id	sid	tid	token	lemma	pos
cana-1179	1	1	communications	communication	NOUN
cana-1179	1	2	on	on	ADP
cana-1179	1	3	applied	apply	VERB
cana-1179	1	4	nonlinear	nonlinear	ADJ
cana-1179	1	5	analysis	analysis	NOUN
cana-1179	1	6	issn	issn	NOUN
cana-1179	1	7	:	:	PUNCT
cana-1179	1	8	1074	1074	NUM
cana-1179	1	9	-	-	PUNCT
cana-1179	1	10	133x	133x	NUM
cana-1179	1	11	vol	vol	NOUN
cana-1179	1	12	31	31	NUM
cana-1179	1	13	no	no	NOUN
cana-1179	1	14	.	.	PUNCT
cana-1179	2	1	6s	6s	NUM
cana-1179	2	2	(	(	PUNCT
cana-1179	2	3	2024	2024	NUM
cana-1179	2	4	)	)	PUNCT
cana-1179	2	5	207	207	NUM
cana-1179	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1179	2	7	on	on	ADP
cana-1179	2	8	beta	beta	ADJ
cana-1179	2	9	generalized	generalize	VERB
cana-1179	2	10	star	star	NOUN
cana-1179	2	11	pre	pre	VERB
cana-1179	2	12	-	-	ADJ
cana-1179	2	13	i	i	ADV
cana-1179	2	14	-	-	PUNCT
cana-1179	2	15	closed	close	VERB
cana-1179	2	16	sets	set	NOUN
cana-1179	2	17	in	in	ADP
cana-1179	2	18	ideal	ideal	ADJ
cana-1179	2	19	topological	topological	ADJ
cana-1179	2	20	spaces	space	NOUN
cana-1179	2	21	s.	s.	PROPN
cana-1179	2	22	gowri	gowri	PROPN
cana-1179	2	23	*	*	PUNCT
cana-1179	2	24	and	and	CCONJ
cana-1179	2	25	v.	v.	ADP
cana-1179	2	26	pankajam	pankajam	NOUN
cana-1179	2	27	*	*	PUNCT
cana-1179	2	28	*	*	PUNCT
cana-1179	2	29	research	research	NOUN
cana-1179	2	30	scholar	scholar	NOUN
cana-1179	2	31	*	*	PUNCT
cana-1179	2	32	and	and	CCONJ
cana-1179	2	33	assistant	assistant	PROPN
cana-1179	2	34	professor	professor	NOUN
cana-1179	2	35	*	*	PROPN
cana-1179	2	36	*	*	PROPN
cana-1179	2	37	sri	sri	PROPN
cana-1179	2	38	gvg	gvg	PROPN
cana-1179	2	39	visalakshi	visalakshi	PROPN
cana-1179	2	40	college	college	PROPN
cana-1179	2	41	for	for	ADP
cana-1179	2	42	women	woman	NOUN
cana-1179	2	43	,	,	PUNCT
cana-1179	2	44	udumalpet	udumalpet	ADJ
cana-1179	2	45	,	,	PUNCT
cana-1179	2	46	india	india	PROPN
cana-1179	2	47	.	.	PUNCT
cana-1179	3	1	gowrisivakumar95@gmail.com	gowrisivakumar95@gmail.com	PROPN
cana-1179	3	2	and	and	CCONJ
cana-1179	3	3	pankajamgurusamy@gmail.com	pankajamgurusamy@gmail.com	NOUN
cana-1179	3	4	article	article	NOUN
cana-1179	3	5	history	history	NOUN
cana-1179	3	6	:	:	PUNCT
cana-1179	3	7	received	receive	VERB
cana-1179	3	8	:	:	PUNCT
cana-1179	3	9	28	28	NUM
cana-1179	3	10	-	-	SYM
cana-1179	3	11	05	05	NUM
cana-1179	3	12	-	-	PUNCT
cana-1179	3	13	2024	2024	NUM
cana-1179	3	14	revised	revise	VERB
cana-1179	3	15	:	:	PUNCT
cana-1179	3	16	20	20	NUM
cana-1179	3	17	-	-	SYM
cana-1179	3	18	07	07	NUM
cana-1179	3	19	-	-	PUNCT
cana-1179	3	20	2024	2024	NUM
cana-1179	3	21	accepted	accept	VERB
cana-1179	3	22	:	:	PUNCT
cana-1179	3	23	31	31	NUM
cana-1179	3	24	-	-	SYM
cana-1179	3	25	07	07	NUM
cana-1179	3	26	-	-	PUNCT
cana-1179	3	27	2024	2024	NUM
cana-1179	3	28	abstract	abstract	NOUN
cana-1179	3	29	:	:	PUNCT
cana-1179	3	30	the	the	DET
cana-1179	3	31	purpose	purpose	NOUN
cana-1179	3	32	of	of	ADP
cana-1179	3	33	this	this	DET
cana-1179	3	34	paper	paper	NOUN
cana-1179	3	35	is	be	AUX
cana-1179	3	36	to	to	PART
cana-1179	3	37	define	define	VERB
cana-1179	3	38	the	the	DET
cana-1179	3	39	new	new	ADJ
cana-1179	3	40	idea	idea	NOUN
cana-1179	3	41	beta	beta	NOUN
cana-1179	3	42	generalized	generalize	VERB
cana-1179	3	43	star	star	NOUN
cana-1179	3	44	pre	pre	ADJ
cana-1179	3	45	-	-	ADJ
cana-1179	3	46	closed	closed	ADJ
cana-1179	3	47	sets	set	NOUN
cana-1179	3	48	,	,	PUNCT
cana-1179	3	49	a	a	DET
cana-1179	3	50	new	new	ADJ
cana-1179	3	51	class	class	NOUN
cana-1179	3	52	of	of	ADP
cana-1179	3	53	closed	closed	ADJ
cana-1179	3	54	and	and	CCONJ
cana-1179	3	55	open	open	ADJ
cana-1179	3	56	sets	set	NOUN
cana-1179	3	57	in	in	ADP
cana-1179	3	58	topological	topological	ADJ
cana-1179	3	59	spaces	space	NOUN
cana-1179	3	60	,	,	PUNCT
cana-1179	3	61	and	and	CCONJ
cana-1179	3	62	to	to	PART
cana-1179	3	63	examine	examine	VERB
cana-1179	3	64	some	some	PRON
cana-1179	3	65	of	of	ADP
cana-1179	3	66	its	its	PRON
cana-1179	3	67	characteristics	characteristic	NOUN
cana-1179	3	68	using	use	VERB
cana-1179	3	69	few	few	ADJ
cana-1179	3	70	examples	example	NOUN
cana-1179	3	71	.	.	PUNCT
cana-1179	4	1	in	in	ADP
cana-1179	4	2	addition	addition	NOUN
cana-1179	4	3	,	,	PUNCT
cana-1179	4	4	we	we	PRON
cana-1179	4	5	define	define	VERB
cana-1179	4	6	beta	beta	ADJ
cana-1179	4	7	generalized	generalize	VERB
cana-1179	4	8	star	star	NOUN
cana-1179	4	9	pre	pre	ADJ
cana-1179	4	10	-	-	ADJ
cana-1179	4	11	iclosed	iclosed	ADJ
cana-1179	4	12	sets	set	NOUN
cana-1179	4	13	,	,	PUNCT
cana-1179	4	14	a	a	DET
cana-1179	4	15	new	new	ADJ
cana-1179	4	16	class	class	NOUN
cana-1179	4	17	of	of	ADP
cana-1179	4	18	closed	closed	ADJ
cana-1179	4	19	and	and	CCONJ
cana-1179	4	20	open	open	ADJ
cana-1179	4	21	sets	set	NOUN
cana-1179	4	22	in	in	ADP
cana-1179	4	23	ideal	ideal	ADJ
cana-1179	4	24	topological	topological	ADJ
cana-1179	4	25	spaces	space	NOUN
cana-1179	4	26	and	and	CCONJ
cana-1179	4	27	discuss	discuss	VERB
cana-1179	4	28	through	through	ADP
cana-1179	4	29	their	their	PRON
cana-1179	4	30	characteristics	characteristic	NOUN
cana-1179	4	31	.	.	PUNCT
cana-1179	5	1	keywords	keyword	NOUN
cana-1179	5	2	:	:	PUNCT
cana-1179	5	3	topological	topological	ADJ
cana-1179	5	4	spaces	space	NOUN
cana-1179	5	5	,	,	PUNCT
cana-1179	5	6	pre	pre	ADJ
cana-1179	5	7	-	-	ADJ
cana-1179	5	8	closed	closed	ADJ
cana-1179	5	9	set	set	NOUN
cana-1179	5	10	,	,	PUNCT
cana-1179	5	11	g*-open	g*-open	INTJ
cana-1179	5	12	set	set	VERB
cana-1179	5	13	,	,	PUNCT
cana-1179	5	14	β	β	NOUN
cana-1179	5	15	-	-	ADJ
cana-1179	5	16	closed	closed	ADJ
cana-1179	5	17	set	set	NOUN
cana-1179	5	18	,	,	PUNCT
cana-1179	5	19	βg*p	βg*p	NUM
cana-1179	5	20	-	-	PUNCT
cana-1179	5	21	closed	closed	ADJ
cana-1179	5	22	set	set	NOUN
cana-1179	5	23	,	,	PUNCT
cana-1179	5	24	βg*p	βg*p	NUM
cana-1179	5	25	-	-	PUNCT
cana-1179	5	26	open	open	ADJ
cana-1179	5	27	set	set	NOUN
cana-1179	5	28	,	,	PUNCT
cana-1179	5	29	βg*p	βg*p	X
cana-1179	5	30	-	-	PUNCT
cana-1179	5	31	i	i	NOUN
cana-1179	5	32	-	-	PUNCT
cana-1179	5	33	closed	close	VERB
cana-1179	5	34	set	set	NOUN
cana-1179	5	35	,	,	PUNCT
cana-1179	5	36	βg*p	βg*p	X
cana-1179	5	37	-	-	PUNCT
cana-1179	5	38	i	i	PRON
cana-1179	5	39	-	-	PUNCT
cana-1179	5	40	open	open	ADJ
cana-1179	5	41	set	set	NOUN
cana-1179	5	42	.	.	PUNCT
cana-1179	6	1	1	1	X
cana-1179	6	2	.	.	X
cana-1179	6	3	introduction	introduction	NOUN
cana-1179	6	4	n.	n.	PROPN
cana-1179	6	5	levine	levine	PROPN
cana-1179	6	6	[	[	X
cana-1179	6	7	11	11	NUM
cana-1179	6	8	]	]	PUNCT
cana-1179	6	9	proposed	propose	VERB
cana-1179	6	10	the	the	DET
cana-1179	6	11	theory	theory	NOUN
cana-1179	6	12	of	of	ADP
cana-1179	6	13	generalized	generalized	ADJ
cana-1179	6	14	closed	closed	ADJ
cana-1179	6	15	sets	set	NOUN
cana-1179	6	16	and	and	CCONJ
cana-1179	6	17	generalized	generalize	VERB
cana-1179	6	18	open	open	ADJ
cana-1179	6	19	sets	set	NOUN
cana-1179	6	20	in	in	ADP
cana-1179	6	21	topological	topological	ADJ
cana-1179	6	22	spaces	space	NOUN
cana-1179	6	23	.	.	PUNCT
cana-1179	7	1	a	a	DET
cana-1179	7	2	new	new	ADJ
cana-1179	7	3	class	class	NOUN
cana-1179	7	4	of	of	ADP
cana-1179	7	5	generalized	generalized	ADJ
cana-1179	7	6	pre	pre	NOUN
cana-1179	7	7	regular	regular	ADJ
cana-1179	7	8	closed	closed	ADJ
cana-1179	7	9	sets	set	NOUN
cana-1179	7	10	in	in	ADP
cana-1179	7	11	topological	topological	ADJ
cana-1179	7	12	spaces	space	NOUN
cana-1179	7	13	was	be	AUX
cana-1179	7	14	presented	present	VERB
cana-1179	7	15	by	by	ADP
cana-1179	7	16	y.	y.	PROPN
cana-1179	7	17	gnanambal	gnanambal	PROPN
cana-1179	8	1	[	[	X
cana-1179	8	2	4	4	NUM
cana-1179	8	3	]	]	PUNCT
cana-1179	8	4	in	in	ADP
cana-1179	8	5	1997	1997	NUM
cana-1179	8	6	.	.	PUNCT
cana-1179	9	1	beta	beta	ADJ
cana-1179	9	2	generalized	generalize	VERB
cana-1179	9	3	closed	closed	ADJ
cana-1179	9	4	sets	set	NOUN
cana-1179	9	5	were	be	AUX
cana-1179	9	6	first	first	ADV
cana-1179	9	7	introduced	introduce	VERB
cana-1179	9	8	in	in	ADP
cana-1179	9	9	2022	2022	NUM
cana-1179	9	10	by	by	ADP
cana-1179	9	11	kavitha	kavitha	PROPN
cana-1179	9	12	and	and	CCONJ
cana-1179	9	13	sasikala	sasikala	VERB
cana-1179	10	1	[	[	X
cana-1179	10	2	9	9	NUM
cana-1179	10	3	]	]	PUNCT
cana-1179	10	4	.	.	PUNCT
cana-1179	11	1	the	the	DET
cana-1179	11	2	notion	notion	NOUN
cana-1179	11	3	of	of	ADP
cana-1179	11	4	βg*-closed	βg*-close	VERB
cana-1179	11	5	sets	set	NOUN
cana-1179	11	6	in	in	ADP
cana-1179	11	7	topological	topological	ADJ
cana-1179	11	8	spaces	space	NOUN
cana-1179	11	9	was	be	AUX
cana-1179	11	10	originated	originate	VERB
cana-1179	11	11	by	by	ADP
cana-1179	11	12	dhanapakyam	dhanapakyam	NOUN
cana-1179	11	13	and	and	CCONJ
cana-1179	11	14	indirani	indirani	NOUN
cana-1179	11	15	[	[	X
cana-1179	11	16	3	3	NUM
cana-1179	11	17	]	]	PUNCT
cana-1179	11	18	.	.	PUNCT
cana-1179	12	1	in	in	ADP
cana-1179	12	2	general	general	ADJ
cana-1179	12	3	topology	topology	NOUN
cana-1179	12	4	,	,	PUNCT
cana-1179	12	5	the	the	DET
cana-1179	12	6	idea	idea	NOUN
cana-1179	12	7	of	of	ADP
cana-1179	12	8	generalized	generalize	VERB
cana-1179	12	9	closed	closed	ADJ
cana-1179	12	10	sets	set	NOUN
cana-1179	12	11	is	be	AUX
cana-1179	12	12	crucial	crucial	ADJ
cana-1179	12	13	.	.	PUNCT
cana-1179	13	1	numerous	numerous	ADJ
cana-1179	13	2	research	research	NOUN
cana-1179	13	3	articles	article	NOUN
cana-1179	13	4	analyzing	analyze	VERB
cana-1179	13	5	plenty	plenty	NOUN
cana-1179	13	6	of	of	ADP
cana-1179	13	7	generalized	generalized	ADJ
cana-1179	13	8	closed	closed	ADJ
cana-1179	13	9	sets	set	NOUN
cana-1179	13	10	were	be	AUX
cana-1179	13	11	produced	produce	VERB
cana-1179	13	12	following	follow	VERB
cana-1179	13	13	the	the	DET
cana-1179	13	14	arrival	arrival	NOUN
cana-1179	13	15	of	of	ADP
cana-1179	13	16	these	these	DET
cana-1179	13	17	sets	set	NOUN
cana-1179	13	18	.	.	PUNCT
cana-1179	14	1	a	a	DET
cana-1179	14	2	non	non	ADJ
cana-1179	14	3	-	-	ADJ
cana-1179	14	4	empty	empty	ADJ
cana-1179	14	5	set	set	NOUN
cana-1179	14	6	of	of	ADP
cana-1179	14	7	x	x	PUNCT
cana-1179	14	8	subsets	subset	NOUN
cana-1179	14	9	that	that	PRON
cana-1179	14	10	is	be	AUX
cana-1179	14	11	closed	close	VERB
cana-1179	14	12	with	with	ADP
cana-1179	14	13	respect	respect	NOUN
cana-1179	14	14	to	to	ADP
cana-1179	14	15	finite	finite	VERB
cana-1179	14	16	union	union	NOUN
cana-1179	14	17	is	be	AUX
cana-1179	14	18	called	call	VERB
cana-1179	14	19	an	an	DET
cana-1179	14	20	ideal	ideal	NOUN
cana-1179	14	21	,	,	PUNCT
cana-1179	14	22	i.	i.	PROPN
cana-1179	14	23	it	it	PRON
cana-1179	14	24	is	be	AUX
cana-1179	14	25	recognized	recognize	VERB
cana-1179	14	26	as	as	ADP
cana-1179	14	27	an	an	DET
cana-1179	14	28	ideal	ideal	ADJ
cana-1179	14	29	space	space	NOUN
cana-1179	14	30	since	since	SCONJ
cana-1179	14	31	(	(	PUNCT
cana-1179	14	32	x	x	X
cana-1179	14	33	,	,	PUNCT
cana-1179	14	34	τ	τ	PROPN
cana-1179	14	35	,	,	PUNCT
cana-1179	14	36	i	i	PROPN
cana-1179	14	37	)	)	PUNCT
cana-1179	14	38	is	be	AUX
cana-1179	14	39	an	an	DET
cana-1179	14	40	ideal	ideal	ADJ
cana-1179	14	41	topological	topological	ADJ
cana-1179	14	42	space	space	NOUN
cana-1179	14	43	.	.	PUNCT
cana-1179	15	1	the	the	DET
cana-1179	15	2	local	local	ADJ
cana-1179	15	3	function	function	NOUN
cana-1179	15	4	of	of	ADP
cana-1179	15	5	a	a	PRON
cana-1179	15	6	for	for	ADP
cana-1179	15	7	a	a	DET
cana-1179	15	8	subset	subset	NOUN
cana-1179	15	9	a	a	PRON
cana-1179	15	10	of	of	ADP
cana-1179	15	11	x	x	PROPN
cana-1179	15	12	is	be	AUX
cana-1179	15	13	given	give	VERB
cana-1179	15	14	by	by	ADP
cana-1179	15	15	a	a	PRON
cana-1179	15	16	*	*	X
cana-1179	15	17	=	=	SYM
cana-1179	15	18	{	{	PUNCT
cana-1179	15	19	x	x	PUNCT
cana-1179	15	20	∈	∈	PROPN
cana-1179	15	21	x	x	NOUN
cana-1179	15	22	:	:	PUNCT
cana-1179	15	23	u	u	NOUN
cana-1179	15	24	∩	∩	NOUN
cana-1179	15	25	a	a	DET
cana-1179	15	26	∉	∉	PROPN
cana-1179	15	27	i	i	PROPN
cana-1179	15	28	for	for	ADP
cana-1179	15	29	each	each	DET
cana-1179	15	30	u	u	PROPN
cana-1179	15	31	∈	∈	NOUN
cana-1179	15	32	τ(x	τ(x	NOUN
cana-1179	15	33	)	)	PUNCT
cana-1179	15	34	}	}	PUNCT
cana-1179	15	35	,	,	PUNCT
cana-1179	15	36	where	where	SCONJ
cana-1179	15	37	τ(x	τ(x	NOUN
cana-1179	15	38	)	)	PUNCT
cana-1179	15	39	is	be	AUX
cana-1179	15	40	the	the	DET
cana-1179	15	41	set	set	NOUN
cana-1179	15	42	of	of	ADP
cana-1179	15	43	all	all	DET
cana-1179	15	44	non	non	ADJ
cana-1179	15	45	-	-	ADJ
cana-1179	15	46	empty	empty	ADJ
cana-1179	15	47	open	open	ADJ
cana-1179	15	48	sets	set	NOUN
cana-1179	15	49	where	where	SCONJ
cana-1179	15	50	x	x	PRON
cana-1179	15	51	occurs	occur	VERB
cana-1179	15	52	.	.	PUNCT
cana-1179	16	1	to	to	PART
cana-1179	16	2	avoid	avoid	VERB
cana-1179	16	3	any	any	DET
cana-1179	16	4	confusion	confusion	NOUN
cana-1179	16	5	,	,	PUNCT
cana-1179	16	6	just	just	ADV
cana-1179	16	7	write	write	VERB
cana-1179	16	8	a	a	PRON
cana-1179	16	9	*	*	PUNCT
cana-1179	16	10	from	from	ADP
cana-1179	16	11	this	this	PRON
cana-1179	16	12	instead	instead	ADV
cana-1179	16	13	of	of	ADP
cana-1179	16	14	a*(i	a*(i	NOUN
cana-1179	16	15	)	)	PUNCT
cana-1179	16	16	.	.	PUNCT
cana-1179	17	1	a	a	DET
cana-1179	17	2	kuratowski	kuratowski	ADJ
cana-1179	17	3	closure	closure	NOUN
cana-1179	17	4	operator	operator	NOUN
cana-1179	17	5	cl	cl	NOUN
cana-1179	17	6	*	*	PUNCT
cana-1179	17	7	(	(	PUNCT
cana-1179	17	8	.	.	PUNCT
cana-1179	17	9	)	)	PUNCT
cana-1179	17	10	for	for	ADP
cana-1179	17	11	a	a	DET
cana-1179	17	12	topology	topology	NOUN
cana-1179	17	13	τ*(i	τ*(i	NOUN
cana-1179	17	14	,	,	PUNCT
cana-1179	17	15	τ	τ	X
cana-1179	17	16	)	)	PUNCT
cana-1179	17	17	is	be	AUX
cana-1179	17	18	established	establish	VERB
cana-1179	17	19	cl*(a	cl*(a	NOUN
cana-1179	17	20	)	)	PUNCT
cana-1179	17	21	=	=	PUNCT
cana-1179	17	22	a	a	PRON
cana-1179	17	23	∪	∪	NOUN
cana-1179	17	24	a	a	PRON
cana-1179	17	25	*	*	PUNCT
cana-1179	17	26	,	,	PUNCT
cana-1179	17	27	which	which	PRON
cana-1179	17	28	is	be	AUX
cana-1179	17	29	finer	fine	ADJ
cana-1179	17	30	than	than	ADP
cana-1179	17	31	τ	τ	PROPN
cana-1179	17	32	.	.	PUNCT
cana-1179	18	1	whenever	whenever	SCONJ
cana-1179	18	2	a	a	PRON
cana-1179	18	3	is	be	AUX
cana-1179	18	4	contained	contain	VERB
cana-1179	18	5	in	in	ADP
cana-1179	18	6	x	x	X
cana-1179	18	7	,	,	PUNCT
cana-1179	18	8	a	a	DET
cana-1179	18	9	’s	’s	NOUN
cana-1179	18	10	closure	closure	NOUN
cana-1179	18	11	and	and	CCONJ
cana-1179	18	12	interior	interior	ADJ
cana-1179	18	13	in	in	ADP
cana-1179	18	14	(	(	PUNCT
cana-1179	18	15	x	x	NOUN
cana-1179	18	16	,	,	PUNCT
cana-1179	18	17	τ	τ	X
cana-1179	18	18	)	)	PUNCT
cana-1179	18	19	are	be	AUX
cana-1179	18	20	indicated	indicate	VERB
cana-1179	18	21	by	by	ADP
cana-1179	18	22	cl(a	cl(a	NUM
cana-1179	18	23	)	)	PUNCT
cana-1179	18	24	and	and	CCONJ
cana-1179	18	25	int(a	int(a	PROPN
cana-1179	18	26	)	)	PUNCT
cana-1179	18	27	,	,	PUNCT
cana-1179	18	28	respectively	respectively	ADV
cana-1179	18	29	and	and	CCONJ
cana-1179	18	30	a	a	PRON
cana-1179	18	31	's	's	PART
cana-1179	18	32	closure	closure	NOUN
cana-1179	18	33	and	and	CCONJ
cana-1179	18	34	interior	interior	ADJ
cana-1179	18	35	in	in	ADP
cana-1179	18	36	(	(	PUNCT
cana-1179	18	37	x	x	X
cana-1179	18	38	*	*	PROPN
cana-1179	18	39	,	,	PUNCT
cana-1179	18	40	τ	τ	X
cana-1179	18	41	)	)	PUNCT
cana-1179	18	42	are	be	AUX
cana-1179	18	43	shown	show	VERB
cana-1179	18	44	by	by	ADP
cana-1179	18	45	cl*(a	cl*(a	PRON
cana-1179	18	46	)	)	PUNCT
cana-1179	18	47	and	and	CCONJ
cana-1179	18	48	int*(a	int*(a	NOUN
cana-1179	18	49	)	)	PUNCT
cana-1179	18	50	.	.	PUNCT
cana-1179	19	1	in	in	ADP
cana-1179	19	2	this	this	DET
cana-1179	19	3	work	work	NOUN
cana-1179	19	4	,	,	PUNCT
cana-1179	19	5	the	the	DET
cana-1179	19	6	concept	concept	NOUN
cana-1179	19	7	of	of	ADP
cana-1179	19	8	βg*p	βg*p	ADJ
cana-1179	19	9	-	-	PUNCT
cana-1179	19	10	closed	closed	ADJ
cana-1179	19	11	sets	set	NOUN
cana-1179	19	12	in	in	ADP
cana-1179	19	13	topological	topological	ADJ
cana-1179	19	14	spaces	space	NOUN
cana-1179	19	15	and	and	CCONJ
cana-1179	19	16	βg*p	βg*p	PROPN
cana-1179	19	17	-	-	PUNCT
cana-1179	19	18	i	i	NOUN
cana-1179	19	19	-	-	PUNCT
cana-1179	19	20	closed	close	VERB
cana-1179	19	21	sets	set	NOUN
cana-1179	19	22	in	in	ADP
cana-1179	19	23	ideal	ideal	ADJ
cana-1179	19	24	topological	topological	ADJ
cana-1179	19	25	spaces	space	NOUN
cana-1179	19	26	were	be	AUX
cana-1179	19	27	presented	present	VERB
cana-1179	19	28	and	and	CCONJ
cana-1179	19	29	examined	examine	VERB
cana-1179	19	30	.	.	PUNCT
cana-1179	20	1	2	2	X
cana-1179	20	2	.	.	X
cana-1179	20	3	preliminaries	preliminary	NOUN
cana-1179	20	4	definition	definition	NOUN
cana-1179	20	5	2.1	2.1	NUM
cana-1179	20	6	[	[	X
cana-1179	20	7	1,2	1,2	NUM
cana-1179	20	8	]	]	PUNCT
cana-1179	20	9	in	in	ADP
cana-1179	20	10	topological	topological	ADJ
cana-1179	20	11	space	space	NOUN
cana-1179	20	12	x	x	NOUN
cana-1179	20	13	,	,	PUNCT
cana-1179	20	14	a	a	DET
cana-1179	20	15	subset	subset	NOUN
cana-1179	20	16	a	a	PRON
cana-1179	20	17	is	be	AUX
cana-1179	20	18	termed	term	VERB
cana-1179	20	19	as	as	ADP
cana-1179	20	20	i.it	i.it	PROPN
cana-1179	20	21	is	be	AUX
cana-1179	20	22	semi	semi	ADJ
cana-1179	20	23	-	-	ADJ
cana-1179	20	24	closed	closed	ADJ
cana-1179	20	25	if	if	SCONJ
cana-1179	20	26	int(cl(a	int(cl(a	PROPN
cana-1179	20	27	)	)	PUNCT
cana-1179	20	28	)	)	PUNCT
cana-1179	21	1	⊆	⊆	NUM
cana-1179	21	2	a	a	PRON
cana-1179	21	3	and	and	CCONJ
cana-1179	21	4	semi	semi	ADJ
cana-1179	21	5	-	-	ADJ
cana-1179	21	6	open	open	ADJ
cana-1179	21	7	if	if	SCONJ
cana-1179	21	8	a	a	DET
cana-1179	21	9	⊆	⊆	NUM
cana-1179	21	10	cl(int(a	cl(int(a	NOUN
cana-1179	21	11	)	)	PUNCT
cana-1179	21	12	)	)	PUNCT
cana-1179	21	13	.	.	PUNCT
cana-1179	22	1	ii.pre	ii.pre	NOUN
cana-1179	22	2	-	-	PUNCT
cana-1179	22	3	closed	close	VERB
cana-1179	22	4	if	if	SCONJ
cana-1179	22	5	cl(int(a	cl(int(a	NOUN
cana-1179	22	6	)	)	PUNCT
cana-1179	22	7	)	)	PUNCT
cana-1179	23	1	⊆	⊆	NUM
cana-1179	23	2	a	a	PRON
cana-1179	23	3	and	and	CCONJ
cana-1179	23	4	pre	pre	ADJ
cana-1179	23	5	-	-	VERB
cana-1179	23	6	open	open	ADJ
cana-1179	23	7	if	if	SCONJ
cana-1179	23	8	a	a	DET
cana-1179	23	9	⊆	⊆	NUM
cana-1179	23	10	int(cl(a	int(cl(a	PROPN
cana-1179	23	11	)	)	PUNCT
cana-1179	23	12	)	)	PUNCT
cana-1179	23	13	.	.	PUNCT
cana-1179	24	1	iii.α	iii.α	NOUN
cana-1179	24	2	-	-	PUNCT
cana-1179	24	3	closed	closed	ADJ
cana-1179	24	4	if	if	SCONJ
cana-1179	24	5	cl(int(cl(a	cl(int(cl(a	NOUN
cana-1179	24	6	)	)	PUNCT
cana-1179	24	7	)	)	PUNCT
cana-1179	24	8	)	)	PUNCT
cana-1179	25	1	⊆	⊆	NUM
cana-1179	25	2	a	a	PRON
cana-1179	25	3	and	and	CCONJ
cana-1179	25	4	α	α	NOUN
cana-1179	25	5	-	-	NOUN
cana-1179	25	6	open	open	ADJ
cana-1179	25	7	if	if	SCONJ
cana-1179	25	8	a	a	DET
cana-1179	25	9	⊆	⊆	NUM
cana-1179	25	10	int(cl(int(a	int(cl(int(a	NOUN
cana-1179	25	11	)	)	PUNCT
cana-1179	25	12	)	)	PUNCT
cana-1179	25	13	)	)	PUNCT
cana-1179	25	14	.	.	PUNCT
cana-1179	26	1	iv.if	iv.if	PROPN
cana-1179	26	2	a	a	PRON
cana-1179	26	3	is	be	AUX
cana-1179	26	4	regular	regular	ADV
cana-1179	26	5	-	-	PUNCT
cana-1179	26	6	closed	close	VERB
cana-1179	26	7	a	a	DET
cana-1179	26	8	=	=	NOUN
cana-1179	26	9	cl(int(a	cl(int(a	NOUN
cana-1179	26	10	)	)	PUNCT
cana-1179	26	11	)	)	PUNCT
cana-1179	27	1	and	and	CCONJ
cana-1179	27	2	if	if	SCONJ
cana-1179	27	3	regular	regular	ADJ
cana-1179	27	4	-	-	PUNCT
cana-1179	27	5	open	open	VERB
cana-1179	27	6	a	a	DET
cana-1179	27	7	=	=	SYM
cana-1179	27	8	int(cl(a	int(cl(a	PROPN
cana-1179	27	9	)	)	PUNCT
cana-1179	27	10	)	)	PUNCT
cana-1179	27	11	.	.	PUNCT
cana-1179	28	1	communications	communication	NOUN
cana-1179	28	2	on	on	ADP
cana-1179	28	3	applied	apply	VERB
cana-1179	28	4	nonlinear	nonlinear	ADJ
cana-1179	28	5	analysis	analysis	NOUN
cana-1179	28	6	issn	issn	NOUN
cana-1179	28	7	:	:	PUNCT
cana-1179	28	8	1074	1074	NUM
cana-1179	28	9	-	-	PUNCT
cana-1179	28	10	133x	133x	NUM
cana-1179	28	11	vol	vol	NOUN
cana-1179	28	12	31	31	NUM
cana-1179	28	13	no	no	NOUN
cana-1179	28	14	.	.	PUNCT
cana-1179	29	1	6s	6s	NUM
cana-1179	29	2	(	(	PUNCT
cana-1179	29	3	2024	2024	NUM
cana-1179	29	4	)	)	PUNCT
cana-1179	29	5	208	208	NUM
cana-1179	29	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1179	29	7	v.β	v.β	PROPN
cana-1179	29	8	-	-	PUNCT
cana-1179	29	9	closed(semi	closed(semi	PROPN
cana-1179	29	10	-	-	PUNCT
cana-1179	29	11	pre	pre	ADJ
cana-1179	29	12	-	-	ADJ
cana-1179	29	13	closed	closed	ADJ
cana-1179	29	14	)	)	PUNCT
cana-1179	29	15	if	if	SCONJ
cana-1179	29	16	int(cl(int(a	int(cl(int(a	PROPN
cana-1179	29	17	)	)	PUNCT
cana-1179	29	18	)	)	PUNCT
cana-1179	29	19	)	)	PUNCT
cana-1179	30	1	⊆	⊆	NUM
cana-1179	30	2	a	a	DET
cana-1179	30	3	and	and	CCONJ
cana-1179	30	4	β	β	NOUN
cana-1179	30	5	-	-	PUNCT
cana-1179	30	6	open(semi	open(semi	PROPN
cana-1179	30	7	-	-	PUNCT
cana-1179	30	8	pre	pre	NOUN
cana-1179	30	9	-	-	ADJ
cana-1179	30	10	open	open	ADJ
cana-1179	30	11	)	)	PUNCT
cana-1179	30	12	if	if	SCONJ
cana-1179	30	13	a	a	DET
cana-1179	30	14	⊆	⊆	NUM
cana-1179	30	15	cl(int(cl(a	cl(int(cl(a	NOUN
cana-1179	30	16	)	)	PUNCT
cana-1179	30	17	)	)	PUNCT
cana-1179	30	18	)	)	PUNCT
cana-1179	30	19	.	.	PUNCT
cana-1179	31	1	definition	definition	NOUN
cana-1179	31	2	2.2	2.2	NUM
cana-1179	31	3	[	[	X
cana-1179	31	4	5,6,17,18,19	5,6,17,18,19	NUM
cana-1179	31	5	]	]	X
cana-1179	31	6	a	a	DET
cana-1179	31	7	topological	topological	ADJ
cana-1179	31	8	space	space	NOUN
cana-1179	31	9	(	(	PUNCT
cana-1179	31	10	x	x	X
cana-1179	31	11	,	,	PUNCT
cana-1179	31	12	τ	τ	X
cana-1179	31	13	)	)	PUNCT
cana-1179	31	14	subset	subset	VERB
cana-1179	31	15	a	a	PRON
cana-1179	31	16	is	be	AUX
cana-1179	31	17	referred	refer	VERB
cana-1179	31	18	to	to	ADP
cana-1179	31	19	as	as	ADP
cana-1179	31	20	i.if	i.if	PROPN
cana-1179	31	21	cl(a	cl(a	X
cana-1179	31	22	)	)	PUNCT
cana-1179	31	23	⊆	⊆	NUM
cana-1179	31	24	u	u	NOUN
cana-1179	31	25	whenever	whenever	SCONJ
cana-1179	31	26	a	a	PRON
cana-1179	31	27	is	be	AUX
cana-1179	31	28	a	a	DET
cana-1179	31	29	subset	subset	NOUN
cana-1179	31	30	of	of	ADP
cana-1179	31	31	u	u	NOUN
cana-1179	31	32	and	and	CCONJ
cana-1179	31	33	u	u	NOUN
cana-1179	31	34	is	be	AUX
cana-1179	31	35	open	open	ADJ
cana-1179	31	36	in	in	ADP
cana-1179	31	37	the	the	DET
cana-1179	31	38	space	space	NOUN
cana-1179	31	39	x	x	NOUN
cana-1179	31	40	,	,	PUNCT
cana-1179	31	41	then	then	ADV
cana-1179	31	42	the	the	DET
cana-1179	31	43	set	set	NOUN
cana-1179	31	44	is	be	AUX
cana-1179	31	45	generalized	generalize	VERB
cana-1179	31	46	closed	close	VERB
cana-1179	31	47	(	(	PUNCT
cana-1179	31	48	g	g	NOUN
cana-1179	31	49	-	-	PUNCT
cana-1179	31	50	closed	closed	ADJ
cana-1179	31	51	)	)	PUNCT
cana-1179	31	52	set	set	NOUN
cana-1179	31	53	.	.	PUNCT
cana-1179	32	1	ii.when	ii.when	ADV
cana-1179	32	2	a	a	PRON
cana-1179	32	3	is	be	AUX
cana-1179	32	4	a	a	DET
cana-1179	32	5	subset	subset	NOUN
cana-1179	32	6	of	of	ADP
cana-1179	32	7	u	u	NOUN
cana-1179	32	8	and	and	CCONJ
cana-1179	32	9	u	u	NOUN
cana-1179	32	10	is	be	AUX
cana-1179	32	11	semi	semi	ADV
cana-1179	32	12	open	open	ADJ
cana-1179	32	13	in	in	ADP
cana-1179	32	14	the	the	DET
cana-1179	32	15	space	space	NOUN
cana-1179	32	16	x	x	NOUN
cana-1179	32	17	,	,	PUNCT
cana-1179	32	18	then	then	ADV
cana-1179	32	19	scl(a	scl(a	PROPN
cana-1179	32	20	)	)	PUNCT
cana-1179	32	21	⊆	⊆	NUM
cana-1179	32	22	u	u	NOUN
cana-1179	32	23	denotes	denote	VERB
cana-1179	32	24	a	a	DET
cana-1179	32	25	semi	semi	ADV
cana-1179	32	26	generalized	generalized	ADJ
cana-1179	32	27	closed	close	VERB
cana-1179	32	28	(	(	PUNCT
cana-1179	32	29	sg	sg	NOUN
cana-1179	32	30	-	-	PUNCT
cana-1179	32	31	closed	closed	ADJ
cana-1179	32	32	)	)	PUNCT
cana-1179	32	33	set	set	NOUN
cana-1179	32	34	.	.	PUNCT
cana-1179	33	1	iii	iii	X
cana-1179	33	2	.	.	PUNCT
cana-1179	34	1	when	when	SCONJ
cana-1179	34	2	a	a	PRON
cana-1179	34	3	is	be	AUX
cana-1179	34	4	a	a	DET
cana-1179	34	5	subset	subset	NOUN
cana-1179	34	6	of	of	ADP
cana-1179	34	7	u	u	NOUN
cana-1179	34	8	and	and	CCONJ
cana-1179	34	9	u	u	NOUN
cana-1179	34	10	is	be	AUX
cana-1179	34	11	open	open	ADJ
cana-1179	34	12	in	in	ADP
cana-1179	34	13	the	the	DET
cana-1179	34	14	space	space	NOUN
cana-1179	34	15	x	x	NOUN
cana-1179	34	16	,	,	PUNCT
cana-1179	34	17	then	then	ADV
cana-1179	34	18	scl(a	scl(a	PROPN
cana-1179	34	19	)	)	PUNCT
cana-1179	34	20	⊆	⊆	NUM
cana-1179	34	21	u	u	NOUN
cana-1179	34	22	denotes	denote	VERB
cana-1179	34	23	a	a	DET
cana-1179	34	24	generalized	generalized	ADJ
cana-1179	34	25	semi	semi	ADV
cana-1179	34	26	closed	closed	ADJ
cana-1179	34	27	(	(	PUNCT
cana-1179	34	28	gs	gs	NOUN
cana-1179	34	29	-	-	PUNCT
cana-1179	34	30	closed	closed	ADJ
cana-1179	34	31	)	)	PUNCT
cana-1179	34	32	set	set	NOUN
cana-1179	34	33	.	.	PUNCT
cana-1179	35	1	iv.if	iv.if	PROPN
cana-1179	35	2	αcl(a	αcl(a	NUM
cana-1179	35	3	)	)	PUNCT
cana-1179	35	4	⊆	⊆	NUM
cana-1179	35	5	u	u	NOUN
cana-1179	35	6	whenever	whenever	SCONJ
cana-1179	35	7	a	a	PRON
cana-1179	35	8	is	be	AUX
cana-1179	35	9	a	a	DET
cana-1179	35	10	subset	subset	NOUN
cana-1179	35	11	of	of	ADP
cana-1179	35	12	u	u	NOUN
cana-1179	35	13	and	and	CCONJ
cana-1179	35	14	u	u	NOUN
cana-1179	35	15	is	be	AUX
cana-1179	35	16	open	open	ADJ
cana-1179	35	17	in	in	ADP
cana-1179	35	18	the	the	DET
cana-1179	35	19	space	space	NOUN
cana-1179	35	20	x	x	NOUN
cana-1179	35	21	,	,	PUNCT
cana-1179	35	22	then	then	ADV
cana-1179	35	23	the	the	DET
cana-1179	35	24	set	set	NOUN
cana-1179	35	25	is	be	AUX
cana-1179	35	26	α	α	PRON
cana-1179	35	27	generalized	generalize	VERB
cana-1179	35	28	closed	close	VERB
cana-1179	35	29	(	(	PUNCT
cana-1179	35	30	αg	αg	NOUN
cana-1179	35	31	-	-	PUNCT
cana-1179	35	32	closed	closed	ADJ
cana-1179	35	33	)	)	PUNCT
cana-1179	35	34	set	set	NOUN
cana-1179	35	35	.	.	PUNCT
cana-1179	36	1	v.if	v.if	PROPN
cana-1179	36	2	αcl(a	αcl(a	PROPN
cana-1179	36	3	)	)	PUNCT
cana-1179	36	4	⊆	⊆	NUM
cana-1179	36	5	u	u	NOUN
cana-1179	36	6	whenever	whenever	SCONJ
cana-1179	36	7	a	a	PRON
cana-1179	36	8	is	be	AUX
cana-1179	36	9	a	a	DET
cana-1179	36	10	subset	subset	NOUN
cana-1179	36	11	of	of	ADP
cana-1179	36	12	u	u	NOUN
cana-1179	36	13	and	and	CCONJ
cana-1179	36	14	u	u	NOUN
cana-1179	36	15	is	be	AUX
cana-1179	36	16	α	α	NOUN
cana-1179	36	17	-	-	NOUN
cana-1179	36	18	open	open	ADJ
cana-1179	36	19	in	in	ADP
cana-1179	36	20	the	the	DET
cana-1179	36	21	space	space	NOUN
cana-1179	36	22	x	x	NOUN
cana-1179	36	23	,	,	PUNCT
cana-1179	36	24	then	then	ADV
cana-1179	36	25	the	the	DET
cana-1179	36	26	set	set	NOUN
cana-1179	36	27	is	be	AUX
cana-1179	36	28	generalized	generalize	VERB
cana-1179	36	29	α	α	PRON
cana-1179	36	30	-	-	ADJ
cana-1179	36	31	closed	closed	ADJ
cana-1179	36	32	(	(	PUNCT
cana-1179	36	33	gα	gα	NOUN
cana-1179	36	34	-	-	PUNCT
cana-1179	36	35	closed	closed	ADJ
cana-1179	36	36	)	)	PUNCT
cana-1179	36	37	set	set	NOUN
cana-1179	36	38	.	.	PUNCT
cana-1179	37	1	vi.when	vi.when	ADV
cana-1179	37	2	a	a	PRON
cana-1179	37	3	is	be	AUX
cana-1179	37	4	a	a	DET
cana-1179	37	5	subset	subset	NOUN
cana-1179	37	6	of	of	ADP
cana-1179	37	7	u	u	NOUN
cana-1179	37	8	and	and	CCONJ
cana-1179	37	9	u	u	NOUN
cana-1179	37	10	is	be	AUX
cana-1179	37	11	open	open	ADJ
cana-1179	37	12	in	in	ADP
cana-1179	37	13	the	the	DET
cana-1179	37	14	space	space	NOUN
cana-1179	37	15	x	x	NOUN
cana-1179	37	16	,	,	PUNCT
cana-1179	37	17	then	then	ADV
cana-1179	37	18	spcl(a	spcl(a	NUM
cana-1179	37	19	)	)	PUNCT
cana-1179	37	20	⊆	⊆	NUM
cana-1179	37	21	u	u	NOUN
cana-1179	37	22	denotes	denote	VERB
cana-1179	37	23	a	a	DET
cana-1179	37	24	generalized	generalized	ADJ
cana-1179	37	25	semi	semi	ADJ
cana-1179	37	26	-	-	ADJ
cana-1179	37	27	pre	pre	ADJ
cana-1179	37	28	-	-	ADJ
cana-1179	37	29	closed	closed	ADJ
cana-1179	37	30	(	(	PUNCT
cana-1179	37	31	gsp	gsp	VERB
cana-1179	37	32	-	-	PUNCT
cana-1179	37	33	closed	closed	ADJ
cana-1179	37	34	)	)	PUNCT
cana-1179	37	35	set	set	NOUN
cana-1179	37	36	.	.	PUNCT
cana-1179	38	1	vii.when	vii.when	ADV
cana-1179	38	2	a	a	PRON
cana-1179	38	3	is	be	AUX
cana-1179	38	4	a	a	DET
cana-1179	38	5	subset	subset	NOUN
cana-1179	38	6	of	of	ADP
cana-1179	38	7	u	u	NOUN
cana-1179	38	8	and	and	CCONJ
cana-1179	38	9	u	u	NOUN
cana-1179	38	10	is	be	AUX
cana-1179	38	11	regular	regular	ADJ
cana-1179	38	12	open	open	ADJ
cana-1179	38	13	in	in	ADP
cana-1179	38	14	the	the	DET
cana-1179	38	15	space	space	NOUN
cana-1179	38	16	x	x	NOUN
cana-1179	38	17	,	,	PUNCT
cana-1179	38	18	then	then	ADV
cana-1179	38	19	pcl(a	pcl(a	NUM
cana-1179	38	20	)	)	PUNCT
cana-1179	39	1	⊆	⊆	NUM
cana-1179	39	2	u	u	NOUN
cana-1179	39	3	denotes	denote	VERB
cana-1179	39	4	a	a	DET
cana-1179	39	5	generalized	generalized	ADJ
cana-1179	39	6	pre	pre	ADJ
cana-1179	39	7	-	-	ADJ
cana-1179	39	8	regular	regular	ADJ
cana-1179	39	9	-	-	PUNCT
cana-1179	39	10	closed	close	VERB
cana-1179	39	11	(	(	PUNCT
cana-1179	39	12	gpr	gpr	NOUN
cana-1179	39	13	-	-	PUNCT
cana-1179	39	14	closed	closed	ADJ
cana-1179	39	15	)	)	PUNCT
cana-1179	39	16	set	set	NOUN
cana-1179	39	17	.	.	PUNCT
cana-1179	40	1	viii	viii	PROPN
cana-1179	40	2	.	.	PUNCT
cana-1179	41	1	if	if	SCONJ
cana-1179	41	2	cl(a	cl(a	NUM
cana-1179	41	3	)	)	PUNCT
cana-1179	41	4	⊆	⊆	NUM
cana-1179	41	5	u	u	NOUN
cana-1179	41	6	whenever	whenever	SCONJ
cana-1179	41	7	a	a	PRON
cana-1179	41	8	is	be	AUX
cana-1179	41	9	a	a	DET
cana-1179	41	10	subset	subset	NOUN
cana-1179	41	11	of	of	ADP
cana-1179	41	12	u	u	NOUN
cana-1179	41	13	and	and	CCONJ
cana-1179	41	14	u	u	NOUN
cana-1179	41	15	is	be	AUX
cana-1179	41	16	regular	regular	ADJ
cana-1179	41	17	open	open	ADJ
cana-1179	41	18	in	in	ADP
cana-1179	41	19	the	the	DET
cana-1179	41	20	space	space	NOUN
cana-1179	41	21	x	x	NOUN
cana-1179	41	22	,	,	PUNCT
cana-1179	41	23	then	then	ADV
cana-1179	41	24	the	the	DET
cana-1179	41	25	set	set	NOUN
cana-1179	41	26	is	be	AUX
cana-1179	41	27	regular	regular	ADJ
cana-1179	41	28	generalized	generalize	VERB
cana-1179	41	29	closed	close	VERB
cana-1179	41	30	(	(	PUNCT
cana-1179	41	31	rg	rg	NOUN
cana-1179	41	32	-	-	PUNCT
cana-1179	41	33	closed	closed	ADJ
cana-1179	41	34	)	)	PUNCT
cana-1179	41	35	set	set	NOUN
cana-1179	41	36	.	.	PUNCT
cana-1179	42	1	ix.if	ix.if	PROPN
cana-1179	42	2	cl(int(a	cl(int(a	PROPN
cana-1179	42	3	)	)	PUNCT
cana-1179	42	4	)	)	PUNCT
cana-1179	43	1	⊆	⊆	X
cana-1179	43	2	u	u	NOUN
cana-1179	43	3	whenever	whenever	SCONJ
cana-1179	43	4	a	a	PRON
cana-1179	43	5	is	be	AUX
cana-1179	43	6	a	a	DET
cana-1179	43	7	subset	subset	NOUN
cana-1179	43	8	of	of	ADP
cana-1179	43	9	u	u	NOUN
cana-1179	43	10	and	and	CCONJ
cana-1179	43	11	u	u	NOUN
cana-1179	43	12	is	be	AUX
cana-1179	43	13	open	open	ADJ
cana-1179	43	14	in	in	ADP
cana-1179	43	15	the	the	DET
cana-1179	43	16	space	space	NOUN
cana-1179	43	17	x	x	NOUN
cana-1179	43	18	,	,	PUNCT
cana-1179	43	19	then	then	ADV
cana-1179	43	20	the	the	DET
cana-1179	43	21	set	set	NOUN
cana-1179	43	22	is	be	AUX
cana-1179	43	23	weakly	weakly	ADV
cana-1179	43	24	generalized	generalized	ADJ
cana-1179	43	25	(	(	PUNCT
cana-1179	43	26	wg	wg	NOUN
cana-1179	43	27	-	-	PUNCT
cana-1179	43	28	closed	closed	ADJ
cana-1179	43	29	)	)	PUNCT
cana-1179	43	30	set	set	NOUN
cana-1179	43	31	.	.	PUNCT
cana-1179	44	1	x.a	x.a	PROPN
cana-1179	44	2	strongly	strongly	ADV
cana-1179	44	3	generalized	generalize	VERB
cana-1179	44	4	closed	close	VERB
cana-1179	44	5	(	(	PUNCT
cana-1179	44	6	g*-closed	g*-closed	ADJ
cana-1179	44	7	)	)	PUNCT
cana-1179	44	8	set	set	VERB
cana-1179	44	9	if	if	SCONJ
cana-1179	44	10	cl(a	cl(a	NUM
cana-1179	44	11	)	)	PUNCT
cana-1179	44	12	⊆	⊆	NUM
cana-1179	44	13	u	u	NOUN
cana-1179	44	14	whenever	whenever	SCONJ
cana-1179	44	15	a	a	PRON
cana-1179	44	16	is	be	AUX
cana-1179	44	17	a	a	DET
cana-1179	44	18	subset	subset	NOUN
cana-1179	44	19	of	of	ADP
cana-1179	44	20	u	u	NOUN
cana-1179	44	21	and	and	CCONJ
cana-1179	44	22	u	u	NOUN
cana-1179	44	23	is	be	AUX
cana-1179	44	24	g	g	NOUN
cana-1179	44	25	-	-	PUNCT
cana-1179	44	26	open	open	ADJ
cana-1179	44	27	in	in	ADP
cana-1179	44	28	the	the	DET
cana-1179	44	29	space	space	NOUN
cana-1179	44	30	x.	x.	NOUN
cana-1179	44	31	xi.when	xi.when	PUNCT
cana-1179	45	1	a	a	PRON
cana-1179	45	2	is	be	AUX
cana-1179	45	3	a	a	DET
cana-1179	45	4	subset	subset	NOUN
cana-1179	45	5	of	of	ADP
cana-1179	45	6	u	u	NOUN
cana-1179	45	7	and	and	CCONJ
cana-1179	45	8	u	u	NOUN
cana-1179	45	9	is	be	AUX
cana-1179	45	10	g	g	NOUN
cana-1179	45	11	-	-	PUNCT
cana-1179	45	12	open	open	ADJ
cana-1179	45	13	in	in	ADP
cana-1179	45	14	the	the	DET
cana-1179	45	15	space	space	NOUN
cana-1179	45	16	x	x	NOUN
cana-1179	45	17	,	,	PUNCT
cana-1179	45	18	then	then	ADV
cana-1179	45	19	(	(	PUNCT
cana-1179	45	20	cl(int(a	cl(int(a	PROPN
cana-1179	45	21	)	)	PUNCT
cana-1179	45	22	)	)	PUNCT
cana-1179	46	1	⊆	⊆	NUM
cana-1179	46	2	u	u	NOUN
cana-1179	46	3	denotes	denote	VERB
cana-1179	46	4	a	a	DET
cana-1179	46	5	mildly	mildly	ADV
cana-1179	46	6	generalized	generalized	ADJ
cana-1179	46	7	closed	close	VERB
cana-1179	46	8	(	(	PUNCT
cana-1179	46	9	mildly	mildly	ADV
cana-1179	46	10	g	g	NOUN
cana-1179	46	11	-	-	PUNCT
cana-1179	46	12	closed	closed	ADJ
cana-1179	46	13	)	)	PUNCT
cana-1179	46	14	set	set	NOUN
cana-1179	46	15	.	.	PUNCT
cana-1179	47	1	xii.when	xii.when	PUNCT
cana-1179	48	1	a	a	PRON
cana-1179	48	2	is	be	AUX
cana-1179	48	3	a	a	DET
cana-1179	48	4	subset	subset	NOUN
cana-1179	48	5	of	of	ADP
cana-1179	48	6	u	u	NOUN
cana-1179	48	7	and	and	CCONJ
cana-1179	48	8	u	u	NOUN
cana-1179	48	9	is	be	AUX
cana-1179	48	10	g	g	NOUN
cana-1179	48	11	-	-	PUNCT
cana-1179	48	12	open	open	ADJ
cana-1179	48	13	in	in	ADP
cana-1179	48	14	the	the	DET
cana-1179	48	15	space	space	NOUN
cana-1179	48	16	x	x	NOUN
cana-1179	48	17	,	,	PUNCT
cana-1179	48	18	then	then	ADV
cana-1179	48	19	pcl(a	pcl(a	NUM
cana-1179	48	20	)	)	PUNCT
cana-1179	48	21	⊆	⊆	NUM
cana-1179	48	22	u	u	NOUN
cana-1179	48	23	denotes	denote	VERB
cana-1179	48	24	a	a	DET
cana-1179	48	25	generalized	generalized	ADJ
cana-1179	48	26	star	star	NOUN
cana-1179	48	27	pre	pre	ADJ
cana-1179	48	28	-	-	ADJ
cana-1179	48	29	closed	closed	ADJ
cana-1179	48	30	(	(	PUNCT
cana-1179	48	31	g*p	g*p	NOUN
cana-1179	48	32	-	-	PUNCT
cana-1179	48	33	closed	close	VERB
cana-1179	48	34	)	)	PUNCT
cana-1179	48	35	set	set	NOUN
cana-1179	48	36	.	.	PUNCT
cana-1179	49	1	xiii	xiii	PROPN
cana-1179	49	2	.	.	PUNCT
cana-1179	50	1	if	if	SCONJ
cana-1179	50	2	βcl(a	βcl(a	NUM
cana-1179	50	3	)	)	PUNCT
cana-1179	50	4	⊆	⊆	NUM
cana-1179	50	5	u	u	NOUN
cana-1179	50	6	whenever	whenever	SCONJ
cana-1179	50	7	a	a	PRON
cana-1179	50	8	is	be	AUX
cana-1179	50	9	a	a	DET
cana-1179	50	10	subset	subset	NOUN
cana-1179	50	11	of	of	ADP
cana-1179	50	12	u	u	NOUN
cana-1179	50	13	and	and	CCONJ
cana-1179	50	14	u	u	NOUN
cana-1179	50	15	is	be	AUX
cana-1179	50	16	g	g	NOUN
cana-1179	50	17	-	-	PUNCT
cana-1179	50	18	open	open	ADJ
cana-1179	50	19	in	in	ADP
cana-1179	50	20	the	the	DET
cana-1179	50	21	space	space	NOUN
cana-1179	50	22	x	x	NOUN
cana-1179	50	23	,	,	PUNCT
cana-1179	50	24	then	then	ADV
cana-1179	50	25	the	the	DET
cana-1179	50	26	set	set	NOUN
cana-1179	50	27	is	be	AUX
cana-1179	50	28	beta	beta	NOUN
cana-1179	50	29	generalized	generalize	VERB
cana-1179	50	30	closed	close	VERB
cana-1179	50	31	(	(	PUNCT
cana-1179	50	32	βg	βg	ADV
cana-1179	50	33	-	-	PUNCT
cana-1179	50	34	closed	closed	ADJ
cana-1179	50	35	)	)	PUNCT
cana-1179	50	36	set	set	NOUN
cana-1179	50	37	.	.	PUNCT
cana-1179	51	1	xiv.if	xiv.if	PROPN
cana-1179	51	2	cl(int(a	cl(int(a	PROPN
cana-1179	51	3	)	)	PUNCT
cana-1179	51	4	)	)	PUNCT
cana-1179	52	1	⊆	⊆	X
cana-1179	52	2	u	u	NOUN
cana-1179	52	3	whenever	whenever	SCONJ
cana-1179	52	4	a	a	PRON
cana-1179	52	5	is	be	AUX
cana-1179	52	6	a	a	DET
cana-1179	52	7	subset	subset	NOUN
cana-1179	52	8	of	of	ADP
cana-1179	52	9	u	u	NOUN
cana-1179	52	10	and	and	CCONJ
cana-1179	52	11	u	u	NOUN
cana-1179	52	12	is	be	AUX
cana-1179	52	13	g	g	NOUN
cana-1179	52	14	-	-	PUNCT
cana-1179	52	15	open	open	ADJ
cana-1179	52	16	in	in	ADP
cana-1179	52	17	the	the	DET
cana-1179	52	18	space	space	NOUN
cana-1179	52	19	x	x	NOUN
cana-1179	52	20	,	,	PUNCT
cana-1179	52	21	then	then	ADV
cana-1179	52	22	the	the	DET
cana-1179	52	23	set	set	NOUN
cana-1179	52	24	is	be	AUX
cana-1179	52	25	beta	beta	NOUN
cana-1179	52	26	star	star	NOUN
cana-1179	52	27	closed	close	VERB
cana-1179	52	28	(	(	PUNCT
cana-1179	52	29	β*-closed	β*-closed	ADJ
cana-1179	52	30	)	)	PUNCT
cana-1179	52	31	set	set	NOUN
cana-1179	52	32	.	.	PUNCT
cana-1179	53	1	xv.if	xv.if	PROPN
cana-1179	53	2	gcl(a	gcl(a	PROPN
cana-1179	53	3	)	)	PUNCT
cana-1179	53	4	⊆	⊆	NUM
cana-1179	53	5	u	u	NOUN
cana-1179	53	6	whenever	whenever	SCONJ
cana-1179	53	7	a	a	PRON
cana-1179	53	8	is	be	AUX
cana-1179	53	9	a	a	DET
cana-1179	53	10	subset	subset	NOUN
cana-1179	53	11	of	of	ADP
cana-1179	53	12	u	u	NOUN
cana-1179	53	13	and	and	CCONJ
cana-1179	53	14	u	u	NOUN
cana-1179	53	15	is	be	AUX
cana-1179	53	16	β	β	X
cana-1179	53	17	-	-	VERB
cana-1179	53	18	open	open	ADJ
cana-1179	53	19	in	in	ADP
cana-1179	53	20	the	the	DET
cana-1179	53	21	space	space	NOUN
cana-1179	53	22	x	x	NOUN
cana-1179	53	23	,	,	PUNCT
cana-1179	53	24	then	then	ADV
cana-1179	53	25	the	the	DET
cana-1179	53	26	set	set	NOUN
cana-1179	53	27	is	be	AUX
cana-1179	53	28	beta	beta	ADJ
cana-1179	53	29	generalized	generalize	VERB
cana-1179	53	30	star	star	NOUN
cana-1179	53	31	closed	close	VERB
cana-1179	53	32	(	(	PUNCT
cana-1179	53	33	βg*-closed	βg*-closed	ADJ
cana-1179	53	34	)	)	PUNCT
cana-1179	53	35	set	set	NOUN
cana-1179	53	36	.	.	PUNCT
cana-1179	54	1	xvi.if	xvi.if	PROPN
cana-1179	54	2	pcl(a	pcl(a	PROPN
cana-1179	54	3	)	)	PUNCT
cana-1179	54	4	⊆	⊆	NUM
cana-1179	54	5	u	u	NOUN
cana-1179	54	6	whenever	whenever	SCONJ
cana-1179	54	7	a	a	PRON
cana-1179	54	8	is	be	AUX
cana-1179	54	9	a	a	DET
cana-1179	54	10	subset	subset	NOUN
cana-1179	54	11	of	of	ADP
cana-1179	54	12	u	u	NOUN
cana-1179	54	13	and	and	CCONJ
cana-1179	54	14	u	u	NOUN
cana-1179	54	15	is	be	AUX
cana-1179	54	16	β	β	X
cana-1179	54	17	-	-	VERB
cana-1179	54	18	open	open	ADJ
cana-1179	54	19	in	in	ADP
cana-1179	54	20	the	the	DET
cana-1179	54	21	space	space	NOUN
cana-1179	54	22	x	x	NOUN
cana-1179	54	23	,	,	PUNCT
cana-1179	54	24	then	then	ADV
cana-1179	54	25	the	the	DET
cana-1179	54	26	set	set	NOUN
cana-1179	54	27	is	be	AUX
cana-1179	54	28	beta	beta	NOUN
cana-1179	54	29	generalized	generalize	VERB
cana-1179	54	30	pre	pre	ADJ
cana-1179	54	31	-	-	ADJ
cana-1179	54	32	closed	closed	ADJ
cana-1179	54	33	(	(	PUNCT
cana-1179	54	34	βgp	βgp	ADV
cana-1179	54	35	-	-	PUNCT
cana-1179	54	36	closed	closed	ADJ
cana-1179	54	37	)	)	PUNCT
cana-1179	54	38	set	set	NOUN
cana-1179	54	39	.	.	PUNCT
cana-1179	55	1	definition	definition	NOUN
cana-1179	55	2	2.3	2.3	NUM
cana-1179	55	3	let	let	AUX
cana-1179	55	4	be	be	AUX
cana-1179	55	5	a	a	DET
cana-1179	55	6	topological	topological	ADJ
cana-1179	55	7	space	space	NOUN
cana-1179	55	8	(	(	PUNCT
cana-1179	55	9	x	x	X
cana-1179	55	10	,	,	PUNCT
cana-1179	55	11	τ	τ	PROPN
cana-1179	55	12	)	)	PUNCT
cana-1179	55	13	.	.	PUNCT
cana-1179	56	1	let	let	VERB
cana-1179	56	2	i	i	PRON
cana-1179	56	3	represent	represent	VERB
cana-1179	56	4	an	an	DET
cana-1179	56	5	ideal	ideal	NOUN
cana-1179	56	6	on	on	ADP
cana-1179	56	7	x.	x.	NOUN
cana-1179	56	8	when	when	SCONJ
cana-1179	56	9	the	the	DET
cana-1179	56	10	space	space	NOUN
cana-1179	56	11	(	(	PUNCT
cana-1179	56	12	x	x	X
cana-1179	56	13	,	,	PUNCT
cana-1179	56	14	τ	τ	PROPN
cana-1179	56	15	,	,	PUNCT
cana-1179	56	16	i	i	NOUN
cana-1179	56	17	)	)	PUNCT
cana-1179	56	18	satisfies	satisfy	VERB
cana-1179	56	19	the	the	DET
cana-1179	56	20	two	two	NUM
cana-1179	56	21	requirements	requirement	NOUN
cana-1179	56	22	,	,	PUNCT
cana-1179	56	23	it	it	PRON
cana-1179	56	24	is	be	AUX
cana-1179	56	25	referred	refer	VERB
cana-1179	56	26	to	to	ADP
cana-1179	56	27	as	as	ADP
cana-1179	56	28	an	an	DET
cana-1179	56	29	ideal	ideal	ADJ
cana-1179	56	30	topological	topological	ADJ
cana-1179	56	31	space	space	NOUN
cana-1179	56	32	,	,	PUNCT
cana-1179	56	33	i.if	i.if	PROPN
cana-1179	56	34	a	a	DET
cana-1179	56	35	∈	∈	PROPN
cana-1179	56	36	i	i	PRON
cana-1179	56	37	and	and	CCONJ
cana-1179	56	38	b	b	X
cana-1179	56	39	⊆	⊆	NUM
cana-1179	56	40	a	a	DET
cana-1179	56	41	⇒	⇒	NOUN
cana-1179	56	42	b	b	PROPN
cana-1179	56	43	∈	∈	PROPN
cana-1179	56	44	i.	i.	NOUN
cana-1179	56	45	ii.if	ii.if	PROPN
cana-1179	57	1	a	a	PRON
cana-1179	57	2	∈	∈	PROPN
cana-1179	58	1	i	i	PRON
cana-1179	58	2	and	and	CCONJ
cana-1179	58	3	b	b	X
cana-1179	58	4	∈	∈	PROPN
cana-1179	59	1	i	i	PRON
cana-1179	59	2	,	,	PUNCT
cana-1179	59	3	then	then	ADV
cana-1179	59	4	a	a	DET
cana-1179	59	5	∪	∪	X
cana-1179	59	6	b	b	PROPN
cana-1179	59	7	∈	∈	PROPN
cana-1179	59	8	i.	i.	NOUN
cana-1179	59	9	definition	definition	NOUN
cana-1179	59	10	2.4	2.4	NUM
cana-1179	59	11	[	[	SYM
cana-1179	59	12	16	16	NUM
cana-1179	59	13	]	]	X
cana-1179	59	14	an	an	DET
cana-1179	59	15	ideal	ideal	ADJ
cana-1179	59	16	topological	topological	ADJ
cana-1179	59	17	space	space	NOUN
cana-1179	59	18	(	(	PUNCT
cana-1179	59	19	x	x	X
cana-1179	59	20	,	,	PUNCT
cana-1179	59	21	τ	τ	PROPN
cana-1179	59	22	,	,	PUNCT
cana-1179	59	23	i	i	NOUN
cana-1179	59	24	)	)	PUNCT
cana-1179	59	25	subset	subset	VERB
cana-1179	59	26	a	a	PRON
cana-1179	59	27	is	be	AUX
cana-1179	59	28	referred	refer	VERB
cana-1179	59	29	to	to	ADP
cana-1179	59	30	as	as	SCONJ
cana-1179	59	31	communications	communication	NOUN
cana-1179	59	32	on	on	ADP
cana-1179	59	33	applied	apply	VERB
cana-1179	59	34	nonlinear	nonlinear	ADJ
cana-1179	59	35	analysis	analysis	NOUN
cana-1179	59	36	issn	issn	NOUN
cana-1179	59	37	:	:	PUNCT
cana-1179	59	38	1074	1074	NUM
cana-1179	59	39	-	-	PUNCT
cana-1179	59	40	133x	133x	NUM
cana-1179	59	41	vol	vol	NOUN
cana-1179	59	42	31	31	NUM
cana-1179	59	43	no	no	NOUN
cana-1179	59	44	.	.	PUNCT
cana-1179	60	1	6s	6s	NUM
cana-1179	60	2	(	(	PUNCT
cana-1179	60	3	2024	2024	NUM
cana-1179	60	4	)	)	PUNCT
cana-1179	60	5	209	209	NUM
cana-1179	60	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1179	61	1	i.if	i.if	PROPN
cana-1179	61	2	cl*(int(a	cl*(int(a	NOUN
cana-1179	61	3	)	)	PUNCT
cana-1179	61	4	)	)	PUNCT
cana-1179	62	1	⊆	⊆	NUM
cana-1179	62	2	a	a	PRON
cana-1179	62	3	,	,	PUNCT
cana-1179	62	4	then	then	ADV
cana-1179	62	5	the	the	DET
cana-1179	62	6	pre	pre	PROPN
cana-1179	62	7	-	-	ADJ
cana-1179	62	8	i	i	ADV
cana-1179	62	9	-	-	PUNCT
cana-1179	62	10	closed	close	VERB
cana-1179	62	11	set	set	NOUN
cana-1179	62	12	.	.	PUNCT
cana-1179	63	1	a	a	PRON
cana-1179	63	2	is	be	AUX
cana-1179	63	3	referred	refer	VERB
cana-1179	63	4	to	to	ADP
cana-1179	63	5	as	as	ADP
cana-1179	63	6	a	a	DET
cana-1179	63	7	pre	pre	ADJ
cana-1179	63	8	-	-	ADJ
cana-1179	63	9	i	i	PRON
cana-1179	63	10	-	-	PUNCT
cana-1179	63	11	open	open	ADJ
cana-1179	63	12	if	if	SCONJ
cana-1179	63	13	a	a	DET
cana-1179	63	14	⊆	⊆	NUM
cana-1179	63	15	(	(	PUNCT
cana-1179	63	16	int(cl*(a	int(cl*(a	NUM
cana-1179	63	17	)	)	PUNCT
cana-1179	63	18	)	)	PUNCT
cana-1179	63	19	.	.	PUNCT
cana-1179	64	1	ii.if	ii.if	PROPN
cana-1179	64	2	int(cl*(a	int(cl*(a	NUM
cana-1179	64	3	)	)	PUNCT
cana-1179	64	4	)	)	PUNCT
cana-1179	65	1	⊆	⊆	NUM
cana-1179	65	2	a	a	DET
cana-1179	65	3	then	then	ADV
cana-1179	65	4	the	the	DET
cana-1179	65	5	semi	semi	ADJ
cana-1179	65	6	-	-	ADJ
cana-1179	65	7	i	i	ADV
cana-1179	65	8	-	-	PUNCT
cana-1179	65	9	closed	close	VERB
cana-1179	65	10	set	set	NOUN
cana-1179	65	11	.	.	PUNCT
cana-1179	66	1	a	a	PRON
cana-1179	66	2	is	be	AUX
cana-1179	66	3	referred	refer	VERB
cana-1179	66	4	to	to	ADP
cana-1179	66	5	as	as	ADP
cana-1179	66	6	a	a	DET
cana-1179	66	7	semi	semi	ADJ
cana-1179	66	8	-	-	ADJ
cana-1179	66	9	i	i	PRON
cana-1179	66	10	-	-	PUNCT
cana-1179	66	11	open	open	ADJ
cana-1179	66	12	if	if	SCONJ
cana-1179	66	13	a	a	DET
cana-1179	66	14	⊆	⊆	NUM
cana-1179	66	15	(	(	PUNCT
cana-1179	66	16	cl*(int(a	cl*(int(a	NOUN
cana-1179	66	17	)	)	PUNCT
cana-1179	66	18	)	)	PUNCT
cana-1179	66	19	.	.	PUNCT
cana-1179	67	1	iii.if	iii.if	PROPN
cana-1179	67	2	cl*(int(cl*(a	cl*(int(cl*(a	PROPN
cana-1179	67	3	)	)	PUNCT
cana-1179	67	4	)	)	PUNCT
cana-1179	67	5	)	)	PUNCT
cana-1179	68	1	⊆	⊆	X
cana-1179	68	2	a	a	DET
cana-1179	68	3	then	then	ADV
cana-1179	68	4	α	α	X
cana-1179	68	5	-	-	PUNCT
cana-1179	68	6	i	i	PRON
cana-1179	68	7	-	-	PUNCT
cana-1179	68	8	closed	close	VERB
cana-1179	68	9	set	set	NOUN
cana-1179	68	10	.	.	PUNCT
cana-1179	69	1	a	a	PRON
cana-1179	69	2	is	be	AUX
cana-1179	69	3	referred	refer	VERB
cana-1179	69	4	to	to	ADP
cana-1179	69	5	as	as	ADP
cana-1179	69	6	a	a	DET
cana-1179	69	7	α	α	NOUN
cana-1179	69	8	-	-	PUNCT
cana-1179	69	9	i	i	PRON
cana-1179	69	10	-	-	PUNCT
cana-1179	69	11	open	open	ADJ
cana-1179	69	12	if	if	SCONJ
cana-1179	69	13	a	a	DET
cana-1179	69	14	⊆	⊆	NUM
cana-1179	69	15	(	(	PUNCT
cana-1179	69	16	int(cl*(int(a	int(cl*(int(a	NOUN
cana-1179	69	17	)	)	PUNCT
cana-1179	69	18	)	)	PUNCT
cana-1179	69	19	)	)	PUNCT
cana-1179	69	20	.	.	PUNCT
cana-1179	70	1	iv.the	iv.the	DET
cana-1179	70	2	closed	closed	ADJ
cana-1179	70	3	set	set	ADJ
cana-1179	70	4	β	β	X
cana-1179	70	5	-	-	PUNCT
cana-1179	70	6	i	i	PRON
cana-1179	70	7	is	be	AUX
cana-1179	70	8	if	if	SCONJ
cana-1179	70	9	(	(	PUNCT
cana-1179	70	10	int(cl*(int(a	int(cl*(int(a	PROPN
cana-1179	70	11	)	)	PUNCT
cana-1179	70	12	)	)	PUNCT
cana-1179	70	13	)	)	PUNCT
cana-1179	71	1	⊆	⊆	NUM
cana-1179	71	2	a.	a.	NOUN
cana-1179	71	3	a	a	NOUN
cana-1179	71	4	is	be	AUX
cana-1179	71	5	referred	refer	VERB
cana-1179	71	6	to	to	PART
cana-1179	71	7	be	be	AUX
cana-1179	71	8	a	a	DET
cana-1179	71	9	β	β	NOUN
cana-1179	71	10	-	-	ADJ
cana-1179	71	11	i	i	PRON
cana-1179	71	12	-	-	PUNCT
cana-1179	71	13	open	open	ADJ
cana-1179	71	14	if	if	SCONJ
cana-1179	71	15	a	a	DET
cana-1179	71	16	⊆	⊆	NUM
cana-1179	71	17	(	(	PUNCT
cana-1179	71	18	cl*(int(cl*(a	cl*(int(cl*(a	PROPN
cana-1179	71	19	)	)	PUNCT
cana-1179	71	20	)	)	PUNCT
cana-1179	71	21	)	)	PUNCT
cana-1179	71	22	.	.	PUNCT
cana-1179	72	1	v.if	v.if	PROPN
cana-1179	72	2	a	a	DET
cana-1179	72	3	=	=	NOUN
cana-1179	72	4	cl*(int(a	cl*(int(a	NOUN
cana-1179	72	5	)	)	PUNCT
cana-1179	72	6	)	)	PUNCT
cana-1179	73	1	then	then	ADV
cana-1179	73	2	the	the	DET
cana-1179	73	3	set	set	NOUN
cana-1179	73	4	is	be	AUX
cana-1179	73	5	regular	regular	ADJ
cana-1179	73	6	-	-	PUNCT
cana-1179	73	7	i	i	NOUN
cana-1179	73	8	-	-	PUNCT
cana-1179	73	9	closed	closed	ADJ
cana-1179	73	10	.	.	PUNCT
cana-1179	74	1	a	a	DET
cana-1179	74	2	set	set	NOUN
cana-1179	74	3	is	be	AUX
cana-1179	74	4	considered	consider	VERB
cana-1179	74	5	regular	regular	ADJ
cana-1179	74	6	-	-	PUNCT
cana-1179	74	7	i	i	PRON
cana-1179	74	8	-	-	PUNCT
cana-1179	74	9	open	open	ADJ
cana-1179	74	10	if	if	SCONJ
cana-1179	74	11	a	a	DET
cana-1179	74	12	=	=	X
cana-1179	74	13	(	(	PUNCT
cana-1179	74	14	int(cl*(a	int(cl*(a	NUM
cana-1179	74	15	)	)	PUNCT
cana-1179	74	16	)	)	PUNCT
cana-1179	74	17	.	.	PUNCT
cana-1179	75	1	lemma	lemma	PROPN
cana-1179	75	2	2.5	2.5	NUM
cana-1179	76	1	[	[	X
cana-1179	76	2	12	12	NUM
cana-1179	76	3	]	]	PUNCT
cana-1179	76	4	let	let	VERB
cana-1179	76	5	x	x	PRON
cana-1179	76	6	has	have	VERB
cana-1179	76	7	two	two	NUM
cana-1179	76	8	subsets	subset	NOUN
cana-1179	76	9	a	a	PRON
cana-1179	76	10	and	and	CCONJ
cana-1179	76	11	b.	b.	PROPN
cana-1179	77	1	an	an	DET
cana-1179	77	2	ideal	ideal	ADJ
cana-1179	77	3	topological	topological	ADJ
cana-1179	77	4	space	space	NOUN
cana-1179	77	5	is	be	AUX
cana-1179	77	6	(	(	PUNCT
cana-1179	77	7	x	x	X
cana-1179	77	8	,	,	PUNCT
cana-1179	77	9	τ	τ	PROPN
cana-1179	77	10	,	,	PUNCT
cana-1179	77	11	i	i	PROPN
cana-1179	77	12	)	)	PUNCT
cana-1179	77	13	.	.	PUNCT
cana-1179	78	1	the	the	DET
cana-1179	78	2	preceding	precede	VERB
cana-1179	78	3	characteristics	characteristic	NOUN
cana-1179	78	4	are	be	AUX
cana-1179	78	5	then	then	ADV
cana-1179	78	6	:	:	PUNCT
cana-1179	78	7	i.a	i.a	PROPN
cana-1179	78	8	⊆	⊆	PROPN
cana-1179	78	9	b	b	PROPN
cana-1179	78	10	⇒	⇒	NOUN
cana-1179	78	11	a	a	DET
cana-1179	78	12	*	*	PUNCT
cana-1179	78	13	⊆	⊆	NUM
cana-1179	78	14	b	b	X
cana-1179	78	15	*	*	NOUN
cana-1179	78	16	,	,	PUNCT
cana-1179	78	17	ii.a	ii.a	PROPN
cana-1179	78	18	*	*	NOUN
cana-1179	78	19	=	=	SYM
cana-1179	78	20	cl(a	cl(a	X
cana-1179	78	21	*	*	NUM
cana-1179	78	22	)	)	PUNCT
cana-1179	78	23	=	=	SYM
cana-1179	78	24	cl(a	cl(a	X
cana-1179	78	25	)	)	PUNCT
cana-1179	78	26	=	=	SYM
cana-1179	78	27	cl*(a	cl*(a	NOUN
cana-1179	78	28	)	)	PUNCT
cana-1179	78	29	,	,	PUNCT
cana-1179	78	30	iii.(a	iii.(a	VERB
cana-1179	78	31	∪	∪	ADJ
cana-1179	78	32	b	b	NOUN
cana-1179	78	33	)	)	PUNCT
cana-1179	78	34	*	*	PUNCT
cana-1179	79	1	=	=	PUNCT
cana-1179	79	2	a	a	PRON
cana-1179	79	3	*	*	PUNCT
cana-1179	79	4	∪	∪	ADP
cana-1179	79	5	b	b	X
cana-1179	79	6	*	*	PROPN
cana-1179	79	7	,	,	PUNCT
cana-1179	79	8	iv.(a	iv.(a	PROPN
cana-1179	79	9	∩	∩	ADJ
cana-1179	79	10	b	b	X
cana-1179	79	11	)	)	PUNCT
cana-1179	79	12	*	*	PUNCT
cana-1179	80	1	⊆	⊆	NUM
cana-1179	80	2	a	a	DET
cana-1179	80	3	*	*	X
cana-1179	80	4	∩	∩	ADJ
cana-1179	80	5	b	b	PROPN
cana-1179	80	6	*	*	PROPN
cana-1179	80	7	,	,	PUNCT
cana-1179	80	8	v.(a	v.(a	PROPN
cana-1179	80	9	*	*	X
cana-1179	80	10	)	)	PUNCT
cana-1179	80	11	*	*	PUNCT
cana-1179	81	1	⊆	⊆	NUM
cana-1179	81	2	a	a	PRON
cana-1179	81	3	*	*	NOUN
cana-1179	81	4	.	.	PROPN
cana-1179	82	1	3	3	NUM
cana-1179	82	2	.	.	X
cana-1179	82	3	βg*p	βg*p	X
cana-1179	82	4	-	-	PUNCT
cana-1179	82	5	closed	closed	ADJ
cana-1179	82	6	sets	set	NOUN
cana-1179	82	7	in	in	ADP
cana-1179	82	8	topological	topological	ADJ
cana-1179	82	9	spaces	space	NOUN
cana-1179	82	10	definition	definition	NOUN
cana-1179	82	11	3.1	3.1	NUM
cana-1179	82	12	if	if	SCONJ
cana-1179	82	13	pcl(b	pcl(b	PROPN
cana-1179	82	14	)	)	PUNCT
cana-1179	83	1	⊆	⊆	NUM
cana-1179	83	2	d	d	NOUN
cana-1179	83	3	whenever	whenever	SCONJ
cana-1179	83	4	b	b	NOUN
cana-1179	83	5	is	be	AUX
cana-1179	83	6	a	a	DET
cana-1179	83	7	subset	subset	NOUN
cana-1179	83	8	of	of	ADP
cana-1179	83	9	d	d	PROPN
cana-1179	83	10	(	(	PUNCT
cana-1179	83	11	b	b	PROPN
cana-1179	83	12	⊆	⊆	NUM
cana-1179	83	13	d	d	NOUN
cana-1179	83	14	)	)	PUNCT
cana-1179	83	15	and	and	CCONJ
cana-1179	83	16	d	d	NOUN
cana-1179	83	17	is	be	AUX
cana-1179	83	18	g*-open	g*-open	ADJ
cana-1179	83	19	in	in	ADP
cana-1179	83	20	x	x	PRON
cana-1179	83	21	,	,	PUNCT
cana-1179	83	22	then	then	ADV
cana-1179	83	23	subset	subset	VERB
cana-1179	83	24	b	b	PROPN
cana-1179	83	25	of	of	ADP
cana-1179	83	26	a	a	DET
cana-1179	83	27	topological	topological	ADJ
cana-1179	83	28	space	space	NOUN
cana-1179	83	29	(	(	PUNCT
cana-1179	83	30	x	x	X
cana-1179	83	31	,	,	PUNCT
cana-1179	83	32	τ	τ	X
cana-1179	83	33	)	)	PUNCT
cana-1179	83	34	is	be	AUX
cana-1179	83	35	called	call	VERB
cana-1179	83	36	beta	beta	ADJ
cana-1179	83	37	generalized	generalize	VERB
cana-1179	83	38	star	star	NOUN
cana-1179	83	39	pre	pre	ADJ
cana-1179	83	40	-	-	ADJ
cana-1179	83	41	closed	closed	ADJ
cana-1179	83	42	set	set	NOUN
cana-1179	83	43	(	(	PUNCT
cana-1179	83	44	briefly	briefly	ADV
cana-1179	83	45	βg*p	βg*p	X
cana-1179	83	46	-	-	PUNCT
cana-1179	83	47	closed	closed	ADJ
cana-1179	83	48	)	)	PUNCT
cana-1179	83	49	.	.	PUNCT
cana-1179	84	1	theorem	theorem	VERB
cana-1179	84	2	3.2	3.2	NUM
cana-1179	84	3	each	each	DET
cana-1179	84	4	closed	closed	ADJ
cana-1179	84	5	set	set	NOUN
cana-1179	84	6	is	be	AUX
cana-1179	84	7	closed	close	VERB
cana-1179	84	8	in	in	ADP
cana-1179	84	9	βg*p	βg*p	ADV
cana-1179	84	10	-	-	PUNCT
cana-1179	84	11	closed	closed	ADJ
cana-1179	84	12	.	.	PUNCT
cana-1179	85	1	proof	proof	NOUN
cana-1179	85	2	:	:	PUNCT
cana-1179	85	3	in	in	ADP
cana-1179	85	4	the	the	DET
cana-1179	85	5	topological	topological	ADJ
cana-1179	85	6	spaces	space	NOUN
cana-1179	85	7	,	,	PUNCT
cana-1179	85	8	let	let	VERB
cana-1179	85	9	b	b	PRON
cana-1179	85	10	represent	represent	VERB
cana-1179	85	11	any	any	DET
cana-1179	85	12	closed	closed	ADJ
cana-1179	85	13	set	set	NOUN
cana-1179	85	14	.	.	PUNCT
cana-1179	86	1	let	let	VERB
cana-1179	86	2	d	d	PRON
cana-1179	86	3	be	be	AUX
cana-1179	86	4	any	any	DET
cana-1179	86	5	open	open	ADJ
cana-1179	86	6	set	set	NOUN
cana-1179	86	7	g	g	PROPN
cana-1179	86	8	*	*	NOUN
cana-1179	86	9	that	that	PRON
cana-1179	86	10	contains	contain	VERB
cana-1179	86	11	b.	b.	PROPN
cana-1179	86	12	pcl(b	pcl(b	PROPN
cana-1179	86	13	)	)	PUNCT
cana-1179	86	14	equals	equal	VERB
cana-1179	86	15	b.	b.	PROPN
cana-1179	87	1	since	since	SCONJ
cana-1179	87	2	b	b	PROPN
cana-1179	87	3	is	be	AUX
cana-1179	87	4	a	a	DET
cana-1179	87	5	closed	closed	ADJ
cana-1179	87	6	set	set	NOUN
cana-1179	87	7	.	.	PUNCT
cana-1179	88	1	consequently	consequently	ADV
cana-1179	88	2	,	,	PUNCT
cana-1179	88	3	pcl(b	pcl(b	PROPN
cana-1179	88	4	)	)	PUNCT
cana-1179	89	1	⊆	⊆	PROPN
cana-1179	89	2	d.	d.	PROPN
cana-1179	89	3	b	b	PROPN
cana-1179	89	4	is	be	AUX
cana-1179	89	5	therefore	therefore	ADV
cana-1179	89	6	βg*p	βg*p	ADJ
cana-1179	89	7	-	-	PUNCT
cana-1179	89	8	closed	closed	ADJ
cana-1179	89	9	in	in	ADP
cana-1179	89	10	x.	x.	NOUN
cana-1179	89	11	the	the	DET
cana-1179	89	12	following	follow	VERB
cana-1179	89	13	example	example	NOUN
cana-1179	89	14	demonstrates	demonstrate	VERB
cana-1179	89	15	why	why	SCONJ
cana-1179	89	16	the	the	DET
cana-1179	89	17	converse	converse	NOUN
cana-1179	89	18	of	of	ADP
cana-1179	89	19	the	the	DET
cana-1179	89	20	preceding	precede	VERB
cana-1179	89	21	theorem	theorem	NOUN
cana-1179	89	22	need	need	AUX
cana-1179	89	23	not	not	PART
cana-1179	89	24	be	be	AUX
cana-1179	89	25	true	true	ADJ
cana-1179	89	26	.	.	PUNCT
cana-1179	90	1	example	example	NOUN
cana-1179	90	2	3.3	3.3	NUM
cana-1179	90	3	assume	assume	VERB
cana-1179	90	4	that	that	SCONJ
cana-1179	90	5	the	the	DET
cana-1179	90	6	set	set	NOUN
cana-1179	90	7	x	x	X
cana-1179	90	8	=	=	PRON
cana-1179	90	9	{	{	PUNCT
cana-1179	90	10	u	u	NOUN
cana-1179	90	11	,	,	PUNCT
cana-1179	90	12	v	v	NOUN
cana-1179	90	13	,	,	PUNCT
cana-1179	90	14	w	w	NOUN
cana-1179	90	15	}	}	PUNCT
cana-1179	90	16	,	,	PUNCT
cana-1179	90	17	considering	consider	VERB
cana-1179	90	18	a	a	DET
cana-1179	90	19	topology	topology	NOUN
cana-1179	90	20	τ	τ	X
cana-1179	90	21	=	=	PUNCT
cana-1179	90	22	{	{	PUNCT
cana-1179	90	23	ϕ	ϕ	NOUN
cana-1179	90	24	,	,	PUNCT
cana-1179	90	25	{	{	PUNCT
cana-1179	90	26	v	v	NOUN
cana-1179	90	27	}	}	PUNCT
cana-1179	90	28	,	,	PUNCT
cana-1179	90	29	{	{	PUNCT
cana-1179	90	30	u	u	NOUN
cana-1179	90	31	,	,	PUNCT
cana-1179	90	32	w	w	PROPN
cana-1179	90	33	}	}	PUNCT
cana-1179	90	34	,	,	PUNCT
cana-1179	90	35	x	x	NOUN
cana-1179	90	36	}	}	PUNCT
cana-1179	90	37	and	and	CCONJ
cana-1179	90	38	its	its	PRON
cana-1179	90	39	closed	closed	ADJ
cana-1179	90	40	form	form	NOUN
cana-1179	90	41	τ	τ	X
cana-1179	90	42	c	c	NOUN
cana-1179	90	43	=	=	PUNCT
cana-1179	90	44	{	{	PUNCT
cana-1179	90	45	ϕ	ϕ	NOUN
cana-1179	90	46	,	,	PUNCT
cana-1179	90	47	{	{	PUNCT
cana-1179	90	48	u	u	NOUN
cana-1179	90	49	,	,	PUNCT
cana-1179	90	50	w	w	PROPN
cana-1179	90	51	}	}	PUNCT
cana-1179	90	52	,	,	PUNCT
cana-1179	90	53	{	{	PUNCT
cana-1179	90	54	v	v	NOUN
cana-1179	90	55	}	}	PUNCT
cana-1179	90	56	,	,	PUNCT
cana-1179	90	57	x	x	NOUN
cana-1179	90	58	}	}	PUNCT
cana-1179	90	59	.	.	PUNCT
cana-1179	91	1	the	the	DET
cana-1179	91	2	closed	close	VERB
cana-1179	91	3	sets	set	NOUN
cana-1179	91	4	of	of	ADP
cana-1179	91	5	βg*p	βg*p	X
cana-1179	91	6	are	be	AUX
cana-1179	91	7	{	{	PUNCT
cana-1179	91	8	ϕ	ϕ	NOUN
cana-1179	91	9	,	,	PUNCT
cana-1179	91	10	{	{	PUNCT
cana-1179	91	11	u	u	NOUN
cana-1179	91	12	}	}	PUNCT
cana-1179	91	13	,	,	PUNCT
cana-1179	91	14	{	{	PUNCT
cana-1179	91	15	v	v	NOUN
cana-1179	91	16	}	}	PUNCT
cana-1179	91	17	,	,	PUNCT
cana-1179	91	18	{	{	PUNCT
cana-1179	91	19	w	w	NOUN
cana-1179	91	20	}	}	PUNCT
cana-1179	91	21	,	,	PUNCT
cana-1179	91	22	{	{	PUNCT
cana-1179	91	23	u	u	NOUN
cana-1179	91	24	,	,	PUNCT
cana-1179	91	25	v	v	NOUN
cana-1179	91	26	}	}	PUNCT
cana-1179	91	27	,	,	PUNCT
cana-1179	91	28	{	{	PUNCT
cana-1179	91	29	v	v	NOUN
cana-1179	91	30	,	,	PUNCT
cana-1179	91	31	w	w	NOUN
cana-1179	91	32	}	}	PUNCT
cana-1179	91	33	,	,	PUNCT
cana-1179	91	34	{	{	PUNCT
cana-1179	91	35	u	u	NOUN
cana-1179	91	36	,	,	PUNCT
cana-1179	91	37	w	w	PROPN
cana-1179	91	38	}	}	PUNCT
cana-1179	91	39	,	,	PUNCT
cana-1179	91	40	x	x	NOUN
cana-1179	91	41	}	}	PUNCT
cana-1179	91	42	.	.	PUNCT
cana-1179	92	1	in	in	ADP
cana-1179	92	2	this	this	DET
cana-1179	92	3	case	case	NOUN
cana-1179	92	4	,	,	PUNCT
cana-1179	92	5	b	b	X
cana-1179	92	6	=	=	PRON
cana-1179	92	7	{	{	PUNCT
cana-1179	92	8	w	w	NOUN
cana-1179	92	9	}	}	PUNCT
cana-1179	92	10	is	be	AUX
cana-1179	92	11	a	a	DET
cana-1179	92	12	closed	closed	ADJ
cana-1179	92	13	set	set	NOUN
cana-1179	92	14	of	of	ADP
cana-1179	92	15	βg*p	βg*p	PUNCT
cana-1179	92	16	but	but	CCONJ
cana-1179	92	17	not	not	PART
cana-1179	92	18	a	a	DET
cana-1179	92	19	closed	closed	ADJ
cana-1179	92	20	set	set	NOUN
cana-1179	92	21	.	.	PUNCT
cana-1179	93	1	theorem	theorem	VERB
cana-1179	93	2	3.4	3.4	NUM
cana-1179	93	3	each	each	PRON
cana-1179	93	4	and	and	CCONJ
cana-1179	93	5	every	every	DET
cana-1179	93	6	βg*-closed	βg*-closed	ADJ
cana-1179	93	7	set	set	NOUN
cana-1179	93	8	is	be	AUX
cana-1179	93	9	a	a	DET
cana-1179	93	10	βg*p	βg*p	ADV
cana-1179	93	11	-	-	PUNCT
cana-1179	93	12	closed	closed	ADJ
cana-1179	93	13	set	set	NOUN
cana-1179	93	14	.	.	PUNCT
cana-1179	94	1	proof	proof	NOUN
cana-1179	94	2	:	:	PUNCT
cana-1179	94	3	suppose	suppose	VERB
cana-1179	94	4	that	that	SCONJ
cana-1179	94	5	d	d	PROPN
cana-1179	94	6	is	be	AUX
cana-1179	94	7	any	any	DET
cana-1179	94	8	g*-open	g*-open	NOUN
cana-1179	94	9	set	set	VERB
cana-1179	94	10	in	in	ADP
cana-1179	94	11	x	x	PUNCT
cana-1179	94	12	such	such	ADJ
cana-1179	94	13	that	that	DET
cana-1179	94	14	b	b	NOUN
cana-1179	94	15	⊆	⊆	NUM
cana-1179	94	16	d	d	NOUN
cana-1179	94	17	and	and	CCONJ
cana-1179	94	18	that	that	PRON
cana-1179	94	19	b	b	NOUN
cana-1179	94	20	is	be	AUX
cana-1179	94	21	a	a	DET
cana-1179	94	22	βg*-closed	βg*-closed	ADJ
cana-1179	94	23	set	set	NOUN
cana-1179	94	24	in	in	ADP
cana-1179	94	25	the	the	DET
cana-1179	94	26	space	space	NOUN
cana-1179	94	27	(	(	PUNCT
cana-1179	94	28	x	x	X
cana-1179	94	29	,	,	PUNCT
cana-1179	94	30	τ	τ	PROPN
cana-1179	94	31	)	)	PUNCT
cana-1179	94	32	.	.	PUNCT
cana-1179	95	1	assume	assume	VERB
cana-1179	95	2	that	that	SCONJ
cana-1179	95	3	each	each	DET
cana-1179	95	4	g*-open	g*-open	NOUN
cana-1179	95	5	set	set	VERB
cana-1179	95	6	is	be	AUX
cana-1179	95	7	β	β	NOUN
cana-1179	95	8	-	-	ADJ
cana-1179	95	9	open	open	ADJ
cana-1179	95	10	.	.	PUNCT
cana-1179	96	1	considering	consider	VERB
cana-1179	96	2	the	the	DET
cana-1179	96	3	closure	closure	NOUN
cana-1179	96	4	of	of	ADP
cana-1179	96	5	b	b	PROPN
cana-1179	96	6	,	,	PUNCT
cana-1179	96	7	pcl(b	pcl(b	PROPN
cana-1179	96	8	)	)	PUNCT
cana-1179	96	9	⊆	⊆	NUM
cana-1179	96	10	cl(b	cl(b	NOUN
cana-1179	96	11	)	)	PUNCT
cana-1179	96	12	⊆	⊆	NUM
cana-1179	96	13	d.	d.	NOUN
cana-1179	96	14	this	this	PRON
cana-1179	96	15	clarifies	clarify	VERB
cana-1179	96	16	that	that	SCONJ
cana-1179	96	17	b	b	NOUN
cana-1179	96	18	is	be	AUX
cana-1179	96	19	a	a	DET
cana-1179	96	20	closed	closed	ADJ
cana-1179	96	21	set	set	NOUN
cana-1179	96	22	in	in	ADP
cana-1179	96	23	(	(	PUNCT
cana-1179	96	24	x	x	NOUN
cana-1179	96	25	,	,	PUNCT
cana-1179	96	26	τ	τ	X
cana-1179	96	27	)	)	PUNCT
cana-1179	96	28	with	with	ADP
cana-1179	96	29	βg*p	βg*p	NOUN
cana-1179	96	30	.	.	PUNCT
cana-1179	97	1	the	the	DET
cana-1179	97	2	following	follow	VERB
cana-1179	97	3	example	example	NOUN
cana-1179	97	4	demonstrates	demonstrate	VERB
cana-1179	97	5	,	,	PUNCT
cana-1179	97	6	the	the	DET
cana-1179	97	7	converse	converse	NOUN
cana-1179	97	8	of	of	ADP
cana-1179	97	9	the	the	DET
cana-1179	97	10	aforementioned	aforementione	VERB
cana-1179	97	11	theorem	theorem	NOUN
cana-1179	97	12	need	need	AUX
cana-1179	97	13	not	not	PART
cana-1179	97	14	be	be	AUX
cana-1179	97	15	true	true	ADJ
cana-1179	97	16	.	.	PUNCT
cana-1179	98	1	example	example	NOUN
cana-1179	98	2	3.5	3.5	NUM
cana-1179	98	3	with	with	ADP
cana-1179	98	4	the	the	DET
cana-1179	98	5	topology	topology	NOUN
cana-1179	98	6	τ	τ	X
cana-1179	98	7	=	=	PUNCT
cana-1179	98	8	{	{	PUNCT
cana-1179	98	9	ϕ	ϕ	NOUN
cana-1179	98	10	,	,	PUNCT
cana-1179	98	11	{	{	PUNCT
cana-1179	98	12	v	v	NOUN
cana-1179	98	13	,	,	PUNCT
cana-1179	98	14	w	w	NOUN
cana-1179	98	15	}	}	PUNCT
cana-1179	98	16	,	,	PUNCT
cana-1179	98	17	x	x	NOUN
cana-1179	98	18	}	}	PUNCT
cana-1179	98	19	,	,	PUNCT
cana-1179	98	20	let	let	VERB
cana-1179	98	21	x	x	PUNCT
cana-1179	98	22	=	=	PRON
cana-1179	98	23	{	{	PUNCT
cana-1179	98	24	u	u	NOUN
cana-1179	98	25	,	,	PUNCT
cana-1179	98	26	v	v	NOUN
cana-1179	98	27	,	,	PUNCT
cana-1179	98	28	w	w	NOUN
cana-1179	98	29	}	}	PUNCT
cana-1179	98	30	.	.	PUNCT
cana-1179	99	1	the	the	DET
cana-1179	99	2	closed	close	VERB
cana-1179	99	3	sets	set	NOUN
cana-1179	99	4	of	of	ADP
cana-1179	99	5	βg*p	βg*p	X
cana-1179	99	6	form	form	NOUN
cana-1179	99	7	are	be	AUX
cana-1179	99	8	{	{	PUNCT
cana-1179	99	9	ϕ	ϕ	NOUN
cana-1179	99	10	,	,	PUNCT
cana-1179	99	11	{	{	PUNCT
cana-1179	99	12	u	u	NOUN
cana-1179	99	13	}	}	PUNCT
cana-1179	99	14	,	,	PUNCT
cana-1179	99	15	{	{	PUNCT
cana-1179	99	16	v	v	NOUN
cana-1179	99	17	}	}	PUNCT
cana-1179	99	18	,	,	PUNCT
cana-1179	99	19	{	{	PUNCT
cana-1179	99	20	w	w	NOUN
cana-1179	99	21	}	}	PUNCT
cana-1179	99	22	,	,	PUNCT
cana-1179	99	23	{	{	PUNCT
cana-1179	99	24	u	u	NOUN
cana-1179	99	25	,	,	PUNCT
cana-1179	99	26	v	v	NOUN
cana-1179	99	27	}	}	PUNCT
cana-1179	99	28	,	,	PUNCT
cana-1179	99	29	{	{	PUNCT
cana-1179	99	30	u	u	NOUN
cana-1179	99	31	,	,	PUNCT
cana-1179	99	32	w	w	PROPN
cana-1179	99	33	}	}	PUNCT
cana-1179	99	34	,	,	PUNCT
cana-1179	99	35	x	x	NOUN
cana-1179	99	36	}	}	PUNCT
cana-1179	99	37	.	.	PUNCT
cana-1179	100	1	it	it	PRON
cana-1179	100	2	is	be	AUX
cana-1179	100	3	not	not	PART
cana-1179	100	4	a	a	DET
cana-1179	100	5	βg*-closed	βg*-closed	ADJ
cana-1179	100	6	set	set	NOUN
cana-1179	100	7	,	,	PUNCT
cana-1179	100	8	although	although	SCONJ
cana-1179	100	9	in	in	ADP
cana-1179	100	10	this	this	DET
cana-1179	100	11	case	case	NOUN
cana-1179	100	12	,	,	PUNCT
cana-1179	100	13	b	b	X
cana-1179	100	14	=	=	PRON
cana-1179	100	15	{	{	PUNCT
cana-1179	100	16	v	v	NOUN
cana-1179	100	17	}	}	PUNCT
cana-1179	100	18	is	be	AUX
cana-1179	100	19	βg*p	βg*p	X
cana-1179	100	20	-	-	PUNCT
cana-1179	100	21	closed	closed	ADJ
cana-1179	100	22	set	set	NOUN
cana-1179	100	23	.	.	PUNCT
cana-1179	101	1	theorem	theorem	VERB
cana-1179	101	2	3.6	3.6	NUM
cana-1179	101	3	any	any	DET
cana-1179	101	4	βg*p	βg*p	ADJ
cana-1179	101	5	-	-	PUNCT
cana-1179	101	6	closed	closed	ADJ
cana-1179	101	7	set	set	NOUN
cana-1179	101	8	can	can	AUX
cana-1179	101	9	also	also	ADV
cana-1179	101	10	be	be	AUX
cana-1179	101	11	a	a	DET
cana-1179	101	12	βg	βg	ADV
cana-1179	101	13	-	-	PUNCT
cana-1179	101	14	closed	closed	ADJ
cana-1179	101	15	.	.	PUNCT
cana-1179	102	1	proof	proof	NOUN
cana-1179	102	2	:	:	PUNCT
cana-1179	102	3	consider	consider	VERB
cana-1179	102	4	b	b	NOUN
cana-1179	102	5	to	to	PART
cana-1179	102	6	be	be	AUX
cana-1179	102	7	a	a	DET
cana-1179	102	8	βg*p	βg*p	ADV
cana-1179	102	9	-	-	PUNCT
cana-1179	102	10	closed	closed	ADJ
cana-1179	102	11	set	set	NOUN
cana-1179	102	12	in	in	ADP
cana-1179	102	13	the	the	DET
cana-1179	102	14	space	space	NOUN
cana-1179	102	15	(	(	PUNCT
cana-1179	102	16	x	x	X
cana-1179	102	17	,	,	PUNCT
cana-1179	102	18	τ	τ	PROPN
cana-1179	102	19	)	)	PUNCT
cana-1179	102	20	.	.	PUNCT
cana-1179	103	1	for	for	ADP
cana-1179	103	2	any	any	DET
cana-1179	103	3	g	g	NOUN
cana-1179	103	4	-	-	PUNCT
cana-1179	103	5	open	open	ADJ
cana-1179	103	6	set	set	NOUN
cana-1179	103	7	which	which	PRON
cana-1179	103	8	has	have	AUX
cana-1179	103	9	b.	b.	AUX
cana-1179	103	10	each	each	DET
cana-1179	103	11	gopen	gopen	NOUN
cana-1179	103	12	set	set	NOUN
cana-1179	103	13	is	be	AUX
cana-1179	103	14	g*-open	g*-open	ADJ
cana-1179	103	15	,	,	PUNCT
cana-1179	103	16	accordingly	accordingly	ADV
cana-1179	103	17	pcl(b	pcl(b	PROPN
cana-1179	103	18	)	)	PUNCT
cana-1179	104	1	⊆	⊆	NUM
cana-1179	104	2	d.	d.	PROPN
cana-1179	104	3	b	b	PROPN
cana-1179	104	4	is	be	AUX
cana-1179	104	5	a	a	DET
cana-1179	104	6	closed	closed	ADJ
cana-1179	104	7	set	set	NOUN
cana-1179	104	8	,	,	PUNCT
cana-1179	104	9	so	so	CCONJ
cana-1179	104	10	it	it	PRON
cana-1179	104	11	is	be	AUX
cana-1179	104	12	βg	βg	ADJ
cana-1179	104	13	.	.	PUNCT
cana-1179	105	1	as	as	SCONJ
cana-1179	105	2	the	the	DET
cana-1179	105	3	following	following	ADJ
cana-1179	105	4	example	example	NOUN
cana-1179	105	5	demonstrates	demonstrate	VERB
cana-1179	105	6	,	,	PUNCT
cana-1179	105	7	the	the	DET
cana-1179	105	8	converse	converse	NOUN
cana-1179	105	9	of	of	ADP
cana-1179	105	10	the	the	DET
cana-1179	105	11	aforementioned	aforementione	VERB
cana-1179	105	12	theorem	theorem	NOUN
cana-1179	105	13	need	need	AUX
cana-1179	105	14	not	not	PART
cana-1179	105	15	be	be	AUX
cana-1179	105	16	true	true	ADJ
cana-1179	105	17	.	.	PUNCT
cana-1179	106	1	communications	communication	NOUN
cana-1179	106	2	on	on	ADP
cana-1179	106	3	applied	apply	VERB
cana-1179	106	4	nonlinear	nonlinear	ADJ
cana-1179	106	5	analysis	analysis	NOUN
cana-1179	106	6	issn	issn	NOUN
cana-1179	106	7	:	:	PUNCT
cana-1179	106	8	1074	1074	NUM
cana-1179	106	9	-	-	PUNCT
cana-1179	106	10	133x	133x	NUM
cana-1179	106	11	vol	vol	NOUN
cana-1179	106	12	31	31	NUM
cana-1179	106	13	no	no	NOUN
cana-1179	106	14	.	.	PUNCT
cana-1179	107	1	6s	6s	NUM
cana-1179	107	2	(	(	PUNCT
cana-1179	107	3	2024	2024	NUM
cana-1179	107	4	)	)	PUNCT
cana-1179	107	5	210	210	NUM
cana-1179	107	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1179	107	7	example	example	NOUN
cana-1179	107	8	3.7	3.7	NUM
cana-1179	107	9	with	with	ADP
cana-1179	107	10	the	the	DET
cana-1179	107	11	topology	topology	NOUN
cana-1179	107	12	τ	τ	X
cana-1179	107	13	=	=	PUNCT
cana-1179	107	14	{	{	PUNCT
cana-1179	107	15	ϕ	ϕ	NOUN
cana-1179	107	16	,	,	PUNCT
cana-1179	107	17	{	{	PUNCT
cana-1179	107	18	u	u	NOUN
cana-1179	107	19	}	}	PUNCT
cana-1179	107	20	,	,	PUNCT
cana-1179	107	21	{	{	PUNCT
cana-1179	107	22	v	v	NOUN
cana-1179	107	23	}	}	PUNCT
cana-1179	107	24	,	,	PUNCT
cana-1179	107	25	{	{	PUNCT
cana-1179	107	26	v	v	NOUN
cana-1179	107	27	,	,	PUNCT
cana-1179	107	28	u	u	NOUN
cana-1179	107	29	}	}	PUNCT
cana-1179	107	30	,	,	PUNCT
cana-1179	107	31	x	x	X
cana-1179	107	32	}	}	PUNCT
cana-1179	107	33	,	,	PUNCT
cana-1179	107	34	let	let	VERB
cana-1179	107	35	x	x	PUNCT
cana-1179	107	36	=	=	PRON
cana-1179	107	37	{	{	PUNCT
cana-1179	107	38	u	u	NOUN
cana-1179	107	39	,	,	PUNCT
cana-1179	107	40	v	v	NOUN
cana-1179	107	41	,	,	PUNCT
cana-1179	107	42	w	w	NOUN
cana-1179	107	43	}	}	PUNCT
cana-1179	107	44	.	.	PUNCT
cana-1179	108	1	the	the	DET
cana-1179	108	2	closed	close	VERB
cana-1179	108	3	sets	set	NOUN
cana-1179	108	4	of	of	ADP
cana-1179	108	5	βg*p	βg*p	X
cana-1179	108	6	structure	structure	NOUN
cana-1179	108	7	are	be	AUX
cana-1179	108	8	{	{	PUNCT
cana-1179	108	9	ϕ	ϕ	NOUN
cana-1179	108	10	,	,	PUNCT
cana-1179	108	11	{	{	PUNCT
cana-1179	108	12	w	w	NOUN
cana-1179	108	13	}	}	PUNCT
cana-1179	108	14	,	,	PUNCT
cana-1179	108	15	{	{	PUNCT
cana-1179	108	16	v	v	NOUN
cana-1179	108	17	,	,	PUNCT
cana-1179	108	18	w	w	NOUN
cana-1179	108	19	}	}	PUNCT
cana-1179	108	20	,	,	PUNCT
cana-1179	108	21	{	{	PUNCT
cana-1179	108	22	u	u	NOUN
cana-1179	108	23	,	,	PUNCT
cana-1179	108	24	w	w	PROPN
cana-1179	108	25	}	}	PUNCT
cana-1179	108	26	,	,	PUNCT
cana-1179	108	27	x	x	NOUN
cana-1179	108	28	}	}	PUNCT
cana-1179	108	29	.	.	PUNCT
cana-1179	109	1	this	this	PRON
cana-1179	109	2	is	be	AUX
cana-1179	109	3	not	not	PART
cana-1179	109	4	a	a	DET
cana-1179	109	5	βg*p	βg*p	ADV
cana-1179	109	6	-	-	PUNCT
cana-1179	109	7	closed	closed	ADJ
cana-1179	109	8	set	set	NOUN
cana-1179	109	9	rather	rather	ADV
cana-1179	109	10	,	,	PUNCT
cana-1179	109	11	b	b	X
cana-1179	109	12	=	=	PRON
cana-1179	109	13	{	{	PUNCT
cana-1179	109	14	u	u	NOUN
cana-1179	109	15	}	}	PUNCT
cana-1179	109	16	is	be	AUX
cana-1179	109	17	a	a	DET
cana-1179	109	18	βg	βg	ADV
cana-1179	109	19	-	-	PUNCT
cana-1179	109	20	closed	close	VERB
cana-1179	109	21	set	set	NOUN
cana-1179	109	22	.	.	PUNCT
cana-1179	110	1	theorem	theorem	VERB
cana-1179	110	2	3.8	3.8	NUM
cana-1179	110	3	in	in	ADP
cana-1179	110	4	the	the	DET
cana-1179	110	5	topological	topological	ADJ
cana-1179	110	6	space	space	NOUN
cana-1179	110	7	(	(	PUNCT
cana-1179	110	8	x	x	X
cana-1179	110	9	,	,	PUNCT
cana-1179	110	10	τ	τ	PROPN
cana-1179	110	11	)	)	PUNCT
cana-1179	110	12	,	,	PUNCT
cana-1179	110	13	every	every	DET
cana-1179	110	14	weakly	weakly	ADJ
cana-1179	110	15	generalized	generalize	VERB
cana-1179	110	16	-	-	PUNCT
cana-1179	110	17	closed	closed	ADJ
cana-1179	110	18	,	,	PUNCT
cana-1179	110	19	regular	regular	ADJ
cana-1179	110	20	weakly	weakly	ADJ
cana-1179	110	21	generalized	generalize	VERB
cana-1179	110	22	-	-	PUNCT
cana-1179	110	23	closed	closed	ADJ
cana-1179	110	24	,	,	PUNCT
cana-1179	110	25	generalized*pre	generalized*pre	NOUN
cana-1179	110	26	-	-	PUNCT
cana-1179	110	27	closed	closed	ADJ
cana-1179	110	28	,	,	PUNCT
cana-1179	110	29	mildly	mildly	ADV
cana-1179	110	30	g	g	NOUN
cana-1179	110	31	-	-	PUNCT
cana-1179	110	32	closed	closed	ADJ
cana-1179	110	33	,	,	PUNCT
cana-1179	110	34	β*-closed	β*-close	VERB
cana-1179	110	35	,	,	PUNCT
cana-1179	110	36	α	α	PROPN
cana-1179	110	37	generalized	generalize	VERB
cana-1179	110	38	-	-	PUNCT
cana-1179	110	39	closed	closed	ADJ
cana-1179	110	40	,	,	PUNCT
cana-1179	110	41	generalized	generalized	ADJ
cana-1179	110	42	α	α	NOUN
cana-1179	110	43	-	-	PUNCT
cana-1179	110	44	closed	closed	ADJ
cana-1179	110	45	sets	set	NOUN
cana-1179	110	46	in	in	ADP
cana-1179	110	47	the	the	DET
cana-1179	110	48	space	space	NOUN
cana-1179	110	49	are	be	AUX
cana-1179	110	50	βg*p	βg*p	X
cana-1179	110	51	-	-	PUNCT
cana-1179	110	52	closed	closed	ADJ
cana-1179	110	53	.	.	PUNCT
cana-1179	111	1	proof	proof	NOUN
cana-1179	111	2	:	:	PUNCT
cana-1179	111	3	as	as	ADP
cana-1179	111	4	a	a	DET
cana-1179	111	5	result	result	NOUN
cana-1179	111	6	,	,	PUNCT
cana-1179	111	7	each	each	DET
cana-1179	111	8	open	open	ADJ
cana-1179	111	9	set	set	NOUN
cana-1179	111	10	in	in	ADP
cana-1179	111	11	the	the	DET
cana-1179	111	12	space	space	NOUN
cana-1179	111	13	(	(	PUNCT
cana-1179	111	14	x	x	X
cana-1179	111	15	,	,	PUNCT
cana-1179	111	16	τ	τ	X
cana-1179	111	17	)	)	PUNCT
cana-1179	111	18	is	be	AUX
cana-1179	111	19	g*-open	g*-open	AUX
cana-1179	111	20	set	set	VERB
cana-1179	111	21	.	.	PUNCT
cana-1179	112	1	the	the	DET
cana-1179	112	2	theorems	theorems	PROPN
cana-1179	112	3	converse	converse	NOUN
cana-1179	112	4	need	need	AUX
cana-1179	112	5	not	not	PART
cana-1179	112	6	always	always	ADV
cana-1179	112	7	be	be	AUX
cana-1179	112	8	true	true	ADJ
cana-1179	112	9	.	.	PUNCT
cana-1179	113	1	the	the	DET
cana-1179	113	2	example	example	NOUN
cana-1179	113	3	that	that	PRON
cana-1179	113	4	follows	follow	VERB
cana-1179	113	5	exemplifies	exemplify	VERB
cana-1179	113	6	it	it	PRON
cana-1179	113	7	.	.	PUNCT
cana-1179	114	1	example	example	NOUN
cana-1179	114	2	3.9	3.9	NUM
cana-1179	114	3	assume	assume	VERB
cana-1179	114	4	that	that	SCONJ
cana-1179	114	5	the	the	DET
cana-1179	114	6	set	set	NOUN
cana-1179	114	7	x	x	X
cana-1179	114	8	=	=	PRON
cana-1179	114	9	{	{	PUNCT
cana-1179	114	10	u	u	NOUN
cana-1179	114	11	,	,	PUNCT
cana-1179	114	12	v	v	NOUN
cana-1179	114	13	,	,	PUNCT
cana-1179	114	14	w	w	NOUN
cana-1179	114	15	}	}	PUNCT
cana-1179	114	16	,	,	PUNCT
cana-1179	114	17	considering	consider	VERB
cana-1179	114	18	a	a	DET
cana-1179	114	19	topology	topology	NOUN
cana-1179	114	20	τ	τ	X
cana-1179	114	21	=	=	PUNCT
cana-1179	114	22	{	{	PUNCT
cana-1179	114	23	ϕ	ϕ	NOUN
cana-1179	114	24	,	,	PUNCT
cana-1179	114	25	{	{	PUNCT
cana-1179	114	26	u	u	NOUN
cana-1179	114	27	}	}	PUNCT
cana-1179	114	28	,	,	PUNCT
cana-1179	114	29	{	{	PUNCT
cana-1179	114	30	w	w	NOUN
cana-1179	114	31	}	}	PUNCT
cana-1179	114	32	,	,	PUNCT
cana-1179	114	33	{	{	PUNCT
cana-1179	114	34	u	u	NOUN
cana-1179	114	35	,	,	PUNCT
cana-1179	114	36	v	v	NOUN
cana-1179	114	37	}	}	PUNCT
cana-1179	114	38	,	,	PUNCT
cana-1179	114	39	{	{	PUNCT
cana-1179	114	40	u	u	NOUN
cana-1179	114	41	,	,	PUNCT
cana-1179	114	42	w	w	PROPN
cana-1179	114	43	}	}	PUNCT
cana-1179	114	44	,	,	PUNCT
cana-1179	114	45	x	x	NOUN
cana-1179	114	46	}	}	PUNCT
cana-1179	114	47	.	.	PUNCT
cana-1179	115	1	the	the	DET
cana-1179	115	2	closed	close	VERB
cana-1179	115	3	sets	set	NOUN
cana-1179	115	4	of	of	ADP
cana-1179	115	5	βg*p	βg*p	X
cana-1179	115	6	are	be	AUX
cana-1179	115	7	{	{	PUNCT
cana-1179	115	8	ϕ	ϕ	NOUN
cana-1179	115	9	,	,	PUNCT
cana-1179	115	10	{	{	PUNCT
cana-1179	115	11	w	w	NOUN
cana-1179	115	12	}	}	PUNCT
cana-1179	115	13	,	,	PUNCT
cana-1179	115	14	{	{	PUNCT
cana-1179	115	15	v	v	NOUN
cana-1179	115	16	}	}	PUNCT
cana-1179	115	17	,	,	PUNCT
cana-1179	115	18	{	{	PUNCT
cana-1179	115	19	v	v	NOUN
cana-1179	115	20	,	,	PUNCT
cana-1179	115	21	w	w	NOUN
cana-1179	115	22	}	}	PUNCT
cana-1179	115	23	,	,	PUNCT
cana-1179	115	24	{	{	PUNCT
cana-1179	115	25	u	u	NOUN
cana-1179	115	26	,	,	PUNCT
cana-1179	115	27	v	v	NOUN
cana-1179	115	28	}	}	PUNCT
cana-1179	115	29	,	,	PUNCT
cana-1179	115	30	x	x	NOUN
cana-1179	115	31	}	}	PUNCT
cana-1179	115	32	.	.	PUNCT
cana-1179	116	1	here	here	ADV
cana-1179	116	2	b	b	X
cana-1179	116	3	=	=	SYM
cana-1179	116	4	{	{	PUNCT
cana-1179	116	5	v	v	NOUN
cana-1179	116	6	,	,	PUNCT
cana-1179	116	7	w	w	NOUN
cana-1179	116	8	}	}	PUNCT
cana-1179	116	9	is	be	AUX
cana-1179	116	10	βg*p	βg*p	X
cana-1179	116	11	-	-	PUNCT
cana-1179	116	12	closed	closed	ADJ
cana-1179	116	13	set	set	NOUN
cana-1179	116	14	but	but	CCONJ
cana-1179	116	15	it	it	PRON
cana-1179	116	16	is	be	AUX
cana-1179	116	17	not	not	PART
cana-1179	116	18	a	a	DET
cana-1179	116	19	wg	wg	NOUN
cana-1179	116	20	-	-	PUNCT
cana-1179	116	21	closed	close	VERB
cana-1179	116	22	and	and	CCONJ
cana-1179	116	23	rwg	rwg	ADV
cana-1179	116	24	-	-	PUNCT
cana-1179	116	25	closed	close	VERB
cana-1179	116	26	sets	set	NOUN
cana-1179	116	27	.	.	PUNCT
cana-1179	117	1	example	example	NOUN
cana-1179	117	2	3.10	3.10	NUM
cana-1179	117	3	assume	assume	VERB
cana-1179	117	4	that	that	SCONJ
cana-1179	117	5	x	x	X
cana-1179	117	6	=	=	PRON
cana-1179	117	7	{	{	PUNCT
cana-1179	117	8	u	u	NOUN
cana-1179	117	9	,	,	PUNCT
cana-1179	117	10	v	v	NOUN
cana-1179	117	11	,	,	PUNCT
cana-1179	117	12	w	w	NOUN
cana-1179	117	13	}	}	PUNCT
cana-1179	117	14	and	and	CCONJ
cana-1179	117	15	its	its	PRON
cana-1179	117	16	topology	topology	NOUN
cana-1179	117	17	τ	τ	X
cana-1179	117	18	=	=	PUNCT
cana-1179	117	19	{	{	PUNCT
cana-1179	117	20	ϕ	ϕ	NOUN
cana-1179	117	21	,	,	PUNCT
cana-1179	117	22	{	{	PUNCT
cana-1179	117	23	w	w	NOUN
cana-1179	117	24	}	}	PUNCT
cana-1179	117	25	,	,	PUNCT
cana-1179	117	26	x	x	NOUN
cana-1179	117	27	}	}	PUNCT
cana-1179	117	28	.	.	PUNCT
cana-1179	118	1	{	{	PUNCT
cana-1179	118	2	ϕ	ϕ	NOUN
cana-1179	118	3	,	,	PUNCT
cana-1179	118	4	{	{	PUNCT
cana-1179	118	5	u	u	NOUN
cana-1179	118	6	}	}	PUNCT
cana-1179	118	7	,	,	PUNCT
cana-1179	118	8	{	{	PUNCT
cana-1179	118	9	v	v	NOUN
cana-1179	118	10	}	}	PUNCT
cana-1179	118	11	,	,	PUNCT
cana-1179	118	12	{	{	PUNCT
cana-1179	118	13	v	v	NOUN
cana-1179	118	14	,	,	PUNCT
cana-1179	118	15	w	w	NOUN
cana-1179	118	16	}	}	PUNCT
cana-1179	118	17	,	,	PUNCT
cana-1179	118	18	{	{	PUNCT
cana-1179	118	19	w	w	NOUN
cana-1179	118	20	,	,	PUNCT
cana-1179	118	21	u	u	NOUN
cana-1179	118	22	}	}	PUNCT
cana-1179	118	23	,	,	PUNCT
cana-1179	118	24	{	{	PUNCT
cana-1179	118	25	u	u	NOUN
cana-1179	118	26	,	,	PUNCT
cana-1179	118	27	v	v	NOUN
cana-1179	118	28	}	}	PUNCT
cana-1179	118	29	,	,	PUNCT
cana-1179	118	30	x	x	X
cana-1179	118	31	}	}	PUNCT
cana-1179	118	32	are	be	AUX
cana-1179	118	33	the	the	DET
cana-1179	118	34	βg*p	βg*p	ADJ
cana-1179	118	35	-	-	PUNCT
cana-1179	118	36	closed	closed	ADJ
cana-1179	118	37	sets	set	NOUN
cana-1179	118	38	of	of	ADP
cana-1179	118	39	x.	x.	NOUN
cana-1179	118	40	although	although	SCONJ
cana-1179	118	41	it	it	PRON
cana-1179	118	42	is	be	AUX
cana-1179	118	43	a	a	DET
cana-1179	118	44	βg*p	βg*p	ADV
cana-1179	118	45	-	-	PUNCT
cana-1179	118	46	closed	closed	ADJ
cana-1179	118	47	set	set	NOUN
cana-1179	118	48	in	in	ADP
cana-1179	118	49	this	this	DET
cana-1179	118	50	case	case	NOUN
cana-1179	118	51	b	b	X
cana-1179	118	52	=	=	PRON
cana-1179	118	53	{	{	PUNCT
cana-1179	118	54	v	v	NOUN
cana-1179	118	55	,	,	PUNCT
cana-1179	118	56	w	w	NOUN
cana-1179	118	57	}	}	PUNCT
cana-1179	118	58	is	be	AUX
cana-1179	118	59	not	not	PART
cana-1179	118	60	a	a	DET
cana-1179	118	61	g*p	g*p	PROPN
cana-1179	118	62	-	-	PUNCT
cana-1179	118	63	closed	closed	ADJ
cana-1179	118	64	and	and	CCONJ
cana-1179	118	65	mildly	mildly	ADV
cana-1179	118	66	g	g	NOUN
cana-1179	118	67	-	-	PUNCT
cana-1179	118	68	closed	close	VERB
cana-1179	118	69	sets	set	NOUN
cana-1179	118	70	.	.	PUNCT
cana-1179	119	1	example	example	NOUN
cana-1179	119	2	3.11	3.11	NUM
cana-1179	119	3	given	give	VERB
cana-1179	119	4	x	x	PROPN
cana-1179	119	5	=	=	PRON
cana-1179	119	6	{	{	PUNCT
cana-1179	119	7	u	u	NOUN
cana-1179	119	8	,	,	PUNCT
cana-1179	119	9	v	v	NOUN
cana-1179	119	10	,	,	PUNCT
cana-1179	119	11	w	w	NOUN
cana-1179	119	12	}	}	PUNCT
cana-1179	119	13	and	and	CCONJ
cana-1179	119	14	topology	topology	NOUN
cana-1179	119	15	τ	τ	PROPN
cana-1179	119	16	=	=	PUNCT
cana-1179	119	17	{	{	PUNCT
cana-1179	119	18	ϕ	ϕ	NOUN
cana-1179	119	19	,	,	PUNCT
cana-1179	119	20	{	{	PUNCT
cana-1179	119	21	u	u	NOUN
cana-1179	119	22	,	,	PUNCT
cana-1179	119	23	v	v	NOUN
cana-1179	119	24	}	}	PUNCT
cana-1179	119	25	,	,	PUNCT
cana-1179	119	26	x	x	NOUN
cana-1179	119	27	}	}	PUNCT
cana-1179	119	28	.	.	PUNCT
cana-1179	120	1	{	{	PUNCT
cana-1179	120	2	ϕ	ϕ	NOUN
cana-1179	120	3	,	,	PUNCT
cana-1179	120	4	{	{	PUNCT
cana-1179	120	5	u	u	NOUN
cana-1179	120	6	}	}	PUNCT
cana-1179	120	7	,	,	PUNCT
cana-1179	120	8	{	{	PUNCT
cana-1179	120	9	v	v	NOUN
cana-1179	120	10	}	}	PUNCT
cana-1179	120	11	,	,	PUNCT
cana-1179	120	12	{	{	PUNCT
cana-1179	120	13	w	w	NOUN
cana-1179	120	14	}	}	PUNCT
cana-1179	120	15	,	,	PUNCT
cana-1179	120	16	{	{	PUNCT
cana-1179	120	17	v	v	NOUN
cana-1179	120	18	,	,	PUNCT
cana-1179	120	19	w	w	NOUN
cana-1179	120	20	}	}	PUNCT
cana-1179	120	21	,	,	PUNCT
cana-1179	120	22	{	{	PUNCT
cana-1179	120	23	w	w	NOUN
cana-1179	120	24	,	,	PUNCT
cana-1179	120	25	u	u	NOUN
cana-1179	120	26	}	}	PUNCT
cana-1179	120	27	,	,	PUNCT
cana-1179	120	28	x	x	X
cana-1179	120	29	}	}	PUNCT
cana-1179	120	30	are	be	AUX
cana-1179	120	31	the	the	DET
cana-1179	120	32	βg*p	βg*p	ADJ
cana-1179	120	33	-	-	PUNCT
cana-1179	120	34	closed	closed	ADJ
cana-1179	120	35	sets	set	NOUN
cana-1179	120	36	of	of	ADP
cana-1179	120	37	x.	x.	NOUN
cana-1179	120	38	while	while	NOUN
cana-1179	120	39	,	,	PUNCT
cana-1179	120	40	b	b	X
cana-1179	120	41	=	=	SYM
cana-1179	120	42	{	{	PUNCT
cana-1179	120	43	u	u	NOUN
cana-1179	120	44	}	}	PUNCT
cana-1179	120	45	is	be	AUX
cana-1179	120	46	not	not	PART
cana-1179	120	47	a	a	DET
cana-1179	120	48	αg	αg	NOUN
cana-1179	120	49	-	-	PUNCT
cana-1179	120	50	closed	closed	ADJ
cana-1179	120	51	and	and	CCONJ
cana-1179	120	52	gα	gα	NOUN
cana-1179	120	53	-	-	PUNCT
cana-1179	120	54	closed	closed	ADJ
cana-1179	120	55	sets	set	NOUN
cana-1179	120	56	it	it	PRON
cana-1179	120	57	is	be	AUX
cana-1179	120	58	βg*pclosed	βg*pclose	VERB
cana-1179	120	59	set	set	VERB
cana-1179	120	60	.	.	PUNCT
cana-1179	121	1	remark	remark	PROPN
cana-1179	121	2	3.12	3.12	NUM
cana-1179	121	3	the	the	DET
cana-1179	121	4	class	class	NOUN
cana-1179	121	5	of	of	ADP
cana-1179	121	6	regular	regular	ADJ
cana-1179	121	7	generalized	generalize	VERB
cana-1179	121	8	-	-	PUNCT
cana-1179	121	9	closed	closed	ADJ
cana-1179	121	10	,	,	PUNCT
cana-1179	121	11	generalized	generalized	ADJ
cana-1179	121	12	pre	pre	ADJ
cana-1179	121	13	-	-	ADJ
cana-1179	121	14	regular	regular	ADJ
cana-1179	121	15	closed	closed	ADJ
cana-1179	121	16	and	and	CCONJ
cana-1179	121	17	generalized	generalize	VERB
cana-1179	121	18	semi	semi	ADJ
cana-1179	121	19	pre	pre	ADJ
cana-1179	121	20	-	-	ADJ
cana-1179	121	21	closed	closed	ADJ
cana-1179	121	22	sets	set	NOUN
cana-1179	121	23	in	in	ADP
cana-1179	121	24	the	the	DET
cana-1179	121	25	topological	topological	ADJ
cana-1179	121	26	space	space	NOUN
cana-1179	121	27	(	(	PUNCT
cana-1179	121	28	x	x	X
cana-1179	121	29	,	,	PUNCT
cana-1179	121	30	τ	τ	X
cana-1179	121	31	)	)	PUNCT
cana-1179	121	32	is	be	AUX
cana-1179	121	33	independent	independent	ADJ
cana-1179	121	34	of	of	ADP
cana-1179	121	35	the	the	DET
cana-1179	121	36	class	class	NOUN
cana-1179	121	37	of	of	ADP
cana-1179	121	38	βg*p	βg*p	ADJ
cana-1179	121	39	-	-	PUNCT
cana-1179	121	40	closed	closed	ADJ
cana-1179	121	41	sets	set	NOUN
cana-1179	121	42	in	in	ADP
cana-1179	121	43	topological	topological	ADJ
cana-1179	121	44	space	space	NOUN
cana-1179	121	45	.	.	PUNCT
cana-1179	122	1	example	example	NOUN
cana-1179	122	2	3.13	3.13	NUM
cana-1179	122	3	with	with	ADP
cana-1179	122	4	the	the	DET
cana-1179	122	5	topology	topology	NOUN
cana-1179	122	6	τ	τ	X
cana-1179	122	7	=	=	PUNCT
cana-1179	122	8	{	{	PUNCT
cana-1179	122	9	ϕ	ϕ	NOUN
cana-1179	122	10	,	,	PUNCT
cana-1179	122	11	{	{	PUNCT
cana-1179	122	12	u	u	NOUN
cana-1179	122	13	}	}	PUNCT
cana-1179	122	14	,	,	PUNCT
cana-1179	122	15	{	{	PUNCT
cana-1179	122	16	w	w	NOUN
cana-1179	122	17	}	}	PUNCT
cana-1179	122	18	,	,	PUNCT
cana-1179	122	19	{	{	PUNCT
cana-1179	122	20	u	u	NOUN
cana-1179	122	21	,	,	PUNCT
cana-1179	122	22	w	w	PROPN
cana-1179	122	23	}	}	PUNCT
cana-1179	122	24	,	,	PUNCT
cana-1179	122	25	x	x	NOUN
cana-1179	122	26	}	}	PUNCT
cana-1179	122	27	,	,	PUNCT
cana-1179	122	28	let	let	VERB
cana-1179	122	29	x	x	PUNCT
cana-1179	122	30	=	=	PRON
cana-1179	122	31	{	{	PUNCT
cana-1179	122	32	u	u	NOUN
cana-1179	122	33	,	,	PUNCT
cana-1179	122	34	v	v	NOUN
cana-1179	122	35	,	,	PUNCT
cana-1179	122	36	w	w	NOUN
cana-1179	122	37	}	}	PUNCT
cana-1179	122	38	.	.	PUNCT
cana-1179	123	1	the	the	DET
cana-1179	123	2	closed	close	VERB
cana-1179	123	3	sets	set	NOUN
cana-1179	123	4	of	of	ADP
cana-1179	123	5	βg*p	βg*p	X
cana-1179	123	6	form	form	NOUN
cana-1179	123	7	are	be	AUX
cana-1179	123	8	{	{	PUNCT
cana-1179	123	9	ϕ	ϕ	NOUN
cana-1179	123	10	,	,	PUNCT
cana-1179	123	11	{	{	PUNCT
cana-1179	123	12	v	v	NOUN
cana-1179	123	13	}	}	PUNCT
cana-1179	123	14	,	,	PUNCT
cana-1179	123	15	{	{	PUNCT
cana-1179	123	16	v	v	NOUN
cana-1179	123	17	,	,	PUNCT
cana-1179	123	18	w	w	NOUN
cana-1179	123	19	}	}	PUNCT
cana-1179	123	20	,	,	PUNCT
cana-1179	123	21	{	{	PUNCT
cana-1179	123	22	u	u	NOUN
cana-1179	123	23	,	,	PUNCT
cana-1179	123	24	v	v	NOUN
cana-1179	123	25	}	}	PUNCT
cana-1179	123	26	,	,	PUNCT
cana-1179	123	27	x	x	NOUN
cana-1179	123	28	}	}	PUNCT
cana-1179	123	29	.	.	PUNCT
cana-1179	124	1	it	it	PRON
cana-1179	124	2	is	be	AUX
cana-1179	124	3	not	not	PART
cana-1179	124	4	a	a	DET
cana-1179	124	5	βg*p	βg*p	ADV
cana-1179	124	6	-	-	PUNCT
cana-1179	124	7	closed	closed	ADJ
cana-1179	124	8	set	set	NOUN
cana-1179	124	9	,	,	PUNCT
cana-1179	124	10	although	although	SCONJ
cana-1179	124	11	in	in	ADP
cana-1179	124	12	this	this	DET
cana-1179	124	13	case	case	NOUN
cana-1179	124	14	,	,	PUNCT
cana-1179	124	15	b	b	X
cana-1179	124	16	=	=	SYM
cana-1179	124	17	{	{	PUNCT
cana-1179	124	18	u	u	NOUN
cana-1179	124	19	,	,	PUNCT
cana-1179	124	20	w	w	NOUN
cana-1179	124	21	}	}	PUNCT
cana-1179	124	22	is	be	AUX
cana-1179	124	23	rg	rg	NOUN
cana-1179	124	24	-	-	PUNCT
cana-1179	124	25	closed	close	VERB
cana-1179	124	26	and	and	CCONJ
cana-1179	124	27	gpr	gpr	NOUN
cana-1179	124	28	-	-	PUNCT
cana-1179	124	29	closed	close	VERB
cana-1179	124	30	sets	set	NOUN
cana-1179	124	31	.	.	PUNCT
cana-1179	125	1	example	example	NOUN
cana-1179	125	2	3.14	3.14	NUM
cana-1179	125	3	with	with	ADP
cana-1179	125	4	the	the	DET
cana-1179	125	5	topology	topology	NOUN
cana-1179	125	6	τ	τ	X
cana-1179	125	7	=	=	PUNCT
cana-1179	125	8	{	{	PUNCT
cana-1179	125	9	ϕ	ϕ	NOUN
cana-1179	125	10	,	,	PUNCT
cana-1179	125	11	{	{	PUNCT
cana-1179	125	12	u	u	NOUN
cana-1179	125	13	}	}	PUNCT
cana-1179	125	14	,	,	PUNCT
cana-1179	125	15	{	{	PUNCT
cana-1179	125	16	v	v	NOUN
cana-1179	125	17	}	}	PUNCT
cana-1179	125	18	,	,	PUNCT
cana-1179	125	19	{	{	PUNCT
cana-1179	125	20	v	v	NOUN
cana-1179	125	21	,	,	PUNCT
cana-1179	125	22	u	u	NOUN
cana-1179	125	23	}	}	PUNCT
cana-1179	125	24	,	,	PUNCT
cana-1179	125	25	x	x	NOUN
cana-1179	125	26	}	}	PUNCT
cana-1179	125	27	and	and	CCONJ
cana-1179	125	28	let	let	VERB
cana-1179	125	29	x	x	PUNCT
cana-1179	125	30	=	=	PRON
cana-1179	125	31	{	{	PUNCT
cana-1179	125	32	u	u	NOUN
cana-1179	125	33	,	,	PUNCT
cana-1179	125	34	v	v	NOUN
cana-1179	125	35	,	,	PUNCT
cana-1179	125	36	w	w	NOUN
cana-1179	125	37	}	}	PUNCT
cana-1179	125	38	.	.	PUNCT
cana-1179	126	1	the	the	DET
cana-1179	126	2	closed	close	VERB
cana-1179	126	3	sets	set	NOUN
cana-1179	126	4	βg*p	βg*p	PUNCT
cana-1179	126	5	form	form	NOUN
cana-1179	126	6	are	be	AUX
cana-1179	126	7	{	{	PUNCT
cana-1179	126	8	ϕ	ϕ	NOUN
cana-1179	126	9	,	,	PUNCT
cana-1179	126	10	{	{	PUNCT
cana-1179	126	11	w	w	NOUN
cana-1179	126	12	}	}	PUNCT
cana-1179	126	13	,	,	PUNCT
cana-1179	126	14	{	{	PUNCT
cana-1179	126	15	v	v	NOUN
cana-1179	126	16	,	,	PUNCT
cana-1179	126	17	w	w	NOUN
cana-1179	126	18	}	}	PUNCT
cana-1179	126	19	,	,	PUNCT
cana-1179	126	20	{	{	PUNCT
cana-1179	126	21	u	u	NOUN
cana-1179	126	22	,	,	PUNCT
cana-1179	126	23	w	w	PROPN
cana-1179	126	24	}	}	PUNCT
cana-1179	126	25	,	,	PUNCT
cana-1179	126	26	x	x	NOUN
cana-1179	126	27	}	}	PUNCT
cana-1179	126	28	.	.	PUNCT
cana-1179	127	1	it	it	PRON
cana-1179	127	2	is	be	AUX
cana-1179	127	3	not	not	PART
cana-1179	127	4	a	a	DET
cana-1179	127	5	βg*p	βg*p	ADV
cana-1179	127	6	-	-	PUNCT
cana-1179	127	7	closed	closed	ADJ
cana-1179	127	8	set	set	NOUN
cana-1179	127	9	,	,	PUNCT
cana-1179	127	10	although	although	SCONJ
cana-1179	127	11	in	in	ADP
cana-1179	127	12	this	this	DET
cana-1179	127	13	case	case	NOUN
cana-1179	127	14	,	,	PUNCT
cana-1179	127	15	b	b	X
cana-1179	127	16	=	=	PRON
cana-1179	127	17	{	{	PUNCT
cana-1179	127	18	v	v	NOUN
cana-1179	127	19	}	}	PUNCT
cana-1179	127	20	is	be	AUX
cana-1179	127	21	gsp	gsp	VERB
cana-1179	127	22	-	-	PUNCT
cana-1179	127	23	closed	close	VERB
cana-1179	127	24	set	set	NOUN
cana-1179	127	25	.	.	PUNCT
cana-1179	128	1	theorem	theorem	VERB
cana-1179	128	2	3.15	3.15	NUM
cana-1179	128	3	union	union	NOUN
cana-1179	128	4	of	of	ADP
cana-1179	128	5	two	two	NUM
cana-1179	128	6	βg*p	βg*p	NUM
cana-1179	128	7	-	-	PUNCT
cana-1179	128	8	closed	close	VERB
cana-1179	128	9	sets	set	NOUN
cana-1179	128	10	is	be	AUX
cana-1179	128	11	βg*p	βg*p	NOUN
cana-1179	128	12	-	-	PUNCT
cana-1179	128	13	closed	closed	ADJ
cana-1179	128	14	set	set	NOUN
cana-1179	128	15	in	in	ADP
cana-1179	128	16	x.	x.	NOUN
cana-1179	128	17	proof	proof	NOUN
cana-1179	128	18	:	:	PUNCT
cana-1179	128	19	take	take	VERB
cana-1179	128	20	m	m	PRON
cana-1179	128	21	and	and	CCONJ
cana-1179	128	22	n	n	ADV
cana-1179	128	23	as	as	ADP
cana-1179	128	24	two	two	NUM
cana-1179	128	25	βg*p	βg*p	X
cana-1179	128	26	-	-	PUNCT
cana-1179	128	27	closed	closed	ADJ
cana-1179	128	28	sets	set	NOUN
cana-1179	128	29	in	in	ADP
cana-1179	128	30	x	x	PUNCT
cana-1179	128	31	to	to	PART
cana-1179	128	32	demonstrate	demonstrate	VERB
cana-1179	128	33	the	the	DET
cana-1179	128	34	proof	proof	NOUN
cana-1179	128	35	.	.	PUNCT
cana-1179	129	1	assume	assume	VERB
cana-1179	129	2	d	d	X
cana-1179	129	3	is	be	AUX
cana-1179	129	4	any	any	DET
cana-1179	129	5	g*-open	g*-open	NOUN
cana-1179	129	6	set	set	VERB
cana-1179	129	7	in	in	ADP
cana-1179	129	8	(	(	PUNCT
cana-1179	129	9	x	x	NOUN
cana-1179	129	10	,	,	PUNCT
cana-1179	129	11	τ	τ	X
cana-1179	129	12	)	)	PUNCT
cana-1179	129	13	such	such	ADJ
cana-1179	129	14	that	that	SCONJ
cana-1179	129	15	m	m	VERB
cana-1179	129	16	∪	∪	VERB
cana-1179	129	17	n	n	PRON
cana-1179	129	18	⊆	⊆	NUM
cana-1179	129	19	d.	d.	NOUN
cana-1179	129	20	following	follow	VERB
cana-1179	129	21	that	that	SCONJ
cana-1179	129	22	d	d	PROPN
cana-1179	129	23	contained	contain	VERB
cana-1179	129	24	in	in	ADP
cana-1179	129	25	both	both	DET
cana-1179	129	26	m	m	PROPN
cana-1179	129	27	and	and	CCONJ
cana-1179	129	28	n.	n.	NOUN
cana-1179	129	29	as	as	ADP
cana-1179	129	30	a	a	DET
cana-1179	129	31	result	result	NOUN
cana-1179	129	32	,	,	PUNCT
cana-1179	129	33	m	m	VERB
cana-1179	129	34	and	and	CCONJ
cana-1179	129	35	n	n	PROPN
cana-1179	129	36	are	be	AUX
cana-1179	129	37	βg*p	βg*p	X
cana-1179	129	38	-	-	PUNCT
cana-1179	129	39	closed	closed	ADJ
cana-1179	129	40	sets	set	NOUN
cana-1179	129	41	and	and	CCONJ
cana-1179	129	42	pcl(m	pcl(m	PROPN
cana-1179	129	43	)	)	PUNCT
cana-1179	130	1	⊆	⊆	NUM
cana-1179	130	2	d	d	PROPN
cana-1179	130	3	and	and	CCONJ
cana-1179	130	4	pcl(n	pcl(n	PROPN
cana-1179	130	5	)	)	PUNCT
cana-1179	130	6	⊆	⊆	NUM
cana-1179	130	7	d.	d.	NOUN
cana-1179	130	8	for	for	ADP
cana-1179	130	9	this	this	DET
cana-1179	130	10	reason	reason	NOUN
cana-1179	130	11	,	,	PUNCT
cana-1179	130	12	pcl(m	pcl(m	PROPN
cana-1179	130	13	)	)	PUNCT
cana-1179	130	14	∪	∪	ADP
cana-1179	130	15	pcl(n	pcl(n	PROPN
cana-1179	130	16	)	)	PUNCT
cana-1179	130	17	⊆	⊆	NUM
cana-1179	130	18	pcl	pcl	PROPN
cana-1179	130	19	(	(	PUNCT
cana-1179	130	20	m	m	PROPN
cana-1179	130	21	∪	∪	NOUN
cana-1179	130	22	n	n	CCONJ
cana-1179	130	23	)	)	PUNCT
cana-1179	130	24	⊆	⊆	PROPN
cana-1179	130	25	d.	d.	PROPN
cana-1179	130	26	therefore	therefore	ADV
cana-1179	130	27	,	,	PUNCT
cana-1179	130	28	m	m	VERB
cana-1179	130	29	∪	∪	ADJ
cana-1179	130	30	n	n	X
cana-1179	130	31	is	be	AUX
cana-1179	130	32	a	a	DET
cana-1179	130	33	closed	closed	ADJ
cana-1179	130	34	set	set	NOUN
cana-1179	130	35	of	of	ADP
cana-1179	130	36	βg*p	βg*p	PROPN
cana-1179	130	37	.	.	PROPN
cana-1179	130	38	example	example	NOUN
cana-1179	130	39	3.16	3.16	NUM
cana-1179	130	40	with	with	ADP
cana-1179	130	41	the	the	DET
cana-1179	130	42	topology	topology	NOUN
cana-1179	130	43	,	,	PUNCT
cana-1179	130	44	τ	τ	PROPN
cana-1179	130	45	=	=	PUNCT
cana-1179	130	46	{	{	PUNCT
cana-1179	130	47	ϕ	ϕ	NOUN
cana-1179	130	48	,	,	PUNCT
cana-1179	130	49	{	{	PUNCT
cana-1179	130	50	w	w	NOUN
cana-1179	130	51	}	}	PUNCT
cana-1179	130	52	,	,	PUNCT
cana-1179	130	53	{	{	PUNCT
cana-1179	130	54	v	v	NOUN
cana-1179	130	55	,	,	PUNCT
cana-1179	130	56	w	w	NOUN
cana-1179	130	57	}	}	PUNCT
cana-1179	130	58	,	,	PUNCT
cana-1179	130	59	{	{	PUNCT
cana-1179	130	60	u	u	NOUN
cana-1179	130	61	,	,	PUNCT
cana-1179	130	62	w	w	PROPN
cana-1179	130	63	}	}	PUNCT
cana-1179	130	64	,	,	PUNCT
cana-1179	130	65	x	x	NOUN
cana-1179	130	66	}	}	PUNCT
cana-1179	130	67	and	and	CCONJ
cana-1179	130	68	let	let	VERB
cana-1179	130	69	x	x	PUNCT
cana-1179	130	70	=	=	PRON
cana-1179	130	71	{	{	PUNCT
cana-1179	130	72	u	u	NOUN
cana-1179	130	73	,	,	PUNCT
cana-1179	130	74	v	v	NOUN
cana-1179	130	75	,	,	PUNCT
cana-1179	130	76	w	w	NOUN
cana-1179	130	77	}	}	PUNCT
cana-1179	130	78	.	.	PUNCT
cana-1179	131	1	the	the	DET
cana-1179	131	2	closed	close	VERB
cana-1179	131	3	sets	set	NOUN
cana-1179	131	4	of	of	ADP
cana-1179	131	5	βg*p	βg*p	X
cana-1179	131	6	=	=	SYM
cana-1179	131	7	{	{	PUNCT
cana-1179	131	8	ϕ	ϕ	NOUN
cana-1179	131	9	,	,	PUNCT
cana-1179	131	10	{	{	PUNCT
cana-1179	131	11	v	v	NOUN
cana-1179	131	12	}	}	PUNCT
cana-1179	131	13	,	,	PUNCT
cana-1179	131	14	{	{	PUNCT
cana-1179	131	15	u	u	NOUN
cana-1179	131	16	}	}	PUNCT
cana-1179	131	17	,	,	PUNCT
cana-1179	131	18	{	{	PUNCT
cana-1179	131	19	u	u	NOUN
cana-1179	131	20	,	,	PUNCT
cana-1179	131	21	v	v	NOUN
cana-1179	131	22	}	}	PUNCT
cana-1179	131	23	,	,	PUNCT
cana-1179	131	24	x	x	NOUN
cana-1179	131	25	}	}	PUNCT
cana-1179	131	26	.	.	PUNCT
cana-1179	132	1	take	take	VERB
cana-1179	132	2	m	m	NOUN
cana-1179	132	3	=	=	SYM
cana-1179	132	4	{	{	PUNCT
cana-1179	132	5	u	u	NOUN
cana-1179	132	6	}	}	PUNCT
cana-1179	132	7	and	and	CCONJ
cana-1179	132	8	n	n	CCONJ
cana-1179	132	9	=	=	NOUN
cana-1179	132	10	{	{	PUNCT
cana-1179	132	11	v	v	NOUN
cana-1179	132	12	}	}	PUNCT
cana-1179	132	13	be	be	AUX
cana-1179	132	14	closed	close	VERB
cana-1179	132	15	sets	set	NOUN
cana-1179	132	16	similarly	similarly	ADV
cana-1179	132	17	their	their	PRON
cana-1179	132	18	union	union	NOUN
cana-1179	132	19	m	m	VERB
cana-1179	132	20	∪	∪	ADJ
cana-1179	132	21	n	n	X
cana-1179	132	22	=	=	SYM
cana-1179	132	23	{	{	PUNCT
cana-1179	132	24	u	u	NOUN
cana-1179	132	25	,	,	PUNCT
cana-1179	132	26	v	v	NOUN
cana-1179	132	27	}	}	PUNCT
cana-1179	132	28	are	be	AUX
cana-1179	132	29	also	also	ADV
cana-1179	132	30	βg*p	βg*p	ADJ
cana-1179	132	31	-	-	PUNCT
cana-1179	132	32	closed	closed	ADJ
cana-1179	132	33	set	set	NOUN
cana-1179	132	34	.	.	PUNCT
cana-1179	133	1	remark	remark	VERB
cana-1179	133	2	3.17	3.17	NUM
cana-1179	133	3	the	the	DET
cana-1179	133	4	intersection	intersection	NOUN
cana-1179	133	5	of	of	ADP
cana-1179	133	6	two	two	NUM
cana-1179	133	7	βg*p	βg*p	NUM
cana-1179	133	8	-	-	PUNCT
cana-1179	133	9	closed	close	VERB
cana-1179	133	10	sets	set	NOUN
cana-1179	133	11	need	need	AUX
cana-1179	133	12	not	not	PART
cana-1179	133	13	be	be	AUX
cana-1179	133	14	βg*p	βg*p	X
cana-1179	133	15	-	-	PUNCT
cana-1179	133	16	closed	closed	ADJ
cana-1179	133	17	set	set	NOUN
cana-1179	133	18	.	.	PUNCT
cana-1179	134	1	communications	communication	NOUN
cana-1179	134	2	on	on	ADP
cana-1179	134	3	applied	apply	VERB
cana-1179	134	4	nonlinear	nonlinear	ADJ
cana-1179	134	5	analysis	analysis	NOUN
cana-1179	134	6	issn	issn	NOUN
cana-1179	134	7	:	:	PUNCT
cana-1179	134	8	1074	1074	NUM
cana-1179	134	9	-	-	PUNCT
cana-1179	134	10	133x	133x	NUM
cana-1179	134	11	vol	vol	NOUN
cana-1179	134	12	31	31	NUM
cana-1179	134	13	no	no	NOUN
cana-1179	134	14	.	.	PUNCT
cana-1179	135	1	6s	6s	NUM
cana-1179	135	2	(	(	PUNCT
cana-1179	135	3	2024	2024	NUM
cana-1179	135	4	)	)	PUNCT
cana-1179	135	5	211	211	NUM
cana-1179	135	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1179	135	7	example	example	NOUN
cana-1179	135	8	3.18	3.18	NUM
cana-1179	135	9	with	with	ADP
cana-1179	135	10	the	the	DET
cana-1179	135	11	topology	topology	NOUN
cana-1179	135	12	τ	τ	X
cana-1179	136	1	=	=	PUNCT
cana-1179	136	2	{	{	PUNCT
cana-1179	136	3	ϕ	ϕ	NOUN
cana-1179	136	4	,	,	PUNCT
cana-1179	136	5	{	{	PUNCT
cana-1179	136	6	w	w	NOUN
cana-1179	136	7	}	}	PUNCT
cana-1179	136	8	,	,	PUNCT
cana-1179	136	9	x	x	NOUN
cana-1179	136	10	}	}	PUNCT
cana-1179	136	11	and	and	CCONJ
cana-1179	136	12	take	take	VERB
cana-1179	136	13	x	x	X
cana-1179	136	14	=	=	PRON
cana-1179	136	15	{	{	PUNCT
cana-1179	136	16	u	u	NOUN
cana-1179	136	17	,	,	PUNCT
cana-1179	136	18	v	v	NOUN
cana-1179	136	19	,	,	PUNCT
cana-1179	136	20	w	w	NOUN
cana-1179	136	21	}	}	PUNCT
cana-1179	136	22	.	.	PUNCT
cana-1179	137	1	{	{	PUNCT
cana-1179	137	2	ϕ	ϕ	NOUN
cana-1179	137	3	,	,	PUNCT
cana-1179	137	4	{	{	PUNCT
cana-1179	137	5	u	u	NOUN
cana-1179	137	6	}	}	PUNCT
cana-1179	137	7	,	,	PUNCT
cana-1179	137	8	{	{	PUNCT
cana-1179	137	9	v	v	NOUN
cana-1179	137	10	}	}	PUNCT
cana-1179	137	11	,	,	PUNCT
cana-1179	137	12	{	{	PUNCT
cana-1179	137	13	u	u	NOUN
cana-1179	137	14	,	,	PUNCT
cana-1179	137	15	v	v	NOUN
cana-1179	137	16	}	}	PUNCT
cana-1179	137	17	,	,	PUNCT
cana-1179	137	18	{	{	PUNCT
cana-1179	137	19	v	v	NOUN
cana-1179	137	20	,	,	PUNCT
cana-1179	137	21	w	w	NOUN
cana-1179	137	22	}	}	PUNCT
cana-1179	137	23	,	,	PUNCT
cana-1179	137	24	{	{	PUNCT
cana-1179	137	25	w	w	NOUN
cana-1179	137	26	,	,	PUNCT
cana-1179	137	27	u	u	NOUN
cana-1179	137	28	}	}	PUNCT
cana-1179	137	29	,	,	PUNCT
cana-1179	137	30	x	x	X
cana-1179	137	31	}	}	PUNCT
cana-1179	137	32	are	be	AUX
cana-1179	137	33	the	the	DET
cana-1179	137	34	closed	closed	ADJ
cana-1179	137	35	sets	set	NOUN
cana-1179	137	36	of	of	ADP
cana-1179	137	37	βg*p	βg*p	NUM
cana-1179	137	38	.	.	PUNCT
cana-1179	138	1	let	let	VERB
cana-1179	138	2	m	m	VERB
cana-1179	138	3	=	=	PUNCT
cana-1179	138	4	{	{	PUNCT
cana-1179	138	5	u	u	NOUN
cana-1179	138	6	,	,	PUNCT
cana-1179	138	7	v	v	NOUN
cana-1179	138	8	}	}	PUNCT
cana-1179	138	9	and	and	CCONJ
cana-1179	138	10	n	n	CCONJ
cana-1179	138	11	=	=	SYM
cana-1179	138	12	{	{	PUNCT
cana-1179	138	13	v	v	NOUN
cana-1179	138	14	,	,	PUNCT
cana-1179	138	15	w	w	NOUN
cana-1179	138	16	}	}	PUNCT
cana-1179	138	17	.	.	PUNCT
cana-1179	139	1	similarly	similarly	ADV
cana-1179	139	2	,	,	PUNCT
cana-1179	139	3	their	their	PRON
cana-1179	139	4	intersection	intersection	NOUN
cana-1179	139	5	m	m	VERB
cana-1179	139	6	∩	∩	ADJ
cana-1179	139	7	n	n	NOUN
cana-1179	139	8	=	=	SYM
cana-1179	139	9	{	{	PUNCT
cana-1179	139	10	v	v	NOUN
cana-1179	139	11	}	}	PUNCT
cana-1179	139	12	is	be	AUX
cana-1179	139	13	also	also	ADV
cana-1179	139	14	in	in	ADP
cana-1179	139	15	βg*p	βg*p	ADJ
cana-1179	139	16	-	-	PUNCT
cana-1179	139	17	closed	closed	ADJ
cana-1179	139	18	sets	set	NOUN
cana-1179	139	19	.	.	PUNCT
cana-1179	140	1	theorem	theorem	VERB
cana-1179	140	2	3.19	3.19	NUM
cana-1179	140	3	a	a	DET
cana-1179	140	4	subset	subset	NOUN
cana-1179	140	5	b	b	NOUN
cana-1179	140	6	of	of	ADP
cana-1179	140	7	x	x	PROPN
cana-1179	140	8	is	be	AUX
cana-1179	140	9	βg*p	βg*p	X
cana-1179	140	10	-	-	PUNCT
cana-1179	140	11	closed	closed	ADJ
cana-1179	140	12	set	set	NOUN
cana-1179	140	13	in	in	ADP
cana-1179	140	14	(	(	PUNCT
cana-1179	140	15	x	x	NOUN
cana-1179	140	16	,	,	PUNCT
cana-1179	140	17	τ	τ	X
cana-1179	140	18	)	)	PUNCT
cana-1179	141	1	if	if	SCONJ
cana-1179	142	1	and	and	CCONJ
cana-1179	142	2	only	only	ADV
cana-1179	142	3	if	if	SCONJ
cana-1179	142	4	b	b	PROPN
cana-1179	142	5	⊆	⊆	NUM
cana-1179	142	6	c	c	PROPN
cana-1179	142	7	⊆	⊆	NUM
cana-1179	142	8	pcl(b	pcl(b	PROPN
cana-1179	142	9	)	)	PUNCT
cana-1179	142	10	.	.	PUNCT
cana-1179	143	1	proof	proof	NOUN
cana-1179	143	2	:	:	PUNCT
cana-1179	143	3	assume	assume	VERB
cana-1179	143	4	that	that	SCONJ
cana-1179	143	5	d	d	NOUN
cana-1179	143	6	is	be	AUX
cana-1179	143	7	a	a	DET
cana-1179	143	8	g*-open	g*-open	NOUN
cana-1179	143	9	set	set	VERB
cana-1179	143	10	in	in	ADP
cana-1179	143	11	x	x	PUNCT
cana-1179	143	12	with	with	ADP
cana-1179	143	13	c	c	PROPN
cana-1179	143	14	⊆	⊆	PROPN
cana-1179	143	15	d.	d.	PROPN
cana-1179	143	16	pcl(b	pcl(b	PROPN
cana-1179	143	17	)	)	PUNCT
cana-1179	144	1	⊆	⊆	NUM
cana-1179	144	2	d	d	NOUN
cana-1179	144	3	,	,	PUNCT
cana-1179	144	4	since	since	SCONJ
cana-1179	144	5	b	b	PROPN
cana-1179	144	6	⊆	⊆	NUM
cana-1179	144	7	d	d	PROPN
cana-1179	144	8	and	and	CCONJ
cana-1179	144	9	b	b	PROPN
cana-1179	144	10	is	be	AUX
cana-1179	144	11	βg*p	βg*p	X
cana-1179	144	12	-	-	PUNCT
cana-1179	144	13	closed	closed	ADJ
cana-1179	144	14	.	.	PUNCT
cana-1179	145	1	we	we	PRON
cana-1179	145	2	have	have	AUX
cana-1179	145	3	pcl(c	pcl(c	VERB
cana-1179	145	4	)	)	PUNCT
cana-1179	145	5	⊆	⊆	NUM
cana-1179	145	6	pcl(pcl(b	pcl(pcl(b	NOUN
cana-1179	145	7	)	)	PUNCT
cana-1179	145	8	)	)	PUNCT
cana-1179	146	1	=	=	SYM
cana-1179	146	2	pcl(b	pcl(b	PROPN
cana-1179	146	3	)	)	PUNCT
cana-1179	146	4	as	as	ADP
cana-1179	146	5	c	c	PROPN
cana-1179	146	6	⊆	⊆	NUM
cana-1179	146	7	pcl(b	pcl(b	PROPN
cana-1179	146	8	)	)	PUNCT
cana-1179	146	9	.	.	PUNCT
cana-1179	147	1	consequently	consequently	ADV
cana-1179	147	2	,	,	PUNCT
cana-1179	147	3	pcl(c	pcl(c	NOUN
cana-1179	147	4	)	)	PUNCT
cana-1179	148	1	⊆	⊆	NUM
cana-1179	148	2	d.	d.	PROPN
cana-1179	148	3	hence	hence	ADV
cana-1179	148	4	,	,	PUNCT
cana-1179	148	5	c	c	PROPN
cana-1179	148	6	is	be	AUX
cana-1179	148	7	a	a	DET
cana-1179	148	8	closed	closed	ADJ
cana-1179	148	9	set	set	NOUN
cana-1179	148	10	of	of	ADP
cana-1179	148	11	(	(	PUNCT
cana-1179	148	12	x	x	PROPN
cana-1179	148	13	,	,	PUNCT
cana-1179	148	14	τ	τ	X
cana-1179	148	15	)	)	PUNCT
cana-1179	148	16	that	that	PRON
cana-1179	148	17	is	be	AUX
cana-1179	148	18	βg*p	βg*p	NUM
cana-1179	148	19	.	.	PUNCT
cana-1179	148	20	theorem	theorem	VERB
cana-1179	148	21	3.20	3.20	NUM
cana-1179	148	22	in	in	ADP
cana-1179	148	23	(	(	PUNCT
cana-1179	148	24	x	x	NOUN
cana-1179	148	25	,	,	PUNCT
cana-1179	148	26	τ	τ	PROPN
cana-1179	148	27	)	)	PUNCT
cana-1179	148	28	,	,	PUNCT
cana-1179	148	29	b	b	PROPN
cana-1179	148	30	is	be	AUX
cana-1179	148	31	βg*p	βg*p	X
cana-1179	148	32	-	-	PUNCT
cana-1179	148	33	closed	closed	ADJ
cana-1179	148	34	if	if	SCONJ
cana-1179	148	35	it	it	PRON
cana-1179	148	36	is	be	AUX
cana-1179	148	37	both	both	CCONJ
cana-1179	148	38	open	open	ADJ
cana-1179	148	39	and	and	CCONJ
cana-1179	148	40	βg	βg	ADV
cana-1179	148	41	-	-	PUNCT
cana-1179	148	42	open	open	ADJ
cana-1179	148	43	.	.	PUNCT
cana-1179	149	1	proof	proof	NOUN
cana-1179	149	2	:	:	PUNCT
cana-1179	149	3	let	let	VERB
cana-1179	149	4	b	b	NOUN
cana-1179	149	5	⊆	⊆	NUM
cana-1179	149	6	d	d	PROPN
cana-1179	149	7	and	and	CCONJ
cana-1179	149	8	d	d	NOUN
cana-1179	149	9	be	be	AUX
cana-1179	149	10	g*-open	g*-open	ADJ
cana-1179	149	11	.	.	PUNCT
cana-1179	150	1	at	at	ADP
cana-1179	150	2	this	this	DET
cana-1179	150	3	point	point	NOUN
cana-1179	150	4	b	b	PROPN
cana-1179	150	5	⊆	⊆	NUM
cana-1179	150	6	b.	b.	NOUN
cana-1179	150	7	according	accord	VERB
cana-1179	150	8	to	to	ADP
cana-1179	150	9	the	the	DET
cana-1179	150	10	hypothesis	hypothesis	NOUN
cana-1179	150	11	,	,	PUNCT
cana-1179	150	12	βcl(b	βcl(b	PROPN
cana-1179	150	13	)	)	PUNCT
cana-1179	150	14	⊆	⊆	NUM
cana-1179	150	15	b.	b.	NOUN
cana-1179	150	16	every	every	DET
cana-1179	150	17	β	β	X
cana-1179	150	18	-	-	ADJ
cana-1179	150	19	closed	closed	ADJ
cana-1179	150	20	set	set	NOUN
cana-1179	150	21	is	be	AUX
cana-1179	150	22	pre	pre	ADJ
cana-1179	150	23	-	-	ADJ
cana-1179	150	24	closed	closed	ADJ
cana-1179	150	25	,	,	PUNCT
cana-1179	150	26	therefore	therefore	ADV
cana-1179	150	27	pcl(b	pcl(b	PROPN
cana-1179	150	28	)	)	PUNCT
cana-1179	150	29	⊆	⊆	NUM
cana-1179	150	30	cl(b	cl(b	NOUN
cana-1179	150	31	)	)	PUNCT
cana-1179	150	32	.	.	PUNCT
cana-1179	151	1	as	as	ADP
cana-1179	151	2	a	a	DET
cana-1179	151	3	result	result	NOUN
cana-1179	151	4	,	,	PUNCT
cana-1179	151	5	pcl(b	pcl(b	PROPN
cana-1179	151	6	)	)	PUNCT
cana-1179	151	7	⊆	⊆	NUM
cana-1179	151	8	b	b	PROPN
cana-1179	151	9	⊆	⊆	NUM
cana-1179	151	10	u.	u.	NOUN
cana-1179	151	11	therefore	therefore	ADV
cana-1179	151	12	,	,	PUNCT
cana-1179	151	13	b	b	PROPN
cana-1179	151	14	is	be	AUX
cana-1179	151	15	βg*p	βg*p	X
cana-1179	151	16	-	-	PUNCT
cana-1179	151	17	closed	closed	ADJ
cana-1179	151	18	set	set	NOUN
cana-1179	151	19	.	.	PUNCT
cana-1179	152	1	theorem	theorem	VERB
cana-1179	152	2	3.21	3.21	NUM
cana-1179	152	3	if	if	SCONJ
cana-1179	152	4	b	b	PROPN
cana-1179	152	5	⊆	⊆	NUM
cana-1179	152	6	x	x	PUNCT
cana-1179	152	7	is	be	AUX
cana-1179	152	8	a	a	DET
cana-1179	152	9	βg*p	βg*p	ADV
cana-1179	152	10	-	-	PUNCT
cana-1179	152	11	closed	closed	ADJ
cana-1179	152	12	set	set	NOUN
cana-1179	152	13	,	,	PUNCT
cana-1179	152	14	then	then	ADV
cana-1179	152	15	there	there	PRON
cana-1179	152	16	is	be	VERB
cana-1179	152	17	no	no	DET
cana-1179	152	18	non	non	ADJ
cana-1179	152	19	-	-	ADJ
cana-1179	152	20	empty	empty	ADJ
cana-1179	152	21	β	β	ADJ
cana-1179	152	22	-	-	ADJ
cana-1179	152	23	closed	closed	ADJ
cana-1179	152	24	set	set	NOUN
cana-1179	152	25	in	in	ADP
cana-1179	152	26	pcl(b)-b	pcl(b)-b	PROPN
cana-1179	152	27	.	.	PUNCT
cana-1179	153	1	proof	proof	NOUN
cana-1179	153	2	:	:	PUNCT
cana-1179	153	3	assume	assume	VERB
cana-1179	153	4	that	that	SCONJ
cana-1179	153	5	b	b	PROPN
cana-1179	153	6	is	be	AUX
cana-1179	153	7	a	a	DET
cana-1179	153	8	βg*p	βg*p	ADV
cana-1179	153	9	-	-	PUNCT
cana-1179	153	10	closed	closed	ADJ
cana-1179	153	11	set	set	NOUN
cana-1179	153	12	and	and	CCONJ
cana-1179	153	13	c	c	NOUN
cana-1179	153	14	is	be	AUX
cana-1179	153	15	a	a	DET
cana-1179	153	16	closed	closed	ADJ
cana-1179	153	17	set	set	NOUN
cana-1179	153	18	in	in	ADP
cana-1179	153	19	x	x	PUNCT
cana-1179	153	20	such	such	ADJ
cana-1179	153	21	that	that	SCONJ
cana-1179	153	22	c	c	PROPN
cana-1179	153	23	⊆	⊆	NUM
cana-1179	153	24	pcl(b	pcl(b	PROPN
cana-1179	153	25	)	)	PUNCT
cana-1179	153	26	b.	b.	NOUN
cana-1179	154	1	then	then	ADV
cana-1179	154	2	c	c	PROPN
cana-1179	154	3	⊆	⊆	NUM
cana-1179	154	4	pcl(b	pcl(b	PROPN
cana-1179	154	5	)	)	PUNCT
cana-1179	154	6	and	and	CCONJ
cana-1179	154	7	c	c	NOUN
cana-1179	154	8	⊆	⊆	NUM
cana-1179	154	9	x	x	SYM
cana-1179	154	10	b	b	NOUN
cana-1179	154	11	implies	imply	VERB
cana-1179	154	12	b	b	NOUN
cana-1179	154	13	⊆	⊆	NUM
cana-1179	154	14	x	x	SYM
cana-1179	154	15	c.	c.	PROPN
cana-1179	154	16	given	give	VERB
cana-1179	154	17	that	that	PRON
cana-1179	154	18	b	b	NOUN
cana-1179	154	19	is	be	AUX
cana-1179	154	20	a	a	DET
cana-1179	154	21	βg*p	βg*p	ADV
cana-1179	154	22	-	-	PUNCT
cana-1179	154	23	closed	closed	ADJ
cana-1179	154	24	set	set	NOUN
cana-1179	154	25	and	and	CCONJ
cana-1179	154	26	x	x	SYM
cana-1179	154	27	c	c	NOUN
cana-1179	154	28	is	be	AUX
cana-1179	154	29	a	a	DET
cana-1179	154	30	β	β	NOUN
cana-1179	154	31	-	-	ADJ
cana-1179	154	32	open	open	ADJ
cana-1179	154	33	set	set	NOUN
cana-1179	154	34	containing	contain	VERB
cana-1179	154	35	b	b	NOUN
cana-1179	154	36	,	,	PUNCT
cana-1179	154	37	it	it	PRON
cana-1179	154	38	can	can	AUX
cana-1179	154	39	be	be	AUX
cana-1179	154	40	proved	prove	VERB
cana-1179	154	41	that	that	SCONJ
cana-1179	154	42	pcl(b	pcl(b	NOUN
cana-1179	154	43	)	)	PUNCT
cana-1179	155	1	⊆	⊆	NUM
cana-1179	155	2	x	x	SYM
cana-1179	155	3	c	c	NOUN
cana-1179	155	4	and	and	CCONJ
cana-1179	155	5	thus	thus	ADV
cana-1179	155	6	c	c	X
cana-1179	155	7	⊆	⊆	NUM
cana-1179	155	8	x	x	PROPN
cana-1179	155	9	-	-	PUNCT
cana-1179	155	10	pcl(b	pcl(b	NOUN
cana-1179	155	11	)	)	PUNCT
cana-1179	155	12	.	.	PUNCT
cana-1179	156	1	as	as	ADP
cana-1179	156	2	a	a	DET
cana-1179	156	3	result	result	NOUN
cana-1179	156	4	,	,	PUNCT
cana-1179	156	5	it	it	PRON
cana-1179	156	6	follows	follow	VERB
cana-1179	156	7	that	that	SCONJ
cana-1179	156	8	c	c	PROPN
cana-1179	156	9	=	=	SYM
cana-1179	156	10	ϕ	ϕ	PROPN
cana-1179	156	11	,	,	PUNCT
cana-1179	156	12	c	c	PROPN
cana-1179	156	13	pcl(b	pcl(b	PROPN
cana-1179	156	14	)	)	PUNCT
cana-1179	156	15	=	=	SYM
cana-1179	157	1	ϕ.	ϕ.	PROPN
cana-1179	157	2	4	4	NUM
cana-1179	157	3	.	.	X
cana-1179	157	4	βg*p	βg*p	X
cana-1179	157	5	-	-	PUNCT
cana-1179	157	6	open	open	ADJ
cana-1179	157	7	sets	set	NOUN
cana-1179	157	8	in	in	ADP
cana-1179	157	9	topological	topological	ADJ
cana-1179	157	10	spaces	space	NOUN
cana-1179	157	11	definition	definition	NOUN
cana-1179	157	12	4.1	4.1	NUM
cana-1179	157	13	if	if	SCONJ
cana-1179	157	14	the	the	DET
cana-1179	157	15	complement	complement	NOUN
cana-1179	157	16	of	of	ADP
cana-1179	157	17	a	a	DET
cana-1179	157	18	subset	subset	NOUN
cana-1179	157	19	b	b	NOUN
cana-1179	157	20	of	of	ADP
cana-1179	157	21	a	a	DET
cana-1179	157	22	topological	topological	ADJ
cana-1179	157	23	space	space	NOUN
cana-1179	157	24	(	(	PUNCT
cana-1179	157	25	x	x	X
cana-1179	157	26	,	,	PUNCT
cana-1179	157	27	τ	τ	X
cana-1179	157	28	)	)	PUNCT
cana-1179	157	29	is	be	AUX
cana-1179	157	30	βg*p	βg*p	NOUN
cana-1179	157	31	-	-	PUNCT
cana-1179	157	32	closed	closed	ADJ
cana-1179	157	33	,	,	PUNCT
cana-1179	157	34	then	then	ADV
cana-1179	157	35	the	the	DET
cana-1179	157	36	subset	subset	NOUN
cana-1179	157	37	is	be	AUX
cana-1179	157	38	referred	refer	VERB
cana-1179	157	39	to	to	ADP
cana-1179	157	40	as	as	ADP
cana-1179	157	41	a	a	DET
cana-1179	157	42	beta	beta	NOUN
cana-1179	157	43	generalized	generalize	VERB
cana-1179	157	44	star	star	NOUN
cana-1179	157	45	pre	pre	X
cana-1179	157	46	(	(	PUNCT
cana-1179	157	47	βg*p)-open	βg*p)-open	ADJ
cana-1179	157	48	set	set	NOUN
cana-1179	157	49	.	.	PUNCT
cana-1179	158	1	theorem	theorem	VERB
cana-1179	158	2	4.2	4.2	NUM
cana-1179	158	3	when	when	SCONJ
cana-1179	158	4	c	c	PROPN
cana-1179	158	5	is	be	AUX
cana-1179	158	6	g*-closed	g*-close	VERB
cana-1179	158	7	in	in	ADP
cana-1179	158	8	x	x	PUNCT
cana-1179	158	9	and	and	CCONJ
cana-1179	158	10	c	c	PROPN
cana-1179	158	11	⊆	⊆	NUM
cana-1179	158	12	b	b	NOUN
cana-1179	158	13	,	,	PUNCT
cana-1179	158	14	a	a	DET
cana-1179	158	15	subset	subset	NOUN
cana-1179	158	16	b	b	NOUN
cana-1179	158	17	of	of	ADP
cana-1179	158	18	a	a	DET
cana-1179	158	19	topological	topological	ADJ
cana-1179	158	20	space	space	NOUN
cana-1179	158	21	(	(	PUNCT
cana-1179	158	22	x	x	X
cana-1179	158	23	,	,	PUNCT
cana-1179	158	24	τ	τ	X
cana-1179	158	25	)	)	PUNCT
cana-1179	158	26	is	be	AUX
cana-1179	158	27	βg*popen	βg*popen	ADJ
cana-1179	158	28	if	if	SCONJ
cana-1179	158	29	and	and	CCONJ
cana-1179	158	30	only	only	ADV
cana-1179	158	31	if	if	SCONJ
cana-1179	158	32	c	c	PROPN
cana-1179	158	33	⊆	⊆	NUM
cana-1179	158	34	pint(b	pint(b	PROPN
cana-1179	158	35	)	)	PUNCT
cana-1179	158	36	.	.	PUNCT
cana-1179	159	1	proof	proof	NOUN
cana-1179	159	2	:	:	PUNCT
cana-1179	159	3	necessity	necessity	NOUN
cana-1179	159	4	:	:	PUNCT
cana-1179	159	5	assume	assume	VERB
cana-1179	159	6	c	c	PROPN
cana-1179	159	7	is	be	AUX
cana-1179	159	8	g*-closed	g*-close	VERB
cana-1179	159	9	in	in	ADP
cana-1179	159	10	(	(	PUNCT
cana-1179	159	11	x	x	X
cana-1179	159	12	,	,	PUNCT
cana-1179	159	13	τ	τ	X
cana-1179	159	14	)	)	PUNCT
cana-1179	159	15	and	and	CCONJ
cana-1179	159	16	c	c	PROPN
cana-1179	159	17	⊆	⊆	PROPN
cana-1179	159	18	b.	b.	NOUN
cana-1179	159	19	let	let	VERB
cana-1179	159	20	c	c	NOUN
cana-1179	159	21	⊆	⊆	NUM
cana-1179	159	22	pint(b	pint(b	PROPN
cana-1179	159	23	)	)	PUNCT
cana-1179	159	24	where	where	SCONJ
cana-1179	159	25	e	e	NOUN
cana-1179	159	26	is	be	AUX
cana-1179	159	27	g*-open	g*-open	ADJ
cana-1179	159	28	,	,	PUNCT
cana-1179	159	29	let	let	VERB
cana-1179	159	30	bc	bc	PROPN
cana-1179	159	31	⊆	⊆	PROPN
cana-1179	159	32	e.	e.	PROPN
cana-1179	159	33	therefore	therefore	ADV
cana-1179	159	34	,	,	PUNCT
cana-1179	159	35	e	e	PROPN
cana-1179	159	36	c	c	PROPN
cana-1179	159	37	is	be	AUX
cana-1179	159	38	g*-closed	g*-close	VERB
cana-1179	159	39	in	in	ADP
cana-1179	159	40	e	e	PROPN
cana-1179	159	41	c	c	PROPN
cana-1179	159	42	⊆	⊆	PROPN
cana-1179	159	43	b.	b.	PROPN
cana-1179	159	44	accordingly	accordingly	ADV
cana-1179	159	45	,	,	PUNCT
cana-1179	159	46	e	e	PROPN
cana-1179	159	47	c	c	NOUN
cana-1179	159	48	⊆	⊆	NUM
cana-1179	159	49	pint(b	pint(b	NOUN
cana-1179	159	50	)	)	PUNCT
cana-1179	159	51	by	by	ADP
cana-1179	159	52	assumption	assumption	NOUN
cana-1179	159	53	which	which	PRON
cana-1179	159	54	implies	imply	VERB
cana-1179	159	55	(	(	PUNCT
cana-1179	159	56	pint(b	pint(b	NOUN
cana-1179	159	57	)	)	PUNCT
cana-1179	159	58	)	)	PUNCT
cana-1179	160	1	c	c	PROPN
cana-1179	161	1	⊆	⊆	NUM
cana-1179	161	2	e.	e.	PROPN
cana-1179	161	3	consequently	consequently	ADV
cana-1179	161	4	,	,	PUNCT
cana-1179	161	5	pcl(bc	pcl(bc	VERB
cana-1179	161	6	⊆	⊆	NUM
cana-1179	161	7	e	e	NOUN
cana-1179	161	8	)	)	PUNCT
cana-1179	161	9	.	.	PUNCT
cana-1179	162	1	since	since	SCONJ
cana-1179	162	2	b	b	PROPN
cana-1179	162	3	is	be	AUX
cana-1179	162	4	implied	imply	VERB
cana-1179	162	5	to	to	PART
cana-1179	162	6	be	be	AUX
cana-1179	162	7	βg*p	βg*p	NOUN
cana-1179	162	8	-	-	ADJ
cana-1179	162	9	open	open	ADJ
cana-1179	162	10	,	,	PUNCT
cana-1179	162	11	b	b	PROPN
cana-1179	162	12	c	c	PROPN
cana-1179	162	13	is	be	AUX
cana-1179	162	14	βg*p	βg*p	X
cana-1179	162	15	-	-	PUNCT
cana-1179	162	16	closed	closed	ADJ
cana-1179	162	17	.	.	PUNCT
cana-1179	163	1	sufficiency	sufficiency	NOUN
cana-1179	163	2	:	:	PUNCT
cana-1179	163	3	let	let	VERB
cana-1179	163	4	f	f	PROPN
cana-1179	163	5	⊆	⊆	NUM
cana-1179	163	6	b	b	PROPN
cana-1179	163	7	,	,	PUNCT
cana-1179	163	8	where	where	SCONJ
cana-1179	163	9	f	f	PROPN
cana-1179	163	10	is	be	AUX
cana-1179	163	11	g*-closed	g*-close	VERB
cana-1179	163	12	and	and	CCONJ
cana-1179	163	13	let	let	VERB
cana-1179	163	14	b	b	PRON
cana-1179	163	15	be	be	AUX
cana-1179	163	16	βg*p	βg*p	NOUN
cana-1179	163	17	-	-	ADJ
cana-1179	163	18	open	open	ADJ
cana-1179	163	19	in	in	ADP
cana-1179	163	20	x.	x.	NOUN
cana-1179	163	21	with	with	ADP
cana-1179	163	22	b	b	PROPN
cana-1179	163	23	c	c	NOUN
cana-1179	163	24	⊆	⊆	NUM
cana-1179	163	25	f	f	SYM
cana-1179	163	26	c	c	NOUN
cana-1179	163	27	,	,	PUNCT
cana-1179	163	28	where	where	SCONJ
cana-1179	163	29	fc	fc	PROPN
cana-1179	163	30	is	be	AUX
cana-1179	163	31	g*-open	g*-open	ADJ
cana-1179	163	32	,	,	PUNCT
cana-1179	163	33	we	we	PRON
cana-1179	163	34	have	have	VERB
cana-1179	163	35	b	b	PROPN
cana-1179	163	36	c	c	PROPN
cana-1179	163	37	is	be	AUX
cana-1179	163	38	βg*p	βg*p	X
cana-1179	163	39	-	-	PUNCT
cana-1179	163	40	closed	closed	ADJ
cana-1179	163	41	.	.	PUNCT
cana-1179	164	1	pcl	pcl	PROPN
cana-1179	164	2	(	(	PUNCT
cana-1179	164	3	b	b	PROPN
cana-1179	164	4	c	c	X
cana-1179	164	5	)	)	PUNCT
cana-1179	164	6	⊆	⊆	NUM
cana-1179	164	7	fc	fc	PROPN
cana-1179	164	8	provides	provide	VERB
cana-1179	164	9	f	f	PROPN
cana-1179	164	10	⊆	⊆	NUM
cana-1179	164	11	x	x	SYM
cana-1179	164	12	pcl(bc	pcl(bc	NUM
cana-1179	164	13	)	)	PUNCT
cana-1179	164	14	=	=	SYM
cana-1179	164	15	pint(x	pint(x	PROPN
cana-1179	164	16	–	–	PUNCT
cana-1179	164	17	b	b	NOUN
cana-1179	164	18	c	c	NOUN
cana-1179	164	19	)	)	PUNCT
cana-1179	164	20	=	=	SYM
cana-1179	164	21	pint(b	pint(b	PROPN
cana-1179	164	22	)	)	PUNCT
cana-1179	164	23	.	.	PUNCT
cana-1179	165	1	theorem	theorem	VERB
cana-1179	165	2	4.3	4.3	NUM
cana-1179	165	3	if	if	SCONJ
cana-1179	165	4	b	b	NOUN
cana-1179	165	5	is	be	AUX
cana-1179	165	6	a	a	DET
cana-1179	165	7	βg*p	βg*p	ADJ
cana-1179	165	8	-	-	PUNCT
cana-1179	165	9	open	open	ADJ
cana-1179	165	10	subsets	subset	NOUN
cana-1179	165	11	of	of	ADP
cana-1179	165	12	(	(	PUNCT
cana-1179	165	13	x	x	X
cana-1179	165	14	,	,	PUNCT
cana-1179	165	15	τ	τ	X
cana-1179	165	16	)	)	PUNCT
cana-1179	165	17	and	and	CCONJ
cana-1179	165	18	pint(b	pint(b	PROPN
cana-1179	165	19	)	)	PUNCT
cana-1179	166	1	⊆	⊆	NUM
cana-1179	166	2	c	c	NOUN
cana-1179	166	3	⊆	⊆	NUM
cana-1179	166	4	b	b	NOUN
cana-1179	166	5	,	,	PUNCT
cana-1179	166	6	then	then	ADV
cana-1179	166	7	b	b	PROPN
cana-1179	166	8	is	be	AUX
cana-1179	166	9	moreover	moreover	ADV
cana-1179	166	10	a	a	DET
cana-1179	166	11	βg*popen	βg*popen	ADJ
cana-1179	166	12	subset	subset	NOUN
cana-1179	166	13	of	of	ADP
cana-1179	166	14	(	(	PUNCT
cana-1179	166	15	x	x	PROPN
cana-1179	166	16	,	,	PUNCT
cana-1179	166	17	τ	τ	PROPN
cana-1179	166	18	)	)	PUNCT
cana-1179	166	19	.	.	PUNCT
cana-1179	167	1	proof	proof	NOUN
cana-1179	167	2	:	:	PUNCT
cana-1179	167	3	given	give	VERB
cana-1179	167	4	that	that	DET
cana-1179	167	5	pint(b	pint(b	NOUN
cana-1179	167	6	)	)	PUNCT
cana-1179	167	7	⊆	⊆	NUM
cana-1179	167	8	c	c	NOUN
cana-1179	167	9	⊆	⊆	NUM
cana-1179	167	10	b	b	NOUN
cana-1179	167	11	,	,	PUNCT
cana-1179	167	12	bc	bc	PROPN
cana-1179	167	13	⊆	⊆	NUM
cana-1179	167	14	cc	cc	ADP
cana-1179	167	15	⊆	⊆	NUM
cana-1179	167	16	pcl(b	pcl(b	PROPN
cana-1179	167	17	c	c	NOUN
cana-1179	167	18	)	)	PUNCT
cana-1179	167	19	.	.	PUNCT
cana-1179	168	1	with	with	ADP
cana-1179	168	2	b	b	PROPN
cana-1179	168	3	c	c	PROPN
cana-1179	168	4	being	be	AUX
cana-1179	168	5	βg*p	βg*p	NOUN
cana-1179	168	6	-	-	PUNCT
cana-1179	168	7	closed	closed	ADJ
cana-1179	168	8	.	.	PUNCT
cana-1179	169	1	cc	cc	PROPN
cana-1179	169	2	is	be	AUX
cana-1179	169	3	βg*p	βg*p	X
cana-1179	169	4	-	-	PUNCT
cana-1179	169	5	closed	closed	ADJ
cana-1179	169	6	by	by	ADP
cana-1179	169	7	theorem	theorem	NOUN
cana-1179	169	8	3.19	3.19	NUM
cana-1179	169	9	.	.	PUNCT
cana-1179	170	1	consequently	consequently	ADV
cana-1179	170	2	,	,	PUNCT
cana-1179	170	3	c	c	PROPN
cana-1179	170	4	is	be	AUX
cana-1179	170	5	βg*p	βg*p	ADV
cana-1179	170	6	-	-	ADJ
cana-1179	170	7	open	open	ADJ
cana-1179	170	8	.	.	PUNCT
cana-1179	171	1	theorem	theorem	VERB
cana-1179	171	2	4.4	4.4	NUM
cana-1179	171	3	if	if	SCONJ
cana-1179	171	4	b	b	PROPN
cana-1179	171	5	⊆	⊆	NUM
cana-1179	171	6	x	x	X
cana-1179	171	7	is	be	AUX
cana-1179	171	8	βg*p	βg*p	X
cana-1179	171	9	-	-	PUNCT
cana-1179	171	10	closed	closed	ADJ
cana-1179	171	11	then	then	ADV
cana-1179	171	12	pcl(b	pcl(b	PROPN
cana-1179	171	13	)	)	PUNCT
cana-1179	172	1	-b	-b	PUNCT
cana-1179	172	2	is	be	AUX
cana-1179	172	3	g*-open	g*-open	ADJ
cana-1179	172	4	.	.	PUNCT
cana-1179	173	1	proof	proof	NOUN
cana-1179	173	2	:	:	PUNCT
cana-1179	173	3	let	let	VERB
cana-1179	173	4	b	b	X
cana-1179	173	5	be	be	AUX
cana-1179	173	6	a	a	DET
cana-1179	173	7	closed	closed	ADJ
cana-1179	173	8	set	set	NOUN
cana-1179	173	9	in	in	ADP
cana-1179	173	10	x	x	SYM
cana-1179	173	11	that	that	PRON
cana-1179	173	12	is	be	AUX
cana-1179	173	13	βg*p	βg*p	NOUN
cana-1179	173	14	.	.	PUNCT
cana-1179	173	15	given	give	VERB
cana-1179	173	16	a	a	DET
cana-1179	173	17	g*-closed	g*-close	VERB
cana-1179	173	18	set	set	NOUN
cana-1179	173	19	e	e	NOUN
cana-1179	173	20	,	,	PUNCT
cana-1179	173	21	assume	assume	VERB
cana-1179	173	22	that	that	SCONJ
cana-1179	173	23	e	e	PROPN
cana-1179	173	24	⊆	⊆	NUM
cana-1179	173	25	pcl(b	pcl(b	PROPN
cana-1179	173	26	)	)	PUNCT
cana-1179	173	27	b.	b.	PROPN
cana-1179	174	1	consequently	consequently	ADV
cana-1179	174	2	,	,	PUNCT
cana-1179	174	3	there	there	PRON
cana-1179	174	4	is	be	VERB
cana-1179	174	5	not	not	PART
cana-1179	174	6	a	a	DET
cana-1179	174	7	single	single	ADJ
cana-1179	174	8	non	non	ADJ
cana-1179	174	9	-	-	ADJ
cana-1179	174	10	empty	empty	ADJ
cana-1179	174	11	g*-closed	g*-closed	ADJ
cana-1179	174	12	set	set	NOUN
cana-1179	174	13	in	in	ADP
cana-1179	174	14	pcl(b	pcl(b	PROPN
cana-1179	174	15	)	)	PUNCT
cana-1179	174	16	b.	b.	NOUN
cana-1179	175	1	therefore	therefore	ADV
cana-1179	175	2	e	e	X
cana-1179	175	3	=	=	PUNCT
cana-1179	175	4	𝜙	𝜙	PROPN
cana-1179	175	5	and	and	CCONJ
cana-1179	175	6	e	e	X
cana-1179	175	7	⊆	⊆	NUM
cana-1179	175	8	int(pcl(b	int(pcl(b	X
cana-1179	175	9	)	)	PUNCT
cana-1179	175	10	b	b	NOUN
cana-1179	175	11	)	)	PUNCT
cana-1179	175	12	.	.	PUNCT
cana-1179	176	1	as	as	SCONJ
cana-1179	176	2	can	can	AUX
cana-1179	176	3	be	be	AUX
cana-1179	176	4	shown	show	VERB
cana-1179	176	5	,	,	PUNCT
cana-1179	176	6	pcl(b	pcl(b	PROPN
cana-1179	176	7	)	)	PUNCT
cana-1179	176	8	b	b	NOUN
cana-1179	176	9	is	be	AUX
cana-1179	176	10	g*-open	g*-open	ADJ
cana-1179	176	11	.	.	PUNCT
cana-1179	177	1	b	b	NOUN
cana-1179	177	2	is	be	AUX
cana-1179	177	3	consequently	consequently	ADV
cana-1179	177	4	g*p	g*p	PROPN
cana-1179	177	5	-	-	PUNCT
cana-1179	177	6	closed	close	VERB
cana-1179	177	7	in	in	ADP
cana-1179	177	8	(	(	PUNCT
cana-1179	177	9	x	x	NOUN
cana-1179	177	10	,	,	PUNCT
cana-1179	177	11	τ	τ	PROPN
cana-1179	177	12	)	)	PUNCT
cana-1179	177	13	.	.	PUNCT
cana-1179	178	1	communications	communication	NOUN
cana-1179	178	2	on	on	ADP
cana-1179	178	3	applied	apply	VERB
cana-1179	178	4	nonlinear	nonlinear	ADJ
cana-1179	178	5	analysis	analysis	NOUN
cana-1179	178	6	issn	issn	NOUN
cana-1179	178	7	:	:	PUNCT
cana-1179	178	8	1074	1074	NUM
cana-1179	178	9	-	-	PUNCT
cana-1179	178	10	133x	133x	NUM
cana-1179	178	11	vol	vol	NOUN
cana-1179	178	12	31	31	NUM
cana-1179	178	13	no	no	NOUN
cana-1179	178	14	.	.	PUNCT
cana-1179	179	1	6s	6s	NUM
cana-1179	179	2	(	(	PUNCT
cana-1179	179	3	2024	2024	NUM
cana-1179	179	4	)	)	PUNCT
cana-1179	179	5	212	212	NUM
cana-1179	179	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1179	179	7	5	5	NUM
cana-1179	179	8	.	.	X
cana-1179	179	9	βg*p	βg*p	X
cana-1179	179	10	-	-	PUNCT
cana-1179	179	11	i	i	NOUN
cana-1179	179	12	-	-	PUNCT
cana-1179	179	13	closed	close	VERB
cana-1179	179	14	sets	set	NOUN
cana-1179	179	15	in	in	ADP
cana-1179	179	16	ideal	ideal	ADJ
cana-1179	179	17	topological	topological	ADJ
cana-1179	179	18	spaces	space	NOUN
cana-1179	179	19	definition	definition	NOUN
cana-1179	179	20	5.1	5.1	NUM
cana-1179	179	21	an	an	DET
cana-1179	179	22	ideal	ideal	ADJ
cana-1179	179	23	topological	topological	ADJ
cana-1179	179	24	space	space	NOUN
cana-1179	179	25	(	(	PUNCT
cana-1179	179	26	x	x	X
cana-1179	179	27	,	,	PUNCT
cana-1179	179	28	τ	τ	PROPN
cana-1179	179	29	,	,	PUNCT
cana-1179	179	30	i	i	NOUN
cana-1179	179	31	)	)	PUNCT
cana-1179	179	32	has	have	VERB
cana-1179	179	33	a	a	DET
cana-1179	179	34	subset	subset	NOUN
cana-1179	179	35	b	b	NOUN
cana-1179	179	36	known	know	VERB
cana-1179	179	37	as	as	ADP
cana-1179	179	38	the	the	DET
cana-1179	179	39	beta	beta	ADJ
cana-1179	179	40	generalized	generalize	VERB
cana-1179	179	41	star	star	NOUN
cana-1179	179	42	pre	pre	VERB
cana-1179	179	43	-	-	ADJ
cana-1179	179	44	i	i	ADV
cana-1179	179	45	-	-	PUNCT
cana-1179	179	46	closed	close	VERB
cana-1179	179	47	set	set	NOUN
cana-1179	179	48	if	if	SCONJ
cana-1179	179	49	b	b	PROPN
cana-1179	179	50	*	*	PUNCT
cana-1179	179	51	⊆	⊆	NUM
cana-1179	179	52	d	d	NOUN
cana-1179	179	53	whenever	whenever	SCONJ
cana-1179	179	54	b	b	PROPN
cana-1179	179	55	⊆	⊆	NUM
cana-1179	179	56	d	d	PROPN
cana-1179	179	57	and	and	CCONJ
cana-1179	179	58	d	d	NOUN
cana-1179	179	59	is	be	AUX
cana-1179	179	60	β*-open	β*-open	PROPN
cana-1179	179	61	.	.	PUNCT
cana-1179	180	1	theorem	theorem	VERB
cana-1179	180	2	5.2	5.2	NUM
cana-1179	180	3	a	a	DET
cana-1179	180	4	closed	closed	ADJ
cana-1179	180	5	set	set	NOUN
cana-1179	180	6	is	be	AUX
cana-1179	180	7	always	always	ADV
cana-1179	180	8	βg*p	βg*p	NOUN
cana-1179	180	9	-	-	PUNCT
cana-1179	180	10	i	i	NOUN
cana-1179	180	11	-	-	PUNCT
cana-1179	180	12	closed	closed	ADJ
cana-1179	180	13	.	.	PUNCT
cana-1179	181	1	proof	proof	NOUN
cana-1179	181	2	:	:	PUNCT
cana-1179	181	3	let	let	VERB
cana-1179	181	4	d	d	PRON
cana-1179	181	5	be	be	AUX
cana-1179	181	6	a	a	DET
cana-1179	181	7	β*-open	β*-open	NOUN
cana-1179	181	8	set	set	NOUN
cana-1179	181	9	in	in	ADP
cana-1179	181	10	the	the	DET
cana-1179	181	11	space	space	NOUN
cana-1179	181	12	(	(	PUNCT
cana-1179	181	13	x	x	X
cana-1179	181	14	,	,	PUNCT
cana-1179	181	15	τ	τ	PROPN
cana-1179	181	16	,	,	PUNCT
cana-1179	181	17	i	i	PROPN
cana-1179	181	18	)	)	PUNCT
cana-1179	181	19	that	that	PRON
cana-1179	181	20	contains	contain	VERB
cana-1179	181	21	b	b	NOUN
cana-1179	181	22	*	*	PUNCT
cana-1179	181	23	.	.	PUNCT
cana-1179	182	1	let	let	VERB
cana-1179	182	2	b	b	NOUN
cana-1179	182	3	be	be	AUX
cana-1179	182	4	closed	close	VERB
cana-1179	182	5	subset	subset	NOUN
cana-1179	182	6	of	of	ADP
cana-1179	182	7	the	the	DET
cana-1179	182	8	ideal	ideal	ADJ
cana-1179	182	9	topological	topological	ADJ
cana-1179	182	10	space	space	NOUN
cana-1179	182	11	.	.	PUNCT
cana-1179	183	1	b	b	X
cana-1179	183	2	=	=	SYM
cana-1179	183	3	b	b	X
cana-1179	183	4	*	*	PUNCT
cana-1179	183	5	is	be	AUX
cana-1179	183	6	what	what	PRON
cana-1179	183	7	we	we	PRON
cana-1179	183	8	obtain	obtain	VERB
cana-1179	183	9	as	as	ADP
cana-1179	183	10	the	the	DET
cana-1179	183	11	impact	impact	NOUN
cana-1179	183	12	of	of	ADP
cana-1179	183	13	b	b	NOUN
cana-1179	183	14	's	's	PART
cana-1179	183	15	closure	closure	NOUN
cana-1179	183	16	.	.	PUNCT
cana-1179	184	1	as	as	ADP
cana-1179	184	2	a	a	DET
cana-1179	184	3	result	result	NOUN
cana-1179	184	4	,	,	PUNCT
cana-1179	184	5	b	b	X
cana-1179	184	6	*	*	PUNCT
cana-1179	184	7	⊆	⊆	NUM
cana-1179	184	8	cl*(b	cl*(b	NOUN
cana-1179	184	9	)	)	PUNCT
cana-1179	184	10	⊆	⊆	NUM
cana-1179	184	11	d.	d.	PROPN
cana-1179	184	12	this	this	PRON
cana-1179	184	13	suggest	suggest	VERB
cana-1179	184	14	a	a	DET
cana-1179	184	15	βg*p	βg*p	ADJ
cana-1179	184	16	-	-	PUNCT
cana-1179	184	17	i	i	NOUN
cana-1179	184	18	-	-	PUNCT
cana-1179	184	19	closed	close	VERB
cana-1179	184	20	set	set	NOUN
cana-1179	184	21	,	,	PUNCT
cana-1179	184	22	designated	designate	VERB
cana-1179	184	23	as	as	ADP
cana-1179	184	24	b.	b.	PROPN
cana-1179	184	25	not	not	PART
cana-1179	184	26	every	every	DET
cana-1179	184	27	time	time	NOUN
cana-1179	184	28	the	the	DET
cana-1179	184	29	preceding	precede	VERB
cana-1179	184	30	theorems	theorem	NOUN
cana-1179	184	31	converse	converse	NOUN
cana-1179	184	32	true	true	ADJ
cana-1179	184	33	.	.	PUNCT
cana-1179	185	1	example	example	NOUN
cana-1179	185	2	5.3	5.3	NUM
cana-1179	185	3	construct	construct	VERB
cana-1179	185	4	a	a	DET
cana-1179	185	5	topology	topology	NOUN
cana-1179	185	6	,	,	PUNCT
cana-1179	185	7	τ	τ	PROPN
cana-1179	185	8	=	=	PUNCT
cana-1179	185	9	{	{	PUNCT
cana-1179	185	10	ϕ	ϕ	NOUN
cana-1179	185	11	,	,	PUNCT
cana-1179	185	12	x	x	NOUN
cana-1179	185	13	}	}	PUNCT
cana-1179	185	14	and	and	CCONJ
cana-1179	185	15	set	set	VERB
cana-1179	185	16	x	x	PUNCT
cana-1179	185	17	=	=	PRON
cana-1179	185	18	{	{	PUNCT
cana-1179	185	19	u	u	NOUN
cana-1179	185	20	,	,	PUNCT
cana-1179	185	21	v	v	NOUN
cana-1179	185	22	,	,	PUNCT
cana-1179	185	23	w	w	NOUN
cana-1179	185	24	}	}	PUNCT
cana-1179	185	25	,	,	PUNCT
cana-1179	185	26	i	i	PRON
cana-1179	185	27	=	=	PUNCT
cana-1179	185	28	{	{	PUNCT
cana-1179	185	29	ϕ	ϕ	NOUN
cana-1179	185	30	,	,	PUNCT
cana-1179	185	31	{	{	PUNCT
cana-1179	185	32	u	u	NOUN
cana-1179	185	33	}	}	PUNCT
cana-1179	185	34	,	,	PUNCT
cana-1179	185	35	{	{	PUNCT
cana-1179	185	36	w	w	NOUN
cana-1179	185	37	}	}	PUNCT
cana-1179	185	38	,	,	PUNCT
cana-1179	185	39	{	{	PUNCT
cana-1179	185	40	u	u	NOUN
cana-1179	185	41	,	,	PUNCT
cana-1179	185	42	w	w	NOUN
cana-1179	185	43	}	}	PUNCT
cana-1179	185	44	}	}	PUNCT
cana-1179	185	45	.	.	PUNCT
cana-1179	186	1	the	the	DET
cana-1179	186	2	closed	close	VERB
cana-1179	186	3	sets	set	NOUN
cana-1179	186	4	consisting	consist	VERB
cana-1179	186	5	of	of	ADP
cana-1179	186	6	βg*p	βg*p	NOUN
cana-1179	186	7	-	-	PUNCT
cana-1179	186	8	i	i	NOUN
cana-1179	186	9	-	-	PUNCT
cana-1179	186	10	closed	close	VERB
cana-1179	186	11	=	=	SYM
cana-1179	186	12	{	{	PUNCT
cana-1179	186	13	ϕ	ϕ	NOUN
cana-1179	186	14	,	,	PUNCT
cana-1179	186	15	{	{	PUNCT
cana-1179	186	16	u	u	NOUN
cana-1179	186	17	}	}	PUNCT
cana-1179	186	18	,	,	PUNCT
cana-1179	186	19	{	{	PUNCT
cana-1179	186	20	v	v	NOUN
cana-1179	186	21	}	}	PUNCT
cana-1179	186	22	,	,	PUNCT
cana-1179	186	23	{	{	PUNCT
cana-1179	186	24	w	w	NOUN
cana-1179	186	25	}	}	PUNCT
cana-1179	186	26	,	,	PUNCT
cana-1179	186	27	{	{	PUNCT
cana-1179	186	28	v	v	NOUN
cana-1179	186	29	,	,	PUNCT
cana-1179	186	30	w	w	NOUN
cana-1179	186	31	}	}	PUNCT
cana-1179	186	32	,	,	PUNCT
cana-1179	186	33	{	{	PUNCT
cana-1179	186	34	w	w	NOUN
cana-1179	186	35	,	,	PUNCT
cana-1179	186	36	u	u	NOUN
cana-1179	186	37	}	}	PUNCT
cana-1179	186	38	,	,	PUNCT
cana-1179	186	39	{	{	PUNCT
cana-1179	186	40	u	u	NOUN
cana-1179	186	41	,	,	PUNCT
cana-1179	186	42	v	v	NOUN
cana-1179	186	43	}	}	PUNCT
cana-1179	186	44	,	,	PUNCT
cana-1179	186	45	x	x	NOUN
cana-1179	186	46	}	}	PUNCT
cana-1179	186	47	.	.	PUNCT
cana-1179	187	1	b	b	X
cana-1179	187	2	=	=	PRON
cana-1179	187	3	{	{	PUNCT
cana-1179	187	4	w	w	NOUN
cana-1179	187	5	}	}	PUNCT
cana-1179	187	6	is	be	AUX
cana-1179	187	7	a	a	DET
cana-1179	187	8	closed	closed	ADJ
cana-1179	187	9	set	set	NOUN
cana-1179	187	10	in	in	ADP
cana-1179	187	11	terms	term	NOUN
cana-1179	187	12	of	of	ADP
cana-1179	187	13	βg*p	βg*p	X
cana-1179	187	14	-	-	PUNCT
cana-1179	187	15	i	i	PROPN
cana-1179	187	16	,	,	PUNCT
cana-1179	187	17	despite	despite	SCONJ
cana-1179	187	18	it	it	PRON
cana-1179	187	19	is	be	AUX
cana-1179	187	20	not	not	PART
cana-1179	187	21	a	a	DET
cana-1179	187	22	closed	closed	ADJ
cana-1179	187	23	set	set	NOUN
cana-1179	187	24	in	in	ADP
cana-1179	187	25	this	this	DET
cana-1179	187	26	particular	particular	ADJ
cana-1179	187	27	case	case	NOUN
cana-1179	187	28	.	.	PUNCT
cana-1179	188	1	theorem	theorem	VERB
cana-1179	188	2	5.4	5.4	NUM
cana-1179	188	3	a	a	DET
cana-1179	188	4	set	set	NOUN
cana-1179	188	5	which	which	PRON
cana-1179	188	6	is	be	AUX
cana-1179	188	7	always	always	ADV
cana-1179	188	8	βg*p	βg*p	NOUN
cana-1179	188	9	-	-	PUNCT
cana-1179	188	10	i	i	PRON
cana-1179	188	11	-	-	PUNCT
cana-1179	188	12	closed	closed	ADJ
cana-1179	188	13	is	be	AUX
cana-1179	188	14	βgp	βgp	NOUN
cana-1179	188	15	-	-	PUNCT
cana-1179	188	16	closed	closed	ADJ
cana-1179	188	17	.	.	PUNCT
cana-1179	189	1	proof	proof	NOUN
cana-1179	189	2	:	:	PUNCT
cana-1179	189	3	let	let	VERB
cana-1179	189	4	d	d	PRON
cana-1179	189	5	be	be	AUX
cana-1179	189	6	a	a	DET
cana-1179	189	7	β*-open	β*-open	NOUN
cana-1179	189	8	set	set	NOUN
cana-1179	189	9	in	in	ADP
cana-1179	189	10	the	the	DET
cana-1179	189	11	space	space	NOUN
cana-1179	189	12	(	(	PUNCT
cana-1179	189	13	x	x	X
cana-1179	189	14	,	,	PUNCT
cana-1179	189	15	τ	τ	PROPN
cana-1179	189	16	,	,	PUNCT
cana-1179	189	17	i	i	PROPN
cana-1179	189	18	)	)	PUNCT
cana-1179	189	19	that	that	PRON
cana-1179	189	20	contains	contain	VERB
cana-1179	189	21	b	b	NOUN
cana-1179	189	22	*	*	PUNCT
cana-1179	189	23	.	.	PUNCT
cana-1179	190	1	let	let	VERB
cana-1179	190	2	b	b	X
cana-1179	190	3	be	be	AUX
cana-1179	190	4	the	the	DET
cana-1179	190	5	ideal	ideal	ADJ
cana-1179	190	6	topological	topological	ADJ
cana-1179	190	7	space	space	NOUN
cana-1179	190	8	's	's	PART
cana-1179	190	9	closed	closed	ADJ
cana-1179	190	10	set	set	NOUN
cana-1179	190	11	.	.	PUNCT
cana-1179	191	1	since	since	SCONJ
cana-1179	191	2	b	b	PROPN
cana-1179	191	3	's	's	PART
cana-1179	191	4	closure	closure	NOUN
cana-1179	191	5	has	have	VERB
cana-1179	191	6	an	an	DET
cana-1179	191	7	impact	impact	NOUN
cana-1179	191	8	,	,	PUNCT
cana-1179	191	9	we	we	PRON
cana-1179	191	10	obtain	obtain	VERB
cana-1179	191	11	b	b	NOUN
cana-1179	191	12	=	=	SYM
cana-1179	191	13	b	b	PROPN
cana-1179	191	14	*	*	PROPN
cana-1179	191	15	.	.	PUNCT
cana-1179	192	1	as	as	ADP
cana-1179	192	2	a	a	DET
cana-1179	192	3	consequence	consequence	NOUN
cana-1179	192	4	,	,	PUNCT
cana-1179	192	5	b	b	X
cana-1179	192	6	*	*	PUNCT
cana-1179	192	7	⊆	⊆	NUM
cana-1179	192	8	cl*(b	cl*(b	NOUN
cana-1179	192	9	)	)	PUNCT
cana-1179	192	10	⊆	⊆	NUM
cana-1179	192	11	d.	d.	PROPN
cana-1179	192	12	this	this	PRON
cana-1179	192	13	indicates	indicate	VERB
cana-1179	192	14	that	that	SCONJ
cana-1179	192	15	b	b	NOUN
cana-1179	192	16	is	be	AUX
cana-1179	192	17	a	a	DET
cana-1179	192	18	βg*p	βg*p	ADJ
cana-1179	192	19	-	-	PUNCT
cana-1179	192	20	i	i	NOUN
cana-1179	192	21	-	-	PUNCT
cana-1179	192	22	closed	close	VERB
cana-1179	192	23	set	set	NOUN
cana-1179	192	24	.	.	PUNCT
cana-1179	193	1	not	not	PART
cana-1179	193	2	every	every	DET
cana-1179	193	3	time	time	NOUN
cana-1179	193	4	is	be	AUX
cana-1179	193	5	the	the	DET
cana-1179	193	6	preceding	precede	VERB
cana-1179	193	7	theorem	theorem	NOUN
cana-1179	193	8	's	's	PART
cana-1179	193	9	converse	converse	NOUN
cana-1179	193	10	is	be	AUX
cana-1179	193	11	true	true	ADJ
cana-1179	193	12	.	.	PUNCT
cana-1179	194	1	example	example	NOUN
cana-1179	194	2	5.5	5.5	NUM
cana-1179	194	3	create	create	VERB
cana-1179	194	4	a	a	DET
cana-1179	194	5	topology	topology	NOUN
cana-1179	194	6	,	,	PUNCT
cana-1179	194	7	τ	τ	PROPN
cana-1179	194	8	=	=	PUNCT
cana-1179	194	9	{	{	PUNCT
cana-1179	194	10	ϕ	ϕ	NOUN
cana-1179	194	11	,	,	PUNCT
cana-1179	194	12	{	{	PUNCT
cana-1179	194	13	u	u	NOUN
cana-1179	194	14	,	,	PUNCT
cana-1179	194	15	w	w	PROPN
cana-1179	194	16	}	}	PUNCT
cana-1179	194	17	,	,	PUNCT
cana-1179	194	18	{	{	PUNCT
cana-1179	194	19	v	v	NOUN
cana-1179	194	20	,	,	PUNCT
cana-1179	194	21	w	w	NOUN
cana-1179	194	22	}	}	PUNCT
cana-1179	194	23	,	,	PUNCT
cana-1179	194	24	{	{	PUNCT
cana-1179	194	25	w	w	NOUN
cana-1179	194	26	}	}	PUNCT
cana-1179	194	27	,	,	PUNCT
cana-1179	194	28	x	x	NOUN
cana-1179	194	29	}	}	PUNCT
cana-1179	194	30	and	and	CCONJ
cana-1179	194	31	set	set	VERB
cana-1179	194	32	x	x	PUNCT
cana-1179	194	33	=	=	PRON
cana-1179	194	34	{	{	PUNCT
cana-1179	194	35	u	u	NOUN
cana-1179	194	36	,	,	PUNCT
cana-1179	194	37	v	v	NOUN
cana-1179	194	38	,	,	PUNCT
cana-1179	194	39	w	w	NOUN
cana-1179	194	40	}	}	PUNCT
cana-1179	194	41	,	,	PUNCT
cana-1179	194	42	i	i	PRON
cana-1179	194	43	=	=	PUNCT
cana-1179	194	44	{	{	PUNCT
cana-1179	194	45	ϕ	ϕ	NOUN
cana-1179	194	46	,	,	PUNCT
cana-1179	194	47	{	{	PUNCT
cana-1179	194	48	w	w	NOUN
cana-1179	194	49	}	}	PUNCT
cana-1179	194	50	}	}	PUNCT
cana-1179	194	51	.	.	PUNCT
cana-1179	195	1	the	the	DET
cana-1179	195	2	closed	close	VERB
cana-1179	195	3	sets	set	NOUN
cana-1179	195	4	consist	consist	VERB
cana-1179	195	5	of	of	ADP
cana-1179	195	6	βg*p	βg*p	X
cana-1179	195	7	-	-	PUNCT
cana-1179	195	8	i	i	NOUN
cana-1179	195	9	-	-	PUNCT
cana-1179	195	10	closed	close	VERB
cana-1179	195	11	=	=	SYM
cana-1179	195	12	{	{	PUNCT
cana-1179	195	13	ϕ	ϕ	NOUN
cana-1179	195	14	,	,	PUNCT
cana-1179	195	15	{	{	PUNCT
cana-1179	195	16	u	u	NOUN
cana-1179	195	17	}	}	PUNCT
cana-1179	195	18	,	,	PUNCT
cana-1179	195	19	{	{	PUNCT
cana-1179	195	20	v	v	NOUN
cana-1179	195	21	}	}	PUNCT
cana-1179	195	22	,	,	PUNCT
cana-1179	195	23	{	{	PUNCT
cana-1179	195	24	w	w	NOUN
cana-1179	195	25	}	}	PUNCT
cana-1179	195	26	,	,	PUNCT
cana-1179	195	27	{	{	PUNCT
cana-1179	195	28	v	v	NOUN
cana-1179	195	29	,	,	PUNCT
cana-1179	195	30	w	w	NOUN
cana-1179	195	31	}	}	PUNCT
cana-1179	195	32	,	,	PUNCT
cana-1179	195	33	{	{	PUNCT
cana-1179	195	34	w	w	NOUN
cana-1179	195	35	,	,	PUNCT
cana-1179	195	36	u	u	NOUN
cana-1179	195	37	}	}	PUNCT
cana-1179	195	38	,	,	PUNCT
cana-1179	195	39	{	{	PUNCT
cana-1179	195	40	u	u	NOUN
cana-1179	195	41	,	,	PUNCT
cana-1179	195	42	v	v	NOUN
cana-1179	195	43	}	}	PUNCT
cana-1179	195	44	,	,	PUNCT
cana-1179	195	45	x	x	NOUN
cana-1179	195	46	}	}	PUNCT
cana-1179	195	47	.	.	PUNCT
cana-1179	196	1	in	in	ADP
cana-1179	196	2	this	this	DET
cana-1179	196	3	case	case	NOUN
cana-1179	196	4	,	,	PUNCT
cana-1179	196	5	b	b	X
cana-1179	196	6	=	=	PRON
cana-1179	196	7	{	{	PUNCT
cana-1179	196	8	w	w	NOUN
cana-1179	196	9	}	}	PUNCT
cana-1179	196	10	is	be	AUX
cana-1179	196	11	a	a	DET
cana-1179	196	12	closed	closed	ADJ
cana-1179	196	13	set	set	NOUN
cana-1179	196	14	in	in	ADP
cana-1179	196	15	terms	term	NOUN
cana-1179	196	16	of	of	ADP
cana-1179	196	17	βg*p	βg*p	X
cana-1179	196	18	-	-	PUNCT
cana-1179	196	19	i	i	NOUN
cana-1179	196	20	,	,	PUNCT
cana-1179	196	21	even	even	ADV
cana-1179	196	22	though	though	SCONJ
cana-1179	196	23	it	it	PRON
cana-1179	196	24	is	be	AUX
cana-1179	196	25	not	not	PART
cana-1179	196	26	a	a	DET
cana-1179	196	27	βgp	βgp	ADV
cana-1179	196	28	-	-	PUNCT
cana-1179	196	29	closed	closed	ADJ
cana-1179	196	30	set	set	NOUN
cana-1179	196	31	.	.	PUNCT
cana-1179	197	1	theorem	theorem	VERB
cana-1179	197	2	5.6	5.6	NUM
cana-1179	197	3	all	all	DET
cana-1179	197	4	βg*p	βg*p	ADJ
cana-1179	197	5	-	-	PUNCT
cana-1179	197	6	closed	closed	ADJ
cana-1179	197	7	sets	set	NOUN
cana-1179	197	8	are	be	AUX
cana-1179	197	9	βg*p	βg*p	NOUN
cana-1179	197	10	-	-	PUNCT
cana-1179	197	11	i	i	NOUN
cana-1179	197	12	-	-	PUNCT
cana-1179	197	13	closed	closed	ADJ
cana-1179	197	14	.	.	PUNCT
cana-1179	198	1	proof	proof	NOUN
cana-1179	198	2	:	:	PUNCT
cana-1179	198	3	let	let	VERB
cana-1179	198	4	b	b	X
cana-1179	198	5	be	be	AUX
cana-1179	198	6	a	a	DET
cana-1179	198	7	βg*p	βg*p	ADJ
cana-1179	198	8	-	-	PUNCT
cana-1179	198	9	i	i	NOUN
cana-1179	198	10	-	-	PUNCT
cana-1179	198	11	closed	close	VERB
cana-1179	198	12	set	set	NOUN
cana-1179	198	13	in	in	ADP
cana-1179	198	14	the	the	DET
cana-1179	198	15	ideal	ideal	ADJ
cana-1179	198	16	topological	topological	ADJ
cana-1179	198	17	space	space	NOUN
cana-1179	198	18	(	(	PUNCT
cana-1179	198	19	x	x	X
cana-1179	198	20	,	,	PUNCT
cana-1179	198	21	τ	τ	PROPN
cana-1179	198	22	,	,	PUNCT
cana-1179	198	23	i	i	PROPN
cana-1179	198	24	)	)	PUNCT
cana-1179	198	25	.	.	PUNCT
cana-1179	199	1	allow	allow	VERB
cana-1179	199	2	d	d	NOUN
cana-1179	199	3	to	to	PART
cana-1179	199	4	be	be	AUX
cana-1179	199	5	an	an	DET
cana-1179	199	6	open	open	ADJ
cana-1179	199	7	set	set	NOUN
cana-1179	199	8	in	in	ADP
cana-1179	199	9	the	the	DET
cana-1179	199	10	space	space	NOUN
cana-1179	199	11	such	such	ADJ
cana-1179	199	12	that	that	DET
cana-1179	199	13	b	b	X
cana-1179	199	14	*	*	PUNCT
cana-1179	199	15	⊆	⊆	NUM
cana-1179	199	16	d.	d.	NOUN
cana-1179	199	17	for	for	ADP
cana-1179	199	18	every	every	DET
cana-1179	199	19	open	open	ADJ
cana-1179	199	20	set	set	NOUN
cana-1179	199	21	to	to	PART
cana-1179	199	22	be	be	AUX
cana-1179	199	23	β*-open	β*-open	PROPN
cana-1179	199	24	,	,	PUNCT
cana-1179	199	25	b	b	NOUN
cana-1179	199	26	must	must	AUX
cana-1179	199	27	be	be	AUX
cana-1179	199	28	βg*p	βg*p	NOUN
cana-1179	199	29	-	-	PUNCT
cana-1179	199	30	i	i	NOUN
cana-1179	199	31	-	-	PUNCT
cana-1179	199	32	closed	closed	ADJ
cana-1179	199	33	.	.	PUNCT
cana-1179	200	1	but	but	CCONJ
cana-1179	200	2	pcl(b	pcl(b	PROPN
cana-1179	200	3	)	)	PUNCT
cana-1179	200	4	⊆	⊆	NUM
cana-1179	200	5	b	b	X
cana-1179	200	6	*	*	PUNCT
cana-1179	200	7	is	be	AUX
cana-1179	200	8	a	a	DET
cana-1179	200	9	constant	constant	ADJ
cana-1179	200	10	.	.	PUNCT
cana-1179	201	1	hence	hence	ADV
cana-1179	201	2	,	,	PUNCT
cana-1179	201	3	pcl(b	pcl(b	PROPN
cana-1179	201	4	)	)	PUNCT
cana-1179	202	1	⊆	⊆	PROPN
cana-1179	202	2	d.	d.	PROPN
cana-1179	202	3	b	b	PROPN
cana-1179	202	4	is	be	AUX
cana-1179	202	5	thus	thus	ADV
cana-1179	202	6	a	a	DET
cana-1179	202	7	βg*p	βg*p	ADV
cana-1179	202	8	-	-	PUNCT
cana-1179	202	9	closed	closed	ADJ
cana-1179	202	10	set	set	NOUN
cana-1179	202	11	in	in	ADP
cana-1179	202	12	ideal	ideal	ADJ
cana-1179	202	13	topological	topological	ADJ
cana-1179	202	14	space	space	NOUN
cana-1179	202	15	.	.	PUNCT
cana-1179	203	1	the	the	DET
cana-1179	203	2	inverse	inverse	NOUN
cana-1179	203	3	of	of	ADP
cana-1179	203	4	this	this	DET
cana-1179	203	5	theorem	theorem	NOUN
cana-1179	203	6	generally	generally	ADV
cana-1179	203	7	proves	prove	VERB
cana-1179	203	8	true	true	ADJ
cana-1179	203	9	.	.	PUNCT
cana-1179	204	1	example	example	NOUN
cana-1179	204	2	5.7	5.7	NUM
cana-1179	204	3	determine	determine	VERB
cana-1179	204	4	a	a	DET
cana-1179	204	5	topology	topology	NOUN
cana-1179	204	6	,	,	PUNCT
cana-1179	204	7	τ	τ	PROPN
cana-1179	204	8	=	=	PUNCT
cana-1179	204	9	{	{	PUNCT
cana-1179	204	10	ϕ	ϕ	NOUN
cana-1179	204	11	,	,	PUNCT
cana-1179	204	12	{	{	PUNCT
cana-1179	204	13	u	u	NOUN
cana-1179	204	14	,	,	PUNCT
cana-1179	204	15	v	v	NOUN
cana-1179	204	16	}	}	PUNCT
cana-1179	204	17	,	,	PUNCT
cana-1179	204	18	{	{	PUNCT
cana-1179	204	19	u	u	NOUN
cana-1179	204	20	,	,	PUNCT
cana-1179	204	21	w	w	PROPN
cana-1179	204	22	}	}	PUNCT
cana-1179	204	23	,	,	PUNCT
cana-1179	204	24	{	{	PUNCT
cana-1179	204	25	w	w	NOUN
cana-1179	204	26	}	}	PUNCT
cana-1179	204	27	,	,	PUNCT
cana-1179	204	28	{	{	PUNCT
cana-1179	204	29	u	u	NOUN
cana-1179	204	30	}	}	PUNCT
cana-1179	204	31	,	,	PUNCT
cana-1179	204	32	x	x	NOUN
cana-1179	204	33	}	}	PUNCT
cana-1179	204	34	and	and	CCONJ
cana-1179	204	35	set	set	VERB
cana-1179	204	36	x	x	PUNCT
cana-1179	204	37	=	=	PRON
cana-1179	204	38	{	{	PUNCT
cana-1179	204	39	u	u	NOUN
cana-1179	204	40	,	,	PUNCT
cana-1179	204	41	v	v	NOUN
cana-1179	204	42	,	,	PUNCT
cana-1179	204	43	w	w	NOUN
cana-1179	204	44	}	}	PUNCT
cana-1179	204	45	,	,	PUNCT
cana-1179	204	46	i	i	PRON
cana-1179	204	47	=	=	PUNCT
cana-1179	204	48	{	{	PUNCT
cana-1179	204	49	ϕ	ϕ	NOUN
cana-1179	204	50	,	,	PUNCT
cana-1179	204	51	{	{	PUNCT
cana-1179	204	52	u	u	NOUN
cana-1179	204	53	}	}	PUNCT
cana-1179	204	54	}	}	PUNCT
cana-1179	204	55	.	.	PUNCT
cana-1179	205	1	the	the	DET
cana-1179	205	2	closed	close	VERB
cana-1179	205	3	sets	set	NOUN
cana-1179	205	4	consist	consist	VERB
cana-1179	205	5	of	of	ADP
cana-1179	205	6	βg*p	βg*p	X
cana-1179	205	7	-	-	PUNCT
cana-1179	205	8	i	i	NOUN
cana-1179	205	9	-	-	PUNCT
cana-1179	205	10	closed	close	VERB
cana-1179	205	11	=	=	SYM
cana-1179	205	12	{	{	PUNCT
cana-1179	205	13	ϕ	ϕ	NOUN
cana-1179	205	14	,	,	PUNCT
cana-1179	205	15	{	{	PUNCT
cana-1179	205	16	u	u	NOUN
cana-1179	205	17	}	}	PUNCT
cana-1179	205	18	,	,	PUNCT
cana-1179	205	19	{	{	PUNCT
cana-1179	205	20	v	v	NOUN
cana-1179	205	21	}	}	PUNCT
cana-1179	205	22	,	,	PUNCT
cana-1179	205	23	{	{	PUNCT
cana-1179	205	24	w	w	NOUN
cana-1179	205	25	}	}	PUNCT
cana-1179	205	26	,	,	PUNCT
cana-1179	205	27	{	{	PUNCT
cana-1179	205	28	v	v	NOUN
cana-1179	205	29	,	,	PUNCT
cana-1179	205	30	w	w	NOUN
cana-1179	205	31	}	}	PUNCT
cana-1179	205	32	,	,	PUNCT
cana-1179	205	33	{	{	PUNCT
cana-1179	205	34	u	u	NOUN
cana-1179	205	35	,	,	PUNCT
cana-1179	205	36	v	v	NOUN
cana-1179	205	37	}	}	PUNCT
cana-1179	205	38	,	,	PUNCT
cana-1179	205	39	x	x	NOUN
cana-1179	205	40	}	}	PUNCT
cana-1179	205	41	.	.	PUNCT
cana-1179	206	1	in	in	ADP
cana-1179	206	2	this	this	DET
cana-1179	206	3	case	case	NOUN
cana-1179	206	4	,	,	PUNCT
cana-1179	206	5	βg*p	βg*p	X
cana-1179	206	6	-	-	PUNCT
cana-1179	206	7	closed	closed	ADJ
cana-1179	206	8	in	in	ADP
cana-1179	206	9	set	set	NOUN
cana-1179	206	10	b	b	NOUN
cana-1179	206	11	=	=	SYM
cana-1179	206	12	{	{	PUNCT
cana-1179	206	13	w	w	PROPN
cana-1179	206	14	,	,	PUNCT
cana-1179	206	15	u	u	NOUN
cana-1179	206	16	}	}	PUNCT
cana-1179	206	17	but	but	CCONJ
cana-1179	206	18	βg*p	βg*p	X
cana-1179	206	19	-	-	PUNCT
cana-1179	206	20	i	i	PRON
cana-1179	206	21	is	be	AUX
cana-1179	206	22	not	not	PART
cana-1179	206	23	closed	closed	ADJ
cana-1179	206	24	.	.	PUNCT
cana-1179	207	1	theorem	theorem	VERB
cana-1179	207	2	5.8	5.8	NUM
cana-1179	207	3	a	a	DET
cana-1179	207	4	pre	pre	ADJ
cana-1179	207	5	-	-	ADJ
cana-1179	207	6	closed	closed	ADJ
cana-1179	207	7	,	,	PUNCT
cana-1179	207	8	α	α	NOUN
cana-1179	207	9	generalized	generalize	VERB
cana-1179	207	10	-	-	PUNCT
cana-1179	207	11	closed	closed	ADJ
cana-1179	207	12	,	,	PUNCT
cana-1179	207	13	g*-closed	g*-closed	ADJ
cana-1179	207	14	,	,	PUNCT
cana-1179	207	15	weakly	weakly	ADJ
cana-1179	207	16	generalized	generalize	VERB
cana-1179	207	17	-	-	PUNCT
cana-1179	207	18	closed	close	VERB
cana-1179	207	19	sets	set	NOUN
cana-1179	207	20	are	be	AUX
cana-1179	207	21	equivalent	equivalent	ADJ
cana-1179	207	22	to	to	ADP
cana-1179	207	23	a	a	DET
cana-1179	207	24	βg*p	βg*p	ADJ
cana-1179	207	25	-	-	PUNCT
cana-1179	207	26	i	i	NOUN
cana-1179	207	27	-	-	PUNCT
cana-1179	207	28	closed	close	VERB
cana-1179	207	29	set	set	NOUN
cana-1179	207	30	in	in	ADP
cana-1179	207	31	the	the	DET
cana-1179	207	32	ideal	ideal	ADJ
cana-1179	207	33	topological	topological	ADJ
cana-1179	207	34	space	space	NOUN
cana-1179	207	35	.	.	PUNCT
cana-1179	208	1	proof	proof	NOUN
cana-1179	208	2	:	:	PUNCT
cana-1179	208	3	every	every	DET
cana-1179	208	4	open	open	ADJ
cana-1179	208	5	set	set	NOUN
cana-1179	208	6	in	in	ADP
cana-1179	208	7	the	the	DET
cana-1179	208	8	space	space	NOUN
cana-1179	208	9	is	be	AUX
cana-1179	208	10	a	a	DET
cana-1179	208	11	β*-open	β*-open	NOUN
cana-1179	208	12	.	.	PUNCT
cana-1179	209	1	the	the	DET
cana-1179	209	2	following	follow	VERB
cana-1179	209	3	examples	example	NOUN
cana-1179	209	4	show	show	VERB
cana-1179	209	5	that	that	SCONJ
cana-1179	209	6	the	the	DET
cana-1179	209	7	contradiction	contradiction	NOUN
cana-1179	209	8	in	in	ADP
cana-1179	209	9	the	the	DET
cana-1179	209	10	preceding	precede	VERB
cana-1179	209	11	theorem	theorem	NOUN
cana-1179	209	12	need	need	AUX
cana-1179	209	13	not	not	PART
cana-1179	209	14	hold	hold	VERB
cana-1179	209	15	.	.	PUNCT
cana-1179	210	1	example	example	NOUN
cana-1179	210	2	5.9	5.9	NUM
cana-1179	210	3	consider	consider	VERB
cana-1179	210	4	a	a	DET
cana-1179	210	5	topology	topology	NOUN
cana-1179	210	6	,	,	PUNCT
cana-1179	210	7	τ	τ	PROPN
cana-1179	210	8	=	=	PUNCT
cana-1179	210	9	{	{	PUNCT
cana-1179	210	10	ϕ	ϕ	NOUN
cana-1179	210	11	,	,	PUNCT
cana-1179	210	12	{	{	PUNCT
cana-1179	210	13	u	u	NOUN
cana-1179	210	14	,	,	PUNCT
cana-1179	210	15	v	v	NOUN
cana-1179	210	16	}	}	PUNCT
cana-1179	210	17	,	,	PUNCT
cana-1179	210	18	{	{	PUNCT
cana-1179	210	19	v	v	NOUN
cana-1179	210	20	,	,	PUNCT
cana-1179	210	21	w	w	NOUN
cana-1179	210	22	}	}	PUNCT
cana-1179	210	23	,	,	PUNCT
cana-1179	210	24	{	{	PUNCT
cana-1179	210	25	v	v	NOUN
cana-1179	210	26	}	}	PUNCT
cana-1179	210	27	,	,	PUNCT
cana-1179	210	28	x	x	NOUN
cana-1179	210	29	}	}	PUNCT
cana-1179	210	30	and	and	CCONJ
cana-1179	210	31	set	set	VERB
cana-1179	210	32	x	x	PUNCT
cana-1179	210	33	=	=	PRON
cana-1179	210	34	{	{	PUNCT
cana-1179	210	35	u	u	NOUN
cana-1179	210	36	,	,	PUNCT
cana-1179	210	37	v	v	NOUN
cana-1179	210	38	,	,	PUNCT
cana-1179	210	39	w	w	NOUN
cana-1179	210	40	}	}	PUNCT
cana-1179	210	41	,	,	PUNCT
cana-1179	210	42	i	i	PRON
cana-1179	210	43	=	=	PUNCT
cana-1179	210	44	{	{	PUNCT
cana-1179	210	45	ϕ	ϕ	NOUN
cana-1179	210	46	,	,	PUNCT
cana-1179	210	47	{	{	PUNCT
cana-1179	210	48	v	v	NOUN
cana-1179	210	49	}	}	PUNCT
cana-1179	210	50	}	}	PUNCT
cana-1179	210	51	.	.	PUNCT
cana-1179	211	1	the	the	DET
cana-1179	211	2	closed	close	VERB
cana-1179	211	3	sets	set	NOUN
cana-1179	211	4	consist	consist	VERB
cana-1179	211	5	of	of	ADP
cana-1179	211	6	βg*p	βg*p	X
cana-1179	211	7	-	-	PUNCT
cana-1179	211	8	i	i	NOUN
cana-1179	211	9	-	-	PUNCT
cana-1179	211	10	closed	close	VERB
cana-1179	211	11	=	=	SYM
cana-1179	211	12	{	{	PUNCT
cana-1179	211	13	ϕ	ϕ	NOUN
cana-1179	211	14	,	,	PUNCT
cana-1179	211	15	{	{	PUNCT
cana-1179	211	16	u	u	NOUN
cana-1179	211	17	}	}	PUNCT
cana-1179	211	18	,	,	PUNCT
cana-1179	211	19	{	{	PUNCT
cana-1179	211	20	v	v	NOUN
cana-1179	211	21	}	}	PUNCT
cana-1179	211	22	,	,	PUNCT
cana-1179	211	23	{	{	PUNCT
cana-1179	211	24	w	w	NOUN
cana-1179	211	25	}	}	PUNCT
cana-1179	211	26	,	,	PUNCT
cana-1179	211	27	{	{	PUNCT
cana-1179	211	28	v	v	NOUN
cana-1179	211	29	,	,	PUNCT
cana-1179	211	30	w	w	NOUN
cana-1179	211	31	}	}	PUNCT
cana-1179	211	32	,	,	PUNCT
cana-1179	211	33	{	{	PUNCT
cana-1179	211	34	w	w	NOUN
cana-1179	211	35	,	,	PUNCT
cana-1179	211	36	u	u	NOUN
cana-1179	211	37	}	}	PUNCT
cana-1179	211	38	,	,	PUNCT
cana-1179	211	39	{	{	PUNCT
cana-1179	211	40	u	u	NOUN
cana-1179	211	41	,	,	PUNCT
cana-1179	211	42	v	v	NOUN
cana-1179	211	43	}	}	PUNCT
cana-1179	211	44	,	,	PUNCT
cana-1179	211	45	x	x	NOUN
cana-1179	211	46	}	}	PUNCT
cana-1179	211	47	.	.	PUNCT
cana-1179	212	1	in	in	ADP
cana-1179	212	2	this	this	DET
cana-1179	212	3	case	case	NOUN
cana-1179	212	4	,	,	PUNCT
cana-1179	212	5	b	b	X
cana-1179	212	6	=	=	PRON
cana-1179	212	7	{	{	PUNCT
cana-1179	212	8	w	w	PROPN
cana-1179	212	9	,	,	PUNCT
cana-1179	212	10	v	v	NOUN
cana-1179	212	11	}	}	PUNCT
cana-1179	212	12	is	be	AUX
cana-1179	212	13	a	a	DET
cana-1179	212	14	closed	closed	ADJ
cana-1179	212	15	set	set	NOUN
cana-1179	212	16	in	in	ADP
cana-1179	212	17	terms	term	NOUN
cana-1179	212	18	of	of	ADP
cana-1179	212	19	βg*p	βg*p	X
cana-1179	212	20	-	-	PUNCT
cana-1179	212	21	i	i	NOUN
cana-1179	212	22	,	,	PUNCT
cana-1179	212	23	even	even	ADV
cana-1179	212	24	though	though	SCONJ
cana-1179	212	25	it	it	PRON
cana-1179	212	26	is	be	AUX
cana-1179	212	27	not	not	PART
cana-1179	212	28	a	a	DET
cana-1179	212	29	pre	pre	ADJ
cana-1179	212	30	-	-	ADJ
cana-1179	212	31	closed	closed	ADJ
cana-1179	212	32	set	set	NOUN
cana-1179	212	33	.	.	PUNCT
cana-1179	213	1	example	example	NOUN
cana-1179	213	2	5.10	5.10	NUM
cana-1179	213	3	construct	construct	VERB
cana-1179	213	4	a	a	DET
cana-1179	213	5	topology	topology	NOUN
cana-1179	213	6	,	,	PUNCT
cana-1179	213	7	τ	τ	PROPN
cana-1179	213	8	=	=	PUNCT
cana-1179	213	9	{	{	PUNCT
cana-1179	213	10	ϕ	ϕ	NOUN
cana-1179	213	11	,	,	PUNCT
cana-1179	213	12	{	{	PUNCT
cana-1179	213	13	u	u	NOUN
cana-1179	213	14	}	}	PUNCT
cana-1179	213	15	,	,	PUNCT
cana-1179	213	16	{	{	PUNCT
cana-1179	213	17	v	v	NOUN
cana-1179	213	18	}	}	PUNCT
cana-1179	213	19	,	,	PUNCT
cana-1179	213	20	{	{	PUNCT
cana-1179	213	21	u	u	NOUN
cana-1179	213	22	,	,	PUNCT
cana-1179	213	23	v	v	NOUN
cana-1179	213	24	}	}	PUNCT
cana-1179	213	25	,	,	PUNCT
cana-1179	213	26	{	{	PUNCT
cana-1179	213	27	u	u	NOUN
cana-1179	213	28	,	,	PUNCT
cana-1179	213	29	w	w	PROPN
cana-1179	213	30	}	}	PUNCT
cana-1179	213	31	,	,	PUNCT
cana-1179	213	32	x	x	NOUN
cana-1179	213	33	}	}	PUNCT
cana-1179	213	34	and	and	CCONJ
cana-1179	213	35	set	set	VERB
cana-1179	213	36	x	x	PUNCT
cana-1179	213	37	=	=	PRON
cana-1179	213	38	{	{	PUNCT
cana-1179	213	39	u	u	NOUN
cana-1179	213	40	,	,	PUNCT
cana-1179	213	41	v	v	NOUN
cana-1179	213	42	,	,	PUNCT
cana-1179	213	43	w	w	NOUN
cana-1179	213	44	}	}	PUNCT
cana-1179	213	45	,	,	PUNCT
cana-1179	213	46	i	i	PRON
cana-1179	213	47	=	=	PUNCT
cana-1179	213	48	{	{	PUNCT
cana-1179	213	49	ϕ	ϕ	NOUN
cana-1179	213	50	,	,	PUNCT
cana-1179	213	51	{	{	PUNCT
cana-1179	213	52	u	u	NOUN
cana-1179	213	53	}	}	PUNCT
cana-1179	213	54	,	,	PUNCT
cana-1179	213	55	{	{	PUNCT
cana-1179	213	56	v	v	NOUN
cana-1179	213	57	}	}	PUNCT
cana-1179	213	58	,	,	PUNCT
cana-1179	213	59	{	{	PUNCT
cana-1179	213	60	u	u	NOUN
cana-1179	213	61	,	,	PUNCT
cana-1179	213	62	v	v	NOUN
cana-1179	213	63	}	}	PUNCT
cana-1179	213	64	}	}	PUNCT
cana-1179	213	65	.	.	PUNCT
cana-1179	214	1	the	the	DET
cana-1179	214	2	closed	close	VERB
cana-1179	214	3	sets	set	NOUN
cana-1179	214	4	consist	consist	VERB
cana-1179	214	5	of	of	ADP
cana-1179	214	6	βg*p	βg*p	X
cana-1179	214	7	-	-	PUNCT
cana-1179	214	8	i	i	NOUN
cana-1179	214	9	-	-	PUNCT
cana-1179	214	10	closed	close	VERB
cana-1179	214	11	=	=	SYM
cana-1179	214	12	{	{	PUNCT
cana-1179	214	13	ϕ	ϕ	NOUN
cana-1179	214	14	,	,	PUNCT
cana-1179	214	15	{	{	PUNCT
cana-1179	214	16	u	u	NOUN
cana-1179	214	17	}	}	PUNCT
cana-1179	214	18	,	,	PUNCT
cana-1179	214	19	{	{	PUNCT
cana-1179	214	20	v	v	NOUN
cana-1179	214	21	}	}	PUNCT
cana-1179	214	22	,	,	PUNCT
cana-1179	214	23	{	{	PUNCT
cana-1179	214	24	w	w	NOUN
cana-1179	214	25	}	}	PUNCT
cana-1179	214	26	,	,	PUNCT
cana-1179	214	27	{	{	PUNCT
cana-1179	214	28	w	w	NOUN
cana-1179	214	29	,	,	PUNCT
cana-1179	214	30	u	u	NOUN
cana-1179	214	31	}	}	PUNCT
cana-1179	214	32	,	,	PUNCT
cana-1179	214	33	{	{	PUNCT
cana-1179	214	34	u	u	NOUN
cana-1179	214	35	,	,	PUNCT
cana-1179	214	36	v	v	NOUN
cana-1179	214	37	}	}	PUNCT
cana-1179	214	38	,	,	PUNCT
cana-1179	214	39	x	x	NOUN
cana-1179	214	40	}	}	PUNCT
cana-1179	214	41	.	.	PUNCT
cana-1179	215	1	in	in	ADP
cana-1179	215	2	this	this	DET
cana-1179	215	3	case	case	NOUN
cana-1179	215	4	,	,	PUNCT
cana-1179	215	5	b	b	X
cana-1179	215	6	=	=	SYM
cana-1179	215	7	{	{	PUNCT
cana-1179	215	8	u	u	NOUN
cana-1179	215	9	,	,	PUNCT
cana-1179	215	10	v	v	NOUN
cana-1179	215	11	}	}	PUNCT
cana-1179	215	12	is	be	AUX
cana-1179	215	13	a	a	DET
cana-1179	215	14	closed	closed	ADJ
cana-1179	215	15	set	set	NOUN
cana-1179	215	16	in	in	ADP
cana-1179	215	17	terms	term	NOUN
cana-1179	215	18	of	of	ADP
cana-1179	215	19	βg*p	βg*p	X
cana-1179	215	20	-	-	PUNCT
cana-1179	215	21	i	i	NOUN
cana-1179	215	22	,	,	PUNCT
cana-1179	215	23	even	even	ADV
cana-1179	215	24	though	though	SCONJ
cana-1179	215	25	it	it	PRON
cana-1179	215	26	is	be	AUX
cana-1179	215	27	not	not	PART
cana-1179	215	28	a	a	DET
cana-1179	215	29	α	α	NOUN
cana-1179	215	30	generalized	generalize	VERB
cana-1179	215	31	-	-	PUNCT
cana-1179	215	32	closed	close	VERB
cana-1179	215	33	set	set	NOUN
cana-1179	215	34	.	.	PUNCT
cana-1179	216	1	communications	communication	NOUN
cana-1179	216	2	on	on	ADP
cana-1179	216	3	applied	apply	VERB
cana-1179	216	4	nonlinear	nonlinear	ADJ
cana-1179	216	5	analysis	analysis	NOUN
cana-1179	216	6	issn	issn	NOUN
cana-1179	216	7	:	:	PUNCT
cana-1179	216	8	1074	1074	NUM
cana-1179	216	9	-	-	PUNCT
cana-1179	216	10	133x	133x	NUM
cana-1179	216	11	vol	vol	NOUN
cana-1179	216	12	31	31	NUM
cana-1179	216	13	no	no	NOUN
cana-1179	216	14	.	.	PUNCT
cana-1179	217	1	6s	6s	NUM
cana-1179	217	2	(	(	PUNCT
cana-1179	217	3	2024	2024	NUM
cana-1179	217	4	)	)	PUNCT
cana-1179	217	5	213	213	NUM
cana-1179	217	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1179	217	7	example	example	NOUN
cana-1179	217	8	5.11	5.11	NUM
cana-1179	217	9	determine	determine	VERB
cana-1179	217	10	a	a	DET
cana-1179	217	11	topology	topology	NOUN
cana-1179	217	12	,	,	PUNCT
cana-1179	217	13	τ	τ	PROPN
cana-1179	217	14	=	=	PUNCT
cana-1179	217	15	{	{	PUNCT
cana-1179	217	16	ϕ	ϕ	NOUN
cana-1179	217	17	,	,	PUNCT
cana-1179	217	18	{	{	PUNCT
cana-1179	217	19	u	u	NOUN
cana-1179	217	20	,	,	PUNCT
cana-1179	217	21	v	v	NOUN
cana-1179	217	22	}	}	PUNCT
cana-1179	217	23	,	,	PUNCT
cana-1179	217	24	{	{	PUNCT
cana-1179	217	25	v	v	NOUN
cana-1179	217	26	,	,	PUNCT
cana-1179	217	27	w	w	NOUN
cana-1179	217	28	}	}	PUNCT
cana-1179	217	29	,	,	PUNCT
cana-1179	217	30	{	{	PUNCT
cana-1179	217	31	w	w	NOUN
cana-1179	217	32	}	}	PUNCT
cana-1179	217	33	,	,	PUNCT
cana-1179	217	34	{	{	PUNCT
cana-1179	217	35	v	v	NOUN
cana-1179	217	36	}	}	PUNCT
cana-1179	217	37	,	,	PUNCT
cana-1179	217	38	x	x	NOUN
cana-1179	217	39	}	}	PUNCT
cana-1179	217	40	and	and	CCONJ
cana-1179	217	41	set	set	VERB
cana-1179	217	42	x	x	PUNCT
cana-1179	217	43	=	=	PRON
cana-1179	217	44	{	{	PUNCT
cana-1179	217	45	u	u	NOUN
cana-1179	217	46	,	,	PUNCT
cana-1179	217	47	v	v	NOUN
cana-1179	217	48	,	,	PUNCT
cana-1179	217	49	w	w	NOUN
cana-1179	217	50	}	}	PUNCT
cana-1179	217	51	,	,	PUNCT
cana-1179	217	52	i	i	PRON
cana-1179	217	53	=	=	PUNCT
cana-1179	217	54	{	{	PUNCT
cana-1179	217	55	ϕ	ϕ	NOUN
cana-1179	217	56	,	,	PUNCT
cana-1179	217	57	{	{	PUNCT
cana-1179	217	58	v	v	NOUN
cana-1179	217	59	}	}	PUNCT
cana-1179	217	60	}	}	PUNCT
cana-1179	217	61	.	.	PUNCT
cana-1179	218	1	the	the	DET
cana-1179	218	2	closed	close	VERB
cana-1179	218	3	sets	set	NOUN
cana-1179	218	4	consist	consist	VERB
cana-1179	218	5	of	of	ADP
cana-1179	218	6	βg*p	βg*p	X
cana-1179	218	7	-	-	PUNCT
cana-1179	218	8	i	i	NOUN
cana-1179	218	9	-	-	PUNCT
cana-1179	218	10	closed	close	VERB
cana-1179	218	11	=	=	SYM
cana-1179	218	12	{	{	PUNCT
cana-1179	218	13	ϕ	ϕ	NOUN
cana-1179	218	14	,	,	PUNCT
cana-1179	218	15	{	{	PUNCT
cana-1179	218	16	u	u	NOUN
cana-1179	218	17	}	}	PUNCT
cana-1179	218	18	,	,	PUNCT
cana-1179	218	19	{	{	PUNCT
cana-1179	218	20	v	v	NOUN
cana-1179	218	21	}	}	PUNCT
cana-1179	218	22	,	,	PUNCT
cana-1179	218	23	{	{	PUNCT
cana-1179	218	24	w	w	NOUN
cana-1179	218	25	}	}	PUNCT
cana-1179	218	26	,	,	PUNCT
cana-1179	218	27	{	{	PUNCT
cana-1179	218	28	w	w	NOUN
cana-1179	218	29	,	,	PUNCT
cana-1179	218	30	u	u	NOUN
cana-1179	218	31	}	}	PUNCT
cana-1179	218	32	,	,	PUNCT
cana-1179	218	33	{	{	PUNCT
cana-1179	218	34	u	u	NOUN
cana-1179	218	35	,	,	PUNCT
cana-1179	218	36	v	v	NOUN
cana-1179	218	37	}	}	PUNCT
cana-1179	218	38	,	,	PUNCT
cana-1179	218	39	x	x	NOUN
cana-1179	218	40	}	}	PUNCT
cana-1179	218	41	.	.	PUNCT
cana-1179	219	1	in	in	ADP
cana-1179	219	2	this	this	DET
cana-1179	219	3	case	case	NOUN
cana-1179	219	4	,	,	PUNCT
cana-1179	219	5	b	b	X
cana-1179	219	6	=	=	PRON
cana-1179	219	7	{	{	PUNCT
cana-1179	219	8	v	v	NOUN
cana-1179	219	9	}	}	PUNCT
cana-1179	219	10	is	be	AUX
cana-1179	219	11	a	a	DET
cana-1179	219	12	closed	closed	ADJ
cana-1179	219	13	set	set	NOUN
cana-1179	219	14	in	in	ADP
cana-1179	219	15	terms	term	NOUN
cana-1179	219	16	of	of	ADP
cana-1179	219	17	βg*p	βg*p	X
cana-1179	219	18	-	-	PUNCT
cana-1179	219	19	i	i	NOUN
cana-1179	219	20	,	,	PUNCT
cana-1179	219	21	even	even	ADV
cana-1179	219	22	though	though	SCONJ
cana-1179	219	23	it	it	PRON
cana-1179	219	24	is	be	AUX
cana-1179	219	25	not	not	PART
cana-1179	219	26	a	a	DET
cana-1179	219	27	g*-closed	g*-close	VERB
cana-1179	219	28	set	set	NOUN
cana-1179	219	29	.	.	PUNCT
cana-1179	220	1	example	example	NOUN
cana-1179	220	2	5.12	5.12	NUM
cana-1179	220	3	set	set	NOUN
cana-1179	220	4	x	x	X
cana-1179	220	5	=	=	PRON
cana-1179	220	6	{	{	PUNCT
cana-1179	220	7	u	u	NOUN
cana-1179	220	8	,	,	PUNCT
cana-1179	220	9	v	v	NOUN
cana-1179	220	10	,	,	PUNCT
cana-1179	220	11	w	w	NOUN
cana-1179	220	12	}	}	PUNCT
cana-1179	220	13	,	,	PUNCT
cana-1179	220	14	i	i	PRON
cana-1179	220	15	=	=	PUNCT
cana-1179	220	16	{	{	PUNCT
cana-1179	220	17	ϕ	ϕ	NOUN
cana-1179	220	18	,	,	PUNCT
cana-1179	220	19	{	{	PUNCT
cana-1179	220	20	v	v	NOUN
cana-1179	220	21	}	}	PUNCT
cana-1179	220	22	}	}	PUNCT
cana-1179	220	23	and	and	CCONJ
cana-1179	220	24	topology	topology	NOUN
cana-1179	220	25	,	,	PUNCT
cana-1179	220	26	τ	τ	PROPN
cana-1179	220	27	=	=	PUNCT
cana-1179	220	28	{	{	PUNCT
cana-1179	220	29	ϕ	ϕ	NOUN
cana-1179	220	30	,	,	PUNCT
cana-1179	220	31	{	{	PUNCT
cana-1179	220	32	u	u	NOUN
cana-1179	220	33	,	,	PUNCT
cana-1179	220	34	v	v	NOUN
cana-1179	220	35	}	}	PUNCT
cana-1179	220	36	,	,	PUNCT
cana-1179	220	37	{	{	PUNCT
cana-1179	220	38	v	v	NOUN
cana-1179	220	39	,	,	PUNCT
cana-1179	220	40	w	w	NOUN
cana-1179	220	41	}	}	PUNCT
cana-1179	220	42	,	,	PUNCT
cana-1179	220	43	{	{	PUNCT
cana-1179	220	44	u	u	NOUN
cana-1179	220	45	}	}	PUNCT
cana-1179	220	46	,	,	PUNCT
cana-1179	220	47	{	{	PUNCT
cana-1179	220	48	v	v	NOUN
cana-1179	220	49	}	}	PUNCT
cana-1179	220	50	,	,	PUNCT
cana-1179	220	51	x	x	NOUN
cana-1179	220	52	}	}	PUNCT
cana-1179	220	53	.	.	PUNCT
cana-1179	221	1	the	the	DET
cana-1179	221	2	closed	close	VERB
cana-1179	221	3	sets	set	NOUN
cana-1179	221	4	consist	consist	VERB
cana-1179	221	5	of	of	ADP
cana-1179	221	6	βg*p	βg*p	X
cana-1179	221	7	-	-	PUNCT
cana-1179	221	8	i	i	NOUN
cana-1179	221	9	-	-	PUNCT
cana-1179	221	10	closed	close	VERB
cana-1179	221	11	=	=	SYM
cana-1179	221	12	{	{	PUNCT
cana-1179	221	13	ϕ	ϕ	NOUN
cana-1179	221	14	,	,	PUNCT
cana-1179	221	15	{	{	PUNCT
cana-1179	221	16	u	u	NOUN
cana-1179	221	17	}	}	PUNCT
cana-1179	221	18	,	,	PUNCT
cana-1179	221	19	{	{	PUNCT
cana-1179	221	20	v	v	NOUN
cana-1179	221	21	}	}	PUNCT
cana-1179	221	22	,	,	PUNCT
cana-1179	221	23	{	{	PUNCT
cana-1179	221	24	w	w	NOUN
cana-1179	221	25	}	}	PUNCT
cana-1179	221	26	,	,	PUNCT
cana-1179	221	27	{	{	PUNCT
cana-1179	221	28	v	v	NOUN
cana-1179	221	29	,	,	PUNCT
cana-1179	221	30	w	w	NOUN
cana-1179	221	31	}	}	PUNCT
cana-1179	221	32	,	,	PUNCT
cana-1179	221	33	{	{	PUNCT
cana-1179	221	34	w	w	NOUN
cana-1179	221	35	,	,	PUNCT
cana-1179	221	36	u	u	NOUN
cana-1179	221	37	}	}	PUNCT
cana-1179	221	38	,	,	PUNCT
cana-1179	221	39	x	x	NOUN
cana-1179	221	40	}	}	PUNCT
cana-1179	221	41	.	.	PUNCT
cana-1179	222	1	in	in	ADP
cana-1179	222	2	this	this	DET
cana-1179	222	3	case	case	NOUN
cana-1179	222	4	,	,	PUNCT
cana-1179	222	5	b	b	X
cana-1179	222	6	=	=	PRON
cana-1179	222	7	{	{	PUNCT
cana-1179	222	8	w	w	PROPN
cana-1179	222	9	,	,	PUNCT
cana-1179	222	10	u	u	NOUN
cana-1179	222	11	}	}	PUNCT
cana-1179	222	12	is	be	AUX
cana-1179	222	13	βg*p	βg*p	NOUN
cana-1179	222	14	-	-	PUNCT
cana-1179	222	15	i	i	NOUN
cana-1179	222	16	-	-	PUNCT
cana-1179	222	17	closed	close	VERB
cana-1179	222	18	set	set	NOUN
cana-1179	222	19	but	but	CCONJ
cana-1179	222	20	it	it	PRON
cana-1179	222	21	is	be	AUX
cana-1179	222	22	not	not	PART
cana-1179	222	23	a	a	DET
cana-1179	222	24	weakly	weakly	ADV
cana-1179	222	25	generalized	generalize	VERB
cana-1179	222	26	-	-	PUNCT
cana-1179	222	27	closed	close	VERB
cana-1179	222	28	set	set	NOUN
cana-1179	222	29	.	.	PUNCT
cana-1179	223	1	remark	remark	NOUN
cana-1179	223	2	5.13	5.13	NUM
cana-1179	223	3	a	a	DET
cana-1179	223	4	generalized	generalized	ADJ
cana-1179	223	5	α	α	PRON
cana-1179	223	6	-	-	VERB
cana-1179	223	7	closed	closed	ADJ
cana-1179	223	8	,	,	PUNCT
cana-1179	223	9	generalized	generalized	ADJ
cana-1179	223	10	star	star	NOUN
cana-1179	223	11	pre	pre	ADJ
cana-1179	223	12	-	-	ADJ
cana-1179	223	13	closed	closed	ADJ
cana-1179	223	14	,	,	PUNCT
cana-1179	223	15	mildly	mildly	ADV
cana-1179	223	16	g	g	NOUN
cana-1179	223	17	-	-	PUNCT
cana-1179	223	18	closed	closed	ADJ
cana-1179	223	19	,	,	PUNCT
cana-1179	223	20	generalized	generalize	VERB
cana-1179	223	21	semi	semi	ADJ
cana-1179	223	22	pre	pre	ADJ
cana-1179	223	23	-	-	ADJ
cana-1179	223	24	closed	closed	ADJ
cana-1179	223	25	,	,	PUNCT
cana-1179	223	26	regular	regular	ADJ
cana-1179	223	27	generalized	generalize	VERB
cana-1179	223	28	-	-	PUNCT
cana-1179	223	29	closed	closed	ADJ
cana-1179	223	30	,	,	PUNCT
cana-1179	223	31	generalized	generalized	ADJ
cana-1179	223	32	pre	pre	ADJ
cana-1179	223	33	-	-	ADJ
cana-1179	223	34	regular	regular	ADJ
cana-1179	223	35	closed	closed	ADJ
cana-1179	223	36	,	,	PUNCT
cana-1179	223	37	regular	regular	ADJ
cana-1179	223	38	weakly	weakly	ADV
cana-1179	223	39	generalizedclosed	generalizedclose	VERB
cana-1179	223	40	,	,	PUNCT
cana-1179	223	41	semi	semi	ADV
cana-1179	223	42	generalized	generalize	VERB
cana-1179	223	43	-	-	PUNCT
cana-1179	223	44	closed	closed	ADJ
cana-1179	223	45	,	,	PUNCT
cana-1179	223	46	generalized	generalized	ADJ
cana-1179	223	47	semi	semi	ADV
cana-1179	223	48	-	-	ADJ
cana-1179	223	49	closed	closed	ADJ
cana-1179	223	50	,	,	PUNCT
cana-1179	223	51	beta	beta	ADJ
cana-1179	223	52	generalized	generalize	VERB
cana-1179	223	53	-	-	PUNCT
cana-1179	223	54	closed	closed	ADJ
cana-1179	223	55	and	and	CCONJ
cana-1179	223	56	beta	beta	ADJ
cana-1179	223	57	generalized	generalized	ADJ
cana-1179	223	58	star	star	NOUN
cana-1179	223	59	-	-	PUNCT
cana-1179	223	60	closed	close	VERB
cana-1179	223	61	sets	set	NOUN
cana-1179	223	62	are	be	AUX
cana-1179	223	63	all	all	PRON
cana-1179	223	64	considered	consider	VERB
cana-1179	223	65	to	to	PART
cana-1179	223	66	be	be	AUX
cana-1179	223	67	beyond	beyond	ADP
cana-1179	223	68	of	of	ADP
cana-1179	223	69	the	the	DET
cana-1179	223	70	boundaries	boundary	NOUN
cana-1179	223	71	of	of	ADP
cana-1179	223	72	the	the	DET
cana-1179	223	73	concept	concept	NOUN
cana-1179	223	74	of	of	ADP
cana-1179	223	75	βg*pi	βg*pi	NOUN
cana-1179	223	76	-	-	PUNCT
cana-1179	223	77	closed	close	VERB
cana-1179	223	78	sets	set	NOUN
cana-1179	223	79	in	in	ADP
cana-1179	223	80	the	the	DET
cana-1179	223	81	space	space	NOUN
cana-1179	223	82	.	.	PUNCT
cana-1179	224	1	example	example	NOUN
cana-1179	224	2	5.14	5.14	NUM
cana-1179	224	3	take	take	VERB
cana-1179	224	4	a	a	DET
cana-1179	224	5	topology	topology	NOUN
cana-1179	224	6	,	,	PUNCT
cana-1179	224	7	τ	τ	PROPN
cana-1179	224	8	=	=	PUNCT
cana-1179	224	9	{	{	PUNCT
cana-1179	224	10	ϕ	ϕ	NOUN
cana-1179	224	11	,	,	PUNCT
cana-1179	224	12	{	{	PUNCT
cana-1179	224	13	u	u	NOUN
cana-1179	224	14	,	,	PUNCT
cana-1179	224	15	w	w	PROPN
cana-1179	224	16	}	}	PUNCT
cana-1179	224	17	,	,	PUNCT
cana-1179	224	18	{	{	PUNCT
cana-1179	224	19	v	v	NOUN
cana-1179	224	20	,	,	PUNCT
cana-1179	224	21	u	u	NOUN
cana-1179	224	22	}	}	PUNCT
cana-1179	224	23	,	,	PUNCT
cana-1179	224	24	{	{	PUNCT
cana-1179	224	25	u	u	NOUN
cana-1179	224	26	}	}	PUNCT
cana-1179	224	27	,	,	PUNCT
cana-1179	224	28	{	{	PUNCT
cana-1179	224	29	v	v	NOUN
cana-1179	224	30	}	}	PUNCT
cana-1179	224	31	,	,	PUNCT
cana-1179	224	32	x	x	NOUN
cana-1179	224	33	}	}	PUNCT
cana-1179	224	34	and	and	CCONJ
cana-1179	224	35	set	set	VERB
cana-1179	224	36	x	x	PUNCT
cana-1179	224	37	=	=	PRON
cana-1179	224	38	{	{	PUNCT
cana-1179	224	39	u	u	NOUN
cana-1179	224	40	,	,	PUNCT
cana-1179	224	41	v	v	NOUN
cana-1179	224	42	,	,	PUNCT
cana-1179	224	43	w	w	NOUN
cana-1179	224	44	}	}	PUNCT
cana-1179	224	45	,	,	PUNCT
cana-1179	224	46	i	i	PRON
cana-1179	224	47	=	=	PUNCT
cana-1179	224	48	{	{	PUNCT
cana-1179	224	49	ϕ	ϕ	NOUN
cana-1179	224	50	,	,	PUNCT
cana-1179	224	51	{	{	PUNCT
cana-1179	224	52	u	u	NOUN
cana-1179	224	53	}	}	PUNCT
cana-1179	224	54	,	,	PUNCT
cana-1179	224	55	{	{	PUNCT
cana-1179	224	56	v	v	NOUN
cana-1179	224	57	}	}	PUNCT
cana-1179	224	58	,	,	PUNCT
cana-1179	224	59	{	{	PUNCT
cana-1179	224	60	u	u	NOUN
cana-1179	224	61	,	,	PUNCT
cana-1179	224	62	v	v	NOUN
cana-1179	224	63	}	}	PUNCT
cana-1179	224	64	}	}	PUNCT
cana-1179	224	65	.	.	PUNCT
cana-1179	225	1	the	the	DET
cana-1179	225	2	closed	close	VERB
cana-1179	225	3	sets	set	NOUN
cana-1179	225	4	consist	consist	VERB
cana-1179	225	5	of	of	ADP
cana-1179	225	6	βg*p	βg*p	X
cana-1179	225	7	-	-	PUNCT
cana-1179	225	8	i	i	NOUN
cana-1179	225	9	-	-	PUNCT
cana-1179	225	10	closed	close	VERB
cana-1179	225	11	=	=	SYM
cana-1179	225	12	{	{	PUNCT
cana-1179	225	13	ϕ	ϕ	NOUN
cana-1179	225	14	,	,	PUNCT
cana-1179	225	15	{	{	PUNCT
cana-1179	225	16	u	u	NOUN
cana-1179	225	17	}	}	PUNCT
cana-1179	225	18	,	,	PUNCT
cana-1179	225	19	{	{	PUNCT
cana-1179	225	20	v	v	NOUN
cana-1179	225	21	}	}	PUNCT
cana-1179	225	22	,	,	PUNCT
cana-1179	225	23	{	{	PUNCT
cana-1179	225	24	w	w	NOUN
cana-1179	225	25	}	}	PUNCT
cana-1179	225	26	,	,	PUNCT
cana-1179	225	27	{	{	PUNCT
cana-1179	225	28	w	w	NOUN
cana-1179	225	29	,	,	PUNCT
cana-1179	225	30	u	u	NOUN
cana-1179	225	31	}	}	PUNCT
cana-1179	225	32	,	,	PUNCT
cana-1179	225	33	{	{	PUNCT
cana-1179	225	34	u	u	NOUN
cana-1179	225	35	,	,	PUNCT
cana-1179	225	36	v	v	NOUN
cana-1179	225	37	}	}	PUNCT
cana-1179	225	38	,	,	PUNCT
cana-1179	225	39	x	x	NOUN
cana-1179	225	40	}	}	PUNCT
cana-1179	225	41	.	.	PUNCT
cana-1179	226	1	in	in	ADP
cana-1179	226	2	this	this	DET
cana-1179	226	3	case	case	NOUN
cana-1179	226	4	,	,	PUNCT
cana-1179	226	5	generalized	generalize	VERB
cana-1179	226	6	α	α	NOUN
cana-1179	226	7	are	be	AUX
cana-1179	226	8	closed	close	VERB
cana-1179	226	9	in	in	ADP
cana-1179	226	10	set	set	NOUN
cana-1179	226	11	b	b	NOUN
cana-1179	226	12	=	=	SYM
cana-1179	226	13	{	{	PUNCT
cana-1179	226	14	v	v	NOUN
cana-1179	226	15	,	,	PUNCT
cana-1179	226	16	w	w	NOUN
cana-1179	226	17	}	}	PUNCT
cana-1179	226	18	but	but	CCONJ
cana-1179	226	19	not	not	PART
cana-1179	226	20	a	a	DET
cana-1179	226	21	βg*p	βg*p	NOUN
cana-1179	226	22	-	-	PUNCT
cana-1179	226	23	i	i	NOUN
cana-1179	226	24	-	-	PUNCT
cana-1179	226	25	closed	close	VERB
cana-1179	226	26	set	set	NOUN
cana-1179	226	27	.	.	PUNCT
cana-1179	227	1	example	example	NOUN
cana-1179	227	2	5.15	5.15	NUM
cana-1179	227	3	construct	construct	VERB
cana-1179	227	4	a	a	DET
cana-1179	227	5	topology	topology	NOUN
cana-1179	227	6	,	,	PUNCT
cana-1179	227	7	τ	τ	PROPN
cana-1179	227	8	=	=	PUNCT
cana-1179	227	9	{	{	PUNCT
cana-1179	227	10	ϕ	ϕ	NOUN
cana-1179	227	11	,	,	PUNCT
cana-1179	227	12	{	{	PUNCT
cana-1179	227	13	u	u	NOUN
cana-1179	227	14	,	,	PUNCT
cana-1179	227	15	w	w	PROPN
cana-1179	227	16	}	}	PUNCT
cana-1179	227	17	,	,	PUNCT
cana-1179	227	18	x	x	NOUN
cana-1179	227	19	}	}	PUNCT
cana-1179	227	20	and	and	CCONJ
cana-1179	227	21	set	set	VERB
cana-1179	227	22	x	x	PUNCT
cana-1179	227	23	=	=	PRON
cana-1179	227	24	{	{	PUNCT
cana-1179	227	25	u	u	NOUN
cana-1179	227	26	,	,	PUNCT
cana-1179	227	27	v	v	NOUN
cana-1179	227	28	,	,	PUNCT
cana-1179	227	29	w	w	NOUN
cana-1179	227	30	}	}	PUNCT
cana-1179	227	31	,	,	PUNCT
cana-1179	227	32	i	i	PRON
cana-1179	227	33	=	=	PUNCT
cana-1179	227	34	{	{	PUNCT
cana-1179	227	35	ϕ	ϕ	NOUN
cana-1179	227	36	,	,	PUNCT
cana-1179	227	37	{	{	PUNCT
cana-1179	227	38	w	w	NOUN
cana-1179	227	39	}	}	PUNCT
cana-1179	227	40	,	,	PUNCT
cana-1179	227	41	{	{	PUNCT
cana-1179	227	42	v	v	NOUN
cana-1179	227	43	}	}	PUNCT
cana-1179	227	44	,	,	PUNCT
cana-1179	227	45	{	{	PUNCT
cana-1179	227	46	w	w	NOUN
cana-1179	227	47	,	,	PUNCT
cana-1179	227	48	v	v	NOUN
cana-1179	227	49	}	}	PUNCT
cana-1179	227	50	}	}	PUNCT
cana-1179	227	51	.	.	PUNCT
cana-1179	228	1	the	the	DET
cana-1179	228	2	closed	close	VERB
cana-1179	228	3	sets	set	NOUN
cana-1179	228	4	consist	consist	VERB
cana-1179	228	5	of	of	ADP
cana-1179	228	6	βg*p	βg*p	X
cana-1179	228	7	-	-	PUNCT
cana-1179	228	8	i	i	NOUN
cana-1179	228	9	-	-	PUNCT
cana-1179	228	10	closed	close	VERB
cana-1179	228	11	=	=	SYM
cana-1179	228	12	{	{	PUNCT
cana-1179	228	13	ϕ	ϕ	NOUN
cana-1179	228	14	,	,	PUNCT
cana-1179	228	15	{	{	PUNCT
cana-1179	228	16	v	v	NOUN
cana-1179	228	17	}	}	PUNCT
cana-1179	228	18	,	,	PUNCT
cana-1179	228	19	{	{	PUNCT
cana-1179	228	20	w	w	NOUN
cana-1179	228	21	}	}	PUNCT
cana-1179	228	22	,	,	PUNCT
cana-1179	228	23	{	{	PUNCT
cana-1179	228	24	v	v	NOUN
cana-1179	228	25	,	,	PUNCT
cana-1179	228	26	w	w	NOUN
cana-1179	228	27	}	}	PUNCT
cana-1179	228	28	,	,	PUNCT
cana-1179	228	29	x	x	NOUN
cana-1179	228	30	}	}	PUNCT
cana-1179	228	31	.	.	PUNCT
cana-1179	229	1	in	in	ADP
cana-1179	229	2	this	this	DET
cana-1179	229	3	case	case	NOUN
cana-1179	229	4	,	,	PUNCT
cana-1179	229	5	g*p	g*p	PROPN
cana-1179	229	6	and	and	CCONJ
cana-1179	229	7	mildly	mildly	ADV
cana-1179	229	8	g	g	NOUN
cana-1179	229	9	are	be	AUX
cana-1179	229	10	closed	close	VERB
cana-1179	229	11	in	in	ADP
cana-1179	229	12	set	set	NOUN
cana-1179	229	13	b	b	NOUN
cana-1179	229	14	=	=	SYM
cana-1179	229	15	{	{	PUNCT
cana-1179	229	16	u	u	NOUN
cana-1179	229	17	,	,	PUNCT
cana-1179	229	18	v	v	NOUN
cana-1179	229	19	}	}	PUNCT
cana-1179	229	20	but	but	CCONJ
cana-1179	229	21	not	not	PART
cana-1179	229	22	a	a	DET
cana-1179	229	23	βg*p	βg*p	NOUN
cana-1179	229	24	-	-	PUNCT
cana-1179	229	25	i	i	NOUN
cana-1179	229	26	-	-	PUNCT
cana-1179	229	27	closed	close	VERB
cana-1179	229	28	set	set	NOUN
cana-1179	229	29	.	.	PUNCT
cana-1179	230	1	example	example	NOUN
cana-1179	230	2	5.16	5.16	NUM
cana-1179	230	3	define	define	VERB
cana-1179	230	4	a	a	DET
cana-1179	230	5	topology	topology	NOUN
cana-1179	230	6	,	,	PUNCT
cana-1179	230	7	τ	τ	PROPN
cana-1179	230	8	=	=	PUNCT
cana-1179	230	9	{	{	PUNCT
cana-1179	230	10	ϕ	ϕ	NOUN
cana-1179	230	11	,	,	PUNCT
cana-1179	230	12	{	{	PUNCT
cana-1179	230	13	u	u	NOUN
cana-1179	230	14	,	,	PUNCT
cana-1179	230	15	v	v	NOUN
cana-1179	230	16	}	}	PUNCT
cana-1179	230	17	,	,	PUNCT
cana-1179	230	18	{	{	PUNCT
cana-1179	230	19	v	v	NOUN
cana-1179	230	20	,	,	PUNCT
cana-1179	230	21	w	w	NOUN
cana-1179	230	22	}	}	PUNCT
cana-1179	230	23	,	,	PUNCT
cana-1179	230	24	{	{	PUNCT
cana-1179	230	25	v	v	NOUN
cana-1179	230	26	}	}	PUNCT
cana-1179	230	27	,	,	PUNCT
cana-1179	230	28	{	{	PUNCT
cana-1179	230	29	w	w	NOUN
cana-1179	230	30	}	}	PUNCT
cana-1179	230	31	,	,	PUNCT
cana-1179	230	32	x	x	NOUN
cana-1179	230	33	}	}	PUNCT
cana-1179	230	34	and	and	CCONJ
cana-1179	230	35	set	set	VERB
cana-1179	230	36	x	x	PUNCT
cana-1179	230	37	=	=	PRON
cana-1179	230	38	{	{	PUNCT
cana-1179	230	39	u	u	NOUN
cana-1179	230	40	,	,	PUNCT
cana-1179	230	41	v	v	NOUN
cana-1179	230	42	,	,	PUNCT
cana-1179	230	43	w	w	NOUN
cana-1179	230	44	}	}	PUNCT
cana-1179	230	45	,	,	PUNCT
cana-1179	230	46	i	i	PRON
cana-1179	230	47	=	=	PUNCT
cana-1179	230	48	{	{	PUNCT
cana-1179	230	49	ϕ	ϕ	NOUN
cana-1179	230	50	,	,	PUNCT
cana-1179	230	51	{	{	PUNCT
cana-1179	230	52	v	v	NOUN
cana-1179	230	53	}	}	PUNCT
cana-1179	230	54	}	}	PUNCT
cana-1179	230	55	.	.	PUNCT
cana-1179	231	1	the	the	DET
cana-1179	231	2	closed	close	VERB
cana-1179	231	3	sets	set	NOUN
cana-1179	231	4	consist	consist	VERB
cana-1179	231	5	of	of	ADP
cana-1179	231	6	βg*p	βg*p	X
cana-1179	231	7	-	-	PUNCT
cana-1179	231	8	i	i	NOUN
cana-1179	231	9	-	-	PUNCT
cana-1179	231	10	closed	close	VERB
cana-1179	231	11	=	=	SYM
cana-1179	231	12	{	{	PUNCT
cana-1179	231	13	ϕ	ϕ	NOUN
cana-1179	231	14	,	,	PUNCT
cana-1179	231	15	{	{	PUNCT
cana-1179	231	16	u	u	NOUN
cana-1179	231	17	}	}	PUNCT
cana-1179	231	18	,	,	PUNCT
cana-1179	231	19	{	{	PUNCT
cana-1179	231	20	v	v	NOUN
cana-1179	231	21	}	}	PUNCT
cana-1179	231	22	,	,	PUNCT
cana-1179	231	23	{	{	PUNCT
cana-1179	231	24	w	w	NOUN
cana-1179	231	25	}	}	PUNCT
cana-1179	231	26	,	,	PUNCT
cana-1179	231	27	{	{	PUNCT
cana-1179	231	28	w	w	NOUN
cana-1179	231	29	,	,	PUNCT
cana-1179	231	30	u	u	NOUN
cana-1179	231	31	}	}	PUNCT
cana-1179	231	32	,	,	PUNCT
cana-1179	231	33	{	{	PUNCT
cana-1179	231	34	u	u	NOUN
cana-1179	231	35	,	,	PUNCT
cana-1179	231	36	v	v	NOUN
cana-1179	231	37	}	}	PUNCT
cana-1179	231	38	,	,	PUNCT
cana-1179	231	39	x	x	NOUN
cana-1179	231	40	}	}	PUNCT
cana-1179	231	41	.	.	PUNCT
cana-1179	232	1	in	in	ADP
cana-1179	232	2	this	this	DET
cana-1179	232	3	case	case	NOUN
cana-1179	232	4	,	,	PUNCT
cana-1179	232	5	regular	regular	ADJ
cana-1179	232	6	generalized	generalize	VERB
cana-1179	232	7	are	be	AUX
cana-1179	232	8	closed	close	VERB
cana-1179	232	9	in	in	ADP
cana-1179	232	10	set	set	NOUN
cana-1179	232	11	b	b	NOUN
cana-1179	232	12	=	=	SYM
cana-1179	232	13	{	{	PUNCT
cana-1179	232	14	v	v	NOUN
cana-1179	232	15	,	,	PUNCT
cana-1179	232	16	w	w	NOUN
cana-1179	232	17	}	}	PUNCT
cana-1179	232	18	but	but	CCONJ
cana-1179	232	19	not	not	PART
cana-1179	232	20	a	a	DET
cana-1179	232	21	βg*p	βg*p	NOUN
cana-1179	232	22	-	-	PUNCT
cana-1179	232	23	i	i	NOUN
cana-1179	232	24	-	-	PUNCT
cana-1179	232	25	closed	close	VERB
cana-1179	232	26	set	set	NOUN
cana-1179	232	27	.	.	PUNCT
cana-1179	233	1	example	example	NOUN
cana-1179	233	2	5.17	5.17	NUM
cana-1179	233	3	consider	consider	VERB
cana-1179	233	4	a	a	DET
cana-1179	233	5	topology	topology	NOUN
cana-1179	233	6	,	,	PUNCT
cana-1179	233	7	τ	τ	PROPN
cana-1179	233	8	=	=	PUNCT
cana-1179	233	9	{	{	PUNCT
cana-1179	233	10	ϕ	ϕ	NOUN
cana-1179	233	11	,	,	PUNCT
cana-1179	233	12	{	{	PUNCT
cana-1179	233	13	u	u	NOUN
cana-1179	233	14	}	}	PUNCT
cana-1179	233	15	,	,	PUNCT
cana-1179	233	16	{	{	PUNCT
cana-1179	233	17	v	v	NOUN
cana-1179	233	18	,	,	PUNCT
cana-1179	233	19	w	w	NOUN
cana-1179	233	20	}	}	PUNCT
cana-1179	233	21	,	,	PUNCT
cana-1179	233	22	x	x	NOUN
cana-1179	233	23	}	}	PUNCT
cana-1179	233	24	and	and	CCONJ
cana-1179	233	25	set	set	VERB
cana-1179	233	26	x	x	PUNCT
cana-1179	233	27	=	=	PRON
cana-1179	233	28	{	{	PUNCT
cana-1179	233	29	u	u	NOUN
cana-1179	233	30	,	,	PUNCT
cana-1179	233	31	v	v	NOUN
cana-1179	233	32	,	,	PUNCT
cana-1179	233	33	w	w	NOUN
cana-1179	233	34	}	}	PUNCT
cana-1179	233	35	,	,	PUNCT
cana-1179	233	36	i	i	PRON
cana-1179	233	37	=	=	PUNCT
cana-1179	233	38	{	{	PUNCT
cana-1179	233	39	ϕ	ϕ	NOUN
cana-1179	233	40	,	,	PUNCT
cana-1179	233	41	{	{	PUNCT
cana-1179	233	42	w	w	NOUN
cana-1179	233	43	}	}	PUNCT
cana-1179	233	44	}	}	PUNCT
cana-1179	233	45	.	.	PUNCT
cana-1179	234	1	the	the	DET
cana-1179	234	2	closed	close	VERB
cana-1179	234	3	sets	set	NOUN
cana-1179	234	4	consist	consist	VERB
cana-1179	234	5	of	of	ADP
cana-1179	234	6	βg*p	βg*p	X
cana-1179	234	7	-	-	PUNCT
cana-1179	234	8	i	i	NOUN
cana-1179	234	9	-	-	PUNCT
cana-1179	234	10	closed	close	VERB
cana-1179	234	11	=	=	SYM
cana-1179	234	12	{	{	PUNCT
cana-1179	234	13	ϕ	ϕ	NOUN
cana-1179	234	14	,	,	PUNCT
cana-1179	234	15	{	{	PUNCT
cana-1179	234	16	u	u	NOUN
cana-1179	234	17	}	}	PUNCT
cana-1179	234	18	,	,	PUNCT
cana-1179	234	19	{	{	PUNCT
cana-1179	234	20	w	w	NOUN
cana-1179	234	21	}	}	PUNCT
cana-1179	234	22	,	,	PUNCT
cana-1179	234	23	{	{	PUNCT
cana-1179	234	24	v	v	NOUN
cana-1179	234	25	,	,	PUNCT
cana-1179	234	26	w	w	NOUN
cana-1179	234	27	}	}	PUNCT
cana-1179	234	28	,	,	PUNCT
cana-1179	234	29	x	x	NOUN
cana-1179	234	30	}	}	PUNCT
cana-1179	234	31	.	.	PUNCT
cana-1179	235	1	in	in	ADP
cana-1179	235	2	this	this	DET
cana-1179	235	3	case	case	NOUN
cana-1179	235	4	,	,	PUNCT
cana-1179	235	5	regular	regular	ADJ
cana-1179	235	6	weakly	weakly	ADV
cana-1179	235	7	generalized	generalized	ADJ
cana-1179	235	8	and	and	CCONJ
cana-1179	235	9	generalized	generalize	VERB
cana-1179	235	10	semi	semi	ADV
cana-1179	235	11	pre	pre	X
cana-1179	235	12	are	be	AUX
cana-1179	235	13	closed	close	VERB
cana-1179	235	14	in	in	ADP
cana-1179	235	15	set	set	NOUN
cana-1179	235	16	b	b	NOUN
cana-1179	235	17	=	=	SYM
cana-1179	235	18	{	{	PUNCT
cana-1179	235	19	v	v	NOUN
cana-1179	235	20	}	}	PUNCT
cana-1179	235	21	but	but	CCONJ
cana-1179	235	22	not	not	PART
cana-1179	235	23	a	a	DET
cana-1179	235	24	βg*p	βg*p	NOUN
cana-1179	235	25	-	-	PUNCT
cana-1179	235	26	i	i	NOUN
cana-1179	235	27	-	-	PUNCT
cana-1179	235	28	closed	close	VERB
cana-1179	235	29	set	set	NOUN
cana-1179	235	30	.	.	PUNCT
cana-1179	236	1	example	example	NOUN
cana-1179	236	2	5.18	5.18	NUM
cana-1179	236	3	determine	determine	VERB
cana-1179	236	4	a	a	DET
cana-1179	236	5	topology	topology	NOUN
cana-1179	236	6	,	,	PUNCT
cana-1179	236	7	τ	τ	PROPN
cana-1179	236	8	=	=	PUNCT
cana-1179	236	9	{	{	PUNCT
cana-1179	236	10	ϕ	ϕ	NOUN
cana-1179	236	11	,	,	PUNCT
cana-1179	236	12	{	{	PUNCT
cana-1179	236	13	u	u	NOUN
cana-1179	236	14	,	,	PUNCT
cana-1179	236	15	v	v	NOUN
cana-1179	236	16	}	}	PUNCT
cana-1179	236	17	,	,	PUNCT
cana-1179	236	18	{	{	PUNCT
cana-1179	236	19	v	v	NOUN
cana-1179	236	20	,	,	PUNCT
cana-1179	236	21	w	w	NOUN
cana-1179	236	22	}	}	PUNCT
cana-1179	236	23	,	,	PUNCT
cana-1179	236	24	{	{	PUNCT
cana-1179	236	25	u	u	NOUN
cana-1179	236	26	}	}	PUNCT
cana-1179	236	27	,	,	PUNCT
cana-1179	236	28	{	{	PUNCT
cana-1179	236	29	v	v	NOUN
cana-1179	236	30	}	}	PUNCT
cana-1179	236	31	,	,	PUNCT
cana-1179	236	32	x	x	NOUN
cana-1179	236	33	}	}	PUNCT
cana-1179	236	34	and	and	CCONJ
cana-1179	236	35	set	set	VERB
cana-1179	236	36	x	x	PUNCT
cana-1179	236	37	=	=	PRON
cana-1179	236	38	{	{	PUNCT
cana-1179	236	39	u	u	NOUN
cana-1179	236	40	,	,	PUNCT
cana-1179	236	41	v	v	NOUN
cana-1179	236	42	,	,	PUNCT
cana-1179	236	43	w	w	NOUN
cana-1179	236	44	}	}	PUNCT
cana-1179	236	45	,	,	PUNCT
cana-1179	236	46	i	i	PRON
cana-1179	236	47	=	=	PUNCT
cana-1179	236	48	{	{	PUNCT
cana-1179	236	49	ϕ	ϕ	NOUN
cana-1179	236	50	,	,	PUNCT
cana-1179	236	51	{	{	PUNCT
cana-1179	236	52	v	v	NOUN
cana-1179	236	53	}	}	PUNCT
cana-1179	236	54	}	}	PUNCT
cana-1179	236	55	.	.	PUNCT
cana-1179	237	1	the	the	DET
cana-1179	237	2	closed	close	VERB
cana-1179	237	3	sets	set	NOUN
cana-1179	237	4	consist	consist	VERB
cana-1179	237	5	of	of	ADP
cana-1179	237	6	βg*p	βg*p	X
cana-1179	237	7	-	-	PUNCT
cana-1179	237	8	i	i	NOUN
cana-1179	237	9	-	-	PUNCT
cana-1179	237	10	closed	close	VERB
cana-1179	237	11	=	=	SYM
cana-1179	237	12	{	{	PUNCT
cana-1179	237	13	ϕ	ϕ	NOUN
cana-1179	237	14	,	,	PUNCT
cana-1179	237	15	{	{	PUNCT
cana-1179	237	16	u	u	NOUN
cana-1179	237	17	}	}	PUNCT
cana-1179	237	18	,	,	PUNCT
cana-1179	237	19	{	{	PUNCT
cana-1179	237	20	v	v	NOUN
cana-1179	237	21	}	}	PUNCT
cana-1179	237	22	,	,	PUNCT
cana-1179	237	23	{	{	PUNCT
cana-1179	237	24	w	w	NOUN
cana-1179	237	25	}	}	PUNCT
cana-1179	237	26	,	,	PUNCT
cana-1179	237	27	{	{	PUNCT
cana-1179	237	28	v	v	NOUN
cana-1179	237	29	,	,	PUNCT
cana-1179	237	30	w	w	NOUN
cana-1179	237	31	}	}	PUNCT
cana-1179	237	32	,	,	PUNCT
cana-1179	237	33	{	{	PUNCT
cana-1179	237	34	w	w	NOUN
cana-1179	237	35	,	,	PUNCT
cana-1179	237	36	u	u	NOUN
cana-1179	237	37	}	}	PUNCT
cana-1179	237	38	,	,	PUNCT
cana-1179	237	39	x	x	NOUN
cana-1179	237	40	}	}	PUNCT
cana-1179	237	41	.	.	PUNCT
cana-1179	238	1	in	in	ADP
cana-1179	238	2	this	this	DET
cana-1179	238	3	case	case	NOUN
cana-1179	238	4	,	,	PUNCT
cana-1179	238	5	generalized	generalize	VERB
cana-1179	238	6	pre	pre	NOUN
cana-1179	238	7	regular	regular	ADJ
cana-1179	238	8	are	be	AUX
cana-1179	238	9	closed	close	VERB
cana-1179	238	10	in	in	ADP
cana-1179	238	11	set	set	NOUN
cana-1179	238	12	b	b	NOUN
cana-1179	238	13	=	=	SYM
cana-1179	238	14	{	{	PUNCT
cana-1179	238	15	u	u	NOUN
cana-1179	238	16	,	,	PUNCT
cana-1179	238	17	v	v	NOUN
cana-1179	238	18	}	}	PUNCT
cana-1179	238	19	but	but	CCONJ
cana-1179	238	20	not	not	PART
cana-1179	238	21	a	a	DET
cana-1179	238	22	βg*p	βg*p	NOUN
cana-1179	238	23	-	-	PUNCT
cana-1179	238	24	i	i	NOUN
cana-1179	238	25	-	-	PUNCT
cana-1179	238	26	closed	close	VERB
cana-1179	238	27	set	set	NOUN
cana-1179	238	28	.	.	PUNCT
cana-1179	239	1	example	example	NOUN
cana-1179	239	2	5.19	5.19	NUM
cana-1179	239	3	create	create	VERB
cana-1179	239	4	a	a	DET
cana-1179	239	5	topology	topology	NOUN
cana-1179	239	6	,	,	PUNCT
cana-1179	239	7	τ	τ	PROPN
cana-1179	239	8	=	=	PUNCT
cana-1179	239	9	{	{	PUNCT
cana-1179	239	10	ϕ	ϕ	NOUN
cana-1179	239	11	,	,	PUNCT
cana-1179	239	12	{	{	PUNCT
cana-1179	239	13	u	u	NOUN
cana-1179	239	14	,	,	PUNCT
cana-1179	239	15	w	w	PROPN
cana-1179	239	16	}	}	PUNCT
cana-1179	239	17	,	,	PUNCT
cana-1179	239	18	{	{	PUNCT
cana-1179	239	19	u	u	NOUN
cana-1179	239	20	}	}	PUNCT
cana-1179	239	21	,	,	PUNCT
cana-1179	239	22	{	{	PUNCT
cana-1179	239	23	w	w	NOUN
cana-1179	239	24	}	}	PUNCT
cana-1179	239	25	,	,	PUNCT
cana-1179	239	26	x	x	NOUN
cana-1179	239	27	}	}	PUNCT
cana-1179	239	28	and	and	CCONJ
cana-1179	239	29	set	set	VERB
cana-1179	239	30	x	x	PUNCT
cana-1179	239	31	=	=	PRON
cana-1179	239	32	{	{	PUNCT
cana-1179	239	33	u	u	NOUN
cana-1179	239	34	,	,	PUNCT
cana-1179	239	35	v	v	NOUN
cana-1179	239	36	,	,	PUNCT
cana-1179	239	37	w	w	NOUN
cana-1179	239	38	}	}	PUNCT
cana-1179	239	39	,	,	PUNCT
cana-1179	239	40	i	i	PRON
cana-1179	239	41	=	=	PUNCT
cana-1179	239	42	{	{	PUNCT
cana-1179	239	43	ϕ	ϕ	NOUN
cana-1179	239	44	,	,	PUNCT
cana-1179	239	45	{	{	PUNCT
cana-1179	239	46	u	u	NOUN
cana-1179	239	47	}	}	PUNCT
cana-1179	239	48	}	}	PUNCT
cana-1179	239	49	.	.	PUNCT
cana-1179	240	1	the	the	DET
cana-1179	240	2	closed	close	VERB
cana-1179	240	3	sets	set	NOUN
cana-1179	240	4	consist	consist	VERB
cana-1179	240	5	of	of	ADP
cana-1179	240	6	βg*p	βg*p	X
cana-1179	240	7	-	-	PUNCT
cana-1179	240	8	i	i	NOUN
cana-1179	240	9	-	-	PUNCT
cana-1179	240	10	closed	close	VERB
cana-1179	240	11	=	=	SYM
cana-1179	240	12	{	{	PUNCT
cana-1179	240	13	ϕ	ϕ	NOUN
cana-1179	240	14	,	,	PUNCT
cana-1179	240	15	{	{	PUNCT
cana-1179	240	16	u	u	NOUN
cana-1179	240	17	}	}	PUNCT
cana-1179	240	18	,	,	PUNCT
cana-1179	240	19	{	{	PUNCT
cana-1179	240	20	v	v	NOUN
cana-1179	240	21	}	}	PUNCT
cana-1179	240	22	,	,	PUNCT
cana-1179	240	23	{	{	PUNCT
cana-1179	240	24	v	v	NOUN
cana-1179	240	25	,	,	PUNCT
cana-1179	240	26	w	w	NOUN
cana-1179	240	27	}	}	PUNCT
cana-1179	240	28	,	,	PUNCT
cana-1179	240	29	{	{	PUNCT
cana-1179	240	30	u	u	NOUN
cana-1179	240	31	,	,	PUNCT
cana-1179	240	32	v	v	NOUN
cana-1179	240	33	}	}	PUNCT
cana-1179	240	34	,	,	PUNCT
cana-1179	240	35	x	x	NOUN
cana-1179	240	36	}	}	PUNCT
cana-1179	240	37	.	.	PUNCT
cana-1179	241	1	in	in	ADP
cana-1179	241	2	this	this	DET
cana-1179	241	3	case	case	NOUN
cana-1179	241	4	,	,	PUNCT
cana-1179	241	5	semi	semi	ADV
cana-1179	241	6	generalized	generalized	ADJ
cana-1179	241	7	and	and	CCONJ
cana-1179	241	8	generalized	generalized	ADJ
cana-1179	241	9	semi	semi	ADV
cana-1179	241	10	are	be	AUX
cana-1179	241	11	closed	close	VERB
cana-1179	241	12	in	in	ADP
cana-1179	241	13	set	set	NOUN
cana-1179	241	14	b	b	NOUN
cana-1179	241	15	=	=	PRON
cana-1179	241	16	{	{	PUNCT
cana-1179	241	17	w	w	NOUN
cana-1179	241	18	}	}	PUNCT
cana-1179	241	19	but	but	CCONJ
cana-1179	241	20	not	not	PART
cana-1179	241	21	a	a	DET
cana-1179	241	22	βg*p	βg*p	NOUN
cana-1179	241	23	-	-	PUNCT
cana-1179	241	24	i	i	NOUN
cana-1179	241	25	-	-	PUNCT
cana-1179	241	26	closed	close	VERB
cana-1179	241	27	set	set	NOUN
cana-1179	241	28	.	.	PUNCT
cana-1179	242	1	example	example	NOUN
cana-1179	242	2	5.20	5.20	NUM
cana-1179	242	3	extract	extract	VERB
cana-1179	242	4	a	a	DET
cana-1179	242	5	topology	topology	NOUN
cana-1179	242	6	,	,	PUNCT
cana-1179	242	7	τ	τ	PROPN
cana-1179	242	8	=	=	PUNCT
cana-1179	242	9	{	{	PUNCT
cana-1179	242	10	ϕ	ϕ	NOUN
cana-1179	242	11	,	,	PUNCT
cana-1179	242	12	{	{	PUNCT
cana-1179	242	13	u	u	NOUN
cana-1179	242	14	,	,	PUNCT
cana-1179	242	15	v	v	NOUN
cana-1179	242	16	}	}	PUNCT
cana-1179	242	17	,	,	PUNCT
cana-1179	242	18	{	{	PUNCT
cana-1179	242	19	w	w	NOUN
cana-1179	242	20	}	}	PUNCT
cana-1179	242	21	,	,	PUNCT
cana-1179	242	22	x	x	NOUN
cana-1179	242	23	}	}	PUNCT
cana-1179	242	24	and	and	CCONJ
cana-1179	242	25	set	set	VERB
cana-1179	242	26	x	x	PUNCT
cana-1179	242	27	=	=	PRON
cana-1179	242	28	{	{	PUNCT
cana-1179	242	29	u	u	NOUN
cana-1179	242	30	,	,	PUNCT
cana-1179	242	31	v	v	NOUN
cana-1179	242	32	,	,	PUNCT
cana-1179	242	33	w	w	NOUN
cana-1179	242	34	}	}	PUNCT
cana-1179	242	35	,	,	PUNCT
cana-1179	242	36	i	i	PRON
cana-1179	242	37	=	=	PUNCT
cana-1179	242	38	{	{	PUNCT
cana-1179	242	39	ϕ	ϕ	NOUN
cana-1179	242	40	,	,	PUNCT
cana-1179	242	41	{	{	PUNCT
cana-1179	242	42	w	w	NOUN
cana-1179	242	43	}	}	PUNCT
cana-1179	242	44	}	}	PUNCT
cana-1179	242	45	.	.	PUNCT
cana-1179	243	1	the	the	DET
cana-1179	243	2	closed	close	VERB
cana-1179	243	3	sets	set	NOUN
cana-1179	243	4	consist	consist	VERB
cana-1179	243	5	of	of	ADP
cana-1179	243	6	βg*p	βg*p	X
cana-1179	243	7	-	-	PUNCT
cana-1179	243	8	i	i	NOUN
cana-1179	243	9	-	-	PUNCT
cana-1179	243	10	closed	close	VERB
cana-1179	243	11	=	=	SYM
cana-1179	243	12	{	{	PUNCT
cana-1179	243	13	ϕ	ϕ	NOUN
cana-1179	243	14	,	,	PUNCT
cana-1179	243	15	{	{	PUNCT
cana-1179	243	16	u	u	NOUN
cana-1179	243	17	}	}	PUNCT
cana-1179	243	18	,	,	PUNCT
cana-1179	243	19	{	{	PUNCT
cana-1179	243	20	w	w	NOUN
cana-1179	243	21	}	}	PUNCT
cana-1179	243	22	,	,	PUNCT
cana-1179	243	23	{	{	PUNCT
cana-1179	243	24	u	u	NOUN
cana-1179	243	25	,	,	PUNCT
cana-1179	243	26	v	v	NOUN
cana-1179	243	27	}	}	PUNCT
cana-1179	243	28	,	,	PUNCT
cana-1179	243	29	x	x	NOUN
cana-1179	243	30	}	}	PUNCT
cana-1179	243	31	.	.	PUNCT
cana-1179	244	1	in	in	ADP
cana-1179	244	2	this	this	DET
cana-1179	244	3	case	case	NOUN
cana-1179	244	4	,	,	PUNCT
cana-1179	244	5	β	β	X
cana-1179	244	6	generalized	generalize	VERB
cana-1179	244	7	and	and	CCONJ
cana-1179	244	8	β	β	X
cana-1179	244	9	generalized	generalize	VERB
cana-1179	244	10	*	*	PUNCT
cana-1179	244	11	are	be	AUX
cana-1179	244	12	closed	close	VERB
cana-1179	244	13	in	in	ADP
cana-1179	244	14	set	set	NOUN
cana-1179	244	15	b	b	NOUN
cana-1179	244	16	=	=	SYM
cana-1179	244	17	{	{	PUNCT
cana-1179	244	18	u	u	NOUN
cana-1179	244	19	,	,	PUNCT
cana-1179	244	20	w	w	NOUN
cana-1179	244	21	}	}	PUNCT
cana-1179	244	22	but	but	CCONJ
cana-1179	244	23	not	not	PART
cana-1179	244	24	a	a	DET
cana-1179	244	25	βg*p	βg*p	NOUN
cana-1179	244	26	-	-	PUNCT
cana-1179	244	27	i	i	NOUN
cana-1179	244	28	-	-	PUNCT
cana-1179	244	29	closed	close	VERB
cana-1179	244	30	set	set	NOUN
cana-1179	244	31	.	.	PUNCT
cana-1179	245	1	theorem	theorem	VERB
cana-1179	245	2	5.21	5.21	NUM
cana-1179	245	3	two	two	NUM
cana-1179	245	4	closed	closed	ADJ
cana-1179	245	5	sets	set	NOUN
cana-1179	245	6	βg*p	βg*p	NOUN
cana-1179	245	7	-	-	PUNCT
cana-1179	245	8	i	i	PRON
cana-1179	245	9	are	be	AUX
cana-1179	245	10	united	unite	VERB
cana-1179	245	11	in	in	ADP
cana-1179	245	12	any	any	DET
cana-1179	245	13	ideal	ideal	ADJ
cana-1179	245	14	topological	topological	ADJ
cana-1179	245	15	space	space	NOUN
cana-1179	245	16	(	(	PUNCT
cana-1179	245	17	x	x	X
cana-1179	245	18	,	,	PUNCT
cana-1179	245	19	τ	τ	PROPN
cana-1179	245	20	,	,	PUNCT
cana-1179	245	21	i	i	PROPN
cana-1179	245	22	)	)	PUNCT
cana-1179	245	23	.	.	PUNCT
cana-1179	246	1	proof	proof	NOUN
cana-1179	246	2	:	:	PUNCT
cana-1179	246	3	suppose	suppose	VERB
cana-1179	246	4	that	that	SCONJ
cana-1179	246	5	m	m	PROPN
cana-1179	246	6	and	and	CCONJ
cana-1179	246	7	n	n	PRON
cana-1179	246	8	are	be	AUX
cana-1179	246	9	two	two	NUM
cana-1179	246	10	closed	closed	ADJ
cana-1179	246	11	sets	set	NOUN
cana-1179	246	12	of	of	ADP
cana-1179	246	13	βg*p	βg*p	NOUN
cana-1179	246	14	-	-	PUNCT
cana-1179	246	15	i	i	PRON
cana-1179	246	16	in	in	ADP
cana-1179	246	17	the	the	DET
cana-1179	246	18	space	space	NOUN
cana-1179	246	19	(	(	PUNCT
cana-1179	246	20	x	x	X
cana-1179	246	21	,	,	PUNCT
cana-1179	246	22	τ	τ	PROPN
cana-1179	246	23	,	,	PUNCT
cana-1179	246	24	i	i	PROPN
cana-1179	246	25	)	)	PUNCT
cana-1179	246	26	.	.	PUNCT
cana-1179	247	1	assume	assume	VERB
cana-1179	247	2	that	that	SCONJ
cana-1179	247	3	any	any	DET
cana-1179	247	4	β*open	β*open	PROPN
cana-1179	247	5	set	set	VERB
cana-1179	247	6	d	d	NOUN
cana-1179	247	7	in	in	ADP
cana-1179	247	8	x	x	X
cana-1179	247	9	such	such	ADJ
cana-1179	247	10	that	that	SCONJ
cana-1179	247	11	m	m	VERB
cana-1179	247	12	∪	∪	ADP
cana-1179	247	13	n	n	PRON
cana-1179	247	14	⊆	⊆	NUM
cana-1179	247	15	d.	d.	NOUN
cana-1179	247	16	since	since	SCONJ
cana-1179	247	17	m	m	PROPN
cana-1179	247	18	⊆	⊆	NUM
cana-1179	247	19	d	d	PROPN
cana-1179	247	20	and	and	CCONJ
cana-1179	247	21	n	n	CCONJ
cana-1179	247	22	⊆	⊆	NUM
cana-1179	247	23	d	d	PROPN
cana-1179	247	24	,	,	PUNCT
cana-1179	247	25	m	m	VERB
cana-1179	247	26	and	and	CCONJ
cana-1179	247	27	n	n	PROPN
cana-1179	247	28	are	be	AUX
cana-1179	247	29	βg*p	βg*p	X
cana-1179	247	30	-	-	PUNCT
cana-1179	247	31	i	i	NOUN
cana-1179	247	32	-	-	PUNCT
cana-1179	247	33	closed	close	VERB
cana-1179	247	34	sets	set	NOUN
cana-1179	247	35	.	.	PUNCT
cana-1179	248	1	being	be	AUX
cana-1179	248	2	that	that	PRON
cana-1179	248	3	d	d	NOUN
cana-1179	248	4	is	be	AUX
cana-1179	248	5	β*-open	β*-open	ADJ
cana-1179	248	6	whenever	whenever	SCONJ
cana-1179	248	7	m	m	VERB
cana-1179	248	8	*	*	NOUN
cana-1179	248	9	∪	∪	X
cana-1179	248	10	n	n	CCONJ
cana-1179	248	11	*	*	PUNCT
cana-1179	248	12	=	=	SYM
cana-1179	248	13	(	(	PUNCT
cana-1179	248	14	m	m	PROPN
cana-1179	248	15	∪	∪	NOUN
cana-1179	248	16	n	n	CCONJ
cana-1179	248	17	)	)	PUNCT
cana-1179	248	18	*	*	PUNCT
cana-1179	249	1	⊆	⊆	NUM
cana-1179	249	2	d	d	PROPN
cana-1179	249	3	,	,	PUNCT
cana-1179	249	4	m	m	PROPN
cana-1179	249	5	*	*	PUNCT
cana-1179	249	6	⊆	⊆	NUM
cana-1179	249	7	d	d	NOUN
cana-1179	249	8	and	and	CCONJ
cana-1179	249	9	n	n	CCONJ
cana-1179	249	10	*	*	PROPN
cana-1179	249	11	⊆	⊆	NUM
cana-1179	249	12	d.	d.	NOUN
cana-1179	249	13	in	in	ADP
cana-1179	249	14	the	the	DET
cana-1179	249	15	ideal	ideal	ADJ
cana-1179	249	16	topological	topological	ADJ
cana-1179	249	17	space	space	NOUN
cana-1179	249	18	,	,	PUNCT
cana-1179	249	19	m	m	VERB
cana-1179	249	20	∪	∪	X
cana-1179	249	21	n	n	X
cana-1179	249	22	is	be	AUX
cana-1179	249	23	a	a	DET
cana-1179	249	24	closed	closed	ADJ
cana-1179	249	25	set	set	VERB
cana-1179	249	26	with	with	ADP
cana-1179	249	27	respect	respect	NOUN
cana-1179	249	28	to	to	ADP
cana-1179	249	29	βg*p	βg*p	ADJ
cana-1179	249	30	-	-	PUNCT
cana-1179	249	31	i.	i.	NOUN
cana-1179	249	32	communications	communication	NOUN
cana-1179	249	33	on	on	ADP
cana-1179	249	34	applied	apply	VERB
cana-1179	249	35	nonlinear	nonlinear	ADJ
cana-1179	249	36	analysis	analysis	NOUN
cana-1179	249	37	issn	issn	NOUN
cana-1179	249	38	:	:	PUNCT
cana-1179	249	39	1074	1074	NUM
cana-1179	249	40	-	-	PUNCT
cana-1179	249	41	133x	133x	NUM
cana-1179	249	42	vol	vol	NOUN
cana-1179	249	43	31	31	NUM
cana-1179	249	44	no	no	NOUN
cana-1179	249	45	.	.	PUNCT
cana-1179	250	1	6s	6s	NUM
cana-1179	250	2	(	(	PUNCT
cana-1179	250	3	2024	2024	NUM
cana-1179	250	4	)	)	PUNCT
cana-1179	250	5	214	214	NUM
cana-1179	250	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1179	250	7	example	example	NOUN
cana-1179	250	8	5.22	5.22	NUM
cana-1179	250	9	set	set	NOUN
cana-1179	250	10	x	x	X
cana-1179	250	11	=	=	PRON
cana-1179	250	12	{	{	PUNCT
cana-1179	250	13	u	u	NOUN
cana-1179	250	14	,	,	PUNCT
cana-1179	250	15	v	v	NOUN
cana-1179	250	16	,	,	PUNCT
cana-1179	250	17	w	w	NOUN
cana-1179	250	18	}	}	PUNCT
cana-1179	250	19	,	,	PUNCT
cana-1179	250	20	i	i	PRON
cana-1179	250	21	=	=	PUNCT
cana-1179	250	22	{	{	PUNCT
cana-1179	250	23	ϕ	ϕ	NOUN
cana-1179	250	24	,	,	PUNCT
cana-1179	250	25	{	{	PUNCT
cana-1179	250	26	u	u	NOUN
cana-1179	250	27	}	}	PUNCT
cana-1179	250	28	,	,	PUNCT
cana-1179	250	29	{	{	PUNCT
cana-1179	250	30	w	w	NOUN
cana-1179	250	31	}	}	PUNCT
cana-1179	250	32	,	,	PUNCT
cana-1179	250	33	{	{	PUNCT
cana-1179	250	34	u	u	NOUN
cana-1179	250	35	,	,	PUNCT
cana-1179	250	36	w	w	NOUN
cana-1179	250	37	}	}	PUNCT
cana-1179	250	38	}	}	PUNCT
cana-1179	250	39	and	and	CCONJ
cana-1179	250	40	topology	topology	NOUN
cana-1179	250	41	,	,	PUNCT
cana-1179	250	42	τ	τ	PROPN
cana-1179	250	43	=	=	PUNCT
cana-1179	250	44	{	{	PUNCT
cana-1179	250	45	ϕ	ϕ	NOUN
cana-1179	250	46	,	,	PUNCT
cana-1179	250	47	x	x	NOUN
cana-1179	250	48	}	}	PUNCT
cana-1179	250	49	.	.	PUNCT
cana-1179	251	1	the	the	DET
cana-1179	251	2	closed	close	VERB
cana-1179	251	3	sets	set	NOUN
cana-1179	251	4	include	include	VERB
cana-1179	251	5	βg*p	βg*p	NOUN
cana-1179	251	6	-	-	SYM
cana-1179	251	7	i	i	PRON
cana-1179	251	8	=	=	PUNCT
cana-1179	251	9	{	{	PUNCT
cana-1179	251	10	ϕ	ϕ	NOUN
cana-1179	251	11	,	,	PUNCT
cana-1179	251	12	{	{	PUNCT
cana-1179	251	13	u	u	NOUN
cana-1179	251	14	}	}	PUNCT
cana-1179	251	15	,	,	PUNCT
cana-1179	251	16	{	{	PUNCT
cana-1179	251	17	v	v	NOUN
cana-1179	251	18	}	}	PUNCT
cana-1179	251	19	,	,	PUNCT
cana-1179	251	20	{	{	PUNCT
cana-1179	251	21	w	w	NOUN
cana-1179	251	22	}	}	PUNCT
cana-1179	251	23	,	,	PUNCT
cana-1179	251	24	{	{	PUNCT
cana-1179	251	25	u	u	NOUN
cana-1179	251	26	,	,	PUNCT
cana-1179	251	27	v	v	NOUN
cana-1179	251	28	}	}	PUNCT
cana-1179	251	29	,	,	PUNCT
cana-1179	251	30	{	{	PUNCT
cana-1179	251	31	v	v	NOUN
cana-1179	251	32	,	,	PUNCT
cana-1179	251	33	w	w	NOUN
cana-1179	251	34	}	}	PUNCT
cana-1179	251	35	,	,	PUNCT
cana-1179	251	36	{	{	PUNCT
cana-1179	251	37	w	w	NOUN
cana-1179	251	38	,	,	PUNCT
cana-1179	251	39	u	u	NOUN
cana-1179	251	40	}	}	PUNCT
cana-1179	251	41	,	,	PUNCT
cana-1179	251	42	x	x	NOUN
cana-1179	251	43	}	}	PUNCT
cana-1179	251	44	.	.	PUNCT
cana-1179	252	1	given	give	VERB
cana-1179	252	2	two	two	NUM
cana-1179	252	3	closed	closed	ADJ
cana-1179	252	4	sets	set	NOUN
cana-1179	252	5	of	of	ADP
cana-1179	252	6	βg*p	βg*p	NOUN
cana-1179	252	7	-	-	PUNCT
cana-1179	252	8	i	i	PROPN
cana-1179	252	9	,	,	PUNCT
cana-1179	252	10	let	let	VERB
cana-1179	252	11	m	m	AUX
cana-1179	252	12	=	=	PUNCT
cana-1179	252	13	{	{	PUNCT
cana-1179	252	14	v	v	NOUN
cana-1179	252	15	,	,	PUNCT
cana-1179	252	16	w	w	NOUN
cana-1179	252	17	}	}	PUNCT
cana-1179	252	18	and	and	CCONJ
cana-1179	252	19	n	n	CCONJ
cana-1179	252	20	=	=	CCONJ
cana-1179	252	21	{	{	PUNCT
cana-1179	252	22	u	u	NOUN
cana-1179	252	23	}	}	PUNCT
cana-1179	252	24	,	,	PUNCT
cana-1179	252	25	then	then	ADV
cana-1179	252	26	their	their	PRON
cana-1179	252	27	union	union	NOUN
cana-1179	252	28	m	m	VERB
cana-1179	252	29	∪	∪	ADJ
cana-1179	252	30	n	n	X
cana-1179	252	31	=	=	SYM
cana-1179	252	32	{	{	PUNCT
cana-1179	252	33	x	x	NOUN
cana-1179	252	34	}	}	PUNCT
cana-1179	252	35	is	be	AUX
cana-1179	252	36	a	a	DET
cana-1179	252	37	closed	closed	ADJ
cana-1179	252	38	set	set	NOUN
cana-1179	252	39	of	of	ADP
cana-1179	252	40	βg*p	βg*p	NOUN
cana-1179	252	41	-	-	PUNCT
cana-1179	252	42	i.	i.	NOUN
cana-1179	252	43	remark	remark	NOUN
cana-1179	252	44	5.23	5.23	NUM
cana-1179	252	45	there	there	PRON
cana-1179	252	46	is	be	VERB
cana-1179	252	47	no	no	DET
cana-1179	252	48	requirement	requirement	NOUN
cana-1179	252	49	for	for	ADP
cana-1179	252	50	the	the	DET
cana-1179	252	51	intersection	intersection	NOUN
cana-1179	252	52	of	of	ADP
cana-1179	252	53	two	two	NUM
cana-1179	252	54	βg*p	βg*p	NOUN
cana-1179	252	55	-	-	PUNCT
cana-1179	252	56	i	i	NOUN
cana-1179	252	57	-	-	PUNCT
cana-1179	252	58	closed	close	VERB
cana-1179	252	59	sets	set	NOUN
cana-1179	252	60	.	.	PUNCT
cana-1179	253	1	theorem	theorem	VERB
cana-1179	253	2	5.24	5.24	NUM
cana-1179	253	3	a	a	DET
cana-1179	253	4	subset	subset	ADJ
cana-1179	253	5	b	b	NOUN
cana-1179	253	6	of	of	ADP
cana-1179	253	7	a	a	DET
cana-1179	253	8	topological	topological	ADJ
cana-1179	253	9	space	space	NOUN
cana-1179	253	10	x	x	PUNCT
cana-1179	253	11	is	be	AUX
cana-1179	253	12	a	a	DET
cana-1179	253	13	g*p	g*p	PROPN
cana-1179	253	14	-	-	PUNCT
cana-1179	253	15	i	i	PROPN
cana-1179	253	16	-	-	PUNCT
cana-1179	253	17	closed	close	VERB
cana-1179	253	18	set	set	NOUN
cana-1179	253	19	in	in	ADP
cana-1179	253	20	the	the	DET
cana-1179	253	21	topological	topological	ADJ
cana-1179	253	22	space	space	NOUN
cana-1179	253	23	(	(	PUNCT
cana-1179	253	24	x	x	X
cana-1179	253	25	,	,	PUNCT
cana-1179	253	26	τ	τ	PROPN
cana-1179	253	27	,	,	PUNCT
cana-1179	253	28	i	i	NOUN
cana-1179	253	29	)	)	PUNCT
cana-1179	253	30	if	if	SCONJ
cana-1179	253	31	it	it	PRON
cana-1179	253	32	is	be	AUX
cana-1179	253	33	both	both	PRON
cana-1179	253	34	pre	pre	ADJ
cana-1179	253	35	-	-	ADJ
cana-1179	253	36	open	open	ADJ
cana-1179	253	37	and	and	CCONJ
cana-1179	253	38	β	β	NOUN
cana-1179	253	39	-	-	VERB
cana-1179	253	40	closed	closed	ADJ
cana-1179	253	41	.	.	PUNCT
cana-1179	254	1	proof	proof	NOUN
cana-1179	254	2	:	:	PUNCT
cana-1179	254	3	let	let	VERB
cana-1179	254	4	b	b	X
cana-1179	254	5	be	be	AUX
cana-1179	254	6	a	a	DET
cana-1179	254	7	pre	pre	ADJ
cana-1179	254	8	-	-	ADJ
cana-1179	254	9	open	open	ADJ
cana-1179	254	10	and	and	CCONJ
cana-1179	254	11	β	β	NOUN
cana-1179	254	12	-	-	ADJ
cana-1179	254	13	closed	closed	ADJ
cana-1179	254	14	set	set	NOUN
cana-1179	254	15	in	in	ADP
cana-1179	254	16	the	the	DET
cana-1179	254	17	space	space	NOUN
cana-1179	254	18	(	(	PUNCT
cana-1179	254	19	x	x	X
cana-1179	254	20	,	,	PUNCT
cana-1179	254	21	τ	τ	PROPN
cana-1179	254	22	,	,	PUNCT
cana-1179	254	23	i	i	PROPN
cana-1179	254	24	)	)	PUNCT
cana-1179	254	25	.	.	PUNCT
cana-1179	255	1	hypothesize	hypothesize	VERB
cana-1179	255	2	that	that	SCONJ
cana-1179	255	3	d	d	PROPN
cana-1179	255	4	is	be	AUX
cana-1179	255	5	a	a	DET
cana-1179	255	6	β*-open	β*-open	NOUN
cana-1179	255	7	set	set	NOUN
cana-1179	255	8	in	in	ADP
cana-1179	255	9	the	the	DET
cana-1179	255	10	space	space	NOUN
cana-1179	255	11	(	(	PUNCT
cana-1179	255	12	x	x	X
cana-1179	255	13	,	,	PUNCT
cana-1179	255	14	τ	τ	PROPN
cana-1179	255	15	,	,	PUNCT
cana-1179	255	16	i	i	PROPN
cana-1179	255	17	)	)	PUNCT
cana-1179	255	18	and	and	CCONJ
cana-1179	255	19	suppose	suppose	VERB
cana-1179	255	20	that	that	SCONJ
cana-1179	255	21	b	b	PROPN
cana-1179	255	22	⊆	⊆	NUM
cana-1179	255	23	d.	d.	NOUN
cana-1179	255	24	considering	consider	VERB
cana-1179	255	25	that	that	SCONJ
cana-1179	255	26	b	b	PROPN
cana-1179	255	27	⊆	⊆	NUM
cana-1179	255	28	b	b	PROPN
cana-1179	255	29	,	,	PUNCT
cana-1179	255	30	b	b	PROPN
cana-1179	255	31	*	*	PUNCT
cana-1179	255	32	⊆	⊆	NUM
cana-1179	255	33	b	b	NOUN
cana-1179	255	34	and	and	CCONJ
cana-1179	255	35	then	then	ADV
cana-1179	255	36	b	b	X
cana-1179	255	37	*	*	PUNCT
cana-1179	255	38	⊆	⊆	NUM
cana-1179	255	39	b	b	PROPN
cana-1179	255	40	⊆	⊆	NUM
cana-1179	255	41	d.	d.	NOUN
cana-1179	255	42	in	in	ADP
cana-1179	255	43	consequence	consequence	PROPN
cana-1179	255	44	,	,	PUNCT
cana-1179	255	45	b	b	PROPN
cana-1179	255	46	is	be	AUX
cana-1179	255	47	βg*p	βg*p	NUM
cana-1179	255	48	-	-	PUNCT
cana-1179	255	49	i	i	NOUN
cana-1179	255	50	-	-	PUNCT
cana-1179	255	51	closed	close	VERB
cana-1179	255	52	set	set	NOUN
cana-1179	255	53	in	in	ADP
cana-1179	255	54	the	the	DET
cana-1179	255	55	ideal	ideal	ADJ
cana-1179	255	56	topological	topological	ADJ
cana-1179	255	57	space	space	NOUN
cana-1179	255	58	.	.	PUNCT
cana-1179	256	1	if	if	SCONJ
cana-1179	256	2	b	b	PROPN
cana-1179	256	3	is	be	AUX
cana-1179	256	4	pre	pre	ADJ
cana-1179	256	5	-	-	ADJ
cana-1179	256	6	open	open	ADJ
cana-1179	256	7	and	and	CCONJ
cana-1179	256	8	βg*p	βg*p	NOUN
cana-1179	256	9	-	-	PUNCT
cana-1179	256	10	iclosed	iclose	VERB
cana-1179	256	11	sets	set	NOUN
cana-1179	256	12	in	in	ADP
cana-1179	256	13	the	the	DET
cana-1179	256	14	space	space	NOUN
cana-1179	256	15	(	(	PUNCT
cana-1179	256	16	x	x	X
cana-1179	256	17	,	,	PUNCT
cana-1179	256	18	τ	τ	PROPN
cana-1179	256	19	,	,	PUNCT
cana-1179	256	20	i	i	PROPN
cana-1179	256	21	)	)	PUNCT
cana-1179	256	22	,	,	PUNCT
cana-1179	256	23	then	then	ADV
cana-1179	256	24	it	it	PRON
cana-1179	256	25	is	be	AUX
cana-1179	256	26	need	need	ADJ
cana-1179	256	27	to	to	PART
cana-1179	256	28	be	be	AUX
cana-1179	256	29	a	a	DET
cana-1179	256	30	β	β	NOUN
cana-1179	256	31	-	-	VERB
cana-1179	256	32	closed	closed	ADJ
cana-1179	256	33	set	set	NOUN
cana-1179	256	34	as	as	SCONJ
cana-1179	256	35	demonstrated	demonstrate	VERB
cana-1179	256	36	by	by	ADP
cana-1179	256	37	the	the	DET
cana-1179	256	38	example	example	NOUN
cana-1179	256	39	below	below	ADV
cana-1179	256	40	.	.	PUNCT
cana-1179	257	1	example	example	NOUN
cana-1179	257	2	5.25	5.25	NUM
cana-1179	257	3	take	take	VERB
cana-1179	257	4	a	a	DET
cana-1179	257	5	topology	topology	NOUN
cana-1179	257	6	,	,	PUNCT
cana-1179	257	7	τ	τ	PROPN
cana-1179	257	8	=	=	PUNCT
cana-1179	257	9	{	{	PUNCT
cana-1179	257	10	ϕ	ϕ	NOUN
cana-1179	257	11	,	,	PUNCT
cana-1179	257	12	{	{	PUNCT
cana-1179	257	13	u	u	NOUN
cana-1179	257	14	}	}	PUNCT
cana-1179	257	15	,	,	PUNCT
cana-1179	257	16	{	{	PUNCT
cana-1179	257	17	v	v	NOUN
cana-1179	257	18	}	}	PUNCT
cana-1179	257	19	,	,	PUNCT
cana-1179	257	20	{	{	PUNCT
cana-1179	257	21	w	w	NOUN
cana-1179	257	22	}	}	PUNCT
cana-1179	257	23	,	,	PUNCT
cana-1179	257	24	{	{	PUNCT
cana-1179	257	25	u	u	NOUN
cana-1179	257	26	,	,	PUNCT
cana-1179	257	27	v	v	NOUN
cana-1179	257	28	}	}	PUNCT
cana-1179	257	29	,	,	PUNCT
cana-1179	257	30	{	{	PUNCT
cana-1179	257	31	v	v	NOUN
cana-1179	257	32	,	,	PUNCT
cana-1179	257	33	w	w	NOUN
cana-1179	257	34	}	}	PUNCT
cana-1179	257	35	,	,	PUNCT
cana-1179	257	36	x	x	NOUN
cana-1179	257	37	}	}	PUNCT
cana-1179	257	38	,	,	PUNCT
cana-1179	257	39	x	x	SYM
cana-1179	257	40	=	=	PRON
cana-1179	257	41	{	{	PUNCT
cana-1179	257	42	u	u	NOUN
cana-1179	257	43	,	,	PUNCT
cana-1179	257	44	v	v	NOUN
cana-1179	257	45	,	,	PUNCT
cana-1179	257	46	w	w	NOUN
cana-1179	257	47	}	}	PUNCT
cana-1179	257	48	and	and	CCONJ
cana-1179	257	49	i	i	PRON
cana-1179	257	50	=	=	PUNCT
cana-1179	257	51	{	{	PUNCT
cana-1179	257	52	ϕ	ϕ	NOUN
cana-1179	257	53	,	,	PUNCT
cana-1179	257	54	{	{	PUNCT
cana-1179	257	55	v	v	NOUN
cana-1179	257	56	}	}	PUNCT
cana-1179	257	57	}	}	PUNCT
cana-1179	257	58	.	.	PUNCT
cana-1179	258	1	the	the	DET
cana-1179	258	2	closed	close	VERB
cana-1179	258	3	sets	set	NOUN
cana-1179	258	4	of	of	ADP
cana-1179	258	5	βg*p	βg*p	NOUN
cana-1179	258	6	-	-	PUNCT
cana-1179	258	7	i	i	PRON
cana-1179	258	8	=	=	PUNCT
cana-1179	258	9	{	{	PUNCT
cana-1179	258	10	ϕ	ϕ	NOUN
cana-1179	258	11	,	,	PUNCT
cana-1179	258	12	{	{	PUNCT
cana-1179	258	13	u	u	NOUN
cana-1179	258	14	}	}	PUNCT
cana-1179	258	15	,	,	PUNCT
cana-1179	258	16	{	{	PUNCT
cana-1179	258	17	v	v	NOUN
cana-1179	258	18	}	}	PUNCT
cana-1179	258	19	,	,	PUNCT
cana-1179	258	20	{	{	PUNCT
cana-1179	258	21	w	w	NOUN
cana-1179	258	22	}	}	PUNCT
cana-1179	258	23	,	,	PUNCT
cana-1179	258	24	{	{	PUNCT
cana-1179	258	25	u	u	NOUN
cana-1179	258	26	,	,	PUNCT
cana-1179	258	27	v	v	NOUN
cana-1179	258	28	}	}	PUNCT
cana-1179	258	29	,	,	PUNCT
cana-1179	258	30	{	{	PUNCT
cana-1179	258	31	v	v	NOUN
cana-1179	258	32	,	,	PUNCT
cana-1179	258	33	w	w	NOUN
cana-1179	258	34	}	}	PUNCT
cana-1179	258	35	,	,	PUNCT
cana-1179	258	36	{	{	PUNCT
cana-1179	258	37	w	w	NOUN
cana-1179	258	38	,	,	PUNCT
cana-1179	258	39	u	u	NOUN
cana-1179	258	40	}	}	PUNCT
cana-1179	258	41	,	,	PUNCT
cana-1179	258	42	x	x	NOUN
cana-1179	258	43	}	}	PUNCT
cana-1179	258	44	.	.	PUNCT
cana-1179	259	1	β	β	X
cana-1179	259	2	=	=	PUNCT
cana-1179	259	3	{	{	PUNCT
cana-1179	259	4	ϕ	ϕ	NOUN
cana-1179	259	5	,	,	PUNCT
cana-1179	259	6	{	{	PUNCT
cana-1179	259	7	u	u	NOUN
cana-1179	259	8	}	}	PUNCT
cana-1179	259	9	,	,	PUNCT
cana-1179	259	10	{	{	PUNCT
cana-1179	259	11	w	w	NOUN
cana-1179	259	12	}	}	PUNCT
cana-1179	259	13	,	,	PUNCT
cana-1179	259	14	{	{	PUNCT
cana-1179	259	15	u	u	NOUN
cana-1179	259	16	,	,	PUNCT
cana-1179	259	17	w	w	PROPN
cana-1179	259	18	}	}	PUNCT
cana-1179	259	19	,	,	PUNCT
cana-1179	259	20	x	x	X
cana-1179	259	21	}	}	PUNCT
cana-1179	259	22	are	be	AUX
cana-1179	259	23	the	the	DET
cana-1179	259	24	closed	closed	ADJ
cana-1179	259	25	sets	set	NOUN
cana-1179	259	26	of	of	ADP
cana-1179	259	27	x.	x.	NOUN
cana-1179	259	28	from	from	ADP
cana-1179	259	29	this	this	PRON
cana-1179	259	30	,	,	PUNCT
cana-1179	259	31	it	it	PRON
cana-1179	259	32	is	be	AUX
cana-1179	259	33	possible	possible	ADJ
cana-1179	259	34	to	to	PART
cana-1179	259	35	seen	see	VERB
cana-1179	259	36	that	that	PRON
cana-1179	259	37	b	b	X
cana-1179	259	38	=	=	PRON
cana-1179	259	39	{	{	PUNCT
cana-1179	259	40	v	v	NOUN
cana-1179	259	41	}	}	PUNCT
cana-1179	259	42	is	be	AUX
cana-1179	259	43	pre	pre	ADJ
cana-1179	259	44	-	-	ADJ
cana-1179	259	45	open	open	ADJ
cana-1179	259	46	and	and	CCONJ
cana-1179	259	47	βg*p	βg*p	NOUN
cana-1179	259	48	-	-	PUNCT
cana-1179	259	49	iclosed	iclose	VERB
cana-1179	259	50	sets	set	NOUN
cana-1179	259	51	,	,	PUNCT
cana-1179	259	52	but	but	CCONJ
cana-1179	259	53	not	not	PART
cana-1179	259	54	a	a	DET
cana-1179	259	55	β	β	NOUN
cana-1179	259	56	-	-	ADJ
cana-1179	259	57	closed	closed	ADJ
cana-1179	259	58	set	set	NOUN
cana-1179	259	59	.	.	PUNCT
cana-1179	260	1	theorem	theorem	VERB
cana-1179	260	2	5.26	5.26	NUM
cana-1179	260	3	in	in	ADP
cana-1179	260	4	the	the	DET
cana-1179	260	5	ideal	ideal	ADJ
cana-1179	260	6	topological	topological	ADJ
cana-1179	260	7	space	space	NOUN
cana-1179	260	8	(	(	PUNCT
cana-1179	260	9	x	x	X
cana-1179	260	10	,	,	PUNCT
cana-1179	260	11	τ	τ	PROPN
cana-1179	260	12	,	,	PUNCT
cana-1179	260	13	i	i	PROPN
cana-1179	260	14	)	)	PUNCT
cana-1179	260	15	,	,	PUNCT
cana-1179	260	16	if	if	SCONJ
cana-1179	260	17	b	b	PROPN
cana-1179	260	18	is	be	AUX
cana-1179	260	19	*	*	NOUN
cana-1179	260	20	β	β	NOUN
cana-1179	260	21	-	-	ADJ
cana-1179	260	22	open	open	ADJ
cana-1179	260	23	and	and	CCONJ
cana-1179	260	24	βg*p	βg*p	NOUN
cana-1179	260	25	-	-	PUNCT
cana-1179	260	26	i	i	NOUN
cana-1179	260	27	-	-	PUNCT
cana-1179	260	28	closed	closed	ADJ
cana-1179	260	29	,	,	PUNCT
cana-1179	260	30	then	then	ADV
cana-1179	260	31	b	b	PROPN
cana-1179	260	32	is	be	AUX
cana-1179	260	33	g*p	g*p	NOUN
cana-1179	260	34	-	-	PUNCT
cana-1179	260	35	closed	closed	ADJ
cana-1179	260	36	.	.	PUNCT
cana-1179	261	1	proof	proof	NOUN
cana-1179	261	2	:	:	PUNCT
cana-1179	261	3	b	b	X
cana-1179	261	4	⊆	⊆	NUM
cana-1179	261	5	b	b	NOUN
cana-1179	261	6	,	,	PUNCT
cana-1179	261	7	since	since	SCONJ
cana-1179	261	8	b	b	NOUN
cana-1179	261	9	is	be	AUX
cana-1179	261	10	both	both	PRON
cana-1179	261	11	β*-open	β*-open	ADJ
cana-1179	261	12	and	and	CCONJ
cana-1179	261	13	βg*p	βg*p	PROPN
cana-1179	261	14	-	-	PUNCT
cana-1179	261	15	i	i	NOUN
cana-1179	261	16	-	-	PUNCT
cana-1179	261	17	closed	closed	ADJ
cana-1179	261	18	,	,	PUNCT
cana-1179	261	19	b	b	X
cana-1179	261	20	*	*	SYM
cana-1179	261	21	⊆	⊆	NUM
cana-1179	261	22	b.	b.	NOUN
cana-1179	261	23	consequently	consequently	ADV
cana-1179	261	24	,	,	PUNCT
cana-1179	261	25	b	b	X
cana-1179	261	26	*	*	PUNCT
cana-1179	261	27	=	=	PROPN
cana-1179	261	28	b.	b.	PROPN
cana-1179	261	29	b	b	PROPN
cana-1179	261	30	has	have	AUX
cana-1179	261	31	become	become	VERB
cana-1179	261	32	an	an	DET
cana-1179	261	33	g*p	g*p	PROPN
cana-1179	261	34	-	-	PUNCT
cana-1179	261	35	closed	close	VERB
cana-1179	261	36	set	set	NOUN
cana-1179	261	37	in	in	ADP
cana-1179	261	38	x.	x.	NOUN
cana-1179	261	39	theorem	theorem	VERB
cana-1179	261	40	5.27	5.27	NUM
cana-1179	261	41	in	in	ADP
cana-1179	261	42	an	an	DET
cana-1179	261	43	ideal	ideal	ADJ
cana-1179	261	44	topological	topological	ADJ
cana-1179	261	45	space	space	NOUN
cana-1179	261	46	(	(	PUNCT
cana-1179	261	47	x	x	X
cana-1179	261	48	,	,	PUNCT
cana-1179	261	49	τ	τ	PROPN
cana-1179	261	50	,	,	PUNCT
cana-1179	261	51	i	i	PROPN
cana-1179	261	52	)	)	PUNCT
cana-1179	261	53	,	,	PUNCT
cana-1179	261	54	let	let	VERB
cana-1179	261	55	b	b	X
cana-1179	261	56	be	be	AUX
cana-1179	261	57	a	a	DET
cana-1179	261	58	βg*p	βg*p	ADJ
cana-1179	261	59	-	-	PUNCT
cana-1179	261	60	i	i	NOUN
cana-1179	261	61	-	-	PUNCT
cana-1179	261	62	closed	close	VERB
cana-1179	261	63	such	such	ADJ
cana-1179	261	64	that	that	SCONJ
cana-1179	261	65	b	b	NOUN
cana-1179	261	66	⊆	⊆	NUM
cana-1179	261	67	c	c	PROPN
cana-1179	261	68	b	b	PROPN
cana-1179	261	69	*	*	PUNCT
cana-1179	261	70	.	.	PUNCT
cana-1179	262	1	thus	thus	ADV
cana-1179	262	2	,	,	PUNCT
cana-1179	262	3	b	b	PROPN
cana-1179	262	4	is	be	AUX
cana-1179	262	5	also	also	ADV
cana-1179	262	6	a	a	DET
cana-1179	262	7	closed	closed	ADJ
cana-1179	262	8	set	set	NOUN
cana-1179	262	9	in	in	ADP
cana-1179	262	10	βg*p	βg*p	ADJ
cana-1179	262	11	-	-	PUNCT
cana-1179	262	12	i.	i.	NOUN
cana-1179	262	13	proof	proof	NOUN
cana-1179	262	14	:	:	PUNCT
cana-1179	262	15	let	let	VERB
cana-1179	262	16	d	d	PRON
cana-1179	262	17	be	be	AUX
cana-1179	262	18	an	an	DET
cana-1179	262	19	β*-open	β*-open	PROPN
cana-1179	262	20	set	set	NOUN
cana-1179	262	21	which	which	PRON
cana-1179	262	22	contains	contain	VERB
cana-1179	262	23	c.	c.	PROPN
cana-1179	262	24	b	b	PROPN
cana-1179	263	1	⊆	⊆	NUM
cana-1179	263	2	c	c	NOUN
cana-1179	263	3	⊆	⊆	NUM
cana-1179	263	4	d	d	NOUN
cana-1179	263	5	⇒	⇒	NOUN
cana-1179	263	6	c	c	NOUN
cana-1179	263	7	*	*	SYM
cana-1179	263	8	⊆	⊆	NUM
cana-1179	263	9	b	b	X
cana-1179	263	10	*	*	PUNCT
cana-1179	263	11	⊆	⊆	NUM
cana-1179	263	12	d.	d.	PROPN
cana-1179	263	13	therefore	therefore	ADV
cana-1179	263	14	,	,	PUNCT
cana-1179	263	15	b	b	PROPN
cana-1179	263	16	is	be	AUX
cana-1179	263	17	closed	close	VERB
cana-1179	263	18	in	in	ADP
cana-1179	263	19	βg*p	βg*p	NOUN
cana-1179	263	20	-	-	PUNCT
cana-1179	263	21	i.	i.	NOUN
cana-1179	263	22	6	6	NUM
cana-1179	263	23	.	.	PUNCT
cana-1179	263	24	βg*p	βg*p	X
cana-1179	263	25	-	-	PUNCT
cana-1179	263	26	i	i	PRON
cana-1179	263	27	-	-	PUNCT
cana-1179	263	28	open	open	ADJ
cana-1179	263	29	sets	set	NOUN
cana-1179	263	30	in	in	ADP
cana-1179	263	31	ideal	ideal	ADJ
cana-1179	263	32	topological	topological	ADJ
cana-1179	263	33	spaces	space	NOUN
cana-1179	263	34	definition	definition	NOUN
cana-1179	263	35	6.1	6.1	NUM
cana-1179	263	36	an	an	DET
cana-1179	263	37	βg*p	βg*p	NUM
cana-1179	263	38	-	-	PUNCT
cana-1179	263	39	i	i	NOUN
cana-1179	263	40	-	-	PUNCT
cana-1179	263	41	open	open	ADJ
cana-1179	263	42	set	set	NOUN
cana-1179	263	43	is	be	AUX
cana-1179	263	44	defined	define	VERB
cana-1179	263	45	as	as	ADP
cana-1179	263	46	the	the	DET
cana-1179	263	47	complement	complement	NOUN
cana-1179	263	48	of	of	ADP
cana-1179	263	49	a	a	DET
cana-1179	263	50	βg*p	βg*p	ADJ
cana-1179	263	51	-	-	PUNCT
cana-1179	263	52	i	i	NOUN
cana-1179	263	53	-	-	PUNCT
cana-1179	263	54	closed	close	VERB
cana-1179	263	55	set	set	NOUN
cana-1179	263	56	.	.	PUNCT
cana-1179	264	1	theorem	theorem	VERB
cana-1179	264	2	6.2	6.2	NUM
cana-1179	264	3	if	if	SCONJ
cana-1179	264	4	m	m	VERB
cana-1179	264	5	and	and	CCONJ
cana-1179	264	6	n	n	PRON
cana-1179	264	7	are	be	AUX
cana-1179	264	8	βg*p	βg*p	X
cana-1179	264	9	-	-	PUNCT
cana-1179	264	10	i	i	PRON
cana-1179	264	11	-	-	PUNCT
cana-1179	264	12	open	open	ADJ
cana-1179	264	13	sets	set	NOUN
cana-1179	264	14	of	of	ADP
cana-1179	264	15	the	the	DET
cana-1179	264	16	ideal	ideal	ADJ
cana-1179	264	17	topological	topological	ADJ
cana-1179	264	18	space	space	NOUN
cana-1179	264	19	(	(	PUNCT
cana-1179	264	20	x	x	X
cana-1179	264	21	,	,	PUNCT
cana-1179	264	22	τ	τ	PROPN
cana-1179	264	23	,	,	PUNCT
cana-1179	264	24	i	i	PROPN
cana-1179	264	25	)	)	PUNCT
cana-1179	264	26	then	then	ADV
cana-1179	264	27	,	,	PUNCT
cana-1179	264	28	m	m	NOUN
cana-1179	264	29	∩	∩	ADJ
cana-1179	264	30	n	n	CCONJ
cana-1179	264	31	also	also	ADV
cana-1179	264	32	βg*p	βg*p	X
cana-1179	264	33	-	-	PUNCT
cana-1179	264	34	i	i	PRON
cana-1179	264	35	-	-	PUNCT
cana-1179	264	36	open	open	ADJ
cana-1179	264	37	set	set	NOUN
cana-1179	264	38	.	.	PUNCT
cana-1179	265	1	proof	proof	NOUN
cana-1179	265	2	:	:	PUNCT
cana-1179	265	3	assuming	assume	VERB
cana-1179	265	4	two	two	NUM
cana-1179	265	5	βg*p	βg*p	NOUN
cana-1179	265	6	-	-	PUNCT
cana-1179	265	7	i	i	NOUN
cana-1179	265	8	-	-	PUNCT
cana-1179	265	9	open	open	ADJ
cana-1179	265	10	sets	set	NOUN
cana-1179	265	11	m	m	PROPN
cana-1179	265	12	*	*	NOUN
cana-1179	265	13	and	and	CCONJ
cana-1179	265	14	n	n	CCONJ
cana-1179	265	15	*	*	VERB
cana-1179	265	16	in	in	ADP
cana-1179	265	17	x.	x.	NOUN
cana-1179	265	18	considering	consider	VERB
cana-1179	265	19	that	that	SCONJ
cana-1179	265	20	(	(	PUNCT
cana-1179	265	21	m	m	NOUN
cana-1179	265	22	*	*	ADJ
cana-1179	265	23	)	)	PUNCT
cana-1179	265	24	c	c	PROPN
cana-1179	265	25	and	and	CCONJ
cana-1179	265	26	(	(	PUNCT
cana-1179	265	27	n	n	X
cana-1179	265	28	*	*	PUNCT
cana-1179	265	29	)	)	PUNCT
cana-1179	266	1	c	c	NOUN
cana-1179	266	2	are	be	AUX
cana-1179	266	3	closed	closed	ADJ
cana-1179	266	4	sets	set	NOUN
cana-1179	266	5	of	of	ADP
cana-1179	266	6	βg*p	βg*p	NOUN
cana-1179	266	7	-	-	PROPN
cana-1179	266	8	i	i	PRON
cana-1179	266	9	in	in	ADP
cana-1179	266	10	x.	x.	PROPN
cana-1179	266	11	(	(	PUNCT
cana-1179	266	12	m	m	PROPN
cana-1179	266	13	*	*	PUNCT
cana-1179	266	14	)	)	PUNCT
cana-1179	266	15	c	c	NOUN
cana-1179	266	16	∪	∪	X
cana-1179	266	17	(	(	PUNCT
cana-1179	266	18	n	n	NOUN
cana-1179	266	19	*	*	PUNCT
cana-1179	266	20	)	)	PUNCT
cana-1179	266	21	c	c	PROPN
cana-1179	266	22	is	be	AUX
cana-1179	266	23	a	a	DET
cana-1179	266	24	βg*p	βg*p	ADJ
cana-1179	266	25	-	-	PUNCT
cana-1179	266	26	i	i	NOUN
cana-1179	266	27	-	-	PUNCT
cana-1179	266	28	closed	close	VERB
cana-1179	266	29	in	in	ADP
cana-1179	266	30	x	x	PUNCT
cana-1179	266	31	by	by	ADP
cana-1179	266	32	theorem	theorem	NOUN
cana-1179	266	33	5.21	5.21	NUM
cana-1179	266	34	.	.	PUNCT
cana-1179	267	1	it	it	PRON
cana-1179	267	2	indicates	indicate	VERB
cana-1179	267	3	βg*p	βg*p	PROPN
cana-1179	267	4	-	-	PUNCT
cana-1179	267	5	i	i	NOUN
cana-1179	267	6	-	-	PUNCT
cana-1179	267	7	closed	close	VERB
cana-1179	267	8	in	in	ADP
cana-1179	267	9	x	x	PUNCT
cana-1179	267	10	is	be	AUX
cana-1179	267	11	(	(	PUNCT
cana-1179	267	12	m	m	NOUN
cana-1179	267	13	*	*	NOUN
cana-1179	267	14	∩	∩	X
cana-1179	267	15	n	n	CCONJ
cana-1179	267	16	*	*	NOUN
cana-1179	267	17	)	)	PUNCT
cana-1179	267	18	c.	c.	PROPN
cana-1179	267	19	m	m	PROPN
cana-1179	267	20	∩	∩	PROPN
cana-1179	267	21	n	n	PRON
cana-1179	267	22	is	be	AUX
cana-1179	267	23	therefore	therefore	ADV
cana-1179	267	24	an	an	DET
cana-1179	267	25	open	open	ADJ
cana-1179	267	26	set	set	NOUN
cana-1179	267	27	in	in	ADP
cana-1179	267	28	x	x	PUNCT
cana-1179	267	29	with	with	ADP
cana-1179	267	30	βg*p	βg*p	ADJ
cana-1179	267	31	-	-	PUNCT
cana-1179	267	32	i.	i.	NOUN
cana-1179	267	33	theorem	theorem	VERB
cana-1179	267	34	6.3	6.3	NUM
cana-1179	267	35	c	c	NOUN
cana-1179	267	36	*	*	PUNCT
cana-1179	267	37	is	be	AUX
cana-1179	267	38	βg*p	βg*p	NOUN
cana-1179	267	39	-	-	PUNCT
cana-1179	267	40	i	i	PRON
cana-1179	267	41	-	-	PUNCT
cana-1179	267	42	open	open	ADJ
cana-1179	267	43	in	in	ADP
cana-1179	267	44	x	x	PUNCT
cana-1179	267	45	if	if	SCONJ
cana-1179	267	46	int*(c	int*(c	NOUN
cana-1179	267	47	)	)	PUNCT
cana-1179	267	48	⊆	⊆	NUM
cana-1179	267	49	c	c	X
cana-1179	267	50	*	*	PUNCT
cana-1179	267	51	⊆	⊆	NUM
cana-1179	267	52	b	b	X
cana-1179	267	53	*	*	PUNCT
cana-1179	267	54	and	and	CCONJ
cana-1179	267	55	if	if	SCONJ
cana-1179	267	56	b	b	X
cana-1179	267	57	*	*	PROPN
cana-1179	267	58	is	be	AUX
cana-1179	267	59	βg*p	βg*p	NOUN
cana-1179	267	60	-	-	PUNCT
cana-1179	267	61	i	i	PRON
cana-1179	267	62	-	-	PUNCT
cana-1179	267	63	open	open	ADJ
cana-1179	267	64	in	in	ADP
cana-1179	267	65	x.	x.	NOUN
cana-1179	267	66	proof	proof	NOUN
cana-1179	267	67	:	:	PUNCT
cana-1179	267	68	assume	assume	VERB
cana-1179	267	69	that	that	SCONJ
cana-1179	267	70	if	if	SCONJ
cana-1179	267	71	b	b	X
cana-1179	267	72	*	*	PROPN
cana-1179	267	73	is	be	AUX
cana-1179	267	74	βg*p	βg*p	NOUN
cana-1179	267	75	-	-	PUNCT
cana-1179	267	76	i	i	PRON
cana-1179	267	77	-	-	PUNCT
cana-1179	267	78	open	open	ADJ
cana-1179	267	79	in	in	ADP
cana-1179	267	80	x	x	NOUN
cana-1179	267	81	and	and	CCONJ
cana-1179	267	82	int*(c	int*(c	NOUN
cana-1179	267	83	)	)	PUNCT
cana-1179	267	84	⊆	⊆	NUM
cana-1179	267	85	c	c	X
cana-1179	267	86	*	*	PUNCT
cana-1179	267	87	⊆	⊆	NUM
cana-1179	267	88	b	b	NOUN
cana-1179	267	89	,	,	PUNCT
cana-1179	267	90	then	then	ADV
cana-1179	267	91	(	(	PUNCT
cana-1179	267	92	b	b	X
cana-1179	267	93	*	*	NOUN
cana-1179	267	94	)	)	PUNCT
cana-1179	267	95	c	c	NOUN
cana-1179	268	1	⊆	⊆	NUM
cana-1179	268	2	(	(	PUNCT
cana-1179	268	3	c	c	NOUN
cana-1179	268	4	*	*	NOUN
cana-1179	268	5	)	)	PUNCT
cana-1179	268	6	c	c	NOUN
cana-1179	268	7	⊆	⊆	NUM
cana-1179	268	8	cl*(b	cl*(b	NOUN
cana-1179	268	9	)	)	PUNCT
cana-1179	268	10	c.	c.	NOUN
cana-1179	268	11	by	by	ADP
cana-1179	268	12	theorem	theorem	NOUN
cana-1179	268	13	6.2	6.2	NUM
cana-1179	268	14	,	,	PUNCT
cana-1179	268	15	c	c	X
cana-1179	268	16	*	*	VERB
cana-1179	268	17	is	be	AUX
cana-1179	268	18	βg*p	βg*p	NOUN
cana-1179	268	19	-	-	PUNCT
cana-1179	268	20	i	i	PRON
cana-1179	268	21	-	-	PUNCT
cana-1179	268	22	open	open	ADJ
cana-1179	268	23	in	in	ADP
cana-1179	268	24	x	x	PUNCT
cana-1179	268	25	since	since	SCONJ
cana-1179	268	26	(	(	PUNCT
cana-1179	268	27	b	b	NOUN
cana-1179	268	28	*	*	NOUN
cana-1179	268	29	)	)	PUNCT
cana-1179	268	30	c	c	PROPN
cana-1179	268	31	is	be	AUX
cana-1179	268	32	βg*p	βg*p	NOUN
cana-1179	268	33	-	-	PUNCT
cana-1179	268	34	i	i	PRON
cana-1179	268	35	-	-	PUNCT
cana-1179	268	36	open	open	ADJ
cana-1179	268	37	in	in	ADP
cana-1179	268	38	x.	x.	NOUN
cana-1179	268	39	theorem	theorem	VERB
cana-1179	268	40	6.4	6.4	NUM
cana-1179	268	41	when	when	SCONJ
cana-1179	268	42	g	g	PROPN
cana-1179	268	43	is	be	AUX
cana-1179	268	44	a	a	DET
cana-1179	268	45	g	g	NOUN
cana-1179	268	46	-	-	PUNCT
cana-1179	268	47	closed	close	VERB
cana-1179	268	48	set	set	NOUN
cana-1179	268	49	and	and	CCONJ
cana-1179	268	50	g	g	PROPN
cana-1179	268	51	⊆	⊆	NUM
cana-1179	268	52	b	b	PROPN
cana-1179	268	53	*	*	PROPN
cana-1179	268	54	,	,	PUNCT
cana-1179	268	55	a	a	DET
cana-1179	268	56	subset	subset	NOUN
cana-1179	268	57	b	b	NOUN
cana-1179	268	58	*	*	PUNCT
cana-1179	268	59	is	be	AUX
cana-1179	268	60	β*-open	β*-open	ADJ
cana-1179	268	61	if	if	SCONJ
cana-1179	268	62	and	and	CCONJ
cana-1179	268	63	only	only	ADV
cana-1179	268	64	if	if	SCONJ
cana-1179	268	65	g	g	PROPN
cana-1179	268	66	⊆	⊆	NUM
cana-1179	268	67	pint*(b	pint*(b	NOUN
cana-1179	268	68	)	)	PUNCT
cana-1179	268	69	.	.	PUNCT
cana-1179	269	1	proof	proof	NOUN
cana-1179	269	2	:	:	PUNCT
cana-1179	269	3	essential	essential	ADJ
cana-1179	269	4	:	:	PUNCT
cana-1179	269	5	assume	assume	VERB
cana-1179	269	6	that	that	SCONJ
cana-1179	269	7	b	b	NOUN
cana-1179	269	8	is	be	AUX
cana-1179	269	9	an	an	DET
cana-1179	269	10	open	open	ADJ
cana-1179	269	11	set	set	NOUN
cana-1179	269	12	along	along	ADP
cana-1179	269	13	with	with	ADP
cana-1179	269	14	g	g	PROPN
cana-1179	269	15	⊆	⊆	NUM
cana-1179	269	16	b	b	PROPN
cana-1179	269	17	*	*	PUNCT
cana-1179	269	18	is	be	AUX
cana-1179	269	19	a	a	DET
cana-1179	269	20	g	g	NOUN
cana-1179	269	21	-	-	PUNCT
cana-1179	269	22	closed	close	VERB
cana-1179	269	23	subset	subset	NOUN
cana-1179	269	24	of	of	ADP
cana-1179	269	25	(	(	PUNCT
cana-1179	269	26	x	x	PROPN
cana-1179	269	27	,	,	PUNCT
cana-1179	269	28	τ	τ	PROPN
cana-1179	269	29	,	,	PUNCT
cana-1179	269	30	i	i	PROPN
cana-1179	269	31	)	)	PUNCT
cana-1179	269	32	.	.	PUNCT
cana-1179	270	1	by	by	ADP
cana-1179	270	2	definition	definition	NOUN
cana-1179	270	3	,	,	PUNCT
cana-1179	270	4	the	the	DET
cana-1179	270	5	set	set	NOUN
cana-1179	270	6	β*-closed	β*-close	VERB
cana-1179	270	7	contains	contain	VERB
cana-1179	270	8	x	x	X
cana-1179	270	9	b	b	X
cana-1179	270	10	*	*	PUNCT
cana-1179	270	11	.	.	PUNCT
cana-1179	271	1	furthermore	furthermore	ADV
cana-1179	271	2	,	,	PUNCT
cana-1179	271	3	the	the	DET
cana-1179	271	4	g	g	NOUN
cana-1179	271	5	-	-	PUNCT
cana-1179	271	6	open	open	ADJ
cana-1179	271	7	set	set	NOUN
cana-1179	271	8	x	x	VERB
cana-1179	271	9	g	g	NOUN
cana-1179	271	10	contains	contain	VERB
cana-1179	271	11	x	x	X
cana-1179	271	12	b	b	X
cana-1179	271	13	*	*	PROPN
cana-1179	271	14	.	.	PUNCT
cana-1179	272	1	βcl*(x	βcl*(x	PROPN
cana-1179	272	2	communications	communication	NOUN
cana-1179	272	3	on	on	ADP
cana-1179	272	4	applied	apply	VERB
cana-1179	272	5	nonlinear	nonlinear	ADJ
cana-1179	272	6	analysis	analysis	NOUN
cana-1179	272	7	issn	issn	NOUN
cana-1179	272	8	:	:	PUNCT
cana-1179	272	9	1074	1074	NUM
cana-1179	272	10	-	-	PUNCT
cana-1179	272	11	133x	133x	NUM
cana-1179	272	12	vol	vol	NOUN
cana-1179	272	13	31	31	NUM
cana-1179	272	14	no	no	NOUN
cana-1179	272	15	.	.	PUNCT
cana-1179	273	1	6s	6s	NUM
cana-1179	273	2	(	(	PUNCT
cana-1179	273	3	2024	2024	NUM
cana-1179	273	4	)	)	PUNCT
cana-1179	273	5	215	215	NUM
cana-1179	273	6	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-1179	273	7	b	b	X
cana-1179	273	8	)	)	PUNCT
cana-1179	273	9	⊆	⊆	NUM
cana-1179	273	10	x	x	SYM
cana-1179	273	11	g	g	NOUN
cana-1179	273	12	is	be	AUX
cana-1179	273	13	implied	imply	VERB
cana-1179	273	14	x	x	X
cana-1179	273	15	βint*(b	βint*(b	PUNCT
cana-1179	273	16	)	)	PUNCT
cana-1179	273	17	=	=	SYM
cana-1179	273	18	βcl*(x	βcl*(x	PROPN
cana-1179	273	19	g	g	NOUN
cana-1179	273	20	)	)	PUNCT
cana-1179	273	21	at	at	ADP
cana-1179	273	22	this	this	DET
cana-1179	273	23	point	point	NOUN
cana-1179	273	24	.	.	PUNCT
cana-1179	274	1	thus	thus	ADV
cana-1179	274	2	,	,	PUNCT
cana-1179	274	3	x	x	PUNCT
cana-1179	274	4	βint*(b	βint*(b	PUNCT
cana-1179	274	5	)	)	PUNCT
cana-1179	274	6	⊆	⊆	NUM
cana-1179	274	7	x	x	SYM
cana-1179	274	8	g	g	NOUN
cana-1179	274	9	or	or	CCONJ
cana-1179	274	10	g	g	NOUN
cana-1179	274	11	⊆	⊆	NUM
cana-1179	274	12	pint*(b	pint*(b	NOUN
cana-1179	274	13	)	)	PUNCT
cana-1179	274	14	follows	follow	VERB
cana-1179	274	15	.	.	PUNCT
cana-1179	275	1	sufficiency	sufficiency	NOUN
cana-1179	275	2	:	:	PUNCT
cana-1179	275	3	in	in	ADP
cana-1179	275	4	the	the	DET
cana-1179	275	5	instance	instance	NOUN
cana-1179	275	6	that	that	SCONJ
cana-1179	275	7	g	g	PROPN
cana-1179	275	8	is	be	AUX
cana-1179	275	9	a	a	DET
cana-1179	275	10	g	g	NOUN
cana-1179	275	11	-	-	PUNCT
cana-1179	275	12	closed	close	VERB
cana-1179	275	13	set	set	NOUN
cana-1179	275	14	and	and	CCONJ
cana-1179	275	15	g	g	NOUN
cana-1179	275	16	⊆	⊆	NUM
cana-1179	275	17	βint*(b	βint*(b	NUM
cana-1179	275	18	)	)	PUNCT
cana-1179	275	19	where	where	SCONJ
cana-1179	275	20	g	g	PROPN
cana-1179	275	21	⊆	⊆	NUM
cana-1179	275	22	b	b	NOUN
cana-1179	275	23	,	,	PUNCT
cana-1179	275	24	x	x	PROPN
cana-1179	275	25	b	b	X
cana-1179	275	26	⊆	⊆	NUM
cana-1179	275	27	x	x	SYM
cana-1179	275	28	g	g	NOUN
cana-1179	275	29	and	and	CCONJ
cana-1179	275	30	βint*(b	βint*(b	NUM
cana-1179	275	31	)	)	PUNCT
cana-1179	275	32	⊆	⊆	NUM
cana-1179	275	33	x	x	SYM
cana-1179	275	34	g	g	NOUN
cana-1179	275	35	follows	follow	VERB
cana-1179	275	36	that	that	SCONJ
cana-1179	275	37	βcl*(x	βcl*(x	PROPN
cana-1179	275	38	b	b	NOUN
cana-1179	275	39	)	)	PUNCT
cana-1179	275	40	⊆	⊆	NUM
cana-1179	275	41	x	x	X
cana-1179	275	42	g.	g.	PROPN
cana-1179	275	43	thus	thus	ADV
cana-1179	275	44	,	,	PUNCT
cana-1179	275	45	x	x	PROPN
cana-1179	275	46	b	b	NOUN
cana-1179	275	47	is	be	AUX
cana-1179	275	48	a	a	DET
cana-1179	275	49	β*-closed	β*-close	VERB
cana-1179	275	50	set	set	NOUN
cana-1179	275	51	and	and	CCONJ
cana-1179	275	52	b	b	NOUN
cana-1179	275	53	turns	turn	VERB
cana-1179	275	54	into	into	ADP
cana-1179	275	55	a	a	DET
cana-1179	275	56	β*-open	β*-open	PROPN
cana-1179	275	57	set	set	NOUN
cana-1179	275	58	.	.	PUNCT
cana-1179	276	1	references	reference	NOUN
cana-1179	276	2	[	[	X
cana-1179	276	3	1	1	NUM
cana-1179	276	4	]	]	PUNCT
cana-1179	276	5	p.	p.	NOUN
cana-1179	276	6	bhattacharyya	bhattacharyya	PROPN
cana-1179	276	7	and	and	CCONJ
cana-1179	276	8	b.	b.	PROPN
cana-1179	276	9	k.	k.	PROPN
cana-1179	276	10	lahiri	lahiri	PROPN
cana-1179	276	11	,	,	PUNCT
cana-1179	276	12	semi	semi	ADV
cana-1179	276	13	generalized	generalized	ADJ
cana-1179	276	14	closed	closed	ADJ
cana-1179	276	15	sets	set	NOUN
cana-1179	276	16	in	in	ADP
cana-1179	276	17	topology	topology	NOUN
cana-1179	276	18	,	,	PUNCT
cana-1179	276	19	indian	indian	PROPN
cana-1179	276	20	j.	j.	PROPN
cana-1179	276	21	math	math	PROPN
cana-1179	276	22	.	.	PUNCT
cana-1179	276	23	,	,	PUNCT
cana-1179	276	24	29(1987	29(1987	NUM
cana-1179	276	25	)	)	PUNCT
cana-1179	276	26	,	,	PUNCT
cana-1179	276	27	376	376	NUM
cana-1179	276	28	-	-	SYM
cana-1179	276	29	382	382	NUM
cana-1179	276	30	.	.	PUNCT
cana-1179	277	1	[	[	X
cana-1179	277	2	2	2	NUM
cana-1179	277	3	]	]	PUNCT
cana-1179	277	4	p.	p.	NOUN
cana-1179	277	5	bhattacharya	bhattacharya	PROPN
cana-1179	277	6	,	,	PUNCT
cana-1179	277	7	on	on	ADP
cana-1179	277	8	generalized	generalize	VERB
cana-1179	277	9	regular	regular	ADJ
cana-1179	277	10	closed	closed	ADJ
cana-1179	277	11	sets	set	NOUN
cana-1179	277	12	,	,	PUNCT
cana-1179	277	13	int	int	NOUN
cana-1179	277	14	.	.	PUNCT
cana-1179	278	1	j	j	NOUN
cana-1179	278	2	,	,	PUNCT
cana-1179	278	3	contemp	contemp	NOUN
cana-1179	278	4	.	.	PUNCT
cana-1179	279	1	math	math	NOUN
cana-1179	279	2	.	.	PUNCT
cana-1179	280	1	sciences	science	NOUN
cana-1179	280	2	.	.	PUNCT
cana-1179	280	3	,	,	PUNCT
cana-1179	280	4	6	6	NUM
cana-1179	280	5	(	(	PUNCT
cana-1179	280	6	2011	2011	NUM
cana-1179	280	7	)	)	PUNCT
cana-1179	280	8	,	,	PUNCT
cana-1179	280	9	145	145	NUM
cana-1179	280	10	-	-	SYM
cana-1179	280	11	152	152	NUM
cana-1179	280	12	.	.	PUNCT
cana-1179	281	1	[	[	X
cana-1179	281	2	3	3	X
cana-1179	281	3	]	]	X
cana-1179	281	4	c.	c.	PROPN
cana-1179	281	5	dhanapakyam	dhanapakyam	PROPN
cana-1179	281	6	and	and	CCONJ
cana-1179	281	7	k.	k.	PROPN
cana-1179	281	8	indirani	indirani	PROPN
cana-1179	281	9	,	,	PUNCT
cana-1179	281	10	on	on	ADP
cana-1179	281	11	βg*-closed	βg*-closed	ADJ
cana-1179	281	12	sets	set	NOUN
cana-1179	281	13	in	in	ADP
cana-1179	281	14	topological	topological	ADJ
cana-1179	281	15	spaces	space	NOUN
cana-1179	281	16	,	,	PUNCT
cana-1179	281	17	international	international	ADJ
cana-1179	281	18	journal	journal	NOUN
cana-1179	281	19	of	of	ADP
cana-1179	281	20	applied	apply	VERB
cana-1179	281	21	research	research	NOUN
cana-1179	281	22	.	.	PUNCT
cana-1179	281	23	,	,	PUNCT
cana-1179	281	24	2(4	2(4	NUM
cana-1179	281	25	)	)	PUNCT
cana-1179	281	26	(	(	PUNCT
cana-1179	281	27	2016	2016	NUM
cana-1179	281	28	)	)	PUNCT
cana-1179	281	29	,	,	PUNCT
cana-1179	281	30	388	388	NUM
cana-1179	281	31	-	-	SYM
cana-1179	281	32	391	391	NUM
cana-1179	281	33	.	.	PUNCT
cana-1179	282	1	[	[	X
cana-1179	282	2	4	4	X
cana-1179	282	3	]	]	X
cana-1179	282	4	y.	y.	PROPN
cana-1179	282	5	gnanambal	gnanambal	PROPN
cana-1179	282	6	,	,	PUNCT
cana-1179	282	7	on	on	ADP
cana-1179	282	8	generalized	generalized	ADJ
cana-1179	282	9	pre	pre	ADJ
cana-1179	282	10	regular	regular	ADJ
cana-1179	282	11	closed	closed	ADJ
cana-1179	282	12	sets	set	NOUN
cana-1179	282	13	in	in	ADP
cana-1179	282	14	topological	topological	ADJ
cana-1179	282	15	spaces	space	NOUN
cana-1179	282	16	,	,	PUNCT
cana-1179	282	17	indian	indian	PROPN
cana-1179	282	18	j.	j.	PROPN
cana-1179	282	19	pure	pure	PROPN
cana-1179	282	20	appl	appl	PROPN
cana-1179	282	21	.	.	PUNCT
cana-1179	282	22	math	math	PROPN
cana-1179	282	23	.	.	PUNCT
cana-1179	282	24	,	,	PUNCT
cana-1179	282	25	28(1997	28(1997	NUM
cana-1179	282	26	)	)	PUNCT
cana-1179	282	27	,	,	PUNCT
cana-1179	282	28	351	351	NUM
cana-1179	282	29	-	-	SYM
cana-1179	282	30	360	360	NUM
cana-1179	282	31	.	.	PUNCT
cana-1179	283	1	[	[	X
cana-1179	283	2	5	5	X
cana-1179	283	3	]	]	PUNCT
cana-1179	283	4	s.	s.	PROPN
cana-1179	283	5	gowri	gowri	PROPN
cana-1179	283	6	and	and	CCONJ
cana-1179	283	7	v.	v.	ADP
cana-1179	283	8	pankajam	pankajam	NOUN
cana-1179	283	9	,	,	PUNCT
cana-1179	283	10	on	on	ADP
cana-1179	283	11	beta	beta	NOUN
cana-1179	283	12	generalized	generalize	VERB
cana-1179	283	13	pre	pre	VERB
cana-1179	283	14	closed	close	VERB
cana-1179	283	15	sets	set	NOUN
cana-1179	283	16	in	in	ADP
cana-1179	283	17	topological	topological	ADJ
cana-1179	283	18	spaces	space	NOUN
cana-1179	283	19	,	,	PUNCT
cana-1179	283	20	journal	journal	NOUN
cana-1179	283	21	of	of	ADP
cana-1179	283	22	southwest	southwest	PROPN
cana-1179	283	23	jiaotong	jiaotong	PROPN
cana-1179	283	24	university	university	PROPN
cana-1179	283	25	.	.	PUNCT
cana-1179	283	26	,	,	PUNCT
cana-1179	283	27	58(2	58(2	NUM
cana-1179	283	28	)	)	PUNCT
cana-1179	283	29	,	,	PUNCT
cana-1179	283	30	(	(	PUNCT
cana-1179	283	31	2023	2023	NUM
cana-1179	283	32	)	)	PUNCT
cana-1179	283	33	,	,	PUNCT
cana-1179	283	34	559	559	NUM
cana-1179	283	35	-	-	SYM
cana-1179	283	36	564	564	NUM
cana-1179	283	37	.	.	PUNCT
cana-1179	284	1	[	[	X
cana-1179	284	2	6	6	NUM
cana-1179	284	3	]	]	PUNCT
cana-1179	284	4	s.	s.	PROPN
cana-1179	284	5	gowri	gowri	PROPN
cana-1179	284	6	and	and	CCONJ
cana-1179	284	7	v.	v.	ADP
cana-1179	284	8	pankajam	pankajam	NOUN
cana-1179	284	9	,	,	PUNCT
cana-1179	284	10	on	on	ADP
cana-1179	284	11	beta	beta	ADJ
cana-1179	284	12	generalized	generalize	VERB
cana-1179	284	13	-	-	PUNCT
cana-1179	284	14	i	i	PRON
cana-1179	284	15	-	-	PUNCT
cana-1179	284	16	closed	close	VERB
cana-1179	284	17	sets	set	NOUN
cana-1179	284	18	in	in	ADP
cana-1179	284	19	ideal	ideal	ADJ
cana-1179	284	20	topological	topological	ADJ
cana-1179	284	21	spaces	space	NOUN
cana-1179	284	22	,	,	PUNCT
cana-1179	284	23	indian	indian	ADJ
cana-1179	284	24	journal	journal	NOUN
cana-1179	284	25	of	of	ADP
cana-1179	284	26	natural	natural	ADJ
cana-1179	284	27	sciences	science	NOUN
cana-1179	284	28	.	.	PUNCT
cana-1179	284	29	,	,	PUNCT
cana-1179	284	30	14(82	14(82	NUM
cana-1179	284	31	)	)	PUNCT
cana-1179	284	32	,	,	PUNCT
cana-1179	284	33	(	(	PUNCT
cana-1179	284	34	2024	2024	NUM
cana-1179	284	35	)	)	PUNCT
cana-1179	284	36	,	,	PUNCT
cana-1179	284	37	67683	67683	NUM
cana-1179	284	38	-	-	SYM
cana-1179	284	39	67688	67688	NUM
cana-1179	284	40	.	.	PUNCT
cana-1179	285	1	[	[	X
cana-1179	285	2	7	7	X
cana-1179	285	3	]	]	PUNCT
cana-1179	285	4	s.	s.	PROPN
cana-1179	285	5	jafari	jafari	PROPN
cana-1179	285	6	,	,	PUNCT
cana-1179	285	7	r.	r.	PROPN
cana-1179	285	8	m.	m.	PROPN
cana-1179	285	9	latif	latif	PROPN
cana-1179	285	10	,	,	PUNCT
cana-1179	285	11	g.	g.	PROPN
cana-1179	285	12	nordo	nordo	PROPN
cana-1179	285	13	and	and	CCONJ
cana-1179	285	14	n.	n.	PROPN
cana-1179	285	15	rajesh	rajesh	PROPN
cana-1179	285	16	,	,	PUNCT
cana-1179	285	17	properties	property	NOUN
cana-1179	285	18	of	of	ADP
cana-1179	285	19	α	α	NOUN
cana-1179	285	20	-	-	ADJ
cana-1179	285	21	open	open	ADJ
cana-1179	285	22	sets	set	NOUN
cana-1179	285	23	in	in	ADP
cana-1179	285	24	ideal	ideal	ADJ
cana-1179	285	25	generalized	generalize	VERB
cana-1179	285	26	topological	topological	ADJ
cana-1179	285	27	spaces	space	NOUN
cana-1179	285	28	,	,	PUNCT
cana-1179	285	29	poincare	poincare	PROPN
cana-1179	285	30	journal	journal	PROPN
cana-1179	285	31	of	of	ADP
cana-1179	285	32	analysis	analysis	NOUN
cana-1179	285	33	and	and	CCONJ
cana-1179	285	34	applications	application	NOUN
cana-1179	285	35	.	.	PUNCT
cana-1179	285	36	,	,	PUNCT
cana-1179	285	37	vol.8	vol.8	PROPN
cana-1179	285	38	,	,	PUNCT
cana-1179	285	39	(	(	PUNCT
cana-1179	285	40	2021	2021	NUM
cana-1179	285	41	)	)	PUNCT
cana-1179	285	42	,	,	PUNCT
cana-1179	285	43	231	231	NUM
cana-1179	285	44	-	-	SYM
cana-1179	285	45	244	244	NUM
cana-1179	285	46	.	.	PUNCT
cana-1179	286	1	[	[	X
cana-1179	286	2	8	8	NUM
cana-1179	286	3	]	]	X
cana-1179	286	4	jose	jose	NOUN
cana-1179	286	5	tormet	tormet	NOUN
cana-1179	286	6	,	,	PUNCT
cana-1179	286	7	rafael	rafael	PROPN
cana-1179	286	8	lozada	lozada	PROPN
cana-1179	286	9	-	-	PUNCT
cana-1179	286	10	yavina	yavina	PROPN
cana-1179	286	11	and	and	CCONJ
cana-1179	286	12	margot	margot	PROPN
cana-1179	286	13	salas	salas	PROPN
cana-1179	286	14	-	-	PUNCT
cana-1179	286	15	brown	brown	PROPN
cana-1179	286	16	,	,	PUNCT
cana-1179	286	17	a	a	DET
cana-1179	286	18	generalized	generalized	ADJ
cana-1179	286	19	local	local	ADJ
cana-1179	286	20	function	function	NOUN
cana-1179	286	21	in	in	ADP
cana-1179	286	22	ideal	ideal	ADJ
cana-1179	286	23	topological	topological	ADJ
cana-1179	286	24	spaces	space	NOUN
cana-1179	286	25	,	,	PUNCT
cana-1179	286	26	poincare	poincare	PROPN
cana-1179	286	27	journal	journal	PROPN
cana-1179	286	28	of	of	ADP
cana-1179	286	29	analysis	analysis	NOUN
cana-1179	286	30	and	and	CCONJ
cana-1179	286	31	applications	application	NOUN
cana-1179	286	32	,	,	PUNCT
cana-1179	286	33	vol.9	vol.9	PROPN
cana-1179	286	34	,	,	PUNCT
cana-1179	286	35	(	(	PUNCT
cana-1179	286	36	2022	2022	NUM
cana-1179	286	37	)	)	PUNCT
cana-1179	286	38	,	,	PUNCT
cana-1179	286	39	77	77	NUM
cana-1179	286	40	-	-	SYM
cana-1179	286	41	89	89	NUM
cana-1179	286	42	.	.	PUNCT
cana-1179	287	1	[	[	X
cana-1179	287	2	9	9	NUM
cana-1179	287	3	]	]	PUNCT
cana-1179	287	4	v.	v.	CCONJ
cana-1179	287	5	kavitha	kavitha	PROPN
cana-1179	287	6	and	and	CCONJ
cana-1179	287	7	v.	v.	PROPN
cana-1179	287	8	e.	e.	PROPN
cana-1179	287	9	sasikala	sasikala	PROPN
cana-1179	287	10	,	,	PUNCT
cana-1179	287	11	beta	beta	ADJ
cana-1179	287	12	generalized	generalize	VERB
cana-1179	287	13	closed	close	VERB
cana-1179	287	14	sets	set	NOUN
cana-1179	287	15	in	in	ADP
cana-1179	287	16	topological	topological	ADJ
cana-1179	287	17	spaces	space	NOUN
cana-1179	287	18	,	,	PUNCT
cana-1179	287	19	journal	journal	NOUN
cana-1179	287	20	of	of	ADP
cana-1179	287	21	algebraic	algebraic	PROPN
cana-1179	287	22	statistics	statistic	NOUN
cana-1179	287	23	.	.	PUNCT
cana-1179	287	24	,	,	PUNCT
cana-1179	287	25	issn:1309	issn:1309	NOUN
cana-1179	287	26	-	-	PUNCT
cana-1179	287	27	3452	3452	NUM
cana-1179	287	28	,	,	PUNCT
cana-1179	287	29	vol.13	vol.13	NOUN
cana-1179	287	30	,	,	PUNCT
cana-1179	287	31	no.3-(2022	no.3-(2022	PROPN
cana-1179	287	32	)	)	PUNCT
cana-1179	287	33	,	,	PUNCT
cana-1179	287	34	891	891	NUM
cana-1179	287	35	-	-	SYM
cana-1179	287	36	898	898	NUM
cana-1179	287	37	.	.	PUNCT
cana-1179	288	1	[	[	X
cana-1179	288	2	10	10	NUM
cana-1179	288	3	]	]	X
cana-1179	288	4	m.	m.	NOUN
cana-1179	288	5	khan	khan	PROPN
cana-1179	288	6	and	and	CCONJ
cana-1179	288	7	t.	t.	PROPN
cana-1179	288	8	noiri	noiri	PROPN
cana-1179	288	9	,	,	PUNCT
cana-1179	288	10	on	on	ADP
cana-1179	288	11	gi	gi	NOUN
cana-1179	288	12	-	-	PUNCT
cana-1179	288	13	closed	close	VERB
cana-1179	288	14	sets	set	NOUN
cana-1179	288	15	in	in	ADP
cana-1179	288	16	ideal	ideal	ADJ
cana-1179	288	17	topological	topological	ADJ
cana-1179	288	18	spaces	space	NOUN
cana-1179	288	19	,	,	PUNCT
cana-1179	288	20	journal	journal	NOUN
cana-1179	288	21	of	of	ADP
cana-1179	288	22	advanced	advanced	ADJ
cana-1179	288	23	studies	study	NOUN
cana-1179	288	24	in	in	ADP
cana-1179	288	25	topology	topology	NOUN
cana-1179	288	26	.	.	PUNCT
cana-1179	288	27	,	,	PUNCT
cana-1179	288	28	vol.1	vol.1	NUM
cana-1179	288	29	,	,	PUNCT
cana-1179	288	30	(	(	PUNCT
cana-1179	288	31	2010	2010	NUM
cana-1179	288	32	)	)	PUNCT
cana-1179	288	33	,	,	PUNCT
cana-1179	288	34	29	29	NUM
cana-1179	288	35	-	-	SYM
cana-1179	288	36	33	33	NUM
cana-1179	288	37	.	.	PUNCT
cana-1179	289	1	[	[	X
cana-1179	289	2	11	11	NUM
cana-1179	289	3	]	]	X
cana-1179	289	4	n.	n.	PROPN
cana-1179	289	5	levine	levine	PROPN
cana-1179	289	6	,	,	PUNCT
cana-1179	289	7	generalized	generalize	VERB
cana-1179	289	8	closed	closed	ADJ
cana-1179	289	9	sets	set	NOUN
cana-1179	289	10	in	in	ADP
cana-1179	289	11	topology	topology	NOUN
cana-1179	289	12	,	,	PUNCT
cana-1179	289	13	rend	rend	VERB
cana-1179	289	14	.	.	PUNCT
cana-1179	290	1	circ	circ	PROPN
cana-1179	290	2	.	.	PUNCT
cana-1179	291	1	math	math	NOUN
cana-1179	291	2	.	.	PUNCT
cana-1179	292	1	palermo	palermo	PROPN
cana-1179	292	2	.	.	PUNCT
cana-1179	292	3	,	,	PUNCT
cana-1179	292	4	19(2	19(2	NUM
cana-1179	292	5	)	)	PUNCT
cana-1179	292	6	(	(	PUNCT
cana-1179	292	7	1970	1970	NUM
cana-1179	292	8	)	)	PUNCT
cana-1179	292	9	,	,	PUNCT
cana-1179	292	10	(	(	PUNCT
cana-1179	292	11	89	89	NUM
cana-1179	292	12	-	-	SYM
cana-1179	292	13	96	96	NUM
cana-1179	292	14	)	)	PUNCT
cana-1179	292	15	.	.	PUNCT
cana-1179	293	1	[	[	X
cana-1179	293	2	12	12	NUM
cana-1179	293	3	]	]	X
cana-1179	293	4	s.	s.	PROPN
cana-1179	293	5	maragathavalli	maragathavalli	PROPN
cana-1179	293	6	and	and	CCONJ
cana-1179	293	7	d.	d.	PROPN
cana-1179	293	8	vinodhini	vinodhini	PROPN
cana-1179	293	9	,	,	PUNCT
cana-1179	293	10	on	on	ADP
cana-1179	293	11	α	α	DET
cana-1179	293	12	generalized	generalize	VERB
cana-1179	293	13	closed	close	VERB
cana-1179	293	14	sets	set	NOUN
cana-1179	293	15	in	in	ADP
cana-1179	293	16	ideal	ideal	ADJ
cana-1179	293	17	topological	topological	ADJ
cana-1179	293	18	spaces	space	NOUN
cana-1179	293	19	,	,	PUNCT
cana-1179	293	20	iosr	iosr	ADJ
cana-1179	293	21	journal	journal	NOUN
cana-1179	293	22	of	of	ADP
cana-1179	293	23	mathematics	mathematics	PROPN
cana-1179	293	24	.	.	PUNCT
cana-1179	293	25	,	,	PUNCT
cana-1179	293	26	vol.10	vol.10	PROPN
cana-1179	293	27	,	,	PUNCT
cana-1179	293	28	(	(	PUNCT
cana-1179	293	29	2014	2014	NUM
cana-1179	293	30	)	)	PUNCT
cana-1179	293	31	,	,	PUNCT
cana-1179	293	32	33	33	NUM
cana-1179	293	33	-	-	SYM
cana-1179	293	34	38	38	NUM
cana-1179	293	35	.	.	PUNCT
cana-1179	294	1	[	[	X
cana-1179	294	2	13	13	NUM
cana-1179	294	3	]	]	PUNCT
cana-1179	294	4	m.	m.	NOUN
cana-1179	294	5	w.	w.	PROPN
cana-1179	294	6	mohammed	mohammed	PROPN
cana-1179	294	7	and	and	CCONJ
cana-1179	294	8	a.	a.	NOUN
cana-1179	294	9	a.	a.	PROPN
cana-1179	294	10	mohammed	mohammed	PROPN
cana-1179	294	11	,	,	PUNCT
cana-1179	294	12	some	some	DET
cana-1179	294	13	classes	class	NOUN
cana-1179	294	14	in	in	ADP
cana-1179	294	15	ideal	ideal	ADJ
cana-1179	294	16	topological	topological	ADJ
cana-1179	294	17	spaces	space	NOUN
cana-1179	294	18	,	,	PUNCT
cana-1179	294	19	journal	journal	NOUN
cana-1179	294	20	of	of	ADP
cana-1179	294	21	education	education	NOUN
cana-1179	294	22	and	and	CCONJ
cana-1179	294	23	sciences	science	NOUN
cana-1179	294	24	.	.	PUNCT
cana-1179	294	25	,	,	PUNCT
cana-1179	294	26	32(2	32(2	NUM
cana-1179	294	27	)	)	PUNCT
cana-1179	294	28	,	,	PUNCT
cana-1179	294	29	(	(	PUNCT
cana-1179	294	30	2023	2023	NUM
cana-1179	294	31	)	)	PUNCT
cana-1179	294	32	,	,	PUNCT
cana-1179	294	33	128	128	NUM
cana-1179	294	34	.	.	PUNCT
cana-1179	295	1	[	[	X
cana-1179	295	2	14	14	NUM
cana-1179	295	3	]	]	X
cana-1179	295	4	o.	o.	PROPN
cana-1179	295	5	najastad	najastad	PROPN
cana-1179	295	6	,	,	PUNCT
cana-1179	295	7	on	on	ADP
cana-1179	295	8	some	some	DET
cana-1179	295	9	classes	class	NOUN
cana-1179	295	10	of	of	ADP
cana-1179	295	11	nearly	nearly	ADV
cana-1179	295	12	open	open	ADJ
cana-1179	295	13	sets	set	NOUN
cana-1179	295	14	,	,	PUNCT
cana-1179	295	15	pacific	pacific	PROPN
cana-1179	295	16	j.	j.	PROPN
cana-1179	295	17	math	math	PROPN
cana-1179	295	18	.	.	PUNCT
cana-1179	295	19	,15(1965	,15(1965	PUNCT
cana-1179	295	20	)	)	PUNCT
cana-1179	295	21	,	,	PUNCT
cana-1179	295	22	961	961	NUM
cana-1179	295	23	-	-	SYM
cana-1179	295	24	970	970	NUM
cana-1179	295	25	.	.	PUNCT
cana-1179	296	1	[	[	X
cana-1179	296	2	15	15	NUM
cana-1179	296	3	]	]	X
cana-1179	296	4	n.	n.	NOUN
cana-1179	296	5	palaniappan	palaniappan	PROPN
cana-1179	296	6	and	and	CCONJ
cana-1179	296	7	k.	k.	PROPN
cana-1179	296	8	c.	c.	PROPN
cana-1179	296	9	rao	rao	PROPN
cana-1179	296	10	,	,	PUNCT
cana-1179	296	11	regular	regular	ADJ
cana-1179	296	12	generalized	generalize	VERB
cana-1179	296	13	closed	close	VERB
cana-1179	296	14	sets	set	NOUN
cana-1179	296	15	,	,	PUNCT
cana-1179	296	16	kyungpook	kyungpook	NOUN
cana-1179	296	17	math	math	NOUN
cana-1179	296	18	.	.	PUNCT
cana-1179	296	19	,	,	PUNCT
cana-1179	296	20	3(2	3(2	NUM
cana-1179	296	21	)	)	PUNCT
cana-1179	296	22	(	(	PUNCT
cana-1179	296	23	1993	1993	NUM
cana-1179	296	24	)	)	PUNCT
cana-1179	296	25	,	,	PUNCT
cana-1179	296	26	211	211	NUM
cana-1179	296	27	.	.	PUNCT
cana-1179	297	1	[	[	X
cana-1179	297	2	16	16	NUM
cana-1179	297	3	]	]	PUNCT
cana-1179	297	4	p.	p.	NOUN
cana-1179	297	5	g.	g.	PROPN
cana-1179	297	6	palanimani	palanimani	PROPN
cana-1179	297	7	and	and	CCONJ
cana-1179	297	8	r.	r.	PROPN
cana-1179	297	9	parimelazhagan	parimelazhagan	PROPN
cana-1179	297	10	,	,	PUNCT
cana-1179	297	11	β*-closed	β*-close	VERB
cana-1179	297	12	sets	set	NOUN
cana-1179	297	13	in	in	ADP
cana-1179	297	14	topological	topological	ADJ
cana-1179	297	15	spaces	space	NOUN
cana-1179	297	16	,	,	PUNCT
cana-1179	297	17	iosr	iosr	ADJ
cana-1179	297	18	journal	journal	NOUN
cana-1179	297	19	of	of	ADP
cana-1179	297	20	mathematics	mathematic	NOUN
cana-1179	297	21	.	.	PUNCT
cana-1179	297	22	,	,	PUNCT
cana-1179	297	23	vol	vol	NOUN
cana-1179	297	24	5	5	NUM
cana-1179	297	25	,	,	PUNCT
cana-1179	297	26	(	(	PUNCT
cana-1179	297	27	2013	2013	NUM
cana-1179	297	28	)	)	PUNCT
cana-1179	297	29	,	,	PUNCT
cana-1179	297	30	47	47	NUM
cana-1179	297	31	-	-	SYM
cana-1179	297	32	50	50	NUM
cana-1179	297	33	.	.	PUNCT
cana-1179	298	1	[	[	X
cana-1179	298	2	17	17	NUM
cana-1179	298	3	]	]	X
cana-1179	298	4	v.	v.	ADP
cana-1179	298	5	pankajam	pankajam	PROPN
cana-1179	298	6	and	and	CCONJ
cana-1179	298	7	s.	s.	PROPN
cana-1179	298	8	gowri	gowri	PROPN
cana-1179	298	9	,	,	PUNCT
cana-1179	298	10	on	on	ADP
cana-1179	298	11	αωi	αωi	NOUN
cana-1179	298	12	-	-	PUNCT
cana-1179	298	13	closed	close	VERB
cana-1179	298	14	sets	set	NOUN
cana-1179	298	15	in	in	ADP
cana-1179	298	16	ideal	ideal	ADJ
cana-1179	298	17	topological	topological	ADJ
cana-1179	298	18	spaces	space	NOUN
cana-1179	298	19	,	,	PUNCT
cana-1179	298	20	journal	journal	NOUN
cana-1179	298	21	of	of	ADP
cana-1179	298	22	xidian	xidian	PROPN
cana-1179	298	23	university	university	PROPN
cana-1179	298	24	.	.	PUNCT
cana-1179	299	1	,15(3	,15(3	PROPN
cana-1179	299	2	)	)	PUNCT
cana-1179	299	3	,	,	PUNCT
cana-1179	299	4	(	(	PUNCT
cana-1179	299	5	2021	2021	NUM
cana-1179	299	6	)	)	PUNCT
cana-1179	299	7	,	,	PUNCT
cana-1179	299	8	30	30	NUM
cana-1179	299	9	-	-	SYM
cana-1179	299	10	36	36	NUM
cana-1179	299	11	.	.	PUNCT
cana-1179	300	1	[	[	X
cana-1179	300	2	18	18	NUM
cana-1179	300	3	]	]	X
cana-1179	300	4	v.	v.	ADP
cana-1179	300	5	pankajam	pankajam	PROPN
cana-1179	300	6	and	and	CCONJ
cana-1179	300	7	s.	s.	PROPN
cana-1179	300	8	gowri	gowri	PROPN
cana-1179	300	9	,	,	PUNCT
cana-1179	300	10	on	on	ADP
cana-1179	300	11	generalized	generalized	ADJ
cana-1179	300	12	star	star	NOUN
cana-1179	300	13	ωαi	ωαi	ADJ
cana-1179	300	14	-	-	PUNCT
cana-1179	300	15	closed	closed	ADJ
cana-1179	300	16	sets	set	NOUN
cana-1179	300	17	in	in	ADP
cana-1179	300	18	ideal	ideal	ADJ
cana-1179	300	19	topological	topological	ADJ
cana-1179	300	20	spaces	space	NOUN
cana-1179	300	21	,	,	PUNCT
cana-1179	300	22	bull	bull	NOUN
cana-1179	300	23	.	.	PUNCT
cana-1179	301	1	pure	pure	ADJ
cana-1179	301	2	appl	appl	PROPN
cana-1179	301	3	.	.	PUNCT
cana-1179	302	1	sci	sci	PROPN
cana-1179	302	2	.	.	PUNCT
cana-1179	302	3	sect	sect	NOUN
cana-1179	302	4	.	.	PUNCT
cana-1179	303	1	e	e	X
cana-1179	303	2	math	math	NOUN
cana-1179	303	3	.	.	PUNCT
cana-1179	304	1	stat	stat	PROPN
cana-1179	304	2	.	.	PUNCT
cana-1179	305	1	40e	40e	NOUN
cana-1179	305	2	(	(	PUNCT
cana-1179	305	3	1	1	NUM
cana-1179	305	4	)	)	PUNCT
cana-1179	305	5	,	,	PUNCT
cana-1179	305	6	(	(	PUNCT
cana-1179	305	7	2021	2021	NUM
cana-1179	305	8	)	)	PUNCT
cana-1179	305	9	,	,	PUNCT
cana-1179	305	10	121	121	NUM
cana-1179	305	11	-	-	SYM
cana-1179	305	12	126	126	NUM
cana-1179	305	13	.	.	PUNCT
cana-1179	306	1	[	[	X
cana-1179	306	2	19	19	NUM
cana-1179	306	3	]	]	PUNCT
cana-1179	306	4	j.	j.	PROPN
cana-1179	306	5	k.	k.	PROPN
cana-1179	306	6	park	park	PROPN
cana-1179	306	7	,	,	PUNCT
cana-1179	306	8	mildly	mildly	ADV
cana-1179	306	9	generalized	generalize	VERB
cana-1179	306	10	closed	closed	ADJ
cana-1179	306	11	sets	set	NOUN
cana-1179	306	12	,	,	PUNCT
cana-1179	306	13	almost	almost	ADV
cana-1179	306	14	normal	normal	ADJ
cana-1179	306	15	and	and	CCONJ
cana-1179	306	16	mildly	mildly	ADV
cana-1179	306	17	normal	normal	ADJ
cana-1179	306	18	spaces	space	NOUN
cana-1179	306	19	,	,	PUNCT
cana-1179	306	20	chaos	chaos	NOUN
cana-1179	306	21	solutions	solution	NOUN
cana-1179	306	22	and	and	CCONJ
cana-1179	306	23	fractals	fractal	NOUN
cana-1179	306	24	.	.	PUNCT
cana-1179	306	25	,	,	PUNCT
cana-1179	306	26	20-(2004	20-(2004	PROPN
cana-1179	306	27	)	)	PUNCT
cana-1179	306	28	,	,	PUNCT
cana-1179	306	29	1103	1103	NUM
cana-1179	306	30	-	-	SYM
cana-1179	306	31	1111	1111	NUM
cana-1179	306	32	.	.	PUNCT
cana-1179	307	1	[	[	X
cana-1179	307	2	20	20	NUM
cana-1179	307	3	]	]	PUNCT
cana-1179	307	4	m.	m.	NOUN
cana-1179	307	5	k.	k.	PROPN
cana-1179	307	6	r.	r.	PROPN
cana-1179	307	7	s.	s.	PROPN
cana-1179	307	8	veerakumar	veerakumar	PROPN
cana-1179	307	9	,	,	PUNCT
cana-1179	307	10	between	between	ADP
cana-1179	307	11	closed	close	VERB
cana-1179	307	12	sets	set	NOUN
cana-1179	307	13	and	and	CCONJ
cana-1179	307	14	g	g	NOUN
cana-1179	307	15	-	-	PUNCT
cana-1179	307	16	closed	close	VERB
cana-1179	307	17	sets	set	NOUN
cana-1179	307	18	,	,	PUNCT
cana-1179	307	19	mem	mem	PROPN
cana-1179	307	20	.	.	PUNCT
cana-1179	307	21	fac	fac	PROPN
cana-1179	307	22	.	.	PUNCT
cana-1179	308	1	sci	sci	PROPN
cana-1179	308	2	.	.	PROPN
cana-1179	308	3	kochi	kochi	PROPN
cana-1179	308	4	univ	univ	PROPN
cana-1179	308	5	.	.	PUNCT
cana-1179	309	1	ser	ser	PROPN
cana-1179	309	2	.	.	PUNCT
cana-1179	309	3	a.	a.	PROPN
cana-1179	309	4	math	math	PROPN
cana-1179	309	5	,	,	PUNCT
cana-1179	309	6	21(2000),1	21(2000),1	PRON
cana-1179	309	7	-	-	SYM
cana-1179	309	8	19	19	NUM
cana-1179	309	9	.	.	PUNCT