id	sid	tid	token	lemma	pos
ejpam-1595	1	1	6_michael.dvi	6_michael.dvi	NUM
ejpam-1595	1	2	european	european	ADJ
ejpam-1595	1	3	journal	journal	NOUN
ejpam-1595	1	4	of	of	ADP
ejpam-1595	1	5	pure	pure	ADJ
ejpam-1595	1	6	and	and	CCONJ
ejpam-1595	1	7	applied	apply	VERB
ejpam-1595	1	8	mathematics	mathematic	NOUN
ejpam-1595	1	9	vol	vol	NOUN
ejpam-1595	1	10	.	.	PROPN
ejpam-1595	1	11	6	6	NUM
ejpam-1595	1	12	,	,	PUNCT
ejpam-1595	1	13	no	no	INTJ
ejpam-1595	1	14	.	.	NOUN
ejpam-1595	1	15	1	1	NUM
ejpam-1595	1	16	,	,	PUNCT
ejpam-1595	1	17	2013	2013	NUM
ejpam-1595	1	18	,	,	PUNCT
ejpam-1595	1	19	53	53	NUM
ejpam-1595	1	20	-	-	SYM
ejpam-1595	1	21	58	58	NUM
ejpam-1595	1	22	issn	issn	PROPN
ejpam-1595	1	23	1307	1307	NUM
ejpam-1595	1	24	-	-	SYM
ejpam-1595	1	25	5543	5543	NUM
ejpam-1595	1	26	–	–	PUNCT
ejpam-1595	1	27	www.ejpam.com	www.ejpam.com	X
ejpam-1595	1	28	on	on	ADP
ejpam-1595	1	29	semi	semi	ADJ
ejpam-1595	1	30	-	-	ADJ
ejpam-1595	1	31	open	open	ADJ
ejpam-1595	1	32	sets	set	NOUN
ejpam-1595	1	33	with	with	ADP
ejpam-1595	1	34	respect	respect	NOUN
ejpam-1595	1	35	to	to	ADP
ejpam-1595	1	36	an	an	DET
ejpam-1595	1	37	ideal	ideal	ADJ
ejpam-1595	1	38	friday	friday	PROPN
ejpam-1595	1	39	ifeanyi	ifeanyi	PROPN
ejpam-1595	1	40	michael	michael	PROPN
ejpam-1595	1	41	department	department	PROPN
ejpam-1595	1	42	of	of	ADP
ejpam-1595	1	43	mathematics	mathematics	PROPN
ejpam-1595	1	44	,	,	PUNCT
ejpam-1595	1	45	faculty	faculty	NOUN
ejpam-1595	1	46	of	of	ADP
ejpam-1595	1	47	science	science	NOUN
ejpam-1595	1	48	,	,	PUNCT
ejpam-1595	1	49	the	the	DET
ejpam-1595	1	50	university	university	NOUN
ejpam-1595	1	51	of	of	ADP
ejpam-1595	1	52	western	western	PROPN
ejpam-1595	1	53	ontario	ontario	PROPN
ejpam-1595	1	54	,	,	PUNCT
ejpam-1595	1	55	london	london	PROPN
ejpam-1595	1	56	,	,	PUNCT
ejpam-1595	1	57	ontario	ontario	PROPN
ejpam-1595	1	58	,	,	PUNCT
ejpam-1595	1	59	canada	canada	PROPN
ejpam-1595	1	60	abstract	abstract	NOUN
ejpam-1595	1	61	.	.	PUNCT
ejpam-1595	2	1	we	we	PRON
ejpam-1595	2	2	introduce	introduce	VERB
ejpam-1595	2	3	a	a	DET
ejpam-1595	2	4	notion	notion	NOUN
ejpam-1595	2	5	of	of	ADP
ejpam-1595	2	6	semi	semi	ADJ
ejpam-1595	2	7	-	-	ADJ
ejpam-1595	2	8	open	open	ADJ
ejpam-1595	2	9	sets	set	NOUN
ejpam-1595	2	10	in	in	ADP
ejpam-1595	2	11	terms	term	NOUN
ejpam-1595	2	12	of	of	ADP
ejpam-1595	2	13	ideals	ideal	NOUN
ejpam-1595	2	14	,	,	PUNCT
ejpam-1595	2	15	which	which	PRON
ejpam-1595	2	16	generalizes	generalize	VERB
ejpam-1595	2	17	the	the	DET
ejpam-1595	2	18	usual	usual	ADJ
ejpam-1595	2	19	notion	notion	NOUN
ejpam-1595	2	20	of	of	ADP
ejpam-1595	2	21	semi	semi	ADJ
ejpam-1595	2	22	-	-	ADJ
ejpam-1595	2	23	open	open	ADJ
ejpam-1595	2	24	sets	set	NOUN
ejpam-1595	2	25	.	.	PUNCT
ejpam-1595	3	1	2010	2010	NUM
ejpam-1595	3	2	mathematics	mathematic	NOUN
ejpam-1595	3	3	subject	subject	NOUN
ejpam-1595	3	4	classifications	classification	NOUN
ejpam-1595	3	5	:	:	PUNCT
ejpam-1595	3	6	54c10	54c10	NUM
ejpam-1595	3	7	key	key	ADJ
ejpam-1595	3	8	words	word	NOUN
ejpam-1595	3	9	and	and	CCONJ
ejpam-1595	3	10	phrases	phrase	NOUN
ejpam-1595	3	11	:	:	PUNCT
ejpam-1595	3	12	semi	semi	ADJ
ejpam-1595	3	13	-	-	ADJ
ejpam-1595	3	14	open	open	ADJ
ejpam-1595	3	15	sets	set	NOUN
ejpam-1595	3	16	,	,	PUNCT
ejpam-1595	3	17	semi	semi	ADJ
ejpam-1595	3	18	-	-	ADJ
ejpam-1595	3	19	closed	closed	ADJ
ejpam-1595	3	20	sets	set	NOUN
ejpam-1595	3	21	,	,	PUNCT
ejpam-1595	3	22	generalized	generalize	VERB
ejpam-1595	3	23	closed	closed	ADJ
ejpam-1595	3	24	sets	set	NOUN
ejpam-1595	3	25	,	,	PUNCT
ejpam-1595	3	26	ideals	ideal	NOUN
ejpam-1595	3	27	1	1	NUM
ejpam-1595	3	28	.	.	PUNCT
ejpam-1595	4	1	introduction	introduction	NOUN
ejpam-1595	4	2	with	with	ADP
ejpam-1595	4	3	the	the	DET
ejpam-1595	4	4	impetus	impetus	NOUN
ejpam-1595	4	5	given	give	VERB
ejpam-1595	4	6	by	by	ADP
ejpam-1595	4	7	levine	levine	PROPN
ejpam-1595	4	8	’s	’s	PART
ejpam-1595	4	9	introduction	introduction	NOUN
ejpam-1595	4	10	of	of	ADP
ejpam-1595	4	11	semi	semi	ADJ
ejpam-1595	4	12	-	-	ADJ
ejpam-1595	4	13	open	open	ADJ
ejpam-1595	4	14	sets	set	NOUN
ejpam-1595	4	15	and	and	CCONJ
ejpam-1595	4	16	generalized	generalize	VERB
ejpam-1595	4	17	closed	closed	ADJ
ejpam-1595	4	18	sets	set	NOUN
ejpam-1595	4	19	[	[	X
ejpam-1595	4	20	11	11	NUM
ejpam-1595	4	21	,	,	PUNCT
ejpam-1595	4	22	12	12	NUM
ejpam-1595	4	23	]	]	PUNCT
ejpam-1595	4	24	,	,	PUNCT
ejpam-1595	4	25	there	there	PRON
ejpam-1595	4	26	have	have	AUX
ejpam-1595	4	27	been	be	AUX
ejpam-1595	4	28	other	other	ADJ
ejpam-1595	4	29	attempts	attempt	NOUN
ejpam-1595	4	30	by	by	ADP
ejpam-1595	4	31	some	some	DET
ejpam-1595	4	32	topologists	topologist	NOUN
ejpam-1595	4	33	to	to	PART
ejpam-1595	4	34	study	study	VERB
ejpam-1595	4	35	closed	closed	ADJ
ejpam-1595	4	36	sets	set	NOUN
ejpam-1595	4	37	together	together	ADV
ejpam-1595	4	38	with	with	ADP
ejpam-1595	4	39	the	the	DET
ejpam-1595	4	40	accompanying	accompanying	ADJ
ejpam-1595	4	41	topological	topological	ADJ
ejpam-1595	4	42	notions	notion	NOUN
ejpam-1595	4	43	from	from	ADP
ejpam-1595	4	44	different	different	ADJ
ejpam-1595	4	45	perspectives	perspective	NOUN
ejpam-1595	4	46	[	[	AUX
ejpam-1595	4	47	see	see	VERB
ejpam-1595	4	48	,	,	PUNCT
ejpam-1595	4	49	for	for	ADP
ejpam-1595	4	50	example	example	NOUN
ejpam-1595	4	51	,	,	PUNCT
ejpam-1595	4	52	1	1	NUM
ejpam-1595	4	53	,	,	PUNCT
ejpam-1595	4	54	2	2	NUM
ejpam-1595	4	55	,	,	PUNCT
ejpam-1595	4	56	3	3	NUM
ejpam-1595	4	57	,	,	PUNCT
ejpam-1595	4	58	5	5	NUM
ejpam-1595	4	59	,	,	PUNCT
ejpam-1595	4	60	6	6	NUM
ejpam-1595	4	61	,	,	PUNCT
ejpam-1595	4	62	4	4	NUM
ejpam-1595	4	63	]	]	PUNCT
ejpam-1595	4	64	.	.	PUNCT
ejpam-1595	5	1	relevant	relevant	ADJ
ejpam-1595	5	2	to	to	ADP
ejpam-1595	5	3	the	the	DET
ejpam-1595	5	4	present	present	ADJ
ejpam-1595	5	5	work	work	NOUN
ejpam-1595	5	6	is	be	AUX
ejpam-1595	5	7	the	the	DET
ejpam-1595	5	8	idea	idea	NOUN
ejpam-1595	5	9	of	of	ADP
ejpam-1595	5	10	using	use	VERB
ejpam-1595	5	11	topological	topological	ADJ
ejpam-1595	5	12	ideals	ideal	NOUN
ejpam-1595	5	13	in	in	ADP
ejpam-1595	5	14	describing	describe	VERB
ejpam-1595	5	15	topological	topological	ADJ
ejpam-1595	5	16	notions	notion	NOUN
ejpam-1595	5	17	,	,	PUNCT
ejpam-1595	5	18	which	which	PRON
ejpam-1595	5	19	,	,	PUNCT
ejpam-1595	5	20	for	for	ADP
ejpam-1595	5	21	some	some	DET
ejpam-1595	5	22	years	year	NOUN
ejpam-1595	5	23	now	now	ADV
ejpam-1595	5	24	,	,	PUNCT
ejpam-1595	5	25	has	have	AUX
ejpam-1595	5	26	been	be	AUX
ejpam-1595	5	27	an	an	DET
ejpam-1595	5	28	interesting	interesting	ADJ
ejpam-1595	5	29	subject	subject	NOUN
ejpam-1595	5	30	for	for	ADP
ejpam-1595	5	31	investigation	investigation	NOUN
ejpam-1595	5	32	[	[	X
ejpam-1595	5	33	see	see	VERB
ejpam-1595	5	34	some	some	PRON
ejpam-1595	5	35	of	of	ADP
ejpam-1595	5	36	the	the	DET
ejpam-1595	5	37	pioneering	pioneering	ADJ
ejpam-1595	5	38	works	work	NOUN
ejpam-1595	5	39	in	in	ADP
ejpam-1595	5	40	7	7	NUM
ejpam-1595	5	41	,	,	PUNCT
ejpam-1595	5	42	8	8	NUM
ejpam-1595	5	43	,	,	PUNCT
ejpam-1595	5	44	9	9	NUM
ejpam-1595	5	45	]	]	PUNCT
ejpam-1595	5	46	.	.	PUNCT
ejpam-1595	6	1	we	we	PRON
ejpam-1595	6	2	recall	recall	VERB
ejpam-1595	6	3	here	here	ADV
ejpam-1595	6	4	that	that	SCONJ
ejpam-1595	6	5	an	an	DET
ejpam-1595	6	6	ideal	ideal	NOUN
ejpam-1595	6	7	i	i	PRON
ejpam-1595	6	8	on	on	ADP
ejpam-1595	6	9	a	a	DET
ejpam-1595	6	10	topological	topological	ADJ
ejpam-1595	6	11	space	space	NOUN
ejpam-1595	6	12	(	(	PUNCT
ejpam-1595	6	13	x	x	X
ejpam-1595	6	14	,	,	PUNCT
ejpam-1595	6	15	τ	τ	X
ejpam-1595	6	16	)	)	PUNCT
ejpam-1595	6	17	is	be	AUX
ejpam-1595	6	18	a	a	DET
ejpam-1595	6	19	non	non	ADJ
ejpam-1595	6	20	-	-	ADJ
ejpam-1595	6	21	empty	empty	ADJ
ejpam-1595	6	22	collection	collection	NOUN
ejpam-1595	6	23	of	of	ADP
ejpam-1595	6	24	subsets	subset	NOUN
ejpam-1595	6	25	of	of	ADP
ejpam-1595	6	26	x	x	PUNCT
ejpam-1595	6	27	having	have	VERB
ejpam-1595	6	28	the	the	DET
ejpam-1595	6	29	heredity	heredity	NOUN
ejpam-1595	6	30	property	property	NOUN
ejpam-1595	6	31	(	(	PUNCT
ejpam-1595	6	32	that	that	ADV
ejpam-1595	6	33	is	is	ADV
ejpam-1595	6	34	,	,	PUNCT
ejpam-1595	6	35	if	if	SCONJ
ejpam-1595	6	36	a∈	a∈	PROPN
ejpam-1595	6	37	i	i	PROPN
ejpam-1595	6	38	and	and	CCONJ
ejpam-1595	6	39	b	b	PROPN
ejpam-1595	6	40	⊂	⊂	PROPN
ejpam-1595	6	41	a	a	PROPN
ejpam-1595	6	42	,	,	PUNCT
ejpam-1595	6	43	then	then	ADV
ejpam-1595	6	44	b	b	X
ejpam-1595	6	45	∈	∈	PROPN
ejpam-1595	6	46	i	i	X
ejpam-1595	6	47	)	)	PUNCT
ejpam-1595	6	48	and	and	CCONJ
ejpam-1595	6	49	also	also	ADV
ejpam-1595	6	50	satisfying	satisfy	VERB
ejpam-1595	6	51	finite	finite	ADJ
ejpam-1595	6	52	additivity	additivity	NOUN
ejpam-1595	6	53	(	(	PUNCT
ejpam-1595	6	54	that	that	PRON
ejpam-1595	6	55	is	is	ADV
ejpam-1595	6	56	,	,	PUNCT
ejpam-1595	6	57	if	if	SCONJ
ejpam-1595	6	58	a	a	PRON
ejpam-1595	6	59	,	,	PUNCT
ejpam-1595	6	60	b	b	X
ejpam-1595	7	1	∈	∈	PROPN
ejpam-1595	8	1	i	i	PRON
ejpam-1595	8	2	,	,	PUNCT
ejpam-1595	8	3	then	then	ADV
ejpam-1595	8	4	a∪	a∪	PROPN
ejpam-1595	8	5	b	b	PROPN
ejpam-1595	8	6	∈	∈	PROPN
ejpam-1595	8	7	i	i	PRON
ejpam-1595	8	8	)	)	PUNCT
ejpam-1595	8	9	.	.	PUNCT
ejpam-1595	9	1	in	in	ADP
ejpam-1595	9	2	this	this	DET
ejpam-1595	9	3	paper	paper	NOUN
ejpam-1595	9	4	,	,	PUNCT
ejpam-1595	9	5	we	we	PRON
ejpam-1595	9	6	define	define	VERB
ejpam-1595	9	7	semi	semi	ADJ
ejpam-1595	9	8	-	-	ADJ
ejpam-1595	9	9	open	open	ADJ
ejpam-1595	9	10	sets	set	NOUN
ejpam-1595	9	11	with	with	ADP
ejpam-1595	9	12	respect	respect	NOUN
ejpam-1595	9	13	to	to	ADP
ejpam-1595	9	14	an	an	DET
ejpam-1595	9	15	ideal	ideal	NOUN
ejpam-1595	9	16	i	i	PRON
ejpam-1595	9	17	,	,	PUNCT
ejpam-1595	9	18	and	and	CCONJ
ejpam-1595	9	19	also	also	ADV
ejpam-1595	9	20	study	study	VERB
ejpam-1595	9	21	some	some	PRON
ejpam-1595	9	22	of	of	ADP
ejpam-1595	9	23	their	their	PRON
ejpam-1595	9	24	properties	property	NOUN
ejpam-1595	9	25	.	.	PUNCT
ejpam-1595	10	1	it	it	PRON
ejpam-1595	10	2	turns	turn	VERB
ejpam-1595	10	3	out	out	ADP
ejpam-1595	10	4	that	that	SCONJ
ejpam-1595	10	5	our	our	PRON
ejpam-1595	10	6	notion	notion	NOUN
ejpam-1595	10	7	of	of	ADP
ejpam-1595	10	8	semi	semi	ADJ
ejpam-1595	10	9	-	-	ADJ
ejpam-1595	10	10	open	open	ADJ
ejpam-1595	10	11	sets	set	NOUN
ejpam-1595	10	12	with	with	ADP
ejpam-1595	10	13	respect	respect	NOUN
ejpam-1595	10	14	to	to	ADP
ejpam-1595	10	15	a	a	DET
ejpam-1595	10	16	given	give	VERB
ejpam-1595	10	17	ideal	ideal	NOUN
ejpam-1595	10	18	i	i	PRON
ejpam-1595	10	19	generalizes	generalize	VERB
ejpam-1595	10	20	both	both	CCONJ
ejpam-1595	10	21	the	the	DET
ejpam-1595	10	22	usual	usual	ADJ
ejpam-1595	10	23	notion	notion	NOUN
ejpam-1595	10	24	of	of	ADP
ejpam-1595	10	25	semi	semi	ADJ
ejpam-1595	10	26	-	-	NOUN
ejpam-1595	10	27	openness	openness	ADJ
ejpam-1595	10	28	[	[	X
ejpam-1595	10	29	11	11	NUM
ejpam-1595	10	30	]	]	PUNCT
ejpam-1595	10	31	and	and	CCONJ
ejpam-1595	10	32	the	the	DET
ejpam-1595	10	33	notion	notion	NOUN
ejpam-1595	10	34	of	of	ADP
ejpam-1595	10	35	semi	semi	ADJ
ejpam-1595	10	36	-	-	ADJ
ejpam-1595	10	37	i	i	PRON
ejpam-1595	10	38	-openness	-openness	NOUN
ejpam-1595	10	39	considered	consider	VERB
ejpam-1595	10	40	in	in	ADP
ejpam-1595	10	41	[	[	X
ejpam-1595	10	42	5	5	NUM
ejpam-1595	10	43	]	]	PUNCT
ejpam-1595	10	44	;	;	PUNCT
ejpam-1595	10	45	in	in	ADP
ejpam-1595	10	46	particular	particular	ADJ
ejpam-1595	10	47	,	,	PUNCT
ejpam-1595	10	48	semi	semi	ADJ
ejpam-1595	10	49	-	-	ADJ
ejpam-1595	10	50	i	i	PRON
ejpam-1595	10	51	-openness	-openness	PROPN
ejpam-1595	10	52	implies	imply	VERB
ejpam-1595	10	53	the	the	DET
ejpam-1595	10	54	usual	usual	ADJ
ejpam-1595	10	55	semi	semi	ADJ
ejpam-1595	10	56	-	-	NOUN
ejpam-1595	10	57	openness	openness	NOUN
ejpam-1595	10	58	,	,	PUNCT
ejpam-1595	10	59	which	which	PRON
ejpam-1595	10	60	in	in	ADP
ejpam-1595	10	61	turn	turn	NOUN
ejpam-1595	10	62	implies	imply	VERB
ejpam-1595	10	63	semi	semi	ADJ
ejpam-1595	10	64	-	-	NOUN
ejpam-1595	10	65	openness	openness	ADJ
ejpam-1595	10	66	in	in	ADP
ejpam-1595	10	67	our	our	PRON
ejpam-1595	10	68	sense	sense	NOUN
ejpam-1595	10	69	.	.	PUNCT
ejpam-1595	11	1	throughout	throughout	SCONJ
ejpam-1595	11	2	we	we	PRON
ejpam-1595	11	3	work	work	VERB
ejpam-1595	11	4	with	with	ADP
ejpam-1595	11	5	a	a	DET
ejpam-1595	11	6	topological	topological	ADJ
ejpam-1595	11	7	space	space	NOUN
ejpam-1595	11	8	(	(	PUNCT
ejpam-1595	11	9	x	x	X
ejpam-1595	11	10	,	,	PUNCT
ejpam-1595	11	11	τ	τ	X
ejpam-1595	11	12	)	)	PUNCT
ejpam-1595	11	13	(	(	PUNCT
ejpam-1595	11	14	or	or	CCONJ
ejpam-1595	11	15	simply	simply	ADV
ejpam-1595	11	16	x	x	X
ejpam-1595	11	17	)	)	PUNCT
ejpam-1595	11	18	,	,	PUNCT
ejpam-1595	11	19	where	where	SCONJ
ejpam-1595	11	20	no	no	DET
ejpam-1595	11	21	separation	separation	NOUN
ejpam-1595	11	22	axioms	axiom	NOUN
ejpam-1595	11	23	are	be	AUX
ejpam-1595	11	24	assumed	assume	VERB
ejpam-1595	11	25	.	.	PUNCT
ejpam-1595	12	1	the	the	DET
ejpam-1595	12	2	usual	usual	ADJ
ejpam-1595	12	3	notation	notation	NOUN
ejpam-1595	12	4	cl(a	cl(a	NUM
ejpam-1595	12	5	)	)	PUNCT
ejpam-1595	12	6	for	for	ADP
ejpam-1595	12	7	the	the	DET
ejpam-1595	12	8	closure	closure	NOUN
ejpam-1595	12	9	,	,	PUNCT
ejpam-1595	12	10	and	and	CCONJ
ejpam-1595	12	11	int(a	int(a	PROPN
ejpam-1595	12	12	)	)	PUNCT
ejpam-1595	12	13	for	for	ADP
ejpam-1595	12	14	the	the	DET
ejpam-1595	12	15	interior	interior	NOUN
ejpam-1595	12	16	,	,	PUNCT
ejpam-1595	12	17	of	of	ADP
ejpam-1595	12	18	a	a	DET
ejpam-1595	12	19	subset	subset	NOUN
ejpam-1595	12	20	a	a	PRON
ejpam-1595	12	21	of	of	ADP
ejpam-1595	12	22	a	a	DET
ejpam-1595	12	23	topological	topological	ADJ
ejpam-1595	12	24	space	space	NOUN
ejpam-1595	12	25	(	(	PUNCT
ejpam-1595	12	26	x	x	X
ejpam-1595	12	27	,	,	PUNCT
ejpam-1595	12	28	τ	τ	PROPN
ejpam-1595	12	29	)	)	PUNCT
ejpam-1595	12	30	,	,	PUNCT
ejpam-1595	12	31	will	will	AUX
ejpam-1595	12	32	be	be	AUX
ejpam-1595	12	33	used	use	VERB
ejpam-1595	12	34	[	[	PUNCT
ejpam-1595	12	35	see	see	VERB
ejpam-1595	12	36	3	3	NUM
ejpam-1595	12	37	,	,	PUNCT
ejpam-1595	12	38	10	10	NUM
ejpam-1595	12	39	,	,	PUNCT
ejpam-1595	12	40	4	4	NUM
ejpam-1595	12	41	,	,	PUNCT
ejpam-1595	12	42	for	for	ADP
ejpam-1595	12	43	example	example	NOUN
ejpam-1595	12	44	]	]	PUNCT
ejpam-1595	12	45	.	.	PUNCT
ejpam-1595	13	1	email	email	NOUN
ejpam-1595	13	2	address	address	PROPN
ejpam-1595	13	3	:	:	PUNCT
ejpam-1595	13	4	fmi	fmi	PROPN
ejpam-1595	13	5	hael	hael	PROPN
ejpam-1595	13	6	�	�	PROPN
ejpam-1595	13	7	uwo	uwo	PROPN
ejpam-1595	13	8	.	.	PUNCT
ejpam-1595	14	1	a	a	DET
ejpam-1595	14	2	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1595	14	3	53	53	NUM
ejpam-1595	14	4	c	c	X
ejpam-1595	14	5	©	©	PROPN
ejpam-1595	14	6	2013	2013	NUM
ejpam-1595	14	7	ejpam	ejpam	NOUN
ejpam-1595	14	8	all	all	DET
ejpam-1595	14	9	rights	right	NOUN
ejpam-1595	14	10	reserved	reserve	VERB
ejpam-1595	14	11	.	.	PUNCT
ejpam-1595	15	1	f.	f.	PROPN
ejpam-1595	15	2	michael	michael	PROPN
ejpam-1595	15	3	/	/	SYM
ejpam-1595	15	4	eur	eur	PROPN
ejpam-1595	15	5	.	.	PUNCT
ejpam-1595	16	1	j.	j.	PROPN
ejpam-1595	16	2	pure	pure	PROPN
ejpam-1595	16	3	appl	appl	PROPN
ejpam-1595	16	4	.	.	PROPN
ejpam-1595	16	5	math	math	PROPN
ejpam-1595	16	6	,	,	PUNCT
ejpam-1595	16	7	6	6	NUM
ejpam-1595	16	8	(	(	PUNCT
ejpam-1595	16	9	2013	2013	NUM
ejpam-1595	16	10	)	)	PUNCT
ejpam-1595	16	11	,	,	PUNCT
ejpam-1595	16	12	53	53	NUM
ejpam-1595	16	13	-	-	SYM
ejpam-1595	16	14	58	58	NUM
ejpam-1595	16	15	54	54	NUM
ejpam-1595	16	16	2	2	NUM
ejpam-1595	16	17	.	.	PUNCT
ejpam-1595	16	18	semi	semi	ADJ
ejpam-1595	16	19	-	-	NOUN
ejpam-1595	16	20	openness	openness	NOUN
ejpam-1595	16	21	with	with	ADP
ejpam-1595	16	22	respect	respect	NOUN
ejpam-1595	16	23	to	to	ADP
ejpam-1595	16	24	an	an	DET
ejpam-1595	16	25	ideal	ideal	NOUN
ejpam-1595	16	26	let	let	VERB
ejpam-1595	16	27	x	x	PRON
ejpam-1595	16	28	be	be	AUX
ejpam-1595	16	29	a	a	DET
ejpam-1595	16	30	topological	topological	ADJ
ejpam-1595	16	31	space	space	NOUN
ejpam-1595	16	32	.	.	PUNCT
ejpam-1595	17	1	recall	recall	VERB
ejpam-1595	17	2	that	that	SCONJ
ejpam-1595	17	3	a	a	DET
ejpam-1595	17	4	subset	subset	NOUN
ejpam-1595	17	5	a	a	PRON
ejpam-1595	17	6	of	of	ADP
ejpam-1595	17	7	x	x	SYM
ejpam-1595	17	8	is	be	AUX
ejpam-1595	17	9	said	say	VERB
ejpam-1595	17	10	to	to	PART
ejpam-1595	17	11	be	be	AUX
ejpam-1595	17	12	semi	semi	ADJ
ejpam-1595	17	13	-	-	ADJ
ejpam-1595	17	14	open	open	ADJ
ejpam-1595	17	15	[	[	X
ejpam-1595	17	16	11	11	NUM
ejpam-1595	17	17	]	]	X
ejpam-1595	17	18	if	if	SCONJ
ejpam-1595	17	19	there	there	PRON
ejpam-1595	17	20	is	be	VERB
ejpam-1595	17	21	an	an	DET
ejpam-1595	17	22	open	open	ADJ
ejpam-1595	17	23	set	set	NOUN
ejpam-1595	17	24	u	u	PRON
ejpam-1595	17	25	such	such	ADJ
ejpam-1595	17	26	that	that	SCONJ
ejpam-1595	17	27	u	u	NOUN
ejpam-1595	17	28	⊆	⊆	NUM
ejpam-1595	17	29	a⊂	a⊂	NOUN
ejpam-1595	17	30	cl(u	cl(u	NOUN
ejpam-1595	17	31	)	)	PUNCT
ejpam-1595	17	32	.	.	PUNCT
ejpam-1595	18	1	this	this	PRON
ejpam-1595	18	2	motivates	motivate	VERB
ejpam-1595	18	3	our	our	PRON
ejpam-1595	18	4	first	first	ADJ
ejpam-1595	18	5	definition	definition	NOUN
ejpam-1595	18	6	.	.	PUNCT
ejpam-1595	19	1	definition	definition	NOUN
ejpam-1595	19	2	1	1	NUM
ejpam-1595	19	3	.	.	PUNCT
ejpam-1595	20	1	a	a	DET
ejpam-1595	20	2	subset	subset	NOUN
ejpam-1595	20	3	a	a	PRON
ejpam-1595	20	4	of	of	ADP
ejpam-1595	20	5	x	x	SYM
ejpam-1595	20	6	is	be	AUX
ejpam-1595	20	7	said	say	VERB
ejpam-1595	20	8	to	to	PART
ejpam-1595	20	9	be	be	AUX
ejpam-1595	20	10	semi	semi	ADJ
ejpam-1595	20	11	-	-	ADJ
ejpam-1595	20	12	open	open	ADJ
ejpam-1595	20	13	with	with	ADP
ejpam-1595	20	14	respect	respect	NOUN
ejpam-1595	20	15	to	to	ADP
ejpam-1595	20	16	an	an	DET
ejpam-1595	20	17	ideal	ideal	NOUN
ejpam-1595	20	18	i	i	PRON
ejpam-1595	20	19	(	(	PUNCT
ejpam-1595	20	20	written	write	VERB
ejpam-1595	20	21	as	as	SCONJ
ejpam-1595	20	22	i	i	PRON
ejpam-1595	20	23	-semi	-semi	VERB
ejpam-1595	20	24	-	-	PUNCT
ejpam-1595	20	25	open	open	ADJ
ejpam-1595	20	26	)	)	PUNCT
ejpam-1595	20	27	if	if	SCONJ
ejpam-1595	20	28	there	there	PRON
ejpam-1595	20	29	exists	exist	VERB
ejpam-1595	20	30	an	an	DET
ejpam-1595	20	31	open	open	ADJ
ejpam-1595	20	32	set	set	NOUN
ejpam-1595	20	33	u	u	PRON
ejpam-1595	20	34	such	such	ADJ
ejpam-1595	20	35	that	that	PRON
ejpam-1595	20	36	u	u	PROPN
ejpam-1595	20	37	−	−	PROPN
ejpam-1595	20	38	a∈	a∈	PROPN
ejpam-1595	20	39	i	i	PROPN
ejpam-1595	20	40	and	and	CCONJ
ejpam-1595	20	41	a−	a−	PROPN
ejpam-1595	20	42	cl(u	cl(u	X
ejpam-1595	20	43	)	)	PUNCT
ejpam-1595	20	44	∈	∈	PROPN
ejpam-1595	21	1	i	i	PRON
ejpam-1595	21	2	.	.	PUNCT
ejpam-1595	22	1	if	if	SCONJ
ejpam-1595	22	2	a	a	DET
ejpam-1595	22	3	∈	∈	X
ejpam-1595	22	4	i	i	PRON
ejpam-1595	22	5	,	,	PUNCT
ejpam-1595	22	6	then	then	ADV
ejpam-1595	22	7	it	it	PRON
ejpam-1595	22	8	is	be	AUX
ejpam-1595	22	9	easy	easy	ADJ
ejpam-1595	22	10	to	to	PART
ejpam-1595	22	11	see	see	VERB
ejpam-1595	22	12	that	that	SCONJ
ejpam-1595	22	13	a	a	PRON
ejpam-1595	22	14	is	be	AUX
ejpam-1595	22	15	i	i	PRON
ejpam-1595	22	16	-semi	-semi	NOUN
ejpam-1595	22	17	-	-	PUNCT
ejpam-1595	22	18	open	open	ADJ
ejpam-1595	22	19	.	.	PUNCT
ejpam-1595	23	1	moreover	moreover	ADV
ejpam-1595	23	2	,	,	PUNCT
ejpam-1595	23	3	every	every	DET
ejpam-1595	23	4	open	open	NOUN
ejpam-1595	23	5	set	set	VERB
ejpam-1595	23	6	a	a	PRON
ejpam-1595	23	7	is	be	AUX
ejpam-1595	23	8	semi	semi	ADJ
ejpam-1595	23	9	-	-	ADJ
ejpam-1595	23	10	open	open	ADJ
ejpam-1595	23	11	,	,	PUNCT
ejpam-1595	23	12	and	and	CCONJ
ejpam-1595	23	13	every	every	DET
ejpam-1595	23	14	semi	semi	ADJ
ejpam-1595	23	15	-	-	ADJ
ejpam-1595	23	16	open	open	ADJ
ejpam-1595	23	17	set	set	NOUN
ejpam-1595	23	18	b	b	PROPN
ejpam-1595	23	19	is	be	AUX
ejpam-1595	23	20	i	i	PRON
ejpam-1595	23	21	-semi	-semi	NOUN
ejpam-1595	23	22	-	-	PUNCT
ejpam-1595	23	23	open	open	ADJ
ejpam-1595	23	24	,	,	PUNCT
ejpam-1595	23	25	for	for	ADP
ejpam-1595	23	26	any	any	DET
ejpam-1595	23	27	ideal	ideal	NOUN
ejpam-1595	23	28	i	i	PRON
ejpam-1595	23	29	on	on	ADP
ejpam-1595	23	30	x	x	X
ejpam-1595	23	31	.	.	PUNCT
ejpam-1595	23	32	example	example	NOUN
ejpam-1595	24	1	1	1	NUM
ejpam-1595	24	2	.	.	X
ejpam-1595	24	3	consider	consider	VERB
ejpam-1595	24	4	a	a	DET
ejpam-1595	24	5	topological	topological	ADJ
ejpam-1595	24	6	space	space	NOUN
ejpam-1595	24	7	(	(	PUNCT
ejpam-1595	24	8	x	x	X
ejpam-1595	24	9	,	,	PUNCT
ejpam-1595	24	10	τ	τ	PROPN
ejpam-1595	24	11	)	)	PUNCT
ejpam-1595	24	12	;	;	PUNCT
ejpam-1595	24	13	x	x	SYM
ejpam-1595	24	14	=	=	X
ejpam-1595	24	15	{	{	PUNCT
ejpam-1595	24	16	a	a	PRON
ejpam-1595	24	17	,	,	PUNCT
ejpam-1595	24	18	b	b	NOUN
ejpam-1595	24	19	,	,	PUNCT
ejpam-1595	24	20	c	c	NOUN
ejpam-1595	24	21	}	}	PUNCT
ejpam-1595	24	22	and	and	CCONJ
ejpam-1595	24	23	τ	τ	PROPN
ejpam-1595	24	24	=	=	SYM
ejpam-1595	24	25	{	{	PUNCT
ejpam-1595	24	26	∅	∅	NOUN
ejpam-1595	24	27	,	,	PUNCT
ejpam-1595	24	28	{	{	PUNCT
ejpam-1595	24	29	a	a	X
ejpam-1595	24	30	}	}	PUNCT
ejpam-1595	24	31	,	,	PUNCT
ejpam-1595	24	32	{	{	PUNCT
ejpam-1595	24	33	a	a	X
ejpam-1595	24	34	,	,	PUNCT
ejpam-1595	24	35	c	c	NOUN
ejpam-1595	24	36	}	}	PUNCT
ejpam-1595	24	37	,	,	PUNCT
ejpam-1595	24	38	x	x	SYM
ejpam-1595	24	39	}	}	PUNCT
ejpam-1595	24	40	.	.	PUNCT
ejpam-1595	25	1	choose	choose	VERB
ejpam-1595	25	2	i	i	PRON
ejpam-1595	25	3	=	=	SYM
ejpam-1595	25	4	{	{	PUNCT
ejpam-1595	25	5	∅	∅	NOUN
ejpam-1595	25	6	,	,	PUNCT
ejpam-1595	25	7	{	{	PUNCT
ejpam-1595	25	8	b	b	NOUN
ejpam-1595	25	9	}	}	PUNCT
ejpam-1595	25	10	,	,	PUNCT
ejpam-1595	25	11	{	{	PUNCT
ejpam-1595	25	12	c	c	X
ejpam-1595	25	13	}	}	PUNCT
ejpam-1595	25	14	,	,	PUNCT
ejpam-1595	25	15	{	{	PUNCT
ejpam-1595	25	16	b	b	X
ejpam-1595	25	17	,	,	PUNCT
ejpam-1595	25	18	c	c	NOUN
ejpam-1595	25	19	}	}	PUNCT
ejpam-1595	25	20	}	}	PUNCT
ejpam-1595	25	21	,	,	PUNCT
ejpam-1595	25	22	and	and	CCONJ
ejpam-1595	25	23	observe	observe	VERB
ejpam-1595	25	24	that	that	SCONJ
ejpam-1595	25	25	{	{	PUNCT
ejpam-1595	25	26	b	b	X
ejpam-1595	25	27	}	}	PUNCT
ejpam-1595	25	28	is	be	AUX
ejpam-1595	25	29	i	i	PRON
ejpam-1595	25	30	-semi	-semi	NOUN
ejpam-1595	25	31	-	-	PUNCT
ejpam-1595	25	32	open	open	ADJ
ejpam-1595	25	33	;	;	PUNCT
ejpam-1595	25	34	however	however	ADV
ejpam-1595	25	35	,	,	PUNCT
ejpam-1595	25	36	{	{	PUNCT
ejpam-1595	25	37	b	b	X
ejpam-1595	25	38	}	}	PUNCT
ejpam-1595	25	39	is	be	AUX
ejpam-1595	25	40	not	not	PART
ejpam-1595	25	41	semiopen	semiopen	ADJ
ejpam-1595	25	42	in	in	ADP
ejpam-1595	25	43	the	the	DET
ejpam-1595	25	44	sense	sense	NOUN
ejpam-1595	25	45	of	of	ADP
ejpam-1595	25	46	[	[	X
ejpam-1595	25	47	11	11	NUM
ejpam-1595	25	48	]	]	PUNCT
ejpam-1595	25	49	as	as	SCONJ
ejpam-1595	25	50	there	there	PRON
ejpam-1595	25	51	is	be	VERB
ejpam-1595	25	52	no	no	DET
ejpam-1595	25	53	open	open	ADJ
ejpam-1595	25	54	set	set	NOUN
ejpam-1595	25	55	u	u	PRON
ejpam-1595	25	56	such	such	ADJ
ejpam-1595	25	57	that	that	SCONJ
ejpam-1595	25	58	u	u	PROPN
ejpam-1595	25	59	⊂	⊂	X
ejpam-1595	25	60	{	{	PUNCT
ejpam-1595	25	61	b	b	X
ejpam-1595	25	62	}	}	PUNCT
ejpam-1595	25	63	⊂	⊂	NOUN
ejpam-1595	25	64	cl(u	cl(u	PROPN
ejpam-1595	25	65	)	)	PUNCT
ejpam-1595	25	66	.	.	PUNCT
ejpam-1595	26	1	thus	thus	ADV
ejpam-1595	26	2	,	,	PUNCT
ejpam-1595	26	3	if	if	SCONJ
ejpam-1595	26	4	a	a	DET
ejpam-1595	26	5	set	set	NOUN
ejpam-1595	26	6	is	be	AUX
ejpam-1595	26	7	i	i	PRON
ejpam-1595	26	8	-semi	-semi	NOUN
ejpam-1595	26	9	-	-	PUNCT
ejpam-1595	26	10	open	open	ADJ
ejpam-1595	26	11	,	,	PUNCT
ejpam-1595	26	12	it	it	PRON
ejpam-1595	26	13	may	may	AUX
ejpam-1595	26	14	not	not	PART
ejpam-1595	26	15	be	be	AUX
ejpam-1595	26	16	semi	semi	ADJ
ejpam-1595	26	17	-	-	ADJ
ejpam-1595	26	18	open	open	ADJ
ejpam-1595	26	19	in	in	ADP
ejpam-1595	26	20	the	the	DET
ejpam-1595	26	21	usual	usual	ADJ
ejpam-1595	26	22	sense	sense	NOUN
ejpam-1595	26	23	.	.	PUNCT
ejpam-1595	27	1	for	for	ADP
ejpam-1595	27	2	an	an	DET
ejpam-1595	27	3	ideal	ideal	NOUN
ejpam-1595	27	4	i	i	PRON
ejpam-1595	27	5	that	that	PRON
ejpam-1595	27	6	is	be	AUX
ejpam-1595	27	7	not	not	PART
ejpam-1595	27	8	countably	countably	ADV
ejpam-1595	27	9	additive	additive	ADJ
ejpam-1595	27	10	,	,	PUNCT
ejpam-1595	27	11	the	the	DET
ejpam-1595	27	12	concepts	concept	NOUN
ejpam-1595	27	13	of	of	ADP
ejpam-1595	27	14	semi	semi	ADJ
ejpam-1595	27	15	-	-	NOUN
ejpam-1595	27	16	openness	openness	NOUN
ejpam-1595	27	17	and	and	CCONJ
ejpam-1595	27	18	i	i	PRON
ejpam-1595	27	19	-semiopenness	-semiopenness	ADJ
ejpam-1595	27	20	coincide	coincide	NOUN
ejpam-1595	27	21	in	in	ADP
ejpam-1595	27	22	the	the	DET
ejpam-1595	27	23	following	follow	VERB
ejpam-1595	27	24	case	case	NOUN
ejpam-1595	27	25	.	.	PUNCT
ejpam-1595	28	1	theorem	theorem	NOUN
ejpam-1595	28	2	1	1	NUM
ejpam-1595	28	3	.	.	X
ejpam-1595	29	1	for	for	ADP
ejpam-1595	29	2	an	an	DET
ejpam-1595	29	3	ideal	ideal	NOUN
ejpam-1595	29	4	i	i	PRON
ejpam-1595	29	5	on	on	ADP
ejpam-1595	29	6	a	a	DET
ejpam-1595	29	7	topological	topological	ADJ
ejpam-1595	29	8	space	space	NOUN
ejpam-1595	29	9	x	x	SYM
ejpam-1595	29	10	,	,	PUNCT
ejpam-1595	29	11	the	the	DET
ejpam-1595	29	12	following	follow	VERB
ejpam-1595	29	13	are	be	AUX
ejpam-1595	29	14	equivalent	equivalent	ADJ
ejpam-1595	29	15	:	:	PUNCT
ejpam-1595	30	1	1	1	X
ejpam-1595	30	2	.	.	X
ejpam-1595	31	1	i	i	PRON
ejpam-1595	31	2	is	be	AUX
ejpam-1595	31	3	the	the	DET
ejpam-1595	31	4	minimal	minimal	ADJ
ejpam-1595	31	5	ideal	ideal	NOUN
ejpam-1595	31	6	on	on	ADP
ejpam-1595	31	7	x	x	SYM
ejpam-1595	31	8	,	,	PUNCT
ejpam-1595	31	9	that	that	ADV
ejpam-1595	31	10	is	is	ADV
ejpam-1595	31	11	,	,	PUNCT
ejpam-1595	31	12	i	i	PRON
ejpam-1595	31	13	=	=	NOUN
ejpam-1595	31	14	{	{	PUNCT
ejpam-1595	31	15	∅	∅	NOUN
ejpam-1595	31	16	}	}	PUNCT
ejpam-1595	31	17	;	;	PUNCT
ejpam-1595	32	1	2	2	X
ejpam-1595	32	2	.	.	X
ejpam-1595	32	3	the	the	DET
ejpam-1595	32	4	concepts	concept	NOUN
ejpam-1595	32	5	of	of	ADP
ejpam-1595	32	6	semi	semi	ADJ
ejpam-1595	32	7	-	-	NOUN
ejpam-1595	32	8	openness	openness	NOUN
ejpam-1595	32	9	and	and	CCONJ
ejpam-1595	32	10	i	i	PRON
ejpam-1595	32	11	-semi	-semi	NOUN
ejpam-1595	32	12	-	-	PUNCT
ejpam-1595	32	13	openness	openness	NOUN
ejpam-1595	32	14	are	be	AUX
ejpam-1595	32	15	the	the	DET
ejpam-1595	32	16	same	same	ADJ
ejpam-1595	32	17	.	.	PUNCT
ejpam-1595	33	1	proof	proof	NOUN
ejpam-1595	33	2	.	.	PUNCT
ejpam-1595	34	1	first	first	ADV
ejpam-1595	34	2	suppose	suppose	VERB
ejpam-1595	34	3	that	that	SCONJ
ejpam-1595	34	4	i	i	PRON
ejpam-1595	34	5	=	=	PUNCT
ejpam-1595	34	6	{	{	PUNCT
ejpam-1595	34	7	∅	∅	NOUN
ejpam-1595	34	8	}	}	PUNCT
ejpam-1595	34	9	.	.	PUNCT
ejpam-1595	35	1	it	it	PRON
ejpam-1595	35	2	suffices	suffice	VERB
ejpam-1595	35	3	to	to	PART
ejpam-1595	35	4	show	show	VERB
ejpam-1595	35	5	that	that	SCONJ
ejpam-1595	35	6	whenever	whenever	SCONJ
ejpam-1595	35	7	a	a	DET
ejpam-1595	35	8	set	set	NOUN
ejpam-1595	35	9	a	a	PRON
ejpam-1595	35	10	is	be	AUX
ejpam-1595	35	11	i	i	PRON
ejpam-1595	35	12	-semiopen	-semiopen	ADJ
ejpam-1595	35	13	,	,	PUNCT
ejpam-1595	35	14	then	then	ADV
ejpam-1595	35	15	it	it	PRON
ejpam-1595	35	16	is	be	AUX
ejpam-1595	35	17	semi	semi	ADJ
ejpam-1595	35	18	-	-	ADJ
ejpam-1595	35	19	open	open	ADJ
ejpam-1595	35	20	in	in	ADP
ejpam-1595	35	21	the	the	DET
ejpam-1595	35	22	usual	usual	ADJ
ejpam-1595	35	23	sense	sense	NOUN
ejpam-1595	35	24	.	.	PUNCT
ejpam-1595	36	1	indeed	indeed	ADV
ejpam-1595	36	2	,	,	PUNCT
ejpam-1595	36	3	if	if	SCONJ
ejpam-1595	36	4	a	a	PRON
ejpam-1595	36	5	is	be	AUX
ejpam-1595	36	6	i	i	PRON
ejpam-1595	36	7	-semi	-semi	NOUN
ejpam-1595	36	8	-	-	PUNCT
ejpam-1595	36	9	open	open	ADJ
ejpam-1595	36	10	,	,	PUNCT
ejpam-1595	36	11	then	then	ADV
ejpam-1595	36	12	there	there	PRON
ejpam-1595	36	13	is	be	VERB
ejpam-1595	36	14	an	an	DET
ejpam-1595	36	15	open	open	ADJ
ejpam-1595	36	16	set	set	NOUN
ejpam-1595	36	17	u	u	PRON
ejpam-1595	36	18	such	such	ADJ
ejpam-1595	36	19	that	that	DET
ejpam-1595	36	20	u	u	NOUN
ejpam-1595	36	21	−	−	PROPN
ejpam-1595	36	22	a	a	PRON
ejpam-1595	36	23	,	,	PUNCT
ejpam-1595	36	24	a−	a−	PROPN
ejpam-1595	36	25	cl(u	cl(u	X
ejpam-1595	36	26	)	)	PUNCT
ejpam-1595	36	27	∈	∈	PROPN
ejpam-1595	37	1	i	i	PRON
ejpam-1595	37	2	=	=	NOUN
ejpam-1595	37	3	{	{	PUNCT
ejpam-1595	37	4	∅	∅	NOUN
ejpam-1595	37	5	}	}	PUNCT
ejpam-1595	37	6	,	,	PUNCT
ejpam-1595	37	7	and	and	CCONJ
ejpam-1595	37	8	so	so	ADV
ejpam-1595	37	9	u	u	PRON
ejpam-1595	37	10	⊂	⊂	PROPN
ejpam-1595	37	11	a	a	DET
ejpam-1595	37	12	⊂	⊂	PROPN
ejpam-1595	37	13	cl(u	cl(u	PROPN
ejpam-1595	37	14	)	)	PUNCT
ejpam-1595	37	15	,	,	PUNCT
ejpam-1595	37	16	proving	prove	VERB
ejpam-1595	37	17	that	that	SCONJ
ejpam-1595	37	18	a	a	PRON
ejpam-1595	37	19	is	be	AUX
ejpam-1595	37	20	semi	semi	ADJ
ejpam-1595	37	21	-	-	ADJ
ejpam-1595	37	22	open	open	ADJ
ejpam-1595	37	23	.	.	PUNCT
ejpam-1595	38	1	conversely	conversely	ADV
ejpam-1595	38	2	,	,	PUNCT
ejpam-1595	38	3	suppose	suppose	VERB
ejpam-1595	38	4	that	that	SCONJ
ejpam-1595	38	5	whenever	whenever	SCONJ
ejpam-1595	38	6	a	a	DET
ejpam-1595	38	7	set	set	NOUN
ejpam-1595	38	8	a	a	PRON
ejpam-1595	38	9	is	be	AUX
ejpam-1595	38	10	i	i	PRON
ejpam-1595	38	11	-semi	-semi	NOUN
ejpam-1595	38	12	-	-	PUNCT
ejpam-1595	38	13	open	open	ADJ
ejpam-1595	38	14	,	,	PUNCT
ejpam-1595	38	15	then	then	ADV
ejpam-1595	38	16	it	it	PRON
ejpam-1595	38	17	is	be	AUX
ejpam-1595	38	18	semi	semi	ADJ
ejpam-1595	38	19	-	-	ADJ
ejpam-1595	38	20	open	open	ADJ
ejpam-1595	38	21	.	.	PUNCT
ejpam-1595	39	1	let	let	VERB
ejpam-1595	39	2	b	b	X
ejpam-1595	39	3	∈	∈	PROPN
ejpam-1595	40	1	i	i	PRON
ejpam-1595	40	2	.	.	PUNCT
ejpam-1595	41	1	then	then	ADV
ejpam-1595	41	2	,	,	PUNCT
ejpam-1595	41	3	b	b	PROPN
ejpam-1595	41	4	is	be	AUX
ejpam-1595	41	5	i	i	PRON
ejpam-1595	41	6	-semi	-semi	NOUN
ejpam-1595	41	7	-	-	PUNCT
ejpam-1595	41	8	open	open	ADJ
ejpam-1595	41	9	,	,	PUNCT
ejpam-1595	41	10	and	and	CCONJ
ejpam-1595	41	11	by	by	ADP
ejpam-1595	41	12	assumption	assumption	NOUN
ejpam-1595	41	13	,	,	PUNCT
ejpam-1595	41	14	b	b	PROPN
ejpam-1595	41	15	is	be	AUX
ejpam-1595	41	16	semi	semi	ADJ
ejpam-1595	41	17	-	-	ADJ
ejpam-1595	41	18	open	open	ADJ
ejpam-1595	41	19	.	.	PUNCT
ejpam-1595	42	1	thus	thus	ADV
ejpam-1595	42	2	,	,	PUNCT
ejpam-1595	42	3	there	there	PRON
ejpam-1595	42	4	is	be	VERB
ejpam-1595	42	5	an	an	DET
ejpam-1595	42	6	open	open	ADJ
ejpam-1595	42	7	set	set	VERB
ejpam-1595	42	8	v1	v1	NOUN
ejpam-1595	42	9	such	such	ADJ
ejpam-1595	42	10	that	that	DET
ejpam-1595	42	11	v1	v1	PROPN
ejpam-1595	42	12	⊂	⊂	PROPN
ejpam-1595	42	13	b	b	X
ejpam-1595	42	14	⊂	⊂	ADJ
ejpam-1595	42	15	cl(v1	cl(v1	NOUN
ejpam-1595	42	16	)	)	PUNCT
ejpam-1595	42	17	.	.	PUNCT
ejpam-1595	43	1	since	since	SCONJ
ejpam-1595	43	2	b	b	PROPN
ejpam-1595	43	3	∈	∈	PROPN
ejpam-1595	43	4	i	i	PRON
ejpam-1595	43	5	and	and	CCONJ
ejpam-1595	43	6	v1	v1	VERB
ejpam-1595	43	7	⊂	⊂	PROPN
ejpam-1595	43	8	b	b	PROPN
ejpam-1595	43	9	,	,	PUNCT
ejpam-1595	43	10	we	we	PRON
ejpam-1595	43	11	have	have	VERB
ejpam-1595	43	12	that	that	DET
ejpam-1595	43	13	v1	v1	NOUN
ejpam-1595	43	14	∈	∈	PROPN
ejpam-1595	44	1	i	i	PRON
ejpam-1595	44	2	,	,	PUNCT
ejpam-1595	44	3	and	and	CCONJ
ejpam-1595	44	4	so	so	ADV
ejpam-1595	44	5	b∪v1	b∪v1	PUNCT
ejpam-1595	44	6	∈	∈	PROPN
ejpam-1595	45	1	i	i	PRON
ejpam-1595	45	2	.	.	PUNCT
ejpam-1595	46	1	as	as	SCONJ
ejpam-1595	46	2	b∪v1	b∪v1	NOUN
ejpam-1595	46	3	is	be	AUX
ejpam-1595	46	4	i	i	PRON
ejpam-1595	46	5	-semi	-semi	NOUN
ejpam-1595	46	6	-	-	PUNCT
ejpam-1595	46	7	open	open	ADJ
ejpam-1595	46	8	,	,	PUNCT
ejpam-1595	46	9	it	it	PRON
ejpam-1595	46	10	is	be	AUX
ejpam-1595	46	11	semi	semi	ADJ
ejpam-1595	46	12	-	-	ADJ
ejpam-1595	46	13	open	open	ADJ
ejpam-1595	46	14	,	,	PUNCT
ejpam-1595	46	15	so	so	SCONJ
ejpam-1595	46	16	that	that	SCONJ
ejpam-1595	46	17	there	there	PRON
ejpam-1595	46	18	is	be	VERB
ejpam-1595	46	19	an	an	DET
ejpam-1595	46	20	open	open	ADJ
ejpam-1595	46	21	set	set	NOUN
ejpam-1595	46	22	v2	v2	NOUN
ejpam-1595	46	23	for	for	ADP
ejpam-1595	46	24	which	which	PRON
ejpam-1595	46	25	v2	v2	X
ejpam-1595	46	26	⊂	⊂	X
ejpam-1595	46	27	(	(	PUNCT
ejpam-1595	46	28	b	b	PROPN
ejpam-1595	46	29	∪	∪	X
ejpam-1595	46	30	v1	v1	NOUN
ejpam-1595	46	31	)	)	PUNCT
ejpam-1595	46	32	⊂	⊂	PROPN
ejpam-1595	46	33	cl(v2	cl(v2	NOUN
ejpam-1595	46	34	)	)	PUNCT
ejpam-1595	46	35	.	.	PUNCT
ejpam-1595	47	1	similarly	similarly	ADV
ejpam-1595	47	2	,	,	PUNCT
ejpam-1595	47	3	there	there	PRON
ejpam-1595	47	4	is	be	VERB
ejpam-1595	47	5	an	an	DET
ejpam-1595	47	6	open	open	ADJ
ejpam-1595	47	7	set	set	VERB
ejpam-1595	47	8	v3	v3	PROPN
ejpam-1595	47	9	such	such	ADJ
ejpam-1595	47	10	that	that	SCONJ
ejpam-1595	47	11	v3	v3	PROPN
ejpam-1595	47	12	⊂	⊂	PROPN
ejpam-1595	47	13	(	(	PUNCT
ejpam-1595	47	14	b	b	X
ejpam-1595	47	15	∪	∪	VERB
ejpam-1595	47	16	v1	v1	PROPN
ejpam-1595	47	17	∪	∪	NOUN
ejpam-1595	47	18	v2)⊂	v2)⊂	NOUN
ejpam-1595	47	19	cl(v3	cl(v3	NOUN
ejpam-1595	47	20	)	)	PUNCT
ejpam-1595	47	21	.	.	PUNCT
ejpam-1595	48	1	continuing	continue	VERB
ejpam-1595	48	2	in	in	ADP
ejpam-1595	48	3	this	this	DET
ejpam-1595	48	4	way	way	NOUN
ejpam-1595	48	5	,	,	PUNCT
ejpam-1595	48	6	we	we	PRON
ejpam-1595	48	7	have	have	VERB
ejpam-1595	48	8	an	an	DET
ejpam-1595	48	9	infinite	infinite	ADJ
ejpam-1595	48	10	collection	collection	NOUN
ejpam-1595	48	11	of	of	ADP
ejpam-1595	48	12	open	open	ADJ
ejpam-1595	48	13	sets	set	NOUN
ejpam-1595	48	14	v1	v1	NOUN
ejpam-1595	48	15	,	,	PUNCT
ejpam-1595	48	16	v2	v2	PROPN
ejpam-1595	48	17	,	,	PUNCT
ejpam-1595	48	18	v3	v3	PROPN
ejpam-1595	48	19	,	,	PUNCT
ejpam-1595	48	20	.	.	PUNCT
ejpam-1595	48	21	.	.	PUNCT
ejpam-1595	49	1	.	.	PUNCT
ejpam-1595	50	1	,	,	PUNCT
ejpam-1595	50	2	such	such	ADJ
ejpam-1595	50	3	that	that	SCONJ
ejpam-1595	50	4	b	b	NOUN
ejpam-1595	50	5	∪	∪	X
ejpam-1595	50	6	v1	v1	NOUN
ejpam-1595	50	7	∪	∪	ADP
ejpam-1595	50	8	v2	v2	PROPN
ejpam-1595	50	9	∪	∪	X
ejpam-1595	50	10	v3	v3	PROPN
ejpam-1595	50	11	∪	∪	NOUN
ejpam-1595	50	12	.	.	PUNCT
ejpam-1595	50	13	.	.	PUNCT
ejpam-1595	50	14	.	.	PUNCT
ejpam-1595	51	1	∈	∈	PROPN
ejpam-1595	52	1	i	i	PRON
ejpam-1595	52	2	,	,	PUNCT
ejpam-1595	52	3	which	which	PRON
ejpam-1595	52	4	is	be	AUX
ejpam-1595	52	5	impossible	impossible	ADJ
ejpam-1595	52	6	,	,	PUNCT
ejpam-1595	52	7	as	as	ADP
ejpam-1595	52	8	the	the	DET
ejpam-1595	52	9	ideal	ideal	NOUN
ejpam-1595	52	10	i	i	PRON
ejpam-1595	52	11	is	be	AUX
ejpam-1595	52	12	not	not	PART
ejpam-1595	52	13	closed	close	VERB
ejpam-1595	52	14	under	under	ADP
ejpam-1595	52	15	countable	countable	ADJ
ejpam-1595	52	16	additivity	additivity	NOUN
ejpam-1595	52	17	.	.	PUNCT
ejpam-1595	53	1	thus	thus	ADV
ejpam-1595	53	2	,	,	PUNCT
ejpam-1595	53	3	it	it	PRON
ejpam-1595	53	4	must	must	AUX
ejpam-1595	53	5	be	be	AUX
ejpam-1595	53	6	the	the	DET
ejpam-1595	53	7	case	case	NOUN
ejpam-1595	53	8	that	that	SCONJ
ejpam-1595	53	9	v1	v1	NOUN
ejpam-1595	53	10	=	=	SYM
ejpam-1595	53	11	∅	∅	NOUN
ejpam-1595	53	12	(	(	PUNCT
ejpam-1595	53	13	similarly	similarly	ADV
ejpam-1595	53	14	for	for	ADP
ejpam-1595	53	15	the	the	DET
ejpam-1595	53	16	other	other	ADJ
ejpam-1595	53	17	vi	vi	PROPN
ejpam-1595	53	18	’s	’s	NOUN
ejpam-1595	53	19	)	)	PUNCT
ejpam-1595	53	20	;	;	PUNCT
ejpam-1595	53	21	therefore	therefore	ADV
ejpam-1595	53	22	,	,	PUNCT
ejpam-1595	53	23	cl(v1	cl(v1	ADJ
ejpam-1595	53	24	)	)	PUNCT
ejpam-1595	53	25	=	=	SYM
ejpam-1595	53	26	∅	∅	NOUN
ejpam-1595	53	27	,	,	PUNCT
ejpam-1595	53	28	and	and	CCONJ
ejpam-1595	53	29	the	the	DET
ejpam-1595	53	30	relations	relation	NOUN
ejpam-1595	53	31	v1	v1	VERB
ejpam-1595	53	32	⊂	⊂	PROPN
ejpam-1595	53	33	b	b	X
ejpam-1595	53	34	⊂	⊂	X
ejpam-1595	53	35	cl(v1	cl(v1	X
ejpam-1595	53	36	)	)	PUNCT
ejpam-1595	53	37	then	then	ADV
ejpam-1595	53	38	give	give	VERB
ejpam-1595	53	39	b	b	NOUN
ejpam-1595	53	40	=	=	NOUN
ejpam-1595	53	41	∅	∅	NOUN
ejpam-1595	53	42	,	,	PUNCT
ejpam-1595	53	43	proving	prove	VERB
ejpam-1595	53	44	that	that	SCONJ
ejpam-1595	53	45	i	i	PRON
ejpam-1595	53	46	=	=	PUNCT
ejpam-1595	53	47	{	{	PUNCT
ejpam-1595	53	48	∅	∅	NOUN
ejpam-1595	53	49	}	}	PUNCT
ejpam-1595	53	50	.	.	PUNCT
ejpam-1595	54	1	proposition	proposition	NOUN
ejpam-1595	54	2	1	1	NUM
ejpam-1595	54	3	.	.	PUNCT
ejpam-1595	55	1	let	let	VERB
ejpam-1595	55	2	i	i	PRON
ejpam-1595	55	3	and	and	CCONJ
ejpam-1595	55	4	i	i	PRON
ejpam-1595	55	5	′	′	VERB
ejpam-1595	55	6	be	be	VERB
ejpam-1595	55	7	two	two	NUM
ejpam-1595	55	8	ideals	ideal	NOUN
ejpam-1595	55	9	on	on	ADP
ejpam-1595	55	10	a	a	DET
ejpam-1595	55	11	topological	topological	ADJ
ejpam-1595	55	12	space	space	NOUN
ejpam-1595	56	1	x	x	X
ejpam-1595	56	2	.	.	PUNCT
ejpam-1595	57	1	1	1	X
ejpam-1595	57	2	.	.	X
ejpam-1595	58	1	if	if	SCONJ
ejpam-1595	58	2	i	i	PRON
ejpam-1595	58	3	⊂	⊂	VERB
ejpam-1595	58	4	i	i	NOUN
ejpam-1595	58	5	′	′	VERB
ejpam-1595	58	6	,	,	PUNCT
ejpam-1595	58	7	then	then	ADV
ejpam-1595	58	8	every	every	DET
ejpam-1595	58	9	i	i	NOUN
ejpam-1595	58	10	-semi	-semi	NOUN
ejpam-1595	58	11	-	-	PUNCT
ejpam-1595	58	12	open	open	ADJ
ejpam-1595	58	13	set	set	NOUN
ejpam-1595	58	14	a	a	PRON
ejpam-1595	58	15	is	be	AUX
ejpam-1595	58	16	i	i	PRON
ejpam-1595	58	17	′-semi	′-semi	NOUN
ejpam-1595	58	18	-	-	PUNCT
ejpam-1595	58	19	open	open	ADJ
ejpam-1595	58	20	;	;	PUNCT
ejpam-1595	58	21	2	2	X
ejpam-1595	58	22	.	.	X
ejpam-1595	59	1	if	if	SCONJ
ejpam-1595	59	2	a	a	PRON
ejpam-1595	59	3	is	be	AUX
ejpam-1595	59	4	(	(	PUNCT
ejpam-1595	59	5	i	i	PRON
ejpam-1595	59	6	∩i	∩i	VERB
ejpam-1595	59	7	′)-semi	′)-semi	ADJ
ejpam-1595	59	8	-	-	ADJ
ejpam-1595	59	9	open	open	ADJ
ejpam-1595	59	10	,	,	PUNCT
ejpam-1595	59	11	then	then	ADV
ejpam-1595	59	12	it	it	PRON
ejpam-1595	59	13	is	be	AUX
ejpam-1595	59	14	simultaneously	simultaneously	ADV
ejpam-1595	59	15	i	i	PRON
ejpam-1595	59	16	-semi	-semi	NOUN
ejpam-1595	59	17	-	-	PUNCT
ejpam-1595	59	18	open	open	ADJ
ejpam-1595	59	19	and	and	CCONJ
ejpam-1595	59	20	i	i	PRON
ejpam-1595	59	21	′-semi	′-semi	PROPN
ejpam-1595	59	22	-	-	PUNCT
ejpam-1595	59	23	open	open	ADJ
ejpam-1595	59	24	.	.	PUNCT
ejpam-1595	60	1	corollary	corollary	ADJ
ejpam-1595	60	2	1	1	NUM
ejpam-1595	60	3	.	.	PUNCT
ejpam-1595	61	1	for	for	ADP
ejpam-1595	61	2	a	a	DET
ejpam-1595	61	3	subset	subset	NOUN
ejpam-1595	61	4	a	a	PRON
ejpam-1595	61	5	of	of	ADP
ejpam-1595	61	6	x	x	X
ejpam-1595	61	7	and	and	CCONJ
ejpam-1595	61	8	an	an	DET
ejpam-1595	61	9	ideal	ideal	NOUN
ejpam-1595	61	10	i	i	PRON
ejpam-1595	61	11	on	on	ADP
ejpam-1595	61	12	x	x	SYM
ejpam-1595	61	13	,	,	PUNCT
ejpam-1595	61	14	recall	recall	VERB
ejpam-1595	61	15	that	that	PRON
ejpam-1595	61	16	ia	ia	PROPN
ejpam-1595	61	17	=	=	PRON
ejpam-1595	61	18	{	{	PUNCT
ejpam-1595	62	1	a∩	a∩	PROPN
ejpam-1595	62	2	s|s	s|s	NOUN
ejpam-1595	62	3	∈	∈	PROPN
ejpam-1595	62	4	i	i	PRON
ejpam-1595	62	5	}	}	PUNCT
ejpam-1595	62	6	is	be	AUX
ejpam-1595	62	7	also	also	ADV
ejpam-1595	62	8	an	an	DET
ejpam-1595	62	9	ideal	ideal	NOUN
ejpam-1595	62	10	on	on	ADP
ejpam-1595	62	11	x	x	X
ejpam-1595	62	12	.	.	PUNCT
ejpam-1595	63	1	1	1	X
ejpam-1595	63	2	.	.	X
ejpam-1595	64	1	if	if	SCONJ
ejpam-1595	64	2	a	a	DET
ejpam-1595	64	3	set	set	NOUN
ejpam-1595	64	4	b	b	NOUN
ejpam-1595	64	5	is	be	AUX
ejpam-1595	64	6	ia	ia	ADJ
ejpam-1595	64	7	-	-	PUNCT
ejpam-1595	64	8	semi	semi	ADV
ejpam-1595	64	9	-	-	ADJ
ejpam-1595	64	10	open	open	ADJ
ejpam-1595	64	11	,	,	PUNCT
ejpam-1595	64	12	then	then	ADV
ejpam-1595	64	13	it	it	PRON
ejpam-1595	64	14	is	be	AUX
ejpam-1595	64	15	i	i	PRON
ejpam-1595	64	16	-semi	-semi	NOUN
ejpam-1595	64	17	-	-	PUNCT
ejpam-1595	64	18	open	open	ADJ
ejpam-1595	64	19	.	.	PUNCT
ejpam-1595	65	1	f.	f.	PROPN
ejpam-1595	65	2	michael	michael	PROPN
ejpam-1595	65	3	/	/	SYM
ejpam-1595	65	4	eur	eur	PROPN
ejpam-1595	65	5	.	.	PUNCT
ejpam-1595	66	1	j.	j.	PROPN
ejpam-1595	66	2	pure	pure	PROPN
ejpam-1595	66	3	appl	appl	PROPN
ejpam-1595	66	4	.	.	PROPN
ejpam-1595	66	5	math	math	PROPN
ejpam-1595	66	6	,	,	PUNCT
ejpam-1595	66	7	6	6	NUM
ejpam-1595	66	8	(	(	PUNCT
ejpam-1595	66	9	2013	2013	NUM
ejpam-1595	66	10	)	)	PUNCT
ejpam-1595	66	11	,	,	PUNCT
ejpam-1595	66	12	53	53	NUM
ejpam-1595	66	13	-	-	SYM
ejpam-1595	66	14	58	58	NUM
ejpam-1595	66	15	55	55	NUM
ejpam-1595	66	16	2	2	NUM
ejpam-1595	66	17	.	.	PUNCT
ejpam-1595	67	1	if	if	SCONJ
ejpam-1595	67	2	a=	a=	NOUN
ejpam-1595	67	3	∅	∅	NOUN
ejpam-1595	67	4	,	,	PUNCT
ejpam-1595	67	5	then	then	ADV
ejpam-1595	67	6	ia	ia	PROPN
ejpam-1595	67	7	=	=	SYM
ejpam-1595	67	8	i∅	i∅	PUNCT
ejpam-1595	67	9	=	=	X
ejpam-1595	67	10	{	{	PUNCT
ejpam-1595	67	11	∅	∅	NOUN
ejpam-1595	67	12	}	}	PUNCT
ejpam-1595	67	13	,	,	PUNCT
ejpam-1595	67	14	the	the	DET
ejpam-1595	67	15	minimal	minimal	ADJ
ejpam-1595	67	16	ideal	ideal	NOUN
ejpam-1595	67	17	.	.	PUNCT
ejpam-1595	68	1	thus	thus	ADV
ejpam-1595	68	2	,	,	PUNCT
ejpam-1595	68	3	if	if	SCONJ
ejpam-1595	68	4	a	a	DET
ejpam-1595	68	5	set	set	NOUN
ejpam-1595	68	6	c	c	NOUN
ejpam-1595	68	7	is	be	AUX
ejpam-1595	68	8	i∅-semi	i∅-semi	NOUN
ejpam-1595	68	9	-	-	PUNCT
ejpam-1595	68	10	open	open	ADJ
ejpam-1595	68	11	,	,	PUNCT
ejpam-1595	68	12	then	then	ADV
ejpam-1595	68	13	c	c	PROPN
ejpam-1595	68	14	is	be	AUX
ejpam-1595	68	15	also	also	ADV
ejpam-1595	68	16	i	i	PRON
ejpam-1595	68	17	-semi	-semi	NOUN
ejpam-1595	68	18	-	-	PUNCT
ejpam-1595	68	19	open	open	ADJ
ejpam-1595	68	20	.	.	PUNCT
ejpam-1595	69	1	proposition	proposition	NOUN
ejpam-1595	69	2	2	2	NUM
ejpam-1595	69	3	.	.	PUNCT
ejpam-1595	70	1	if	if	SCONJ
ejpam-1595	70	2	a	a	PRON
ejpam-1595	70	3	and	and	CCONJ
ejpam-1595	70	4	b	b	NOUN
ejpam-1595	70	5	are	be	AUX
ejpam-1595	70	6	both	both	PRON
ejpam-1595	70	7	i	i	PRON
ejpam-1595	70	8	-semi	-semi	VERB
ejpam-1595	70	9	-	-	PUNCT
ejpam-1595	70	10	open	open	ADJ
ejpam-1595	70	11	,	,	PUNCT
ejpam-1595	70	12	then	then	ADV
ejpam-1595	70	13	so	so	ADV
ejpam-1595	70	14	is	be	AUX
ejpam-1595	70	15	their	their	PRON
ejpam-1595	70	16	union	union	NOUN
ejpam-1595	70	17	a∪	a∪	PROPN
ejpam-1595	70	18	b.	b.	PROPN
ejpam-1595	70	19	proof	proof	PROPN
ejpam-1595	70	20	.	.	PUNCT
ejpam-1595	71	1	let	let	VERB
ejpam-1595	71	2	the	the	DET
ejpam-1595	71	3	given	give	VERB
ejpam-1595	71	4	conditions	condition	NOUN
ejpam-1595	71	5	hold	hold	VERB
ejpam-1595	71	6	.	.	PUNCT
ejpam-1595	72	1	to	to	PART
ejpam-1595	72	2	show	show	VERB
ejpam-1595	72	3	that	that	SCONJ
ejpam-1595	72	4	a	a	DET
ejpam-1595	72	5	∪	∪	ADJ
ejpam-1595	72	6	b	b	NOUN
ejpam-1595	72	7	is	be	AUX
ejpam-1595	72	8	i	i	PRON
ejpam-1595	72	9	-semi	-semi	NOUN
ejpam-1595	72	10	-	-	PUNCT
ejpam-1595	72	11	open	open	ADJ
ejpam-1595	72	12	,	,	PUNCT
ejpam-1595	72	13	we	we	PRON
ejpam-1595	72	14	need	need	VERB
ejpam-1595	72	15	to	to	PART
ejpam-1595	72	16	produce	produce	VERB
ejpam-1595	72	17	an	an	DET
ejpam-1595	72	18	open	open	ADJ
ejpam-1595	72	19	set	set	NOUN
ejpam-1595	72	20	u	u	PRON
ejpam-1595	72	21	such	such	ADJ
ejpam-1595	72	22	that	that	DET
ejpam-1595	72	23	u	u	NOUN
ejpam-1595	72	24	−	−	PROPN
ejpam-1595	73	1	(	(	PUNCT
ejpam-1595	73	2	a∪	a∪	NOUN
ejpam-1595	73	3	b	b	X
ejpam-1595	73	4	)	)	PUNCT
ejpam-1595	73	5	∈	∈	PROPN
ejpam-1595	74	1	i	i	PRON
ejpam-1595	74	2	and	and	CCONJ
ejpam-1595	74	3	(	(	PUNCT
ejpam-1595	74	4	a∪	a∪	INTJ
ejpam-1595	74	5	b)−	b)−	PROPN
ejpam-1595	74	6	cl(u	cl(u	NOUN
ejpam-1595	74	7	)	)	PUNCT
ejpam-1595	74	8	∈	∈	PROPN
ejpam-1595	75	1	i	i	PRON
ejpam-1595	75	2	.	.	PUNCT
ejpam-1595	76	1	since	since	SCONJ
ejpam-1595	76	2	a	a	PRON
ejpam-1595	76	3	and	and	CCONJ
ejpam-1595	76	4	b	b	NOUN
ejpam-1595	76	5	are	be	AUX
ejpam-1595	76	6	both	both	PRON
ejpam-1595	76	7	i	i	PRON
ejpam-1595	76	8	-semi	-semi	VERB
ejpam-1595	76	9	-	-	PUNCT
ejpam-1595	76	10	open	open	ADJ
ejpam-1595	76	11	,	,	PUNCT
ejpam-1595	76	12	there	there	PRON
ejpam-1595	76	13	are	be	VERB
ejpam-1595	76	14	open	open	ADJ
ejpam-1595	76	15	sets	set	NOUN
ejpam-1595	76	16	u1	u1	NOUN
ejpam-1595	76	17	and	and	CCONJ
ejpam-1595	76	18	u2	u2	NOUN
ejpam-1595	76	19	such	such	ADJ
ejpam-1595	76	20	that	that	DET
ejpam-1595	76	21	u1	u1	PROPN
ejpam-1595	76	22	−	−	PROPN
ejpam-1595	76	23	a∈	a∈	PROPN
ejpam-1595	77	1	i	i	PRON
ejpam-1595	77	2	,	,	PUNCT
ejpam-1595	77	3	a−	a−	PROPN
ejpam-1595	77	4	cl(u1	cl(u1	NOUN
ejpam-1595	77	5	)	)	PUNCT
ejpam-1595	77	6	∈	∈	PROPN
ejpam-1595	78	1	i	i	PRON
ejpam-1595	78	2	,	,	PUNCT
ejpam-1595	78	3	u2	u2	PROPN
ejpam-1595	78	4	−	−	PROPN
ejpam-1595	78	5	b	b	PROPN
ejpam-1595	78	6	∈	∈	PROPN
ejpam-1595	79	1	i	i	PRON
ejpam-1595	79	2	,	,	PUNCT
ejpam-1595	79	3	b−	b−	PROPN
ejpam-1595	79	4	cl(u2	cl(u2	PROPN
ejpam-1595	79	5	)	)	PUNCT
ejpam-1595	79	6	∈	∈	PROPN
ejpam-1595	80	1	i	i	PRON
ejpam-1595	80	2	.	.	PUNCT
ejpam-1595	81	1	choose	choose	VERB
ejpam-1595	81	2	u	u	NOUN
ejpam-1595	81	3	=	=	NOUN
ejpam-1595	81	4	u1	u1	PROPN
ejpam-1595	81	5	∪	∪	NOUN
ejpam-1595	81	6	u2	u2	NOUN
ejpam-1595	81	7	,	,	PUNCT
ejpam-1595	81	8	and	and	CCONJ
ejpam-1595	81	9	observe	observe	VERB
ejpam-1595	81	10	that	that	SCONJ
ejpam-1595	81	11	(	(	PUNCT
ejpam-1595	81	12	u1	u1	NOUN
ejpam-1595	81	13	∪	∪	VERB
ejpam-1595	81	14	u2)−	u2)−	PROPN
ejpam-1595	81	15	(	(	PUNCT
ejpam-1595	81	16	a∪	a∪	NOUN
ejpam-1595	81	17	b	b	X
ejpam-1595	81	18	)	)	PUNCT
ejpam-1595	81	19	=	=	SYM
ejpam-1595	81	20	(	(	PUNCT
ejpam-1595	81	21	(	(	PUNCT
ejpam-1595	81	22	u1−	u1−	PROPN
ejpam-1595	81	23	a)−	a)−	PROPN
ejpam-1595	81	24	b)∪	b)∪	NOUN
ejpam-1595	81	25	(	(	PUNCT
ejpam-1595	81	26	(	(	PUNCT
ejpam-1595	81	27	u2−	u2−	NOUN
ejpam-1595	81	28	b)−	b)−	PROPN
ejpam-1595	81	29	a	a	PRON
ejpam-1595	81	30	)	)	PUNCT
ejpam-1595	81	31	∈	∈	PROPN
ejpam-1595	82	1	i	i	PRON
ejpam-1595	82	2	.	.	PUNCT
ejpam-1595	83	1	also	also	ADV
ejpam-1595	83	2	,	,	PUNCT
ejpam-1595	83	3	(	(	PUNCT
ejpam-1595	83	4	a∪	a∪	X
ejpam-1595	83	5	b)−	b)−	PROPN
ejpam-1595	83	6	cl(u1	cl(u1	NOUN
ejpam-1595	83	7	∪	∪	X
ejpam-1595	83	8	u2	u2	NOUN
ejpam-1595	83	9	)	)	PUNCT
ejpam-1595	83	10	=	=	SYM
ejpam-1595	83	11	(	(	PUNCT
ejpam-1595	83	12	(	(	PUNCT
ejpam-1595	83	13	a−	a−	ADJ
ejpam-1595	83	14	cl(u1))−	cl(u1))−	NOUN
ejpam-1595	83	15	cl(u2))∪	cl(u2))∪	NOUN
ejpam-1595	83	16	(	(	PUNCT
ejpam-1595	83	17	(	(	PUNCT
ejpam-1595	83	18	b−	b−	PROPN
ejpam-1595	83	19	cl(u2))−	cl(u2))−	NOUN
ejpam-1595	83	20	cl(u1	cl(u1	NOUN
ejpam-1595	83	21	)	)	PUNCT
ejpam-1595	83	22	)	)	PUNCT
ejpam-1595	84	1	∈	∈	PROPN
ejpam-1595	85	1	i	i	PRON
ejpam-1595	85	2	.	.	PUNCT
ejpam-1595	86	1	therefore	therefore	ADV
ejpam-1595	86	2	,	,	PUNCT
ejpam-1595	86	3	by	by	ADP
ejpam-1595	86	4	definition	definition	NOUN
ejpam-1595	86	5	,	,	PUNCT
ejpam-1595	86	6	a∪	a∪	PROPN
ejpam-1595	86	7	b	b	NOUN
ejpam-1595	86	8	is	be	AUX
ejpam-1595	86	9	i	i	PRON
ejpam-1595	86	10	-semi	-semi	NOUN
ejpam-1595	86	11	-	-	PUNCT
ejpam-1595	86	12	open	open	ADJ
ejpam-1595	86	13	.	.	PUNCT
ejpam-1595	87	1	proposition	proposition	NOUN
ejpam-1595	87	2	3	3	X
ejpam-1595	87	3	.	.	PUNCT
ejpam-1595	88	1	let	let	VERB
ejpam-1595	88	2	x	x	PRON
ejpam-1595	88	3	be	be	AUX
ejpam-1595	88	4	a	a	DET
ejpam-1595	88	5	topological	topological	ADJ
ejpam-1595	88	6	space	space	NOUN
ejpam-1595	88	7	in	in	ADP
ejpam-1595	88	8	which	which	PRON
ejpam-1595	88	9	there	there	PRON
ejpam-1595	88	10	is	be	VERB
ejpam-1595	88	11	an	an	DET
ejpam-1595	88	12	open	open	ADJ
ejpam-1595	88	13	singleton	singleton	NOUN
ejpam-1595	88	14	subset	subset	NOUN
ejpam-1595	88	15	{	{	PUNCT
ejpam-1595	88	16	a	a	PRON
ejpam-1595	88	17	}	}	PUNCT
ejpam-1595	88	18	satisfying	satisfy	VERB
ejpam-1595	88	19	cl({a	cl({a	ADJ
ejpam-1595	88	20	}	}	PUNCT
ejpam-1595	88	21	)	)	PUNCT
ejpam-1595	89	1	=	=	PUNCT
ejpam-1595	89	2	x	x	X
ejpam-1595	89	3	.	.	PUNCT
ejpam-1595	90	1	for	for	ADP
ejpam-1595	90	2	any	any	DET
ejpam-1595	90	3	ideal	ideal	NOUN
ejpam-1595	90	4	i	i	PRON
ejpam-1595	90	5	on	on	ADP
ejpam-1595	90	6	x	x	PUNCT
ejpam-1595	90	7	with	with	ADP
ejpam-1595	90	8	{	{	PUNCT
ejpam-1595	90	9	a	a	PRON
ejpam-1595	90	10	}	}	PUNCT
ejpam-1595	90	11	∈	∈	NOUN
ejpam-1595	90	12	i	i	PRON
ejpam-1595	90	13	,	,	PUNCT
ejpam-1595	90	14	we	we	PRON
ejpam-1595	90	15	have	have	VERB
ejpam-1595	90	16	that	that	PRON
ejpam-1595	90	17	:	:	PUNCT
ejpam-1595	91	1	1	1	X
ejpam-1595	91	2	.	.	X
ejpam-1595	92	1	every	every	DET
ejpam-1595	92	2	singleton	singleton	NOUN
ejpam-1595	92	3	subset	subset	NOUN
ejpam-1595	92	4	of	of	ADP
ejpam-1595	92	5	x	x	SYM
ejpam-1595	92	6	is	be	AUX
ejpam-1595	92	7	i	i	PRON
ejpam-1595	92	8	-semi	-semi	ADV
ejpam-1595	92	9	-	-	PUNCT
ejpam-1595	92	10	open	open	ADJ
ejpam-1595	92	11	;	;	PUNCT
ejpam-1595	92	12	2	2	X
ejpam-1595	92	13	.	.	X
ejpam-1595	93	1	every	every	DET
ejpam-1595	93	2	finite	finite	PROPN
ejpam-1595	93	3	subset	subset	NOUN
ejpam-1595	93	4	of	of	ADP
ejpam-1595	93	5	x	x	SYM
ejpam-1595	93	6	is	be	AUX
ejpam-1595	93	7	i	i	PRON
ejpam-1595	93	8	-semi	-semi	NOUN
ejpam-1595	93	9	-	-	PUNCT
ejpam-1595	93	10	open	open	ADJ
ejpam-1595	93	11	.	.	PUNCT
ejpam-1595	94	1	proof	proof	NOUN
ejpam-1595	94	2	.	.	PUNCT
ejpam-1595	95	1	let	let	VERB
ejpam-1595	95	2	the	the	DET
ejpam-1595	95	3	given	give	VERB
ejpam-1595	95	4	conditions	condition	NOUN
ejpam-1595	95	5	hold	hold	VERB
ejpam-1595	95	6	.	.	PUNCT
ejpam-1595	96	1	suppose	suppose	VERB
ejpam-1595	96	2	that	that	SCONJ
ejpam-1595	96	3	{	{	PUNCT
ejpam-1595	96	4	s	s	X
ejpam-1595	96	5	}	}	PUNCT
ejpam-1595	96	6	is	be	AUX
ejpam-1595	96	7	a	a	DET
ejpam-1595	96	8	singleton	singleton	NOUN
ejpam-1595	96	9	subset	subset	NOUN
ejpam-1595	96	10	of	of	ADP
ejpam-1595	96	11	x	x	X
ejpam-1595	96	12	.	.	PUNCT
ejpam-1595	97	1	since	since	SCONJ
ejpam-1595	97	2	(	(	PUNCT
ejpam-1595	97	3	{	{	PUNCT
ejpam-1595	97	4	a	a	PRON
ejpam-1595	97	5	}	}	PUNCT
ejpam-1595	97	6	is	be	AUX
ejpam-1595	97	7	open	open	ADJ
ejpam-1595	97	8	and	and	CCONJ
ejpam-1595	97	9	)	)	PUNCT
ejpam-1595	97	10	{	{	PUNCT
ejpam-1595	97	11	a	a	NOUN
ejpam-1595	97	12	}	}	PUNCT
ejpam-1595	97	13	−	−	PROPN
ejpam-1595	97	14	{	{	PUNCT
ejpam-1595	97	15	s	s	NOUN
ejpam-1595	97	16	}	}	PUNCT
ejpam-1595	97	17	=	=	SYM
ejpam-1595	97	18	{	{	PUNCT
ejpam-1595	97	19	a	a	PRON
ejpam-1595	97	20	}	}	PUNCT
ejpam-1595	97	21	∈	∈	NOUN
ejpam-1595	97	22	i	i	PRON
ejpam-1595	97	23	,	,	PUNCT
ejpam-1595	97	24	and	and	CCONJ
ejpam-1595	97	25	{	{	PUNCT
ejpam-1595	97	26	s	s	NOUN
ejpam-1595	97	27	}	}	PUNCT
ejpam-1595	97	28	−	−	PROPN
ejpam-1595	97	29	cl({a	cl({a	ADJ
ejpam-1595	97	30	}	}	PUNCT
ejpam-1595	97	31	)	)	PUNCT
ejpam-1595	97	32	=	=	PRON
ejpam-1595	97	33	{	{	PUNCT
ejpam-1595	97	34	s	s	NOUN
ejpam-1595	97	35	}	}	PUNCT
ejpam-1595	98	1	−	−	NOUN
ejpam-1595	98	2	x	x	SYM
ejpam-1595	98	3	=	=	NOUN
ejpam-1595	98	4	∅	∅	NOUN
ejpam-1595	98	5	∈	∈	PROPN
ejpam-1595	98	6	i	i	PRON
ejpam-1595	98	7	,	,	PUNCT
ejpam-1595	98	8	it	it	PRON
ejpam-1595	98	9	follows	follow	VERB
ejpam-1595	98	10	that	that	SCONJ
ejpam-1595	98	11	{	{	PUNCT
ejpam-1595	98	12	s	s	X
ejpam-1595	98	13	}	}	PUNCT
ejpam-1595	98	14	is	be	AUX
ejpam-1595	98	15	i	i	PRON
ejpam-1595	98	16	-semi	-semi	NOUN
ejpam-1595	98	17	-	-	PUNCT
ejpam-1595	98	18	open	open	ADJ
ejpam-1595	98	19	;	;	PUNCT
ejpam-1595	98	20	this	this	PRON
ejpam-1595	98	21	proves	prove	VERB
ejpam-1595	98	22	(	(	PUNCT
ejpam-1595	98	23	1	1	NUM
ejpam-1595	98	24	)	)	PUNCT
ejpam-1595	98	25	.	.	PUNCT
ejpam-1595	99	1	to	to	PART
ejpam-1595	99	2	see	see	VERB
ejpam-1595	99	3	that	that	PRON
ejpam-1595	99	4	(	(	PUNCT
ejpam-1595	99	5	2	2	X
ejpam-1595	99	6	)	)	PUNCT
ejpam-1595	99	7	holds	hold	NOUN
ejpam-1595	99	8	,	,	PUNCT
ejpam-1595	99	9	let	let	VERB
ejpam-1595	99	10	a=	a=	VERB
ejpam-1595	99	11	{	{	PUNCT
ejpam-1595	99	12	s1	s1	NOUN
ejpam-1595	99	13	,	,	PUNCT
ejpam-1595	99	14	s2	s2	PROPN
ejpam-1595	99	15	,	,	PUNCT
ejpam-1595	99	16	s3	s3	PROPN
ejpam-1595	99	17	,	,	PUNCT
ejpam-1595	99	18	.	.	PUNCT
ejpam-1595	99	19	.	.	PUNCT
ejpam-1595	100	1	.	.	PUNCT
ejpam-1595	101	1	,	,	PUNCT
ejpam-1595	101	2	sn	sn	AUX
ejpam-1595	101	3	}	}	PUNCT
ejpam-1595	101	4	be	be	AUX
ejpam-1595	101	5	a	a	DET
ejpam-1595	101	6	finite	finite	NOUN
ejpam-1595	101	7	subset	subset	NOUN
ejpam-1595	101	8	of	of	ADP
ejpam-1595	101	9	x	x	X
ejpam-1595	101	10	.	.	PUNCT
ejpam-1595	102	1	since	since	SCONJ
ejpam-1595	102	2	a=	a=	ADV
ejpam-1595	102	3	{	{	PUNCT
ejpam-1595	102	4	s1	s1	NOUN
ejpam-1595	102	5	}	}	PUNCT
ejpam-1595	102	6	∪	∪	NOUN
ejpam-1595	102	7	{	{	PUNCT
ejpam-1595	102	8	s2	s2	NOUN
ejpam-1595	102	9	}	}	PUNCT
ejpam-1595	102	10	∪	∪	NOUN
ejpam-1595	102	11	{	{	PUNCT
ejpam-1595	102	12	s3	s3	NOUN
ejpam-1595	102	13	}	}	PUNCT
ejpam-1595	102	14	∪	∪	NOUN
ejpam-1595	102	15	.	.	PUNCT
ejpam-1595	102	16	.	.	PUNCT
ejpam-1595	102	17	.	.	PUNCT
ejpam-1595	103	1	∪	∪	PROPN
ejpam-1595	103	2	{	{	PUNCT
ejpam-1595	103	3	sn	sn	NOUN
ejpam-1595	103	4	}	}	PUNCT
ejpam-1595	103	5	,	,	PUNCT
ejpam-1595	103	6	the	the	DET
ejpam-1595	103	7	result	result	NOUN
ejpam-1595	103	8	follows	follow	VERB
ejpam-1595	103	9	from	from	ADP
ejpam-1595	103	10	the	the	DET
ejpam-1595	103	11	fact	fact	NOUN
ejpam-1595	103	12	that	that	SCONJ
ejpam-1595	103	13	each	each	DET
ejpam-1595	103	14	singleton	singleton	NOUN
ejpam-1595	103	15	subset	subset	VERB
ejpam-1595	103	16	{	{	PUNCT
ejpam-1595	103	17	si	si	NOUN
ejpam-1595	103	18	}	}	PUNCT
ejpam-1595	103	19	(	(	PUNCT
ejpam-1595	103	20	i	i	NOUN
ejpam-1595	103	21	=	=	NOUN
ejpam-1595	103	22	1,2,3	1,2,3	NUM
ejpam-1595	103	23	,	,	PUNCT
ejpam-1595	103	24	.	.	PUNCT
ejpam-1595	103	25	.	.	PUNCT
ejpam-1595	104	1	.	.	PUNCT
ejpam-1595	105	1	,	,	PUNCT
ejpam-1595	105	2	n	n	CCONJ
ejpam-1595	105	3	)	)	PUNCT
ejpam-1595	105	4	is	be	AUX
ejpam-1595	105	5	i	i	PRON
ejpam-1595	105	6	-semi	-semi	NOUN
ejpam-1595	105	7	-	-	PUNCT
ejpam-1595	105	8	open	open	ADJ
ejpam-1595	105	9	and	and	CCONJ
ejpam-1595	105	10	a	a	DET
ejpam-1595	105	11	repeated	repeat	VERB
ejpam-1595	105	12	use	use	NOUN
ejpam-1595	105	13	of	of	ADP
ejpam-1595	105	14	proposition	proposition	NOUN
ejpam-1595	105	15	2	2	NUM
ejpam-1595	105	16	above	above	ADV
ejpam-1595	105	17	.	.	PUNCT
ejpam-1595	106	1	proposition	proposition	NOUN
ejpam-1595	106	2	3	3	NUM
ejpam-1595	106	3	does	do	AUX
ejpam-1595	106	4	not	not	PART
ejpam-1595	106	5	hold	hold	VERB
ejpam-1595	106	6	for	for	ADP
ejpam-1595	106	7	any	any	DET
ejpam-1595	106	8	choice	choice	NOUN
ejpam-1595	106	9	of	of	ADP
ejpam-1595	106	10	ideal	ideal	NOUN
ejpam-1595	106	11	.	.	PUNCT
ejpam-1595	107	1	example	example	NOUN
ejpam-1595	108	1	2	2	NUM
ejpam-1595	108	2	.	.	X
ejpam-1595	108	3	consider	consider	VERB
ejpam-1595	108	4	x	x	PUNCT
ejpam-1595	108	5	=	=	PRON
ejpam-1595	108	6	{	{	PUNCT
ejpam-1595	108	7	a	a	DET
ejpam-1595	108	8	,	,	PUNCT
ejpam-1595	108	9	b	b	NOUN
ejpam-1595	108	10	,	,	PUNCT
ejpam-1595	108	11	c	c	NOUN
ejpam-1595	108	12	}	}	PUNCT
ejpam-1595	108	13	,	,	PUNCT
ejpam-1595	108	14	τ	τ	X
ejpam-1595	108	15	=	=	PUNCT
ejpam-1595	108	16	{	{	PUNCT
ejpam-1595	108	17	∅	∅	NOUN
ejpam-1595	108	18	,	,	PUNCT
ejpam-1595	108	19	{	{	PUNCT
ejpam-1595	108	20	a	a	X
ejpam-1595	108	21	}	}	PUNCT
ejpam-1595	108	22	,	,	PUNCT
ejpam-1595	108	23	{	{	PUNCT
ejpam-1595	108	24	a	a	X
ejpam-1595	108	25	,	,	PUNCT
ejpam-1595	108	26	c	c	NOUN
ejpam-1595	108	27	}	}	PUNCT
ejpam-1595	108	28	,	,	PUNCT
ejpam-1595	108	29	x	x	SYM
ejpam-1595	108	30	}	}	PUNCT
ejpam-1595	108	31	,	,	PUNCT
ejpam-1595	108	32	and	and	CCONJ
ejpam-1595	108	33	observe	observe	VERB
ejpam-1595	108	34	that	that	SCONJ
ejpam-1595	108	35	cl({a	cl({a	ADJ
ejpam-1595	108	36	}	}	PUNCT
ejpam-1595	108	37	)	)	PUNCT
ejpam-1595	109	1	=	=	PUNCT
ejpam-1595	110	1	x	x	X
ejpam-1595	110	2	.	.	PUNCT
ejpam-1595	111	1	if	if	SCONJ
ejpam-1595	111	2	we	we	PRON
ejpam-1595	111	3	choose	choose	VERB
ejpam-1595	111	4	the	the	DET
ejpam-1595	111	5	minimal	minimal	ADJ
ejpam-1595	111	6	ideal	ideal	NOUN
ejpam-1595	111	7	i	i	PRON
ejpam-1595	111	8	=	=	SYM
ejpam-1595	111	9	{	{	PUNCT
ejpam-1595	111	10	∅	∅	NOUN
ejpam-1595	111	11	}	}	PUNCT
ejpam-1595	111	12	on	on	ADP
ejpam-1595	111	13	x	x	X
ejpam-1595	111	14	,	,	PUNCT
ejpam-1595	111	15	then	then	ADV
ejpam-1595	111	16	the	the	DET
ejpam-1595	111	17	singleton	singleton	PROPN
ejpam-1595	111	18	subset	subset	NOUN
ejpam-1595	111	19	{	{	PUNCT
ejpam-1595	111	20	b	b	NOUN
ejpam-1595	111	21	}	}	PUNCT
ejpam-1595	111	22	is	be	AUX
ejpam-1595	111	23	not	not	PART
ejpam-1595	111	24	i	i	PRON
ejpam-1595	111	25	-semi	-semi	ADV
ejpam-1595	111	26	-	-	PUNCT
ejpam-1595	111	27	open	open	ADJ
ejpam-1595	111	28	,	,	PUNCT
ejpam-1595	111	29	as	as	SCONJ
ejpam-1595	111	30	there	there	PRON
ejpam-1595	111	31	is	be	VERB
ejpam-1595	111	32	no	no	DET
ejpam-1595	111	33	open	open	ADJ
ejpam-1595	111	34	set	set	NOUN
ejpam-1595	111	35	u	u	NOUN
ejpam-1595	111	36	satisfying	satisfy	VERB
ejpam-1595	111	37	u	u	NOUN
ejpam-1595	111	38	−	−	PROPN
ejpam-1595	111	39	{	{	PUNCT
ejpam-1595	111	40	b	b	NOUN
ejpam-1595	111	41	}	}	PUNCT
ejpam-1595	111	42	∈	∈	NOUN
ejpam-1595	112	1	i	i	PRON
ejpam-1595	112	2	and	and	CCONJ
ejpam-1595	112	3	{	{	PUNCT
ejpam-1595	112	4	b	b	NOUN
ejpam-1595	112	5	}	}	PUNCT
ejpam-1595	112	6	−	−	NOUN
ejpam-1595	112	7	cl(u	cl(u	NOUN
ejpam-1595	112	8	)	)	PUNCT
ejpam-1595	112	9	∈	∈	NOUN
ejpam-1595	112	10	i	i	PRON
ejpam-1595	112	11	simultaneously	simultaneously	ADV
ejpam-1595	112	12	.	.	PUNCT
ejpam-1595	113	1	proposition	proposition	NOUN
ejpam-1595	113	2	4	4	NUM
ejpam-1595	113	3	.	.	PUNCT
ejpam-1595	114	1	let	let	VERB
ejpam-1595	114	2	a	a	PRON
ejpam-1595	114	3	and	and	CCONJ
ejpam-1595	114	4	b	b	NOUN
ejpam-1595	114	5	be	be	AUX
ejpam-1595	114	6	subsets	subset	NOUN
ejpam-1595	114	7	of	of	ADP
ejpam-1595	114	8	a	a	DET
ejpam-1595	114	9	topological	topological	ADJ
ejpam-1595	114	10	space	space	NOUN
ejpam-1595	114	11	x	x	PUNCT
ejpam-1595	114	12	such	such	ADJ
ejpam-1595	114	13	that	that	SCONJ
ejpam-1595	114	14	a	a	PRON
ejpam-1595	114	15	is	be	AUX
ejpam-1595	114	16	open	open	ADJ
ejpam-1595	114	17	,	,	PUNCT
ejpam-1595	114	18	a⊂	a⊂	NOUN
ejpam-1595	114	19	b	b	NOUN
ejpam-1595	114	20	,	,	PUNCT
ejpam-1595	114	21	and	and	CCONJ
ejpam-1595	114	22	a	a	PRON
ejpam-1595	114	23	is	be	AUX
ejpam-1595	114	24	dense	dense	ADJ
ejpam-1595	114	25	in	in	ADP
ejpam-1595	114	26	b	b	NOUN
ejpam-1595	114	27	(	(	PUNCT
ejpam-1595	114	28	that	that	PRON
ejpam-1595	114	29	is	is	ADV
ejpam-1595	114	30	,	,	PUNCT
ejpam-1595	114	31	b	b	PROPN
ejpam-1595	114	32	⊂	⊂	PROPN
ejpam-1595	114	33	cl(a	cl(a	X
ejpam-1595	114	34	)	)	PUNCT
ejpam-1595	114	35	)	)	PUNCT
ejpam-1595	114	36	.	.	PUNCT
ejpam-1595	115	1	then	then	ADV
ejpam-1595	115	2	b	b	X
ejpam-1595	115	3	is	be	AUX
ejpam-1595	115	4	i	i	PRON
ejpam-1595	115	5	-semi	-semi	NOUN
ejpam-1595	115	6	-	-	PUNCT
ejpam-1595	115	7	open	open	ADJ
ejpam-1595	115	8	for	for	ADP
ejpam-1595	115	9	any	any	DET
ejpam-1595	115	10	ideal	ideal	NOUN
ejpam-1595	115	11	i	i	PRON
ejpam-1595	115	12	on	on	ADP
ejpam-1595	115	13	x	x	X
ejpam-1595	115	14	.	.	PUNCT
ejpam-1595	116	1	in	in	ADP
ejpam-1595	116	2	particular	particular	ADJ
ejpam-1595	116	3	,	,	PUNCT
ejpam-1595	116	4	the	the	DET
ejpam-1595	116	5	conclusion	conclusion	NOUN
ejpam-1595	116	6	holds	hold	VERB
ejpam-1595	116	7	in	in	ADP
ejpam-1595	116	8	the	the	DET
ejpam-1595	116	9	special	special	ADJ
ejpam-1595	116	10	case	case	NOUN
ejpam-1595	116	11	when	when	SCONJ
ejpam-1595	116	12	b	b	PROPN
ejpam-1595	116	13	=	=	SYM
ejpam-1595	116	14	cl(a	cl(a	X
ejpam-1595	116	15	)	)	PUNCT
ejpam-1595	116	16	.	.	PUNCT
ejpam-1595	117	1	remark	remark	PROPN
ejpam-1595	117	2	1	1	NUM
ejpam-1595	117	3	.	.	PUNCT
ejpam-1595	118	1	if	if	SCONJ
ejpam-1595	118	2	two	two	NUM
ejpam-1595	118	3	sets	set	VERB
ejpam-1595	118	4	a	a	PRON
ejpam-1595	118	5	and	and	CCONJ
ejpam-1595	118	6	b	b	NOUN
ejpam-1595	118	7	are	be	AUX
ejpam-1595	118	8	i	i	PRON
ejpam-1595	118	9	-semi	-semi	NOUN
ejpam-1595	118	10	-	-	PUNCT
ejpam-1595	118	11	open	open	ADJ
ejpam-1595	118	12	,	,	PUNCT
ejpam-1595	118	13	then	then	ADV
ejpam-1595	118	14	their	their	PRON
ejpam-1595	118	15	intersection	intersection	NOUN
ejpam-1595	118	16	a∩	a∩	PROPN
ejpam-1595	118	17	b	b	NOUN
ejpam-1595	118	18	need	need	AUX
ejpam-1595	118	19	not	not	PART
ejpam-1595	118	20	be	be	AUX
ejpam-1595	118	21	i	i	PRON
ejpam-1595	118	22	semi	semi	ADJ
ejpam-1595	118	23	-	-	ADJ
ejpam-1595	118	24	open	open	ADJ
ejpam-1595	118	25	.	.	PUNCT
ejpam-1595	119	1	for	for	ADP
ejpam-1595	119	2	example	example	NOUN
ejpam-1595	119	3	,	,	PUNCT
ejpam-1595	119	4	let	let	VERB
ejpam-1595	119	5	x	x	PUNCT
ejpam-1595	119	6	=	=	PRON
ejpam-1595	119	7	{	{	PUNCT
ejpam-1595	119	8	a	a	PRON
ejpam-1595	119	9	,	,	PUNCT
ejpam-1595	119	10	b	b	NOUN
ejpam-1595	119	11	,	,	PUNCT
ejpam-1595	119	12	c	c	AUX
ejpam-1595	119	13	}	}	PUNCT
ejpam-1595	119	14	be	be	AUX
ejpam-1595	119	15	equipped	equip	VERB
ejpam-1595	119	16	with	with	ADP
ejpam-1595	119	17	a	a	DET
ejpam-1595	119	18	topologyσ	topologyσ	NOUN
ejpam-1595	119	19	=	=	SYM
ejpam-1595	119	20	{	{	PUNCT
ejpam-1595	119	21	∅	∅	NOUN
ejpam-1595	119	22	,	,	PUNCT
ejpam-1595	119	23	{	{	PUNCT
ejpam-1595	119	24	a	a	X
ejpam-1595	119	25	}	}	PUNCT
ejpam-1595	119	26	,	,	PUNCT
ejpam-1595	119	27	{	{	PUNCT
ejpam-1595	119	28	c	c	X
ejpam-1595	119	29	}	}	PUNCT
ejpam-1595	119	30	,	,	PUNCT
ejpam-1595	119	31	{	{	PUNCT
ejpam-1595	119	32	a	a	X
ejpam-1595	119	33	,	,	PUNCT
ejpam-1595	119	34	c	c	NOUN
ejpam-1595	119	35	}	}	PUNCT
ejpam-1595	119	36	,	,	PUNCT
ejpam-1595	119	37	x	x	SYM
ejpam-1595	119	38	}	}	PUNCT
ejpam-1595	119	39	.	.	PUNCT
ejpam-1595	120	1	note	note	VERB
ejpam-1595	120	2	that	that	SCONJ
ejpam-1595	120	3	cl({a	cl({a	ADJ
ejpam-1595	120	4	}	}	PUNCT
ejpam-1595	120	5	)	)	PUNCT
ejpam-1595	121	1	=	=	PRON
ejpam-1595	121	2	{	{	PUNCT
ejpam-1595	121	3	a	a	DET
ejpam-1595	121	4	,	,	PUNCT
ejpam-1595	121	5	b	b	NOUN
ejpam-1595	121	6	}	}	PUNCT
ejpam-1595	121	7	and	and	CCONJ
ejpam-1595	121	8	cl({c	cl({c	NOUN
ejpam-1595	121	9	}	}	PUNCT
ejpam-1595	121	10	)	)	PUNCT
ejpam-1595	122	1	=	=	PRON
ejpam-1595	122	2	{	{	PUNCT
ejpam-1595	122	3	b	b	NOUN
ejpam-1595	122	4	,	,	PUNCT
ejpam-1595	122	5	c	c	NOUN
ejpam-1595	122	6	}	}	PUNCT
ejpam-1595	122	7	;	;	PUNCT
ejpam-1595	122	8	moreover	moreover	ADV
ejpam-1595	122	9	,	,	PUNCT
ejpam-1595	122	10	the	the	DET
ejpam-1595	122	11	subsets	subset	NOUN
ejpam-1595	122	12	{	{	PUNCT
ejpam-1595	122	13	a	a	PRON
ejpam-1595	122	14	,	,	PUNCT
ejpam-1595	122	15	b	b	NOUN
ejpam-1595	122	16	}	}	PUNCT
ejpam-1595	122	17	and	and	CCONJ
ejpam-1595	122	18	{	{	PUNCT
ejpam-1595	122	19	b	b	NOUN
ejpam-1595	122	20	,	,	PUNCT
ejpam-1595	122	21	c	c	AUX
ejpam-1595	122	22	}	}	PUNCT
ejpam-1595	122	23	are	be	AUX
ejpam-1595	122	24	semiopen	semiopen	VERB
ejpam-1595	122	25	with	with	ADP
ejpam-1595	122	26	respect	respect	NOUN
ejpam-1595	122	27	to	to	ADP
ejpam-1595	122	28	the	the	DET
ejpam-1595	122	29	minimal	minimal	ADJ
ejpam-1595	122	30	ideal	ideal	NOUN
ejpam-1595	122	31	i	i	PRON
ejpam-1595	122	32	=	=	SYM
ejpam-1595	122	33	{	{	PUNCT
ejpam-1595	122	34	∅	∅	NOUN
ejpam-1595	122	35	}	}	PUNCT
ejpam-1595	122	36	,	,	PUNCT
ejpam-1595	122	37	in	in	ADP
ejpam-1595	122	38	view	view	NOUN
ejpam-1595	122	39	of	of	ADP
ejpam-1595	122	40	proposition	proposition	NOUN
ejpam-1595	122	41	4	4	NUM
ejpam-1595	122	42	above	above	ADV
ejpam-1595	122	43	.	.	PUNCT
ejpam-1595	123	1	however	however	ADV
ejpam-1595	123	2	,	,	PUNCT
ejpam-1595	123	3	the	the	DET
ejpam-1595	123	4	singleton	singleton	PROPN
ejpam-1595	123	5	subset	subset	NOUN
ejpam-1595	123	6	{	{	PUNCT
ejpam-1595	123	7	b	b	NOUN
ejpam-1595	123	8	}	}	PUNCT
ejpam-1595	123	9	=	=	SYM
ejpam-1595	123	10	{	{	PUNCT
ejpam-1595	123	11	a	a	PRON
ejpam-1595	123	12	,	,	PUNCT
ejpam-1595	123	13	b}∩{b	b}∩{b	PROPN
ejpam-1595	123	14	,	,	PUNCT
ejpam-1595	123	15	c	c	NOUN
ejpam-1595	123	16	}	}	PUNCT
ejpam-1595	123	17	is	be	AUX
ejpam-1595	123	18	not	not	PART
ejpam-1595	123	19	semi	semi	ADJ
ejpam-1595	123	20	-	-	ADJ
ejpam-1595	123	21	open	open	ADJ
ejpam-1595	123	22	with	with	ADP
ejpam-1595	123	23	respect	respect	NOUN
ejpam-1595	123	24	to	to	ADP
ejpam-1595	123	25	the	the	DET
ejpam-1595	123	26	minimal	minimal	ADJ
ejpam-1595	123	27	ideal	ideal	NOUN
ejpam-1595	124	1	i	i	PRON
ejpam-1595	124	2	=	=	SYM
ejpam-1595	124	3	{	{	PUNCT
ejpam-1595	124	4	∅	∅	NOUN
ejpam-1595	124	5	}	}	PUNCT
ejpam-1595	124	6	.	.	PUNCT
ejpam-1595	125	1	f.	f.	PROPN
ejpam-1595	125	2	michael	michael	PROPN
ejpam-1595	125	3	/	/	SYM
ejpam-1595	125	4	eur	eur	PROPN
ejpam-1595	125	5	.	.	PUNCT
ejpam-1595	126	1	j.	j.	PROPN
ejpam-1595	126	2	pure	pure	PROPN
ejpam-1595	126	3	appl	appl	PROPN
ejpam-1595	126	4	.	.	PROPN
ejpam-1595	126	5	math	math	PROPN
ejpam-1595	126	6	,	,	PUNCT
ejpam-1595	126	7	6	6	NUM
ejpam-1595	126	8	(	(	PUNCT
ejpam-1595	126	9	2013	2013	NUM
ejpam-1595	126	10	)	)	PUNCT
ejpam-1595	126	11	,	,	PUNCT
ejpam-1595	126	12	53	53	NUM
ejpam-1595	126	13	-	-	SYM
ejpam-1595	126	14	58	58	NUM
ejpam-1595	126	15	56	56	NUM
ejpam-1595	126	16	obviously	obviously	ADV
ejpam-1595	126	17	,	,	PUNCT
ejpam-1595	126	18	if	if	SCONJ
ejpam-1595	126	19	a	a	PRON
ejpam-1595	126	20	,	,	PUNCT
ejpam-1595	126	21	b	b	X
ejpam-1595	126	22	∈	∈	PROPN
ejpam-1595	127	1	i	i	PRON
ejpam-1595	127	2	,	,	PUNCT
ejpam-1595	127	3	then	then	ADV
ejpam-1595	127	4	a∩	a∩	PROPN
ejpam-1595	127	5	b	b	PROPN
ejpam-1595	127	6	will	will	AUX
ejpam-1595	127	7	be	be	AUX
ejpam-1595	127	8	semi	semi	ADJ
ejpam-1595	127	9	-	-	ADJ
ejpam-1595	127	10	open	open	ADJ
ejpam-1595	127	11	with	with	ADP
ejpam-1595	127	12	respect	respect	NOUN
ejpam-1595	127	13	to	to	ADP
ejpam-1595	127	14	the	the	DET
ejpam-1595	127	15	ideal	ideal	NOUN
ejpam-1595	127	16	i	i	PRON
ejpam-1595	127	17	.	.	PUNCT
ejpam-1595	128	1	for	for	ADP
ejpam-1595	128	2	those	those	DET
ejpam-1595	128	3	subsets	subset	NOUN
ejpam-1595	128	4	that	that	PRON
ejpam-1595	128	5	are	be	AUX
ejpam-1595	128	6	not	not	PART
ejpam-1595	128	7	members	member	NOUN
ejpam-1595	128	8	of	of	ADP
ejpam-1595	128	9	the	the	DET
ejpam-1595	128	10	ideal	ideal	NOUN
ejpam-1595	128	11	i	i	PRON
ejpam-1595	128	12	,	,	PUNCT
ejpam-1595	128	13	one	one	NUM
ejpam-1595	128	14	rather	rather	ADV
ejpam-1595	128	15	strong	strong	ADJ
ejpam-1595	128	16	condition	condition	NOUN
ejpam-1595	128	17	for	for	ADP
ejpam-1595	128	18	their	their	PRON
ejpam-1595	128	19	intersection	intersection	NOUN
ejpam-1595	128	20	to	to	PART
ejpam-1595	128	21	be	be	AUX
ejpam-1595	128	22	semi	semi	ADJ
ejpam-1595	128	23	-	-	ADJ
ejpam-1595	128	24	open	open	ADJ
ejpam-1595	128	25	with	with	ADP
ejpam-1595	128	26	respect	respect	NOUN
ejpam-1595	128	27	to	to	ADP
ejpam-1595	128	28	the	the	DET
ejpam-1595	128	29	ideal	ideal	NOUN
ejpam-1595	128	30	i	i	PRON
ejpam-1595	128	31	is	be	AUX
ejpam-1595	128	32	given	give	VERB
ejpam-1595	128	33	below	below	ADV
ejpam-1595	128	34	.	.	PUNCT
ejpam-1595	129	1	proposition	proposition	NOUN
ejpam-1595	129	2	5	5	NUM
ejpam-1595	129	3	.	.	PUNCT
ejpam-1595	130	1	let	let	VERB
ejpam-1595	130	2	i	i	PRON
ejpam-1595	130	3	be	be	AUX
ejpam-1595	130	4	an	an	DET
ejpam-1595	130	5	ideal	ideal	NOUN
ejpam-1595	130	6	on	on	ADP
ejpam-1595	130	7	a	a	DET
ejpam-1595	130	8	topological	topological	ADJ
ejpam-1595	130	9	space	space	NOUN
ejpam-1595	130	10	x	x	SYM
ejpam-1595	130	11	,	,	PUNCT
ejpam-1595	130	12	where	where	SCONJ
ejpam-1595	130	13	every	every	DET
ejpam-1595	130	14	non	non	ADJ
ejpam-1595	130	15	-	-	ADJ
ejpam-1595	130	16	empty	empty	ADJ
ejpam-1595	130	17	open	open	ADJ
ejpam-1595	130	18	subset	subset	NOUN
ejpam-1595	130	19	of	of	ADP
ejpam-1595	130	20	x	x	PUNCT
ejpam-1595	130	21	is	be	AUX
ejpam-1595	130	22	dense	dense	ADJ
ejpam-1595	130	23	,	,	PUNCT
ejpam-1595	130	24	and	and	CCONJ
ejpam-1595	130	25	the	the	DET
ejpam-1595	130	26	collection	collection	NOUN
ejpam-1595	130	27	of	of	ADP
ejpam-1595	130	28	open	open	ADJ
ejpam-1595	130	29	subsets	subset	NOUN
ejpam-1595	130	30	of	of	ADP
ejpam-1595	130	31	x	x	PRON
ejpam-1595	130	32	satisfies	satisfy	VERB
ejpam-1595	130	33	the	the	DET
ejpam-1595	130	34	finite	finite	ADJ
ejpam-1595	130	35	intersection	intersection	NOUN
ejpam-1595	130	36	property	property	NOUN
ejpam-1595	130	37	.	.	PUNCT
ejpam-1595	131	1	1	1	X
ejpam-1595	131	2	.	.	X
ejpam-1595	132	1	if	if	SCONJ
ejpam-1595	132	2	a	a	PRON
ejpam-1595	132	3	is	be	AUX
ejpam-1595	132	4	i	i	PRON
ejpam-1595	132	5	-semi	-semi	NOUN
ejpam-1595	132	6	-	-	PUNCT
ejpam-1595	132	7	open	open	ADJ
ejpam-1595	132	8	and	and	CCONJ
ejpam-1595	132	9	a⊂	a⊂	SYM
ejpam-1595	132	10	b	b	NOUN
ejpam-1595	132	11	,	,	PUNCT
ejpam-1595	132	12	then	then	ADV
ejpam-1595	132	13	b	b	PROPN
ejpam-1595	132	14	is	be	AUX
ejpam-1595	132	15	i	i	PRON
ejpam-1595	132	16	-semi	-semi	ADV
ejpam-1595	132	17	-	-	PUNCT
ejpam-1595	132	18	open	open	ADJ
ejpam-1595	132	19	;	;	PUNCT
ejpam-1595	132	20	2	2	X
ejpam-1595	132	21	.	.	X
ejpam-1595	133	1	if	if	SCONJ
ejpam-1595	133	2	a	a	PRON
ejpam-1595	133	3	is	be	AUX
ejpam-1595	133	4	i	i	PRON
ejpam-1595	133	5	-semi	-semi	NOUN
ejpam-1595	133	6	-	-	PUNCT
ejpam-1595	133	7	open	open	ADJ
ejpam-1595	133	8	,	,	PUNCT
ejpam-1595	133	9	then	then	ADV
ejpam-1595	133	10	so	so	ADV
ejpam-1595	133	11	is	be	AUX
ejpam-1595	133	12	a∪	a∪	ADP
ejpam-1595	133	13	b	b	NOUN
ejpam-1595	133	14	,	,	PUNCT
ejpam-1595	133	15	for	for	ADP
ejpam-1595	133	16	any	any	DET
ejpam-1595	133	17	subset	subset	NOUN
ejpam-1595	133	18	b	b	PROPN
ejpam-1595	133	19	of	of	ADP
ejpam-1595	133	20	x	x	SYM
ejpam-1595	133	21	;	;	PUNCT
ejpam-1595	133	22	3	3	X
ejpam-1595	133	23	.	.	X
ejpam-1595	134	1	if	if	SCONJ
ejpam-1595	134	2	both	both	PRON
ejpam-1595	134	3	a	a	PRON
ejpam-1595	134	4	and	and	CCONJ
ejpam-1595	134	5	b	b	NOUN
ejpam-1595	134	6	are	be	AUX
ejpam-1595	134	7	i	i	PRON
ejpam-1595	134	8	-semi	-semi	NOUN
ejpam-1595	134	9	-	-	PUNCT
ejpam-1595	134	10	open	open	ADJ
ejpam-1595	134	11	,	,	PUNCT
ejpam-1595	134	12	then	then	ADV
ejpam-1595	134	13	so	so	ADV
ejpam-1595	134	14	is	be	AUX
ejpam-1595	134	15	their	their	PRON
ejpam-1595	134	16	intersection	intersection	NOUN
ejpam-1595	134	17	a∩	a∩	PROPN
ejpam-1595	134	18	b.	b.	PROPN
ejpam-1595	134	19	proof	proof	PROPN
ejpam-1595	134	20	.	.	PUNCT
ejpam-1595	135	1	(	(	PUNCT
ejpam-1595	135	2	1	1	X
ejpam-1595	135	3	)	)	PUNCT
ejpam-1595	135	4	suppose	suppose	VERB
ejpam-1595	135	5	that	that	SCONJ
ejpam-1595	135	6	a	a	PRON
ejpam-1595	135	7	is	be	AUX
ejpam-1595	135	8	i	i	PRON
ejpam-1595	135	9	-semi	-semi	NOUN
ejpam-1595	135	10	-	-	PUNCT
ejpam-1595	135	11	open	open	ADJ
ejpam-1595	135	12	,	,	PUNCT
ejpam-1595	135	13	and	and	CCONJ
ejpam-1595	135	14	that	that	SCONJ
ejpam-1595	135	15	a⊂	a⊂	VERB
ejpam-1595	135	16	b.	b.	NOUN
ejpam-1595	135	17	there	there	PRON
ejpam-1595	135	18	is	be	VERB
ejpam-1595	135	19	an	an	DET
ejpam-1595	135	20	open	open	ADJ
ejpam-1595	135	21	set	set	NOUN
ejpam-1595	135	22	u	u	PRON
ejpam-1595	135	23	such	such	ADJ
ejpam-1595	135	24	that	that	PRON
ejpam-1595	135	25	u	u	PROPN
ejpam-1595	135	26	−	−	PROPN
ejpam-1595	135	27	a∈	a∈	PROPN
ejpam-1595	135	28	i	i	PROPN
ejpam-1595	135	29	and	and	CCONJ
ejpam-1595	135	30	a−	a−	PROPN
ejpam-1595	135	31	cl(u	cl(u	X
ejpam-1595	135	32	)	)	PUNCT
ejpam-1595	135	33	∈	∈	PROPN
ejpam-1595	136	1	i	i	PRON
ejpam-1595	136	2	.	.	PUNCT
ejpam-1595	137	1	notice	notice	VERB
ejpam-1595	137	2	that	that	SCONJ
ejpam-1595	137	3	such	such	DET
ejpam-1595	137	4	an	an	DET
ejpam-1595	137	5	open	open	ADJ
ejpam-1595	137	6	set	set	NOUN
ejpam-1595	137	7	u	u	NOUN
ejpam-1595	137	8	is	be	AUX
ejpam-1595	137	9	necessarily	necessarily	ADV
ejpam-1595	137	10	non	non	ADJ
ejpam-1595	137	11	-	-	ADJ
ejpam-1595	137	12	empty	empty	ADJ
ejpam-1595	137	13	,	,	PUNCT
ejpam-1595	137	14	since	since	SCONJ
ejpam-1595	137	15	we	we	PRON
ejpam-1595	137	16	are	be	AUX
ejpam-1595	137	17	dealing	deal	VERB
ejpam-1595	137	18	with	with	ADP
ejpam-1595	137	19	those	those	DET
ejpam-1595	137	20	subsets	subset	NOUN
ejpam-1595	137	21	of	of	ADP
ejpam-1595	137	22	x	x	PRON
ejpam-1595	137	23	that	that	PRON
ejpam-1595	137	24	do	do	AUX
ejpam-1595	137	25	not	not	PART
ejpam-1595	137	26	belong	belong	VERB
ejpam-1595	137	27	to	to	ADP
ejpam-1595	137	28	the	the	DET
ejpam-1595	137	29	ideal	ideal	NOUN
ejpam-1595	137	30	i	i	PRON
ejpam-1595	137	31	.	.	PUNCT
ejpam-1595	138	1	since	since	SCONJ
ejpam-1595	138	2	a	a	DET
ejpam-1595	138	3	⊂	⊂	PROPN
ejpam-1595	138	4	b	b	PROPN
ejpam-1595	138	5	,	,	PUNCT
ejpam-1595	138	6	we	we	PRON
ejpam-1595	138	7	have	have	VERB
ejpam-1595	138	8	that	that	DET
ejpam-1595	138	9	u	u	PROPN
ejpam-1595	138	10	−	−	PROPN
ejpam-1595	138	11	b	b	PROPN
ejpam-1595	138	12	⊂	⊂	PUNCT
ejpam-1595	138	13	u	u	PROPN
ejpam-1595	138	14	−a∈	−a∈	PROPN
ejpam-1595	138	15	i	i	PRON
ejpam-1595	138	16	;	;	PUNCT
ejpam-1595	138	17	moreover	moreover	ADV
ejpam-1595	138	18	,	,	PUNCT
ejpam-1595	138	19	b−	b−	NOUN
ejpam-1595	138	20	cl(u	cl(u	X
ejpam-1595	138	21	)	)	PUNCT
ejpam-1595	139	1	=	=	SYM
ejpam-1595	139	2	b−	b−	NOUN
ejpam-1595	139	3	x	x	PUNCT
ejpam-1595	139	4	=	=	NOUN
ejpam-1595	139	5	∅	∅	NOUN
ejpam-1595	139	6	∈	∈	PROPN
ejpam-1595	140	1	i	i	PRON
ejpam-1595	140	2	.	.	PUNCT
ejpam-1595	141	1	thus	thus	ADV
ejpam-1595	141	2	,	,	PUNCT
ejpam-1595	141	3	b	b	PROPN
ejpam-1595	141	4	is	be	AUX
ejpam-1595	141	5	i	i	PRON
ejpam-1595	141	6	-semi	-semi	NOUN
ejpam-1595	141	7	-	-	PUNCT
ejpam-1595	141	8	open	open	ADJ
ejpam-1595	141	9	.	.	PUNCT
ejpam-1595	142	1	(	(	PUNCT
ejpam-1595	142	2	2	2	X
ejpam-1595	142	3	)	)	PUNCT
ejpam-1595	142	4	since	since	SCONJ
ejpam-1595	142	5	a⊂	a⊂	NOUN
ejpam-1595	142	6	b⇔	b⇔	NOUN
ejpam-1595	142	7	a∪	a∪	ADP
ejpam-1595	142	8	b	b	NOUN
ejpam-1595	142	9	=	=	SYM
ejpam-1595	142	10	b	b	PROPN
ejpam-1595	142	11	,	,	PUNCT
ejpam-1595	142	12	(	(	PUNCT
ejpam-1595	142	13	2	2	X
ejpam-1595	142	14	)	)	PUNCT
ejpam-1595	142	15	immediately	immediately	ADV
ejpam-1595	142	16	follows	follow	VERB
ejpam-1595	142	17	from	from	ADP
ejpam-1595	142	18	(	(	PUNCT
ejpam-1595	142	19	1	1	NUM
ejpam-1595	142	20	)	)	PUNCT
ejpam-1595	142	21	.	.	PUNCT
ejpam-1595	143	1	(	(	PUNCT
ejpam-1595	143	2	3	3	X
ejpam-1595	143	3	)	)	PUNCT
ejpam-1595	143	4	suppose	suppose	VERB
ejpam-1595	143	5	that	that	SCONJ
ejpam-1595	143	6	both	both	PRON
ejpam-1595	143	7	a	a	PRON
ejpam-1595	143	8	and	and	CCONJ
ejpam-1595	143	9	b	b	NOUN
ejpam-1595	143	10	are	be	AUX
ejpam-1595	143	11	i	i	PRON
ejpam-1595	143	12	-semi	-semi	NOUN
ejpam-1595	143	13	-	-	PUNCT
ejpam-1595	143	14	open	open	ADJ
ejpam-1595	143	15	.	.	PUNCT
ejpam-1595	144	1	without	without	ADP
ejpam-1595	144	2	loss	loss	NOUN
ejpam-1595	144	3	of	of	ADP
ejpam-1595	144	4	generality	generality	NOUN
ejpam-1595	144	5	,	,	PUNCT
ejpam-1595	144	6	suppose	suppose	VERB
ejpam-1595	144	7	that	that	SCONJ
ejpam-1595	144	8	a∩	a∩	PROPN
ejpam-1595	144	9	b	b	PROPN
ejpam-1595	144	10	6=	6=	NUM
ejpam-1595	144	11	∅	∅	NOUN
ejpam-1595	144	12	;	;	PUNCT
ejpam-1595	144	13	otherwise	otherwise	ADV
ejpam-1595	144	14	,	,	PUNCT
ejpam-1595	144	15	a∩	a∩	PROPN
ejpam-1595	144	16	b	b	PROPN
ejpam-1595	144	17	will	will	AUX
ejpam-1595	144	18	be	be	AUX
ejpam-1595	144	19	trivially	trivially	ADV
ejpam-1595	144	20	i	i	PRON
ejpam-1595	144	21	-semi	-semi	NOUN
ejpam-1595	144	22	-	-	PUNCT
ejpam-1595	144	23	open	open	ADJ
ejpam-1595	144	24	.	.	PUNCT
ejpam-1595	145	1	by	by	ADP
ejpam-1595	145	2	assumption	assumption	NOUN
ejpam-1595	145	3	,	,	PUNCT
ejpam-1595	145	4	there	there	PRON
ejpam-1595	145	5	are	be	VERB
ejpam-1595	145	6	open	open	ADJ
ejpam-1595	145	7	sets	set	NOUN
ejpam-1595	145	8	u	u	NOUN
ejpam-1595	145	9	and	and	CCONJ
ejpam-1595	145	10	v	v	ADP
ejpam-1595	145	11	such	such	ADJ
ejpam-1595	145	12	that	that	DET
ejpam-1595	145	13	u	u	NOUN
ejpam-1595	146	1	−	−	PROPN
ejpam-1595	146	2	a	a	PRON
ejpam-1595	146	3	,	,	PUNCT
ejpam-1595	146	4	a−	a−	PROPN
ejpam-1595	146	5	cl(u	cl(u	X
ejpam-1595	146	6	)	)	PUNCT
ejpam-1595	146	7	∈	∈	PROPN
ejpam-1595	147	1	i	i	PRON
ejpam-1595	147	2	and	and	CCONJ
ejpam-1595	147	3	v	v	ADP
ejpam-1595	147	4	−	−	PROPN
ejpam-1595	147	5	b	b	PROPN
ejpam-1595	147	6	,	,	PUNCT
ejpam-1595	147	7	b	b	NOUN
ejpam-1595	147	8	−	−	NOUN
ejpam-1595	147	9	cl(v	cl(v	NOUN
ejpam-1595	147	10	)	)	PUNCT
ejpam-1595	147	11	∈	∈	PROPN
ejpam-1595	148	1	i	i	PRON
ejpam-1595	148	2	.	.	PUNCT
ejpam-1595	149	1	consider	consider	VERB
ejpam-1595	149	2	the	the	DET
ejpam-1595	149	3	open	open	ADJ
ejpam-1595	149	4	set	set	NOUN
ejpam-1595	149	5	u	u	PROPN
ejpam-1595	149	6	∩	∩	ADJ
ejpam-1595	149	7	v	v	NOUN
ejpam-1595	149	8	,	,	PUNCT
ejpam-1595	149	9	which	which	PRON
ejpam-1595	149	10	is	be	AUX
ejpam-1595	149	11	non	non	ADJ
ejpam-1595	149	12	-	-	ADJ
ejpam-1595	149	13	empty	empty	ADJ
ejpam-1595	149	14	(	(	PUNCT
ejpam-1595	149	15	by	by	ADP
ejpam-1595	149	16	the	the	DET
ejpam-1595	149	17	finite	finite	ADJ
ejpam-1595	149	18	intersection	intersection	NOUN
ejpam-1595	149	19	property	property	NOUN
ejpam-1595	149	20	)	)	PUNCT
ejpam-1595	149	21	.	.	PUNCT
ejpam-1595	150	1	since	since	SCONJ
ejpam-1595	150	2	(	(	PUNCT
ejpam-1595	150	3	u∩v	u∩v	PROPN
ejpam-1595	150	4	)	)	PUNCT
ejpam-1595	150	5	−(a∩b	−(a∩b	PROPN
ejpam-1595	150	6	)	)	PUNCT
ejpam-1595	150	7	=	=	SYM
ejpam-1595	150	8	(	(	PUNCT
ejpam-1595	150	9	(	(	PUNCT
ejpam-1595	150	10	u−a)∩v)∪(u∩(v−b	u−a)∩v)∪(u∩(v−b	PROPN
ejpam-1595	150	11	)	)	PUNCT
ejpam-1595	150	12	)	)	PUNCT
ejpam-1595	151	1	∈	∈	PROPN
ejpam-1595	152	1	i	i	PRON
ejpam-1595	152	2	and	and	CCONJ
ejpam-1595	152	3	(	(	PUNCT
ejpam-1595	152	4	a∩b)−cl(u∩v	a∩b)−cl(u∩v	PROPN
ejpam-1595	152	5	)	)	PUNCT
ejpam-1595	152	6	=	=	PUNCT
ejpam-1595	152	7	(	(	PUNCT
ejpam-1595	152	8	a∩b)−x	a∩b)−x	NOUN
ejpam-1595	152	9	=	=	SYM
ejpam-1595	152	10	∅	∅	NOUN
ejpam-1595	152	11	∈	∈	PROPN
ejpam-1595	153	1	i	i	PRON
ejpam-1595	153	2	,	,	PUNCT
ejpam-1595	153	3	it	it	PRON
ejpam-1595	153	4	follows	follow	VERB
ejpam-1595	153	5	that	that	SCONJ
ejpam-1595	153	6	a∩	a∩	PROPN
ejpam-1595	153	7	b	b	PROPN
ejpam-1595	153	8	is	be	AUX
ejpam-1595	153	9	i	i	PRON
ejpam-1595	153	10	-semi	-semi	NOUN
ejpam-1595	153	11	-	-	PUNCT
ejpam-1595	153	12	open	open	ADJ
ejpam-1595	153	13	.	.	PUNCT
ejpam-1595	154	1	remark	remark	NOUN
ejpam-1595	154	2	2	2	NUM
ejpam-1595	154	3	.	.	PUNCT
ejpam-1595	155	1	in	in	ADP
ejpam-1595	155	2	example	example	NOUN
ejpam-1595	155	3	2	2	NUM
ejpam-1595	155	4	,	,	PUNCT
ejpam-1595	155	5	we	we	PRON
ejpam-1595	155	6	saw	see	VERB
ejpam-1595	155	7	that	that	SCONJ
ejpam-1595	155	8	the	the	DET
ejpam-1595	155	9	singleton	singleton	PROPN
ejpam-1595	155	10	subset	subset	NOUN
ejpam-1595	155	11	{	{	PUNCT
ejpam-1595	155	12	b	b	NOUN
ejpam-1595	155	13	}	}	PUNCT
ejpam-1595	155	14	was	be	AUX
ejpam-1595	155	15	not	not	PART
ejpam-1595	155	16	semi	semi	ADJ
ejpam-1595	155	17	-	-	ADJ
ejpam-1595	155	18	open	open	ADJ
ejpam-1595	155	19	with	with	ADP
ejpam-1595	155	20	respect	respect	NOUN
ejpam-1595	155	21	to	to	ADP
ejpam-1595	155	22	the	the	DET
ejpam-1595	155	23	minimal	minimal	ADJ
ejpam-1595	155	24	ideal	ideal	NOUN
ejpam-1595	156	1	i	i	PRON
ejpam-1595	156	2	=	=	SYM
ejpam-1595	156	3	{	{	PUNCT
ejpam-1595	156	4	∅	∅	NOUN
ejpam-1595	156	5	}	}	PUNCT
ejpam-1595	156	6	.	.	PUNCT
ejpam-1595	157	1	notice	notice	VERB
ejpam-1595	157	2	that	that	SCONJ
ejpam-1595	157	3	the	the	DET
ejpam-1595	157	4	set	set	NOUN
ejpam-1595	157	5	{	{	PUNCT
ejpam-1595	157	6	a	a	PRON
ejpam-1595	157	7	,	,	PUNCT
ejpam-1595	157	8	b	b	NOUN
ejpam-1595	157	9	}	}	PUNCT
ejpam-1595	157	10	=	=	SYM
ejpam-1595	157	11	{	{	PUNCT
ejpam-1595	157	12	a	a	PRON
ejpam-1595	157	13	}	}	PUNCT
ejpam-1595	157	14	∪	∪	NOUN
ejpam-1595	157	15	{	{	PUNCT
ejpam-1595	157	16	b	b	NOUN
ejpam-1595	157	17	}	}	PUNCT
ejpam-1595	157	18	is	be	AUX
ejpam-1595	157	19	semi	semi	ADJ
ejpam-1595	157	20	-	-	ADJ
ejpam-1595	157	21	open	open	ADJ
ejpam-1595	157	22	with	with	ADP
ejpam-1595	157	23	respect	respect	NOUN
ejpam-1595	157	24	to	to	ADP
ejpam-1595	157	25	i	i	PRON
ejpam-1595	157	26	=	=	PUNCT
ejpam-1595	157	27	{	{	PUNCT
ejpam-1595	157	28	∅	∅	NOUN
ejpam-1595	157	29	}	}	PUNCT
ejpam-1595	157	30	,	,	PUNCT
ejpam-1595	157	31	simply	simply	ADV
ejpam-1595	157	32	because	because	SCONJ
ejpam-1595	157	33	the	the	DET
ejpam-1595	157	34	non	non	ADJ
ejpam-1595	157	35	-	-	ADJ
ejpam-1595	157	36	empty	empty	ADJ
ejpam-1595	157	37	open	open	ADJ
ejpam-1595	157	38	singleton	singleton	NOUN
ejpam-1595	157	39	subset	subset	NOUN
ejpam-1595	157	40	{	{	PUNCT
ejpam-1595	157	41	a	a	PRON
ejpam-1595	157	42	}	}	PUNCT
ejpam-1595	157	43	is	be	AUX
ejpam-1595	157	44	dense	dense	ADJ
ejpam-1595	157	45	in	in	ADP
ejpam-1595	157	46	x	x	SYM
ejpam-1595	157	47	;	;	PUNCT
ejpam-1595	157	48	this	this	PRON
ejpam-1595	157	49	is	be	AUX
ejpam-1595	157	50	an	an	DET
ejpam-1595	157	51	instance	instance	NOUN
ejpam-1595	157	52	of	of	ADP
ejpam-1595	157	53	proposition	proposition	NOUN
ejpam-1595	157	54	5(2	5(2	NUM
ejpam-1595	157	55	)	)	PUNCT
ejpam-1595	157	56	above	above	ADV
ejpam-1595	157	57	.	.	PUNCT
ejpam-1595	158	1	proposition	proposition	NOUN
ejpam-1595	158	2	6	6	NUM
ejpam-1595	158	3	.	.	PUNCT
ejpam-1595	159	1	under	under	ADP
ejpam-1595	159	2	the	the	DET
ejpam-1595	159	3	conditions	condition	NOUN
ejpam-1595	159	4	of	of	ADP
ejpam-1595	159	5	proposition	proposition	NOUN
ejpam-1595	159	6	5	5	NUM
ejpam-1595	159	7	,	,	PUNCT
ejpam-1595	159	8	we	we	PRON
ejpam-1595	159	9	have	have	VERB
ejpam-1595	159	10	that	that	SCONJ
ejpam-1595	159	11	a	a	PRON
ejpam-1595	159	12	is	be	AUX
ejpam-1595	159	13	i	i	PRON
ejpam-1595	159	14	-semi	-semi	NOUN
ejpam-1595	159	15	-	-	PUNCT
ejpam-1595	159	16	open	open	ADJ
ejpam-1595	159	17	if	if	SCONJ
ejpam-1595	159	18	and	and	CCONJ
ejpam-1595	159	19	only	only	ADV
ejpam-1595	159	20	if	if	SCONJ
ejpam-1595	159	21	cl(a	cl(a	NUM
ejpam-1595	159	22	)	)	PUNCT
ejpam-1595	159	23	is	be	AUX
ejpam-1595	159	24	i	i	PRON
ejpam-1595	159	25	-semi	-semi	NOUN
ejpam-1595	159	26	-	-	PUNCT
ejpam-1595	159	27	open	open	ADJ
ejpam-1595	159	28	.	.	PUNCT
ejpam-1595	160	1	proof	proof	NOUN
ejpam-1595	160	2	.	.	PUNCT
ejpam-1595	161	1	if	if	SCONJ
ejpam-1595	161	2	a	a	PRON
ejpam-1595	161	3	is	be	AUX
ejpam-1595	161	4	i	i	PRON
ejpam-1595	161	5	-semi	-semi	NOUN
ejpam-1595	161	6	-	-	PUNCT
ejpam-1595	161	7	open	open	ADJ
ejpam-1595	161	8	,	,	PUNCT
ejpam-1595	161	9	then	then	ADV
ejpam-1595	161	10	because	because	SCONJ
ejpam-1595	161	11	a	a	DET
ejpam-1595	161	12	⊂	⊂	ADJ
ejpam-1595	161	13	cl(a	cl(a	NUM
ejpam-1595	161	14	)	)	PUNCT
ejpam-1595	161	15	so	so	ADV
ejpam-1595	161	16	is	be	AUX
ejpam-1595	161	17	cl(a	cl(a	NUM
ejpam-1595	161	18	)	)	PUNCT
ejpam-1595	161	19	,	,	PUNCT
ejpam-1595	161	20	by	by	ADP
ejpam-1595	161	21	proposition	proposition	NOUN
ejpam-1595	161	22	5(2	5(2	NUM
ejpam-1595	161	23	)	)	PUNCT
ejpam-1595	161	24	.	.	PUNCT
ejpam-1595	162	1	conversely	conversely	ADV
ejpam-1595	162	2	,	,	PUNCT
ejpam-1595	162	3	suppose	suppose	VERB
ejpam-1595	162	4	that	that	SCONJ
ejpam-1595	162	5	cl(a	cl(a	X
ejpam-1595	162	6	)	)	PUNCT
ejpam-1595	162	7	is	be	AUX
ejpam-1595	162	8	i	i	PRON
ejpam-1595	162	9	-semi	-semi	NOUN
ejpam-1595	162	10	-	-	PUNCT
ejpam-1595	162	11	open	open	ADJ
ejpam-1595	162	12	.	.	PUNCT
ejpam-1595	163	1	then	then	ADV
ejpam-1595	163	2	there	there	PRON
ejpam-1595	163	3	is	be	VERB
ejpam-1595	163	4	an	an	DET
ejpam-1595	163	5	open	open	ADJ
ejpam-1595	163	6	set	set	NOUN
ejpam-1595	163	7	u	u	PRON
ejpam-1595	163	8	such	such	ADJ
ejpam-1595	163	9	that	that	DET
ejpam-1595	163	10	u	u	NOUN
ejpam-1595	163	11	−	−	PROPN
ejpam-1595	163	12	cl(a	cl(a	X
ejpam-1595	163	13	)	)	PUNCT
ejpam-1595	163	14	∈	∈	PROPN
ejpam-1595	164	1	i	i	PRON
ejpam-1595	164	2	and	and	CCONJ
ejpam-1595	164	3	cl(a)−	cl(a)−	VERB
ejpam-1595	164	4	cl(u	cl(u	NOUN
ejpam-1595	164	5	)	)	PUNCT
ejpam-1595	164	6	∈	∈	PROPN
ejpam-1595	165	1	i	i	PRON
ejpam-1595	165	2	.	.	PUNCT
ejpam-1595	166	1	notice	notice	VERB
ejpam-1595	166	2	that	that	SCONJ
ejpam-1595	166	3	u	u	NOUN
ejpam-1595	166	4	is	be	AUX
ejpam-1595	166	5	necessarily	necessarily	ADV
ejpam-1595	166	6	non	non	ADJ
ejpam-1595	166	7	-	-	ADJ
ejpam-1595	166	8	empty	empty	ADJ
ejpam-1595	166	9	;	;	PUNCT
ejpam-1595	166	10	otherwise	otherwise	ADV
ejpam-1595	166	11	,	,	PUNCT
ejpam-1595	166	12	we	we	PRON
ejpam-1595	166	13	would	would	AUX
ejpam-1595	166	14	have	have	VERB
ejpam-1595	166	15	cl(u	cl(u	NOUN
ejpam-1595	166	16	)	)	PUNCT
ejpam-1595	166	17	=	=	SYM
ejpam-1595	166	18	∅	∅	NOUN
ejpam-1595	166	19	,	,	PUNCT
ejpam-1595	166	20	which	which	PRON
ejpam-1595	166	21	forces	force	VERB
ejpam-1595	166	22	a∈	a∈	PROPN
ejpam-1595	166	23	i	i	PRON
ejpam-1595	166	24	,	,	PUNCT
ejpam-1595	166	25	which	which	PRON
ejpam-1595	166	26	we	we	PRON
ejpam-1595	166	27	do	do	AUX
ejpam-1595	166	28	n’t	not	PART
ejpam-1595	166	29	want	want	VERB
ejpam-1595	166	30	(	(	PUNCT
ejpam-1595	166	31	as	as	SCONJ
ejpam-1595	166	32	we	we	PRON
ejpam-1595	166	33	’re	’re	AUX
ejpam-1595	166	34	dealing	deal	VERB
ejpam-1595	166	35	with	with	ADP
ejpam-1595	166	36	those	those	DET
ejpam-1595	166	37	subsets	subset	NOUN
ejpam-1595	166	38	that	that	PRON
ejpam-1595	166	39	do	do	AUX
ejpam-1595	166	40	not	not	PART
ejpam-1595	166	41	belong	belong	VERB
ejpam-1595	166	42	to	to	ADP
ejpam-1595	166	43	the	the	DET
ejpam-1595	166	44	ideal	ideal	NOUN
ejpam-1595	166	45	i	i	NOUN
ejpam-1595	166	46	)	)	PUNCT
ejpam-1595	166	47	.	.	PUNCT
ejpam-1595	167	1	to	to	PART
ejpam-1595	167	2	show	show	VERB
ejpam-1595	167	3	that	that	SCONJ
ejpam-1595	167	4	a	a	PRON
ejpam-1595	167	5	is	be	AUX
ejpam-1595	167	6	i	i	PRON
ejpam-1595	167	7	-semi	-semi	NOUN
ejpam-1595	167	8	-	-	PUNCT
ejpam-1595	167	9	open	open	ADJ
ejpam-1595	167	10	,	,	PUNCT
ejpam-1595	167	11	consider	consider	VERB
ejpam-1595	167	12	the	the	DET
ejpam-1595	167	13	open	open	ADJ
ejpam-1595	167	14	set	set	NOUN
ejpam-1595	167	15	v	v	NOUN
ejpam-1595	167	16	=	=	SYM
ejpam-1595	167	17	u−cl(a	u−cl(a	ADJ
ejpam-1595	167	18	)	)	PUNCT
ejpam-1595	167	19	=	=	SYM
ejpam-1595	168	1	u∩(cl(a))c	u∩(cl(a))c	PROPN
ejpam-1595	168	2	∈	∈	PROPN
ejpam-1595	168	3	i	i	PRON
ejpam-1595	168	4	,	,	PUNCT
ejpam-1595	168	5	by	by	ADP
ejpam-1595	168	6	assumption	assumption	NOUN
ejpam-1595	168	7	.	.	PUNCT
ejpam-1595	169	1	we	we	PRON
ejpam-1595	169	2	have	have	VERB
ejpam-1595	169	3	that	that	DET
ejpam-1595	169	4	v−a=	v−a=	NOUN
ejpam-1595	169	5	u∩(cl(a))c∩ac	u∩(cl(a))c∩ac	X
ejpam-1595	169	6	∈	∈	PROPN
ejpam-1595	170	1	i	i	PRON
ejpam-1595	170	2	,	,	PUNCT
ejpam-1595	170	3	because	because	SCONJ
ejpam-1595	170	4	of	of	ADP
ejpam-1595	170	5	the	the	DET
ejpam-1595	170	6	heredity	heredity	NOUN
ejpam-1595	170	7	property	property	NOUN
ejpam-1595	170	8	;	;	PUNCT
ejpam-1595	170	9	moreover	moreover	ADV
ejpam-1595	170	10	,	,	PUNCT
ejpam-1595	170	11	a−	a−	PROPN
ejpam-1595	170	12	cl(v	cl(v	NOUN
ejpam-1595	170	13	)	)	PUNCT
ejpam-1595	170	14	=	=	SYM
ejpam-1595	170	15	a−	a−	PROPN
ejpam-1595	170	16	cl(u	cl(u	NOUN
ejpam-1595	170	17	∩	∩	NOUN
ejpam-1595	170	18	(	(	PUNCT
ejpam-1595	170	19	cl(a))c	cl(a))c	X
ejpam-1595	170	20	)	)	PUNCT
ejpam-1595	170	21	=	=	SYM
ejpam-1595	170	22	a−	a−	NOUN
ejpam-1595	170	23	x	x	PUNCT
ejpam-1595	170	24	=	=	PUNCT
ejpam-1595	170	25	∅	∅	NOUN
ejpam-1595	170	26	∈	∈	NOUN
ejpam-1595	170	27	i	i	PRON
ejpam-1595	170	28	.	.	PUNCT
ejpam-1595	171	1	this	this	PRON
ejpam-1595	171	2	shows	show	VERB
ejpam-1595	171	3	that	that	SCONJ
ejpam-1595	171	4	a	a	PRON
ejpam-1595	171	5	is	be	AUX
ejpam-1595	171	6	i	i	PRON
ejpam-1595	171	7	-semi	-semi	NOUN
ejpam-1595	171	8	-	-	PUNCT
ejpam-1595	171	9	open	open	ADJ
ejpam-1595	171	10	.	.	PUNCT
ejpam-1595	172	1	theorem	theorem	NOUN
ejpam-1595	172	2	2	2	NUM
ejpam-1595	172	3	.	.	PUNCT
ejpam-1595	173	1	the	the	DET
ejpam-1595	173	2	following	follow	VERB
ejpam-1595	173	3	are	be	AUX
ejpam-1595	173	4	equivalent	equivalent	ADJ
ejpam-1595	173	5	for	for	ADP
ejpam-1595	173	6	a	a	DET
ejpam-1595	173	7	subset	subset	NOUN
ejpam-1595	173	8	a	a	PRON
ejpam-1595	173	9	of	of	ADP
ejpam-1595	173	10	x	x	PRON
ejpam-1595	173	11	:	:	PUNCT
ejpam-1595	173	12	references	reference	NOUN
ejpam-1595	173	13	57	57	NUM
ejpam-1595	173	14	1	1	NUM
ejpam-1595	173	15	.	.	PUNCT
ejpam-1595	174	1	x	x	PUNCT
ejpam-1595	174	2	−	−	NOUN
ejpam-1595	174	3	a	a	PRON
ejpam-1595	174	4	is	be	AUX
ejpam-1595	174	5	i	i	PRON
ejpam-1595	174	6	-semi	-semi	NOUN
ejpam-1595	174	7	-	-	PUNCT
ejpam-1595	174	8	open	open	ADJ
ejpam-1595	174	9	.	.	PUNCT
ejpam-1595	175	1	2	2	X
ejpam-1595	175	2	.	.	X
ejpam-1595	175	3	there	there	PRON
ejpam-1595	175	4	exists	exist	VERB
ejpam-1595	175	5	a	a	DET
ejpam-1595	175	6	closed	closed	ADJ
ejpam-1595	175	7	set	set	NOUN
ejpam-1595	175	8	f	f	PROPN
ejpam-1595	175	9	such	such	ADJ
ejpam-1595	175	10	that	that	PRON
ejpam-1595	175	11	int(f)−	int(f)−	VERB
ejpam-1595	175	12	a∈	a∈	PROPN
ejpam-1595	175	13	i	i	PROPN
ejpam-1595	175	14	and	and	CCONJ
ejpam-1595	175	15	a−	a−	PROPN
ejpam-1595	175	16	f	f	PROPN
ejpam-1595	176	1	∈	∈	PROPN
ejpam-1595	177	1	i	i	PRON
ejpam-1595	177	2	.	.	PUNCT
ejpam-1595	178	1	proof	proof	NOUN
ejpam-1595	178	2	.	.	PUNCT
ejpam-1595	179	1	first	first	ADV
ejpam-1595	179	2	suppose	suppose	VERB
ejpam-1595	179	3	that	that	SCONJ
ejpam-1595	179	4	x	x	X
ejpam-1595	179	5	−	−	NOUN
ejpam-1595	179	6	a	a	PRON
ejpam-1595	179	7	is	be	AUX
ejpam-1595	179	8	i	i	PRON
ejpam-1595	179	9	-semi	-semi	NOUN
ejpam-1595	179	10	-	-	PUNCT
ejpam-1595	179	11	open	open	ADJ
ejpam-1595	179	12	.	.	PUNCT
ejpam-1595	180	1	then	then	ADV
ejpam-1595	180	2	there	there	PRON
ejpam-1595	180	3	exists	exist	VERB
ejpam-1595	180	4	an	an	DET
ejpam-1595	180	5	open	open	ADJ
ejpam-1595	180	6	set	set	NOUN
ejpam-1595	180	7	u	u	PRON
ejpam-1595	180	8	such	such	ADJ
ejpam-1595	180	9	that	that	DET
ejpam-1595	180	10	u	u	NOUN
ejpam-1595	180	11	−	−	PROPN
ejpam-1595	181	1	(	(	PUNCT
ejpam-1595	181	2	x	x	X
ejpam-1595	181	3	−	−	NOUN
ejpam-1595	181	4	a	a	X
ejpam-1595	181	5	)	)	PUNCT
ejpam-1595	181	6	∈	∈	PROPN
ejpam-1595	181	7	i	i	PRON
ejpam-1595	181	8	and	and	CCONJ
ejpam-1595	181	9	(	(	PUNCT
ejpam-1595	181	10	x	x	X
ejpam-1595	181	11	−	−	NOUN
ejpam-1595	181	12	a	a	X
ejpam-1595	181	13	)	)	PUNCT
ejpam-1595	181	14	−	−	NOUN
ejpam-1595	181	15	cl(u	cl(u	NOUN
ejpam-1595	181	16	)	)	PUNCT
ejpam-1595	181	17	∈	∈	PROPN
ejpam-1595	182	1	i	i	PRON
ejpam-1595	182	2	.	.	PUNCT
ejpam-1595	183	1	since	since	SCONJ
ejpam-1595	183	2	u	u	PRON
ejpam-1595	183	3	−	−	PROPN
ejpam-1595	183	4	(	(	PUNCT
ejpam-1595	183	5	x	x	X
ejpam-1595	183	6	−	−	NOUN
ejpam-1595	183	7	a	a	X
ejpam-1595	183	8	)	)	PUNCT
ejpam-1595	183	9	=	=	SYM
ejpam-1595	183	10	a−	a−	PROPN
ejpam-1595	183	11	(	(	PUNCT
ejpam-1595	183	12	x	x	SYM
ejpam-1595	183	13	−	−	PROPN
ejpam-1595	183	14	u	u	NOUN
ejpam-1595	183	15	)	)	PUNCT
ejpam-1595	183	16	and	and	CCONJ
ejpam-1595	183	17	(	(	PUNCT
ejpam-1595	183	18	x−a)−cl(u	x−a)−cl(u	PROPN
ejpam-1595	183	19	)	)	PUNCT
ejpam-1595	183	20	=	=	VERB
ejpam-1595	183	21	int(x−u)−a	int(x−u)−a	PROPN
ejpam-1595	183	22	,	,	PUNCT
ejpam-1595	183	23	we	we	PRON
ejpam-1595	183	24	have	have	VERB
ejpam-1595	183	25	(	(	PUNCT
ejpam-1595	183	26	2	2	NUM
ejpam-1595	183	27	)	)	PUNCT
ejpam-1595	183	28	by	by	ADP
ejpam-1595	183	29	choosing	choose	VERB
ejpam-1595	183	30	the	the	DET
ejpam-1595	183	31	closed	closed	ADJ
ejpam-1595	183	32	set	set	NOUN
ejpam-1595	183	33	x−u	x−u	PROPN
ejpam-1595	183	34	as	as	ADP
ejpam-1595	183	35	f	f	PROPN
ejpam-1595	183	36	.	.	PUNCT
ejpam-1595	184	1	conversely	conversely	ADV
ejpam-1595	184	2	,	,	PUNCT
ejpam-1595	184	3	if	if	SCONJ
ejpam-1595	184	4	we	we	PRON
ejpam-1595	184	5	suppose	suppose	VERB
ejpam-1595	184	6	that	that	SCONJ
ejpam-1595	184	7	(	(	PUNCT
ejpam-1595	184	8	2	2	X
ejpam-1595	184	9	)	)	PUNCT
ejpam-1595	184	10	holds	hold	VERB
ejpam-1595	184	11	,	,	PUNCT
ejpam-1595	184	12	then	then	ADV
ejpam-1595	184	13	the	the	DET
ejpam-1595	184	14	choice	choice	NOUN
ejpam-1595	184	15	of	of	ADP
ejpam-1595	184	16	the	the	DET
ejpam-1595	184	17	open	open	ADJ
ejpam-1595	184	18	set	set	NOUN
ejpam-1595	184	19	u	u	NOUN
ejpam-1595	185	1	=	=	NOUN
ejpam-1595	185	2	x	x	SYM
ejpam-1595	185	3	−	−	PROPN
ejpam-1595	185	4	f	f	PROPN
ejpam-1595	185	5	shows	show	VERB
ejpam-1595	185	6	that	that	SCONJ
ejpam-1595	185	7	x	x	PUNCT
ejpam-1595	186	1	−	−	NOUN
ejpam-1595	186	2	a	a	PRON
ejpam-1595	186	3	is	be	AUX
ejpam-1595	186	4	i	i	PRON
ejpam-1595	186	5	-semi	-semi	NOUN
ejpam-1595	186	6	-	-	PUNCT
ejpam-1595	186	7	open	open	ADJ
ejpam-1595	186	8	.	.	PUNCT
ejpam-1595	187	1	definition	definition	NOUN
ejpam-1595	187	2	2	2	NUM
ejpam-1595	187	3	.	.	PUNCT
ejpam-1595	188	1	a	a	DET
ejpam-1595	188	2	subset	subset	NOUN
ejpam-1595	188	3	a	a	PRON
ejpam-1595	188	4	of	of	ADP
ejpam-1595	188	5	x	x	SYM
ejpam-1595	188	6	is	be	AUX
ejpam-1595	188	7	said	say	VERB
ejpam-1595	188	8	to	to	PART
ejpam-1595	188	9	be	be	AUX
ejpam-1595	188	10	semi	semi	ADJ
ejpam-1595	188	11	-	-	ADJ
ejpam-1595	188	12	closed	closed	ADJ
ejpam-1595	188	13	with	with	ADP
ejpam-1595	188	14	respect	respect	NOUN
ejpam-1595	188	15	to	to	ADP
ejpam-1595	188	16	i	i	PRON
ejpam-1595	188	17	(	(	PUNCT
ejpam-1595	188	18	written	write	VERB
ejpam-1595	188	19	as	as	SCONJ
ejpam-1595	188	20	i	i	PRON
ejpam-1595	188	21	-semiclosed	-semiclosed	ADJ
ejpam-1595	188	22	)	)	PUNCT
ejpam-1595	189	1	if	if	SCONJ
ejpam-1595	189	2	and	and	CCONJ
ejpam-1595	189	3	only	only	ADV
ejpam-1595	189	4	if	if	SCONJ
ejpam-1595	189	5	x	x	X
ejpam-1595	189	6	−	−	NOUN
ejpam-1595	190	1	a	a	PRON
ejpam-1595	190	2	is	be	AUX
ejpam-1595	190	3	i	i	PRON
ejpam-1595	190	4	-semi	-semi	NOUN
ejpam-1595	190	5	-	-	PUNCT
ejpam-1595	190	6	open	open	ADJ
ejpam-1595	190	7	.	.	PUNCT
ejpam-1595	191	1	proposition	proposition	NOUN
ejpam-1595	191	2	7	7	NUM
ejpam-1595	191	3	.	.	PUNCT
ejpam-1595	192	1	if	if	SCONJ
ejpam-1595	192	2	both	both	PRON
ejpam-1595	192	3	a	a	PRON
ejpam-1595	192	4	and	and	CCONJ
ejpam-1595	192	5	b	b	NOUN
ejpam-1595	192	6	are	be	AUX
ejpam-1595	192	7	i	i	PRON
ejpam-1595	192	8	-semi	-semi	NOUN
ejpam-1595	192	9	-	-	PUNCT
ejpam-1595	192	10	closed	closed	ADJ
ejpam-1595	192	11	,	,	PUNCT
ejpam-1595	192	12	then	then	ADV
ejpam-1595	192	13	so	so	ADV
ejpam-1595	192	14	is	be	AUX
ejpam-1595	192	15	their	their	PRON
ejpam-1595	192	16	intersection	intersection	NOUN
ejpam-1595	192	17	a∩	a∩	PROPN
ejpam-1595	192	18	b.	b.	PROPN
ejpam-1595	192	19	proof	proof	PROPN
ejpam-1595	192	20	.	.	PUNCT
ejpam-1595	193	1	let	let	VERB
ejpam-1595	193	2	the	the	DET
ejpam-1595	193	3	given	give	VERB
ejpam-1595	193	4	conditions	condition	NOUN
ejpam-1595	193	5	hold	hold	VERB
ejpam-1595	193	6	.	.	PUNCT
ejpam-1595	194	1	there	there	PRON
ejpam-1595	194	2	are	be	VERB
ejpam-1595	194	3	closed	closed	ADJ
ejpam-1595	194	4	sets	set	NOUN
ejpam-1595	194	5	f1	f1	NOUN
ejpam-1595	194	6	and	and	CCONJ
ejpam-1595	194	7	f2	f2	NOUN
ejpam-1595	195	1	such	such	ADJ
ejpam-1595	195	2	that	that	SCONJ
ejpam-1595	195	3	int(f1)−	int(f1)−	VERB
ejpam-1595	195	4	a	a	DET
ejpam-1595	195	5	,	,	PUNCT
ejpam-1595	195	6	a−	a−	PROPN
ejpam-1595	195	7	f1	f1	NOUN
ejpam-1595	195	8	∈	∈	PROPN
ejpam-1595	195	9	i	i	PRON
ejpam-1595	195	10	and	and	CCONJ
ejpam-1595	195	11	int(f2)−	int(f2)−	VERB
ejpam-1595	195	12	b	b	NOUN
ejpam-1595	195	13	,	,	PUNCT
ejpam-1595	195	14	b	b	NOUN
ejpam-1595	195	15	−	−	NOUN
ejpam-1595	195	16	f2	f2	PROPN
ejpam-1595	195	17	∈	∈	PROPN
ejpam-1595	196	1	i	i	PRON
ejpam-1595	196	2	.	.	PUNCT
ejpam-1595	197	1	with	with	ADP
ejpam-1595	197	2	f	f	PROPN
ejpam-1595	197	3	=	=	SYM
ejpam-1595	197	4	f1	f1	PROPN
ejpam-1595	197	5	∩	∩	NOUN
ejpam-1595	197	6	f2	f2	PROPN
ejpam-1595	197	7	,	,	PUNCT
ejpam-1595	197	8	we	we	PRON
ejpam-1595	197	9	have	have	VERB
ejpam-1595	197	10	that	that	PRON
ejpam-1595	197	11	int(f1	int(f1	CCONJ
ejpam-1595	197	12	∩	∩	NOUN
ejpam-1595	197	13	f2)−	f2)−	X
ejpam-1595	197	14	(	(	PUNCT
ejpam-1595	197	15	a∩	a∩	PROPN
ejpam-1595	197	16	b	b	X
ejpam-1595	197	17	)	)	PUNCT
ejpam-1595	197	18	=	=	SYM
ejpam-1595	197	19	(	(	PUNCT
ejpam-1595	197	20	(	(	PUNCT
ejpam-1595	197	21	int(f1)−	int(f1)−	ADJ
ejpam-1595	197	22	a)∩	a)∩	PROPN
ejpam-1595	197	23	int(f2))∪	int(f2))∪	NOUN
ejpam-1595	197	24	(	(	PUNCT
ejpam-1595	197	25	int(f1)∩	int(f1)∩	PROPN
ejpam-1595	197	26	(	(	PUNCT
ejpam-1595	197	27	int(f2)−	int(f2)−	ADJ
ejpam-1595	197	28	b	b	NOUN
ejpam-1595	197	29	)	)	PUNCT
ejpam-1595	197	30	)	)	PUNCT
ejpam-1595	198	1	∈	∈	PROPN
ejpam-1595	199	1	i	i	PRON
ejpam-1595	199	2	,	,	PUNCT
ejpam-1595	199	3	and	and	CCONJ
ejpam-1595	199	4	(	(	PUNCT
ejpam-1595	199	5	a∩	a∩	PROPN
ejpam-1595	199	6	b)−	b)−	PROPN
ejpam-1595	199	7	(	(	PUNCT
ejpam-1595	199	8	f1	f1	PROPN
ejpam-1595	199	9	∩	∩	NOUN
ejpam-1595	199	10	f2	f2	PROPN
ejpam-1595	199	11	)	)	PUNCT
ejpam-1595	199	12	=	=	SYM
ejpam-1595	199	13	(	(	PUNCT
ejpam-1595	199	14	(	(	PUNCT
ejpam-1595	199	15	a−	a−	X
ejpam-1595	199	16	f1)∩	f1)∩	VERB
ejpam-1595	199	17	b)∪	b)∪	NOUN
ejpam-1595	199	18	(	(	PUNCT
ejpam-1595	199	19	a∩	a∩	PROPN
ejpam-1595	199	20	(	(	PUNCT
ejpam-1595	199	21	b−	b−	NOUN
ejpam-1595	199	22	f2	f2	PROPN
ejpam-1595	199	23	)	)	PUNCT
ejpam-1595	199	24	)	)	PUNCT
ejpam-1595	200	1	∈	∈	PROPN
ejpam-1595	201	1	i	i	PRON
ejpam-1595	201	2	;	;	PUNCT
ejpam-1595	201	3	therefore	therefore	ADV
ejpam-1595	201	4	,	,	PUNCT
ejpam-1595	201	5	a∩	a∩	PROPN
ejpam-1595	201	6	b	b	PROPN
ejpam-1595	201	7	is	be	AUX
ejpam-1595	201	8	i	i	PRON
ejpam-1595	201	9	-semi	-semi	NOUN
ejpam-1595	201	10	-	-	PUNCT
ejpam-1595	201	11	closed	closed	ADJ
ejpam-1595	201	12	.	.	PUNCT
ejpam-1595	202	1	references	reference	NOUN
ejpam-1595	202	2	[	[	X
ejpam-1595	202	3	1	1	NUM
ejpam-1595	202	4	]	]	PUNCT
ejpam-1595	202	5	k.	k.	PROPN
ejpam-1595	202	6	al	al	PROPN
ejpam-1595	202	7	-	-	PROPN
ejpam-1595	202	8	zoubi	zoubi	PROPN
ejpam-1595	202	9	.	.	PUNCT
ejpam-1595	203	1	on	on	ADP
ejpam-1595	203	2	generalized	generalized	ADJ
ejpam-1595	203	3	ω	ω	VERB
ejpam-1595	203	4	-	-	PUNCT
ejpam-1595	203	5	closed	closed	ADJ
ejpam-1595	203	6	sets	set	NOUN
ejpam-1595	203	7	.	.	PUNCT
ejpam-1595	204	1	international	international	ADJ
ejpam-1595	204	2	journal	journal	NOUN
ejpam-1595	204	3	of	of	ADP
ejpam-1595	204	4	mathematics	mathematics	PROPN
ejpam-1595	204	5	and	and	CCONJ
ejpam-1595	204	6	mathematical	mathematical	ADJ
ejpam-1595	204	7	sciences	science	NOUN
ejpam-1595	204	8	,	,	PUNCT
ejpam-1595	204	9	2005(13):2011–2021	2005(13):2011–2021	NUM
ejpam-1595	204	10	,	,	PUNCT
ejpam-1595	204	11	2005	2005	NUM
ejpam-1595	204	12	.	.	PUNCT
ejpam-1595	205	1	[	[	X
ejpam-1595	205	2	2	2	NUM
ejpam-1595	205	3	]	]	X
ejpam-1595	205	4	s.p	s.p	PROPN
ejpam-1595	205	5	.	.	PROPN
ejpam-1595	205	6	arya	arya	PROPN
ejpam-1595	205	7	and	and	CCONJ
ejpam-1595	205	8	t.m	t.m	PROPN
ejpam-1595	205	9	.	.	PROPN
ejpam-1595	205	10	nour	nour	PROPN
ejpam-1595	205	11	.	.	PUNCT
ejpam-1595	206	1	characterization	characterization	NOUN
ejpam-1595	206	2	of	of	ADP
ejpam-1595	206	3	s	s	NOUN
ejpam-1595	206	4	-	-	ADJ
ejpam-1595	206	5	normal	normal	ADJ
ejpam-1595	206	6	spaces	space	NOUN
ejpam-1595	206	7	.	.	PUNCT
ejpam-1595	207	1	indian	indian	ADJ
ejpam-1595	207	2	journal	journal	PROPN
ejpam-1595	207	3	of	of	ADP
ejpam-1595	207	4	pure	pure	ADJ
ejpam-1595	207	5	and	and	CCONJ
ejpam-1595	207	6	applied	applied	ADJ
ejpam-1595	207	7	mathematics	mathematic	NOUN
ejpam-1595	207	8	,	,	PUNCT
ejpam-1595	207	9	21(8):717–719	21(8):717–719	NUM
ejpam-1595	207	10	,	,	PUNCT
ejpam-1595	207	11	1990	1990	NUM
ejpam-1595	207	12	.	.	PUNCT
ejpam-1595	208	1	[	[	X
ejpam-1595	208	2	3	3	X
ejpam-1595	208	3	]	]	PUNCT
ejpam-1595	208	4	c.	c.	PROPN
ejpam-1595	208	5	boonpok	boonpok	PROPN
ejpam-1595	208	6	.	.	PUNCT
ejpam-1595	209	1	generalized	generalize	VERB
ejpam-1595	209	2	closed	close	VERB
ejpam-1595	209	3	sets	set	NOUN
ejpam-1595	209	4	in	in	ADP
ejpam-1595	209	5	isotonic	isotonic	ADJ
ejpam-1595	209	6	spaces	space	NOUN
ejpam-1595	209	7	.	.	PUNCT
ejpam-1595	210	1	international	international	ADJ
ejpam-1595	210	2	journal	journal	PROPN
ejpam-1595	210	3	of	of	ADP
ejpam-1595	210	4	mathematical	mathematical	ADJ
ejpam-1595	210	5	analysis	analysis	NOUN
ejpam-1595	210	6	,	,	PUNCT
ejpam-1595	210	7	5:241–256	5:241–256	NUM
ejpam-1595	210	8	,	,	PUNCT
ejpam-1595	210	9	2011	2011	NUM
ejpam-1595	210	10	.	.	PUNCT
ejpam-1595	211	1	[	[	X
ejpam-1595	211	2	4	4	NUM
ejpam-1595	211	3	]	]	X
ejpam-1595	211	4	r.	r.	PROPN
ejpam-1595	211	5	davi	davi	PROPN
ejpam-1595	211	6	,	,	PUNCT
ejpam-1595	211	7	k.	k.	PROPN
ejpam-1595	211	8	balachandran	balachandran	PROPN
ejpam-1595	211	9	,	,	PUNCT
ejpam-1595	211	10	and	and	CCONJ
ejpam-1595	211	11	h.	h.	PROPN
ejpam-1595	211	12	maki	maki	PROPN
ejpam-1595	211	13	.	.	PUNCT
ejpam-1595	212	1	semi	semi	ADJ
ejpam-1595	212	2	-	-	ADJ
ejpam-1595	212	3	generalized	generalized	ADJ
ejpam-1595	212	4	homeomorphisms	homeomorphism	NOUN
ejpam-1595	212	5	and	and	CCONJ
ejpam-1595	212	6	generalized	generalize	VERB
ejpam-1595	212	7	semi	semi	NOUN
ejpam-1595	212	8	-	-	NOUN
ejpam-1595	212	9	homeomorphisms	homeomorphism	NOUN
ejpam-1595	212	10	in	in	ADP
ejpam-1595	212	11	topological	topological	ADJ
ejpam-1595	212	12	spaces	space	NOUN
ejpam-1595	212	13	.	.	PUNCT
ejpam-1595	213	1	indian	indian	ADJ
ejpam-1595	213	2	journal	journal	PROPN
ejpam-1595	213	3	of	of	ADP
ejpam-1595	213	4	pure	pure	ADJ
ejpam-1595	213	5	and	and	CCONJ
ejpam-1595	213	6	applied	applied	ADJ
ejpam-1595	213	7	mathematics	mathematic	NOUN
ejpam-1595	213	8	,	,	PUNCT
ejpam-1595	213	9	26(3):271–284	26(3):271–284	NOUN
ejpam-1595	213	10	,	,	PUNCT
ejpam-1595	213	11	1995	1995	NUM
ejpam-1595	213	12	.	.	PUNCT
ejpam-1595	214	1	[	[	X
ejpam-1595	214	2	5	5	X
ejpam-1595	214	3	]	]	PUNCT
ejpam-1595	214	4	e.	e.	PROPN
ejpam-1595	214	5	hatir	hatir	PROPN
ejpam-1595	214	6	and	and	CCONJ
ejpam-1595	214	7	t.	t.	PROPN
ejpam-1595	214	8	noiri	noiri	PROPN
ejpam-1595	214	9	.	.	PUNCT
ejpam-1595	215	1	on	on	ADP
ejpam-1595	215	2	semi	semi	ADJ
ejpam-1595	215	3	-	-	ADJ
ejpam-1595	215	4	i	i	PRON
ejpam-1595	215	5	-open	-open	NOUN
ejpam-1595	215	6	sets	set	NOUN
ejpam-1595	215	7	and	and	CCONJ
ejpam-1595	215	8	semi	semi	ADJ
ejpam-1595	215	9	-	-	ADJ
ejpam-1595	215	10	i	i	ADV
ejpam-1595	215	11	-continuous	-continuous	ADJ
ejpam-1595	215	12	functions	function	NOUN
ejpam-1595	215	13	.	.	PUNCT
ejpam-1595	216	1	acta	acta	PROPN
ejpam-1595	216	2	mathematica	mathematica	PROPN
ejpam-1595	216	3	hungarica	hungarica	PROPN
ejpam-1595	216	4	,	,	PUNCT
ejpam-1595	216	5	107(4):345–353	107(4):345–353	NUM
ejpam-1595	216	6	,	,	PUNCT
ejpam-1595	216	7	2005	2005	NUM
ejpam-1595	216	8	.	.	PUNCT
ejpam-1595	217	1	[	[	X
ejpam-1595	217	2	6	6	NUM
ejpam-1595	217	3	]	]	PUNCT
ejpam-1595	217	4	s.	s.	PROPN
ejpam-1595	217	5	jafari	jafari	PROPN
ejpam-1595	217	6	and	and	CCONJ
ejpam-1595	217	7	n.	n.	PROPN
ejpam-1595	217	8	rajesh	rajesh	PROPN
ejpam-1595	217	9	.	.	PUNCT
ejpam-1595	218	1	generalized	generalize	VERB
ejpam-1595	218	2	closed	close	VERB
ejpam-1595	218	3	sets	set	NOUN
ejpam-1595	218	4	with	with	ADP
ejpam-1595	218	5	respect	respect	NOUN
ejpam-1595	218	6	to	to	ADP
ejpam-1595	218	7	ideals	ideal	NOUN
ejpam-1595	218	8	.	.	PUNCT
ejpam-1595	219	1	european	european	ADJ
ejpam-1595	219	2	journal	journal	PROPN
ejpam-1595	219	3	of	of	ADP
ejpam-1595	219	4	pure	pure	ADJ
ejpam-1595	219	5	and	and	CCONJ
ejpam-1595	219	6	applied	applied	ADJ
ejpam-1595	219	7	mathematics	mathematic	NOUN
ejpam-1595	219	8	,	,	PUNCT
ejpam-1595	219	9	4(2):147–151	4(2):147–151	NUM
ejpam-1595	219	10	,	,	PUNCT
ejpam-1595	219	11	2011	2011	NUM
ejpam-1595	219	12	.	.	PUNCT
ejpam-1595	220	1	[	[	X
ejpam-1595	220	2	7	7	X
ejpam-1595	220	3	]	]	X
ejpam-1595	220	4	d.	d.	PROPN
ejpam-1595	220	5	jankovic	jankovic	PROPN
ejpam-1595	220	6	and	and	CCONJ
ejpam-1595	220	7	t.r	t.r	PROPN
ejpam-1595	220	8	.	.	PROPN
ejpam-1595	220	9	hamlett	hamlett	PROPN
ejpam-1595	220	10	.	.	PUNCT
ejpam-1595	221	1	ideals	ideal	NOUN
ejpam-1595	221	2	in	in	ADP
ejpam-1595	221	3	topology	topology	NOUN
ejpam-1595	221	4	and	and	CCONJ
ejpam-1595	221	5	applications	application	NOUN
ejpam-1595	221	6	.	.	PUNCT
ejpam-1595	222	1	lecture	lecture	NOUN
ejpam-1595	222	2	notes	note	NOUN
ejpam-1595	222	3	in	in	ADP
ejpam-1595	222	4	pure	pure	ADJ
ejpam-1595	222	5	and	and	CCONJ
ejpam-1595	222	6	applied	applied	ADJ
ejpam-1595	222	7	mathematics	mathematic	NOUN
ejpam-1595	222	8	,	,	PUNCT
ejpam-1595	222	9	pages	page	NOUN
ejpam-1595	222	10	115–125	115–125	NUM
ejpam-1595	222	11	,	,	PUNCT
ejpam-1595	222	12	1990	1990	NUM
ejpam-1595	222	13	.	.	PUNCT
ejpam-1595	223	1	references	reference	NOUN
ejpam-1595	223	2	58	58	NUM
ejpam-1595	224	1	[	[	X
ejpam-1595	224	2	8	8	NUM
ejpam-1595	224	3	]	]	X
ejpam-1595	224	4	d.	d.	PROPN
ejpam-1595	224	5	jankovic	jankovic	PROPN
ejpam-1595	224	6	and	and	CCONJ
ejpam-1595	224	7	t.r	t.r	PROPN
ejpam-1595	224	8	.	.	PROPN
ejpam-1595	224	9	hamlett	hamlett	PROPN
ejpam-1595	224	10	.	.	PUNCT
ejpam-1595	225	1	ideals	ideal	NOUN
ejpam-1595	225	2	in	in	ADP
ejpam-1595	225	3	topology	topology	NOUN
ejpam-1595	225	4	and	and	CCONJ
ejpam-1595	225	5	the	the	DET
ejpam-1595	225	6	set	set	NOUN
ejpam-1595	225	7	operator	operator	NOUN
ejpam-1595	225	8	.	.	PUNCT
ejpam-1595	226	1	bollettino	bollettino	PROPN
ejpam-1595	226	2	della	della	PROPN
ejpam-1595	226	3	unione	unione	PROPN
ejpam-1595	226	4	matematica	matematica	PROPN
ejpam-1595	226	5	italiana	italiana	PROPN
ejpam-1595	226	6	,	,	PUNCT
ejpam-1595	226	7	7:863–894	7:863–894	NUM
ejpam-1595	226	8	,	,	PUNCT
ejpam-1595	226	9	1990	1990	NUM
ejpam-1595	226	10	.	.	PUNCT
ejpam-1595	227	1	[	[	X
ejpam-1595	227	2	9	9	NUM
ejpam-1595	227	3	]	]	X
ejpam-1595	227	4	d.	d.	PROPN
ejpam-1595	227	5	jankovic	jankovic	PROPN
ejpam-1595	227	6	and	and	CCONJ
ejpam-1595	227	7	t.r	t.r	PROPN
ejpam-1595	227	8	.	.	PROPN
ejpam-1595	227	9	hamlett	hamlett	PROPN
ejpam-1595	227	10	.	.	PUNCT
ejpam-1595	228	1	new	new	ADJ
ejpam-1595	228	2	topologies	topology	NOUN
ejpam-1595	228	3	from	from	ADP
ejpam-1595	228	4	old	old	ADJ
ejpam-1595	228	5	via	via	ADP
ejpam-1595	228	6	ideals	ideal	NOUN
ejpam-1595	228	7	.	.	PUNCT
ejpam-1595	229	1	american	american	PROPN
ejpam-1595	229	2	mathematical	mathematical	PROPN
ejpam-1595	229	3	monthly	monthly	PROPN
ejpam-1595	229	4	,	,	PUNCT
ejpam-1595	229	5	97:295–310	97:295–310	PROPN
ejpam-1595	229	6	,	,	PUNCT
ejpam-1595	229	7	1990	1990	NUM
ejpam-1595	229	8	.	.	PUNCT
ejpam-1595	230	1	[	[	X
ejpam-1595	230	2	10	10	NUM
ejpam-1595	230	3	]	]	X
ejpam-1595	230	4	j.l	j.l	PROPN
ejpam-1595	230	5	.	.	PROPN
ejpam-1595	230	6	kelly	kelly	PROPN
ejpam-1595	230	7	.	.	PUNCT
ejpam-1595	230	8	general	general	ADJ
ejpam-1595	230	9	topology	topology	NOUN
ejpam-1595	230	10	.	.	PUNCT
ejpam-1595	231	1	d	d	X
ejpam-1595	231	2	van	van	PROPN
ejpam-1595	231	3	nostrand	nostrand	PROPN
ejpam-1595	231	4	company	company	NOUN
ejpam-1595	231	5	incorporated	incorporate	VERB
ejpam-1595	231	6	,	,	PUNCT
ejpam-1595	231	7	princeton	princeton	PROPN
ejpam-1595	231	8	,	,	PUNCT
ejpam-1595	231	9	n.j	n.j	PROPN
ejpam-1595	231	10	.	.	PROPN
ejpam-1595	231	11	,	,	PUNCT
ejpam-1595	231	12	1955	1955	NUM
ejpam-1595	231	13	.	.	PUNCT
ejpam-1595	232	1	[	[	X
ejpam-1595	232	2	11	11	NUM
ejpam-1595	232	3	]	]	X
ejpam-1595	232	4	n.	n.	PROPN
ejpam-1595	232	5	levine	levine	PROPN
ejpam-1595	232	6	.	.	PUNCT
ejpam-1595	233	1	semi	semi	ADJ
ejpam-1595	233	2	-	-	ADJ
ejpam-1595	233	3	open	open	ADJ
ejpam-1595	233	4	sets	set	NOUN
ejpam-1595	233	5	and	and	CCONJ
ejpam-1595	233	6	semi	semi	ADJ
ejpam-1595	233	7	-	-	NOUN
ejpam-1595	233	8	continuity	continuity	NOUN
ejpam-1595	233	9	in	in	ADP
ejpam-1595	233	10	topological	topological	ADJ
ejpam-1595	233	11	spaces	space	NOUN
ejpam-1595	233	12	.	.	PUNCT
ejpam-1595	234	1	american	american	PROPN
ejpam-1595	234	2	mathematical	mathematical	PROPN
ejpam-1595	234	3	monthly	monthly	ADV
ejpam-1595	234	4	,	,	PUNCT
ejpam-1595	234	5	70:36–41	70:36–41	NUM
ejpam-1595	234	6	,	,	PUNCT
ejpam-1595	234	7	1963	1963	NUM
ejpam-1595	234	8	.	.	PUNCT
ejpam-1595	235	1	[	[	X
ejpam-1595	235	2	12	12	NUM
ejpam-1595	235	3	]	]	X
ejpam-1595	235	4	n.	n.	PROPN
ejpam-1595	235	5	levine	levine	PROPN
ejpam-1595	235	6	.	.	PUNCT
ejpam-1595	236	1	generalized	generalize	VERB
ejpam-1595	236	2	closed	closed	ADJ
ejpam-1595	236	3	sets	set	NOUN
ejpam-1595	236	4	in	in	ADP
ejpam-1595	236	5	topology	topology	NOUN
ejpam-1595	236	6	.	.	PUNCT
ejpam-1595	237	1	rendiconti	rendiconti	VERB
ejpam-1595	237	2	del	del	PROPN
ejpam-1595	237	3	circolo	circolo	PROPN
ejpam-1595	237	4	matematico	matematico	NOUN
ejpam-1595	237	5	di	di	NOUN
ejpam-1595	237	6	palermo	palermo	NOUN
ejpam-1595	237	7	,	,	PUNCT
ejpam-1595	237	8	19:89–96	19:89–96	NUM
ejpam-1595	237	9	,	,	PUNCT
ejpam-1595	237	10	1970	1970	NUM
ejpam-1595	237	11	.	.	PUNCT