id	sid	tid	token	lemma	pos
ejpam-726	1	1	3_726_acikgoz.dvi	3_726_acikgoz.dvi	NUM
ejpam-726	1	2	european	european	ADJ
ejpam-726	1	3	journal	journal	NOUN
ejpam-726	1	4	of	of	ADP
ejpam-726	1	5	pure	pure	ADJ
ejpam-726	1	6	and	and	CCONJ
ejpam-726	1	7	applied	apply	VERB
ejpam-726	1	8	mathematics	mathematic	NOUN
ejpam-726	1	9	vol	vol	NOUN
ejpam-726	1	10	.	.	PROPN
ejpam-726	1	11	4	4	NUM
ejpam-726	1	12	,	,	PUNCT
ejpam-726	1	13	no	no	INTJ
ejpam-726	1	14	.	.	NOUN
ejpam-726	1	15	1	1	NUM
ejpam-726	1	16	,	,	PUNCT
ejpam-726	1	17	2011	2011	NUM
ejpam-726	1	18	,	,	PUNCT
ejpam-726	1	19	20	20	NUM
ejpam-726	1	20	-	-	SYM
ejpam-726	1	21	33	33	NUM
ejpam-726	1	22	issn	issn	PROPN
ejpam-726	1	23	1307	1307	NUM
ejpam-726	1	24	-	-	SYM
ejpam-726	1	25	5543	5543	NUM
ejpam-726	1	26	–	–	PUNCT
ejpam-726	1	27	www.ejpam.com	www.ejpam.com	X
ejpam-726	1	28	on	on	ADP
ejpam-726	1	29	β∗g−closed	β∗g−close	VERB
ejpam-726	1	30	sets	set	NOUN
ejpam-726	1	31	and	and	CCONJ
ejpam-726	1	32	new	new	ADJ
ejpam-726	1	33	separation	separation	NOUN
ejpam-726	1	34	axioms	axiom	VERB
ejpam-726	1	35	ahu	ahu	PROPN
ejpam-726	1	36	açıkgöz	açıkgöz	PROPN
ejpam-726	1	37	department	department	PROPN
ejpam-726	1	38	of	of	ADP
ejpam-726	1	39	mathematics	mathematic	NOUN
ejpam-726	1	40	,	,	PUNCT
ejpam-726	1	41	aksaray	aksaray	VERB
ejpam-726	1	42	university	university	NOUN
ejpam-726	1	43	,	,	PUNCT
ejpam-726	1	44	68100	68100	NUM
ejpam-726	1	45	aksaray	aksaray	VERB
ejpam-726	1	46	abstract	abstract	ADJ
ejpam-726	1	47	.	.	PUNCT
ejpam-726	2	1	in	in	ADP
ejpam-726	2	2	this	this	DET
ejpam-726	2	3	paper	paper	NOUN
ejpam-726	2	4	,	,	PUNCT
ejpam-726	2	5	by	by	ADP
ejpam-726	2	6	using	use	VERB
ejpam-726	2	7	β∗−set	β∗−set	PROPN
ejpam-726	2	8	[	[	X
ejpam-726	2	9	24	24	NUM
ejpam-726	2	10	]	]	X
ejpam-726	2	11	we	we	PRON
ejpam-726	2	12	introduce	introduce	VERB
ejpam-726	2	13	a	a	DET
ejpam-726	2	14	new	new	ADJ
ejpam-726	2	15	class	class	NOUN
ejpam-726	2	16	of	of	ADP
ejpam-726	2	17	sets	set	NOUN
ejpam-726	2	18	called	call	VERB
ejpam-726	2	19	β∗g−closed	β∗g−close	VERB
ejpam-726	2	20	sets	set	NOUN
ejpam-726	2	21	,	,	PUNCT
ejpam-726	2	22	which	which	PRON
ejpam-726	2	23	is	be	AUX
ejpam-726	2	24	stronger	strong	ADJ
ejpam-726	2	25	than	than	ADP
ejpam-726	2	26	g−closed	g−close	VERB
ejpam-726	2	27	sets	set	NOUN
ejpam-726	2	28	and	and	CCONJ
ejpam-726	2	29	weaker	weak	ADJ
ejpam-726	2	30	than	than	ADP
ejpam-726	2	31	closed	closed	ADJ
ejpam-726	2	32	sets	set	NOUN
ejpam-726	2	33	.	.	PUNCT
ejpam-726	3	1	we	we	PRON
ejpam-726	3	2	define	define	VERB
ejpam-726	3	3	two	two	NUM
ejpam-726	3	4	new	new	ADJ
ejpam-726	3	5	separation	separation	NOUN
ejpam-726	3	6	axioms	axiom	NOUN
ejpam-726	3	7	called	call	VERB
ejpam-726	3	8	β∗t1/2	β∗t1/2	PROPN
ejpam-726	3	9	and	and	CCONJ
ejpam-726	3	10	β∗∗t1/2	β∗∗t1/2	PUNCT
ejpam-726	3	11	spaces	space	NOUN
ejpam-726	3	12	as	as	ADP
ejpam-726	3	13	applications	application	NOUN
ejpam-726	3	14	of	of	ADP
ejpam-726	3	15	β∗g−closed	β∗g−close	VERB
ejpam-726	3	16	sets	set	NOUN
ejpam-726	3	17	.	.	PUNCT
ejpam-726	4	1	the	the	DET
ejpam-726	4	2	notions	notion	NOUN
ejpam-726	4	3	β∗g−continuity	β∗g−continuity	PROPN
ejpam-726	4	4	and	and	CCONJ
ejpam-726	4	5	β∗g−irresoluteness	β∗g−irresoluteness	NOUN
ejpam-726	4	6	are	be	AUX
ejpam-726	4	7	also	also	ADV
ejpam-726	4	8	introduced	introduce	VERB
ejpam-726	4	9	.	.	PUNCT
ejpam-726	5	1	2000	2000	NUM
ejpam-726	5	2	mathematics	mathematic	NOUN
ejpam-726	5	3	subject	subject	NOUN
ejpam-726	5	4	classifications	classification	NOUN
ejpam-726	5	5	:	:	PUNCT
ejpam-726	5	6	54c08	54c08	NUM
ejpam-726	5	7	,	,	PUNCT
ejpam-726	5	8	54a05	54a05	NUM
ejpam-726	5	9	key	key	ADJ
ejpam-726	5	10	words	word	NOUN
ejpam-726	5	11	and	and	CCONJ
ejpam-726	5	12	phrases	phrase	NOUN
ejpam-726	5	13	:	:	PUNCT
ejpam-726	5	14	β∗-set	β∗-set	PROPN
ejpam-726	5	15	,	,	PUNCT
ejpam-726	5	16	β∗g−closed	β∗g−close	VERB
ejpam-726	5	17	set	set	NOUN
ejpam-726	5	18	,	,	PUNCT
ejpam-726	5	19	β∗g−continuous	β∗g−continuous	NUM
ejpam-726	5	20	,	,	PUNCT
ejpam-726	5	21	β∗t1/2	β∗t1/2	PROPN
ejpam-726	5	22	space	space	NOUN
ejpam-726	5	23	1	1	NUM
ejpam-726	5	24	.	.	PUNCT
ejpam-726	5	25	introduction	introduction	NOUN
ejpam-726	5	26	and	and	CCONJ
ejpam-726	5	27	preliminaries	preliminary	NOUN
ejpam-726	5	28	to	to	ADP
ejpam-726	5	29	date	date	NOUN
ejpam-726	5	30	,	,	PUNCT
ejpam-726	5	31	many	many	ADJ
ejpam-726	5	32	studies	study	NOUN
ejpam-726	5	33	have	have	AUX
ejpam-726	5	34	been	be	AUX
ejpam-726	5	35	made	make	VERB
ejpam-726	5	36	on	on	ADP
ejpam-726	5	37	closed	closed	ADJ
ejpam-726	5	38	sets	set	NOUN
ejpam-726	5	39	and	and	CCONJ
ejpam-726	5	40	set	set	VERB
ejpam-726	5	41	concepts	concept	NOUN
ejpam-726	5	42	derived	derive	VERB
ejpam-726	5	43	from	from	ADP
ejpam-726	5	44	this	this	DET
ejpam-726	5	45	set	set	NOUN
ejpam-726	5	46	.	.	PUNCT
ejpam-726	6	1	the	the	DET
ejpam-726	6	2	concept	concept	NOUN
ejpam-726	6	3	of	of	ADP
ejpam-726	6	4	g−closed	g−close	VERB
ejpam-726	6	5	sets	set	NOUN
ejpam-726	6	6	was	be	AUX
ejpam-726	6	7	introduced	introduce	VERB
ejpam-726	6	8	by	by	ADP
ejpam-726	6	9	levine	levine	PROPN
ejpam-726	6	10	[	[	X
ejpam-726	6	11	17	17	NUM
ejpam-726	6	12	]	]	PUNCT
ejpam-726	6	13	and	and	CCONJ
ejpam-726	6	14	was	be	AUX
ejpam-726	6	15	used	use	VERB
ejpam-726	6	16	to	to	PART
ejpam-726	6	17	obtain	obtain	VERB
ejpam-726	6	18	a	a	DET
ejpam-726	6	19	t1/2	t1/2	ADJ
ejpam-726	6	20	space	space	NOUN
ejpam-726	6	21	in	in	ADP
ejpam-726	6	22	which	which	PRON
ejpam-726	6	23	the	the	DET
ejpam-726	6	24	closed	closed	ADJ
ejpam-726	6	25	sets	set	NOUN
ejpam-726	6	26	and	and	CCONJ
ejpam-726	6	27	g−closed	g−close	VERB
ejpam-726	6	28	sets	set	NOUN
ejpam-726	6	29	coincide	coincide	NOUN
ejpam-726	6	30	.	.	PUNCT
ejpam-726	7	1	this	this	DET
ejpam-726	7	2	natural	natural	ADJ
ejpam-726	7	3	generalization	generalization	NOUN
ejpam-726	7	4	of	of	ADP
ejpam-726	7	5	a	a	DET
ejpam-726	7	6	closed	closed	ADJ
ejpam-726	7	7	set	set	NOUN
ejpam-726	7	8	concept	concept	NOUN
ejpam-726	7	9	has	have	AUX
ejpam-726	7	10	made	make	VERB
ejpam-726	7	11	it	it	PRON
ejpam-726	7	12	possible	possible	ADJ
ejpam-726	7	13	to	to	PART
ejpam-726	7	14	use	use	VERB
ejpam-726	7	15	the	the	DET
ejpam-726	7	16	concept	concept	NOUN
ejpam-726	7	17	in	in	ADP
ejpam-726	7	18	many	many	ADJ
ejpam-726	7	19	areas	area	NOUN
ejpam-726	7	20	,	,	PUNCT
ejpam-726	7	21	especially	especially	ADV
ejpam-726	7	22	in	in	ADP
ejpam-726	7	23	quantum	quantum	ADJ
ejpam-726	7	24	physics	physics	NOUN
ejpam-726	7	25	[	[	X
ejpam-726	7	26	13	13	NUM
ejpam-726	7	27	]	]	PUNCT
ejpam-726	7	28	and	and	CCONJ
ejpam-726	7	29	computer	computer	NOUN
ejpam-726	7	30	graphics	graphic	NOUN
ejpam-726	8	1	[	[	X
ejpam-726	8	2	13	13	NUM
ejpam-726	8	3	-	-	SYM
ejpam-726	8	4	15	15	NUM
ejpam-726	8	5	]	]	PUNCT
ejpam-726	8	6	.	.	PUNCT
ejpam-726	9	1	the	the	DET
ejpam-726	9	2	notion	notion	NOUN
ejpam-726	9	3	has	have	AUX
ejpam-726	9	4	been	be	AUX
ejpam-726	9	5	studied	study	VERB
ejpam-726	9	6	extensively	extensively	ADV
ejpam-726	9	7	in	in	ADP
ejpam-726	9	8	recent	recent	ADJ
ejpam-726	9	9	years	year	NOUN
ejpam-726	9	10	by	by	ADP
ejpam-726	9	11	many	many	ADJ
ejpam-726	9	12	topologists	topologist	NOUN
ejpam-726	9	13	.	.	PUNCT
ejpam-726	10	1	more	more	ADV
ejpam-726	10	2	importantly	importantly	ADV
ejpam-726	10	3	several	several	ADJ
ejpam-726	10	4	new	new	ADJ
ejpam-726	10	5	separations	separation	NOUN
ejpam-726	10	6	which	which	PRON
ejpam-726	10	7	are	be	AUX
ejpam-726	10	8	between	between	ADP
ejpam-726	10	9	t0	t0	PROPN
ejpam-726	10	10	and	and	CCONJ
ejpam-726	10	11	t1	t1	NOUN
ejpam-726	10	12	such	such	ADJ
ejpam-726	10	13	as	as	ADP
ejpam-726	10	14	t1/2	t1/2	PROPN
ejpam-726	10	15	,	,	PUNCT
ejpam-726	10	16	tgs	tgs	PROPN
ejpam-726	10	17	,	,	PUNCT
ejpam-726	10	18	πgp−	πgp−	PROPN
ejpam-726	10	19	t1/2	t1/2	PROPN
ejpam-726	10	20	and	and	CCONJ
ejpam-726	10	21	t3/4	t3/4	NOUN
ejpam-726	10	22	are	be	AUX
ejpam-726	10	23	suggested	suggest	VERB
ejpam-726	10	24	.	.	PUNCT
ejpam-726	11	1	some	some	PRON
ejpam-726	11	2	of	of	ADP
ejpam-726	11	3	these	these	PRON
ejpam-726	11	4	have	have	AUX
ejpam-726	11	5	been	be	AUX
ejpam-726	11	6	found	find	VERB
ejpam-726	11	7	to	to	PART
ejpam-726	11	8	be	be	AUX
ejpam-726	11	9	useful	useful	ADJ
ejpam-726	11	10	in	in	ADP
ejpam-726	11	11	computer	computer	NOUN
ejpam-726	11	12	science	science	NOUN
ejpam-726	11	13	and	and	CCONJ
ejpam-726	11	14	digital	digital	ADJ
ejpam-726	11	15	topology	topology	NOUN
ejpam-726	11	16	(	(	PUNCT
ejpam-726	11	17	see	see	VERB
ejpam-726	11	18	[	[	X
ejpam-726	11	19	7	7	NUM
ejpam-726	11	20	,	,	PUNCT
ejpam-726	11	21	12	12	NUM
ejpam-726	11	22	-	-	SYM
ejpam-726	11	23	15	15	NUM
ejpam-726	11	24	]	]	PUNCT
ejpam-726	11	25	,	,	PUNCT
ejpam-726	11	26	for	for	ADP
ejpam-726	11	27	example	example	NOUN
ejpam-726	11	28	)	)	PUNCT
ejpam-726	11	29	.	.	PUNCT
ejpam-726	12	1	as	as	ADP
ejpam-726	12	2	a	a	DET
ejpam-726	12	3	brief	brief	ADJ
ejpam-726	12	4	literature	literature	NOUN
ejpam-726	12	5	review	review	PROPN
ejpam-726	12	6	,	,	PUNCT
ejpam-726	12	7	related	related	ADJ
ejpam-726	12	8	studies	study	NOUN
ejpam-726	12	9	of	of	ADP
ejpam-726	12	10	g−closed	g−close	VERB
ejpam-726	12	11	sets	set	NOUN
ejpam-726	12	12	can	can	AUX
ejpam-726	12	13	be	be	AUX
ejpam-726	12	14	summarized	summarize	VERB
ejpam-726	12	15	as	as	SCONJ
ejpam-726	12	16	follows	follow	VERB
ejpam-726	12	17	.	.	PUNCT
ejpam-726	13	1	dontchev	dontchev	NOUN
ejpam-726	13	2	and	and	CCONJ
ejpam-726	13	3	noiri	noiri	ADV
ejpam-726	14	1	[	[	X
ejpam-726	14	2	8	8	NUM
ejpam-726	14	3	]	]	PUNCT
ejpam-726	14	4	introduced	introduce	VERB
ejpam-726	14	5	the	the	DET
ejpam-726	14	6	notion	notion	NOUN
ejpam-726	14	7	of	of	ADP
ejpam-726	14	8	r	r	NOUN
ejpam-726	14	9	g−closed	g−close	VERB
ejpam-726	14	10	sets	set	NOUN
ejpam-726	14	11	which	which	PRON
ejpam-726	14	12	are	be	AUX
ejpam-726	14	13	weaker	weak	ADJ
ejpam-726	14	14	than	than	ADP
ejpam-726	14	15	that	that	PRON
ejpam-726	14	16	of	of	ADP
ejpam-726	14	17	g−closed	g−close	VERB
ejpam-726	14	18	sets	set	NOUN
ejpam-726	14	19	.	.	PUNCT
ejpam-726	15	1	kumar	kumar	PROPN
ejpam-726	16	1	[	[	X
ejpam-726	16	2	16	16	NUM
ejpam-726	16	3	]	]	PUNCT
ejpam-726	16	4	defined	define	VERB
ejpam-726	16	5	the	the	DET
ejpam-726	16	6	notion	notion	NOUN
ejpam-726	16	7	of	of	ADP
ejpam-726	16	8	g∗-closed	g∗-close	VERB
ejpam-726	16	9	sets	set	NOUN
ejpam-726	16	10	that	that	PRON
ejpam-726	16	11	are	be	AUX
ejpam-726	16	12	generalizations	generalization	NOUN
ejpam-726	16	13	of	of	ADP
ejpam-726	16	14	g−closed	g−close	VERB
ejpam-726	16	15	sets	set	NOUN
ejpam-726	16	16	and	and	CCONJ
ejpam-726	16	17	introduced	introduce	VERB
ejpam-726	16	18	t	t	PROPN
ejpam-726	16	19	∗	∗	X
ejpam-726	16	20	1/2	1/2	NUM
ejpam-726	16	21	and	and	CCONJ
ejpam-726	16	22	∗t1/2	∗t1/2	NOUN
ejpam-726	16	23	spaces	space	NOUN
ejpam-726	16	24	as	as	ADP
ejpam-726	16	25	applications	application	NOUN
ejpam-726	16	26	of	of	ADP
ejpam-726	16	27	g∗-closed	g∗-close	VERB
ejpam-726	16	28	sets	set	NOUN
ejpam-726	16	29	.	.	PUNCT
ejpam-726	17	1	devi	devi	PROPN
ejpam-726	17	2	et	et	PROPN
ejpam-726	17	3	al	al	PROPN
ejpam-726	17	4	.	.	PUNCT
ejpam-726	18	1	[	[	X
ejpam-726	18	2	6	6	NUM
ejpam-726	18	3	]	]	PUNCT
ejpam-726	18	4	introduced	introduce	VERB
ejpam-726	18	5	and	and	CCONJ
ejpam-726	18	6	studied	study	VERB
ejpam-726	18	7	gs−closed	gs−close	VERB
ejpam-726	18	8	and	and	CCONJ
ejpam-726	18	9	sg−closed	sg−close	VERB
ejpam-726	18	10	sets	set	NOUN
ejpam-726	18	11	which	which	PRON
ejpam-726	18	12	are	be	AUX
ejpam-726	18	13	weaker	weak	ADJ
ejpam-726	18	14	than	than	ADP
ejpam-726	18	15	g−closed	g−close	VERB
ejpam-726	18	16	sets	set	NOUN
ejpam-726	18	17	.	.	PUNCT
ejpam-726	19	1	arya	arya	PROPN
ejpam-726	19	2	and	and	CCONJ
ejpam-726	19	3	nour	nour	PROPN
ejpam-726	20	1	[	[	X
ejpam-726	20	2	1	1	NUM
ejpam-726	20	3	]	]	PUNCT
ejpam-726	20	4	gave	give	VERB
ejpam-726	20	5	some	some	DET
ejpam-726	20	6	properties	property	NOUN
ejpam-726	20	7	of	of	ADP
ejpam-726	20	8	s	s	NOUN
ejpam-726	20	9	-	-	ADJ
ejpam-726	20	10	normal	normal	ADJ
ejpam-726	20	11	spaces	space	NOUN
ejpam-726	20	12	by	by	ADP
ejpam-726	20	13	using	use	VERB
ejpam-726	20	14	gs−open	gs−open	NOUN
ejpam-726	20	15	sets	set	NOUN
ejpam-726	20	16	.	.	PUNCT
ejpam-726	21	1	the	the	DET
ejpam-726	21	2	notion	notion	NOUN
ejpam-726	21	3	of	of	ADP
ejpam-726	21	4	s	s	NOUN
ejpam-726	21	5	-	-	ADJ
ejpam-726	21	6	normal	normal	ADJ
ejpam-726	21	7	space	space	NOUN
ejpam-726	21	8	was	be	AUX
ejpam-726	21	9	studied	study	VERB
ejpam-726	21	10	extensively	extensively	ADV
ejpam-726	21	11	by	by	ADP
ejpam-726	21	12	noiri	noiri	NOUN
ejpam-726	21	13	[	[	X
ejpam-726	21	14	20	20	NUM
ejpam-726	21	15	]	]	PUNCT
ejpam-726	21	16	.	.	PUNCT
ejpam-726	22	1	zaitsev	zaitsev	NOUN
ejpam-726	23	1	[	[	X
ejpam-726	23	2	25	25	NUM
ejpam-726	23	3	]	]	PUNCT
ejpam-726	23	4	introduced	introduce	VERB
ejpam-726	23	5	the	the	DET
ejpam-726	23	6	notions	notion	NOUN
ejpam-726	23	7	of	of	ADP
ejpam-726	23	8	π	π	PROPN
ejpam-726	23	9	-	-	ADJ
ejpam-726	23	10	closed	closed	ADJ
ejpam-726	23	11	sets	set	NOUN
ejpam-726	23	12	and	and	CCONJ
ejpam-726	23	13	guasi	guasi	NOUN
ejpam-726	23	14	normal	normal	ADJ
ejpam-726	23	15	spaces	space	NOUN
ejpam-726	23	16	.	.	PUNCT
ejpam-726	24	1	dontchev	dontchev	NOUN
ejpam-726	24	2	and	and	CCONJ
ejpam-726	24	3	noiri	noiri	ADV
ejpam-726	25	1	[	[	X
ejpam-726	25	2	8	8	NUM
ejpam-726	25	3	]	]	PUNCT
ejpam-726	25	4	introduced	introduce	VERB
ejpam-726	25	5	the	the	DET
ejpam-726	25	6	notion	notion	NOUN
ejpam-726	25	7	of	of	ADP
ejpam-726	25	8	πg	πg	PRON
ejpam-726	25	9	-	-	PUNCT
ejpam-726	25	10	closed	closed	ADJ
ejpam-726	25	11	sets	set	NOUN
ejpam-726	25	12	and	and	CCONJ
ejpam-726	25	13	obtained	obtain	VERB
ejpam-726	25	14	some	some	DET
ejpam-726	25	15	theorems	theorem	NOUN
ejpam-726	25	16	in	in	ADP
ejpam-726	25	17	quasi	quasi	ADJ
ejpam-726	25	18	normal	normal	ADJ
ejpam-726	25	19	spaces	space	NOUN
ejpam-726	25	20	by	by	ADP
ejpam-726	25	21	using	use	VERB
ejpam-726	25	22	this	this	DET
ejpam-726	25	23	notion	notion	NOUN
ejpam-726	25	24	.	.	PUNCT
ejpam-726	26	1	of	of	ADP
ejpam-726	26	2	course	course	NOUN
ejpam-726	26	3	,	,	PUNCT
ejpam-726	26	4	these	these	DET
ejpam-726	26	5	studies	study	NOUN
ejpam-726	26	6	of	of	ADP
ejpam-726	26	7	general	general	ADJ
ejpam-726	26	8	topology	topology	NOUN
ejpam-726	26	9	email	email	NOUN
ejpam-726	26	10	address	address	NOUN
ejpam-726	26	11	:	:	PUNCT
ejpam-726	26	12	ahua	ahua	PROPN
ejpam-726	26	13	ikgoz�aksaray.edu.tr	ikgoz�aksaray.edu.tr	PROPN
ejpam-726	26	14	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-726	27	1	20	20	NUM
ejpam-726	27	2	c	c	X
ejpam-726	27	3	©	©	NOUN
ejpam-726	27	4	2010	2010	NUM
ejpam-726	27	5	ejpam	ejpam	NOUN
ejpam-726	27	6	all	all	DET
ejpam-726	27	7	rights	right	NOUN
ejpam-726	27	8	reserved	reserve	VERB
ejpam-726	27	9	.	.	PUNCT
ejpam-726	28	1	a.	a.	NOUN
ejpam-726	28	2	açıkgöz	açıkgöz	PROPN
ejpam-726	28	3	/	/	SYM
ejpam-726	28	4	eur	eur	PROPN
ejpam-726	28	5	.	.	PUNCT
ejpam-726	29	1	j.	j.	PROPN
ejpam-726	29	2	pure	pure	PROPN
ejpam-726	29	3	appl	appl	PROPN
ejpam-726	29	4	.	.	PROPN
ejpam-726	29	5	math	math	PROPN
ejpam-726	29	6	,	,	PUNCT
ejpam-726	29	7	4	4	NUM
ejpam-726	29	8	(	(	PUNCT
ejpam-726	29	9	2011	2011	NUM
ejpam-726	29	10	)	)	PUNCT
ejpam-726	29	11	,	,	PUNCT
ejpam-726	29	12	20	20	NUM
ejpam-726	29	13	-	-	SYM
ejpam-726	29	14	33	33	NUM
ejpam-726	29	15	21	21	NUM
ejpam-726	29	16	are	be	AUX
ejpam-726	29	17	not	not	PART
ejpam-726	29	18	limited	limit	VERB
ejpam-726	29	19	with	with	ADP
ejpam-726	29	20	these	these	PRON
ejpam-726	29	21	[	[	X
ejpam-726	29	22	4,9	4,9	NUM
ejpam-726	29	23	-	-	PUNCT
ejpam-726	29	24	10].more	10].more	PROPN
ejpam-726	29	25	recently	recently	ADV
ejpam-726	29	26	,	,	PUNCT
ejpam-726	29	27	several	several	ADJ
ejpam-726	29	28	topologists	topologist	NOUN
ejpam-726	29	29	have	have	AUX
ejpam-726	29	30	defined	define	VERB
ejpam-726	29	31	new	new	ADJ
ejpam-726	29	32	separation	separation	NOUN
ejpam-726	29	33	axioms	axiom	NOUN
ejpam-726	29	34	at	at	ADP
ejpam-726	29	35	topological	topological	ADJ
ejpam-726	29	36	space	space	NOUN
ejpam-726	29	37	by	by	ADP
ejpam-726	29	38	giving	give	VERB
ejpam-726	29	39	some	some	DET
ejpam-726	29	40	convenient	convenient	ADJ
ejpam-726	29	41	definitions	definition	NOUN
ejpam-726	29	42	of	of	ADP
ejpam-726	29	43	variety	variety	NOUN
ejpam-726	29	44	.	.	PUNCT
ejpam-726	30	1	park	park	NOUN
ejpam-726	31	1	[	[	X
ejpam-726	31	2	21	21	NUM
ejpam-726	31	3	]	]	PUNCT
ejpam-726	31	4	has	have	AUX
ejpam-726	31	5	introduced	introduce	VERB
ejpam-726	31	6	the	the	DET
ejpam-726	31	7	class	class	NOUN
ejpam-726	31	8	of	of	ADP
ejpam-726	31	9	πgp−closed	πgp−close	VERB
ejpam-726	31	10	sets	set	NOUN
ejpam-726	31	11	which	which	PRON
ejpam-726	31	12	is	be	AUX
ejpam-726	31	13	weaker	weak	ADJ
ejpam-726	31	14	than	than	SCONJ
ejpam-726	31	15	gp−closed	gp−close	VERB
ejpam-726	31	16	and	and	CCONJ
ejpam-726	31	17	stronger	strong	ADJ
ejpam-726	31	18	than	than	ADP
ejpam-726	31	19	gpr	gpr	NOUN
ejpam-726	31	20	-	-	PUNCT
ejpam-726	31	21	closed	close	VERB
ejpam-726	31	22	sets	set	NOUN
ejpam-726	31	23	.	.	PUNCT
ejpam-726	32	1	park	park	NOUN
ejpam-726	32	2	and	and	CCONJ
ejpam-726	32	3	park	park	NOUN
ejpam-726	33	1	[	[	X
ejpam-726	33	2	22	22	NUM
ejpam-726	33	3	]	]	PUNCT
ejpam-726	33	4	further	far	ADV
ejpam-726	33	5	studied	study	VERB
ejpam-726	33	6	the	the	DET
ejpam-726	33	7	class	class	NOUN
ejpam-726	33	8	of	of	ADP
ejpam-726	33	9	πgp−closed	πgp−close	VERB
ejpam-726	33	10	sets	set	NOUN
ejpam-726	33	11	and	and	CCONJ
ejpam-726	33	12	defined	define	VERB
ejpam-726	33	13	the	the	DET
ejpam-726	33	14	concepts	concept	NOUN
ejpam-726	33	15	πgp	πgp	VERB
ejpam-726	33	16	-	-	PUNCT
ejpam-726	33	17	compactness	compactness	NOUN
ejpam-726	33	18	and	and	CCONJ
ejpam-726	33	19	πgp	πgp	VERB
ejpam-726	33	20	-	-	PUNCT
ejpam-726	33	21	connectedness	connectedness	NOUN
ejpam-726	33	22	.	.	PUNCT
ejpam-726	34	1	aslım	aslım	PROPN
ejpam-726	34	2	et	et	PROPN
ejpam-726	34	3	al	al	PROPN
ejpam-726	34	4	.	.	PUNCT
ejpam-726	35	1	[	[	X
ejpam-726	35	2	2	2	X
ejpam-726	35	3	]	]	PUNCT
ejpam-726	35	4	have	have	AUX
ejpam-726	35	5	introduced	introduce	VERB
ejpam-726	35	6	the	the	DET
ejpam-726	35	7	notions	notion	NOUN
ejpam-726	35	8	of	of	ADP
ejpam-726	35	9	πgs−closed	πgs−close	VERB
ejpam-726	35	10	sets	set	NOUN
ejpam-726	35	11	which	which	PRON
ejpam-726	35	12	are	be	AUX
ejpam-726	35	13	implied	imply	VERB
ejpam-726	35	14	by	by	ADP
ejpam-726	35	15	that	that	PRON
ejpam-726	35	16	of	of	ADP
ejpam-726	35	17	gs−closed	gs−close	VERB
ejpam-726	35	18	sets	set	NOUN
ejpam-726	35	19	and	and	CCONJ
ejpam-726	35	20	πgs−	πgs−	NOUN
ejpam-726	35	21	t1/2	t1/2	NOUN
ejpam-726	35	22	-	-	NOUN
ejpam-726	35	23	spaces	space	NOUN
ejpam-726	35	24	.	.	PUNCT
ejpam-726	36	1	on	on	ADP
ejpam-726	36	2	the	the	DET
ejpam-726	36	3	other	other	ADJ
ejpam-726	36	4	hand	hand	NOUN
ejpam-726	36	5	,	,	PUNCT
ejpam-726	36	6	recently	recently	ADV
ejpam-726	36	7	yuksel	yuksel	ADJ
ejpam-726	36	8	and	and	CCONJ
ejpam-726	36	9	beceren	beceren	NOUN
ejpam-726	36	10	[	[	X
ejpam-726	36	11	24	24	NUM
ejpam-726	36	12	]	]	PUNCT
ejpam-726	36	13	have	have	AUX
ejpam-726	36	14	defined	define	VERB
ejpam-726	36	15	the	the	DET
ejpam-726	36	16	notion	notion	NOUN
ejpam-726	36	17	of	of	ADP
ejpam-726	36	18	β∗-set	β∗-set	NOUN
ejpam-726	36	19	and	and	CCONJ
ejpam-726	36	20	established	establish	VERB
ejpam-726	36	21	a	a	DET
ejpam-726	36	22	decomposition	decomposition	NOUN
ejpam-726	36	23	of	of	ADP
ejpam-726	36	24	continuity	continuity	NOUN
ejpam-726	36	25	.	.	PUNCT
ejpam-726	37	1	at	at	ADP
ejpam-726	37	2	this	this	DET
ejpam-726	37	3	point	point	NOUN
ejpam-726	37	4	,	,	PUNCT
ejpam-726	37	5	we	we	PRON
ejpam-726	37	6	shall	shall	AUX
ejpam-726	37	7	introduce	introduce	VERB
ejpam-726	37	8	and	and	CCONJ
ejpam-726	37	9	study	study	VERB
ejpam-726	37	10	the	the	DET
ejpam-726	37	11	notions	notion	NOUN
ejpam-726	37	12	of	of	ADP
ejpam-726	37	13	β∗g−closed	β∗g−close	VERB
ejpam-726	37	14	sets	set	NOUN
ejpam-726	37	15	which	which	PRON
ejpam-726	37	16	are	be	AUX
ejpam-726	37	17	situated	situate	VERB
ejpam-726	37	18	between	between	ADP
ejpam-726	37	19	the	the	DET
ejpam-726	37	20	class	class	NOUN
ejpam-726	37	21	of	of	ADP
ejpam-726	37	22	closed	closed	ADJ
ejpam-726	37	23	sets	set	NOUN
ejpam-726	37	24	and	and	CCONJ
ejpam-726	37	25	g−closed	g−close	VERB
ejpam-726	37	26	sets	set	NOUN
ejpam-726	37	27	.	.	PUNCT
ejpam-726	38	1	using	use	VERB
ejpam-726	38	2	these	these	DET
ejpam-726	38	3	sets	set	NOUN
ejpam-726	38	4	,	,	PUNCT
ejpam-726	38	5	we	we	PRON
ejpam-726	38	6	introduce	introduce	VERB
ejpam-726	38	7	two	two	NUM
ejpam-726	38	8	new	new	ADJ
ejpam-726	38	9	separation	separation	NOUN
ejpam-726	38	10	axioms	axiom	NOUN
ejpam-726	38	11	called	call	VERB
ejpam-726	38	12	β∗t1/2	β∗t1/2	PROPN
ejpam-726	38	13	and	and	CCONJ
ejpam-726	38	14	β∗∗t1/2	β∗∗t1/2	PROPN
ejpam-726	38	15	.	.	PUNCT
ejpam-726	39	1	(	(	PUNCT
ejpam-726	39	2	both	both	CCONJ
ejpam-726	39	3	β∗t1/2	β∗t1/2	PROPN
ejpam-726	39	4	and	and	CCONJ
ejpam-726	39	5	β∗∗t1/2	β∗∗t1/2	PUNCT
ejpam-726	39	6	contain	contain	VERB
ejpam-726	39	7	the	the	DET
ejpam-726	39	8	class	class	NOUN
ejpam-726	39	9	of	of	ADP
ejpam-726	39	10	t1/2	t1/2	ADJ
ejpam-726	39	11	spaces	space	NOUN
ejpam-726	39	12	.	.	PUNCT
ejpam-726	39	13	)	)	PUNCT
ejpam-726	40	1	we	we	PRON
ejpam-726	40	2	show	show	VERB
ejpam-726	40	3	that	that	SCONJ
ejpam-726	40	4	the	the	DET
ejpam-726	40	5	class	class	NOUN
ejpam-726	40	6	of	of	ADP
ejpam-726	40	7	β∗∗t1/2	β∗∗t1/2	PUNCT
ejpam-726	40	8	spaces	space	NOUN
ejpam-726	40	9	is	be	AUX
ejpam-726	40	10	the	the	DET
ejpam-726	40	11	dual	dual	ADJ
ejpam-726	40	12	of	of	ADP
ejpam-726	40	13	the	the	DET
ejpam-726	40	14	class	class	NOUN
ejpam-726	40	15	of	of	ADP
ejpam-726	40	16	β∗t1/2	β∗t1/2	PROPN
ejpam-726	40	17	spaces	space	VERB
ejpam-726	40	18	to	to	ADP
ejpam-726	40	19	the	the	DET
ejpam-726	40	20	class	class	NOUN
ejpam-726	40	21	of	of	ADP
ejpam-726	40	22	t1/2	t1/2	ADJ
ejpam-726	40	23	spaces	space	NOUN
ejpam-726	40	24	.	.	PUNCT
ejpam-726	41	1	we	we	PRON
ejpam-726	41	2	also	also	ADV
ejpam-726	41	3	introduce	introduce	VERB
ejpam-726	41	4	β∗g−continuity	β∗g−continuity	NUM
ejpam-726	41	5	and	and	CCONJ
ejpam-726	41	6	β∗g−irresolute	β∗g−irresolute	NUM
ejpam-726	41	7	functions	function	NOUN
ejpam-726	41	8	for	for	ADP
ejpam-726	41	9	preservation	preservation	NOUN
ejpam-726	41	10	theorems	theorem	NOUN
ejpam-726	41	11	.	.	PUNCT
ejpam-726	42	1	it	it	PRON
ejpam-726	42	2	should	should	AUX
ejpam-726	42	3	be	be	AUX
ejpam-726	42	4	mentioned	mention	VERB
ejpam-726	42	5	that	that	SCONJ
ejpam-726	42	6	the	the	DET
ejpam-726	42	7	present	present	ADJ
ejpam-726	42	8	work	work	NOUN
ejpam-726	42	9	may	may	AUX
ejpam-726	42	10	be	be	AUX
ejpam-726	42	11	found	find	VERB
ejpam-726	42	12	relevant	relevant	ADJ
ejpam-726	42	13	to	to	AUX
ejpam-726	42	14	work	work	VERB
ejpam-726	42	15	of	of	ADP
ejpam-726	42	16	witten	witten	PROPN
ejpam-726	43	1	[	[	X
ejpam-726	43	2	23	23	NUM
ejpam-726	43	3	]	]	PUNCT
ejpam-726	43	4	.	.	PUNCT
ejpam-726	44	1	throughout	throughout	ADP
ejpam-726	44	2	this	this	DET
ejpam-726	44	3	paper	paper	NOUN
ejpam-726	44	4	,	,	PUNCT
ejpam-726	44	5	spaces	space	VERB
ejpam-726	44	6	(	(	PUNCT
ejpam-726	44	7	x	x	X
ejpam-726	44	8	,	,	PUNCT
ejpam-726	44	9	τ	τ	PROPN
ejpam-726	44	10	)	)	PUNCT
ejpam-726	44	11	and	and	CCONJ
ejpam-726	44	12	(	(	PUNCT
ejpam-726	44	13	y	y	PROPN
ejpam-726	44	14	,	,	PUNCT
ejpam-726	44	15	σ	σ	PROPN
ejpam-726	44	16	)	)	PUNCT
ejpam-726	44	17	(	(	PUNCT
ejpam-726	44	18	or	or	CCONJ
ejpam-726	44	19	simply	simply	ADV
ejpam-726	44	20	x	x	X
ejpam-726	44	21	and	and	CCONJ
ejpam-726	44	22	y	y	PROPN
ejpam-726	44	23	)	)	PUNCT
ejpam-726	44	24	always	always	ADV
ejpam-726	44	25	mean	mean	VERB
ejpam-726	44	26	topological	topological	ADJ
ejpam-726	44	27	spaces	space	NOUN
ejpam-726	44	28	on	on	ADP
ejpam-726	44	29	which	which	PRON
ejpam-726	44	30	no	no	DET
ejpam-726	44	31	separation	separation	NOUN
ejpam-726	44	32	axioms	axiom	NOUN
ejpam-726	44	33	are	be	AUX
ejpam-726	44	34	assumed	assume	VERB
ejpam-726	44	35	unless	unless	SCONJ
ejpam-726	44	36	explicitly	explicitly	ADV
ejpam-726	44	37	stated	state	VERB
ejpam-726	44	38	.	.	PUNCT
ejpam-726	45	1	let	let	VERB
ejpam-726	45	2	a	a	DET
ejpam-726	45	3	be	be	AUX
ejpam-726	45	4	a	a	DET
ejpam-726	45	5	subset	subset	NOUN
ejpam-726	45	6	of	of	ADP
ejpam-726	45	7	a	a	DET
ejpam-726	45	8	space	space	NOUN
ejpam-726	45	9	x	x	NOUN
ejpam-726	45	10	.	.	PUNCT
ejpam-726	46	1	the	the	DET
ejpam-726	46	2	closure	closure	NOUN
ejpam-726	46	3	of	of	ADP
ejpam-726	46	4	a	a	PRON
ejpam-726	46	5	and	and	CCONJ
ejpam-726	46	6	the	the	DET
ejpam-726	46	7	interior	interior	NOUN
ejpam-726	46	8	of	of	ADP
ejpam-726	46	9	a	a	PRON
ejpam-726	46	10	are	be	AUX
ejpam-726	46	11	denoted	denote	VERB
ejpam-726	46	12	by	by	ADP
ejpam-726	46	13	cl(a	cl(a	NOUN
ejpam-726	46	14	)	)	PUNCT
ejpam-726	46	15	and	and	CCONJ
ejpam-726	46	16	int(a	int(a	PROPN
ejpam-726	46	17	)	)	PUNCT
ejpam-726	46	18	,	,	PUNCT
ejpam-726	46	19	respectively	respectively	ADV
ejpam-726	46	20	.	.	PUNCT
ejpam-726	47	1	a	a	DET
ejpam-726	47	2	subset	subset	NOUN
ejpam-726	47	3	a	a	PRON
ejpam-726	47	4	is	be	AUX
ejpam-726	47	5	said	say	VERB
ejpam-726	47	6	to	to	PART
ejpam-726	47	7	be	be	AUX
ejpam-726	47	8	locally	locally	ADV
ejpam-726	47	9	closed	close	VERB
ejpam-726	47	10	(	(	PUNCT
ejpam-726	47	11	briefly	briefly	ADV
ejpam-726	47	12	,	,	PUNCT
ejpam-726	47	13	lc	lc	NOUN
ejpam-726	47	14	-	-	PUNCT
ejpam-726	47	15	set	set	NOUN
ejpam-726	47	16	)	)	PUNCT
ejpam-726	48	1	[	[	X
ejpam-726	48	2	3	3	X
ejpam-726	48	3	]	]	X
ejpam-726	48	4	if	if	SCONJ
ejpam-726	48	5	a	a	DET
ejpam-726	48	6	=	=	X
ejpam-726	48	7	u	u	NOUN
ejpam-726	48	8	∩	∩	NOUN
ejpam-726	48	9	v	v	NOUN
ejpam-726	48	10	,	,	PUNCT
ejpam-726	48	11	where	where	SCONJ
ejpam-726	48	12	u	u	NOUN
ejpam-726	48	13	is	be	AUX
ejpam-726	48	14	open	open	ADJ
ejpam-726	48	15	and	and	CCONJ
ejpam-726	48	16	v	v	NOUN
ejpam-726	48	17	is	be	AUX
ejpam-726	48	18	closed	closed	ADJ
ejpam-726	48	19	.	.	PUNCT
ejpam-726	49	1	a	a	DET
ejpam-726	49	2	subset	subset	NOUN
ejpam-726	49	3	a	a	PRON
ejpam-726	49	4	is	be	AUX
ejpam-726	49	5	said	say	VERB
ejpam-726	49	6	to	to	PART
ejpam-726	49	7	be	be	AUX
ejpam-726	49	8	regular	regular	ADJ
ejpam-726	49	9	open	open	ADJ
ejpam-726	49	10	(	(	PUNCT
ejpam-726	49	11	resp	resp	NOUN
ejpam-726	49	12	.	.	PUNCT
ejpam-726	50	1	regular	regular	ADJ
ejpam-726	50	2	closed	closed	ADJ
ejpam-726	50	3	)	)	PUNCT
ejpam-726	50	4	if	if	SCONJ
ejpam-726	50	5	a	a	DET
ejpam-726	50	6	=	=	SYM
ejpam-726	50	7	int(cl(a	int(cl(a	PROPN
ejpam-726	50	8	)	)	PUNCT
ejpam-726	50	9	(	(	PUNCT
ejpam-726	50	10	resp	resp	NOUN
ejpam-726	50	11	.	.	PUNCT
ejpam-726	51	1	a	a	DET
ejpam-726	51	2	=	=	NOUN
ejpam-726	51	3	cl(int(a	cl(int(a	PROPN
ejpam-726	51	4	)	)	PUNCT
ejpam-726	51	5	)	)	PUNCT
ejpam-726	51	6	.	.	PUNCT
ejpam-726	52	1	the	the	DET
ejpam-726	52	2	finite	finite	PROPN
ejpam-726	52	3	union	union	NOUN
ejpam-726	52	4	of	of	ADP
ejpam-726	52	5	regular	regular	ADJ
ejpam-726	52	6	open	open	ADJ
ejpam-726	52	7	sets	set	NOUN
ejpam-726	52	8	is	be	AUX
ejpam-726	52	9	said	say	VERB
ejpam-726	52	10	to	to	PART
ejpam-726	52	11	be	be	AUX
ejpam-726	52	12	π	π	NOUN
ejpam-726	52	13	-	-	NOUN
ejpam-726	52	14	open	open	ADJ
ejpam-726	52	15	.	.	PUNCT
ejpam-726	53	1	the	the	DET
ejpam-726	53	2	complement	complement	NOUN
ejpam-726	53	3	of	of	ADP
ejpam-726	53	4	a	a	DET
ejpam-726	53	5	π	π	NOUN
ejpam-726	53	6	-	-	ADJ
ejpam-726	53	7	open	open	ADJ
ejpam-726	53	8	set	set	NOUN
ejpam-726	53	9	is	be	AUX
ejpam-726	53	10	said	say	VERB
ejpam-726	53	11	to	to	PART
ejpam-726	53	12	be	be	AUX
ejpam-726	53	13	π	π	NOUN
ejpam-726	53	14	-	-	VERB
ejpam-726	53	15	closed	closed	ADJ
ejpam-726	53	16	.	.	PUNCT
ejpam-726	54	1	a	a	DET
ejpam-726	54	2	subset	subset	NOUN
ejpam-726	54	3	a	a	PRON
ejpam-726	54	4	is	be	AUX
ejpam-726	54	5	said	say	VERB
ejpam-726	54	6	to	to	PART
ejpam-726	54	7	be	be	AUX
ejpam-726	54	8	semiopen	semiopen	ADJ
ejpam-726	55	1	[	[	PUNCT
ejpam-726	55	2	5	5	NUM
ejpam-726	55	3	]	]	PUNCT
ejpam-726	55	4	if	if	SCONJ
ejpam-726	55	5	a	a	DET
ejpam-726	55	6	⊂	⊂	PROPN
ejpam-726	55	7	cl(int(a	cl(int(a	PROPN
ejpam-726	55	8	)	)	PUNCT
ejpam-726	55	9	)	)	PUNCT
ejpam-726	56	1	and	and	CCONJ
ejpam-726	56	2	the	the	DET
ejpam-726	56	3	complement	complement	NOUN
ejpam-726	56	4	of	of	ADP
ejpam-726	56	5	a	a	DET
ejpam-726	56	6	semiopen	semiopen	ADJ
ejpam-726	56	7	set	set	NOUN
ejpam-726	56	8	is	be	AUX
ejpam-726	56	9	called	call	VERB
ejpam-726	56	10	semiclosed	semiclose	VERB
ejpam-726	56	11	.	.	PUNCT
ejpam-726	57	1	the	the	DET
ejpam-726	57	2	intersection	intersection	NOUN
ejpam-726	57	3	of	of	ADP
ejpam-726	57	4	all	all	DET
ejpam-726	57	5	semiclosed	semiclose	VERB
ejpam-726	57	6	sets	set	NOUN
ejpam-726	57	7	containing	contain	VERB
ejpam-726	57	8	a	a	PRON
ejpam-726	57	9	is	be	AUX
ejpam-726	57	10	called	call	VERB
ejpam-726	57	11	the	the	DET
ejpam-726	57	12	semiclosure	semiclosure	NOUN
ejpam-726	57	13	[	[	X
ejpam-726	57	14	5	5	NUM
ejpam-726	57	15	]	]	PUNCT
ejpam-726	57	16	of	of	ADP
ejpam-726	57	17	a	a	PRON
ejpam-726	57	18	and	and	CCONJ
ejpam-726	57	19	is	be	AUX
ejpam-726	57	20	denoted	denote	VERB
ejpam-726	57	21	by	by	ADP
ejpam-726	57	22	sc	sc	PROPN
ejpam-726	57	23	l(a	l(a	PROPN
ejpam-726	57	24	)	)	PUNCT
ejpam-726	57	25	.	.	PUNCT
ejpam-726	58	1	dually	dually	PROPN
ejpam-726	58	2	the	the	DET
ejpam-726	58	3	semiinterior	semiinterior	NOUN
ejpam-726	59	1	[	[	X
ejpam-726	59	2	5	5	NUM
ejpam-726	59	3	]	]	PUNCT
ejpam-726	59	4	of	of	ADP
ejpam-726	59	5	a	a	PRON
ejpam-726	59	6	is	be	AUX
ejpam-726	59	7	defined	define	VERB
ejpam-726	59	8	to	to	PART
ejpam-726	59	9	be	be	AUX
ejpam-726	59	10	the	the	DET
ejpam-726	59	11	union	union	NOUN
ejpam-726	59	12	of	of	ADP
ejpam-726	59	13	all	all	DET
ejpam-726	59	14	semiopen	semiopen	ADJ
ejpam-726	59	15	sets	set	NOUN
ejpam-726	59	16	contained	contain	VERB
ejpam-726	59	17	in	in	ADP
ejpam-726	59	18	a	a	PRON
ejpam-726	59	19	and	and	CCONJ
ejpam-726	59	20	is	be	AUX
ejpam-726	59	21	denoted	denote	VERB
ejpam-726	59	22	by	by	ADP
ejpam-726	59	23	sint(a	sint(a	PROPN
ejpam-726	59	24	)	)	PUNCT
ejpam-726	59	25	.	.	PUNCT
ejpam-726	60	1	a	a	DET
ejpam-726	60	2	subset	subset	NOUN
ejpam-726	60	3	a	a	PRON
ejpam-726	60	4	is	be	AUX
ejpam-726	60	5	said	say	VERB
ejpam-726	60	6	to	to	PART
ejpam-726	60	7	be	be	AUX
ejpam-726	60	8	pre	pre	VERB
ejpam-726	60	9	open	open	ADJ
ejpam-726	61	1	[	[	X
ejpam-726	61	2	19	19	NUM
ejpam-726	61	3	]	]	X
ejpam-726	61	4	if	if	SCONJ
ejpam-726	61	5	a	a	DET
ejpam-726	61	6	⊂	⊂	PROPN
ejpam-726	61	7	int(cl(a	int(cl(a	PROPN
ejpam-726	61	8	)	)	PUNCT
ejpam-726	61	9	)	)	PUNCT
ejpam-726	62	1	and	and	CCONJ
ejpam-726	62	2	the	the	DET
ejpam-726	62	3	complement	complement	NOUN
ejpam-726	62	4	of	of	ADP
ejpam-726	62	5	a	a	DET
ejpam-726	62	6	pre	pre	ADJ
ejpam-726	62	7	open	open	ADJ
ejpam-726	62	8	set	set	NOUN
ejpam-726	62	9	is	be	AUX
ejpam-726	62	10	called	call	VERB
ejpam-726	62	11	pre	pre	NOUN
ejpam-726	62	12	closed	closed	ADJ
ejpam-726	62	13	.	.	PUNCT
ejpam-726	63	1	the	the	DET
ejpam-726	63	2	intersection	intersection	NOUN
ejpam-726	63	3	of	of	ADP
ejpam-726	63	4	all	all	DET
ejpam-726	63	5	preclosed	preclose	VERB
ejpam-726	63	6	sets	set	NOUN
ejpam-726	63	7	containing	contain	VERB
ejpam-726	63	8	a	a	PRON
ejpam-726	63	9	is	be	AUX
ejpam-726	63	10	called	call	VERB
ejpam-726	63	11	the	the	DET
ejpam-726	63	12	preclosure	preclosure	ADJ
ejpam-726	63	13	[	[	X
ejpam-726	63	14	19	19	NUM
ejpam-726	63	15	]	]	PUNCT
ejpam-726	63	16	of	of	ADP
ejpam-726	63	17	a	a	PRON
ejpam-726	63	18	and	and	CCONJ
ejpam-726	63	19	is	be	AUX
ejpam-726	63	20	denoted	denote	VERB
ejpam-726	63	21	by	by	ADP
ejpam-726	63	22	pcl(a	pcl(a	PROPN
ejpam-726	63	23	)	)	PUNCT
ejpam-726	63	24	.	.	PUNCT
ejpam-726	64	1	dually	dually	PROPN
ejpam-726	64	2	the	the	DET
ejpam-726	64	3	preinterior	preinterior	PROPN
ejpam-726	64	4	[	[	X
ejpam-726	64	5	19	19	NUM
ejpam-726	64	6	]	]	PUNCT
ejpam-726	64	7	of	of	ADP
ejpam-726	64	8	a	a	PRON
ejpam-726	64	9	is	be	AUX
ejpam-726	64	10	defined	define	VERB
ejpam-726	64	11	to	to	PART
ejpam-726	64	12	be	be	AUX
ejpam-726	64	13	the	the	DET
ejpam-726	64	14	union	union	NOUN
ejpam-726	64	15	of	of	ADP
ejpam-726	64	16	all	all	PRON
ejpam-726	64	17	pre	pre	ADJ
ejpam-726	64	18	open	open	ADJ
ejpam-726	64	19	sets	set	NOUN
ejpam-726	64	20	contained	contain	VERB
ejpam-726	64	21	in	in	ADP
ejpam-726	64	22	a	a	PRON
ejpam-726	64	23	and	and	CCONJ
ejpam-726	64	24	is	be	AUX
ejpam-726	64	25	denoted	denote	VERB
ejpam-726	64	26	by	by	ADP
ejpam-726	64	27	pint(a	pint(a	PROPN
ejpam-726	64	28	)	)	PUNCT
ejpam-726	64	29	.	.	PUNCT
ejpam-726	65	1	note	note	VERB
ejpam-726	65	2	that	that	SCONJ
ejpam-726	65	3	sc	sc	PROPN
ejpam-726	65	4	l(a	l(a	PROPN
ejpam-726	65	5	)	)	PUNCT
ejpam-726	65	6	=	=	PUNCT
ejpam-726	65	7	a∪	a∪	PROPN
ejpam-726	65	8	int(cl(a	int(cl(a	PROPN
ejpam-726	65	9	)	)	PUNCT
ejpam-726	65	10	)	)	PUNCT
ejpam-726	65	11	,	,	PUNCT
ejpam-726	65	12	sint(a	sint(a	NOUN
ejpam-726	65	13	)	)	PUNCT
ejpam-726	65	14	=	=	SYM
ejpam-726	65	15	a∩	a∩	PROPN
ejpam-726	65	16	cl(int(a	cl(int(a	PROPN
ejpam-726	65	17	)	)	PUNCT
ejpam-726	65	18	)	)	PUNCT
ejpam-726	65	19	,	,	PUNCT
ejpam-726	65	20	pcl(a	pcl(a	PROPN
ejpam-726	65	21	)	)	PUNCT
ejpam-726	65	22	=	=	PUNCT
ejpam-726	65	23	a∪	a∪	PRON
ejpam-726	65	24	cl(int(a	cl(int(a	PROPN
ejpam-726	65	25	)	)	PUNCT
ejpam-726	65	26	)	)	PUNCT
ejpam-726	65	27	and	and	CCONJ
ejpam-726	65	28	pint(a	pint(a	NOUN
ejpam-726	65	29	)	)	PUNCT
ejpam-726	65	30	=	=	SYM
ejpam-726	65	31	a∩	a∩	PROPN
ejpam-726	65	32	int(cl(a	int(cl(a	PROPN
ejpam-726	65	33	)	)	PUNCT
ejpam-726	65	34	)	)	PUNCT
ejpam-726	65	35	.	.	PUNCT
ejpam-726	66	1	2	2	X
ejpam-726	66	2	.	.	X
ejpam-726	66	3	β∗g−-closed	β∗g−-closed	ADJ
ejpam-726	66	4	sets	set	NOUN
ejpam-726	66	5	definition	definition	NOUN
ejpam-726	66	6	1	1	NUM
ejpam-726	66	7	.	.	PUNCT
ejpam-726	67	1	a	a	DET
ejpam-726	67	2	subset	subset	NOUN
ejpam-726	67	3	a	a	PRON
ejpam-726	67	4	of	of	ADP
ejpam-726	67	5	a	a	DET
ejpam-726	67	6	space	space	NOUN
ejpam-726	67	7	(	(	PUNCT
ejpam-726	67	8	x	x	X
ejpam-726	67	9	,	,	PUNCT
ejpam-726	67	10	τ	τ	X
ejpam-726	67	11	)	)	PUNCT
ejpam-726	67	12	is	be	AUX
ejpam-726	67	13	called	call	VERB
ejpam-726	67	14	a	a	DET
ejpam-726	67	15	generalized	generalize	VERB
ejpam-726	67	16	closed	close	VERB
ejpam-726	67	17	set	set	NOUN
ejpam-726	67	18	(	(	PUNCT
ejpam-726	67	19	briefly	briefly	ADV
ejpam-726	67	20	,	,	PUNCT
ejpam-726	67	21	g−closed	g−closed	ADJ
ejpam-726	67	22	)	)	PUNCT
ejpam-726	68	1	[	[	X
ejpam-726	68	2	17	17	NUM
ejpam-726	68	3	]	]	X
ejpam-726	68	4	if	if	SCONJ
ejpam-726	68	5	c	c	PROPN
ejpam-726	68	6	l(a	l(a	X
ejpam-726	68	7	)	)	PUNCT
ejpam-726	69	1	⊂	⊂	PROPN
ejpam-726	69	2	u	u	NOUN
ejpam-726	69	3	whenever	whenever	SCONJ
ejpam-726	69	4	a⊂	a⊂	PUNCT
ejpam-726	69	5	u	u	NOUN
ejpam-726	69	6	and	and	CCONJ
ejpam-726	69	7	u	u	NOUN
ejpam-726	69	8	is	be	AUX
ejpam-726	69	9	open	open	ADJ
ejpam-726	69	10	.	.	PUNCT
ejpam-726	70	1	the	the	DET
ejpam-726	70	2	complement	complement	NOUN
ejpam-726	70	3	of	of	ADP
ejpam-726	70	4	a	a	DET
ejpam-726	70	5	g−closed	g−close	VERB
ejpam-726	70	6	set	set	NOUN
ejpam-726	70	7	is	be	AUX
ejpam-726	70	8	called	call	VERB
ejpam-726	70	9	a	a	DET
ejpam-726	70	10	g−open	g−open	NOUN
ejpam-726	70	11	set	set	NOUN
ejpam-726	70	12	.	.	PUNCT
ejpam-726	71	1	(	(	PUNCT
ejpam-726	71	2	a	a	X
ejpam-726	71	3	)	)	PUNCT
ejpam-726	71	4	a	a	DET
ejpam-726	71	5	regular	regular	ADJ
ejpam-726	71	6	generalized	generalize	VERB
ejpam-726	71	7	closed	close	VERB
ejpam-726	71	8	set	set	NOUN
ejpam-726	71	9	(	(	PUNCT
ejpam-726	71	10	for	for	ADP
ejpam-726	71	11	short	short	ADJ
ejpam-726	71	12	,	,	PUNCT
ejpam-726	71	13	r	r	NOUN
ejpam-726	71	14	g−closed	g−close	VERB
ejpam-726	71	15	)	)	PUNCT
ejpam-726	72	1	[	[	X
ejpam-726	72	2	8	8	X
ejpam-726	72	3	]	]	X
ejpam-726	72	4	if	if	SCONJ
ejpam-726	72	5	c	c	PROPN
ejpam-726	72	6	l(a	l(a	X
ejpam-726	72	7	)	)	PUNCT
ejpam-726	73	1	⊂	⊂	PROPN
ejpam-726	73	2	u	u	NOUN
ejpam-726	73	3	whenever	whenever	SCONJ
ejpam-726	73	4	a	a	DET
ejpam-726	73	5	⊂	⊂	PROPN
ejpam-726	73	6	u	u	NOUN
ejpam-726	73	7	and	and	CCONJ
ejpam-726	73	8	u	u	NOUN
ejpam-726	73	9	is	be	AUX
ejpam-726	73	10	regular	regular	ADJ
ejpam-726	73	11	open	open	ADJ
ejpam-726	73	12	in	in	ADP
ejpam-726	73	13	x	x	ADP
ejpam-726	73	14	;	;	PUNCT
ejpam-726	73	15	(	(	PUNCT
ejpam-726	73	16	b	b	X
ejpam-726	73	17	)	)	PUNCT
ejpam-726	73	18	g∗-closed	g∗-close	VERB
ejpam-726	74	1	[	[	X
ejpam-726	74	2	16	16	NUM
ejpam-726	74	3	]	]	PUNCT
ejpam-726	74	4	if	if	SCONJ
ejpam-726	74	5	c	c	PROPN
ejpam-726	74	6	l(a)⊂	l(a)⊂	PROPN
ejpam-726	74	7	u	u	NOUN
ejpam-726	74	8	whenever	whenever	SCONJ
ejpam-726	74	9	a⊂	a⊂	PUNCT
ejpam-726	74	10	u	u	NOUN
ejpam-726	74	11	and	and	CCONJ
ejpam-726	74	12	u	u	NOUN
ejpam-726	74	13	is	be	AUX
ejpam-726	74	14	g−open	g−open	VERB
ejpam-726	74	15	;	;	PUNCT
ejpam-726	74	16	(	(	PUNCT
ejpam-726	74	17	c	c	X
ejpam-726	74	18	)	)	PUNCT
ejpam-726	74	19	πg−closed	πg−close	VERB
ejpam-726	75	1	[	[	X
ejpam-726	75	2	8	8	X
ejpam-726	75	3	]	]	X
ejpam-726	75	4	if	if	SCONJ
ejpam-726	75	5	c	c	PROPN
ejpam-726	75	6	l(a)⊂	l(a)⊂	PROPN
ejpam-726	75	7	u	u	NOUN
ejpam-726	75	8	whenever	whenever	SCONJ
ejpam-726	75	9	a⊂	a⊂	PUNCT
ejpam-726	75	10	u	u	NOUN
ejpam-726	75	11	and	and	CCONJ
ejpam-726	75	12	u	u	NOUN
ejpam-726	75	13	is	be	AUX
ejpam-726	75	14	π	π	NOUN
ejpam-726	75	15	-	-	NOUN
ejpam-726	75	16	open	open	ADJ
ejpam-726	75	17	in	in	ADP
ejpam-726	75	18	x	x	PRON
ejpam-726	75	19	;	;	PUNCT
ejpam-726	75	20	(	(	PUNCT
ejpam-726	75	21	d	d	X
ejpam-726	75	22	)	)	PUNCT
ejpam-726	75	23	gp−closed	gp−close	VERB
ejpam-726	75	24	[	[	X
ejpam-726	75	25	18	18	NUM
ejpam-726	75	26	]	]	X
ejpam-726	75	27	if	if	SCONJ
ejpam-726	75	28	pc	pc	NOUN
ejpam-726	75	29	l(a)⊂	l(a)⊂	NOUN
ejpam-726	75	30	u	u	NOUN
ejpam-726	75	31	whenever	whenever	SCONJ
ejpam-726	75	32	a⊂	a⊂	PUNCT
ejpam-726	75	33	u	u	NOUN
ejpam-726	75	34	and	and	CCONJ
ejpam-726	75	35	u	u	NOUN
ejpam-726	75	36	is	be	AUX
ejpam-726	75	37	open	open	ADJ
ejpam-726	75	38	in	in	ADP
ejpam-726	75	39	x	x	NOUN
ejpam-726	75	40	;	;	PUNCT
ejpam-726	75	41	a.	a.	NOUN
ejpam-726	75	42	açıkgöz	açıkgöz	PROPN
ejpam-726	75	43	/	/	SYM
ejpam-726	75	44	eur	eur	PROPN
ejpam-726	75	45	.	.	PUNCT
ejpam-726	76	1	j.	j.	PROPN
ejpam-726	76	2	pure	pure	PROPN
ejpam-726	76	3	appl	appl	PROPN
ejpam-726	76	4	.	.	PROPN
ejpam-726	76	5	math	math	PROPN
ejpam-726	76	6	,	,	PUNCT
ejpam-726	76	7	4	4	NUM
ejpam-726	76	8	(	(	PUNCT
ejpam-726	76	9	2011	2011	NUM
ejpam-726	76	10	)	)	PUNCT
ejpam-726	76	11	,	,	PUNCT
ejpam-726	76	12	20	20	NUM
ejpam-726	76	13	-	-	SYM
ejpam-726	76	14	33	33	NUM
ejpam-726	76	15	22	22	NUM
ejpam-726	76	16	(	(	PUNCT
ejpam-726	76	17	e	e	NOUN
ejpam-726	76	18	)	)	PUNCT
ejpam-726	76	19	gs−closed	gs−close	VERB
ejpam-726	77	1	[	[	X
ejpam-726	77	2	1	1	X
ejpam-726	77	3	]	]	PUNCT
ejpam-726	77	4	if	if	SCONJ
ejpam-726	77	5	sc	sc	PROPN
ejpam-726	77	6	l(a)⊂	l(a)⊂	PROPN
ejpam-726	77	7	u	u	NOUN
ejpam-726	77	8	whenever	whenever	SCONJ
ejpam-726	77	9	a⊂	a⊂	PUNCT
ejpam-726	77	10	u	u	NOUN
ejpam-726	77	11	and	and	CCONJ
ejpam-726	77	12	u	u	NOUN
ejpam-726	77	13	is	be	AUX
ejpam-726	77	14	open	open	ADJ
ejpam-726	77	15	in	in	ADP
ejpam-726	77	16	x	x	PRON
ejpam-726	77	17	;	;	PUNCT
ejpam-726	77	18	(	(	PUNCT
ejpam-726	77	19	f	f	X
ejpam-726	77	20	)	)	PUNCT
ejpam-726	77	21	πgp−closed	πgp−close	VERB
ejpam-726	78	1	[	[	X
ejpam-726	78	2	21	21	NUM
ejpam-726	78	3	]	]	X
ejpam-726	78	4	if	if	SCONJ
ejpam-726	78	5	pc	pc	NOUN
ejpam-726	78	6	l(a	l(a	X
ejpam-726	78	7	)	)	PUNCT
ejpam-726	79	1	⊂	⊂	PROPN
ejpam-726	79	2	u	u	NOUN
ejpam-726	79	3	whenever	whenever	SCONJ
ejpam-726	79	4	a⊂	a⊂	PUNCT
ejpam-726	79	5	u	u	NOUN
ejpam-726	79	6	and	and	CCONJ
ejpam-726	79	7	u	u	NOUN
ejpam-726	79	8	is	be	AUX
ejpam-726	79	9	π	π	X
ejpam-726	79	10	-	-	ADJ
ejpam-726	79	11	open	open	ADJ
ejpam-726	79	12	inx	inx	NOUN
ejpam-726	79	13	;	;	PUNCT
ejpam-726	79	14	(	(	PUNCT
ejpam-726	79	15	g	g	NOUN
ejpam-726	79	16	)	)	PUNCT
ejpam-726	79	17	πgs−closed	πgs−closed	PUNCT
ejpam-726	80	1	[	[	X
ejpam-726	80	2	2	2	X
ejpam-726	80	3	]	]	PUNCT
ejpam-726	80	4	if	if	SCONJ
ejpam-726	80	5	sc	sc	PROPN
ejpam-726	80	6	l(a)⊂	l(a)⊂	PROPN
ejpam-726	80	7	u	u	NOUN
ejpam-726	80	8	whenever	whenever	SCONJ
ejpam-726	80	9	a⊂	a⊂	PUNCT
ejpam-726	80	10	u	u	NOUN
ejpam-726	80	11	and	and	CCONJ
ejpam-726	80	12	u	u	NOUN
ejpam-726	80	13	is	be	AUX
ejpam-726	80	14	π	π	NOUN
ejpam-726	80	15	-	-	NOUN
ejpam-726	80	16	open	open	ADJ
ejpam-726	80	17	in	in	ADP
ejpam-726	80	18	x	x	PRON
ejpam-726	80	19	;	;	PUNCT
ejpam-726	80	20	(	(	PUNCT
ejpam-726	80	21	h	h	NOUN
ejpam-726	80	22	)	)	PUNCT
ejpam-726	80	23	πgs−open	πgs−open	PUNCT
ejpam-726	80	24	(	(	PUNCT
ejpam-726	80	25	resp	resp	NOUN
ejpam-726	80	26	.	.	PUNCT
ejpam-726	81	1	g∗-open	g∗-open	VERB
ejpam-726	81	2	,	,	PUNCT
ejpam-726	81	3	πg−open	πg−open	ADJ
ejpam-726	81	4	,	,	PUNCT
ejpam-726	81	5	gp−open	gp−open	ADJ
ejpam-726	81	6	,	,	PUNCT
ejpam-726	81	7	πgp−open	πgp−open	NOUN
ejpam-726	81	8	,	,	PUNCT
ejpam-726	81	9	gs−open	gs−open	PROPN
ejpam-726	81	10	)	)	PUNCT
ejpam-726	81	11	if	if	SCONJ
ejpam-726	81	12	the	the	DET
ejpam-726	81	13	complement	complement	NOUN
ejpam-726	81	14	of	of	ADP
ejpam-726	81	15	a	a	PRON
ejpam-726	81	16	is	be	AUX
ejpam-726	81	17	πgs−closed	πgs−close	VERB
ejpam-726	81	18	(	(	PUNCT
ejpam-726	81	19	resp	resp	NOUN
ejpam-726	81	20	.	.	PUNCT
ejpam-726	82	1	g∗-closed	g∗-close	VERB
ejpam-726	82	2	,	,	PUNCT
ejpam-726	82	3	πg−closed	πg−close	VERB
ejpam-726	82	4	,	,	PUNCT
ejpam-726	82	5	gp−closed	gp−close	VERB
ejpam-726	82	6	,	,	PUNCT
ejpam-726	82	7	πgp−closed	πgp−closed	ADJ
ejpam-726	82	8	,	,	PUNCT
ejpam-726	82	9	gs−closed	gs−close	VERB
ejpam-726	82	10	)	)	PUNCT
ejpam-726	82	11	.	.	PUNCT
ejpam-726	83	1	definition	definition	NOUN
ejpam-726	83	2	2	2	NUM
ejpam-726	83	3	.	.	PUNCT
ejpam-726	84	1	a	a	DET
ejpam-726	84	2	subset	subset	NOUN
ejpam-726	84	3	a	a	PRON
ejpam-726	84	4	of	of	ADP
ejpam-726	84	5	a	a	DET
ejpam-726	84	6	space	space	NOUN
ejpam-726	84	7	(	(	PUNCT
ejpam-726	84	8	x	x	X
ejpam-726	84	9	,	,	PUNCT
ejpam-726	84	10	τ	τ	X
ejpam-726	84	11	)	)	PUNCT
ejpam-726	84	12	is	be	AUX
ejpam-726	84	13	called	call	VERB
ejpam-726	84	14	(	(	PUNCT
ejpam-726	84	15	a	a	X
ejpam-726	84	16	)	)	PUNCT
ejpam-726	84	17	a	a	DET
ejpam-726	84	18	β∗-set	β∗-set	NOUN
ejpam-726	85	1	[	[	X
ejpam-726	85	2	24	24	NUM
ejpam-726	85	3	]	]	X
ejpam-726	85	4	if	if	SCONJ
ejpam-726	85	5	a=	a=	VERB
ejpam-726	85	6	u	u	NOUN
ejpam-726	85	7	∩	∩	NOUN
ejpam-726	85	8	v	v	NOUN
ejpam-726	85	9	,	,	PUNCT
ejpam-726	85	10	where	where	SCONJ
ejpam-726	85	11	u	u	NOUN
ejpam-726	85	12	is	be	AUX
ejpam-726	85	13	open	open	ADJ
ejpam-726	85	14	and	and	CCONJ
ejpam-726	85	15	int(v	int(v	NOUN
ejpam-726	85	16	)	)	PUNCT
ejpam-726	85	17	=	=	NOUN
ejpam-726	85	18	cl(int(v	cl(int(v	NOUN
ejpam-726	85	19	)	)	PUNCT
ejpam-726	85	20	)	)	PUNCT
ejpam-726	85	21	.	.	PUNCT
ejpam-726	86	1	(	(	PUNCT
ejpam-726	86	2	b	b	X
ejpam-726	86	3	)	)	PUNCT
ejpam-726	86	4	β∗g−closed	β∗g−close	VERB
ejpam-726	86	5	if	if	SCONJ
ejpam-726	86	6	c	c	PROPN
ejpam-726	86	7	l(a)⊂	l(a)⊂	PROPN
ejpam-726	86	8	u	u	NOUN
ejpam-726	86	9	whenever	whenever	SCONJ
ejpam-726	86	10	a⊂	a⊂	PUNCT
ejpam-726	86	11	u	u	NOUN
ejpam-726	86	12	and	and	CCONJ
ejpam-726	86	13	u	u	NOUN
ejpam-726	86	14	is	be	AUX
ejpam-726	86	15	a	a	DET
ejpam-726	86	16	β∗-set	β∗-set	NOUN
ejpam-726	86	17	.	.	PUNCT
ejpam-726	87	1	(	(	PUNCT
ejpam-726	87	2	c	c	X
ejpam-726	87	3	)	)	PUNCT
ejpam-726	87	4	β∗sg−closed	β∗sg−close	VERB
ejpam-726	87	5	if	if	SCONJ
ejpam-726	87	6	sc	sc	PROPN
ejpam-726	87	7	l(a)⊂	l(a)⊂	PROPN
ejpam-726	87	8	u	u	NOUN
ejpam-726	87	9	whenever	whenever	SCONJ
ejpam-726	87	10	a⊂	a⊂	PUNCT
ejpam-726	87	11	u	u	NOUN
ejpam-726	87	12	and	and	CCONJ
ejpam-726	87	13	u	u	NOUN
ejpam-726	87	14	is	be	AUX
ejpam-726	87	15	a	a	DET
ejpam-726	87	16	β∗-set	β∗-set	NOUN
ejpam-726	87	17	.	.	PUNCT
ejpam-726	88	1	(	(	PUNCT
ejpam-726	88	2	d	d	X
ejpam-726	88	3	)	)	PUNCT
ejpam-726	88	4	β∗pg−closed	β∗pg−close	VERB
ejpam-726	88	5	if	if	SCONJ
ejpam-726	88	6	pcl(a)⊂	pcl(a)⊂	PROPN
ejpam-726	88	7	u	u	NOUN
ejpam-726	88	8	whenever	whenever	SCONJ
ejpam-726	88	9	a⊂	a⊂	PUNCT
ejpam-726	88	10	u	u	NOUN
ejpam-726	88	11	and	and	CCONJ
ejpam-726	88	12	u	u	NOUN
ejpam-726	88	13	is	be	AUX
ejpam-726	88	14	a	a	DET
ejpam-726	88	15	β∗-set	β∗-set	NOUN
ejpam-726	88	16	.	.	PUNCT
ejpam-726	89	1	(	(	PUNCT
ejpam-726	89	2	e	e	X
ejpam-726	89	3	)	)	PUNCT
ejpam-726	89	4	β∗pg−open	β∗pg−open	PROPN
ejpam-726	89	5	(	(	PUNCT
ejpam-726	89	6	resp	resp	NOUN
ejpam-726	89	7	.	.	PUNCT
ejpam-726	90	1	β∗g−open	β∗g−open	NOUN
ejpam-726	90	2	,	,	PUNCT
ejpam-726	90	3	β∗sg−open	β∗sg−open	PROPN
ejpam-726	90	4	)	)	PUNCT
ejpam-726	90	5	if	if	SCONJ
ejpam-726	90	6	the	the	DET
ejpam-726	90	7	complement	complement	NOUN
ejpam-726	90	8	of	of	ADP
ejpam-726	90	9	a	a	DET
ejpam-726	90	10	is	be	AUX
ejpam-726	90	11	β∗pg−closed	β∗pg−closed	ADJ
ejpam-726	90	12	(	(	PUNCT
ejpam-726	90	13	resp	resp	NOUN
ejpam-726	90	14	.	.	PUNCT
ejpam-726	91	1	β∗g−closed	β∗g−close	VERB
ejpam-726	91	2	,	,	PUNCT
ejpam-726	91	3	β∗sg−closed	β∗sg−closed	PROPN
ejpam-726	91	4	)	)	PUNCT
ejpam-726	91	5	.	.	PUNCT
ejpam-726	92	1	the	the	DET
ejpam-726	92	2	class	class	NOUN
ejpam-726	92	3	of	of	ADP
ejpam-726	92	4	all	all	DET
ejpam-726	92	5	β∗g−closed	β∗g−close	VERB
ejpam-726	92	6	subsets	subset	NOUN
ejpam-726	92	7	of	of	ADP
ejpam-726	92	8	(	(	PUNCT
ejpam-726	92	9	x	x	INTJ
ejpam-726	92	10	,	,	PUNCT
ejpam-726	92	11	τ	τ	X
ejpam-726	92	12	)	)	PUNCT
ejpam-726	92	13	is	be	AUX
ejpam-726	92	14	denoted	denote	VERB
ejpam-726	92	15	by	by	ADP
ejpam-726	92	16	β∗gc(x	β∗gc(x	NOUN
ejpam-726	92	17	,	,	PUNCT
ejpam-726	92	18	τ	τ	PROPN
ejpam-726	92	19	)	)	PUNCT
ejpam-726	92	20	.	.	PUNCT
ejpam-726	93	1	levine	levine	PROPN
ejpam-726	94	1	[	[	X
ejpam-726	94	2	17	17	NUM
ejpam-726	94	3	]	]	PUNCT
ejpam-726	94	4	and	and	CCONJ
ejpam-726	94	5	kumar	kumar	PROPN
ejpam-726	95	1	[	[	X
ejpam-726	95	2	16	16	NUM
ejpam-726	95	3	]	]	PUNCT
ejpam-726	95	4	gave	give	VERB
ejpam-726	95	5	the	the	DET
ejpam-726	95	6	following	follow	VERB
ejpam-726	95	7	diagrams	diagram	NOUN
ejpam-726	95	8	using	use	VERB
ejpam-726	95	9	some	some	PRON
ejpam-726	95	10	of	of	ADP
ejpam-726	95	11	the	the	DET
ejpam-726	95	12	expressions	expression	NOUN
ejpam-726	95	13	,	,	PUNCT
ejpam-726	95	14	respectively	respectively	ADV
ejpam-726	95	15	.	.	PUNCT
ejpam-726	96	1	diagram	diagram	PROPN
ejpam-726	96	2	i.	i.	PROPN
ejpam-726	96	3	closed	close	VERB
ejpam-726	96	4	set	set	VERB
ejpam-726	96	5	−→	−→	NOUN
ejpam-726	96	6	g−closed	g−close	VERB
ejpam-726	96	7	set	set	VERB
ejpam-726	96	8	−→	−→	NOUN
ejpam-726	96	9	r	r	NOUN
ejpam-726	96	10	g−closed	g−close	VERB
ejpam-726	96	11	set	set	VERB
ejpam-726	96	12	diagram	diagram	PROPN
ejpam-726	96	13	ii	ii	PROPN
ejpam-726	96	14	.	.	PROPN
ejpam-726	97	1	closed	close	VERB
ejpam-726	97	2	set	set	VERB
ejpam-726	97	3	−→	−→	NOUN
ejpam-726	97	4	g∗−closed	g∗−closed	AUX
ejpam-726	97	5	set	set	VERB
ejpam-726	97	6	−→	−→	NOUN
ejpam-726	97	7	g−closed	g−close	VERB
ejpam-726	97	8	s	s	PART
ejpam-726	97	9	furhermore	furhermore	NOUN
ejpam-726	97	10	,	,	PUNCT
ejpam-726	97	11	aslım	aslım	VERB
ejpam-726	97	12	et	et	PROPN
ejpam-726	97	13	al	al	PROPN
ejpam-726	97	14	.	.	PUNCT
ejpam-726	98	1	[	[	X
ejpam-726	98	2	2	2	X
ejpam-726	98	3	]	]	PUNCT
ejpam-726	98	4	indicated	indicate	VERB
ejpam-726	98	5	that	that	SCONJ
ejpam-726	98	6	every	every	DET
ejpam-726	98	7	gs−closed	gs−close	VERB
ejpam-726	98	8	set	set	NOUN
ejpam-726	98	9	is	be	AUX
ejpam-726	98	10	a	a	DET
ejpam-726	98	11	πgs−closed	πgs−close	VERB
ejpam-726	98	12	set	set	NOUN
ejpam-726	98	13	and	and	CCONJ
ejpam-726	98	14	every	every	DET
ejpam-726	98	15	πg−closed	πg−close	VERB
ejpam-726	98	16	set	set	NOUN
ejpam-726	98	17	is	be	AUX
ejpam-726	98	18	a	a	DET
ejpam-726	98	19	πgs−closed	πgs−close	VERB
ejpam-726	98	20	set	set	NOUN
ejpam-726	98	21	.	.	PUNCT
ejpam-726	99	1	they	they	PRON
ejpam-726	99	2	gave	give	VERB
ejpam-726	99	3	the	the	DET
ejpam-726	99	4	following	follow	VERB
ejpam-726	99	5	diagram	diagram	NOUN
ejpam-726	99	6	using	use	VERB
ejpam-726	99	7	these	these	DET
ejpam-726	99	8	properties	property	NOUN
ejpam-726	99	9	.	.	PUNCT
ejpam-726	100	1	diagram	diagram	PROPN
ejpam-726	100	2	iii	iii	PROPN
ejpam-726	100	3	.	.	PUNCT
ejpam-726	101	1	pre	pre	VERB
ejpam-726	101	2	-	-	ADJ
ejpam-726	101	3	closed	closed	ADJ
ejpam-726	101	4	−→	−→	NOUN
ejpam-726	101	5	gp−	gp−	PUNCT
ejpam-726	101	6	closed	close	VERB
ejpam-726	101	7	−→	−→	NOUN
ejpam-726	101	8	πgp−	πgp−	PROPN
ejpam-726	101	9	closed	close	VERB
ejpam-726	101	10	↑	↑	PROPN
ejpam-726	101	11	↑	↑	PROPN
ejpam-726	101	12	↑	↑	PROPN
ejpam-726	102	1	π−	π−	PROPN
ejpam-726	102	2	closed	close	VERB
ejpam-726	102	3	−→	−→	ADV
ejpam-726	102	4	closed	close	VERB
ejpam-726	102	5	−→	−→	ADJ
ejpam-726	102	6	g	g	PROPN
ejpam-726	102	7	−	−	PROPN
ejpam-726	102	8	closed	close	VERB
ejpam-726	102	9	−→	−→	ADJ
ejpam-726	102	10	πg	πg	ADP
ejpam-726	102	11	−	−	PROPN
ejpam-726	102	12	closed	close	VERB
ejpam-726	102	13	↓	↓	PROPN
ejpam-726	102	14	↓	↓	PROPN
ejpam-726	102	15	↓	↓	PROPN
ejpam-726	102	16	semi	semi	ADJ
ejpam-726	102	17	-	-	ADJ
ejpam-726	102	18	closed	closed	ADJ
ejpam-726	102	19	−→	−→	NOUN
ejpam-726	102	20	gs−	gs−	NUM
ejpam-726	102	21	closed	close	VERB
ejpam-726	102	22	−→	−→	ADJ
ejpam-726	102	23	πgs−	πgs−	NOUN
ejpam-726	102	24	closed	closed	ADJ
ejpam-726	102	25	remark	remark	NOUN
ejpam-726	102	26	1	1	NUM
ejpam-726	102	27	.	.	PUNCT
ejpam-726	103	1	a	a	DET
ejpam-726	103	2	lc	lc	NOUN
ejpam-726	103	3	-	-	PUNCT
ejpam-726	103	4	set	set	NOUN
ejpam-726	103	5	is	be	AUX
ejpam-726	103	6	independent	independent	ADJ
ejpam-726	103	7	from	from	ADP
ejpam-726	103	8	a	a	DET
ejpam-726	103	9	g−closed	g−close	VERB
ejpam-726	103	10	set	set	NOUN
ejpam-726	103	11	as	as	SCONJ
ejpam-726	103	12	it	it	PRON
ejpam-726	103	13	can	can	AUX
ejpam-726	103	14	be	be	AUX
ejpam-726	103	15	seen	see	VERB
ejpam-726	103	16	from	from	ADP
ejpam-726	103	17	the	the	DET
ejpam-726	103	18	next	next	ADJ
ejpam-726	103	19	two	two	NUM
ejpam-726	103	20	examples	example	NOUN
ejpam-726	103	21	.	.	PUNCT
ejpam-726	104	1	a.	a.	NOUN
ejpam-726	104	2	açıkgöz	açıkgöz	PROPN
ejpam-726	104	3	/	/	SYM
ejpam-726	104	4	eur	eur	PROPN
ejpam-726	104	5	.	.	PUNCT
ejpam-726	105	1	j.	j.	PROPN
ejpam-726	105	2	pure	pure	PROPN
ejpam-726	105	3	appl	appl	PROPN
ejpam-726	105	4	.	.	PROPN
ejpam-726	105	5	math	math	PROPN
ejpam-726	105	6	,	,	PUNCT
ejpam-726	105	7	4	4	NUM
ejpam-726	105	8	(	(	PUNCT
ejpam-726	105	9	2011	2011	NUM
ejpam-726	105	10	)	)	PUNCT
ejpam-726	105	11	,	,	PUNCT
ejpam-726	105	12	20	20	NUM
ejpam-726	105	13	-	-	SYM
ejpam-726	105	14	33	33	NUM
ejpam-726	105	15	23	23	NUM
ejpam-726	105	16	remark	remark	NOUN
ejpam-726	105	17	2	2	NUM
ejpam-726	105	18	.	.	PUNCT
ejpam-726	106	1	let	let	VERB
ejpam-726	106	2	x	x	PUNCT
ejpam-726	106	3	=	=	PRON
ejpam-726	106	4	{	{	PUNCT
ejpam-726	106	5	a	a	PRON
ejpam-726	106	6	,	,	PUNCT
ejpam-726	106	7	b	b	NOUN
ejpam-726	106	8	,	,	PUNCT
ejpam-726	106	9	c	c	NOUN
ejpam-726	106	10	}	}	PUNCT
ejpam-726	106	11	and	and	CCONJ
ejpam-726	106	12	τ	τ	PROPN
ejpam-726	106	13	=	=	PUNCT
ejpam-726	106	14	{	{	PUNCT
ejpam-726	106	15	x	x	X
ejpam-726	106	16	,	,	PUNCT
ejpam-726	106	17	;	;	PUNCT
ejpam-726	106	18	,	,	PUNCT
ejpam-726	106	19	{	{	PUNCT
ejpam-726	106	20	a	a	X
ejpam-726	106	21	}	}	PUNCT
ejpam-726	106	22	}	}	PUNCT
ejpam-726	106	23	.	.	PUNCT
ejpam-726	107	1	then	then	ADV
ejpam-726	107	2	{	{	PUNCT
ejpam-726	107	3	a	a	PRON
ejpam-726	107	4	}	}	PUNCT
ejpam-726	107	5	is	be	AUX
ejpam-726	107	6	a	a	DET
ejpam-726	107	7	lc	lc	NOUN
ejpam-726	107	8	-	-	PUNCT
ejpam-726	107	9	set	set	NOUN
ejpam-726	107	10	,	,	PUNCT
ejpam-726	107	11	but	but	CCONJ
ejpam-726	107	12	it	it	PRON
ejpam-726	107	13	is	be	AUX
ejpam-726	107	14	not	not	PART
ejpam-726	107	15	a	a	DET
ejpam-726	107	16	g−closed	g−close	VERB
ejpam-726	107	17	set	set	NOUN
ejpam-726	107	18	.	.	PUNCT
ejpam-726	108	1	remark	remark	PROPN
ejpam-726	108	2	3	3	NUM
ejpam-726	108	3	.	.	PUNCT
ejpam-726	109	1	let	let	VERB
ejpam-726	109	2	x	x	PUNCT
ejpam-726	109	3	=	=	PRON
ejpam-726	109	4	{	{	PUNCT
ejpam-726	109	5	a	a	PRON
ejpam-726	109	6	,	,	PUNCT
ejpam-726	109	7	b	b	NOUN
ejpam-726	109	8	,	,	PUNCT
ejpam-726	109	9	c	c	NOUN
ejpam-726	109	10	}	}	PUNCT
ejpam-726	109	11	and	and	CCONJ
ejpam-726	109	12	τ	τ	PROPN
ejpam-726	109	13	=	=	PUNCT
ejpam-726	109	14	{	{	PUNCT
ejpam-726	109	15	x	x	X
ejpam-726	109	16	,	,	PUNCT
ejpam-726	109	17	;	;	PUNCT
ejpam-726	109	18	,	,	PUNCT
ejpam-726	109	19	{	{	PUNCT
ejpam-726	109	20	a	a	X
ejpam-726	109	21	}	}	PUNCT
ejpam-726	109	22	}	}	PUNCT
ejpam-726	109	23	.	.	PUNCT
ejpam-726	110	1	then	then	ADV
ejpam-726	110	2	{	{	PUNCT
ejpam-726	110	3	a	a	PRON
ejpam-726	110	4	,	,	PUNCT
ejpam-726	110	5	b	b	NOUN
ejpam-726	110	6	}	}	PUNCT
ejpam-726	110	7	is	be	AUX
ejpam-726	110	8	a	a	DET
ejpam-726	110	9	g−closed	g−close	VERB
ejpam-726	110	10	set	set	NOUN
ejpam-726	110	11	,	,	PUNCT
ejpam-726	110	12	but	but	CCONJ
ejpam-726	110	13	it	it	PRON
ejpam-726	110	14	is	be	AUX
ejpam-726	110	15	not	not	PART
ejpam-726	110	16	a	a	DET
ejpam-726	110	17	lc	lc	NOUN
ejpam-726	110	18	-	-	PUNCT
ejpam-726	110	19	set	set	NOUN
ejpam-726	110	20	.	.	PUNCT
ejpam-726	111	1	theorem	theorem	NOUN
ejpam-726	111	2	1	1	NUM
ejpam-726	111	3	.	.	X
ejpam-726	111	4	for	for	ADP
ejpam-726	111	5	a	a	DET
ejpam-726	111	6	subset	subset	NOUN
ejpam-726	111	7	a	a	PRON
ejpam-726	111	8	of	of	ADP
ejpam-726	111	9	a	a	DET
ejpam-726	111	10	topological	topological	ADJ
ejpam-726	111	11	space	space	NOUN
ejpam-726	111	12	(	(	PUNCT
ejpam-726	111	13	x	x	X
ejpam-726	111	14	,	,	PUNCT
ejpam-726	111	15	τ	τ	PROPN
ejpam-726	111	16	)	)	PUNCT
ejpam-726	111	17	,	,	PUNCT
ejpam-726	111	18	the	the	DET
ejpam-726	111	19	following	follow	VERB
ejpam-726	111	20	are	be	AUX
ejpam-726	111	21	equivalent	equivalent	ADJ
ejpam-726	111	22	:	:	PUNCT
ejpam-726	111	23	(	(	PUNCT
ejpam-726	111	24	a	a	X
ejpam-726	111	25	)	)	PUNCT
ejpam-726	111	26	a	a	PRON
ejpam-726	111	27	is	be	AUX
ejpam-726	111	28	a	a	DET
ejpam-726	111	29	lc	lc	NOUN
ejpam-726	111	30	-	-	PUNCT
ejpam-726	111	31	set	set	NOUN
ejpam-726	111	32	.	.	PUNCT
ejpam-726	112	1	(	(	PUNCT
ejpam-726	112	2	b	b	X
ejpam-726	112	3	)	)	PUNCT
ejpam-726	112	4	a=	a=	PROPN
ejpam-726	112	5	u	u	NOUN
ejpam-726	112	6	∩	∩	NOUN
ejpam-726	112	7	cl(a	cl(a	NUM
ejpam-726	112	8	)	)	PUNCT
ejpam-726	112	9	for	for	ADP
ejpam-726	112	10	some	some	DET
ejpam-726	112	11	u	u	NOUN
ejpam-726	112	12	open	open	ADJ
ejpam-726	112	13	set	set	NOUN
ejpam-726	112	14	.	.	PUNCT
ejpam-726	113	1	proof	proof	NOUN
ejpam-726	113	2	.	.	PUNCT
ejpam-726	114	1	(	(	PUNCT
ejpam-726	114	2	a	a	X
ejpam-726	114	3	)	)	PUNCT
ejpam-726	114	4	→	→	SYM
ejpam-726	114	5	(	(	PUNCT
ejpam-726	114	6	b	b	NOUN
ejpam-726	114	7	):	):	PUNCT
ejpam-726	114	8	since	since	SCONJ
ejpam-726	114	9	a	a	PRON
ejpam-726	114	10	is	be	AUX
ejpam-726	114	11	a	a	DET
ejpam-726	114	12	lc	lc	NOUN
ejpam-726	114	13	-	-	PUNCT
ejpam-726	114	14	set	set	NOUN
ejpam-726	114	15	,	,	PUNCT
ejpam-726	114	16	then	then	ADV
ejpam-726	114	17	a=	a=	VERB
ejpam-726	114	18	u	u	NOUN
ejpam-726	114	19	∩	∩	NOUN
ejpam-726	114	20	v	v	NOUN
ejpam-726	114	21	,	,	PUNCT
ejpam-726	114	22	where	where	SCONJ
ejpam-726	114	23	u	u	NOUN
ejpam-726	114	24	is	be	AUX
ejpam-726	114	25	open	open	ADJ
ejpam-726	114	26	and	and	CCONJ
ejpam-726	114	27	v	v	NOUN
ejpam-726	114	28	is	be	AUX
ejpam-726	114	29	closed	closed	ADJ
ejpam-726	114	30	.	.	PUNCT
ejpam-726	115	1	so	so	ADV
ejpam-726	115	2	,	,	PUNCT
ejpam-726	115	3	a⊂	a⊂	VERB
ejpam-726	115	4	u	u	NOUN
ejpam-726	115	5	and	and	CCONJ
ejpam-726	115	6	a⊂	a⊂	SYM
ejpam-726	115	7	v	v	NOUN
ejpam-726	115	8	.	.	PUNCT
ejpam-726	116	1	hence	hence	ADV
ejpam-726	116	2	,	,	PUNCT
ejpam-726	116	3	cl(a	cl(a	X
ejpam-726	116	4	)	)	PUNCT
ejpam-726	117	1	⊂	⊂	PROPN
ejpam-726	117	2	cl(v	cl(v	NOUN
ejpam-726	117	3	)	)	PUNCT
ejpam-726	117	4	.	.	PUNCT
ejpam-726	118	1	therefore	therefore	ADV
ejpam-726	118	2	,	,	PUNCT
ejpam-726	118	3	a⊂	a⊂	PUNCT
ejpam-726	118	4	u	u	NOUN
ejpam-726	118	5	∩	∩	NOUN
ejpam-726	118	6	cl(a)⊂	cl(a)⊂	ADJ
ejpam-726	118	7	u	u	NOUN
ejpam-726	118	8	∩	∩	NOUN
ejpam-726	118	9	cl(v	cl(v	X
ejpam-726	118	10	)	)	PUNCT
ejpam-726	119	1	=	=	SYM
ejpam-726	119	2	u	u	NOUN
ejpam-726	119	3	∩	∩	NOUN
ejpam-726	119	4	v	v	NOUN
ejpam-726	119	5	=	=	PUNCT
ejpam-726	119	6	a.	a.	NOUN
ejpam-726	119	7	thus	thus	ADV
ejpam-726	119	8	,	,	PUNCT
ejpam-726	119	9	a=	a=	VERB
ejpam-726	119	10	u	u	NOUN
ejpam-726	119	11	∩	∩	NOUN
ejpam-726	119	12	cl(a	cl(a	NUM
ejpam-726	119	13	)	)	PUNCT
ejpam-726	119	14	.	.	PUNCT
ejpam-726	120	1	(	(	PUNCT
ejpam-726	120	2	b	b	X
ejpam-726	120	3	)	)	PUNCT
ejpam-726	120	4	→	→	PUNCT
ejpam-726	120	5	(	(	PUNCT
ejpam-726	120	6	a	a	X
ejpam-726	120	7	):	):	PUNCT
ejpam-726	120	8	it	it	PRON
ejpam-726	120	9	is	be	AUX
ejpam-726	120	10	obvious	obvious	ADJ
ejpam-726	120	11	because	because	SCONJ
ejpam-726	120	12	cl(a	cl(a	NUM
ejpam-726	120	13	)	)	PUNCT
ejpam-726	120	14	is	be	AUX
ejpam-726	120	15	closed	close	VERB
ejpam-726	120	16	.	.	PUNCT
ejpam-726	121	1	theorem	theorem	NOUN
ejpam-726	121	2	2	2	NUM
ejpam-726	121	3	.	.	X
ejpam-726	121	4	for	for	ADP
ejpam-726	121	5	a	a	DET
ejpam-726	121	6	subset	subset	NOUN
ejpam-726	121	7	a	a	PRON
ejpam-726	121	8	of	of	ADP
ejpam-726	121	9	a	a	DET
ejpam-726	121	10	topological	topological	ADJ
ejpam-726	121	11	space	space	NOUN
ejpam-726	121	12	(	(	PUNCT
ejpam-726	121	13	x	x	X
ejpam-726	121	14	,	,	PUNCT
ejpam-726	121	15	τ	τ	PROPN
ejpam-726	121	16	)	)	PUNCT
ejpam-726	121	17	,	,	PUNCT
ejpam-726	121	18	the	the	DET
ejpam-726	121	19	following	follow	VERB
ejpam-726	121	20	are	be	AUX
ejpam-726	121	21	equivalent	equivalent	ADJ
ejpam-726	121	22	:	:	PUNCT
ejpam-726	121	23	(	(	PUNCT
ejpam-726	121	24	a	a	X
ejpam-726	121	25	)	)	PUNCT
ejpam-726	121	26	a	a	PRON
ejpam-726	121	27	is	be	AUX
ejpam-726	121	28	closed	closed	ADJ
ejpam-726	121	29	.	.	PUNCT
ejpam-726	122	1	(	(	PUNCT
ejpam-726	122	2	b	b	X
ejpam-726	122	3	)	)	PUNCT
ejpam-726	122	4	a	a	PRON
ejpam-726	122	5	is	be	AUX
ejpam-726	122	6	a	a	DET
ejpam-726	122	7	lc	lc	NOUN
ejpam-726	122	8	-	-	PUNCT
ejpam-726	122	9	set	set	VERB
ejpam-726	122	10	and	and	CCONJ
ejpam-726	122	11	g−closed	g−closed	ADJ
ejpam-726	122	12	.	.	PUNCT
ejpam-726	123	1	proof	proof	NOUN
ejpam-726	123	2	.	.	PUNCT
ejpam-726	124	1	(	(	PUNCT
ejpam-726	124	2	a	a	X
ejpam-726	124	3	)	)	PUNCT
ejpam-726	124	4	→	→	SYM
ejpam-726	124	5	(	(	PUNCT
ejpam-726	124	6	b	b	X
ejpam-726	124	7	):	):	PUNCT
ejpam-726	124	8	this	this	PRON
ejpam-726	124	9	is	be	AUX
ejpam-726	124	10	obvious	obvious	ADJ
ejpam-726	124	11	.	.	PUNCT
ejpam-726	125	1	(	(	PUNCT
ejpam-726	125	2	b	b	X
ejpam-726	125	3	)	)	PUNCT
ejpam-726	125	4	→	→	PUNCT
ejpam-726	125	5	(	(	PUNCT
ejpam-726	125	6	a	a	X
ejpam-726	125	7	):	):	PUNCT
ejpam-726	125	8	since	since	SCONJ
ejpam-726	125	9	a	a	PRON
ejpam-726	125	10	is	be	AUX
ejpam-726	125	11	a	a	DET
ejpam-726	125	12	lc	lc	NOUN
ejpam-726	125	13	-	-	PUNCT
ejpam-726	125	14	set	set	NOUN
ejpam-726	125	15	,	,	PUNCT
ejpam-726	125	16	then	then	ADV
ejpam-726	125	17	a=	a=	VERB
ejpam-726	125	18	u	u	NOUN
ejpam-726	125	19	∩	∩	NOUN
ejpam-726	125	20	cl(a	cl(a	NUM
ejpam-726	125	21	)	)	PUNCT
ejpam-726	125	22	,	,	PUNCT
ejpam-726	125	23	where	where	SCONJ
ejpam-726	125	24	u	u	NOUN
ejpam-726	125	25	is	be	AUX
ejpam-726	125	26	an	an	DET
ejpam-726	125	27	open	open	ADJ
ejpam-726	125	28	set	set	NOUN
ejpam-726	125	29	in	in	ADP
ejpam-726	125	30	x.	x.	NOUN
ejpam-726	125	31	so	so	ADV
ejpam-726	125	32	,	,	PUNCT
ejpam-726	125	33	a⊂	a⊂	PUNCT
ejpam-726	125	34	u	u	NOUN
ejpam-726	125	35	and	and	CCONJ
ejpam-726	125	36	since	since	SCONJ
ejpam-726	125	37	a	a	PRON
ejpam-726	125	38	is	be	AUX
ejpam-726	125	39	g−closed	g−close	VERB
ejpam-726	125	40	,	,	PUNCT
ejpam-726	125	41	then	then	ADV
ejpam-726	125	42	cl(a	cl(a	NUM
ejpam-726	125	43	)	)	PUNCT
ejpam-726	126	1	⊂	⊂	PROPN
ejpam-726	126	2	u	u	PROPN
ejpam-726	126	3	.	.	PUNCT
ejpam-726	127	1	therefore	therefore	ADV
ejpam-726	127	2	,	,	PUNCT
ejpam-726	127	3	cl(a	cl(a	X
ejpam-726	127	4	)	)	PUNCT
ejpam-726	127	5	⊂u∩cl(a	⊂u∩cl(a	PROPN
ejpam-726	127	6	)	)	PUNCT
ejpam-726	127	7	=	=	SYM
ejpam-726	127	8	a.	a.	NOUN
ejpam-726	127	9	hence	hence	ADV
ejpam-726	127	10	,	,	PUNCT
ejpam-726	127	11	a	a	PRON
ejpam-726	127	12	is	be	AUX
ejpam-726	127	13	closed	closed	ADJ
ejpam-726	127	14	.	.	PUNCT
ejpam-726	128	1	theorem	theorem	NOUN
ejpam-726	128	2	3	3	X
ejpam-726	128	3	.	.	PUNCT
ejpam-726	129	1	let	let	AUX
ejpam-726	129	2	(	(	PUNCT
ejpam-726	129	3	x	x	X
ejpam-726	129	4	,	,	PUNCT
ejpam-726	129	5	τ	τ	X
ejpam-726	129	6	)	)	PUNCT
ejpam-726	129	7	be	be	VERB
ejpam-726	129	8	a	a	DET
ejpam-726	129	9	topological	topological	ADJ
ejpam-726	129	10	space	space	NOUN
ejpam-726	129	11	.	.	PUNCT
ejpam-726	130	1	then	then	ADV
ejpam-726	130	2	we	we	PRON
ejpam-726	130	3	have	have	VERB
ejpam-726	130	4	(	(	PUNCT
ejpam-726	130	5	a	a	X
ejpam-726	130	6	)	)	PUNCT
ejpam-726	130	7	every	every	DET
ejpam-726	130	8	closed	closed	ADJ
ejpam-726	130	9	set	set	NOUN
ejpam-726	130	10	is	be	AUX
ejpam-726	130	11	a	a	DET
ejpam-726	130	12	β∗g−closed	β∗g−close	VERB
ejpam-726	130	13	set	set	NOUN
ejpam-726	130	14	.	.	PUNCT
ejpam-726	131	1	(	(	PUNCT
ejpam-726	131	2	b	b	X
ejpam-726	131	3	)	)	PUNCT
ejpam-726	131	4	every	every	DET
ejpam-726	131	5	β∗g−closed	β∗g−close	VERB
ejpam-726	131	6	set	set	NOUN
ejpam-726	131	7	is	be	AUX
ejpam-726	131	8	a	a	DET
ejpam-726	131	9	g−closed	g−close	VERB
ejpam-726	131	10	set	set	NOUN
ejpam-726	131	11	.	.	PUNCT
ejpam-726	132	1	proof	proof	NOUN
ejpam-726	132	2	.	.	PUNCT
ejpam-726	133	1	(	(	PUNCT
ejpam-726	133	2	a	a	X
ejpam-726	133	3	)	)	PUNCT
ejpam-726	133	4	this	this	PRON
ejpam-726	133	5	is	be	AUX
ejpam-726	133	6	obvious	obvious	ADJ
ejpam-726	133	7	.	.	PUNCT
ejpam-726	134	1	(	(	PUNCT
ejpam-726	134	2	b	b	X
ejpam-726	134	3	)	)	PUNCT
ejpam-726	134	4	let	let	VERB
ejpam-726	134	5	a	a	PRON
ejpam-726	134	6	be	be	AUX
ejpam-726	134	7	a	a	DET
ejpam-726	134	8	β∗g−closed	β∗g−close	VERB
ejpam-726	134	9	set	set	NOUN
ejpam-726	134	10	of	of	ADP
ejpam-726	134	11	(	(	PUNCT
ejpam-726	134	12	x	x	PROPN
ejpam-726	134	13	,	,	PUNCT
ejpam-726	134	14	τ	τ	PROPN
ejpam-726	134	15	)	)	PUNCT
ejpam-726	134	16	and	and	CCONJ
ejpam-726	134	17	a⊂	a⊂	VERB
ejpam-726	134	18	u	u	NOUN
ejpam-726	134	19	where	where	SCONJ
ejpam-726	134	20	u	u	PROPN
ejpam-726	134	21	∈	∈	PROPN
ejpam-726	134	22	τ	τ	X
ejpam-726	134	23	.	.	PUNCT
ejpam-726	135	1	since	since	SCONJ
ejpam-726	135	2	every	every	DET
ejpam-726	135	3	open	open	ADJ
ejpam-726	135	4	set	set	NOUN
ejpam-726	135	5	is	be	AUX
ejpam-726	135	6	a	a	DET
ejpam-726	135	7	β∗-set	β∗-set	NOUN
ejpam-726	135	8	,	,	PUNCT
ejpam-726	135	9	so	so	CCONJ
ejpam-726	135	10	u	u	NOUN
ejpam-726	135	11	is	be	AUX
ejpam-726	135	12	a	a	DET
ejpam-726	135	13	β∗-set	β∗-set	NOUN
ejpam-726	135	14	of	of	ADP
ejpam-726	135	15	(	(	PUNCT
ejpam-726	135	16	x	x	PROPN
ejpam-726	135	17	,	,	PUNCT
ejpam-726	135	18	τ	τ	PROPN
ejpam-726	135	19	)	)	PUNCT
ejpam-726	135	20	.	.	PUNCT
ejpam-726	136	1	since	since	SCONJ
ejpam-726	136	2	a	a	PRON
ejpam-726	136	3	is	be	AUX
ejpam-726	136	4	a	a	DET
ejpam-726	136	5	β∗g−closed	β∗g−close	VERB
ejpam-726	136	6	set	set	NOUN
ejpam-726	136	7	,	,	PUNCT
ejpam-726	136	8	we	we	PRON
ejpam-726	136	9	obtain	obtain	VERB
ejpam-726	136	10	that	that	PRON
ejpam-726	136	11	cl(a)⊂	cl(a)⊂	PROPN
ejpam-726	136	12	u	u	NOUN
ejpam-726	136	13	,	,	PUNCT
ejpam-726	136	14	hence	hence	ADV
ejpam-726	136	15	a	a	PRON
ejpam-726	136	16	is	be	AUX
ejpam-726	136	17	a	a	DET
ejpam-726	136	18	g−closed	g−close	VERB
ejpam-726	136	19	set	set	NOUN
ejpam-726	136	20	of	of	ADP
ejpam-726	136	21	(	(	PUNCT
ejpam-726	136	22	x	x	PROPN
ejpam-726	136	23	,	,	PUNCT
ejpam-726	136	24	τ	τ	PROPN
ejpam-726	136	25	)	)	PUNCT
ejpam-726	136	26	.	.	PUNCT
ejpam-726	137	1	a.	a.	NOUN
ejpam-726	137	2	açıkgöz	açıkgöz	PROPN
ejpam-726	137	3	/	/	SYM
ejpam-726	137	4	eur	eur	PROPN
ejpam-726	137	5	.	.	PUNCT
ejpam-726	138	1	j.	j.	PROPN
ejpam-726	138	2	pure	pure	PROPN
ejpam-726	138	3	appl	appl	PROPN
ejpam-726	138	4	.	.	PROPN
ejpam-726	138	5	math	math	PROPN
ejpam-726	138	6	,	,	PUNCT
ejpam-726	138	7	4	4	NUM
ejpam-726	138	8	(	(	PUNCT
ejpam-726	138	9	2011	2011	NUM
ejpam-726	138	10	)	)	PUNCT
ejpam-726	138	11	,	,	PUNCT
ejpam-726	138	12	20	20	NUM
ejpam-726	138	13	-	-	SYM
ejpam-726	138	14	33	33	NUM
ejpam-726	138	15	24	24	NUM
ejpam-726	138	16	remark	remark	NOUN
ejpam-726	138	17	4	4	NUM
ejpam-726	138	18	.	.	PUNCT
ejpam-726	139	1	the	the	DET
ejpam-726	139	2	converses	converse	NOUN
ejpam-726	139	3	of	of	ADP
ejpam-726	139	4	theorem	theorem	NOUN
ejpam-726	139	5	3	3	NUM
ejpam-726	139	6	need	need	AUX
ejpam-726	139	7	not	not	PART
ejpam-726	139	8	be	be	AUX
ejpam-726	139	9	true	true	ADJ
ejpam-726	139	10	as	as	SCONJ
ejpam-726	139	11	shown	show	VERB
ejpam-726	139	12	in	in	ADP
ejpam-726	139	13	the	the	DET
ejpam-726	139	14	following	follow	VERB
ejpam-726	139	15	examples	example	NOUN
ejpam-726	139	16	.	.	PUNCT
ejpam-726	140	1	example	example	NOUN
ejpam-726	141	1	1	1	NUM
ejpam-726	141	2	.	.	PUNCT
ejpam-726	141	3	let	let	VERB
ejpam-726	141	4	x	x	PUNCT
ejpam-726	141	5	=	=	PRON
ejpam-726	141	6	{	{	PUNCT
ejpam-726	141	7	a	a	PRON
ejpam-726	141	8	,	,	PUNCT
ejpam-726	141	9	b	b	NOUN
ejpam-726	141	10	,	,	PUNCT
ejpam-726	141	11	c	c	NOUN
ejpam-726	141	12	,	,	PUNCT
ejpam-726	141	13	d	d	NOUN
ejpam-726	141	14	}	}	PUNCT
ejpam-726	141	15	and	and	CCONJ
ejpam-726	141	16	τ	τ	PROPN
ejpam-726	141	17	=	=	PUNCT
ejpam-726	141	18	{	{	PUNCT
ejpam-726	141	19	x	x	X
ejpam-726	141	20	,	,	PUNCT
ejpam-726	141	21	;	;	PUNCT
ejpam-726	141	22	,	,	PUNCT
ejpam-726	141	23	{	{	PUNCT
ejpam-726	141	24	b	b	NOUN
ejpam-726	141	25	}	}	PUNCT
ejpam-726	141	26	,	,	PUNCT
ejpam-726	141	27	{	{	PUNCT
ejpam-726	141	28	c	c	X
ejpam-726	141	29	}	}	PUNCT
ejpam-726	141	30	,	,	PUNCT
ejpam-726	141	31	{	{	PUNCT
ejpam-726	141	32	a	a	DET
ejpam-726	141	33	,	,	PUNCT
ejpam-726	141	34	b	b	NOUN
ejpam-726	141	35	}	}	PUNCT
ejpam-726	141	36	,	,	PUNCT
ejpam-726	141	37	{	{	PUNCT
ejpam-726	141	38	b	b	X
ejpam-726	141	39	,	,	PUNCT
ejpam-726	141	40	c	c	NOUN
ejpam-726	141	41	}	}	PUNCT
ejpam-726	141	42	,	,	PUNCT
ejpam-726	141	43	{	{	PUNCT
ejpam-726	141	44	a	a	DET
ejpam-726	141	45	,	,	PUNCT
ejpam-726	141	46	b	b	NOUN
ejpam-726	141	47	,	,	PUNCT
ejpam-726	141	48	c	c	NOUN
ejpam-726	141	49	}	}	PUNCT
ejpam-726	141	50	,	,	PUNCT
ejpam-726	141	51	{	{	PUNCT
ejpam-726	141	52	a	a	DET
ejpam-726	141	53	,	,	PUNCT
ejpam-726	141	54	b	b	NOUN
ejpam-726	141	55	,	,	PUNCT
ejpam-726	141	56	d	d	NOUN
ejpam-726	141	57	}	}	PUNCT
ejpam-726	141	58	}	}	PUNCT
ejpam-726	141	59	.	.	PUNCT
ejpam-726	142	1	then	then	ADV
ejpam-726	142	2	{	{	PUNCT
ejpam-726	142	3	a	a	PRON
ejpam-726	142	4	,	,	PUNCT
ejpam-726	142	5	b	b	NOUN
ejpam-726	142	6	}	}	PUNCT
ejpam-726	142	7	is	be	AUX
ejpam-726	142	8	a	a	DET
ejpam-726	142	9	β∗g−closed	β∗g−close	VERB
ejpam-726	142	10	set	set	NOUN
ejpam-726	142	11	,	,	PUNCT
ejpam-726	142	12	but	but	CCONJ
ejpam-726	142	13	it	it	PRON
ejpam-726	142	14	is	be	AUX
ejpam-726	142	15	not	not	PART
ejpam-726	142	16	a	a	DET
ejpam-726	142	17	closed	closed	ADJ
ejpam-726	142	18	set	set	NOUN
ejpam-726	142	19	.	.	PUNCT
ejpam-726	142	20	example	example	NOUN
ejpam-726	143	1	2	2	NUM
ejpam-726	143	2	.	.	PUNCT
ejpam-726	143	3	let	let	VERB
ejpam-726	143	4	x	x	PUNCT
ejpam-726	143	5	=	=	PRON
ejpam-726	143	6	{	{	PUNCT
ejpam-726	143	7	a	a	PRON
ejpam-726	143	8	,	,	PUNCT
ejpam-726	143	9	b	b	NOUN
ejpam-726	143	10	,	,	PUNCT
ejpam-726	143	11	c	c	NOUN
ejpam-726	143	12	}	}	PUNCT
ejpam-726	143	13	and	and	CCONJ
ejpam-726	143	14	τ	τ	PROPN
ejpam-726	143	15	=	=	PUNCT
ejpam-726	143	16	{	{	PUNCT
ejpam-726	143	17	x	x	X
ejpam-726	143	18	,	,	PUNCT
ejpam-726	143	19	;	;	PUNCT
ejpam-726	143	20	,	,	PUNCT
ejpam-726	143	21	{	{	PUNCT
ejpam-726	143	22	a	a	X
ejpam-726	143	23	}	}	PUNCT
ejpam-726	143	24	,	,	PUNCT
ejpam-726	143	25	{	{	PUNCT
ejpam-726	143	26	c	c	X
ejpam-726	143	27	}	}	PUNCT
ejpam-726	143	28	,	,	PUNCT
ejpam-726	143	29	{	{	PUNCT
ejpam-726	143	30	a	a	DET
ejpam-726	143	31	,	,	PUNCT
ejpam-726	143	32	b	b	NOUN
ejpam-726	143	33	}	}	PUNCT
ejpam-726	143	34	,	,	PUNCT
ejpam-726	143	35	{	{	PUNCT
ejpam-726	143	36	a	a	PRON
ejpam-726	143	37	,	,	PUNCT
ejpam-726	143	38	c	c	NOUN
ejpam-726	143	39	}	}	PUNCT
ejpam-726	143	40	}	}	PUNCT
ejpam-726	143	41	.	.	PUNCT
ejpam-726	144	1	then	then	ADV
ejpam-726	144	2	{	{	PUNCT
ejpam-726	144	3	c	c	X
ejpam-726	144	4	}	}	PUNCT
ejpam-726	144	5	is	be	AUX
ejpam-726	144	6	a	a	DET
ejpam-726	144	7	g−closed	g−close	VERB
ejpam-726	144	8	set	set	NOUN
ejpam-726	144	9	,	,	PUNCT
ejpam-726	144	10	but	but	CCONJ
ejpam-726	144	11	it	it	PRON
ejpam-726	144	12	is	be	AUX
ejpam-726	144	13	not	not	PART
ejpam-726	144	14	a	a	DET
ejpam-726	144	15	β∗g−closed	β∗g−close	VERB
ejpam-726	144	16	set	set	NOUN
ejpam-726	144	17	.	.	PUNCT
ejpam-726	145	1	theorem	theorem	ADJ
ejpam-726	145	2	4	4	NUM
ejpam-726	145	3	.	.	PUNCT
ejpam-726	146	1	let	let	AUX
ejpam-726	146	2	(	(	PUNCT
ejpam-726	146	3	x	x	X
ejpam-726	146	4	,	,	PUNCT
ejpam-726	146	5	τ	τ	X
ejpam-726	146	6	)	)	PUNCT
ejpam-726	146	7	be	be	VERB
ejpam-726	146	8	a	a	DET
ejpam-726	146	9	topological	topological	ADJ
ejpam-726	146	10	space	space	NOUN
ejpam-726	146	11	.	.	PUNCT
ejpam-726	147	1	then	then	ADV
ejpam-726	147	2	we	we	PRON
ejpam-726	147	3	have	have	VERB
ejpam-726	147	4	(	(	PUNCT
ejpam-726	147	5	a	a	X
ejpam-726	147	6	)	)	PUNCT
ejpam-726	147	7	every	every	DET
ejpam-726	147	8	β∗g−closed	β∗g−close	VERB
ejpam-726	147	9	set	set	NOUN
ejpam-726	147	10	is	be	AUX
ejpam-726	147	11	a	a	DET
ejpam-726	147	12	β∗pg−closed	β∗pg−closed	ADJ
ejpam-726	147	13	set	set	NOUN
ejpam-726	147	14	.	.	PUNCT
ejpam-726	148	1	(	(	PUNCT
ejpam-726	148	2	b	b	X
ejpam-726	148	3	)	)	PUNCT
ejpam-726	148	4	every	every	DET
ejpam-726	148	5	β∗g−closed	β∗g−close	VERB
ejpam-726	148	6	set	set	NOUN
ejpam-726	148	7	is	be	AUX
ejpam-726	148	8	a	a	DET
ejpam-726	148	9	β∗sg−closed	β∗sg−closed	PROPN
ejpam-726	148	10	set	set	NOUN
ejpam-726	148	11	.	.	PUNCT
ejpam-726	149	1	proof	proof	NOUN
ejpam-726	149	2	.	.	PUNCT
ejpam-726	150	1	this	this	PRON
ejpam-726	150	2	is	be	AUX
ejpam-726	150	3	obvious	obvious	ADJ
ejpam-726	150	4	.	.	PUNCT
ejpam-726	151	1	remark	remark	NOUN
ejpam-726	151	2	5	5	NUM
ejpam-726	151	3	.	.	PUNCT
ejpam-726	152	1	the	the	DET
ejpam-726	152	2	converses	converse	NOUN
ejpam-726	152	3	of	of	ADP
ejpam-726	152	4	theorem	theorem	NOUN
ejpam-726	152	5	4	4	NUM
ejpam-726	152	6	need	need	AUX
ejpam-726	152	7	not	not	PART
ejpam-726	152	8	be	be	AUX
ejpam-726	152	9	true	true	ADJ
ejpam-726	152	10	as	as	SCONJ
ejpam-726	152	11	shown	show	VERB
ejpam-726	152	12	in	in	ADP
ejpam-726	152	13	the	the	DET
ejpam-726	152	14	following	follow	VERB
ejpam-726	152	15	examples	example	NOUN
ejpam-726	152	16	.	.	PUNCT
ejpam-726	153	1	example	example	NOUN
ejpam-726	154	1	3	3	X
ejpam-726	154	2	.	.	PUNCT
ejpam-726	154	3	let	let	VERB
ejpam-726	154	4	x	x	PUNCT
ejpam-726	154	5	=	=	PRON
ejpam-726	154	6	{	{	PUNCT
ejpam-726	154	7	a	a	PRON
ejpam-726	154	8	,	,	PUNCT
ejpam-726	154	9	b	b	NOUN
ejpam-726	154	10	,	,	PUNCT
ejpam-726	154	11	c	c	NOUN
ejpam-726	154	12	}	}	PUNCT
ejpam-726	154	13	and	and	CCONJ
ejpam-726	154	14	τ	τ	PROPN
ejpam-726	154	15	=	=	PUNCT
ejpam-726	154	16	{	{	PUNCT
ejpam-726	154	17	x	x	X
ejpam-726	154	18	,	,	PUNCT
ejpam-726	154	19	;	;	PUNCT
ejpam-726	154	20	,	,	PUNCT
ejpam-726	154	21	{	{	PUNCT
ejpam-726	154	22	a	a	X
ejpam-726	154	23	}	}	PUNCT
ejpam-726	154	24	,	,	PUNCT
ejpam-726	154	25	{	{	PUNCT
ejpam-726	154	26	c	c	X
ejpam-726	154	27	}	}	PUNCT
ejpam-726	154	28	,	,	PUNCT
ejpam-726	154	29	{	{	PUNCT
ejpam-726	154	30	a	a	DET
ejpam-726	154	31	,	,	PUNCT
ejpam-726	154	32	b	b	NOUN
ejpam-726	154	33	}	}	PUNCT
ejpam-726	154	34	,	,	PUNCT
ejpam-726	154	35	{	{	PUNCT
ejpam-726	154	36	a	a	PRON
ejpam-726	154	37	,	,	PUNCT
ejpam-726	154	38	c	c	NOUN
ejpam-726	154	39	}	}	PUNCT
ejpam-726	154	40	}	}	PUNCT
ejpam-726	154	41	.	.	PUNCT
ejpam-726	155	1	then	then	ADV
ejpam-726	155	2	{	{	PUNCT
ejpam-726	155	3	a	a	PRON
ejpam-726	155	4	,	,	PUNCT
ejpam-726	155	5	b	b	NOUN
ejpam-726	155	6	}	}	PUNCT
ejpam-726	155	7	is	be	AUX
ejpam-726	155	8	a	a	DET
ejpam-726	155	9	β∗pg−closed	β∗pg−closed	PROPN
ejpam-726	155	10	set	set	NOUN
ejpam-726	155	11	which	which	PRON
ejpam-726	155	12	is	be	AUX
ejpam-726	155	13	not	not	PART
ejpam-726	155	14	a	a	DET
ejpam-726	155	15	β∗g−closed	β∗g−close	VERB
ejpam-726	155	16	set	set	NOUN
ejpam-726	155	17	.	.	PUNCT
ejpam-726	155	18	example	example	NOUN
ejpam-726	156	1	4	4	X
ejpam-726	156	2	.	.	PUNCT
ejpam-726	156	3	let	let	VERB
ejpam-726	156	4	x	x	PUNCT
ejpam-726	156	5	=	=	PRON
ejpam-726	156	6	{	{	PUNCT
ejpam-726	156	7	a	a	PRON
ejpam-726	156	8	,	,	PUNCT
ejpam-726	156	9	b	b	NOUN
ejpam-726	156	10	,	,	PUNCT
ejpam-726	156	11	c	c	NOUN
ejpam-726	156	12	}	}	PUNCT
ejpam-726	156	13	and	and	CCONJ
ejpam-726	156	14	τ	τ	PROPN
ejpam-726	156	15	=	=	PUNCT
ejpam-726	156	16	{	{	PUNCT
ejpam-726	156	17	x	x	X
ejpam-726	156	18	,	,	PUNCT
ejpam-726	156	19	;	;	PUNCT
ejpam-726	156	20	,	,	PUNCT
ejpam-726	156	21	{	{	PUNCT
ejpam-726	156	22	a	a	X
ejpam-726	156	23	}	}	PUNCT
ejpam-726	156	24	,	,	PUNCT
ejpam-726	156	25	{	{	PUNCT
ejpam-726	156	26	c	c	X
ejpam-726	156	27	}	}	PUNCT
ejpam-726	156	28	,	,	PUNCT
ejpam-726	156	29	{	{	PUNCT
ejpam-726	156	30	a	a	DET
ejpam-726	156	31	,	,	PUNCT
ejpam-726	156	32	b	b	NOUN
ejpam-726	156	33	}	}	PUNCT
ejpam-726	156	34	,	,	PUNCT
ejpam-726	156	35	{	{	PUNCT
ejpam-726	156	36	b	b	X
ejpam-726	156	37	,	,	PUNCT
ejpam-726	156	38	c	c	NOUN
ejpam-726	156	39	}	}	PUNCT
ejpam-726	156	40	}	}	PUNCT
ejpam-726	156	41	.	.	PUNCT
ejpam-726	157	1	then	then	ADV
ejpam-726	157	2	{	{	PUNCT
ejpam-726	157	3	b	b	X
ejpam-726	157	4	,	,	PUNCT
ejpam-726	157	5	c	c	NOUN
ejpam-726	157	6	}	}	PUNCT
ejpam-726	157	7	is	be	AUX
ejpam-726	157	8	a	a	DET
ejpam-726	157	9	β∗sg−closed	β∗sg−closed	PROPN
ejpam-726	157	10	set	set	NOUN
ejpam-726	157	11	which	which	PRON
ejpam-726	157	12	is	be	AUX
ejpam-726	157	13	not	not	PART
ejpam-726	157	14	a	a	DET
ejpam-726	157	15	β∗g−closed	β∗g−close	VERB
ejpam-726	157	16	set	set	NOUN
ejpam-726	157	17	.	.	PUNCT
ejpam-726	158	1	it	it	PRON
ejpam-726	158	2	can	can	AUX
ejpam-726	158	3	be	be	AUX
ejpam-726	158	4	expanded	expand	VERB
ejpam-726	158	5	to	to	ADP
ejpam-726	158	6	the	the	DET
ejpam-726	158	7	following	follow	VERB
ejpam-726	158	8	diagram	diagram	NOUN
ejpam-726	158	9	using	use	VERB
ejpam-726	158	10	diagrams	diagram	NOUN
ejpam-726	158	11	i	i	PRON
ejpam-726	158	12	,	,	PUNCT
ejpam-726	158	13	ii	ii	PROPN
ejpam-726	158	14	and	and	CCONJ
ejpam-726	158	15	iii	iii	NUM
ejpam-726	158	16	diagram	diagram	NOUN
ejpam-726	158	17	iv	iv	PROPN
ejpam-726	158	18	.	.	PUNCT
ejpam-726	158	19	remark	remark	PROPN
ejpam-726	158	20	6	6	NUM
ejpam-726	158	21	.	.	PUNCT
ejpam-726	159	1	by	by	ADP
ejpam-726	159	2	the	the	DET
ejpam-726	159	3	two	two	NUM
ejpam-726	159	4	examples	example	NOUN
ejpam-726	159	5	stated	state	VERB
ejpam-726	159	6	below	below	ADV
ejpam-726	159	7	,	,	PUNCT
ejpam-726	159	8	we	we	PRON
ejpam-726	159	9	show	show	VERB
ejpam-726	159	10	that	that	SCONJ
ejpam-726	159	11	β∗g−closed	β∗g−close	VERB
ejpam-726	159	12	and	and	CCONJ
ejpam-726	159	13	g∗-closed	g∗-close	VERB
ejpam-726	159	14	are	be	AUX
ejpam-726	159	15	independent	independent	ADJ
ejpam-726	159	16	of	of	ADP
ejpam-726	159	17	each	each	DET
ejpam-726	159	18	other	other	ADJ
ejpam-726	159	19	.	.	PUNCT
ejpam-726	159	20	example	example	NOUN
ejpam-726	160	1	5	5	NUM
ejpam-726	160	2	.	.	PUNCT
ejpam-726	160	3	let	let	VERB
ejpam-726	160	4	x	x	PUNCT
ejpam-726	160	5	=	=	PRON
ejpam-726	160	6	{	{	PUNCT
ejpam-726	160	7	a	a	PRON
ejpam-726	160	8	,	,	PUNCT
ejpam-726	160	9	b	b	NOUN
ejpam-726	160	10	,	,	PUNCT
ejpam-726	160	11	c	c	NOUN
ejpam-726	160	12	,	,	PUNCT
ejpam-726	160	13	d	d	NOUN
ejpam-726	160	14	}	}	PUNCT
ejpam-726	160	15	and	and	CCONJ
ejpam-726	160	16	τ	τ	PROPN
ejpam-726	160	17	=	=	PUNCT
ejpam-726	160	18	{	{	PUNCT
ejpam-726	160	19	x	x	X
ejpam-726	160	20	,	,	PUNCT
ejpam-726	160	21	;	;	PUNCT
ejpam-726	160	22	,	,	PUNCT
ejpam-726	160	23	{	{	PUNCT
ejpam-726	160	24	b	b	NOUN
ejpam-726	160	25	}	}	PUNCT
ejpam-726	160	26	,	,	PUNCT
ejpam-726	160	27	{	{	PUNCT
ejpam-726	160	28	c	c	X
ejpam-726	160	29	}	}	PUNCT
ejpam-726	160	30	,	,	PUNCT
ejpam-726	160	31	{	{	PUNCT
ejpam-726	160	32	a	a	DET
ejpam-726	160	33	,	,	PUNCT
ejpam-726	160	34	b	b	NOUN
ejpam-726	160	35	}	}	PUNCT
ejpam-726	160	36	,	,	PUNCT
ejpam-726	160	37	{	{	PUNCT
ejpam-726	160	38	b	b	X
ejpam-726	160	39	,	,	PUNCT
ejpam-726	160	40	c	c	NOUN
ejpam-726	160	41	}	}	PUNCT
ejpam-726	160	42	,	,	PUNCT
ejpam-726	160	43	{	{	PUNCT
ejpam-726	160	44	a	a	DET
ejpam-726	160	45	,	,	PUNCT
ejpam-726	160	46	b	b	NOUN
ejpam-726	160	47	,	,	PUNCT
ejpam-726	160	48	c	c	NOUN
ejpam-726	160	49	}	}	PUNCT
ejpam-726	160	50	,	,	PUNCT
ejpam-726	160	51	{	{	PUNCT
ejpam-726	160	52	a	a	DET
ejpam-726	160	53	,	,	PUNCT
ejpam-726	160	54	b	b	NOUN
ejpam-726	160	55	,	,	PUNCT
ejpam-726	160	56	d	d	NOUN
ejpam-726	160	57	}	}	PUNCT
ejpam-726	160	58	}	}	PUNCT
ejpam-726	160	59	.	.	PUNCT
ejpam-726	161	1	then	then	ADV
ejpam-726	161	2	{	{	PUNCT
ejpam-726	161	3	a	a	PRON
ejpam-726	161	4	,	,	PUNCT
ejpam-726	161	5	b	b	NOUN
ejpam-726	161	6	}	}	PUNCT
ejpam-726	161	7	is	be	AUX
ejpam-726	161	8	a	a	DET
ejpam-726	161	9	β∗g−closed	β∗g−close	VERB
ejpam-726	161	10	set	set	NOUN
ejpam-726	161	11	,	,	PUNCT
ejpam-726	161	12	but	but	CCONJ
ejpam-726	161	13	it	it	PRON
ejpam-726	161	14	is	be	AUX
ejpam-726	161	15	not	not	PART
ejpam-726	161	16	a	a	DET
ejpam-726	161	17	g∗-closed	g∗-close	VERB
ejpam-726	161	18	set	set	NOUN
ejpam-726	161	19	.	.	PUNCT
ejpam-726	161	20	example	example	NOUN
ejpam-726	162	1	6	6	NUM
ejpam-726	162	2	.	.	PUNCT
ejpam-726	163	1	let	let	VERB
ejpam-726	163	2	x	x	PUNCT
ejpam-726	163	3	=	=	PRON
ejpam-726	163	4	{	{	PUNCT
ejpam-726	163	5	a	a	PRON
ejpam-726	163	6	,	,	PUNCT
ejpam-726	163	7	b	b	NOUN
ejpam-726	163	8	,	,	PUNCT
ejpam-726	163	9	c	c	NOUN
ejpam-726	163	10	}	}	PUNCT
ejpam-726	163	11	and	and	CCONJ
ejpam-726	163	12	τ	τ	PROPN
ejpam-726	163	13	=	=	PUNCT
ejpam-726	163	14	{	{	PUNCT
ejpam-726	163	15	x	x	X
ejpam-726	163	16	,	,	PUNCT
ejpam-726	163	17	;	;	PUNCT
ejpam-726	163	18	,	,	PUNCT
ejpam-726	163	19	{	{	PUNCT
ejpam-726	163	20	a	a	X
ejpam-726	163	21	}	}	PUNCT
ejpam-726	163	22	,	,	PUNCT
ejpam-726	163	23	{	{	PUNCT
ejpam-726	163	24	c	c	X
ejpam-726	163	25	}	}	PUNCT
ejpam-726	163	26	,	,	PUNCT
ejpam-726	163	27	{	{	PUNCT
ejpam-726	163	28	a	a	DET
ejpam-726	163	29	,	,	PUNCT
ejpam-726	163	30	b	b	NOUN
ejpam-726	163	31	}	}	PUNCT
ejpam-726	163	32	,	,	PUNCT
ejpam-726	163	33	{	{	PUNCT
ejpam-726	163	34	a	a	PRON
ejpam-726	163	35	,	,	PUNCT
ejpam-726	163	36	c	c	NOUN
ejpam-726	163	37	}	}	PUNCT
ejpam-726	163	38	}	}	PUNCT
ejpam-726	163	39	.	.	PUNCT
ejpam-726	164	1	then	then	ADV
ejpam-726	164	2	{	{	PUNCT
ejpam-726	164	3	c	c	X
ejpam-726	164	4	}	}	PUNCT
ejpam-726	164	5	is	be	AUX
ejpam-726	164	6	a	a	DET
ejpam-726	164	7	g∗-closed	g∗-close	VERB
ejpam-726	164	8	set	set	NOUN
ejpam-726	164	9	,	,	PUNCT
ejpam-726	164	10	but	but	CCONJ
ejpam-726	164	11	it	it	PRON
ejpam-726	164	12	is	be	AUX
ejpam-726	164	13	not	not	PART
ejpam-726	164	14	a	a	DET
ejpam-726	164	15	β∗g−closed	β∗g−close	VERB
ejpam-726	164	16	set	set	NOUN
ejpam-726	164	17	.	.	PUNCT
ejpam-726	165	1	a.	a.	NOUN
ejpam-726	165	2	açıkgöz	açıkgöz	PROPN
ejpam-726	165	3	/	/	SYM
ejpam-726	165	4	eur	eur	PROPN
ejpam-726	165	5	.	.	PUNCT
ejpam-726	166	1	j.	j.	PROPN
ejpam-726	166	2	pure	pure	PROPN
ejpam-726	166	3	appl	appl	PROPN
ejpam-726	166	4	.	.	PROPN
ejpam-726	166	5	math	math	PROPN
ejpam-726	166	6	,	,	PUNCT
ejpam-726	166	7	4	4	NUM
ejpam-726	166	8	(	(	PUNCT
ejpam-726	166	9	2011	2011	NUM
ejpam-726	166	10	)	)	PUNCT
ejpam-726	166	11	,	,	PUNCT
ejpam-726	166	12	20	20	NUM
ejpam-726	166	13	-	-	SYM
ejpam-726	166	14	33	33	NUM
ejpam-726	166	15	25	25	NUM
ejpam-726	166	16	remark	remark	NOUN
ejpam-726	166	17	7	7	NUM
ejpam-726	166	18	.	.	PUNCT
ejpam-726	167	1	a	a	DET
ejpam-726	167	2	β∗-set	β∗-set	NOUN
ejpam-726	167	3	is	be	AUX
ejpam-726	167	4	independent	independent	ADJ
ejpam-726	167	5	from	from	ADP
ejpam-726	167	6	β∗g−closed	β∗g−close	VERB
ejpam-726	167	7	as	as	SCONJ
ejpam-726	167	8	it	it	PRON
ejpam-726	167	9	can	can	AUX
ejpam-726	167	10	be	be	AUX
ejpam-726	167	11	seen	see	VERB
ejpam-726	167	12	from	from	ADP
ejpam-726	167	13	the	the	DET
ejpam-726	167	14	next	next	ADJ
ejpam-726	167	15	two	two	NUM
ejpam-726	167	16	examples	example	NOUN
ejpam-726	167	17	.	.	PUNCT
ejpam-726	168	1	example	example	NOUN
ejpam-726	169	1	7	7	NUM
ejpam-726	169	2	.	.	PUNCT
ejpam-726	170	1	let	let	VERB
ejpam-726	170	2	x	x	PUNCT
ejpam-726	170	3	=	=	PRON
ejpam-726	170	4	{	{	PUNCT
ejpam-726	170	5	a	a	PRON
ejpam-726	170	6	,	,	PUNCT
ejpam-726	170	7	b	b	NOUN
ejpam-726	170	8	,	,	PUNCT
ejpam-726	170	9	c	c	NOUN
ejpam-726	170	10	}	}	PUNCT
ejpam-726	170	11	and	and	CCONJ
ejpam-726	170	12	τ	τ	PROPN
ejpam-726	170	13	=	=	PUNCT
ejpam-726	170	14	{	{	PUNCT
ejpam-726	170	15	x	x	X
ejpam-726	170	16	,	,	PUNCT
ejpam-726	170	17	;	;	PUNCT
ejpam-726	170	18	,	,	PUNCT
ejpam-726	170	19	{	{	PUNCT
ejpam-726	170	20	a	a	X
ejpam-726	170	21	}	}	PUNCT
ejpam-726	170	22	}	}	PUNCT
ejpam-726	170	23	.	.	PUNCT
ejpam-726	171	1	then	then	ADV
ejpam-726	171	2	{	{	PUNCT
ejpam-726	171	3	a	a	PRON
ejpam-726	171	4	}	}	PUNCT
ejpam-726	171	5	is	be	AUX
ejpam-726	171	6	a	a	DET
ejpam-726	171	7	β∗-set	β∗-set	NOUN
ejpam-726	171	8	,	,	PUNCT
ejpam-726	171	9	but	but	CCONJ
ejpam-726	171	10	it	it	PRON
ejpam-726	171	11	is	be	AUX
ejpam-726	171	12	not	not	PART
ejpam-726	171	13	a	a	DET
ejpam-726	171	14	β∗g−closed	β∗g−close	VERB
ejpam-726	171	15	set	set	NOUN
ejpam-726	171	16	.	.	PUNCT
ejpam-726	171	17	example	example	NOUN
ejpam-726	172	1	8	8	NUM
ejpam-726	172	2	.	.	PUNCT
ejpam-726	173	1	let	let	VERB
ejpam-726	173	2	x	x	PUNCT
ejpam-726	173	3	=	=	PRON
ejpam-726	173	4	{	{	PUNCT
ejpam-726	173	5	a	a	PRON
ejpam-726	173	6	,	,	PUNCT
ejpam-726	173	7	b	b	NOUN
ejpam-726	173	8	,	,	PUNCT
ejpam-726	173	9	c	c	NOUN
ejpam-726	173	10	}	}	PUNCT
ejpam-726	173	11	and	and	CCONJ
ejpam-726	173	12	τ	τ	PROPN
ejpam-726	173	13	=	=	PUNCT
ejpam-726	173	14	{	{	PUNCT
ejpam-726	173	15	x	x	X
ejpam-726	173	16	,	,	PUNCT
ejpam-726	173	17	;	;	PUNCT
ejpam-726	173	18	,	,	PUNCT
ejpam-726	173	19	{	{	PUNCT
ejpam-726	173	20	b	b	NOUN
ejpam-726	173	21	}	}	PUNCT
ejpam-726	173	22	,	,	PUNCT
ejpam-726	173	23	{	{	PUNCT
ejpam-726	173	24	c	c	X
ejpam-726	173	25	}	}	PUNCT
ejpam-726	173	26	,	,	PUNCT
ejpam-726	173	27	{	{	PUNCT
ejpam-726	173	28	a	a	DET
ejpam-726	173	29	,	,	PUNCT
ejpam-726	173	30	b	b	NOUN
ejpam-726	173	31	}	}	PUNCT
ejpam-726	173	32	,	,	PUNCT
ejpam-726	173	33	{	{	PUNCT
ejpam-726	173	34	b	b	X
ejpam-726	173	35	,	,	PUNCT
ejpam-726	173	36	c	c	NOUN
ejpam-726	173	37	}	}	PUNCT
ejpam-726	173	38	}	}	PUNCT
ejpam-726	173	39	.	.	PUNCT
ejpam-726	174	1	then	then	ADV
ejpam-726	174	2	{	{	PUNCT
ejpam-726	174	3	a	a	PRON
ejpam-726	174	4	,	,	PUNCT
ejpam-726	174	5	b	b	NOUN
ejpam-726	174	6	}	}	PUNCT
ejpam-726	174	7	is	be	AUX
ejpam-726	174	8	a	a	DET
ejpam-726	174	9	β∗g−closed	β∗g−close	VERB
ejpam-726	174	10	set	set	NOUN
ejpam-726	174	11	,	,	PUNCT
ejpam-726	174	12	but	but	CCONJ
ejpam-726	174	13	it	it	PRON
ejpam-726	174	14	is	be	AUX
ejpam-726	174	15	not	not	PART
ejpam-726	174	16	a	a	DET
ejpam-726	174	17	β∗-set	β∗-set	NOUN
ejpam-726	174	18	.	.	PUNCT
ejpam-726	175	1	theorem	theorem	NOUN
ejpam-726	175	2	5	5	NUM
ejpam-726	175	3	.	.	PUNCT
ejpam-726	176	1	if	if	SCONJ
ejpam-726	176	2	a	a	PRON
ejpam-726	176	3	is	be	AUX
ejpam-726	176	4	both	both	PRON
ejpam-726	176	5	β∗-set	β∗-set	NOUN
ejpam-726	176	6	and	and	CCONJ
ejpam-726	176	7	β∗g−closed	β∗g−close	VERB
ejpam-726	176	8	set	set	NOUN
ejpam-726	176	9	of	of	ADP
ejpam-726	176	10	(	(	PUNCT
ejpam-726	176	11	x	x	PROPN
ejpam-726	176	12	,	,	PUNCT
ejpam-726	176	13	τ	τ	PROPN
ejpam-726	176	14	)	)	PUNCT
ejpam-726	176	15	,	,	PUNCT
ejpam-726	176	16	then	then	ADV
ejpam-726	176	17	a	a	PRON
ejpam-726	176	18	is	be	AUX
ejpam-726	176	19	closed	closed	ADJ
ejpam-726	176	20	.	.	PUNCT
ejpam-726	177	1	proof	proof	NOUN
ejpam-726	177	2	.	.	PUNCT
ejpam-726	178	1	let	let	VERB
ejpam-726	178	2	a	a	DET
ejpam-726	178	3	be	be	AUX
ejpam-726	178	4	both	both	PRON
ejpam-726	178	5	β∗-set	β∗-set	NOUN
ejpam-726	178	6	and	and	CCONJ
ejpam-726	178	7	β∗g−closed	β∗g−close	VERB
ejpam-726	178	8	set	set	NOUN
ejpam-726	178	9	of	of	ADP
ejpam-726	178	10	(	(	PUNCT
ejpam-726	178	11	x	x	PROPN
ejpam-726	178	12	,	,	PUNCT
ejpam-726	178	13	τ	τ	PROPN
ejpam-726	178	14	)	)	PUNCT
ejpam-726	178	15	.	.	PUNCT
ejpam-726	179	1	then	then	ADV
ejpam-726	179	2	cl(a	cl(a	PUNCT
ejpam-726	179	3	)	)	PUNCT
ejpam-726	179	4	⊂	⊂	PROPN
ejpam-726	179	5	a	a	X
ejpam-726	179	6	,	,	PUNCT
ejpam-726	179	7	whenever	whenever	SCONJ
ejpam-726	179	8	a	a	PRON
ejpam-726	179	9	is	be	AUX
ejpam-726	179	10	a	a	DET
ejpam-726	179	11	β∗-set	β∗-set	NOUN
ejpam-726	179	12	and	and	CCONJ
ejpam-726	180	1	a⊂	a⊂	NOUN
ejpam-726	180	2	a.	a.	NOUN
ejpam-726	180	3	so	so	ADV
ejpam-726	180	4	we	we	PRON
ejpam-726	180	5	obtain	obtain	VERB
ejpam-726	180	6	that	that	DET
ejpam-726	180	7	a=	a=	NOUN
ejpam-726	180	8	cl(a	cl(a	X
ejpam-726	180	9	)	)	PUNCT
ejpam-726	180	10	and	and	CCONJ
ejpam-726	180	11	hence	hence	ADV
ejpam-726	180	12	a	a	PRON
ejpam-726	180	13	is	be	AUX
ejpam-726	180	14	closed	closed	ADJ
ejpam-726	180	15	.	.	PUNCT
ejpam-726	181	1	proposition	proposition	NOUN
ejpam-726	181	2	1	1	NUM
ejpam-726	181	3	.	.	PUNCT
ejpam-726	182	1	if	if	SCONJ
ejpam-726	182	2	a	a	PRON
ejpam-726	182	3	and	and	CCONJ
ejpam-726	182	4	b	b	NOUN
ejpam-726	182	5	are	be	AUX
ejpam-726	182	6	β∗g−closed	β∗g−close	VERB
ejpam-726	182	7	sets	set	NOUN
ejpam-726	182	8	,	,	PUNCT
ejpam-726	182	9	then	then	ADV
ejpam-726	182	10	a∪	a∪	PROPN
ejpam-726	182	11	b	b	PROPN
ejpam-726	182	12	is	be	AUX
ejpam-726	182	13	β∗g−closed	β∗g−close	VERB
ejpam-726	182	14	.	.	PUNCT
ejpam-726	183	1	proof	proof	NOUN
ejpam-726	183	2	.	.	PUNCT
ejpam-726	184	1	let	let	VERB
ejpam-726	184	2	a∪	a∪	PROPN
ejpam-726	184	3	b	b	NOUN
ejpam-726	184	4	⊆	⊆	NUM
ejpam-726	184	5	u	u	NOUN
ejpam-726	184	6	,	,	PUNCT
ejpam-726	184	7	where	where	SCONJ
ejpam-726	184	8	u	u	NOUN
ejpam-726	184	9	is	be	AUX
ejpam-726	184	10	a	a	DET
ejpam-726	184	11	β∗-set	β∗-set	NOUN
ejpam-726	184	12	.	.	PUNCT
ejpam-726	185	1	since	since	SCONJ
ejpam-726	185	2	a	a	DET
ejpam-726	185	3	,	,	PUNCT
ejpam-726	185	4	b	b	PROPN
ejpam-726	185	5	are	be	AUX
ejpam-726	185	6	β∗g−closed	β∗g−close	VERB
ejpam-726	185	7	sets	set	NOUN
ejpam-726	185	8	,	,	PUNCT
ejpam-726	185	9	cl(a)⊆	cl(a)⊆	PROPN
ejpam-726	185	10	u	u	NOUN
ejpam-726	185	11	and	and	CCONJ
ejpam-726	185	12	cl(b	cl(b	NOUN
ejpam-726	185	13	)	)	PUNCT
ejpam-726	185	14	⊆	⊆	NUM
ejpam-726	185	15	u	u	NOUN
ejpam-726	185	16	,	,	PUNCT
ejpam-726	185	17	whenever	whenever	SCONJ
ejpam-726	185	18	a⊆	a⊆	VERB
ejpam-726	185	19	u	u	PROPN
ejpam-726	185	20	,	,	PUNCT
ejpam-726	185	21	b	b	PROPN
ejpam-726	185	22	⊆	⊆	NUM
ejpam-726	185	23	u	u	NOUN
ejpam-726	185	24	and	and	CCONJ
ejpam-726	185	25	u	u	NOUN
ejpam-726	185	26	is	be	AUX
ejpam-726	185	27	a	a	DET
ejpam-726	185	28	β∗-set	β∗-set	NOUN
ejpam-726	185	29	.	.	PUNCT
ejpam-726	186	1	therefore	therefore	ADV
ejpam-726	186	2	,	,	PUNCT
ejpam-726	186	3	cl(a∪	cl(a∪	PROPN
ejpam-726	186	4	b	b	NOUN
ejpam-726	186	5	)	)	PUNCT
ejpam-726	186	6	=	=	SYM
ejpam-726	186	7	cl(a)∪	cl(a)∪	NOUN
ejpam-726	186	8	cl(b	cl(b	NOUN
ejpam-726	186	9	)	)	PUNCT
ejpam-726	186	10	⊆	⊆	NUM
ejpam-726	186	11	u	u	NOUN
ejpam-726	186	12	.	.	PUNCT
ejpam-726	187	1	hence	hence	ADV
ejpam-726	187	2	we	we	PRON
ejpam-726	187	3	obtain	obtain	VERB
ejpam-726	187	4	that	that	PRON
ejpam-726	187	5	a∪	a∪	PROPN
ejpam-726	188	1	b	b	NOUN
ejpam-726	188	2	is	be	AUX
ejpam-726	188	3	a	a	DET
ejpam-726	188	4	β∗g−closed	β∗g−close	VERB
ejpam-726	188	5	set	set	NOUN
ejpam-726	188	6	of	of	ADP
ejpam-726	188	7	(	(	PUNCT
ejpam-726	188	8	x	x	PROPN
ejpam-726	188	9	,	,	PUNCT
ejpam-726	188	10	τ	τ	PROPN
ejpam-726	188	11	)	)	PUNCT
ejpam-726	188	12	.	.	PUNCT
ejpam-726	189	1	remark	remark	PROPN
ejpam-726	189	2	8	8	NUM
ejpam-726	189	3	.	.	PUNCT
ejpam-726	190	1	the	the	DET
ejpam-726	190	2	intersection	intersection	NOUN
ejpam-726	190	3	of	of	ADP
ejpam-726	190	4	two	two	NUM
ejpam-726	190	5	β∗g−closed	β∗g−close	VERB
ejpam-726	190	6	sets	set	NOUN
ejpam-726	190	7	are	be	AUX
ejpam-726	190	8	not	not	PART
ejpam-726	190	9	always	always	ADV
ejpam-726	190	10	a	a	DET
ejpam-726	190	11	β∗g−closed	β∗g−close	VERB
ejpam-726	190	12	set	set	NOUN
ejpam-726	190	13	.	.	PUNCT
ejpam-726	190	14	example	example	NOUN
ejpam-726	191	1	9	9	NUM
ejpam-726	191	2	.	.	PUNCT
ejpam-726	192	1	let	let	VERB
ejpam-726	192	2	x	x	PUNCT
ejpam-726	192	3	=	=	PRON
ejpam-726	192	4	{	{	PUNCT
ejpam-726	192	5	a	a	PRON
ejpam-726	192	6	,	,	PUNCT
ejpam-726	192	7	b	b	NOUN
ejpam-726	192	8	,	,	PUNCT
ejpam-726	192	9	c	c	NOUN
ejpam-726	192	10	,	,	PUNCT
ejpam-726	192	11	d	d	NOUN
ejpam-726	192	12	}	}	PUNCT
ejpam-726	192	13	and	and	CCONJ
ejpam-726	192	14	τ	τ	PROPN
ejpam-726	192	15	=	=	PUNCT
ejpam-726	192	16	{	{	PUNCT
ejpam-726	192	17	x	x	X
ejpam-726	192	18	,	,	PUNCT
ejpam-726	192	19	;	;	PUNCT
ejpam-726	192	20	,	,	PUNCT
ejpam-726	192	21	{	{	PUNCT
ejpam-726	192	22	b	b	NOUN
ejpam-726	192	23	}	}	PUNCT
ejpam-726	192	24	,	,	PUNCT
ejpam-726	192	25	{	{	PUNCT
ejpam-726	192	26	c	c	X
ejpam-726	192	27	}	}	PUNCT
ejpam-726	192	28	,	,	PUNCT
ejpam-726	192	29	{	{	PUNCT
ejpam-726	192	30	a	a	DET
ejpam-726	192	31	,	,	PUNCT
ejpam-726	192	32	b	b	NOUN
ejpam-726	192	33	}	}	PUNCT
ejpam-726	192	34	,	,	PUNCT
ejpam-726	192	35	{	{	PUNCT
ejpam-726	192	36	b	b	X
ejpam-726	192	37	,	,	PUNCT
ejpam-726	192	38	c	c	NOUN
ejpam-726	192	39	}	}	PUNCT
ejpam-726	192	40	,	,	PUNCT
ejpam-726	192	41	{	{	PUNCT
ejpam-726	192	42	a	a	DET
ejpam-726	192	43	,	,	PUNCT
ejpam-726	192	44	b	b	NOUN
ejpam-726	192	45	,	,	PUNCT
ejpam-726	192	46	c	c	NOUN
ejpam-726	192	47	}	}	PUNCT
ejpam-726	192	48	,	,	PUNCT
ejpam-726	192	49	{	{	PUNCT
ejpam-726	192	50	a	a	DET
ejpam-726	192	51	,	,	PUNCT
ejpam-726	192	52	b	b	NOUN
ejpam-726	192	53	,	,	PUNCT
ejpam-726	192	54	d	d	NOUN
ejpam-726	192	55	}	}	PUNCT
ejpam-726	192	56	}	}	PUNCT
ejpam-726	192	57	.	.	PUNCT
ejpam-726	193	1	then	then	ADV
ejpam-726	193	2	{	{	PUNCT
ejpam-726	193	3	a	a	PRON
ejpam-726	193	4	,	,	PUNCT
ejpam-726	193	5	b	b	NOUN
ejpam-726	193	6	}	}	PUNCT
ejpam-726	193	7	and	and	CCONJ
ejpam-726	193	8	{	{	PUNCT
ejpam-726	193	9	b	b	NOUN
ejpam-726	193	10	,	,	PUNCT
ejpam-726	193	11	c	c	NOUN
ejpam-726	193	12	}	}	PUNCT
ejpam-726	193	13	are	be	AUX
ejpam-726	193	14	β∗g−closed	β∗g−close	VERB
ejpam-726	193	15	sets	set	NOUN
ejpam-726	193	16	,	,	PUNCT
ejpam-726	193	17	but	but	CCONJ
ejpam-726	193	18	{	{	PUNCT
ejpam-726	193	19	a	a	PRON
ejpam-726	193	20	,	,	PUNCT
ejpam-726	193	21	b	b	NOUN
ejpam-726	193	22	}	}	PUNCT
ejpam-726	193	23	⋂	⋂	PROPN
ejpam-726	193	24	{	{	PUNCT
ejpam-726	193	25	b	b	PROPN
ejpam-726	193	26	,	,	PUNCT
ejpam-726	193	27	c	c	NOUN
ejpam-726	193	28	}	}	PUNCT
ejpam-726	193	29	=	=	SYM
ejpam-726	193	30	{	{	PUNCT
ejpam-726	193	31	b	b	NOUN
ejpam-726	193	32	}	}	PUNCT
ejpam-726	193	33	is	be	AUX
ejpam-726	193	34	not	not	PART
ejpam-726	193	35	β∗g−closed	β∗g−close	VERB
ejpam-726	193	36	.	.	PUNCT
ejpam-726	194	1	theorem	theorem	VERB
ejpam-726	194	2	6	6	NUM
ejpam-726	194	3	.	.	PUNCT
ejpam-726	195	1	if	if	SCONJ
ejpam-726	195	2	a	a	PRON
ejpam-726	195	3	is	be	AUX
ejpam-726	195	4	a	a	DET
ejpam-726	195	5	β∗g−closed	β∗g−close	VERB
ejpam-726	195	6	set	set	NOUN
ejpam-726	195	7	of	of	ADP
ejpam-726	195	8	(	(	PUNCT
ejpam-726	195	9	x	x	PROPN
ejpam-726	195	10	,	,	PUNCT
ejpam-726	195	11	τ	τ	X
ejpam-726	195	12	)	)	PUNCT
ejpam-726	195	13	such	such	ADJ
ejpam-726	195	14	that	that	SCONJ
ejpam-726	195	15	a	a	DET
ejpam-726	195	16	⊂	⊂	X
ejpam-726	195	17	b	b	X
ejpam-726	195	18	⊂	⊂	PROPN
ejpam-726	195	19	cl(a	cl(a	X
ejpam-726	195	20	)	)	PUNCT
ejpam-726	195	21	,	,	PUNCT
ejpam-726	195	22	then	then	ADV
ejpam-726	195	23	b	b	PROPN
ejpam-726	195	24	is	be	AUX
ejpam-726	195	25	also	also	ADV
ejpam-726	195	26	a	a	DET
ejpam-726	195	27	β∗g−closed	β∗g−close	VERB
ejpam-726	195	28	set	set	NOUN
ejpam-726	195	29	of	of	ADP
ejpam-726	195	30	(	(	PUNCT
ejpam-726	195	31	x	x	PROPN
ejpam-726	195	32	,	,	PUNCT
ejpam-726	195	33	τ	τ	PROPN
ejpam-726	195	34	)	)	PUNCT
ejpam-726	195	35	.	.	PUNCT
ejpam-726	196	1	proof	proof	NOUN
ejpam-726	196	2	.	.	PUNCT
ejpam-726	197	1	let	let	VERB
ejpam-726	197	2	u	u	PRON
ejpam-726	197	3	be	be	AUX
ejpam-726	197	4	a	a	DET
ejpam-726	197	5	β∗-set	β∗-set	NOUN
ejpam-726	197	6	of	of	ADP
ejpam-726	197	7	(	(	PUNCT
ejpam-726	197	8	x	x	INTJ
ejpam-726	197	9	,	,	PUNCT
ejpam-726	197	10	τ	τ	X
ejpam-726	197	11	)	)	PUNCT
ejpam-726	197	12	such	such	ADJ
ejpam-726	197	13	that	that	SCONJ
ejpam-726	197	14	b	b	PROPN
ejpam-726	197	15	⊂	⊂	PROPN
ejpam-726	197	16	u	u	PROPN
ejpam-726	197	17	.	.	PUNCT
ejpam-726	198	1	then	then	ADV
ejpam-726	198	2	a	a	DET
ejpam-726	198	3	⊂	⊂	PROPN
ejpam-726	198	4	u	u	PROPN
ejpam-726	198	5	.	.	PUNCT
ejpam-726	199	1	since	since	SCONJ
ejpam-726	199	2	a	a	PRON
ejpam-726	199	3	is	be	AUX
ejpam-726	199	4	β∗g−closed	β∗g−close	VERB
ejpam-726	199	5	,	,	PUNCT
ejpam-726	199	6	we	we	PRON
ejpam-726	199	7	have	have	VERB
ejpam-726	199	8	cl(a	cl(a	PUNCT
ejpam-726	199	9	)	)	PUNCT
ejpam-726	200	1	⊂	⊂	PROPN
ejpam-726	200	2	u	u	PROPN
ejpam-726	200	3	.	.	PUNCT
ejpam-726	201	1	now	now	ADV
ejpam-726	201	2	cl(b	cl(b	NOUN
ejpam-726	201	3	)	)	PUNCT
ejpam-726	201	4	⊂	⊂	PROPN
ejpam-726	201	5	cl(cl(a	cl(cl(a	PROPN
ejpam-726	201	6	)	)	PUNCT
ejpam-726	201	7	)	)	PUNCT
ejpam-726	201	8	=	=	SYM
ejpam-726	201	9	cl(a	cl(a	X
ejpam-726	201	10	)	)	PUNCT
ejpam-726	201	11	⊂	⊂	PROPN
ejpam-726	201	12	u	u	PROPN
ejpam-726	201	13	.	.	PUNCT
ejpam-726	202	1	therefore	therefore	ADV
ejpam-726	202	2	,	,	PUNCT
ejpam-726	202	3	b	b	PROPN
ejpam-726	202	4	is	be	AUX
ejpam-726	202	5	also	also	ADV
ejpam-726	202	6	a	a	DET
ejpam-726	202	7	β∗g−closed	β∗g−close	VERB
ejpam-726	202	8	set	set	NOUN
ejpam-726	202	9	of	of	ADP
ejpam-726	202	10	(	(	PUNCT
ejpam-726	202	11	x	x	PROPN
ejpam-726	202	12	,	,	PUNCT
ejpam-726	202	13	τ	τ	PROPN
ejpam-726	202	14	)	)	PUNCT
ejpam-726	202	15	.	.	PUNCT
ejpam-726	203	1	theorem	theorem	VERB
ejpam-726	203	2	7	7	NUM
ejpam-726	203	3	.	.	X
ejpam-726	203	4	for	for	ADP
ejpam-726	203	5	any	any	DET
ejpam-726	203	6	topological	topological	ADJ
ejpam-726	203	7	space	space	NOUN
ejpam-726	203	8	(	(	PUNCT
ejpam-726	203	9	x	x	X
ejpam-726	203	10	,	,	PUNCT
ejpam-726	203	11	τ	τ	PROPN
ejpam-726	203	12	)	)	PUNCT
ejpam-726	203	13	,	,	PUNCT
ejpam-726	203	14	every	every	DET
ejpam-726	203	15	singleton	singleton	NOUN
ejpam-726	203	16	{	{	PUNCT
ejpam-726	203	17	x	x	NOUN
ejpam-726	203	18	}	}	PUNCT
ejpam-726	203	19	of	of	ADP
ejpam-726	203	20	x	x	PRON
ejpam-726	203	21	is	be	AUX
ejpam-726	203	22	a	a	DET
ejpam-726	203	23	β∗-set	β∗-set	NOUN
ejpam-726	203	24	.	.	PUNCT
ejpam-726	204	1	proof	proof	NOUN
ejpam-726	204	2	.	.	PUNCT
ejpam-726	205	1	let	let	VERB
ejpam-726	205	2	x	x	SYM
ejpam-726	205	3	∈	∈	PROPN
ejpam-726	205	4	x	x	X
ejpam-726	205	5	,	,	PUNCT
ejpam-726	205	6	if	if	SCONJ
ejpam-726	205	7	{	{	PUNCT
ejpam-726	205	8	x	x	NOUN
ejpam-726	205	9	}	}	PUNCT
ejpam-726	205	10	∈	∈	PROPN
ejpam-726	205	11	τ	τ	NOUN
ejpam-726	205	12	,	,	PUNCT
ejpam-726	205	13	then	then	ADV
ejpam-726	205	14	{	{	PUNCT
ejpam-726	205	15	x	x	X
ejpam-726	205	16	}	}	PUNCT
ejpam-726	205	17	is	be	AUX
ejpam-726	205	18	a	a	DET
ejpam-726	205	19	β∗-set	β∗-set	NOUN
ejpam-726	206	1	[	[	X
ejpam-726	206	2	3	3	NUM
ejpam-726	206	3	]	]	PUNCT
ejpam-726	206	4	.	.	PUNCT
ejpam-726	207	1	if	if	SCONJ
ejpam-726	207	2	{	{	PUNCT
ejpam-726	207	3	x	x	NOUN
ejpam-726	207	4	}	}	PUNCT
ejpam-726	207	5	/∈	/∈	PUNCT
ejpam-726	208	1	τ	τ	PROPN
ejpam-726	208	2	,	,	PUNCT
ejpam-726	208	3	then	then	ADV
ejpam-726	208	4	int({x	int({x	PROPN
ejpam-726	208	5	}	}	PUNCT
ejpam-726	208	6	)	)	PUNCT
ejpam-726	209	1	=	=	SYM
ejpam-726	209	2	;	;	PUNCT
ejpam-726	209	3	=	=	PUNCT
ejpam-726	209	4	cl(int({x	cl(int({x	NOUN
ejpam-726	209	5	}	}	PUNCT
ejpam-726	209	6	)	)	PUNCT
ejpam-726	209	7	)	)	PUNCT
ejpam-726	209	8	,	,	PUNCT
ejpam-726	209	9	so	so	CCONJ
ejpam-726	209	10	{	{	PUNCT
ejpam-726	209	11	x	x	X
ejpam-726	209	12	}	}	PUNCT
ejpam-726	209	13	is	be	AUX
ejpam-726	209	14	a	a	DET
ejpam-726	209	15	β∗-set	β∗-set	NOUN
ejpam-726	209	16	.	.	PUNCT
ejpam-726	210	1	corollary	corollary	ADJ
ejpam-726	210	2	1	1	NUM
ejpam-726	210	3	.	.	PUNCT
ejpam-726	211	1	for	for	SCONJ
ejpam-726	211	2	every	every	DET
ejpam-726	211	3	x	x	SYM
ejpam-726	211	4	∈	∈	PROPN
ejpam-726	211	5	x	x	X
ejpam-726	211	6	,	,	PUNCT
ejpam-726	211	7	{	{	PUNCT
ejpam-726	211	8	x	x	NOUN
ejpam-726	211	9	}	}	PUNCT
ejpam-726	211	10	is	be	AUX
ejpam-726	211	11	a	a	DET
ejpam-726	211	12	β∗g−closed	β∗g−close	VERB
ejpam-726	211	13	set	set	NOUN
ejpam-726	211	14	of	of	ADP
ejpam-726	211	15	(	(	PUNCT
ejpam-726	211	16	x	x	PROPN
ejpam-726	211	17	,	,	PUNCT
ejpam-726	211	18	τ	τ	X
ejpam-726	211	19	)	)	PUNCT
ejpam-726	212	1	if	if	SCONJ
ejpam-726	212	2	and	and	CCONJ
ejpam-726	212	3	only	only	ADV
ejpam-726	212	4	if	if	SCONJ
ejpam-726	212	5	{	{	PUNCT
ejpam-726	212	6	x	x	NOUN
ejpam-726	212	7	}	}	PUNCT
ejpam-726	212	8	is	be	AUX
ejpam-726	212	9	a	a	DET
ejpam-726	212	10	closed	closed	ADJ
ejpam-726	212	11	set	set	NOUN
ejpam-726	212	12	of	of	ADP
ejpam-726	212	13	x.	x.	NOUN
ejpam-726	212	14	proof	proof	NOUN
ejpam-726	212	15	.	.	PUNCT
ejpam-726	213	1	necessity	necessity	NOUN
ejpam-726	213	2	:	:	PUNCT
ejpam-726	213	3	let	let	VERB
ejpam-726	213	4	{	{	PUNCT
ejpam-726	213	5	x	x	VERB
ejpam-726	213	6	}	}	PUNCT
ejpam-726	213	7	be	be	AUX
ejpam-726	213	8	β∗g−closed	β∗g−close	VERB
ejpam-726	213	9	.	.	PUNCT
ejpam-726	214	1	then	then	ADV
ejpam-726	214	2	,	,	PUNCT
ejpam-726	214	3	by	by	ADP
ejpam-726	214	4	theorem	theorem	VERB
ejpam-726	214	5	7	7	NUM
ejpam-726	214	6	{	{	PUNCT
ejpam-726	214	7	x	x	NOUN
ejpam-726	214	8	}	}	PUNCT
ejpam-726	214	9	is	be	AUX
ejpam-726	214	10	closed	closed	ADJ
ejpam-726	214	11	.	.	PUNCT
ejpam-726	215	1	sufficiency	sufficiency	NOUN
ejpam-726	215	2	:	:	PUNCT
ejpam-726	215	3	let	let	VERB
ejpam-726	215	4	{	{	PUNCT
ejpam-726	215	5	x	x	AUX
ejpam-726	215	6	}	}	PUNCT
ejpam-726	215	7	be	be	AUX
ejpam-726	215	8	a	a	DET
ejpam-726	215	9	closed	closed	ADJ
ejpam-726	215	10	set	set	NOUN
ejpam-726	215	11	.	.	PUNCT
ejpam-726	216	1	by	by	ADP
ejpam-726	216	2	theorem	theorem	ADJ
ejpam-726	216	3	3	3	NUM
ejpam-726	216	4	{	{	PUNCT
ejpam-726	216	5	x	x	NOUN
ejpam-726	216	6	}	}	PUNCT
ejpam-726	216	7	is	be	AUX
ejpam-726	216	8	β∗g−closed	β∗g−close	VERB
ejpam-726	216	9	.	.	PUNCT
ejpam-726	217	1	theorem	theorem	NOUN
ejpam-726	217	2	8	8	NUM
ejpam-726	217	3	.	.	PUNCT
ejpam-726	218	1	let	let	VERB
ejpam-726	218	2	a	a	PRON
ejpam-726	218	3	be	be	AUX
ejpam-726	218	4	β∗g−closed	β∗g−close	VERB
ejpam-726	218	5	in	in	ADP
ejpam-726	218	6	(	(	PUNCT
ejpam-726	218	7	x	x	INTJ
ejpam-726	218	8	,	,	PUNCT
ejpam-726	218	9	τ	τ	PROPN
ejpam-726	218	10	)	)	PUNCT
ejpam-726	218	11	.	.	PUNCT
ejpam-726	219	1	then	then	ADV
ejpam-726	219	2	cl(a)−	cl(a)−	VERB
ejpam-726	219	3	a	a	PRON
ejpam-726	219	4	does	do	AUX
ejpam-726	219	5	not	not	PART
ejpam-726	219	6	contain	contain	VERB
ejpam-726	219	7	any	any	DET
ejpam-726	219	8	non	non	ADJ
ejpam-726	219	9	-	-	ADJ
ejpam-726	219	10	empty	empty	ADJ
ejpam-726	219	11	complement	complement	NOUN
ejpam-726	219	12	of	of	ADP
ejpam-726	219	13	a	a	DET
ejpam-726	219	14	β∗-set	β∗-set	NOUN
ejpam-726	219	15	.	.	PUNCT
ejpam-726	220	1	proof	proof	NOUN
ejpam-726	220	2	.	.	PUNCT
ejpam-726	221	1	let	let	VERB
ejpam-726	221	2	a	a	DET
ejpam-726	221	3	be	be	AUX
ejpam-726	221	4	a	a	DET
ejpam-726	221	5	β∗g−closed	β∗g−close	VERB
ejpam-726	221	6	set	set	NOUN
ejpam-726	221	7	.	.	PUNCT
ejpam-726	222	1	suppose	suppose	VERB
ejpam-726	222	2	that	that	SCONJ
ejpam-726	222	3	f	f	PROPN
ejpam-726	222	4	is	be	AUX
ejpam-726	222	5	the	the	DET
ejpam-726	222	6	complement	complement	NOUN
ejpam-726	222	7	of	of	ADP
ejpam-726	222	8	a	a	DET
ejpam-726	222	9	β∗-set	β∗-set	NOUN
ejpam-726	222	10	and	and	CCONJ
ejpam-726	222	11	f	f	PROPN
ejpam-726	222	12	⊂	⊂	AUX
ejpam-726	222	13	cl(a)−	cl(a)−	VERB
ejpam-726	222	14	a.	a.	NOUN
ejpam-726	222	15	since	since	SCONJ
ejpam-726	222	16	f	f	PROPN
ejpam-726	222	17	⊂	⊂	PROPN
ejpam-726	222	18	cl(a)−	cl(a)−	VERB
ejpam-726	222	19	a	a	DET
ejpam-726	222	20	⊂	⊂	PROPN
ejpam-726	222	21	x	x	PUNCT
ejpam-726	222	22	−	−	NOUN
ejpam-726	222	23	a	a	X
ejpam-726	222	24	,	,	PUNCT
ejpam-726	222	25	a	a	DET
ejpam-726	222	26	⊂	⊂	X
ejpam-726	222	27	x	x	PUNCT
ejpam-726	222	28	−	−	PROPN
ejpam-726	222	29	f	f	PROPN
ejpam-726	222	30	and	and	CCONJ
ejpam-726	222	31	x	x	SYM
ejpam-726	222	32	−	−	PROPN
ejpam-726	222	33	f	f	PROPN
ejpam-726	222	34	is	be	AUX
ejpam-726	222	35	a	a	DET
ejpam-726	222	36	β∗-set	β∗-set	NOUN
ejpam-726	222	37	.	.	PUNCT
ejpam-726	223	1	therefore	therefore	ADV
ejpam-726	223	2	,	,	PUNCT
ejpam-726	223	3	cl(a)⊂	cl(a)⊂	PROPN
ejpam-726	223	4	x	x	X
ejpam-726	223	5	−	−	PROPN
ejpam-726	223	6	f	f	PROPN
ejpam-726	223	7	and	and	CCONJ
ejpam-726	223	8	f	f	PROPN
ejpam-726	223	9	⊂	⊂	PROPN
ejpam-726	223	10	x	x	PUNCT
ejpam-726	223	11	−	−	NOUN
ejpam-726	223	12	cl(a	cl(a	NUM
ejpam-726	223	13	)	)	PUNCT
ejpam-726	223	14	.	.	PUNCT
ejpam-726	224	1	however	however	ADV
ejpam-726	224	2	,	,	PUNCT
ejpam-726	224	3	since	since	SCONJ
ejpam-726	224	4	f	f	PROPN
ejpam-726	224	5	⊂	⊂	PROPN
ejpam-726	224	6	cl(a)−a	cl(a)−a	PROPN
ejpam-726	224	7	,	,	PUNCT
ejpam-726	224	8	f	f	PROPN
ejpam-726	224	9	=	=	PUNCT
ejpam-726	224	10	;	;	PUNCT
ejpam-726	224	11	.	.	PUNCT
ejpam-726	224	12	a.	a.	PROPN
ejpam-726	224	13	açıkgöz	açıkgöz	PROPN
ejpam-726	224	14	/	/	SYM
ejpam-726	224	15	eur	eur	PROPN
ejpam-726	224	16	.	.	PUNCT
ejpam-726	225	1	j.	j.	PROPN
ejpam-726	225	2	pure	pure	PROPN
ejpam-726	225	3	appl	appl	PROPN
ejpam-726	225	4	.	.	PROPN
ejpam-726	225	5	math	math	PROPN
ejpam-726	225	6	,	,	PUNCT
ejpam-726	225	7	4	4	NUM
ejpam-726	225	8	(	(	PUNCT
ejpam-726	225	9	2011	2011	NUM
ejpam-726	225	10	)	)	PUNCT
ejpam-726	225	11	,	,	PUNCT
ejpam-726	225	12	20	20	NUM
ejpam-726	225	13	-	-	SYM
ejpam-726	225	14	33	33	NUM
ejpam-726	225	15	26	26	NUM
ejpam-726	225	16	3	3	NUM
ejpam-726	225	17	.	.	PUNCT
ejpam-726	225	18	β∗g−closures	β∗g−closures	PROPN
ejpam-726	225	19	in	in	ADP
ejpam-726	225	20	this	this	DET
ejpam-726	225	21	section	section	NOUN
ejpam-726	225	22	,	,	PUNCT
ejpam-726	225	23	the	the	DET
ejpam-726	225	24	notion	notion	NOUN
ejpam-726	225	25	of	of	ADP
ejpam-726	225	26	the	the	PRON
ejpam-726	225	27	β∗g−closure	β∗g−closure	PUNCT
ejpam-726	225	28	is	be	AUX
ejpam-726	225	29	defined	define	VERB
ejpam-726	225	30	and	and	CCONJ
ejpam-726	225	31	some	some	PRON
ejpam-726	225	32	of	of	ADP
ejpam-726	225	33	its	its	PRON
ejpam-726	225	34	basic	basic	ADJ
ejpam-726	225	35	properties	property	NOUN
ejpam-726	225	36	are	be	AUX
ejpam-726	225	37	studied	study	VERB
ejpam-726	225	38	.	.	PUNCT
ejpam-726	226	1	definition	definition	NOUN
ejpam-726	226	2	3	3	NUM
ejpam-726	226	3	.	.	PUNCT
ejpam-726	227	1	for	for	ADP
ejpam-726	227	2	a	a	DET
ejpam-726	227	3	subset	subset	NOUN
ejpam-726	227	4	a	a	PRON
ejpam-726	227	5	of	of	ADP
ejpam-726	227	6	(	(	PUNCT
ejpam-726	227	7	x	x	PROPN
ejpam-726	227	8	,	,	PUNCT
ejpam-726	227	9	τ	τ	PROPN
ejpam-726	227	10	)	)	PUNCT
ejpam-726	227	11	,	,	PUNCT
ejpam-726	227	12	we	we	PRON
ejpam-726	227	13	define	define	VERB
ejpam-726	227	14	the	the	DET
ejpam-726	227	15	β∗g−closure	β∗g−closure	NUM
ejpam-726	227	16	of	of	ADP
ejpam-726	227	17	as	as	SCONJ
ejpam-726	227	18	follows	follow	VERB
ejpam-726	227	19	:	:	PUNCT
ejpam-726	227	20	β∗g	β∗g	PUNCT
ejpam-726	227	21	−	−	NOUN
ejpam-726	227	22	cl(a	cl(a	PUNCT
ejpam-726	227	23	)	)	PUNCT
ejpam-726	227	24	=	=	SYM
ejpam-726	228	1	⋂	⋂	PROPN
ejpam-726	228	2	f	f	PROPN
ejpam-726	228	3	is	be	AUX
ejpam-726	228	4	β∗g−closed	β∗g−close	VERB
ejpam-726	228	5	in	in	ADP
ejpam-726	228	6	x	x	SYM
ejpam-726	228	7	,	,	PUNCT
ejpam-726	228	8	a⊂	a⊂	X
ejpam-726	228	9	f	f	NOUN
ejpam-726	228	10	}	}	PUNCT
ejpam-726	228	11	.	.	PUNCT
ejpam-726	229	1	lemma	lemma	PROPN
ejpam-726	229	2	1	1	X
ejpam-726	229	3	.	.	PUNCT
ejpam-726	230	1	let	let	VERB
ejpam-726	230	2	a	a	DET
ejpam-726	230	3	be	be	AUX
ejpam-726	230	4	a	a	DET
ejpam-726	230	5	subset	subset	NOUN
ejpam-726	230	6	of	of	ADP
ejpam-726	230	7	(	(	PUNCT
ejpam-726	230	8	x	x	PROPN
ejpam-726	230	9	,	,	PUNCT
ejpam-726	230	10	τ	τ	PROPN
ejpam-726	230	11	)	)	PUNCT
ejpam-726	230	12	and	and	CCONJ
ejpam-726	230	13	x	x	PUNCT
ejpam-726	230	14	∈	∈	NOUN
ejpam-726	230	15	x	x	X
ejpam-726	230	16	.	.	PUNCT
ejpam-726	231	1	then	then	ADV
ejpam-726	231	2	x	x	PUNCT
ejpam-726	231	3	∈	∈	PROPN
ejpam-726	231	4	β∗g	β∗g	NUM
ejpam-726	231	5	−	−	NOUN
ejpam-726	231	6	cl(a	cl(a	PUNCT
ejpam-726	231	7	)	)	PUNCT
ejpam-726	231	8	if	if	SCONJ
ejpam-726	231	9	and	and	CCONJ
ejpam-726	231	10	only	only	ADV
ejpam-726	231	11	if	if	SCONJ
ejpam-726	231	12	v	v	PRON
ejpam-726	231	13	∩a	∩a	PROPN
ejpam-726	231	14	6=	6=	PROPN
ejpam-726	231	15	;	;	PUNCT
ejpam-726	231	16	for	for	ADP
ejpam-726	231	17	every	every	DET
ejpam-726	231	18	β∗g−open	β∗g−open	NUM
ejpam-726	231	19	set	set	VERB
ejpam-726	231	20	v	v	NOUN
ejpam-726	231	21	containing	contain	VERB
ejpam-726	231	22	x.	x.	NOUN
ejpam-726	231	23	proof	proof	NOUN
ejpam-726	231	24	.	.	PUNCT
ejpam-726	232	1	suppose	suppose	VERB
ejpam-726	232	2	that	that	SCONJ
ejpam-726	232	3	there	there	PRON
ejpam-726	232	4	exists	exist	VERB
ejpam-726	232	5	a	a	DET
ejpam-726	232	6	β∗g−open	β∗g−open	NOUN
ejpam-726	232	7	set	set	VERB
ejpam-726	232	8	v	v	NOUN
ejpam-726	232	9	containing	contain	VERB
ejpam-726	232	10	x	x	PUNCT
ejpam-726	232	11	such	such	ADJ
ejpam-726	232	12	that	that	PRON
ejpam-726	232	13	v	v	ADP
ejpam-726	232	14	∩	∩	NOUN
ejpam-726	232	15	a	a	NOUN
ejpam-726	232	16	=	=	X
ejpam-726	232	17	;	;	PUNCT
ejpam-726	232	18	.	.	PUNCT
ejpam-726	233	1	since	since	SCONJ
ejpam-726	233	2	a⊂	a⊂	NOUN
ejpam-726	233	3	x	x	PUNCT
ejpam-726	233	4	−	−	PROPN
ejpam-726	233	5	v	v	NUM
ejpam-726	233	6	,	,	PUNCT
ejpam-726	233	7	β∗g	β∗g	NUM
ejpam-726	233	8	−	−	NOUN
ejpam-726	233	9	cl(a	cl(a	NUM
ejpam-726	233	10	)	)	PUNCT
ejpam-726	233	11	⊂	⊂	X
ejpam-726	233	12	x	x	PUNCT
ejpam-726	234	1	−	−	NOUN
ejpam-726	234	2	v	v	X
ejpam-726	234	3	and	and	CCONJ
ejpam-726	234	4	then	then	ADV
ejpam-726	234	5	x	x	X
ejpam-726	234	6	/∈	/∈	SYM
ejpam-726	234	7	β∗g	β∗g	NUM
ejpam-726	234	8	−	−	NOUN
ejpam-726	234	9	cl(a	cl(a	NUM
ejpam-726	234	10	)	)	PUNCT
ejpam-726	234	11	.	.	PUNCT
ejpam-726	235	1	conversely	conversely	ADV
ejpam-726	235	2	,	,	PUNCT
ejpam-726	235	3	suppose	suppose	VERB
ejpam-726	235	4	that	that	SCONJ
ejpam-726	235	5	x	x	SYM
ejpam-726	235	6	/∈	/∈	SYM
ejpam-726	235	7	β∗g	β∗g	NUM
ejpam-726	235	8	−	−	NOUN
ejpam-726	235	9	cl(a	cl(a	NUM
ejpam-726	235	10	)	)	PUNCT
ejpam-726	235	11	.	.	PUNCT
ejpam-726	236	1	then	then	ADV
ejpam-726	236	2	there	there	PRON
ejpam-726	236	3	exists	exist	VERB
ejpam-726	236	4	a	a	DET
ejpam-726	236	5	β∗g−closed	β∗g−close	VERB
ejpam-726	236	6	set	set	NOUN
ejpam-726	236	7	f	f	PROPN
ejpam-726	236	8	containing	contain	VERB
ejpam-726	236	9	a	a	DET
ejpam-726	236	10	such	such	ADJ
ejpam-726	236	11	that	that	PRON
ejpam-726	236	12	x	x	PROPN
ejpam-726	236	13	/∈	/∈	PROPN
ejpam-726	237	1	f	f	PROPN
ejpam-726	237	2	.	.	PUNCT
ejpam-726	238	1	since	since	SCONJ
ejpam-726	238	2	x	x	PROPN
ejpam-726	238	3	∈	∈	PROPN
ejpam-726	238	4	x	x	X
ejpam-726	238	5	−	−	PROPN
ejpam-726	238	6	f	f	PROPN
ejpam-726	238	7	and	and	CCONJ
ejpam-726	238	8	x	x	SYM
ejpam-726	238	9	−	−	PROPN
ejpam-726	238	10	f	f	PROPN
ejpam-726	238	11	is	be	AUX
ejpam-726	238	12	β∗g−open	β∗g−open	ADJ
ejpam-726	238	13	,	,	PUNCT
ejpam-726	238	14	(	(	PUNCT
ejpam-726	238	15	x	x	SYM
ejpam-726	238	16	−	−	PROPN
ejpam-726	238	17	f)∩	f)∩	NOUN
ejpam-726	238	18	a=	a=	VERB
ejpam-726	238	19	;	;	PUNCT
ejpam-726	238	20	.	.	PUNCT
ejpam-726	239	1	lemma	lemma	PROPN
ejpam-726	239	2	2	2	X
ejpam-726	239	3	.	.	PUNCT
ejpam-726	240	1	let	let	VERB
ejpam-726	240	2	a	a	PRON
ejpam-726	240	3	and	and	CCONJ
ejpam-726	240	4	b	b	NOUN
ejpam-726	240	5	be	be	AUX
ejpam-726	240	6	subsets	subset	NOUN
ejpam-726	240	7	of	of	ADP
ejpam-726	240	8	(	(	PUNCT
ejpam-726	240	9	x	x	PROPN
ejpam-726	240	10	,	,	PUNCT
ejpam-726	240	11	τ	τ	PROPN
ejpam-726	240	12	)	)	PUNCT
ejpam-726	240	13	.	.	PUNCT
ejpam-726	241	1	then	then	ADV
ejpam-726	241	2	we	we	PRON
ejpam-726	241	3	have	have	VERB
ejpam-726	241	4	(	(	PUNCT
ejpam-726	241	5	a	a	X
ejpam-726	241	6	)	)	PUNCT
ejpam-726	241	7	β∗g	β∗g	NUM
ejpam-726	241	8	−	−	NOUN
ejpam-726	241	9	cl	cl	NOUN
ejpam-726	241	10	(;	(;	PUNCT
ejpam-726	241	11	)	)	PUNCT
ejpam-726	241	12	=	=	SYM
ejpam-726	241	13	;	;	PUNCT
ejpam-726	241	14	and	and	CCONJ
ejpam-726	241	15	β∗g	β∗g	NUM
ejpam-726	241	16	−	−	NOUN
ejpam-726	241	17	cl(x	cl(x	NUM
ejpam-726	241	18	)	)	PUNCT
ejpam-726	242	1	=	=	PUNCT
ejpam-726	243	1	x	x	X
ejpam-726	243	2	.	.	PUNCT
ejpam-726	244	1	(	(	PUNCT
ejpam-726	244	2	b	b	X
ejpam-726	244	3	)	)	PUNCT
ejpam-726	244	4	if	if	SCONJ
ejpam-726	244	5	a⊂	a⊂	NOUN
ejpam-726	244	6	b	b	NOUN
ejpam-726	244	7	,	,	PUNCT
ejpam-726	244	8	then	then	ADV
ejpam-726	244	9	β∗g	β∗g	NUM
ejpam-726	244	10	−	−	PROPN
ejpam-726	244	11	cl(a)⊂	cl(a)⊂	PROPN
ejpam-726	244	12	β∗g	β∗g	NUM
ejpam-726	244	13	−	−	PROPN
ejpam-726	244	14	cl(b	cl(b	NOUN
ejpam-726	244	15	)	)	PUNCT
ejpam-726	244	16	.	.	PUNCT
ejpam-726	245	1	(	(	PUNCT
ejpam-726	245	2	c	c	X
ejpam-726	245	3	)	)	PUNCT
ejpam-726	245	4	β∗g	β∗g	NUM
ejpam-726	245	5	−	−	NOUN
ejpam-726	245	6	cl(a	cl(a	NUM
ejpam-726	245	7	)	)	PUNCT
ejpam-726	245	8	=	=	SYM
ejpam-726	245	9	β∗g	β∗g	PUNCT
ejpam-726	246	1	−	−	NOUN
ejpam-726	246	2	cl(β∗g	cl(β∗g	NOUN
ejpam-726	246	3	−	−	NOUN
ejpam-726	246	4	cl(a	cl(a	NUM
ejpam-726	246	5	)	)	PUNCT
ejpam-726	246	6	)	)	PUNCT
ejpam-726	246	7	.	.	PUNCT
ejpam-726	247	1	(	(	PUNCT
ejpam-726	247	2	d	d	X
ejpam-726	247	3	)	)	PUNCT
ejpam-726	247	4	β∗g	β∗g	NUM
ejpam-726	247	5	−	−	PROPN
ejpam-726	247	6	cl(a∪	cl(a∪	PROPN
ejpam-726	247	7	b	b	NOUN
ejpam-726	247	8	)	)	PUNCT
ejpam-726	247	9	=	=	SYM
ejpam-726	247	10	β∗g	β∗g	PUNCT
ejpam-726	247	11	−	−	NOUN
ejpam-726	247	12	cl(a)∪	cl(a)∪	NOUN
ejpam-726	247	13	β∗g	β∗g	NUM
ejpam-726	247	14	−	−	NOUN
ejpam-726	247	15	cl(b	cl(b	NOUN
ejpam-726	247	16	)	)	PUNCT
ejpam-726	247	17	.	.	PUNCT
ejpam-726	248	1	(	(	PUNCT
ejpam-726	248	2	e	e	X
ejpam-726	248	3	)	)	PUNCT
ejpam-726	248	4	β∗g	β∗g	PUNCT
ejpam-726	248	5	−	−	PROPN
ejpam-726	248	6	cl(a∩	cl(a∩	PROPN
ejpam-726	248	7	b	b	X
ejpam-726	248	8	)	)	PUNCT
ejpam-726	248	9	⊂	⊂	PROPN
ejpam-726	248	10	β∗g	β∗g	PUNCT
ejpam-726	248	11	−	−	X
ejpam-726	248	12	cl(a)∩	cl(a)∩	SYM
ejpam-726	248	13	β∗g	β∗g	NUM
ejpam-726	248	14	−	−	NOUN
ejpam-726	248	15	cl(b	cl(b	NOUN
ejpam-726	248	16	)	)	PUNCT
ejpam-726	248	17	.	.	PUNCT
ejpam-726	249	1	proof	proof	NOUN
ejpam-726	249	2	.	.	PUNCT
ejpam-726	250	1	straightforward	straightforward	ADJ
ejpam-726	250	2	.	.	PUNCT
ejpam-726	251	1	remark	remark	NOUN
ejpam-726	251	2	9	9	NUM
ejpam-726	251	3	.	.	PUNCT
ejpam-726	252	1	(	(	PUNCT
ejpam-726	252	2	a	a	X
ejpam-726	252	3	)	)	PUNCT
ejpam-726	252	4	if	if	SCONJ
ejpam-726	252	5	a	a	PRON
ejpam-726	252	6	is	be	AUX
ejpam-726	252	7	β∗g−closed	β∗g−close	VERB
ejpam-726	252	8	in	in	ADP
ejpam-726	252	9	(	(	PUNCT
ejpam-726	252	10	x	x	INTJ
ejpam-726	252	11	,	,	PUNCT
ejpam-726	252	12	τ	τ	PROPN
ejpam-726	252	13	)	)	PUNCT
ejpam-726	252	14	,	,	PUNCT
ejpam-726	252	15	then	then	ADV
ejpam-726	252	16	β∗g	β∗g	NUM
ejpam-726	252	17	−	−	NOUN
ejpam-726	252	18	cl(a	cl(a	NUM
ejpam-726	252	19	)	)	PUNCT
ejpam-726	252	20	=	=	SYM
ejpam-726	253	1	a.	a.	NOUN
ejpam-726	254	1	but	but	CCONJ
ejpam-726	254	2	the	the	DET
ejpam-726	254	3	converse	converse	NOUN
ejpam-726	254	4	is	be	AUX
ejpam-726	254	5	not	not	PART
ejpam-726	254	6	true	true	ADJ
ejpam-726	254	7	as	as	SCONJ
ejpam-726	254	8	seen	see	VERB
ejpam-726	254	9	by	by	ADP
ejpam-726	254	10	the	the	DET
ejpam-726	254	11	following	follow	VERB
ejpam-726	254	12	example	example	NOUN
ejpam-726	254	13	:	:	PUNCT
ejpam-726	254	14	(	(	PUNCT
ejpam-726	254	15	b	b	X
ejpam-726	254	16	)	)	PUNCT
ejpam-726	254	17	in	in	ADP
ejpam-726	254	18	general	general	ADJ
ejpam-726	254	19	,	,	PUNCT
ejpam-726	254	20	β∗g	β∗g	NUM
ejpam-726	254	21	−	−	NOUN
ejpam-726	254	22	cl(a)∩	cl(a)∩	SYM
ejpam-726	254	23	β∗g	β∗g	NUM
ejpam-726	254	24	−	−	NOUN
ejpam-726	254	25	cl(b	cl(b	NOUN
ejpam-726	254	26	)	)	PUNCT
ejpam-726	254	27	6⊂	6⊂	NUM
ejpam-726	254	28	β∗g	β∗g	PUNCT
ejpam-726	254	29	−	−	PROPN
ejpam-726	254	30	cl(a∩	cl(a∩	PROPN
ejpam-726	254	31	b	b	PROPN
ejpam-726	254	32	)	)	PUNCT
ejpam-726	254	33	.	.	PUNCT
ejpam-726	255	1	for	for	ADP
ejpam-726	255	2	example	example	NOUN
ejpam-726	255	3	,	,	PUNCT
ejpam-726	255	4	example	example	NOUN
ejpam-726	255	5	10	10	NUM
ejpam-726	255	6	.	.	PUNCT
ejpam-726	256	1	let	let	VERB
ejpam-726	256	2	x	x	PUNCT
ejpam-726	256	3	=	=	PRON
ejpam-726	256	4	{	{	PUNCT
ejpam-726	256	5	a	a	PRON
ejpam-726	256	6	,	,	PUNCT
ejpam-726	256	7	b	b	NOUN
ejpam-726	256	8	,	,	PUNCT
ejpam-726	256	9	c	c	NOUN
ejpam-726	256	10	,	,	PUNCT
ejpam-726	256	11	d	d	NOUN
ejpam-726	256	12	}	}	PUNCT
ejpam-726	256	13	and	and	CCONJ
ejpam-726	256	14	τ	τ	PROPN
ejpam-726	256	15	=	=	PUNCT
ejpam-726	256	16	{	{	PUNCT
ejpam-726	256	17	x	x	X
ejpam-726	256	18	,	,	PUNCT
ejpam-726	256	19	;	;	PUNCT
ejpam-726	256	20	,	,	PUNCT
ejpam-726	256	21	{	{	PUNCT
ejpam-726	256	22	b	b	NOUN
ejpam-726	256	23	}	}	PUNCT
ejpam-726	256	24	,	,	PUNCT
ejpam-726	256	25	{	{	PUNCT
ejpam-726	256	26	c	c	X
ejpam-726	256	27	}	}	PUNCT
ejpam-726	256	28	,	,	PUNCT
ejpam-726	256	29	{	{	PUNCT
ejpam-726	256	30	a	a	DET
ejpam-726	256	31	,	,	PUNCT
ejpam-726	256	32	b	b	NOUN
ejpam-726	256	33	}	}	PUNCT
ejpam-726	256	34	,	,	PUNCT
ejpam-726	256	35	{	{	PUNCT
ejpam-726	256	36	b	b	X
ejpam-726	256	37	,	,	PUNCT
ejpam-726	256	38	c	c	NOUN
ejpam-726	256	39	}	}	PUNCT
ejpam-726	256	40	,	,	PUNCT
ejpam-726	256	41	{	{	PUNCT
ejpam-726	256	42	a	a	DET
ejpam-726	256	43	,	,	PUNCT
ejpam-726	256	44	b	b	NOUN
ejpam-726	256	45	,	,	PUNCT
ejpam-726	256	46	c	c	NOUN
ejpam-726	256	47	}	}	PUNCT
ejpam-726	256	48	,	,	PUNCT
ejpam-726	256	49	{	{	PUNCT
ejpam-726	256	50	a	a	DET
ejpam-726	256	51	,	,	PUNCT
ejpam-726	256	52	b	b	NOUN
ejpam-726	256	53	,	,	PUNCT
ejpam-726	256	54	d	d	NOUN
ejpam-726	256	55	}	}	PUNCT
ejpam-726	256	56	}	}	PUNCT
ejpam-726	256	57	.	.	PUNCT
ejpam-726	257	1	let	let	VERB
ejpam-726	257	2	a=	a=	VERB
ejpam-726	257	3	{	{	PUNCT
ejpam-726	257	4	b	b	NOUN
ejpam-726	257	5	}	}	PUNCT
ejpam-726	257	6	then	then	ADV
ejpam-726	257	7	β∗g	β∗g	PUNCT
ejpam-726	257	8	−	−	NOUN
ejpam-726	257	9	cl(a	cl(a	NUM
ejpam-726	257	10	)	)	PUNCT
ejpam-726	257	11	=	=	SYM
ejpam-726	257	12	β∗g	β∗g	PUNCT
ejpam-726	257	13	−	−	NOUN
ejpam-726	257	14	cl({b	cl({b	NOUN
ejpam-726	257	15	}	}	PUNCT
ejpam-726	257	16	)	)	PUNCT
ejpam-726	258	1	=	=	PRON
ejpam-726	258	2	{	{	PUNCT
ejpam-726	258	3	b	b	NOUN
ejpam-726	258	4	}	}	PUNCT
ejpam-726	258	5	but	but	CCONJ
ejpam-726	258	6	{	{	PUNCT
ejpam-726	258	7	b	b	X
ejpam-726	258	8	}	}	PUNCT
ejpam-726	258	9	is	be	AUX
ejpam-726	258	10	not	not	PART
ejpam-726	258	11	β∗g−closed	β∗g−close	VERB
ejpam-726	258	12	set	set	NOUN
ejpam-726	258	13	.	.	PUNCT
ejpam-726	259	1	example	example	NOUN
ejpam-726	260	1	11	11	NUM
ejpam-726	260	2	.	.	PUNCT
ejpam-726	261	1	let	let	VERB
ejpam-726	261	2	x	x	PUNCT
ejpam-726	261	3	=	=	PRON
ejpam-726	261	4	{	{	PUNCT
ejpam-726	261	5	a	a	PRON
ejpam-726	261	6	,	,	PUNCT
ejpam-726	261	7	b	b	NOUN
ejpam-726	261	8	,	,	PUNCT
ejpam-726	261	9	c	c	NOUN
ejpam-726	261	10	,	,	PUNCT
ejpam-726	261	11	d	d	NOUN
ejpam-726	261	12	}	}	PUNCT
ejpam-726	261	13	and	and	CCONJ
ejpam-726	261	14	τ	τ	PROPN
ejpam-726	261	15	=	=	PUNCT
ejpam-726	261	16	{	{	PUNCT
ejpam-726	261	17	x	x	X
ejpam-726	261	18	,	,	PUNCT
ejpam-726	261	19	;	;	PUNCT
ejpam-726	261	20	,	,	PUNCT
ejpam-726	261	21	{	{	PUNCT
ejpam-726	261	22	b	b	NOUN
ejpam-726	261	23	}	}	PUNCT
ejpam-726	261	24	,	,	PUNCT
ejpam-726	261	25	{	{	PUNCT
ejpam-726	261	26	c	c	X
ejpam-726	261	27	}	}	PUNCT
ejpam-726	261	28	,	,	PUNCT
ejpam-726	261	29	{	{	PUNCT
ejpam-726	261	30	a	a	DET
ejpam-726	261	31	,	,	PUNCT
ejpam-726	261	32	b	b	NOUN
ejpam-726	261	33	}	}	PUNCT
ejpam-726	261	34	,	,	PUNCT
ejpam-726	261	35	{	{	PUNCT
ejpam-726	261	36	b	b	X
ejpam-726	261	37	,	,	PUNCT
ejpam-726	261	38	c	c	NOUN
ejpam-726	261	39	}	}	PUNCT
ejpam-726	261	40	,	,	PUNCT
ejpam-726	261	41	{	{	PUNCT
ejpam-726	261	42	a	a	DET
ejpam-726	261	43	,	,	PUNCT
ejpam-726	261	44	b	b	NOUN
ejpam-726	261	45	,	,	PUNCT
ejpam-726	261	46	c	c	NOUN
ejpam-726	261	47	}	}	PUNCT
ejpam-726	261	48	,	,	PUNCT
ejpam-726	261	49	{	{	PUNCT
ejpam-726	261	50	a	a	DET
ejpam-726	261	51	,	,	PUNCT
ejpam-726	261	52	b	b	NOUN
ejpam-726	261	53	,	,	PUNCT
ejpam-726	261	54	d	d	NOUN
ejpam-726	261	55	}	}	PUNCT
ejpam-726	261	56	}	}	PUNCT
ejpam-726	261	57	.	.	PUNCT
ejpam-726	262	1	let	let	VERB
ejpam-726	262	2	a=	a=	VERB
ejpam-726	262	3	{	{	PUNCT
ejpam-726	262	4	a	a	X
ejpam-726	262	5	,	,	PUNCT
ejpam-726	262	6	c	c	NOUN
ejpam-726	262	7	}	}	PUNCT
ejpam-726	262	8	and	and	CCONJ
ejpam-726	262	9	b	b	X
ejpam-726	262	10	=	=	NOUN
ejpam-726	262	11	{	{	PUNCT
ejpam-726	262	12	a	a	PROPN
ejpam-726	262	13	,	,	PUNCT
ejpam-726	262	14	b	b	NOUN
ejpam-726	262	15	}	}	PUNCT
ejpam-726	262	16	.	.	PUNCT
ejpam-726	263	1	then	then	ADV
ejpam-726	263	2	β∗g	β∗g	NUM
ejpam-726	263	3	−	−	PROPN
ejpam-726	263	4	cl(a)∩β∗g	cl(a)∩β∗g	PROPN
ejpam-726	263	5	−	−	NOUN
ejpam-726	263	6	cl(b	cl(b	NOUN
ejpam-726	263	7	)	)	PUNCT
ejpam-726	264	1	=	=	PRON
ejpam-726	264	2	{	{	PUNCT
ejpam-726	264	3	a	a	X
ejpam-726	264	4	,	,	PUNCT
ejpam-726	264	5	d	d	NOUN
ejpam-726	264	6	}	}	PUNCT
ejpam-726	264	7	6⊂	6⊂	NUM
ejpam-726	264	8	{	{	PUNCT
ejpam-726	264	9	a	a	NOUN
ejpam-726	264	10	}	}	PUNCT
ejpam-726	264	11	=	=	SYM
ejpam-726	264	12	β∗g	β∗g	PUNCT
ejpam-726	264	13	−	−	PROPN
ejpam-726	264	14	cl(a∩	cl(a∩	PROPN
ejpam-726	264	15	b	b	PROPN
ejpam-726	264	16	)	)	PUNCT
ejpam-726	264	17	.	.	PUNCT
ejpam-726	265	1	definition	definition	NOUN
ejpam-726	265	2	4	4	NUM
ejpam-726	265	3	.	.	PUNCT
ejpam-726	266	1	for	for	ADP
ejpam-726	266	2	a	a	DET
ejpam-726	266	3	subset	subset	NOUN
ejpam-726	266	4	a	a	PRON
ejpam-726	266	5	of	of	ADP
ejpam-726	266	6	(	(	PUNCT
ejpam-726	266	7	x	x	PROPN
ejpam-726	266	8	,	,	PUNCT
ejpam-726	266	9	τ	τ	PROPN
ejpam-726	266	10	)	)	PUNCT
ejpam-726	266	11	,	,	PUNCT
ejpam-726	266	12	(	(	PUNCT
ejpam-726	266	13	a	a	X
ejpam-726	266	14	)	)	PUNCT
ejpam-726	266	15	c∗(a	c∗(a	PROPN
ejpam-726	266	16	)	)	PUNCT
ejpam-726	266	17	=	=	SYM
ejpam-726	266	18	⋂	⋂	PROPN
ejpam-726	266	19	{	{	PUNCT
ejpam-726	266	20	f	f	NOUN
ejpam-726	266	21	:	:	PUNCT
ejpam-726	266	22	f	f	PROPN
ejpam-726	266	23	is	be	AUX
ejpam-726	266	24	g−closed	g−close	VERB
ejpam-726	266	25	,	,	PUNCT
ejpam-726	266	26	a⊂	a⊂	X
ejpam-726	266	27	f	f	NOUN
ejpam-726	266	28	}	}	PUNCT
ejpam-726	266	29	:	:	PUNCT
ejpam-726	266	30	g	g	NOUN
ejpam-726	266	31	-	-	PUNCT
ejpam-726	266	32	closure	closure	NOUN
ejpam-726	266	33	of	of	ADP
ejpam-726	266	34	a	a	PRON
ejpam-726	266	35	[	[	X
ejpam-726	266	36	18	18	NUM
ejpam-726	266	37	]	]	X
ejpam-726	266	38	;	;	PUNCT
ejpam-726	266	39	(	(	PUNCT
ejpam-726	266	40	b	b	X
ejpam-726	266	41	)	)	PUNCT
ejpam-726	266	42	πg	πg	ADP
ejpam-726	266	43	−	−	PROPN
ejpam-726	266	44	cl(a	cl(a	PUNCT
ejpam-726	266	45	)	)	PUNCT
ejpam-726	266	46	=	=	SYM
ejpam-726	267	1	⋂	⋂	PROPN
ejpam-726	267	2	{	{	PUNCT
ejpam-726	267	3	f	f	NOUN
ejpam-726	267	4	:	:	PUNCT
ejpam-726	267	5	f	f	PROPN
ejpam-726	267	6	is	be	AUX
ejpam-726	267	7	πg−closed	πg−close	VERB
ejpam-726	267	8	,	,	PUNCT
ejpam-726	267	9	a⊂	a⊂	PRON
ejpam-726	267	10	f	f	NOUN
ejpam-726	267	11	}	}	PUNCT
ejpam-726	267	12	:	:	PUNCT
ejpam-726	267	13	πg−closure	πg−closure	NOUN
ejpam-726	267	14	of	of	ADP
ejpam-726	267	15	a	a	PRON
ejpam-726	267	16	[	[	X
ejpam-726	267	17	11	11	NUM
ejpam-726	267	18	]	]	PUNCT
ejpam-726	267	19	.	.	PUNCT
ejpam-726	268	1	a.	a.	PROPN
ejpam-726	268	2	açıkgöz	açıkgöz	PROPN
ejpam-726	268	3	/	/	SYM
ejpam-726	268	4	eur	eur	PROPN
ejpam-726	268	5	.	.	PUNCT
ejpam-726	269	1	j.	j.	PROPN
ejpam-726	269	2	pure	pure	PROPN
ejpam-726	269	3	appl	appl	PROPN
ejpam-726	269	4	.	.	PROPN
ejpam-726	269	5	math	math	PROPN
ejpam-726	269	6	,	,	PUNCT
ejpam-726	269	7	4	4	NUM
ejpam-726	269	8	(	(	PUNCT
ejpam-726	269	9	2011	2011	NUM
ejpam-726	269	10	)	)	PUNCT
ejpam-726	269	11	,	,	PUNCT
ejpam-726	269	12	20	20	NUM
ejpam-726	269	13	-	-	SYM
ejpam-726	269	14	33	33	NUM
ejpam-726	269	15	27	27	NUM
ejpam-726	269	16	definition	definition	NOUN
ejpam-726	269	17	5	5	NUM
ejpam-726	269	18	.	.	PUNCT
ejpam-726	270	1	for	for	ADP
ejpam-726	270	2	a	a	DET
ejpam-726	270	3	topological	topological	ADJ
ejpam-726	270	4	space	space	NOUN
ejpam-726	270	5	(	(	PUNCT
ejpam-726	270	6	x	x	X
ejpam-726	270	7	,	,	PUNCT
ejpam-726	270	8	τ	τ	PROPN
ejpam-726	270	9	)	)	PUNCT
ejpam-726	270	10	,	,	PUNCT
ejpam-726	270	11	(	(	PUNCT
ejpam-726	270	12	a	a	X
ejpam-726	270	13	)	)	PUNCT
ejpam-726	270	14	cτ∗	cτ∗	NOUN
ejpam-726	270	15	=	=	PUNCT
ejpam-726	270	16	{	{	PUNCT
ejpam-726	270	17	u	u	X
ejpam-726	270	18	⊂	⊂	PROPN
ejpam-726	270	19	x	x	X
ejpam-726	270	20	:	:	PUNCT
ejpam-726	270	21	c∗(x	c∗(x	NOUN
ejpam-726	270	22	−	−	PROPN
ejpam-726	270	23	u	u	NOUN
ejpam-726	270	24	)	)	PUNCT
ejpam-726	270	25	=	=	SYM
ejpam-726	270	26	(	(	PUNCT
ejpam-726	270	27	x	x	X
ejpam-726	270	28	−	−	PROPN
ejpam-726	270	29	u	u	NOUN
ejpam-726	270	30	)	)	PUNCT
ejpam-726	270	31	}	}	PUNCT
ejpam-726	271	1	[	[	X
ejpam-726	271	2	18	18	NUM
ejpam-726	271	3	]	]	X
ejpam-726	271	4	;	;	PUNCT
ejpam-726	271	5	(	(	PUNCT
ejpam-726	271	6	b	b	X
ejpam-726	271	7	)	)	PUNCT
ejpam-726	271	8	βτ∗	βτ∗	NOUN
ejpam-726	271	9	=	=	PRON
ejpam-726	271	10	{	{	PUNCT
ejpam-726	271	11	u	u	X
ejpam-726	271	12	⊂	⊂	PROPN
ejpam-726	271	13	x	x	X
ejpam-726	271	14	:	:	PUNCT
ejpam-726	271	15	β∗g	β∗g	PUNCT
ejpam-726	271	16	−	−	NOUN
ejpam-726	271	17	cl(x	cl(x	PUNCT
ejpam-726	271	18	−	−	PROPN
ejpam-726	271	19	u	u	NOUN
ejpam-726	271	20	)	)	PUNCT
ejpam-726	271	21	=	=	SYM
ejpam-726	271	22	(	(	PUNCT
ejpam-726	271	23	x	x	X
ejpam-726	271	24	−	−	PROPN
ejpam-726	271	25	u	u	NOUN
ejpam-726	271	26	)	)	PUNCT
ejpam-726	271	27	}	}	PUNCT
ejpam-726	271	28	;	;	PUNCT
ejpam-726	271	29	(	(	PUNCT
ejpam-726	271	30	c	c	X
ejpam-726	271	31	)	)	PUNCT
ejpam-726	271	32	πgτ∗	πgτ∗	NOUN
ejpam-726	271	33	=	=	SYM
ejpam-726	271	34	{	{	PUNCT
ejpam-726	271	35	u	u	X
ejpam-726	271	36	⊂	⊂	PROPN
ejpam-726	271	37	x	x	X
ejpam-726	271	38	:	:	PUNCT
ejpam-726	271	39	πg	πg	ADP
ejpam-726	271	40	−	−	PROPN
ejpam-726	271	41	cl(x	cl(x	PUNCT
ejpam-726	271	42	−	−	PROPN
ejpam-726	271	43	u	u	NOUN
ejpam-726	271	44	)	)	PUNCT
ejpam-726	271	45	=	=	SYM
ejpam-726	271	46	(	(	PUNCT
ejpam-726	271	47	x	x	X
ejpam-726	271	48	−	−	PROPN
ejpam-726	271	49	u	u	NOUN
ejpam-726	271	50	)	)	PUNCT
ejpam-726	271	51	}	}	PUNCT
ejpam-726	272	1	[	[	X
ejpam-726	272	2	11	11	NUM
ejpam-726	272	3	]	]	PUNCT
ejpam-726	272	4	;	;	PUNCT
ejpam-726	272	5	proposition	proposition	NOUN
ejpam-726	272	6	2	2	NUM
ejpam-726	272	7	.	.	X
ejpam-726	272	8	for	for	ADP
ejpam-726	272	9	a	a	DET
ejpam-726	272	10	subset	subset	NOUN
ejpam-726	272	11	a	a	PRON
ejpam-726	272	12	of	of	ADP
ejpam-726	272	13	(	(	PUNCT
ejpam-726	272	14	x	x	PROPN
ejpam-726	272	15	,	,	PUNCT
ejpam-726	272	16	τ	τ	PROPN
ejpam-726	272	17	)	)	PUNCT
ejpam-726	272	18	,	,	PUNCT
ejpam-726	272	19	the	the	DET
ejpam-726	272	20	following	follow	VERB
ejpam-726	272	21	statements	statement	NOUN
ejpam-726	272	22	hold	hold	VERB
ejpam-726	272	23	:	:	PUNCT
ejpam-726	272	24	(	(	PUNCT
ejpam-726	272	25	a	a	X
ejpam-726	272	26	)	)	PUNCT
ejpam-726	272	27	a⊂	a⊂	NOUN
ejpam-726	272	28	πg	πg	ADP
ejpam-726	272	29	−	−	PROPN
ejpam-726	272	30	cl(a	cl(a	PUNCT
ejpam-726	272	31	)	)	PUNCT
ejpam-726	273	1	⊂	⊂	PROPN
ejpam-726	273	2	c∗(a)⊂	c∗(a)⊂	PROPN
ejpam-726	273	3	β∗g	β∗g	PUNCT
ejpam-726	273	4	−	−	NOUN
ejpam-726	273	5	cl(a	cl(a	NUM
ejpam-726	273	6	)	)	PUNCT
ejpam-726	273	7	.	.	PUNCT
ejpam-726	274	1	(	(	PUNCT
ejpam-726	274	2	b	b	X
ejpam-726	274	3	)	)	PUNCT
ejpam-726	274	4	τ	τ	PROPN
ejpam-726	275	1	⊂	⊂	PROPN
ejpam-726	275	2	βτ∗	βτ∗	PROPN
ejpam-726	275	3	⊂	⊂	PROPN
ejpam-726	275	4	cτ∗	cτ∗	PROPN
ejpam-726	275	5	⊂	⊂	PROPN
ejpam-726	275	6	πgτ∗.	πgτ∗.	X
ejpam-726	275	7	proof	proof	NOUN
ejpam-726	275	8	.	.	PUNCT
ejpam-726	276	1	the	the	DET
ejpam-726	276	2	proof	proof	NOUN
ejpam-726	276	3	follows	follow	VERB
ejpam-726	276	4	from	from	ADP
ejpam-726	276	5	definitions	definition	NOUN
ejpam-726	276	6	.	.	PUNCT
ejpam-726	277	1	definition	definition	NOUN
ejpam-726	277	2	6	6	NUM
ejpam-726	277	3	.	.	PUNCT
ejpam-726	278	1	a	a	DET
ejpam-726	278	2	topological	topological	ADJ
ejpam-726	278	3	space	space	NOUN
ejpam-726	278	4	(	(	PUNCT
ejpam-726	278	5	x	x	X
ejpam-726	278	6	,	,	PUNCT
ejpam-726	278	7	τ	τ	X
ejpam-726	278	8	)	)	PUNCT
ejpam-726	278	9	is	be	AUX
ejpam-726	278	10	said	say	VERB
ejpam-726	278	11	to	to	PART
ejpam-726	278	12	be	be	AUX
ejpam-726	278	13	(	(	PUNCT
ejpam-726	278	14	a	a	X
ejpam-726	278	15	)	)	PUNCT
ejpam-726	278	16	t1/2	t1/2	ADJ
ejpam-726	278	17	space	space	NOUN
ejpam-726	278	18	[	[	X
ejpam-726	278	19	17	17	NUM
ejpam-726	278	20	]	]	PUNCT
ejpam-726	278	21	if	if	SCONJ
ejpam-726	278	22	every	every	DET
ejpam-726	278	23	g−closed	g−close	VERB
ejpam-726	278	24	set	set	NOUN
ejpam-726	278	25	is	be	AUX
ejpam-726	278	26	closed	closed	ADJ
ejpam-726	278	27	.	.	PUNCT
ejpam-726	279	1	(	(	PUNCT
ejpam-726	279	2	b	b	X
ejpam-726	279	3	)	)	PUNCT
ejpam-726	279	4	t	t	NOUN
ejpam-726	279	5	∗	∗	NOUN
ejpam-726	279	6	1/2	1/2	NUM
ejpam-726	279	7	space	space	NOUN
ejpam-726	279	8	[	[	X
ejpam-726	279	9	16	16	NUM
ejpam-726	279	10	]	]	PUNCT
ejpam-726	279	11	if	if	SCONJ
ejpam-726	279	12	every	every	DET
ejpam-726	279	13	g∗-closed	g∗-close	VERB
ejpam-726	279	14	set	set	NOUN
ejpam-726	279	15	is	be	AUX
ejpam-726	279	16	closed	closed	ADJ
ejpam-726	279	17	.	.	PUNCT
ejpam-726	280	1	(	(	PUNCT
ejpam-726	280	2	c	c	X
ejpam-726	280	3	)	)	PUNCT
ejpam-726	280	4	∗t1/2	∗t1/2	NOUN
ejpam-726	280	5	space	space	NOUN
ejpam-726	281	1	[	[	X
ejpam-726	281	2	16	16	NUM
ejpam-726	281	3	]	]	PUNCT
ejpam-726	281	4	if	if	SCONJ
ejpam-726	281	5	every	every	DET
ejpam-726	281	6	g−closed	g−close	VERB
ejpam-726	281	7	set	set	NOUN
ejpam-726	281	8	is	be	AUX
ejpam-726	281	9	g∗-closed	g∗-close	VERB
ejpam-726	281	10	.	.	PUNCT
ejpam-726	282	1	theorem	theorem	NOUN
ejpam-726	282	2	9	9	NUM
ejpam-726	282	3	.	.	PUNCT
ejpam-726	283	1	let	let	AUX
ejpam-726	283	2	(	(	PUNCT
ejpam-726	283	3	x	x	X
ejpam-726	283	4	,	,	PUNCT
ejpam-726	283	5	τ	τ	X
ejpam-726	283	6	)	)	PUNCT
ejpam-726	283	7	be	be	AUX
ejpam-726	283	8	a	a	DET
ejpam-726	283	9	space	space	NOUN
ejpam-726	283	10	.	.	PUNCT
ejpam-726	284	1	then	then	ADV
ejpam-726	284	2	(	(	PUNCT
ejpam-726	284	3	a	a	X
ejpam-726	284	4	)	)	PUNCT
ejpam-726	284	5	every	every	DET
ejpam-726	284	6	g−closed	g−close	VERB
ejpam-726	284	7	set	set	NOUN
ejpam-726	284	8	is	be	AUX
ejpam-726	284	9	closed	close	VERB
ejpam-726	284	10	(	(	PUNCT
ejpam-726	284	11	i.e.	i.e.	X
ejpam-726	284	12	(	(	PUNCT
ejpam-726	284	13	x	x	X
ejpam-726	284	14	,	,	PUNCT
ejpam-726	284	15	τ	τ	X
ejpam-726	284	16	)	)	PUNCT
ejpam-726	284	17	is	be	AUX
ejpam-726	284	18	t1/2	t1/2	NOUN
ejpam-726	284	19	)	)	PUNCT
ejpam-726	284	20	if	if	SCONJ
ejpam-726	284	21	and	and	CCONJ
ejpam-726	284	22	only	only	ADV
ejpam-726	284	23	if	if	SCONJ
ejpam-726	284	24	cτ∗	cτ∗	ADJ
ejpam-726	284	25	=	=	SYM
ejpam-726	284	26	τ	τ	PROPN
ejpam-726	284	27	.	.	PUNCT
ejpam-726	284	28	(	(	PUNCT
ejpam-726	284	29	b	b	X
ejpam-726	284	30	)	)	PUNCT
ejpam-726	284	31	every	every	DET
ejpam-726	284	32	β∗g−closed	β∗g−close	VERB
ejpam-726	284	33	set	set	NOUN
ejpam-726	284	34	is	be	AUX
ejpam-726	284	35	closed	close	VERB
ejpam-726	284	36	(	(	PUNCT
ejpam-726	284	37	i.e.	i.e.	X
ejpam-726	284	38	(	(	PUNCT
ejpam-726	284	39	x	x	X
ejpam-726	284	40	,	,	PUNCT
ejpam-726	284	41	τ	τ	X
ejpam-726	284	42	)	)	PUNCT
ejpam-726	284	43	is	be	AUX
ejpam-726	284	44	β∗t1/2	β∗t1/2	PROPN
ejpam-726	284	45	)	)	PUNCT
ejpam-726	285	1	if	if	SCONJ
ejpam-726	285	2	and	and	CCONJ
ejpam-726	285	3	only	only	ADV
ejpam-726	285	4	if	if	SCONJ
ejpam-726	285	5	βτ∗	βτ∗	NOUN
ejpam-726	285	6	=	=	SYM
ejpam-726	285	7	τ	τ	PROPN
ejpam-726	285	8	.	.	PUNCT
ejpam-726	285	9	(	(	PUNCT
ejpam-726	285	10	c	c	X
ejpam-726	285	11	)	)	PUNCT
ejpam-726	285	12	every	every	DET
ejpam-726	285	13	g−closed	g−close	VERB
ejpam-726	285	14	set	set	NOUN
ejpam-726	285	15	is	be	AUX
ejpam-726	285	16	β∗g−closed	β∗g−close	VERB
ejpam-726	285	17	(	(	PUNCT
ejpam-726	285	18	i.e.	i.e.	X
ejpam-726	285	19	(	(	PUNCT
ejpam-726	285	20	x	x	X
ejpam-726	285	21	,	,	PUNCT
ejpam-726	285	22	τ	τ	X
ejpam-726	285	23	)	)	PUNCT
ejpam-726	285	24	is	be	AUX
ejpam-726	285	25	β∗∗t1/2	β∗∗t1/2	PROPN
ejpam-726	285	26	)	)	PUNCT
ejpam-726	286	1	if	if	SCONJ
ejpam-726	286	2	and	and	CCONJ
ejpam-726	286	3	only	only	ADV
ejpam-726	286	4	if	if	SCONJ
ejpam-726	286	5	cτ∗	cτ∗	ADJ
ejpam-726	286	6	=	=	NOUN
ejpam-726	286	7	βτ∗.	βτ∗.	NOUN
ejpam-726	286	8	proof	proof	NOUN
ejpam-726	286	9	.	.	PUNCT
ejpam-726	287	1	(	(	PUNCT
ejpam-726	287	2	a	a	X
ejpam-726	287	3	)	)	PUNCT
ejpam-726	287	4	let	let	VERB
ejpam-726	287	5	a	a	DET
ejpam-726	287	6	∈	∈	NOUN
ejpam-726	287	7	cτ∗.	cτ∗.	NOUN
ejpam-726	287	8	then	then	ADV
ejpam-726	287	9	c∗(x	c∗(x	VERB
ejpam-726	287	10	−	−	PROPN
ejpam-726	287	11	a	a	X
ejpam-726	287	12	)	)	PUNCT
ejpam-726	287	13	=	=	SYM
ejpam-726	288	1	(	(	PUNCT
ejpam-726	288	2	x	x	X
ejpam-726	288	3	−	−	NOUN
ejpam-726	288	4	a	a	NOUN
ejpam-726	288	5	)	)	PUNCT
ejpam-726	288	6	.	.	PUNCT
ejpam-726	289	1	by	by	ADP
ejpam-726	289	2	hypothesis	hypothesis	NOUN
ejpam-726	289	3	,	,	PUNCT
ejpam-726	289	4	cl(x	cl(x	PUNCT
ejpam-726	289	5	−	−	PROPN
ejpam-726	289	6	a	a	X
ejpam-726	289	7	)	)	PUNCT
ejpam-726	289	8	=	=	SYM
ejpam-726	289	9	c∗(x	c∗(x	NOUN
ejpam-726	289	10	−	−	NOUN
ejpam-726	289	11	a	a	NOUN
ejpam-726	289	12	)	)	PUNCT
ejpam-726	289	13	=	=	PUNCT
ejpam-726	290	1	x	x	X
ejpam-726	290	2	−	−	NOUN
ejpam-726	290	3	a	a	PRON
ejpam-726	290	4	and	and	CCONJ
ejpam-726	290	5	hence	hence	ADV
ejpam-726	290	6	a	a	DET
ejpam-726	290	7	∈	∈	NOUN
ejpam-726	290	8	τ	τ	X
ejpam-726	290	9	.	.	PUNCT
ejpam-726	291	1	conversely	conversely	ADV
ejpam-726	291	2	,	,	PUNCT
ejpam-726	291	3	let	let	VERB
ejpam-726	291	4	a	a	PRON
ejpam-726	291	5	be	be	AUX
ejpam-726	291	6	a	a	DET
ejpam-726	291	7	g−closed	g−close	VERB
ejpam-726	291	8	set	set	NOUN
ejpam-726	291	9	.	.	PUNCT
ejpam-726	292	1	then	then	ADV
ejpam-726	292	2	c∗(a	c∗(a	PROPN
ejpam-726	292	3	)	)	PUNCT
ejpam-726	292	4	=	=	NOUN
ejpam-726	293	1	a	a	PRON
ejpam-726	293	2	and	and	CCONJ
ejpam-726	293	3	hence	hence	ADV
ejpam-726	293	4	x	x	NOUN
ejpam-726	293	5	−	−	PROPN
ejpam-726	293	6	a∈	a∈	PROPN
ejpam-726	293	7	cτ∗	cτ∗	NOUN
ejpam-726	293	8	=	=	SYM
ejpam-726	293	9	τ	τ	PROPN
ejpam-726	293	10	,	,	PUNCT
ejpam-726	293	11	i.e.	i.e.	X
ejpam-726	293	12	a	a	PRON
ejpam-726	293	13	is	be	AUX
ejpam-726	293	14	closed	closed	ADJ
ejpam-726	293	15	.	.	PUNCT
ejpam-726	294	1	(	(	PUNCT
ejpam-726	294	2	b	b	X
ejpam-726	294	3	)	)	PUNCT
ejpam-726	294	4	let	let	VERB
ejpam-726	294	5	a∈	a∈	PROPN
ejpam-726	294	6	βτ∗.	βτ∗.	PROPN
ejpam-726	294	7	then	then	ADV
ejpam-726	294	8	β∗g	β∗g	NUM
ejpam-726	294	9	−	−	NOUN
ejpam-726	294	10	cl(x	cl(x	NUM
ejpam-726	294	11	−	−	PROPN
ejpam-726	294	12	a	a	X
ejpam-726	294	13	)	)	PUNCT
ejpam-726	294	14	=	=	PUNCT
ejpam-726	295	1	x	x	X
ejpam-726	295	2	−	−	NOUN
ejpam-726	295	3	a	a	PRON
ejpam-726	295	4	and	and	CCONJ
ejpam-726	295	5	by	by	ADP
ejpam-726	295	6	hypothesis	hypothesis	NOUN
ejpam-726	295	7	,	,	PUNCT
ejpam-726	295	8	cl(x	cl(x	PUNCT
ejpam-726	295	9	−	−	PROPN
ejpam-726	295	10	a	a	X
ejpam-726	295	11	)	)	PUNCT
ejpam-726	295	12	=	=	SYM
ejpam-726	295	13	β∗g	β∗g	NUM
ejpam-726	295	14	−	−	NOUN
ejpam-726	295	15	cl(x	cl(x	NUM
ejpam-726	295	16	−	−	PROPN
ejpam-726	295	17	a	a	X
ejpam-726	295	18	)	)	PUNCT
ejpam-726	296	1	=	=	PUNCT
ejpam-726	296	2	x	x	SYM
ejpam-726	296	3	−	−	NOUN
ejpam-726	296	4	a.	a.	NOUN
ejpam-726	296	5	hence	hence	ADV
ejpam-726	296	6	a∈	a∈	PROPN
ejpam-726	296	7	τ	τ	PROPN
ejpam-726	296	8	.	.	PUNCT
ejpam-726	297	1	(	(	PUNCT
ejpam-726	297	2	c	c	X
ejpam-726	297	3	)	)	PUNCT
ejpam-726	297	4	similar	similar	ADJ
ejpam-726	297	5	to	to	ADP
ejpam-726	297	6	(	(	PUNCT
ejpam-726	297	7	a	a	NOUN
ejpam-726	297	8	)	)	PUNCT
ejpam-726	297	9	.	.	PUNCT
ejpam-726	298	1	definition	definition	NOUN
ejpam-726	298	2	7	7	NUM
ejpam-726	298	3	.	.	PUNCT
ejpam-726	299	1	a	a	DET
ejpam-726	299	2	topological	topological	ADJ
ejpam-726	299	3	space	space	NOUN
ejpam-726	299	4	(	(	PUNCT
ejpam-726	299	5	x	x	X
ejpam-726	299	6	,	,	PUNCT
ejpam-726	299	7	τ	τ	X
ejpam-726	299	8	)	)	PUNCT
ejpam-726	299	9	is	be	AUX
ejpam-726	299	10	called	call	VERB
ejpam-726	299	11	a	a	DET
ejpam-726	299	12	β∗t1/2	β∗t1/2	PROPN
ejpam-726	299	13	space	space	NOUN
ejpam-726	299	14	if	if	SCONJ
ejpam-726	299	15	every	every	DET
ejpam-726	299	16	β∗g−closed	β∗g−close	VERB
ejpam-726	299	17	set	set	NOUN
ejpam-726	299	18	is	be	AUX
ejpam-726	299	19	closed	close	VERB
ejpam-726	299	20	.	.	PUNCT
ejpam-726	299	21	theorem	theorem	VERB
ejpam-726	299	22	10	10	NUM
ejpam-726	299	23	.	.	PUNCT
ejpam-726	300	1	a	a	DET
ejpam-726	300	2	topological	topological	ADJ
ejpam-726	300	3	space	space	NOUN
ejpam-726	300	4	(	(	PUNCT
ejpam-726	300	5	x	x	X
ejpam-726	300	6	,	,	PUNCT
ejpam-726	300	7	τ	τ	X
ejpam-726	300	8	)	)	PUNCT
ejpam-726	300	9	is	be	AUX
ejpam-726	300	10	β∗t1/2	β∗t1/2	PROPN
ejpam-726	300	11	if	if	SCONJ
ejpam-726	301	1	and	and	CCONJ
ejpam-726	301	2	only	only	ADV
ejpam-726	301	3	if	if	SCONJ
ejpam-726	301	4	each	each	DET
ejpam-726	301	5	singleton	singleton	NOUN
ejpam-726	301	6	of	of	ADP
ejpam-726	301	7	x	x	SYM
ejpam-726	301	8	is	be	AUX
ejpam-726	301	9	open	open	ADJ
ejpam-726	301	10	or	or	CCONJ
ejpam-726	301	11	x	x	SYM
ejpam-726	301	12	−	−	PROPN
ejpam-726	301	13	{	{	PUNCT
ejpam-726	301	14	x	x	NOUN
ejpam-726	301	15	}	}	PUNCT
ejpam-726	301	16	is	be	AUX
ejpam-726	301	17	a	a	DET
ejpam-726	301	18	β∗-set	β∗-set	NOUN
ejpam-726	301	19	for	for	ADP
ejpam-726	301	20	each	each	DET
ejpam-726	301	21	x	x	SYM
ejpam-726	301	22	∈	∈	PROPN
ejpam-726	301	23	x	x	X
ejpam-726	301	24	.	.	PUNCT
ejpam-726	302	1	proof	proof	NOUN
ejpam-726	302	2	.	.	PUNCT
ejpam-726	303	1	a.	a.	NOUN
ejpam-726	303	2	açıkgöz	açıkgöz	PROPN
ejpam-726	303	3	/	/	SYM
ejpam-726	303	4	eur	eur	PROPN
ejpam-726	303	5	.	.	PUNCT
ejpam-726	304	1	j.	j.	PROPN
ejpam-726	304	2	pure	pure	PROPN
ejpam-726	304	3	appl	appl	PROPN
ejpam-726	304	4	.	.	PROPN
ejpam-726	304	5	math	math	PROPN
ejpam-726	304	6	,	,	PUNCT
ejpam-726	304	7	4	4	NUM
ejpam-726	304	8	(	(	PUNCT
ejpam-726	304	9	2011	2011	NUM
ejpam-726	304	10	)	)	PUNCT
ejpam-726	304	11	,	,	PUNCT
ejpam-726	304	12	20	20	NUM
ejpam-726	304	13	-	-	SYM
ejpam-726	304	14	33	33	NUM
ejpam-726	304	15	28	28	NUM
ejpam-726	304	16	necessity	necessity	NOUN
ejpam-726	304	17	:	:	PUNCT
ejpam-726	304	18	let	let	VERB
ejpam-726	304	19	x	x	PRON
ejpam-726	304	20	be	be	AUX
ejpam-726	304	21	a	a	DET
ejpam-726	304	22	point	point	NOUN
ejpam-726	304	23	of	of	ADP
ejpam-726	304	24	x.	x.	NOUN
ejpam-726	304	25	suppose	suppose	VERB
ejpam-726	304	26	that	that	SCONJ
ejpam-726	304	27	x	x	PRON
ejpam-726	304	28	−	−	X
ejpam-726	304	29	{	{	PUNCT
ejpam-726	304	30	x	x	NOUN
ejpam-726	304	31	}	}	PUNCT
ejpam-726	304	32	is	be	AUX
ejpam-726	304	33	not	not	PART
ejpam-726	304	34	a	a	DET
ejpam-726	304	35	β∗-set	β∗-set	NOUN
ejpam-726	304	36	.	.	PUNCT
ejpam-726	305	1	then	then	ADV
ejpam-726	305	2	x	x	X
ejpam-726	305	3	−	−	PROPN
ejpam-726	305	4	{	{	PUNCT
ejpam-726	305	5	x	x	NOUN
ejpam-726	305	6	}	}	PUNCT
ejpam-726	305	7	is	be	AUX
ejpam-726	305	8	β∗g−closed	β∗g−close	VERB
ejpam-726	305	9	.	.	PUNCT
ejpam-726	306	1	since	since	SCONJ
ejpam-726	306	2	(	(	PUNCT
ejpam-726	306	3	x	x	X
ejpam-726	306	4	,	,	PUNCT
ejpam-726	306	5	τ	τ	X
ejpam-726	306	6	)	)	PUNCT
ejpam-726	306	7	is	be	AUX
ejpam-726	306	8	β∗t1/2	β∗t1/2	PROPN
ejpam-726	306	9	,	,	PUNCT
ejpam-726	306	10	x	x	PUNCT
ejpam-726	306	11	−	−	PROPN
ejpam-726	306	12	{	{	PUNCT
ejpam-726	306	13	x	x	NOUN
ejpam-726	306	14	}	}	PUNCT
ejpam-726	306	15	is	be	AUX
ejpam-726	306	16	closed	close	VERB
ejpam-726	306	17	and	and	CCONJ
ejpam-726	306	18	thus	thus	ADV
ejpam-726	306	19	{	{	PUNCT
ejpam-726	306	20	x	x	X
ejpam-726	306	21	}	}	PUNCT
ejpam-726	306	22	is	be	AUX
ejpam-726	306	23	open	open	ADJ
ejpam-726	306	24	in	in	ADP
ejpam-726	306	25	(	(	PUNCT
ejpam-726	306	26	x	x	INTJ
ejpam-726	306	27	,	,	PUNCT
ejpam-726	306	28	τ	τ	PROPN
ejpam-726	306	29	)	)	PUNCT
ejpam-726	306	30	.	.	PUNCT
ejpam-726	307	1	sufficiency	sufficiency	NOUN
ejpam-726	307	2	:	:	PUNCT
ejpam-726	307	3	suppose	suppose	VERB
ejpam-726	307	4	that	that	SCONJ
ejpam-726	307	5	a	a	PRON
ejpam-726	307	6	is	be	AUX
ejpam-726	307	7	β∗g−closed	β∗g−close	VERB
ejpam-726	307	8	.	.	PUNCT
ejpam-726	308	1	we	we	PRON
ejpam-726	308	2	shall	shall	AUX
ejpam-726	308	3	show	show	VERB
ejpam-726	308	4	that	that	SCONJ
ejpam-726	308	5	cl(a	cl(a	PUNCT
ejpam-726	308	6	)	)	PUNCT
ejpam-726	308	7	⊂	⊂	PROPN
ejpam-726	308	8	a.	a.	NOUN
ejpam-726	308	9	let	let	VERB
ejpam-726	308	10	x	x	PRON
ejpam-726	308	11	be	be	AUX
ejpam-726	308	12	any	any	DET
ejpam-726	308	13	point	point	NOUN
ejpam-726	308	14	of	of	ADP
ejpam-726	308	15	cl(a	cl(a	NUM
ejpam-726	308	16	)	)	PUNCT
ejpam-726	308	17	.	.	PUNCT
ejpam-726	309	1	then	then	ADV
ejpam-726	309	2	{	{	PUNCT
ejpam-726	309	3	x	x	X
ejpam-726	309	4	}	}	PUNCT
ejpam-726	309	5	is	be	AUX
ejpam-726	309	6	open	open	ADJ
ejpam-726	309	7	in	in	ADP
ejpam-726	309	8	(	(	PUNCT
ejpam-726	309	9	x	x	INTJ
ejpam-726	309	10	,	,	PUNCT
ejpam-726	309	11	τ	τ	PROPN
ejpam-726	309	12	)	)	PUNCT
ejpam-726	309	13	or	or	CCONJ
ejpam-726	309	14	x	x	ADJ
ejpam-726	309	15	−	−	PROPN
ejpam-726	309	16	β∗-set	β∗-set	NOUN
ejpam-726	309	17	.	.	PUNCT
ejpam-726	310	1	(	(	PUNCT
ejpam-726	310	2	i	i	NOUN
ejpam-726	310	3	)	)	PUNCT
ejpam-726	310	4	in	in	ADP
ejpam-726	310	5	case	case	NOUN
ejpam-726	310	6	{	{	PUNCT
ejpam-726	310	7	x	x	X
ejpam-726	310	8	}	}	PUNCT
ejpam-726	310	9	is	be	AUX
ejpam-726	310	10	open	open	ADJ
ejpam-726	310	11	:	:	PUNCT
ejpam-726	310	12	since	since	SCONJ
ejpam-726	310	13	x	x	PROPN
ejpam-726	310	14	∈	∈	PROPN
ejpam-726	310	15	cl(a	cl(a	NUM
ejpam-726	310	16	)	)	PUNCT
ejpam-726	310	17	,	,	PUNCT
ejpam-726	310	18	{	{	PUNCT
ejpam-726	310	19	x	x	X
ejpam-726	310	20	}	}	PUNCT
ejpam-726	310	21	∩	∩	NOUN
ejpam-726	310	22	a	a	DET
ejpam-726	310	23	6=	6=	NOUN
ejpam-726	310	24	;	;	PUNCT
ejpam-726	310	25	and	and	CCONJ
ejpam-726	310	26	hence	hence	ADV
ejpam-726	310	27	x	x	PART
ejpam-726	310	28	∈	∈	PROPN
ejpam-726	310	29	a.	a.	NOUN
ejpam-726	310	30	(	(	PUNCT
ejpam-726	310	31	ii	ii	NOUN
ejpam-726	310	32	)	)	PUNCT
ejpam-726	310	33	in	in	ADP
ejpam-726	310	34	case	case	NOUN
ejpam-726	310	35	x−{x	x−{x	PROPN
ejpam-726	310	36	}	}	PUNCT
ejpam-726	310	37	is	be	AUX
ejpam-726	310	38	a	a	DET
ejpam-726	310	39	β∗-set	β∗-set	NOUN
ejpam-726	310	40	:	:	PUNCT
ejpam-726	310	41	by	by	ADP
ejpam-726	310	42	theorem	theorem	NOUN
ejpam-726	310	43	8	8	NUM
ejpam-726	310	44	,	,	PUNCT
ejpam-726	310	45	cl(a)−a	cl(a)−a	NOUN
ejpam-726	310	46	does	do	AUX
ejpam-726	310	47	not	not	PART
ejpam-726	310	48	contain	contain	VERB
ejpam-726	310	49	any	any	DET
ejpam-726	310	50	nonempty	nonempty	ADJ
ejpam-726	310	51	complement	complement	NOUN
ejpam-726	310	52	of	of	ADP
ejpam-726	310	53	a	a	DET
ejpam-726	310	54	β∗-set	β∗-set	NOUN
ejpam-726	310	55	.	.	PUNCT
ejpam-726	311	1	therefore	therefore	ADV
ejpam-726	311	2	,	,	PUNCT
ejpam-726	311	3	x	x	PUNCT
ejpam-726	311	4	/∈	/∈	PUNCT
ejpam-726	311	5	cl(a)−	cl(a)−	VERB
ejpam-726	311	6	a	a	PRON
ejpam-726	311	7	but	but	CCONJ
ejpam-726	311	8	x	x	SYM
ejpam-726	311	9	∈	∈	NOUN
ejpam-726	311	10	cl(a	cl(a	NUM
ejpam-726	311	11	)	)	PUNCT
ejpam-726	311	12	.	.	PUNCT
ejpam-726	312	1	thus	thus	ADV
ejpam-726	312	2	,	,	PUNCT
ejpam-726	312	3	x	x	PUNCT
ejpam-726	312	4	∈	∈	NOUN
ejpam-726	312	5	a.	a.	NOUN
ejpam-726	312	6	by	by	ADP
ejpam-726	312	7	(	(	PUNCT
ejpam-726	312	8	i	i	NOUN
ejpam-726	312	9	)	)	PUNCT
ejpam-726	312	10	and	and	CCONJ
ejpam-726	312	11	(	(	PUNCT
ejpam-726	312	12	ii	ii	NOUN
ejpam-726	312	13	)	)	PUNCT
ejpam-726	312	14	,	,	PUNCT
ejpam-726	312	15	we	we	PRON
ejpam-726	312	16	obtain	obtain	VERB
ejpam-726	312	17	cl(a)⊂	cl(a)⊂	PROPN
ejpam-726	312	18	a	a	PRON
ejpam-726	312	19	and	and	CCONJ
ejpam-726	312	20	hence	hence	ADV
ejpam-726	312	21	a	a	PRON
ejpam-726	312	22	is	be	AUX
ejpam-726	312	23	closed	closed	ADJ
ejpam-726	312	24	.	.	PUNCT
ejpam-726	313	1	theorem	theorem	VERB
ejpam-726	313	2	11	11	NUM
ejpam-726	313	3	.	.	PUNCT
ejpam-726	314	1	every	every	DET
ejpam-726	314	2	t1/2	t1/2	ADJ
ejpam-726	314	3	space	space	NOUN
ejpam-726	314	4	is	be	AUX
ejpam-726	314	5	a	a	DET
ejpam-726	314	6	β∗t1/2	β∗t1/2	PROPN
ejpam-726	314	7	space	space	NOUN
ejpam-726	314	8	.	.	PUNCT
ejpam-726	315	1	proof	proof	NOUN
ejpam-726	315	2	.	.	PUNCT
ejpam-726	316	1	let	let	VERB
ejpam-726	316	2	(	(	PUNCT
ejpam-726	316	3	x	x	X
ejpam-726	316	4	,	,	PUNCT
ejpam-726	316	5	τ	τ	X
ejpam-726	316	6	)	)	PUNCT
ejpam-726	316	7	be	be	VERB
ejpam-726	316	8	a	a	DET
ejpam-726	316	9	t1/2	t1/2	ADJ
ejpam-726	316	10	space	space	NOUN
ejpam-726	316	11	and	and	CCONJ
ejpam-726	316	12	a	a	DET
ejpam-726	316	13	a	a	DET
ejpam-726	316	14	β∗g−closed	β∗g−close	VERB
ejpam-726	316	15	set	set	NOUN
ejpam-726	316	16	of	of	ADP
ejpam-726	316	17	(	(	PUNCT
ejpam-726	316	18	x	x	PROPN
ejpam-726	316	19	,	,	PUNCT
ejpam-726	316	20	τ	τ	PROPN
ejpam-726	316	21	)	)	PUNCT
ejpam-726	316	22	.	.	PUNCT
ejpam-726	317	1	by	by	ADP
ejpam-726	317	2	theorem	theorem	NOUN
ejpam-726	317	3	3	3	NUM
ejpam-726	317	4	,	,	PUNCT
ejpam-726	317	5	a	a	PRON
ejpam-726	317	6	is	be	AUX
ejpam-726	317	7	a	a	DET
ejpam-726	317	8	g−closed	g−close	VERB
ejpam-726	317	9	set	set	NOUN
ejpam-726	317	10	of	of	ADP
ejpam-726	317	11	(	(	PUNCT
ejpam-726	317	12	x	x	PROPN
ejpam-726	317	13	,	,	PUNCT
ejpam-726	317	14	τ	τ	PROPN
ejpam-726	317	15	)	)	PUNCT
ejpam-726	317	16	.	.	PUNCT
ejpam-726	318	1	since	since	SCONJ
ejpam-726	318	2	x	x	PRON
ejpam-726	318	3	is	be	AUX
ejpam-726	318	4	a	a	DET
ejpam-726	318	5	t1/2	t1/2	ADJ
ejpam-726	318	6	space	space	NOUN
ejpam-726	318	7	,	,	PUNCT
ejpam-726	318	8	a	a	PRON
ejpam-726	318	9	is	be	AUX
ejpam-726	318	10	closed	closed	ADJ
ejpam-726	318	11	.	.	PUNCT
ejpam-726	319	1	therefore	therefore	ADV
ejpam-726	319	2	,	,	PUNCT
ejpam-726	319	3	x	x	X
ejpam-726	319	4	is	be	AUX
ejpam-726	319	5	a	a	DET
ejpam-726	319	6	β∗t1/2	β∗t1/2	PROPN
ejpam-726	319	7	space	space	NOUN
ejpam-726	319	8	.	.	PUNCT
ejpam-726	320	1	definition	definition	NOUN
ejpam-726	320	2	8	8	NUM
ejpam-726	320	3	.	.	PUNCT
ejpam-726	321	1	a	a	DET
ejpam-726	321	2	topological	topological	ADJ
ejpam-726	321	3	space	space	NOUN
ejpam-726	321	4	(	(	PUNCT
ejpam-726	321	5	x	x	X
ejpam-726	321	6	,	,	PUNCT
ejpam-726	321	7	τ	τ	X
ejpam-726	321	8	)	)	PUNCT
ejpam-726	321	9	is	be	AUX
ejpam-726	321	10	called	call	VERB
ejpam-726	321	11	a	a	DET
ejpam-726	321	12	β∗∗t1/2	β∗∗t1/2	PUNCT
ejpam-726	321	13	space	space	NOUN
ejpam-726	321	14	if	if	SCONJ
ejpam-726	321	15	every	every	DET
ejpam-726	321	16	g−closed	g−close	VERB
ejpam-726	321	17	set	set	NOUN
ejpam-726	321	18	is	be	AUX
ejpam-726	321	19	β∗g−closed	β∗g−close	VERB
ejpam-726	321	20	.	.	PUNCT
ejpam-726	322	1	theorem	theorem	NOUN
ejpam-726	322	2	12	12	NUM
ejpam-726	322	3	.	.	PUNCT
ejpam-726	323	1	every	every	DET
ejpam-726	323	2	t1/2	t1/2	ADJ
ejpam-726	323	3	space	space	NOUN
ejpam-726	323	4	is	be	AUX
ejpam-726	323	5	a	a	DET
ejpam-726	323	6	β∗∗t1/2	β∗∗t1/2	ADJ
ejpam-726	323	7	space	space	NOUN
ejpam-726	323	8	.	.	PUNCT
ejpam-726	324	1	proof	proof	NOUN
ejpam-726	324	2	.	.	PUNCT
ejpam-726	325	1	let	let	VERB
ejpam-726	325	2	(	(	PUNCT
ejpam-726	325	3	x	x	X
ejpam-726	325	4	,	,	PUNCT
ejpam-726	325	5	τ	τ	X
ejpam-726	325	6	)	)	PUNCT
ejpam-726	325	7	be	be	VERB
ejpam-726	325	8	a	a	DET
ejpam-726	325	9	t1/2	t1/2	ADJ
ejpam-726	325	10	space	space	NOUN
ejpam-726	325	11	and	and	CCONJ
ejpam-726	325	12	a	a	DET
ejpam-726	325	13	a	a	DET
ejpam-726	325	14	g−closed	g−close	VERB
ejpam-726	325	15	set	set	NOUN
ejpam-726	325	16	of	of	ADP
ejpam-726	325	17	(	(	PUNCT
ejpam-726	325	18	x	x	PROPN
ejpam-726	325	19	,	,	PUNCT
ejpam-726	325	20	τ	τ	PROPN
ejpam-726	325	21	)	)	PUNCT
ejpam-726	325	22	.	.	PUNCT
ejpam-726	326	1	since	since	SCONJ
ejpam-726	326	2	x	x	PRON
ejpam-726	326	3	is	be	AUX
ejpam-726	326	4	a	a	DET
ejpam-726	326	5	t1/2	t1/2	ADJ
ejpam-726	326	6	space	space	NOUN
ejpam-726	326	7	,	,	PUNCT
ejpam-726	326	8	a	a	PRON
ejpam-726	326	9	is	be	AUX
ejpam-726	326	10	closed	closed	ADJ
ejpam-726	326	11	.	.	PUNCT
ejpam-726	327	1	by	by	ADP
ejpam-726	327	2	theorem	theorem	NOUN
ejpam-726	327	3	3	3	NUM
ejpam-726	327	4	,	,	PUNCT
ejpam-726	327	5	a	a	PRON
ejpam-726	327	6	is	be	AUX
ejpam-726	327	7	a	a	DET
ejpam-726	327	8	β∗g−closed	β∗g−close	VERB
ejpam-726	327	9	set	set	NOUN
ejpam-726	327	10	of	of	ADP
ejpam-726	327	11	(	(	PUNCT
ejpam-726	327	12	x	x	PROPN
ejpam-726	327	13	,	,	PUNCT
ejpam-726	327	14	τ	τ	PROPN
ejpam-726	327	15	)	)	PUNCT
ejpam-726	327	16	.	.	PUNCT
ejpam-726	328	1	therefore	therefore	ADV
ejpam-726	328	2	,	,	PUNCT
ejpam-726	328	3	x	x	X
ejpam-726	328	4	is	be	AUX
ejpam-726	328	5	a	a	DET
ejpam-726	328	6	β∗∗t1/2	β∗∗t1/2	PUNCT
ejpam-726	328	7	space	space	NOUN
ejpam-726	328	8	.	.	PUNCT
ejpam-726	329	1	theorem	theorem	VERB
ejpam-726	329	2	13	13	NUM
ejpam-726	329	3	.	.	PUNCT
ejpam-726	330	1	a	a	DET
ejpam-726	330	2	space	space	NOUN
ejpam-726	330	3	(	(	PUNCT
ejpam-726	330	4	x	x	X
ejpam-726	330	5	,	,	PUNCT
ejpam-726	330	6	τ	τ	X
ejpam-726	330	7	)	)	PUNCT
ejpam-726	330	8	is	be	AUX
ejpam-726	330	9	t1/2	t1/2	ADJ
ejpam-726	330	10	space	space	NOUN
ejpam-726	330	11	if	if	SCONJ
ejpam-726	330	12	and	and	CCONJ
ejpam-726	330	13	only	only	ADV
ejpam-726	330	14	if	if	SCONJ
ejpam-726	330	15	it	it	PRON
ejpam-726	330	16	is	be	AUX
ejpam-726	330	17	β∗t1/2	β∗t1/2	PROPN
ejpam-726	330	18	and	and	CCONJ
ejpam-726	330	19	β∗∗t1/2	β∗∗t1/2	PROPN
ejpam-726	330	20	.	.	PUNCT
ejpam-726	331	1	proof	proof	NOUN
ejpam-726	331	2	.	.	PUNCT
ejpam-726	332	1	necessity	necessity	NOUN
ejpam-726	332	2	:	:	PUNCT
ejpam-726	332	3	it	it	PRON
ejpam-726	332	4	follows	follow	VERB
ejpam-726	332	5	from	from	ADP
ejpam-726	332	6	the	the	DET
ejpam-726	332	7	theorems	theorem	NOUN
ejpam-726	332	8	11	11	NUM
ejpam-726	332	9	and	and	CCONJ
ejpam-726	332	10	12	12	NUM
ejpam-726	332	11	.	.	PUNCT
ejpam-726	333	1	sufficiency	sufficiency	NOUN
ejpam-726	333	2	:	:	PUNCT
ejpam-726	333	3	suppose	suppose	VERB
ejpam-726	333	4	that	that	SCONJ
ejpam-726	333	5	x	x	PRON
ejpam-726	333	6	is	be	AUX
ejpam-726	333	7	both	both	PRON
ejpam-726	333	8	β∗t1/2	β∗t1/2	PROPN
ejpam-726	333	9	and	and	CCONJ
ejpam-726	333	10	β∗∗t1/2	β∗∗t1/2	PROPN
ejpam-726	333	11	.	.	PUNCT
ejpam-726	334	1	let	let	VERB
ejpam-726	334	2	a	a	DET
ejpam-726	334	3	be	be	AUX
ejpam-726	334	4	a	a	DET
ejpam-726	334	5	g−closed	g−close	VERB
ejpam-726	334	6	set	set	NOUN
ejpam-726	334	7	of	of	ADP
ejpam-726	334	8	x.	x.	NOUN
ejpam-726	334	9	since	since	SCONJ
ejpam-726	334	10	x	x	PRON
ejpam-726	334	11	is	be	AUX
ejpam-726	334	12	β∗∗t1/2	β∗∗t1/2	PROPN
ejpam-726	334	13	,	,	PUNCT
ejpam-726	334	14	then	then	ADV
ejpam-726	334	15	a	a	PRON
ejpam-726	334	16	is	be	AUX
ejpam-726	334	17	β∗g−closed	β∗g−close	VERB
ejpam-726	334	18	.	.	PUNCT
ejpam-726	335	1	since	since	SCONJ
ejpam-726	335	2	x	x	PRON
ejpam-726	335	3	is	be	AUX
ejpam-726	335	4	a	a	DET
ejpam-726	335	5	β∗t1/2	β∗t1/2	PROPN
ejpam-726	335	6	space	space	NOUN
ejpam-726	335	7	,	,	PUNCT
ejpam-726	335	8	then	then	ADV
ejpam-726	335	9	a	a	PRON
ejpam-726	335	10	is	be	AUX
ejpam-726	335	11	a	a	DET
ejpam-726	335	12	closed	closed	ADJ
ejpam-726	335	13	set	set	NOUN
ejpam-726	335	14	of	of	ADP
ejpam-726	335	15	x.	x.	NOUN
ejpam-726	335	16	thus	thus	ADV
ejpam-726	335	17	x	x	PRON
ejpam-726	335	18	is	be	AUX
ejpam-726	335	19	a	a	DET
ejpam-726	335	20	t1/2	t1/2	ADJ
ejpam-726	335	21	space	space	NOUN
ejpam-726	335	22	.	.	PUNCT
ejpam-726	336	1	4	4	X
ejpam-726	336	2	.	.	X
ejpam-726	336	3	β∗g−open	β∗g−open	NOUN
ejpam-726	336	4	sets	set	NOUN
ejpam-726	336	5	theorem	theorem	VERB
ejpam-726	336	6	14	14	NUM
ejpam-726	336	7	.	.	PUNCT
ejpam-726	337	1	let	let	AUX
ejpam-726	337	2	(	(	PUNCT
ejpam-726	337	3	x	x	X
ejpam-726	337	4	,	,	PUNCT
ejpam-726	337	5	τ	τ	X
ejpam-726	337	6	)	)	PUNCT
ejpam-726	337	7	be	be	VERB
ejpam-726	337	8	a	a	DET
ejpam-726	337	9	topological	topological	ADJ
ejpam-726	337	10	space	space	NOUN
ejpam-726	337	11	.	.	PUNCT
ejpam-726	338	1	a	a	DET
ejpam-726	338	2	⊂	⊂	X
ejpam-726	338	3	x	x	X
ejpam-726	338	4	is	be	AUX
ejpam-726	338	5	β∗g−open	β∗g−open	ADJ
ejpam-726	338	6	if	if	SCONJ
ejpam-726	338	7	and	and	CCONJ
ejpam-726	338	8	only	only	ADV
ejpam-726	338	9	if	if	SCONJ
ejpam-726	338	10	f	f	PROPN
ejpam-726	338	11	⊂	⊂	PROPN
ejpam-726	338	12	int(a	int(a	PROPN
ejpam-726	338	13	)	)	PUNCT
ejpam-726	338	14	whenever	whenever	SCONJ
ejpam-726	338	15	x	x	PUNCT
ejpam-726	338	16	−	−	PROPN
ejpam-726	338	17	f	f	PROPN
ejpam-726	338	18	is	be	AUX
ejpam-726	338	19	a	a	DET
ejpam-726	338	20	β∗-set	β∗-set	NOUN
ejpam-726	338	21	and	and	CCONJ
ejpam-726	338	22	f	f	PROPN
ejpam-726	338	23	⊂	⊂	PROPN
ejpam-726	338	24	a.	a.	NOUN
ejpam-726	338	25	proof	proof	NOUN
ejpam-726	338	26	.	.	PUNCT
ejpam-726	339	1	necessity	necessity	NOUN
ejpam-726	339	2	:	:	PUNCT
ejpam-726	339	3	let	let	VERB
ejpam-726	339	4	a	a	PRON
ejpam-726	339	5	be	be	AUX
ejpam-726	339	6	β∗g−open	β∗g−open	ADJ
ejpam-726	339	7	.	.	PUNCT
ejpam-726	340	1	let	let	VERB
ejpam-726	340	2	x	x	PRON
ejpam-726	340	3	−	−	PROPN
ejpam-726	340	4	f	f	PRON
ejpam-726	340	5	be	be	AUX
ejpam-726	340	6	a	a	DET
ejpam-726	340	7	β∗-set	β∗-set	NOUN
ejpam-726	340	8	and	and	CCONJ
ejpam-726	340	9	f	f	PROPN
ejpam-726	340	10	⊂	⊂	PROPN
ejpam-726	340	11	a.	a.	NOUN
ejpam-726	341	1	then	then	ADV
ejpam-726	341	2	x	x	X
ejpam-726	341	3	−	−	PROPN
ejpam-726	341	4	a⊂	a⊂	VERB
ejpam-726	341	5	x	x	X
ejpam-726	341	6	−	−	PROPN
ejpam-726	341	7	f	f	X
ejpam-726	342	1	where	where	SCONJ
ejpam-726	342	2	x	x	PUNCT
ejpam-726	342	3	−	−	PROPN
ejpam-726	343	1	f	f	PROPN
ejpam-726	343	2	is	be	AUX
ejpam-726	343	3	a	a	DET
ejpam-726	343	4	β∗-set	β∗-set	NOUN
ejpam-726	343	5	.	.	PUNCT
ejpam-726	344	1	β∗g−closedness	β∗g−closedness	PRON
ejpam-726	345	1	of	of	ADP
ejpam-726	345	2	x	x	SYM
ejpam-726	345	3	−	−	PROPN
ejpam-726	345	4	a	a	DET
ejpam-726	345	5	implies	implie	NOUN
ejpam-726	345	6	cl(x	cl(x	PUNCT
ejpam-726	345	7	−	−	PROPN
ejpam-726	345	8	a	a	X
ejpam-726	345	9	)	)	PUNCT
ejpam-726	345	10	⊂	⊂	PROPN
ejpam-726	345	11	x	x	X
ejpam-726	346	1	−	−	PROPN
ejpam-726	346	2	f	f	X
ejpam-726	346	3	.	.	PUNCT
ejpam-726	347	1	so	so	ADV
ejpam-726	347	2	f	f	PROPN
ejpam-726	347	3	⊂	⊂	PROPN
ejpam-726	347	4	int(a	int(a	PROPN
ejpam-726	347	5	)	)	PUNCT
ejpam-726	347	6	.	.	PUNCT
ejpam-726	348	1	a.	a.	NOUN
ejpam-726	348	2	açıkgöz	açıkgöz	PROPN
ejpam-726	348	3	/	/	SYM
ejpam-726	348	4	eur	eur	PROPN
ejpam-726	348	5	.	.	PUNCT
ejpam-726	349	1	j.	j.	PROPN
ejpam-726	349	2	pure	pure	PROPN
ejpam-726	349	3	appl	appl	PROPN
ejpam-726	349	4	.	.	PROPN
ejpam-726	349	5	math	math	PROPN
ejpam-726	349	6	,	,	PUNCT
ejpam-726	349	7	4	4	NUM
ejpam-726	349	8	(	(	PUNCT
ejpam-726	349	9	2011	2011	NUM
ejpam-726	349	10	)	)	PUNCT
ejpam-726	349	11	,	,	PUNCT
ejpam-726	349	12	20	20	NUM
ejpam-726	349	13	-	-	SYM
ejpam-726	349	14	33	33	NUM
ejpam-726	349	15	29	29	NUM
ejpam-726	349	16	sufficiency	sufficiency	NOUN
ejpam-726	349	17	:	:	PUNCT
ejpam-726	349	18	suppose	suppose	VERB
ejpam-726	349	19	x−f	x−f	PROPN
ejpam-726	349	20	is	be	AUX
ejpam-726	349	21	a	a	DET
ejpam-726	349	22	β∗-set	β∗-set	NOUN
ejpam-726	349	23	and	and	CCONJ
ejpam-726	349	24	f	f	PROPN
ejpam-726	349	25	⊂	⊂	PROPN
ejpam-726	349	26	a	a	DET
ejpam-726	349	27	imply	imply	VERB
ejpam-726	349	28	f	f	PROPN
ejpam-726	349	29	⊂	⊂	PROPN
ejpam-726	349	30	int(a	int(a	PROPN
ejpam-726	349	31	)	)	PUNCT
ejpam-726	349	32	.	.	PUNCT
ejpam-726	350	1	let	let	VERB
ejpam-726	350	2	x−a⊂	x−a⊂	PROPN
ejpam-726	350	3	u	u	PRON
ejpam-726	350	4	where	where	SCONJ
ejpam-726	350	5	u	u	NOUN
ejpam-726	350	6	is	be	AUX
ejpam-726	350	7	a	a	DET
ejpam-726	350	8	β∗-set	β∗-set	NOUN
ejpam-726	350	9	.	.	PUNCT
ejpam-726	351	1	then	then	ADV
ejpam-726	351	2	x−u	x−u	PROPN
ejpam-726	351	3	⊂	⊂	PROPN
ejpam-726	351	4	a	a	PRON
ejpam-726	351	5	and	and	CCONJ
ejpam-726	351	6	x−(x−u	x−(x−u	NOUN
ejpam-726	351	7	)	)	PUNCT
ejpam-726	351	8	is	be	AUX
ejpam-726	351	9	a	a	DET
ejpam-726	351	10	β∗-set	β∗-set	NOUN
ejpam-726	351	11	.	.	PUNCT
ejpam-726	352	1	by	by	ADP
ejpam-726	352	2	hypothesis	hypothesis	NOUN
ejpam-726	352	3	x−u	x−u	X
ejpam-726	352	4	⊂	⊂	PROPN
ejpam-726	352	5	int(a	int(a	PROPN
ejpam-726	352	6	)	)	PUNCT
ejpam-726	352	7	,	,	PUNCT
ejpam-726	352	8	that	that	PRON
ejpam-726	352	9	is	be	AUX
ejpam-726	352	10	x	x	X
ejpam-726	352	11	−	−	PROPN
ejpam-726	352	12	int(a	int(a	PROPN
ejpam-726	352	13	)	)	PUNCT
ejpam-726	352	14	⊂	⊂	PROPN
ejpam-726	352	15	u	u	NOUN
ejpam-726	352	16	and	and	CCONJ
ejpam-726	352	17	cl(x	cl(x	NUM
ejpam-726	352	18	−a)⊂	−a)⊂	X
ejpam-726	352	19	u	u	PROPN
ejpam-726	352	20	.	.	PUNCT
ejpam-726	353	1	so	so	ADV
ejpam-726	353	2	,	,	PUNCT
ejpam-726	353	3	x	x	PRON
ejpam-726	353	4	−a	−a	NOUN
ejpam-726	353	5	is	be	AUX
ejpam-726	353	6	β∗g−closed	β∗g−close	VERB
ejpam-726	353	7	and	and	CCONJ
ejpam-726	353	8	a	a	PRON
ejpam-726	353	9	is	be	AUX
ejpam-726	353	10	β∗g−open	β∗g−open	NOUN
ejpam-726	353	11	.	.	PUNCT
ejpam-726	354	1	theorem	theorem	NOUN
ejpam-726	354	2	15	15	NUM
ejpam-726	354	3	.	.	PUNCT
ejpam-726	355	1	if	if	SCONJ
ejpam-726	355	2	a	a	PRON
ejpam-726	355	3	is	be	AUX
ejpam-726	355	4	a	a	DET
ejpam-726	355	5	β∗g−open	β∗g−open	ADJ
ejpam-726	355	6	set	set	NOUN
ejpam-726	355	7	of	of	ADP
ejpam-726	355	8	(	(	PUNCT
ejpam-726	355	9	x	x	PROPN
ejpam-726	355	10	,	,	PUNCT
ejpam-726	355	11	τ	τ	X
ejpam-726	355	12	)	)	PUNCT
ejpam-726	355	13	such	such	ADJ
ejpam-726	355	14	that	that	SCONJ
ejpam-726	355	15	int(a	int(a	AUX
ejpam-726	355	16	)	)	PUNCT
ejpam-726	355	17	⊂	⊂	PROPN
ejpam-726	356	1	b	b	X
ejpam-726	356	2	⊂	⊂	PROPN
ejpam-726	356	3	a	a	X
ejpam-726	356	4	,	,	PUNCT
ejpam-726	356	5	then	then	ADV
ejpam-726	356	6	b	b	PROPN
ejpam-726	356	7	is	be	AUX
ejpam-726	356	8	also	also	ADV
ejpam-726	356	9	a	a	DET
ejpam-726	356	10	β∗g−open	β∗g−open	ADJ
ejpam-726	356	11	set	set	NOUN
ejpam-726	356	12	of	of	ADP
ejpam-726	356	13	(	(	PUNCT
ejpam-726	356	14	x	x	PROPN
ejpam-726	356	15	,	,	PUNCT
ejpam-726	356	16	τ	τ	PROPN
ejpam-726	356	17	)	)	PUNCT
ejpam-726	356	18	.	.	PUNCT
ejpam-726	357	1	proof	proof	NOUN
ejpam-726	357	2	.	.	PUNCT
ejpam-726	358	1	this	this	PRON
ejpam-726	358	2	is	be	AUX
ejpam-726	358	3	an	an	DET
ejpam-726	358	4	immediate	immediate	ADJ
ejpam-726	358	5	consequence	consequence	NOUN
ejpam-726	358	6	of	of	ADP
ejpam-726	358	7	theorem	theorem	ADJ
ejpam-726	358	8	6	6	NUM
ejpam-726	358	9	.	.	PUNCT
ejpam-726	358	10	definition	definition	NOUN
ejpam-726	358	11	9	9	NUM
ejpam-726	358	12	.	.	PUNCT
ejpam-726	359	1	a	a	DET
ejpam-726	359	2	function	function	NOUN
ejpam-726	359	3	f	f	NOUN
ejpam-726	359	4	:	:	PUNCT
ejpam-726	359	5	(	(	PUNCT
ejpam-726	359	6	x	x	X
ejpam-726	359	7	,	,	PUNCT
ejpam-726	359	8	τ)→	τ)→	PROPN
ejpam-726	359	9	(	(	PUNCT
ejpam-726	359	10	y	y	PROPN
ejpam-726	359	11	,	,	PUNCT
ejpam-726	359	12	σ	σ	PROPN
ejpam-726	359	13	)	)	PUNCT
ejpam-726	359	14	is	be	AUX
ejpam-726	359	15	said	say	VERB
ejpam-726	359	16	to	to	PART
ejpam-726	359	17	be	be	AUX
ejpam-726	359	18	(	(	PUNCT
ejpam-726	359	19	a	a	X
ejpam-726	359	20	)	)	PUNCT
ejpam-726	359	21	β∗g−open	β∗g−open	NOUN
ejpam-726	359	22	if	if	SCONJ
ejpam-726	359	23	f	f	PROPN
ejpam-726	359	24	(	(	PUNCT
ejpam-726	359	25	v	v	NOUN
ejpam-726	359	26	)	)	PUNCT
ejpam-726	359	27	is	be	AUX
ejpam-726	359	28	β∗g−open	β∗g−open	ADJ
ejpam-726	359	29	in	in	ADP
ejpam-726	359	30	y	y	NOUN
ejpam-726	359	31	for	for	ADP
ejpam-726	359	32	every	every	DET
ejpam-726	359	33	open	open	ADJ
ejpam-726	359	34	set	set	VERB
ejpam-726	359	35	v	v	NOUN
ejpam-726	359	36	of	of	ADP
ejpam-726	359	37	x	x	X
ejpam-726	359	38	.	.	PUNCT
ejpam-726	360	1	(	(	PUNCT
ejpam-726	360	2	b	b	X
ejpam-726	360	3	)	)	PUNCT
ejpam-726	360	4	β∗g−closed	β∗g−close	VERB
ejpam-726	360	5	if	if	SCONJ
ejpam-726	360	6	f	f	PROPN
ejpam-726	360	7	(	(	PUNCT
ejpam-726	360	8	f	f	X
ejpam-726	360	9	)	)	PUNCT
ejpam-726	360	10	is	be	AUX
ejpam-726	360	11	β∗g−closed	β∗g−close	VERB
ejpam-726	360	12	in	in	ADP
ejpam-726	360	13	y	y	PROPN
ejpam-726	360	14	for	for	ADP
ejpam-726	360	15	every	every	DET
ejpam-726	360	16	closed	close	VERB
ejpam-726	360	17	set	set	VERB
ejpam-726	360	18	f	f	PROPN
ejpam-726	360	19	of	of	ADP
ejpam-726	360	20	x	x	PROPN
ejpam-726	360	21	.	.	PUNCT
ejpam-726	361	1	(	(	PUNCT
ejpam-726	361	2	c	c	X
ejpam-726	361	3	)	)	PUNCT
ejpam-726	361	4	β∗g−preserving	β∗g−preserve	VERB
ejpam-726	361	5	(	(	PUNCT
ejpam-726	361	6	resp	resp	NOUN
ejpam-726	361	7	.	.	PUNCT
ejpam-726	362	1	contra	contra	PROPN
ejpam-726	362	2	β∗g−open	β∗g−open	PROPN
ejpam-726	362	3	)	)	PUNCT
ejpam-726	363	1	if	if	SCONJ
ejpam-726	363	2	f	f	PROPN
ejpam-726	363	3	(	(	PUNCT
ejpam-726	363	4	f	f	X
ejpam-726	363	5	)	)	PUNCT
ejpam-726	363	6	is	be	AUX
ejpam-726	363	7	β∗g−closed	β∗g−close	VERB
ejpam-726	363	8	(	(	PUNCT
ejpam-726	363	9	resp	resp	NOUN
ejpam-726	363	10	.	.	PUNCT
ejpam-726	364	1	β∗g−closed	β∗g−close	VERB
ejpam-726	364	2	)	)	PUNCT
ejpam-726	365	1	in	in	ADP
ejpam-726	365	2	y	y	PROPN
ejpam-726	365	3	for	for	ADP
ejpam-726	365	4	every	every	DET
ejpam-726	365	5	β∗g−closed	β∗g−close	VERB
ejpam-726	365	6	(	(	PUNCT
ejpam-726	365	7	resp	resp	NOUN
ejpam-726	365	8	.	.	PUNCT
ejpam-726	366	1	open	open	ADJ
ejpam-726	366	2	)	)	PUNCT
ejpam-726	366	3	set	set	VERB
ejpam-726	366	4	f	f	PROPN
ejpam-726	366	5	of	of	ADP
ejpam-726	366	6	x	x	PROPN
ejpam-726	366	7	.	.	PUNCT
ejpam-726	367	1	5	5	NUM
ejpam-726	367	2	.	.	X
ejpam-726	367	3	β∗g−continuity	β∗g−continuity	NOUN
ejpam-726	367	4	and	and	CCONJ
ejpam-726	367	5	β∗g−irresoluteness	β∗g−irresoluteness	NOUN
ejpam-726	367	6	definition	definition	NOUN
ejpam-726	367	7	10	10	NUM
ejpam-726	367	8	.	.	PUNCT
ejpam-726	368	1	a	a	DET
ejpam-726	368	2	function	function	NOUN
ejpam-726	368	3	f	f	NOUN
ejpam-726	368	4	:	:	PUNCT
ejpam-726	368	5	(	(	PUNCT
ejpam-726	368	6	x	x	X
ejpam-726	368	7	,	,	PUNCT
ejpam-726	368	8	τ)→	τ)→	PROPN
ejpam-726	368	9	(	(	PUNCT
ejpam-726	368	10	y	y	PROPN
ejpam-726	368	11	,	,	PUNCT
ejpam-726	368	12	σ	σ	PROPN
ejpam-726	368	13	)	)	PUNCT
ejpam-726	368	14	is	be	AUX
ejpam-726	368	15	said	say	VERB
ejpam-726	368	16	to	to	PART
ejpam-726	368	17	be	be	AUX
ejpam-726	368	18	(	(	PUNCT
ejpam-726	368	19	a	a	X
ejpam-726	368	20	)	)	PUNCT
ejpam-726	368	21	π	π	NOUN
ejpam-726	368	22	-	-	ADJ
ejpam-726	368	23	continuous	continuous	ADJ
ejpam-726	368	24	[	[	X
ejpam-726	368	25	9	9	NUM
ejpam-726	368	26	]	]	PUNCT
ejpam-726	368	27	(	(	PUNCT
ejpam-726	368	28	resp	resp	NOUN
ejpam-726	368	29	.	.	PUNCT
ejpam-726	369	1	πg−continuous	πg−continuous	ADJ
ejpam-726	370	1	[	[	X
ejpam-726	370	2	8	8	NUM
ejpam-726	370	3	]	]	PUNCT
ejpam-726	370	4	,	,	PUNCT
ejpam-726	370	5	πgp	πgp	X
ejpam-726	370	6	-	-	ADJ
ejpam-726	370	7	continuous	continuous	ADJ
ejpam-726	370	8	[	[	X
ejpam-726	370	9	22	22	NUM
ejpam-726	370	10	]	]	PUNCT
ejpam-726	370	11	,	,	PUNCT
ejpam-726	370	12	πgs	πgs	ADV
ejpam-726	370	13	-	-	ADJ
ejpam-726	370	14	continuous	continuous	ADJ
ejpam-726	370	15	[	[	X
ejpam-726	370	16	2	2	NUM
ejpam-726	370	17	]	]	PUNCT
ejpam-726	370	18	)	)	PUNCT
ejpam-726	370	19	if	if	SCONJ
ejpam-726	370	20	f	f	PROPN
ejpam-726	370	21	−1(f	−1(f	PROPN
ejpam-726	370	22	)	)	PUNCT
ejpam-726	370	23	is	be	AUX
ejpam-726	370	24	π	π	PROPN
ejpam-726	370	25	-	-	ADJ
ejpam-726	370	26	closed	closed	ADJ
ejpam-726	370	27	(	(	PUNCT
ejpam-726	370	28	resp	resp	NOUN
ejpam-726	370	29	.	.	PUNCT
ejpam-726	370	30	πg−closed	πg−close	VERB
ejpam-726	370	31	,	,	PUNCT
ejpam-726	370	32	πgp−closed	πgp−close	VERB
ejpam-726	370	33	,	,	PUNCT
ejpam-726	370	34	πgs−closed	πgs−closed	PUNCT
ejpam-726	370	35	)	)	PUNCT
ejpam-726	370	36	in	in	ADP
ejpam-726	370	37	(	(	PUNCT
ejpam-726	370	38	x	x	INTJ
ejpam-726	370	39	,	,	PUNCT
ejpam-726	370	40	τ	τ	PROPN
ejpam-726	370	41	)	)	PUNCT
ejpam-726	370	42	for	for	ADP
ejpam-726	370	43	every	every	DET
ejpam-726	370	44	closed	close	VERB
ejpam-726	370	45	set	set	VERB
ejpam-726	370	46	f	f	PROPN
ejpam-726	370	47	of	of	ADP
ejpam-726	370	48	(	(	PUNCT
ejpam-726	370	49	y	y	PROPN
ejpam-726	370	50	,	,	PUNCT
ejpam-726	370	51	σ	σ	PROPN
ejpam-726	370	52	)	)	PUNCT
ejpam-726	370	53	;	;	PUNCT
ejpam-726	370	54	(	(	PUNCT
ejpam-726	370	55	b	b	X
ejpam-726	370	56	)	)	PUNCT
ejpam-726	370	57	lc	lc	NOUN
ejpam-726	370	58	-	-	PUNCT
ejpam-726	370	59	continuous	continuous	ADJ
ejpam-726	370	60	[	[	X
ejpam-726	370	61	4	4	NUM
ejpam-726	370	62	]	]	X
ejpam-726	370	63	if	if	SCONJ
ejpam-726	370	64	f	f	PROPN
ejpam-726	370	65	−1(f	−1(f	PROPN
ejpam-726	370	66	)	)	PUNCT
ejpam-726	370	67	is	be	AUX
ejpam-726	370	68	a	a	DET
ejpam-726	370	69	lc	lc	NOUN
ejpam-726	370	70	-	-	PUNCT
ejpam-726	370	71	set	set	NOUN
ejpam-726	370	72	in	in	ADP
ejpam-726	370	73	(	(	PUNCT
ejpam-726	370	74	x	x	INTJ
ejpam-726	370	75	,	,	PUNCT
ejpam-726	370	76	τ	τ	PROPN
ejpam-726	370	77	)	)	PUNCT
ejpam-726	370	78	for	for	ADP
ejpam-726	370	79	every	every	DET
ejpam-726	370	80	closed	close	VERB
ejpam-726	370	81	set	set	VERB
ejpam-726	370	82	f	f	PROPN
ejpam-726	370	83	of	of	ADP
ejpam-726	370	84	(	(	PUNCT
ejpam-726	370	85	y	y	PROPN
ejpam-726	370	86	,	,	PUNCT
ejpam-726	370	87	σ	σ	PROPN
ejpam-726	370	88	)	)	PUNCT
ejpam-726	370	89	;	;	PUNCT
ejpam-726	370	90	(	(	PUNCT
ejpam-726	370	91	c	c	X
ejpam-726	370	92	)	)	PUNCT
ejpam-726	370	93	g∗-continuous	g∗-continuous	ADJ
ejpam-726	370	94	[	[	X
ejpam-726	370	95	17	17	NUM
ejpam-726	370	96	]	]	PUNCT
ejpam-726	370	97	if	if	SCONJ
ejpam-726	370	98	f	f	PROPN
ejpam-726	370	99	−1(f	−1(f	PROPN
ejpam-726	370	100	)	)	PUNCT
ejpam-726	370	101	is	be	AUX
ejpam-726	370	102	g∗-closed	g∗-close	VERB
ejpam-726	370	103	in	in	ADP
ejpam-726	370	104	(	(	PUNCT
ejpam-726	370	105	x	x	INTJ
ejpam-726	370	106	,	,	PUNCT
ejpam-726	370	107	τ	τ	PROPN
ejpam-726	370	108	)	)	PUNCT
ejpam-726	370	109	for	for	ADP
ejpam-726	370	110	every	every	DET
ejpam-726	370	111	closed	close	VERB
ejpam-726	370	112	set	set	VERB
ejpam-726	370	113	f	f	PROPN
ejpam-726	370	114	of	of	ADP
ejpam-726	370	115	(	(	PUNCT
ejpam-726	370	116	y	y	PROPN
ejpam-726	370	117	,	,	PUNCT
ejpam-726	370	118	σ	σ	PROPN
ejpam-726	370	119	)	)	PUNCT
ejpam-726	370	120	;	;	PUNCT
ejpam-726	370	121	(	(	PUNCT
ejpam-726	370	122	d	d	X
ejpam-726	370	123	)	)	PUNCT
ejpam-726	370	124	g−continuous	g−continuous	PROPN
ejpam-726	371	1	[	[	X
ejpam-726	371	2	18	18	NUM
ejpam-726	371	3	]	]	PUNCT
ejpam-726	371	4	(	(	PUNCT
ejpam-726	371	5	resp	resp	NOUN
ejpam-726	371	6	.	.	PUNCT
ejpam-726	372	1	gp	gp	NOUN
ejpam-726	372	2	-	-	NOUN
ejpam-726	372	3	continuous	continuous	ADJ
ejpam-726	372	4	[	[	X
ejpam-726	372	5	19	19	NUM
ejpam-726	372	6	]	]	PUNCT
ejpam-726	372	7	,	,	PUNCT
ejpam-726	372	8	gs	gs	NOUN
ejpam-726	372	9	-	-	ADJ
ejpam-726	372	10	continuous	continuous	ADJ
ejpam-726	372	11	[	[	X
ejpam-726	372	12	1	1	NUM
ejpam-726	372	13	]	]	PUNCT
ejpam-726	372	14	)	)	PUNCT
ejpam-726	372	15	if	if	SCONJ
ejpam-726	372	16	f	f	PROPN
ejpam-726	372	17	−1(f	−1(f	PROPN
ejpam-726	372	18	)	)	PUNCT
ejpam-726	372	19	is	be	AUX
ejpam-726	372	20	g−closed	g−close	VERB
ejpam-726	372	21	(	(	PUNCT
ejpam-726	372	22	resp	resp	NOUN
ejpam-726	372	23	.	.	PUNCT
ejpam-726	373	1	gp−closed	gp−close	VERB
ejpam-726	373	2	,	,	PUNCT
ejpam-726	373	3	gs−closed	gs−close	VERB
ejpam-726	373	4	)	)	PUNCT
ejpam-726	373	5	in	in	ADP
ejpam-726	373	6	(	(	PUNCT
ejpam-726	373	7	x	x	INTJ
ejpam-726	373	8	,	,	PUNCT
ejpam-726	373	9	τ	τ	PROPN
ejpam-726	373	10	)	)	PUNCT
ejpam-726	373	11	for	for	ADP
ejpam-726	373	12	every	every	DET
ejpam-726	373	13	closed	close	VERB
ejpam-726	373	14	set	set	VERB
ejpam-726	373	15	f	f	PROPN
ejpam-726	373	16	of	of	ADP
ejpam-726	373	17	(	(	PUNCT
ejpam-726	373	18	y	y	PROPN
ejpam-726	373	19	,	,	PUNCT
ejpam-726	373	20	σ	σ	PROPN
ejpam-726	373	21	)	)	PUNCT
ejpam-726	373	22	.	.	PUNCT
ejpam-726	374	1	definition	definition	NOUN
ejpam-726	374	2	11	11	NUM
ejpam-726	374	3	.	.	PUNCT
ejpam-726	375	1	a	a	DET
ejpam-726	375	2	function	function	NOUN
ejpam-726	375	3	f	f	NOUN
ejpam-726	375	4	:	:	PUNCT
ejpam-726	375	5	(	(	PUNCT
ejpam-726	375	6	x	x	X
ejpam-726	375	7	,	,	PUNCT
ejpam-726	375	8	τ)→	τ)→	PROPN
ejpam-726	375	9	(	(	PUNCT
ejpam-726	375	10	y	y	PROPN
ejpam-726	375	11	,	,	PUNCT
ejpam-726	375	12	σ	σ	PROPN
ejpam-726	375	13	)	)	PUNCT
ejpam-726	375	14	is	be	AUX
ejpam-726	375	15	said	say	VERB
ejpam-726	375	16	to	to	PART
ejpam-726	375	17	be	be	AUX
ejpam-726	375	18	β∗g−continuous	β∗g−continuous	PROPN
ejpam-726	375	19	(	(	PUNCT
ejpam-726	375	20	resp	resp	NOUN
ejpam-726	375	21	.	.	PUNCT
ejpam-726	376	1	β∗gp	β∗gp	ADJ
ejpam-726	376	2	-	-	PUNCT
ejpam-726	376	3	continuous	continuous	ADJ
ejpam-726	376	4	,	,	PUNCT
ejpam-726	376	5	β∗gs	β∗gs	ADJ
ejpam-726	376	6	-	-	ADJ
ejpam-726	376	7	continuous	continuous	ADJ
ejpam-726	376	8	)	)	PUNCT
ejpam-726	376	9	if	if	SCONJ
ejpam-726	376	10	f	f	PROPN
ejpam-726	376	11	−1(f	−1(f	PROPN
ejpam-726	376	12	)	)	PUNCT
ejpam-726	376	13	is	be	AUX
ejpam-726	376	14	β∗g−closed	β∗g−close	VERB
ejpam-726	376	15	(	(	PUNCT
ejpam-726	376	16	resp	resp	NOUN
ejpam-726	376	17	.	.	PUNCT
ejpam-726	377	1	β∗gp−closed	β∗gp−closed	PROPN
ejpam-726	377	2	,	,	PUNCT
ejpam-726	377	3	β∗gs−closed	β∗gs−closed	PROPN
ejpam-726	377	4	)	)	PUNCT
ejpam-726	377	5	in	in	ADP
ejpam-726	377	6	(	(	PUNCT
ejpam-726	377	7	x	x	INTJ
ejpam-726	377	8	,	,	PUNCT
ejpam-726	377	9	τ	τ	PROPN
ejpam-726	377	10	)	)	PUNCT
ejpam-726	377	11	for	for	ADP
ejpam-726	377	12	every	every	DET
ejpam-726	377	13	closed	close	VERB
ejpam-726	377	14	set	set	VERB
ejpam-726	377	15	f	f	PROPN
ejpam-726	377	16	of	of	ADP
ejpam-726	377	17	(	(	PUNCT
ejpam-726	377	18	y	y	PROPN
ejpam-726	377	19	,	,	PUNCT
ejpam-726	377	20	σ	σ	PROPN
ejpam-726	377	21	)	)	PUNCT
ejpam-726	377	22	.	.	PUNCT
ejpam-726	378	1	definition	definition	NOUN
ejpam-726	378	2	12	12	NUM
ejpam-726	378	3	.	.	PUNCT
ejpam-726	379	1	a	a	DET
ejpam-726	379	2	function	function	NOUN
ejpam-726	379	3	f	f	NOUN
ejpam-726	379	4	:	:	PUNCT
ejpam-726	379	5	(	(	PUNCT
ejpam-726	379	6	x	x	X
ejpam-726	379	7	,	,	PUNCT
ejpam-726	379	8	τ)→	τ)→	PROPN
ejpam-726	379	9	(	(	PUNCT
ejpam-726	379	10	y	y	PROPN
ejpam-726	379	11	,	,	PUNCT
ejpam-726	379	12	σ	σ	PROPN
ejpam-726	379	13	)	)	PUNCT
ejpam-726	379	14	is	be	AUX
ejpam-726	379	15	said	say	VERB
ejpam-726	379	16	to	to	PART
ejpam-726	379	17	be	be	AUX
ejpam-726	379	18	β∗g−irresolute	β∗g−irresolute	NOUN
ejpam-726	379	19	(	(	PUNCT
ejpam-726	379	20	resp	resp	NOUN
ejpam-726	379	21	.	.	PUNCT
ejpam-726	380	1	β∗-irresolute	β∗-irresolute	VERB
ejpam-726	380	2	)	)	PUNCT
ejpam-726	380	3	if	if	SCONJ
ejpam-726	380	4	f	f	PROPN
ejpam-726	380	5	−1(v	−1(v	PROPN
ejpam-726	380	6	)	)	PUNCT
ejpam-726	380	7	is	be	AUX
ejpam-726	380	8	β∗g−closed	β∗g−close	VERB
ejpam-726	380	9	(	(	PUNCT
ejpam-726	380	10	resp	resp	NOUN
ejpam-726	380	11	.	.	PUNCT
ejpam-726	381	1	β∗-set	β∗-set	PROPN
ejpam-726	381	2	)	)	PUNCT
ejpam-726	382	1	in	in	SCONJ
ejpam-726	382	2	x	x	PUNCT
ejpam-726	382	3	for	for	ADP
ejpam-726	382	4	every	every	DET
ejpam-726	382	5	β∗g−closed	β∗g−close	VERB
ejpam-726	382	6	(	(	PUNCT
ejpam-726	382	7	resp	resp	NOUN
ejpam-726	382	8	.	.	PUNCT
ejpam-726	382	9	β∗-set	β∗-set	PROPN
ejpam-726	382	10	)	)	PUNCT
ejpam-726	382	11	set	set	VERB
ejpam-726	382	12	v	v	NOUN
ejpam-726	382	13	of	of	ADP
ejpam-726	382	14	y	y	PROPN
ejpam-726	382	15	.	.	PUNCT
ejpam-726	383	1	definition	definition	NOUN
ejpam-726	383	2	13	13	NUM
ejpam-726	383	3	.	.	PUNCT
ejpam-726	384	1	a	a	DET
ejpam-726	384	2	function	function	NOUN
ejpam-726	384	3	f	f	NOUN
ejpam-726	384	4	:	:	PUNCT
ejpam-726	384	5	(	(	PUNCT
ejpam-726	384	6	x	x	X
ejpam-726	384	7	,	,	PUNCT
ejpam-726	384	8	τ	τ	PROPN
ejpam-726	384	9	)	)	PUNCT
ejpam-726	384	10	→	→	SYM
ejpam-726	384	11	(	(	PUNCT
ejpam-726	384	12	y	y	PROPN
ejpam-726	384	13	,	,	PUNCT
ejpam-726	384	14	σ	σ	PROPN
ejpam-726	384	15	)	)	PUNCT
ejpam-726	384	16	is	be	AUX
ejpam-726	384	17	said	say	VERB
ejpam-726	384	18	to	to	PART
ejpam-726	384	19	be	be	AUX
ejpam-726	384	20	perfectly	perfectly	ADV
ejpam-726	384	21	β∗g−continuous	β∗g−continuous	PRON
ejpam-726	384	22	(	(	PUNCT
ejpam-726	384	23	resp	resp	NOUN
ejpam-726	384	24	.	.	PUNCT
ejpam-726	385	1	strongly	strongly	ADV
ejpam-726	385	2	β∗g−continuous	β∗g−continuous	NUM
ejpam-726	385	3	)	)	PUNCT
ejpam-726	385	4	if	if	SCONJ
ejpam-726	385	5	f	f	PROPN
ejpam-726	385	6	−1(v	−1(v	PROPN
ejpam-726	385	7	)	)	PUNCT
ejpam-726	385	8	is	be	AUX
ejpam-726	385	9	clopen	clopen	ADJ
ejpam-726	385	10	(	(	PUNCT
ejpam-726	385	11	resp	resp	NOUN
ejpam-726	385	12	.	.	PUNCT
ejpam-726	386	1	open	open	ADJ
ejpam-726	386	2	)	)	PUNCT
ejpam-726	386	3	in	in	ADP
ejpam-726	386	4	(	(	PUNCT
ejpam-726	386	5	x	x	INTJ
ejpam-726	386	6	,	,	PUNCT
ejpam-726	386	7	τ	τ	PROPN
ejpam-726	386	8	)	)	PUNCT
ejpam-726	386	9	for	for	ADP
ejpam-726	386	10	every	every	DET
ejpam-726	386	11	β∗g−open	β∗g−open	NUM
ejpam-726	386	12	set	set	VERB
ejpam-726	386	13	v	v	NUM
ejpam-726	386	14	of	of	ADP
ejpam-726	386	15	(	(	PUNCT
ejpam-726	386	16	y	y	PROPN
ejpam-726	386	17	,	,	PUNCT
ejpam-726	386	18	σ	σ	PROPN
ejpam-726	386	19	)	)	PUNCT
ejpam-726	386	20	.	.	PUNCT
ejpam-726	387	1	definition	definition	NOUN
ejpam-726	387	2	14	14	NUM
ejpam-726	387	3	.	.	PUNCT
ejpam-726	388	1	a	a	DET
ejpam-726	388	2	function	function	NOUN
ejpam-726	388	3	f	f	NOUN
ejpam-726	388	4	:	:	PUNCT
ejpam-726	388	5	(	(	PUNCT
ejpam-726	388	6	x	x	X
ejpam-726	388	7	,	,	PUNCT
ejpam-726	388	8	τ)→	τ)→	PROPN
ejpam-726	388	9	(	(	PUNCT
ejpam-726	388	10	y	y	PROPN
ejpam-726	388	11	,	,	PUNCT
ejpam-726	388	12	σ	σ	PROPN
ejpam-726	388	13	)	)	PUNCT
ejpam-726	388	14	is	be	AUX
ejpam-726	388	15	said	say	VERB
ejpam-726	388	16	to	to	PART
ejpam-726	388	17	be	be	AUX
ejpam-726	388	18	almost	almost	ADV
ejpam-726	388	19	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	388	20	if	if	SCONJ
ejpam-726	388	21	f	f	PROPN
ejpam-726	388	22	−1(v	−1(v	PROPN
ejpam-726	388	23	)	)	PUNCT
ejpam-726	388	24	is	be	AUX
ejpam-726	388	25	β∗g−open	β∗g−open	ADJ
ejpam-726	388	26	in	in	X
ejpam-726	388	27	(	(	PUNCT
ejpam-726	388	28	x	x	INTJ
ejpam-726	388	29	,	,	PUNCT
ejpam-726	388	30	τ	τ	PROPN
ejpam-726	388	31	)	)	PUNCT
ejpam-726	388	32	for	for	ADP
ejpam-726	388	33	every	every	DET
ejpam-726	388	34	regular	regular	ADJ
ejpam-726	388	35	open	open	ADJ
ejpam-726	388	36	set	set	NOUN
ejpam-726	388	37	v	v	NUM
ejpam-726	388	38	of	of	ADP
ejpam-726	388	39	(	(	PUNCT
ejpam-726	388	40	y	y	PROPN
ejpam-726	388	41	,	,	PUNCT
ejpam-726	388	42	σ	σ	PROPN
ejpam-726	388	43	)	)	PUNCT
ejpam-726	388	44	.	.	PUNCT
ejpam-726	389	1	a.	a.	NOUN
ejpam-726	389	2	açıkgöz	açıkgöz	PROPN
ejpam-726	389	3	/	/	SYM
ejpam-726	389	4	eur	eur	PROPN
ejpam-726	389	5	.	.	PUNCT
ejpam-726	390	1	j.	j.	PROPN
ejpam-726	390	2	pure	pure	PROPN
ejpam-726	390	3	appl	appl	PROPN
ejpam-726	390	4	.	.	PROPN
ejpam-726	390	5	math	math	PROPN
ejpam-726	390	6	,	,	PUNCT
ejpam-726	390	7	4	4	NUM
ejpam-726	390	8	(	(	PUNCT
ejpam-726	390	9	2011	2011	NUM
ejpam-726	390	10	)	)	PUNCT
ejpam-726	390	11	,	,	PUNCT
ejpam-726	390	12	20	20	NUM
ejpam-726	390	13	-	-	SYM
ejpam-726	390	14	33	33	NUM
ejpam-726	390	15	30	30	NUM
ejpam-726	390	16	definition	definition	NOUN
ejpam-726	390	17	15	15	NUM
ejpam-726	390	18	.	.	PUNCT
ejpam-726	391	1	a	a	DET
ejpam-726	391	2	function	function	NOUN
ejpam-726	391	3	f	f	NOUN
ejpam-726	391	4	:	:	PUNCT
ejpam-726	391	5	(	(	PUNCT
ejpam-726	391	6	x	x	X
ejpam-726	391	7	,	,	PUNCT
ejpam-726	391	8	τ)→	τ)→	PROPN
ejpam-726	391	9	(	(	PUNCT
ejpam-726	391	10	y	y	PROPN
ejpam-726	391	11	,	,	PUNCT
ejpam-726	391	12	σ	σ	PROPN
ejpam-726	391	13	)	)	PUNCT
ejpam-726	391	14	is	be	AUX
ejpam-726	391	15	said	say	VERB
ejpam-726	391	16	to	to	PART
ejpam-726	391	17	be	be	AUX
ejpam-726	391	18	contra	contra	PROPN
ejpam-726	391	19	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	391	20	if	if	SCONJ
ejpam-726	391	21	f	f	PROPN
ejpam-726	391	22	−1(v	−1(v	PROPN
ejpam-726	391	23	)	)	PUNCT
ejpam-726	391	24	is	be	AUX
ejpam-726	391	25	β∗g−closed	β∗g−close	VERB
ejpam-726	391	26	in	in	ADP
ejpam-726	391	27	(	(	PUNCT
ejpam-726	391	28	x	x	INTJ
ejpam-726	391	29	,	,	PUNCT
ejpam-726	391	30	τ	τ	PROPN
ejpam-726	391	31	)	)	PUNCT
ejpam-726	391	32	for	for	ADP
ejpam-726	391	33	every	every	DET
ejpam-726	391	34	open	open	ADJ
ejpam-726	391	35	set	set	VERB
ejpam-726	391	36	v	v	NUM
ejpam-726	391	37	of	of	ADP
ejpam-726	391	38	(	(	PUNCT
ejpam-726	391	39	y	y	PROPN
ejpam-726	391	40	,	,	PUNCT
ejpam-726	391	41	σ	σ	PROPN
ejpam-726	391	42	)	)	PUNCT
ejpam-726	391	43	.	.	PUNCT
ejpam-726	392	1	theorem	theorem	VERB
ejpam-726	392	2	16	16	NUM
ejpam-726	392	3	.	.	PUNCT
ejpam-726	393	1	for	for	ADP
ejpam-726	393	2	a	a	DET
ejpam-726	393	3	function	function	NOUN
ejpam-726	393	4	f	f	NOUN
ejpam-726	393	5	:	:	PUNCT
ejpam-726	393	6	(	(	PUNCT
ejpam-726	393	7	x	x	X
ejpam-726	393	8	,	,	PUNCT
ejpam-726	393	9	τ)→	τ)→	PROPN
ejpam-726	393	10	(	(	PUNCT
ejpam-726	393	11	y	y	PROPN
ejpam-726	393	12	,	,	PUNCT
ejpam-726	393	13	σ	σ	PROPN
ejpam-726	393	14	)	)	PUNCT
ejpam-726	393	15	,	,	PUNCT
ejpam-726	393	16	the	the	DET
ejpam-726	393	17	following	follow	VERB
ejpam-726	393	18	properties	property	NOUN
ejpam-726	393	19	are	be	AUX
ejpam-726	393	20	equivalent	equivalent	ADJ
ejpam-726	393	21	:	:	PUNCT
ejpam-726	393	22	(	(	PUNCT
ejpam-726	393	23	a	a	X
ejpam-726	393	24	)	)	PUNCT
ejpam-726	393	25	f	f	PROPN
ejpam-726	393	26	is	be	AUX
ejpam-726	393	27	continuous	continuous	ADJ
ejpam-726	393	28	,	,	PUNCT
ejpam-726	393	29	(	(	PUNCT
ejpam-726	393	30	b	b	X
ejpam-726	393	31	)	)	PUNCT
ejpam-726	393	32	f	f	PROPN
ejpam-726	393	33	is	be	AUX
ejpam-726	393	34	lc	lc	NOUN
ejpam-726	393	35	-	-	PUNCT
ejpam-726	393	36	continuous	continuous	ADJ
ejpam-726	393	37	and	and	CCONJ
ejpam-726	393	38	g−continuous	g−continuous	ADJ
ejpam-726	393	39	.	.	PUNCT
ejpam-726	393	40	theorem	theorem	VERB
ejpam-726	393	41	17	17	NUM
ejpam-726	393	42	.	.	PUNCT
ejpam-726	394	1	for	for	ADP
ejpam-726	394	2	a	a	DET
ejpam-726	394	3	function	function	NOUN
ejpam-726	394	4	f	f	NOUN
ejpam-726	394	5	:	:	PUNCT
ejpam-726	394	6	(	(	PUNCT
ejpam-726	394	7	x	x	X
ejpam-726	394	8	,	,	PUNCT
ejpam-726	394	9	τ)→	τ)→	PROPN
ejpam-726	394	10	(	(	PUNCT
ejpam-726	394	11	y	y	PROPN
ejpam-726	394	12	,	,	PUNCT
ejpam-726	394	13	σ	σ	PROPN
ejpam-726	394	14	)	)	PUNCT
ejpam-726	394	15	,	,	PUNCT
ejpam-726	394	16	the	the	DET
ejpam-726	394	17	following	follow	VERB
ejpam-726	394	18	properties	property	NOUN
ejpam-726	394	19	are	be	AUX
ejpam-726	394	20	hold	hold	ADJ
ejpam-726	394	21	:	:	PUNCT
ejpam-726	394	22	(	(	PUNCT
ejpam-726	394	23	a	a	X
ejpam-726	394	24	)	)	PUNCT
ejpam-726	394	25	if	if	SCONJ
ejpam-726	394	26	f	f	PROPN
ejpam-726	394	27	is	be	AUX
ejpam-726	394	28	continuous	continuous	ADJ
ejpam-726	394	29	,	,	PUNCT
ejpam-726	394	30	then	then	ADV
ejpam-726	394	31	f	f	PROPN
ejpam-726	394	32	is	be	AUX
ejpam-726	394	33	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	394	34	.	.	PUNCT
ejpam-726	395	1	(	(	PUNCT
ejpam-726	395	2	b	b	X
ejpam-726	395	3	)	)	PUNCT
ejpam-726	395	4	if	if	SCONJ
ejpam-726	395	5	β∗g−continuous	β∗g−continuous	NUM
ejpam-726	395	6	,	,	PUNCT
ejpam-726	395	7	then	then	ADV
ejpam-726	395	8	f	f	PROPN
ejpam-726	395	9	is	be	AUX
ejpam-726	395	10	g−continuous	g−continuous	PROPN
ejpam-726	395	11	.	.	PUNCT
ejpam-726	396	1	theorem	theorem	NOUN
ejpam-726	396	2	18	18	NUM
ejpam-726	396	3	.	.	PUNCT
ejpam-726	397	1	let	let	AUX
ejpam-726	397	2	(	(	PUNCT
ejpam-726	397	3	x	x	X
ejpam-726	397	4	,	,	PUNCT
ejpam-726	397	5	τ	τ	X
ejpam-726	397	6	)	)	PUNCT
ejpam-726	397	7	be	be	VERB
ejpam-726	397	8	a	a	DET
ejpam-726	397	9	topological	topological	ADJ
ejpam-726	397	10	space	space	NOUN
ejpam-726	397	11	.	.	PUNCT
ejpam-726	398	1	then	then	ADV
ejpam-726	398	2	we	we	PRON
ejpam-726	398	3	have	have	VERB
ejpam-726	398	4	(	(	PUNCT
ejpam-726	398	5	a	a	X
ejpam-726	398	6	)	)	PUNCT
ejpam-726	398	7	if	if	SCONJ
ejpam-726	398	8	f	f	PROPN
ejpam-726	398	9	is	be	AUX
ejpam-726	398	10	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	398	11	,	,	PUNCT
ejpam-726	398	12	then	then	ADV
ejpam-726	398	13	f	f	PROPN
ejpam-726	398	14	is	be	AUX
ejpam-726	398	15	β∗pg−continuous	β∗pg−continuous	PROPN
ejpam-726	398	16	.	.	PUNCT
ejpam-726	399	1	(	(	PUNCT
ejpam-726	399	2	b	b	X
ejpam-726	399	3	)	)	PUNCT
ejpam-726	399	4	if	if	SCONJ
ejpam-726	399	5	f	f	PROPN
ejpam-726	399	6	is	be	AUX
ejpam-726	399	7	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	399	8	,	,	PUNCT
ejpam-726	399	9	then	then	ADV
ejpam-726	399	10	f	f	PROPN
ejpam-726	399	11	is	be	AUX
ejpam-726	399	12	β∗sg−continuous	β∗sg−continuous	PROPN
ejpam-726	399	13	.	.	PUNCT
ejpam-726	400	1	theorem	theorem	NOUN
ejpam-726	400	2	19	19	NUM
ejpam-726	400	3	.	.	PUNCT
ejpam-726	401	1	for	for	ADP
ejpam-726	401	2	a	a	DET
ejpam-726	401	3	function	function	NOUN
ejpam-726	401	4	f	f	NOUN
ejpam-726	401	5	:	:	PUNCT
ejpam-726	401	6	(	(	PUNCT
ejpam-726	401	7	x	x	X
ejpam-726	401	8	,	,	PUNCT
ejpam-726	401	9	τ)→	τ)→	PROPN
ejpam-726	401	10	(	(	PUNCT
ejpam-726	401	11	y	y	PROPN
ejpam-726	401	12	,	,	PUNCT
ejpam-726	401	13	σ	σ	PROPN
ejpam-726	401	14	)	)	PUNCT
ejpam-726	401	15	,	,	PUNCT
ejpam-726	401	16	the	the	DET
ejpam-726	401	17	following	follow	VERB
ejpam-726	401	18	properties	property	NOUN
ejpam-726	401	19	hold	hold	VERB
ejpam-726	401	20	:	:	PUNCT
ejpam-726	401	21	figure	figure	NOUN
ejpam-726	401	22	1	1	NUM
ejpam-726	401	23	:	:	PUNCT
ejpam-726	401	24	*	*	PUNCT
ejpam-726	401	25	diagram	diagram	NOUN
ejpam-726	401	26	iv	iv	X
ejpam-726	401	27	(	(	PUNCT
ejpam-726	401	28	repeated	repeat	VERB
ejpam-726	401	29	)	)	PUNCT
ejpam-726	401	30	.	.	PUNCT
ejpam-726	402	1	proof	proof	NOUN
ejpam-726	402	2	.	.	PUNCT
ejpam-726	403	1	obvious	obvious	ADJ
ejpam-726	403	2	by	by	ADP
ejpam-726	403	3	diagram	diagram	NOUN
ejpam-726	403	4	4	4	NUM
ejpam-726	403	5	.	.	PUNCT
ejpam-726	404	1	theorem	theorem	VERB
ejpam-726	404	2	20	20	NUM
ejpam-726	404	3	.	.	PUNCT
ejpam-726	405	1	if	if	SCONJ
ejpam-726	405	2	f	f	PROPN
ejpam-726	405	3	:	:	PUNCT
ejpam-726	405	4	(	(	PUNCT
ejpam-726	405	5	x	x	X
ejpam-726	405	6	,	,	PUNCT
ejpam-726	405	7	τ)→	τ)→	PROPN
ejpam-726	405	8	(	(	PUNCT
ejpam-726	405	9	y	y	PROPN
ejpam-726	405	10	,	,	PUNCT
ejpam-726	405	11	σ	σ	PROPN
ejpam-726	405	12	)	)	PUNCT
ejpam-726	405	13	is	be	AUX
ejpam-726	405	14	a	a	DET
ejpam-726	405	15	β∗-irresolute	β∗-irresolute	NOUN
ejpam-726	405	16	and	and	CCONJ
ejpam-726	405	17	closed	closed	ADJ
ejpam-726	405	18	function	function	NOUN
ejpam-726	405	19	,	,	PUNCT
ejpam-726	405	20	then	then	ADV
ejpam-726	405	21	f	f	X
ejpam-726	405	22	(	(	PUNCT
ejpam-726	405	23	a	a	PRON
ejpam-726	405	24	)	)	PUNCT
ejpam-726	405	25	is	be	AUX
ejpam-726	405	26	β∗g−closed	β∗g−close	VERB
ejpam-726	405	27	in	in	ADP
ejpam-726	405	28	y	y	PROPN
ejpam-726	405	29	for	for	ADP
ejpam-726	405	30	every	every	DET
ejpam-726	405	31	β∗g−closed	β∗g−close	VERB
ejpam-726	405	32	set	set	VERB
ejpam-726	405	33	a	a	PRON
ejpam-726	405	34	of	of	ADP
ejpam-726	405	35	x	x	SYM
ejpam-726	405	36	.	.	PUNCT
ejpam-726	406	1	proof	proof	NOUN
ejpam-726	406	2	.	.	PUNCT
ejpam-726	407	1	let	let	VERB
ejpam-726	407	2	a	a	DET
ejpam-726	407	3	be	be	AUX
ejpam-726	407	4	any	any	DET
ejpam-726	407	5	β∗g−closed	β∗g−close	VERB
ejpam-726	407	6	set	set	NOUN
ejpam-726	407	7	of	of	ADP
ejpam-726	407	8	x	x	PUNCT
ejpam-726	407	9	and	and	CCONJ
ejpam-726	407	10	u	u	NOUN
ejpam-726	407	11	be	be	VERB
ejpam-726	407	12	any	any	DET
ejpam-726	407	13	β∗-set	β∗-set	NOUN
ejpam-726	407	14	of	of	ADP
ejpam-726	407	15	y	y	PROPN
ejpam-726	407	16	containing	contain	VERB
ejpam-726	407	17	f	f	PROPN
ejpam-726	407	18	(	(	PUNCT
ejpam-726	407	19	a	a	NOUN
ejpam-726	407	20	)	)	PUNCT
ejpam-726	407	21	.	.	PUNCT
ejpam-726	408	1	since	since	SCONJ
ejpam-726	408	2	f	f	PROPN
ejpam-726	408	3	is	be	AUX
ejpam-726	408	4	β∗-irresolute	β∗-irresolute	PROPN
ejpam-726	408	5	,	,	PUNCT
ejpam-726	408	6	f	f	PROPN
ejpam-726	408	7	−1(u	−1(u	X
ejpam-726	408	8	)	)	PUNCT
ejpam-726	408	9	is	be	AUX
ejpam-726	408	10	a	a	DET
ejpam-726	408	11	β∗-set	β∗-set	NOUN
ejpam-726	408	12	in	in	ADP
ejpam-726	408	13	x	x	PUNCT
ejpam-726	408	14	and	and	CCONJ
ejpam-726	408	15	a⊂	a⊂	VERB
ejpam-726	408	16	f	f	NOUN
ejpam-726	408	17	−1(u	−1(u	NOUN
ejpam-726	408	18	)	)	PUNCT
ejpam-726	408	19	.	.	PUNCT
ejpam-726	409	1	therefore	therefore	ADV
ejpam-726	409	2	,	,	PUNCT
ejpam-726	409	3	we	we	PRON
ejpam-726	409	4	have	have	VERB
ejpam-726	409	5	cl(a)⊂	cl(a)⊂	PROPN
ejpam-726	409	6	f	f	X
ejpam-726	409	7	−1(u	−1(u	NOUN
ejpam-726	409	8	)	)	PUNCT
ejpam-726	409	9	and	and	CCONJ
ejpam-726	409	10	hence	hence	ADV
ejpam-726	409	11	f	f	X
ejpam-726	409	12	(	(	PUNCT
ejpam-726	409	13	cl(a))⊂	cl(a))⊂	NOUN
ejpam-726	409	14	u	u	NOUN
ejpam-726	409	15	.	.	PUNCT
ejpam-726	410	1	since	since	SCONJ
ejpam-726	410	2	f	f	PROPN
ejpam-726	410	3	is	be	AUX
ejpam-726	410	4	closed	closed	ADJ
ejpam-726	410	5	,	,	PUNCT
ejpam-726	410	6	cl	cl	NOUN
ejpam-726	410	7	(	(	PUNCT
ejpam-726	410	8	f	f	PROPN
ejpam-726	410	9	(	(	PUNCT
ejpam-726	410	10	a))⊂	a))⊂	NOUN
ejpam-726	410	11	f	f	NOUN
ejpam-726	410	12	(	(	PUNCT
ejpam-726	410	13	cl(a))⊂	cl(a))⊂	NOUN
ejpam-726	410	14	u	u	NOUN
ejpam-726	410	15	.	.	PUNCT
ejpam-726	411	1	hence	hence	ADV
ejpam-726	411	2	f	f	PROPN
ejpam-726	411	3	(	(	PUNCT
ejpam-726	411	4	a	a	NOUN
ejpam-726	411	5	)	)	PUNCT
ejpam-726	411	6	is	be	AUX
ejpam-726	411	7	β∗g−closed	β∗g−close	VERB
ejpam-726	411	8	in	in	ADP
ejpam-726	411	9	y	y	PROPN
ejpam-726	411	10	.	.	PUNCT
ejpam-726	412	1	the	the	DET
ejpam-726	412	2	composition	composition	NOUN
ejpam-726	412	3	of	of	ADP
ejpam-726	412	4	two	two	NUM
ejpam-726	412	5	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	412	6	functions	function	NOUN
ejpam-726	412	7	need	need	AUX
ejpam-726	412	8	not	not	PART
ejpam-726	412	9	be	be	AUX
ejpam-726	412	10	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	412	11	.	.	PUNCT
ejpam-726	413	1	for	for	ADP
ejpam-726	413	2	,	,	PUNCT
ejpam-726	413	3	consider	consider	VERB
ejpam-726	413	4	the	the	DET
ejpam-726	413	5	following	follow	VERB
ejpam-726	413	6	example	example	NOUN
ejpam-726	413	7	:	:	PUNCT
ejpam-726	413	8	a.	a.	NOUN
ejpam-726	413	9	açıkgöz	açıkgöz	PROPN
ejpam-726	413	10	/	/	SYM
ejpam-726	413	11	eur	eur	PROPN
ejpam-726	413	12	.	.	PUNCT
ejpam-726	414	1	j.	j.	PROPN
ejpam-726	414	2	pure	pure	PROPN
ejpam-726	414	3	appl	appl	PROPN
ejpam-726	414	4	.	.	PROPN
ejpam-726	414	5	math	math	PROPN
ejpam-726	414	6	,	,	PUNCT
ejpam-726	414	7	4	4	NUM
ejpam-726	414	8	(	(	PUNCT
ejpam-726	414	9	2011	2011	NUM
ejpam-726	414	10	)	)	PUNCT
ejpam-726	414	11	,	,	PUNCT
ejpam-726	414	12	20	20	NUM
ejpam-726	414	13	-	-	SYM
ejpam-726	414	14	33	33	NUM
ejpam-726	414	15	31	31	NUM
ejpam-726	414	16	example	example	NOUN
ejpam-726	414	17	12	12	NUM
ejpam-726	414	18	.	.	PUNCT
ejpam-726	415	1	let	let	VERB
ejpam-726	415	2	x	x	PUNCT
ejpam-726	415	3	=	=	PRON
ejpam-726	415	4	{	{	PUNCT
ejpam-726	415	5	a	a	PRON
ejpam-726	415	6	,	,	PUNCT
ejpam-726	415	7	b	b	NOUN
ejpam-726	415	8	,	,	PUNCT
ejpam-726	415	9	c	c	NOUN
ejpam-726	415	10	,	,	PUNCT
ejpam-726	415	11	d	d	NOUN
ejpam-726	415	12	}	}	PUNCT
ejpam-726	415	13	,	,	PUNCT
ejpam-726	415	14	τ	τ	X
ejpam-726	415	15	=	=	PUNCT
ejpam-726	415	16	{	{	PUNCT
ejpam-726	415	17	x	x	X
ejpam-726	415	18	,	,	PUNCT
ejpam-726	415	19	;	;	PUNCT
ejpam-726	415	20	,	,	PUNCT
ejpam-726	415	21	{	{	PUNCT
ejpam-726	415	22	b	b	NOUN
ejpam-726	415	23	}	}	PUNCT
ejpam-726	415	24	,	,	PUNCT
ejpam-726	415	25	{	{	PUNCT
ejpam-726	415	26	c	c	X
ejpam-726	415	27	}	}	PUNCT
ejpam-726	415	28	,	,	PUNCT
ejpam-726	415	29	{	{	PUNCT
ejpam-726	415	30	b	b	NOUN
ejpam-726	415	31	,	,	PUNCT
ejpam-726	415	32	c}},σ	c}},σ	NOUN
ejpam-726	415	33	=	=	SYM
ejpam-726	415	34	{	{	PUNCT
ejpam-726	415	35	x	x	X
ejpam-726	415	36	,	,	PUNCT
ejpam-726	415	37	;	;	PUNCT
ejpam-726	415	38	,	,	PUNCT
ejpam-726	415	39	{	{	PUNCT
ejpam-726	415	40	a	a	PRON
ejpam-726	415	41	,	,	PUNCT
ejpam-726	415	42	b	b	NOUN
ejpam-726	415	43	,	,	PUNCT
ejpam-726	415	44	d	d	NOUN
ejpam-726	415	45	}	}	PUNCT
ejpam-726	415	46	}	}	PUNCT
ejpam-726	415	47	,	,	PUNCT
ejpam-726	415	48	η	η	PROPN
ejpam-726	415	49	=	=	PRON
ejpam-726	415	50	{	{	PUNCT
ejpam-726	415	51	x	x	X
ejpam-726	415	52	,	,	PUNCT
ejpam-726	415	53	;	;	PUNCT
ejpam-726	415	54	,	,	PUNCT
ejpam-726	415	55	{	{	PUNCT
ejpam-726	415	56	a	a	PRON
ejpam-726	415	57	,	,	PUNCT
ejpam-726	415	58	d	d	NOUN
ejpam-726	415	59	}	}	PUNCT
ejpam-726	415	60	}	}	PUNCT
ejpam-726	415	61	.	.	PUNCT
ejpam-726	416	1	define	define	VERB
ejpam-726	416	2	f	f	NOUN
ejpam-726	416	3	:	:	PUNCT
ejpam-726	416	4	(	(	PUNCT
ejpam-726	416	5	x	x	X
ejpam-726	416	6	,	,	PUNCT
ejpam-726	416	7	τ)→	τ)→	PROPN
ejpam-726	416	8	(	(	PUNCT
ejpam-726	416	9	x	x	X
ejpam-726	416	10	,	,	PUNCT
ejpam-726	416	11	σ	σ	PROPN
ejpam-726	416	12	)	)	PUNCT
ejpam-726	416	13	by	by	ADP
ejpam-726	416	14	f	f	PROPN
ejpam-726	416	15	(	(	PUNCT
ejpam-726	416	16	a	a	NOUN
ejpam-726	416	17	)	)	PUNCT
ejpam-726	416	18	=	=	SYM
ejpam-726	416	19	a	a	PROPN
ejpam-726	416	20	,	,	PUNCT
ejpam-726	416	21	f	f	PROPN
ejpam-726	416	22	(	(	PUNCT
ejpam-726	416	23	b	b	NOUN
ejpam-726	416	24	)	)	PUNCT
ejpam-726	417	1	=	=	SYM
ejpam-726	417	2	c	c	X
ejpam-726	417	3	,	,	PUNCT
ejpam-726	417	4	f	f	PROPN
ejpam-726	417	5	(	(	PUNCT
ejpam-726	417	6	c	c	NOUN
ejpam-726	417	7	)	)	PUNCT
ejpam-726	417	8	=	=	SYM
ejpam-726	417	9	b	b	PROPN
ejpam-726	417	10	,	,	PUNCT
ejpam-726	417	11	f	f	PROPN
ejpam-726	417	12	(	(	PUNCT
ejpam-726	417	13	d	d	NOUN
ejpam-726	417	14	)	)	PUNCT
ejpam-726	418	1	=	=	SYM
ejpam-726	418	2	d	d	PROPN
ejpam-726	418	3	and	and	CCONJ
ejpam-726	418	4	g	g	PROPN
ejpam-726	418	5	:	:	PUNCT
ejpam-726	418	6	(	(	PUNCT
ejpam-726	418	7	x	x	X
ejpam-726	418	8	,	,	PUNCT
ejpam-726	418	9	σ	σ	PROPN
ejpam-726	418	10	)	)	PUNCT
ejpam-726	418	11	→	→	SYM
ejpam-726	418	12	(	(	PUNCT
ejpam-726	418	13	x	x	X
ejpam-726	418	14	,	,	PUNCT
ejpam-726	418	15	η	η	PROPN
ejpam-726	418	16	)	)	PUNCT
ejpam-726	418	17	by	by	ADP
ejpam-726	418	18	g(a	g(a	PROPN
ejpam-726	418	19	)	)	PUNCT
ejpam-726	419	1	=	=	SYM
ejpam-726	419	2	d	d	PROPN
ejpam-726	419	3	,	,	PUNCT
ejpam-726	419	4	g(b	g(b	NOUN
ejpam-726	419	5	)	)	PUNCT
ejpam-726	419	6	=	=	SYM
ejpam-726	419	7	c	c	X
ejpam-726	419	8	,	,	PUNCT
ejpam-726	419	9	g(c	g(c	NOUN
ejpam-726	419	10	)	)	PUNCT
ejpam-726	419	11	=	=	SYM
ejpam-726	419	12	b	b	PROPN
ejpam-726	419	13	,	,	PUNCT
ejpam-726	419	14	g(d	g(d	PROPN
ejpam-726	419	15	)	)	PUNCT
ejpam-726	419	16	=	=	PUNCT
ejpam-726	420	1	a.	a.	NOUN
ejpam-726	420	2	then	then	ADV
ejpam-726	420	3	f	f	PROPN
ejpam-726	420	4	and	and	CCONJ
ejpam-726	420	5	g	g	PROPN
ejpam-726	420	6	are	be	AUX
ejpam-726	420	7	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	420	8	.	.	PUNCT
ejpam-726	421	1	{	{	PUNCT
ejpam-726	421	2	b	b	X
ejpam-726	421	3	,	,	PUNCT
ejpam-726	421	4	c	c	AUX
ejpam-726	421	5	}	}	PUNCT
ejpam-726	421	6	is	be	AUX
ejpam-726	421	7	closed	close	VERB
ejpam-726	421	8	in	in	ADP
ejpam-726	421	9	(	(	PUNCT
ejpam-726	421	10	x	x	INTJ
ejpam-726	421	11	,	,	PUNCT
ejpam-726	421	12	η	η	PROPN
ejpam-726	421	13	)	)	PUNCT
ejpam-726	421	14	.	.	PUNCT
ejpam-726	422	1	(	(	PUNCT
ejpam-726	422	2	g	g	X
ejpam-726	422	3	◦	◦	NOUN
ejpam-726	422	4	f	f	PROPN
ejpam-726	422	5	)	)	PUNCT
ejpam-726	422	6	−1({b	−1({b	PROPN
ejpam-726	422	7	,	,	PUNCT
ejpam-726	422	8	c	c	NOUN
ejpam-726	422	9	}	}	PUNCT
ejpam-726	422	10	)	)	PUNCT
ejpam-726	423	1	=	=	SYM
ejpam-726	423	2	f	f	PROPN
ejpam-726	423	3	−1(g−1({b	−1(g−1({b	PROPN
ejpam-726	423	4	,	,	PUNCT
ejpam-726	423	5	c	c	NOUN
ejpam-726	423	6	}	}	PUNCT
ejpam-726	423	7	)	)	PUNCT
ejpam-726	423	8	)	)	PUNCT
ejpam-726	424	1	=	=	PUNCT
ejpam-726	425	1	f	f	X
ejpam-726	425	2	−1({b	−1({b	PROPN
ejpam-726	425	3	,	,	PUNCT
ejpam-726	425	4	c	c	NOUN
ejpam-726	425	5	}	}	PUNCT
ejpam-726	425	6	)	)	PUNCT
ejpam-726	426	1	=	=	PRON
ejpam-726	426	2	{	{	PUNCT
ejpam-726	426	3	b	b	NOUN
ejpam-726	426	4	,	,	PUNCT
ejpam-726	426	5	c	c	NOUN
ejpam-726	426	6	}	}	PUNCT
ejpam-726	426	7	which	which	PRON
ejpam-726	426	8	is	be	AUX
ejpam-726	426	9	not	not	PART
ejpam-726	426	10	β∗g−closed	β∗g−close	VERB
ejpam-726	426	11	in	in	ADP
ejpam-726	426	12	(	(	PUNCT
ejpam-726	426	13	x	x	INTJ
ejpam-726	426	14	,	,	PUNCT
ejpam-726	426	15	τ	τ	PROPN
ejpam-726	426	16	)	)	PUNCT
ejpam-726	426	17	.	.	PUNCT
ejpam-726	427	1	hence	hence	ADV
ejpam-726	427	2	g	g	PROPN
ejpam-726	427	3	◦	◦	NOUN
ejpam-726	427	4	f	f	X
ejpam-726	427	5	is	be	AUX
ejpam-726	427	6	not	not	PART
ejpam-726	427	7	β∗g−continuous	β∗g−continuous	NUM
ejpam-726	427	8	.	.	PUNCT
ejpam-726	428	1	theorem	theorem	NOUN
ejpam-726	428	2	21	21	NUM
ejpam-726	428	3	.	.	PUNCT
ejpam-726	429	1	let	let	VERB
ejpam-726	429	2	f	f	NOUN
ejpam-726	429	3	:	:	PUNCT
ejpam-726	429	4	(	(	PUNCT
ejpam-726	429	5	x	x	X
ejpam-726	429	6	,	,	PUNCT
ejpam-726	429	7	τ)→	τ)→	PROPN
ejpam-726	429	8	(	(	PUNCT
ejpam-726	429	9	y	y	PROPN
ejpam-726	429	10	,	,	PUNCT
ejpam-726	429	11	σ	σ	PROPN
ejpam-726	429	12	)	)	PUNCT
ejpam-726	429	13	and	and	CCONJ
ejpam-726	429	14	g	g	NOUN
ejpam-726	429	15	:	:	PUNCT
ejpam-726	429	16	(	(	PUNCT
ejpam-726	429	17	y	y	NOUN
ejpam-726	429	18	,	,	PUNCT
ejpam-726	429	19	σ)→	σ)→	PROPN
ejpam-726	429	20	(	(	PUNCT
ejpam-726	429	21	z	z	PROPN
ejpam-726	429	22	,	,	PUNCT
ejpam-726	429	23	η	η	PROPN
ejpam-726	429	24	)	)	PUNCT
ejpam-726	429	25	be	be	VERB
ejpam-726	429	26	any	any	DET
ejpam-726	429	27	two	two	NUM
ejpam-726	429	28	functions	function	NOUN
ejpam-726	429	29	.	.	PUNCT
ejpam-726	430	1	then	then	ADV
ejpam-726	430	2	(	(	PUNCT
ejpam-726	430	3	a	a	X
ejpam-726	430	4	)	)	PUNCT
ejpam-726	430	5	g	g	NOUN
ejpam-726	430	6	◦	◦	NOUN
ejpam-726	430	7	f	f	PROPN
ejpam-726	430	8	is	be	AUX
ejpam-726	430	9	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	430	10	,	,	PUNCT
ejpam-726	430	11	if	if	SCONJ
ejpam-726	430	12	g	g	PROPN
ejpam-726	430	13	is	be	AUX
ejpam-726	430	14	continuous	continuous	ADJ
ejpam-726	430	15	and	and	CCONJ
ejpam-726	430	16	f	f	PROPN
ejpam-726	430	17	is	be	AUX
ejpam-726	430	18	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	430	19	.	.	PUNCT
ejpam-726	431	1	(	(	PUNCT
ejpam-726	431	2	b	b	X
ejpam-726	431	3	)	)	PUNCT
ejpam-726	431	4	g	g	NOUN
ejpam-726	431	5	◦	◦	NOUN
ejpam-726	431	6	f	f	PROPN
ejpam-726	431	7	is	be	AUX
ejpam-726	431	8	β∗g−irresolute	β∗g−irresolute	PROPN
ejpam-726	431	9	,	,	PUNCT
ejpam-726	431	10	if	if	SCONJ
ejpam-726	431	11	g	g	PROPN
ejpam-726	431	12	is	be	AUX
ejpam-726	431	13	β∗g−irresolute	β∗g−irresolute	NUM
ejpam-726	431	14	and	and	CCONJ
ejpam-726	431	15	f	f	PROPN
ejpam-726	431	16	is	be	AUX
ejpam-726	431	17	β∗g−irresolute	β∗g−irresolute	PROPN
ejpam-726	431	18	.	.	PUNCT
ejpam-726	432	1	(	(	PUNCT
ejpam-726	432	2	c	c	X
ejpam-726	432	3	)	)	PUNCT
ejpam-726	432	4	g	g	NOUN
ejpam-726	432	5	◦	◦	NOUN
ejpam-726	432	6	f	f	PROPN
ejpam-726	432	7	is	be	AUX
ejpam-726	432	8	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	432	9	,	,	PUNCT
ejpam-726	432	10	if	if	SCONJ
ejpam-726	432	11	g	g	PROPN
ejpam-726	432	12	is	be	AUX
ejpam-726	432	13	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	432	14	and	and	CCONJ
ejpam-726	432	15	f	f	PROPN
ejpam-726	432	16	is	be	AUX
ejpam-726	432	17	β∗g−irresolute	β∗g−irresolute	PROPN
ejpam-726	432	18	.	.	PUNCT
ejpam-726	433	1	(	(	PUNCT
ejpam-726	433	2	d	d	X
ejpam-726	433	3	)	)	PUNCT
ejpam-726	433	4	g	g	NOUN
ejpam-726	433	5	◦	◦	NOUN
ejpam-726	433	6	f	f	PROPN
ejpam-726	433	7	is	be	AUX
ejpam-726	433	8	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	433	9	,	,	PUNCT
ejpam-726	433	10	if	if	SCONJ
ejpam-726	433	11	f	f	PROPN
ejpam-726	433	12	is	be	AUX
ejpam-726	433	13	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	433	14	and	and	CCONJ
ejpam-726	433	15	g	g	PROPN
ejpam-726	433	16	is	be	AUX
ejpam-726	433	17	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	433	18	and	and	CCONJ
ejpam-726	433	19	y	y	PROPN
ejpam-726	433	20	is	be	AUX
ejpam-726	433	21	a	a	DET
ejpam-726	433	22	β∗t1/2	β∗t1/2	PROPN
ejpam-726	433	23	-	-	PUNCT
ejpam-726	433	24	space	space	NOUN
ejpam-726	433	25	.	.	PUNCT
ejpam-726	434	1	proof	proof	NOUN
ejpam-726	434	2	.	.	PUNCT
ejpam-726	435	1	(	(	PUNCT
ejpam-726	435	2	a	a	X
ejpam-726	435	3	)	)	PUNCT
ejpam-726	435	4	let	let	VERB
ejpam-726	435	5	v	v	PART
ejpam-726	435	6	be	be	AUX
ejpam-726	435	7	closed	close	VERB
ejpam-726	435	8	in	in	ADP
ejpam-726	435	9	(	(	PUNCT
ejpam-726	435	10	z	z	PROPN
ejpam-726	435	11	,	,	PUNCT
ejpam-726	435	12	η	η	PROPN
ejpam-726	435	13	)	)	PUNCT
ejpam-726	435	14	.	.	PUNCT
ejpam-726	436	1	then	then	ADV
ejpam-726	436	2	g−1(v	g−1(v	PROPN
ejpam-726	436	3	)	)	PUNCT
ejpam-726	436	4	is	be	AUX
ejpam-726	436	5	closed	close	VERB
ejpam-726	436	6	in	in	ADP
ejpam-726	436	7	(	(	PUNCT
ejpam-726	436	8	y	y	PROPN
ejpam-726	436	9	,	,	PUNCT
ejpam-726	436	10	σ	σ	PROPN
ejpam-726	436	11	)	)	PUNCT
ejpam-726	436	12	,	,	PUNCT
ejpam-726	436	13	since	since	SCONJ
ejpam-726	436	14	g	g	PROPN
ejpam-726	436	15	is	be	AUX
ejpam-726	436	16	continuous	continuous	ADJ
ejpam-726	436	17	.	.	PUNCT
ejpam-726	437	1	β∗g−continuity	β∗g−continuity	NOUN
ejpam-726	437	2	of	of	ADP
ejpam-726	437	3	f	f	PROPN
ejpam-726	437	4	implies	imply	VERB
ejpam-726	437	5	that	that	SCONJ
ejpam-726	437	6	f	f	PROPN
ejpam-726	437	7	−1(g−1(v	−1(g−1(v	NOUN
ejpam-726	437	8	)	)	PUNCT
ejpam-726	437	9	)	)	PUNCT
ejpam-726	437	10	is	be	AUX
ejpam-726	437	11	β∗g−closed	β∗g−close	VERB
ejpam-726	437	12	in	in	ADP
ejpam-726	437	13	(	(	PUNCT
ejpam-726	437	14	x	x	INTJ
ejpam-726	437	15	,	,	PUNCT
ejpam-726	437	16	τ	τ	PROPN
ejpam-726	437	17	)	)	PUNCT
ejpam-726	437	18	.	.	PUNCT
ejpam-726	438	1	hence	hence	ADV
ejpam-726	438	2	g	g	PROPN
ejpam-726	438	3	◦	◦	NOUN
ejpam-726	438	4	f	f	PROPN
ejpam-726	438	5	is	be	AUX
ejpam-726	438	6	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	438	7	.	.	PUNCT
ejpam-726	439	1	(	(	PUNCT
ejpam-726	439	2	b	b	X
ejpam-726	439	3	)	)	PUNCT
ejpam-726	439	4	let	let	VERB
ejpam-726	439	5	v	v	PART
ejpam-726	439	6	be	be	AUX
ejpam-726	439	7	β∗g−closed	β∗g−close	VERB
ejpam-726	439	8	in	in	ADP
ejpam-726	439	9	(	(	PUNCT
ejpam-726	439	10	z	z	PROPN
ejpam-726	439	11	,	,	PUNCT
ejpam-726	439	12	η	η	PROPN
ejpam-726	439	13	)	)	PUNCT
ejpam-726	439	14	.	.	PUNCT
ejpam-726	440	1	then	then	ADV
ejpam-726	440	2	g−1(v	g−1(v	PROPN
ejpam-726	440	3	)	)	PUNCT
ejpam-726	440	4	is	be	AUX
ejpam-726	440	5	β∗g−closed	β∗g−close	VERB
ejpam-726	440	6	in	in	ADP
ejpam-726	440	7	(	(	PUNCT
ejpam-726	440	8	y	y	PROPN
ejpam-726	440	9	,	,	PUNCT
ejpam-726	440	10	σ	σ	PROPN
ejpam-726	440	11	)	)	PUNCT
ejpam-726	440	12	,	,	PUNCT
ejpam-726	440	13	since	since	SCONJ
ejpam-726	440	14	g	g	PROPN
ejpam-726	440	15	is	be	AUX
ejpam-726	440	16	β∗g−irresolute	β∗g−irresolute	NOUN
ejpam-726	440	17	.	.	PUNCT
ejpam-726	441	1	since	since	SCONJ
ejpam-726	441	2	f	f	PROPN
ejpam-726	441	3	is	be	AUX
ejpam-726	441	4	β∗g−irresolute	β∗g−irresolute	PROPN
ejpam-726	441	5	,	,	PUNCT
ejpam-726	441	6	f	f	PROPN
ejpam-726	441	7	−1(g−1(v	−1(g−1(v	PROPN
ejpam-726	441	8	)	)	PUNCT
ejpam-726	441	9	)	)	PUNCT
ejpam-726	441	10	is	be	AUX
ejpam-726	441	11	β∗g−closed	β∗g−close	VERB
ejpam-726	441	12	in	in	ADP
ejpam-726	441	13	(	(	PUNCT
ejpam-726	441	14	x	x	INTJ
ejpam-726	441	15	,	,	PUNCT
ejpam-726	441	16	τ	τ	PROPN
ejpam-726	441	17	)	)	PUNCT
ejpam-726	441	18	.	.	PUNCT
ejpam-726	442	1	hence	hence	ADV
ejpam-726	442	2	g	g	PROPN
ejpam-726	442	3	◦	◦	PROPN
ejpam-726	442	4	f	f	PROPN
ejpam-726	442	5	is	be	AUX
ejpam-726	442	6	β∗g−irresolute	β∗g−irresolute	PROPN
ejpam-726	442	7	.	.	PUNCT
ejpam-726	443	1	(	(	PUNCT
ejpam-726	443	2	c	c	X
ejpam-726	443	3	)	)	PUNCT
ejpam-726	443	4	let	let	VERB
ejpam-726	443	5	v	v	PART
ejpam-726	443	6	be	be	AUX
ejpam-726	443	7	closed	close	VERB
ejpam-726	443	8	in	in	ADP
ejpam-726	443	9	(	(	PUNCT
ejpam-726	443	10	z	z	PROPN
ejpam-726	443	11	,	,	PUNCT
ejpam-726	443	12	η	η	PROPN
ejpam-726	443	13	)	)	PUNCT
ejpam-726	443	14	.	.	PUNCT
ejpam-726	444	1	since	since	SCONJ
ejpam-726	444	2	g	g	PROPN
ejpam-726	444	3	is	be	AUX
ejpam-726	444	4	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	444	5	,	,	PUNCT
ejpam-726	444	6	g−1(v	g−1(v	NOUN
ejpam-726	444	7	)	)	PUNCT
ejpam-726	444	8	is	be	AUX
ejpam-726	444	9	β∗g−closed	β∗g−close	VERB
ejpam-726	444	10	in	in	ADP
ejpam-726	444	11	(	(	PUNCT
ejpam-726	444	12	y	y	PROPN
ejpam-726	444	13	,	,	PUNCT
ejpam-726	444	14	σ	σ	PROPN
ejpam-726	444	15	)	)	PUNCT
ejpam-726	444	16	.	.	PUNCT
ejpam-726	445	1	as	as	SCONJ
ejpam-726	445	2	f	f	PROPN
ejpam-726	445	3	is	be	AUX
ejpam-726	445	4	β∗g−irresolute	β∗g−irresolute	PROPN
ejpam-726	445	5	,	,	PUNCT
ejpam-726	445	6	f	f	PROPN
ejpam-726	445	7	−1(g−1(v	−1(g−1(v	PROPN
ejpam-726	445	8	)	)	PUNCT
ejpam-726	445	9	)	)	PUNCT
ejpam-726	445	10	is	be	AUX
ejpam-726	445	11	β∗g−closed	β∗g−close	VERB
ejpam-726	445	12	in	in	ADP
ejpam-726	445	13	(	(	PUNCT
ejpam-726	445	14	x	x	INTJ
ejpam-726	445	15	,	,	PUNCT
ejpam-726	445	16	τ	τ	PROPN
ejpam-726	445	17	)	)	PUNCT
ejpam-726	445	18	.	.	PUNCT
ejpam-726	446	1	hence	hence	ADV
ejpam-726	446	2	g	g	ADP
ejpam-726	446	3	◦	◦	NOUN
ejpam-726	446	4	f	f	PROPN
ejpam-726	446	5	is	be	AUX
ejpam-726	446	6	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	446	7	.	.	PUNCT
ejpam-726	447	1	(	(	PUNCT
ejpam-726	447	2	d	d	X
ejpam-726	447	3	)	)	PUNCT
ejpam-726	447	4	let	let	VERB
ejpam-726	447	5	v	v	PART
ejpam-726	447	6	be	be	AUX
ejpam-726	447	7	closed	close	VERB
ejpam-726	447	8	in	in	ADP
ejpam-726	447	9	(	(	PUNCT
ejpam-726	447	10	z	z	PROPN
ejpam-726	447	11	,	,	PUNCT
ejpam-726	447	12	η	η	PROPN
ejpam-726	447	13	)	)	PUNCT
ejpam-726	447	14	.	.	PUNCT
ejpam-726	448	1	then	then	ADV
ejpam-726	448	2	g−1(v	g−1(v	PROPN
ejpam-726	448	3	)	)	PUNCT
ejpam-726	448	4	is	be	AUX
ejpam-726	448	5	β∗g−closed	β∗g−close	VERB
ejpam-726	448	6	in	in	ADP
ejpam-726	448	7	(	(	PUNCT
ejpam-726	448	8	y	y	PROPN
ejpam-726	448	9	,	,	PUNCT
ejpam-726	448	10	σ	σ	PROPN
ejpam-726	448	11	)	)	PUNCT
ejpam-726	448	12	,	,	PUNCT
ejpam-726	448	13	since	since	SCONJ
ejpam-726	448	14	g	g	PROPN
ejpam-726	448	15	is	be	AUX
ejpam-726	448	16	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	448	17	.	.	PUNCT
ejpam-726	449	1	as	as	SCONJ
ejpam-726	449	2	(	(	PUNCT
ejpam-726	449	3	y	y	PROPN
ejpam-726	449	4	,	,	PUNCT
ejpam-726	449	5	σ	σ	PROPN
ejpam-726	449	6	)	)	PUNCT
ejpam-726	449	7	is	be	AUX
ejpam-726	449	8	a	a	DET
ejpam-726	449	9	β∗t1/2	β∗t1/2	PROPN
ejpam-726	449	10	space	space	NOUN
ejpam-726	449	11	,	,	PUNCT
ejpam-726	449	12	g−1(v	g−1(v	PROPN
ejpam-726	449	13	)	)	PUNCT
ejpam-726	449	14	is	be	AUX
ejpam-726	449	15	closed	close	VERB
ejpam-726	449	16	in	in	ADP
ejpam-726	449	17	(	(	PUNCT
ejpam-726	449	18	y	y	PROPN
ejpam-726	449	19	,	,	PUNCT
ejpam-726	449	20	σ	σ	PROPN
ejpam-726	449	21	)	)	PUNCT
ejpam-726	449	22	.	.	PUNCT
ejpam-726	450	1	β∗g−continuity	β∗g−continuity	NOUN
ejpam-726	450	2	of	of	ADP
ejpam-726	450	3	f	f	PROPN
ejpam-726	450	4	implies	imply	VERB
ejpam-726	450	5	that	that	SCONJ
ejpam-726	450	6	f	f	PROPN
ejpam-726	450	7	−1(g−1(v	−1(g−1(v	NOUN
ejpam-726	450	8	)	)	PUNCT
ejpam-726	450	9	)	)	PUNCT
ejpam-726	450	10	is	be	AUX
ejpam-726	450	11	β∗g−closed	β∗g−close	VERB
ejpam-726	450	12	in	in	ADP
ejpam-726	450	13	(	(	PUNCT
ejpam-726	450	14	x	x	INTJ
ejpam-726	450	15	,	,	PUNCT
ejpam-726	450	16	τ	τ	PROPN
ejpam-726	450	17	)	)	PUNCT
ejpam-726	450	18	.	.	PUNCT
ejpam-726	451	1	hence	hence	ADV
ejpam-726	451	2	g	g	PROPN
ejpam-726	451	3	◦	◦	NOUN
ejpam-726	451	4	f	f	PROPN
ejpam-726	451	5	is	be	AUX
ejpam-726	451	6	β∗g−continuous	β∗g−continuous	NUM
ejpam-726	451	7	.	.	PUNCT
ejpam-726	452	1	theorem	theorem	NOUN
ejpam-726	452	2	22	22	NUM
ejpam-726	452	3	.	.	PUNCT
ejpam-726	453	1	let	let	VERB
ejpam-726	453	2	f	f	NOUN
ejpam-726	453	3	:	:	PUNCT
ejpam-726	453	4	(	(	PUNCT
ejpam-726	453	5	x	x	X
ejpam-726	453	6	,	,	PUNCT
ejpam-726	453	7	τ	τ	PROPN
ejpam-726	453	8	)	)	PUNCT
ejpam-726	453	9	→	→	SYM
ejpam-726	453	10	(	(	PUNCT
ejpam-726	453	11	y	y	PROPN
ejpam-726	453	12	,	,	PUNCT
ejpam-726	453	13	σ	σ	PROPN
ejpam-726	453	14	)	)	PUNCT
ejpam-726	453	15	be	be	AUX
ejpam-726	453	16	a	a	DET
ejpam-726	453	17	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	453	18	function	function	NOUN
ejpam-726	453	19	.	.	PUNCT
ejpam-726	454	1	if	if	SCONJ
ejpam-726	454	2	(	(	PUNCT
ejpam-726	454	3	x	x	X
ejpam-726	454	4	,	,	PUNCT
ejpam-726	454	5	τ	τ	X
ejpam-726	454	6	)	)	PUNCT
ejpam-726	454	7	is	be	AUX
ejpam-726	454	8	a	a	DET
ejpam-726	454	9	β∗-t1/2	β∗-t1/2	ADJ
ejpam-726	454	10	space	space	NOUN
ejpam-726	454	11	,	,	PUNCT
ejpam-726	454	12	then	then	ADV
ejpam-726	454	13	f	f	PROPN
ejpam-726	454	14	is	be	AUX
ejpam-726	454	15	continuous	continuous	ADJ
ejpam-726	454	16	.	.	PUNCT
ejpam-726	455	1	proof	proof	NOUN
ejpam-726	455	2	.	.	PUNCT
ejpam-726	456	1	let	let	VERB
ejpam-726	456	2	f	f	PRON
ejpam-726	456	3	be	be	AUX
ejpam-726	456	4	a	a	DET
ejpam-726	456	5	β∗g−continuous	β∗g−continuous	ADJ
ejpam-726	456	6	function	function	NOUN
ejpam-726	456	7	.	.	PUNCT
ejpam-726	457	1	then	then	ADV
ejpam-726	457	2	f	f	PROPN
ejpam-726	457	3	−1(v	−1(v	PROPN
ejpam-726	457	4	)	)	PUNCT
ejpam-726	457	5	is	be	AUX
ejpam-726	457	6	a	a	DET
ejpam-726	457	7	β∗g−closed	β∗g−close	VERB
ejpam-726	457	8	set	set	NOUN
ejpam-726	457	9	of	of	ADP
ejpam-726	457	10	x	x	PUNCT
ejpam-726	457	11	for	for	ADP
ejpam-726	457	12	every	every	DET
ejpam-726	457	13	closed	close	VERB
ejpam-726	457	14	set	set	VERB
ejpam-726	457	15	v	v	NOUN
ejpam-726	457	16	of	of	ADP
ejpam-726	457	17	y	y	PROPN
ejpam-726	457	18	.	.	PUNCT
ejpam-726	458	1	since	since	SCONJ
ejpam-726	458	2	x	x	PRON
ejpam-726	458	3	is	be	AUX
ejpam-726	458	4	a	a	DET
ejpam-726	458	5	β∗t1/2	β∗t1/2	PROPN
ejpam-726	458	6	space	space	NOUN
ejpam-726	458	7	,	,	PUNCT
ejpam-726	458	8	β∗gc(x	β∗gc(x	NOUN
ejpam-726	458	9	,	,	PUNCT
ejpam-726	458	10	τ	τ	X
ejpam-726	458	11	)	)	PUNCT
ejpam-726	458	12	=	=	SYM
ejpam-726	458	13	c(x	c(x	NOUN
ejpam-726	458	14	,	,	PUNCT
ejpam-726	458	15	τ	τ	PROPN
ejpam-726	458	16	)	)	PUNCT
ejpam-726	458	17	.	.	PUNCT
ejpam-726	459	1	hence	hence	ADV
ejpam-726	459	2	,	,	PUNCT
ejpam-726	459	3	for	for	ADP
ejpam-726	459	4	every	every	DET
ejpam-726	459	5	closed	close	VERB
ejpam-726	459	6	set	set	VERB
ejpam-726	459	7	v	v	NOUN
ejpam-726	459	8	of	of	ADP
ejpam-726	459	9	y	y	PROPN
ejpam-726	459	10	,	,	PUNCT
ejpam-726	459	11	f	f	PROPN
ejpam-726	459	12	−1(v	−1(v	PROPN
ejpam-726	459	13	)	)	PUNCT
ejpam-726	459	14	is	be	AUX
ejpam-726	459	15	a	a	DET
ejpam-726	459	16	closed	closed	ADJ
ejpam-726	459	17	set	set	NOUN
ejpam-726	459	18	of	of	ADP
ejpam-726	459	19	x	x	PUNCT
ejpam-726	459	20	and	and	CCONJ
ejpam-726	459	21	so	so	ADV
ejpam-726	459	22	f	f	PROPN
ejpam-726	459	23	is	be	AUX
ejpam-726	459	24	continuous	continuous	ADJ
ejpam-726	459	25	.	.	PUNCT
ejpam-726	460	1	theorem	theorem	NOUN
ejpam-726	460	2	23	23	NUM
ejpam-726	460	3	.	.	PUNCT
ejpam-726	461	1	let	let	VERB
ejpam-726	461	2	f	f	NOUN
ejpam-726	461	3	:	:	PUNCT
ejpam-726	461	4	(	(	PUNCT
ejpam-726	461	5	x	x	X
ejpam-726	461	6	,	,	PUNCT
ejpam-726	461	7	τ)→	τ)→	PROPN
ejpam-726	461	8	(	(	PUNCT
ejpam-726	461	9	y	y	PROPN
ejpam-726	461	10	,	,	PUNCT
ejpam-726	461	11	σ	σ	PROPN
ejpam-726	461	12	)	)	PUNCT
ejpam-726	461	13	be	be	AUX
ejpam-726	461	14	onto	onto	ADP
ejpam-726	461	15	,	,	PUNCT
ejpam-726	461	16	β∗g−irresolute	β∗g−irresolute	NOUN
ejpam-726	461	17	and	and	CCONJ
ejpam-726	461	18	closed	closed	ADJ
ejpam-726	461	19	.	.	PUNCT
ejpam-726	462	1	if	if	SCONJ
ejpam-726	462	2	(	(	PUNCT
ejpam-726	462	3	x	x	X
ejpam-726	462	4	,	,	PUNCT
ejpam-726	462	5	τ	τ	X
ejpam-726	462	6	)	)	PUNCT
ejpam-726	462	7	is	be	AUX
ejpam-726	462	8	a	a	DET
ejpam-726	462	9	β∗t1/2	β∗t1/2	PROPN
ejpam-726	462	10	space	space	NOUN
ejpam-726	462	11	,	,	PUNCT
ejpam-726	462	12	then	then	ADV
ejpam-726	462	13	(	(	PUNCT
ejpam-726	462	14	y	y	PROPN
ejpam-726	462	15	,	,	PUNCT
ejpam-726	462	16	σ	σ	PROPN
ejpam-726	462	17	)	)	PUNCT
ejpam-726	462	18	is	be	AUX
ejpam-726	462	19	also	also	ADV
ejpam-726	462	20	a	a	DET
ejpam-726	462	21	β∗t1/2	β∗t1/2	PROPN
ejpam-726	462	22	space	space	NOUN
ejpam-726	462	23	.	.	PUNCT
ejpam-726	463	1	proof	proof	NOUN
ejpam-726	463	2	.	.	PUNCT
ejpam-726	464	1	let	let	VERB
ejpam-726	464	2	f	f	PRON
ejpam-726	464	3	be	be	AUX
ejpam-726	464	4	any	any	DET
ejpam-726	464	5	β∗g−closed	β∗g−close	VERB
ejpam-726	464	6	set	set	NOUN
ejpam-726	464	7	of	of	ADP
ejpam-726	464	8	y	y	PROPN
ejpam-726	464	9	.	.	PUNCT
ejpam-726	465	1	since	since	SCONJ
ejpam-726	465	2	f	f	PROPN
ejpam-726	465	3	is	be	AUX
ejpam-726	465	4	β∗g−irresolute	β∗g−irresolute	PROPN
ejpam-726	465	5	,	,	PUNCT
ejpam-726	465	6	f	f	PROPN
ejpam-726	465	7	−1(f	−1(f	PROPN
ejpam-726	465	8	)	)	PUNCT
ejpam-726	465	9	is	be	AUX
ejpam-726	465	10	β∗g−closed	β∗g−close	VERB
ejpam-726	465	11	in	in	ADP
ejpam-726	465	12	x	x	X
ejpam-726	465	13	.	.	PUNCT
ejpam-726	466	1	since	since	SCONJ
ejpam-726	466	2	x	x	PROPN
ejpam-726	466	3	is	be	AUX
ejpam-726	466	4	β∗t1/2	β∗t1/2	PROPN
ejpam-726	466	5	,	,	PUNCT
ejpam-726	466	6	f	f	PROPN
ejpam-726	466	7	−1(f	−1(f	PROPN
ejpam-726	466	8	)	)	PUNCT
ejpam-726	466	9	is	be	AUX
ejpam-726	466	10	closed	close	VERB
ejpam-726	466	11	in	in	ADP
ejpam-726	466	12	x	x	PUNCT
ejpam-726	466	13	and	and	CCONJ
ejpam-726	466	14	hence	hence	ADV
ejpam-726	466	15	f	f	PROPN
ejpam-726	466	16	(	(	PUNCT
ejpam-726	466	17	f	f	PROPN
ejpam-726	466	18	−1(f	−1(f	PROPN
ejpam-726	466	19	)	)	PUNCT
ejpam-726	466	20	)	)	PUNCT
ejpam-726	467	1	=	=	PUNCT
ejpam-726	467	2	f	f	PROPN
ejpam-726	467	3	is	be	AUX
ejpam-726	467	4	closed	close	VERB
ejpam-726	467	5	in	in	ADP
ejpam-726	467	6	y	y	PROPN
ejpam-726	467	7	.	.	PUNCT
ejpam-726	468	1	this	this	PRON
ejpam-726	468	2	shows	show	VERB
ejpam-726	468	3	that	that	SCONJ
ejpam-726	468	4	(	(	PUNCT
ejpam-726	468	5	y	y	PROPN
ejpam-726	468	6	,	,	PUNCT
ejpam-726	468	7	σ	σ	PROPN
ejpam-726	468	8	)	)	PUNCT
ejpam-726	468	9	is	be	AUX
ejpam-726	468	10	also	also	ADV
ejpam-726	468	11	a	a	DET
ejpam-726	468	12	β∗t1/2	β∗t1/2	PROPN
ejpam-726	468	13	space	space	NOUN
ejpam-726	468	14	.	.	PUNCT
ejpam-726	469	1	references	reference	NOUN
ejpam-726	469	2	32	32	NUM
ejpam-726	469	3	references	reference	NOUN
ejpam-726	469	4	[	[	X
ejpam-726	469	5	1	1	NUM
ejpam-726	469	6	]	]	X
ejpam-726	469	7	s.p	s.p	PROPN
ejpam-726	469	8	.	.	PROPN
ejpam-726	469	9	arya	arya	PROPN
ejpam-726	469	10	,	,	PUNCT
ejpam-726	469	11	t.	t.	PROPN
ejpam-726	469	12	nour	nour	PROPN
ejpam-726	469	13	.	.	PUNCT
ejpam-726	470	1	characterizations	characterization	NOUN
ejpam-726	470	2	of	of	ADP
ejpam-726	470	3	s	s	NOUN
ejpam-726	470	4	-	-	ADJ
ejpam-726	470	5	normal	normal	ADJ
ejpam-726	470	6	spaces	space	NOUN
ejpam-726	470	7	.	.	PUNCT
ejpam-726	471	1	indian	indian	PROPN
ejpam-726	471	2	j.	j.	PROPN
ejpam-726	471	3	pure	pure	PROPN
ejpam-726	471	4	appl	appl	PROPN
ejpam-726	471	5	.	.	PUNCT
ejpam-726	471	6	math	math	PROPN
ejpam-726	471	7	.	.	PUNCT
ejpam-726	471	8	,	,	PUNCT
ejpam-726	471	9	21	21	NUM
ejpam-726	471	10	,	,	PUNCT
ejpam-726	471	11	717	717	NUM
ejpam-726	471	12	-	-	SYM
ejpam-726	471	13	719	719	NUM
ejpam-726	471	14	,	,	PUNCT
ejpam-726	471	15	1990	1990	NUM
ejpam-726	471	16	.	.	PUNCT
ejpam-726	472	1	[	[	X
ejpam-726	472	2	2	2	X
ejpam-726	472	3	]	]	X
ejpam-726	472	4	g.	g.	PROPN
ejpam-726	472	5	aslım	aslım	PROPN
ejpam-726	472	6	,	,	PUNCT
ejpam-726	472	7	c.	c.	PROPN
ejpam-726	472	8	guler	guler	NOUN
ejpam-726	472	9	and	and	CCONJ
ejpam-726	472	10	t.	t.	PROPN
ejpam-726	472	11	noiri	noiri	PROPN
ejpam-726	472	12	.	.	PUNCT
ejpam-726	473	1	on	on	ADP
ejpam-726	473	2	πgs−closed	πgs−closed	PUNCT
ejpam-726	473	3	sets	set	NOUN
ejpam-726	473	4	in	in	ADP
ejpam-726	473	5	topological	topological	ADJ
ejpam-726	473	6	spaces	space	NOUN
ejpam-726	473	7	.	.	PUNCT
ejpam-726	474	1	acta	acta	PROPN
ejpam-726	474	2	math	math	PROPN
ejpam-726	474	3	.	.	PUNCT
ejpam-726	475	1	hungar	hungar	PROPN
ejpam-726	475	2	.	.	PUNCT
ejpam-726	475	3	,	,	PUNCT
ejpam-726	475	4	112	112	NUM
ejpam-726	475	5	,	,	PUNCT
ejpam-726	475	6	4	4	NUM
ejpam-726	475	7	,	,	PUNCT
ejpam-726	475	8	275	275	NUM
ejpam-726	475	9	-	-	SYM
ejpam-726	475	10	283	283	NUM
ejpam-726	475	11	,	,	PUNCT
ejpam-726	475	12	2006	2006	NUM
ejpam-726	475	13	.	.	PUNCT
ejpam-726	476	1	[	[	X
ejpam-726	476	2	3	3	X
ejpam-726	476	3	]	]	X
ejpam-726	476	4	n.	n.	NOUN
ejpam-726	476	5	bourbaki	bourbaki	PROPN
ejpam-726	476	6	.	.	PUNCT
ejpam-726	477	1	general	general	ADJ
ejpam-726	477	2	topology	topology	PROPN
ejpam-726	477	3	.	.	PUNCT
ejpam-726	478	1	part	part	PROPN
ejpam-726	478	2	i.	i.	PROPN
ejpam-726	478	3	addison	addison	PROPN
ejpam-726	478	4	-	-	PUNCT
ejpam-726	478	5	wesley	wesley	PROPN
ejpam-726	478	6	.	.	PUNCT
ejpam-726	479	1	reading	read	VERB
ejpam-726	479	2	mass	mass	PROPN
ejpam-726	479	3	.	.	PROPN
ejpam-726	479	4	,	,	PUNCT
ejpam-726	479	5	1966	1966	NUM
ejpam-726	479	6	.	.	PUNCT
ejpam-726	480	1	[	[	X
ejpam-726	480	2	4	4	NUM
ejpam-726	480	3	]	]	PUNCT
ejpam-726	480	4	m.	m.	NOUN
ejpam-726	480	5	caldas	caldas	PROPN
ejpam-726	480	6	,	,	PUNCT
ejpam-726	480	7	s.	s.	PROPN
ejpam-726	480	8	jafari	jafari	PROPN
ejpam-726	480	9	and	and	CCONJ
ejpam-726	480	10	t.	t.	PROPN
ejpam-726	480	11	noiri	noiri	PROPN
ejpam-726	480	12	and	and	CCONJ
ejpam-726	480	13	m.	m.	NOUN
ejpam-726	480	14	simoes	simoe	NOUN
ejpam-726	480	15	.	.	PUNCT
ejpam-726	481	1	a	a	DET
ejpam-726	481	2	new	new	ADJ
ejpam-726	481	3	generalization	generalization	NOUN
ejpam-726	481	4	of	of	ADP
ejpam-726	481	5	contracontinuity	contracontinuity	NOUN
ejpam-726	481	6	via	via	ADP
ejpam-726	481	7	levine	levine	PROPN
ejpam-726	481	8	’s	’s	PART
ejpam-726	481	9	g−closed	g−close	VERB
ejpam-726	481	10	sets	set	NOUN
ejpam-726	481	11	.	.	PUNCT
ejpam-726	482	1	chaos	chaos	NOUN
ejpam-726	482	2	solitions	solition	NOUN
ejpam-726	482	3	fractals	fractal	NOUN
ejpam-726	482	4	,	,	PUNCT
ejpam-726	482	5	in	in	ADP
ejpam-726	482	6	press	press	NOUN
ejpam-726	482	7	,	,	PUNCT
ejpam-726	482	8	doi	doi	PROPN
ejpam-726	482	9	:	:	PUNCT
ejpam-726	482	10	10.1016	10.1016	NUM
ejpam-726	482	11	/	/	SYM
ejpam-726	482	12	j.chaos.2005.12.032	j.chaos.2005.12.032	PROPN
ejpam-726	482	13	.	.	PUNCT
ejpam-726	483	1	[	[	X
ejpam-726	483	2	5	5	NUM
ejpam-726	483	3	]	]	X
ejpam-726	483	4	s.g	s.g	PROPN
ejpam-726	483	5	.	.	PROPN
ejpam-726	483	6	crossley	crossley	PROPN
ejpam-726	483	7	and	and	CCONJ
ejpam-726	483	8	s.	s.	PROPN
ejpam-726	483	9	k.	k.	PROPN
ejpam-726	483	10	hildebrand	hildebrand	PROPN
ejpam-726	483	11	.	.	PUNCT
ejpam-726	483	12	semi	semi	ADJ
ejpam-726	483	13	topological	topological	ADJ
ejpam-726	483	14	properties	property	NOUN
ejpam-726	483	15	.	.	PUNCT
ejpam-726	484	1	fund	fund	NOUN
ejpam-726	484	2	math	math	PROPN
ejpam-726	484	3	.	.	PUNCT
ejpam-726	484	4	,	,	PUNCT
ejpam-726	484	5	74	74	NUM
ejpam-726	484	6	,	,	PUNCT
ejpam-726	484	7	233254	233254	NUM
ejpam-726	484	8	,	,	PUNCT
ejpam-726	484	9	1972	1972	NUM
ejpam-726	484	10	.	.	PUNCT
ejpam-726	485	1	[	[	X
ejpam-726	485	2	6	6	NUM
ejpam-726	485	3	]	]	PUNCT
ejpam-726	485	4	r.	r.	PROPN
ejpam-726	485	5	devi	devi	PROPN
ejpam-726	485	6	,	,	PUNCT
ejpam-726	485	7	h.	h.	PROPN
ejpam-726	485	8	maki	maki	PROPN
ejpam-726	485	9	and	and	CCONJ
ejpam-726	485	10	k.balachandran	k.balachandran	VERB
ejpam-726	485	11	.	.	PUNCT
ejpam-726	486	1	semi	semi	ADJ
ejpam-726	486	2	-	-	ADJ
ejpam-726	486	3	generalized	generalized	ADJ
ejpam-726	486	4	closed	closed	ADJ
ejpam-726	486	5	maps	map	NOUN
ejpam-726	486	6	and	and	CCONJ
ejpam-726	486	7	generalized	generalized	ADJ
ejpam-726	486	8	semi	semi	ADJ
ejpam-726	486	9	-	-	ADJ
ejpam-726	486	10	closed	closed	ADJ
ejpam-726	486	11	maps	map	NOUN
ejpam-726	486	12	.	.	PUNCT
ejpam-726	487	1	mem	mem	PROPN
ejpam-726	487	2	.	.	PUNCT
ejpam-726	487	3	fac	fac	PROPN
ejpam-726	487	4	.	.	PUNCT
ejpam-726	488	1	sci	sci	PROPN
ejpam-726	488	2	.	.	PROPN
ejpam-726	488	3	kochi	kochi	PROPN
ejpam-726	488	4	univ	univ	PROPN
ejpam-726	488	5	.	.	PUNCT
ejpam-726	489	1	ser	ser	PROPN
ejpam-726	489	2	.	.	PUNCT
ejpam-726	490	1	a	a	DET
ejpam-726	490	2	math	math	NOUN
ejpam-726	490	3	.	.	PUNCT
ejpam-726	491	1	,	,	PUNCT
ejpam-726	491	2	14	14	NUM
ejpam-726	491	3	,	,	PUNCT
ejpam-726	491	4	41	41	NUM
ejpam-726	491	5	-	-	SYM
ejpam-726	491	6	54	54	NUM
ejpam-726	491	7	,	,	PUNCT
ejpam-726	491	8	1993	1993	NUM
ejpam-726	491	9	.	.	PUNCT
ejpam-726	492	1	[	[	X
ejpam-726	492	2	7	7	X
ejpam-726	492	3	]	]	X
ejpam-726	492	4	j.	j.	PROPN
ejpam-726	492	5	dontchev	dontchev	PROPN
ejpam-726	492	6	and	and	CCONJ
ejpam-726	492	7	m.	m.	NOUN
ejpam-726	492	8	ganster	ganster	NOUN
ejpam-726	492	9	.	.	PUNCT
ejpam-726	493	1	on	on	ADP
ejpam-726	493	2	δ	δ	PROPN
ejpam-726	493	3	-	-	PUNCT
ejpam-726	493	4	generalized	generalize	VERB
ejpam-726	493	5	closed	closed	ADJ
ejpam-726	493	6	sets	set	NOUN
ejpam-726	493	7	and	and	CCONJ
ejpam-726	493	8	t3/4	t3/4	NOUN
ejpam-726	493	9	-	-	PUNCT
ejpam-726	493	10	spaces	space	NOUN
ejpam-726	493	11	.	.	PUNCT
ejpam-726	494	1	mem	mem	PROPN
ejpam-726	494	2	.	.	PUNCT
ejpam-726	495	1	fac	fac	PROPN
ejpam-726	495	2	.	.	PUNCT
ejpam-726	496	1	sci	sci	PROPN
ejpam-726	496	2	.	.	PROPN
ejpam-726	496	3	kochi	kochi	PROPN
ejpam-726	496	4	univ	univ	PROPN
ejpam-726	496	5	.	.	PUNCT
ejpam-726	497	1	ser	ser	PROPN
ejpam-726	497	2	.	.	PUNCT
ejpam-726	498	1	a	a	DET
ejpam-726	498	2	math	math	NOUN
ejpam-726	498	3	.	.	PUNCT
ejpam-726	498	4	,	,	PUNCT
ejpam-726	498	5	17	17	NUM
ejpam-726	498	6	,	,	PUNCT
ejpam-726	498	7	15	15	NUM
ejpam-726	498	8	-	-	SYM
ejpam-726	498	9	31	31	NUM
ejpam-726	498	10	,	,	PUNCT
ejpam-726	498	11	1996	1996	NUM
ejpam-726	498	12	.	.	PUNCT
ejpam-726	499	1	[	[	X
ejpam-726	499	2	8	8	X
ejpam-726	499	3	]	]	X
ejpam-726	499	4	j.	j.	PROPN
ejpam-726	499	5	dontchev	dontchev	PROPN
ejpam-726	499	6	and	and	CCONJ
ejpam-726	499	7	t.	t.	PROPN
ejpam-726	499	8	noiri	noiri	PROPN
ejpam-726	499	9	.	.	PUNCT
ejpam-726	500	1	quasi	quasi	ADJ
ejpam-726	500	2	-	-	ADJ
ejpam-726	500	3	normal	normal	ADJ
ejpam-726	500	4	spaces	space	NOUN
ejpam-726	500	5	and	and	CCONJ
ejpam-726	500	6	πg−closed	πg−close	VERB
ejpam-726	500	7	sets	set	NOUN
ejpam-726	500	8	.	.	PUNCT
ejpam-726	501	1	acta	acta	PROPN
ejpam-726	501	2	math	math	PROPN
ejpam-726	501	3	hungar	hungar	PROPN
ejpam-726	501	4	.	.	PUNCT
ejpam-726	502	1	,	,	PUNCT
ejpam-726	502	2	89	89	NUM
ejpam-726	502	3	,	,	PUNCT
ejpam-726	502	4	211	211	NUM
ejpam-726	502	5	-	-	SYM
ejpam-726	502	6	219	219	NUM
ejpam-726	502	7	,	,	PUNCT
ejpam-726	502	8	2000	2000	NUM
ejpam-726	502	9	.	.	PUNCT
ejpam-726	503	1	[	[	X
ejpam-726	503	2	9	9	NUM
ejpam-726	503	3	]	]	PUNCT
ejpam-726	503	4	e.	e.	PROPN
ejpam-726	503	5	ekici	ekici	PROPN
ejpam-726	503	6	.	.	PUNCT
ejpam-726	504	1	on	on	ADP
ejpam-726	504	2	almost	almost	ADV
ejpam-726	504	3	πgp	πgp	VERB
ejpam-726	504	4	-	-	PUNCT
ejpam-726	504	5	continuous	continuous	ADJ
ejpam-726	504	6	functions	function	NOUN
ejpam-726	504	7	,	,	PUNCT
ejpam-726	504	8	chaos	chaos	NOUN
ejpam-726	504	9	solitions	solition	NOUN
ejpam-726	504	10	fractals	fractal	NOUN
ejpam-726	504	11	,	,	PUNCT
ejpam-726	504	12	doi:10.1016	doi:10.1016	PROPN
ejpam-726	504	13	/	/	SYM
ejpam-726	504	14	j.chaos.2005.12.056	j.chaos.2005.12.056	PROPN
ejpam-726	504	15	.	.	PROPN
ejpam-726	504	16	,	,	PUNCT
ejpam-726	504	17	2006	2006	NUM
ejpam-726	504	18	.	.	PUNCT
ejpam-726	505	1	[	[	X
ejpam-726	505	2	10	10	NUM
ejpam-726	505	3	]	]	X
ejpam-726	505	4	e.	e.	PROPN
ejpam-726	505	5	ekici	ekici	PROPN
ejpam-726	505	6	.	.	PUNCT
ejpam-726	506	1	on	on	ADP
ejpam-726	506	2	contra	contra	PROPN
ejpam-726	506	3	πg−continuous	πg−continuous	ADJ
ejpam-726	506	4	functions	function	NOUN
ejpam-726	506	5	.	.	PUNCT
ejpam-726	507	1	chaos	chaos	NOUN
ejpam-726	507	2	solitions	solition	NOUN
ejpam-726	507	3	fractals	fractal	NOUN
ejpam-726	507	4	,	,	PUNCT
ejpam-726	507	5	doi	doi	NOUN
ejpam-726	507	6	:	:	PUNCT
ejpam-726	507	7	10.1016	10.1016	NUM
ejpam-726	507	8	/	/	SYM
ejpam-726	507	9	j.chaos.2006.05.016	j.chaos.2006.05.016	PROPN
ejpam-726	507	10	.	.	PROPN
ejpam-726	507	11	,	,	PUNCT
ejpam-726	507	12	2006	2006	NUM
ejpam-726	508	1	[	[	X
ejpam-726	508	2	11	11	NUM
ejpam-726	508	3	]	]	X
ejpam-726	508	4	e.	e.	PROPN
ejpam-726	508	5	ekici	ekici	PROPN
ejpam-726	508	6	and	and	CCONJ
ejpam-726	508	7	c.w	c.w	PROPN
ejpam-726	508	8	.	.	PROPN
ejpam-726	508	9	baker	baker	PROPN
ejpam-726	508	10	.	.	PUNCT
ejpam-726	509	1	on	on	ADP
ejpam-726	509	2	πg−closed	πg−close	VERB
ejpam-726	509	3	sets	set	NOUN
ejpam-726	509	4	and	and	CCONJ
ejpam-726	509	5	continuity	continuity	NOUN
ejpam-726	509	6	.	.	PUNCT
ejpam-726	510	1	kochi	kochi	PROPN
ejpam-726	510	2	j.	j.	PROPN
ejpam-726	510	3	math	math	PROPN
ejpam-726	510	4	.	.	PUNCT
ejpam-726	510	5	,	,	PUNCT
ejpam-726	510	6	in	in	ADP
ejpam-726	510	7	press	press	NOUN
ejpam-726	510	8	.	.	PUNCT
ejpam-726	511	1	[	[	X
ejpam-726	511	2	12	12	NUM
ejpam-726	511	3	]	]	X
ejpam-726	511	4	e.d	e.d	PROPN
ejpam-726	511	5	.	.	PROPN
ejpam-726	511	6	khalimsky	khalimsky	PROPN
ejpam-726	511	7	.	.	PUNCT
ejpam-726	512	1	applications	application	NOUN
ejpam-726	512	2	of	of	ADP
ejpam-726	512	3	connected	connected	ADJ
ejpam-726	512	4	ordered	order	VERB
ejpam-726	512	5	topological	topological	ADJ
ejpam-726	512	6	spaces	space	NOUN
ejpam-726	512	7	in	in	ADP
ejpam-726	512	8	topology	topology	NOUN
ejpam-726	512	9	.	.	PUNCT
ejpam-726	513	1	in	in	ADP
ejpam-726	513	2	:	:	PUNCT
ejpam-726	513	3	conference	conference	NOUN
ejpam-726	513	4	of	of	ADP
ejpam-726	513	5	math	math	PROPN
ejpam-726	513	6	department	department	PROPN
ejpam-726	513	7	of	of	ADP
ejpam-726	513	8	povolsia	povolsia	PROPN
ejpam-726	513	9	,	,	PUNCT
ejpam-726	513	10	1970	1970	NUM
ejpam-726	513	11	.	.	PUNCT
ejpam-726	514	1	[	[	X
ejpam-726	514	2	13	13	NUM
ejpam-726	514	3	]	]	X
ejpam-726	514	4	e.d	e.d	PROPN
ejpam-726	514	5	.	.	PROPN
ejpam-726	514	6	khalimsky	khalimsky	PROPN
ejpam-726	514	7	,	,	PUNCT
ejpam-726	514	8	r.	r.	PROPN
ejpam-726	514	9	kopperman	kopperman	PROPN
ejpam-726	514	10	and	and	CCONJ
ejpam-726	514	11	p.r	p.r	PROPN
ejpam-726	514	12	.	.	PROPN
ejpam-726	514	13	meyer	meyer	PROPN
ejpam-726	514	14	.	.	PROPN
ejpam-726	514	15	computer	computer	NOUN
ejpam-726	514	16	graphics	graphic	NOUN
ejpam-726	514	17	and	and	CCONJ
ejpam-726	514	18	connected	connect	VERB
ejpam-726	514	19	ordered	order	VERB
ejpam-726	514	20	topological	topological	ADJ
ejpam-726	514	21	topologies	topology	NOUN
ejpam-726	514	22	on	on	ADP
ejpam-726	514	23	finite	finite	NOUN
ejpam-726	514	24	ordered	order	VERB
ejpam-726	514	25	sets	set	NOUN
ejpam-726	514	26	.	.	PUNCT
ejpam-726	515	1	topol	topol	NOUN
ejpam-726	515	2	.	.	PUNCT
ejpam-726	516	1	apply	apply	VERB
ejpam-726	516	2	.	.	PUNCT
ejpam-726	517	1	,	,	PUNCT
ejpam-726	517	2	36	36	NUM
ejpam-726	517	3	,	,	PUNCT
ejpam-726	517	4	1	1	NUM
ejpam-726	517	5	-	-	SYM
ejpam-726	517	6	17	17	NUM
ejpam-726	517	7	,	,	PUNCT
ejpam-726	517	8	1990	1990	NUM
ejpam-726	517	9	.	.	PUNCT
ejpam-726	518	1	[	[	X
ejpam-726	518	2	14	14	NUM
ejpam-726	518	3	]	]	X
ejpam-726	518	4	t.y	t.y	PROPN
ejpam-726	518	5	.	.	PROPN
ejpam-726	518	6	kong	kong	PROPN
ejpam-726	518	7	,	,	PUNCT
ejpam-726	518	8	r.	r.	PROPN
ejpam-726	518	9	kopperman	kopperman	PROPN
ejpam-726	518	10	and	and	CCONJ
ejpam-726	518	11	p.r	p.r	PROPN
ejpam-726	518	12	.	.	PROPN
ejpam-726	518	13	meyer	meyer	PROPN
ejpam-726	518	14	.	.	PUNCT
ejpam-726	519	1	a	a	DET
ejpam-726	519	2	topological	topological	ADJ
ejpam-726	519	3	approach	approach	NOUN
ejpam-726	519	4	to	to	ADP
ejpam-726	519	5	digital	digital	ADJ
ejpam-726	519	6	topology	topology	NOUN
ejpam-726	519	7	.	.	PUNCT
ejpam-726	520	1	am	be	AUX
ejpam-726	520	2	.	.	PUNCT
ejpam-726	521	1	math	math	NOUN
ejpam-726	521	2	.	.	PUNCT
ejpam-726	522	1	monthly	monthly	ADJ
ejpam-726	522	2	,	,	PUNCT
ejpam-726	522	3	98	98	NUM
ejpam-726	522	4	,	,	PUNCT
ejpam-726	522	5	901	901	NUM
ejpam-726	522	6	-	-	SYM
ejpam-726	522	7	917	917	NUM
ejpam-726	522	8	,	,	PUNCT
ejpam-726	522	9	1991	1991	NUM
ejpam-726	522	10	.	.	PUNCT
ejpam-726	523	1	[	[	X
ejpam-726	523	2	15	15	NUM
ejpam-726	523	3	]	]	X
ejpam-726	523	4	v.	v.	ADP
ejpam-726	523	5	kovalevsky	kovalevsky	PROPN
ejpam-726	523	6	and	and	CCONJ
ejpam-726	523	7	r.	r.	PROPN
ejpam-726	523	8	kopperman	kopperman	PROPN
ejpam-726	523	9	.	.	PUNCT
ejpam-726	524	1	some	some	DET
ejpam-726	524	2	topology	topology	NOUN
ejpam-726	524	3	-	-	PUNCT
ejpam-726	524	4	based	base	VERB
ejpam-726	524	5	image	image	NOUN
ejpam-726	524	6	processing	processing	NOUN
ejpam-726	524	7	algorithims	algorithim	NOUN
ejpam-726	524	8	.	.	PUNCT
ejpam-726	525	1	ann	ann	PROPN
ejpam-726	525	2	ny	ny	PROPN
ejpam-726	525	3	acad	acad	PROPN
ejpam-726	525	4	.	.	PUNCT
ejpam-726	526	1	sci	sci	PROPN
ejpam-726	526	2	.	.	PROPN
ejpam-726	526	3	,	,	PUNCT
ejpam-726	526	4	728	728	NUM
ejpam-726	526	5	,	,	PUNCT
ejpam-726	526	6	174	174	NUM
ejpam-726	526	7	-	-	SYM
ejpam-726	526	8	182	182	NUM
ejpam-726	526	9	,	,	PUNCT
ejpam-726	526	10	1994	1994	NUM
ejpam-726	526	11	.	.	PUNCT
ejpam-726	527	1	[	[	X
ejpam-726	527	2	16	16	NUM
ejpam-726	527	3	]	]	PUNCT
ejpam-726	527	4	m.	m.	PROPN
ejpam-726	527	5	k.	k.	PROPN
ejpam-726	527	6	r.	r.	PROPN
ejpam-726	527	7	s.	s.	PROPN
ejpam-726	527	8	v.	v.	PROPN
ejpam-726	527	9	kumar	kumar	PROPN
ejpam-726	527	10	.	.	PUNCT
ejpam-726	528	1	between	between	ADP
ejpam-726	528	2	closed	close	VERB
ejpam-726	528	3	sets	set	NOUN
ejpam-726	528	4	and	and	CCONJ
ejpam-726	528	5	g−closed	g−close	VERB
ejpam-726	528	6	sets	set	NOUN
ejpam-726	528	7	.	.	PUNCT
ejpam-726	529	1	mem	mem	PROPN
ejpam-726	529	2	.	.	PUNCT
ejpam-726	529	3	fac	fac	PROPN
ejpam-726	529	4	.	.	PUNCT
ejpam-726	530	1	sci	sci	PROPN
ejpam-726	530	2	.	.	PROPN
ejpam-726	530	3	kochi	kochi	PROPN
ejpam-726	530	4	univ	univ	PROPN
ejpam-726	530	5	.	.	PUNCT
ejpam-726	531	1	ser	ser	PROPN
ejpam-726	531	2	a	a	DET
ejpam-726	531	3	math	math	NOUN
ejpam-726	531	4	.	.	PUNCT
ejpam-726	532	1	,	,	PUNCT
ejpam-726	532	2	.	.	PUNCT
ejpam-726	533	1	21	21	NUM
ejpam-726	533	2	,	,	PUNCT
ejpam-726	533	3	1	1	NUM
ejpam-726	533	4	-	-	SYM
ejpam-726	533	5	19	19	NUM
ejpam-726	533	6	,	,	PUNCT
ejpam-726	533	7	2000	2000	NUM
ejpam-726	533	8	.	.	PUNCT
ejpam-726	534	1	references	reference	NOUN
ejpam-726	534	2	33	33	NUM
ejpam-726	534	3	[	[	SYM
ejpam-726	534	4	17	17	NUM
ejpam-726	534	5	]	]	X
ejpam-726	534	6	n.	n.	PROPN
ejpam-726	534	7	levine	levine	PROPN
ejpam-726	534	8	.	.	PUNCT
ejpam-726	535	1	generalized	generalize	VERB
ejpam-726	535	2	closed	closed	ADJ
ejpam-726	535	3	sets	set	NOUN
ejpam-726	535	4	in	in	ADP
ejpam-726	535	5	topology	topology	NOUN
ejpam-726	535	6	.	.	PUNCT
ejpam-726	536	1	rend	rend	VERB
ejpam-726	536	2	.	.	PUNCT
ejpam-726	537	1	circ	circ	PROPN
ejpam-726	537	2	.	.	PUNCT
ejpam-726	538	1	mat	mat	PROPN
ejpam-726	538	2	.	.	PUNCT
ejpam-726	538	3	palermo	palermo	PROPN
ejpam-726	538	4	,	,	PUNCT
ejpam-726	538	5	19	19	NUM
ejpam-726	538	6	,	,	PUNCT
ejpam-726	538	7	89	89	NUM
ejpam-726	538	8	-	-	SYM
ejpam-726	538	9	96	96	NUM
ejpam-726	538	10	,	,	PUNCT
ejpam-726	538	11	1970	1970	NUM
ejpam-726	538	12	.	.	PUNCT
ejpam-726	539	1	[	[	X
ejpam-726	539	2	18	18	NUM
ejpam-726	539	3	]	]	X
ejpam-726	539	4	h.	h.	PROPN
ejpam-726	539	5	maki	maki	PROPN
ejpam-726	539	6	,	,	PUNCT
ejpam-726	539	7	j.	j.	PROPN
ejpam-726	539	8	umehara	umehara	PROPN
ejpam-726	539	9	and	and	CCONJ
ejpam-726	539	10	t.	t.	PROPN
ejpam-726	539	11	noiri	noiri	PROPN
ejpam-726	539	12	.	.	PUNCT
ejpam-726	540	1	every	every	DET
ejpam-726	540	2	topological	topological	ADJ
ejpam-726	540	3	space	space	NOUN
ejpam-726	540	4	is	be	AUX
ejpam-726	540	5	pre	pre	ADJ
ejpam-726	540	6	-	-	VERB
ejpam-726	540	7	t1/2	t1/2	ADJ
ejpam-726	540	8	.	.	PUNCT
ejpam-726	541	1	mem	mem	PROPN
ejpam-726	541	2	.	.	PUNCT
ejpam-726	541	3	fac	fac	PROPN
ejpam-726	541	4	.	.	PUNCT
ejpam-726	542	1	sci	sci	PROPN
ejpam-726	542	2	.	.	PROPN
ejpam-726	542	3	kochi	kochi	PROPN
ejpam-726	542	4	univ	univ	PROPN
ejpam-726	542	5	.	.	PUNCT
ejpam-726	543	1	ser	ser	PROPN
ejpam-726	543	2	.	.	PUNCT
ejpam-726	544	1	a	a	DET
ejpam-726	544	2	(	(	PUNCT
ejpam-726	544	3	math	math	NOUN
ejpam-726	544	4	)	)	PUNCT
ejpam-726	544	5	.	.	PUNCT
ejpam-726	544	6	,	,	PUNCT
ejpam-726	544	7	17	17	NUM
ejpam-726	544	8	,	,	PUNCT
ejpam-726	544	9	33	33	NUM
ejpam-726	544	10	-	-	SYM
ejpam-726	544	11	42	42	NUM
ejpam-726	544	12	,	,	PUNCT
ejpam-726	544	13	1996	1996	NUM
ejpam-726	544	14	.	.	PUNCT
ejpam-726	545	1	[	[	X
ejpam-726	545	2	19	19	NUM
ejpam-726	545	3	]	]	X
ejpam-726	545	4	a.s	a.s	PROPN
ejpam-726	545	5	.	.	PROPN
ejpam-726	545	6	mashhour	mashhour	PROPN
ejpam-726	545	7	,	,	PUNCT
ejpam-726	545	8	m.e	m.e	PROPN
ejpam-726	545	9	.	.	PROPN
ejpam-726	545	10	abd	abd	PROPN
ejpam-726	545	11	el	el	PROPN
ejpam-726	545	12	-	-	PROPN
ejpam-726	545	13	monsef	monsef	PROPN
ejpam-726	545	14	and	and	CCONJ
ejpam-726	545	15	s.n	s.n	PROPN
ejpam-726	545	16	.	.	PROPN
ejpam-726	546	1	el	el	PROPN
ejpam-726	546	2	-	-	PUNCT
ejpam-726	546	3	deep	deep	ADJ
ejpam-726	546	4	.	.	PUNCT
ejpam-726	547	1	on	on	ADP
ejpam-726	547	2	precontinuous	precontinuous	ADJ
ejpam-726	547	3	and	and	CCONJ
ejpam-726	547	4	weak	weak	ADJ
ejpam-726	547	5	precontinuous	precontinuous	ADJ
ejpam-726	547	6	mappings	mapping	NOUN
ejpam-726	547	7	.	.	PUNCT
ejpam-726	548	1	proc	proc	NOUN
ejpam-726	548	2	.	.	PUNCT
ejpam-726	549	1	math	math	NOUN
ejpam-726	549	2	.	.	PUNCT
ejpam-726	550	1	phys	phy	NOUN
ejpam-726	550	2	.	.	PUNCT
ejpam-726	551	1	soc	soc	PROPN
ejpam-726	551	2	.	.	PUNCT
ejpam-726	552	1	egypt	egypt	PROPN
ejpam-726	552	2	,	,	PUNCT
ejpam-726	552	3	53	53	NUM
ejpam-726	552	4	,	,	PUNCT
ejpam-726	552	5	47	47	NUM
ejpam-726	552	6	-	-	SYM
ejpam-726	552	7	53	53	NUM
ejpam-726	552	8	,	,	PUNCT
ejpam-726	552	9	1982	1982	NUM
ejpam-726	552	10	.	.	PUNCT
ejpam-726	553	1	[	[	X
ejpam-726	553	2	20	20	NUM
ejpam-726	553	3	]	]	PUNCT
ejpam-726	553	4	t.	t.	PROPN
ejpam-726	553	5	noiri	noiri	PROPN
ejpam-726	553	6	.	.	PUNCT
ejpam-726	554	1	on	on	ADP
ejpam-726	554	2	s	s	NOUN
ejpam-726	554	3	-	-	ADJ
ejpam-726	554	4	normal	normal	ADJ
ejpam-726	554	5	spaces	space	NOUN
ejpam-726	554	6	and	and	CCONJ
ejpam-726	554	7	pre	pre	ADJ
ejpam-726	554	8	gs−closed	gs−close	VERB
ejpam-726	554	9	functions	function	NOUN
ejpam-726	554	10	.	.	PUNCT
ejpam-726	555	1	acta	acta	PROPN
ejpam-726	555	2	math	math	PROPN
ejpam-726	555	3	.	.	PUNCT
ejpam-726	556	1	hungar	hungar	PROPN
ejpam-726	556	2	.	.	PUNCT
ejpam-726	556	3	,	,	PUNCT
ejpam-726	556	4	80	80	NUM
ejpam-726	556	5	,	,	PUNCT
ejpam-726	556	6	105113	105113	NUM
ejpam-726	556	7	,	,	PUNCT
ejpam-726	556	8	1998	1998	NUM
ejpam-726	556	9	.	.	PUNCT
ejpam-726	557	1	[	[	X
ejpam-726	557	2	21	21	NUM
ejpam-726	557	3	]	]	X
ejpam-726	557	4	j.h	j.h	PROPN
ejpam-726	557	5	.	.	PROPN
ejpam-726	557	6	park	park	PROPN
ejpam-726	557	7	.	.	PUNCT
ejpam-726	558	1	on	on	ADP
ejpam-726	558	2	pgp−closed	pgp−close	VERB
ejpam-726	558	3	sets	set	NOUN
ejpam-726	558	4	in	in	ADP
ejpam-726	558	5	topological	topological	ADJ
ejpam-726	558	6	spaces	space	NOUN
ejpam-726	558	7	.	.	PUNCT
ejpam-726	559	1	indian	indian	PROPN
ejpam-726	559	2	j.	j.	PROPN
ejpam-726	559	3	pure	pure	PROPN
ejpam-726	559	4	apply	apply	VERB
ejpam-726	559	5	math	math	NOUN
ejpam-726	559	6	.	.	PUNCT
ejpam-726	559	7	,	,	PUNCT
ejpam-726	559	8	(	(	PUNCT
ejpam-726	559	9	to	to	PART
ejpam-726	559	10	appear	appear	VERB
ejpam-726	559	11	)	)	PUNCT
ejpam-726	560	1	[	[	X
ejpam-726	560	2	22	22	NUM
ejpam-726	560	3	]	]	X
ejpam-726	560	4	j.h	j.h	PROPN
ejpam-726	560	5	.	.	PROPN
ejpam-726	560	6	park	park	PROPN
ejpam-726	560	7	and	and	CCONJ
ejpam-726	560	8	j.k	j.k	PROPN
ejpam-726	560	9	.	.	PROPN
ejpam-726	560	10	park	park	PROPN
ejpam-726	560	11	.	.	PUNCT
ejpam-726	561	1	on	on	ADP
ejpam-726	561	2	pgp	pgp	NOUN
ejpam-726	561	3	-	-	PUNCT
ejpam-726	561	4	continuous	continuous	ADJ
ejpam-726	561	5	functions	function	NOUN
ejpam-726	561	6	in	in	ADP
ejpam-726	561	7	topological	topological	ADJ
ejpam-726	561	8	spaces	space	NOUN
ejpam-726	561	9	.	.	PUNCT
ejpam-726	562	1	chaos	chaos	NOUN
ejpam-726	562	2	solitions	solition	NOUN
ejpam-726	562	3	fractals	fractal	NOUN
ejpam-726	562	4	,	,	PUNCT
ejpam-726	562	5	20	20	NUM
ejpam-726	562	6	,	,	PUNCT
ejpam-726	562	7	467–77	467–77	NUM
ejpam-726	562	8	,	,	PUNCT
ejpam-726	562	9	2004	2004	NUM
ejpam-726	562	10	.	.	PUNCT
ejpam-726	563	1	[	[	X
ejpam-726	563	2	23	23	NUM
ejpam-726	563	3	]	]	X
ejpam-726	563	4	w.	w.	PROPN
ejpam-726	563	5	witten	witten	PROPN
ejpam-726	563	6	.	.	PUNCT
ejpam-726	564	1	phys	phy	NOUN
ejpam-726	564	2	.	.	PUNCT
ejpam-726	565	1	today	today	NOUN
ejpam-726	565	2	,	,	PUNCT
ejpam-726	565	3	24–30	24–30	NUM
ejpam-726	565	4	,	,	PUNCT
ejpam-726	565	5	1996	1996	NUM
ejpam-726	565	6	.	.	PUNCT
ejpam-726	566	1	[	[	X
ejpam-726	566	2	24	24	NUM
ejpam-726	566	3	]	]	SYM
ejpam-726	566	4	yuksel	yuksel	PROPN
ejpam-726	566	5	,	,	PUNCT
ejpam-726	566	6	s.	s.	PROPN
ejpam-726	566	7	and	and	CCONJ
ejpam-726	566	8	beceren	beceren	PROPN
ejpam-726	566	9	,	,	PUNCT
ejpam-726	566	10	y.	y.	NOUN
ejpam-726	566	11	:	:	PUNCT
ejpam-726	566	12	a	a	DET
ejpam-726	566	13	decomposition	decomposition	NOUN
ejpam-726	566	14	of	of	ADP
ejpam-726	566	15	continuity	continuity	NOUN
ejpam-726	566	16	.	.	PUNCT
ejpam-726	567	1	selcuk	selcuk	PROPN
ejpam-726	567	2	univ	univ	PROPN
ejpam-726	567	3	.	.	PUNCT
ejpam-726	568	1	fac	fac	PROPN
ejpam-726	568	2	.	.	PROPN
ejpam-726	568	3	of	of	ADP
ejpam-726	568	4	arts	arts	PROPN
ejpam-726	568	5	science	science	PROPN
ejpam-726	568	6	j.	j.	PROPN
ejpam-726	568	7	,	,	PUNCT
ejpam-726	568	8	14	14	NUM
ejpam-726	568	9	,	,	PUNCT
ejpam-726	568	10	1	1	NUM
ejpam-726	568	11	,	,	PUNCT
ejpam-726	568	12	79	79	NUM
ejpam-726	568	13	–	–	PUNCT
ejpam-726	568	14	83	83	NUM
ejpam-726	568	15	,	,	PUNCT
ejpam-726	568	16	1997	1997	NUM
ejpam-726	568	17	.	.	PUNCT
ejpam-726	569	1	[	[	X
ejpam-726	569	2	25	25	NUM
ejpam-726	569	3	]	]	X
ejpam-726	569	4	v.	v.	NOUN
ejpam-726	569	5	zaitsev	zaitsev	NOUN
ejpam-726	569	6	.	.	PUNCT
ejpam-726	570	1	on	on	ADP
ejpam-726	570	2	certain	certain	ADJ
ejpam-726	570	3	classes	class	NOUN
ejpam-726	570	4	of	of	ADP
ejpam-726	570	5	topological	topological	ADJ
ejpam-726	570	6	spaces	space	NOUN
ejpam-726	570	7	and	and	CCONJ
ejpam-726	570	8	their	their	PRON
ejpam-726	570	9	bicompactifications	bicompactification	NOUN
ejpam-726	570	10	.	.	PUNCT
ejpam-726	571	1	dokl	dokl	NOUN
ejpam-726	571	2	.	.	PUNCT
ejpam-726	571	3	akad	akad	PROPN
ejpam-726	571	4	.	.	PUNCT
ejpam-726	572	1	nauk	nauk	PROPN
ejpam-726	572	2	.	.	PROPN
ejpam-726	572	3	sssr	sssr	PROPN
ejpam-726	572	4	.	.	PUNCT
ejpam-726	572	5	,	,	PUNCT
ejpam-726	572	6	178	178	NUM
ejpam-726	572	7	,	,	PUNCT
ejpam-726	572	8	778–789	778–789	NUM
ejpam-726	572	9	,	,	PUNCT
ejpam-726	572	10	1968	1968	NUM
ejpam-726	572	11	.	.	PUNCT