id	sid	tid	token	lemma	pos
ejpam-2240	1	1	compile	compile	NOUN
ejpam-2240	1	2	/	/	SYM
ejpam-2240	1	3	output.dvi	output.dvi	NOUN
ejpam-2240	1	4	european	european	ADJ
ejpam-2240	1	5	journal	journal	NOUN
ejpam-2240	1	6	of	of	ADP
ejpam-2240	1	7	pure	pure	ADJ
ejpam-2240	1	8	and	and	CCONJ
ejpam-2240	1	9	applied	apply	VERB
ejpam-2240	1	10	mathematics	mathematic	NOUN
ejpam-2240	1	11	vol	vol	NOUN
ejpam-2240	1	12	.	.	PROPN
ejpam-2240	1	13	8	8	NUM
ejpam-2240	1	14	,	,	PUNCT
ejpam-2240	1	15	no	no	INTJ
ejpam-2240	1	16	.	.	NOUN
ejpam-2240	1	17	3	3	NUM
ejpam-2240	1	18	,	,	PUNCT
ejpam-2240	1	19	2015	2015	NUM
ejpam-2240	1	20	,	,	PUNCT
ejpam-2240	1	21	389	389	NUM
ejpam-2240	1	22	-	-	SYM
ejpam-2240	1	23	394	394	NUM
ejpam-2240	1	24	issn	issn	PROPN
ejpam-2240	1	25	1307	1307	NUM
ejpam-2240	1	26	-	-	SYM
ejpam-2240	1	27	5543	5543	NUM
ejpam-2240	1	28	–	–	PUNCT
ejpam-2240	1	29	www.ejpam.com	www.ejpam.com	X
ejpam-2240	1	30	idealization	idealization	NOUN
ejpam-2240	1	31	of	of	ADP
ejpam-2240	1	32	some	some	DET
ejpam-2240	1	33	topological	topological	ADJ
ejpam-2240	1	34	concepts	concept	NOUN
ejpam-2240	1	35	arafa	arafa	PROPN
ejpam-2240	1	36	a.	a.	PROPN
ejpam-2240	1	37	nasef1	nasef1	PROPN
ejpam-2240	1	38	,	,	PUNCT
ejpam-2240	1	39	r.	r.	PROPN
ejpam-2240	1	40	mareay2,∗	mareay2,∗	PROPN
ejpam-2240	1	41	,	,	PUNCT
ejpam-2240	1	42	f.i	f.i	PROPN
ejpam-2240	1	43	.	.	PROPN
ejpam-2240	1	44	michael3	michael3	PROPN
ejpam-2240	2	1	1	1	NUM
ejpam-2240	2	2	department	department	PROPN
ejpam-2240	2	3	of	of	ADP
ejpam-2240	2	4	physics	physics	PROPN
ejpam-2240	2	5	and	and	CCONJ
ejpam-2240	2	6	engineering	engineering	NOUN
ejpam-2240	2	7	mathematics	mathematic	NOUN
ejpam-2240	2	8	,	,	PUNCT
ejpam-2240	2	9	faculty	faculty	NOUN
ejpam-2240	2	10	of	of	ADP
ejpam-2240	2	11	engineering	engineering	PROPN
ejpam-2240	2	12	,	,	PUNCT
ejpam-2240	2	13	kafrelsheikh	kafrelsheikh	PROPN
ejpam-2240	2	14	university	university	PROPN
ejpam-2240	2	15	,	,	PUNCT
ejpam-2240	2	16	kafr	kafr	PROPN
ejpam-2240	2	17	el	el	PROPN
ejpam-2240	2	18	-	-	PUNCT
ejpam-2240	2	19	sheikh	sheikh	PROPN
ejpam-2240	2	20	33516	33516	NUM
ejpam-2240	2	21	,	,	PUNCT
ejpam-2240	2	22	egypt	egypt	PROPN
ejpam-2240	2	23	2	2	NUM
ejpam-2240	2	24	department	department	NOUN
ejpam-2240	2	25	of	of	ADP
ejpam-2240	2	26	mathematics	mathematic	NOUN
ejpam-2240	2	27	,	,	PUNCT
ejpam-2240	2	28	faculty	faculty	NOUN
ejpam-2240	2	29	of	of	ADP
ejpam-2240	2	30	science	science	NOUN
ejpam-2240	2	31	,	,	PUNCT
ejpam-2240	2	32	kafrelsheikh	kafrelsheikh	PROPN
ejpam-2240	2	33	university	university	PROPN
ejpam-2240	2	34	,	,	PUNCT
ejpam-2240	2	35	kafr	kafr	PROPN
ejpam-2240	2	36	el	el	PROPN
ejpam-2240	2	37	-	-	PUNCT
ejpam-2240	2	38	sheikh	sheikh	PROPN
ejpam-2240	2	39	33516	33516	NUM
ejpam-2240	2	40	,	,	PUNCT
ejpam-2240	2	41	egypt	egypt	PROPN
ejpam-2240	2	42	3	3	NUM
ejpam-2240	2	43	department	department	NOUN
ejpam-2240	2	44	of	of	ADP
ejpam-2240	2	45	mathematics	mathematic	NOUN
ejpam-2240	2	46	,	,	PUNCT
ejpam-2240	2	47	faculty	faculty	NOUN
ejpam-2240	2	48	of	of	ADP
ejpam-2240	2	49	science	science	NOUN
ejpam-2240	2	50	,	,	PUNCT
ejpam-2240	2	51	university	university	NOUN
ejpam-2240	2	52	of	of	ADP
ejpam-2240	2	53	western	western	PROPN
ejpam-2240	2	54	ontario	ontario	PROPN
ejpam-2240	2	55	,	,	PUNCT
ejpam-2240	2	56	london	london	PROPN
ejpam-2240	2	57	,	,	PUNCT
ejpam-2240	2	58	ontario	ontario	PROPN
ejpam-2240	2	59	,	,	PUNCT
ejpam-2240	2	60	canada	canada	PROPN
ejpam-2240	2	61	abstract	abstract	NOUN
ejpam-2240	2	62	.	.	PUNCT
ejpam-2240	3	1	we	we	PRON
ejpam-2240	3	2	introduce	introduce	VERB
ejpam-2240	3	3	a	a	DET
ejpam-2240	3	4	notion	notion	NOUN
ejpam-2240	3	5	of	of	ADP
ejpam-2240	3	6	β	β	X
ejpam-2240	3	7	-open	-open	NOUN
ejpam-2240	3	8	sets	set	NOUN
ejpam-2240	3	9	in	in	ADP
ejpam-2240	3	10	terms	term	NOUN
ejpam-2240	3	11	of	of	ADP
ejpam-2240	3	12	ideals	ideal	NOUN
ejpam-2240	3	13	,	,	PUNCT
ejpam-2240	3	14	which	which	PRON
ejpam-2240	3	15	generalizes	generalize	VERB
ejpam-2240	3	16	the	the	DET
ejpam-2240	3	17	usual	usual	ADJ
ejpam-2240	3	18	notion	notion	NOUN
ejpam-2240	3	19	of	of	ADP
ejpam-2240	3	20	β	β	X
ejpam-2240	3	21	-open	-open	NOUN
ejpam-2240	3	22	sets	set	NOUN
ejpam-2240	3	23	.	.	PUNCT
ejpam-2240	4	1	2010	2010	NUM
ejpam-2240	4	2	mathematics	mathematic	NOUN
ejpam-2240	4	3	subject	subject	NOUN
ejpam-2240	4	4	classifications	classification	NOUN
ejpam-2240	4	5	:	:	PUNCT
ejpam-2240	4	6	primary	primary	ADJ
ejpam-2240	4	7	:	:	PUNCT
ejpam-2240	4	8	54c10	54c10	NUM
ejpam-2240	4	9	;	;	PUNCT
ejpam-2240	4	10	54c08	54c08	NUM
ejpam-2240	4	11	,	,	PUNCT
ejpam-2240	4	12	secondary	secondary	ADJ
ejpam-2240	4	13	:	:	PUNCT
ejpam-2240	4	14	54h05	54h05	NUM
ejpam-2240	4	15	;	;	PUNCT
ejpam-2240	4	16	54c05	54c05	NUM
ejpam-2240	4	17	key	key	ADJ
ejpam-2240	4	18	words	word	NOUN
ejpam-2240	4	19	and	and	CCONJ
ejpam-2240	4	20	phrases	phrase	NOUN
ejpam-2240	4	21	:	:	PUNCT
ejpam-2240	4	22	β	β	NOUN
ejpam-2240	4	23	-open	-open	NOUN
ejpam-2240	4	24	sets	set	NOUN
ejpam-2240	4	25	,	,	PUNCT
ejpam-2240	4	26	β	β	X
ejpam-2240	4	27	-closed	-close	VERB
ejpam-2240	4	28	sets	set	NOUN
ejpam-2240	4	29	,	,	PUNCT
ejpam-2240	4	30	generalized	generalize	VERB
ejpam-2240	4	31	closed	closed	ADJ
ejpam-2240	4	32	sets	set	NOUN
ejpam-2240	4	33	,	,	PUNCT
ejpam-2240	4	34	ideals	ideal	NOUN
ejpam-2240	4	35	.	.	PUNCT
ejpam-2240	5	1	1	1	X
ejpam-2240	5	2	.	.	X
ejpam-2240	5	3	introduction	introduction	NOUN
ejpam-2240	5	4	the	the	DET
ejpam-2240	5	5	notion	notion	NOUN
ejpam-2240	5	6	of	of	ADP
ejpam-2240	5	7	ideal	ideal	ADJ
ejpam-2240	5	8	topological	topological	ADJ
ejpam-2240	5	9	spaces	space	NOUN
ejpam-2240	5	10	was	be	AUX
ejpam-2240	5	11	studied	study	VERB
ejpam-2240	5	12	by	by	ADP
ejpam-2240	5	13	kuratowski	kuratowski	ADJ
ejpam-2240	6	1	[	[	X
ejpam-2240	6	2	6	6	NUM
ejpam-2240	6	3	]	]	PUNCT
ejpam-2240	6	4	and	and	CCONJ
ejpam-2240	6	5	vaidyanathasmamy	vaidyanathasmamy	ADJ
ejpam-2240	6	6	[	[	X
ejpam-2240	6	7	11	11	NUM
ejpam-2240	6	8	]	]	PUNCT
ejpam-2240	6	9	.	.	PUNCT
ejpam-2240	7	1	applications	application	NOUN
ejpam-2240	7	2	to	to	ADP
ejpam-2240	7	3	various	various	ADJ
ejpam-2240	7	4	fields	field	NOUN
ejpam-2240	7	5	were	be	AUX
ejpam-2240	7	6	further	far	ADV
ejpam-2240	7	7	investigated	investigate	VERB
ejpam-2240	7	8	by	by	ADP
ejpam-2240	7	9	jankovic	jankovic	PROPN
ejpam-2240	7	10	and	and	CCONJ
ejpam-2240	7	11	hamlett	hamlett	PROPN
ejpam-2240	8	1	[	[	X
ejpam-2240	8	2	5	5	NUM
ejpam-2240	8	3	]	]	PUNCT
ejpam-2240	8	4	;	;	PUNCT
ejpam-2240	8	5	dontchev	dontchev	PROPN
ejpam-2240	8	6	et	et	PROPN
ejpam-2240	8	7	al	al	PROPN
ejpam-2240	8	8	.	.	PUNCT
ejpam-2240	9	1	[	[	X
ejpam-2240	9	2	3	3	NUM
ejpam-2240	9	3	]	]	PUNCT
ejpam-2240	9	4	,	,	PUNCT
ejpam-2240	9	5	mukherjee	mukherjee	PROPN
ejpam-2240	9	6	et	et	PROPN
ejpam-2240	9	7	al	al	PROPN
ejpam-2240	9	8	.	.	PUNCT
ejpam-2240	10	1	[	[	X
ejpam-2240	10	2	9	9	NUM
ejpam-2240	10	3	]	]	PUNCT
ejpam-2240	10	4	;	;	PUNCT
ejpam-2240	10	5	arenos	arenos	PROPN
ejpam-2240	10	6	et	et	PROPN
ejpam-2240	10	7	al	al	PROPN
ejpam-2240	10	8	.	.	PUNCT
ejpam-2240	11	1	[	[	X
ejpam-2240	11	2	2	2	NUM
ejpam-2240	11	3	]	]	PUNCT
ejpam-2240	11	4	;	;	PUNCT
ejpam-2240	11	5	nasef	nasef	PROPN
ejpam-2240	11	6	and	and	CCONJ
ejpam-2240	11	7	mahmoud	mahmoud	PROPN
ejpam-2240	12	1	[	[	X
ejpam-2240	12	2	10	10	NUM
ejpam-2240	12	3	]	]	PUNCT
ejpam-2240	12	4	,	,	PUNCT
ejpam-2240	12	5	etc	etc	X
ejpam-2240	12	6	.	.	X
ejpam-2240	13	1	the	the	DET
ejpam-2240	13	2	interest	interest	NOUN
ejpam-2240	13	3	in	in	ADP
ejpam-2240	13	4	the	the	DET
ejpam-2240	13	5	idealized	idealized	ADJ
ejpam-2240	13	6	version	version	NOUN
ejpam-2240	13	7	of	of	ADP
ejpam-2240	13	8	many	many	ADJ
ejpam-2240	13	9	general	general	ADJ
ejpam-2240	13	10	topological	topological	ADJ
ejpam-2240	13	11	properties	property	NOUN
ejpam-2240	13	12	has	have	AUX
ejpam-2240	13	13	grown	grow	VERB
ejpam-2240	13	14	drastically	drastically	ADV
ejpam-2240	13	15	in	in	ADP
ejpam-2240	13	16	the	the	DET
ejpam-2240	13	17	past	past	ADJ
ejpam-2240	13	18	20	20	NUM
ejpam-2240	13	19	years	year	NOUN
ejpam-2240	13	20	.	.	PUNCT
ejpam-2240	14	1	in	in	ADP
ejpam-2240	14	2	this	this	DET
ejpam-2240	14	3	paper	paper	NOUN
ejpam-2240	14	4	,	,	PUNCT
ejpam-2240	14	5	we	we	PRON
ejpam-2240	14	6	define	define	VERB
ejpam-2240	14	7	β	β	X
ejpam-2240	14	8	-open	-open	NOUN
ejpam-2240	14	9	sets	set	NOUN
ejpam-2240	14	10	with	with	ADP
ejpam-2240	14	11	respect	respect	NOUN
ejpam-2240	14	12	to	to	ADP
ejpam-2240	14	13	an	an	DET
ejpam-2240	14	14	ideal	ideal	NOUN
ejpam-2240	14	15	i	i	PRON
ejpam-2240	14	16	,	,	PUNCT
ejpam-2240	14	17	and	and	CCONJ
ejpam-2240	14	18	also	also	ADV
ejpam-2240	14	19	study	study	VERB
ejpam-2240	14	20	some	some	PRON
ejpam-2240	14	21	of	of	ADP
ejpam-2240	14	22	their	their	PRON
ejpam-2240	14	23	properties	property	NOUN
ejpam-2240	14	24	.	.	PUNCT
ejpam-2240	15	1	it	it	PRON
ejpam-2240	15	2	turns	turn	VERB
ejpam-2240	15	3	out	out	ADP
ejpam-2240	15	4	that	that	SCONJ
ejpam-2240	15	5	our	our	PRON
ejpam-2240	15	6	notion	notion	NOUN
ejpam-2240	15	7	of	of	ADP
ejpam-2240	15	8	β	β	X
ejpam-2240	15	9	-open	-open	NOUN
ejpam-2240	15	10	sets	set	NOUN
ejpam-2240	15	11	with	with	ADP
ejpam-2240	15	12	respect	respect	NOUN
ejpam-2240	15	13	to	to	ADP
ejpam-2240	15	14	a	a	DET
ejpam-2240	15	15	given	give	VERB
ejpam-2240	15	16	ideal	ideal	NOUN
ejpam-2240	15	17	i	i	PRON
ejpam-2240	15	18	generalizes	generalize	VERB
ejpam-2240	15	19	both	both	CCONJ
ejpam-2240	15	20	the	the	DET
ejpam-2240	15	21	usual	usual	ADJ
ejpam-2240	15	22	notion	notion	NOUN
ejpam-2240	15	23	of	of	ADP
ejpam-2240	15	24	β	β	X
ejpam-2240	15	25	-openness	-openness	PROPN
ejpam-2240	15	26	[	[	X
ejpam-2240	15	27	1	1	NUM
ejpam-2240	15	28	]	]	PUNCT
ejpam-2240	15	29	and	and	CCONJ
ejpam-2240	15	30	the	the	DET
ejpam-2240	15	31	notion	notion	NOUN
ejpam-2240	15	32	of	of	ADP
ejpam-2240	15	33	β	β	X
ejpam-2240	15	34	−	−	PROPN
ejpam-2240	15	35	i	i	PRON
ejpam-2240	15	36	-openness	-openness	PROPN
ejpam-2240	15	37	considered	consider	VERB
ejpam-2240	15	38	in	in	ADP
ejpam-2240	15	39	[	[	X
ejpam-2240	15	40	4	4	NUM
ejpam-2240	15	41	]	]	PUNCT
ejpam-2240	15	42	,	,	PUNCT
ejpam-2240	15	43	in	in	ADP
ejpam-2240	15	44	particular	particular	ADJ
ejpam-2240	15	45	,	,	PUNCT
ejpam-2240	15	46	β	β	PROPN
ejpam-2240	15	47	−	−	NOUN
ejpam-2240	15	48	i	i	PRON
ejpam-2240	15	49	-openness	-openness	PROPN
ejpam-2240	15	50	implies	imply	VERB
ejpam-2240	15	51	the	the	DET
ejpam-2240	15	52	usual	usual	ADJ
ejpam-2240	15	53	β	β	NOUN
ejpam-2240	15	54	-openness	-openness	PROPN
ejpam-2240	15	55	,	,	PUNCT
ejpam-2240	15	56	which	which	PRON
ejpam-2240	15	57	in	in	ADP
ejpam-2240	15	58	turn	turn	NOUN
ejpam-2240	15	59	implies	imply	VERB
ejpam-2240	15	60	β	β	NOUN
ejpam-2240	15	61	-openness	-openness	NOUN
ejpam-2240	15	62	in	in	ADP
ejpam-2240	15	63	our	our	PRON
ejpam-2240	15	64	sense	sense	NOUN
ejpam-2240	15	65	.	.	PUNCT
ejpam-2240	16	1	this	this	DET
ejpam-2240	16	2	paper	paper	NOUN
ejpam-2240	16	3	is	be	AUX
ejpam-2240	16	4	related	relate	VERB
ejpam-2240	16	5	to	to	ADP
ejpam-2240	16	6	[	[	X
ejpam-2240	16	7	8	8	NUM
ejpam-2240	16	8	]	]	PUNCT
ejpam-2240	16	9	.	.	PUNCT
ejpam-2240	17	1	∗corresponding	∗corresponde	VERB
ejpam-2240	17	2	author	author	NOUN
ejpam-2240	17	3	.	.	PUNCT
ejpam-2240	18	1	email	email	NOUN
ejpam-2240	18	2	address	address	NOUN
ejpam-2240	18	3	:	:	PUNCT
ejpam-2240	18	4	roshdeymareay@yahoo.com	roshdeymareay@yahoo.com	X
ejpam-2240	18	5	(	(	PUNCT
ejpam-2240	18	6	r.	r.	PROPN
ejpam-2240	18	7	mareay	mareay	PROPN
ejpam-2240	18	8	)	)	PUNCT
ejpam-2240	18	9	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2240	19	1	389	389	NUM
ejpam-2240	19	2	c	c	NOUN
ejpam-2240	19	3	©	©	PROPN
ejpam-2240	19	4	2015	2015	NUM
ejpam-2240	19	5	ejpam	ejpam	NOUN
ejpam-2240	19	6	all	all	DET
ejpam-2240	19	7	rights	right	NOUN
ejpam-2240	19	8	reserved	reserve	VERB
ejpam-2240	19	9	.	.	PUNCT
ejpam-2240	20	1	a.	a.	PROPN
ejpam-2240	20	2	nasef	nasef	PROPN
ejpam-2240	20	3	,	,	PUNCT
ejpam-2240	20	4	r.	r.	PROPN
ejpam-2240	20	5	mareay	mareay	PROPN
ejpam-2240	20	6	,	,	PUNCT
ejpam-2240	20	7	f.	f.	PROPN
ejpam-2240	20	8	michael	michael	PROPN
ejpam-2240	20	9	/	/	SYM
ejpam-2240	20	10	eur	eur	PROPN
ejpam-2240	20	11	.	.	PUNCT
ejpam-2240	21	1	j.	j.	PROPN
ejpam-2240	21	2	pure	pure	PROPN
ejpam-2240	21	3	appl	appl	PROPN
ejpam-2240	21	4	.	.	PROPN
ejpam-2240	21	5	math	math	PROPN
ejpam-2240	21	6	,	,	PUNCT
ejpam-2240	21	7	8	8	NUM
ejpam-2240	21	8	(	(	PUNCT
ejpam-2240	21	9	2015	2015	NUM
ejpam-2240	21	10	)	)	PUNCT
ejpam-2240	21	11	,	,	PUNCT
ejpam-2240	21	12	389	389	NUM
ejpam-2240	21	13	-	-	SYM
ejpam-2240	21	14	394	394	NUM
ejpam-2240	21	15	390	390	NUM
ejpam-2240	21	16	2	2	NUM
ejpam-2240	21	17	.	.	PUNCT
ejpam-2240	21	18	preliminaries	preliminary	NOUN
ejpam-2240	21	19	throughout	throughout	ADP
ejpam-2240	21	20	this	this	DET
ejpam-2240	21	21	paper	paper	NOUN
ejpam-2240	21	22	(	(	PUNCT
ejpam-2240	21	23	x	x	X
ejpam-2240	21	24	,	,	PUNCT
ejpam-2240	21	25	τ	τ	X
ejpam-2240	21	26	)	)	PUNCT
ejpam-2240	21	27	(	(	PUNCT
ejpam-2240	21	28	or	or	CCONJ
ejpam-2240	21	29	simply	simply	ADV
ejpam-2240	21	30	x	x	X
ejpam-2240	21	31	)	)	PUNCT
ejpam-2240	21	32	represent	represent	VERB
ejpam-2240	21	33	topological	topological	ADJ
ejpam-2240	21	34	spaces	space	NOUN
ejpam-2240	21	35	on	on	ADP
ejpam-2240	21	36	which	which	PRON
ejpam-2240	21	37	no	no	DET
ejpam-2240	21	38	separation	separation	NOUN
ejpam-2240	21	39	axioms	axiom	NOUN
ejpam-2240	21	40	are	be	AUX
ejpam-2240	21	41	assumed	assume	VERB
ejpam-2240	21	42	unless	unless	SCONJ
ejpam-2240	21	43	otherwise	otherwise	ADV
ejpam-2240	21	44	mentioned	mention	VERB
ejpam-2240	21	45	.	.	PUNCT
ejpam-2240	22	1	for	for	ADP
ejpam-2240	22	2	a	a	DET
ejpam-2240	22	3	subset	subset	NOUN
ejpam-2240	22	4	a	a	PRON
ejpam-2240	22	5	of	of	ADP
ejpam-2240	22	6	x	x	PRON
ejpam-2240	22	7	,	,	PUNCT
ejpam-2240	22	8	cl(a	cl(a	NUM
ejpam-2240	22	9	)	)	PUNCT
ejpam-2240	22	10	,	,	PUNCT
ejpam-2240	22	11	int(a	int(a	PROPN
ejpam-2240	22	12	)	)	PUNCT
ejpam-2240	22	13	and	and	CCONJ
ejpam-2240	22	14	ac	ac	PROPN
ejpam-2240	22	15	denote	denote	VERB
ejpam-2240	22	16	the	the	DET
ejpam-2240	22	17	closure	closure	NOUN
ejpam-2240	22	18	of	of	ADP
ejpam-2240	22	19	a	a	PRON
ejpam-2240	22	20	,	,	PUNCT
ejpam-2240	22	21	the	the	DET
ejpam-2240	22	22	interior	interior	NOUN
ejpam-2240	22	23	of	of	ADP
ejpam-2240	22	24	a	a	PRON
ejpam-2240	22	25	and	and	CCONJ
ejpam-2240	22	26	the	the	DET
ejpam-2240	22	27	complement	complement	NOUN
ejpam-2240	22	28	of	of	ADP
ejpam-2240	22	29	a	a	PRON
ejpam-2240	22	30	,	,	PUNCT
ejpam-2240	22	31	respectively	respectively	ADV
ejpam-2240	22	32	.	.	PUNCT
ejpam-2240	23	1	let	let	VERB
ejpam-2240	23	2	us	we	PRON
ejpam-2240	23	3	recall	recall	VERB
ejpam-2240	23	4	the	the	DET
ejpam-2240	23	5	following	follow	VERB
ejpam-2240	23	6	definitions	definition	NOUN
ejpam-2240	23	7	,	,	PUNCT
ejpam-2240	23	8	which	which	PRON
ejpam-2240	23	9	are	be	AUX
ejpam-2240	23	10	useful	useful	ADJ
ejpam-2240	23	11	in	in	ADP
ejpam-2240	23	12	the	the	DET
ejpam-2240	23	13	sequel	sequel	NOUN
ejpam-2240	23	14	.	.	PUNCT
ejpam-2240	24	1	definition	definition	NOUN
ejpam-2240	24	2	1	1	NUM
ejpam-2240	24	3	.	.	PUNCT
ejpam-2240	25	1	a	a	DET
ejpam-2240	25	2	subset	subset	NOUN
ejpam-2240	25	3	a	a	PRON
ejpam-2240	25	4	of	of	ADP
ejpam-2240	25	5	a	a	DET
ejpam-2240	25	6	space	space	NOUN
ejpam-2240	25	7	(	(	PUNCT
ejpam-2240	25	8	x	x	X
ejpam-2240	25	9	,	,	PUNCT
ejpam-2240	25	10	τ	τ	X
ejpam-2240	25	11	)	)	PUNCT
ejpam-2240	25	12	is	be	AUX
ejpam-2240	25	13	called	call	VERB
ejpam-2240	25	14	a	a	DET
ejpam-2240	25	15	(	(	PUNCT
ejpam-2240	25	16	i	i	NOUN
ejpam-2240	25	17	)	)	PUNCT
ejpam-2240	25	18	preopen	preopen	NOUN
ejpam-2240	25	19	set	set	VERB
ejpam-2240	25	20	[	[	X
ejpam-2240	25	21	7	7	X
ejpam-2240	25	22	]	]	X
ejpam-2240	25	23	if	if	SCONJ
ejpam-2240	25	24	a⊆	a⊆	PROPN
ejpam-2240	25	25	int(cl(a	int(cl(a	PROPN
ejpam-2240	25	26	)	)	PUNCT
ejpam-2240	25	27	)	)	PUNCT
ejpam-2240	26	1	,	,	PUNCT
ejpam-2240	26	2	(	(	PUNCT
ejpam-2240	26	3	ii	ii	NOUN
ejpam-2240	26	4	)	)	PUNCT
ejpam-2240	26	5	β	β	X
ejpam-2240	26	6	-open	-open	NOUN
ejpam-2240	26	7	set	set	VERB
ejpam-2240	26	8	[	[	X
ejpam-2240	26	9	1	1	NUM
ejpam-2240	26	10	]	]	X
ejpam-2240	26	11	if	if	SCONJ
ejpam-2240	26	12	a⊆	a⊆	NOUN
ejpam-2240	26	13	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-2240	26	14	)	)	PUNCT
ejpam-2240	26	15	)	)	PUNCT
ejpam-2240	26	16	)	)	PUNCT
ejpam-2240	26	17	.	.	PUNCT
ejpam-2240	27	1	the	the	DET
ejpam-2240	27	2	complement	complement	NOUN
ejpam-2240	27	3	of	of	ADP
ejpam-2240	27	4	a	a	DET
ejpam-2240	27	5	preopen	preopen	ADJ
ejpam-2240	27	6	(	(	PUNCT
ejpam-2240	27	7	resp	resp	NOUN
ejpam-2240	27	8	.	.	PUNCT
ejpam-2240	28	1	β	β	X
ejpam-2240	28	2	-open	-open	NOUN
ejpam-2240	28	3	)	)	PUNCT
ejpam-2240	28	4	set	set	NOUN
ejpam-2240	28	5	is	be	AUX
ejpam-2240	28	6	called	call	VERB
ejpam-2240	28	7	preclosed	preclose	VERB
ejpam-2240	28	8	(	(	PUNCT
ejpam-2240	28	9	resp	resp	NOUN
ejpam-2240	28	10	.	.	PUNCT
ejpam-2240	29	1	β	β	X
ejpam-2240	29	2	-closed	-close	VERB
ejpam-2240	29	3	)	)	PUNCT
ejpam-2240	29	4	.	.	PUNCT
ejpam-2240	30	1	definition	definition	NOUN
ejpam-2240	30	2	2	2	NUM
ejpam-2240	30	3	.	.	PUNCT
ejpam-2240	31	1	an	an	DET
ejpam-2240	31	2	ideal	ideal	NOUN
ejpam-2240	31	3	i	i	PRON
ejpam-2240	31	4	on	on	ADP
ejpam-2240	31	5	a	a	DET
ejpam-2240	31	6	topological	topological	ADJ
ejpam-2240	31	7	space	space	NOUN
ejpam-2240	31	8	(	(	PUNCT
ejpam-2240	31	9	x	x	X
ejpam-2240	31	10	,	,	PUNCT
ejpam-2240	31	11	τ	τ	X
ejpam-2240	31	12	)	)	PUNCT
ejpam-2240	31	13	is	be	AUX
ejpam-2240	31	14	a	a	DET
ejpam-2240	31	15	non	non	X
ejpam-2240	31	16	empty	empty	ADJ
ejpam-2240	31	17	collection	collection	NOUN
ejpam-2240	31	18	of	of	ADP
ejpam-2240	31	19	subsets	subset	NOUN
ejpam-2240	31	20	of	of	ADP
ejpam-2240	31	21	x	x	PUNCT
ejpam-2240	31	22	which	which	PRON
ejpam-2240	31	23	satisfies	satisfy	VERB
ejpam-2240	31	24	the	the	DET
ejpam-2240	31	25	following	follow	VERB
ejpam-2240	31	26	conditions	condition	NOUN
ejpam-2240	31	27	:	:	PUNCT
ejpam-2240	31	28	(	(	PUNCT
ejpam-2240	31	29	i	i	NOUN
ejpam-2240	31	30	)	)	PUNCT
ejpam-2240	31	31	a∈	a∈	PROPN
ejpam-2240	32	1	i	i	PROPN
ejpam-2240	32	2	and	and	CCONJ
ejpam-2240	32	3	b	b	X
ejpam-2240	32	4	⊆	⊆	NUM
ejpam-2240	32	5	a	a	PRON
ejpam-2240	32	6	implies	imply	VERB
ejpam-2240	32	7	b	b	X
ejpam-2240	32	8	∈	∈	X
ejpam-2240	32	9	i	i	PRON
ejpam-2240	32	10	(	(	PUNCT
ejpam-2240	32	11	heredity	heredity	NOUN
ejpam-2240	32	12	)	)	PUNCT
ejpam-2240	32	13	,	,	PUNCT
ejpam-2240	32	14	(	(	PUNCT
ejpam-2240	32	15	ii	ii	NOUN
ejpam-2240	32	16	)	)	PUNCT
ejpam-2240	32	17	a∈	a∈	PROPN
ejpam-2240	32	18	i	i	PROPN
ejpam-2240	32	19	and	and	CCONJ
ejpam-2240	32	20	b	b	X
ejpam-2240	32	21	∈	∈	PROPN
ejpam-2240	32	22	i	i	PRON
ejpam-2240	32	23	implies	imply	VERB
ejpam-2240	32	24	a∪	a∪	PROPN
ejpam-2240	32	25	b	b	X
ejpam-2240	32	26	∈	∈	PROPN
ejpam-2240	33	1	i	i	PRON
ejpam-2240	33	2	(	(	PUNCT
ejpam-2240	33	3	finite	finite	PROPN
ejpam-2240	33	4	additinity	additinity	NOUN
ejpam-2240	33	5	)	)	PUNCT
ejpam-2240	33	6	.	.	PUNCT
ejpam-2240	34	1	an	an	DET
ejpam-2240	34	2	ideal	ideal	ADJ
ejpam-2240	34	3	topological	topological	ADJ
ejpam-2240	34	4	space	space	NOUN
ejpam-2240	34	5	(	(	PUNCT
ejpam-2240	34	6	x	x	X
ejpam-2240	34	7	,	,	PUNCT
ejpam-2240	34	8	τ	τ	PROPN
ejpam-2240	34	9	)	)	PUNCT
ejpam-2240	34	10	with	with	ADP
ejpam-2240	34	11	an	an	DET
ejpam-2240	34	12	ideal	ideal	ADJ
ejpam-2240	34	13	i	i	PRON
ejpam-2240	34	14	on	on	ADP
ejpam-2240	34	15	x	x	VERB
ejpam-2240	34	16	is	be	AUX
ejpam-2240	34	17	denoted	denote	VERB
ejpam-2240	34	18	by	by	ADP
ejpam-2240	34	19	(	(	PUNCT
ejpam-2240	34	20	x	x	INTJ
ejpam-2240	34	21	,	,	PUNCT
ejpam-2240	34	22	τ	τ	PROPN
ejpam-2240	34	23	,	,	PUNCT
ejpam-2240	34	24	i	i	PROPN
ejpam-2240	34	25	)	)	PUNCT
ejpam-2240	34	26	.	.	PUNCT
ejpam-2240	35	1	definition	definition	NOUN
ejpam-2240	35	2	3	3	NUM
ejpam-2240	35	3	.	.	PUNCT
ejpam-2240	36	1	for	for	ADP
ejpam-2240	36	2	a	a	DET
ejpam-2240	36	3	subset	subset	NOUN
ejpam-2240	36	4	a	a	DET
ejpam-2240	36	5	⊆	⊆	NUM
ejpam-2240	36	6	x	x	SYM
ejpam-2240	36	7	,	,	PUNCT
ejpam-2240	36	8	a∗(i	a∗(i	PROPN
ejpam-2240	36	9	)	)	PUNCT
ejpam-2240	36	10	=	=	PRON
ejpam-2240	36	11	{	{	PUNCT
ejpam-2240	36	12	x	x	PUNCT
ejpam-2240	36	13	∈	∈	PROPN
ejpam-2240	36	14	x	x	X
ejpam-2240	36	15	:	:	PUNCT
ejpam-2240	36	16	g	g	PROPN
ejpam-2240	36	17	∩	∩	PROPN
ejpam-2240	36	18	a	a	X
ejpam-2240	36	19	/∈	/∈	PUNCT
ejpam-2240	36	20	i	i	PRON
ejpam-2240	36	21	for	for	ADP
ejpam-2240	36	22	each	each	DET
ejpam-2240	36	23	neighborhood	neighborhood	NOUN
ejpam-2240	36	24	g	g	NOUN
ejpam-2240	36	25	of	of	ADP
ejpam-2240	36	26	x	x	PRON
ejpam-2240	36	27	}	}	PUNCT
ejpam-2240	36	28	is	be	AUX
ejpam-2240	36	29	called	call	VERB
ejpam-2240	36	30	the	the	DET
ejpam-2240	36	31	local	local	ADJ
ejpam-2240	36	32	function	function	NOUN
ejpam-2240	36	33	of	of	ADP
ejpam-2240	36	34	a	a	PRON
ejpam-2240	36	35	with	with	ADP
ejpam-2240	36	36	respect	respect	NOUN
ejpam-2240	36	37	to	to	ADP
ejpam-2240	36	38	i	i	PRON
ejpam-2240	36	39	and	and	CCONJ
ejpam-2240	36	40	τ	τ	X
ejpam-2240	37	1	[	[	X
ejpam-2240	37	2	11	11	NUM
ejpam-2240	37	3	]	]	PUNCT
ejpam-2240	37	4	.	.	PUNCT
ejpam-2240	38	1	we	we	PRON
ejpam-2240	38	2	simply	simply	ADV
ejpam-2240	38	3	write	write	VERB
ejpam-2240	38	4	a∗	a∗	PROPN
ejpam-2240	38	5	instead	instead	ADV
ejpam-2240	38	6	of	of	ADP
ejpam-2240	38	7	a∗(i	a∗(i	PROPN
ejpam-2240	38	8	)	)	PUNCT
ejpam-2240	38	9	in	in	ADP
ejpam-2240	38	10	case	case	NOUN
ejpam-2240	38	11	there	there	PRON
ejpam-2240	38	12	is	be	VERB
ejpam-2240	38	13	no	no	DET
ejpam-2240	38	14	chance	chance	NOUN
ejpam-2240	38	15	for	for	ADP
ejpam-2240	38	16	confusion	confusion	NOUN
ejpam-2240	38	17	.	.	PUNCT
ejpam-2240	39	1	definition	definition	NOUN
ejpam-2240	39	2	4	4	NUM
ejpam-2240	39	3	.	.	PUNCT
ejpam-2240	40	1	a	a	DET
ejpam-2240	40	2	subset	subset	NOUN
ejpam-2240	40	3	a	a	PRON
ejpam-2240	40	4	of	of	ADP
ejpam-2240	40	5	an	an	DET
ejpam-2240	40	6	ideal	ideal	ADJ
ejpam-2240	40	7	topological	topological	ADJ
ejpam-2240	40	8	space	space	NOUN
ejpam-2240	40	9	(	(	PUNCT
ejpam-2240	40	10	x	x	X
ejpam-2240	40	11	,	,	PUNCT
ejpam-2240	40	12	τ	τ	PROPN
ejpam-2240	40	13	,	,	PUNCT
ejpam-2240	40	14	i	i	PROPN
ejpam-2240	40	15	)	)	PUNCT
ejpam-2240	40	16	is	be	AUX
ejpam-2240	40	17	said	say	VERB
ejpam-2240	40	18	to	to	PART
ejpam-2240	40	19	be	be	AUX
ejpam-2240	40	20	β	β	VERB
ejpam-2240	40	21	−	−	PROPN
ejpam-2240	41	1	i	i	PRON
ejpam-2240	41	2	-open	-open	VERB
ejpam-2240	42	1	[	[	X
ejpam-2240	42	2	4	4	NUM
ejpam-2240	42	3	]	]	X
ejpam-2240	42	4	if	if	SCONJ
ejpam-2240	42	5	a⊆	a⊆	NOUN
ejpam-2240	42	6	cl(int(cl∗(a	cl(int(cl∗(a	NOUN
ejpam-2240	42	7	)	)	PUNCT
ejpam-2240	42	8	)	)	PUNCT
ejpam-2240	42	9	)	)	PUNCT
ejpam-2240	42	10	.	.	PUNCT
ejpam-2240	43	1	3	3	X
ejpam-2240	43	2	.	.	X
ejpam-2240	43	3	β	β	X
ejpam-2240	43	4	-	-	NOUN
ejpam-2240	43	5	openness	openness	NOUN
ejpam-2240	43	6	with	with	ADP
ejpam-2240	43	7	respect	respect	NOUN
ejpam-2240	43	8	to	to	ADP
ejpam-2240	43	9	an	an	DET
ejpam-2240	43	10	ideal	ideal	NOUN
ejpam-2240	43	11	let	let	VERB
ejpam-2240	43	12	x	x	PRON
ejpam-2240	43	13	be	be	AUX
ejpam-2240	43	14	a	a	DET
ejpam-2240	43	15	topological	topological	ADJ
ejpam-2240	43	16	space	space	NOUN
ejpam-2240	43	17	.	.	PUNCT
ejpam-2240	44	1	recall	recall	VERB
ejpam-2240	44	2	that	that	SCONJ
ejpam-2240	44	3	a	a	DET
ejpam-2240	44	4	subset	subset	NOUN
ejpam-2240	44	5	a	a	PRON
ejpam-2240	44	6	of	of	ADP
ejpam-2240	44	7	x	x	SYM
ejpam-2240	44	8	is	be	AUX
ejpam-2240	44	9	said	say	VERB
ejpam-2240	44	10	to	to	PART
ejpam-2240	44	11	be	be	AUX
ejpam-2240	44	12	β	β	X
ejpam-2240	44	13	-open	-open	NOUN
ejpam-2240	45	1	[	[	X
ejpam-2240	45	2	1	1	NUM
ejpam-2240	45	3	]	]	PUNCT
ejpam-2240	45	4	if	if	SCONJ
ejpam-2240	45	5	there	there	PRON
ejpam-2240	45	6	is	be	VERB
ejpam-2240	45	7	a	a	DET
ejpam-2240	45	8	preopen	preopen	ADJ
ejpam-2240	45	9	set	set	VERB
ejpam-2240	45	10	g	g	PROPN
ejpam-2240	45	11	such	such	ADJ
ejpam-2240	45	12	that	that	SCONJ
ejpam-2240	45	13	g	g	PROPN
ejpam-2240	45	14	⊆	⊆	NUM
ejpam-2240	45	15	a⊆	a⊆	NOUN
ejpam-2240	45	16	cl(g	cl(g	NOUN
ejpam-2240	45	17	)	)	PUNCT
ejpam-2240	45	18	.	.	PUNCT
ejpam-2240	46	1	this	this	PRON
ejpam-2240	46	2	motivates	motivate	VERB
ejpam-2240	46	3	our	our	PRON
ejpam-2240	46	4	first	first	ADJ
ejpam-2240	46	5	definition	definition	NOUN
ejpam-2240	46	6	.	.	PUNCT
ejpam-2240	47	1	definition	definition	NOUN
ejpam-2240	47	2	5	5	NUM
ejpam-2240	47	3	.	.	PUNCT
ejpam-2240	48	1	a	a	DET
ejpam-2240	48	2	subset	subset	NOUN
ejpam-2240	48	3	a	a	PRON
ejpam-2240	48	4	of	of	ADP
ejpam-2240	48	5	an	an	DET
ejpam-2240	48	6	ideal	ideal	ADJ
ejpam-2240	48	7	topological	topological	ADJ
ejpam-2240	48	8	space	space	NOUN
ejpam-2240	48	9	(	(	PUNCT
ejpam-2240	48	10	x	x	X
ejpam-2240	48	11	,	,	PUNCT
ejpam-2240	48	12	τ	τ	X
ejpam-2240	48	13	)	)	PUNCT
ejpam-2240	48	14	is	be	AUX
ejpam-2240	48	15	said	say	VERB
ejpam-2240	48	16	to	to	PART
ejpam-2240	48	17	be	be	AUX
ejpam-2240	48	18	β	β	X
ejpam-2240	48	19	-open	-open	NOUN
ejpam-2240	48	20	with	with	ADP
ejpam-2240	48	21	respect	respect	NOUN
ejpam-2240	48	22	to	to	ADP
ejpam-2240	48	23	an	an	DET
ejpam-2240	48	24	ideal	ideal	NOUN
ejpam-2240	48	25	i	i	PRON
ejpam-2240	48	26	(	(	PUNCT
ejpam-2240	48	27	written	write	VERB
ejpam-2240	48	28	as	as	ADP
ejpam-2240	48	29	i−β	i−β	VERB
ejpam-2240	48	30	-open	-open	NOUN
ejpam-2240	48	31	)	)	PUNCT
ejpam-2240	48	32	if	if	SCONJ
ejpam-2240	48	33	there	there	PRON
ejpam-2240	48	34	exists	exist	VERB
ejpam-2240	48	35	a	a	DET
ejpam-2240	48	36	preopen	preopen	NOUN
ejpam-2240	48	37	set	set	VERB
ejpam-2240	48	38	g	g	PROPN
ejpam-2240	48	39	such	such	DET
ejpam-2240	48	40	that	that	SCONJ
ejpam-2240	48	41	g	g	PROPN
ejpam-2240	49	1	\a∈	\a∈	INTJ
ejpam-2240	50	1	i	i	PRON
ejpam-2240	50	2	and	and	CCONJ
ejpam-2240	50	3	a\	a\	ADJ
ejpam-2240	50	4	cl(g	cl(g	X
ejpam-2240	50	5	)	)	PUNCT
ejpam-2240	50	6	∈	∈	PROPN
ejpam-2240	51	1	i	i	PRON
ejpam-2240	51	2	.	.	PUNCT
ejpam-2240	52	1	if	if	SCONJ
ejpam-2240	52	2	a	a	DET
ejpam-2240	52	3	∈	∈	X
ejpam-2240	52	4	i	i	PRON
ejpam-2240	52	5	,	,	PUNCT
ejpam-2240	52	6	then	then	ADV
ejpam-2240	52	7	it	it	PRON
ejpam-2240	52	8	is	be	AUX
ejpam-2240	52	9	easy	easy	ADJ
ejpam-2240	52	10	to	to	PART
ejpam-2240	52	11	see	see	VERB
ejpam-2240	52	12	that	that	SCONJ
ejpam-2240	52	13	a	a	PRON
ejpam-2240	52	14	is	be	AUX
ejpam-2240	52	15	i	i	PRON
ejpam-2240	52	16	−	−	PROPN
ejpam-2240	52	17	β	β	X
ejpam-2240	52	18	-open	-open	NOUN
ejpam-2240	52	19	.	.	PUNCT
ejpam-2240	53	1	moreover	moreover	ADV
ejpam-2240	53	2	,	,	PUNCT
ejpam-2240	53	3	every	every	DET
ejpam-2240	53	4	open	open	NOUN
ejpam-2240	53	5	set	set	VERB
ejpam-2240	53	6	a	a	PRON
ejpam-2240	53	7	is	be	AUX
ejpam-2240	53	8	β	β	NOUN
ejpam-2240	53	9	-open	-open	NOUN
ejpam-2240	53	10	,	,	PUNCT
ejpam-2240	53	11	and	and	CCONJ
ejpam-2240	53	12	every	every	DET
ejpam-2240	53	13	β	β	X
ejpam-2240	53	14	-open	-open	NOUN
ejpam-2240	53	15	set	set	NOUN
ejpam-2240	53	16	b	b	NOUN
ejpam-2240	53	17	is	be	AUX
ejpam-2240	53	18	i	i	PRON
ejpam-2240	53	19	−	−	PROPN
ejpam-2240	53	20	β	β	X
ejpam-2240	53	21	-open	-open	PROPN
ejpam-2240	53	22	,	,	PUNCT
ejpam-2240	53	23	for	for	ADP
ejpam-2240	53	24	any	any	DET
ejpam-2240	53	25	ideal	ideal	NOUN
ejpam-2240	53	26	on	on	ADP
ejpam-2240	53	27	x	x	PROPN
ejpam-2240	53	28	.	.	PUNCT
ejpam-2240	53	29	example	example	NOUN
ejpam-2240	54	1	1	1	NUM
ejpam-2240	54	2	.	.	X
ejpam-2240	54	3	consider	consider	VERB
ejpam-2240	54	4	a	a	DET
ejpam-2240	54	5	topological	topological	ADJ
ejpam-2240	54	6	space	space	NOUN
ejpam-2240	54	7	(	(	PUNCT
ejpam-2240	54	8	x	x	X
ejpam-2240	54	9	,	,	PUNCT
ejpam-2240	54	10	τ	τ	PROPN
ejpam-2240	54	11	)	)	PUNCT
ejpam-2240	54	12	;	;	PUNCT
ejpam-2240	54	13	x	x	SYM
ejpam-2240	54	14	=	=	X
ejpam-2240	54	15	{	{	PUNCT
ejpam-2240	54	16	a	a	PRON
ejpam-2240	54	17	,	,	PUNCT
ejpam-2240	54	18	b	b	NOUN
ejpam-2240	54	19	,	,	PUNCT
ejpam-2240	54	20	c	c	NOUN
ejpam-2240	54	21	}	}	PUNCT
ejpam-2240	54	22	and	and	CCONJ
ejpam-2240	54	23	τ=	τ=	X
ejpam-2240	54	24	{	{	PUNCT
ejpam-2240	54	25	;	;	PUNCT
ejpam-2240	54	26	,	,	PUNCT
ejpam-2240	54	27	{	{	PUNCT
ejpam-2240	54	28	a	a	X
ejpam-2240	54	29	}	}	PUNCT
ejpam-2240	54	30	,	,	PUNCT
ejpam-2240	54	31	{	{	PUNCT
ejpam-2240	54	32	a	a	X
ejpam-2240	54	33	,	,	PUNCT
ejpam-2240	54	34	c	c	NOUN
ejpam-2240	54	35	}	}	PUNCT
ejpam-2240	54	36	,	,	PUNCT
ejpam-2240	54	37	x	x	SYM
ejpam-2240	54	38	}	}	PUNCT
ejpam-2240	54	39	.	.	PUNCT
ejpam-2240	55	1	choose	choose	VERB
ejpam-2240	55	2	i	i	PRON
ejpam-2240	55	3	=	=	PUNCT
ejpam-2240	55	4	{	{	PUNCT
ejpam-2240	55	5	;	;	PUNCT
ejpam-2240	55	6	,	,	PUNCT
ejpam-2240	55	7	{	{	PUNCT
ejpam-2240	55	8	b	b	NOUN
ejpam-2240	55	9	}	}	PUNCT
ejpam-2240	55	10	,	,	PUNCT
ejpam-2240	55	11	{	{	PUNCT
ejpam-2240	55	12	c	c	X
ejpam-2240	55	13	}	}	PUNCT
ejpam-2240	55	14	,	,	PUNCT
ejpam-2240	55	15	{	{	PUNCT
ejpam-2240	55	16	b	b	X
ejpam-2240	55	17	,	,	PUNCT
ejpam-2240	55	18	c	c	NOUN
ejpam-2240	55	19	}	}	PUNCT
ejpam-2240	55	20	}	}	PUNCT
ejpam-2240	55	21	,	,	PUNCT
ejpam-2240	55	22	and	and	CCONJ
ejpam-2240	55	23	observe	observe	VERB
ejpam-2240	55	24	that	that	SCONJ
ejpam-2240	55	25	{	{	PUNCT
ejpam-2240	55	26	b	b	X
ejpam-2240	55	27	}	}	PUNCT
ejpam-2240	55	28	is	be	AUX
ejpam-2240	55	29	i−β	i−β	VERB
ejpam-2240	55	30	-open	-open	NOUN
ejpam-2240	55	31	,	,	PUNCT
ejpam-2240	55	32	however	however	ADV
ejpam-2240	55	33	{	{	PUNCT
ejpam-2240	55	34	b	b	NOUN
ejpam-2240	55	35	}	}	PUNCT
ejpam-2240	55	36	is	be	AUX
ejpam-2240	55	37	not	not	PART
ejpam-2240	55	38	β	β	PART
ejpam-2240	55	39	-open	-open	NOUN
ejpam-2240	55	40	in	in	ADP
ejpam-2240	55	41	the	the	DET
ejpam-2240	55	42	sense	sense	NOUN
ejpam-2240	55	43	of	of	ADP
ejpam-2240	55	44	[	[	X
ejpam-2240	55	45	1	1	NUM
ejpam-2240	55	46	]	]	PUNCT
ejpam-2240	55	47	as	as	SCONJ
ejpam-2240	55	48	there	there	PRON
ejpam-2240	55	49	is	be	VERB
ejpam-2240	55	50	no	no	DET
ejpam-2240	55	51	preopen	preopen	NOUN
ejpam-2240	55	52	set	set	VERB
ejpam-2240	55	53	g	g	PROPN
ejpam-2240	56	1	such	such	ADJ
ejpam-2240	56	2	that	that	SCONJ
ejpam-2240	56	3	g	g	PROPN
ejpam-2240	56	4	⊆	⊆	NUM
ejpam-2240	56	5	{	{	PUNCT
ejpam-2240	56	6	b	b	NOUN
ejpam-2240	56	7	}	}	PUNCT
ejpam-2240	56	8	⊆	⊆	NUM
ejpam-2240	56	9	cl(g	cl(g	NOUN
ejpam-2240	56	10	)	)	PUNCT
ejpam-2240	57	1	.	.	PUNCT
ejpam-2240	58	1	thus	thus	ADV
ejpam-2240	58	2	,	,	PUNCT
ejpam-2240	58	3	if	if	SCONJ
ejpam-2240	58	4	a	a	DET
ejpam-2240	58	5	set	set	NOUN
ejpam-2240	58	6	is	be	AUX
ejpam-2240	58	7	i	i	PRON
ejpam-2240	58	8	−β	−β	PROPN
ejpam-2240	58	9	-open	-open	PROPN
ejpam-2240	58	10	,	,	PUNCT
ejpam-2240	58	11	it	it	PRON
ejpam-2240	58	12	may	may	AUX
ejpam-2240	58	13	not	not	PART
ejpam-2240	58	14	be	be	AUX
ejpam-2240	58	15	β	β	NOUN
ejpam-2240	58	16	-open	-open	NOUN
ejpam-2240	58	17	in	in	ADP
ejpam-2240	58	18	the	the	DET
ejpam-2240	58	19	usual	usual	ADJ
ejpam-2240	58	20	sense	sense	NOUN
ejpam-2240	58	21	.	.	PUNCT
ejpam-2240	59	1	for	for	ADP
ejpam-2240	59	2	an	an	DET
ejpam-2240	59	3	ideal	ideal	NOUN
ejpam-2240	59	4	i	i	PRON
ejpam-2240	59	5	that	that	PRON
ejpam-2240	59	6	is	be	AUX
ejpam-2240	59	7	not	not	PART
ejpam-2240	59	8	countably	countably	ADV
ejpam-2240	59	9	additive	additive	ADJ
ejpam-2240	59	10	,	,	PUNCT
ejpam-2240	59	11	the	the	DET
ejpam-2240	59	12	concepts	concept	NOUN
ejpam-2240	59	13	of	of	ADP
ejpam-2240	59	14	β	β	X
ejpam-2240	59	15	-openness	-openness	PROPN
ejpam-2240	59	16	and	and	CCONJ
ejpam-2240	59	17	i−β	i−β	VERB
ejpam-2240	59	18	-openness	-openness	ADJ
ejpam-2240	59	19	coincide	coincide	NOUN
ejpam-2240	59	20	in	in	ADP
ejpam-2240	59	21	the	the	DET
ejpam-2240	59	22	following	follow	VERB
ejpam-2240	59	23	case	case	NOUN
ejpam-2240	59	24	.	.	PUNCT
ejpam-2240	60	1	a.	a.	PROPN
ejpam-2240	60	2	nasef	nasef	PROPN
ejpam-2240	60	3	,	,	PUNCT
ejpam-2240	60	4	r.	r.	PROPN
ejpam-2240	60	5	mareay	mareay	PROPN
ejpam-2240	60	6	,	,	PUNCT
ejpam-2240	60	7	f.	f.	PROPN
ejpam-2240	60	8	michael	michael	PROPN
ejpam-2240	60	9	/	/	SYM
ejpam-2240	60	10	eur	eur	PROPN
ejpam-2240	60	11	.	.	PUNCT
ejpam-2240	61	1	j.	j.	PROPN
ejpam-2240	61	2	pure	pure	PROPN
ejpam-2240	61	3	appl	appl	PROPN
ejpam-2240	61	4	.	.	PROPN
ejpam-2240	61	5	math	math	PROPN
ejpam-2240	61	6	,	,	PUNCT
ejpam-2240	61	7	8	8	NUM
ejpam-2240	61	8	(	(	PUNCT
ejpam-2240	61	9	2015	2015	NUM
ejpam-2240	61	10	)	)	PUNCT
ejpam-2240	61	11	,	,	PUNCT
ejpam-2240	61	12	389	389	NUM
ejpam-2240	61	13	-	-	SYM
ejpam-2240	61	14	394	394	NUM
ejpam-2240	61	15	391	391	NUM
ejpam-2240	61	16	theorem	theorem	NOUN
ejpam-2240	61	17	1	1	NUM
ejpam-2240	61	18	.	.	X
ejpam-2240	62	1	for	for	ADP
ejpam-2240	62	2	an	an	DET
ejpam-2240	62	3	ideal	ideal	NOUN
ejpam-2240	62	4	i	i	PRON
ejpam-2240	62	5	on	on	ADP
ejpam-2240	62	6	a	a	DET
ejpam-2240	62	7	topological	topological	ADJ
ejpam-2240	62	8	space	space	NOUN
ejpam-2240	62	9	(	(	PUNCT
ejpam-2240	62	10	x	x	X
ejpam-2240	62	11	,	,	PUNCT
ejpam-2240	62	12	τ	τ	PROPN
ejpam-2240	62	13	)	)	PUNCT
ejpam-2240	62	14	,	,	PUNCT
ejpam-2240	62	15	the	the	DET
ejpam-2240	62	16	following	follow	VERB
ejpam-2240	62	17	are	be	AUX
ejpam-2240	62	18	equivalent	equivalent	ADJ
ejpam-2240	62	19	:	:	PUNCT
ejpam-2240	62	20	(	(	PUNCT
ejpam-2240	62	21	i	i	NOUN
ejpam-2240	62	22	)	)	PUNCT
ejpam-2240	63	1	i	i	PRON
ejpam-2240	63	2	is	be	AUX
ejpam-2240	63	3	the	the	DET
ejpam-2240	63	4	minimal	minimal	ADJ
ejpam-2240	63	5	ideal	ideal	NOUN
ejpam-2240	63	6	on	on	ADP
ejpam-2240	63	7	x	x	SYM
ejpam-2240	63	8	,	,	PUNCT
ejpam-2240	63	9	that	that	PRON
ejpam-2240	63	10	is	is	ADV
ejpam-2240	63	11	i	i	PRON
ejpam-2240	63	12	=	=	PUNCT
ejpam-2240	63	13	{	{	PUNCT
ejpam-2240	63	14	;	;	PUNCT
ejpam-2240	63	15	}	}	PUNCT
ejpam-2240	63	16	,	,	PUNCT
ejpam-2240	63	17	(	(	PUNCT
ejpam-2240	63	18	ii	ii	NOUN
ejpam-2240	63	19	)	)	PUNCT
ejpam-2240	63	20	the	the	DET
ejpam-2240	63	21	concepts	concept	NOUN
ejpam-2240	63	22	of	of	ADP
ejpam-2240	63	23	β	β	X
ejpam-2240	63	24	-openness	-openness	PROPN
ejpam-2240	63	25	and	and	CCONJ
ejpam-2240	63	26	i	i	PRON
ejpam-2240	63	27	−	−	PROPN
ejpam-2240	63	28	β	β	NOUN
ejpam-2240	63	29	-openness	-openness	NOUN
ejpam-2240	63	30	are	be	AUX
ejpam-2240	63	31	the	the	DET
ejpam-2240	63	32	same	same	ADJ
ejpam-2240	63	33	.	.	PUNCT
ejpam-2240	64	1	proof	proof	NOUN
ejpam-2240	64	2	.	.	PUNCT
ejpam-2240	65	1	first	first	ADV
ejpam-2240	65	2	suppose	suppose	VERB
ejpam-2240	65	3	that	that	SCONJ
ejpam-2240	65	4	i	i	PRON
ejpam-2240	65	5	=	=	PRON
ejpam-2240	65	6	{	{	PUNCT
ejpam-2240	65	7	;	;	PUNCT
ejpam-2240	65	8	}	}	PUNCT
ejpam-2240	65	9	.	.	PUNCT
ejpam-2240	66	1	it	it	PRON
ejpam-2240	66	2	suffices	suffice	VERB
ejpam-2240	66	3	to	to	PART
ejpam-2240	66	4	show	show	VERB
ejpam-2240	66	5	that	that	SCONJ
ejpam-2240	66	6	whenever	whenever	SCONJ
ejpam-2240	66	7	a	a	DET
ejpam-2240	66	8	set	set	NOUN
ejpam-2240	66	9	a	a	PRON
ejpam-2240	66	10	is	be	AUX
ejpam-2240	66	11	i	i	PRON
ejpam-2240	66	12	−	−	PROPN
ejpam-2240	66	13	β	β	X
ejpam-2240	66	14	-open	-open	PROPN
ejpam-2240	66	15	,	,	PUNCT
ejpam-2240	66	16	then	then	ADV
ejpam-2240	66	17	it	it	PRON
ejpam-2240	66	18	is	be	AUX
ejpam-2240	66	19	β	β	PART
ejpam-2240	66	20	-open	-open	NOUN
ejpam-2240	66	21	in	in	ADP
ejpam-2240	66	22	the	the	DET
ejpam-2240	66	23	usual	usual	ADJ
ejpam-2240	66	24	sense	sense	NOUN
ejpam-2240	66	25	.	.	PUNCT
ejpam-2240	67	1	indeed	indeed	ADV
ejpam-2240	67	2	,	,	PUNCT
ejpam-2240	67	3	if	if	SCONJ
ejpam-2240	67	4	a	a	PRON
ejpam-2240	67	5	is	be	AUX
ejpam-2240	67	6	i	i	PRON
ejpam-2240	67	7	−	−	NOUN
ejpam-2240	67	8	β	β	X
ejpam-2240	67	9	-open	-open	PROPN
ejpam-2240	67	10	,	,	PUNCT
ejpam-2240	67	11	then	then	ADV
ejpam-2240	67	12	there	there	PRON
ejpam-2240	67	13	is	be	VERB
ejpam-2240	67	14	a	a	DET
ejpam-2240	67	15	preopen	preopen	ADJ
ejpam-2240	67	16	set	set	VERB
ejpam-2240	67	17	g	g	PROPN
ejpam-2240	68	1	such	such	ADJ
ejpam-2240	68	2	that	that	SCONJ
ejpam-2240	68	3	g	g	PROPN
ejpam-2240	68	4	\	\	PROPN
ejpam-2240	68	5	a	a	PRON
ejpam-2240	68	6	,	,	PUNCT
ejpam-2240	68	7	and	and	CCONJ
ejpam-2240	68	8	a\	a\	NOUN
ejpam-2240	68	9	cl(g	cl(g	X
ejpam-2240	68	10	)	)	PUNCT
ejpam-2240	68	11	∈	∈	PROPN
ejpam-2240	69	1	i	i	PRON
ejpam-2240	69	2	=	=	PUNCT
ejpam-2240	69	3	{	{	PUNCT
ejpam-2240	69	4	;	;	PUNCT
ejpam-2240	69	5	}	}	PUNCT
ejpam-2240	69	6	,	,	PUNCT
ejpam-2240	69	7	and	and	CCONJ
ejpam-2240	69	8	so	so	ADV
ejpam-2240	69	9	g	g	PROPN
ejpam-2240	69	10	⊆	⊆	NUM
ejpam-2240	69	11	a⊆	a⊆	NOUN
ejpam-2240	69	12	cl(g	cl(g	NOUN
ejpam-2240	69	13	)	)	PUNCT
ejpam-2240	69	14	,	,	PUNCT
ejpam-2240	69	15	proving	prove	VERB
ejpam-2240	69	16	that	that	SCONJ
ejpam-2240	69	17	a	a	PRON
ejpam-2240	69	18	is	be	AUX
ejpam-2240	69	19	β	β	NOUN
ejpam-2240	69	20	-open	-open	NOUN
ejpam-2240	69	21	.	.	PUNCT
ejpam-2240	70	1	conversely	conversely	ADV
ejpam-2240	70	2	,	,	PUNCT
ejpam-2240	70	3	suppose	suppose	VERB
ejpam-2240	70	4	that	that	SCONJ
ejpam-2240	70	5	whenever	whenever	SCONJ
ejpam-2240	70	6	a	a	DET
ejpam-2240	70	7	set	set	NOUN
ejpam-2240	70	8	a	a	PRON
ejpam-2240	70	9	is	be	AUX
ejpam-2240	70	10	i	i	PRON
ejpam-2240	70	11	−β	−β	PROPN
ejpam-2240	70	12	-open	-open	PROPN
ejpam-2240	70	13	,	,	PUNCT
ejpam-2240	70	14	then	then	ADV
ejpam-2240	70	15	it	it	PRON
ejpam-2240	70	16	is	be	AUX
ejpam-2240	70	17	β	β	NOUN
ejpam-2240	70	18	-open	-open	PROPN
ejpam-2240	70	19	.	.	PUNCT
ejpam-2240	71	1	let	let	VERB
ejpam-2240	71	2	b	b	X
ejpam-2240	71	3	∈	∈	PROPN
ejpam-2240	72	1	i	i	PRON
ejpam-2240	72	2	.	.	PUNCT
ejpam-2240	73	1	then	then	ADV
ejpam-2240	73	2	,	,	PUNCT
ejpam-2240	73	3	b	b	PROPN
ejpam-2240	73	4	is	be	AUX
ejpam-2240	73	5	i	i	PRON
ejpam-2240	73	6	−	−	PROPN
ejpam-2240	73	7	β	β	X
ejpam-2240	73	8	-open	-open	PROPN
ejpam-2240	73	9	,	,	PUNCT
ejpam-2240	73	10	and	and	CCONJ
ejpam-2240	73	11	by	by	ADP
ejpam-2240	73	12	assumption	assumption	NOUN
ejpam-2240	73	13	,	,	PUNCT
ejpam-2240	73	14	b	b	PROPN
ejpam-2240	73	15	is	be	AUX
ejpam-2240	73	16	β	β	X
ejpam-2240	73	17	-open	-open	NOUN
ejpam-2240	73	18	.	.	PUNCT
ejpam-2240	74	1	thus	thus	ADV
ejpam-2240	74	2	,	,	PUNCT
ejpam-2240	74	3	there	there	PRON
ejpam-2240	74	4	is	be	VERB
ejpam-2240	74	5	a	a	DET
ejpam-2240	74	6	preopen	preopen	ADJ
ejpam-2240	74	7	set	set	NOUN
ejpam-2240	74	8	h1	h1	NOUN
ejpam-2240	74	9	,	,	PUNCT
ejpam-2240	74	10	such	such	ADJ
ejpam-2240	74	11	that	that	PRON
ejpam-2240	74	12	h1	h1	VERB
ejpam-2240	74	13	⊆	⊆	NUM
ejpam-2240	74	14	b	b	NOUN
ejpam-2240	74	15	⊆	⊆	NUM
ejpam-2240	74	16	cl(h1	cl(h1	NOUN
ejpam-2240	74	17	)	)	PUNCT
ejpam-2240	74	18	.	.	PUNCT
ejpam-2240	75	1	since	since	SCONJ
ejpam-2240	75	2	b	b	PROPN
ejpam-2240	75	3	∈	∈	PROPN
ejpam-2240	75	4	i	i	PRON
ejpam-2240	75	5	and	and	CCONJ
ejpam-2240	75	6	h1	h1	VERB
ejpam-2240	75	7	⊆	⊆	NUM
ejpam-2240	75	8	b	b	NOUN
ejpam-2240	75	9	,	,	PUNCT
ejpam-2240	75	10	we	we	PRON
ejpam-2240	75	11	have	have	VERB
ejpam-2240	75	12	that	that	DET
ejpam-2240	75	13	h1	h1	PROPN
ejpam-2240	75	14	∈	∈	PROPN
ejpam-2240	75	15	i	i	PRON
ejpam-2240	75	16	,	,	PUNCT
ejpam-2240	75	17	and	and	CCONJ
ejpam-2240	75	18	so	so	ADV
ejpam-2240	75	19	b∪h1	b∪h1	NOUN
ejpam-2240	75	20	∈	∈	NOUN
ejpam-2240	75	21	i	i	PRON
ejpam-2240	75	22	.	.	PUNCT
ejpam-2240	76	1	as	as	SCONJ
ejpam-2240	76	2	b∪h1	b∪h1	NOUN
ejpam-2240	76	3	is	be	AUX
ejpam-2240	76	4	i	i	PRON
ejpam-2240	76	5	−	−	NOUN
ejpam-2240	76	6	β	β	X
ejpam-2240	76	7	-open	-open	PROPN
ejpam-2240	76	8	,	,	PUNCT
ejpam-2240	76	9	it	it	PRON
ejpam-2240	76	10	is	be	AUX
ejpam-2240	76	11	β	β	NOUN
ejpam-2240	76	12	-open	-open	PROPN
ejpam-2240	76	13	,	,	PUNCT
ejpam-2240	76	14	so	so	SCONJ
ejpam-2240	76	15	that	that	SCONJ
ejpam-2240	76	16	there	there	PRON
ejpam-2240	76	17	is	be	VERB
ejpam-2240	76	18	a	a	DET
ejpam-2240	76	19	preopen	preopen	ADJ
ejpam-2240	76	20	set	set	VERB
ejpam-2240	76	21	h2	h2	NOUN
ejpam-2240	76	22	for	for	ADP
ejpam-2240	76	23	which	which	PRON
ejpam-2240	76	24	h2	h2	NOUN
ejpam-2240	76	25	⊆	⊆	NUM
ejpam-2240	76	26	(	(	PUNCT
ejpam-2240	76	27	b	b	NOUN
ejpam-2240	76	28	∪	∪	X
ejpam-2240	76	29	h1	h1	PROPN
ejpam-2240	76	30	)	)	PUNCT
ejpam-2240	76	31	⊆	⊆	NUM
ejpam-2240	76	32	cl(h2	cl(h2	NOUN
ejpam-2240	76	33	)	)	PUNCT
ejpam-2240	76	34	.	.	PUNCT
ejpam-2240	77	1	similarly	similarly	ADV
ejpam-2240	77	2	,	,	PUNCT
ejpam-2240	77	3	there	there	PRON
ejpam-2240	77	4	is	be	VERB
ejpam-2240	77	5	a	a	DET
ejpam-2240	77	6	preopen	preopen	ADJ
ejpam-2240	77	7	set	set	NOUN
ejpam-2240	77	8	h3	h3	NOUN
ejpam-2240	77	9	such	such	ADJ
ejpam-2240	77	10	that	that	SCONJ
ejpam-2240	77	11	h3	h3	NOUN
ejpam-2240	77	12	⊂	⊂	X
ejpam-2240	77	13	(	(	PUNCT
ejpam-2240	77	14	b	b	X
ejpam-2240	77	15	∪h1	∪h1	NOUN
ejpam-2240	77	16	∪h2	∪h2	PROPN
ejpam-2240	77	17	)	)	PUNCT
ejpam-2240	77	18	⊂	⊂	PROPN
ejpam-2240	77	19	cl(h3	cl(h3	NOUN
ejpam-2240	77	20	)	)	PUNCT
ejpam-2240	77	21	.	.	PUNCT
ejpam-2240	78	1	continuing	continue	VERB
ejpam-2240	78	2	in	in	ADP
ejpam-2240	78	3	this	this	DET
ejpam-2240	78	4	way	way	NOUN
ejpam-2240	78	5	,	,	PUNCT
ejpam-2240	78	6	we	we	PRON
ejpam-2240	78	7	have	have	VERB
ejpam-2240	78	8	an	an	DET
ejpam-2240	78	9	infinite	infinite	ADJ
ejpam-2240	78	10	collection	collection	NOUN
ejpam-2240	78	11	of	of	ADP
ejpam-2240	78	12	preopen	preopen	ADJ
ejpam-2240	78	13	sets	set	NOUN
ejpam-2240	78	14	h1	h1	PROPN
ejpam-2240	78	15	,	,	PUNCT
ejpam-2240	78	16	h2	h2	PROPN
ejpam-2240	78	17	,	,	PUNCT
ejpam-2240	78	18	h3	h3	NOUN
ejpam-2240	78	19	,	,	PUNCT
ejpam-2240	78	20	.	.	PUNCT
ejpam-2240	78	21	.	.	PUNCT
ejpam-2240	79	1	.	.	PUNCT
ejpam-2240	80	1	,	,	PUNCT
ejpam-2240	80	2	such	such	ADJ
ejpam-2240	80	3	that	that	SCONJ
ejpam-2240	80	4	b	b	NOUN
ejpam-2240	80	5	∪	∪	X
ejpam-2240	80	6	h1	h1	PROPN
ejpam-2240	80	7	∪	∪	PROPN
ejpam-2240	80	8	h2	h2	NOUN
ejpam-2240	80	9	∪	∪	ADP
ejpam-2240	80	10	h3	h3	NOUN
ejpam-2240	80	11	∪	∪	ADJ
ejpam-2240	80	12	.	.	PUNCT
ejpam-2240	80	13	.	.	PUNCT
ejpam-2240	80	14	.	.	PUNCT
ejpam-2240	81	1	∈	∈	PROPN
ejpam-2240	82	1	i	i	PRON
ejpam-2240	82	2	,	,	PUNCT
ejpam-2240	82	3	which	which	PRON
ejpam-2240	82	4	is	be	AUX
ejpam-2240	82	5	impossible	impossible	ADJ
ejpam-2240	82	6	,	,	PUNCT
ejpam-2240	82	7	as	as	ADP
ejpam-2240	82	8	the	the	DET
ejpam-2240	82	9	ideal	ideal	NOUN
ejpam-2240	82	10	i	i	PRON
ejpam-2240	82	11	is	be	AUX
ejpam-2240	82	12	not	not	PART
ejpam-2240	82	13	closed	close	VERB
ejpam-2240	82	14	under	under	ADP
ejpam-2240	82	15	countable	countable	ADJ
ejpam-2240	82	16	additivity	additivity	NOUN
ejpam-2240	82	17	.	.	PUNCT
ejpam-2240	83	1	thus	thus	ADV
ejpam-2240	83	2	,	,	PUNCT
ejpam-2240	83	3	it	it	PRON
ejpam-2240	83	4	must	must	AUX
ejpam-2240	83	5	be	be	AUX
ejpam-2240	83	6	the	the	DET
ejpam-2240	83	7	case	case	NOUN
ejpam-2240	83	8	that	that	PRON
ejpam-2240	83	9	h1	h1	VERB
ejpam-2240	83	10	=	=	PRON
ejpam-2240	83	11	;	;	PUNCT
ejpam-2240	83	12	(	(	PUNCT
ejpam-2240	83	13	similarly	similarly	ADV
ejpam-2240	83	14	for	for	ADP
ejpam-2240	83	15	the	the	DET
ejpam-2240	83	16	other	other	ADJ
ejpam-2240	83	17	h	h	NOUN
ejpam-2240	83	18	′	′	NUM
ejpam-2240	84	1	i	i	PRON
ejpam-2240	84	2	s	s	VERB
ejpam-2240	84	3	,	,	PUNCT
ejpam-2240	84	4	i	i	NOUN
ejpam-2240	84	5	=	=	NOUN
ejpam-2240	84	6	1,2,3	1,2,3	NUM
ejpam-2240	84	7	,	,	PUNCT
ejpam-2240	84	8	.	.	PUNCT
ejpam-2240	84	9	.	.	PUNCT
ejpam-2240	84	10	.	.	PUNCT
ejpam-2240	84	11	)	)	PUNCT
ejpam-2240	85	1	,	,	PUNCT
ejpam-2240	85	2	therefore	therefore	ADV
ejpam-2240	85	3	,	,	PUNCT
ejpam-2240	85	4	cl(h1	cl(h1	NOUN
ejpam-2240	85	5	)	)	PUNCT
ejpam-2240	85	6	=	=	SYM
ejpam-2240	85	7	;	;	PUNCT
ejpam-2240	85	8	,	,	PUNCT
ejpam-2240	85	9	and	and	CCONJ
ejpam-2240	85	10	the	the	DET
ejpam-2240	85	11	relations	relation	NOUN
ejpam-2240	85	12	h1	h1	VERB
ejpam-2240	85	13	⊆	⊆	NUM
ejpam-2240	85	14	b	b	PROPN
ejpam-2240	85	15	⊆	⊆	NUM
ejpam-2240	85	16	cl(h1	cl(h1	NOUN
ejpam-2240	85	17	)	)	PUNCT
ejpam-2240	85	18	,	,	PUNCT
ejpam-2240	85	19	then	then	ADV
ejpam-2240	85	20	give	give	VERB
ejpam-2240	85	21	b	b	NOUN
ejpam-2240	85	22	=	=	PUNCT
ejpam-2240	85	23	;	;	PUNCT
ejpam-2240	85	24	,	,	PUNCT
ejpam-2240	85	25	proving	prove	VERB
ejpam-2240	85	26	that	that	SCONJ
ejpam-2240	85	27	i	i	PRON
ejpam-2240	85	28	=	=	PRON
ejpam-2240	85	29	{	{	PUNCT
ejpam-2240	85	30	;	;	PUNCT
ejpam-2240	85	31	}	}	PUNCT
ejpam-2240	85	32	.	.	PUNCT
ejpam-2240	86	1	proposition	proposition	NOUN
ejpam-2240	86	2	1	1	NUM
ejpam-2240	86	3	.	.	PUNCT
ejpam-2240	87	1	let	let	VERB
ejpam-2240	87	2	i	i	PRON
ejpam-2240	87	3	and	and	CCONJ
ejpam-2240	87	4	jbe	jbe	PRON
ejpam-2240	87	5	two	two	NUM
ejpam-2240	87	6	ideals	ideal	NOUN
ejpam-2240	87	7	on	on	ADP
ejpam-2240	87	8	a	a	DET
ejpam-2240	87	9	topological	topological	ADJ
ejpam-2240	87	10	space	space	NOUN
ejpam-2240	87	11	(	(	PUNCT
ejpam-2240	87	12	x	x	X
ejpam-2240	87	13	,	,	PUNCT
ejpam-2240	87	14	τ	τ	PROPN
ejpam-2240	87	15	)	)	PUNCT
ejpam-2240	87	16	,	,	PUNCT
ejpam-2240	87	17	(	(	PUNCT
ejpam-2240	87	18	i	i	NOUN
ejpam-2240	87	19	)	)	PUNCT
ejpam-2240	87	20	if	if	SCONJ
ejpam-2240	87	21	i	i	PRON
ejpam-2240	87	22	⊆	⊆	NUM
ejpam-2240	87	23	j	j	PROPN
ejpam-2240	87	24	,	,	PUNCT
ejpam-2240	87	25	then	then	ADV
ejpam-2240	87	26	every	every	DET
ejpam-2240	87	27	i	i	PRON
ejpam-2240	87	28	−	−	VERB
ejpam-2240	87	29	β	β	X
ejpam-2240	87	30	-open	-open	NOUN
ejpam-2240	87	31	set	set	VERB
ejpam-2240	87	32	a	a	PRON
ejpam-2240	87	33	is	be	AUX
ejpam-2240	87	34	j	j	PROPN
ejpam-2240	87	35	−	−	NOUN
ejpam-2240	87	36	β	β	X
ejpam-2240	87	37	-open	-open	PROPN
ejpam-2240	87	38	;	;	PUNCT
ejpam-2240	87	39	(	(	PUNCT
ejpam-2240	87	40	ii	ii	NOUN
ejpam-2240	87	41	)	)	PUNCT
ejpam-2240	87	42	if	if	SCONJ
ejpam-2240	87	43	a	a	PRON
ejpam-2240	87	44	is	be	AUX
ejpam-2240	87	45	(	(	PUNCT
ejpam-2240	87	46	i	i	NOUN
ejpam-2240	87	47	∩	∩	NOUN
ejpam-2240	87	48	j)−	j)−	PROPN
ejpam-2240	87	49	β	β	PROPN
ejpam-2240	87	50	-open	-open	PROPN
ejpam-2240	87	51	,	,	PUNCT
ejpam-2240	87	52	then	then	ADV
ejpam-2240	87	53	it	it	PRON
ejpam-2240	87	54	is	be	AUX
ejpam-2240	87	55	simultaneously	simultaneously	ADV
ejpam-2240	87	56	i	i	PRON
ejpam-2240	87	57	−	−	NOUN
ejpam-2240	87	58	β	β	PART
ejpam-2240	87	59	-open	-open	NOUN
ejpam-2240	87	60	and	and	CCONJ
ejpam-2240	87	61	j	j	PROPN
ejpam-2240	87	62	−	−	X
ejpam-2240	87	63	β	β	X
ejpam-2240	87	64	-open	-open	PROPN
ejpam-2240	87	65	.	.	PUNCT
ejpam-2240	88	1	corollary	corollary	ADJ
ejpam-2240	88	2	1	1	NUM
ejpam-2240	88	3	.	.	PUNCT
ejpam-2240	89	1	for	for	ADP
ejpam-2240	89	2	a	a	DET
ejpam-2240	89	3	subset	subset	NOUN
ejpam-2240	89	4	a	a	PRON
ejpam-2240	89	5	of	of	ADP
ejpam-2240	89	6	an	an	DET
ejpam-2240	89	7	ideal	ideal	ADJ
ejpam-2240	89	8	topological	topological	ADJ
ejpam-2240	89	9	space	space	NOUN
ejpam-2240	89	10	(	(	PUNCT
ejpam-2240	89	11	x	x	X
ejpam-2240	89	12	,	,	PUNCT
ejpam-2240	89	13	τ	τ	PROPN
ejpam-2240	89	14	)	)	PUNCT
ejpam-2240	89	15	,	,	PUNCT
ejpam-2240	89	16	recall	recall	VERB
ejpam-2240	89	17	that	that	PRON
ejpam-2240	89	18	ia	ia	PROPN
ejpam-2240	89	19	=	=	PRON
ejpam-2240	89	20	{	{	PUNCT
ejpam-2240	89	21	a∩	a∩	PROPN
ejpam-2240	89	22	e	e	NOUN
ejpam-2240	89	23	:	:	PUNCT
ejpam-2240	89	24	e	e	X
ejpam-2240	89	25	∈	∈	PROPN
ejpam-2240	89	26	i	i	PRON
ejpam-2240	89	27	}	}	PUNCT
ejpam-2240	89	28	is	be	AUX
ejpam-2240	89	29	also	also	ADV
ejpam-2240	89	30	an	an	DET
ejpam-2240	89	31	ideal	ideal	NOUN
ejpam-2240	89	32	on	on	ADP
ejpam-2240	89	33	x	x	X
ejpam-2240	89	34	.	.	PUNCT
ejpam-2240	90	1	(	(	PUNCT
ejpam-2240	90	2	i	i	NOUN
ejpam-2240	90	3	)	)	PUNCT
ejpam-2240	90	4	if	if	SCONJ
ejpam-2240	90	5	a	a	DET
ejpam-2240	90	6	set	set	NOUN
ejpam-2240	90	7	b	b	NOUN
ejpam-2240	90	8	is	be	AUX
ejpam-2240	90	9	ia−	ia−	PROPN
ejpam-2240	90	10	β	β	X
ejpam-2240	90	11	-open	-open	NOUN
ejpam-2240	90	12	,	,	PUNCT
ejpam-2240	90	13	then	then	ADV
ejpam-2240	90	14	it	it	PRON
ejpam-2240	90	15	is	be	AUX
ejpam-2240	90	16	i	i	PRON
ejpam-2240	90	17	−	−	PROPN
ejpam-2240	90	18	β	β	X
ejpam-2240	90	19	-open	-open	PROPN
ejpam-2240	90	20	,	,	PUNCT
ejpam-2240	90	21	(	(	PUNCT
ejpam-2240	90	22	ii	ii	NOUN
ejpam-2240	90	23	)	)	PUNCT
ejpam-2240	90	24	if	if	SCONJ
ejpam-2240	90	25	a	a	PRON
ejpam-2240	90	26	=	=	X
ejpam-2240	90	27	;	;	PUNCT
ejpam-2240	90	28	,	,	PUNCT
ejpam-2240	90	29	then	then	ADV
ejpam-2240	90	30	ia	ia	PROPN
ejpam-2240	90	31	=	=	PUNCT
ejpam-2240	90	32	i	i	PROPN
ejpam-2240	90	33	;	;	PUNCT
ejpam-2240	90	34	=	=	SYM
ejpam-2240	90	35	{	{	PUNCT
ejpam-2240	90	36	;	;	PUNCT
ejpam-2240	90	37	}	}	PUNCT
ejpam-2240	90	38	,	,	PUNCT
ejpam-2240	90	39	vthe	vthe	DET
ejpam-2240	90	40	minimal	minimal	ADJ
ejpam-2240	90	41	ideal	ideal	NOUN
ejpam-2240	90	42	.	.	PUNCT
ejpam-2240	91	1	thus	thus	ADV
ejpam-2240	91	2	,	,	PUNCT
ejpam-2240	91	3	if	if	SCONJ
ejpam-2240	91	4	a	a	DET
ejpam-2240	91	5	set	set	NOUN
ejpam-2240	91	6	c	c	NOUN
ejpam-2240	91	7	is	be	AUX
ejpam-2240	91	8	i	i	PRON
ejpam-2240	91	9	;	;	PUNCT
ejpam-2240	91	10	−	−	PROPN
ejpam-2240	91	11	β	β	X
ejpam-2240	91	12	-open	-open	PROPN
ejpam-2240	91	13	,	,	PUNCT
ejpam-2240	91	14	then	then	ADV
ejpam-2240	91	15	c	c	PROPN
ejpam-2240	91	16	is	be	AUX
ejpam-2240	91	17	also	also	ADV
ejpam-2240	91	18	i	i	PRON
ejpam-2240	91	19	−	−	NOUN
ejpam-2240	91	20	β	β	X
ejpam-2240	91	21	-open	-open	PROPN
ejpam-2240	91	22	.	.	PUNCT
ejpam-2240	92	1	proposition	proposition	NOUN
ejpam-2240	92	2	2	2	NUM
ejpam-2240	92	3	.	.	PUNCT
ejpam-2240	93	1	if	if	SCONJ
ejpam-2240	93	2	a	a	PRON
ejpam-2240	93	3	and	and	CCONJ
ejpam-2240	93	4	b	b	NOUN
ejpam-2240	93	5	are	be	AUX
ejpam-2240	93	6	both	both	PRON
ejpam-2240	93	7	i	i	PRON
ejpam-2240	93	8	−	−	VERB
ejpam-2240	93	9	β	β	X
ejpam-2240	93	10	-open	-open	PROPN
ejpam-2240	93	11	,	,	PUNCT
ejpam-2240	93	12	then	then	ADV
ejpam-2240	93	13	so	so	ADV
ejpam-2240	93	14	is	be	AUX
ejpam-2240	93	15	their	their	PRON
ejpam-2240	93	16	union	union	NOUN
ejpam-2240	93	17	a∪	a∪	PROPN
ejpam-2240	93	18	b.	b.	PROPN
ejpam-2240	93	19	proof	proof	PROPN
ejpam-2240	93	20	.	.	PUNCT
ejpam-2240	94	1	let	let	VERB
ejpam-2240	94	2	the	the	DET
ejpam-2240	94	3	given	give	VERB
ejpam-2240	94	4	conditions	condition	NOUN
ejpam-2240	94	5	hold	hold	VERB
ejpam-2240	94	6	.	.	PUNCT
ejpam-2240	95	1	to	to	PART
ejpam-2240	95	2	show	show	VERB
ejpam-2240	95	3	that	that	SCONJ
ejpam-2240	95	4	a∪b	a∪b	NOUN
ejpam-2240	95	5	is	be	AUX
ejpam-2240	95	6	i−β	i−β	VERB
ejpam-2240	95	7	-open	-open	NOUN
ejpam-2240	95	8	,	,	PUNCT
ejpam-2240	95	9	we	we	PRON
ejpam-2240	95	10	need	need	VERB
ejpam-2240	95	11	to	to	PART
ejpam-2240	95	12	produce	produce	VERB
ejpam-2240	95	13	a	a	DET
ejpam-2240	95	14	preopen	preopen	ADJ
ejpam-2240	95	15	set	set	VERB
ejpam-2240	95	16	g	g	PROPN
ejpam-2240	96	1	such	such	ADJ
ejpam-2240	96	2	that	that	PRON
ejpam-2240	96	3	g\(a∪b	g\(a∪b	ADJ
ejpam-2240	96	4	)	)	PUNCT
ejpam-2240	96	5	∈	∈	PROPN
ejpam-2240	97	1	i	i	PRON
ejpam-2240	97	2	and	and	CCONJ
ejpam-2240	97	3	(	(	PUNCT
ejpam-2240	97	4	a∪b)\cl(g	a∪b)\cl(g	PROPN
ejpam-2240	97	5	)	)	PUNCT
ejpam-2240	97	6	∈	∈	PROPN
ejpam-2240	98	1	i	i	PRON
ejpam-2240	98	2	.	.	PUNCT
ejpam-2240	99	1	since	since	SCONJ
ejpam-2240	99	2	a	a	PRON
ejpam-2240	99	3	and	and	CCONJ
ejpam-2240	99	4	b	b	NOUN
ejpam-2240	99	5	are	be	AUX
ejpam-2240	99	6	both	both	CCONJ
ejpam-2240	99	7	i−β	i−β	VERB
ejpam-2240	99	8	-open	-open	NOUN
ejpam-2240	99	9	,	,	PUNCT
ejpam-2240	99	10	there	there	PRON
ejpam-2240	99	11	are	be	VERB
ejpam-2240	99	12	two	two	NUM
ejpam-2240	99	13	preopen	preopen	ADJ
ejpam-2240	99	14	sets	set	NOUN
ejpam-2240	99	15	g1	g1	NOUN
ejpam-2240	99	16	and	and	CCONJ
ejpam-2240	99	17	g2	g2	PROPN
ejpam-2240	99	18	such	such	ADJ
ejpam-2240	99	19	that	that	SCONJ
ejpam-2240	99	20	g1\a∈	g1\a∈	VERB
ejpam-2240	99	21	i	i	PRON
ejpam-2240	99	22	,	,	PUNCT
ejpam-2240	99	23	a\cl(g1	a\cl(g1	PROPN
ejpam-2240	99	24	)	)	PUNCT
ejpam-2240	99	25	∈	∈	PROPN
ejpam-2240	100	1	i	i	PRON
ejpam-2240	100	2	,	,	PUNCT
ejpam-2240	100	3	g2\b	g2\b	PROPN
ejpam-2240	100	4	∈	∈	PROPN
ejpam-2240	100	5	i	i	PRON
ejpam-2240	100	6	,	,	PUNCT
ejpam-2240	100	7	b\cl(g2	b\cl(g2	X
ejpam-2240	100	8	)	)	PUNCT
ejpam-2240	100	9	∈	∈	PROPN
ejpam-2240	101	1	i	i	PRON
ejpam-2240	101	2	.	.	PUNCT
ejpam-2240	102	1	choose	choose	VERB
ejpam-2240	102	2	g	g	PROPN
ejpam-2240	102	3	=	=	PROPN
ejpam-2240	102	4	g1	g1	PROPN
ejpam-2240	102	5	∪	∪	ADP
ejpam-2240	102	6	g2	g2	PROPN
ejpam-2240	102	7	,	,	PUNCT
ejpam-2240	102	8	and	and	CCONJ
ejpam-2240	102	9	observe	observe	VERB
ejpam-2240	102	10	that	that	SCONJ
ejpam-2240	102	11	(	(	PUNCT
ejpam-2240	102	12	g1	g1	PROPN
ejpam-2240	102	13	∪	∪	ADP
ejpam-2240	102	14	g2	g2	PROPN
ejpam-2240	102	15	)	)	PUNCT
ejpam-2240	102	16	\	\	PUNCT
ejpam-2240	103	1	(	(	PUNCT
ejpam-2240	103	2	a∪	a∪	NOUN
ejpam-2240	103	3	b	b	X
ejpam-2240	103	4	)	)	PUNCT
ejpam-2240	103	5	=	=	SYM
ejpam-2240	103	6	(	(	PUNCT
ejpam-2240	103	7	(	(	PUNCT
ejpam-2240	103	8	g1	g1	PROPN
ejpam-2240	103	9	\	\	PROPN
ejpam-2240	103	10	a	a	DET
ejpam-2240	103	11	)	)	PUNCT
ejpam-2240	103	12	\	\	NOUN
ejpam-2240	103	13	b)∪	b)∪	NOUN
ejpam-2240	103	14	(	(	PUNCT
ejpam-2240	103	15	(	(	PUNCT
ejpam-2240	103	16	g2	g2	PROPN
ejpam-2240	103	17	\	\	PROPN
ejpam-2240	103	18	b	b	X
ejpam-2240	103	19	)	)	PUNCT
ejpam-2240	103	20	\	\	PROPN
ejpam-2240	103	21	a	a	DET
ejpam-2240	103	22	)	)	PUNCT
ejpam-2240	103	23	∈	∈	PROPN
ejpam-2240	104	1	i	i	PRON
ejpam-2240	104	2	.	.	PUNCT
ejpam-2240	105	1	also	also	ADV
ejpam-2240	105	2	,	,	PUNCT
ejpam-2240	105	3	(	(	PUNCT
ejpam-2240	105	4	a∪	a∪	NOUN
ejpam-2240	105	5	b	b	X
ejpam-2240	105	6	)	)	PUNCT
ejpam-2240	105	7	\	\	NOUN
ejpam-2240	105	8	cl(g1	cl(g1	NOUN
ejpam-2240	105	9	∪	∪	PROPN
ejpam-2240	105	10	g2	g2	PROPN
ejpam-2240	105	11	)	)	PUNCT
ejpam-2240	106	1	=	=	SYM
ejpam-2240	107	1	(	(	PUNCT
ejpam-2240	107	2	(	(	PUNCT
ejpam-2240	107	3	a\	a\	NOUN
ejpam-2240	107	4	cl(g1	cl(g1	NOUN
ejpam-2240	107	5	)	)	PUNCT
ejpam-2240	107	6	)	)	PUNCT
ejpam-2240	107	7	\	\	PROPN
ejpam-2240	108	1	cl(g2))∪	cl(g2))∪	PROPN
ejpam-2240	108	2	(	(	PUNCT
ejpam-2240	108	3	(	(	PUNCT
ejpam-2240	108	4	b	b	PROPN
ejpam-2240	108	5	\	\	PROPN
ejpam-2240	108	6	cl(g2	cl(g2	PROPN
ejpam-2240	108	7	)	)	PUNCT
ejpam-2240	108	8	)	)	PUNCT
ejpam-2240	108	9	\	\	NOUN
ejpam-2240	108	10	cl(g1	cl(g1	NOUN
ejpam-2240	108	11	)	)	PUNCT
ejpam-2240	108	12	)	)	PUNCT
ejpam-2240	109	1	∈	∈	PROPN
ejpam-2240	110	1	i	i	PRON
ejpam-2240	110	2	.	.	PUNCT
ejpam-2240	111	1	therefore	therefore	ADV
ejpam-2240	111	2	,	,	PUNCT
ejpam-2240	111	3	by	by	ADP
ejpam-2240	111	4	definition	definition	NOUN
ejpam-2240	111	5	,	,	PUNCT
ejpam-2240	111	6	a∪	a∪	PROPN
ejpam-2240	111	7	b	b	NOUN
ejpam-2240	111	8	is	be	AUX
ejpam-2240	111	9	i	i	PRON
ejpam-2240	111	10	−	−	PROPN
ejpam-2240	111	11	β	β	X
ejpam-2240	111	12	-open	-open	PROPN
ejpam-2240	111	13	.	.	PUNCT
ejpam-2240	112	1	proposition	proposition	NOUN
ejpam-2240	112	2	3	3	X
ejpam-2240	112	3	.	.	PUNCT
ejpam-2240	113	1	let	let	AUX
ejpam-2240	113	2	(	(	PUNCT
ejpam-2240	113	3	x	x	X
ejpam-2240	113	4	,	,	PUNCT
ejpam-2240	113	5	τ	τ	X
ejpam-2240	113	6	)	)	PUNCT
ejpam-2240	113	7	be	be	VERB
ejpam-2240	113	8	a	a	DET
ejpam-2240	113	9	topological	topological	ADJ
ejpam-2240	113	10	space	space	NOUN
ejpam-2240	113	11	in	in	ADP
ejpam-2240	113	12	which	which	PRON
ejpam-2240	113	13	there	there	PRON
ejpam-2240	113	14	is	be	VERB
ejpam-2240	113	15	a	a	DET
ejpam-2240	113	16	preopen	preopen	ADJ
ejpam-2240	113	17	singleton	singleton	NOUN
ejpam-2240	113	18	subset	subset	NOUN
ejpam-2240	113	19	{	{	PUNCT
ejpam-2240	113	20	a	a	PRON
ejpam-2240	113	21	}	}	PUNCT
ejpam-2240	113	22	satisfying	satisfy	VERB
ejpam-2240	113	23	cl({a	cl({a	ADJ
ejpam-2240	113	24	}	}	PUNCT
ejpam-2240	113	25	)	)	PUNCT
ejpam-2240	114	1	=	=	PUNCT
ejpam-2240	115	1	x	x	X
ejpam-2240	115	2	.for	.for	ADP
ejpam-2240	115	3	any	any	DET
ejpam-2240	115	4	ideal	ideal	NOUN
ejpam-2240	116	1	i	i	PRON
ejpam-2240	116	2	on	on	ADP
ejpam-2240	116	3	x	x	PUNCT
ejpam-2240	116	4	with	with	ADP
ejpam-2240	116	5	{	{	PUNCT
ejpam-2240	116	6	a	a	PRON
ejpam-2240	116	7	}	}	PUNCT
ejpam-2240	116	8	∈	∈	NOUN
ejpam-2240	116	9	i	i	PRON
ejpam-2240	116	10	,	,	PUNCT
ejpam-2240	116	11	we	we	PRON
ejpam-2240	116	12	have	have	VERB
ejpam-2240	116	13	that	that	PRON
ejpam-2240	116	14	:	:	PUNCT
ejpam-2240	116	15	(	(	PUNCT
ejpam-2240	116	16	i	i	NOUN
ejpam-2240	116	17	)	)	PUNCT
ejpam-2240	116	18	every	every	DET
ejpam-2240	116	19	singleton	singleton	NOUN
ejpam-2240	116	20	subset	subset	NOUN
ejpam-2240	116	21	of	of	ADP
ejpam-2240	116	22	x	x	SYM
ejpam-2240	116	23	is	be	AUX
ejpam-2240	116	24	i	i	PRON
ejpam-2240	116	25	−	−	NOUN
ejpam-2240	116	26	β	β	X
ejpam-2240	116	27	-open	-open	PROPN
ejpam-2240	116	28	;	;	PUNCT
ejpam-2240	116	29	a.	a.	NOUN
ejpam-2240	116	30	nasef	nasef	PROPN
ejpam-2240	116	31	,	,	PUNCT
ejpam-2240	116	32	r.	r.	PROPN
ejpam-2240	116	33	mareay	mareay	PROPN
ejpam-2240	116	34	,	,	PUNCT
ejpam-2240	116	35	f.	f.	PROPN
ejpam-2240	116	36	michael	michael	PROPN
ejpam-2240	116	37	/	/	SYM
ejpam-2240	116	38	eur	eur	PROPN
ejpam-2240	116	39	.	.	PUNCT
ejpam-2240	117	1	j.	j.	PROPN
ejpam-2240	117	2	pure	pure	PROPN
ejpam-2240	117	3	appl	appl	PROPN
ejpam-2240	117	4	.	.	PROPN
ejpam-2240	117	5	math	math	PROPN
ejpam-2240	117	6	,	,	PUNCT
ejpam-2240	117	7	8	8	NUM
ejpam-2240	117	8	(	(	PUNCT
ejpam-2240	117	9	2015	2015	NUM
ejpam-2240	117	10	)	)	PUNCT
ejpam-2240	117	11	,	,	PUNCT
ejpam-2240	117	12	389	389	NUM
ejpam-2240	117	13	-	-	SYM
ejpam-2240	117	14	394	394	NUM
ejpam-2240	117	15	392	392	NUM
ejpam-2240	117	16	(	(	PUNCT
ejpam-2240	117	17	ii	ii	NOUN
ejpam-2240	117	18	)	)	PUNCT
ejpam-2240	117	19	every	every	DET
ejpam-2240	117	20	finite	finite	PROPN
ejpam-2240	117	21	subset	subset	NOUN
ejpam-2240	117	22	of	of	ADP
ejpam-2240	117	23	x	x	SYM
ejpam-2240	117	24	is	be	AUX
ejpam-2240	117	25	i	i	PRON
ejpam-2240	117	26	−	−	PROPN
ejpam-2240	117	27	β	β	X
ejpam-2240	117	28	-open	-open	NOUN
ejpam-2240	117	29	.	.	PUNCT
ejpam-2240	118	1	proof	proof	NOUN
ejpam-2240	118	2	.	.	PUNCT
ejpam-2240	119	1	let	let	VERB
ejpam-2240	119	2	the	the	DET
ejpam-2240	119	3	given	give	VERB
ejpam-2240	119	4	conditions	condition	NOUN
ejpam-2240	119	5	hold	hold	VERB
ejpam-2240	119	6	.	.	PUNCT
ejpam-2240	120	1	suppose	suppose	VERB
ejpam-2240	120	2	that	that	SCONJ
ejpam-2240	120	3	{	{	PUNCT
ejpam-2240	120	4	b	b	X
ejpam-2240	120	5	}	}	PUNCT
ejpam-2240	120	6	is	be	AUX
ejpam-2240	120	7	a	a	DET
ejpam-2240	120	8	singleton	singleton	NOUN
ejpam-2240	120	9	subset	subset	NOUN
ejpam-2240	120	10	of	of	ADP
ejpam-2240	120	11	x	x	X
ejpam-2240	120	12	.	.	PUNCT
ejpam-2240	121	1	since	since	SCONJ
ejpam-2240	121	2	(	(	PUNCT
ejpam-2240	121	3	{	{	PUNCT
ejpam-2240	121	4	a	a	PRON
ejpam-2240	121	5	}	}	PUNCT
ejpam-2240	121	6	is	be	AUX
ejpam-2240	121	7	preopen	preopen	ADJ
ejpam-2240	121	8	)	)	PUNCT
ejpam-2240	121	9	,	,	PUNCT
ejpam-2240	121	10	{	{	PUNCT
ejpam-2240	121	11	a	a	PRON
ejpam-2240	121	12	}	}	PUNCT
ejpam-2240	121	13	\	\	NOUN
ejpam-2240	121	14	{	{	PUNCT
ejpam-2240	121	15	b	b	NOUN
ejpam-2240	121	16	}	}	PUNCT
ejpam-2240	121	17	=	=	SYM
ejpam-2240	121	18	{	{	PUNCT
ejpam-2240	121	19	a	a	PRON
ejpam-2240	121	20	}	}	PUNCT
ejpam-2240	121	21	∈	∈	NOUN
ejpam-2240	121	22	i	i	PRON
ejpam-2240	121	23	,	,	PUNCT
ejpam-2240	121	24	and	and	CCONJ
ejpam-2240	121	25	{	{	PUNCT
ejpam-2240	121	26	b	b	NOUN
ejpam-2240	121	27	}	}	PUNCT
ejpam-2240	121	28	\	\	NOUN
ejpam-2240	121	29	cl({a	cl({a	ADJ
ejpam-2240	121	30	}	}	PUNCT
ejpam-2240	121	31	)	)	PUNCT
ejpam-2240	122	1	=	=	PRON
ejpam-2240	122	2	{	{	PUNCT
ejpam-2240	122	3	b	b	NOUN
ejpam-2240	122	4	}	}	PUNCT
ejpam-2240	122	5	\	\	NOUN
ejpam-2240	122	6	x	x	X
ejpam-2240	123	1	=	=	PUNCT
ejpam-2240	123	2	;	;	PUNCT
ejpam-2240	123	3	∈	∈	PROPN
ejpam-2240	124	1	i	i	PRON
ejpam-2240	124	2	,	,	PUNCT
ejpam-2240	124	3	it	it	PRON
ejpam-2240	124	4	follows	follow	VERB
ejpam-2240	124	5	that	that	SCONJ
ejpam-2240	124	6	{	{	PUNCT
ejpam-2240	124	7	b	b	X
ejpam-2240	124	8	}	}	PUNCT
ejpam-2240	124	9	is	be	AUX
ejpam-2240	124	10	i	i	PRON
ejpam-2240	124	11	−	−	NOUN
ejpam-2240	124	12	β	β	X
ejpam-2240	124	13	-open	-open	PROPN
ejpam-2240	124	14	,	,	PUNCT
ejpam-2240	124	15	this	this	PRON
ejpam-2240	124	16	proves	prove	VERB
ejpam-2240	124	17	that	that	SCONJ
ejpam-2240	124	18	(	(	PUNCT
ejpam-2240	124	19	i	i	NOUN
ejpam-2240	124	20	)	)	PUNCT
ejpam-2240	124	21	.	.	PUNCT
ejpam-2240	125	1	to	to	PART
ejpam-2240	125	2	see	see	VERB
ejpam-2240	125	3	that	that	DET
ejpam-2240	125	4	(	(	PUNCT
ejpam-2240	125	5	ii	ii	NOUN
ejpam-2240	125	6	)	)	PUNCT
ejpam-2240	125	7	holds	hold	VERB
ejpam-2240	125	8	,	,	PUNCT
ejpam-2240	125	9	let	let	VERB
ejpam-2240	125	10	a=	a=	ADV
ejpam-2240	125	11	{	{	PUNCT
ejpam-2240	125	12	b1	b1	NOUN
ejpam-2240	125	13	,	,	PUNCT
ejpam-2240	125	14	b2	b2	NOUN
ejpam-2240	125	15	,	,	PUNCT
ejpam-2240	125	16	b3	b3	PROPN
ejpam-2240	125	17	,	,	PUNCT
ejpam-2240	125	18	.	.	PUNCT
ejpam-2240	125	19	.	.	PUNCT
ejpam-2240	126	1	.	.	PUNCT
ejpam-2240	127	1	,	,	PUNCT
ejpam-2240	127	2	bn	bn	AUX
ejpam-2240	127	3	}	}	PUNCT
ejpam-2240	127	4	be	be	AUX
ejpam-2240	127	5	a	a	DET
ejpam-2240	127	6	finite	finite	NOUN
ejpam-2240	127	7	subset	subset	NOUN
ejpam-2240	127	8	of	of	ADP
ejpam-2240	127	9	x	x	X
ejpam-2240	127	10	.	.	PUNCT
ejpam-2240	128	1	since	since	SCONJ
ejpam-2240	128	2	a	a	DET
ejpam-2240	128	3	=	=	PUNCT
ejpam-2240	128	4	{	{	PUNCT
ejpam-2240	128	5	b1	b1	PROPN
ejpam-2240	128	6	}	}	PUNCT
ejpam-2240	128	7	∪	∪	NOUN
ejpam-2240	128	8	{	{	PUNCT
ejpam-2240	128	9	b2	b2	NOUN
ejpam-2240	128	10	}	}	PUNCT
ejpam-2240	128	11	∪	∪	NOUN
ejpam-2240	128	12	{	{	PUNCT
ejpam-2240	128	13	b3	b3	NOUN
ejpam-2240	128	14	}	}	PUNCT
ejpam-2240	128	15	∪	∪	NOUN
ejpam-2240	128	16	.	.	PUNCT
ejpam-2240	128	17	.	.	PUNCT
ejpam-2240	128	18	.	.	PUNCT
ejpam-2240	129	1	∪	∪	PROPN
ejpam-2240	129	2	{	{	PUNCT
ejpam-2240	129	3	bn	bn	NOUN
ejpam-2240	129	4	}	}	PUNCT
ejpam-2240	129	5	,	,	PUNCT
ejpam-2240	129	6	the	the	DET
ejpam-2240	129	7	result	result	NOUN
ejpam-2240	129	8	follows	follow	VERB
ejpam-2240	129	9	from	from	ADP
ejpam-2240	129	10	the	the	DET
ejpam-2240	129	11	fact	fact	NOUN
ejpam-2240	129	12	that	that	SCONJ
ejpam-2240	129	13	each	each	DET
ejpam-2240	129	14	singleton	singleton	NOUN
ejpam-2240	129	15	subset	subset	VERB
ejpam-2240	129	16	{	{	PUNCT
ejpam-2240	129	17	bi	bi	NOUN
ejpam-2240	129	18	}	}	PUNCT
ejpam-2240	129	19	,	,	PUNCT
ejpam-2240	129	20	(	(	PUNCT
ejpam-2240	129	21	i	i	NOUN
ejpam-2240	129	22	=	=	NOUN
ejpam-2240	129	23	1,2,3	1,2,3	NUM
ejpam-2240	129	24	,	,	PUNCT
ejpam-2240	129	25	.	.	PUNCT
ejpam-2240	129	26	.	.	PUNCT
ejpam-2240	130	1	.	.	PUNCT
ejpam-2240	131	1	,	,	PUNCT
ejpam-2240	131	2	n	n	CCONJ
ejpam-2240	131	3	)	)	PUNCT
ejpam-2240	131	4	is	be	AUX
ejpam-2240	131	5	i	i	PRON
ejpam-2240	131	6	−	−	NOUN
ejpam-2240	131	7	β	β	PUNCT
ejpam-2240	131	8	-open	-open	NOUN
ejpam-2240	131	9	and	and	CCONJ
ejpam-2240	131	10	a	a	DET
ejpam-2240	131	11	repeated	repeat	VERB
ejpam-2240	131	12	use	use	NOUN
ejpam-2240	131	13	of	of	ADP
ejpam-2240	131	14	proposition	proposition	NOUN
ejpam-2240	131	15	2	2	NUM
ejpam-2240	131	16	above	above	ADV
ejpam-2240	131	17	.	.	PUNCT
ejpam-2240	132	1	proposition	proposition	NOUN
ejpam-2240	132	2	3	3	NUM
ejpam-2240	132	3	does	do	AUX
ejpam-2240	132	4	not	not	PART
ejpam-2240	132	5	hold	hold	VERB
ejpam-2240	132	6	for	for	ADP
ejpam-2240	132	7	any	any	DET
ejpam-2240	132	8	choice	choice	NOUN
ejpam-2240	132	9	of	of	ADP
ejpam-2240	132	10	ideal	ideal	NOUN
ejpam-2240	132	11	.	.	PUNCT
ejpam-2240	133	1	example	example	NOUN
ejpam-2240	134	1	2	2	NUM
ejpam-2240	134	2	.	.	X
ejpam-2240	134	3	consider	consider	VERB
ejpam-2240	134	4	x	x	PUNCT
ejpam-2240	134	5	=	=	PRON
ejpam-2240	134	6	{	{	PUNCT
ejpam-2240	134	7	a	a	PRON
ejpam-2240	134	8	,	,	PUNCT
ejpam-2240	134	9	b	b	NOUN
ejpam-2240	134	10	,	,	PUNCT
ejpam-2240	134	11	c	c	NOUN
ejpam-2240	134	12	}	}	PUNCT
ejpam-2240	134	13	and	and	CCONJ
ejpam-2240	134	14	τ	τ	PROPN
ejpam-2240	134	15	=	=	PUNCT
ejpam-2240	134	16	{	{	PUNCT
ejpam-2240	134	17	;	;	PUNCT
ejpam-2240	134	18	,	,	PUNCT
ejpam-2240	134	19	{	{	PUNCT
ejpam-2240	134	20	a	a	X
ejpam-2240	134	21	}	}	PUNCT
ejpam-2240	134	22	,	,	PUNCT
ejpam-2240	134	23	{	{	PUNCT
ejpam-2240	134	24	a	a	X
ejpam-2240	134	25	,	,	PUNCT
ejpam-2240	134	26	c	c	NOUN
ejpam-2240	134	27	}	}	PUNCT
ejpam-2240	134	28	,	,	PUNCT
ejpam-2240	134	29	x	x	SYM
ejpam-2240	134	30	}	}	PUNCT
ejpam-2240	134	31	and	and	CCONJ
ejpam-2240	134	32	observe	observe	VERB
ejpam-2240	134	33	that	that	SCONJ
ejpam-2240	134	34	cl({a	cl({a	ADJ
ejpam-2240	134	35	}	}	PUNCT
ejpam-2240	134	36	)	)	PUNCT
ejpam-2240	135	1	=	=	PUNCT
ejpam-2240	136	1	x	x	X
ejpam-2240	136	2	.	.	PUNCT
ejpam-2240	137	1	if	if	SCONJ
ejpam-2240	137	2	we	we	PRON
ejpam-2240	137	3	choose	choose	VERB
ejpam-2240	137	4	the	the	DET
ejpam-2240	137	5	minimal	minimal	ADJ
ejpam-2240	137	6	ideal	ideal	NOUN
ejpam-2240	137	7	i	i	PRON
ejpam-2240	137	8	=	=	PUNCT
ejpam-2240	137	9	{	{	PUNCT
ejpam-2240	137	10	;	;	PUNCT
ejpam-2240	137	11	}	}	PUNCT
ejpam-2240	137	12	on	on	ADP
ejpam-2240	137	13	x	x	SYM
ejpam-2240	137	14	,	,	PUNCT
ejpam-2240	137	15	then	then	ADV
ejpam-2240	137	16	the	the	DET
ejpam-2240	137	17	singleton	singleton	PROPN
ejpam-2240	137	18	subset	subset	NOUN
ejpam-2240	137	19	{	{	PUNCT
ejpam-2240	137	20	b	b	NOUN
ejpam-2240	137	21	}	}	PUNCT
ejpam-2240	137	22	is	be	AUX
ejpam-2240	137	23	not	not	PART
ejpam-2240	137	24	i	i	PRON
ejpam-2240	137	25	−	−	NOUN
ejpam-2240	137	26	β	β	X
ejpam-2240	137	27	-open	-open	PROPN
ejpam-2240	137	28	,	,	PUNCT
ejpam-2240	137	29	as	as	SCONJ
ejpam-2240	137	30	there	there	PRON
ejpam-2240	137	31	is	be	VERB
ejpam-2240	137	32	no	no	DET
ejpam-2240	137	33	preopen	preopen	NOUN
ejpam-2240	137	34	set	set	VERB
ejpam-2240	137	35	g	g	NOUN
ejpam-2240	137	36	satisfying	satisfy	VERB
ejpam-2240	137	37	g	g	PROPN
ejpam-2240	137	38	\	\	PROPN
ejpam-2240	137	39	{	{	PUNCT
ejpam-2240	137	40	b	b	X
ejpam-2240	137	41	}	}	PUNCT
ejpam-2240	137	42	∈	∈	NOUN
ejpam-2240	137	43	i	i	PRON
ejpam-2240	137	44	and	and	CCONJ
ejpam-2240	137	45	{	{	PUNCT
ejpam-2240	137	46	b	b	NOUN
ejpam-2240	137	47	}	}	PUNCT
ejpam-2240	137	48	\	\	NOUN
ejpam-2240	137	49	cl(g	cl(g	X
ejpam-2240	137	50	)	)	PUNCT
ejpam-2240	137	51	∈	∈	PROPN
ejpam-2240	138	1	i	i	PRON
ejpam-2240	138	2	,	,	PUNCT
ejpam-2240	138	3	simultaneously	simultaneously	ADV
ejpam-2240	138	4	.	.	PUNCT
ejpam-2240	139	1	proposition	proposition	NOUN
ejpam-2240	139	2	4	4	NUM
ejpam-2240	139	3	.	.	PUNCT
ejpam-2240	140	1	let	let	VERB
ejpam-2240	140	2	a	a	PRON
ejpam-2240	140	3	and	and	CCONJ
ejpam-2240	140	4	b	b	NOUN
ejpam-2240	140	5	be	be	AUX
ejpam-2240	140	6	subsets	subset	NOUN
ejpam-2240	140	7	of	of	ADP
ejpam-2240	140	8	a	a	DET
ejpam-2240	140	9	topological	topological	ADJ
ejpam-2240	140	10	space	space	NOUN
ejpam-2240	140	11	(	(	PUNCT
ejpam-2240	140	12	x	x	X
ejpam-2240	140	13	,	,	PUNCT
ejpam-2240	140	14	τ	τ	X
ejpam-2240	140	15	)	)	PUNCT
ejpam-2240	140	16	such	such	ADJ
ejpam-2240	140	17	that	that	SCONJ
ejpam-2240	140	18	a	a	PRON
ejpam-2240	140	19	is	be	AUX
ejpam-2240	140	20	preopen	preopen	ADJ
ejpam-2240	140	21	,	,	PUNCT
ejpam-2240	140	22	a⊆	a⊆	NOUN
ejpam-2240	140	23	b	b	NOUN
ejpam-2240	140	24	,	,	PUNCT
ejpam-2240	140	25	and	and	CCONJ
ejpam-2240	140	26	a	a	PRON
ejpam-2240	140	27	is	be	AUX
ejpam-2240	140	28	dense	dense	ADJ
ejpam-2240	140	29	in	in	ADP
ejpam-2240	140	30	b	b	NOUN
ejpam-2240	140	31	(	(	PUNCT
ejpam-2240	140	32	that	that	PRON
ejpam-2240	140	33	is	be	AUX
ejpam-2240	140	34	,	,	PUNCT
ejpam-2240	140	35	b	b	PROPN
ejpam-2240	140	36	⊆	⊆	NUM
ejpam-2240	140	37	cl(a	cl(a	NUM
ejpam-2240	140	38	)	)	PUNCT
ejpam-2240	140	39	)	)	PUNCT
ejpam-2240	140	40	.	.	PUNCT
ejpam-2240	141	1	then	then	ADV
ejpam-2240	141	2	b	b	X
ejpam-2240	141	3	is	be	AUX
ejpam-2240	141	4	i	i	PRON
ejpam-2240	141	5	−β	−β	NOUN
ejpam-2240	141	6	-open	-open	ADJ
ejpam-2240	141	7	for	for	ADP
ejpam-2240	141	8	any	any	DET
ejpam-2240	141	9	ideal	ideal	NOUN
ejpam-2240	141	10	i	i	PRON
ejpam-2240	141	11	on	on	ADP
ejpam-2240	141	12	x	x	X
ejpam-2240	141	13	.	.	PUNCT
ejpam-2240	142	1	in	in	ADP
ejpam-2240	142	2	particular	particular	ADJ
ejpam-2240	142	3	,	,	PUNCT
ejpam-2240	142	4	the	the	DET
ejpam-2240	142	5	conclusion	conclusion	NOUN
ejpam-2240	142	6	holds	hold	VERB
ejpam-2240	142	7	in	in	ADP
ejpam-2240	142	8	the	the	DET
ejpam-2240	142	9	special	special	ADJ
ejpam-2240	142	10	case	case	NOUN
ejpam-2240	142	11	when	when	SCONJ
ejpam-2240	142	12	b	b	PROPN
ejpam-2240	142	13	=	=	SYM
ejpam-2240	142	14	cl(a	cl(a	X
ejpam-2240	142	15	)	)	PUNCT
ejpam-2240	142	16	.	.	PUNCT
ejpam-2240	143	1	remark	remark	PROPN
ejpam-2240	143	2	1	1	NUM
ejpam-2240	143	3	.	.	PUNCT
ejpam-2240	144	1	if	if	SCONJ
ejpam-2240	144	2	a	a	PRON
ejpam-2240	144	3	and	and	CCONJ
ejpam-2240	144	4	b	b	NOUN
ejpam-2240	144	5	are	be	AUX
ejpam-2240	144	6	two	two	NUM
ejpam-2240	145	1	i	i	PRON
ejpam-2240	145	2	−	−	NOUN
ejpam-2240	145	3	β	β	NOUN
ejpam-2240	145	4	-open	-open	NOUN
ejpam-2240	145	5	sets	set	NOUN
ejpam-2240	145	6	,	,	PUNCT
ejpam-2240	145	7	then	then	ADV
ejpam-2240	145	8	their	their	PRON
ejpam-2240	145	9	intersection	intersection	NOUN
ejpam-2240	145	10	a∩	a∩	PROPN
ejpam-2240	145	11	b	b	NOUN
ejpam-2240	145	12	need	need	AUX
ejpam-2240	145	13	not	not	PART
ejpam-2240	145	14	be	be	AUX
ejpam-2240	145	15	i	i	PRON
ejpam-2240	145	16	−	−	NOUN
ejpam-2240	145	17	β	β	AUX
ejpam-2240	145	18	open	open	ADJ
ejpam-2240	145	19	.	.	PUNCT
ejpam-2240	146	1	for	for	ADP
ejpam-2240	146	2	example	example	NOUN
ejpam-2240	146	3	,	,	PUNCT
ejpam-2240	146	4	let	let	VERB
ejpam-2240	146	5	x	x	PUNCT
ejpam-2240	146	6	=	=	PRON
ejpam-2240	146	7	{	{	PUNCT
ejpam-2240	146	8	a	a	PRON
ejpam-2240	146	9	,	,	PUNCT
ejpam-2240	146	10	b	b	NOUN
ejpam-2240	146	11	,	,	PUNCT
ejpam-2240	146	12	c	c	AUX
ejpam-2240	146	13	}	}	PUNCT
ejpam-2240	146	14	be	be	AUX
ejpam-2240	146	15	equipped	equip	VERB
ejpam-2240	146	16	with	with	ADP
ejpam-2240	146	17	a	a	DET
ejpam-2240	146	18	topology	topology	NOUN
ejpam-2240	146	19	τ=	τ=	PRON
ejpam-2240	146	20	{	{	PUNCT
ejpam-2240	146	21	;	;	PUNCT
ejpam-2240	146	22	,	,	PUNCT
ejpam-2240	146	23	{	{	PUNCT
ejpam-2240	146	24	a	a	X
ejpam-2240	146	25	}	}	PUNCT
ejpam-2240	146	26	,	,	PUNCT
ejpam-2240	146	27	{	{	PUNCT
ejpam-2240	146	28	c	c	X
ejpam-2240	146	29	}	}	PUNCT
ejpam-2240	146	30	,	,	PUNCT
ejpam-2240	146	31	{	{	PUNCT
ejpam-2240	146	32	a	a	X
ejpam-2240	146	33	,	,	PUNCT
ejpam-2240	146	34	c	c	NOUN
ejpam-2240	146	35	}	}	PUNCT
ejpam-2240	146	36	,	,	PUNCT
ejpam-2240	146	37	x	x	SYM
ejpam-2240	146	38	}	}	PUNCT
ejpam-2240	146	39	.	.	PUNCT
ejpam-2240	147	1	note	note	VERB
ejpam-2240	147	2	that	that	SCONJ
ejpam-2240	147	3	cl({a	cl({a	ADJ
ejpam-2240	147	4	}	}	PUNCT
ejpam-2240	147	5	)	)	PUNCT
ejpam-2240	148	1	=	=	PRON
ejpam-2240	148	2	{	{	PUNCT
ejpam-2240	148	3	a	a	DET
ejpam-2240	148	4	,	,	PUNCT
ejpam-2240	148	5	b	b	NOUN
ejpam-2240	148	6	}	}	PUNCT
ejpam-2240	148	7	and	and	CCONJ
ejpam-2240	148	8	cl({c	cl({c	NOUN
ejpam-2240	148	9	}	}	PUNCT
ejpam-2240	148	10	)	)	PUNCT
ejpam-2240	149	1	=	=	PRON
ejpam-2240	149	2	{	{	PUNCT
ejpam-2240	149	3	b	b	NOUN
ejpam-2240	149	4	,	,	PUNCT
ejpam-2240	149	5	c	c	NOUN
ejpam-2240	149	6	}	}	PUNCT
ejpam-2240	149	7	;	;	PUNCT
ejpam-2240	149	8	moreover	moreover	ADV
ejpam-2240	149	9	,	,	PUNCT
ejpam-2240	149	10	the	the	DET
ejpam-2240	149	11	subsets	subset	NOUN
ejpam-2240	149	12	{	{	PUNCT
ejpam-2240	149	13	a	a	PRON
ejpam-2240	149	14	,	,	PUNCT
ejpam-2240	149	15	b	b	NOUN
ejpam-2240	149	16	}	}	PUNCT
ejpam-2240	149	17	and	and	CCONJ
ejpam-2240	149	18	{	{	PUNCT
ejpam-2240	149	19	b	b	NOUN
ejpam-2240	149	20	,	,	PUNCT
ejpam-2240	149	21	c	c	NOUN
ejpam-2240	149	22	}	}	PUNCT
ejpam-2240	149	23	are	be	AUX
ejpam-2240	149	24	β	β	X
ejpam-2240	149	25	-open	-open	NOUN
ejpam-2240	149	26	with	with	ADP
ejpam-2240	149	27	respect	respect	NOUN
ejpam-2240	149	28	to	to	ADP
ejpam-2240	149	29	the	the	DET
ejpam-2240	149	30	minimal	minimal	ADJ
ejpam-2240	149	31	ideal	ideal	NOUN
ejpam-2240	150	1	i	i	PRON
ejpam-2240	150	2	=	=	PUNCT
ejpam-2240	150	3	{	{	PUNCT
ejpam-2240	150	4	;	;	PUNCT
ejpam-2240	150	5	}	}	PUNCT
ejpam-2240	150	6	,	,	PUNCT
ejpam-2240	150	7	in	in	ADP
ejpam-2240	150	8	view	view	NOUN
ejpam-2240	150	9	of	of	ADP
ejpam-2240	150	10	proposition	proposition	NOUN
ejpam-2240	150	11	4	4	NUM
ejpam-2240	150	12	above	above	ADV
ejpam-2240	150	13	.	.	PUNCT
ejpam-2240	151	1	however	however	ADV
ejpam-2240	151	2	,	,	PUNCT
ejpam-2240	151	3	the	the	DET
ejpam-2240	151	4	singleton	singleton	PROPN
ejpam-2240	151	5	subset	subset	NOUN
ejpam-2240	151	6	{	{	PUNCT
ejpam-2240	151	7	b	b	NOUN
ejpam-2240	151	8	}	}	PUNCT
ejpam-2240	151	9	=	=	SYM
ejpam-2240	151	10	{	{	PUNCT
ejpam-2240	151	11	a	a	PRON
ejpam-2240	151	12	,	,	PUNCT
ejpam-2240	151	13	b	b	NOUN
ejpam-2240	151	14	}	}	PUNCT
ejpam-2240	151	15	∩	∩	ADJ
ejpam-2240	151	16	{	{	PUNCT
ejpam-2240	151	17	b	b	NOUN
ejpam-2240	151	18	,	,	PUNCT
ejpam-2240	151	19	c	c	NOUN
ejpam-2240	151	20	}	}	PUNCT
ejpam-2240	151	21	is	be	AUX
ejpam-2240	151	22	not	not	PART
ejpam-2240	151	23	β	β	PART
ejpam-2240	151	24	-open	-open	NOUN
ejpam-2240	151	25	with	with	ADP
ejpam-2240	151	26	respect	respect	NOUN
ejpam-2240	151	27	to	to	ADP
ejpam-2240	151	28	the	the	DET
ejpam-2240	151	29	minimal	minimal	ADJ
ejpam-2240	151	30	ideal	ideal	NOUN
ejpam-2240	152	1	i	i	PRON
ejpam-2240	152	2	=	=	PUNCT
ejpam-2240	152	3	{	{	PUNCT
ejpam-2240	152	4	;	;	PUNCT
ejpam-2240	152	5	}	}	PUNCT
ejpam-2240	152	6	.	.	PUNCT
ejpam-2240	153	1	obviously	obviously	ADV
ejpam-2240	153	2	,	,	PUNCT
ejpam-2240	153	3	if	if	SCONJ
ejpam-2240	153	4	a	a	PRON
ejpam-2240	153	5	,	,	PUNCT
ejpam-2240	153	6	b	b	X
ejpam-2240	153	7	∈	∈	PROPN
ejpam-2240	154	1	i	i	PRON
ejpam-2240	154	2	,	,	PUNCT
ejpam-2240	154	3	then	then	ADV
ejpam-2240	154	4	a∩	a∩	PROPN
ejpam-2240	154	5	b	b	PROPN
ejpam-2240	154	6	will	will	AUX
ejpam-2240	154	7	be	be	AUX
ejpam-2240	154	8	β	β	NOUN
ejpam-2240	154	9	-open	-open	NOUN
ejpam-2240	154	10	with	with	ADP
ejpam-2240	154	11	respect	respect	NOUN
ejpam-2240	154	12	to	to	ADP
ejpam-2240	154	13	the	the	DET
ejpam-2240	154	14	ideal	ideal	NOUN
ejpam-2240	154	15	i	i	PRON
ejpam-2240	154	16	.	.	PUNCT
ejpam-2240	155	1	for	for	ADP
ejpam-2240	155	2	those	those	DET
ejpam-2240	155	3	subsets	subset	NOUN
ejpam-2240	155	4	that	that	PRON
ejpam-2240	155	5	are	be	AUX
ejpam-2240	155	6	not	not	PART
ejpam-2240	155	7	members	member	NOUN
ejpam-2240	155	8	of	of	ADP
ejpam-2240	155	9	the	the	DET
ejpam-2240	155	10	ideal	ideal	NOUN
ejpam-2240	155	11	i	i	PRON
ejpam-2240	155	12	,	,	PUNCT
ejpam-2240	155	13	one	one	NUM
ejpam-2240	155	14	rather	rather	ADV
ejpam-2240	155	15	strong	strong	ADJ
ejpam-2240	155	16	condition	condition	NOUN
ejpam-2240	155	17	for	for	ADP
ejpam-2240	155	18	their	their	PRON
ejpam-2240	155	19	intersection	intersection	NOUN
ejpam-2240	155	20	to	to	PART
ejpam-2240	155	21	be	be	AUX
ejpam-2240	155	22	β	β	X
ejpam-2240	155	23	-open	-open	NOUN
ejpam-2240	155	24	with	with	ADP
ejpam-2240	155	25	respect	respect	NOUN
ejpam-2240	155	26	to	to	ADP
ejpam-2240	155	27	the	the	DET
ejpam-2240	155	28	ideal	ideal	NOUN
ejpam-2240	155	29	i	i	PRON
ejpam-2240	155	30	is	be	AUX
ejpam-2240	155	31	given	give	VERB
ejpam-2240	155	32	below	below	ADV
ejpam-2240	155	33	.	.	PUNCT
ejpam-2240	156	1	proposition	proposition	NOUN
ejpam-2240	156	2	5	5	NUM
ejpam-2240	156	3	.	.	PUNCT
ejpam-2240	157	1	let	let	VERB
ejpam-2240	157	2	i	i	PRON
ejpam-2240	157	3	be	be	AUX
ejpam-2240	157	4	an	an	DET
ejpam-2240	157	5	ideal	ideal	NOUN
ejpam-2240	157	6	on	on	ADP
ejpam-2240	157	7	a	a	DET
ejpam-2240	157	8	topological	topological	ADJ
ejpam-2240	157	9	space	space	NOUN
ejpam-2240	157	10	(	(	PUNCT
ejpam-2240	157	11	x	x	X
ejpam-2240	157	12	,	,	PUNCT
ejpam-2240	157	13	τ	τ	PROPN
ejpam-2240	157	14	)	)	PUNCT
ejpam-2240	157	15	,	,	PUNCT
ejpam-2240	157	16	where	where	SCONJ
ejpam-2240	157	17	every	every	DET
ejpam-2240	157	18	non	non	ADJ
ejpam-2240	157	19	-	-	ADJ
ejpam-2240	157	20	empty	empty	ADJ
ejpam-2240	157	21	preopen	preopen	ADJ
ejpam-2240	157	22	subset	subset	NOUN
ejpam-2240	157	23	of	of	ADP
ejpam-2240	157	24	x	x	PUNCT
ejpam-2240	157	25	is	be	AUX
ejpam-2240	157	26	dense	dense	ADJ
ejpam-2240	157	27	and	and	CCONJ
ejpam-2240	157	28	the	the	DET
ejpam-2240	157	29	collection	collection	NOUN
ejpam-2240	157	30	of	of	ADP
ejpam-2240	157	31	preopen	preopen	ADJ
ejpam-2240	157	32	subsets	subset	NOUN
ejpam-2240	157	33	of	of	ADP
ejpam-2240	157	34	x	x	PRON
ejpam-2240	157	35	satisfies	satisfy	VERB
ejpam-2240	157	36	the	the	DET
ejpam-2240	157	37	finite	finite	ADJ
ejpam-2240	157	38	intersection	intersection	NOUN
ejpam-2240	157	39	property	property	NOUN
ejpam-2240	157	40	:	:	PUNCT
ejpam-2240	157	41	(	(	PUNCT
ejpam-2240	157	42	i	i	NOUN
ejpam-2240	157	43	)	)	PUNCT
ejpam-2240	157	44	if	if	SCONJ
ejpam-2240	157	45	a	a	PRON
ejpam-2240	157	46	is	be	AUX
ejpam-2240	157	47	i	i	PRON
ejpam-2240	157	48	−	−	NOUN
ejpam-2240	157	49	β	β	PART
ejpam-2240	157	50	-open	-open	NOUN
ejpam-2240	157	51	and	and	CCONJ
ejpam-2240	157	52	a⊆	a⊆	NOUN
ejpam-2240	157	53	b	b	NOUN
ejpam-2240	157	54	,	,	PUNCT
ejpam-2240	157	55	then	then	ADV
ejpam-2240	157	56	b	b	PROPN
ejpam-2240	157	57	is	be	AUX
ejpam-2240	157	58	i	i	PRON
ejpam-2240	157	59	−	−	PROPN
ejpam-2240	157	60	β	β	X
ejpam-2240	157	61	-open	-open	PROPN
ejpam-2240	157	62	,	,	PUNCT
ejpam-2240	157	63	(	(	PUNCT
ejpam-2240	157	64	ii	ii	NOUN
ejpam-2240	157	65	)	)	PUNCT
ejpam-2240	157	66	if	if	SCONJ
ejpam-2240	157	67	a	a	PRON
ejpam-2240	157	68	is	be	AUX
ejpam-2240	157	69	i	i	PRON
ejpam-2240	157	70	−	−	NOUN
ejpam-2240	157	71	β	β	X
ejpam-2240	157	72	-open	-open	PROPN
ejpam-2240	157	73	,	,	PUNCT
ejpam-2240	157	74	then	then	ADV
ejpam-2240	157	75	so	so	ADV
ejpam-2240	157	76	is	be	AUX
ejpam-2240	157	77	a∪	a∪	ADP
ejpam-2240	157	78	b	b	NOUN
ejpam-2240	157	79	,	,	PUNCT
ejpam-2240	157	80	for	for	ADP
ejpam-2240	157	81	any	any	DET
ejpam-2240	157	82	subset	subset	NOUN
ejpam-2240	157	83	b	b	PROPN
ejpam-2240	157	84	of	of	ADP
ejpam-2240	157	85	x	x	SYM
ejpam-2240	157	86	,	,	PUNCT
ejpam-2240	157	87	(	(	PUNCT
ejpam-2240	157	88	iii	iii	NOUN
ejpam-2240	157	89	)	)	PUNCT
ejpam-2240	157	90	if	if	SCONJ
ejpam-2240	157	91	both	both	PRON
ejpam-2240	157	92	a	a	PRON
ejpam-2240	157	93	and	and	CCONJ
ejpam-2240	157	94	b	b	NOUN
ejpam-2240	157	95	are	be	AUX
ejpam-2240	157	96	i	i	PRON
ejpam-2240	157	97	−	−	NOUN
ejpam-2240	157	98	β	β	X
ejpam-2240	157	99	-open	-open	PROPN
ejpam-2240	157	100	,	,	PUNCT
ejpam-2240	157	101	then	then	ADV
ejpam-2240	157	102	so	so	ADV
ejpam-2240	157	103	is	be	AUX
ejpam-2240	157	104	their	their	PRON
ejpam-2240	157	105	intersection	intersection	NOUN
ejpam-2240	157	106	a∩	a∩	PROPN
ejpam-2240	157	107	b	b	NOUN
ejpam-2240	157	108	,	,	PUNCT
ejpam-2240	157	109	proof	proof	NOUN
ejpam-2240	157	110	.	.	PUNCT
ejpam-2240	158	1	(	(	PUNCT
ejpam-2240	158	2	i	i	NOUN
ejpam-2240	158	3	)	)	PUNCT
ejpam-2240	158	4	suppose	suppose	VERB
ejpam-2240	158	5	that	that	SCONJ
ejpam-2240	158	6	a	a	PRON
ejpam-2240	158	7	is	be	AUX
ejpam-2240	158	8	i	i	PRON
ejpam-2240	158	9	−	−	NOUN
ejpam-2240	158	10	β	β	PUNCT
ejpam-2240	158	11	-open	-open	NOUN
ejpam-2240	158	12	and	and	CCONJ
ejpam-2240	158	13	that	that	PRON
ejpam-2240	158	14	a⊆	a⊆	VERB
ejpam-2240	159	1	b.	b.	PROPN
ejpam-2240	159	2	there	there	PRON
ejpam-2240	159	3	is	be	VERB
ejpam-2240	159	4	a	a	DET
ejpam-2240	159	5	preopen	preopen	ADJ
ejpam-2240	159	6	set	set	VERB
ejpam-2240	159	7	g	g	PROPN
ejpam-2240	159	8	such	such	ADJ
ejpam-2240	159	9	that	that	SCONJ
ejpam-2240	159	10	g	g	PROPN
ejpam-2240	159	11	\	\	PROPN
ejpam-2240	159	12	a	a	DET
ejpam-2240	159	13	∈	∈	PROPN
ejpam-2240	160	1	i	i	PRON
ejpam-2240	160	2	and	and	CCONJ
ejpam-2240	160	3	a	a	DET
ejpam-2240	160	4	\	\	NOUN
ejpam-2240	160	5	cl(g	cl(g	X
ejpam-2240	160	6	)	)	PUNCT
ejpam-2240	160	7	∈	∈	PROPN
ejpam-2240	161	1	i	i	PRON
ejpam-2240	161	2	.	.	PUNCT
ejpam-2240	162	1	notice	notice	VERB
ejpam-2240	162	2	that	that	SCONJ
ejpam-2240	162	3	such	such	DET
ejpam-2240	162	4	a	a	DET
ejpam-2240	162	5	preopen	preopen	ADJ
ejpam-2240	162	6	set	set	NOUN
ejpam-2240	162	7	g	g	NOUN
ejpam-2240	162	8	is	be	AUX
ejpam-2240	162	9	necessarily	necessarily	ADV
ejpam-2240	162	10	non	non	ADJ
ejpam-2240	162	11	-	-	ADJ
ejpam-2240	162	12	empty	empty	ADJ
ejpam-2240	162	13	,	,	PUNCT
ejpam-2240	162	14	since	since	SCONJ
ejpam-2240	162	15	we	we	PRON
ejpam-2240	162	16	are	be	AUX
ejpam-2240	162	17	dealing	deal	VERB
ejpam-2240	162	18	with	with	ADP
ejpam-2240	162	19	those	those	DET
ejpam-2240	162	20	subsets	subset	NOUN
ejpam-2240	162	21	of	of	ADP
ejpam-2240	162	22	x	x	PRON
ejpam-2240	162	23	that	that	PRON
ejpam-2240	162	24	do	do	AUX
ejpam-2240	162	25	not	not	PART
ejpam-2240	162	26	belong	belong	VERB
ejpam-2240	162	27	to	to	ADP
ejpam-2240	162	28	the	the	DET
ejpam-2240	162	29	ideal	ideal	NOUN
ejpam-2240	162	30	i	i	PRON
ejpam-2240	162	31	.	.	PUNCT
ejpam-2240	163	1	since	since	SCONJ
ejpam-2240	163	2	a⊆	a⊆	PROPN
ejpam-2240	163	3	b	b	X
ejpam-2240	163	4	,	,	PUNCT
ejpam-2240	163	5	we	we	PRON
ejpam-2240	163	6	have	have	VERB
ejpam-2240	163	7	that	that	PRON
ejpam-2240	163	8	g	g	PROPN
ejpam-2240	163	9	\	\	PROPN
ejpam-2240	163	10	b	b	PROPN
ejpam-2240	163	11	⊆	⊆	NUM
ejpam-2240	163	12	g	g	NOUN
ejpam-2240	163	13	\	\	PROPN
ejpam-2240	163	14	a∈	a∈	PROPN
ejpam-2240	163	15	i	i	PRON
ejpam-2240	163	16	;	;	PUNCT
ejpam-2240	163	17	moreover	moreover	ADV
ejpam-2240	163	18	,	,	PUNCT
ejpam-2240	163	19	b	b	NOUN
ejpam-2240	163	20	\	\	X
ejpam-2240	163	21	cl(g	cl(g	X
ejpam-2240	163	22	)	)	PUNCT
ejpam-2240	163	23	=	=	SYM
ejpam-2240	164	1	b	b	X
ejpam-2240	164	2	\	\	X
ejpam-2240	164	3	x	x	X
ejpam-2240	164	4	=	=	PUNCT
ejpam-2240	164	5	;	;	PUNCT
ejpam-2240	164	6	∈	∈	PROPN
ejpam-2240	164	7	i	i	PRON
ejpam-2240	164	8	.	.	PUNCT
ejpam-2240	165	1	thus	thus	ADV
ejpam-2240	165	2	,	,	PUNCT
ejpam-2240	165	3	b	b	PROPN
ejpam-2240	165	4	is	be	AUX
ejpam-2240	165	5	i	i	PRON
ejpam-2240	165	6	−	−	PROPN
ejpam-2240	165	7	β	β	X
ejpam-2240	165	8	-open	-open	PROPN
ejpam-2240	165	9	.	.	PUNCT
ejpam-2240	166	1	(	(	PUNCT
ejpam-2240	166	2	ii	ii	NOUN
ejpam-2240	166	3	)	)	PUNCT
ejpam-2240	166	4	follows	follow	VERB
ejpam-2240	166	5	immediately	immediately	ADV
ejpam-2240	166	6	from	from	ADP
ejpam-2240	166	7	(	(	PUNCT
ejpam-2240	166	8	i	i	NOUN
ejpam-2240	166	9	)	)	PUNCT
ejpam-2240	166	10	and	and	CCONJ
ejpam-2240	166	11	the	the	DET
ejpam-2240	166	12	observation	observation	NOUN
ejpam-2240	166	13	since	since	SCONJ
ejpam-2240	166	14	a⊆	a⊆	PROPN
ejpam-2240	166	15	b⇔	b⇔	PROPN
ejpam-2240	166	16	a∪	a∪	ADP
ejpam-2240	166	17	b	b	PROPN
ejpam-2240	166	18	=	=	PROPN
ejpam-2240	166	19	b.	b.	PROPN
ejpam-2240	166	20	(	(	PUNCT
ejpam-2240	166	21	iii	iii	NOUN
ejpam-2240	166	22	)	)	PUNCT
ejpam-2240	166	23	suppose	suppose	VERB
ejpam-2240	166	24	that	that	SCONJ
ejpam-2240	166	25	both	both	PRON
ejpam-2240	166	26	a	a	PRON
ejpam-2240	166	27	and	and	CCONJ
ejpam-2240	166	28	b	b	NOUN
ejpam-2240	166	29	are	be	AUX
ejpam-2240	166	30	i	i	PRON
ejpam-2240	166	31	−	−	NOUN
ejpam-2240	166	32	β	β	X
ejpam-2240	166	33	-open	-open	PROPN
ejpam-2240	166	34	.	.	PUNCT
ejpam-2240	167	1	without	without	ADP
ejpam-2240	167	2	loss	loss	NOUN
ejpam-2240	167	3	of	of	ADP
ejpam-2240	167	4	generality	generality	NOUN
ejpam-2240	167	5	,	,	PUNCT
ejpam-2240	167	6	suppose	suppose	VERB
ejpam-2240	167	7	that	that	SCONJ
ejpam-2240	167	8	a∩	a∩	PROPN
ejpam-2240	167	9	b	b	PROPN
ejpam-2240	167	10	6=	6=	PROPN
ejpam-2240	167	11	;	;	PUNCT
ejpam-2240	167	12	;	;	PUNCT
ejpam-2240	167	13	otherwise	otherwise	ADV
ejpam-2240	167	14	,	,	PUNCT
ejpam-2240	167	15	a∩	a∩	PROPN
ejpam-2240	167	16	b	b	PROPN
ejpam-2240	167	17	will	will	AUX
ejpam-2240	167	18	be	be	AUX
ejpam-2240	167	19	trivially	trivially	ADV
ejpam-2240	167	20	i	i	PRON
ejpam-2240	167	21	−β	−β	PROPN
ejpam-2240	167	22	-open	-open	PROPN
ejpam-2240	167	23	.	.	PUNCT
ejpam-2240	168	1	by	by	ADP
ejpam-2240	168	2	assumption	assumption	NOUN
ejpam-2240	168	3	,	,	PUNCT
ejpam-2240	168	4	there	there	PRON
ejpam-2240	168	5	are	be	VERB
ejpam-2240	168	6	preopen	preopen	ADJ
ejpam-2240	168	7	sets	set	NOUN
ejpam-2240	168	8	g	g	PROPN
ejpam-2240	168	9	,	,	PUNCT
ejpam-2240	168	10	h	h	NOUN
ejpam-2240	168	11	such	such	ADJ
ejpam-2240	168	12	that	that	SCONJ
ejpam-2240	168	13	g	g	PROPN
ejpam-2240	168	14	\	\	PROPN
ejpam-2240	168	15	a	a	DET
ejpam-2240	168	16	,	,	PUNCT
ejpam-2240	168	17	a\	a\	NOUN
ejpam-2240	168	18	cl(g	cl(g	X
ejpam-2240	168	19	)	)	PUNCT
ejpam-2240	168	20	∈	∈	PROPN
ejpam-2240	169	1	i	i	PRON
ejpam-2240	169	2	and	and	CCONJ
ejpam-2240	169	3	h	h	NOUN
ejpam-2240	169	4	\	\	PROPN
ejpam-2240	169	5	b	b	PROPN
ejpam-2240	169	6	,	,	PUNCT
ejpam-2240	169	7	b	b	NOUN
ejpam-2240	169	8	\	\	PROPN
ejpam-2240	169	9	cl(h	cl(h	CCONJ
ejpam-2240	169	10	)	)	PUNCT
ejpam-2240	169	11	∈	∈	PROPN
ejpam-2240	170	1	i	i	PRON
ejpam-2240	170	2	.	.	PUNCT
ejpam-2240	171	1	consider	consider	VERB
ejpam-2240	171	2	the	the	DET
ejpam-2240	171	3	preopen	preopen	NOUN
ejpam-2240	171	4	set	set	VERB
ejpam-2240	171	5	g	g	PROPN
ejpam-2240	171	6	∩	∩	ADJ
ejpam-2240	171	7	h	h	NOUN
ejpam-2240	171	8	,	,	PUNCT
ejpam-2240	171	9	which	which	PRON
ejpam-2240	171	10	is	be	AUX
ejpam-2240	171	11	non	non	ADJ
ejpam-2240	171	12	-	-	ADJ
ejpam-2240	171	13	empty	empty	ADJ
ejpam-2240	171	14	(	(	PUNCT
ejpam-2240	171	15	by	by	ADP
ejpam-2240	171	16	the	the	DET
ejpam-2240	171	17	finite	finite	ADJ
ejpam-2240	171	18	intersection	intersection	NOUN
ejpam-2240	171	19	property	property	NOUN
ejpam-2240	171	20	)	)	PUNCT
ejpam-2240	171	21	.	.	PUNCT
ejpam-2240	172	1	since	since	SCONJ
ejpam-2240	172	2	(	(	PUNCT
ejpam-2240	172	3	g	g	PROPN
ejpam-2240	172	4	∩	∩	ADJ
ejpam-2240	172	5	h	h	NOUN
ejpam-2240	172	6	)	)	PUNCT
ejpam-2240	172	7	\	\	PUNCT
ejpam-2240	172	8	(	(	PUNCT
ejpam-2240	172	9	a∩	a∩	PROPN
ejpam-2240	172	10	b	b	X
ejpam-2240	172	11	)	)	PUNCT
ejpam-2240	172	12	=	=	SYM
ejpam-2240	172	13	(	(	PUNCT
ejpam-2240	172	14	(	(	PUNCT
ejpam-2240	172	15	g	g	PROPN
ejpam-2240	172	16	\	\	PROPN
ejpam-2240	172	17	a)∩	a)∩	X
ejpam-2240	172	18	h)∪	h)∪	PROPN
ejpam-2240	172	19	(	(	PUNCT
ejpam-2240	172	20	g	g	PROPN
ejpam-2240	172	21	∩	∩	NOUN
ejpam-2240	172	22	(	(	PUNCT
ejpam-2240	172	23	h	h	NOUN
ejpam-2240	172	24	\	\	PROPN
ejpam-2240	172	25	b	b	X
ejpam-2240	172	26	)	)	PUNCT
ejpam-2240	172	27	)	)	PUNCT
ejpam-2240	173	1	∈	∈	PROPN
ejpam-2240	173	2	i	i	PRON
ejpam-2240	173	3	a.	a.	NOUN
ejpam-2240	173	4	nasef	nasef	PROPN
ejpam-2240	173	5	,	,	PUNCT
ejpam-2240	173	6	r.	r.	PROPN
ejpam-2240	173	7	mareay	mareay	PROPN
ejpam-2240	173	8	,	,	PUNCT
ejpam-2240	173	9	f.	f.	PROPN
ejpam-2240	173	10	michael	michael	PROPN
ejpam-2240	173	11	/	/	SYM
ejpam-2240	173	12	eur	eur	PROPN
ejpam-2240	173	13	.	.	PUNCT
ejpam-2240	174	1	j.	j.	PROPN
ejpam-2240	174	2	pure	pure	PROPN
ejpam-2240	174	3	appl	appl	PROPN
ejpam-2240	174	4	.	.	PROPN
ejpam-2240	174	5	math	math	PROPN
ejpam-2240	174	6	,	,	PUNCT
ejpam-2240	174	7	8	8	NUM
ejpam-2240	174	8	(	(	PUNCT
ejpam-2240	174	9	2015	2015	NUM
ejpam-2240	174	10	)	)	PUNCT
ejpam-2240	174	11	,	,	PUNCT
ejpam-2240	174	12	389	389	NUM
ejpam-2240	174	13	-	-	SYM
ejpam-2240	174	14	394	394	NUM
ejpam-2240	174	15	393	393	NUM
ejpam-2240	174	16	and	and	CCONJ
ejpam-2240	174	17	(	(	PUNCT
ejpam-2240	174	18	a∩	a∩	PROPN
ejpam-2240	174	19	b	b	X
ejpam-2240	174	20	)	)	PUNCT
ejpam-2240	174	21	\	\	NOUN
ejpam-2240	174	22	cl(g	cl(g	NOUN
ejpam-2240	174	23	∩	∩	ADJ
ejpam-2240	174	24	h	h	NOUN
ejpam-2240	174	25	)	)	PUNCT
ejpam-2240	174	26	=	=	SYM
ejpam-2240	174	27	(	(	PUNCT
ejpam-2240	174	28	a∩	a∩	PROPN
ejpam-2240	174	29	b	b	X
ejpam-2240	174	30	)	)	PUNCT
ejpam-2240	174	31	\	\	NOUN
ejpam-2240	175	1	x	x	X
ejpam-2240	175	2	=	=	PUNCT
ejpam-2240	175	3	;	;	PUNCT
ejpam-2240	175	4	∈	∈	PROPN
ejpam-2240	176	1	i	i	PRON
ejpam-2240	176	2	,	,	PUNCT
ejpam-2240	176	3	it	it	PRON
ejpam-2240	176	4	follows	follow	VERB
ejpam-2240	176	5	that	that	SCONJ
ejpam-2240	176	6	a∩	a∩	PROPN
ejpam-2240	176	7	b	b	PROPN
ejpam-2240	176	8	is	be	AUX
ejpam-2240	176	9	i	i	PRON
ejpam-2240	176	10	−	−	PROPN
ejpam-2240	176	11	β	β	X
ejpam-2240	176	12	-open	-open	PROPN
ejpam-2240	176	13	.	.	PUNCT
ejpam-2240	177	1	remark	remark	NOUN
ejpam-2240	177	2	2	2	NUM
ejpam-2240	177	3	.	.	PUNCT
ejpam-2240	178	1	in	in	ADP
ejpam-2240	178	2	example	example	NOUN
ejpam-2240	178	3	2	2	NUM
ejpam-2240	178	4	,	,	PUNCT
ejpam-2240	178	5	we	we	PRON
ejpam-2240	178	6	saw	see	VERB
ejpam-2240	178	7	that	that	SCONJ
ejpam-2240	178	8	the	the	DET
ejpam-2240	178	9	singleton	singleton	PROPN
ejpam-2240	178	10	subset	subset	NOUN
ejpam-2240	178	11	{	{	PUNCT
ejpam-2240	178	12	b	b	NOUN
ejpam-2240	178	13	}	}	PUNCT
ejpam-2240	178	14	was	be	AUX
ejpam-2240	178	15	not	not	PART
ejpam-2240	178	16	β	β	PART
ejpam-2240	178	17	-open	-open	NOUN
ejpam-2240	178	18	with	with	ADP
ejpam-2240	178	19	respect	respect	NOUN
ejpam-2240	178	20	to	to	ADP
ejpam-2240	178	21	the	the	DET
ejpam-2240	178	22	minimal	minimal	ADJ
ejpam-2240	178	23	ideal	ideal	NOUN
ejpam-2240	179	1	i	i	PRON
ejpam-2240	179	2	=	=	PUNCT
ejpam-2240	179	3	{	{	PUNCT
ejpam-2240	179	4	;	;	PUNCT
ejpam-2240	179	5	}	}	PUNCT
ejpam-2240	179	6	.	.	PUNCT
ejpam-2240	180	1	notice	notice	VERB
ejpam-2240	180	2	that	that	SCONJ
ejpam-2240	180	3	the	the	DET
ejpam-2240	180	4	set	set	NOUN
ejpam-2240	180	5	{	{	PUNCT
ejpam-2240	180	6	a	a	PRON
ejpam-2240	180	7	,	,	PUNCT
ejpam-2240	180	8	b	b	NOUN
ejpam-2240	180	9	}	}	PUNCT
ejpam-2240	180	10	=	=	SYM
ejpam-2240	180	11	{	{	PUNCT
ejpam-2240	180	12	a	a	DET
ejpam-2240	180	13	}	}	PUNCT
ejpam-2240	180	14	∪	∪	NOUN
ejpam-2240	180	15	{	{	PUNCT
ejpam-2240	180	16	b	b	NOUN
ejpam-2240	180	17	}	}	PUNCT
ejpam-2240	180	18	is	be	AUX
ejpam-2240	180	19	β	β	PART
ejpam-2240	180	20	-open	-open	NOUN
ejpam-2240	180	21	with	with	ADP
ejpam-2240	180	22	respect	respect	NOUN
ejpam-2240	180	23	to	to	ADP
ejpam-2240	180	24	i	i	PRON
ejpam-2240	180	25	=	=	PUNCT
ejpam-2240	180	26	{	{	PUNCT
ejpam-2240	180	27	;	;	PUNCT
ejpam-2240	180	28	}	}	PUNCT
ejpam-2240	180	29	,	,	PUNCT
ejpam-2240	180	30	simply	simply	ADV
ejpam-2240	180	31	because	because	SCONJ
ejpam-2240	180	32	the	the	DET
ejpam-2240	180	33	non	non	ADJ
ejpam-2240	180	34	-	-	ADJ
ejpam-2240	180	35	empty	empty	ADJ
ejpam-2240	180	36	preopen	preopen	ADJ
ejpam-2240	180	37	singleton	singleton	PROPN
ejpam-2240	180	38	subset	subset	NOUN
ejpam-2240	180	39	{	{	PUNCT
ejpam-2240	180	40	a	a	PRON
ejpam-2240	180	41	}	}	PUNCT
ejpam-2240	180	42	is	be	AUX
ejpam-2240	180	43	dense	dense	ADJ
ejpam-2240	180	44	in	in	ADP
ejpam-2240	180	45	x	x	SYM
ejpam-2240	180	46	,	,	PUNCT
ejpam-2240	180	47	this	this	PRON
ejpam-2240	180	48	is	be	AUX
ejpam-2240	180	49	an	an	DET
ejpam-2240	180	50	instance	instance	NOUN
ejpam-2240	180	51	of	of	ADP
ejpam-2240	180	52	proposition	proposition	NOUN
ejpam-2240	180	53	5	5	NUM
ejpam-2240	180	54	,	,	PUNCT
ejpam-2240	180	55	(	(	PUNCT
ejpam-2240	180	56	ii	ii	NOUN
ejpam-2240	180	57	)	)	PUNCT
ejpam-2240	180	58	above	above	ADV
ejpam-2240	180	59	.	.	PUNCT
ejpam-2240	181	1	proposition	proposition	NOUN
ejpam-2240	181	2	6	6	NUM
ejpam-2240	181	3	.	.	PUNCT
ejpam-2240	182	1	under	under	ADP
ejpam-2240	182	2	the	the	DET
ejpam-2240	182	3	conditions	condition	NOUN
ejpam-2240	182	4	of	of	ADP
ejpam-2240	182	5	proposition	proposition	NOUN
ejpam-2240	182	6	5	5	NUM
ejpam-2240	182	7	,	,	PUNCT
ejpam-2240	182	8	we	we	PRON
ejpam-2240	182	9	have	have	VERB
ejpam-2240	182	10	that	that	SCONJ
ejpam-2240	182	11	a	a	PRON
ejpam-2240	182	12	is	be	AUX
ejpam-2240	182	13	i	i	PRON
ejpam-2240	182	14	−β	−β	ADV
ejpam-2240	182	15	-open	-open	ADJ
ejpam-2240	182	16	if	if	SCONJ
ejpam-2240	182	17	and	and	CCONJ
ejpam-2240	182	18	only	only	ADV
ejpam-2240	182	19	if	if	SCONJ
ejpam-2240	182	20	cl(a	cl(a	NUM
ejpam-2240	182	21	)	)	PUNCT
ejpam-2240	182	22	is	be	AUX
ejpam-2240	182	23	i	i	PRON
ejpam-2240	182	24	−	−	NOUN
ejpam-2240	182	25	β	β	X
ejpam-2240	182	26	-open	-open	NOUN
ejpam-2240	182	27	.	.	PUNCT
ejpam-2240	183	1	proof	proof	NOUN
ejpam-2240	183	2	.	.	PUNCT
ejpam-2240	184	1	if	if	SCONJ
ejpam-2240	184	2	a	a	PRON
ejpam-2240	184	3	is	be	AUX
ejpam-2240	184	4	i	i	PRON
ejpam-2240	184	5	−	−	PROPN
ejpam-2240	184	6	β	β	X
ejpam-2240	184	7	-open	-open	PROPN
ejpam-2240	184	8	,	,	PUNCT
ejpam-2240	184	9	then	then	ADV
ejpam-2240	184	10	,	,	PUNCT
ejpam-2240	184	11	because	because	SCONJ
ejpam-2240	184	12	a	a	DET
ejpam-2240	184	13	⊆	⊆	NUM
ejpam-2240	184	14	cl(a),so	cl(a),so	NOUN
ejpam-2240	184	15	is	be	AUX
ejpam-2240	184	16	cl(a	cl(a	NUM
ejpam-2240	184	17	)	)	PUNCT
ejpam-2240	184	18	,	,	PUNCT
ejpam-2240	184	19	by	by	ADP
ejpam-2240	184	20	proposition	proposition	NOUN
ejpam-2240	184	21	5	5	NUM
ejpam-2240	184	22	,	,	PUNCT
ejpam-2240	184	23	(	(	PUNCT
ejpam-2240	184	24	ii	ii	NOUN
ejpam-2240	184	25	)	)	PUNCT
ejpam-2240	184	26	.	.	PUNCT
ejpam-2240	185	1	conversely	conversely	ADV
ejpam-2240	185	2	,	,	PUNCT
ejpam-2240	185	3	suppose	suppose	VERB
ejpam-2240	185	4	that	that	SCONJ
ejpam-2240	185	5	cl(a	cl(a	X
ejpam-2240	185	6	)	)	PUNCT
ejpam-2240	185	7	is	be	AUX
ejpam-2240	185	8	i	i	PRON
ejpam-2240	185	9	−β	−β	PROPN
ejpam-2240	185	10	-open	-open	PROPN
ejpam-2240	185	11	.	.	PUNCT
ejpam-2240	186	1	then	then	ADV
ejpam-2240	186	2	there	there	PRON
ejpam-2240	186	3	is	be	VERB
ejpam-2240	186	4	a	a	DET
ejpam-2240	186	5	preopen	preopen	ADJ
ejpam-2240	186	6	set	set	VERB
ejpam-2240	186	7	g	g	PROPN
ejpam-2240	186	8	such	such	ADJ
ejpam-2240	186	9	that	that	SCONJ
ejpam-2240	186	10	g	g	PROPN
ejpam-2240	186	11	\	\	PROPN
ejpam-2240	186	12	cl(a	cl(a	X
ejpam-2240	186	13	)	)	PUNCT
ejpam-2240	186	14	∈	∈	PROPN
ejpam-2240	186	15	i	i	PRON
ejpam-2240	186	16	and	and	CCONJ
ejpam-2240	186	17	cl(a	cl(a	NUM
ejpam-2240	186	18	)	)	PUNCT
ejpam-2240	186	19	\	\	NOUN
ejpam-2240	186	20	cl(g	cl(g	X
ejpam-2240	186	21	)	)	PUNCT
ejpam-2240	186	22	∈	∈	PROPN
ejpam-2240	187	1	i	i	PRON
ejpam-2240	187	2	.	.	PUNCT
ejpam-2240	188	1	notice	notice	VERB
ejpam-2240	188	2	that	that	SCONJ
ejpam-2240	188	3	g	g	PROPN
ejpam-2240	188	4	is	be	AUX
ejpam-2240	188	5	necessarily	necessarily	ADV
ejpam-2240	188	6	non	non	ADJ
ejpam-2240	188	7	-	-	ADJ
ejpam-2240	188	8	empty	empty	ADJ
ejpam-2240	188	9	,	,	PUNCT
ejpam-2240	188	10	otherwise	otherwise	ADV
ejpam-2240	188	11	,	,	PUNCT
ejpam-2240	188	12	we	we	PRON
ejpam-2240	188	13	would	would	AUX
ejpam-2240	188	14	have	have	VERB
ejpam-2240	188	15	cl(g	cl(g	NOUN
ejpam-2240	188	16	)	)	PUNCT
ejpam-2240	188	17	=	=	SYM
ejpam-2240	188	18	;	;	PUNCT
ejpam-2240	188	19	,	,	PUNCT
ejpam-2240	188	20	which	which	PRON
ejpam-2240	188	21	forces	force	VERB
ejpam-2240	188	22	a	a	DET
ejpam-2240	188	23	∈	∈	NOUN
ejpam-2240	188	24	i	i	PRON
ejpam-2240	188	25	,	,	PUNCT
ejpam-2240	188	26	which	which	PRON
ejpam-2240	188	27	we	we	PRON
ejpam-2240	188	28	do	do	AUX
ejpam-2240	188	29	not	not	PART
ejpam-2240	188	30	want	want	VERB
ejpam-2240	188	31	(	(	PUNCT
ejpam-2240	188	32	as	as	SCONJ
ejpam-2240	188	33	we	we	PRON
ejpam-2240	188	34	are	be	AUX
ejpam-2240	188	35	dealing	deal	VERB
ejpam-2240	188	36	with	with	ADP
ejpam-2240	188	37	those	those	DET
ejpam-2240	188	38	subsets	subset	NOUN
ejpam-2240	188	39	that	that	PRON
ejpam-2240	188	40	do	do	AUX
ejpam-2240	188	41	not	not	PART
ejpam-2240	188	42	belong	belong	VERB
ejpam-2240	188	43	to	to	ADP
ejpam-2240	188	44	the	the	DET
ejpam-2240	188	45	ideal	ideal	NOUN
ejpam-2240	188	46	i	i	PROPN
ejpam-2240	188	47	)	)	PUNCT
ejpam-2240	188	48	.	.	PUNCT
ejpam-2240	189	1	to	to	PART
ejpam-2240	189	2	show	show	VERB
ejpam-2240	189	3	that	that	SCONJ
ejpam-2240	189	4	a	a	PRON
ejpam-2240	189	5	is	be	AUX
ejpam-2240	189	6	i	i	PRON
ejpam-2240	189	7	−	−	NOUN
ejpam-2240	189	8	β	β	X
ejpam-2240	189	9	-open	-open	PROPN
ejpam-2240	189	10	,	,	PUNCT
ejpam-2240	189	11	consider	consider	VERB
ejpam-2240	189	12	the	the	DET
ejpam-2240	189	13	preopen	preopen	ADJ
ejpam-2240	189	14	set	set	NOUN
ejpam-2240	189	15	h	h	NOUN
ejpam-2240	189	16	=	=	SYM
ejpam-2240	189	17	g	g	PROPN
ejpam-2240	189	18	\	\	X
ejpam-2240	189	19	cl(a	cl(a	X
ejpam-2240	189	20	)	)	PUNCT
ejpam-2240	189	21	=	=	SYM
ejpam-2240	189	22	g	g	PROPN
ejpam-2240	189	23	∩	∩	NOUN
ejpam-2240	189	24	(	(	PUNCT
ejpam-2240	189	25	cl(a))c	cl(a))c	PROPN
ejpam-2240	189	26	∈	∈	PROPN
ejpam-2240	190	1	i	i	PRON
ejpam-2240	190	2	,	,	PUNCT
ejpam-2240	190	3	by	by	ADP
ejpam-2240	190	4	assumption	assumption	NOUN
ejpam-2240	190	5	.	.	PUNCT
ejpam-2240	191	1	we	we	PRON
ejpam-2240	191	2	have	have	VERB
ejpam-2240	191	3	that	that	DET
ejpam-2240	191	4	h	h	NOUN
ejpam-2240	191	5	\	\	PROPN
ejpam-2240	191	6	a=	a=	PROPN
ejpam-2240	191	7	g	g	PROPN
ejpam-2240	191	8	∩	∩	NOUN
ejpam-2240	191	9	(	(	PUNCT
ejpam-2240	191	10	cl(a))c	cl(a))c	PROPN
ejpam-2240	191	11	∩	∩	PROPN
ejpam-2240	191	12	ac	ac	PROPN
ejpam-2240	191	13	∈	∈	PROPN
ejpam-2240	192	1	i	i	PRON
ejpam-2240	192	2	,	,	PUNCT
ejpam-2240	192	3	because	because	SCONJ
ejpam-2240	192	4	of	of	ADP
ejpam-2240	192	5	the	the	DET
ejpam-2240	192	6	heredity	heredity	NOUN
ejpam-2240	192	7	property	property	NOUN
ejpam-2240	192	8	,	,	PUNCT
ejpam-2240	192	9	moreover	moreover	ADV
ejpam-2240	192	10	,	,	PUNCT
ejpam-2240	192	11	a	a	DET
ejpam-2240	192	12	\	\	NOUN
ejpam-2240	192	13	cl(h	cl(h	NUM
ejpam-2240	192	14	)	)	PUNCT
ejpam-2240	192	15	=	=	PUNCT
ejpam-2240	192	16	a	a	DET
ejpam-2240	192	17	\	\	NOUN
ejpam-2240	192	18	cl(g	cl(g	NOUN
ejpam-2240	192	19	∩	∩	NOUN
ejpam-2240	192	20	(	(	PUNCT
ejpam-2240	192	21	cl(a)c	cl(a)c	NOUN
ejpam-2240	192	22	)	)	PUNCT
ejpam-2240	192	23	)	)	PUNCT
ejpam-2240	192	24	=	=	PUNCT
ejpam-2240	193	1	a	a	DET
ejpam-2240	193	2	\	\	NOUN
ejpam-2240	193	3	x	x	X
ejpam-2240	193	4	=	=	PUNCT
ejpam-2240	193	5	;	;	PUNCT
ejpam-2240	193	6	∈	∈	PROPN
ejpam-2240	193	7	i	i	PRON
ejpam-2240	193	8	.	.	PUNCT
ejpam-2240	194	1	this	this	PRON
ejpam-2240	194	2	shows	show	VERB
ejpam-2240	194	3	that	that	SCONJ
ejpam-2240	194	4	a	a	PRON
ejpam-2240	194	5	is	be	AUX
ejpam-2240	194	6	i	i	PRON
ejpam-2240	194	7	−	−	PROPN
ejpam-2240	194	8	β	β	X
ejpam-2240	194	9	-open	-open	PROPN
ejpam-2240	194	10	.	.	PUNCT
ejpam-2240	195	1	theorem	theorem	NOUN
ejpam-2240	195	2	2	2	NUM
ejpam-2240	195	3	.	.	PUNCT
ejpam-2240	196	1	the	the	DET
ejpam-2240	196	2	following	follow	VERB
ejpam-2240	196	3	are	be	AUX
ejpam-2240	196	4	equivalent	equivalent	ADJ
ejpam-2240	196	5	for	for	ADP
ejpam-2240	196	6	a	a	DET
ejpam-2240	196	7	subset	subset	NOUN
ejpam-2240	196	8	a	a	PRON
ejpam-2240	196	9	of	of	ADP
ejpam-2240	196	10	an	an	DET
ejpam-2240	196	11	ideal	ideal	ADJ
ejpam-2240	196	12	topological	topological	ADJ
ejpam-2240	196	13	space	space	NOUN
ejpam-2240	196	14	(	(	PUNCT
ejpam-2240	196	15	x	x	X
ejpam-2240	196	16	,	,	PUNCT
ejpam-2240	196	17	τ	τ	X
ejpam-2240	196	18	):	):	PUNCT
ejpam-2240	196	19	(	(	PUNCT
ejpam-2240	196	20	i	i	NOUN
ejpam-2240	196	21	)	)	PUNCT
ejpam-2240	196	22	x	x	SYM
ejpam-2240	196	23	\	\	PROPN
ejpam-2240	197	1	a	a	PRON
ejpam-2240	197	2	is	be	AUX
ejpam-2240	197	3	i	i	PRON
ejpam-2240	197	4	−	−	PROPN
ejpam-2240	197	5	β	β	X
ejpam-2240	197	6	-open	-open	PROPN
ejpam-2240	197	7	,	,	PUNCT
ejpam-2240	197	8	(	(	PUNCT
ejpam-2240	197	9	ii	ii	NOUN
ejpam-2240	197	10	)	)	PUNCT
ejpam-2240	197	11	there	there	PRON
ejpam-2240	197	12	exists	exist	VERB
ejpam-2240	197	13	a	a	DET
ejpam-2240	197	14	preclosed	preclose	VERB
ejpam-2240	197	15	set	set	NOUN
ejpam-2240	197	16	f	f	PROPN
ejpam-2240	197	17	such	such	ADJ
ejpam-2240	197	18	that	that	PRON
ejpam-2240	197	19	int(f	int(f	PROPN
ejpam-2240	197	20	)	)	PUNCT
ejpam-2240	197	21	\	\	PROPN
ejpam-2240	198	1	a∈	a∈	PROPN
ejpam-2240	198	2	i	i	PROPN
ejpam-2240	198	3	and	and	CCONJ
ejpam-2240	198	4	a\	a\	PROPN
ejpam-2240	198	5	f	f	PROPN
ejpam-2240	198	6	∈	∈	PROPN
ejpam-2240	199	1	i	i	PRON
ejpam-2240	199	2	.	.	PUNCT
ejpam-2240	200	1	proof	proof	NOUN
ejpam-2240	200	2	.	.	PUNCT
ejpam-2240	201	1	first	first	ADV
ejpam-2240	201	2	suppose	suppose	VERB
ejpam-2240	201	3	that	that	SCONJ
ejpam-2240	201	4	x	x	X
ejpam-2240	201	5	\	\	PROPN
ejpam-2240	201	6	a	a	PRON
ejpam-2240	201	7	is	be	AUX
ejpam-2240	201	8	i	i	PRON
ejpam-2240	201	9	−	−	PROPN
ejpam-2240	201	10	β	β	X
ejpam-2240	201	11	-open	-open	PROPN
ejpam-2240	201	12	.	.	PUNCT
ejpam-2240	202	1	then	then	ADV
ejpam-2240	202	2	there	there	PRON
ejpam-2240	202	3	exists	exist	VERB
ejpam-2240	202	4	a	a	DET
ejpam-2240	202	5	preopen	preopen	NOUN
ejpam-2240	202	6	set	set	VERB
ejpam-2240	202	7	g	g	PROPN
ejpam-2240	202	8	such	such	ADJ
ejpam-2240	202	9	that	that	SCONJ
ejpam-2240	202	10	g	g	NOUN
ejpam-2240	202	11	\	\	PUNCT
ejpam-2240	203	1	(	(	PUNCT
ejpam-2240	203	2	x	x	SYM
ejpam-2240	203	3	\	\	PROPN
ejpam-2240	203	4	a	a	DET
ejpam-2240	203	5	)	)	PUNCT
ejpam-2240	203	6	∈	∈	PROPN
ejpam-2240	203	7	i	i	PRON
ejpam-2240	203	8	and	and	CCONJ
ejpam-2240	203	9	(	(	PUNCT
ejpam-2240	203	10	x	x	SYM
ejpam-2240	203	11	\	\	PROPN
ejpam-2240	203	12	a	a	PRON
ejpam-2240	203	13	)	)	PUNCT
ejpam-2240	203	14	\	\	NOUN
ejpam-2240	203	15	cl(g	cl(g	X
ejpam-2240	203	16	)	)	PUNCT
ejpam-2240	203	17	∈	∈	PROPN
ejpam-2240	204	1	i	i	PRON
ejpam-2240	204	2	.	.	PUNCT
ejpam-2240	205	1	since	since	SCONJ
ejpam-2240	205	2	g	g	PROPN
ejpam-2240	205	3	\	\	PUNCT
ejpam-2240	205	4	(	(	PUNCT
ejpam-2240	205	5	x	x	SYM
ejpam-2240	205	6	\	\	PROPN
ejpam-2240	205	7	a	a	PRON
ejpam-2240	205	8	)	)	PUNCT
ejpam-2240	205	9	=	=	SYM
ejpam-2240	205	10	a\	a\	NOUN
ejpam-2240	205	11	(	(	PUNCT
ejpam-2240	205	12	x	x	SYM
ejpam-2240	205	13	\	\	PROPN
ejpam-2240	205	14	g	g	NOUN
ejpam-2240	205	15	)	)	PUNCT
ejpam-2240	205	16	and	and	CCONJ
ejpam-2240	205	17	(	(	PUNCT
ejpam-2240	205	18	x	x	SYM
ejpam-2240	205	19	\	\	PROPN
ejpam-2240	205	20	a	a	PRON
ejpam-2240	205	21	)	)	PUNCT
ejpam-2240	205	22	\	\	NOUN
ejpam-2240	205	23	cl(g	cl(g	X
ejpam-2240	205	24	)	)	PUNCT
ejpam-2240	205	25	=	=	PUNCT
ejpam-2240	205	26	int(x	int(x	PROPN
ejpam-2240	205	27	\	\	PROPN
ejpam-2240	205	28	g	g	NOUN
ejpam-2240	205	29	)	)	PUNCT
ejpam-2240	205	30	\	\	PROPN
ejpam-2240	206	1	a	a	PRON
ejpam-2240	206	2	,	,	PUNCT
ejpam-2240	206	3	we	we	PRON
ejpam-2240	206	4	have	have	VERB
ejpam-2240	206	5	(	(	PUNCT
ejpam-2240	206	6	ii	ii	NOUN
ejpam-2240	206	7	)	)	PUNCT
ejpam-2240	206	8	by	by	ADP
ejpam-2240	206	9	choosing	choose	VERB
ejpam-2240	206	10	the	the	DET
ejpam-2240	206	11	closed	closed	ADJ
ejpam-2240	206	12	set	set	NOUN
ejpam-2240	206	13	x	x	NOUN
ejpam-2240	206	14	\	\	PROPN
ejpam-2240	206	15	g	g	PROPN
ejpam-2240	206	16	as	as	ADP
ejpam-2240	206	17	f	f	PROPN
ejpam-2240	206	18	.	.	PUNCT
ejpam-2240	207	1	conversely	conversely	ADV
ejpam-2240	207	2	,	,	PUNCT
ejpam-2240	207	3	if	if	SCONJ
ejpam-2240	207	4	we	we	PRON
ejpam-2240	207	5	suppose	suppose	VERB
ejpam-2240	207	6	that	that	SCONJ
ejpam-2240	207	7	(	(	PUNCT
ejpam-2240	207	8	ii	ii	NOUN
ejpam-2240	207	9	)	)	PUNCT
ejpam-2240	207	10	holds	hold	VERB
ejpam-2240	207	11	,	,	PUNCT
ejpam-2240	207	12	then	then	ADV
ejpam-2240	207	13	the	the	DET
ejpam-2240	207	14	choice	choice	NOUN
ejpam-2240	207	15	of	of	ADP
ejpam-2240	207	16	the	the	DET
ejpam-2240	207	17	preopen	preopen	NOUN
ejpam-2240	207	18	set	set	VERB
ejpam-2240	207	19	g	g	PROPN
ejpam-2240	207	20	=	=	PUNCT
ejpam-2240	207	21	x	x	SYM
ejpam-2240	207	22	\	\	PROPN
ejpam-2240	207	23	f	f	PROPN
ejpam-2240	207	24	shows	show	VERB
ejpam-2240	207	25	that	that	SCONJ
ejpam-2240	207	26	x	x	X
ejpam-2240	207	27	\	\	PROPN
ejpam-2240	208	1	a	a	PRON
ejpam-2240	208	2	is	be	AUX
ejpam-2240	208	3	i	i	PRON
ejpam-2240	208	4	−	−	PROPN
ejpam-2240	208	5	β	β	X
ejpam-2240	208	6	-open	-open	PROPN
ejpam-2240	208	7	.	.	PUNCT
ejpam-2240	209	1	definition	definition	NOUN
ejpam-2240	209	2	6	6	NUM
ejpam-2240	209	3	.	.	PUNCT
ejpam-2240	210	1	a	a	DET
ejpam-2240	210	2	subset	subset	NOUN
ejpam-2240	210	3	a	a	PRON
ejpam-2240	210	4	of	of	ADP
ejpam-2240	210	5	an	an	DET
ejpam-2240	210	6	ideal	ideal	ADJ
ejpam-2240	210	7	topological	topological	ADJ
ejpam-2240	210	8	space	space	NOUN
ejpam-2240	210	9	(	(	PUNCT
ejpam-2240	210	10	x	x	X
ejpam-2240	210	11	,	,	PUNCT
ejpam-2240	210	12	τ	τ	X
ejpam-2240	210	13	)	)	PUNCT
ejpam-2240	210	14	is	be	AUX
ejpam-2240	210	15	said	say	VERB
ejpam-2240	210	16	to	to	PART
ejpam-2240	210	17	be	be	AUX
ejpam-2240	210	18	β	β	AUX
ejpam-2240	210	19	-closed	-close	VERB
ejpam-2240	210	20	with	with	ADP
ejpam-2240	210	21	respect	respect	NOUN
ejpam-2240	210	22	to	to	ADP
ejpam-2240	210	23	i	i	PRON
ejpam-2240	210	24	(	(	PUNCT
ejpam-2240	210	25	written	write	VERB
ejpam-2240	210	26	as	as	ADP
ejpam-2240	210	27	i	i	PRON
ejpam-2240	210	28	−	−	PROPN
ejpam-2240	210	29	β	β	X
ejpam-2240	210	30	-closed	-closed	ADJ
ejpam-2240	210	31	)	)	PUNCT
ejpam-2240	210	32	if	if	SCONJ
ejpam-2240	211	1	and	and	CCONJ
ejpam-2240	211	2	only	only	ADV
ejpam-2240	211	3	if	if	SCONJ
ejpam-2240	211	4	x	x	SYM
ejpam-2240	211	5	\	\	PROPN
ejpam-2240	211	6	a	a	PRON
ejpam-2240	211	7	is	be	AUX
ejpam-2240	211	8	i	i	PRON
ejpam-2240	211	9	−	−	PROPN
ejpam-2240	211	10	β	β	X
ejpam-2240	211	11	-open	-open	PROPN
ejpam-2240	211	12	.	.	PUNCT
ejpam-2240	212	1	proposition	proposition	NOUN
ejpam-2240	212	2	7	7	NUM
ejpam-2240	212	3	.	.	PUNCT
ejpam-2240	213	1	if	if	SCONJ
ejpam-2240	213	2	both	both	PRON
ejpam-2240	213	3	a	a	PRON
ejpam-2240	213	4	and	and	CCONJ
ejpam-2240	213	5	b	b	NOUN
ejpam-2240	213	6	are	be	AUX
ejpam-2240	213	7	i	i	PRON
ejpam-2240	213	8	−	−	PROPN
ejpam-2240	213	9	β	β	X
ejpam-2240	213	10	-closed	-closed	PROPN
ejpam-2240	213	11	,	,	PUNCT
ejpam-2240	213	12	then	then	ADV
ejpam-2240	213	13	so	so	ADV
ejpam-2240	213	14	is	be	AUX
ejpam-2240	213	15	their	their	PRON
ejpam-2240	213	16	intersection	intersection	NOUN
ejpam-2240	213	17	a∩	a∩	PROPN
ejpam-2240	213	18	b.	b.	PROPN
ejpam-2240	213	19	proof	proof	PROPN
ejpam-2240	213	20	.	.	PUNCT
ejpam-2240	214	1	let	let	VERB
ejpam-2240	214	2	the	the	DET
ejpam-2240	214	3	given	give	VERB
ejpam-2240	214	4	conditions	condition	NOUN
ejpam-2240	214	5	hold	hold	VERB
ejpam-2240	214	6	.	.	PUNCT
ejpam-2240	215	1	there	there	PRON
ejpam-2240	215	2	are	be	VERB
ejpam-2240	215	3	preclosed	preclose	VERB
ejpam-2240	215	4	sets	set	NOUN
ejpam-2240	215	5	f1	f1	NOUN
ejpam-2240	215	6	,	,	PUNCT
ejpam-2240	215	7	f2	f2	NOUN
ejpam-2240	215	8	such	such	ADJ
ejpam-2240	215	9	that	that	SCONJ
ejpam-2240	215	10	int(f1	int(f1	NOUN
ejpam-2240	215	11	)	)	PUNCT
ejpam-2240	215	12	\	\	PROPN
ejpam-2240	216	1	a	a	DET
ejpam-2240	216	2	,	,	PUNCT
ejpam-2240	216	3	(	(	PUNCT
ejpam-2240	216	4	a\	a\	ADJ
ejpam-2240	216	5	f1	f1	NOUN
ejpam-2240	216	6	)	)	PUNCT
ejpam-2240	216	7	∈	∈	PROPN
ejpam-2240	216	8	i	i	PROPN
ejpam-2240	216	9	and	and	CCONJ
ejpam-2240	216	10	int(f2	int(f2	PROPN
ejpam-2240	216	11	)	)	PUNCT
ejpam-2240	216	12	\	\	PROPN
ejpam-2240	216	13	b	b	PROPN
ejpam-2240	216	14	,	,	PUNCT
ejpam-2240	216	15	(	(	PUNCT
ejpam-2240	216	16	b	b	NOUN
ejpam-2240	216	17	\	\	PROPN
ejpam-2240	216	18	f2	f2	PROPN
ejpam-2240	216	19	)	)	PUNCT
ejpam-2240	216	20	∈	∈	PROPN
ejpam-2240	216	21	i	i	PRON
ejpam-2240	216	22	.	.	PUNCT
ejpam-2240	217	1	with	with	ADP
ejpam-2240	217	2	f	f	PROPN
ejpam-2240	217	3	=	=	SYM
ejpam-2240	217	4	f1	f1	PROPN
ejpam-2240	217	5	∩	∩	NOUN
ejpam-2240	217	6	f2	f2	PROPN
ejpam-2240	217	7	,	,	PUNCT
ejpam-2240	217	8	we	we	PRON
ejpam-2240	217	9	have	have	VERB
ejpam-2240	217	10	that	that	DET
ejpam-2240	217	11	int(f1	int(f1	PROPN
ejpam-2240	217	12	∩	∩	NOUN
ejpam-2240	217	13	f2	f2	PROPN
ejpam-2240	217	14	)	)	PUNCT
ejpam-2240	217	15	\	\	PUNCT
ejpam-2240	218	1	(	(	PUNCT
ejpam-2240	218	2	a∩	a∩	PROPN
ejpam-2240	218	3	b	b	X
ejpam-2240	218	4	)	)	PUNCT
ejpam-2240	218	5	=	=	SYM
ejpam-2240	218	6	(	(	PUNCT
ejpam-2240	218	7	(	(	PUNCT
ejpam-2240	218	8	int(f1	int(f1	NOUN
ejpam-2240	218	9	)	)	PUNCT
ejpam-2240	218	10	\	\	NOUN
ejpam-2240	218	11	a)∩	a)∩	X
ejpam-2240	218	12	int(f2))∪	int(f2))∪	ADJ
ejpam-2240	218	13	(	(	PUNCT
ejpam-2240	218	14	(	(	PUNCT
ejpam-2240	218	15	int(f1))∩	int(f1))∩	PROPN
ejpam-2240	218	16	(	(	PUNCT
ejpam-2240	218	17	int(f2	int(f2	PROPN
ejpam-2240	218	18	)	)	PUNCT
ejpam-2240	218	19	\	\	PROPN
ejpam-2240	218	20	b	b	X
ejpam-2240	218	21	)	)	PUNCT
ejpam-2240	218	22	)	)	PUNCT
ejpam-2240	219	1	∈	∈	PROPN
ejpam-2240	220	1	i	i	PRON
ejpam-2240	220	2	,	,	PUNCT
ejpam-2240	220	3	and	and	CCONJ
ejpam-2240	220	4	(	(	PUNCT
ejpam-2240	220	5	a∩	a∩	PROPN
ejpam-2240	220	6	b	b	X
ejpam-2240	220	7	)	)	PUNCT
ejpam-2240	220	8	\	\	NOUN
ejpam-2240	220	9	(	(	PUNCT
ejpam-2240	220	10	f1∩	f1∩	VERB
ejpam-2240	220	11	f2	f2	NOUN
ejpam-2240	220	12	)	)	PUNCT
ejpam-2240	220	13	=	=	PUNCT
ejpam-2240	220	14	(	(	PUNCT
ejpam-2240	220	15	(	(	PUNCT
ejpam-2240	220	16	a\	a\	NOUN
ejpam-2240	220	17	f1)∩	f1)∩	VERB
ejpam-2240	220	18	b)∪	b)∪	NOUN
ejpam-2240	220	19	(	(	PUNCT
ejpam-2240	220	20	a∩	a∩	PROPN
ejpam-2240	220	21	(	(	PUNCT
ejpam-2240	220	22	b	b	NOUN
ejpam-2240	220	23	\	\	PROPN
ejpam-2240	220	24	f2	f2	PROPN
ejpam-2240	220	25	)	)	PUNCT
ejpam-2240	220	26	)	)	PUNCT
ejpam-2240	221	1	∈	∈	PROPN
ejpam-2240	222	1	i	i	PRON
ejpam-2240	222	2	.	.	PUNCT
ejpam-2240	223	1	therefore	therefore	ADV
ejpam-2240	223	2	,	,	PUNCT
ejpam-2240	223	3	a∩	a∩	PROPN
ejpam-2240	223	4	b	b	PROPN
ejpam-2240	223	5	is	be	AUX
ejpam-2240	223	6	i	i	PRON
ejpam-2240	223	7	−β	−β	PROPN
ejpam-2240	223	8	-closed	-closed	PROPN
ejpam-2240	223	9	.	.	PUNCT
ejpam-2240	224	1	references	reference	NOUN
ejpam-2240	224	2	394	394	NUM
ejpam-2240	224	3	4	4	NUM
ejpam-2240	224	4	.	.	PUNCT
ejpam-2240	225	1	conclusion	conclusion	NOUN
ejpam-2240	225	2	topology	topology	NOUN
ejpam-2240	225	3	plays	play	VERB
ejpam-2240	225	4	a	a	DET
ejpam-2240	225	5	significant	significant	ADJ
ejpam-2240	225	6	role	role	NOUN
ejpam-2240	225	7	in	in	ADP
ejpam-2240	225	8	quantum	quantum	ADJ
ejpam-2240	225	9	physics	physics	NOUN
ejpam-2240	225	10	,	,	PUNCT
ejpam-2240	225	11	high	high	ADJ
ejpam-2240	225	12	energy	energy	NOUN
ejpam-2240	225	13	physics	physics	NOUN
ejpam-2240	225	14	and	and	CCONJ
ejpam-2240	225	15	superstring	superstring	NOUN
ejpam-2240	225	16	theory	theory	NOUN
ejpam-2240	225	17	.	.	PUNCT
ejpam-2240	226	1	in	in	ADP
ejpam-2240	226	2	this	this	DET
ejpam-2240	226	3	paper	paper	NOUN
ejpam-2240	226	4	,	,	PUNCT
ejpam-2240	226	5	new	new	ADJ
ejpam-2240	226	6	types	type	NOUN
ejpam-2240	226	7	of	of	ADP
ejpam-2240	226	8	sets	set	NOUN
ejpam-2240	226	9	via	via	ADP
ejpam-2240	226	10	ideals	ideal	NOUN
ejpam-2240	226	11	were	be	AUX
ejpam-2240	226	12	investigated	investigate	VERB
ejpam-2240	226	13	and	and	CCONJ
ejpam-2240	226	14	some	some	PRON
ejpam-2240	226	15	of	of	ADP
ejpam-2240	226	16	their	their	PRON
ejpam-2240	226	17	properties	property	NOUN
ejpam-2240	226	18	were	be	AUX
ejpam-2240	226	19	obtained	obtain	VERB
ejpam-2240	226	20	.	.	PUNCT
ejpam-2240	227	1	the	the	DET
ejpam-2240	227	2	notions	notion	NOUN
ejpam-2240	227	3	of	of	ADP
ejpam-2240	227	4	the	the	DET
ejpam-2240	227	5	sets	set	NOUN
ejpam-2240	227	6	and	and	CCONJ
ejpam-2240	227	7	functions	function	NOUN
ejpam-2240	227	8	in	in	ADP
ejpam-2240	227	9	topological	topological	ADJ
ejpam-2240	227	10	and	and	CCONJ
ejpam-2240	227	11	fuzzy	fuzzy	ADJ
ejpam-2240	227	12	topological	topological	ADJ
ejpam-2240	227	13	spaces	space	NOUN
ejpam-2240	227	14	are	be	AUX
ejpam-2240	227	15	highly	highly	ADV
ejpam-2240	227	16	developed	develop	VERB
ejpam-2240	227	17	and	and	CCONJ
ejpam-2240	227	18	used	use	VERB
ejpam-2240	227	19	extensively	extensively	ADV
ejpam-2240	227	20	in	in	ADP
ejpam-2240	227	21	many	many	ADJ
ejpam-2240	227	22	practical	practical	ADJ
ejpam-2240	227	23	and	and	CCONJ
ejpam-2240	227	24	engineering	engineering	NOUN
ejpam-2240	227	25	problems	problem	NOUN
ejpam-2240	227	26	,	,	PUNCT
ejpam-2240	227	27	computational	computational	ADJ
ejpam-2240	227	28	topology	topology	NOUN
ejpam-2240	227	29	for	for	ADP
ejpam-2240	227	30	geometric	geometric	ADJ
ejpam-2240	227	31	design	design	NOUN
ejpam-2240	227	32	,	,	PUNCT
ejpam-2240	227	33	computer	computer	NOUN
ejpam-2240	227	34	-	-	PUNCT
ejpam-2240	227	35	aided	aid	VERB
ejpam-2240	227	36	geometric	geometric	ADJ
ejpam-2240	227	37	design	design	NOUN
ejpam-2240	227	38	,	,	PUNCT
ejpam-2240	227	39	engineering	engineering	NOUN
ejpam-2240	227	40	design	design	NOUN
ejpam-2240	227	41	research	research	NOUN
ejpam-2240	227	42	and	and	CCONJ
ejpam-2240	227	43	mathematical	mathematical	ADJ
ejpam-2240	227	44	sciences	science	NOUN
ejpam-2240	227	45	.	.	PUNCT
ejpam-2240	228	1	references	reference	NOUN
ejpam-2240	228	2	[	[	X
ejpam-2240	228	3	1	1	NUM
ejpam-2240	228	4	]	]	X
ejpam-2240	228	5	m.e	m.e	PROPN
ejpam-2240	228	6	.	.	PROPN
ejpam-2240	228	7	abd	abd	PROPN
ejpam-2240	228	8	el	el	PROPN
ejpam-2240	228	9	-	-	PROPN
ejpam-2240	228	10	monsef	monsef	ADJ
ejpam-2240	228	11	,	,	PUNCT
ejpam-2240	228	12	s.n	s.n	PROPN
ejpam-2240	228	13	.	.	PROPN
ejpam-2240	228	14	el	el	PROPN
ejpam-2240	228	15	-	-	PUNCT
ejpam-2240	228	16	deeb	deeb	PROPN
ejpam-2240	228	17	,	,	PUNCT
ejpam-2240	228	18	and	and	CCONJ
ejpam-2240	228	19	r.a	r.a	PROPN
ejpam-2240	228	20	.	.	PROPN
ejpam-2240	228	21	mahmoud	mahmoud	PROPN
ejpam-2240	228	22	.	.	PUNCT
ejpam-2240	229	1	β	β	PROPN
ejpam-2240	229	2	-open	-open	PROPN
ejpam-2240	229	3	sets	set	NOUN
ejpam-2240	229	4	and	and	CCONJ
ejpam-2240	229	5	β	β	PRON
ejpam-2240	229	6	-continuous	-continuous	ADJ
ejpam-2240	229	7	mappings	mapping	NOUN
ejpam-2240	229	8	,	,	PUNCT
ejpam-2240	229	9	bulletin	bulletin	NOUN
ejpam-2240	229	10	of	of	ADP
ejpam-2240	229	11	the	the	DET
ejpam-2240	229	12	faculty	faculty	NOUN
ejpam-2240	229	13	science	science	NOUN
ejpam-2240	229	14	assiut	assiut	PROPN
ejpam-2240	229	15	university	university	PROPN
ejpam-2240	229	16	,	,	PUNCT
ejpam-2240	229	17	12	12	NUM
ejpam-2240	229	18	,	,	PUNCT
ejpam-2240	229	19	77	77	NUM
ejpam-2240	229	20	-	-	SYM
ejpam-2240	229	21	90	90	NUM
ejpam-2240	229	22	.	.	PUNCT
ejpam-2240	229	23	1983	1983	NUM
ejpam-2240	229	24	.	.	PUNCT
ejpam-2240	230	1	[	[	X
ejpam-2240	230	2	2	2	NUM
ejpam-2240	230	3	]	]	SYM
ejpam-2240	230	4	f.g	f.g	NOUN
ejpam-2240	230	5	.	.	PROPN
ejpam-2240	230	6	arenos	arenos	PROPN
ejpam-2240	230	7	,	,	PUNCT
ejpam-2240	230	8	j.	j.	PROPN
ejpam-2240	230	9	dontchev	dontchev	PROPN
ejpam-2240	230	10	,	,	PUNCT
ejpam-2240	230	11	and	and	CCONJ
ejpam-2240	230	12	m.l	m.l	PROPN
ejpam-2240	230	13	.	.	PROPN
ejpam-2240	230	14	puertas	puertas	PROPN
ejpam-2240	230	15	.	.	PUNCT
ejpam-2240	231	1	idealization	idealization	NOUN
ejpam-2240	231	2	of	of	ADP
ejpam-2240	231	3	some	some	DET
ejpam-2240	231	4	weak	weak	ADJ
ejpam-2240	231	5	seperation	seperation	NOUN
ejpam-2240	231	6	axioms	axiom	NOUN
ejpam-2240	231	7	,	,	PUNCT
ejpam-2240	231	8	acta	acta	PROPN
ejpam-2240	231	9	mathematica	mathematica	PROPN
ejpam-2240	231	10	hungarica	hungarica	PROPN
ejpam-2240	231	11	,	,	PUNCT
ejpam-2240	231	12	89(1	89(1	ADJ
ejpam-2240	231	13	-	-	ADJ
ejpam-2240	231	14	2	2	NUM
ejpam-2240	231	15	)	)	PUNCT
ejpam-2240	231	16	,	,	PUNCT
ejpam-2240	231	17	47	47	NUM
ejpam-2240	231	18	-	-	SYM
ejpam-2240	231	19	53	53	NUM
ejpam-2240	231	20	.	.	PUNCT
ejpam-2240	231	21	2000	2000	NUM
ejpam-2240	231	22	.	.	PUNCT
ejpam-2240	232	1	[	[	X
ejpam-2240	232	2	3	3	X
ejpam-2240	232	3	]	]	PUNCT
ejpam-2240	232	4	j.	j.	PROPN
ejpam-2240	232	5	dontchev	dontchev	PROPN
ejpam-2240	232	6	,	,	PUNCT
ejpam-2240	232	7	m.	m.	NOUN
ejpam-2240	232	8	ganster	ganster	NOUN
ejpam-2240	232	9	,	,	PUNCT
ejpam-2240	232	10	and	and	CCONJ
ejpam-2240	232	11	d.	d.	PROPN
ejpam-2240	232	12	rose	rise	VERB
ejpam-2240	232	13	.	.	PUNCT
ejpam-2240	233	1	ideal	ideal	ADJ
ejpam-2240	233	2	resolvability	resolvability	NOUN
ejpam-2240	233	3	,	,	PUNCT
ejpam-2240	233	4	topology	topology	NOUN
ejpam-2240	233	5	and	and	CCONJ
ejpam-2240	233	6	its	its	PRON
ejpam-2240	233	7	applications	application	NOUN
ejpam-2240	233	8	,	,	PUNCT
ejpam-2240	233	9	93	93	NUM
ejpam-2240	233	10	,	,	PUNCT
ejpam-2240	233	11	1	1	NUM
ejpam-2240	233	12	-	-	SYM
ejpam-2240	233	13	16	16	NUM
ejpam-2240	233	14	.	.	PUNCT
ejpam-2240	233	15	1999	1999	NUM
ejpam-2240	233	16	.	.	PUNCT
ejpam-2240	234	1	[	[	X
ejpam-2240	234	2	4	4	X
ejpam-2240	234	3	]	]	X
ejpam-2240	234	4	e.	e.	PROPN
ejpam-2240	234	5	hatir	hatir	PROPN
ejpam-2240	234	6	and	and	CCONJ
ejpam-2240	234	7	t.	t.	PROPN
ejpam-2240	234	8	noiri	noiri	PROPN
ejpam-2240	234	9	on	on	ADP
ejpam-2240	234	10	decompositions	decomposition	NOUN
ejpam-2240	234	11	of	of	ADP
ejpam-2240	234	12	continuity	continuity	NOUN
ejpam-2240	234	13	via	via	ADP
ejpam-2240	234	14	idealization	idealization	NOUN
ejpam-2240	234	15	,	,	PUNCT
ejpam-2240	234	16	acta	acta	PROPN
ejpam-2240	234	17	mathematica	mathematica	PROPN
ejpam-2240	234	18	hungarica	hungarica	PROPN
ejpam-2240	234	19	,	,	PUNCT
ejpam-2240	234	20	96(4	96(4	NOUN
ejpam-2240	234	21	)	)	PUNCT
ejpam-2240	234	22	,	,	PUNCT
ejpam-2240	234	23	341	341	NUM
ejpam-2240	234	24	-	-	SYM
ejpam-2240	234	25	349	349	NUM
ejpam-2240	234	26	.	.	PUNCT
ejpam-2240	234	27	2002	2002	NUM
ejpam-2240	234	28	.	.	PUNCT
ejpam-2240	235	1	[	[	X
ejpam-2240	235	2	5	5	X
ejpam-2240	235	3	]	]	PUNCT
ejpam-2240	235	4	d.	d.	PROPN
ejpam-2240	235	5	jankovic	jankovic	PROPN
ejpam-2240	235	6	and	and	CCONJ
ejpam-2240	235	7	t.r	t.r	PROPN
ejpam-2240	235	8	.	.	PROPN
ejpam-2240	235	9	hamlett	hamlett	PROPN
ejpam-2240	235	10	.	.	PUNCT
ejpam-2240	236	1	ideal	ideal	PROPN
ejpam-2240	236	2	in	in	ADP
ejpam-2240	236	3	topology	topology	NOUN
ejpam-2240	236	4	and	and	CCONJ
ejpam-2240	236	5	the	the	DET
ejpam-2240	236	6	set	set	NOUN
ejpam-2240	236	7	operator	operator	NOUN
ejpam-2240	236	8	,	,	PUNCT
ejpam-2240	236	9	bollettino	bollettino	PROPN
ejpam-2240	236	10	dell’unione	dell’unione	PROPN
ejpam-2240	236	11	matematica	matematica	PROPN
ejpam-2240	236	12	italiana	italiana	PROPN
ejpam-2240	236	13	,	,	PUNCT
ejpam-2240	236	14	7	7	NUM
ejpam-2240	236	15	,	,	PUNCT
ejpam-2240	236	16	863	863	NUM
ejpam-2240	236	17	-	-	SYM
ejpam-2240	236	18	894	894	NUM
ejpam-2240	236	19	.	.	NUM
ejpam-2240	236	20	1990	1990	NUM
ejpam-2240	236	21	.	.	PUNCT
ejpam-2240	237	1	[	[	X
ejpam-2240	237	2	6	6	NUM
ejpam-2240	237	3	]	]	PUNCT
ejpam-2240	237	4	k.	k.	PROPN
ejpam-2240	237	5	kuratowski	kuratowski	PROPN
ejpam-2240	237	6	.	.	PUNCT
ejpam-2240	238	1	topology	topology	PROPN
ejpam-2240	238	2	,	,	PUNCT
ejpam-2240	238	3	vol.i	vol.i	PROPN
ejpam-2240	238	4	.	.	PUNCT
ejpam-2240	239	1	new	new	PROPN
ejpam-2240	239	2	york	york	PROPN
ejpam-2240	239	3	:	:	PUNCT
ejpam-2240	239	4	academic	academic	ADJ
ejpam-2240	239	5	press	press	NOUN
ejpam-2240	239	6	,	,	PUNCT
ejpam-2240	239	7	1966	1966	NUM
ejpam-2240	239	8	.	.	PUNCT
ejpam-2240	240	1	[	[	X
ejpam-2240	240	2	7	7	X
ejpam-2240	240	3	]	]	X
ejpam-2240	240	4	a.s	a.s	PROPN
ejpam-2240	240	5	.	.	PROPN
ejpam-2240	240	6	mashhour	mashhour	PROPN
ejpam-2240	240	7	,	,	PUNCT
ejpam-2240	240	8	m.e	m.e	PROPN
ejpam-2240	240	9	.	.	PROPN
ejpam-2240	240	10	abd	abd	PROPN
ejpam-2240	240	11	el	el	PROPN
ejpam-2240	240	12	-	-	PROPN
ejpam-2240	240	13	monsef	monsef	ADJ
ejpam-2240	240	14	,	,	PUNCT
ejpam-2240	240	15	and	and	CCONJ
ejpam-2240	240	16	s.n	s.n	PROPN
ejpam-2240	240	17	.	.	PROPN
ejpam-2240	240	18	el	el	PROPN
ejpam-2240	240	19	-	-	PUNCT
ejpam-2240	240	20	deeb	deeb	PROPN
ejpam-2240	240	21	.	.	PUNCT
ejpam-2240	241	1	on	on	ADP
ejpam-2240	241	2	precontinuous	precontinuous	ADJ
ejpam-2240	241	3	and	and	CCONJ
ejpam-2240	241	4	weak	weak	ADJ
ejpam-2240	241	5	precontinuous	precontinuous	ADJ
ejpam-2240	241	6	mappings	mapping	NOUN
ejpam-2240	241	7	,	,	PUNCT
ejpam-2240	241	8	proceedings	proceeding	NOUN
ejpam-2240	241	9	of	of	ADP
ejpam-2240	241	10	mathematical	mathematical	ADJ
ejpam-2240	241	11	physics	physics	PROPN
ejpam-2240	241	12	society	society	PROPN
ejpam-2240	241	13	egyptian	egyptian	PROPN
ejpam-2240	241	14	,	,	PUNCT
ejpam-2240	241	15	53	53	NUM
ejpam-2240	241	16	,	,	PUNCT
ejpam-2240	241	17	47	47	NUM
ejpam-2240	241	18	-	-	SYM
ejpam-2240	241	19	53	53	NUM
ejpam-2240	241	20	.	.	NUM
ejpam-2240	241	21	1982	1982	NUM
ejpam-2240	241	22	.	.	PUNCT
ejpam-2240	242	1	[	[	X
ejpam-2240	242	2	8	8	NUM
ejpam-2240	242	3	]	]	X
ejpam-2240	242	4	f.i	f.i	PROPN
ejpam-2240	242	5	.	.	PROPN
ejpam-2240	242	6	michael	michael	PROPN
ejpam-2240	242	7	.	.	PUNCT
ejpam-2240	243	1	on	on	ADP
ejpam-2240	243	2	semi	semi	ADJ
ejpam-2240	243	3	-	-	ADJ
ejpam-2240	243	4	open	open	ADJ
ejpam-2240	243	5	sets	set	NOUN
ejpam-2240	243	6	with	with	ADP
ejpam-2240	243	7	respect	respect	NOUN
ejpam-2240	243	8	to	to	ADP
ejpam-2240	243	9	an	an	DET
ejpam-2240	243	10	ideal	ideal	ADJ
ejpam-2240	243	11	,	,	PUNCT
ejpam-2240	243	12	european	european	ADJ
ejpam-2240	243	13	journal	journal	PROPN
ejpam-2240	243	14	of	of	ADP
ejpam-2240	243	15	pure	pure	ADJ
ejpam-2240	243	16	and	and	CCONJ
ejpam-2240	243	17	applied	applied	ADJ
ejpam-2240	243	18	mathematics	mathematic	NOUN
ejpam-2240	243	19	,	,	PUNCT
ejpam-2240	243	20	6(1	6(1	NUM
ejpam-2240	243	21	)	)	PUNCT
ejpam-2240	243	22	,	,	PUNCT
ejpam-2240	243	23	53	53	NUM
ejpam-2240	243	24	-	-	SYM
ejpam-2240	243	25	58	58	NUM
ejpam-2240	243	26	.	.	PUNCT
ejpam-2240	243	27	2013	2013	NUM
ejpam-2240	243	28	.	.	PUNCT
ejpam-2240	244	1	[	[	X
ejpam-2240	244	2	9	9	NUM
ejpam-2240	244	3	]	]	X
ejpam-2240	244	4	m.n	m.n	PROPN
ejpam-2240	244	5	.	.	PROPN
ejpam-2240	244	6	mukharjee	mukharjee	PROPN
ejpam-2240	244	7	,	,	PUNCT
ejpam-2240	244	8	r.	r.	PROPN
ejpam-2240	244	9	bishwambhar	bishwambhar	PROPN
ejpam-2240	244	10	,	,	PUNCT
ejpam-2240	244	11	and	and	CCONJ
ejpam-2240	244	12	r.	r.	PROPN
ejpam-2240	244	13	sen	sen	PROPN
ejpam-2240	244	14	.	.	PROPN
ejpam-2240	244	15	on	on	ADP
ejpam-2240	244	16	extension	extension	NOUN
ejpam-2240	244	17	of	of	ADP
ejpam-2240	244	18	topological	topological	ADJ
ejpam-2240	244	19	spaces	space	NOUN
ejpam-2240	244	20	in	in	ADP
ejpam-2240	244	21	terms	term	NOUN
ejpam-2240	244	22	of	of	ADP
ejpam-2240	244	23	ideals	ideal	NOUN
ejpam-2240	244	24	,	,	PUNCT
ejpam-2240	244	25	topology	topology	NOUN
ejpam-2240	244	26	and	and	CCONJ
ejpam-2240	244	27	its	its	PRON
ejpam-2240	244	28	applications	application	NOUN
ejpam-2240	244	29	,	,	PUNCT
ejpam-2240	244	30	154	154	NUM
ejpam-2240	244	31	,	,	PUNCT
ejpam-2240	244	32	3167	3167	NUM
ejpam-2240	244	33	-	-	SYM
ejpam-2240	244	34	3172	3172	NUM
ejpam-2240	244	35	.	.	PUNCT
ejpam-2240	245	1	2007	2007	NUM
ejpam-2240	245	2	.	.	PUNCT
ejpam-2240	246	1	[	[	X
ejpam-2240	246	2	10	10	NUM
ejpam-2240	246	3	]	]	X
ejpam-2240	246	4	a.a	a.a	PROPN
ejpam-2240	246	5	.	.	PROPN
ejpam-2240	246	6	nasef	nasef	PROPN
ejpam-2240	246	7	and	and	CCONJ
ejpam-2240	246	8	r.a	r.a	PROPN
ejpam-2240	246	9	.	.	PROPN
ejpam-2240	246	10	mahmoud	mahmoud	PROPN
ejpam-2240	246	11	.	.	PUNCT
ejpam-2240	247	1	some	some	DET
ejpam-2240	247	2	applications	application	NOUN
ejpam-2240	247	3	via	via	ADP
ejpam-2240	247	4	fuzzy	fuzzy	ADJ
ejpam-2240	247	5	ideals	ideal	NOUN
ejpam-2240	247	6	,	,	PUNCT
ejpam-2240	247	7	chaos	chaos	NOUN
ejpam-2240	247	8	,	,	PUNCT
ejpam-2240	247	9	solitons	soliton	NOUN
ejpam-2240	247	10	and	and	CCONJ
ejpam-2240	247	11	fractals	fractal	NOUN
ejpam-2240	247	12	,	,	PUNCT
ejpam-2240	247	13	13	13	NUM
ejpam-2240	247	14	,	,	PUNCT
ejpam-2240	247	15	825	825	NUM
ejpam-2240	247	16	-	-	SYM
ejpam-2240	247	17	831	831	NUM
ejpam-2240	247	18	.	.	PUNCT
ejpam-2240	247	19	2002	2002	NUM
ejpam-2240	247	20	.	.	PUNCT
ejpam-2240	248	1	[	[	X
ejpam-2240	248	2	11	11	NUM
ejpam-2240	248	3	]	]	X
ejpam-2240	248	4	r.	r.	PROPN
ejpam-2240	248	5	vaidyanathasmamy	vaidyanathasmamy	PROPN
ejpam-2240	248	6	.	.	PUNCT
ejpam-2240	249	1	the	the	DET
ejpam-2240	249	2	localization	localization	NOUN
ejpam-2240	249	3	theory	theory	NOUN
ejpam-2240	249	4	in	in	ADP
ejpam-2240	249	5	set	set	NOUN
ejpam-2240	249	6	-	-	PUNCT
ejpam-2240	249	7	topology	topology	NOUN
ejpam-2240	249	8	,	,	PUNCT
ejpam-2240	249	9	proceedings	proceeding	NOUN
ejpam-2240	249	10	of	of	ADP
ejpam-2240	249	11	the	the	DET
ejpam-2240	249	12	indian	indian	PROPN
ejpam-2240	249	13	academy	academy	PROPN
ejpam-2240	249	14	of	of	ADP
ejpam-2240	249	15	science	science	PROPN
ejpam-2240	249	16	,	,	PUNCT
ejpam-2240	249	17	20	20	NUM
ejpam-2240	249	18	,	,	PUNCT
ejpam-2240	249	19	15	15	NUM
ejpam-2240	249	20	-	-	SYM
ejpam-2240	249	21	51	51	NUM
ejpam-2240	249	22	.	.	PUNCT
ejpam-2240	249	23	1945	1945	NUM
ejpam-2240	249	24	.	.	PUNCT