id	sid	tid	token	lemma	pos
ejpam-2214	1	1	european	european	PROPN
ejpam-2214	1	2	journal	journal	PROPN
ejpam-2214	1	3	of	of	ADP
ejpam-2214	1	4	pure	pure	ADJ
ejpam-2214	1	5	and	and	CCONJ
ejpam-2214	1	6	applied	apply	VERB
ejpam-2214	1	7	mathematics	mathematic	NOUN
ejpam-2214	1	8	vol	vol	NOUN
ejpam-2214	1	9	.	.	PUNCT
ejpam-2214	2	1	7	7	NUM
ejpam-2214	2	2	,	,	PUNCT
ejpam-2214	2	3	no	no	INTJ
ejpam-2214	2	4	.	.	NOUN
ejpam-2214	2	5	4	4	NUM
ejpam-2214	2	6	,	,	PUNCT
ejpam-2214	2	7	2014	2014	NUM
ejpam-2214	2	8	,	,	PUNCT
ejpam-2214	2	9	437	437	NUM
ejpam-2214	2	10	-	-	SYM
ejpam-2214	2	11	441	441	NUM
ejpam-2214	2	12	issn	issn	PROPN
ejpam-2214	2	13	1307	1307	NUM
ejpam-2214	2	14	-	-	SYM
ejpam-2214	2	15	5543	5543	NUM
ejpam-2214	2	16	–	–	PUNCT
ejpam-2214	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2214	2	18	some	some	DET
ejpam-2214	2	19	remarks	remark	VERB
ejpam-2214	2	20	on	on	ADP
ejpam-2214	2	21	semi	semi	ADJ
ejpam-2214	2	22	open	open	ADJ
ejpam-2214	2	23	sets	set	NOUN
ejpam-2214	2	24	with	with	ADP
ejpam-2214	2	25	respect	respect	NOUN
ejpam-2214	2	26	to	to	ADP
ejpam-2214	2	27	an	an	DET
ejpam-2214	2	28	ideal	ideal	NOUN
ejpam-2214	2	29	carlos	carlos	PROPN
ejpam-2214	2	30	carpintero1	carpintero1	PROPN
ejpam-2214	2	31	,	,	PUNCT
ejpam-2214	2	32	alvaro	alvaro	PROPN
ejpam-2214	2	33	muñoz2	muñoz2	PROPN
ejpam-2214	2	34	,	,	PUNCT
ejpam-2214	2	35	jackeline	jackeline	PROPN
ejpam-2214	2	36	pacheco3	pacheco3	PROPN
ejpam-2214	2	37	,	,	PUNCT
ejpam-2214	2	38	ennis	ennis	PROPN
ejpam-2214	2	39	rosas4∗	rosas4∗	NUM
ejpam-2214	2	40	1,4	1,4	NUM
ejpam-2214	2	41	departamento	departamento	NOUN
ejpam-2214	2	42	de	de	PROPN
ejpam-2214	2	43	matemáticas	matemáticas	NOUN
ejpam-2214	2	44	,	,	PUNCT
ejpam-2214	2	45	universidad	universidad	PROPN
ejpam-2214	2	46	de	de	X
ejpam-2214	2	47	oriente	oriente	PROPN
ejpam-2214	2	48	,	,	PUNCT
ejpam-2214	2	49	núcleo	núcleo	X
ejpam-2214	2	50	de	de	PROPN
ejpam-2214	2	51	sucre	sucre	PROPN
ejpam-2214	2	52	cumaná	cumaná	PROPN
ejpam-2214	2	53	,	,	PUNCT
ejpam-2214	2	54	venezuela	venezuela	PROPN
ejpam-2214	2	55	and	and	CCONJ
ejpam-2214	2	56	universidad	universidad	PROPN
ejpam-2214	2	57	del	del	PROPN
ejpam-2214	2	58	atlántico	atlántico	PROPN
ejpam-2214	2	59	,	,	PUNCT
ejpam-2214	2	60	facultad	facultad	PROPN
ejpam-2214	2	61	de	de	PROPN
ejpam-2214	2	62	ciencias	ciencias	PROPN
ejpam-2214	2	63	básicas	básicas	PROPN
ejpam-2214	2	64	,	,	PUNCT
ejpam-2214	2	65	barranquilla	barranquilla	NOUN
ejpam-2214	2	66	,	,	PUNCT
ejpam-2214	2	67	colombia	colombia	PROPN
ejpam-2214	2	68	2,3	2,3	NUM
ejpam-2214	2	69	programa	programa	X
ejpam-2214	2	70	de	de	X
ejpam-2214	2	71	matemáticas	matemáticas	NOUN
ejpam-2214	2	72	,	,	PUNCT
ejpam-2214	2	73	facultad	facultad	PROPN
ejpam-2214	2	74	de	de	PROPN
ejpam-2214	2	75	ciencias	ciencias	PROPN
ejpam-2214	2	76	básicas	básicas	PROPN
ejpam-2214	2	77	,	,	PUNCT
ejpam-2214	2	78	universidad	universidad	PROPN
ejpam-2214	2	79	del	del	PROPN
ejpam-2214	2	80	atlántico	atlántico	PROPN
ejpam-2214	2	81	,	,	PUNCT
ejpam-2214	2	82	barranquilla	barranquilla	PROPN
ejpam-2214	2	83	,	,	PUNCT
ejpam-2214	2	84	colombia	colombia	PROPN
ejpam-2214	2	85	abstract	abstract	NOUN
ejpam-2214	2	86	.	.	PUNCT
ejpam-2214	3	1	in	in	ADP
ejpam-2214	3	2	this	this	DET
ejpam-2214	3	3	article	article	NOUN
ejpam-2214	3	4	we	we	PRON
ejpam-2214	3	5	introduce	introduce	VERB
ejpam-2214	3	6	the	the	DET
ejpam-2214	3	7	notions	notion	NOUN
ejpam-2214	3	8	of	of	ADP
ejpam-2214	3	9	weakly	weakly	ADJ
ejpam-2214	3	10	semi	semi	ADJ
ejpam-2214	3	11	open	open	ADJ
ejpam-2214	3	12	sets	set	NOUN
ejpam-2214	3	13	with	with	ADP
ejpam-2214	3	14	respect	respect	NOUN
ejpam-2214	3	15	to	to	ADP
ejpam-2214	3	16	an	an	DET
ejpam-2214	3	17	ideal	ideal	NOUN
ejpam-2214	3	18	,	,	PUNCT
ejpam-2214	3	19	characterize	characterize	VERB
ejpam-2214	3	20	them	they	PRON
ejpam-2214	3	21	and	and	CCONJ
ejpam-2214	3	22	find	find	VERB
ejpam-2214	3	23	some	some	DET
ejpam-2214	3	24	properties	property	NOUN
ejpam-2214	3	25	2010	2010	NUM
ejpam-2214	3	26	mathematics	mathematic	NOUN
ejpam-2214	3	27	subject	subject	NOUN
ejpam-2214	3	28	classifications	classification	NOUN
ejpam-2214	3	29	:	:	PUNCT
ejpam-2214	3	30	54c10	54c10	NUM
ejpam-2214	3	31	key	key	ADJ
ejpam-2214	3	32	words	word	NOUN
ejpam-2214	3	33	and	and	CCONJ
ejpam-2214	3	34	phrases	phrase	NOUN
ejpam-2214	3	35	:	:	PUNCT
ejpam-2214	3	36	weakly	weakly	ADJ
ejpam-2214	3	37	semi	semi	ADV
ejpam-2214	3	38	open	open	ADJ
ejpam-2214	3	39	set	set	VERB
ejpam-2214	3	40	with	with	ADP
ejpam-2214	3	41	respect	respect	NOUN
ejpam-2214	3	42	to	to	ADP
ejpam-2214	3	43	an	an	DET
ejpam-2214	3	44	ideal	ideal	ADJ
ejpam-2214	3	45	,	,	PUNCT
ejpam-2214	3	46	semi	semi	ADV
ejpam-2214	3	47	open	open	ADJ
ejpam-2214	3	48	set	set	VERB
ejpam-2214	3	49	with	with	ADP
ejpam-2214	3	50	respect	respect	NOUN
ejpam-2214	3	51	to	to	ADP
ejpam-2214	3	52	an	an	DET
ejpam-2214	3	53	ideal	ideal	ADJ
ejpam-2214	3	54	,	,	PUNCT
ejpam-2214	3	55	semi	semi	ADV
ejpam-2214	3	56	open	open	ADJ
ejpam-2214	3	57	set	set	NOUN
ejpam-2214	3	58	1	1	NUM
ejpam-2214	3	59	.	.	PUNCT
ejpam-2214	4	1	introduction	introduction	NOUN
ejpam-2214	4	2	currently	currently	ADV
ejpam-2214	4	3	many	many	ADJ
ejpam-2214	4	4	mathematicians	mathematician	NOUN
ejpam-2214	4	5	have	have	AUX
ejpam-2214	4	6	worked	work	VERB
ejpam-2214	4	7	in	in	ADP
ejpam-2214	4	8	generalized	generalized	ADJ
ejpam-2214	4	9	mathematical	mathematical	ADJ
ejpam-2214	4	10	notions	notion	NOUN
ejpam-2214	4	11	,	,	PUNCT
ejpam-2214	4	12	see	see	VERB
ejpam-2214	4	13	[	[	X
ejpam-2214	4	14	1	1	NUM
ejpam-2214	4	15	,	,	PUNCT
ejpam-2214	4	16	2	2	NUM
ejpam-2214	4	17	,	,	PUNCT
ejpam-2214	4	18	4	4	NUM
ejpam-2214	4	19	]	]	PUNCT
ejpam-2214	4	20	,	,	PUNCT
ejpam-2214	4	21	also	also	ADV
ejpam-2214	4	22	provided	provide	VERB
ejpam-2214	4	23	some	some	DET
ejpam-2214	4	24	characterizations	characterization	NOUN
ejpam-2214	4	25	of	of	ADP
ejpam-2214	4	26	these	these	DET
ejpam-2214	4	27	notions	notion	NOUN
ejpam-2214	4	28	,	,	PUNCT
ejpam-2214	4	29	this	this	PRON
ejpam-2214	4	30	is	be	AUX
ejpam-2214	4	31	the	the	DET
ejpam-2214	4	32	case	case	NOUN
ejpam-2214	4	33	of	of	ADP
ejpam-2214	4	34	friday	friday	PROPN
ejpam-2214	4	35	ifeanyi	ifeanyi	PROPN
ejpam-2214	4	36	michael	michael	PROPN
ejpam-2214	4	37	k.	k.	PROPN
ejpam-2214	5	1	[	[	X
ejpam-2214	5	2	1	1	X
ejpam-2214	5	3	]	]	PUNCT
ejpam-2214	5	4	in	in	ADP
ejpam-2214	5	5	the	the	DET
ejpam-2214	5	6	article	article	NOUN
ejpam-2214	5	7	“	"	PUNCT
ejpam-2214	5	8	on	on	ADP
ejpam-2214	5	9	semi	semi	ADV
ejpam-2214	5	10	open	open	ADJ
ejpam-2214	5	11	sets	set	NOUN
ejpam-2214	5	12	with	with	ADP
ejpam-2214	5	13	respect	respect	NOUN
ejpam-2214	5	14	to	to	ADP
ejpam-2214	5	15	an	an	DET
ejpam-2214	5	16	ideal	ideal	NOUN
ejpam-2214	5	17	”	"	PUNCT
ejpam-2214	5	18	in	in	ADP
ejpam-2214	5	19	which	which	PRON
ejpam-2214	5	20	it	it	PRON
ejpam-2214	5	21	generalizes	generalize	VERB
ejpam-2214	5	22	the	the	DET
ejpam-2214	5	23	notion	notion	NOUN
ejpam-2214	5	24	of	of	ADP
ejpam-2214	5	25	semi	semi	ADJ
ejpam-2214	5	26	open	open	ADJ
ejpam-2214	5	27	sets	set	NOUN
ejpam-2214	5	28	defined	define	VERB
ejpam-2214	5	29	by	by	ADP
ejpam-2214	5	30	norman	norman	PROPN
ejpam-2214	5	31	levine	levine	PROPN
ejpam-2214	5	32	in	in	ADP
ejpam-2214	5	33	[	[	X
ejpam-2214	5	34	3	3	NUM
ejpam-2214	5	35	]	]	PUNCT
ejpam-2214	5	36	.	.	PUNCT
ejpam-2214	6	1	in	in	ADP
ejpam-2214	6	2	this	this	DET
ejpam-2214	6	3	article	article	NOUN
ejpam-2214	6	4	the	the	DET
ejpam-2214	6	5	author	author	NOUN
ejpam-2214	6	6	define	define	VERB
ejpam-2214	6	7	the	the	DET
ejpam-2214	6	8	notion	notion	NOUN
ejpam-2214	6	9	of	of	ADP
ejpam-2214	6	10	i	i	PRON
ejpam-2214	6	11	-semi	-semi	VERB
ejpam-2214	6	12	open	open	ADJ
ejpam-2214	6	13	set	set	VERB
ejpam-2214	6	14	as	as	SCONJ
ejpam-2214	6	15	follows	follow	VERB
ejpam-2214	6	16	:	:	PUNCT
ejpam-2214	6	17	let	let	VERB
ejpam-2214	6	18	x	x	PRON
ejpam-2214	6	19	be	be	AUX
ejpam-2214	6	20	a	a	DET
ejpam-2214	6	21	topological	topological	ADJ
ejpam-2214	6	22	space	space	NOUN
ejpam-2214	6	23	and	and	CCONJ
ejpam-2214	6	24	i	i	PRON
ejpam-2214	6	25	an	an	DET
ejpam-2214	6	26	ideal	ideal	NOUN
ejpam-2214	6	27	on	on	ADP
ejpam-2214	6	28	x	x	X
ejpam-2214	6	29	,	,	PUNCT
ejpam-2214	6	30	a⊆	a⊆	PROPN
ejpam-2214	6	31	x	x	VERB
ejpam-2214	6	32	is	be	AUX
ejpam-2214	6	33	said	say	VERB
ejpam-2214	6	34	to	to	PART
ejpam-2214	6	35	be	be	AUX
ejpam-2214	6	36	i	i	PRON
ejpam-2214	6	37	-semi	-semi	VERB
ejpam-2214	6	38	open	open	ADJ
ejpam-2214	6	39	set	set	VERB
ejpam-2214	6	40	if	if	SCONJ
ejpam-2214	6	41	there	there	PRON
ejpam-2214	6	42	exists	exist	VERB
ejpam-2214	6	43	an	an	DET
ejpam-2214	6	44	open	open	ADJ
ejpam-2214	6	45	set	set	NOUN
ejpam-2214	6	46	u	u	PRON
ejpam-2214	6	47	such	such	ADJ
ejpam-2214	6	48	that	that	SCONJ
ejpam-2214	6	49	u	u	NOUN
ejpam-2214	6	50	\a∈	\a∈	INTJ
ejpam-2214	6	51	i	i	PRON
ejpam-2214	6	52	and	and	CCONJ
ejpam-2214	6	53	a\	a\	PRON
ejpam-2214	6	54	cl(u	cl(u	NOUN
ejpam-2214	6	55	)	)	PUNCT
ejpam-2214	6	56	∈	∈	PROPN
ejpam-2214	7	1	i	i	PRON
ejpam-2214	7	2	.	.	PUNCT
ejpam-2214	8	1	in	in	ADP
ejpam-2214	8	2	this	this	DET
ejpam-2214	8	3	article	article	NOUN
ejpam-2214	8	4	the	the	DET
ejpam-2214	8	5	following	follow	VERB
ejpam-2214	8	6	properties	property	NOUN
ejpam-2214	8	7	were	be	AUX
ejpam-2214	8	8	proved	prove	VERB
ejpam-2214	8	9	:	:	PUNCT
ejpam-2214	8	10	proposition	proposition	NOUN
ejpam-2214	8	11	5	5	NUM
ejpam-2214	8	12	.	.	PUNCT
ejpam-2214	9	1	let	let	VERB
ejpam-2214	9	2	i	i	PRON
ejpam-2214	9	3	be	be	AUX
ejpam-2214	9	4	an	an	DET
ejpam-2214	9	5	ideal	ideal	NOUN
ejpam-2214	9	6	on	on	ADP
ejpam-2214	9	7	a	a	DET
ejpam-2214	9	8	topological	topological	ADJ
ejpam-2214	9	9	space	space	NOUN
ejpam-2214	9	10	x	x	SYM
ejpam-2214	9	11	,	,	PUNCT
ejpam-2214	9	12	where	where	SCONJ
ejpam-2214	9	13	every	every	DET
ejpam-2214	9	14	subset	subset	NOUN
ejpam-2214	9	15	of	of	ADP
ejpam-2214	9	16	x	x	PUNCT
ejpam-2214	9	17	is	be	AUX
ejpam-2214	9	18	dense	dense	ADJ
ejpam-2214	9	19	and	and	CCONJ
ejpam-2214	9	20	the	the	DET
ejpam-2214	9	21	collection	collection	NOUN
ejpam-2214	9	22	of	of	ADP
ejpam-2214	9	23	open	open	ADJ
ejpam-2214	9	24	subsets	subset	NOUN
ejpam-2214	9	25	of	of	ADP
ejpam-2214	9	26	x	x	PRON
ejpam-2214	9	27	satisfies	satisfy	VERB
ejpam-2214	9	28	the	the	DET
ejpam-2214	9	29	finite	finite	ADJ
ejpam-2214	9	30	intersection	intersection	NOUN
ejpam-2214	9	31	property	property	NOUN
ejpam-2214	9	32	:	:	PUNCT
ejpam-2214	10	1	1	1	X
ejpam-2214	10	2	.	.	X
ejpam-2214	10	3	if	if	SCONJ
ejpam-2214	10	4	a	a	PRON
ejpam-2214	10	5	is	be	AUX
ejpam-2214	10	6	i	i	PRON
ejpam-2214	10	7	-semi	-semi	VERB
ejpam-2214	10	8	open	open	ADJ
ejpam-2214	10	9	and	and	CCONJ
ejpam-2214	10	10	a⊆	a⊆	VERB
ejpam-2214	10	11	b	b	X
ejpam-2214	10	12	,	,	PUNCT
ejpam-2214	10	13	then	then	ADV
ejpam-2214	10	14	b	b	PROPN
ejpam-2214	10	15	is	be	AUX
ejpam-2214	10	16	i	i	PRON
ejpam-2214	10	17	-semi	-semi	VERB
ejpam-2214	10	18	open	open	ADJ
ejpam-2214	10	19	.	.	PUNCT
ejpam-2214	11	1	2	2	X
ejpam-2214	11	2	.	.	X
ejpam-2214	11	3	if	if	SCONJ
ejpam-2214	11	4	a	a	PRON
ejpam-2214	11	5	is	be	AUX
ejpam-2214	11	6	i	i	PRON
ejpam-2214	11	7	semi	semi	ADV
ejpam-2214	11	8	open	open	ADJ
ejpam-2214	11	9	,	,	PUNCT
ejpam-2214	11	10	then	then	ADV
ejpam-2214	11	11	so	so	ADV
ejpam-2214	11	12	is	be	AUX
ejpam-2214	11	13	a∪	a∪	PROPN
ejpam-2214	11	14	b	b	NOUN
ejpam-2214	11	15	for	for	ADP
ejpam-2214	11	16	any	any	DET
ejpam-2214	11	17	subset	subset	NOUN
ejpam-2214	11	18	b	b	PROPN
ejpam-2214	11	19	of	of	ADP
ejpam-2214	11	20	x	x	PROPN
ejpam-2214	11	21	.	.	PUNCT
ejpam-2214	12	1	3	3	X
ejpam-2214	12	2	.	.	X
ejpam-2214	13	1	if	if	SCONJ
ejpam-2214	13	2	both	both	PRON
ejpam-2214	13	3	a	a	PRON
ejpam-2214	13	4	and	and	CCONJ
ejpam-2214	13	5	b	b	NOUN
ejpam-2214	13	6	are	be	AUX
ejpam-2214	13	7	i	i	PRON
ejpam-2214	13	8	-semi	-semi	VERB
ejpam-2214	13	9	open	open	ADJ
ejpam-2214	13	10	,	,	PUNCT
ejpam-2214	13	11	then	then	ADV
ejpam-2214	13	12	so	so	ADV
ejpam-2214	13	13	is	be	AUX
ejpam-2214	13	14	a∩	a∩	PROPN
ejpam-2214	13	15	b.	b.	PROPN
ejpam-2214	13	16	∗corresponding	∗corresponde	VERB
ejpam-2214	13	17	author	author	NOUN
ejpam-2214	13	18	.	.	PUNCT
ejpam-2214	14	1	email	email	NOUN
ejpam-2214	14	2	addresses	address	NOUN
ejpam-2214	14	3	:	:	PUNCT
ejpam-2214	14	4	carpintero.carlos@gmail.com	carpintero.carlos@gmail.com	X
ejpam-2214	14	5	(	(	PUNCT
ejpam-2214	14	6	c.	c.	PROPN
ejpam-2214	14	7	carpintero	carpintero	PROPN
ejpam-2214	14	8	)	)	PUNCT
ejpam-2214	14	9	,	,	PUNCT
ejpam-2214	14	10	almuoz@hotmail.com	almuoz@hotmail.com	X
ejpam-2214	15	1	(	(	PUNCT
ejpam-2214	15	2	a.	a.	PROPN
ejpam-2214	15	3	muñoz	muñoz	PROPN
ejpam-2214	15	4	)	)	PUNCT
ejpam-2214	15	5	,	,	PUNCT
ejpam-2214	15	6	jackelinepacheco25@gmail.com	jackelinepacheco25@gmail.com	PROPN
ejpam-2214	15	7	(	(	PUNCT
ejpam-2214	15	8	j.	j.	PROPN
ejpam-2214	15	9	pacheco	pacheco	PROPN
ejpam-2214	15	10	)	)	PUNCT
ejpam-2214	15	11	,	,	PUNCT
ejpam-2214	15	12	ennisrafael@gmail.com	ennisrafael@gmail.com	X
ejpam-2214	16	1	(	(	PUNCT
ejpam-2214	16	2	e.	e.	PROPN
ejpam-2214	16	3	rosas	rosas	PROPN
ejpam-2214	16	4	)	)	PUNCT
ejpam-2214	16	5	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2214	17	1	437	437	NUM
ejpam-2214	18	1	c	c	X
ejpam-2214	18	2	©	©	PROPN
ejpam-2214	18	3	2014	2014	NUM
ejpam-2214	18	4	ejpam	ejpam	NOUN
ejpam-2214	18	5	all	all	DET
ejpam-2214	18	6	rights	right	NOUN
ejpam-2214	18	7	reserved	reserve	VERB
ejpam-2214	18	8	.	.	PUNCT
ejpam-2214	19	1	c.	c.	PROPN
ejpam-2214	19	2	carpintero	carpintero	PROPN
ejpam-2214	19	3	,	,	PUNCT
ejpam-2214	19	4	a.	a.	PROPN
ejpam-2214	19	5	muñoz	muñoz	PROPN
ejpam-2214	19	6	,	,	PUNCT
ejpam-2214	19	7	j.	j.	PROPN
ejpam-2214	19	8	pacheco	pacheco	PROPN
ejpam-2214	19	9	,	,	PUNCT
ejpam-2214	19	10	e.	e.	PROPN
ejpam-2214	19	11	rosas	rosas	PROPN
ejpam-2214	19	12	/	/	SYM
ejpam-2214	19	13	eur	eur	PROPN
ejpam-2214	19	14	.	.	PUNCT
ejpam-2214	20	1	j.	j.	PROPN
ejpam-2214	20	2	pure	pure	PROPN
ejpam-2214	20	3	appl	appl	PROPN
ejpam-2214	20	4	.	.	PROPN
ejpam-2214	20	5	math	math	PROPN
ejpam-2214	20	6	,	,	PUNCT
ejpam-2214	20	7	7	7	NUM
ejpam-2214	20	8	(	(	PUNCT
ejpam-2214	20	9	2014	2014	NUM
ejpam-2214	20	10	)	)	PUNCT
ejpam-2214	20	11	,	,	PUNCT
ejpam-2214	20	12	437	437	NUM
ejpam-2214	20	13	-	-	SYM
ejpam-2214	20	14	441	441	NUM
ejpam-2214	20	15	438	438	NUM
ejpam-2214	20	16	proposition	proposition	NOUN
ejpam-2214	20	17	6	6	NUM
ejpam-2214	20	18	.	.	PUNCT
ejpam-2214	21	1	under	under	ADP
ejpam-2214	21	2	the	the	DET
ejpam-2214	21	3	condition	condition	NOUN
ejpam-2214	21	4	of	of	ADP
ejpam-2214	21	5	proposition	proposition	NOUN
ejpam-2214	21	6	5	5	NUM
ejpam-2214	21	7	,	,	PUNCT
ejpam-2214	21	8	we	we	PRON
ejpam-2214	21	9	have	have	VERB
ejpam-2214	21	10	that	that	SCONJ
ejpam-2214	21	11	a	a	PRON
ejpam-2214	21	12	is	be	AUX
ejpam-2214	21	13	i	i	PRON
ejpam-2214	21	14	-semi	-semi	VERB
ejpam-2214	21	15	open	open	ADJ
ejpam-2214	21	16	if	if	SCONJ
ejpam-2214	21	17	and	and	CCONJ
ejpam-2214	22	1	only	only	ADV
ejpam-2214	22	2	if	if	SCONJ
ejpam-2214	22	3	cl(a	cl(a	NUM
ejpam-2214	22	4	)	)	PUNCT
ejpam-2214	22	5	is	be	AUX
ejpam-2214	22	6	i	i	PRON
ejpam-2214	22	7	-semi	-semi	VERB
ejpam-2214	22	8	open	open	ADJ
ejpam-2214	22	9	.	.	PUNCT
ejpam-2214	23	1	if	if	SCONJ
ejpam-2214	23	2	we	we	PRON
ejpam-2214	23	3	analyze	analyze	VERB
ejpam-2214	23	4	with	with	ADP
ejpam-2214	23	5	more	more	ADJ
ejpam-2214	23	6	detail	detail	NOUN
ejpam-2214	23	7	,	,	PUNCT
ejpam-2214	23	8	the	the	DET
ejpam-2214	23	9	proposition	proposition	NOUN
ejpam-2214	23	10	6	6	NUM
ejpam-2214	23	11	,	,	PUNCT
ejpam-2214	23	12	observe	observe	VERB
ejpam-2214	23	13	the	the	DET
ejpam-2214	23	14	following	follow	VERB
ejpam-2214	23	15	example	example	NOUN
ejpam-2214	23	16	.	.	PUNCT
ejpam-2214	24	1	example	example	NOUN
ejpam-2214	25	1	1	1	NUM
ejpam-2214	25	2	.	.	PUNCT
ejpam-2214	25	3	let	let	VERB
ejpam-2214	25	4	x	x	PUNCT
ejpam-2214	25	5	=	=	PRON
ejpam-2214	25	6	{	{	PUNCT
ejpam-2214	25	7	a	a	PRON
ejpam-2214	25	8	,	,	PUNCT
ejpam-2214	25	9	b	b	NOUN
ejpam-2214	25	10	,	,	PUNCT
ejpam-2214	25	11	c	c	NOUN
ejpam-2214	25	12	,	,	PUNCT
ejpam-2214	25	13	d	d	NOUN
ejpam-2214	25	14	}	}	PUNCT
ejpam-2214	25	15	with	with	ADP
ejpam-2214	25	16	topology	topology	NOUN
ejpam-2214	25	17	τ	τ	X
ejpam-2214	25	18	=	=	PUNCT
ejpam-2214	25	19	{	{	PUNCT
ejpam-2214	25	20	x	x	X
ejpam-2214	25	21	,	,	PUNCT
ejpam-2214	25	22	;	;	PUNCT
ejpam-2214	25	23	,	,	PUNCT
ejpam-2214	25	24	{	{	PUNCT
ejpam-2214	25	25	a	a	NOUN
ejpam-2214	25	26	,	,	PUNCT
ejpam-2214	25	27	c	c	NOUN
ejpam-2214	25	28	}	}	PUNCT
ejpam-2214	25	29	,	,	PUNCT
ejpam-2214	25	30	{	{	PUNCT
ejpam-2214	25	31	a	a	PRON
ejpam-2214	25	32	,	,	PUNCT
ejpam-2214	25	33	b	b	NOUN
ejpam-2214	25	34	,	,	PUNCT
ejpam-2214	25	35	c	c	NOUN
ejpam-2214	25	36	}	}	PUNCT
ejpam-2214	25	37	}	}	PUNCT
ejpam-2214	25	38	and	and	CCONJ
ejpam-2214	25	39	i	i	PRON
ejpam-2214	25	40	=	=	PUNCT
ejpam-2214	25	41	{	{	PUNCT
ejpam-2214	25	42	;	;	PUNCT
ejpam-2214	25	43	}	}	PUNCT
ejpam-2214	25	44	.	.	PUNCT
ejpam-2214	26	1	notice	notice	VERB
ejpam-2214	26	2	that	that	SCONJ
ejpam-2214	26	3	the	the	DET
ejpam-2214	26	4	collection	collection	NOUN
ejpam-2214	26	5	of	of	ADP
ejpam-2214	26	6	nonempty	nonempty	ADJ
ejpam-2214	26	7	subsets	subset	NOUN
ejpam-2214	26	8	of	of	ADP
ejpam-2214	26	9	x	x	PRON
ejpam-2214	26	10	satisfies	satisfy	VERB
ejpam-2214	26	11	the	the	DET
ejpam-2214	26	12	finite	finite	ADJ
ejpam-2214	26	13	intersection	intersection	NOUN
ejpam-2214	26	14	property	property	NOUN
ejpam-2214	26	15	and	and	CCONJ
ejpam-2214	26	16	every	every	DET
ejpam-2214	26	17	nonempty	nonempty	ADJ
ejpam-2214	26	18	open	open	ADJ
ejpam-2214	26	19	subset	subset	NOUN
ejpam-2214	26	20	of	of	ADP
ejpam-2214	26	21	x	x	PUNCT
ejpam-2214	26	22	is	be	AUX
ejpam-2214	26	23	dense	dense	ADJ
ejpam-2214	26	24	.	.	PUNCT
ejpam-2214	27	1	now	now	ADV
ejpam-2214	27	2	consider	consider	VERB
ejpam-2214	27	3	a=	a=	ADJ
ejpam-2214	27	4	{	{	PUNCT
ejpam-2214	27	5	b	b	NOUN
ejpam-2214	27	6	,	,	PUNCT
ejpam-2214	27	7	c	c	NOUN
ejpam-2214	27	8	}	}	PUNCT
ejpam-2214	27	9	,	,	PUNCT
ejpam-2214	27	10	it	it	PRON
ejpam-2214	27	11	is	be	AUX
ejpam-2214	27	12	easy	easy	ADJ
ejpam-2214	27	13	to	to	PART
ejpam-2214	27	14	see	see	VERB
ejpam-2214	27	15	that	that	PRON
ejpam-2214	27	16	cl(a	cl(a	PUNCT
ejpam-2214	27	17	)	)	PUNCT
ejpam-2214	27	18	=	=	PUNCT
ejpam-2214	28	1	x	x	X
ejpam-2214	28	2	is	be	AUX
ejpam-2214	28	3	i	i	NOUN
ejpam-2214	28	4	-	-	PUNCT
ejpam-2214	28	5	semi	semi	ADV
ejpam-2214	28	6	open	open	ADJ
ejpam-2214	28	7	but	but	CCONJ
ejpam-2214	28	8	a	a	PRON
ejpam-2214	28	9	is	be	AUX
ejpam-2214	28	10	not	not	PART
ejpam-2214	28	11	i	i	NOUN
ejpam-2214	28	12	-	-	PUNCT
ejpam-2214	28	13	semi	semi	ADV
ejpam-2214	28	14	open	open	ADJ
ejpam-2214	28	15	.	.	PUNCT
ejpam-2214	29	1	this	this	DET
ejpam-2214	29	2	example	example	NOUN
ejpam-2214	29	3	shows	show	VERB
ejpam-2214	29	4	that	that	SCONJ
ejpam-2214	29	5	the	the	DET
ejpam-2214	29	6	proposition	proposition	NOUN
ejpam-2214	29	7	6	6	NUM
ejpam-2214	29	8	given	give	VERB
ejpam-2214	29	9	in	in	ADP
ejpam-2214	29	10	[	[	NOUN
ejpam-2214	29	11	1	1	NUM
ejpam-2214	29	12	]	]	PUNCT
ejpam-2214	29	13	is	be	AUX
ejpam-2214	29	14	not	not	PART
ejpam-2214	29	15	necessarily	necessarily	ADV
ejpam-2214	29	16	true	true	ADJ
ejpam-2214	29	17	.	.	PUNCT
ejpam-2214	30	1	using	use	VERB
ejpam-2214	30	2	this	this	DET
ejpam-2214	30	3	fact	fact	NOUN
ejpam-2214	30	4	,	,	PUNCT
ejpam-2214	30	5	our	our	PRON
ejpam-2214	30	6	interest	interest	NOUN
ejpam-2214	30	7	is	be	AUX
ejpam-2214	30	8	to	to	PART
ejpam-2214	30	9	find	find	VERB
ejpam-2214	30	10	some	some	DET
ejpam-2214	30	11	weaker	weak	ADJ
ejpam-2214	30	12	condition	condition	NOUN
ejpam-2214	30	13	of	of	ADP
ejpam-2214	30	14	semi	semi	ADJ
ejpam-2214	30	15	open	open	ADJ
ejpam-2214	30	16	set	set	VERB
ejpam-2214	30	17	with	with	ADP
ejpam-2214	30	18	respect	respect	NOUN
ejpam-2214	30	19	to	to	ADP
ejpam-2214	30	20	an	an	DET
ejpam-2214	30	21	ideal	ideal	NOUN
ejpam-2214	30	22	in	in	SCONJ
ejpam-2214	30	23	order	order	NOUN
ejpam-2214	30	24	to	to	PART
ejpam-2214	30	25	prove	prove	VERB
ejpam-2214	30	26	that	that	PRON
ejpam-2214	30	27	:	:	PUNCT
ejpam-2214	30	28	let	let	VERB
ejpam-2214	30	29	i	i	PRON
ejpam-2214	30	30	6=	6=	NUM
ejpam-2214	30	31	;	;	PUNCT
ejpam-2214	30	32	be	be	AUX
ejpam-2214	30	33	an	an	DET
ejpam-2214	30	34	ideal	ideal	NOUN
ejpam-2214	30	35	,	,	PUNCT
ejpam-2214	30	36	a	a	DET
ejpam-2214	30	37	⊆	⊆	NUM
ejpam-2214	30	38	x	x	SYM
ejpam-2214	30	39	satisfies	satisfie	NOUN
ejpam-2214	30	40	the	the	DET
ejpam-2214	30	41	weaker	weak	ADJ
ejpam-2214	30	42	condition	condition	NOUN
ejpam-2214	30	43	if	if	SCONJ
ejpam-2214	30	44	and	and	CCONJ
ejpam-2214	30	45	only	only	ADV
ejpam-2214	30	46	if	if	SCONJ
ejpam-2214	30	47	cl(a	cl(a	NOUN
ejpam-2214	30	48	)	)	PUNCT
ejpam-2214	30	49	satisfies	satisfy	VERB
ejpam-2214	30	50	the	the	DET
ejpam-2214	30	51	weaker	weak	ADJ
ejpam-2214	30	52	condition	condition	NOUN
ejpam-2214	30	53	.	.	PUNCT
ejpam-2214	31	1	2	2	X
ejpam-2214	31	2	.	.	X
ejpam-2214	31	3	weakly	weakly	ADJ
ejpam-2214	31	4	semi	semi	ADV
ejpam-2214	31	5	open	open	ADJ
ejpam-2214	31	6	sets	set	NOUN
ejpam-2214	31	7	with	with	ADP
ejpam-2214	31	8	respect	respect	NOUN
ejpam-2214	31	9	to	to	ADP
ejpam-2214	31	10	an	an	DET
ejpam-2214	31	11	ideal	ideal	NOUN
ejpam-2214	31	12	let	let	VERB
ejpam-2214	31	13	x	x	PRON
ejpam-2214	31	14	be	be	AUX
ejpam-2214	31	15	a	a	DET
ejpam-2214	31	16	topological	topological	ADJ
ejpam-2214	31	17	space	space	NOUN
ejpam-2214	31	18	.	.	PUNCT
ejpam-2214	32	1	recall	recall	VERB
ejpam-2214	32	2	that	that	SCONJ
ejpam-2214	32	3	a	a	DET
ejpam-2214	32	4	⊂	⊂	X
ejpam-2214	32	5	x	x	X
ejpam-2214	32	6	is	be	AUX
ejpam-2214	32	7	a	a	DET
ejpam-2214	32	8	semi	semi	ADJ
ejpam-2214	32	9	open	open	ADJ
ejpam-2214	32	10	set	set	NOUN
ejpam-2214	32	11	[	[	X
ejpam-2214	32	12	3	3	NUM
ejpam-2214	32	13	]	]	PUNCT
ejpam-2214	32	14	,	,	PUNCT
ejpam-2214	32	15	if	if	SCONJ
ejpam-2214	32	16	there	there	PRON
ejpam-2214	32	17	exists	exist	VERB
ejpam-2214	32	18	an	an	DET
ejpam-2214	32	19	open	open	ADJ
ejpam-2214	32	20	set	set	NOUN
ejpam-2214	32	21	u	u	PRON
ejpam-2214	32	22	such	such	ADJ
ejpam-2214	32	23	that	that	SCONJ
ejpam-2214	32	24	u	u	PROPN
ejpam-2214	32	25	⊂	⊂	X
ejpam-2214	32	26	a⊂	a⊂	X
ejpam-2214	32	27	cl(u	cl(u	NOUN
ejpam-2214	32	28	)	)	PUNCT
ejpam-2214	32	29	.	.	PUNCT
ejpam-2214	33	1	definition	definition	NOUN
ejpam-2214	33	2	1	1	NUM
ejpam-2214	33	3	.	.	PUNCT
ejpam-2214	34	1	a	a	DET
ejpam-2214	34	2	subset	subset	NOUN
ejpam-2214	34	3	a	a	PRON
ejpam-2214	34	4	of	of	ADP
ejpam-2214	34	5	x	x	SYM
ejpam-2214	34	6	is	be	AUX
ejpam-2214	34	7	said	say	VERB
ejpam-2214	34	8	to	to	PART
ejpam-2214	34	9	be	be	AUX
ejpam-2214	34	10	weakly	weakly	ADV
ejpam-2214	34	11	semi	semi	ADV
ejpam-2214	34	12	open	open	ADJ
ejpam-2214	34	13	set	set	VERB
ejpam-2214	34	14	with	with	ADP
ejpam-2214	34	15	respect	respect	NOUN
ejpam-2214	34	16	to	to	ADP
ejpam-2214	34	17	an	an	DET
ejpam-2214	34	18	ideal	ideal	NOUN
ejpam-2214	35	1	i	i	PRON
ejpam-2214	35	2	(	(	PUNCT
ejpam-2214	35	3	denoted	denote	VERB
ejpam-2214	35	4	by	by	ADP
ejpam-2214	35	5	weakly	weakly	ADJ
ejpam-2214	35	6	i	i	NOUN
ejpam-2214	35	7	-	-	PUNCT
ejpam-2214	35	8	semi	semi	ADV
ejpam-2214	35	9	open	open	ADJ
ejpam-2214	35	10	)	)	PUNCT
ejpam-2214	35	11	if	if	SCONJ
ejpam-2214	35	12	a=	a=	ADV
ejpam-2214	35	13	;	;	PUNCT
ejpam-2214	35	14	or	or	CCONJ
ejpam-2214	35	15	if	if	SCONJ
ejpam-2214	35	16	a	a	PRON
ejpam-2214	35	17	6=	6=	NUM
ejpam-2214	35	18	;	;	PUNCT
ejpam-2214	35	19	there	there	PRON
ejpam-2214	35	20	exists	exist	VERB
ejpam-2214	35	21	an	an	DET
ejpam-2214	35	22	open	open	ADJ
ejpam-2214	35	23	set	set	VERB
ejpam-2214	35	24	u	u	PROPN
ejpam-2214	35	25	6=	6=	PROPN
ejpam-2214	35	26	;	;	PUNCT
ejpam-2214	35	27	such	such	ADJ
ejpam-2214	35	28	that	that	SCONJ
ejpam-2214	35	29	u	u	PROPN
ejpam-2214	35	30	\	\	PROPN
ejpam-2214	35	31	a∈	a∈	PROPN
ejpam-2214	35	32	i	i	PRON
ejpam-2214	35	33	.	.	PUNCT
ejpam-2214	36	1	observe	observe	VERB
ejpam-2214	36	2	that	that	SCONJ
ejpam-2214	36	3	(	(	PUNCT
ejpam-2214	36	4	i	i	NOUN
ejpam-2214	36	5	)	)	PUNCT
ejpam-2214	36	6	for	for	ADP
ejpam-2214	36	7	any	any	DET
ejpam-2214	36	8	ideal	ideal	NOUN
ejpam-2214	36	9	i	i	PRON
ejpam-2214	36	10	any	any	DET
ejpam-2214	36	11	semi	semi	ADV
ejpam-2214	36	12	open	open	ADJ
ejpam-2214	36	13	set	set	NOUN
ejpam-2214	36	14	is	be	AUX
ejpam-2214	36	15	weakly	weakly	ADJ
ejpam-2214	36	16	i	i	PRON
ejpam-2214	36	17	-semi	-semi	VERB
ejpam-2214	36	18	open	open	ADJ
ejpam-2214	36	19	.	.	PUNCT
ejpam-2214	37	1	(	(	PUNCT
ejpam-2214	37	2	ii	ii	NOUN
ejpam-2214	37	3	)	)	PUNCT
ejpam-2214	37	4	for	for	ADP
ejpam-2214	37	5	any	any	DET
ejpam-2214	37	6	ideal	ideal	NOUN
ejpam-2214	38	1	i	i	PRON
ejpam-2214	38	2	,	,	PUNCT
ejpam-2214	38	3	if	if	SCONJ
ejpam-2214	38	4	a	a	DET
ejpam-2214	38	5	⊆	⊆	NUM
ejpam-2214	38	6	x	x	SYM
ejpam-2214	38	7	is	be	AUX
ejpam-2214	38	8	i	i	PRON
ejpam-2214	38	9	-semi	-semi	VERB
ejpam-2214	38	10	open	open	ADJ
ejpam-2214	38	11	then	then	ADV
ejpam-2214	38	12	a	a	PRON
ejpam-2214	38	13	is	be	AUX
ejpam-2214	38	14	weakly	weakly	ADJ
ejpam-2214	38	15	i	i	PRON
ejpam-2214	38	16	-semi	-semi	VERB
ejpam-2214	38	17	open	open	ADJ
ejpam-2214	38	18	.	.	PUNCT
ejpam-2214	39	1	observe	observe	VERB
ejpam-2214	39	2	that	that	SCONJ
ejpam-2214	39	3	if	if	SCONJ
ejpam-2214	39	4	a∈	a∈	PROPN
ejpam-2214	39	5	i	i	PRON
ejpam-2214	39	6	not	not	PART
ejpam-2214	39	7	necessarily	necessarily	ADV
ejpam-2214	39	8	a	a	PRON
ejpam-2214	39	9	is	be	AUX
ejpam-2214	39	10	weakly	weakly	ADJ
ejpam-2214	39	11	i	i	PRON
ejpam-2214	39	12	-semi	-semi	VERB
ejpam-2214	39	13	open	open	ADJ
ejpam-2214	39	14	.	.	PUNCT
ejpam-2214	40	1	(	(	PUNCT
ejpam-2214	40	2	iii	iii	X
ejpam-2214	40	3	)	)	PUNCT
ejpam-2214	40	4	there	there	PRON
ejpam-2214	40	5	exists	exist	VERB
ejpam-2214	40	6	weakly	weakly	ADV
ejpam-2214	40	7	i	i	PRON
ejpam-2214	40	8	-semi	-semi	VERB
ejpam-2214	40	9	open	open	ADJ
ejpam-2214	40	10	sets	set	NOUN
ejpam-2214	40	11	that	that	PRON
ejpam-2214	40	12	are	be	AUX
ejpam-2214	40	13	neither	neither	CCONJ
ejpam-2214	40	14	semi	semi	ADV
ejpam-2214	40	15	open	open	ADJ
ejpam-2214	40	16	nor	nor	CCONJ
ejpam-2214	40	17	i	i	PRON
ejpam-2214	40	18	-semi	-semi	VERB
ejpam-2214	40	19	open	open	ADJ
ejpam-2214	40	20	.	.	PUNCT
ejpam-2214	41	1	example	example	NOUN
ejpam-2214	42	1	2	2	NUM
ejpam-2214	42	2	.	.	PUNCT
ejpam-2214	42	3	let	let	VERB
ejpam-2214	42	4	x	x	PUNCT
ejpam-2214	42	5	=	=	PRON
ejpam-2214	42	6	{	{	PUNCT
ejpam-2214	42	7	a	a	DET
ejpam-2214	42	8	,	,	PUNCT
ejpam-2214	42	9	b	b	NOUN
ejpam-2214	42	10	,	,	PUNCT
ejpam-2214	42	11	c	c	NOUN
ejpam-2214	42	12	}	}	PUNCT
ejpam-2214	42	13	,	,	PUNCT
ejpam-2214	42	14	with	with	ADP
ejpam-2214	42	15	topology	topology	NOUN
ejpam-2214	42	16	τ=	τ=	X
ejpam-2214	42	17	{	{	PUNCT
ejpam-2214	42	18	;	;	PUNCT
ejpam-2214	42	19	,	,	PUNCT
ejpam-2214	42	20	x	x	X
ejpam-2214	42	21	,	,	PUNCT
ejpam-2214	42	22	{	{	PUNCT
ejpam-2214	42	23	a	a	NOUN
ejpam-2214	42	24	}	}	PUNCT
ejpam-2214	42	25	,	,	PUNCT
ejpam-2214	42	26	{	{	PUNCT
ejpam-2214	42	27	b	b	X
ejpam-2214	42	28	,	,	PUNCT
ejpam-2214	42	29	c	c	NOUN
ejpam-2214	42	30	}	}	PUNCT
ejpam-2214	42	31	}	}	PUNCT
ejpam-2214	42	32	and	and	CCONJ
ejpam-2214	42	33	i	i	PRON
ejpam-2214	42	34	=	=	PUNCT
ejpam-2214	42	35	{	{	PUNCT
ejpam-2214	42	36	;	;	PUNCT
ejpam-2214	42	37	}	}	PUNCT
ejpam-2214	42	38	.	.	PUNCT
ejpam-2214	43	1	the	the	DET
ejpam-2214	43	2	set	set	NOUN
ejpam-2214	43	3	a=	a=	NOUN
ejpam-2214	43	4	{	{	PUNCT
ejpam-2214	43	5	a	a	DET
ejpam-2214	43	6	,	,	PUNCT
ejpam-2214	43	7	b	b	NOUN
ejpam-2214	43	8	}	}	PUNCT
ejpam-2214	43	9	is	be	AUX
ejpam-2214	43	10	weakly	weakly	ADJ
ejpam-2214	43	11	i	i	PRON
ejpam-2214	43	12	-	-	PUNCT
ejpam-2214	43	13	semi	semi	ADV
ejpam-2214	43	14	open	open	ADJ
ejpam-2214	43	15	but	but	CCONJ
ejpam-2214	43	16	is	be	AUX
ejpam-2214	43	17	neither	neither	CCONJ
ejpam-2214	43	18	semi	semi	ADV
ejpam-2214	43	19	open	open	ADJ
ejpam-2214	43	20	nor	nor	CCONJ
ejpam-2214	43	21	i	i	PRON
ejpam-2214	43	22	-	-	PUNCT
ejpam-2214	43	23	semi	semi	ADV
ejpam-2214	43	24	open	open	ADJ
ejpam-2214	43	25	.	.	PUNCT
ejpam-2214	44	1	now	now	ADV
ejpam-2214	44	2	we	we	PRON
ejpam-2214	44	3	characterize	characterize	VERB
ejpam-2214	44	4	the	the	DET
ejpam-2214	44	5	weakly	weakly	ADJ
ejpam-2214	44	6	i	i	PRON
ejpam-2214	44	7	-semi	-semi	VERB
ejpam-2214	44	8	open	open	ADJ
ejpam-2214	44	9	sets	set	NOUN
ejpam-2214	44	10	.	.	PUNCT
ejpam-2214	45	1	theorem	theorem	NOUN
ejpam-2214	45	2	1	1	NUM
ejpam-2214	45	3	.	.	PUNCT
ejpam-2214	46	1	let	let	VERB
ejpam-2214	46	2	a	a	DET
ejpam-2214	46	3	6=	6=	NOUN
ejpam-2214	46	4	;	;	PUNCT
ejpam-2214	46	5	a	a	DET
ejpam-2214	46	6	subset	subset	NOUN
ejpam-2214	46	7	of	of	ADP
ejpam-2214	46	8	x	x	PUNCT
ejpam-2214	46	9	and	and	CCONJ
ejpam-2214	46	10	i	i	PRON
ejpam-2214	46	11	an	an	DET
ejpam-2214	46	12	ideal	ideal	NOUN
ejpam-2214	46	13	.	.	PUNCT
ejpam-2214	47	1	a	a	PRON
ejpam-2214	47	2	is	be	AUX
ejpam-2214	47	3	weakly	weakly	ADJ
ejpam-2214	47	4	i	i	PRON
ejpam-2214	47	5	-	-	PUNCT
ejpam-2214	47	6	semi	semi	ADV
ejpam-2214	47	7	open	open	ADJ
ejpam-2214	47	8	if	if	SCONJ
ejpam-2214	47	9	and	and	CCONJ
ejpam-2214	47	10	only	only	ADV
ejpam-2214	47	11	if	if	SCONJ
ejpam-2214	47	12	there	there	PRON
ejpam-2214	47	13	exists	exist	VERB
ejpam-2214	47	14	an	an	DET
ejpam-2214	47	15	open	open	ADJ
ejpam-2214	47	16	set	set	NOUN
ejpam-2214	47	17	u	u	NOUN
ejpam-2214	47	18	and	and	CCONJ
ejpam-2214	47	19	c	c	NOUN
ejpam-2214	47	20	∈	∈	PROPN
ejpam-2214	48	1	i	i	PRON
ejpam-2214	48	2	such	such	ADJ
ejpam-2214	48	3	that	that	SCONJ
ejpam-2214	48	4	(	(	PUNCT
ejpam-2214	48	5	u	u	NOUN
ejpam-2214	48	6	\	\	PROPN
ejpam-2214	48	7	c	c	X
ejpam-2214	48	8	)	)	PUNCT
ejpam-2214	48	9	⊂	⊂	PROPN
ejpam-2214	48	10	a	a	DET
ejpam-2214	48	11	proof	proof	NOUN
ejpam-2214	48	12	.	.	PUNCT
ejpam-2214	48	13	suppose	suppose	VERB
ejpam-2214	48	14	that	that	SCONJ
ejpam-2214	48	15	a	a	DET
ejpam-2214	48	16	6=	6=	NUM
ejpam-2214	48	17	;	;	PUNCT
ejpam-2214	48	18	is	be	AUX
ejpam-2214	48	19	weakly	weakly	ADJ
ejpam-2214	48	20	i	i	PRON
ejpam-2214	48	21	-semi	-semi	VERB
ejpam-2214	48	22	open	open	ADJ
ejpam-2214	48	23	,	,	PUNCT
ejpam-2214	48	24	then	then	ADV
ejpam-2214	48	25	there	there	PRON
ejpam-2214	48	26	exists	exist	VERB
ejpam-2214	48	27	an	an	DET
ejpam-2214	48	28	open	open	ADJ
ejpam-2214	48	29	set	set	VERB
ejpam-2214	48	30	u	u	PROPN
ejpam-2214	48	31	6=	6=	PROPN
ejpam-2214	48	32	;	;	PUNCT
ejpam-2214	49	1	such	such	ADJ
ejpam-2214	49	2	that	that	SCONJ
ejpam-2214	49	3	u	u	NOUN
ejpam-2214	49	4	\	\	VERB
ejpam-2214	49	5	a	a	DET
ejpam-2214	49	6	∈	∈	PROPN
ejpam-2214	49	7	i	i	PRON
ejpam-2214	49	8	.	.	PUNCT
ejpam-2214	50	1	take	take	VERB
ejpam-2214	50	2	c	c	NOUN
ejpam-2214	50	3	=	=	SYM
ejpam-2214	50	4	u	u	NOUN
ejpam-2214	50	5	\	\	PROPN
ejpam-2214	50	6	a	a	DET
ejpam-2214	50	7	=	=	SYM
ejpam-2214	50	8	u	u	NOUN
ejpam-2214	50	9	∩	∩	NOUN
ejpam-2214	50	10	(	(	PUNCT
ejpam-2214	50	11	x	x	SYM
ejpam-2214	50	12	\	\	PROPN
ejpam-2214	50	13	a	a	PRON
ejpam-2214	50	14	)	)	PUNCT
ejpam-2214	50	15	.	.	PUNCT
ejpam-2214	51	1	then	then	ADV
ejpam-2214	51	2	u	u	X
ejpam-2214	51	3	\	\	PROPN
ejpam-2214	51	4	c	c	PROPN
ejpam-2214	51	5	⊂	⊂	PROPN
ejpam-2214	51	6	a.	a.	PROPN
ejpam-2214	51	7	reciprocally	reciprocally	ADV
ejpam-2214	51	8	suppose	suppose	VERB
ejpam-2214	51	9	that	that	SCONJ
ejpam-2214	51	10	there	there	PRON
ejpam-2214	51	11	exists	exist	VERB
ejpam-2214	51	12	an	an	DET
ejpam-2214	51	13	open	open	ADJ
ejpam-2214	51	14	set	set	NOUN
ejpam-2214	51	15	u	u	NOUN
ejpam-2214	51	16	and	and	CCONJ
ejpam-2214	51	17	c	c	NOUN
ejpam-2214	51	18	∈	∈	PROPN
ejpam-2214	51	19	i	i	PRON
ejpam-2214	51	20	such	such	ADJ
ejpam-2214	51	21	that	that	SCONJ
ejpam-2214	51	22	(	(	PUNCT
ejpam-2214	51	23	u	u	NOUN
ejpam-2214	51	24	\	\	PROPN
ejpam-2214	51	25	c	c	X
ejpam-2214	51	26	)	)	PUNCT
ejpam-2214	51	27	⊂	⊂	PROPN
ejpam-2214	51	28	a	a	X
ejpam-2214	51	29	,	,	PUNCT
ejpam-2214	51	30	then	then	ADV
ejpam-2214	51	31	(	(	PUNCT
ejpam-2214	51	32	u	u	NOUN
ejpam-2214	51	33	\	\	PROPN
ejpam-2214	51	34	a	a	PRON
ejpam-2214	51	35	)	)	PUNCT
ejpam-2214	51	36	⊂	⊂	PROPN
ejpam-2214	51	37	c	c	PROPN
ejpam-2214	51	38	,	,	PUNCT
ejpam-2214	51	39	follows	follow	VERB
ejpam-2214	51	40	that	that	SCONJ
ejpam-2214	51	41	u	u	PROPN
ejpam-2214	51	42	\	\	PROPN
ejpam-2214	51	43	a∈	a∈	PROPN
ejpam-2214	51	44	i	i	PRON
ejpam-2214	51	45	.	.	PUNCT
ejpam-2214	52	1	definition	definition	NOUN
ejpam-2214	52	2	2	2	NUM
ejpam-2214	52	3	.	.	PUNCT
ejpam-2214	53	1	a	a	DET
ejpam-2214	53	2	subset	subset	NOUN
ejpam-2214	53	3	a	a	PRON
ejpam-2214	53	4	of	of	ADP
ejpam-2214	53	5	x	x	SYM
ejpam-2214	53	6	is	be	AUX
ejpam-2214	53	7	said	say	VERB
ejpam-2214	53	8	to	to	PART
ejpam-2214	53	9	be	be	AUX
ejpam-2214	53	10	weakly	weakly	ADJ
ejpam-2214	53	11	i	i	PRON
ejpam-2214	53	12	-	-	PUNCT
ejpam-2214	53	13	semi	semi	ADV
ejpam-2214	53	14	closed	close	VERB
ejpam-2214	53	15	if	if	SCONJ
ejpam-2214	53	16	x	x	ADP
ejpam-2214	53	17	\	\	PROPN
ejpam-2214	53	18	a	a	PRON
ejpam-2214	53	19	is	be	AUX
ejpam-2214	53	20	weakly	weakly	ADJ
ejpam-2214	53	21	i	i	PRON
ejpam-2214	53	22	-	-	PUNCT
ejpam-2214	53	23	semi	semi	ADV
ejpam-2214	53	24	open	open	ADJ
ejpam-2214	53	25	.	.	PUNCT
ejpam-2214	54	1	theorem	theorem	NOUN
ejpam-2214	54	2	2	2	NUM
ejpam-2214	54	3	.	.	X
ejpam-2214	55	1	let	let	AUX
ejpam-2214	55	2	(	(	PUNCT
ejpam-2214	55	3	x	x	X
ejpam-2214	55	4	,	,	PUNCT
ejpam-2214	55	5	τ	τ	X
ejpam-2214	55	6	)	)	PUNCT
ejpam-2214	55	7	be	be	VERB
ejpam-2214	55	8	a	a	DET
ejpam-2214	55	9	topological	topological	ADJ
ejpam-2214	55	10	space	space	NOUN
ejpam-2214	55	11	,	,	PUNCT
ejpam-2214	55	12	i	i	PRON
ejpam-2214	55	13	an	an	DET
ejpam-2214	55	14	ideal	ideal	NOUN
ejpam-2214	55	15	and	and	CCONJ
ejpam-2214	55	16	a⊆	a⊆	NOUN
ejpam-2214	55	17	x	x	X
ejpam-2214	55	18	.	.	PUNCT
ejpam-2214	56	1	if	if	SCONJ
ejpam-2214	56	2	a	a	PRON
ejpam-2214	56	3	is	be	AUX
ejpam-2214	56	4	weakly	weakly	ADJ
ejpam-2214	56	5	i	i	PRON
ejpam-2214	56	6	-	-	PUNCT
ejpam-2214	56	7	semi	semi	ADV
ejpam-2214	56	8	closed	closed	ADJ
ejpam-2214	56	9	then	then	ADV
ejpam-2214	56	10	a⊂	a⊂	X
ejpam-2214	56	11	(	(	PUNCT
ejpam-2214	56	12	k	k	X
ejpam-2214	56	13	∪	∪	X
ejpam-2214	56	14	b	b	NOUN
ejpam-2214	56	15	)	)	PUNCT
ejpam-2214	56	16	for	for	ADP
ejpam-2214	56	17	some	some	DET
ejpam-2214	56	18	closed	close	VERB
ejpam-2214	56	19	subset	subset	NOUN
ejpam-2214	57	1	k	k	PROPN
ejpam-2214	57	2	of	of	ADP
ejpam-2214	57	3	x	x	PROPN
ejpam-2214	57	4	and	and	CCONJ
ejpam-2214	57	5	b	b	X
ejpam-2214	57	6	∈	∈	PROPN
ejpam-2214	58	1	i	i	PRON
ejpam-2214	58	2	.	.	PUNCT
ejpam-2214	59	1	c.	c.	PROPN
ejpam-2214	59	2	carpintero	carpintero	PROPN
ejpam-2214	59	3	,	,	PUNCT
ejpam-2214	59	4	a.	a.	PROPN
ejpam-2214	59	5	muñoz	muñoz	PROPN
ejpam-2214	59	6	,	,	PUNCT
ejpam-2214	59	7	j.	j.	PROPN
ejpam-2214	59	8	pacheco	pacheco	PROPN
ejpam-2214	59	9	,	,	PUNCT
ejpam-2214	59	10	e.	e.	PROPN
ejpam-2214	59	11	rosas	rosas	PROPN
ejpam-2214	59	12	/	/	SYM
ejpam-2214	59	13	eur	eur	PROPN
ejpam-2214	59	14	.	.	PUNCT
ejpam-2214	60	1	j.	j.	PROPN
ejpam-2214	60	2	pure	pure	PROPN
ejpam-2214	60	3	appl	appl	PROPN
ejpam-2214	60	4	.	.	PROPN
ejpam-2214	60	5	math	math	PROPN
ejpam-2214	60	6	,	,	PUNCT
ejpam-2214	60	7	7	7	NUM
ejpam-2214	60	8	(	(	PUNCT
ejpam-2214	60	9	2014	2014	NUM
ejpam-2214	60	10	)	)	PUNCT
ejpam-2214	60	11	,	,	PUNCT
ejpam-2214	60	12	437	437	NUM
ejpam-2214	60	13	-	-	SYM
ejpam-2214	60	14	441	441	NUM
ejpam-2214	60	15	439	439	NUM
ejpam-2214	60	16	proof	proof	NOUN
ejpam-2214	60	17	.	.	PUNCT
ejpam-2214	61	1	if	if	SCONJ
ejpam-2214	61	2	a	a	PRON
ejpam-2214	61	3	is	be	AUX
ejpam-2214	61	4	weakly	weakly	ADJ
ejpam-2214	61	5	i	i	PRON
ejpam-2214	61	6	-semi	-semi	NOUN
ejpam-2214	61	7	closed	closed	ADJ
ejpam-2214	61	8	,	,	PUNCT
ejpam-2214	61	9	then	then	ADV
ejpam-2214	61	10	x	x	PUNCT
ejpam-2214	61	11	\a	\a	ADJ
ejpam-2214	61	12	is	be	AUX
ejpam-2214	61	13	weakly	weakly	ADJ
ejpam-2214	61	14	i	i	PRON
ejpam-2214	61	15	-semi	-semi	VERB
ejpam-2214	61	16	open	open	ADJ
ejpam-2214	61	17	.	.	PUNCT
ejpam-2214	62	1	if	if	SCONJ
ejpam-2214	62	2	x	x	X
ejpam-2214	62	3	\a=	\a=	NOUN
ejpam-2214	62	4	;	;	PUNCT
ejpam-2214	62	5	,	,	PUNCT
ejpam-2214	62	6	then	then	ADV
ejpam-2214	62	7	a=	a=	VERB
ejpam-2214	62	8	x	x	SYM
ejpam-2214	62	9	,	,	PUNCT
ejpam-2214	62	10	in	in	ADP
ejpam-2214	62	11	consequence	consequence	NOUN
ejpam-2214	62	12	,	,	PUNCT
ejpam-2214	62	13	the	the	PRON
ejpam-2214	62	14	;	;	PUNCT
ejpam-2214	62	15	is	be	AUX
ejpam-2214	62	16	weakly	weakly	ADJ
ejpam-2214	62	17	i	i	PRON
ejpam-2214	62	18	-semi	-semi	NOUN
ejpam-2214	62	19	closed	closed	ADJ
ejpam-2214	62	20	.	.	PUNCT
ejpam-2214	63	1	if	if	SCONJ
ejpam-2214	63	2	x	x	PRON
ejpam-2214	63	3	\a	\a	PROPN
ejpam-2214	63	4	6=	6=	NUM
ejpam-2214	63	5	;	;	PUNCT
ejpam-2214	63	6	,	,	PUNCT
ejpam-2214	63	7	then	then	ADV
ejpam-2214	63	8	there	there	PRON
ejpam-2214	63	9	exists	exist	VERB
ejpam-2214	63	10	an	an	DET
ejpam-2214	63	11	open	open	ADJ
ejpam-2214	63	12	set	set	NOUN
ejpam-2214	63	13	u	u	NOUN
ejpam-2214	63	14	and	and	CCONJ
ejpam-2214	63	15	b	b	NOUN
ejpam-2214	63	16	∈	∈	PROPN
ejpam-2214	63	17	i	i	PRON
ejpam-2214	63	18	such	such	ADJ
ejpam-2214	63	19	that	that	SCONJ
ejpam-2214	63	20	(	(	PUNCT
ejpam-2214	63	21	u	u	NOUN
ejpam-2214	63	22	\b	\b	NOUN
ejpam-2214	63	23	)	)	PUNCT
ejpam-2214	64	1	⊂	⊂	PROPN
ejpam-2214	64	2	(	(	PUNCT
ejpam-2214	64	3	x	x	SYM
ejpam-2214	64	4	\a	\a	NUM
ejpam-2214	64	5	)	)	PUNCT
ejpam-2214	64	6	,	,	PUNCT
ejpam-2214	64	7	follows	follow	VERB
ejpam-2214	64	8	that	that	SCONJ
ejpam-2214	64	9	a⊂	a⊂	ADP
ejpam-2214	64	10	x	x	SYM
ejpam-2214	64	11	\	\	PUNCT
ejpam-2214	64	12	(	(	PUNCT
ejpam-2214	64	13	u	u	NOUN
ejpam-2214	64	14	\b	\b	NOUN
ejpam-2214	64	15	)	)	PUNCT
ejpam-2214	64	16	=	=	PUNCT
ejpam-2214	65	1	x	x	X
ejpam-2214	65	2	⊂	⊂	PROPN
ejpam-2214	65	3	(	(	PUNCT
ejpam-2214	65	4	u	u	NOUN
ejpam-2214	65	5	∩	∩	NOUN
ejpam-2214	65	6	(	(	PUNCT
ejpam-2214	65	7	x	x	NOUN
ejpam-2214	65	8	\b	\b	ADJ
ejpam-2214	65	9	)	)	PUNCT
ejpam-2214	65	10	)	)	PUNCT
ejpam-2214	66	1	=	=	PUNCT
ejpam-2214	66	2	(	(	PUNCT
ejpam-2214	66	3	x	x	SYM
ejpam-2214	66	4	\u)∩b	\u)∩b	PROPN
ejpam-2214	66	5	.	.	PUNCT
ejpam-2214	67	1	take	take	VERB
ejpam-2214	67	2	k	k	NOUN
ejpam-2214	67	3	=	=	PUNCT
ejpam-2214	67	4	(	(	PUNCT
ejpam-2214	67	5	x	x	PUNCT
ejpam-2214	67	6	⊂	⊂	X
ejpam-2214	67	7	u	u	NOUN
ejpam-2214	67	8	)	)	PUNCT
ejpam-2214	67	9	then	then	ADV
ejpam-2214	67	10	a⊂	a⊂	VERB
ejpam-2214	67	11	k	k	PROPN
ejpam-2214	67	12	∪	∪	X
ejpam-2214	67	13	b.	b.	PROPN
ejpam-2214	67	14	the	the	DET
ejpam-2214	67	15	converse	converse	NOUN
ejpam-2214	67	16	of	of	ADP
ejpam-2214	67	17	the	the	DET
ejpam-2214	67	18	above	above	ADJ
ejpam-2214	67	19	theorem	theorem	NOUN
ejpam-2214	67	20	is	be	AUX
ejpam-2214	67	21	not	not	PART
ejpam-2214	67	22	necessarily	necessarily	ADV
ejpam-2214	67	23	true	true	ADJ
ejpam-2214	67	24	,	,	PUNCT
ejpam-2214	67	25	as	as	SCONJ
ejpam-2214	67	26	we	we	PRON
ejpam-2214	67	27	see	see	VERB
ejpam-2214	67	28	in	in	ADP
ejpam-2214	67	29	the	the	DET
ejpam-2214	67	30	following	follow	VERB
ejpam-2214	67	31	example	example	NOUN
ejpam-2214	67	32	.	.	PUNCT
ejpam-2214	68	1	example	example	NOUN
ejpam-2214	69	1	3	3	X
ejpam-2214	69	2	.	.	PUNCT
ejpam-2214	69	3	let	let	VERB
ejpam-2214	69	4	x	x	PUNCT
ejpam-2214	69	5	=	=	PRON
ejpam-2214	69	6	{	{	PUNCT
ejpam-2214	69	7	a	a	PRON
ejpam-2214	69	8	,	,	PUNCT
ejpam-2214	69	9	b	b	NOUN
ejpam-2214	69	10	,	,	PUNCT
ejpam-2214	69	11	c	c	NOUN
ejpam-2214	69	12	,	,	PUNCT
ejpam-2214	69	13	d	d	NOUN
ejpam-2214	69	14	}	}	PUNCT
ejpam-2214	69	15	with	with	ADP
ejpam-2214	69	16	topology	topology	NOUN
ejpam-2214	69	17	τ	τ	X
ejpam-2214	69	18	=	=	PUNCT
ejpam-2214	69	19	{	{	PUNCT
ejpam-2214	69	20	;	;	PUNCT
ejpam-2214	69	21	,	,	PUNCT
ejpam-2214	69	22	x	x	X
ejpam-2214	69	23	,	,	PUNCT
ejpam-2214	69	24	{	{	PUNCT
ejpam-2214	69	25	a	a	DET
ejpam-2214	69	26	,	,	PUNCT
ejpam-2214	69	27	b	b	NOUN
ejpam-2214	69	28	}	}	PUNCT
ejpam-2214	69	29	,	,	PUNCT
ejpam-2214	69	30	{	{	PUNCT
ejpam-2214	69	31	c	c	X
ejpam-2214	69	32	,	,	PUNCT
ejpam-2214	69	33	d	d	NOUN
ejpam-2214	69	34	}	}	PUNCT
ejpam-2214	69	35	}	}	PUNCT
ejpam-2214	69	36	.	.	PUNCT
ejpam-2214	70	1	take	take	VERB
ejpam-2214	70	2	i	i	PRON
ejpam-2214	70	3	=	=	PUNCT
ejpam-2214	70	4	{	{	PUNCT
ejpam-2214	70	5	;	;	PUNCT
ejpam-2214	70	6	}	}	PUNCT
ejpam-2214	70	7	and	and	CCONJ
ejpam-2214	70	8	a=	a=	VERB
ejpam-2214	70	9	{	{	PUNCT
ejpam-2214	70	10	a	a	X
ejpam-2214	70	11	,	,	PUNCT
ejpam-2214	70	12	c	c	NOUN
ejpam-2214	70	13	}	}	PUNCT
ejpam-2214	70	14	.	.	PUNCT
ejpam-2214	71	1	if	if	SCONJ
ejpam-2214	71	2	k	k	PROPN
ejpam-2214	71	3	=	=	PUNCT
ejpam-2214	71	4	x	x	PROPN
ejpam-2214	71	5	and	and	CCONJ
ejpam-2214	71	6	b	b	X
ejpam-2214	71	7	=	=	X
ejpam-2214	71	8	;	;	PUNCT
ejpam-2214	71	9	,	,	PUNCT
ejpam-2214	71	10	a⊂	a⊂	X
ejpam-2214	71	11	k	k	PROPN
ejpam-2214	71	12	∪	∪	PROPN
ejpam-2214	71	13	b	b	PROPN
ejpam-2214	71	14	but	but	CCONJ
ejpam-2214	71	15	a	a	PRON
ejpam-2214	71	16	is	be	AUX
ejpam-2214	71	17	not	not	PART
ejpam-2214	71	18	weakly	weakly	ADJ
ejpam-2214	71	19	i	i	PRON
ejpam-2214	71	20	-	-	PUNCT
ejpam-2214	71	21	semi	semi	ADV
ejpam-2214	71	22	closed	closed	ADJ
ejpam-2214	71	23	,	,	PUNCT
ejpam-2214	71	24	because	because	SCONJ
ejpam-2214	71	25	x	x	SYM
ejpam-2214	71	26	\	\	PROPN
ejpam-2214	71	27	a	a	PRON
ejpam-2214	71	28	is	be	AUX
ejpam-2214	71	29	not	not	PART
ejpam-2214	71	30	weakly	weakly	ADJ
ejpam-2214	71	31	i	i	PRON
ejpam-2214	71	32	-	-	PUNCT
ejpam-2214	71	33	semi	semi	ADV
ejpam-2214	71	34	open	open	ADJ
ejpam-2214	71	35	.	.	PUNCT
ejpam-2214	72	1	theorem	theorem	NOUN
ejpam-2214	72	2	3	3	NUM
ejpam-2214	72	3	.	.	PUNCT
ejpam-2214	73	1	the	the	DET
ejpam-2214	73	2	arbitrary	arbitrary	ADJ
ejpam-2214	73	3	union	union	NOUN
ejpam-2214	73	4	of	of	ADP
ejpam-2214	73	5	weakly	weakly	ADJ
ejpam-2214	73	6	i	i	NOUN
ejpam-2214	73	7	-	-	PUNCT
ejpam-2214	73	8	semi	semi	ADV
ejpam-2214	73	9	open	open	ADJ
ejpam-2214	73	10	sets	set	NOUN
ejpam-2214	73	11	is	be	AUX
ejpam-2214	73	12	weakly	weakly	ADJ
ejpam-2214	73	13	i	i	PRON
ejpam-2214	73	14	-	-	PUNCT
ejpam-2214	73	15	semi	semi	ADV
ejpam-2214	73	16	open	open	ADJ
ejpam-2214	73	17	.	.	PUNCT
ejpam-2214	74	1	proof	proof	NOUN
ejpam-2214	74	2	.	.	PUNCT
ejpam-2214	75	1	let	let	VERB
ejpam-2214	75	2	{	{	PUNCT
ejpam-2214	75	3	aα}α∈j	aα}α∈j	VERB
ejpam-2214	75	4	be	be	AUX
ejpam-2214	75	5	a	a	DET
ejpam-2214	75	6	collection	collection	NOUN
ejpam-2214	75	7	of	of	ADP
ejpam-2214	75	8	weakly	weakly	ADJ
ejpam-2214	75	9	i	i	PRON
ejpam-2214	75	10	semi	semi	ADV
ejpam-2214	75	11	open	open	ADJ
ejpam-2214	75	12	sets	set	NOUN
ejpam-2214	75	13	,	,	PUNCT
ejpam-2214	75	14	then	then	ADV
ejpam-2214	75	15	for	for	ADP
ejpam-2214	75	16	each	each	DET
ejpam-2214	75	17	aα	aα	NOUN
ejpam-2214	75	18	with	with	ADP
ejpam-2214	75	19	α	α	PROPN
ejpam-2214	75	20	∈	∈	PROPN
ejpam-2214	75	21	j	j	PROPN
ejpam-2214	75	22	,	,	PUNCT
ejpam-2214	75	23	there	there	PRON
ejpam-2214	75	24	exists	exist	VERB
ejpam-2214	75	25	uα	uα	PROPN
ejpam-2214	75	26	,	,	PUNCT
ejpam-2214	75	27	α	α	PROPN
ejpam-2214	75	28	∈	∈	PROPN
ejpam-2214	75	29	j	j	NOUN
ejpam-2214	75	30	such	such	ADJ
ejpam-2214	75	31	that	that	SCONJ
ejpam-2214	75	32	uα	uα	PROPN
ejpam-2214	75	33	\	\	PROPN
ejpam-2214	75	34	aα	aα	NOUN
ejpam-2214	75	35	∈	∈	PROPN
ejpam-2214	76	1	i	i	PRON
ejpam-2214	76	2	.	.	PUNCT
ejpam-2214	77	1	now	now	ADV
ejpam-2214	77	2	if	if	SCONJ
ejpam-2214	77	3	we	we	PRON
ejpam-2214	77	4	take	take	VERB
ejpam-2214	77	5	a	a	DET
ejpam-2214	77	6	fixed	fix	VERB
ejpam-2214	77	7	α′	α′	NOUN
ejpam-2214	77	8	in	in	ADP
ejpam-2214	77	9	j	j	PROPN
ejpam-2214	77	10	then	then	ADV
ejpam-2214	77	11	u	u	NOUN
ejpam-2214	77	12	′α	′α	NOUN
ejpam-2214	77	13	\	\	PROPN
ejpam-2214	78	1	⋃	⋃	PUNCT
ejpam-2214	78	2	α∈j	α∈j	NOUN
ejpam-2214	78	3	aα	aα	NOUN
ejpam-2214	78	4	⊂	⊂	PROPN
ejpam-2214	78	5	u	u	NOUN
ejpam-2214	78	6	′α	′α	PROPN
ejpam-2214	78	7	\	\	PROPN
ejpam-2214	78	8	a′α	a′α	ADP
ejpam-2214	78	9	∈	∈	PROPN
ejpam-2214	79	1	i	i	PRON
ejpam-2214	79	2	.	.	PUNCT
ejpam-2214	80	1	in	in	ADP
ejpam-2214	80	2	consequence	consequence	NOUN
ejpam-2214	80	3	,	,	PUNCT
ejpam-2214	80	4	⋃	⋃	SCONJ
ejpam-2214	80	5	α∈j	α∈j	ADJ
ejpam-2214	80	6	aα	aα	NOUN
ejpam-2214	80	7	is	be	AUX
ejpam-2214	80	8	weakly	weakly	ADJ
ejpam-2214	80	9	i	i	PRON
ejpam-2214	80	10	-semi	-semi	VERB
ejpam-2214	80	11	open	open	ADJ
ejpam-2214	80	12	.	.	PUNCT
ejpam-2214	81	1	the	the	DET
ejpam-2214	81	2	intersection	intersection	NOUN
ejpam-2214	81	3	of	of	ADP
ejpam-2214	81	4	weakly	weakly	ADJ
ejpam-2214	81	5	i	i	PRON
ejpam-2214	81	6	-semi	-semi	VERB
ejpam-2214	81	7	open	open	ADJ
ejpam-2214	81	8	sets	set	NOUN
ejpam-2214	81	9	is	be	AUX
ejpam-2214	81	10	not	not	PART
ejpam-2214	81	11	necessarily	necessarily	ADV
ejpam-2214	81	12	weakly	weakly	ADJ
ejpam-2214	81	13	i	i	PRON
ejpam-2214	81	14	-semi	-semi	VERB
ejpam-2214	81	15	open	open	ADJ
ejpam-2214	81	16	as	as	SCONJ
ejpam-2214	81	17	we	we	PRON
ejpam-2214	81	18	can	can	AUX
ejpam-2214	81	19	see	see	VERB
ejpam-2214	81	20	in	in	ADP
ejpam-2214	81	21	the	the	DET
ejpam-2214	81	22	following	follow	VERB
ejpam-2214	81	23	example	example	NOUN
ejpam-2214	81	24	.	.	PUNCT
ejpam-2214	82	1	example	example	NOUN
ejpam-2214	83	1	4	4	NUM
ejpam-2214	83	2	.	.	PUNCT
ejpam-2214	83	3	let	let	VERB
ejpam-2214	83	4	x	x	PUNCT
ejpam-2214	83	5	=	=	PRON
ejpam-2214	83	6	{	{	PUNCT
ejpam-2214	83	7	a	a	PRON
ejpam-2214	83	8	,	,	PUNCT
ejpam-2214	83	9	b	b	NOUN
ejpam-2214	83	10	,	,	PUNCT
ejpam-2214	83	11	c	c	NOUN
ejpam-2214	83	12	}	}	PUNCT
ejpam-2214	83	13	with	with	ADP
ejpam-2214	83	14	topology	topology	NOUN
ejpam-2214	83	15	τ	τ	X
ejpam-2214	83	16	=	=	PUNCT
ejpam-2214	83	17	{	{	PUNCT
ejpam-2214	83	18	x	x	X
ejpam-2214	83	19	,	,	PUNCT
ejpam-2214	83	20	;	;	PUNCT
ejpam-2214	83	21	,	,	PUNCT
ejpam-2214	83	22	{	{	PUNCT
ejpam-2214	83	23	a	a	X
ejpam-2214	83	24	}	}	PUNCT
ejpam-2214	83	25	,	,	PUNCT
ejpam-2214	83	26	{	{	PUNCT
ejpam-2214	83	27	c	c	X
ejpam-2214	83	28	}	}	PUNCT
ejpam-2214	83	29	,	,	PUNCT
ejpam-2214	83	30	{	{	PUNCT
ejpam-2214	83	31	a	a	PRON
ejpam-2214	83	32	,	,	PUNCT
ejpam-2214	83	33	c	c	NOUN
ejpam-2214	83	34	}	}	PUNCT
ejpam-2214	83	35	}	}	PUNCT
ejpam-2214	83	36	and	and	CCONJ
ejpam-2214	83	37	i	i	PRON
ejpam-2214	83	38	=	=	PUNCT
ejpam-2214	83	39	{	{	PUNCT
ejpam-2214	83	40	;	;	PUNCT
ejpam-2214	83	41	}	}	PUNCT
ejpam-2214	83	42	.	.	PUNCT
ejpam-2214	84	1	consider	consider	VERB
ejpam-2214	84	2	a=	a=	VERB
ejpam-2214	84	3	{	{	PUNCT
ejpam-2214	84	4	a	a	DET
ejpam-2214	84	5	,	,	PUNCT
ejpam-2214	84	6	b	b	NOUN
ejpam-2214	84	7	}	}	PUNCT
ejpam-2214	84	8	and	and	CCONJ
ejpam-2214	84	9	b	b	X
ejpam-2214	84	10	=	=	SYM
ejpam-2214	84	11	{	{	PUNCT
ejpam-2214	84	12	b	b	NOUN
ejpam-2214	84	13	,	,	PUNCT
ejpam-2214	84	14	c	c	NOUN
ejpam-2214	84	15	}	}	PUNCT
ejpam-2214	84	16	.	.	PUNCT
ejpam-2214	85	1	it	it	PRON
ejpam-2214	85	2	is	be	AUX
ejpam-2214	85	3	easy	easy	ADJ
ejpam-2214	85	4	to	to	PART
ejpam-2214	85	5	see	see	VERB
ejpam-2214	85	6	that	that	SCONJ
ejpam-2214	85	7	a	a	PRON
ejpam-2214	85	8	and	and	CCONJ
ejpam-2214	85	9	b	b	NOUN
ejpam-2214	85	10	are	be	AUX
ejpam-2214	85	11	weakly	weakly	ADJ
ejpam-2214	85	12	i	i	PRON
ejpam-2214	85	13	-	-	PUNCT
ejpam-2214	85	14	semi	semi	ADV
ejpam-2214	85	15	open	open	ADJ
ejpam-2214	85	16	sets	set	NOUN
ejpam-2214	85	17	but	but	CCONJ
ejpam-2214	85	18	a∩b	a∩b	PROPN
ejpam-2214	85	19	=	=	PUNCT
ejpam-2214	85	20	{	{	PUNCT
ejpam-2214	85	21	b	b	NOUN
ejpam-2214	85	22	}	}	PUNCT
ejpam-2214	85	23	is	be	AUX
ejpam-2214	85	24	not	not	PART
ejpam-2214	85	25	weakly	weakly	ADJ
ejpam-2214	85	26	i	i	PRON
ejpam-2214	85	27	-	-	PUNCT
ejpam-2214	85	28	semi	semi	ADV
ejpam-2214	85	29	open	open	ADJ
ejpam-2214	85	30	.	.	PUNCT
ejpam-2214	86	1	remark	remark	NOUN
ejpam-2214	86	2	1	1	NUM
ejpam-2214	86	3	.	.	PUNCT
ejpam-2214	87	1	we	we	PRON
ejpam-2214	87	2	denote	denote	VERB
ejpam-2214	87	3	by	by	ADP
ejpam-2214	87	4	soi(x	soi(x	PROPN
ejpam-2214	87	5	,	,	PUNCT
ejpam-2214	87	6	τ	τ	PROPN
ejpam-2214	87	7	)	)	PUNCT
ejpam-2214	87	8	as	as	ADP
ejpam-2214	87	9	the	the	DET
ejpam-2214	87	10	family	family	NOUN
ejpam-2214	87	11	of	of	ADP
ejpam-2214	87	12	all	all	DET
ejpam-2214	87	13	weakly	weakly	ADJ
ejpam-2214	87	14	i	i	PRON
ejpam-2214	87	15	-	-	PUNCT
ejpam-2214	87	16	semi	semi	ADV
ejpam-2214	87	17	open	open	ADJ
ejpam-2214	87	18	sets	set	NOUN
ejpam-2214	87	19	in	in	ADP
ejpam-2214	87	20	the	the	DET
ejpam-2214	87	21	topological	topological	ADJ
ejpam-2214	87	22	space	space	NOUN
ejpam-2214	87	23	x	x	SYM
ejpam-2214	87	24	,	,	PUNCT
ejpam-2214	87	25	then	then	ADV
ejpam-2214	87	26	soi(x	soi(x	PROPN
ejpam-2214	87	27	,	,	PUNCT
ejpam-2214	87	28	τ	τ	X
ejpam-2214	87	29	)	)	PUNCT
ejpam-2214	87	30	is	be	AUX
ejpam-2214	87	31	a	a	DET
ejpam-2214	87	32	minimal	minimal	ADJ
ejpam-2214	87	33	structure	structure	NOUN
ejpam-2214	87	34	that	that	PRON
ejpam-2214	87	35	satisfies	satisfy	VERB
ejpam-2214	87	36	the	the	DET
ejpam-2214	87	37	maki	maki	NOUN
ejpam-2214	87	38	condition	condition	NOUN
ejpam-2214	88	1	[	[	X
ejpam-2214	88	2	4	4	NUM
ejpam-2214	88	3	]	]	PUNCT
ejpam-2214	88	4	.	.	PUNCT
ejpam-2214	89	1	from	from	ADP
ejpam-2214	89	2	definition	definition	NOUN
ejpam-2214	89	3	1	1	NUM
ejpam-2214	89	4	,	,	PUNCT
ejpam-2214	89	5	we	we	PRON
ejpam-2214	89	6	obtain	obtain	VERB
ejpam-2214	89	7	that	that	PRON
ejpam-2214	89	8	,	,	PUNCT
ejpam-2214	89	9	if	if	SCONJ
ejpam-2214	89	10	;	;	PUNCT
ejpam-2214	89	11	6=	6=	ADP
ejpam-2214	89	12	a	a	DET
ejpam-2214	89	13	⊂	⊂	PROPN
ejpam-2214	89	14	b	b	PROPN
ejpam-2214	89	15	and	and	CCONJ
ejpam-2214	89	16	a	a	PRON
ejpam-2214	89	17	is	be	AUX
ejpam-2214	89	18	weakly	weakly	ADJ
ejpam-2214	89	19	i	i	PRON
ejpam-2214	89	20	-	-	PUNCT
ejpam-2214	89	21	semi	semi	ADV
ejpam-2214	89	22	open	open	ADJ
ejpam-2214	89	23	,	,	PUNCT
ejpam-2214	89	24	then	then	ADV
ejpam-2214	89	25	b	b	NOUN
ejpam-2214	89	26	is	be	AUX
ejpam-2214	89	27	also	also	ADV
ejpam-2214	89	28	weakly	weakly	ADJ
ejpam-2214	89	29	i	i	PRON
ejpam-2214	89	30	-	-	PUNCT
ejpam-2214	89	31	semi	semi	ADV
ejpam-2214	89	32	open	open	ADJ
ejpam-2214	89	33	in	in	ADP
ejpam-2214	89	34	consequence	consequence	NOUN
ejpam-2214	89	35	we	we	PRON
ejpam-2214	89	36	have	have	VERB
ejpam-2214	89	37	the	the	DET
ejpam-2214	89	38	following	follow	VERB
ejpam-2214	89	39	corollary	corollary	NOUN
ejpam-2214	89	40	.	.	PUNCT
ejpam-2214	90	1	corollary	corollary	ADJ
ejpam-2214	90	2	1	1	NUM
ejpam-2214	90	3	.	.	PUNCT
ejpam-2214	91	1	if	if	SCONJ
ejpam-2214	91	2	a	a	PRON
ejpam-2214	91	3	is	be	AUX
ejpam-2214	91	4	weakly	weakly	ADJ
ejpam-2214	91	5	i	i	PRON
ejpam-2214	91	6	-	-	PUNCT
ejpam-2214	91	7	semi	semi	ADV
ejpam-2214	91	8	open	open	ADJ
ejpam-2214	91	9	,	,	PUNCT
ejpam-2214	91	10	then	then	ADV
ejpam-2214	91	11	so	so	ADV
ejpam-2214	91	12	is	be	AUX
ejpam-2214	91	13	a∪b	a∪b	ADJ
ejpam-2214	91	14	,	,	PUNCT
ejpam-2214	91	15	for	for	ADP
ejpam-2214	91	16	any	any	DET
ejpam-2214	91	17	subset	subset	NOUN
ejpam-2214	91	18	b	b	PROPN
ejpam-2214	91	19	of	of	ADP
ejpam-2214	91	20	x	x	PRON
ejpam-2214	91	21	,	,	PUNCT
ejpam-2214	91	22	in	in	ADP
ejpam-2214	91	23	particular	particular	ADJ
ejpam-2214	91	24	cl(a	cl(a	X
ejpam-2214	91	25	)	)	PUNCT
ejpam-2214	91	26	is	be	AUX
ejpam-2214	91	27	weakly	weakly	ADJ
ejpam-2214	91	28	i	i	PRON
ejpam-2214	91	29	-	-	PUNCT
ejpam-2214	91	30	semi	semi	ADV
ejpam-2214	91	31	open	open	ADJ
ejpam-2214	91	32	.	.	PUNCT
ejpam-2214	92	1	the	the	DET
ejpam-2214	92	2	converse	converse	NOUN
ejpam-2214	92	3	of	of	ADP
ejpam-2214	92	4	the	the	DET
ejpam-2214	92	5	above	above	ADJ
ejpam-2214	92	6	corollary	corollary	NOUN
ejpam-2214	92	7	is	be	AUX
ejpam-2214	92	8	not	not	PART
ejpam-2214	92	9	necessarily	necessarily	ADV
ejpam-2214	92	10	true	true	ADJ
ejpam-2214	92	11	as	as	SCONJ
ejpam-2214	92	12	we	we	PRON
ejpam-2214	92	13	see	see	VERB
ejpam-2214	92	14	in	in	ADP
ejpam-2214	92	15	the	the	DET
ejpam-2214	92	16	following	follow	VERB
ejpam-2214	92	17	example	example	NOUN
ejpam-2214	92	18	.	.	PUNCT
ejpam-2214	93	1	example	example	NOUN
ejpam-2214	94	1	5	5	NUM
ejpam-2214	94	2	.	.	PUNCT
ejpam-2214	94	3	let	let	VERB
ejpam-2214	94	4	x	x	PUNCT
ejpam-2214	94	5	=	=	PRON
ejpam-2214	94	6	{	{	PUNCT
ejpam-2214	94	7	a	a	PRON
ejpam-2214	94	8	,	,	PUNCT
ejpam-2214	94	9	b	b	NOUN
ejpam-2214	94	10	,	,	PUNCT
ejpam-2214	94	11	c	c	NOUN
ejpam-2214	94	12	,	,	PUNCT
ejpam-2214	94	13	d	d	NOUN
ejpam-2214	94	14	}	}	PUNCT
ejpam-2214	94	15	with	with	ADP
ejpam-2214	94	16	topology	topology	NOUN
ejpam-2214	94	17	τ=	τ=	PUNCT
ejpam-2214	94	18	{	{	PUNCT
ejpam-2214	94	19	;	;	PUNCT
ejpam-2214	94	20	,	,	PUNCT
ejpam-2214	94	21	x	x	X
ejpam-2214	94	22	,	,	PUNCT
ejpam-2214	94	23	{	{	PUNCT
ejpam-2214	94	24	a	a	DET
ejpam-2214	94	25	,	,	PUNCT
ejpam-2214	94	26	b	b	NOUN
ejpam-2214	94	27	}	}	PUNCT
ejpam-2214	94	28	,	,	PUNCT
ejpam-2214	94	29	{	{	PUNCT
ejpam-2214	94	30	c	c	X
ejpam-2214	94	31	,	,	PUNCT
ejpam-2214	94	32	d	d	NOUN
ejpam-2214	94	33	}	}	PUNCT
ejpam-2214	94	34	}	}	PUNCT
ejpam-2214	94	35	and	and	CCONJ
ejpam-2214	94	36	i	i	PRON
ejpam-2214	94	37	any	any	DET
ejpam-2214	94	38	ideal	ideal	NOUN
ejpam-2214	94	39	such	such	ADJ
ejpam-2214	94	40	that	that	SCONJ
ejpam-2214	94	41	{	{	PUNCT
ejpam-2214	94	42	b	b	NOUN
ejpam-2214	94	43	}	}	PUNCT
ejpam-2214	94	44	/∈	/∈	PUNCT
ejpam-2214	95	1	i	i	PRON
ejpam-2214	95	2	.	.	PUNCT
ejpam-2214	96	1	take	take	VERB
ejpam-2214	96	2	a=	a=	ADV
ejpam-2214	96	3	{	{	PUNCT
ejpam-2214	96	4	a	a	NOUN
ejpam-2214	96	5	}	}	PUNCT
ejpam-2214	96	6	,	,	PUNCT
ejpam-2214	96	7	cl(a	cl(a	NUM
ejpam-2214	96	8	)	)	PUNCT
ejpam-2214	96	9	=	=	PRON
ejpam-2214	96	10	{	{	PUNCT
ejpam-2214	96	11	a	a	DET
ejpam-2214	96	12	,	,	PUNCT
ejpam-2214	96	13	b	b	NOUN
ejpam-2214	96	14	}	}	PUNCT
ejpam-2214	96	15	is	be	AUX
ejpam-2214	96	16	weakly	weakly	ADJ
ejpam-2214	97	1	i	i	PRON
ejpam-2214	97	2	-	-	PUNCT
ejpam-2214	97	3	semi	semi	ADV
ejpam-2214	97	4	open	open	ADJ
ejpam-2214	97	5	but	but	CCONJ
ejpam-2214	97	6	a	a	PRON
ejpam-2214	97	7	is	be	AUX
ejpam-2214	97	8	not	not	PART
ejpam-2214	97	9	weakly	weakly	ADJ
ejpam-2214	97	10	isemi	isemi	NOUN
ejpam-2214	97	11	open	open	ADJ
ejpam-2214	97	12	.	.	PUNCT
ejpam-2214	98	1	the	the	DET
ejpam-2214	98	2	following	follow	VERB
ejpam-2214	98	3	theorem	theorem	NOUN
ejpam-2214	98	4	give	give	VERB
ejpam-2214	98	5	to	to	ADP
ejpam-2214	98	6	us	we	PRON
ejpam-2214	98	7	a	a	DET
ejpam-2214	98	8	sufficient	sufficient	ADJ
ejpam-2214	98	9	condition	condition	NOUN
ejpam-2214	98	10	in	in	ADP
ejpam-2214	98	11	order	order	NOUN
ejpam-2214	98	12	to	to	PART
ejpam-2214	98	13	obtain	obtain	VERB
ejpam-2214	98	14	that	that	DET
ejpam-2214	98	15	soi(x	soi(x	PROPN
ejpam-2214	98	16	,	,	PUNCT
ejpam-2214	98	17	τ	τ	NOUN
ejpam-2214	98	18	)	)	PUNCT
ejpam-2214	98	19	=	=	SYM
ejpam-2214	98	20	p(x	p(x	PROPN
ejpam-2214	98	21	)	)	PUNCT
ejpam-2214	98	22	.	.	PUNCT
ejpam-2214	99	1	theorem	theorem	ADJ
ejpam-2214	99	2	4	4	NUM
ejpam-2214	99	3	.	.	PUNCT
ejpam-2214	100	1	let	let	AUX
ejpam-2214	100	2	(	(	PUNCT
ejpam-2214	100	3	x	x	X
ejpam-2214	100	4	,	,	PUNCT
ejpam-2214	100	5	τ	τ	X
ejpam-2214	100	6	)	)	PUNCT
ejpam-2214	100	7	be	be	VERB
ejpam-2214	100	8	a	a	DET
ejpam-2214	100	9	topological	topological	ADJ
ejpam-2214	100	10	space	space	NOUN
ejpam-2214	100	11	and	and	CCONJ
ejpam-2214	100	12	i	i	PRON
ejpam-2214	100	13	an	an	DET
ejpam-2214	100	14	ideal	ideal	NOUN
ejpam-2214	100	15	such	such	ADJ
ejpam-2214	100	16	that	that	SCONJ
ejpam-2214	100	17	there	there	PRON
ejpam-2214	100	18	exist	exist	VERB
ejpam-2214	100	19	an	an	DET
ejpam-2214	100	20	unitary	unitary	ADJ
ejpam-2214	100	21	set	set	NOUN
ejpam-2214	100	22	that	that	PRON
ejpam-2214	100	23	belongs	belong	VERB
ejpam-2214	100	24	to	to	ADP
ejpam-2214	100	25	the	the	DET
ejpam-2214	100	26	topology	topology	NOUN
ejpam-2214	100	27	and	and	CCONJ
ejpam-2214	100	28	the	the	DET
ejpam-2214	100	29	ideal	ideal	NOUN
ejpam-2214	100	30	,	,	PUNCT
ejpam-2214	100	31	then	then	ADV
ejpam-2214	100	32	soi(x	soi(x	PROPN
ejpam-2214	100	33	,	,	PUNCT
ejpam-2214	100	34	τ	τ	NOUN
ejpam-2214	100	35	)	)	PUNCT
ejpam-2214	100	36	=	=	SYM
ejpam-2214	100	37	p(x	p(x	PROPN
ejpam-2214	100	38	)	)	PUNCT
ejpam-2214	100	39	.	.	PUNCT
ejpam-2214	101	1	c.	c.	PROPN
ejpam-2214	101	2	carpintero	carpintero	PROPN
ejpam-2214	101	3	,	,	PUNCT
ejpam-2214	101	4	a.	a.	PROPN
ejpam-2214	101	5	muñoz	muñoz	PROPN
ejpam-2214	101	6	,	,	PUNCT
ejpam-2214	101	7	j.	j.	PROPN
ejpam-2214	101	8	pacheco	pacheco	PROPN
ejpam-2214	101	9	,	,	PUNCT
ejpam-2214	101	10	e.	e.	PROPN
ejpam-2214	101	11	rosas	rosas	PROPN
ejpam-2214	101	12	/	/	SYM
ejpam-2214	101	13	eur	eur	PROPN
ejpam-2214	101	14	.	.	PUNCT
ejpam-2214	102	1	j.	j.	PROPN
ejpam-2214	102	2	pure	pure	PROPN
ejpam-2214	102	3	appl	appl	PROPN
ejpam-2214	102	4	.	.	PROPN
ejpam-2214	102	5	math	math	PROPN
ejpam-2214	102	6	,	,	PUNCT
ejpam-2214	102	7	7	7	NUM
ejpam-2214	102	8	(	(	PUNCT
ejpam-2214	102	9	2014	2014	NUM
ejpam-2214	102	10	)	)	PUNCT
ejpam-2214	102	11	,	,	PUNCT
ejpam-2214	102	12	437	437	NUM
ejpam-2214	102	13	-	-	SYM
ejpam-2214	102	14	441	441	NUM
ejpam-2214	102	15	440	440	NUM
ejpam-2214	102	16	proof	proof	NOUN
ejpam-2214	102	17	.	.	PUNCT
ejpam-2214	102	18	suppose	suppose	VERB
ejpam-2214	102	19	that	that	SCONJ
ejpam-2214	102	20	the	the	DET
ejpam-2214	102	21	unitary	unitary	ADJ
ejpam-2214	102	22	set	set	NOUN
ejpam-2214	102	23	{	{	PUNCT
ejpam-2214	102	24	a	a	PRON
ejpam-2214	102	25	}	}	PUNCT
ejpam-2214	102	26	∈	∈	NOUN
ejpam-2214	102	27	i	i	PRON
ejpam-2214	102	28	.	.	PUNCT
ejpam-2214	103	1	let	let	VERB
ejpam-2214	103	2	{	{	PUNCT
ejpam-2214	103	3	b	b	NOUN
ejpam-2214	103	4	}	}	PUNCT
ejpam-2214	103	5	any	any	DET
ejpam-2214	103	6	unitary	unitary	ADJ
ejpam-2214	103	7	set	set	NOUN
ejpam-2214	103	8	in	in	ADP
ejpam-2214	103	9	x	x	SYM
ejpam-2214	103	10	,	,	PUNCT
ejpam-2214	103	11	then	then	ADV
ejpam-2214	103	12	{	{	PUNCT
ejpam-2214	103	13	b	b	X
ejpam-2214	103	14	}	}	PUNCT
ejpam-2214	103	15	∈	∈	PROPN
ejpam-2214	103	16	soi(x	soi(x	PROPN
ejpam-2214	103	17	,	,	PUNCT
ejpam-2214	103	18	τ	τ	PROPN
ejpam-2214	103	19	)	)	PUNCT
ejpam-2214	103	20	,	,	PUNCT
ejpam-2214	103	21	because	because	SCONJ
ejpam-2214	103	22	{	{	PUNCT
ejpam-2214	103	23	a	a	PRON
ejpam-2214	103	24	}	}	PUNCT
ejpam-2214	103	25	\	\	NOUN
ejpam-2214	103	26	{	{	PUNCT
ejpam-2214	103	27	b	b	X
ejpam-2214	103	28	}	}	PUNCT
ejpam-2214	103	29	∈	∈	PROPN
ejpam-2214	103	30	i	i	PRON
ejpam-2214	103	31	.	.	PUNCT
ejpam-2214	104	1	now	now	ADV
ejpam-2214	104	2	using	use	VERB
ejpam-2214	104	3	theorem	theorem	NOUN
ejpam-2214	104	4	3	3	NUM
ejpam-2214	104	5	,	,	PUNCT
ejpam-2214	104	6	we	we	PRON
ejpam-2214	104	7	obtain	obtain	VERB
ejpam-2214	104	8	that	that	SCONJ
ejpam-2214	104	9	any	any	DET
ejpam-2214	104	10	subset	subset	NOUN
ejpam-2214	104	11	a	a	PRON
ejpam-2214	104	12	of	of	ADP
ejpam-2214	104	13	x	x	PUNCT
ejpam-2214	104	14	belongs	belong	VERB
ejpam-2214	104	15	to	to	ADP
ejpam-2214	104	16	soi(x	soi(x	PROPN
ejpam-2214	104	17	,	,	PUNCT
ejpam-2214	104	18	τ	τ	PROPN
ejpam-2214	104	19	)	)	PUNCT
ejpam-2214	104	20	.	.	PUNCT
ejpam-2214	105	1	at	at	ADP
ejpam-2214	105	2	this	this	DET
ejpam-2214	105	3	point	point	NOUN
ejpam-2214	105	4	we	we	PRON
ejpam-2214	105	5	want	want	VERB
ejpam-2214	105	6	to	to	PART
ejpam-2214	105	7	determinate	determinate	VERB
ejpam-2214	105	8	under	under	ADP
ejpam-2214	105	9	what	what	DET
ejpam-2214	105	10	conditions	condition	NOUN
ejpam-2214	105	11	,	,	PUNCT
ejpam-2214	105	12	if	if	SCONJ
ejpam-2214	105	13	a⊆	a⊆	NOUN
ejpam-2214	105	14	x	x	VERB
ejpam-2214	105	15	is	be	AUX
ejpam-2214	105	16	cl(a)weakly	cl(a)weakly	ADV
ejpam-2214	105	17	i	i	PRON
ejpam-2214	105	18	-semi	-semi	VERB
ejpam-2214	105	19	open	open	ADJ
ejpam-2214	105	20	set	set	VERB
ejpam-2214	105	21	then	then	ADV
ejpam-2214	105	22	a	a	PRON
ejpam-2214	105	23	is	be	AUX
ejpam-2214	105	24	weakly	weakly	ADJ
ejpam-2214	105	25	i	i	PRON
ejpam-2214	105	26	-semi	-semi	VERB
ejpam-2214	105	27	open	open	ADJ
ejpam-2214	105	28	.	.	PUNCT
ejpam-2214	106	1	observe	observe	VERB
ejpam-2214	106	2	the	the	DET
ejpam-2214	106	3	following	follow	VERB
ejpam-2214	106	4	facts	fact	NOUN
ejpam-2214	106	5	:	:	PUNCT
ejpam-2214	106	6	(	(	PUNCT
ejpam-2214	106	7	i	i	NOUN
ejpam-2214	106	8	)	)	PUNCT
ejpam-2214	106	9	if	if	SCONJ
ejpam-2214	106	10	cl(a	cl(a	VERB
ejpam-2214	106	11	)	)	PUNCT
ejpam-2214	106	12	=	=	PUNCT
ejpam-2214	107	1	x	x	X
ejpam-2214	107	2	then	then	ADV
ejpam-2214	107	3	a	a	PRON
ejpam-2214	107	4	is	be	AUX
ejpam-2214	107	5	not	not	PART
ejpam-2214	107	6	necessarily	necessarily	ADV
ejpam-2214	107	7	weakly	weakly	ADJ
ejpam-2214	107	8	i	i	PRON
ejpam-2214	107	9	-semi	-semi	VERB
ejpam-2214	107	10	open	open	ADJ
ejpam-2214	107	11	.	.	PUNCT
ejpam-2214	108	1	(	(	PUNCT
ejpam-2214	108	2	ii	ii	NOUN
ejpam-2214	108	3	)	)	PUNCT
ejpam-2214	108	4	if	if	SCONJ
ejpam-2214	108	5	there	there	PRON
ejpam-2214	108	6	exists	exist	VERB
ejpam-2214	108	7	a	a	DET
ejpam-2214	108	8	⊂	⊂	PROPN
ejpam-2214	108	9	x	x	X
ejpam-2214	108	10	,	,	PUNCT
ejpam-2214	108	11	such	such	ADJ
ejpam-2214	108	12	that	that	SCONJ
ejpam-2214	108	13	cl(a	cl(a	PUNCT
ejpam-2214	108	14	)	)	PUNCT
ejpam-2214	108	15	is	be	AUX
ejpam-2214	108	16	a	a	DET
ejpam-2214	108	17	clopen	clopen	ADJ
ejpam-2214	108	18	set	set	NOUN
ejpam-2214	108	19	then	then	ADV
ejpam-2214	108	20	a	a	PRON
ejpam-2214	108	21	is	be	AUX
ejpam-2214	108	22	not	not	PART
ejpam-2214	108	23	necessarily	necessarily	ADV
ejpam-2214	108	24	weakly	weakly	ADJ
ejpam-2214	108	25	i	i	PRON
ejpam-2214	108	26	-semi	-semi	VERB
ejpam-2214	108	27	open	open	ADJ
ejpam-2214	108	28	.	.	PUNCT
ejpam-2214	109	1	example	example	NOUN
ejpam-2214	110	1	6	6	NUM
ejpam-2214	110	2	.	.	PUNCT
ejpam-2214	111	1	let	let	VERB
ejpam-2214	111	2	x	x	PUNCT
ejpam-2214	111	3	=	=	PRON
ejpam-2214	111	4	{	{	PUNCT
ejpam-2214	111	5	a	a	PRON
ejpam-2214	111	6	,	,	PUNCT
ejpam-2214	111	7	b	b	NOUN
ejpam-2214	111	8	,	,	PUNCT
ejpam-2214	111	9	c	c	NOUN
ejpam-2214	111	10	,	,	PUNCT
ejpam-2214	111	11	d	d	NOUN
ejpam-2214	111	12	}	}	PUNCT
ejpam-2214	111	13	with	with	ADP
ejpam-2214	111	14	topology	topology	NOUN
ejpam-2214	111	15	τ=	τ=	PUNCT
ejpam-2214	111	16	{	{	PUNCT
ejpam-2214	111	17	;	;	PUNCT
ejpam-2214	111	18	,	,	PUNCT
ejpam-2214	111	19	x	x	X
ejpam-2214	111	20	,	,	PUNCT
ejpam-2214	111	21	{	{	PUNCT
ejpam-2214	111	22	a	a	DET
ejpam-2214	111	23	,	,	PUNCT
ejpam-2214	111	24	b	b	NOUN
ejpam-2214	111	25	}	}	PUNCT
ejpam-2214	111	26	,	,	PUNCT
ejpam-2214	111	27	{	{	PUNCT
ejpam-2214	111	28	c	c	X
ejpam-2214	111	29	,	,	PUNCT
ejpam-2214	111	30	d	d	NOUN
ejpam-2214	111	31	}	}	PUNCT
ejpam-2214	111	32	}	}	PUNCT
ejpam-2214	111	33	.	.	PUNCT
ejpam-2214	112	1	(	(	PUNCT
ejpam-2214	112	2	i	i	NOUN
ejpam-2214	112	3	)	)	PUNCT
ejpam-2214	112	4	if	if	SCONJ
ejpam-2214	112	5	we	we	PRON
ejpam-2214	112	6	take	take	VERB
ejpam-2214	112	7	i	i	PRON
ejpam-2214	112	8	=	=	PUNCT
ejpam-2214	112	9	{	{	PUNCT
ejpam-2214	112	10	;	;	PUNCT
ejpam-2214	112	11	}	}	PUNCT
ejpam-2214	112	12	and	and	CCONJ
ejpam-2214	112	13	a=	a=	VERB
ejpam-2214	112	14	{	{	PUNCT
ejpam-2214	112	15	b	b	NOUN
ejpam-2214	112	16	,	,	PUNCT
ejpam-2214	112	17	d	d	NOUN
ejpam-2214	112	18	}	}	PUNCT
ejpam-2214	112	19	,	,	PUNCT
ejpam-2214	112	20	cl(a	cl(a	X
ejpam-2214	112	21	)	)	PUNCT
ejpam-2214	112	22	=	=	PUNCT
ejpam-2214	113	1	x	x	X
ejpam-2214	113	2	is	be	AUX
ejpam-2214	113	3	weakly	weakly	ADJ
ejpam-2214	113	4	i	i	PRON
ejpam-2214	113	5	-	-	PUNCT
ejpam-2214	113	6	semi	semi	ADV
ejpam-2214	113	7	open	open	ADJ
ejpam-2214	113	8	but	but	CCONJ
ejpam-2214	113	9	a	a	PRON
ejpam-2214	113	10	is	be	AUX
ejpam-2214	113	11	not	not	PART
ejpam-2214	113	12	.	.	PUNCT
ejpam-2214	114	1	(	(	PUNCT
ejpam-2214	114	2	ii	ii	NOUN
ejpam-2214	114	3	)	)	PUNCT
ejpam-2214	114	4	if	if	SCONJ
ejpam-2214	114	5	we	we	PRON
ejpam-2214	114	6	take	take	VERB
ejpam-2214	114	7	i	i	PRON
ejpam-2214	114	8	=	=	PUNCT
ejpam-2214	114	9	{	{	PUNCT
ejpam-2214	114	10	;	;	PUNCT
ejpam-2214	114	11	,	,	PUNCT
ejpam-2214	114	12	{	{	PUNCT
ejpam-2214	114	13	c	c	X
ejpam-2214	114	14	}	}	PUNCT
ejpam-2214	114	15	}	}	PUNCT
ejpam-2214	114	16	and	and	CCONJ
ejpam-2214	114	17	a=	a=	VERB
ejpam-2214	114	18	{	{	PUNCT
ejpam-2214	114	19	a	a	NOUN
ejpam-2214	114	20	}	}	PUNCT
ejpam-2214	114	21	,	,	PUNCT
ejpam-2214	114	22	cl(a	cl(a	NUM
ejpam-2214	114	23	)	)	PUNCT
ejpam-2214	114	24	=	=	PRON
ejpam-2214	114	25	{	{	PUNCT
ejpam-2214	114	26	a	a	DET
ejpam-2214	114	27	,	,	PUNCT
ejpam-2214	114	28	b	b	NOUN
ejpam-2214	114	29	}	}	PUNCT
ejpam-2214	114	30	is	be	AUX
ejpam-2214	114	31	weakly	weakly	ADJ
ejpam-2214	114	32	i	i	PRON
ejpam-2214	114	33	-	-	PUNCT
ejpam-2214	114	34	semi	semi	ADV
ejpam-2214	114	35	open	open	ADJ
ejpam-2214	114	36	but	but	CCONJ
ejpam-2214	114	37	a	a	PRON
ejpam-2214	114	38	is	be	AUX
ejpam-2214	114	39	not	not	PART
ejpam-2214	114	40	.	.	PUNCT
ejpam-2214	115	1	theorem	theorem	ADJ
ejpam-2214	115	2	5	5	NUM
ejpam-2214	115	3	.	.	PUNCT
ejpam-2214	116	1	let	let	VERB
ejpam-2214	116	2	x	x	PRON
ejpam-2214	116	3	be	be	AUX
ejpam-2214	116	4	a	a	DET
ejpam-2214	116	5	topological	topological	ADJ
ejpam-2214	116	6	space	space	NOUN
ejpam-2214	116	7	and	and	CCONJ
ejpam-2214	116	8	i	i	PRON
ejpam-2214	116	9	an	an	DET
ejpam-2214	116	10	ideal	ideal	NOUN
ejpam-2214	116	11	such	such	ADJ
ejpam-2214	116	12	that	that	SCONJ
ejpam-2214	116	13	the	the	DET
ejpam-2214	116	14	collection	collection	NOUN
ejpam-2214	116	15	of	of	ADP
ejpam-2214	116	16	open	open	ADJ
ejpam-2214	116	17	sets	set	NOUN
ejpam-2214	116	18	satisfies	satisfy	VERB
ejpam-2214	116	19	the	the	DET
ejpam-2214	116	20	finite	finite	ADJ
ejpam-2214	116	21	intersection	intersection	NOUN
ejpam-2214	116	22	property	property	NOUN
ejpam-2214	116	23	,	,	PUNCT
ejpam-2214	116	24	if	if	SCONJ
ejpam-2214	116	25	a	a	PRON
ejpam-2214	116	26	and	and	CCONJ
ejpam-2214	116	27	b	b	NOUN
ejpam-2214	116	28	are	be	AUX
ejpam-2214	116	29	weakly	weakly	ADJ
ejpam-2214	116	30	i	i	PRON
ejpam-2214	116	31	-	-	PUNCT
ejpam-2214	116	32	semi	semi	ADV
ejpam-2214	116	33	open	open	ADJ
ejpam-2214	116	34	,	,	PUNCT
ejpam-2214	116	35	then	then	ADV
ejpam-2214	116	36	so	so	ADV
ejpam-2214	116	37	is	be	AUX
ejpam-2214	116	38	a∩	a∩	PROPN
ejpam-2214	116	39	b.	b.	PROPN
ejpam-2214	116	40	proof	proof	NOUN
ejpam-2214	116	41	.	.	PUNCT
ejpam-2214	117	1	since	since	SCONJ
ejpam-2214	117	2	a	a	PRON
ejpam-2214	117	3	and	and	CCONJ
ejpam-2214	117	4	b	b	NOUN
ejpam-2214	117	5	are	be	AUX
ejpam-2214	117	6	weakly	weakly	ADJ
ejpam-2214	117	7	i	i	PRON
ejpam-2214	117	8	-semi	-semi	VERB
ejpam-2214	117	9	open	open	ADJ
ejpam-2214	117	10	sets	set	NOUN
ejpam-2214	117	11	,	,	PUNCT
ejpam-2214	117	12	there	there	PRON
ejpam-2214	117	13	exist	exist	VERB
ejpam-2214	117	14	open	open	ADJ
ejpam-2214	117	15	sets	set	NOUN
ejpam-2214	117	16	u	u	NOUN
ejpam-2214	117	17	,	,	PUNCT
ejpam-2214	117	18	v	v	ADP
ejpam-2214	117	19	such	such	ADJ
ejpam-2214	117	20	that	that	PRON
ejpam-2214	117	21	u	u	PROPN
ejpam-2214	117	22	\	\	PROPN
ejpam-2214	117	23	a∈	a∈	PROPN
ejpam-2214	117	24	i	i	PROPN
ejpam-2214	117	25	and	and	CCONJ
ejpam-2214	117	26	v	v	VERB
ejpam-2214	117	27	\	\	PROPN
ejpam-2214	117	28	b	b	PROPN
ejpam-2214	117	29	∈	∈	PROPN
ejpam-2214	118	1	i	i	PRON
ejpam-2214	118	2	,	,	PUNCT
ejpam-2214	118	3	therefore	therefore	ADV
ejpam-2214	118	4	,	,	PUNCT
ejpam-2214	118	5	(	(	PUNCT
ejpam-2214	118	6	u	u	NOUN
ejpam-2214	118	7	∩	∩	NOUN
ejpam-2214	118	8	v	v	X
ejpam-2214	118	9	)	)	PUNCT
ejpam-2214	118	10	\	\	PUNCT
ejpam-2214	119	1	(	(	PUNCT
ejpam-2214	119	2	a∩	a∩	PROPN
ejpam-2214	119	3	b	b	X
ejpam-2214	119	4	)	)	PUNCT
ejpam-2214	119	5	=	=	SYM
ejpam-2214	119	6	(	(	PUNCT
ejpam-2214	119	7	u	u	NOUN
ejpam-2214	119	8	\	\	PROPN
ejpam-2214	119	9	a)∩	a)∩	X
ejpam-2214	119	10	v	v	ADP
ejpam-2214	119	11	∪	∪	PROPN
ejpam-2214	119	12	u	u	NOUN
ejpam-2214	119	13	∩	∩	NOUN
ejpam-2214	119	14	(	(	PUNCT
ejpam-2214	119	15	v	v	NOUN
ejpam-2214	119	16	\	\	PROPN
ejpam-2214	119	17	b	b	NOUN
ejpam-2214	119	18	)	)	PUNCT
ejpam-2214	119	19	∈	∈	PROPN
ejpam-2214	119	20	i	i	PRON
ejpam-2214	119	21	.	.	PUNCT
ejpam-2214	120	1	remark	remark	PROPN
ejpam-2214	120	2	2	2	NUM
ejpam-2214	120	3	.	.	PUNCT
ejpam-2214	121	1	the	the	DET
ejpam-2214	121	2	following	follow	VERB
ejpam-2214	121	3	theorem	theorem	NOUN
ejpam-2214	121	4	characterizes	characterize	VERB
ejpam-2214	121	5	the	the	DET
ejpam-2214	121	6	subsets	subset	NOUN
ejpam-2214	121	7	a⊆	a⊆	NOUN
ejpam-2214	121	8	x	x	PUNCT
ejpam-2214	121	9	such	such	ADJ
ejpam-2214	121	10	that	that	SCONJ
ejpam-2214	121	11	the	the	DET
ejpam-2214	121	12	cl(a	cl(a	X
ejpam-2214	121	13	)	)	PUNCT
ejpam-2214	121	14	is	be	AUX
ejpam-2214	121	15	weakly	weakly	ADJ
ejpam-2214	121	16	isemi	isemi	ADV
ejpam-2214	121	17	open	open	ADJ
ejpam-2214	121	18	under	under	ADP
ejpam-2214	121	19	some	some	DET
ejpam-2214	121	20	conditions	condition	NOUN
ejpam-2214	121	21	of	of	ADP
ejpam-2214	121	22	the	the	DET
ejpam-2214	121	23	ideal	ideal	NOUN
ejpam-2214	121	24	and	and	CCONJ
ejpam-2214	121	25	the	the	DET
ejpam-2214	121	26	collections	collection	NOUN
ejpam-2214	121	27	of	of	ADP
ejpam-2214	121	28	open	open	ADJ
ejpam-2214	121	29	sets	set	NOUN
ejpam-2214	121	30	of	of	ADP
ejpam-2214	121	31	x	x	X
ejpam-2214	121	32	.	.	PUNCT
ejpam-2214	122	1	theorem	theorem	NOUN
ejpam-2214	122	2	6	6	NUM
ejpam-2214	122	3	.	.	PUNCT
ejpam-2214	123	1	let	let	VERB
ejpam-2214	123	2	x	x	PRON
ejpam-2214	123	3	be	be	AUX
ejpam-2214	123	4	a	a	DET
ejpam-2214	123	5	topological	topological	ADJ
ejpam-2214	123	6	space	space	NOUN
ejpam-2214	123	7	,	,	PUNCT
ejpam-2214	123	8	i	i	PRON
ejpam-2214	123	9	6=	6=	NUM
ejpam-2214	123	10	;	;	PUNCT
ejpam-2214	123	11	an	an	DET
ejpam-2214	123	12	ideal	ideal	NOUN
ejpam-2214	123	13	on	on	ADP
ejpam-2214	123	14	x	x	PUNCT
ejpam-2214	123	15	and	and	CCONJ
ejpam-2214	123	16	the	the	DET
ejpam-2214	123	17	collection	collection	NOUN
ejpam-2214	123	18	of	of	ADP
ejpam-2214	123	19	open	open	ADJ
ejpam-2214	123	20	subsets	subset	NOUN
ejpam-2214	123	21	of	of	ADP
ejpam-2214	123	22	x	x	PRON
ejpam-2214	123	23	satisfies	satisfy	VERB
ejpam-2214	123	24	the	the	DET
ejpam-2214	123	25	finite	finite	ADJ
ejpam-2214	123	26	intersection	intersection	NOUN
ejpam-2214	123	27	property	property	NOUN
ejpam-2214	123	28	.	.	PUNCT
ejpam-2214	124	1	if	if	SCONJ
ejpam-2214	124	2	a⊂	a⊂	PRON
ejpam-2214	124	3	x	x	SYM
ejpam-2214	124	4	such	such	ADJ
ejpam-2214	124	5	that	that	DET
ejpam-2214	124	6	cl(a	cl(a	NUM
ejpam-2214	124	7	)	)	PUNCT
ejpam-2214	124	8	6=	6=	ADP
ejpam-2214	124	9	x	x	SYM
ejpam-2214	124	10	.	.	PUNCT
ejpam-2214	124	11	cl(a	cl(a	NUM
ejpam-2214	124	12	)	)	PUNCT
ejpam-2214	124	13	is	be	AUX
ejpam-2214	124	14	weakly	weakly	ADJ
ejpam-2214	124	15	i	i	PRON
ejpam-2214	124	16	-	-	PUNCT
ejpam-2214	124	17	semi	semi	ADV
ejpam-2214	124	18	open	open	ADJ
ejpam-2214	124	19	if	if	SCONJ
ejpam-2214	124	20	and	and	CCONJ
ejpam-2214	124	21	only	only	ADV
ejpam-2214	124	22	if	if	SCONJ
ejpam-2214	124	23	a	a	PRON
ejpam-2214	124	24	is	be	AUX
ejpam-2214	124	25	weakly	weakly	ADJ
ejpam-2214	124	26	i	i	PRON
ejpam-2214	124	27	-	-	PUNCT
ejpam-2214	124	28	semi	semi	ADV
ejpam-2214	124	29	open	open	ADJ
ejpam-2214	124	30	.	.	PUNCT
ejpam-2214	125	1	proof	proof	NOUN
ejpam-2214	125	2	.	.	PUNCT
ejpam-2214	126	1	if	if	SCONJ
ejpam-2214	126	2	a	a	PRON
ejpam-2214	126	3	is	be	AUX
ejpam-2214	126	4	weakly	weakly	ADJ
ejpam-2214	126	5	i	i	PRON
ejpam-2214	126	6	-semi	-semi	VERB
ejpam-2214	126	7	open	open	ADJ
ejpam-2214	126	8	,	,	PUNCT
ejpam-2214	126	9	then	then	ADV
ejpam-2214	126	10	cl(a	cl(a	NUM
ejpam-2214	126	11	)	)	PUNCT
ejpam-2214	126	12	is	be	AUX
ejpam-2214	126	13	weakly	weakly	ADJ
ejpam-2214	126	14	i	i	PRON
ejpam-2214	126	15	-semi	-semi	VERB
ejpam-2214	126	16	open	open	ADJ
ejpam-2214	126	17	by	by	ADP
ejpam-2214	126	18	corollary	corollary	ADJ
ejpam-2214	126	19	1	1	NUM
ejpam-2214	126	20	.	.	PUNCT
ejpam-2214	127	1	conversely	conversely	ADV
ejpam-2214	127	2	,	,	PUNCT
ejpam-2214	127	3	suppose	suppose	VERB
ejpam-2214	127	4	that	that	SCONJ
ejpam-2214	127	5	cl(a	cl(a	X
ejpam-2214	127	6	)	)	PUNCT
ejpam-2214	127	7	is	be	AUX
ejpam-2214	127	8	weakly	weakly	ADJ
ejpam-2214	127	9	i	i	PRON
ejpam-2214	127	10	-semi	-semi	VERB
ejpam-2214	127	11	open	open	ADJ
ejpam-2214	127	12	,	,	PUNCT
ejpam-2214	127	13	then	then	ADV
ejpam-2214	127	14	cl(a	cl(a	NUM
ejpam-2214	127	15	)	)	PUNCT
ejpam-2214	127	16	=	=	SYM
ejpam-2214	127	17	;	;	PUNCT
ejpam-2214	127	18	or	or	CCONJ
ejpam-2214	127	19	cl(a	cl(a	NUM
ejpam-2214	127	20	)	)	PUNCT
ejpam-2214	127	21	6=	6=	NUM
ejpam-2214	127	22	;	;	PUNCT
ejpam-2214	127	23	.	.	PUNCT
ejpam-2214	128	1	if	if	SCONJ
ejpam-2214	128	2	cl(a	cl(a	NUM
ejpam-2214	128	3	)	)	PUNCT
ejpam-2214	128	4	=	=	SYM
ejpam-2214	128	5	;	;	PUNCT
ejpam-2214	128	6	,	,	PUNCT
ejpam-2214	128	7	then	then	ADV
ejpam-2214	128	8	a	a	DET
ejpam-2214	128	9	∈	∈	ADJ
ejpam-2214	128	10	soi(x	soi(x	PROPN
ejpam-2214	128	11	,	,	PUNCT
ejpam-2214	128	12	τ	τ	PROPN
ejpam-2214	128	13	)	)	PUNCT
ejpam-2214	128	14	.	.	PUNCT
ejpam-2214	129	1	if	if	SCONJ
ejpam-2214	129	2	cl(a	cl(a	NUM
ejpam-2214	129	3	)	)	PUNCT
ejpam-2214	129	4	6=	6=	NUM
ejpam-2214	129	5	;	;	PUNCT
ejpam-2214	129	6	,	,	PUNCT
ejpam-2214	129	7	there	there	PRON
ejpam-2214	129	8	exists	exist	VERB
ejpam-2214	129	9	an	an	DET
ejpam-2214	129	10	open	open	ADJ
ejpam-2214	129	11	set	set	VERB
ejpam-2214	129	12	u	u	PROPN
ejpam-2214	129	13	6=	6=	PROPN
ejpam-2214	129	14	;	;	PUNCT
ejpam-2214	129	15	such	such	ADJ
ejpam-2214	129	16	that	that	SCONJ
ejpam-2214	129	17	u	u	PROPN
ejpam-2214	129	18	\	\	NOUN
ejpam-2214	129	19	cl(a	cl(a	X
ejpam-2214	129	20	)	)	PUNCT
ejpam-2214	129	21	∈	∈	PROPN
ejpam-2214	129	22	i	i	PRON
ejpam-2214	129	23	,	,	PUNCT
ejpam-2214	129	24	take	take	VERB
ejpam-2214	129	25	the	the	DET
ejpam-2214	129	26	open	open	ADJ
ejpam-2214	129	27	set	set	NOUN
ejpam-2214	129	28	v	v	NOUN
ejpam-2214	129	29	=	=	SYM
ejpam-2214	129	30	u	u	NOUN
ejpam-2214	129	31	\	\	NOUN
ejpam-2214	129	32	cl(a	cl(a	NUM
ejpam-2214	129	33	)	)	PUNCT
ejpam-2214	129	34	.	.	PUNCT
ejpam-2214	130	1	using	use	VERB
ejpam-2214	130	2	the	the	DET
ejpam-2214	130	3	hypothesis	hypothesis	NOUN
ejpam-2214	130	4	v	v	NOUN
ejpam-2214	130	5	6=	6=	NUM
ejpam-2214	130	6	;	;	PUNCT
ejpam-2214	130	7	and	and	CCONJ
ejpam-2214	130	8	v	v	X
ejpam-2214	130	9	∈	∈	NOUN
ejpam-2214	131	1	i	i	PRON
ejpam-2214	131	2	.	.	PUNCT
ejpam-2214	132	1	observe	observe	VERB
ejpam-2214	132	2	that	that	SCONJ
ejpam-2214	132	3	v	v	NOUN
ejpam-2214	132	4	\	\	PROPN
ejpam-2214	132	5	a=	a=	X
ejpam-2214	132	6	(	(	PUNCT
ejpam-2214	132	7	u	u	NOUN
ejpam-2214	132	8	\	\	PROPN
ejpam-2214	132	9	cl(a	cl(a	NUM
ejpam-2214	132	10	)	)	PUNCT
ejpam-2214	132	11	)	)	PUNCT
ejpam-2214	132	12	\	\	PROPN
ejpam-2214	132	13	a=	a=	PROPN
ejpam-2214	132	14	u	u	NOUN
ejpam-2214	132	15	\	\	PROPN
ejpam-2214	132	16	cl(a	cl(a	X
ejpam-2214	132	17	)	)	PUNCT
ejpam-2214	132	18	∈	∈	PROPN
ejpam-2214	133	1	i	i	PRON
ejpam-2214	133	2	.	.	PUNCT
ejpam-2214	134	1	in	in	ADP
ejpam-2214	134	2	consequence	consequence	NOUN
ejpam-2214	134	3	,	,	PUNCT
ejpam-2214	134	4	a	a	PRON
ejpam-2214	134	5	is	be	AUX
ejpam-2214	134	6	weakly	weakly	ADJ
ejpam-2214	134	7	i	i	PRON
ejpam-2214	134	8	-semi	-semi	VERB
ejpam-2214	134	9	open	open	ADJ
ejpam-2214	134	10	.	.	PUNCT
ejpam-2214	135	1	remark	remark	NOUN
ejpam-2214	135	2	3	3	NUM
ejpam-2214	135	3	.	.	PUNCT
ejpam-2214	135	4	observe	observe	VERB
ejpam-2214	135	5	that	that	SCONJ
ejpam-2214	135	6	if	if	SCONJ
ejpam-2214	135	7	in	in	ADP
ejpam-2214	135	8	the	the	DET
ejpam-2214	135	9	theorem	theorem	NOUN
ejpam-2214	135	10	6	6	NUM
ejpam-2214	135	11	:	:	PUNCT
ejpam-2214	135	12	(	(	PUNCT
ejpam-2214	135	13	i	i	NOUN
ejpam-2214	135	14	)	)	PUNCT
ejpam-2214	135	15	i	i	PROPN
ejpam-2214	135	16	6=	6=	NUM
ejpam-2214	135	17	;	;	PUNCT
ejpam-2214	135	18	and	and	CCONJ
ejpam-2214	135	19	cl(a	cl(a	NUM
ejpam-2214	135	20	)	)	PUNCT
ejpam-2214	135	21	6=	6=	NUM
ejpam-2214	135	22	x	x	PRON
ejpam-2214	135	23	are	be	AUX
ejpam-2214	135	24	omitted	omit	VERB
ejpam-2214	135	25	,	,	PUNCT
ejpam-2214	135	26	then	then	ADV
ejpam-2214	135	27	the	the	DET
ejpam-2214	135	28	result	result	NOUN
ejpam-2214	135	29	may	may	AUX
ejpam-2214	135	30	be	be	AUX
ejpam-2214	135	31	false	false	ADJ
ejpam-2214	135	32	,	,	PUNCT
ejpam-2214	135	33	(	(	PUNCT
ejpam-2214	135	34	see	see	VERB
ejpam-2214	135	35	example	example	NOUN
ejpam-2214	135	36	1	1	NUM
ejpam-2214	135	37	)	)	PUNCT
ejpam-2214	135	38	.	.	PUNCT
ejpam-2214	136	1	(	(	PUNCT
ejpam-2214	136	2	ii	ii	NOUN
ejpam-2214	136	3	)	)	PUNCT
ejpam-2214	136	4	if	if	SCONJ
ejpam-2214	136	5	we	we	PRON
ejpam-2214	136	6	change	change	VERB
ejpam-2214	136	7	cl(a	cl(a	NOUN
ejpam-2214	136	8	)	)	PUNCT
ejpam-2214	136	9	6=	6=	NUM
ejpam-2214	136	10	x	x	PUNCT
ejpam-2214	136	11	by	by	ADP
ejpam-2214	136	12	cl(a	cl(a	NOUN
ejpam-2214	136	13	)	)	PUNCT
ejpam-2214	136	14	=	=	SYM
ejpam-2214	137	1	x	x	NOUN
ejpam-2214	137	2	,	,	PUNCT
ejpam-2214	137	3	the	the	DET
ejpam-2214	137	4	result	result	NOUN
ejpam-2214	137	5	may	may	AUX
ejpam-2214	137	6	be	be	AUX
ejpam-2214	137	7	false	false	ADJ
ejpam-2214	137	8	.	.	PUNCT
ejpam-2214	138	1	if	if	SCONJ
ejpam-2214	138	2	in	in	ADP
ejpam-2214	138	3	the	the	DET
ejpam-2214	138	4	example	example	NOUN
ejpam-2214	138	5	1	1	NUM
ejpam-2214	138	6	,	,	PUNCT
ejpam-2214	138	7	i	i	PRON
ejpam-2214	138	8	=	=	PUNCT
ejpam-2214	138	9	{	{	PUNCT
ejpam-2214	138	10	;	;	PUNCT
ejpam-2214	138	11	,	,	PUNCT
ejpam-2214	138	12	{	{	PUNCT
ejpam-2214	138	13	c	c	X
ejpam-2214	138	14	}	}	PUNCT
ejpam-2214	138	15	}	}	PUNCT
ejpam-2214	138	16	and	and	CCONJ
ejpam-2214	138	17	a	a	DET
ejpam-2214	138	18	=	=	X
ejpam-2214	138	19	{	{	PUNCT
ejpam-2214	138	20	a	a	NOUN
ejpam-2214	138	21	,	,	PUNCT
ejpam-2214	138	22	d	d	NOUN
ejpam-2214	138	23	}	}	PUNCT
ejpam-2214	138	24	,	,	PUNCT
ejpam-2214	138	25	then	then	ADV
ejpam-2214	138	26	cl(a	cl(a	NUM
ejpam-2214	138	27	)	)	PUNCT
ejpam-2214	138	28	is	be	AUX
ejpam-2214	138	29	weakly	weakly	ADJ
ejpam-2214	138	30	i	i	PRON
ejpam-2214	138	31	-	-	PUNCT
ejpam-2214	138	32	semi	semi	ADV
ejpam-2214	138	33	open	open	ADJ
ejpam-2214	138	34	but	but	CCONJ
ejpam-2214	138	35	a	a	PRON
ejpam-2214	138	36	is	be	AUX
ejpam-2214	138	37	not	not	PART
ejpam-2214	138	38	weakly	weakly	ADJ
ejpam-2214	138	39	i	i	PRON
ejpam-2214	138	40	-	-	PUNCT
ejpam-2214	138	41	semi	semi	ADV
ejpam-2214	138	42	open	open	ADJ
ejpam-2214	138	43	.	.	PUNCT
ejpam-2214	139	1	(	(	PUNCT
ejpam-2214	139	2	iii	iii	X
ejpam-2214	139	3	)	)	PUNCT
ejpam-2214	139	4	the	the	DET
ejpam-2214	139	5	case	case	NOUN
ejpam-2214	139	6	i	i	PRON
ejpam-2214	139	7	=	=	PUNCT
ejpam-2214	139	8	;	;	PUNCT
ejpam-2214	139	9	and	and	CCONJ
ejpam-2214	139	10	cl(a	cl(a	NUM
ejpam-2214	139	11	)	)	PUNCT
ejpam-2214	139	12	6=	6=	NUM
ejpam-2214	139	13	x	x	X
ejpam-2214	139	14	never	never	ADV
ejpam-2214	139	15	happens	happen	VERB
ejpam-2214	139	16	.	.	PUNCT
ejpam-2214	140	1	references	reference	NOUN
ejpam-2214	140	2	441	441	NUM
ejpam-2214	140	3	references	reference	NOUN
ejpam-2214	140	4	[	[	X
ejpam-2214	140	5	1	1	NUM
ejpam-2214	140	6	]	]	X
ejpam-2214	140	7	f	f	PROPN
ejpam-2214	140	8	ifeanyi	ifeanyi	PROPN
ejpam-2214	140	9	and	and	CCONJ
ejpam-2214	140	10	k	k	PROPN
ejpam-2214	140	11	michael	michael	PROPN
ejpam-2214	140	12	.	.	PUNCT
ejpam-2214	141	1	on	on	ADP
ejpam-2214	141	2	semi	semi	ADV
ejpam-2214	141	3	open	open	ADJ
ejpam-2214	141	4	sets	set	NOUN
ejpam-2214	141	5	with	with	ADP
ejpam-2214	141	6	respect	respect	NOUN
ejpam-2214	141	7	to	to	ADP
ejpam-2214	141	8	an	an	DET
ejpam-2214	141	9	ideal	ideal	NOUN
ejpam-2214	141	10	.	.	PUNCT
ejpam-2214	142	1	european	european	ADJ
ejpam-2214	142	2	journal	journal	PROPN
ejpam-2214	142	3	of	of	ADP
ejpam-2214	142	4	pure	pure	ADJ
ejpam-2214	142	5	and	and	CCONJ
ejpam-2214	142	6	applied	applied	ADJ
ejpam-2214	142	7	mathemetics	mathemetic	NOUN
ejpam-2214	142	8	,	,	PUNCT
ejpam-2214	142	9	6(1):53	6(1):53	PROPN
ejpam-2214	142	10	-	-	SYM
ejpam-2214	142	11	58	58	NUM
ejpam-2214	142	12	,	,	PUNCT
ejpam-2214	142	13	2013	2013	NUM
ejpam-2214	142	14	.	.	PUNCT
ejpam-2214	143	1	[	[	X
ejpam-2214	143	2	2	2	NUM
ejpam-2214	143	3	]	]	X
ejpam-2214	143	4	s	s	VERB
ejpam-2214	143	5	jafari	jafari	ADJ
ejpam-2214	143	6	and	and	CCONJ
ejpam-2214	143	7	n	n	PRON
ejpam-2214	143	8	rajesh	rajesh	PROPN
ejpam-2214	143	9	.	.	PUNCT
ejpam-2214	144	1	generalized	generalize	VERB
ejpam-2214	144	2	closed	close	VERB
ejpam-2214	144	3	sets	set	NOUN
ejpam-2214	144	4	with	with	ADP
ejpam-2214	144	5	respect	respect	NOUN
ejpam-2214	144	6	to	to	ADP
ejpam-2214	144	7	and	and	CCONJ
ejpam-2214	144	8	ideal	ideal	ADJ
ejpam-2214	144	9	.	.	PUNCT
ejpam-2214	145	1	european	european	ADJ
ejpam-2214	145	2	journal	journal	PROPN
ejpam-2214	145	3	of	of	ADP
ejpam-2214	145	4	pure	pure	ADJ
ejpam-2214	145	5	and	and	CCONJ
ejpam-2214	145	6	applied	applied	ADJ
ejpam-2214	145	7	mathemetics	mathemetic	NOUN
ejpam-2214	145	8	,	,	PUNCT
ejpam-2214	145	9	4(2):147	4(2):147	NUM
ejpam-2214	145	10	-	-	SYM
ejpam-2214	145	11	151	151	NUM
ejpam-2214	145	12	,	,	PUNCT
ejpam-2214	145	13	2011	2011	NUM
ejpam-2214	145	14	.	.	PUNCT
ejpam-2214	146	1	[	[	X
ejpam-2214	146	2	3	3	NUM
ejpam-2214	146	3	]	]	PUNCT
ejpam-2214	146	4	n	n	DET
ejpam-2214	146	5	levine	levine	PROPN
ejpam-2214	146	6	.	.	PUNCT
ejpam-2214	147	1	semi	semi	ADV
ejpam-2214	147	2	open	open	ADJ
ejpam-2214	147	3	sets	set	NOUN
ejpam-2214	147	4	and	and	CCONJ
ejpam-2214	147	5	semi	semi	ADJ
ejpam-2214	147	6	continuity	continuity	NOUN
ejpam-2214	147	7	in	in	ADP
ejpam-2214	147	8	topological	topological	ADJ
ejpam-2214	147	9	spaces	space	NOUN
ejpam-2214	147	10	.	.	PUNCT
ejpam-2214	148	1	american	american	PROPN
ejpam-2214	148	2	mathematical	mathematical	PROPN
ejpam-2214	148	3	monthly	monthly	ADJ
ejpam-2214	148	4	,	,	PUNCT
ejpam-2214	148	5	70:36	70:36	NUM
ejpam-2214	148	6	-	-	SYM
ejpam-2214	148	7	41	41	NUM
ejpam-2214	148	8	,	,	PUNCT
ejpam-2214	148	9	1963	1963	NUM
ejpam-2214	148	10	.	.	PUNCT
ejpam-2214	149	1	[	[	X
ejpam-2214	149	2	4	4	NUM
ejpam-2214	149	3	]	]	X
ejpam-2214	149	4	h	h	NOUN
ejpam-2214	149	5	maki	maki	NOUN
ejpam-2214	149	6	,	,	PUNCT
ejpam-2214	149	7	r	r	NOUN
ejpam-2214	149	8	chandrasekhara	chandrasekhara	PROPN
ejpam-2214	149	9	,	,	PUNCT
ejpam-2214	149	10	and	and	CCONJ
ejpam-2214	149	11	a	a	DET
ejpam-2214	149	12	nagoor	nagoor	NOUN
ejpam-2214	149	13	gani	gani	PROPN
ejpam-2214	149	14	.	.	PUNCT
ejpam-2214	150	1	on	on	ADP
ejpam-2214	150	2	generalizing	generalize	VERB
ejpam-2214	150	3	semi	semi	ADJ
ejpam-2214	150	4	-	-	ADJ
ejpam-2214	150	5	open	open	ADJ
ejpam-2214	150	6	sets	set	NOUN
ejpam-2214	150	7	and	and	CCONJ
ejpam-2214	150	8	preopen	preopen	ADJ
ejpam-2214	150	9	sets	set	NOUN
ejpam-2214	150	10	.	.	PUNCT
ejpam-2214	151	1	pure	pure	ADJ
ejpam-2214	151	2	and	and	CCONJ
ejpam-2214	151	3	applied	applied	ADJ
ejpam-2214	151	4	mathematical	mathematical	ADJ
ejpam-2214	151	5	sciences	science	NOUN
ejpam-2214	151	6	,	,	PUNCT
ejpam-2214	151	7	49:17	49:17	NUM
ejpam-2214	151	8	-	-	SYM
ejpam-2214	151	9	29	29	NUM
ejpam-2214	151	10	,	,	PUNCT
ejpam-2214	151	11	1999	1999	NUM
ejpam-2214	151	12	.	.	PUNCT