id	sid	tid	token	lemma	pos
cana-1313	1	1	communications	communication	NOUN
cana-1313	1	2	on	on	ADP
cana-1313	1	3	applied	apply	VERB
cana-1313	1	4	nonlinear	nonlinear	ADJ
cana-1313	1	5	analysis	analysis	NOUN
cana-1313	1	6	issn	issn	NOUN
cana-1313	1	7	:	:	PUNCT
cana-1313	1	8	1074	1074	NUM
cana-1313	1	9	-	-	PUNCT
cana-1313	1	10	133x	133x	NUM
cana-1313	1	11	vol	vol	NOUN
cana-1313	1	12	31	31	NUM
cana-1313	1	13	no	no	NOUN
cana-1313	1	14	.	.	PUNCT
cana-1313	2	1	7s	7	NOUN
cana-1313	2	2	(	(	PUNCT
cana-1313	2	3	2024	2024	NUM
cana-1313	2	4	)	)	PUNCT
cana-1313	2	5	349	349	NUM
cana-1313	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1313	2	7	introduction	introduction	NOUN
cana-1313	2	8	to	to	ADP
cana-1313	2	9	rg	rg	VERB
cana-1313	2	10	-	-	PUNCT
cana-1313	2	11	closed	close	VERB
cana-1313	2	12	type	type	NOUN
cana-1313	2	13	sets	set	NOUN
cana-1313	2	14	in	in	ADP
cana-1313	2	15	topological	topological	ADJ
cana-1313	2	16	ordered	order	VERB
cana-1313	2	17	spaces	space	NOUN
cana-1313	2	18	1	1	NUM
cana-1313	2	19	g.	g.	NOUN
cana-1313	2	20	sravani	sravani	PROPN
cana-1313	2	21	,	,	PUNCT
cana-1313	2	22	*	*	PROPN
cana-1313	2	23	2	2	NUM
cana-1313	2	24	g.	g.	PROPN
cana-1313	2	25	srinivasa	srinivasa	PROPN
cana-1313	2	26	rao	rao	PROPN
cana-1313	2	27	1	1	NUM
cana-1313	2	28	research	research	NOUN
cana-1313	2	29	scholar	scholar	NOUN
cana-1313	2	30	,	,	PUNCT
cana-1313	2	31	department	department	NOUN
cana-1313	2	32	of	of	ADP
cana-1313	2	33	mathematics	mathematic	NOUN
cana-1313	2	34	,	,	PUNCT
cana-1313	2	35	school	school	NOUN
cana-1313	2	36	of	of	ADP
cana-1313	2	37	applied	apply	VERB
cana-1313	2	38	science	science	NOUN
cana-1313	2	39	&	&	CCONJ
cana-1313	2	40	humanities	humanities	PROPN
cana-1313	2	41	,	,	PUNCT
cana-1313	2	42	vfstr	vfstr	NOUN
cana-1313	2	43	deemed	deem	VERB
cana-1313	2	44	to	to	PART
cana-1313	2	45	be	be	AUX
cana-1313	2	46	university	university	NOUN
cana-1313	2	47	,	,	PUNCT
cana-1313	2	48	vadlamudi	vadlamudi	NOUN
cana-1313	2	49	,	,	PUNCT
cana-1313	2	50	guntur	guntur	PROPN
cana-1313	2	51	(	(	PUNCT
cana-1313	2	52	dt	dt	PROPN
cana-1313	2	53	.	.	PUNCT
cana-1313	2	54	)	)	PUNCT
cana-1313	2	55	,	,	PUNCT
cana-1313	2	56	andhra	andhra	PROPN
cana-1313	2	57	pradesh	pradesh	PROPN
cana-1313	2	58	,	,	PUNCT
cana-1313	2	59	india	india	PROPN
cana-1313	2	60	.	.	PUNCT
cana-1313	3	1	email:sravanigandikota1998@gmail.com	email:sravanigandikota1998@gmail.com	PROPN
cana-1313	3	2	*	*	SYM
cana-1313	3	3	2	2	NUM
cana-1313	3	4	department	department	NOUN
cana-1313	3	5	of	of	ADP
cana-1313	3	6	mathematics	mathematic	NOUN
cana-1313	3	7	,	,	PUNCT
cana-1313	3	8	school	school	NOUN
cana-1313	3	9	of	of	ADP
cana-1313	3	10	applied	apply	VERB
cana-1313	3	11	science	science	NOUN
cana-1313	3	12	&	&	CCONJ
cana-1313	3	13	humanities	humanities	PROPN
cana-1313	3	14	,	,	PUNCT
cana-1313	3	15	vfstr	vfstr	NOUN
cana-1313	3	16	deemed	deem	VERB
cana-1313	3	17	to	to	PART
cana-1313	3	18	be	be	AUX
cana-1313	3	19	university	university	NOUN
cana-1313	3	20	,	,	PUNCT
cana-1313	3	21	vadlamudi	vadlamudi	NOUN
cana-1313	3	22	,	,	PUNCT
cana-1313	3	23	guntur	guntur	PROPN
cana-1313	3	24	(	(	PUNCT
cana-1313	3	25	dt	dt	PROPN
cana-1313	3	26	.	.	PUNCT
cana-1313	3	27	)	)	PUNCT
cana-1313	3	28	,	,	PUNCT
cana-1313	3	29	andhra	andhra	PROPN
cana-1313	3	30	pradesh	pradesh	PROPN
cana-1313	3	31	,	,	PUNCT
cana-1313	3	32	india	india	PROPN
cana-1313	3	33	.	.	PUNCT
cana-1313	4	1	email:gsrinulakshmi77@gmail.com	email:gsrinulakshmi77@gmail.com	PROPN
cana-1313	4	2	article	article	PROPN
cana-1313	4	3	history	history	NOUN
cana-1313	4	4	:	:	PUNCT
cana-1313	4	5	received	receive	VERB
cana-1313	4	6	:	:	PUNCT
cana-1313	4	7	01	01	NUM
cana-1313	4	8	-	-	PUNCT
cana-1313	4	9	06	06	NUM
cana-1313	4	10	-	-	PUNCT
cana-1313	4	11	2024	2024	NUM
cana-1313	4	12	revised	revise	VERB
cana-1313	4	13	:	:	PUNCT
cana-1313	4	14	03	03	NUM
cana-1313	4	15	-	-	PUNCT
cana-1313	4	16	07	07	NUM
cana-1313	4	17	-	-	PUNCT
cana-1313	4	18	2024	2024	NUM
cana-1313	4	19	accepted	accept	VERB
cana-1313	4	20	:	:	PUNCT
cana-1313	4	21	29	29	NUM
cana-1313	4	22	-	-	SYM
cana-1313	4	23	07	07	NUM
cana-1313	4	24	-	-	PUNCT
cana-1313	4	25	2024	2024	NUM
cana-1313	4	26	abstract	abstract	NOUN
cana-1313	4	27	:	:	PUNCT
cana-1313	4	28	this	this	DET
cana-1313	4	29	research	research	NOUN
cana-1313	4	30	paper	paper	NOUN
cana-1313	4	31	presents	present	VERB
cana-1313	4	32	a	a	DET
cana-1313	4	33	novel	novel	ADJ
cana-1313	4	34	concept	concept	NOUN
cana-1313	4	35	in	in	ADP
cana-1313	4	36	the	the	DET
cana-1313	4	37	field	field	NOUN
cana-1313	4	38	of	of	ADP
cana-1313	4	39	topological	topological	ADJ
cana-1313	4	40	ordered	order	VERB
cana-1313	4	41	spaces	space	NOUN
cana-1313	4	42	,	,	PUNCT
cana-1313	4	43	which	which	PRON
cana-1313	4	44	is	be	AUX
cana-1313	4	45	the	the	DET
cana-1313	4	46	introduction	introduction	NOUN
cana-1313	4	47	of	of	ADP
cana-1313	4	48	a	a	DET
cana-1313	4	49	new	new	ADJ
cana-1313	4	50	class	class	NOUN
cana-1313	4	51	of	of	ADP
cana-1313	4	52	sets	set	NOUN
cana-1313	4	53	called	call	VERB
cana-1313	4	54	"	"	PUNCT
cana-1313	4	55	rg	rg	NOUN
cana-1313	4	56	-	-	PUNCT
cana-1313	4	57	closed	closed	ADJ
cana-1313	4	58	sets	set	NOUN
cana-1313	4	59	"	"	PUNCT
cana-1313	4	60	.	.	PUNCT
cana-1313	5	1	this	this	DET
cana-1313	5	2	new	new	ADJ
cana-1313	5	3	class	class	NOUN
cana-1313	5	4	of	of	ADP
cana-1313	5	5	sets	set	NOUN
cana-1313	5	6	is	be	AUX
cana-1313	5	7	formed	form	VERB
cana-1313	5	8	by	by	ADP
cana-1313	5	9	generalizing	generalize	VERB
cana-1313	5	10	closed	closed	ADJ
cana-1313	5	11	sets	set	NOUN
cana-1313	5	12	using	use	VERB
cana-1313	5	13	rg	rg	NOUN
cana-1313	5	14	-	-	PUNCT
cana-1313	5	15	open	open	ADJ
cana-1313	5	16	sets	set	NOUN
cana-1313	5	17	in	in	ADP
cana-1313	5	18	topological	topological	ADJ
cana-1313	5	19	ordered	order	VERB
cana-1313	5	20	spaces	space	NOUN
cana-1313	5	21	.	.	PUNCT
cana-1313	6	1	notably	notably	ADV
cana-1313	6	2	,	,	PUNCT
cana-1313	6	3	rg	rg	NOUN
cana-1313	6	4	-	-	PUNCT
cana-1313	6	5	closed	close	VERB
cana-1313	6	6	sets	set	NOUN
cana-1313	6	7	strictly	strictly	ADV
cana-1313	6	8	lie	lie	VERB
cana-1313	6	9	between	between	ADP
cana-1313	6	10	the	the	DET
cana-1313	6	11	classes	class	NOUN
cana-1313	6	12	of	of	ADP
cana-1313	6	13	closed	closed	ADJ
cana-1313	6	14	sets	set	NOUN
cana-1313	6	15	and	and	CCONJ
cana-1313	6	16	rg	rg	NOUN
cana-1313	6	17	-	-	PUNCT
cana-1313	6	18	closed	closed	ADJ
cana-1313	6	19	collections	collection	NOUN
cana-1313	6	20	in	in	ADP
cana-1313	6	21	topological	topological	ADJ
cana-1313	6	22	ordered	order	VERB
cana-1313	6	23	spaces	space	NOUN
cana-1313	6	24	.	.	PUNCT
cana-1313	7	1	additionally	additionally	ADV
cana-1313	7	2	,	,	PUNCT
cana-1313	7	3	the	the	DET
cana-1313	7	4	article	article	NOUN
cana-1313	7	5	also	also	ADV
cana-1313	7	6	covers	cover	VERB
cana-1313	7	7	a	a	DET
cana-1313	7	8	discussion	discussion	NOUN
cana-1313	7	9	on	on	ADP
cana-1313	7	10	rg	rg	PROPN
cana-1313	7	11	*	*	ADJ
cana-1313	7	12	closed	closed	ADJ
cana-1313	7	13	sets	set	NOUN
cana-1313	7	14	.	.	PUNCT
cana-1313	8	1	keywords	keyword	NOUN
cana-1313	8	2	:	:	PUNCT
cana-1313	8	3	topological	topological	ADJ
cana-1313	8	4	ordered	order	VERB
cana-1313	8	5	space	space	NOUN
cana-1313	8	6	,	,	PUNCT
cana-1313	8	7	rg	rg	NOUN
cana-1313	8	8	-	-	PUNCT
cana-1313	8	9	closed	closed	ADJ
cana-1313	8	10	set	set	NOUN
cana-1313	8	11	(	(	PUNCT
cana-1313	8	12	irg	irg	PROPN
cana-1313	8	13	,	,	PUNCT
cana-1313	8	14	drg	drg	PROPN
cana-1313	8	15	,	,	PUNCT
cana-1313	8	16	brg	brg	PROPN
cana-1313	8	17	-	-	PUNCT
cana-1313	8	18	closed	close	VERB
cana-1313	8	19	sets	set	NOUN
cana-1313	8	20	)	)	PUNCT
cana-1313	8	21	,	,	PUNCT
cana-1313	8	22	r*g*closed	r*g*close	VERB
cana-1313	8	23	set	set	NOUN
cana-1313	8	24	(	(	PUNCT
cana-1313	8	25	ir*g	ir*g	NOUN
cana-1313	8	26	*	*	NOUN
cana-1313	8	27	,	,	PUNCT
cana-1313	8	28	dr*g	dr*g	PROPN
cana-1313	8	29	*	*	NUM
cana-1313	8	30	,	,	PUNCT
cana-1313	8	31	br*g*-closed	br*g*-close	VERB
cana-1313	8	32	sets	set	NOUN
cana-1313	8	33	)	)	PUNCT
cana-1313	8	34	.	.	PUNCT
cana-1313	9	1	ams	am	NOUN
cana-1313	9	2	classification	classification	NOUN
cana-1313	9	3	:	:	PUNCT
cana-1313	9	4	55xx22	55xx22	NUM
cana-1313	9	5	1	1	NUM
cana-1313	9	6	.	.	PUNCT
cana-1313	10	1	introduction	introduction	NOUN
cana-1313	10	2	the	the	DET
cana-1313	10	3	first	first	ADJ
cana-1313	10	4	research	research	NOUN
cana-1313	10	5	on	on	ADP
cana-1313	10	6	topological	topological	ADJ
cana-1313	10	7	ordered	order	VERB
cana-1313	10	8	spaces	space	NOUN
cana-1313	10	9	was	be	AUX
cana-1313	10	10	conducted	conduct	VERB
cana-1313	10	11	by	by	ADP
cana-1313	10	12	leopoldo	leopoldo	NOUN
cana-1313	10	13	nachbin	nachbin	PROPN
cana-1313	11	1	[	[	X
cana-1313	11	2	1	1	NUM
cana-1313	11	3	]	]	PUNCT
cana-1313	11	4	.	.	PUNCT
cana-1313	12	1	in	in	ADP
cana-1313	12	2	1970	1970	NUM
cana-1313	12	3	,	,	PUNCT
cana-1313	12	4	levine	levine	PROPN
cana-1313	12	5	[	[	X
cana-1313	12	6	19	19	NUM
cana-1313	12	7	]	]	PUNCT
cana-1313	12	8	invented	invent	VERB
cana-1313	12	9	a	a	DET
cana-1313	12	10	superclass	superclass	NOUN
cana-1313	12	11	of	of	ADP
cana-1313	12	12	sets	set	NOUN
cana-1313	12	13	known	know	VERB
cana-1313	12	14	as	as	ADP
cana-1313	12	15	rg	rg	NOUN
cana-1313	12	16	-	-	PUNCT
cana-1313	12	17	closed	close	VERB
cana-1313	12	18	sets	set	NOUN
cana-1313	12	19	.	.	PUNCT
cana-1313	13	1	later	later	ADV
cana-1313	13	2	,	,	PUNCT
cana-1313	13	3	m.	m.	PROPN
cana-1313	13	4	k.	k.	PROPN
cana-1313	13	5	r.	r.	PROPN
cana-1313	13	6	s.	s.	PROPN
cana-1313	13	7	veera	veera	PROPN
cana-1313	13	8	kumar	kumar	PROPN
cana-1313	13	9	introduced	introduce	VERB
cana-1313	13	10	a	a	DET
cana-1313	13	11	novel	novel	ADJ
cana-1313	13	12	category	category	NOUN
cana-1313	13	13	of	of	ADP
cana-1313	13	14	sets	set	NOUN
cana-1313	13	15	[	[	X
cana-1313	13	16	14	14	NUM
cana-1313	13	17	]	]	PUNCT
cana-1313	13	18	,	,	PUNCT
cana-1313	13	19	in	in	ADP
cana-1313	13	20	the	the	DET
cana-1313	13	21	year	year	NOUN
cana-1313	13	22	2014	2014	NUM
cana-1313	13	23	g.	g.	PROPN
cana-1313	13	24	srinivasa	srinivasa	PROPN
cana-1313	13	25	rao	rao	PROPN
cana-1313	13	26	et.al	et.al	PROPN
cana-1313	14	1	[	[	X
cana-1313	14	2	4	4	NUM
cana-1313	14	3	-	-	SYM
cana-1313	14	4	6	6	NUM
cana-1313	14	5	&	&	CCONJ
cana-1313	14	6	19	19	NUM
cana-1313	14	7	-	-	SYM
cana-1313	14	8	25	25	NUM
cana-1313	14	9	]	]	PUNCT
cana-1313	14	10	studied	study	VERB
cana-1313	14	11	and	and	CCONJ
cana-1313	14	12	explained	explain	VERB
cana-1313	14	13	g	g	NOUN
cana-1313	14	14	-	-	PUNCT
cana-1313	14	15	closed	closed	ADJ
cana-1313	14	16	and	and	CCONJ
cana-1313	14	17	g	g	NOUN
cana-1313	14	18	*	*	PUNCT
cana-1313	14	19	-closed	-close	VERB
cana-1313	14	20	sets	set	NOUN
cana-1313	14	21	in	in	ADP
cana-1313	14	22	topological	topological	ADJ
cana-1313	14	23	ordered	order	VERB
cana-1313	14	24	space	space	NOUN
cana-1313	14	25	,	,	PUNCT
cana-1313	14	26	which	which	PRON
cana-1313	14	27	should	should	AUX
cana-1313	14	28	be	be	AUX
cana-1313	14	29	placed	place	VERB
cana-1313	14	30	before	before	ADP
cana-1313	14	31	the	the	DET
cana-1313	14	32	rg	rg	NOUN
cana-1313	14	33	-	-	PUNCT
cana-1313	14	34	closed	closed	ADJ
cana-1313	14	35	sets	set	NOUN
cana-1313	14	36	and	and	CCONJ
cana-1313	14	37	closed	close	VERB
cana-1313	14	38	set	set	ADJ
cana-1313	14	39	classes	class	NOUN
cana-1313	14	40	.	.	PUNCT
cana-1313	15	1	this	this	DET
cana-1313	15	2	new	new	ADJ
cana-1313	15	3	class	class	NOUN
cana-1313	15	4	of	of	ADP
cana-1313	15	5	sets	set	NOUN
cana-1313	15	6	was	be	AUX
cana-1313	15	7	not	not	PART
cana-1313	15	8	only	only	ADV
cana-1313	15	9	different	different	ADJ
cana-1313	15	10	but	but	CCONJ
cana-1313	15	11	also	also	ADV
cana-1313	15	12	significant	significant	ADJ
cana-1313	15	13	.	.	PUNCT
cana-1313	16	1	in	in	ADP
cana-1313	16	2	2001	2001	NUM
cana-1313	16	3	,	,	PUNCT
cana-1313	16	4	m.	m.	PROPN
cana-1313	16	5	k.	k.	PROPN
cana-1313	16	6	r.	r.	PROPN
cana-1313	16	7	s.	s.	PROPN
cana-1313	16	8	veera	veera	PROPN
cana-1313	16	9	kumar	kumar	PROPN
cana-1313	16	10	presented	present	VERB
cana-1313	16	11	research	research	NOUN
cana-1313	16	12	on	on	ADP
cana-1313	16	13	i	i	PROPN
cana-1313	16	14	-	-	PUNCT
cana-1313	16	15	closed	closed	ADJ
cana-1313	16	16	,	,	PUNCT
cana-1313	16	17	d	d	ADJ
cana-1313	16	18	-	-	PUNCT
cana-1313	16	19	closed	closed	ADJ
cana-1313	16	20	,	,	PUNCT
cana-1313	16	21	and	and	CCONJ
cana-1313	16	22	b	b	X
cana-1313	16	23	-	-	PUNCT
cana-1313	16	24	closed	closed	ADJ
cana-1313	16	25	sets	set	NOUN
cana-1313	16	26	,	,	PUNCT
cana-1313	16	27	which	which	PRON
cana-1313	16	28	were	be	AUX
cana-1313	16	29	introduced	introduce	VERB
cana-1313	16	30	for	for	ADP
cana-1313	16	31	the	the	DET
cana-1313	16	32	first	first	ADJ
cana-1313	16	33	time	time	NOUN
cana-1313	16	34	.	.	PUNCT
cana-1313	17	1	a	a	DET
cana-1313	17	2	topological	topological	ADJ
cana-1313	17	3	ordered	order	VERB
cana-1313	17	4	space	space	NOUN
cana-1313	17	5	is	be	AUX
cana-1313	17	6	referred	refer	VERB
cana-1313	17	7	to	to	ADP
cana-1313	17	8	as	as	ADP
cana-1313	17	9	a	a	DET
cana-1313	17	10	triple	triple	ADJ
cana-1313	17	11	(	(	PUNCT
cana-1313	17	12	)	)	PUNCT
cana-1313	17	13	where	where	SCONJ
cana-1313	17	14	x	x	PRON
cana-1313	17	15	is	be	AUX
cana-1313	17	16	a	a	DET
cana-1313	17	17	non	non	ADJ
cana-1313	17	18	-	-	ADJ
cana-1313	17	19	empty	empty	ADJ
cana-1313	17	20	set	set	NOUN
cana-1313	17	21	,	,	PUNCT
cana-1313	17	22	τ	τ	PROPN
cana-1313	17	23	is	be	AUX
cana-1313	17	24	a	a	DET
cana-1313	17	25	topology	topology	NOUN
cana-1313	17	26	on	on	ADP
cana-1313	17	27	x	x	PUNCT
cana-1313	17	28	and	and	CCONJ
cana-1313	17	29	is	be	AUX
cana-1313	17	30	a	a	DET
cana-1313	17	31	partial	partial	ADJ
cana-1313	17	32	order	order	NOUN
cana-1313	17	33	on	on	ADP
cana-1313	17	34	x.	x.	NOUN
cana-1313	17	35	definition	definition	NOUN
cana-1313	17	36	1.1[5	1.1[5	NUM
cana-1313	17	37	]	]	X
cana-1313	17	38	:	:	PUNCT
cana-1313	17	39	for	for	ADP
cana-1313	17	40	any	any	PRON
cana-1313	17	41	*	*	X
cana-1313	17	42	⁄	⁄	PROPN
cana-1313	17	43	+	+	CCONJ
cana-1313	17	44	will	will	AUX
cana-1313	17	45	be	be	AUX
cana-1313	17	46	represented	represent	VERB
cana-1313	17	47	by	by	ADP
cana-1313	17	48	,	,	PUNCT
cana-1313	17	49	-	-	PUNCT
cana-1313	17	50	.	.	PUNCT
cana-1313	18	1	if	if	SCONJ
cana-1313	18	2	p	p	NOUN
cana-1313	18	3	=	=	NOUN
cana-1313	18	4	i(p	i(p	NOUN
cana-1313	18	5	)	)	PUNCT
cana-1313	18	6	,	,	PUNCT
cana-1313	18	7	where	where	SCONJ
cana-1313	18	8	i(p	i(p	NOUN
cana-1313	18	9	)	)	PUNCT
cana-1313	18	10	=	=	SYM
cana-1313	18	11	⋃	⋃	NOUN
cana-1313	18	12	,	,	PUNCT
cana-1313	18	13	,	,	PUNCT
cana-1313	18	14	then	then	ADV
cana-1313	18	15	a	a	DET
cana-1313	18	16	subset	subset	NOUN
cana-1313	18	17	p	p	NOUN
cana-1313	18	18	of	of	ADP
cana-1313	18	19	a	a	DET
cana-1313	18	20	topological	topological	ADJ
cana-1313	18	21	ordered	order	VERB
cana-1313	18	22	space	space	NOUN
cana-1313	18	23	(	(	PUNCT
cana-1313	18	24	)	)	PUNCT
cana-1313	18	25	is	be	AUX
cana-1313	18	26	known	know	VERB
cana-1313	18	27	to	to	PART
cana-1313	18	28	be	be	AUX
cana-1313	18	29	increasing	increase	VERB
cana-1313	18	30	.	.	PUNCT
cana-1313	19	1	definition	definition	NOUN
cana-1313	19	2	1.2[5	1.2[5	NUM
cana-1313	19	3	]	]	PUNCT
cana-1313	19	4	:	:	PUNCT
cana-1313	19	5	for	for	ADP
cana-1313	19	6	any	any	PRON
cana-1313	19	7	*	*	X
cana-1313	19	8	⁄	⁄	PROPN
cana-1313	19	9	+	+	CCONJ
cana-1313	19	10	will	will	AUX
cana-1313	19	11	be	be	AUX
cana-1313	19	12	represented	represent	VERB
cana-1313	19	13	by	by	ADP
cana-1313	19	14	,	,	PUNCT
cana-1313	19	15	-	-	PUNCT
cana-1313	19	16	.	.	PUNCT
cana-1313	20	1	if	if	SCONJ
cana-1313	20	2	p	p	NOUN
cana-1313	20	3	=	=	NOUN
cana-1313	20	4	d(p	d(p	PROPN
cana-1313	20	5	)	)	PUNCT
cana-1313	20	6	,	,	PUNCT
cana-1313	20	7	where	where	SCONJ
cana-1313	20	8	d(p	d(p	PROPN
cana-1313	20	9	)	)	PUNCT
cana-1313	21	1	=	=	PUNCT
cana-1313	21	2	⋃	⋃	PROPN
cana-1313	21	3	,	,	PUNCT
cana-1313	21	4	,	,	PUNCT
cana-1313	21	5	then	then	ADV
cana-1313	21	6	a	a	DET
cana-1313	21	7	subset	subset	NOUN
cana-1313	21	8	p	p	NOUN
cana-1313	21	9	of	of	ADP
cana-1313	21	10	a	a	DET
cana-1313	21	11	topological	topological	ADJ
cana-1313	21	12	ordered	order	VERB
cana-1313	21	13	space	space	NOUN
cana-1313	21	14	(	(	PUNCT
cana-1313	21	15	)	)	PUNCT
cana-1313	21	16	is	be	AUX
cana-1313	21	17	known	know	VERB
cana-1313	21	18	to	to	PART
cana-1313	21	19	be	be	AUX
cana-1313	21	20	decreasing	decrease	VERB
cana-1313	21	21	.	.	PUNCT
cana-1313	22	1	an	an	DET
cana-1313	22	2	increasing	increase	VERB
cana-1313	22	3	(	(	PUNCT
cana-1313	22	4	resp	resp	NOUN
cana-1313	22	5	.	.	PUNCT
cana-1313	23	1	a	a	DET
cana-1313	23	2	decreasing	decrease	VERB
cana-1313	23	3	)	)	PUNCT
cana-1313	23	4	set	set	NOUN
cana-1313	23	5	is	be	AUX
cana-1313	23	6	the	the	DET
cana-1313	23	7	complement	complement	NOUN
cana-1313	23	8	of	of	ADP
cana-1313	23	9	a	a	DET
cana-1313	23	10	decreasing	decrease	VERB
cana-1313	23	11	(	(	PUNCT
cana-1313	23	12	resp	resp	NOUN
cana-1313	23	13	.	.	PUNCT
cana-1313	24	1	an	an	DET
cana-1313	24	2	increasing	increase	VERB
cana-1313	24	3	)	)	PUNCT
cana-1313	24	4	set	set	NOUN
cana-1313	24	5	.	.	PUNCT
cana-1313	25	1	c(p	c(p	NOUN
cana-1313	25	2	)	)	PUNCT
cana-1313	25	3	denotes	denote	VERB
cana-1313	25	4	the	the	DET
cana-1313	25	5	complement	complement	NOUN
cana-1313	25	6	of	of	ADP
cana-1313	25	7	„	„	PUNCT
cana-1313	25	8	p‟	p‟	NUM
cana-1313	25	9	in	in	ADP
cana-1313	25	10	x.	x.	PROPN
cana-1313	25	11	dcl	dcl	PROPN
cana-1313	25	12	(	(	PUNCT
cana-1313	25	13	p	p	NOUN
cana-1313	25	14	)	)	PUNCT
cana-1313	25	15	=	=	SYM
cana-1313	26	1	{	{	PUNCT
cana-1313	26	2	f	f	NOUN
cana-1313	26	3	f	f	PROPN
cana-1313	26	4	is	be	AUX
cana-1313	26	5	a	a	DET
cana-1313	26	6	decreasing	decrease	VERB
cana-1313	26	7	closed	closed	ADJ
cana-1313	26	8	subset	subset	NOUN
cana-1313	26	9	of	of	ADP
cana-1313	26	10	x	x	PUNCT
cana-1313	26	11	containing	contain	VERB
cana-1313	26	12	p	p	NOUN
cana-1313	26	13	with	with	ADP
cana-1313	26	14	f	f	NOUN
cana-1313	26	15	=	=	PUNCT
cana-1313	26	16	d(f	d(f	NOUN
cana-1313	26	17	)	)	PUNCT
cana-1313	26	18	}	}	PUNCT
cana-1313	26	19	.	.	PUNCT
cana-1313	27	1	icl	icl	PROPN
cana-1313	27	2	(	(	PUNCT
cana-1313	27	3	p	p	X
cana-1313	27	4	)	)	PUNCT
cana-1313	27	5	=	=	SYM
cana-1313	28	1	{	{	PUNCT
cana-1313	28	2	f	f	NOUN
cana-1313	28	3	f	f	PROPN
cana-1313	28	4	is	be	AUX
cana-1313	28	5	an	an	DET
cana-1313	28	6	increasing	increase	VERB
cana-1313	28	7	closed	closed	ADJ
cana-1313	28	8	subset	subset	NOUN
cana-1313	28	9	of	of	ADP
cana-1313	28	10	x	x	PUNCT
cana-1313	28	11	containing	contain	VERB
cana-1313	28	12	p	p	NOUN
cana-1313	28	13	with	with	ADP
cana-1313	28	14	f	f	PROPN
cana-1313	28	15	=	=	SYM
cana-1313	28	16	i(f	i(f	PROPN
cana-1313	28	17	)	)	PUNCT
cana-1313	28	18	}	}	PUNCT
cana-1313	28	19	.	.	PUNCT
cana-1313	29	1	bcl	bcl	NOUN
cana-1313	29	2	(	(	PUNCT
cana-1313	29	3	p	p	NOUN
cana-1313	29	4	)	)	PUNCT
cana-1313	29	5	=	=	SYM
cana-1313	30	1	{	{	PUNCT
cana-1313	30	2	f	f	NOUN
cana-1313	30	3	f	f	PROPN
cana-1313	30	4	is	be	AUX
cana-1313	30	5	a	a	DET
cana-1313	30	6	closed	closed	ADJ
cana-1313	30	7	subset	subset	NOUN
cana-1313	30	8	of	of	ADP
cana-1313	30	9	x	x	PUNCT
cana-1313	30	10	containing	contain	VERB
cana-1313	30	11	p	p	NOUN
cana-1313	30	12	with	with	ADP
cana-1313	30	13	f	f	PROPN
cana-1313	30	14	=	=	SYM
cana-1313	30	15	i(f)=d(f	i(f)=d(f	PROPN
cana-1313	30	16	)	)	PUNCT
cana-1313	30	17	}	}	PUNCT
cana-1313	30	18	.	.	PUNCT
cana-1313	31	1	io(x	io(x	PUNCT
cana-1313	31	2	)	)	PUNCT
cana-1313	31	3	(	(	PUNCT
cana-1313	31	4	resp	resp	NOUN
cana-1313	31	5	.	.	PUNCT
cana-1313	31	6	do(x	do(x	NUM
cana-1313	31	7	)	)	PUNCT
cana-1313	31	8	,	,	PUNCT
cana-1313	31	9	bo(x	bo(x	NUM
cana-1313	31	10	)	)	PUNCT
cana-1313	31	11	)	)	PUNCT
cana-1313	31	12	represents	represent	VERB
cana-1313	31	13	the	the	DET
cana-1313	31	14	set	set	NOUN
cana-1313	31	15	of	of	ADP
cana-1313	31	16	all	all	DET
cana-1313	31	17	decreasing	decrease	VERB
cana-1313	31	18	(	(	PUNCT
cana-1313	31	19	or	or	CCONJ
cana-1313	31	20	increasing	increase	VERB
cana-1313	31	21	,	,	PUNCT
cana-1313	31	22	both	both	PRON
cana-1313	31	23	increasing	increase	VERB
cana-1313	31	24	and	and	CCONJ
cana-1313	31	25	decreasing	decrease	VERB
cana-1313	31	26	)	)	PUNCT
cana-1313	31	27	open	open	ADJ
cana-1313	31	28	subsets	subset	NOUN
cana-1313	31	29	of	of	ADP
cana-1313	31	30	a	a	DET
cana-1313	31	31	topological	topological	ADJ
cana-1313	31	32	ordered	order	VERB
cana-1313	31	33	space	space	NOUN
cana-1313	31	34	(	(	PUNCT
cana-1313	31	35	)	)	PUNCT
cana-1313	31	36	.	.	PUNCT
cana-1313	32	1	communications	communication	NOUN
cana-1313	32	2	on	on	ADP
cana-1313	32	3	applied	apply	VERB
cana-1313	32	4	nonlinear	nonlinear	ADJ
cana-1313	32	5	analysis	analysis	NOUN
cana-1313	32	6	issn	issn	NOUN
cana-1313	32	7	:	:	PUNCT
cana-1313	32	8	1074	1074	NUM
cana-1313	32	9	-	-	PUNCT
cana-1313	32	10	133x	133x	NUM
cana-1313	32	11	vol	vol	NOUN
cana-1313	32	12	31	31	NUM
cana-1313	32	13	no	no	NOUN
cana-1313	32	14	.	.	PUNCT
cana-1313	33	1	7s	7	NOUN
cana-1313	33	2	(	(	PUNCT
cana-1313	33	3	2024	2024	NUM
cana-1313	33	4	)	)	PUNCT
cana-1313	33	5	350	350	NUM
cana-1313	33	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1313	33	7	for	for	ADP
cana-1313	33	8	a	a	DET
cana-1313	33	9	subset	subset	NOUN
cana-1313	33	10	p	p	NOUN
cana-1313	33	11	of	of	ADP
cana-1313	33	12	a	a	DET
cana-1313	33	13	space	space	NOUN
cana-1313	33	14	(	(	PUNCT
cana-1313	33	15	)	)	PUNCT
cana-1313	33	16	,	,	PUNCT
cana-1313	33	17	cl(p	cl(p	NOUN
cana-1313	33	18	)	)	PUNCT
cana-1313	33	19	(	(	PUNCT
cana-1313	33	20	resp	resp	NOUN
cana-1313	33	21	.	.	PUNCT
cana-1313	33	22	icl(p	icl(p	PROPN
cana-1313	33	23	)	)	PUNCT
cana-1313	33	24	,	,	PUNCT
cana-1313	33	25	bcl(p	bcl(p	NOUN
cana-1313	33	26	)	)	PUNCT
cana-1313	33	27	)	)	PUNCT
cana-1313	33	28	denote	denote	VERB
cana-1313	33	29	the	the	DET
cana-1313	33	30	decreasing	decrease	VERB
cana-1313	33	31	(	(	PUNCT
cana-1313	33	32	resp	resp	NOUN
cana-1313	33	33	.	.	PUNCT
cana-1313	34	1	increasing	increase	VERB
cana-1313	34	2	,	,	PUNCT
cana-1313	34	3	both	both	PRON
cana-1313	34	4	increasing	increase	VERB
cana-1313	34	5	and	and	CCONJ
cana-1313	34	6	decreasing	decrease	VERB
cana-1313	34	7	)	)	PUNCT
cana-1313	34	8	closure	closure	NOUN
cana-1313	34	9	of	of	ADP
cana-1313	34	10	p.	p.	NOUN
cana-1313	34	11	2	2	NUM
cana-1313	34	12	.	.	PUNCT
cana-1313	35	1	topological	topological	ADJ
cana-1313	35	2	ordered	order	VERB
cana-1313	35	3	space	space	NOUN
cana-1313	35	4	with	with	ADP
cana-1313	35	5	rg	rg	NOUN
cana-1313	35	6	-	-	PUNCT
cana-1313	35	7	closed	closed	ADJ
cana-1313	35	8	sets	set	NOUN
cana-1313	35	9	definition	definition	NOUN
cana-1313	35	10	2.1	2.1	NUM
cana-1313	35	11	:	:	PUNCT
cana-1313	35	12	a	a	DET
cana-1313	35	13	topological	topological	ADJ
cana-1313	35	14	space	space	NOUN
cana-1313	35	15	(	(	PUNCT
cana-1313	35	16	)	)	PUNCT
cana-1313	35	17	has	have	VERB
cana-1313	35	18	a	a	DET
cana-1313	35	19	subset	subset	NOUN
cana-1313	35	20	p	p	X
cana-1313	35	21	is	be	AUX
cana-1313	35	22	known	know	VERB
cana-1313	35	23	as	as	ADP
cana-1313	35	24	rg	rg	NOUN
cana-1313	35	25	-	-	PUNCT
cana-1313	35	26	closed	closed	ADJ
cana-1313	35	27	[	[	X
cana-1313	35	28	29	29	NUM
cana-1313	35	29	]	]	PUNCT
cana-1313	35	30	set	set	NOUN
cana-1313	35	31	,	,	PUNCT
cana-1313	35	32	if	if	SCONJ
cana-1313	35	33	cl(p	cl(p	PUNCT
cana-1313	35	34	)	)	PUNCT
cana-1313	35	35	⊆	⊆	NUM
cana-1313	35	36	r	r	NOUN
cana-1313	35	37	whenever	whenever	SCONJ
cana-1313	35	38	p	p	NOUN
cana-1313	35	39	⊆	⊆	NUM
cana-1313	35	40	r	r	NOUN
cana-1313	35	41	and	and	CCONJ
cana-1313	35	42	r	r	NOUN
cana-1313	35	43	is	be	AUX
cana-1313	35	44	regular	regular	ADV
cana-1313	35	45	open	open	ADJ
cana-1313	35	46	in	in	ADP
cana-1313	35	47	(	(	PUNCT
cana-1313	35	48	)	)	PUNCT
cana-1313	35	49	.	.	PUNCT
cana-1313	36	1	definition	definition	NOUN
cana-1313	36	2	2.2	2.2	NUM
cana-1313	36	3	:	:	PUNCT
cana-1313	36	4	a	a	DET
cana-1313	36	5	topological	topological	ADJ
cana-1313	36	6	space	space	NOUN
cana-1313	36	7	(	(	PUNCT
cana-1313	36	8	)	)	PUNCT
cana-1313	36	9	has	have	VERB
cana-1313	36	10	a	a	DET
cana-1313	36	11	subset	subset	NOUN
cana-1313	36	12	p	p	NOUN
cana-1313	36	13	is	be	AUX
cana-1313	36	14	called	call	VERB
cana-1313	36	15	r*g*-closed	r*g*-close	VERB
cana-1313	36	16	set	set	NOUN
cana-1313	36	17	[	[	X
cana-1313	36	18	29	29	NUM
cana-1313	36	19	]	]	PUNCT
cana-1313	36	20	,	,	PUNCT
cana-1313	36	21	if	if	SCONJ
cana-1313	36	22	rcl(p	rcl(p	ADJ
cana-1313	36	23	)	)	PUNCT
cana-1313	36	24	⊆	⊆	NUM
cana-1313	36	25	r	r	NOUN
cana-1313	36	26	whenever	whenever	SCONJ
cana-1313	36	27	p	p	NOUN
cana-1313	36	28	⊆	⊆	NUM
cana-1313	36	29	r	r	NOUN
cana-1313	36	30	and	and	CCONJ
cana-1313	36	31	r	r	NOUN
cana-1313	36	32	is	be	AUX
cana-1313	36	33	g	g	NOUN
cana-1313	36	34	-	-	PUNCT
cana-1313	36	35	open	open	ADJ
cana-1313	36	36	in	in	ADP
cana-1313	36	37	(	(	PUNCT
cana-1313	36	38	)	)	PUNCT
cana-1313	36	39	.	.	PUNCT
cana-1313	37	1	theorem	theorem	VERB
cana-1313	37	2	2.3	2.3	NUM
cana-1313	37	3	:	:	PUNCT
cana-1313	37	4	every	every	DET
cana-1313	37	5	r*g*-closed	r*g*-close	VERB
cana-1313	37	6	set	set	NOUN
cana-1313	37	7	is	be	AUX
cana-1313	37	8	a	a	DET
cana-1313	37	9	rg	rg	NOUN
cana-1313	37	10	-	-	PUNCT
cana-1313	37	11	closed	closed	ADJ
cana-1313	37	12	set	set	NOUN
cana-1313	37	13	.	.	PUNCT
cana-1313	38	1	proof	proof	NOUN
cana-1313	38	2	:	:	PUNCT
cana-1313	38	3	suppose	suppose	VERB
cana-1313	38	4	p	p	ADP
cana-1313	38	5	⊆	⊆	NUM
cana-1313	38	6	r	r	NOUN
cana-1313	38	7	and	and	CCONJ
cana-1313	38	8	r	r	NOUN
cana-1313	38	9	is	be	AUX
cana-1313	38	10	regular	regular	ADJ
cana-1313	38	11	open	open	ADJ
cana-1313	38	12	.	.	PUNCT
cana-1313	39	1	now	now	ADV
cana-1313	39	2	,	,	PUNCT
cana-1313	39	3	r	r	NOUN
cana-1313	39	4	is	be	AUX
cana-1313	39	5	regular	regular	ADJ
cana-1313	39	6	open	open	ADJ
cana-1313	39	7	r	r	NOUN
cana-1313	39	8	is	be	AUX
cana-1313	39	9	open	open	ADJ
cana-1313	39	10	.	.	PUNCT
cana-1313	40	1	w.	w.	PROPN
cana-1313	40	2	k.	k.	PROPN
cana-1313	40	3	t	t	PROPN
cana-1313	41	1	every	every	DET
cana-1313	41	2	closed	closed	ADJ
cana-1313	41	3	set	set	NOUN
cana-1313	41	4	is	be	AUX
cana-1313	41	5	a	a	DET
cana-1313	41	6	g	g	NOUN
cana-1313	41	7	-	-	PUNCT
cana-1313	41	8	closed	close	VERB
cana-1313	41	9	set	set	NOUN
cana-1313	41	10	.	.	PUNCT
cana-1313	42	1	so	so	ADV
cana-1313	42	2	,	,	PUNCT
cana-1313	42	3	every	every	DET
cana-1313	42	4	open	open	ADJ
cana-1313	42	5	set	set	NOUN
cana-1313	42	6	is	be	AUX
cana-1313	42	7	a	a	DET
cana-1313	42	8	g	g	NOUN
cana-1313	42	9	-	-	PUNCT
cana-1313	42	10	open	open	ADJ
cana-1313	42	11	set	set	NOUN
cana-1313	42	12	.	.	PUNCT
cana-1313	43	1	since	since	ADV
cana-1313	43	2	,	,	PUNCT
cana-1313	43	3	r	r	NOUN
cana-1313	43	4	is	be	AUX
cana-1313	43	5	open	open	ADJ
cana-1313	43	6	we	we	PRON
cana-1313	43	7	have	have	VERB
cana-1313	43	8	r	r	NOUN
cana-1313	43	9	is	be	AUX
cana-1313	43	10	g	g	NOUN
cana-1313	43	11	-	-	PUNCT
cana-1313	43	12	open	open	ADJ
cana-1313	43	13	.	.	PUNCT
cana-1313	44	1	therefore	therefore	ADV
cana-1313	44	2	,	,	PUNCT
cana-1313	44	3	rcl(p	rcl(p	NOUN
cana-1313	44	4	)	)	PUNCT
cana-1313	44	5	⊆	⊆	NUM
cana-1313	44	6	r	r	NOUN
cana-1313	44	7	,	,	PUNCT
cana-1313	44	8	whenever	whenever	SCONJ
cana-1313	44	9	p	p	NOUN
cana-1313	44	10	⊆	⊆	NUM
cana-1313	44	11	r	r	NOUN
cana-1313	44	12	and	and	CCONJ
cana-1313	44	13	r	r	NOUN
cana-1313	44	14	is	be	AUX
cana-1313	44	15	g	g	NOUN
cana-1313	44	16	-	-	PUNCT
cana-1313	44	17	open	open	ADJ
cana-1313	44	18	.	.	PUNCT
cana-1313	45	1	since	since	SCONJ
cana-1313	45	2	p	p	NOUN
cana-1313	45	3	⊆	⊆	NUM
cana-1313	45	4	r	r	NOUN
cana-1313	45	5	,	,	PUNCT
cana-1313	45	6	there	there	PRON
cana-1313	45	7	exist	exist	VERB
cana-1313	45	8	an	an	DET
cana-1313	45	9	open	open	ADJ
cana-1313	45	10	set	set	NOUN
cana-1313	45	11	g	g	NOUN
cana-1313	45	12	we	we	PRON
cana-1313	45	13	have	have	VERB
cana-1313	45	14	p	p	NOUN
cana-1313	45	15	⊆	⊆	NUM
cana-1313	45	16	g	g	NOUN
cana-1313	45	17	⊆	⊆	NUM
cana-1313	45	18	cl(g	cl(g	NOUN
cana-1313	45	19	)	)	PUNCT
cana-1313	45	20	⊆	⊆	NUM
cana-1313	45	21	r.	r.	NOUN
cana-1313	45	22	since	since	SCONJ
cana-1313	45	23	p	p	PROPN
cana-1313	45	24	⊆	⊆	NUM
cana-1313	45	25	cl(g	cl(g	NOUN
cana-1313	45	26	)	)	PUNCT
cana-1313	45	27	cl(p	cl(p	NOUN
cana-1313	45	28	)	)	PUNCT
cana-1313	45	29	⊆	⊆	NUM
cana-1313	45	30	cl(cl(g	cl(cl(g	NOUN
cana-1313	45	31	)	)	PUNCT
cana-1313	45	32	)	)	PUNCT
cana-1313	45	33	=	=	SYM
cana-1313	46	1	cl(g	cl(g	X
cana-1313	46	2	)	)	PUNCT
cana-1313	46	3	⊆	⊆	NUM
cana-1313	46	4	r.	r.	NOUN
cana-1313	46	5	cl(p	cl(p	NOUN
cana-1313	46	6	)	)	PUNCT
cana-1313	46	7	⊆	⊆	X
cana-1313	46	8	r.	r.	PROPN
cana-1313	46	9	therefore	therefore	ADV
cana-1313	46	10	cl	cl	PROPN
cana-1313	46	11	(	(	PUNCT
cana-1313	46	12	p	p	NOUN
cana-1313	46	13	)	)	PUNCT
cana-1313	46	14	⊆	⊆	NUM
cana-1313	46	15	r	r	NOUN
cana-1313	46	16	,	,	PUNCT
cana-1313	46	17	whenever	whenever	SCONJ
cana-1313	46	18	p	p	NOUN
cana-1313	46	19	⊆	⊆	NUM
cana-1313	46	20	r	r	NOUN
cana-1313	46	21	and	and	CCONJ
cana-1313	46	22	r	r	NOUN
cana-1313	46	23	is	be	AUX
cana-1313	46	24	regular	regular	ADJ
cana-1313	46	25	open	open	ADJ
cana-1313	46	26	.	.	PUNCT
cana-1313	47	1	the	the	DET
cana-1313	47	2	following	follow	VERB
cana-1313	47	3	illustration	illustration	NOUN
cana-1313	47	4	demonstrates	demonstrate	VERB
cana-1313	47	5	that	that	SCONJ
cana-1313	47	6	a	a	DET
cana-1313	47	7	rg	rg	NOUN
cana-1313	47	8	-	-	PUNCT
cana-1313	47	9	closed	closed	ADJ
cana-1313	47	10	set	set	NOUN
cana-1313	47	11	does	do	AUX
cana-1313	47	12	not	not	PART
cana-1313	47	13	always	always	ADV
cana-1313	47	14	have	have	VERB
cana-1313	47	15	to	to	PART
cana-1313	47	16	be	be	AUX
cana-1313	47	17	a	a	DET
cana-1313	47	18	r*g*-closed	r*g*-close	VERB
cana-1313	47	19	set	set	NOUN
cana-1313	47	20	.	.	PUNCT
cana-1313	47	21	example	example	NOUN
cana-1313	47	22	2.4	2.4	NUM
cana-1313	47	23	:	:	PUNCT
cana-1313	47	24	let	let	VERB
cana-1313	47	25	x	x	PUNCT
cana-1313	47	26	=	=	PUNCT
cana-1313	47	27	{	{	PUNCT
cana-1313	47	28	p	p	X
cana-1313	47	29	,	,	PUNCT
cana-1313	47	30	q	q	ADJ
cana-1313	47	31	,	,	PUNCT
cana-1313	47	32	r	r	NOUN
cana-1313	47	33	}	}	PUNCT
cana-1313	47	34	,	,	PUNCT
cana-1313	47	35	=	=	PRON
cana-1313	47	36	{	{	PUNCT
cana-1313	47	37	,	,	PUNCT
cana-1313	47	38	x	x	X
cana-1313	47	39	,	,	PUNCT
cana-1313	47	40	{	{	PUNCT
cana-1313	47	41	p},{q},{p	p},{q},{p	ADV
cana-1313	47	42	,	,	PUNCT
cana-1313	47	43	q	q	NOUN
cana-1313	47	44	}	}	PUNCT
cana-1313	47	45	}	}	PUNCT
cana-1313	47	46	and	and	CCONJ
cana-1313	47	47	=	=	PRON
cana-1313	47	48	{	{	PUNCT
cana-1313	47	49	(	(	PUNCT
cana-1313	47	50	p	p	X
cana-1313	47	51	,	,	PUNCT
cana-1313	47	52	p	p	NOUN
cana-1313	47	53	)	)	PUNCT
cana-1313	47	54	,	,	PUNCT
cana-1313	47	55	(	(	PUNCT
cana-1313	47	56	q	q	INTJ
cana-1313	47	57	,	,	PUNCT
cana-1313	47	58	q),(r	q),(r	NOUN
cana-1313	47	59	,	,	PUNCT
cana-1313	47	60	r),(p	r),(p	NOUN
cana-1313	47	61	,	,	PUNCT
cana-1313	47	62	q),(q	q),(q	NOUN
cana-1313	47	63	,	,	PUNCT
cana-1313	47	64	r	r	NOUN
cana-1313	47	65	)	)	PUNCT
cana-1313	47	66	,	,	PUNCT
cana-1313	47	67	(	(	PUNCT
cana-1313	47	68	p	p	X
cana-1313	47	69	,	,	PUNCT
cana-1313	47	70	r	r	NOUN
cana-1313	47	71	)	)	PUNCT
cana-1313	47	72	}	}	PUNCT
cana-1313	47	73	.	.	PUNCT
cana-1313	48	1	clearly	clearly	ADV
cana-1313	48	2	,	,	PUNCT
cana-1313	48	3	a	a	DET
cana-1313	48	4	topological	topological	ADJ
cana-1313	48	5	ordered	order	VERB
cana-1313	48	6	space	space	NOUN
cana-1313	48	7	is	be	AUX
cana-1313	48	8	(	(	PUNCT
cana-1313	48	9	)	)	PUNCT
cana-1313	48	10	.	.	PUNCT
cana-1313	49	1	r*g*-closed	r*g*-close	VERB
cana-1313	49	2	sets	set	NOUN
cana-1313	49	3	are	be	AUX
cana-1313	49	4	,	,	PUNCT
cana-1313	49	5	x	x	X
cana-1313	49	6	,	,	PUNCT
cana-1313	49	7	{	{	PUNCT
cana-1313	49	8	r	r	NOUN
cana-1313	49	9	}	}	PUNCT
cana-1313	49	10	,	,	PUNCT
cana-1313	49	11	{	{	PUNCT
cana-1313	49	12	q	q	X
cana-1313	49	13	,	,	PUNCT
cana-1313	49	14	r	r	NOUN
cana-1313	49	15	}	}	PUNCT
cana-1313	49	16	,	,	PUNCT
cana-1313	49	17	{	{	PUNCT
cana-1313	49	18	p	p	X
cana-1313	49	19	,	,	PUNCT
cana-1313	49	20	r	r	NOUN
cana-1313	49	21	}	}	PUNCT
cana-1313	49	22	.	.	PUNCT
cana-1313	50	1	rg	rg	NOUN
cana-1313	50	2	-	-	PUNCT
cana-1313	50	3	closed	close	VERB
cana-1313	50	4	sets	set	NOUN
cana-1313	50	5	are	be	AUX
cana-1313	50	6	,	,	PUNCT
cana-1313	50	7	x	x	X
cana-1313	50	8	,	,	PUNCT
cana-1313	50	9	{	{	PUNCT
cana-1313	50	10	r	r	NOUN
cana-1313	50	11	}	}	PUNCT
cana-1313	50	12	,	,	PUNCT
cana-1313	50	13	{	{	PUNCT
cana-1313	50	14	p	p	X
cana-1313	50	15	,	,	PUNCT
cana-1313	50	16	q	q	NOUN
cana-1313	50	17	}	}	PUNCT
cana-1313	50	18	,	,	PUNCT
cana-1313	50	19	{	{	PUNCT
cana-1313	50	20	q	q	X
cana-1313	50	21	,	,	PUNCT
cana-1313	50	22	r	r	NOUN
cana-1313	50	23	}	}	PUNCT
cana-1313	50	24	,	,	PUNCT
cana-1313	50	25	{	{	PUNCT
cana-1313	50	26	p	p	X
cana-1313	50	27	,	,	PUNCT
cana-1313	50	28	r	r	NOUN
cana-1313	50	29	}	}	PUNCT
cana-1313	50	30	.	.	PUNCT
cana-1313	51	1	let	let	VERB
cana-1313	51	2	p	p	NOUN
cana-1313	51	3	=	=	X
cana-1313	51	4	{	{	PUNCT
cana-1313	51	5	p	p	X
cana-1313	51	6	,	,	PUNCT
cana-1313	51	7	q	q	NOUN
cana-1313	51	8	}	}	PUNCT
cana-1313	51	9	.	.	PUNCT
cana-1313	52	1	clearly	clearly	ADV
cana-1313	52	2	,	,	PUNCT
cana-1313	52	3	a	a	PRON
cana-1313	52	4	is	be	AUX
cana-1313	52	5	rg	rg	NOUN
cana-1313	52	6	-	-	PUNCT
cana-1313	52	7	closed	closed	ADJ
cana-1313	52	8	set	set	NOUN
cana-1313	52	9	but	but	CCONJ
cana-1313	52	10	not	not	PART
cana-1313	52	11	r*g*-closed	r*g*-close	VERB
cana-1313	52	12	set	set	NOUN
cana-1313	52	13	.	.	PUNCT
cana-1313	53	1	3	3	X
cana-1313	53	2	.	.	X
cana-1313	53	3	results	result	NOUN
cana-1313	53	4	between	between	ADP
cana-1313	53	5	i(r*g	i(r*g	NOUN
cana-1313	53	6	*	*	NUM
cana-1313	53	7	)	)	PUNCT
cana-1313	53	8	,	,	PUNCT
cana-1313	53	9	d(r*g	d(r*g	NOUN
cana-1313	53	10	*	*	PUNCT
cana-1313	53	11	)	)	PUNCT
cana-1313	53	12	and	and	CCONJ
cana-1313	53	13	b(r*g	b(r*g	NOUN
cana-1313	53	14	*	*	PUNCT
cana-1313	53	15	)	)	PUNCT
cana-1313	53	16	closed	closed	ADJ
cana-1313	53	17	type	type	NOUN
cana-1313	53	18	sets	set	NOUN
cana-1313	53	19	here	here	ADV
cana-1313	53	20	are	be	AUX
cana-1313	53	21	some	some	DET
cana-1313	53	22	definitions	definition	NOUN
cana-1313	53	23	that	that	PRON
cana-1313	53	24	we	we	PRON
cana-1313	53	25	introduce	introduce	VERB
cana-1313	53	26	:	:	PUNCT
cana-1313	53	27	definition	definition	NOUN
cana-1313	53	28	3.1	3.1	NUM
cana-1313	53	29	:	:	PUNCT
cana-1313	53	30	if	if	SCONJ
cana-1313	53	31	ircl(p	ircl(p	NOUN
cana-1313	53	32	)	)	PUNCT
cana-1313	53	33	⊆	⊆	NUM
cana-1313	53	34	r	r	NOUN
cana-1313	53	35	whenever	whenever	SCONJ
cana-1313	53	36	p	p	NOUN
cana-1313	53	37	⊆	⊆	NUM
cana-1313	53	38	r	r	NOUN
cana-1313	53	39	and	and	CCONJ
cana-1313	53	40	r	r	NOUN
cana-1313	53	41	is	be	AUX
cana-1313	53	42	g	g	NOUN
cana-1313	53	43	-	-	PUNCT
cana-1313	53	44	open	open	ADJ
cana-1313	53	45	in	in	ADP
cana-1313	53	46	(	(	PUNCT
cana-1313	53	47	)	)	PUNCT
cana-1313	53	48	,	,	PUNCT
cana-1313	53	49	then	then	ADV
cana-1313	53	50	a	a	DET
cana-1313	53	51	subset	subset	NOUN
cana-1313	53	52	p	p	NOUN
cana-1313	53	53	of	of	ADP
cana-1313	53	54	(	(	PUNCT
cana-1313	53	55	)	)	PUNCT
cana-1313	53	56	is	be	AUX
cana-1313	53	57	called	call	VERB
cana-1313	53	58	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	53	59	set	set	NOUN
cana-1313	53	60	.	.	PUNCT
cana-1313	54	1	definition	definition	NOUN
cana-1313	54	2	3.2	3.2	NUM
cana-1313	54	3	:	:	PUNCT
cana-1313	54	4	if	if	SCONJ
cana-1313	54	5	drcl	drcl	NOUN
cana-1313	54	6	(	(	PUNCT
cana-1313	54	7	p	p	NOUN
cana-1313	54	8	)	)	PUNCT
cana-1313	54	9	⊆	⊆	NUM
cana-1313	54	10	r	r	NOUN
cana-1313	54	11	whenever	whenever	SCONJ
cana-1313	54	12	p	p	NOUN
cana-1313	54	13	⊆	⊆	NUM
cana-1313	54	14	r	r	NOUN
cana-1313	54	15	and	and	CCONJ
cana-1313	54	16	r	r	NOUN
cana-1313	54	17	is	be	AUX
cana-1313	54	18	g	g	NOUN
cana-1313	54	19	-	-	PUNCT
cana-1313	54	20	open	open	ADJ
cana-1313	54	21	in	in	ADP
cana-1313	54	22	(	(	PUNCT
cana-1313	54	23	)	)	PUNCT
cana-1313	54	24	,	,	PUNCT
cana-1313	54	25	then	then	ADV
cana-1313	54	26	a	a	DET
cana-1313	54	27	subset	subset	NOUN
cana-1313	54	28	p	p	NOUN
cana-1313	54	29	of	of	ADP
cana-1313	54	30	(	(	PUNCT
cana-1313	54	31	)	)	PUNCT
cana-1313	54	32	is	be	AUX
cana-1313	54	33	called	call	VERB
cana-1313	54	34	d(r*g*)-closed	d(r*g*)-close	VERB
cana-1313	54	35	set	set	NOUN
cana-1313	54	36	.	.	PUNCT
cana-1313	55	1	definition	definition	NOUN
cana-1313	55	2	3.3	3.3	NUM
cana-1313	55	3	:	:	PUNCT
cana-1313	55	4	if	if	SCONJ
cana-1313	55	5	brcl	brcl	NOUN
cana-1313	55	6	(	(	PUNCT
cana-1313	55	7	p	p	NOUN
cana-1313	55	8	)	)	PUNCT
cana-1313	55	9	⊆	⊆	NUM
cana-1313	55	10	r	r	NOUN
cana-1313	55	11	whenever	whenever	SCONJ
cana-1313	55	12	p	p	NOUN
cana-1313	55	13	⊆	⊆	NUM
cana-1313	55	14	r	r	NOUN
cana-1313	55	15	and	and	CCONJ
cana-1313	55	16	r	r	NOUN
cana-1313	55	17	is	be	AUX
cana-1313	55	18	g	g	NOUN
cana-1313	55	19	-	-	PUNCT
cana-1313	55	20	open	open	ADJ
cana-1313	55	21	in	in	ADP
cana-1313	55	22	(	(	PUNCT
cana-1313	55	23	)	)	PUNCT
cana-1313	55	24	,	,	PUNCT
cana-1313	55	25	then	then	ADV
cana-1313	55	26	a	a	DET
cana-1313	55	27	subset	subset	NOUN
cana-1313	55	28	p	p	NOUN
cana-1313	55	29	of	of	ADP
cana-1313	55	30	(	(	PUNCT
cana-1313	55	31	)	)	PUNCT
cana-1313	55	32	is	be	AUX
cana-1313	55	33	called	call	VERB
cana-1313	55	34	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	55	35	set	set	NOUN
cana-1313	55	36	.	.	PUNCT
cana-1313	56	1	theorem	theorem	VERB
cana-1313	56	2	3.4	3.4	NUM
cana-1313	56	3	:	:	PUNCT
cana-1313	56	4	every	every	DET
cana-1313	56	5	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	56	6	set	set	NOUN
cana-1313	56	7	is	be	AUX
cana-1313	56	8	an	an	DET
cana-1313	56	9	i(rg)-closed	i(rg)-closed	ADJ
cana-1313	56	10	set	set	NOUN
cana-1313	56	11	.	.	PUNCT
cana-1313	57	1	proof	proof	NOUN
cana-1313	57	2	:	:	PUNCT
cana-1313	57	3	as	as	ADV
cana-1313	57	4	far	far	ADV
cana-1313	57	5	as	as	SCONJ
cana-1313	57	6	we	we	PRON
cana-1313	57	7	know	know	VERB
cana-1313	57	8	,	,	PUNCT
cana-1313	57	9	every	every	DET
cana-1313	57	10	r*g*-closed	r*g*-close	VERB
cana-1313	57	11	set	set	NOUN
cana-1313	57	12	is	be	AUX
cana-1313	57	13	a	a	DET
cana-1313	57	14	rg	rg	NOUN
cana-1313	57	15	-	-	PUNCT
cana-1313	57	16	closed	closed	ADJ
cana-1313	57	17	set	set	NOUN
cana-1313	57	18	.	.	PUNCT
cana-1313	58	1	therefore	therefore	ADV
cana-1313	58	2	,	,	PUNCT
cana-1313	58	3	every	every	DET
cana-1313	58	4	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	58	5	set	set	NOUN
cana-1313	58	6	is	be	AUX
cana-1313	58	7	a	a	DET
cana-1313	58	8	i(rg)-closed	i(rg)-closed	ADJ
cana-1313	58	9	set	set	NOUN
cana-1313	58	10	.	.	PUNCT
cana-1313	59	1	in	in	ADP
cana-1313	59	2	general	general	ADJ
cana-1313	59	3	,	,	PUNCT
cana-1313	59	4	the	the	DET
cana-1313	59	5	following	follow	VERB
cana-1313	59	6	illustration	illustration	NOUN
cana-1313	59	7	demonstrates	demonstrate	VERB
cana-1313	59	8	that	that	SCONJ
cana-1313	59	9	,	,	PUNCT
cana-1313	59	10	a	a	DET
cana-1313	59	11	i(rg)-closed	i(rg)-closed	ADJ
cana-1313	59	12	set	set	NOUN
cana-1313	59	13	need	need	AUX
cana-1313	59	14	not	not	PART
cana-1313	59	15	be	be	AUX
cana-1313	59	16	an	an	DET
cana-1313	59	17	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	59	18	set	set	NOUN
cana-1313	59	19	.	.	PUNCT
cana-1313	60	1	example	example	NOUN
cana-1313	60	2	3.5	3.5	NUM
cana-1313	60	3	:	:	PUNCT
cana-1313	60	4	let	let	VERB
cana-1313	60	5	x	x	PUNCT
cana-1313	60	6	=	=	PUNCT
cana-1313	60	7	{	{	PUNCT
cana-1313	60	8	p	p	X
cana-1313	60	9	,	,	PUNCT
cana-1313	60	10	q	q	ADJ
cana-1313	60	11	,	,	PUNCT
cana-1313	60	12	r	r	NOUN
cana-1313	60	13	}	}	PUNCT
cana-1313	60	14	,	,	PUNCT
cana-1313	60	15	=	=	PRON
cana-1313	60	16	{	{	PUNCT
cana-1313	60	17	,	,	PUNCT
cana-1313	60	18	x	x	X
cana-1313	60	19	,	,	PUNCT
cana-1313	60	20	{	{	PUNCT
cana-1313	60	21	p	p	X
cana-1313	60	22	}	}	PUNCT
cana-1313	60	23	,	,	PUNCT
cana-1313	60	24	{	{	PUNCT
cana-1313	60	25	p	p	X
cana-1313	60	26	,	,	PUNCT
cana-1313	60	27	q	q	NOUN
cana-1313	60	28	}	}	PUNCT
cana-1313	60	29	,	,	PUNCT
cana-1313	60	30	{	{	PUNCT
cana-1313	60	31	p	p	X
cana-1313	60	32	,	,	PUNCT
cana-1313	60	33	r	r	NOUN
cana-1313	60	34	}	}	PUNCT
cana-1313	60	35	}	}	PUNCT
cana-1313	60	36	and	and	CCONJ
cana-1313	60	37	=	=	PRON
cana-1313	60	38	{	{	PUNCT
cana-1313	60	39	(	(	PUNCT
cana-1313	60	40	p	p	X
cana-1313	60	41	,	,	PUNCT
cana-1313	60	42	p	p	NOUN
cana-1313	60	43	)	)	PUNCT
cana-1313	60	44	,	,	PUNCT
cana-1313	60	45	(	(	PUNCT
cana-1313	60	46	q	q	X
cana-1313	60	47	,	,	PUNCT
cana-1313	60	48	q	q	NOUN
cana-1313	60	49	)	)	PUNCT
cana-1313	60	50	,	,	PUNCT
cana-1313	60	51	(	(	PUNCT
cana-1313	60	52	r	r	NOUN
cana-1313	60	53	,	,	PUNCT
cana-1313	60	54	r	r	NOUN
cana-1313	60	55	)	)	PUNCT
cana-1313	60	56	,	,	PUNCT
cana-1313	60	57	(	(	PUNCT
cana-1313	60	58	p	p	X
cana-1313	60	59	,	,	PUNCT
cana-1313	60	60	q	q	NOUN
cana-1313	60	61	)	)	PUNCT
cana-1313	60	62	,	,	PUNCT
cana-1313	60	63	(	(	PUNCT
cana-1313	60	64	q	q	X
cana-1313	60	65	,	,	PUNCT
cana-1313	60	66	r	r	NOUN
cana-1313	60	67	)	)	PUNCT
cana-1313	60	68	}	}	PUNCT
cana-1313	60	69	.	.	PUNCT
cana-1313	61	1	clearly	clearly	ADV
cana-1313	61	2	(	(	PUNCT
cana-1313	61	3	)	)	PUNCT
cana-1313	61	4	is	be	AUX
cana-1313	61	5	a	a	DET
cana-1313	61	6	topological	topological	ADJ
cana-1313	61	7	ordered	order	VERB
cana-1313	61	8	space	space	NOUN
cana-1313	61	9	.	.	PUNCT
cana-1313	62	1	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	62	2	sets	set	NOUN
cana-1313	62	3	are	be	AUX
cana-1313	62	4	,	,	PUNCT
cana-1313	62	5	x	x	X
cana-1313	62	6	,	,	PUNCT
cana-1313	62	7	{	{	PUNCT
cana-1313	62	8	q	q	X
cana-1313	62	9	,	,	PUNCT
cana-1313	62	10	r	r	NOUN
cana-1313	62	11	}	}	PUNCT
cana-1313	62	12	.	.	PUNCT
cana-1313	63	1	i(rg)closed	i(rg)close	VERB
cana-1313	63	2	sets	set	NOUN
cana-1313	63	3	are	be	AUX
cana-1313	63	4	,	,	PUNCT
cana-1313	63	5	x	x	X
cana-1313	63	6	,	,	PUNCT
cana-1313	63	7	{	{	PUNCT
cana-1313	63	8	r	r	NOUN
cana-1313	63	9	}	}	PUNCT
cana-1313	63	10	,	,	PUNCT
cana-1313	63	11	{	{	PUNCT
cana-1313	63	12	q	q	NOUN
cana-1313	63	13	,	,	PUNCT
cana-1313	63	14	r	r	NOUN
cana-1313	63	15	}	}	PUNCT
cana-1313	63	16	.	.	PUNCT
cana-1313	64	1	let	let	VERB
cana-1313	64	2	p	p	NOUN
cana-1313	64	3	=	=	X
cana-1313	64	4	{	{	PUNCT
cana-1313	64	5	r	r	NOUN
cana-1313	64	6	}	}	PUNCT
cana-1313	64	7	.	.	PUNCT
cana-1313	65	1	clearly	clearly	ADV
cana-1313	65	2	p	p	X
cana-1313	65	3	is	be	AUX
cana-1313	65	4	i(rg)-closed	i(rg)-close	VERB
cana-1313	65	5	set	set	ADJ
cana-1313	65	6	but	but	CCONJ
cana-1313	65	7	not	not	PART
cana-1313	65	8	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	65	9	set	set	NOUN
cana-1313	65	10	.	.	PUNCT
cana-1313	66	1	theorem	theorem	VERB
cana-1313	66	2	3.6	3.6	NUM
cana-1313	66	3	:	:	PUNCT
cana-1313	66	4	every	every	DET
cana-1313	66	5	d(r*g*)-closed	d(r*g*)-closed	NOUN
cana-1313	66	6	set	set	NOUN
cana-1313	66	7	is	be	AUX
cana-1313	66	8	an	an	DET
cana-1313	66	9	d(rg)-closed	d(rg)-close	VERB
cana-1313	66	10	set	set	NOUN
cana-1313	66	11	.	.	PUNCT
cana-1313	67	1	proof	proof	NOUN
cana-1313	67	2	:	:	PUNCT
cana-1313	67	3	we	we	PRON
cana-1313	67	4	know	know	VERB
cana-1313	67	5	,	,	PUNCT
cana-1313	67	6	every	every	DET
cana-1313	67	7	r*g*-closed	r*g*-close	VERB
cana-1313	67	8	set	set	NOUN
cana-1313	67	9	is	be	AUX
cana-1313	67	10	an	an	DET
cana-1313	67	11	rg	rg	NOUN
cana-1313	67	12	-	-	PUNCT
cana-1313	67	13	closed	closed	ADJ
cana-1313	67	14	set	set	NOUN
cana-1313	67	15	.	.	PUNCT
cana-1313	68	1	thus	thus	ADV
cana-1313	68	2	,	,	PUNCT
cana-1313	68	3	every	every	DET
cana-1313	68	4	d(r*g*)-closed	d(r*g*)-closed	NOUN
cana-1313	68	5	set	set	NOUN
cana-1313	68	6	is	be	AUX
cana-1313	68	7	an	an	DET
cana-1313	68	8	d(rg)closed	d(rg)closed	ADJ
cana-1313	68	9	set	set	NOUN
cana-1313	68	10	.	.	PUNCT
cana-1313	69	1	the	the	DET
cana-1313	69	2	following	follow	VERB
cana-1313	69	3	illustration	illustration	NOUN
cana-1313	69	4	demonstrates	demonstrate	VERB
cana-1313	69	5	that	that	SCONJ
cana-1313	69	6	,	,	PUNCT
cana-1313	69	7	a	a	DET
cana-1313	69	8	d(rg)-closed	d(rg)-close	VERB
cana-1313	69	9	set	set	NOUN
cana-1313	69	10	need	need	AUX
cana-1313	69	11	not	not	PART
cana-1313	69	12	always	always	ADV
cana-1313	69	13	be	be	AUX
cana-1313	69	14	an	an	DET
cana-1313	69	15	d(r*g*)-closed	d(r*g*)-close	VERB
cana-1313	69	16	set	set	NOUN
cana-1313	69	17	.	.	PUNCT
cana-1313	70	1	example	example	NOUN
cana-1313	70	2	3.7	3.7	NUM
cana-1313	70	3	:	:	PUNCT
cana-1313	70	4	let	let	VERB
cana-1313	70	5	x	x	PUNCT
cana-1313	70	6	=	=	PUNCT
cana-1313	70	7	{	{	PUNCT
cana-1313	70	8	p	p	X
cana-1313	70	9	,	,	PUNCT
cana-1313	70	10	q	q	ADJ
cana-1313	70	11	,	,	PUNCT
cana-1313	70	12	r	r	NOUN
cana-1313	70	13	}	}	PUNCT
cana-1313	70	14	,	,	PUNCT
cana-1313	70	15	=	=	PRON
cana-1313	70	16	{	{	PUNCT
cana-1313	70	17	,	,	PUNCT
cana-1313	70	18	x	x	X
cana-1313	70	19	,	,	PUNCT
cana-1313	70	20	{	{	PUNCT
cana-1313	70	21	p	p	X
cana-1313	70	22	,	,	PUNCT
cana-1313	70	23	q	q	NOUN
cana-1313	70	24	}	}	PUNCT
cana-1313	70	25	}	}	PUNCT
cana-1313	70	26	and	and	CCONJ
cana-1313	70	27	=	=	PRON
cana-1313	70	28	{	{	PUNCT
cana-1313	70	29	(	(	PUNCT
cana-1313	70	30	p	p	X
cana-1313	70	31	,	,	PUNCT
cana-1313	70	32	p	p	NOUN
cana-1313	70	33	)	)	PUNCT
cana-1313	70	34	,	,	PUNCT
cana-1313	70	35	(	(	PUNCT
cana-1313	70	36	q	q	X
cana-1313	70	37	,	,	PUNCT
cana-1313	70	38	q	q	NOUN
cana-1313	70	39	)	)	PUNCT
cana-1313	70	40	,	,	PUNCT
cana-1313	70	41	(	(	PUNCT
cana-1313	70	42	r	r	NOUN
cana-1313	70	43	,	,	PUNCT
cana-1313	70	44	r	r	NOUN
cana-1313	70	45	)	)	PUNCT
cana-1313	70	46	,	,	PUNCT
cana-1313	70	47	(	(	PUNCT
cana-1313	70	48	p	p	X
cana-1313	70	49	,	,	PUNCT
cana-1313	70	50	q	q	NOUN
cana-1313	70	51	)	)	PUNCT
cana-1313	70	52	,	,	PUNCT
cana-1313	70	53	(	(	PUNCT
cana-1313	70	54	p	p	X
cana-1313	70	55	,	,	PUNCT
cana-1313	70	56	r	r	NOUN
cana-1313	70	57	)	)	PUNCT
cana-1313	70	58	}	}	PUNCT
cana-1313	70	59	.	.	PUNCT
cana-1313	71	1	clearly	clearly	ADV
cana-1313	71	2	,	,	PUNCT
cana-1313	71	3	a	a	DET
cana-1313	71	4	topological	topological	ADJ
cana-1313	71	5	ordered	order	VERB
cana-1313	71	6	space	space	NOUN
cana-1313	71	7	is	be	AUX
cana-1313	71	8	(	(	PUNCT
cana-1313	71	9	)	)	PUNCT
cana-1313	71	10	.	.	PUNCT
cana-1313	72	1	,	,	PUNCT
cana-1313	72	2	x	x	X
cana-1313	72	3	,	,	PUNCT
cana-1313	72	4	{	{	PUNCT
cana-1313	72	5	p	p	X
cana-1313	72	6	,	,	PUNCT
cana-1313	72	7	r	r	NOUN
cana-1313	72	8	}	}	PUNCT
cana-1313	72	9	are	be	AUX
cana-1313	72	10	d(r*g*)-closed	d(r*g*)-close	VERB
cana-1313	72	11	sets	set	NOUN
cana-1313	72	12	.	.	PUNCT
cana-1313	73	1	,	,	PUNCT
cana-1313	73	2	x	x	X
cana-1313	73	3	,	,	PUNCT
cana-1313	73	4	{	{	PUNCT
cana-1313	73	5	p	p	X
cana-1313	73	6	}	}	PUNCT
cana-1313	73	7	,	,	PUNCT
cana-1313	73	8	{	{	PUNCT
cana-1313	73	9	p	p	X
cana-1313	73	10	,	,	PUNCT
cana-1313	73	11	communications	communication	NOUN
cana-1313	73	12	on	on	ADP
cana-1313	73	13	applied	apply	VERB
cana-1313	73	14	nonlinear	nonlinear	ADJ
cana-1313	73	15	analysis	analysis	NOUN
cana-1313	73	16	issn	issn	NOUN
cana-1313	73	17	:	:	PUNCT
cana-1313	73	18	1074	1074	NUM
cana-1313	73	19	-	-	PUNCT
cana-1313	73	20	133x	133x	NUM
cana-1313	73	21	vol	vol	NOUN
cana-1313	73	22	31	31	NUM
cana-1313	73	23	no	no	NOUN
cana-1313	73	24	.	.	PUNCT
cana-1313	74	1	7s	7	NOUN
cana-1313	74	2	(	(	PUNCT
cana-1313	74	3	2024	2024	NUM
cana-1313	74	4	)	)	PUNCT
cana-1313	74	5	351	351	NUM
cana-1313	74	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1313	74	7	q},{p	q},{p	NOUN
cana-1313	74	8	,	,	PUNCT
cana-1313	74	9	r	r	NOUN
cana-1313	74	10	}	}	PUNCT
cana-1313	74	11	are	be	AUX
cana-1313	74	12	d(rg)-closed	d(rg)-close	VERB
cana-1313	74	13	sets	set	NOUN
cana-1313	74	14	.	.	PUNCT
cana-1313	75	1	let	let	VERB
cana-1313	75	2	p	p	NOUN
cana-1313	75	3	=	=	X
cana-1313	75	4	{	{	PUNCT
cana-1313	75	5	p	p	X
cana-1313	75	6	}	}	PUNCT
cana-1313	75	7	.	.	PUNCT
cana-1313	76	1	clearly	clearly	ADV
cana-1313	76	2	,	,	PUNCT
cana-1313	76	3	p	p	PRON
cana-1313	76	4	is	be	AUX
cana-1313	76	5	a	a	DET
cana-1313	76	6	d(rg)-closed	d(rg)-close	VERB
cana-1313	76	7	set	set	NOUN
cana-1313	76	8	but	but	CCONJ
cana-1313	76	9	not	not	PART
cana-1313	76	10	a	a	DET
cana-1313	76	11	d(r*g*)-closed	d(r*g*)-close	VERB
cana-1313	76	12	set	set	NOUN
cana-1313	76	13	.	.	PUNCT
cana-1313	77	1	theorem	theorem	VERB
cana-1313	77	2	3.8	3.8	NUM
cana-1313	77	3	:	:	PUNCT
cana-1313	77	4	every	every	DET
cana-1313	77	5	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	77	6	set	set	NOUN
cana-1313	77	7	is	be	AUX
cana-1313	77	8	a	a	DET
cana-1313	77	9	b(rg)-closed	b(rg)-close	VERB
cana-1313	77	10	set	set	NOUN
cana-1313	77	11	.	.	PUNCT
cana-1313	78	1	proof	proof	NOUN
cana-1313	78	2	:	:	PUNCT
cana-1313	78	3	as	as	ADV
cana-1313	78	4	far	far	ADV
cana-1313	78	5	as	as	SCONJ
cana-1313	78	6	we	we	PRON
cana-1313	78	7	know	know	VERB
cana-1313	78	8	,	,	PUNCT
cana-1313	78	9	every	every	DET
cana-1313	78	10	r*g*-closed	r*g*-close	VERB
cana-1313	78	11	set	set	NOUN
cana-1313	78	12	is	be	AUX
cana-1313	78	13	a	a	DET
cana-1313	78	14	rg	rg	NOUN
cana-1313	78	15	-	-	PUNCT
cana-1313	78	16	closed	closed	ADJ
cana-1313	78	17	set	set	NOUN
cana-1313	78	18	.	.	PUNCT
cana-1313	79	1	thus	thus	ADV
cana-1313	79	2	,	,	PUNCT
cana-1313	79	3	every	every	DET
cana-1313	79	4	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	79	5	set	set	NOUN
cana-1313	79	6	is	be	AUX
cana-1313	79	7	a	a	DET
cana-1313	79	8	b(rg)-closed	b(rg)-close	VERB
cana-1313	79	9	set	set	NOUN
cana-1313	79	10	.	.	PUNCT
cana-1313	80	1	the	the	DET
cana-1313	80	2	following	follow	VERB
cana-1313	80	3	illustration	illustration	NOUN
cana-1313	80	4	demonstrates	demonstrate	VERB
cana-1313	80	5	that	that	SCONJ
cana-1313	80	6	,	,	PUNCT
cana-1313	80	7	a	a	DET
cana-1313	80	8	b(rg)-closed	b(rg)-close	VERB
cana-1313	80	9	set	set	NOUN
cana-1313	80	10	does	do	AUX
cana-1313	80	11	not	not	PART
cana-1313	80	12	always	always	ADV
cana-1313	80	13	have	have	VERB
cana-1313	80	14	to	to	PART
cana-1313	80	15	be	be	AUX
cana-1313	80	16	a	a	DET
cana-1313	80	17	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	80	18	set	set	NOUN
cana-1313	80	19	.	.	PUNCT
cana-1313	81	1	example	example	NOUN
cana-1313	81	2	3.9	3.9	NUM
cana-1313	81	3	:	:	PUNCT
cana-1313	81	4	let	let	VERB
cana-1313	81	5	x	x	PUNCT
cana-1313	81	6	=	=	PUNCT
cana-1313	81	7	{	{	PUNCT
cana-1313	81	8	p	p	X
cana-1313	81	9	,	,	PUNCT
cana-1313	81	10	q	q	ADJ
cana-1313	81	11	,	,	PUNCT
cana-1313	81	12	r	r	NOUN
cana-1313	81	13	}	}	PUNCT
cana-1313	81	14	,	,	PUNCT
cana-1313	81	15	=	=	PRON
cana-1313	81	16	{	{	PUNCT
cana-1313	81	17	,	,	PUNCT
cana-1313	81	18	x	x	X
cana-1313	81	19	,	,	PUNCT
cana-1313	81	20	{	{	PUNCT
cana-1313	81	21	p	p	X
cana-1313	81	22	}	}	PUNCT
cana-1313	81	23	,	,	PUNCT
cana-1313	81	24	{	{	PUNCT
cana-1313	81	25	p	p	X
cana-1313	81	26	,	,	PUNCT
cana-1313	81	27	q	q	NOUN
cana-1313	81	28	}	}	PUNCT
cana-1313	81	29	,	,	PUNCT
cana-1313	81	30	{	{	PUNCT
cana-1313	81	31	p	p	X
cana-1313	81	32	,	,	PUNCT
cana-1313	81	33	r	r	NOUN
cana-1313	81	34	}	}	PUNCT
cana-1313	81	35	}	}	PUNCT
cana-1313	81	36	and	and	CCONJ
cana-1313	81	37	=	=	PRON
cana-1313	81	38	{	{	PUNCT
cana-1313	81	39	(	(	PUNCT
cana-1313	81	40	p	p	X
cana-1313	81	41	,	,	PUNCT
cana-1313	81	42	p	p	NOUN
cana-1313	81	43	)	)	PUNCT
cana-1313	81	44	,	,	PUNCT
cana-1313	81	45	(	(	PUNCT
cana-1313	81	46	q	q	X
cana-1313	81	47	,	,	PUNCT
cana-1313	81	48	q	q	NOUN
cana-1313	81	49	)	)	PUNCT
cana-1313	81	50	,	,	PUNCT
cana-1313	81	51	(	(	PUNCT
cana-1313	81	52	r	r	NOUN
cana-1313	81	53	,	,	PUNCT
cana-1313	81	54	r	r	NOUN
cana-1313	81	55	)	)	PUNCT
cana-1313	81	56	,	,	PUNCT
cana-1313	81	57	(	(	PUNCT
cana-1313	81	58	p	p	X
cana-1313	81	59	,	,	PUNCT
cana-1313	81	60	q	q	NOUN
cana-1313	81	61	)	)	PUNCT
cana-1313	81	62	,	,	PUNCT
cana-1313	81	63	(	(	PUNCT
cana-1313	81	64	r	r	NOUN
cana-1313	81	65	,	,	PUNCT
cana-1313	81	66	q	q	NOUN
cana-1313	81	67	)	)	PUNCT
cana-1313	81	68	}	}	PUNCT
cana-1313	81	69	.	.	PUNCT
cana-1313	82	1	clearly	clearly	ADV
cana-1313	82	2	,	,	PUNCT
cana-1313	82	3	(	(	PUNCT
cana-1313	82	4	)	)	PUNCT
cana-1313	82	5	is	be	AUX
cana-1313	82	6	a	a	DET
cana-1313	82	7	topological	topological	ADJ
cana-1313	82	8	ordered	order	VERB
cana-1313	82	9	space	space	NOUN
cana-1313	82	10	.	.	PUNCT
cana-1313	83	1	,	,	PUNCT
cana-1313	83	2	x	x	PRON
cana-1313	83	3	are	be	AUX
cana-1313	83	4	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	83	5	sets	set	NOUN
cana-1313	83	6	.	.	PUNCT
cana-1313	84	1	,	,	PUNCT
cana-1313	84	2	x	x	X
cana-1313	84	3	,	,	PUNCT
cana-1313	84	4	{	{	PUNCT
cana-1313	84	5	p	p	X
cana-1313	84	6	,	,	PUNCT
cana-1313	84	7	q	q	ADJ
cana-1313	84	8	}	}	PUNCT
cana-1313	84	9	are	be	AUX
cana-1313	84	10	b(rg)-closed	b(rg)-close	VERB
cana-1313	84	11	sets	set	NOUN
cana-1313	84	12	.	.	PUNCT
cana-1313	85	1	let	let	VERB
cana-1313	85	2	p	p	NOUN
cana-1313	85	3	=	=	X
cana-1313	85	4	{	{	PUNCT
cana-1313	85	5	p	p	X
cana-1313	85	6	,	,	PUNCT
cana-1313	85	7	q	q	NOUN
cana-1313	85	8	}	}	PUNCT
cana-1313	85	9	.	.	PUNCT
cana-1313	86	1	clearly	clearly	ADV
cana-1313	86	2	,	,	PUNCT
cana-1313	86	3	p	p	PRON
cana-1313	86	4	is	be	AUX
cana-1313	86	5	a	a	DET
cana-1313	86	6	b(rg)-closed	b(rg)-close	VERB
cana-1313	86	7	set	set	NOUN
cana-1313	86	8	but	but	CCONJ
cana-1313	86	9	not	not	PART
cana-1313	86	10	a	a	DET
cana-1313	86	11	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	86	12	set	set	NOUN
cana-1313	86	13	.	.	PUNCT
cana-1313	87	1	theorem	theorem	VERB
cana-1313	87	2	3.10	3.10	NUM
cana-1313	87	3	:	:	PUNCT
cana-1313	87	4	every	every	DET
cana-1313	87	5	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	87	6	set	set	NOUN
cana-1313	87	7	is	be	AUX
cana-1313	87	8	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	87	9	set	set	NOUN
cana-1313	87	10	.	.	PUNCT
cana-1313	88	1	proof	proof	NOUN
cana-1313	88	2	:	:	PUNCT
cana-1313	88	3	as	as	ADV
cana-1313	88	4	far	far	ADV
cana-1313	88	5	as	as	SCONJ
cana-1313	88	6	we	we	PRON
cana-1313	88	7	know	know	VERB
cana-1313	88	8	,	,	PUNCT
cana-1313	88	9	a	a	DET
cana-1313	88	10	balanced	balanced	ADJ
cana-1313	88	11	set	set	NOUN
cana-1313	88	12	is	be	AUX
cana-1313	88	13	always	always	ADV
cana-1313	88	14	an	an	DET
cana-1313	88	15	increasing	increase	VERB
cana-1313	88	16	set	set	NOUN
cana-1313	88	17	.	.	PUNCT
cana-1313	89	1	then	then	ADV
cana-1313	89	2	,	,	PUNCT
cana-1313	89	3	all	all	DET
cana-1313	89	4	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	89	5	sets	set	NOUN
cana-1313	89	6	are	be	AUX
cana-1313	89	7	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	89	8	sets	set	NOUN
cana-1313	89	9	.	.	PUNCT
cana-1313	90	1	the	the	DET
cana-1313	90	2	following	follow	VERB
cana-1313	90	3	illustration	illustration	NOUN
cana-1313	90	4	demonstrates	demonstrate	VERB
cana-1313	90	5	that	that	SCONJ
cana-1313	90	6	,	,	PUNCT
cana-1313	90	7	an	an	DET
cana-1313	90	8	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	90	9	set	set	NOUN
cana-1313	90	10	need	need	AUX
cana-1313	90	11	not	not	PART
cana-1313	90	12	always	always	ADV
cana-1313	90	13	be	be	AUX
cana-1313	90	14	a	a	DET
cana-1313	90	15	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	90	16	set	set	NOUN
cana-1313	90	17	.	.	PUNCT
cana-1313	91	1	example	example	NOUN
cana-1313	92	1	3.11	3.11	NUM
cana-1313	92	2	:	:	PUNCT
cana-1313	92	3	let	let	VERB
cana-1313	92	4	x	x	PUNCT
cana-1313	92	5	=	=	PUNCT
cana-1313	92	6	{	{	PUNCT
cana-1313	92	7	p	p	X
cana-1313	92	8	,	,	PUNCT
cana-1313	92	9	q	q	ADJ
cana-1313	92	10	,	,	PUNCT
cana-1313	92	11	r	r	NOUN
cana-1313	92	12	}	}	PUNCT
cana-1313	92	13	,	,	PUNCT
cana-1313	92	14	=	=	PRON
cana-1313	92	15	{	{	PUNCT
cana-1313	92	16	,	,	PUNCT
cana-1313	92	17	x	x	X
cana-1313	92	18	,	,	PUNCT
cana-1313	92	19	{	{	PUNCT
cana-1313	92	20	p	p	X
cana-1313	92	21	}	}	PUNCT
cana-1313	92	22	,	,	PUNCT
cana-1313	92	23	{	{	PUNCT
cana-1313	92	24	p	p	X
cana-1313	92	25	,	,	PUNCT
cana-1313	92	26	r	r	NOUN
cana-1313	92	27	}	}	PUNCT
cana-1313	92	28	}	}	PUNCT
cana-1313	92	29	and	and	CCONJ
cana-1313	92	30	=	=	PRON
cana-1313	92	31	{	{	PUNCT
cana-1313	92	32	(	(	PUNCT
cana-1313	92	33	p	p	X
cana-1313	92	34	,	,	PUNCT
cana-1313	92	35	p	p	NOUN
cana-1313	92	36	)	)	PUNCT
cana-1313	92	37	,	,	PUNCT
cana-1313	92	38	(	(	PUNCT
cana-1313	92	39	q	q	X
cana-1313	92	40	,	,	PUNCT
cana-1313	92	41	q	q	NOUN
cana-1313	92	42	)	)	PUNCT
cana-1313	92	43	,	,	PUNCT
cana-1313	92	44	(	(	PUNCT
cana-1313	92	45	r	r	NOUN
cana-1313	92	46	,	,	PUNCT
cana-1313	92	47	r	r	NOUN
cana-1313	92	48	)	)	PUNCT
cana-1313	92	49	,	,	PUNCT
cana-1313	92	50	(	(	PUNCT
cana-1313	92	51	q	q	X
cana-1313	92	52	,	,	PUNCT
cana-1313	92	53	p	p	NOUN
cana-1313	92	54	)	)	PUNCT
cana-1313	92	55	,	,	PUNCT
cana-1313	92	56	(	(	PUNCT
cana-1313	92	57	r	r	NOUN
cana-1313	92	58	,	,	PUNCT
cana-1313	92	59	q	q	NOUN
cana-1313	92	60	)	)	PUNCT
cana-1313	92	61	,	,	PUNCT
cana-1313	92	62	(	(	PUNCT
cana-1313	92	63	r	r	NOUN
cana-1313	92	64	,	,	PUNCT
cana-1313	92	65	p	p	NOUN
cana-1313	92	66	)	)	PUNCT
cana-1313	92	67	}	}	PUNCT
cana-1313	92	68	.	.	PUNCT
cana-1313	93	1	clearly	clearly	ADV
cana-1313	93	2	,	,	PUNCT
cana-1313	93	3	a	a	DET
cana-1313	93	4	topological	topological	ADJ
cana-1313	93	5	ordered	order	VERB
cana-1313	93	6	space	space	NOUN
cana-1313	93	7	is	be	AUX
cana-1313	93	8	(	(	PUNCT
cana-1313	93	9	)	)	PUNCT
cana-1313	93	10	.	.	PUNCT
cana-1313	94	1	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	94	2	sets	set	NOUN
cana-1313	94	3	are	be	AUX
cana-1313	94	4	,	,	PUNCT
cana-1313	94	5	x.	x.	PROPN
cana-1313	94	6	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	94	7	sets	set	NOUN
cana-1313	94	8	are	be	AUX
cana-1313	94	9	,	,	PUNCT
cana-1313	94	10	x	x	X
cana-1313	94	11	,	,	PUNCT
cana-1313	94	12	{	{	PUNCT
cana-1313	94	13	p	p	X
cana-1313	94	14	,	,	PUNCT
cana-1313	94	15	q	q	NOUN
cana-1313	94	16	}	}	PUNCT
cana-1313	94	17	.	.	PUNCT
cana-1313	95	1	let	let	VERB
cana-1313	95	2	p	p	NOUN
cana-1313	95	3	=	=	X
cana-1313	95	4	{	{	PUNCT
cana-1313	95	5	p	p	X
cana-1313	95	6	,	,	PUNCT
cana-1313	95	7	q	q	NOUN
cana-1313	95	8	}	}	PUNCT
cana-1313	95	9	.	.	PUNCT
cana-1313	96	1	clearly	clearly	ADV
cana-1313	96	2	,	,	PUNCT
cana-1313	96	3	p	p	NOUN
cana-1313	96	4	is	be	AUX
cana-1313	96	5	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	96	6	set	set	VERB
cana-1313	96	7	but	but	CCONJ
cana-1313	96	8	not	not	PART
cana-1313	96	9	be	be	AUX
cana-1313	96	10	a	a	DET
cana-1313	96	11	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	96	12	set	set	NOUN
cana-1313	96	13	.	.	PUNCT
cana-1313	97	1	theorem	theorem	VERB
cana-1313	97	2	3.12	3.12	NUM
cana-1313	97	3	:	:	PUNCT
cana-1313	97	4	every	every	DET
cana-1313	97	5	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	97	6	set	set	NOUN
cana-1313	97	7	is	be	AUX
cana-1313	97	8	a	a	DET
cana-1313	97	9	d(r*g*)-closed	d(r*g*)-close	VERB
cana-1313	97	10	set	set	NOUN
cana-1313	97	11	.	.	PUNCT
cana-1313	98	1	proof	proof	NOUN
cana-1313	98	2	:	:	PUNCT
cana-1313	98	3	as	as	SCONJ
cana-1313	98	4	we	we	PRON
cana-1313	98	5	know	know	VERB
cana-1313	98	6	,	,	PUNCT
cana-1313	98	7	every	every	DET
cana-1313	98	8	balanced	balanced	ADJ
cana-1313	98	9	set	set	NOUN
cana-1313	98	10	is	be	AUX
cana-1313	98	11	a	a	DET
cana-1313	98	12	decreasing	decrease	VERB
cana-1313	98	13	set	set	NOUN
cana-1313	98	14	.	.	PUNCT
cana-1313	99	1	any	any	DET
cana-1313	99	2	set	set	NOUN
cana-1313	99	3	that	that	PRON
cana-1313	99	4	is	be	AUX
cana-1313	99	5	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	99	6	is	be	AUX
cana-1313	99	7	also	also	ADV
cana-1313	99	8	be	be	AUX
cana-1313	99	9	an	an	DET
cana-1313	99	10	d(r*g*)-closed	d(r*g*)-closed	PROPN
cana-1313	99	11	.	.	PUNCT
cana-1313	100	1	the	the	DET
cana-1313	100	2	following	follow	VERB
cana-1313	100	3	illustration	illustration	NOUN
cana-1313	100	4	demonstrates	demonstrate	VERB
cana-1313	100	5	that	that	SCONJ
cana-1313	100	6	,	,	PUNCT
cana-1313	100	7	a	a	DET
cana-1313	100	8	d(r*g*)-closed	d(r*g*)-close	VERB
cana-1313	100	9	set	set	NOUN
cana-1313	100	10	does	do	AUX
cana-1313	100	11	not	not	PART
cana-1313	100	12	always	always	ADV
cana-1313	100	13	have	have	VERB
cana-1313	100	14	to	to	PART
cana-1313	100	15	be	be	AUX
cana-1313	100	16	a	a	DET
cana-1313	100	17	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	100	18	set	set	NOUN
cana-1313	100	19	.	.	PUNCT
cana-1313	101	1	example	example	NOUN
cana-1313	101	2	3.13	3.13	NUM
cana-1313	101	3	:	:	PUNCT
cana-1313	101	4	let	let	VERB
cana-1313	101	5	x	x	PUNCT
cana-1313	101	6	=	=	PUNCT
cana-1313	101	7	{	{	PUNCT
cana-1313	101	8	p	p	X
cana-1313	101	9	,	,	PUNCT
cana-1313	101	10	q	q	ADJ
cana-1313	101	11	,	,	PUNCT
cana-1313	101	12	r	r	NOUN
cana-1313	101	13	}	}	PUNCT
cana-1313	101	14	,	,	PUNCT
cana-1313	101	15	=	=	PRON
cana-1313	101	16	{	{	PUNCT
cana-1313	101	17	,	,	PUNCT
cana-1313	101	18	x	x	X
cana-1313	101	19	,	,	PUNCT
cana-1313	101	20	{	{	PUNCT
cana-1313	101	21	p	p	X
cana-1313	101	22	}	}	PUNCT
cana-1313	101	23	,	,	PUNCT
cana-1313	101	24	{	{	PUNCT
cana-1313	101	25	p	p	X
cana-1313	101	26	,	,	PUNCT
cana-1313	101	27	r	r	NOUN
cana-1313	101	28	}	}	PUNCT
cana-1313	101	29	}	}	PUNCT
cana-1313	101	30	and	and	CCONJ
cana-1313	101	31	=	=	PRON
cana-1313	101	32	{	{	PUNCT
cana-1313	101	33	(	(	PUNCT
cana-1313	101	34	p	p	X
cana-1313	101	35	,	,	PUNCT
cana-1313	101	36	p	p	NOUN
cana-1313	101	37	)	)	PUNCT
cana-1313	101	38	,	,	PUNCT
cana-1313	101	39	(	(	PUNCT
cana-1313	101	40	q	q	X
cana-1313	101	41	,	,	PUNCT
cana-1313	101	42	q	q	NOUN
cana-1313	101	43	)	)	PUNCT
cana-1313	101	44	,	,	PUNCT
cana-1313	101	45	(	(	PUNCT
cana-1313	101	46	r	r	NOUN
cana-1313	101	47	,	,	PUNCT
cana-1313	101	48	r	r	NOUN
cana-1313	101	49	)	)	PUNCT
cana-1313	101	50	,	,	PUNCT
cana-1313	101	51	(	(	PUNCT
cana-1313	101	52	q	q	X
cana-1313	101	53	,	,	PUNCT
cana-1313	101	54	p	p	NOUN
cana-1313	101	55	)	)	PUNCT
cana-1313	101	56	,	,	PUNCT
cana-1313	101	57	(	(	PUNCT
cana-1313	101	58	r	r	NOUN
cana-1313	101	59	,	,	PUNCT
cana-1313	101	60	q	q	NOUN
cana-1313	101	61	)	)	PUNCT
cana-1313	101	62	,	,	PUNCT
cana-1313	101	63	(	(	PUNCT
cana-1313	101	64	r	r	NOUN
cana-1313	101	65	,	,	PUNCT
cana-1313	101	66	p	p	NOUN
cana-1313	101	67	)	)	PUNCT
cana-1313	101	68	}	}	PUNCT
cana-1313	101	69	.	.	PUNCT
cana-1313	102	1	clearly	clearly	ADV
cana-1313	102	2	,	,	PUNCT
cana-1313	102	3	(	(	PUNCT
cana-1313	102	4	)	)	PUNCT
cana-1313	102	5	is	be	AUX
cana-1313	102	6	a	a	DET
cana-1313	102	7	topological	topological	ADJ
cana-1313	102	8	ordered	order	VERB
cana-1313	102	9	space	space	NOUN
cana-1313	102	10	.	.	PUNCT
cana-1313	103	1	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	103	2	sets	set	NOUN
cana-1313	103	3	are	be	AUX
cana-1313	103	4	,	,	PUNCT
cana-1313	103	5	x.	x.	NOUN
cana-1313	103	6	d(r*g*)-closed	d(r*g*)-closed	PROPN
cana-1313	103	7	sets	set	NOUN
cana-1313	103	8	are	be	AUX
cana-1313	103	9	,	,	PUNCT
cana-1313	103	10	x	x	X
cana-1313	103	11	,	,	PUNCT
cana-1313	103	12	{	{	PUNCT
cana-1313	103	13	q	q	X
cana-1313	103	14	,	,	PUNCT
cana-1313	103	15	r	r	NOUN
cana-1313	103	16	}	}	PUNCT
cana-1313	103	17	.	.	PUNCT
cana-1313	104	1	let	let	VERB
cana-1313	104	2	p	p	NOUN
cana-1313	104	3	=	=	X
cana-1313	104	4	{	{	PUNCT
cana-1313	104	5	q	q	NOUN
cana-1313	104	6	,	,	PUNCT
cana-1313	104	7	r	r	NOUN
cana-1313	104	8	}	}	PUNCT
cana-1313	104	9	.	.	PUNCT
cana-1313	105	1	clearly	clearly	ADV
cana-1313	105	2	,	,	PUNCT
cana-1313	105	3	p	p	PRON
cana-1313	105	4	is	be	AUX
cana-1313	105	5	a	a	DET
cana-1313	105	6	d(r*g*)-closed	d(r*g*)-close	VERB
cana-1313	105	7	set	set	NOUN
cana-1313	105	8	but	but	CCONJ
cana-1313	105	9	not	not	PART
cana-1313	105	10	a	a	DET
cana-1313	105	11	b(r*g*)-closed	b(r*g*)-close	VERB
cana-1313	105	12	set	set	NOUN
cana-1313	105	13	.	.	PUNCT
cana-1313	106	1	theorem	theorem	VERB
cana-1313	106	2	3.14	3.14	NUM
cana-1313	106	3	:	:	PUNCT
cana-1313	106	4	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	106	5	and	and	CCONJ
cana-1313	106	6	d(r*g*)-closed	d(r*g*)-close	VERB
cana-1313	106	7	are	be	AUX
cana-1313	106	8	independent	independent	ADJ
cana-1313	106	9	notions	notion	NOUN
cana-1313	106	10	.	.	PUNCT
cana-1313	107	1	the	the	DET
cana-1313	107	2	following	follow	VERB
cana-1313	107	3	example	example	NOUN
cana-1313	107	4	will	will	AUX
cana-1313	107	5	demonstrate	demonstrate	VERB
cana-1313	107	6	this	this	PRON
cana-1313	107	7	.	.	PUNCT
cana-1313	108	1	example	example	NOUN
cana-1313	109	1	3.15	3.15	NUM
cana-1313	109	2	:	:	PUNCT
cana-1313	109	3	let	let	VERB
cana-1313	109	4	x	x	PUNCT
cana-1313	109	5	=	=	PUNCT
cana-1313	109	6	{	{	PUNCT
cana-1313	109	7	p	p	X
cana-1313	109	8	,	,	PUNCT
cana-1313	109	9	q	q	ADJ
cana-1313	109	10	,	,	PUNCT
cana-1313	109	11	r	r	NOUN
cana-1313	109	12	}	}	PUNCT
cana-1313	109	13	,	,	PUNCT
cana-1313	109	14	=	=	PRON
cana-1313	109	15	{	{	PUNCT
cana-1313	109	16	,	,	PUNCT
cana-1313	109	17	x	x	X
cana-1313	109	18	,	,	PUNCT
cana-1313	109	19	{	{	PUNCT
cana-1313	109	20	p	p	X
cana-1313	109	21	}	}	PUNCT
cana-1313	109	22	,	,	PUNCT
cana-1313	109	23	{	{	PUNCT
cana-1313	109	24	p	p	X
cana-1313	109	25	,	,	PUNCT
cana-1313	109	26	q	q	NOUN
cana-1313	109	27	}	}	PUNCT
cana-1313	109	28	,	,	PUNCT
cana-1313	109	29	{	{	PUNCT
cana-1313	109	30	p	p	X
cana-1313	109	31	,	,	PUNCT
cana-1313	109	32	r	r	NOUN
cana-1313	109	33	}	}	PUNCT
cana-1313	109	34	}	}	PUNCT
cana-1313	109	35	and	and	CCONJ
cana-1313	109	36	=	=	PRON
cana-1313	109	37	{	{	PUNCT
cana-1313	109	38	(	(	PUNCT
cana-1313	109	39	p	p	X
cana-1313	109	40	,	,	PUNCT
cana-1313	109	41	p	p	NOUN
cana-1313	109	42	)	)	PUNCT
cana-1313	109	43	,	,	PUNCT
cana-1313	109	44	(	(	PUNCT
cana-1313	109	45	q	q	X
cana-1313	109	46	,	,	PUNCT
cana-1313	109	47	q	q	NOUN
cana-1313	109	48	)	)	PUNCT
cana-1313	109	49	,	,	PUNCT
cana-1313	109	50	(	(	PUNCT
cana-1313	109	51	r	r	NOUN
cana-1313	109	52	,	,	PUNCT
cana-1313	109	53	r	r	NOUN
cana-1313	109	54	)	)	PUNCT
cana-1313	109	55	,	,	PUNCT
cana-1313	109	56	(	(	PUNCT
cana-1313	109	57	p	p	X
cana-1313	109	58	,	,	PUNCT
cana-1313	109	59	q	q	NOUN
cana-1313	109	60	)	)	PUNCT
cana-1313	109	61	,	,	PUNCT
cana-1313	109	62	(	(	PUNCT
cana-1313	109	63	p	p	X
cana-1313	109	64	,	,	PUNCT
cana-1313	109	65	r	r	NOUN
cana-1313	109	66	)	)	PUNCT
cana-1313	109	67	}	}	PUNCT
cana-1313	109	68	.	.	PUNCT
cana-1313	110	1	clearly	clearly	ADV
cana-1313	110	2	,	,	PUNCT
cana-1313	110	3	(	(	PUNCT
cana-1313	110	4	)	)	PUNCT
cana-1313	110	5	is	be	AUX
cana-1313	110	6	a	a	DET
cana-1313	110	7	topological	topological	ADJ
cana-1313	110	8	ordered	order	VERB
cana-1313	110	9	space	space	NOUN
cana-1313	110	10	.	.	PUNCT
cana-1313	111	1	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	111	2	sets	set	NOUN
cana-1313	111	3	are	be	AUX
cana-1313	111	4	,	,	PUNCT
cana-1313	111	5	x.	x.	NOUN
cana-1313	111	6	d(r*g*)-closed	d(r*g*)-closed	PROPN
cana-1313	111	7	sets	set	NOUN
cana-1313	111	8	are	be	AUX
cana-1313	111	9	,	,	PUNCT
cana-1313	111	10	x	x	X
cana-1313	111	11	,	,	PUNCT
cana-1313	111	12	{	{	PUNCT
cana-1313	111	13	q	q	X
cana-1313	111	14	,	,	PUNCT
cana-1313	111	15	r	r	NOUN
cana-1313	111	16	}	}	PUNCT
cana-1313	111	17	.	.	PUNCT
cana-1313	112	1	let	let	VERB
cana-1313	112	2	p	p	NOUN
cana-1313	112	3	=	=	X
cana-1313	112	4	{	{	PUNCT
cana-1313	112	5	q	q	NOUN
cana-1313	112	6	,	,	PUNCT
cana-1313	112	7	r	r	NOUN
cana-1313	112	8	}	}	PUNCT
cana-1313	112	9	.	.	PUNCT
cana-1313	113	1	clearly	clearly	ADV
cana-1313	113	2	,	,	PUNCT
cana-1313	113	3	p	p	PRON
cana-1313	113	4	is	be	AUX
cana-1313	113	5	a	a	DET
cana-1313	113	6	d(r*g*)-closed	d(r*g*)-close	VERB
cana-1313	113	7	set	set	NOUN
cana-1313	113	8	but	but	CCONJ
cana-1313	113	9	not	not	PART
cana-1313	113	10	be	be	AUX
cana-1313	113	11	an	an	DET
cana-1313	113	12	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	113	13	set	set	NOUN
cana-1313	113	14	.	.	PUNCT
cana-1313	113	15	example	example	NOUN
cana-1313	114	1	3.16	3.16	NUM
cana-1313	114	2	:	:	PUNCT
cana-1313	114	3	let	let	VERB
cana-1313	114	4	x	x	PUNCT
cana-1313	114	5	=	=	PUNCT
cana-1313	114	6	{	{	PUNCT
cana-1313	114	7	p	p	X
cana-1313	114	8	,	,	PUNCT
cana-1313	114	9	q	q	ADJ
cana-1313	114	10	,	,	PUNCT
cana-1313	114	11	r	r	NOUN
cana-1313	114	12	}	}	PUNCT
cana-1313	114	13	,	,	PUNCT
cana-1313	114	14	=	=	PRON
cana-1313	114	15	{	{	PUNCT
cana-1313	114	16	,	,	PUNCT
cana-1313	114	17	x	x	X
cana-1313	114	18	,	,	PUNCT
cana-1313	114	19	{	{	PUNCT
cana-1313	114	20	p	p	X
cana-1313	114	21	}	}	PUNCT
cana-1313	114	22	,	,	PUNCT
cana-1313	114	23	{	{	PUNCT
cana-1313	114	24	p	p	X
cana-1313	114	25	,	,	PUNCT
cana-1313	114	26	q	q	NOUN
cana-1313	114	27	}	}	PUNCT
cana-1313	114	28	,	,	PUNCT
cana-1313	114	29	{	{	PUNCT
cana-1313	114	30	p	p	X
cana-1313	114	31	,	,	PUNCT
cana-1313	114	32	r	r	NOUN
cana-1313	114	33	}	}	PUNCT
cana-1313	114	34	}	}	PUNCT
cana-1313	114	35	and	and	CCONJ
cana-1313	114	36	=	=	PRON
cana-1313	114	37	{	{	PUNCT
cana-1313	114	38	(	(	PUNCT
cana-1313	114	39	p	p	X
cana-1313	114	40	,	,	PUNCT
cana-1313	114	41	p	p	NOUN
cana-1313	114	42	)	)	PUNCT
cana-1313	114	43	,	,	PUNCT
cana-1313	114	44	(	(	PUNCT
cana-1313	114	45	q	q	X
cana-1313	114	46	,	,	PUNCT
cana-1313	114	47	q	q	NOUN
cana-1313	114	48	)	)	PUNCT
cana-1313	114	49	,	,	PUNCT
cana-1313	114	50	(	(	PUNCT
cana-1313	114	51	r	r	NOUN
cana-1313	114	52	,	,	PUNCT
cana-1313	114	53	r	r	NOUN
cana-1313	114	54	)	)	PUNCT
cana-1313	114	55	,	,	PUNCT
cana-1313	114	56	(	(	PUNCT
cana-1313	114	57	q	q	X
cana-1313	114	58	,	,	PUNCT
cana-1313	114	59	p	p	NOUN
cana-1313	114	60	)	)	PUNCT
cana-1313	114	61	,	,	PUNCT
cana-1313	114	62	(	(	PUNCT
cana-1313	114	63	r	r	NOUN
cana-1313	114	64	,	,	PUNCT
cana-1313	114	65	q	q	NOUN
cana-1313	114	66	)	)	PUNCT
cana-1313	114	67	,	,	PUNCT
cana-1313	114	68	(	(	PUNCT
cana-1313	114	69	r	r	NOUN
cana-1313	114	70	,	,	PUNCT
cana-1313	114	71	p	p	NOUN
cana-1313	114	72	)	)	PUNCT
cana-1313	114	73	}	}	PUNCT
cana-1313	114	74	.	.	PUNCT
cana-1313	115	1	clearly	clearly	ADV
cana-1313	115	2	,	,	PUNCT
cana-1313	115	3	a	a	DET
cana-1313	115	4	topological	topological	ADJ
cana-1313	115	5	ordered	order	VERB
cana-1313	115	6	space	space	NOUN
cana-1313	115	7	is	be	AUX
cana-1313	115	8	(	(	PUNCT
cana-1313	115	9	)	)	PUNCT
cana-1313	115	10	.	.	PUNCT
cana-1313	116	1	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	116	2	sets	set	NOUN
cana-1313	116	3	are	be	AUX
cana-1313	116	4	,	,	PUNCT
cana-1313	116	5	x	x	X
cana-1313	116	6	,	,	PUNCT
cana-1313	116	7	{	{	PUNCT
cana-1313	116	8	q	q	X
cana-1313	116	9	,	,	PUNCT
cana-1313	116	10	r	r	NOUN
cana-1313	116	11	}	}	PUNCT
cana-1313	116	12	.	.	PUNCT
cana-1313	117	1	d(r*g*)-closed	d(r*g*)-close	VERB
cana-1313	117	2	sets	set	NOUN
cana-1313	117	3	are	be	AUX
cana-1313	117	4	,	,	PUNCT
cana-1313	117	5	x.	x.	PROPN
cana-1313	117	6	le	le	PROPN
cana-1313	117	7	t	t	PROPN
cana-1313	118	1	p	p	X
cana-1313	118	2	=	=	X
cana-1313	118	3	{	{	PUNCT
cana-1313	118	4	q	q	NOUN
cana-1313	118	5	,	,	PUNCT
cana-1313	118	6	r	r	NOUN
cana-1313	118	7	}	}	PUNCT
cana-1313	118	8	.	.	PUNCT
cana-1313	119	1	clearly	clearly	ADV
cana-1313	119	2	,	,	PUNCT
cana-1313	119	3	p	p	NOUN
cana-1313	119	4	is	be	AUX
cana-1313	119	5	i(r*g*)-closed	i(r*g*)-close	VERB
cana-1313	119	6	set	set	VERB
cana-1313	119	7	but	but	CCONJ
cana-1313	119	8	not	not	PART
cana-1313	119	9	be	be	AUX
cana-1313	119	10	a	a	DET
cana-1313	119	11	d(r*g*)closed	d(r*g*)close	VERB
cana-1313	119	12	set	set	NOUN
cana-1313	119	13	.	.	PUNCT
cana-1313	120	1	theorem	theorem	VERB
cana-1313	120	2	3.17	3.17	NUM
cana-1313	120	3	:	:	PUNCT
cana-1313	120	4	every	every	DET
cana-1313	120	5	b(rg)-closed	b(rg)-close	VERB
cana-1313	120	6	set	set	NOUN
cana-1313	120	7	is	be	AUX
cana-1313	120	8	an	an	DET
cana-1313	120	9	i(rg)-closed	i(rg)-closed	ADJ
cana-1313	120	10	set	set	NOUN
cana-1313	120	11	.	.	PUNCT
cana-1313	121	1	proof	proof	NOUN
cana-1313	121	2	:	:	PUNCT
cana-1313	121	3	as	as	SCONJ
cana-1313	121	4	we	we	PRON
cana-1313	121	5	know	know	VERB
cana-1313	121	6	,	,	PUNCT
cana-1313	121	7	a	a	DET
cana-1313	121	8	balanced	balanced	ADJ
cana-1313	121	9	set	set	NOUN
cana-1313	121	10	is	be	AUX
cana-1313	121	11	always	always	ADV
cana-1313	121	12	an	an	DET
cana-1313	121	13	increasing	increase	VERB
cana-1313	121	14	set	set	NOUN
cana-1313	121	15	.	.	PUNCT
cana-1313	122	1	hence	hence	ADV
cana-1313	122	2	,	,	PUNCT
cana-1313	122	3	all	all	DET
cana-1313	122	4	b(rg)-closed	b(rg)-close	VERB
cana-1313	122	5	sets	set	NOUN
cana-1313	122	6	are	be	AUX
cana-1313	122	7	i(rg)closed	i(rg)close	VERB
cana-1313	122	8	sets	set	NOUN
cana-1313	122	9	.	.	PUNCT
cana-1313	123	1	generally	generally	ADV
cana-1313	123	2	the	the	DET
cana-1313	123	3	next	next	ADJ
cana-1313	123	4	example	example	NOUN
cana-1313	123	5	demonstrates	demonstrate	VERB
cana-1313	123	6	that	that	SCONJ
cana-1313	123	7	,	,	PUNCT
cana-1313	123	8	i(rg)-closed	i(rg)-closed	ADJ
cana-1313	123	9	sets	set	NOUN
cana-1313	123	10	do	do	AUX
cana-1313	123	11	not	not	PART
cana-1313	123	12	have	have	VERB
cana-1313	123	13	to	to	PART
cana-1313	123	14	be	be	AUX
cana-1313	123	15	b(rg)closed	b(rg)close	VERB
cana-1313	123	16	sets	set	NOUN
cana-1313	123	17	.	.	PUNCT
cana-1313	124	1	example	example	NOUN
cana-1313	124	2	3.18	3.18	NUM
cana-1313	124	3	:	:	PUNCT
cana-1313	124	4	let	let	VERB
cana-1313	124	5	x	x	PUNCT
cana-1313	124	6	=	=	PUNCT
cana-1313	124	7	{	{	PUNCT
cana-1313	124	8	p	p	X
cana-1313	124	9	,	,	PUNCT
cana-1313	124	10	q	q	ADJ
cana-1313	124	11	,	,	PUNCT
cana-1313	124	12	r	r	NOUN
cana-1313	124	13	}	}	PUNCT
cana-1313	124	14	,	,	PUNCT
cana-1313	124	15	=	=	PRON
cana-1313	124	16	{	{	PUNCT
cana-1313	124	17	,	,	PUNCT
cana-1313	124	18	x	x	X
cana-1313	124	19	,	,	PUNCT
cana-1313	124	20	{	{	PUNCT
cana-1313	124	21	p	p	X
cana-1313	124	22	}	}	PUNCT
cana-1313	124	23	,	,	PUNCT
cana-1313	124	24	{	{	PUNCT
cana-1313	124	25	p	p	X
cana-1313	124	26	,	,	PUNCT
cana-1313	124	27	r	r	NOUN
cana-1313	124	28	}	}	PUNCT
cana-1313	124	29	}	}	PUNCT
cana-1313	124	30	and	and	CCONJ
cana-1313	124	31	=	=	PRON
cana-1313	124	32	{	{	PUNCT
cana-1313	124	33	(	(	PUNCT
cana-1313	124	34	p	p	X
cana-1313	124	35	,	,	PUNCT
cana-1313	124	36	p	p	NOUN
cana-1313	124	37	)	)	PUNCT
cana-1313	124	38	,	,	PUNCT
cana-1313	124	39	(	(	PUNCT
cana-1313	124	40	q	q	X
cana-1313	124	41	,	,	PUNCT
cana-1313	124	42	q	q	NOUN
cana-1313	124	43	)	)	PUNCT
cana-1313	124	44	,	,	PUNCT
cana-1313	124	45	(	(	PUNCT
cana-1313	124	46	r	r	NOUN
cana-1313	124	47	,	,	PUNCT
cana-1313	124	48	r	r	NOUN
cana-1313	124	49	)	)	PUNCT
cana-1313	124	50	,	,	PUNCT
cana-1313	124	51	(	(	PUNCT
cana-1313	124	52	p	p	X
cana-1313	124	53	,	,	PUNCT
cana-1313	124	54	q	q	NOUN
cana-1313	124	55	)	)	PUNCT
cana-1313	124	56	,	,	PUNCT
cana-1313	124	57	(	(	PUNCT
cana-1313	124	58	r	r	NOUN
cana-1313	124	59	,	,	PUNCT
cana-1313	124	60	q	q	NOUN
cana-1313	124	61	)	)	PUNCT
cana-1313	124	62	}	}	PUNCT
cana-1313	124	63	.	.	PUNCT
cana-1313	125	1	clearly	clearly	ADV
cana-1313	125	2	,	,	PUNCT
cana-1313	125	3	(	(	PUNCT
cana-1313	125	4	)	)	PUNCT
cana-1313	125	5	is	be	AUX
cana-1313	125	6	a	a	DET
cana-1313	125	7	topological	topological	ADJ
cana-1313	125	8	ordered	order	VERB
cana-1313	125	9	space	space	NOUN
cana-1313	125	10	.	.	PUNCT
cana-1313	126	1	b(rg)-closed	b(rg)-close	VERB
cana-1313	126	2	sets	set	NOUN
cana-1313	126	3	are	be	AUX
cana-1313	126	4	,	,	PUNCT
cana-1313	126	5	x	x	X
cana-1313	126	6	,	,	PUNCT
cana-1313	126	7	{	{	PUNCT
cana-1313	126	8	p	p	X
cana-1313	126	9	,	,	PUNCT
cana-1313	126	10	q	q	NOUN
cana-1313	126	11	}	}	PUNCT
cana-1313	126	12	.	.	PUNCT
cana-1313	127	1	i(rg)-closed	i(rg)-closed	ADJ
cana-1313	127	2	sets	set	NOUN
cana-1313	127	3	are	be	AUX
cana-1313	127	4	,	,	PUNCT
cana-1313	127	5	x	x	X
cana-1313	127	6	,	,	PUNCT
cana-1313	127	7	{	{	PUNCT
cana-1313	127	8	q	q	X
cana-1313	127	9	}	}	PUNCT
cana-1313	127	10	,	,	PUNCT
cana-1313	127	11	{	{	PUNCT
cana-1313	127	12	p	p	X
cana-1313	127	13	,	,	PUNCT
cana-1313	127	14	q	q	NOUN
cana-1313	127	15	}	}	PUNCT
cana-1313	127	16	,	,	PUNCT
cana-1313	127	17	{	{	PUNCT
cana-1313	127	18	q	q	NOUN
cana-1313	127	19	,	,	PUNCT
cana-1313	127	20	r	r	NOUN
cana-1313	127	21	}	}	PUNCT
cana-1313	127	22	.	.	PUNCT
cana-1313	128	1	let	let	VERB
cana-1313	128	2	p	p	NOUN
cana-1313	128	3	=	=	X
cana-1313	128	4	{	{	PUNCT
cana-1313	128	5	p	p	X
cana-1313	128	6	}	}	PUNCT
cana-1313	128	7	.	.	PUNCT
cana-1313	129	1	clearly	clearly	ADV
cana-1313	129	2	,	,	PUNCT
cana-1313	129	3	p	p	PROPN
cana-1313	129	4	is	be	AUX
cana-1313	129	5	i(rg)-closed	i(rg)-close	VERB
cana-1313	129	6	set	set	ADJ
cana-1313	129	7	but	but	CCONJ
cana-1313	129	8	not	not	PART
cana-1313	129	9	a	a	DET
cana-1313	129	10	b(rg)-closed	b(rg)-close	VERB
cana-1313	129	11	set	set	NOUN
cana-1313	129	12	.	.	PUNCT
cana-1313	130	1	theorem	theorem	VERB
cana-1313	130	2	3.19	3.19	NUM
cana-1313	130	3	:	:	PUNCT
cana-1313	130	4	every	every	DET
cana-1313	130	5	b(rg)-closed	b(rg)-close	VERB
cana-1313	130	6	set	set	NOUN
cana-1313	130	7	is	be	AUX
cana-1313	130	8	a	a	DET
cana-1313	130	9	d(rg)-closed	d(rg)-close	VERB
cana-1313	130	10	set	set	NOUN
cana-1313	130	11	.	.	PUNCT
cana-1313	131	1	communications	communication	NOUN
cana-1313	131	2	on	on	ADP
cana-1313	131	3	applied	apply	VERB
cana-1313	131	4	nonlinear	nonlinear	ADJ
cana-1313	131	5	analysis	analysis	NOUN
cana-1313	131	6	issn	issn	NOUN
cana-1313	131	7	:	:	PUNCT
cana-1313	131	8	1074	1074	NUM
cana-1313	131	9	-	-	PUNCT
cana-1313	131	10	133x	133x	NUM
cana-1313	131	11	vol	vol	NOUN
cana-1313	131	12	31	31	NUM
cana-1313	131	13	no	no	NOUN
cana-1313	131	14	.	.	PUNCT
cana-1313	132	1	7s	7	NOUN
cana-1313	132	2	(	(	PUNCT
cana-1313	132	3	2024	2024	NUM
cana-1313	132	4	)	)	PUNCT
cana-1313	132	5	352	352	NUM
cana-1313	132	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1313	132	7	proof	proof	NOUN
cana-1313	132	8	:	:	PUNCT
cana-1313	132	9	every	every	DET
cana-1313	132	10	balanced	balanced	ADJ
cana-1313	132	11	set	set	NOUN
cana-1313	132	12	is	be	AUX
cana-1313	132	13	a	a	DET
cana-1313	132	14	decreasing	decrease	VERB
cana-1313	132	15	set	set	NOUN
cana-1313	132	16	,	,	PUNCT
cana-1313	132	17	as	as	SCONJ
cana-1313	132	18	we	we	PRON
cana-1313	132	19	are	be	AUX
cana-1313	132	20	aware	aware	ADJ
cana-1313	132	21	.	.	PUNCT
cana-1313	133	1	every	every	DET
cana-1313	133	2	b(rg)-closed	b(rg)-close	VERB
cana-1313	133	3	set	set	NOUN
cana-1313	133	4	is	be	AUX
cana-1313	133	5	a	a	DET
cana-1313	133	6	d(rg)closed	d(rg)closed	ADJ
cana-1313	133	7	set	set	NOUN
cana-1313	133	8	,	,	PUNCT
cana-1313	133	9	hence	hence	ADV
cana-1313	133	10	this	this	PRON
cana-1313	133	11	is	be	AUX
cana-1313	133	12	true	true	ADJ
cana-1313	133	13	.	.	PUNCT
cana-1313	134	1	the	the	DET
cana-1313	134	2	following	follow	VERB
cana-1313	134	3	illustration	illustration	NOUN
cana-1313	134	4	demonstrates	demonstrate	VERB
cana-1313	134	5	that	that	SCONJ
cana-1313	134	6	,	,	PUNCT
cana-1313	134	7	a	a	DET
cana-1313	134	8	d(rg)-closed	d(rg)-close	VERB
cana-1313	134	9	set	set	NOUN
cana-1313	134	10	does	do	AUX
cana-1313	134	11	not	not	PART
cana-1313	134	12	always	always	ADV
cana-1313	134	13	have	have	VERB
cana-1313	134	14	to	to	PART
cana-1313	134	15	be	be	AUX
cana-1313	134	16	a	a	DET
cana-1313	134	17	b(rg)-closed	b(rg)-close	VERB
cana-1313	134	18	set	set	NOUN
cana-1313	134	19	.	.	PUNCT
cana-1313	135	1	example	example	NOUN
cana-1313	135	2	3.20	3.20	NUM
cana-1313	135	3	:	:	PUNCT
cana-1313	135	4	let	let	VERB
cana-1313	135	5	x	x	PUNCT
cana-1313	135	6	=	=	PUNCT
cana-1313	135	7	{	{	PUNCT
cana-1313	135	8	p	p	X
cana-1313	135	9	,	,	PUNCT
cana-1313	135	10	q	q	ADJ
cana-1313	135	11	,	,	PUNCT
cana-1313	135	12	r	r	NOUN
cana-1313	135	13	}	}	PUNCT
cana-1313	135	14	,	,	PUNCT
cana-1313	135	15	=	=	PRON
cana-1313	135	16	{	{	PUNCT
cana-1313	135	17	,	,	PUNCT
cana-1313	135	18	x	x	X
cana-1313	135	19	,	,	PUNCT
cana-1313	135	20	{	{	PUNCT
cana-1313	135	21	p	p	X
cana-1313	135	22	}	}	PUNCT
cana-1313	135	23	,	,	PUNCT
cana-1313	135	24	{	{	PUNCT
cana-1313	135	25	p	p	X
cana-1313	135	26	,	,	PUNCT
cana-1313	135	27	r	r	NOUN
cana-1313	135	28	}	}	PUNCT
cana-1313	135	29	}	}	PUNCT
cana-1313	135	30	and	and	CCONJ
cana-1313	135	31	=	=	PRON
cana-1313	135	32	{	{	PUNCT
cana-1313	135	33	(	(	PUNCT
cana-1313	135	34	p	p	X
cana-1313	135	35	,	,	PUNCT
cana-1313	135	36	p	p	NOUN
cana-1313	135	37	)	)	PUNCT
cana-1313	135	38	,	,	PUNCT
cana-1313	135	39	(	(	PUNCT
cana-1313	135	40	q	q	X
cana-1313	135	41	,	,	PUNCT
cana-1313	135	42	q	q	NOUN
cana-1313	135	43	)	)	PUNCT
cana-1313	135	44	,	,	PUNCT
cana-1313	135	45	(	(	PUNCT
cana-1313	135	46	r	r	NOUN
cana-1313	135	47	,	,	PUNCT
cana-1313	135	48	r	r	NOUN
cana-1313	135	49	)	)	PUNCT
cana-1313	135	50	,	,	PUNCT
cana-1313	135	51	(	(	PUNCT
cana-1313	135	52	q	q	X
cana-1313	135	53	,	,	PUNCT
cana-1313	135	54	r	r	NOUN
cana-1313	135	55	)	)	PUNCT
cana-1313	135	56	,	,	PUNCT
cana-1313	135	57	(	(	PUNCT
cana-1313	135	58	p	p	X
cana-1313	135	59	,	,	PUNCT
cana-1313	135	60	r	r	NOUN
cana-1313	135	61	)	)	PUNCT
cana-1313	135	62	}	}	PUNCT
cana-1313	135	63	.	.	PUNCT
cana-1313	136	1	clearly	clearly	ADV
cana-1313	136	2	(	(	PUNCT
cana-1313	136	3	)	)	PUNCT
cana-1313	136	4	is	be	AUX
cana-1313	136	5	a	a	DET
cana-1313	136	6	topological	topological	ADJ
cana-1313	136	7	ordered	order	VERB
cana-1313	136	8	space	space	NOUN
cana-1313	136	9	.	.	PUNCT
cana-1313	137	1	b(rg)-closed	b(rg)-close	VERB
cana-1313	137	2	sets	set	NOUN
cana-1313	137	3	are	be	AUX
cana-1313	137	4	,	,	PUNCT
cana-1313	137	5	x.	x.	NOUN
cana-1313	137	6	d(rg)-closed	d(rg)-close	VERB
cana-1313	137	7	sets	set	NOUN
cana-1313	137	8	are	be	AUX
cana-1313	137	9	,	,	PUNCT
cana-1313	137	10	x	x	X
cana-1313	137	11	,	,	PUNCT
cana-1313	137	12	{	{	PUNCT
cana-1313	137	13	q	q	X
cana-1313	137	14	}	}	PUNCT
cana-1313	137	15	,	,	PUNCT
cana-1313	137	16	{	{	PUNCT
cana-1313	137	17	p	p	X
cana-1313	137	18	,	,	PUNCT
cana-1313	137	19	q	q	NOUN
cana-1313	137	20	}	}	PUNCT
cana-1313	137	21	.	.	PUNCT
cana-1313	138	1	let	let	VERB
cana-1313	138	2	p	p	NOUN
cana-1313	138	3	=	=	X
cana-1313	138	4	{	{	PUNCT
cana-1313	138	5	q	q	X
cana-1313	138	6	}	}	PUNCT
cana-1313	138	7	.	.	PUNCT
cana-1313	139	1	clearly	clearly	ADV
cana-1313	139	2	p	p	X
cana-1313	139	3	is	be	AUX
cana-1313	139	4	a	a	DET
cana-1313	139	5	d(rg)-closed	d(rg)-close	VERB
cana-1313	139	6	set	set	NOUN
cana-1313	139	7	but	but	CCONJ
cana-1313	139	8	not	not	PART
cana-1313	139	9	a	a	DET
cana-1313	139	10	b(rg)-closed	b(rg)-close	VERB
cana-1313	139	11	set	set	NOUN
cana-1313	139	12	.	.	PUNCT
cana-1313	140	1	theorem	theorem	VERB
cana-1313	140	2	3.21	3.21	NUM
cana-1313	140	3	:	:	PUNCT
cana-1313	140	4	i(rg)-closed	i(rg)-closed	ADJ
cana-1313	140	5	and	and	CCONJ
cana-1313	140	6	d(rg)-closed	d(rg)-close	VERB
cana-1313	140	7	are	be	AUX
cana-1313	140	8	independent	independent	ADJ
cana-1313	140	9	notions	notion	NOUN
cana-1313	140	10	.	.	PUNCT
cana-1313	141	1	this	this	PRON
cana-1313	141	2	will	will	AUX
cana-1313	141	3	be	be	AUX
cana-1313	141	4	demonstrated	demonstrate	VERB
cana-1313	141	5	by	by	ADP
cana-1313	141	6	the	the	DET
cana-1313	141	7	example	example	NOUN
cana-1313	141	8	that	that	PRON
cana-1313	141	9	follows	follow	VERB
cana-1313	141	10	.	.	PUNCT
cana-1313	142	1	example	example	NOUN
cana-1313	142	2	3.22	3.22	NUM
cana-1313	142	3	:	:	PUNCT
cana-1313	142	4	let	let	VERB
cana-1313	142	5	x	x	PUNCT
cana-1313	142	6	=	=	PUNCT
cana-1313	142	7	{	{	PUNCT
cana-1313	142	8	p	p	X
cana-1313	142	9	,	,	PUNCT
cana-1313	142	10	q	q	ADJ
cana-1313	142	11	,	,	PUNCT
cana-1313	142	12	r	r	NOUN
cana-1313	142	13	}	}	PUNCT
cana-1313	142	14	,	,	PUNCT
cana-1313	142	15	=	=	PRON
cana-1313	142	16	{	{	PUNCT
cana-1313	142	17	,	,	PUNCT
cana-1313	142	18	x	x	X
cana-1313	142	19	,	,	PUNCT
cana-1313	142	20	{	{	PUNCT
cana-1313	142	21	p	p	X
cana-1313	142	22	}	}	PUNCT
cana-1313	142	23	,	,	PUNCT
cana-1313	142	24	{	{	PUNCT
cana-1313	142	25	q	q	X
cana-1313	142	26	}	}	PUNCT
cana-1313	142	27	,	,	PUNCT
cana-1313	142	28	{	{	PUNCT
cana-1313	142	29	p	p	X
cana-1313	142	30	,	,	PUNCT
cana-1313	142	31	q	q	NOUN
cana-1313	142	32	}	}	PUNCT
cana-1313	142	33	}	}	PUNCT
cana-1313	142	34	and	and	CCONJ
cana-1313	142	35	=	=	PRON
cana-1313	142	36	{	{	PUNCT
cana-1313	142	37	(	(	PUNCT
cana-1313	142	38	p	p	X
cana-1313	142	39	,	,	PUNCT
cana-1313	142	40	p	p	NOUN
cana-1313	142	41	)	)	PUNCT
cana-1313	142	42	,	,	PUNCT
cana-1313	142	43	(	(	PUNCT
cana-1313	142	44	q	q	X
cana-1313	142	45	,	,	PUNCT
cana-1313	142	46	q	q	NOUN
cana-1313	142	47	)	)	PUNCT
cana-1313	142	48	,	,	PUNCT
cana-1313	142	49	(	(	PUNCT
cana-1313	142	50	r	r	NOUN
cana-1313	142	51	,	,	PUNCT
cana-1313	142	52	r	r	NOUN
cana-1313	142	53	)	)	PUNCT
cana-1313	142	54	,	,	PUNCT
cana-1313	142	55	(	(	PUNCT
cana-1313	142	56	p	p	X
cana-1313	142	57	,	,	PUNCT
cana-1313	142	58	q	q	NOUN
cana-1313	142	59	)	)	PUNCT
cana-1313	142	60	,	,	PUNCT
cana-1313	142	61	(	(	PUNCT
cana-1313	142	62	p	p	X
cana-1313	142	63	,	,	PUNCT
cana-1313	142	64	r	r	NOUN
cana-1313	142	65	)	)	PUNCT
cana-1313	142	66	}	}	PUNCT
cana-1313	142	67	.	.	PUNCT
cana-1313	143	1	clearly	clearly	ADV
cana-1313	143	2	,	,	PUNCT
cana-1313	143	3	a	a	DET
cana-1313	143	4	topological	topological	ADJ
cana-1313	143	5	ordered	order	VERB
cana-1313	143	6	space	space	NOUN
cana-1313	143	7	is	be	AUX
cana-1313	143	8	(	(	PUNCT
cana-1313	143	9	)	)	PUNCT
cana-1313	143	10	.	.	PUNCT
cana-1313	144	1	i(rg)-closed	i(rg)-closed	ADJ
cana-1313	144	2	sets	set	NOUN
cana-1313	144	3	are	be	AUX
cana-1313	144	4	,	,	PUNCT
cana-1313	144	5	x	x	X
cana-1313	144	6	,	,	PUNCT
cana-1313	144	7	{	{	PUNCT
cana-1313	144	8	r	r	NOUN
cana-1313	144	9	}	}	PUNCT
cana-1313	144	10	,	,	PUNCT
cana-1313	144	11	{	{	PUNCT
cana-1313	144	12	q	q	NOUN
cana-1313	144	13	,	,	PUNCT
cana-1313	144	14	r	r	NOUN
cana-1313	144	15	}	}	PUNCT
cana-1313	144	16	.	.	PUNCT
cana-1313	145	1	d(rg)closed	d(rg)closed	ADJ
cana-1313	145	2	sets	set	NOUN
cana-1313	145	3	are	be	AUX
cana-1313	145	4	,	,	PUNCT
cana-1313	145	5	x	x	X
cana-1313	145	6	,	,	PUNCT
cana-1313	145	7	{	{	PUNCT
cana-1313	145	8	p	p	X
cana-1313	145	9	,	,	PUNCT
cana-1313	145	10	q	q	NOUN
cana-1313	145	11	}	}	PUNCT
cana-1313	145	12	,	,	PUNCT
cana-1313	145	13	{	{	PUNCT
cana-1313	145	14	p	p	X
cana-1313	145	15	,	,	PUNCT
cana-1313	145	16	r	r	NOUN
cana-1313	145	17	}	}	PUNCT
cana-1313	145	18	.	.	PUNCT
cana-1313	146	1	let	let	VERB
cana-1313	146	2	p	p	NOUN
cana-1313	146	3	=	=	X
cana-1313	146	4	{	{	PUNCT
cana-1313	146	5	p	p	X
cana-1313	146	6	,	,	PUNCT
cana-1313	146	7	r	r	NOUN
cana-1313	146	8	}	}	PUNCT
cana-1313	146	9	.	.	PUNCT
cana-1313	147	1	clearly	clearly	ADV
cana-1313	147	2	,	,	PUNCT
cana-1313	147	3	p	p	PRON
cana-1313	147	4	is	be	AUX
cana-1313	147	5	a	a	DET
cana-1313	147	6	d(rg)-closed	d(rg)-close	VERB
cana-1313	147	7	set	set	NOUN
cana-1313	147	8	but	but	CCONJ
cana-1313	147	9	not	not	PART
cana-1313	147	10	i(rg)-closed	i(rg)-closed	ADJ
cana-1313	147	11	set	set	NOUN
cana-1313	147	12	.	.	PUNCT
cana-1313	148	1	let	let	VERB
cana-1313	148	2	q	q	NOUN
cana-1313	149	1	=	=	PUNCT
cana-1313	149	2	{	{	PUNCT
cana-1313	149	3	r	r	NOUN
cana-1313	149	4	}	}	PUNCT
cana-1313	149	5	.	.	PUNCT
cana-1313	150	1	clearly	clearly	ADV
cana-1313	150	2	,	,	PUNCT
cana-1313	150	3	q	q	PROPN
cana-1313	150	4	is	be	AUX
cana-1313	150	5	i(rg)-closed	i(rg)-close	VERB
cana-1313	150	6	set	set	ADJ
cana-1313	150	7	but	but	CCONJ
cana-1313	150	8	not	not	PART
cana-1313	150	9	a	a	DET
cana-1313	150	10	d(rg)-closed	d(rg)-close	VERB
cana-1313	150	11	set	set	NOUN
cana-1313	150	12	.	.	PUNCT
cana-1313	151	1	4	4	X
cana-1313	151	2	.	.	X
cana-1313	151	3	topological	topological	ADJ
cana-1313	151	4	ordered	order	VERB
cana-1313	151	5	space	space	NOUN
cana-1313	151	6	with	with	ADP
cana-1313	151	7	irg	irg	ADJ
cana-1313	151	8	-	-	PUNCT
cana-1313	151	9	closed	close	VERB
cana-1313	151	10	type	type	NOUN
cana-1313	151	11	sets	set	NOUN
cana-1313	151	12	theorem	theorem	VERB
cana-1313	151	13	4.1	4.1	NUM
cana-1313	151	14	:	:	PUNCT
cana-1313	151	15	a	a	DET
cana-1313	151	16	set	set	NOUN
cana-1313	152	1	p	p	X
cana-1313	152	2	q	q	PROPN
cana-1313	152	3	is	be	AUX
cana-1313	152	4	irg	irg	NOUN
cana-1313	152	5	-	-	ADJ
cana-1313	152	6	closed	closed	ADJ
cana-1313	152	7	if	if	SCONJ
cana-1313	152	8	p	p	NOUN
cana-1313	152	9	and	and	CCONJ
cana-1313	152	10	q	q	PROPN
cana-1313	152	11	are	be	AUX
cana-1313	152	12	irg	irg	ADJ
cana-1313	152	13	-	-	PUNCT
cana-1313	152	14	closed	close	VERB
cana-1313	152	15	sets	set	NOUN
cana-1313	152	16	.	.	PUNCT
cana-1313	153	1	proof	proof	NOUN
cana-1313	153	2	:	:	PUNCT
cana-1313	153	3	if	if	SCONJ
cana-1313	153	4	p	p	NOUN
cana-1313	153	5	q	q	X
cana-1313	153	6	⊆	⊆	NUM
cana-1313	153	7	r	r	NOUN
cana-1313	153	8	and	and	CCONJ
cana-1313	153	9	r	r	NOUN
cana-1313	153	10	is	be	AUX
cana-1313	153	11	regular	regular	ADV
cana-1313	153	12	-	-	PUNCT
cana-1313	153	13	open	open	ADJ
cana-1313	153	14	,	,	PUNCT
cana-1313	153	15	then	then	ADV
cana-1313	153	16	p	p	NOUN
cana-1313	153	17	⊆	⊆	NUM
cana-1313	153	18	r	r	NOUN
cana-1313	153	19	and	and	CCONJ
cana-1313	153	20	q	q	NOUN
cana-1313	153	21	⊆	⊆	NUM
cana-1313	153	22	r.	r.	NOUN
cana-1313	153	23	but	but	CCONJ
cana-1313	153	24	p	p	NOUN
cana-1313	153	25	and	and	CCONJ
cana-1313	153	26	q	q	PROPN
cana-1313	153	27	are	be	AUX
cana-1313	153	28	irg	irg	ADJ
cana-1313	153	29	-	-	PUNCT
cana-1313	153	30	closed	closed	ADJ
cana-1313	153	31	and	and	CCONJ
cana-1313	153	32	therefore	therefore	ADV
cana-1313	153	33	icl(p	icl(p	PROPN
cana-1313	153	34	)	)	PUNCT
cana-1313	153	35	⊆	⊆	NUM
cana-1313	153	36	r	r	NOUN
cana-1313	153	37	and	and	CCONJ
cana-1313	153	38	icl(q	icl(q	PROPN
cana-1313	153	39	)	)	PUNCT
cana-1313	153	40	⊆	⊆	PROPN
cana-1313	153	41	r.	r.	PROPN
cana-1313	153	42	therefore	therefore	ADV
cana-1313	153	43	,	,	PUNCT
cana-1313	153	44	(	(	PUNCT
cana-1313	153	45	icl	icl	PROPN
cana-1313	153	46	(	(	PUNCT
cana-1313	153	47	p	p	NOUN
cana-1313	153	48	)	)	PUNCT
cana-1313	153	49	icl	icl	PROPN
cana-1313	153	50	(	(	PUNCT
cana-1313	153	51	q	q	NOUN
cana-1313	153	52	)	)	PUNCT
cana-1313	153	53	)	)	PUNCT
cana-1313	154	1	⊆	⊆	NUM
cana-1313	154	2	r	r	NOUN
cana-1313	154	3	icl	icl	PROPN
cana-1313	154	4	(	(	PUNCT
cana-1313	154	5	p	p	NOUN
cana-1313	154	6	q	q	NOUN
cana-1313	154	7	)	)	PUNCT
cana-1313	154	8	⊆	⊆	NUM
cana-1313	154	9	r.	r.	PROPN
cana-1313	154	10	hence	hence	ADV
cana-1313	154	11	,	,	PUNCT
cana-1313	154	12	p	p	DET
cana-1313	154	13	q	q	PROPN
cana-1313	154	14	is	be	AUX
cana-1313	154	15	irg	irg	ADJ
cana-1313	154	16	-	-	PUNCT
cana-1313	154	17	closed	closed	ADJ
cana-1313	154	18	.	.	PUNCT
cana-1313	155	1	example	example	NOUN
cana-1313	155	2	4.2	4.2	NUM
cana-1313	155	3	:	:	PUNCT
cana-1313	155	4	let	let	VERB
cana-1313	155	5	x	x	PUNCT
cana-1313	155	6	=	=	PUNCT
cana-1313	155	7	{	{	PUNCT
cana-1313	155	8	p	p	X
cana-1313	155	9	,	,	PUNCT
cana-1313	155	10	q	q	ADJ
cana-1313	155	11	,	,	PUNCT
cana-1313	155	12	r	r	NOUN
cana-1313	155	13	}	}	PUNCT
cana-1313	155	14	,	,	PUNCT
cana-1313	155	15	=	=	PRON
cana-1313	155	16	{	{	PUNCT
cana-1313	155	17	,	,	PUNCT
cana-1313	155	18	x	x	X
cana-1313	155	19	,	,	PUNCT
cana-1313	155	20	{	{	PUNCT
cana-1313	155	21	p	p	X
cana-1313	155	22	}	}	PUNCT
cana-1313	155	23	,	,	PUNCT
cana-1313	155	24	{	{	PUNCT
cana-1313	155	25	q	q	X
cana-1313	155	26	}	}	PUNCT
cana-1313	155	27	,	,	PUNCT
cana-1313	155	28	{	{	PUNCT
cana-1313	155	29	p	p	X
cana-1313	155	30	,	,	PUNCT
cana-1313	155	31	q	q	NOUN
cana-1313	155	32	}	}	PUNCT
cana-1313	155	33	}	}	PUNCT
cana-1313	155	34	and	and	CCONJ
cana-1313	155	35	=	=	PRON
cana-1313	155	36	{	{	PUNCT
cana-1313	155	37	(	(	PUNCT
cana-1313	155	38	p	p	X
cana-1313	155	39	,	,	PUNCT
cana-1313	155	40	p	p	NOUN
cana-1313	155	41	)	)	PUNCT
cana-1313	155	42	,	,	PUNCT
cana-1313	155	43	(	(	PUNCT
cana-1313	155	44	q	q	X
cana-1313	155	45	,	,	PUNCT
cana-1313	155	46	q	q	NOUN
cana-1313	155	47	)	)	PUNCT
cana-1313	155	48	,	,	PUNCT
cana-1313	155	49	(	(	PUNCT
cana-1313	155	50	r	r	NOUN
cana-1313	155	51	,	,	PUNCT
cana-1313	155	52	r	r	NOUN
cana-1313	155	53	)	)	PUNCT
cana-1313	155	54	,	,	PUNCT
cana-1313	155	55	(	(	PUNCT
cana-1313	155	56	p	p	X
cana-1313	155	57	,	,	PUNCT
cana-1313	155	58	q	q	NOUN
cana-1313	155	59	)	)	PUNCT
cana-1313	155	60	}	}	PUNCT
cana-1313	155	61	.	.	PUNCT
cana-1313	156	1	clearly	clearly	ADV
cana-1313	156	2	,	,	PUNCT
cana-1313	156	3	a	a	DET
cana-1313	156	4	topological	topological	ADJ
cana-1313	156	5	ordered	order	VERB
cana-1313	156	6	space	space	NOUN
cana-1313	156	7	is	be	AUX
cana-1313	156	8	(	(	PUNCT
cana-1313	156	9	)	)	PUNCT
cana-1313	156	10	.	.	PUNCT
cana-1313	157	1	take	take	VERB
cana-1313	157	2	,	,	PUNCT
cana-1313	157	3	p	p	NOUN
cana-1313	157	4	=	=	X
cana-1313	157	5	{	{	PUNCT
cana-1313	157	6	r	r	NOUN
cana-1313	157	7	}	}	PUNCT
cana-1313	157	8	,	,	PUNCT
cana-1313	157	9	q	q	NOUN
cana-1313	157	10	=	=	PUNCT
cana-1313	157	11	{	{	PUNCT
cana-1313	157	12	p	p	X
cana-1313	157	13	,	,	PUNCT
cana-1313	157	14	q	q	NOUN
cana-1313	157	15	}	}	PUNCT
cana-1313	157	16	.	.	PUNCT
cana-1313	158	1	if	if	SCONJ
cana-1313	158	2	(	(	PUNCT
cana-1313	158	3	p	p	NOUN
cana-1313	158	4	q	q	NOUN
cana-1313	158	5	)	)	PUNCT
cana-1313	158	6	=	=	SYM
cana-1313	158	7	{	{	PUNCT
cana-1313	158	8	r	r	NOUN
cana-1313	158	9	}	}	PUNCT
cana-1313	158	10	{	{	PUNCT
cana-1313	158	11	p	p	NOUN
cana-1313	158	12	,	,	PUNCT
cana-1313	158	13	q	q	NOUN
cana-1313	158	14	}	}	PUNCT
cana-1313	158	15	=	=	SYM
cana-1313	158	16	{	{	PUNCT
cana-1313	158	17	p	p	X
cana-1313	158	18	,	,	PUNCT
cana-1313	158	19	q	q	ADJ
cana-1313	158	20	,	,	PUNCT
cana-1313	158	21	r	r	NOUN
cana-1313	158	22	}	}	PUNCT
cana-1313	158	23	⊆	⊆	NUM
cana-1313	158	24	r	r	NOUN
cana-1313	158	25	=	=	SYM
cana-1313	158	26	x	x	X
cana-1313	158	27	and	and	CCONJ
cana-1313	158	28	r	r	NOUN
cana-1313	158	29	is	be	AUX
cana-1313	158	30	regular	regular	ADV
cana-1313	158	31	-	-	PUNCT
cana-1313	158	32	open	open	ADJ
cana-1313	158	33	,	,	PUNCT
cana-1313	158	34	then	then	ADV
cana-1313	158	35	{	{	PUNCT
cana-1313	158	36	r}⊆	r}⊆	NOUN
cana-1313	158	37	r	r	NOUN
cana-1313	158	38	and	and	CCONJ
cana-1313	158	39	{	{	PUNCT
cana-1313	158	40	p	p	X
cana-1313	158	41	,	,	PUNCT
cana-1313	158	42	q	q	ADJ
cana-1313	158	43	}	}	PUNCT
cana-1313	158	44	⊆	⊆	NUM
cana-1313	158	45	r.	r.	NOUN
cana-1313	158	46	but	but	CCONJ
cana-1313	158	47	p	p	NOUN
cana-1313	158	48	and	and	CCONJ
cana-1313	158	49	q	q	PROPN
cana-1313	158	50	are	be	AUX
cana-1313	158	51	irg	irg	ADJ
cana-1313	158	52	-	-	PUNCT
cana-1313	158	53	closed	closed	ADJ
cana-1313	158	54	and	and	CCONJ
cana-1313	158	55	therefore	therefore	ADV
cana-1313	158	56	icl(p	icl(p	PROPN
cana-1313	158	57	)	)	PUNCT
cana-1313	158	58	⊆	⊆	NUM
cana-1313	158	59	r	r	NOUN
cana-1313	158	60	and	and	CCONJ
cana-1313	158	61	icl(q	icl(q	PROPN
cana-1313	158	62	)	)	PUNCT
cana-1313	159	1	⊆	⊆	PROPN
cana-1313	159	2	r.	r.	PROPN
cana-1313	159	3	therefore	therefore	ADV
cana-1313	159	4	,	,	PUNCT
cana-1313	159	5	(	(	PUNCT
cana-1313	159	6	icl(p	icl(p	PROPN
cana-1313	159	7	)	)	PUNCT
cana-1313	159	8	icl	icl	PROPN
cana-1313	159	9	(	(	PUNCT
cana-1313	159	10	q	q	NOUN
cana-1313	159	11	)	)	PUNCT
cana-1313	159	12	)	)	PUNCT
cana-1313	160	1	⊆	⊆	NUM
cana-1313	160	2	r	r	NOUN
cana-1313	160	3	,	,	PUNCT
cana-1313	160	4	and	and	CCONJ
cana-1313	160	5	hence	hence	ADV
cana-1313	160	6	icl(p	icl(p	PROPN
cana-1313	160	7	q	q	NOUN
cana-1313	160	8	)	)	PUNCT
cana-1313	160	9	⊆	⊆	NUM
cana-1313	160	10	r.	r.	NOUN
cana-1313	160	11	hence	hence	ADV
cana-1313	160	12	p	p	PROPN
cana-1313	160	13	q	q	PROPN
cana-1313	160	14	is	be	AUX
cana-1313	160	15	irg	irg	ADJ
cana-1313	160	16	-	-	ADJ
cana-1313	160	17	closed	closed	ADJ
cana-1313	160	18	.	.	PUNCT
cana-1313	161	1	theorem	theorem	VERB
cana-1313	161	2	4.3	4.3	NUM
cana-1313	161	3	:	:	PUNCT
cana-1313	161	4	suppose	suppose	VERB
cana-1313	161	5	that	that	SCONJ
cana-1313	161	6	q	q	PROPN
cana-1313	161	7	⊆	⊆	NUM
cana-1313	161	8	p	p	ADP
cana-1313	161	9	⊆	⊆	NUM
cana-1313	161	10	x	x	NOUN
cana-1313	161	11	,	,	PUNCT
cana-1313	161	12	p	p	PRON
cana-1313	161	13	is	be	AUX
cana-1313	161	14	an	an	DET
cana-1313	161	15	ig	ig	NOUN
cana-1313	161	16	-	-	PUNCT
cana-1313	161	17	closed	closed	ADJ
cana-1313	161	18	open	open	ADJ
cana-1313	161	19	subset	subset	NOUN
cana-1313	161	20	of	of	ADP
cana-1313	161	21	x	x	PUNCT
cana-1313	161	22	and	and	CCONJ
cana-1313	161	23	q	q	PROPN
cana-1313	161	24	is	be	AUX
cana-1313	161	25	an	an	DET
cana-1313	161	26	irg	irg	NOUN
cana-1313	161	27	-	-	PUNCT
cana-1313	161	28	closed	close	VERB
cana-1313	161	29	set	set	NOUN
cana-1313	161	30	in	in	ADP
cana-1313	161	31	relation	relation	NOUN
cana-1313	161	32	to	to	ADP
cana-1313	161	33	p.	p.	NOUN
cana-1313	161	34	then	then	ADV
cana-1313	161	35	,	,	PUNCT
cana-1313	161	36	q	q	PROPN
cana-1313	161	37	is	be	AUX
cana-1313	161	38	irg	irg	PROPN
cana-1313	161	39	-	-	PUNCT
cana-1313	161	40	closed	close	VERB
cana-1313	161	41	with	with	ADP
cana-1313	161	42	respect	respect	NOUN
cana-1313	161	43	to	to	ADP
cana-1313	161	44	x.	x.	NOUN
cana-1313	161	45	proof	proof	NOUN
cana-1313	161	46	:	:	PUNCT
cana-1313	161	47	let	let	VERB
cana-1313	161	48	q	q	PRON
cana-1313	161	49	⊆	⊆	NUM
cana-1313	161	50	r	r	NOUN
cana-1313	161	51	and	and	CCONJ
cana-1313	161	52	let	let	VERB
cana-1313	161	53	r	r	PRON
cana-1313	161	54	be	be	AUX
cana-1313	161	55	regular	regular	ADV
cana-1313	161	56	-	-	PUNCT
cana-1313	161	57	open	open	ADJ
cana-1313	161	58	.	.	PUNCT
cana-1313	162	1	we	we	PRON
cana-1313	162	2	have	have	VERB
cana-1313	162	3	q	q	NOUN
cana-1313	162	4	⊆	⊆	NUM
cana-1313	162	5	(	(	PUNCT
cana-1313	162	6	p	p	NOUN
cana-1313	162	7	r	r	NOUN
cana-1313	162	8	)	)	PUNCT
cana-1313	162	9	.	.	PUNCT
cana-1313	163	1	but	but	CCONJ
cana-1313	163	2	q	q	NOUN
cana-1313	163	3	is	be	AUX
cana-1313	163	4	an	an	DET
cana-1313	163	5	irg	irg	ADJ
cana-1313	163	6	-	-	PUNCT
cana-1313	163	7	closed	closed	ADJ
cana-1313	163	8	set	set	NOUN
cana-1313	163	9	relative	relative	ADJ
cana-1313	163	10	to	to	ADP
cana-1313	163	11	p.	p.	NOUN
cana-1313	164	1	hence	hence	ADV
cana-1313	164	2	i	i	PRON
cana-1313	164	3	(	(	PUNCT
cana-1313	164	4	)	)	PUNCT
cana-1313	164	5	⊆	⊆	X
cana-1313	164	6	(	(	PUNCT
cana-1313	164	7	p	p	NOUN
cana-1313	164	8	r	r	NOUN
cana-1313	164	9	)	)	PUNCT
cana-1313	164	10	.	.	PUNCT
cana-1313	165	1	(	(	PUNCT
cana-1313	165	2	1	1	X
cana-1313	165	3	)	)	PUNCT
cana-1313	165	4	note	note	NOUN
cana-1313	165	5	that	that	SCONJ
cana-1313	165	6	p	p	PROPN
cana-1313	165	7	r	r	NOUN
cana-1313	165	8	is	be	AUX
cana-1313	165	9	regular	regular	ADV
cana-1313	165	10	-	-	PUNCT
cana-1313	165	11	open	open	ADJ
cana-1313	165	12	in	in	ADP
cana-1313	165	13	p.	p.	NOUN
cana-1313	166	1	but	but	CCONJ
cana-1313	166	2	i	i	PRON
cana-1313	166	3	(	(	PUNCT
cana-1313	166	4	)	)	PUNCT
cana-1313	166	5	=	=	SYM
cana-1313	166	6	icl(q	icl(q	PROPN
cana-1313	166	7	)	)	PUNCT
cana-1313	166	8	p	p	NOUN
cana-1313	166	9	(	(	PUNCT
cana-1313	166	10	2	2	NUM
cana-1313	166	11	)	)	PUNCT
cana-1313	166	12	from	from	ADP
cana-1313	166	13	(	(	PUNCT
cana-1313	166	14	1	1	NUM
cana-1313	166	15	)	)	PUNCT
cana-1313	166	16	and	and	CCONJ
cana-1313	166	17	(	(	PUNCT
cana-1313	166	18	2	2	NUM
cana-1313	166	19	)	)	PUNCT
cana-1313	166	20	,	,	PUNCT
cana-1313	166	21	(	(	PUNCT
cana-1313	166	22	p	p	NOUN
cana-1313	166	23	icl(q	icl(q	PROPN
cana-1313	166	24	)	)	PUNCT
cana-1313	166	25	)	)	PUNCT
cana-1313	167	1	⊆	⊆	NUM
cana-1313	167	2	(	(	PUNCT
cana-1313	167	3	p	p	NOUN
cana-1313	167	4	r	r	NOUN
cana-1313	167	5	)	)	PUNCT
cana-1313	167	6	consequently	consequently	ADV
cana-1313	167	7	p	p	X
cana-1313	167	8	icl	icl	PROPN
cana-1313	167	9	(	(	PUNCT
cana-1313	167	10	q	q	X
cana-1313	167	11	)	)	PUNCT
cana-1313	167	12	⊆	⊆	NUM
cana-1313	167	13	r.	r.	PROPN
cana-1313	167	14	hence	hence	ADV
cana-1313	167	15	,	,	PUNCT
cana-1313	167	16	p	p	X
cana-1313	167	17	(	(	PUNCT
cana-1313	167	18	icl(q	icl(q	PROPN
cana-1313	167	19	)	)	PUNCT
cana-1313	167	20	c(icl(q	c(icl(q	ADJ
cana-1313	167	21	)	)	PUNCT
cana-1313	167	22	)	)	PUNCT
cana-1313	167	23	)	)	PUNCT
cana-1313	168	1	⊆	⊆	NUM
cana-1313	168	2	r	r	NOUN
cana-1313	168	3	c(icl(q	c(icl(q	ADJ
cana-1313	168	4	)	)	PUNCT
cana-1313	168	5	)	)	PUNCT
cana-1313	168	6	.	.	PUNCT
cana-1313	169	1	that	that	PRON
cana-1313	169	2	is	be	AUX
cana-1313	169	3	p	p	NOUN
cana-1313	169	4	x	x	X
cana-1313	169	5	⊆	⊆	NUM
cana-1313	169	6	(	(	PUNCT
cana-1313	169	7	r	r	NOUN
cana-1313	169	8	c(icl(q	c(icl(q	ADJ
cana-1313	169	9	)	)	PUNCT
cana-1313	169	10	)	)	PUNCT
cana-1313	169	11	)	)	PUNCT
cana-1313	169	12	.	.	PUNCT
cana-1313	170	1	so	so	ADV
cana-1313	170	2	p	p	ADP
cana-1313	170	3	⊆	⊆	NUM
cana-1313	170	4	(	(	PUNCT
cana-1313	170	5	r	r	NOUN
cana-1313	170	6	c(icl(q	c(icl(q	ADJ
cana-1313	170	7	)	)	PUNCT
cana-1313	170	8	)	)	PUNCT
cana-1313	170	9	)	)	PUNCT
cana-1313	171	1	=	=	SYM
cana-1313	171	2	g	g	NOUN
cana-1313	171	3	,	,	PUNCT
cana-1313	171	4	say	say	VERB
cana-1313	171	5	(	(	PUNCT
cana-1313	171	6	3	3	NUM
cana-1313	171	7	)	)	PUNCT
cana-1313	171	8	but	but	CCONJ
cana-1313	171	9	then	then	ADV
cana-1313	171	10	g	g	PROPN
cana-1313	171	11	is	be	AUX
cana-1313	171	12	an	an	DET
cana-1313	171	13	open	open	ADJ
cana-1313	171	14	set	set	NOUN
cana-1313	171	15	.	.	PUNCT
cana-1313	172	1	since	since	SCONJ
cana-1313	172	2	p	p	NOUN
cana-1313	172	3	is	be	AUX
cana-1313	172	4	ig	ig	NOUN
cana-1313	172	5	-	-	VERB
cana-1313	172	6	closed	closed	ADJ
cana-1313	172	7	in	in	ADP
cana-1313	172	8	x	x	NOUN
cana-1313	172	9	,	,	PUNCT
cana-1313	172	10	from	from	ADP
cana-1313	172	11	(	(	PUNCT
cana-1313	172	12	3	3	X
cana-1313	172	13	)	)	PUNCT
cana-1313	172	14	we	we	PRON
cana-1313	172	15	have	have	AUX
cana-1313	172	16	icl(p	icl(p	NOUN
cana-1313	172	17	)	)	PUNCT
cana-1313	172	18	⊆	⊆	NUM
cana-1313	172	19	(	(	PUNCT
cana-1313	172	20	r	r	NOUN
cana-1313	172	21	c(icl(q	c(icl(q	ADJ
cana-1313	172	22	)	)	PUNCT
cana-1313	172	23	)	)	PUNCT
cana-1313	172	24	)	)	PUNCT
cana-1313	173	1	=	=	SYM
cana-1313	173	2	g	g	NOUN
cana-1313	173	3	(	(	PUNCT
cana-1313	173	4	4	4	NUM
cana-1313	173	5	)	)	PUNCT
cana-1313	173	6	but	but	CCONJ
cana-1313	173	7	icl(q	icl(q	X
cana-1313	173	8	)	)	PUNCT
cana-1313	173	9	⊆	⊆	NUM
cana-1313	173	10	icl(p	icl(p	X
cana-1313	173	11	)	)	PUNCT
cana-1313	173	12	(	(	PUNCT
cana-1313	173	13	5	5	NUM
cana-1313	173	14	)	)	PUNCT
cana-1313	173	15	from	from	ADP
cana-1313	173	16	(	(	PUNCT
cana-1313	173	17	4	4	NUM
cana-1313	173	18	)	)	PUNCT
cana-1313	173	19	and	and	CCONJ
cana-1313	173	20	(	(	PUNCT
cana-1313	173	21	5	5	X
cana-1313	173	22	)	)	PUNCT
cana-1313	173	23	we	we	PRON
cana-1313	173	24	have	have	VERB
cana-1313	173	25	icl(q	icl(q	NOUN
cana-1313	173	26	)	)	PUNCT
cana-1313	173	27	⊆	⊆	NUM
cana-1313	173	28	(	(	PUNCT
cana-1313	173	29	r	r	NOUN
cana-1313	173	30	c(icl(q	c(icl(q	ADJ
cana-1313	173	31	)	)	PUNCT
cana-1313	173	32	)	)	PUNCT
cana-1313	173	33	)	)	PUNCT
cana-1313	173	34	.	.	PUNCT
cana-1313	174	1	hence	hence	ADV
cana-1313	174	2	icl(q	icl(q	PROPN
cana-1313	174	3	)	)	PUNCT
cana-1313	174	4	⊆	⊆	NUM
cana-1313	174	5	r	r	NOUN
cana-1313	174	6	because	because	SCONJ
cana-1313	174	7	icl(q	icl(q	PROPN
cana-1313	174	8	)	)	PUNCT
cana-1313	174	9	c(icl(q	c(icl(q	ADJ
cana-1313	174	10	)	)	PUNCT
cana-1313	174	11	)	)	PUNCT
cana-1313	175	1	=	=	PUNCT
cana-1313	175	2	q	q	PROPN
cana-1313	175	3	is	be	AUX
cana-1313	175	4	irg	irg	ADJ
cana-1313	175	5	-	-	ADJ
cana-1313	175	6	closed	closed	ADJ
cana-1313	175	7	relative	relative	ADJ
cana-1313	175	8	to	to	ADP
cana-1313	175	9	x.	x.	NOUN
cana-1313	175	10	corollary	corollary	PROPN
cana-1313	175	11	4.4	4.4	NUM
cana-1313	175	12	:	:	PUNCT
cana-1313	175	13	let	let	VERB
cana-1313	175	14	p	p	PRON
cana-1313	175	15	be	be	AUX
cana-1313	175	16	an	an	DET
cana-1313	175	17	ig	ig	NOUN
cana-1313	175	18	-	-	PUNCT
cana-1313	175	19	closed	closed	ADJ
cana-1313	175	20	,	,	PUNCT
cana-1313	175	21	open	open	ADJ
cana-1313	175	22	set	set	NOUN
cana-1313	175	23	.	.	PUNCT
cana-1313	176	1	suppose	suppose	VERB
cana-1313	176	2	that	that	SCONJ
cana-1313	176	3	q	q	NOUN
cana-1313	176	4	is	be	AUX
cana-1313	176	5	an	an	DET
cana-1313	176	6	i	i	NOUN
cana-1313	176	7	-	-	PUNCT
cana-1313	176	8	closed	close	VERB
cana-1313	176	9	set	set	NOUN
cana-1313	176	10	.	.	PUNCT
cana-1313	177	1	then	then	ADV
cana-1313	177	2	p	p	X
cana-1313	177	3	q	q	PROPN
cana-1313	177	4	is	be	AUX
cana-1313	177	5	an	an	DET
cana-1313	177	6	irg	irg	ADJ
cana-1313	177	7	-	-	PUNCT
cana-1313	177	8	closed	closed	ADJ
cana-1313	177	9	set	set	NOUN
cana-1313	177	10	relative	relative	ADJ
cana-1313	177	11	to	to	ADP
cana-1313	177	12	x.	x.	NOUN
cana-1313	177	13	communications	communication	NOUN
cana-1313	177	14	on	on	ADP
cana-1313	177	15	applied	apply	VERB
cana-1313	177	16	nonlinear	nonlinear	ADJ
cana-1313	177	17	analysis	analysis	NOUN
cana-1313	177	18	issn	issn	NOUN
cana-1313	177	19	:	:	PUNCT
cana-1313	177	20	1074	1074	NUM
cana-1313	177	21	-	-	PUNCT
cana-1313	177	22	133x	133x	NUM
cana-1313	177	23	vol	vol	NOUN
cana-1313	177	24	31	31	NUM
cana-1313	177	25	no	no	NOUN
cana-1313	177	26	.	.	PUNCT
cana-1313	178	1	7s	7	NOUN
cana-1313	178	2	(	(	PUNCT
cana-1313	178	3	2024	2024	NUM
cana-1313	178	4	)	)	PUNCT
cana-1313	178	5	353	353	NUM
cana-1313	178	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1313	178	7	proof	proof	NOUN
cana-1313	178	8	:	:	PUNCT
cana-1313	178	9	we	we	PRON
cana-1313	178	10	have	have	VERB
cana-1313	178	11	that	that	DET
cana-1313	178	12	p	p	PRON
cana-1313	178	13	q	q	NOUN
cana-1313	178	14	is	be	AUX
cana-1313	178	15	closed	close	VERB
cana-1313	178	16	in	in	ADP
cana-1313	178	17	p.	p.	NOUN
cana-1313	179	1	hence	hence	ADV
cana-1313	179	2	icl(p	icl(p	PROPN
cana-1313	179	3	q	q	NOUN
cana-1313	179	4	)	)	PUNCT
cana-1313	179	5	=	=	PUNCT
cana-1313	180	1	p	p	NOUN
cana-1313	180	2	q	q	NOUN
cana-1313	180	3	in	in	ADP
cana-1313	180	4	p.	p.	NOUN
cana-1313	180	5	let	let	VERB
cana-1313	180	6	p	p	NOUN
cana-1313	180	7	q	q	NOUN
cana-1313	180	8	⊆	⊆	NUM
cana-1313	180	9	r	r	NOUN
cana-1313	180	10	,	,	PUNCT
cana-1313	180	11	where	where	SCONJ
cana-1313	180	12	r	r	NOUN
cana-1313	180	13	is	be	AUX
cana-1313	180	14	regular	regular	ADV
cana-1313	180	15	-	-	PUNCT
cana-1313	180	16	open	open	ADJ
cana-1313	180	17	in	in	ADP
cana-1313	180	18	p.	p.	NOUN
cana-1313	180	19	that	that	PRON
cana-1313	180	20	is	be	AUX
cana-1313	180	21	icl(p	icl(p	PROPN
cana-1313	180	22	q	q	NOUN
cana-1313	180	23	)	)	PUNCT
cana-1313	180	24	⊆	⊆	NUM
cana-1313	180	25	r.	r.	NOUN
cana-1313	180	26	hence	hence	ADV
cana-1313	180	27	p	p	PROPN
cana-1313	180	28	q	q	PROPN
cana-1313	180	29	is	be	AUX
cana-1313	180	30	an	an	DET
cana-1313	180	31	irg	irg	NOUN
cana-1313	180	32	-	-	PUNCT
cana-1313	180	33	closed	closed	ADJ
cana-1313	180	34	set	set	NOUN
cana-1313	180	35	in	in	ADP
cana-1313	180	36	the	the	DET
cana-1313	180	37	ig	ig	NOUN
cana-1313	180	38	-	-	PUNCT
cana-1313	180	39	closed	closed	ADJ
cana-1313	180	40	p.	p.	NOUN
cana-1313	180	41	by	by	ADP
cana-1313	180	42	the	the	DET
cana-1313	180	43	theorem	theorem	NOUN
cana-1313	181	1	[	[	X
cana-1313	181	2	4.3	4.3	NUM
cana-1313	181	3	]	]	PUNCT
cana-1313	181	4	,	,	PUNCT
cana-1313	181	5	p	p	DET
cana-1313	181	6	q	q	PROPN
cana-1313	181	7	is	be	AUX
cana-1313	181	8	an	an	DET
cana-1313	181	9	irg	irg	ADJ
cana-1313	181	10	-	-	PUNCT
cana-1313	181	11	closed	closed	ADJ
cana-1313	181	12	set	set	NOUN
cana-1313	181	13	relative	relative	ADJ
cana-1313	181	14	to	to	ADP
cana-1313	181	15	x.	x.	NOUN
cana-1313	181	16	theorem	theorem	VERB
cana-1313	181	17	4.5	4.5	NUM
cana-1313	181	18	:	:	PUNCT
cana-1313	181	19	if	if	SCONJ
cana-1313	181	20	a	a	DET
cana-1313	181	21	set	set	NOUN
cana-1313	181	22	p	p	NOUN
cana-1313	181	23	is	be	AUX
cana-1313	181	24	irg	irg	NOUN
cana-1313	181	25	-	-	PUNCT
cana-1313	181	26	closed	closed	ADJ
cana-1313	181	27	then	then	ADV
cana-1313	181	28	icl(p)\p	icl(p)\p	ADJ
cana-1313	181	29	contains	contain	VERB
cana-1313	181	30	no	no	DET
cana-1313	181	31	nonempty	nonempty	ADJ
cana-1313	181	32	regular	regular	ADJ
cana-1313	181	33	-	-	PUNCT
cana-1313	181	34	closed	close	VERB
cana-1313	181	35	set	set	NOUN
cana-1313	181	36	.	.	PUNCT
cana-1313	182	1	proof	proof	NOUN
cana-1313	182	2	:	:	PUNCT
cana-1313	182	3	suppose	suppose	VERB
cana-1313	182	4	that	that	SCONJ
cana-1313	182	5	p	p	PROPN
cana-1313	182	6	is	be	AUX
cana-1313	182	7	irg	irg	NOUN
cana-1313	182	8	-	-	PUNCT
cana-1313	182	9	closed	closed	ADJ
cana-1313	182	10	.	.	PUNCT
cana-1313	183	1	let	let	VERB
cana-1313	183	2	s	s	PRON
cana-1313	183	3	be	be	AUX
cana-1313	183	4	a	a	DET
cana-1313	183	5	regular	regular	ADJ
cana-1313	183	6	-	-	PUNCT
cana-1313	183	7	closed	closed	ADJ
cana-1313	183	8	subset	subset	NOUN
cana-1313	183	9	of	of	ADP
cana-1313	183	10	icl(p)\p	icl(p)\p	PROPN
cana-1313	183	11	.	.	PUNCT
cana-1313	184	1	then	then	ADV
cana-1313	184	2	s	s	VERB
cana-1313	184	3	⊆	⊆	NUM
cana-1313	184	4	(	(	PUNCT
cana-1313	184	5	icl(p	icl(p	NOUN
cana-1313	184	6	)	)	PUNCT
cana-1313	184	7	c(p	c(p	NOUN
cana-1313	184	8	)	)	PUNCT
cana-1313	184	9	)	)	PUNCT
cana-1313	185	1	and	and	CCONJ
cana-1313	185	2	so	so	ADV
cana-1313	185	3	p	p	PRON
cana-1313	185	4	⊆	⊆	NUM
cana-1313	185	5	c(s	c(	NOUN
cana-1313	185	6	)	)	PUNCT
cana-1313	185	7	.	.	PUNCT
cana-1313	186	1	but	but	CCONJ
cana-1313	186	2	p	p	PROPN
cana-1313	186	3	is	be	AUX
cana-1313	186	4	irg	irg	NOUN
cana-1313	186	5	-	-	PUNCT
cana-1313	186	6	closed	closed	ADJ
cana-1313	186	7	.	.	PUNCT
cana-1313	187	1	therefore	therefore	ADV
cana-1313	187	2	icl(p	icl(p	PROPN
cana-1313	187	3	)	)	PUNCT
cana-1313	187	4	⊆	⊆	NUM
cana-1313	187	5	c(s	c(	NOUN
cana-1313	187	6	)	)	PUNCT
cana-1313	187	7	.	.	PUNCT
cana-1313	188	1	(	(	PUNCT
cana-1313	188	2	1	1	X
cana-1313	188	3	)	)	PUNCT
cana-1313	188	4	consequently	consequently	ADV
cana-1313	188	5	s	s	VERB
cana-1313	188	6	⊆	⊆	NUM
cana-1313	188	7	c(icl(p	c(icl(p	NOUN
cana-1313	188	8	)	)	PUNCT
cana-1313	188	9	)	)	PUNCT
cana-1313	189	1	(	(	PUNCT
cana-1313	189	2	2	2	X
cana-1313	189	3	)	)	PUNCT
cana-1313	189	4	we	we	PRON
cana-1313	189	5	have	have	AUX
cana-1313	189	6	already	already	ADV
cana-1313	189	7	s	s	VERB
cana-1313	189	8	⊆	⊆	NUM
cana-1313	189	9	icl(p	icl(p	X
cana-1313	189	10	)	)	PUNCT
cana-1313	189	11	(	(	PUNCT
cana-1313	189	12	3	3	X
cana-1313	189	13	)	)	PUNCT
cana-1313	189	14	from	from	ADP
cana-1313	189	15	(	(	PUNCT
cana-1313	189	16	2	2	NUM
cana-1313	189	17	)	)	PUNCT
cana-1313	189	18	and	and	CCONJ
cana-1313	189	19	(	(	PUNCT
cana-1313	189	20	3	3	NUM
cana-1313	189	21	)	)	PUNCT
cana-1313	189	22	,	,	PUNCT
cana-1313	189	23	s	s	VERB
cana-1313	189	24	⊆	⊆	NUM
cana-1313	189	25	(	(	PUNCT
cana-1313	189	26	icl(p	icl(p	NOUN
cana-1313	189	27	)	)	PUNCT
cana-1313	189	28	c(icl(p	c(icl(p	NOUN
cana-1313	189	29	)	)	PUNCT
cana-1313	189	30	)	)	PUNCT
cana-1313	189	31	)	)	PUNCT
cana-1313	190	1	=	=	PUNCT
cana-1313	190	2	thus	thus	ADV
cana-1313	190	3	s	s	VERB
cana-1313	190	4	=	=	X
cana-1313	190	5	.	.	PUNCT
cana-1313	191	1	therefore	therefore	ADV
cana-1313	191	2	icl(p)\p	icl(p)\p	PROPN
cana-1313	191	3	contains	contain	VERB
cana-1313	191	4	no	no	DET
cana-1313	191	5	nonempty	nonempty	ADJ
cana-1313	191	6	regular	regular	ADJ
cana-1313	191	7	-	-	PUNCT
cana-1313	191	8	closed	close	VERB
cana-1313	191	9	set	set	NOUN
cana-1313	191	10	.	.	PUNCT
cana-1313	192	1	corollary	corollary	ADJ
cana-1313	192	2	4.6	4.6	NUM
cana-1313	192	3	:	:	PUNCT
cana-1313	192	4	let	let	VERB
cana-1313	192	5	p	p	PRON
cana-1313	192	6	be	be	AUX
cana-1313	192	7	an	an	DET
cana-1313	192	8	irg	irg	ADJ
cana-1313	192	9	-	-	PUNCT
cana-1313	192	10	closed	closed	ADJ
cana-1313	192	11	set	set	NOUN
cana-1313	192	12	.	.	PUNCT
cana-1313	193	1	if	if	SCONJ
cana-1313	193	2	p	p	NOUN
cana-1313	193	3	is	be	AUX
cana-1313	193	4	regular	regular	ADV
cana-1313	193	5	-	-	PUNCT
cana-1313	193	6	closed	closed	ADJ
cana-1313	193	7	then	then	ADV
cana-1313	193	8	cl(int(p))\p	cl(int(p))\p	NOUN
cana-1313	193	9	is	be	AUX
cana-1313	193	10	regular	regular	ADV
cana-1313	193	11	-	-	PUNCT
cana-1313	193	12	closed	closed	ADJ
cana-1313	193	13	.	.	PUNCT
cana-1313	194	1	proof	proof	NOUN
cana-1313	194	2	:	:	PUNCT
cana-1313	194	3	let	let	VERB
cana-1313	194	4	p	p	PRON
cana-1313	194	5	be	be	AUX
cana-1313	194	6	an	an	DET
cana-1313	194	7	irg	irg	NOUN
cana-1313	194	8	-	-	PUNCT
cana-1313	194	9	closed	closed	ADJ
cana-1313	194	10	.	.	PUNCT
cana-1313	195	1	if	if	SCONJ
cana-1313	195	2	p	p	NOUN
cana-1313	195	3	is	be	AUX
cana-1313	195	4	regular	regular	ADV
cana-1313	195	5	-	-	PUNCT
cana-1313	195	6	closed	closed	ADJ
cana-1313	195	7	i.e.	i.e.	X
cana-1313	195	8	,	,	PUNCT
cana-1313	195	9	cl(int(p	cl(int(p	NOUN
cana-1313	195	10	)	)	PUNCT
cana-1313	195	11	)	)	PUNCT
cana-1313	196	1	=	=	SYM
cana-1313	197	1	p.	p.	NOUN
cana-1313	197	2	then	then	ADV
cana-1313	197	3	cl(int(p))\p	cl(int(p))\p	VERB
cana-1313	197	4	=	=	X
cana-1313	197	5	p\p	p\p	PROPN
cana-1313	197	6	=	=	X
cana-1313	197	7	.	.	PUNCT
cana-1313	198	1	but	but	CCONJ
cana-1313	198	2	,	,	PUNCT
cana-1313	198	3	is	be	AUX
cana-1313	198	4	always	always	ADV
cana-1313	198	5	regular	regular	ADV
cana-1313	198	6	-	-	PUNCT
cana-1313	198	7	closed	closed	ADJ
cana-1313	198	8	.	.	PUNCT
cana-1313	199	1	as	as	ADP
cana-1313	199	2	a	a	DET
cana-1313	199	3	result	result	NOUN
cana-1313	199	4	,	,	PUNCT
cana-1313	199	5	cl(int(p))\p	cl(int(p))\p	NOUN
cana-1313	199	6	is	be	AUX
cana-1313	199	7	regular	regular	ADV
cana-1313	199	8	-	-	PUNCT
cana-1313	199	9	closed	closed	ADJ
cana-1313	199	10	.	.	PUNCT
cana-1313	200	1	on	on	ADP
cana-1313	200	2	the	the	DET
cana-1313	200	3	other	other	ADJ
cana-1313	200	4	hand	hand	NOUN
cana-1313	200	5	,	,	PUNCT
cana-1313	200	6	imagine	imagine	VERB
cana-1313	200	7	that	that	DET
cana-1313	200	8	cl(int(p))\p	cl(int(p))\p	NOUN
cana-1313	200	9	is	be	AUX
cana-1313	200	10	regular	regular	ADV
cana-1313	200	11	-	-	PUNCT
cana-1313	200	12	closed	closed	ADJ
cana-1313	200	13	.	.	PUNCT
cana-1313	201	1	however	however	ADV
cana-1313	201	2	,	,	PUNCT
cana-1313	201	3	p	p	PROPN
cana-1313	201	4	is	be	AUX
cana-1313	201	5	irg	irg	NOUN
cana-1313	201	6	-	-	PUNCT
cana-1313	201	7	closed	closed	ADJ
cana-1313	201	8	.	.	PUNCT
cana-1313	202	1	additionally	additionally	ADV
cana-1313	202	2	,	,	PUNCT
cana-1313	202	3	the	the	DET
cana-1313	202	4	regular	regular	ADJ
cana-1313	202	5	-	-	PUNCT
cana-1313	202	6	closed	close	VERB
cana-1313	202	7	set	set	ADJ
cana-1313	202	8	cl(int(p))\p	cl(int(p))\p	NOUN
cana-1313	202	9	is	be	AUX
cana-1313	202	10	contained	contain	VERB
cana-1313	202	11	in	in	ADP
cana-1313	202	12	icl(p)\p	icl(p)\p	ADJ
cana-1313	202	13	.	.	PUNCT
cana-1313	203	1	by	by	ADP
cana-1313	203	2	above	above	ADP
cana-1313	203	3	theorem	theorem	NOUN
cana-1313	203	4	[	[	PUNCT
cana-1313	203	5	4.5	4.5	NUM
cana-1313	203	6	]	]	PUNCT
cana-1313	203	7	,	,	PUNCT
cana-1313	203	8	cl(int(p))\p	cl(int(p))\p	ADJ
cana-1313	204	1	=	=	SYM
cana-1313	204	2	.	.	PUNCT
cana-1313	204	3	hence	hence	ADV
cana-1313	204	4	cl(int(p	cl(int(p	NOUN
cana-1313	204	5	)	)	PUNCT
cana-1313	204	6	)	)	PUNCT
cana-1313	205	1	=	=	VERB
cana-1313	206	1	p.	p.	NOUN
cana-1313	206	2	therefore	therefore	ADV
cana-1313	206	3	p	p	PROPN
cana-1313	206	4	is	be	AUX
cana-1313	206	5	regular	regular	ADV
cana-1313	206	6	-	-	PUNCT
cana-1313	206	7	closed	closed	ADJ
cana-1313	206	8	.	.	PUNCT
cana-1313	207	1	theorem	theorem	VERB
cana-1313	207	2	4.7	4.7	NUM
cana-1313	207	3	:	:	PUNCT
cana-1313	207	4	if	if	SCONJ
cana-1313	207	5	p	p	NOUN
cana-1313	207	6	is	be	AUX
cana-1313	207	7	ig	ig	PROPN
cana-1313	207	8	-	-	VERB
cana-1313	207	9	closed	closed	ADJ
cana-1313	207	10	then	then	ADV
cana-1313	207	11	p	p	NOUN
cana-1313	207	12	is	be	AUX
cana-1313	207	13	irg	irg	NOUN
cana-1313	207	14	-	-	ADJ
cana-1313	207	15	closed	closed	ADJ
cana-1313	207	16	.	.	PUNCT
cana-1313	208	1	proof	proof	NOUN
cana-1313	208	2	:	:	PUNCT
cana-1313	208	3	suppose	suppose	VERB
cana-1313	208	4	that	that	SCONJ
cana-1313	208	5	p	p	PROPN
cana-1313	208	6	⊆	⊆	NUM
cana-1313	208	7	r	r	NOUN
cana-1313	208	8	,	,	PUNCT
cana-1313	208	9	where	where	SCONJ
cana-1313	208	10	r	r	NOUN
cana-1313	208	11	is	be	AUX
cana-1313	208	12	regular	regular	ADV
cana-1313	208	13	-	-	PUNCT
cana-1313	208	14	open	open	ADJ
cana-1313	208	15	.	.	PUNCT
cana-1313	209	1	now	now	ADV
cana-1313	209	2	r	r	VERB
cana-1313	209	3	regular	regular	ADJ
cana-1313	209	4	-	-	PUNCT
cana-1313	209	5	open	open	NOUN
cana-1313	209	6	implies	imply	VERB
cana-1313	209	7	that	that	SCONJ
cana-1313	209	8	r	r	NOUN
cana-1313	209	9	is	be	AUX
cana-1313	209	10	open	open	ADJ
cana-1313	209	11	.	.	PUNCT
cana-1313	210	1	thus	thus	ADV
cana-1313	210	2	p	p	X
cana-1313	210	3	⊆	⊆	NUM
cana-1313	210	4	r	r	NOUN
cana-1313	210	5	and	and	CCONJ
cana-1313	210	6	r	r	NOUN
cana-1313	210	7	is	be	AUX
cana-1313	210	8	open	open	ADJ
cana-1313	210	9	.	.	PUNCT
cana-1313	211	1	but	but	CCONJ
cana-1313	211	2	p	p	NOUN
cana-1313	211	3	is	be	AUX
cana-1313	211	4	ig	ig	PRON
cana-1313	211	5	-	-	PUNCT
cana-1313	211	6	cosed	cosed	ADJ
cana-1313	211	7	.	.	PUNCT
cana-1313	212	1	hence	hence	ADV
cana-1313	212	2	icl(p	icl(p	PROPN
cana-1313	212	3	)	)	PUNCT
cana-1313	212	4	⊆	⊆	PROPN
cana-1313	212	5	r.	r.	PROPN
cana-1313	212	6	therefore	therefore	ADV
cana-1313	212	7	,	,	PUNCT
cana-1313	212	8	p	p	PROPN
cana-1313	212	9	is	be	AUX
cana-1313	212	10	irg	irg	NOUN
cana-1313	212	11	-	-	PUNCT
cana-1313	212	12	closed	closed	ADJ
cana-1313	212	13	.	.	PUNCT
cana-1313	213	1	the	the	DET
cana-1313	213	2	following	follow	VERB
cana-1313	213	3	illustration	illustration	NOUN
cana-1313	213	4	demonstrates	demonstrate	VERB
cana-1313	213	5	that	that	SCONJ
cana-1313	213	6	an	an	DET
cana-1313	213	7	irg	irg	PROPN
cana-1313	213	8	-	-	PUNCT
cana-1313	213	9	closed	closed	ADJ
cana-1313	213	10	set	set	NOUN
cana-1313	213	11	need	need	AUX
cana-1313	213	12	not	not	PART
cana-1313	213	13	always	always	ADV
cana-1313	213	14	be	be	AUX
cana-1313	213	15	an	an	DET
cana-1313	213	16	ig	ig	NOUN
cana-1313	213	17	-	-	PUNCT
cana-1313	213	18	closed	closed	ADJ
cana-1313	213	19	set	set	NOUN
cana-1313	213	20	.	.	PUNCT
cana-1313	214	1	example	example	NOUN
cana-1313	214	2	4.8	4.8	NUM
cana-1313	214	3	:	:	PUNCT
cana-1313	214	4	let	let	VERB
cana-1313	214	5	x	x	PUNCT
cana-1313	214	6	=	=	PUNCT
cana-1313	214	7	{	{	PUNCT
cana-1313	214	8	p	p	X
cana-1313	214	9	,	,	PUNCT
cana-1313	214	10	q	q	ADJ
cana-1313	214	11	,	,	PUNCT
cana-1313	214	12	r	r	NOUN
cana-1313	214	13	}	}	PUNCT
cana-1313	214	14	,	,	PUNCT
cana-1313	214	15	=	=	PRON
cana-1313	214	16	{	{	PUNCT
cana-1313	214	17	,	,	PUNCT
cana-1313	214	18	x	x	X
cana-1313	214	19	,	,	PUNCT
cana-1313	214	20	{	{	PUNCT
cana-1313	214	21	p	p	X
cana-1313	214	22	}	}	PUNCT
cana-1313	214	23	,	,	PUNCT
cana-1313	214	24	{	{	PUNCT
cana-1313	214	25	q	q	X
cana-1313	214	26	}	}	PUNCT
cana-1313	214	27	,	,	PUNCT
cana-1313	214	28	{	{	PUNCT
cana-1313	214	29	p	p	X
cana-1313	214	30	,	,	PUNCT
cana-1313	214	31	q	q	NOUN
cana-1313	214	32	}	}	PUNCT
cana-1313	214	33	}	}	PUNCT
cana-1313	214	34	and	and	CCONJ
cana-1313	214	35	=	=	PRON
cana-1313	214	36	{	{	PUNCT
cana-1313	214	37	(	(	PUNCT
cana-1313	214	38	p	p	X
cana-1313	214	39	,	,	PUNCT
cana-1313	214	40	p	p	NOUN
cana-1313	214	41	)	)	PUNCT
cana-1313	214	42	,	,	PUNCT
cana-1313	214	43	(	(	PUNCT
cana-1313	214	44	q	q	X
cana-1313	214	45	,	,	PUNCT
cana-1313	214	46	q	q	NOUN
cana-1313	214	47	)	)	PUNCT
cana-1313	214	48	,	,	PUNCT
cana-1313	214	49	(	(	PUNCT
cana-1313	214	50	r	r	NOUN
cana-1313	214	51	,	,	PUNCT
cana-1313	214	52	r	r	NOUN
cana-1313	214	53	)	)	PUNCT
cana-1313	214	54	,	,	PUNCT
cana-1313	214	55	(	(	PUNCT
cana-1313	214	56	p	p	X
cana-1313	214	57	,	,	PUNCT
cana-1313	214	58	q	q	NOUN
cana-1313	214	59	)	)	PUNCT
cana-1313	214	60	,	,	PUNCT
cana-1313	214	61	(	(	PUNCT
cana-1313	214	62	r	r	NOUN
cana-1313	214	63	,	,	PUNCT
cana-1313	214	64	q	q	NOUN
cana-1313	214	65	)	)	PUNCT
cana-1313	214	66	}	}	PUNCT
cana-1313	214	67	.	.	PUNCT
cana-1313	215	1	clearly	clearly	ADV
cana-1313	215	2	(	(	PUNCT
cana-1313	215	3	)	)	PUNCT
cana-1313	215	4	is	be	AUX
cana-1313	215	5	a	a	DET
cana-1313	215	6	topological	topological	ADJ
cana-1313	215	7	ordered	order	VERB
cana-1313	215	8	space	space	NOUN
cana-1313	215	9	.	.	PUNCT
cana-1313	216	1	irg	irg	ADJ
cana-1313	216	2	-	-	PUNCT
cana-1313	216	3	closed	close	VERB
cana-1313	216	4	sets	set	NOUN
cana-1313	216	5	are	be	AUX
cana-1313	216	6	,	,	PUNCT
cana-1313	216	7	x	x	X
cana-1313	216	8	,	,	PUNCT
cana-1313	216	9	{	{	PUNCT
cana-1313	216	10	p	p	X
cana-1313	216	11	,	,	PUNCT
cana-1313	216	12	q	q	NOUN
cana-1313	216	13	}	}	PUNCT
cana-1313	216	14	,	,	PUNCT
cana-1313	216	15	{	{	PUNCT
cana-1313	216	16	q	q	NOUN
cana-1313	216	17	,	,	PUNCT
cana-1313	216	18	r	r	NOUN
cana-1313	216	19	}	}	PUNCT
cana-1313	216	20	.	.	PUNCT
cana-1313	217	1	igclosed	igclose	VERB
cana-1313	217	2	sets	set	NOUN
cana-1313	217	3	are	be	AUX
cana-1313	217	4	,	,	PUNCT
cana-1313	217	5	x	x	X
cana-1313	217	6	,	,	PUNCT
cana-1313	217	7	{	{	PUNCT
cana-1313	217	8	q	q	X
cana-1313	217	9	,	,	PUNCT
cana-1313	217	10	r	r	NOUN
cana-1313	217	11	}	}	PUNCT
cana-1313	217	12	.	.	PUNCT
cana-1313	218	1	let	let	VERB
cana-1313	218	2	p	p	NOUN
cana-1313	218	3	=	=	X
cana-1313	218	4	{	{	PUNCT
cana-1313	218	5	p	p	X
cana-1313	218	6	,	,	PUNCT
cana-1313	218	7	q	q	NOUN
cana-1313	218	8	}	}	PUNCT
cana-1313	218	9	.	.	PUNCT
cana-1313	219	1	clearly	clearly	ADV
cana-1313	219	2	p	p	PROPN
cana-1313	219	3	is	be	AUX
cana-1313	219	4	an	an	DET
cana-1313	219	5	irg	irg	ADJ
cana-1313	219	6	-	-	PUNCT
cana-1313	219	7	closed	closed	ADJ
cana-1313	219	8	set	set	NOUN
cana-1313	219	9	but	but	CCONJ
cana-1313	219	10	not	not	PART
cana-1313	219	11	an	an	DET
cana-1313	219	12	ig	ig	ADV
cana-1313	219	13	-	-	PUNCT
cana-1313	219	14	closed	closed	ADJ
cana-1313	219	15	set	set	NOUN
cana-1313	219	16	.	.	PUNCT
cana-1313	220	1	theorem	theorem	VERB
cana-1313	220	2	4.9	4.9	NUM
cana-1313	220	3	:	:	PUNCT
cana-1313	220	4	if	if	SCONJ
cana-1313	220	5	p	p	NOUN
cana-1313	220	6	is	be	AUX
cana-1313	220	7	irg	irg	NOUN
cana-1313	220	8	-	-	ADJ
cana-1313	220	9	closed	closed	ADJ
cana-1313	220	10	and	and	CCONJ
cana-1313	220	11	p	p	ADP
cana-1313	220	12	⊆	⊆	NUM
cana-1313	220	13	q	q	NOUN
cana-1313	220	14	⊆	⊆	NUM
cana-1313	220	15	icl(p	icl(p	NOUN
cana-1313	220	16	)	)	PUNCT
cana-1313	220	17	then	then	ADV
cana-1313	220	18	icl(q)\q	icl(q)\q	PROPN
cana-1313	220	19	contains	contain	VERB
cana-1313	220	20	no	no	DET
cana-1313	220	21	nonempty	nonempty	ADV
cana-1313	220	22	regularclosed	regularclose	VERB
cana-1313	220	23	set	set	NOUN
cana-1313	220	24	.	.	PUNCT
cana-1313	221	1	proof	proof	NOUN
cana-1313	221	2	:	:	PUNCT
cana-1313	221	3	suppose	suppose	VERB
cana-1313	221	4	p	p	PROPN
cana-1313	221	5	is	be	AUX
cana-1313	221	6	irg	irg	NOUN
cana-1313	221	7	-	-	ADJ
cana-1313	221	8	closed	closed	ADJ
cana-1313	221	9	and	and	CCONJ
cana-1313	221	10	p	p	ADP
cana-1313	221	11	⊆	⊆	NUM
cana-1313	221	12	q	q	NOUN
cana-1313	221	13	⊆	⊆	NUM
cana-1313	221	14	icl(p	icl(p	NUM
cana-1313	221	15	)	)	PUNCT
cana-1313	221	16	.	.	PUNCT
cana-1313	222	1	since	since	SCONJ
cana-1313	222	2	p	p	NOUN
cana-1313	222	3	⊆	⊆	NUM
cana-1313	222	4	q	q	NOUN
cana-1313	222	5	c(q	c(q	PROPN
cana-1313	222	6	)	)	PUNCT
cana-1313	222	7	⊆	⊆	NUM
cana-1313	222	8	c(p	c(p	NOUN
cana-1313	222	9	)	)	PUNCT
cana-1313	222	10	(	(	PUNCT
cana-1313	222	11	1	1	X
cana-1313	222	12	)	)	PUNCT
cana-1313	222	13	since	since	SCONJ
cana-1313	222	14	q	q	PROPN
cana-1313	222	15	⊆	⊆	NUM
cana-1313	222	16	icl(p	icl(p	SYM
cana-1313	222	17	)	)	PUNCT
cana-1313	222	18	icl(q	icl(q	PROPN
cana-1313	222	19	)	)	PUNCT
cana-1313	222	20	⊆	⊆	NUM
cana-1313	222	21	icl(icl(p	icl(icl(p	NUM
cana-1313	222	22	)	)	PUNCT
cana-1313	222	23	)	)	PUNCT
cana-1313	222	24	⊆	⊆	NUM
cana-1313	222	25	icl(p	icl(p	X
cana-1313	222	26	)	)	PUNCT
cana-1313	222	27	(	(	PUNCT
cana-1313	222	28	2	2	X
cana-1313	222	29	)	)	PUNCT
cana-1313	222	30	that	that	PRON
cana-1313	222	31	is	be	AUX
cana-1313	222	32	icl(q	icl(q	PROPN
cana-1313	222	33	)	)	PUNCT
cana-1313	222	34	⊆	⊆	NUM
cana-1313	222	35	icl(p	icl(p	X
cana-1313	222	36	)	)	PUNCT
cana-1313	222	37	.	.	PUNCT
cana-1313	223	1	from	from	ADP
cana-1313	223	2	(	(	PUNCT
cana-1313	223	3	1	1	NUM
cana-1313	223	4	)	)	PUNCT
cana-1313	223	5	and	and	CCONJ
cana-1313	223	6	(	(	PUNCT
cana-1313	223	7	2	2	NUM
cana-1313	223	8	)	)	PUNCT
cana-1313	223	9	,	,	PUNCT
cana-1313	223	10	(	(	PUNCT
cana-1313	223	11	icl(q	icl(q	PROPN
cana-1313	223	12	)	)	PUNCT
cana-1313	223	13	c(q	c(q	PROPN
cana-1313	223	14	)	)	PUNCT
cana-1313	223	15	)	)	PUNCT
cana-1313	224	1	⊆	⊆	NUM
cana-1313	224	2	(	(	PUNCT
cana-1313	224	3	icl(p	icl(p	NOUN
cana-1313	224	4	)	)	PUNCT
cana-1313	224	5	c(p	c(p	NOUN
cana-1313	224	6	)	)	PUNCT
cana-1313	224	7	)	)	PUNCT
cana-1313	224	8	.	.	PUNCT
cana-1313	225	1	which	which	PRON
cana-1313	225	2	implies	imply	VERB
cana-1313	225	3	(	(	PUNCT
cana-1313	225	4	icl(q)\q	icl(q)\q	ADJ
cana-1313	225	5	)	)	PUNCT
cana-1313	225	6	⊆	⊆	NUM
cana-1313	225	7	(	(	PUNCT
cana-1313	225	8	icl(p)\p	icl(p)\p	ADJ
cana-1313	225	9	)	)	PUNCT
cana-1313	225	10	.	.	PUNCT
cana-1313	226	1	now	now	ADV
cana-1313	226	2	p	p	NOUN
cana-1313	226	3	is	be	AUX
cana-1313	226	4	irg	irg	NOUN
cana-1313	226	5	-	-	PUNCT
cana-1313	226	6	closed	closed	ADJ
cana-1313	226	7	.	.	PUNCT
cana-1313	227	1	hence	hence	ADV
cana-1313	227	2	,	,	PUNCT
cana-1313	227	3	icl(p)\p	icl(p)\p	ADJ
cana-1313	227	4	has	have	VERB
cana-1313	227	5	no	no	DET
cana-1313	227	6	nonempty	nonempty	ADJ
cana-1313	227	7	regular	regular	ADJ
cana-1313	227	8	-	-	PUNCT
cana-1313	227	9	closed	closed	ADJ
cana-1313	227	10	subsets	subset	NOUN
cana-1313	227	11	neither	neither	CCONJ
cana-1313	227	12	does	do	VERB
cana-1313	227	13	icl(q)\q	icl(q)\q	PROPN
cana-1313	227	14	.	.	PUNCT
cana-1313	228	1	theorem	theorem	VERB
cana-1313	228	2	4.10	4.10	NUM
cana-1313	228	3	:	:	PUNCT
cana-1313	229	1	assume	assume	VERB
cana-1313	229	2	that	that	SCONJ
cana-1313	229	3	p	p	PROPN
cana-1313	229	4	is	be	AUX
cana-1313	229	5	irg	irg	NOUN
cana-1313	229	6	-	-	ADJ
cana-1313	229	7	closed	closed	ADJ
cana-1313	229	8	in	in	ADP
cana-1313	229	9	x	x	X
cana-1313	229	10	and	and	CCONJ
cana-1313	229	11	p	p	ADJ
cana-1313	229	12	⊆	⊆	NUM
cana-1313	229	13	y	y	SYM
cana-1313	229	14	⊆	⊆	NUM
cana-1313	229	15	x.	x.	NOUN
cana-1313	230	1	if	if	SCONJ
cana-1313	230	2	y	y	PROPN
cana-1313	230	3	is	be	AUX
cana-1313	230	4	open	open	ADJ
cana-1313	230	5	in	in	ADP
cana-1313	230	6	x	x	NOUN
cana-1313	230	7	,	,	PUNCT
cana-1313	230	8	then	then	ADV
cana-1313	230	9	p	p	NOUN
cana-1313	230	10	is	be	AUX
cana-1313	230	11	irgclosed	irgclose	VERB
cana-1313	230	12	relative	relative	ADJ
cana-1313	230	13	to	to	ADP
cana-1313	230	14	y.	y.	NOUN
cana-1313	230	15	proof	proof	NOUN
cana-1313	230	16	:	:	PUNCT
cana-1313	230	17	assume	assume	VERB
cana-1313	230	18	that	that	SCONJ
cana-1313	230	19	r	r	NOUN
cana-1313	230	20	is	be	AUX
cana-1313	230	21	regular	regular	ADV
cana-1313	230	22	-	-	PUNCT
cana-1313	230	23	open	open	ADJ
cana-1313	230	24	in	in	ADP
cana-1313	230	25	x	x	X
cana-1313	230	26	and	and	CCONJ
cana-1313	231	1	that	that	SCONJ
cana-1313	231	2	p	p	PROPN
cana-1313	231	3	⊆	⊆	NUM
cana-1313	231	4	y	y	PROPN
cana-1313	231	5	r.	r.	PROPN
cana-1313	231	6	therefore	therefore	ADV
cana-1313	231	7	,	,	PUNCT
cana-1313	231	8	p	p	NOUN
cana-1313	231	9	⊆	⊆	NUM
cana-1313	231	10	r	r	NOUN
cana-1313	231	11	and	and	CCONJ
cana-1313	231	12	hence	hence	ADV
cana-1313	231	13	icl(p	icl(p	PROPN
cana-1313	231	14	)	)	PUNCT
cana-1313	231	15	⊆	⊆	NUM
cana-1313	231	16	r.	r.	NOUN
cana-1313	231	17	it	it	PRON
cana-1313	231	18	follows	follow	VERB
cana-1313	231	19	from	from	ADP
cana-1313	231	20	this	this	PRON
cana-1313	231	21	that	that	SCONJ
cana-1313	231	22	(	(	PUNCT
cana-1313	231	23	y	y	PROPN
cana-1313	231	24	icl(p	icl(p	PROPN
cana-1313	231	25	)	)	PUNCT
cana-1313	231	26	)	)	PUNCT
cana-1313	232	1	⊆	⊆	NUM
cana-1313	232	2	y	y	PROPN
cana-1313	232	3	r.	r.	PROPN
cana-1313	232	4	therefore	therefore	ADV
cana-1313	232	5	p	p	PROPN
cana-1313	232	6	is	be	AUX
cana-1313	232	7	irg	irg	NOUN
cana-1313	232	8	-	-	PUNCT
cana-1313	232	9	closed	close	VERB
cana-1313	232	10	with	with	ADP
cana-1313	232	11	respect	respect	NOUN
cana-1313	232	12	to	to	ADP
cana-1313	232	13	y.	y.	PROPN
cana-1313	232	14	theorem	theorem	PROPN
cana-1313	232	15	4.11	4.11	NUM
cana-1313	232	16	:	:	PUNCT
cana-1313	232	17	let	let	VERB
cana-1313	232	18	x	x	PRON
cana-1313	232	19	be	be	AUX
cana-1313	232	20	a	a	DET
cana-1313	232	21	regular	regular	ADJ
cana-1313	232	22	space	space	NOUN
cana-1313	232	23	.	.	PUNCT
cana-1313	233	1	prove	prove	VERB
cana-1313	233	2	that	that	SCONJ
cana-1313	233	3	every	every	DET
cana-1313	233	4	compact	compact	ADJ
cana-1313	233	5	subset	subset	NOUN
cana-1313	233	6	of	of	ADP
cana-1313	233	7	x	x	PUNCT
cana-1313	233	8	is	be	AUX
cana-1313	233	9	an	an	DET
cana-1313	233	10	irg	irg	ADJ
cana-1313	233	11	-	-	PUNCT
cana-1313	233	12	closed	closed	ADJ
cana-1313	233	13	set	set	NOUN
cana-1313	233	14	.	.	PUNCT
cana-1313	234	1	proof	proof	NOUN
cana-1313	234	2	:	:	PUNCT
cana-1313	234	3	assume	assume	VERB
cana-1313	234	4	p	p	X
cana-1313	234	5	r	r	NOUN
cana-1313	234	6	,	,	PUNCT
cana-1313	234	7	where	where	SCONJ
cana-1313	234	8	r	r	NOUN
cana-1313	234	9	is	be	AUX
cana-1313	234	10	regular	regular	ADV
cana-1313	234	11	-	-	PUNCT
cana-1313	234	12	open	open	ADJ
cana-1313	234	13	.	.	PUNCT
cana-1313	235	1	r	r	NOUN
cana-1313	235	2	is	be	AUX
cana-1313	235	3	open	open	ADJ
cana-1313	235	4	since	since	SCONJ
cana-1313	235	5	it	it	PRON
cana-1313	235	6	is	be	AUX
cana-1313	235	7	regular	regular	ADJ
cana-1313	235	8	-	-	PUNCT
cana-1313	235	9	open	open	ADJ
cana-1313	235	10	right	right	ADV
cana-1313	235	11	now	now	ADV
cana-1313	235	12	.	.	PUNCT
cana-1313	236	1	but	but	CCONJ
cana-1313	236	2	in	in	ADP
cana-1313	236	3	the	the	DET
cana-1313	236	4	typical	typical	ADJ
cana-1313	236	5	space	space	NOUN
cana-1313	236	6	x	x	NOUN
cana-1313	236	7	,	,	PUNCT
cana-1313	236	8	p	p	NOUN
cana-1313	236	9	is	be	AUX
cana-1313	236	10	compact	compact	ADJ
cana-1313	236	11	.	.	PUNCT
cana-1313	237	1	consequently	consequently	ADV
cana-1313	237	2	,	,	PUNCT
cana-1313	237	3	there	there	PRON
cana-1313	237	4	exists	exist	VERB
cana-1313	237	5	an	an	DET
cana-1313	237	6	open	open	ADJ
cana-1313	237	7	set	set	NOUN
cana-1313	237	8	o	o	NOUN
cana-1313	237	9	in	in	ADP
cana-1313	237	10	which	which	PRON
cana-1313	237	11	p	p	NOUN
cana-1313	237	12	⊆	⊆	NUM
cana-1313	237	13	o	o	NOUN
cana-1313	237	14	⊆	⊆	NUM
cana-1313	237	15	cl(o	cl(o	NOUN
cana-1313	237	16	)	)	PUNCT
cana-1313	237	17	⊆	⊆	NUM
cana-1313	237	18	r.	r.	NOUN
cana-1313	237	19	communications	communication	NOUN
cana-1313	237	20	on	on	ADP
cana-1313	237	21	applied	apply	VERB
cana-1313	237	22	nonlinear	nonlinear	ADJ
cana-1313	237	23	analysis	analysis	NOUN
cana-1313	237	24	issn	issn	NOUN
cana-1313	237	25	:	:	PUNCT
cana-1313	237	26	1074	1074	NUM
cana-1313	237	27	-	-	PUNCT
cana-1313	237	28	133x	133x	NUM
cana-1313	237	29	vol	vol	NOUN
cana-1313	237	30	31	31	NUM
cana-1313	237	31	no	no	NOUN
cana-1313	237	32	.	.	PUNCT
cana-1313	238	1	7s	7	NOUN
cana-1313	238	2	(	(	PUNCT
cana-1313	238	3	2024	2024	NUM
cana-1313	238	4	)	)	PUNCT
cana-1313	238	5	354	354	NUM
cana-1313	238	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1313	238	7	since	since	SCONJ
cana-1313	238	8	p	p	PROPN
cana-1313	238	9	⊆	⊆	NUM
cana-1313	238	10	cl(o	cl(o	SYM
cana-1313	238	11	)	)	PUNCT
cana-1313	238	12	icl(p	icl(p	PROPN
cana-1313	238	13	)	)	PUNCT
cana-1313	238	14	⊆	⊆	NUM
cana-1313	238	15	icl(cl(o	icl(cl(o	NOUN
cana-1313	238	16	)	)	PUNCT
cana-1313	238	17	)	)	PUNCT
cana-1313	239	1	=	=	SYM
cana-1313	239	2	cl(cl(o	cl(cl(o	NOUN
cana-1313	239	3	)	)	PUNCT
cana-1313	239	4	)	)	PUNCT
cana-1313	240	1	=	=	SYM
cana-1313	240	2	cl(o	cl(o	X
cana-1313	240	3	)	)	PUNCT
cana-1313	240	4	⊆	⊆	NUM
cana-1313	240	5	r.	r.	NOUN
cana-1313	240	6	that	that	PRON
cana-1313	240	7	is	be	AUX
cana-1313	240	8	icl(p	icl(p	PROPN
cana-1313	240	9	)	)	PUNCT
cana-1313	240	10	⊆	⊆	NUM
cana-1313	240	11	r.	r.	NOUN
cana-1313	240	12	hence	hence	ADV
cana-1313	240	13	p	p	NOUN
cana-1313	240	14	is	be	AUX
cana-1313	240	15	irgclosed	irgclose	VERB
cana-1313	240	16	in	in	ADP
cana-1313	240	17	x.	x.	NOUN
cana-1313	240	18	5	5	NUM
cana-1313	240	19	.	.	PUNCT
cana-1313	241	1	ir*g*closed	ir*g*closed	ADJ
cana-1313	241	2	type	type	NOUN
cana-1313	241	3	sets	set	NOUN
cana-1313	241	4	in	in	ADP
cana-1313	241	5	topological	topological	ADJ
cana-1313	241	6	ordered	order	VERB
cana-1313	241	7	spaces	space	NOUN
cana-1313	241	8	:	:	PUNCT
cana-1313	241	9	theorem	theorem	VERB
cana-1313	241	10	5.1	5.1	NUM
cana-1313	241	11	:	:	PUNCT
cana-1313	241	12	a	a	DET
cana-1313	241	13	set	set	NOUN
cana-1313	242	1	p	p	NOUN
cana-1313	242	2	q	q	X
cana-1313	242	3	is	be	AUX
cana-1313	242	4	ir*g*-closed	ir*g*-close	VERB
cana-1313	242	5	if	if	SCONJ
cana-1313	242	6	p	p	NOUN
cana-1313	242	7	and	and	CCONJ
cana-1313	242	8	q	q	NOUN
cana-1313	242	9	are	be	AUX
cana-1313	242	10	ir*g*-closed	ir*g*-closed	ADJ
cana-1313	242	11	sets	set	NOUN
cana-1313	242	12	.	.	PUNCT
cana-1313	243	1	proof	proof	NOUN
cana-1313	243	2	:	:	PUNCT
cana-1313	243	3	if	if	SCONJ
cana-1313	243	4	p	p	NOUN
cana-1313	243	5	q	q	X
cana-1313	243	6	⊆	⊆	NUM
cana-1313	243	7	r	r	NOUN
cana-1313	243	8	and	and	CCONJ
cana-1313	243	9	r	r	NOUN
cana-1313	243	10	is	be	AUX
cana-1313	243	11	g	g	NOUN
cana-1313	243	12	-	-	PUNCT
cana-1313	243	13	open	open	ADJ
cana-1313	243	14	,	,	PUNCT
cana-1313	243	15	then	then	ADV
cana-1313	243	16	p	p	NOUN
cana-1313	243	17	⊆	⊆	NUM
cana-1313	243	18	r	r	NOUN
cana-1313	243	19	and	and	CCONJ
cana-1313	243	20	q	q	NOUN
cana-1313	243	21	⊆	⊆	NUM
cana-1313	243	22	r.	r.	NOUN
cana-1313	243	23	but	but	CCONJ
cana-1313	243	24	p	p	NOUN
cana-1313	243	25	and	and	CCONJ
cana-1313	243	26	q	q	NOUN
cana-1313	243	27	are	be	AUX
cana-1313	243	28	ir*g*-closed	ir*g*-close	VERB
cana-1313	243	29	and	and	CCONJ
cana-1313	243	30	therefore	therefore	ADV
cana-1313	243	31	ircl	ircl	ADJ
cana-1313	243	32	(	(	PUNCT
cana-1313	243	33	p	p	NOUN
cana-1313	243	34	)	)	PUNCT
cana-1313	243	35	⊆	⊆	NUM
cana-1313	243	36	r	r	NOUN
cana-1313	243	37	and	and	CCONJ
cana-1313	243	38	ircl	ircl	NOUN
cana-1313	243	39	(	(	PUNCT
cana-1313	243	40	q	q	X
cana-1313	243	41	)	)	PUNCT
cana-1313	243	42	⊆	⊆	PROPN
cana-1313	243	43	r.	r.	PROPN
cana-1313	243	44	therefore	therefore	ADV
cana-1313	243	45	,	,	PUNCT
cana-1313	243	46	(	(	PUNCT
cana-1313	243	47	ircl(p	ircl(p	NOUN
cana-1313	243	48	)	)	PUNCT
cana-1313	243	49	ircl(q	ircl(q	NOUN
cana-1313	243	50	)	)	PUNCT
cana-1313	243	51	)	)	PUNCT
cana-1313	244	1	⊆	⊆	NUM
cana-1313	244	2	r	r	NOUN
cana-1313	244	3	,	,	PUNCT
cana-1313	244	4	and	and	CCONJ
cana-1313	244	5	hence	hence	ADV
cana-1313	244	6	ircl(p	ircl(p	PRON
cana-1313	244	7	q	q	NOUN
cana-1313	244	8	)	)	PUNCT
cana-1313	244	9	⊆	⊆	NUM
cana-1313	244	10	r.	r.	NOUN
cana-1313	244	11	hence	hence	ADV
cana-1313	244	12	p	p	PROPN
cana-1313	244	13	q	q	PROPN
cana-1313	244	14	is	be	AUX
cana-1313	244	15	ir*g*-closed	ir*g*-close	VERB
cana-1313	244	16	.	.	PUNCT
cana-1313	245	1	example	example	NOUN
cana-1313	245	2	5.2	5.2	NUM
cana-1313	245	3	:	:	PUNCT
cana-1313	245	4	let	let	VERB
cana-1313	245	5	x	x	PUNCT
cana-1313	245	6	=	=	PUNCT
cana-1313	245	7	{	{	PUNCT
cana-1313	245	8	p	p	X
cana-1313	245	9	,	,	PUNCT
cana-1313	245	10	q	q	ADJ
cana-1313	245	11	,	,	PUNCT
cana-1313	245	12	r	r	NOUN
cana-1313	245	13	}	}	PUNCT
cana-1313	245	14	,	,	PUNCT
cana-1313	245	15	=	=	PRON
cana-1313	245	16	{	{	PUNCT
cana-1313	245	17	,	,	PUNCT
cana-1313	245	18	x	x	X
cana-1313	245	19	,	,	PUNCT
cana-1313	245	20	{	{	PUNCT
cana-1313	245	21	p	p	X
cana-1313	245	22	}	}	PUNCT
cana-1313	245	23	,	,	PUNCT
cana-1313	245	24	{	{	PUNCT
cana-1313	245	25	q	q	X
cana-1313	245	26	}	}	PUNCT
cana-1313	245	27	,	,	PUNCT
cana-1313	245	28	{	{	PUNCT
cana-1313	245	29	p	p	X
cana-1313	245	30	,	,	PUNCT
cana-1313	245	31	q	q	NOUN
cana-1313	245	32	}	}	PUNCT
cana-1313	245	33	}	}	PUNCT
cana-1313	245	34	and	and	CCONJ
cana-1313	245	35	=	=	PRON
cana-1313	245	36	{	{	PUNCT
cana-1313	245	37	(	(	PUNCT
cana-1313	245	38	p	p	X
cana-1313	245	39	,	,	PUNCT
cana-1313	245	40	p	p	NOUN
cana-1313	245	41	)	)	PUNCT
cana-1313	245	42	,	,	PUNCT
cana-1313	245	43	(	(	PUNCT
cana-1313	245	44	q	q	X
cana-1313	245	45	,	,	PUNCT
cana-1313	245	46	q	q	NOUN
cana-1313	245	47	)	)	PUNCT
cana-1313	245	48	,	,	PUNCT
cana-1313	245	49	(	(	PUNCT
cana-1313	245	50	r	r	NOUN
cana-1313	245	51	,	,	PUNCT
cana-1313	245	52	r	r	NOUN
cana-1313	245	53	)	)	PUNCT
cana-1313	245	54	,	,	PUNCT
cana-1313	245	55	(	(	PUNCT
cana-1313	245	56	p	p	X
cana-1313	245	57	,	,	PUNCT
cana-1313	245	58	q	q	NOUN
cana-1313	245	59	)	)	PUNCT
cana-1313	245	60	,	,	PUNCT
cana-1313	245	61	(	(	PUNCT
cana-1313	245	62	q	q	X
cana-1313	245	63	,	,	PUNCT
cana-1313	245	64	r	r	NOUN
cana-1313	245	65	)	)	PUNCT
cana-1313	245	66	}	}	PUNCT
cana-1313	245	67	.	.	PUNCT
cana-1313	246	1	clearly	clearly	ADV
cana-1313	246	2	(	(	PUNCT
cana-1313	246	3	)	)	PUNCT
cana-1313	246	4	is	be	AUX
cana-1313	246	5	a	a	DET
cana-1313	246	6	topological	topological	ADJ
cana-1313	246	7	ordered	order	VERB
cana-1313	246	8	space	space	NOUN
cana-1313	246	9	.	.	PUNCT
cana-1313	247	1	take	take	VERB
cana-1313	247	2	p	p	NOUN
cana-1313	247	3	=	=	NOUN
cana-1313	247	4	{	{	PUNCT
cana-1313	247	5	r	r	NOUN
cana-1313	247	6	}	}	PUNCT
cana-1313	247	7	,	,	PUNCT
cana-1313	247	8	q	q	NOUN
cana-1313	248	1	=	=	PUNCT
cana-1313	248	2	{	{	PUNCT
cana-1313	248	3	q	q	NOUN
cana-1313	248	4	,	,	PUNCT
cana-1313	248	5	r	r	NOUN
cana-1313	248	6	}	}	PUNCT
cana-1313	248	7	.	.	PUNCT
cana-1313	249	1	if	if	SCONJ
cana-1313	249	2	(	(	PUNCT
cana-1313	249	3	p	p	NOUN
cana-1313	249	4	q	q	NOUN
cana-1313	249	5	)	)	PUNCT
cana-1313	249	6	=	=	NOUN
cana-1313	249	7	{	{	PUNCT
cana-1313	249	8	r	r	NOUN
cana-1313	249	9	}	}	PUNCT
cana-1313	249	10	{	{	PUNCT
cana-1313	249	11	q	q	NOUN
cana-1313	249	12	,	,	PUNCT
cana-1313	249	13	r	r	NOUN
cana-1313	249	14	}	}	PUNCT
cana-1313	249	15	=	=	SYM
cana-1313	249	16	{	{	PUNCT
cana-1313	249	17	q	q	NOUN
cana-1313	249	18	,	,	PUNCT
cana-1313	249	19	r	r	NOUN
cana-1313	249	20	}	}	PUNCT
cana-1313	249	21	⊆	⊆	NUM
cana-1313	249	22	r	r	NOUN
cana-1313	249	23	=	=	SYM
cana-1313	249	24	x	x	X
cana-1313	249	25	and	and	CCONJ
cana-1313	249	26	r	r	NOUN
cana-1313	249	27	is	be	AUX
cana-1313	249	28	g	g	NOUN
cana-1313	249	29	-	-	PUNCT
cana-1313	249	30	open	open	ADJ
cana-1313	249	31	,	,	PUNCT
cana-1313	249	32	then	then	ADV
cana-1313	249	33	{	{	PUNCT
cana-1313	249	34	r	r	NOUN
cana-1313	249	35	}	}	PUNCT
cana-1313	249	36	⊆	⊆	NUM
cana-1313	249	37	r	r	NOUN
cana-1313	249	38	and	and	CCONJ
cana-1313	249	39	{	{	PUNCT
cana-1313	249	40	p	p	X
cana-1313	249	41	,	,	PUNCT
cana-1313	249	42	q	q	ADJ
cana-1313	249	43	}	}	PUNCT
cana-1313	249	44	⊆	⊆	NUM
cana-1313	249	45	r.	r.	NOUN
cana-1313	249	46	but	but	CCONJ
cana-1313	249	47	p	p	NOUN
cana-1313	249	48	and	and	CCONJ
cana-1313	249	49	q	q	NOUN
cana-1313	249	50	are	be	AUX
cana-1313	249	51	ir*g*-closed	ir*g*-close	VERB
cana-1313	249	52	and	and	CCONJ
cana-1313	249	53	therefore	therefore	ADV
cana-1313	249	54	ircl	ircl	ADJ
cana-1313	249	55	(	(	PUNCT
cana-1313	249	56	p	p	NOUN
cana-1313	249	57	)	)	PUNCT
cana-1313	249	58	⊆	⊆	NUM
cana-1313	249	59	r	r	NOUN
cana-1313	249	60	and	and	CCONJ
cana-1313	249	61	ircl	ircl	NOUN
cana-1313	249	62	(	(	PUNCT
cana-1313	249	63	q	q	X
cana-1313	249	64	)	)	PUNCT
cana-1313	249	65	⊆	⊆	PROPN
cana-1313	249	66	r.	r.	PROPN
cana-1313	249	67	therefore	therefore	ADV
cana-1313	249	68	,	,	PUNCT
cana-1313	249	69	(	(	PUNCT
cana-1313	249	70	ircl(p	ircl(p	NOUN
cana-1313	249	71	)	)	PUNCT
cana-1313	249	72	ircl(q	ircl(q	NOUN
cana-1313	249	73	)	)	PUNCT
cana-1313	249	74	)	)	PUNCT
cana-1313	250	1	⊆	⊆	NUM
cana-1313	250	2	r	r	NOUN
cana-1313	250	3	,	,	PUNCT
cana-1313	250	4	and	and	CCONJ
cana-1313	250	5	hence	hence	ADV
cana-1313	250	6	ircl(p	ircl(p	PRON
cana-1313	250	7	q	q	NOUN
cana-1313	250	8	)	)	PUNCT
cana-1313	250	9	⊆	⊆	NUM
cana-1313	250	10	r.	r.	NOUN
cana-1313	250	11	hence	hence	ADV
cana-1313	250	12	p	p	PROPN
cana-1313	250	13	q	q	PROPN
cana-1313	250	14	is	be	AUX
cana-1313	250	15	ir*g*-closed	ir*g*-close	VERB
cana-1313	250	16	.	.	PUNCT
cana-1313	251	1	theorem	theorem	VERB
cana-1313	251	2	5.3	5.3	NUM
cana-1313	251	3	:	:	PUNCT
cana-1313	251	4	if	if	SCONJ
cana-1313	251	5	a	a	DET
cana-1313	251	6	set	set	NOUN
cana-1313	251	7	p	p	NOUN
cana-1313	251	8	is	be	AUX
cana-1313	251	9	ir*g*-closed	ir*g*-close	VERB
cana-1313	251	10	then	then	ADV
cana-1313	251	11	ircl(p)\p	ircl(p)\p	ADV
cana-1313	251	12	contains	contain	VERB
cana-1313	251	13	no	no	DET
cana-1313	251	14	nonempty	nonempty	ADJ
cana-1313	251	15	regular	regular	ADJ
cana-1313	251	16	-	-	PUNCT
cana-1313	251	17	closed	close	VERB
cana-1313	251	18	set	set	NOUN
cana-1313	251	19	.	.	PUNCT
cana-1313	252	1	proof	proof	NOUN
cana-1313	252	2	:	:	PUNCT
cana-1313	252	3	suppose	suppose	VERB
cana-1313	252	4	that	that	SCONJ
cana-1313	252	5	p	p	NOUN
cana-1313	252	6	is	be	AUX
cana-1313	252	7	ir*g*-closed	ir*g*-close	VERB
cana-1313	252	8	.	.	PUNCT
cana-1313	253	1	let	let	VERB
cana-1313	253	2	s	s	PRON
cana-1313	253	3	be	be	AUX
cana-1313	253	4	a	a	DET
cana-1313	253	5	regular	regular	ADJ
cana-1313	253	6	-	-	PUNCT
cana-1313	253	7	closed	closed	ADJ
cana-1313	253	8	subset	subset	NOUN
cana-1313	253	9	of	of	ADP
cana-1313	253	10	ircl(p)\p	ircl(p)\p	NOUN
cana-1313	253	11	.	.	PUNCT
cana-1313	254	1	then	then	ADV
cana-1313	254	2	s	s	VERB
cana-1313	254	3	⊆	⊆	NUM
cana-1313	254	4	(	(	PUNCT
cana-1313	254	5	ircl(p	ircl(p	NOUN
cana-1313	254	6	)	)	PUNCT
cana-1313	254	7	c(p	c(p	NOUN
cana-1313	254	8	)	)	PUNCT
cana-1313	254	9	)	)	PUNCT
cana-1313	255	1	and	and	CCONJ
cana-1313	255	2	so	so	ADV
cana-1313	255	3	p	p	PRON
cana-1313	255	4	⊆	⊆	NUM
cana-1313	255	5	c(s	c(	NOUN
cana-1313	255	6	)	)	PUNCT
cana-1313	255	7	.	.	PUNCT
cana-1313	256	1	but	but	CCONJ
cana-1313	256	2	p	p	NOUN
cana-1313	256	3	is	be	AUX
cana-1313	256	4	ir*g*-closed	ir*g*-close	VERB
cana-1313	256	5	.	.	PUNCT
cana-1313	257	1	therefore	therefore	ADV
cana-1313	257	2	ircl(p	ircl(p	NOUN
cana-1313	257	3	)	)	PUNCT
cana-1313	257	4	⊆	⊆	NUM
cana-1313	257	5	c(s	c(	NOUN
cana-1313	257	6	)	)	PUNCT
cana-1313	257	7	.	.	PUNCT
cana-1313	258	1	(	(	PUNCT
cana-1313	258	2	1	1	X
cana-1313	258	3	)	)	PUNCT
cana-1313	258	4	consequently	consequently	ADV
cana-1313	258	5	s	s	VERB
cana-1313	258	6	⊆	⊆	NUM
cana-1313	258	7	c(ircl(p	c(ircl(p	NUM
cana-1313	258	8	)	)	PUNCT
cana-1313	258	9	)	)	PUNCT
cana-1313	259	1	(	(	PUNCT
cana-1313	259	2	2	2	X
cana-1313	259	3	)	)	PUNCT
cana-1313	259	4	we	we	PRON
cana-1313	259	5	have	have	AUX
cana-1313	259	6	already	already	ADV
cana-1313	259	7	s	s	VERB
cana-1313	259	8	⊆	⊆	NUM
cana-1313	259	9	ircl(p	ircl(p	NOUN
cana-1313	259	10	)	)	PUNCT
cana-1313	259	11	(	(	PUNCT
cana-1313	259	12	3	3	NUM
cana-1313	259	13	)	)	PUNCT
cana-1313	259	14	.	.	PUNCT
cana-1313	260	1	from	from	ADP
cana-1313	260	2	(	(	PUNCT
cana-1313	260	3	2	2	NUM
cana-1313	260	4	)	)	PUNCT
cana-1313	260	5	and	and	CCONJ
cana-1313	260	6	(	(	PUNCT
cana-1313	260	7	3	3	X
cana-1313	260	8	)	)	PUNCT
cana-1313	260	9	s	s	PART
cana-1313	260	10	⊆	⊆	NUM
cana-1313	260	11	(	(	PUNCT
cana-1313	260	12	ircl(p	ircl(p	NOUN
cana-1313	260	13	)	)	PUNCT
cana-1313	260	14	c(ircl(p	c(ircl(p	NUM
cana-1313	260	15	)	)	PUNCT
cana-1313	260	16	)	)	PUNCT
cana-1313	260	17	)	)	PUNCT
cana-1313	261	1	=	=	PUNCT
cana-1313	261	2	.	.	PUNCT
cana-1313	262	1	thus	thus	ADV
cana-1313	262	2	s	s	VERB
cana-1313	262	3	=	=	X
cana-1313	262	4	.	.	PUNCT
cana-1313	263	1	therefore	therefore	ADV
cana-1313	263	2	,	,	PUNCT
cana-1313	263	3	ircl	ircl	PROPN
cana-1313	263	4	(	(	PUNCT
cana-1313	263	5	p)\p	p)\p	PROPN
cana-1313	263	6	contains	contain	VERB
cana-1313	263	7	no	no	DET
cana-1313	263	8	nonempty	nonempty	ADJ
cana-1313	263	9	regular	regular	ADJ
cana-1313	263	10	-	-	PUNCT
cana-1313	263	11	closed	close	VERB
cana-1313	263	12	set	set	NOUN
cana-1313	263	13	.	.	PUNCT
cana-1313	264	1	corollary	corollary	ADJ
cana-1313	264	2	5.4	5.4	NUM
cana-1313	264	3	:	:	PUNCT
cana-1313	264	4	if	if	SCONJ
cana-1313	264	5	p	p	NOUN
cana-1313	264	6	is	be	AUX
cana-1313	264	7	an	an	DET
cana-1313	264	8	ir*g*-closed	ir*g*-closed	ADJ
cana-1313	264	9	set	set	NOUN
cana-1313	264	10	,	,	PUNCT
cana-1313	264	11	then	then	ADV
cana-1313	264	12	p	p	PROPN
cana-1313	264	13	is	be	AUX
cana-1313	264	14	regular	regular	ADV
cana-1313	264	15	-	-	PUNCT
cana-1313	264	16	closed	closed	ADJ
cana-1313	264	17	if	if	SCONJ
cana-1313	264	18	and	and	CCONJ
cana-1313	264	19	only	only	ADV
cana-1313	264	20	if	if	SCONJ
cana-1313	264	21	cl(int	cl(int	PROPN
cana-1313	264	22	(	(	PUNCT
cana-1313	264	23	p))\p	p))\p	PRON
cana-1313	264	24	is	be	AUX
cana-1313	264	25	regular	regular	ADJ
cana-1313	264	26	closed	closed	ADJ
cana-1313	264	27	.	.	PUNCT
cana-1313	265	1	proof	proof	NOUN
cana-1313	265	2	:	:	PUNCT
cana-1313	265	3	make	make	VERB
cana-1313	265	4	p	p	PRON
cana-1313	265	5	an	an	DET
cana-1313	265	6	irg	irg	NOUN
cana-1313	265	7	-	-	PUNCT
cana-1313	265	8	closed	closed	ADJ
cana-1313	265	9	.	.	PUNCT
cana-1313	266	1	if	if	SCONJ
cana-1313	266	2	cl(int(p	cl(int(p	NOUN
cana-1313	266	3	)	)	PUNCT
cana-1313	266	4	)	)	PUNCT
cana-1313	267	1	=	=	PUNCT
cana-1313	268	1	p	p	X
cana-1313	268	2	,	,	PUNCT
cana-1313	268	3	then	then	ADV
cana-1313	268	4	p	p	PROPN
cana-1313	268	5	is	be	AUX
cana-1313	268	6	regular	regular	ADV
cana-1313	268	7	-	-	PUNCT
cana-1313	268	8	closed	closed	ADJ
cana-1313	268	9	.	.	PUNCT
cana-1313	269	1	if	if	SCONJ
cana-1313	269	2	so	so	ADV
cana-1313	269	3	,	,	PUNCT
cana-1313	269	4	cl(int(p))\p	cl(int(p))\p	NOUN
cana-1313	269	5	=	=	X
cana-1313	269	6	p\p	p\p	PROPN
cana-1313	269	7	=	=	NOUN
cana-1313	269	8	.	.	PUNCT
cana-1313	270	1	however	however	ADV
cana-1313	270	2	,	,	PUNCT
cana-1313	270	3	is	be	AUX
cana-1313	270	4	always	always	ADV
cana-1313	270	5	regular	regular	ADJ
cana-1313	270	6	closed	closed	ADJ
cana-1313	270	7	.	.	PUNCT
cana-1313	271	1	cl(int(p))\p	cl(int(p))\p	PROPN
cana-1313	271	2	is	be	AUX
cana-1313	271	3	hence	hence	ADV
cana-1313	271	4	regular	regular	ADV
cana-1313	271	5	-	-	PUNCT
cana-1313	271	6	closed	closed	ADJ
cana-1313	271	7	.	.	PUNCT
cana-1313	272	1	assume	assume	VERB
cana-1313	272	2	,	,	PUNCT
cana-1313	272	3	on	on	ADP
cana-1313	272	4	the	the	DET
cana-1313	272	5	other	other	ADJ
cana-1313	272	6	hand	hand	NOUN
cana-1313	272	7	,	,	PUNCT
cana-1313	272	8	that	that	DET
cana-1313	272	9	cl(int(p))\p	cl(int(p))\p	NOUN
cana-1313	272	10	is	be	AUX
cana-1313	272	11	regular	regular	ADV
cana-1313	272	12	-	-	PUNCT
cana-1313	272	13	closed	closed	ADJ
cana-1313	272	14	.	.	PUNCT
cana-1313	273	1	p	p	NOUN
cana-1313	273	2	is	be	AUX
cana-1313	273	3	,	,	PUNCT
cana-1313	273	4	however	however	ADV
cana-1313	273	5	,	,	PUNCT
cana-1313	273	6	irg	irg	PROPN
cana-1313	273	7	-	-	PUNCT
cana-1313	273	8	closed	closed	ADJ
cana-1313	273	9	.	.	PUNCT
cana-1313	274	1	the	the	DET
cana-1313	274	2	regular	regular	ADJ
cana-1313	274	3	-	-	PUNCT
cana-1313	274	4	closed	close	VERB
cana-1313	274	5	set	set	ADJ
cana-1313	274	6	cl(int(p))\p	cl(int(p))\p	NOUN
cana-1313	274	7	is	be	AUX
cana-1313	274	8	also	also	ADV
cana-1313	274	9	contained	contain	VERB
cana-1313	274	10	in	in	ADP
cana-1313	274	11	ircl(p)\p	ircl(p)\p	NOUN
cana-1313	274	12	.	.	PUNCT
cana-1313	275	1	the	the	DET
cana-1313	275	2	statement	statement	NOUN
cana-1313	275	3	"	"	PUNCT
cana-1313	275	4	cl(int(p))\p	cl(int(p))\p	NOUN
cana-1313	275	5	=	=	X
cana-1313	275	6	.	.	PUNCT
cana-1313	275	7	"	"	PUNCT
cana-1313	275	8	is	be	AUX
cana-1313	275	9	based	base	VERB
cana-1313	275	10	on	on	ADP
cana-1313	275	11	the	the	DET
cana-1313	275	12	aforementioned	aforementione	VERB
cana-1313	275	13	theorem	theorem	NOUN
cana-1313	275	14	.	.	PROPN
cana-1313	276	1	as	as	ADP
cana-1313	276	2	a	a	DET
cana-1313	276	3	result	result	NOUN
cana-1313	276	4	,	,	PUNCT
cana-1313	276	5	cl(int(p	cl(int(p	NOUN
cana-1313	276	6	)	)	PUNCT
cana-1313	276	7	)	)	PUNCT
cana-1313	277	1	=	=	SYM
cana-1313	278	1	p.	p.	NOUN
cana-1313	278	2	as	as	ADP
cana-1313	278	3	a	a	DET
cana-1313	278	4	result	result	NOUN
cana-1313	278	5	,	,	PUNCT
cana-1313	278	6	p	p	NOUN
cana-1313	278	7	is	be	AUX
cana-1313	278	8	regular	regular	ADV
cana-1313	278	9	closed	closed	ADJ
cana-1313	278	10	.	.	PUNCT
cana-1313	279	1	theorem	theorem	VERB
cana-1313	279	2	5.5	5.5	NUM
cana-1313	279	3	:	:	PUNCT
cana-1313	279	4	in	in	ADP
cana-1313	279	5	the	the	DET
cana-1313	279	6	event	event	NOUN
cana-1313	279	7	that	that	PRON
cana-1313	279	8	p	p	NOUN
cana-1313	279	9	is	be	AUX
cana-1313	279	10	ir*g*-closed	ir*g*-close	VERB
cana-1313	279	11	and	and	CCONJ
cana-1313	279	12	p	p	ADP
cana-1313	279	13	⊆	⊆	NUM
cana-1313	279	14	q	q	PUNCT
cana-1313	279	15	⊆	⊆	NUM
cana-1313	279	16	ircl(p	ircl(p	NOUN
cana-1313	279	17	)	)	PUNCT
cana-1313	279	18	,	,	PUNCT
cana-1313	279	19	then	then	ADV
cana-1313	279	20	ircl(q)\q	ircl(q)\q	PRON
cana-1313	279	21	does	do	AUX
cana-1313	279	22	not	not	PART
cana-1313	279	23	contain	contain	VERB
cana-1313	279	24	any	any	DET
cana-1313	279	25	nonempty	nonempty	ADJ
cana-1313	279	26	regular	regular	ADJ
cana-1313	279	27	-	-	PUNCT
cana-1313	279	28	closed	close	VERB
cana-1313	279	29	sets	set	NOUN
cana-1313	279	30	.	.	PUNCT
cana-1313	280	1	proof	proof	NOUN
cana-1313	280	2	:	:	PUNCT
cana-1313	280	3	if	if	SCONJ
cana-1313	280	4	p	p	NOUN
cana-1313	280	5	is	be	AUX
cana-1313	280	6	ir*g*-closed	ir*g*-closed	ADJ
cana-1313	280	7	and	and	CCONJ
cana-1313	280	8	p	p	ADP
cana-1313	280	9	⊆	⊆	NUM
cana-1313	280	10	q	q	PUNCT
cana-1313	280	11	⊆	⊆	NUM
cana-1313	280	12	ircl(p	ircl(p	NOUN
cana-1313	280	13	)	)	PUNCT
cana-1313	280	14	.	.	PUNCT
cana-1313	281	1	since	since	SCONJ
cana-1313	281	2	c(p)⊆c(q	c(p)⊆c(q	PROPN
cana-1313	281	3	)	)	PUNCT
cana-1313	281	4	follows	follow	VERB
cana-1313	281	5	from	from	ADP
cana-1313	281	6	p⊆q	p⊆q	PROPN
cana-1313	281	7	(	(	PUNCT
cana-1313	281	8	1	1	NUM
cana-1313	281	9	)	)	PUNCT
cana-1313	281	10	q	q	NOUN
cana-1313	281	11	⊆	⊆	NUM
cana-1313	281	12	ircl(p	ircl(p	NOUN
cana-1313	281	13	)	)	PUNCT
cana-1313	281	14	implies	imply	VERB
cana-1313	281	15	that	that	PRON
cana-1313	281	16	ircl(q	ircl(q	NOUN
cana-1313	281	17	)	)	PUNCT
cana-1313	281	18	⊆	⊆	NUM
cana-1313	281	19	ircl(ircl(p	ircl(ircl(p	NOUN
cana-1313	281	20	)	)	PUNCT
cana-1313	281	21	)	)	PUNCT
cana-1313	282	1	=	=	SYM
cana-1313	282	2	ircl(p	ircl(p	NOUN
cana-1313	282	3	)	)	PUNCT
cana-1313	282	4	.	.	PUNCT
cana-1313	283	1	in	in	ADP
cana-1313	283	2	this	this	DET
cana-1313	283	3	case	case	NOUN
cana-1313	283	4	ircl(q	ircl(q	VERB
cana-1313	283	5	)	)	PUNCT
cana-1313	283	6	⊆	⊆	NUM
cana-1313	283	7	ircl(p	ircl(p	NOUN
cana-1313	283	8	)	)	PUNCT
cana-1313	283	9	(	(	PUNCT
cana-1313	283	10	2	2	NUM
cana-1313	283	11	)	)	PUNCT
cana-1313	283	12	from	from	ADP
cana-1313	283	13	(	(	PUNCT
cana-1313	283	14	1	1	NUM
cana-1313	283	15	)	)	PUNCT
cana-1313	283	16	&	&	CCONJ
cana-1313	283	17	(	(	PUNCT
cana-1313	283	18	2	2	NUM
cana-1313	283	19	)	)	PUNCT
cana-1313	283	20	(	(	PUNCT
cana-1313	283	21	ircl(q	ircl(q	PROPN
cana-1313	283	22	)	)	PUNCT
cana-1313	283	23	c(q	c(q	PROPN
cana-1313	283	24	)	)	PUNCT
cana-1313	283	25	)	)	PUNCT
cana-1313	284	1	⊆	⊆	X
cana-1313	284	2	(	(	PUNCT
cana-1313	284	3	ircl(p	ircl(p	NOUN
cana-1313	284	4	)	)	PUNCT
cana-1313	284	5	c(p	c(p	NOUN
cana-1313	284	6	)	)	PUNCT
cana-1313	284	7	)	)	PUNCT
cana-1313	284	8	implies	imply	VERB
cana-1313	284	9	(	(	PUNCT
cana-1313	284	10	ircl(q)\q	ircl(q)\q	NOUN
cana-1313	284	11	)	)	PUNCT
cana-1313	284	12	⊆	⊆	NUM
cana-1313	284	13	(	(	PUNCT
cana-1313	284	14	ircl(p)\p	ircl(p)\p	NOUN
cana-1313	284	15	)	)	PUNCT
cana-1313	284	16	.	.	PUNCT
cana-1313	285	1	p	p	NOUN
cana-1313	285	2	is	be	AUX
cana-1313	285	3	now	now	ADV
cana-1313	285	4	ir*g*-closed	ir*g*-closed	ADJ
cana-1313	285	5	.	.	PUNCT
cana-1313	286	1	as	as	ADP
cana-1313	286	2	a	a	DET
cana-1313	286	3	result	result	NOUN
cana-1313	286	4	,	,	PUNCT
cana-1313	286	5	neither	neither	CCONJ
cana-1313	286	6	ircl(p)\p	ircl(p)\p	ADJ
cana-1313	286	7	nor	nor	CCONJ
cana-1313	286	8	ircl(q)\q	ircl(q)\q	PRON
cana-1313	286	9	have	have	VERB
cana-1313	286	10	any	any	DET
cana-1313	286	11	nonempty	nonempty	ADJ
cana-1313	286	12	regular	regular	ADJ
cana-1313	286	13	-	-	PUNCT
cana-1313	286	14	closed	closed	ADJ
cana-1313	286	15	subsets	subset	NOUN
cana-1313	286	16	.	.	PUNCT
cana-1313	287	1	theorem	theorem	VERB
cana-1313	287	2	5.6	5.6	NUM
cana-1313	287	3	:	:	PUNCT
cana-1313	287	4	let	let	VERB
cana-1313	287	5	p	p	PRON
cana-1313	287	6	⊆	⊆	NUM
cana-1313	287	7	y	y	NOUN
cana-1313	287	8	⊆	⊆	NUM
cana-1313	287	9	x	x	PUNCT
cana-1313	287	10	and	and	CCONJ
cana-1313	287	11	suppose	suppose	VERB
cana-1313	287	12	that	that	SCONJ
cana-1313	287	13	p	p	NOUN
cana-1313	287	14	is	be	AUX
cana-1313	287	15	ir*g*-closed	ir*g*-close	VERB
cana-1313	287	16	in	in	ADP
cana-1313	287	17	x.	x.	NOUN
cana-1313	287	18	then	then	ADV
cana-1313	287	19	p	p	NOUN
cana-1313	287	20	is	be	AUX
cana-1313	287	21	ir*g*-closed	ir*g*-close	VERB
cana-1313	287	22	relative	relative	ADJ
cana-1313	287	23	to	to	ADP
cana-1313	287	24	y	y	PROPN
cana-1313	287	25	,	,	PUNCT
cana-1313	287	26	provided	provide	VERB
cana-1313	287	27	y	y	PROPN
cana-1313	287	28	is	be	AUX
cana-1313	287	29	open	open	ADJ
cana-1313	287	30	in	in	ADP
cana-1313	287	31	x.	x.	NOUN
cana-1313	287	32	proof	proof	NOUN
cana-1313	287	33	:	:	PUNCT
cana-1313	287	34	let	let	VERB
cana-1313	287	35	p	p	PRON
cana-1313	287	36	⊆	⊆	NUM
cana-1313	287	37	y	y	NOUN
cana-1313	287	38	r	r	NOUN
cana-1313	287	39	and	and	CCONJ
cana-1313	287	40	suppose	suppose	VERB
cana-1313	287	41	that	that	SCONJ
cana-1313	287	42	r	r	NOUN
cana-1313	287	43	is	be	AUX
cana-1313	287	44	g	g	NOUN
cana-1313	287	45	-	-	PUNCT
cana-1313	287	46	open	open	ADJ
cana-1313	287	47	in	in	ADP
cana-1313	287	48	x.	x.	NOUN
cana-1313	287	49	then	then	ADV
cana-1313	287	50	p	p	NOUN
cana-1313	287	51	⊆	⊆	NUM
cana-1313	287	52	r	r	NOUN
cana-1313	287	53	and	and	CCONJ
cana-1313	287	54	hence	hence	ADV
cana-1313	287	55	ircl(p	ircl(p	NOUN
cana-1313	287	56	)	)	PUNCT
cana-1313	287	57	⊆	⊆	NUM
cana-1313	287	58	r.	r.	NOUN
cana-1313	287	59	this	this	PRON
cana-1313	287	60	implies	imply	VERB
cana-1313	287	61	that	that	SCONJ
cana-1313	287	62	(	(	PUNCT
cana-1313	287	63	y	y	PROPN
cana-1313	287	64	ircl(p	ircl(p	PROPN
cana-1313	287	65	)	)	PUNCT
cana-1313	287	66	)	)	PUNCT
cana-1313	288	1	⊆	⊆	NUM
cana-1313	288	2	y	y	PROPN
cana-1313	288	3	r.	r.	PROPN
cana-1313	288	4	thus	thus	ADV
cana-1313	288	5	p	p	X
cana-1313	288	6	is	be	AUX
cana-1313	288	7	ir*g*-closed	ir*g*-close	VERB
cana-1313	288	8	relative	relative	ADJ
cana-1313	288	9	to	to	ADP
cana-1313	288	10	y.	y.	NOUN
cana-1313	288	11	communications	communication	NOUN
cana-1313	288	12	on	on	ADP
cana-1313	288	13	applied	apply	VERB
cana-1313	288	14	nonlinear	nonlinear	ADJ
cana-1313	288	15	analysis	analysis	NOUN
cana-1313	288	16	issn	issn	NOUN
cana-1313	288	17	:	:	PUNCT
cana-1313	288	18	1074	1074	NUM
cana-1313	288	19	-	-	PUNCT
cana-1313	288	20	133x	133x	NUM
cana-1313	288	21	vol	vol	NOUN
cana-1313	288	22	31	31	NUM
cana-1313	288	23	no	no	NOUN
cana-1313	288	24	.	.	PUNCT
cana-1313	289	1	7s	7	NOUN
cana-1313	289	2	(	(	PUNCT
cana-1313	289	3	2024	2024	NUM
cana-1313	289	4	)	)	PUNCT
cana-1313	289	5	355	355	NUM
cana-1313	289	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1313	289	7	conclusion	conclusion	NOUN
cana-1313	289	8	:	:	PUNCT
cana-1313	289	9	in	in	ADP
cana-1313	289	10	this	this	DET
cana-1313	289	11	study	study	NOUN
cana-1313	289	12	,	,	PUNCT
cana-1313	289	13	we	we	PRON
cana-1313	289	14	provided	provide	VERB
cana-1313	289	15	(	(	PUNCT
cana-1313	289	16	r*g*)*closed	r*g*)*close	VERB
cana-1313	289	17	sets	set	NOUN
cana-1313	289	18	and	and	CCONJ
cana-1313	289	19	(	(	PUNCT
cana-1313	289	20	r*g*)*open	r*g*)*open	PROPN
cana-1313	289	21	sets	set	NOUN
cana-1313	289	22	and	and	CCONJ
cana-1313	289	23	examined	examine	VERB
cana-1313	289	24	some	some	PRON
cana-1313	289	25	of	of	ADP
cana-1313	289	26	their	their	PRON
cana-1313	289	27	properties	property	NOUN
cana-1313	289	28	.	.	PUNCT
cana-1313	290	1	this	this	DET
cana-1313	290	2	class	class	NOUN
cana-1313	290	3	of	of	ADP
cana-1313	290	4	sets	set	NOUN
cana-1313	290	5	can	can	AUX
cana-1313	290	6	be	be	AUX
cana-1313	290	7	used	use	VERB
cana-1313	290	8	to	to	PART
cana-1313	290	9	study	study	VERB
cana-1313	290	10	the	the	DET
cana-1313	290	11	ideas	idea	NOUN
cana-1313	290	12	of	of	ADP
cana-1313	290	13	continuity	continuity	NOUN
cana-1313	290	14	,	,	PUNCT
cana-1313	290	15	compactness	compactness	NOUN
cana-1313	290	16	,	,	PUNCT
cana-1313	290	17	and	and	CCONJ
cana-1313	290	18	connectedness	connectedness	NOUN
cana-1313	290	19	in	in	ADP
cana-1313	290	20	different	different	ADJ
cana-1313	290	21	topological	topological	ADJ
cana-1313	290	22	spaces	space	NOUN
cana-1313	290	23	,	,	PUNCT
cana-1313	290	24	such	such	ADJ
cana-1313	290	25	as	as	ADP
cana-1313	290	26	fuzzy	fuzzy	ADJ
cana-1313	290	27	and	and	CCONJ
cana-1313	290	28	bi	bi	ADJ
cana-1313	290	29	-	-	ADJ
cana-1313	290	30	topological	topological	ADJ
cana-1313	290	31	ones	one	NOUN
cana-1313	290	32	.	.	PUNCT
cana-1313	291	1	references	reference	NOUN
cana-1313	291	2	:	:	PUNCT
cana-1313	292	1	[	[	X
cana-1313	292	2	1	1	NUM
cana-1313	292	3	]	]	PUNCT
cana-1313	292	4	a.narmadha	a.narmadha	NOUN
cana-1313	292	5	&	&	CCONJ
cana-1313	292	6	nagaveni	nagaveni	PROPN
cana-1313	292	7	,	,	PUNCT
cana-1313	292	8	on	on	ADP
cana-1313	292	9	regular	regular	ADJ
cana-1313	292	10	b	b	X
cana-1313	292	11	-	-	PUNCT
cana-1313	292	12	open	open	ADJ
cana-1313	292	13	sets	set	NOUN
cana-1313	292	14	in	in	ADP
cana-1313	292	15	topological	topological	ADJ
cana-1313	292	16	spaces	space	NOUN
cana-1313	292	17	,	,	PUNCT
cana-1313	292	18	int.journal	int.journal	PROPN
cana-1313	292	19	of	of	ADP
cana-1313	292	20	math	math	NOUN
cana-1313	292	21	.	.	PUNCT
cana-1313	293	1	analysis	analysis	NOUN
cana-1313	293	2	,	,	PUNCT
cana-1313	293	3	vol	vol	NOUN
cana-1313	293	4	.	.	PROPN
cana-1313	293	5	7	7	NUM
cana-1313	293	6	,	,	PUNCT
cana-1313	293	7	2013	2013	NUM
cana-1313	293	8	,	,	PUNCT
cana-1313	293	9	no.19	no.19	NOUN
cana-1313	293	10	,	,	PUNCT
cana-1313	293	11	937	937	NUM
cana-1313	293	12	-	-	SYM
cana-1313	293	13	948	948	NUM
cana-1313	293	14	.	.	PUNCT
cana-1313	294	1	[	[	X
cana-1313	294	2	2	2	NUM
cana-1313	294	3	]	]	PUNCT
cana-1313	294	4	c.	c.	PROPN
cana-1313	294	5	mugundan	mugundan	PROPN
cana-1313	294	6	,	,	PUNCT
cana-1313	294	7	n.	n.	PROPN
cana-1313	294	8	nagaveni	nagaveni	PROPN
cana-1313	294	9	,	,	PUNCT
cana-1313	294	10	a	a	DET
cana-1313	294	11	weaker	weak	ADJ
cana-1313	294	12	form	form	NOUN
cana-1313	294	13	of	of	ADP
cana-1313	294	14	closed	closed	ADJ
cana-1313	294	15	sets	set	NOUN
cana-1313	294	16	,	,	PUNCT
cana-1313	294	17	2011	2011	NUM
cana-1313	294	18	,	,	PUNCT
cana-1313	294	19	949	949	NUM
cana-1313	294	20	-	-	SYM
cana-1313	294	21	961	961	NUM
cana-1313	294	22	.	.	PUNCT
cana-1313	295	1	[	[	X
cana-1313	295	2	3	3	X
cana-1313	295	3	]	]	X
cana-1313	295	4	y.	y.	PROPN
cana-1313	295	5	gnanambal	gnanambal	PROPN
cana-1313	295	6	,	,	PUNCT
cana-1313	295	7	“	"	PUNCT
cana-1313	295	8	on	on	ADP
cana-1313	295	9	generalized	generalized	ADJ
cana-1313	295	10	pre	pre	ADJ
cana-1313	295	11	-	-	ADJ
cana-1313	295	12	regular	regular	ADJ
cana-1313	295	13	closed	closed	ADJ
cana-1313	295	14	sets	set	NOUN
cana-1313	295	15	in	in	ADP
cana-1313	295	16	topological	topological	ADJ
cana-1313	295	17	spaces	space	NOUN
cana-1313	295	18	”	"	PUNCT
cana-1313	295	19	,	,	PUNCT
cana-1313	295	20	indian	indian	PROPN
cana-1313	295	21	j.	j.	PROPN
cana-1313	295	22	pure	pure	PROPN
cana-1313	295	23	app	app	PROPN
cana-1313	295	24	.	.	PUNCT
cana-1313	296	1	maths	maths	PROPN
cana-1313	296	2	,	,	PUNCT
cana-1313	296	3	28(1997	28(1997	NUM
cana-1313	296	4	)	)	PUNCT
cana-1313	296	5	,	,	PUNCT
cana-1313	296	6	351	351	NUM
cana-1313	296	7	-	-	SYM
cana-1313	296	8	360	360	NUM
cana-1313	296	9	.	.	PUNCT
cana-1313	297	1	[	[	X
cana-1313	297	2	4	4	X
cana-1313	297	3	]	]	PUNCT
cana-1313	297	4	g.	g.	PROPN
cana-1313	297	5	srinivasa	srinivasa	PROPN
cana-1313	297	6	rao	rao	PROPN
cana-1313	297	7	,	,	PUNCT
cana-1313	297	8	et	et	PROPN
cana-1313	297	9	al	al	PROPN
cana-1313	297	10	.	.	PUNCT
cana-1313	298	1	"	"	PUNCT
cana-1313	298	2	g	g	NOUN
cana-1313	298	3	-	-	PUNCT
cana-1313	298	4	closed	close	VERB
cana-1313	298	5	type	type	NOUN
cana-1313	298	6	sets	set	NOUN
cana-1313	298	7	and	and	CCONJ
cana-1313	298	8	g*-closed	g*-closed	ADJ
cana-1313	298	9	type	type	NOUN
cana-1313	298	10	sets	set	NOUN
cana-1313	298	11	in	in	ADP
cana-1313	298	12	topological	topological	ADJ
cana-1313	298	13	ordered	order	VERB
cana-1313	298	14	spaces	space	NOUN
cana-1313	298	15	.	.	PUNCT
cana-1313	299	1	"	"	PUNCT
cana-1313	299	2	,	,	PUNCT
cana-1313	299	3	5(6)2014	5(6)2014	NUM
cana-1313	299	4	,	,	PUNCT
cana-1313	299	5	1276	1276	NUM
cana-1313	299	6	-	-	SYM
cana-1313	299	7	1285	1285	NUM
cana-1313	299	8	.	.	PUNCT
cana-1313	300	1	[	[	X
cana-1313	300	2	5	5	X
cana-1313	300	3	]	]	X
cana-1313	300	4	g.	g.	PROPN
cana-1313	300	5	srinivasarao	srinivasarao	PROPN
cana-1313	300	6	.	.	PUNCT
cana-1313	300	7	,	,	PUNCT
cana-1313	300	8	d.	d.	PROPN
cana-1313	300	9	madhusudanrao	madhusudanrao	PROPN
cana-1313	300	10	,	,	PUNCT
cana-1313	300	11	and	and	CCONJ
cana-1313	300	12	n.	n.	PROPN
cana-1313	300	13	srinivasarao	srinivasarao	PROPN
cana-1313	300	14	.	.	PUNCT
cana-1313	301	1	“	"	PUNCT
cana-1313	301	2	applications	application	NOUN
cana-1313	301	3	of	of	ADP
cana-1313	301	4	ig	ig	PROPN
cana-1313	301	5	,	,	PUNCT
cana-1313	301	6	dg	dg	PROPN
cana-1313	301	7	,	,	PUNCT
cana-1313	301	8	bg	bg	PROPN
cana-1313	301	9	-	-	PUNCT
cana-1313	301	10	closed	close	VERB
cana-1313	301	11	type	type	NOUN
cana-1313	301	12	sets	set	NOUN
cana-1313	301	13	in	in	ADP
cana-1313	301	14	topological	topological	ADJ
cana-1313	301	15	ordered	order	VERB
cana-1313	301	16	spaces”.8(1	spaces”.8(1	NOUN
cana-1313	301	17	)	)	PUNCT
cana-1313	301	18	(	(	PUNCT
cana-1313	301	19	2015	2015	NUM
cana-1313	301	20	)	)	PUNCT
cana-1313	301	21	,	,	PUNCT
cana-1313	301	22	12	12	NUM
cana-1313	301	23	-	-	SYM
cana-1313	301	24	22	22	NUM
cana-1313	301	25	[	[	X
cana-1313	301	26	6	6	NUM
cana-1313	301	27	]	]	X
cana-1313	301	28	g.	g.	PROPN
cana-1313	301	29	srinivasarao	srinivasarao	PROPN
cana-1313	301	30	,	,	PUNCT
cana-1313	301	31	d.	d.	PROPN
cana-1313	301	32	madhusudanarao	madhusudanarao	PROPN
cana-1313	301	33	,	,	PUNCT
cana-1313	301	34	and	and	CCONJ
cana-1313	301	35	n.	n.	PROPN
cana-1313	301	36	srinivasarao	srinivasarao	PROPN
cana-1313	301	37	.	.	PUNCT
cana-1313	302	1	“	"	PUNCT
cana-1313	302	2	separation	separation	NOUN
cana-1313	302	3	axioms	axiom	NOUN
cana-1313	302	4	using	use	VERB
cana-1313	302	5	ig	ig	PRON
cana-1313	302	6	*	*	PROPN
cana-1313	302	7	,	,	PUNCT
cana-1313	302	8	dg	dg	PROPN
cana-1313	302	9	*	*	PUNCT
cana-1313	302	10	,	,	PUNCT
cana-1313	302	11	bg*-closed	bg*-closed	ADJ
cana-1313	302	12	type	type	NOUN
cana-1313	302	13	sets	set	NOUN
cana-1313	302	14	in	in	ADP
cana-1313	302	15	topological	topological	ADJ
cana-1313	302	16	ordered	order	VERB
cana-1313	302	17	spaces	space	NOUN
cana-1313	302	18	”	"	PUNCT
cana-1313	302	19	.	.	PUNCT
cana-1313	303	1	international	international	ADJ
cana-1313	303	2	journal	journal	NOUN
cana-1313	303	3	of	of	ADP
cana-1313	303	4	advances	advance	NOUN
cana-1313	303	5	in	in	ADP
cana-1313	303	6	engineering	engineering	NOUN
cana-1313	303	7	&	&	CCONJ
cana-1313	303	8	technology	technology	PROPN
cana-1313	303	9	7.6	7.6	NUM
cana-1313	303	10	(	(	PUNCT
cana-1313	303	11	2015	2015	NUM
cana-1313	303	12	):	):	PUNCT
cana-1313	303	13	1840	1840	NUM
cana-1313	303	14	-	-	SYM
cana-1313	303	15	1850	1850	NUM
cana-1313	303	16	.	.	PUNCT
cana-1313	304	1	[	[	X
cana-1313	304	2	7	7	X
cana-1313	304	3	]	]	PUNCT
cana-1313	304	4	k.	k.	NOUN
cana-1313	304	5	mariappa	mariappa	PROPN
cana-1313	304	6	and	and	CCONJ
cana-1313	304	7	s.	s.	PROPN
cana-1313	304	8	sekar	sekar	PROPN
cana-1313	304	9	,	,	PUNCT
cana-1313	304	10	on	on	ADP
cana-1313	304	11	regular	regular	ADJ
cana-1313	304	12	generalised	generalised	ADJ
cana-1313	304	13	b	b	NOUN
cana-1313	304	14	-	-	PUNCT
cana-1313	304	15	closed	closed	ADJ
cana-1313	304	16	set	set	NOUN
cana-1313	304	17	,	,	PUNCT
cana-1313	304	18	int	int	NOUN
cana-1313	304	19	.	.	PUNCT
cana-1313	305	1	journal	journal	PROPN
cana-1313	305	2	of	of	ADP
cana-1313	305	3	math	math	NOUN
cana-1313	305	4	.	.	PUNCT
cana-1313	306	1	analysis	analysis	NOUN
cana-1313	306	2	,	,	PUNCT
cana-1313	306	3	vol	vol	NOUN
cana-1313	306	4	,	,	PUNCT
cana-1313	306	5	2013	2013	NUM
cana-1313	306	6	,	,	PUNCT
cana-1313	306	7	no.13	no.13	PROPN
cana-1313	306	8	,	,	PUNCT
cana-1313	306	9	613	613	NUM
cana-1313	306	10	-	-	SYM
cana-1313	306	11	624	624	NUM
cana-1313	306	12	.	.	PUNCT
cana-1313	307	1	[	[	X
cana-1313	307	2	8	8	NUM
cana-1313	307	3	]	]	PUNCT
cana-1313	307	4	m.	m.	NOUN
cana-1313	307	5	e.	e.	PROPN
cana-1313	307	6	abd	abd	PROPN
cana-1313	307	7	el	el	PROPN
cana-1313	307	8	-	-	PROPN
cana-1313	307	9	monsef	monsef	PROPN
cana-1313	307	10	,	,	PUNCT
cana-1313	307	11	s.	s.	PROPN
cana-1313	307	12	n.	n.	PROPN
cana-1313	307	13	el	el	PROPN
cana-1313	307	14	.	.	PUNCT
cana-1313	307	15	deeb	deeb	PROPN
cana-1313	307	16	and	and	CCONJ
cana-1313	307	17	r.	r.	PROPN
cana-1313	307	18	a.	a.	PROPN
cana-1313	307	19	mohamoud	mohamoud	PROPN
cana-1313	307	20	,	,	PUNCT
cana-1313	307	21	β	β	X
cana-1313	307	22	open	open	ADJ
cana-1313	307	23	sets	set	NOUN
cana-1313	307	24	and	and	CCONJ
cana-1313	307	25	β	β	X
cana-1313	307	26	continuous	continuous	ADJ
cana-1313	307	27	mappings	mapping	NOUN
cana-1313	307	28	,	,	PUNCT
cana-1313	307	29	bull	bull	NOUN
cana-1313	307	30	.	.	PUNCT
cana-1313	308	1	fac	fac	PROPN
cana-1313	308	2	.	.	PUNCT
cana-1313	309	1	sci	sci	PROPN
cana-1313	309	2	.	.	PUNCT
cana-1313	309	3	assiut	assiut	PROPN
cana-1313	309	4	univ	univ	PROPN
cana-1313	309	5	.	.	PROPN
cana-1313	309	6	,	,	PUNCT
cana-1313	309	7	12(1983	12(1983	NUM
cana-1313	309	8	)	)	PUNCT
cana-1313	309	9	,	,	PUNCT
cana-1313	309	10	77	77	NUM
cana-1313	309	11	-	-	SYM
cana-1313	309	12	80	80	NUM
cana-1313	309	13	.	.	PUNCT
cana-1313	310	1	[	[	X
cana-1313	310	2	9	9	NUM
cana-1313	310	3	]	]	PUNCT
cana-1313	310	4	m.	m.	NOUN
cana-1313	310	5	k.	k.	PROPN
cana-1313	310	6	r.	r.	PROPN
cana-1313	310	7	s.	s.	PROPN
cana-1313	310	8	veerakumar	veerakumar	PROPN
cana-1313	310	9	,	,	PUNCT
cana-1313	310	10	between	between	ADP
cana-1313	310	11	closed	close	VERB
cana-1313	310	12	sets	set	NOUN
cana-1313	310	13	and	and	CCONJ
cana-1313	310	14	g	g	NOUN
cana-1313	310	15	closed	closed	ADJ
cana-1313	310	16	sets	set	NOUN
cana-1313	310	17	,	,	PUNCT
cana-1313	310	18	mem	mem	PROPN
cana-1313	310	19	.	.	PUNCT
cana-1313	310	20	fac	fac	PROPN
cana-1313	310	21	.	.	PUNCT
cana-1313	310	22	sci	sci	PROPN
cana-1313	310	23	.	.	PROPN
cana-1313	310	24	kochi	kochi	PROPN
cana-1313	310	25	univ	univ	PROPN
cana-1313	310	26	.	.	PUNCT
cana-1313	311	1	ser	ser	PROPN
cana-1313	311	2	.	.	PUNCT
cana-1313	311	3	a.	a.	PROPN
cana-1313	311	4	math	math	PROPN
cana-1313	311	5	.	.	PUNCT
cana-1313	311	6	,	,	PUNCT
cana-1313	311	7	21	21	NUM
cana-1313	311	8	(	(	PUNCT
cana-1313	311	9	2000	2000	NUM
cana-1313	311	10	)	)	PUNCT
cana-1313	311	11	1	1	NUM
cana-1313	311	12	-	-	SYM
cana-1313	311	13	19	19	NUM
cana-1313	311	14	.	.	PUNCT
cana-1313	312	1	[	[	X
cana-1313	312	2	10	10	NUM
cana-1313	312	3	]	]	PUNCT
cana-1313	312	4	m.	m.	NOUN
cana-1313	312	5	k.	k.	PROPN
cana-1313	312	6	r.	r.	PROPN
cana-1313	312	7	s.	s.	PROPN
cana-1313	312	8	veera	veera	PROPN
cana-1313	312	9	kumar	kumar	PROPN
cana-1313	312	10	,	,	PUNCT
cana-1313	312	11	g#-closed	g#-close	VERB
cana-1313	312	12	sets	set	NOUN
cana-1313	312	13	in	in	ADP
cana-1313	312	14	topological	topological	ADJ
cana-1313	312	15	spaces	space	NOUN
cana-1313	312	16	,	,	PUNCT
cana-1313	312	17	mem	mem	PROPN
cana-1313	312	18	.	.	PUNCT
cana-1313	313	1	fac	fac	PROPN
cana-1313	313	2	.	.	PUNCT
cana-1313	314	1	sci	sci	PROPN
cana-1313	314	2	.	.	PROPN
cana-1313	314	3	kochi	kochi	PROPN
cana-1313	314	4	univ	univ	PROPN
cana-1313	314	5	ser	ser	PROPN
cana-1313	314	6	.	.	PUNCT
cana-1313	315	1	a.	a.	PROPN
cana-1313	315	2	,	,	PUNCT
cana-1313	315	3	math	math	NOUN
cana-1313	315	4	.	.	PUNCT
cana-1313	315	5	,	,	PUNCT
cana-1313	315	6	24(2003),1	24(2003),1	PROPN
cana-1313	315	7	-	-	PUNCT
cana-1313	315	8	13	13	NUM
cana-1313	315	9	.	.	PUNCT
cana-1313	316	1	[	[	X
cana-1313	316	2	11	11	NUM
cana-1313	316	3	]	]	X
cana-1313	316	4	n.	n.	PROPN
cana-1313	316	5	levine	levine	PROPN
cana-1313	316	6	,	,	PUNCT
cana-1313	316	7	generalized	generalize	VERB
cana-1313	316	8	closed	closed	ADJ
cana-1313	316	9	sets	set	NOUN
cana-1313	316	10	in	in	ADP
cana-1313	316	11	topology	topology	NOUN
cana-1313	316	12	,	,	PUNCT
cana-1313	316	13	rend	rend	VERB
cana-1313	316	14	.	.	PUNCT
cana-1313	317	1	circ	circ	PROPN
cana-1313	317	2	.	.	PUNCT
cana-1313	318	1	math	math	NOUN
cana-1313	318	2	.	.	PUNCT
cana-1313	319	1	palermo	palermo	PROPN
cana-1313	319	2	,	,	PUNCT
cana-1313	319	3	19(2)(1970),89	19(2)(1970),89	NUM
cana-1313	319	4	-	-	SYM
cana-1313	319	5	96	96	NUM
cana-1313	319	6	.	.	PUNCT
cana-1313	320	1	[	[	X
cana-1313	320	2	12	12	NUM
cana-1313	320	3	]	]	X
cana-1313	320	4	n.	n.	NOUN
cana-1313	320	5	meenakumari	meenakumari	NOUN
cana-1313	320	6	and	and	CCONJ
cana-1313	320	7	t.	t.	PROPN
cana-1313	320	8	indira	indira	PROPN
cana-1313	320	9	,	,	PUNCT
cana-1313	320	10	r*g	r*g	PROPN
cana-1313	320	11	*	*	PUNCT
cana-1313	320	12	closed	closed	ADJ
cana-1313	320	13	sets	set	NOUN
cana-1313	320	14	in	in	ADP
cana-1313	320	15	topological	topological	ADJ
cana-1313	320	16	spaces	space	NOUN
cana-1313	320	17	,	,	PUNCT
cana-1313	320	18	annals	annal	NOUN
cana-1313	320	19	of	of	ADP
cana-1313	320	20	pure	pure	ADJ
cana-1313	320	21	and	and	CCONJ
cana-1313	320	22	applied	apply	VERB
cana-1313	320	23	mathematics	mathematics	PROPN
cana-1313	320	24	vol.6	vol.6	PROPN
cana-1313	320	25	,	,	PUNCT
cana-1313	320	26	no	no	INTJ
cana-1313	320	27	.	.	NOUN
cana-1313	320	28	2	2	NUM
cana-1313	320	29	,	,	PUNCT
cana-1313	320	30	2014	2014	NUM
cana-1313	320	31	,	,	PUNCT
cana-1313	320	32	125	125	NUM
cana-1313	320	33	-	-	SYM
cana-1313	320	34	132	132	NUM
cana-1313	320	35	.	.	PUNCT
cana-1313	321	1	[	[	X
cana-1313	321	2	13	13	NUM
cana-1313	321	3	]	]	X
cana-1313	321	4	n.	n.	PROPN
cana-1313	321	5	palaniappan	palaniappan	PROPN
cana-1313	321	6	&	&	CCONJ
cana-1313	321	7	k.	k.	PROPN
cana-1313	321	8	c.	c.	PROPN
cana-1313	321	9	rao	rao	PROPN
cana-1313	321	10	,	,	PUNCT
cana-1313	321	11	regular	regular	ADJ
cana-1313	321	12	generalized	generalize	VERB
cana-1313	321	13	closed	close	VERB
cana-1313	321	14	sets	set	NOUN
cana-1313	321	15	,	,	PUNCT
cana-1313	321	16	kyungpook	kyungpook	NOUN
cana-1313	321	17	math	math	NOUN
cana-1313	321	18	.	.	PUNCT
cana-1313	322	1	3	3	NUM
cana-1313	322	2	(	(	PUNCT
cana-1313	322	3	2	2	NUM
cana-1313	322	4	)	)	PUNCT
cana-1313	322	5	(	(	PUNCT
cana-1313	322	6	1993	1993	NUM
cana-1313	322	7	)	)	PUNCT
cana-1313	322	8	,	,	PUNCT
cana-1313	322	9	211	211	NUM
cana-1313	322	10	.	.	PUNCT
cana-1313	323	1	[	[	X
cana-1313	323	2	14	14	NUM
cana-1313	323	3	]	]	X
cana-1313	323	4	pauline	pauline	PROPN
cana-1313	323	5	mary	mary	PROPN
cana-1313	323	6	helen	helen	PROPN
cana-1313	323	7	.	.	PUNCT
cana-1313	324	1	m	m	PROPN
cana-1313	324	2	,	,	PUNCT
cana-1313	324	3	veronica	veronica	PROPN
cana-1313	324	4	vijayan	vijayan	PROPN
cana-1313	324	5	,	,	PUNCT
cana-1313	324	6	ponnuthai	ponnuthai	ADJ
cana-1313	324	7	selvarani	selvarani	NOUN
cana-1313	324	8	,	,	PUNCT
cana-1313	324	9	g	g	PROPN
cana-1313	324	10	*	*	PROPN
cana-1313	324	11	*	*	NOUN
cana-1313	324	12	closed	closed	ADJ
cana-1313	324	13	sets	set	NOUN
cana-1313	324	14	in	in	ADP
cana-1313	324	15	topological	topological	ADJ
cana-1313	324	16	spaces	space	NOUN
cana-1313	324	17	,	,	PUNCT
cana-1313	324	18	ijma	ijma	ADJ
cana-1313	324	19	3	3	NUM
cana-1313	324	20	(	(	PUNCT
cana-1313	324	21	5	5	NUM
cana-1313	324	22	)	)	PUNCT
cana-1313	324	23	,	,	PUNCT
cana-1313	324	24	(	(	PUNCT
cana-1313	324	25	2012	2012	NUM
cana-1313	324	26	)	)	PUNCT
cana-1313	324	27	,	,	PUNCT
cana-1313	324	28	1	1	NUM
cana-1313	324	29	-	-	SYM
cana-1313	324	30	15	15	NUM
cana-1313	324	31	.	.	PUNCT
cana-1313	325	1	[	[	X
cana-1313	325	2	15	15	NUM
cana-1313	325	3	]	]	X
cana-1313	325	4	r.	r.	PROPN
cana-1313	325	5	devi	devi	PROPN
cana-1313	325	6	,	,	PUNCT
cana-1313	325	7	h.	h.	PROPN
cana-1313	325	8	maki	maki	PROPN
cana-1313	325	9	,	,	PUNCT
cana-1313	325	10	and	and	CCONJ
cana-1313	325	11	k.	k.	PROPN
cana-1313	325	12	balachandran	balachandran	PROPN
cana-1313	325	13	,	,	PUNCT
cana-1313	325	14	generalized	generalize	VERB
cana-1313	325	15	αclosed	αclose	VERB
cana-1313	325	16	maps	map	NOUN
cana-1313	325	17	and	and	CCONJ
cana-1313	325	18	α	α	PRON
cana-1313	325	19	generalized	generalize	VERB
cana-1313	325	20	closed	closed	ADJ
cana-1313	325	21	maps	map	NOUN
cana-1313	325	22	,	,	PUNCT
cana-1313	325	23	indian	indian	ADJ
cana-1313	325	24	j.	j.	PROPN
cana-1313	325	25	pure	pure	PROPN
cana-1313	325	26	.	.	PUNCT
cana-1313	326	1	appl	appl	PROPN
cana-1313	326	2	.	.	PROPN
cana-1313	326	3	math	math	PROPN
cana-1313	326	4	,	,	PUNCT
cana-1313	326	5	29(1)(1998	29(1)(1998	NUM
cana-1313	326	6	)	)	PUNCT
cana-1313	326	7	,	,	PUNCT
cana-1313	326	8	37	37	NUM
cana-1313	326	9	-	-	SYM
cana-1313	326	10	49	49	NUM
cana-1313	326	11	.	.	PUNCT
cana-1313	327	1	[	[	X
cana-1313	327	2	16	16	NUM
cana-1313	327	3	]	]	X
cana-1313	327	4	savithri	savithri	NOUN
cana-1313	327	5	.	.	PUNCT
cana-1313	328	1	d	d	PROPN
cana-1313	328	2	&	&	CCONJ
cana-1313	328	3	janaki	janaki	PROPN
cana-1313	328	4	.	.	PUNCT
cana-1313	329	1	c	c	X
cana-1313	329	2	,	,	PUNCT
cana-1313	329	3	on	on	ADP
cana-1313	329	4	regular	regular	ADJ
cana-1313	329	5	generalized	generalize	VERB
cana-1313	329	6	closed	closed	ADJ
cana-1313	329	7	sets	set	NOUN
cana-1313	329	8	in	in	ADP
cana-1313	329	9	topological	topological	ADJ
cana-1313	329	10	spaces	space	NOUN
cana-1313	329	11	,	,	PUNCT
cana-1313	329	12	ijma4(4)2013	ijma4(4)2013	NOUN
cana-1313	329	13	,	,	PUNCT
cana-1313	329	14	162	162	NUM
cana-1313	329	15	-	-	SYM
cana-1313	329	16	169	169	NUM
cana-1313	329	17	.	.	PUNCT
cana-1313	330	1	[	[	X
cana-1313	330	2	17	17	NUM
cana-1313	330	3	]	]	PUNCT
cana-1313	330	4	s.	s.	PROPN
cana-1313	330	5	p.	p.	PROPN
cana-1313	330	6	arya	arya	PROPN
cana-1313	330	7	and	and	CCONJ
cana-1313	330	8	t.	t.	PROPN
cana-1313	330	9	m.	m.	PROPN
cana-1313	330	10	nour	nour	PROPN
cana-1313	330	11	,	,	PUNCT
cana-1313	330	12	characterizations	characterization	NOUN
cana-1313	330	13	of	of	ADP
cana-1313	330	14	s	s	VERB
cana-1313	330	15	normal	normal	ADJ
cana-1313	330	16	spaces	space	NOUN
cana-1313	330	17	,	,	PUNCT
cana-1313	330	18	indian	indian	PROPN
cana-1313	330	19	j.	j.	PROPN
cana-1313	330	20	pure	pure	PROPN
cana-1313	330	21	app	app	PROPN
cana-1313	330	22	.	.	PROPN
cana-1313	330	23	math	math	PROPN
cana-1313	330	24	,	,	PUNCT
cana-1313	330	25	21(1990	21(1990	NUM
cana-1313	330	26	)	)	PUNCT
cana-1313	331	1	[	[	X
cana-1313	331	2	18	18	NUM
cana-1313	331	3	]	]	PUNCT
cana-1313	331	4	s.	s.	PROPN
cana-1313	331	5	s.	s.	PROPN
cana-1313	331	6	benchelli	benchelli	PROPN
cana-1313	331	7	and	and	CCONJ
cana-1313	331	8	r.	r.	PROPN
cana-1313	331	9	s.	s.	PROPN
cana-1313	331	10	wali	wali	PROPN
cana-1313	331	11	,	,	PUNCT
cana-1313	331	12	on	on	ADP
cana-1313	331	13	rw	rw	NOUN
cana-1313	331	14	-	-	PUNCT
cana-1313	331	15	closed	close	VERB
cana-1313	331	16	sets	set	NOUN
cana-1313	331	17	in	in	ADP
cana-1313	331	18	topological	topological	ADJ
cana-1313	331	19	spaces	space	NOUN
cana-1313	331	20	,	,	PUNCT
cana-1313	331	21	bull.malayas	bull.malayas	PROPN
cana-1313	331	22	.	.	PUNCT
cana-1313	331	23	math.soc.(2007),99	math.soc.(2007),99	PROPN
cana-1313	331	24	-	-	PUNCT
cana-1313	331	25	110	110	NUM
cana-1313	331	26	.	.	PUNCT
cana-1313	332	1	[	[	X
cana-1313	332	2	19	19	NUM
cana-1313	332	3	]	]	PUNCT
cana-1313	332	4	g.	g.	PROPN
cana-1313	332	5	srinivasa	srinivasa	PROPN
cana-1313	332	6	rao	rao	PROPN
cana-1313	332	7	,	,	PUNCT
cana-1313	332	8	d.	d.	PROPN
cana-1313	332	9	madhusudhanarao	madhusudhanarao	PROPN
cana-1313	332	10	and	and	CCONJ
cana-1313	332	11	p.	p.	PROPN
cana-1313	332	12	siva	siva	PROPN
cana-1313	332	13	prasad	prasad	PROPN
cana-1313	332	14	,	,	PUNCT
cana-1313	332	15	simple	simple	ADJ
cana-1313	332	16	ternary	ternary	ADJ
cana-1313	332	17	semi	semi	NOUN
cana-1313	332	18	-	-	NOUN
cana-1313	332	19	rings	ring	NOUN
cana-1313	332	20	,	,	PUNCT
cana-1313	332	21	the	the	DET
cana-1313	332	22	global	global	ADJ
cana-1313	332	23	journal	journal	NOUN
cana-1313	332	24	of	of	ADP
cana-1313	332	25	mathematics	mathematics	PROPN
cana-1313	332	26	&	&	CCONJ
cana-1313	332	27	mathematical	mathematical	PROPN
cana-1313	332	28	sciences	sciences	PROPN
cana-1313	332	29	,	,	PUNCT
cana-1313	332	30	9(2	9(2	NUM
cana-1313	332	31	)	)	PUNCT
cana-1313	332	32	(	(	PUNCT
cana-1313	332	33	2016	2016	NUM
cana-1313	332	34	)	)	PUNCT
cana-1313	332	35	,	,	PUNCT
cana-1313	332	36	185	185	NUM
cana-1313	332	37	-	-	SYM
cana-1313	332	38	196	196	NUM
cana-1313	332	39	.	.	PUNCT
cana-1313	333	1	[	[	X
cana-1313	333	2	20	20	NUM
cana-1313	333	3	]	]	X
cana-1313	333	4	d.	d.	PROPN
cana-1313	333	5	madhusudhana	madhusudhana	PROPN
cana-1313	333	6	rao	rao	PROPN
cana-1313	333	7	,	,	PUNCT
cana-1313	333	8	g.	g.	PROPN
cana-1313	333	9	srinivasa	srinivasa	PROPN
cana-1313	333	10	rao	rao	PROPN
cana-1313	333	11	,	,	PUNCT
cana-1313	333	12	special	special	ADJ
cana-1313	333	13	elements	element	NOUN
cana-1313	333	14	in	in	ADP
cana-1313	333	15	ternary	ternary	ADJ
cana-1313	333	16	semi	semi	ADJ
cana-1313	333	17	rings	ring	NOUN
cana-1313	333	18	,	,	PUNCT
cana-1313	333	19	international	international	ADJ
cana-1313	333	20	journal	journal	NOUN
cana-1313	333	21	of	of	ADP
cana-1313	333	22	engineering	engineering	NOUN
cana-1313	333	23	research	research	NOUN
cana-1313	333	24	and	and	CCONJ
cana-1313	333	25	applications	application	NOUN
cana-1313	333	26	,	,	PUNCT
cana-1313	333	27	4(11	4(11	NUM
cana-1313	333	28	)	)	PUNCT
cana-1313	333	29	(	(	PUNCT
cana-1313	333	30	2014	2014	NUM
cana-1313	333	31	)	)	PUNCT
cana-1313	333	32	,	,	PUNCT
cana-1313	333	33	123	123	NUM
cana-1313	333	34	-	-	SYM
cana-1313	333	35	130	130	NUM
cana-1313	333	36	.	.	PUNCT
cana-1313	334	1	[	[	X
cana-1313	334	2	21	21	NUM
cana-1313	334	3	]	]	X
cana-1313	334	4	g.	g.	PROPN
cana-1313	334	5	srinivasa	srinivasa	PROPN
cana-1313	334	6	rao	rao	PROPN
cana-1313	334	7	,	,	PUNCT
cana-1313	334	8	d.	d.	PROPN
cana-1313	334	9	madhusudhana	madhusudhana	PROPN
cana-1313	334	10	rao	rao	PROPN
cana-1313	334	11	,	,	PUNCT
cana-1313	334	12	structure	structure	NOUN
cana-1313	334	13	of	of	ADP
cana-1313	334	14	certain	certain	ADJ
cana-1313	334	15	ideals	ideal	NOUN
cana-1313	334	16	in	in	ADP
cana-1313	334	17	ternary	ternary	ADJ
cana-1313	334	18	semi	semi	ADJ
cana-1313	334	19	rings	ring	NOUN
cana-1313	334	20	,	,	PUNCT
cana-1313	334	21	int	int	NOUN
cana-1313	334	22	.	.	PUNCT
cana-1313	335	1	j.	j.	PROPN
cana-1313	335	2	of	of	ADP
cana-1313	335	3	innovative	innovative	ADJ
cana-1313	335	4	science	science	NOUN
cana-1313	335	5	and	and	CCONJ
cana-1313	335	6	modern	modern	ADJ
cana-1313	335	7	engg	engg	PROPN
cana-1313	335	8	.	.	PUNCT
cana-1313	335	9	,	,	PUNCT
cana-1313	335	10	3(3	3(3	NUM
cana-1313	335	11	)	)	PUNCT
cana-1313	335	12	(	(	PUNCT
cana-1313	335	13	2015	2015	NUM
cana-1313	335	14	)	)	PUNCT
cana-1313	335	15	,	,	PUNCT
cana-1313	335	16	49	49	NUM
cana-1313	335	17	-	-	SYM
cana-1313	335	18	56	56	NUM
cana-1313	335	19	.	.	PUNCT
cana-1313	336	1	brg	brg	PROPN
cana-1313	336	2	-	-	PUNCT
cana-1313	336	3	closed	close	VERB
cana-1313	336	4	br*g*-closed	br*g*-close	VERB
cana-1313	336	5	irg	irg	PROPN
cana-1313	336	6	-	-	ADJ
cana-1313	336	7	closed	close	VERB
cana-1313	336	8	ir*g*-closed	ir*g*-closed	ADJ
cana-1313	336	9	drg	drg	NOUN
cana-1313	336	10	-	-	PUNCT
cana-1313	336	11	closed	close	VERB
cana-1313	336	12	dr*g*-closed	dr*g*-close	VERB
cana-1313	336	13	brg	brg	NOUN
cana-1313	336	14	-	-	PUNCT
cana-1313	336	15	closed	close	VERB
cana-1313	336	16	communications	communication	NOUN
cana-1313	336	17	on	on	ADP
cana-1313	336	18	applied	apply	VERB
cana-1313	336	19	nonlinear	nonlinear	ADJ
cana-1313	336	20	analysis	analysis	NOUN
cana-1313	336	21	issn	issn	NOUN
cana-1313	336	22	:	:	PUNCT
cana-1313	336	23	1074	1074	NUM
cana-1313	336	24	-	-	PUNCT
cana-1313	336	25	133x	133x	NUM
cana-1313	336	26	vol	vol	NOUN
cana-1313	336	27	31	31	NUM
cana-1313	336	28	no	no	NOUN
cana-1313	336	29	.	.	PUNCT
cana-1313	337	1	7s	7	NOUN
cana-1313	337	2	(	(	PUNCT
cana-1313	337	3	2024	2024	NUM
cana-1313	337	4	)	)	PUNCT
cana-1313	337	5	356	356	NUM
cana-1313	337	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1313	338	1	[	[	X
cana-1313	338	2	22	22	NUM
cana-1313	338	3	]	]	PUNCT
cana-1313	338	4	g.	g.	PROPN
cana-1313	338	5	srinivasa	srinivasa	PROPN
cana-1313	338	6	rao	rao	PROPN
cana-1313	338	7	,	,	PUNCT
cana-1313	338	8	d.	d.	PROPN
cana-1313	338	9	madhusudhana	madhusudhana	PROPN
cana-1313	338	10	rao	rao	PROPN
cana-1313	338	11	,	,	PUNCT
cana-1313	338	12	a	a	DET
cana-1313	338	13	study	study	NOUN
cana-1313	338	14	on	on	ADP
cana-1313	338	15	ternary	ternary	ADJ
cana-1313	338	16	semi	semi	ADJ
cana-1313	338	17	rings	ring	NOUN
cana-1313	338	18	,	,	PUNCT
cana-1313	338	19	int	int	NOUN
cana-1313	338	20	.	.	PUNCT
cana-1313	339	1	j.	j.	PROPN
cana-1313	339	2	of	of	ADP
cana-1313	339	3	math	math	PROPN
cana-1313	339	4	.	.	PUNCT
cana-1313	340	1	archive	archive	NOUN
cana-1313	340	2	,	,	PUNCT
cana-1313	340	3	5(12	5(12	NUM
cana-1313	340	4	)	)	PUNCT
cana-1313	340	5	(	(	PUNCT
cana-1313	340	6	2014	2014	NUM
cana-1313	340	7	)	)	PUNCT
cana-1313	340	8	,	,	PUNCT
cana-1313	340	9	24	24	NUM
cana-1313	340	10	-	-	SYM
cana-1313	340	11	30	30	NUM
cana-1313	340	12	.	.	PUNCT
cana-1313	341	1	[	[	X
cana-1313	341	2	23	23	NUM
cana-1313	341	3	]	]	PUNCT
cana-1313	341	4	g.	g.	PROPN
cana-1313	341	5	srinivasa	srinivasa	PROPN
cana-1313	341	6	rao	rao	PROPN
cana-1313	341	7	,	,	PUNCT
cana-1313	341	8	d.	d.	PROPN
cana-1313	341	9	madhusudhana	madhusudhana	PROPN
cana-1313	341	10	rao	rao	PROPN
cana-1313	341	11	,	,	PUNCT
cana-1313	341	12	characteristics	characteristic	NOUN
cana-1313	341	13	of	of	ADP
cana-1313	341	14	ternary	ternary	ADJ
cana-1313	341	15	semi	semi	ADJ
cana-1313	341	16	rings	ring	NOUN
cana-1313	341	17	,	,	PUNCT
cana-1313	341	18	int.j	int.j	PROPN
cana-1313	341	19	.	.	PROPN
cana-1313	341	20	of	of	ADP
cana-1313	341	21	engg	engg	PROPN
cana-1313	341	22	.	.	PUNCT
cana-1313	342	1	res	re	NOUN
cana-1313	342	2	.	.	PUNCT
cana-1313	342	3	and	and	CCONJ
cana-1313	342	4	mgt	mgt	PROPN
cana-1313	342	5	.	.	PUNCT
cana-1313	342	6	,	,	PUNCT
cana-1313	342	7	2(1	2(1	NUM
cana-1313	342	8	)	)	PUNCT
cana-1313	342	9	(	(	PUNCT
cana-1313	342	10	2015	2015	NUM
cana-1313	342	11	)	)	PUNCT
cana-1313	342	12	,	,	PUNCT
cana-1313	342	13	3	3	NUM
cana-1313	342	14	-	-	SYM
cana-1313	342	15	6	6	NUM
cana-1313	342	16	.	.	PUNCT
cana-1313	343	1	[	[	X
cana-1313	343	2	24	24	NUM
cana-1313	343	3	]	]	PUNCT
cana-1313	343	4	g.	g.	PROPN
cana-1313	343	5	srinivasa	srinivasa	PROPN
cana-1313	343	6	rao	rao	PROPN
cana-1313	343	7	,	,	PUNCT
cana-1313	343	8	a.	a.	PROPN
cana-1313	343	9	nagamalleswara	nagamalleswara	PROPN
cana-1313	343	10	rao	rao	PROPN
cana-1313	343	11	,	,	PUNCT
cana-1313	343	12	p.l.n	p.l.n	PROPN
cana-1313	343	13	.	.	PROPN
cana-1313	343	14	varma	varma	PROPN
cana-1313	343	15	,	,	PUNCT
cana-1313	343	16	d.madhusudhana	d.madhusudhana	PROPN
cana-1313	343	17	rao	rao	PROPN
cana-1313	343	18	,	,	PUNCT
cana-1313	343	19	ch	ch	NOUN
cana-1313	343	20	.	.	PROPN
cana-1313	343	21	ramprasad	ramprasad	ADJ
cana-1313	343	22	,	,	PUNCT
cana-1313	343	23	prime	prime	ADJ
cana-1313	343	24	biinterior	biinterior	PROPN
cana-1313	343	25	ideals	ideal	NOUN
cana-1313	343	26	in	in	ADP
cana-1313	343	27	tgsr	tgsr	ADJ
cana-1313	343	28	,	,	PUNCT
cana-1313	343	29	malaya	malaya	PROPN
cana-1313	343	30	journal	journal	PROPN
cana-1313	343	31	of	of	ADP
cana-1313	343	32	mathematika	mathematika	NOUN
cana-1313	343	33	,	,	PUNCT
cana-1313	343	34	vol.9	vol.9	PROPN
cana-1313	343	35	,	,	PUNCT
cana-1313	343	36	no.1	no.1	NUM
cana-1313	343	37	,	,	PUNCT
cana-1313	343	38	pp:542	pp:542	ADV
cana-1313	343	39	-	-	PUNCT
cana-1313	343	40	546	546	NUM
cana-1313	343	41	,	,	PUNCT
cana-1313	343	42	2021	2021	NUM
cana-1313	343	43	.	.	PUNCT
cana-1313	344	1	[	[	X
cana-1313	344	2	25	25	NUM
cana-1313	344	3	]	]	PUNCT
cana-1313	344	4	g.	g.	PROPN
cana-1313	344	5	srinivasa	srinivasa	PROPN
cana-1313	344	6	rao	rao	PROPN
cana-1313	344	7	,	,	PUNCT
cana-1313	344	8	a.	a.	PROPN
cana-1313	344	9	nagamalleswara	nagamalleswara	PROPN
cana-1313	344	10	rao	rao	PROPN
cana-1313	344	11	,	,	PUNCT
cana-1313	344	12	p.l.n	p.l.n	PROPN
cana-1313	344	13	.	.	PROPN
cana-1313	344	14	varma	varma	PROPN
cana-1313	344	15	,	,	PUNCT
cana-1313	344	16	d.	d.	PROPN
cana-1313	344	17	madhusudhana	madhusudhana	PROPN
cana-1313	344	18	rao	rao	PROPN
cana-1313	344	19	,	,	PUNCT
cana-1313	344	20	ch	ch	NOUN
cana-1313	344	21	.	.	PROPN
cana-1313	344	22	ramprasad	ramprasad	ADJ
cana-1313	344	23	,	,	PUNCT
cana-1313	344	24	bi	bi	ADJ
cana-1313	344	25	-	-	ADJ
cana-1313	344	26	interior	interior	ADJ
cana-1313	344	27	ideals	ideal	NOUN
cana-1313	344	28	in	in	ADP
cana-1313	344	29	tgsr	tgsr	ADJ
cana-1313	344	30	,	,	PUNCT
cana-1313	344	31	advances	advance	NOUN
cana-1313	344	32	in	in	ADP
cana-1313	344	33	mathematics	mathematics	NOUN
cana-1313	344	34	scientific	scientific	ADJ
cana-1313	344	35	journal	journal	NOUN
cana-1313	344	36	,	,	PUNCT
cana-1313	344	37	10	10	NUM
cana-1313	344	38	(	(	PUNCT
cana-1313	344	39	2021	2021	NUM
cana-1313	344	40	)	)	PUNCT
cana-1313	344	41	,	,	PUNCT
cana-1313	344	42	no.3	no.3	VERB
cana-1313	344	43	,	,	PUNCT
cana-1313	344	44	pp	pp	CCONJ
cana-1313	344	45	:	:	PUNCT
cana-1313	344	46	1183	1183	NUM
cana-1313	344	47	-	-	SYM
cana-1313	344	48	1195	1195	NUM
cana-1313	344	49	.	.	PUNCT