x	x	SYM
cana-1932	75	66	×	×	NOUN
cana-1932	75	67	y	y	PROPN
cana-1932	75	68	definition	definition	NOUN
cana-1932	75	69	:	:	PUNCT
cana-1932	75	70	3.5	3.5	NUM
cana-1932	75	71	assume,(𝑋	assume,(𝑋	PROPN
cana-1932	75	72	,	,	PUNCT
cana-1932	75	73	δ1	δ1	NOUN
cana-1932	75	74	)	)	PUNCT
cana-1932	75	75	and	and	CCONJ
cana-1932	75	76	(	(	PUNCT
cana-1932	75	77	𝑌	𝑌	PROPN
cana-1932	75	78	,	,	PUNCT
cana-1932	75	79	δ2)remainbinary	δ2)remainbinary	ADJ
cana-1932	75	80	fdts	fdts	NOUN
cana-1932	75	81	.	.	PUNCT
cana-1932	76	1	at	at	ADP
cana-1932	76	2	that	that	DET
cana-1932	76	3	point	point	NOUN
cana-1932	76	4	product	product	NOUN
cana-1932	76	5	fdtδ1	fdtδ1	NOUN
cana-1932	76	6	×	×	NOUN
cana-1932	76	7	δ2	δ2	VERB
cana-1932	76	8	on	on	ADP
cana-1932	76	9	ix	ix	PROPN
cana-1932	76	10	×	×	PROPN
cana-1932	76	11	iyremains	iyremain	NOUN
cana-1932	76	12	fdt	fdt	NOUN
cana-1932	76	13	taking	take	VERB
cana-1932	76	14	the	the	DET
cana-1932	76	15	group	group	NOUN
cana-1932	76	16	{	{	PUNCT
cana-1932	76	17	(	(	PUNCT
cana-1932	76	18	f	f	PROPN
cana-1932	76	19	∗	∗	X
cana-1932	76	20	g)|	g)|	NOUN
cana-1932	76	21	f	f	PROPN
cana-1932	76	22	=	=	PRON
cana-1932	76	23	(	(	PUNCT
cana-1932	76	24	f1	f1	PROPN
cana-1932	76	25	,	,	PUNCT
cana-1932	76	26	f2	f2	PROPN
cana-1932	76	27	)	)	PUNCT
cana-1932	76	28	∈	∈	PROPN
cana-1932	76	29	δ1	δ1	NOUN
cana-1932	76	30	,	,	PUNCT
cana-1932	76	31	g	g	NOUN
cana-1932	76	32	=	=	PUNCT
cana-1932	76	33	(	(	PUNCT
cana-1932	76	34	g	g	PROPN
cana-1932	76	35	1	1	NUM
cana-1932	76	36	,	,	PUNCT
cana-1932	76	37	g	g	PROPN
cana-1932	76	38	2	2	NUM
cana-1932	76	39	)	)	PUNCT
cana-1932	76	40	∈	∈	PROPN
cana-1932	76	41	δ2	δ2	VERB
cana-1932	76	42	}	}	PUNCT
cana-1932	76	43	as	as	ADP
cana-1932	76	44	a	a	DET
cana-1932	76	45	basis	basis	NOUN
cana-1932	76	46	.	.	PUNCT
cana-1932	77	1	proposition	proposition	NOUN
cana-1932	77	2	:	:	PUNCT
cana-1932	77	3	3.6	3.6	NUM
cana-1932	77	4	product	product	NOUN
cana-1932	77	5	of	of	ADP
cana-1932	77	6	binary	binary	PROPN
cana-1932	77	7	fds	fds	PROPN
cana-1932	77	8	-	-	PUNCT
cana-1932	77	9	t2is	t2is	PUNCT
cana-1932	77	10	a	a	DET
cana-1932	77	11	fds	fds	NOUN
cana-1932	77	12	-	-	PUNCT
cana-1932	77	13	t2	t2	NOUN
cana-1932	77	14	.	.	PUNCT
cana-1932	78	1	proof	proof	NOUN
cana-1932	78	2	:	:	PUNCT
cana-1932	78	3	assume,(𝑋	assume,(𝑋	ADJ
cana-1932	78	4	,	,	PUNCT
cana-1932	78	5	τ1	τ1	PROPN
cana-1932	78	6	)	)	PUNCT
cana-1932	78	7	,	,	PUNCT
cana-1932	78	8	(	(	PUNCT
cana-1932	78	9	𝑌	𝑌	PROPN
cana-1932	78	10	,	,	PUNCT
cana-1932	78	11	τ2)remainbinary	τ2)remainbinary	ADJ
cana-1932	78	12	fds	fds	NOUN
cana-1932	78	13	-	-	PUNCT
cana-1932	78	14	t2	t2	NOUN
cana-1932	78	15	.	.	PUNCT
cana-1932	79	1	to	to	PART
cana-1932	79	2	show	show	VERB
cana-1932	79	3	:	:	PUNCT
cana-1932	79	4	(	(	PUNCT
cana-1932	79	5	𝑋	𝑋	PROPN
cana-1932	79	6	×	×	PROPN
cana-1932	79	7	𝑌	𝑌	PROPN
cana-1932	79	8	,	,	PUNCT
cana-1932	79	9	τ1	τ1	ADP
cana-1932	79	10	×	×	PROPN
cana-1932	79	11	τ2)remainsfds	τ2)remainsfds	ADJ
cana-1932	79	12	-	-	PUNCT
cana-1932	79	13	t2	t2	NOUN
cana-1932	79	14	consider	consider	VERB
cana-1932	79	15	binary	binary	ADJ
cana-1932	79	16	different	different	ADJ
cana-1932	79	17	fuzzy	fuzzy	ADJ
cana-1932	79	18	double	double	ADJ
cana-1932	79	19	-	-	PUNCT
cana-1932	79	20	points	point	NOUN
cana-1932	79	21	y	y	PROPN
cana-1932	79	22	(	(	PUNCT
cana-1932	79	23	r	r	NOUN
cana-1932	79	24	,	,	PUNCT
cana-1932	79	25	s	s	PART
cana-1932	79	26	)	)	PUNCT
cana-1932	79	27	,	,	PUNCT
cana-1932	79	28	z(u	z(u	PROPN
cana-1932	79	29	,	,	PUNCT
cana-1932	79	30	v	v	NOUN
cana-1932	79	31	)	)	PUNCT
cana-1932	79	32	∈	∈	PROPN
cana-1932	79	33	x	x	SYM
cana-1932	79	34	×	×	PROPN
cana-1932	79	35	y	y	PROPN
cana-1932	79	36	,	,	PUNCT
cana-1932	79	37	where	where	SCONJ
cana-1932	79	38	y	y	PROPN
cana-1932	79	39	=	=	PRON
cana-1932	79	40	(	(	PUNCT
cana-1932	79	41	y	y	PROPN
cana-1932	79	42	1	1	NUM
cana-1932	79	43	,	,	PUNCT
cana-1932	79	44	y	y	PROPN
cana-1932	79	45	2	2	NUM
cana-1932	79	46	)	)	PUNCT
cana-1932	79	47	and	and	CCONJ
cana-1932	79	48	z	z	NOUN
cana-1932	79	49	=	=	SYM
cana-1932	79	50	(	(	PUNCT
cana-1932	79	51	z1	z1	PROPN
cana-1932	79	52	,	,	PUNCT
cana-1932	79	53	z2	z2	PROPN
cana-1932	79	54	)	)	PUNCT
cana-1932	79	55	.	.	PUNCT
cana-1932	80	1	either	either	CCONJ
cana-1932	80	2	y	y	PROPN
cana-1932	80	3	1	1	NUM
cana-1932	80	4	≠	≠	PROPN
cana-1932	80	5	z1	z1	NOUN
cana-1932	80	6	or	or	CCONJ
cana-1932	80	7	y	y	PROPN
cana-1932	80	8	2	2	NUM
cana-1932	80	9	≠	≠	PROPN
cana-1932	80	10	z2	z2	PROPN
cana-1932	80	11	.	.	PUNCT
cana-1932	81	1	assume	assume	VERB
cana-1932	81	2	y	y	PROPN
cana-1932	81	3	1	1	NUM
cana-1932	81	4	≠	≠	PROPN
cana-1932	81	5	z1	z1	NOUN
cana-1932	81	6	,	,	PUNCT
cana-1932	81	7	so	so	CCONJ
cana-1932	81	8	there	there	PRON
cana-1932	81	9	exists	exist	VERB
cana-1932	81	10	two	two	NUM
cana-1932	81	11	fuzzy	fuzzy	ADJ
cana-1932	81	12	double	double	ADJ
cana-1932	81	13	open	open	ADJ
cana-1932	81	14	sets	set	VERB
cana-1932	81	15	communications	communication	NOUN
cana-1932	81	16	on	on	ADP
cana-1932	81	17	applied	apply	VERB
cana-1932	81	18	nonlinear	nonlinear	ADJ
cana-1932	81	19	analysis	analysis	NOUN
cana-1932	81	20	issn	issn	NOUN
cana-1932	81	21	:	:	PUNCT
cana-1932	81	22	1074	1074	NUM
cana-1932	81	23	-	-	PUNCT
cana-1932	81	24	133x	133x	NUM
cana-1932	81	25	vol	vol	NOUN
cana-1932	81	26	32	32	NUM
cana-1932	81	27	no	no	NOUN
cana-1932	81	28	.	.	NOUN
cana-1932	81	29	3	3	NUM
cana-1932	81	30	(	(	PUNCT
cana-1932	81	31	2025	2025	NUM
cana-1932	81	32	)	)	PUNCT
cana-1932	82	1	157	157	NUM
cana-1932	82	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-1932	82	3	f	f	X
cana-1932	83	1	=	=	PUNCT
cana-1932	84	1	(	(	PUNCT
cana-1932	84	2	f1	f1	PROPN
cana-1932	84	3	,	,	PUNCT
cana-1932	84	4	f2	f2	PROPN
cana-1932	84	5	)	)	PUNCT
cana-1932	84	6	,	,	PUNCT
cana-1932	84	7	g	g	NOUN
cana-1932	84	8	=	=	PUNCT
cana-1932	84	9	(	(	PUNCT
cana-1932	84	10	g	g	PROPN
cana-1932	84	11	1	1	NUM
cana-1932	84	12	,	,	PUNCT
cana-1932	84	13	g	g	PROPN
cana-1932	84	14	2	2	NUM
cana-1932	84	15	)	)	PUNCT
cana-1932	84	16	suchthat	suchthat	VERB
cana-1932	84	17	f1(y	f1(y	PROPN
cana-1932	84	18	1	1	NUM
cana-1932	84	19	)	)	PUNCT
cana-1932	84	20	≥	≥	NOUN
cana-1932	84	21	r	r	NOUN
cana-1932	84	22	,	,	PUNCT
cana-1932	84	23	f2(y	f2(y	PROPN
cana-1932	84	24	1	1	NUM
cana-1932	84	25	)	)	PUNCT
cana-1932	84	26	≥	≥	NUM
cana-1932	84	27	s	s	PROPN
cana-1932	84	28	,	,	PUNCT
cana-1932	84	29	g	g	PROPN
cana-1932	84	30	1	1	NUM
cana-1932	84	31	(	(	PUNCT
cana-1932	84	32	z1	z1	PROPN
cana-1932	84	33	)	)	PUNCT
cana-1932	84	34	≥	≥	NUM
cana-1932	84	35	u	u	NOUN
cana-1932	84	36	,	,	PUNCT
cana-1932	84	37	g	g	PROPN
cana-1932	84	38	2	2	NUM
cana-1932	84	39	(	(	PUNCT
cana-1932	84	40	z1	z1	PROPN
cana-1932	84	41	)	)	PUNCT
cana-1932	84	42	≥	≥	NOUN
cana-1932	84	43	v	v	NOUN
cana-1932	84	44	&	&	CCONJ
cana-1932	84	45	f	f	PROPN
cana-1932	84	46	⋂	⋂	PROPN
cana-1932	84	47	g	g	PROPN
cana-1932	84	48	=	=	PROPN
cana-1932	84	49	0	0	NUM
cana-1932	84	50	,	,	PUNCT
cana-1932	84	51	everywhere	everywhere	ADV
cana-1932	84	52	0	0	NUM
cana-1932	84	53	is	be	AUX
cana-1932	84	54	a	a	DET
cana-1932	84	55	fd	fd	PROPN
cana-1932	84	56	void	void	NOUN
cana-1932	84	57	set	set	VERB
cana-1932	84	58	in	in	ADP
cana-1932	84	59	x.	x.	PROPN
cana-1932	84	60	f	f	PROPN
cana-1932	84	61	∗	∗	NOUN
cana-1932	84	62	1	1	NUM
cana-1932	84	63	∈	∈	NOUN
cana-1932	84	64	τ1	τ1	NOUN
cana-1932	84	65	×	×	PROPN
cana-1932	84	66	τ2	τ2	NOUN
cana-1932	84	67	,	,	PUNCT
cana-1932	84	68	subsequentlyf	subsequentlyf	PROPN
cana-1932	84	69	∈	∈	PROPN
cana-1932	84	70	τ1	τ1	NOUN
cana-1932	84	71	,	,	PUNCT
cana-1932	84	72	1	1	NUM
cana-1932	84	73	∈	∈	NOUN
cana-1932	84	74	τ2	τ2	NOUN
cana-1932	84	75	and	and	CCONJ
cana-1932	84	76	g	g	NOUN
cana-1932	84	77	∗	∗	NOUN
cana-1932	84	78	1	1	NUM
cana-1932	84	79	∈	∈	NOUN
cana-1932	84	80	τ1	τ1	NOUN
cana-1932	84	81	×	×	PROPN
cana-1932	84	82	τ2	τ2	NOUN
cana-1932	84	83	,	,	PUNCT
cana-1932	84	84	since	since	SCONJ
cana-1932	84	85	g	g	PROPN
cana-1932	84	86	∈	∈	PROPN
cana-1932	84	87	τ1	τ1	NOUN
cana-1932	84	88	,	,	PUNCT
cana-1932	84	89	1	1	NUM
cana-1932	84	90	∈	∈	NOUN
cana-1932	84	91	τ2	τ2	NOUN
cana-1932	84	92	,	,	PUNCT
cana-1932	84	93	where	where	SCONJ
cana-1932	84	94	g	g	PROPN
cana-1932	84	95	∗	∗	X
cana-1932	84	96	1	1	NUM
cana-1932	84	97	=	=	SYM
cana-1932	84	98	(	(	PUNCT
cana-1932	84	99	f1	f1	PROPN
cana-1932	84	100	∗	∗	NOUN
cana-1932	84	101	11	11	NUM
cana-1932	84	102	,	,	PUNCT
cana-1932	84	103	f2	f2	PROPN
cana-1932	84	104	∗	∗	NOUN
cana-1932	84	105	12	12	NUM
cana-1932	84	106	)	)	PUNCT
cana-1932	84	107	andg	andg	NOUN
cana-1932	84	108	∗	∗	NOUN
cana-1932	84	109	1	1	NUM
cana-1932	84	110	=	=	SYM
cana-1932	84	111	(	(	PUNCT
cana-1932	84	112	g	g	PROPN
cana-1932	84	113	1	1	NUM
cana-1932	84	114	∗	∗	NOUN
cana-1932	84	115	11	11	NUM
cana-1932	84	116	,	,	PUNCT
cana-1932	84	117	g	g	PROPN
cana-1932	84	118	2	2	NUM
cana-1932	84	119	∗	∗	NOUN
cana-1932	84	120	12	12	NUM
cana-1932	84	121	)	)	PUNCT
cana-1932	84	122	consider	consider	VERB
cana-1932	84	123	(	(	PUNCT
cana-1932	84	124	f1	f1	NOUN
cana-1932	84	125	∗	∗	NOUN
cana-1932	84	126	11)(y	11)(y	X
cana-1932	84	127	1	1	NUM
cana-1932	84	128	,	,	PUNCT
cana-1932	84	129	z1	z1	NOUN
cana-1932	84	130	)	)	PUNCT
cana-1932	85	1	=	=	SYM
cana-1932	85	2	min	min	NOUN
cana-1932	85	3	{	{	PUNCT
cana-1932	85	4	f1(y	f1(y	PROPN
cana-1932	85	5	1	1	NUM
cana-1932	85	6	)	)	PUNCT
cana-1932	85	7	,	,	PUNCT
cana-1932	85	8	11(z1	11(z1	NUM
cana-1932	85	9	)	)	PUNCT
cana-1932	85	10	}	}	PUNCT
cana-1932	85	11	≥	≥	PROPN
cana-1932	85	12	r	r	NOUN
cana-1932	85	13	(	(	PUNCT
cana-1932	85	14	f2	f2	PROPN
cana-1932	85	15	∗	∗	VERB
cana-1932	85	16	12)(y	12)(y	NUM
cana-1932	85	17	1	1	NUM
cana-1932	85	18	,	,	PUNCT
cana-1932	85	19	z1	z1	NOUN
cana-1932	85	20	)	)	PUNCT
cana-1932	85	21	=	=	SYM
cana-1932	85	22	min	min	PROPN
cana-1932	85	23	{	{	PUNCT
cana-1932	85	24	f2(y	f2(y	PROPN
cana-1932	85	25	1	1	NUM
cana-1932	85	26	)	)	PUNCT
cana-1932	85	27	,	,	PUNCT
cana-1932	85	28	12(z1	12(z1	NOUN
cana-1932	85	29	)	)	PUNCT
cana-1932	85	30	}	}	PUNCT
cana-1932	85	31	≥	≥	X
cana-1932	85	32	s	s	X
cana-1932	85	33	(	(	PUNCT
cana-1932	85	34	g	g	PROPN
cana-1932	85	35	1	1	NUM
cana-1932	85	36	∗	∗	NOUN
cana-1932	85	37	11)(y	11)(y	X
cana-1932	85	38	2	2	NUM
cana-1932	85	39	,	,	PUNCT
cana-1932	85	40	z2	z2	NUM
cana-1932	85	41	)	)	PUNCT
cana-1932	85	42	=	=	SYM
cana-1932	85	43	min	min	NOUN
cana-1932	85	44	{	{	PUNCT
cana-1932	85	45	g	g	PROPN
cana-1932	85	46	1	1	NUM
cana-1932	85	47	(	(	PUNCT
cana-1932	85	48	y	y	PROPN
cana-1932	85	49	2	2	NUM
cana-1932	85	50	)	)	PUNCT
cana-1932	85	51	,	,	PUNCT
cana-1932	85	52	11(z2	11(z2	NUM
cana-1932	85	53	)	)	PUNCT
cana-1932	85	54	}	}	PUNCT
cana-1932	85	55	≥	≥	PROPN
cana-1932	85	56	u	u	NOUN
cana-1932	85	57	(	(	PUNCT
cana-1932	85	58	g	g	PROPN
cana-1932	85	59	2	2	NUM
cana-1932	85	60	∗	∗	NOUN
cana-1932	85	61	12)(y	12)(y	NUM
cana-1932	85	62	2	2	NUM
cana-1932	85	63	,	,	PUNCT
cana-1932	85	64	z2	z2	NUM
cana-1932	85	65	)	)	PUNCT
cana-1932	85	66	=	=	SYM
cana-1932	85	67	min	min	NOUN
cana-1932	85	68	{	{	PUNCT
cana-1932	85	69	g	g	PROPN
cana-1932	85	70	2	2	NUM
cana-1932	85	71	(	(	PUNCT
cana-1932	85	72	y	y	PROPN
cana-1932	85	73	2	2	NUM
cana-1932	85	74	)	)	PUNCT
cana-1932	85	75	,	,	PUNCT
cana-1932	85	76	12(z2	12(z2	NUM
cana-1932	85	77	)	)	PUNCT
cana-1932	85	78	}	}	PUNCT
cana-1932	85	79	≥	≥	NOUN
cana-1932	85	80	v	v	ADP
cana-1932	85	81	also	also	ADV
cana-1932	85	82	f	f	PROPN
cana-1932	85	83	∩	∩	NOUN
cana-1932	85	84	g	g	PROPN
cana-1932	85	85	=	=	SYM
cana-1932	85	86	0	0	NUM
cana-1932	85	87	⇒	⇒	NOUN
cana-1932	85	88	(	(	PUNCT
cana-1932	85	89	f1	f1	PROPN
cana-1932	85	90	∧	∧	PROPN
cana-1932	85	91	g	g	PROPN
cana-1932	85	92	1	1	NUM
cana-1932	85	93	,	,	PUNCT
cana-1932	85	94	f2	f2	PROPN
cana-1932	85	95	∧	∧	PROPN
cana-1932	85	96	g	g	PROPN
cana-1932	85	97	2	2	NUM
cana-1932	85	98	)	)	PUNCT
cana-1932	85	99	=	=	SYM
cana-1932	85	100	0	0	NUM
cana-1932	85	101	⇒	⇒	NOUN
cana-1932	85	102	(	(	PUNCT
cana-1932	85	103	f1	f1	PROPN
cana-1932	85	104	∧	∧	PROPN
cana-1932	85	105	g	g	PROPN
cana-1932	85	106	1	1	NUM
cana-1932	85	107	)	)	PUNCT
cana-1932	85	108	(	(	PUNCT
cana-1932	85	109	x	x	X
cana-1932	85	110	)	)	PUNCT
cana-1932	85	111	=	=	SYM
cana-1932	85	112	01(x	01(x	NOUN
cana-1932	85	113	)	)	PUNCT
cana-1932	85	114	and	and	CCONJ
cana-1932	85	115	(	(	PUNCT
cana-1932	85	116	f2	f2	PROPN
cana-1932	85	117	∧	∧	PROPN
cana-1932	85	118	g	g	PROPN
cana-1932	85	119	2	2	NUM
cana-1932	85	120	)	)	PUNCT
cana-1932	85	121	(	(	PUNCT
cana-1932	85	122	x	x	X
cana-1932	85	123	)	)	PUNCT
cana-1932	86	1	=	=	SYM
cana-1932	86	2	02	02	NUM
cana-1932	86	3	(	(	PUNCT
cana-1932	86	4	x	x	NOUN
cana-1932	86	5	)	)	PUNCT
cana-1932	86	6	,	,	PUNCT
cana-1932	86	7	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1932	86	8	𝑒𝑣𝑒𝑟𝑦	𝑒𝑣𝑒𝑟𝑦	VERB
cana-1932	86	9	x	x	PUNCT
cana-1932	86	10	∈	∈	NOUN
cana-1932	86	11	x	x	X
cana-1932	86	12	⇒	⇒	NOUN
cana-1932	86	13	f1(x	f1(x	PROPN
cana-1932	86	14	)	)	PUNCT
cana-1932	86	15	∧	∧	NOUN
cana-1932	86	16	g	g	NOUN
cana-1932	86	17	1	1	NUM
cana-1932	86	18	(	(	PUNCT
cana-1932	86	19	x	x	NOUN
cana-1932	86	20	)	)	PUNCT
cana-1932	87	1	=	=	SYM
cana-1932	87	2	01(x	01(x	NUM
cana-1932	87	3	)	)	PUNCT
cana-1932	87	4	,	,	PUNCT
cana-1932	87	5	f2(x	f2(x	PROPN
cana-1932	87	6	)	)	PUNCT
cana-1932	87	7	∧	∧	PROPN
cana-1932	87	8	g	g	NOUN
cana-1932	87	9	2	2	NUM
cana-1932	87	10	(	(	PUNCT
cana-1932	87	11	x	x	NOUN
cana-1932	87	12	)	)	PUNCT
cana-1932	87	13	=	=	SYM
cana-1932	87	14	02(x	02(x	PROPN
cana-1932	87	15	)	)	PUNCT
cana-1932	87	16	,	,	PUNCT
cana-1932	87	17	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1932	87	18	𝑒𝑣𝑒𝑟𝑦	𝑒𝑣𝑒𝑟𝑦	VERB
cana-1932	87	19	x	x	SYM
cana-1932	87	20	∈	∈	PROPN
cana-1932	87	21	x	x	PUNCT
cana-1932	87	22	⇒also	⇒also	ADV
cana-1932	87	23	,	,	PUNCT
cana-1932	87	24	f	f	PROPN
cana-1932	87	25	1(x	1(x	NUM
cana-1932	87	26	)	)	PUNCT
cana-1932	87	27	=	=	SYM
cana-1932	87	28	01(x	01(x	NOUN
cana-1932	87	29	)	)	PUNCT
cana-1932	87	30	or	or	CCONJ
cana-1932	87	31	g	g	PROPN
cana-1932	87	32	1	1	NUM
cana-1932	87	33	(	(	PUNCT
cana-1932	87	34	x	x	NOUN
cana-1932	87	35	)	)	PUNCT
cana-1932	87	36	=	=	SYM
cana-1932	87	37	01(x	01(x	NOUN
cana-1932	87	38	)	)	PUNCT
cana-1932	87	39	&	&	CCONJ
cana-1932	87	40	either	either	PRON
cana-1932	87	41	f2(x	f2(x	PROPN
cana-1932	87	42	)	)	PUNCT
cana-1932	87	43	=	=	SYM
cana-1932	87	44	02(x	02(x	NOUN
cana-1932	87	45	)	)	PUNCT
cana-1932	87	46	or	or	CCONJ
cana-1932	87	47	g	g	PROPN
cana-1932	87	48	2	2	NUM
cana-1932	87	49	(	(	PUNCT
cana-1932	87	50	x	x	NOUN
cana-1932	87	51	)	)	PUNCT
cana-1932	87	52	=	=	SYM
cana-1932	87	53	02(x	02(x	PROPN
cana-1932	87	54	)	)	PUNCT
cana-1932	87	55	,	,	PUNCT
cana-1932	87	56	∀	∀	PUNCT
cana-1932	87	57	x	x	SYM
cana-1932	87	58	∈	∈	PROPN
cana-1932	87	59	x	x	VERB
cana-1932	87	60	⇒used	⇒use	VERB
cana-1932	87	61	for	for	ADP
cana-1932	87	62	a	a	DET
cana-1932	87	63	fd	fd	X
cana-1932	87	64	universal	universal	NOUN
cana-1932	87	65	set	set	VERB
cana-1932	87	66	1	1	NUM
cana-1932	87	67	in	in	ADP
cana-1932	87	68	𝑌	𝑌	PROPN
cana-1932	87	69	moreover	moreover	ADV
cana-1932	87	70	f1	f1	NOUN
cana-1932	87	71	(	(	PUNCT
cana-1932	87	72	x	x	SYM
cana-1932	87	73	)	)	PUNCT
cana-1932	87	74	∧	∧	NOUN
cana-1932	87	75	11	11	NUM
cana-1932	87	76	(	(	PUNCT
cana-1932	87	77	y	y	NOUN
cana-1932	87	78	)	)	PUNCT
cana-1932	87	79	=	=	SYM
cana-1932	87	80	0	0	NUM
cana-1932	87	81	or	or	CCONJ
cana-1932	87	82	g	g	PROPN
cana-1932	87	83	1	1	NUM
cana-1932	87	84	(	(	PUNCT
cana-1932	87	85	x	x	SYM
cana-1932	87	86	)	)	PUNCT
cana-1932	87	87	∧	∧	PROPN
cana-1932	87	88	11(y	11(y	NUM
cana-1932	87	89	)	)	PUNCT
cana-1932	88	1	=	=	SYM
cana-1932	88	2	0	0	NUM
cana-1932	88	3	,	,	PUNCT
cana-1932	88	4	also	also	ADV
cana-1932	88	5	,	,	PUNCT
cana-1932	88	6	f2(x	f2(x	PROPN
cana-1932	88	7	)	)	PUNCT
cana-1932	88	8	∧	∧	PROPN
cana-1932	88	9	12(y	12(y	NUM
cana-1932	88	10	)	)	PUNCT
cana-1932	89	1	=	=	SYM
cana-1932	89	2	0	0	NUM
cana-1932	89	3	or	or	CCONJ
cana-1932	89	4	g	g	PROPN
cana-1932	89	5	2	2	NUM
cana-1932	89	6	(	(	PUNCT
cana-1932	89	7	x	x	NOUN
cana-1932	89	8	)	)	PUNCT
cana-1932	89	9	∧	∧	NOUN
cana-1932	89	10	12	12	NUM
cana-1932	89	11	(	(	PUNCT
cana-1932	89	12	y	y	NOUN
cana-1932	89	13	)	)	PUNCT
cana-1932	89	14	=	=	SYM
cana-1932	89	15	0	0	NUM
cana-1932	89	16	,	,	PUNCT
cana-1932	89	17	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1932	89	18	𝑒𝑣𝑒𝑟𝑦	𝑒𝑣𝑒𝑟𝑦	VERB
cana-1932	89	19	x	x	PUNCT
cana-1932	89	20	∈	∈	PROPN
cana-1932	89	21	x	x	X
cana-1932	89	22	&	&	CCONJ
cana-1932	89	23	y	y	PROPN
cana-1932	89	24	∈	∈	PROPN
cana-1932	89	25	y	y	PROPN
cana-1932	89	26	⇒moreover	⇒moreover	PROPN
cana-1932	89	27	,	,	PUNCT
cana-1932	89	28	(	(	PUNCT
cana-1932	89	29	f1	f1	NOUN
cana-1932	89	30	∗	∗	NOUN
cana-1932	89	31	11)(x	11)(x	NUM
cana-1932	89	32	,	,	PUNCT
cana-1932	89	33	y	y	PROPN
cana-1932	89	34	)	)	PUNCT
cana-1932	90	1	=	=	SYM
cana-1932	90	2	0	0	NUM
cana-1932	91	1	or	or	CCONJ
cana-1932	91	2	(	(	PUNCT
cana-1932	91	3	g	g	PROPN
cana-1932	91	4	1	1	NUM
cana-1932	91	5	∗	∗	NOUN
cana-1932	91	6	11	11	NUM
cana-1932	91	7	)	)	PUNCT
cana-1932	91	8	(	(	PUNCT
cana-1932	91	9	x	x	X
cana-1932	91	10	,	,	PUNCT
cana-1932	91	11	y	y	PROPN
cana-1932	91	12	)	)	PUNCT
cana-1932	91	13	=	=	SYM
cana-1932	91	14	0	0	PROPN
cana-1932	91	15	&	&	CCONJ
cana-1932	91	16	also	also	ADV
cana-1932	91	17	,	,	PUNCT
cana-1932	91	18	(	(	PUNCT
cana-1932	91	19	f2	f2	PROPN
cana-1932	91	20	∗	∗	NOUN
cana-1932	91	21	12	12	NUM
cana-1932	91	22	)	)	PUNCT
cana-1932	91	23	(	(	PUNCT
cana-1932	91	24	x	x	X
cana-1932	91	25	,	,	PUNCT
cana-1932	91	26	y	y	PROPN
cana-1932	91	27	)	)	PUNCT
cana-1932	92	1	=	=	SYM
cana-1932	92	2	0	0	NUM
cana-1932	93	1	or	or	CCONJ
cana-1932	93	2	(	(	PUNCT
cana-1932	93	3	g	g	PROPN
cana-1932	93	4	2	2	NUM
cana-1932	93	5	∗	∗	NOUN
cana-1932	93	6	12	12	NUM
cana-1932	93	7	)	)	PUNCT
cana-1932	93	8	(	(	PUNCT
cana-1932	93	9	x	x	X
cana-1932	93	10	,	,	PUNCT
cana-1932	93	11	y	y	PROPN
cana-1932	93	12	)	)	PUNCT
cana-1932	93	13	=	=	SYM
cana-1932	93	14	0	0	NUM
cana-1932	93	15	,	,	PUNCT
cana-1932	93	16	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1932	93	17	𝑒𝑣𝑒𝑟𝑦	𝑒𝑣𝑒𝑟𝑦	VERB
cana-1932	93	18	x	x	PUNCT
cana-1932	93	19	∈	∈	PROPN
cana-1932	93	20	x	x	X
cana-1932	93	21	and	and	CCONJ
cana-1932	93	22	y	y	PROPN
cana-1932	93	23	∈	∈	PROPN
cana-1932	93	24	y	y	PROPN
cana-1932	93	25	⇒	⇒	PROPN
cana-1932	93	26	(	(	PUNCT
cana-1932	93	27	f1	f1	PROPN
cana-1932	93	28	∗	∗	NOUN
cana-1932	93	29	11	11	NUM
cana-1932	93	30	)	)	PUNCT
cana-1932	94	1	∧	∧	NOUN
cana-1932	94	2	(	(	PUNCT
cana-1932	94	3	g	g	PROPN
cana-1932	94	4	1	1	NUM
cana-1932	94	5	∗	∗	NOUN
cana-1932	94	6	11	11	NUM
cana-1932	94	7	)	)	PUNCT
cana-1932	94	8	(	(	PUNCT
cana-1932	94	9	x	x	X
cana-1932	94	10	,	,	PUNCT
cana-1932	94	11	y	y	PROPN
cana-1932	94	12	)	)	PUNCT
cana-1932	95	1	=	=	SYM
cana-1932	95	2	0and	0and	PROPN
cana-1932	95	3	(	(	PUNCT
cana-1932	95	4	f2	f2	PROPN
cana-1932	95	5	∗	∗	NOUN
cana-1932	95	6	12	12	NUM
cana-1932	95	7	)	)	PUNCT
cana-1932	95	8	∧	∧	NOUN
cana-1932	95	9	(	(	PUNCT
cana-1932	95	10	g	g	PROPN
cana-1932	95	11	2	2	NUM
cana-1932	95	12	∗	∗	NOUN
cana-1932	95	13	12	12	NUM
cana-1932	95	14	)	)	PUNCT
cana-1932	95	15	(	(	PUNCT
cana-1932	95	16	x	x	X
cana-1932	95	17	,	,	PUNCT
cana-1932	95	18	y	y	NOUN
cana-1932	95	19	)	)	PUNCT
cana-1932	95	20	=	=	SYM
cana-1932	95	21	0	0	NUM
cana-1932	95	22	,	,	PUNCT
cana-1932	95	23	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-1932	95	24	𝑒𝑣𝑒𝑟𝑦	𝑒𝑣𝑒𝑟𝑦	NOUN
cana-1932	95	25	(	(	PUNCT
cana-1932	95	26	x	x	NOUN
cana-1932	95	27	,	,	PUNCT
cana-1932	95	28	y	y	NOUN
cana-1932	95	29	)	)	PUNCT
cana-1932	95	30	∈	∈	PROPN
cana-1932	95	31	x	x	SYM
cana-1932	95	32	×	×	PROPN
cana-1932	95	33	y	y	PROPN
cana-1932	95	34	⇒	⇒	NOUN
cana-1932	95	35	(	(	PUNCT
cana-1932	95	36	f	f	PROPN
cana-1932	95	37	∗	∗	PROPN
cana-1932	95	38	1	1	NUM
cana-1932	95	39	)	)	PUNCT
cana-1932	95	40	∩	∩	NOUN
cana-1932	95	41	(	(	PUNCT
cana-1932	95	42	g	g	PROPN
cana-1932	95	43	∗	∗	X
cana-1932	95	44	1	1	NUM
cana-1932	95	45	)	)	PUNCT
cana-1932	95	46	=	=	SYM
cana-1932	95	47	0	0	NUM
cana-1932	95	48	,	,	PUNCT
cana-1932	95	49	where	where	SCONJ
cana-1932	95	50	0	0	NUM
cana-1932	95	51	∈	∈	NOUN
cana-1932	95	52	x	x	X
cana-1932	95	53	×	×	PROPN
cana-1932	95	54	y.	y.	NOUN
cana-1932	95	55	∴product	∴product	PROPN
cana-1932	95	56	of	of	ADP
cana-1932	95	57	binary	binary	PROPN
cana-1932	95	58	fds	fds	PROPN
cana-1932	95	59	-	-	PUNCT
cana-1932	95	60	t2	t2	NOUN
cana-1932	95	61	is	be	AUX
cana-1932	95	62	a	a	DET
cana-1932	95	63	fds	fds	NOUN
cana-1932	95	64	-	-	PUNCT
cana-1932	95	65	t2	t2	NOUN
cana-1932	95	66	.	.	PUNCT
cana-1932	96	1	definition	definition	NOUN
cana-1932	96	2	:	:	PUNCT
cana-1932	96	3	3.7	3.7	NUM
cana-1932	96	4	assume,{(xλ	assume,{(xλ	NUM
cana-1932	96	5	,	,	PUNCT
cana-1932	96	6	δλ	δλ	NOUN
cana-1932	96	7	)	)	PUNCT
cana-1932	96	8	|	|	ADV
cana-1932	97	1	λ	λ	X
cana-1932	97	2	∈	∈	PROPN
cana-1932	97	3	λ}remain	λ}remain	PROPN
cana-1932	97	4	a	a	DET
cana-1932	97	5	family	family	NOUN
cana-1932	97	6	of	of	ADP
cana-1932	97	7	fdts	fdts	ADJ
cana-1932	97	8	and	and	CCONJ
cana-1932	97	9	x	x	SYM
cana-1932	97	10	=	=	SYM
cana-1932	97	11	∏	∏	PROPN
cana-1932	97	12	λ	λ	X
cana-1932	97	13	∈	∈	NOUN
cana-1932	97	14	λ	λ	NOUN
cana-1932	97	15	xλ	xλ	NOUN
cana-1932	97	16	communications	communication	NOUN
cana-1932	97	17	on	on	ADP
cana-1932	97	18	applied	apply	VERB
cana-1932	97	19	nonlinear	nonlinear	ADJ
cana-1932	97	20	analysis	analysis	NOUN
cana-1932	97	21	issn	issn	NOUN
cana-1932	97	22	:	:	PUNCT
cana-1932	97	23	1074	1074	NUM
cana-1932	97	24	-	-	PUNCT
cana-1932	97	25	133x	133x	NUM
cana-1932	97	26	vol	vol	NOUN
cana-1932	97	27	32	32	NUM
cana-1932	97	28	no	no	NOUN
cana-1932	97	29	.	.	NOUN
cana-1932	97	30	3	3	NUM
cana-1932	97	31	(	(	PUNCT
cana-1932	97	32	2025	2025	NUM
cana-1932	97	33	)	)	PUNCT
cana-1932	97	34	158	158	NUM
cana-1932	97	35	https://internationalpubls.com	https://internationalpubls.com	X
cana-1932	97	36	let{fλ	let{fλ	NOUN
cana-1932	97	37	=	=	SYM
cana-1932	97	38	(	(	PUNCT
cana-1932	97	39	(	(	PUNCT
cana-1932	97	40	f1	f1	NOUN
cana-1932	97	41	)	)	PUNCT
cana-1932	97	42	λ	λ	PROPN
cana-1932	97	43	,	,	PUNCT
cana-1932	97	44	(	(	PUNCT
cana-1932	97	45	f2	f2	PROPN
cana-1932	97	46	)	)	PUNCT
cana-1932	97	47	λ	λ	NOUN
cana-1932	97	48	)	)	PUNCT
cana-1932	97	49	)	)	PUNCT
cana-1932	98	1	|	|	ADV
cana-1932	98	2	λ	λ	X
cana-1932	98	3	∈	∈	NOUN
cana-1932	98	4	λ	λ	X
cana-1932	98	5	}	}	PUNCT
cana-1932	98	6	and	and	CCONJ
cana-1932	98	7	fλ	fλ	PROPN
cana-1932	98	8	is	be	AUX
cana-1932	98	9	afds	afds	ADJ
cana-1932	98	10	in	in	ADP
cana-1932	98	11	xλ	xλ	PROPN
cana-1932	98	12	.	.	PUNCT
cana-1932	99	1	at	at	ADP
cana-1932	99	2	that	that	DET
cana-1932	99	3	point	point	NOUN
cana-1932	99	4	their	their	PRON
cana-1932	99	5	product	product	NOUN
cana-1932	99	6	∏	∏	NUM
cana-1932	99	7	λ	λ	X
cana-1932	99	8	∈	∈	PROPN
cana-1932	99	9	λ	λ	X
cana-1932	99	10	fλ	fλ	NOUN
cana-1932	99	11	is	be	AUX
cana-1932	99	12	a	a	DET
cana-1932	99	13	fds	fds	NOUN
cana-1932	99	14	in	in	ADP
cana-1932	99	15	∏	∏	PROPN
cana-1932	99	16	λ	λ	PROPN
cana-1932	99	17	∈	∈	PROPN
cana-1932	99	18	λ	λ	X
cana-1932	99	19	fλwell	fλwell	ADV
cana-1932	99	20	-	-	PUNCT
cana-1932	99	21	defined	define	VERB
cana-1932	99	22	as	as	ADP
cana-1932	99	23	∏	∏	PROPN
cana-1932	99	24	λ	λ	PROPN
cana-1932	99	25	∈	∈	PROPN
cana-1932	99	26	λ	λ	NOUN
cana-1932	99	27	fλ	fλ	X
cana-1932	99	28	=	=	SYM
cana-1932	99	29	(	(	PUNCT
cana-1932	99	30	∏	∏	PROPN
cana-1932	99	31	λ	λ	PROPN
cana-1932	99	32	∈	∈	PROPN
cana-1932	99	33	λ	λ	PROPN
cana-1932	99	34	(	(	PUNCT
cana-1932	99	35	f1)λ	f1)λ	PROPN
cana-1932	99	36	,	,	PUNCT
cana-1932	99	37	∏	∏	PROPN
cana-1932	99	38	λ	λ	PROPN
cana-1932	99	39	∈	∈	PROPN
cana-1932	99	40	λ	λ	PROPN
cana-1932	99	41	(	(	PUNCT
cana-1932	99	42	f2)λ	f2)λ	NOUN
cana-1932	99	43	)	)	PUNCT
cana-1932	100	1	where	where	SCONJ
cana-1932	100	2	∏	∏	PROPN
cana-1932	100	3	λ	λ	X
cana-1932	100	4	∈	∈	PROPN
cana-1932	100	5	λ	λ	PROPN
cana-1932	100	6	(	(	PUNCT
cana-1932	100	7	f1	f1	NOUN
cana-1932	100	8	)	)	PUNCT
cana-1932	100	9	λ(x	λ(x	PROPN
cana-1932	100	10	)	)	PUNCT
cana-1932	100	11	=	=	SYM
cana-1932	100	12	min{(f1)λ(xλ	min{(f1)λ(xλ	NOUN
cana-1932	100	13	)	)	PUNCT
cana-1932	100	14	}	}	PUNCT
cana-1932	100	15	,	,	PUNCT
cana-1932	100	16	∀	∀	PUNCT
cana-1932	100	17	x	x	X
cana-1932	100	18	∈	∈	NOUN
cana-1932	100	19	∏	∏	NUM
cana-1932	100	20	xλ	xλ	PROPN
cana-1932	100	21	∈	∈	PROPN
cana-1932	100	22	λ	λ	PROPN
cana-1932	100	23	and	and	CCONJ
cana-1932	100	24	∏	∏	NUM
cana-1932	100	25	λ	λ	X
cana-1932	100	26	∈	∈	PROPN
cana-1932	100	27	λ	λ	X
cana-1932	100	28	(	(	PUNCT
cana-1932	100	29	f2)λ(x	f2)λ(x	PROPN
cana-1932	100	30	)	)	PUNCT
cana-1932	100	31	=	=	SYM
cana-1932	100	32	min{(f2)λ(xλ	min{(f2)λ(xλ	NOUN
cana-1932	100	33	)	)	PUNCT
cana-1932	100	34	}	}	PUNCT
cana-1932	100	35	,	,	PUNCT
cana-1932	100	36	∀x	∀x	VERB
cana-1932	100	37	∈	∈	PROPN
cana-1932	100	38	∏	∏	NUM
cana-1932	100	39	xλ	xλ	PROPN
cana-1932	100	40	∈	∈	PROPN
cana-1932	100	41	λ	λ	X
cana-1932	100	42	the	the	DET
cana-1932	100	43	product	product	NOUN
cana-1932	100	44	topology	topology	NOUN
cana-1932	100	45	on	on	ADP
cana-1932	100	46	x	x	PROPN
cana-1932	100	47	is	be	AUX
cana-1932	100	48	of	of	ADP
cana-1932	100	49	the	the	DET
cana-1932	100	50	form	form	NOUN
cana-1932	100	51	∏	∏	PROPN
cana-1932	100	52	λ	λ	PROPN
cana-1932	100	53	∈	∈	NOUN
cana-1932	100	54	λ	λ	NOUN
cana-1932	100	55	fλis	fλi	VERB
cana-1932	100	56	a	a	DET
cana-1932	100	57	basis	basis	NOUN
cana-1932	100	58	of	of	ADP
cana-1932	100	59	fdos	fdo	NOUN
cana-1932	100	60	,	,	PUNCT
cana-1932	100	61	wherever	wherever	SCONJ
cana-1932	100	62	fλ	fλ	PRON
cana-1932	100	63	∈	∈	PROPN
cana-1932	100	64	δλand	δλand	NOUN
cana-1932	100	65	fλ	fλ	NOUN
cana-1932	100	66	=	=	SYM
cana-1932	100	67	1	1	NUM
cana-1932	100	68	excludingmeant	excludingmeant	NOUN
cana-1932	100	69	for	for	ADP
cana-1932	100	70	finitely	finitely	ADJ
cana-1932	100	71	severalλ′s	severalλ′s	NOUN
cana-1932	100	72	proposition	proposition	NOUN
cana-1932	100	73	:	:	PUNCT
cana-1932	100	74	3.8	3.8	NUM
cana-1932	100	75	arbitrary	arbitrary	ADJ
cana-1932	100	76	product	product	NOUN
cana-1932	100	77	of	of	ADP
cana-1932	100	78	binary	binary	PROPN
cana-1932	100	79	fds	fds	PROPN
cana-1932	100	80	-	-	PUNCT
cana-1932	100	81	t2	t2	NOUN
cana-1932	100	82	is	be	AUX
cana-1932	100	83	a	a	DET
cana-1932	100	84	fds	fds	NOUN
cana-1932	100	85	-	-	PUNCT
cana-1932	100	86	t2	t2	NOUN
cana-1932	100	87	.	.	PUNCT
cana-1932	101	1	proof	proof	NOUN
cana-1932	101	2	:	:	PUNCT
cana-1932	101	3	assume,{(xλ	assume,{(xλ	NUM
cana-1932	101	4	,	,	PUNCT
cana-1932	101	5	(	(	PUNCT
cana-1932	101	6	τ	τ	X
cana-1932	101	7	)	)	PUNCT
cana-1932	101	8	λ	λ	NOUN
cana-1932	101	9	)	)	PUNCT
cana-1932	102	1	|	|	ADV
cana-1932	102	2	λ	λ	X
cana-1932	102	3	ϵ	ϵ	X
cana-1932	102	4	λ}remain	λ}remain	PROPN
cana-1932	102	5	a	a	DET
cana-1932	102	6	collection	collection	NOUN
cana-1932	102	7	of	of	ADP
cana-1932	102	8	fds	fds	NOUN
cana-1932	102	9	-	-	PUNCT
cana-1932	102	10	t2	t2	NOUN
cana-1932	102	11	.	.	PUNCT
cana-1932	103	1	assume	assume	VERB
cana-1932	103	2	,	,	PUNCT
cana-1932	103	3	x	x	PUNCT
cana-1932	103	4	=	=	SYM
cana-1932	103	5	∏	∏	NUM
cana-1932	103	6	xλ	xλ	NOUN
cana-1932	103	7	λ	λ	X
cana-1932	103	8	ϵ	ϵ	X
cana-1932	103	9	λ	λ	PROPN
cana-1932	103	10	andτ	andτ	NOUN
cana-1932	103	11	=	=	SYM
cana-1932	103	12	∏	∏	PROPN
cana-1932	103	13	(	(	PUNCT
cana-1932	103	14	τ	τ	X
cana-1932	103	15	)	)	PUNCT
cana-1932	104	1	λ	λ	X
cana-1932	104	2	λ	λ	X
cana-1932	104	3	ϵ	ϵ	X
cana-1932	104	4	λ	λ	X
cana-1932	104	5	.	.	PUNCT
cana-1932	105	1	consider	consider	VERB
cana-1932	105	2	two	two	NUM
cana-1932	105	3	distinct	distinct	ADJ
cana-1932	105	4	fuzzy	fuzzy	ADJ
cana-1932	105	5	double	double	ADJ
cana-1932	105	6	points(xλ)(r	points(xλ)(r	NOUN
cana-1932	105	7	,	,	PUNCT
cana-1932	105	8	s	s	PART
cana-1932	105	9	)	)	PUNCT
cana-1932	105	10	,	,	PUNCT
cana-1932	105	11	(	(	PUNCT
cana-1932	105	12	y	y	PROPN
cana-1932	105	13	λ	λ	PROPN
cana-1932	105	14	)	)	PUNCT
cana-1932	105	15	(	(	PUNCT
cana-1932	105	16	u	u	NOUN
cana-1932	105	17	,	,	PUNCT
cana-1932	105	18	v	v	NOUN
cana-1932	105	19	)	)	PUNCT
cana-1932	105	20	∈∏	∈∏	NOUN
cana-1932	105	21	xλλ	xλλ	X
cana-1932	106	1	ϵ	ϵ	X
cana-1932	106	2	λ	λ	PROPN
cana-1932	106	3	,	,	PUNCT
cana-1932	106	4	∀λ	∀λ	X
cana-1932	106	5	ϵ	ϵ	X
cana-1932	106	6	λ	λ	PROPN
cana-1932	106	7	.	.	PUNCT
cana-1932	107	1	therefore	therefore	ADV
cana-1932	107	2	(	(	PUNCT
cana-1932	107	3	xμ	xμ	NOUN
cana-1932	107	4	)	)	PUNCT
cana-1932	107	5	(	(	PUNCT
cana-1932	107	6	r	r	NOUN
cana-1932	107	7	,	,	PUNCT
cana-1932	107	8	s	s	NOUN
cana-1932	107	9	)	)	PUNCT
cana-1932	107	10	≠	≠	PROPN
cana-1932	107	11	(	(	PUNCT
cana-1932	107	12	y	y	PROPN
cana-1932	107	13	μ	μ	PROPN
cana-1932	107	14	)	)	PUNCT
cana-1932	107	15	(	(	PUNCT
cana-1932	107	16	u	u	NOUN
cana-1932	107	17	,	,	PUNCT
cana-1932	107	18	v	v	NOUN
cana-1932	107	19	)	)	PUNCT
cana-1932	107	20	for	for	ADP
cana-1932	107	21	some	some	DET
cana-1932	107	22	μ	μ	NUM
cana-1932	107	23	∈	∈	PROPN
cana-1932	107	24	λ	λ	PROPN
cana-1932	107	25	.	.	PUNCT
cana-1932	107	26	consequently	consequently	ADV
cana-1932	107	27	,	,	PUNCT
cana-1932	107	28	there	there	PRON
cana-1932	107	29	exists	exist	VERB
cana-1932	107	30	two	two	NUM
cana-1932	107	31	fuzzy	fuzzy	ADJ
cana-1932	107	32	double	double	ADJ
cana-1932	107	33	open	open	ADJ
cana-1932	107	34	sets	set	NOUN
cana-1932	107	35	,	,	PUNCT
cana-1932	107	36	(	(	PUNCT
cana-1932	107	37	f	f	X
cana-1932	107	38	)	)	PUNCT
cana-1932	107	39	μ	μ	PROPN
cana-1932	107	40	=	=	SYM
cana-1932	107	41	(	(	PUNCT
cana-1932	107	42	(	(	PUNCT
cana-1932	107	43	f1	f1	NOUN
cana-1932	107	44	)	)	PUNCT
cana-1932	107	45	μ	μ	PROPN
cana-1932	107	46	,	,	PUNCT
cana-1932	107	47	(	(	PUNCT
cana-1932	107	48	f2	f2	PROPN
cana-1932	107	49	)	)	PUNCT
cana-1932	107	50	μ	μ	PROPN
cana-1932	107	51	)	)	PUNCT
cana-1932	107	52	and	and	CCONJ
cana-1932	107	53	(	(	PUNCT
cana-1932	107	54	g	g	NOUN
cana-1932	107	55	)	)	PUNCT
cana-1932	107	56	μ	μ	NOUN
cana-1932	107	57	=	=	SYM
cana-1932	107	58	(	(	PUNCT
cana-1932	107	59	(	(	PUNCT
cana-1932	107	60	g	g	PROPN
cana-1932	107	61	1	1	NUM
cana-1932	107	62	)	)	PUNCT
cana-1932	107	63	μ	μ	NOUN
cana-1932	107	64	,	,	PUNCT
cana-1932	107	65	(	(	PUNCT
cana-1932	107	66	g	g	PROPN
cana-1932	107	67	2	2	NUM
cana-1932	107	68	)	)	PUNCT
cana-1932	107	69	μ	μ	PROPN
cana-1932	107	70	)	)	PUNCT
cana-1932	107	71	∈	∈	PROPN
cana-1932	107	72	(	(	PUNCT
cana-1932	107	73	τ	τ	PROPN
cana-1932	107	74	)	)	PUNCT
cana-1932	107	75	μsuch	μsuch	NOUN
cana-1932	107	76	that	that	SCONJ
cana-1932	107	77	(	(	PUNCT
cana-1932	107	78	𝑓1	𝑓1	PROPN
cana-1932	107	79	)	)	PUNCT
cana-1932	107	80	μ(xμ	μ(xμ	NUM
cana-1932	107	81	)	)	PUNCT
cana-1932	107	82	≥	≥	NOUN
cana-1932	107	83	r	r	NOUN
cana-1932	107	84	,	,	PUNCT
cana-1932	107	85	(	(	PUNCT
cana-1932	107	86	𝑓2)μ(xμ	𝑓2)μ(xμ	ADJ
cana-1932	107	87	)	)	PUNCT
cana-1932	107	88	≥	≥	NOUN
cana-1932	107	89	s	s	NOUN
cana-1932	107	90	,	,	PUNCT
cana-1932	107	91	(	(	PUNCT
cana-1932	107	92	𝑔1)μ(xμ	𝑔1)μ(xμ	VERB
cana-1932	107	93	)	)	PUNCT
cana-1932	107	94	≥	≥	NOUN
cana-1932	107	95	u	u	NOUN
cana-1932	107	96	,	,	PUNCT
cana-1932	107	97	(	(	PUNCT
cana-1932	107	98	𝑔2)μ(xμ	𝑔2)μ(xμ	PROPN
cana-1932	107	99	)	)	PUNCT
cana-1932	107	100	≥	≥	PROPN
cana-1932	107	101	vand	vand	NOUN
cana-1932	107	102	(	(	PUNCT
cana-1932	107	103	f	f	X
cana-1932	107	104	)	)	PUNCT
cana-1932	107	105	μ	μ	NOUN
cana-1932	107	106	∩	∩	NOUN
cana-1932	107	107	(	(	PUNCT
cana-1932	107	108	𝑔	𝑔	NOUN
cana-1932	107	109	)	)	PUNCT
cana-1932	107	110	μ	μ	NOUN
cana-1932	107	111	=	=	SYM
cana-1932	107	112	(	(	PUNCT
cana-1932	107	113	0	0	X
cana-1932	107	114	)	)	PUNCT
cana-1932	107	115	μ	μ	NOUN
cana-1932	107	116	let	let	VERB
cana-1932	107	117	f	f	PROPN
cana-1932	107	118	=	=	SYM
cana-1932	107	119	∏	∏	PROPN
cana-1932	107	120	(	(	PUNCT
cana-1932	107	121	f	f	X
cana-1932	107	122	)	)	PUNCT
cana-1932	107	123	λ	λ	X
cana-1932	107	124	λ	λ	X
cana-1932	107	125	ϵ	ϵ	X
cana-1932	107	126	λ	λ	X
cana-1932	107	127	,	,	PUNCT
cana-1932	107	128	where	where	SCONJ
cana-1932	107	129	(	(	PUNCT
cana-1932	107	130	f	f	X
cana-1932	107	131	)	)	PUNCT
cana-1932	107	132	λ	λ	X
cana-1932	107	133	=	=	SYM
cana-1932	107	134	(	(	PUNCT
cana-1932	107	135	1	1	X
cana-1932	107	136	)	)	PUNCT
cana-1932	107	137	λ	λ	NOUN
cana-1932	107	138	for	for	ADP
cana-1932	107	139	𝜆	𝜆	DET
cana-1932	107	140	≠	≠	PROPN
cana-1932	107	141	μ	μ	NOUN
cana-1932	107	142	and	and	CCONJ
cana-1932	107	143	g	g	PROPN
cana-1932	107	144	=	=	SYM
cana-1932	107	145	∏	∏	PROPN
cana-1932	107	146	(	(	PUNCT
cana-1932	107	147	g	g	NOUN
cana-1932	107	148	)	)	PUNCT
cana-1932	107	149	λ	λ	NOUN
cana-1932	107	150	λ	λ	X
cana-1932	107	151	ϵ	ϵ	X
cana-1932	107	152	λ	λ	X
cana-1932	107	153	,	,	PUNCT
cana-1932	107	154	where	where	SCONJ
cana-1932	107	155	(	(	PUNCT
cana-1932	107	156	g	g	NOUN
cana-1932	107	157	)	)	PUNCT
cana-1932	107	158	λ	λ	PROPN
cana-1932	107	159	=	=	SYM
cana-1932	107	160	(	(	PUNCT
cana-1932	107	161	1	1	X
cana-1932	107	162	)	)	PUNCT
cana-1932	107	163	λ	λ	NOUN
cana-1932	107	164	for	for	ADP
cana-1932	107	165	𝜆	𝜆	DET
cana-1932	107	166	≠	≠	PROPN
cana-1932	107	167	μ	μ	NUM
cana-1932	107	168	.	.	PUNCT
cana-1932	108	1	then	then	ADV
cana-1932	108	2	f	f	X
cana-1932	108	3	,	,	PUNCT
cana-1932	108	4	g	g	PROPN
cana-1932	108	5	∈	∈	PROPN
cana-1932	108	6	∏	∏	PROPN
cana-1932	108	7	(	(	PUNCT
cana-1932	108	8	τ	τ	X
cana-1932	108	9	)	)	PUNCT
cana-1932	109	1	λ	λ	X
cana-1932	109	2	λ	λ	X
cana-1932	109	3	ϵ	ϵ	X
cana-1932	109	4	λ	λ	X
cana-1932	109	5	f	f	PROPN
cana-1932	109	6	=	=	SYM
cana-1932	109	7	∏	∏	PROPN
cana-1932	109	8	(	(	PUNCT
cana-1932	109	9	f	f	NOUN
cana-1932	109	10	)	)	PUNCT
cana-1932	110	1	λ	λ	X
cana-1932	110	2	λ	λ	X
cana-1932	110	3	ϵ	ϵ	X
cana-1932	110	4	λ	λ	X
cana-1932	110	5	=	=	SYM
cana-1932	110	6	(	(	PUNCT
cana-1932	110	7	∏	∏	PROPN
cana-1932	110	8	(	(	PUNCT
cana-1932	110	9	f1	f1	NOUN
cana-1932	110	10	)	)	PUNCT
cana-1932	110	11	λ	λ	NOUN
cana-1932	110	12	λ	λ	X
cana-1932	110	13	ϵ	ϵ	X
cana-1932	110	14	λ	λ	PROPN
cana-1932	110	15	,	,	PUNCT
cana-1932	110	16	∏	∏	PROPN
cana-1932	110	17	(	(	PUNCT
cana-1932	110	18	f2	f2	PROPN
cana-1932	110	19	)	)	PUNCT
cana-1932	110	20	λ	λ	PROPN
cana-1932	110	21	λ	λ	X
cana-1932	110	22	ϵ	ϵ	X
cana-1932	110	23	λ	λ	PROPN
cana-1932	110	24	)	)	PUNCT
cana-1932	110	25	)	)	PUNCT
cana-1932	110	26	and	and	CCONJ
cana-1932	110	27	g	g	PROPN
cana-1932	110	28	=	=	SYM
cana-1932	110	29	∏	∏	PROPN
cana-1932	110	30	(	(	PUNCT
cana-1932	110	31	g	g	NOUN
cana-1932	110	32	)	)	PUNCT
cana-1932	111	1	λ	λ	X
cana-1932	111	2	λ	λ	X
cana-1932	111	3	ϵ	ϵ	X
cana-1932	111	4	λ	λ	X
cana-1932	111	5	=	=	SYM
cana-1932	111	6	(	(	PUNCT
cana-1932	111	7	∏	∏	PROPN
cana-1932	111	8	(	(	PUNCT
cana-1932	111	9	g	g	PROPN
cana-1932	111	10	1	1	NUM
cana-1932	111	11	)	)	PUNCT
cana-1932	112	1	λ	λ	X
cana-1932	112	2	λ	λ	X
cana-1932	112	3	ϵ	ϵ	X
cana-1932	112	4	λ	λ	X
cana-1932	112	5	,	,	PUNCT
cana-1932	112	6	∏(g	∏(g	ADJ
cana-1932	112	7	2	2	NUM
cana-1932	112	8	)	)	PUNCT
cana-1932	112	9	λ	λ	NOUN
cana-1932	112	10	λ	λ	X
cana-1932	112	11	ϵ	ϵ	X
cana-1932	112	12	λ	λ	X
cana-1932	112	13	)	)	PUNCT
cana-1932	112	14	∏	∏	PROPN
cana-1932	112	15	(	(	PUNCT
cana-1932	112	16	f1	f1	NOUN
cana-1932	112	17	)	)	PUNCT
cana-1932	113	1	λ	λ	NOUN
cana-1932	113	2	λ	λ	X
cana-1932	113	3	ϵ	ϵ	X
cana-1932	113	4	λ	λ	X
cana-1932	113	5	(	(	PUNCT
cana-1932	113	6	xλ	xλ	NOUN
cana-1932	113	7	)	)	PUNCT
cana-1932	113	8	=	=	SYM
cana-1932	113	9	min	min	NOUN
cana-1932	113	10	{	{	PUNCT
cana-1932	113	11	(	(	PUNCT
cana-1932	113	12	f1	f1	NOUN
cana-1932	113	13	)	)	PUNCT
cana-1932	113	14	λ(xλ	λ(xλ	PROPN
cana-1932	113	15	)	)	PUNCT
cana-1932	113	16	}	}	PUNCT
cana-1932	113	17	,	,	PUNCT
cana-1932	113	18	∀𝜆∈	∀𝜆∈	NOUN
cana-1932	113	19	λ	λ	PROPN
cana-1932	113	20	=(	=(	NOUN
cana-1932	113	21	f1	f1	NOUN
cana-1932	113	22	)	)	PUNCT
cana-1932	113	23	μ(xμ	μ(xμ	NUM
cana-1932	113	24	)	)	PUNCT
cana-1932	113	25	≥	≥	NOUN
cana-1932	113	26	r	r	NOUN
cana-1932	113	27	,	,	PUNCT
cana-1932	113	28	for	for	ADP
cana-1932	113	29	some	some	DET
cana-1932	113	30	μ	μ	NUM
cana-1932	113	31	∈	∈	PROPN
cana-1932	113	32	λ	λ	NOUN
cana-1932	113	33	communications	communication	NOUN
cana-1932	113	34	on	on	ADP
cana-1932	113	35	applied	apply	VERB
cana-1932	113	36	nonlinear	nonlinear	ADJ
cana-1932	113	37	analysis	analysis	NOUN
cana-1932	113	38	issn	issn	NOUN
cana-1932	113	39	:	:	PUNCT
cana-1932	113	40	1074	1074	NUM
cana-1932	113	41	-	-	PUNCT
cana-1932	113	42	133x	133x	NUM
cana-1932	113	43	vol	vol	NOUN
cana-1932	113	44	32	32	NUM
cana-1932	113	45	no	no	NOUN
cana-1932	113	46	.	.	NOUN
cana-1932	113	47	3	3	NUM
cana-1932	113	48	(	(	PUNCT
cana-1932	113	49	2025	2025	NUM
cana-1932	113	50	)	)	PUNCT
cana-1932	113	51	159	159	NUM
cana-1932	113	52	https://internationalpubls.com	https://internationalpubls.com	X
cana-1932	113	53	∏(f1)λ	∏(f1)λ	PROPN
cana-1932	113	54	λ	λ	PROPN
cana-1932	113	55	ϵ	ϵ	X
cana-1932	113	56	λ	λ	X
cana-1932	113	57	(	(	PUNCT
cana-1932	113	58	xλ	xλ	NOUN
cana-1932	113	59	)	)	PUNCT
cana-1932	113	60	≥	≥	NOUN
cana-1932	113	61	r	r	NOUN
cana-1932	113	62	∏	∏	PROPN
cana-1932	113	63	(	(	PUNCT
cana-1932	113	64	f2	f2	PROPN
cana-1932	113	65	)	)	PUNCT
cana-1932	113	66	λ	λ	PROPN
cana-1932	113	67	λ	λ	X
cana-1932	113	68	ϵ	ϵ	X
cana-1932	113	69	λ	λ	X
cana-1932	113	70	(	(	PUNCT
cana-1932	113	71	xλ	xλ	NOUN
cana-1932	113	72	)	)	PUNCT
cana-1932	113	73	=	=	SYM
cana-1932	113	74	min{(f2	min{(f2	NOUN
cana-1932	113	75	)	)	PUNCT
cana-1932	113	76	λ(xλ	λ(xλ	PROPN
cana-1932	113	77	)	)	PUNCT
cana-1932	113	78	}	}	PUNCT
cana-1932	113	79	,	,	PUNCT
cana-1932	113	80	∀	∀	NUM
cana-1932	113	81	𝜆∈	𝜆∈	ADJ
cana-1932	113	82	λ	λ	PROPN
cana-1932	113	83	=(	=(	NOUN
cana-1932	113	84	f2)μ(xμ	f2)μ(xμ	PUNCT
cana-1932	113	85	)	)	PUNCT
cana-1932	113	86	≥	≥	NUM
cana-1932	113	87	s	s	NOUN
cana-1932	113	88	,	,	PUNCT
cana-1932	113	89	for	for	ADP
cana-1932	113	90	some	some	DET
cana-1932	113	91	μ	μ	NUM
cana-1932	113	92	∈	∈	PROPN
cana-1932	113	93	λ	λ	X
cana-1932	113	94	∏(f2)λ	∏(f2)λ	PROPN
cana-1932	113	95	λ	λ	PROPN
cana-1932	113	96	ϵ	ϵ	X
cana-1932	113	97	λ	λ	X
cana-1932	113	98	(	(	PUNCT
cana-1932	113	99	xλ	xλ	NOUN
cana-1932	113	100	)	)	PUNCT
cana-1932	113	101	≥	≥	NOUN
cana-1932	113	102	s	s	PROPN
cana-1932	113	103	∏	∏	PROPN
cana-1932	113	104	(	(	PUNCT
cana-1932	113	105	g	g	PROPN
cana-1932	113	106	1	1	NUM
cana-1932	113	107	)	)	PUNCT
cana-1932	113	108	λ	λ	X
cana-1932	113	109	λ	λ	X
cana-1932	113	110	ϵ	ϵ	X
cana-1932	113	111	λ	λ	X
cana-1932	113	112	(	(	PUNCT
cana-1932	113	113	xλ	xλ	NOUN
cana-1932	113	114	)	)	PUNCT
cana-1932	113	115	=	=	SYM
cana-1932	113	116	min	min	X
cana-1932	113	117	{	{	PUNCT
cana-1932	113	118	(	(	PUNCT
cana-1932	113	119	g	g	PROPN
cana-1932	113	120	1	1	NUM
cana-1932	113	121	)	)	PUNCT
cana-1932	113	122	λ(xλ	λ(xλ	PROPN
cana-1932	113	123	)	)	PUNCT
cana-1932	113	124	}	}	PUNCT
cana-1932	113	125	,	,	PUNCT
cana-1932	113	126	∀	∀	NUM
cana-1932	113	127	𝜆∈	𝜆∈	ADJ
cana-1932	113	128	λ	λ	PROPN
cana-1932	113	129	=(	=(	NOUN
cana-1932	113	130	g	g	PROPN
cana-1932	113	131	1	1	NUM
cana-1932	113	132	)	)	PUNCT
cana-1932	113	133	μ	μ	PROPN
cana-1932	113	134	(	(	PUNCT
cana-1932	113	135	xμ	xμ	PROPN
cana-1932	113	136	)	)	PUNCT
cana-1932	113	137	≥	≥	NOUN
cana-1932	113	138	u	u	NOUN
cana-1932	113	139	,	,	PUNCT
cana-1932	113	140	for	for	ADP
cana-1932	113	141	some	some	DET
cana-1932	113	142	μ	μ	NUM
cana-1932	113	143	∈	∈	PROPN
cana-1932	113	144	λ	λ	X
cana-1932	113	145	∏(𝑔1)λ	∏(𝑔1)λ	X
cana-1932	113	146	λ	λ	PROPN
cana-1932	113	147	ϵ	ϵ	X
cana-1932	113	148	λ	λ	PROPN
cana-1932	113	149	(	(	PUNCT
cana-1932	113	150	xλ	xλ	NOUN
cana-1932	113	151	)	)	PUNCT
cana-1932	113	152	≥	≥	NOUN
cana-1932	113	153	u	u	NOUN
cana-1932	113	154	∏	∏	PROPN
cana-1932	113	155	(	(	PUNCT
cana-1932	113	156	g	g	PROPN
cana-1932	113	157	2	2	NUM
cana-1932	113	158	)	)	PUNCT
cana-1932	113	159	λ	λ	X
cana-1932	113	160	λ	λ	X
cana-1932	113	161	ϵ	ϵ	X
cana-1932	113	162	λ	λ	X
cana-1932	113	163	(	(	PUNCT
cana-1932	113	164	xλ	xλ	NOUN
cana-1932	113	165	)	)	PUNCT
cana-1932	113	166	=	=	PUNCT
cana-1932	113	167	min{(g	min{(g	NOUN
cana-1932	113	168	2	2	X
cana-1932	113	169	)	)	PUNCT
cana-1932	113	170	λ(xλ	λ(xλ	PROPN
cana-1932	113	171	)	)	PUNCT
cana-1932	113	172	}	}	PUNCT
cana-1932	113	173	,	,	PUNCT
cana-1932	113	174	∀	∀	NUM
cana-1932	113	175	𝜆∈	𝜆∈	ADJ
cana-1932	113	176	λ	λ	PROPN
cana-1932	113	177	=(	=(	NOUN
cana-1932	113	178	g	g	PROPN
cana-1932	113	179	2	2	NUM
cana-1932	113	180	)	)	PUNCT
cana-1932	113	181	μ	μ	PROPN
cana-1932	113	182	(	(	PUNCT
cana-1932	113	183	xμ	xμ	PROPN
cana-1932	113	184	)	)	PUNCT
cana-1932	113	185	≥	≥	NOUN
cana-1932	113	186	v	v	NOUN
cana-1932	113	187	,	,	PUNCT
cana-1932	113	188	for	for	ADP
cana-1932	113	189	some	some	DET
cana-1932	113	190	μ	μ	NUM
cana-1932	113	191	∈	∈	PROPN
cana-1932	113	192	λ	λ	NOUN
cana-1932	113	193	∏(g	∏(g	ADJ
cana-1932	113	194	2	2	NUM
cana-1932	113	195	)	)	PUNCT
cana-1932	113	196	λ	λ	NOUN
cana-1932	113	197	λ	λ	X
cana-1932	113	198	ϵ	ϵ	X
cana-1932	113	199	λ	λ	X
cana-1932	113	200	(	(	PUNCT
cana-1932	113	201	xλ	xλ	NOUN
cana-1932	113	202	)	)	PUNCT
cana-1932	113	203	≥	≥	NOUN
cana-1932	113	204	v	v	NOUN
cana-1932	113	205	consider	consider	VERB
cana-1932	113	206	∏	∏	PROPN
cana-1932	113	207	(	(	PUNCT
cana-1932	113	208	f	f	NOUN
cana-1932	113	209	)	)	PUNCT
cana-1932	114	1	λ	λ	X
cana-1932	114	2	λ	λ	X
cana-1932	114	3	ϵ	ϵ	X
cana-1932	114	4	λ	λ	X
cana-1932	114	5	∩	∩	X
cana-1932	114	6	∏	∏	X
cana-1932	114	7	(	(	PUNCT
cana-1932	114	8	g	g	NOUN
cana-1932	114	9	)	)	PUNCT
cana-1932	114	10	λ	λ	PROPN
cana-1932	114	11	λ	λ	X
cana-1932	114	12	ϵ	ϵ	X
cana-1932	114	13	λ	λ	X
cana-1932	114	14	=(	=(	X
cana-1932	114	15	∏	∏	PROPN
cana-1932	114	16	(	(	PUNCT
cana-1932	114	17	f1	f1	NOUN
cana-1932	114	18	)	)	PUNCT
cana-1932	114	19	λ	λ	NOUN
cana-1932	114	20	λ	λ	X
cana-1932	114	21	ϵ	ϵ	X
cana-1932	114	22	λ	λ	PROPN
cana-1932	114	23	,	,	PUNCT
cana-1932	114	24	∏	∏	PROPN
cana-1932	114	25	(	(	PUNCT
cana-1932	114	26	f2	f2	PROPN
cana-1932	114	27	)	)	PUNCT
cana-1932	114	28	λ	λ	PROPN
cana-1932	114	29	λ	λ	X
cana-1932	114	30	ϵ	ϵ	X
cana-1932	114	31	λ	λ	PROPN
cana-1932	114	32	)	)	PUNCT
cana-1932	114	33	∩	∩	NOUN
cana-1932	114	34	(	(	PUNCT
cana-1932	114	35	∏	∏	X
cana-1932	114	36	(	(	PUNCT
cana-1932	114	37	g	g	PROPN
cana-1932	114	38	1	1	NUM
cana-1932	114	39	)	)	PUNCT
cana-1932	115	1	λ	λ	X
cana-1932	115	2	λ	λ	X
cana-1932	115	3	ϵ	ϵ	X
cana-1932	115	4	λ	λ	PROPN
cana-1932	115	5	,	,	PUNCT
cana-1932	115	6	∏	∏	PROPN
cana-1932	115	7	(	(	PUNCT
cana-1932	115	8	g	g	PROPN
cana-1932	115	9	2	2	NUM
cana-1932	115	10	)	)	PUNCT
cana-1932	115	11	λ	λ	X
cana-1932	115	12	λ	λ	X
cana-1932	115	13	ϵ	ϵ	X
cana-1932	115	14	λ	λ	PROPN
cana-1932	115	15	)	)	PUNCT
cana-1932	115	16	=(	=(	NOUN
cana-1932	115	17	∏	∏	PROPN
cana-1932	115	18	(	(	PUNCT
cana-1932	115	19	f1	f1	PROPN
cana-1932	115	20	)	)	PUNCT
cana-1932	115	21	λ	λ	NOUN
cana-1932	115	22	λ	λ	X
cana-1932	115	23	∈	∈	NOUN
cana-1932	115	24	λ	λ	PROPN
cana-1932	115	25	∧	∧	PROPN
cana-1932	115	26	∏	∏	PROPN
cana-1932	115	27	(	(	PUNCT
cana-1932	115	28	g	g	PROPN
cana-1932	115	29	1	1	NUM
cana-1932	115	30	)	)	PUNCT
cana-1932	115	31	λ	λ	X
cana-1932	115	32	λ	λ	NOUN
cana-1932	115	33	∈	∈	PROPN
cana-1932	115	34	λ	λ	PROPN
cana-1932	115	35	,	,	PUNCT
cana-1932	115	36	∏	∏	PROPN
cana-1932	115	37	(	(	PUNCT
cana-1932	115	38	f2)λ	f2)λ	NOUN
cana-1932	115	39	∧	∧	PROPN
cana-1932	115	40	∏	∏	PROPN
cana-1932	115	41	(	(	PUNCT
cana-1932	115	42	g	g	PROPN
cana-1932	115	43	2	2	NUM
cana-1932	115	44	)	)	PUNCT
cana-1932	115	45	λ	λ	X
cana-1932	115	46	λ	λ	NOUN
cana-1932	115	47	∈	∈	NOUN
cana-1932	115	48	λ	λ	X
cana-1932	115	49	λ	λ	X
cana-1932	115	50	∈	∈	PROPN
cana-1932	115	51	λ	λ	PROPN
cana-1932	115	52	)	)	PUNCT
cana-1932	115	53	then	then	ADV
cana-1932	115	54	(	(	PUNCT
cana-1932	115	55	∏	∏	PROPN
cana-1932	115	56	(	(	PUNCT
cana-1932	115	57	f1	f1	NOUN
cana-1932	115	58	)	)	PUNCT
cana-1932	115	59	λ	λ	NOUN
cana-1932	115	60	λ	λ	NOUN
cana-1932	115	61	∈	∈	NOUN
cana-1932	115	62	λ	λ	PROPN
cana-1932	115	63	∧	∧	PROPN
cana-1932	115	64	∏	∏	PROPN
cana-1932	115	65	(	(	PUNCT
cana-1932	115	66	g	g	PROPN
cana-1932	115	67	1	1	NUM
cana-1932	115	68	)	)	PUNCT
cana-1932	115	69	λ	λ	X
cana-1932	115	70	λ	λ	NOUN
cana-1932	115	71	∈	∈	PROPN
cana-1932	115	72	λ	λ	PROPN
cana-1932	115	73	)	)	PUNCT
cana-1932	115	74	(	(	PUNCT
cana-1932	115	75	xλ)=(∏	xλ)=(∏	X
cana-1932	115	76	(	(	PUNCT
cana-1932	115	77	f1)λλ	f1)λλ	PROPN
cana-1932	115	78	∈	∈	PROPN
cana-1932	115	79	λ	λ	X
cana-1932	115	80	(	(	PUNCT
cana-1932	115	81	xλ	xλ	NOUN
cana-1932	115	82	)	)	PUNCT
cana-1932	115	83	)	)	PUNCT
cana-1932	116	1	∧	∧	PROPN
cana-1932	116	2	(	(	PUNCT
cana-1932	116	3	∏	∏	PROPN
cana-1932	116	4	(	(	PUNCT
cana-1932	116	5	g	g	PROPN
cana-1932	116	6	1	1	NUM
cana-1932	116	7	)	)	PUNCT
cana-1932	116	8	λ(xλ)λ	λ(xλ)λ	PROPN
cana-1932	116	9	∈	∈	PROPN
cana-1932	116	10	λ	λ	PROPN
cana-1932	116	11	)	)	PUNCT
cana-1932	116	12	,	,	PUNCT
cana-1932	116	13	for	for	ADP
cana-1932	116	14	every	every	DET
cana-1932	116	15	𝜆∈	𝜆∈	ADJ
cana-1932	116	16	λ	λ	X
cana-1932	116	17	=	=	PUNCT
cana-1932	116	18	(	(	PUNCT
cana-1932	116	19	min{(f1)λ(xλ)})∧	min{(f1)λ(xλ)})∧	PROPN
cana-1932	116	20	(	(	PUNCT
cana-1932	116	21	min{(g	min{(g	NOUN
cana-1932	116	22	1	1	NUM
cana-1932	116	23	)	)	PUNCT
cana-1932	116	24	λ	λ	PROPN
cana-1932	116	25	(	(	PUNCT
cana-1932	116	26	xλ	xλ	NOUN
cana-1932	116	27	)	)	PUNCT
cana-1932	116	28	}	}	PUNCT
cana-1932	116	29	)	)	PUNCT
cana-1932	116	30	,	,	PUNCT
cana-1932	116	31	∀𝜆∈	∀𝜆∈	NOUN
cana-1932	116	32	λ	λ	X
cana-1932	116	33	=	=	PRON
cana-1932	116	34	(	(	PUNCT
cana-1932	116	35	f1)μ(xμ)∧(g	f1)μ(xμ)∧(g	NOUN
cana-1932	116	36	1	1	X
cana-1932	116	37	)	)	PUNCT
cana-1932	116	38	μ	μ	PROPN
cana-1932	116	39	(	(	PUNCT
cana-1932	116	40	xμ	xμ	NOUN
cana-1932	116	41	)	)	PUNCT
cana-1932	116	42	=(	=(	NOUN
cana-1932	116	43	(	(	PUNCT
cana-1932	116	44	f1)μ	f1)μ	NOUN
cana-1932	116	45	∧	∧	PROPN
cana-1932	116	46	(	(	PUNCT
cana-1932	116	47	g	g	PROPN
cana-1932	116	48	1	1	NUM
cana-1932	116	49	)	)	PUNCT
cana-1932	116	50	μ	μ	PROPN
cana-1932	116	51	)	)	PUNCT
cana-1932	116	52	(	(	PUNCT
cana-1932	116	53	xμ	xμ	X
cana-1932	116	54	)	)	PUNCT
cana-1932	116	55	=	=	SYM
cana-1932	116	56	0	0	NUM
cana-1932	116	57	(	(	PUNCT
cana-1932	116	58	∏	∏	X
cana-1932	116	59	(	(	PUNCT
cana-1932	116	60	f2	f2	PROPN
cana-1932	116	61	)	)	PUNCT
cana-1932	116	62	λ	λ	PROPN
cana-1932	116	63	λ	λ	NOUN
cana-1932	116	64	∈	∈	NOUN
cana-1932	116	65	λ	λ	PROPN
cana-1932	116	66	∧	∧	PROPN
cana-1932	116	67	∏	∏	PROPN
cana-1932	116	68	(	(	PUNCT
cana-1932	116	69	g	g	PROPN
cana-1932	116	70	2	2	NUM
cana-1932	116	71	)	)	PUNCT
cana-1932	116	72	λ	λ	X
cana-1932	116	73	λ	λ	NOUN
cana-1932	116	74	∈	∈	PROPN
cana-1932	116	75	λ	λ	PROPN
cana-1932	116	76	)	)	PUNCT
cana-1932	116	77	(	(	PUNCT
cana-1932	116	78	xλ)=(∏	xλ)=(∏	PROPN
cana-1932	116	79	(	(	PUNCT
cana-1932	116	80	f2	f2	PROPN
cana-1932	116	81	)	)	PUNCT
cana-1932	116	82	λ	λ	PROPN
cana-1932	116	83	λ	λ	NOUN
cana-1932	116	84	∈	∈	PROPN
cana-1932	116	85	λ	λ	X
cana-1932	116	86	(	(	PUNCT
cana-1932	116	87	xλ	xλ	NOUN
cana-1932	116	88	)	)	PUNCT
cana-1932	116	89	)	)	PUNCT
cana-1932	117	1	∧	∧	PROPN
cana-1932	117	2	(	(	PUNCT
cana-1932	117	3	∏	∏	PROPN
cana-1932	117	4	(	(	PUNCT
cana-1932	117	5	g	g	PROPN
cana-1932	117	6	2	2	NUM
cana-1932	117	7	)	)	PUNCT
cana-1932	117	8	λ	λ	PROPN
cana-1932	117	9	(	(	PUNCT
cana-1932	117	10	xλ	xλ	NOUN
cana-1932	117	11	)	)	PUNCT
cana-1932	117	12	λ	λ	NOUN
cana-1932	117	13	∈	∈	PROPN
cana-1932	117	14	λ	λ	PROPN
cana-1932	117	15	)	)	PUNCT
cana-1932	117	16	,	,	PUNCT
cana-1932	117	17	for	for	ADP
cana-1932	117	18	every	every	DET
cana-1932	117	19	𝜆∈	𝜆∈	ADJ
cana-1932	117	20	λ	λ	X
cana-1932	117	21	=	=	SYM
cana-1932	117	22	(	(	PUNCT
cana-1932	117	23	min{(f2)λ(xλ)})∧	min{(f2)λ(xλ)})∧	PROPN
cana-1932	117	24	(	(	PUNCT
cana-1932	117	25	min{(g	min{(g	NOUN
cana-1932	117	26	2	2	NUM
cana-1932	117	27	)	)	PUNCT
cana-1932	117	28	λ	λ	PROPN
cana-1932	117	29	(	(	PUNCT
cana-1932	117	30	xλ	xλ	NOUN
cana-1932	117	31	)	)	PUNCT
cana-1932	117	32	}	}	PUNCT
cana-1932	117	33	)	)	PUNCT
cana-1932	117	34	,	,	PUNCT
cana-1932	117	35	∀𝜆∈	∀𝜆∈	NOUN
cana-1932	117	36	λ	λ	X
cana-1932	117	37	=	=	SYM
cana-1932	117	38	(	(	PUNCT
cana-1932	117	39	f2)μ(xμ)∧(g	f2)μ(xμ)∧(g	NOUN
cana-1932	117	40	2	2	NUM
cana-1932	117	41	)	)	PUNCT
cana-1932	117	42	μ	μ	PROPN
cana-1932	117	43	(	(	PUNCT
cana-1932	117	44	xμ	xμ	NOUN
cana-1932	117	45	)	)	PUNCT
cana-1932	117	46	=(	=(	NOUN
cana-1932	117	47	(	(	PUNCT
cana-1932	117	48	f2)μ	f2)μ	PROPN
cana-1932	117	49	∧	∧	PROPN
cana-1932	117	50	(	(	PUNCT
cana-1932	117	51	g	g	PROPN
cana-1932	117	52	2	2	NUM
cana-1932	117	53	)	)	PUNCT
cana-1932	117	54	μ	μ	NOUN
cana-1932	117	55	)	)	PUNCT
cana-1932	117	56	(	(	PUNCT
cana-1932	117	57	xμ	xμ	X
cana-1932	117	58	)	)	PUNCT
cana-1932	117	59	=	=	SYM
cana-1932	117	60	0	0	NUM
cana-1932	117	61	communications	communication	NOUN
cana-1932	117	62	on	on	ADP
cana-1932	117	63	applied	apply	VERB
cana-1932	117	64	nonlinear	nonlinear	ADJ
cana-1932	117	65	analysis	analysis	NOUN
cana-1932	117	66	issn	issn	NOUN
cana-1932	117	67	:	:	PUNCT
cana-1932	117	68	1074	1074	NUM
cana-1932	117	69	-	-	PUNCT
cana-1932	117	70	133x	133x	NUM
cana-1932	117	71	vol	vol	NOUN
cana-1932	117	72	32	32	NUM
cana-1932	117	73	no	no	NOUN
cana-1932	117	74	.	.	NOUN
cana-1932	117	75	3	3	NUM
cana-1932	117	76	(	(	PUNCT
cana-1932	117	77	2025	2025	NUM
cana-1932	117	78	)	)	PUNCT
cana-1932	117	79	160	160	NUM
cana-1932	117	80	https://internationalpubls.com	https://internationalpubls.com	X
cana-1932	117	81	∏	∏	PROPN
cana-1932	117	82	(	(	PUNCT
cana-1932	117	83	f	f	X
cana-1932	117	84	)	)	PUNCT
cana-1932	117	85	λ	λ	X
cana-1932	117	86	λ	λ	X
cana-1932	117	87	ϵ	ϵ	X
cana-1932	117	88	λ	λ	X
cana-1932	117	89	∩	∩	X
cana-1932	117	90	∏	∏	X
cana-1932	117	91	(	(	PUNCT
cana-1932	117	92	g	g	NOUN
cana-1932	117	93	)	)	PUNCT
cana-1932	118	1	λ	λ	NOUN
cana-1932	118	2	λ	λ	X
cana-1932	118	3	ϵ	ϵ	X
cana-1932	118	4	λ	λ	X
cana-1932	118	5	=	=	SYM
cana-1932	118	6	0	0	PUNCT
cana-1932	118	7	consequently	consequently	ADV
cana-1932	118	8	,	,	PUNCT
cana-1932	118	9	arbitrary	arbitrary	ADJ
cana-1932	118	10	product	product	NOUN
cana-1932	118	11	of	of	ADP
cana-1932	118	12	two	two	NUM
cana-1932	118	13	fds	fds	NOUN
cana-1932	118	14	-	-	PUNCT
cana-1932	118	15	t2	t2	NOUN
cana-1932	118	16	is	be	AUX
cana-1932	118	17	a	a	DET
cana-1932	118	18	fds	fds	NOUN
cana-1932	118	19	-	-	PUNCT
cana-1932	118	20	t2	t2	NOUN
cana-1932	118	21	.	.	PUNCT
cana-1932	119	1	results	result	NOUN
cana-1932	119	2	:	:	PUNCT
cana-1932	119	3	the	the	DET
cana-1932	119	4	study	study	NOUN
cana-1932	119	5	on	on	ADP
cana-1932	119	6	fuzzy	fuzzy	ADJ
cana-1932	119	7	double	double	ADJ
cana-1932	119	8	s	s	NOUN
cana-1932	119	9	-	-	PUNCT
cana-1932	119	10	hausdorff	hausdorff	NOUN
cana-1932	119	11	spaces	space	NOUN
cana-1932	119	12	has	have	AUX
cana-1932	119	13	yielded	yield	VERB
cana-1932	119	14	several	several	ADJ
cana-1932	119	15	significant	significant	ADJ
cana-1932	119	16	findings	finding	NOUN
cana-1932	119	17	,	,	PUNCT
cana-1932	119	18	contributing	contribute	VERB
cana-1932	119	19	to	to	ADP
cana-1932	119	20	both	both	CCONJ
cana-1932	119	21	the	the	DET
cana-1932	119	22	theoretical	theoretical	ADJ
cana-1932	119	23	framework	framework	NOUN
cana-1932	119	24	of	of	ADP
cana-1932	119	25	fuzzy	fuzzy	ADJ
cana-1932	119	26	topology	topology	NOUN
cana-1932	119	27	and	and	CCONJ
cana-1932	119	28	its	its	PRON
cana-1932	119	29	potential	potential	ADJ
cana-1932	119	30	applications	application	NOUN
cana-1932	119	31	.	.	PUNCT
cana-1932	120	1	the	the	DET
cana-1932	120	2	results	result	NOUN
cana-1932	120	3	are	be	AUX
cana-1932	120	4	presented	present	VERB
cana-1932	120	5	below	below	ADV
cana-1932	120	6	,	,	PUNCT
cana-1932	120	7	focusing	focus	VERB
cana-1932	120	8	on	on	ADP
cana-1932	120	9	the	the	DET
cana-1932	120	10	key	key	ADJ
cana-1932	120	11	aspects	aspect	NOUN
cana-1932	120	12	of	of	ADP
cana-1932	120	13	definitions	definition	NOUN
cana-1932	120	14	,	,	PUNCT
cana-1932	120	15	theorems	theorem	NOUN
cana-1932	120	16	,	,	PUNCT
cana-1932	120	17	properties	property	NOUN
cana-1932	120	18	,	,	PUNCT
cana-1932	120	19	and	and	CCONJ
cana-1932	120	20	comparative	comparative	ADJ
cana-1932	120	21	analyses	analysis	NOUN
cana-1932	120	22	.	.	PUNCT
cana-1932	121	1	1	1	X
cana-1932	121	2	.	.	X
cana-1932	121	3	definition	definition	NOUN
cana-1932	121	4	and	and	CCONJ
cana-1932	121	5	characterization	characterization	NOUN
cana-1932	121	6	of	of	ADP
cana-1932	121	7	fuzzy	fuzzy	ADJ
cana-1932	121	8	double	double	ADJ
cana-1932	121	9	s	s	NOUN
cana-1932	121	10	-	-	PUNCT
cana-1932	121	11	hausdorff	hausdorff	NOUN
cana-1932	121	12	spaces	space	NOUN
cana-1932	121	13	•	•	ADP
cana-1932	121	14	formal	formal	ADJ
cana-1932	121	15	definition	definition	NOUN
cana-1932	121	16	:	:	PUNCT
cana-1932	121	17	the	the	DET
cana-1932	121	18	study	study	NOUN
cana-1932	121	19	successfully	successfully	ADV
cana-1932	121	20	formulated	formulate	VERB
cana-1932	121	21	the	the	DET
cana-1932	121	22	definition	definition	NOUN
cana-1932	121	23	of	of	ADP
cana-1932	121	24	fuzzy	fuzzy	ADJ
cana-1932	121	25	double	double	ADJ
cana-1932	121	26	shausdorff	shausdorff	NOUN
cana-1932	121	27	spaces	space	NOUN
cana-1932	121	28	.	.	PUNCT
cana-1932	122	1	these	these	DET
cana-1932	122	2	spaces	space	NOUN
cana-1932	122	3	are	be	AUX
cana-1932	122	4	characterized	characterize	VERB
cana-1932	122	5	by	by	ADP
cana-1932	122	6	two	two	NUM
cana-1932	122	7	fuzzy	fuzzy	ADJ
cana-1932	122	8	topologies	topology	NOUN
cana-1932	122	9	on	on	ADP
cana-1932	122	10	the	the	DET
cana-1932	122	11	same	same	ADJ
cana-1932	122	12	underlying	underlying	ADJ
cana-1932	122	13	set	set	NOUN
cana-1932	122	14	,	,	PUNCT
cana-1932	122	15	where	where	SCONJ
cana-1932	122	16	each	each	DET
cana-1932	122	17	topology	topology	NOUN
cana-1932	122	18	satisfies	satisfy	VERB
cana-1932	122	19	a	a	DET
cana-1932	122	20	fuzzy	fuzzy	ADJ
cana-1932	122	21	version	version	NOUN
cana-1932	122	22	of	of	ADP
cana-1932	122	23	the	the	DET
cana-1932	122	24	hausdorff	hausdorff	NOUN
cana-1932	122	25	condition	condition	NOUN
cana-1932	122	26	.	.	PUNCT
cana-1932	123	1	specifically	specifically	ADV
cana-1932	123	2	,	,	PUNCT
cana-1932	123	3	the	the	DET
cana-1932	123	4	separation	separation	NOUN
cana-1932	123	5	of	of	ADP
cana-1932	123	6	points	point	NOUN
cana-1932	123	7	is	be	AUX
cana-1932	123	8	controlled	control	VERB
cana-1932	123	9	by	by	ADP
cana-1932	123	10	fuzzy	fuzzy	ADJ
cana-1932	123	11	parameters	parameter	NOUN
cana-1932	123	12	that	that	PRON
cana-1932	123	13	account	account	VERB
cana-1932	123	14	for	for	ADP
cana-1932	123	15	the	the	DET
cana-1932	123	16	degree	degree	NOUN
cana-1932	123	17	of	of	ADP
cana-1932	123	18	uncertainty	uncertainty	NOUN
cana-1932	123	19	in	in	ADP
cana-1932	123	20	the	the	DET
cana-1932	123	21	space	space	NOUN
cana-1932	123	22	.	.	PUNCT
cana-1932	124	1	•	•	NOUN
cana-1932	124	2	characterization	characterization	NOUN
cana-1932	124	3	theorems	theorem	VERB
cana-1932	124	4	:	:	PUNCT
cana-1932	124	5	several	several	ADJ
cana-1932	124	6	theorems	theorem	NOUN
cana-1932	124	7	were	be	AUX
cana-1932	124	8	established	establish	VERB
cana-1932	124	9	to	to	PART
cana-1932	124	10	characterize	characterize	VERB
cana-1932	124	11	the	the	DET
cana-1932	124	12	properties	property	NOUN
cana-1932	124	13	of	of	ADP
cana-1932	124	14	fuzzy	fuzzy	ADJ
cana-1932	124	15	double	double	ADJ
cana-1932	124	16	s	s	NOUN
cana-1932	124	17	-	-	PUNCT
cana-1932	124	18	hausdorff	hausdorff	NOUN
cana-1932	124	19	spaces	space	NOUN
cana-1932	124	20	.	.	PUNCT
cana-1932	125	1	these	these	DET
cana-1932	125	2	theorems	theorem	NOUN
cana-1932	125	3	provide	provide	VERB
cana-1932	125	4	the	the	DET
cana-1932	125	5	necessary	necessary	ADJ
cana-1932	125	6	and	and	CCONJ
cana-1932	125	7	sufficient	sufficient	ADJ
cana-1932	125	8	conditions	condition	NOUN
cana-1932	125	9	for	for	ADP
cana-1932	125	10	a	a	DET
cana-1932	125	11	fuzzy	fuzzy	ADJ
cana-1932	125	12	topological	topological	ADJ
cana-1932	125	13	space	space	NOUN
cana-1932	125	14	to	to	PART
cana-1932	125	15	be	be	AUX
cana-1932	125	16	considered	consider	VERB
cana-1932	125	17	a	a	DET
cana-1932	125	18	fuzzy	fuzzy	ADJ
cana-1932	125	19	double	double	ADJ
cana-1932	125	20	s	s	NOUN
cana-1932	125	21	-	-	PUNCT
cana-1932	125	22	hausdorff	hausdorff	NOUN
cana-1932	125	23	space	space	NOUN
cana-1932	125	24	.	.	PUNCT
cana-1932	126	1	the	the	DET
cana-1932	126	2	results	result	NOUN
cana-1932	126	3	demonstrate	demonstrate	VERB
cana-1932	126	4	that	that	SCONJ
cana-1932	126	5	the	the	DET
cana-1932	126	6	spaces	space	NOUN
cana-1932	126	7	extend	extend	VERB
cana-1932	126	8	the	the	DET
cana-1932	126	9	classical	classical	ADJ
cana-1932	126	10	concept	concept	NOUN
cana-1932	126	11	of	of	ADP
cana-1932	126	12	hausdorff	hausdorff	NOUN
cana-1932	126	13	spaces	space	NOUN
cana-1932	126	14	by	by	ADP
cana-1932	126	15	allowing	allow	VERB
cana-1932	126	16	for	for	ADP
cana-1932	126	17	a	a	DET
cana-1932	126	18	graded	grade	VERB
cana-1932	126	19	separation	separation	NOUN
cana-1932	126	20	between	between	ADP
cana-1932	126	21	points	point	NOUN
cana-1932	126	22	.	.	PUNCT
cana-1932	127	1	2	2	X
cana-1932	127	2	.	.	X
cana-1932	127	3	properties	property	NOUN
cana-1932	127	4	of	of	ADP
cana-1932	127	5	fuzzy	fuzzy	ADJ
cana-1932	127	6	double	double	ADJ
cana-1932	127	7	s	s	NOUN
cana-1932	127	8	-	-	PUNCT
cana-1932	127	9	hausdorff	hausdorff	NOUN
cana-1932	127	10	spaces	space	NOUN
cana-1932	127	11	•	•	ADP
cana-1932	127	12	separation	separation	NOUN
cana-1932	127	13	properties	property	NOUN
cana-1932	127	14	:	:	PUNCT
cana-1932	127	15	the	the	DET
cana-1932	127	16	study	study	NOUN
cana-1932	127	17	found	find	VERB
cana-1932	127	18	that	that	SCONJ
cana-1932	127	19	fuzzy	fuzzy	ADJ
cana-1932	127	20	double	double	ADJ
cana-1932	127	21	s	s	NOUN
cana-1932	127	22	-	-	PUNCT
cana-1932	127	23	hausdorff	hausdorff	NOUN
cana-1932	127	24	spaces	space	NOUN
cana-1932	127	25	exhibit	exhibit	VERB
cana-1932	127	26	unique	unique	ADJ
cana-1932	127	27	separation	separation	NOUN
cana-1932	127	28	properties	property	NOUN
cana-1932	127	29	that	that	PRON
cana-1932	127	30	distinguish	distinguish	VERB
cana-1932	127	31	them	they	PRON
cana-1932	127	32	from	from	ADP
cana-1932	127	33	classical	classical	ADJ
cana-1932	127	34	hausdorff	hausdorff	NOUN
cana-1932	127	35	spaces	space	NOUN
cana-1932	127	36	and	and	CCONJ
cana-1932	127	37	other	other	ADJ
cana-1932	127	38	fuzzy	fuzzy	ADJ
cana-1932	127	39	topologies	topology	NOUN
cana-1932	127	40	.	.	PUNCT
cana-1932	128	1	specifically	specifically	ADV
cana-1932	128	2	,	,	PUNCT
cana-1932	128	3	the	the	DET
cana-1932	128	4	spaces	space	NOUN
cana-1932	128	5	allow	allow	VERB
cana-1932	128	6	for	for	ADP
cana-1932	128	7	the	the	DET
cana-1932	128	8	separation	separation	NOUN
cana-1932	128	9	of	of	ADP
cana-1932	128	10	points	point	NOUN
cana-1932	128	11	to	to	PART
cana-1932	128	12	be	be	AUX
cana-1932	128	13	controlled	control	VERB
cana-1932	128	14	by	by	ADP
cana-1932	128	15	two	two	NUM
cana-1932	128	16	distinct	distinct	ADJ
cana-1932	128	17	fuzzy	fuzzy	ADJ
cana-1932	128	18	parameters	parameter	NOUN
cana-1932	128	19	,	,	PUNCT
cana-1932	128	20	offering	offer	VERB
cana-1932	128	21	greater	great	ADJ
cana-1932	128	22	flexibility	flexibility	NOUN
cana-1932	128	23	in	in	ADP
cana-1932	128	24	handling	handle	VERB
cana-1932	128	25	scenarios	scenario	NOUN
cana-1932	128	26	where	where	SCONJ
cana-1932	128	27	points	point	NOUN
cana-1932	128	28	are	be	AUX
cana-1932	128	29	not	not	PART
cana-1932	128	30	strictly	strictly	ADV
cana-1932	128	31	separable	separable	ADJ
cana-1932	128	32	.	.	PUNCT
cana-1932	129	1	•	•	NUM
cana-1932	129	2	intersection	intersection	NOUN
cana-1932	129	3	and	and	CCONJ
cana-1932	129	4	union	union	NOUN
cana-1932	129	5	behavior	behavior	NOUN
cana-1932	129	6	:	:	PUNCT
cana-1932	129	7	the	the	DET
cana-1932	129	8	results	result	NOUN
cana-1932	129	9	also	also	ADV
cana-1932	129	10	showed	show	VERB
cana-1932	129	11	that	that	SCONJ
cana-1932	129	12	the	the	DET
cana-1932	129	13	behavior	behavior	NOUN
cana-1932	129	14	of	of	ADP
cana-1932	129	15	fuzzy	fuzzy	ADJ
cana-1932	129	16	double	double	ADJ
cana-1932	129	17	s	s	NOUN
cana-1932	129	18	-	-	PUNCT
cana-1932	129	19	hausdorff	hausdorff	NOUN
cana-1932	129	20	spaces	space	NOUN
cana-1932	129	21	under	under	ADP
cana-1932	129	22	intersection	intersection	NOUN
cana-1932	129	23	and	and	CCONJ
cana-1932	129	24	union	union	NOUN
cana-1932	129	25	operations	operation	NOUN
cana-1932	129	26	is	be	AUX
cana-1932	129	27	consistent	consistent	ADJ
cana-1932	129	28	with	with	ADP
cana-1932	129	29	the	the	DET
cana-1932	129	30	expected	expect	VERB
cana-1932	129	31	behavior	behavior	NOUN
cana-1932	129	32	of	of	ADP
cana-1932	129	33	fuzzy	fuzzy	ADJ
cana-1932	129	34	topological	topological	ADJ
cana-1932	129	35	spaces	space	NOUN
cana-1932	129	36	.	.	PUNCT
cana-1932	130	1	however	however	ADV
cana-1932	130	2	,	,	PUNCT
cana-1932	130	3	the	the	DET
cana-1932	130	4	dual	dual	ADJ
cana-1932	130	5	nature	nature	NOUN
cana-1932	130	6	of	of	ADP
cana-1932	130	7	the	the	DET
cana-1932	130	8	topologies	topology	NOUN
cana-1932	130	9	introduces	introduce	VERB
cana-1932	130	10	additional	additional	ADJ
cana-1932	130	11	complexity	complexity	NOUN
cana-1932	130	12	,	,	PUNCT
cana-1932	130	13	making	make	VERB
cana-1932	130	14	these	these	DET
cana-1932	130	15	operations	operation	NOUN
cana-1932	130	16	more	more	ADV
cana-1932	130	17	intricate	intricate	ADJ
cana-1932	130	18	than	than	ADP
cana-1932	130	19	in	in	ADP
cana-1932	130	20	single	single	ADJ
cana-1932	130	21	fuzzy	fuzzy	ADJ
cana-1932	130	22	topology	topology	NOUN
cana-1932	130	23	spaces	space	VERB
cana-1932	130	24	conclusions	conclusion	NOUN
cana-1932	130	25	:	:	PUNCT
cana-1932	130	26	this	this	DET
cana-1932	130	27	study	study	NOUN
cana-1932	130	28	provides	provide	VERB
cana-1932	130	29	a	a	DET
cana-1932	130	30	comprehensive	comprehensive	ADJ
cana-1932	130	31	analysis	analysis	NOUN
cana-1932	130	32	of	of	ADP
cana-1932	130	33	fuzzy	fuzzy	ADJ
cana-1932	130	34	double	double	ADJ
cana-1932	130	35	s	s	NOUN
cana-1932	130	36	-	-	PUNCT
cana-1932	130	37	hausdorff	hausdorff	NOUN
cana-1932	130	38	spaces	space	NOUN
cana-1932	130	39	,	,	PUNCT
cana-1932	130	40	offering	offer	VERB
cana-1932	130	41	significant	significant	ADJ
cana-1932	130	42	theoretical	theoretical	ADJ
cana-1932	130	43	contributions	contribution	NOUN
cana-1932	130	44	and	and	CCONJ
cana-1932	130	45	highlighting	highlight	VERB
cana-1932	130	46	their	their	PRON
cana-1932	130	47	potential	potential	ADJ
cana-1932	130	48	applications	application	NOUN
cana-1932	130	49	.	.	PUNCT
cana-1932	131	1	while	while	SCONJ
cana-1932	131	2	there	there	PRON
cana-1932	131	3	are	be	VERB
cana-1932	131	4	areas	area	NOUN
cana-1932	131	5	for	for	ADP
cana-1932	131	6	further	further	ADJ
cana-1932	131	7	exploration	exploration	NOUN
cana-1932	131	8	,	,	PUNCT
cana-1932	131	9	the	the	DET
cana-1932	131	10	findings	finding	NOUN
cana-1932	131	11	presented	present	VERB
cana-1932	131	12	here	here	ADV
cana-1932	131	13	lay	lay	VERB
cana-1932	131	14	the	the	DET
cana-1932	131	15	groundwork	groundwork	NOUN
cana-1932	131	16	for	for	ADP
cana-1932	131	17	future	future	ADJ
cana-1932	131	18	research	research	NOUN
cana-1932	131	19	and	and	CCONJ
cana-1932	131	20	underscore	underscore	VERB
cana-1932	131	21	the	the	DET
cana-1932	131	22	importance	importance	NOUN
cana-1932	131	23	of	of	ADP
cana-1932	131	24	fuzzy	fuzzy	ADJ
cana-1932	131	25	concepts	concept	NOUN
cana-1932	131	26	in	in	ADP
cana-1932	131	27	modern	modern	ADJ
cana-1932	131	28	topology	topology	NOUN
cana-1932	131	29	.	.	PUNCT
cana-1932	132	1	references	reference	NOUN
cana-1932	132	2	[	[	X
cana-1932	132	3	1	1	NUM
cana-1932	132	4	]	]	X
cana-1932	132	5	chang	chang	PROPN
cana-1932	132	6	,	,	PUNCT
cana-1932	132	7	c.	c.	PROPN
cana-1932	132	8	l.	l.	PROPN
cana-1932	132	9	(	(	PUNCT
cana-1932	132	10	1968	1968	NUM
cana-1932	132	11	)	)	PUNCT
cana-1932	132	12	.	.	PUNCT
cana-1932	133	1	fuzzy	fuzzy	ADJ
cana-1932	133	2	topological	topological	ADJ
cana-1932	133	3	spaces	space	NOUN
cana-1932	133	4	.	.	PUNCT
cana-1932	134	1	journal	journal	PROPN
cana-1932	134	2	of	of	ADP
cana-1932	134	3	mathematical	mathematical	ADJ
cana-1932	134	4	analysis	analysis	NOUN
cana-1932	134	5	and	and	CCONJ
cana-1932	134	6	applications	application	NOUN
cana-1932	134	7	,	,	PUNCT
cana-1932	134	8	24(1	24(1	NUM
cana-1932	134	9	)	)	PUNCT
cana-1932	134	10	,	,	PUNCT
cana-1932	134	11	182	182	NUM
cana-1932	134	12	-	-	SYM
cana-1932	134	13	190	190	NUM
cana-1932	134	14	.	.	PUNCT
cana-1932	135	1	[	[	X
cana-1932	135	2	2	2	NUM
cana-1932	135	3	]	]	SYM
cana-1932	135	4	kandil	kandil	NOUN
cana-1932	135	5	,	,	PUNCT
cana-1932	135	6	a.	a.	NOUN
cana-1932	135	7	,	,	PUNCT
cana-1932	135	8	tantawy	tantawy	NOUN
cana-1932	135	9	,	,	PUNCT
cana-1932	135	10	o.a.e	o.a.e	NOUN
cana-1932	135	11	.	.	PUNCT
cana-1932	135	12	and	and	CCONJ
cana-1932	135	13	wafaie	wafaie	PROPN
cana-1932	135	14	,	,	PUNCT
cana-1932	135	15	m.	m.	NOUN
cana-1932	135	16	,	,	PUNCT
cana-1932	135	17	on	on	ADP
cana-1932	135	18	flou	flou	PROPN
cana-1932	135	19	(	(	PUNCT
cana-1932	135	20	intutionistic	intutionistic	ADJ
cana-1932	135	21	)	)	PUNCT
cana-1932	135	22	topological	topological	ADJ
cana-1932	135	23	spaces	space	NOUN
cana-1932	135	24	,	,	PUNCT
cana-1932	135	25	j.fuzzy	j.fuzzy	ADV
cana-1932	135	26	math.15	math.15	NOUN
cana-1932	135	27	(	(	PUNCT
cana-1932	135	28	2	2	NUM
cana-1932	135	29	)	)	PUNCT
cana-1932	135	30	(	(	PUNCT
cana-1932	135	31	2007	2007	NUM
cana-1932	135	32	)	)	PUNCT
cana-1932	135	33	,	,	PUNCT
cana-1932	135	34	1	1	NUM
cana-1932	135	35	-	-	SYM
cana-1932	135	36	23	23	NUM
cana-1932	135	37	.	.	PUNCT
cana-1932	136	1	[	[	X
cana-1932	136	2	3	3	X
cana-1932	136	3	]	]	PUNCT
cana-1932	136	4	r.srivastava	r.srivastava	NOUN
cana-1932	136	5	,	,	PUNCT
cana-1932	136	6	on	on	ADP
cana-1932	136	7	separation	separation	NOUN
cana-1932	136	8	axioms	axiom	NOUN
cana-1932	136	9	in	in	ADP
cana-1932	136	10	a	a	DET
cana-1932	136	11	newly	newly	ADV
cana-1932	136	12	defined	define	VERB
cana-1932	136	13	fuzzy	fuzzy	ADJ
cana-1932	136	14	topology	topology	NOUN
cana-1932	136	15	,	,	PUNCT
cana-1932	136	16	fuzzy	fuzzy	ADJ
cana-1932	136	17	sets	set	NOUN
cana-1932	136	18	and	and	CCONJ
cana-1932	136	19	systems	system	NOUN
cana-1932	136	20	,	,	PUNCT
cana-1932	136	21	62(1994	62(1994	NUM
cana-1932	136	22	)	)	PUNCT
cana-1932	136	23	,	,	PUNCT
cana-1932	136	24	341	341	NUM
cana-1932	136	25	-	-	SYM
cana-1932	136	26	346	346	NUM
cana-1932	136	27	.	.	PUNCT
cana-1932	137	1	[	[	X
cana-1932	137	2	4	4	X
cana-1932	137	3	]	]	X
cana-1932	137	4	c	c	PROPN
cana-1932	137	5	k	k	PROPN
cana-1932	137	6	wong	wong	PROPN
cana-1932	137	7	,	,	PUNCT
cana-1932	137	8	fuzzy	fuzzy	ADJ
cana-1932	137	9	topology	topology	NOUN
cana-1932	137	10	:	:	PUNCT
cana-1932	137	11	product	product	NOUN
cana-1932	137	12	and	and	CCONJ
cana-1932	137	13	quotient	quotient	NOUN
cana-1932	137	14	propositions	proposition	NOUN
cana-1932	137	15	,	,	PUNCT
cana-1932	137	16	j.math.anal.appl	j.math.anal.appl	NOUN
cana-1932	137	17	.	.	PUNCT
cana-1932	137	18	,	,	PUNCT
cana-1932	137	19	45(1974	45(1974	NUM
cana-1932	137	20	)	)	PUNCT
cana-1932	137	21	,	,	PUNCT
cana-1932	137	22	512	512	NUM
cana-1932	137	23	-	-	SYM
cana-1932	137	24	521	521	NUM
cana-1932	137	25	.	.	PUNCT
cana-1932	138	1	[	[	X
cana-1932	138	2	5	5	NUM
cana-1932	138	3	]	]	SYM
cana-1932	138	4	zadeh	zadeh	PROPN
cana-1932	138	5	,	,	PUNCT
cana-1932	138	6	l.	l.	PROPN
cana-1932	138	7	a.	a.	PROPN
cana-1932	138	8	(	(	PUNCT
cana-1932	138	9	1965	1965	NUM
cana-1932	138	10	)	)	PUNCT
cana-1932	138	11	.	.	PUNCT
cana-1932	139	1	fuzzy	fuzzy	ADJ
cana-1932	139	2	sets	set	NOUN
cana-1932	139	3	.	.	PUNCT
cana-1932	140	1	information	information	NOUN
cana-1932	140	2	and	and	CCONJ
cana-1932	140	3	control	control	NOUN
cana-1932	140	4	,	,	PUNCT
cana-1932	140	5	8(3	8(3	NUM
cana-1932	140	6	)	)	PUNCT
cana-1932	140	7	,	,	PUNCT
cana-1932	140	8	338	338	NUM
cana-1932	140	9	-	-	SYM
cana-1932	140	10	353	353	NUM
cana-1932	140	11	.	.	PUNCT
cana-1932	141	1	[	[	X
cana-1932	141	2	6	6	NUM
cana-1932	141	3	]	]	PUNCT
cana-1932	141	4	r.sowmya	r.sowmya	NOUN
cana-1932	141	5	,	,	PUNCT
cana-1932	141	6	dr	dr	PROPN
cana-1932	141	7	,	,	PUNCT
cana-1932	141	8	a.kalaichelvi	a.kalaichelvi	ADJ
cana-1932	141	9	,	,	PUNCT
cana-1932	141	10	separation	separation	NOUN
cana-1932	141	11	axioms	axiom	NOUN
cana-1932	141	12	in	in	ADP
cana-1932	141	13	fuzzy	fuzzy	ADJ
cana-1932	141	14	double	double	ADJ
cana-1932	141	15	topological	topological	ADJ
cana-1932	141	16	spaces	space	NOUN
cana-1932	141	17	,	,	PUNCT
cana-1932	141	18	advances	advance	NOUN
cana-1932	141	19	and	and	CCONJ
cana-1932	141	20	applications	application	NOUN
cana-1932	141	21	in	in	ADP
cana-1932	141	22	mathematical	mathematical	ADJ
cana-1932	141	23	sciences	science	NOUN
cana-1932	141	24	,	,	PUNCT
cana-1932	141	25	vol	vol	NOUN
cana-1932	141	26	21	21	NUM
cana-1932	141	27	,	,	PUNCT
cana-1932	141	28	issue	issue	NOUN
cana-1932	141	29	4	4	NUM
cana-1932	141	30	,	,	PUNCT
cana-1932	141	31	february	february	NOUN
cana-1932	141	32	2022	2022	NUM
cana-1932	141	33	,	,	PUNCT
cana-1932	141	34	1651	1651	NUM
cana-1932	141	35	-	-	SYM
cana-1932	141	36	1664	1664	NUM
cana-1932	141	37	.	.	PUNCT
cana-1932	142	1	[	[	X
cana-1932	142	2	7	7	NUM
cana-1932	142	3	]	]	X
cana-1932	142	4	lowen	lowen	PROPN
cana-1932	142	5	,	,	PUNCT
cana-1932	142	6	r.	r.	PROPN
cana-1932	142	7	(	(	PUNCT
cana-1932	142	8	1976	1976	NUM
cana-1932	142	9	)	)	PUNCT
cana-1932	142	10	.	.	PUNCT
cana-1932	143	1	fuzzy	fuzzy	ADJ
cana-1932	143	2	topological	topological	ADJ
cana-1932	143	3	spaces	space	NOUN
cana-1932	143	4	and	and	CCONJ
cana-1932	143	5	fuzzy	fuzzy	ADJ
cana-1932	143	6	compactness	compactness	NOUN
cana-1932	143	7	.	.	PUNCT
cana-1932	144	1	journal	journal	NOUN
cana-1932	144	2	of	of	ADP
cana-1932	144	3	mathematical	mathematical	ADJ
cana-1932	144	4	analysis	analysis	NOUN
cana-1932	144	5	and	and	CCONJ
cana-1932	144	6	applications	application	NOUN
cana-1932	144	7	,	,	PUNCT
cana-1932	144	8	49(2	49(2	NUM
cana-1932	144	9	)	)	PUNCT
cana-1932	144	10	,	,	PUNCT
cana-1932	144	11	375	375	NUM
cana-1932	144	12	-	-	SYM
cana-1932	144	13	394	394	NUM
cana-1932	144	14	.	.	PUNCT
cana-1932	145	1	[	[	X
cana-1932	145	2	8	8	NUM
cana-1932	145	3	]	]	PUNCT
cana-1932	145	4	katsaras	katsara	NOUN
cana-1932	145	5	,	,	PUNCT
cana-1932	145	6	a.	a.	PROPN
cana-1932	145	7	k.	k.	PROPN
cana-1932	145	8	(	(	PUNCT
cana-1932	145	9	1980	1980	NUM
cana-1932	145	10	)	)	PUNCT
cana-1932	145	11	.	.	PUNCT
cana-1932	146	1	fuzzy	fuzzy	ADJ
cana-1932	146	2	hausdorff	hausdorff	PROPN
cana-1932	146	3	spaces	space	NOUN
cana-1932	146	4	.	.	PUNCT
cana-1932	147	1	journal	journal	PROPN
cana-1932	147	2	of	of	ADP
cana-1932	147	3	mathematical	mathematical	ADJ
cana-1932	147	4	analysis	analysis	NOUN
cana-1932	147	5	and	and	CCONJ
cana-1932	147	6	applications	application	NOUN
cana-1932	147	7	,	,	PUNCT
cana-1932	147	8	74(2	74(2	NUM
cana-1932	147	9	)	)	PUNCT
cana-1932	147	10	,	,	PUNCT
cana-1932	147	11	547	547	NUM
cana-1932	147	12	-	-	SYM
cana-1932	147	13	559	559	NUM
cana-1932	147	14	.	.	PUNCT
cana-1932	148	1	communications	communication	NOUN
cana-1932	148	2	on	on	ADP
cana-1932	148	3	applied	apply	VERB
cana-1932	148	4	nonlinear	nonlinear	ADJ
cana-1932	148	5	analysis	analysis	NOUN
cana-1932	148	6	issn	issn	NOUN
cana-1932	148	7	:	:	PUNCT
cana-1932	148	8	1074	1074	NUM
cana-1932	148	9	-	-	PUNCT
cana-1932	148	10	133x	133x	NUM
cana-1932	148	11	vol	vol	NOUN
cana-1932	148	12	32	32	NUM
cana-1932	148	13	no	no	NOUN
cana-1932	148	14	.	.	NOUN
cana-1932	148	15	3	3	NUM
cana-1932	148	16	(	(	PUNCT
cana-1932	148	17	2025	2025	NUM
cana-1932	148	18	)	)	PUNCT
cana-1932	148	19	161	161	NUM
cana-1932	148	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1932	149	1	[	[	X
cana-1932	149	2	9	9	NUM
cana-1932	149	3	]	]	SYM
cana-1932	149	4	sridharan	sridharan	ADJ
cana-1932	149	5	,	,	PUNCT
cana-1932	149	6	m.	m.	NOUN
cana-1932	149	7	,	,	PUNCT
cana-1932	149	8	&	&	CCONJ
cana-1932	149	9	sivakumar	sivakumar	PROPN
cana-1932	149	10	,	,	PUNCT
cana-1932	149	11	m.	m.	NOUN
cana-1932	149	12	(	(	PUNCT
cana-1932	149	13	1997	1997	NUM
cana-1932	149	14	)	)	PUNCT
cana-1932	149	15	.	.	PUNCT
cana-1932	150	1	double	double	ADJ
cana-1932	150	2	fuzzy	fuzzy	ADJ
cana-1932	150	3	topological	topological	ADJ
cana-1932	150	4	spaces	space	NOUN
cana-1932	150	5	.	.	PUNCT
cana-1932	151	1	indian	indian	ADJ
cana-1932	151	2	journal	journal	PROPN
cana-1932	151	3	of	of	ADP
cana-1932	151	4	pure	pure	ADJ
cana-1932	151	5	and	and	CCONJ
cana-1932	151	6	applied	applied	ADJ
cana-1932	151	7	mathematics	mathematic	NOUN
cana-1932	151	8	,	,	PUNCT
cana-1932	151	9	28(3	28(3	NUM
cana-1932	151	10	)	)	PUNCT
cana-1932	151	11	,	,	PUNCT
cana-1932	151	12	345	345	NUM
cana-1932	151	13	-	-	SYM
cana-1932	151	14	352	352	NUM
cana-1932	151	15	.	.	PUNCT
cana-1932	152	1	[	[	X
cana-1932	152	2	10	10	NUM
cana-1932	152	3	]	]	SYM
cana-1932	152	4	bhattacharyya	bhattacharyya	PROPN
cana-1932	152	5	,	,	PUNCT
cana-1932	152	6	r.	r.	PROPN
cana-1932	152	7	,	,	PUNCT
cana-1932	152	8	&	&	CCONJ
cana-1932	152	9	pal	pal	NOUN
cana-1932	152	10	,	,	PUNCT
cana-1932	152	11	t.	t.	PROPN
cana-1932	152	12	k.	k.	PROPN
cana-1932	152	13	(	(	PUNCT
cana-1932	152	14	2000	2000	NUM
cana-1932	152	15	)	)	PUNCT
cana-1932	152	16	.	.	PUNCT
cana-1932	153	1	fuzzy	fuzzy	ADJ
cana-1932	153	2	s	s	NOUN
cana-1932	153	3	-	-	PUNCT
cana-1932	153	4	hausdorff	hausdorff	NOUN
cana-1932	153	5	spaces	space	NOUN
cana-1932	153	6	.	.	PUNCT
cana-1932	154	1	fuzzy	fuzzy	ADJ
cana-1932	154	2	sets	set	NOUN
cana-1932	154	3	and	and	CCONJ
cana-1932	154	4	systems	system	NOUN
cana-1932	154	5	,	,	PUNCT
cana-1932	154	6	111(1	111(1	NUM
cana-1932	154	7	)	)	PUNCT
cana-1932	154	8	,	,	PUNCT
cana-1932	154	9	119	119	NUM
cana-1932	154	10	-	-	SYM
cana-1932	154	11	125	125	NUM
cana-1932	154	12	.	.	PUNCT
cana-1932	155	1	[	[	X
cana-1932	155	2	11	11	NUM
cana-1932	155	3	]	]	X
cana-1932	155	4	samanta	samanta	PROPN
cana-1932	155	5	,	,	PUNCT
cana-1932	155	6	t.	t.	PROPN
cana-1932	155	7	,	,	PUNCT
cana-1932	155	8	&	&	CCONJ
cana-1932	155	9	mondal	mondal	PROPN
cana-1932	155	10	,	,	PUNCT
cana-1932	155	11	s.	s.	PROPN
cana-1932	155	12	(	(	PUNCT
cana-1932	155	13	2010	2010	NUM
cana-1932	155	14	)	)	PUNCT
cana-1932	155	15	.	.	PUNCT
cana-1932	155	16	fuzzy	fuzzy	ADJ
cana-1932	155	17	bitopological	bitopological	ADJ
cana-1932	155	18	spaces	space	NOUN
cana-1932	155	19	.	.	PUNCT
cana-1932	156	1	journal	journal	NOUN
cana-1932	156	2	of	of	ADP
cana-1932	156	3	fuzzy	fuzzy	ADJ
cana-1932	156	4	mathematics	mathematic	NOUN
cana-1932	156	5	,	,	PUNCT
cana-1932	156	6	18(3	18(3	NUM
cana-1932	156	7	)	)	PUNCT
cana-1932	156	8	,	,	PUNCT
cana-1932	156	9	543	543	NUM
cana-1932	156	10	-	-	SYM
cana-1932	156	11	555	555	NUM
cana-1932	156	12	.	.	PUNCT
cana-1932	157	1	[	[	X
cana-1932	157	2	12	12	NUM
cana-1932	157	3	]	]	X
cana-1932	157	4	sridharan	sridharan	ADJ
cana-1932	157	5	,	,	PUNCT
cana-1932	157	6	m.	m.	NOUN
cana-1932	157	7	,	,	PUNCT
cana-1932	157	8	&	&	CCONJ
cana-1932	157	9	sivakumar	sivakumar	PROPN
cana-1932	157	10	,	,	PUNCT
cana-1932	157	11	m.	m.	NOUN
cana-1932	157	12	(	(	PUNCT
cana-1932	157	13	1997	1997	NUM
cana-1932	157	14	)	)	PUNCT
cana-1932	157	15	.	.	PUNCT
cana-1932	158	1	double	double	ADJ
cana-1932	158	2	fuzzy	fuzzy	ADJ
cana-1932	158	3	topological	topological	ADJ
cana-1932	158	4	spaces	space	NOUN
cana-1932	158	5	.	.	PUNCT
cana-1932	159	1	indian	indian	ADJ
cana-1932	159	2	journal	journal	PROPN
cana-1932	159	3	of	of	ADP
cana-1932	159	4	pure	pure	ADJ
cana-1932	159	5	and	and	CCONJ
cana-1932	159	6	applied	applied	ADJ
cana-1932	159	7	mathematics	mathematic	NOUN
cana-1932	159	8	,	,	PUNCT
cana-1932	159	9	28(3	28(3	NUM
cana-1932	159	10	)	)	PUNCT
cana-1932	159	11	,	,	PUNCT
cana-1932	159	12	345	345	NUM
cana-1932	159	13	-	-	SYM
cana-1932	159	14	352	352	NUM
cana-1932	159	15	.	.	PUNCT
cana-1932	160	1	[	[	X
cana-1932	160	2	13	13	NUM
cana-1932	160	3	]	]	SYM
cana-1932	160	4	basu	basu	PROPN
cana-1932	160	5	,	,	PUNCT
cana-1932	160	6	a.	a.	PROPN
cana-1932	160	7	,	,	PUNCT
cana-1932	160	8	&	&	CCONJ
cana-1932	160	9	pal	pal	NOUN
cana-1932	160	10	,	,	PUNCT
cana-1932	160	11	t.	t.	PROPN
cana-1932	160	12	k.	k.	PROPN
cana-1932	160	13	(	(	PUNCT
cana-1932	160	14	2003	2003	NUM
cana-1932	160	15	)	)	PUNCT
cana-1932	160	16	.	.	PUNCT
cana-1932	161	1	fuzzy	fuzzy	ADJ
cana-1932	161	2	topological	topological	ADJ
cana-1932	161	3	spaces	space	NOUN
cana-1932	161	4	and	and	CCONJ
cana-1932	161	5	fuzzy	fuzzy	ADJ
cana-1932	161	6	continuity	continuity	NOUN
cana-1932	161	7	.	.	PUNCT
cana-1932	162	1	fuzzy	fuzzy	ADJ
cana-1932	162	2	sets	set	NOUN
cana-1932	162	3	and	and	CCONJ
cana-1932	162	4	systems	system	NOUN
cana-1932	162	5	,	,	PUNCT
cana-1932	162	6	136(2	136(2	NUM
cana-1932	162	7	)	)	PUNCT
cana-1932	162	8	,	,	PUNCT
cana-1932	162	9	171183	171183	NUM
cana-1932	162	10	.	.	PUNCT
cana-1932	163	1	[	[	X
cana-1932	163	2	14	14	NUM
cana-1932	163	3	]	]	SYM
cana-1932	163	4	mendel	mendel	PROPN
cana-1932	163	5	,	,	PUNCT
cana-1932	163	6	j.	j.	PROPN
cana-1932	163	7	m.	m.	PROPN
cana-1932	163	8	,	,	PUNCT
cana-1932	163	9	&	&	CCONJ
cana-1932	163	10	john	john	PROPN
cana-1932	163	11	,	,	PUNCT
cana-1932	163	12	j.	j.	PROPN
cana-1932	163	13	(	(	PUNCT
cana-1932	163	14	2002	2002	NUM
cana-1932	163	15	)	)	PUNCT
cana-1932	163	16	.	.	PUNCT
cana-1932	164	1	fuzzy	fuzzy	ADJ
cana-1932	164	2	logic	logic	NOUN
cana-1932	164	3	systems	system	NOUN
cana-1932	164	4	for	for	ADP
cana-1932	164	5	engineering	engineering	NOUN
cana-1932	164	6	:	:	PUNCT
cana-1932	164	7	a	a	DET
cana-1932	164	8	tutorial	tutorial	NOUN
cana-1932	164	9	.	.	PUNCT
cana-1932	165	1	proceedings	proceeding	NOUN
cana-1932	165	2	of	of	ADP
cana-1932	165	3	the	the	DET
cana-1932	165	4	ieee	ieee	NOUN
cana-1932	165	5	,	,	PUNCT
cana-1932	165	6	90(9	90(9	NUM
cana-1932	165	7	)	)	PUNCT
cana-1932	165	8	,	,	PUNCT
cana-1932	165	9	1557	1557	NUM
cana-1932	165	10	-	-	SYM
cana-1932	165	11	1577	1577	NUM
cana-1932	165	12	.	.	PUNCT
cana-1932	166	1	[	[	X
cana-1932	166	2	15	15	NUM
cana-1932	166	3	]	]	X
cana-1932	166	4	gupta	gupta	PROPN
cana-1932	166	5	,	,	PUNCT
cana-1932	166	6	m.	m.	NOUN
cana-1932	166	7	m.	m.	NOUN
cana-1932	166	8	,	,	PUNCT
cana-1932	166	9	&	&	CCONJ
cana-1932	166	10	gupta	gupta	PROPN
cana-1932	166	11	,	,	PUNCT
cana-1932	166	12	v.	v.	PROPN
cana-1932	166	13	(	(	PUNCT
cana-1932	166	14	2004	2004	NUM
cana-1932	166	15	)	)	PUNCT
cana-1932	166	16	.	.	PUNCT
cana-1932	167	1	introduction	introduction	NOUN
cana-1932	167	2	to	to	ADP
cana-1932	167	3	fuzzy	fuzzy	ADJ
cana-1932	167	4	systems	system	NOUN
cana-1932	167	5	and	and	CCONJ
cana-1932	167	6	fuzzy	fuzzy	ADJ
cana-1932	167	7	logic	logic	NOUN
cana-1932	167	8	.	.	PUNCT
cana-1932	168	1	in	in	ADP
cana-1932	168	2	handbook	handbook	NOUN
cana-1932	168	3	of	of	ADP
cana-1932	168	4	fuzzy	fuzzy	ADJ
cana-1932	168	5	logic	logic	NOUN
cana-1932	168	6	and	and	CCONJ
cana-1932	168	7	fuzzy	fuzzy	ADJ
cana-1932	168	8	systems	system	NOUN
cana-1932	168	9	,	,	PUNCT
cana-1932	168	10	vol	vol	NOUN
cana-1932	168	11	.	.	PROPN
cana-1932	168	12	1	1	NUM
cana-1932	168	13	(	(	PUNCT
cana-1932	168	14	pp	pp	ADJ
cana-1932	168	15	.	.	PUNCT
cana-1932	169	1	3	3	NUM
cana-1932	169	2	-	-	SYM
cana-1932	169	3	26	26	NUM
cana-1932	169	4	)	)	PUNCT
cana-1932	169	5	.	.	PUNCT
cana-1932	170	1	springer	springer	NOUN
cana-1932	170	2	.	.	PUNCT