id	sid	tid	token	lemma	pos
cana-1225	1	1	communications	communication	NOUN
cana-1225	1	2	on	on	ADP
cana-1225	1	3	applied	apply	VERB
cana-1225	1	4	nonlinear	nonlinear	ADJ
cana-1225	1	5	analysis	analysis	NOUN
cana-1225	1	6	issn	issn	NOUN
cana-1225	1	7	:	:	PUNCT
cana-1225	1	8	1074	1074	NUM
cana-1225	1	9	-	-	PUNCT
cana-1225	1	10	133x	133x	NUM
cana-1225	1	11	vol	vol	NOUN
cana-1225	1	12	31	31	NUM
cana-1225	1	13	no	no	NOUN
cana-1225	1	14	.	.	PUNCT
cana-1225	2	1	6s	6s	NUM
cana-1225	2	2	(	(	PUNCT
cana-1225	2	3	2024	2024	NUM
cana-1225	2	4	)	)	PUNCT
cana-1225	2	5	318	318	NUM
cana-1225	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1225	2	7	bipolar	bipolar	ADJ
cana-1225	2	8	vague	vague	NOUN
cana-1225	2	9	𝜶	𝜶	ADP
cana-1225	2	10	generalized	generalize	VERB
cana-1225	2	11	continuous	continuous	ADJ
cana-1225	2	12	mappings	mapping	NOUN
cana-1225	2	13	in	in	ADP
cana-1225	2	14	topological	topological	ADJ
cana-1225	2	15	spaces	space	NOUN
cana-1225	2	16	f.	f.	PROPN
cana-1225	2	17	prishka1	prishka1	PROPN
cana-1225	2	18	and	and	CCONJ
cana-1225	2	19	dr	dr	PROPN
cana-1225	2	20	.	.	PROPN
cana-1225	2	21	l.	l.	PROPN
cana-1225	2	22	mariapresenti2	mariapresenti2	PUNCT
cana-1225	3	1	1research	1research	NUM
cana-1225	3	2	scholar	scholar	NOUN
cana-1225	3	3	,	,	PUNCT
cana-1225	3	4	department	department	NOUN
cana-1225	3	5	of	of	ADP
cana-1225	3	6	mathematics	mathematics	PROPN
cana-1225	3	7	,	,	PUNCT
cana-1225	3	8	nirmala	nirmala	PROPN
cana-1225	3	9	college	college	PROPN
cana-1225	3	10	for	for	ADP
cana-1225	3	11	women	woman	NOUN
cana-1225	3	12	,	,	PUNCT
cana-1225	3	13	redfield	redfield	PROPN
cana-1225	3	14	’s	’s	PROPN
cana-1225	3	15	,	,	PUNCT
cana-1225	3	16	coimbatore	coimbatore	PROPN
cana-1225	3	17	,	,	PUNCT
cana-1225	3	18	tamil	tamil	PROPN
cana-1225	3	19	nadu	nadu	PROPN
cana-1225	3	20	,	,	PUNCT
cana-1225	3	21	india	india	PROPN
cana-1225	3	22	.	.	PUNCT
cana-1225	4	1	2assistant	2assistant	NUM
cana-1225	4	2	professor	professor	NOUN
cana-1225	4	3	,	,	PUNCT
cana-1225	4	4	department	department	NOUN
cana-1225	4	5	of	of	ADP
cana-1225	4	6	mathematics	mathematics	PROPN
cana-1225	4	7	,	,	PUNCT
cana-1225	4	8	nirmala	nirmala	PROPN
cana-1225	4	9	college	college	PROPN
cana-1225	4	10	for	for	ADP
cana-1225	4	11	women	woman	NOUN
cana-1225	4	12	,	,	PUNCT
cana-1225	4	13	redfield	redfield	PROPN
cana-1225	4	14	’s	’s	PROPN
cana-1225	4	15	,	,	PUNCT
cana-1225	4	16	coimbatore	coimbatore	PROPN
cana-1225	4	17	,	,	PUNCT
cana-1225	4	18	tamil	tamil	PROPN
cana-1225	4	19	nadu	nadu	PROPN
cana-1225	4	20	,	,	PUNCT
cana-1225	4	21	india	india	PROPN
cana-1225	4	22	.	.	PUNCT
cana-1225	5	1	email1	email1	PROPN
cana-1225	5	2	:	:	PUNCT
cana-1225	5	3	prishkamaths@gmail.com	prishkamaths@gmail.com	X
cana-1225	5	4	and	and	CCONJ
cana-1225	5	5	email2	email2	NOUN
cana-1225	5	6	:	:	PUNCT
cana-1225	5	7	presentimaria88@gmail.com	presentimaria88@gmail.com	X
cana-1225	5	8	article	article	NOUN
cana-1225	5	9	history	history	NOUN
cana-1225	5	10	:	:	PUNCT
cana-1225	5	11	received	receive	VERB
cana-1225	5	12	:	:	PUNCT
cana-1225	5	13	10	10	NUM
cana-1225	5	14	-	-	SYM
cana-1225	5	15	06	06	NUM
cana-1225	5	16	-	-	PUNCT
cana-1225	5	17	2024	2024	NUM
cana-1225	5	18	revised	revise	VERB
cana-1225	5	19	:	:	PUNCT
cana-1225	5	20	10	10	NUM
cana-1225	5	21	-	-	SYM
cana-1225	5	22	07	07	NUM
cana-1225	5	23	-	-	PUNCT
cana-1225	5	24	2024	2024	NUM
cana-1225	5	25	accepted	accept	VERB
cana-1225	5	26	:	:	PUNCT
cana-1225	5	27	30	30	NUM
cana-1225	5	28	-	-	SYM
cana-1225	5	29	07	07	NUM
cana-1225	5	30	-	-	PUNCT
cana-1225	5	31	2024	2024	NUM
cana-1225	5	32	abstract	abstract	NOUN
cana-1225	5	33	:	:	PUNCT
cana-1225	5	34	in	in	ADP
cana-1225	5	35	this	this	DET
cana-1225	5	36	paper	paper	NOUN
cana-1225	5	37	we	we	PRON
cana-1225	5	38	have	have	AUX
cana-1225	5	39	introduced	introduce	VERB
cana-1225	5	40	bipolar	bipolar	ADJ
cana-1225	5	41	vague	vague	NOUN
cana-1225	5	42	𝛼	𝛼	ADP
cana-1225	5	43	generalized	generalize	VERB
cana-1225	5	44	continuous	continuous	ADJ
cana-1225	5	45	mappings	mapping	NOUN
cana-1225	5	46	in	in	ADP
cana-1225	5	47	topological	topological	ADJ
cana-1225	5	48	spaces	space	NOUN
cana-1225	5	49	and	and	CCONJ
cana-1225	5	50	investigated	investigate	VERB
cana-1225	5	51	some	some	PRON
cana-1225	5	52	of	of	ADP
cana-1225	5	53	their	their	PRON
cana-1225	5	54	properties	property	NOUN
cana-1225	5	55	.	.	PUNCT
cana-1225	6	1	also	also	ADV
cana-1225	6	2	,	,	PUNCT
cana-1225	6	3	we	we	PRON
cana-1225	6	4	have	have	AUX
cana-1225	6	5	provided	provide	VERB
cana-1225	6	6	some	some	DET
cana-1225	6	7	characterization	characterization	NOUN
cana-1225	6	8	of	of	ADP
cana-1225	6	9	bipolar	bipolar	ADJ
cana-1225	6	10	vague	vague	NOUN
cana-1225	6	11	𝛼	𝛼	ADP
cana-1225	6	12	generalized	generalize	VERB
cana-1225	6	13	continuous	continuous	ADJ
cana-1225	6	14	mappings	mapping	NOUN
cana-1225	6	15	in	in	ADP
cana-1225	6	16	topological	topological	ADJ
cana-1225	6	17	spaces	space	NOUN
cana-1225	6	18	.	.	PUNCT
cana-1225	7	1	keywords	keyword	NOUN
cana-1225	7	2	:	:	PUNCT
cana-1225	7	3	bipolar	bipolar	ADJ
cana-1225	7	4	vague	vague	ADJ
cana-1225	7	5	sets	set	NOUN
cana-1225	7	6	,	,	PUNCT
cana-1225	7	7	bipolar	bipolar	ADJ
cana-1225	7	8	vague	vague	ADJ
cana-1225	7	9	topology	topology	NOUN
cana-1225	7	10	,	,	PUNCT
cana-1225	7	11	bipolar	bipolar	ADJ
cana-1225	7	12	vague	vague	NOUN
cana-1225	7	13	𝛼	𝛼	PRON
cana-1225	7	14	generalized	generalize	VERB
cana-1225	7	15	closed	close	VERB
cana-1225	7	16	sets	set	NOUN
cana-1225	7	17	,	,	PUNCT
cana-1225	7	18	bipolar	bipolar	ADJ
cana-1225	7	19	vague	vague	NOUN
cana-1225	8	1	𝛼	𝛼	ADP
cana-1225	8	2	generalized	generalize	VERB
cana-1225	8	3	continuous	continuous	ADJ
cana-1225	8	4	mappings	mapping	NOUN
cana-1225	8	5	and	and	CCONJ
cana-1225	8	6	bipolar	bipolar	ADJ
cana-1225	8	7	vague	vague	NOUN
cana-1225	8	8	𝛼	𝛼	ADP
cana-1225	8	9	generalized	generalize	VERB
cana-1225	8	10	irresolute	irresolute	ADJ
cana-1225	8	11	mappings	mapping	NOUN
cana-1225	8	12	.	.	PUNCT
cana-1225	9	1	1	1	X
cana-1225	9	2	.	.	X
cana-1225	9	3	introduction	introduction	NOUN
cana-1225	9	4	fuzzy	fuzzy	ADJ
cana-1225	9	5	set	set	NOUN
cana-1225	9	6	was	be	AUX
cana-1225	9	7	introduced	introduce	VERB
cana-1225	9	8	by	by	ADP
cana-1225	9	9	l.a.zadeh	l.a.zadeh	NOUN
cana-1225	9	10	[	[	X
cana-1225	9	11	11	11	NUM
cana-1225	9	12	]	]	PUNCT
cana-1225	9	13	in	in	ADP
cana-1225	9	14	1965	1965	NUM
cana-1225	9	15	.	.	PUNCT
cana-1225	10	1	the	the	DET
cana-1225	10	2	concept	concept	NOUN
cana-1225	10	3	of	of	ADP
cana-1225	10	4	fuzzy	fuzzy	ADJ
cana-1225	10	5	topology	topology	NOUN
cana-1225	10	6	was	be	AUX
cana-1225	10	7	introduced	introduce	VERB
cana-1225	10	8	by	by	ADP
cana-1225	10	9	c.l.chang	c.l.chang	NOUN
cana-1225	10	10	[	[	X
cana-1225	10	11	3	3	NUM
cana-1225	10	12	]	]	PUNCT
cana-1225	10	13	in	in	ADP
cana-1225	10	14	1968	1968	NUM
cana-1225	10	15	.	.	PUNCT
cana-1225	11	1	the	the	DET
cana-1225	11	2	generalized	generalize	VERB
cana-1225	11	3	closed	close	VERB
cana-1225	11	4	sets	set	NOUN
cana-1225	11	5	in	in	ADP
cana-1225	11	6	general	general	ADJ
cana-1225	11	7	topology	topology	NOUN
cana-1225	11	8	were	be	AUX
cana-1225	11	9	first	first	ADV
cana-1225	11	10	introduced	introduce	VERB
cana-1225	11	11	by	by	ADP
cana-1225	11	12	n.levine	n.levine	PRON
cana-1225	11	13	[	[	X
cana-1225	11	14	9	9	NUM
cana-1225	11	15	]	]	PUNCT
cana-1225	11	16	in	in	ADP
cana-1225	11	17	1970	1970	NUM
cana-1225	11	18	.	.	PUNCT
cana-1225	12	1	k.atanassov	k.atanassov	X
cana-1225	13	1	[	[	X
cana-1225	13	2	2	2	X
cana-1225	13	3	]	]	PUNCT
cana-1225	13	4	in	in	ADP
cana-1225	13	5	1986	1986	NUM
cana-1225	13	6	introduced	introduce	VERB
cana-1225	13	7	the	the	DET
cana-1225	13	8	concept	concept	NOUN
cana-1225	13	9	of	of	ADP
cana-1225	13	10	intuitionistic	intuitionistic	ADJ
cana-1225	13	11	fuzzy	fuzzy	ADJ
cana-1225	13	12	sets	set	NOUN
cana-1225	13	13	.	.	PUNCT
cana-1225	14	1	the	the	DET
cana-1225	14	2	notion	notion	NOUN
cana-1225	14	3	of	of	ADP
cana-1225	14	4	vague	vague	ADJ
cana-1225	14	5	set	set	NOUN
cana-1225	14	6	theory	theory	NOUN
cana-1225	14	7	was	be	AUX
cana-1225	14	8	introduced	introduce	VERB
cana-1225	14	9	by	by	ADP
cana-1225	14	10	w.l.gau	w.l.gau	PROPN
cana-1225	14	11	and	and	CCONJ
cana-1225	14	12	d.j.buehrer	d.j.buehrer	NOUN
cana-1225	14	13	[	[	X
cana-1225	14	14	7	7	X
cana-1225	14	15	]	]	PUNCT
cana-1225	14	16	in	in	ADP
cana-1225	14	17	1993	1993	NUM
cana-1225	14	18	.	.	PUNCT
cana-1225	15	1	d.coker	d.coker	NOUN
cana-1225	16	1	[	[	X
cana-1225	16	2	6	6	NUM
cana-1225	16	3	]	]	PUNCT
cana-1225	16	4	in	in	ADP
cana-1225	16	5	1997	1997	NUM
cana-1225	16	6	introduced	introduce	VERB
cana-1225	16	7	intuitionistic	intuitionistic	ADJ
cana-1225	16	8	fuzzy	fuzzy	ADJ
cana-1225	16	9	topological	topological	ADJ
cana-1225	16	10	spaces	space	NOUN
cana-1225	16	11	.	.	PUNCT
cana-1225	17	1	bipolarvalued	bipolarvalue	VERB
cana-1225	17	2	fuzzy	fuzzy	ADJ
cana-1225	17	3	sets	set	NOUN
cana-1225	17	4	,	,	PUNCT
cana-1225	17	5	which	which	PRON
cana-1225	17	6	was	be	AUX
cana-1225	17	7	introduced	introduce	VERB
cana-1225	17	8	by	by	ADP
cana-1225	17	9	k.m.lee	k.m.lee	PROPN
cana-1225	18	1	[	[	X
cana-1225	18	2	8	8	NUM
cana-1225	18	3	]	]	PUNCT
cana-1225	18	4	in	in	ADP
cana-1225	18	5	2000	2000	NUM
cana-1225	18	6	is	be	AUX
cana-1225	18	7	an	an	DET
cana-1225	18	8	extension	extension	NOUN
cana-1225	18	9	of	of	ADP
cana-1225	18	10	fuzzy	fuzzy	ADJ
cana-1225	18	11	sets	set	NOUN
cana-1225	18	12	whose	whose	DET
cana-1225	18	13	membership	membership	NOUN
cana-1225	18	14	degree	degree	NOUN
cana-1225	18	15	range	range	NOUN
cana-1225	18	16	is	be	AUX
cana-1225	18	17	enlarged	enlarge	VERB
cana-1225	18	18	from	from	ADP
cana-1225	18	19	the	the	DET
cana-1225	18	20	interval	interval	NOUN
cana-1225	18	21	[	[	X
cana-1225	18	22	0	0	NUM
cana-1225	18	23	,	,	PUNCT
cana-1225	18	24	1	1	NUM
cana-1225	18	25	]	]	PUNCT
cana-1225	18	26	to	to	ADP
cana-1225	18	27	[	[	X
cana-1225	18	28	-1,1	-1,1	X
cana-1225	18	29	]	]	X
cana-1225	18	30	.	.	PUNCT
cana-1225	19	1	a	a	DET
cana-1225	19	2	new	new	ADJ
cana-1225	19	3	class	class	NOUN
cana-1225	19	4	of	of	ADP
cana-1225	19	5	generalized	generalized	ADJ
cana-1225	19	6	bipolar	bipolar	ADJ
cana-1225	19	7	vague	vague	ADJ
cana-1225	19	8	sets	set	NOUN
cana-1225	19	9	was	be	AUX
cana-1225	19	10	introduced	introduce	VERB
cana-1225	19	11	by	by	ADP
cana-1225	19	12	s.cicily	s.cicily	ADV
cana-1225	19	13	flora	flora	NOUN
cana-1225	19	14	and	and	CCONJ
cana-1225	19	15	i.arockiarani	i.arockiarani	NOUN
cana-1225	20	1	[	[	X
cana-1225	20	2	4	4	X
cana-1225	20	3	]	]	PUNCT
cana-1225	20	4	in	in	ADP
cana-1225	20	5	2016	2016	NUM
cana-1225	20	6	.	.	PUNCT
cana-1225	21	1	f.prishka	f.prishka	ADJ
cana-1225	21	2	and	and	CCONJ
cana-1225	21	3	l.mariapresenti	l.mariapresenti	ADJ
cana-1225	21	4	[	[	X
cana-1225	21	5	10	10	NUM
cana-1225	21	6	]	]	PUNCT
cana-1225	21	7	introduced	introduce	VERB
cana-1225	21	8	bipolar	bipolar	ADJ
cana-1225	21	9	vague	vague	NOUN
cana-1225	21	10	𝛼	𝛼	DET
cana-1225	21	11	generalized	generalize	VERB
cana-1225	21	12	closed	close	VERB
cana-1225	21	13	sets	set	NOUN
cana-1225	21	14	in	in	ADP
cana-1225	21	15	topological	topological	ADJ
cana-1225	21	16	spaces	space	NOUN
cana-1225	21	17	.	.	PUNCT
cana-1225	22	1	in	in	ADP
cana-1225	22	2	continuation	continuation	NOUN
cana-1225	22	3	of	of	ADP
cana-1225	22	4	our	our	PRON
cana-1225	22	5	research	research	NOUN
cana-1225	22	6	work	work	NOUN
cana-1225	22	7	we	we	PRON
cana-1225	22	8	have	have	AUX
cana-1225	22	9	introduced	introduce	VERB
cana-1225	22	10	bipolar	bipolar	ADJ
cana-1225	22	11	vague	vague	NOUN
cana-1225	22	12	𝛼	𝛼	ADP
cana-1225	22	13	generalized	generalize	VERB
cana-1225	22	14	continuous	continuous	ADJ
cana-1225	22	15	mappings	mapping	NOUN
cana-1225	22	16	in	in	ADP
cana-1225	22	17	topological	topological	ADJ
cana-1225	22	18	spaces	space	NOUN
cana-1225	22	19	,	,	PUNCT
cana-1225	22	20	bipolar	bipolar	ADJ
cana-1225	22	21	vague	vague	NOUN
cana-1225	22	22	𝛼	𝛼	ADP
cana-1225	22	23	generalized	generalize	VERB
cana-1225	22	24	irresolute	irresolute	ADJ
cana-1225	22	25	mappings	mapping	NOUN
cana-1225	22	26	in	in	ADP
cana-1225	22	27	topological	topological	ADJ
cana-1225	22	28	spaces	space	NOUN
cana-1225	22	29	and	and	CCONJ
cana-1225	22	30	investigated	investigate	VERB
cana-1225	22	31	some	some	PRON
cana-1225	22	32	of	of	ADP
cana-1225	22	33	their	their	PRON
cana-1225	22	34	properties	property	NOUN
cana-1225	22	35	.	.	PUNCT
cana-1225	23	1	also	also	ADV
cana-1225	23	2	,	,	PUNCT
cana-1225	23	3	we	we	PRON
cana-1225	23	4	have	have	AUX
cana-1225	23	5	provided	provide	VERB
cana-1225	23	6	some	some	DET
cana-1225	23	7	characterization	characterization	NOUN
cana-1225	23	8	of	of	ADP
cana-1225	23	9	bipolar	bipolar	ADJ
cana-1225	23	10	vague	vague	NOUN
cana-1225	23	11	𝛼	𝛼	ADP
cana-1225	23	12	generalized	generalize	VERB
cana-1225	23	13	continuous	continuous	ADJ
cana-1225	23	14	mappings	mapping	NOUN
cana-1225	23	15	in	in	ADP
cana-1225	23	16	topological	topological	ADJ
cana-1225	23	17	spaces	space	NOUN
cana-1225	23	18	.	.	PUNCT
cana-1225	24	1	2	2	X
cana-1225	24	2	.	.	X
cana-1225	24	3	preliminaries	preliminary	NOUN
cana-1225	24	4	here	here	ADV
cana-1225	24	5	in	in	ADP
cana-1225	24	6	this	this	DET
cana-1225	24	7	paper	paper	NOUN
cana-1225	24	8	the	the	DET
cana-1225	24	9	bipolar	bipolar	ADJ
cana-1225	24	10	vague	vague	ADJ
cana-1225	24	11	topological	topological	ADJ
cana-1225	24	12	spaces	space	NOUN
cana-1225	24	13	are	be	AUX
cana-1225	24	14	denoted	denote	VERB
cana-1225	24	15	by	by	ADP
cana-1225	24	16	(	(	PUNCT
cana-1225	24	17	x	x	NOUN
cana-1225	24	18	,	,	PUNCT
cana-1225	24	19	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	24	20	)	)	PUNCT
cana-1225	24	21	.	.	PUNCT
cana-1225	25	1	also	also	ADV
cana-1225	25	2	,	,	PUNCT
cana-1225	25	3	the	the	DET
cana-1225	25	4	bipolar	bipolar	ADJ
cana-1225	25	5	vague	vague	ADJ
cana-1225	25	6	interior	interior	NOUN
cana-1225	25	7	,	,	PUNCT
cana-1225	25	8	bipolar	bipolar	ADJ
cana-1225	25	9	vague	vague	ADJ
cana-1225	25	10	closure	closure	NOUN
cana-1225	25	11	of	of	ADP
cana-1225	25	12	a	a	DET
cana-1225	25	13	bipolar	bipolar	ADJ
cana-1225	25	14	vague	vague	NOUN
cana-1225	25	15	set	set	NOUN
cana-1225	25	16	a	a	PRON
cana-1225	25	17	are	be	AUX
cana-1225	25	18	denoted	denote	VERB
cana-1225	25	19	by	by	ADP
cana-1225	25	20	bvint(a	bvint(a	PROPN
cana-1225	25	21	)	)	PUNCT
cana-1225	25	22	and	and	CCONJ
cana-1225	25	23	bvcl(a	bvcl(a	NUM
cana-1225	25	24	)	)	PUNCT
cana-1225	25	25	.	.	PUNCT
cana-1225	26	1	the	the	DET
cana-1225	26	2	complement	complement	NOUN
cana-1225	26	3	of	of	ADP
cana-1225	26	4	a	a	DET
cana-1225	26	5	bipolar	bipolar	ADJ
cana-1225	26	6	vague	vague	NOUN
cana-1225	26	7	set	set	NOUN
cana-1225	26	8	a	a	PRON
cana-1225	26	9	is	be	AUX
cana-1225	26	10	denoted	denote	VERB
cana-1225	26	11	by	by	ADP
cana-1225	26	12	ac	ac	PROPN
cana-1225	26	13	and	and	CCONJ
cana-1225	26	14	the	the	DET
cana-1225	26	15	empty	empty	ADJ
cana-1225	26	16	set	set	NOUN
cana-1225	26	17	and	and	CCONJ
cana-1225	26	18	whole	whole	ADJ
cana-1225	26	19	sets	set	NOUN
cana-1225	26	20	are	be	AUX
cana-1225	26	21	denoted	denote	VERB
cana-1225	26	22	by	by	ADP
cana-1225	26	23	0~	0~	NOUN
cana-1225	26	24	and	and	CCONJ
cana-1225	26	25	1~	1~	NUM
cana-1225	26	26	respectively	respectively	ADV
cana-1225	26	27	.	.	PUNCT
cana-1225	27	1	definition	definition	NOUN
cana-1225	27	2	2.1	2.1	NUM
cana-1225	27	3	:	:	PUNCT
cana-1225	28	1	[	[	X
cana-1225	28	2	8	8	NUM
cana-1225	28	3	]	]	PUNCT
cana-1225	28	4	let	let	VERB
cana-1225	28	5	x	x	PRON
cana-1225	28	6	be	be	AUX
cana-1225	28	7	the	the	DET
cana-1225	28	8	universe	universe	NOUN
cana-1225	28	9	.	.	PUNCT
cana-1225	29	1	then	then	ADV
cana-1225	29	2	a	a	DET
cana-1225	29	3	bipolar	bipolar	ADJ
cana-1225	29	4	valued	value	VERB
cana-1225	29	5	fuzzy	fuzzy	ADJ
cana-1225	29	6	sets	set	NOUN
cana-1225	29	7	,	,	PUNCT
cana-1225	29	8	a	a	DET
cana-1225	29	9	on	on	NOUN
cana-1225	29	10	x	x	VERB
cana-1225	29	11	is	be	AUX
cana-1225	29	12	defined	define	VERB
cana-1225	29	13	by	by	ADP
cana-1225	29	14	positive	positive	ADJ
cana-1225	29	15	membership	membership	NOUN
cana-1225	29	16	function	function	NOUN
cana-1225	29	17	𝜇𝐴	𝜇𝐴	ADP
cana-1225	29	18	+	+	PROPN
cana-1225	29	19	,	,	PUNCT
cana-1225	29	20	that	that	PRON
cana-1225	29	21	is	be	AUX
cana-1225	29	22	𝜇𝐴	𝜇𝐴	ADP
cana-1225	29	23	+	+	PROPN
cana-1225	29	24	:	:	PUNCT
cana-1225	29	25	x→	x→	PUNCT
cana-1225	30	1	[	[	X
cana-1225	30	2	0,1	0,1	NUM
cana-1225	30	3	]	]	PUNCT
cana-1225	30	4	,	,	PUNCT
cana-1225	30	5	and	and	CCONJ
cana-1225	30	6	a	a	DET
cana-1225	30	7	negative	negative	ADJ
cana-1225	30	8	membership	membership	NOUN
cana-1225	30	9	function	function	NOUN
cana-1225	30	10	𝜇𝐴	𝜇𝐴	ADP
cana-1225	30	11	−	−	PROPN
cana-1225	30	12	,	,	PUNCT
cana-1225	30	13	that	that	SCONJ
cana-1225	30	14	communications	communication	NOUN
cana-1225	30	15	on	on	ADP
cana-1225	30	16	applied	apply	VERB
cana-1225	30	17	nonlinear	nonlinear	ADJ
cana-1225	30	18	analysis	analysis	NOUN
cana-1225	30	19	issn	issn	NOUN
cana-1225	30	20	:	:	PUNCT
cana-1225	30	21	1074	1074	NUM
cana-1225	30	22	-	-	PUNCT
cana-1225	30	23	133x	133x	NUM
cana-1225	30	24	vol	vol	NOUN
cana-1225	30	25	31	31	NUM
cana-1225	30	26	no	no	NOUN
cana-1225	30	27	.	.	PUNCT
cana-1225	31	1	6s	6s	NUM
cana-1225	31	2	(	(	PUNCT
cana-1225	31	3	2024	2024	NUM
cana-1225	31	4	)	)	PUNCT
cana-1225	31	5	319	319	NUM
cana-1225	31	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1225	31	7	is	be	AUX
cana-1225	31	8	𝜇𝐴	𝜇𝐴	ADP
cana-1225	31	9	−	−	PROPN
cana-1225	31	10	:	:	PUNCT
cana-1225	31	11	x→	x→	PUNCT
cana-1225	32	1	[	[	X
cana-1225	32	2	-1,0	-1,0	X
cana-1225	32	3	]	]	X
cana-1225	32	4	.	.	PUNCT
cana-1225	33	1	for	for	ADP
cana-1225	33	2	the	the	DET
cana-1225	33	3	sake	sake	NOUN
cana-1225	33	4	of	of	ADP
cana-1225	33	5	simplicity	simplicity	NOUN
cana-1225	33	6	,	,	PUNCT
cana-1225	33	7	we	we	PRON
cana-1225	33	8	shall	shall	AUX
cana-1225	33	9	use	use	VERB
cana-1225	33	10	the	the	DET
cana-1225	33	11	symbol	symbol	NOUN
cana-1225	33	12	a	a	PRON
cana-1225	33	13	=	=	PUNCT
cana-1225	33	14	{	{	PUNCT
cana-1225	33	15	𝑥	𝑥	NOUN
cana-1225	33	16	,	,	PUNCT
cana-1225	33	17	𝜇𝐴	𝜇𝐴	ADP
cana-1225	33	18	+	+	PROPN
cana-1225	33	19	(	(	PUNCT
cana-1225	33	20	𝑥	𝑥	NOUN
cana-1225	33	21	)	)	PUNCT
cana-1225	33	22	,	,	PUNCT
cana-1225	33	23	𝜇𝐴	𝜇𝐴	ADP
cana-1225	33	24	−(x)	−(x)	NOUN
cana-1225	33	25	:	:	PUNCT
cana-1225	33	26	𝑥	𝑥	PROPN
cana-1225	33	27	∈	∈	PROPN
cana-1225	33	28	𝑋	𝑋	PROPN
cana-1225	33	29	}	}	PUNCT
cana-1225	33	30	.	.	PUNCT
cana-1225	34	1	definition	definition	NOUN
cana-1225	34	2	2.2	2.2	NUM
cana-1225	34	3	:	:	PUNCT
cana-1225	35	1	[	[	X
cana-1225	35	2	8	8	NUM
cana-1225	35	3	]	]	PUNCT
cana-1225	35	4	let	let	VERB
cana-1225	35	5	a	a	PRON
cana-1225	35	6	and	and	CCONJ
cana-1225	35	7	b	b	NOUN
cana-1225	35	8	be	be	AUX
cana-1225	35	9	two	two	NUM
cana-1225	35	10	bipolar	bipolar	ADJ
cana-1225	35	11	valued	value	VERB
cana-1225	35	12	fuzzy	fuzzy	ADJ
cana-1225	35	13	sets	set	NOUN
cana-1225	35	14	then	then	ADV
cana-1225	35	15	their	their	PRON
cana-1225	35	16	union	union	NOUN
cana-1225	35	17	,	,	PUNCT
cana-1225	35	18	intersection	intersection	NOUN
cana-1225	35	19	and	and	CCONJ
cana-1225	35	20	complement	complement	NOUN
cana-1225	35	21	are	be	AUX
cana-1225	35	22	defined	define	VERB
cana-1225	35	23	as	as	SCONJ
cana-1225	35	24	follows	follow	VERB
cana-1225	35	25	:	:	PUNCT
cana-1225	35	26	(	(	PUNCT
cana-1225	35	27	i	i	NOUN
cana-1225	35	28	)	)	PUNCT
cana-1225	35	29	𝜇𝐴∪𝐵	𝜇𝐴∪𝐵	NOUN
cana-1225	36	1	+	+	CCONJ
cana-1225	36	2	=	=	SYM
cana-1225	36	3	max	max	NOUN
cana-1225	36	4	{	{	PUNCT
cana-1225	36	5	𝜇𝐴	𝜇𝐴	ADP
cana-1225	36	6	+	+	ADJ
cana-1225	36	7	(	(	PUNCT
cana-1225	36	8	𝑥	𝑥	NOUN
cana-1225	36	9	)	)	PUNCT
cana-1225	36	10	,	,	PUNCT
cana-1225	36	11	𝜇𝐵	𝜇𝐵	VERB
cana-1225	36	12	+	+	NOUN
cana-1225	36	13	𝑥	𝑥	NOUN
cana-1225	36	14	)	)	PUNCT
cana-1225	36	15	}	}	PUNCT
cana-1225	36	16	(	(	PUNCT
cana-1225	36	17	ii	ii	NOUN
cana-1225	36	18	)	)	PUNCT
cana-1225	36	19	𝜇𝐴∪𝐵	𝜇𝐴∪𝐵	NOUN
cana-1225	36	20	−	−	PROPN
cana-1225	36	21	=	=	SYM
cana-1225	36	22	min	min	NOUN
cana-1225	36	23	{	{	PUNCT
cana-1225	36	24	𝜇𝐴	𝜇𝐴	ADP
cana-1225	36	25	−(𝑥	−(𝑥	NOUN
cana-1225	36	26	)	)	PUNCT
cana-1225	36	27	,	,	PUNCT
cana-1225	36	28	𝜇𝐵	𝜇𝐵	NOUN
cana-1225	36	29	−𝑥	−𝑥	NOUN
cana-1225	36	30	)	)	PUNCT
cana-1225	36	31	}	}	PUNCT
cana-1225	36	32	(	(	PUNCT
cana-1225	36	33	iii	iii	NOUN
cana-1225	36	34	)	)	PUNCT
cana-1225	36	35	𝜇𝐴∩𝐵	𝜇𝐴∩𝐵	NOUN
cana-1225	36	36	+	+	CCONJ
cana-1225	36	37	=	=	SYM
cana-1225	36	38	min	min	NOUN
cana-1225	36	39	{	{	PUNCT
cana-1225	36	40	𝜇𝐴	𝜇𝐴	ADP
cana-1225	36	41	+	+	ADJ
cana-1225	36	42	(	(	PUNCT
cana-1225	36	43	𝑥	𝑥	NOUN
cana-1225	36	44	)	)	PUNCT
cana-1225	36	45	,	,	PUNCT
cana-1225	36	46	𝜇𝐵	𝜇𝐵	VERB
cana-1225	36	47	+	+	NOUN
cana-1225	36	48	𝑥	𝑥	NOUN
cana-1225	36	49	)	)	PUNCT
cana-1225	36	50	}	}	PUNCT
cana-1225	36	51	(	(	PUNCT
cana-1225	36	52	iv	iv	X
cana-1225	36	53	)	)	PUNCT
cana-1225	36	54	𝜇𝐴∩𝐵	𝜇𝐴∩𝐵	NOUN
cana-1225	36	55	−	−	NOUN
cana-1225	37	1	=	=	SYM
cana-1225	37	2	max	max	PROPN
cana-1225	37	3	{	{	PUNCT
cana-1225	37	4	𝜇𝐴	𝜇𝐴	ADP
cana-1225	37	5	−(𝑥	−(𝑥	NOUN
cana-1225	37	6	)	)	PUNCT
cana-1225	37	7	,	,	PUNCT
cana-1225	37	8	𝜇𝐵	𝜇𝐵	NOUN
cana-1225	37	9	−𝑥	−𝑥	NOUN
cana-1225	37	10	)	)	PUNCT
cana-1225	37	11	}	}	PUNCT
cana-1225	37	12	(	(	PUNCT
cana-1225	37	13	v	v	NOUN
cana-1225	37	14	)	)	PUNCT
cana-1225	37	15	𝜇𝐴𝑐	𝜇𝐴𝑐	NOUN
cana-1225	37	16	+	+	CCONJ
cana-1225	37	17	(	(	PUNCT
cana-1225	37	18	x	x	X
cana-1225	37	19	)	)	PUNCT
cana-1225	37	20	=	=	SYM
cana-1225	38	1	1-𝜇𝐴	1-𝜇𝐴	NUM
cana-1225	39	1	+	+	ADJ
cana-1225	39	2	(	(	PUNCT
cana-1225	39	3	𝑥	𝑥	NOUN
cana-1225	39	4	)	)	PUNCT
cana-1225	39	5	and	and	CCONJ
cana-1225	39	6	𝜇𝐴𝑐	𝜇𝐴𝑐	VERB
cana-1225	39	7	−	−	PROPN
cana-1225	39	8	(	(	PUNCT
cana-1225	39	9	x	x	X
cana-1225	39	10	)	)	PUNCT
cana-1225	39	11	=	=	SYM
cana-1225	39	12	-1-𝜇𝐴	-1-𝜇𝐴	PUNCT
cana-1225	39	13	−(𝑥	−(𝑥	NOUN
cana-1225	39	14	)	)	PUNCT
cana-1225	39	15	for	for	ADP
cana-1225	39	16	all	all	DET
cana-1225	39	17	𝑥	𝑥	DET
cana-1225	39	18	∈	∈	PROPN
cana-1225	39	19	𝑋.	𝑋.	PROPN
cana-1225	39	20	definition	definition	NOUN
cana-1225	39	21	2.3	2.3	NUM
cana-1225	39	22	:	:	PUNCT
cana-1225	40	1	[	[	X
cana-1225	40	2	7	7	X
cana-1225	40	3	]	]	PUNCT
cana-1225	40	4	a	a	DET
cana-1225	40	5	vague	vague	NOUN
cana-1225	40	6	set	set	VERB
cana-1225	40	7	a	a	PRON
cana-1225	40	8	in	in	ADP
cana-1225	40	9	the	the	DET
cana-1225	40	10	universe	universe	NOUN
cana-1225	40	11	of	of	ADP
cana-1225	40	12	discourse	discourse	NOUN
cana-1225	40	13	u	u	NOUN
cana-1225	40	14	is	be	AUX
cana-1225	40	15	a	a	DET
cana-1225	40	16	pair	pair	NOUN
cana-1225	40	17	of	of	ADP
cana-1225	40	18	(	(	PUNCT
cana-1225	40	19	𝑡𝐴	𝑡𝐴	NOUN
cana-1225	40	20	,	,	PUNCT
cana-1225	40	21	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	40	22	)	)	PUNCT
cana-1225	40	23	where	where	SCONJ
cana-1225	40	24	𝑡𝐴	𝑡𝐴	ADV
cana-1225	40	25	:	:	PUNCT
cana-1225	40	26	u→[0,1	u→[0,1	ADV
cana-1225	40	27	]	]	PUNCT
cana-1225	40	28	,	,	PUNCT
cana-1225	40	29	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	40	30	:	:	PUNCT
cana-1225	40	31	u→[0,1	u→[0,1	ADV
cana-1225	40	32	]	]	PUNCT
cana-1225	40	33	are	be	AUX
cana-1225	40	34	the	the	DET
cana-1225	40	35	mapping	mapping	NOUN
cana-1225	40	36	such	such	ADJ
cana-1225	40	37	that	that	PRON
cana-1225	40	38	𝑡𝐴	𝑡𝐴	PROPN
cana-1225	40	39	+	+	NOUN
cana-1225	40	40	𝑓𝐴	𝑓𝐴	VERB
cana-1225	40	41	≤	≤	NOUN
cana-1225	40	42	1	1	NUM
cana-1225	40	43	for	for	ADP
cana-1225	40	44	all	all	DET
cana-1225	40	45	𝑢	𝑢	PRON
cana-1225	40	46	∈	∈	PROPN
cana-1225	40	47	𝑈.	𝑈.	NOUN
cana-1225	40	48	the	the	DET
cana-1225	40	49	function	function	NOUN
cana-1225	40	50	𝑡𝐴	𝑡𝐴	NOUN
cana-1225	40	51	and	and	CCONJ
cana-1225	40	52	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	40	53	are	be	AUX
cana-1225	40	54	called	call	VERB
cana-1225	40	55	true	true	ADJ
cana-1225	40	56	membership	membership	NOUN
cana-1225	40	57	function	function	NOUN
cana-1225	40	58	and	and	CCONJ
cana-1225	40	59	false	false	ADJ
cana-1225	40	60	membership	membership	NOUN
cana-1225	40	61	function	function	NOUN
cana-1225	40	62	respectively	respectively	ADV
cana-1225	40	63	.	.	PUNCT
cana-1225	41	1	the	the	DET
cana-1225	41	2	interval	interval	NOUN
cana-1225	41	3	[	[	X
cana-1225	41	4	𝑡𝐴	𝑡𝐴	NOUN
cana-1225	41	5	,	,	PUNCT
cana-1225	41	6	1	1	NUM
cana-1225	41	7	−	−	NOUN
cana-1225	41	8	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	41	9	]	]	PUNCT
cana-1225	41	10	is	be	AUX
cana-1225	41	11	called	call	VERB
cana-1225	41	12	the	the	DET
cana-1225	41	13	vague	vague	ADJ
cana-1225	41	14	value	value	NOUN
cana-1225	41	15	of	of	ADP
cana-1225	41	16	u	u	NOUN
cana-1225	41	17	in	in	ADP
cana-1225	41	18	a	a	PRON
cana-1225	41	19	,	,	PUNCT
cana-1225	41	20	and	and	CCONJ
cana-1225	41	21	denoted	denote	VERB
cana-1225	41	22	by	by	ADP
cana-1225	41	23	𝜈𝐴(𝑢	𝜈𝐴(𝑢	NOUN
cana-1225	41	24	)	)	PUNCT
cana-1225	41	25	,	,	PUNCT
cana-1225	41	26	that	that	PRON
cana-1225	41	27	is	be	AUX
cana-1225	41	28	𝜈𝐴(𝑢	𝜈𝐴(𝑢	NOUN
cana-1225	41	29	)	)	PUNCT
cana-1225	41	30	=	=	PUNCT
cana-1225	42	1	[	[	X
cana-1225	42	2	𝑡𝐴(𝑢	𝑡𝐴(𝑢	NUM
cana-1225	42	3	)	)	PUNCT
cana-1225	42	4	,	,	PUNCT
cana-1225	42	5	1	1	NUM
cana-1225	42	6	−	−	PROPN
cana-1225	42	7	𝑓(𝑢	𝑓(𝑢	PROPN
cana-1225	42	8	)	)	PUNCT
cana-1225	42	9	]	]	PUNCT
cana-1225	42	10	.	.	PUNCT
cana-1225	43	1	definition	definition	NOUN
cana-1225	43	2	2.4	2.4	NUM
cana-1225	43	3	:	:	PUNCT
cana-1225	44	1	[	[	X
cana-1225	44	2	7	7	X
cana-1225	44	3	]	]	PUNCT
cana-1225	44	4	let	let	VERB
cana-1225	44	5	a	a	PRON
cana-1225	44	6	be	be	AUX
cana-1225	44	7	a	a	DET
cana-1225	44	8	non	non	ADJ
cana-1225	44	9	-	-	ADJ
cana-1225	44	10	empty	empty	ADJ
cana-1225	44	11	set	set	NOUN
cana-1225	44	12	and	and	CCONJ
cana-1225	44	13	the	the	DET
cana-1225	44	14	vague	vague	NOUN
cana-1225	44	15	set	set	VERB
cana-1225	44	16	a	a	PRON
cana-1225	44	17	and	and	CCONJ
cana-1225	44	18	b	b	NOUN
cana-1225	44	19	in	in	ADP
cana-1225	44	20	the	the	DET
cana-1225	44	21	form	form	NOUN
cana-1225	44	22	a	a	PRON
cana-1225	44	23	=	=	SYM
cana-1225	44	24	{	{	PUNCT
cana-1225	44	25	𝑥	𝑥	PROPN
cana-1225	44	26	,	,	PUNCT
cana-1225	44	27	𝑡𝐴(𝑥	𝑡𝐴(𝑥	NUM
cana-1225	44	28	)	)	PUNCT
cana-1225	44	29	,	,	PUNCT
cana-1225	44	30	1	1	NUM
cana-1225	44	31	−	−	NOUN
cana-1225	44	32	𝑓𝐴(𝑥)	𝑓𝐴(𝑥)	NOUN
cana-1225	44	33	:	:	PUNCT
cana-1225	44	34	x	x	X
cana-1225	44	35	∈	∈	NOUN
cana-1225	44	36	x	x	PUNCT
cana-1225	44	37	}	}	PUNCT
cana-1225	44	38	,	,	PUNCT
cana-1225	44	39	b	b	X
cana-1225	44	40	=	=	SYM
cana-1225	44	41	{	{	PUNCT
cana-1225	44	42	𝑥	𝑥	PROPN
cana-1225	44	43	,	,	PUNCT
cana-1225	44	44	𝑡𝐵(𝑥	𝑡𝐵(𝑥	PROPN
cana-1225	44	45	)	)	PUNCT
cana-1225	44	46	,	,	PUNCT
cana-1225	44	47	1	1	NUM
cana-1225	44	48	−	−	NOUN
cana-1225	44	49	𝑓𝐵(𝑥)	𝑓𝐵(𝑥)	NOUN
cana-1225	44	50	:	:	PUNCT
cana-1225	44	51	x	x	X
cana-1225	44	52	∈	∈	NOUN
cana-1225	44	53	x	x	PUNCT
cana-1225	44	54	}	}	PUNCT
cana-1225	44	55	.	.	PUNCT
cana-1225	45	1	then	then	ADV
cana-1225	45	2	(	(	PUNCT
cana-1225	45	3	i	i	NOUN
cana-1225	45	4	)	)	PUNCT
cana-1225	45	5	a	a	DET
cana-1225	45	6	⊆	⊆	NUM
cana-1225	45	7	b	b	NOUN
cana-1225	45	8	if	if	SCONJ
cana-1225	45	9	and	and	CCONJ
cana-1225	45	10	only	only	ADV
cana-1225	45	11	if	if	SCONJ
cana-1225	45	12	𝑡𝐴(𝑥	𝑡𝐴(𝑥	NUM
cana-1225	45	13	)	)	PUNCT
cana-1225	45	14	≤	≤	NOUN
cana-1225	45	15	𝑡𝐵(𝑥	𝑡𝐵(𝑥	PROPN
cana-1225	45	16	)	)	PUNCT
cana-1225	45	17	and	and	CCONJ
cana-1225	45	18	1	1	NUM
cana-1225	45	19	−	−	PROPN
cana-1225	45	20	𝑓𝐴(𝑥	𝑓𝐴(𝑥	NUM
cana-1225	45	21	)	)	PUNCT
cana-1225	45	22	≤	≤	NOUN
cana-1225	45	23	1	1	NUM
cana-1225	45	24	−	−	PROPN
cana-1225	45	25	𝑓𝐵(𝑥	𝑓𝐵(𝑥	PROPN
cana-1225	45	26	)	)	PUNCT
cana-1225	45	27	(	(	PUNCT
cana-1225	45	28	ii	ii	NOUN
cana-1225	45	29	)	)	PUNCT
cana-1225	45	30	a	a	DET
cana-1225	45	31	∪	∪	X
cana-1225	45	32	b	b	NOUN
cana-1225	45	33	=	=	SYM
cana-1225	45	34	{	{	PUNCT
cana-1225	45	35			X
cana-1225	45	36	max	max	PROPN
cana-1225	45	37	(	(	PUNCT
cana-1225	45	38	𝑡𝐴(𝑥	𝑡𝐴(𝑥	PROPN
cana-1225	45	39	)	)	PUNCT
cana-1225	45	40	,	,	PUNCT
cana-1225	45	41	𝑡𝐵(𝑥	𝑡𝐵(𝑥	PROPN
cana-1225	45	42	)	)	PUNCT
cana-1225	45	43	)	)	PUNCT
cana-1225	45	44	,	,	PUNCT
cana-1225	45	45	max	max	PROPN
cana-1225	45	46	(	(	PUNCT
cana-1225	45	47	1−𝑓𝐴(𝑥),1−𝑓𝐵(𝑥))	1−𝑓𝐴(𝑥),1−𝑓𝐵(𝑥))	NUM
cana-1225	45	48	x	x	SYM
cana-1225	45	49	∈	∈	NOUN
cana-1225	45	50	x	x	PUNCT
cana-1225	45	51	}	}	PUNCT
cana-1225	45	52	.	.	PUNCT
cana-1225	46	1	(	(	PUNCT
cana-1225	46	2	iii	iii	X
cana-1225	46	3	)	)	PUNCT
cana-1225	46	4	a	a	DET
cana-1225	46	5	∩	∩	ADJ
cana-1225	46	6	b	b	X
cana-1225	46	7	=	=	SYM
cana-1225	46	8	{	{	PUNCT
cana-1225	46	9			X
cana-1225	46	10	min	min	NOUN
cana-1225	46	11	(	(	PUNCT
cana-1225	46	12	𝑡𝐴(𝑥	𝑡𝐴(𝑥	NUM
cana-1225	46	13	)	)	PUNCT
cana-1225	46	14	,	,	PUNCT
cana-1225	46	15	𝑡𝐵(𝑥	𝑡𝐵(𝑥	PROPN
cana-1225	46	16	)	)	PUNCT
cana-1225	46	17	)	)	PUNCT
cana-1225	46	18	,	,	PUNCT
cana-1225	46	19	min	min	PROPN
cana-1225	46	20	(	(	PUNCT
cana-1225	46	21	1−𝑓𝐴(𝑥),1−𝑓𝐵(𝑥))	1−𝑓𝐴(𝑥),1−𝑓𝐵(𝑥))	NUM
cana-1225	46	22	x	x	SYM
cana-1225	46	23	∈	∈	NOUN
cana-1225	46	24	x	x	PUNCT
cana-1225	46	25	}	}	PUNCT
cana-1225	46	26	.	.	PUNCT
cana-1225	47	1	(	(	PUNCT
cana-1225	47	2	iv	iv	X
cana-1225	47	3	)	)	PUNCT
cana-1225	47	4	ac	ac	PROPN
cana-1225	48	1	=	=	SYM
cana-1225	48	2	{	{	PUNCT
cana-1225	48	3	𝑥	𝑥	PROPN
cana-1225	48	4	,	,	PUNCT
cana-1225	48	5	𝑓𝐴(𝑥	𝑓𝐴(𝑥	NUM
cana-1225	48	6	)	)	PUNCT
cana-1225	48	7	,	,	PUNCT
cana-1225	48	8	1	1	NUM
cana-1225	48	9	−	−	NOUN
cana-1225	48	10	𝑡𝐴(𝑥)	𝑡𝐴(𝑥)	NUM
cana-1225	48	11	:	:	PUNCT
cana-1225	48	12	x	x	X
cana-1225	48	13	∈	∈	NOUN
cana-1225	48	14	x	x	PUNCT
cana-1225	48	15	}	}	PUNCT
cana-1225	48	16	.	.	PUNCT
cana-1225	49	1	definition	definition	NOUN
cana-1225	49	2	2.5	2.5	NUM
cana-1225	49	3	:	:	PUNCT
cana-1225	50	1	[	[	X
cana-1225	50	2	1	1	X
cana-1225	50	3	]	]	PUNCT
cana-1225	50	4	let	let	VERB
cana-1225	50	5	x	x	PRON
cana-1225	50	6	be	be	AUX
cana-1225	50	7	the	the	DET
cana-1225	50	8	universe	universe	NOUN
cana-1225	50	9	of	of	ADP
cana-1225	50	10	discourse	discourse	NOUN
cana-1225	50	11	.	.	PUNCT
cana-1225	51	1	a	a	DET
cana-1225	51	2	bipolar	bipolar	ADV
cana-1225	51	3	-	-	PUNCT
cana-1225	51	4	valued	value	VERB
cana-1225	51	5	vague	vague	NOUN
cana-1225	51	6	set	set	VERB
cana-1225	51	7	a	a	DET
cana-1225	51	8	in	in	ADP
cana-1225	51	9	x	x	SYM
cana-1225	51	10	is	be	AUX
cana-1225	51	11	an	an	DET
cana-1225	51	12	object	object	NOUN
cana-1225	51	13	having	have	VERB
cana-1225	51	14	the	the	DET
cana-1225	51	15	form	form	NOUN
cana-1225	51	16	a	a	DET
cana-1225	51	17	=	=	SYM
cana-1225	51	18	{	{	PUNCT
cana-1225	51	19	x	x	NUM
cana-1225	51	20	,	,	PUNCT
cana-1225	51	21	[	[	X
cana-1225	51	22	𝑡𝐴	𝑡𝐴	ADP
cana-1225	51	23	+	+	ADJ
cana-1225	51	24	(	(	PUNCT
cana-1225	51	25	𝑥	𝑥	NOUN
cana-1225	51	26	)	)	PUNCT
cana-1225	51	27	,	,	PUNCT
cana-1225	51	28	1	1	NUM
cana-1225	51	29	−	−	NOUN
cana-1225	51	30	𝑓𝐴	𝑓𝐴	VERB
cana-1225	51	31	+	+	NOUN
cana-1225	51	32	(	(	PUNCT
cana-1225	51	33	𝑥	𝑥	NOUN
cana-1225	51	34	)	)	PUNCT
cana-1225	51	35	]	]	PUNCT
cana-1225	51	36	,	,	PUNCT
cana-1225	51	37	[	[	X
cana-1225	51	38	−1	−1	NOUN
cana-1225	51	39	−	−	ADP
cana-1225	51	40	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	51	41	−(𝑥	−(𝑥	NOUN
cana-1225	51	42	)	)	PUNCT
cana-1225	51	43	,	,	PUNCT
cana-1225	51	44	𝑡𝐴	𝑡𝐴	VERB
cana-1225	51	45	−(𝑥)]	−(𝑥)]	NUM
cana-1225	51	46	∶	∶	NOUN
cana-1225	51	47	x	x	X
cana-1225	51	48	∈	∈	NOUN
cana-1225	51	49	x	x	PUNCT
cana-1225	51	50	}	}	PUNCT
cana-1225	51	51	where	where	SCONJ
cana-1225	51	52	[	[	X
cana-1225	51	53	𝑡𝐴	𝑡𝐴	NOUN
cana-1225	51	54	+	+	ADJ
cana-1225	51	55	,	,	PUNCT
cana-1225	51	56	1	1	NUM
cana-1225	51	57	−	−	NOUN
cana-1225	51	58	𝑓𝐴	𝑓𝐴	VERB
cana-1225	51	59	+	+	PROPN
cana-1225	51	60	]	]	X
cana-1225	51	61	:	:	PUNCT
cana-1225	52	1	x→[0,1	x→[0,1	ADP
cana-1225	52	2	]	]	PUNCT
cana-1225	52	3	and	and	CCONJ
cana-1225	52	4	[	[	X
cana-1225	52	5	−1	−1	NOUN
cana-1225	52	6	−	−	NOUN
cana-1225	52	7	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	52	8	−	−	NOUN
cana-1225	52	9	,	,	PUNCT
cana-1225	52	10	𝑡𝐴	𝑡𝐴	VERB
cana-1225	52	11	−	−	NOUN
cana-1225	52	12	]	]	X
cana-1225	52	13	:	:	PUNCT
cana-1225	52	14	x→[-1,0	x→[-1,0	X
cana-1225	52	15	]	]	X
cana-1225	52	16	are	be	AUX
cana-1225	52	17	the	the	DET
cana-1225	52	18	mapping	mapping	NOUN
cana-1225	52	19	such	such	ADJ
cana-1225	52	20	that	that	SCONJ
cana-1225	53	1	𝑡𝐴	𝑡𝐴	ADP
cana-1225	53	2	+	+	PROPN
cana-1225	53	3	(	(	PUNCT
cana-1225	53	4	𝑥	𝑥	NOUN
cana-1225	53	5	)	)	PUNCT
cana-1225	53	6	+	+	CCONJ
cana-1225	53	7	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	53	8	+	+	ADJ
cana-1225	53	9	(	(	PUNCT
cana-1225	53	10	𝑥	𝑥	NOUN
cana-1225	53	11	)	)	PUNCT
cana-1225	53	12	≤	≤	NOUN
cana-1225	53	13	1	1	NUM
cana-1225	53	14	and	and	CCONJ
cana-1225	53	15	-1≤	-1≤	NOUN
cana-1225	53	16	𝑡𝐴	𝑡𝐴	PROPN
cana-1225	53	17	−+	−+	NOUN
cana-1225	53	18	𝑓𝐴	𝑓𝐴	VERB
cana-1225	53	19	−.	−.	ADV
cana-1225	53	20	the	the	DET
cana-1225	53	21	positive	positive	ADJ
cana-1225	53	22	membership	membership	NOUN
cana-1225	53	23	degree	degree	NOUN
cana-1225	54	1	[	[	X
cana-1225	54	2	𝑡𝐴	𝑡𝐴	ADP
cana-1225	54	3	+	+	ADJ
cana-1225	54	4	(	(	PUNCT
cana-1225	54	5	𝑥	𝑥	NOUN
cana-1225	54	6	)	)	PUNCT
cana-1225	54	7	,	,	PUNCT
cana-1225	54	8	1	1	NUM
cana-1225	54	9	−	−	NOUN
cana-1225	54	10	𝑓𝐴	𝑓𝐴	VERB
cana-1225	54	11	+	+	NOUN
cana-1225	54	12	(	(	PUNCT
cana-1225	54	13	𝑥	𝑥	NOUN
cana-1225	54	14	)	)	PUNCT
cana-1225	54	15	]	]	PUNCT
cana-1225	54	16	denotes	denote	VERB
cana-1225	54	17	the	the	DET
cana-1225	54	18	satisfaction	satisfaction	NOUN
cana-1225	54	19	region	region	NOUN
cana-1225	54	20	of	of	ADP
cana-1225	54	21	an	an	DET
cana-1225	54	22	element	element	NOUN
cana-1225	54	23	x	x	X
cana-1225	54	24	to	to	ADP
cana-1225	54	25	the	the	DET
cana-1225	54	26	property	property	NOUN
cana-1225	54	27	corresponding	correspond	VERB
cana-1225	54	28	to	to	ADP
cana-1225	54	29	a	a	DET
cana-1225	54	30	bipolar	bipolar	ADV
cana-1225	54	31	-	-	PUNCT
cana-1225	54	32	valued	value	VERB
cana-1225	54	33	set	set	NOUN
cana-1225	54	34	a	a	PRON
cana-1225	54	35	and	and	CCONJ
cana-1225	54	36	the	the	DET
cana-1225	54	37	negative	negative	ADJ
cana-1225	54	38	membership	membership	NOUN
cana-1225	54	39	degree	degree	NOUN
cana-1225	55	1	[	[	X
cana-1225	55	2	−1	−1	NOUN
cana-1225	55	3	−	−	ADP
cana-1225	55	4	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	55	5	−(𝑥	−(𝑥	NOUN
cana-1225	55	6	)	)	PUNCT
cana-1225	55	7	,	,	PUNCT
cana-1225	55	8	𝑡𝐴	𝑡𝐴	VERB
cana-1225	55	9	−(𝑥	−(𝑥	NOUN
cana-1225	55	10	)	)	PUNCT
cana-1225	55	11	]	]	PUNCT
cana-1225	55	12	denotes	denote	VERB
cana-1225	55	13	the	the	DET
cana-1225	55	14	satisfaction	satisfaction	NOUN
cana-1225	55	15	region	region	NOUN
cana-1225	55	16	of	of	ADP
cana-1225	55	17	x	x	INTJ
cana-1225	55	18	to	to	ADP
cana-1225	55	19	some	some	DET
cana-1225	55	20	implicit	implicit	ADJ
cana-1225	55	21	counter	counter	ADJ
cana-1225	55	22	property	property	NOUN
cana-1225	55	23	of	of	ADP
cana-1225	55	24	a.	a.	NOUN
cana-1225	55	25	for	for	ADP
cana-1225	55	26	a	a	DET
cana-1225	55	27	sake	sake	NOUN
cana-1225	55	28	of	of	ADP
cana-1225	55	29	simplicity	simplicity	NOUN
cana-1225	55	30	,	,	PUNCT
cana-1225	55	31	we	we	PRON
cana-1225	55	32	shall	shall	AUX
cana-1225	55	33	use	use	VERB
cana-1225	55	34	the	the	DET
cana-1225	55	35	notion	notion	NOUN
cana-1225	55	36	of	of	ADP
cana-1225	55	37	bipolar	bipolar	ADJ
cana-1225	55	38	vague	vague	ADJ
cana-1225	55	39	set	set	NOUN
cana-1225	55	40	𝜈𝐴	𝜈𝐴	ADV
cana-1225	56	1	+	+	PUNCT
cana-1225	56	2	=	=	PUNCT
cana-1225	57	1	[	[	X
cana-1225	57	2	𝑡𝐴	𝑡𝐴	NOUN
cana-1225	57	3	+	+	ADJ
cana-1225	57	4	,	,	PUNCT
cana-1225	57	5	1	1	NUM
cana-1225	57	6	−	−	NOUN
cana-1225	57	7	𝑓𝐴	𝑓𝐴	VERB
cana-1225	57	8	+	+	NOUN
cana-1225	57	9	]	]	X
cana-1225	57	10	and	and	CCONJ
cana-1225	57	11	𝜈𝐴	𝜈𝐴	ADP
cana-1225	57	12	−	−	PROPN
cana-1225	57	13	=	=	SYM
cana-1225	58	1	[	[	X
cana-1225	58	2	−1	−1	NOUN
cana-1225	58	3	−	−	NOUN
cana-1225	58	4	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	58	5	−	−	NOUN
cana-1225	58	6	,	,	PUNCT
cana-1225	58	7	𝑡𝐴	𝑡𝐴	VERB
cana-1225	58	8	−	−	NOUN
cana-1225	58	9	]	]	PUNCT
cana-1225	58	10	.	.	PUNCT
cana-1225	59	1	definition	definition	NOUN
cana-1225	59	2	2.6	2.6	NUM
cana-1225	59	3	:	:	PUNCT
cana-1225	60	1	[	[	X
cana-1225	60	2	5	5	X
cana-1225	60	3	]	]	PUNCT
cana-1225	60	4	a	a	DET
cana-1225	60	5	bipolar	bipolar	ADJ
cana-1225	60	6	vague	vague	NOUN
cana-1225	60	7	set	set	VERB
cana-1225	60	8	a	a	PRON
cana-1225	60	9	=	=	PRON
cana-1225	61	1	[	[	X
cana-1225	61	2	𝜈𝐴	𝜈𝐴	PROPN
cana-1225	61	3	+	+	ADJ
cana-1225	61	4	,	,	PUNCT
cana-1225	61	5	𝜈𝐴	𝜈𝐴	ADP
cana-1225	61	6	−	−	NOUN
cana-1225	61	7	]	]	PUNCT
cana-1225	61	8	of	of	ADP
cana-1225	61	9	a	a	DET
cana-1225	61	10	set	set	ADJ
cana-1225	61	11	u	u	NOUN
cana-1225	61	12	with	with	ADP
cana-1225	61	13	𝜈𝐴	𝜈𝐴	PROPN
cana-1225	61	14	+	+	PROPN
cana-1225	61	15	=	=	SYM
cana-1225	61	16	0	0	NUM
cana-1225	61	17	implies	imply	VERB
cana-1225	61	18	that	that	SCONJ
cana-1225	61	19	𝑡𝐴	𝑡𝐴	ADP
cana-1225	61	20	+	+	SYM
cana-1225	61	21	=	=	SYM
cana-1225	61	22	0	0	NUM
cana-1225	61	23	,	,	PUNCT
cana-1225	61	24	1	1	NUM
cana-1225	61	25	−	−	NOUN
cana-1225	61	26	𝑓𝐴	𝑓𝐴	VERB
cana-1225	61	27	+	+	NOUN
cana-1225	61	28	=	=	SYM
cana-1225	61	29	0	0	NUM
cana-1225	61	30	and	and	CCONJ
cana-1225	61	31	𝜈𝐴	𝜈𝐴	PROPN
cana-1225	61	32	−=	−=	ADP
cana-1225	61	33	0	0	NUM
cana-1225	61	34	implies	imply	VERB
cana-1225	61	35	that	that	SCONJ
cana-1225	61	36	𝑡𝐴	𝑡𝐴	VERB
cana-1225	61	37	−	−	PROPN
cana-1225	61	38	=	=	SYM
cana-1225	61	39	0	0	NUM
cana-1225	61	40	,	,	PUNCT
cana-1225	61	41	−1	−1	NOUN
cana-1225	61	42	−	−	NOUN
cana-1225	61	43	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	61	44	−	−	NOUN
cana-1225	61	45	=	=	SYM
cana-1225	61	46	0	0	NUM
cana-1225	61	47	for	for	ADP
cana-1225	61	48	all	all	PRON
cana-1225	61	49	x	x	SYM
cana-1225	61	50	∈	∈	NOUN
cana-1225	61	51	u	u	NOUN
cana-1225	61	52	is	be	AUX
cana-1225	61	53	called	call	VERB
cana-1225	61	54	zero	zero	NUM
cana-1225	61	55	bipolar	bipolar	ADJ
cana-1225	61	56	vague	vague	ADJ
cana-1225	61	57	set	set	NOUN
cana-1225	61	58	and	and	CCONJ
cana-1225	61	59	it	it	PRON
cana-1225	61	60	is	be	AUX
cana-1225	61	61	denoted	denote	VERB
cana-1225	61	62	by	by	ADP
cana-1225	61	63	0	0	NUM
cana-1225	61	64	.	.	PUNCT
cana-1225	61	65	definition	definition	NOUN
cana-1225	61	66	2.7	2.7	NUM
cana-1225	61	67	:	:	PUNCT
cana-1225	62	1	[	[	X
cana-1225	62	2	5	5	X
cana-1225	62	3	]	]	PUNCT
cana-1225	62	4	a	a	DET
cana-1225	62	5	bipolar	bipolar	ADJ
cana-1225	62	6	vague	vague	NOUN
cana-1225	62	7	set	set	VERB
cana-1225	62	8	a	a	PRON
cana-1225	62	9	=	=	PRON
cana-1225	63	1	[	[	X
cana-1225	63	2	𝜈𝐴	𝜈𝐴	PROPN
cana-1225	63	3	+	+	ADJ
cana-1225	63	4	,	,	PUNCT
cana-1225	63	5	𝜈𝐴	𝜈𝐴	ADP
cana-1225	63	6	−	−	NOUN
cana-1225	63	7	]	]	PUNCT
cana-1225	63	8	of	of	ADP
cana-1225	63	9	a	a	DET
cana-1225	63	10	set	set	ADJ
cana-1225	63	11	u	u	NOUN
cana-1225	63	12	with	with	ADP
cana-1225	63	13	𝜈𝐴	𝜈𝐴	PROPN
cana-1225	63	14	+	+	PROPN
cana-1225	63	15	=	=	SYM
cana-1225	63	16	1	1	NUM
cana-1225	63	17	implies	imply	VERB
cana-1225	63	18	that	that	SCONJ
cana-1225	63	19	𝑡𝐴	𝑡𝐴	ADP
cana-1225	63	20	+	+	SYM
cana-1225	63	21	=	=	SYM
cana-1225	63	22	1	1	NUM
cana-1225	63	23	,	,	PUNCT
cana-1225	63	24	1	1	NUM
cana-1225	63	25	−	−	NOUN
cana-1225	63	26	𝑓𝐴	𝑓𝐴	VERB
cana-1225	63	27	+	+	NOUN
cana-1225	63	28	=	=	SYM
cana-1225	63	29	1	1	NUM
cana-1225	63	30	and	and	CCONJ
cana-1225	63	31	𝜈𝐴	𝜈𝐴	PROPN
cana-1225	63	32	−=	−=	PUNCT
cana-1225	63	33	-1	-1	PROPN
cana-1225	63	34	implies	imply	VERB
cana-1225	63	35	that	that	SCONJ
cana-1225	63	36	𝑡𝐴	𝑡𝐴	NOUN
cana-1225	63	37	−	−	PROPN
cana-1225	63	38	=	=	SYM
cana-1225	63	39	-1	-1	ADJ
cana-1225	63	40	,	,	PUNCT
cana-1225	63	41	−1	−1	NOUN
cana-1225	63	42	−	−	NOUN
cana-1225	63	43	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	63	44	−	−	NOUN
cana-1225	63	45	=	=	SYM
cana-1225	63	46	-1	-1	ADP
cana-1225	63	47	for	for	ADP
cana-1225	63	48	all	all	PRON
cana-1225	63	49	x	x	SYM
cana-1225	63	50	∈	∈	NOUN
cana-1225	63	51	u	u	NOUN
cana-1225	63	52	is	be	AUX
cana-1225	63	53	called	call	VERB
cana-1225	63	54	unit	unit	ADJ
cana-1225	63	55	bipolar	bipolar	ADJ
cana-1225	63	56	vague	vague	NOUN
cana-1225	63	57	set	set	NOUN
cana-1225	63	58	and	and	CCONJ
cana-1225	63	59	it	it	PRON
cana-1225	63	60	is	be	AUX
cana-1225	63	61	denoted	denote	VERB
cana-1225	63	62	by	by	ADP
cana-1225	63	63	1	1	NUM
cana-1225	63	64	.	.	PUNCT
cana-1225	63	65	definition	definition	NOUN
cana-1225	63	66	2.8	2.8	NUM
cana-1225	63	67	:	:	PUNCT
cana-1225	64	1	[	[	X
cana-1225	64	2	4	4	X
cana-1225	64	3	]	]	PUNCT
cana-1225	64	4	let	let	VERB
cana-1225	64	5	a	a	DET
cana-1225	64	6	=	=	SYM
cana-1225	64	7	x	x	NOUN
cana-1225	64	8	,	,	PUNCT
cana-1225	64	9	[	[	X
cana-1225	64	10	𝑡𝐴	𝑡𝐴	SYM
cana-1225	64	11	+	+	ADJ
cana-1225	64	12	,	,	PUNCT
cana-1225	64	13	1	1	NUM
cana-1225	64	14	−	−	NOUN
cana-1225	64	15	𝑓𝐴	𝑓𝐴	VERB
cana-1225	65	1	+	+	PROPN
cana-1225	65	2	]	]	X
cana-1225	65	3	,	,	PUNCT
cana-1225	65	4	[	[	X
cana-1225	65	5	−1	−1	NOUN
cana-1225	65	6	−	−	VERB
cana-1225	65	7	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	65	8	−	−	NOUN
cana-1225	65	9	,	,	PUNCT
cana-1225	65	10	𝑡𝐴	𝑡𝐴	NOUN
cana-1225	65	11	−]	−]	PROPN
cana-1225	65	12	and	and	CCONJ
cana-1225	65	13	x	x	NUM
cana-1225	65	14	,	,	PUNCT
cana-1225	66	1	[	[	X
cana-1225	66	2	𝑡𝐵	𝑡𝐵	ADJ
cana-1225	66	3	+	+	PROPN
cana-1225	66	4	,	,	PUNCT
cana-1225	66	5	1	1	NUM
cana-1225	66	6	−	−	PROPN
cana-1225	66	7	𝑓𝐵	𝑓𝐵	PROPN
cana-1225	66	8	+	+	PROPN
cana-1225	66	9	]	]	X
cana-1225	66	10	,	,	PUNCT
cana-1225	66	11	[	[	X
cana-1225	66	12	−1	−1	NOUN
cana-1225	66	13	−	−	PROPN
cana-1225	66	14	𝑓𝐵	𝑓𝐵	PROPN
cana-1225	66	15	−	−	PROPN
cana-1225	66	16	,	,	PUNCT
cana-1225	66	17	𝑡𝐵	𝑡𝐵	ADJ
cana-1225	66	18	−]	−]	PRON
cana-1225	66	19	be	be	AUX
cana-1225	66	20	two	two	NUM
cana-1225	66	21	bipolar	bipolar	ADJ
cana-1225	66	22	vague	vague	ADJ
cana-1225	66	23	sets	set	NOUN
cana-1225	66	24	then	then	ADV
cana-1225	66	25	their	their	PRON
cana-1225	66	26	union	union	NOUN
cana-1225	66	27	,	,	PUNCT
cana-1225	66	28	intersection	intersection	NOUN
cana-1225	66	29	and	and	CCONJ
cana-1225	66	30	complement	complement	NOUN
cana-1225	66	31	are	be	AUX
cana-1225	66	32	defined	define	VERB
cana-1225	66	33	as	as	SCONJ
cana-1225	66	34	follows	follow	VERB
cana-1225	66	35	:	:	PUNCT
cana-1225	66	36	communications	communication	NOUN
cana-1225	66	37	on	on	ADP
cana-1225	66	38	applied	apply	VERB
cana-1225	66	39	nonlinear	nonlinear	ADJ
cana-1225	66	40	analysis	analysis	NOUN
cana-1225	66	41	issn	issn	NOUN
cana-1225	66	42	:	:	PUNCT
cana-1225	66	43	1074	1074	NUM
cana-1225	66	44	-	-	PUNCT
cana-1225	66	45	133x	133x	NUM
cana-1225	66	46	vol	vol	NOUN
cana-1225	66	47	31	31	NUM
cana-1225	66	48	no	no	NOUN
cana-1225	66	49	.	.	PUNCT
cana-1225	67	1	6s	6s	NUM
cana-1225	67	2	(	(	PUNCT
cana-1225	67	3	2024	2024	NUM
cana-1225	67	4	)	)	PUNCT
cana-1225	67	5	320	320	NUM
cana-1225	67	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1225	67	7	(	(	PUNCT
cana-1225	67	8	i	i	NOUN
cana-1225	67	9	)	)	PUNCT
cana-1225	67	10	a	a	DET
cana-1225	67	11	∪	∪	X
cana-1225	67	12	b	b	NOUN
cana-1225	67	13	=	=	SYM
cana-1225	67	14	{	{	PUNCT
cana-1225	67	15	x	x	NUM
cana-1225	67	16	,	,	PUNCT
cana-1225	68	1	[	[	X
cana-1225	68	2	𝑡𝐴∪𝐵	𝑡𝐴∪𝐵	X
cana-1225	68	3	+	+	CCONJ
cana-1225	68	4	(	(	PUNCT
cana-1225	68	5	𝑥	𝑥	NOUN
cana-1225	68	6	)	)	PUNCT
cana-1225	68	7	,	,	PUNCT
cana-1225	68	8	1	1	NUM
cana-1225	68	9	−	−	NOUN
cana-1225	68	10	𝑓𝐴∪𝐵	𝑓𝐴∪𝐵	NOUN
cana-1225	69	1	+	+	CCONJ
cana-1225	69	2	(	(	PUNCT
cana-1225	69	3	𝑥	𝑥	NOUN
cana-1225	69	4	)	)	PUNCT
cana-1225	69	5	]	]	PUNCT
cana-1225	69	6	,	,	PUNCT
cana-1225	69	7	[	[	X
cana-1225	69	8	−1−𝑓𝐴∪𝐵	−1−𝑓𝐴∪𝐵	ADV
cana-1225	69	9	−	−	PROPN
cana-1225	69	10	(	(	PUNCT
cana-1225	69	11	𝑥),𝑡𝐴∪𝐵	𝑥),𝑡𝐴∪𝐵	NOUN
cana-1225	69	12	−	−	PROPN
cana-1225	69	13	(	(	PUNCT
cana-1225	69	14	𝑥)]	𝑥)]	X
cana-1225	69	15	x	x	SYM
cana-1225	69	16	∈	∈	PROPN
cana-1225	69	17	x	x	PUNCT
cana-1225	69	18	}	}	PUNCT
cana-1225	69	19	where	where	SCONJ
cana-1225	69	20	𝑡𝐴∪𝐵	𝑡𝐴∪𝐵	ADP
cana-1225	69	21	+	+	CCONJ
cana-1225	69	22	(	(	PUNCT
cana-1225	69	23	𝑥	𝑥	NOUN
cana-1225	69	24	)	)	PUNCT
cana-1225	69	25	=	=	SYM
cana-1225	69	26	max	max	NOUN
cana-1225	69	27	{	{	PUNCT
cana-1225	69	28	𝑡𝐴	𝑡𝐴	ADP
cana-1225	69	29	+	+	ADJ
cana-1225	69	30	(	(	PUNCT
cana-1225	69	31	𝑥	𝑥	NOUN
cana-1225	69	32	)	)	PUNCT
cana-1225	69	33	,	,	PUNCT
cana-1225	69	34	𝑡𝐵	𝑡𝐵	ADJ
cana-1225	69	35	+	+	PROPN
cana-1225	69	36	(	(	PUNCT
cana-1225	69	37	𝑥	𝑥	NOUN
cana-1225	69	38	)	)	PUNCT
cana-1225	69	39	}	}	PUNCT
cana-1225	69	40	,	,	PUNCT
cana-1225	69	41	𝑡𝐴∪𝐵	𝑡𝐴∪𝐵	ADP
cana-1225	69	42	−	−	PROPN
cana-1225	69	43	(	(	PUNCT
cana-1225	69	44	𝑥	𝑥	NOUN
cana-1225	69	45	)	)	PUNCT
cana-1225	69	46	=	=	SYM
cana-1225	69	47	min	min	NOUN
cana-1225	69	48	{	{	PUNCT
cana-1225	69	49	𝑡𝐴	𝑡𝐴	NOUN
cana-1225	69	50	−(𝑥	−(𝑥	NOUN
cana-1225	69	51	)	)	PUNCT
cana-1225	69	52	,	,	PUNCT
cana-1225	69	53	𝑡𝐵	𝑡𝐵	ADJ
cana-1225	69	54	−(𝑥	−(𝑥	NOUN
cana-1225	69	55	)	)	PUNCT
cana-1225	69	56	}	}	PUNCT
cana-1225	69	57	and	and	CCONJ
cana-1225	69	58	1	1	NUM
cana-1225	69	59	−	−	NOUN
cana-1225	69	60	𝑓𝐴∪𝐵	𝑓𝐴∪𝐵	NOUN
cana-1225	70	1	+	+	CCONJ
cana-1225	70	2	(	(	PUNCT
cana-1225	70	3	𝑥	𝑥	NOUN
cana-1225	70	4	)	)	PUNCT
cana-1225	70	5	=	=	SYM
cana-1225	70	6	max	max	PROPN
cana-1225	70	7	{	{	PUNCT
cana-1225	70	8	1	1	NUM
cana-1225	70	9	−	−	NOUN
cana-1225	70	10	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	70	11	+	+	PROPN
cana-1225	70	12	(	(	PUNCT
cana-1225	70	13	𝑥	𝑥	NOUN
cana-1225	70	14	)	)	PUNCT
cana-1225	70	15	,	,	PUNCT
cana-1225	70	16	1	1	NUM
cana-1225	70	17	−	−	PROPN
cana-1225	70	18	𝑓𝐵	𝑓𝐵	PROPN
cana-1225	70	19	+	+	ADJ
cana-1225	70	20	(	(	PUNCT
cana-1225	70	21	𝑥	𝑥	NOUN
cana-1225	70	22	)	)	PUNCT
cana-1225	70	23	}	}	PUNCT
cana-1225	70	24	,	,	PUNCT
cana-1225	70	25	−1	−1	NOUN
cana-1225	70	26	−	−	NOUN
cana-1225	70	27	𝑓𝐴∪𝐵	𝑓𝐴∪𝐵	NOUN
cana-1225	70	28	−	−	PROPN
cana-1225	70	29	(	(	PUNCT
cana-1225	70	30	𝑥	𝑥	NOUN
cana-1225	70	31	)	)	PUNCT
cana-1225	70	32	=	=	SYM
cana-1225	70	33	min	min	NOUN
cana-1225	70	34	{	{	PUNCT
cana-1225	70	35	−1	−1	NOUN
cana-1225	70	36	−	−	ADP
cana-1225	70	37	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	70	38	−(𝑥	−(𝑥	NOUN
cana-1225	70	39	)	)	PUNCT
cana-1225	70	40	,	,	PUNCT
cana-1225	70	41	−1	−1	NOUN
cana-1225	70	42	−	−	PROPN
cana-1225	70	43	𝑓𝐵	𝑓𝐵	PROPN
cana-1225	70	44	−(𝑥	−(𝑥	NOUN
cana-1225	70	45	)	)	PUNCT
cana-1225	70	46	}	}	PUNCT
cana-1225	70	47	.	.	PUNCT
cana-1225	71	1	(	(	PUNCT
cana-1225	71	2	ii	ii	X
cana-1225	71	3	)	)	PUNCT
cana-1225	71	4	a	a	DET
cana-1225	71	5	∩	∩	ADJ
cana-1225	71	6	b	b	NOUN
cana-1225	71	7	=	=	SYM
cana-1225	71	8	{	{	PUNCT
cana-1225	71	9	x	x	NUM
cana-1225	71	10	,	,	PUNCT
cana-1225	71	11	[	[	X
cana-1225	71	12	𝑡𝐴∩𝐵	𝑡𝐴∩𝐵	NOUN
cana-1225	71	13	+	+	CCONJ
cana-1225	71	14	(	(	PUNCT
cana-1225	71	15	𝑥	𝑥	NOUN
cana-1225	71	16	)	)	PUNCT
cana-1225	71	17	,	,	PUNCT
cana-1225	71	18	1	1	NUM
cana-1225	71	19	−	−	NOUN
cana-1225	71	20	𝑓𝐴∩𝐵	𝑓𝐴∩𝐵	NOUN
cana-1225	71	21	+	+	CCONJ
cana-1225	71	22	(	(	PUNCT
cana-1225	71	23	𝑥	𝑥	NOUN
cana-1225	71	24	)	)	PUNCT
cana-1225	71	25	]	]	PUNCT
cana-1225	71	26	,	,	PUNCT
cana-1225	71	27	[	[	X
cana-1225	71	28	−1−𝑓𝐴∩𝐵	−1−𝑓𝐴∩𝐵	NOUN
cana-1225	71	29	−	−	NOUN
cana-1225	71	30	(	(	PUNCT
cana-1225	71	31	𝑥),𝑡𝐴∩𝐵	𝑥),𝑡𝐴∩𝐵	NOUN
cana-1225	71	32	−	−	PROPN
cana-1225	71	33	(	(	PUNCT
cana-1225	71	34	𝑥)]	𝑥)]	X
cana-1225	71	35	x	x	SYM
cana-1225	71	36	∈	∈	PROPN
cana-1225	71	37	x	x	PUNCT
cana-1225	71	38	}	}	PUNCT
cana-1225	71	39	where	where	SCONJ
cana-1225	71	40	𝑡𝐴∩𝐵	𝑡𝐴∩𝐵	NOUN
cana-1225	71	41	+	+	CCONJ
cana-1225	71	42	(	(	PUNCT
cana-1225	71	43	𝑥	𝑥	NOUN
cana-1225	71	44	)	)	PUNCT
cana-1225	71	45	=	=	SYM
cana-1225	71	46	min	min	NOUN
cana-1225	71	47	{	{	PUNCT
cana-1225	71	48	𝑡𝐴	𝑡𝐴	ADJ
cana-1225	71	49	+	+	ADJ
cana-1225	71	50	(	(	PUNCT
cana-1225	71	51	𝑥	𝑥	NOUN
cana-1225	71	52	)	)	PUNCT
cana-1225	71	53	,	,	PUNCT
cana-1225	71	54	𝑡𝐵	𝑡𝐵	ADJ
cana-1225	71	55	+	+	PROPN
cana-1225	71	56	(	(	PUNCT
cana-1225	71	57	𝑥	𝑥	NOUN
cana-1225	71	58	)	)	PUNCT
cana-1225	71	59	}	}	PUNCT
cana-1225	71	60	,	,	PUNCT
cana-1225	71	61	𝑡𝐴∩𝐵	𝑡𝐴∩𝐵	X
cana-1225	71	62	−	−	PROPN
cana-1225	71	63	(	(	PUNCT
cana-1225	71	64	𝑥	𝑥	NOUN
cana-1225	71	65	)	)	PUNCT
cana-1225	71	66	=	=	SYM
cana-1225	71	67	max	max	NOUN
cana-1225	71	68	{	{	PUNCT
cana-1225	71	69	𝑡𝐴	𝑡𝐴	NOUN
cana-1225	71	70	−(𝑥	−(𝑥	NOUN
cana-1225	71	71	)	)	PUNCT
cana-1225	71	72	,	,	PUNCT
cana-1225	71	73	𝑡𝐵	𝑡𝐵	ADJ
cana-1225	71	74	−(𝑥	−(𝑥	NOUN
cana-1225	71	75	)	)	PUNCT
cana-1225	71	76	}	}	PUNCT
cana-1225	71	77	and	and	CCONJ
cana-1225	71	78	1	1	NUM
cana-1225	71	79	−	−	NOUN
cana-1225	71	80	𝑓𝐴∩𝐵	𝑓𝐴∩𝐵	NOUN
cana-1225	71	81	+	+	CCONJ
cana-1225	71	82	(	(	PUNCT
cana-1225	71	83	𝑥	𝑥	NOUN
cana-1225	71	84	)	)	PUNCT
cana-1225	71	85	=	=	SYM
cana-1225	71	86	min	min	NOUN
cana-1225	71	87	{	{	PUNCT
cana-1225	71	88	1	1	NUM
cana-1225	71	89	−	−	NOUN
cana-1225	71	90	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	71	91	+	+	PROPN
cana-1225	71	92	(	(	PUNCT
cana-1225	71	93	𝑥	𝑥	NOUN
cana-1225	71	94	)	)	PUNCT
cana-1225	71	95	,	,	PUNCT
cana-1225	71	96	1	1	NUM
cana-1225	71	97	−	−	PROPN
cana-1225	71	98	𝑓𝐵	𝑓𝐵	PROPN
cana-1225	71	99	+	+	ADJ
cana-1225	71	100	(	(	PUNCT
cana-1225	71	101	𝑥	𝑥	NOUN
cana-1225	71	102	)	)	PUNCT
cana-1225	71	103	}	}	PUNCT
cana-1225	71	104	,	,	PUNCT
cana-1225	71	105	−1	−1	NOUN
cana-1225	71	106	−	−	NOUN
cana-1225	71	107	𝑓𝐴∪𝐵	𝑓𝐴∪𝐵	NOUN
cana-1225	71	108	−	−	PROPN
cana-1225	71	109	(	(	PUNCT
cana-1225	71	110	𝑥	𝑥	NOUN
cana-1225	71	111	)	)	PUNCT
cana-1225	71	112	=	=	SYM
cana-1225	71	113	max	max	PROPN
cana-1225	71	114	{	{	PUNCT
cana-1225	71	115	−1	−1	NOUN
cana-1225	71	116	−	−	ADP
cana-1225	71	117	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	71	118	−(𝑥	−(𝑥	NOUN
cana-1225	71	119	)	)	PUNCT
cana-1225	71	120	,	,	PUNCT
cana-1225	71	121	−1	−1	NOUN
cana-1225	71	122	−	−	PROPN
cana-1225	71	123	𝑓𝐵	𝑓𝐵	PROPN
cana-1225	71	124	−(𝑥	−(𝑥	NOUN
cana-1225	71	125	)	)	PUNCT
cana-1225	71	126	}	}	PUNCT
cana-1225	71	127	.	.	PUNCT
cana-1225	72	1	(	(	PUNCT
cana-1225	72	2	iii	iii	X
cana-1225	72	3	)	)	PUNCT
cana-1225	72	4	ac	ac	NOUN
cana-1225	72	5	=	=	PUNCT
cana-1225	72	6	{	{	PUNCT
cana-1225	72	7	x	x	NUM
cana-1225	72	8	,	,	PUNCT
cana-1225	72	9	[	[	X
cana-1225	72	10	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	72	11	+	+	NOUN
cana-1225	72	12	(	(	PUNCT
cana-1225	72	13	𝑥	𝑥	NOUN
cana-1225	72	14	)	)	PUNCT
cana-1225	72	15	,	,	PUNCT
cana-1225	72	16	1	1	NUM
cana-1225	72	17	−	−	NOUN
cana-1225	72	18	𝑡𝐴	𝑡𝐴	ADP
cana-1225	72	19	+	+	PROPN
cana-1225	72	20	(	(	PUNCT
cana-1225	72	21	𝑥	𝑥	NOUN
cana-1225	72	22	)	)	PUNCT
cana-1225	72	23	]	]	PUNCT
cana-1225	72	24	,	,	PUNCT
cana-1225	73	1	[	[	X
cana-1225	73	2	−1	−1	NOUN
cana-1225	73	3	−	−	ADP
cana-1225	73	4	𝑡𝐴	𝑡𝐴	NOUN
cana-1225	73	5	−(x	−(x	NOUN
cana-1225	73	6	)	)	PUNCT
cana-1225	73	7	,	,	PUNCT
cana-1225	73	8	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	73	9	−(x)]/	−(x)]/	X
cana-1225	73	10	x	x	SYM
cana-1225	73	11	∈	∈	NOUN
cana-1225	73	12	x	x	X
cana-1225	73	13	}	}	PUNCT
cana-1225	73	14	.	.	PUNCT
cana-1225	74	1	definition	definition	NOUN
cana-1225	74	2	2.9	2.9	NUM
cana-1225	74	3	:	:	PUNCT
cana-1225	75	1	[	[	X
cana-1225	75	2	4	4	X
cana-1225	75	3	]	]	PUNCT
cana-1225	75	4	let	let	VERB
cana-1225	75	5	a	a	PRON
cana-1225	75	6	and	and	CCONJ
cana-1225	75	7	b	b	NOUN
cana-1225	75	8	be	be	AUX
cana-1225	75	9	two	two	NUM
cana-1225	75	10	bipolar	bipolar	ADJ
cana-1225	75	11	vague	vague	ADJ
cana-1225	75	12	sets	set	NOUN
cana-1225	75	13	defined	define	VERB
cana-1225	75	14	over	over	ADP
cana-1225	75	15	a	a	DET
cana-1225	75	16	universe	universe	NOUN
cana-1225	75	17	of	of	ADP
cana-1225	75	18	discourse	discourse	NOUN
cana-1225	75	19	x.	x.	NOUN
cana-1225	76	1	we	we	PRON
cana-1225	76	2	say	say	VERB
cana-1225	76	3	that	that	SCONJ
cana-1225	76	4	a	a	DET
cana-1225	76	5	⊆	⊆	NUM
cana-1225	76	6	b	b	NOUN
cana-1225	76	7	if	if	SCONJ
cana-1225	76	8	and	and	CCONJ
cana-1225	76	9	only	only	ADV
cana-1225	76	10	if	if	SCONJ
cana-1225	76	11	𝑡𝐴	𝑡𝐴	VERB
cana-1225	76	12	+	+	ADJ
cana-1225	76	13	(	(	PUNCT
cana-1225	76	14	𝑥	𝑥	NOUN
cana-1225	76	15	)	)	PUNCT
cana-1225	76	16	≤	≤	NOUN
cana-1225	77	1	𝑡𝐵	𝑡𝐵	ADP
cana-1225	77	2	+	+	PROPN
cana-1225	77	3	(	(	PUNCT
cana-1225	77	4	𝑥	𝑥	NOUN
cana-1225	77	5	)	)	PUNCT
cana-1225	77	6	,	,	PUNCT
cana-1225	77	7	1	1	NUM
cana-1225	77	8	−	−	NOUN
cana-1225	77	9	𝑓𝐴	𝑓𝐴	VERB
cana-1225	77	10	+	+	PROPN
cana-1225	77	11	(	(	PUNCT
cana-1225	77	12	𝑥	𝑥	NOUN
cana-1225	77	13	)	)	PUNCT
cana-1225	77	14	≤	≤	NOUN
cana-1225	77	15	1	1	NUM
cana-1225	77	16	−	−	PROPN
cana-1225	77	17	𝑓𝐵	𝑓𝐵	PROPN
cana-1225	77	18	+	+	ADJ
cana-1225	77	19	(	(	PUNCT
cana-1225	77	20	𝑥	𝑥	NOUN
cana-1225	77	21	)	)	PUNCT
cana-1225	77	22	and	and	CCONJ
cana-1225	77	23	𝑡𝐴	𝑡𝐴	ADP
cana-1225	77	24	−(𝑥	−(𝑥	NOUN
cana-1225	77	25	)	)	PUNCT
cana-1225	77	26	≥	≥	NOUN
cana-1225	77	27	𝑡𝐵	𝑡𝐵	PROPN
cana-1225	77	28	−(𝑥	−(𝑥	PROPN
cana-1225	77	29	)	)	PUNCT
cana-1225	77	30	,	,	PUNCT
cana-1225	77	31	−1	−1	NOUN
cana-1225	77	32	−	−	NOUN
cana-1225	77	33	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	77	34	−(𝑥	−(𝑥	NOUN
cana-1225	77	35	)	)	PUNCT
cana-1225	77	36	≥	≥	NOUN
cana-1225	77	37	1	1	NUM
cana-1225	77	38	−	−	PROPN
cana-1225	77	39	𝑓𝐵	𝑓𝐵	PROPN
cana-1225	77	40	−(𝑥	−(𝑥	NOUN
cana-1225	77	41	)	)	PUNCT
cana-1225	77	42	for	for	ADP
cana-1225	77	43	all	all	DET
cana-1225	77	44	x	x	SYM
cana-1225	77	45	∈	∈	NOUN
cana-1225	77	46	x.	x.	NOUN
cana-1225	77	47	definition	definition	NOUN
cana-1225	77	48	2.10	2.10	NUM
cana-1225	77	49	:	:	PUNCT
cana-1225	78	1	[	[	X
cana-1225	78	2	4	4	X
cana-1225	78	3	]	]	PUNCT
cana-1225	78	4	a	a	DET
cana-1225	78	5	bipolar	bipolar	ADJ
cana-1225	78	6	vague	vague	ADJ
cana-1225	78	7	topology	topology	NOUN
cana-1225	78	8	(	(	PUNCT
cana-1225	78	9	bvt	bvt	PROPN
cana-1225	78	10	)	)	PUNCT
cana-1225	78	11	on	on	ADP
cana-1225	78	12	a	a	DET
cana-1225	78	13	non	non	ADJ
cana-1225	78	14	-	-	ADJ
cana-1225	78	15	empty	empty	ADJ
cana-1225	78	16	set	set	NOUN
cana-1225	78	17	x	x	PUNCT
cana-1225	78	18	is	be	AUX
cana-1225	78	19	a	a	DET
cana-1225	78	20	family	family	NOUN
cana-1225	78	21	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	78	22	of	of	ADP
cana-1225	78	23	bipolar	bipolar	ADJ
cana-1225	78	24	vague	vague	NOUN
cana-1225	78	25	set	set	NOUN
cana-1225	78	26	in	in	ADP
cana-1225	78	27	x	x	PUNCT
cana-1225	78	28	satisfying	satisfy	VERB
cana-1225	78	29	the	the	DET
cana-1225	78	30	following	follow	VERB
cana-1225	78	31	axioms	axiom	NOUN
cana-1225	78	32	:	:	PUNCT
cana-1225	78	33	(	(	PUNCT
cana-1225	78	34	i	i	NOUN
cana-1225	78	35	)	)	PUNCT
cana-1225	78	36	0~,1~	0~,1~	PRON
cana-1225	79	1	∈	∈	PROPN
cana-1225	79	2	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	79	3	(	(	PUNCT
cana-1225	79	4	ii	ii	PROPN
cana-1225	79	5	)	)	PUNCT
cana-1225	79	6	𝐺1	𝐺1	NOUN
cana-1225	79	7	∩	∩	PROPN
cana-1225	79	8	𝐺2	𝐺2	PROPN
cana-1225	79	9	∈	∈	PROPN
cana-1225	79	10	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	79	11	,	,	PUNCT
cana-1225	79	12	for	for	ADP
cana-1225	79	13	any	any	DET
cana-1225	79	14	𝐺1	𝐺1	NOUN
cana-1225	79	15	,	,	PUNCT
cana-1225	79	16	𝐺2	𝐺2	ADJ
cana-1225	79	17	∈	∈	PROPN
cana-1225	79	18	b𝑉𝜏	b𝑉𝜏	ADJ
cana-1225	79	19	(	(	PUNCT
cana-1225	79	20	iii	iii	NOUN
cana-1225	79	21	)	)	PUNCT
cana-1225	79	22	∪	∪	ADP
cana-1225	79	23	𝐺𝑖	𝐺𝑖	PROPN
cana-1225	79	24	∈	∈	PROPN
cana-1225	79	25	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	79	26	,	,	PUNCT
cana-1225	79	27	for	for	ADP
cana-1225	79	28	any	any	DET
cana-1225	79	29	arbitrary	arbitrary	ADJ
cana-1225	79	30	family	family	NOUN
cana-1225	79	31	{	{	PUNCT
cana-1225	79	32	𝐺𝑖	𝐺𝑖	PROPN
cana-1225	79	33	:	:	PUNCT
cana-1225	79	34	𝐺𝑖	𝐺𝑖	VERB
cana-1225	79	35	∈	∈	PROPN
cana-1225	79	36	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	79	37	,	,	PUNCT
cana-1225	79	38	i	i	PRON
cana-1225	79	39	∈	∈	VERB
cana-1225	79	40	i	i	X
cana-1225	79	41	}	}	PUNCT
cana-1225	79	42	.	.	PUNCT
cana-1225	80	1	in	in	ADP
cana-1225	80	2	this	this	DET
cana-1225	80	3	case	case	NOUN
cana-1225	80	4	the	the	DET
cana-1225	80	5	pair	pair	NOUN
cana-1225	80	6	(	(	PUNCT
cana-1225	80	7	x	x	NOUN
cana-1225	80	8	,	,	PUNCT
cana-1225	80	9	b𝑉𝜏	b𝑉𝜏	NUM
cana-1225	80	10	)	)	PUNCT
cana-1225	80	11	is	be	AUX
cana-1225	80	12	called	call	VERB
cana-1225	80	13	a	a	DET
cana-1225	80	14	bipolar	bipolar	ADJ
cana-1225	80	15	vague	vague	ADJ
cana-1225	80	16	topological	topological	ADJ
cana-1225	80	17	space	space	NOUN
cana-1225	80	18	and	and	CCONJ
cana-1225	80	19	any	any	DET
cana-1225	80	20	bipolar	bipolar	ADJ
cana-1225	80	21	vague	vague	ADJ
cana-1225	80	22	set	set	NOUN
cana-1225	80	23	(	(	PUNCT
cana-1225	80	24	bvs	bvs	PROPN
cana-1225	80	25	)	)	PUNCT
cana-1225	80	26	in	in	ADP
cana-1225	80	27	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	80	28	is	be	AUX
cana-1225	80	29	known	know	VERB
cana-1225	80	30	as	as	ADP
cana-1225	80	31	bipolar	bipolar	ADJ
cana-1225	80	32	vague	vague	ADJ
cana-1225	80	33	open	open	ADJ
cana-1225	80	34	set	set	VERB
cana-1225	80	35	in	in	ADP
cana-1225	80	36	x.	x.	NOUN
cana-1225	80	37	the	the	DET
cana-1225	80	38	complement	complement	NOUN
cana-1225	80	39	ac	ac	INTJ
cana-1225	80	40	of	of	ADP
cana-1225	80	41	a	a	DET
cana-1225	80	42	bipolar	bipolar	ADJ
cana-1225	80	43	vague	vague	ADJ
cana-1225	80	44	open	open	ADJ
cana-1225	80	45	set	set	NOUN
cana-1225	80	46	(	(	PUNCT
cana-1225	80	47	bvos	bvos	PROPN
cana-1225	80	48	)	)	PUNCT
cana-1225	80	49	a	a	PRON
cana-1225	80	50	in	in	ADP
cana-1225	80	51	a	a	DET
cana-1225	80	52	bipolar	bipolar	ADJ
cana-1225	80	53	vague	vague	ADJ
cana-1225	80	54	topological	topological	ADJ
cana-1225	80	55	space	space	NOUN
cana-1225	80	56	(	(	PUNCT
cana-1225	80	57	x	x	NOUN
cana-1225	80	58	,	,	PUNCT
cana-1225	80	59	b𝑉𝜏	b𝑉𝜏	NUM
cana-1225	80	60	)	)	PUNCT
cana-1225	80	61	is	be	AUX
cana-1225	80	62	called	call	VERB
cana-1225	80	63	a	a	DET
cana-1225	80	64	bipolar	bipolar	ADJ
cana-1225	80	65	vague	vague	NOUN
cana-1225	80	66	closed	close	VERB
cana-1225	80	67	set	set	NOUN
cana-1225	80	68	(	(	PUNCT
cana-1225	80	69	bvcs	bvcs	NOUN
cana-1225	80	70	)	)	PUNCT
cana-1225	80	71	in	in	ADP
cana-1225	80	72	x.	x.	NOUN
cana-1225	80	73	definition	definition	NOUN
cana-1225	80	74	2.11	2.11	NUM
cana-1225	80	75	:	:	PUNCT
cana-1225	81	1	[	[	X
cana-1225	81	2	4	4	X
cana-1225	81	3	]	]	X
cana-1225	81	4	let	let	VERB
cana-1225	81	5	(	(	PUNCT
cana-1225	81	6	x	x	NOUN
cana-1225	81	7	,	,	PUNCT
cana-1225	81	8	b𝑉𝜏	b𝑉𝜏	NUM
cana-1225	81	9	)	)	PUNCT
cana-1225	81	10	be	be	AUX
cana-1225	81	11	a	a	DET
cana-1225	81	12	bipolar	bipolar	ADJ
cana-1225	81	13	vague	vague	ADJ
cana-1225	81	14	topological	topological	ADJ
cana-1225	81	15	space	space	NOUN
cana-1225	81	16	a	a	DET
cana-1225	81	17	=	=	SYM
cana-1225	81	18	x	x	NOUN
cana-1225	81	19	,	,	PUNCT
cana-1225	81	20	[	[	X
cana-1225	81	21	𝑡𝐴	𝑡𝐴	SYM
cana-1225	81	22	+	+	ADJ
cana-1225	81	23	,	,	PUNCT
cana-1225	81	24	1	1	NUM
cana-1225	81	25	−	−	NOUN
cana-1225	81	26	𝑓𝐴	𝑓𝐴	VERB
cana-1225	81	27	+	+	PROPN
cana-1225	81	28	]	]	X
cana-1225	81	29	,	,	PUNCT
cana-1225	81	30	[	[	X
cana-1225	81	31	−1	−1	NOUN
cana-1225	81	32	−	−	VERB
cana-1225	81	33	𝑓𝐴	𝑓𝐴	NOUN
cana-1225	81	34	−	−	NOUN
cana-1225	81	35	,	,	PUNCT
cana-1225	81	36	𝑡𝐴	𝑡𝐴	PROPN
cana-1225	81	37	−]	−]	PRON
cana-1225	81	38	be	be	AUX
cana-1225	81	39	a	a	DET
cana-1225	81	40	bipolar	bipolar	ADJ
cana-1225	81	41	vague	vague	NOUN
cana-1225	81	42	set	set	NOUN
cana-1225	81	43	in	in	ADP
cana-1225	81	44	x.	x.	NOUN
cana-1225	81	45	then	then	ADV
cana-1225	81	46	the	the	DET
cana-1225	81	47	bipolar	bipolar	ADJ
cana-1225	81	48	vague	vague	ADJ
cana-1225	81	49	interior	interior	ADJ
cana-1225	81	50	and	and	CCONJ
cana-1225	81	51	bipolar	bipolar	ADJ
cana-1225	81	52	vague	vague	ADJ
cana-1225	81	53	closure	closure	NOUN
cana-1225	81	54	of	of	ADP
cana-1225	81	55	a	a	PRON
cana-1225	81	56	are	be	AUX
cana-1225	81	57	defined	define	VERB
cana-1225	81	58	by	by	ADP
cana-1225	81	59	,	,	PUNCT
cana-1225	81	60	bvint(a	bvint(a	NOUN
cana-1225	81	61	)	)	PUNCT
cana-1225	81	62	=	=	SYM
cana-1225	81	63	∪	∪	X
cana-1225	81	64	{	{	PUNCT
cana-1225	81	65	g	g	NOUN
cana-1225	81	66	:	:	PUNCT
cana-1225	81	67	g	g	PROPN
cana-1225	81	68	is	be	AUX
cana-1225	81	69	a	a	DET
cana-1225	81	70	bipolar	bipolar	ADJ
cana-1225	81	71	vague	vague	ADJ
cana-1225	81	72	open	open	ADJ
cana-1225	81	73	set	set	VERB
cana-1225	81	74	in	in	ADP
cana-1225	81	75	x	x	X
cana-1225	81	76	and	and	CCONJ
cana-1225	81	77	g	g	PROPN
cana-1225	81	78	⊆	⊆	NUM
cana-1225	81	79	a	a	PRON
cana-1225	81	80	}	}	PUNCT
cana-1225	81	81	,	,	PUNCT
cana-1225	81	82	bvcl(a	bvcl(a	NUM
cana-1225	81	83	)	)	PUNCT
cana-1225	81	84	=	=	NOUN
cana-1225	81	85	∩	∩	NOUN
cana-1225	81	86	{	{	PUNCT
cana-1225	81	87	k	k	NOUN
cana-1225	81	88	:	:	PUNCT
cana-1225	81	89	k	k	X
cana-1225	81	90	is	be	AUX
cana-1225	81	91	a	a	DET
cana-1225	81	92	bipolar	bipolar	ADJ
cana-1225	81	93	vague	vague	NOUN
cana-1225	81	94	closed	close	VERB
cana-1225	81	95	set	set	VERB
cana-1225	81	96	in	in	ADP
cana-1225	81	97	x	x	PUNCT
cana-1225	81	98	and	and	CCONJ
cana-1225	81	99	a⊆	a⊆	VERB
cana-1225	81	100	k	k	X
cana-1225	81	101	}	}	PUNCT
cana-1225	81	102	.	.	PUNCT
cana-1225	82	1	note	note	VERB
cana-1225	82	2	that	that	SCONJ
cana-1225	82	3	bvcl(a	bvcl(a	NOUN
cana-1225	82	4	)	)	PUNCT
cana-1225	82	5	is	be	AUX
cana-1225	82	6	a	a	DET
cana-1225	82	7	bipolar	bipolar	ADJ
cana-1225	82	8	vague	vague	NOUN
cana-1225	82	9	closed	close	VERB
cana-1225	82	10	set	set	NOUN
cana-1225	82	11	and	and	CCONJ
cana-1225	82	12	bvint(a	bvint(a	PROPN
cana-1225	82	13	)	)	PUNCT
cana-1225	82	14	is	be	AUX
cana-1225	82	15	a	a	DET
cana-1225	82	16	bipolar	bipolar	ADJ
cana-1225	82	17	vague	vague	ADJ
cana-1225	82	18	open	open	ADJ
cana-1225	82	19	set	set	VERB
cana-1225	82	20	in	in	ADP
cana-1225	82	21	x.	x.	NOUN
cana-1225	82	22	further	far	ADV
cana-1225	82	23	,	,	PUNCT
cana-1225	82	24	(	(	PUNCT
cana-1225	82	25	i	i	NOUN
cana-1225	82	26	)	)	PUNCT
cana-1225	82	27	a	a	PRON
cana-1225	82	28	is	be	AUX
cana-1225	82	29	a	a	DET
cana-1225	82	30	bipolar	bipolar	ADJ
cana-1225	82	31	vague	vague	NOUN
cana-1225	82	32	closed	close	VERB
cana-1225	82	33	set	set	VERB
cana-1225	82	34	in	in	ADP
cana-1225	82	35	x	x	PUNCT
cana-1225	82	36	if	if	SCONJ
cana-1225	83	1	and	and	CCONJ
cana-1225	83	2	only	only	ADV
cana-1225	83	3	if	if	SCONJ
cana-1225	83	4	bvcl(a	bvcl(a	NUM
cana-1225	83	5	)	)	PUNCT
cana-1225	83	6	=	=	SYM
cana-1225	84	1	a	a	DET
cana-1225	84	2	,	,	PUNCT
cana-1225	84	3	(	(	PUNCT
cana-1225	84	4	ii	ii	NOUN
cana-1225	84	5	)	)	PUNCT
cana-1225	84	6	a	a	PRON
cana-1225	84	7	is	be	AUX
cana-1225	84	8	a	a	DET
cana-1225	84	9	bipolar	bipolar	ADJ
cana-1225	84	10	vague	vague	ADJ
cana-1225	84	11	open	open	ADJ
cana-1225	84	12	set	set	VERB
cana-1225	84	13	in	in	ADP
cana-1225	84	14	x	x	PUNCT
cana-1225	84	15	if	if	SCONJ
cana-1225	84	16	and	and	CCONJ
cana-1225	84	17	only	only	ADV
cana-1225	84	18	if	if	SCONJ
cana-1225	84	19	bvint(a	bvint(a	PROPN
cana-1225	84	20	)	)	PUNCT
cana-1225	84	21	=	=	SYM
cana-1225	84	22	a.	a.	NOUN
cana-1225	84	23	definition	definition	NOUN
cana-1225	84	24	2.12	2.12	NUM
cana-1225	84	25	:	:	PUNCT
cana-1225	85	1	[	[	X
cana-1225	85	2	4	4	X
cana-1225	85	3	]	]	X
cana-1225	85	4	let	let	VERB
cana-1225	85	5	(	(	PUNCT
cana-1225	85	6	x	x	NOUN
cana-1225	85	7	,	,	PUNCT
cana-1225	85	8	b𝑉𝜏	b𝑉𝜏	NUM
cana-1225	85	9	)	)	PUNCT
cana-1225	85	10	be	be	AUX
cana-1225	85	11	a	a	DET
cana-1225	85	12	bipolar	bipolar	ADJ
cana-1225	85	13	vague	vague	ADJ
cana-1225	85	14	topological	topological	ADJ
cana-1225	85	15	space	space	NOUN
cana-1225	85	16	.	.	PUNCT
cana-1225	86	1	a	a	DET
cana-1225	86	2	bipolar	bipolar	ADJ
cana-1225	86	3	vague	vague	NOUN
cana-1225	86	4	set	set	VERB
cana-1225	86	5	a	a	PRON
cana-1225	86	6	in	in	ADP
cana-1225	86	7	(	(	PUNCT
cana-1225	86	8	x	x	NOUN
cana-1225	86	9	,	,	PUNCT
cana-1225	86	10	b𝑉𝜏	b𝑉𝜏	NUM
cana-1225	86	11	)	)	PUNCT
cana-1225	86	12	is	be	AUX
cana-1225	86	13	said	say	VERB
cana-1225	86	14	to	to	PART
cana-1225	86	15	be	be	AUX
cana-1225	86	16	a	a	DET
cana-1225	86	17	generalized	generalized	ADJ
cana-1225	86	18	bipolar	bipolar	ADJ
cana-1225	86	19	vague	vague	NOUN
cana-1225	86	20	closed	close	VERB
cana-1225	86	21	set	set	VERB
cana-1225	86	22	if	if	SCONJ
cana-1225	86	23	bvcl(a	bvcl(a	NUM
cana-1225	86	24	)	)	PUNCT
cana-1225	86	25	⊆	⊆	NUM
cana-1225	86	26	g	g	NOUN
cana-1225	86	27	whenever	whenever	SCONJ
cana-1225	86	28	a⊆	a⊆	VERB
cana-1225	86	29	g	g	NOUN
cana-1225	86	30	and	and	CCONJ
cana-1225	86	31	g	g	PROPN
cana-1225	86	32	is	be	AUX
cana-1225	86	33	bipolar	bipolar	ADJ
cana-1225	86	34	vague	vague	ADJ
cana-1225	86	35	open	open	ADJ
cana-1225	86	36	.	.	PUNCT
cana-1225	87	1	the	the	DET
cana-1225	87	2	complement	complement	NOUN
cana-1225	87	3	of	of	ADP
cana-1225	87	4	a	a	DET
cana-1225	87	5	generalized	generalized	ADJ
cana-1225	87	6	bipolar	bipolar	ADJ
cana-1225	87	7	vague	vague	NOUN
cana-1225	87	8	closed	close	VERB
cana-1225	87	9	set	set	VERB
cana-1225	87	10	is	be	AUX
cana-1225	87	11	generalized	generalize	VERB
cana-1225	87	12	bipolar	bipolar	ADJ
cana-1225	87	13	vague	vague	ADJ
cana-1225	87	14	open	open	ADJ
cana-1225	87	15	set	set	NOUN
cana-1225	87	16	.	.	PUNCT
cana-1225	88	1	definition	definition	NOUN
cana-1225	88	2	2.13	2.13	NUM
cana-1225	88	3	:	:	PUNCT
cana-1225	89	1	[	[	X
cana-1225	89	2	4	4	X
cana-1225	89	3	]	]	X
cana-1225	89	4	let	let	VERB
cana-1225	89	5	(	(	PUNCT
cana-1225	89	6	x	x	NOUN
cana-1225	89	7	,	,	PUNCT
cana-1225	89	8	b𝑉𝜏	b𝑉𝜏	NUM
cana-1225	89	9	)	)	PUNCT
cana-1225	89	10	be	be	AUX
cana-1225	89	11	a	a	DET
cana-1225	89	12	bipolar	bipolar	ADJ
cana-1225	89	13	vague	vague	ADJ
cana-1225	89	14	topological	topological	ADJ
cana-1225	89	15	space	space	NOUN
cana-1225	89	16	and	and	CCONJ
cana-1225	89	17	a	a	DET
cana-1225	89	18	be	be	AUX
cana-1225	89	19	a	a	DET
cana-1225	89	20	bipolar	bipolar	ADJ
cana-1225	89	21	vague	vague	NOUN
cana-1225	89	22	set	set	NOUN
cana-1225	89	23	in	in	ADP
cana-1225	89	24	x.	x.	NOUN
cana-1225	89	25	then	then	ADV
cana-1225	89	26	the	the	DET
cana-1225	89	27	generalized	generalized	ADJ
cana-1225	89	28	bipolar	bipolar	ADJ
cana-1225	89	29	vague	vague	ADJ
cana-1225	89	30	closure	closure	NOUN
cana-1225	89	31	and	and	CCONJ
cana-1225	89	32	generalized	generalize	VERB
cana-1225	89	33	bipolar	bipolar	ADJ
cana-1225	89	34	vague	vague	ADJ
cana-1225	89	35	interior	interior	NOUN
cana-1225	89	36	of	of	ADP
cana-1225	89	37	a	a	PRON
cana-1225	89	38	are	be	AUX
cana-1225	89	39	defined	define	VERB
cana-1225	89	40	by	by	ADP
cana-1225	89	41	,	,	PUNCT
cana-1225	89	42	communications	communication	NOUN
cana-1225	89	43	on	on	ADP
cana-1225	89	44	applied	apply	VERB
cana-1225	89	45	nonlinear	nonlinear	ADJ
cana-1225	89	46	analysis	analysis	NOUN
cana-1225	89	47	issn	issn	NOUN
cana-1225	89	48	:	:	PUNCT
cana-1225	89	49	1074	1074	NUM
cana-1225	89	50	-	-	PUNCT
cana-1225	89	51	133x	133x	NUM
cana-1225	89	52	vol	vol	NOUN
cana-1225	89	53	31	31	NUM
cana-1225	89	54	no	no	NOUN
cana-1225	89	55	.	.	PUNCT
cana-1225	90	1	6s	6s	NUM
cana-1225	90	2	(	(	PUNCT
cana-1225	90	3	2024	2024	NUM
cana-1225	90	4	)	)	PUNCT
cana-1225	90	5	321	321	NUM
cana-1225	90	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1225	90	7	gbvcl(a	gbvcl(a	NOUN
cana-1225	90	8	)	)	PUNCT
cana-1225	90	9	=	=	NOUN
cana-1225	90	10	∩	∩	NOUN
cana-1225	90	11	{	{	PUNCT
cana-1225	90	12	g	g	NOUN
cana-1225	90	13	:	:	PUNCT
cana-1225	90	14	g	g	PROPN
cana-1225	90	15	is	be	AUX
cana-1225	90	16	a	a	DET
cana-1225	90	17	generalized	generalized	ADJ
cana-1225	90	18	bipolar	bipolar	ADJ
cana-1225	90	19	vague	vague	NOUN
cana-1225	90	20	closed	close	VERB
cana-1225	90	21	set	set	VERB
cana-1225	90	22	in	in	ADP
cana-1225	90	23	x	x	PUNCT
cana-1225	90	24	and	and	CCONJ
cana-1225	90	25	a⊆	a⊆	VERB
cana-1225	90	26	g	g	NOUN
cana-1225	90	27	}	}	PUNCT
cana-1225	90	28	,	,	PUNCT
cana-1225	90	29	gbint(a	gbint(a	NOUN
cana-1225	90	30	)	)	PUNCT
cana-1225	90	31	=	=	SYM
cana-1225	90	32	∪	∪	X
cana-1225	90	33	{	{	PUNCT
cana-1225	90	34	g	g	NOUN
cana-1225	90	35	:	:	PUNCT
cana-1225	90	36	g	g	PROPN
cana-1225	90	37	is	be	AUX
cana-1225	90	38	a	a	DET
cana-1225	90	39	generalized	generalized	ADJ
cana-1225	90	40	bipolar	bipolar	ADJ
cana-1225	90	41	vague	vague	ADJ
cana-1225	90	42	open	open	ADJ
cana-1225	90	43	set	set	VERB
cana-1225	90	44	in	in	ADP
cana-1225	90	45	x	x	PUNCT
cana-1225	90	46	and	and	CCONJ
cana-1225	90	47	a	a	DET
cana-1225	90	48	⊇	⊇	NOUN
cana-1225	90	49	g	g	NOUN
cana-1225	90	50	}	}	PUNCT
cana-1225	90	51	.	.	PUNCT
cana-1225	91	1	definition	definition	NOUN
cana-1225	91	2	2.14	2.14	NUM
cana-1225	91	3	:	:	PUNCT
cana-1225	92	1	[	[	X
cana-1225	92	2	10	10	NUM
cana-1225	92	3	]	]	X
cana-1225	92	4	a	a	DET
cana-1225	92	5	bipolar	bipolar	ADJ
cana-1225	92	6	vague	vague	NOUN
cana-1225	92	7	set	set	VERB
cana-1225	92	8	a	a	PRON
cana-1225	92	9	of	of	ADP
cana-1225	92	10	a	a	DET
cana-1225	92	11	bipolar	bipolar	ADJ
cana-1225	92	12	vague	vague	ADJ
cana-1225	92	13	topological	topological	ADJ
cana-1225	92	14	space	space	NOUN
cana-1225	92	15	x	x	NOUN
cana-1225	92	16	,	,	PUNCT
cana-1225	92	17	is	be	AUX
cana-1225	92	18	said	say	VERB
cana-1225	92	19	to	to	PART
cana-1225	92	20	be	be	AUX
cana-1225	92	21	(	(	PUNCT
cana-1225	92	22	i	i	NOUN
cana-1225	92	23	)	)	PUNCT
cana-1225	92	24	a	a	DET
cana-1225	92	25	bipolar	bipolar	ADJ
cana-1225	92	26	vague	vague	ADJ
cana-1225	92	27	𝛼-open	𝛼-open	NOUN
cana-1225	92	28	set	set	NOUN
cana-1225	92	29	if	if	SCONJ
cana-1225	92	30	a	a	DET
cana-1225	92	31	⊆	⊆	NUM
cana-1225	92	32	bvint(bvcl(bvint(a	bvint(bvcl(bvint(a	NOUN
cana-1225	92	33	)	)	PUNCT
cana-1225	92	34	)	)	PUNCT
cana-1225	92	35	)	)	PUNCT
cana-1225	93	1	(	(	PUNCT
cana-1225	93	2	ii	ii	X
cana-1225	93	3	)	)	PUNCT
cana-1225	93	4	a	a	DET
cana-1225	93	5	bipolar	bipolar	ADJ
cana-1225	93	6	vague	vague	ADJ
cana-1225	93	7	pre	pre	ADJ
cana-1225	93	8	-	-	ADJ
cana-1225	93	9	open	open	ADJ
cana-1225	93	10	set	set	NOUN
cana-1225	93	11	if	if	SCONJ
cana-1225	93	12	a	a	DET
cana-1225	93	13	⊆	⊆	NUM
cana-1225	93	14	bvint(bvcl(a	bvint(bvcl(a	NUM
cana-1225	93	15	)	)	PUNCT
cana-1225	93	16	)	)	PUNCT
cana-1225	93	17	(	(	PUNCT
cana-1225	93	18	iii	iii	X
cana-1225	93	19	)	)	PUNCT
cana-1225	93	20	a	a	DET
cana-1225	93	21	bipolar	bipolar	ADJ
cana-1225	93	22	vague	vague	ADJ
cana-1225	93	23	semi	semi	ADJ
cana-1225	93	24	-	-	ADJ
cana-1225	93	25	open	open	ADJ
cana-1225	93	26	set	set	NOUN
cana-1225	93	27	if	if	SCONJ
cana-1225	93	28	a	a	DET
cana-1225	93	29	⊆	⊆	NUM
cana-1225	93	30	bvcl(bvint(a	bvcl(bvint(a	NUM
cana-1225	93	31	)	)	PUNCT
cana-1225	93	32	)	)	PUNCT
cana-1225	93	33	(	(	PUNCT
cana-1225	93	34	iv	iv	X
cana-1225	93	35	)	)	PUNCT
cana-1225	93	36	a	a	DET
cana-1225	93	37	bipolar	bipolar	ADJ
cana-1225	93	38	vague	vague	ADJ
cana-1225	93	39	semi-𝛼-open	semi-𝛼-open	NOUN
cana-1225	93	40	set	set	VERB
cana-1225	93	41	if	if	SCONJ
cana-1225	93	42	a	a	DET
cana-1225	93	43	⊆	⊆	NUM
cana-1225	93	44	bvcl(𝛼bvint(a	bvcl(𝛼bvint(a	NUM
cana-1225	93	45	)	)	PUNCT
cana-1225	93	46	)	)	PUNCT
cana-1225	94	1	(	(	PUNCT
cana-1225	94	2	v	v	NOUN
cana-1225	94	3	)	)	PUNCT
cana-1225	94	4	a	a	DET
cana-1225	94	5	bipolar	bipolar	ADJ
cana-1225	94	6	vague	vague	ADJ
cana-1225	94	7	regular	regular	ADJ
cana-1225	94	8	-	-	PUNCT
cana-1225	94	9	open	open	NOUN
cana-1225	94	10	set	set	NOUN
cana-1225	94	11	bvint(bvcl(a	bvint(bvcl(a	NUM
cana-1225	94	12	)	)	PUNCT
cana-1225	94	13	)	)	PUNCT
cana-1225	95	1	=	=	SYM
cana-1225	95	2	a	a	DET
cana-1225	95	3	(	(	PUNCT
cana-1225	95	4	vi	vi	NOUN
cana-1225	95	5	)	)	PUNCT
cana-1225	95	6	a	a	DET
cana-1225	95	7	bipolar	bipolar	ADJ
cana-1225	95	8	vague	vague	ADJ
cana-1225	95	9	𝛽-open	𝛽-open	NOUN
cana-1225	95	10	set	set	VERB
cana-1225	95	11	a	a	DET
cana-1225	95	12	⊆	⊆	NUM
cana-1225	95	13	bvcl(bvint(bvcl(a	bvcl(bvint(bvcl(a	NOUN
cana-1225	95	14	)	)	PUNCT
cana-1225	95	15	)	)	PUNCT
cana-1225	95	16	)	)	PUNCT
cana-1225	95	17	.	.	PUNCT
cana-1225	96	1	definition	definition	NOUN
cana-1225	96	2	2.15	2.15	NUM
cana-1225	96	3	:	:	PUNCT
cana-1225	97	1	[	[	X
cana-1225	97	2	10	10	NUM
cana-1225	97	3	]	]	X
cana-1225	97	4	a	a	DET
cana-1225	97	5	bipolar	bipolar	ADJ
cana-1225	97	6	vague	vague	NOUN
cana-1225	97	7	set	set	VERB
cana-1225	97	8	a	a	PRON
cana-1225	97	9	of	of	ADP
cana-1225	97	10	a	a	DET
cana-1225	97	11	bipolar	bipolar	ADJ
cana-1225	97	12	vague	vague	ADJ
cana-1225	97	13	topological	topological	ADJ
cana-1225	97	14	space	space	NOUN
cana-1225	97	15	x	x	NOUN
cana-1225	97	16	,	,	PUNCT
cana-1225	97	17	is	be	AUX
cana-1225	97	18	said	say	VERB
cana-1225	97	19	to	to	PART
cana-1225	97	20	be	be	AUX
cana-1225	97	21	(	(	PUNCT
cana-1225	97	22	i	i	NOUN
cana-1225	97	23	)	)	PUNCT
cana-1225	97	24	a	a	DET
cana-1225	97	25	bipolar	bipolar	ADJ
cana-1225	97	26	vague	vague	NOUN
cana-1225	97	27	𝛼-closed	𝛼-close	VERB
cana-1225	97	28	set	set	VERB
cana-1225	97	29	if	if	SCONJ
cana-1225	97	30	bvcl(bvint(bvcl(a	bvcl(bvint(bvcl(a	PROPN
cana-1225	97	31	)	)	PUNCT
cana-1225	97	32	)	)	PUNCT
cana-1225	97	33	)	)	PUNCT
cana-1225	98	1	⊆	⊆	X
cana-1225	98	2	a	a	DET
cana-1225	98	3	(	(	PUNCT
cana-1225	98	4	ii	ii	NOUN
cana-1225	98	5	)	)	PUNCT
cana-1225	98	6	a	a	DET
cana-1225	98	7	bipolar	bipolar	ADJ
cana-1225	98	8	vague	vague	ADJ
cana-1225	98	9	pre	pre	ADJ
cana-1225	98	10	-	-	ADJ
cana-1225	98	11	closed	closed	ADJ
cana-1225	98	12	set	set	NOUN
cana-1225	98	13	if	if	SCONJ
cana-1225	98	14	bvcl(bvint(a	bvcl(bvint(a	NOUN
cana-1225	98	15	)	)	PUNCT
cana-1225	98	16	)	)	PUNCT
cana-1225	99	1	⊆	⊆	NUM
cana-1225	99	2	a	a	DET
cana-1225	99	3	(	(	PUNCT
cana-1225	99	4	iii	iii	NOUN
cana-1225	99	5	)	)	PUNCT
cana-1225	99	6	a	a	DET
cana-1225	99	7	bipolar	bipolar	ADJ
cana-1225	99	8	vague	vague	ADJ
cana-1225	99	9	semi	semi	ADJ
cana-1225	99	10	-	-	ADJ
cana-1225	99	11	closed	closed	ADJ
cana-1225	99	12	set	set	NOUN
cana-1225	99	13	if	if	SCONJ
cana-1225	99	14	bvint(bvcl(a	bvint(bvcl(a	NUM
cana-1225	99	15	)	)	PUNCT
cana-1225	99	16	)	)	PUNCT
cana-1225	100	1	⊆	⊆	NUM
cana-1225	100	2	a	a	DET
cana-1225	100	3	(	(	PUNCT
cana-1225	100	4	iv	iv	NOUN
cana-1225	100	5	)	)	PUNCT
cana-1225	100	6	a	a	DET
cana-1225	100	7	bipolar	bipolar	ADJ
cana-1225	100	8	vague	vague	ADJ
cana-1225	100	9	semi-𝛼-closed	semi-𝛼-close	VERB
cana-1225	100	10	set	set	NOUN
cana-1225	100	11	if	if	SCONJ
cana-1225	100	12	bvint(𝛼bvcl(a	bvint(𝛼bvcl(a	NUM
cana-1225	100	13	)	)	PUNCT
cana-1225	100	14	)	)	PUNCT
cana-1225	101	1	⊆	⊆	NUM
cana-1225	101	2	a	a	DET
cana-1225	101	3	(	(	PUNCT
cana-1225	101	4	v	v	NOUN
cana-1225	101	5	)	)	PUNCT
cana-1225	101	6	a	a	DET
cana-1225	101	7	bipolar	bipolar	ADJ
cana-1225	101	8	vague	vague	ADJ
cana-1225	101	9	regular	regular	ADJ
cana-1225	101	10	-	-	PUNCT
cana-1225	101	11	closed	close	VERB
cana-1225	101	12	set	set	NOUN
cana-1225	101	13	if	if	SCONJ
cana-1225	101	14	bvcl(bvint(a	bvcl(bvint(a	NOUN
cana-1225	101	15	)	)	PUNCT
cana-1225	101	16	)	)	PUNCT
cana-1225	102	1	=	=	SYM
cana-1225	102	2	a	a	DET
cana-1225	102	3	(	(	PUNCT
cana-1225	102	4	vi	vi	NOUN
cana-1225	102	5	)	)	PUNCT
cana-1225	102	6	a	a	DET
cana-1225	102	7	bipolar	bipolar	ADJ
cana-1225	102	8	vague	vague	NOUN
cana-1225	102	9	𝛽-closed	𝛽-close	VERB
cana-1225	102	10	set	set	VERB
cana-1225	102	11	if	if	SCONJ
cana-1225	102	12	bvint(bvcl(bvint(a	bvint(bvcl(bvint(a	PROPN
cana-1225	102	13	)	)	PUNCT
cana-1225	102	14	)	)	PUNCT
cana-1225	102	15	)	)	PUNCT
cana-1225	103	1	⊆	⊆	NUM
cana-1225	103	2	a.	a.	NOUN
cana-1225	103	3	definition	definition	NOUN
cana-1225	103	4	2.16	2.16	NUM
cana-1225	103	5	:	:	PUNCT
cana-1225	104	1	[	[	X
cana-1225	104	2	10	10	NUM
cana-1225	104	3	]	]	PUNCT
cana-1225	104	4	let	let	VERB
cana-1225	104	5	a	a	PRON
cana-1225	104	6	be	be	AUX
cana-1225	104	7	a	a	DET
cana-1225	104	8	bipolar	bipolar	ADJ
cana-1225	104	9	vague	vague	ADJ
cana-1225	104	10	set	set	NOUN
cana-1225	104	11	of	of	ADP
cana-1225	104	12	a	a	DET
cana-1225	104	13	bipolar	bipolar	ADJ
cana-1225	104	14	vague	vague	ADJ
cana-1225	104	15	topological	topological	ADJ
cana-1225	104	16	space	space	NOUN
cana-1225	104	17	(	(	PUNCT
cana-1225	104	18	x	x	NOUN
cana-1225	104	19	,	,	PUNCT
cana-1225	104	20	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	104	21	)	)	PUNCT
cana-1225	104	22	.	.	PUNCT
cana-1225	105	1	then	then	ADV
cana-1225	105	2	the	the	DET
cana-1225	105	3	bipolar	bipolar	ADJ
cana-1225	105	4	vague	vague	NOUN
cana-1225	105	5	𝛼	𝛼	NOUN
cana-1225	105	6	interior	interior	ADJ
cana-1225	105	7	and	and	CCONJ
cana-1225	105	8	bipolar	bipolar	ADJ
cana-1225	105	9	vague	vague	ADJ
cana-1225	105	10	𝛼	𝛼	NOUN
cana-1225	105	11	closure	closure	NOUN
cana-1225	105	12	are	be	AUX
cana-1225	105	13	defined	define	VERB
cana-1225	105	14	as	as	ADP
cana-1225	105	15	b𝑉𝛼int(a	b𝑉𝛼int(a	NUM
cana-1225	105	16	)	)	PUNCT
cana-1225	106	1	=	=	SYM
cana-1225	106	2	∪	∪	X
cana-1225	106	3	{	{	PUNCT
cana-1225	106	4	g	g	NOUN
cana-1225	106	5	:	:	PUNCT
cana-1225	106	6	g	g	PROPN
cana-1225	106	7	is	be	AUX
cana-1225	106	8	a	a	DET
cana-1225	106	9	bipolar	bipolar	ADJ
cana-1225	106	10	vague	vague	ADJ
cana-1225	106	11	𝛼-open	𝛼-open	NOUN
cana-1225	106	12	set	set	NOUN
cana-1225	106	13	in	in	ADP
cana-1225	106	14	x	x	X
cana-1225	106	15	and	and	CCONJ
cana-1225	106	16	g	g	PROPN
cana-1225	106	17	⊆	⊆	NUM
cana-1225	106	18	a	a	PRON
cana-1225	106	19	}	}	PUNCT
cana-1225	106	20	,	,	PUNCT
cana-1225	106	21	b𝑉𝛼cl(a	b𝑉𝛼cl(a	ADJ
cana-1225	106	22	)	)	PUNCT
cana-1225	106	23	=	=	NOUN
cana-1225	106	24	∩	∩	NOUN
cana-1225	106	25	{	{	PUNCT
cana-1225	106	26	k	k	NOUN
cana-1225	106	27	:	:	PUNCT
cana-1225	106	28	k	k	X
cana-1225	106	29	is	be	AUX
cana-1225	106	30	a	a	DET
cana-1225	106	31	bipolar	bipolar	ADJ
cana-1225	106	32	vague	vague	NOUN
cana-1225	106	33	𝛼-closed	𝛼-close	VERB
cana-1225	106	34	set	set	VERB
cana-1225	106	35	in	in	ADP
cana-1225	106	36	x	x	PUNCT
cana-1225	106	37	and	and	CCONJ
cana-1225	106	38	a⊆	a⊆	VERB
cana-1225	106	39	k	k	X
cana-1225	106	40	}	}	PUNCT
cana-1225	106	41	.	.	PUNCT
cana-1225	107	1	definition	definition	NOUN
cana-1225	107	2	2.17	2.17	NUM
cana-1225	107	3	:	:	PUNCT
cana-1225	108	1	[	[	X
cana-1225	108	2	10	10	NUM
cana-1225	108	3	]	]	X
cana-1225	108	4	a	a	DET
cana-1225	108	5	bipolar	bipolar	ADJ
cana-1225	108	6	vague	vague	NOUN
cana-1225	108	7	set	set	VERB
cana-1225	108	8	a	a	PRON
cana-1225	108	9	in	in	ADP
cana-1225	108	10	a	a	DET
cana-1225	108	11	bipolar	bipolar	ADJ
cana-1225	108	12	vague	vague	ADJ
cana-1225	108	13	topological	topological	ADJ
cana-1225	108	14	space	space	NOUN
cana-1225	108	15	x	x	NOUN
cana-1225	108	16	,	,	PUNCT
cana-1225	108	17	is	be	AUX
cana-1225	108	18	said	say	VERB
cana-1225	108	19	to	to	PART
cana-1225	108	20	be	be	AUX
cana-1225	108	21	a	a	DET
cana-1225	108	22	bipolar	bipolar	ADJ
cana-1225	108	23	vague	vague	NOUN
cana-1225	109	1	𝛼	𝛼	PRON
cana-1225	109	2	generalized	generalize	VERB
cana-1225	109	3	closed	close	VERB
cana-1225	109	4	set	set	VERB
cana-1225	109	5	if	if	SCONJ
cana-1225	109	6	b𝑉𝛼cl(a	b𝑉𝛼cl(a	PROPN
cana-1225	109	7	)	)	PUNCT
cana-1225	109	8	⊆	⊆	NUM
cana-1225	109	9	u	u	NOUN
cana-1225	109	10	whenever	whenever	SCONJ
cana-1225	109	11	a⊆	a⊆	VERB
cana-1225	109	12	u	u	NOUN
cana-1225	109	13	and	and	CCONJ
cana-1225	109	14	u	u	NOUN
cana-1225	109	15	is	be	AUX
cana-1225	109	16	a	a	DET
cana-1225	109	17	bipolar	bipolar	ADJ
cana-1225	109	18	vague	vague	ADJ
cana-1225	109	19	open	open	ADJ
cana-1225	109	20	set	set	VERB
cana-1225	109	21	in	in	ADP
cana-1225	109	22	x.	x.	NOUN
cana-1225	109	23	the	the	DET
cana-1225	109	24	complement	complement	NOUN
cana-1225	109	25	ac	ac	INTJ
cana-1225	109	26	of	of	ADP
cana-1225	109	27	a	a	DET
cana-1225	109	28	bipolar	bipolar	ADJ
cana-1225	109	29	vague	vague	NOUN
cana-1225	110	1	𝛼	𝛼	DET
cana-1225	110	2	generalized	generalize	VERB
cana-1225	110	3	closed	close	VERB
cana-1225	110	4	set	set	VERB
cana-1225	110	5	a	a	PRON
cana-1225	110	6	is	be	AUX
cana-1225	110	7	a	a	DET
cana-1225	110	8	bipolar	bipolar	ADJ
cana-1225	110	9	vague	vague	NOUN
cana-1225	110	10	𝛼	𝛼	ADP
cana-1225	110	11	generalized	generalize	VERB
cana-1225	110	12	open	open	ADJ
cana-1225	110	13	set	set	VERB
cana-1225	110	14	in	in	ADP
cana-1225	110	15	x.	x.	NOUN
cana-1225	110	16	definition	definition	NOUN
cana-1225	110	17	2.18	2.18	NUM
cana-1225	110	18	:	:	PUNCT
cana-1225	110	19	[	[	X
cana-1225	110	20	4	4	X
cana-1225	110	21	]	]	X
cana-1225	110	22	let	let	VERB
cana-1225	110	23	(	(	PUNCT
cana-1225	110	24	x	x	NOUN
cana-1225	110	25	,	,	PUNCT
cana-1225	110	26	b𝑉𝜏	b𝑉𝜏	NUM
cana-1225	110	27	)	)	PUNCT
cana-1225	110	28	and	and	CCONJ
cana-1225	110	29	(	(	PUNCT
cana-1225	110	30	y	y	NOUN
cana-1225	110	31	,	,	PUNCT
cana-1225	110	32	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	110	33	)	)	PUNCT
cana-1225	110	34	be	be	VERB
cana-1225	110	35	two	two	NUM
cana-1225	110	36	bipolar	bipolar	ADJ
cana-1225	110	37	vague	vague	ADJ
cana-1225	110	38	topological	topological	ADJ
cana-1225	110	39	spaces	space	NOUN
cana-1225	110	40	and	and	CCONJ
cana-1225	110	41	𝑓	𝑓	PRON
cana-1225	110	42	:	:	PUNCT
cana-1225	110	43	x→	x→	PUNCT
cana-1225	111	1	y	y	NOUN
cana-1225	111	2	be	be	AUX
cana-1225	111	3	a	a	DET
cana-1225	111	4	function	function	NOUN
cana-1225	111	5	.	.	PUNCT
cana-1225	112	1	then	then	ADV
cana-1225	112	2	𝜑	𝜑	PROPN
cana-1225	112	3	is	be	AUX
cana-1225	112	4	said	say	VERB
cana-1225	112	5	to	to	PART
cana-1225	112	6	be	be	AUX
cana-1225	112	7	bipolar	bipolar	ADJ
cana-1225	112	8	vague	vague	ADJ
cana-1225	112	9	continuous	continuous	ADJ
cana-1225	112	10	if	if	SCONJ
cana-1225	112	11	and	and	CCONJ
cana-1225	112	12	only	only	ADV
cana-1225	112	13	if	if	SCONJ
cana-1225	112	14	the	the	DET
cana-1225	112	15	preimage	preimage	NOUN
cana-1225	112	16	of	of	ADP
cana-1225	112	17	each	each	DET
cana-1225	112	18	bipolar	bipolar	ADJ
cana-1225	112	19	vague	vague	ADJ
cana-1225	112	20	open	open	ADJ
cana-1225	112	21	set	set	VERB
cana-1225	112	22	in	in	ADP
cana-1225	112	23	y	y	PROPN
cana-1225	112	24	is	be	AUX
cana-1225	112	25	a	a	DET
cana-1225	112	26	bipolar	bipolar	ADJ
cana-1225	112	27	vague	vague	ADJ
cana-1225	112	28	open	open	ADJ
cana-1225	112	29	set	set	VERB
cana-1225	112	30	in	in	ADP
cana-1225	112	31	x.	x.	NOUN
cana-1225	112	32	definition	definition	NOUN
cana-1225	112	33	2.19	2.19	NUM
cana-1225	112	34	:	:	PUNCT
cana-1225	113	1	[	[	X
cana-1225	113	2	4	4	X
cana-1225	113	3	]	]	PUNCT
cana-1225	113	4	a	a	DET
cana-1225	113	5	map	map	NOUN
cana-1225	113	6	𝑓	𝑓	X
cana-1225	113	7	:	:	PUNCT
cana-1225	113	8	(	(	PUNCT
cana-1225	113	9	x	x	NOUN
cana-1225	113	10	,	,	PUNCT
cana-1225	113	11	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	113	12	)	)	PUNCT
cana-1225	113	13	→	→	SYM
cana-1225	113	14	(	(	PUNCT
cana-1225	113	15	y	y	NOUN
cana-1225	113	16	,	,	PUNCT
cana-1225	113	17	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	113	18	)	)	PUNCT
cana-1225	113	19	is	be	AUX
cana-1225	113	20	said	say	VERB
cana-1225	113	21	to	to	PART
cana-1225	113	22	be	be	AUX
cana-1225	113	23	generalized	generalize	VERB
cana-1225	113	24	bipolar	bipolar	ADJ
cana-1225	113	25	vague	vague	ADJ
cana-1225	113	26	continuous	continuous	ADJ
cana-1225	113	27	if	if	SCONJ
cana-1225	113	28	the	the	DET
cana-1225	113	29	inverse	inverse	ADJ
cana-1225	113	30	image	image	NOUN
cana-1225	113	31	of	of	ADP
cana-1225	113	32	every	every	DET
cana-1225	113	33	bipolar	bipolar	ADJ
cana-1225	113	34	vague	vague	ADJ
cana-1225	113	35	open	open	ADJ
cana-1225	113	36	set	set	VERB
cana-1225	113	37	in	in	ADP
cana-1225	113	38	(	(	PUNCT
cana-1225	113	39	y	y	NOUN
cana-1225	113	40	,	,	PUNCT
cana-1225	113	41	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	113	42	)	)	PUNCT
cana-1225	113	43	is	be	AUX
cana-1225	113	44	a	a	DET
cana-1225	113	45	generalized	generalized	ADJ
cana-1225	113	46	vague	vague	ADJ
cana-1225	113	47	open	open	ADJ
cana-1225	113	48	set	set	VERB
cana-1225	113	49	in	in	ADP
cana-1225	113	50	(	(	PUNCT
cana-1225	113	51	x	x	NOUN
cana-1225	113	52	,	,	PUNCT
cana-1225	113	53	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	113	54	)	)	PUNCT
cana-1225	113	55	.	.	PUNCT
cana-1225	114	1	definition	definition	NOUN
cana-1225	114	2	2.20	2.20	NUM
cana-1225	114	3	:	:	PUNCT
cana-1225	115	1	[	[	X
cana-1225	115	2	4	4	X
cana-1225	115	3	]	]	PUNCT
cana-1225	115	4	let	let	VERB
cana-1225	115	5	𝑓	𝑓	PRON
cana-1225	115	6	be	be	AUX
cana-1225	115	7	a	a	DET
cana-1225	115	8	mapping	mapping	NOUN
cana-1225	115	9	from	from	ADP
cana-1225	115	10	a	a	DET
cana-1225	115	11	bipolar	bipolar	ADJ
cana-1225	115	12	vague	vague	ADJ
cana-1225	115	13	topological	topological	ADJ
cana-1225	115	14	space	space	NOUN
cana-1225	115	15	(	(	PUNCT
cana-1225	115	16	x	x	NOUN
cana-1225	115	17	,	,	PUNCT
cana-1225	115	18	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	115	19	)	)	PUNCT
cana-1225	115	20	into	into	ADP
cana-1225	115	21	a	a	DET
cana-1225	115	22	bipolar	bipolar	ADJ
cana-1225	115	23	vague	vague	ADJ
cana-1225	115	24	topological	topological	ADJ
cana-1225	115	25	space	space	NOUN
cana-1225	115	26	(	(	PUNCT
cana-1225	115	27	y	y	NOUN
cana-1225	115	28	,	,	PUNCT
cana-1225	115	29	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	115	30	)	)	PUNCT
cana-1225	115	31	.	.	PUNCT
cana-1225	116	1	then	then	ADV
cana-1225	116	2	𝑓	𝑓	PRON
cana-1225	116	3	is	be	AUX
cana-1225	116	4	said	say	VERB
cana-1225	116	5	to	to	PART
cana-1225	116	6	be	be	AUX
cana-1225	116	7	a	a	DET
cana-1225	116	8	bipolar	bipolar	ADJ
cana-1225	116	9	vague	vague	ADJ
cana-1225	116	10	generalized	generalized	ADJ
cana-1225	116	11	irresolute	irresolute	ADJ
cana-1225	116	12	mapping	mapping	NOUN
cana-1225	116	13	if	if	SCONJ
cana-1225	116	14	the	the	DET
cana-1225	116	15	inverse	inverse	ADJ
cana-1225	116	16	image	image	NOUN
cana-1225	116	17	of	of	ADP
cana-1225	116	18	every	every	DET
cana-1225	116	19	bipolar	bipolar	ADJ
cana-1225	116	20	vague	vague	ADJ
cana-1225	116	21	generalized	generalize	VERB
cana-1225	116	22	closed	close	VERB
cana-1225	116	23	set	set	VERB
cana-1225	116	24	in	in	ADP
cana-1225	116	25	(	(	PUNCT
cana-1225	116	26	y	y	NOUN
cana-1225	116	27	,	,	PUNCT
cana-1225	116	28	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	116	29	)	)	PUNCT
cana-1225	116	30	is	be	AUX
cana-1225	116	31	a	a	DET
cana-1225	116	32	bipolar	bipolar	ADJ
cana-1225	116	33	vague	vague	ADJ
cana-1225	116	34	generalized	generalize	VERB
cana-1225	116	35	closed	close	VERB
cana-1225	116	36	set	set	VERB
cana-1225	116	37	in	in	ADP
cana-1225	116	38	(	(	PUNCT
cana-1225	116	39	x	x	NOUN
cana-1225	116	40	,	,	PUNCT
cana-1225	116	41	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	116	42	)	)	PUNCT
cana-1225	116	43	.	.	PUNCT
cana-1225	117	1	3	3	X
cana-1225	117	2	.	.	X
cana-1225	117	3	bipolar	bipolar	ADJ
cana-1225	117	4	vague	vague	NOUN
cana-1225	117	5	𝜶	𝜶	ADP
cana-1225	117	6	generalized	generalize	VERB
cana-1225	117	7	continuous	continuous	ADJ
cana-1225	117	8	mappings	mapping	NOUN
cana-1225	117	9	in	in	ADP
cana-1225	117	10	topological	topological	ADJ
cana-1225	117	11	spaces	space	NOUN
cana-1225	117	12	communications	communication	NOUN
cana-1225	117	13	on	on	ADP
cana-1225	117	14	applied	apply	VERB
cana-1225	117	15	nonlinear	nonlinear	ADJ
cana-1225	117	16	analysis	analysis	NOUN
cana-1225	117	17	issn	issn	NOUN
cana-1225	117	18	:	:	PUNCT
cana-1225	117	19	1074	1074	NUM
cana-1225	117	20	-	-	PUNCT
cana-1225	117	21	133x	133x	NUM
cana-1225	117	22	vol	vol	NOUN
cana-1225	117	23	31	31	NUM
cana-1225	117	24	no	no	NOUN
cana-1225	117	25	.	.	PUNCT
cana-1225	118	1	6s	6s	NUM
cana-1225	118	2	(	(	PUNCT
cana-1225	118	3	2024	2024	NUM
cana-1225	118	4	)	)	PUNCT
cana-1225	118	5	322	322	NUM
cana-1225	118	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1225	118	7	in	in	ADP
cana-1225	118	8	this	this	DET
cana-1225	118	9	section	section	NOUN
cana-1225	118	10	we	we	PRON
cana-1225	118	11	have	have	AUX
cana-1225	118	12	introduced	introduce	VERB
cana-1225	118	13	bipolar	bipolar	ADJ
cana-1225	118	14	vague	vague	NOUN
cana-1225	118	15	𝛼	𝛼	ADP
cana-1225	118	16	generalized	generalize	VERB
cana-1225	118	17	continuous	continuous	ADJ
cana-1225	118	18	mappings	mapping	NOUN
cana-1225	118	19	and	and	CCONJ
cana-1225	118	20	investigated	investigate	VERB
cana-1225	118	21	some	some	PRON
cana-1225	118	22	of	of	ADP
cana-1225	118	23	their	their	PRON
cana-1225	118	24	properties	property	NOUN
cana-1225	118	25	.	.	PUNCT
cana-1225	119	1	also	also	ADV
cana-1225	119	2	,	,	PUNCT
cana-1225	119	3	we	we	PRON
cana-1225	119	4	have	have	AUX
cana-1225	119	5	established	establish	VERB
cana-1225	119	6	the	the	DET
cana-1225	119	7	relation	relation	NOUN
cana-1225	119	8	between	between	ADP
cana-1225	119	9	the	the	DET
cana-1225	119	10	newly	newly	ADV
cana-1225	119	11	introduced	introduce	VERB
cana-1225	119	12	mappings	mapping	NOUN
cana-1225	119	13	and	and	CCONJ
cana-1225	119	14	already	already	ADV
cana-1225	119	15	existing	exist	VERB
cana-1225	119	16	mappings	mapping	NOUN
cana-1225	119	17	.	.	PUNCT
cana-1225	120	1	definition	definition	NOUN
cana-1225	120	2	3.1	3.1	NUM
cana-1225	120	3	:	:	PUNCT
cana-1225	120	4	let	let	VERB
cana-1225	120	5	(	(	PUNCT
cana-1225	120	6	x	x	NOUN
cana-1225	120	7	,	,	PUNCT
cana-1225	120	8	b𝑉𝜏	b𝑉𝜏	NUM
cana-1225	120	9	)	)	PUNCT
cana-1225	120	10	and	and	CCONJ
cana-1225	120	11	(	(	PUNCT
cana-1225	120	12	y	y	NOUN
cana-1225	120	13	,	,	PUNCT
cana-1225	120	14	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	120	15	)	)	PUNCT
cana-1225	120	16	be	be	VERB
cana-1225	120	17	two	two	NUM
cana-1225	120	18	bipolar	bipolar	ADJ
cana-1225	120	19	vague	vague	ADJ
cana-1225	120	20	topological	topological	ADJ
cana-1225	120	21	spaces	space	NOUN
cana-1225	120	22	.	.	PUNCT
cana-1225	121	1	then	then	ADV
cana-1225	121	2	the	the	DET
cana-1225	121	3	mapping	mapping	NOUN
cana-1225	121	4	𝑓	𝑓	X
cana-1225	121	5	:	:	PUNCT
cana-1225	121	6	(	(	PUNCT
cana-1225	121	7	x	x	NOUN
cana-1225	121	8	,	,	PUNCT
cana-1225	121	9	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	121	10	)	)	PUNCT
cana-1225	121	11	→	→	SYM
cana-1225	121	12	(	(	PUNCT
cana-1225	121	13	y	y	NOUN
cana-1225	121	14	,	,	PUNCT
cana-1225	121	15	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	121	16	)	)	PUNCT
cana-1225	121	17	is	be	AUX
cana-1225	121	18	called	call	VERB
cana-1225	121	19	(	(	PUNCT
cana-1225	121	20	i	i	NOUN
cana-1225	121	21	)	)	PUNCT
cana-1225	121	22	a	a	DET
cana-1225	121	23	bipolar	bipolar	ADJ
cana-1225	121	24	vague	vague	NOUN
cana-1225	121	25	𝛼	𝛼	NOUN
cana-1225	121	26	continuous	continuous	ADJ
cana-1225	121	27	if	if	SCONJ
cana-1225	121	28	the	the	DET
cana-1225	121	29	inverse	inverse	ADJ
cana-1225	121	30	image	image	NOUN
cana-1225	121	31	of	of	ADP
cana-1225	121	32	every	every	DET
cana-1225	121	33	bipolar	bipolar	ADJ
cana-1225	121	34	vague	vague	NOUN
cana-1225	121	35	closed	close	VERB
cana-1225	121	36	set	set	VERB
cana-1225	121	37	in	in	ADP
cana-1225	121	38	(	(	PUNCT
cana-1225	121	39	y	y	NOUN
cana-1225	121	40	,	,	PUNCT
cana-1225	121	41	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	121	42	)	)	PUNCT
cana-1225	121	43	is	be	AUX
cana-1225	121	44	a	a	DET
cana-1225	121	45	bipolar	bipolar	ADJ
cana-1225	121	46	vague	vague	NOUN
cana-1225	121	47	𝛼-closed	𝛼-close	VERB
cana-1225	121	48	set	set	VERB
cana-1225	121	49	in	in	ADP
cana-1225	121	50	(	(	PUNCT
cana-1225	121	51	x	x	NOUN
cana-1225	121	52	,	,	PUNCT
cana-1225	121	53	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	121	54	)	)	PUNCT
cana-1225	121	55	.	.	PUNCT
cana-1225	122	1	(	(	PUNCT
cana-1225	122	2	ii	ii	X
cana-1225	122	3	)	)	PUNCT
cana-1225	122	4	a	a	DET
cana-1225	122	5	bipolar	bipolar	ADJ
cana-1225	122	6	vague	vague	ADJ
cana-1225	122	7	pre	pre	NOUN
cana-1225	122	8	continuous	continuous	ADJ
cana-1225	122	9	if	if	SCONJ
cana-1225	122	10	the	the	DET
cana-1225	122	11	inverse	inverse	ADJ
cana-1225	122	12	image	image	NOUN
cana-1225	122	13	of	of	ADP
cana-1225	122	14	every	every	DET
cana-1225	122	15	bipolar	bipolar	ADJ
cana-1225	122	16	vague	vague	NOUN
cana-1225	122	17	closed	close	VERB
cana-1225	122	18	set	set	VERB
cana-1225	122	19	in	in	ADP
cana-1225	122	20	(	(	PUNCT
cana-1225	122	21	y	y	NOUN
cana-1225	122	22	,	,	PUNCT
cana-1225	122	23	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	122	24	)	)	PUNCT
cana-1225	122	25	is	be	AUX
cana-1225	122	26	a	a	DET
cana-1225	122	27	bipolar	bipolar	ADJ
cana-1225	122	28	vague	vague	ADJ
cana-1225	122	29	pre	pre	ADJ
cana-1225	122	30	-	-	ADJ
cana-1225	122	31	closed	closed	ADJ
cana-1225	122	32	set	set	NOUN
cana-1225	122	33	in	in	ADP
cana-1225	122	34	(	(	PUNCT
cana-1225	122	35	x	x	NOUN
cana-1225	122	36	,	,	PUNCT
cana-1225	122	37	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	122	38	)	)	PUNCT
cana-1225	122	39	.	.	PUNCT
cana-1225	123	1	(	(	PUNCT
cana-1225	123	2	iii	iii	X
cana-1225	123	3	)	)	PUNCT
cana-1225	123	4	a	a	DET
cana-1225	123	5	bipolar	bipolar	ADJ
cana-1225	123	6	vague	vague	NOUN
cana-1225	123	7	semi	semi	ADV
cana-1225	123	8	continuous	continuous	ADJ
cana-1225	123	9	if	if	SCONJ
cana-1225	123	10	the	the	DET
cana-1225	123	11	inverse	inverse	ADJ
cana-1225	123	12	image	image	NOUN
cana-1225	123	13	of	of	ADP
cana-1225	123	14	every	every	DET
cana-1225	123	15	bipolar	bipolar	ADJ
cana-1225	123	16	vague	vague	NOUN
cana-1225	123	17	closed	close	VERB
cana-1225	123	18	set	set	VERB
cana-1225	123	19	in	in	ADP
cana-1225	123	20	(	(	PUNCT
cana-1225	123	21	y	y	NOUN
cana-1225	123	22	,	,	PUNCT
cana-1225	123	23	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	123	24	)	)	PUNCT
cana-1225	123	25	is	be	AUX
cana-1225	123	26	a	a	DET
cana-1225	123	27	bipolar	bipolar	ADJ
cana-1225	123	28	vague	vague	ADJ
cana-1225	123	29	semi	semi	ADJ
cana-1225	123	30	-	-	ADJ
cana-1225	123	31	closed	closed	ADJ
cana-1225	123	32	set	set	NOUN
cana-1225	123	33	in	in	ADP
cana-1225	123	34	(	(	PUNCT
cana-1225	123	35	x	x	NOUN
cana-1225	123	36	,	,	PUNCT
cana-1225	123	37	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	123	38	)	)	PUNCT
cana-1225	123	39	.	.	PUNCT
cana-1225	124	1	definition	definition	NOUN
cana-1225	124	2	3.2	3.2	NUM
cana-1225	124	3	:	:	PUNCT
cana-1225	124	4	let	let	VERB
cana-1225	124	5	(	(	PUNCT
cana-1225	124	6	x	x	NOUN
cana-1225	124	7	,	,	PUNCT
cana-1225	124	8	b𝑉𝜏	b𝑉𝜏	NUM
cana-1225	124	9	)	)	PUNCT
cana-1225	124	10	and	and	CCONJ
cana-1225	124	11	(	(	PUNCT
cana-1225	124	12	y	y	NOUN
cana-1225	124	13	,	,	PUNCT
cana-1225	124	14	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	124	15	)	)	PUNCT
cana-1225	124	16	be	be	VERB
cana-1225	124	17	two	two	NUM
cana-1225	124	18	bipolar	bipolar	ADJ
cana-1225	124	19	vague	vague	ADJ
cana-1225	124	20	topological	topological	ADJ
cana-1225	124	21	spaces	space	NOUN
cana-1225	124	22	.	.	PUNCT
cana-1225	125	1	a	a	DET
cana-1225	125	2	mapping	mapping	NOUN
cana-1225	125	3	𝑓	𝑓	X
cana-1225	125	4	:	:	PUNCT
cana-1225	125	5	(	(	PUNCT
cana-1225	125	6	x	x	NOUN
cana-1225	125	7	,	,	PUNCT
cana-1225	125	8	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	125	9	)	)	PUNCT
cana-1225	125	10	→	→	SYM
cana-1225	125	11	(	(	PUNCT
cana-1225	125	12	y	y	NOUN
cana-1225	125	13	,	,	PUNCT
cana-1225	125	14	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	125	15	)	)	PUNCT
cana-1225	125	16	is	be	AUX
cana-1225	125	17	called	call	VERB
cana-1225	125	18	a	a	DET
cana-1225	125	19	bipolar	bipolar	ADJ
cana-1225	125	20	vague	vague	NOUN
cana-1225	126	1	𝛼	𝛼	ADP
cana-1225	126	2	generalized	generalize	VERB
cana-1225	126	3	continuous	continuous	ADJ
cana-1225	126	4	mapping	mapping	NOUN
cana-1225	126	5	if	if	SCONJ
cana-1225	126	6	𝑓−1(b	𝑓−1(b	PROPN
cana-1225	126	7	)	)	PUNCT
cana-1225	126	8	is	be	AUX
cana-1225	126	9	a	a	DET
cana-1225	126	10	bipolar	bipolar	ADJ
cana-1225	126	11	vague	vague	NOUN
cana-1225	126	12	𝛼	𝛼	DET
cana-1225	126	13	generalized	generalize	VERB
cana-1225	126	14	closed	close	VERB
cana-1225	126	15	set	set	VERB
cana-1225	126	16	in	in	ADP
cana-1225	126	17	(	(	PUNCT
cana-1225	126	18	x	x	NOUN
cana-1225	126	19	,	,	PUNCT
cana-1225	126	20	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	126	21	)	)	PUNCT
cana-1225	126	22	for	for	ADP
cana-1225	126	23	every	every	DET
cana-1225	126	24	bipolar	bipolar	ADJ
cana-1225	126	25	vague	vague	NOUN
cana-1225	126	26	closed	close	VERB
cana-1225	126	27	set	set	ADJ
cana-1225	126	28	b	b	PROPN
cana-1225	126	29	of	of	ADP
cana-1225	126	30	(	(	PUNCT
cana-1225	126	31	y	y	PROPN
cana-1225	126	32	,	,	PUNCT
cana-1225	126	33	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	126	34	)	)	PUNCT
cana-1225	126	35	.	.	PUNCT
cana-1225	127	1	example	example	NOUN
cana-1225	127	2	3.3	3.3	NUM
cana-1225	127	3	:	:	PUNCT
cana-1225	127	4	let	let	VERB
cana-1225	127	5	x	x	PUNCT
cana-1225	127	6	=	=	PRON
cana-1225	127	7	{	{	PUNCT
cana-1225	127	8	a	a	DET
cana-1225	127	9	,	,	PUNCT
cana-1225	127	10	b	b	NOUN
cana-1225	127	11	}	}	PUNCT
cana-1225	127	12	and	and	CCONJ
cana-1225	127	13	y	y	PROPN
cana-1225	127	14	=	=	SYM
cana-1225	127	15	{	{	PUNCT
cana-1225	127	16	u	u	NOUN
cana-1225	127	17	,	,	PUNCT
cana-1225	127	18	v	v	NOUN
cana-1225	127	19	}	}	PUNCT
cana-1225	127	20	.	.	PUNCT
cana-1225	128	1	then	then	ADV
cana-1225	128	2	𝜏	𝜏	X
cana-1225	128	3	=	=	PUNCT
cana-1225	128	4	{	{	PUNCT
cana-1225	128	5	0~	0~	NOUN
cana-1225	128	6	,	,	PUNCT
cana-1225	128	7	a	a	DET
cana-1225	128	8	,	,	PUNCT
cana-1225	128	9	1~	1~	NUM
cana-1225	128	10	}	}	PUNCT
cana-1225	128	11	and	and	CCONJ
cana-1225	128	12	𝜎	𝜎	NOUN
cana-1225	128	13	=	=	SYM
cana-1225	128	14	{	{	PUNCT
cana-1225	128	15	0~	0~	NOUN
cana-1225	128	16	,	,	PUNCT
cana-1225	128	17	b	b	NOUN
cana-1225	128	18	,	,	PUNCT
cana-1225	128	19	1~	1~	NUM
cana-1225	128	20	}	}	PUNCT
cana-1225	128	21	are	be	AUX
cana-1225	128	22	bipolar	bipolar	ADJ
cana-1225	128	23	vague	vague	ADJ
cana-1225	128	24	topologies	topology	NOUN
cana-1225	128	25	on	on	ADP
cana-1225	128	26	x	x	X
cana-1225	128	27	and	and	CCONJ
cana-1225	128	28	y	y	PROPN
cana-1225	128	29	respectively	respectively	ADV
cana-1225	128	30	,	,	PUNCT
cana-1225	128	31	where	where	SCONJ
cana-1225	128	32	a	a	DET
cana-1225	128	33	=	=	SYM
cana-1225	128	34	x	x	NOUN
cana-1225	128	35	,	,	PUNCT
cana-1225	128	36	[	[	X
cana-1225	128	37	0.5	0.5	NUM
cana-1225	128	38	,	,	PUNCT
cana-1225	128	39	0.5	0.5	NUM
cana-1225	128	40	]	]	PUNCT
cana-1225	128	41	[	[	X
cana-1225	128	42	-0.5	-0.5	X
cana-1225	128	43	,	,	PUNCT
cana-1225	128	44	-0.5	-0.5	PROPN
cana-1225	128	45	]	]	PUNCT
cana-1225	128	46	,	,	PUNCT
cana-1225	128	47	[	[	X
cana-1225	128	48	0.5	0.5	NUM
cana-1225	128	49	,	,	PUNCT
cana-1225	128	50	0.5	0.5	NUM
cana-1225	128	51	]	]	PUNCT
cana-1225	129	1	[	[	X
cana-1225	129	2	0.5	0.5	NUM
cana-1225	129	3	,	,	PUNCT
cana-1225	129	4	-0.5]	-0.5]	NOUN
cana-1225	129	5	and	and	CCONJ
cana-1225	129	6	b	b	X
cana-1225	129	7	=	=	SYM
cana-1225	129	8	y	y	PROPN
cana-1225	129	9	,	,	PUNCT
cana-1225	129	10	[	[	X
cana-1225	129	11	0.7	0.7	NUM
cana-1225	129	12	,	,	PUNCT
cana-1225	129	13	0.6	0.6	NUM
cana-1225	129	14	]	]	PUNCT
cana-1225	130	1	[	[	X
cana-1225	130	2	-0.9	-0.9	NOUN
cana-1225	130	3	,	,	PUNCT
cana-1225	130	4	-0.9	-0.9	NOUN
cana-1225	130	5	]	]	X
cana-1225	130	6	,	,	PUNCT
cana-1225	130	7	[	[	X
cana-1225	130	8	0.6	0.6	NUM
cana-1225	130	9	,	,	PUNCT
cana-1225	130	10	0.6	0.6	NUM
cana-1225	130	11	]	]	PUNCT
cana-1225	131	1	[	[	X
cana-1225	131	2	-0.5	-0.5	NOUN
cana-1225	131	3	,	,	PUNCT
cana-1225	131	4	-0.5].	-0.5].	VERB
cana-1225	131	5	define	define	VERB
cana-1225	131	6	a	a	DET
cana-1225	131	7	mapping	mapping	NOUN
cana-1225	131	8	𝑓	𝑓	X
cana-1225	131	9	:	:	PUNCT
cana-1225	131	10	(	(	PUNCT
cana-1225	131	11	x	x	NOUN
cana-1225	131	12	,	,	PUNCT
cana-1225	131	13	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	131	14	)	)	PUNCT
cana-1225	131	15	→	→	SYM
cana-1225	131	16	(	(	PUNCT
cana-1225	131	17	y	y	NOUN
cana-1225	131	18	,	,	PUNCT
cana-1225	131	19	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	131	20	)	)	PUNCT
cana-1225	131	21	by	by	ADP
cana-1225	131	22	f(a	f(a	PROPN
cana-1225	131	23	)	)	PUNCT
cana-1225	131	24	=	=	SYM
cana-1225	131	25	u	u	NOUN
cana-1225	131	26	and	and	CCONJ
cana-1225	131	27	f(b	f(b	PROPN
cana-1225	131	28	)	)	PUNCT
cana-1225	132	1	=	=	PUNCT
cana-1225	133	1	v.	v.	CCONJ
cana-1225	133	2	here	here	ADV
cana-1225	133	3	the	the	DET
cana-1225	133	4	bipolar	bipolar	ADJ
cana-1225	133	5	vague	vague	NOUN
cana-1225	133	6	set	set	VERB
cana-1225	133	7	bc	bc	PROPN
cana-1225	133	8	=	=	PUNCT
cana-1225	133	9	y	y	PROPN
cana-1225	133	10	,	,	PUNCT
cana-1225	133	11	[	[	X
cana-1225	133	12	0.4	0.4	NUM
cana-1225	133	13	,	,	PUNCT
cana-1225	133	14	0.3	0.3	NUM
cana-1225	133	15	]	]	PUNCT
cana-1225	134	1	[	[	X
cana-1225	134	2	-0.1	-0.1	PROPN
cana-1225	134	3	,	,	PUNCT
cana-1225	134	4	-0.1	-0.1	PROPN
cana-1225	134	5	]	]	X
cana-1225	134	6	,	,	PUNCT
cana-1225	134	7	[	[	X
cana-1225	134	8	0.4	0.4	NUM
cana-1225	134	9	,	,	PUNCT
cana-1225	134	10	0.4	0.4	NUM
cana-1225	134	11	]	]	PUNCT
cana-1225	135	1	[	[	X
cana-1225	135	2	-0.5	-0.5	NOUN
cana-1225	135	3	,	,	PUNCT
cana-1225	135	4	-0.5]	-0.5]	NOUN
cana-1225	135	5	is	be	AUX
cana-1225	135	6	a	a	DET
cana-1225	135	7	bipolar	bipolar	ADJ
cana-1225	135	8	vague	vague	NOUN
cana-1225	135	9	closed	close	VERB
cana-1225	135	10	set	set	VERB
cana-1225	135	11	in	in	ADP
cana-1225	135	12	y.	y.	NOUN
cana-1225	135	13	then	then	ADV
cana-1225	135	14	𝑓−1	𝑓−1	PROPN
cana-1225	135	15	(	(	PUNCT
cana-1225	135	16	bc	bc	PROPN
cana-1225	135	17	)	)	PUNCT
cana-1225	135	18	=	=	SYM
cana-1225	136	1	x	x	PUNCT
cana-1225	136	2	,	,	PUNCT
cana-1225	136	3	[	[	X
cana-1225	136	4	0.4	0.4	NUM
cana-1225	136	5	,	,	PUNCT
cana-1225	136	6	0.3	0.3	NUM
cana-1225	136	7	]	]	PUNCT
cana-1225	137	1	[	[	X
cana-1225	137	2	-0.1	-0.1	PROPN
cana-1225	137	3	,	,	PUNCT
cana-1225	137	4	-0.1	-0.1	PROPN
cana-1225	137	5	]	]	X
cana-1225	137	6	,	,	PUNCT
cana-1225	137	7	[	[	X
cana-1225	137	8	0.4	0.4	NUM
cana-1225	137	9	,	,	PUNCT
cana-1225	137	10	0.4	0.4	NUM
cana-1225	137	11	]	]	PUNCT
cana-1225	138	1	[	[	X
cana-1225	138	2	-0.5	-0.5	NOUN
cana-1225	138	3	,	,	PUNCT
cana-1225	138	4	-0.5]	-0.5]	NOUN
cana-1225	138	5	is	be	AUX
cana-1225	138	6	a	a	DET
cana-1225	138	7	bipolar	bipolar	ADJ
cana-1225	138	8	vague	vague	NOUN
cana-1225	138	9	𝛼	𝛼	DET
cana-1225	138	10	generalized	generalize	VERB
cana-1225	138	11	closed	close	VERB
cana-1225	138	12	set	set	VERB
cana-1225	138	13	in	in	ADP
cana-1225	138	14	(	(	PUNCT
cana-1225	138	15	x	x	NOUN
cana-1225	138	16	,	,	PUNCT
cana-1225	138	17	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	138	18	)	)	PUNCT
cana-1225	138	19	as	as	ADP
cana-1225	138	20	𝑓−1	𝑓−1	PROPN
cana-1225	138	21	(	(	PUNCT
cana-1225	138	22	bc	bc	PROPN
cana-1225	138	23	)	)	PUNCT
cana-1225	138	24	⊆	⊆	PROPN
cana-1225	138	25	a	a	PRON
cana-1225	138	26	and	and	CCONJ
cana-1225	138	27	b𝑉𝛼cl(𝑓−1	b𝑉𝛼cl(𝑓−1	PROPN
cana-1225	138	28	(	(	PUNCT
cana-1225	138	29	bc	bc	PROPN
cana-1225	138	30	)	)	PUNCT
cana-1225	138	31	)	)	PUNCT
cana-1225	139	1	=	=	SYM
cana-1225	140	1	𝑓−1	𝑓−1	PROPN
cana-1225	140	2	(	(	PUNCT
cana-1225	140	3	bc	bc	PROPN
cana-1225	140	4	)	)	PUNCT
cana-1225	140	5	∪	∪	NOUN
cana-1225	140	6	bvcl(bvint(bvcl(𝑓−1	bvcl(bvint(bvcl(𝑓−1	PROPN
cana-1225	140	7	(	(	PUNCT
cana-1225	140	8	bc	bc	PROPN
cana-1225	140	9	)	)	PUNCT
cana-1225	140	10	)	)	PUNCT
cana-1225	140	11	)	)	PUNCT
cana-1225	140	12	)	)	PUNCT
cana-1225	141	1	=	=	PRON
cana-1225	141	2	ac	ac	PROPN
cana-1225	141	3	⊆	⊆	NUM
cana-1225	141	4	a	a	PRON
cana-1225	141	5	,	,	PUNCT
cana-1225	141	6	where	where	SCONJ
cana-1225	141	7	a	a	PRON
cana-1225	141	8	is	be	AUX
cana-1225	141	9	a	a	DET
cana-1225	141	10	bipolar	bipolar	ADJ
cana-1225	141	11	vague	vague	ADJ
cana-1225	141	12	open	open	ADJ
cana-1225	141	13	set	set	VERB
cana-1225	141	14	in	in	ADP
cana-1225	141	15	x.	x.	NOUN
cana-1225	141	16	therefore	therefore	ADV
cana-1225	141	17	,	,	PUNCT
cana-1225	141	18	𝑓	𝑓	PRON
cana-1225	141	19	is	be	AUX
cana-1225	141	20	a	a	DET
cana-1225	141	21	bipolar	bipolar	ADJ
cana-1225	141	22	vague	vague	NOUN
cana-1225	141	23	𝛼	𝛼	ADP
cana-1225	141	24	generalized	generalize	VERB
cana-1225	141	25	continuous	continuous	ADJ
cana-1225	141	26	mapping	mapping	NOUN
cana-1225	141	27	.	.	PUNCT
cana-1225	142	1	proposition	proposition	NOUN
cana-1225	142	2	3.4	3.4	NUM
cana-1225	142	3	:	:	PUNCT
cana-1225	142	4	every	every	DET
cana-1225	142	5	bipolar	bipolar	ADJ
cana-1225	142	6	vague	vague	ADJ
cana-1225	142	7	continuous	continuous	ADJ
cana-1225	142	8	mapping	mapping	NOUN
cana-1225	142	9	is	be	AUX
cana-1225	142	10	a	a	DET
cana-1225	142	11	bipolar	bipolar	ADJ
cana-1225	142	12	vague	vague	NOUN
cana-1225	142	13	𝛼	𝛼	ADP
cana-1225	142	14	generalized	generalize	VERB
cana-1225	142	15	continuous	continuous	ADJ
cana-1225	142	16	mapping	mapping	NOUN
cana-1225	142	17	but	but	CCONJ
cana-1225	142	18	not	not	PART
cana-1225	142	19	conversely	conversely	ADV
cana-1225	142	20	in	in	ADP
cana-1225	142	21	general	general	ADJ
cana-1225	142	22	.	.	PUNCT
cana-1225	143	1	proof	proof	NOUN
cana-1225	143	2	:	:	PUNCT
cana-1225	143	3	let	let	VERB
cana-1225	143	4	𝑓	𝑓	PRON
cana-1225	143	5	:	:	PUNCT
cana-1225	143	6	(	(	PUNCT
cana-1225	143	7	x	x	NOUN
cana-1225	143	8	,	,	PUNCT
cana-1225	143	9	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	143	10	)	)	PUNCT
cana-1225	143	11	→	→	SYM
cana-1225	143	12	(	(	PUNCT
cana-1225	143	13	y	y	NOUN
cana-1225	143	14	,	,	PUNCT
cana-1225	143	15	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	143	16	)	)	PUNCT
cana-1225	143	17	be	be	VERB
cana-1225	143	18	a	a	DET
cana-1225	143	19	bipolar	bipolar	ADJ
cana-1225	143	20	vague	vague	ADJ
cana-1225	143	21	continuous	continuous	ADJ
cana-1225	143	22	mapping	mapping	NOUN
cana-1225	143	23	.	.	PUNCT
cana-1225	144	1	let	let	VERB
cana-1225	144	2	a	a	DET
cana-1225	144	3	be	be	AUX
cana-1225	144	4	a	a	DET
cana-1225	144	5	bipolar	bipolar	ADJ
cana-1225	144	6	vague	vague	NOUN
cana-1225	144	7	closed	close	VERB
cana-1225	144	8	set	set	VERB
cana-1225	144	9	in	in	ADP
cana-1225	144	10	y.	y.	PROPN
cana-1225	144	11	then	then	ADV
cana-1225	144	12	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	144	13	)	)	PUNCT
cana-1225	144	14	is	be	AUX
cana-1225	144	15	a	a	DET
cana-1225	144	16	bipolar	bipolar	ADJ
cana-1225	144	17	vague	vague	NOUN
cana-1225	144	18	closed	close	VERB
cana-1225	144	19	set	set	VERB
cana-1225	144	20	in	in	ADP
cana-1225	144	21	x.	x.	NOUN
cana-1225	144	22	since	since	SCONJ
cana-1225	144	23	every	every	DET
cana-1225	144	24	bipolar	bipolar	ADJ
cana-1225	144	25	vague	vague	NOUN
cana-1225	144	26	closed	close	VERB
cana-1225	144	27	set	set	NOUN
cana-1225	144	28	is	be	AUX
cana-1225	144	29	a	a	DET
cana-1225	144	30	bipolar	bipolar	ADJ
cana-1225	144	31	vague	vague	NOUN
cana-1225	144	32	𝛼	𝛼	DET
cana-1225	144	33	generalized	generalize	VERB
cana-1225	144	34	closed	close	VERB
cana-1225	144	35	set	set	VERB
cana-1225	144	36	in	in	ADP
cana-1225	144	37	x	x	PUNCT
cana-1225	145	1	[	[	X
cana-1225	145	2	10	10	NUM
cana-1225	145	3	]	]	PUNCT
cana-1225	145	4	,	,	PUNCT
cana-1225	145	5	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	145	6	)	)	PUNCT
cana-1225	145	7	is	be	AUX
cana-1225	145	8	a	a	DET
cana-1225	145	9	bipolar	bipolar	ADJ
cana-1225	145	10	vague	vague	NOUN
cana-1225	145	11	𝛼	𝛼	DET
cana-1225	145	12	generalized	generalize	VERB
cana-1225	145	13	closed	close	VERB
cana-1225	145	14	set	set	VERB
cana-1225	145	15	in	in	ADP
cana-1225	145	16	x.	x.	NOUN
cana-1225	145	17	hence	hence	ADV
cana-1225	145	18	𝑓	𝑓	PROPN
cana-1225	145	19	is	be	AUX
cana-1225	145	20	a	a	DET
cana-1225	145	21	bipolar	bipolar	ADJ
cana-1225	145	22	vague	vague	NOUN
cana-1225	145	23	𝛼	𝛼	ADP
cana-1225	145	24	generalized	generalize	VERB
cana-1225	145	25	continuous	continuous	ADJ
cana-1225	145	26	mapping	mapping	NOUN
cana-1225	145	27	.	.	PUNCT
cana-1225	146	1	example	example	NOUN
cana-1225	146	2	3.5	3.5	NUM
cana-1225	146	3	:	:	PUNCT
cana-1225	146	4	let	let	VERB
cana-1225	146	5	x	x	PUNCT
cana-1225	146	6	=	=	PRON
cana-1225	146	7	{	{	PUNCT
cana-1225	146	8	a	a	DET
cana-1225	146	9	,	,	PUNCT
cana-1225	146	10	b	b	NOUN
cana-1225	146	11	}	}	PUNCT
cana-1225	146	12	and	and	CCONJ
cana-1225	146	13	y	y	PROPN
cana-1225	146	14	=	=	SYM
cana-1225	146	15	{	{	PUNCT
cana-1225	146	16	u	u	NOUN
cana-1225	146	17	,	,	PUNCT
cana-1225	146	18	v	v	NOUN
cana-1225	146	19	}	}	PUNCT
cana-1225	146	20	.	.	PUNCT
cana-1225	147	1	then	then	ADV
cana-1225	147	2	𝜏	𝜏	X
cana-1225	147	3	=	=	PUNCT
cana-1225	147	4	{	{	PUNCT
cana-1225	147	5	0~	0~	NOUN
cana-1225	147	6	,	,	PUNCT
cana-1225	147	7	a	a	DET
cana-1225	147	8	,	,	PUNCT
cana-1225	147	9	1~	1~	NUM
cana-1225	147	10	}	}	PUNCT
cana-1225	147	11	and	and	CCONJ
cana-1225	147	12	𝜎	𝜎	NOUN
cana-1225	147	13	=	=	SYM
cana-1225	147	14	{	{	PUNCT
cana-1225	147	15	0~	0~	NOUN
cana-1225	147	16	,	,	PUNCT
cana-1225	147	17	b	b	NOUN
cana-1225	147	18	,	,	PUNCT
cana-1225	147	19	1~	1~	NUM
cana-1225	147	20	}	}	PUNCT
cana-1225	147	21	are	be	AUX
cana-1225	147	22	bipolar	bipolar	ADJ
cana-1225	147	23	vague	vague	ADJ
cana-1225	147	24	topologies	topology	NOUN
cana-1225	147	25	on	on	ADP
cana-1225	147	26	x	x	X
cana-1225	147	27	and	and	CCONJ
cana-1225	147	28	y	y	PROPN
cana-1225	147	29	respectively	respectively	ADV
cana-1225	147	30	,	,	PUNCT
cana-1225	147	31	where	where	SCONJ
cana-1225	147	32	a	a	DET
cana-1225	147	33	=	=	SYM
cana-1225	147	34	x	x	NOUN
cana-1225	147	35	,	,	PUNCT
cana-1225	147	36	[	[	X
cana-1225	147	37	0.2	0.2	NUM
cana-1225	147	38	,	,	PUNCT
cana-1225	147	39	0.3	0.3	NUM
cana-1225	147	40	]	]	PUNCT
cana-1225	148	1	[	[	X
cana-1225	148	2	-0.4	-0.4	X
cana-1225	148	3	,	,	PUNCT
cana-1225	148	4	-0.4	-0.4	NOUN
cana-1225	148	5	]	]	PUNCT
cana-1225	148	6	,	,	PUNCT
cana-1225	148	7	[	[	X
cana-1225	148	8	0.5	0.5	NUM
cana-1225	148	9	,	,	PUNCT
cana-1225	148	10	0.5	0.5	NUM
cana-1225	148	11	]	]	PUNCT
cana-1225	148	12	[	[	X
cana-1225	148	13	0.4	0.4	NUM
cana-1225	148	14	,	,	PUNCT
cana-1225	148	15	-0.4]	-0.4]	PUNCT
cana-1225	148	16	and	and	CCONJ
cana-1225	148	17	b	b	X
cana-1225	148	18	=	=	SYM
cana-1225	148	19	y	y	PROPN
cana-1225	148	20	,	,	PUNCT
cana-1225	148	21	[	[	X
cana-1225	148	22	0.4	0.4	NUM
cana-1225	148	23	,	,	PUNCT
cana-1225	148	24	0.4	0.4	NUM
cana-1225	148	25	]	]	PUNCT
cana-1225	149	1	[	[	X
cana-1225	149	2	-0.4	-0.4	X
cana-1225	149	3	,	,	PUNCT
cana-1225	149	4	-0.4	-0.4	NOUN
cana-1225	149	5	]	]	PUNCT
cana-1225	149	6	,	,	PUNCT
cana-1225	149	7	[	[	X
cana-1225	149	8	0.6	0.6	NUM
cana-1225	149	9	,	,	PUNCT
cana-1225	149	10	0.6	0.6	NUM
cana-1225	149	11	]	]	PUNCT
cana-1225	150	1	[	[	X
cana-1225	150	2	-0.4	-0.4	NOUN
cana-1225	150	3	,	,	PUNCT
cana-1225	150	4	-0.4].	-0.4].	VERB
cana-1225	150	5	define	define	VERB
cana-1225	150	6	a	a	DET
cana-1225	150	7	mapping	mapping	NOUN
cana-1225	150	8	𝑓	𝑓	X
cana-1225	150	9	:	:	PUNCT
cana-1225	150	10	(	(	PUNCT
cana-1225	150	11	x	x	NOUN
cana-1225	150	12	,	,	PUNCT
cana-1225	150	13	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	150	14	)	)	PUNCT
cana-1225	150	15	→	→	SYM
cana-1225	150	16	(	(	PUNCT
cana-1225	150	17	y	y	NOUN
cana-1225	150	18	,	,	PUNCT
cana-1225	150	19	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	150	20	)	)	PUNCT
cana-1225	150	21	by	by	ADP
cana-1225	150	22	f(a	f(a	PROPN
cana-1225	150	23	)	)	PUNCT
cana-1225	150	24	=	=	SYM
cana-1225	150	25	u	u	NOUN
cana-1225	150	26	and	and	CCONJ
cana-1225	150	27	f(b	f(b	PROPN
cana-1225	150	28	)	)	PUNCT
cana-1225	150	29	=	=	PUNCT
cana-1225	151	1	v.	v.	CCONJ
cana-1225	151	2	here	here	ADV
cana-1225	151	3	the	the	DET
cana-1225	151	4	bipolar	bipolar	ADJ
cana-1225	151	5	vague	vague	NOUN
cana-1225	151	6	set	set	VERB
cana-1225	151	7	bc	bc	PROPN
cana-1225	151	8	=	=	PUNCT
cana-1225	151	9	y	y	PROPN
cana-1225	151	10	,	,	PUNCT
cana-1225	151	11	[	[	X
cana-1225	151	12	0.6	0.6	NUM
cana-1225	151	13	,	,	PUNCT
cana-1225	151	14	0.6	0.6	NUM
cana-1225	151	15	]	]	PUNCT
cana-1225	152	1	[	[	X
cana-1225	152	2	-0.6	-0.6	X
cana-1225	152	3	,	,	PUNCT
cana-1225	152	4	-0.6	-0.6	NOUN
cana-1225	152	5	]	]	PUNCT
cana-1225	152	6	,	,	PUNCT
cana-1225	152	7	[	[	X
cana-1225	152	8	0.4	0.4	NUM
cana-1225	152	9	,	,	PUNCT
cana-1225	152	10	0.4	0.4	NUM
cana-1225	152	11	]	]	PUNCT
cana-1225	153	1	[	[	X
cana-1225	153	2	-0.6	-0.6	X
cana-1225	153	3	,	,	PUNCT
cana-1225	153	4	-0.6]	-0.6]	PRON
cana-1225	153	5	is	be	AUX
cana-1225	153	6	a	a	DET
cana-1225	153	7	bipolar	bipolar	ADJ
cana-1225	153	8	vague	vague	NOUN
cana-1225	153	9	closed	close	VERB
cana-1225	153	10	set	set	VERB
cana-1225	153	11	in	in	ADP
cana-1225	153	12	y.	y.	NOUN
cana-1225	153	13	then	then	ADV
cana-1225	154	1	𝑓−1	𝑓−1	PROPN
cana-1225	154	2	(	(	PUNCT
cana-1225	154	3	bc	bc	PROPN
cana-1225	154	4	)	)	PUNCT
cana-1225	154	5	=	=	SYM
cana-1225	154	6	x	x	PUNCT
cana-1225	154	7	,	,	PUNCT
cana-1225	154	8	[	[	X
cana-1225	154	9	0.6	0.6	NUM
cana-1225	154	10	,	,	PUNCT
cana-1225	154	11	0.6	0.6	NUM
cana-1225	154	12	]	]	PUNCT
cana-1225	155	1	[	[	X
cana-1225	155	2	-0.6	-0.6	X
cana-1225	155	3	,	,	PUNCT
cana-1225	155	4	-0.6	-0.6	NOUN
cana-1225	155	5	]	]	PUNCT
cana-1225	155	6	,	,	PUNCT
cana-1225	155	7	[	[	X
cana-1225	155	8	0.4	0.4	NUM
cana-1225	155	9	,	,	PUNCT
cana-1225	155	10	0.4	0.4	NUM
cana-1225	155	11	]	]	PUNCT
cana-1225	156	1	[	[	X
cana-1225	156	2	-0.6	-0.6	X
cana-1225	156	3	,	,	PUNCT
cana-1225	156	4	-0.6]	-0.6]	PRON
cana-1225	156	5	is	be	AUX
cana-1225	156	6	a	a	DET
cana-1225	156	7	bipolar	bipolar	ADJ
cana-1225	156	8	vague	vague	NOUN
cana-1225	156	9	𝛼	𝛼	DET
cana-1225	156	10	generalized	generalize	VERB
cana-1225	156	11	closed	close	VERB
cana-1225	156	12	set	set	VERB
cana-1225	156	13	in	in	ADP
cana-1225	156	14	(	(	PUNCT
cana-1225	156	15	x	x	NOUN
cana-1225	156	16	,	,	PUNCT
cana-1225	156	17	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	156	18	)	)	PUNCT
cana-1225	156	19	as	as	ADP
cana-1225	156	20	𝑓−1	𝑓−1	PROPN
cana-1225	156	21	(	(	PUNCT
cana-1225	156	22	bc	bc	PROPN
cana-1225	156	23	)	)	PUNCT
cana-1225	156	24	⊆	⊆	NUM
cana-1225	156	25	1~	1~	NUM
cana-1225	156	26	and	and	CCONJ
cana-1225	156	27	b𝑉𝛼cl(𝑓−1	b𝑉𝛼cl(𝑓−1	PROPN
cana-1225	156	28	(	(	PUNCT
cana-1225	156	29	bc	bc	PROPN
cana-1225	156	30	)	)	PUNCT
cana-1225	156	31	)	)	PUNCT
cana-1225	157	1	=	=	SYM
cana-1225	158	1	𝑓−1	𝑓−1	PROPN
cana-1225	158	2	(	(	PUNCT
cana-1225	158	3	bc	bc	PROPN
cana-1225	158	4	)	)	PUNCT
cana-1225	158	5	∪	∪	NOUN
cana-1225	158	6	bvcl(bvint(bvcl(𝑓−1	bvcl(bvint(bvcl(𝑓−1	PROPN
cana-1225	158	7	(	(	PUNCT
cana-1225	158	8	bc	bc	PROPN
cana-1225	158	9	)	)	PUNCT
cana-1225	158	10	)	)	PUNCT
cana-1225	158	11	)	)	PUNCT
cana-1225	158	12	)	)	PUNCT
cana-1225	159	1	=	=	PRON
cana-1225	159	2	ac	ac	PROPN
cana-1225	159	3	⊆	⊆	NUM
cana-1225	159	4	1~	1~	NUM
cana-1225	159	5	,	,	PUNCT
cana-1225	159	6	where	where	SCONJ
cana-1225	159	7	ac	ac	PROPN
cana-1225	159	8	is	be	AUX
cana-1225	159	9	a	a	DET
cana-1225	159	10	bipolar	bipolar	ADJ
cana-1225	159	11	vague	vague	NOUN
cana-1225	159	12	closed	close	VERB
cana-1225	159	13	set	set	VERB
cana-1225	159	14	in	in	ADP
cana-1225	159	15	x.	x.	NOUN
cana-1225	159	16	therefore	therefore	ADV
cana-1225	159	17	,	,	PUNCT
cana-1225	159	18	𝑓	𝑓	PRON
cana-1225	159	19	is	be	AUX
cana-1225	159	20	a	a	DET
cana-1225	159	21	bipolar	bipolar	ADJ
cana-1225	159	22	vague	vague	NOUN
cana-1225	159	23	𝛼	𝛼	ADP
cana-1225	159	24	generalized	generalize	VERB
cana-1225	159	25	continuous	continuous	ADJ
cana-1225	159	26	mapping	mapping	NOUN
cana-1225	159	27	but	but	CCONJ
cana-1225	159	28	since	since	SCONJ
cana-1225	159	29	𝑓−1	𝑓−1	PROPN
cana-1225	159	30	(	(	PUNCT
cana-1225	159	31	bc	bc	PROPN
cana-1225	159	32	)	)	PUNCT
cana-1225	159	33	is	be	AUX
cana-1225	159	34	not	not	PART
cana-1225	159	35	a	a	DET
cana-1225	159	36	bipolar	bipolar	ADJ
cana-1225	159	37	vague	vague	NOUN
cana-1225	159	38	closed	close	VERB
cana-1225	159	39	set	set	VERB
cana-1225	159	40	in	in	ADP
cana-1225	159	41	x	x	PUNCT
cana-1225	159	42	as	as	ADP
cana-1225	159	43	bvcl(𝑓−1	bvcl(𝑓−1	PROPN
cana-1225	159	44	(	(	PUNCT
cana-1225	159	45	bc	bc	PROPN
cana-1225	159	46	)	)	PUNCT
cana-1225	159	47	)	)	PUNCT
cana-1225	160	1	=	=	SYM
cana-1225	160	2	ac	ac	PROPN
cana-1225	160	3	≠	≠	PROPN
cana-1225	160	4	𝑓−1	𝑓−1	PROPN
cana-1225	160	5	(	(	PUNCT
cana-1225	160	6	bc	bc	PROPN
cana-1225	160	7	)	)	PUNCT
cana-1225	160	8	,	,	PUNCT
cana-1225	160	9	𝑓	𝑓	PRON
cana-1225	160	10	is	be	AUX
cana-1225	160	11	not	not	PART
cana-1225	160	12	a	a	DET
cana-1225	160	13	bipolar	bipolar	ADJ
cana-1225	160	14	vague	vague	ADJ
cana-1225	160	15	continuous	continuous	ADJ
cana-1225	160	16	mapping	mapping	NOUN
cana-1225	160	17	.	.	PUNCT
cana-1225	161	1	communications	communication	NOUN
cana-1225	161	2	on	on	ADP
cana-1225	161	3	applied	apply	VERB
cana-1225	161	4	nonlinear	nonlinear	ADJ
cana-1225	161	5	analysis	analysis	NOUN
cana-1225	161	6	issn	issn	NOUN
cana-1225	161	7	:	:	PUNCT
cana-1225	161	8	1074	1074	NUM
cana-1225	161	9	-	-	PUNCT
cana-1225	161	10	133x	133x	NUM
cana-1225	161	11	vol	vol	NOUN
cana-1225	161	12	31	31	NUM
cana-1225	161	13	no	no	NOUN
cana-1225	161	14	.	.	PUNCT
cana-1225	162	1	6s	6s	NUM
cana-1225	162	2	(	(	PUNCT
cana-1225	162	3	2024	2024	NUM
cana-1225	162	4	)	)	PUNCT
cana-1225	162	5	323	323	NUM
cana-1225	162	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1225	162	7	proposition	proposition	NOUN
cana-1225	162	8	3.6	3.6	NUM
cana-1225	162	9	:	:	PUNCT
cana-1225	162	10	every	every	DET
cana-1225	162	11	bipolar	bipolar	ADJ
cana-1225	162	12	vague	vague	NOUN
cana-1225	162	13	𝛼	𝛼	ADP
cana-1225	162	14	continuous	continuous	ADJ
cana-1225	162	15	mapping	mapping	NOUN
cana-1225	162	16	is	be	AUX
cana-1225	162	17	a	a	DET
cana-1225	162	18	bipolar	bipolar	ADJ
cana-1225	162	19	vague	vague	NOUN
cana-1225	162	20	𝛼	𝛼	ADP
cana-1225	162	21	generalized	generalize	VERB
cana-1225	162	22	continuous	continuous	ADJ
cana-1225	162	23	mapping	mapping	NOUN
cana-1225	162	24	but	but	CCONJ
cana-1225	162	25	not	not	PART
cana-1225	162	26	conversely	conversely	ADV
cana-1225	162	27	in	in	ADP
cana-1225	162	28	general	general	ADJ
cana-1225	162	29	.	.	PUNCT
cana-1225	163	1	proof	proof	NOUN
cana-1225	163	2	:	:	PUNCT
cana-1225	163	3	let	let	VERB
cana-1225	163	4	𝑓	𝑓	PRON
cana-1225	163	5	:	:	PUNCT
cana-1225	163	6	(	(	PUNCT
cana-1225	163	7	x	x	NOUN
cana-1225	163	8	,	,	PUNCT
cana-1225	163	9	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	163	10	)	)	PUNCT
cana-1225	163	11	→	→	SYM
cana-1225	163	12	(	(	PUNCT
cana-1225	163	13	y	y	NOUN
cana-1225	163	14	,	,	PUNCT
cana-1225	163	15	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	163	16	)	)	PUNCT
cana-1225	163	17	be	be	VERB
cana-1225	163	18	a	a	DET
cana-1225	163	19	bipolar	bipolar	ADJ
cana-1225	163	20	vague	vague	NOUN
cana-1225	163	21	𝛼	𝛼	ADP
cana-1225	163	22	continuous	continuous	ADJ
cana-1225	163	23	mapping	mapping	NOUN
cana-1225	163	24	.	.	PUNCT
cana-1225	164	1	let	let	VERB
cana-1225	164	2	a	a	DET
cana-1225	164	3	be	be	AUX
cana-1225	164	4	a	a	DET
cana-1225	164	5	bipolar	bipolar	ADJ
cana-1225	164	6	vague	vague	NOUN
cana-1225	164	7	closed	close	VERB
cana-1225	164	8	set	set	VERB
cana-1225	164	9	in	in	ADP
cana-1225	164	10	y.	y.	PROPN
cana-1225	164	11	then	then	ADV
cana-1225	164	12	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	164	13	)	)	PUNCT
cana-1225	164	14	is	be	AUX
cana-1225	164	15	a	a	DET
cana-1225	164	16	bipolar	bipolar	ADJ
cana-1225	164	17	vague	vague	NOUN
cana-1225	164	18	𝛼-closed	𝛼-close	VERB
cana-1225	164	19	set	set	VERB
cana-1225	164	20	in	in	ADP
cana-1225	164	21	x.	x.	NOUN
cana-1225	164	22	since	since	SCONJ
cana-1225	164	23	every	every	DET
cana-1225	164	24	bipolar	bipolar	ADJ
cana-1225	164	25	vague	vague	NOUN
cana-1225	164	26	𝛼-closed	𝛼-close	VERB
cana-1225	164	27	set	set	VERB
cana-1225	164	28	is	be	AUX
cana-1225	164	29	a	a	DET
cana-1225	164	30	bipolar	bipolar	ADJ
cana-1225	164	31	vague	vague	NOUN
cana-1225	164	32	𝛼	𝛼	DET
cana-1225	164	33	generalized	generalize	VERB
cana-1225	164	34	closed	close	VERB
cana-1225	164	35	set	set	VERB
cana-1225	164	36	in	in	ADP
cana-1225	164	37	x	x	PUNCT
cana-1225	165	1	[	[	X
cana-1225	165	2	10	10	NUM
cana-1225	165	3	]	]	PUNCT
cana-1225	165	4	,	,	PUNCT
cana-1225	165	5	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	165	6	)	)	PUNCT
cana-1225	165	7	is	be	AUX
cana-1225	165	8	a	a	DET
cana-1225	165	9	bipolar	bipolar	ADJ
cana-1225	165	10	vague	vague	NOUN
cana-1225	165	11	𝛼	𝛼	DET
cana-1225	165	12	generalized	generalize	VERB
cana-1225	165	13	closed	close	VERB
cana-1225	165	14	set	set	VERB
cana-1225	165	15	in	in	ADP
cana-1225	165	16	x.	x.	NOUN
cana-1225	165	17	hence	hence	ADV
cana-1225	165	18	𝑓	𝑓	PROPN
cana-1225	165	19	is	be	AUX
cana-1225	165	20	a	a	DET
cana-1225	165	21	bipolar	bipolar	ADJ
cana-1225	165	22	vague	vague	NOUN
cana-1225	165	23	𝛼	𝛼	ADP
cana-1225	165	24	generalized	generalize	VERB
cana-1225	165	25	continuous	continuous	ADJ
cana-1225	165	26	mapping	mapping	NOUN
cana-1225	165	27	.	.	PUNCT
cana-1225	166	1	example	example	NOUN
cana-1225	166	2	3.7	3.7	NUM
cana-1225	166	3	:	:	PUNCT
cana-1225	166	4	let	let	VERB
cana-1225	166	5	x	x	PUNCT
cana-1225	166	6	=	=	PRON
cana-1225	166	7	{	{	PUNCT
cana-1225	166	8	a	a	DET
cana-1225	166	9	,	,	PUNCT
cana-1225	166	10	b	b	NOUN
cana-1225	166	11	}	}	PUNCT
cana-1225	166	12	and	and	CCONJ
cana-1225	166	13	y	y	PROPN
cana-1225	166	14	=	=	SYM
cana-1225	166	15	{	{	PUNCT
cana-1225	166	16	u	u	NOUN
cana-1225	166	17	,	,	PUNCT
cana-1225	166	18	v	v	NOUN
cana-1225	166	19	}	}	PUNCT
cana-1225	166	20	.	.	PUNCT
cana-1225	167	1	then	then	ADV
cana-1225	167	2	𝜏	𝜏	X
cana-1225	167	3	=	=	PUNCT
cana-1225	167	4	{	{	PUNCT
cana-1225	167	5	0~	0~	NOUN
cana-1225	167	6	,	,	PUNCT
cana-1225	167	7	a	a	DET
cana-1225	167	8	,	,	PUNCT
cana-1225	167	9	1~	1~	NUM
cana-1225	167	10	}	}	PUNCT
cana-1225	167	11	and	and	CCONJ
cana-1225	167	12	𝜎	𝜎	NOUN
cana-1225	167	13	=	=	SYM
cana-1225	167	14	{	{	PUNCT
cana-1225	167	15	0~	0~	NOUN
cana-1225	167	16	,	,	PUNCT
cana-1225	167	17	b	b	NOUN
cana-1225	167	18	,	,	PUNCT
cana-1225	167	19	1~	1~	NUM
cana-1225	167	20	}	}	PUNCT
cana-1225	167	21	are	be	AUX
cana-1225	167	22	bipolar	bipolar	ADJ
cana-1225	167	23	vague	vague	ADJ
cana-1225	167	24	topologies	topology	NOUN
cana-1225	167	25	on	on	ADP
cana-1225	167	26	x	x	X
cana-1225	167	27	and	and	CCONJ
cana-1225	167	28	y	y	PROPN
cana-1225	167	29	respectively	respectively	ADV
cana-1225	167	30	,	,	PUNCT
cana-1225	167	31	where	where	SCONJ
cana-1225	167	32	a	a	DET
cana-1225	167	33	=	=	SYM
cana-1225	167	34	x	x	NOUN
cana-1225	167	35	,	,	PUNCT
cana-1225	167	36	[	[	X
cana-1225	167	37	0.2	0.2	NUM
cana-1225	167	38	,	,	PUNCT
cana-1225	167	39	0.3	0.3	NUM
cana-1225	167	40	]	]	PUNCT
cana-1225	168	1	[	[	X
cana-1225	168	2	-0.3	-0.3	PROPN
cana-1225	168	3	,	,	PUNCT
cana-1225	168	4	-0.3	-0.3	PROPN
cana-1225	168	5	]	]	X
cana-1225	168	6	,	,	PUNCT
cana-1225	168	7	[	[	X
cana-1225	168	8	0.5	0.5	NUM
cana-1225	168	9	,	,	PUNCT
cana-1225	168	10	0.5	0.5	NUM
cana-1225	168	11	]	]	PUNCT
cana-1225	168	12	[	[	X
cana-1225	168	13	0.4	0.4	NUM
cana-1225	168	14	,	,	PUNCT
cana-1225	168	15	-0.4]	-0.4]	PUNCT
cana-1225	168	16	and	and	CCONJ
cana-1225	168	17	b	b	X
cana-1225	168	18	=	=	SYM
cana-1225	168	19	y	y	PROPN
cana-1225	168	20	,	,	PUNCT
cana-1225	168	21	[	[	X
cana-1225	168	22	0.3	0.3	NUM
cana-1225	168	23	,	,	PUNCT
cana-1225	168	24	0.3	0.3	NUM
cana-1225	168	25	]	]	PUNCT
cana-1225	169	1	[	[	X
cana-1225	169	2	-0.3	-0.3	PROPN
cana-1225	169	3	,	,	PUNCT
cana-1225	169	4	-0.3	-0.3	PROPN
cana-1225	169	5	]	]	X
cana-1225	169	6	,	,	PUNCT
cana-1225	169	7	[	[	X
cana-1225	169	8	0.7	0.7	NUM
cana-1225	169	9	,	,	PUNCT
cana-1225	169	10	0.7	0.7	NUM
cana-1225	169	11	]	]	PUNCT
cana-1225	170	1	[	[	X
cana-1225	170	2	-0.5	-0.5	NOUN
cana-1225	170	3	,	,	PUNCT
cana-1225	170	4	-0.5].	-0.5].	VERB
cana-1225	170	5	define	define	VERB
cana-1225	170	6	a	a	DET
cana-1225	170	7	mapping	mapping	NOUN
cana-1225	170	8	𝑓	𝑓	X
cana-1225	170	9	:	:	PUNCT
cana-1225	170	10	(	(	PUNCT
cana-1225	170	11	x	x	NOUN
cana-1225	170	12	,	,	PUNCT
cana-1225	170	13	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	170	14	)	)	PUNCT
cana-1225	170	15	→	→	SYM
cana-1225	170	16	(	(	PUNCT
cana-1225	170	17	y	y	NOUN
cana-1225	170	18	,	,	PUNCT
cana-1225	170	19	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	170	20	)	)	PUNCT
cana-1225	170	21	by	by	ADP
cana-1225	170	22	f(a	f(a	PROPN
cana-1225	170	23	)	)	PUNCT
cana-1225	170	24	=	=	SYM
cana-1225	170	25	u	u	NOUN
cana-1225	170	26	and	and	CCONJ
cana-1225	170	27	f(b	f(b	PROPN
cana-1225	170	28	)	)	PUNCT
cana-1225	171	1	=	=	PUNCT
cana-1225	172	1	v.	v.	CCONJ
cana-1225	172	2	here	here	ADV
cana-1225	172	3	the	the	DET
cana-1225	172	4	bipolar	bipolar	ADJ
cana-1225	172	5	vague	vague	NOUN
cana-1225	172	6	set	set	VERB
cana-1225	172	7	bc	bc	PROPN
cana-1225	172	8	=	=	PUNCT
cana-1225	172	9	y	y	PROPN
cana-1225	172	10	,	,	PUNCT
cana-1225	172	11	[	[	X
cana-1225	172	12	0.7	0.7	NUM
cana-1225	172	13	,	,	PUNCT
cana-1225	172	14	0.7	0.7	NUM
cana-1225	172	15	]	]	PUNCT
cana-1225	173	1	[	[	X
cana-1225	173	2	-0.7	-0.7	X
cana-1225	173	3	,	,	PUNCT
cana-1225	173	4	-0.7	-0.7	PROPN
cana-1225	173	5	]	]	X
cana-1225	173	6	,	,	PUNCT
cana-1225	173	7	[	[	X
cana-1225	173	8	0.3	0.3	NUM
cana-1225	173	9	,	,	PUNCT
cana-1225	173	10	0.3	0.3	NUM
cana-1225	173	11	]	]	PUNCT
cana-1225	173	12	[	[	X
cana-1225	173	13	-0.5	-0.5	NOUN
cana-1225	173	14	,	,	PUNCT
cana-1225	173	15	-0.5]	-0.5]	NOUN
cana-1225	173	16	is	be	AUX
cana-1225	173	17	a	a	DET
cana-1225	173	18	bipolar	bipolar	ADJ
cana-1225	173	19	vague	vague	NOUN
cana-1225	173	20	closed	close	VERB
cana-1225	173	21	set	set	VERB
cana-1225	173	22	in	in	ADP
cana-1225	173	23	y.	y.	NOUN
cana-1225	173	24	then	then	ADV
cana-1225	173	25	𝑓−1	𝑓−1	PROPN
cana-1225	173	26	(	(	PUNCT
cana-1225	173	27	bc	bc	PROPN
cana-1225	173	28	)	)	PUNCT
cana-1225	173	29	=	=	SYM
cana-1225	174	1	x	x	PUNCT
cana-1225	174	2	,	,	PUNCT
cana-1225	174	3	[	[	X
cana-1225	174	4	0.7	0.7	NUM
cana-1225	174	5	,	,	PUNCT
cana-1225	174	6	0.7	0.7	NUM
cana-1225	174	7	]	]	PUNCT
cana-1225	175	1	[	[	X
cana-1225	175	2	-0.7	-0.7	X
cana-1225	175	3	,	,	PUNCT
cana-1225	175	4	-0.7	-0.7	PROPN
cana-1225	175	5	]	]	X
cana-1225	175	6	,	,	PUNCT
cana-1225	175	7	[	[	X
cana-1225	175	8	0.3	0.3	NUM
cana-1225	175	9	,	,	PUNCT
cana-1225	175	10	0.3	0.3	NUM
cana-1225	175	11	]	]	PUNCT
cana-1225	175	12	[	[	X
cana-1225	175	13	-0.5	-0.5	NOUN
cana-1225	175	14	,	,	PUNCT
cana-1225	175	15	-0.5]	-0.5]	NOUN
cana-1225	175	16	is	be	AUX
cana-1225	175	17	a	a	DET
cana-1225	175	18	bipolar	bipolar	ADJ
cana-1225	175	19	vague	vague	NOUN
cana-1225	175	20	𝛼	𝛼	DET
cana-1225	175	21	generalized	generalize	VERB
cana-1225	175	22	closed	close	VERB
cana-1225	175	23	set	set	VERB
cana-1225	175	24	in	in	ADP
cana-1225	175	25	(	(	PUNCT
cana-1225	175	26	x	x	NOUN
cana-1225	175	27	,	,	PUNCT
cana-1225	175	28	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	175	29	)	)	PUNCT
cana-1225	175	30	as	as	ADP
cana-1225	175	31	𝑓−1	𝑓−1	PROPN
cana-1225	175	32	(	(	PUNCT
cana-1225	175	33	bc	bc	PROPN
cana-1225	175	34	)	)	PUNCT
cana-1225	175	35	⊆	⊆	NUM
cana-1225	175	36	1~	1~	NUM
cana-1225	175	37	and	and	CCONJ
cana-1225	175	38	b𝑉𝛼cl(𝑓−1	b𝑉𝛼cl(𝑓−1	PROPN
cana-1225	175	39	(	(	PUNCT
cana-1225	175	40	bc	bc	PROPN
cana-1225	175	41	)	)	PUNCT
cana-1225	175	42	)	)	PUNCT
cana-1225	176	1	=	=	SYM
cana-1225	177	1	𝑓−1	𝑓−1	PROPN
cana-1225	177	2	(	(	PUNCT
cana-1225	177	3	bc	bc	PROPN
cana-1225	177	4	)	)	PUNCT
cana-1225	177	5	∪	∪	NOUN
cana-1225	177	6	bvcl(bvint(bvcl(𝑓−1	bvcl(bvint(bvcl(𝑓−1	PROPN
cana-1225	177	7	(	(	PUNCT
cana-1225	177	8	bc	bc	PROPN
cana-1225	177	9	)	)	PUNCT
cana-1225	177	10	)	)	PUNCT
cana-1225	177	11	)	)	PUNCT
cana-1225	177	12	)	)	PUNCT
cana-1225	178	1	=	=	PRON
cana-1225	178	2	ac	ac	PROPN
cana-1225	178	3	⊆	⊆	NUM
cana-1225	178	4	1~	1~	NUM
cana-1225	178	5	,	,	PUNCT
cana-1225	178	6	where	where	SCONJ
cana-1225	178	7	ac	ac	PROPN
cana-1225	178	8	is	be	AUX
cana-1225	178	9	a	a	DET
cana-1225	178	10	bipolar	bipolar	ADJ
cana-1225	178	11	vague	vague	NOUN
cana-1225	178	12	closed	close	VERB
cana-1225	178	13	set	set	VERB
cana-1225	178	14	in	in	ADP
cana-1225	178	15	x.	x.	NOUN
cana-1225	178	16	therefore	therefore	ADV
cana-1225	178	17	,	,	PUNCT
cana-1225	178	18	𝑓	𝑓	PRON
cana-1225	178	19	is	be	AUX
cana-1225	178	20	a	a	DET
cana-1225	178	21	bipolar	bipolar	ADJ
cana-1225	178	22	vague	vague	NOUN
cana-1225	178	23	𝛼	𝛼	ADP
cana-1225	178	24	generalized	generalize	VERB
cana-1225	178	25	continuous	continuous	ADJ
cana-1225	178	26	mapping	mapping	NOUN
cana-1225	178	27	but	but	CCONJ
cana-1225	178	28	since	since	SCONJ
cana-1225	178	29	bvcl	bvcl	PROPN
cana-1225	178	30	(	(	PUNCT
cana-1225	178	31	bvint	bvint	NOUN
cana-1225	178	32	(	(	PUNCT
cana-1225	178	33	bvcl(𝑓−1	bvcl(𝑓−1	PROPN
cana-1225	178	34	(	(	PUNCT
cana-1225	178	35	𝐵𝑐	𝐵𝑐	PROPN
cana-1225	178	36	)	)	PUNCT
cana-1225	178	37	)	)	PUNCT
cana-1225	178	38	)	)	PUNCT
cana-1225	178	39	)	)	PUNCT
cana-1225	179	1	=	=	PUNCT
cana-1225	179	2	ac	ac	ADV
cana-1225	179	3	⊄	⊄	NOUN
cana-1225	179	4	𝑓−1	𝑓−1	NUM
cana-1225	179	5	(	(	PUNCT
cana-1225	179	6	bc	bc	PROPN
cana-1225	179	7	)	)	PUNCT
cana-1225	179	8	.	.	PUNCT
cana-1225	180	1	hence	hence	ADV
cana-1225	180	2	𝑓	𝑓	PRON
cana-1225	180	3	is	be	AUX
cana-1225	180	4	not	not	PART
cana-1225	180	5	a	a	DET
cana-1225	180	6	bipolar	bipolar	ADJ
cana-1225	180	7	vague	vague	NOUN
cana-1225	180	8	𝛼	𝛼	ADP
cana-1225	180	9	continuous	continuous	ADJ
cana-1225	180	10	mapping	mapping	NOUN
cana-1225	180	11	.	.	PUNCT
cana-1225	181	1	remark	remark	VERB
cana-1225	181	2	3.8	3.8	NUM
cana-1225	181	3	:	:	PUNCT
cana-1225	181	4	every	every	DET
cana-1225	181	5	bipolar	bipolar	ADJ
cana-1225	181	6	vague	vague	NOUN
cana-1225	181	7	semi	semi	ADJ
cana-1225	181	8	continuous	continuous	ADJ
cana-1225	181	9	mapping	mapping	NOUN
cana-1225	181	10	and	and	CCONJ
cana-1225	181	11	bipolar	bipolar	ADJ
cana-1225	181	12	vague	vague	NOUN
cana-1225	181	13	𝛼	𝛼	ADP
cana-1225	181	14	generalized	generalized	ADJ
cana-1225	181	15	continuous	continuous	ADJ
cana-1225	181	16	mapping	mapping	NOUN
cana-1225	181	17	are	be	AUX
cana-1225	181	18	independent	independent	ADJ
cana-1225	181	19	to	to	ADP
cana-1225	181	20	each	each	DET
cana-1225	181	21	other	other	ADJ
cana-1225	181	22	in	in	ADP
cana-1225	181	23	general	general	ADJ
cana-1225	181	24	.	.	PUNCT
cana-1225	182	1	example	example	NOUN
cana-1225	182	2	3.9	3.9	NUM
cana-1225	182	3	:	:	PUNCT
cana-1225	182	4	in	in	ADP
cana-1225	182	5	example	example	NOUN
cana-1225	182	6	3.4	3.4	NUM
cana-1225	182	7	,	,	PUNCT
cana-1225	182	8	𝑓	𝑓	PRON
cana-1225	182	9	is	be	AUX
cana-1225	182	10	a	a	DET
cana-1225	182	11	bipolar	bipolar	ADJ
cana-1225	182	12	vague	vague	NOUN
cana-1225	182	13	𝛼	𝛼	ADP
cana-1225	182	14	generalized	generalize	VERB
cana-1225	182	15	continuous	continuous	ADJ
cana-1225	182	16	mapping	mapping	NOUN
cana-1225	182	17	but	but	CCONJ
cana-1225	182	18	since	since	SCONJ
cana-1225	182	19	bvint	bvint	NOUN
cana-1225	182	20	(	(	PUNCT
cana-1225	182	21	bvcl(𝑓−1	bvcl(𝑓−1	PROPN
cana-1225	182	22	(	(	PUNCT
cana-1225	182	23	𝐵𝑐	𝐵𝑐	PROPN
cana-1225	182	24	)	)	PUNCT
cana-1225	182	25	)	)	PUNCT
cana-1225	182	26	)	)	PUNCT
cana-1225	183	1	=	=	PUNCT
cana-1225	183	2	bvint(ac	bvint(ac	PROPN
cana-1225	183	3	)	)	PUNCT
cana-1225	183	4	=	=	PUNCT
cana-1225	183	5	a	a	DET
cana-1225	183	6	⊄	⊄	NOUN
cana-1225	183	7	𝑓−1	𝑓−1	PROPN
cana-1225	183	8	(	(	PUNCT
cana-1225	183	9	bc	bc	PROPN
cana-1225	183	10	)	)	PUNCT
cana-1225	183	11	=	=	SYM
cana-1225	183	12	x	x	PUNCT
cana-1225	183	13	,	,	PUNCT
cana-1225	183	14	[	[	X
cana-1225	183	15	0.4	0.4	NUM
cana-1225	183	16	,	,	PUNCT
cana-1225	183	17	0.3	0.3	NUM
cana-1225	183	18	]	]	PUNCT
cana-1225	184	1	[	[	X
cana-1225	184	2	-0.1	-0.1	PROPN
cana-1225	184	3	,	,	PUNCT
cana-1225	184	4	-0.1	-0.1	PROPN
cana-1225	184	5	]	]	X
cana-1225	184	6	,	,	PUNCT
cana-1225	184	7	[	[	X
cana-1225	184	8	0.4	0.4	NUM
cana-1225	184	9	,	,	PUNCT
cana-1225	184	10	0.4	0.4	NUM
cana-1225	184	11	]	]	PUNCT
cana-1225	185	1	[	[	X
cana-1225	185	2	0.5	0.5	NUM
cana-1225	185	3	,	,	PUNCT
cana-1225	185	4	-0.5]	-0.5]	NOUN
cana-1225	185	5	,	,	PUNCT
cana-1225	185	6	𝑓−1	𝑓−1	NUM
cana-1225	185	7	(	(	PUNCT
cana-1225	185	8	bc	bc	PROPN
cana-1225	185	9	)	)	PUNCT
cana-1225	185	10	is	be	AUX
cana-1225	185	11	not	not	PART
cana-1225	185	12	a	a	DET
cana-1225	185	13	bipolar	bipolar	ADJ
cana-1225	185	14	vague	vague	ADJ
cana-1225	185	15	semi	semi	ADJ
cana-1225	185	16	-	-	ADJ
cana-1225	185	17	closed	closed	ADJ
cana-1225	185	18	set	set	NOUN
cana-1225	185	19	in	in	ADP
cana-1225	185	20	x.	x.	NOUN
cana-1225	185	21	hence	hence	ADV
cana-1225	185	22	𝑓	𝑓	PROPN
cana-1225	185	23	is	be	AUX
cana-1225	185	24	not	not	PART
cana-1225	185	25	a	a	DET
cana-1225	185	26	bipolar	bipolar	ADJ
cana-1225	185	27	vague	vague	NOUN
cana-1225	185	28	semi	semi	ADJ
cana-1225	185	29	continuous	continuous	ADJ
cana-1225	185	30	mapping	mapping	NOUN
cana-1225	185	31	.	.	PUNCT
cana-1225	186	1	example	example	NOUN
cana-1225	187	1	3.10	3.10	NUM
cana-1225	187	2	:	:	PUNCT
cana-1225	187	3	let	let	VERB
cana-1225	187	4	x	x	PUNCT
cana-1225	187	5	=	=	PRON
cana-1225	187	6	{	{	PUNCT
cana-1225	187	7	a	a	DET
cana-1225	187	8	,	,	PUNCT
cana-1225	187	9	b	b	NOUN
cana-1225	187	10	}	}	PUNCT
cana-1225	187	11	and	and	CCONJ
cana-1225	187	12	y	y	PROPN
cana-1225	187	13	=	=	SYM
cana-1225	187	14	{	{	PUNCT
cana-1225	187	15	u	u	NOUN
cana-1225	187	16	,	,	PUNCT
cana-1225	187	17	v	v	NOUN
cana-1225	187	18	}	}	PUNCT
cana-1225	187	19	.	.	PUNCT
cana-1225	188	1	then	then	ADV
cana-1225	188	2	𝜏	𝜏	X
cana-1225	188	3	=	=	PUNCT
cana-1225	188	4	{	{	PUNCT
cana-1225	188	5	0~	0~	NOUN
cana-1225	188	6	,	,	PUNCT
cana-1225	188	7	a	a	DET
cana-1225	188	8	,	,	PUNCT
cana-1225	188	9	1~	1~	NUM
cana-1225	188	10	}	}	PUNCT
cana-1225	188	11	and	and	CCONJ
cana-1225	188	12	𝜎	𝜎	NOUN
cana-1225	188	13	=	=	SYM
cana-1225	188	14	{	{	PUNCT
cana-1225	188	15	0~	0~	NOUN
cana-1225	188	16	,	,	PUNCT
cana-1225	188	17	b	b	NOUN
cana-1225	188	18	,	,	PUNCT
cana-1225	188	19	1~	1~	NUM
cana-1225	188	20	}	}	PUNCT
cana-1225	188	21	are	be	AUX
cana-1225	188	22	bipolar	bipolar	ADJ
cana-1225	188	23	vague	vague	ADJ
cana-1225	188	24	topologies	topology	NOUN
cana-1225	188	25	on	on	ADP
cana-1225	188	26	x	x	X
cana-1225	188	27	and	and	CCONJ
cana-1225	188	28	y	y	PROPN
cana-1225	188	29	respectively	respectively	ADV
cana-1225	188	30	,	,	PUNCT
cana-1225	188	31	where	where	SCONJ
cana-1225	188	32	a	a	DET
cana-1225	188	33	=	=	SYM
cana-1225	188	34	x	x	NOUN
cana-1225	188	35	,	,	PUNCT
cana-1225	188	36	[	[	X
cana-1225	188	37	0.4	0.4	NUM
cana-1225	188	38	,	,	PUNCT
cana-1225	188	39	0.3	0.3	NUM
cana-1225	188	40	]	]	PUNCT
cana-1225	189	1	[	[	X
cana-1225	189	2	-0.2	-0.2	NOUN
cana-1225	189	3	,	,	PUNCT
cana-1225	189	4	-0.2	-0.2	PROPN
cana-1225	189	5	]	]	X
cana-1225	189	6	,	,	PUNCT
cana-1225	189	7	[	[	X
cana-1225	189	8	0.5	0.5	NUM
cana-1225	189	9	,	,	PUNCT
cana-1225	189	10	0.5	0.5	NUM
cana-1225	189	11	]	]	PUNCT
cana-1225	189	12	[	[	X
cana-1225	189	13	0.5	0.5	NUM
cana-1225	189	14	,	,	PUNCT
cana-1225	189	15	-0.5]	-0.5]	NOUN
cana-1225	189	16	and	and	CCONJ
cana-1225	189	17	b	b	X
cana-1225	189	18	=	=	SYM
cana-1225	189	19	y	y	PROPN
cana-1225	189	20	,	,	PUNCT
cana-1225	189	21	[	[	X
cana-1225	189	22	0.7	0.7	NUM
cana-1225	189	23	,	,	PUNCT
cana-1225	189	24	0.6	0.6	NUM
cana-1225	189	25	]	]	PUNCT
cana-1225	190	1	[	[	X
cana-1225	190	2	-0.8	-0.8	ADJ
cana-1225	190	3	,	,	PUNCT
cana-1225	190	4	-0.8	-0.8	PROPN
cana-1225	190	5	]	]	PUNCT
cana-1225	190	6	,	,	PUNCT
cana-1225	190	7	[	[	X
cana-1225	190	8	0.5	0.5	NUM
cana-1225	190	9	,	,	PUNCT
cana-1225	190	10	0.5	0.5	NUM
cana-1225	190	11	]	]	PUNCT
cana-1225	190	12	[	[	X
cana-1225	190	13	-0.5	-0.5	NOUN
cana-1225	190	14	,	,	PUNCT
cana-1225	190	15	-0.5].	-0.5].	VERB
cana-1225	190	16	define	define	VERB
cana-1225	190	17	a	a	DET
cana-1225	190	18	mapping	mapping	NOUN
cana-1225	190	19	𝑓	𝑓	X
cana-1225	190	20	:	:	PUNCT
cana-1225	190	21	(	(	PUNCT
cana-1225	190	22	x	x	NOUN
cana-1225	190	23	,	,	PUNCT
cana-1225	190	24	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	190	25	)	)	PUNCT
cana-1225	190	26	→	→	SYM
cana-1225	190	27	(	(	PUNCT
cana-1225	190	28	y	y	NOUN
cana-1225	190	29	,	,	PUNCT
cana-1225	190	30	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	190	31	)	)	PUNCT
cana-1225	190	32	by	by	ADP
cana-1225	190	33	f(a	f(a	PROPN
cana-1225	190	34	)	)	PUNCT
cana-1225	190	35	=	=	SYM
cana-1225	190	36	u	u	NOUN
cana-1225	190	37	and	and	CCONJ
cana-1225	190	38	f(b	f(b	PROPN
cana-1225	190	39	)	)	PUNCT
cana-1225	190	40	=	=	PUNCT
cana-1225	191	1	v.	v.	CCONJ
cana-1225	191	2	here	here	ADV
cana-1225	191	3	the	the	DET
cana-1225	191	4	bipolar	bipolar	ADJ
cana-1225	191	5	vague	vague	NOUN
cana-1225	191	6	set	set	VERB
cana-1225	191	7	bc	bc	PROPN
cana-1225	191	8	=	=	PUNCT
cana-1225	191	9	y	y	PROPN
cana-1225	191	10	,	,	PUNCT
cana-1225	191	11	[	[	X
cana-1225	191	12	0.4	0.4	NUM
cana-1225	191	13	,	,	PUNCT
cana-1225	191	14	0.3	0.3	NUM
cana-1225	191	15	]	]	PUNCT
cana-1225	192	1	[	[	X
cana-1225	192	2	-0.2	-0.2	NOUN
cana-1225	192	3	,	,	PUNCT
cana-1225	192	4	-0.2	-0.2	PROPN
cana-1225	192	5	]	]	X
cana-1225	192	6	,	,	PUNCT
cana-1225	192	7	[	[	X
cana-1225	192	8	0.5	0.5	NUM
cana-1225	192	9	,	,	PUNCT
cana-1225	192	10	0.5	0.5	NUM
cana-1225	192	11	]	]	PUNCT
cana-1225	192	12	[	[	X
cana-1225	192	13	-0.5	-0.5	NOUN
cana-1225	192	14	,	,	PUNCT
cana-1225	192	15	-0.5]	-0.5]	NOUN
cana-1225	192	16	is	be	AUX
cana-1225	192	17	a	a	DET
cana-1225	192	18	bipolar	bipolar	ADJ
cana-1225	192	19	vague	vague	NOUN
cana-1225	192	20	closed	close	VERB
cana-1225	192	21	set	set	VERB
cana-1225	192	22	in	in	ADP
cana-1225	192	23	y.	y.	PROPN
cana-1225	192	24	but	but	CCONJ
cana-1225	192	25	𝑓−1	𝑓−1	NUM
cana-1225	192	26	(	(	PUNCT
cana-1225	192	27	bc	bc	PROPN
cana-1225	192	28	)	)	PUNCT
cana-1225	192	29	=	=	SYM
cana-1225	192	30	x	x	PUNCT
cana-1225	192	31	,	,	PUNCT
cana-1225	192	32	[	[	X
cana-1225	192	33	0.4	0.4	NUM
cana-1225	192	34	,	,	PUNCT
cana-1225	192	35	0.3	0.3	NUM
cana-1225	192	36	]	]	PUNCT
cana-1225	193	1	[	[	X
cana-1225	193	2	-0.2	-0.2	NOUN
cana-1225	193	3	,	,	PUNCT
cana-1225	193	4	0.2	0.2	NUM
cana-1225	193	5	]	]	PUNCT
cana-1225	193	6	,	,	PUNCT
cana-1225	193	7	[	[	X
cana-1225	193	8	0.5	0.5	NUM
cana-1225	193	9	,	,	PUNCT
cana-1225	193	10	0.5	0.5	NUM
cana-1225	193	11	]	]	PUNCT
cana-1225	194	1	[	[	X
cana-1225	194	2	-0.5	-0.5	NOUN
cana-1225	194	3	,	,	PUNCT
cana-1225	194	4	-0.5]	-0.5]	NOUN
cana-1225	194	5	is	be	AUX
cana-1225	194	6	not	not	PART
cana-1225	194	7	a	a	DET
cana-1225	194	8	bipolar	bipolar	ADJ
cana-1225	194	9	vague	vague	NOUN
cana-1225	194	10	𝛼	𝛼	DET
cana-1225	194	11	generalized	generalize	VERB
cana-1225	194	12	closed	close	VERB
cana-1225	194	13	set	set	VERB
cana-1225	194	14	in	in	ADP
cana-1225	194	15	(	(	PUNCT
cana-1225	194	16	x	x	NOUN
cana-1225	194	17	,	,	PUNCT
cana-1225	194	18	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	194	19	)	)	PUNCT
cana-1225	194	20	as	as	ADP
cana-1225	194	21	𝑓−1	𝑓−1	PROPN
cana-1225	194	22	(	(	PUNCT
cana-1225	194	23	bc	bc	PROPN
cana-1225	194	24	)	)	PUNCT
cana-1225	194	25	⊆	⊆	PROPN
cana-1225	194	26	a	a	PRON
cana-1225	194	27	and	and	CCONJ
cana-1225	194	28	b𝑉𝛼cl(𝑓−1	b𝑉𝛼cl(𝑓−1	PROPN
cana-1225	194	29	(	(	PUNCT
cana-1225	194	30	bc	bc	PROPN
cana-1225	194	31	)	)	PUNCT
cana-1225	194	32	)	)	PUNCT
cana-1225	195	1	=	=	SYM
cana-1225	196	1	𝑓−1	𝑓−1	PROPN
cana-1225	196	2	(	(	PUNCT
cana-1225	196	3	bc	bc	PROPN
cana-1225	196	4	)	)	PUNCT
cana-1225	196	5	∪	∪	NOUN
cana-1225	196	6	bvcl(bvint(bvcl(𝑓−1	bvcl(bvint(bvcl(𝑓−1	PROPN
cana-1225	196	7	(	(	PUNCT
cana-1225	196	8	bc	bc	PROPN
cana-1225	196	9	)	)	PUNCT
cana-1225	196	10	)	)	PUNCT
cana-1225	196	11	)	)	PUNCT
cana-1225	196	12	)	)	PUNCT
cana-1225	197	1	=	=	PUNCT
cana-1225	197	2	ac	ac	PROPN
cana-1225	197	3	⊄	⊄	NOUN
cana-1225	197	4	a	a	X
cana-1225	197	5	,	,	PUNCT
cana-1225	197	6	where	where	SCONJ
cana-1225	197	7	ac	ac	PROPN
cana-1225	197	8	is	be	AUX
cana-1225	197	9	a	a	DET
cana-1225	197	10	bipolar	bipolar	ADJ
cana-1225	197	11	vague	vague	NOUN
cana-1225	197	12	closed	close	VERB
cana-1225	197	13	set	set	VERB
cana-1225	197	14	in	in	ADP
cana-1225	197	15	x.	x.	NOUN
cana-1225	197	16	therefore	therefore	ADV
cana-1225	197	17	,	,	PUNCT
cana-1225	197	18	𝑓	𝑓	PRON
cana-1225	197	19	is	be	AUX
cana-1225	197	20	not	not	PART
cana-1225	197	21	a	a	DET
cana-1225	197	22	bipolar	bipolar	ADJ
cana-1225	197	23	vague	vague	NOUN
cana-1225	197	24	𝛼	𝛼	ADP
cana-1225	197	25	generalized	generalize	VERB
cana-1225	197	26	continuous	continuous	ADJ
cana-1225	197	27	mapping	mapping	NOUN
cana-1225	197	28	but	but	CCONJ
cana-1225	197	29	since	since	SCONJ
cana-1225	197	30	bvint	bvint	NOUN
cana-1225	197	31	(	(	PUNCT
cana-1225	197	32	bvcl(𝑓−1	bvcl(𝑓−1	PROPN
cana-1225	197	33	(	(	PUNCT
cana-1225	197	34	𝐵𝑐	𝐵𝑐	PROPN
cana-1225	197	35	)	)	PUNCT
cana-1225	197	36	)	)	PUNCT
cana-1225	197	37	)	)	PUNCT
cana-1225	198	1	=	=	PUNCT
cana-1225	198	2	a	a	DET
cana-1225	198	3	⊆	⊆	NUM
cana-1225	198	4	𝑓−1	𝑓−1	PROPN
cana-1225	198	5	(	(	PUNCT
cana-1225	198	6	bc	bc	PROPN
cana-1225	198	7	)	)	PUNCT
cana-1225	198	8	is	be	AUX
cana-1225	198	9	a	a	DET
cana-1225	198	10	bipolar	bipolar	ADJ
cana-1225	198	11	vague	vague	ADJ
cana-1225	198	12	semi	semi	ADJ
cana-1225	198	13	-	-	ADJ
cana-1225	198	14	closed	closed	ADJ
cana-1225	198	15	set	set	NOUN
cana-1225	198	16	in	in	ADP
cana-1225	198	17	x.	x.	NOUN
cana-1225	198	18	hence	hence	ADV
cana-1225	198	19	𝑓	𝑓	PROPN
cana-1225	198	20	is	be	AUX
cana-1225	198	21	a	a	DET
cana-1225	198	22	bipolar	bipolar	ADJ
cana-1225	198	23	vague	vague	NOUN
cana-1225	198	24	semi	semi	ADJ
cana-1225	198	25	continuous	continuous	ADJ
cana-1225	198	26	mapping	mapping	NOUN
cana-1225	198	27	.	.	PUNCT
cana-1225	199	1	remark	remark	NOUN
cana-1225	199	2	3.11	3.11	NUM
cana-1225	199	3	:	:	PUNCT
cana-1225	199	4	every	every	DET
cana-1225	199	5	bipolar	bipolar	ADJ
cana-1225	199	6	vague	vague	ADJ
cana-1225	199	7	pre	pre	ADJ
cana-1225	199	8	continuous	continuous	ADJ
cana-1225	199	9	mapping	mapping	NOUN
cana-1225	199	10	and	and	CCONJ
cana-1225	199	11	bipolar	bipolar	ADJ
cana-1225	199	12	vague	vague	NOUN
cana-1225	199	13	𝛼	𝛼	ADP
cana-1225	199	14	generalized	generalized	ADJ
cana-1225	199	15	continuous	continuous	ADJ
cana-1225	199	16	mapping	mapping	NOUN
cana-1225	199	17	are	be	AUX
cana-1225	199	18	independent	independent	ADJ
cana-1225	199	19	to	to	ADP
cana-1225	199	20	each	each	DET
cana-1225	199	21	other	other	ADJ
cana-1225	199	22	in	in	ADP
cana-1225	199	23	general	general	ADJ
cana-1225	199	24	.	.	PUNCT
cana-1225	200	1	example	example	NOUN
cana-1225	201	1	3.12	3.12	NUM
cana-1225	201	2	:	:	PUNCT
cana-1225	201	3	let	let	VERB
cana-1225	201	4	x	x	PUNCT
cana-1225	201	5	=	=	PRON
cana-1225	201	6	{	{	PUNCT
cana-1225	201	7	a	a	DET
cana-1225	201	8	,	,	PUNCT
cana-1225	201	9	b	b	NOUN
cana-1225	201	10	}	}	PUNCT
cana-1225	201	11	and	and	CCONJ
cana-1225	201	12	y	y	PROPN
cana-1225	201	13	=	=	SYM
cana-1225	201	14	{	{	PUNCT
cana-1225	201	15	u	u	NOUN
cana-1225	201	16	,	,	PUNCT
cana-1225	201	17	v	v	NOUN
cana-1225	201	18	}	}	PUNCT
cana-1225	201	19	.	.	PUNCT
cana-1225	202	1	then	then	ADV
cana-1225	202	2	𝜏	𝜏	X
cana-1225	202	3	=	=	PUNCT
cana-1225	202	4	{	{	PUNCT
cana-1225	202	5	0~	0~	NOUN
cana-1225	202	6	,	,	PUNCT
cana-1225	202	7	a	a	DET
cana-1225	202	8	,	,	PUNCT
cana-1225	202	9	1~	1~	NUM
cana-1225	202	10	}	}	PUNCT
cana-1225	202	11	and	and	CCONJ
cana-1225	202	12	𝜎	𝜎	NOUN
cana-1225	202	13	=	=	SYM
cana-1225	202	14	{	{	PUNCT
cana-1225	202	15	0~	0~	NOUN
cana-1225	202	16	,	,	PUNCT
cana-1225	202	17	b	b	NOUN
cana-1225	202	18	,	,	PUNCT
cana-1225	202	19	1~	1~	NUM
cana-1225	202	20	}	}	PUNCT
cana-1225	202	21	are	be	AUX
cana-1225	202	22	bipolar	bipolar	ADJ
cana-1225	202	23	vague	vague	ADJ
cana-1225	202	24	topologies	topology	NOUN
cana-1225	202	25	on	on	ADP
cana-1225	202	26	x	x	X
cana-1225	202	27	and	and	CCONJ
cana-1225	202	28	y	y	PROPN
cana-1225	202	29	respectively	respectively	ADV
cana-1225	202	30	,	,	PUNCT
cana-1225	202	31	where	where	SCONJ
cana-1225	202	32	a	a	DET
cana-1225	202	33	=	=	SYM
cana-1225	202	34	x	x	NOUN
cana-1225	202	35	,	,	PUNCT
cana-1225	202	36	[	[	X
cana-1225	202	37	0.1	0.1	NUM
cana-1225	202	38	,	,	PUNCT
cana-1225	202	39	0.1	0.1	NUM
cana-1225	202	40	]	]	PUNCT
cana-1225	203	1	[	[	X
cana-1225	203	2	-0.4	-0.4	X
cana-1225	203	3	,	,	PUNCT
cana-1225	203	4	-0.4	-0.4	NOUN
cana-1225	203	5	]	]	PUNCT
cana-1225	203	6	,	,	PUNCT
cana-1225	203	7	[	[	X
cana-1225	203	8	0.6	0.6	NUM
cana-1225	203	9	,	,	PUNCT
cana-1225	203	10	0.3	0.3	NUM
cana-1225	203	11	]	]	PUNCT
cana-1225	204	1	[	[	X
cana-1225	204	2	0.5	0.5	NUM
cana-1225	204	3	,	,	PUNCT
cana-1225	204	4	-0.5]	-0.5]	NOUN
cana-1225	204	5	and	and	CCONJ
cana-1225	204	6	b	b	X
cana-1225	204	7	=	=	SYM
cana-1225	204	8	y	y	PROPN
cana-1225	204	9	,	,	PUNCT
cana-1225	204	10	[	[	X
cana-1225	204	11	0.2	0.2	NUM
cana-1225	204	12	,	,	PUNCT
cana-1225	204	13	0.2	0.2	NUM
cana-1225	204	14	]	]	PUNCT
cana-1225	205	1	[	[	X
cana-1225	205	2	-0.5	-0.5	X
cana-1225	205	3	,	,	PUNCT
cana-1225	205	4	-0.5	-0.5	PROPN
cana-1225	205	5	]	]	PUNCT
cana-1225	205	6	,	,	PUNCT
cana-1225	205	7	[	[	X
cana-1225	205	8	0.7	0.7	NUM
cana-1225	205	9	,	,	PUNCT
cana-1225	205	10	0.3	0.3	NUM
cana-1225	205	11	]	]	PUNCT
cana-1225	206	1	[	[	X
cana-1225	206	2	-0.5	-0.5	NOUN
cana-1225	206	3	,	,	PUNCT
cana-1225	206	4	-0.5].	-0.5].	VERB
cana-1225	206	5	define	define	VERB
cana-1225	206	6	a	a	DET
cana-1225	206	7	mapping	mapping	NOUN
cana-1225	206	8	𝑓	𝑓	DET
cana-1225	206	9	communications	communication	NOUN
cana-1225	206	10	on	on	ADP
cana-1225	206	11	applied	apply	VERB
cana-1225	206	12	nonlinear	nonlinear	ADJ
cana-1225	206	13	analysis	analysis	NOUN
cana-1225	206	14	issn	issn	NOUN
cana-1225	206	15	:	:	PUNCT
cana-1225	206	16	1074	1074	NUM
cana-1225	206	17	-	-	PUNCT
cana-1225	206	18	133x	133x	NUM
cana-1225	206	19	vol	vol	NOUN
cana-1225	206	20	31	31	NUM
cana-1225	206	21	no	no	NOUN
cana-1225	206	22	.	.	PUNCT
cana-1225	207	1	6s	6s	NUM
cana-1225	207	2	(	(	PUNCT
cana-1225	207	3	2024	2024	NUM
cana-1225	207	4	)	)	PUNCT
cana-1225	207	5	324	324	NUM
cana-1225	207	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1225	207	7	:	:	PUNCT
cana-1225	207	8	(	(	PUNCT
cana-1225	207	9	x	x	X
cana-1225	207	10	,	,	PUNCT
cana-1225	207	11	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	207	12	)	)	PUNCT
cana-1225	207	13	→	→	SYM
cana-1225	207	14	(	(	PUNCT
cana-1225	207	15	y	y	NOUN
cana-1225	207	16	,	,	PUNCT
cana-1225	207	17	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	207	18	)	)	PUNCT
cana-1225	207	19	by	by	ADP
cana-1225	207	20	f(a	f(a	PROPN
cana-1225	207	21	)	)	PUNCT
cana-1225	207	22	=	=	SYM
cana-1225	207	23	u	u	NOUN
cana-1225	207	24	and	and	CCONJ
cana-1225	207	25	f(b	f(b	PROPN
cana-1225	207	26	)	)	PUNCT
cana-1225	208	1	=	=	PUNCT
cana-1225	209	1	v.	v.	CCONJ
cana-1225	209	2	here	here	ADV
cana-1225	209	3	the	the	DET
cana-1225	209	4	bipolar	bipolar	ADJ
cana-1225	209	5	vague	vague	NOUN
cana-1225	209	6	set	set	VERB
cana-1225	209	7	bc	bc	PROPN
cana-1225	209	8	=	=	PUNCT
cana-1225	209	9	y	y	PROPN
cana-1225	209	10	,	,	PUNCT
cana-1225	209	11	[	[	X
cana-1225	209	12	0.8	0.8	NUM
cana-1225	209	13	,	,	PUNCT
cana-1225	209	14	0.8	0.8	NUM
cana-1225	209	15	]	]	PUNCT
cana-1225	210	1	[	[	X
cana-1225	210	2	-0.5	-0.5	X
cana-1225	210	3	,	,	PUNCT
cana-1225	210	4	-0.5	-0.5	PROPN
cana-1225	210	5	]	]	PUNCT
cana-1225	210	6	,	,	PUNCT
cana-1225	210	7	[	[	X
cana-1225	210	8	0.7	0.7	NUM
cana-1225	210	9	,	,	PUNCT
cana-1225	210	10	0.3	0.3	NUM
cana-1225	210	11	]	]	PUNCT
cana-1225	211	1	[	[	X
cana-1225	211	2	-0.5	-0.5	NOUN
cana-1225	211	3	,	,	PUNCT
cana-1225	211	4	-0.5]	-0.5]	NOUN
cana-1225	211	5	is	be	AUX
cana-1225	211	6	a	a	DET
cana-1225	211	7	bipolar	bipolar	ADJ
cana-1225	211	8	vague	vague	NOUN
cana-1225	211	9	closed	close	VERB
cana-1225	211	10	set	set	VERB
cana-1225	211	11	in	in	ADP
cana-1225	211	12	y.	y.	NOUN
cana-1225	211	13	then	then	ADV
cana-1225	211	14	𝑓−1	𝑓−1	PROPN
cana-1225	211	15	(	(	PUNCT
cana-1225	211	16	bc	bc	PROPN
cana-1225	211	17	)	)	PUNCT
cana-1225	211	18	=	=	SYM
cana-1225	211	19	x	x	PUNCT
cana-1225	211	20	,	,	PUNCT
cana-1225	211	21	[	[	X
cana-1225	211	22	0.8	0.8	NUM
cana-1225	211	23	,	,	PUNCT
cana-1225	211	24	0.8	0.8	NUM
cana-1225	211	25	]	]	PUNCT
cana-1225	212	1	[	[	X
cana-1225	212	2	-0.5	-0.5	X
cana-1225	212	3	,	,	PUNCT
cana-1225	212	4	-0.5	-0.5	PROPN
cana-1225	212	5	]	]	PUNCT
cana-1225	212	6	,	,	PUNCT
cana-1225	212	7	[	[	X
cana-1225	212	8	0.7	0.7	NUM
cana-1225	212	9	,	,	PUNCT
cana-1225	212	10	0.3	0.3	NUM
cana-1225	212	11	]	]	PUNCT
cana-1225	213	1	[	[	X
cana-1225	213	2	-0.5	-0.5	NOUN
cana-1225	213	3	,	,	PUNCT
cana-1225	213	4	-0.5]	-0.5]	NOUN
cana-1225	213	5	is	be	AUX
cana-1225	213	6	a	a	DET
cana-1225	213	7	bipolar	bipolar	ADJ
cana-1225	213	8	vague	vague	NOUN
cana-1225	213	9	𝛼	𝛼	DET
cana-1225	213	10	generalized	generalize	VERB
cana-1225	213	11	closed	close	VERB
cana-1225	213	12	set	set	VERB
cana-1225	213	13	in	in	ADP
cana-1225	213	14	(	(	PUNCT
cana-1225	213	15	x	x	NOUN
cana-1225	213	16	,	,	PUNCT
cana-1225	213	17	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	213	18	)	)	PUNCT
cana-1225	213	19	as	as	ADP
cana-1225	213	20	𝑓−1	𝑓−1	PROPN
cana-1225	213	21	(	(	PUNCT
cana-1225	213	22	bc	bc	PROPN
cana-1225	213	23	)	)	PUNCT
cana-1225	213	24	⊆	⊆	NUM
cana-1225	213	25	1~	1~	NUM
cana-1225	213	26	and	and	CCONJ
cana-1225	213	27	b𝑉𝛼cl(𝑓−1	b𝑉𝛼cl(𝑓−1	PROPN
cana-1225	213	28	(	(	PUNCT
cana-1225	213	29	bc	bc	PROPN
cana-1225	213	30	)	)	PUNCT
cana-1225	213	31	)	)	PUNCT
cana-1225	214	1	=	=	SYM
cana-1225	215	1	𝑓−1	𝑓−1	PROPN
cana-1225	215	2	(	(	PUNCT
cana-1225	215	3	bc	bc	PROPN
cana-1225	215	4	)	)	PUNCT
cana-1225	215	5	∪	∪	NOUN
cana-1225	215	6	bvcl(bvint(bvcl(𝑓−1	bvcl(bvint(bvcl(𝑓−1	PROPN
cana-1225	215	7	(	(	PUNCT
cana-1225	215	8	bc	bc	PROPN
cana-1225	215	9	)	)	PUNCT
cana-1225	215	10	)	)	PUNCT
cana-1225	215	11	)	)	PUNCT
cana-1225	215	12	)	)	PUNCT
cana-1225	216	1	=	=	PRON
cana-1225	216	2	ac	ac	PROPN
cana-1225	216	3	⊆	⊆	NUM
cana-1225	216	4	1~	1~	NUM
cana-1225	216	5	,	,	PUNCT
cana-1225	216	6	where	where	SCONJ
cana-1225	216	7	ac	ac	PROPN
cana-1225	216	8	is	be	AUX
cana-1225	216	9	a	a	DET
cana-1225	216	10	bipolar	bipolar	ADJ
cana-1225	216	11	vague	vague	NOUN
cana-1225	216	12	closed	close	VERB
cana-1225	216	13	set	set	VERB
cana-1225	216	14	in	in	ADP
cana-1225	216	15	x.	x.	NOUN
cana-1225	216	16	therefore	therefore	ADV
cana-1225	216	17	,	,	PUNCT
cana-1225	216	18	𝑓	𝑓	PRON
cana-1225	216	19	is	be	AUX
cana-1225	216	20	a	a	DET
cana-1225	216	21	bipolar	bipolar	ADJ
cana-1225	216	22	vague	vague	NOUN
cana-1225	216	23	𝛼	𝛼	ADP
cana-1225	216	24	generalized	generalize	VERB
cana-1225	216	25	continuous	continuous	ADJ
cana-1225	216	26	mapping	mapping	NOUN
cana-1225	216	27	.	.	PUNCT
cana-1225	217	1	since	since	SCONJ
cana-1225	217	2	bvcl	bvcl	PROPN
cana-1225	217	3	(	(	PUNCT
cana-1225	217	4	bvint(𝑓−1	bvint(𝑓−1	PROPN
cana-1225	217	5	(	(	PUNCT
cana-1225	217	6	𝐵𝑐	𝐵𝑐	PROPN
cana-1225	217	7	)	)	PUNCT
cana-1225	217	8	)	)	PUNCT
cana-1225	217	9	)	)	PUNCT
cana-1225	218	1	=	=	PUNCT
cana-1225	218	2	ac	ac	ADV
cana-1225	218	3	⊄	⊄	NOUN
cana-1225	218	4	𝑓−1	𝑓−1	NUM
cana-1225	218	5	(	(	PUNCT
cana-1225	218	6	bc	bc	PROPN
cana-1225	218	7	)	)	PUNCT
cana-1225	218	8	,	,	PUNCT
cana-1225	218	9	𝑓−1	𝑓−1	PROPN
cana-1225	218	10	(	(	PUNCT
cana-1225	218	11	bc	bc	PROPN
cana-1225	218	12	)	)	PUNCT
cana-1225	218	13	is	be	AUX
cana-1225	218	14	not	not	PART
cana-1225	218	15	a	a	DET
cana-1225	218	16	bipolar	bipolar	ADJ
cana-1225	218	17	vague	vague	ADJ
cana-1225	218	18	pre	pre	ADJ
cana-1225	218	19	-	-	ADJ
cana-1225	218	20	closed	closed	ADJ
cana-1225	218	21	set	set	NOUN
cana-1225	218	22	in	in	ADP
cana-1225	218	23	x.	x.	NOUN
cana-1225	218	24	hence	hence	ADV
cana-1225	218	25	𝑓	𝑓	PROPN
cana-1225	218	26	is	be	AUX
cana-1225	218	27	not	not	PART
cana-1225	218	28	a	a	DET
cana-1225	218	29	bipolar	bipolar	ADJ
cana-1225	218	30	vague	vague	ADJ
cana-1225	218	31	pre	pre	ADJ
cana-1225	218	32	continuous	continuous	ADJ
cana-1225	218	33	mapping	mapping	NOUN
cana-1225	218	34	.	.	PUNCT
cana-1225	219	1	example	example	NOUN
cana-1225	219	2	3.13	3.13	NUM
cana-1225	219	3	:	:	PUNCT
cana-1225	219	4	let	let	VERB
cana-1225	219	5	x	x	PUNCT
cana-1225	219	6	=	=	PRON
cana-1225	219	7	{	{	PUNCT
cana-1225	219	8	a	a	DET
cana-1225	219	9	,	,	PUNCT
cana-1225	219	10	b	b	NOUN
cana-1225	219	11	}	}	PUNCT
cana-1225	219	12	and	and	CCONJ
cana-1225	219	13	y	y	PROPN
cana-1225	219	14	=	=	SYM
cana-1225	219	15	{	{	PUNCT
cana-1225	219	16	u	u	NOUN
cana-1225	219	17	,	,	PUNCT
cana-1225	219	18	v	v	NOUN
cana-1225	219	19	}	}	PUNCT
cana-1225	219	20	.	.	PUNCT
cana-1225	220	1	then	then	ADV
cana-1225	220	2	𝜏	𝜏	X
cana-1225	220	3	=	=	PUNCT
cana-1225	220	4	{	{	PUNCT
cana-1225	220	5	0~	0~	NOUN
cana-1225	220	6	,	,	PUNCT
cana-1225	220	7	a	a	DET
cana-1225	220	8	,	,	PUNCT
cana-1225	220	9	1~	1~	NUM
cana-1225	220	10	}	}	PUNCT
cana-1225	220	11	and	and	CCONJ
cana-1225	220	12	𝜎	𝜎	NOUN
cana-1225	220	13	=	=	SYM
cana-1225	220	14	{	{	PUNCT
cana-1225	220	15	0~	0~	NOUN
cana-1225	220	16	,	,	PUNCT
cana-1225	220	17	b	b	NOUN
cana-1225	220	18	,	,	PUNCT
cana-1225	220	19	1~	1~	NUM
cana-1225	220	20	}	}	PUNCT
cana-1225	220	21	are	be	AUX
cana-1225	220	22	bipolar	bipolar	ADJ
cana-1225	220	23	vague	vague	ADJ
cana-1225	220	24	topologies	topology	NOUN
cana-1225	220	25	on	on	ADP
cana-1225	220	26	x	x	X
cana-1225	220	27	and	and	CCONJ
cana-1225	220	28	y	y	PROPN
cana-1225	220	29	respectively	respectively	ADV
cana-1225	220	30	,	,	PUNCT
cana-1225	220	31	where	where	SCONJ
cana-1225	220	32	a	a	DET
cana-1225	220	33	=	=	SYM
cana-1225	220	34	x	x	NOUN
cana-1225	220	35	,	,	PUNCT
cana-1225	220	36	[	[	X
cana-1225	220	37	0.5	0.5	NUM
cana-1225	220	38	,	,	PUNCT
cana-1225	220	39	0.4	0.4	NUM
cana-1225	220	40	]	]	PUNCT
cana-1225	221	1	[	[	X
cana-1225	221	2	-0.3	-0.3	PROPN
cana-1225	221	3	,	,	PUNCT
cana-1225	221	4	-0.2	-0.2	NOUN
cana-1225	221	5	]	]	X
cana-1225	221	6	,	,	PUNCT
cana-1225	221	7	[	[	X
cana-1225	221	8	0.5	0.5	NUM
cana-1225	221	9	,	,	PUNCT
cana-1225	221	10	0.5	0.5	NUM
cana-1225	221	11	]	]	PUNCT
cana-1225	221	12	[	[	X
cana-1225	221	13	0.3	0.3	NUM
cana-1225	221	14	,	,	PUNCT
cana-1225	221	15	-0.2]	-0.2]	PUNCT
cana-1225	221	16	and	and	CCONJ
cana-1225	221	17	b	b	X
cana-1225	221	18	=	=	SYM
cana-1225	221	19	y	y	PROPN
cana-1225	221	20	,	,	PUNCT
cana-1225	221	21	[	[	X
cana-1225	221	22	0.8	0.8	NUM
cana-1225	221	23	,	,	PUNCT
cana-1225	221	24	0.7	0.7	NUM
cana-1225	221	25	]	]	PUNCT
cana-1225	222	1	[	[	X
cana-1225	222	2	-0.8	-0.8	ADJ
cana-1225	222	3	,	,	PUNCT
cana-1225	222	4	-0.8	-0.8	PROPN
cana-1225	222	5	]	]	PUNCT
cana-1225	222	6	,	,	PUNCT
cana-1225	222	7	[	[	X
cana-1225	222	8	0.5	0.5	NUM
cana-1225	222	9	,	,	PUNCT
cana-1225	222	10	0.5	0.5	NUM
cana-1225	222	11	]	]	PUNCT
cana-1225	222	12	[	[	X
cana-1225	222	13	-0.8	-0.8	ADJ
cana-1225	222	14	,	,	PUNCT
cana-1225	222	15	-0.8].	-0.8].	NOUN
cana-1225	222	16	define	define	VERB
cana-1225	222	17	a	a	DET
cana-1225	222	18	mapping	mapping	NOUN
cana-1225	222	19	𝑓	𝑓	X
cana-1225	222	20	:	:	PUNCT
cana-1225	222	21	(	(	PUNCT
cana-1225	222	22	x	x	NOUN
cana-1225	222	23	,	,	PUNCT
cana-1225	222	24	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	222	25	)	)	PUNCT
cana-1225	222	26	→	→	SYM
cana-1225	222	27	(	(	PUNCT
cana-1225	222	28	y	y	NOUN
cana-1225	222	29	,	,	PUNCT
cana-1225	222	30	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	222	31	)	)	PUNCT
cana-1225	222	32	by	by	ADP
cana-1225	222	33	f(a	f(a	PROPN
cana-1225	222	34	)	)	PUNCT
cana-1225	222	35	=	=	SYM
cana-1225	222	36	u	u	NOUN
cana-1225	222	37	and	and	CCONJ
cana-1225	222	38	f(b	f(b	PROPN
cana-1225	222	39	)	)	PUNCT
cana-1225	223	1	=	=	PUNCT
cana-1225	224	1	v.	v.	CCONJ
cana-1225	224	2	here	here	ADV
cana-1225	224	3	the	the	DET
cana-1225	224	4	bipolar	bipolar	ADJ
cana-1225	224	5	vague	vague	NOUN
cana-1225	224	6	set	set	VERB
cana-1225	224	7	bc	bc	PROPN
cana-1225	224	8	=	=	PUNCT
cana-1225	224	9	y	y	PROPN
cana-1225	224	10	,	,	PUNCT
cana-1225	224	11	[	[	X
cana-1225	224	12	0.3	0.3	NUM
cana-1225	224	13	,	,	PUNCT
cana-1225	224	14	0.2	0.2	NUM
cana-1225	224	15	]	]	PUNCT
cana-1225	225	1	[	[	X
cana-1225	225	2	-0.2	-0.2	NOUN
cana-1225	225	3	,	,	PUNCT
cana-1225	225	4	-0.2	-0.2	PROPN
cana-1225	225	5	]	]	X
cana-1225	225	6	,	,	PUNCT
cana-1225	225	7	[	[	X
cana-1225	225	8	0.5	0.5	NUM
cana-1225	225	9	,	,	PUNCT
cana-1225	225	10	0.5	0.5	NUM
cana-1225	225	11	]	]	PUNCT
cana-1225	225	12	[	[	X
cana-1225	225	13	-0.2	-0.2	NOUN
cana-1225	225	14	,	,	PUNCT
cana-1225	225	15	-0.2]	-0.2]	PUNCT
cana-1225	225	16	is	be	AUX
cana-1225	225	17	a	a	DET
cana-1225	225	18	bipolar	bipolar	ADJ
cana-1225	225	19	vague	vague	NOUN
cana-1225	225	20	closed	close	VERB
cana-1225	225	21	set	set	VERB
cana-1225	225	22	in	in	ADP
cana-1225	225	23	y.	y.	NOUN
cana-1225	225	24	then	then	ADV
cana-1225	226	1	𝑓−1	𝑓−1	PROPN
cana-1225	226	2	(	(	PUNCT
cana-1225	226	3	bc	bc	PROPN
cana-1225	226	4	)	)	PUNCT
cana-1225	226	5	=	=	SYM
cana-1225	226	6	x	x	PUNCT
cana-1225	226	7	,	,	PUNCT
cana-1225	226	8	[	[	X
cana-1225	226	9	0.3	0.3	NUM
cana-1225	226	10	,	,	PUNCT
cana-1225	226	11	0.2	0.2	NUM
cana-1225	226	12	]	]	PUNCT
cana-1225	227	1	[	[	X
cana-1225	227	2	-0.2	-0.2	NOUN
cana-1225	227	3	,	,	PUNCT
cana-1225	227	4	-0.2	-0.2	PROPN
cana-1225	227	5	]	]	X
cana-1225	227	6	,	,	PUNCT
cana-1225	227	7	[	[	X
cana-1225	227	8	0.5	0.5	NUM
cana-1225	227	9	,	,	PUNCT
cana-1225	227	10	0.5	0.5	NUM
cana-1225	227	11	]	]	PUNCT
cana-1225	227	12	[	[	X
cana-1225	227	13	-0.2	-0.2	NOUN
cana-1225	227	14	,	,	PUNCT
cana-1225	227	15	-0.2]	-0.2]	PUNCT
cana-1225	227	16	is	be	AUX
cana-1225	227	17	not	not	PART
cana-1225	227	18	a	a	DET
cana-1225	227	19	bipolar	bipolar	ADJ
cana-1225	227	20	vague	vague	NOUN
cana-1225	227	21	𝛼	𝛼	DET
cana-1225	227	22	generalized	generalize	VERB
cana-1225	227	23	closed	close	VERB
cana-1225	227	24	set	set	VERB
cana-1225	227	25	in	in	ADP
cana-1225	227	26	(	(	PUNCT
cana-1225	227	27	x	x	NOUN
cana-1225	227	28	,	,	PUNCT
cana-1225	227	29	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	227	30	)	)	PUNCT
cana-1225	227	31	as	as	ADP
cana-1225	227	32	𝑓−1	𝑓−1	PROPN
cana-1225	227	33	(	(	PUNCT
cana-1225	227	34	bc	bc	PROPN
cana-1225	227	35	)	)	PUNCT
cana-1225	227	36	⊆	⊆	PROPN
cana-1225	227	37	a	a	PRON
cana-1225	227	38	and	and	CCONJ
cana-1225	227	39	b𝑉𝛼cl(𝑓−1	b𝑉𝛼cl(𝑓−1	PROPN
cana-1225	227	40	(	(	PUNCT
cana-1225	227	41	bc	bc	PROPN
cana-1225	227	42	)	)	PUNCT
cana-1225	227	43	)	)	PUNCT
cana-1225	228	1	=	=	SYM
cana-1225	229	1	𝑓−1	𝑓−1	PROPN
cana-1225	229	2	(	(	PUNCT
cana-1225	229	3	bc	bc	PROPN
cana-1225	229	4	)	)	PUNCT
cana-1225	229	5	∪	∪	NOUN
cana-1225	229	6	bvcl(bvint(bvcl(𝑓−1	bvcl(bvint(bvcl(𝑓−1	PROPN
cana-1225	229	7	(	(	PUNCT
cana-1225	229	8	bc	bc	PROPN
cana-1225	229	9	)	)	PUNCT
cana-1225	229	10	)	)	PUNCT
cana-1225	229	11	)	)	PUNCT
cana-1225	229	12	)	)	PUNCT
cana-1225	230	1	=	=	PUNCT
cana-1225	230	2	ac	ac	PROPN
cana-1225	230	3	⊄	⊄	NOUN
cana-1225	230	4	a	a	X
cana-1225	230	5	,	,	PUNCT
cana-1225	230	6	where	where	SCONJ
cana-1225	230	7	ac	ac	PROPN
cana-1225	230	8	is	be	AUX
cana-1225	230	9	a	a	DET
cana-1225	230	10	bipolar	bipolar	ADJ
cana-1225	230	11	vague	vague	NOUN
cana-1225	230	12	closed	close	VERB
cana-1225	230	13	set	set	VERB
cana-1225	230	14	in	in	ADP
cana-1225	230	15	x.	x.	NOUN
cana-1225	230	16	therefore	therefore	ADV
cana-1225	230	17	,	,	PUNCT
cana-1225	230	18	𝑓	𝑓	PRON
cana-1225	230	19	is	be	AUX
cana-1225	230	20	not	not	PART
cana-1225	230	21	a	a	DET
cana-1225	230	22	bipolar	bipolar	ADJ
cana-1225	230	23	vague	vague	NOUN
cana-1225	230	24	𝛼	𝛼	ADP
cana-1225	230	25	generalized	generalize	VERB
cana-1225	230	26	continuous	continuous	ADJ
cana-1225	230	27	mapping	mapping	NOUN
cana-1225	230	28	.	.	PUNCT
cana-1225	231	1	since	since	SCONJ
cana-1225	231	2	bvcl	bvcl	PROPN
cana-1225	231	3	(	(	PUNCT
cana-1225	231	4	bvint(𝑓−1	bvint(𝑓−1	PROPN
cana-1225	231	5	(	(	PUNCT
cana-1225	231	6	𝐵𝑐	𝐵𝑐	PROPN
cana-1225	231	7	)	)	PUNCT
cana-1225	231	8	)	)	PUNCT
cana-1225	231	9	)	)	PUNCT
cana-1225	232	1	=	=	SYM
cana-1225	232	2	0~	0~	NOUN
cana-1225	232	3	⊆	⊆	NUM
cana-1225	232	4	𝑓−1	𝑓−1	PROPN
cana-1225	232	5	(	(	PUNCT
cana-1225	232	6	bc	bc	PROPN
cana-1225	232	7	)	)	PUNCT
cana-1225	232	8	,	,	PUNCT
cana-1225	232	9	𝑓−1	𝑓−1	PROPN
cana-1225	232	10	(	(	PUNCT
cana-1225	232	11	bc	bc	PROPN
cana-1225	232	12	)	)	PUNCT
cana-1225	232	13	is	be	AUX
cana-1225	232	14	a	a	DET
cana-1225	232	15	bipolar	bipolar	ADJ
cana-1225	232	16	vague	vague	ADJ
cana-1225	232	17	pre	pre	ADJ
cana-1225	232	18	-	-	ADJ
cana-1225	232	19	closed	closed	ADJ
cana-1225	232	20	set	set	NOUN
cana-1225	232	21	in	in	ADP
cana-1225	232	22	x.	x.	NOUN
cana-1225	232	23	hence	hence	ADV
cana-1225	232	24	𝑓	𝑓	PROPN
cana-1225	232	25	is	be	AUX
cana-1225	232	26	a	a	DET
cana-1225	232	27	bipolar	bipolar	ADJ
cana-1225	232	28	vague	vague	ADJ
cana-1225	232	29	pre	pre	ADJ
cana-1225	232	30	continuous	continuous	ADJ
cana-1225	232	31	mapping	mapping	NOUN
cana-1225	232	32	.	.	PUNCT
cana-1225	233	1	the	the	DET
cana-1225	233	2	relation	relation	NOUN
cana-1225	233	3	between	between	ADP
cana-1225	233	4	various	various	ADJ
cana-1225	233	5	types	type	NOUN
cana-1225	233	6	of	of	ADP
cana-1225	233	7	bipolar	bipolar	ADJ
cana-1225	233	8	vague	vague	ADJ
cana-1225	233	9	continuity	continuity	NOUN
cana-1225	233	10	is	be	AUX
cana-1225	233	11	given	give	VERB
cana-1225	233	12	in	in	ADP
cana-1225	233	13	the	the	DET
cana-1225	233	14	following	follow	VERB
cana-1225	233	15	diagram	diagram	NOUN
cana-1225	233	16	:	:	PUNCT
cana-1225	233	17	proposition	proposition	NOUN
cana-1225	233	18	3.14	3.14	NUM
cana-1225	233	19	:	:	PUNCT
cana-1225	233	20	a	a	DET
cana-1225	233	21	mapping	mapping	NOUN
cana-1225	233	22	𝑓	𝑓	X
cana-1225	233	23	:	:	PUNCT
cana-1225	233	24	(	(	PUNCT
cana-1225	233	25	x	x	NOUN
cana-1225	233	26	,	,	PUNCT
cana-1225	233	27	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	233	28	)	)	PUNCT
cana-1225	233	29	→	→	SYM
cana-1225	233	30	(	(	PUNCT
cana-1225	233	31	y	y	NOUN
cana-1225	233	32	,	,	PUNCT
cana-1225	233	33	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	233	34	)	)	PUNCT
cana-1225	233	35	is	be	AUX
cana-1225	233	36	a	a	DET
cana-1225	233	37	bipolar	bipolar	ADJ
cana-1225	233	38	vague	vague	NOUN
cana-1225	233	39	𝛼	𝛼	ADP
cana-1225	233	40	generalized	generalize	VERB
cana-1225	233	41	continuous	continuous	ADJ
cana-1225	233	42	if	if	SCONJ
cana-1225	233	43	and	and	CCONJ
cana-1225	233	44	only	only	ADV
cana-1225	233	45	if	if	SCONJ
cana-1225	233	46	the	the	DET
cana-1225	233	47	inverse	inverse	ADJ
cana-1225	233	48	image	image	NOUN
cana-1225	233	49	of	of	ADP
cana-1225	233	50	each	each	DET
cana-1225	233	51	bipolar	bipolar	ADJ
cana-1225	233	52	vague	vague	ADJ
cana-1225	233	53	open	open	ADJ
cana-1225	233	54	set	set	VERB
cana-1225	233	55	in	in	ADP
cana-1225	233	56	y	y	PROPN
cana-1225	233	57	is	be	AUX
cana-1225	233	58	a	a	DET
cana-1225	233	59	bipolar	bipolar	ADJ
cana-1225	233	60	vague	vague	NOUN
cana-1225	233	61	𝛼	𝛼	ADP
cana-1225	233	62	generalized	generalize	VERB
cana-1225	233	63	open	open	ADJ
cana-1225	233	64	set	set	NOUN
cana-1225	233	65	in	in	ADP
cana-1225	233	66	x.	x.	NOUN
cana-1225	233	67	proof	proof	NOUN
cana-1225	233	68	:	:	PUNCT
cana-1225	233	69	necessity	necessity	NOUN
cana-1225	233	70	:	:	PUNCT
cana-1225	233	71	let	let	VERB
cana-1225	233	72	a	a	PRON
cana-1225	233	73	be	be	AUX
cana-1225	233	74	a	a	DET
cana-1225	233	75	bipolar	bipolar	ADJ
cana-1225	233	76	vague	vague	ADJ
cana-1225	233	77	open	open	ADJ
cana-1225	233	78	set	set	VERB
cana-1225	233	79	in	in	ADP
cana-1225	233	80	y.	y.	PROPN
cana-1225	233	81	this	this	PRON
cana-1225	233	82	implies	imply	VERB
cana-1225	233	83	ac	ac	PROPN
cana-1225	233	84	is	be	AUX
cana-1225	233	85	a	a	DET
cana-1225	233	86	bipolar	bipolar	ADJ
cana-1225	233	87	vague	vague	NOUN
cana-1225	233	88	closed	close	VERB
cana-1225	233	89	set	set	VERB
cana-1225	233	90	in	in	ADP
cana-1225	233	91	y.	y.	NOUN
cana-1225	233	92	since	since	SCONJ
cana-1225	233	93	𝑓	𝑓	PROPN
cana-1225	233	94	is	be	AUX
cana-1225	233	95	a	a	DET
cana-1225	233	96	bipolar	bipolar	ADJ
cana-1225	233	97	vague	vague	NOUN
cana-1225	234	1	𝛼	𝛼	ADP
cana-1225	234	2	generalized	generalized	ADJ
cana-1225	234	3	continuous	continuous	ADJ
cana-1225	234	4	,	,	PUNCT
cana-1225	234	5	𝑓−1(ac	𝑓−1(ac	PROPN
cana-1225	234	6	)	)	PUNCT
cana-1225	234	7	is	be	AUX
cana-1225	234	8	a	a	DET
cana-1225	234	9	bipolar	bipolar	ADJ
cana-1225	234	10	vague	vague	NOUN
cana-1225	234	11	𝛼	𝛼	DET
cana-1225	234	12	generalized	generalize	VERB
cana-1225	234	13	closed	close	VERB
cana-1225	234	14	set	set	VERB
cana-1225	234	15	in	in	ADP
cana-1225	234	16	x.	x.	NOUN
cana-1225	234	17	since	since	SCONJ
cana-1225	234	18	𝑓−1(ac	𝑓−1(ac	NOUN
cana-1225	234	19	)	)	PUNCT
cana-1225	234	20	=	=	PUNCT
cana-1225	234	21	(	(	PUNCT
cana-1225	234	22	𝑓−1(a))c	𝑓−1(a))c	NUM
cana-1225	234	23	,	,	PUNCT
cana-1225	234	24	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	234	25	)	)	PUNCT
cana-1225	234	26	is	be	AUX
cana-1225	234	27	a	a	DET
cana-1225	234	28	bipolar	bipolar	ADJ
cana-1225	234	29	vague	vague	NOUN
cana-1225	234	30	𝛼	𝛼	ADP
cana-1225	234	31	generalized	generalize	VERB
cana-1225	234	32	open	open	ADJ
cana-1225	234	33	set	set	NOUN
cana-1225	234	34	in	in	ADP
cana-1225	234	35	x.	x.	PROPN
cana-1225	234	36	sufficiency	sufficiency	PROPN
cana-1225	234	37	:	:	PUNCT
cana-1225	234	38	let	let	VERB
cana-1225	234	39	a	a	PRON
cana-1225	234	40	be	be	AUX
cana-1225	234	41	a	a	DET
cana-1225	234	42	bipolar	bipolar	ADJ
cana-1225	234	43	vague	vague	NOUN
cana-1225	234	44	closed	close	VERB
cana-1225	234	45	set	set	VERB
cana-1225	234	46	in	in	ADP
cana-1225	234	47	y.	y.	PROPN
cana-1225	234	48	this	this	PRON
cana-1225	234	49	implies	imply	VERB
cana-1225	234	50	ac	ac	PROPN
cana-1225	234	51	is	be	AUX
cana-1225	234	52	a	a	DET
cana-1225	234	53	bipolar	bipolar	ADJ
cana-1225	234	54	vague	vague	ADJ
cana-1225	234	55	open	open	ADJ
cana-1225	234	56	set	set	VERB
cana-1225	234	57	in	in	ADP
cana-1225	234	58	y.	y.	NOUN
cana-1225	234	59	by	by	ADP
cana-1225	234	60	hypothesis	hypothesis	NOUN
cana-1225	234	61	,	,	PUNCT
cana-1225	234	62	𝑓−1(ac	𝑓−1(ac	PROPN
cana-1225	234	63	)	)	PUNCT
cana-1225	234	64	is	be	AUX
cana-1225	234	65	a	a	DET
cana-1225	234	66	bipolar	bipolar	ADJ
cana-1225	234	67	vague	vague	NOUN
cana-1225	234	68	𝛼	𝛼	ADP
cana-1225	234	69	generalized	generalize	VERB
cana-1225	234	70	open	open	ADJ
cana-1225	234	71	set	set	VERB
cana-1225	234	72	in	in	ADP
cana-1225	234	73	x.	x.	NOUN
cana-1225	234	74	since	since	SCONJ
cana-1225	234	75	𝑓−1(ac	𝑓−1(ac	NOUN
cana-1225	234	76	)	)	PUNCT
cana-1225	234	77	=	=	PUNCT
cana-1225	234	78	(	(	PUNCT
cana-1225	234	79	𝑓−1(a))c	𝑓−1(a))c	NUM
cana-1225	234	80	,	,	PUNCT
cana-1225	234	81	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	234	82	)	)	PUNCT
cana-1225	234	83	is	be	AUX
cana-1225	234	84	a	a	DET
cana-1225	234	85	bipolar	bipolar	ADJ
cana-1225	234	86	vague	vague	NOUN
cana-1225	234	87	𝛼	𝛼	DET
cana-1225	234	88	generalized	generalize	VERB
cana-1225	234	89	closed	close	VERB
cana-1225	234	90	set	set	VERB
cana-1225	234	91	in	in	ADP
cana-1225	234	92	x.	x.	NOUN
cana-1225	234	93	hence	hence	ADV
cana-1225	234	94	𝑓	𝑓	PROPN
cana-1225	234	95	is	be	AUX
cana-1225	234	96	a	a	DET
cana-1225	234	97	bipolar	bipolar	ADJ
cana-1225	234	98	vague	vague	NOUN
cana-1225	235	1	𝛼	𝛼	ADP
cana-1225	235	2	generalized	generalize	VERB
cana-1225	235	3	continuous	continuous	ADJ
cana-1225	235	4	mapping	mapping	NOUN
cana-1225	235	5	.	.	PUNCT
cana-1225	236	1	bipolar	bipolar	ADJ
cana-1225	236	2	vague	vague	ADJ
cana-1225	236	3	continuous	continuous	ADJ
cana-1225	236	4	bipolar	bipolar	ADJ
cana-1225	236	5	vague	vague	NOUN
cana-1225	236	6	𝜶	𝜶	ADP
cana-1225	236	7	generalized	generalize	VERB
cana-1225	236	8	continuous	continuous	ADJ
cana-1225	236	9	bipolar	bipolar	ADJ
cana-1225	236	10	vague	vague	NOUN
cana-1225	236	11	𝛼	𝛼	ADP
cana-1225	236	12	continuous	continuous	ADJ
cana-1225	236	13	bipolar	bipolar	ADJ
cana-1225	236	14	vague	vague	ADJ
cana-1225	236	15	pre	pre	ADJ
cana-1225	236	16	continuous	continuous	ADJ
cana-1225	236	17	bipolar	bipolar	ADJ
cana-1225	236	18	vague	vague	ADJ
cana-1225	236	19	semi	semi	ADJ
cana-1225	236	20	continuous	continuous	ADJ
cana-1225	236	21	communications	communication	NOUN
cana-1225	236	22	on	on	ADP
cana-1225	236	23	applied	apply	VERB
cana-1225	236	24	nonlinear	nonlinear	ADJ
cana-1225	236	25	analysis	analysis	NOUN
cana-1225	236	26	issn	issn	NOUN
cana-1225	236	27	:	:	PUNCT
cana-1225	236	28	1074	1074	NUM
cana-1225	236	29	-	-	PUNCT
cana-1225	236	30	133x	133x	NUM
cana-1225	236	31	vol	vol	NOUN
cana-1225	236	32	31	31	NUM
cana-1225	236	33	no	no	NOUN
cana-1225	236	34	.	.	PUNCT
cana-1225	237	1	6s	6s	NUM
cana-1225	237	2	(	(	PUNCT
cana-1225	237	3	2024	2024	NUM
cana-1225	237	4	)	)	PUNCT
cana-1225	237	5	325	325	NUM
cana-1225	237	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1225	237	7	proposition	proposition	NOUN
cana-1225	237	8	3.15	3.15	NUM
cana-1225	237	9	:	:	PUNCT
cana-1225	237	10	if	if	SCONJ
cana-1225	237	11	𝑓	𝑓	X
cana-1225	237	12	:	:	PUNCT
cana-1225	237	13	(	(	PUNCT
cana-1225	237	14	x	x	NOUN
cana-1225	237	15	,	,	PUNCT
cana-1225	237	16	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	237	17	)	)	PUNCT
cana-1225	237	18	→	→	SYM
cana-1225	237	19	(	(	PUNCT
cana-1225	237	20	y	y	NOUN
cana-1225	237	21	,	,	PUNCT
cana-1225	237	22	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	237	23	)	)	PUNCT
cana-1225	237	24	is	be	AUX
cana-1225	237	25	a	a	DET
cana-1225	237	26	bipolar	bipolar	ADJ
cana-1225	237	27	vague	vague	NOUN
cana-1225	237	28	𝛼	𝛼	ADP
cana-1225	237	29	generalized	generalize	VERB
cana-1225	237	30	continuous	continuous	ADJ
cana-1225	237	31	mapping	mapping	NOUN
cana-1225	237	32	and	and	CCONJ
cana-1225	237	33	𝑔	𝑔	PROPN
cana-1225	237	34	:	:	PUNCT
cana-1225	237	35	(	(	PUNCT
cana-1225	237	36	y	y	NOUN
cana-1225	237	37	,	,	PUNCT
cana-1225	237	38	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	237	39	)	)	PUNCT
cana-1225	237	40	→	→	SYM
cana-1225	237	41	(	(	PUNCT
cana-1225	237	42	z	z	NOUN
cana-1225	237	43	,	,	PUNCT
cana-1225	237	44	b𝑉𝛿	b𝑉𝛿	PROPN
cana-1225	237	45	)	)	PUNCT
cana-1225	237	46	is	be	AUX
cana-1225	237	47	a	a	DET
cana-1225	237	48	bipolar	bipolar	ADJ
cana-1225	237	49	vague	vague	ADJ
cana-1225	237	50	continuous	continuous	ADJ
cana-1225	237	51	mapping	mapping	NOUN
cana-1225	237	52	,	,	PUNCT
cana-1225	237	53	then	then	ADV
cana-1225	237	54	𝑔	𝑔	PROPN
cana-1225	237	55	∘	∘	PROPN
cana-1225	237	56	𝑓	𝑓	PROPN
cana-1225	237	57	:	:	PUNCT
cana-1225	237	58	(	(	PUNCT
cana-1225	237	59	x	x	NOUN
cana-1225	237	60	,	,	PUNCT
cana-1225	237	61	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	237	62	)	)	PUNCT
cana-1225	237	63	→	→	SYM
cana-1225	237	64	(	(	PUNCT
cana-1225	237	65	z	z	NOUN
cana-1225	237	66	,	,	PUNCT
cana-1225	237	67	b𝑉𝛿	b𝑉𝛿	PROPN
cana-1225	237	68	)	)	PUNCT
cana-1225	237	69	is	be	AUX
cana-1225	237	70	a	a	DET
cana-1225	237	71	bipolar	bipolar	ADJ
cana-1225	237	72	vague	vague	NOUN
cana-1225	237	73	𝛼	𝛼	ADP
cana-1225	237	74	generalized	generalize	VERB
cana-1225	237	75	continuous	continuous	ADJ
cana-1225	237	76	mapping	mapping	NOUN
cana-1225	237	77	.	.	PUNCT
cana-1225	238	1	proof	proof	NOUN
cana-1225	238	2	:	:	PUNCT
cana-1225	238	3	let	let	VERB
cana-1225	238	4	a	a	PRON
cana-1225	238	5	be	be	AUX
cana-1225	238	6	a	a	DET
cana-1225	238	7	bipolar	bipolar	ADJ
cana-1225	238	8	vague	vague	NOUN
cana-1225	238	9	closed	close	VERB
cana-1225	238	10	set	set	VERB
cana-1225	238	11	in	in	ADP
cana-1225	238	12	z.	z.	PROPN
cana-1225	238	13	then	then	ADV
cana-1225	238	14	𝑔−1(a	𝑔−1(a	ADV
cana-1225	238	15	)	)	PUNCT
cana-1225	238	16	be	be	AUX
cana-1225	238	17	a	a	DET
cana-1225	238	18	bipolar	bipolar	ADJ
cana-1225	238	19	vague	vague	NOUN
cana-1225	238	20	closed	close	VERB
cana-1225	238	21	set	set	VERB
cana-1225	238	22	in	in	ADP
cana-1225	238	23	y	y	PROPN
cana-1225	238	24	,	,	PUNCT
cana-1225	238	25	by	by	ADP
cana-1225	238	26	hypothesis	hypothesis	NOUN
cana-1225	238	27	.	.	PUNCT
cana-1225	239	1	since	since	SCONJ
cana-1225	239	2	𝑓	𝑓	PRON
cana-1225	239	3	is	be	AUX
cana-1225	239	4	a	a	DET
cana-1225	239	5	bipolar	bipolar	ADJ
cana-1225	239	6	vague	vague	NOUN
cana-1225	239	7	𝛼	𝛼	ADP
cana-1225	239	8	generalized	generalize	VERB
cana-1225	239	9	continuous	continuous	ADJ
cana-1225	239	10	mapping	mapping	NOUN
cana-1225	239	11	,	,	PUNCT
cana-1225	239	12	𝑓−1	𝑓−1	PROPN
cana-1225	239	13	(	(	PUNCT
cana-1225	239	14	𝑔−1(a	𝑔−1(a	NOUN
cana-1225	239	15	)	)	PUNCT
cana-1225	239	16	)	)	PUNCT
cana-1225	239	17	is	be	AUX
cana-1225	239	18	a	a	DET
cana-1225	239	19	bipolar	bipolar	ADJ
cana-1225	239	20	vague	vague	NOUN
cana-1225	239	21	𝛼	𝛼	DET
cana-1225	239	22	generalized	generalize	VERB
cana-1225	239	23	closed	close	VERB
cana-1225	239	24	set	set	VERB
cana-1225	239	25	in	in	ADP
cana-1225	239	26	x.	x.	NOUN
cana-1225	239	27	hence	hence	ADV
cana-1225	239	28	𝑔	𝑔	PROPN
cana-1225	239	29	∘	∘	PROPN
cana-1225	239	30	𝑓	𝑓	PRON
cana-1225	239	31	is	be	AUX
cana-1225	239	32	a	a	DET
cana-1225	239	33	bipolar	bipolar	ADJ
cana-1225	239	34	vague	vague	NOUN
cana-1225	239	35	𝛼	𝛼	ADP
cana-1225	239	36	generalized	generalize	VERB
cana-1225	239	37	continuous	continuous	ADJ
cana-1225	239	38	mapping	mapping	NOUN
cana-1225	239	39	.	.	PUNCT
cana-1225	240	1	definition	definition	NOUN
cana-1225	240	2	3.16	3.16	NUM
cana-1225	240	3	:	:	PUNCT
cana-1225	240	4	let	let	VERB
cana-1225	240	5	(	(	PUNCT
cana-1225	240	6	x	x	NOUN
cana-1225	240	7	,	,	PUNCT
cana-1225	240	8	b𝑉𝜏	b𝑉𝜏	NUM
cana-1225	240	9	)	)	PUNCT
cana-1225	240	10	be	be	AUX
cana-1225	240	11	a	a	DET
cana-1225	240	12	bipolar	bipolar	ADJ
cana-1225	240	13	vague	vague	ADJ
cana-1225	240	14	topological	topological	ADJ
cana-1225	240	15	space	space	NOUN
cana-1225	240	16	.	.	PUNCT
cana-1225	241	1	the	the	DET
cana-1225	241	2	bipolar	bipolar	ADJ
cana-1225	241	3	vague	vague	ADJ
cana-1225	241	4	alpha	alpha	NOUN
cana-1225	241	5	generalized	generalize	VERB
cana-1225	241	6	closure	closure	NOUN
cana-1225	241	7	(	(	PUNCT
cana-1225	241	8	b𝑉𝛼𝑔cl(a	b𝑉𝛼𝑔cl(a	NOUN
cana-1225	241	9	)	)	PUNCT
cana-1225	241	10	)	)	PUNCT
cana-1225	241	11	for	for	ADP
cana-1225	241	12	any	any	DET
cana-1225	241	13	bipolar	bipolar	ADJ
cana-1225	241	14	vague	vague	NOUN
cana-1225	241	15	set	set	NOUN
cana-1225	241	16	a	a	PRON
cana-1225	241	17	is	be	AUX
cana-1225	241	18	defined	define	VERB
cana-1225	241	19	as	as	SCONJ
cana-1225	241	20	follows	follow	VERB
cana-1225	241	21	:	:	PUNCT
cana-1225	241	22	b𝑉𝛼cl(a	b𝑉𝛼cl(a	ADJ
cana-1225	241	23	)	)	PUNCT
cana-1225	241	24	=	=	NOUN
cana-1225	241	25	∩	∩	NOUN
cana-1225	241	26	{	{	PUNCT
cana-1225	241	27	k	k	NOUN
cana-1225	241	28	:	:	PUNCT
cana-1225	241	29	k	k	X
cana-1225	241	30	is	be	AUX
cana-1225	241	31	a	a	DET
cana-1225	241	32	bipolar	bipolar	ADJ
cana-1225	241	33	vague	vague	NOUN
cana-1225	241	34	𝛼	𝛼	DET
cana-1225	241	35	generalized	generalize	VERB
cana-1225	241	36	closed	close	VERB
cana-1225	241	37	set	set	VERB
cana-1225	241	38	in	in	ADP
cana-1225	241	39	x	x	PUNCT
cana-1225	241	40	and	and	CCONJ
cana-1225	241	41	a⊆	a⊆	VERB
cana-1225	241	42	k	k	X
cana-1225	241	43	}	}	PUNCT
cana-1225	241	44	.	.	PUNCT
cana-1225	242	1	if	if	SCONJ
cana-1225	242	2	a	a	PRON
cana-1225	242	3	is	be	AUX
cana-1225	242	4	a	a	DET
cana-1225	242	5	bipolar	bipolar	ADJ
cana-1225	242	6	vague	vague	NOUN
cana-1225	242	7	𝛼	𝛼	DET
cana-1225	242	8	generalized	generalize	VERB
cana-1225	242	9	closed	close	VERB
cana-1225	242	10	set	set	NOUN
cana-1225	242	11	,	,	PUNCT
cana-1225	242	12	then	then	ADV
cana-1225	242	13	b𝑉𝛼𝑔cl(a	b𝑉𝛼𝑔cl(a	NUM
cana-1225	242	14	)	)	PUNCT
cana-1225	242	15	=	=	SYM
cana-1225	242	16	a.	a.	NOUN
cana-1225	242	17	proposition	proposition	NOUN
cana-1225	242	18	3.17	3.17	NUM
cana-1225	242	19	:	:	PUNCT
cana-1225	242	20	let	let	VERB
cana-1225	242	21	𝑓	𝑓	PRON
cana-1225	242	22	:	:	PUNCT
cana-1225	242	23	(	(	PUNCT
cana-1225	242	24	x	x	NOUN
cana-1225	242	25	,	,	PUNCT
cana-1225	242	26	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	242	27	)	)	PUNCT
cana-1225	242	28	→	→	SYM
cana-1225	242	29	(	(	PUNCT
cana-1225	242	30	y	y	NOUN
cana-1225	242	31	,	,	PUNCT
cana-1225	242	32	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	242	33	)	)	PUNCT
cana-1225	242	34	be	be	VERB
cana-1225	242	35	a	a	DET
cana-1225	242	36	bipolar	bipolar	ADJ
cana-1225	242	37	vague	vague	NOUN
cana-1225	242	38	𝛼	𝛼	ADP
cana-1225	242	39	generalized	generalize	VERB
cana-1225	242	40	continuous	continuous	ADJ
cana-1225	242	41	mapping	mapping	NOUN
cana-1225	242	42	.	.	PUNCT
cana-1225	243	1	then	then	ADV
cana-1225	243	2	the	the	DET
cana-1225	243	3	following	follow	VERB
cana-1225	243	4	conditions	condition	NOUN
cana-1225	243	5	are	be	AUX
cana-1225	243	6	hold	hold	ADJ
cana-1225	243	7	:	:	PUNCT
cana-1225	243	8	(	(	PUNCT
cana-1225	243	9	i	i	NOUN
cana-1225	243	10	)	)	PUNCT
cana-1225	243	11	𝑓(b𝑉𝛼𝑔cl(a	𝑓(b𝑉𝛼𝑔cl(a	PROPN
cana-1225	243	12	)	)	PUNCT
cana-1225	243	13	)	)	PUNCT
cana-1225	244	1	⊆	⊆	NUM
cana-1225	244	2	bvcl(𝑓(a	bvcl(𝑓(a	NUM
cana-1225	244	3	)	)	PUNCT
cana-1225	244	4	)	)	PUNCT
cana-1225	244	5	,	,	PUNCT
cana-1225	244	6	for	for	SCONJ
cana-1225	244	7	every	every	DET
cana-1225	244	8	bipolar	bipolar	ADJ
cana-1225	244	9	vague	vague	NOUN
cana-1225	244	10	set	set	VERB
cana-1225	244	11	a	a	PRON
cana-1225	244	12	in	in	ADP
cana-1225	244	13	x.	x.	PROPN
cana-1225	244	14	(	(	PUNCT
cana-1225	244	15	ii	ii	PROPN
cana-1225	244	16	)	)	PUNCT
cana-1225	244	17	b𝑉𝛼𝑔cl(𝑓−1(b	b𝑉𝛼𝑔cl(𝑓−1(b	PROPN
cana-1225	244	18	)	)	PUNCT
cana-1225	244	19	)	)	PUNCT
cana-1225	245	1	⊆	⊆	X
cana-1225	245	2	𝑓−1(bvcl(b	𝑓−1(bvcl(b	PROPN
cana-1225	245	3	)	)	PUNCT
cana-1225	245	4	)	)	PUNCT
cana-1225	245	5	,	,	PUNCT
cana-1225	245	6	for	for	ADP
cana-1225	245	7	every	every	DET
cana-1225	245	8	bipolar	bipolar	ADJ
cana-1225	245	9	vague	vague	ADJ
cana-1225	245	10	set	set	NOUN
cana-1225	245	11	b	b	NOUN
cana-1225	245	12	in	in	ADP
cana-1225	245	13	y.	y.	PROPN
cana-1225	245	14	proof	proof	NOUN
cana-1225	245	15	:	:	PUNCT
cana-1225	245	16	(	(	PUNCT
cana-1225	245	17	i	i	NOUN
cana-1225	245	18	)	)	PUNCT
cana-1225	245	19	since	since	SCONJ
cana-1225	245	20	bvcl(𝑓(a	bvcl(𝑓(a	NUM
cana-1225	245	21	)	)	PUNCT
cana-1225	245	22	)	)	PUNCT
cana-1225	245	23	is	be	AUX
cana-1225	245	24	a	a	DET
cana-1225	245	25	bipolar	bipolar	ADJ
cana-1225	245	26	vague	vague	NOUN
cana-1225	245	27	closed	close	VERB
cana-1225	245	28	set	set	VERB
cana-1225	245	29	in	in	ADP
cana-1225	245	30	y	y	PROPN
cana-1225	245	31	and	and	CCONJ
cana-1225	245	32	𝑓	𝑓	PRON
cana-1225	245	33	is	be	AUX
cana-1225	245	34	a	a	DET
cana-1225	245	35	bipolar	bipolar	ADJ
cana-1225	245	36	vague	vague	NOUN
cana-1225	246	1	𝛼	𝛼	ADP
cana-1225	246	2	generalized	generalize	VERB
cana-1225	246	3	continuous	continuous	ADJ
cana-1225	246	4	mapping	mapping	NOUN
cana-1225	246	5	,	,	PUNCT
cana-1225	246	6	then	then	ADV
cana-1225	246	7	𝑓−1(bvcl(f(a	𝑓−1(bvcl(f(a	NOUN
cana-1225	246	8	)	)	PUNCT
cana-1225	246	9	)	)	PUNCT
cana-1225	246	10	)	)	PUNCT
cana-1225	247	1	is	be	AUX
cana-1225	247	2	a	a	DET
cana-1225	247	3	bipolar	bipolar	ADJ
cana-1225	247	4	vague	vague	NOUN
cana-1225	247	5	𝛼	𝛼	DET
cana-1225	247	6	generalized	generalize	VERB
cana-1225	247	7	closed	close	VERB
cana-1225	247	8	set	set	VERB
cana-1225	247	9	in	in	ADP
cana-1225	247	10	x.	x.	NOUN
cana-1225	247	11	that	that	PRON
cana-1225	247	12	is	be	AUX
cana-1225	247	13	b𝑉𝛼𝑔cl(𝑓−1(bvcl(f(a	b𝑉𝛼𝑔cl(𝑓−1(bvcl(f(a	NOUN
cana-1225	247	14	)	)	PUNCT
cana-1225	247	15	)	)	PUNCT
cana-1225	247	16	)	)	PUNCT
cana-1225	247	17	)	)	PUNCT
cana-1225	248	1	=	=	SYM
cana-1225	248	2	𝑓−1(bvcl(f(a	𝑓−1(bvcl(f(a	NOUN
cana-1225	248	3	)	)	PUNCT
cana-1225	248	4	)	)	PUNCT
cana-1225	248	5	)	)	PUNCT
cana-1225	248	6	.	.	PUNCT
cana-1225	249	1	now	now	ADV
cana-1225	249	2	,	,	PUNCT
cana-1225	249	3	𝑓(b𝑉𝛼𝑔cl(𝑓−1(bvcl(f(a	𝑓(b𝑉𝛼𝑔cl(𝑓−1(bvcl(f(a	PROPN
cana-1225	249	4	)	)	PUNCT
cana-1225	249	5	)	)	PUNCT
cana-1225	249	6	)	)	PUNCT
cana-1225	249	7	)	)	PUNCT
cana-1225	250	1	=	=	SYM
cana-1225	250	2	𝑓𝑓−1(bvcl(f(a	𝑓𝑓−1(bvcl(f(a	NOUN
cana-1225	250	3	)	)	PUNCT
cana-1225	250	4	)	)	PUNCT
cana-1225	250	5	)	)	PUNCT
cana-1225	251	1	⊆	⊆	NUM
cana-1225	251	2	bvcl(𝑓(a	bvcl(𝑓(a	NUM
cana-1225	251	3	)	)	PUNCT
cana-1225	251	4	)	)	PUNCT
cana-1225	251	5	.	.	PUNCT
cana-1225	252	1	then	then	ADV
cana-1225	252	2	𝑓(b𝑉𝛼𝑔cl(a	𝑓(b𝑉𝛼𝑔cl(a	PROPN
cana-1225	252	3	)	)	PUNCT
cana-1225	252	4	)	)	PUNCT
cana-1225	252	5	⊆	⊆	NUM
cana-1225	252	6	𝑓(b𝑉𝛼𝑔cl	𝑓(b𝑉𝛼𝑔cl	ADJ
cana-1225	252	7	(	(	PUNCT
cana-1225	252	8	𝑓−1𝑓(a	𝑓−1𝑓(a	NOUN
cana-1225	252	9	)	)	PUNCT
cana-1225	252	10	)	)	PUNCT
cana-1225	252	11	)	)	PUNCT
cana-1225	253	1	⊆	⊆	NUM
cana-1225	253	2	𝑓(b𝑉𝛼𝑔cl(𝑓−1(bvcl(f(a	𝑓(b𝑉𝛼𝑔cl(𝑓−1(bvcl(f(a	NOUN
cana-1225	253	3	)	)	PUNCT
cana-1225	253	4	)	)	PUNCT
cana-1225	253	5	)	)	PUNCT
cana-1225	253	6	)	)	PUNCT
cana-1225	254	1	⊆	⊆	NUM
cana-1225	254	2	bvcl(𝑓(a	bvcl(𝑓(a	NUM
cana-1225	254	3	)	)	PUNCT
cana-1225	254	4	)	)	PUNCT
cana-1225	254	5	.	.	PUNCT
cana-1225	255	1	therefore	therefore	ADV
cana-1225	255	2	𝑓(b𝑉𝛼𝑔cl(a	𝑓(b𝑉𝛼𝑔cl(a	NOUN
cana-1225	255	3	)	)	PUNCT
cana-1225	255	4	)	)	PUNCT
cana-1225	256	1	⊆	⊆	NUM
cana-1225	256	2	bvcl(𝑓(a	bvcl(𝑓(a	NUM
cana-1225	256	3	)	)	PUNCT
cana-1225	256	4	)	)	PUNCT
cana-1225	256	5	,	,	PUNCT
cana-1225	256	6	for	for	SCONJ
cana-1225	256	7	every	every	DET
cana-1225	256	8	bipolar	bipolar	ADJ
cana-1225	256	9	vague	vague	NOUN
cana-1225	256	10	set	set	VERB
cana-1225	256	11	a	a	PRON
cana-1225	256	12	in	in	ADP
cana-1225	256	13	x.	x.	PROPN
cana-1225	256	14	(	(	PUNCT
cana-1225	256	15	ii	ii	NOUN
cana-1225	256	16	)	)	PUNCT
cana-1225	256	17	replacing	replace	VERB
cana-1225	256	18	a	a	PRON
cana-1225	256	19	by	by	ADP
cana-1225	256	20	𝑓−1(b	𝑓−1(b	PROPN
cana-1225	256	21	)	)	PUNCT
cana-1225	256	22	in	in	ADP
cana-1225	256	23	(	(	PUNCT
cana-1225	256	24	i	i	NOUN
cana-1225	256	25	)	)	PUNCT
cana-1225	256	26	,	,	PUNCT
cana-1225	256	27	we	we	PRON
cana-1225	256	28	get	get	VERB
cana-1225	256	29	𝑓(b𝑉𝛼𝑔cl	𝑓(b𝑉𝛼𝑔cl	ADV
cana-1225	256	30	(	(	PUNCT
cana-1225	256	31	𝑓−1(b	𝑓−1(b	PROPN
cana-1225	256	32	)	)	PUNCT
cana-1225	256	33	)	)	PUNCT
cana-1225	256	34	)	)	PUNCT
cana-1225	257	1	⊆	⊆	NUM
cana-1225	257	2	bvcl(𝑓(𝑓−1(b	bvcl(𝑓(𝑓−1(b	NUM
cana-1225	257	3	)	)	PUNCT
cana-1225	257	4	)	)	PUNCT
cana-1225	257	5	)	)	PUNCT
cana-1225	258	1	⊆	⊆	NUM
cana-1225	258	2	bvcl(b	bvcl(b	NOUN
cana-1225	258	3	)	)	PUNCT
cana-1225	258	4	.	.	PUNCT
cana-1225	259	1	hence	hence	ADV
cana-1225	259	2	b𝑉𝛼𝑔cl(𝑓−1(b	b𝑉𝛼𝑔cl(𝑓−1(b	NUM
cana-1225	259	3	)	)	PUNCT
cana-1225	259	4	)	)	PUNCT
cana-1225	260	1	⊆	⊆	X
cana-1225	260	2	𝑓−1(𝑓	𝑓−1(𝑓	NOUN
cana-1225	260	3	(	(	PUNCT
cana-1225	260	4	b𝑉𝛼𝑔cl(𝑓−1(b	b𝑉𝛼𝑔cl(𝑓−1(b	NUM
cana-1225	260	5	)	)	PUNCT
cana-1225	260	6	)	)	PUNCT
cana-1225	260	7	)	)	PUNCT
cana-1225	260	8	)	)	PUNCT
cana-1225	261	1	⊆	⊆	X
cana-1225	261	2	𝑓−1(bvcl(b	𝑓−1(bvcl(b	PROPN
cana-1225	261	3	)	)	PUNCT
cana-1225	261	4	)	)	PUNCT
cana-1225	261	5	,	,	PUNCT
cana-1225	261	6	for	for	ADP
cana-1225	261	7	every	every	DET
cana-1225	261	8	bipolar	bipolar	ADJ
cana-1225	261	9	vague	vague	ADJ
cana-1225	261	10	set	set	NOUN
cana-1225	261	11	b	b	PROPN
cana-1225	261	12	in	in	ADP
cana-1225	261	13	y.	y.	PROPN
cana-1225	261	14	definition	definition	NOUN
cana-1225	261	15	3.18	3.18	NUM
cana-1225	261	16	:	:	PUNCT
cana-1225	261	17	a	a	DET
cana-1225	261	18	bipolar	bipolar	ADJ
cana-1225	261	19	vague	vague	ADJ
cana-1225	261	20	topological	topological	ADJ
cana-1225	261	21	space	space	NOUN
cana-1225	261	22	(	(	PUNCT
cana-1225	261	23	x	x	NOUN
cana-1225	261	24	,	,	PUNCT
cana-1225	261	25	b𝑉𝜏	b𝑉𝜏	NUM
cana-1225	261	26	)	)	PUNCT
cana-1225	261	27	is	be	AUX
cana-1225	261	28	said	say	VERB
cana-1225	261	29	to	to	PART
cana-1225	261	30	be	be	AUX
cana-1225	261	31	bipolar	bipolar	ADJ
cana-1225	261	32	vague	vague	NOUN
cana-1225	261	33	𝛼𝑎	𝛼𝑎	ADP
cana-1225	261	34	𝑇1/2(b𝑉𝛼𝑎𝑇1/2	𝑇1/2(b𝑉𝛼𝑎𝑇1/2	NOUN
cana-1225	261	35	)	)	PUNCT
cana-1225	261	36	space	space	NOUN
cana-1225	261	37	if	if	SCONJ
cana-1225	261	38	every	every	DET
cana-1225	261	39	bipolar	bipolar	ADJ
cana-1225	261	40	vague	vague	NOUN
cana-1225	261	41	𝛼	𝛼	PRON
cana-1225	261	42	generalized	generalize	VERB
cana-1225	261	43	closed	close	VERB
cana-1225	261	44	set	set	VERB
cana-1225	261	45	in	in	ADP
cana-1225	261	46	x	x	PRON
cana-1225	261	47	is	be	AUX
cana-1225	261	48	a	a	DET
cana-1225	261	49	bipolar	bipolar	ADJ
cana-1225	261	50	vague	vague	NOUN
cana-1225	261	51	closed	close	VERB
cana-1225	261	52	set	set	VERB
cana-1225	261	53	in	in	ADP
cana-1225	261	54	x.	x.	NOUN
cana-1225	261	55	definition	definition	NOUN
cana-1225	261	56	3.19	3.19	NUM
cana-1225	261	57	:	:	PUNCT
cana-1225	261	58	a	a	DET
cana-1225	261	59	bipolar	bipolar	ADJ
cana-1225	261	60	vague	vague	ADJ
cana-1225	261	61	topological	topological	ADJ
cana-1225	261	62	space	space	NOUN
cana-1225	261	63	(	(	PUNCT
cana-1225	261	64	x	x	NOUN
cana-1225	261	65	,	,	PUNCT
cana-1225	261	66	b𝑉𝜏	b𝑉𝜏	NUM
cana-1225	261	67	)	)	PUNCT
cana-1225	261	68	is	be	AUX
cana-1225	261	69	said	say	VERB
cana-1225	261	70	to	to	PART
cana-1225	261	71	be	be	AUX
cana-1225	261	72	bipolar	bipolar	ADJ
cana-1225	261	73	vague	vague	ADJ
cana-1225	261	74	𝛼𝑏	𝛼𝑏	NOUN
cana-1225	261	75	𝑇1/2(b𝑉𝛼𝑏𝑇1/2	𝑇1/2(b𝑉𝛼𝑏𝑇1/2	NOUN
cana-1225	261	76	)	)	PUNCT
cana-1225	261	77	space	space	NOUN
cana-1225	261	78	if	if	SCONJ
cana-1225	261	79	every	every	DET
cana-1225	261	80	bipolar	bipolar	ADJ
cana-1225	261	81	vague	vague	NOUN
cana-1225	261	82	𝛼	𝛼	PRON
cana-1225	261	83	generalized	generalize	VERB
cana-1225	261	84	closed	close	VERB
cana-1225	261	85	set	set	VERB
cana-1225	261	86	in	in	ADP
cana-1225	261	87	x	x	PRON
cana-1225	261	88	is	be	AUX
cana-1225	261	89	a	a	DET
cana-1225	261	90	bipolar	bipolar	ADJ
cana-1225	261	91	vague	vague	ADJ
cana-1225	261	92	generalized	generalize	VERB
cana-1225	261	93	closed	close	VERB
cana-1225	261	94	set	set	VERB
cana-1225	261	95	in	in	ADP
cana-1225	261	96	x.	x.	NOUN
cana-1225	261	97	proposition	proposition	NOUN
cana-1225	261	98	3.20	3.20	NUM
cana-1225	261	99	:	:	PUNCT
cana-1225	261	100	let	let	VERB
cana-1225	261	101	𝑓	𝑓	PRON
cana-1225	261	102	:	:	PUNCT
cana-1225	261	103	(	(	PUNCT
cana-1225	261	104	x	x	NOUN
cana-1225	261	105	,	,	PUNCT
cana-1225	261	106	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	261	107	)	)	PUNCT
cana-1225	261	108	→	→	SYM
cana-1225	261	109	(	(	PUNCT
cana-1225	261	110	y	y	NOUN
cana-1225	261	111	,	,	PUNCT
cana-1225	261	112	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	261	113	)	)	PUNCT
cana-1225	261	114	be	be	VERB
cana-1225	261	115	a	a	DET
cana-1225	261	116	bipolar	bipolar	ADJ
cana-1225	261	117	vague	vague	NOUN
cana-1225	261	118	𝛼	𝛼	ADP
cana-1225	261	119	generalized	generalize	VERB
cana-1225	261	120	continuous	continuous	ADJ
cana-1225	261	121	mapping	mapping	NOUN
cana-1225	261	122	,	,	PUNCT
cana-1225	261	123	then	then	ADV
cana-1225	261	124	𝑓	𝑓	PRON
cana-1225	261	125	is	be	AUX
cana-1225	261	126	a	a	DET
cana-1225	261	127	bipolar	bipolar	ADJ
cana-1225	261	128	vague	vague	ADJ
cana-1225	261	129	continuous	continuous	ADJ
cana-1225	261	130	mapping	mapping	NOUN
cana-1225	261	131	,	,	PUNCT
cana-1225	261	132	if	if	SCONJ
cana-1225	261	133	x	x	PRON
cana-1225	261	134	is	be	AUX
cana-1225	261	135	a	a	DET
cana-1225	261	136	b𝑉𝛼𝑎𝑇1/2	b𝑉𝛼𝑎𝑇1/2	NOUN
cana-1225	261	137	space	space	NOUN
cana-1225	261	138	.	.	PUNCT
cana-1225	262	1	proof	proof	NOUN
cana-1225	262	2	:	:	PUNCT
cana-1225	262	3	let	let	VERB
cana-1225	262	4	a	a	PRON
cana-1225	262	5	be	be	AUX
cana-1225	262	6	a	a	DET
cana-1225	262	7	bipolar	bipolar	ADJ
cana-1225	262	8	vague	vague	NOUN
cana-1225	262	9	closed	close	VERB
cana-1225	262	10	set	set	VERB
cana-1225	262	11	in	in	ADP
cana-1225	262	12	y.	y.	PROPN
cana-1225	262	13	then	then	ADV
cana-1225	262	14	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	262	15	)	)	PUNCT
cana-1225	262	16	is	be	AUX
cana-1225	262	17	a	a	DET
cana-1225	262	18	bipolar	bipolar	ADJ
cana-1225	262	19	vague	vague	NOUN
cana-1225	262	20	𝛼	𝛼	DET
cana-1225	262	21	generalized	generalize	VERB
cana-1225	262	22	closed	close	VERB
cana-1225	262	23	set	set	VERB
cana-1225	262	24	in	in	ADP
cana-1225	262	25	x	x	NOUN
cana-1225	262	26	,	,	PUNCT
cana-1225	262	27	by	by	ADP
cana-1225	262	28	hypothesis	hypothesis	NOUN
cana-1225	262	29	.	.	PUNCT
cana-1225	263	1	since	since	SCONJ
cana-1225	263	2	x	x	PRON
cana-1225	263	3	is	be	AUX
cana-1225	263	4	a	a	DET
cana-1225	263	5	b𝑉𝛼𝑎𝑇1/2	b𝑉𝛼𝑎𝑇1/2	NOUN
cana-1225	263	6	space	space	NOUN
cana-1225	263	7	,	,	PUNCT
cana-1225	263	8	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	263	9	)	)	PUNCT
cana-1225	263	10	is	be	AUX
cana-1225	263	11	a	a	DET
cana-1225	263	12	bipolar	bipolar	ADJ
cana-1225	263	13	vague	vague	NOUN
cana-1225	263	14	closed	close	VERB
cana-1225	263	15	set	set	VERB
cana-1225	263	16	in	in	ADP
cana-1225	263	17	x.	x.	NOUN
cana-1225	263	18	hence	hence	ADV
cana-1225	263	19	𝑓	𝑓	PROPN
cana-1225	263	20	is	be	AUX
cana-1225	263	21	a	a	DET
cana-1225	263	22	bipolar	bipolar	ADJ
cana-1225	263	23	vague	vague	ADJ
cana-1225	263	24	continuous	continuous	ADJ
cana-1225	263	25	mapping	mapping	NOUN
cana-1225	263	26	.	.	PUNCT
cana-1225	264	1	proposition	proposition	NOUN
cana-1225	264	2	3.21	3.21	NUM
cana-1225	264	3	:	:	PUNCT
cana-1225	264	4	let	let	VERB
cana-1225	264	5	𝑓	𝑓	PRON
cana-1225	264	6	:	:	PUNCT
cana-1225	264	7	(	(	PUNCT
cana-1225	264	8	x	x	NOUN
cana-1225	264	9	,	,	PUNCT
cana-1225	264	10	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	264	11	)	)	PUNCT
cana-1225	264	12	→	→	SYM
cana-1225	264	13	(	(	PUNCT
cana-1225	264	14	y	y	NOUN
cana-1225	264	15	,	,	PUNCT
cana-1225	264	16	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	264	17	)	)	PUNCT
cana-1225	264	18	be	be	VERB
cana-1225	264	19	a	a	DET
cana-1225	264	20	bipolar	bipolar	ADJ
cana-1225	264	21	vague	vague	NOUN
cana-1225	264	22	𝛼	𝛼	ADP
cana-1225	264	23	generalized	generalize	VERB
cana-1225	264	24	continuous	continuous	ADJ
cana-1225	264	25	mapping	mapping	NOUN
cana-1225	264	26	,	,	PUNCT
cana-1225	264	27	then	then	ADV
cana-1225	264	28	𝑓	𝑓	PRON
cana-1225	264	29	is	be	AUX
cana-1225	264	30	a	a	DET
cana-1225	264	31	bipolar	bipolar	ADJ
cana-1225	264	32	vague	vague	NOUN
cana-1225	264	33	generalized	generalize	VERB
cana-1225	264	34	continuous	continuous	ADJ
cana-1225	264	35	mapping	mapping	NOUN
cana-1225	264	36	,	,	PUNCT
cana-1225	264	37	if	if	SCONJ
cana-1225	264	38	x	x	PRON
cana-1225	264	39	is	be	AUX
cana-1225	264	40	a	a	DET
cana-1225	264	41	b𝑉𝛼𝑏𝑇1/2	b𝑉𝛼𝑏𝑇1/2	NOUN
cana-1225	264	42	space	space	NOUN
cana-1225	264	43	.	.	PUNCT
cana-1225	265	1	communications	communication	NOUN
cana-1225	265	2	on	on	ADP
cana-1225	265	3	applied	apply	VERB
cana-1225	265	4	nonlinear	nonlinear	ADJ
cana-1225	265	5	analysis	analysis	NOUN
cana-1225	265	6	issn	issn	NOUN
cana-1225	265	7	:	:	PUNCT
cana-1225	265	8	1074	1074	NUM
cana-1225	265	9	-	-	PUNCT
cana-1225	265	10	133x	133x	NUM
cana-1225	265	11	vol	vol	NOUN
cana-1225	265	12	31	31	NUM
cana-1225	265	13	no	no	NOUN
cana-1225	265	14	.	.	PUNCT
cana-1225	266	1	6s	6s	NUM
cana-1225	266	2	(	(	PUNCT
cana-1225	266	3	2024	2024	NUM
cana-1225	266	4	)	)	PUNCT
cana-1225	266	5	326	326	NUM
cana-1225	267	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1225	267	2	proof	proof	NOUN
cana-1225	267	3	:	:	PUNCT
cana-1225	267	4	let	let	VERB
cana-1225	267	5	a	a	PRON
cana-1225	267	6	be	be	AUX
cana-1225	267	7	a	a	DET
cana-1225	267	8	bipolar	bipolar	ADJ
cana-1225	267	9	vague	vague	NOUN
cana-1225	267	10	closed	close	VERB
cana-1225	267	11	set	set	VERB
cana-1225	267	12	in	in	ADP
cana-1225	267	13	y.	y.	PROPN
cana-1225	267	14	then	then	ADV
cana-1225	267	15	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	267	16	)	)	PUNCT
cana-1225	267	17	is	be	AUX
cana-1225	267	18	a	a	DET
cana-1225	267	19	bipolar	bipolar	ADJ
cana-1225	267	20	vague	vague	NOUN
cana-1225	267	21	𝛼	𝛼	DET
cana-1225	267	22	generalized	generalize	VERB
cana-1225	267	23	closed	close	VERB
cana-1225	267	24	set	set	VERB
cana-1225	267	25	in	in	ADP
cana-1225	267	26	x	x	NOUN
cana-1225	267	27	,	,	PUNCT
cana-1225	267	28	by	by	ADP
cana-1225	267	29	hypothesis	hypothesis	NOUN
cana-1225	267	30	.	.	PUNCT
cana-1225	268	1	since	since	SCONJ
cana-1225	268	2	x	x	PRON
cana-1225	268	3	is	be	AUX
cana-1225	268	4	a	a	DET
cana-1225	268	5	b𝑉𝛼𝑏𝑇1/2	b𝑉𝛼𝑏𝑇1/2	NOUN
cana-1225	268	6	space	space	NOUN
cana-1225	268	7	,	,	PUNCT
cana-1225	268	8	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	268	9	)	)	PUNCT
cana-1225	268	10	is	be	AUX
cana-1225	268	11	a	a	DET
cana-1225	268	12	bipolar	bipolar	ADJ
cana-1225	268	13	vague	vague	ADJ
cana-1225	268	14	generalized	generalize	VERB
cana-1225	268	15	closed	close	VERB
cana-1225	268	16	set	set	VERB
cana-1225	268	17	in	in	ADP
cana-1225	268	18	x.	x.	NOUN
cana-1225	268	19	hence	hence	ADV
cana-1225	268	20	𝑓	𝑓	PROPN
cana-1225	268	21	is	be	AUX
cana-1225	268	22	a	a	DET
cana-1225	268	23	bipolar	bipolar	ADJ
cana-1225	268	24	vague	vague	NOUN
cana-1225	268	25	generalized	generalize	VERB
cana-1225	268	26	continuous	continuous	ADJ
cana-1225	268	27	mapping	mapping	NOUN
cana-1225	268	28	.	.	PUNCT
cana-1225	269	1	proposition	proposition	NOUN
cana-1225	269	2	3.22	3.22	NUM
cana-1225	269	3	:	:	PUNCT
cana-1225	269	4	let	let	VERB
cana-1225	269	5	𝑓	𝑓	PRON
cana-1225	269	6	:	:	PUNCT
cana-1225	269	7	(	(	PUNCT
cana-1225	269	8	x	x	NOUN
cana-1225	269	9	,	,	PUNCT
cana-1225	269	10	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	269	11	)	)	PUNCT
cana-1225	269	12	→	→	SYM
cana-1225	269	13	(	(	PUNCT
cana-1225	269	14	y	y	NOUN
cana-1225	269	15	,	,	PUNCT
cana-1225	269	16	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	269	17	)	)	PUNCT
cana-1225	269	18	be	be	VERB
cana-1225	269	19	a	a	DET
cana-1225	269	20	mapping	mapping	NOUN
cana-1225	269	21	from	from	ADP
cana-1225	269	22	a	a	DET
cana-1225	269	23	bipolar	bipolar	ADJ
cana-1225	269	24	vague	vague	ADJ
cana-1225	269	25	topological	topological	ADJ
cana-1225	269	26	space	space	NOUN
cana-1225	269	27	x	x	PUNCT
cana-1225	269	28	into	into	ADP
cana-1225	269	29	a	a	DET
cana-1225	269	30	bipolar	bipolar	ADJ
cana-1225	269	31	vague	vague	ADJ
cana-1225	269	32	topological	topological	ADJ
cana-1225	269	33	space	space	NOUN
cana-1225	270	1	y.	y.	NOUN
cana-1225	270	2	then	then	ADV
cana-1225	270	3	the	the	DET
cana-1225	270	4	following	follow	VERB
cana-1225	270	5	conditions	condition	NOUN
cana-1225	270	6	are	be	AUX
cana-1225	270	7	equivalent	equivalent	ADJ
cana-1225	270	8	if	if	SCONJ
cana-1225	270	9	x	x	PRON
cana-1225	270	10	is	be	AUX
cana-1225	270	11	a	a	DET
cana-1225	270	12	b𝑉𝛼𝑎𝑇1/2	b𝑉𝛼𝑎𝑇1/2	NOUN
cana-1225	270	13	space	space	NOUN
cana-1225	270	14	:	:	PUNCT
cana-1225	270	15	(	(	PUNCT
cana-1225	270	16	i	i	NOUN
cana-1225	270	17	)	)	PUNCT
cana-1225	270	18	𝑓	𝑓	PRON
cana-1225	270	19	is	be	AUX
cana-1225	270	20	a	a	DET
cana-1225	270	21	bipolar	bipolar	ADJ
cana-1225	270	22	vague	vague	NOUN
cana-1225	270	23	𝛼	𝛼	ADP
cana-1225	270	24	generalized	generalize	VERB
cana-1225	270	25	continuous	continuous	ADJ
cana-1225	270	26	mapping	mapping	NOUN
cana-1225	270	27	.	.	PUNCT
cana-1225	271	1	(	(	PUNCT
cana-1225	271	2	ii	ii	NOUN
cana-1225	271	3	)	)	PUNCT
cana-1225	271	4	if	if	SCONJ
cana-1225	271	5	b	b	PROPN
cana-1225	271	6	is	be	AUX
cana-1225	271	7	a	a	DET
cana-1225	271	8	bipolar	bipolar	ADJ
cana-1225	271	9	vague	vague	ADJ
cana-1225	271	10	open	open	ADJ
cana-1225	271	11	set	set	NOUN
cana-1225	271	12	in	in	ADP
cana-1225	271	13	y	y	PROPN
cana-1225	271	14	,	,	PUNCT
cana-1225	271	15	𝑓−1(b	𝑓−1(b	PROPN
cana-1225	271	16	)	)	PUNCT
cana-1225	271	17	is	be	AUX
cana-1225	271	18	a	a	DET
cana-1225	271	19	bipolar	bipolar	ADJ
cana-1225	271	20	vague	vague	NOUN
cana-1225	271	21	𝛼	𝛼	DET
cana-1225	271	22	generalized	generalize	VERB
cana-1225	271	23	closed	close	VERB
cana-1225	271	24	set	set	VERB
cana-1225	271	25	in	in	ADP
cana-1225	271	26	x.	x.	PROPN
cana-1225	271	27	(	(	PUNCT
cana-1225	271	28	iii	iii	NOUN
cana-1225	271	29	)	)	PUNCT
cana-1225	271	30	𝑓−1(bvint(b	𝑓−1(bvint(b	PROPN
cana-1225	271	31	)	)	PUNCT
cana-1225	271	32	)	)	PUNCT
cana-1225	272	1	⊆	⊆	NUM
cana-1225	272	2	bvint(bvcl(bvint(𝑓−1(b	bvint(bvcl(bvint(𝑓−1(b	NOUN
cana-1225	272	3	)	)	PUNCT
cana-1225	272	4	)	)	PUNCT
cana-1225	272	5	)	)	PUNCT
cana-1225	272	6	)	)	PUNCT
cana-1225	273	1	for	for	ADP
cana-1225	273	2	every	every	DET
cana-1225	273	3	bipolar	bipolar	ADJ
cana-1225	273	4	vague	vague	ADJ
cana-1225	273	5	set	set	NOUN
cana-1225	273	6	b	b	NOUN
cana-1225	273	7	in	in	ADP
cana-1225	273	8	y.	y.	PROPN
cana-1225	273	9	proof	proof	NOUN
cana-1225	273	10	:	:	PUNCT
cana-1225	273	11	(	(	PUNCT
cana-1225	273	12	i	i	NOUN
cana-1225	273	13	)	)	PUNCT
cana-1225	273	14	⟹	⟹	PROPN
cana-1225	273	15	(	(	PUNCT
cana-1225	273	16	ii	ii	NOUN
cana-1225	273	17	)	)	PUNCT
cana-1225	273	18	is	be	AUX
cana-1225	273	19	obviously	obviously	ADV
cana-1225	273	20	true	true	ADJ
cana-1225	273	21	.	.	PUNCT
cana-1225	274	1	(	(	PUNCT
cana-1225	274	2	ii	ii	NOUN
cana-1225	274	3	)	)	PUNCT
cana-1225	274	4	⟹	⟹	PROPN
cana-1225	274	5	(	(	PUNCT
cana-1225	274	6	iii	iii	NOUN
cana-1225	274	7	)	)	PUNCT
cana-1225	274	8	.	.	PUNCT
cana-1225	275	1	let	let	VERB
cana-1225	275	2	b	b	X
cana-1225	275	3	be	be	AUX
cana-1225	275	4	any	any	DET
cana-1225	275	5	bipolar	bipolar	ADJ
cana-1225	275	6	vague	vague	ADJ
cana-1225	275	7	open	open	ADJ
cana-1225	275	8	set	set	VERB
cana-1225	275	9	in	in	ADP
cana-1225	275	10	y.	y.	NOUN
cana-1225	275	11	the	the	DET
cana-1225	275	12	bvint(b	bvint(b	NOUN
cana-1225	275	13	)	)	PUNCT
cana-1225	275	14	is	be	AUX
cana-1225	275	15	a	a	DET
cana-1225	275	16	bipolar	bipolar	ADJ
cana-1225	275	17	vague	vague	ADJ
cana-1225	275	18	open	open	ADJ
cana-1225	275	19	set	set	VERB
cana-1225	275	20	in	in	ADP
cana-1225	275	21	y.	y.	PROPN
cana-1225	275	22	then	then	ADV
cana-1225	275	23	𝑓−1(bvint(b	𝑓−1(bvint(b	PROPN
cana-1225	275	24	)	)	PUNCT
cana-1225	275	25	)	)	PUNCT
cana-1225	275	26	is	be	AUX
cana-1225	275	27	a	a	DET
cana-1225	275	28	bipolar	bipolar	ADJ
cana-1225	275	29	vague	vague	NOUN
cana-1225	275	30	𝛼	𝛼	ADP
cana-1225	275	31	generalized	generalize	VERB
cana-1225	275	32	open	open	ADJ
cana-1225	275	33	set	set	VERB
cana-1225	275	34	in	in	ADP
cana-1225	275	35	x.	x.	NOUN
cana-1225	275	36	since	since	SCONJ
cana-1225	275	37	x	x	PRON
cana-1225	275	38	is	be	AUX
cana-1225	275	39	a	a	DET
cana-1225	275	40	b𝑉𝛼𝑎𝑇1/2	b𝑉𝛼𝑎𝑇1/2	NOUN
cana-1225	275	41	space	space	NOUN
cana-1225	275	42	,	,	PUNCT
cana-1225	275	43	𝑓−1(bvint(b	𝑓−1(bvint(b	PROPN
cana-1225	275	44	)	)	PUNCT
cana-1225	275	45	)	)	PUNCT
cana-1225	275	46	is	be	AUX
cana-1225	275	47	a	a	DET
cana-1225	275	48	bipolar	bipolar	ADJ
cana-1225	275	49	vague	vague	ADJ
cana-1225	275	50	open	open	ADJ
cana-1225	275	51	set	set	VERB
cana-1225	275	52	in	in	ADP
cana-1225	275	53	x.	x.	NOUN
cana-1225	275	54	therefore	therefore	ADV
cana-1225	275	55	,	,	PUNCT
cana-1225	275	56	𝑓−1(bvint(b	𝑓−1(bvint(b	PROPN
cana-1225	275	57	)	)	PUNCT
cana-1225	275	58	)	)	PUNCT
cana-1225	276	1	=	=	SYM
cana-1225	276	2	bvint(𝑓−1(bvint(b	bvint(𝑓−1(bvint(b	NOUN
cana-1225	276	3	)	)	PUNCT
cana-1225	276	4	)	)	PUNCT
cana-1225	276	5	)	)	PUNCT
cana-1225	277	1	⊆	⊆	NUM
cana-1225	277	2	bvint(bvcl(bvint(𝑓−1(b	bvint(bvcl(bvint(𝑓−1(b	NOUN
cana-1225	277	3	)	)	PUNCT
cana-1225	277	4	)	)	PUNCT
cana-1225	277	5	)	)	PUNCT
cana-1225	277	6	)	)	PUNCT
cana-1225	277	7	.	.	PUNCT
cana-1225	278	1	(	(	PUNCT
cana-1225	278	2	iii	iii	X
cana-1225	278	3	)	)	PUNCT
cana-1225	278	4	⟹	⟹	PUNCT
cana-1225	279	1	(	(	PUNCT
cana-1225	279	2	i	i	NOUN
cana-1225	279	3	)	)	PUNCT
cana-1225	279	4	.	.	PUNCT
cana-1225	280	1	let	let	VERB
cana-1225	280	2	b	b	X
cana-1225	280	3	be	be	AUX
cana-1225	280	4	a	a	DET
cana-1225	280	5	bipolar	bipolar	ADJ
cana-1225	280	6	vague	vague	NOUN
cana-1225	280	7	closed	close	VERB
cana-1225	280	8	set	set	VERB
cana-1225	280	9	in	in	ADP
cana-1225	280	10	y.	y.	PROPN
cana-1225	280	11	then	then	ADV
cana-1225	280	12	its	its	PRON
cana-1225	280	13	complement	complement	NOUN
cana-1225	280	14	bc	bc	PROPN
cana-1225	280	15	is	be	AUX
cana-1225	280	16	a	a	DET
cana-1225	280	17	bipolar	bipolar	ADJ
cana-1225	280	18	vague	vague	ADJ
cana-1225	280	19	open	open	ADJ
cana-1225	280	20	set	set	VERB
cana-1225	280	21	in	in	ADP
cana-1225	280	22	y.	y.	NOUN
cana-1225	280	23	by	by	ADP
cana-1225	280	24	hypothesis	hypothesis	NOUN
cana-1225	280	25	,	,	PUNCT
cana-1225	280	26	𝑓−1(bvint(bc	𝑓−1(bvint(bc	ADJ
cana-1225	280	27	)	)	PUNCT
cana-1225	280	28	)	)	PUNCT
cana-1225	281	1	⊆	⊆	NUM
cana-1225	281	2	bvint(bvcl(bvint(𝑓−1(bc	bvint(bvcl(bvint(𝑓−1(bc	PROPN
cana-1225	281	3	)	)	PUNCT
cana-1225	281	4	)	)	PUNCT
cana-1225	281	5	)	)	PUNCT
cana-1225	281	6	)	)	PUNCT
cana-1225	281	7	.	.	PUNCT
cana-1225	282	1	this	this	PRON
cana-1225	282	2	implies	imply	VERB
cana-1225	282	3	𝑓−1	𝑓−1	NUM
cana-1225	282	4	(	(	PUNCT
cana-1225	282	5	bc	bc	PROPN
cana-1225	282	6	)	)	PUNCT
cana-1225	282	7	⊆	⊆	NUM
cana-1225	282	8	bvint(bvcl(bvint(𝑓−1(bc	bvint(bvcl(bvint(𝑓−1(bc	PROPN
cana-1225	282	9	)	)	PUNCT
cana-1225	282	10	)	)	PUNCT
cana-1225	282	11	)	)	PUNCT
cana-1225	282	12	)	)	PUNCT
cana-1225	282	13	.	.	PUNCT
cana-1225	283	1	hence	hence	ADV
cana-1225	283	2	𝑓−1	𝑓−1	NUM
cana-1225	283	3	(	(	PUNCT
cana-1225	283	4	bc	bc	PROPN
cana-1225	283	5	)	)	PUNCT
cana-1225	283	6	is	be	AUX
cana-1225	283	7	a	a	DET
cana-1225	283	8	bipolar	bipolar	ADJ
cana-1225	283	9	vague	vague	ADJ
cana-1225	283	10	𝛼-open	𝛼-open	NOUN
cana-1225	283	11	set	set	NOUN
cana-1225	283	12	in	in	ADP
cana-1225	283	13	x.	x.	NOUN
cana-1225	283	14	since	since	SCONJ
cana-1225	283	15	every	every	DET
cana-1225	283	16	bipolar	bipolar	ADJ
cana-1225	283	17	vague	vague	ADJ
cana-1225	283	18	𝛼-open	𝛼-open	NOUN
cana-1225	283	19	set	set	NOUN
cana-1225	283	20	is	be	AUX
cana-1225	283	21	a	a	DET
cana-1225	283	22	bipolar	bipolar	ADJ
cana-1225	283	23	vague	vague	NOUN
cana-1225	283	24	𝛼	𝛼	ADP
cana-1225	283	25	generalized	generalize	VERB
cana-1225	283	26	open	open	ADJ
cana-1225	283	27	set	set	NOUN
cana-1225	283	28	,	,	PUNCT
cana-1225	283	29	𝑓−1	𝑓−1	PROPN
cana-1225	283	30	(	(	PUNCT
cana-1225	283	31	bc	bc	PROPN
cana-1225	283	32	)	)	PUNCT
cana-1225	283	33	is	be	AUX
cana-1225	283	34	a	a	DET
cana-1225	283	35	bipolar	bipolar	ADJ
cana-1225	283	36	vague	vague	NOUN
cana-1225	283	37	𝛼	𝛼	ADP
cana-1225	283	38	generalized	generalize	VERB
cana-1225	283	39	open	open	ADJ
cana-1225	283	40	set	set	NOUN
cana-1225	283	41	in	in	ADP
cana-1225	283	42	x.	x.	NOUN
cana-1225	283	43	therefore	therefore	ADV
cana-1225	283	44	,	,	PUNCT
cana-1225	283	45	𝑓−1(b	𝑓−1(b	PROPN
cana-1225	283	46	)	)	PUNCT
cana-1225	283	47	is	be	AUX
cana-1225	283	48	a	a	DET
cana-1225	283	49	bipolar	bipolar	ADJ
cana-1225	283	50	vague	vague	NOUN
cana-1225	284	1	𝛼	𝛼	DET
cana-1225	284	2	generalized	generalize	VERB
cana-1225	284	3	closed	close	VERB
cana-1225	284	4	set	set	VERB
cana-1225	284	5	in	in	ADP
cana-1225	284	6	x.	x.	NOUN
cana-1225	284	7	hence	hence	ADV
cana-1225	284	8	𝑓	𝑓	PROPN
cana-1225	284	9	is	be	AUX
cana-1225	284	10	a	a	DET
cana-1225	284	11	bipolar	bipolar	ADJ
cana-1225	284	12	vague	vague	NOUN
cana-1225	284	13	𝛼	𝛼	ADP
cana-1225	284	14	generalized	generalize	VERB
cana-1225	284	15	continuous	continuous	ADJ
cana-1225	284	16	mapping	mapping	NOUN
cana-1225	284	17	.	.	PUNCT
cana-1225	285	1	proposition	proposition	NOUN
cana-1225	285	2	3.23	3.23	NUM
cana-1225	285	3	:	:	PUNCT
cana-1225	285	4	let	let	VERB
cana-1225	285	5	𝑓	𝑓	PRON
cana-1225	285	6	:	:	PUNCT
cana-1225	285	7	(	(	PUNCT
cana-1225	285	8	x	x	NOUN
cana-1225	285	9	,	,	PUNCT
cana-1225	285	10	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	285	11	)	)	PUNCT
cana-1225	285	12	→	→	SYM
cana-1225	285	13	(	(	PUNCT
cana-1225	285	14	y	y	NOUN
cana-1225	285	15	,	,	PUNCT
cana-1225	285	16	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	285	17	)	)	PUNCT
cana-1225	285	18	be	be	VERB
cana-1225	285	19	a	a	DET
cana-1225	285	20	mapping	mapping	NOUN
cana-1225	285	21	from	from	ADP
cana-1225	285	22	a	a	DET
cana-1225	285	23	bipolar	bipolar	ADJ
cana-1225	285	24	vague	vague	ADJ
cana-1225	285	25	topological	topological	ADJ
cana-1225	285	26	space	space	NOUN
cana-1225	285	27	x	x	PUNCT
cana-1225	285	28	into	into	ADP
cana-1225	285	29	a	a	DET
cana-1225	285	30	bipolar	bipolar	ADJ
cana-1225	285	31	vague	vague	ADJ
cana-1225	285	32	topological	topological	ADJ
cana-1225	285	33	space	space	NOUN
cana-1225	286	1	y.	y.	NOUN
cana-1225	286	2	then	then	ADV
cana-1225	286	3	the	the	DET
cana-1225	286	4	following	follow	VERB
cana-1225	286	5	conditions	condition	NOUN
cana-1225	286	6	are	be	AUX
cana-1225	286	7	equivalent	equivalent	ADJ
cana-1225	286	8	if	if	SCONJ
cana-1225	286	9	x	x	PRON
cana-1225	286	10	is	be	AUX
cana-1225	286	11	a	a	DET
cana-1225	286	12	b𝑉𝛼𝑎𝑇1/2	b𝑉𝛼𝑎𝑇1/2	NOUN
cana-1225	286	13	space	space	NOUN
cana-1225	286	14	:	:	PUNCT
cana-1225	286	15	(	(	PUNCT
cana-1225	286	16	i	i	NOUN
cana-1225	286	17	)	)	PUNCT
cana-1225	286	18	𝑓	𝑓	PRON
cana-1225	286	19	is	be	AUX
cana-1225	286	20	a	a	DET
cana-1225	286	21	bipolar	bipolar	ADJ
cana-1225	286	22	vague	vague	NOUN
cana-1225	286	23	𝛼	𝛼	ADP
cana-1225	286	24	generalized	generalize	VERB
cana-1225	286	25	continuous	continuous	ADJ
cana-1225	286	26	mapping	mapping	NOUN
cana-1225	286	27	.	.	PUNCT
cana-1225	287	1	(	(	PUNCT
cana-1225	287	2	ii	ii	NOUN
cana-1225	287	3	)	)	PUNCT
cana-1225	287	4	if	if	SCONJ
cana-1225	287	5	𝑓−1(b	𝑓−1(b	PROPN
cana-1225	287	6	)	)	PUNCT
cana-1225	287	7	is	be	AUX
cana-1225	287	8	a	a	DET
cana-1225	287	9	bipolar	bipolar	ADJ
cana-1225	287	10	vague	vague	NOUN
cana-1225	287	11	𝛼	𝛼	DET
cana-1225	287	12	generalized	generalize	VERB
cana-1225	287	13	closed	close	VERB
cana-1225	287	14	set	set	VERB
cana-1225	287	15	in	in	ADP
cana-1225	287	16	x	x	NOUN
cana-1225	287	17	,	,	PUNCT
cana-1225	287	18	for	for	SCONJ
cana-1225	287	19	every	every	DET
cana-1225	287	20	bipolar	bipolar	ADJ
cana-1225	287	21	vague	vague	NOUN
cana-1225	287	22	closed	close	VERB
cana-1225	287	23	set	set	ADJ
cana-1225	287	24	b	b	NOUN
cana-1225	287	25	in	in	ADP
cana-1225	287	26	y.	y.	PROPN
cana-1225	287	27	(	(	PUNCT
cana-1225	287	28	iii	iii	PROPN
cana-1225	287	29	)	)	PUNCT
cana-1225	287	30	bvcl(bvint(bvcl(𝑓−1(a	bvcl(bvint(bvcl(𝑓−1(a	ADV
cana-1225	287	31	)	)	PUNCT
cana-1225	287	32	)	)	PUNCT
cana-1225	287	33	)	)	PUNCT
cana-1225	287	34	)	)	PUNCT
cana-1225	288	1	⊆	⊆	X
cana-1225	288	2	𝑓−1(bvcl(a	𝑓−1(bvcl(a	PROPN
cana-1225	288	3	)	)	PUNCT
cana-1225	288	4	)	)	PUNCT
cana-1225	288	5	for	for	ADP
cana-1225	288	6	every	every	DET
cana-1225	288	7	bipolar	bipolar	ADJ
cana-1225	288	8	vague	vague	ADJ
cana-1225	288	9	set	set	NOUN
cana-1225	288	10	b	b	NOUN
cana-1225	288	11	in	in	ADP
cana-1225	288	12	y.	y.	PROPN
cana-1225	288	13	proof	proof	NOUN
cana-1225	288	14	:	:	PUNCT
cana-1225	288	15	(	(	PUNCT
cana-1225	288	16	i	i	NOUN
cana-1225	288	17	)	)	PUNCT
cana-1225	288	18	⟹	⟹	PROPN
cana-1225	288	19	(	(	PUNCT
cana-1225	288	20	ii	ii	NOUN
cana-1225	288	21	)	)	PUNCT
cana-1225	288	22	is	be	AUX
cana-1225	288	23	obviously	obviously	ADV
cana-1225	288	24	true	true	ADJ
cana-1225	288	25	.	.	PUNCT
cana-1225	289	1	(	(	PUNCT
cana-1225	289	2	ii	ii	NOUN
cana-1225	289	3	)	)	PUNCT
cana-1225	289	4	⟹	⟹	PROPN
cana-1225	289	5	(	(	PUNCT
cana-1225	289	6	iii	iii	NOUN
cana-1225	289	7	)	)	PUNCT
cana-1225	289	8	.	.	PUNCT
cana-1225	290	1	let	let	VERB
cana-1225	290	2	a	a	DET
cana-1225	290	3	be	be	AUX
cana-1225	290	4	any	any	DET
cana-1225	290	5	bipolar	bipolar	ADJ
cana-1225	290	6	vague	vague	NOUN
cana-1225	290	7	set	set	NOUN
cana-1225	290	8	in	in	ADP
cana-1225	290	9	y.	y.	PROPN
cana-1225	290	10	then	then	ADV
cana-1225	290	11	bvcl(a	bvcl(a	NUM
cana-1225	290	12	)	)	PUNCT
cana-1225	290	13	is	be	AUX
cana-1225	290	14	a	a	DET
cana-1225	290	15	bipolar	bipolar	ADJ
cana-1225	290	16	vague	vague	NOUN
cana-1225	290	17	closed	close	VERB
cana-1225	290	18	set	set	VERB
cana-1225	290	19	in	in	ADP
cana-1225	290	20	y.	y.	NOUN
cana-1225	290	21	by	by	ADP
cana-1225	290	22	hypothesis	hypothesis	NOUN
cana-1225	290	23	,	,	PUNCT
cana-1225	290	24	𝑓−1(bvcl(a	𝑓−1(bvcl(a	PROPN
cana-1225	290	25	)	)	PUNCT
cana-1225	290	26	)	)	PUNCT
cana-1225	290	27	is	be	AUX
cana-1225	290	28	a	a	DET
cana-1225	290	29	bipolar	bipolar	ADJ
cana-1225	290	30	vague	vague	NOUN
cana-1225	290	31	𝛼	𝛼	DET
cana-1225	290	32	generalized	generalize	VERB
cana-1225	290	33	closed	close	VERB
cana-1225	290	34	set	set	VERB
cana-1225	290	35	in	in	ADP
cana-1225	290	36	x.	x.	NOUN
cana-1225	290	37	since	since	SCONJ
cana-1225	290	38	x	x	PRON
cana-1225	290	39	is	be	AUX
cana-1225	290	40	a	a	DET
cana-1225	290	41	b𝑉𝛼𝑎𝑇1/2	b𝑉𝛼𝑎𝑇1/2	NOUN
cana-1225	290	42	space	space	NOUN
cana-1225	290	43	,	,	PUNCT
cana-1225	290	44	𝑓−1(bvcl(a	𝑓−1(bvcl(a	PROPN
cana-1225	290	45	)	)	PUNCT
cana-1225	290	46	)	)	PUNCT
cana-1225	290	47	is	be	AUX
cana-1225	290	48	a	a	DET
cana-1225	290	49	bipolar	bipolar	ADJ
cana-1225	290	50	vague	vague	NOUN
cana-1225	290	51	closed	close	VERB
cana-1225	290	52	set	set	VERB
cana-1225	290	53	in	in	ADP
cana-1225	290	54	x.	x.	NOUN
cana-1225	290	55	therefore	therefore	ADV
cana-1225	290	56	,	,	PUNCT
cana-1225	290	57	bvcl(𝑓−1(bvcl(a	bvcl(𝑓−1(bvcl(a	ADJ
cana-1225	290	58	)	)	PUNCT
cana-1225	290	59	)	)	PUNCT
cana-1225	290	60	)	)	PUNCT
cana-1225	291	1	=	=	SYM
cana-1225	291	2	𝑓−1(bvcl(a	𝑓−1(bvcl(a	PROPN
cana-1225	291	3	)	)	PUNCT
cana-1225	291	4	)	)	PUNCT
cana-1225	291	5	.	.	PUNCT
cana-1225	292	1	now	now	ADV
cana-1225	292	2	bvcl(bvint(bvcl((𝑓−1(a	bvcl(bvint(bvcl((𝑓−1(a	ADP
cana-1225	292	3	)	)	PUNCT
cana-1225	292	4	)	)	PUNCT
cana-1225	292	5	)	)	PUNCT
cana-1225	292	6	)	)	PUNCT
cana-1225	293	1	⊆	⊆	NUM
cana-1225	293	2	bvcl(bvint(bvcl((𝑓−1(bvcl(a	bvcl(bvint(bvcl((𝑓−1(bvcl(a	NOUN
cana-1225	293	3	)	)	PUNCT
cana-1225	293	4	)	)	PUNCT
cana-1225	293	5	)	)	PUNCT
cana-1225	293	6	)	)	PUNCT
cana-1225	294	1	⊆	⊆	X
cana-1225	294	2	𝑓−1(bvcl(a	𝑓−1(bvcl(a	PROPN
cana-1225	294	3	)	)	PUNCT
cana-1225	294	4	)	)	PUNCT
cana-1225	294	5	.	.	PUNCT
cana-1225	295	1	(	(	PUNCT
cana-1225	295	2	iii	iii	X
cana-1225	295	3	)	)	PUNCT
cana-1225	295	4	⟹	⟹	PUNCT
cana-1225	296	1	(	(	PUNCT
cana-1225	296	2	i	i	NOUN
cana-1225	296	3	)	)	PUNCT
cana-1225	296	4	.	.	PUNCT
cana-1225	297	1	let	let	VERB
cana-1225	297	2	a	a	DET
cana-1225	297	3	be	be	AUX
cana-1225	297	4	a	a	DET
cana-1225	297	5	bipolar	bipolar	ADJ
cana-1225	297	6	vague	vague	NOUN
cana-1225	297	7	closed	close	VERB
cana-1225	297	8	set	set	VERB
cana-1225	297	9	in	in	ADP
cana-1225	297	10	y.	y.	PROPN
cana-1225	297	11	then	then	ADV
cana-1225	297	12	by	by	ADP
cana-1225	297	13	hypothesis	hypothesis	NOUN
cana-1225	297	14	,	,	PUNCT
cana-1225	297	15	bvcl(bvint(bvcl(𝑓−1(bvcl(a	bvcl(bvint(bvcl(𝑓−1(bvcl(a	NOUN
cana-1225	297	16	)	)	PUNCT
cana-1225	297	17	)	)	PUNCT
cana-1225	297	18	)	)	PUNCT
cana-1225	297	19	)	)	PUNCT
cana-1225	297	20	)	)	PUNCT
cana-1225	298	1	⊆	⊆	X
cana-1225	298	2	𝑓−1(bvcl(a	𝑓−1(bvcl(a	PROPN
cana-1225	298	3	)	)	PUNCT
cana-1225	298	4	)	)	PUNCT
cana-1225	299	1	=	=	SYM
cana-1225	299	2	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	299	3	)	)	PUNCT
cana-1225	299	4	.	.	PUNCT
cana-1225	300	1	this	this	PRON
cana-1225	300	2	implies	imply	VERB
cana-1225	300	3	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	300	4	)	)	PUNCT
cana-1225	300	5	is	be	AUX
cana-1225	300	6	a	a	DET
cana-1225	300	7	bipolar	bipolar	ADJ
cana-1225	300	8	vague	vague	NOUN
cana-1225	300	9	𝛼-closed	𝛼-close	VERB
cana-1225	300	10	set	set	VERB
cana-1225	300	11	in	in	ADP
cana-1225	300	12	x	x	PUNCT
cana-1225	301	1	and	and	CCONJ
cana-1225	301	2	hence	hence	ADV
cana-1225	301	3	it	it	PRON
cana-1225	301	4	is	be	AUX
cana-1225	301	5	a	a	DET
cana-1225	301	6	bipolar	bipolar	ADJ
cana-1225	301	7	vague	vague	NOUN
cana-1225	301	8	𝛼	𝛼	DET
cana-1225	301	9	generalized	generalize	VERB
cana-1225	301	10	closed	close	VERB
cana-1225	301	11	set	set	VERB
cana-1225	301	12	in	in	ADP
cana-1225	301	13	x.	x.	NOUN
cana-1225	301	14	therefore	therefore	ADV
cana-1225	301	15	,	,	PUNCT
cana-1225	301	16	𝑓	𝑓	PRON
cana-1225	301	17	is	be	AUX
cana-1225	301	18	a	a	DET
cana-1225	301	19	bipolar	bipolar	ADJ
cana-1225	301	20	vague	vague	NOUN
cana-1225	301	21	𝛼	𝛼	ADP
cana-1225	301	22	generalized	generalize	VERB
cana-1225	301	23	continuous	continuous	ADJ
cana-1225	301	24	mapping	mapping	NOUN
cana-1225	301	25	.	.	PUNCT
cana-1225	302	1	communications	communication	NOUN
cana-1225	302	2	on	on	ADP
cana-1225	302	3	applied	apply	VERB
cana-1225	302	4	nonlinear	nonlinear	ADJ
cana-1225	302	5	analysis	analysis	NOUN
cana-1225	302	6	issn	issn	NOUN
cana-1225	302	7	:	:	PUNCT
cana-1225	302	8	1074	1074	NUM
cana-1225	302	9	-	-	PUNCT
cana-1225	302	10	133x	133x	NUM
cana-1225	302	11	vol	vol	NOUN
cana-1225	302	12	31	31	NUM
cana-1225	302	13	no	no	NOUN
cana-1225	302	14	.	.	PUNCT
cana-1225	303	1	6s	6s	NUM
cana-1225	303	2	(	(	PUNCT
cana-1225	303	3	2024	2024	NUM
cana-1225	303	4	)	)	PUNCT
cana-1225	303	5	327	327	NUM
cana-1225	303	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1225	303	7	4	4	X
cana-1225	303	8	.	.	X
cana-1225	303	9	bipolar	bipolar	ADJ
cana-1225	303	10	vague	vague	NOUN
cana-1225	303	11	𝜶	𝜶	ADP
cana-1225	303	12	generalized	generalize	VERB
cana-1225	303	13	irresolute	irresolute	ADJ
cana-1225	303	14	mappings	mapping	NOUN
cana-1225	303	15	in	in	ADP
cana-1225	303	16	topological	topological	ADJ
cana-1225	303	17	spaces	space	NOUN
cana-1225	303	18	in	in	ADP
cana-1225	303	19	this	this	DET
cana-1225	303	20	section	section	NOUN
cana-1225	303	21	we	we	PRON
cana-1225	303	22	have	have	AUX
cana-1225	303	23	introduced	introduce	VERB
cana-1225	303	24	bipolar	bipolar	ADJ
cana-1225	303	25	vague	vague	NOUN
cana-1225	303	26	𝛼	𝛼	ADP
cana-1225	303	27	generalized	generalize	VERB
cana-1225	303	28	irresolute	irresolute	ADJ
cana-1225	303	29	mappings	mapping	NOUN
cana-1225	303	30	and	and	CCONJ
cana-1225	303	31	studied	study	VERB
cana-1225	303	32	some	some	PRON
cana-1225	303	33	of	of	ADP
cana-1225	303	34	their	their	PRON
cana-1225	303	35	properties	property	NOUN
cana-1225	303	36	.	.	PUNCT
cana-1225	304	1	definition	definition	NOUN
cana-1225	304	2	4.1	4.1	NUM
cana-1225	304	3	:	:	PUNCT
cana-1225	304	4	a	a	DET
cana-1225	304	5	mapping	mapping	NOUN
cana-1225	304	6	𝑓	𝑓	X
cana-1225	304	7	:	:	PUNCT
cana-1225	304	8	(	(	PUNCT
cana-1225	304	9	x	x	NOUN
cana-1225	304	10	,	,	PUNCT
cana-1225	304	11	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	304	12	)	)	PUNCT
cana-1225	304	13	→	→	SYM
cana-1225	304	14	(	(	PUNCT
cana-1225	304	15	y	y	NOUN
cana-1225	304	16	,	,	PUNCT
cana-1225	304	17	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	304	18	)	)	PUNCT
cana-1225	304	19	is	be	AUX
cana-1225	304	20	called	call	VERB
cana-1225	304	21	a	a	DET
cana-1225	304	22	bipolar	bipolar	ADJ
cana-1225	304	23	vague	vague	NOUN
cana-1225	304	24	𝛼	𝛼	DET
cana-1225	304	25	generalized	generalized	ADJ
cana-1225	304	26	irresolute	irresolute	ADJ
cana-1225	304	27	mapping	mapping	NOUN
cana-1225	304	28	if	if	SCONJ
cana-1225	304	29	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	304	30	)	)	PUNCT
cana-1225	304	31	is	be	AUX
cana-1225	304	32	a	a	DET
cana-1225	304	33	bipolar	bipolar	ADJ
cana-1225	304	34	vague	vague	NOUN
cana-1225	305	1	𝛼	𝛼	DET
cana-1225	305	2	generalized	generalize	VERB
cana-1225	305	3	closed	close	VERB
cana-1225	305	4	set	set	VERB
cana-1225	305	5	in	in	ADP
cana-1225	305	6	(	(	PUNCT
cana-1225	305	7	x	x	NOUN
cana-1225	305	8	,	,	PUNCT
cana-1225	305	9	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	305	10	)	)	PUNCT
cana-1225	305	11	for	for	ADP
cana-1225	305	12	every	every	DET
cana-1225	305	13	bipolar	bipolar	ADJ
cana-1225	305	14	vague	vague	NOUN
cana-1225	305	15	𝛼	𝛼	DET
cana-1225	305	16	generalized	generalize	VERB
cana-1225	305	17	closed	close	VERB
cana-1225	305	18	set	set	VERB
cana-1225	305	19	a	a	PRON
cana-1225	305	20	of	of	ADP
cana-1225	305	21	(	(	PUNCT
cana-1225	305	22	y	y	NOUN
cana-1225	305	23	,	,	PUNCT
cana-1225	305	24	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	305	25	)	)	PUNCT
cana-1225	305	26	.	.	PUNCT
cana-1225	306	1	proposition	proposition	NOUN
cana-1225	306	2	4.2	4.2	NUM
cana-1225	306	3	:	:	PUNCT
cana-1225	306	4	let	let	VERB
cana-1225	306	5	𝑓	𝑓	PRON
cana-1225	306	6	:	:	PUNCT
cana-1225	306	7	(	(	PUNCT
cana-1225	306	8	x	x	NOUN
cana-1225	306	9	,	,	PUNCT
cana-1225	306	10	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	306	11	)	)	PUNCT
cana-1225	306	12	→	→	SYM
cana-1225	306	13	(	(	PUNCT
cana-1225	306	14	y	y	NOUN
cana-1225	306	15	,	,	PUNCT
cana-1225	306	16	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	306	17	)	)	PUNCT
cana-1225	306	18	be	be	VERB
cana-1225	306	19	a	a	DET
cana-1225	306	20	bipolar	bipolar	ADJ
cana-1225	306	21	vague	vague	NOUN
cana-1225	306	22	𝛼	𝛼	ADP
cana-1225	306	23	generalized	generalized	ADJ
cana-1225	306	24	irresolute	irresolute	ADJ
cana-1225	306	25	mapping	mapping	NOUN
cana-1225	306	26	,	,	PUNCT
cana-1225	306	27	then	then	ADV
cana-1225	306	28	𝑓	𝑓	PRON
cana-1225	306	29	is	be	AUX
cana-1225	306	30	a	a	DET
cana-1225	306	31	bipolar	bipolar	ADJ
cana-1225	306	32	vague	vague	NOUN
cana-1225	306	33	𝛼	𝛼	ADP
cana-1225	306	34	generalized	generalize	VERB
cana-1225	306	35	continuous	continuous	ADJ
cana-1225	306	36	mapping	mapping	NOUN
cana-1225	306	37	but	but	CCONJ
cana-1225	306	38	not	not	PART
cana-1225	306	39	conversely	conversely	ADV
cana-1225	306	40	.	.	PUNCT
cana-1225	307	1	proof	proof	NOUN
cana-1225	307	2	:	:	PUNCT
cana-1225	307	3	let	let	VERB
cana-1225	307	4	𝑓	𝑓	PRON
cana-1225	307	5	be	be	AUX
cana-1225	307	6	a	a	DET
cana-1225	307	7	bipolar	bipolar	ADJ
cana-1225	307	8	vague	vague	NOUN
cana-1225	307	9	𝛼	𝛼	ADP
cana-1225	307	10	generalized	generalize	VERB
cana-1225	307	11	irresolute	irresolute	ADJ
cana-1225	307	12	mapping	mapping	NOUN
cana-1225	307	13	.	.	PUNCT
cana-1225	308	1	let	let	VERB
cana-1225	308	2	a	a	DET
cana-1225	308	3	be	be	AUX
cana-1225	308	4	any	any	DET
cana-1225	308	5	bipolar	bipolar	ADJ
cana-1225	308	6	vague	vague	NOUN
cana-1225	308	7	closed	close	VERB
cana-1225	308	8	set	set	VERB
cana-1225	308	9	in	in	ADP
cana-1225	308	10	y.	y.	NOUN
cana-1225	308	11	since	since	SCONJ
cana-1225	308	12	every	every	DET
cana-1225	308	13	bipolar	bipolar	ADJ
cana-1225	308	14	vague	vague	NOUN
cana-1225	308	15	closed	close	VERB
cana-1225	308	16	set	set	NOUN
cana-1225	308	17	is	be	AUX
cana-1225	308	18	a	a	DET
cana-1225	308	19	bipolar	bipolar	ADJ
cana-1225	308	20	vague	vague	NOUN
cana-1225	308	21	𝛼	𝛼	DET
cana-1225	308	22	generalized	generalize	VERB
cana-1225	308	23	closed	close	VERB
cana-1225	308	24	set	set	NOUN
cana-1225	309	1	[	[	X
cana-1225	309	2	10	10	NUM
cana-1225	309	3	]	]	PUNCT
cana-1225	309	4	,	,	PUNCT
cana-1225	309	5	a	a	PRON
cana-1225	309	6	is	be	AUX
cana-1225	309	7	a	a	DET
cana-1225	309	8	bipolar	bipolar	ADJ
cana-1225	309	9	vague	vague	NOUN
cana-1225	309	10	𝛼	𝛼	DET
cana-1225	309	11	generalized	generalize	VERB
cana-1225	309	12	closed	close	VERB
cana-1225	309	13	set	set	VERB
cana-1225	309	14	in	in	ADP
cana-1225	309	15	y.	y.	NOUN
cana-1225	309	16	by	by	ADP
cana-1225	309	17	hypothesis	hypothesis	NOUN
cana-1225	309	18	,	,	PUNCT
cana-1225	309	19	𝑓−1(a	𝑓−1(a	PROPN
cana-1225	309	20	)	)	PUNCT
cana-1225	309	21	is	be	AUX
cana-1225	309	22	a	a	DET
cana-1225	309	23	bipolar	bipolar	ADJ
cana-1225	309	24	vague	vague	NOUN
cana-1225	309	25	𝛼	𝛼	DET
cana-1225	309	26	generalized	generalize	VERB
cana-1225	309	27	closed	close	VERB
cana-1225	309	28	set	set	VERB
cana-1225	309	29	in	in	ADP
cana-1225	309	30	x.	x.	NOUN
cana-1225	309	31	hence	hence	ADV
cana-1225	309	32	𝑓	𝑓	PROPN
cana-1225	309	33	is	be	AUX
cana-1225	309	34	a	a	DET
cana-1225	309	35	bipolar	bipolar	ADJ
cana-1225	309	36	vague	vague	NOUN
cana-1225	309	37	𝛼	𝛼	ADP
cana-1225	309	38	generalized	generalize	VERB
cana-1225	309	39	continuous	continuous	ADJ
cana-1225	309	40	mapping	mapping	NOUN
cana-1225	309	41	.	.	PUNCT
cana-1225	310	1	example	example	NOUN
cana-1225	310	2	4.3	4.3	NUM
cana-1225	310	3	:	:	PUNCT
cana-1225	310	4	let	let	VERB
cana-1225	310	5	x	x	PUNCT
cana-1225	310	6	=	=	PRON
cana-1225	310	7	{	{	PUNCT
cana-1225	310	8	a	a	DET
cana-1225	310	9	,	,	PUNCT
cana-1225	310	10	b	b	NOUN
cana-1225	310	11	}	}	PUNCT
cana-1225	310	12	and	and	CCONJ
cana-1225	310	13	y	y	PROPN
cana-1225	310	14	=	=	SYM
cana-1225	310	15	{	{	PUNCT
cana-1225	310	16	u	u	NOUN
cana-1225	310	17	,	,	PUNCT
cana-1225	310	18	v	v	NOUN
cana-1225	310	19	}	}	PUNCT
cana-1225	310	20	.	.	PUNCT
cana-1225	311	1	then	then	ADV
cana-1225	311	2	𝜏	𝜏	X
cana-1225	311	3	=	=	PUNCT
cana-1225	311	4	{	{	PUNCT
cana-1225	311	5	0~	0~	NOUN
cana-1225	311	6	,	,	PUNCT
cana-1225	311	7	a	a	DET
cana-1225	311	8	,	,	PUNCT
cana-1225	311	9	1~	1~	NUM
cana-1225	311	10	}	}	PUNCT
cana-1225	311	11	and	and	CCONJ
cana-1225	311	12	𝜎	𝜎	NOUN
cana-1225	311	13	=	=	SYM
cana-1225	311	14	{	{	PUNCT
cana-1225	311	15	0~	0~	NOUN
cana-1225	311	16	,	,	PUNCT
cana-1225	311	17	b	b	NOUN
cana-1225	311	18	,	,	PUNCT
cana-1225	311	19	1~	1~	NUM
cana-1225	311	20	}	}	PUNCT
cana-1225	311	21	are	be	AUX
cana-1225	311	22	bipolar	bipolar	ADJ
cana-1225	311	23	vague	vague	ADJ
cana-1225	311	24	topologies	topology	NOUN
cana-1225	311	25	on	on	ADP
cana-1225	311	26	x	x	X
cana-1225	311	27	and	and	CCONJ
cana-1225	311	28	y	y	PROPN
cana-1225	311	29	respectively	respectively	ADV
cana-1225	311	30	,	,	PUNCT
cana-1225	311	31	where	where	SCONJ
cana-1225	311	32	a	a	DET
cana-1225	311	33	=	=	SYM
cana-1225	311	34	x	x	NOUN
cana-1225	311	35	,	,	PUNCT
cana-1225	311	36	[	[	X
cana-1225	311	37	0.1	0.1	NUM
cana-1225	311	38	,	,	PUNCT
cana-1225	311	39	0.3	0.3	NUM
cana-1225	311	40	]	]	PUNCT
cana-1225	312	1	[	[	X
cana-1225	312	2	-0.3	-0.3	PROPN
cana-1225	312	3	,	,	PUNCT
cana-1225	312	4	-0.3	-0.3	PROPN
cana-1225	312	5	]	]	X
cana-1225	312	6	,	,	PUNCT
cana-1225	312	7	[	[	X
cana-1225	312	8	0.6	0.6	NUM
cana-1225	312	9	,	,	PUNCT
cana-1225	312	10	0.3	0.3	NUM
cana-1225	312	11	]	]	PUNCT
cana-1225	313	1	[	[	X
cana-1225	313	2	0.3	0.3	NUM
cana-1225	313	3	,	,	PUNCT
cana-1225	313	4	-0.3]	-0.3]	ADJ
cana-1225	313	5	and	and	CCONJ
cana-1225	313	6	b	b	X
cana-1225	313	7	=	=	SYM
cana-1225	313	8	y	y	PROPN
cana-1225	313	9	,	,	PUNCT
cana-1225	313	10	[	[	X
cana-1225	313	11	0.3	0.3	NUM
cana-1225	313	12	,	,	PUNCT
cana-1225	313	13	0.1	0.1	NUM
cana-1225	313	14	]	]	PUNCT
cana-1225	314	1	[	[	X
cana-1225	314	2	-0.1	-0.1	PROPN
cana-1225	314	3	,	,	PUNCT
cana-1225	314	4	-0.1	-0.1	PROPN
cana-1225	314	5	]	]	X
cana-1225	314	6	,	,	PUNCT
cana-1225	314	7	[	[	X
cana-1225	314	8	0.5	0.5	NUM
cana-1225	314	9	,	,	PUNCT
cana-1225	314	10	0.6	0.6	NUM
cana-1225	314	11	]	]	PUNCT
cana-1225	314	12	[	[	X
cana-1225	314	13	-0.1	-0.1	NOUN
cana-1225	314	14	,	,	PUNCT
cana-1225	314	15	-0.1].	-0.1].	NOUN
cana-1225	314	16	define	define	VERB
cana-1225	314	17	a	a	DET
cana-1225	314	18	mapping	mapping	NOUN
cana-1225	314	19	𝑓	𝑓	X
cana-1225	314	20	:	:	PUNCT
cana-1225	314	21	(	(	PUNCT
cana-1225	314	22	x	x	NOUN
cana-1225	314	23	,	,	PUNCT
cana-1225	314	24	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	314	25	)	)	PUNCT
cana-1225	314	26	→	→	SYM
cana-1225	314	27	(	(	PUNCT
cana-1225	314	28	y	y	NOUN
cana-1225	314	29	,	,	PUNCT
cana-1225	314	30	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	314	31	)	)	PUNCT
cana-1225	314	32	by	by	ADP
cana-1225	314	33	f(a	f(a	PROPN
cana-1225	314	34	)	)	PUNCT
cana-1225	314	35	=	=	SYM
cana-1225	314	36	u	u	NOUN
cana-1225	314	37	and	and	CCONJ
cana-1225	314	38	f(b	f(b	PROPN
cana-1225	314	39	)	)	PUNCT
cana-1225	315	1	=	=	VERB
cana-1225	316	1	v.	v.	CCONJ
cana-1225	316	2	then	then	ADV
cana-1225	316	3	𝑓	𝑓	PRON
cana-1225	316	4	is	be	AUX
cana-1225	316	5	a	a	DET
cana-1225	316	6	bipolar	bipolar	ADJ
cana-1225	316	7	vague	vague	NOUN
cana-1225	316	8	𝛼	𝛼	ADP
cana-1225	316	9	generalized	generalize	VERB
cana-1225	316	10	continuous	continuous	ADJ
cana-1225	316	11	mapping	mapping	NOUN
cana-1225	316	12	but	but	CCONJ
cana-1225	316	13	not	not	PART
cana-1225	316	14	a	a	DET
cana-1225	316	15	bipolar	bipolar	ADJ
cana-1225	316	16	vague	vague	NOUN
cana-1225	316	17	𝛼	𝛼	ADP
cana-1225	316	18	generalized	generalize	VERB
cana-1225	316	19	irresolute	irresolute	ADJ
cana-1225	316	20	mapping	mapping	NOUN
cana-1225	316	21	.	.	PUNCT
cana-1225	317	1	since	since	SCONJ
cana-1225	317	2	the	the	DET
cana-1225	317	3	bipolar	bipolar	ADJ
cana-1225	317	4	vague	vague	NOUN
cana-1225	317	5	set	set	NOUN
cana-1225	317	6	m	m	NOUN
cana-1225	317	7	=	=	SYM
cana-1225	317	8	y	y	SYM
cana-1225	317	9	,	,	PUNCT
cana-1225	317	10	[	[	X
cana-1225	317	11	0.1	0.1	NUM
cana-1225	317	12	,	,	PUNCT
cana-1225	317	13	0.3	0.3	NUM
cana-1225	317	14	]	]	PUNCT
cana-1225	318	1	[	[	X
cana-1225	318	2	-0.2	-0.2	NOUN
cana-1225	318	3	,	,	PUNCT
cana-1225	318	4	-0.2	-0.2	PROPN
cana-1225	318	5	]	]	X
cana-1225	318	6	,	,	PUNCT
cana-1225	318	7	[	[	X
cana-1225	318	8	0.6	0.6	NUM
cana-1225	318	9	,	,	PUNCT
cana-1225	318	10	0.2	0.2	NUM
cana-1225	318	11	]	]	PUNCT
cana-1225	319	1	[	[	X
cana-1225	319	2	-0.2	-0.2	NOUN
cana-1225	319	3	,	,	PUNCT
cana-1225	319	4	-0.2]	-0.2]	PUNCT
cana-1225	319	5	is	be	AUX
cana-1225	319	6	a	a	DET
cana-1225	319	7	bipolar	bipolar	ADJ
cana-1225	319	8	vague	vague	NOUN
cana-1225	319	9	𝛼	𝛼	DET
cana-1225	319	10	generalized	generalize	VERB
cana-1225	319	11	closed	close	VERB
cana-1225	319	12	set	set	VERB
cana-1225	319	13	in	in	ADP
cana-1225	319	14	y	y	PROPN
cana-1225	319	15	but	but	CCONJ
cana-1225	319	16	𝑓−1	𝑓−1	NUM
cana-1225	319	17	(	(	PUNCT
cana-1225	319	18	m	m	NOUN
cana-1225	319	19	)	)	PUNCT
cana-1225	319	20	is	be	AUX
cana-1225	319	21	not	not	PART
cana-1225	319	22	a	a	DET
cana-1225	319	23	bipolar	bipolar	ADJ
cana-1225	319	24	vague	vague	NOUN
cana-1225	319	25	𝛼	𝛼	DET
cana-1225	319	26	generalized	generalize	VERB
cana-1225	319	27	closed	close	VERB
cana-1225	319	28	set	set	VERB
cana-1225	319	29	in	in	ADP
cana-1225	319	30	x	x	PUNCT
cana-1225	319	31	as	as	ADP
cana-1225	319	32	𝑓−1	𝑓−1	NUM
cana-1225	319	33	(	(	PUNCT
cana-1225	319	34	m	m	NOUN
cana-1225	319	35	)	)	PUNCT
cana-1225	319	36	=	=	SYM
cana-1225	320	1	x	x	PUNCT
cana-1225	320	2	,	,	PUNCT
cana-1225	320	3	[	[	X
cana-1225	320	4	0.1	0.1	NUM
cana-1225	320	5	,	,	PUNCT
cana-1225	320	6	0.3	0.3	NUM
cana-1225	320	7	]	]	PUNCT
cana-1225	321	1	[	[	X
cana-1225	321	2	-0.2	-0.2	NOUN
cana-1225	321	3	,	,	PUNCT
cana-1225	321	4	0.2	0.2	NUM
cana-1225	321	5	]	]	PUNCT
cana-1225	321	6	,	,	PUNCT
cana-1225	321	7	[	[	X
cana-1225	321	8	0.6	0.6	NUM
cana-1225	321	9	,	,	PUNCT
cana-1225	321	10	0.2	0.2	NUM
cana-1225	321	11	]	]	PUNCT
cana-1225	322	1	[	[	X
cana-1225	322	2	-0.2	-0.2	NOUN
cana-1225	322	3	,	,	PUNCT
cana-1225	322	4	-0.2]	-0.2]	PROPN
cana-1225	322	5	⊆	⊆	NUM
cana-1225	322	6	a	a	PRON
cana-1225	322	7	but	but	CCONJ
cana-1225	322	8	b𝑉𝛼cl(𝑓−1	b𝑉𝛼cl(𝑓−1	NUM
cana-1225	322	9	(	(	PUNCT
cana-1225	322	10	m	m	NOUN
cana-1225	322	11	)	)	PUNCT
cana-1225	322	12	)	)	PUNCT
cana-1225	323	1	=	=	SYM
cana-1225	323	2	𝑓−1	𝑓−1	NUM
cana-1225	323	3	(	(	PUNCT
cana-1225	323	4	m	m	NOUN
cana-1225	323	5	)	)	PUNCT
cana-1225	323	6	∪	∪	ADJ
cana-1225	323	7	bvcl(bvint(bvcl	bvcl(bvint(bvcl	X
cana-1225	323	8	(	(	PUNCT
cana-1225	323	9	𝑓−1	𝑓−1	NUM
cana-1225	323	10	(	(	PUNCT
cana-1225	323	11	m	m	NOUN
cana-1225	323	12	)	)	PUNCT
cana-1225	323	13	)	)	PUNCT
cana-1225	323	14	)	)	PUNCT
cana-1225	323	15	)	)	PUNCT
cana-1225	324	1	=	=	PUNCT
cana-1225	324	2	ac	ac	PROPN
cana-1225	324	3	⊄	⊄	NOUN
cana-1225	324	4	a.	a.	NOUN
cana-1225	324	5	hence	hence	ADV
cana-1225	324	6	𝑓	𝑓	PROPN
cana-1225	324	7	is	be	AUX
cana-1225	324	8	not	not	PART
cana-1225	324	9	a	a	DET
cana-1225	324	10	bipolar	bipolar	ADJ
cana-1225	324	11	vague	vague	NOUN
cana-1225	324	12	𝛼	𝛼	ADP
cana-1225	324	13	generalized	generalize	VERB
cana-1225	324	14	irresolute	irresolute	ADJ
cana-1225	324	15	mapping	mapping	NOUN
cana-1225	324	16	.	.	PUNCT
cana-1225	325	1	proposition	proposition	NOUN
cana-1225	325	2	4.4	4.4	NUM
cana-1225	325	3	:	:	PUNCT
cana-1225	325	4	let	let	VERB
cana-1225	325	5	𝑓	𝑓	PRON
cana-1225	325	6	:	:	PUNCT
cana-1225	325	7	(	(	PUNCT
cana-1225	325	8	x	x	NOUN
cana-1225	325	9	,	,	PUNCT
cana-1225	325	10	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	325	11	)	)	PUNCT
cana-1225	325	12	→	→	SYM
cana-1225	325	13	(	(	PUNCT
cana-1225	325	14	y	y	NOUN
cana-1225	325	15	,	,	PUNCT
cana-1225	325	16	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	325	17	)	)	PUNCT
cana-1225	325	18	and	and	CCONJ
cana-1225	325	19	𝑔	𝑔	PROPN
cana-1225	325	20	:	:	PUNCT
cana-1225	325	21	(	(	PUNCT
cana-1225	325	22	y	y	NOUN
cana-1225	325	23	,	,	PUNCT
cana-1225	325	24	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	325	25	)	)	PUNCT
cana-1225	325	26	→	→	SYM
cana-1225	325	27	(	(	PUNCT
cana-1225	325	28	z	z	NOUN
cana-1225	325	29	,	,	PUNCT
cana-1225	325	30	b𝑉𝛿	b𝑉𝛿	PROPN
cana-1225	325	31	)	)	PUNCT
cana-1225	325	32	be	be	VERB
cana-1225	325	33	any	any	DET
cana-1225	325	34	two	two	NUM
cana-1225	325	35	bipolar	bipolar	ADJ
cana-1225	325	36	vague	vague	NOUN
cana-1225	325	37	𝛼	𝛼	ADP
cana-1225	325	38	generalized	generalize	VERB
cana-1225	325	39	irresolute	irresolute	ADJ
cana-1225	325	40	mappings	mapping	NOUN
cana-1225	325	41	,	,	PUNCT
cana-1225	325	42	then	then	ADV
cana-1225	325	43	𝑔	𝑔	PROPN
cana-1225	325	44	∘	∘	PROPN
cana-1225	325	45	𝑓	𝑓	X
cana-1225	325	46	:	:	PUNCT
cana-1225	325	47	(	(	PUNCT
cana-1225	325	48	x	x	NOUN
cana-1225	325	49	,	,	PUNCT
cana-1225	325	50	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	325	51	)	)	PUNCT
cana-1225	325	52	→	→	SYM
cana-1225	325	53	(	(	PUNCT
cana-1225	325	54	z	z	NOUN
cana-1225	325	55	,	,	PUNCT
cana-1225	325	56	b𝑉𝛿	b𝑉𝛿	PROPN
cana-1225	325	57	)	)	PUNCT
cana-1225	325	58	is	be	AUX
cana-1225	325	59	a	a	DET
cana-1225	325	60	bipolar	bipolar	ADJ
cana-1225	325	61	vague	vague	NOUN
cana-1225	325	62	𝛼	𝛼	ADP
cana-1225	325	63	generalized	generalize	VERB
cana-1225	325	64	irresolute	irresolute	ADJ
cana-1225	325	65	mapping	mapping	NOUN
cana-1225	325	66	.	.	PUNCT
cana-1225	326	1	proof	proof	NOUN
cana-1225	326	2	:	:	PUNCT
cana-1225	326	3	let	let	VERB
cana-1225	326	4	a	a	PRON
cana-1225	326	5	be	be	AUX
cana-1225	326	6	a	a	DET
cana-1225	326	7	bipolar	bipolar	ADJ
cana-1225	326	8	vague	vague	NOUN
cana-1225	326	9	𝛼	𝛼	DET
cana-1225	326	10	generalized	generalize	VERB
cana-1225	326	11	closed	close	VERB
cana-1225	326	12	set	set	VERB
cana-1225	326	13	in	in	ADP
cana-1225	326	14	z.	z.	PROPN
cana-1225	326	15	then	then	ADV
cana-1225	327	1	𝑔−1	𝑔−1	PROPN
cana-1225	327	2	(	(	PUNCT
cana-1225	327	3	a	a	PRON
cana-1225	327	4	)	)	PUNCT
cana-1225	327	5	is	be	AUX
cana-1225	327	6	a	a	DET
cana-1225	327	7	bipolar	bipolar	ADJ
cana-1225	327	8	vague	vague	NOUN
cana-1225	327	9	𝛼	𝛼	DET
cana-1225	327	10	generalized	generalize	VERB
cana-1225	327	11	closed	close	VERB
cana-1225	327	12	set	set	VERB
cana-1225	327	13	in	in	ADP
cana-1225	327	14	y.	y.	NOUN
cana-1225	327	15	since	since	SCONJ
cana-1225	327	16	𝑓	𝑓	PROPN
cana-1225	327	17	is	be	AUX
cana-1225	327	18	a	a	DET
cana-1225	327	19	bipolar	bipolar	ADJ
cana-1225	327	20	vague	vague	NOUN
cana-1225	327	21	𝛼	𝛼	ADP
cana-1225	327	22	generalized	generalize	VERB
cana-1225	327	23	irresolute	irresolute	ADJ
cana-1225	327	24	mapping	mapping	NOUN
cana-1225	327	25	,	,	PUNCT
cana-1225	327	26	𝑓−1(𝑔−1	𝑓−1(𝑔−1	PROPN
cana-1225	327	27	(	(	PUNCT
cana-1225	327	28	a	a	NOUN
cana-1225	327	29	)	)	PUNCT
cana-1225	327	30	)	)	PUNCT
cana-1225	327	31	is	be	AUX
cana-1225	327	32	a	a	DET
cana-1225	327	33	bipolar	bipolar	ADJ
cana-1225	327	34	vague	vague	NOUN
cana-1225	328	1	𝛼	𝛼	DET
cana-1225	328	2	generalized	generalize	VERB
cana-1225	328	3	closed	close	VERB
cana-1225	328	4	set	set	VERB
cana-1225	328	5	in	in	ADP
cana-1225	328	6	x.	x.	NOUN
cana-1225	328	7	hence	hence	ADV
cana-1225	328	8	𝑔	𝑔	PROPN
cana-1225	328	9	∘	∘	PROPN
cana-1225	328	10	𝑓	𝑓	PRON
cana-1225	328	11	is	be	AUX
cana-1225	328	12	a	a	DET
cana-1225	328	13	bipolar	bipolar	ADJ
cana-1225	328	14	vague	vague	NOUN
cana-1225	328	15	𝛼	𝛼	ADP
cana-1225	328	16	generalized	generalize	VERB
cana-1225	328	17	irresolute	irresolute	ADJ
cana-1225	328	18	mapping	mapping	NOUN
cana-1225	328	19	.	.	PUNCT
cana-1225	329	1	proposition	proposition	NOUN
cana-1225	329	2	4.5	4.5	NUM
cana-1225	329	3	:	:	PUNCT
cana-1225	329	4	let	let	VERB
cana-1225	329	5	𝑓	𝑓	PRON
cana-1225	329	6	:	:	PUNCT
cana-1225	329	7	(	(	PUNCT
cana-1225	329	8	x	x	NOUN
cana-1225	329	9	,	,	PUNCT
cana-1225	329	10	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	329	11	)	)	PUNCT
cana-1225	329	12	→	→	SYM
cana-1225	329	13	(	(	PUNCT
cana-1225	329	14	y	y	NOUN
cana-1225	329	15	,	,	PUNCT
cana-1225	329	16	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	329	17	)	)	PUNCT
cana-1225	329	18	be	be	VERB
cana-1225	329	19	a	a	DET
cana-1225	329	20	bipolar	bipolar	ADJ
cana-1225	329	21	vague	vague	NOUN
cana-1225	329	22	𝛼	𝛼	DET
cana-1225	329	23	generalized	generalize	VERB
cana-1225	329	24	irresolute	irresolute	ADJ
cana-1225	329	25	mapping	mapping	NOUN
cana-1225	329	26	and	and	CCONJ
cana-1225	329	27	𝑔	𝑔	PROPN
cana-1225	329	28	:	:	PUNCT
cana-1225	329	29	(	(	PUNCT
cana-1225	329	30	y	y	NOUN
cana-1225	329	31	,	,	PUNCT
cana-1225	329	32	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	329	33	)	)	PUNCT
cana-1225	329	34	→	→	SYM
cana-1225	329	35	(	(	PUNCT
cana-1225	329	36	z	z	NOUN
cana-1225	329	37	,	,	PUNCT
cana-1225	329	38	b𝑉𝛿	b𝑉𝛿	PROPN
cana-1225	329	39	)	)	PUNCT
cana-1225	329	40	be	be	VERB
cana-1225	329	41	a	a	DET
cana-1225	329	42	bipolar	bipolar	ADJ
cana-1225	329	43	vague	vague	NOUN
cana-1225	329	44	𝛼	𝛼	ADP
cana-1225	329	45	generalized	generalize	VERB
cana-1225	329	46	continuous	continuous	ADJ
cana-1225	329	47	mapping	mapping	NOUN
cana-1225	329	48	,	,	PUNCT
cana-1225	329	49	then	then	ADV
cana-1225	329	50	𝑔	𝑔	PROPN
cana-1225	329	51	∘	∘	PROPN
cana-1225	329	52	𝑓	𝑓	X
cana-1225	329	53	:	:	PUNCT
cana-1225	329	54	(	(	PUNCT
cana-1225	329	55	x	x	NOUN
cana-1225	329	56	,	,	PUNCT
cana-1225	329	57	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	329	58	)	)	PUNCT
cana-1225	329	59	→	→	SYM
cana-1225	329	60	(	(	PUNCT
cana-1225	329	61	z	z	NOUN
cana-1225	329	62	,	,	PUNCT
cana-1225	329	63	b𝑉𝛿	b𝑉𝛿	PROPN
cana-1225	329	64	)	)	PUNCT
cana-1225	329	65	is	be	AUX
cana-1225	329	66	a	a	DET
cana-1225	329	67	bipolar	bipolar	ADJ
cana-1225	329	68	vague	vague	NOUN
cana-1225	330	1	𝛼	𝛼	ADP
cana-1225	330	2	generalized	generalize	VERB
cana-1225	330	3	continuous	continuous	ADJ
cana-1225	330	4	mapping	mapping	NOUN
cana-1225	330	5	.	.	PUNCT
cana-1225	331	1	proof	proof	NOUN
cana-1225	331	2	:	:	PUNCT
cana-1225	331	3	let	let	VERB
cana-1225	331	4	a	a	PRON
cana-1225	331	5	be	be	AUX
cana-1225	331	6	a	a	DET
cana-1225	331	7	bipolar	bipolar	ADJ
cana-1225	331	8	vague	vague	NOUN
cana-1225	331	9	closed	close	VERB
cana-1225	331	10	set	set	VERB
cana-1225	331	11	in	in	ADP
cana-1225	331	12	z.	z.	PROPN
cana-1225	331	13	then	then	ADV
cana-1225	331	14	𝑔−1	𝑔−1	PROPN
cana-1225	331	15	(	(	PUNCT
cana-1225	331	16	a	a	PRON
cana-1225	331	17	)	)	PUNCT
cana-1225	331	18	is	be	AUX
cana-1225	331	19	a	a	DET
cana-1225	331	20	bipolar	bipolar	ADJ
cana-1225	331	21	vague	vague	NOUN
cana-1225	331	22	𝛼	𝛼	DET
cana-1225	331	23	generalized	generalize	VERB
cana-1225	331	24	closed	close	VERB
cana-1225	331	25	set	set	VERB
cana-1225	331	26	in	in	ADP
cana-1225	331	27	y	y	PROPN
cana-1225	331	28	,	,	PUNCT
cana-1225	331	29	by	by	ADP
cana-1225	331	30	hypothesis	hypothesis	NOUN
cana-1225	331	31	.	.	PUNCT
cana-1225	332	1	since	since	SCONJ
cana-1225	332	2	f	f	PROPN
cana-1225	332	3	is	be	AUX
cana-1225	332	4	a	a	DET
cana-1225	332	5	bipolar	bipolar	ADJ
cana-1225	332	6	vague	vague	NOUN
cana-1225	332	7	𝛼	𝛼	ADP
cana-1225	332	8	generalized	generalize	VERB
cana-1225	332	9	irresolute	irresolute	ADJ
cana-1225	332	10	mapping	mapping	NOUN
cana-1225	332	11	,	,	PUNCT
cana-1225	332	12	𝑓−1(𝑔−1	𝑓−1(𝑔−1	PROPN
cana-1225	332	13	(	(	PUNCT
cana-1225	332	14	a	a	NOUN
cana-1225	332	15	)	)	PUNCT
cana-1225	332	16	)	)	PUNCT
cana-1225	332	17	is	be	AUX
cana-1225	332	18	a	a	DET
cana-1225	332	19	bipolar	bipolar	ADJ
cana-1225	332	20	vague	vague	NOUN
cana-1225	332	21	𝛼	𝛼	DET
cana-1225	332	22	generalized	generalize	VERB
cana-1225	332	23	closed	close	VERB
cana-1225	332	24	set	set	VERB
cana-1225	332	25	in	in	ADP
cana-1225	332	26	x.	x.	NOUN
cana-1225	332	27	hence	hence	ADV
cana-1225	332	28	𝑔	𝑔	PROPN
cana-1225	332	29	∘	∘	PROPN
cana-1225	332	30	𝑓	𝑓	PRON
cana-1225	332	31	is	be	AUX
cana-1225	332	32	a	a	DET
cana-1225	332	33	bipolar	bipolar	ADJ
cana-1225	332	34	vague	vague	NOUN
cana-1225	332	35	𝛼	𝛼	ADP
cana-1225	332	36	generalized	generalize	VERB
cana-1225	332	37	continuous	continuous	ADJ
cana-1225	332	38	mapping	mapping	NOUN
cana-1225	332	39	.	.	PUNCT
cana-1225	333	1	proposition	proposition	NOUN
cana-1225	333	2	4.6	4.6	NUM
cana-1225	333	3	:	:	PUNCT
cana-1225	333	4	let	let	VERB
cana-1225	333	5	𝑓	𝑓	PRON
cana-1225	333	6	:	:	PUNCT
cana-1225	333	7	(	(	PUNCT
cana-1225	333	8	x	x	NOUN
cana-1225	333	9	,	,	PUNCT
cana-1225	333	10	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	333	11	)	)	PUNCT
cana-1225	333	12	→	→	SYM
cana-1225	333	13	(	(	PUNCT
cana-1225	333	14	y	y	NOUN
cana-1225	333	15	,	,	PUNCT
cana-1225	333	16	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	333	17	)	)	PUNCT
cana-1225	333	18	be	be	VERB
cana-1225	333	19	a	a	DET
cana-1225	333	20	bipolar	bipolar	ADJ
cana-1225	333	21	vague	vague	NOUN
cana-1225	333	22	𝛼	𝛼	DET
cana-1225	333	23	generalized	generalize	VERB
cana-1225	333	24	irresolute	irresolute	ADJ
cana-1225	333	25	mapping	mapping	NOUN
cana-1225	333	26	and	and	CCONJ
cana-1225	333	27	𝑔	𝑔	PROPN
cana-1225	333	28	:	:	PUNCT
cana-1225	333	29	(	(	PUNCT
cana-1225	333	30	y	y	NOUN
cana-1225	333	31	,	,	PUNCT
cana-1225	333	32	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	333	33	)	)	PUNCT
cana-1225	333	34	→	→	SYM
cana-1225	333	35	(	(	PUNCT
cana-1225	333	36	z	z	NOUN
cana-1225	333	37	,	,	PUNCT
cana-1225	333	38	b𝑉𝛿	b𝑉𝛿	PROPN
cana-1225	333	39	)	)	PUNCT
cana-1225	333	40	be	be	VERB
cana-1225	333	41	a	a	DET
cana-1225	333	42	bipolar	bipolar	ADJ
cana-1225	333	43	vague	vague	ADJ
cana-1225	333	44	continuous	continuous	ADJ
cana-1225	333	45	mapping	mapping	NOUN
cana-1225	333	46	,	,	PUNCT
cana-1225	333	47	then	then	ADV
cana-1225	333	48	𝑔	𝑔	PROPN
cana-1225	333	49	∘	∘	PROPN
cana-1225	333	50	𝑓	𝑓	X
cana-1225	333	51	:	:	PUNCT
cana-1225	333	52	(	(	PUNCT
cana-1225	333	53	x	x	NOUN
cana-1225	333	54	,	,	PUNCT
cana-1225	333	55	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	333	56	)	)	PUNCT
cana-1225	333	57	→	→	SYM
cana-1225	333	58	(	(	PUNCT
cana-1225	333	59	z	z	NOUN
cana-1225	333	60	,	,	PUNCT
cana-1225	333	61	b𝑉𝛿	b𝑉𝛿	PROPN
cana-1225	333	62	)	)	PUNCT
cana-1225	333	63	is	be	AUX
cana-1225	333	64	a	a	DET
cana-1225	333	65	bipolar	bipolar	ADJ
cana-1225	333	66	vague	vague	NOUN
cana-1225	334	1	𝛼	𝛼	ADP
cana-1225	334	2	generalized	generalize	VERB
cana-1225	334	3	continuous	continuous	ADJ
cana-1225	334	4	mapping	mapping	NOUN
cana-1225	334	5	.	.	PUNCT
cana-1225	335	1	communications	communication	NOUN
cana-1225	335	2	on	on	ADP
cana-1225	335	3	applied	apply	VERB
cana-1225	335	4	nonlinear	nonlinear	ADJ
cana-1225	335	5	analysis	analysis	NOUN
cana-1225	335	6	issn	issn	NOUN
cana-1225	335	7	:	:	PUNCT
cana-1225	335	8	1074	1074	NUM
cana-1225	335	9	-	-	PUNCT
cana-1225	335	10	133x	133x	NUM
cana-1225	335	11	vol	vol	NOUN
cana-1225	335	12	31	31	NUM
cana-1225	335	13	no	no	NOUN
cana-1225	335	14	.	.	PUNCT
cana-1225	336	1	6s	6s	NUM
cana-1225	336	2	(	(	PUNCT
cana-1225	336	3	2024	2024	NUM
cana-1225	336	4	)	)	PUNCT
cana-1225	336	5	328	328	NUM
cana-1225	337	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1225	337	2	proof	proof	NOUN
cana-1225	337	3	:	:	PUNCT
cana-1225	337	4	let	let	VERB
cana-1225	337	5	a	a	PRON
cana-1225	337	6	be	be	AUX
cana-1225	337	7	a	a	DET
cana-1225	337	8	bipolar	bipolar	ADJ
cana-1225	337	9	vague	vague	NOUN
cana-1225	337	10	closed	close	VERB
cana-1225	337	11	set	set	VERB
cana-1225	337	12	in	in	ADP
cana-1225	337	13	z.	z.	PROPN
cana-1225	337	14	then	then	ADV
cana-1225	337	15	𝑔−1	𝑔−1	PROPN
cana-1225	337	16	(	(	PUNCT
cana-1225	337	17	a	a	PRON
cana-1225	337	18	)	)	PUNCT
cana-1225	337	19	is	be	AUX
cana-1225	337	20	a	a	DET
cana-1225	337	21	bipolar	bipolar	ADJ
cana-1225	337	22	vague	vague	NOUN
cana-1225	337	23	closed	close	VERB
cana-1225	337	24	set	set	VERB
cana-1225	337	25	in	in	ADP
cana-1225	337	26	y.	y.	NOUN
cana-1225	337	27	since	since	SCONJ
cana-1225	337	28	every	every	DET
cana-1225	337	29	bipolar	bipolar	ADJ
cana-1225	337	30	vague	vague	NOUN
cana-1225	337	31	closed	close	VERB
cana-1225	337	32	set	set	NOUN
cana-1225	337	33	is	be	AUX
cana-1225	337	34	a	a	DET
cana-1225	337	35	bipolar	bipolar	ADJ
cana-1225	337	36	vague	vague	NOUN
cana-1225	337	37	𝛼	𝛼	DET
cana-1225	337	38	generalized	generalize	VERB
cana-1225	337	39	closed	close	VERB
cana-1225	337	40	set	set	NOUN
cana-1225	337	41	[	[	X
cana-1225	337	42	10	10	NUM
cana-1225	337	43	]	]	PUNCT
cana-1225	337	44	,	,	PUNCT
cana-1225	337	45	𝑔−1	𝑔−1	PROPN
cana-1225	337	46	(	(	PUNCT
cana-1225	337	47	a	a	PRON
cana-1225	337	48	)	)	PUNCT
cana-1225	337	49	is	be	AUX
cana-1225	337	50	a	a	DET
cana-1225	337	51	bipolar	bipolar	ADJ
cana-1225	337	52	vague	vague	NOUN
cana-1225	337	53	𝛼	𝛼	DET
cana-1225	337	54	generalized	generalize	VERB
cana-1225	337	55	closed	close	VERB
cana-1225	337	56	set	set	VERB
cana-1225	337	57	in	in	ADP
cana-1225	337	58	y.	y.	PROPN
cana-1225	337	59	therefore	therefore	ADV
cana-1225	337	60	𝑓−1(𝑔−1	𝑓−1(𝑔−1	PROPN
cana-1225	337	61	(	(	PUNCT
cana-1225	337	62	a	a	NOUN
cana-1225	337	63	)	)	PUNCT
cana-1225	337	64	)	)	PUNCT
cana-1225	337	65	is	be	AUX
cana-1225	337	66	a	a	DET
cana-1225	337	67	bipolar	bipolar	ADJ
cana-1225	337	68	vague	vague	NOUN
cana-1225	337	69	𝛼	𝛼	DET
cana-1225	337	70	generalized	generalize	VERB
cana-1225	337	71	closed	close	VERB
cana-1225	337	72	set	set	VERB
cana-1225	337	73	in	in	ADP
cana-1225	337	74	x	x	NOUN
cana-1225	337	75	,	,	PUNCT
cana-1225	337	76	by	by	ADP
cana-1225	337	77	hypothesis	hypothesis	NOUN
cana-1225	337	78	.	.	PUNCT
cana-1225	338	1	hence	hence	ADV
cana-1225	338	2	𝑔	𝑔	PROPN
cana-1225	338	3	∘	∘	PROPN
cana-1225	338	4	𝑓	𝑓	PRON
cana-1225	338	5	is	be	AUX
cana-1225	338	6	a	a	DET
cana-1225	338	7	bipolar	bipolar	ADJ
cana-1225	338	8	vague	vague	NOUN
cana-1225	338	9	𝛼	𝛼	ADP
cana-1225	338	10	generalized	generalize	VERB
cana-1225	338	11	continuous	continuous	ADJ
cana-1225	338	12	mapping	mapping	NOUN
cana-1225	338	13	.	.	PUNCT
cana-1225	339	1	proposition	proposition	NOUN
cana-1225	339	2	4.7	4.7	NUM
cana-1225	339	3	:	:	PUNCT
cana-1225	339	4	a	a	DET
cana-1225	339	5	mapping	mapping	NOUN
cana-1225	339	6	𝑓	𝑓	X
cana-1225	339	7	:	:	PUNCT
cana-1225	339	8	(	(	PUNCT
cana-1225	339	9	x	x	NOUN
cana-1225	339	10	,	,	PUNCT
cana-1225	339	11	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	339	12	)	)	PUNCT
cana-1225	339	13	→	→	SYM
cana-1225	339	14	(	(	PUNCT
cana-1225	339	15	y	y	NOUN
cana-1225	339	16	,	,	PUNCT
cana-1225	339	17	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	339	18	)	)	PUNCT
cana-1225	339	19	is	be	AUX
cana-1225	339	20	a	a	DET
cana-1225	339	21	bipolar	bipolar	ADJ
cana-1225	339	22	vague	vague	NOUN
cana-1225	339	23	𝛼	𝛼	DET
cana-1225	339	24	generalized	generalize	VERB
cana-1225	339	25	irresolute	irresolute	ADJ
cana-1225	339	26	mapping	mapping	NOUN
cana-1225	339	27	if	if	SCONJ
cana-1225	339	28	and	and	CCONJ
cana-1225	339	29	only	only	ADV
cana-1225	339	30	if	if	SCONJ
cana-1225	339	31	the	the	DET
cana-1225	339	32	inverse	inverse	ADJ
cana-1225	339	33	image	image	NOUN
cana-1225	339	34	of	of	ADP
cana-1225	339	35	each	each	DET
cana-1225	339	36	bipolar	bipolar	ADJ
cana-1225	339	37	vague	vague	NOUN
cana-1225	339	38	𝛼	𝛼	ADP
cana-1225	339	39	generalized	generalize	VERB
cana-1225	339	40	open	open	ADJ
cana-1225	339	41	set	set	NOUN
cana-1225	339	42	in	in	ADP
cana-1225	339	43	y	y	PROPN
cana-1225	339	44	is	be	AUX
cana-1225	339	45	a	a	DET
cana-1225	339	46	bipolar	bipolar	ADJ
cana-1225	339	47	vague	vague	NOUN
cana-1225	339	48	𝛼	𝛼	ADP
cana-1225	339	49	generalized	generalize	VERB
cana-1225	339	50	open	open	ADJ
cana-1225	339	51	set	set	NOUN
cana-1225	339	52	in	in	ADP
cana-1225	339	53	x.	x.	NOUN
cana-1225	339	54	proof	proof	NOUN
cana-1225	339	55	:	:	PUNCT
cana-1225	339	56	necessity	necessity	NOUN
cana-1225	339	57	:	:	PUNCT
cana-1225	339	58	let	let	VERB
cana-1225	339	59	a	a	PRON
cana-1225	339	60	be	be	AUX
cana-1225	339	61	a	a	DET
cana-1225	339	62	bipolar	bipolar	ADJ
cana-1225	339	63	vague	vague	NOUN
cana-1225	339	64	𝛼	𝛼	ADP
cana-1225	339	65	generalized	generalize	VERB
cana-1225	339	66	open	open	ADJ
cana-1225	339	67	set	set	NOUN
cana-1225	339	68	in	in	ADP
cana-1225	339	69	y.	y.	PROPN
cana-1225	339	70	then	then	ADV
cana-1225	339	71	ac	ac	PROPN
cana-1225	339	72	is	be	AUX
cana-1225	339	73	a	a	DET
cana-1225	339	74	bipolar	bipolar	ADJ
cana-1225	339	75	vague	vague	NOUN
cana-1225	339	76	𝛼	𝛼	DET
cana-1225	339	77	generalized	generalize	VERB
cana-1225	339	78	closed	close	VERB
cana-1225	339	79	set	set	VERB
cana-1225	339	80	in	in	ADP
cana-1225	339	81	y.	y.	NOUN
cana-1225	339	82	since	since	SCONJ
cana-1225	339	83	𝑓	𝑓	PROPN
cana-1225	339	84	is	be	AUX
cana-1225	339	85	a	a	DET
cana-1225	339	86	bipolar	bipolar	ADJ
cana-1225	339	87	vague	vague	NOUN
cana-1225	340	1	𝛼	𝛼	PRON
cana-1225	340	2	generalized	generalize	VERB
cana-1225	340	3	irresolute	irresolute	NOUN
cana-1225	340	4	,	,	PUNCT
cana-1225	340	5	𝑓−1(ac	𝑓−1(ac	PROPN
cana-1225	340	6	)	)	PUNCT
cana-1225	340	7	is	be	AUX
cana-1225	340	8	a	a	DET
cana-1225	340	9	bipolar	bipolar	ADJ
cana-1225	340	10	vague	vague	NOUN
cana-1225	340	11	𝛼	𝛼	DET
cana-1225	340	12	generalized	generalize	VERB
cana-1225	340	13	closed	close	VERB
cana-1225	340	14	set	set	VERB
cana-1225	340	15	in	in	ADP
cana-1225	340	16	x.	x.	NOUN
cana-1225	340	17	since	since	SCONJ
cana-1225	340	18	𝑓−1(ac	𝑓−1(ac	NOUN
cana-1225	340	19	)	)	PUNCT
cana-1225	340	20	=	=	PUNCT
cana-1225	340	21	(	(	PUNCT
cana-1225	340	22	𝑓−1(a))c	𝑓−1(a))c	NUM
cana-1225	340	23	,	,	PUNCT
cana-1225	340	24	𝑓−1	𝑓−1	NUM
cana-1225	340	25	(	(	PUNCT
cana-1225	340	26	a	a	NOUN
cana-1225	340	27	)	)	PUNCT
cana-1225	340	28	is	be	AUX
cana-1225	340	29	a	a	DET
cana-1225	340	30	bipolar	bipolar	ADJ
cana-1225	340	31	vague	vague	NOUN
cana-1225	340	32	𝛼	𝛼	ADP
cana-1225	340	33	generalized	generalize	VERB
cana-1225	340	34	open	open	ADJ
cana-1225	340	35	set	set	NOUN
cana-1225	340	36	in	in	ADP
cana-1225	340	37	x.	x.	PROPN
cana-1225	340	38	sufficiency	sufficiency	PROPN
cana-1225	340	39	:	:	PUNCT
cana-1225	340	40	let	let	VERB
cana-1225	340	41	a	a	PRON
cana-1225	340	42	be	be	AUX
cana-1225	340	43	a	a	DET
cana-1225	340	44	bipolar	bipolar	ADJ
cana-1225	340	45	vague	vague	NOUN
cana-1225	340	46	𝛼	𝛼	DET
cana-1225	340	47	generalized	generalize	VERB
cana-1225	340	48	closed	close	VERB
cana-1225	340	49	set	set	VERB
cana-1225	340	50	in	in	ADP
cana-1225	340	51	y.	y.	PROPN
cana-1225	340	52	this	this	PRON
cana-1225	340	53	implies	imply	VERB
cana-1225	340	54	ac	ac	PROPN
cana-1225	340	55	is	be	AUX
cana-1225	340	56	a	a	DET
cana-1225	340	57	bipolar	bipolar	ADJ
cana-1225	340	58	vague	vague	NOUN
cana-1225	340	59	𝛼	𝛼	ADP
cana-1225	340	60	generalized	generalize	VERB
cana-1225	340	61	open	open	ADJ
cana-1225	340	62	set	set	VERB
cana-1225	340	63	in	in	ADP
cana-1225	340	64	y.	y.	NOUN
cana-1225	340	65	by	by	ADP
cana-1225	340	66	hypothesis	hypothesis	NOUN
cana-1225	340	67	,	,	PUNCT
cana-1225	340	68	𝑓−1(ac	𝑓−1(ac	PROPN
cana-1225	340	69	)	)	PUNCT
cana-1225	340	70	is	be	AUX
cana-1225	340	71	a	a	DET
cana-1225	340	72	bipolar	bipolar	ADJ
cana-1225	340	73	vague	vague	NOUN
cana-1225	340	74	𝛼	𝛼	ADP
cana-1225	340	75	generalized	generalize	VERB
cana-1225	340	76	open	open	ADJ
cana-1225	340	77	set	set	VERB
cana-1225	340	78	in	in	ADP
cana-1225	340	79	x.	x.	NOUN
cana-1225	340	80	since	since	SCONJ
cana-1225	340	81	𝑓−1(ac	𝑓−1(ac	NOUN
cana-1225	340	82	)	)	PUNCT
cana-1225	341	1	=	=	PUNCT
cana-1225	341	2	(	(	PUNCT
cana-1225	341	3	𝑓−1(a))c	𝑓−1(a))c	NUM
cana-1225	341	4	,	,	PUNCT
cana-1225	341	5	𝑓−1	𝑓−1	NUM
cana-1225	341	6	(	(	PUNCT
cana-1225	341	7	a	a	NOUN
cana-1225	341	8	)	)	PUNCT
cana-1225	341	9	is	be	AUX
cana-1225	341	10	a	a	DET
cana-1225	341	11	bipolar	bipolar	ADJ
cana-1225	341	12	vague	vague	NOUN
cana-1225	341	13	𝛼	𝛼	DET
cana-1225	341	14	generalized	generalize	VERB
cana-1225	341	15	closed	close	VERB
cana-1225	341	16	set	set	VERB
cana-1225	341	17	in	in	ADP
cana-1225	341	18	x.	x.	NOUN
cana-1225	341	19	hence	hence	ADV
cana-1225	341	20	𝑓	𝑓	PROPN
cana-1225	341	21	is	be	AUX
cana-1225	341	22	a	a	DET
cana-1225	341	23	bipolar	bipolar	ADJ
cana-1225	341	24	vague	vague	NOUN
cana-1225	341	25	𝛼	𝛼	ADP
cana-1225	341	26	generalized	generalize	VERB
cana-1225	341	27	irresolute	irresolute	ADJ
cana-1225	341	28	mapping	mapping	NOUN
cana-1225	341	29	.	.	PUNCT
cana-1225	342	1	proposition	proposition	NOUN
cana-1225	342	2	4.8	4.8	NUM
cana-1225	342	3	:	:	PUNCT
cana-1225	342	4	let	let	VERB
cana-1225	342	5	𝑓	𝑓	PRON
cana-1225	342	6	:	:	PUNCT
cana-1225	342	7	(	(	PUNCT
cana-1225	342	8	x	x	NOUN
cana-1225	342	9	,	,	PUNCT
cana-1225	342	10	b𝑉𝜏	b𝑉𝜏	NOUN
cana-1225	342	11	)	)	PUNCT
cana-1225	342	12	→	→	SYM
cana-1225	342	13	(	(	PUNCT
cana-1225	342	14	y	y	NOUN
cana-1225	342	15	,	,	PUNCT
cana-1225	342	16	b𝑉𝜎	b𝑉𝜎	NOUN
cana-1225	342	17	)	)	PUNCT
cana-1225	342	18	be	be	VERB
cana-1225	342	19	a	a	DET
cana-1225	342	20	mapping	mapping	NOUN
cana-1225	342	21	from	from	ADP
cana-1225	342	22	a	a	DET
cana-1225	342	23	bipolar	bipolar	ADJ
cana-1225	342	24	vague	vague	ADJ
cana-1225	342	25	topological	topological	ADJ
cana-1225	342	26	space	space	NOUN
cana-1225	342	27	x	x	PUNCT
cana-1225	342	28	into	into	ADP
cana-1225	342	29	a	a	DET
cana-1225	342	30	bipolar	bipolar	ADJ
cana-1225	342	31	vague	vague	ADJ
cana-1225	342	32	topological	topological	ADJ
cana-1225	342	33	space	space	NOUN
cana-1225	342	34	y.	y.	NOUN
cana-1225	342	35	then	then	ADV
cana-1225	342	36	the	the	DET
cana-1225	342	37	following	follow	VERB
cana-1225	342	38	conditions	condition	NOUN
cana-1225	342	39	are	be	AUX
cana-1225	342	40	equivalent	equivalent	ADJ
cana-1225	342	41	if	if	SCONJ
cana-1225	342	42	x	x	PROPN
cana-1225	342	43	and	and	CCONJ
cana-1225	342	44	y	y	PROPN
cana-1225	342	45	are	be	AUX
cana-1225	342	46	b𝑉𝛼𝑎𝑇1/2	b𝑉𝛼𝑎𝑇1/2	NOUN
cana-1225	342	47	spaces	space	NOUN
cana-1225	342	48	:	:	PUNCT
cana-1225	342	49	(	(	PUNCT
cana-1225	342	50	i	i	NOUN
cana-1225	342	51	)	)	PUNCT
cana-1225	342	52	𝑓	𝑓	PRON
cana-1225	342	53	is	be	AUX
cana-1225	342	54	a	a	DET
cana-1225	342	55	bipolar	bipolar	ADJ
cana-1225	342	56	vague	vague	NOUN
cana-1225	342	57	𝛼	𝛼	ADP
cana-1225	342	58	generalized	generalize	VERB
cana-1225	342	59	irresolute	irresolute	ADJ
cana-1225	342	60	mapping	mapping	NOUN
cana-1225	342	61	.	.	PUNCT
cana-1225	343	1	(	(	PUNCT
cana-1225	343	2	ii	ii	NOUN
cana-1225	343	3	)	)	PUNCT
cana-1225	343	4	𝑓−1	𝑓−1	NUM
cana-1225	343	5	(	(	PUNCT
cana-1225	343	6	b	b	NOUN
cana-1225	343	7	)	)	PUNCT
cana-1225	343	8	is	be	AUX
cana-1225	343	9	a	a	DET
cana-1225	343	10	bipolar	bipolar	ADJ
cana-1225	343	11	vague	vague	NOUN
cana-1225	343	12	𝛼	𝛼	ADP
cana-1225	343	13	generalized	generalize	VERB
cana-1225	343	14	open	open	ADJ
cana-1225	343	15	set	set	NOUN
cana-1225	343	16	in	in	ADP
cana-1225	343	17	x	x	PUNCT
cana-1225	343	18	for	for	ADP
cana-1225	343	19	each	each	DET
cana-1225	343	20	bipolar	bipolar	ADJ
cana-1225	343	21	vague	vague	NOUN
cana-1225	343	22	𝛼	𝛼	ADP
cana-1225	343	23	generalized	generalize	VERB
cana-1225	343	24	open	open	ADJ
cana-1225	343	25	set	set	NOUN
cana-1225	343	26	in	in	ADP
cana-1225	343	27	y.	y.	PROPN
cana-1225	343	28	(	(	PUNCT
cana-1225	343	29	iii	iii	PROPN
cana-1225	343	30	)	)	PUNCT
cana-1225	343	31	bvcl(𝑓−1	bvcl(𝑓−1	PROPN
cana-1225	343	32	(	(	PUNCT
cana-1225	343	33	b	b	NOUN
cana-1225	343	34	)	)	PUNCT
cana-1225	343	35	)	)	PUNCT
cana-1225	344	1	⊆	⊆	X
cana-1225	344	2	𝑓−1(bvcl(b	𝑓−1(bvcl(b	PROPN
cana-1225	344	3	)	)	PUNCT
cana-1225	344	4	)	)	PUNCT
cana-1225	344	5	for	for	ADP
cana-1225	344	6	each	each	DET
cana-1225	344	7	bipolar	bipolar	ADJ
cana-1225	344	8	vague	vague	ADJ
cana-1225	344	9	set	set	NOUN
cana-1225	344	10	b	b	PROPN
cana-1225	344	11	of	of	ADP
cana-1225	344	12	y.	y.	PROPN
cana-1225	344	13	proof	proof	PROPN
cana-1225	344	14	:	:	PUNCT
cana-1225	344	15	(	(	PUNCT
cana-1225	344	16	i	i	NOUN
cana-1225	344	17	)	)	PUNCT
cana-1225	344	18	⟹	⟹	PROPN
cana-1225	344	19	(	(	PUNCT
cana-1225	344	20	ii	ii	NOUN
cana-1225	344	21	)	)	PUNCT
cana-1225	344	22	is	be	AUX
cana-1225	344	23	obviously	obviously	ADV
cana-1225	344	24	true	true	ADJ
cana-1225	344	25	from	from	ADP
cana-1225	344	26	the	the	DET
cana-1225	344	27	proposition	proposition	NOUN
cana-1225	344	28	4.7	4.7	NUM
cana-1225	344	29	.	.	PUNCT
cana-1225	345	1	(	(	PUNCT
cana-1225	345	2	ii	ii	PROPN
cana-1225	345	3	)	)	PUNCT
cana-1225	345	4	⟹	⟹	PROPN
cana-1225	345	5	(	(	PUNCT
cana-1225	345	6	iii	iii	NOUN
cana-1225	345	7	)	)	PUNCT
cana-1225	345	8	.	.	PUNCT
cana-1225	346	1	let	let	VERB
cana-1225	346	2	b	b	X
cana-1225	346	3	be	be	AUX
cana-1225	346	4	any	any	DET
cana-1225	346	5	bipolar	bipolar	ADJ
cana-1225	346	6	vague	vague	NOUN
cana-1225	346	7	set	set	NOUN
cana-1225	346	8	in	in	ADP
cana-1225	346	9	y	y	PROPN
cana-1225	346	10	and	and	CCONJ
cana-1225	346	11	b	b	NOUN
cana-1225	346	12	⊆	⊆	NUM
cana-1225	346	13	bvcl(b	bvcl(b	NOUN
cana-1225	346	14	)	)	PUNCT
cana-1225	346	15	.	.	PUNCT
cana-1225	347	1	then	then	ADV
cana-1225	347	2	𝑓−1	𝑓−1	NUM
cana-1225	347	3	(	(	PUNCT
cana-1225	347	4	b	b	NOUN
cana-1225	347	5	)	)	PUNCT
cana-1225	347	6	⊆	⊆	PROPN
cana-1225	347	7	𝑓−1(bvcl(b	𝑓−1(bvcl(b	PROPN
cana-1225	347	8	)	)	PUNCT
cana-1225	347	9	)	)	PUNCT
cana-1225	347	10	.	.	PUNCT
cana-1225	348	1	since	since	SCONJ
cana-1225	348	2	bvcl(b	bvcl(b	NOUN
cana-1225	348	3	)	)	PUNCT
cana-1225	348	4	is	be	AUX
cana-1225	348	5	a	a	DET
cana-1225	348	6	bipolar	bipolar	ADJ
cana-1225	348	7	vague	vague	NOUN
cana-1225	348	8	closed	close	VERB
cana-1225	348	9	set	set	VERB
cana-1225	348	10	in	in	ADP
cana-1225	348	11	y	y	PROPN
cana-1225	348	12	,	,	PUNCT
cana-1225	348	13	𝑓−1(bvcl(b	𝑓−1(bvcl(b	PROPN
cana-1225	348	14	)	)	PUNCT
cana-1225	348	15	)	)	PUNCT
cana-1225	348	16	is	be	AUX
cana-1225	348	17	a	a	DET
cana-1225	348	18	bipolar	bipolar	ADJ
cana-1225	348	19	vague	vague	NOUN
cana-1225	348	20	𝛼	𝛼	DET
cana-1225	348	21	generalized	generalize	VERB
cana-1225	348	22	closed	close	VERB
cana-1225	348	23	set	set	VERB
cana-1225	348	24	in	in	ADP
cana-1225	348	25	x	x	NOUN
cana-1225	348	26	,	,	PUNCT
cana-1225	348	27	by	by	ADP
cana-1225	348	28	hypothesis	hypothesis	NOUN
cana-1225	348	29	.	.	PUNCT
cana-1225	349	1	since	since	SCONJ
cana-1225	349	2	x	x	PRON
cana-1225	349	3	is	be	AUX
cana-1225	349	4	a	a	DET
cana-1225	349	5	b𝑉𝛼𝑎𝑇1/2	b𝑉𝛼𝑎𝑇1/2	NOUN
cana-1225	349	6	space	space	NOUN
cana-1225	349	7	,	,	PUNCT
cana-1225	349	8	𝑓−1(bvcl(b	𝑓−1(bvcl(b	PROPN
cana-1225	349	9	)	)	PUNCT
cana-1225	349	10	)	)	PUNCT
cana-1225	349	11	is	be	AUX
cana-1225	349	12	a	a	DET
cana-1225	349	13	bipolar	bipolar	ADJ
cana-1225	349	14	vague	vague	NOUN
cana-1225	349	15	closed	close	VERB
cana-1225	349	16	set	set	VERB
cana-1225	349	17	in	in	ADP
cana-1225	349	18	x.	x.	NOUN
cana-1225	349	19	hence	hence	ADV
cana-1225	349	20	bvcl(𝑓−1	bvcl(𝑓−1	PROPN
cana-1225	349	21	(	(	PUNCT
cana-1225	349	22	b	b	NOUN
cana-1225	349	23	)	)	PUNCT
cana-1225	349	24	)	)	PUNCT
cana-1225	350	1	⊆	⊆	NUM
cana-1225	350	2	bvcl(𝑓−1(bvcl(b	bvcl(𝑓−1(bvcl(b	NUM
cana-1225	350	3	)	)	PUNCT
cana-1225	350	4	)	)	PUNCT
cana-1225	350	5	)	)	PUNCT
cana-1225	351	1	=	=	PUNCT
cana-1225	351	2	𝑓−1(bvcl(b	𝑓−1(bvcl(b	PROPN
cana-1225	351	3	)	)	PUNCT
cana-1225	351	4	)	)	PUNCT
cana-1225	351	5	.	.	PUNCT
cana-1225	352	1	(	(	PUNCT
cana-1225	352	2	iii	iii	X
cana-1225	352	3	)	)	PUNCT
cana-1225	352	4	⟹	⟹	PUNCT
cana-1225	353	1	(	(	PUNCT
cana-1225	353	2	i	i	NOUN
cana-1225	353	3	)	)	PUNCT
cana-1225	353	4	.	.	PUNCT
cana-1225	354	1	let	let	VERB
cana-1225	354	2	b	b	X
cana-1225	354	3	be	be	AUX
cana-1225	354	4	a	a	DET
cana-1225	354	5	bipolar	bipolar	ADJ
cana-1225	354	6	vague	vague	NOUN
cana-1225	354	7	𝛼	𝛼	DET
cana-1225	354	8	generalized	generalize	VERB
cana-1225	354	9	closed	close	VERB
cana-1225	354	10	set	set	VERB
cana-1225	354	11	in	in	ADP
cana-1225	354	12	y.	y.	NOUN
cana-1225	354	13	since	since	SCONJ
cana-1225	354	14	y	y	PROPN
cana-1225	354	15	is	be	AUX
cana-1225	354	16	a	a	DET
cana-1225	354	17	b𝑉𝛼𝑎𝑇1/2	b𝑉𝛼𝑎𝑇1/2	NOUN
cana-1225	354	18	space	space	NOUN
cana-1225	354	19	,	,	PUNCT
cana-1225	354	20	b	b	PROPN
cana-1225	354	21	is	be	AUX
cana-1225	354	22	a	a	DET
cana-1225	354	23	bipolar	bipolar	ADJ
cana-1225	354	24	vague	vague	NOUN
cana-1225	354	25	closed	close	VERB
cana-1225	354	26	set	set	VERB
cana-1225	354	27	in	in	ADP
cana-1225	354	28	y	y	PROPN
cana-1225	354	29	and	and	CCONJ
cana-1225	354	30	bvcl(b	bvcl(b	PROPN
cana-1225	354	31	)	)	PUNCT
cana-1225	354	32	=	=	SYM
cana-1225	354	33	b.	b.	PROPN
cana-1225	355	1	hence	hence	ADV
cana-1225	355	2	𝑓−1	𝑓−1	NUM
cana-1225	355	3	(	(	PUNCT
cana-1225	355	4	b	b	NOUN
cana-1225	355	5	)	)	PUNCT
cana-1225	355	6	=	=	SYM
cana-1225	355	7	𝑓−1(bvcl(b	𝑓−1(bvcl(b	PROPN
cana-1225	355	8	)	)	PUNCT
cana-1225	355	9	)	)	PUNCT
cana-1225	355	10	⊇	⊇	PROPN
cana-1225	355	11	bvcl(𝑓−1	bvcl(𝑓−1	PROPN
cana-1225	355	12	(	(	PUNCT
cana-1225	355	13	b	b	NOUN
cana-1225	355	14	)	)	PUNCT
cana-1225	355	15	)	)	PUNCT
cana-1225	355	16	.	.	PUNCT
cana-1225	356	1	but	but	CCONJ
cana-1225	356	2	𝑓−1	𝑓−1	NUM
cana-1225	356	3	(	(	PUNCT
cana-1225	356	4	b	b	NOUN
cana-1225	356	5	)	)	PUNCT
cana-1225	356	6	⊆	⊆	NUM
cana-1225	356	7	bvcl(𝑓−1	bvcl(𝑓−1	PROPN
cana-1225	356	8	(	(	PUNCT
cana-1225	356	9	b	b	NOUN
cana-1225	356	10	)	)	PUNCT
cana-1225	356	11	)	)	PUNCT
cana-1225	356	12	.	.	PUNCT
cana-1225	357	1	therefore	therefore	ADV
cana-1225	357	2	,	,	PUNCT
cana-1225	357	3	bvcl(𝑓−1	bvcl(𝑓−1	PROPN
cana-1225	357	4	(	(	PUNCT
cana-1225	357	5	b	b	NOUN
cana-1225	357	6	)	)	PUNCT
cana-1225	357	7	)	)	PUNCT
cana-1225	357	8	=	=	SYM
cana-1225	357	9	𝑓−1	𝑓−1	NUM
cana-1225	357	10	(	(	PUNCT
cana-1225	357	11	b	b	NOUN
cana-1225	357	12	)	)	PUNCT
cana-1225	357	13	.	.	PUNCT
cana-1225	358	1	this	this	PRON
cana-1225	358	2	implies	imply	VERB
cana-1225	358	3	𝑓−1	𝑓−1	NUM
cana-1225	358	4	(	(	PUNCT
cana-1225	358	5	b	b	NOUN
cana-1225	358	6	)	)	PUNCT
cana-1225	358	7	is	be	AUX
cana-1225	358	8	a	a	DET
cana-1225	358	9	bipolar	bipolar	ADJ
cana-1225	358	10	vague	vague	NOUN
cana-1225	358	11	closed	close	VERB
cana-1225	358	12	set	set	VERB
cana-1225	358	13	and	and	CCONJ
cana-1225	358	14	hence	hence	ADV
cana-1225	358	15	it	it	PRON
cana-1225	358	16	is	be	AUX
cana-1225	358	17	a	a	DET
cana-1225	358	18	bipolar	bipolar	ADJ
cana-1225	358	19	vague	vague	NOUN
cana-1225	358	20	𝛼	𝛼	DET
cana-1225	358	21	generalized	generalize	VERB
cana-1225	358	22	closed	close	VERB
cana-1225	358	23	set	set	VERB
cana-1225	358	24	in	in	ADP
cana-1225	358	25	x.	x.	NOUN
cana-1225	358	26	thus	thus	ADV
cana-1225	358	27	𝑓	𝑓	PRON
cana-1225	358	28	is	be	AUX
cana-1225	358	29	a	a	DET
cana-1225	358	30	bipolar	bipolar	ADJ
cana-1225	358	31	vague	vague	NOUN
cana-1225	358	32	𝛼	𝛼	ADP
cana-1225	358	33	generalized	generalize	VERB
cana-1225	358	34	irresolute	irresolute	ADJ
cana-1225	358	35	mapping	mapping	NOUN
cana-1225	358	36	.	.	PUNCT
cana-1225	359	1	references	reference	NOUN
cana-1225	359	2	:	:	PUNCT
cana-1225	360	1	[	[	X
cana-1225	360	2	1	1	NUM
cana-1225	360	3	]	]	PUNCT
cana-1225	360	4	arockiarani.i	arockiarani.i	NOUN
cana-1225	360	5	and	and	CCONJ
cana-1225	360	6	cicily	cicily	ADV
cana-1225	360	7	flora.s	flora.	NOUN
cana-1225	360	8	.	.	PUNCT
cana-1225	360	9	,	,	PUNCT
cana-1225	360	10	positive	positive	ADJ
cana-1225	360	11	implicative	implicative	ADJ
cana-1225	360	12	bipolar	bipolar	ADJ
cana-1225	360	13	vague	vague	ADJ
cana-1225	360	14	ideals	ideal	NOUN
cana-1225	360	15	in	in	ADP
cana-1225	360	16	bck	bck	NOUN
cana-1225	360	17	-	-	PUNCT
cana-1225	360	18	algebras	algebras	PROPN
cana-1225	360	19	,	,	PUNCT
cana-1225	360	20	international	international	ADJ
cana-1225	360	21	research	research	NOUN
cana-1225	360	22	journal	journal	NOUN
cana-1225	360	23	of	of	ADP
cana-1225	360	24	pure	pure	ADJ
cana-1225	360	25	algebra	algebra	NOUN
cana-1225	360	26	,	,	PUNCT
cana-1225	360	27	2016	2016	NUM
cana-1225	360	28	,	,	PUNCT
cana-1225	360	29	1	1	NUM
cana-1225	360	30	-	-	SYM
cana-1225	360	31	7	7	NUM
cana-1225	360	32	.	.	PUNCT
cana-1225	360	33	communications	communication	NOUN
cana-1225	360	34	on	on	ADP
cana-1225	360	35	applied	apply	VERB
cana-1225	360	36	nonlinear	nonlinear	ADJ
cana-1225	360	37	analysis	analysis	NOUN
cana-1225	360	38	issn	issn	NOUN
cana-1225	360	39	:	:	PUNCT
cana-1225	360	40	1074	1074	NUM
cana-1225	360	41	-	-	PUNCT
cana-1225	360	42	133x	133x	NUM
cana-1225	360	43	vol	vol	NOUN
cana-1225	360	44	31	31	NUM
cana-1225	360	45	no	no	NOUN
cana-1225	360	46	.	.	PUNCT
cana-1225	361	1	6s	6s	NUM
cana-1225	361	2	(	(	PUNCT
cana-1225	361	3	2024	2024	NUM
cana-1225	361	4	)	)	PUNCT
cana-1225	361	5	329	329	NUM
cana-1225	361	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1225	362	1	[	[	X
cana-1225	362	2	2	2	NUM
cana-1225	362	3	]	]	PUNCT
cana-1225	362	4	atanassov.k	atanassov.k	PROPN
cana-1225	362	5	.	.	PROPN
cana-1225	362	6	,	,	PUNCT
cana-1225	362	7	intuitionistic	intuitionistic	ADJ
cana-1225	362	8	fuzzy	fuzzy	ADJ
cana-1225	362	9	sets	set	NOUN
cana-1225	362	10	,	,	PUNCT
cana-1225	362	11	fuzzy	fuzzy	ADJ
cana-1225	362	12	sets	set	NOUN
cana-1225	362	13	and	and	CCONJ
cana-1225	362	14	systems	system	NOUN
cana-1225	362	15	,	,	PUNCT
cana-1225	362	16	1986	1986	NUM
cana-1225	362	17	,	,	PUNCT
cana-1225	362	18	87	87	NUM
cana-1225	362	19	-	-	SYM
cana-1225	362	20	96	96	NUM
cana-1225	362	21	.	.	PUNCT
cana-1225	363	1	[	[	X
cana-1225	363	2	3	3	NUM
cana-1225	363	3	]	]	SYM
cana-1225	363	4	chang.c.l	chang.c.l	NOUN
cana-1225	363	5	.	.	PUNCT
cana-1225	363	6	,	,	PUNCT
cana-1225	363	7	fuzzy	fuzzy	ADJ
cana-1225	363	8	topological	topological	ADJ
cana-1225	363	9	spaces	space	NOUN
cana-1225	363	10	,	,	PUNCT
cana-1225	363	11	j	j	PROPN
cana-1225	363	12	math	math	NOUN
cana-1225	363	13	.	.	PUNCT
cana-1225	364	1	anal	anal	PROPN
cana-1225	364	2	.	.	PUNCT
cana-1225	364	3	appl	appl	PROPN
cana-1225	364	4	,	,	PUNCT
cana-1225	364	5	1968	1968	NUM
cana-1225	364	6	,	,	PUNCT
cana-1225	364	7	182	182	NUM
cana-1225	364	8	-	-	SYM
cana-1225	364	9	190	190	NUM
cana-1225	364	10	.	.	PUNCT
cana-1225	365	1	[	[	X
cana-1225	365	2	4	4	X
cana-1225	365	3	]	]	PUNCT
cana-1225	365	4	cicily	cicily	ADV
cana-1225	365	5	flora.s	flora.s	PUNCT
cana-1225	365	6	and	and	CCONJ
cana-1225	365	7	arockiarani.i	arockiarani.i	PROPN
cana-1225	365	8	.	.	PROPN
cana-1225	365	9	,	,	PUNCT
cana-1225	365	10	a	a	DET
cana-1225	365	11	new	new	ADJ
cana-1225	365	12	class	class	NOUN
cana-1225	365	13	of	of	ADP
cana-1225	365	14	generalized	generalized	ADJ
cana-1225	365	15	bipolar	bipolar	ADJ
cana-1225	365	16	vague	vague	ADJ
cana-1225	365	17	sets	set	NOUN
cana-1225	365	18	,	,	PUNCT
cana-1225	365	19	international	international	ADJ
cana-1225	365	20	journal	journal	NOUN
cana-1225	365	21	of	of	ADP
cana-1225	365	22	information	information	NOUN
cana-1225	365	23	research	research	NOUN
cana-1225	365	24	and	and	CCONJ
cana-1225	365	25	review	review	NOUN
cana-1225	365	26	,	,	PUNCT
cana-1225	365	27	2016	2016	NUM
cana-1225	365	28	,	,	PUNCT
cana-1225	365	29	3058	3058	NUM
cana-1225	365	30	-	-	SYM
cana-1225	365	31	3065	3065	NUM
cana-1225	365	32	.	.	PUNCT
cana-1225	366	1	[	[	X
cana-1225	366	2	5	5	NUM
cana-1225	366	3	]	]	PUNCT
cana-1225	366	4	cicily	cicily	ADV
cana-1225	366	5	flora.s	flora.s	PUNCT
cana-1225	366	6	and	and	CCONJ
cana-1225	366	7	arockiarani.i	arockiarani.i	PROPN
cana-1225	366	8	.	.	PROPN
cana-1225	366	9	,	,	PUNCT
cana-1225	366	10	on	on	ADP
cana-1225	366	11	bipolar	bipolar	ADJ
cana-1225	366	12	vague	vague	ADJ
cana-1225	366	13	ring	ring	NOUN
cana-1225	366	14	in	in	ADP
cana-1225	366	15	baire	baire	NOUN
cana-1225	366	16	spaces	space	NOUN
cana-1225	366	17	,	,	PUNCT
cana-1225	366	18	bulletin	bulletin	NOUN
cana-1225	366	19	of	of	ADP
cana-1225	366	20	mathematics	mathematic	NOUN
cana-1225	366	21	and	and	CCONJ
cana-1225	366	22	statistics	statistic	NOUN
cana-1225	366	23	research	research	NOUN
cana-1225	366	24	,	,	PUNCT
cana-1225	366	25	2017	2017	NUM
cana-1225	366	26	,	,	PUNCT
cana-1225	366	27	1	1	NUM
cana-1225	366	28	-	-	SYM
cana-1225	366	29	9	9	NUM
cana-1225	366	30	.	.	PUNCT
cana-1225	367	1	[	[	X
cana-1225	367	2	6	6	NUM
cana-1225	367	3	]	]	PUNCT
cana-1225	367	4	coker.d	coker.d	PROPN
cana-1225	367	5	.	.	PROPN
cana-1225	367	6	,	,	PUNCT
cana-1225	367	7	an	an	DET
cana-1225	367	8	introduction	introduction	NOUN
cana-1225	367	9	to	to	ADP
cana-1225	367	10	intuitionistic	intuitionistic	ADJ
cana-1225	367	11	fuzzy	fuzzy	ADJ
cana-1225	367	12	topological	topological	ADJ
cana-1225	367	13	spaces	space	NOUN
cana-1225	367	14	,	,	PUNCT
cana-1225	367	15	fuzzy	fuzzy	ADJ
cana-1225	367	16	sets	set	NOUN
cana-1225	367	17	and	and	CCONJ
cana-1225	367	18	systems	system	NOUN
cana-1225	367	19	,	,	PUNCT
cana-1225	367	20	1997	1997	NUM
cana-1225	367	21	,	,	PUNCT
cana-1225	367	22	81	81	NUM
cana-1225	367	23	-	-	SYM
cana-1225	367	24	89	89	NUM
cana-1225	367	25	.	.	PUNCT
cana-1225	368	1	[	[	X
cana-1225	368	2	7	7	NUM
cana-1225	368	3	]	]	X
cana-1225	368	4	gau.w.l	gau.w.l	NOUN
cana-1225	368	5	and	and	CCONJ
cana-1225	368	6	d.j.buehrer	d.j.buehrer	NOUN
cana-1225	368	7	.	.	PUNCT
cana-1225	369	1	,	,	PUNCT
cana-1225	369	2	vague	vague	ADJ
cana-1225	369	3	sets	set	NOUN
cana-1225	369	4	,	,	PUNCT
cana-1225	369	5	ieee	ieee	NOUN
cana-1225	369	6	trans	tran	NOUN
cana-1225	369	7	.	.	PUNCT
cana-1225	370	1	systems	system	NOUN
cana-1225	370	2	man	man	NOUN
cana-1225	370	3	and	and	CCONJ
cana-1225	370	4	cybernet	cybernet	NOUN
cana-1225	370	5	,	,	PUNCT
cana-1225	370	6	1993	1993	NUM
cana-1225	370	7	,	,	PUNCT
cana-1225	370	8	610	610	NUM
cana-1225	370	9	-	-	SYM
cana-1225	370	10	614	614	NUM
cana-1225	370	11	.	.	PUNCT
cana-1225	371	1	[	[	X
cana-1225	371	2	8	8	NUM
cana-1225	371	3	]	]	SYM
cana-1225	371	4	lee.k.m	lee.k.m	NOUN
cana-1225	371	5	.	.	PUNCT
cana-1225	371	6	,	,	PUNCT
cana-1225	371	7	bipolar	bipolar	ADV
cana-1225	371	8	-	-	PUNCT
cana-1225	371	9	valued	value	VERB
cana-1225	371	10	fuzzy	fuzzy	ADJ
cana-1225	371	11	sets	set	NOUN
cana-1225	371	12	and	and	CCONJ
cana-1225	371	13	their	their	PRON
cana-1225	371	14	operations	operation	NOUN
cana-1225	371	15	,	,	PUNCT
cana-1225	371	16	proc	proc	NOUN
cana-1225	371	17	.	.	PUNCT
cana-1225	372	1	int	int	NOUN
cana-1225	372	2	.	.	PUNCT
cana-1225	372	3	conf	conf	PROPN
cana-1225	372	4	.	.	PUNCT
cana-1225	373	1	on	on	ADP
cana-1225	373	2	intelligent	intelligent	ADJ
cana-1225	373	3	technologies	technology	NOUN
cana-1225	373	4	,	,	PUNCT
cana-1225	373	5	bangkok	bangkok	PROPN
cana-1225	373	6	,	,	PUNCT
cana-1225	373	7	thailand	thailand	PROPN
cana-1225	373	8	,	,	PUNCT
cana-1225	373	9	2000	2000	NUM
cana-1225	373	10	,	,	PUNCT
cana-1225	373	11	307	307	NUM
cana-1225	373	12	-	-	SYM
cana-1225	373	13	312	312	NUM
cana-1225	373	14	.	.	PUNCT
cana-1225	374	1	[	[	X
cana-1225	374	2	9	9	NUM
cana-1225	374	3	]	]	SYM
cana-1225	374	4	levine.n	levine.n	PROPN
cana-1225	374	5	.	.	PROPN
cana-1225	374	6	,	,	PUNCT
cana-1225	374	7	generalized	generalize	VERB
cana-1225	374	8	closed	closed	ADJ
cana-1225	374	9	sets	set	NOUN
cana-1225	374	10	in	in	ADP
cana-1225	374	11	topological	topological	ADJ
cana-1225	374	12	spaces	space	NOUN
cana-1225	374	13	,	,	PUNCT
cana-1225	374	14	rend	rend	VERB
cana-1225	374	15	.	.	PUNCT
cana-1225	375	1	circ	circ	PROPN
cana-1225	375	2	.	.	PUNCT
cana-1225	376	1	mat	mat	PROPN
cana-1225	376	2	.	.	PUNCT
cana-1225	376	3	palermo	palermo	PROPN
cana-1225	376	4	,	,	PUNCT
cana-1225	376	5	1970	1970	NUM
cana-1225	376	6	,	,	PUNCT
cana-1225	376	7	89	89	NUM
cana-1225	376	8	-	-	SYM
cana-1225	376	9	96	96	NUM
cana-1225	376	10	.	.	PUNCT
cana-1225	377	1	[	[	X
cana-1225	377	2	10	10	NUM
cana-1225	377	3	]	]	X
cana-1225	377	4	prishka.f	prishka.f	PROPN
cana-1225	377	5	and	and	CCONJ
cana-1225	377	6	mariapresenti.l	mariapresenti.l	PROPN
cana-1225	377	7	.	.	PROPN
cana-1225	378	1	,	,	PUNCT
cana-1225	378	2	bipolar	bipolar	ADJ
cana-1225	378	3	vague	vague	NOUN
cana-1225	378	4	𝛼	𝛼	DET
cana-1225	378	5	generalized	generalize	VERB
cana-1225	378	6	closed	close	VERB
cana-1225	378	7	sets	set	NOUN
cana-1225	378	8	in	in	ADP
cana-1225	378	9	topological	topological	ADJ
cana-1225	378	10	spaces	space	NOUN
cana-1225	378	11	,	,	PUNCT
cana-1225	378	12	journal	journal	NOUN
cana-1225	378	13	of	of	ADP
cana-1225	378	14	basic	basic	ADJ
cana-1225	378	15	science	science	NOUN
cana-1225	378	16	and	and	CCONJ
cana-1225	378	17	engineering	engineering	NOUN
cana-1225	378	18	,	,	PUNCT
cana-1225	378	19	1390	1390	NUM
cana-1225	378	20	-	-	SYM
cana-1225	378	21	1398	1398	NUM
cana-1225	378	22	.	.	PUNCT
cana-1225	379	1	[	[	X
cana-1225	379	2	11	11	NUM
cana-1225	379	3	]	]	PUNCT
cana-1225	379	4	zadeh.l.a	zadeh.l.a	NOUN
cana-1225	379	5	.	.	PROPN
cana-1225	379	6	,	,	PUNCT
cana-1225	379	7	fuzzy	fuzzy	ADJ
cana-1225	379	8	sets	set	NOUN
cana-1225	379	9	,	,	PUNCT
cana-1225	379	10	informartion	informartion	NOUN
cana-1225	379	11	and	and	CCONJ
cana-1225	379	12	control	control	NOUN
cana-1225	379	13	,	,	PUNCT
cana-1225	379	14	1965	1965	NUM
cana-1225	379	15	,	,	PUNCT
cana-1225	379	16	338	338	NUM
cana-1225	379	17	-	-	SYM
cana-1225	379	18	335	335	NUM
cana-1225	379	19	.	.	PUNCT