id	sid	tid	token	lemma	pos
ejpam-2469	1	1	european	european	PROPN
ejpam-2469	1	2	journal	journal	PROPN
ejpam-2469	1	3	of	of	ADP
ejpam-2469	1	4	pure	pure	ADJ
ejpam-2469	1	5	and	and	CCONJ
ejpam-2469	1	6	applied	apply	VERB
ejpam-2469	1	7	mathematics	mathematic	NOUN
ejpam-2469	1	8	vol	vol	NOUN
ejpam-2469	1	9	.	.	PROPN
ejpam-2469	2	1	9	9	NUM
ejpam-2469	2	2	,	,	PUNCT
ejpam-2469	2	3	no	no	INTJ
ejpam-2469	2	4	.	.	NOUN
ejpam-2469	2	5	4	4	NUM
ejpam-2469	2	6	,	,	PUNCT
ejpam-2469	2	7	2016	2016	NUM
ejpam-2469	2	8	,	,	PUNCT
ejpam-2469	2	9	434	434	NUM
ejpam-2469	2	10	-	-	SYM
ejpam-2469	2	11	442	442	NUM
ejpam-2469	2	12	issn	issn	PROPN
ejpam-2469	2	13	1307	1307	NUM
ejpam-2469	2	14	-	-	SYM
ejpam-2469	2	15	5543	5543	NUM
ejpam-2469	2	16	–	–	PUNCT
ejpam-2469	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2469	2	18	some	some	DET
ejpam-2469	2	19	new	new	ADJ
ejpam-2469	2	20	regular	regular	ADJ
ejpam-2469	2	21	generalized	generalize	VERB
ejpam-2469	2	22	closed	close	VERB
ejpam-2469	2	23	sets	set	NOUN
ejpam-2469	2	24	in	in	ADP
ejpam-2469	2	25	ideal	ideal	ADJ
ejpam-2469	2	26	topological	topological	ADJ
ejpam-2469	2	27	spaces	space	NOUN
ejpam-2469	2	28	ümit	ümit	PROPN
ejpam-2469	2	29	karabıyık1	karabıyık1	PROPN
ejpam-2469	2	30	,	,	PUNCT
ejpam-2469	2	31	aynur	aynur	NOUN
ejpam-2469	2	32	keskin	keskin	PROPN
ejpam-2469	2	33	kaymakcı	kaymakcı	PROPN
ejpam-2469	2	34	2,∗	2,∗	NUM
ejpam-2469	2	35	1	1	NUM
ejpam-2469	2	36	department	department	NOUN
ejpam-2469	2	37	of	of	ADP
ejpam-2469	2	38	mathematics	mathematic	NOUN
ejpam-2469	2	39	–	–	PUNCT
ejpam-2469	2	40	computer	computer	NOUN
ejpam-2469	2	41	sciences	science	NOUN
ejpam-2469	2	42	,	,	PUNCT
ejpam-2469	2	43	faculty	faculty	NOUN
ejpam-2469	2	44	of	of	ADP
ejpam-2469	2	45	science	science	NOUN
ejpam-2469	2	46	,	,	PUNCT
ejpam-2469	2	47	necmettin	necmettin	PROPN
ejpam-2469	2	48	erbakan	erbakan	PROPN
ejpam-2469	2	49	university	university	PROPN
ejpam-2469	2	50	,	,	PUNCT
ejpam-2469	2	51	konya	konya	PROPN
ejpam-2469	2	52	,	,	PUNCT
ejpam-2469	2	53	turkey	turkey	PROPN
ejpam-2469	2	54	2	2	NUM
ejpam-2469	2	55	department	department	NOUN
ejpam-2469	2	56	of	of	ADP
ejpam-2469	2	57	mathematics	mathematic	NOUN
ejpam-2469	2	58	,	,	PUNCT
ejpam-2469	2	59	faculty	faculty	NOUN
ejpam-2469	2	60	of	of	ADP
ejpam-2469	2	61	science	science	NOUN
ejpam-2469	2	62	,	,	PUNCT
ejpam-2469	2	63	selcuk	selcuk	PROPN
ejpam-2469	2	64	university	university	PROPN
ejpam-2469	2	65	,	,	PUNCT
ejpam-2469	2	66	konya	konya	PROPN
ejpam-2469	2	67	,	,	PUNCT
ejpam-2469	2	68	turkey	turkey	PROPN
ejpam-2469	2	69	abstract	abstract	NOUN
ejpam-2469	2	70	.	.	PUNCT
ejpam-2469	3	1	we	we	PRON
ejpam-2469	3	2	introduce	introduce	VERB
ejpam-2469	3	3	the	the	DET
ejpam-2469	3	4	notions	notion	NOUN
ejpam-2469	3	5	of	of	ADP
ejpam-2469	3	6	saw	saw	NOUN
ejpam-2469	3	7	-	-	PUNCT
ejpam-2469	3	8	ir	ir	NOUN
ejpam-2469	3	9	g	g	NOUN
ejpam-2469	3	10	-closed	-close	VERB
ejpam-2469	3	11	sets	set	NOUN
ejpam-2469	3	12	and	and	CCONJ
ejpam-2469	3	13	weakly	weakly	ADJ
ejpam-2469	3	14	-	-	PUNCT
ejpam-2469	3	15	r	r	NOUN
ejpam-2469	3	16	gi	gi	NOUN
ejpam-2469	3	17	-closed	-close	VERB
ejpam-2469	3	18	by	by	ADP
ejpam-2469	3	19	using	use	VERB
ejpam-2469	3	20	the	the	DET
ejpam-2469	3	21	notion	notion	NOUN
ejpam-2469	3	22	of	of	ADP
ejpam-2469	3	23	regular	regular	ADJ
ejpam-2469	3	24	open	open	ADJ
ejpam-2469	3	25	sets	set	NOUN
ejpam-2469	3	26	.	.	PUNCT
ejpam-2469	4	1	further	far	ADV
ejpam-2469	4	2	,	,	PUNCT
ejpam-2469	4	3	we	we	PRON
ejpam-2469	4	4	study	study	VERB
ejpam-2469	4	5	the	the	DET
ejpam-2469	4	6	concept	concept	NOUN
ejpam-2469	4	7	of	of	ADP
ejpam-2469	4	8	saw	saw	NOUN
ejpam-2469	4	9	-	-	PUNCT
ejpam-2469	4	10	ir	ir	NOUN
ejpam-2469	4	11	g	g	NOUN
ejpam-2469	4	12	-closed	-close	VERB
ejpam-2469	4	13	sets	set	NOUN
ejpam-2469	4	14	and	and	CCONJ
ejpam-2469	4	15	their	their	PRON
ejpam-2469	4	16	relationships	relationship	NOUN
ejpam-2469	4	17	in	in	ADP
ejpam-2469	4	18	ideal	ideal	ADJ
ejpam-2469	4	19	topological	topological	ADJ
ejpam-2469	4	20	spaces	space	NOUN
ejpam-2469	4	21	by	by	ADP
ejpam-2469	4	22	using	use	VERB
ejpam-2469	4	23	these	these	DET
ejpam-2469	4	24	new	new	ADJ
ejpam-2469	4	25	notions	notion	NOUN
ejpam-2469	4	26	.	.	PUNCT
ejpam-2469	5	1	furthermore	furthermore	ADV
ejpam-2469	5	2	,	,	PUNCT
ejpam-2469	5	3	we	we	PRON
ejpam-2469	5	4	introduce	introduce	VERB
ejpam-2469	5	5	and	and	CCONJ
ejpam-2469	5	6	examine	examine	VERB
ejpam-2469	5	7	some	some	DET
ejpam-2469	5	8	properties	property	NOUN
ejpam-2469	5	9	of	of	ADP
ejpam-2469	5	10	αi	αi	PRON
ejpam-2469	5	11	-∗-normal	-∗-normal	ADJ
ejpam-2469	5	12	space	space	NOUN
ejpam-2469	5	13	.	.	PUNCT
ejpam-2469	6	1	2010	2010	NUM
ejpam-2469	6	2	mathematics	mathematic	NOUN
ejpam-2469	6	3	subject	subject	NOUN
ejpam-2469	6	4	classifications	classification	NOUN
ejpam-2469	6	5	:	:	PUNCT
ejpam-2469	6	6	54a05	54a05	NUM
ejpam-2469	6	7	,	,	PUNCT
ejpam-2469	6	8	54c05	54c05	NUM
ejpam-2469	6	9	key	key	ADJ
ejpam-2469	6	10	words	word	NOUN
ejpam-2469	6	11	and	and	CCONJ
ejpam-2469	6	12	phrases	phrase	NOUN
ejpam-2469	6	13	:	:	PUNCT
ejpam-2469	6	14	ir	ir	PROPN
ejpam-2469	6	15	g	g	PROPN
ejpam-2469	6	16	-closed	-close	VERB
ejpam-2469	6	17	,	,	PUNCT
ejpam-2469	6	18	saw	saw	NOUN
ejpam-2469	6	19	-	-	PUNCT
ejpam-2469	6	20	ir	ir	NOUN
ejpam-2469	6	21	g	g	PROPN
ejpam-2469	6	22	-closed	-close	VERB
ejpam-2469	6	23	,	,	PUNCT
ejpam-2469	6	24	w	w	NOUN
ejpam-2469	6	25	-	-	PUNCT
ejpam-2469	6	26	r	r	NOUN
ejpam-2469	6	27	gi	gi	NOUN
ejpam-2469	6	28	-closed	-close	VERB
ejpam-2469	6	29	,	,	PUNCT
ejpam-2469	6	30	ideal	ideal	ADJ
ejpam-2469	6	31	topological	topological	ADJ
ejpam-2469	6	32	spaces	space	NOUN
ejpam-2469	6	33	1	1	NUM
ejpam-2469	6	34	.	.	PUNCT
ejpam-2469	6	35	introduction	introduction	NOUN
ejpam-2469	6	36	in	in	ADP
ejpam-2469	6	37	1990	1990	NUM
ejpam-2469	6	38	,	,	PUNCT
ejpam-2469	6	39	jankovic	jankovic	PROPN
ejpam-2469	6	40	and	and	CCONJ
ejpam-2469	6	41	hamlett	hamlett	PROPN
ejpam-2469	7	1	[	[	X
ejpam-2469	7	2	3	3	NUM
ejpam-2469	7	3	]	]	PUNCT
ejpam-2469	7	4	,	,	PUNCT
ejpam-2469	7	5	have	have	AUX
ejpam-2469	7	6	initiated	initiate	VERB
ejpam-2469	7	7	the	the	DET
ejpam-2469	7	8	application	application	NOUN
ejpam-2469	7	9	ideal	ideal	NOUN
ejpam-2469	7	10	topological	topological	ADJ
ejpam-2469	7	11	spaces	space	NOUN
ejpam-2469	7	12	.	.	PUNCT
ejpam-2469	8	1	khan	khan	PROPN
ejpam-2469	8	2	and	and	CCONJ
ejpam-2469	8	3	noiri	noiri	ADV
ejpam-2469	9	1	[	[	X
ejpam-2469	9	2	3	3	X
ejpam-2469	9	3	]	]	PUNCT
ejpam-2469	9	4	have	have	AUX
ejpam-2469	9	5	introduced	introduce	VERB
ejpam-2469	9	6	semi	semi	ADJ
ejpam-2469	9	7	-	-	ADJ
ejpam-2469	9	8	local	local	ADJ
ejpam-2469	9	9	functions	function	NOUN
ejpam-2469	9	10	in	in	ADP
ejpam-2469	9	11	ideal	ideal	ADJ
ejpam-2469	9	12	topological	topological	ADJ
ejpam-2469	9	13	spaces	space	NOUN
ejpam-2469	9	14	.	.	PUNCT
ejpam-2469	10	1	firstly	firstly	ADV
ejpam-2469	10	2	the	the	DET
ejpam-2469	10	3	notion	notion	NOUN
ejpam-2469	10	4	of	of	ADP
ejpam-2469	10	5	ig	ig	PROPN
ejpam-2469	10	6	-closed	-close	VERB
ejpam-2469	10	7	set	set	NOUN
ejpam-2469	10	8	is	be	AUX
ejpam-2469	10	9	given	give	VERB
ejpam-2469	10	10	by	by	ADP
ejpam-2469	10	11	dontchev	dontchev	PROPN
ejpam-2469	10	12	et	et	PROPN
ejpam-2469	10	13	al	al	PROPN
ejpam-2469	10	14	.	.	PUNCT
ejpam-2469	11	1	[	[	X
ejpam-2469	11	2	1	1	NUM
ejpam-2469	11	3	]	]	PUNCT
ejpam-2469	11	4	.	.	PUNCT
ejpam-2469	12	1	in	in	ADP
ejpam-2469	12	2	2007	2007	NUM
ejpam-2469	12	3	,	,	PUNCT
ejpam-2469	12	4	navaneethakrishnan	navaneethakrishnan	NOUN
ejpam-2469	12	5	and	and	CCONJ
ejpam-2469	12	6	joseph	joseph	PROPN
ejpam-2469	12	7	[	[	X
ejpam-2469	12	8	9	9	NUM
ejpam-2469	12	9	]	]	PUNCT
ejpam-2469	12	10	have	have	AUX
ejpam-2469	12	11	introduced	introduce	VERB
ejpam-2469	12	12	some	some	PRON
ejpam-2469	12	13	of	of	ADP
ejpam-2469	12	14	properties	property	NOUN
ejpam-2469	12	15	of	of	ADP
ejpam-2469	12	16	ig	ig	PROPN
ejpam-2469	12	17	-closed	-close	VERB
ejpam-2469	12	18	sets	set	NOUN
ejpam-2469	12	19	and	and	CCONJ
ejpam-2469	12	20	ig	ig	PRON
ejpam-2469	12	21	-open	-open	NOUN
ejpam-2469	12	22	sets	set	NOUN
ejpam-2469	12	23	by	by	ADP
ejpam-2469	12	24	using	use	VERB
ejpam-2469	12	25	local	local	ADJ
ejpam-2469	12	26	function	function	NOUN
ejpam-2469	12	27	.	.	PUNCT
ejpam-2469	13	1	recently	recently	ADV
ejpam-2469	13	2	karabiyik	karabiyik	VERB
ejpam-2469	13	3	[	[	X
ejpam-2469	13	4	5	5	NUM
ejpam-2469	13	5	]	]	PUNCT
ejpam-2469	13	6	,	,	PUNCT
ejpam-2469	13	7	has	have	AUX
ejpam-2469	13	8	defined	define	VERB
ejpam-2469	13	9	concept	concept	NOUN
ejpam-2469	13	10	of	of	ADP
ejpam-2469	13	11	r	r	NOUN
ejpam-2469	13	12	gi	gi	NOUN
ejpam-2469	13	13	-closed	-close	VERB
ejpam-2469	13	14	set	set	NOUN
ejpam-2469	13	15	which	which	PRON
ejpam-2469	13	16	is	be	AUX
ejpam-2469	13	17	weaker	weak	ADJ
ejpam-2469	13	18	than	than	SCONJ
ejpam-2469	13	19	ig	ig	PROPN
ejpam-2469	13	20	-closed	-close	VERB
ejpam-2469	13	21	set	set	NOUN
ejpam-2469	13	22	and	and	CCONJ
ejpam-2469	13	23	examined	examine	VERB
ejpam-2469	13	24	some	some	DET
ejpam-2469	13	25	properties	property	NOUN
ejpam-2469	13	26	.	.	PUNCT
ejpam-2469	14	1	also	also	ADV
ejpam-2469	14	2	,	,	PUNCT
ejpam-2469	14	3	he	he	PRON
ejpam-2469	14	4	has	have	AUX
ejpam-2469	14	5	given	give	VERB
ejpam-2469	14	6	some	some	DET
ejpam-2469	14	7	characterization	characterization	NOUN
ejpam-2469	14	8	of	of	ADP
ejpam-2469	14	9	this	this	DET
ejpam-2469	14	10	set	set	NOUN
ejpam-2469	14	11	.	.	PUNCT
ejpam-2469	15	1	in	in	ADP
ejpam-2469	15	2	2013	2013	NUM
ejpam-2469	15	3	,	,	PUNCT
ejpam-2469	15	4	ekici	ekici	NOUN
ejpam-2469	15	5	and	and	CCONJ
ejpam-2469	15	6	ozen	ozen	NOUN
ejpam-2469	15	7	[	[	X
ejpam-2469	15	8	2	2	NUM
ejpam-2469	15	9	]	]	PUNCT
ejpam-2469	15	10	introduced	introduce	VERB
ejpam-2469	15	11	weakly	weakly	ADJ
ejpam-2469	15	12	-	-	PUNCT
ejpam-2469	15	13	ir	ir	NOUN
ejpam-2469	15	14	g	g	NOUN
ejpam-2469	15	15	-closed	-close	VERB
ejpam-2469	15	16	sets	set	NOUN
ejpam-2469	15	17	which	which	PRON
ejpam-2469	15	18	is	be	AUX
ejpam-2469	15	19	a	a	DET
ejpam-2469	15	20	generalized	generalized	ADJ
ejpam-2469	15	21	class	class	NOUN
ejpam-2469	15	22	of	of	ADP
ejpam-2469	15	23	τ∗.	τ∗.	NOUN
ejpam-2469	15	24	in	in	ADP
ejpam-2469	15	25	this	this	DET
ejpam-2469	15	26	paper	paper	NOUN
ejpam-2469	15	27	,	,	PUNCT
ejpam-2469	15	28	we	we	PRON
ejpam-2469	15	29	define	define	VERB
ejpam-2469	15	30	saw	saw	NOUN
ejpam-2469	15	31	-	-	PUNCT
ejpam-2469	15	32	ir	ir	NOUN
ejpam-2469	15	33	g	g	NOUN
ejpam-2469	15	34	-closed	-close	VERB
ejpam-2469	15	35	sets	set	NOUN
ejpam-2469	15	36	and	and	CCONJ
ejpam-2469	15	37	weakly	weakly	ADJ
ejpam-2469	15	38	-	-	PUNCT
ejpam-2469	15	39	r	r	NOUN
ejpam-2469	15	40	gi	gi	NOUN
ejpam-2469	15	41	-closed	-close	VERB
ejpam-2469	15	42	by	by	ADP
ejpam-2469	15	43	have	have	AUX
ejpam-2469	15	44	using	use	VERB
ejpam-2469	15	45	the	the	DET
ejpam-2469	15	46	notion	notion	NOUN
ejpam-2469	15	47	of	of	ADP
ejpam-2469	15	48	regular	regular	ADJ
ejpam-2469	15	49	open	open	ADJ
ejpam-2469	15	50	sets	set	NOUN
ejpam-2469	15	51	.	.	PUNCT
ejpam-2469	16	1	we	we	PRON
ejpam-2469	16	2	investigated	investigate	VERB
ejpam-2469	16	3	some	some	PRON
ejpam-2469	16	4	of	of	ADP
ejpam-2469	16	5	their	their	PRON
ejpam-2469	16	6	properties	property	NOUN
ejpam-2469	16	7	.	.	PUNCT
ejpam-2469	17	1	also	also	ADV
ejpam-2469	17	2	,	,	PUNCT
ejpam-2469	17	3	we	we	PRON
ejpam-2469	17	4	generalized	generalize	VERB
ejpam-2469	17	5	many	many	ADJ
ejpam-2469	17	6	concepts	concept	NOUN
ejpam-2469	17	7	which	which	PRON
ejpam-2469	17	8	is	be	AUX
ejpam-2469	17	9	defined	define	VERB
ejpam-2469	17	10	in	in	ADP
ejpam-2469	17	11	ideal	ideal	ADJ
ejpam-2469	17	12	topological	topological	ADJ
ejpam-2469	17	13	spaces	space	NOUN
ejpam-2469	17	14	.	.	PUNCT
ejpam-2469	18	1	the	the	DET
ejpam-2469	18	2	relationships	relationship	NOUN
ejpam-2469	18	3	of	of	ADP
ejpam-2469	18	4	generalized	generalized	ADJ
ejpam-2469	18	5	class	class	NOUN
ejpam-2469	18	6	and	and	CCONJ
ejpam-2469	18	7	various	various	ADJ
ejpam-2469	18	8	properties	property	NOUN
ejpam-2469	18	9	are	be	AUX
ejpam-2469	18	10	examined	examine	VERB
ejpam-2469	18	11	.	.	PUNCT
ejpam-2469	19	1	∗corresponding	∗corresponde	VERB
ejpam-2469	19	2	author	author	NOUN
ejpam-2469	19	3	.	.	PUNCT
ejpam-2469	20	1	email	email	NOUN
ejpam-2469	20	2	addresses	address	NOUN
ejpam-2469	20	3	:	:	PUNCT
ejpam-2469	20	4	ukarabiyik@konya.edu.tr	ukarabiyik@konya.edu.tr	PROPN
ejpam-2469	20	5	(	(	PUNCT
ejpam-2469	20	6	ü	ü	PROPN
ejpam-2469	20	7	karabıyık	karabıyık	PROPN
ejpam-2469	20	8	)	)	PUNCT
ejpam-2469	20	9	,	,	PUNCT
ejpam-2469	20	10	akeskin@selcuk.edu.tr	akeskin@selcuk.edu.tr	PRON
ejpam-2469	20	11	(	(	PUNCT
ejpam-2469	20	12	a	a	DET
ejpam-2469	20	13	kaymakcı	kaymakcı	NOUN
ejpam-2469	20	14	)	)	PUNCT
ejpam-2469	20	15	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2469	21	1	434	434	NUM
ejpam-2469	22	1	c	c	X
ejpam-2469	22	2	©	©	PROPN
ejpam-2469	22	3	2016	2016	NUM
ejpam-2469	22	4	ejpam	ejpam	VERB
ejpam-2469	22	5	all	all	DET
ejpam-2469	22	6	rights	right	NOUN
ejpam-2469	22	7	reserved	reserve	VERB
ejpam-2469	22	8	.	.	PUNCT
ejpam-2469	23	1	ü	ü	DET
ejpam-2469	23	2	karabıyık	karabıyık	PROPN
ejpam-2469	23	3	,	,	PUNCT
ejpam-2469	23	4	a	a	DET
ejpam-2469	23	5	kaymakcı	kaymakcı	NOUN
ejpam-2469	23	6	/	/	SYM
ejpam-2469	23	7	eur	eur	NOUN
ejpam-2469	23	8	.	.	PUNCT
ejpam-2469	24	1	j.	j.	PROPN
ejpam-2469	24	2	pure	pure	PROPN
ejpam-2469	24	3	appl	appl	PROPN
ejpam-2469	24	4	.	.	PROPN
ejpam-2469	24	5	math	math	PROPN
ejpam-2469	24	6	,	,	PUNCT
ejpam-2469	24	7	9	9	NUM
ejpam-2469	24	8	(	(	PUNCT
ejpam-2469	24	9	2016	2016	NUM
ejpam-2469	24	10	)	)	PUNCT
ejpam-2469	24	11	,	,	PUNCT
ejpam-2469	24	12	434	434	NUM
ejpam-2469	24	13	-	-	SYM
ejpam-2469	24	14	442	442	NUM
ejpam-2469	24	15	435	435	NUM
ejpam-2469	24	16	2	2	NUM
ejpam-2469	24	17	.	.	PUNCT
ejpam-2469	24	18	preliminaries	preliminary	NOUN
ejpam-2469	24	19	in	in	ADP
ejpam-2469	24	20	this	this	DET
ejpam-2469	24	21	paper	paper	NOUN
ejpam-2469	24	22	,	,	PUNCT
ejpam-2469	24	23	(	(	PUNCT
ejpam-2469	24	24	x	x	X
ejpam-2469	24	25	,	,	PUNCT
ejpam-2469	24	26	τ	τ	PROPN
ejpam-2469	24	27	)	)	PUNCT
ejpam-2469	24	28	symbolize	symbolize	PROPN
ejpam-2469	24	29	topological	topological	ADJ
ejpam-2469	24	30	spaces	space	NOUN
ejpam-2469	24	31	on	on	ADP
ejpam-2469	24	32	which	which	PRON
ejpam-2469	24	33	no	no	DET
ejpam-2469	24	34	separation	separation	NOUN
ejpam-2469	24	35	axioms	axiom	NOUN
ejpam-2469	24	36	are	be	AUX
ejpam-2469	24	37	assumed	assume	VERB
ejpam-2469	24	38	unless	unless	SCONJ
ejpam-2469	24	39	clearly	clearly	ADV
ejpam-2469	24	40	stated	state	VERB
ejpam-2469	24	41	.	.	PUNCT
ejpam-2469	25	1	for	for	ADP
ejpam-2469	25	2	a	a	DET
ejpam-2469	25	3	subset	subset	NOUN
ejpam-2469	25	4	a	a	PRON
ejpam-2469	25	5	of	of	ADP
ejpam-2469	25	6	x	x	SYM
ejpam-2469	25	7	,	,	PUNCT
ejpam-2469	25	8	cl∗(a	cl∗(a	NOUN
ejpam-2469	25	9	)	)	PUNCT
ejpam-2469	25	10	and	and	CCONJ
ejpam-2469	25	11	int∗(a	int∗(a	NOUN
ejpam-2469	25	12	)	)	PUNCT
ejpam-2469	25	13	will	will	AUX
ejpam-2469	25	14	represent	represent	VERB
ejpam-2469	25	15	the	the	DET
ejpam-2469	25	16	closure	closure	NOUN
ejpam-2469	25	17	of	of	ADP
ejpam-2469	25	18	a	a	PRON
ejpam-2469	25	19	and	and	CCONJ
ejpam-2469	25	20	the	the	DET
ejpam-2469	25	21	interior	interior	NOUN
ejpam-2469	25	22	of	of	ADP
ejpam-2469	25	23	a	a	DET
ejpam-2469	25	24	in	in	ADP
ejpam-2469	25	25	(	(	PUNCT
ejpam-2469	25	26	x	x	INTJ
ejpam-2469	25	27	,	,	PUNCT
ejpam-2469	25	28	τ	τ	PROPN
ejpam-2469	25	29	)	)	PUNCT
ejpam-2469	25	30	.	.	PUNCT
ejpam-2469	26	1	a	a	DET
ejpam-2469	26	2	subset	subset	NOUN
ejpam-2469	26	3	a	a	PRON
ejpam-2469	26	4	of	of	ADP
ejpam-2469	26	5	a	a	DET
ejpam-2469	26	6	topological	topological	ADJ
ejpam-2469	26	7	spaces	space	NOUN
ejpam-2469	26	8	(	(	PUNCT
ejpam-2469	26	9	x	x	SYM
ejpam-2469	26	10	,	,	PUNCT
ejpam-2469	26	11	τ	τ	X
ejpam-2469	26	12	)	)	PUNCT
ejpam-2469	26	13	is	be	AUX
ejpam-2469	26	14	said	say	VERB
ejpam-2469	26	15	to	to	PART
ejpam-2469	26	16	be	be	AUX
ejpam-2469	26	17	regular	regular	ADJ
ejpam-2469	26	18	open	open	ADJ
ejpam-2469	26	19	[	[	X
ejpam-2469	26	20	11	11	NUM
ejpam-2469	26	21	]	]	PUNCT
ejpam-2469	26	22	(	(	PUNCT
ejpam-2469	26	23	resp	resp	NOUN
ejpam-2469	26	24	.	.	PUNCT
ejpam-2469	27	1	regular	regular	ADJ
ejpam-2469	27	2	closed	closed	ADJ
ejpam-2469	27	3	)	)	PUNCT
ejpam-2469	27	4	if	if	SCONJ
ejpam-2469	27	5	a=	a=	ADV
ejpam-2469	27	6	int(cl(a	int(cl(a	PROPN
ejpam-2469	27	7	)	)	PUNCT
ejpam-2469	27	8	)	)	PUNCT
ejpam-2469	27	9	(	(	PUNCT
ejpam-2469	27	10	resp	resp	NOUN
ejpam-2469	27	11	.	.	PUNCT
ejpam-2469	28	1	a=	a=	PROPN
ejpam-2469	28	2	cl(int(a	cl(int(a	PROPN
ejpam-2469	28	3	)	)	PUNCT
ejpam-2469	28	4	)	)	PUNCT
ejpam-2469	28	5	)	)	PUNCT
ejpam-2469	28	6	.	.	PUNCT
ejpam-2469	29	1	an	an	DET
ejpam-2469	29	2	ideal	ideal	NOUN
ejpam-2469	29	3	i	i	PRON
ejpam-2469	29	4	on	on	ADP
ejpam-2469	29	5	a	a	DET
ejpam-2469	29	6	nonempty	nonempty	ADV
ejpam-2469	29	7	set	set	VERB
ejpam-2469	29	8	x	x	PUNCT
ejpam-2469	29	9	is	be	AUX
ejpam-2469	29	10	a	a	DET
ejpam-2469	29	11	collection	collection	NOUN
ejpam-2469	29	12	of	of	ADP
ejpam-2469	29	13	subsets	subset	NOUN
ejpam-2469	29	14	of	of	ADP
ejpam-2469	29	15	x	x	PUNCT
ejpam-2469	29	16	which	which	PRON
ejpam-2469	29	17	satisfies	satisfy	VERB
ejpam-2469	29	18	the	the	DET
ejpam-2469	29	19	following	follow	VERB
ejpam-2469	29	20	properties	property	NOUN
ejpam-2469	29	21	[	[	X
ejpam-2469	29	22	7	7	NUM
ejpam-2469	29	23	]	]	SYM
ejpam-2469	29	24	(	(	PUNCT
ejpam-2469	29	25	i	i	NOUN
ejpam-2469	29	26	)	)	PUNCT
ejpam-2469	29	27	a∈	a∈	PROPN
ejpam-2469	30	1	i	i	PROPN
ejpam-2469	30	2	and	and	CCONJ
ejpam-2469	30	3	b	b	X
ejpam-2469	30	4	⊆	⊆	NUM
ejpam-2469	30	5	a	a	PRON
ejpam-2469	30	6	implies	imply	VERB
ejpam-2469	30	7	b	b	X
ejpam-2469	30	8	∈	∈	ADV
ejpam-2469	30	9	i	i	PRON
ejpam-2469	30	10	and	and	CCONJ
ejpam-2469	30	11	(	(	PUNCT
ejpam-2469	30	12	ii	ii	NOUN
ejpam-2469	30	13	)	)	PUNCT
ejpam-2469	30	14	a∈	a∈	PROPN
ejpam-2469	30	15	i	i	PROPN
ejpam-2469	30	16	and	and	CCONJ
ejpam-2469	30	17	b	b	X
ejpam-2469	30	18	∈	∈	PROPN
ejpam-2469	30	19	i	i	PRON
ejpam-2469	30	20	implies	imply	VERB
ejpam-2469	30	21	a∪	a∪	PROPN
ejpam-2469	31	1	b	b	X
ejpam-2469	31	2	∈	∈	PROPN
ejpam-2469	32	1	i	i	PRON
ejpam-2469	32	2	.	.	PUNCT
ejpam-2469	33	1	a	a	DET
ejpam-2469	33	2	topological	topological	ADJ
ejpam-2469	33	3	spaces	space	NOUN
ejpam-2469	33	4	(	(	PUNCT
ejpam-2469	33	5	x	x	X
ejpam-2469	33	6	,	,	PUNCT
ejpam-2469	33	7	τ	τ	PROPN
ejpam-2469	33	8	)	)	PUNCT
ejpam-2469	33	9	with	with	ADP
ejpam-2469	33	10	an	an	DET
ejpam-2469	33	11	ideal	ideal	ADJ
ejpam-2469	33	12	i	i	PRON
ejpam-2469	33	13	on	on	ADP
ejpam-2469	33	14	x	x	SYM
ejpam-2469	33	15	is	be	AUX
ejpam-2469	33	16	called	call	VERB
ejpam-2469	33	17	an	an	DET
ejpam-2469	33	18	ideal	ideal	ADJ
ejpam-2469	33	19	topological	topological	ADJ
ejpam-2469	33	20	spaces	space	NOUN
ejpam-2469	33	21	and	and	CCONJ
ejpam-2469	33	22	is	be	AUX
ejpam-2469	33	23	denoted	denote	VERB
ejpam-2469	33	24	by	by	ADP
ejpam-2469	33	25	(	(	PUNCT
ejpam-2469	33	26	x	x	INTJ
ejpam-2469	33	27	,	,	PUNCT
ejpam-2469	33	28	τ	τ	PROPN
ejpam-2469	33	29	,	,	PUNCT
ejpam-2469	33	30	i	i	PROPN
ejpam-2469	33	31	)	)	PUNCT
ejpam-2469	33	32	.	.	PUNCT
ejpam-2469	34	1	if	if	SCONJ
ejpam-2469	34	2	y	y	PROPN
ejpam-2469	34	3	is	be	AUX
ejpam-2469	34	4	a	a	DET
ejpam-2469	34	5	subset	subset	NOUN
ejpam-2469	34	6	of	of	ADP
ejpam-2469	34	7	x	x	PRON
ejpam-2469	34	8	then	then	ADV
ejpam-2469	34	9	iy	iy	INTJ
ejpam-2469	35	1	=	=	PUNCT
ejpam-2469	35	2	{	{	PUNCT
ejpam-2469	35	3	g	g	PROPN
ejpam-2469	35	4	∩	∩	ADJ
ejpam-2469	35	5	y	y	NOUN
ejpam-2469	35	6	:	:	PUNCT
ejpam-2469	35	7	g	g	PROPN
ejpam-2469	35	8	∈	∈	PROPN
ejpam-2469	36	1	i	i	PRON
ejpam-2469	36	2	}	}	PUNCT
ejpam-2469	36	3	is	be	AUX
ejpam-2469	36	4	an	an	DET
ejpam-2469	36	5	ideal	ideal	NOUN
ejpam-2469	36	6	on	on	ADP
ejpam-2469	36	7	y	y	PROPN
ejpam-2469	36	8	and	and	CCONJ
ejpam-2469	36	9	(	(	PUNCT
ejpam-2469	36	10	y	y	PROPN
ejpam-2469	36	11	,	,	PUNCT
ejpam-2469	36	12	τ	τ	PROPN
ejpam-2469	36	13	/	/	SYM
ejpam-2469	36	14	y	y	PROPN
ejpam-2469	36	15	,	,	PUNCT
ejpam-2469	36	16	i	i	PROPN
ejpam-2469	36	17	/	/	SYM
ejpam-2469	36	18	y	y	PROPN
ejpam-2469	36	19	)	)	PUNCT
ejpam-2469	36	20	denote	denote	VERB
ejpam-2469	36	21	the	the	DET
ejpam-2469	36	22	ideal	ideal	ADJ
ejpam-2469	36	23	topological	topological	ADJ
ejpam-2469	36	24	subspaces	subspace	NOUN
ejpam-2469	36	25	.	.	PUNCT
ejpam-2469	37	1	let	let	VERB
ejpam-2469	37	2	p(x	p(x	PROPN
ejpam-2469	37	3	)	)	PUNCT
ejpam-2469	37	4	is	be	AUX
ejpam-2469	37	5	the	the	DET
ejpam-2469	37	6	set	set	NOUN
ejpam-2469	37	7	of	of	ADP
ejpam-2469	37	8	all	all	DET
ejpam-2469	37	9	subset	subset	NOUN
ejpam-2469	37	10	of	of	ADP
ejpam-2469	37	11	x	x	PRON
ejpam-2469	37	12	,	,	PUNCT
ejpam-2469	37	13	a	a	DET
ejpam-2469	37	14	set	set	NOUN
ejpam-2469	37	15	operator	operator	NOUN
ejpam-2469	37	16	∗	∗	NOUN
ejpam-2469	37	17	:	:	PUNCT
ejpam-2469	37	18	p(x	p(x	PROPN
ejpam-2469	37	19	)	)	PUNCT
ejpam-2469	37	20	→	→	SYM
ejpam-2469	37	21	p(x	p(x	PROPN
ejpam-2469	37	22	)	)	PUNCT
ejpam-2469	37	23	,	,	PUNCT
ejpam-2469	37	24	called	call	VERB
ejpam-2469	37	25	a	a	DET
ejpam-2469	37	26	local	local	ADJ
ejpam-2469	37	27	function	function	NOUN
ejpam-2469	38	1	[	[	X
ejpam-2469	38	2	6	6	NUM
ejpam-2469	38	3	]	]	PUNCT
ejpam-2469	38	4	of	of	ADP
ejpam-2469	38	5	a	a	PRON
ejpam-2469	38	6	with	with	ADP
ejpam-2469	38	7	respect	respect	NOUN
ejpam-2469	38	8	to	to	ADP
ejpam-2469	38	9	τ	τ	PROPN
ejpam-2469	38	10	and	and	CCONJ
ejpam-2469	38	11	i	i	PRON
ejpam-2469	38	12	,	,	PUNCT
ejpam-2469	38	13	which	which	PRON
ejpam-2469	38	14	is	be	AUX
ejpam-2469	38	15	defined	define	VERB
ejpam-2469	38	16	as	as	ADP
ejpam-2469	38	17	:	:	PUNCT
ejpam-2469	38	18	for	for	ADP
ejpam-2469	38	19	a	a	DET
ejpam-2469	38	20	⊂	⊂	PROPN
ejpam-2469	38	21	x	x	X
ejpam-2469	38	22	,	,	PUNCT
ejpam-2469	38	23	a∗(i	a∗(i	PROPN
ejpam-2469	38	24	,	,	PUNCT
ejpam-2469	38	25	τ	τ	X
ejpam-2469	38	26	)	)	PUNCT
ejpam-2469	38	27	=	=	PRON
ejpam-2469	39	1	{	{	PUNCT
ejpam-2469	39	2	x	x	PUNCT
ejpam-2469	39	3	∈	∈	PROPN
ejpam-2469	39	4	x	x	X
ejpam-2469	39	5	:	:	PUNCT
ejpam-2469	39	6	u	u	NOUN
ejpam-2469	39	7	∩	∩	NOUN
ejpam-2469	39	8	a	a	DET
ejpam-2469	39	9	6∈	6∈	NUM
ejpam-2469	39	10	i	i	PRON
ejpam-2469	39	11	for	for	ADP
ejpam-2469	39	12	every	every	DET
ejpam-2469	39	13	u	u	PROPN
ejpam-2469	39	14	∈	∈	PROPN
ejpam-2469	39	15	τ(x	τ(x	PUNCT
ejpam-2469	39	16	,	,	PUNCT
ejpam-2469	39	17	x	x	NOUN
ejpam-2469	39	18	)	)	PUNCT
ejpam-2469	39	19	}	}	PUNCT
ejpam-2469	39	20	.	.	PUNCT
ejpam-2469	40	1	we	we	PRON
ejpam-2469	40	2	simply	simply	ADV
ejpam-2469	40	3	write	write	VERB
ejpam-2469	40	4	a∗	a∗	PROPN
ejpam-2469	40	5	instead	instead	ADV
ejpam-2469	40	6	of	of	ADP
ejpam-2469	40	7	a∗(i	a∗(i	PROPN
ejpam-2469	40	8	,	,	PUNCT
ejpam-2469	40	9	τ	τ	PROPN
ejpam-2469	40	10	)	)	PUNCT
ejpam-2469	40	11	in	in	ADP
ejpam-2469	40	12	case	case	NOUN
ejpam-2469	40	13	there	there	PRON
ejpam-2469	40	14	is	be	VERB
ejpam-2469	40	15	no	no	DET
ejpam-2469	40	16	concision	concision	NOUN
ejpam-2469	40	17	.	.	PUNCT
ejpam-2469	41	1	for	for	ADP
ejpam-2469	41	2	every	every	DET
ejpam-2469	41	3	ideal	ideal	ADJ
ejpam-2469	41	4	topological	topological	ADJ
ejpam-2469	41	5	spaces	space	NOUN
ejpam-2469	41	6	(	(	PUNCT
ejpam-2469	41	7	x	x	SYM
ejpam-2469	41	8	,	,	PUNCT
ejpam-2469	41	9	τ	τ	PROPN
ejpam-2469	41	10	,	,	PUNCT
ejpam-2469	41	11	i	i	PROPN
ejpam-2469	41	12	)	)	PUNCT
ejpam-2469	41	13	there	there	PRON
ejpam-2469	41	14	exists	exist	VERB
ejpam-2469	41	15	a	a	DET
ejpam-2469	41	16	topology	topology	NOUN
ejpam-2469	41	17	τ∗	τ∗	NOUN
ejpam-2469	41	18	finer	fine	ADJ
ejpam-2469	41	19	than	than	SCONJ
ejpam-2469	41	20	τ	τ	PROPN
ejpam-2469	41	21	defined	define	VERB
ejpam-2469	41	22	as	as	ADP
ejpam-2469	41	23	τ∗	τ∗	NOUN
ejpam-2469	41	24	=	=	SYM
ejpam-2469	41	25	{	{	PUNCT
ejpam-2469	41	26	u	u	NOUN
ejpam-2469	41	27	⊆	⊆	NUM
ejpam-2469	41	28	x	x	SYM
ejpam-2469	41	29	:	:	PUNCT
ejpam-2469	41	30	cl∗(x	cl∗(x	NOUN
ejpam-2469	41	31	−a	−a	NOUN
ejpam-2469	41	32	)	)	PUNCT
ejpam-2469	41	33	=	=	PUNCT
ejpam-2469	41	34	x	x	SYM
ejpam-2469	41	35	−a	−a	ADV
ejpam-2469	41	36	}	}	PUNCT
ejpam-2469	41	37	generated	generate	VERB
ejpam-2469	41	38	by	by	ADP
ejpam-2469	41	39	the	the	DET
ejpam-2469	41	40	base	base	NOUN
ejpam-2469	41	41	β(i	β(i	PUNCT
ejpam-2469	41	42	,	,	PUNCT
ejpam-2469	41	43	τ	τ	X
ejpam-2469	41	44	)	)	PUNCT
ejpam-2469	41	45	=	=	PRON
ejpam-2469	41	46	{	{	PUNCT
ejpam-2469	41	47	u	u	NOUN
ejpam-2469	41	48	⊆	⊆	NUM
ejpam-2469	41	49	j	j	NOUN
ejpam-2469	41	50	:	:	PUNCT
ejpam-2469	41	51	u	u	PROPN
ejpam-2469	41	52	∈	∈	PROPN
ejpam-2469	41	53	τ	τ	X
ejpam-2469	41	54	and	and	CCONJ
ejpam-2469	41	55	j	j	PROPN
ejpam-2469	41	56	∈	∈	PROPN
ejpam-2469	41	57	i	i	X
ejpam-2469	41	58	}	}	PUNCT
ejpam-2469	41	59	.	.	PUNCT
ejpam-2469	42	1	a	a	DET
ejpam-2469	42	2	kuratowski	kuratowski	ADJ
ejpam-2469	42	3	closure	closure	NOUN
ejpam-2469	42	4	operator	operator	NOUN
ejpam-2469	42	5	cl∗	cl∗	PROPN
ejpam-2469	42	6	(	(	PUNCT
ejpam-2469	42	7	·	·	PUNCT
ejpam-2469	42	8	)	)	PUNCT
ejpam-2469	42	9	for	for	ADP
ejpam-2469	42	10	a	a	DET
ejpam-2469	42	11	topology	topology	NOUN
ejpam-2469	42	12	τ∗(i	τ∗(i	PROPN
ejpam-2469	42	13	,	,	PUNCT
ejpam-2469	42	14	τ	τ	PROPN
ejpam-2469	42	15	)	)	PUNCT
ejpam-2469	42	16	called	call	VERB
ejpam-2469	42	17	the	the	DET
ejpam-2469	42	18	∗-topology	∗-topology	NOUN
ejpam-2469	42	19	,	,	PUNCT
ejpam-2469	42	20	finer	fine	ADJ
ejpam-2469	42	21	than	than	SCONJ
ejpam-2469	42	22	τ	τ	PROPN
ejpam-2469	42	23	is	be	AUX
ejpam-2469	42	24	defined	define	VERB
ejpam-2469	42	25	by	by	ADP
ejpam-2469	42	26	cl∗(a	cl∗(a	NOUN
ejpam-2469	42	27	)	)	PUNCT
ejpam-2469	42	28	=	=	PUNCT
ejpam-2469	43	1	a∪	a∪	DET
ejpam-2469	43	2	a∗	a∗	PROPN
ejpam-2469	44	1	[	[	X
ejpam-2469	44	2	12	12	NUM
ejpam-2469	44	3	]	]	PUNCT
ejpam-2469	44	4	.	.	PUNCT
ejpam-2469	45	1	for	for	ADP
ejpam-2469	45	2	a	a	DET
ejpam-2469	45	3	subset	subset	NOUN
ejpam-2469	45	4	a	a	PRON
ejpam-2469	45	5	of	of	ADP
ejpam-2469	45	6	x	x	SYM
ejpam-2469	45	7	,	,	PUNCT
ejpam-2469	45	8	cl∗(a	cl∗(a	NOUN
ejpam-2469	45	9	)	)	PUNCT
ejpam-2469	45	10	and	and	CCONJ
ejpam-2469	45	11	int∗(a	int∗(a	NOUN
ejpam-2469	45	12	)	)	PUNCT
ejpam-2469	45	13	will	will	AUX
ejpam-2469	45	14	represent	represent	VERB
ejpam-2469	45	15	the	the	DET
ejpam-2469	45	16	closure	closure	NOUN
ejpam-2469	45	17	of	of	ADP
ejpam-2469	45	18	a	a	PRON
ejpam-2469	45	19	and	and	CCONJ
ejpam-2469	45	20	the	the	DET
ejpam-2469	45	21	interior	interior	NOUN
ejpam-2469	45	22	of	of	ADP
ejpam-2469	45	23	a	a	DET
ejpam-2469	45	24	in	in	ADP
ejpam-2469	45	25	(	(	PUNCT
ejpam-2469	45	26	x	x	INTJ
ejpam-2469	45	27	,	,	PUNCT
ejpam-2469	45	28	τ∗	τ∗	NOUN
ejpam-2469	45	29	)	)	PUNCT
ejpam-2469	45	30	,	,	PUNCT
ejpam-2469	45	31	respectively	respectively	ADV
ejpam-2469	45	32	.	.	PUNCT
ejpam-2469	46	1	a	a	DET
ejpam-2469	46	2	subset	subset	NOUN
ejpam-2469	46	3	a	a	PRON
ejpam-2469	46	4	of	of	ADP
ejpam-2469	46	5	an	an	DET
ejpam-2469	46	6	ideal	ideal	ADJ
ejpam-2469	46	7	topological	topological	ADJ
ejpam-2469	46	8	spaces	space	NOUN
ejpam-2469	46	9	(	(	PUNCT
ejpam-2469	46	10	x	x	SYM
ejpam-2469	46	11	,	,	PUNCT
ejpam-2469	46	12	τ	τ	PROPN
ejpam-2469	46	13	,	,	PUNCT
ejpam-2469	46	14	i	i	PROPN
ejpam-2469	46	15	)	)	PUNCT
ejpam-2469	46	16	is	be	AUX
ejpam-2469	46	17	said	say	VERB
ejpam-2469	46	18	to	to	PART
ejpam-2469	46	19	be	be	AUX
ejpam-2469	46	20	a	a	DET
ejpam-2469	46	21	τ∗-closed	τ∗-close	VERB
ejpam-2469	46	22	[	[	PUNCT
ejpam-2469	46	23	3	3	NUM
ejpam-2469	46	24	]	]	PUNCT
ejpam-2469	46	25	,	,	PUNCT
ejpam-2469	46	26	if	if	SCONJ
ejpam-2469	46	27	a∗	a∗	PROPN
ejpam-2469	46	28	⊂	⊂	PROPN
ejpam-2469	46	29	a.	a.	NOUN
ejpam-2469	46	30	a	a	PRON
ejpam-2469	46	31	subset	subset	VERB
ejpam-2469	46	32	a	a	PRON
ejpam-2469	46	33	of	of	ADP
ejpam-2469	46	34	an	an	DET
ejpam-2469	46	35	ideal	ideal	ADJ
ejpam-2469	46	36	topological	topological	ADJ
ejpam-2469	46	37	spaces	space	NOUN
ejpam-2469	46	38	(	(	PUNCT
ejpam-2469	46	39	x	x	SYM
ejpam-2469	46	40	,	,	PUNCT
ejpam-2469	46	41	τ	τ	PROPN
ejpam-2469	46	42	,	,	PUNCT
ejpam-2469	46	43	i	i	PROPN
ejpam-2469	46	44	)	)	PUNCT
ejpam-2469	46	45	is	be	AUX
ejpam-2469	46	46	said	say	VERB
ejpam-2469	46	47	to	to	PART
ejpam-2469	46	48	be	be	AUX
ejpam-2469	46	49	a	a	DET
ejpam-2469	46	50	i	i	NOUN
ejpam-2469	46	51	-open	-open	PROPN
ejpam-2469	47	1	[	[	X
ejpam-2469	47	2	4	4	NUM
ejpam-2469	47	3	]	]	PUNCT
ejpam-2469	47	4	,	,	PUNCT
ejpam-2469	47	5	if	if	SCONJ
ejpam-2469	47	6	a	a	DET
ejpam-2469	47	7	⊂	⊂	PROPN
ejpam-2469	47	8	int(a∗	int(a∗	PART
ejpam-2469	47	9	)	)	PUNCT
ejpam-2469	47	10	.	.	PUNCT
ejpam-2469	48	1	a	a	DET
ejpam-2469	48	2	subset	subset	NOUN
ejpam-2469	48	3	a	a	PRON
ejpam-2469	48	4	of	of	ADP
ejpam-2469	48	5	an	an	DET
ejpam-2469	48	6	ideal	ideal	ADJ
ejpam-2469	48	7	topological	topological	ADJ
ejpam-2469	48	8	spaces	space	NOUN
ejpam-2469	48	9	(	(	PUNCT
ejpam-2469	48	10	x	x	SYM
ejpam-2469	48	11	,	,	PUNCT
ejpam-2469	48	12	τ	τ	PROPN
ejpam-2469	48	13	,	,	PUNCT
ejpam-2469	48	14	i	i	PROPN
ejpam-2469	48	15	)	)	PUNCT
ejpam-2469	48	16	is	be	AUX
ejpam-2469	48	17	said	say	VERB
ejpam-2469	48	18	to	to	PART
ejpam-2469	48	19	be	be	AUX
ejpam-2469	48	20	a	a	DET
ejpam-2469	48	21	i	i	NOUN
ejpam-2469	48	22	-regular	-regular	ADJ
ejpam-2469	48	23	open	open	ADJ
ejpam-2469	48	24	(	(	PUNCT
ejpam-2469	48	25	resp	resp	NOUN
ejpam-2469	48	26	.	.	PUNCT
ejpam-2469	49	1	i	i	PRON
ejpam-2469	49	2	-regular	-regular	VERB
ejpam-2469	49	3	closed	close	VERB
ejpam-2469	49	4	)	)	PUNCT
ejpam-2469	50	1	[	[	X
ejpam-2469	50	2	8	8	NUM
ejpam-2469	50	3	]	]	PUNCT
ejpam-2469	50	4	,	,	PUNCT
ejpam-2469	50	5	if	if	SCONJ
ejpam-2469	50	6	a	a	DET
ejpam-2469	50	7	=	=	NOUN
ejpam-2469	50	8	int∗(cl∗(a	int∗(cl∗(a	NOUN
ejpam-2469	50	9	)	)	PUNCT
ejpam-2469	50	10	)	)	PUNCT
ejpam-2469	50	11	(	(	PUNCT
ejpam-2469	50	12	resp	resp	NOUN
ejpam-2469	50	13	.	.	PUNCT
ejpam-2469	51	1	a	a	DET
ejpam-2469	51	2	=	=	PUNCT
ejpam-2469	51	3	cl∗(int∗(a	cl∗(int∗(a	NOUN
ejpam-2469	51	4	)	)	PUNCT
ejpam-2469	51	5	)	)	PUNCT
ejpam-2469	51	6	)	)	PUNCT
ejpam-2469	51	7	.	.	PUNCT
ejpam-2469	52	1	a	a	DET
ejpam-2469	52	2	subset	subset	NOUN
ejpam-2469	52	3	a	a	PRON
ejpam-2469	52	4	of	of	ADP
ejpam-2469	52	5	an	an	DET
ejpam-2469	52	6	(	(	PUNCT
ejpam-2469	52	7	x	x	SYM
ejpam-2469	52	8	,	,	PUNCT
ejpam-2469	52	9	τ	τ	PROPN
ejpam-2469	52	10	,	,	PUNCT
ejpam-2469	52	11	i	i	PRON
ejpam-2469	52	12	)	)	PUNCT
ejpam-2469	52	13	be	be	VERB
ejpam-2469	52	14	a	a	DET
ejpam-2469	52	15	ideal	ideal	ADJ
ejpam-2469	52	16	topological	topological	ADJ
ejpam-2469	52	17	spaces	space	NOUN
ejpam-2469	52	18	is	be	AUX
ejpam-2469	52	19	said	say	VERB
ejpam-2469	52	20	to	to	PART
ejpam-2469	52	21	be	be	AUX
ejpam-2469	52	22	a	a	DET
ejpam-2469	52	23	ir	ir	NOUN
ejpam-2469	52	24	g	g	NOUN
ejpam-2469	52	25	-closed	-closed	ADJ
ejpam-2469	53	1	[	[	PUNCT
ejpam-2469	53	2	10	10	NUM
ejpam-2469	53	3	]	]	PUNCT
ejpam-2469	53	4	,	,	PUNCT
ejpam-2469	53	5	a∗	a∗	PROPN
ejpam-2469	53	6	⊂	⊂	PROPN
ejpam-2469	53	7	u	u	PROPN
ejpam-2469	53	8	whenever	whenever	SCONJ
ejpam-2469	53	9	a⊆	a⊆	VERB
ejpam-2469	53	10	u	u	NOUN
ejpam-2469	53	11	and	and	CCONJ
ejpam-2469	53	12	is	be	AUX
ejpam-2469	53	13	regular	regular	ADJ
ejpam-2469	53	14	open	open	ADJ
ejpam-2469	53	15	set	set	NOUN
ejpam-2469	53	16	in	in	ADP
ejpam-2469	53	17	x	x	X
ejpam-2469	53	18	.	.	PUNCT
ejpam-2469	54	1	a	a	DET
ejpam-2469	54	2	subset	subset	NOUN
ejpam-2469	54	3	a	a	PRON
ejpam-2469	54	4	of	of	ADP
ejpam-2469	54	5	an	an	DET
ejpam-2469	54	6	(	(	PUNCT
ejpam-2469	54	7	x	x	SYM
ejpam-2469	54	8	,	,	PUNCT
ejpam-2469	54	9	τ	τ	PROPN
ejpam-2469	54	10	,	,	PUNCT
ejpam-2469	54	11	i	i	PRON
ejpam-2469	54	12	)	)	PUNCT
ejpam-2469	54	13	be	be	VERB
ejpam-2469	54	14	a	a	DET
ejpam-2469	54	15	ideal	ideal	ADJ
ejpam-2469	54	16	topological	topological	ADJ
ejpam-2469	54	17	spaces	space	NOUN
ejpam-2469	54	18	is	be	AUX
ejpam-2469	54	19	said	say	VERB
ejpam-2469	54	20	to	to	PART
ejpam-2469	54	21	be	be	AUX
ejpam-2469	54	22	a	a	DET
ejpam-2469	54	23	r	r	NOUN
ejpam-2469	54	24	gi	gi	NOUN
ejpam-2469	54	25	-closed	-close	VERB
ejpam-2469	55	1	[	[	X
ejpam-2469	55	2	5	5	NUM
ejpam-2469	55	3	]	]	PUNCT
ejpam-2469	55	4	,	,	PUNCT
ejpam-2469	55	5	if	if	SCONJ
ejpam-2469	55	6	cl∗(a	cl∗(a	ADJ
ejpam-2469	55	7	)	)	PUNCT
ejpam-2469	55	8	⊂	⊂	PROPN
ejpam-2469	55	9	u	u	NOUN
ejpam-2469	55	10	whenever	whenever	SCONJ
ejpam-2469	55	11	a⊆	a⊆	VERB
ejpam-2469	55	12	u	u	NOUN
ejpam-2469	55	13	and	and	CCONJ
ejpam-2469	55	14	is	be	AUX
ejpam-2469	55	15	regular	regular	ADJ
ejpam-2469	55	16	open	open	ADJ
ejpam-2469	55	17	set	set	NOUN
ejpam-2469	55	18	in	in	ADP
ejpam-2469	55	19	x	x	X
ejpam-2469	55	20	.	.	PUNCT
ejpam-2469	56	1	a	a	DET
ejpam-2469	56	2	subset	subset	NOUN
ejpam-2469	56	3	a	a	PRON
ejpam-2469	56	4	of	of	ADP
ejpam-2469	56	5	an	an	DET
ejpam-2469	56	6	(	(	PUNCT
ejpam-2469	56	7	x	x	SYM
ejpam-2469	56	8	,	,	PUNCT
ejpam-2469	56	9	τ	τ	PROPN
ejpam-2469	56	10	,	,	PUNCT
ejpam-2469	56	11	i	i	PRON
ejpam-2469	56	12	)	)	PUNCT
ejpam-2469	56	13	be	be	VERB
ejpam-2469	56	14	a	a	DET
ejpam-2469	56	15	ideal	ideal	ADJ
ejpam-2469	56	16	topological	topological	ADJ
ejpam-2469	56	17	spaces	space	NOUN
ejpam-2469	56	18	is	be	AUX
ejpam-2469	56	19	said	say	VERB
ejpam-2469	56	20	to	to	PART
ejpam-2469	56	21	be	be	AUX
ejpam-2469	56	22	a	a	DET
ejpam-2469	56	23	weakly	weakly	ADJ
ejpam-2469	56	24	-	-	PUNCT
ejpam-2469	56	25	ir	ir	NOUN
ejpam-2469	56	26	g	g	PROPN
ejpam-2469	56	27	-closed(briefly	-closed(briefly	PROPN
ejpam-2469	56	28	w	w	PROPN
ejpam-2469	56	29	-	-	PUNCT
ejpam-2469	56	30	ir	ir	NOUN
ejpam-2469	56	31	g	g	NOUN
ejpam-2469	56	32	-closed	-close	VERB
ejpam-2469	56	33	)	)	PUNCT
ejpam-2469	57	1	[	[	X
ejpam-2469	57	2	2	2	NUM
ejpam-2469	57	3	]	]	PUNCT
ejpam-2469	57	4	,	,	PUNCT
ejpam-2469	57	5	if	if	SCONJ
ejpam-2469	57	6	(	(	PUNCT
ejpam-2469	57	7	int(a))∗	int(a))∗	PROPN
ejpam-2469	57	8	⊂	⊂	PROPN
ejpam-2469	57	9	u	u	PROPN
ejpam-2469	57	10	whenever	whenever	SCONJ
ejpam-2469	57	11	a⊆	a⊆	VERB
ejpam-2469	57	12	u	u	NOUN
ejpam-2469	57	13	and	and	CCONJ
ejpam-2469	57	14	is	be	AUX
ejpam-2469	57	15	regular	regular	ADJ
ejpam-2469	57	16	open	open	ADJ
ejpam-2469	57	17	set	set	NOUN
ejpam-2469	57	18	in	in	ADP
ejpam-2469	57	19	x	x	PROPN
ejpam-2469	57	20	.	.	PUNCT
ejpam-2469	58	1	3	3	X
ejpam-2469	58	2	.	.	X
ejpam-2469	58	3	saw	saw	NOUN
ejpam-2469	58	4	-	-	PUNCT
ejpam-2469	58	5	ir	ir	NOUN
ejpam-2469	58	6	g	g	NOUN
ejpam-2469	58	7	-	-	PUNCT
ejpam-2469	58	8	closed	close	VERB
ejpam-2469	58	9	sets	set	NOUN
ejpam-2469	58	10	in	in	ADP
ejpam-2469	58	11	this	this	DET
ejpam-2469	58	12	section	section	NOUN
ejpam-2469	58	13	,	,	PUNCT
ejpam-2469	58	14	fist	fist	NOUN
ejpam-2469	58	15	of	of	ADP
ejpam-2469	58	16	all	all	PRON
ejpam-2469	58	17	we	we	PRON
ejpam-2469	58	18	introduce	introduce	VERB
ejpam-2469	58	19	the	the	DET
ejpam-2469	58	20	notion	notion	NOUN
ejpam-2469	58	21	called	call	VERB
ejpam-2469	58	22	saw	saw	NOUN
ejpam-2469	58	23	-	-	PUNCT
ejpam-2469	58	24	ir	ir	NOUN
ejpam-2469	58	25	g	g	NOUN
ejpam-2469	58	26	-closed	-close	VERB
ejpam-2469	58	27	and	and	CCONJ
ejpam-2469	58	28	give	give	VERB
ejpam-2469	58	29	some	some	DET
ejpam-2469	58	30	characterizations	characterization	NOUN
ejpam-2469	58	31	of	of	ADP
ejpam-2469	58	32	this	this	DET
ejpam-2469	58	33	sets	set	NOUN
ejpam-2469	58	34	.	.	PUNCT
ejpam-2469	59	1	definition	definition	NOUN
ejpam-2469	59	2	1	1	NUM
ejpam-2469	59	3	.	.	PUNCT
ejpam-2469	60	1	a	a	DET
ejpam-2469	60	2	subset	subset	NOUN
ejpam-2469	60	3	a	a	PRON
ejpam-2469	60	4	of	of	ADP
ejpam-2469	60	5	an	an	DET
ejpam-2469	60	6	(	(	PUNCT
ejpam-2469	60	7	x	x	SYM
ejpam-2469	60	8	,	,	PUNCT
ejpam-2469	60	9	τ	τ	PROPN
ejpam-2469	60	10	,	,	PUNCT
ejpam-2469	60	11	i	i	PRON
ejpam-2469	60	12	)	)	PUNCT
ejpam-2469	60	13	be	be	VERB
ejpam-2469	60	14	a	a	DET
ejpam-2469	60	15	ideal	ideal	ADJ
ejpam-2469	60	16	topological	topological	ADJ
ejpam-2469	60	17	spaces	space	NOUN
ejpam-2469	60	18	is	be	AUX
ejpam-2469	60	19	said	say	VERB
ejpam-2469	60	20	to	to	PART
ejpam-2469	60	21	be	be	AUX
ejpam-2469	60	22	αi	αi	PRON
ejpam-2469	60	23	-∗-closed	-∗-close	VERB
ejpam-2469	60	24	if	if	SCONJ
ejpam-2469	60	25	cl∗(int(cl(a	cl∗(int(cl(a	PROPN
ejpam-2469	60	26	)	)	PUNCT
ejpam-2469	60	27	)	)	PUNCT
ejpam-2469	60	28	)	)	PUNCT
ejpam-2469	61	1	⊂	⊂	PROPN
ejpam-2469	61	2	a.	a.	NOUN
ejpam-2469	61	3	the	the	DET
ejpam-2469	61	4	complement	complement	NOUN
ejpam-2469	61	5	of	of	ADP
ejpam-2469	61	6	αi	αi	PROPN
ejpam-2469	61	7	-∗-closed	-∗-close	VERB
ejpam-2469	61	8	set	set	NOUN
ejpam-2469	61	9	is	be	AUX
ejpam-2469	61	10	said	say	VERB
ejpam-2469	61	11	to	to	PART
ejpam-2469	61	12	be	be	AUX
ejpam-2469	61	13	αi	αi	X
ejpam-2469	61	14	-∗-open	-∗-open	PROPN
ejpam-2469	61	15	.	.	PUNCT
ejpam-2469	62	1	definition	definition	NOUN
ejpam-2469	62	2	2	2	NUM
ejpam-2469	62	3	.	.	PUNCT
ejpam-2469	63	1	a	a	DET
ejpam-2469	63	2	subset	subset	NOUN
ejpam-2469	63	3	a	a	PRON
ejpam-2469	63	4	of	of	ADP
ejpam-2469	63	5	an	an	DET
ejpam-2469	63	6	(	(	PUNCT
ejpam-2469	63	7	x	x	SYM
ejpam-2469	63	8	,	,	PUNCT
ejpam-2469	63	9	τ	τ	PROPN
ejpam-2469	63	10	,	,	PUNCT
ejpam-2469	63	11	i	i	PRON
ejpam-2469	63	12	)	)	PUNCT
ejpam-2469	63	13	be	be	VERB
ejpam-2469	63	14	a	a	DET
ejpam-2469	63	15	ideal	ideal	ADJ
ejpam-2469	63	16	topological	topological	ADJ
ejpam-2469	63	17	spaces	space	NOUN
ejpam-2469	63	18	is	be	AUX
ejpam-2469	63	19	said	say	VERB
ejpam-2469	63	20	to	to	PART
ejpam-2469	63	21	be	be	AUX
ejpam-2469	63	22	:	:	PUNCT
ejpam-2469	63	23	(	(	PUNCT
ejpam-2469	63	24	1	1	X
ejpam-2469	63	25	)	)	PUNCT
ejpam-2469	63	26	strongly	strongly	ADV
ejpam-2469	63	27	almost	almost	ADV
ejpam-2469	63	28	weakly	weakly	ADV
ejpam-2469	63	29	-	-	PUNCT
ejpam-2469	63	30	ir	ir	NOUN
ejpam-2469	63	31	g	g	NOUN
ejpam-2469	63	32	-closed(briefly	-closed(briefly	PUNCT
ejpam-2469	63	33	saw	see	VERB
ejpam-2469	63	34	-	-	PUNCT
ejpam-2469	63	35	ir	ir	NOUN
ejpam-2469	63	36	g	g	PROPN
ejpam-2469	63	37	-closed	-close	VERB
ejpam-2469	63	38	)	)	PUNCT
ejpam-2469	63	39	if	if	SCONJ
ejpam-2469	63	40	(	(	PUNCT
ejpam-2469	63	41	int(cl(a)))∗	int(cl(a)))∗	PROPN
ejpam-2469	63	42	⊂	⊂	PROPN
ejpam-2469	63	43	u	u	PROPN
ejpam-2469	63	44	whenever	whenever	SCONJ
ejpam-2469	63	45	a⊆	a⊆	VERB
ejpam-2469	63	46	u	u	NOUN
ejpam-2469	63	47	and	and	CCONJ
ejpam-2469	63	48	u	u	NOUN
ejpam-2469	63	49	is	be	AUX
ejpam-2469	63	50	a	a	DET
ejpam-2469	63	51	regular	regular	ADJ
ejpam-2469	63	52	open	open	NOUN
ejpam-2469	63	53	in	in	ADP
ejpam-2469	63	54	x	x	X
ejpam-2469	63	55	.	.	PUNCT
ejpam-2469	64	1	ü	ü	PROPN
ejpam-2469	64	2	karabıyık	karabıyık	PROPN
ejpam-2469	64	3	,	,	PUNCT
ejpam-2469	64	4	a	a	DET
ejpam-2469	64	5	kaymakcı	kaymakcı	NOUN
ejpam-2469	64	6	/	/	SYM
ejpam-2469	64	7	eur	eur	NOUN
ejpam-2469	64	8	.	.	PUNCT
ejpam-2469	65	1	j.	j.	PROPN
ejpam-2469	65	2	pure	pure	PROPN
ejpam-2469	65	3	appl	appl	PROPN
ejpam-2469	65	4	.	.	PROPN
ejpam-2469	65	5	math	math	PROPN
ejpam-2469	65	6	,	,	PUNCT
ejpam-2469	65	7	9	9	NUM
ejpam-2469	65	8	(	(	PUNCT
ejpam-2469	65	9	2016	2016	NUM
ejpam-2469	65	10	)	)	PUNCT
ejpam-2469	65	11	,	,	PUNCT
ejpam-2469	65	12	434	434	NUM
ejpam-2469	65	13	-	-	SYM
ejpam-2469	65	14	442	442	NUM
ejpam-2469	65	15	436	436	NUM
ejpam-2469	65	16	(	(	PUNCT
ejpam-2469	65	17	2	2	NUM
ejpam-2469	65	18	)	)	PUNCT
ejpam-2469	65	19	almost	almost	ADV
ejpam-2469	65	20	weakly	weakly	ADV
ejpam-2469	65	21	-	-	PUNCT
ejpam-2469	65	22	ir	ir	NOUN
ejpam-2469	65	23	g	g	NOUN
ejpam-2469	65	24	-closed(briefly	-closed(briefly	PROPN
ejpam-2469	65	25	aw	aw	INTJ
ejpam-2469	65	26	-	-	NOUN
ejpam-2469	65	27	ir	ir	NOUN
ejpam-2469	65	28	g	g	PROPN
ejpam-2469	65	29	-closed	-close	VERB
ejpam-2469	65	30	)	)	PUNCT
ejpam-2469	65	31	if	if	SCONJ
ejpam-2469	65	32	(	(	PUNCT
ejpam-2469	65	33	int(a∗))∗	int(a∗))∗	NOUN
ejpam-2469	65	34	⊂	⊂	PROPN
ejpam-2469	65	35	u	u	NOUN
ejpam-2469	65	36	whenever	whenever	SCONJ
ejpam-2469	65	37	a	a	DET
ejpam-2469	65	38	⊆	⊆	NUM
ejpam-2469	65	39	u	u	NOUN
ejpam-2469	65	40	and	and	CCONJ
ejpam-2469	65	41	u	u	NOUN
ejpam-2469	65	42	is	be	AUX
ejpam-2469	65	43	a	a	DET
ejpam-2469	65	44	regular	regular	ADJ
ejpam-2469	65	45	open	open	NOUN
ejpam-2469	65	46	in	in	ADP
ejpam-2469	65	47	x	x	X
ejpam-2469	65	48	.	.	PUNCT
ejpam-2469	66	1	(	(	PUNCT
ejpam-2469	66	2	3	3	X
ejpam-2469	66	3	)	)	PUNCT
ejpam-2469	66	4	almost	almost	ADV
ejpam-2469	66	5	weakly	weakly	ADJ
ejpam-2469	66	6	-	-	PUNCT
ejpam-2469	66	7	r	r	NOUN
ejpam-2469	66	8	gi	gi	NOUN
ejpam-2469	66	9	-closed(briefly	-closed(briefly	PUNCT
ejpam-2469	66	10	aw	aw	INTJ
ejpam-2469	66	11	-	-	PUNCT
ejpam-2469	66	12	r	r	NOUN
ejpam-2469	66	13	gi	gi	NOUN
ejpam-2469	66	14	-closed	-close	VERB
ejpam-2469	66	15	)	)	PUNCT
ejpam-2469	66	16	if	if	SCONJ
ejpam-2469	66	17	(	(	PUNCT
ejpam-2469	66	18	(	(	PUNCT
ejpam-2469	66	19	int(cl∗(a))))∗	int(cl∗(a))))∗	PROPN
ejpam-2469	66	20	⊂	⊂	PROPN
ejpam-2469	66	21	u	u	NOUN
ejpam-2469	66	22	whenever	whenever	SCONJ
ejpam-2469	66	23	a	a	DET
ejpam-2469	66	24	⊆	⊆	NUM
ejpam-2469	66	25	u	u	NOUN
ejpam-2469	66	26	and	and	CCONJ
ejpam-2469	66	27	u	u	NOUN
ejpam-2469	66	28	is	be	AUX
ejpam-2469	66	29	a	a	DET
ejpam-2469	66	30	regular	regular	ADJ
ejpam-2469	66	31	open	open	NOUN
ejpam-2469	66	32	in	in	ADP
ejpam-2469	66	33	x	x	X
ejpam-2469	66	34	.	.	PUNCT
ejpam-2469	67	1	theorem	theorem	NOUN
ejpam-2469	67	2	1	1	X
ejpam-2469	67	3	.	.	PUNCT
ejpam-2469	68	1	let	let	VERB
ejpam-2469	68	2	(	(	PUNCT
ejpam-2469	68	3	x	x	X
ejpam-2469	68	4	,	,	PUNCT
ejpam-2469	68	5	τ	τ	PROPN
ejpam-2469	68	6	,	,	PUNCT
ejpam-2469	68	7	i	i	PRON
ejpam-2469	68	8	)	)	PUNCT
ejpam-2469	68	9	be	be	VERB
ejpam-2469	68	10	a	a	DET
ejpam-2469	68	11	ideal	ideal	ADJ
ejpam-2469	68	12	topological	topological	ADJ
ejpam-2469	68	13	spaces	space	NOUN
ejpam-2469	68	14	and	and	CCONJ
ejpam-2469	68	15	a	a	DET
ejpam-2469	68	16	⊂	⊂	PROPN
ejpam-2469	68	17	x	x	X
ejpam-2469	68	18	,	,	PUNCT
ejpam-2469	68	19	the	the	DET
ejpam-2469	68	20	following	follow	VERB
ejpam-2469	68	21	properties	property	NOUN
ejpam-2469	68	22	are	be	AUX
ejpam-2469	68	23	equivalent	equivalent	ADJ
ejpam-2469	68	24	:	:	PUNCT
ejpam-2469	68	25	(	(	PUNCT
ejpam-2469	68	26	1	1	X
ejpam-2469	68	27	)	)	PUNCT
ejpam-2469	68	28	a	a	PRON
ejpam-2469	68	29	is	be	AUX
ejpam-2469	68	30	a	a	DET
ejpam-2469	68	31	saw	saw	NOUN
ejpam-2469	68	32	-	-	PUNCT
ejpam-2469	68	33	ir	ir	NOUN
ejpam-2469	68	34	g	g	NOUN
ejpam-2469	68	35	-closed	-close	VERB
ejpam-2469	68	36	sets	set	NOUN
ejpam-2469	68	37	(	(	PUNCT
ejpam-2469	68	38	2	2	NUM
ejpam-2469	68	39	)	)	PUNCT
ejpam-2469	68	40	cl∗(int(cl(a	cl∗(int(cl(a	NOUN
ejpam-2469	68	41	)	)	PUNCT
ejpam-2469	68	42	)	)	PUNCT
ejpam-2469	68	43	)	)	PUNCT
ejpam-2469	69	1	⊂	⊂	PROPN
ejpam-2469	69	2	u	u	NOUN
ejpam-2469	69	3	whenever	whenever	SCONJ
ejpam-2469	69	4	a⊆	a⊆	VERB
ejpam-2469	69	5	u	u	NOUN
ejpam-2469	69	6	and	and	CCONJ
ejpam-2469	69	7	u	u	NOUN
ejpam-2469	69	8	is	be	AUX
ejpam-2469	69	9	a	a	DET
ejpam-2469	69	10	regular	regular	ADJ
ejpam-2469	69	11	open	open	NOUN
ejpam-2469	69	12	in	in	ADP
ejpam-2469	69	13	x	x	X
ejpam-2469	69	14	.	.	PUNCT
ejpam-2469	70	1	proof	proof	NOUN
ejpam-2469	70	2	.	.	PUNCT
ejpam-2469	71	1	(	(	PUNCT
ejpam-2469	71	2	1)⇒	1)⇒	NUM
ejpam-2469	71	3	(	(	PUNCT
ejpam-2469	71	4	2	2	NUM
ejpam-2469	71	5	)	)	PUNCT
ejpam-2469	71	6	let	let	VERB
ejpam-2469	71	7	a	a	PRON
ejpam-2469	71	8	is	be	AUX
ejpam-2469	71	9	a	a	DET
ejpam-2469	71	10	saw	saw	NOUN
ejpam-2469	71	11	-	-	PUNCT
ejpam-2469	71	12	ir	ir	NOUN
ejpam-2469	71	13	g	g	NOUN
ejpam-2469	71	14	-closed	-close	VERB
ejpam-2469	71	15	set	set	NOUN
ejpam-2469	71	16	.	.	PUNCT
ejpam-2469	72	1	assume	assume	VERB
ejpam-2469	72	2	that	that	SCONJ
ejpam-2469	72	3	a⊆	a⊆	VERB
ejpam-2469	72	4	u	u	NOUN
ejpam-2469	72	5	and	and	CCONJ
ejpam-2469	72	6	u	u	NOUN
ejpam-2469	72	7	is	be	AUX
ejpam-2469	72	8	a	a	DET
ejpam-2469	72	9	regular	regular	ADJ
ejpam-2469	72	10	open	open	NOUN
ejpam-2469	72	11	in	in	ADP
ejpam-2469	72	12	x	x	X
ejpam-2469	72	13	.then	.then	VERB
ejpam-2469	72	14	we	we	PRON
ejpam-2469	72	15	have	have	VERB
ejpam-2469	72	16	(	(	PUNCT
ejpam-2469	72	17	int(cl(a)))∗	int(cl(a)))∗	PROPN
ejpam-2469	72	18	⊂	⊂	PROPN
ejpam-2469	72	19	u	u	PROPN
ejpam-2469	72	20	.	.	PUNCT
ejpam-2469	73	1	since	since	SCONJ
ejpam-2469	73	2	int(cl(a	int(cl(a	PROPN
ejpam-2469	73	3	)	)	PUNCT
ejpam-2469	73	4	)	)	PUNCT
ejpam-2469	73	5	⊂	⊂	PROPN
ejpam-2469	73	6	cl(a	cl(a	X
ejpam-2469	73	7	)	)	PUNCT
ejpam-2469	73	8	⊂	⊂	PROPN
ejpam-2469	73	9	a	a	DET
ejpam-2469	73	10	⊂	⊂	PROPN
ejpam-2469	73	11	u	u	PROPN
ejpam-2469	73	12	.	.	PUNCT
ejpam-2469	74	1	this	this	PRON
ejpam-2469	74	2	implies	imply	VERB
ejpam-2469	74	3	that	that	DET
ejpam-2469	74	4	int(cl(a))∪	int(cl(a))∪	ADV
ejpam-2469	74	5	(	(	PUNCT
ejpam-2469	74	6	int(cl(a)))∗	int(cl(a)))∗	PROPN
ejpam-2469	74	7	=	=	SYM
ejpam-2469	74	8	cl∗(int(cl(a	cl∗(int(cl(a	PROPN
ejpam-2469	74	9	)	)	PUNCT
ejpam-2469	74	10	)	)	PUNCT
ejpam-2469	74	11	)	)	PUNCT
ejpam-2469	75	1	⊂	⊂	PROPN
ejpam-2469	75	2	u	u	PROPN
ejpam-2469	75	3	.	.	PUNCT
ejpam-2469	76	1	(	(	PUNCT
ejpam-2469	76	2	2	2	X
ejpam-2469	76	3	)	)	PUNCT
ejpam-2469	76	4	⇒	⇒	NOUN
ejpam-2469	76	5	(	(	PUNCT
ejpam-2469	76	6	1	1	X
ejpam-2469	76	7	)	)	PUNCT
ejpam-2469	76	8	let	let	VERB
ejpam-2469	76	9	cl∗(int(cl(a	cl∗(int(cl(a	PROPN
ejpam-2469	76	10	)	)	PUNCT
ejpam-2469	76	11	)	)	PUNCT
ejpam-2469	76	12	)	)	PUNCT
ejpam-2469	77	1	⊂	⊂	PROPN
ejpam-2469	77	2	u	u	NOUN
ejpam-2469	77	3	whenever	whenever	SCONJ
ejpam-2469	77	4	a	a	DET
ejpam-2469	77	5	⊆	⊆	NUM
ejpam-2469	77	6	u	u	NOUN
ejpam-2469	77	7	and	and	CCONJ
ejpam-2469	77	8	u	u	NOUN
ejpam-2469	77	9	is	be	AUX
ejpam-2469	77	10	a	a	DET
ejpam-2469	77	11	regular	regular	ADJ
ejpam-2469	77	12	open	open	NOUN
ejpam-2469	77	13	in	in	ADP
ejpam-2469	77	14	x	x	X
ejpam-2469	77	15	.	.	PUNCT
ejpam-2469	78	1	since	since	SCONJ
ejpam-2469	78	2	(	(	PUNCT
ejpam-2469	78	3	int(cl(a)))∗	int(cl(a)))∗	PROPN
ejpam-2469	78	4	∪	∪	X
ejpam-2469	78	5	(	(	PUNCT
ejpam-2469	78	6	int(cl(a	int(cl(a	PROPN
ejpam-2469	78	7	)	)	PUNCT
ejpam-2469	78	8	)	)	PUNCT
ejpam-2469	78	9	)	)	PUNCT
ejpam-2469	79	1	⊂	⊂	PROPN
ejpam-2469	79	2	u	u	NOUN
ejpam-2469	79	3	then	then	ADV
ejpam-2469	79	4	(	(	PUNCT
ejpam-2469	79	5	int(cl(a)))∗	int(cl(a)))∗	PROPN
ejpam-2469	79	6	⊂	⊂	PROPN
ejpam-2469	79	7	u	u	NOUN
ejpam-2469	79	8	whenever	whenever	SCONJ
ejpam-2469	79	9	a	a	DET
ejpam-2469	79	10	⊆	⊆	NUM
ejpam-2469	79	11	u	u	NOUN
ejpam-2469	79	12	and	and	CCONJ
ejpam-2469	79	13	is	be	AUX
ejpam-2469	79	14	u	u	NOUN
ejpam-2469	79	15	is	be	AUX
ejpam-2469	79	16	regular	regular	ADJ
ejpam-2469	79	17	open	open	ADJ
ejpam-2469	79	18	in	in	ADP
ejpam-2469	79	19	x	x	X
ejpam-2469	79	20	.	.	PUNCT
ejpam-2469	80	1	theorem	theorem	NOUN
ejpam-2469	80	2	2	2	NUM
ejpam-2469	80	3	.	.	X
ejpam-2469	80	4	for	for	ADP
ejpam-2469	80	5	a	a	DET
ejpam-2469	80	6	subset	subset	NOUN
ejpam-2469	80	7	a	a	PRON
ejpam-2469	80	8	of	of	ADP
ejpam-2469	80	9	x	x	PRON
ejpam-2469	80	10	the	the	DET
ejpam-2469	80	11	following	follow	VERB
ejpam-2469	80	12	properties	property	NOUN
ejpam-2469	80	13	hold	hold	VERB
ejpam-2469	80	14	:	:	PUNCT
ejpam-2469	80	15	(	(	PUNCT
ejpam-2469	80	16	1	1	X
ejpam-2469	80	17	)	)	PUNCT
ejpam-2469	80	18	a	a	PRON
ejpam-2469	80	19	is	be	AUX
ejpam-2469	80	20	open	open	ADJ
ejpam-2469	80	21	and	and	CCONJ
ejpam-2469	80	22	saw	see	VERB
ejpam-2469	80	23	-	-	PUNCT
ejpam-2469	80	24	ir	ir	NOUN
ejpam-2469	81	1	g	g	PROPN
ejpam-2469	81	2	-closed	-close	VERB
ejpam-2469	81	3	then	then	ADV
ejpam-2469	81	4	a	a	PRON
ejpam-2469	81	5	is	be	AUX
ejpam-2469	81	6	ir	ir	PROPN
ejpam-2469	81	7	g	g	NOUN
ejpam-2469	81	8	-closed	-close	VERB
ejpam-2469	81	9	,	,	PUNCT
ejpam-2469	81	10	(	(	PUNCT
ejpam-2469	81	11	2	2	X
ejpam-2469	81	12	)	)	PUNCT
ejpam-2469	81	13	a	a	PRON
ejpam-2469	81	14	is	be	AUX
ejpam-2469	81	15	open	open	ADJ
ejpam-2469	81	16	and	and	CCONJ
ejpam-2469	81	17	w	w	NOUN
ejpam-2469	81	18	-	-	PUNCT
ejpam-2469	81	19	ir	ir	NOUN
ejpam-2469	81	20	g	g	PROPN
ejpam-2469	81	21	-closed	-close	VERB
ejpam-2469	81	22	then	then	ADV
ejpam-2469	81	23	a	a	PRON
ejpam-2469	81	24	is	be	AUX
ejpam-2469	81	25	ir	ir	PROPN
ejpam-2469	81	26	g	g	NOUN
ejpam-2469	81	27	-closed	-close	VERB
ejpam-2469	81	28	,	,	PUNCT
ejpam-2469	81	29	(	(	PUNCT
ejpam-2469	81	30	3	3	X
ejpam-2469	81	31	)	)	PUNCT
ejpam-2469	81	32	a	a	PRON
ejpam-2469	81	33	is	be	AUX
ejpam-2469	81	34	i	i	PRON
ejpam-2469	81	35	-	-	PUNCT
ejpam-2469	81	36	open	open	ADJ
ejpam-2469	81	37	and	and	CCONJ
ejpam-2469	82	1	aw	aw	INTJ
ejpam-2469	82	2	-	-	PUNCT
ejpam-2469	82	3	ir	ir	NOUN
ejpam-2469	82	4	g	g	PROPN
ejpam-2469	82	5	-closed	-close	VERB
ejpam-2469	82	6	then	then	ADV
ejpam-2469	82	7	a	a	PRON
ejpam-2469	82	8	is	be	AUX
ejpam-2469	82	9	ir	ir	PROPN
ejpam-2469	82	10	g	g	NOUN
ejpam-2469	82	11	-closed	-close	VERB
ejpam-2469	82	12	.	.	PUNCT
ejpam-2469	83	1	proof	proof	NOUN
ejpam-2469	83	2	.	.	PUNCT
ejpam-2469	84	1	(	(	PUNCT
ejpam-2469	84	2	1	1	X
ejpam-2469	84	3	)	)	PUNCT
ejpam-2469	84	4	let	let	VERB
ejpam-2469	84	5	a	a	PRON
ejpam-2469	84	6	be	be	AUX
ejpam-2469	84	7	a	a	DET
ejpam-2469	84	8	open	open	ADJ
ejpam-2469	84	9	and	and	CCONJ
ejpam-2469	84	10	saw	saw	NOUN
ejpam-2469	84	11	-	-	PUNCT
ejpam-2469	84	12	ir	ir	NOUN
ejpam-2469	84	13	g	g	NOUN
ejpam-2469	84	14	-closed	-close	VERB
ejpam-2469	84	15	set	set	VERB
ejpam-2469	84	16	in	in	ADP
ejpam-2469	84	17	(	(	PUNCT
ejpam-2469	84	18	x	x	INTJ
ejpam-2469	84	19	,	,	PUNCT
ejpam-2469	84	20	τ	τ	PROPN
ejpam-2469	84	21	,	,	PUNCT
ejpam-2469	84	22	i	i	PROPN
ejpam-2469	84	23	)	)	PUNCT
ejpam-2469	84	24	.	.	PUNCT
ejpam-2469	85	1	since	since	SCONJ
ejpam-2469	85	2	a	a	PRON
ejpam-2469	85	3	is	be	AUX
ejpam-2469	85	4	a	a	DET
ejpam-2469	85	5	open	open	ADJ
ejpam-2469	85	6	,	,	PUNCT
ejpam-2469	85	7	a	a	DET
ejpam-2469	85	8	⊂	⊂	PROPN
ejpam-2469	85	9	int(a	int(a	PROPN
ejpam-2469	85	10	)	)	PUNCT
ejpam-2469	85	11	⊂	⊂	PROPN
ejpam-2469	85	12	int(cl(a	int(cl(a	PROPN
ejpam-2469	85	13	)	)	PUNCT
ejpam-2469	85	14	)	)	PUNCT
ejpam-2469	85	15	.	.	PUNCT
ejpam-2469	86	1	hence	hence	ADV
ejpam-2469	86	2	,	,	PUNCT
ejpam-2469	86	3	a∗	a∗	PROPN
ejpam-2469	86	4	⊂	⊂	PROPN
ejpam-2469	86	5	(	(	PUNCT
ejpam-2469	86	6	int(cl(a)))∗	int(cl(a)))∗	PROPN
ejpam-2469	86	7	⊂	⊂	PROPN
ejpam-2469	86	8	u	u	PROPN
ejpam-2469	86	9	,	,	PUNCT
ejpam-2469	86	10	a	a	DET
ejpam-2469	86	11	⊆	⊆	NUM
ejpam-2469	86	12	u	u	NOUN
ejpam-2469	86	13	and	and	CCONJ
ejpam-2469	86	14	u	u	NOUN
ejpam-2469	86	15	is	be	AUX
ejpam-2469	86	16	regular	regular	ADJ
ejpam-2469	86	17	open	open	ADJ
ejpam-2469	86	18	in	in	ADP
ejpam-2469	86	19	x	x	X
ejpam-2469	86	20	.	.	PUNCT
ejpam-2469	87	1	so	so	ADV
ejpam-2469	87	2	,	,	PUNCT
ejpam-2469	87	3	we	we	PRON
ejpam-2469	87	4	have	have	VERB
ejpam-2469	87	5	a	a	DET
ejpam-2469	87	6	is	be	AUX
ejpam-2469	87	7	a	a	DET
ejpam-2469	87	8	ir	ir	NOUN
ejpam-2469	87	9	g	g	NOUN
ejpam-2469	87	10	-closed	-close	VERB
ejpam-2469	87	11	.	.	PUNCT
ejpam-2469	88	1	(	(	PUNCT
ejpam-2469	88	2	2	2	X
ejpam-2469	88	3	)	)	PUNCT
ejpam-2469	88	4	let	let	VERB
ejpam-2469	88	5	a	a	PRON
ejpam-2469	88	6	be	be	AUX
ejpam-2469	88	7	a	a	DET
ejpam-2469	88	8	open	open	ADJ
ejpam-2469	88	9	and	and	CCONJ
ejpam-2469	88	10	w	w	NOUN
ejpam-2469	88	11	-	-	PUNCT
ejpam-2469	88	12	ir	ir	NOUN
ejpam-2469	88	13	g	g	NOUN
ejpam-2469	88	14	-closed	-close	VERB
ejpam-2469	88	15	set	set	VERB
ejpam-2469	88	16	in	in	ADP
ejpam-2469	88	17	(	(	PUNCT
ejpam-2469	88	18	x	x	INTJ
ejpam-2469	88	19	,	,	PUNCT
ejpam-2469	88	20	τ	τ	PROPN
ejpam-2469	88	21	,	,	PUNCT
ejpam-2469	88	22	i	i	PROPN
ejpam-2469	88	23	)	)	PUNCT
ejpam-2469	88	24	.	.	PUNCT
ejpam-2469	89	1	hence	hence	ADV
ejpam-2469	89	2	we	we	PRON
ejpam-2469	89	3	have	have	VERB
ejpam-2469	89	4	a⊂	a⊂	NOUN
ejpam-2469	89	5	int(a	int(a	NOUN
ejpam-2469	89	6	)	)	PUNCT
ejpam-2469	89	7	and	and	CCONJ
ejpam-2469	89	8	a∗	a∗	PROPN
ejpam-2469	89	9	⊂	⊂	PROPN
ejpam-2469	89	10	(	(	PUNCT
ejpam-2469	89	11	int(a))∗	int(a))∗	PROPN
ejpam-2469	89	12	⊂	⊂	PROPN
ejpam-2469	89	13	u	u	PROPN
ejpam-2469	89	14	.	.	PUNCT
ejpam-2469	90	1	this	this	PRON
ejpam-2469	90	2	implies	imply	VERB
ejpam-2469	90	3	that	that	SCONJ
ejpam-2469	90	4	a	a	PRON
ejpam-2469	90	5	is	be	AUX
ejpam-2469	90	6	a	a	DET
ejpam-2469	90	7	ir	ir	NOUN
ejpam-2469	90	8	g	g	NOUN
ejpam-2469	90	9	-closed	-close	VERB
ejpam-2469	90	10	.	.	PUNCT
ejpam-2469	91	1	(	(	PUNCT
ejpam-2469	91	2	3	3	X
ejpam-2469	91	3	)	)	PUNCT
ejpam-2469	91	4	let	let	VERB
ejpam-2469	91	5	a	a	PRON
ejpam-2469	91	6	be	be	AUX
ejpam-2469	91	7	a	a	DET
ejpam-2469	91	8	i	i	NOUN
ejpam-2469	91	9	-open	-open	ADJ
ejpam-2469	91	10	and	and	CCONJ
ejpam-2469	91	11	aw	aw	INTJ
ejpam-2469	91	12	-	-	PUNCT
ejpam-2469	91	13	ir	ir	INTJ
ejpam-2469	91	14	g	g	PROPN
ejpam-2469	91	15	-closed	-close	VERB
ejpam-2469	91	16	set	set	VERB
ejpam-2469	91	17	in	in	ADP
ejpam-2469	91	18	(	(	PUNCT
ejpam-2469	91	19	x	x	INTJ
ejpam-2469	91	20	,	,	PUNCT
ejpam-2469	91	21	τ	τ	PROPN
ejpam-2469	91	22	,	,	PUNCT
ejpam-2469	91	23	i	i	PROPN
ejpam-2469	91	24	)	)	PUNCT
ejpam-2469	91	25	.	.	PUNCT
ejpam-2469	92	1	hence	hence	ADV
ejpam-2469	92	2	we	we	PRON
ejpam-2469	92	3	have	have	VERB
ejpam-2469	92	4	a	a	DET
ejpam-2469	92	5	⊂	⊂	PROPN
ejpam-2469	92	6	int(a∗	int(a∗	PART
ejpam-2469	92	7	)	)	PUNCT
ejpam-2469	92	8	and	and	CCONJ
ejpam-2469	92	9	a∗	a∗	PROPN
ejpam-2469	92	10	⊂	⊂	PROPN
ejpam-2469	92	11	(	(	PUNCT
ejpam-2469	92	12	int(a∗))∗	int(a∗))∗	PROPN
ejpam-2469	92	13	⊂	⊂	PROPN
ejpam-2469	92	14	u	u	PROPN
ejpam-2469	92	15	.	.	PUNCT
ejpam-2469	93	1	this	this	PRON
ejpam-2469	93	2	shows	show	VERB
ejpam-2469	93	3	that	that	SCONJ
ejpam-2469	93	4	a	a	PRON
ejpam-2469	93	5	is	be	AUX
ejpam-2469	93	6	a	a	DET
ejpam-2469	93	7	ir	ir	NOUN
ejpam-2469	93	8	g	g	NOUN
ejpam-2469	93	9	-closed	-close	VERB
ejpam-2469	93	10	.	.	PUNCT
ejpam-2469	94	1	theorem	theorem	NOUN
ejpam-2469	94	2	3	3	NUM
ejpam-2469	94	3	.	.	PUNCT
ejpam-2469	95	1	every	every	DET
ejpam-2469	95	2	αi	αi	NOUN
ejpam-2469	95	3	-∗-closed	-∗-close	VERB
ejpam-2469	95	4	set	set	NOUN
ejpam-2469	95	5	is	be	AUX
ejpam-2469	95	6	saw	saw	NOUN
ejpam-2469	95	7	-	-	PUNCT
ejpam-2469	95	8	ir	ir	NOUN
ejpam-2469	95	9	g	g	PROPN
ejpam-2469	95	10	-closed	-close	VERB
ejpam-2469	95	11	set	set	NOUN
ejpam-2469	95	12	.	.	PUNCT
ejpam-2469	96	1	proof	proof	NOUN
ejpam-2469	96	2	.	.	PUNCT
ejpam-2469	97	1	let	let	VERB
ejpam-2469	97	2	a⊆	a⊆	PROPN
ejpam-2469	97	3	u	u	NOUN
ejpam-2469	97	4	and	and	CCONJ
ejpam-2469	97	5	u	u	NOUN
ejpam-2469	97	6	is	be	AUX
ejpam-2469	97	7	a	a	DET
ejpam-2469	97	8	regular	regular	ADJ
ejpam-2469	97	9	open	open	NOUN
ejpam-2469	97	10	in	in	ADP
ejpam-2469	97	11	x	x	X
ejpam-2469	97	12	.	.	PUNCT
ejpam-2469	98	1	since	since	SCONJ
ejpam-2469	98	2	a	a	PRON
ejpam-2469	98	3	is	be	AUX
ejpam-2469	98	4	a	a	DET
ejpam-2469	98	5	αi	αi	ADV
ejpam-2469	98	6	-∗-closed	-∗-close	VERB
ejpam-2469	98	7	,	,	PUNCT
ejpam-2469	98	8	cl∗(int(cl(a	cl∗(int(cl(a	PROPN
ejpam-2469	98	9	)	)	PUNCT
ejpam-2469	98	10	)	)	PUNCT
ejpam-2469	98	11	)	)	PUNCT
ejpam-2469	99	1	⊂	⊂	PROPN
ejpam-2469	99	2	cl∗(int(cl(u	cl∗(int(cl(u	PROPN
ejpam-2469	99	3	)	)	PUNCT
ejpam-2469	99	4	)	)	PUNCT
ejpam-2469	99	5	)	)	PUNCT
ejpam-2469	100	1	⊂	⊂	PROPN
ejpam-2469	100	2	u	u	PROPN
ejpam-2469	100	3	.	.	PUNCT
ejpam-2469	101	1	thus	thus	ADV
ejpam-2469	101	2	,	,	PUNCT
ejpam-2469	101	3	a	a	PRON
ejpam-2469	101	4	is	be	AUX
ejpam-2469	101	5	a	a	DET
ejpam-2469	101	6	saw	saw	NOUN
ejpam-2469	101	7	-	-	PUNCT
ejpam-2469	101	8	ir	ir	NOUN
ejpam-2469	101	9	g	g	NOUN
ejpam-2469	101	10	-closed	-close	VERB
ejpam-2469	101	11	set	set	VERB
ejpam-2469	101	12	in	in	ADP
ejpam-2469	101	13	(	(	PUNCT
ejpam-2469	101	14	x	x	INTJ
ejpam-2469	101	15	,	,	PUNCT
ejpam-2469	101	16	τ	τ	PROPN
ejpam-2469	101	17	,	,	PUNCT
ejpam-2469	101	18	i	i	PROPN
ejpam-2469	101	19	)	)	PUNCT
ejpam-2469	101	20	.	.	PUNCT
ejpam-2469	102	1	the	the	DET
ejpam-2469	102	2	following	follow	VERB
ejpam-2469	102	3	example	example	NOUN
ejpam-2469	102	4	shows	show	VERB
ejpam-2469	102	5	that	that	SCONJ
ejpam-2469	102	6	the	the	DET
ejpam-2469	102	7	reverse	reverse	NOUN
ejpam-2469	102	8	of	of	ADP
ejpam-2469	102	9	theorem	theorem	NOUN
ejpam-2469	102	10	3	3	NUM
ejpam-2469	102	11	is	be	AUX
ejpam-2469	102	12	not	not	PART
ejpam-2469	102	13	true	true	ADJ
ejpam-2469	102	14	.	.	PUNCT
ejpam-2469	103	1	ü	ü	DET
ejpam-2469	103	2	karabıyık	karabıyık	PROPN
ejpam-2469	103	3	,	,	PUNCT
ejpam-2469	103	4	a	a	DET
ejpam-2469	103	5	kaymakcı	kaymakcı	NOUN
ejpam-2469	103	6	/	/	SYM
ejpam-2469	103	7	eur	eur	NOUN
ejpam-2469	103	8	.	.	PUNCT
ejpam-2469	104	1	j.	j.	PROPN
ejpam-2469	104	2	pure	pure	PROPN
ejpam-2469	104	3	appl	appl	PROPN
ejpam-2469	104	4	.	.	PROPN
ejpam-2469	104	5	math	math	PROPN
ejpam-2469	104	6	,	,	PUNCT
ejpam-2469	104	7	9	9	NUM
ejpam-2469	104	8	(	(	PUNCT
ejpam-2469	104	9	2016	2016	NUM
ejpam-2469	104	10	)	)	PUNCT
ejpam-2469	104	11	,	,	PUNCT
ejpam-2469	104	12	434	434	NUM
ejpam-2469	104	13	-	-	SYM
ejpam-2469	104	14	442	442	NUM
ejpam-2469	104	15	437	437	NUM
ejpam-2469	104	16	example	example	NOUN
ejpam-2469	104	17	1	1	NUM
ejpam-2469	104	18	.	.	X
ejpam-2469	105	1	let	let	VERB
ejpam-2469	105	2	(	(	PUNCT
ejpam-2469	105	3	x	x	X
ejpam-2469	105	4	,	,	PUNCT
ejpam-2469	105	5	τ	τ	PROPN
ejpam-2469	105	6	,	,	PUNCT
ejpam-2469	105	7	i	i	PRON
ejpam-2469	105	8	)	)	PUNCT
ejpam-2469	105	9	be	be	VERB
ejpam-2469	105	10	a	a	DET
ejpam-2469	105	11	ideal	ideal	ADJ
ejpam-2469	105	12	topological	topological	ADJ
ejpam-2469	105	13	space	space	NOUN
ejpam-2469	105	14	such	such	ADJ
ejpam-2469	105	15	that	that	SCONJ
ejpam-2469	105	16	x	x	X
ejpam-2469	105	17	=	=	X
ejpam-2469	105	18	{	{	PUNCT
ejpam-2469	105	19	a	a	PRON
ejpam-2469	105	20	,	,	PUNCT
ejpam-2469	105	21	b	b	NOUN
ejpam-2469	105	22	,	,	PUNCT
ejpam-2469	105	23	c	c	NOUN
ejpam-2469	105	24	,	,	PUNCT
ejpam-2469	105	25	d	d	NOUN
ejpam-2469	105	26	}	}	PUNCT
ejpam-2469	105	27	,	,	PUNCT
ejpam-2469	105	28	τ	τ	X
ejpam-2469	105	29	=	=	PUNCT
ejpam-2469	105	30	{	{	PUNCT
ejpam-2469	105	31	;	;	PUNCT
ejpam-2469	105	32	,	,	PUNCT
ejpam-2469	105	33	x	x	X
ejpam-2469	105	34	,	,	PUNCT
ejpam-2469	105	35	{	{	PUNCT
ejpam-2469	105	36	b	b	NOUN
ejpam-2469	105	37	}	}	PUNCT
ejpam-2469	105	38	,	,	PUNCT
ejpam-2469	105	39	{	{	PUNCT
ejpam-2469	105	40	c	c	X
ejpam-2469	105	41	}	}	PUNCT
ejpam-2469	105	42	,	,	PUNCT
ejpam-2469	105	43	{	{	PUNCT
ejpam-2469	105	44	b	b	X
ejpam-2469	105	45	,	,	PUNCT
ejpam-2469	105	46	c	c	NOUN
ejpam-2469	105	47	}	}	PUNCT
ejpam-2469	105	48	,	,	PUNCT
ejpam-2469	105	49	{	{	PUNCT
ejpam-2469	105	50	c	c	X
ejpam-2469	105	51	,	,	PUNCT
ejpam-2469	105	52	d	d	NOUN
ejpam-2469	105	53	}	}	PUNCT
ejpam-2469	105	54	,	,	PUNCT
ejpam-2469	105	55	{	{	PUNCT
ejpam-2469	105	56	b	b	X
ejpam-2469	105	57	,	,	PUNCT
ejpam-2469	105	58	c	c	NOUN
ejpam-2469	105	59	,	,	PUNCT
ejpam-2469	105	60	d	d	NOUN
ejpam-2469	105	61	}	}	PUNCT
ejpam-2469	105	62	,	,	PUNCT
ejpam-2469	105	63	{	{	PUNCT
ejpam-2469	105	64	a	a	PRON
ejpam-2469	105	65	,	,	PUNCT
ejpam-2469	105	66	c	c	NOUN
ejpam-2469	105	67	,	,	PUNCT
ejpam-2469	105	68	d	d	NOUN
ejpam-2469	105	69	}	}	PUNCT
ejpam-2469	105	70	}	}	PUNCT
ejpam-2469	105	71	and	and	CCONJ
ejpam-2469	105	72	i	i	PRON
ejpam-2469	105	73	=	=	PUNCT
ejpam-2469	105	74	{	{	PUNCT
ejpam-2469	105	75	;	;	PUNCT
ejpam-2469	105	76	}	}	PUNCT
ejpam-2469	105	77	.	.	PUNCT
ejpam-2469	106	1	then	then	ADV
ejpam-2469	106	2	,	,	PUNCT
ejpam-2469	106	3	a	a	DET
ejpam-2469	106	4	=	=	X
ejpam-2469	106	5	{	{	PUNCT
ejpam-2469	106	6	a	a	X
ejpam-2469	106	7	,	,	PUNCT
ejpam-2469	106	8	c	c	NOUN
ejpam-2469	106	9	}	}	PUNCT
ejpam-2469	106	10	⊂	⊂	PROPN
ejpam-2469	106	11	x	x	X
ejpam-2469	106	12	is	be	AUX
ejpam-2469	106	13	sawir	sawir	VERB
ejpam-2469	106	14	g	g	PROPN
ejpam-2469	106	15	-closed	-close	VERB
ejpam-2469	106	16	set	set	NOUN
ejpam-2469	106	17	but	but	CCONJ
ejpam-2469	106	18	is	be	AUX
ejpam-2469	106	19	not	not	PART
ejpam-2469	106	20	αi	αi	PRON
ejpam-2469	106	21	-∗-closed	-∗-close	VERB
ejpam-2469	106	22	set	set	NOUN
ejpam-2469	106	23	.	.	PUNCT
ejpam-2469	107	1	remark	remark	PROPN
ejpam-2469	107	2	1	1	NUM
ejpam-2469	107	3	.	.	PUNCT
ejpam-2469	108	1	the	the	DET
ejpam-2469	108	2	intersection	intersection	NOUN
ejpam-2469	108	3	of	of	ADP
ejpam-2469	108	4	two	two	NUM
ejpam-2469	108	5	saw	saw	NOUN
ejpam-2469	108	6	-	-	PUNCT
ejpam-2469	108	7	ir	ir	NOUN
ejpam-2469	108	8	g	g	PROPN
ejpam-2469	108	9	-closed	-close	VERB
ejpam-2469	108	10	set	set	VERB
ejpam-2469	108	11	in	in	ADP
ejpam-2469	108	12	ideal	ideal	ADJ
ejpam-2469	108	13	topological	topological	ADJ
ejpam-2469	108	14	spaces	space	NOUN
ejpam-2469	108	15	need	need	AUX
ejpam-2469	108	16	not	not	PART
ejpam-2469	108	17	be	be	AUX
ejpam-2469	108	18	a	a	DET
ejpam-2469	108	19	saw	saw	NOUN
ejpam-2469	108	20	-	-	PUNCT
ejpam-2469	108	21	ir	ir	NOUN
ejpam-2469	108	22	g	g	NOUN
ejpam-2469	108	23	-closed	-close	VERB
ejpam-2469	108	24	set	set	NOUN
ejpam-2469	108	25	.	.	PUNCT
ejpam-2469	108	26	example	example	NOUN
ejpam-2469	109	1	2	2	NUM
ejpam-2469	109	2	.	.	X
ejpam-2469	110	1	let	let	AUX
ejpam-2469	110	2	(	(	PUNCT
ejpam-2469	110	3	x	x	X
ejpam-2469	110	4	,	,	PUNCT
ejpam-2469	110	5	τ	τ	PROPN
ejpam-2469	110	6	,	,	PUNCT
ejpam-2469	110	7	i	i	PRON
ejpam-2469	110	8	)	)	PUNCT
ejpam-2469	110	9	be	be	VERB
ejpam-2469	110	10	a	a	DET
ejpam-2469	110	11	ideal	ideal	ADJ
ejpam-2469	110	12	topological	topological	ADJ
ejpam-2469	110	13	space	space	NOUN
ejpam-2469	110	14	such	such	ADJ
ejpam-2469	110	15	that	that	SCONJ
ejpam-2469	110	16	x	x	X
ejpam-2469	110	17	=	=	X
ejpam-2469	110	18	{	{	PUNCT
ejpam-2469	110	19	a	a	PRON
ejpam-2469	110	20	,	,	PUNCT
ejpam-2469	110	21	b	b	NOUN
ejpam-2469	110	22	,	,	PUNCT
ejpam-2469	110	23	c	c	NOUN
ejpam-2469	110	24	}	}	PUNCT
ejpam-2469	110	25	,	,	PUNCT
ejpam-2469	110	26	i	i	PRON
ejpam-2469	110	27	=	=	PUNCT
ejpam-2469	110	28	{	{	PUNCT
ejpam-2469	110	29	;	;	PUNCT
ejpam-2469	110	30	,	,	PUNCT
ejpam-2469	110	31	{	{	PUNCT
ejpam-2469	110	32	b	b	NOUN
ejpam-2469	110	33	}	}	PUNCT
ejpam-2469	110	34	}	}	PUNCT
ejpam-2469	110	35	,	,	PUNCT
ejpam-2469	110	36	and	and	CCONJ
ejpam-2469	110	37	τ=	τ=	X
ejpam-2469	110	38	{	{	PUNCT
ejpam-2469	110	39	;	;	PUNCT
ejpam-2469	110	40	,	,	PUNCT
ejpam-2469	110	41	x	x	X
ejpam-2469	110	42	,	,	PUNCT
ejpam-2469	110	43	{	{	PUNCT
ejpam-2469	110	44	a	a	NOUN
ejpam-2469	110	45	}	}	PUNCT
ejpam-2469	110	46	,	,	PUNCT
ejpam-2469	110	47	{	{	PUNCT
ejpam-2469	110	48	b	b	NOUN
ejpam-2469	110	49	}	}	PUNCT
ejpam-2469	110	50	,	,	PUNCT
ejpam-2469	110	51	{	{	PUNCT
ejpam-2469	110	52	a	a	PRON
ejpam-2469	110	53	,	,	PUNCT
ejpam-2469	110	54	b	b	NOUN
ejpam-2469	110	55	}	}	PUNCT
ejpam-2469	110	56	}	}	PUNCT
ejpam-2469	110	57	.	.	PUNCT
ejpam-2469	111	1	let	let	VERB
ejpam-2469	111	2	a=	a=	VERB
ejpam-2469	111	3	{	{	PUNCT
ejpam-2469	111	4	a	a	X
ejpam-2469	111	5	,	,	PUNCT
ejpam-2469	111	6	c	c	NOUN
ejpam-2469	111	7	}	}	PUNCT
ejpam-2469	111	8	and	and	CCONJ
ejpam-2469	111	9	b	b	X
ejpam-2469	111	10	=	=	NOUN
ejpam-2469	111	11	{	{	PUNCT
ejpam-2469	111	12	a	a	PROPN
ejpam-2469	111	13	,	,	PUNCT
ejpam-2469	111	14	b	b	NOUN
ejpam-2469	111	15	}	}	PUNCT
ejpam-2469	111	16	.	.	PUNCT
ejpam-2469	112	1	from	from	ADP
ejpam-2469	112	2	here	here	ADV
ejpam-2469	112	3	a	a	PRON
ejpam-2469	112	4	and	and	CCONJ
ejpam-2469	112	5	b	b	NOUN
ejpam-2469	112	6	are	be	AUX
ejpam-2469	112	7	saw	see	VERB
ejpam-2469	112	8	-	-	PUNCT
ejpam-2469	112	9	ir	ir	NOUN
ejpam-2469	112	10	g	g	NOUN
ejpam-2469	112	11	-closed	-close	VERB
ejpam-2469	112	12	set	set	NOUN
ejpam-2469	113	1	but	but	CCONJ
ejpam-2469	113	2	a∩	a∩	PROPN
ejpam-2469	113	3	b	b	PROPN
ejpam-2469	113	4	=	=	X
ejpam-2469	113	5	{	{	PUNCT
ejpam-2469	113	6	a	a	PRON
ejpam-2469	113	7	}	}	PUNCT
ejpam-2469	113	8	is	be	AUX
ejpam-2469	113	9	not	not	PART
ejpam-2469	113	10	saw	see	VERB
ejpam-2469	113	11	-	-	PUNCT
ejpam-2469	113	12	ir	ir	NOUN
ejpam-2469	113	13	g	g	PROPN
ejpam-2469	113	14	-closed	-close	VERB
ejpam-2469	113	15	set	set	NOUN
ejpam-2469	113	16	.	.	PUNCT
ejpam-2469	114	1	theorem	theorem	ADJ
ejpam-2469	114	2	4	4	NUM
ejpam-2469	114	3	.	.	PUNCT
ejpam-2469	115	1	let	let	VERB
ejpam-2469	115	2	(	(	PUNCT
ejpam-2469	115	3	x	x	X
ejpam-2469	115	4	,	,	PUNCT
ejpam-2469	115	5	τ	τ	PROPN
ejpam-2469	115	6	,	,	PUNCT
ejpam-2469	115	7	i	i	PRON
ejpam-2469	115	8	)	)	PUNCT
ejpam-2469	115	9	be	be	VERB
ejpam-2469	115	10	a	a	DET
ejpam-2469	115	11	ideal	ideal	ADJ
ejpam-2469	115	12	topological	topological	ADJ
ejpam-2469	115	13	spaces	space	NOUN
ejpam-2469	115	14	a	a	DET
ejpam-2469	115	15	⊂	⊂	PROPN
ejpam-2469	115	16	x	x	X
ejpam-2469	115	17	.	.	PUNCT
ejpam-2469	116	1	if	if	SCONJ
ejpam-2469	116	2	a	a	PRON
ejpam-2469	116	3	is	be	AUX
ejpam-2469	116	4	a	a	DET
ejpam-2469	116	5	saw	saw	NOUN
ejpam-2469	116	6	-	-	PUNCT
ejpam-2469	116	7	ir	ir	NOUN
ejpam-2469	116	8	g	g	PROPN
ejpam-2469	116	9	-closed	-close	VERB
ejpam-2469	116	10	set	set	NOUN
ejpam-2469	116	11	then	then	ADV
ejpam-2469	116	12	(	(	PUNCT
ejpam-2469	116	13	int(cl(a)))∗	int(cl(a)))∗	X
ejpam-2469	116	14	−	−	PUNCT
ejpam-2469	117	1	a	a	PRON
ejpam-2469	117	2	contains	contain	VERB
ejpam-2469	117	3	no	no	DET
ejpam-2469	117	4	any	any	PRON
ejpam-2469	117	5	nonempty	nonempty	ADJ
ejpam-2469	117	6	regular	regular	ADJ
ejpam-2469	117	7	closed	close	VERB
ejpam-2469	117	8	set	set	NOUN
ejpam-2469	117	9	.	.	PUNCT
ejpam-2469	118	1	proof	proof	NOUN
ejpam-2469	118	2	.	.	PUNCT
ejpam-2469	119	1	let	let	VERB
ejpam-2469	119	2	a	a	PRON
ejpam-2469	119	3	is	be	AUX
ejpam-2469	119	4	a	a	DET
ejpam-2469	119	5	saw	saw	NOUN
ejpam-2469	119	6	-	-	PUNCT
ejpam-2469	119	7	ir	ir	NOUN
ejpam-2469	119	8	g	g	NOUN
ejpam-2469	119	9	-closed	-close	VERB
ejpam-2469	119	10	set	set	VERB
ejpam-2469	119	11	in	in	ADP
ejpam-2469	119	12	(	(	PUNCT
ejpam-2469	119	13	x	x	INTJ
ejpam-2469	119	14	,	,	PUNCT
ejpam-2469	119	15	τ	τ	PROPN
ejpam-2469	119	16	,	,	PUNCT
ejpam-2469	119	17	i	i	PROPN
ejpam-2469	119	18	)	)	PUNCT
ejpam-2469	119	19	.	.	PUNCT
ejpam-2469	120	1	suppose	suppose	VERB
ejpam-2469	120	2	that	that	SCONJ
ejpam-2469	120	3	u	u	PROPN
ejpam-2469	120	4	is	be	AUX
ejpam-2469	120	5	a	a	DET
ejpam-2469	120	6	closed	closed	ADJ
ejpam-2469	120	7	set	set	NOUN
ejpam-2469	120	8	.	.	PUNCT
ejpam-2469	121	1	such	such	ADJ
ejpam-2469	121	2	that	that	SCONJ
ejpam-2469	121	3	u	u	PROPN
ejpam-2469	121	4	⊆	⊆	NUM
ejpam-2469	121	5	(	(	PUNCT
ejpam-2469	121	6	int(cl(a)))∗	int(cl(a)))∗	X
ejpam-2469	121	7	−	−	PROPN
ejpam-2469	121	8	a.	a.	NOUN
ejpam-2469	121	9	since	since	SCONJ
ejpam-2469	121	10	x	x	SYM
ejpam-2469	121	11	−	−	PROPN
ejpam-2469	121	12	u	u	NOUN
ejpam-2469	121	13	is	be	AUX
ejpam-2469	121	14	open	open	ADJ
ejpam-2469	121	15	and	and	CCONJ
ejpam-2469	121	16	a⊂	a⊂	PRON
ejpam-2469	121	17	x	x	SYM
ejpam-2469	121	18	−	−	PROPN
ejpam-2469	121	19	u	u	NOUN
ejpam-2469	121	20	,	,	PUNCT
ejpam-2469	121	21	then	then	ADV
ejpam-2469	121	22	(	(	PUNCT
ejpam-2469	121	23	int(cl(a)))∗	int(cl(a)))∗	PROPN
ejpam-2469	121	24	⊂	⊂	PROPN
ejpam-2469	121	25	x	x	PUNCT
ejpam-2469	122	1	−	−	PROPN
ejpam-2469	122	2	u	u	NOUN
ejpam-2469	122	3	.	.	PUNCT
ejpam-2469	123	1	then	then	ADV
ejpam-2469	123	2	,	,	PUNCT
ejpam-2469	123	3	we	we	PRON
ejpam-2469	123	4	get	get	VERB
ejpam-2469	123	5	u	u	NOUN
ejpam-2469	123	6	⊂	⊂	NOUN
ejpam-2469	123	7	x	x	PUNCT
ejpam-2469	123	8	−	−	PROPN
ejpam-2469	123	9	(	(	PUNCT
ejpam-2469	123	10	int(cl(a)))∗.	int(cl(a)))∗.	NOUN
ejpam-2469	123	11	hence	hence	ADV
ejpam-2469	123	12	u	u	PROPN
ejpam-2469	123	13	⊂	⊂	PROPN
ejpam-2469	123	14	(	(	PUNCT
ejpam-2469	123	15	int(cl(a)))∗.	int(cl(a)))∗.	PROPN
ejpam-2469	123	16	thus	thus	ADV
ejpam-2469	123	17	,	,	PUNCT
ejpam-2469	123	18	u	u	PROPN
ejpam-2469	123	19	⊂	⊂	PROPN
ejpam-2469	123	20	(	(	PUNCT
ejpam-2469	123	21	int(cl(a)))∗	int(cl(a)))∗	PROPN
ejpam-2469	123	22	∩	∩	PROPN
ejpam-2469	123	23	x	x	X
ejpam-2469	123	24	−	−	PROPN
ejpam-2469	123	25	(	(	PUNCT
ejpam-2469	123	26	int(cl(a)))∗	int(cl(a)))∗	X
ejpam-2469	123	27	=	=	PUNCT
ejpam-2469	123	28	;	;	PUNCT
ejpam-2469	123	29	and	and	CCONJ
ejpam-2469	123	30	(	(	PUNCT
ejpam-2469	123	31	int(cl(a)))∗	int(cl(a)))∗	X
ejpam-2469	123	32	−	−	PUNCT
ejpam-2469	124	1	a	a	PRON
ejpam-2469	124	2	contains	contain	VERB
ejpam-2469	124	3	no	no	DET
ejpam-2469	124	4	any	any	DET
ejpam-2469	124	5	nonempty	nonempty	ADV
ejpam-2469	124	6	closed	close	VERB
ejpam-2469	124	7	set	set	NOUN
ejpam-2469	124	8	.	.	PUNCT
ejpam-2469	125	1	proposition	proposition	NOUN
ejpam-2469	125	2	1	1	NUM
ejpam-2469	125	3	.	.	PUNCT
ejpam-2469	126	1	let	let	VERB
ejpam-2469	126	2	(	(	PUNCT
ejpam-2469	126	3	x	x	X
ejpam-2469	126	4	,	,	PUNCT
ejpam-2469	126	5	τ	τ	PROPN
ejpam-2469	126	6	,	,	PUNCT
ejpam-2469	126	7	i	i	PRON
ejpam-2469	126	8	)	)	PUNCT
ejpam-2469	126	9	be	be	VERB
ejpam-2469	126	10	an	an	DET
ejpam-2469	126	11	ideal	ideal	ADJ
ejpam-2469	126	12	topological	topological	ADJ
ejpam-2469	126	13	spaces	space	NOUN
ejpam-2469	126	14	.	.	PUNCT
ejpam-2469	127	1	if	if	SCONJ
ejpam-2469	127	2	a	a	DET
ejpam-2469	127	3	⊂	⊂	X
ejpam-2469	127	4	b	b	PROPN
ejpam-2469	127	5	⊂	⊂	PROPN
ejpam-2469	127	6	cl∗(int(cl(a	cl∗(int(cl(a	PROPN
ejpam-2469	127	7	)	)	PUNCT
ejpam-2469	127	8	)	)	PUNCT
ejpam-2469	127	9	)	)	PUNCT
ejpam-2469	127	10	and	and	CCONJ
ejpam-2469	127	11	a	a	PRON
ejpam-2469	127	12	is	be	AUX
ejpam-2469	127	13	saw	saw	NOUN
ejpam-2469	127	14	-	-	PUNCT
ejpam-2469	127	15	ir	ir	NOUN
ejpam-2469	127	16	g	g	PROPN
ejpam-2469	127	17	-closed	-close	VERB
ejpam-2469	127	18	,	,	PUNCT
ejpam-2469	127	19	then	then	ADV
ejpam-2469	127	20	b	b	PROPN
ejpam-2469	127	21	is	be	AUX
ejpam-2469	127	22	saw	see	VERB
ejpam-2469	127	23	-	-	PUNCT
ejpam-2469	127	24	ir	ir	NOUN
ejpam-2469	127	25	g	g	PROPN
ejpam-2469	127	26	-closed	-close	VERB
ejpam-2469	127	27	.	.	PUNCT
ejpam-2469	128	1	proof	proof	NOUN
ejpam-2469	128	2	.	.	PUNCT
ejpam-2469	129	1	let	let	VERB
ejpam-2469	129	2	b	b	NOUN
ejpam-2469	129	3	⊂	⊂	PRON
ejpam-2469	129	4	u	u	PROPN
ejpam-2469	129	5	and	and	CCONJ
ejpam-2469	129	6	u	u	NOUN
ejpam-2469	129	7	is	be	AUX
ejpam-2469	129	8	a	a	DET
ejpam-2469	129	9	regular	regular	ADJ
ejpam-2469	129	10	open	open	NOUN
ejpam-2469	129	11	in	in	ADP
ejpam-2469	129	12	x	x	X
ejpam-2469	129	13	.	.	PUNCT
ejpam-2469	130	1	since	since	SCONJ
ejpam-2469	130	2	a	a	DET
ejpam-2469	130	3	⊂	⊂	PROPN
ejpam-2469	130	4	u	u	NOUN
ejpam-2469	130	5	and	and	CCONJ
ejpam-2469	130	6	a	a	PRON
ejpam-2469	130	7	is	be	AUX
ejpam-2469	130	8	a	a	DET
ejpam-2469	130	9	saw	saw	NOUN
ejpam-2469	130	10	-	-	PUNCT
ejpam-2469	130	11	ir	ir	NOUN
ejpam-2469	130	12	g	g	PROPN
ejpam-2469	130	13	-closed	-close	VERB
ejpam-2469	130	14	set	set	NOUN
ejpam-2469	130	15	then	then	ADV
ejpam-2469	130	16	cl∗(int(cl(a	cl∗(int(cl(a	PROPN
ejpam-2469	130	17	)	)	PUNCT
ejpam-2469	130	18	)	)	PUNCT
ejpam-2469	130	19	)	)	PUNCT
ejpam-2469	131	1	⊂	⊂	PROPN
ejpam-2469	131	2	u	u	PROPN
ejpam-2469	131	3	.	.	PUNCT
ejpam-2469	132	1	since	since	SCONJ
ejpam-2469	132	2	,	,	PUNCT
ejpam-2469	132	3	b	b	PROPN
ejpam-2469	132	4	⊂	⊂	PROPN
ejpam-2469	132	5	cl∗(int(cl(a	cl∗(int(cl(a	PROPN
ejpam-2469	132	6	)	)	PUNCT
ejpam-2469	132	7	)	)	PUNCT
ejpam-2469	132	8	)	)	PUNCT
ejpam-2469	133	1	⊂	⊂	PROPN
ejpam-2469	133	2	u	u	INTJ
ejpam-2469	133	3	,	,	PUNCT
ejpam-2469	133	4	we	we	PRON
ejpam-2469	133	5	obtain	obtain	VERB
ejpam-2469	133	6	cl∗(int(cl(b	cl∗(int(cl(b	NOUN
ejpam-2469	133	7	)	)	PUNCT
ejpam-2469	133	8	)	)	PUNCT
ejpam-2469	133	9	)	)	PUNCT
ejpam-2469	134	1	⊂	⊂	PROPN
ejpam-2469	134	2	cl∗(int(cl(cl∗(int(cl(a	cl∗(int(cl(cl∗(int(cl(a	NUM
ejpam-2469	134	3	)	)	PUNCT
ejpam-2469	134	4	)	)	PUNCT
ejpam-2469	134	5	)	)	PUNCT
ejpam-2469	134	6	)	)	PUNCT
ejpam-2469	134	7	)	)	PUNCT
ejpam-2469	134	8	)	)	PUNCT
ejpam-2469	135	1	⊂	⊂	PROPN
ejpam-2469	135	2	cl∗(int(cl(a	cl∗(int(cl(a	PROPN
ejpam-2469	135	3	)	)	PUNCT
ejpam-2469	135	4	)	)	PUNCT
ejpam-2469	135	5	)	)	PUNCT
ejpam-2469	136	1	⊂	⊂	PROPN
ejpam-2469	136	2	u	u	PROPN
ejpam-2469	136	3	.	.	PUNCT
ejpam-2469	137	1	therefore	therefore	ADV
ejpam-2469	137	2	cl∗(int(cl(b	cl∗(int(cl(b	PROPN
ejpam-2469	137	3	)	)	PUNCT
ejpam-2469	137	4	)	)	PUNCT
ejpam-2469	137	5	)	)	PUNCT
ejpam-2469	138	1	⊂	⊂	PROPN
ejpam-2469	138	2	u	u	PROPN
ejpam-2469	138	3	,	,	PUNCT
ejpam-2469	138	4	b	b	PROPN
ejpam-2469	138	5	is	be	AUX
ejpam-2469	138	6	a	a	DET
ejpam-2469	138	7	saw	saw	NOUN
ejpam-2469	138	8	-	-	PUNCT
ejpam-2469	138	9	ir	ir	NOUN
ejpam-2469	138	10	g	g	PROPN
ejpam-2469	138	11	-closed	-close	VERB
ejpam-2469	138	12	.	.	PUNCT
ejpam-2469	139	1	corollary	corollary	ADJ
ejpam-2469	139	2	1	1	NUM
ejpam-2469	139	3	.	.	PUNCT
ejpam-2469	140	1	let	let	VERB
ejpam-2469	140	2	(	(	PUNCT
ejpam-2469	140	3	x	x	X
ejpam-2469	140	4	,	,	PUNCT
ejpam-2469	140	5	τ	τ	PROPN
ejpam-2469	140	6	,	,	PUNCT
ejpam-2469	140	7	i	i	PRON
ejpam-2469	140	8	)	)	PUNCT
ejpam-2469	140	9	be	be	VERB
ejpam-2469	140	10	an	an	DET
ejpam-2469	140	11	ideal	ideal	ADJ
ejpam-2469	140	12	topological	topological	ADJ
ejpam-2469	140	13	spaces	space	NOUN
ejpam-2469	140	14	and	and	CCONJ
ejpam-2469	140	15	a⊂	a⊂	PRON
ejpam-2469	140	16	x	x	X
ejpam-2469	140	17	.	.	PUNCT
ejpam-2469	141	1	if	if	SCONJ
ejpam-2469	141	2	a	a	PRON
ejpam-2469	141	3	is	be	AUX
ejpam-2469	141	4	a	a	DET
ejpam-2469	141	5	saw	saw	NOUN
ejpam-2469	141	6	-	-	PUNCT
ejpam-2469	141	7	ir	ir	NOUN
ejpam-2469	141	8	g	g	NOUN
ejpam-2469	141	9	-closed	-close	VERB
ejpam-2469	141	10	and	and	CCONJ
ejpam-2469	141	11	regular	regular	ADJ
ejpam-2469	141	12	open	open	ADJ
ejpam-2469	141	13	set	set	NOUN
ejpam-2469	141	14	,	,	PUNCT
ejpam-2469	141	15	then	then	ADV
ejpam-2469	141	16	cl∗(a	cl∗(a	NOUN
ejpam-2469	141	17	)	)	PUNCT
ejpam-2469	141	18	is	be	AUX
ejpam-2469	141	19	a	a	DET
ejpam-2469	141	20	saw	saw	NOUN
ejpam-2469	141	21	-	-	PUNCT
ejpam-2469	141	22	ir	ir	NOUN
ejpam-2469	141	23	g	g	PROPN
ejpam-2469	141	24	-closed	-close	VERB
ejpam-2469	141	25	.	.	PUNCT
ejpam-2469	142	1	proof	proof	NOUN
ejpam-2469	142	2	.	.	PUNCT
ejpam-2469	143	1	let	let	VERB
ejpam-2469	143	2	a	a	PRON
ejpam-2469	143	3	is	be	AUX
ejpam-2469	143	4	saw	see	VERB
ejpam-2469	143	5	-	-	PUNCT
ejpam-2469	143	6	ir	ir	NOUN
ejpam-2469	143	7	g	g	NOUN
ejpam-2469	143	8	-closed	-close	VERB
ejpam-2469	143	9	set	set	NOUN
ejpam-2469	143	10	and	and	CCONJ
ejpam-2469	143	11	regular	regular	ADJ
ejpam-2469	143	12	open	open	ADJ
ejpam-2469	143	13	set	set	NOUN
ejpam-2469	143	14	in	in	ADP
ejpam-2469	143	15	(	(	PUNCT
ejpam-2469	143	16	x	x	INTJ
ejpam-2469	143	17	,	,	PUNCT
ejpam-2469	143	18	τ	τ	PROPN
ejpam-2469	143	19	,	,	PUNCT
ejpam-2469	143	20	i	i	PROPN
ejpam-2469	143	21	)	)	PUNCT
ejpam-2469	143	22	.	.	PUNCT
ejpam-2469	144	1	then	then	ADV
ejpam-2469	144	2	we	we	PRON
ejpam-2469	144	3	have	have	VERB
ejpam-2469	144	4	a⊂	a⊂	NOUN
ejpam-2469	144	5	cl∗(a	cl∗(a	NOUN
ejpam-2469	144	6	)	)	PUNCT
ejpam-2469	144	7	⊂	⊂	PROPN
ejpam-2469	144	8	cl∗(a	cl∗(a	NOUN
ejpam-2469	144	9	)	)	PUNCT
ejpam-2469	144	10	=	=	SYM
ejpam-2469	144	11	cl∗(int(cl(a	cl∗(int(cl(a	PROPN
ejpam-2469	144	12	)	)	PUNCT
ejpam-2469	144	13	)	)	PUNCT
ejpam-2469	144	14	)	)	PUNCT
ejpam-2469	144	15	.	.	PUNCT
ejpam-2469	145	1	hence	hence	ADV
ejpam-2469	145	2	by	by	ADP
ejpam-2469	145	3	proposition	proposition	NOUN
ejpam-2469	145	4	1	1	NUM
ejpam-2469	145	5	cl∗(a	cl∗(a	NOUN
ejpam-2469	145	6	)	)	PUNCT
ejpam-2469	145	7	is	be	AUX
ejpam-2469	145	8	a	a	DET
ejpam-2469	145	9	saw	saw	NOUN
ejpam-2469	145	10	-	-	PUNCT
ejpam-2469	145	11	ir	ir	NOUN
ejpam-2469	145	12	g	g	NOUN
ejpam-2469	145	13	-closed	-close	VERB
ejpam-2469	145	14	set	set	VERB
ejpam-2469	145	15	in	in	ADP
ejpam-2469	145	16	(	(	PUNCT
ejpam-2469	145	17	x	x	INTJ
ejpam-2469	145	18	,	,	PUNCT
ejpam-2469	145	19	τ	τ	PROPN
ejpam-2469	145	20	,	,	PUNCT
ejpam-2469	145	21	i	i	PROPN
ejpam-2469	145	22	)	)	PUNCT
ejpam-2469	145	23	.	.	PUNCT
ejpam-2469	146	1	theorem	theorem	NOUN
ejpam-2469	146	2	5	5	NUM
ejpam-2469	146	3	.	.	PUNCT
ejpam-2469	147	1	let	let	VERB
ejpam-2469	147	2	(	(	PUNCT
ejpam-2469	147	3	x	x	X
ejpam-2469	147	4	,	,	PUNCT
ejpam-2469	147	5	τ	τ	PROPN
ejpam-2469	147	6	,	,	PUNCT
ejpam-2469	147	7	i	i	PRON
ejpam-2469	147	8	)	)	PUNCT
ejpam-2469	147	9	be	be	VERB
ejpam-2469	147	10	an	an	DET
ejpam-2469	147	11	ideal	ideal	ADJ
ejpam-2469	147	12	topological	topological	ADJ
ejpam-2469	147	13	spaces	space	NOUN
ejpam-2469	147	14	and	and	CCONJ
ejpam-2469	147	15	a⊂	a⊂	NOUN
ejpam-2469	147	16	x	x	SYM
ejpam-2469	147	17	.	.	PUNCT
ejpam-2469	148	1	assume	assume	VERB
ejpam-2469	148	2	that	that	SCONJ
ejpam-2469	148	3	a	a	PRON
ejpam-2469	148	4	is	be	AUX
ejpam-2469	148	5	a	a	DET
ejpam-2469	148	6	saw	saw	NOUN
ejpam-2469	148	7	-	-	PUNCT
ejpam-2469	148	8	ir	ir	NOUN
ejpam-2469	148	9	g	g	PROPN
ejpam-2469	148	10	closed	close	VERB
ejpam-2469	148	11	set	set	NOUN
ejpam-2469	148	12	.	.	PUNCT
ejpam-2469	149	1	the	the	DET
ejpam-2469	149	2	following	follow	VERB
ejpam-2469	149	3	properties	property	NOUN
ejpam-2469	149	4	are	be	AUX
ejpam-2469	149	5	equivalent	equivalent	ADJ
ejpam-2469	149	6	:	:	PUNCT
ejpam-2469	149	7	(	(	PUNCT
ejpam-2469	149	8	1	1	X
ejpam-2469	149	9	)	)	PUNCT
ejpam-2469	149	10	a	a	PRON
ejpam-2469	149	11	is	be	AUX
ejpam-2469	149	12	a	a	DET
ejpam-2469	149	13	αi	αi	NOUN
ejpam-2469	149	14	-∗-closed	-∗-close	VERB
ejpam-2469	149	15	,	,	PUNCT
ejpam-2469	149	16	(	(	PUNCT
ejpam-2469	149	17	2	2	NUM
ejpam-2469	149	18	)	)	PUNCT
ejpam-2469	149	19	(	(	PUNCT
ejpam-2469	149	20	int(cl(a)))∗	int(cl(a)))∗	X
ejpam-2469	150	1	−	−	PROPN
ejpam-2469	151	1	a	a	PRON
ejpam-2469	151	2	is	be	AUX
ejpam-2469	151	3	a	a	DET
ejpam-2469	151	4	regular	regular	ADJ
ejpam-2469	151	5	closed	closed	ADJ
ejpam-2469	151	6	set	set	NOUN
ejpam-2469	151	7	.	.	PUNCT
ejpam-2469	152	1	ü	ü	DET
ejpam-2469	152	2	karabıyık	karabıyık	PROPN
ejpam-2469	152	3	,	,	PUNCT
ejpam-2469	152	4	a	a	DET
ejpam-2469	152	5	kaymakcı	kaymakcı	NOUN
ejpam-2469	152	6	/	/	SYM
ejpam-2469	152	7	eur	eur	NOUN
ejpam-2469	152	8	.	.	PUNCT
ejpam-2469	153	1	j.	j.	PROPN
ejpam-2469	153	2	pure	pure	PROPN
ejpam-2469	153	3	appl	appl	PROPN
ejpam-2469	153	4	.	.	PROPN
ejpam-2469	153	5	math	math	PROPN
ejpam-2469	153	6	,	,	PUNCT
ejpam-2469	153	7	9	9	NUM
ejpam-2469	153	8	(	(	PUNCT
ejpam-2469	153	9	2016	2016	NUM
ejpam-2469	153	10	)	)	PUNCT
ejpam-2469	153	11	,	,	PUNCT
ejpam-2469	153	12	434	434	NUM
ejpam-2469	153	13	-	-	SYM
ejpam-2469	153	14	442	442	NUM
ejpam-2469	153	15	438	438	NUM
ejpam-2469	153	16	proof	proof	NOUN
ejpam-2469	153	17	.	.	PUNCT
ejpam-2469	154	1	(	(	PUNCT
ejpam-2469	154	2	1	1	X
ejpam-2469	154	3	)	)	PUNCT
ejpam-2469	154	4	⇒	⇒	NOUN
ejpam-2469	154	5	(	(	PUNCT
ejpam-2469	154	6	2	2	X
ejpam-2469	154	7	)	)	PUNCT
ejpam-2469	154	8	let	let	VERB
ejpam-2469	154	9	a	a	DET
ejpam-2469	154	10	be	be	AUX
ejpam-2469	154	11	a	a	DET
ejpam-2469	154	12	αi	αi	NOUN
ejpam-2469	154	13	-∗-closed	-∗-close	VERB
ejpam-2469	154	14	set	set	NOUN
ejpam-2469	154	15	.	.	PUNCT
ejpam-2469	155	1	which	which	PRON
ejpam-2469	155	2	means	mean	VERB
ejpam-2469	155	3	that	that	SCONJ
ejpam-2469	155	4	cl∗(int(cl(a	cl∗(int(cl(a	NOUN
ejpam-2469	155	5	)	)	PUNCT
ejpam-2469	155	6	)	)	PUNCT
ejpam-2469	155	7	)	)	PUNCT
ejpam-2469	156	1	⊂	⊂	PROPN
ejpam-2469	156	2	a.	a.	NOUN
ejpam-2469	156	3	this	this	PRON
ejpam-2469	156	4	implies	imply	VERB
ejpam-2469	156	5	(	(	PUNCT
ejpam-2469	156	6	int(cl(a)))∗	int(cl(a)))∗	PROPN
ejpam-2469	156	7	⊂	⊂	PROPN
ejpam-2469	156	8	a	a	PRON
ejpam-2469	156	9	and	and	CCONJ
ejpam-2469	156	10	(	(	PUNCT
ejpam-2469	156	11	int(cl(a)))∗−a=	int(cl(a)))∗−a=	NOUN
ejpam-2469	156	12	;	;	PUNCT
ejpam-2469	156	13	.	.	PUNCT
ejpam-2469	157	1	thus	thus	ADV
ejpam-2469	157	2	,	,	PUNCT
ejpam-2469	157	3	cl∗(int(cl(a)))−a	cl∗(int(cl(a)))−a	PROPN
ejpam-2469	157	4	is	be	AUX
ejpam-2469	157	5	a	a	DET
ejpam-2469	157	6	regular	regular	ADJ
ejpam-2469	157	7	closed	closed	ADJ
ejpam-2469	157	8	set	set	NOUN
ejpam-2469	157	9	.	.	PUNCT
ejpam-2469	158	1	(	(	PUNCT
ejpam-2469	158	2	2)⇒	2)⇒	NUM
ejpam-2469	158	3	(	(	PUNCT
ejpam-2469	158	4	1	1	X
ejpam-2469	158	5	)	)	PUNCT
ejpam-2469	158	6	let	let	VERB
ejpam-2469	158	7	cl∗(int(cl(a)))−	cl∗(int(cl(a)))−	PUNCT
ejpam-2469	158	8	a	a	DET
ejpam-2469	158	9	be	be	AUX
ejpam-2469	158	10	a	a	DET
ejpam-2469	158	11	regular	regular	ADJ
ejpam-2469	158	12	closed	closed	ADJ
ejpam-2469	158	13	set	set	NOUN
ejpam-2469	158	14	.	.	PUNCT
ejpam-2469	159	1	since	since	SCONJ
ejpam-2469	159	2	a	a	PRON
ejpam-2469	159	3	is	be	AUX
ejpam-2469	159	4	a	a	DET
ejpam-2469	159	5	saw	saw	NOUN
ejpam-2469	159	6	-	-	PUNCT
ejpam-2469	159	7	ir	ir	NOUN
ejpam-2469	159	8	g	g	NOUN
ejpam-2469	159	9	-closed	-close	VERB
ejpam-2469	159	10	set	set	VERB
ejpam-2469	159	11	in	in	ADP
ejpam-2469	159	12	(	(	PUNCT
ejpam-2469	159	13	x	x	INTJ
ejpam-2469	159	14	,	,	PUNCT
ejpam-2469	159	15	τ	τ	PROPN
ejpam-2469	159	16	,	,	PUNCT
ejpam-2469	159	17	i	i	PROPN
ejpam-2469	159	18	)	)	PUNCT
ejpam-2469	159	19	,	,	PUNCT
ejpam-2469	159	20	by	by	ADP
ejpam-2469	159	21	theorem	theorem	NOUN
ejpam-2469	159	22	3	3	NUM
ejpam-2469	159	23	(	(	PUNCT
ejpam-2469	159	24	int(cl(a)))∗	int(cl(a)))∗	X
ejpam-2469	159	25	−	−	PROPN
ejpam-2469	159	26	a=	a=	PROPN
ejpam-2469	159	27	;	;	PUNCT
ejpam-2469	159	28	.	.	PUNCT
ejpam-2469	160	1	hence	hence	ADV
ejpam-2469	160	2	,	,	PUNCT
ejpam-2469	160	3	we	we	PRON
ejpam-2469	160	4	get	get	VERB
ejpam-2469	160	5	cl∗(int(cl(a	cl∗(int(cl(a	NOUN
ejpam-2469	160	6	)	)	PUNCT
ejpam-2469	160	7	)	)	PUNCT
ejpam-2469	160	8	)	)	PUNCT
ejpam-2469	161	1	⊂	⊂	PROPN
ejpam-2469	161	2	a.	a.	NOUN
ejpam-2469	161	3	thus	thus	ADV
ejpam-2469	161	4	a	a	PRON
ejpam-2469	161	5	is	be	AUX
ejpam-2469	161	6	a	a	DET
ejpam-2469	161	7	αi	αi	ADV
ejpam-2469	161	8	-∗-closed	-∗-close	VERB
ejpam-2469	161	9	.	.	PUNCT
ejpam-2469	162	1	corollary	corollary	ADJ
ejpam-2469	162	2	2	2	NUM
ejpam-2469	162	3	.	.	PUNCT
ejpam-2469	163	1	let	let	AUX
ejpam-2469	163	2	(	(	PUNCT
ejpam-2469	163	3	x	x	X
ejpam-2469	163	4	,	,	PUNCT
ejpam-2469	163	5	τ	τ	PROPN
ejpam-2469	163	6	,	,	PUNCT
ejpam-2469	163	7	i	i	PRON
ejpam-2469	163	8	)	)	PUNCT
ejpam-2469	163	9	be	be	VERB
ejpam-2469	163	10	an	an	DET
ejpam-2469	163	11	ideal	ideal	ADJ
ejpam-2469	163	12	topological	topological	ADJ
ejpam-2469	163	13	spaces	space	NOUN
ejpam-2469	163	14	and	and	CCONJ
ejpam-2469	163	15	a	a	DET
ejpam-2469	163	16	⊂	⊂	PROPN
ejpam-2469	163	17	x	x	X
ejpam-2469	163	18	.	.	PUNCT
ejpam-2469	164	1	if	if	SCONJ
ejpam-2469	164	2	a	a	PRON
ejpam-2469	164	3	is	be	AUX
ejpam-2469	164	4	a	a	DET
ejpam-2469	164	5	aw	aw	INTJ
ejpam-2469	164	6	-	-	PUNCT
ejpam-2469	164	7	ir	ir	NOUN
ejpam-2469	164	8	g	g	NOUN
ejpam-2469	164	9	-closed	-close	VERB
ejpam-2469	164	10	and	and	CCONJ
ejpam-2469	164	11	τ∗-closed	τ∗-close	VERB
ejpam-2469	164	12	set	set	NOUN
ejpam-2469	164	13	,	,	PUNCT
ejpam-2469	164	14	then	then	ADV
ejpam-2469	164	15	cl∗(int(a∗	cl∗(int(a∗	PROPN
ejpam-2469	164	16	)	)	PUNCT
ejpam-2469	164	17	)	)	PUNCT
ejpam-2469	165	1	⊂	⊂	PROPN
ejpam-2469	165	2	u	u	NOUN
ejpam-2469	165	3	whenever	whenever	SCONJ
ejpam-2469	165	4	a⊂	a⊂	PUNCT
ejpam-2469	165	5	u	u	NOUN
ejpam-2469	165	6	and	and	CCONJ
ejpam-2469	165	7	u	u	NOUN
ejpam-2469	165	8	is	be	AUX
ejpam-2469	165	9	a	a	DET
ejpam-2469	165	10	regular	regular	ADJ
ejpam-2469	165	11	open	open	ADJ
ejpam-2469	165	12	set	set	NOUN
ejpam-2469	165	13	in	in	ADP
ejpam-2469	165	14	x	x	X
ejpam-2469	165	15	.	.	PUNCT
ejpam-2469	166	1	proof	proof	NOUN
ejpam-2469	166	2	.	.	PUNCT
ejpam-2469	167	1	let	let	VERB
ejpam-2469	167	2	a	a	DET
ejpam-2469	167	3	be	be	AUX
ejpam-2469	167	4	a	a	DET
ejpam-2469	167	5	aw	aw	INTJ
ejpam-2469	167	6	-	-	PUNCT
ejpam-2469	167	7	ir	ir	NOUN
ejpam-2469	167	8	g	g	PROPN
ejpam-2469	167	9	-closed	-close	VERB
ejpam-2469	167	10	set	set	VERB
ejpam-2469	167	11	in	in	ADP
ejpam-2469	167	12	x	x	X
ejpam-2469	167	13	.	.	PUNCT
ejpam-2469	167	14	suppose	suppose	VERB
ejpam-2469	167	15	that	that	SCONJ
ejpam-2469	167	16	a⊂	a⊂	PUNCT
ejpam-2469	167	17	u	u	NOUN
ejpam-2469	167	18	and	and	CCONJ
ejpam-2469	167	19	u	u	NOUN
ejpam-2469	167	20	is	be	AUX
ejpam-2469	167	21	a	a	DET
ejpam-2469	167	22	regular	regular	ADJ
ejpam-2469	167	23	open	open	ADJ
ejpam-2469	167	24	set	set	NOUN
ejpam-2469	167	25	in	in	ADP
ejpam-2469	167	26	x	x	X
ejpam-2469	167	27	.	.	PUNCT
ejpam-2469	168	1	we	we	PRON
ejpam-2469	168	2	have	have	VERB
ejpam-2469	168	3	(	(	PUNCT
ejpam-2469	168	4	int(a∗))∗	int(a∗))∗	NOUN
ejpam-2469	168	5	⊂	⊂	PROPN
ejpam-2469	168	6	u	u	PROPN
ejpam-2469	168	7	.	.	PUNCT
ejpam-2469	169	1	on	on	ADP
ejpam-2469	169	2	the	the	DET
ejpam-2469	169	3	other	other	ADJ
ejpam-2469	169	4	hand	hand	NOUN
ejpam-2469	169	5	since	since	SCONJ
ejpam-2469	169	6	a	a	PRON
ejpam-2469	169	7	is	be	AUX
ejpam-2469	169	8	a	a	DET
ejpam-2469	169	9	τ∗-closed	τ∗-close	VERB
ejpam-2469	169	10	,	,	PUNCT
ejpam-2469	169	11	we	we	PRON
ejpam-2469	169	12	have	have	VERB
ejpam-2469	169	13	int(a∗	int(a∗	X
ejpam-2469	169	14	)	)	PUNCT
ejpam-2469	169	15	⊂	⊂	PROPN
ejpam-2469	169	16	int(a	int(a	PROPN
ejpam-2469	169	17	)	)	PUNCT
ejpam-2469	169	18	⊂	⊂	PROPN
ejpam-2469	169	19	a⊂	a⊂	X
ejpam-2469	169	20	u	u	NOUN
ejpam-2469	169	21	.	.	PUNCT
ejpam-2469	170	1	hence	hence	ADV
ejpam-2469	170	2	(	(	PUNCT
ejpam-2469	170	3	int(a∗))∗	int(a∗))∗	NOUN
ejpam-2469	170	4	∪	∪	NOUN
ejpam-2469	170	5	int(a∗	int(a∗	PART
ejpam-2469	170	6	)	)	PUNCT
ejpam-2469	170	7	⊂	⊂	PROPN
ejpam-2469	170	8	u	u	PROPN
ejpam-2469	170	9	.	.	PUNCT
ejpam-2469	171	1	this	this	PRON
ejpam-2469	171	2	implies	imply	VERB
ejpam-2469	171	3	cl∗(int(a∗	cl∗(int(a∗	NOUN
ejpam-2469	171	4	)	)	PUNCT
ejpam-2469	171	5	)	)	PUNCT
ejpam-2469	172	1	⊂	⊂	PROPN
ejpam-2469	172	2	u	u	PROPN
ejpam-2469	172	3	.	.	PUNCT
ejpam-2469	173	1	theorem	theorem	VERB
ejpam-2469	173	2	6	6	NUM
ejpam-2469	173	3	.	.	PUNCT
ejpam-2469	174	1	if	if	SCONJ
ejpam-2469	174	2	(	(	PUNCT
ejpam-2469	174	3	x	x	X
ejpam-2469	174	4	,	,	PUNCT
ejpam-2469	174	5	τ	τ	PROPN
ejpam-2469	174	6	,	,	PUNCT
ejpam-2469	174	7	i	i	PROPN
ejpam-2469	174	8	)	)	PUNCT
ejpam-2469	174	9	is	be	AUX
ejpam-2469	174	10	any	any	DET
ejpam-2469	174	11	ideal	ideal	ADJ
ejpam-2469	174	12	topological	topological	ADJ
ejpam-2469	174	13	spaces	space	NOUN
ejpam-2469	174	14	where	where	SCONJ
ejpam-2469	174	15	i	i	PRON
ejpam-2469	174	16	=	=	X
ejpam-2469	174	17	{	{	PUNCT
ejpam-2469	174	18	;	;	PUNCT
ejpam-2469	174	19	}	}	PUNCT
ejpam-2469	174	20	,	,	PUNCT
ejpam-2469	174	21	then	then	ADV
ejpam-2469	174	22	a	a	PRON
ejpam-2469	174	23	is	be	AUX
ejpam-2469	174	24	a	a	DET
ejpam-2469	174	25	aw	aw	INTJ
ejpam-2469	174	26	-	-	PUNCT
ejpam-2469	174	27	ir	ir	NOUN
ejpam-2469	174	28	g	g	NOUN
ejpam-2469	174	29	-closed	-close	VERB
ejpam-2469	174	30	if	if	SCONJ
ejpam-2469	174	31	and	and	CCONJ
ejpam-2469	174	32	only	only	ADV
ejpam-2469	174	33	if	if	SCONJ
ejpam-2469	174	34	g	g	PROPN
ejpam-2469	174	35	⊂	⊂	PROPN
ejpam-2469	174	36	int∗(cl(a∗	int∗(cl(a∗	PROPN
ejpam-2469	174	37	)	)	PUNCT
ejpam-2469	174	38	)	)	PUNCT
ejpam-2469	174	39	whenever	whenever	SCONJ
ejpam-2469	174	40	g	g	PROPN
ejpam-2469	174	41	⊂	⊂	PROPN
ejpam-2469	174	42	a	a	PROPN
ejpam-2469	174	43	and	and	CCONJ
ejpam-2469	174	44	g	g	PROPN
ejpam-2469	174	45	is	be	AUX
ejpam-2469	174	46	a	a	DET
ejpam-2469	174	47	regular	regular	ADJ
ejpam-2469	174	48	closed	closed	ADJ
ejpam-2469	174	49	set	set	NOUN
ejpam-2469	174	50	.	.	PUNCT
ejpam-2469	175	1	proof	proof	NOUN
ejpam-2469	175	2	.	.	PUNCT
ejpam-2469	176	1	necessity	necessity	NOUN
ejpam-2469	176	2	:	:	PUNCT
ejpam-2469	176	3	let	let	VERB
ejpam-2469	176	4	g	g	PRON
ejpam-2469	176	5	be	be	AUX
ejpam-2469	176	6	a	a	DET
ejpam-2469	176	7	regular	regular	ADJ
ejpam-2469	176	8	closed	close	VERB
ejpam-2469	176	9	set	set	NOUN
ejpam-2469	176	10	in	in	ADP
ejpam-2469	176	11	x	x	X
ejpam-2469	176	12	and	and	CCONJ
ejpam-2469	176	13	g	g	PROPN
ejpam-2469	176	14	⊂	⊂	PROPN
ejpam-2469	176	15	a.	a.	NOUN
ejpam-2469	177	1	then	then	ADV
ejpam-2469	177	2	it	it	PRON
ejpam-2469	177	3	is	be	AUX
ejpam-2469	177	4	well	well	ADV
ejpam-2469	177	5	-	-	PUNCT
ejpam-2469	177	6	known	know	VERB
ejpam-2469	177	7	that	that	SCONJ
ejpam-2469	177	8	x	x	X
ejpam-2469	178	1	−	−	NOUN
ejpam-2469	178	2	g	g	NOUN
ejpam-2469	178	3	is	be	AUX
ejpam-2469	178	4	regular	regular	ADJ
ejpam-2469	178	5	open	open	ADJ
ejpam-2469	178	6	set	set	NOUN
ejpam-2469	178	7	and	and	CCONJ
ejpam-2469	178	8	(	(	PUNCT
ejpam-2469	178	9	x	x	PART
ejpam-2469	178	10	−	−	PROPN
ejpam-2469	178	11	a	a	X
ejpam-2469	178	12	)	)	PUNCT
ejpam-2469	178	13	⊂	⊂	PROPN
ejpam-2469	178	14	(	(	PUNCT
ejpam-2469	178	15	x	x	X
ejpam-2469	178	16	−	−	NOUN
ejpam-2469	178	17	g	g	NOUN
ejpam-2469	178	18	)	)	PUNCT
ejpam-2469	178	19	.	.	PUNCT
ejpam-2469	179	1	since	since	SCONJ
ejpam-2469	179	2	x	x	X
ejpam-2469	179	3	−	−	PROPN
ejpam-2469	179	4	a	a	PRON
ejpam-2469	179	5	is	be	AUX
ejpam-2469	179	6	a	a	DET
ejpam-2469	179	7	aw	aw	INTJ
ejpam-2469	179	8	-	-	PUNCT
ejpam-2469	179	9	ir	ir	NOUN
ejpam-2469	179	10	g	g	PROPN
ejpam-2469	179	11	-closed	-close	VERB
ejpam-2469	179	12	,	,	PUNCT
ejpam-2469	179	13	then	then	ADV
ejpam-2469	179	14	cl∗(int((x	cl∗(int((x	ADJ
ejpam-2469	179	15	−	−	PROPN
ejpam-2469	179	16	a)∗	a)∗	PROPN
ejpam-2469	179	17	)	)	PUNCT
ejpam-2469	179	18	)	)	PUNCT
ejpam-2469	180	1	⊂	⊂	PRON
ejpam-2469	180	2	(	(	PUNCT
ejpam-2469	180	3	x	x	X
ejpam-2469	180	4	−	−	NOUN
ejpam-2469	180	5	g	g	NOUN
ejpam-2469	180	6	)	)	PUNCT
ejpam-2469	180	7	.	.	PUNCT
ejpam-2469	181	1	from	from	ADP
ejpam-2469	181	2	the	the	DET
ejpam-2469	181	3	fact	fact	NOUN
ejpam-2469	181	4	that	that	SCONJ
ejpam-2469	181	5	for	for	ADP
ejpam-2469	181	6	i	i	PRON
ejpam-2469	181	7	=	=	PUNCT
ejpam-2469	181	8	{	{	PUNCT
ejpam-2469	181	9	;	;	PUNCT
ejpam-2469	181	10	}	}	PUNCT
ejpam-2469	181	11	,	,	PUNCT
ejpam-2469	181	12	a∗	a∗	PROPN
ejpam-2469	181	13	=	=	SYM
ejpam-2469	181	14	cl(a	cl(a	X
ejpam-2469	181	15	)	)	PUNCT
ejpam-2469	181	16	.	.	PUNCT
ejpam-2469	182	1	therefore	therefore	ADV
ejpam-2469	182	2	x	x	X
ejpam-2469	182	3	−	−	NOUN
ejpam-2469	182	4	int∗(cl(a∗	int∗(cl(a∗	NOUN
ejpam-2469	182	5	)	)	PUNCT
ejpam-2469	182	6	)	)	PUNCT
ejpam-2469	183	1	⊂	⊂	PRON
ejpam-2469	183	2	(	(	PUNCT
ejpam-2469	183	3	x	x	X
ejpam-2469	183	4	−	−	NOUN
ejpam-2469	183	5	g	g	NOUN
ejpam-2469	183	6	)	)	PUNCT
ejpam-2469	183	7	.	.	PUNCT
ejpam-2469	184	1	so	so	ADV
ejpam-2469	184	2	,	,	PUNCT
ejpam-2469	184	3	we	we	PRON
ejpam-2469	184	4	have	have	VERB
ejpam-2469	184	5	g	g	PROPN
ejpam-2469	184	6	⊂	⊂	PROPN
ejpam-2469	184	7	int∗(cl(a∗	int∗(cl(a∗	NOUN
ejpam-2469	184	8	)	)	PUNCT
ejpam-2469	184	9	)	)	PUNCT
ejpam-2469	184	10	.	.	PUNCT
ejpam-2469	185	1	sufficiency	sufficiency	NOUN
ejpam-2469	185	2	:	:	PUNCT
ejpam-2469	185	3	let	let	VERB
ejpam-2469	185	4	h	h	NOUN
ejpam-2469	185	5	be	be	AUX
ejpam-2469	185	6	a	a	DET
ejpam-2469	185	7	regular	regular	ADJ
ejpam-2469	185	8	open	open	ADJ
ejpam-2469	185	9	set	set	NOUN
ejpam-2469	185	10	in	in	ADP
ejpam-2469	185	11	x	x	PUNCT
ejpam-2469	185	12	and	and	CCONJ
ejpam-2469	185	13	(	(	PUNCT
ejpam-2469	185	14	x	x	PART
ejpam-2469	185	15	−	−	NOUN
ejpam-2469	185	16	a	a	X
ejpam-2469	185	17	)	)	PUNCT
ejpam-2469	186	1	⊂	⊂	PROPN
ejpam-2469	187	1	h.	h.	PROPN
ejpam-2469	188	1	since	since	SCONJ
ejpam-2469	188	2	(	(	PUNCT
ejpam-2469	188	3	x	x	X
ejpam-2469	188	4	−	−	PROPN
ejpam-2469	188	5	h	h	NOUN
ejpam-2469	188	6	)	)	PUNCT
ejpam-2469	188	7	is	be	AUX
ejpam-2469	188	8	a	a	DET
ejpam-2469	188	9	regular	regular	ADJ
ejpam-2469	188	10	open	open	ADJ
ejpam-2469	188	11	set	set	NOUN
ejpam-2469	188	12	such	such	ADJ
ejpam-2469	188	13	that	that	SCONJ
ejpam-2469	188	14	(	(	PUNCT
ejpam-2469	188	15	x	x	NOUN
ejpam-2469	188	16	−h	−h	ADV
ejpam-2469	188	17	)	)	PUNCT
ejpam-2469	188	18	⊂	⊂	PROPN
ejpam-2469	188	19	a	a	X
ejpam-2469	188	20	,	,	PUNCT
ejpam-2469	188	21	then	then	ADV
ejpam-2469	188	22	(	(	PUNCT
ejpam-2469	188	23	x	x	NOUN
ejpam-2469	188	24	−h	−h	ADV
ejpam-2469	188	25	)	)	PUNCT
ejpam-2469	188	26	⊂	⊂	PROPN
ejpam-2469	188	27	int∗(cl(a∗	int∗(cl(a∗	PROPN
ejpam-2469	188	28	)	)	PUNCT
ejpam-2469	188	29	)	)	PUNCT
ejpam-2469	188	30	.	.	PUNCT
ejpam-2469	189	1	we	we	PRON
ejpam-2469	189	2	have	have	VERB
ejpam-2469	189	3	x	x	NOUN
ejpam-2469	189	4	−	−	NOUN
ejpam-2469	189	5	int∗(cl(a∗	int∗(cl(a∗	NOUN
ejpam-2469	189	6	)	)	PUNCT
ejpam-2469	189	7	)	)	PUNCT
ejpam-2469	190	1	=	=	SYM
ejpam-2469	190	2	cl∗(int(a∗	cl∗(int(a∗	NOUN
ejpam-2469	190	3	)	)	PUNCT
ejpam-2469	190	4	)	)	PUNCT
ejpam-2469	191	1	⊂	⊂	PROPN
ejpam-2469	191	2	h.	h.	PROPN
ejpam-2469	192	1	thus	thus	ADV
ejpam-2469	192	2	,	,	PUNCT
ejpam-2469	192	3	(	(	PUNCT
ejpam-2469	192	4	x	x	X
ejpam-2469	192	5	−	−	NOUN
ejpam-2469	192	6	a	a	NOUN
ejpam-2469	192	7	)	)	PUNCT
ejpam-2469	192	8	is	be	AUX
ejpam-2469	192	9	a	a	DET
ejpam-2469	192	10	aw	aw	INTJ
ejpam-2469	192	11	-	-	PUNCT
ejpam-2469	192	12	ir	ir	NOUN
ejpam-2469	192	13	g	g	PROPN
ejpam-2469	192	14	-closed	-close	VERB
ejpam-2469	192	15	set	set	NOUN
ejpam-2469	192	16	.	.	PUNCT
ejpam-2469	193	1	hence	hence	ADV
ejpam-2469	193	2	,	,	PUNCT
ejpam-2469	193	3	a	a	PRON
ejpam-2469	193	4	is	be	AUX
ejpam-2469	193	5	a	a	DET
ejpam-2469	193	6	aw	aw	INTJ
ejpam-2469	193	7	-	-	PUNCT
ejpam-2469	193	8	ir	ir	NOUN
ejpam-2469	193	9	g	g	PROPN
ejpam-2469	193	10	-open	-open	NOUN
ejpam-2469	193	11	set	set	VERB
ejpam-2469	193	12	in	in	ADP
ejpam-2469	193	13	x	x	PROPN
ejpam-2469	193	14	.	.	PUNCT
ejpam-2469	194	1	4	4	X
ejpam-2469	194	2	.	.	X
ejpam-2469	194	3	weakly	weakly	ADJ
ejpam-2469	194	4	-	-	PUNCT
ejpam-2469	194	5	r	r	NOUN
ejpam-2469	194	6	gi	gi	NOUN
ejpam-2469	194	7	-closed	-close	VERB
ejpam-2469	194	8	sets	set	NOUN
ejpam-2469	194	9	in	in	ADP
ejpam-2469	194	10	this	this	DET
ejpam-2469	194	11	section	section	NOUN
ejpam-2469	194	12	,	,	PUNCT
ejpam-2469	194	13	secondly	secondly	ADV
ejpam-2469	194	14	we	we	PRON
ejpam-2469	194	15	introduce	introduce	VERB
ejpam-2469	194	16	weakly	weakly	ADJ
ejpam-2469	194	17	-	-	PUNCT
ejpam-2469	194	18	r	r	NOUN
ejpam-2469	194	19	gi	gi	NOUN
ejpam-2469	194	20	-closed	-close	VERB
ejpam-2469	194	21	sets	set	NOUN
ejpam-2469	194	22	and	and	CCONJ
ejpam-2469	194	23	investigate	investigate	VERB
ejpam-2469	194	24	their	their	PRON
ejpam-2469	194	25	basic	basic	ADJ
ejpam-2469	194	26	properties	property	NOUN
ejpam-2469	194	27	.	.	PUNCT
ejpam-2469	195	1	definition	definition	NOUN
ejpam-2469	195	2	3	3	NUM
ejpam-2469	195	3	.	.	PUNCT
ejpam-2469	196	1	a	a	DET
ejpam-2469	196	2	subset	subset	NOUN
ejpam-2469	196	3	a	a	PRON
ejpam-2469	196	4	of	of	ADP
ejpam-2469	196	5	an	an	DET
ejpam-2469	196	6	(	(	PUNCT
ejpam-2469	196	7	x	x	SYM
ejpam-2469	196	8	,	,	PUNCT
ejpam-2469	196	9	τ	τ	PROPN
ejpam-2469	196	10	,	,	PUNCT
ejpam-2469	196	11	i	i	PRON
ejpam-2469	196	12	)	)	PUNCT
ejpam-2469	196	13	be	be	VERB
ejpam-2469	196	14	a	a	DET
ejpam-2469	196	15	ideal	ideal	ADJ
ejpam-2469	196	16	topological	topological	ADJ
ejpam-2469	196	17	spaces	space	NOUN
ejpam-2469	196	18	is	be	AUX
ejpam-2469	196	19	said	say	VERB
ejpam-2469	196	20	to	to	PART
ejpam-2469	196	21	be	be	AUX
ejpam-2469	196	22	(	(	PUNCT
ejpam-2469	196	23	i	i	NOUN
ejpam-2469	196	24	)	)	PUNCT
ejpam-2469	196	25	weakly	weakly	ADJ
ejpam-2469	196	26	-	-	PUNCT
ejpam-2469	196	27	r	r	NOUN
ejpam-2469	196	28	gi	gi	NOUN
ejpam-2469	196	29	-closed(briefly	-closed(briefly	PUNCT
ejpam-2469	196	30	w	w	NOUN
ejpam-2469	196	31	-	-	PUNCT
ejpam-2469	196	32	r	r	NOUN
ejpam-2469	196	33	gi	gi	NOUN
ejpam-2469	196	34	-closed	-close	VERB
ejpam-2469	196	35	)	)	PUNCT
ejpam-2469	196	36	set	set	VERB
ejpam-2469	197	1	if	if	SCONJ
ejpam-2469	197	2	(	(	PUNCT
ejpam-2469	197	3	int(cl∗(a	int(cl∗(a	NOUN
ejpam-2469	197	4	)	)	PUNCT
ejpam-2469	197	5	)	)	PUNCT
ejpam-2469	197	6	)	)	PUNCT
ejpam-2469	198	1	⊂	⊂	PROPN
ejpam-2469	198	2	u	u	NOUN
ejpam-2469	198	3	whenever	whenever	SCONJ
ejpam-2469	198	4	a⊆	a⊆	VERB
ejpam-2469	198	5	u	u	NOUN
ejpam-2469	198	6	and	and	CCONJ
ejpam-2469	198	7	u	u	NOUN
ejpam-2469	198	8	is	be	AUX
ejpam-2469	198	9	a	a	DET
ejpam-2469	198	10	regular	regular	ADJ
ejpam-2469	198	11	open	open	NOUN
ejpam-2469	198	12	in	in	ADP
ejpam-2469	198	13	x	x	X
ejpam-2469	198	14	.	.	PUNCT
ejpam-2469	199	1	(	(	PUNCT
ejpam-2469	199	2	ii	ii	NOUN
ejpam-2469	199	3	)	)	PUNCT
ejpam-2469	199	4	weakly	weakly	ADV
ejpam-2469	199	5	-	-	PUNCT
ejpam-2469	199	6	i	i	PRON
ejpam-2469	199	7	r	r	VERB
ejpam-2469	199	8	g	g	NOUN
ejpam-2469	199	9	-	-	PUNCT
ejpam-2469	199	10	closed(briefly	closed(briefly	NOUN
ejpam-2469	199	11	w	w	NOUN
ejpam-2469	199	12	-	-	PUNCT
ejpam-2469	199	13	i	i	PRON
ejpam-2469	199	14	r	r	VERB
ejpam-2469	199	15	g	g	NOUN
ejpam-2469	199	16	-	-	PUNCT
ejpam-2469	199	17	closed	closed	ADJ
ejpam-2469	199	18	)	)	PUNCT
ejpam-2469	199	19	set	set	VERB
ejpam-2469	199	20	if	if	SCONJ
ejpam-2469	199	21	(	(	PUNCT
ejpam-2469	199	22	int(cl∗(a	int(cl∗(a	NOUN
ejpam-2469	199	23	)	)	PUNCT
ejpam-2469	199	24	)	)	PUNCT
ejpam-2469	199	25	)	)	PUNCT
ejpam-2469	200	1	⊂	⊂	PROPN
ejpam-2469	200	2	u	u	NOUN
ejpam-2469	200	3	whenever	whenever	SCONJ
ejpam-2469	200	4	a⊆	a⊆	VERB
ejpam-2469	200	5	u	u	NOUN
ejpam-2469	200	6	and	and	CCONJ
ejpam-2469	200	7	u	u	NOUN
ejpam-2469	200	8	is	be	AUX
ejpam-2469	200	9	i	i	NOUN
ejpam-2469	200	10	-	-	PUNCT
ejpam-2469	200	11	regular	regular	ADV
ejpam-2469	200	12	open	open	NOUN
ejpam-2469	200	13	in	in	ADP
ejpam-2469	200	14	x	x	X
ejpam-2469	200	15	.	.	PUNCT
ejpam-2469	201	1	theorem	theorem	ADJ
ejpam-2469	201	2	7	7	NUM
ejpam-2469	201	3	.	.	PUNCT
ejpam-2469	202	1	every	every	DET
ejpam-2469	202	2	w	w	NOUN
ejpam-2469	202	3	-	-	PUNCT
ejpam-2469	202	4	i	i	NOUN
ejpam-2469	202	5	r	r	NOUN
ejpam-2469	202	6	g	g	NOUN
ejpam-2469	202	7	-	-	PUNCT
ejpam-2469	202	8	closed	close	VERB
ejpam-2469	202	9	set	set	NOUN
ejpam-2469	202	10	is	be	AUX
ejpam-2469	202	11	a	a	DET
ejpam-2469	202	12	w	w	NOUN
ejpam-2469	202	13	-	-	PUNCT
ejpam-2469	202	14	r	r	NOUN
ejpam-2469	202	15	gi	gi	NOUN
ejpam-2469	202	16	-closed	-close	VERB
ejpam-2469	202	17	set	set	NOUN
ejpam-2469	202	18	.	.	PUNCT
ejpam-2469	203	1	proof	proof	NOUN
ejpam-2469	203	2	.	.	PUNCT
ejpam-2469	204	1	let	let	VERB
ejpam-2469	204	2	a	a	PRON
ejpam-2469	204	3	be	be	AUX
ejpam-2469	204	4	a	a	DET
ejpam-2469	204	5	w	w	NOUN
ejpam-2469	204	6	-	-	PUNCT
ejpam-2469	204	7	i	i	NOUN
ejpam-2469	204	8	r	r	NOUN
ejpam-2469	204	9	g	g	NOUN
ejpam-2469	204	10	-	-	PUNCT
ejpam-2469	204	11	closed	close	VERB
ejpam-2469	204	12	set	set	NOUN
ejpam-2469	204	13	.	.	PUNCT
ejpam-2469	205	1	then	then	ADV
ejpam-2469	205	2	(	(	PUNCT
ejpam-2469	205	3	int(cl∗(a	int(cl∗(a	NOUN
ejpam-2469	205	4	)	)	PUNCT
ejpam-2469	205	5	)	)	PUNCT
ejpam-2469	205	6	)	)	PUNCT
ejpam-2469	206	1	⊂	⊂	PROPN
ejpam-2469	206	2	u	u	NOUN
ejpam-2469	206	3	whenever	whenever	SCONJ
ejpam-2469	206	4	a	a	DET
ejpam-2469	206	5	⊆	⊆	NUM
ejpam-2469	206	6	u	u	NOUN
ejpam-2469	206	7	and	and	CCONJ
ejpam-2469	206	8	u	u	NOUN
ejpam-2469	206	9	is	be	AUX
ejpam-2469	206	10	a	a	DET
ejpam-2469	206	11	i	i	NOUN
ejpam-2469	206	12	-regular	-regular	ADJ
ejpam-2469	206	13	open	open	ADJ
ejpam-2469	206	14	in	in	ADP
ejpam-2469	206	15	x	x	X
ejpam-2469	206	16	.	.	PUNCT
ejpam-2469	207	1	since	since	SCONJ
ejpam-2469	207	2	u	u	NOUN
ejpam-2469	207	3	is	be	AUX
ejpam-2469	207	4	i	i	PRON
ejpam-2469	207	5	-regular	-regular	ADJ
ejpam-2469	207	6	open	open	ADJ
ejpam-2469	207	7	,	,	PUNCT
ejpam-2469	207	8	we	we	PRON
ejpam-2469	207	9	have	have	VERB
ejpam-2469	207	10	u	u	NOUN
ejpam-2469	207	11	=	=	PROPN
ejpam-2469	207	12	int∗(cl∗((u	int∗(cl∗((u	PROPN
ejpam-2469	207	13	)	)	PUNCT
ejpam-2469	207	14	)	)	PUNCT
ejpam-2469	207	15	)	)	PUNCT
ejpam-2469	207	16	and	and	CCONJ
ejpam-2469	207	17	int(cl(u	int(cl(u	PROPN
ejpam-2469	207	18	)	)	PUNCT
ejpam-2469	207	19	)	)	PUNCT
ejpam-2469	208	1	⊂	⊂	PROPN
ejpam-2469	208	2	int∗(cl∗((u	int∗(cl∗((u	PROPN
ejpam-2469	208	3	)	)	PUNCT
ejpam-2469	208	4	)	)	PUNCT
ejpam-2469	208	5	)	)	PUNCT
ejpam-2469	208	6	.	.	PUNCT
ejpam-2469	209	1	therefore	therefore	ADV
ejpam-2469	209	2	,	,	PUNCT
ejpam-2469	209	3	u	u	NOUN
ejpam-2469	209	4	is	be	AUX
ejpam-2469	209	5	a	a	DET
ejpam-2469	209	6	regular	regular	ADJ
ejpam-2469	209	7	open	open	NOUN
ejpam-2469	209	8	in	in	ADP
ejpam-2469	209	9	x	x	X
ejpam-2469	209	10	.	.	PUNCT
ejpam-2469	210	1	this	this	PRON
ejpam-2469	210	2	shows	show	VERB
ejpam-2469	210	3	that	that	SCONJ
ejpam-2469	210	4	a	a	PRON
ejpam-2469	210	5	is	be	AUX
ejpam-2469	210	6	a	a	DET
ejpam-2469	210	7	w	w	NOUN
ejpam-2469	210	8	-	-	PUNCT
ejpam-2469	210	9	r	r	NOUN
ejpam-2469	210	10	gi	gi	NOUN
ejpam-2469	210	11	-closed	-close	VERB
ejpam-2469	210	12	set	set	NOUN
ejpam-2469	210	13	.	.	PUNCT
ejpam-2469	211	1	the	the	DET
ejpam-2469	211	2	following	follow	VERB
ejpam-2469	211	3	example	example	NOUN
ejpam-2469	211	4	shows	show	VERB
ejpam-2469	211	5	that	that	SCONJ
ejpam-2469	211	6	the	the	DET
ejpam-2469	211	7	reverse	reverse	NOUN
ejpam-2469	211	8	of	of	ADP
ejpam-2469	211	9	theorem	theorem	NOUN
ejpam-2469	211	10	7	7	NUM
ejpam-2469	211	11	is	be	AUX
ejpam-2469	211	12	not	not	PART
ejpam-2469	211	13	true	true	ADJ
ejpam-2469	211	14	.	.	PUNCT
ejpam-2469	212	1	ü	ü	DET
ejpam-2469	212	2	karabıyık	karabıyık	PROPN
ejpam-2469	212	3	,	,	PUNCT
ejpam-2469	212	4	a	a	DET
ejpam-2469	212	5	kaymakcı	kaymakcı	NOUN
ejpam-2469	212	6	/	/	SYM
ejpam-2469	212	7	eur	eur	NOUN
ejpam-2469	212	8	.	.	PUNCT
ejpam-2469	213	1	j.	j.	PROPN
ejpam-2469	213	2	pure	pure	PROPN
ejpam-2469	213	3	appl	appl	PROPN
ejpam-2469	213	4	.	.	PROPN
ejpam-2469	213	5	math	math	PROPN
ejpam-2469	213	6	,	,	PUNCT
ejpam-2469	213	7	9	9	NUM
ejpam-2469	213	8	(	(	PUNCT
ejpam-2469	213	9	2016	2016	NUM
ejpam-2469	213	10	)	)	PUNCT
ejpam-2469	213	11	,	,	PUNCT
ejpam-2469	213	12	434	434	NUM
ejpam-2469	213	13	-	-	SYM
ejpam-2469	213	14	442	442	NUM
ejpam-2469	213	15	439	439	NUM
ejpam-2469	213	16	example	example	NOUN
ejpam-2469	213	17	3	3	NUM
ejpam-2469	213	18	.	.	X
ejpam-2469	214	1	let	let	AUX
ejpam-2469	214	2	(	(	PUNCT
ejpam-2469	214	3	x	x	X
ejpam-2469	214	4	,	,	PUNCT
ejpam-2469	214	5	τ	τ	PROPN
ejpam-2469	214	6	,	,	PUNCT
ejpam-2469	214	7	i	i	PRON
ejpam-2469	214	8	)	)	PUNCT
ejpam-2469	214	9	be	be	VERB
ejpam-2469	214	10	a	a	DET
ejpam-2469	214	11	ideal	ideal	ADJ
ejpam-2469	214	12	topological	topological	ADJ
ejpam-2469	214	13	space	space	NOUN
ejpam-2469	214	14	such	such	ADJ
ejpam-2469	214	15	that	that	SCONJ
ejpam-2469	215	1	x	x	X
ejpam-2469	215	2	=	=	X
ejpam-2469	215	3	{	{	PUNCT
ejpam-2469	215	4	a	a	PRON
ejpam-2469	215	5	,	,	PUNCT
ejpam-2469	215	6	b	b	NOUN
ejpam-2469	215	7	,	,	PUNCT
ejpam-2469	215	8	c	c	NOUN
ejpam-2469	215	9	}	}	PUNCT
ejpam-2469	215	10	,	,	PUNCT
ejpam-2469	215	11	i	i	PRON
ejpam-2469	215	12	=	=	PUNCT
ejpam-2469	215	13	{	{	PUNCT
ejpam-2469	215	14	;	;	PUNCT
ejpam-2469	215	15	,	,	PUNCT
ejpam-2469	215	16	{	{	PUNCT
ejpam-2469	215	17	c	c	NOUN
ejpam-2469	215	18	}	}	PUNCT
ejpam-2469	215	19	}	}	PUNCT
ejpam-2469	215	20	,	,	PUNCT
ejpam-2469	215	21	and	and	CCONJ
ejpam-2469	215	22	τ	τ	PROPN
ejpam-2469	215	23	=	=	PUNCT
ejpam-2469	215	24	{	{	PUNCT
ejpam-2469	215	25	;	;	PUNCT
ejpam-2469	215	26	,	,	PUNCT
ejpam-2469	215	27	x	x	X
ejpam-2469	215	28	,	,	PUNCT
ejpam-2469	215	29	{	{	PUNCT
ejpam-2469	215	30	a	a	NOUN
ejpam-2469	215	31	}	}	PUNCT
ejpam-2469	215	32	,	,	PUNCT
ejpam-2469	215	33	{	{	PUNCT
ejpam-2469	215	34	c	c	X
ejpam-2469	215	35	}	}	PUNCT
ejpam-2469	215	36	,	,	PUNCT
ejpam-2469	215	37	{	{	PUNCT
ejpam-2469	215	38	a	a	PRON
ejpam-2469	215	39	,	,	PUNCT
ejpam-2469	215	40	c	c	NOUN
ejpam-2469	215	41	}	}	PUNCT
ejpam-2469	215	42	}	}	PUNCT
ejpam-2469	215	43	.	.	PUNCT
ejpam-2469	216	1	then	then	ADV
ejpam-2469	216	2	a	a	X
ejpam-2469	216	3	=	=	X
ejpam-2469	216	4	{	{	PUNCT
ejpam-2469	216	5	a	a	NOUN
ejpam-2469	216	6	}	}	PUNCT
ejpam-2469	216	7	⊂	⊂	PROPN
ejpam-2469	216	8	x	x	X
ejpam-2469	216	9	is	be	AUX
ejpam-2469	216	10	a	a	DET
ejpam-2469	216	11	w	w	NOUN
ejpam-2469	216	12	-	-	PUNCT
ejpam-2469	216	13	r	r	NOUN
ejpam-2469	216	14	gi	gi	NOUN
ejpam-2469	216	15	-closed	-close	VERB
ejpam-2469	216	16	set	set	NOUN
ejpam-2469	216	17	but	but	CCONJ
ejpam-2469	216	18	is	be	AUX
ejpam-2469	216	19	not	not	PART
ejpam-2469	216	20	w	w	NOUN
ejpam-2469	216	21	-	-	ADJ
ejpam-2469	216	22	i	i	NOUN
ejpam-2469	216	23	r	r	VERB
ejpam-2469	216	24	g	g	NOUN
ejpam-2469	216	25	-	-	PUNCT
ejpam-2469	216	26	closed	close	VERB
ejpam-2469	216	27	set	set	NOUN
ejpam-2469	216	28	,	,	PUNCT
ejpam-2469	216	29	since	since	SCONJ
ejpam-2469	216	30	int(cl∗(a	int(cl∗(a	NOUN
ejpam-2469	216	31	)	)	PUNCT
ejpam-2469	216	32	)	)	PUNCT
ejpam-2469	217	1	=	=	PUNCT
ejpam-2469	217	2	int(cl∗({a	int(cl∗({a	NOUN
ejpam-2469	217	3	}	}	PUNCT
ejpam-2469	217	4	)	)	PUNCT
ejpam-2469	217	5	)	)	PUNCT
ejpam-2469	218	1	=	=	PRON
ejpam-2469	218	2	{	{	PUNCT
ejpam-2469	218	3	a	a	NOUN
ejpam-2469	218	4	}	}	PUNCT
ejpam-2469	218	5	where	where	SCONJ
ejpam-2469	218	6	a	a	PRON
ejpam-2469	218	7	is	be	AUX
ejpam-2469	218	8	contained	contain	VERB
ejpam-2469	218	9	in	in	ADP
ejpam-2469	218	10	the	the	DET
ejpam-2469	218	11	regular	regular	ADJ
ejpam-2469	218	12	open	open	ADJ
ejpam-2469	218	13	set	set	NOUN
ejpam-2469	218	14	u.	u.	NOUN
ejpam-2469	218	15	but	but	CCONJ
ejpam-2469	218	16	int∗(cl∗(a	int∗(cl∗(a	NOUN
ejpam-2469	218	17	)	)	PUNCT
ejpam-2469	218	18	)	)	PUNCT
ejpam-2469	219	1	=	=	PUNCT
ejpam-2469	220	1	int∗(cl∗({a	int∗(cl∗({a	PROPN
ejpam-2469	220	2	}	}	PUNCT
ejpam-2469	220	3	)	)	PUNCT
ejpam-2469	220	4	)	)	PUNCT
ejpam-2469	221	1	6=	6=	X
ejpam-2469	221	2	{	{	PUNCT
ejpam-2469	221	3	a	a	PRON
ejpam-2469	221	4	}	}	PUNCT
ejpam-2469	221	5	which	which	PRON
ejpam-2469	221	6	means	mean	VERB
ejpam-2469	221	7	a	a	PRON
ejpam-2469	221	8	is	be	AUX
ejpam-2469	221	9	contained	contain	VERB
ejpam-2469	221	10	there	there	PRON
ejpam-2469	221	11	is	be	VERB
ejpam-2469	221	12	not	not	PART
ejpam-2469	221	13	i	i	NOUN
ejpam-2469	221	14	-	-	PUNCT
ejpam-2469	221	15	regular	regular	ADJ
ejpam-2469	221	16	open	open	ADJ
ejpam-2469	221	17	set	set	NOUN
ejpam-2469	221	18	u.	u.	PROPN
ejpam-2469	221	19	remark	remark	PROPN
ejpam-2469	221	20	2	2	NUM
ejpam-2469	221	21	.	.	PUNCT
ejpam-2469	222	1	let	let	AUX
ejpam-2469	222	2	be	be	AUX
ejpam-2469	222	3	a	a	DET
ejpam-2469	222	4	(	(	PUNCT
ejpam-2469	222	5	x	x	SYM
ejpam-2469	222	6	,	,	PUNCT
ejpam-2469	222	7	τ	τ	PROPN
ejpam-2469	222	8	,	,	PUNCT
ejpam-2469	222	9	i	i	PRON
ejpam-2469	222	10	)	)	PUNCT
ejpam-2469	222	11	be	be	VERB
ejpam-2469	222	12	a	a	DET
ejpam-2469	222	13	ideal	ideal	ADJ
ejpam-2469	222	14	topological	topological	ADJ
ejpam-2469	222	15	spaces	space	NOUN
ejpam-2469	222	16	.	.	PUNCT
ejpam-2469	223	1	the	the	DET
ejpam-2469	223	2	following	follow	VERB
ejpam-2469	223	3	diagram	diagram	NOUN
ejpam-2469	223	4	holds	hold	VERB
ejpam-2469	223	5	for	for	ADP
ejpam-2469	223	6	a	a	DET
ejpam-2469	223	7	subset	subset	NOUN
ejpam-2469	223	8	a⊂	a⊂	NOUN
ejpam-2469	223	9	x	x	NOUN
ejpam-2469	223	10	:	:	PUNCT
ejpam-2469	223	11	r	r	NOUN
ejpam-2469	223	12	gi	gi	NOUN
ejpam-2469	223	13	−	−	PROPN
ejpam-2469	223	14	closed	closed	ADJ
ejpam-2469	223	15	//	//	SYM
ejpam-2469	223	16	�	�	PROPN
ejpam-2469	223	17	�	�	PROPN
ejpam-2469	223	18	w−	w−	PROPN
ejpam-2469	223	19	r	r	NOUN
ejpam-2469	223	20	gi	gi	NOUN
ejpam-2469	223	21	−	−	PROPN
ejpam-2469	223	22	closed	closed	ADJ
ejpam-2469	223	23	�	�	PROPN
ejpam-2469	223	24	�	�	NOUN
ejpam-2469	223	25	w−	w−	NOUN
ejpam-2469	224	1	i	i	PRON
ejpam-2469	224	2	r	r	VERB
ejpam-2469	224	3	g	g	NOUN
ejpam-2469	224	4	−	−	PROPN
ejpam-2469	224	5	closedoo	closedoo	NOUN
ejpam-2469	224	6	ir	ir	PROPN
ejpam-2469	224	7	g	g	PROPN
ejpam-2469	224	8	−	−	PROPN
ejpam-2469	224	9	closed	closed	ADJ
ejpam-2469	224	10	//	//	NUM
ejpam-2469	224	11	w−	w−	PROPN
ejpam-2469	224	12	ir	ir	PROPN
ejpam-2469	224	13	g	g	PROPN
ejpam-2469	224	14	−	−	PROPN
ejpam-2469	224	15	closed	close	VERB
ejpam-2469	224	16	aw−	aw−	NOUN
ejpam-2469	224	17	r	r	NOUN
ejpam-2469	224	18	gi	gi	NOUN
ejpam-2469	224	19	−	−	PROPN
ejpam-2469	224	20	closedoo	closedoo	PROPN
ejpam-2469	224	21	�	�	PROPN
ejpam-2469	224	22	�	�	PROPN
ejpam-2469	224	23	αi	αi	PART
ejpam-2469	224	24	−	−	PROPN
ejpam-2469	224	25	∗−	∗−	NOUN
ejpam-2469	224	26	closed	closed	ADJ
ejpam-2469	224	27	//	//	X
ejpam-2469	224	28	saw−	saw−	VERB
ejpam-2469	224	29	ir	ir	PROPN
ejpam-2469	224	30	g	g	PROPN
ejpam-2469	224	31	−	−	PROPN
ejpam-2469	224	32	closed	closed	ADJ
ejpam-2469	224	33	//	//	NUM
ejpam-2469	224	34	oo	oo	NOUN
ejpam-2469	224	35	aw−	aw−	PROPN
ejpam-2469	224	36	ir	ir	PROPN
ejpam-2469	224	37	g	g	PROPN
ejpam-2469	224	38	−	−	PROPN
ejpam-2469	224	39	closed	closed	ADJ
ejpam-2469	224	40	theorem	theorem	NOUN
ejpam-2469	224	41	8	8	X
ejpam-2469	224	42	.	.	PUNCT
ejpam-2469	225	1	let	let	AUX
ejpam-2469	225	2	be	be	AUX
ejpam-2469	225	3	a	a	DET
ejpam-2469	225	4	(	(	PUNCT
ejpam-2469	225	5	x	x	SYM
ejpam-2469	225	6	,	,	PUNCT
ejpam-2469	225	7	τ	τ	PROPN
ejpam-2469	225	8	,	,	PUNCT
ejpam-2469	225	9	i	i	PRON
ejpam-2469	225	10	)	)	PUNCT
ejpam-2469	225	11	be	be	VERB
ejpam-2469	225	12	a	a	DET
ejpam-2469	225	13	ideal	ideal	ADJ
ejpam-2469	225	14	topological	topological	ADJ
ejpam-2469	225	15	spaces	space	NOUN
ejpam-2469	225	16	.	.	PUNCT
ejpam-2469	226	1	for	for	ADP
ejpam-2469	226	2	every	every	DET
ejpam-2469	226	3	subset	subset	NOUN
ejpam-2469	226	4	a∈	a∈	PROPN
ejpam-2469	227	1	i	i	PRON
ejpam-2469	227	2	,	,	PUNCT
ejpam-2469	227	3	a	a	PRON
ejpam-2469	227	4	is	be	AUX
ejpam-2469	227	5	a	a	DET
ejpam-2469	227	6	r	r	NOUN
ejpam-2469	227	7	gi	gi	NOUN
ejpam-2469	227	8	-closed	-close	VERB
ejpam-2469	227	9	sets	set	NOUN
ejpam-2469	227	10	.	.	PUNCT
ejpam-2469	228	1	proof	proof	NOUN
ejpam-2469	228	2	.	.	PUNCT
ejpam-2469	229	1	let	let	VERB
ejpam-2469	229	2	a	a	DET
ejpam-2469	229	3	⊂	⊂	PROPN
ejpam-2469	229	4	u	u	NOUN
ejpam-2469	229	5	,	,	PUNCT
ejpam-2469	229	6	where	where	SCONJ
ejpam-2469	229	7	u	u	NOUN
ejpam-2469	229	8	is	be	AUX
ejpam-2469	229	9	regular	regular	ADJ
ejpam-2469	229	10	open	open	ADJ
ejpam-2469	229	11	.	.	PUNCT
ejpam-2469	230	1	since	since	SCONJ
ejpam-2469	230	2	a∗	a∗	PROPN
ejpam-2469	230	3	=	=	SYM
ejpam-2469	230	4	;	;	PUNCT
ejpam-2469	230	5	for	for	ADP
ejpam-2469	230	6	every	every	DET
ejpam-2469	230	7	a	a	DET
ejpam-2469	230	8	∈	∈	NOUN
ejpam-2469	230	9	i	i	PRON
ejpam-2469	230	10	,	,	PUNCT
ejpam-2469	230	11	we	we	PRON
ejpam-2469	230	12	obtain	obtain	VERB
ejpam-2469	230	13	cl∗(a	cl∗(a	NOUN
ejpam-2469	230	14	)	)	PUNCT
ejpam-2469	230	15	=	=	PUNCT
ejpam-2469	230	16	a∪	a∪	DET
ejpam-2469	230	17	a∗	a∗	NOUN
ejpam-2469	230	18	=	=	SYM
ejpam-2469	230	19	a⊂	a⊂	PUNCT
ejpam-2469	230	20	u	u	NOUN
ejpam-2469	230	21	.	.	PUNCT
ejpam-2469	231	1	therefore	therefore	ADV
ejpam-2469	231	2	a	a	PRON
ejpam-2469	231	3	is	be	AUX
ejpam-2469	231	4	a	a	DET
ejpam-2469	231	5	r	r	NOUN
ejpam-2469	231	6	gi	gi	NOUN
ejpam-2469	231	7	-closed	-close	VERB
ejpam-2469	231	8	set	set	NOUN
ejpam-2469	231	9	.	.	PUNCT
ejpam-2469	232	1	theorem	theorem	VERB
ejpam-2469	232	2	9	9	NUM
ejpam-2469	232	3	.	.	PUNCT
ejpam-2469	233	1	let	let	AUX
ejpam-2469	233	2	be	be	AUX
ejpam-2469	233	3	a	a	DET
ejpam-2469	233	4	(	(	PUNCT
ejpam-2469	233	5	x	x	SYM
ejpam-2469	233	6	,	,	PUNCT
ejpam-2469	233	7	τ	τ	PROPN
ejpam-2469	233	8	,	,	PUNCT
ejpam-2469	233	9	i	i	PRON
ejpam-2469	233	10	)	)	PUNCT
ejpam-2469	233	11	be	be	VERB
ejpam-2469	233	12	a	a	DET
ejpam-2469	233	13	ideal	ideal	ADJ
ejpam-2469	233	14	topological	topological	ADJ
ejpam-2469	233	15	spaces	space	NOUN
ejpam-2469	233	16	.	.	PUNCT
ejpam-2469	234	1	for	for	ADP
ejpam-2469	234	2	every	every	DET
ejpam-2469	234	3	subset	subset	NOUN
ejpam-2469	234	4	a	a	PRON
ejpam-2469	234	5	of	of	ADP
ejpam-2469	234	6	x	x	PRON
ejpam-2469	234	7	,	,	PUNCT
ejpam-2469	234	8	a∗	a∗	PROPN
ejpam-2469	234	9	is	be	AUX
ejpam-2469	234	10	a	a	DET
ejpam-2469	234	11	r	r	NOUN
ejpam-2469	234	12	gi	gi	NOUN
ejpam-2469	234	13	-closed	-close	VERB
ejpam-2469	234	14	sets	set	NOUN
ejpam-2469	234	15	.	.	PUNCT
ejpam-2469	235	1	proof	proof	NOUN
ejpam-2469	235	2	.	.	PUNCT
ejpam-2469	236	1	let	let	VERB
ejpam-2469	236	2	a∗	a∗	PROPN
ejpam-2469	236	3	⊂	⊂	PROPN
ejpam-2469	236	4	u	u	PROPN
ejpam-2469	236	5	,	,	PUNCT
ejpam-2469	236	6	where	where	SCONJ
ejpam-2469	236	7	u	u	NOUN
ejpam-2469	236	8	is	be	AUX
ejpam-2469	236	9	a	a	DET
ejpam-2469	236	10	regular	regular	ADJ
ejpam-2469	236	11	open	open	NOUN
ejpam-2469	236	12	.	.	PUNCT
ejpam-2469	237	1	since	since	SCONJ
ejpam-2469	237	2	(	(	PUNCT
ejpam-2469	237	3	a∗)∗	a∗)∗	PROPN
ejpam-2469	237	4	⊂	⊂	PROPN
ejpam-2469	237	5	a∗	a∗	PROPN
ejpam-2469	237	6	,	,	PUNCT
ejpam-2469	237	7	we	we	PRON
ejpam-2469	237	8	have	have	VERB
ejpam-2469	237	9	cl∗(a∗	cl∗(a∗	NUM
ejpam-2469	237	10	)	)	PUNCT
ejpam-2469	237	11	⊂	⊂	PROPN
ejpam-2469	237	12	u	u	PROPN
ejpam-2469	237	13	.	.	PUNCT
ejpam-2469	238	1	hence	hence	ADV
ejpam-2469	238	2	a∗	a∗	PROPN
ejpam-2469	238	3	is	be	AUX
ejpam-2469	238	4	a	a	DET
ejpam-2469	238	5	r	r	NOUN
ejpam-2469	238	6	gi	gi	NOUN
ejpam-2469	238	7	-closed	-close	VERB
ejpam-2469	238	8	set	set	NOUN
ejpam-2469	238	9	.	.	PUNCT
ejpam-2469	239	1	theorem	theorem	VERB
ejpam-2469	239	2	10	10	NUM
ejpam-2469	239	3	.	.	PUNCT
ejpam-2469	240	1	let	let	AUX
ejpam-2469	240	2	be	be	AUX
ejpam-2469	240	3	a	a	DET
ejpam-2469	240	4	(	(	PUNCT
ejpam-2469	240	5	x	x	SYM
ejpam-2469	240	6	,	,	PUNCT
ejpam-2469	240	7	τ	τ	PROPN
ejpam-2469	240	8	,	,	PUNCT
ejpam-2469	240	9	i	i	PRON
ejpam-2469	240	10	)	)	PUNCT
ejpam-2469	240	11	be	be	VERB
ejpam-2469	240	12	a	a	DET
ejpam-2469	240	13	ideal	ideal	ADJ
ejpam-2469	240	14	topological	topological	ADJ
ejpam-2469	240	15	spaces	space	NOUN
ejpam-2469	240	16	.	.	PUNCT
ejpam-2469	241	1	if	if	SCONJ
ejpam-2469	241	2	a	a	PRON
ejpam-2469	241	3	is	be	AUX
ejpam-2469	241	4	a	a	DET
ejpam-2469	241	5	r	r	NOUN
ejpam-2469	241	6	gi	gi	NOUN
ejpam-2469	241	7	-closed	-close	VERB
ejpam-2469	241	8	set	set	NOUN
ejpam-2469	241	9	,	,	PUNCT
ejpam-2469	241	10	then	then	ADV
ejpam-2469	241	11	cl∗(a)−a	cl∗(a)−a	NOUN
ejpam-2469	241	12	does	do	AUX
ejpam-2469	241	13	not	not	PART
ejpam-2469	241	14	contain	contain	VERB
ejpam-2469	241	15	any	any	DET
ejpam-2469	241	16	nonempty	nonempty	ADJ
ejpam-2469	241	17	regular	regular	ADJ
ejpam-2469	241	18	closed	close	VERB
ejpam-2469	241	19	set	set	NOUN
ejpam-2469	241	20	.	.	PUNCT
ejpam-2469	242	1	proof	proof	NOUN
ejpam-2469	242	2	.	.	PUNCT
ejpam-2469	243	1	let	let	VERB
ejpam-2469	243	2	f	f	PRON
ejpam-2469	243	3	be	be	AUX
ejpam-2469	243	4	a	a	DET
ejpam-2469	243	5	regular	regular	ADJ
ejpam-2469	243	6	closed	closed	ADJ
ejpam-2469	243	7	subset	subset	NOUN
ejpam-2469	243	8	of	of	ADP
ejpam-2469	243	9	x	x	PRON
ejpam-2469	243	10	,	,	PUNCT
ejpam-2469	243	11	such	such	ADJ
ejpam-2469	243	12	that	that	SCONJ
ejpam-2469	243	13	f	f	PROPN
ejpam-2469	243	14	⊂	⊂	PROPN
ejpam-2469	243	15	cl∗(a)−a	cl∗(a)−a	PROPN
ejpam-2469	243	16	where	where	SCONJ
ejpam-2469	243	17	a	a	PRON
ejpam-2469	243	18	is	be	AUX
ejpam-2469	243	19	r	r	NOUN
ejpam-2469	243	20	gi	gi	NOUN
ejpam-2469	243	21	-closed	-close	VERB
ejpam-2469	243	22	set	set	NOUN
ejpam-2469	243	23	.	.	PUNCT
ejpam-2469	244	1	we	we	PRON
ejpam-2469	244	2	get	get	VERB
ejpam-2469	244	3	cl∗(a	cl∗(a	NOUN
ejpam-2469	244	4	)	)	PUNCT
ejpam-2469	244	5	⊂	⊂	PROPN
ejpam-2469	244	6	(	(	PUNCT
ejpam-2469	245	1	x	x	X
ejpam-2469	245	2	−	−	PROPN
ejpam-2469	245	3	f	f	X
ejpam-2469	245	4	)	)	PUNCT
ejpam-2469	245	5	.	.	PUNCT
ejpam-2469	246	1	this	this	PRON
ejpam-2469	246	2	shows	show	VERB
ejpam-2469	246	3	that	that	SCONJ
ejpam-2469	246	4	f	f	PROPN
ejpam-2469	246	5	⊂	⊂	PROPN
ejpam-2469	246	6	(	(	PUNCT
ejpam-2469	246	7	x	x	X
ejpam-2469	246	8	−	−	PROPN
ejpam-2469	246	9	cl∗(a))∩	cl∗(a))∩	NOUN
ejpam-2469	246	10	cl∗(a	cl∗(a	NOUN
ejpam-2469	246	11	)	)	PUNCT
ejpam-2469	246	12	.	.	PUNCT
ejpam-2469	247	1	hence	hence	ADV
ejpam-2469	247	2	f	f	PROPN
ejpam-2469	247	3	=	=	X
ejpam-2469	247	4	;	;	PUNCT
ejpam-2469	247	5	.	.	PUNCT
ejpam-2469	248	1	theorem	theorem	NOUN
ejpam-2469	248	2	11	11	NUM
ejpam-2469	248	3	.	.	PUNCT
ejpam-2469	249	1	(	(	PUNCT
ejpam-2469	249	2	x	x	X
ejpam-2469	249	3	,	,	PUNCT
ejpam-2469	249	4	τ	τ	PROPN
ejpam-2469	249	5	,	,	PUNCT
ejpam-2469	249	6	i	i	PRON
ejpam-2469	249	7	)	)	PUNCT
ejpam-2469	249	8	be	be	VERB
ejpam-2469	249	9	a	a	DET
ejpam-2469	249	10	ideal	ideal	ADJ
ejpam-2469	249	11	topological	topological	ADJ
ejpam-2469	249	12	spaces	space	NOUN
ejpam-2469	249	13	and	and	CCONJ
ejpam-2469	249	14	a	a	DET
ejpam-2469	249	15	⊂	⊂	PROPN
ejpam-2469	249	16	x	x	X
ejpam-2469	249	17	.	.	PUNCT
ejpam-2469	250	1	if	if	SCONJ
ejpam-2469	250	2	a	a	PRON
ejpam-2469	250	3	is	be	AUX
ejpam-2469	250	4	w	w	NOUN
ejpam-2469	250	5	-	-	PUNCT
ejpam-2469	250	6	r	r	NOUN
ejpam-2469	250	7	gi	gi	NOUN
ejpam-2469	250	8	-closed	-close	VERB
ejpam-2469	250	9	set	set	NOUN
ejpam-2469	250	10	and	and	CCONJ
ejpam-2469	250	11	τ∗-closed	τ∗-close	VERB
ejpam-2469	250	12	,	,	PUNCT
ejpam-2469	250	13	then	then	ADV
ejpam-2469	250	14	a	a	PRON
ejpam-2469	250	15	is	be	AUX
ejpam-2469	250	16	a	a	DET
ejpam-2469	250	17	w	w	PROPN
ejpam-2469	250	18	-	-	PUNCT
ejpam-2469	250	19	ir	ir	NOUN
ejpam-2469	250	20	g	g	PROPN
ejpam-2469	250	21	-closed	-close	VERB
ejpam-2469	250	22	set	set	NOUN
ejpam-2469	250	23	.	.	PUNCT
ejpam-2469	251	1	proof	proof	NOUN
ejpam-2469	251	2	.	.	PUNCT
ejpam-2469	252	1	let	let	VERB
ejpam-2469	252	2	a	a	DET
ejpam-2469	252	3	be	be	AUX
ejpam-2469	252	4	a	a	DET
ejpam-2469	252	5	τ∗-closed	τ∗-close	VERB
ejpam-2469	252	6	and	and	CCONJ
ejpam-2469	252	7	w	w	NOUN
ejpam-2469	252	8	-	-	PUNCT
ejpam-2469	252	9	r	r	NOUN
ejpam-2469	252	10	gi	gi	NOUN
ejpam-2469	252	11	-closed	-close	VERB
ejpam-2469	252	12	set	set	NOUN
ejpam-2469	252	13	in	in	ADP
ejpam-2469	252	14	(	(	PUNCT
ejpam-2469	252	15	x	x	INTJ
ejpam-2469	252	16	,	,	PUNCT
ejpam-2469	252	17	τ	τ	PROPN
ejpam-2469	252	18	,	,	PUNCT
ejpam-2469	252	19	i	i	PROPN
ejpam-2469	252	20	)	)	PUNCT
ejpam-2469	252	21	.	.	PUNCT
ejpam-2469	253	1	then	then	ADV
ejpam-2469	253	2	,	,	PUNCT
ejpam-2469	253	3	we	we	PRON
ejpam-2469	253	4	have	have	VERB
ejpam-2469	253	5	a∗	a∗	PROPN
ejpam-2469	253	6	⊂	⊂	PROPN
ejpam-2469	253	7	a	a	PROPN
ejpam-2469	253	8	and	and	CCONJ
ejpam-2469	253	9	we	we	PRON
ejpam-2469	253	10	have	have	VERB
ejpam-2469	253	11	int(a∗	int(a∗	X
ejpam-2469	253	12	)	)	PUNCT
ejpam-2469	253	13	⊂	⊂	PROPN
ejpam-2469	253	14	int(a	int(a	PROPN
ejpam-2469	253	15	)	)	PUNCT
ejpam-2469	253	16	.	.	PUNCT
ejpam-2469	254	1	on	on	ADP
ejpam-2469	254	2	the	the	DET
ejpam-2469	254	3	other	other	ADJ
ejpam-2469	254	4	hand	hand	NOUN
ejpam-2469	254	5	,	,	PUNCT
ejpam-2469	254	6	int(cl∗(a	int(cl∗(a	NOUN
ejpam-2469	254	7	)	)	PUNCT
ejpam-2469	254	8	)	)	PUNCT
ejpam-2469	255	1	⊂	⊂	PROPN
ejpam-2469	255	2	u	u	NOUN
ejpam-2469	255	3	whenever	whenever	SCONJ
ejpam-2469	255	4	a	a	DET
ejpam-2469	255	5	⊂	⊂	PROPN
ejpam-2469	255	6	u	u	NOUN
ejpam-2469	255	7	and	and	CCONJ
ejpam-2469	255	8	u	u	NOUN
ejpam-2469	255	9	regular	regular	ADJ
ejpam-2469	255	10	open	open	ADJ
ejpam-2469	255	11	in	in	ADP
ejpam-2469	255	12	x	x	X
ejpam-2469	255	13	.	.	PUNCT
ejpam-2469	256	1	hence	hence	ADV
ejpam-2469	256	2	,	,	PUNCT
ejpam-2469	256	3	int(a	int(a	PROPN
ejpam-2469	256	4	)	)	PUNCT
ejpam-2469	256	5	⊂	⊂	PROPN
ejpam-2469	257	1	int(a	int(a	PROPN
ejpam-2469	257	2	)	)	PUNCT
ejpam-2469	257	3	∪	∪	NOUN
ejpam-2469	257	4	int(a∗	int(a∗	X
ejpam-2469	257	5	)	)	PUNCT
ejpam-2469	257	6	⊂	⊂	PRON
ejpam-2469	257	7	int(a∪	int(a∪	NOUN
ejpam-2469	257	8	a∗	a∗	NOUN
ejpam-2469	257	9	)	)	PUNCT
ejpam-2469	257	10	=	=	SYM
ejpam-2469	257	11	int(cl∗(a	int(cl∗(a	NOUN
ejpam-2469	257	12	)	)	PUNCT
ejpam-2469	257	13	)	)	PUNCT
ejpam-2469	257	14	.	.	PUNCT
ejpam-2469	258	1	this	this	PRON
ejpam-2469	258	2	implies	imply	VERB
ejpam-2469	258	3	that	that	SCONJ
ejpam-2469	258	4	(	(	PUNCT
ejpam-2469	258	5	int(a))∗	int(a))∗	PROPN
ejpam-2469	258	6	⊂	⊂	PROPN
ejpam-2469	258	7	int(cl∗(a	int(cl∗(a	PROPN
ejpam-2469	258	8	)	)	PUNCT
ejpam-2469	258	9	)	)	PUNCT
ejpam-2469	259	1	⊂	⊂	PROPN
ejpam-2469	259	2	u	u	NOUN
ejpam-2469	259	3	and	and	CCONJ
ejpam-2469	259	4	so	so	ADV
ejpam-2469	259	5	a	a	PRON
ejpam-2469	259	6	is	be	AUX
ejpam-2469	259	7	a	a	DET
ejpam-2469	259	8	w	w	PROPN
ejpam-2469	259	9	-	-	PUNCT
ejpam-2469	259	10	ir	ir	NOUN
ejpam-2469	259	11	g	g	PROPN
ejpam-2469	259	12	-closed	-close	VERB
ejpam-2469	259	13	set	set	NOUN
ejpam-2469	259	14	.	.	PUNCT
ejpam-2469	260	1	theorem	theorem	PROPN
ejpam-2469	260	2	12	12	NUM
ejpam-2469	260	3	.	.	PUNCT
ejpam-2469	261	1	in	in	ADP
ejpam-2469	261	2	an	an	DET
ejpam-2469	261	3	ideal	ideal	ADJ
ejpam-2469	261	4	spaces	space	NOUN
ejpam-2469	261	5	(	(	PUNCT
ejpam-2469	261	6	x	x	X
ejpam-2469	261	7	,	,	PUNCT
ejpam-2469	261	8	τ	τ	PROPN
ejpam-2469	261	9	,	,	PUNCT
ejpam-2469	261	10	i	i	PROPN
ejpam-2469	261	11	)	)	PUNCT
ejpam-2469	261	12	,	,	PUNCT
ejpam-2469	261	13	the	the	DET
ejpam-2469	261	14	union	union	NOUN
ejpam-2469	261	15	two	two	NUM
ejpam-2469	261	16	r	r	NOUN
ejpam-2469	261	17	gi	gi	NOUN
ejpam-2469	261	18	-closed	-close	VERB
ejpam-2469	261	19	set	set	NOUN
ejpam-2469	261	20	in	in	ADP
ejpam-2469	261	21	an	an	DET
ejpam-2469	261	22	r	r	NOUN
ejpam-2469	261	23	gi	gi	NOUN
ejpam-2469	261	24	-closed	-close	VERB
ejpam-2469	261	25	set	set	NOUN
ejpam-2469	261	26	.	.	PUNCT
ejpam-2469	262	1	ü	ü	DET
ejpam-2469	262	2	karabıyık	karabıyık	PROPN
ejpam-2469	262	3	,	,	PUNCT
ejpam-2469	262	4	a	a	DET
ejpam-2469	262	5	kaymakcı	kaymakcı	NOUN
ejpam-2469	262	6	/	/	SYM
ejpam-2469	262	7	eur	eur	NOUN
ejpam-2469	262	8	.	.	PUNCT
ejpam-2469	263	1	j.	j.	PROPN
ejpam-2469	263	2	pure	pure	PROPN
ejpam-2469	263	3	appl	appl	PROPN
ejpam-2469	263	4	.	.	PROPN
ejpam-2469	263	5	math	math	PROPN
ejpam-2469	263	6	,	,	PUNCT
ejpam-2469	263	7	9	9	NUM
ejpam-2469	263	8	(	(	PUNCT
ejpam-2469	263	9	2016	2016	NUM
ejpam-2469	263	10	)	)	PUNCT
ejpam-2469	263	11	,	,	PUNCT
ejpam-2469	263	12	434	434	NUM
ejpam-2469	263	13	-	-	SYM
ejpam-2469	263	14	442	442	NUM
ejpam-2469	263	15	440	440	NUM
ejpam-2469	263	16	proof	proof	NOUN
ejpam-2469	263	17	.	.	PUNCT
ejpam-2469	264	1	let	let	VERB
ejpam-2469	264	2	a	a	DET
ejpam-2469	264	3	and	and	CCONJ
ejpam-2469	264	4	b	b	NOUN
ejpam-2469	264	5	r	r	NOUN
ejpam-2469	264	6	gi	gi	NOUN
ejpam-2469	264	7	-closed	-close	VERB
ejpam-2469	264	8	set	set	NOUN
ejpam-2469	264	9	.	.	PUNCT
ejpam-2469	265	1	suppose	suppose	VERB
ejpam-2469	265	2	a∪b	a∪b	ADJ
ejpam-2469	265	3	⊂	⊂	X
ejpam-2469	265	4	u	u	NOUN
ejpam-2469	265	5	and	and	CCONJ
ejpam-2469	265	6	u	u	NOUN
ejpam-2469	265	7	is	be	AUX
ejpam-2469	265	8	regular	regular	ADJ
ejpam-2469	265	9	open	open	ADJ
ejpam-2469	265	10	.	.	PUNCT
ejpam-2469	266	1	then	then	ADV
ejpam-2469	266	2	a⊂	a⊂	VERB
ejpam-2469	266	3	u	u	NOUN
ejpam-2469	266	4	and	and	CCONJ
ejpam-2469	266	5	b	b	NOUN
ejpam-2469	266	6	⊂	⊂	X
ejpam-2469	266	7	u	u	NOUN
ejpam-2469	266	8	by	by	ADP
ejpam-2469	266	9	hypothesis	hypothesis	NOUN
ejpam-2469	266	10	,	,	PUNCT
ejpam-2469	266	11	cl∗(a	cl∗(a	NOUN
ejpam-2469	266	12	)	)	PUNCT
ejpam-2469	266	13	⊂	⊂	PROPN
ejpam-2469	266	14	u	u	NOUN
ejpam-2469	266	15	and	and	CCONJ
ejpam-2469	266	16	cl∗(b	cl∗(b	ADJ
ejpam-2469	266	17	)	)	PUNCT
ejpam-2469	266	18	⊂	⊂	PROPN
ejpam-2469	266	19	u	u	PROPN
ejpam-2469	266	20	.	.	PUNCT
ejpam-2469	267	1	therefore	therefore	ADV
ejpam-2469	267	2	cl∗(a∪	cl∗(a∪	PROPN
ejpam-2469	267	3	b	b	PROPN
ejpam-2469	267	4	)	)	PUNCT
ejpam-2469	267	5	=	=	PUNCT
ejpam-2469	267	6	cl∗(a)∪	cl∗(a)∪	VERB
ejpam-2469	267	7	cl∗(b	cl∗(b	PROPN
ejpam-2469	267	8	)	)	PUNCT
ejpam-2469	267	9	⊂	⊂	PROPN
ejpam-2469	267	10	u	u	PROPN
ejpam-2469	267	11	.	.	PUNCT
ejpam-2469	268	1	this	this	PRON
ejpam-2469	268	2	shows	show	VERB
ejpam-2469	268	3	that	that	SCONJ
ejpam-2469	268	4	a∪	a∪	PROPN
ejpam-2469	268	5	b	b	NOUN
ejpam-2469	268	6	is	be	AUX
ejpam-2469	268	7	r	r	NOUN
ejpam-2469	268	8	gi	gi	NOUN
ejpam-2469	268	9	-closed	-close	VERB
ejpam-2469	268	10	set	set	NOUN
ejpam-2469	268	11	.	.	PUNCT
ejpam-2469	268	12	example	example	NOUN
ejpam-2469	269	1	4	4	X
ejpam-2469	269	2	.	.	PUNCT
ejpam-2469	269	3	let	let	VERB
ejpam-2469	269	4	x	x	PUNCT
ejpam-2469	269	5	=	=	PRON
ejpam-2469	269	6	{	{	PUNCT
ejpam-2469	269	7	a	a	PRON
ejpam-2469	269	8	,	,	PUNCT
ejpam-2469	269	9	b	b	PROPN
ejpam-2469	269	10	,	,	PUNCT
ejpam-2469	269	11	c},τ	c},τ	PROPN
ejpam-2469	269	12	=	=	PUNCT
ejpam-2469	269	13	{	{	PUNCT
ejpam-2469	269	14	;	;	PUNCT
ejpam-2469	269	15	,	,	PUNCT
ejpam-2469	269	16	x	x	X
ejpam-2469	269	17	,	,	PUNCT
ejpam-2469	269	18	{	{	PUNCT
ejpam-2469	269	19	a	a	NOUN
ejpam-2469	269	20	}	}	PUNCT
ejpam-2469	269	21	,	,	PUNCT
ejpam-2469	269	22	{	{	PUNCT
ejpam-2469	269	23	b	b	NOUN
ejpam-2469	269	24	}	}	PUNCT
ejpam-2469	269	25	,	,	PUNCT
ejpam-2469	269	26	{	{	PUNCT
ejpam-2469	269	27	a	a	PRON
ejpam-2469	269	28	,	,	PUNCT
ejpam-2469	269	29	b	b	NOUN
ejpam-2469	269	30	}	}	PUNCT
ejpam-2469	269	31	}	}	PUNCT
ejpam-2469	269	32	and	and	CCONJ
ejpam-2469	269	33	i	i	PRON
ejpam-2469	269	34	=	=	PUNCT
ejpam-2469	269	35	{	{	PUNCT
ejpam-2469	269	36	;	;	PUNCT
ejpam-2469	269	37	,	,	PUNCT
ejpam-2469	269	38	{	{	PUNCT
ejpam-2469	269	39	b	b	NOUN
ejpam-2469	269	40	}	}	PUNCT
ejpam-2469	269	41	}	}	PUNCT
ejpam-2469	269	42	.	.	PUNCT
ejpam-2469	270	1	for	for	ADP
ejpam-2469	270	2	a	a	DET
ejpam-2469	270	3	=	=	X
ejpam-2469	270	4	{	{	PUNCT
ejpam-2469	270	5	a	a	PROPN
ejpam-2469	270	6	,	,	PUNCT
ejpam-2469	270	7	b	b	NOUN
ejpam-2469	270	8	}	}	PUNCT
ejpam-2469	270	9	and	and	CCONJ
ejpam-2469	270	10	b	b	X
ejpam-2469	270	11	=	=	NOUN
ejpam-2469	270	12	{	{	PUNCT
ejpam-2469	270	13	a	a	X
ejpam-2469	270	14	,	,	PUNCT
ejpam-2469	270	15	c	c	NOUN
ejpam-2469	270	16	}	}	PUNCT
ejpam-2469	270	17	since	since	SCONJ
ejpam-2469	270	18	x	x	PRON
ejpam-2469	270	19	is	be	AUX
ejpam-2469	270	20	the	the	DET
ejpam-2469	270	21	only	only	ADJ
ejpam-2469	270	22	regular	regular	ADJ
ejpam-2469	270	23	open	open	ADJ
ejpam-2469	270	24	set	set	NOUN
ejpam-2469	270	25	containing	contain	VERB
ejpam-2469	270	26	a	a	PRON
ejpam-2469	270	27	and	and	CCONJ
ejpam-2469	270	28	b.	b.	PROPN
ejpam-2469	270	29	therefore	therefore	ADV
ejpam-2469	270	30	a	a	PROPN
ejpam-2469	270	31	and	and	CCONJ
ejpam-2469	270	32	b	b	NOUN
ejpam-2469	270	33	are	be	AUX
ejpam-2469	270	34	r	r	NOUN
ejpam-2469	270	35	gi	gi	NOUN
ejpam-2469	270	36	closed	closed	ADJ
ejpam-2469	270	37	sets	set	NOUN
ejpam-2469	270	38	.	.	PUNCT
ejpam-2469	271	1	now	now	ADV
ejpam-2469	271	2	,	,	PUNCT
ejpam-2469	271	3	a∩b	a∩b	PROPN
ejpam-2469	271	4	=	=	PRON
ejpam-2469	271	5	{	{	PUNCT
ejpam-2469	271	6	a	a	PRON
ejpam-2469	271	7	}	}	PUNCT
ejpam-2469	271	8	is	be	AUX
ejpam-2469	271	9	regular	regular	ADJ
ejpam-2469	271	10	open	open	ADJ
ejpam-2469	271	11	and	and	CCONJ
ejpam-2469	271	12	cl∗(a∩b	cl∗(a∩b	NOUN
ejpam-2469	271	13	)	)	PUNCT
ejpam-2469	272	1	=	=	PRON
ejpam-2469	272	2	{	{	PUNCT
ejpam-2469	272	3	a	a	X
ejpam-2469	272	4	,	,	PUNCT
ejpam-2469	272	5	c	c	NOUN
ejpam-2469	272	6	}	}	PUNCT
ejpam-2469	272	7	6⊆	6⊆	NOUN
ejpam-2469	272	8	{	{	PUNCT
ejpam-2469	272	9	a	a	X
ejpam-2469	272	10	}	}	PUNCT
ejpam-2469	272	11	.	.	PUNCT
ejpam-2469	273	1	this	this	PRON
ejpam-2469	273	2	shows	show	VERB
ejpam-2469	273	3	that	that	SCONJ
ejpam-2469	273	4	a∩b	a∩b	PROPN
ejpam-2469	273	5	is	be	AUX
ejpam-2469	273	6	not	not	PART
ejpam-2469	273	7	an	an	DET
ejpam-2469	273	8	r	r	NOUN
ejpam-2469	273	9	gi	gi	NOUN
ejpam-2469	273	10	-closed	-close	VERB
ejpam-2469	273	11	set	set	NOUN
ejpam-2469	273	12	.	.	PUNCT
ejpam-2469	274	1	theorem	theorem	VERB
ejpam-2469	274	2	13	13	NUM
ejpam-2469	274	3	.	.	PUNCT
ejpam-2469	275	1	(	(	PUNCT
ejpam-2469	275	2	x	x	X
ejpam-2469	275	3	,	,	PUNCT
ejpam-2469	275	4	τ	τ	PROPN
ejpam-2469	275	5	,	,	PUNCT
ejpam-2469	275	6	i	i	PRON
ejpam-2469	275	7	)	)	PUNCT
ejpam-2469	275	8	be	be	VERB
ejpam-2469	275	9	a	a	DET
ejpam-2469	275	10	ideal	ideal	ADJ
ejpam-2469	275	11	topological	topological	ADJ
ejpam-2469	275	12	spaces	space	NOUN
ejpam-2469	275	13	and	and	CCONJ
ejpam-2469	275	14	a⊂	a⊂	PRON
ejpam-2469	275	15	x	x	X
ejpam-2469	275	16	.	.	PUNCT
ejpam-2469	276	1	if	if	SCONJ
ejpam-2469	276	2	a	a	PRON
ejpam-2469	276	3	is	be	AUX
ejpam-2469	276	4	w	w	NOUN
ejpam-2469	276	5	-	-	PUNCT
ejpam-2469	276	6	r	r	NOUN
ejpam-2469	276	7	gi	gi	NOUN
ejpam-2469	276	8	-closed	-close	VERB
ejpam-2469	276	9	,	,	PUNCT
ejpam-2469	276	10	b	b	NOUN
ejpam-2469	276	11	is	be	AUX
ejpam-2469	276	12	regular	regular	ADJ
ejpam-2469	276	13	closed	closed	ADJ
ejpam-2469	276	14	and	and	CCONJ
ejpam-2469	276	15	τ∗-closed[3	τ∗-closed[3	NOUN
ejpam-2469	276	16	]	]	PUNCT
ejpam-2469	276	17	then	then	ADV
ejpam-2469	276	18	,	,	PUNCT
ejpam-2469	276	19	a∩	a∩	PROPN
ejpam-2469	276	20	b	b	PROPN
ejpam-2469	276	21	is	be	AUX
ejpam-2469	276	22	a	a	DET
ejpam-2469	276	23	w	w	NOUN
ejpam-2469	276	24	-	-	PUNCT
ejpam-2469	276	25	r	r	NOUN
ejpam-2469	276	26	gi	gi	NOUN
ejpam-2469	276	27	-closed	-close	VERB
ejpam-2469	276	28	.	.	PUNCT
ejpam-2469	277	1	proof	proof	NOUN
ejpam-2469	277	2	.	.	PUNCT
ejpam-2469	278	1	let	let	VERB
ejpam-2469	278	2	u	u	PRON
ejpam-2469	278	3	be	be	AUX
ejpam-2469	278	4	a	a	DET
ejpam-2469	278	5	regular	regular	ADJ
ejpam-2469	278	6	open	open	NOUN
ejpam-2469	278	7	such	such	ADJ
ejpam-2469	278	8	that	that	SCONJ
ejpam-2469	278	9	a∩	a∩	PROPN
ejpam-2469	278	10	b	b	PROPN
ejpam-2469	278	11	⊂	⊂	PROPN
ejpam-2469	278	12	u	u	PROPN
ejpam-2469	278	13	.	.	PUNCT
ejpam-2469	279	1	then	then	ADV
ejpam-2469	279	2	we	we	PRON
ejpam-2469	279	3	have	have	VERB
ejpam-2469	279	4	a⊂	a⊂	NOUN
ejpam-2469	279	5	u	u	NOUN
ejpam-2469	279	6	∩	∩	NOUN
ejpam-2469	279	7	(	(	PUNCT
ejpam-2469	279	8	x	x	SYM
ejpam-2469	279	9	−	−	PROPN
ejpam-2469	279	10	b	b	NOUN
ejpam-2469	279	11	)	)	PUNCT
ejpam-2469	279	12	.	.	PUNCT
ejpam-2469	280	1	since	since	SCONJ
ejpam-2469	280	2	a	a	PRON
ejpam-2469	280	3	is	be	AUX
ejpam-2469	280	4	a	a	DET
ejpam-2469	280	5	w	w	NOUN
ejpam-2469	280	6	-	-	PUNCT
ejpam-2469	280	7	r	r	NOUN
ejpam-2469	280	8	gi	gi	NOUN
ejpam-2469	280	9	-closed	-close	VERB
ejpam-2469	280	10	and	and	CCONJ
ejpam-2469	280	11	b	b	NOUN
ejpam-2469	280	12	is	be	AUX
ejpam-2469	280	13	a	a	DET
ejpam-2469	280	14	τ∗-closed	τ∗-close	VERB
ejpam-2469	280	15	,	,	PUNCT
ejpam-2469	280	16	then	then	ADV
ejpam-2469	280	17	int(cl∗(a	int(cl∗(a	PROPN
ejpam-2469	280	18	)	)	PUNCT
ejpam-2469	280	19	)	)	PUNCT
ejpam-2469	281	1	⊂	⊂	PROPN
ejpam-2469	281	2	u	u	PROPN
ejpam-2469	281	3	∩	∩	NOUN
ejpam-2469	281	4	(	(	PUNCT
ejpam-2469	281	5	x	x	SYM
ejpam-2469	281	6	−	−	PROPN
ejpam-2469	281	7	b	b	NOUN
ejpam-2469	281	8	)	)	PUNCT
ejpam-2469	281	9	.	.	PUNCT
ejpam-2469	282	1	also	also	ADV
ejpam-2469	282	2	,	,	PUNCT
ejpam-2469	282	3	cl∗(a∩	cl∗(a∩	PROPN
ejpam-2469	282	4	b	b	X
ejpam-2469	282	5	)	)	PUNCT
ejpam-2469	282	6	⊂	⊂	PROPN
ejpam-2469	282	7	cl∗(a)∩	cl∗(a)∩	PROPN
ejpam-2469	282	8	cl∗(b	cl∗(b	PROPN
ejpam-2469	282	9	)	)	PUNCT
ejpam-2469	282	10	.	.	PUNCT
ejpam-2469	283	1	therefore	therefore	ADV
ejpam-2469	283	2	cl∗(a∩	cl∗(a∩	PROPN
ejpam-2469	283	3	b	b	X
ejpam-2469	283	4	)	)	PUNCT
ejpam-2469	283	5	⊂	⊂	PROPN
ejpam-2469	283	6	u	u	PROPN
ejpam-2469	283	7	∩	∩	NOUN
ejpam-2469	283	8	(	(	PUNCT
ejpam-2469	283	9	x	x	SYM
ejpam-2469	283	10	−	−	PROPN
ejpam-2469	283	11	b	b	NOUN
ejpam-2469	283	12	)	)	PUNCT
ejpam-2469	283	13	.	.	PUNCT
ejpam-2469	284	1	since	since	SCONJ
ejpam-2469	284	2	b	b	PROPN
ejpam-2469	284	3	is	be	AUX
ejpam-2469	284	4	τ∗-closed	τ∗-close	VERB
ejpam-2469	284	5	,	,	PUNCT
ejpam-2469	284	6	cl∗(a∩	cl∗(a∩	PROPN
ejpam-2469	284	7	b	b	X
ejpam-2469	284	8	)	)	PUNCT
ejpam-2469	284	9	⊂	⊂	PROPN
ejpam-2469	284	10	u	u	PROPN
ejpam-2469	284	11	.	.	PUNCT
ejpam-2469	285	1	this	this	PRON
ejpam-2469	285	2	shows	show	VERB
ejpam-2469	285	3	that	that	SCONJ
ejpam-2469	285	4	a∩	a∩	PROPN
ejpam-2469	285	5	b	b	PROPN
ejpam-2469	285	6	is	be	AUX
ejpam-2469	285	7	a	a	DET
ejpam-2469	285	8	w	w	NOUN
ejpam-2469	285	9	-	-	PUNCT
ejpam-2469	285	10	r	r	NOUN
ejpam-2469	285	11	gi	gi	NOUN
ejpam-2469	285	12	-closed	-close	VERB
ejpam-2469	285	13	.	.	PUNCT
ejpam-2469	286	1	theorem	theorem	VERB
ejpam-2469	286	2	14	14	NUM
ejpam-2469	286	3	.	.	PUNCT
ejpam-2469	287	1	let	let	VERB
ejpam-2469	287	2	(	(	PUNCT
ejpam-2469	287	3	x	x	X
ejpam-2469	287	4	,	,	PUNCT
ejpam-2469	287	5	τ	τ	PROPN
ejpam-2469	287	6	,	,	PUNCT
ejpam-2469	287	7	i	i	PRON
ejpam-2469	287	8	)	)	PUNCT
ejpam-2469	287	9	be	be	VERB
ejpam-2469	287	10	a	a	DET
ejpam-2469	287	11	ideal	ideal	ADJ
ejpam-2469	287	12	topological	topological	ADJ
ejpam-2469	287	13	spaces	space	NOUN
ejpam-2469	287	14	a	a	DET
ejpam-2469	287	15	⊂	⊂	PROPN
ejpam-2469	287	16	x	x	X
ejpam-2469	287	17	.	.	PUNCT
ejpam-2469	288	1	if	if	SCONJ
ejpam-2469	288	2	a	a	PRON
ejpam-2469	288	3	is	be	AUX
ejpam-2469	288	4	w	w	NOUN
ejpam-2469	288	5	-	-	PUNCT
ejpam-2469	288	6	r	r	NOUN
ejpam-2469	288	7	gi	gi	NOUN
ejpam-2469	288	8	-closed	-close	VERB
ejpam-2469	288	9	set	set	NOUN
ejpam-2469	288	10	then	then	ADV
ejpam-2469	288	11	(	(	PUNCT
ejpam-2469	288	12	int(cl∗(a)))−	int(cl∗(a)))−	ADP
ejpam-2469	288	13	a	a	PRON
ejpam-2469	288	14	contains	contain	VERB
ejpam-2469	288	15	no	no	DET
ejpam-2469	288	16	any	any	PRON
ejpam-2469	288	17	nonempty	nonempty	ADJ
ejpam-2469	288	18	regular	regular	ADJ
ejpam-2469	288	19	closed	close	VERB
ejpam-2469	288	20	set	set	NOUN
ejpam-2469	288	21	.	.	PUNCT
ejpam-2469	289	1	proof	proof	NOUN
ejpam-2469	289	2	.	.	PUNCT
ejpam-2469	290	1	this	this	DET
ejpam-2469	290	2	theorem	theorem	NOUN
ejpam-2469	290	3	can	can	AUX
ejpam-2469	290	4	be	be	AUX
ejpam-2469	290	5	proved	prove	VERB
ejpam-2469	290	6	similar	similar	ADJ
ejpam-2469	290	7	way	way	NOUN
ejpam-2469	290	8	of	of	ADP
ejpam-2469	290	9	theorem	theorem	ADJ
ejpam-2469	290	10	4	4	NUM
ejpam-2469	290	11	.	.	PUNCT
ejpam-2469	290	12	theorem	theorem	NOUN
ejpam-2469	290	13	15	15	NUM
ejpam-2469	290	14	.	.	PUNCT
ejpam-2469	291	1	let	let	VERB
ejpam-2469	291	2	(	(	PUNCT
ejpam-2469	291	3	x	x	X
ejpam-2469	291	4	,	,	PUNCT
ejpam-2469	291	5	τ	τ	PROPN
ejpam-2469	291	6	,	,	PUNCT
ejpam-2469	291	7	i	i	PRON
ejpam-2469	291	8	)	)	PUNCT
ejpam-2469	291	9	be	be	VERB
ejpam-2469	291	10	a	a	DET
ejpam-2469	291	11	ideal	ideal	ADJ
ejpam-2469	291	12	topological	topological	ADJ
ejpam-2469	291	13	spaces	space	NOUN
ejpam-2469	291	14	a	a	DET
ejpam-2469	291	15	subset	subset	NOUN
ejpam-2469	291	16	of	of	ADP
ejpam-2469	291	17	x	x	X
ejpam-2469	291	18	.	.	PUNCT
ejpam-2469	292	1	a	a	PRON
ejpam-2469	292	2	is	be	AUX
ejpam-2469	292	3	w	w	NOUN
ejpam-2469	292	4	-	-	PUNCT
ejpam-2469	292	5	r	r	NOUN
ejpam-2469	292	6	gi	gi	NOUN
ejpam-2469	292	7	-open	-open	NOUN
ejpam-2469	292	8	set	set	VERB
ejpam-2469	292	9	if	if	SCONJ
ejpam-2469	292	10	and	and	CCONJ
ejpam-2469	292	11	only	only	ADV
ejpam-2469	292	12	if	if	SCONJ
ejpam-2469	292	13	g	g	PROPN
ejpam-2469	292	14	⊂	⊂	PROPN
ejpam-2469	292	15	cl(int∗(a	cl(int∗(a	PROPN
ejpam-2469	292	16	)	)	PUNCT
ejpam-2469	292	17	)	)	PUNCT
ejpam-2469	293	1	whenever	whenever	SCONJ
ejpam-2469	293	2	g	g	PROPN
ejpam-2469	293	3	⊂	⊂	PROPN
ejpam-2469	293	4	a	a	PROPN
ejpam-2469	293	5	and	and	CCONJ
ejpam-2469	293	6	g	g	PROPN
ejpam-2469	293	7	is	be	AUX
ejpam-2469	293	8	regular	regular	ADV
ejpam-2469	293	9	closed	closed	ADJ
ejpam-2469	293	10	.	.	PUNCT
ejpam-2469	294	1	proof	proof	NOUN
ejpam-2469	294	2	.	.	PUNCT
ejpam-2469	295	1	necessity	necessity	NOUN
ejpam-2469	295	2	:	:	PUNCT
ejpam-2469	295	3	let	let	VERB
ejpam-2469	295	4	g	g	PROPN
ejpam-2469	295	5	⊂	⊂	PROPN
ejpam-2469	295	6	a	a	PROPN
ejpam-2469	295	7	and	and	CCONJ
ejpam-2469	295	8	g	g	PROPN
ejpam-2469	295	9	be	be	AUX
ejpam-2469	295	10	regular	regular	ADV
ejpam-2469	295	11	closed	closed	ADJ
ejpam-2469	295	12	.	.	PUNCT
ejpam-2469	296	1	then	then	ADV
ejpam-2469	296	2	(	(	PUNCT
ejpam-2469	296	3	x	x	X
ejpam-2469	296	4	−	−	NOUN
ejpam-2469	296	5	a	a	X
ejpam-2469	296	6	)	)	PUNCT
ejpam-2469	296	7	⊂	⊂	PROPN
ejpam-2469	296	8	(	(	PUNCT
ejpam-2469	296	9	x	x	X
ejpam-2469	296	10	−	−	PROPN
ejpam-2469	296	11	g	g	NOUN
ejpam-2469	296	12	)	)	PUNCT
ejpam-2469	296	13	and	and	CCONJ
ejpam-2469	296	14	(	(	PUNCT
ejpam-2469	296	15	x	x	X
ejpam-2469	296	16	−	−	PROPN
ejpam-2469	296	17	g	g	NOUN
ejpam-2469	296	18	)	)	PUNCT
ejpam-2469	296	19	regular	regular	ADJ
ejpam-2469	296	20	open	open	NOUN
ejpam-2469	296	21	.	.	PUNCT
ejpam-2469	297	1	since	since	SCONJ
ejpam-2469	297	2	(	(	PUNCT
ejpam-2469	297	3	x	x	X
ejpam-2469	297	4	−	−	NOUN
ejpam-2469	297	5	a	a	NOUN
ejpam-2469	297	6	)	)	PUNCT
ejpam-2469	297	7	is	be	AUX
ejpam-2469	297	8	a	a	DET
ejpam-2469	297	9	w	w	NOUN
ejpam-2469	297	10	-	-	PUNCT
ejpam-2469	297	11	r	r	NOUN
ejpam-2469	297	12	gi	gi	NOUN
ejpam-2469	297	13	-closed	-close	VERB
ejpam-2469	297	14	,	,	PUNCT
ejpam-2469	297	15	(	(	PUNCT
ejpam-2469	297	16	int(cl∗(x	int(cl∗(x	VERB
ejpam-2469	297	17	−	−	PROPN
ejpam-2469	297	18	a	a	NOUN
ejpam-2469	297	19	)	)	PUNCT
ejpam-2469	297	20	)	)	PUNCT
ejpam-2469	297	21	)	)	PUNCT
ejpam-2469	298	1	⊂	⊂	PRON
ejpam-2469	298	2	(	(	PUNCT
ejpam-2469	298	3	x	x	X
ejpam-2469	298	4	−	−	ADP
ejpam-2469	298	5	g	g	NOUN
ejpam-2469	298	6	)	)	PUNCT
ejpam-2469	298	7	.	.	PUNCT
ejpam-2469	299	1	hence	hence	ADV
ejpam-2469	299	2	,	,	PUNCT
ejpam-2469	299	3	we	we	PRON
ejpam-2469	299	4	get	get	VERB
ejpam-2469	299	5	g	g	PROPN
ejpam-2469	299	6	⊂	⊂	PROPN
ejpam-2469	299	7	cl(int∗(a	cl(int∗(a	PROPN
ejpam-2469	299	8	)	)	PUNCT
ejpam-2469	299	9	)	)	PUNCT
ejpam-2469	299	10	.	.	PUNCT
ejpam-2469	300	1	sufficiency	sufficiency	NOUN
ejpam-2469	300	2	:	:	PUNCT
ejpam-2469	300	3	suppose	suppose	VERB
ejpam-2469	300	4	that	that	SCONJ
ejpam-2469	300	5	g	g	PROPN
ejpam-2469	300	6	⊂	⊂	PROPN
ejpam-2469	300	7	cl(int∗(a	cl(int∗(a	PROPN
ejpam-2469	300	8	)	)	PUNCT
ejpam-2469	300	9	)	)	PUNCT
ejpam-2469	300	10	whenever	whenever	SCONJ
ejpam-2469	300	11	g	g	PROPN
ejpam-2469	300	12	⊂	⊂	PROPN
ejpam-2469	300	13	a	a	PROPN
ejpam-2469	300	14	and	and	CCONJ
ejpam-2469	300	15	g	g	PROPN
ejpam-2469	300	16	is	be	AUX
ejpam-2469	300	17	regular	regular	ADV
ejpam-2469	300	18	closed	closed	ADJ
ejpam-2469	300	19	.	.	PUNCT
ejpam-2469	301	1	let	let	VERB
ejpam-2469	301	2	(	(	PUNCT
ejpam-2469	301	3	x	x	X
ejpam-2469	301	4	−	−	PROPN
ejpam-2469	301	5	a	a	X
ejpam-2469	301	6	)	)	PUNCT
ejpam-2469	301	7	⊂	⊂	PROPN
ejpam-2469	301	8	u	u	NOUN
ejpam-2469	301	9	where	where	SCONJ
ejpam-2469	301	10	u	u	NOUN
ejpam-2469	301	11	is	be	AUX
ejpam-2469	301	12	regular	regular	ADJ
ejpam-2469	301	13	open	open	ADJ
ejpam-2469	301	14	.	.	PUNCT
ejpam-2469	302	1	then	then	ADV
ejpam-2469	302	2	(	(	PUNCT
ejpam-2469	302	3	x	x	X
ejpam-2469	302	4	−	−	PROPN
ejpam-2469	302	5	u	u	NOUN
ejpam-2469	302	6	)	)	PUNCT
ejpam-2469	302	7	⊂	⊂	PROPN
ejpam-2469	302	8	a.	a.	NOUN
ejpam-2469	302	9	by	by	ADP
ejpam-2469	302	10	hypothesis	hypothesis	NOUN
ejpam-2469	302	11	(	(	PUNCT
ejpam-2469	302	12	x	x	NOUN
ejpam-2469	302	13	−	−	PROPN
ejpam-2469	302	14	u	u	NOUN
ejpam-2469	302	15	)	)	PUNCT
ejpam-2469	302	16	⊂	⊂	PROPN
ejpam-2469	302	17	cl(int∗(a	cl(int∗(a	PROPN
ejpam-2469	302	18	)	)	PUNCT
ejpam-2469	302	19	)	)	PUNCT
ejpam-2469	303	1	and	and	CCONJ
ejpam-2469	303	2	cl(int∗(x	cl(int∗(x	VERB
ejpam-2469	303	3	−	−	PROPN
ejpam-2469	303	4	a	a	X
ejpam-2469	303	5	)	)	PUNCT
ejpam-2469	303	6	)	)	PUNCT
ejpam-2469	304	1	⊂	⊂	PROPN
ejpam-2469	304	2	u	u	PROPN
ejpam-2469	304	3	.	.	PUNCT
ejpam-2469	305	1	therefore	therefore	ADV
ejpam-2469	305	2	a	a	PRON
ejpam-2469	305	3	is	be	AUX
ejpam-2469	305	4	w	w	NOUN
ejpam-2469	305	5	-	-	PUNCT
ejpam-2469	305	6	r	r	NOUN
ejpam-2469	305	7	gi	gi	NOUN
ejpam-2469	305	8	-open	-open	NOUN
ejpam-2469	305	9	.	.	PUNCT
ejpam-2469	306	1	5	5	NUM
ejpam-2469	306	2	.	.	X
ejpam-2469	306	3	αi	αi	PROPN
ejpam-2469	306	4	-∗-normal	-∗-normal	PROPN
ejpam-2469	307	1	spaces	space	NOUN
ejpam-2469	307	2	in	in	ADP
ejpam-2469	307	3	this	this	DET
ejpam-2469	307	4	section	section	NOUN
ejpam-2469	307	5	,	,	PUNCT
ejpam-2469	307	6	we	we	PRON
ejpam-2469	307	7	talk	talk	VERB
ejpam-2469	307	8	about	about	ADP
ejpam-2469	307	9	the	the	DET
ejpam-2469	307	10	αi	αi	PROPN
ejpam-2469	307	11	-∗-normal	-∗-normal	PROPN
ejpam-2469	307	12	space	space	NOUN
ejpam-2469	307	13	and	and	CCONJ
ejpam-2469	307	14	investigate	investigate	VERB
ejpam-2469	307	15	some	some	DET
ejpam-2469	307	16	properties	property	NOUN
ejpam-2469	307	17	.	.	PUNCT
ejpam-2469	308	1	definition	definition	NOUN
ejpam-2469	308	2	4	4	NUM
ejpam-2469	308	3	.	.	PUNCT
ejpam-2469	309	1	an	an	DET
ejpam-2469	309	2	ideal	ideal	ADJ
ejpam-2469	309	3	topological	topological	ADJ
ejpam-2469	309	4	spaces	space	NOUN
ejpam-2469	309	5	(	(	PUNCT
ejpam-2469	309	6	x	x	SYM
ejpam-2469	309	7	,	,	PUNCT
ejpam-2469	309	8	τ	τ	PROPN
ejpam-2469	309	9	,	,	PUNCT
ejpam-2469	309	10	i	i	PROPN
ejpam-2469	309	11	)	)	PUNCT
ejpam-2469	309	12	is	be	AUX
ejpam-2469	309	13	said	say	VERB
ejpam-2469	309	14	to	to	PART
ejpam-2469	309	15	be	be	AUX
ejpam-2469	309	16	αi	αi	PRON
ejpam-2469	309	17	-∗-normal	-∗-normal	PUNCT
ejpam-2469	309	18	if	if	SCONJ
ejpam-2469	309	19	for	for	ADP
ejpam-2469	309	20	every	every	DET
ejpam-2469	309	21	pair	pair	NOUN
ejpam-2469	309	22	of	of	ADP
ejpam-2469	309	23	disjoint	disjoint	NOUN
ejpam-2469	309	24	regular	regular	ADJ
ejpam-2469	309	25	closed	closed	ADJ
ejpam-2469	309	26	subsets	subset	NOUN
ejpam-2469	309	27	a	a	PRON
ejpam-2469	309	28	,	,	PUNCT
ejpam-2469	309	29	b	b	PROPN
ejpam-2469	309	30	of	of	ADP
ejpam-2469	309	31	x	x	SYM
ejpam-2469	309	32	,	,	PUNCT
ejpam-2469	309	33	there	there	PRON
ejpam-2469	309	34	exist	exist	VERB
ejpam-2469	309	35	disjoint	disjoint	NOUN
ejpam-2469	309	36	αi	αi	NOUN
ejpam-2469	309	37	-∗-open	-∗-open	PUNCT
ejpam-2469	309	38	sets	set	VERB
ejpam-2469	309	39	u	u	PROPN
ejpam-2469	309	40	,	,	PUNCT
ejpam-2469	309	41	v	v	NOUN
ejpam-2469	309	42	of	of	ADP
ejpam-2469	309	43	x	x	INTJ
ejpam-2469	309	44	such	such	ADJ
ejpam-2469	309	45	that	that	PRON
ejpam-2469	309	46	a⊆	a⊆	PROPN
ejpam-2469	309	47	u	u	NOUN
ejpam-2469	309	48	and	and	CCONJ
ejpam-2469	309	49	b	b	NOUN
ejpam-2469	309	50	⊆	⊆	NUM
ejpam-2469	309	51	v	v	NOUN
ejpam-2469	309	52	.	.	PUNCT
ejpam-2469	310	1	theorem	theorem	NOUN
ejpam-2469	310	2	16	16	NUM
ejpam-2469	310	3	.	.	PUNCT
ejpam-2469	311	1	let	let	VERB
ejpam-2469	311	2	(	(	PUNCT
ejpam-2469	311	3	x	x	X
ejpam-2469	311	4	,	,	PUNCT
ejpam-2469	311	5	τ	τ	PROPN
ejpam-2469	311	6	,	,	PUNCT
ejpam-2469	311	7	i	i	PRON
ejpam-2469	311	8	)	)	PUNCT
ejpam-2469	311	9	be	be	VERB
ejpam-2469	311	10	a	a	DET
ejpam-2469	311	11	ideal	ideal	ADJ
ejpam-2469	311	12	topological	topological	ADJ
ejpam-2469	311	13	spaces	space	NOUN
ejpam-2469	311	14	and	and	CCONJ
ejpam-2469	311	15	a	a	DET
ejpam-2469	311	16	⊂	⊂	PROPN
ejpam-2469	311	17	x	x	X
ejpam-2469	311	18	,	,	PUNCT
ejpam-2469	311	19	the	the	DET
ejpam-2469	311	20	following	follow	VERB
ejpam-2469	311	21	properties	property	NOUN
ejpam-2469	311	22	are	be	AUX
ejpam-2469	311	23	equivalent	equivalent	ADJ
ejpam-2469	311	24	:	:	PUNCT
ejpam-2469	311	25	(	(	PUNCT
ejpam-2469	311	26	1	1	X
ejpam-2469	311	27	)	)	PUNCT
ejpam-2469	311	28	x	x	X
ejpam-2469	311	29	is	be	AUX
ejpam-2469	311	30	a	a	DET
ejpam-2469	311	31	αi	αi	X
ejpam-2469	311	32	-∗-normal	-∗-normal	NOUN
ejpam-2469	311	33	,	,	PUNCT
ejpam-2469	311	34	ü	ü	PROPN
ejpam-2469	311	35	karabıyık	karabıyık	PROPN
ejpam-2469	311	36	,	,	PUNCT
ejpam-2469	311	37	a	a	DET
ejpam-2469	311	38	kaymakcı	kaymakcı	NOUN
ejpam-2469	311	39	/	/	SYM
ejpam-2469	311	40	eur	eur	NOUN
ejpam-2469	311	41	.	.	PUNCT
ejpam-2469	312	1	j.	j.	PROPN
ejpam-2469	312	2	pure	pure	PROPN
ejpam-2469	312	3	appl	appl	PROPN
ejpam-2469	312	4	.	.	PROPN
ejpam-2469	312	5	math	math	PROPN
ejpam-2469	312	6	,	,	PUNCT
ejpam-2469	312	7	9	9	NUM
ejpam-2469	312	8	(	(	PUNCT
ejpam-2469	312	9	2016	2016	NUM
ejpam-2469	312	10	)	)	PUNCT
ejpam-2469	312	11	,	,	PUNCT
ejpam-2469	312	12	434	434	NUM
ejpam-2469	312	13	-	-	SYM
ejpam-2469	312	14	442	442	NUM
ejpam-2469	312	15	441	441	NUM
ejpam-2469	312	16	(	(	PUNCT
ejpam-2469	312	17	2	2	NUM
ejpam-2469	312	18	)	)	PUNCT
ejpam-2469	312	19	for	for	ADP
ejpam-2469	312	20	any	any	DET
ejpam-2469	312	21	disjoint	disjoint	ADJ
ejpam-2469	312	22	regular	regular	ADJ
ejpam-2469	312	23	closed	closed	ADJ
ejpam-2469	312	24	sets	set	NOUN
ejpam-2469	312	25	a	a	PRON
ejpam-2469	312	26	and	and	CCONJ
ejpam-2469	312	27	b	b	NOUN
ejpam-2469	312	28	,	,	PUNCT
ejpam-2469	312	29	there	there	PRON
ejpam-2469	312	30	exist	exist	VERB
ejpam-2469	312	31	disjoint	disjoint	NOUN
ejpam-2469	312	32	saw	saw	NOUN
ejpam-2469	312	33	-	-	PUNCT
ejpam-2469	312	34	ir	ir	NOUN
ejpam-2469	312	35	g	g	PROPN
ejpam-2469	312	36	-open	-open	PROPN
ejpam-2469	312	37	sets	set	VERB
ejpam-2469	312	38	u	u	NOUN
ejpam-2469	312	39	and	and	CCONJ
ejpam-2469	312	40	v	v	NOUN
ejpam-2469	312	41	of	of	ADP
ejpam-2469	312	42	x	x	PUNCT
ejpam-2469	312	43	such	such	ADJ
ejpam-2469	312	44	that	that	PRON
ejpam-2469	312	45	a⊆	a⊆	PROPN
ejpam-2469	312	46	u	u	NOUN
ejpam-2469	312	47	and	and	CCONJ
ejpam-2469	312	48	b	b	NOUN
ejpam-2469	312	49	⊆	⊆	NUM
ejpam-2469	312	50	v	v	NOUN
ejpam-2469	312	51	,	,	PUNCT
ejpam-2469	312	52	(	(	PUNCT
ejpam-2469	312	53	3	3	X
ejpam-2469	312	54	)	)	PUNCT
ejpam-2469	312	55	for	for	ADP
ejpam-2469	312	56	any	any	DET
ejpam-2469	312	57	regular	regular	ADJ
ejpam-2469	312	58	closed	closed	ADJ
ejpam-2469	312	59	set	set	NOUN
ejpam-2469	312	60	a	a	PRON
ejpam-2469	312	61	and	and	CCONJ
ejpam-2469	312	62	any	any	DET
ejpam-2469	312	63	regular	regular	ADJ
ejpam-2469	312	64	open	open	ADJ
ejpam-2469	312	65	set	set	NOUN
ejpam-2469	312	66	b	b	PROPN
ejpam-2469	312	67	containing	contain	VERB
ejpam-2469	312	68	a	a	PRON
ejpam-2469	312	69	,	,	PUNCT
ejpam-2469	312	70	there	there	PRON
ejpam-2469	312	71	exists	exist	VERB
ejpam-2469	312	72	a	a	DET
ejpam-2469	312	73	saw	saw	NOUN
ejpam-2469	312	74	-	-	PUNCT
ejpam-2469	312	75	ir	ir	NOUN
ejpam-2469	312	76	g	g	PROPN
ejpam-2469	312	77	open	open	ADJ
ejpam-2469	312	78	set	set	VERB
ejpam-2469	312	79	u	u	PRON
ejpam-2469	312	80	such	such	ADJ
ejpam-2469	312	81	that	that	SCONJ
ejpam-2469	312	82	a⊆	a⊆	PROPN
ejpam-2469	312	83	u	u	NOUN
ejpam-2469	312	84	⊆	⊆	NUM
ejpam-2469	312	85	cl∗(int(cl(a	cl∗(int(cl(a	NOUN
ejpam-2469	312	86	)	)	PUNCT
ejpam-2469	312	87	)	)	PUNCT
ejpam-2469	312	88	)	)	PUNCT
ejpam-2469	313	1	⊆	⊆	NUM
ejpam-2469	313	2	b.	b.	NOUN
ejpam-2469	313	3	proof	proof	NOUN
ejpam-2469	313	4	.	.	PUNCT
ejpam-2469	314	1	(	(	PUNCT
ejpam-2469	314	2	1)⇒	1)⇒	NUM
ejpam-2469	314	3	(	(	PUNCT
ejpam-2469	314	4	2	2	NUM
ejpam-2469	314	5	)	)	PUNCT
ejpam-2469	314	6	the	the	DET
ejpam-2469	314	7	proof	proof	NOUN
ejpam-2469	314	8	is	be	AUX
ejpam-2469	314	9	obvious	obvious	ADJ
ejpam-2469	314	10	.	.	PUNCT
ejpam-2469	315	1	(	(	PUNCT
ejpam-2469	315	2	2)⇒	2)⇒	NUM
ejpam-2469	315	3	(	(	PUNCT
ejpam-2469	315	4	3	3	X
ejpam-2469	315	5	)	)	PUNCT
ejpam-2469	315	6	let	let	VERB
ejpam-2469	315	7	a	a	PRON
ejpam-2469	315	8	be	be	AUX
ejpam-2469	315	9	a	a	DET
ejpam-2469	315	10	regular	regular	ADJ
ejpam-2469	315	11	closed	closed	ADJ
ejpam-2469	315	12	and	and	CCONJ
ejpam-2469	315	13	b	b	NOUN
ejpam-2469	315	14	be	be	AUX
ejpam-2469	315	15	a	a	DET
ejpam-2469	315	16	regular	regular	ADJ
ejpam-2469	315	17	open	open	ADJ
ejpam-2469	315	18	subset	subset	NOUN
ejpam-2469	315	19	of	of	ADP
ejpam-2469	315	20	x	x	PRON
ejpam-2469	315	21	,	,	PUNCT
ejpam-2469	316	1	such	such	ADJ
ejpam-2469	316	2	that	that	SCONJ
ejpam-2469	316	3	a⊆	a⊆	PROPN
ejpam-2469	316	4	b.	b.	PROPN
ejpam-2469	316	5	then	then	ADV
ejpam-2469	316	6	a	a	PRON
ejpam-2469	316	7	and	and	CCONJ
ejpam-2469	316	8	(	(	PUNCT
ejpam-2469	316	9	x	x	NOUN
ejpam-2469	316	10	−	−	PROPN
ejpam-2469	316	11	b	b	X
ejpam-2469	316	12	)	)	PUNCT
ejpam-2469	316	13	are	be	AUX
ejpam-2469	316	14	disjoint	disjoint	NOUN
ejpam-2469	316	15	regular	regular	ADJ
ejpam-2469	316	16	closed	close	VERB
ejpam-2469	316	17	set	set	NOUN
ejpam-2469	316	18	of	of	ADP
ejpam-2469	316	19	x	x	X
ejpam-2469	316	20	.	.	PUNCT
ejpam-2469	317	1	by	by	ADP
ejpam-2469	317	2	the	the	DET
ejpam-2469	317	3	hypothesis	hypothesis	NOUN
ejpam-2469	317	4	,	,	PUNCT
ejpam-2469	317	5	there	there	PRON
ejpam-2469	317	6	exist	exist	VERB
ejpam-2469	317	7	a	a	DET
ejpam-2469	317	8	saw	saw	NOUN
ejpam-2469	317	9	-	-	PUNCT
ejpam-2469	317	10	ir	ir	NOUN
ejpam-2469	317	11	g	g	PROPN
ejpam-2469	317	12	-open	-open	PROPN
ejpam-2469	317	13	sets	set	VERB
ejpam-2469	317	14	u	u	NOUN
ejpam-2469	317	15	and	and	CCONJ
ejpam-2469	317	16	v	v	NOUN
ejpam-2469	317	17	of	of	ADP
ejpam-2469	317	18	x	x	PUNCT
ejpam-2469	317	19	such	such	ADJ
ejpam-2469	317	20	that	that	SCONJ
ejpam-2469	317	21	a	a	DET
ejpam-2469	317	22	⊆	⊆	NUM
ejpam-2469	317	23	u	u	NOUN
ejpam-2469	317	24	and	and	CCONJ
ejpam-2469	317	25	(	(	PUNCT
ejpam-2469	317	26	x	x	PROPN
ejpam-2469	317	27	−	−	PROPN
ejpam-2469	317	28	b	b	NUM
ejpam-2469	317	29	)	)	PUNCT
ejpam-2469	317	30	⊆	⊆	NUM
ejpam-2469	317	31	v	v	NOUN
ejpam-2469	317	32	.	.	PUNCT
ejpam-2469	318	1	since	since	SCONJ
ejpam-2469	318	2	v	v	NOUN
ejpam-2469	318	3	is	be	AUX
ejpam-2469	318	4	saw	saw	NOUN
ejpam-2469	318	5	-	-	PUNCT
ejpam-2469	318	6	ir	ir	NOUN
ejpam-2469	318	7	g	g	PROPN
ejpam-2469	318	8	-open	-open	NOUN
ejpam-2469	318	9	set	set	NOUN
ejpam-2469	318	10	,	,	PUNCT
ejpam-2469	318	11	(	(	PUNCT
ejpam-2469	318	12	x	x	X
ejpam-2469	318	13	−	−	PROPN
ejpam-2469	318	14	b	b	X
ejpam-2469	318	15	)	)	PUNCT
ejpam-2469	318	16	⊆	⊆	NUM
ejpam-2469	318	17	int∗(cl(int(v	int∗(cl(int(v	NOUN
ejpam-2469	318	18	)	)	PUNCT
ejpam-2469	318	19	)	)	PUNCT
ejpam-2469	318	20	)	)	PUNCT
ejpam-2469	318	21	.	.	PUNCT
ejpam-2469	319	1	hence	hence	ADV
ejpam-2469	319	2	we	we	PRON
ejpam-2469	319	3	have	have	VERB
ejpam-2469	319	4	u	u	NOUN
ejpam-2469	319	5	∩	∩	NOUN
ejpam-2469	319	6	int∗(cl(int(v	int∗(cl(int(v	X
ejpam-2469	319	7	)	)	PUNCT
ejpam-2469	319	8	)	)	PUNCT
ejpam-2469	319	9	)	)	PUNCT
ejpam-2469	320	1	=	=	PUNCT
ejpam-2469	320	2	;	;	PUNCT
ejpam-2469	320	3	.	.	PUNCT
ejpam-2469	321	1	so	so	ADV
ejpam-2469	321	2	,	,	PUNCT
ejpam-2469	321	3	we	we	PRON
ejpam-2469	321	4	obtain	obtain	VERB
ejpam-2469	321	5	cl∗(int(cl(u	cl∗(int(cl(u	NOUN
ejpam-2469	321	6	)	)	PUNCT
ejpam-2469	321	7	)	)	PUNCT
ejpam-2469	321	8	)	)	PUNCT
ejpam-2469	322	1	⊆	⊆	NUM
ejpam-2469	322	2	cl∗(int(cl(x	cl∗(int(cl(x	PROPN
ejpam-2469	322	3	−	−	PROPN
ejpam-2469	322	4	v	v	NOUN
ejpam-2469	322	5	)	)	PUNCT
ejpam-2469	322	6	)	)	PUNCT
ejpam-2469	322	7	)	)	PUNCT
ejpam-2469	322	8	.	.	PUNCT
ejpam-2469	323	1	this	this	PRON
ejpam-2469	323	2	shows	show	VERB
ejpam-2469	323	3	that	that	PRON
ejpam-2469	323	4	a⊆	a⊆	VERB
ejpam-2469	323	5	u	u	NOUN
ejpam-2469	323	6	⊆	⊆	NUM
ejpam-2469	323	7	cl∗(int(cl(u	cl∗(int(cl(u	NOUN
ejpam-2469	323	8	)	)	PUNCT
ejpam-2469	323	9	)	)	PUNCT
ejpam-2469	323	10	)	)	PUNCT
ejpam-2469	324	1	⊆	⊆	NUM
ejpam-2469	324	2	b.	b.	NOUN
ejpam-2469	324	3	(	(	PUNCT
ejpam-2469	324	4	3)⇒	3)⇒	NUM
ejpam-2469	324	5	(	(	PUNCT
ejpam-2469	324	6	1	1	NUM
ejpam-2469	324	7	)	)	PUNCT
ejpam-2469	324	8	let	let	VERB
ejpam-2469	324	9	a	a	DET
ejpam-2469	324	10	and	and	CCONJ
ejpam-2469	324	11	b	b	NOUN
ejpam-2469	324	12	any	any	DET
ejpam-2469	324	13	disjoint	disjoint	ADJ
ejpam-2469	324	14	regular	regular	ADJ
ejpam-2469	324	15	closed	close	VERB
ejpam-2469	324	16	set	set	NOUN
ejpam-2469	324	17	of	of	ADP
ejpam-2469	324	18	x	x	X
ejpam-2469	324	19	.	.	PUNCT
ejpam-2469	325	1	then	then	ADV
ejpam-2469	325	2	,	,	PUNCT
ejpam-2469	325	3	a⊆	a⊆	VERB
ejpam-2469	325	4	(	(	PUNCT
ejpam-2469	325	5	x	x	NOUN
ejpam-2469	325	6	−	−	PROPN
ejpam-2469	325	7	b	b	NOUN
ejpam-2469	325	8	)	)	PUNCT
ejpam-2469	325	9	and	and	CCONJ
ejpam-2469	325	10	(	(	PUNCT
ejpam-2469	325	11	x	x	X
ejpam-2469	325	12	−	−	PROPN
ejpam-2469	325	13	b	b	X
ejpam-2469	325	14	)	)	PUNCT
ejpam-2469	325	15	is	be	AUX
ejpam-2469	325	16	regular	regular	ADJ
ejpam-2469	325	17	open	open	ADJ
ejpam-2469	325	18	set	set	NOUN
ejpam-2469	325	19	.	.	PUNCT
ejpam-2469	326	1	hence	hence	ADV
ejpam-2469	326	2	there	there	PRON
ejpam-2469	326	3	exist	exist	VERB
ejpam-2469	326	4	a	a	DET
ejpam-2469	326	5	saw	saw	NOUN
ejpam-2469	326	6	-	-	PUNCT
ejpam-2469	326	7	ir	ir	NOUN
ejpam-2469	326	8	g	g	PROPN
ejpam-2469	326	9	-open	-open	NOUN
ejpam-2469	326	10	set	set	VERB
ejpam-2469	326	11	g	g	NOUN
ejpam-2469	326	12	of	of	ADP
ejpam-2469	326	13	x	x	INTJ
ejpam-2469	326	14	such	such	ADJ
ejpam-2469	326	15	that	that	PRON
ejpam-2469	326	16	a⊆	a⊆	VERB
ejpam-2469	326	17	g	g	ADP
ejpam-2469	326	18	⊆	⊆	NUM
ejpam-2469	326	19	cl∗(int(cl(g	cl∗(int(cl(g	NOUN
ejpam-2469	326	20	)	)	PUNCT
ejpam-2469	326	21	)	)	PUNCT
ejpam-2469	326	22	)	)	PUNCT
ejpam-2469	327	1	⊆	⊆	X
ejpam-2469	327	2	(	(	PUNCT
ejpam-2469	327	3	x	x	NOUN
ejpam-2469	327	4	−	−	PROPN
ejpam-2469	327	5	b	b	NOUN
ejpam-2469	327	6	)	)	PUNCT
ejpam-2469	327	7	.	.	PUNCT
ejpam-2469	328	1	definition	definition	NOUN
ejpam-2469	328	2	5	5	NUM
ejpam-2469	328	3	.	.	PUNCT
ejpam-2469	329	1	a	a	DET
ejpam-2469	329	2	function	function	NOUN
ejpam-2469	329	3	f	f	NOUN
ejpam-2469	329	4	:	:	PUNCT
ejpam-2469	329	5	(	(	PUNCT
ejpam-2469	329	6	x	x	X
ejpam-2469	329	7	,	,	PUNCT
ejpam-2469	329	8	τ	τ	PROPN
ejpam-2469	329	9	,	,	PUNCT
ejpam-2469	329	10	i)→	i)→	PROPN
ejpam-2469	329	11	(	(	PUNCT
ejpam-2469	329	12	y,ϕ	y,ϕ	NOUN
ejpam-2469	329	13	)	)	PUNCT
ejpam-2469	329	14	is	be	AUX
ejpam-2469	329	15	said	say	VERB
ejpam-2469	329	16	to	to	PART
ejpam-2469	329	17	be	be	AUX
ejpam-2469	329	18	saw	see	VERB
ejpam-2469	329	19	-	-	PUNCT
ejpam-2469	329	20	ir	ir	NOUN
ejpam-2469	329	21	g	g	NOUN
ejpam-2469	329	22	-continuous	-continuous	ADJ
ejpam-2469	329	23	if	if	SCONJ
ejpam-2469	329	24	for	for	ADP
ejpam-2469	329	25	every	every	DET
ejpam-2469	329	26	closed	close	VERB
ejpam-2469	329	27	set	set	VERB
ejpam-2469	329	28	f	f	PROPN
ejpam-2469	329	29	in	in	ADP
ejpam-2469	329	30	y	y	PROPN
ejpam-2469	329	31	,	,	PUNCT
ejpam-2469	329	32	f	f	PROPN
ejpam-2469	329	33	−1(f	−1(f	PROPN
ejpam-2469	329	34	)	)	PUNCT
ejpam-2469	329	35	saw	saw	NOUN
ejpam-2469	329	36	-	-	PUNCT
ejpam-2469	329	37	ir	ir	NOUN
ejpam-2469	329	38	g	g	NOUN
ejpam-2469	329	39	-closed	-close	VERB
ejpam-2469	329	40	in	in	ADP
ejpam-2469	329	41	x	x	PROPN
ejpam-2469	329	42	.	.	PUNCT
ejpam-2469	330	1	definition	definition	NOUN
ejpam-2469	330	2	6	6	NUM
ejpam-2469	330	3	.	.	PUNCT
ejpam-2469	331	1	a	a	DET
ejpam-2469	331	2	function	function	NOUN
ejpam-2469	331	3	f	f	NOUN
ejpam-2469	331	4	:	:	PUNCT
ejpam-2469	331	5	(	(	PUNCT
ejpam-2469	331	6	x	x	X
ejpam-2469	331	7	,	,	PUNCT
ejpam-2469	331	8	τ	τ	PROPN
ejpam-2469	331	9	,	,	PUNCT
ejpam-2469	331	10	i)→	i)→	PROPN
ejpam-2469	331	11	(	(	PUNCT
ejpam-2469	331	12	y,ϕ	y,ϕ	PROPN
ejpam-2469	331	13	,	,	PUNCT
ejpam-2469	331	14	j	j	NOUN
ejpam-2469	331	15	)	)	PUNCT
ejpam-2469	331	16	is	be	AUX
ejpam-2469	331	17	called	call	VERB
ejpam-2469	331	18	saw	saw	NOUN
ejpam-2469	331	19	-	-	PUNCT
ejpam-2469	331	20	ir	ir	NOUN
ejpam-2469	331	21	g	g	PROPN
ejpam-2469	331	22	-irresolute	-irresolute	PROPN
ejpam-2469	331	23	if	if	SCONJ
ejpam-2469	331	24	for	for	SCONJ
ejpam-2469	331	25	every	every	DET
ejpam-2469	331	26	saw	saw	NOUN
ejpam-2469	331	27	-	-	PUNCT
ejpam-2469	331	28	jr	jr	PROPN
ejpam-2469	331	29	g	g	PROPN
ejpam-2469	331	30	closed	close	VERB
ejpam-2469	331	31	in	in	ADP
ejpam-2469	331	32	y	y	PROPN
ejpam-2469	331	33	,	,	PUNCT
ejpam-2469	331	34	f	f	PROPN
ejpam-2469	331	35	−1(f	−1(f	PROPN
ejpam-2469	331	36	)	)	PUNCT
ejpam-2469	331	37	saw	saw	NOUN
ejpam-2469	331	38	-	-	PUNCT
ejpam-2469	331	39	ir	ir	NOUN
ejpam-2469	331	40	g	g	NOUN
ejpam-2469	331	41	-closed	-close	VERB
ejpam-2469	331	42	in	in	ADP
ejpam-2469	331	43	x	x	X
ejpam-2469	331	44	.	.	PUNCT
ejpam-2469	331	45	theorem	theorem	PROPN
ejpam-2469	331	46	17	17	NUM
ejpam-2469	331	47	.	.	PUNCT
ejpam-2469	332	1	let	let	VERB
ejpam-2469	332	2	f	f	NOUN
ejpam-2469	332	3	:	:	PUNCT
ejpam-2469	332	4	x	x	X
ejpam-2469	332	5	→	→	SYM
ejpam-2469	332	6	y	y	X
ejpam-2469	332	7	be	be	AUX
ejpam-2469	332	8	a	a	DET
ejpam-2469	332	9	saw	saw	NOUN
ejpam-2469	332	10	-	-	PUNCT
ejpam-2469	332	11	ir	ir	NOUN
ejpam-2469	332	12	g	g	PROPN
ejpam-2469	332	13	-continuous	-continuous	ADJ
ejpam-2469	332	14	regular	regular	ADJ
ejpam-2469	332	15	closed	closed	ADJ
ejpam-2469	332	16	and	and	CCONJ
ejpam-2469	332	17	injective	injective	ADJ
ejpam-2469	332	18	function	function	NOUN
ejpam-2469	332	19	.	.	PUNCT
ejpam-2469	333	1	if	if	SCONJ
ejpam-2469	333	2	y	y	PROPN
ejpam-2469	333	3	is	be	AUX
ejpam-2469	333	4	normal	normal	ADJ
ejpam-2469	333	5	,	,	PUNCT
ejpam-2469	333	6	then	then	ADV
ejpam-2469	333	7	x	x	PUNCT
ejpam-2469	333	8	is	be	AUX
ejpam-2469	333	9	αi	αi	X
ejpam-2469	333	10	-∗-normal	-∗-normal	ADJ
ejpam-2469	333	11	.	.	PUNCT
ejpam-2469	334	1	proof	proof	NOUN
ejpam-2469	334	2	.	.	PUNCT
ejpam-2469	335	1	let	let	VERB
ejpam-2469	335	2	a	a	DET
ejpam-2469	335	3	and	and	CCONJ
ejpam-2469	335	4	b	b	NOUN
ejpam-2469	335	5	any	any	DET
ejpam-2469	335	6	disjoint	disjoint	ADJ
ejpam-2469	335	7	regular	regular	ADJ
ejpam-2469	335	8	closed	close	VERB
ejpam-2469	335	9	set	set	NOUN
ejpam-2469	335	10	of	of	ADP
ejpam-2469	335	11	x	x	X
ejpam-2469	335	12	.	.	PUNCT
ejpam-2469	336	1	since	since	SCONJ
ejpam-2469	336	2	f	f	PROPN
ejpam-2469	336	3	is	be	AUX
ejpam-2469	336	4	regular	regular	ADJ
ejpam-2469	336	5	closed	closed	ADJ
ejpam-2469	336	6	injection	injection	NOUN
ejpam-2469	336	7	,	,	PUNCT
ejpam-2469	336	8	f	f	PROPN
ejpam-2469	336	9	(	(	PUNCT
ejpam-2469	336	10	a	a	NOUN
ejpam-2469	336	11	)	)	PUNCT
ejpam-2469	336	12	and	and	CCONJ
ejpam-2469	336	13	f	f	PROPN
ejpam-2469	336	14	(	(	PUNCT
ejpam-2469	336	15	b	b	NOUN
ejpam-2469	336	16	)	)	PUNCT
ejpam-2469	336	17	are	be	AUX
ejpam-2469	336	18	disjoint	disjoint	NOUN
ejpam-2469	336	19	regular	regular	ADJ
ejpam-2469	336	20	closed	closed	ADJ
ejpam-2469	336	21	sets	set	NOUN
ejpam-2469	336	22	of	of	ADP
ejpam-2469	336	23	y	y	PROPN
ejpam-2469	336	24	.	.	PUNCT
ejpam-2469	337	1	by	by	ADP
ejpam-2469	337	2	the	the	DET
ejpam-2469	337	3	normality	normality	NOUN
ejpam-2469	337	4	of	of	ADP
ejpam-2469	337	5	y	y	PROPN
ejpam-2469	337	6	,	,	PUNCT
ejpam-2469	337	7	there	there	PRON
ejpam-2469	337	8	exist	exist	VERB
ejpam-2469	337	9	disjoint	disjoint	ADJ
ejpam-2469	337	10	open	open	ADJ
ejpam-2469	337	11	sets	set	NOUN
ejpam-2469	337	12	u	u	NOUN
ejpam-2469	337	13	and	and	CCONJ
ejpam-2469	337	14	v	v	ADP
ejpam-2469	337	15	such	such	ADJ
ejpam-2469	337	16	that	that	SCONJ
ejpam-2469	337	17	f	f	PROPN
ejpam-2469	337	18	(	(	PUNCT
ejpam-2469	337	19	a	a	NOUN
ejpam-2469	337	20	)	)	PUNCT
ejpam-2469	337	21	⊆	⊆	NUM
ejpam-2469	337	22	u	u	NOUN
ejpam-2469	337	23	and	and	CCONJ
ejpam-2469	337	24	f	f	PROPN
ejpam-2469	337	25	(	(	PUNCT
ejpam-2469	337	26	b	b	NOUN
ejpam-2469	337	27	)	)	PUNCT
ejpam-2469	337	28	⊆	⊆	NUM
ejpam-2469	337	29	v	v	NOUN
ejpam-2469	337	30	.	.	PUNCT
ejpam-2469	338	1	since	since	SCONJ
ejpam-2469	338	2	f	f	PROPN
ejpam-2469	338	3	is	be	AUX
ejpam-2469	338	4	saw	see	VERB
ejpam-2469	338	5	-	-	PUNCT
ejpam-2469	338	6	ir	ir	NOUN
ejpam-2469	338	7	g	g	PROPN
ejpam-2469	338	8	-continuous	-continuous	ADJ
ejpam-2469	338	9	,	,	PUNCT
ejpam-2469	338	10	then	then	ADV
ejpam-2469	338	11	f	f	PROPN
ejpam-2469	338	12	−1(u	−1(u	NOUN
ejpam-2469	338	13	)	)	PUNCT
ejpam-2469	338	14	and	and	CCONJ
ejpam-2469	338	15	f	f	PROPN
ejpam-2469	338	16	−1(v	−1(v	PROPN
ejpam-2469	338	17	)	)	PUNCT
ejpam-2469	338	18	are	be	AUX
ejpam-2469	338	19	saw	see	VERB
ejpam-2469	338	20	-	-	PUNCT
ejpam-2469	338	21	ir	ir	NOUN
ejpam-2469	338	22	g	g	PROPN
ejpam-2469	338	23	-open	-open	NOUN
ejpam-2469	338	24	sets	set	NOUN
ejpam-2469	338	25	such	such	ADJ
ejpam-2469	338	26	that	that	PRON
ejpam-2469	338	27	a⊆	a⊆	PROPN
ejpam-2469	338	28	f	f	NOUN
ejpam-2469	338	29	−1(u	−1(u	NOUN
ejpam-2469	338	30	)	)	PUNCT
ejpam-2469	338	31	and	and	CCONJ
ejpam-2469	338	32	b	b	X
ejpam-2469	338	33	⊆	⊆	NUM
ejpam-2469	338	34	f	f	PROPN
ejpam-2469	338	35	−1(v	−1(v	PROPN
ejpam-2469	338	36	)	)	PUNCT
ejpam-2469	338	37	.	.	PUNCT
ejpam-2469	339	1	therefore	therefore	ADV
ejpam-2469	339	2	x	x	X
ejpam-2469	339	3	is	be	AUX
ejpam-2469	339	4	αi	αi	PRON
ejpam-2469	339	5	-∗-normal	-∗-normal	NUM
ejpam-2469	339	6	by	by	ADP
ejpam-2469	339	7	theorem	theorem	NOUN
ejpam-2469	339	8	16	16	NUM
ejpam-2469	339	9	.	.	PUNCT
ejpam-2469	340	1	theorem	theorem	NOUN
ejpam-2469	340	2	18	18	NUM
ejpam-2469	340	3	.	.	PUNCT
ejpam-2469	341	1	let	let	VERB
ejpam-2469	341	2	f	f	NOUN
ejpam-2469	341	3	:	:	PUNCT
ejpam-2469	341	4	x	x	X
ejpam-2469	341	5	→	→	SYM
ejpam-2469	341	6	y	y	X
ejpam-2469	341	7	be	be	AUX
ejpam-2469	341	8	a	a	DET
ejpam-2469	341	9	saw	saw	NOUN
ejpam-2469	341	10	-	-	PUNCT
ejpam-2469	341	11	ir	ir	NOUN
ejpam-2469	341	12	g	g	PROPN
ejpam-2469	341	13	-irresolute	-irresolute	PROPN
ejpam-2469	341	14	regular	regular	ADJ
ejpam-2469	341	15	closed	closed	ADJ
ejpam-2469	341	16	injection	injection	NOUN
ejpam-2469	341	17	.	.	PUNCT
ejpam-2469	342	1	if	if	SCONJ
ejpam-2469	342	2	y	y	PROPN
ejpam-2469	342	3	is	be	AUX
ejpam-2469	342	4	αi	αi	ADV
ejpam-2469	342	5	-∗-normal	-∗-normal	ADJ
ejpam-2469	342	6	,	,	PUNCT
ejpam-2469	342	7	then	then	ADV
ejpam-2469	342	8	x	x	PUNCT
ejpam-2469	342	9	is	be	AUX
ejpam-2469	342	10	αi	αi	X
ejpam-2469	342	11	-∗-normal	-∗-normal	ADJ
ejpam-2469	342	12	.	.	PUNCT
ejpam-2469	343	1	proof	proof	NOUN
ejpam-2469	343	2	.	.	PUNCT
ejpam-2469	344	1	let	let	VERB
ejpam-2469	344	2	a	a	DET
ejpam-2469	344	3	and	and	CCONJ
ejpam-2469	344	4	b	b	NOUN
ejpam-2469	344	5	any	any	DET
ejpam-2469	344	6	disjoint	disjoint	ADJ
ejpam-2469	344	7	regular	regular	ADJ
ejpam-2469	344	8	closed	close	VERB
ejpam-2469	344	9	set	set	NOUN
ejpam-2469	344	10	of	of	ADP
ejpam-2469	344	11	x	x	X
ejpam-2469	344	12	.	.	PUNCT
ejpam-2469	345	1	since	since	SCONJ
ejpam-2469	345	2	f	f	PROPN
ejpam-2469	345	3	is	be	AUX
ejpam-2469	345	4	regular	regular	ADJ
ejpam-2469	345	5	closed	closed	ADJ
ejpam-2469	345	6	injection	injection	NOUN
ejpam-2469	345	7	,	,	PUNCT
ejpam-2469	345	8	f	f	PROPN
ejpam-2469	345	9	(	(	PUNCT
ejpam-2469	345	10	a	a	NOUN
ejpam-2469	345	11	)	)	PUNCT
ejpam-2469	345	12	and	and	CCONJ
ejpam-2469	345	13	f	f	PROPN
ejpam-2469	345	14	(	(	PUNCT
ejpam-2469	345	15	b	b	NOUN
ejpam-2469	345	16	)	)	PUNCT
ejpam-2469	345	17	are	be	AUX
ejpam-2469	345	18	disjoint	disjoint	NOUN
ejpam-2469	345	19	regular	regular	ADJ
ejpam-2469	345	20	closed	closed	ADJ
ejpam-2469	345	21	sets	set	NOUN
ejpam-2469	345	22	of	of	ADP
ejpam-2469	345	23	y	y	PROPN
ejpam-2469	345	24	.	.	PUNCT
ejpam-2469	346	1	since	since	SCONJ
ejpam-2469	346	2	y	y	PROPN
ejpam-2469	346	3	is	be	AUX
ejpam-2469	346	4	αi	αi	X
ejpam-2469	346	5	-∗-normal	-∗-normal	ADJ
ejpam-2469	346	6	,	,	PUNCT
ejpam-2469	346	7	by	by	ADP
ejpam-2469	346	8	theorem	theorem	NOUN
ejpam-2469	346	9	16	16	NUM
ejpam-2469	346	10	there	there	PRON
ejpam-2469	346	11	exist	exist	VERB
ejpam-2469	346	12	disjoint	disjoint	NOUN
ejpam-2469	346	13	saw	saw	NOUN
ejpam-2469	346	14	-	-	PUNCT
ejpam-2469	346	15	ir	ir	NOUN
ejpam-2469	346	16	g	g	PROPN
ejpam-2469	346	17	-open	-open	PROPN
ejpam-2469	346	18	u	u	NOUN
ejpam-2469	346	19	and	and	CCONJ
ejpam-2469	346	20	v	v	ADP
ejpam-2469	346	21	such	such	ADJ
ejpam-2469	346	22	that	that	SCONJ
ejpam-2469	346	23	f	f	PROPN
ejpam-2469	346	24	(	(	PUNCT
ejpam-2469	346	25	a	a	NOUN
ejpam-2469	346	26	)	)	PUNCT
ejpam-2469	346	27	⊆	⊆	NUM
ejpam-2469	346	28	u	u	NOUN
ejpam-2469	346	29	and	and	CCONJ
ejpam-2469	346	30	f	f	PROPN
ejpam-2469	346	31	(	(	PUNCT
ejpam-2469	346	32	b	b	NOUN
ejpam-2469	346	33	)	)	PUNCT
ejpam-2469	346	34	⊆	⊆	NUM
ejpam-2469	346	35	v	v	NOUN
ejpam-2469	346	36	since	since	SCONJ
ejpam-2469	346	37	f	f	PROPN
ejpam-2469	346	38	is	be	AUX
ejpam-2469	346	39	saw	see	VERB
ejpam-2469	346	40	-	-	PUNCT
ejpam-2469	346	41	ir	ir	NOUN
ejpam-2469	346	42	g	g	PROPN
ejpam-2469	346	43	irresolute	irresolute	ADJ
ejpam-2469	346	44	,	,	PUNCT
ejpam-2469	346	45	then	then	ADV
ejpam-2469	346	46	f	f	PROPN
ejpam-2469	346	47	−1(u	−1(u	NOUN
ejpam-2469	346	48	)	)	PUNCT
ejpam-2469	346	49	and	and	CCONJ
ejpam-2469	346	50	f	f	PROPN
ejpam-2469	346	51	−1(v	−1(v	PROPN
ejpam-2469	346	52	)	)	PUNCT
ejpam-2469	346	53	are	be	AUX
ejpam-2469	346	54	saw	see	VERB
ejpam-2469	346	55	-	-	PUNCT
ejpam-2469	346	56	ir	ir	NOUN
ejpam-2469	346	57	g	g	PROPN
ejpam-2469	346	58	-open	-open	NOUN
ejpam-2469	346	59	sets	set	NOUN
ejpam-2469	346	60	such	such	ADJ
ejpam-2469	346	61	that	that	PRON
ejpam-2469	346	62	a⊆	a⊆	PROPN
ejpam-2469	346	63	f	f	NOUN
ejpam-2469	346	64	−1(u	−1(u	NOUN
ejpam-2469	346	65	)	)	PUNCT
ejpam-2469	346	66	and	and	CCONJ
ejpam-2469	346	67	b	b	X
ejpam-2469	346	68	⊆	⊆	NUM
ejpam-2469	346	69	f	f	PROPN
ejpam-2469	346	70	−1(v	−1(v	PROPN
ejpam-2469	346	71	)	)	PUNCT
ejpam-2469	346	72	.	.	PUNCT
ejpam-2469	347	1	therefore	therefore	ADV
ejpam-2469	347	2	x	x	X
ejpam-2469	347	3	is	be	AUX
ejpam-2469	347	4	αi	αi	PROPN
ejpam-2469	347	5	-∗-normal	-∗-normal	PROPN
ejpam-2469	347	6	.	.	PUNCT
ejpam-2469	348	1	theorem	theorem	VERB
ejpam-2469	348	2	19	19	NUM
ejpam-2469	348	3	.	.	PUNCT
ejpam-2469	349	1	let	let	VERB
ejpam-2469	349	2	f	f	NOUN
ejpam-2469	349	3	:	:	PUNCT
ejpam-2469	349	4	x	x	X
ejpam-2469	349	5	→	→	SYM
ejpam-2469	349	6	y	y	X
ejpam-2469	349	7	be	be	AUX
ejpam-2469	349	8	a	a	DET
ejpam-2469	349	9	saw	saw	NOUN
ejpam-2469	349	10	-	-	PUNCT
ejpam-2469	349	11	ir	ir	NOUN
ejpam-2469	349	12	g	g	PROPN
ejpam-2469	349	13	-irresolute	-irresolute	PROPN
ejpam-2469	349	14	regular	regular	ADJ
ejpam-2469	349	15	closed	closed	ADJ
ejpam-2469	349	16	injection	injection	NOUN
ejpam-2469	349	17	.	.	PUNCT
ejpam-2469	350	1	if	if	SCONJ
ejpam-2469	350	2	x	x	PRON
ejpam-2469	350	3	is	be	AUX
ejpam-2469	350	4	αi	αi	X
ejpam-2469	350	5	-∗-normal	-∗-normal	PUNCT
ejpam-2469	350	6	and	and	CCONJ
ejpam-2469	350	7	y	y	PROPN
ejpam-2469	350	8	⊂	⊂	PROPN
ejpam-2469	350	9	x	x	PROPN
ejpam-2469	351	1	regular	regular	ADJ
ejpam-2469	351	2	closed	close	VERB
ejpam-2469	351	3	,	,	PUNCT
ejpam-2469	351	4	then	then	ADV
ejpam-2469	351	5	y	y	PROPN
ejpam-2469	351	6	is	be	AUX
ejpam-2469	351	7	αi	αi	PROPN
ejpam-2469	351	8	/	/	SYM
ejpam-2469	351	9	y	y	PROPN
ejpam-2469	351	10	-∗-normal	-∗-normal	NOUN
ejpam-2469	351	11	spaces	space	NOUN
ejpam-2469	351	12	.	.	PUNCT
ejpam-2469	352	1	proof	proof	NOUN
ejpam-2469	352	2	.	.	PUNCT
ejpam-2469	353	1	let	let	VERB
ejpam-2469	353	2	a	a	DET
ejpam-2469	353	3	and	and	CCONJ
ejpam-2469	353	4	b	b	NOUN
ejpam-2469	353	5	any	any	DET
ejpam-2469	353	6	disjoint	disjoint	ADJ
ejpam-2469	353	7	regular	regular	ADJ
ejpam-2469	353	8	closed	close	VERB
ejpam-2469	353	9	set	set	NOUN
ejpam-2469	353	10	of	of	ADP
ejpam-2469	353	11	y	y	PROPN
ejpam-2469	353	12	.	.	PUNCT
ejpam-2469	354	1	since	since	SCONJ
ejpam-2469	354	2	y	y	PROPN
ejpam-2469	354	3	is	be	AUX
ejpam-2469	354	4	regular	regular	ADV
ejpam-2469	354	5	closed	closed	ADJ
ejpam-2469	354	6	,	,	PUNCT
ejpam-2469	354	7	we	we	PRON
ejpam-2469	354	8	have	have	VERB
ejpam-2469	354	9	a	a	PRON
ejpam-2469	354	10	and	and	CCONJ
ejpam-2469	354	11	b	b	NOUN
ejpam-2469	354	12	are	be	AUX
ejpam-2469	354	13	disjoint	disjoint	ADJ
ejpam-2469	354	14	regular	regular	ADJ
ejpam-2469	354	15	closed	closed	ADJ
ejpam-2469	354	16	sets	set	NOUN
ejpam-2469	354	17	of	of	ADP
ejpam-2469	354	18	x	x	X
ejpam-2469	354	19	.	.	PUNCT
ejpam-2469	355	1	since	since	SCONJ
ejpam-2469	355	2	x	x	PRON
ejpam-2469	355	3	is	be	AUX
ejpam-2469	355	4	αi	αi	X
ejpam-2469	355	5	-∗-normal	-∗-normal	ADJ
ejpam-2469	355	6	,	,	PUNCT
ejpam-2469	355	7	there	there	PRON
ejpam-2469	355	8	exist	exist	VERB
ejpam-2469	355	9	disjoint	disjoint	NOUN
ejpam-2469	355	10	saw	saw	NOUN
ejpam-2469	355	11	-	-	PUNCT
ejpam-2469	355	12	ir	ir	NOUN
ejpam-2469	355	13	g	g	PROPN
ejpam-2469	355	14	-open	-open	PROPN
ejpam-2469	355	15	u	u	NOUN
ejpam-2469	355	16	and	and	CCONJ
ejpam-2469	355	17	v	v	ADP
ejpam-2469	355	18	such	such	ADJ
ejpam-2469	355	19	that	that	PRON
ejpam-2469	355	20	a⊆	a⊆	PROPN
ejpam-2469	355	21	u	u	NOUN
ejpam-2469	355	22	and	and	CCONJ
ejpam-2469	355	23	b	b	NOUN
ejpam-2469	356	1	⊆	⊆	NUM
ejpam-2469	356	2	v	v	NOUN
ejpam-2469	356	3	.	.	PUNCT
ejpam-2469	357	1	if	if	SCONJ
ejpam-2469	357	2	h	h	PROPN
ejpam-2469	357	3	∈	∈	PROPN
ejpam-2469	357	4	i	i	PRON
ejpam-2469	357	5	and	and	CCONJ
ejpam-2469	357	6	g	g	PROPN
ejpam-2469	357	7	∈	∈	PROPN
ejpam-2469	358	1	i	i	PRON
ejpam-2469	358	2	,	,	PUNCT
ejpam-2469	358	3	then	then	ADV
ejpam-2469	358	4	a⊂	a⊂	PRON
ejpam-2469	358	5	(	(	PUNCT
ejpam-2469	358	6	u	u	NOUN
ejpam-2469	358	7	∩	∩	ADJ
ejpam-2469	358	8	h	h	NOUN
ejpam-2469	358	9	)	)	PUNCT
ejpam-2469	358	10	and	and	CCONJ
ejpam-2469	358	11	references	reference	NOUN
ejpam-2469	358	12	442	442	NUM
ejpam-2469	358	13	b	b	X
ejpam-2469	358	14	⊂	⊂	PROPN
ejpam-2469	358	15	(	(	PUNCT
ejpam-2469	358	16	v	v	PROPN
ejpam-2469	358	17	∩	∩	NOUN
ejpam-2469	358	18	g	g	NOUN
ejpam-2469	358	19	)	)	PUNCT
ejpam-2469	358	20	.	.	PUNCT
ejpam-2469	359	1	since	since	SCONJ
ejpam-2469	359	2	a⊂	a⊂	NOUN
ejpam-2469	359	3	y	y	PROPN
ejpam-2469	359	4	and	and	CCONJ
ejpam-2469	359	5	b	b	PROPN
ejpam-2469	359	6	⊂	⊂	PROPN
ejpam-2469	359	7	y	y	PROPN
ejpam-2469	359	8	,	,	PUNCT
ejpam-2469	359	9	we	we	PRON
ejpam-2469	359	10	have	have	VERB
ejpam-2469	359	11	a⊂	a⊂	NOUN
ejpam-2469	359	12	y	y	PROPN
ejpam-2469	359	13	∩	∩	NOUN
ejpam-2469	359	14	(	(	PUNCT
ejpam-2469	359	15	u	u	PROPN
ejpam-2469	359	16	∩	∩	ADJ
ejpam-2469	359	17	h	h	NOUN
ejpam-2469	359	18	)	)	PUNCT
ejpam-2469	359	19	and	and	CCONJ
ejpam-2469	359	20	b	b	X
ejpam-2469	359	21	⊂	⊂	PROPN
ejpam-2469	359	22	y	y	PROPN
ejpam-2469	359	23	∩	∩	X
ejpam-2469	359	24	(	(	PUNCT
ejpam-2469	359	25	v	v	NUM
ejpam-2469	359	26	∩	∩	NOUN
ejpam-2469	359	27	g	g	NOUN
ejpam-2469	359	28	)	)	PUNCT
ejpam-2469	359	29	.	.	PUNCT
ejpam-2469	360	1	hence	hence	ADV
ejpam-2469	360	2	,	,	PUNCT
ejpam-2469	360	3	a⊂	a⊂	PRON
ejpam-2469	360	4	(	(	PUNCT
ejpam-2469	360	5	y	y	PROPN
ejpam-2469	360	6	∩u)∩h	∩u)∩h	PROPN
ejpam-2469	360	7	and	and	CCONJ
ejpam-2469	360	8	b	b	X
ejpam-2469	360	9	⊂	⊂	PROPN
ejpam-2469	360	10	(	(	PUNCT
ejpam-2469	360	11	y	y	PROPN
ejpam-2469	360	12	∩v	∩v	NOUN
ejpam-2469	360	13	)	)	PUNCT
ejpam-2469	360	14	∩g	∩g	PROPN
ejpam-2469	360	15	.	.	PUNCT
ejpam-2469	361	1	if	if	SCONJ
ejpam-2469	361	2	we	we	PRON
ejpam-2469	361	3	take	take	VERB
ejpam-2469	361	4	(	(	PUNCT
ejpam-2469	361	5	y	y	PROPN
ejpam-2469	361	6	∩u	∩u	PROPN
ejpam-2469	361	7	)	)	PUNCT
ejpam-2469	362	1	=	=	SYM
ejpam-2469	362	2	u1	u1	NOUN
ejpam-2469	362	3	and	and	CCONJ
ejpam-2469	362	4	(	(	PUNCT
ejpam-2469	362	5	y	y	PROPN
ejpam-2469	362	6	∩v	∩v	NOUN
ejpam-2469	362	7	)	)	PUNCT
ejpam-2469	363	1	=	=	SYM
ejpam-2469	363	2	v1	v1	NOUN
ejpam-2469	363	3	,	,	PUNCT
ejpam-2469	363	4	then	then	ADV
ejpam-2469	363	5	u1∩v1	u1∩v1	PROPN
ejpam-2469	363	6	=	=	SYM
ejpam-2469	363	7	;	;	PUNCT
ejpam-2469	363	8	such	such	ADJ
ejpam-2469	363	9	that	that	DET
ejpam-2469	363	10	u1	u1	NOUN
ejpam-2469	363	11	and	and	CCONJ
ejpam-2469	363	12	v1	v1	NOUN
ejpam-2469	363	13	is	be	AUX
ejpam-2469	363	14	saw	saw	NOUN
ejpam-2469	363	15	-	-	PUNCT
ejpam-2469	363	16	ir	ir	NOUN
ejpam-2469	363	17	g	g	PROPN
ejpam-2469	363	18	-open	-open	NOUN
ejpam-2469	363	19	.	.	PUNCT
ejpam-2469	364	1	this	this	PRON
ejpam-2469	364	2	shows	show	VERB
ejpam-2469	364	3	that	that	SCONJ
ejpam-2469	364	4	y	y	PROPN
ejpam-2469	364	5	is	be	AUX
ejpam-2469	364	6	αi	αi	PROPN
ejpam-2469	364	7	/	/	SYM
ejpam-2469	364	8	y	y	PROPN
ejpam-2469	364	9	-∗-normal	-∗-normal	NOUN
ejpam-2469	364	10	spaces	space	NOUN
ejpam-2469	364	11	.	.	PUNCT
ejpam-2469	365	1	acknowledgements	acknowledgement	NOUN
ejpam-2469	365	2	we	we	PRON
ejpam-2469	365	3	would	would	AUX
ejpam-2469	365	4	like	like	VERB
ejpam-2469	365	5	to	to	PART
ejpam-2469	365	6	express	express	VERB
ejpam-2469	365	7	our	our	PRON
ejpam-2469	365	8	sincere	sincere	ADJ
ejpam-2469	365	9	gratitude	gratitude	NOUN
ejpam-2469	365	10	to	to	ADP
ejpam-2469	365	11	the	the	DET
ejpam-2469	365	12	referees	referee	NOUN
ejpam-2469	365	13	.	.	PUNCT
ejpam-2469	366	1	references	reference	NOUN
ejpam-2469	366	2	[	[	X
ejpam-2469	366	3	1	1	NUM
ejpam-2469	366	4	]	]	PUNCT
ejpam-2469	366	5	j	j	PROPN
ejpam-2469	366	6	dontchev	dontchev	PROPN
ejpam-2469	366	7	,	,	PUNCT
ejpam-2469	366	8	m	m	NOUN
ejpam-2469	366	9	ganster	ganster	NOUN
ejpam-2469	366	10	,	,	PUNCT
ejpam-2469	366	11	and	and	CCONJ
ejpam-2469	366	12	t	t	PROPN
ejpam-2469	366	13	noiri	noiri	PROPN
ejpam-2469	366	14	.	.	PUNCT
ejpam-2469	367	1	unified	unified	ADJ
ejpam-2469	367	2	operation	operation	NOUN
ejpam-2469	367	3	approach	approach	NOUN
ejpam-2469	367	4	of	of	ADP
ejpam-2469	367	5	generalized	generalized	ADJ
ejpam-2469	367	6	closed	close	VERB
ejpam-2469	367	7	sets	set	NOUN
ejpam-2469	367	8	via	via	ADP
ejpam-2469	367	9	topological	topological	ADJ
ejpam-2469	367	10	ideals	ideal	NOUN
ejpam-2469	367	11	.	.	PUNCT
ejpam-2469	368	1	mathematica	mathematica	PROPN
ejpam-2469	368	2	japonica	japonica	PROPN
ejpam-2469	368	3	,	,	PUNCT
ejpam-2469	368	4	49(3):395–401	49(3):395–401	PROPN
ejpam-2469	368	5	,	,	PUNCT
ejpam-2469	368	6	1999	1999	NUM
ejpam-2469	368	7	.	.	PUNCT
ejpam-2469	369	1	[	[	X
ejpam-2469	369	2	2	2	NUM
ejpam-2469	369	3	]	]	PUNCT
ejpam-2469	369	4	e	e	NOUN
ejpam-2469	369	5	ekici	ekici	NOUN
ejpam-2469	369	6	and	and	CCONJ
ejpam-2469	369	7	s	s	VERB
ejpam-2469	369	8	ozen	ozen	ADJ
ejpam-2469	369	9	.	.	PUNCT
ejpam-2469	370	1	a	a	DET
ejpam-2469	370	2	generalized	generalized	ADJ
ejpam-2469	370	3	class	class	NOUN
ejpam-2469	370	4	of	of	ADP
ejpam-2469	370	5	τ∗	τ∗	NOUN
ejpam-2469	370	6	in	in	ADP
ejpam-2469	370	7	ideal	ideal	ADJ
ejpam-2469	370	8	spaces	space	NOUN
ejpam-2469	370	9	.	.	PUNCT
ejpam-2469	371	1	filomat	filomat	NOUN
ejpam-2469	371	2	,	,	PUNCT
ejpam-2469	371	3	27(4):529–535	27(4):529–535	PROPN
ejpam-2469	371	4	,	,	PUNCT
ejpam-2469	371	5	2013	2013	NUM
ejpam-2469	371	6	.	.	PUNCT
ejpam-2469	372	1	[	[	X
ejpam-2469	372	2	3	3	NUM
ejpam-2469	372	3	]	]	X
ejpam-2469	372	4	d	d	X
ejpam-2469	372	5	jankovic	jankovic	PROPN
ejpam-2469	372	6	and	and	CCONJ
ejpam-2469	372	7	t	t	PROPN
ejpam-2469	372	8	r	r	PROPN
ejpam-2469	372	9	hamlett	hamlett	PROPN
ejpam-2469	372	10	.	.	PUNCT
ejpam-2469	373	1	new	new	ADJ
ejpam-2469	373	2	topologies	topology	NOUN
ejpam-2469	373	3	from	from	ADP
ejpam-2469	373	4	old	old	ADJ
ejpam-2469	373	5	via	via	ADP
ejpam-2469	373	6	ideals	ideal	NOUN
ejpam-2469	373	7	.	.	PUNCT
ejpam-2469	374	1	the	the	DET
ejpam-2469	374	2	american	american	PROPN
ejpam-2469	374	3	mathematical	mathematical	PROPN
ejpam-2469	374	4	monthly	monthly	PROPN
ejpam-2469	374	5	,	,	PUNCT
ejpam-2469	374	6	97(4):295–310	97(4):295–310	PROPN
ejpam-2469	374	7	,	,	PUNCT
ejpam-2469	374	8	1990	1990	NUM
ejpam-2469	374	9	.	.	PUNCT
ejpam-2469	375	1	[	[	X
ejpam-2469	375	2	4	4	NUM
ejpam-2469	375	3	]	]	X
ejpam-2469	375	4	d	d	X
ejpam-2469	375	5	jankovic	jankovic	PROPN
ejpam-2469	375	6	and	and	CCONJ
ejpam-2469	375	7	t	t	PROPN
ejpam-2469	375	8	r	r	PROPN
ejpam-2469	375	9	hamlett	hamlett	PROPN
ejpam-2469	375	10	.	.	PUNCT
ejpam-2469	376	1	compatible	compatible	ADJ
ejpam-2469	376	2	extensions	extension	NOUN
ejpam-2469	376	3	of	of	ADP
ejpam-2469	376	4	ideals	ideal	NOUN
ejpam-2469	376	5	.	.	PUNCT
ejpam-2469	377	1	bollettino	bollettino	PROPN
ejpam-2469	377	2	della	della	PROPN
ejpam-2469	377	3	unione	unione	PROPN
ejpam-2469	377	4	matematica	matematica	PROPN
ejpam-2469	377	5	italiana	italiana	PROPN
ejpam-2469	377	6	,	,	PUNCT
ejpam-2469	377	7	7(6):453–465	7(6):453–465	NUM
ejpam-2469	377	8	,	,	PUNCT
ejpam-2469	377	9	1992	1992	NUM
ejpam-2469	377	10	.	.	PUNCT
ejpam-2469	378	1	[	[	X
ejpam-2469	378	2	5	5	NUM
ejpam-2469	378	3	]	]	PUNCT
ejpam-2469	378	4	u	u	NOUN
ejpam-2469	378	5	karabiyik	karabiyik	NOUN
ejpam-2469	378	6	.	.	PUNCT
ejpam-2469	379	1	some	some	DET
ejpam-2469	379	2	generalized	generalize	VERB
ejpam-2469	379	3	closed	closed	ADJ
ejpam-2469	379	4	sets	set	NOUN
ejpam-2469	379	5	and	and	CCONJ
ejpam-2469	379	6	continuous	continuous	ADJ
ejpam-2469	379	7	functions	function	NOUN
ejpam-2469	379	8	in	in	ADP
ejpam-2469	379	9	ideal	ideal	ADJ
ejpam-2469	379	10	topological	topological	ADJ
ejpam-2469	379	11	spaces	space	NOUN
ejpam-2469	379	12	.	.	PUNCT
ejpam-2469	380	1	master	master	NOUN
ejpam-2469	380	2	thesis	thesis	NOUN
ejpam-2469	380	3	,	,	PUNCT
ejpam-2469	380	4	selcuk	selcuk	PROPN
ejpam-2469	380	5	university	university	NOUN
ejpam-2469	380	6	,	,	PUNCT
ejpam-2469	380	7	department	department	NOUN
ejpam-2469	380	8	of	of	ADP
ejpam-2469	380	9	mathematics	mathematic	NOUN
ejpam-2469	380	10	,	,	PUNCT
ejpam-2469	380	11	2008	2008	NUM
ejpam-2469	380	12	.	.	PUNCT
ejpam-2469	381	1	[	[	X
ejpam-2469	381	2	6	6	NUM
ejpam-2469	381	3	]	]	X
ejpam-2469	381	4	m	m	VERB
ejpam-2469	381	5	khan	khan	PROPN
ejpam-2469	381	6	and	and	CCONJ
ejpam-2469	381	7	t	t	PROPN
ejpam-2469	381	8	noiri	noiri	PROPN
ejpam-2469	381	9	.	.	PUNCT
ejpam-2469	382	1	semi	semi	ADJ
ejpam-2469	382	2	-	-	ADJ
ejpam-2469	382	3	local	local	ADJ
ejpam-2469	382	4	function	function	NOUN
ejpam-2469	382	5	in	in	ADP
ejpam-2469	382	6	ideal	ideal	ADJ
ejpam-2469	382	7	topological	topological	ADJ
ejpam-2469	382	8	spaces	space	NOUN
ejpam-2469	382	9	.	.	PUNCT
ejpam-2469	383	1	journal	journal	NOUN
ejpam-2469	383	2	of	of	ADP
ejpam-2469	383	3	advanced	advanced	ADJ
ejpam-2469	383	4	research	research	NOUN
ejpam-2469	383	5	in	in	ADP
ejpam-2469	383	6	pure	pure	ADJ
ejpam-2469	383	7	mathematics	mathematic	NOUN
ejpam-2469	383	8	,	,	PUNCT
ejpam-2469	383	9	2(1):36–42	2(1):36–42	NUM
ejpam-2469	383	10	,	,	PUNCT
ejpam-2469	383	11	2010	2010	NUM
ejpam-2469	383	12	.	.	PUNCT
ejpam-2469	384	1	[	[	X
ejpam-2469	384	2	7	7	X
ejpam-2469	384	3	]	]	X
ejpam-2469	384	4	k	k	PROPN
ejpam-2469	384	5	kuratowski	kuratowski	PROPN
ejpam-2469	384	6	.	.	PUNCT
ejpam-2469	385	1	topologies	topology	NOUN
ejpam-2469	385	2	i.	i.	PROPN
ejpam-2469	385	3	academic	academic	PROPN
ejpam-2469	385	4	press	press	PROPN
ejpam-2469	385	5	,	,	PUNCT
ejpam-2469	385	6	polish	polish	ADJ
ejpam-2469	385	7	scientific	scientific	ADJ
ejpam-2469	385	8	publishers	publisher	NOUN
ejpam-2469	385	9	,	,	PUNCT
ejpam-2469	385	10	new	new	PROPN
ejpam-2469	385	11	york	york	PROPN
ejpam-2469	385	12	,	,	PUNCT
ejpam-2469	385	13	london	london	PROPN
ejpam-2469	385	14	,	,	PUNCT
ejpam-2469	385	15	warszawa	warszawa	PROPN
ejpam-2469	385	16	,	,	PUNCT
ejpam-2469	385	17	1961	1961	NUM
ejpam-2469	385	18	.	.	PUNCT
ejpam-2469	386	1	[	[	X
ejpam-2469	386	2	8	8	NUM
ejpam-2469	386	3	]	]	SYM
ejpam-2469	386	4	v	v	ADP
ejpam-2469	386	5	inthumathi	inthumathi	NOUN
ejpam-2469	386	6	m	m	NOUN
ejpam-2469	386	7	rajamani	rajamani	NOUN
ejpam-2469	386	8	and	and	CCONJ
ejpam-2469	386	9	v	v	ADJ
ejpam-2469	386	10	chitra	chitra	NOUN
ejpam-2469	386	11	.	.	PUNCT
ejpam-2469	387	1	on	on	ADP
ejpam-2469	387	2	decompositions	decomposition	NOUN
ejpam-2469	387	3	of	of	ADP
ejpam-2469	387	4	i	i	PROPN
ejpam-2469	387	5	-	-	PUNCT
ejpam-2469	387	6	rg	rg	NOUN
ejpam-2469	387	7	-	-	PUNCT
ejpam-2469	387	8	continuity	continuity	NOUN
ejpam-2469	387	9	.	.	PUNCT
ejpam-2469	388	1	boletim	boletim	PROPN
ejpam-2469	388	2	da	da	PROPN
ejpam-2469	388	3	sociedade	sociedade	PROPN
ejpam-2469	388	4	paranaense	paranaense	PROPN
ejpam-2469	388	5	de	de	PROPN
ejpam-2469	388	6	matematica	matematica	PROPN
ejpam-2469	388	7	,	,	PUNCT
ejpam-2469	388	8	3(2):33–38	3(2):33–38	NUM
ejpam-2469	388	9	,	,	PUNCT
ejpam-2469	388	10	2012	2012	NUM
ejpam-2469	388	11	.	.	PUNCT
ejpam-2469	389	1	[	[	X
ejpam-2469	389	2	9	9	NUM
ejpam-2469	389	3	]	]	SYM
ejpam-2469	389	4	m	m	NOUN
ejpam-2469	389	5	navaneethakrishnan	navaneethakrishnan	ADJ
ejpam-2469	389	6	and	and	CCONJ
ejpam-2469	389	7	j	j	PROPN
ejpam-2469	389	8	joseph	joseph	PROPN
ejpam-2469	389	9	.	.	PUNCT
ejpam-2469	390	1	g	g	NOUN
ejpam-2469	390	2	-	-	PUNCT
ejpam-2469	390	3	closed	close	VERB
ejpam-2469	390	4	sets	set	NOUN
ejpam-2469	390	5	in	in	ADP
ejpam-2469	390	6	ideal	ideal	ADJ
ejpam-2469	390	7	topological	topological	ADJ
ejpam-2469	390	8	spaces	space	NOUN
ejpam-2469	390	9	.	.	PUNCT
ejpam-2469	391	1	acta	acta	PROPN
ejpam-2469	391	2	mathematica	mathematica	PROPN
ejpam-2469	391	3	hungarica	hungarica	PROPN
ejpam-2469	391	4	,	,	PUNCT
ejpam-2469	391	5	119(4):365–371	119(4):365–371	NUM
ejpam-2469	391	6	,	,	PUNCT
ejpam-2469	391	7	2008	2008	NUM
ejpam-2469	391	8	.	.	PUNCT
ejpam-2469	392	1	[	[	X
ejpam-2469	392	2	10	10	NUM
ejpam-2469	392	3	]	]	X
ejpam-2469	392	4	m	m	NOUN
ejpam-2469	392	5	navaneethakrishnan	navaneethakrishnan	ADJ
ejpam-2469	392	6	and	and	CCONJ
ejpam-2469	392	7	d	d	ADP
ejpam-2469	392	8	sivaroj	sivaroj	NOUN
ejpam-2469	392	9	.	.	PUNCT
ejpam-2469	393	1	regular	regular	ADJ
ejpam-2469	393	2	generalized	generalize	VERB
ejpam-2469	393	3	closed	close	VERB
ejpam-2469	393	4	sets	set	NOUN
ejpam-2469	393	5	in	in	ADP
ejpam-2469	393	6	ideal	ideal	ADJ
ejpam-2469	393	7	topological	topological	ADJ
ejpam-2469	393	8	spaces	space	NOUN
ejpam-2469	393	9	.	.	PUNCT
ejpam-2469	394	1	journal	journal	NOUN
ejpam-2469	394	2	of	of	ADP
ejpam-2469	394	3	advanced	advanced	ADJ
ejpam-2469	394	4	research	research	NOUN
ejpam-2469	394	5	in	in	ADP
ejpam-2469	394	6	pure	pure	ADJ
ejpam-2469	394	7	mathematics	mathematic	NOUN
ejpam-2469	394	8	,	,	PUNCT
ejpam-2469	394	9	2(3):24–33	2(3):24–33	NUM
ejpam-2469	394	10	,	,	PUNCT
ejpam-2469	394	11	2010	2010	NUM
ejpam-2469	394	12	.	.	PUNCT
ejpam-2469	395	1	[	[	X
ejpam-2469	395	2	11	11	NUM
ejpam-2469	395	3	]	]	X
ejpam-2469	395	4	m	m	VERB
ejpam-2469	395	5	h	h	NOUN
ejpam-2469	395	6	stone	stone	NOUN
ejpam-2469	395	7	.	.	PUNCT
ejpam-2469	396	1	applications	application	NOUN
ejpam-2469	396	2	of	of	ADP
ejpam-2469	396	3	the	the	DET
ejpam-2469	396	4	theory	theory	NOUN
ejpam-2469	396	5	of	of	ADP
ejpam-2469	396	6	boolean	boolean	ADJ
ejpam-2469	396	7	rings	ring	NOUN
ejpam-2469	396	8	to	to	ADP
ejpam-2469	396	9	general	general	ADJ
ejpam-2469	396	10	topology	topology	NOUN
ejpam-2469	396	11	.	.	PUNCT
ejpam-2469	397	1	transactions	transaction	NOUN
ejpam-2469	397	2	of	of	ADP
ejpam-2469	397	3	the	the	DET
ejpam-2469	397	4	american	american	PROPN
ejpam-2469	397	5	mathematical	mathematical	PROPN
ejpam-2469	397	6	society	society	NOUN
ejpam-2469	397	7	,	,	PUNCT
ejpam-2469	397	8	41(3):375–381	41(3):375–381	PROPN
ejpam-2469	397	9	,	,	PUNCT
ejpam-2469	397	10	1937	1937	NUM
ejpam-2469	397	11	.	.	PUNCT
ejpam-2469	398	1	[	[	X
ejpam-2469	398	2	12	12	NUM
ejpam-2469	398	3	]	]	X
ejpam-2469	398	4	r	r	NOUN
ejpam-2469	398	5	vaidyanathaswamy	vaidyanathaswamy	NOUN
ejpam-2469	398	6	.	.	PUNCT
ejpam-2469	399	1	set	set	VERB
ejpam-2469	399	2	topology	topology	NOUN
ejpam-2469	399	3	.	.	PUNCT
ejpam-2469	400	1	published	publish	VERB
ejpam-2469	400	2	by	by	ADP
ejpam-2469	400	3	chelsea	chelsea	PROPN
ejpam-2469	400	4	publishing	publishing	PROPN
ejpam-2469	400	5	company	company	NOUN
ejpam-2469	400	6	,	,	PUNCT
ejpam-2469	400	7	new	new	PROPN
ejpam-2469	400	8	york	york	PROPN
ejpam-2469	400	9	,	,	PUNCT
ejpam-2469	400	10	1960	1960	NUM
ejpam-2469	400	11	.	.	PUNCT