id	sid	tid	token	lemma	pos
ejpam-2488	1	1	compile	compile	NOUN
ejpam-2488	1	2	/	/	SYM
ejpam-2488	1	3	output.dvi	output.dvi	NOUN
ejpam-2488	1	4	european	european	ADJ
ejpam-2488	1	5	journal	journal	NOUN
ejpam-2488	1	6	of	of	ADP
ejpam-2488	1	7	pure	pure	ADJ
ejpam-2488	1	8	and	and	CCONJ
ejpam-2488	1	9	applied	apply	VERB
ejpam-2488	1	10	mathematics	mathematic	NOUN
ejpam-2488	1	11	vol	vol	NOUN
ejpam-2488	1	12	.	.	PROPN
ejpam-2488	1	13	8	8	NUM
ejpam-2488	1	14	,	,	PUNCT
ejpam-2488	1	15	no	no	INTJ
ejpam-2488	1	16	.	.	NOUN
ejpam-2488	1	17	4	4	NUM
ejpam-2488	1	18	,	,	PUNCT
ejpam-2488	1	19	2015	2015	NUM
ejpam-2488	1	20	,	,	PUNCT
ejpam-2488	1	21	502	502	NUM
ejpam-2488	1	22	-	-	SYM
ejpam-2488	1	23	513	513	NUM
ejpam-2488	1	24	issn	issn	PROPN
ejpam-2488	1	25	1307	1307	NUM
ejpam-2488	1	26	-	-	SYM
ejpam-2488	1	27	5543	5543	NUM
ejpam-2488	1	28	–	–	PUNCT
ejpam-2488	1	29	www.ejpam.com	www.ejpam.com	X
ejpam-2488	1	30	weak	weak	ADJ
ejpam-2488	1	31	separation	separation	NOUN
ejpam-2488	1	32	axioms	axiom	NOUN
ejpam-2488	1	33	via	via	ADP
ejpam-2488	1	34	e	e	NOUN
ejpam-2488	1	35	-	-	PUNCT
ejpam-2488	1	36	i	i	PRON
ejpam-2488	1	37	sets	set	VERB
ejpam-2488	1	38	in	in	ADP
ejpam-2488	1	39	ideal	ideal	ADJ
ejpam-2488	1	40	topological	topological	ADJ
ejpam-2488	1	41	spaces	space	NOUN
ejpam-2488	1	42	wadei	wadei	VERB
ejpam-2488	1	43	faris	faris	PROPN
ejpam-2488	1	44	al	al	PROPN
ejpam-2488	1	45	-	-	PUNCT
ejpam-2488	1	46	omeri1	omeri1	PROPN
ejpam-2488	1	47	,	,	PUNCT
ejpam-2488	1	48	m.s	m.s	PROPN
ejpam-2488	1	49	.	.	PROPN
ejpam-2488	1	50	md	md	PROPN
ejpam-2488	1	51	.	.	PROPN
ejpam-2488	2	1	noorani1	noorani1	PROPN
ejpam-2488	2	2	,	,	PUNCT
ejpam-2488	2	3	a.	a.	PROPN
ejpam-2488	2	4	al	al	PROPN
ejpam-2488	2	5	-	-	PUNCT
ejpam-2488	2	6	omari2	omari2	PROPN
ejpam-2488	2	7	,	,	PUNCT
ejpam-2488	2	8	t.	t.	PROPN
ejpam-2488	2	9	noiri3,∗	noiri3,∗	PROPN
ejpam-2488	2	10	1	1	NUM
ejpam-2488	2	11	school	school	NOUN
ejpam-2488	2	12	of	of	ADP
ejpam-2488	2	13	mathematical	mathematical	ADJ
ejpam-2488	2	14	sciences	science	NOUN
ejpam-2488	2	15	,	,	PUNCT
ejpam-2488	2	16	faculty	faculty	NOUN
ejpam-2488	2	17	of	of	ADP
ejpam-2488	2	18	science	science	NOUN
ejpam-2488	2	19	and	and	CCONJ
ejpam-2488	2	20	technology	technology	PROPN
ejpam-2488	2	21	universiti	universiti	PROPN
ejpam-2488	2	22	kebangsaan	kebangsaan	PROPN
ejpam-2488	2	23	malaysia	malaysia	PROPN
ejpam-2488	2	24	,	,	PUNCT
ejpam-2488	2	25	43600	43600	NUM
ejpam-2488	2	26	ukm	ukm	PROPN
ejpam-2488	2	27	bangi	bangi	PROPN
ejpam-2488	2	28	,	,	PUNCT
ejpam-2488	2	29	selangor	selangor	PROPN
ejpam-2488	2	30	de	de	PROPN
ejpam-2488	2	31	,	,	PUNCT
ejpam-2488	2	32	malaysia	malaysia	PROPN
ejpam-2488	2	33	2	2	NUM
ejpam-2488	2	34	department	department	NOUN
ejpam-2488	2	35	of	of	ADP
ejpam-2488	2	36	mathematics	mathematic	NOUN
ejpam-2488	2	37	,	,	PUNCT
ejpam-2488	2	38	faculty	faculty	NOUN
ejpam-2488	2	39	of	of	ADP
ejpam-2488	2	40	science	science	PROPN
ejpam-2488	2	41	al	al	PROPN
ejpam-2488	2	42	al	al	PROPN
ejpam-2488	2	43	-	-	PUNCT
ejpam-2488	2	44	bayat	bayat	PROPN
ejpam-2488	2	45	university	university	NOUN
ejpam-2488	2	46	,	,	PUNCT
ejpam-2488	2	47	p.o.box	p.o.box	PROPN
ejpam-2488	2	48	130095	130095	NUM
ejpam-2488	2	49	,	,	PUNCT
ejpam-2488	2	50	mafraq25113	mafraq25113	PROPN
ejpam-2488	2	51	,	,	PUNCT
ejpam-2488	2	52	jordan	jordan	PROPN
ejpam-2488	2	53	3	3	NUM
ejpam-2488	2	54	2949	2949	NUM
ejpam-2488	2	55	-	-	SYM
ejpam-2488	2	56	1	1	NUM
ejpam-2488	2	57	shiokita	shiokita	NOUN
ejpam-2488	2	58	-	-	PUNCT
ejpam-2488	2	59	cho	cho	ADJ
ejpam-2488	2	60	,	,	PUNCT
ejpam-2488	2	61	hinagu	hinagu	ADJ
ejpam-2488	2	62	,	,	PUNCT
ejpam-2488	2	63	yatsushiro	yatsushiro	PROPN
ejpam-2488	2	64	-	-	PUNCT
ejpam-2488	2	65	shi	shi	PROPN
ejpam-2488	2	66	,	,	PUNCT
ejpam-2488	2	67	kumamoto	kumamoto	PROPN
ejpam-2488	2	68	-	-	PUNCT
ejpam-2488	2	69	ken	ken	PROPN
ejpam-2488	2	70	869	869	NUM
ejpam-2488	2	71	-	-	SYM
ejpam-2488	2	72	5142	5142	NUM
ejpam-2488	2	73	,	,	PUNCT
ejpam-2488	2	74	japan	japan	PROPN
ejpam-2488	2	75	.	.	PUNCT
ejpam-2488	3	1	abstract	abstract	PROPN
ejpam-2488	3	2	.	.	PUNCT
ejpam-2488	4	1	in	in	ADP
ejpam-2488	4	2	this	this	DET
ejpam-2488	4	3	paper	paper	NOUN
ejpam-2488	4	4	,	,	PUNCT
ejpam-2488	4	5	we	we	PRON
ejpam-2488	4	6	use	use	VERB
ejpam-2488	4	7	the	the	DET
ejpam-2488	4	8	notion	notion	NOUN
ejpam-2488	4	9	of	of	ADP
ejpam-2488	4	10	e	e	NOUN
ejpam-2488	4	11	-	-	ADJ
ejpam-2488	4	12	i	i	PRON
ejpam-2488	4	13	-open	-open	NOUN
ejpam-2488	4	14	sets	set	NOUN
ejpam-2488	4	15	to	to	PART
ejpam-2488	4	16	introduce	introduce	VERB
ejpam-2488	4	17	and	and	CCONJ
ejpam-2488	4	18	define	define	VERB
ejpam-2488	4	19	some	some	DET
ejpam-2488	4	20	new	new	ADJ
ejpam-2488	4	21	weak	weak	ADJ
ejpam-2488	4	22	separation	separation	NOUN
ejpam-2488	4	23	axioms	axiom	NOUN
ejpam-2488	4	24	.	.	PUNCT
ejpam-2488	5	1	also	also	ADV
ejpam-2488	5	2	we	we	PRON
ejpam-2488	5	3	study	study	VERB
ejpam-2488	5	4	some	some	PRON
ejpam-2488	5	5	of	of	ADP
ejpam-2488	5	6	their	their	PRON
ejpam-2488	5	7	basic	basic	ADJ
ejpam-2488	5	8	properties	property	NOUN
ejpam-2488	5	9	.	.	PUNCT
ejpam-2488	6	1	additionally	additionally	ADV
ejpam-2488	6	2	,	,	PUNCT
ejpam-2488	6	3	we	we	PRON
ejpam-2488	6	4	investigate	investigate	VERB
ejpam-2488	6	5	the	the	DET
ejpam-2488	6	6	relationship	relationship	NOUN
ejpam-2488	6	7	and	and	CCONJ
ejpam-2488	6	8	implications	implication	NOUN
ejpam-2488	6	9	of	of	ADP
ejpam-2488	6	10	these	these	DET
ejpam-2488	6	11	axioms	axiom	NOUN
ejpam-2488	6	12	among	among	ADP
ejpam-2488	6	13	themselves	themselves	PRON
ejpam-2488	6	14	and	and	CCONJ
ejpam-2488	6	15	with	with	ADP
ejpam-2488	6	16	other	other	ADJ
ejpam-2488	6	17	known	know	VERB
ejpam-2488	6	18	axioms	axiom	NOUN
ejpam-2488	6	19	.	.	PUNCT
ejpam-2488	7	1	2010	2010	NUM
ejpam-2488	7	2	mathematics	mathematic	NOUN
ejpam-2488	7	3	subject	subject	NOUN
ejpam-2488	7	4	classifications	classification	NOUN
ejpam-2488	7	5	:	:	PUNCT
ejpam-2488	7	6	54a05	54a05	NUM
ejpam-2488	7	7	key	key	ADJ
ejpam-2488	7	8	words	word	NOUN
ejpam-2488	7	9	and	and	CCONJ
ejpam-2488	7	10	phrases	phrase	NOUN
ejpam-2488	7	11	:	:	PUNCT
ejpam-2488	7	12	ideal	ideal	ADJ
ejpam-2488	7	13	topological	topological	ADJ
ejpam-2488	7	14	space	space	NOUN
ejpam-2488	7	15	,	,	PUNCT
ejpam-2488	7	16	e	e	X
ejpam-2488	7	17	-	-	PUNCT
ejpam-2488	7	18	i	i	PRON
ejpam-2488	7	19	-r0	-r0	NOUN
ejpam-2488	7	20	space	space	NOUN
ejpam-2488	7	21	,	,	PUNCT
ejpam-2488	7	22	e	e	X
ejpam-2488	7	23	-	-	NOUN
ejpam-2488	7	24	i	i	PRON
ejpam-2488	7	25	-r1	-r1	NOUN
ejpam-2488	7	26	space	space	NOUN
ejpam-2488	7	27	,	,	PUNCT
ejpam-2488	7	28	e	e	NOUN
ejpam-2488	7	29	-	-	NOUN
ejpam-2488	7	30	i	i	PRON
ejpam-2488	7	31	-open	-open	NOUN
ejpam-2488	7	32	set	set	NOUN
ejpam-2488	7	33	,	,	PUNCT
ejpam-2488	7	34	e	e	X
ejpam-2488	7	35	-	-	NOUN
ejpam-2488	7	36	i	i	PRON
ejpam-2488	7	37	-r2	-r2	NOUN
ejpam-2488	7	38	space	space	NOUN
ejpam-2488	7	39	1	1	NUM
ejpam-2488	7	40	.	.	PUNCT
ejpam-2488	7	41	introduction	introduction	NOUN
ejpam-2488	7	42	the	the	DET
ejpam-2488	7	43	notion	notion	NOUN
ejpam-2488	7	44	of	of	ADP
ejpam-2488	7	45	r0	r0	NOUN
ejpam-2488	7	46	topological	topological	ADJ
ejpam-2488	7	47	spaces	space	NOUN
ejpam-2488	7	48	is	be	AUX
ejpam-2488	7	49	introduced	introduce	VERB
ejpam-2488	7	50	by	by	ADP
ejpam-2488	7	51	shanin	shanin	PROPN
ejpam-2488	8	1	[	[	X
ejpam-2488	8	2	15	15	NUM
ejpam-2488	8	3	]	]	X
ejpam-2488	8	4	in	in	ADP
ejpam-2488	8	5	1943	1943	NUM
ejpam-2488	8	6	.	.	PUNCT
ejpam-2488	9	1	later	later	ADV
ejpam-2488	9	2	,	,	PUNCT
ejpam-2488	9	3	davis	davis	PROPN
ejpam-2488	9	4	[	[	X
ejpam-2488	9	5	4	4	X
ejpam-2488	9	6	]	]	PUNCT
ejpam-2488	9	7	rediscovered	rediscover	VERB
ejpam-2488	9	8	it	it	PRON
ejpam-2488	9	9	and	and	CCONJ
ejpam-2488	9	10	studied	study	VERB
ejpam-2488	9	11	some	some	DET
ejpam-2488	9	12	properties	property	NOUN
ejpam-2488	9	13	of	of	ADP
ejpam-2488	9	14	this	this	DET
ejpam-2488	9	15	weak	weak	ADJ
ejpam-2488	9	16	separation	separation	NOUN
ejpam-2488	9	17	axiom	axiom	NOUN
ejpam-2488	9	18	.	.	PUNCT
ejpam-2488	10	1	several	several	ADJ
ejpam-2488	10	2	topologists	topologist	NOUN
ejpam-2488	10	3	(	(	PUNCT
ejpam-2488	10	4	e.g.	e.g.	ADV
ejpam-2488	10	5	[	[	X
ejpam-2488	10	6	6	6	NUM
ejpam-2488	10	7	,	,	PUNCT
ejpam-2488	10	8	10	10	NUM
ejpam-2488	10	9	,	,	PUNCT
ejpam-2488	10	10	13	13	NUM
ejpam-2488	10	11	]	]	PUNCT
ejpam-2488	10	12	)	)	PUNCT
ejpam-2488	10	13	further	far	ADV
ejpam-2488	10	14	investigated	investigate	VERB
ejpam-2488	10	15	properties	property	NOUN
ejpam-2488	10	16	of	of	ADP
ejpam-2488	10	17	r0	r0	PROPN
ejpam-2488	10	18	topological	topological	ADJ
ejpam-2488	10	19	spaces	space	NOUN
ejpam-2488	10	20	and	and	CCONJ
ejpam-2488	10	21	many	many	ADJ
ejpam-2488	10	22	interesting	interesting	ADJ
ejpam-2488	10	23	results	result	NOUN
ejpam-2488	10	24	have	have	AUX
ejpam-2488	10	25	been	be	AUX
ejpam-2488	10	26	obtained	obtain	VERB
ejpam-2488	10	27	in	in	ADP
ejpam-2488	10	28	various	various	ADJ
ejpam-2488	10	29	contexts	contexts	NOUN
ejpam-2488	10	30	.	.	PUNCT
ejpam-2488	11	1	in	in	ADP
ejpam-2488	11	2	the	the	DET
ejpam-2488	11	3	same	same	ADJ
ejpam-2488	11	4	paper	paper	NOUN
ejpam-2488	11	5	,	,	PUNCT
ejpam-2488	11	6	davis	davis	PROPN
ejpam-2488	11	7	also	also	ADV
ejpam-2488	11	8	introduced	introduce	VERB
ejpam-2488	11	9	the	the	DET
ejpam-2488	11	10	notion	notion	NOUN
ejpam-2488	11	11	of	of	ADP
ejpam-2488	11	12	r1	r1	PROPN
ejpam-2488	11	13	topological	topological	ADJ
ejpam-2488	11	14	spaces	space	NOUN
ejpam-2488	11	15	which	which	PRON
ejpam-2488	11	16	are	be	AUX
ejpam-2488	11	17	independent	independent	ADJ
ejpam-2488	11	18	of	of	ADP
ejpam-2488	11	19	both	both	DET
ejpam-2488	11	20	t0	t0	PROPN
ejpam-2488	11	21	and	and	CCONJ
ejpam-2488	11	22	t1	t1	NOUN
ejpam-2488	11	23	but	but	CCONJ
ejpam-2488	11	24	strictly	strictly	ADV
ejpam-2488	11	25	weaker	weak	ADJ
ejpam-2488	11	26	than	than	ADP
ejpam-2488	11	27	t2	t2	NOUN
ejpam-2488	11	28	.	.	PUNCT
ejpam-2488	12	1	a	a	DET
ejpam-2488	12	2	subset	subset	NOUN
ejpam-2488	12	3	a	a	PRON
ejpam-2488	12	4	of	of	ADP
ejpam-2488	12	5	a	a	DET
ejpam-2488	12	6	space	space	NOUN
ejpam-2488	12	7	(	(	PUNCT
ejpam-2488	12	8	x	x	X
ejpam-2488	12	9	,	,	PUNCT
ejpam-2488	12	10	τ	τ	X
ejpam-2488	12	11	)	)	PUNCT
ejpam-2488	12	12	is	be	AUX
ejpam-2488	12	13	said	say	VERB
ejpam-2488	12	14	to	to	PART
ejpam-2488	12	15	be	be	AUX
ejpam-2488	12	16	regular	regular	ADJ
ejpam-2488	12	17	open	open	ADJ
ejpam-2488	12	18	(	(	PUNCT
ejpam-2488	12	19	resp	resp	NOUN
ejpam-2488	12	20	.	.	PUNCT
ejpam-2488	13	1	regular	regular	ADJ
ejpam-2488	13	2	closed	closed	ADJ
ejpam-2488	13	3	)	)	PUNCT
ejpam-2488	14	1	[	[	X
ejpam-2488	14	2	16	16	NUM
ejpam-2488	14	3	]	]	X
ejpam-2488	14	4	if	if	SCONJ
ejpam-2488	14	5	a	a	PRON
ejpam-2488	14	6	=	=	SYM
ejpam-2488	14	7	int(cl(a	int(cl(a	PROPN
ejpam-2488	14	8	)	)	PUNCT
ejpam-2488	14	9	)	)	PUNCT
ejpam-2488	15	1	(	(	PUNCT
ejpam-2488	15	2	resp	resp	NOUN
ejpam-2488	15	3	.	.	PUNCT
ejpam-2488	16	1	a	a	DET
ejpam-2488	16	2	=	=	NOUN
ejpam-2488	16	3	cl(int(a	cl(int(a	PROPN
ejpam-2488	16	4	)	)	PUNCT
ejpam-2488	16	5	)	)	PUNCT
ejpam-2488	16	6	)	)	PUNCT
ejpam-2488	16	7	.	.	PUNCT
ejpam-2488	17	1	a	a	PRON
ejpam-2488	17	2	is	be	AUX
ejpam-2488	17	3	said	say	VERB
ejpam-2488	17	4	to	to	PART
ejpam-2488	17	5	be	be	AUX
ejpam-2488	17	6	δ	δ	NOUN
ejpam-2488	17	7	-	-	ADJ
ejpam-2488	17	8	open	open	ADJ
ejpam-2488	17	9	[	[	X
ejpam-2488	17	10	18	18	NUM
ejpam-2488	17	11	]	]	X
ejpam-2488	17	12	if	if	SCONJ
ejpam-2488	17	13	for	for	ADP
ejpam-2488	17	14	each	each	DET
ejpam-2488	17	15	x	x	SYM
ejpam-2488	17	16	∈	∈	PROPN
ejpam-2488	17	17	a	a	PRON
ejpam-2488	17	18	,	,	PUNCT
ejpam-2488	17	19	there	there	PRON
ejpam-2488	17	20	exists	exist	VERB
ejpam-2488	17	21	a	a	DET
ejpam-2488	17	22	regular	regular	ADJ
ejpam-2488	17	23	open	open	ADJ
ejpam-2488	17	24	set	set	NOUN
ejpam-2488	17	25	g	g	PROPN
ejpam-2488	17	26	such	such	ADJ
ejpam-2488	17	27	that	that	SCONJ
ejpam-2488	17	28	x	x	SYM
ejpam-2488	17	29	∈	∈	PROPN
ejpam-2488	18	1	g	g	PROPN
ejpam-2488	18	2	⊂	⊂	PROPN
ejpam-2488	18	3	a.	a.	NOUN
ejpam-2488	18	4	the	the	DET
ejpam-2488	18	5	complement	complement	NOUN
ejpam-2488	18	6	of	of	ADP
ejpam-2488	18	7	a	a	DET
ejpam-2488	18	8	δ	δ	NOUN
ejpam-2488	18	9	-	-	ADJ
ejpam-2488	18	10	open	open	ADJ
ejpam-2488	18	11	set	set	NOUN
ejpam-2488	18	12	is	be	AUX
ejpam-2488	18	13	said	say	VERB
ejpam-2488	18	14	to	to	PART
ejpam-2488	18	15	be	be	AUX
ejpam-2488	18	16	δ	δ	NOUN
ejpam-2488	18	17	-	-	PUNCT
ejpam-2488	18	18	closed	closed	ADJ
ejpam-2488	18	19	.	.	PUNCT
ejpam-2488	19	1	a	a	DET
ejpam-2488	19	2	point	point	NOUN
ejpam-2488	19	3	x	x	X
ejpam-2488	19	4	∈	∈	NOUN
ejpam-2488	19	5	x	x	PUNCT
ejpam-2488	19	6	is	be	AUX
ejpam-2488	19	7	called	call	VERB
ejpam-2488	19	8	a	a	DET
ejpam-2488	19	9	δ	δ	NOUN
ejpam-2488	19	10	-	-	PUNCT
ejpam-2488	19	11	cluster	cluster	NOUN
ejpam-2488	19	12	point	point	NOUN
ejpam-2488	19	13	of	of	ADP
ejpam-2488	19	14	a	a	DET
ejpam-2488	19	15	if	if	NOUN
ejpam-2488	19	16	int(cl(u))∩a	int(cl(u))∩a	PROPN
ejpam-2488	19	17	6=	6=	NUM
ejpam-2488	19	18	;	;	PUNCT
ejpam-2488	19	19	for	for	ADP
ejpam-2488	19	20	each	each	DET
ejpam-2488	19	21	open	open	ADJ
ejpam-2488	19	22	set	set	VERB
ejpam-2488	19	23	u	u	NOUN
ejpam-2488	19	24	containing	contain	VERB
ejpam-2488	19	25	x	x	X
ejpam-2488	19	26	.	.	PUNCT
ejpam-2488	20	1	the	the	DET
ejpam-2488	20	2	set	set	NOUN
ejpam-2488	20	3	of	of	ADP
ejpam-2488	20	4	all	all	DET
ejpam-2488	20	5	δ	δ	NOUN
ejpam-2488	20	6	-	-	PUNCT
ejpam-2488	20	7	cluster	cluster	NOUN
ejpam-2488	20	8	points	point	NOUN
ejpam-2488	20	9	of	of	ADP
ejpam-2488	20	10	a	a	PRON
ejpam-2488	20	11	is	be	AUX
ejpam-2488	20	12	called	call	VERB
ejpam-2488	20	13	the	the	DET
ejpam-2488	20	14	δ	δ	NOUN
ejpam-2488	20	15	-	-	NOUN
ejpam-2488	20	16	closure	closure	NOUN
ejpam-2488	20	17	of	of	ADP
ejpam-2488	20	18	a	a	PRON
ejpam-2488	20	19	and	and	CCONJ
ejpam-2488	20	20	is	be	AUX
ejpam-2488	20	21	denoted	denote	VERB
ejpam-2488	20	22	by	by	ADP
ejpam-2488	20	23	clδ(a	clδ(a	NOUN
ejpam-2488	20	24	)	)	PUNCT
ejpam-2488	21	1	[	[	X
ejpam-2488	21	2	18	18	NUM
ejpam-2488	21	3	]	]	PUNCT
ejpam-2488	21	4	.	.	PUNCT
ejpam-2488	22	1	the	the	DET
ejpam-2488	22	2	set	set	VERB
ejpam-2488	22	3	δ	δ	PROPN
ejpam-2488	22	4	-	-	NOUN
ejpam-2488	22	5	interior	interior	NOUN
ejpam-2488	22	6	of	of	ADP
ejpam-2488	22	7	a	a	DET
ejpam-2488	22	8	[	[	X
ejpam-2488	22	9	18	18	NUM
ejpam-2488	22	10	]	]	PUNCT
ejpam-2488	22	11	is	be	AUX
ejpam-2488	22	12	the	the	DET
ejpam-2488	22	13	union	union	NOUN
ejpam-2488	22	14	of	of	ADP
ejpam-2488	22	15	all	all	DET
ejpam-2488	22	16	regular	regular	ADJ
ejpam-2488	22	17	open	open	ADJ
ejpam-2488	22	18	sets	set	NOUN
ejpam-2488	22	19	of	of	ADP
ejpam-2488	22	20	x	x	PUNCT
ejpam-2488	22	21	contained	contain	VERB
ejpam-2488	22	22	in	in	ADP
ejpam-2488	22	23	a	a	PRON
ejpam-2488	22	24	and	and	CCONJ
ejpam-2488	22	25	is	be	AUX
ejpam-2488	22	26	denoted	denote	VERB
ejpam-2488	22	27	by	by	ADP
ejpam-2488	22	28	intδ(a	intδ(a	NOUN
ejpam-2488	22	29	)	)	PUNCT
ejpam-2488	22	30	.	.	PUNCT
ejpam-2488	23	1	a	a	PRON
ejpam-2488	23	2	is	be	AUX
ejpam-2488	23	3	δ	δ	NOUN
ejpam-2488	23	4	-	-	ADJ
ejpam-2488	23	5	open	open	ADJ
ejpam-2488	23	6	if	if	SCONJ
ejpam-2488	23	7	intδ(a	intδ(a	NOUN
ejpam-2488	23	8	)	)	PUNCT
ejpam-2488	23	9	=	=	PUNCT
ejpam-2488	23	10	a.	a.	NOUN
ejpam-2488	23	11	the	the	DET
ejpam-2488	23	12	collection	collection	NOUN
ejpam-2488	23	13	of	of	ADP
ejpam-2488	23	14	all	all	DET
ejpam-2488	23	15	δ	δ	NOUN
ejpam-2488	23	16	-	-	ADJ
ejpam-2488	23	17	open	open	ADJ
ejpam-2488	23	18	sets	set	NOUN
ejpam-2488	23	19	of	of	ADP
ejpam-2488	23	20	(	(	PUNCT
ejpam-2488	23	21	x	x	INTJ
ejpam-2488	23	22	,	,	PUNCT
ejpam-2488	23	23	τ	τ	X
ejpam-2488	23	24	)	)	PUNCT
ejpam-2488	23	25	is	be	AUX
ejpam-2488	23	26	denoted	denote	VERB
ejpam-2488	23	27	by	by	ADP
ejpam-2488	23	28	δo(x	δo(x	NUM
ejpam-2488	23	29	)	)	PUNCT
ejpam-2488	23	30	and	and	CCONJ
ejpam-2488	23	31	forms	form	VERB
ejpam-2488	23	32	a	a	DET
ejpam-2488	23	33	topology	topology	NOUN
ejpam-2488	23	34	τδ	τδ	ADP
ejpam-2488	23	35	.	.	PUNCT
ejpam-2488	24	1	∗corresponding	∗corresponde	VERB
ejpam-2488	24	2	author	author	NOUN
ejpam-2488	24	3	.	.	PUNCT
ejpam-2488	25	1	email	email	NOUN
ejpam-2488	25	2	addresses	address	NOUN
ejpam-2488	25	3	:	:	PUNCT
ejpam-2488	25	4	wadeimoon1@hotmail.com	wadeimoon1@hotmail.com	X
ejpam-2488	25	5	(	(	PUNCT
ejpam-2488	25	6	w.	w.	PROPN
ejpam-2488	25	7	al	al	PROPN
ejpam-2488	25	8	-	-	PUNCT
ejpam-2488	25	9	omeri	omeri	NOUN
ejpam-2488	25	10	)	)	PUNCT
ejpam-2488	25	11	,	,	PUNCT
ejpam-2488	25	12	msn@ukm.my	msn@ukm.my	X
ejpam-2488	25	13	(	(	PUNCT
ejpam-2488	25	14	m.	m.	NOUN
ejpam-2488	25	15	noorani	noorani	PROPN
ejpam-2488	25	16	)	)	PUNCT
ejpam-2488	25	17	,	,	PUNCT
ejpam-2488	25	18	omarimutah1@yahoo.com	omarimutah1@yahoo.com	PROPN
ejpam-2488	25	19	(	(	PUNCT
ejpam-2488	25	20	a.	a.	PROPN
ejpam-2488	25	21	al	al	PROPN
ejpam-2488	25	22	-	-	PUNCT
ejpam-2488	25	23	omari	omari	PROPN
ejpam-2488	25	24	)	)	PUNCT
ejpam-2488	25	25	,	,	PUNCT
ejpam-2488	25	26	t.noiri@nifty.com	t.noiri@nifty.com	X
ejpam-2488	25	27	(	(	PUNCT
ejpam-2488	25	28	t.	t.	PROPN
ejpam-2488	25	29	noiri	noiri	PROPN
ejpam-2488	25	30	)	)	PUNCT
ejpam-2488	25	31	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2488	26	1	502	502	NUM
ejpam-2488	26	2	c	c	X
ejpam-2488	26	3	©	©	PROPN
ejpam-2488	26	4	2015	2015	NUM
ejpam-2488	26	5	ejpam	ejpam	NOUN
ejpam-2488	26	6	all	all	DET
ejpam-2488	26	7	rights	right	NOUN
ejpam-2488	26	8	reserved	reserve	VERB
ejpam-2488	26	9	.	.	PUNCT
ejpam-2488	27	1	w.	w.	PROPN
ejpam-2488	27	2	al	al	PROPN
ejpam-2488	27	3	-	-	PUNCT
ejpam-2488	27	4	omeri	omeri	ADJ
ejpam-2488	27	5	,	,	PUNCT
ejpam-2488	27	6	m.	m.	NOUN
ejpam-2488	27	7	noorani	noorani	PROPN
ejpam-2488	27	8	,	,	PUNCT
ejpam-2488	27	9	a.	a.	PROPN
ejpam-2488	27	10	al	al	PROPN
ejpam-2488	27	11	-	-	PUNCT
ejpam-2488	27	12	omari	omari	PROPN
ejpam-2488	27	13	,	,	PUNCT
ejpam-2488	27	14	and	and	CCONJ
ejpam-2488	27	15	t.	t.	PROPN
ejpam-2488	27	16	noiri	noiri	PROPN
ejpam-2488	27	17	/	/	SYM
ejpam-2488	27	18	eur	eur	PROPN
ejpam-2488	27	19	.	.	PUNCT
ejpam-2488	28	1	j.	j.	PROPN
ejpam-2488	28	2	pure	pure	PROPN
ejpam-2488	28	3	appl	appl	PROPN
ejpam-2488	28	4	.	.	PROPN
ejpam-2488	28	5	math	math	PROPN
ejpam-2488	28	6	,	,	PUNCT
ejpam-2488	28	7	8	8	NUM
ejpam-2488	28	8	(	(	PUNCT
ejpam-2488	28	9	2015	2015	NUM
ejpam-2488	28	10	)	)	PUNCT
ejpam-2488	28	11	,	,	PUNCT
ejpam-2488	28	12	502	502	NUM
ejpam-2488	28	13	-	-	SYM
ejpam-2488	28	14	513	513	NUM
ejpam-2488	28	15	503	503	NUM
ejpam-2488	28	16	an	an	DET
ejpam-2488	28	17	ideal	ideal	NOUN
ejpam-2488	29	1	i	i	PRON
ejpam-2488	29	2	on	on	ADP
ejpam-2488	29	3	a	a	DET
ejpam-2488	29	4	topological	topological	ADJ
ejpam-2488	29	5	space	space	NOUN
ejpam-2488	29	6	(	(	PUNCT
ejpam-2488	29	7	x	x	X
ejpam-2488	29	8	,	,	PUNCT
ejpam-2488	29	9	i	i	PROPN
ejpam-2488	29	10	)	)	PUNCT
ejpam-2488	29	11	is	be	AUX
ejpam-2488	29	12	a	a	DET
ejpam-2488	29	13	nonempty	nonempty	ADJ
ejpam-2488	29	14	collection	collection	NOUN
ejpam-2488	29	15	of	of	ADP
ejpam-2488	29	16	subsets	subset	NOUN
ejpam-2488	29	17	of	of	ADP
ejpam-2488	29	18	x	x	PUNCT
ejpam-2488	29	19	which	which	PRON
ejpam-2488	29	20	satisfies	satisfy	VERB
ejpam-2488	29	21	the	the	DET
ejpam-2488	29	22	following	follow	VERB
ejpam-2488	29	23	conditions	condition	NOUN
ejpam-2488	29	24	:	:	PUNCT
ejpam-2488	29	25	a	a	DET
ejpam-2488	29	26	∈	∈	ADJ
ejpam-2488	30	1	i	i	PRON
ejpam-2488	30	2	and	and	CCONJ
ejpam-2488	30	3	b	b	PROPN
ejpam-2488	30	4	⊂	⊂	PROPN
ejpam-2488	30	5	a	a	PRON
ejpam-2488	30	6	implies	imply	VERB
ejpam-2488	30	7	b	b	X
ejpam-2488	30	8	∈	∈	PROPN
ejpam-2488	30	9	i	i	PRON
ejpam-2488	30	10	;	;	PUNCT
ejpam-2488	30	11	a	a	DET
ejpam-2488	30	12	∈	∈	NOUN
ejpam-2488	30	13	i	i	PRON
ejpam-2488	30	14	and	and	CCONJ
ejpam-2488	30	15	b	b	X
ejpam-2488	30	16	∈	∈	PROPN
ejpam-2488	30	17	i	i	PRON
ejpam-2488	30	18	implies	imply	VERB
ejpam-2488	30	19	a∪	a∪	PROPN
ejpam-2488	31	1	b	b	X
ejpam-2488	31	2	∈	∈	PROPN
ejpam-2488	32	1	i	i	PRON
ejpam-2488	32	2	.	.	PUNCT
ejpam-2488	33	1	applications	application	NOUN
ejpam-2488	33	2	to	to	ADP
ejpam-2488	33	3	various	various	ADJ
ejpam-2488	33	4	fields	field	NOUN
ejpam-2488	33	5	were	be	AUX
ejpam-2488	33	6	further	far	ADV
ejpam-2488	33	7	investigated	investigate	VERB
ejpam-2488	33	8	by	by	ADP
ejpam-2488	33	9	jankovic	jankovic	PROPN
ejpam-2488	33	10	and	and	CCONJ
ejpam-2488	33	11	hamlett	hamlett	PROPN
ejpam-2488	34	1	[	[	X
ejpam-2488	34	2	11	11	NUM
ejpam-2488	34	3	]	]	X
ejpam-2488	34	4	;	;	PUNCT
ejpam-2488	34	5	dontchev	dontchev	PROPN
ejpam-2488	34	6	[	[	X
ejpam-2488	34	7	5	5	NUM
ejpam-2488	34	8	]	]	PUNCT
ejpam-2488	34	9	;	;	PUNCT
ejpam-2488	34	10	mukherjee	mukherjee	PROPN
ejpam-2488	34	11	et	et	PROPN
ejpam-2488	34	12	al	al	PROPN
ejpam-2488	34	13	.	.	PUNCT
ejpam-2488	35	1	[	[	X
ejpam-2488	35	2	12	12	NUM
ejpam-2488	35	3	]	]	X
ejpam-2488	35	4	;	;	PUNCT
ejpam-2488	35	5	arenas	arenas	PROPN
ejpam-2488	35	6	et	et	PROPN
ejpam-2488	35	7	al	al	PROPN
ejpam-2488	35	8	.	.	PUNCT
ejpam-2488	36	1	[	[	X
ejpam-2488	36	2	3	3	NUM
ejpam-2488	36	3	]	]	PUNCT
ejpam-2488	36	4	;	;	PUNCT
ejpam-2488	36	5	nasef	nasef	PROPN
ejpam-2488	36	6	and	and	CCONJ
ejpam-2488	36	7	mahmoud	mahmoud	PROPN
ejpam-2488	37	1	[	[	X
ejpam-2488	37	2	14	14	NUM
ejpam-2488	37	3	]	]	PUNCT
ejpam-2488	37	4	,	,	PUNCT
ejpam-2488	37	5	etc	etc	X
ejpam-2488	37	6	.	.	X
ejpam-2488	37	7	given	give	VERB
ejpam-2488	37	8	a	a	DET
ejpam-2488	37	9	topological	topological	ADJ
ejpam-2488	37	10	space	space	NOUN
ejpam-2488	37	11	(	(	PUNCT
ejpam-2488	37	12	x	x	X
ejpam-2488	37	13	,	,	PUNCT
ejpam-2488	37	14	i	i	PROPN
ejpam-2488	37	15	)	)	PUNCT
ejpam-2488	37	16	with	with	ADP
ejpam-2488	37	17	an	an	DET
ejpam-2488	37	18	ideal	ideal	ADJ
ejpam-2488	37	19	i	i	PRON
ejpam-2488	37	20	on	on	ADP
ejpam-2488	37	21	x	x	X
ejpam-2488	37	22	and	and	CCONJ
ejpam-2488	37	23	if	if	SCONJ
ejpam-2488	37	24	℘(x	℘(x	VERB
ejpam-2488	37	25	)	)	PUNCT
ejpam-2488	37	26	is	be	AUX
ejpam-2488	37	27	the	the	DET
ejpam-2488	37	28	set	set	NOUN
ejpam-2488	37	29	of	of	ADP
ejpam-2488	37	30	all	all	DET
ejpam-2488	37	31	subsets	subset	NOUN
ejpam-2488	37	32	of	of	ADP
ejpam-2488	37	33	x	x	PRON
ejpam-2488	37	34	,	,	PUNCT
ejpam-2488	37	35	a	a	DET
ejpam-2488	37	36	set	set	NOUN
ejpam-2488	37	37	operator	operator	NOUN
ejpam-2488	37	38	(	(	PUNCT
ejpam-2488	37	39	.)∗	.)∗	X
ejpam-2488	37	40	:	:	PUNCT
ejpam-2488	37	41	℘(x	℘(x	ADJ
ejpam-2488	37	42	)	)	PUNCT
ejpam-2488	37	43	→	→	SYM
ejpam-2488	37	44	℘(x	℘(x	ADJ
ejpam-2488	37	45	)	)	PUNCT
ejpam-2488	37	46	,	,	PUNCT
ejpam-2488	37	47	called	call	VERB
ejpam-2488	37	48	a	a	DET
ejpam-2488	37	49	local	local	ADJ
ejpam-2488	37	50	function	function	NOUN
ejpam-2488	37	51	[	[	X
ejpam-2488	37	52	11	11	NUM
ejpam-2488	37	53	,	,	PUNCT
ejpam-2488	37	54	17	17	NUM
ejpam-2488	37	55	]	]	PUNCT
ejpam-2488	37	56	of	of	ADP
ejpam-2488	37	57	a	a	PRON
ejpam-2488	37	58	with	with	ADP
ejpam-2488	37	59	respect	respect	NOUN
ejpam-2488	37	60	to	to	ADP
ejpam-2488	37	61	τ	τ	PROPN
ejpam-2488	37	62	and	and	CCONJ
ejpam-2488	37	63	i	i	PRON
ejpam-2488	37	64	is	be	AUX
ejpam-2488	37	65	defined	define	VERB
ejpam-2488	37	66	as	as	SCONJ
ejpam-2488	37	67	follows	follow	VERB
ejpam-2488	37	68	:	:	PUNCT
ejpam-2488	37	69	for	for	ADP
ejpam-2488	37	70	a⊆	a⊆	PROPN
ejpam-2488	37	71	x	x	SYM
ejpam-2488	37	72	,	,	PUNCT
ejpam-2488	37	73	a∗(i	a∗(i	PROPN
ejpam-2488	37	74	,	,	PUNCT
ejpam-2488	37	75	τ	τ	X
ejpam-2488	37	76	)	)	PUNCT
ejpam-2488	38	1	=	=	SYM
ejpam-2488	38	2	�	�	PROPN
ejpam-2488	38	3	x	x	SYM
ejpam-2488	38	4	∈	∈	PROPN
ejpam-2488	38	5	x	x	X
ejpam-2488	38	6	|	|	ADV
ejpam-2488	38	7	u	u	X
ejpam-2488	38	8	∩	∩	NOUN
ejpam-2488	38	9	a	a	X
ejpam-2488	38	10	/∈	/∈	PUNCT
ejpam-2488	39	1	i	i	PRON
ejpam-2488	39	2	for	for	ADP
ejpam-2488	39	3	every	every	DET
ejpam-2488	39	4	u	u	PROPN
ejpam-2488	39	5	∈	∈	PROPN
ejpam-2488	39	6	τ(x	τ(x	NOUN
ejpam-2488	39	7	)	)	PUNCT
ejpam-2488	39	8	where	where	SCONJ
ejpam-2488	39	9	τ(x	τ(x	NOUN
ejpam-2488	39	10	)	)	PUNCT
ejpam-2488	39	11	=	=	PRON
ejpam-2488	39	12	{	{	PUNCT
ejpam-2488	39	13	u	u	X
ejpam-2488	39	14	∈	∈	PROPN
ejpam-2488	39	15	τ	τ	X
ejpam-2488	40	1	|	|	ADV
ejpam-2488	40	2	x	x	X
ejpam-2488	40	3	∈	∈	PROPN
ejpam-2488	40	4	u	u	NOUN
ejpam-2488	40	5	}	}	PUNCT
ejpam-2488	40	6	.	.	PUNCT
ejpam-2488	41	1	furthermore	furthermore	ADV
ejpam-2488	41	2	cl∗(a	cl∗(a	NOUN
ejpam-2488	41	3	)	)	PUNCT
ejpam-2488	41	4	=	=	PUNCT
ejpam-2488	42	1	a∪	a∪	PROPN
ejpam-2488	42	2	a∗(i	a∗(i	PROPN
ejpam-2488	42	3	,	,	PUNCT
ejpam-2488	42	4	τ	τ	PROPN
ejpam-2488	42	5	)	)	PUNCT
ejpam-2488	42	6	defines	define	VERB
ejpam-2488	42	7	a	a	DET
ejpam-2488	42	8	kuratowski	kuratowski	ADJ
ejpam-2488	42	9	closure	closure	NOUN
ejpam-2488	42	10	operator	operator	NOUN
ejpam-2488	42	11	for	for	ADP
ejpam-2488	42	12	the	the	DET
ejpam-2488	42	13	topology	topology	NOUN
ejpam-2488	42	14	τ∗.	τ∗.	VERB
ejpam-2488	42	15	when	when	SCONJ
ejpam-2488	42	16	there	there	PRON
ejpam-2488	42	17	is	be	VERB
ejpam-2488	42	18	no	no	DET
ejpam-2488	42	19	chance	chance	NOUN
ejpam-2488	42	20	for	for	ADP
ejpam-2488	42	21	confusion	confusion	NOUN
ejpam-2488	42	22	,	,	PUNCT
ejpam-2488	42	23	we	we	PRON
ejpam-2488	42	24	will	will	AUX
ejpam-2488	42	25	simply	simply	ADV
ejpam-2488	42	26	write	write	VERB
ejpam-2488	42	27	a∗	a∗	NOUN
ejpam-2488	42	28	for	for	ADP
ejpam-2488	42	29	a∗(i	a∗(i	PROPN
ejpam-2488	42	30	,	,	PUNCT
ejpam-2488	42	31	τ	τ	PROPN
ejpam-2488	42	32	)	)	PUNCT
ejpam-2488	42	33	.	.	PUNCT
ejpam-2488	43	1	x	x	PUNCT
ejpam-2488	43	2	∗	∗	NOUN
ejpam-2488	43	3	is	be	AUX
ejpam-2488	43	4	often	often	ADV
ejpam-2488	43	5	a	a	DET
ejpam-2488	43	6	proper	proper	ADJ
ejpam-2488	43	7	subset	subset	NOUN
ejpam-2488	43	8	of	of	ADP
ejpam-2488	43	9	x	x	X
ejpam-2488	43	10	.	.	PUNCT
ejpam-2488	44	1	by	by	ADP
ejpam-2488	44	2	a	a	DET
ejpam-2488	44	3	space	space	NOUN
ejpam-2488	44	4	,	,	PUNCT
ejpam-2488	44	5	we	we	PRON
ejpam-2488	44	6	always	always	ADV
ejpam-2488	44	7	mean	mean	VERB
ejpam-2488	44	8	a	a	DET
ejpam-2488	44	9	topological	topological	ADJ
ejpam-2488	44	10	space	space	NOUN
ejpam-2488	44	11	(	(	PUNCT
ejpam-2488	44	12	x	x	NOUN
ejpam-2488	44	13	,	,	PUNCT
ejpam-2488	44	14	τ)with	τ)with	PUNCT
ejpam-2488	45	1	no	no	DET
ejpam-2488	45	2	separation	separation	NOUN
ejpam-2488	45	3	properties	property	NOUN
ejpam-2488	45	4	assumed	assume	VERB
ejpam-2488	45	5	.	.	PUNCT
ejpam-2488	46	1	if	if	SCONJ
ejpam-2488	46	2	a⊂	a⊂	DET
ejpam-2488	46	3	x	x	SYM
ejpam-2488	46	4	,	,	PUNCT
ejpam-2488	46	5	cl(a	cl(a	NUM
ejpam-2488	46	6	)	)	PUNCT
ejpam-2488	46	7	and	and	CCONJ
ejpam-2488	46	8	int(a)will	int(a)will	ADV
ejpam-2488	46	9	denote	denote	VERB
ejpam-2488	46	10	the	the	DET
ejpam-2488	46	11	closure	closure	NOUN
ejpam-2488	46	12	and	and	CCONJ
ejpam-2488	46	13	interior	interior	NOUN
ejpam-2488	46	14	of	of	ADP
ejpam-2488	46	15	a	a	DET
ejpam-2488	46	16	in	in	ADP
ejpam-2488	46	17	(	(	PUNCT
ejpam-2488	46	18	x	x	INTJ
ejpam-2488	46	19	,	,	PUNCT
ejpam-2488	46	20	τ	τ	PROPN
ejpam-2488	46	21	)	)	PUNCT
ejpam-2488	46	22	,	,	PUNCT
ejpam-2488	46	23	respectively	respectively	ADV
ejpam-2488	46	24	.	.	PUNCT
ejpam-2488	47	1	a	a	DET
ejpam-2488	47	2	subset	subset	NOUN
ejpam-2488	47	3	a	a	PRON
ejpam-2488	47	4	of	of	ADP
ejpam-2488	47	5	a	a	DET
ejpam-2488	47	6	topological	topological	ADJ
ejpam-2488	47	7	space	space	NOUN
ejpam-2488	47	8	(	(	PUNCT
ejpam-2488	47	9	x	x	X
ejpam-2488	47	10	,	,	PUNCT
ejpam-2488	47	11	τ	τ	X
ejpam-2488	47	12	)	)	PUNCT
ejpam-2488	47	13	is	be	AUX
ejpam-2488	47	14	said	say	VERB
ejpam-2488	47	15	to	to	PART
ejpam-2488	47	16	be	be	AUX
ejpam-2488	47	17	e	e	VERB
ejpam-2488	47	18	-	-	NOUN
ejpam-2488	47	19	open	open	ADJ
ejpam-2488	48	1	[	[	X
ejpam-2488	48	2	9	9	NUM
ejpam-2488	48	3	]	]	X
ejpam-2488	48	4	if	if	SCONJ
ejpam-2488	48	5	a⊂	a⊂	PRON
ejpam-2488	48	6	int(δcl(a))∪	int(δcl(a))∪	NOUN
ejpam-2488	48	7	cl(δint(a	cl(δint(a	NUM
ejpam-2488	48	8	)	)	PUNCT
ejpam-2488	48	9	)	)	PUNCT
ejpam-2488	48	10	.	.	PUNCT
ejpam-2488	49	1	the	the	DET
ejpam-2488	49	2	notion	notion	NOUN
ejpam-2488	49	3	of	of	ADP
ejpam-2488	49	4	e	e	NOUN
ejpam-2488	49	5	-	-	ADJ
ejpam-2488	49	6	open	open	ADJ
ejpam-2488	49	7	sets	set	NOUN
ejpam-2488	49	8	has	have	AUX
ejpam-2488	49	9	been	be	AUX
ejpam-2488	49	10	study	study	VERB
ejpam-2488	49	11	extensively	extensively	ADV
ejpam-2488	49	12	in	in	ADP
ejpam-2488	49	13	recent	recent	ADJ
ejpam-2488	49	14	years	year	NOUN
ejpam-2488	49	15	by	by	ADP
ejpam-2488	49	16	many	many	ADJ
ejpam-2488	49	17	topologists	topologist	NOUN
ejpam-2488	49	18	.	.	PUNCT
ejpam-2488	50	1	in	in	ADP
ejpam-2488	50	2	this	this	DET
ejpam-2488	50	3	paper	paper	NOUN
ejpam-2488	50	4	,	,	PUNCT
ejpam-2488	50	5	we	we	PRON
ejpam-2488	50	6	use	use	VERB
ejpam-2488	50	7	the	the	DET
ejpam-2488	50	8	notion	notion	NOUN
ejpam-2488	50	9	of	of	ADP
ejpam-2488	50	10	e	e	NOUN
ejpam-2488	50	11	-	-	ADJ
ejpam-2488	50	12	i	i	PRON
ejpam-2488	50	13	-open	-open	NOUN
ejpam-2488	50	14	sets	set	NOUN
ejpam-2488	50	15	to	to	PART
ejpam-2488	50	16	introduce	introduce	VERB
ejpam-2488	50	17	and	and	CCONJ
ejpam-2488	50	18	define	define	VERB
ejpam-2488	50	19	some	some	DET
ejpam-2488	50	20	new	new	ADJ
ejpam-2488	50	21	weak	weak	ADJ
ejpam-2488	50	22	separation	separation	NOUN
ejpam-2488	50	23	axioms	axiom	NOUN
ejpam-2488	50	24	.	.	PUNCT
ejpam-2488	51	1	also	also	ADV
ejpam-2488	51	2	we	we	PRON
ejpam-2488	51	3	study	study	VERB
ejpam-2488	51	4	some	some	PRON
ejpam-2488	51	5	of	of	ADP
ejpam-2488	51	6	their	their	PRON
ejpam-2488	51	7	basic	basic	ADJ
ejpam-2488	51	8	properties	property	NOUN
ejpam-2488	51	9	.	.	PUNCT
ejpam-2488	52	1	additionally	additionally	ADV
ejpam-2488	52	2	,	,	PUNCT
ejpam-2488	52	3	we	we	PRON
ejpam-2488	52	4	investigate	investigate	VERB
ejpam-2488	52	5	the	the	DET
ejpam-2488	52	6	relationship	relationship	NOUN
ejpam-2488	52	7	and	and	CCONJ
ejpam-2488	52	8	implications	implication	NOUN
ejpam-2488	52	9	of	of	ADP
ejpam-2488	52	10	these	these	DET
ejpam-2488	52	11	axioms	axiom	NOUN
ejpam-2488	52	12	among	among	ADP
ejpam-2488	52	13	themselves	themselves	PRON
ejpam-2488	52	14	and	and	CCONJ
ejpam-2488	52	15	with	with	ADP
ejpam-2488	52	16	other	other	ADJ
ejpam-2488	52	17	known	know	VERB
ejpam-2488	52	18	axioms	axiom	NOUN
ejpam-2488	52	19	.	.	PUNCT
ejpam-2488	53	1	2	2	X
ejpam-2488	53	2	.	.	NUM
ejpam-2488	53	3	preliminaries	preliminary	NOUN
ejpam-2488	53	4	a	a	DET
ejpam-2488	53	5	subset	subset	NOUN
ejpam-2488	53	6	a	a	PRON
ejpam-2488	53	7	of	of	ADP
ejpam-2488	53	8	an	an	DET
ejpam-2488	53	9	ideal	ideal	ADJ
ejpam-2488	53	10	topological	topological	ADJ
ejpam-2488	53	11	space	space	NOUN
ejpam-2488	53	12	(	(	PUNCT
ejpam-2488	53	13	x	x	X
ejpam-2488	53	14	,	,	PUNCT
ejpam-2488	53	15	τ	τ	PROPN
ejpam-2488	53	16	,	,	PUNCT
ejpam-2488	53	17	i	i	PROPN
ejpam-2488	53	18	)	)	PUNCT
ejpam-2488	53	19	is	be	AUX
ejpam-2488	53	20	said	say	VERB
ejpam-2488	53	21	to	to	PART
ejpam-2488	53	22	be	be	AUX
ejpam-2488	53	23	e	e	NOUN
ejpam-2488	53	24	-	-	ADJ
ejpam-2488	53	25	i	i	PRON
ejpam-2488	53	26	-open	-open	NOUN
ejpam-2488	54	1	[	[	X
ejpam-2488	54	2	2	2	NUM
ejpam-2488	54	3	]	]	PUNCT
ejpam-2488	54	4	if	if	SCONJ
ejpam-2488	54	5	a⊂	a⊂	PRON
ejpam-2488	54	6	cl(δint	cl(δint	NOUN
ejpam-2488	54	7	i(a))∪	i(a))∪	PROPN
ejpam-2488	54	8	int(δcli(a	int(δcli(a	NOUN
ejpam-2488	54	9	)	)	PUNCT
ejpam-2488	54	10	)	)	PUNCT
ejpam-2488	54	11	.	.	PUNCT
ejpam-2488	55	1	the	the	DET
ejpam-2488	55	2	complement	complement	NOUN
ejpam-2488	55	3	of	of	ADP
ejpam-2488	55	4	an	an	DET
ejpam-2488	55	5	e	e	NOUN
ejpam-2488	55	6	-	-	NOUN
ejpam-2488	55	7	i	i	PRON
ejpam-2488	55	8	-open	-open	ADJ
ejpam-2488	55	9	set	set	NOUN
ejpam-2488	55	10	is	be	AUX
ejpam-2488	55	11	called	call	VERB
ejpam-2488	55	12	an	an	DET
ejpam-2488	55	13	e	e	NOUN
ejpam-2488	55	14	-	-	NOUN
ejpam-2488	55	15	i	i	PRON
ejpam-2488	55	16	-closed	-close	VERB
ejpam-2488	55	17	set	set	NOUN
ejpam-2488	55	18	[	[	X
ejpam-2488	55	19	2	2	NUM
ejpam-2488	55	20	]	]	PUNCT
ejpam-2488	55	21	.	.	PUNCT
ejpam-2488	56	1	the	the	DET
ejpam-2488	56	2	intersection	intersection	NOUN
ejpam-2488	56	3	of	of	ADP
ejpam-2488	56	4	all	all	DET
ejpam-2488	56	5	e	e	NOUN
ejpam-2488	56	6	-	-	ADJ
ejpam-2488	56	7	i	i	PRON
ejpam-2488	56	8	-closed	-closed	ADJ
ejpam-2488	56	9	sets	set	NOUN
ejpam-2488	56	10	containing	contain	VERB
ejpam-2488	56	11	a	a	PRON
ejpam-2488	56	12	is	be	AUX
ejpam-2488	56	13	called	call	VERB
ejpam-2488	56	14	the	the	DET
ejpam-2488	56	15	e	e	NOUN
ejpam-2488	56	16	-	-	PROPN
ejpam-2488	56	17	i	i	NOUN
ejpam-2488	56	18	-closure	-closure	NOUN
ejpam-2488	56	19	of	of	ADP
ejpam-2488	56	20	a	a	PRON
ejpam-2488	56	21	and	and	CCONJ
ejpam-2488	56	22	is	be	AUX
ejpam-2488	56	23	denoted	denote	VERB
ejpam-2488	56	24	by	by	ADP
ejpam-2488	56	25	cl∗e	cl∗e	PROPN
ejpam-2488	56	26	(	(	PUNCT
ejpam-2488	56	27	a	a	NOUN
ejpam-2488	56	28	)	)	PUNCT
ejpam-2488	56	29	.	.	PUNCT
ejpam-2488	57	1	the	the	DET
ejpam-2488	57	2	ei	ei	NOUN
ejpam-2488	57	3	-interior	-interior	NOUN
ejpam-2488	57	4	of	of	ADP
ejpam-2488	57	5	a	a	PRON
ejpam-2488	57	6	is	be	AUX
ejpam-2488	57	7	defined	define	VERB
ejpam-2488	57	8	by	by	ADP
ejpam-2488	57	9	the	the	DET
ejpam-2488	57	10	union	union	NOUN
ejpam-2488	57	11	of	of	ADP
ejpam-2488	57	12	all	all	DET
ejpam-2488	57	13	e	e	NOUN
ejpam-2488	57	14	-	-	ADJ
ejpam-2488	57	15	i	i	PRON
ejpam-2488	57	16	-open	-open	NOUN
ejpam-2488	57	17	sets	set	NOUN
ejpam-2488	57	18	contained	contain	VERB
ejpam-2488	57	19	in	in	ADP
ejpam-2488	57	20	a	a	PRON
ejpam-2488	57	21	and	and	CCONJ
ejpam-2488	57	22	is	be	AUX
ejpam-2488	57	23	denoted	denote	VERB
ejpam-2488	57	24	by	by	ADP
ejpam-2488	57	25	int∗e	int∗e	NOUN
ejpam-2488	57	26	(	(	PUNCT
ejpam-2488	57	27	a	a	NOUN
ejpam-2488	57	28	)	)	PUNCT
ejpam-2488	57	29	.	.	PUNCT
ejpam-2488	58	1	the	the	DET
ejpam-2488	58	2	family	family	NOUN
ejpam-2488	58	3	of	of	ADP
ejpam-2488	58	4	all	all	DET
ejpam-2488	58	5	e	e	NOUN
ejpam-2488	58	6	-	-	ADJ
ejpam-2488	58	7	i	i	PRON
ejpam-2488	58	8	-open	-open	ADJ
ejpam-2488	58	9	(	(	PUNCT
ejpam-2488	58	10	resp	resp	NOUN
ejpam-2488	58	11	.	.	PUNCT
ejpam-2488	59	1	e	e	X
ejpam-2488	59	2	-	-	PUNCT
ejpam-2488	59	3	i	i	PRON
ejpam-2488	59	4	-closed	-closed	ADJ
ejpam-2488	59	5	)	)	PUNCT
ejpam-2488	59	6	sets	set	NOUN
ejpam-2488	59	7	of	of	ADP
ejpam-2488	59	8	(	(	PUNCT
ejpam-2488	59	9	x	x	INTJ
ejpam-2488	59	10	,	,	PUNCT
ejpam-2488	59	11	τ	τ	PROPN
ejpam-2488	59	12	,	,	PUNCT
ejpam-2488	59	13	i	i	PROPN
ejpam-2488	59	14	)	)	PUNCT
ejpam-2488	59	15	containing	contain	VERB
ejpam-2488	59	16	a	a	DET
ejpam-2488	59	17	point	point	NOUN
ejpam-2488	59	18	x	x	SYM
ejpam-2488	59	19	∈	∈	NOUN
ejpam-2488	59	20	x	x	PUNCT
ejpam-2488	59	21	is	be	AUX
ejpam-2488	59	22	denoted	denote	VERB
ejpam-2488	59	23	by	by	ADP
ejpam-2488	59	24	eio(x	eio(x	X
ejpam-2488	59	25	,	,	PUNCT
ejpam-2488	59	26	x	x	NOUN
ejpam-2488	59	27	)	)	PUNCT
ejpam-2488	59	28	(	(	PUNCT
ejpam-2488	59	29	resp	resp	NOUN
ejpam-2488	59	30	.	.	PUNCT
ejpam-2488	60	1	ei	ei	ADP
ejpam-2488	60	2	c(x	c(x	NOUN
ejpam-2488	60	3	,	,	PUNCT
ejpam-2488	60	4	x	x	NOUN
ejpam-2488	60	5	)	)	PUNCT
ejpam-2488	60	6	)	)	PUNCT
ejpam-2488	60	7	.	.	PUNCT
ejpam-2488	61	1	a	a	DET
ejpam-2488	61	2	subset	subset	ADJ
ejpam-2488	61	3	u	u	NOUN
ejpam-2488	61	4	of	of	ADP
ejpam-2488	61	5	x	x	PRON
ejpam-2488	61	6	is	be	AUX
ejpam-2488	61	7	called	call	VERB
ejpam-2488	61	8	an	an	DET
ejpam-2488	61	9	e	e	NOUN
ejpam-2488	61	10	-	-	NOUN
ejpam-2488	61	11	i	i	PRON
ejpam-2488	61	12	neighborhood	neighborhood	NOUN
ejpam-2488	61	13	of	of	ADP
ejpam-2488	61	14	a	a	DET
ejpam-2488	61	15	point	point	NOUN
ejpam-2488	61	16	x	x	SYM
ejpam-2488	61	17	∈	∈	NOUN
ejpam-2488	61	18	x	x	INTJ
ejpam-2488	61	19	if	if	SCONJ
ejpam-2488	61	20	there	there	PRON
ejpam-2488	61	21	exists	exist	VERB
ejpam-2488	61	22	an	an	DET
ejpam-2488	61	23	e	e	NOUN
ejpam-2488	61	24	-	-	NOUN
ejpam-2488	61	25	i	i	PRON
ejpam-2488	61	26	-open	-open	NOUN
ejpam-2488	61	27	set	set	VERB
ejpam-2488	61	28	v	v	NOUN
ejpam-2488	61	29	of	of	ADP
ejpam-2488	61	30	(	(	PUNCT
ejpam-2488	61	31	x	x	INTJ
ejpam-2488	61	32	,	,	PUNCT
ejpam-2488	61	33	τ	τ	PROPN
ejpam-2488	61	34	,	,	PUNCT
ejpam-2488	61	35	i	i	PROPN
ejpam-2488	61	36	)	)	PUNCT
ejpam-2488	61	37	such	such	ADJ
ejpam-2488	61	38	that	that	SCONJ
ejpam-2488	61	39	x	x	SYM
ejpam-2488	61	40	∈	∈	PROPN
ejpam-2488	61	41	v	v	ADP
ejpam-2488	61	42	⊂	⊂	PROPN
ejpam-2488	61	43	u	u	PROPN
ejpam-2488	61	44	.	.	PUNCT
ejpam-2488	62	1	a	a	DET
ejpam-2488	62	2	function	function	NOUN
ejpam-2488	62	3	f	f	NOUN
ejpam-2488	62	4	:	:	PUNCT
ejpam-2488	62	5	(	(	PUNCT
ejpam-2488	62	6	x	x	X
ejpam-2488	62	7	,	,	PUNCT
ejpam-2488	62	8	τ	τ	PROPN
ejpam-2488	62	9	,	,	PUNCT
ejpam-2488	62	10	i	i	NOUN
ejpam-2488	62	11	)	)	PUNCT
ejpam-2488	62	12	→	→	SYM
ejpam-2488	62	13	(	(	PUNCT
ejpam-2488	62	14	y	y	PROPN
ejpam-2488	62	15	,	,	PUNCT
ejpam-2488	62	16	σ	σ	PROPN
ejpam-2488	62	17	)	)	PUNCT
ejpam-2488	62	18	is	be	AUX
ejpam-2488	62	19	said	say	VERB
ejpam-2488	62	20	to	to	PART
ejpam-2488	62	21	be	be	AUX
ejpam-2488	62	22	e	e	NOUN
ejpam-2488	62	23	-	-	ADJ
ejpam-2488	62	24	i	i	PRON
ejpam-2488	62	25	-continuous	-continuous	ADJ
ejpam-2488	62	26	if	if	SCONJ
ejpam-2488	62	27	f	f	PROPN
ejpam-2488	62	28	−1(v	−1(v	X
ejpam-2488	62	29	)	)	PUNCT
ejpam-2488	62	30	∈	∈	PROPN
ejpam-2488	62	31	eio(x	eio(x	PROPN
ejpam-2488	62	32	)	)	PUNCT
ejpam-2488	62	33	for	for	ADP
ejpam-2488	62	34	every	every	DET
ejpam-2488	62	35	open	open	ADJ
ejpam-2488	62	36	set	set	VERB
ejpam-2488	62	37	v	v	NOUN
ejpam-2488	62	38	of	of	ADP
ejpam-2488	62	39	y	y	PROPN
ejpam-2488	62	40	.	.	PUNCT
ejpam-2488	63	1	definition	definition	NOUN
ejpam-2488	63	2	1	1	NUM
ejpam-2488	63	3	.	.	PUNCT
ejpam-2488	64	1	a	a	DET
ejpam-2488	64	2	topological	topological	ADJ
ejpam-2488	64	3	space	space	NOUN
ejpam-2488	64	4	(	(	PUNCT
ejpam-2488	64	5	x	x	X
ejpam-2488	64	6	,	,	PUNCT
ejpam-2488	64	7	τ	τ	X
ejpam-2488	64	8	)	)	PUNCT
ejpam-2488	64	9	is	be	AUX
ejpam-2488	64	10	said	say	VERB
ejpam-2488	64	11	to	to	PART
ejpam-2488	64	12	be	be	AUX
ejpam-2488	64	13	:	:	PUNCT
ejpam-2488	64	14	(	(	PUNCT
ejpam-2488	64	15	i	i	NOUN
ejpam-2488	64	16	)	)	PUNCT
ejpam-2488	64	17	r0	r0	NOUN
ejpam-2488	64	18	[	[	X
ejpam-2488	64	19	4	4	X
ejpam-2488	64	20	]	]	X
ejpam-2488	64	21	if	if	SCONJ
ejpam-2488	64	22	every	every	DET
ejpam-2488	64	23	open	open	ADJ
ejpam-2488	64	24	set	set	NOUN
ejpam-2488	64	25	contains	contain	VERB
ejpam-2488	64	26	the	the	DET
ejpam-2488	64	27	closure	closure	NOUN
ejpam-2488	64	28	of	of	ADP
ejpam-2488	64	29	each	each	PRON
ejpam-2488	64	30	of	of	ADP
ejpam-2488	64	31	its	its	PRON
ejpam-2488	64	32	singletons	singleton	NOUN
ejpam-2488	64	33	.	.	PUNCT
ejpam-2488	65	1	(	(	PUNCT
ejpam-2488	65	2	ii	ii	NOUN
ejpam-2488	65	3	)	)	PUNCT
ejpam-2488	65	4	r1	r1	NOUN
ejpam-2488	66	1	[	[	X
ejpam-2488	66	2	4	4	X
ejpam-2488	66	3	]	]	X
ejpam-2488	66	4	if	if	SCONJ
ejpam-2488	66	5	for	for	ADP
ejpam-2488	66	6	x	x	X
ejpam-2488	66	7	,	,	PUNCT
ejpam-2488	66	8	y	y	PROPN
ejpam-2488	66	9	in	in	ADP
ejpam-2488	66	10	x	x	PUNCT
ejpam-2488	66	11	with	with	ADP
ejpam-2488	66	12	cl({x	cl({x	NOUN
ejpam-2488	66	13	}	}	PUNCT
ejpam-2488	66	14	)	)	PUNCT
ejpam-2488	66	15	6=	6=	ADP
ejpam-2488	66	16	cl({y	cl({y	X
ejpam-2488	66	17	}	}	PUNCT
ejpam-2488	66	18	)	)	PUNCT
ejpam-2488	66	19	,	,	PUNCT
ejpam-2488	66	20	there	there	PRON
ejpam-2488	66	21	exist	exist	VERB
ejpam-2488	66	22	disjoint	disjoint	ADJ
ejpam-2488	66	23	open	open	ADJ
ejpam-2488	66	24	sets	set	NOUN
ejpam-2488	66	25	u	u	NOUN
ejpam-2488	66	26	and	and	CCONJ
ejpam-2488	66	27	v	v	ADP
ejpam-2488	66	28	such	such	ADJ
ejpam-2488	66	29	that	that	DET
ejpam-2488	66	30	cl({x	cl({x	NOUN
ejpam-2488	66	31	}	}	PUNCT
ejpam-2488	66	32	)	)	PUNCT
ejpam-2488	67	1	⊂	⊂	PROPN
ejpam-2488	67	2	u	u	PROPN
ejpam-2488	67	3	and	and	CCONJ
ejpam-2488	67	4	cl({y	cl({y	NOUN
ejpam-2488	67	5	}	}	PUNCT
ejpam-2488	67	6	⊂	⊂	PROPN
ejpam-2488	67	7	v	v	PROPN
ejpam-2488	67	8	.	.	PUNCT
ejpam-2488	68	1	definition	definition	NOUN
ejpam-2488	68	2	2	2	NUM
ejpam-2488	68	3	.	.	PUNCT
ejpam-2488	69	1	a	a	DET
ejpam-2488	69	2	topological	topological	ADJ
ejpam-2488	69	3	space	space	NOUN
ejpam-2488	69	4	(	(	PUNCT
ejpam-2488	69	5	x	x	X
ejpam-2488	69	6	,	,	PUNCT
ejpam-2488	69	7	τ	τ	X
ejpam-2488	69	8	)	)	PUNCT
ejpam-2488	69	9	is	be	AUX
ejpam-2488	69	10	said	say	VERB
ejpam-2488	69	11	to	to	PART
ejpam-2488	69	12	be	be	AUX
ejpam-2488	69	13	:	:	PUNCT
ejpam-2488	69	14	(	(	PUNCT
ejpam-2488	69	15	i	i	NOUN
ejpam-2488	69	16	)	)	PUNCT
ejpam-2488	69	17	e	e	X
ejpam-2488	69	18	-	-	NOUN
ejpam-2488	69	19	t1	t1	NOUN
ejpam-2488	69	20	[	[	X
ejpam-2488	69	21	7	7	NUM
ejpam-2488	69	22	,	,	PUNCT
ejpam-2488	69	23	8	8	NUM
ejpam-2488	69	24	]	]	X
ejpam-2488	69	25	if	if	SCONJ
ejpam-2488	69	26	for	for	ADP
ejpam-2488	69	27	each	each	DET
ejpam-2488	69	28	pair	pair	NOUN
ejpam-2488	69	29	of	of	ADP
ejpam-2488	69	30	distinct	distinct	ADJ
ejpam-2488	69	31	points	point	NOUN
ejpam-2488	69	32	x	x	PUNCT
ejpam-2488	69	33	and	and	CCONJ
ejpam-2488	69	34	y	y	PROPN
ejpam-2488	69	35	in	in	ADP
ejpam-2488	69	36	x	x	SYM
ejpam-2488	69	37	,	,	PUNCT
ejpam-2488	69	38	there	there	PRON
ejpam-2488	69	39	exist	exist	VERB
ejpam-2488	69	40	e	e	ADJ
ejpam-2488	69	41	-	-	ADJ
ejpam-2488	69	42	open	open	ADJ
ejpam-2488	69	43	sets	set	VERB
ejpam-2488	69	44	u	u	NOUN
ejpam-2488	69	45	and	and	CCONJ
ejpam-2488	69	46	v	v	ADP
ejpam-2488	69	47	containing	contain	VERB
ejpam-2488	69	48	x	x	PROPN
ejpam-2488	69	49	and	and	CCONJ
ejpam-2488	69	50	y	y	PROPN
ejpam-2488	69	51	,	,	PUNCT
ejpam-2488	69	52	respectively	respectively	ADV
ejpam-2488	69	53	,	,	PUNCT
ejpam-2488	69	54	such	such	ADJ
ejpam-2488	69	55	that	that	SCONJ
ejpam-2488	69	56	y	y	PROPN
ejpam-2488	69	57	/∈	/∈	PUNCT
ejpam-2488	69	58	u	u	PROPN
ejpam-2488	69	59	and	and	CCONJ
ejpam-2488	69	60	x	x	NOUN
ejpam-2488	69	61	/∈	/∈	NOUN
ejpam-2488	70	1	v	v	NOUN
ejpam-2488	70	2	.	.	PUNCT
ejpam-2488	71	1	w.	w.	PROPN
ejpam-2488	71	2	al	al	PROPN
ejpam-2488	71	3	-	-	PUNCT
ejpam-2488	71	4	omeri	omeri	ADJ
ejpam-2488	71	5	,	,	PUNCT
ejpam-2488	71	6	m.	m.	NOUN
ejpam-2488	71	7	noorani	noorani	PROPN
ejpam-2488	71	8	,	,	PUNCT
ejpam-2488	71	9	a.	a.	PROPN
ejpam-2488	71	10	al	al	PROPN
ejpam-2488	71	11	-	-	PUNCT
ejpam-2488	71	12	omari	omari	PROPN
ejpam-2488	71	13	,	,	PUNCT
ejpam-2488	71	14	and	and	CCONJ
ejpam-2488	71	15	t.	t.	PROPN
ejpam-2488	71	16	noiri	noiri	PROPN
ejpam-2488	71	17	/	/	SYM
ejpam-2488	71	18	eur	eur	PROPN
ejpam-2488	71	19	.	.	PUNCT
ejpam-2488	72	1	j.	j.	PROPN
ejpam-2488	72	2	pure	pure	PROPN
ejpam-2488	72	3	appl	appl	PROPN
ejpam-2488	72	4	.	.	PROPN
ejpam-2488	72	5	math	math	PROPN
ejpam-2488	72	6	,	,	PUNCT
ejpam-2488	72	7	8	8	NUM
ejpam-2488	72	8	(	(	PUNCT
ejpam-2488	72	9	2015	2015	NUM
ejpam-2488	72	10	)	)	PUNCT
ejpam-2488	72	11	,	,	PUNCT
ejpam-2488	72	12	502	502	NUM
ejpam-2488	72	13	-	-	SYM
ejpam-2488	72	14	513	513	NUM
ejpam-2488	72	15	504	504	NUM
ejpam-2488	72	16	(	(	PUNCT
ejpam-2488	72	17	ii	ii	NOUN
ejpam-2488	72	18	)	)	PUNCT
ejpam-2488	72	19	e	e	NOUN
ejpam-2488	72	20	-	-	NOUN
ejpam-2488	72	21	t2	t2	NOUN
ejpam-2488	72	22	[	[	X
ejpam-2488	72	23	7	7	NUM
ejpam-2488	72	24	,	,	PUNCT
ejpam-2488	72	25	8	8	NUM
ejpam-2488	72	26	]	]	X
ejpam-2488	72	27	if	if	SCONJ
ejpam-2488	72	28	for	for	ADP
ejpam-2488	72	29	each	each	DET
ejpam-2488	72	30	pair	pair	NOUN
ejpam-2488	72	31	of	of	ADP
ejpam-2488	72	32	distinct	distinct	ADJ
ejpam-2488	72	33	points	point	NOUN
ejpam-2488	72	34	x	x	PUNCT
ejpam-2488	72	35	and	and	CCONJ
ejpam-2488	72	36	y	y	PROPN
ejpam-2488	72	37	in	in	ADP
ejpam-2488	72	38	x	x	SYM
ejpam-2488	72	39	,	,	PUNCT
ejpam-2488	72	40	there	there	PRON
ejpam-2488	72	41	exist	exist	VERB
ejpam-2488	72	42	disjoint	disjoint	NOUN
ejpam-2488	72	43	e	e	NOUN
ejpam-2488	72	44	-	-	ADJ
ejpam-2488	72	45	open	open	ADJ
ejpam-2488	72	46	sets	set	VERB
ejpam-2488	72	47	u	u	NOUN
ejpam-2488	72	48	and	and	CCONJ
ejpam-2488	72	49	v	v	ADP
ejpam-2488	72	50	such	such	ADJ
ejpam-2488	72	51	that	that	SCONJ
ejpam-2488	72	52	x	x	SYM
ejpam-2488	72	53	∈	∈	PROPN
ejpam-2488	72	54	u	u	NOUN
ejpam-2488	72	55	and	and	CCONJ
ejpam-2488	72	56	y	y	PROPN
ejpam-2488	72	57	∈	∈	PROPN
ejpam-2488	72	58	v	v	NOUN
ejpam-2488	72	59	.	.	PUNCT
ejpam-2488	73	1	definition	definition	NOUN
ejpam-2488	73	2	3	3	NUM
ejpam-2488	73	3	.	.	PUNCT
ejpam-2488	74	1	an	an	DET
ejpam-2488	74	2	ideal	ideal	ADJ
ejpam-2488	74	3	topological	topological	ADJ
ejpam-2488	74	4	space	space	NOUN
ejpam-2488	74	5	(	(	PUNCT
ejpam-2488	74	6	x	x	X
ejpam-2488	74	7	,	,	PUNCT
ejpam-2488	74	8	τ	τ	PROPN
ejpam-2488	74	9	,	,	PUNCT
ejpam-2488	74	10	i	i	PROPN
ejpam-2488	74	11	)	)	PUNCT
ejpam-2488	74	12	is	be	AUX
ejpam-2488	74	13	said	say	VERB
ejpam-2488	74	14	to	to	PART
ejpam-2488	74	15	be	be	AUX
ejpam-2488	74	16	:	:	PUNCT
ejpam-2488	74	17	(	(	PUNCT
ejpam-2488	74	18	i	i	NOUN
ejpam-2488	74	19	)	)	PUNCT
ejpam-2488	74	20	e	e	X
ejpam-2488	74	21	-	-	NOUN
ejpam-2488	74	22	i	i	PRON
ejpam-2488	74	23	-t1	-t1	VERB
ejpam-2488	75	1	[	[	X
ejpam-2488	75	2	1	1	X
ejpam-2488	75	3	]	]	X
ejpam-2488	75	4	if	if	SCONJ
ejpam-2488	75	5	for	for	ADP
ejpam-2488	75	6	each	each	DET
ejpam-2488	75	7	pair	pair	NOUN
ejpam-2488	75	8	of	of	ADP
ejpam-2488	75	9	distinct	distinct	ADJ
ejpam-2488	75	10	points	point	NOUN
ejpam-2488	75	11	x	x	PUNCT
ejpam-2488	75	12	and	and	CCONJ
ejpam-2488	75	13	y	y	PROPN
ejpam-2488	75	14	in	in	ADP
ejpam-2488	75	15	x	x	SYM
ejpam-2488	75	16	,	,	PUNCT
ejpam-2488	75	17	there	there	PRON
ejpam-2488	75	18	exist	exist	VERB
ejpam-2488	75	19	e	e	NOUN
ejpam-2488	75	20	-	-	ADJ
ejpam-2488	75	21	i	i	PRON
ejpam-2488	75	22	-open	-open	NOUN
ejpam-2488	75	23	sets	set	VERB
ejpam-2488	75	24	u	u	NOUN
ejpam-2488	75	25	and	and	CCONJ
ejpam-2488	75	26	v	v	NOUN
ejpam-2488	75	27	of	of	ADP
ejpam-2488	75	28	x	x	PRON
ejpam-2488	75	29	,	,	PUNCT
ejpam-2488	75	30	such	such	ADJ
ejpam-2488	75	31	that	that	SCONJ
ejpam-2488	75	32	x	x	SYM
ejpam-2488	75	33	∈	∈	PROPN
ejpam-2488	75	34	u	u	NOUN
ejpam-2488	75	35	and	and	CCONJ
ejpam-2488	75	36	y	y	PROPN
ejpam-2488	75	37	/∈	/∈	PUNCT
ejpam-2488	76	1	u	u	PROPN
ejpam-2488	76	2	,	,	PUNCT
ejpam-2488	76	3	y	y	PROPN
ejpam-2488	76	4	∈	∈	PROPN
ejpam-2488	76	5	v	v	NOUN
ejpam-2488	76	6	and	and	CCONJ
ejpam-2488	76	7	x	x	NOUN
ejpam-2488	76	8	/∈	/∈	NOUN
ejpam-2488	76	9	v	v	INTJ
ejpam-2488	76	10	.	.	PUNCT
ejpam-2488	77	1	(	(	PUNCT
ejpam-2488	77	2	ii	ii	NOUN
ejpam-2488	77	3	)	)	PUNCT
ejpam-2488	77	4	e	e	X
ejpam-2488	77	5	-	-	NOUN
ejpam-2488	77	6	i	i	PRON
ejpam-2488	77	7	-t2	-t2	VERB
ejpam-2488	78	1	[	[	X
ejpam-2488	78	2	1	1	X
ejpam-2488	78	3	]	]	X
ejpam-2488	78	4	if	if	SCONJ
ejpam-2488	78	5	for	for	ADP
ejpam-2488	78	6	each	each	DET
ejpam-2488	78	7	pair	pair	NOUN
ejpam-2488	78	8	of	of	ADP
ejpam-2488	78	9	distinct	distinct	ADJ
ejpam-2488	78	10	points	point	NOUN
ejpam-2488	78	11	x	x	PUNCT
ejpam-2488	78	12	and	and	CCONJ
ejpam-2488	78	13	y	y	PROPN
ejpam-2488	78	14	in	in	ADP
ejpam-2488	78	15	x	x	SYM
ejpam-2488	78	16	,	,	PUNCT
ejpam-2488	78	17	there	there	PRON
ejpam-2488	78	18	exist	exist	VERB
ejpam-2488	78	19	disjoint	disjoint	NOUN
ejpam-2488	78	20	e	e	NOUN
ejpam-2488	78	21	-	-	ADJ
ejpam-2488	78	22	i	i	PRON
ejpam-2488	78	23	-open	-open	NOUN
ejpam-2488	78	24	sets	set	VERB
ejpam-2488	78	25	u	u	NOUN
ejpam-2488	78	26	and	and	CCONJ
ejpam-2488	78	27	v	v	NOUN
ejpam-2488	78	28	in	in	ADP
ejpam-2488	78	29	x	x	PUNCT
ejpam-2488	78	30	such	such	ADJ
ejpam-2488	78	31	that	that	SCONJ
ejpam-2488	78	32	x	x	SYM
ejpam-2488	78	33	∈	∈	PROPN
ejpam-2488	78	34	u	u	NOUN
ejpam-2488	78	35	and	and	CCONJ
ejpam-2488	78	36	y	y	PROPN
ejpam-2488	78	37	∈	∈	PROPN
ejpam-2488	78	38	v	v	NOUN
ejpam-2488	78	39	.	.	PUNCT
ejpam-2488	79	1	3	3	X
ejpam-2488	79	2	.	.	X
ejpam-2488	80	1	on	on	ADP
ejpam-2488	80	2	e	e	X
ejpam-2488	80	3	-	-	PROPN
ejpam-2488	80	4	i	i	PRON
ejpam-2488	80	5	-r0	-r0	PROPN
ejpam-2488	80	6	spaces	space	VERB
ejpam-2488	80	7	definition	definition	NOUN
ejpam-2488	80	8	4	4	X
ejpam-2488	80	9	.	.	PUNCT
ejpam-2488	81	1	let	let	AUX
ejpam-2488	81	2	(	(	PUNCT
ejpam-2488	81	3	x	x	X
ejpam-2488	81	4	,	,	PUNCT
ejpam-2488	81	5	τ	τ	PROPN
ejpam-2488	81	6	,	,	PUNCT
ejpam-2488	81	7	i	i	PRON
ejpam-2488	81	8	)	)	PUNCT
ejpam-2488	81	9	be	be	AUX
ejpam-2488	81	10	an	an	DET
ejpam-2488	81	11	ideal	ideal	ADJ
ejpam-2488	81	12	topological	topological	ADJ
ejpam-2488	81	13	space	space	NOUN
ejpam-2488	81	14	and	and	CCONJ
ejpam-2488	81	15	a	a	DET
ejpam-2488	81	16	⊂	⊂	PROPN
ejpam-2488	81	17	x	x	X
ejpam-2488	81	18	.	.	PUNCT
ejpam-2488	82	1	then	then	ADV
ejpam-2488	82	2	the	the	DET
ejpam-2488	82	3	e	e	NOUN
ejpam-2488	82	4	-	-	PROPN
ejpam-2488	82	5	i	i	PRON
ejpam-2488	82	6	-kernel	-kernel	NOUN
ejpam-2488	82	7	of	of	ADP
ejpam-2488	82	8	a	a	PRON
ejpam-2488	82	9	,	,	PUNCT
ejpam-2488	82	10	denoted	denote	VERB
ejpam-2488	82	11	by	by	ADP
ejpam-2488	82	12	ieker(a	ieker(a	NOUN
ejpam-2488	82	13	)	)	PUNCT
ejpam-2488	82	14	,	,	PUNCT
ejpam-2488	82	15	is	be	AUX
ejpam-2488	82	16	defined	define	VERB
ejpam-2488	82	17	to	to	PART
ejpam-2488	82	18	be	be	AUX
ejpam-2488	82	19	the	the	DET
ejpam-2488	82	20	set	set	NOUN
ejpam-2488	82	21	ieker(a	ieker(a	VERB
ejpam-2488	82	22	)	)	PUNCT
ejpam-2488	82	23	=	=	SYM
ejpam-2488	82	24	∩{g	∩{g	PROPN
ejpam-2488	82	25	∈	∈	PROPN
ejpam-2488	82	26	eio(x	eio(x	X
ejpam-2488	82	27	)	)	PUNCT
ejpam-2488	82	28	|a⊂	|a⊂	NOUN
ejpam-2488	82	29	g	g	NOUN
ejpam-2488	82	30	}	}	PUNCT
ejpam-2488	82	31	.	.	PUNCT
ejpam-2488	83	1	lemma	lemma	PROPN
ejpam-2488	83	2	1	1	X
ejpam-2488	83	3	.	.	PUNCT
ejpam-2488	84	1	let	let	AUX
ejpam-2488	84	2	(	(	PUNCT
ejpam-2488	84	3	x	x	X
ejpam-2488	84	4	,	,	PUNCT
ejpam-2488	84	5	τ	τ	PROPN
ejpam-2488	84	6	,	,	PUNCT
ejpam-2488	84	7	i	i	PRON
ejpam-2488	84	8	)	)	PUNCT
ejpam-2488	84	9	be	be	AUX
ejpam-2488	84	10	an	an	DET
ejpam-2488	84	11	ideal	ideal	ADJ
ejpam-2488	84	12	topological	topological	ADJ
ejpam-2488	84	13	space	space	NOUN
ejpam-2488	84	14	and	and	CCONJ
ejpam-2488	84	15	x	x	NOUN
ejpam-2488	84	16	,	,	PUNCT
ejpam-2488	84	17	y	y	PROPN
ejpam-2488	84	18	∈	∈	PROPN
ejpam-2488	84	19	x	x	X
ejpam-2488	84	20	.	.	PUNCT
ejpam-2488	85	1	then	then	ADV
ejpam-2488	85	2	,	,	PUNCT
ejpam-2488	85	3	y	y	PROPN
ejpam-2488	85	4	∈	∈	PROPN
ejpam-2488	85	5	ieker({x	ieker({x	PROPN
ejpam-2488	85	6	}	}	PUNCT
ejpam-2488	85	7	)	)	PUNCT
ejpam-2488	86	1	if	if	SCONJ
ejpam-2488	86	2	and	and	CCONJ
ejpam-2488	86	3	only	only	ADV
ejpam-2488	86	4	if	if	SCONJ
ejpam-2488	86	5	x	x	PROPN
ejpam-2488	86	6	∈	∈	PROPN
ejpam-2488	86	7	cl∗e	cl∗e	X
ejpam-2488	86	8	(	(	PUNCT
ejpam-2488	86	9	{	{	PUNCT
ejpam-2488	86	10	y	y	NOUN
ejpam-2488	86	11	}	}	PUNCT
ejpam-2488	86	12	)	)	PUNCT
ejpam-2488	86	13	.	.	PUNCT
ejpam-2488	87	1	proof	proof	NOUN
ejpam-2488	87	2	.	.	PUNCT
ejpam-2488	88	1	suppose	suppose	VERB
ejpam-2488	88	2	that	that	SCONJ
ejpam-2488	88	3	y	y	PROPN
ejpam-2488	88	4	/∈	/∈	PUNCT
ejpam-2488	88	5	ieker({x	ieker({x	PROPN
ejpam-2488	88	6	}	}	PUNCT
ejpam-2488	88	7	)	)	PUNCT
ejpam-2488	88	8	.	.	PUNCT
ejpam-2488	89	1	then	then	ADV
ejpam-2488	89	2	there	there	PRON
ejpam-2488	89	3	exists	exist	VERB
ejpam-2488	89	4	u	u	PROPN
ejpam-2488	89	5	∈	∈	PROPN
ejpam-2488	89	6	eio(x	eio(x	X
ejpam-2488	89	7	,	,	PUNCT
ejpam-2488	89	8	x	x	X
ejpam-2488	89	9	)	)	PUNCT
ejpam-2488	89	10	such	such	ADJ
ejpam-2488	89	11	that	that	SCONJ
ejpam-2488	89	12	y	y	PROPN
ejpam-2488	89	13	/∈	/∈	PUNCT
ejpam-2488	89	14	u	u	PROPN
ejpam-2488	89	15	.	.	PUNCT
ejpam-2488	90	1	therefore	therefore	ADV
ejpam-2488	90	2	,	,	PUNCT
ejpam-2488	90	3	we	we	PRON
ejpam-2488	90	4	have	have	VERB
ejpam-2488	90	5	x	x	X
ejpam-2488	90	6	/∈	/∈	PUNCT
ejpam-2488	90	7	cl∗e	cl∗e	PROPN
ejpam-2488	90	8	(	(	PUNCT
ejpam-2488	90	9	{	{	PUNCT
ejpam-2488	90	10	y	y	NOUN
ejpam-2488	90	11	}	}	PUNCT
ejpam-2488	90	12	)	)	PUNCT
ejpam-2488	90	13	.	.	PUNCT
ejpam-2488	91	1	the	the	DET
ejpam-2488	91	2	proof	proof	NOUN
ejpam-2488	91	3	of	of	ADP
ejpam-2488	91	4	the	the	DET
ejpam-2488	91	5	converse	converse	NOUN
ejpam-2488	91	6	case	case	NOUN
ejpam-2488	91	7	can	can	AUX
ejpam-2488	91	8	be	be	AUX
ejpam-2488	91	9	done	do	VERB
ejpam-2488	91	10	similarly	similarly	ADV
ejpam-2488	91	11	.	.	PUNCT
ejpam-2488	92	1	lemma	lemma	PROPN
ejpam-2488	92	2	2	2	X
ejpam-2488	92	3	.	.	PUNCT
ejpam-2488	93	1	let	let	AUX
ejpam-2488	93	2	(	(	PUNCT
ejpam-2488	93	3	x	x	X
ejpam-2488	93	4	,	,	PUNCT
ejpam-2488	93	5	τ	τ	PROPN
ejpam-2488	93	6	,	,	PUNCT
ejpam-2488	93	7	i	i	PRON
ejpam-2488	93	8	)	)	PUNCT
ejpam-2488	93	9	be	be	AUX
ejpam-2488	93	10	an	an	DET
ejpam-2488	93	11	ideal	ideal	ADJ
ejpam-2488	93	12	topological	topological	ADJ
ejpam-2488	93	13	space	space	NOUN
ejpam-2488	93	14	and	and	CCONJ
ejpam-2488	93	15	s	s	VERB
ejpam-2488	93	16	a	a	DET
ejpam-2488	93	17	subset	subset	NOUN
ejpam-2488	93	18	of	of	ADP
ejpam-2488	93	19	x	x	X
ejpam-2488	93	20	.	.	PUNCT
ejpam-2488	94	1	then	then	ADV
ejpam-2488	94	2	,	,	PUNCT
ejpam-2488	94	3	ieker(s	ieker(s	X
ejpam-2488	94	4	)	)	PUNCT
ejpam-2488	94	5	=	=	SYM
ejpam-2488	94	6	{	{	PUNCT
ejpam-2488	94	7	x	x	SYM
ejpam-2488	94	8	∈	∈	PROPN
ejpam-2488	94	9	x	x	X
ejpam-2488	94	10	|cl∗e	|cl∗e	X
ejpam-2488	94	11	(	(	PUNCT
ejpam-2488	94	12	{	{	PUNCT
ejpam-2488	94	13	x})∩	x})∩	PROPN
ejpam-2488	94	14	s	s	PROPN
ejpam-2488	94	15	6=	6=	NUM
ejpam-2488	94	16	;	;	PUNCT
ejpam-2488	94	17	}	}	PUNCT
ejpam-2488	94	18	.	.	PUNCT
ejpam-2488	95	1	proof	proof	NOUN
ejpam-2488	95	2	.	.	PUNCT
ejpam-2488	96	1	let	let	VERB
ejpam-2488	96	2	x	x	PUNCT
ejpam-2488	96	3	∈	∈	PROPN
ejpam-2488	96	4	ieker(s	ieker(s	NOUN
ejpam-2488	96	5	)	)	PUNCT
ejpam-2488	96	6	.	.	PUNCT
ejpam-2488	97	1	suppose	suppose	VERB
ejpam-2488	97	2	that	that	SCONJ
ejpam-2488	97	3	cl∗e	cl∗e	PROPN
ejpam-2488	97	4	(	(	PUNCT
ejpam-2488	97	5	{	{	PUNCT
ejpam-2488	97	6	x})∩s	x})∩s	PROPN
ejpam-2488	97	7	=	=	PUNCT
ejpam-2488	97	8	;	;	PUNCT
ejpam-2488	97	9	.	.	PUNCT
ejpam-2488	98	1	hence	hence	ADV
ejpam-2488	98	2	x	x	X
ejpam-2488	98	3	/∈	/∈	PUNCT
ejpam-2488	99	1	x\cl∗e	x\cl∗e	PROPN
ejpam-2488	99	2	(	(	PUNCT
ejpam-2488	99	3	{	{	PUNCT
ejpam-2488	99	4	x})which	x})which	PROPN
ejpam-2488	99	5	is	be	AUX
ejpam-2488	99	6	an	an	DET
ejpam-2488	99	7	e	e	NOUN
ejpam-2488	99	8	-	-	ADJ
ejpam-2488	99	9	i	i	PRON
ejpam-2488	99	10	-open	-open	NOUN
ejpam-2488	99	11	set	set	VERB
ejpam-2488	99	12	containing	contain	VERB
ejpam-2488	99	13	s.	s.	PROPN
ejpam-2488	99	14	since	since	SCONJ
ejpam-2488	99	15	x	x	PROPN
ejpam-2488	99	16	/∈	/∈	PUNCT
ejpam-2488	100	1	ieker(s	ieker(s	NOUN
ejpam-2488	100	2	)	)	PUNCT
ejpam-2488	100	3	,	,	PUNCT
ejpam-2488	100	4	this	this	PRON
ejpam-2488	100	5	is	be	AUX
ejpam-2488	100	6	a	a	DET
ejpam-2488	100	7	contradiction	contradiction	NOUN
ejpam-2488	100	8	.	.	PUNCT
ejpam-2488	101	1	hence	hence	ADV
ejpam-2488	101	2	cl∗e	cl∗e	PROPN
ejpam-2488	101	3	(	(	PUNCT
ejpam-2488	101	4	{	{	PUNCT
ejpam-2488	101	5	x})∩s	x})∩s	PROPN
ejpam-2488	101	6	6=	6=	NUM
ejpam-2488	101	7	;	;	PUNCT
ejpam-2488	101	8	.	.	PUNCT
ejpam-2488	102	1	conversely	conversely	ADV
ejpam-2488	102	2	,	,	PUNCT
ejpam-2488	102	3	suppose	suppose	VERB
ejpam-2488	102	4	that	that	SCONJ
ejpam-2488	102	5	cl∗e	cl∗e	PROPN
ejpam-2488	102	6	(	(	PUNCT
ejpam-2488	102	7	{	{	PUNCT
ejpam-2488	102	8	x	x	NOUN
ejpam-2488	102	9	}	}	PUNCT
ejpam-2488	102	10	)	)	PUNCT
ejpam-2488	102	11	∩	∩	PROPN
ejpam-2488	102	12	s	s	PART
ejpam-2488	102	13	6=	6=	NUM
ejpam-2488	102	14	;	;	PUNCT
ejpam-2488	102	15	.	.	PUNCT
ejpam-2488	103	1	next	next	ADV
ejpam-2488	103	2	,	,	PUNCT
ejpam-2488	103	3	let	let	VERB
ejpam-2488	103	4	x	x	PUNCT
ejpam-2488	103	5	∈	∈	PROPN
ejpam-2488	103	6	x	x	X
ejpam-2488	103	7	such	such	ADJ
ejpam-2488	103	8	that	that	SCONJ
ejpam-2488	103	9	cl∗e	cl∗e	PROPN
ejpam-2488	103	10	(	(	PUNCT
ejpam-2488	103	11	{	{	PUNCT
ejpam-2488	103	12	x	x	NOUN
ejpam-2488	103	13	}	}	PUNCT
ejpam-2488	103	14	)	)	PUNCT
ejpam-2488	103	15	∩	∩	PROPN
ejpam-2488	103	16	s	s	PART
ejpam-2488	103	17	6=	6=	NUM
ejpam-2488	103	18	;	;	PUNCT
ejpam-2488	103	19	and	and	CCONJ
ejpam-2488	103	20	suppose	suppose	VERB
ejpam-2488	103	21	that	that	SCONJ
ejpam-2488	103	22	x	x	X
ejpam-2488	103	23	/∈	/∈	PUNCT
ejpam-2488	104	1	ieker(s	ieker(s	NOUN
ejpam-2488	104	2	)	)	PUNCT
ejpam-2488	104	3	.	.	PUNCT
ejpam-2488	105	1	then	then	ADV
ejpam-2488	105	2	,	,	PUNCT
ejpam-2488	105	3	there	there	PRON
ejpam-2488	105	4	exists	exist	VERB
ejpam-2488	105	5	an	an	DET
ejpam-2488	105	6	e	e	NOUN
ejpam-2488	105	7	-	-	NOUN
ejpam-2488	105	8	i	i	PRON
ejpam-2488	105	9	-open	-open	VERB
ejpam-2488	105	10	set	set	VERB
ejpam-2488	105	11	u	u	NOUN
ejpam-2488	105	12	containing	contain	VERB
ejpam-2488	105	13	s	s	PRON
ejpam-2488	105	14	and	and	CCONJ
ejpam-2488	105	15	x	x	ADJ
ejpam-2488	105	16	/∈	/∈	PUNCT
ejpam-2488	105	17	u	u	INTJ
ejpam-2488	105	18	.	.	PUNCT
ejpam-2488	106	1	let	let	VERB
ejpam-2488	106	2	y	y	PROPN
ejpam-2488	106	3	∈	∈	PROPN
ejpam-2488	106	4	cl∗e	cl∗e	PROPN
ejpam-2488	106	5	(	(	PUNCT
ejpam-2488	106	6	{	{	PUNCT
ejpam-2488	106	7	x})∩	x})∩	PROPN
ejpam-2488	106	8	s.	s.	PROPN
ejpam-2488	106	9	hence	hence	ADV
ejpam-2488	106	10	,	,	PUNCT
ejpam-2488	106	11	u	u	PROPN
ejpam-2488	106	12	is	be	AUX
ejpam-2488	106	13	an	an	DET
ejpam-2488	106	14	e	e	NOUN
ejpam-2488	106	15	-	-	NOUN
ejpam-2488	106	16	i	i	PRON
ejpam-2488	106	17	-neighborhood	-neighborhood	NOUN
ejpam-2488	106	18	of	of	ADP
ejpam-2488	106	19	y	y	PRON
ejpam-2488	106	20	which	which	PRON
ejpam-2488	106	21	does	do	AUX
ejpam-2488	106	22	not	not	PART
ejpam-2488	106	23	contains	contain	VERB
ejpam-2488	106	24	x	x	X
ejpam-2488	106	25	.	.	PUNCT
ejpam-2488	107	1	by	by	ADP
ejpam-2488	107	2	this	this	DET
ejpam-2488	107	3	contradiction	contradiction	NOUN
ejpam-2488	107	4	x	x	PUNCT
ejpam-2488	107	5	∈	∈	PROPN
ejpam-2488	107	6	ieker(s	ieker(s	NOUN
ejpam-2488	107	7	)	)	PUNCT
ejpam-2488	107	8	and	and	CCONJ
ejpam-2488	107	9	hence	hence	ADV
ejpam-2488	107	10	the	the	DET
ejpam-2488	107	11	claim	claim	NOUN
ejpam-2488	107	12	.	.	PUNCT
ejpam-2488	108	1	definition	definition	NOUN
ejpam-2488	108	2	5	5	NUM
ejpam-2488	108	3	.	.	PUNCT
ejpam-2488	109	1	an	an	DET
ejpam-2488	109	2	ideal	ideal	ADJ
ejpam-2488	109	3	topological	topological	ADJ
ejpam-2488	109	4	space	space	NOUN
ejpam-2488	109	5	(	(	PUNCT
ejpam-2488	109	6	x	x	X
ejpam-2488	109	7	,	,	PUNCT
ejpam-2488	109	8	τ	τ	PROPN
ejpam-2488	109	9	,	,	PUNCT
ejpam-2488	109	10	i	i	PROPN
ejpam-2488	109	11	)	)	PUNCT
ejpam-2488	109	12	is	be	AUX
ejpam-2488	109	13	called	call	VERB
ejpam-2488	109	14	an	an	DET
ejpam-2488	109	15	e	e	NOUN
ejpam-2488	109	16	-	-	PROPN
ejpam-2488	109	17	i	i	PRON
ejpam-2488	109	18	-r0	-r0	NOUN
ejpam-2488	109	19	space	space	NOUN
ejpam-2488	109	20	if	if	SCONJ
ejpam-2488	109	21	every	every	DET
ejpam-2488	109	22	e	e	NOUN
ejpam-2488	109	23	-	-	NOUN
ejpam-2488	109	24	i	i	PRON
ejpam-2488	109	25	-open	-open	ADJ
ejpam-2488	109	26	set	set	NOUN
ejpam-2488	109	27	contains	contain	VERB
ejpam-2488	109	28	the	the	DET
ejpam-2488	109	29	e	e	NOUN
ejpam-2488	109	30	-	-	NOUN
ejpam-2488	109	31	i	i	NOUN
ejpam-2488	109	32	-closure	-closure	NOUN
ejpam-2488	109	33	of	of	ADP
ejpam-2488	109	34	each	each	PRON
ejpam-2488	109	35	of	of	ADP
ejpam-2488	109	36	its	its	PRON
ejpam-2488	109	37	singletons	singleton	NOUN
ejpam-2488	109	38	.	.	PUNCT
ejpam-2488	110	1	definition	definition	NOUN
ejpam-2488	110	2	6	6	NUM
ejpam-2488	110	3	.	.	PUNCT
ejpam-2488	111	1	an	an	DET
ejpam-2488	111	2	ideal	ideal	ADJ
ejpam-2488	111	3	topological	topological	ADJ
ejpam-2488	111	4	space	space	NOUN
ejpam-2488	111	5	(	(	PUNCT
ejpam-2488	111	6	x	x	X
ejpam-2488	111	7	,	,	PUNCT
ejpam-2488	111	8	τ	τ	PROPN
ejpam-2488	111	9	,	,	PUNCT
ejpam-2488	111	10	i	i	PROPN
ejpam-2488	111	11	)	)	PUNCT
ejpam-2488	111	12	is	be	AUX
ejpam-2488	111	13	said	say	VERB
ejpam-2488	111	14	to	to	PART
ejpam-2488	111	15	be	be	AUX
ejpam-2488	111	16	e	e	NOUN
ejpam-2488	111	17	-	-	PUNCT
ejpam-2488	111	18	i	i	PRON
ejpam-2488	111	19	-t0	-t0	VERB
ejpam-2488	111	20	if	if	SCONJ
ejpam-2488	111	21	for	for	ADP
ejpam-2488	111	22	each	each	DET
ejpam-2488	111	23	pair	pair	NOUN
ejpam-2488	111	24	of	of	ADP
ejpam-2488	111	25	distinct	distinct	ADJ
ejpam-2488	111	26	points	point	NOUN
ejpam-2488	111	27	x	x	PUNCT
ejpam-2488	111	28	and	and	CCONJ
ejpam-2488	111	29	y	y	PROPN
ejpam-2488	111	30	in	in	ADP
ejpam-2488	111	31	x	x	X
ejpam-2488	111	32	,	,	PUNCT
ejpam-2488	111	33	there	there	PRON
ejpam-2488	111	34	exists	exist	VERB
ejpam-2488	111	35	an	an	DET
ejpam-2488	111	36	e	e	NOUN
ejpam-2488	111	37	-	-	NOUN
ejpam-2488	111	38	i	i	PRON
ejpam-2488	111	39	-open	-open	NOUN
ejpam-2488	111	40	set	set	VERB
ejpam-2488	111	41	u	u	PRON
ejpam-2488	111	42	such	such	ADJ
ejpam-2488	111	43	that	that	SCONJ
ejpam-2488	111	44	x	x	SYM
ejpam-2488	111	45	∈	∈	PROPN
ejpam-2488	111	46	u	u	NOUN
ejpam-2488	111	47	and	and	CCONJ
ejpam-2488	111	48	y	y	PROPN
ejpam-2488	111	49	/∈	/∈	PUNCT
ejpam-2488	111	50	u	u	NOUN
ejpam-2488	111	51	,	,	PUNCT
ejpam-2488	111	52	or	or	CCONJ
ejpam-2488	111	53	there	there	PRON
ejpam-2488	111	54	exists	exist	VERB
ejpam-2488	111	55	an	an	DET
ejpam-2488	111	56	e	e	NOUN
ejpam-2488	111	57	-	-	NOUN
ejpam-2488	111	58	i	i	PRON
ejpam-2488	111	59	-open	-open	NOUN
ejpam-2488	111	60	set	set	VERB
ejpam-2488	111	61	v	v	ADP
ejpam-2488	111	62	such	such	ADJ
ejpam-2488	111	63	that	that	SCONJ
ejpam-2488	111	64	y	y	PROPN
ejpam-2488	111	65	∈	∈	PROPN
ejpam-2488	111	66	v	v	NOUN
ejpam-2488	111	67	and	and	CCONJ
ejpam-2488	111	68	x	x	NOUN
ejpam-2488	111	69	/∈	/∈	NOUN
ejpam-2488	111	70	v	v	X
ejpam-2488	111	71	.	.	PUNCT
ejpam-2488	112	1	theorem	theorem	NOUN
ejpam-2488	112	2	1	1	NUM
ejpam-2488	112	3	.	.	PUNCT
ejpam-2488	113	1	let	let	AUX
ejpam-2488	113	2	(	(	PUNCT
ejpam-2488	113	3	x	x	X
ejpam-2488	113	4	,	,	PUNCT
ejpam-2488	113	5	τ	τ	PROPN
ejpam-2488	113	6	,	,	PUNCT
ejpam-2488	113	7	i	i	PRON
ejpam-2488	113	8	)	)	PUNCT
ejpam-2488	113	9	be	be	AUX
ejpam-2488	113	10	an	an	DET
ejpam-2488	113	11	ideal	ideal	ADJ
ejpam-2488	113	12	topological	topological	ADJ
ejpam-2488	113	13	space	space	NOUN
ejpam-2488	113	14	.	.	PUNCT
ejpam-2488	114	1	then	then	ADV
ejpam-2488	114	2	x	x	X
ejpam-2488	114	3	is	be	AUX
ejpam-2488	114	4	e	e	NOUN
ejpam-2488	114	5	-	-	PUNCT
ejpam-2488	114	6	i	i	PRON
ejpam-2488	114	7	-t1	-t1	VERB
ejpam-2488	115	1	if	if	SCONJ
ejpam-2488	116	1	and	and	CCONJ
ejpam-2488	116	2	only	only	ADV
ejpam-2488	116	3	if	if	SCONJ
ejpam-2488	116	4	it	it	PRON
ejpam-2488	116	5	is	be	AUX
ejpam-2488	116	6	e	e	NOUN
ejpam-2488	116	7	-	-	PUNCT
ejpam-2488	116	8	i	i	PRON
ejpam-2488	116	9	-t0	-t0	NOUN
ejpam-2488	116	10	and	and	CCONJ
ejpam-2488	116	11	e	e	X
ejpam-2488	116	12	-	-	PROPN
ejpam-2488	116	13	i	i	PRON
ejpam-2488	116	14	-r0	-r0	NOUN
ejpam-2488	116	15	.	.	PUNCT
ejpam-2488	117	1	proof	proof	NOUN
ejpam-2488	117	2	.	.	PUNCT
ejpam-2488	118	1	let	let	VERB
ejpam-2488	118	2	x	x	PRON
ejpam-2488	118	3	be	be	AUX
ejpam-2488	118	4	an	an	DET
ejpam-2488	118	5	e	e	NOUN
ejpam-2488	118	6	-	-	NOUN
ejpam-2488	118	7	i	i	PRON
ejpam-2488	118	8	-t1	-t1	VERB
ejpam-2488	118	9	space	space	NOUN
ejpam-2488	118	10	.	.	PUNCT
ejpam-2488	119	1	by	by	ADP
ejpam-2488	119	2	the	the	DET
ejpam-2488	119	3	definition	definition	NOUN
ejpam-2488	119	4	of	of	ADP
ejpam-2488	119	5	an	an	DET
ejpam-2488	119	6	e	e	NOUN
ejpam-2488	119	7	-	-	NOUN
ejpam-2488	119	8	i	i	PRON
ejpam-2488	119	9	-t1	-t1	VERB
ejpam-2488	119	10	space	space	NOUN
ejpam-2488	119	11	,	,	PUNCT
ejpam-2488	119	12	it	it	PRON
ejpam-2488	119	13	is	be	AUX
ejpam-2488	119	14	an	an	DET
ejpam-2488	119	15	e	e	NOUN
ejpam-2488	119	16	-	-	NOUN
ejpam-2488	119	17	i	i	PRON
ejpam-2488	119	18	-t0	-t0	NOUN
ejpam-2488	119	19	and	and	CCONJ
ejpam-2488	119	20	e	e	X
ejpam-2488	119	21	-	-	NOUN
ejpam-2488	119	22	i	i	PRON
ejpam-2488	119	23	-r0	-r0	PROPN
ejpam-2488	119	24	space	space	NOUN
ejpam-2488	119	25	.	.	PUNCT
ejpam-2488	120	1	conversely	conversely	ADV
ejpam-2488	120	2	,	,	PUNCT
ejpam-2488	120	3	let	let	VERB
ejpam-2488	120	4	x	x	PRON
ejpam-2488	120	5	be	be	AUX
ejpam-2488	120	6	an	an	DET
ejpam-2488	120	7	e	e	NOUN
ejpam-2488	120	8	-	-	NOUN
ejpam-2488	120	9	i	i	PRON
ejpam-2488	120	10	-t0	-t0	NOUN
ejpam-2488	120	11	and	and	CCONJ
ejpam-2488	120	12	e	e	X
ejpam-2488	120	13	-	-	NOUN
ejpam-2488	120	14	i	i	PRON
ejpam-2488	120	15	-r0	-r0	PROPN
ejpam-2488	120	16	space	space	NOUN
ejpam-2488	120	17	.	.	PUNCT
ejpam-2488	121	1	let	let	VERB
ejpam-2488	121	2	x	x	PRON
ejpam-2488	121	3	,	,	PUNCT
ejpam-2488	121	4	y	y	PROPN
ejpam-2488	121	5	be	be	VERB
ejpam-2488	121	6	any	any	DET
ejpam-2488	121	7	two	two	NUM
ejpam-2488	121	8	distinct	distinct	ADJ
ejpam-2488	121	9	points	point	NOUN
ejpam-2488	121	10	of	of	ADP
ejpam-2488	121	11	x	x	X
ejpam-2488	121	12	.	.	PUNCT
ejpam-2488	122	1	since	since	SCONJ
ejpam-2488	122	2	x	x	PROPN
ejpam-2488	122	3	is	be	AUX
ejpam-2488	122	4	e	e	NOUN
ejpam-2488	122	5	-	-	PUNCT
ejpam-2488	122	6	i	i	PRON
ejpam-2488	122	7	-t0	-t0	VERB
ejpam-2488	122	8	,	,	PUNCT
ejpam-2488	122	9	then	then	ADV
ejpam-2488	122	10	there	there	PRON
ejpam-2488	122	11	exists	exist	VERB
ejpam-2488	122	12	an	an	DET
ejpam-2488	122	13	e	e	NOUN
ejpam-2488	122	14	-	-	NOUN
ejpam-2488	122	15	i	i	PRON
ejpam-2488	122	16	-open	-open	NOUN
ejpam-2488	122	17	set	set	VERB
ejpam-2488	122	18	u	u	PRON
ejpam-2488	122	19	such	such	ADJ
ejpam-2488	122	20	that	that	SCONJ
ejpam-2488	122	21	x	x	SYM
ejpam-2488	122	22	∈	∈	PROPN
ejpam-2488	122	23	u	u	NOUN
ejpam-2488	122	24	and	and	CCONJ
ejpam-2488	122	25	y	y	PROPN
ejpam-2488	122	26	/∈	/∈	PUNCT
ejpam-2488	123	1	u	u	PROPN
ejpam-2488	123	2	or	or	CCONJ
ejpam-2488	123	3	w.	w.	PROPN
ejpam-2488	123	4	al	al	PROPN
ejpam-2488	123	5	-	-	PUNCT
ejpam-2488	123	6	omeri	omeri	ADJ
ejpam-2488	123	7	,	,	PUNCT
ejpam-2488	123	8	m.	m.	NOUN
ejpam-2488	123	9	noorani	noorani	PROPN
ejpam-2488	123	10	,	,	PUNCT
ejpam-2488	123	11	a.	a.	PROPN
ejpam-2488	123	12	al	al	PROPN
ejpam-2488	123	13	-	-	PUNCT
ejpam-2488	123	14	omari	omari	PROPN
ejpam-2488	123	15	,	,	PUNCT
ejpam-2488	123	16	and	and	CCONJ
ejpam-2488	123	17	t.	t.	PROPN
ejpam-2488	123	18	noiri	noiri	PROPN
ejpam-2488	123	19	/	/	SYM
ejpam-2488	123	20	eur	eur	PROPN
ejpam-2488	123	21	.	.	PUNCT
ejpam-2488	124	1	j.	j.	PROPN
ejpam-2488	124	2	pure	pure	PROPN
ejpam-2488	124	3	appl	appl	PROPN
ejpam-2488	124	4	.	.	PROPN
ejpam-2488	124	5	math	math	PROPN
ejpam-2488	124	6	,	,	PUNCT
ejpam-2488	124	7	8	8	NUM
ejpam-2488	124	8	(	(	PUNCT
ejpam-2488	124	9	2015	2015	NUM
ejpam-2488	124	10	)	)	PUNCT
ejpam-2488	124	11	,	,	PUNCT
ejpam-2488	124	12	502	502	NUM
ejpam-2488	124	13	-	-	SYM
ejpam-2488	124	14	513	513	NUM
ejpam-2488	124	15	505	505	NUM
ejpam-2488	124	16	there	there	PRON
ejpam-2488	124	17	exists	exist	VERB
ejpam-2488	124	18	an	an	DET
ejpam-2488	124	19	e	e	NOUN
ejpam-2488	124	20	-	-	NOUN
ejpam-2488	124	21	i	i	PRON
ejpam-2488	124	22	-open	-open	NOUN
ejpam-2488	124	23	set	set	VERB
ejpam-2488	124	24	v	v	ADP
ejpam-2488	124	25	such	such	ADJ
ejpam-2488	124	26	that	that	SCONJ
ejpam-2488	124	27	y	y	PROPN
ejpam-2488	124	28	∈	∈	PROPN
ejpam-2488	124	29	v	v	NOUN
ejpam-2488	124	30	and	and	CCONJ
ejpam-2488	124	31	x	x	NOUN
ejpam-2488	124	32	/∈	/∈	NOUN
ejpam-2488	125	1	v	v	INTJ
ejpam-2488	125	2	.	.	PUNCT
ejpam-2488	126	1	let	let	VERB
ejpam-2488	126	2	x	x	PUNCT
ejpam-2488	126	3	∈	∈	PROPN
ejpam-2488	126	4	u	u	NOUN
ejpam-2488	126	5	and	and	CCONJ
ejpam-2488	126	6	y	y	PROPN
ejpam-2488	126	7	/∈	/∈	PUNCT
ejpam-2488	127	1	u	u	PROPN
ejpam-2488	127	2	.	.	PUNCT
ejpam-2488	128	1	since	since	SCONJ
ejpam-2488	128	2	x	x	PROPN
ejpam-2488	128	3	is	be	AUX
ejpam-2488	128	4	e	e	NOUN
ejpam-2488	128	5	-	-	PUNCT
ejpam-2488	128	6	i	i	PRON
ejpam-2488	128	7	-r0	-r0	NOUN
ejpam-2488	128	8	,	,	PUNCT
ejpam-2488	128	9	then	then	ADV
ejpam-2488	128	10	cl∗e	cl∗e	PROPN
ejpam-2488	128	11	(	(	PUNCT
ejpam-2488	128	12	x	x	X
ejpam-2488	128	13	)	)	PUNCT
ejpam-2488	128	14	⊂	⊂	PROPN
ejpam-2488	128	15	u	u	PROPN
ejpam-2488	128	16	.	.	PUNCT
ejpam-2488	129	1	we	we	PRON
ejpam-2488	129	2	have	have	VERB
ejpam-2488	129	3	y	y	PROPN
ejpam-2488	129	4	/∈	/∈	PUNCT
ejpam-2488	129	5	u	u	NOUN
ejpam-2488	130	1	and	and	CCONJ
ejpam-2488	130	2	then	then	ADV
ejpam-2488	130	3	y	y	PROPN
ejpam-2488	130	4	/∈	/∈	PUNCT
ejpam-2488	131	1	cl∗e	cl∗e	PROPN
ejpam-2488	131	2	(	(	PUNCT
ejpam-2488	131	3	x	x	NOUN
ejpam-2488	131	4	)	)	PUNCT
ejpam-2488	131	5	.	.	PUNCT
ejpam-2488	132	1	we	we	PRON
ejpam-2488	132	2	obtain	obtain	VERB
ejpam-2488	132	3	y	y	PROPN
ejpam-2488	132	4	∈	∈	PROPN
ejpam-2488	132	5	x\cl∗e	x\cl∗e	PROPN
ejpam-2488	132	6	(	(	PUNCT
ejpam-2488	132	7	x	x	NOUN
ejpam-2488	132	8	)	)	PUNCT
ejpam-2488	132	9	.	.	PUNCT
ejpam-2488	133	1	take	take	VERB
ejpam-2488	133	2	s	s	PART
ejpam-2488	133	3	=	=	SYM
ejpam-2488	133	4	x\cl∗e	x\cl∗e	PROPN
ejpam-2488	133	5	(	(	PUNCT
ejpam-2488	133	6	x	x	NOUN
ejpam-2488	133	7	)	)	PUNCT
ejpam-2488	133	8	.	.	PUNCT
ejpam-2488	134	1	thus	thus	ADV
ejpam-2488	134	2	,	,	PUNCT
ejpam-2488	134	3	u	u	NOUN
ejpam-2488	134	4	and	and	CCONJ
ejpam-2488	134	5	s	s	NOUN
ejpam-2488	134	6	are	be	AUX
ejpam-2488	134	7	e	e	NOUN
ejpam-2488	134	8	-	-	ADJ
ejpam-2488	134	9	i	i	PRON
ejpam-2488	134	10	-open	-open	NOUN
ejpam-2488	134	11	sets	set	NOUN
ejpam-2488	134	12	containing	contain	VERB
ejpam-2488	134	13	x	x	X
ejpam-2488	134	14	and	and	CCONJ
ejpam-2488	134	15	y	y	PROPN
ejpam-2488	134	16	,	,	PUNCT
ejpam-2488	134	17	respectively	respectively	ADV
ejpam-2488	134	18	,	,	PUNCT
ejpam-2488	134	19	such	such	ADJ
ejpam-2488	134	20	that	that	SCONJ
ejpam-2488	134	21	y	y	PROPN
ejpam-2488	134	22	/∈	/∈	PUNCT
ejpam-2488	134	23	u	u	PROPN
ejpam-2488	134	24	and	and	CCONJ
ejpam-2488	134	25	x	x	PROPN
ejpam-2488	134	26	/∈	/∈	PROPN
ejpam-2488	135	1	s.	s.	PROPN
ejpam-2488	135	2	hence	hence	ADV
ejpam-2488	135	3	,	,	PUNCT
ejpam-2488	135	4	x	x	X
ejpam-2488	135	5	is	be	AUX
ejpam-2488	135	6	e	e	VERB
ejpam-2488	135	7	-	-	NOUN
ejpam-2488	135	8	i	i	PRON
ejpam-2488	135	9	-t1	-t1	VERB
ejpam-2488	135	10	.	.	PUNCT
ejpam-2488	136	1	remark	remark	PROPN
ejpam-2488	136	2	1	1	NUM
ejpam-2488	136	3	.	.	PUNCT
ejpam-2488	137	1	since	since	SCONJ
ejpam-2488	137	2	an	an	DET
ejpam-2488	137	3	ideal	ideal	ADJ
ejpam-2488	137	4	topological	topological	ADJ
ejpam-2488	137	5	space	space	NOUN
ejpam-2488	137	6	(	(	PUNCT
ejpam-2488	137	7	x	x	X
ejpam-2488	137	8	,	,	PUNCT
ejpam-2488	137	9	τ	τ	PROPN
ejpam-2488	137	10	,	,	PUNCT
ejpam-2488	137	11	i	i	PROPN
ejpam-2488	137	12	)	)	PUNCT
ejpam-2488	137	13	is	be	AUX
ejpam-2488	137	14	e	e	NOUN
ejpam-2488	137	15	-	-	PUNCT
ejpam-2488	137	16	i	i	PRON
ejpam-2488	137	17	-t1	-t1	VERB
ejpam-2488	137	18	if	if	SCONJ
ejpam-2488	137	19	and	and	CCONJ
ejpam-2488	137	20	only	only	ADV
ejpam-2488	137	21	if	if	SCONJ
ejpam-2488	137	22	the	the	DET
ejpam-2488	137	23	singletons	singleton	NOUN
ejpam-2488	137	24	are	be	AUX
ejpam-2488	137	25	e	e	ADJ
ejpam-2488	137	26	-	-	PUNCT
ejpam-2488	137	27	i	i	PRON
ejpam-2488	137	28	-closed	-close	VERB
ejpam-2488	137	29	,	,	PUNCT
ejpam-2488	137	30	it	it	PRON
ejpam-2488	137	31	is	be	AUX
ejpam-2488	137	32	clear	clear	ADJ
ejpam-2488	137	33	that	that	SCONJ
ejpam-2488	137	34	every	every	DET
ejpam-2488	137	35	e	e	NOUN
ejpam-2488	137	36	-	-	PROPN
ejpam-2488	137	37	i	i	PRON
ejpam-2488	137	38	-t1	-t1	VERB
ejpam-2488	137	39	space	space	NOUN
ejpam-2488	137	40	e	e	NOUN
ejpam-2488	137	41	-	-	PROPN
ejpam-2488	137	42	i	i	PRON
ejpam-2488	137	43	-r0	-r0	PROPN
ejpam-2488	137	44	.	.	PUNCT
ejpam-2488	138	1	but	but	CCONJ
ejpam-2488	138	2	the	the	DET
ejpam-2488	138	3	converse	converse	NOUN
ejpam-2488	138	4	is	be	AUX
ejpam-2488	138	5	not	not	PART
ejpam-2488	138	6	true	true	ADJ
ejpam-2488	138	7	in	in	ADP
ejpam-2488	138	8	general	general	ADJ
ejpam-2488	138	9	.	.	PUNCT
ejpam-2488	138	10	example	example	NOUN
ejpam-2488	139	1	1	1	NUM
ejpam-2488	139	2	.	.	PUNCT
ejpam-2488	139	3	let	let	VERB
ejpam-2488	139	4	x	x	PUNCT
ejpam-2488	139	5	=	=	PRON
ejpam-2488	139	6	{	{	PUNCT
ejpam-2488	139	7	a	a	DET
ejpam-2488	139	8	,	,	PUNCT
ejpam-2488	139	9	b	b	NOUN
ejpam-2488	139	10	,	,	PUNCT
ejpam-2488	139	11	c}with	c}with	ADP
ejpam-2488	139	12	a	a	DET
ejpam-2488	139	13	topologyτ=	topologyτ=	NOUN
ejpam-2488	139	14	{	{	PUNCT
ejpam-2488	139	15	;	;	PUNCT
ejpam-2488	139	16	,	,	PUNCT
ejpam-2488	139	17	x	x	X
ejpam-2488	139	18	,	,	PUNCT
ejpam-2488	139	19	{	{	PUNCT
ejpam-2488	139	20	a	a	NOUN
ejpam-2488	139	21	}	}	PUNCT
ejpam-2488	139	22	,	,	PUNCT
ejpam-2488	139	23	{	{	PUNCT
ejpam-2488	139	24	b	b	X
ejpam-2488	139	25	,	,	PUNCT
ejpam-2488	139	26	c	c	NOUN
ejpam-2488	139	27	}	}	PUNCT
ejpam-2488	139	28	}	}	PUNCT
ejpam-2488	139	29	andi	andi	PROPN
ejpam-2488	139	30	=	=	PUNCT
ejpam-2488	139	31	{	{	PUNCT
ejpam-2488	139	32	o	o	NOUN
ejpam-2488	139	33	,	,	PUNCT
ejpam-2488	139	34	{	{	PUNCT
ejpam-2488	139	35	c	c	NOUN
ejpam-2488	139	36	}	}	PUNCT
ejpam-2488	139	37	,	,	PUNCT
ejpam-2488	139	38	{	{	PUNCT
ejpam-2488	139	39	b	b	X
ejpam-2488	139	40	}	}	PUNCT
ejpam-2488	139	41	,	,	PUNCT
ejpam-2488	139	42	{	{	PUNCT
ejpam-2488	139	43	b	b	X
ejpam-2488	139	44	,	,	PUNCT
ejpam-2488	139	45	c	c	NOUN
ejpam-2488	139	46	}	}	PUNCT
ejpam-2488	139	47	}	}	PUNCT
ejpam-2488	139	48	.	.	PUNCT
ejpam-2488	140	1	since	since	SCONJ
ejpam-2488	140	2	e	e	PROPN
ejpam-2488	140	3	-	-	PROPN
ejpam-2488	140	4	i	i	PRON
ejpam-2488	140	5	-open={φ	-open={φ	PROPN
ejpam-2488	140	6	,	,	PUNCT
ejpam-2488	140	7	x	x	INTJ
ejpam-2488	140	8	,	,	PUNCT
ejpam-2488	140	9	{	{	PUNCT
ejpam-2488	140	10	a	a	NOUN
ejpam-2488	140	11	}	}	PUNCT
ejpam-2488	140	12	,	,	PUNCT
ejpam-2488	140	13	{	{	PUNCT
ejpam-2488	140	14	b	b	X
ejpam-2488	140	15	,	,	PUNCT
ejpam-2488	140	16	c	c	NOUN
ejpam-2488	140	17	}	}	PUNCT
ejpam-2488	140	18	}	}	PUNCT
ejpam-2488	140	19	.	.	PUNCT
ejpam-2488	141	1	it	it	PRON
ejpam-2488	141	2	is	be	AUX
ejpam-2488	141	3	clear	clear	ADJ
ejpam-2488	141	4	that	that	SCONJ
ejpam-2488	141	5	every	every	DET
ejpam-2488	141	6	e	e	NOUN
ejpam-2488	141	7	-	-	ADJ
ejpam-2488	141	8	i	i	PRON
ejpam-2488	141	9	-open	-open	ADJ
ejpam-2488	141	10	set	set	NOUN
ejpam-2488	141	11	contains	contain	VERB
ejpam-2488	141	12	the	the	DET
ejpam-2488	141	13	e	e	NOUN
ejpam-2488	141	14	-	-	NOUN
ejpam-2488	141	15	i	i	NOUN
ejpam-2488	141	16	-closure	-closure	NOUN
ejpam-2488	141	17	of	of	ADP
ejpam-2488	141	18	each	each	PRON
ejpam-2488	141	19	of	of	ADP
ejpam-2488	141	20	its	its	PRON
ejpam-2488	141	21	singletons	singleton	NOUN
ejpam-2488	141	22	so	so	SCONJ
ejpam-2488	141	23	the	the	DET
ejpam-2488	141	24	ideal	ideal	ADJ
ejpam-2488	141	25	topological	topological	ADJ
ejpam-2488	141	26	space	space	NOUN
ejpam-2488	141	27	is	be	AUX
ejpam-2488	141	28	e	e	NOUN
ejpam-2488	141	29	-	-	PUNCT
ejpam-2488	141	30	i	i	PRON
ejpam-2488	141	31	-r0	-r0	NOUN
ejpam-2488	141	32	,	,	PUNCT
ejpam-2488	141	33	but	but	CCONJ
ejpam-2488	141	34	none	none	NOUN
ejpam-2488	141	35	of	of	ADP
ejpam-2488	141	36	e	e	NOUN
ejpam-2488	141	37	-	-	PROPN
ejpam-2488	141	38	i	i	PRON
ejpam-2488	141	39	-t0	-t0	NOUN
ejpam-2488	141	40	and	and	CCONJ
ejpam-2488	141	41	e	e	X
ejpam-2488	141	42	-	-	NOUN
ejpam-2488	141	43	i	i	PRON
ejpam-2488	141	44	-t1	-t1	VERB
ejpam-2488	141	45	.	.	PUNCT
ejpam-2488	142	1	remark	remark	PROPN
ejpam-2488	142	2	2	2	NUM
ejpam-2488	142	3	.	.	PUNCT
ejpam-2488	143	1	the	the	DET
ejpam-2488	143	2	following	follow	VERB
ejpam-2488	143	3	example	example	NOUN
ejpam-2488	143	4	and	and	CCONJ
ejpam-2488	143	5	example	example	NOUN
ejpam-2488	143	6	1	1	NUM
ejpam-2488	143	7	show	show	VERB
ejpam-2488	143	8	that	that	SCONJ
ejpam-2488	143	9	the	the	DET
ejpam-2488	143	10	notions	notion	NOUN
ejpam-2488	143	11	e	e	NOUN
ejpam-2488	143	12	-	-	VERB
ejpam-2488	143	13	i	i	PRON
ejpam-2488	143	14	-t0	-t0	NOUN
ejpam-2488	143	15	-	-	NOUN
ejpam-2488	143	16	ness	ness	NOUN
ejpam-2488	143	17	and	and	CCONJ
ejpam-2488	143	18	e	e	X
ejpam-2488	143	19	-	-	PROPN
ejpam-2488	143	20	i	i	PRON
ejpam-2488	143	21	-r0	-r0	PUNCT
ejpam-2488	143	22	-	-	ADJ
ejpam-2488	143	23	ness	ness	ADV
ejpam-2488	143	24	are	be	AUX
ejpam-2488	143	25	independent	independent	ADJ
ejpam-2488	143	26	.	.	PUNCT
ejpam-2488	143	27	example	example	NOUN
ejpam-2488	144	1	2	2	NUM
ejpam-2488	144	2	.	.	PUNCT
ejpam-2488	144	3	let	let	VERB
ejpam-2488	144	4	x	x	PUNCT
ejpam-2488	144	5	=	=	PRON
ejpam-2488	144	6	{	{	PUNCT
ejpam-2488	144	7	a	a	PRON
ejpam-2488	144	8	,	,	PUNCT
ejpam-2488	144	9	b	b	NOUN
ejpam-2488	144	10	,	,	PUNCT
ejpam-2488	144	11	c	c	NOUN
ejpam-2488	144	12	}	}	PUNCT
ejpam-2488	144	13	with	with	ADP
ejpam-2488	144	14	a	a	DET
ejpam-2488	144	15	topology	topology	NOUN
ejpam-2488	144	16	τ	τ	X
ejpam-2488	144	17	=	=	PUNCT
ejpam-2488	144	18	{	{	PUNCT
ejpam-2488	144	19	;	;	PUNCT
ejpam-2488	144	20	,	,	PUNCT
ejpam-2488	144	21	x	x	X
ejpam-2488	144	22	,	,	PUNCT
ejpam-2488	144	23	{	{	PUNCT
ejpam-2488	144	24	a	a	X
ejpam-2488	144	25	}	}	PUNCT
ejpam-2488	144	26	}	}	PUNCT
ejpam-2488	144	27	and	and	CCONJ
ejpam-2488	144	28	i	i	PRON
ejpam-2488	144	29	=	=	PUNCT
ejpam-2488	144	30	{	{	PUNCT
ejpam-2488	144	31	o	o	NOUN
ejpam-2488	144	32	,	,	PUNCT
ejpam-2488	144	33	{	{	PUNCT
ejpam-2488	144	34	a	a	X
ejpam-2488	144	35	}	}	PUNCT
ejpam-2488	144	36	}	}	PUNCT
ejpam-2488	144	37	.	.	PUNCT
ejpam-2488	145	1	now	now	ADV
ejpam-2488	145	2	,	,	PUNCT
ejpam-2488	145	3	we	we	PRON
ejpam-2488	145	4	determine	determine	VERB
ejpam-2488	145	5	e	e	NOUN
ejpam-2488	145	6	-	-	PROPN
ejpam-2488	145	7	i	i	PRON
ejpam-2488	145	8	-open={φ	-open={φ	PROPN
ejpam-2488	145	9	,	,	PUNCT
ejpam-2488	145	10	x	x	INTJ
ejpam-2488	145	11	,	,	PUNCT
ejpam-2488	145	12	{	{	PUNCT
ejpam-2488	145	13	a	a	X
ejpam-2488	145	14	}	}	PUNCT
ejpam-2488	145	15	}	}	PUNCT
ejpam-2488	145	16	.	.	PUNCT
ejpam-2488	146	1	then	then	ADV
ejpam-2488	146	2	(	(	PUNCT
ejpam-2488	146	3	x	x	X
ejpam-2488	146	4	,	,	PUNCT
ejpam-2488	146	5	τ	τ	PROPN
ejpam-2488	146	6	,	,	PUNCT
ejpam-2488	146	7	i	i	PROPN
ejpam-2488	146	8	)	)	PUNCT
ejpam-2488	146	9	is	be	AUX
ejpam-2488	146	10	e	e	NOUN
ejpam-2488	146	11	-	-	NOUN
ejpam-2488	146	12	i	i	PRON
ejpam-2488	146	13	-t0	-t0	NOUN
ejpam-2488	147	1	but	but	CCONJ
ejpam-2488	147	2	it	it	PRON
ejpam-2488	147	3	is	be	AUX
ejpam-2488	147	4	not	not	PART
ejpam-2488	147	5	e	e	NOUN
ejpam-2488	147	6	-	-	NOUN
ejpam-2488	147	7	i	i	PRON
ejpam-2488	147	8	-r0	-r0	PROPN
ejpam-2488	147	9	.	.	PUNCT
ejpam-2488	148	1	lemma	lemma	PROPN
ejpam-2488	148	2	3	3	X
ejpam-2488	148	3	.	.	PUNCT
ejpam-2488	149	1	let	let	AUX
ejpam-2488	149	2	(	(	PUNCT
ejpam-2488	149	3	x	x	X
ejpam-2488	149	4	,	,	PUNCT
ejpam-2488	149	5	τ	τ	PROPN
ejpam-2488	149	6	,	,	PUNCT
ejpam-2488	149	7	i	i	PRON
ejpam-2488	149	8	)	)	PUNCT
ejpam-2488	149	9	be	be	AUX
ejpam-2488	149	10	an	an	DET
ejpam-2488	149	11	ideal	ideal	ADJ
ejpam-2488	149	12	topological	topological	ADJ
ejpam-2488	149	13	space	space	NOUN
ejpam-2488	149	14	.	.	PUNCT
ejpam-2488	150	1	then	then	ADV
ejpam-2488	150	2	for	for	ADP
ejpam-2488	150	3	any	any	DET
ejpam-2488	150	4	points	point	NOUN
ejpam-2488	150	5	x	x	PUNCT
ejpam-2488	150	6	and	and	CCONJ
ejpam-2488	150	7	y	y	PROPN
ejpam-2488	150	8	in	in	ADP
ejpam-2488	150	9	x	x	X
ejpam-2488	150	10	,	,	PUNCT
ejpam-2488	150	11	the	the	DET
ejpam-2488	150	12	following	follow	VERB
ejpam-2488	150	13	statements	statement	NOUN
ejpam-2488	150	14	are	be	AUX
ejpam-2488	150	15	equivalent	equivalent	ADJ
ejpam-2488	150	16	:	:	PUNCT
ejpam-2488	150	17	(	(	PUNCT
ejpam-2488	150	18	i	i	NOUN
ejpam-2488	150	19	)	)	PUNCT
ejpam-2488	150	20	ieker({x	ieker({x	PROPN
ejpam-2488	150	21	}	}	PUNCT
ejpam-2488	150	22	)	)	PUNCT
ejpam-2488	150	23	6=	6=	ADP
ejpam-2488	150	24	ieker({y	ieker({y	PROPN
ejpam-2488	150	25	}	}	PUNCT
ejpam-2488	150	26	)	)	PUNCT
ejpam-2488	150	27	.	.	PUNCT
ejpam-2488	151	1	(	(	PUNCT
ejpam-2488	151	2	ii	ii	X
ejpam-2488	151	3	)	)	PUNCT
ejpam-2488	151	4	c	c	PROPN
ejpam-2488	151	5	l∗e	l∗e	PUNCT
ejpam-2488	151	6	(	(	PUNCT
ejpam-2488	151	7	{	{	PUNCT
ejpam-2488	151	8	x	x	NOUN
ejpam-2488	151	9	}	}	PUNCT
ejpam-2488	151	10	)	)	PUNCT
ejpam-2488	152	1	6=	6=	ADP
ejpam-2488	152	2	cl∗e	cl∗e	X
ejpam-2488	152	3	(	(	PUNCT
ejpam-2488	152	4	{	{	PUNCT
ejpam-2488	152	5	y	y	NOUN
ejpam-2488	152	6	}	}	PUNCT
ejpam-2488	152	7	)	)	PUNCT
ejpam-2488	152	8	.	.	PUNCT
ejpam-2488	153	1	proof	proof	NOUN
ejpam-2488	153	2	.	.	PUNCT
ejpam-2488	154	1	(	(	PUNCT
ejpam-2488	154	2	i)⇒	i)⇒	PROPN
ejpam-2488	154	3	(	(	PUNCT
ejpam-2488	154	4	ii	ii	NOUN
ejpam-2488	154	5	):	):	PUNCT
ejpam-2488	154	6	let	let	VERB
ejpam-2488	154	7	ieker({x	ieker({x	PRON
ejpam-2488	154	8	}	}	PUNCT
ejpam-2488	154	9	)	)	PUNCT
ejpam-2488	154	10	6=	6=	ADP
ejpam-2488	154	11	ieker({y	ieker({y	PROPN
ejpam-2488	154	12	}	}	PUNCT
ejpam-2488	154	13	)	)	PUNCT
ejpam-2488	154	14	,	,	PUNCT
ejpam-2488	154	15	then	then	ADV
ejpam-2488	154	16	there	there	PRON
ejpam-2488	154	17	exists	exist	VERB
ejpam-2488	154	18	a	a	DET
ejpam-2488	154	19	point	point	NOUN
ejpam-2488	154	20	k	k	X
ejpam-2488	154	21	in	in	ADP
ejpam-2488	154	22	x	x	PUNCT
ejpam-2488	154	23	such	such	ADJ
ejpam-2488	154	24	that	that	SCONJ
ejpam-2488	154	25	k	k	PROPN
ejpam-2488	154	26	∈	∈	PROPN
ejpam-2488	154	27	ieker({x	ieker({x	PROPN
ejpam-2488	154	28	}	}	PUNCT
ejpam-2488	154	29	)	)	PUNCT
ejpam-2488	154	30	and	and	CCONJ
ejpam-2488	154	31	k	k	PROPN
ejpam-2488	154	32	/∈	/∈	PUNCT
ejpam-2488	154	33	ieker({y	ieker({y	NUM
ejpam-2488	154	34	}	}	PUNCT
ejpam-2488	154	35	)	)	PUNCT
ejpam-2488	154	36	.	.	PUNCT
ejpam-2488	155	1	by	by	ADP
ejpam-2488	155	2	lemma	lemma	PROPN
ejpam-2488	155	3	1	1	NUM
ejpam-2488	155	4	,	,	PUNCT
ejpam-2488	155	5	x	x	SYM
ejpam-2488	155	6	∈	∈	PROPN
ejpam-2488	155	7	cl∗e	cl∗e	X
ejpam-2488	155	8	(	(	PUNCT
ejpam-2488	155	9	{	{	PUNCT
ejpam-2488	155	10	x	x	NOUN
ejpam-2488	155	11	}	}	PUNCT
ejpam-2488	155	12	)	)	PUNCT
ejpam-2488	155	13	and	and	CCONJ
ejpam-2488	155	14	y	y	PROPN
ejpam-2488	155	15	/∈	/∈	PUNCT
ejpam-2488	155	16	cl∗e	cl∗e	PROPN
ejpam-2488	155	17	(	(	PUNCT
ejpam-2488	155	18	{	{	PUNCT
ejpam-2488	155	19	x	x	NOUN
ejpam-2488	155	20	}	}	PUNCT
ejpam-2488	155	21	)	)	PUNCT
ejpam-2488	155	22	.	.	PUNCT
ejpam-2488	156	1	therefore	therefore	ADV
ejpam-2488	156	2	,	,	PUNCT
ejpam-2488	156	3	cl∗e	cl∗e	PROPN
ejpam-2488	156	4	(	(	PUNCT
ejpam-2488	156	5	{	{	PUNCT
ejpam-2488	156	6	x	x	NOUN
ejpam-2488	156	7	}	}	PUNCT
ejpam-2488	156	8	)	)	PUNCT
ejpam-2488	156	9	⊂	⊂	PROPN
ejpam-2488	156	10	cl∗e	cl∗e	PROPN
ejpam-2488	156	11	(	(	PUNCT
ejpam-2488	156	12	cl∗e	cl∗e	PROPN
ejpam-2488	156	13	(	(	PUNCT
ejpam-2488	156	14	{	{	PUNCT
ejpam-2488	156	15	k	k	NOUN
ejpam-2488	156	16	}	}	PUNCT
ejpam-2488	156	17	)	)	PUNCT
ejpam-2488	156	18	)	)	PUNCT
ejpam-2488	156	19	=	=	SYM
ejpam-2488	156	20	cl∗e	cl∗e	PROPN
ejpam-2488	156	21	(	(	PUNCT
ejpam-2488	156	22	{	{	PUNCT
ejpam-2488	156	23	k	k	NOUN
ejpam-2488	156	24	}	}	PUNCT
ejpam-2488	156	25	)	)	PUNCT
ejpam-2488	156	26	and	and	CCONJ
ejpam-2488	156	27	hence	hence	ADV
ejpam-2488	156	28	y	y	PROPN
ejpam-2488	156	29	/∈	/∈	PUNCT
ejpam-2488	156	30	cl∗e	cl∗e	PROPN
ejpam-2488	156	31	(	(	PUNCT
ejpam-2488	156	32	{	{	PUNCT
ejpam-2488	156	33	x	x	NOUN
ejpam-2488	156	34	}	}	PUNCT
ejpam-2488	156	35	)	)	PUNCT
ejpam-2488	156	36	.	.	PUNCT
ejpam-2488	157	1	hence	hence	ADV
ejpam-2488	157	2	cl∗e	cl∗e	PROPN
ejpam-2488	157	3	(	(	PUNCT
ejpam-2488	157	4	{	{	PUNCT
ejpam-2488	157	5	x	x	NOUN
ejpam-2488	157	6	}	}	PUNCT
ejpam-2488	157	7	)	)	PUNCT
ejpam-2488	157	8	6=	6=	ADP
ejpam-2488	157	9	cl∗e	cl∗e	X
ejpam-2488	157	10	(	(	PUNCT
ejpam-2488	157	11	{	{	PUNCT
ejpam-2488	157	12	y	y	NOUN
ejpam-2488	157	13	}	}	PUNCT
ejpam-2488	157	14	)	)	PUNCT
ejpam-2488	157	15	.	.	PUNCT
ejpam-2488	158	1	by	by	ADP
ejpam-2488	158	2	using	use	VERB
ejpam-2488	158	3	ieker({x	ieker({x	PRON
ejpam-2488	158	4	}	}	PUNCT
ejpam-2488	158	5	)	)	PUNCT
ejpam-2488	158	6	6=	6=	ADP
ejpam-2488	158	7	ieker({y	ieker({y	PROPN
ejpam-2488	158	8	}	}	PUNCT
ejpam-2488	158	9	)	)	PUNCT
ejpam-2488	158	10	,	,	PUNCT
ejpam-2488	158	11	we	we	PRON
ejpam-2488	158	12	obtain	obtain	VERB
ejpam-2488	158	13	cl∗e	cl∗e	PROPN
ejpam-2488	158	14	(	(	PUNCT
ejpam-2488	158	15	{	{	PUNCT
ejpam-2488	158	16	x	x	NOUN
ejpam-2488	158	17	}	}	PUNCT
ejpam-2488	158	18	)	)	PUNCT
ejpam-2488	158	19	6=	6=	ADP
ejpam-2488	159	1	cl∗e	cl∗e	X
ejpam-2488	159	2	(	(	PUNCT
ejpam-2488	159	3	{	{	PUNCT
ejpam-2488	159	4	y	y	NOUN
ejpam-2488	159	5	}	}	PUNCT
ejpam-2488	159	6	)	)	PUNCT
ejpam-2488	159	7	.	.	PUNCT
ejpam-2488	160	1	(	(	PUNCT
ejpam-2488	160	2	ii)⇒	ii)⇒	PROPN
ejpam-2488	160	3	(	(	PUNCT
ejpam-2488	160	4	i	i	NOUN
ejpam-2488	160	5	):	):	PUNCT
ejpam-2488	160	6	let	let	VERB
ejpam-2488	160	7	cl∗e	cl∗e	X
ejpam-2488	160	8	(	(	PUNCT
ejpam-2488	160	9	{	{	PUNCT
ejpam-2488	160	10	x	x	NOUN
ejpam-2488	160	11	}	}	PUNCT
ejpam-2488	160	12	)	)	PUNCT
ejpam-2488	160	13	6=	6=	ADP
ejpam-2488	160	14	cl∗e	cl∗e	X
ejpam-2488	160	15	(	(	PUNCT
ejpam-2488	160	16	{	{	PUNCT
ejpam-2488	160	17	y	y	NOUN
ejpam-2488	160	18	}	}	PUNCT
ejpam-2488	160	19	)	)	PUNCT
ejpam-2488	160	20	,	,	PUNCT
ejpam-2488	160	21	then	then	ADV
ejpam-2488	160	22	there	there	PRON
ejpam-2488	160	23	exists	exist	VERB
ejpam-2488	160	24	a	a	DET
ejpam-2488	160	25	point	point	NOUN
ejpam-2488	160	26	k	k	X
ejpam-2488	160	27	in	in	ADP
ejpam-2488	160	28	x	x	PUNCT
ejpam-2488	160	29	such	such	ADJ
ejpam-2488	160	30	that	that	SCONJ
ejpam-2488	160	31	k	k	PROPN
ejpam-2488	160	32	∈	∈	PROPN
ejpam-2488	160	33	cl∗e	cl∗e	PROPN
ejpam-2488	160	34	(	(	PUNCT
ejpam-2488	160	35	{	{	PUNCT
ejpam-2488	160	36	x	x	NOUN
ejpam-2488	160	37	}	}	PUNCT
ejpam-2488	160	38	)	)	PUNCT
ejpam-2488	160	39	and	and	CCONJ
ejpam-2488	160	40	k	k	PROPN
ejpam-2488	160	41	/∈	/∈	PUNCT
ejpam-2488	160	42	cl∗e	cl∗e	PROPN
ejpam-2488	160	43	(	(	PUNCT
ejpam-2488	160	44	{	{	PUNCT
ejpam-2488	160	45	y	y	NOUN
ejpam-2488	160	46	}	}	PUNCT
ejpam-2488	160	47	)	)	PUNCT
ejpam-2488	160	48	and	and	CCONJ
ejpam-2488	160	49	then	then	ADV
ejpam-2488	160	50	there	there	PRON
ejpam-2488	160	51	exists	exist	VERB
ejpam-2488	160	52	an	an	DET
ejpam-2488	160	53	e	e	NOUN
ejpam-2488	160	54	-	-	NOUN
ejpam-2488	160	55	i	i	PRON
ejpam-2488	160	56	-open	-open	NOUN
ejpam-2488	160	57	set	set	VERB
ejpam-2488	160	58	containing	contain	VERB
ejpam-2488	160	59	k	k	PROPN
ejpam-2488	160	60	and	and	CCONJ
ejpam-2488	160	61	therefore	therefore	ADV
ejpam-2488	160	62	x	x	X
ejpam-2488	160	63	but	but	CCONJ
ejpam-2488	160	64	not	not	PART
ejpam-2488	160	65	y	y	PROPN
ejpam-2488	160	66	,	,	PUNCT
ejpam-2488	160	67	namely	namely	ADV
ejpam-2488	160	68	,	,	PUNCT
ejpam-2488	160	69	y	y	PROPN
ejpam-2488	160	70	/∈	/∈	PUNCT
ejpam-2488	160	71	ieker({x	ieker({x	PROPN
ejpam-2488	160	72	}	}	PUNCT
ejpam-2488	160	73	)	)	PUNCT
ejpam-2488	160	74	and	and	CCONJ
ejpam-2488	160	75	thus	thus	ADV
ejpam-2488	160	76	ieker({x	ieker({x	NOUN
ejpam-2488	160	77	}	}	PUNCT
ejpam-2488	160	78	)	)	PUNCT
ejpam-2488	160	79	6=	6=	ADP
ejpam-2488	160	80	ieker({y	ieker({y	PROPN
ejpam-2488	160	81	}	}	PUNCT
ejpam-2488	160	82	)	)	PUNCT
ejpam-2488	160	83	.	.	PUNCT
ejpam-2488	161	1	proposition	proposition	NOUN
ejpam-2488	161	2	1	1	NUM
ejpam-2488	161	3	.	.	PUNCT
ejpam-2488	162	1	for	for	ADP
ejpam-2488	162	2	an	an	DET
ejpam-2488	162	3	ideal	ideal	ADJ
ejpam-2488	162	4	topological	topological	ADJ
ejpam-2488	162	5	space	space	NOUN
ejpam-2488	162	6	(	(	PUNCT
ejpam-2488	162	7	x	x	X
ejpam-2488	162	8	,	,	PUNCT
ejpam-2488	162	9	τ	τ	PROPN
ejpam-2488	162	10	,	,	PUNCT
ejpam-2488	162	11	i	i	PROPN
ejpam-2488	162	12	)	)	PUNCT
ejpam-2488	162	13	,	,	PUNCT
ejpam-2488	162	14	the	the	DET
ejpam-2488	162	15	following	follow	VERB
ejpam-2488	162	16	properties	property	NOUN
ejpam-2488	162	17	are	be	AUX
ejpam-2488	162	18	equivalent	equivalent	ADJ
ejpam-2488	162	19	:	:	PUNCT
ejpam-2488	162	20	(	(	PUNCT
ejpam-2488	162	21	i	i	NOUN
ejpam-2488	162	22	)	)	PUNCT
ejpam-2488	162	23	(	(	PUNCT
ejpam-2488	162	24	x	x	X
ejpam-2488	162	25	,	,	PUNCT
ejpam-2488	162	26	τ	τ	PROPN
ejpam-2488	162	27	,	,	PUNCT
ejpam-2488	162	28	i	i	PROPN
ejpam-2488	162	29	)	)	PUNCT
ejpam-2488	162	30	is	be	AUX
ejpam-2488	162	31	an	an	DET
ejpam-2488	162	32	e	e	NOUN
ejpam-2488	162	33	-	-	NOUN
ejpam-2488	162	34	i	i	PRON
ejpam-2488	162	35	-r0	-r0	PROPN
ejpam-2488	162	36	space	space	NOUN
ejpam-2488	162	37	,	,	PUNCT
ejpam-2488	162	38	(	(	PUNCT
ejpam-2488	162	39	ii	ii	NOUN
ejpam-2488	162	40	)	)	PUNCT
ejpam-2488	162	41	for	for	ADP
ejpam-2488	162	42	any	any	DET
ejpam-2488	162	43	k	k	PROPN
ejpam-2488	162	44	∈	∈	PROPN
ejpam-2488	162	45	ei	ei	ADP
ejpam-2488	162	46	c(x	c(x	NOUN
ejpam-2488	162	47	)	)	PUNCT
ejpam-2488	162	48	,	,	PUNCT
ejpam-2488	162	49	x	x	X
ejpam-2488	162	50	/∈	/∈	PUNCT
ejpam-2488	163	1	k	k	PROPN
ejpam-2488	163	2	implies	imply	VERB
ejpam-2488	163	3	k	k	PROPN
ejpam-2488	163	4	⊂	⊂	PUNCT
ejpam-2488	163	5	u	u	PROPN
ejpam-2488	163	6	and	and	CCONJ
ejpam-2488	163	7	x	x	NOUN
ejpam-2488	163	8	/∈	/∈	PUNCT
ejpam-2488	163	9	u	u	NOUN
ejpam-2488	163	10	for	for	ADP
ejpam-2488	163	11	some	some	DET
ejpam-2488	163	12	u	u	NOUN
ejpam-2488	163	13	∈	∈	PROPN
ejpam-2488	163	14	eio(x	eio(x	PROPN
ejpam-2488	163	15	)	)	PUNCT
ejpam-2488	163	16	,	,	PUNCT
ejpam-2488	163	17	(	(	PUNCT
ejpam-2488	163	18	iii	iii	X
ejpam-2488	163	19	)	)	PUNCT
ejpam-2488	163	20	for	for	ADP
ejpam-2488	163	21	any	any	DET
ejpam-2488	163	22	k	k	PROPN
ejpam-2488	163	23	∈	∈	PROPN
ejpam-2488	163	24	ei	ei	ADP
ejpam-2488	163	25	c(x	c(x	NOUN
ejpam-2488	163	26	)	)	PUNCT
ejpam-2488	163	27	,	,	PUNCT
ejpam-2488	163	28	x	x	X
ejpam-2488	163	29	/∈	/∈	PUNCT
ejpam-2488	164	1	k	k	PROPN
ejpam-2488	164	2	implies	imply	VERB
ejpam-2488	164	3	k	k	PROPN
ejpam-2488	164	4	∩	∩	X
ejpam-2488	164	5	cl∗e	cl∗e	X
ejpam-2488	164	6	(	(	PUNCT
ejpam-2488	164	7	{	{	PUNCT
ejpam-2488	164	8	x	x	NOUN
ejpam-2488	164	9	}	}	PUNCT
ejpam-2488	164	10	)	)	PUNCT
ejpam-2488	164	11	=	=	SYM
ejpam-2488	165	1	;	;	PUNCT
ejpam-2488	165	2	,	,	PUNCT
ejpam-2488	165	3	(	(	PUNCT
ejpam-2488	165	4	iv	iv	X
ejpam-2488	165	5	)	)	PUNCT
ejpam-2488	165	6	for	for	ADP
ejpam-2488	165	7	any	any	DET
ejpam-2488	165	8	distinct	distinct	ADJ
ejpam-2488	165	9	points	point	NOUN
ejpam-2488	165	10	x	x	PUNCT
ejpam-2488	165	11	and	and	CCONJ
ejpam-2488	165	12	y	y	PROPN
ejpam-2488	165	13	of	of	ADP
ejpam-2488	165	14	x	x	PRON
ejpam-2488	165	15	,	,	PUNCT
ejpam-2488	165	16	either	either	CCONJ
ejpam-2488	165	17	c	c	NOUN
ejpam-2488	165	18	l∗e	l∗e	ADV
ejpam-2488	165	19	(	(	PUNCT
ejpam-2488	165	20	{	{	PUNCT
ejpam-2488	165	21	x	x	NOUN
ejpam-2488	165	22	}	}	PUNCT
ejpam-2488	165	23	)	)	PUNCT
ejpam-2488	165	24	=	=	SYM
ejpam-2488	165	25	cl∗e	cl∗e	PROPN
ejpam-2488	165	26	(	(	PUNCT
ejpam-2488	165	27	{	{	PUNCT
ejpam-2488	165	28	y	y	NOUN
ejpam-2488	165	29	}	}	PUNCT
ejpam-2488	165	30	)	)	PUNCT
ejpam-2488	165	31	or	or	CCONJ
ejpam-2488	165	32	cl∗e	cl∗e	X
ejpam-2488	165	33	(	(	PUNCT
ejpam-2488	165	34	{	{	PUNCT
ejpam-2488	165	35	x})∩	x})∩	PROPN
ejpam-2488	165	36	cl∗e	cl∗e	PROPN
ejpam-2488	165	37	(	(	PUNCT
ejpam-2488	165	38	{	{	PUNCT
ejpam-2488	165	39	y	y	NOUN
ejpam-2488	165	40	}	}	PUNCT
ejpam-2488	165	41	)	)	PUNCT
ejpam-2488	165	42	=	=	SYM
ejpam-2488	165	43	;	;	PUNCT
ejpam-2488	165	44	.	.	PUNCT
ejpam-2488	165	45	proof	proof	NOUN
ejpam-2488	165	46	.	.	PUNCT
ejpam-2488	166	1	(	(	PUNCT
ejpam-2488	166	2	i	i	NOUN
ejpam-2488	166	3	)	)	PUNCT
ejpam-2488	166	4	⇒	⇒	PROPN
ejpam-2488	166	5	(	(	PUNCT
ejpam-2488	166	6	ii	ii	PROPN
ejpam-2488	166	7	):	):	PUNCT
ejpam-2488	166	8	let	let	VERB
ejpam-2488	166	9	k	k	PROPN
ejpam-2488	166	10	∈	∈	PROPN
ejpam-2488	166	11	ei	ei	ADP
ejpam-2488	166	12	c(x	c(x	NOUN
ejpam-2488	166	13	)	)	PUNCT
ejpam-2488	166	14	and	and	CCONJ
ejpam-2488	166	15	x	x	X
ejpam-2488	166	16	/∈	/∈	PUNCT
ejpam-2488	167	1	k	k	INTJ
ejpam-2488	167	2	.	.	PUNCT
ejpam-2488	168	1	then	then	ADV
ejpam-2488	168	2	by	by	ADP
ejpam-2488	168	3	(	(	PUNCT
ejpam-2488	168	4	i	i	NOUN
ejpam-2488	168	5	)	)	PUNCT
ejpam-2488	168	6	,	,	PUNCT
ejpam-2488	168	7	cl∗e	cl∗e	PROPN
ejpam-2488	168	8	(	(	PUNCT
ejpam-2488	168	9	{	{	PUNCT
ejpam-2488	168	10	x	x	NOUN
ejpam-2488	168	11	}	}	PUNCT
ejpam-2488	168	12	)	)	PUNCT
ejpam-2488	168	13	⊂	⊂	PROPN
ejpam-2488	168	14	x\k	x\k	PROPN
ejpam-2488	168	15	.	.	PUNCT
ejpam-2488	169	1	set	set	VERB
ejpam-2488	169	2	u	u	NOUN
ejpam-2488	169	3	=	=	SYM
ejpam-2488	169	4	x\cl∗e	x\cl∗e	PROPN
ejpam-2488	169	5	(	(	PUNCT
ejpam-2488	169	6	{	{	PUNCT
ejpam-2488	169	7	x	x	NOUN
ejpam-2488	169	8	}	}	PUNCT
ejpam-2488	169	9	)	)	PUNCT
ejpam-2488	169	10	,	,	PUNCT
ejpam-2488	169	11	then	then	ADV
ejpam-2488	169	12	u	u	X
ejpam-2488	169	13	∈	∈	PROPN
ejpam-2488	169	14	eio(x	eio(x	PROPN
ejpam-2488	169	15	)	)	PUNCT
ejpam-2488	169	16	,	,	PUNCT
ejpam-2488	170	1	k	k	PROPN
ejpam-2488	170	2	⊂	⊂	PUNCT
ejpam-2488	170	3	u	u	PROPN
ejpam-2488	170	4	and	and	CCONJ
ejpam-2488	170	5	x	x	NOUN
ejpam-2488	170	6	/∈	/∈	PUNCT
ejpam-2488	170	7	u	u	INTJ
ejpam-2488	170	8	.	.	PUNCT
ejpam-2488	171	1	(	(	PUNCT
ejpam-2488	171	2	ii	ii	NOUN
ejpam-2488	171	3	)	)	PUNCT
ejpam-2488	171	4	⇒	⇒	NOUN
ejpam-2488	171	5	(	(	PUNCT
ejpam-2488	171	6	iii	iii	NOUN
ejpam-2488	171	7	):	):	PUNCT
ejpam-2488	171	8	let	let	VERB
ejpam-2488	171	9	k	k	PROPN
ejpam-2488	171	10	∈	∈	PROPN
ejpam-2488	171	11	ei	ei	ADP
ejpam-2488	171	12	c(x	c(x	NOUN
ejpam-2488	171	13	)	)	PUNCT
ejpam-2488	171	14	and	and	CCONJ
ejpam-2488	171	15	x	x	X
ejpam-2488	171	16	/∈	/∈	PUNCT
ejpam-2488	172	1	k	k	X
ejpam-2488	172	2	.	.	PUNCT
ejpam-2488	173	1	there	there	PRON
ejpam-2488	173	2	exists	exist	VERB
ejpam-2488	173	3	u	u	PROPN
ejpam-2488	173	4	∈	∈	PROPN
ejpam-2488	173	5	eio(x	eio(x	X
ejpam-2488	173	6	)	)	PUNCT
ejpam-2488	173	7	such	such	ADJ
ejpam-2488	173	8	that	that	SCONJ
ejpam-2488	173	9	k	k	PROPN
ejpam-2488	173	10	⊂	⊂	PROPN
ejpam-2488	173	11	u	u	PROPN
ejpam-2488	173	12	and	and	CCONJ
ejpam-2488	173	13	w.	w.	PROPN
ejpam-2488	173	14	al	al	PROPN
ejpam-2488	173	15	-	-	PUNCT
ejpam-2488	173	16	omeri	omeri	ADJ
ejpam-2488	173	17	,	,	PUNCT
ejpam-2488	173	18	m.	m.	NOUN
ejpam-2488	173	19	noorani	noorani	PROPN
ejpam-2488	173	20	,	,	PUNCT
ejpam-2488	173	21	a.	a.	PROPN
ejpam-2488	173	22	al	al	PROPN
ejpam-2488	173	23	-	-	PUNCT
ejpam-2488	173	24	omari	omari	PROPN
ejpam-2488	173	25	,	,	PUNCT
ejpam-2488	173	26	and	and	CCONJ
ejpam-2488	173	27	t.	t.	PROPN
ejpam-2488	173	28	noiri	noiri	PROPN
ejpam-2488	173	29	/	/	SYM
ejpam-2488	173	30	eur	eur	PROPN
ejpam-2488	173	31	.	.	PUNCT
ejpam-2488	174	1	j.	j.	PROPN
ejpam-2488	174	2	pure	pure	PROPN
ejpam-2488	174	3	appl	appl	PROPN
ejpam-2488	174	4	.	.	PROPN
ejpam-2488	174	5	math	math	PROPN
ejpam-2488	174	6	,	,	PUNCT
ejpam-2488	174	7	8	8	NUM
ejpam-2488	174	8	(	(	PUNCT
ejpam-2488	174	9	2015	2015	NUM
ejpam-2488	174	10	)	)	PUNCT
ejpam-2488	174	11	,	,	PUNCT
ejpam-2488	174	12	502	502	NUM
ejpam-2488	174	13	-	-	SYM
ejpam-2488	174	14	513	513	NUM
ejpam-2488	174	15	506	506	NUM
ejpam-2488	174	16	x	x	NOUN
ejpam-2488	174	17	/∈	/∈	PUNCT
ejpam-2488	174	18	u	u	INTJ
ejpam-2488	174	19	.	.	PUNCT
ejpam-2488	175	1	since	since	SCONJ
ejpam-2488	175	2	u	u	NOUN
ejpam-2488	175	3	∈	∈	PROPN
ejpam-2488	175	4	eio(x	eio(x	PROPN
ejpam-2488	175	5	)	)	PUNCT
ejpam-2488	175	6	,	,	PUNCT
ejpam-2488	175	7	u	u	PROPN
ejpam-2488	175	8	∩	∩	PROPN
ejpam-2488	175	9	cl∗e	cl∗e	X
ejpam-2488	175	10	(	(	PUNCT
ejpam-2488	175	11	{	{	PUNCT
ejpam-2488	175	12	x	x	NOUN
ejpam-2488	175	13	}	}	PUNCT
ejpam-2488	175	14	)	)	PUNCT
ejpam-2488	175	15	=	=	SYM
ejpam-2488	175	16	;	;	PUNCT
ejpam-2488	175	17	and	and	CCONJ
ejpam-2488	175	18	k	k	PROPN
ejpam-2488	175	19	∩	∩	PROPN
ejpam-2488	175	20	cl∗e	cl∗e	X
ejpam-2488	175	21	(	(	PUNCT
ejpam-2488	175	22	{	{	PUNCT
ejpam-2488	175	23	x	x	NOUN
ejpam-2488	175	24	}	}	PUNCT
ejpam-2488	175	25	)	)	PUNCT
ejpam-2488	175	26	=	=	SYM
ejpam-2488	175	27	;	;	PUNCT
ejpam-2488	175	28	.	.	PUNCT
ejpam-2488	175	29	(	(	PUNCT
ejpam-2488	175	30	iii	iii	X
ejpam-2488	175	31	)	)	PUNCT
ejpam-2488	175	32	⇒	⇒	NOUN
ejpam-2488	175	33	(	(	PUNCT
ejpam-2488	175	34	iv	iv	NUM
ejpam-2488	175	35	):	):	PUNCT
ejpam-2488	175	36	suppose	suppose	VERB
ejpam-2488	175	37	that	that	SCONJ
ejpam-2488	175	38	cl∗e	cl∗e	PROPN
ejpam-2488	175	39	(	(	PUNCT
ejpam-2488	175	40	{	{	PUNCT
ejpam-2488	175	41	x	x	NOUN
ejpam-2488	175	42	}	}	PUNCT
ejpam-2488	175	43	)	)	PUNCT
ejpam-2488	175	44	6=	6=	ADP
ejpam-2488	175	45	cl∗e	cl∗e	X
ejpam-2488	175	46	(	(	PUNCT
ejpam-2488	175	47	{	{	PUNCT
ejpam-2488	175	48	y	y	NOUN
ejpam-2488	175	49	}	}	PUNCT
ejpam-2488	175	50	)	)	PUNCT
ejpam-2488	175	51	for	for	ADP
ejpam-2488	175	52	distinct	distinct	ADJ
ejpam-2488	175	53	points	point	NOUN
ejpam-2488	175	54	x	x	X
ejpam-2488	175	55	,	,	PUNCT
ejpam-2488	175	56	y	y	PROPN
ejpam-2488	175	57	∈	∈	PROPN
ejpam-2488	175	58	x	x	X
ejpam-2488	175	59	.	.	PUNCT
ejpam-2488	176	1	there	there	PRON
ejpam-2488	176	2	exists	exist	VERB
ejpam-2488	176	3	k	k	PROPN
ejpam-2488	176	4	∈	∈	PROPN
ejpam-2488	176	5	cl∗e	cl∗e	PROPN
ejpam-2488	176	6	(	(	PUNCT
ejpam-2488	176	7	{	{	PUNCT
ejpam-2488	176	8	x	x	NOUN
ejpam-2488	176	9	}	}	PUNCT
ejpam-2488	176	10	)	)	PUNCT
ejpam-2488	176	11	such	such	ADJ
ejpam-2488	176	12	that	that	SCONJ
ejpam-2488	176	13	k	k	PROPN
ejpam-2488	176	14	/∈	/∈	PUNCT
ejpam-2488	176	15	cl∗e	cl∗e	PROPN
ejpam-2488	176	16	(	(	PUNCT
ejpam-2488	176	17	{	{	PUNCT
ejpam-2488	176	18	y	y	NOUN
ejpam-2488	176	19	}	}	PUNCT
ejpam-2488	176	20	)	)	PUNCT
ejpam-2488	176	21	(	(	PUNCT
ejpam-2488	176	22	or	or	CCONJ
ejpam-2488	176	23	k	k	PROPN
ejpam-2488	176	24	∈	∈	PROPN
ejpam-2488	176	25	cl∗e	cl∗e	PROPN
ejpam-2488	176	26	(	(	PUNCT
ejpam-2488	176	27	{	{	PUNCT
ejpam-2488	176	28	y	y	NOUN
ejpam-2488	176	29	}	}	PUNCT
ejpam-2488	176	30	)	)	PUNCT
ejpam-2488	176	31	such	such	ADJ
ejpam-2488	176	32	that	that	SCONJ
ejpam-2488	176	33	k	k	PROPN
ejpam-2488	176	34	/∈	/∈	PUNCT
ejpam-2488	176	35	cl∗e	cl∗e	PROPN
ejpam-2488	176	36	(	(	PUNCT
ejpam-2488	176	37	{	{	PUNCT
ejpam-2488	176	38	x	x	NOUN
ejpam-2488	176	39	}	}	PUNCT
ejpam-2488	176	40	)	)	PUNCT
ejpam-2488	176	41	.	.	PUNCT
ejpam-2488	177	1	there	there	PRON
ejpam-2488	177	2	exists	exist	VERB
ejpam-2488	177	3	v	v	ADP
ejpam-2488	177	4	∈	∈	PROPN
ejpam-2488	177	5	eio(x	eio(x	X
ejpam-2488	177	6	)	)	PUNCT
ejpam-2488	177	7	such	such	ADJ
ejpam-2488	177	8	that	that	SCONJ
ejpam-2488	177	9	y	y	PROPN
ejpam-2488	177	10	/∈	/∈	PUNCT
ejpam-2488	177	11	v	v	NOUN
ejpam-2488	178	1	and	and	CCONJ
ejpam-2488	178	2	k	k	PROPN
ejpam-2488	178	3	∈	∈	PROPN
ejpam-2488	178	4	v	v	NOUN
ejpam-2488	178	5	;	;	PUNCT
ejpam-2488	178	6	hence	hence	ADV
ejpam-2488	178	7	x	x	SYM
ejpam-2488	178	8	∈	∈	PROPN
ejpam-2488	178	9	v	v	NOUN
ejpam-2488	178	10	.	.	PUNCT
ejpam-2488	179	1	therefore	therefore	ADV
ejpam-2488	179	2	,	,	PUNCT
ejpam-2488	179	3	we	we	PRON
ejpam-2488	179	4	have	have	VERB
ejpam-2488	179	5	x	x	X
ejpam-2488	179	6	/∈	/∈	PUNCT
ejpam-2488	179	7	cl∗e	cl∗e	PROPN
ejpam-2488	179	8	(	(	PUNCT
ejpam-2488	179	9	{	{	PUNCT
ejpam-2488	179	10	y	y	NOUN
ejpam-2488	179	11	}	}	PUNCT
ejpam-2488	179	12	)	)	PUNCT
ejpam-2488	179	13	.	.	PUNCT
ejpam-2488	180	1	by	by	ADP
ejpam-2488	180	2	(	(	PUNCT
ejpam-2488	180	3	iii	iii	NOUN
ejpam-2488	180	4	)	)	PUNCT
ejpam-2488	180	5	,	,	PUNCT
ejpam-2488	180	6	we	we	PRON
ejpam-2488	180	7	obtain	obtain	VERB
ejpam-2488	180	8	cl∗e	cl∗e	PROPN
ejpam-2488	180	9	(	(	PUNCT
ejpam-2488	180	10	{	{	PUNCT
ejpam-2488	180	11	x})∩	x})∩	PROPN
ejpam-2488	180	12	cl∗e	cl∗e	PROPN
ejpam-2488	180	13	(	(	PUNCT
ejpam-2488	180	14	{	{	PUNCT
ejpam-2488	180	15	y	y	NOUN
ejpam-2488	180	16	}	}	PUNCT
ejpam-2488	180	17	)	)	PUNCT
ejpam-2488	180	18	=	=	SYM
ejpam-2488	180	19	;	;	PUNCT
ejpam-2488	180	20	.	.	PUNCT
ejpam-2488	181	1	the	the	DET
ejpam-2488	181	2	proof	proof	NOUN
ejpam-2488	181	3	for	for	ADP
ejpam-2488	181	4	otherwise	otherwise	ADV
ejpam-2488	181	5	is	be	AUX
ejpam-2488	181	6	similar	similar	ADJ
ejpam-2488	181	7	.	.	PUNCT
ejpam-2488	182	1	(	(	PUNCT
ejpam-2488	182	2	iv	iv	X
ejpam-2488	182	3	)	)	PUNCT
ejpam-2488	182	4	⇒	⇒	NOUN
ejpam-2488	182	5	(	(	PUNCT
ejpam-2488	182	6	i	i	NOUN
ejpam-2488	182	7	):	):	PUNCT
ejpam-2488	182	8	let	let	VERB
ejpam-2488	182	9	v	v	X
ejpam-2488	182	10	∈	∈	PROPN
ejpam-2488	182	11	eio(x	eio(x	X
ejpam-2488	182	12	,	,	PUNCT
ejpam-2488	182	13	x	x	NOUN
ejpam-2488	182	14	)	)	PUNCT
ejpam-2488	182	15	.	.	PUNCT
ejpam-2488	183	1	for	for	ADP
ejpam-2488	183	2	each	each	DET
ejpam-2488	183	3	y	y	PROPN
ejpam-2488	183	4	/∈	/∈	PUNCT
ejpam-2488	183	5	v	v	INTJ
ejpam-2488	183	6	,	,	PUNCT
ejpam-2488	183	7	x	x	PROPN
ejpam-2488	183	8	6=	6=	ADP
ejpam-2488	183	9	y	y	PROPN
ejpam-2488	183	10	and	and	CCONJ
ejpam-2488	183	11	x	x	PROPN
ejpam-2488	183	12	/∈	/∈	PUNCT
ejpam-2488	183	13	cl∗e	cl∗e	PROPN
ejpam-2488	183	14	(	(	PUNCT
ejpam-2488	183	15	{	{	PUNCT
ejpam-2488	183	16	y	y	NOUN
ejpam-2488	183	17	}	}	PUNCT
ejpam-2488	183	18	)	)	PUNCT
ejpam-2488	183	19	.	.	PUNCT
ejpam-2488	184	1	this	this	PRON
ejpam-2488	184	2	shows	show	VERB
ejpam-2488	184	3	that	that	SCONJ
ejpam-2488	184	4	cl∗e	cl∗e	PROPN
ejpam-2488	184	5	(	(	PUNCT
ejpam-2488	184	6	{	{	PUNCT
ejpam-2488	184	7	x	x	NOUN
ejpam-2488	184	8	}	}	PUNCT
ejpam-2488	184	9	)	)	PUNCT
ejpam-2488	184	10	6=	6=	ADP
ejpam-2488	185	1	cl∗e	cl∗e	X
ejpam-2488	185	2	(	(	PUNCT
ejpam-2488	185	3	{	{	PUNCT
ejpam-2488	185	4	y	y	NOUN
ejpam-2488	185	5	}	}	PUNCT
ejpam-2488	185	6	)	)	PUNCT
ejpam-2488	185	7	.	.	PUNCT
ejpam-2488	186	1	by	by	ADP
ejpam-2488	186	2	(	(	PUNCT
ejpam-2488	186	3	iv	iv	X
ejpam-2488	186	4	)	)	PUNCT
ejpam-2488	186	5	,	,	PUNCT
ejpam-2488	186	6	cl∗e	cl∗e	PROPN
ejpam-2488	186	7	(	(	PUNCT
ejpam-2488	186	8	{	{	PUNCT
ejpam-2488	186	9	x	x	NOUN
ejpam-2488	186	10	}	}	PUNCT
ejpam-2488	186	11	)	)	PUNCT
ejpam-2488	186	12	∩	∩	PROPN
ejpam-2488	186	13	cl∗e	cl∗e	X
ejpam-2488	186	14	(	(	PUNCT
ejpam-2488	186	15	{	{	PUNCT
ejpam-2488	186	16	y	y	NOUN
ejpam-2488	186	17	}	}	PUNCT
ejpam-2488	186	18	)	)	PUNCT
ejpam-2488	186	19	=	=	SYM
ejpam-2488	186	20	;	;	PUNCT
ejpam-2488	186	21	for	for	ADP
ejpam-2488	186	22	each	each	DET
ejpam-2488	186	23	y	y	PROPN
ejpam-2488	186	24	∈	∈	PROPN
ejpam-2488	186	25	x\v	x\v	PROPN
ejpam-2488	186	26	and	and	CCONJ
ejpam-2488	186	27	hence	hence	ADV
ejpam-2488	186	28	cl∗e	cl∗e	PROPN
ejpam-2488	186	29	(	(	PUNCT
ejpam-2488	186	30	{	{	PUNCT
ejpam-2488	186	31	x	x	NOUN
ejpam-2488	186	32	}	}	PUNCT
ejpam-2488	186	33	)	)	PUNCT
ejpam-2488	186	34	∩	∩	NOUN
ejpam-2488	186	35	(	(	PUNCT
ejpam-2488	186	36	∪y∈x\v	∪y∈x\v	PROPN
ejpam-2488	186	37	cl∗e	cl∗e	PROPN
ejpam-2488	186	38	(	(	PUNCT
ejpam-2488	186	39	{	{	PUNCT
ejpam-2488	186	40	y	y	NOUN
ejpam-2488	186	41	}	}	PUNCT
ejpam-2488	186	42	)	)	PUNCT
ejpam-2488	186	43	)	)	PUNCT
ejpam-2488	186	44	=	=	SYM
ejpam-2488	186	45	;	;	PUNCT
ejpam-2488	186	46	.	.	PUNCT
ejpam-2488	187	1	on	on	ADP
ejpam-2488	187	2	the	the	DET
ejpam-2488	187	3	other	other	ADJ
ejpam-2488	187	4	hand	hand	NOUN
ejpam-2488	187	5	,	,	PUNCT
ejpam-2488	187	6	since	since	SCONJ
ejpam-2488	187	7	v	v	NUM
ejpam-2488	187	8	∈	∈	PROPN
ejpam-2488	187	9	eio(x	eio(x	X
ejpam-2488	187	10	)	)	PUNCT
ejpam-2488	187	11	and	and	CCONJ
ejpam-2488	187	12	y	y	PROPN
ejpam-2488	187	13	∈	∈	PROPN
ejpam-2488	187	14	x\v	x\v	PROPN
ejpam-2488	187	15	,	,	PUNCT
ejpam-2488	187	16	we	we	PRON
ejpam-2488	187	17	have	have	VERB
ejpam-2488	187	18	cl∗e	cl∗e	PROPN
ejpam-2488	187	19	(	(	PUNCT
ejpam-2488	187	20	{	{	PUNCT
ejpam-2488	187	21	y	y	NOUN
ejpam-2488	187	22	}	}	PUNCT
ejpam-2488	187	23	)	)	PUNCT
ejpam-2488	188	1	⊂	⊂	PROPN
ejpam-2488	188	2	x\v	x\v	PROPN
ejpam-2488	188	3	and	and	CCONJ
ejpam-2488	188	4	hence	hence	ADV
ejpam-2488	188	5	x\v	x\v	PROPN
ejpam-2488	188	6	=	=	PROPN
ejpam-2488	188	7	∪y∈x\v	∪y∈x\v	PROPN
ejpam-2488	188	8	cl∗e	cl∗e	PROPN
ejpam-2488	188	9	(	(	PUNCT
ejpam-2488	188	10	{	{	PUNCT
ejpam-2488	188	11	y	y	NOUN
ejpam-2488	188	12	}	}	PUNCT
ejpam-2488	188	13	)	)	PUNCT
ejpam-2488	188	14	.	.	PUNCT
ejpam-2488	189	1	therefore	therefore	ADV
ejpam-2488	189	2	,	,	PUNCT
ejpam-2488	189	3	we	we	PRON
ejpam-2488	189	4	obtain	obtain	VERB
ejpam-2488	189	5	(	(	PUNCT
ejpam-2488	189	6	x\v	x\v	PROPN
ejpam-2488	189	7	)	)	PUNCT
ejpam-2488	189	8	∩	∩	PROPN
ejpam-2488	189	9	cl∗e	cl∗e	X
ejpam-2488	189	10	(	(	PUNCT
ejpam-2488	189	11	{	{	PUNCT
ejpam-2488	189	12	x	x	NOUN
ejpam-2488	189	13	}	}	PUNCT
ejpam-2488	189	14	)	)	PUNCT
ejpam-2488	189	15	=	=	SYM
ejpam-2488	189	16	;	;	PUNCT
ejpam-2488	189	17	and	and	CCONJ
ejpam-2488	189	18	cl∗e	cl∗e	X
ejpam-2488	189	19	(	(	PUNCT
ejpam-2488	189	20	{	{	PUNCT
ejpam-2488	189	21	x	x	NOUN
ejpam-2488	189	22	}	}	PUNCT
ejpam-2488	189	23	)	)	PUNCT
ejpam-2488	190	1	⊂	⊂	PROPN
ejpam-2488	190	2	v	v	X
ejpam-2488	190	3	.	.	PUNCT
ejpam-2488	191	1	this	this	PRON
ejpam-2488	191	2	shows	show	VERB
ejpam-2488	191	3	that	that	SCONJ
ejpam-2488	191	4	(	(	PUNCT
ejpam-2488	191	5	x	x	X
ejpam-2488	191	6	,	,	PUNCT
ejpam-2488	191	7	τ	τ	PROPN
ejpam-2488	191	8	,	,	PUNCT
ejpam-2488	191	9	i	i	PROPN
ejpam-2488	191	10	)	)	PUNCT
ejpam-2488	191	11	is	be	AUX
ejpam-2488	191	12	an	an	DET
ejpam-2488	191	13	e	e	NOUN
ejpam-2488	191	14	-	-	NOUN
ejpam-2488	191	15	i	i	PRON
ejpam-2488	191	16	-r0	-r0	PROPN
ejpam-2488	191	17	space	space	NOUN
ejpam-2488	191	18	.	.	PUNCT
ejpam-2488	192	1	theorem	theorem	NOUN
ejpam-2488	192	2	2	2	NUM
ejpam-2488	192	3	.	.	PUNCT
ejpam-2488	193	1	an	an	DET
ejpam-2488	193	2	ideal	ideal	ADJ
ejpam-2488	193	3	topological	topological	ADJ
ejpam-2488	193	4	space	space	NOUN
ejpam-2488	193	5	(	(	PUNCT
ejpam-2488	193	6	x	x	X
ejpam-2488	193	7	,	,	PUNCT
ejpam-2488	193	8	τ	τ	PROPN
ejpam-2488	193	9	,	,	PUNCT
ejpam-2488	193	10	i	i	PROPN
ejpam-2488	193	11	)	)	PUNCT
ejpam-2488	193	12	is	be	AUX
ejpam-2488	193	13	e	e	NOUN
ejpam-2488	193	14	-	-	PUNCT
ejpam-2488	193	15	i	i	PRON
ejpam-2488	193	16	-r0	-r0	NOUN
ejpam-2488	193	17	space	space	NOUN
ejpam-2488	194	1	if	if	SCONJ
ejpam-2488	194	2	and	and	CCONJ
ejpam-2488	194	3	only	only	ADV
ejpam-2488	194	4	if	if	SCONJ
ejpam-2488	194	5	for	for	ADP
ejpam-2488	194	6	any	any	DET
ejpam-2488	194	7	x	x	NOUN
ejpam-2488	194	8	and	and	CCONJ
ejpam-2488	194	9	y	y	PROPN
ejpam-2488	194	10	in	in	ADP
ejpam-2488	194	11	x	x	X
ejpam-2488	194	12	,	,	PUNCT
ejpam-2488	194	13	c	c	PROPN
ejpam-2488	194	14	l∗e	l∗e	PUNCT
ejpam-2488	194	15	(	(	PUNCT
ejpam-2488	194	16	{	{	PUNCT
ejpam-2488	194	17	x	x	NOUN
ejpam-2488	194	18	}	}	PUNCT
ejpam-2488	194	19	)	)	PUNCT
ejpam-2488	194	20	6=	6=	ADP
ejpam-2488	194	21	cl∗e	cl∗e	X
ejpam-2488	194	22	(	(	PUNCT
ejpam-2488	194	23	{	{	PUNCT
ejpam-2488	194	24	y	y	NOUN
ejpam-2488	194	25	}	}	PUNCT
ejpam-2488	194	26	)	)	PUNCT
ejpam-2488	194	27	implies	imply	VERB
ejpam-2488	194	28	c	c	NOUN
ejpam-2488	194	29	l∗e	l∗e	ADV
ejpam-2488	194	30	(	(	PUNCT
ejpam-2488	194	31	{	{	PUNCT
ejpam-2488	194	32	x})∩	x})∩	PROPN
ejpam-2488	194	33	cl∗e	cl∗e	X
ejpam-2488	194	34	(	(	PUNCT
ejpam-2488	194	35	{	{	PUNCT
ejpam-2488	194	36	y	y	NOUN
ejpam-2488	194	37	}	}	PUNCT
ejpam-2488	194	38	)	)	PUNCT
ejpam-2488	194	39	=	=	SYM
ejpam-2488	194	40	;	;	PUNCT
ejpam-2488	194	41	.	.	PUNCT
ejpam-2488	195	1	proof	proof	NOUN
ejpam-2488	195	2	.	.	PUNCT
ejpam-2488	196	1	let	let	VERB
ejpam-2488	196	2	(	(	PUNCT
ejpam-2488	196	3	x	x	X
ejpam-2488	196	4	,	,	PUNCT
ejpam-2488	196	5	τ	τ	PROPN
ejpam-2488	196	6	,	,	PUNCT
ejpam-2488	196	7	i	i	PROPN
ejpam-2488	196	8	)	)	PUNCT
ejpam-2488	196	9	is	be	AUX
ejpam-2488	196	10	e	e	NOUN
ejpam-2488	196	11	-	-	PUNCT
ejpam-2488	196	12	i	i	PRON
ejpam-2488	196	13	-r0	-r0	NOUN
ejpam-2488	196	14	.	.	PUNCT
ejpam-2488	197	1	by	by	ADP
ejpam-2488	197	2	proposition	proposition	NOUN
ejpam-2488	197	3	1	1	NUM
ejpam-2488	197	4	,	,	PUNCT
ejpam-2488	197	5	we	we	PRON
ejpam-2488	197	6	obtain	obtain	VERB
ejpam-2488	197	7	the	the	DET
ejpam-2488	197	8	assertion	assertion	NOUN
ejpam-2488	197	9	.	.	PUNCT
ejpam-2488	198	1	conversely	conversely	ADV
ejpam-2488	198	2	,	,	PUNCT
ejpam-2488	198	3	let	let	VERB
ejpam-2488	198	4	v	v	PRON
ejpam-2488	198	5	∈	∈	NOUN
ejpam-2488	198	6	eio(x	eio(x	X
ejpam-2488	198	7	;	;	PUNCT
ejpam-2488	198	8	x	x	X
ejpam-2488	198	9	)	)	PUNCT
ejpam-2488	198	10	.	.	PUNCT
ejpam-2488	199	1	we	we	PRON
ejpam-2488	199	2	will	will	AUX
ejpam-2488	199	3	show	show	VERB
ejpam-2488	199	4	that	that	SCONJ
ejpam-2488	199	5	cl∗e	cl∗e	PROPN
ejpam-2488	199	6	(	(	PUNCT
ejpam-2488	199	7	{	{	PUNCT
ejpam-2488	199	8	x	x	NOUN
ejpam-2488	199	9	}	}	PUNCT
ejpam-2488	199	10	)	)	PUNCT
ejpam-2488	200	1	⊂	⊂	PROPN
ejpam-2488	200	2	v	v	X
ejpam-2488	200	3	.	.	PUNCT
ejpam-2488	201	1	let	let	VERB
ejpam-2488	201	2	y	y	PROPN
ejpam-2488	201	3	∈	∈	PROPN
ejpam-2488	201	4	x\v	x\v	PROPN
ejpam-2488	201	5	.	.	PUNCT
ejpam-2488	202	1	then	then	ADV
ejpam-2488	202	2	x	x	X
ejpam-2488	202	3	6=	6=	PROPN
ejpam-2488	202	4	y	y	PROPN
ejpam-2488	202	5	and	and	CCONJ
ejpam-2488	202	6	x	x	PROPN
ejpam-2488	202	7	/∈	/∈	PUNCT
ejpam-2488	202	8	cl∗e	cl∗e	PROPN
ejpam-2488	202	9	(	(	PUNCT
ejpam-2488	202	10	{	{	PUNCT
ejpam-2488	202	11	y	y	NOUN
ejpam-2488	202	12	}	}	PUNCT
ejpam-2488	202	13	)	)	PUNCT
ejpam-2488	202	14	.	.	PUNCT
ejpam-2488	203	1	this	this	PRON
ejpam-2488	203	2	shows	show	VERB
ejpam-2488	203	3	that	that	SCONJ
ejpam-2488	203	4	cl∗e	cl∗e	PROPN
ejpam-2488	203	5	(	(	PUNCT
ejpam-2488	203	6	{	{	PUNCT
ejpam-2488	203	7	x	x	NOUN
ejpam-2488	203	8	}	}	PUNCT
ejpam-2488	203	9	)	)	PUNCT
ejpam-2488	203	10	6=	6=	ADP
ejpam-2488	204	1	cl∗e	cl∗e	X
ejpam-2488	204	2	(	(	PUNCT
ejpam-2488	204	3	{	{	PUNCT
ejpam-2488	204	4	y	y	NOUN
ejpam-2488	204	5	}	}	PUNCT
ejpam-2488	204	6	)	)	PUNCT
ejpam-2488	204	7	.	.	PUNCT
ejpam-2488	205	1	by	by	ADP
ejpam-2488	205	2	assumption	assumption	NOUN
ejpam-2488	205	3	,	,	PUNCT
ejpam-2488	205	4	cl∗e	cl∗e	PROPN
ejpam-2488	205	5	(	(	PUNCT
ejpam-2488	205	6	{	{	PUNCT
ejpam-2488	205	7	x})∩cl∗e	x})∩cl∗e	X
ejpam-2488	205	8	(	(	PUNCT
ejpam-2488	205	9	{	{	PUNCT
ejpam-2488	205	10	y	y	NOUN
ejpam-2488	205	11	}	}	PUNCT
ejpam-2488	205	12	)	)	PUNCT
ejpam-2488	205	13	=	=	SYM
ejpam-2488	205	14	;	;	PUNCT
ejpam-2488	205	15	.	.	PUNCT
ejpam-2488	206	1	hence	hence	ADV
ejpam-2488	206	2	y	y	PROPN
ejpam-2488	206	3	/∈	/∈	PUNCT
ejpam-2488	206	4	cl∗e	cl∗e	PROPN
ejpam-2488	206	5	(	(	PUNCT
ejpam-2488	206	6	{	{	PUNCT
ejpam-2488	206	7	x	x	NOUN
ejpam-2488	206	8	}	}	PUNCT
ejpam-2488	206	9	)	)	PUNCT
ejpam-2488	206	10	and	and	CCONJ
ejpam-2488	206	11	therefore	therefore	ADV
ejpam-2488	206	12	cl∗e	cl∗e	PROPN
ejpam-2488	206	13	(	(	PUNCT
ejpam-2488	206	14	{	{	PUNCT
ejpam-2488	206	15	x	x	NOUN
ejpam-2488	206	16	}	}	PUNCT
ejpam-2488	206	17	)	)	PUNCT
ejpam-2488	206	18	⊂	⊂	PROPN
ejpam-2488	206	19	v	v	X
ejpam-2488	206	20	.	.	PUNCT
ejpam-2488	207	1	theorem	theorem	NOUN
ejpam-2488	207	2	3	3	X
ejpam-2488	207	3	.	.	PUNCT
ejpam-2488	208	1	let	let	AUX
ejpam-2488	208	2	(	(	PUNCT
ejpam-2488	208	3	x	x	X
ejpam-2488	208	4	,	,	PUNCT
ejpam-2488	208	5	τ	τ	PROPN
ejpam-2488	208	6	,	,	PUNCT
ejpam-2488	208	7	i	i	PRON
ejpam-2488	208	8	)	)	PUNCT
ejpam-2488	208	9	be	be	AUX
ejpam-2488	208	10	an	an	DET
ejpam-2488	208	11	ideal	ideal	ADJ
ejpam-2488	208	12	topological	topological	ADJ
ejpam-2488	208	13	space	space	NOUN
ejpam-2488	208	14	.	.	PUNCT
ejpam-2488	209	1	then	then	ADV
ejpam-2488	209	2	the	the	DET
ejpam-2488	209	3	following	follow	VERB
ejpam-2488	209	4	properties	property	NOUN
ejpam-2488	209	5	are	be	AUX
ejpam-2488	209	6	equivalent	equivalent	ADJ
ejpam-2488	209	7	:	:	PUNCT
ejpam-2488	209	8	(	(	PUNCT
ejpam-2488	209	9	i	i	NOUN
ejpam-2488	209	10	)	)	PUNCT
ejpam-2488	209	11	(	(	PUNCT
ejpam-2488	209	12	x	x	X
ejpam-2488	209	13	,	,	PUNCT
ejpam-2488	209	14	τ	τ	PROPN
ejpam-2488	209	15	,	,	PUNCT
ejpam-2488	209	16	i	i	PROPN
ejpam-2488	209	17	)	)	PUNCT
ejpam-2488	209	18	is	be	AUX
ejpam-2488	209	19	an	an	DET
ejpam-2488	209	20	e	e	NOUN
ejpam-2488	209	21	-	-	NOUN
ejpam-2488	209	22	i	i	PRON
ejpam-2488	209	23	-r0	-r0	PROPN
ejpam-2488	209	24	space	space	NOUN
ejpam-2488	209	25	,	,	PUNCT
ejpam-2488	209	26	(	(	PUNCT
ejpam-2488	209	27	ii	ii	NOUN
ejpam-2488	209	28	)	)	PUNCT
ejpam-2488	210	1	x	x	SYM
ejpam-2488	210	2	∈	∈	PROPN
ejpam-2488	210	3	cl∗e	cl∗e	X
ejpam-2488	210	4	(	(	PUNCT
ejpam-2488	210	5	{	{	PUNCT
ejpam-2488	210	6	y	y	NOUN
ejpam-2488	210	7	}	}	PUNCT
ejpam-2488	210	8	)	)	PUNCT
ejpam-2488	210	9	if	if	SCONJ
ejpam-2488	210	10	and	and	CCONJ
ejpam-2488	210	11	only	only	ADV
ejpam-2488	210	12	if	if	SCONJ
ejpam-2488	210	13	y	y	PROPN
ejpam-2488	210	14	∈	∈	PROPN
ejpam-2488	210	15	cl∗e	cl∗e	PROPN
ejpam-2488	210	16	(	(	PUNCT
ejpam-2488	210	17	{	{	PUNCT
ejpam-2488	210	18	x	x	NOUN
ejpam-2488	210	19	}	}	PUNCT
ejpam-2488	210	20	)	)	PUNCT
ejpam-2488	210	21	for	for	ADP
ejpam-2488	210	22	any	any	DET
ejpam-2488	210	23	points	point	NOUN
ejpam-2488	210	24	x	x	PUNCT
ejpam-2488	210	25	and	and	CCONJ
ejpam-2488	210	26	y	y	PROPN
ejpam-2488	210	27	in	in	ADP
ejpam-2488	210	28	x	x	X
ejpam-2488	210	29	.	.	PUNCT
ejpam-2488	211	1	proof	proof	NOUN
ejpam-2488	211	2	.	.	PUNCT
ejpam-2488	212	1	(	(	PUNCT
ejpam-2488	212	2	i)⇒	i)⇒	PROPN
ejpam-2488	212	3	(	(	PUNCT
ejpam-2488	212	4	ii	ii	NOUN
ejpam-2488	212	5	):	):	PUNCT
ejpam-2488	212	6	assume	assume	VERB
ejpam-2488	212	7	that	that	SCONJ
ejpam-2488	212	8	(	(	PUNCT
ejpam-2488	212	9	x	x	X
ejpam-2488	212	10	,	,	PUNCT
ejpam-2488	212	11	τ	τ	PROPN
ejpam-2488	212	12	,	,	PUNCT
ejpam-2488	212	13	i	i	PROPN
ejpam-2488	212	14	)	)	PUNCT
ejpam-2488	212	15	is	be	AUX
ejpam-2488	212	16	e	e	NOUN
ejpam-2488	212	17	-	-	PUNCT
ejpam-2488	212	18	i	i	PRON
ejpam-2488	212	19	-r0	-r0	INTJ
ejpam-2488	212	20	.	.	PUNCT
ejpam-2488	213	1	let	let	VERB
ejpam-2488	213	2	x	x	X
ejpam-2488	213	3	∈	∈	PROPN
ejpam-2488	213	4	cl∗e	cl∗e	X
ejpam-2488	213	5	(	(	PUNCT
ejpam-2488	213	6	{	{	PUNCT
ejpam-2488	213	7	y	y	NOUN
ejpam-2488	213	8	}	}	PUNCT
ejpam-2488	213	9	)	)	PUNCT
ejpam-2488	213	10	and	and	CCONJ
ejpam-2488	213	11	a	a	DET
ejpam-2488	213	12	∈	∈	PROPN
ejpam-2488	213	13	eio(x	eio(x	X
ejpam-2488	213	14	,	,	PUNCT
ejpam-2488	213	15	y	y	PROPN
ejpam-2488	213	16	)	)	PUNCT
ejpam-2488	213	17	.	.	PUNCT
ejpam-2488	214	1	now	now	ADV
ejpam-2488	214	2	by	by	ADP
ejpam-2488	214	3	hypothesis	hypothesis	NOUN
ejpam-2488	214	4	,	,	PUNCT
ejpam-2488	214	5	x	x	X
ejpam-2488	214	6	∈	∈	PROPN
ejpam-2488	214	7	cl∗e	cl∗e	X
ejpam-2488	214	8	(	(	PUNCT
ejpam-2488	214	9	{	{	PUNCT
ejpam-2488	214	10	y	y	NOUN
ejpam-2488	214	11	}	}	PUNCT
ejpam-2488	214	12	)	)	PUNCT
ejpam-2488	214	13	⊂	⊂	PROPN
ejpam-2488	214	14	a	a	PRON
ejpam-2488	214	15	and	and	CCONJ
ejpam-2488	214	16	x	x	SYM
ejpam-2488	214	17	∈	∈	NOUN
ejpam-2488	214	18	a.	a.	NOUN
ejpam-2488	214	19	therefore	therefore	ADV
ejpam-2488	214	20	,	,	PUNCT
ejpam-2488	214	21	every	every	DET
ejpam-2488	214	22	e	e	NOUN
ejpam-2488	214	23	-	-	ADJ
ejpam-2488	214	24	i	i	PRON
ejpam-2488	214	25	-open	-open	NOUN
ejpam-2488	214	26	set	set	VERB
ejpam-2488	214	27	containing	contain	VERB
ejpam-2488	214	28	y	y	PROPN
ejpam-2488	214	29	contains	contain	VERB
ejpam-2488	214	30	x	x	X
ejpam-2488	214	31	.	.	PUNCT
ejpam-2488	215	1	hence	hence	ADV
ejpam-2488	215	2	y	y	PROPN
ejpam-2488	215	3	∈	∈	PROPN
ejpam-2488	215	4	cl∗e	cl∗e	X
ejpam-2488	215	5	(	(	PUNCT
ejpam-2488	215	6	{	{	PUNCT
ejpam-2488	215	7	x	x	NOUN
ejpam-2488	215	8	}	}	PUNCT
ejpam-2488	215	9	)	)	PUNCT
ejpam-2488	215	10	.	.	PUNCT
ejpam-2488	216	1	(	(	PUNCT
ejpam-2488	216	2	ii	ii	NOUN
ejpam-2488	216	3	)	)	PUNCT
ejpam-2488	216	4	⇒	⇒	NOUN
ejpam-2488	216	5	(	(	PUNCT
ejpam-2488	216	6	i	i	NOUN
ejpam-2488	216	7	):	):	PUNCT
ejpam-2488	216	8	let	let	VERB
ejpam-2488	216	9	u	u	PRON
ejpam-2488	216	10	∈	∈	PROPN
ejpam-2488	216	11	eio(x	eio(x	X
ejpam-2488	216	12	,	,	PUNCT
ejpam-2488	216	13	x	x	NOUN
ejpam-2488	216	14	)	)	PUNCT
ejpam-2488	216	15	.	.	PUNCT
ejpam-2488	217	1	if	if	SCONJ
ejpam-2488	217	2	y	y	PROPN
ejpam-2488	217	3	/∈	/∈	PUNCT
ejpam-2488	217	4	u	u	PROPN
ejpam-2488	217	5	,	,	PUNCT
ejpam-2488	217	6	then	then	ADV
ejpam-2488	217	7	x	x	X
ejpam-2488	217	8	/∈	/∈	PUNCT
ejpam-2488	217	9	cl∗e	cl∗e	PROPN
ejpam-2488	217	10	(	(	PUNCT
ejpam-2488	217	11	{	{	PUNCT
ejpam-2488	217	12	y	y	NOUN
ejpam-2488	217	13	}	}	PUNCT
ejpam-2488	217	14	)	)	PUNCT
ejpam-2488	217	15	and	and	CCONJ
ejpam-2488	217	16	hence	hence	ADV
ejpam-2488	217	17	y	y	PROPN
ejpam-2488	217	18	/∈	/∈	PUNCT
ejpam-2488	217	19	cl∗e	cl∗e	PROPN
ejpam-2488	217	20	(	(	PUNCT
ejpam-2488	217	21	{	{	PUNCT
ejpam-2488	217	22	x	x	NOUN
ejpam-2488	217	23	}	}	PUNCT
ejpam-2488	217	24	)	)	PUNCT
ejpam-2488	217	25	.	.	PUNCT
ejpam-2488	218	1	this	this	PRON
ejpam-2488	218	2	implies	imply	VERB
ejpam-2488	218	3	that	that	SCONJ
ejpam-2488	218	4	cl∗e	cl∗e	PROPN
ejpam-2488	218	5	(	(	PUNCT
ejpam-2488	218	6	{	{	PUNCT
ejpam-2488	218	7	x	x	NOUN
ejpam-2488	218	8	}	}	PUNCT
ejpam-2488	218	9	)	)	PUNCT
ejpam-2488	219	1	⊂	⊂	PROPN
ejpam-2488	219	2	u	u	PROPN
ejpam-2488	219	3	.	.	PUNCT
ejpam-2488	220	1	hence	hence	ADV
ejpam-2488	220	2	(	(	PUNCT
ejpam-2488	220	3	x	x	X
ejpam-2488	220	4	,	,	PUNCT
ejpam-2488	220	5	τ	τ	PROPN
ejpam-2488	220	6	,	,	PUNCT
ejpam-2488	220	7	i	i	PROPN
ejpam-2488	220	8	)	)	PUNCT
ejpam-2488	220	9	is	be	AUX
ejpam-2488	220	10	e	e	NOUN
ejpam-2488	220	11	-	-	PUNCT
ejpam-2488	220	12	i	i	PRON
ejpam-2488	220	13	-r0	-r0	PROPN
ejpam-2488	220	14	theorem	theorem	VERB
ejpam-2488	220	15	4	4	NUM
ejpam-2488	220	16	.	.	X
ejpam-2488	220	17	for	for	ADP
ejpam-2488	220	18	an	an	DET
ejpam-2488	220	19	ideal	ideal	ADJ
ejpam-2488	220	20	topological	topological	ADJ
ejpam-2488	220	21	space	space	NOUN
ejpam-2488	220	22	(	(	PUNCT
ejpam-2488	220	23	x	x	X
ejpam-2488	220	24	,	,	PUNCT
ejpam-2488	220	25	τ	τ	PROPN
ejpam-2488	220	26	,	,	PUNCT
ejpam-2488	220	27	i	i	PROPN
ejpam-2488	220	28	)	)	PUNCT
ejpam-2488	220	29	,	,	PUNCT
ejpam-2488	220	30	the	the	DET
ejpam-2488	220	31	following	follow	VERB
ejpam-2488	220	32	properties	property	NOUN
ejpam-2488	220	33	are	be	AUX
ejpam-2488	220	34	equivalent	equivalent	ADJ
ejpam-2488	220	35	:	:	PUNCT
ejpam-2488	220	36	(	(	PUNCT
ejpam-2488	220	37	i	i	NOUN
ejpam-2488	220	38	)	)	PUNCT
ejpam-2488	220	39	(	(	PUNCT
ejpam-2488	220	40	x	x	X
ejpam-2488	220	41	,	,	PUNCT
ejpam-2488	220	42	τ	τ	PROPN
ejpam-2488	220	43	,	,	PUNCT
ejpam-2488	220	44	i	i	PROPN
ejpam-2488	220	45	)	)	PUNCT
ejpam-2488	220	46	is	be	AUX
ejpam-2488	220	47	an	an	DET
ejpam-2488	220	48	e	e	NOUN
ejpam-2488	220	49	-	-	NOUN
ejpam-2488	220	50	i	i	PRON
ejpam-2488	220	51	-r0	-r0	NOUN
ejpam-2488	220	52	space	space	NOUN
ejpam-2488	220	53	;	;	PUNCT
ejpam-2488	220	54	(	(	PUNCT
ejpam-2488	220	55	ii	ii	NOUN
ejpam-2488	220	56	)	)	PUNCT
ejpam-2488	220	57	for	for	ADP
ejpam-2488	220	58	any	any	DET
ejpam-2488	220	59	nonempty	nonempty	ADJ
ejpam-2488	220	60	set	set	VERB
ejpam-2488	220	61	s	s	PROPN
ejpam-2488	220	62	of	of	ADP
ejpam-2488	220	63	x	x	X
ejpam-2488	220	64	and	and	CCONJ
ejpam-2488	220	65	any	any	DET
ejpam-2488	220	66	g	g	PROPN
ejpam-2488	220	67	∈	∈	PROPN
ejpam-2488	220	68	eio(x	eio(x	X
ejpam-2488	220	69	)	)	PUNCT
ejpam-2488	220	70	such	such	ADJ
ejpam-2488	220	71	that	that	DET
ejpam-2488	220	72	s	s	PART
ejpam-2488	220	73	∩	∩	NOUN
ejpam-2488	220	74	g	g	PROPN
ejpam-2488	220	75	6=	6=	PROPN
ejpam-2488	220	76	;	;	PUNCT
ejpam-2488	220	77	,	,	PUNCT
ejpam-2488	220	78	there	there	PRON
ejpam-2488	220	79	exists	exist	VERB
ejpam-2488	220	80	k	k	PROPN
ejpam-2488	220	81	∈	∈	PROPN
ejpam-2488	220	82	ei	ei	X
ejpam-2488	220	83	c(x	c(x	NOUN
ejpam-2488	220	84	)	)	PUNCT
ejpam-2488	220	85	such	such	ADJ
ejpam-2488	220	86	that	that	PRON
ejpam-2488	220	87	s	s	NOUN
ejpam-2488	220	88	∩	∩	PROPN
ejpam-2488	220	89	k	k	PROPN
ejpam-2488	220	90	6=	6=	PROPN
ejpam-2488	220	91	;	;	PUNCT
ejpam-2488	220	92	and	and	CCONJ
ejpam-2488	220	93	k	k	PROPN
ejpam-2488	220	94	⊂	⊂	PROPN
ejpam-2488	220	95	g	g	PROPN
ejpam-2488	220	96	;	;	PUNCT
ejpam-2488	220	97	(	(	PUNCT
ejpam-2488	220	98	iii	iii	NOUN
ejpam-2488	220	99	)	)	PUNCT
ejpam-2488	220	100	for	for	ADP
ejpam-2488	220	101	any	any	DET
ejpam-2488	220	102	g	g	PROPN
ejpam-2488	220	103	∈	∈	PROPN
ejpam-2488	220	104	eio(x	eio(x	PROPN
ejpam-2488	220	105	)	)	PUNCT
ejpam-2488	220	106	,	,	PUNCT
ejpam-2488	220	107	g	g	PROPN
ejpam-2488	220	108	=	=	SYM
ejpam-2488	220	109	∪{k	∪{k	PROPN
ejpam-2488	220	110	∈	∈	NOUN
ejpam-2488	220	111	ei	ei	ADP
ejpam-2488	220	112	c(x	c(x	NOUN
ejpam-2488	220	113	)	)	PUNCT
ejpam-2488	220	114	|k	|k	VERB
ejpam-2488	221	1	⊂	⊂	X
ejpam-2488	221	2	g	g	NOUN
ejpam-2488	221	3	}	}	PUNCT
ejpam-2488	221	4	;	;	PUNCT
ejpam-2488	221	5	(	(	PUNCT
ejpam-2488	221	6	iv	iv	X
ejpam-2488	221	7	)	)	PUNCT
ejpam-2488	221	8	for	for	ADP
ejpam-2488	221	9	any	any	DET
ejpam-2488	221	10	k	k	PROPN
ejpam-2488	221	11	∈	∈	PROPN
ejpam-2488	221	12	ei	ei	ADP
ejpam-2488	221	13	c(x	c(x	NOUN
ejpam-2488	221	14	)	)	PUNCT
ejpam-2488	221	15	,	,	PUNCT
ejpam-2488	221	16	k	k	X
ejpam-2488	221	17	=	=	X
ejpam-2488	221	18	∩{g	∩{g	PROPN
ejpam-2488	221	19	∈	∈	PROPN
ejpam-2488	221	20	eio(x	eio(x	X
ejpam-2488	221	21	)	)	PUNCT
ejpam-2488	221	22	|k	|k	NOUN
ejpam-2488	222	1	⊂	⊂	X
ejpam-2488	222	2	g	g	NOUN
ejpam-2488	222	3	}	}	PUNCT
ejpam-2488	222	4	;	;	PUNCT
ejpam-2488	222	5	(	(	PUNCT
ejpam-2488	222	6	v	v	NOUN
ejpam-2488	222	7	)	)	PUNCT
ejpam-2488	222	8	for	for	ADP
ejpam-2488	222	9	any	any	DET
ejpam-2488	222	10	x	x	SYM
ejpam-2488	222	11	∈	∈	PROPN
ejpam-2488	222	12	x	x	X
ejpam-2488	222	13	,	,	PUNCT
ejpam-2488	222	14	cl∗e	cl∗e	PROPN
ejpam-2488	222	15	(	(	PUNCT
ejpam-2488	222	16	{	{	PUNCT
ejpam-2488	222	17	x	x	NOUN
ejpam-2488	222	18	}	}	PUNCT
ejpam-2488	222	19	)	)	PUNCT
ejpam-2488	223	1	⊂	⊂	PROPN
ejpam-2488	223	2	ieker({x	ieker({x	PROPN
ejpam-2488	223	3	}	}	PUNCT
ejpam-2488	223	4	)	)	PUNCT
ejpam-2488	223	5	.	.	PUNCT
ejpam-2488	224	1	w.	w.	PROPN
ejpam-2488	224	2	al	al	PROPN
ejpam-2488	224	3	-	-	PUNCT
ejpam-2488	224	4	omeri	omeri	ADJ
ejpam-2488	224	5	,	,	PUNCT
ejpam-2488	224	6	m.	m.	NOUN
ejpam-2488	224	7	noorani	noorani	PROPN
ejpam-2488	224	8	,	,	PUNCT
ejpam-2488	224	9	a.	a.	PROPN
ejpam-2488	224	10	al	al	PROPN
ejpam-2488	224	11	-	-	PUNCT
ejpam-2488	224	12	omari	omari	PROPN
ejpam-2488	224	13	,	,	PUNCT
ejpam-2488	224	14	and	and	CCONJ
ejpam-2488	224	15	t.	t.	PROPN
ejpam-2488	224	16	noiri	noiri	PROPN
ejpam-2488	224	17	/	/	SYM
ejpam-2488	224	18	eur	eur	PROPN
ejpam-2488	224	19	.	.	PUNCT
ejpam-2488	225	1	j.	j.	PROPN
ejpam-2488	225	2	pure	pure	PROPN
ejpam-2488	225	3	appl	appl	PROPN
ejpam-2488	225	4	.	.	PROPN
ejpam-2488	225	5	math	math	PROPN
ejpam-2488	225	6	,	,	PUNCT
ejpam-2488	225	7	8	8	NUM
ejpam-2488	225	8	(	(	PUNCT
ejpam-2488	225	9	2015	2015	NUM
ejpam-2488	225	10	)	)	PUNCT
ejpam-2488	225	11	,	,	PUNCT
ejpam-2488	225	12	502	502	NUM
ejpam-2488	225	13	-	-	SYM
ejpam-2488	225	14	513	513	NUM
ejpam-2488	225	15	507	507	NUM
ejpam-2488	225	16	proof	proof	NOUN
ejpam-2488	225	17	.	.	PUNCT
ejpam-2488	226	1	(	(	PUNCT
ejpam-2488	226	2	i)⇒	i)⇒	PROPN
ejpam-2488	226	3	(	(	PUNCT
ejpam-2488	226	4	ii):let	ii):let	PROPN
ejpam-2488	226	5	s	s	PRON
ejpam-2488	226	6	be	be	AUX
ejpam-2488	226	7	a	a	DET
ejpam-2488	226	8	nonempty	nonempty	ADJ
ejpam-2488	226	9	set	set	NOUN
ejpam-2488	226	10	of	of	ADP
ejpam-2488	226	11	x	x	X
ejpam-2488	226	12	and	and	CCONJ
ejpam-2488	226	13	g	g	PROPN
ejpam-2488	226	14	∈	∈	PROPN
ejpam-2488	226	15	eio(x	eio(x	X
ejpam-2488	226	16	)	)	PUNCT
ejpam-2488	226	17	such	such	ADJ
ejpam-2488	226	18	that	that	SCONJ
ejpam-2488	226	19	s∩g	s∩g	NOUN
ejpam-2488	226	20	6=	6=	NUM
ejpam-2488	226	21	;	;	PUNCT
ejpam-2488	226	22	.	.	PUNCT
ejpam-2488	227	1	there	there	PRON
ejpam-2488	227	2	exists	exist	VERB
ejpam-2488	227	3	x	x	X
ejpam-2488	227	4	∈	∈	PROPN
ejpam-2488	227	5	s	s	PART
ejpam-2488	227	6	∩	∩	ADJ
ejpam-2488	227	7	g.	g.	NOUN
ejpam-2488	227	8	since	since	SCONJ
ejpam-2488	227	9	x	x	PROPN
ejpam-2488	227	10	∈	∈	PROPN
ejpam-2488	227	11	g	g	PROPN
ejpam-2488	227	12	∈	∈	PROPN
ejpam-2488	227	13	eio(x	eio(x	PROPN
ejpam-2488	227	14	)	)	PUNCT
ejpam-2488	227	15	,	,	PUNCT
ejpam-2488	227	16	it	it	PRON
ejpam-2488	227	17	follows	follow	VERB
ejpam-2488	227	18	that	that	SCONJ
ejpam-2488	227	19	cl∗e	cl∗e	PROPN
ejpam-2488	227	20	(	(	PUNCT
ejpam-2488	227	21	{	{	PUNCT
ejpam-2488	227	22	x	x	NOUN
ejpam-2488	227	23	}	}	PUNCT
ejpam-2488	227	24	)	)	PUNCT
ejpam-2488	228	1	⊂	⊂	PROPN
ejpam-2488	228	2	g.	g.	PROPN
ejpam-2488	228	3	take	take	VERB
ejpam-2488	228	4	k	k	PROPN
ejpam-2488	228	5	=	=	PUNCT
ejpam-2488	228	6	cl∗e	cl∗e	PROPN
ejpam-2488	228	7	(	(	PUNCT
ejpam-2488	228	8	{	{	PUNCT
ejpam-2488	228	9	x	x	NOUN
ejpam-2488	228	10	}	}	PUNCT
ejpam-2488	228	11	)	)	PUNCT
ejpam-2488	228	12	,	,	PUNCT
ejpam-2488	228	13	then	then	ADV
ejpam-2488	228	14	k	k	PROPN
ejpam-2488	228	15	∈	∈	PROPN
ejpam-2488	228	16	ei	ei	ADP
ejpam-2488	228	17	c(x	c(x	NOUN
ejpam-2488	228	18	)	)	PUNCT
ejpam-2488	228	19	,	,	PUNCT
ejpam-2488	228	20	k	k	PROPN
ejpam-2488	228	21	⊂	⊂	PROPN
ejpam-2488	228	22	g	g	PROPN
ejpam-2488	228	23	and	and	CCONJ
ejpam-2488	228	24	s	s	X
ejpam-2488	228	25	∩	∩	PROPN
ejpam-2488	228	26	k	k	PROPN
ejpam-2488	228	27	6=	6=	PROPN
ejpam-2488	228	28	;	;	PUNCT
ejpam-2488	228	29	.	.	PUNCT
ejpam-2488	229	1	(	(	PUNCT
ejpam-2488	229	2	ii)⇒	ii)⇒	X
ejpam-2488	229	3	(	(	PUNCT
ejpam-2488	229	4	iii	iii	NOUN
ejpam-2488	229	5	):	):	PUNCT
ejpam-2488	229	6	let	let	VERB
ejpam-2488	229	7	g	g	PROPN
ejpam-2488	229	8	∈	∈	PROPN
ejpam-2488	229	9	eio(x	eio(x	PROPN
ejpam-2488	229	10	)	)	PUNCT
ejpam-2488	229	11	.	.	PUNCT
ejpam-2488	230	1	we	we	PRON
ejpam-2488	230	2	have	have	VERB
ejpam-2488	230	3	g	g	PROPN
ejpam-2488	230	4	⊃	⊃	NOUN
ejpam-2488	230	5	∪{k	∪{k	ADP
ejpam-2488	230	6	∈	∈	PROPN
ejpam-2488	230	7	ei	ei	ADP
ejpam-2488	230	8	c(x	c(x	NOUN
ejpam-2488	230	9	)	)	PUNCT
ejpam-2488	230	10	|k	|k	VERB
ejpam-2488	231	1	⊂	⊂	X
ejpam-2488	231	2	g	g	NOUN
ejpam-2488	231	3	}	}	PUNCT
ejpam-2488	231	4	.	.	PUNCT
ejpam-2488	232	1	let	let	VERB
ejpam-2488	232	2	x	x	PRON
ejpam-2488	232	3	be	be	AUX
ejpam-2488	232	4	any	any	DET
ejpam-2488	232	5	point	point	NOUN
ejpam-2488	232	6	of	of	ADP
ejpam-2488	232	7	g.	g.	PROPN
ejpam-2488	232	8	by	by	ADP
ejpam-2488	232	9	(	(	PUNCT
ejpam-2488	232	10	ii	ii	NOUN
ejpam-2488	232	11	)	)	PUNCT
ejpam-2488	232	12	there	there	PRON
ejpam-2488	232	13	exists	exist	VERB
ejpam-2488	232	14	k	k	PROPN
ejpam-2488	232	15	∈	∈	PROPN
ejpam-2488	232	16	ei	ei	X
ejpam-2488	232	17	c(x	c(x	NOUN
ejpam-2488	232	18	)	)	PUNCT
ejpam-2488	232	19	such	such	ADJ
ejpam-2488	232	20	that	that	SCONJ
ejpam-2488	232	21	x	x	SYM
ejpam-2488	232	22	∈	∈	PROPN
ejpam-2488	232	23	k	k	PROPN
ejpam-2488	232	24	and	and	CCONJ
ejpam-2488	232	25	k	k	PROPN
ejpam-2488	232	26	⊂	⊂	PROPN
ejpam-2488	232	27	g.	g.	PROPN
ejpam-2488	232	28	thus	thus	ADV
ejpam-2488	232	29	,	,	PUNCT
ejpam-2488	232	30	we	we	PRON
ejpam-2488	232	31	have	have	VERB
ejpam-2488	232	32	x	x	X
ejpam-2488	232	33	∈	∈	PROPN
ejpam-2488	232	34	k	k	PROPN
ejpam-2488	232	35	⊂	⊂	PUNCT
ejpam-2488	232	36	∪{k	∪{k	PROPN
ejpam-2488	232	37	∈	∈	PROPN
ejpam-2488	232	38	ei	ei	ADP
ejpam-2488	232	39	c(x	c(x	NOUN
ejpam-2488	232	40	)	)	PUNCT
ejpam-2488	232	41	|k	|k	VERB
ejpam-2488	233	1	⊂	⊂	X
ejpam-2488	233	2	g	g	NOUN
ejpam-2488	233	3	}	}	PUNCT
ejpam-2488	233	4	and	and	CCONJ
ejpam-2488	233	5	hence	hence	ADV
ejpam-2488	233	6	g	g	PROPN
ejpam-2488	233	7	=	=	SYM
ejpam-2488	233	8	∪{k	∪{k	PROPN
ejpam-2488	233	9	∈	∈	NOUN
ejpam-2488	233	10	ei	ei	ADP
ejpam-2488	233	11	c(x	c(x	NOUN
ejpam-2488	233	12	)	)	PUNCT
ejpam-2488	233	13	|k	|k	VERB
ejpam-2488	234	1	⊂	⊂	X
ejpam-2488	234	2	g	g	NOUN
ejpam-2488	234	3	}	}	PUNCT
ejpam-2488	234	4	.	.	PUNCT
ejpam-2488	235	1	(	(	PUNCT
ejpam-2488	235	2	iii)⇒	iii)⇒	PROPN
ejpam-2488	235	3	(	(	PUNCT
ejpam-2488	235	4	iv	iv	NUM
ejpam-2488	235	5	):	):	PUNCT
ejpam-2488	235	6	this	this	PRON
ejpam-2488	235	7	is	be	AUX
ejpam-2488	235	8	obvious	obvious	ADJ
ejpam-2488	235	9	.	.	PUNCT
ejpam-2488	236	1	(	(	PUNCT
ejpam-2488	236	2	iv)⇒	iv)⇒	X
ejpam-2488	236	3	(	(	PUNCT
ejpam-2488	236	4	v	v	NOUN
ejpam-2488	236	5	):	):	PUNCT
ejpam-2488	236	6	let	let	VERB
ejpam-2488	236	7	x	x	PRON
ejpam-2488	236	8	be	be	AUX
ejpam-2488	236	9	any	any	DET
ejpam-2488	236	10	point	point	NOUN
ejpam-2488	236	11	of	of	ADP
ejpam-2488	236	12	x	x	PUNCT
ejpam-2488	236	13	and	and	CCONJ
ejpam-2488	236	14	y	y	PROPN
ejpam-2488	236	15	/∈	/∈	PUNCT
ejpam-2488	237	1	ieker({x	ieker({x	PROPN
ejpam-2488	237	2	}	}	PUNCT
ejpam-2488	237	3	)	)	PUNCT
ejpam-2488	237	4	.	.	PUNCT
ejpam-2488	238	1	there	there	PRON
ejpam-2488	238	2	exists	exist	VERB
ejpam-2488	238	3	v	v	ADP
ejpam-2488	238	4	∈	∈	PROPN
ejpam-2488	238	5	eio(x	eio(x	X
ejpam-2488	238	6	)	)	PUNCT
ejpam-2488	238	7	such	such	ADJ
ejpam-2488	238	8	that	that	SCONJ
ejpam-2488	238	9	x	x	SYM
ejpam-2488	238	10	∈	∈	PROPN
ejpam-2488	238	11	v	v	NOUN
ejpam-2488	238	12	and	and	CCONJ
ejpam-2488	238	13	y	y	PROPN
ejpam-2488	238	14	/∈	/∈	PUNCT
ejpam-2488	239	1	v	v	ADJ
ejpam-2488	239	2	;	;	PUNCT
ejpam-2488	239	3	hence	hence	ADV
ejpam-2488	239	4	cl∗e	cl∗e	PROPN
ejpam-2488	239	5	(	(	PUNCT
ejpam-2488	239	6	{	{	PUNCT
ejpam-2488	239	7	y})∩	y})∩	NOUN
ejpam-2488	239	8	v	v	NOUN
ejpam-2488	239	9	=	=	PUNCT
ejpam-2488	239	10	;	;	PUNCT
ejpam-2488	239	11	.	.	PUNCT
ejpam-2488	240	1	by	by	ADP
ejpam-2488	240	2	(	(	PUNCT
ejpam-2488	240	3	iv	iv	X
ejpam-2488	240	4	)	)	PUNCT
ejpam-2488	240	5	,	,	PUNCT
ejpam-2488	241	1	[	[	X
ejpam-2488	241	2	∩{g	∩{g	X
ejpam-2488	241	3	∈	∈	PROPN
ejpam-2488	241	4	eio(x	eio(x	X
ejpam-2488	241	5	)	)	PUNCT
ejpam-2488	241	6	|cl∗e	|cl∗e	PROPN
ejpam-2488	241	7	(	(	PUNCT
ejpam-2488	241	8	{	{	PUNCT
ejpam-2488	241	9	y	y	NOUN
ejpam-2488	241	10	}	}	PUNCT
ejpam-2488	241	11	)	)	PUNCT
ejpam-2488	242	1	⊂	⊂	PROPN
ejpam-2488	242	2	g}]∩	g}]∩	X
ejpam-2488	242	3	v	v	X
ejpam-2488	242	4	=	=	PUNCT
ejpam-2488	242	5	;	;	PUNCT
ejpam-2488	242	6	and	and	CCONJ
ejpam-2488	242	7	there	there	PRON
ejpam-2488	242	8	exists	exist	VERB
ejpam-2488	242	9	g	g	PROPN
ejpam-2488	242	10	∈	∈	PROPN
ejpam-2488	242	11	eio(x	eio(x	X
ejpam-2488	242	12	)	)	PUNCT
ejpam-2488	242	13	such	such	ADJ
ejpam-2488	242	14	that	that	DET
ejpam-2488	242	15	x	x	PROPN
ejpam-2488	242	16	/∈	/∈	NOUN
ejpam-2488	242	17	g	g	PROPN
ejpam-2488	242	18	and	and	CCONJ
ejpam-2488	242	19	cl∗e	cl∗e	PROPN
ejpam-2488	242	20	(	(	PUNCT
ejpam-2488	242	21	{	{	PUNCT
ejpam-2488	242	22	y	y	NOUN
ejpam-2488	242	23	}	}	PUNCT
ejpam-2488	242	24	)	)	PUNCT
ejpam-2488	243	1	⊂	⊂	PROPN
ejpam-2488	243	2	g.	g.	PROPN
ejpam-2488	243	3	hence	hence	ADV
ejpam-2488	243	4	,	,	PUNCT
ejpam-2488	243	5	cl∗e	cl∗e	PROPN
ejpam-2488	243	6	(	(	PUNCT
ejpam-2488	243	7	{	{	PUNCT
ejpam-2488	243	8	x	x	NOUN
ejpam-2488	243	9	}	}	PUNCT
ejpam-2488	243	10	)	)	PUNCT
ejpam-2488	243	11	∩	∩	PROPN
ejpam-2488	243	12	g	g	NOUN
ejpam-2488	243	13	=	=	PUNCT
ejpam-2488	243	14	;	;	PUNCT
ejpam-2488	243	15	and	and	CCONJ
ejpam-2488	243	16	y	y	PROPN
ejpam-2488	243	17	/∈	/∈	PUNCT
ejpam-2488	244	1	cl∗e	cl∗e	PROPN
ejpam-2488	244	2	(	(	PUNCT
ejpam-2488	244	3	{	{	PUNCT
ejpam-2488	244	4	x	x	NOUN
ejpam-2488	244	5	}	}	PUNCT
ejpam-2488	244	6	)	)	PUNCT
ejpam-2488	244	7	.	.	PUNCT
ejpam-2488	245	1	thus	thus	ADV
ejpam-2488	245	2	,	,	PUNCT
ejpam-2488	245	3	cl∗e	cl∗e	PROPN
ejpam-2488	245	4	(	(	PUNCT
ejpam-2488	245	5	{	{	PUNCT
ejpam-2488	245	6	x	x	NOUN
ejpam-2488	245	7	}	}	PUNCT
ejpam-2488	245	8	)	)	PUNCT
ejpam-2488	245	9	⊂	⊂	PROPN
ejpam-2488	245	10	ieker({x	ieker({x	PROPN
ejpam-2488	245	11	}	}	PUNCT
ejpam-2488	245	12	)	)	PUNCT
ejpam-2488	245	13	.	.	PUNCT
ejpam-2488	246	1	(	(	PUNCT
ejpam-2488	246	2	v	v	NOUN
ejpam-2488	246	3	)	)	PUNCT
ejpam-2488	246	4	⇒	⇒	NOUN
ejpam-2488	246	5	(	(	PUNCT
ejpam-2488	246	6	i	i	NOUN
ejpam-2488	246	7	):	):	PUNCT
ejpam-2488	246	8	let	let	VERB
ejpam-2488	246	9	g	g	PROPN
ejpam-2488	246	10	∈	∈	PROPN
ejpam-2488	246	11	eio(x	eio(x	X
ejpam-2488	246	12	)	)	PUNCT
ejpam-2488	246	13	and	and	CCONJ
ejpam-2488	246	14	x	x	PUNCT
ejpam-2488	246	15	∈	∈	PROPN
ejpam-2488	246	16	g.	g.	NOUN
ejpam-2488	246	17	let	let	VERB
ejpam-2488	246	18	y	y	PROPN
ejpam-2488	246	19	∈	∈	VERB
ejpam-2488	246	20	ieker({x	ieker({x	PROPN
ejpam-2488	246	21	}	}	PUNCT
ejpam-2488	246	22	)	)	PUNCT
ejpam-2488	246	23	.	.	PUNCT
ejpam-2488	247	1	we	we	PRON
ejpam-2488	247	2	have	have	VERB
ejpam-2488	247	3	x	x	X
ejpam-2488	247	4	∈	∈	PROPN
ejpam-2488	247	5	cl∗e	cl∗e	X
ejpam-2488	247	6	(	(	PUNCT
ejpam-2488	247	7	{	{	PUNCT
ejpam-2488	247	8	y	y	NOUN
ejpam-2488	247	9	}	}	PUNCT
ejpam-2488	247	10	)	)	PUNCT
ejpam-2488	247	11	and	and	CCONJ
ejpam-2488	247	12	y	y	PROPN
ejpam-2488	247	13	∈	∈	PROPN
ejpam-2488	247	14	g.	g.	NOUN
ejpam-2488	248	1	it	it	PRON
ejpam-2488	248	2	follows	follow	VERB
ejpam-2488	248	3	that	that	SCONJ
ejpam-2488	248	4	ieker({x	ieker({x	PRON
ejpam-2488	248	5	}	}	PUNCT
ejpam-2488	248	6	)	)	PUNCT
ejpam-2488	249	1	⊂	⊂	PROPN
ejpam-2488	250	1	g.	g.	PROPN
ejpam-2488	251	1	thus	thus	ADV
ejpam-2488	251	2	,	,	PUNCT
ejpam-2488	251	3	we	we	PRON
ejpam-2488	251	4	obtain	obtain	VERB
ejpam-2488	251	5	x	x	PUNCT
ejpam-2488	251	6	∈	∈	PROPN
ejpam-2488	251	7	cl∗e	cl∗e	X
ejpam-2488	251	8	(	(	PUNCT
ejpam-2488	251	9	{	{	PUNCT
ejpam-2488	251	10	x	x	NOUN
ejpam-2488	251	11	}	}	PUNCT
ejpam-2488	251	12	)	)	PUNCT
ejpam-2488	251	13	⊂	⊂	PROPN
ejpam-2488	251	14	ieker({x	ieker({x	PROPN
ejpam-2488	251	15	}	}	PUNCT
ejpam-2488	251	16	)	)	PUNCT
ejpam-2488	252	1	⊂	⊂	PROPN
ejpam-2488	252	2	g.	g.	PROPN
ejpam-2488	252	3	this	this	PRON
ejpam-2488	252	4	shows	show	VERB
ejpam-2488	252	5	that	that	SCONJ
ejpam-2488	252	6	(	(	PUNCT
ejpam-2488	252	7	x	x	X
ejpam-2488	252	8	,	,	PUNCT
ejpam-2488	252	9	τ	τ	PROPN
ejpam-2488	252	10	,	,	PUNCT
ejpam-2488	252	11	i	i	PROPN
ejpam-2488	252	12	)	)	PUNCT
ejpam-2488	252	13	is	be	AUX
ejpam-2488	252	14	an	an	DET
ejpam-2488	252	15	e	e	NOUN
ejpam-2488	252	16	-	-	NOUN
ejpam-2488	252	17	i	i	PRON
ejpam-2488	252	18	-r0	-r0	PROPN
ejpam-2488	252	19	space	space	NOUN
ejpam-2488	252	20	.	.	PUNCT
ejpam-2488	253	1	theorem	theorem	VERB
ejpam-2488	253	2	5	5	NUM
ejpam-2488	253	3	.	.	PUNCT
ejpam-2488	254	1	an	an	DET
ejpam-2488	254	2	ideal	ideal	ADJ
ejpam-2488	254	3	topological	topological	ADJ
ejpam-2488	254	4	space	space	NOUN
ejpam-2488	254	5	(	(	PUNCT
ejpam-2488	254	6	x	x	X
ejpam-2488	254	7	,	,	PUNCT
ejpam-2488	254	8	τ	τ	PROPN
ejpam-2488	254	9	,	,	PUNCT
ejpam-2488	254	10	i	i	PROPN
ejpam-2488	254	11	)	)	PUNCT
ejpam-2488	254	12	is	be	AUX
ejpam-2488	254	13	e	e	NOUN
ejpam-2488	254	14	-	-	PUNCT
ejpam-2488	254	15	i	i	PRON
ejpam-2488	254	16	-r0	-r0	INTJ
ejpam-2488	255	1	if	if	SCONJ
ejpam-2488	255	2	and	and	CCONJ
ejpam-2488	255	3	only	only	ADV
ejpam-2488	255	4	if	if	SCONJ
ejpam-2488	255	5	for	for	ADP
ejpam-2488	255	6	any	any	DET
ejpam-2488	255	7	pair	pair	NOUN
ejpam-2488	255	8	of	of	ADP
ejpam-2488	255	9	points	point	NOUN
ejpam-2488	255	10	x	x	PUNCT
ejpam-2488	255	11	and	and	CCONJ
ejpam-2488	255	12	y	y	PROPN
ejpam-2488	255	13	in	in	ADP
ejpam-2488	255	14	x	x	X
ejpam-2488	255	15	,	,	PUNCT
ejpam-2488	255	16	ieker({x	ieker({x	PROPN
ejpam-2488	255	17	}	}	PUNCT
ejpam-2488	255	18	)	)	PUNCT
ejpam-2488	255	19	6=	6=	ADP
ejpam-2488	255	20	ieker({y	ieker({y	PROPN
ejpam-2488	255	21	}	}	PUNCT
ejpam-2488	255	22	)	)	PUNCT
ejpam-2488	255	23	implies	imply	VERB
ejpam-2488	255	24	ieker({x})∩ieker({y	ieker({x})∩ieker({y	X
ejpam-2488	255	25	}	}	PUNCT
ejpam-2488	255	26	)	)	PUNCT
ejpam-2488	255	27	=	=	SYM
ejpam-2488	255	28	;	;	PUNCT
ejpam-2488	255	29	.	.	PUNCT
ejpam-2488	256	1	proof	proof	NOUN
ejpam-2488	256	2	.	.	PUNCT
ejpam-2488	257	1	suppose	suppose	VERB
ejpam-2488	257	2	that	that	SCONJ
ejpam-2488	257	3	(	(	PUNCT
ejpam-2488	257	4	x	x	X
ejpam-2488	257	5	,	,	PUNCT
ejpam-2488	257	6	τ	τ	PROPN
ejpam-2488	257	7	,	,	PUNCT
ejpam-2488	257	8	i	i	PROPN
ejpam-2488	257	9	)	)	PUNCT
ejpam-2488	257	10	is	be	AUX
ejpam-2488	257	11	an	an	DET
ejpam-2488	257	12	e	e	NOUN
ejpam-2488	257	13	-	-	NOUN
ejpam-2488	257	14	i	i	PRON
ejpam-2488	257	15	-r0	-r0	PROPN
ejpam-2488	257	16	space	space	NOUN
ejpam-2488	257	17	.	.	PUNCT
ejpam-2488	258	1	thus	thus	ADV
ejpam-2488	258	2	by	by	ADP
ejpam-2488	258	3	lemma	lemma	PROPN
ejpam-2488	258	4	3	3	NUM
ejpam-2488	258	5	,	,	PUNCT
ejpam-2488	258	6	for	for	ADP
ejpam-2488	258	7	any	any	DET
ejpam-2488	258	8	points	point	NOUN
ejpam-2488	258	9	x	x	PUNCT
ejpam-2488	258	10	and	and	CCONJ
ejpam-2488	258	11	y	y	PROPN
ejpam-2488	258	12	in	in	ADP
ejpam-2488	258	13	x	x	PROPN
ejpam-2488	258	14	if	if	SCONJ
ejpam-2488	258	15	ieker({x	ieker({x	NOUN
ejpam-2488	258	16	}	}	PUNCT
ejpam-2488	258	17	)	)	PUNCT
ejpam-2488	258	18	6=	6=	ADP
ejpam-2488	258	19	ieker({y	ieker({y	PROPN
ejpam-2488	258	20	}	}	PUNCT
ejpam-2488	258	21	)	)	PUNCT
ejpam-2488	258	22	,	,	PUNCT
ejpam-2488	258	23	then	then	ADV
ejpam-2488	258	24	cl∗e	cl∗e	PROPN
ejpam-2488	258	25	(	(	PUNCT
ejpam-2488	258	26	{	{	PUNCT
ejpam-2488	258	27	x	x	NOUN
ejpam-2488	258	28	}	}	PUNCT
ejpam-2488	258	29	)	)	PUNCT
ejpam-2488	258	30	6=	6=	ADP
ejpam-2488	258	31	cl∗e	cl∗e	X
ejpam-2488	258	32	(	(	PUNCT
ejpam-2488	258	33	{	{	PUNCT
ejpam-2488	258	34	y	y	NOUN
ejpam-2488	258	35	}	}	PUNCT
ejpam-2488	258	36	)	)	PUNCT
ejpam-2488	258	37	.	.	PUNCT
ejpam-2488	259	1	now	now	ADV
ejpam-2488	259	2	we	we	PRON
ejpam-2488	259	3	prove	prove	VERB
ejpam-2488	259	4	that	that	SCONJ
ejpam-2488	259	5	ieker({x})∩ieker({y	ieker({x})∩ieker({y	VERB
ejpam-2488	259	6	}	}	PUNCT
ejpam-2488	259	7	)	)	PUNCT
ejpam-2488	259	8	=	=	SYM
ejpam-2488	259	9	;	;	PUNCT
ejpam-2488	259	10	.	.	PUNCT
ejpam-2488	260	1	assume	assume	VERB
ejpam-2488	260	2	that	that	SCONJ
ejpam-2488	260	3	z	z	PROPN
ejpam-2488	260	4	∈	∈	PROPN
ejpam-2488	260	5	ieker({x})∩ieker({y	ieker({x})∩ieker({y	NOUN
ejpam-2488	260	6	}	}	PUNCT
ejpam-2488	260	7	)	)	PUNCT
ejpam-2488	260	8	.	.	PUNCT
ejpam-2488	261	1	by	by	ADP
ejpam-2488	261	2	z	z	PROPN
ejpam-2488	261	3	∈	∈	PROPN
ejpam-2488	261	4	ieker({x	ieker({x	PROPN
ejpam-2488	261	5	}	}	PUNCT
ejpam-2488	261	6	)	)	PUNCT
ejpam-2488	261	7	and	and	CCONJ
ejpam-2488	261	8	lemma	lemma	PROPN
ejpam-2488	261	9	1	1	NUM
ejpam-2488	261	10	,	,	PUNCT
ejpam-2488	261	11	it	it	PRON
ejpam-2488	261	12	follows	follow	VERB
ejpam-2488	261	13	that	that	SCONJ
ejpam-2488	261	14	x	x	PUNCT
ejpam-2488	261	15	∈	∈	PRON
ejpam-2488	261	16	cl∗e	cl∗e	X
ejpam-2488	261	17	(	(	PUNCT
ejpam-2488	261	18	{	{	PUNCT
ejpam-2488	261	19	z	z	NOUN
ejpam-2488	261	20	}	}	PUNCT
ejpam-2488	261	21	)	)	PUNCT
ejpam-2488	261	22	.	.	PUNCT
ejpam-2488	262	1	since	since	SCONJ
ejpam-2488	262	2	x	x	PROPN
ejpam-2488	262	3	∈	∈	PROPN
ejpam-2488	262	4	cl∗e	cl∗e	X
ejpam-2488	262	5	(	(	PUNCT
ejpam-2488	262	6	{	{	PUNCT
ejpam-2488	262	7	x	x	NOUN
ejpam-2488	262	8	}	}	PUNCT
ejpam-2488	262	9	)	)	PUNCT
ejpam-2488	262	10	,	,	PUNCT
ejpam-2488	262	11	by	by	ADP
ejpam-2488	262	12	theorem	theorem	NOUN
ejpam-2488	262	13	2	2	NUM
ejpam-2488	262	14	,	,	PUNCT
ejpam-2488	262	15	cl∗e	cl∗e	PROPN
ejpam-2488	262	16	(	(	PUNCT
ejpam-2488	262	17	{	{	PUNCT
ejpam-2488	262	18	x	x	NOUN
ejpam-2488	262	19	}	}	PUNCT
ejpam-2488	262	20	)	)	PUNCT
ejpam-2488	262	21	=	=	SYM
ejpam-2488	262	22	cl∗e	cl∗e	PROPN
ejpam-2488	262	23	(	(	PUNCT
ejpam-2488	262	24	{	{	PUNCT
ejpam-2488	262	25	z	z	NOUN
ejpam-2488	262	26	}	}	PUNCT
ejpam-2488	262	27	)	)	PUNCT
ejpam-2488	262	28	.	.	PUNCT
ejpam-2488	263	1	similarly	similarly	ADV
ejpam-2488	263	2	,	,	PUNCT
ejpam-2488	263	3	we	we	PRON
ejpam-2488	263	4	have	have	VERB
ejpam-2488	263	5	cl∗e	cl∗e	PROPN
ejpam-2488	263	6	(	(	PUNCT
ejpam-2488	263	7	{	{	PUNCT
ejpam-2488	263	8	x	x	NOUN
ejpam-2488	263	9	}	}	PUNCT
ejpam-2488	263	10	)	)	PUNCT
ejpam-2488	263	11	=	=	SYM
ejpam-2488	264	1	cl∗e	cl∗e	PROPN
ejpam-2488	264	2	(	(	PUNCT
ejpam-2488	264	3	{	{	PUNCT
ejpam-2488	264	4	z	z	NOUN
ejpam-2488	264	5	}	}	PUNCT
ejpam-2488	264	6	)	)	PUNCT
ejpam-2488	264	7	=	=	SYM
ejpam-2488	264	8	cl∗e	cl∗e	PROPN
ejpam-2488	264	9	(	(	PUNCT
ejpam-2488	264	10	{	{	PUNCT
ejpam-2488	264	11	y	y	NOUN
ejpam-2488	264	12	}	}	PUNCT
ejpam-2488	264	13	)	)	PUNCT
ejpam-2488	264	14	.	.	PUNCT
ejpam-2488	265	1	this	this	PRON
ejpam-2488	265	2	is	be	AUX
ejpam-2488	265	3	a	a	DET
ejpam-2488	265	4	contradiction	contradiction	NOUN
ejpam-2488	265	5	.	.	PUNCT
ejpam-2488	266	1	therefore	therefore	ADV
ejpam-2488	266	2	,	,	PUNCT
ejpam-2488	266	3	we	we	PRON
ejpam-2488	266	4	have	have	VERB
ejpam-2488	266	5	ieker({x})∩ieker({y	ieker({x})∩ieker({y	NOUN
ejpam-2488	266	6	}	}	PUNCT
ejpam-2488	266	7	)	)	PUNCT
ejpam-2488	266	8	=	=	SYM
ejpam-2488	266	9	;	;	PUNCT
ejpam-2488	266	10	.	.	PUNCT
ejpam-2488	267	1	conversely	conversely	ADV
ejpam-2488	267	2	,	,	PUNCT
ejpam-2488	267	3	let	let	VERB
ejpam-2488	267	4	(	(	PUNCT
ejpam-2488	267	5	x	x	X
ejpam-2488	267	6	,	,	PUNCT
ejpam-2488	267	7	τ	τ	PROPN
ejpam-2488	267	8	,	,	PUNCT
ejpam-2488	267	9	i	i	PRON
ejpam-2488	267	10	)	)	PUNCT
ejpam-2488	267	11	be	be	AUX
ejpam-2488	267	12	an	an	DET
ejpam-2488	267	13	ideal	ideal	ADJ
ejpam-2488	267	14	topological	topological	ADJ
ejpam-2488	267	15	space	space	NOUN
ejpam-2488	267	16	such	such	ADJ
ejpam-2488	267	17	that	that	PRON
ejpam-2488	267	18	for	for	ADP
ejpam-2488	267	19	any	any	DET
ejpam-2488	267	20	points	point	NOUN
ejpam-2488	267	21	x	x	PUNCT
ejpam-2488	267	22	and	and	CCONJ
ejpam-2488	267	23	y	y	PROPN
ejpam-2488	267	24	in	in	ADP
ejpam-2488	267	25	x	x	X
ejpam-2488	267	26	,	,	PUNCT
ejpam-2488	267	27	ieker({x	ieker({x	PROPN
ejpam-2488	267	28	}	}	PUNCT
ejpam-2488	267	29	)	)	PUNCT
ejpam-2488	267	30	6=	6=	ADP
ejpam-2488	267	31	ieker({y	ieker({y	PROPN
ejpam-2488	267	32	}	}	PUNCT
ejpam-2488	267	33	)	)	PUNCT
ejpam-2488	268	1	implies	imply	VERB
ejpam-2488	268	2	ieker({x})∩ieker({y	ieker({x})∩ieker({y	X
ejpam-2488	268	3	}	}	PUNCT
ejpam-2488	268	4	)	)	PUNCT
ejpam-2488	268	5	=	=	SYM
ejpam-2488	269	1	;	;	PUNCT
ejpam-2488	269	2	.	.	PUNCT
ejpam-2488	270	1	if	if	SCONJ
ejpam-2488	270	2	cl∗e	cl∗e	PROPN
ejpam-2488	270	3	(	(	PUNCT
ejpam-2488	270	4	{	{	PUNCT
ejpam-2488	270	5	x	x	NOUN
ejpam-2488	270	6	}	}	PUNCT
ejpam-2488	270	7	)	)	PUNCT
ejpam-2488	270	8	6=	6=	ADP
ejpam-2488	270	9	cl∗e	cl∗e	X
ejpam-2488	270	10	(	(	PUNCT
ejpam-2488	270	11	{	{	PUNCT
ejpam-2488	270	12	y	y	NOUN
ejpam-2488	270	13	}	}	PUNCT
ejpam-2488	270	14	)	)	PUNCT
ejpam-2488	270	15	,	,	PUNCT
ejpam-2488	270	16	then	then	ADV
ejpam-2488	270	17	by	by	ADP
ejpam-2488	270	18	lemma	lemma	PROPN
ejpam-2488	270	19	3	3	NUM
ejpam-2488	270	20	,	,	PUNCT
ejpam-2488	270	21	ieker({x	ieker({x	PROPN
ejpam-2488	270	22	}	}	PUNCT
ejpam-2488	270	23	)	)	PUNCT
ejpam-2488	270	24	6=	6=	ADP
ejpam-2488	270	25	ieker({y	ieker({y	PROPN
ejpam-2488	270	26	}	}	PUNCT
ejpam-2488	270	27	)	)	PUNCT
ejpam-2488	270	28	.	.	PUNCT
ejpam-2488	271	1	hence	hence	ADV
ejpam-2488	271	2	,	,	PUNCT
ejpam-2488	271	3	ieker({x})∩ieker({y	ieker({x})∩ieker({y	ADV
ejpam-2488	271	4	}	}	PUNCT
ejpam-2488	271	5	)	)	PUNCT
ejpam-2488	272	1	=	=	SYM
ejpam-2488	272	2	;	;	PUNCT
ejpam-2488	272	3	which	which	PRON
ejpam-2488	272	4	implies	imply	VERB
ejpam-2488	272	5	cl∗e	cl∗e	PROPN
ejpam-2488	272	6	(	(	PUNCT
ejpam-2488	272	7	{	{	PUNCT
ejpam-2488	272	8	x	x	NOUN
ejpam-2488	272	9	}	}	PUNCT
ejpam-2488	272	10	)	)	PUNCT
ejpam-2488	272	11	∩	∩	PROPN
ejpam-2488	272	12	cl∗e	cl∗e	X
ejpam-2488	272	13	(	(	PUNCT
ejpam-2488	272	14	{	{	PUNCT
ejpam-2488	272	15	y	y	NOUN
ejpam-2488	272	16	}	}	PUNCT
ejpam-2488	272	17	)	)	PUNCT
ejpam-2488	272	18	=	=	SYM
ejpam-2488	272	19	;	;	PUNCT
ejpam-2488	272	20	.	.	PUNCT
ejpam-2488	273	1	because	because	SCONJ
ejpam-2488	273	2	z	z	PROPN
ejpam-2488	273	3	∈	∈	PROPN
ejpam-2488	273	4	cl∗e	cl∗e	X
ejpam-2488	273	5	(	(	PUNCT
ejpam-2488	273	6	{	{	PUNCT
ejpam-2488	273	7	x	x	NOUN
ejpam-2488	273	8	}	}	PUNCT
ejpam-2488	273	9	)	)	PUNCT
ejpam-2488	273	10	implies	imply	VERB
ejpam-2488	273	11	that	that	SCONJ
ejpam-2488	273	12	x	x	PUNCT
ejpam-2488	273	13	∈	∈	NOUN
ejpam-2488	273	14	ieker({z	ieker({z	PROPN
ejpam-2488	273	15	}	}	PUNCT
ejpam-2488	273	16	)	)	PUNCT
ejpam-2488	273	17	and	and	CCONJ
ejpam-2488	273	18	therefore	therefore	ADV
ejpam-2488	273	19	ieker({x	ieker({x	PROPN
ejpam-2488	273	20	}	}	PUNCT
ejpam-2488	273	21	)	)	PUNCT
ejpam-2488	273	22	∩	∩	NOUN
ejpam-2488	273	23	ieker({z	ieker({z	PROPN
ejpam-2488	273	24	}	}	PUNCT
ejpam-2488	273	25	)	)	PUNCT
ejpam-2488	273	26	6=	6=	NUM
ejpam-2488	273	27	;	;	PUNCT
ejpam-2488	273	28	.	.	PUNCT
ejpam-2488	274	1	by	by	ADP
ejpam-2488	274	2	hypothesis	hypothesis	NOUN
ejpam-2488	274	3	,	,	PUNCT
ejpam-2488	274	4	we	we	PRON
ejpam-2488	274	5	have	have	VERB
ejpam-2488	274	6	ieker({x	ieker({x	NOUN
ejpam-2488	274	7	}	}	PUNCT
ejpam-2488	274	8	)	)	PUNCT
ejpam-2488	274	9	=	=	SYM
ejpam-2488	274	10	ieker({z	ieker({z	PROPN
ejpam-2488	274	11	}	}	PUNCT
ejpam-2488	274	12	)	)	PUNCT
ejpam-2488	274	13	.	.	PUNCT
ejpam-2488	275	1	then	then	ADV
ejpam-2488	275	2	z	z	PROPN
ejpam-2488	275	3	∈	∈	PROPN
ejpam-2488	275	4	cl∗e	cl∗e	X
ejpam-2488	275	5	(	(	PUNCT
ejpam-2488	275	6	{	{	PUNCT
ejpam-2488	275	7	x	x	NOUN
ejpam-2488	275	8	}	}	PUNCT
ejpam-2488	275	9	)	)	PUNCT
ejpam-2488	275	10	∩	∩	PROPN
ejpam-2488	275	11	cl∗e	cl∗e	X
ejpam-2488	275	12	(	(	PUNCT
ejpam-2488	275	13	{	{	PUNCT
ejpam-2488	275	14	y	y	NOUN
ejpam-2488	275	15	}	}	PUNCT
ejpam-2488	275	16	)	)	PUNCT
ejpam-2488	275	17	implies	imply	VERB
ejpam-2488	275	18	that	that	SCONJ
ejpam-2488	275	19	ieker({x	ieker({x	PRON
ejpam-2488	275	20	}	}	PUNCT
ejpam-2488	275	21	)	)	PUNCT
ejpam-2488	275	22	=	=	SYM
ejpam-2488	275	23	ieker({z	ieker({z	PROPN
ejpam-2488	275	24	}	}	PUNCT
ejpam-2488	275	25	)	)	PUNCT
ejpam-2488	275	26	=	=	PUNCT
ejpam-2488	275	27	ieker({y	ieker({y	PROPN
ejpam-2488	275	28	}	}	PUNCT
ejpam-2488	275	29	)	)	PUNCT
ejpam-2488	275	30	.	.	PUNCT
ejpam-2488	276	1	this	this	PRON
ejpam-2488	276	2	is	be	AUX
ejpam-2488	276	3	a	a	DET
ejpam-2488	276	4	contradiction	contradiction	NOUN
ejpam-2488	276	5	.	.	PUNCT
ejpam-2488	277	1	therefore	therefore	ADV
ejpam-2488	277	2	,	,	PUNCT
ejpam-2488	277	3	cl∗e	cl∗e	PROPN
ejpam-2488	277	4	(	(	PUNCT
ejpam-2488	277	5	{	{	PUNCT
ejpam-2488	277	6	x	x	NOUN
ejpam-2488	277	7	}	}	PUNCT
ejpam-2488	277	8	)	)	PUNCT
ejpam-2488	277	9	∩	∩	PROPN
ejpam-2488	277	10	cl∗e	cl∗e	X
ejpam-2488	277	11	(	(	PUNCT
ejpam-2488	277	12	{	{	PUNCT
ejpam-2488	277	13	y	y	NOUN
ejpam-2488	277	14	}	}	PUNCT
ejpam-2488	277	15	)	)	PUNCT
ejpam-2488	277	16	=	=	SYM
ejpam-2488	277	17	;	;	PUNCT
ejpam-2488	277	18	and	and	CCONJ
ejpam-2488	277	19	by	by	ADP
ejpam-2488	277	20	theorem	theorem	NOUN
ejpam-2488	277	21	2	2	NUM
ejpam-2488	277	22	(	(	PUNCT
ejpam-2488	277	23	x	x	PROPN
ejpam-2488	277	24	,	,	PUNCT
ejpam-2488	277	25	τ	τ	PROPN
ejpam-2488	277	26	,	,	PUNCT
ejpam-2488	277	27	i	i	PROPN
ejpam-2488	277	28	)	)	PUNCT
ejpam-2488	277	29	is	be	AUX
ejpam-2488	277	30	an	an	DET
ejpam-2488	277	31	e	e	NOUN
ejpam-2488	277	32	-	-	NOUN
ejpam-2488	277	33	i	i	PRON
ejpam-2488	277	34	-r0	-r0	PROPN
ejpam-2488	277	35	space	space	NOUN
ejpam-2488	277	36	.	.	PUNCT
ejpam-2488	278	1	theorem	theorem	VERB
ejpam-2488	278	2	6	6	NUM
ejpam-2488	278	3	.	.	PUNCT
ejpam-2488	278	4	for	for	ADP
ejpam-2488	278	5	an	an	DET
ejpam-2488	278	6	ideal	ideal	ADJ
ejpam-2488	278	7	topological	topological	ADJ
ejpam-2488	278	8	space	space	NOUN
ejpam-2488	278	9	(	(	PUNCT
ejpam-2488	278	10	x	x	X
ejpam-2488	278	11	,	,	PUNCT
ejpam-2488	278	12	τ	τ	PROPN
ejpam-2488	278	13	,	,	PUNCT
ejpam-2488	278	14	i	i	PROPN
ejpam-2488	278	15	)	)	PUNCT
ejpam-2488	278	16	,	,	PUNCT
ejpam-2488	278	17	the	the	DET
ejpam-2488	278	18	following	follow	VERB
ejpam-2488	278	19	properties	property	NOUN
ejpam-2488	278	20	are	be	AUX
ejpam-2488	278	21	equivalent	equivalent	ADJ
ejpam-2488	278	22	:	:	PUNCT
ejpam-2488	278	23	(	(	PUNCT
ejpam-2488	278	24	i	i	NOUN
ejpam-2488	278	25	)	)	PUNCT
ejpam-2488	278	26	(	(	PUNCT
ejpam-2488	278	27	x	x	X
ejpam-2488	278	28	,	,	PUNCT
ejpam-2488	278	29	τ	τ	PROPN
ejpam-2488	278	30	,	,	PUNCT
ejpam-2488	278	31	i	i	PROPN
ejpam-2488	278	32	)	)	PUNCT
ejpam-2488	278	33	is	be	AUX
ejpam-2488	278	34	an	an	DET
ejpam-2488	278	35	e	e	NOUN
ejpam-2488	278	36	-	-	NOUN
ejpam-2488	278	37	i	i	PRON
ejpam-2488	278	38	-r0	-r0	PROPN
ejpam-2488	278	39	space	space	NOUN
ejpam-2488	278	40	,	,	PUNCT
ejpam-2488	278	41	(	(	PUNCT
ejpam-2488	278	42	ii	ii	NOUN
ejpam-2488	278	43	)	)	PUNCT
ejpam-2488	278	44	if	if	SCONJ
ejpam-2488	278	45	f	f	PROPN
ejpam-2488	278	46	is	be	AUX
ejpam-2488	278	47	an	an	DET
ejpam-2488	278	48	e	e	NOUN
ejpam-2488	278	49	-	-	NOUN
ejpam-2488	278	50	i	i	PRON
ejpam-2488	278	51	-closed	-close	VERB
ejpam-2488	278	52	subset	subset	NOUN
ejpam-2488	278	53	of	of	ADP
ejpam-2488	278	54	x	x	SYM
ejpam-2488	278	55	,	,	PUNCT
ejpam-2488	278	56	then	then	ADV
ejpam-2488	278	57	f	f	PROPN
ejpam-2488	278	58	=	=	SYM
ejpam-2488	278	59	ieker(f	ieker(f	PROPN
ejpam-2488	278	60	)	)	PUNCT
ejpam-2488	278	61	,	,	PUNCT
ejpam-2488	278	62	(	(	PUNCT
ejpam-2488	278	63	iii	iii	X
ejpam-2488	278	64	)	)	PUNCT
ejpam-2488	278	65	if	if	SCONJ
ejpam-2488	278	66	f	f	PROPN
ejpam-2488	278	67	is	be	AUX
ejpam-2488	278	68	an	an	DET
ejpam-2488	278	69	e	e	NOUN
ejpam-2488	278	70	-	-	NOUN
ejpam-2488	278	71	i	i	PRON
ejpam-2488	278	72	-closed	-close	VERB
ejpam-2488	278	73	subset	subset	NOUN
ejpam-2488	278	74	of	of	ADP
ejpam-2488	278	75	x	x	PUNCT
ejpam-2488	278	76	and	and	CCONJ
ejpam-2488	279	1	x	x	SYM
ejpam-2488	279	2	∈	∈	PROPN
ejpam-2488	279	3	f	f	X
ejpam-2488	279	4	,	,	PUNCT
ejpam-2488	279	5	then	then	ADV
ejpam-2488	279	6	ieker({x	ieker({x	PROPN
ejpam-2488	279	7	}	}	PUNCT
ejpam-2488	279	8	)	)	PUNCT
ejpam-2488	280	1	⊂	⊂	PROPN
ejpam-2488	281	1	f	f	X
ejpam-2488	281	2	,	,	PUNCT
ejpam-2488	281	3	(	(	PUNCT
ejpam-2488	281	4	iv	iv	X
ejpam-2488	281	5	)	)	PUNCT
ejpam-2488	281	6	if	if	SCONJ
ejpam-2488	281	7	x	x	PUNCT
ejpam-2488	281	8	∈	∈	PROPN
ejpam-2488	281	9	x	x	X
ejpam-2488	281	10	,	,	PUNCT
ejpam-2488	281	11	then	then	ADV
ejpam-2488	281	12	ieker({x	ieker({x	PROPN
ejpam-2488	281	13	}	}	PUNCT
ejpam-2488	281	14	)	)	PUNCT
ejpam-2488	282	1	⊂	⊂	PROPN
ejpam-2488	282	2	cl∗e	cl∗e	PROPN
ejpam-2488	282	3	(	(	PUNCT
ejpam-2488	282	4	{	{	PUNCT
ejpam-2488	282	5	x	x	NOUN
ejpam-2488	282	6	}	}	PUNCT
ejpam-2488	282	7	)	)	PUNCT
ejpam-2488	282	8	.	.	PUNCT
ejpam-2488	283	1	proof	proof	NOUN
ejpam-2488	283	2	.	.	PUNCT
ejpam-2488	284	1	(	(	PUNCT
ejpam-2488	284	2	i)⇒	i)⇒	PROPN
ejpam-2488	284	3	(	(	PUNCT
ejpam-2488	284	4	ii	ii	NOUN
ejpam-2488	284	5	):	):	PUNCT
ejpam-2488	284	6	let	let	VERB
ejpam-2488	284	7	f	f	PRON
ejpam-2488	284	8	be	be	AUX
ejpam-2488	284	9	an	an	DET
ejpam-2488	284	10	e	e	NOUN
ejpam-2488	284	11	-	-	NOUN
ejpam-2488	284	12	i	i	PRON
ejpam-2488	284	13	-closed	-close	VERB
ejpam-2488	284	14	subset	subset	NOUN
ejpam-2488	284	15	of	of	ADP
ejpam-2488	284	16	x	x	PUNCT
ejpam-2488	284	17	and	and	CCONJ
ejpam-2488	284	18	x	x	SYM
ejpam-2488	284	19	/∈	/∈	PROPN
ejpam-2488	285	1	f	f	PROPN
ejpam-2488	285	2	.	.	PUNCT
ejpam-2488	286	1	thus	thus	ADV
ejpam-2488	286	2	x\f	x\f	PRON
ejpam-2488	286	3	∈	∈	PROPN
ejpam-2488	286	4	eio(x	eio(x	PROPN
ejpam-2488	286	5	x	x	NOUN
ejpam-2488	286	6	)	)	PUNCT
ejpam-2488	286	7	.	.	PUNCT
ejpam-2488	287	1	since	since	SCONJ
ejpam-2488	287	2	(	(	PUNCT
ejpam-2488	287	3	x	x	X
ejpam-2488	287	4	,	,	PUNCT
ejpam-2488	287	5	τ	τ	PROPN
ejpam-2488	287	6	,	,	PUNCT
ejpam-2488	287	7	i	i	PROPN
ejpam-2488	287	8	)	)	PUNCT
ejpam-2488	287	9	is	be	AUX
ejpam-2488	287	10	e	e	NOUN
ejpam-2488	287	11	-	-	PUNCT
ejpam-2488	287	12	i	i	PRON
ejpam-2488	287	13	-r0	-r0	NOUN
ejpam-2488	287	14	,	,	PUNCT
ejpam-2488	287	15	cl∗e	cl∗e	PROPN
ejpam-2488	287	16	(	(	PUNCT
ejpam-2488	287	17	{	{	PUNCT
ejpam-2488	287	18	x	x	NOUN
ejpam-2488	287	19	}	}	PUNCT
ejpam-2488	287	20	)	)	PUNCT
ejpam-2488	288	1	⊂	⊂	PROPN
ejpam-2488	289	1	x\f	x\f	PROPN
ejpam-2488	289	2	.	.	PUNCT
ejpam-2488	290	1	since	since	SCONJ
ejpam-2488	290	2	f	f	PROPN
ejpam-2488	290	3	⊂	⊂	PROPN
ejpam-2488	290	4	x	x	PUNCT
ejpam-2488	290	5	\	\	X
ejpam-2488	290	6	cl∗e	cl∗e	X
ejpam-2488	290	7	(	(	PUNCT
ejpam-2488	290	8	{	{	PUNCT
ejpam-2488	290	9	x	x	NOUN
ejpam-2488	290	10	}	}	PUNCT
ejpam-2488	290	11	)	)	PUNCT
ejpam-2488	290	12	,	,	PUNCT
ejpam-2488	290	13	ieker(f	ieker(f	NOUN
ejpam-2488	290	14	)	)	PUNCT
ejpam-2488	290	15	⊂	⊂	X
ejpam-2488	291	1	x	x	X
ejpam-2488	291	2	−	−	X
ejpam-2488	291	3	cl∗e	cl∗e	SYM
ejpam-2488	291	4	(	(	PUNCT
ejpam-2488	291	5	{	{	PUNCT
ejpam-2488	291	6	x	x	NOUN
ejpam-2488	291	7	}	}	PUNCT
ejpam-2488	291	8	)	)	PUNCT
ejpam-2488	291	9	w.	w.	PROPN
ejpam-2488	291	10	al	al	PROPN
ejpam-2488	291	11	-	-	PUNCT
ejpam-2488	291	12	omeri	omeri	ADJ
ejpam-2488	291	13	,	,	PUNCT
ejpam-2488	291	14	m.	m.	NOUN
ejpam-2488	291	15	noorani	noorani	PROPN
ejpam-2488	291	16	,	,	PUNCT
ejpam-2488	291	17	a.	a.	PROPN
ejpam-2488	291	18	al	al	PROPN
ejpam-2488	291	19	-	-	PUNCT
ejpam-2488	291	20	omari	omari	PROPN
ejpam-2488	291	21	,	,	PUNCT
ejpam-2488	291	22	and	and	CCONJ
ejpam-2488	291	23	t.	t.	PROPN
ejpam-2488	291	24	noiri	noiri	PROPN
ejpam-2488	291	25	/	/	SYM
ejpam-2488	291	26	eur	eur	PROPN
ejpam-2488	291	27	.	.	PUNCT
ejpam-2488	292	1	j.	j.	PROPN
ejpam-2488	292	2	pure	pure	PROPN
ejpam-2488	292	3	appl	appl	PROPN
ejpam-2488	292	4	.	.	PROPN
ejpam-2488	292	5	math	math	PROPN
ejpam-2488	292	6	,	,	PUNCT
ejpam-2488	292	7	8	8	NUM
ejpam-2488	292	8	(	(	PUNCT
ejpam-2488	292	9	2015	2015	NUM
ejpam-2488	292	10	)	)	PUNCT
ejpam-2488	292	11	,	,	PUNCT
ejpam-2488	292	12	502	502	NUM
ejpam-2488	292	13	-	-	SYM
ejpam-2488	292	14	513	513	NUM
ejpam-2488	292	15	508	508	NUM
ejpam-2488	292	16	and	and	CCONJ
ejpam-2488	292	17	x	x	NOUN
ejpam-2488	292	18	/∈	/∈	PUNCT
ejpam-2488	292	19	ieker(f	ieker(f	NOUN
ejpam-2488	292	20	)	)	PUNCT
ejpam-2488	292	21	.	.	PUNCT
ejpam-2488	293	1	therefore	therefore	ADV
ejpam-2488	293	2	,	,	PUNCT
ejpam-2488	293	3	ieker(f	ieker(f	NOUN
ejpam-2488	293	4	)	)	PUNCT
ejpam-2488	294	1	=	=	SYM
ejpam-2488	294	2	f	f	PROPN
ejpam-2488	294	3	.	.	PUNCT
ejpam-2488	295	1	(	(	PUNCT
ejpam-2488	295	2	ii)⇒	ii)⇒	PROPN
ejpam-2488	295	3	(	(	PUNCT
ejpam-2488	295	4	iii	iii	NOUN
ejpam-2488	295	5	):	):	PUNCT
ejpam-2488	295	6	in	in	ADP
ejpam-2488	295	7	general	general	ADJ
ejpam-2488	295	8	,	,	PUNCT
ejpam-2488	295	9	a	a	DET
ejpam-2488	295	10	⊂	⊂	PROPN
ejpam-2488	295	11	b	b	PROPN
ejpam-2488	295	12	implies	imply	VERB
ejpam-2488	295	13	ieker(a	ieker(a	VERB
ejpam-2488	295	14	)	)	PUNCT
ejpam-2488	295	15	⊂	⊂	PROPN
ejpam-2488	295	16	ieker(b	ieker(b	PROPN
ejpam-2488	295	17	)	)	PUNCT
ejpam-2488	295	18	.	.	PUNCT
ejpam-2488	296	1	therefore	therefore	ADV
ejpam-2488	296	2	,	,	PUNCT
ejpam-2488	296	3	it	it	PRON
ejpam-2488	296	4	follows	follow	VERB
ejpam-2488	296	5	from	from	ADP
ejpam-2488	296	6	(	(	PUNCT
ejpam-2488	296	7	ii	ii	NOUN
ejpam-2488	296	8	)	)	PUNCT
ejpam-2488	296	9	that	that	SCONJ
ejpam-2488	296	10	ieker({x	ieker({x	PROPN
ejpam-2488	296	11	}	}	PUNCT
ejpam-2488	296	12	)	)	PUNCT
ejpam-2488	296	13	⊂	⊂	PROPN
ejpam-2488	297	1	ieker(f	ieker(f	NOUN
ejpam-2488	297	2	)	)	PUNCT
ejpam-2488	298	1	=	=	SYM
ejpam-2488	298	2	f	f	PROPN
ejpam-2488	298	3	.	.	PUNCT
ejpam-2488	299	1	(	(	PUNCT
ejpam-2488	299	2	iii)⇒	iii)⇒	PROPN
ejpam-2488	299	3	(	(	PUNCT
ejpam-2488	299	4	iv	iv	NUM
ejpam-2488	299	5	):	):	PUNCT
ejpam-2488	299	6	since	since	SCONJ
ejpam-2488	299	7	x	x	PROPN
ejpam-2488	299	8	∈	∈	PROPN
ejpam-2488	299	9	cl∗e	cl∗e	X
ejpam-2488	299	10	(	(	PUNCT
ejpam-2488	299	11	{	{	PUNCT
ejpam-2488	299	12	x	x	NOUN
ejpam-2488	299	13	}	}	PUNCT
ejpam-2488	299	14	)	)	PUNCT
ejpam-2488	299	15	and	and	CCONJ
ejpam-2488	299	16	cl∗e	cl∗e	PROPN
ejpam-2488	299	17	(	(	PUNCT
ejpam-2488	299	18	{	{	PUNCT
ejpam-2488	299	19	x	x	NOUN
ejpam-2488	299	20	}	}	PUNCT
ejpam-2488	299	21	)	)	PUNCT
ejpam-2488	299	22	is	be	AUX
ejpam-2488	299	23	e	e	NOUN
ejpam-2488	299	24	-	-	PUNCT
ejpam-2488	299	25	i	i	PRON
ejpam-2488	299	26	-closed	-close	VERB
ejpam-2488	299	27	,	,	PUNCT
ejpam-2488	299	28	by	by	ADP
ejpam-2488	299	29	(	(	PUNCT
ejpam-2488	299	30	iii	iii	NOUN
ejpam-2488	299	31	)	)	PUNCT
ejpam-2488	299	32	ieker({x	ieker({x	PROPN
ejpam-2488	299	33	}	}	PUNCT
ejpam-2488	299	34	)	)	PUNCT
ejpam-2488	300	1	⊂	⊂	PROPN
ejpam-2488	300	2	cl∗e	cl∗e	PROPN
ejpam-2488	300	3	(	(	PUNCT
ejpam-2488	300	4	{	{	PUNCT
ejpam-2488	300	5	x	x	NOUN
ejpam-2488	300	6	}	}	PUNCT
ejpam-2488	300	7	)	)	PUNCT
ejpam-2488	300	8	.	.	PUNCT
ejpam-2488	301	1	(	(	PUNCT
ejpam-2488	301	2	iv)⇒	iv)⇒	X
ejpam-2488	301	3	(	(	PUNCT
ejpam-2488	301	4	i	i	NOUN
ejpam-2488	301	5	):	):	PUNCT
ejpam-2488	301	6	we	we	PRON
ejpam-2488	301	7	show	show	VERB
ejpam-2488	301	8	the	the	DET
ejpam-2488	301	9	implication	implication	NOUN
ejpam-2488	301	10	by	by	ADP
ejpam-2488	301	11	using	use	VERB
ejpam-2488	301	12	theorem	theorem	NOUN
ejpam-2488	301	13	3	3	X
ejpam-2488	301	14	.	.	PUNCT
ejpam-2488	302	1	let	let	VERB
ejpam-2488	302	2	x	x	X
ejpam-2488	302	3	∈	∈	PROPN
ejpam-2488	302	4	cl∗e	cl∗e	X
ejpam-2488	302	5	(	(	PUNCT
ejpam-2488	302	6	{	{	PUNCT
ejpam-2488	302	7	y	y	NOUN
ejpam-2488	302	8	}	}	PUNCT
ejpam-2488	302	9	)	)	PUNCT
ejpam-2488	302	10	.	.	PUNCT
ejpam-2488	303	1	then	then	ADV
ejpam-2488	303	2	by	by	ADP
ejpam-2488	303	3	lemma	lemma	PROPN
ejpam-2488	303	4	1	1	NUM
ejpam-2488	303	5	y	y	PROPN
ejpam-2488	303	6	∈	∈	PROPN
ejpam-2488	303	7	ieker({x	ieker({x	PROPN
ejpam-2488	303	8	}	}	PUNCT
ejpam-2488	303	9	)	)	PUNCT
ejpam-2488	303	10	.	.	PUNCT
ejpam-2488	304	1	by	by	ADP
ejpam-2488	304	2	(	(	PUNCT
ejpam-2488	304	3	iv	iv	X
ejpam-2488	304	4	)	)	PUNCT
ejpam-2488	304	5	,	,	PUNCT
ejpam-2488	304	6	we	we	PRON
ejpam-2488	304	7	obtain	obtain	VERB
ejpam-2488	304	8	y	y	PROPN
ejpam-2488	304	9	∈	∈	PROPN
ejpam-2488	304	10	ieker({x	ieker({x	PROPN
ejpam-2488	304	11	}	}	PUNCT
ejpam-2488	304	12	)	)	PUNCT
ejpam-2488	305	1	⊂	⊂	PROPN
ejpam-2488	305	2	cl∗e	cl∗e	PROPN
ejpam-2488	305	3	(	(	PUNCT
ejpam-2488	305	4	{	{	PUNCT
ejpam-2488	305	5	x	x	NOUN
ejpam-2488	305	6	}	}	PUNCT
ejpam-2488	305	7	)	)	PUNCT
ejpam-2488	305	8	.	.	PUNCT
ejpam-2488	306	1	therefore	therefore	ADV
ejpam-2488	306	2	,	,	PUNCT
ejpam-2488	306	3	x	x	PUNCT
ejpam-2488	306	4	∈	∈	PROPN
ejpam-2488	306	5	cl∗e	cl∗e	X
ejpam-2488	306	6	(	(	PUNCT
ejpam-2488	306	7	{	{	PUNCT
ejpam-2488	306	8	y	y	NOUN
ejpam-2488	306	9	}	}	PUNCT
ejpam-2488	306	10	)	)	PUNCT
ejpam-2488	306	11	implies	imply	VERB
ejpam-2488	306	12	y	y	PROPN
ejpam-2488	306	13	∈	∈	PROPN
ejpam-2488	306	14	cl∗e	cl∗e	PROPN
ejpam-2488	306	15	(	(	PUNCT
ejpam-2488	306	16	{	{	PUNCT
ejpam-2488	306	17	x	x	NOUN
ejpam-2488	306	18	}	}	PUNCT
ejpam-2488	306	19	)	)	PUNCT
ejpam-2488	306	20	.	.	PUNCT
ejpam-2488	307	1	the	the	DET
ejpam-2488	307	2	converse	converse	NOUN
ejpam-2488	307	3	is	be	AUX
ejpam-2488	307	4	obvious	obvious	ADJ
ejpam-2488	307	5	and	and	CCONJ
ejpam-2488	307	6	(	(	PUNCT
ejpam-2488	307	7	x	x	X
ejpam-2488	307	8	,	,	PUNCT
ejpam-2488	307	9	τ	τ	PROPN
ejpam-2488	307	10	,	,	PUNCT
ejpam-2488	307	11	i	i	PROPN
ejpam-2488	307	12	)	)	PUNCT
ejpam-2488	307	13	is	be	AUX
ejpam-2488	307	14	an	an	DET
ejpam-2488	307	15	e	e	NOUN
ejpam-2488	307	16	-	-	NOUN
ejpam-2488	307	17	i	i	PRON
ejpam-2488	307	18	-r0	-r0	PROPN
ejpam-2488	307	19	space	space	NOUN
ejpam-2488	307	20	.	.	PUNCT
ejpam-2488	308	1	corollary	corollary	ADJ
ejpam-2488	308	2	1	1	NUM
ejpam-2488	308	3	.	.	PUNCT
ejpam-2488	309	1	for	for	ADP
ejpam-2488	309	2	an	an	DET
ejpam-2488	309	3	ideal	ideal	ADJ
ejpam-2488	309	4	topological	topological	ADJ
ejpam-2488	309	5	space	space	NOUN
ejpam-2488	309	6	(	(	PUNCT
ejpam-2488	309	7	x	x	X
ejpam-2488	309	8	,	,	PUNCT
ejpam-2488	309	9	τ	τ	PROPN
ejpam-2488	309	10	,	,	PUNCT
ejpam-2488	309	11	i	i	PROPN
ejpam-2488	309	12	)	)	PUNCT
ejpam-2488	309	13	,	,	PUNCT
ejpam-2488	309	14	the	the	DET
ejpam-2488	309	15	following	follow	VERB
ejpam-2488	309	16	properties	property	NOUN
ejpam-2488	309	17	are	be	AUX
ejpam-2488	309	18	equivalent	equivalent	ADJ
ejpam-2488	309	19	:	:	PUNCT
ejpam-2488	309	20	(	(	PUNCT
ejpam-2488	309	21	i	i	NOUN
ejpam-2488	309	22	)	)	PUNCT
ejpam-2488	309	23	(	(	PUNCT
ejpam-2488	309	24	x	x	X
ejpam-2488	309	25	,	,	PUNCT
ejpam-2488	309	26	τ	τ	PROPN
ejpam-2488	309	27	,	,	PUNCT
ejpam-2488	309	28	i	i	PROPN
ejpam-2488	309	29	)	)	PUNCT
ejpam-2488	309	30	is	be	AUX
ejpam-2488	309	31	an	an	DET
ejpam-2488	309	32	e	e	NOUN
ejpam-2488	309	33	-	-	NOUN
ejpam-2488	309	34	i	i	PRON
ejpam-2488	309	35	-r0	-r0	PROPN
ejpam-2488	309	36	space	space	NOUN
ejpam-2488	309	37	,	,	PUNCT
ejpam-2488	309	38	(	(	PUNCT
ejpam-2488	309	39	ii	ii	NOUN
ejpam-2488	309	40	)	)	PUNCT
ejpam-2488	309	41	c	c	PROPN
ejpam-2488	309	42	l∗e	l∗e	PUNCT
ejpam-2488	309	43	(	(	PUNCT
ejpam-2488	309	44	{	{	PUNCT
ejpam-2488	309	45	x	x	NOUN
ejpam-2488	309	46	}	}	PUNCT
ejpam-2488	309	47	)	)	PUNCT
ejpam-2488	309	48	=	=	SYM
ejpam-2488	309	49	ieker({x	ieker({x	PROPN
ejpam-2488	309	50	}	}	PUNCT
ejpam-2488	309	51	)	)	PUNCT
ejpam-2488	309	52	for	for	ADP
ejpam-2488	309	53	all	all	PRON
ejpam-2488	309	54	x	x	SYM
ejpam-2488	309	55	∈	∈	PROPN
ejpam-2488	309	56	x	x	X
ejpam-2488	309	57	.	.	PUNCT
ejpam-2488	310	1	proof	proof	NOUN
ejpam-2488	310	2	.	.	PUNCT
ejpam-2488	311	1	(	(	PUNCT
ejpam-2488	311	2	i)⇒	i)⇒	PROPN
ejpam-2488	311	3	(	(	PUNCT
ejpam-2488	311	4	ii	ii	PROPN
ejpam-2488	311	5	):	):	PUNCT
ejpam-2488	311	6	suppose	suppose	VERB
ejpam-2488	311	7	that	that	SCONJ
ejpam-2488	311	8	(	(	PUNCT
ejpam-2488	311	9	x	x	X
ejpam-2488	311	10	,	,	PUNCT
ejpam-2488	311	11	τ	τ	PROPN
ejpam-2488	311	12	,	,	PUNCT
ejpam-2488	311	13	i	i	PROPN
ejpam-2488	311	14	)	)	PUNCT
ejpam-2488	311	15	is	be	AUX
ejpam-2488	311	16	an	an	DET
ejpam-2488	311	17	e	e	NOUN
ejpam-2488	311	18	-	-	NOUN
ejpam-2488	311	19	i	i	PRON
ejpam-2488	311	20	-r0	-r0	PROPN
ejpam-2488	311	21	space	space	NOUN
ejpam-2488	311	22	.	.	PUNCT
ejpam-2488	312	1	by	by	ADP
ejpam-2488	312	2	theorem	theorem	NOUN
ejpam-2488	312	3	4	4	NUM
ejpam-2488	312	4	,	,	PUNCT
ejpam-2488	312	5	cl∗e	cl∗e	PROPN
ejpam-2488	312	6	(	(	PUNCT
ejpam-2488	312	7	{	{	PUNCT
ejpam-2488	312	8	x	x	NOUN
ejpam-2488	312	9	}	}	PUNCT
ejpam-2488	312	10	)	)	PUNCT
ejpam-2488	312	11	⊂	⊂	PROPN
ejpam-2488	312	12	ieker({x	ieker({x	PROPN
ejpam-2488	312	13	}	}	PUNCT
ejpam-2488	312	14	)	)	PUNCT
ejpam-2488	312	15	for	for	ADP
ejpam-2488	312	16	each	each	DET
ejpam-2488	312	17	x	x	SYM
ejpam-2488	312	18	∈	∈	PROPN
ejpam-2488	312	19	x	x	X
ejpam-2488	312	20	.	.	PUNCT
ejpam-2488	313	1	by	by	ADP
ejpam-2488	313	2	theorem	theorem	NOUN
ejpam-2488	313	3	6	6	NUM
ejpam-2488	313	4	,	,	PUNCT
ejpam-2488	313	5	ieker({x	ieker({x	PROPN
ejpam-2488	313	6	}	}	PUNCT
ejpam-2488	313	7	)	)	PUNCT
ejpam-2488	313	8	⊂	⊂	PROPN
ejpam-2488	313	9	cl∗e	cl∗e	PROPN
ejpam-2488	313	10	(	(	PUNCT
ejpam-2488	313	11	{	{	PUNCT
ejpam-2488	313	12	x	x	NOUN
ejpam-2488	313	13	}	}	PUNCT
ejpam-2488	313	14	)	)	PUNCT
ejpam-2488	313	15	.	.	PUNCT
ejpam-2488	314	1	this	this	PRON
ejpam-2488	314	2	shows	show	VERB
ejpam-2488	314	3	that	that	SCONJ
ejpam-2488	314	4	cl∗e	cl∗e	PROPN
ejpam-2488	314	5	(	(	PUNCT
ejpam-2488	314	6	{	{	PUNCT
ejpam-2488	314	7	x	x	NOUN
ejpam-2488	314	8	}	}	PUNCT
ejpam-2488	314	9	)	)	PUNCT
ejpam-2488	314	10	=	=	SYM
ejpam-2488	314	11	ieker({x	ieker({x	PROPN
ejpam-2488	314	12	}	}	PUNCT
ejpam-2488	314	13	)	)	PUNCT
ejpam-2488	314	14	.	.	PUNCT
ejpam-2488	315	1	(	(	PUNCT
ejpam-2488	315	2	ii)⇒	ii)⇒	PROPN
ejpam-2488	315	3	(	(	PUNCT
ejpam-2488	315	4	i	i	NOUN
ejpam-2488	315	5	):	):	PUNCT
ejpam-2488	315	6	this	this	PRON
ejpam-2488	315	7	is	be	AUX
ejpam-2488	315	8	obvious	obvious	ADJ
ejpam-2488	315	9	by	by	ADP
ejpam-2488	315	10	theorem	theorem	ADJ
ejpam-2488	315	11	6	6	NUM
ejpam-2488	315	12	.	.	PUNCT
ejpam-2488	315	13	corollary	corollary	ADJ
ejpam-2488	315	14	2	2	NUM
ejpam-2488	315	15	.	.	PUNCT
ejpam-2488	316	1	let	let	AUX
ejpam-2488	316	2	(	(	PUNCT
ejpam-2488	316	3	x	x	X
ejpam-2488	316	4	,	,	PUNCT
ejpam-2488	316	5	τ	τ	PROPN
ejpam-2488	316	6	,	,	PUNCT
ejpam-2488	316	7	i	i	PRON
ejpam-2488	316	8	)	)	PUNCT
ejpam-2488	316	9	be	be	VERB
ejpam-2488	316	10	e	e	NOUN
ejpam-2488	316	11	-	-	PUNCT
ejpam-2488	316	12	i	i	PRON
ejpam-2488	316	13	-r0	-r0	PROPN
ejpam-2488	317	1	and	and	CCONJ
ejpam-2488	317	2	x	x	PUNCT
ejpam-2488	317	3	∈	∈	PROPN
ejpam-2488	317	4	x	x	X
ejpam-2488	317	5	.	.	PUNCT
ejpam-2488	318	1	if	if	SCONJ
ejpam-2488	318	2	c	c	PROPN
ejpam-2488	318	3	l∗e	l∗e	ADV
ejpam-2488	318	4	(	(	PUNCT
ejpam-2488	318	5	{	{	PUNCT
ejpam-2488	318	6	x	x	NOUN
ejpam-2488	318	7	}	}	PUNCT
ejpam-2488	318	8	)	)	PUNCT
ejpam-2488	318	9	∩	∩	PROPN
ejpam-2488	318	10	ieker({x	ieker({x	NOUN
ejpam-2488	318	11	}	}	PUNCT
ejpam-2488	318	12	)	)	PUNCT
ejpam-2488	318	13	=	=	PRON
ejpam-2488	318	14	{	{	PUNCT
ejpam-2488	318	15	x	x	NOUN
ejpam-2488	318	16	}	}	PUNCT
ejpam-2488	318	17	,	,	PUNCT
ejpam-2488	318	18	then	then	ADV
ejpam-2488	318	19	ieker({x	ieker({x	PROPN
ejpam-2488	318	20	}	}	PUNCT
ejpam-2488	318	21	)	)	PUNCT
ejpam-2488	319	1	=	=	PRON
ejpam-2488	319	2	{	{	PUNCT
ejpam-2488	319	3	x	x	NOUN
ejpam-2488	319	4	}	}	PUNCT
ejpam-2488	319	5	.	.	PUNCT
ejpam-2488	320	1	proof	proof	NOUN
ejpam-2488	320	2	.	.	PUNCT
ejpam-2488	321	1	the	the	DET
ejpam-2488	321	2	proof	proof	NOUN
ejpam-2488	321	3	follows	follow	VERB
ejpam-2488	321	4	from	from	ADP
ejpam-2488	321	5	theorem	theorem	ADJ
ejpam-2488	321	6	6	6	NUM
ejpam-2488	321	7	(	(	PUNCT
ejpam-2488	321	8	iv	iv	NUM
ejpam-2488	321	9	)	)	PUNCT
ejpam-2488	321	10	.	.	PUNCT
ejpam-2488	322	1	definition	definition	NOUN
ejpam-2488	322	2	7	7	NUM
ejpam-2488	322	3	.	.	PUNCT
ejpam-2488	323	1	a	a	DET
ejpam-2488	323	2	net	net	NOUN
ejpam-2488	323	3	{	{	PUNCT
ejpam-2488	323	4	xλ}λ∈∧	xλ}λ∈∧	PROPN
ejpam-2488	323	5	is	be	AUX
ejpam-2488	323	6	said	say	VERB
ejpam-2488	323	7	to	to	PART
ejpam-2488	323	8	be	be	AUX
ejpam-2488	323	9	e	e	NOUN
ejpam-2488	323	10	-	-	NOUN
ejpam-2488	323	11	i	i	PRON
ejpam-2488	323	12	-convergent	-convergent	ADJ
ejpam-2488	323	13	to	to	ADP
ejpam-2488	323	14	a	a	DET
ejpam-2488	323	15	point	point	NOUN
ejpam-2488	323	16	x	x	PUNCT
ejpam-2488	323	17	in	in	ADP
ejpam-2488	323	18	x	x	SYM
ejpam-2488	323	19	,	,	PUNCT
ejpam-2488	323	20	if	if	SCONJ
ejpam-2488	323	21	for	for	ADP
ejpam-2488	323	22	any	any	DET
ejpam-2488	323	23	u	u	NOUN
ejpam-2488	323	24	∈	∈	PROPN
ejpam-2488	323	25	eio(x	eio(x	X
ejpam-2488	323	26	,	,	PUNCT
ejpam-2488	323	27	x	x	X
ejpam-2488	323	28	)	)	PUNCT
ejpam-2488	323	29	,	,	PUNCT
ejpam-2488	323	30	there	there	PRON
ejpam-2488	323	31	exists	exist	VERB
ejpam-2488	323	32	λ0	λ0	NOUN
ejpam-2488	323	33	∈	∈	NOUN
ejpam-2488	323	34	∧	∧	NOUN
ejpam-2488	323	35	such	such	ADJ
ejpam-2488	323	36	that	that	SCONJ
ejpam-2488	323	37	xλ	xλ	PROPN
ejpam-2488	323	38	∈	∈	PROPN
ejpam-2488	323	39	u	u	NOUN
ejpam-2488	323	40	for	for	ADP
ejpam-2488	323	41	any	any	DET
ejpam-2488	323	42	λ	λ	PROPN
ejpam-2488	323	43	∈	∈	PROPN
ejpam-2488	323	44	∧	∧	NOUN
ejpam-2488	323	45	such	such	ADJ
ejpam-2488	323	46	that	that	SCONJ
ejpam-2488	323	47	λ	λ	PROPN
ejpam-2488	323	48	≥	≥	PRON
ejpam-2488	323	49	λo	λo	NOUN
ejpam-2488	323	50	.	.	PUNCT
ejpam-2488	324	1	lemma	lemma	PROPN
ejpam-2488	324	2	4	4	X
ejpam-2488	324	3	.	.	PUNCT
ejpam-2488	325	1	let	let	AUX
ejpam-2488	325	2	(	(	PUNCT
ejpam-2488	325	3	x	x	X
ejpam-2488	325	4	,	,	PUNCT
ejpam-2488	325	5	τ	τ	PROPN
ejpam-2488	325	6	,	,	PUNCT
ejpam-2488	325	7	i	i	PRON
ejpam-2488	325	8	)	)	PUNCT
ejpam-2488	325	9	be	be	AUX
ejpam-2488	325	10	an	an	DET
ejpam-2488	325	11	ideal	ideal	ADJ
ejpam-2488	325	12	topological	topological	ADJ
ejpam-2488	325	13	space	space	NOUN
ejpam-2488	325	14	and	and	CCONJ
ejpam-2488	325	15	let	let	VERB
ejpam-2488	325	16	x	x	PRON
ejpam-2488	325	17	and	and	CCONJ
ejpam-2488	325	18	y	y	PROPN
ejpam-2488	325	19	be	be	AUX
ejpam-2488	325	20	any	any	DET
ejpam-2488	325	21	two	two	NUM
ejpam-2488	325	22	points	point	NOUN
ejpam-2488	325	23	in	in	ADP
ejpam-2488	325	24	x	x	SYM
ejpam-2488	325	25	such	such	ADJ
ejpam-2488	325	26	that	that	SCONJ
ejpam-2488	325	27	every	every	DET
ejpam-2488	325	28	net	net	NOUN
ejpam-2488	325	29	in	in	ADP
ejpam-2488	325	30	x	x	X
ejpam-2488	325	31	e	e	X
ejpam-2488	325	32	-	-	PROPN
ejpam-2488	325	33	i	i	PRON
ejpam-2488	325	34	-converging	-converge	VERB
ejpam-2488	325	35	to	to	ADP
ejpam-2488	325	36	y	y	PROPN
ejpam-2488	325	37	e	e	PROPN
ejpam-2488	325	38	-	-	PROPN
ejpam-2488	325	39	i	i	PRON
ejpam-2488	325	40	-converges	-converge	NOUN
ejpam-2488	325	41	to	to	PART
ejpam-2488	325	42	x.	x.	NOUN
ejpam-2488	325	43	then	then	ADV
ejpam-2488	325	44	x	x	SYM
ejpam-2488	325	45	∈	∈	PROPN
ejpam-2488	325	46	cl∗e	cl∗e	X
ejpam-2488	325	47	(	(	PUNCT
ejpam-2488	325	48	{	{	PUNCT
ejpam-2488	325	49	y	y	NOUN
ejpam-2488	325	50	}	}	PUNCT
ejpam-2488	325	51	)	)	PUNCT
ejpam-2488	325	52	.	.	PUNCT
ejpam-2488	326	1	proof	proof	NOUN
ejpam-2488	326	2	.	.	PUNCT
ejpam-2488	327	1	suppose	suppose	VERB
ejpam-2488	327	2	that	that	SCONJ
ejpam-2488	327	3	xn	xn	PROPN
ejpam-2488	327	4	=	=	SYM
ejpam-2488	327	5	y	y	PROPN
ejpam-2488	327	6	for	for	ADP
ejpam-2488	327	7	each	each	DET
ejpam-2488	327	8	n	n	PRON
ejpam-2488	327	9	∈	∈	PROPN
ejpam-2488	327	10	n	n	NOUN
ejpam-2488	327	11	.	.	PUNCT
ejpam-2488	328	1	then	then	ADV
ejpam-2488	328	2	{	{	PUNCT
ejpam-2488	328	3	xn}n∈n	xn}n∈n	X
ejpam-2488	328	4	is	be	AUX
ejpam-2488	328	5	a	a	DET
ejpam-2488	328	6	net	net	NOUN
ejpam-2488	328	7	in	in	ADP
ejpam-2488	328	8	cl∗e	cl∗e	PROPN
ejpam-2488	328	9	(	(	PUNCT
ejpam-2488	328	10	{	{	PUNCT
ejpam-2488	328	11	y	y	NOUN
ejpam-2488	328	12	}	}	PUNCT
ejpam-2488	328	13	)	)	PUNCT
ejpam-2488	328	14	.	.	PUNCT
ejpam-2488	329	1	since	since	SCONJ
ejpam-2488	329	2	{	{	PUNCT
ejpam-2488	329	3	xn}n∈n	xn}n∈n	PART
ejpam-2488	329	4	e	e	X
ejpam-2488	329	5	-	-	NOUN
ejpam-2488	329	6	i	i	PRON
ejpam-2488	329	7	-converges	-converge	NOUN
ejpam-2488	329	8	to	to	ADP
ejpam-2488	329	9	y	y	PROPN
ejpam-2488	329	10	,	,	PUNCT
ejpam-2488	329	11	then	then	ADV
ejpam-2488	329	12	{	{	PUNCT
ejpam-2488	329	13	xn}n∈n	xn}n∈n	PUNCT
ejpam-2488	329	14	e	e	X
ejpam-2488	329	15	-	-	NOUN
ejpam-2488	329	16	i	i	PRON
ejpam-2488	329	17	-converges	-converge	NOUN
ejpam-2488	329	18	to	to	ADP
ejpam-2488	329	19	x	x	PUNCT
ejpam-2488	329	20	and	and	CCONJ
ejpam-2488	329	21	this	this	PRON
ejpam-2488	329	22	implies	imply	VERB
ejpam-2488	329	23	that	that	SCONJ
ejpam-2488	329	24	x	x	PUNCT
ejpam-2488	329	25	∈	∈	PRON
ejpam-2488	329	26	cl∗e	cl∗e	X
ejpam-2488	329	27	(	(	PUNCT
ejpam-2488	329	28	{	{	PUNCT
ejpam-2488	329	29	y	y	NOUN
ejpam-2488	329	30	}	}	PUNCT
ejpam-2488	329	31	)	)	PUNCT
ejpam-2488	329	32	.	.	PUNCT
ejpam-2488	330	1	theorem	theorem	VERB
ejpam-2488	330	2	7	7	NUM
ejpam-2488	330	3	.	.	X
ejpam-2488	330	4	for	for	ADP
ejpam-2488	330	5	an	an	DET
ejpam-2488	330	6	ideal	ideal	ADJ
ejpam-2488	330	7	topological	topological	ADJ
ejpam-2488	330	8	space	space	NOUN
ejpam-2488	330	9	(	(	PUNCT
ejpam-2488	330	10	x	x	X
ejpam-2488	330	11	,	,	PUNCT
ejpam-2488	330	12	τ	τ	PROPN
ejpam-2488	330	13	,	,	PUNCT
ejpam-2488	330	14	i	i	PROPN
ejpam-2488	330	15	)	)	PUNCT
ejpam-2488	330	16	,	,	PUNCT
ejpam-2488	330	17	the	the	DET
ejpam-2488	330	18	following	follow	VERB
ejpam-2488	330	19	properties	property	NOUN
ejpam-2488	330	20	are	be	AUX
ejpam-2488	330	21	equivalent	equivalent	ADJ
ejpam-2488	330	22	:	:	PUNCT
ejpam-2488	330	23	(	(	PUNCT
ejpam-2488	330	24	i	i	NOUN
ejpam-2488	330	25	)	)	PUNCT
ejpam-2488	330	26	(	(	PUNCT
ejpam-2488	330	27	x	x	X
ejpam-2488	330	28	,	,	PUNCT
ejpam-2488	330	29	τ	τ	PROPN
ejpam-2488	330	30	,	,	PUNCT
ejpam-2488	330	31	i	i	PROPN
ejpam-2488	330	32	)	)	PUNCT
ejpam-2488	330	33	is	be	AUX
ejpam-2488	330	34	an	an	DET
ejpam-2488	330	35	e	e	NOUN
ejpam-2488	330	36	-	-	NOUN
ejpam-2488	330	37	i	i	PRON
ejpam-2488	330	38	-r0	-r0	PROPN
ejpam-2488	330	39	space	space	NOUN
ejpam-2488	330	40	,	,	PUNCT
ejpam-2488	330	41	(	(	PUNCT
ejpam-2488	330	42	ii	ii	NOUN
ejpam-2488	330	43	)	)	PUNCT
ejpam-2488	330	44	if	if	SCONJ
ejpam-2488	330	45	x	x	PRON
ejpam-2488	330	46	,	,	PUNCT
ejpam-2488	330	47	y	y	PROPN
ejpam-2488	330	48	∈	∈	PROPN
ejpam-2488	331	1	x	x	INTJ
ejpam-2488	331	2	,	,	PUNCT
ejpam-2488	331	3	then	then	ADV
ejpam-2488	331	4	y	y	PROPN
ejpam-2488	331	5	∈	∈	PROPN
ejpam-2488	331	6	cl∗e	cl∗e	PROPN
ejpam-2488	331	7	(	(	PUNCT
ejpam-2488	331	8	{	{	PUNCT
ejpam-2488	331	9	x	x	NOUN
ejpam-2488	331	10	}	}	PUNCT
ejpam-2488	331	11	)	)	PUNCT
ejpam-2488	332	1	if	if	SCONJ
ejpam-2488	332	2	and	and	CCONJ
ejpam-2488	332	3	only	only	ADV
ejpam-2488	332	4	if	if	SCONJ
ejpam-2488	332	5	every	every	DET
ejpam-2488	332	6	net	net	NOUN
ejpam-2488	332	7	in	in	ADP
ejpam-2488	332	8	x	x	X
ejpam-2488	332	9	e	e	X
ejpam-2488	332	10	-	-	PROPN
ejpam-2488	332	11	i	i	PRON
ejpam-2488	332	12	-converging	-converge	VERB
ejpam-2488	332	13	to	to	ADP
ejpam-2488	332	14	y	y	PROPN
ejpam-2488	332	15	e	e	PROPN
ejpam-2488	332	16	-	-	PROPN
ejpam-2488	332	17	i	i	PRON
ejpam-2488	332	18	converges	converge	VERB
ejpam-2488	332	19	to	to	ADP
ejpam-2488	332	20	x.	x.	NOUN
ejpam-2488	332	21	proof	proof	NOUN
ejpam-2488	332	22	.	.	PUNCT
ejpam-2488	333	1	(	(	PUNCT
ejpam-2488	333	2	i	i	NOUN
ejpam-2488	333	3	)	)	PUNCT
ejpam-2488	333	4	⇒	⇒	PROPN
ejpam-2488	333	5	(	(	PUNCT
ejpam-2488	333	6	ii	ii	PROPN
ejpam-2488	333	7	):	):	PUNCT
ejpam-2488	333	8	let	let	VERB
ejpam-2488	333	9	x	x	PRON
ejpam-2488	333	10	,	,	PUNCT
ejpam-2488	333	11	y	y	PROPN
ejpam-2488	333	12	∈	∈	PROPN
ejpam-2488	333	13	x	x	PUNCT
ejpam-2488	333	14	such	such	ADJ
ejpam-2488	333	15	that	that	SCONJ
ejpam-2488	333	16	y	y	PROPN
ejpam-2488	333	17	∈	∈	PROPN
ejpam-2488	333	18	cl∗e	cl∗e	PROPN
ejpam-2488	333	19	(	(	PUNCT
ejpam-2488	333	20	{	{	PUNCT
ejpam-2488	333	21	x	x	NOUN
ejpam-2488	333	22	}	}	PUNCT
ejpam-2488	333	23	)	)	PUNCT
ejpam-2488	333	24	.	.	PUNCT
ejpam-2488	334	1	suppose	suppose	VERB
ejpam-2488	334	2	that	that	SCONJ
ejpam-2488	334	3	{	{	PUNCT
ejpam-2488	334	4	xα}α∈n	xα}α∈n	AUX
ejpam-2488	334	5	be	be	AUX
ejpam-2488	334	6	a	a	DET
ejpam-2488	334	7	net	net	NOUN
ejpam-2488	334	8	in	in	ADP
ejpam-2488	334	9	x	x	INTJ
ejpam-2488	334	10	such	such	ADJ
ejpam-2488	334	11	that	that	SCONJ
ejpam-2488	334	12	{	{	PUNCT
ejpam-2488	334	13	xα}α∈n	xα}α∈n	NUM
ejpam-2488	334	14	e	e	X
ejpam-2488	334	15	-	-	NOUN
ejpam-2488	334	16	i	i	PRON
ejpam-2488	334	17	-converges	-converge	NOUN
ejpam-2488	334	18	to	to	ADP
ejpam-2488	334	19	y	y	PROPN
ejpam-2488	334	20	.	.	PUNCT
ejpam-2488	335	1	since	since	SCONJ
ejpam-2488	335	2	y	y	PROPN
ejpam-2488	335	3	∈	∈	PROPN
ejpam-2488	335	4	cl∗e	cl∗e	PROPN
ejpam-2488	335	5	(	(	PUNCT
ejpam-2488	335	6	{	{	PUNCT
ejpam-2488	335	7	x	x	NOUN
ejpam-2488	335	8	}	}	PUNCT
ejpam-2488	335	9	)	)	PUNCT
ejpam-2488	335	10	,	,	PUNCT
ejpam-2488	335	11	by	by	ADP
ejpam-2488	335	12	theorem	theorem	NOUN
ejpam-2488	335	13	2	2	NUM
ejpam-2488	335	14	we	we	PRON
ejpam-2488	335	15	have	have	VERB
ejpam-2488	335	16	cl∗e	cl∗e	PROPN
ejpam-2488	335	17	(	(	PUNCT
ejpam-2488	335	18	{	{	PUNCT
ejpam-2488	335	19	x	x	NOUN
ejpam-2488	335	20	}	}	PUNCT
ejpam-2488	335	21	)	)	PUNCT
ejpam-2488	335	22	=	=	SYM
ejpam-2488	335	23	cl∗e	cl∗e	PROPN
ejpam-2488	335	24	(	(	PUNCT
ejpam-2488	335	25	{	{	PUNCT
ejpam-2488	335	26	y	y	NOUN
ejpam-2488	335	27	}	}	PUNCT
ejpam-2488	335	28	)	)	PUNCT
ejpam-2488	335	29	.	.	PUNCT
ejpam-2488	336	1	therefore	therefore	ADV
ejpam-2488	336	2	x	x	X
ejpam-2488	336	3	∈	∈	PROPN
ejpam-2488	336	4	cl∗e	cl∗e	X
ejpam-2488	336	5	(	(	PUNCT
ejpam-2488	336	6	{	{	PUNCT
ejpam-2488	336	7	y	y	NOUN
ejpam-2488	336	8	}	}	PUNCT
ejpam-2488	336	9	)	)	PUNCT
ejpam-2488	336	10	.	.	PUNCT
ejpam-2488	337	1	this	this	PRON
ejpam-2488	337	2	means	mean	VERB
ejpam-2488	337	3	that	that	SCONJ
ejpam-2488	337	4	{	{	PUNCT
ejpam-2488	337	5	xα}α∈n	xα}α∈n	NUM
ejpam-2488	337	6	e	e	X
ejpam-2488	337	7	-	-	NOUN
ejpam-2488	337	8	i	i	PRON
ejpam-2488	337	9	-converges	-converge	NOUN
ejpam-2488	337	10	to	to	ADP
ejpam-2488	337	11	x	x	X
ejpam-2488	337	12	.	.	PUNCT
ejpam-2488	338	1	conversely	conversely	ADV
ejpam-2488	338	2	,	,	PUNCT
ejpam-2488	338	3	let	let	VERB
ejpam-2488	338	4	x	x	PRON
ejpam-2488	338	5	,	,	PUNCT
ejpam-2488	338	6	y	y	PROPN
ejpam-2488	338	7	∈	∈	PROPN
ejpam-2488	338	8	x	x	PUNCT
ejpam-2488	338	9	such	such	ADJ
ejpam-2488	338	10	that	that	SCONJ
ejpam-2488	338	11	every	every	DET
ejpam-2488	338	12	net	net	NOUN
ejpam-2488	338	13	in	in	ADP
ejpam-2488	338	14	x	x	X
ejpam-2488	338	15	e	e	X
ejpam-2488	338	16	-	-	PROPN
ejpam-2488	338	17	i	i	PRON
ejpam-2488	338	18	-converging	-converge	VERB
ejpam-2488	338	19	to	to	ADP
ejpam-2488	338	20	y	y	PROPN
ejpam-2488	338	21	e	e	PROPN
ejpam-2488	338	22	-	-	PROPN
ejpam-2488	338	23	i	i	PRON
ejpam-2488	338	24	converges	converge	VERB
ejpam-2488	338	25	to	to	ADP
ejpam-2488	338	26	x	x	X
ejpam-2488	338	27	.	.	PUNCT
ejpam-2488	339	1	then	then	ADV
ejpam-2488	339	2	x	x	SYM
ejpam-2488	339	3	∈	∈	PROPN
ejpam-2488	339	4	cl∗e	cl∗e	X
ejpam-2488	339	5	(	(	PUNCT
ejpam-2488	339	6	{	{	PUNCT
ejpam-2488	339	7	y	y	NOUN
ejpam-2488	339	8	}	}	PUNCT
ejpam-2488	339	9	)	)	PUNCT
ejpam-2488	339	10	by	by	ADP
ejpam-2488	339	11	lemma	lemma	PROPN
ejpam-2488	339	12	4	4	NUM
ejpam-2488	339	13	.	.	PUNCT
ejpam-2488	339	14	by	by	ADP
ejpam-2488	339	15	theorem	theorem	NOUN
ejpam-2488	339	16	2	2	NUM
ejpam-2488	339	17	,	,	PUNCT
ejpam-2488	339	18	we	we	PRON
ejpam-2488	339	19	have	have	VERB
ejpam-2488	339	20	cl∗e	cl∗e	PROPN
ejpam-2488	339	21	(	(	PUNCT
ejpam-2488	339	22	{	{	PUNCT
ejpam-2488	339	23	x	x	NOUN
ejpam-2488	339	24	}	}	PUNCT
ejpam-2488	339	25	)	)	PUNCT
ejpam-2488	339	26	=	=	SYM
ejpam-2488	340	1	cl∗e	cl∗e	PROPN
ejpam-2488	340	2	(	(	PUNCT
ejpam-2488	340	3	{	{	PUNCT
ejpam-2488	340	4	y	y	NOUN
ejpam-2488	340	5	}	}	PUNCT
ejpam-2488	340	6	)	)	PUNCT
ejpam-2488	340	7	.	.	PUNCT
ejpam-2488	341	1	therefore	therefore	ADV
ejpam-2488	341	2	w.	w.	PROPN
ejpam-2488	341	3	al	al	PROPN
ejpam-2488	341	4	-	-	PUNCT
ejpam-2488	341	5	omeri	omeri	ADJ
ejpam-2488	341	6	,	,	PUNCT
ejpam-2488	341	7	m.	m.	NOUN
ejpam-2488	341	8	noorani	noorani	PROPN
ejpam-2488	341	9	,	,	PUNCT
ejpam-2488	341	10	a.	a.	PROPN
ejpam-2488	341	11	al	al	PROPN
ejpam-2488	341	12	-	-	PUNCT
ejpam-2488	341	13	omari	omari	PROPN
ejpam-2488	341	14	,	,	PUNCT
ejpam-2488	341	15	and	and	CCONJ
ejpam-2488	341	16	t.	t.	PROPN
ejpam-2488	341	17	noiri	noiri	PROPN
ejpam-2488	341	18	/	/	SYM
ejpam-2488	341	19	eur	eur	PROPN
ejpam-2488	341	20	.	.	PUNCT
ejpam-2488	342	1	j.	j.	PROPN
ejpam-2488	342	2	pure	pure	PROPN
ejpam-2488	342	3	appl	appl	PROPN
ejpam-2488	342	4	.	.	PROPN
ejpam-2488	342	5	math	math	PROPN
ejpam-2488	342	6	,	,	PUNCT
ejpam-2488	342	7	8	8	NUM
ejpam-2488	342	8	(	(	PUNCT
ejpam-2488	342	9	2015	2015	NUM
ejpam-2488	342	10	)	)	PUNCT
ejpam-2488	342	11	,	,	PUNCT
ejpam-2488	342	12	502	502	NUM
ejpam-2488	342	13	-	-	SYM
ejpam-2488	342	14	513	513	NUM
ejpam-2488	342	15	509	509	NUM
ejpam-2488	342	16	y	y	PROPN
ejpam-2488	342	17	∈	∈	PROPN
ejpam-2488	342	18	cl∗e	cl∗e	X
ejpam-2488	342	19	(	(	PUNCT
ejpam-2488	342	20	{	{	PUNCT
ejpam-2488	342	21	x	x	NOUN
ejpam-2488	342	22	}	}	PUNCT
ejpam-2488	342	23	)	)	PUNCT
ejpam-2488	342	24	.	.	PUNCT
ejpam-2488	343	1	(	(	PUNCT
ejpam-2488	343	2	ii)⇒	ii)⇒	PROPN
ejpam-2488	343	3	(	(	PUNCT
ejpam-2488	343	4	i	i	NOUN
ejpam-2488	343	5	):	):	PUNCT
ejpam-2488	343	6	assume	assume	VERB
ejpam-2488	343	7	that	that	SCONJ
ejpam-2488	343	8	x	x	PRON
ejpam-2488	343	9	and	and	CCONJ
ejpam-2488	343	10	y	y	PROPN
ejpam-2488	343	11	are	be	AUX
ejpam-2488	343	12	any	any	DET
ejpam-2488	343	13	two	two	NUM
ejpam-2488	343	14	points	point	NOUN
ejpam-2488	343	15	of	of	ADP
ejpam-2488	343	16	x	x	SYM
ejpam-2488	343	17	such	such	ADJ
ejpam-2488	343	18	that	that	SCONJ
ejpam-2488	343	19	cl∗e	cl∗e	PROPN
ejpam-2488	343	20	(	(	PUNCT
ejpam-2488	343	21	{	{	PUNCT
ejpam-2488	343	22	x	x	NOUN
ejpam-2488	343	23	}	}	PUNCT
ejpam-2488	343	24	)	)	PUNCT
ejpam-2488	343	25	∩	∩	PROPN
ejpam-2488	343	26	cl∗e	cl∗e	X
ejpam-2488	343	27	(	(	PUNCT
ejpam-2488	343	28	{	{	PUNCT
ejpam-2488	343	29	y	y	NOUN
ejpam-2488	343	30	}	}	PUNCT
ejpam-2488	343	31	)	)	PUNCT
ejpam-2488	343	32	6=	6=	NUM
ejpam-2488	343	33	;	;	PUNCT
ejpam-2488	343	34	.	.	PUNCT
ejpam-2488	344	1	let	let	VERB
ejpam-2488	344	2	z	z	NOUN
ejpam-2488	344	3	∈	∈	PROPN
ejpam-2488	344	4	cl∗e	cl∗e	X
ejpam-2488	344	5	(	(	PUNCT
ejpam-2488	344	6	{	{	PUNCT
ejpam-2488	344	7	x	x	NOUN
ejpam-2488	344	8	}	}	PUNCT
ejpam-2488	344	9	)	)	PUNCT
ejpam-2488	344	10	∩	∩	PROPN
ejpam-2488	344	11	cl∗e	cl∗e	X
ejpam-2488	344	12	(	(	PUNCT
ejpam-2488	344	13	{	{	PUNCT
ejpam-2488	344	14	y	y	NOUN
ejpam-2488	344	15	}	}	PUNCT
ejpam-2488	344	16	)	)	PUNCT
ejpam-2488	344	17	.	.	PUNCT
ejpam-2488	345	1	so	so	ADV
ejpam-2488	345	2	there	there	PRON
ejpam-2488	345	3	exists	exist	VERB
ejpam-2488	345	4	a	a	DET
ejpam-2488	345	5	net	net	NOUN
ejpam-2488	345	6	{	{	PUNCT
ejpam-2488	345	7	xα}α∈n	xα}α∈n	NUM
ejpam-2488	345	8	in	in	ADP
ejpam-2488	345	9	cl∗e	cl∗e	PROPN
ejpam-2488	345	10	(	(	PUNCT
ejpam-2488	345	11	{	{	PUNCT
ejpam-2488	345	12	x	x	NOUN
ejpam-2488	345	13	}	}	PUNCT
ejpam-2488	345	14	)	)	PUNCT
ejpam-2488	345	15	such	such	ADJ
ejpam-2488	345	16	that	that	SCONJ
ejpam-2488	345	17	{	{	PUNCT
ejpam-2488	345	18	xα}α∈n	xα}α∈n	NUM
ejpam-2488	345	19	e	e	X
ejpam-2488	345	20	-	-	NOUN
ejpam-2488	345	21	i	i	PRON
ejpam-2488	345	22	-converges	-converge	NOUN
ejpam-2488	345	23	to	to	ADP
ejpam-2488	345	24	z.	z.	PROPN
ejpam-2488	345	25	since	since	SCONJ
ejpam-2488	345	26	z	z	PROPN
ejpam-2488	345	27	∈	∈	PROPN
ejpam-2488	345	28	cl∗e	cl∗e	X
ejpam-2488	345	29	(	(	PUNCT
ejpam-2488	345	30	{	{	PUNCT
ejpam-2488	345	31	y	y	NOUN
ejpam-2488	345	32	}	}	PUNCT
ejpam-2488	345	33	)	)	PUNCT
ejpam-2488	345	34	,	,	PUNCT
ejpam-2488	345	35	then	then	ADV
ejpam-2488	345	36	{	{	PUNCT
ejpam-2488	345	37	xα}α∈n	xα}α∈n	PROPN
ejpam-2488	345	38	e	e	X
ejpam-2488	345	39	-	-	NOUN
ejpam-2488	345	40	i	i	PRON
ejpam-2488	345	41	-converges	-converge	NOUN
ejpam-2488	345	42	to	to	ADP
ejpam-2488	345	43	y	y	PROPN
ejpam-2488	345	44	.	.	PUNCT
ejpam-2488	346	1	it	it	PRON
ejpam-2488	346	2	follows	follow	VERB
ejpam-2488	346	3	that	that	SCONJ
ejpam-2488	346	4	y	y	PROPN
ejpam-2488	346	5	∈	∈	PROPN
ejpam-2488	346	6	cl∗e	cl∗e	PROPN
ejpam-2488	346	7	(	(	PUNCT
ejpam-2488	346	8	{	{	PUNCT
ejpam-2488	346	9	x	x	NOUN
ejpam-2488	346	10	}	}	PUNCT
ejpam-2488	346	11	)	)	PUNCT
ejpam-2488	346	12	.	.	PUNCT
ejpam-2488	347	1	by	by	ADP
ejpam-2488	347	2	the	the	DET
ejpam-2488	347	3	same	same	ADJ
ejpam-2488	347	4	token	token	NOUN
ejpam-2488	347	5	we	we	PRON
ejpam-2488	347	6	obtain	obtain	VERB
ejpam-2488	347	7	x	x	SYM
ejpam-2488	347	8	∈	∈	PROPN
ejpam-2488	347	9	cl∗e	cl∗e	X
ejpam-2488	347	10	(	(	PUNCT
ejpam-2488	347	11	{	{	PUNCT
ejpam-2488	347	12	y	y	NOUN
ejpam-2488	347	13	}	}	PUNCT
ejpam-2488	347	14	)	)	PUNCT
ejpam-2488	347	15	.	.	PUNCT
ejpam-2488	348	1	therefore	therefore	ADV
ejpam-2488	348	2	cl∗e	cl∗e	PROPN
ejpam-2488	348	3	(	(	PUNCT
ejpam-2488	348	4	{	{	PUNCT
ejpam-2488	348	5	x	x	NOUN
ejpam-2488	348	6	}	}	PUNCT
ejpam-2488	348	7	)	)	PUNCT
ejpam-2488	348	8	=	=	SYM
ejpam-2488	348	9	cl∗e	cl∗e	PROPN
ejpam-2488	348	10	(	(	PUNCT
ejpam-2488	348	11	{	{	PUNCT
ejpam-2488	348	12	y	y	NOUN
ejpam-2488	348	13	}	}	PUNCT
ejpam-2488	348	14	)	)	PUNCT
ejpam-2488	348	15	and	and	CCONJ
ejpam-2488	348	16	by	by	ADP
ejpam-2488	348	17	theorem	theorem	NOUN
ejpam-2488	348	18	2	2	NUM
ejpam-2488	348	19	(	(	PUNCT
ejpam-2488	348	20	x	x	PROPN
ejpam-2488	348	21	,	,	PUNCT
ejpam-2488	348	22	τ	τ	PROPN
ejpam-2488	348	23	,	,	PUNCT
ejpam-2488	348	24	i	i	PROPN
ejpam-2488	348	25	)	)	PUNCT
ejpam-2488	348	26	is	be	AUX
ejpam-2488	348	27	an	an	DET
ejpam-2488	348	28	e	e	NOUN
ejpam-2488	348	29	-	-	NOUN
ejpam-2488	348	30	i	i	PRON
ejpam-2488	348	31	-r0	-r0	PROPN
ejpam-2488	348	32	space	space	NOUN
ejpam-2488	348	33	.	.	PUNCT
ejpam-2488	349	1	4	4	X
ejpam-2488	349	2	.	.	X
ejpam-2488	349	3	on	on	ADP
ejpam-2488	349	4	e	e	PROPN
ejpam-2488	349	5	-	-	PROPN
ejpam-2488	349	6	i	i	PRON
ejpam-2488	349	7	-r1	-r1	NOUN
ejpam-2488	349	8	spaces	space	VERB
ejpam-2488	349	9	definition	definition	NOUN
ejpam-2488	349	10	8	8	NUM
ejpam-2488	349	11	.	.	PUNCT
ejpam-2488	350	1	an	an	DET
ejpam-2488	350	2	ideal	ideal	ADJ
ejpam-2488	350	3	topological	topological	ADJ
ejpam-2488	350	4	space	space	NOUN
ejpam-2488	350	5	(	(	PUNCT
ejpam-2488	350	6	x	x	X
ejpam-2488	350	7	,	,	PUNCT
ejpam-2488	350	8	τ	τ	PROPN
ejpam-2488	350	9	,	,	PUNCT
ejpam-2488	350	10	i	i	PROPN
ejpam-2488	350	11	)	)	PUNCT
ejpam-2488	350	12	is	be	AUX
ejpam-2488	350	13	said	say	VERB
ejpam-2488	350	14	to	to	PART
ejpam-2488	350	15	be	be	AUX
ejpam-2488	350	16	e	e	NOUN
ejpam-2488	350	17	-	-	NOUN
ejpam-2488	350	18	i	i	PRON
ejpam-2488	350	19	-r1	-r1	NOUN
ejpam-2488	350	20	if	if	SCONJ
ejpam-2488	350	21	for	for	ADP
ejpam-2488	350	22	x	x	X
ejpam-2488	350	23	,	,	PUNCT
ejpam-2488	350	24	y	y	PROPN
ejpam-2488	350	25	in	in	ADP
ejpam-2488	350	26	x	x	PUNCT
ejpam-2488	350	27	with	with	ADP
ejpam-2488	350	28	cl∗e	cl∗e	PROPN
ejpam-2488	350	29	(	(	PUNCT
ejpam-2488	350	30	{	{	PUNCT
ejpam-2488	350	31	x	x	NOUN
ejpam-2488	350	32	}	}	PUNCT
ejpam-2488	350	33	)	)	PUNCT
ejpam-2488	351	1	6=	6=	ADP
ejpam-2488	351	2	cl∗e	cl∗e	X
ejpam-2488	351	3	(	(	PUNCT
ejpam-2488	351	4	{	{	PUNCT
ejpam-2488	351	5	y	y	NOUN
ejpam-2488	351	6	}	}	PUNCT
ejpam-2488	351	7	)	)	PUNCT
ejpam-2488	351	8	,	,	PUNCT
ejpam-2488	351	9	there	there	PRON
ejpam-2488	351	10	exist	exist	VERB
ejpam-2488	351	11	disjoint	disjoint	NOUN
ejpam-2488	351	12	e	e	NOUN
ejpam-2488	351	13	-	-	ADJ
ejpam-2488	351	14	i	i	PRON
ejpam-2488	351	15	-open	-open	NOUN
ejpam-2488	351	16	sets	set	VERB
ejpam-2488	351	17	u	u	NOUN
ejpam-2488	351	18	and	and	CCONJ
ejpam-2488	351	19	v	v	ADP
ejpam-2488	351	20	such	such	ADJ
ejpam-2488	351	21	that	that	SCONJ
ejpam-2488	351	22	cl∗e	cl∗e	PROPN
ejpam-2488	351	23	(	(	PUNCT
ejpam-2488	351	24	{	{	PUNCT
ejpam-2488	351	25	x	x	NOUN
ejpam-2488	351	26	}	}	PUNCT
ejpam-2488	351	27	)	)	PUNCT
ejpam-2488	351	28	is	be	AUX
ejpam-2488	351	29	a	a	DET
ejpam-2488	351	30	subset	subset	NOUN
ejpam-2488	351	31	of	of	ADP
ejpam-2488	351	32	u	u	NOUN
ejpam-2488	351	33	and	and	CCONJ
ejpam-2488	351	34	cl∗e	cl∗e	PROPN
ejpam-2488	351	35	(	(	PUNCT
ejpam-2488	351	36	{	{	PUNCT
ejpam-2488	351	37	y	y	NOUN
ejpam-2488	351	38	}	}	PUNCT
ejpam-2488	351	39	)	)	PUNCT
ejpam-2488	351	40	is	be	AUX
ejpam-2488	351	41	a	a	DET
ejpam-2488	351	42	subset	subset	NOUN
ejpam-2488	351	43	of	of	ADP
ejpam-2488	351	44	v	v	NOUN
ejpam-2488	351	45	.	.	PUNCT
ejpam-2488	352	1	proposition	proposition	NOUN
ejpam-2488	352	2	2	2	NUM
ejpam-2488	352	3	.	.	PUNCT
ejpam-2488	353	1	if	if	SCONJ
ejpam-2488	353	2	(	(	PUNCT
ejpam-2488	353	3	x	x	X
ejpam-2488	353	4	,	,	PUNCT
ejpam-2488	353	5	τ	τ	PROPN
ejpam-2488	353	6	,	,	PUNCT
ejpam-2488	353	7	i	i	PROPN
ejpam-2488	353	8	)	)	PUNCT
ejpam-2488	353	9	is	be	AUX
ejpam-2488	353	10	e	e	NOUN
ejpam-2488	353	11	-	-	NOUN
ejpam-2488	353	12	i	i	PRON
ejpam-2488	353	13	-r1	-r1	NOUN
ejpam-2488	353	14	,	,	PUNCT
ejpam-2488	353	15	then	then	ADV
ejpam-2488	353	16	it	it	PRON
ejpam-2488	353	17	is	be	AUX
ejpam-2488	353	18	e	e	NOUN
ejpam-2488	353	19	-	-	PUNCT
ejpam-2488	353	20	i	i	PRON
ejpam-2488	353	21	-r0	-r0	NOUN
ejpam-2488	353	22	.	.	PUNCT
ejpam-2488	354	1	proof	proof	NOUN
ejpam-2488	354	2	.	.	PUNCT
ejpam-2488	355	1	let	let	VERB
ejpam-2488	355	2	u	u	PRON
ejpam-2488	355	3	∈	∈	PROPN
ejpam-2488	355	4	eio(x	eio(x	X
ejpam-2488	355	5	,	,	PUNCT
ejpam-2488	355	6	x	x	NOUN
ejpam-2488	355	7	)	)	PUNCT
ejpam-2488	355	8	.	.	PUNCT
ejpam-2488	356	1	if	if	SCONJ
ejpam-2488	356	2	y	y	PROPN
ejpam-2488	356	3	/∈	/∈	PUNCT
ejpam-2488	356	4	u	u	PROPN
ejpam-2488	356	5	,	,	PUNCT
ejpam-2488	356	6	since	since	SCONJ
ejpam-2488	356	7	x	x	PROPN
ejpam-2488	356	8	/∈	/∈	PROPN
ejpam-2488	356	9	cl∗e	cl∗e	PROPN
ejpam-2488	356	10	(	(	PUNCT
ejpam-2488	356	11	{	{	PUNCT
ejpam-2488	356	12	y	y	NOUN
ejpam-2488	356	13	}	}	PUNCT
ejpam-2488	356	14	)	)	PUNCT
ejpam-2488	356	15	,	,	PUNCT
ejpam-2488	356	16	we	we	PRON
ejpam-2488	356	17	have	have	VERB
ejpam-2488	356	18	cl∗e	cl∗e	PROPN
ejpam-2488	356	19	(	(	PUNCT
ejpam-2488	356	20	{	{	PUNCT
ejpam-2488	356	21	x	x	NOUN
ejpam-2488	356	22	}	}	PUNCT
ejpam-2488	356	23	)	)	PUNCT
ejpam-2488	356	24	6=	6=	ADP
ejpam-2488	357	1	cl∗e	cl∗e	X
ejpam-2488	357	2	(	(	PUNCT
ejpam-2488	357	3	{	{	PUNCT
ejpam-2488	357	4	y	y	NOUN
ejpam-2488	357	5	}	}	PUNCT
ejpam-2488	357	6	)	)	PUNCT
ejpam-2488	357	7	.	.	PUNCT
ejpam-2488	358	1	so	so	ADV
ejpam-2488	358	2	,	,	PUNCT
ejpam-2488	358	3	there	there	PRON
ejpam-2488	358	4	exists	exist	VERB
ejpam-2488	358	5	an	an	DET
ejpam-2488	358	6	e	e	NOUN
ejpam-2488	358	7	-	-	NOUN
ejpam-2488	358	8	i	i	PRON
ejpam-2488	358	9	-open	-open	NOUN
ejpam-2488	358	10	set	set	VERB
ejpam-2488	358	11	vy	vy	ADP
ejpam-2488	359	1	such	such	ADJ
ejpam-2488	359	2	that	that	SCONJ
ejpam-2488	359	3	cl∗e	cl∗e	PROPN
ejpam-2488	359	4	(	(	PUNCT
ejpam-2488	359	5	{	{	PUNCT
ejpam-2488	359	6	y	y	NOUN
ejpam-2488	359	7	}	}	PUNCT
ejpam-2488	359	8	)	)	PUNCT
ejpam-2488	359	9	⊂	⊂	PROPN
ejpam-2488	359	10	vy	vy	PROPN
ejpam-2488	359	11	and	and	CCONJ
ejpam-2488	359	12	x	x	PROPN
ejpam-2488	359	13	/∈	/∈	PUNCT
ejpam-2488	359	14	vy	vy	INTJ
ejpam-2488	359	15	,	,	PUNCT
ejpam-2488	359	16	which	which	PRON
ejpam-2488	359	17	implies	imply	VERB
ejpam-2488	359	18	y	y	PROPN
ejpam-2488	359	19	/∈	/∈	PUNCT
ejpam-2488	359	20	cl∗e	cl∗e	PROPN
ejpam-2488	359	21	(	(	PUNCT
ejpam-2488	359	22	{	{	PUNCT
ejpam-2488	359	23	x	x	NOUN
ejpam-2488	359	24	}	}	PUNCT
ejpam-2488	359	25	)	)	PUNCT
ejpam-2488	359	26	.	.	PUNCT
ejpam-2488	360	1	thus	thus	ADV
ejpam-2488	360	2	cl∗e	cl∗e	PRON
ejpam-2488	360	3	(	(	PUNCT
ejpam-2488	360	4	{	{	PUNCT
ejpam-2488	360	5	x	x	NOUN
ejpam-2488	360	6	}	}	PUNCT
ejpam-2488	360	7	)	)	PUNCT
ejpam-2488	360	8	⊂	⊂	PROPN
ejpam-2488	360	9	u	u	PROPN
ejpam-2488	360	10	.	.	PUNCT
ejpam-2488	361	1	therefore	therefore	ADV
ejpam-2488	361	2	(	(	PUNCT
ejpam-2488	361	3	x	x	X
ejpam-2488	361	4	,	,	PUNCT
ejpam-2488	361	5	τ	τ	PROPN
ejpam-2488	361	6	,	,	PUNCT
ejpam-2488	361	7	i	i	PROPN
ejpam-2488	361	8	)	)	PUNCT
ejpam-2488	361	9	is	be	AUX
ejpam-2488	361	10	e	e	NOUN
ejpam-2488	361	11	-	-	PUNCT
ejpam-2488	361	12	i	i	PRON
ejpam-2488	361	13	-r0	-r0	PROPN
ejpam-2488	361	14	.	.	PUNCT
ejpam-2488	362	1	theorem	theorem	VERB
ejpam-2488	362	2	8	8	NUM
ejpam-2488	362	3	.	.	PUNCT
ejpam-2488	363	1	an	an	DET
ejpam-2488	363	2	ideal	ideal	ADJ
ejpam-2488	363	3	topological	topological	ADJ
ejpam-2488	363	4	space	space	NOUN
ejpam-2488	363	5	(	(	PUNCT
ejpam-2488	363	6	x	x	X
ejpam-2488	363	7	,	,	PUNCT
ejpam-2488	363	8	τ	τ	PROPN
ejpam-2488	363	9	,	,	PUNCT
ejpam-2488	363	10	i	i	PROPN
ejpam-2488	363	11	)	)	PUNCT
ejpam-2488	363	12	is	be	AUX
ejpam-2488	363	13	e	e	NOUN
ejpam-2488	363	14	-	-	NOUN
ejpam-2488	363	15	i	i	PRON
ejpam-2488	363	16	-r1	-r1	NOUN
ejpam-2488	363	17	if	if	SCONJ
ejpam-2488	363	18	and	and	CCONJ
ejpam-2488	363	19	only	only	ADV
ejpam-2488	363	20	if	if	SCONJ
ejpam-2488	363	21	for	for	ADP
ejpam-2488	363	22	x	x	X
ejpam-2488	363	23	,	,	PUNCT
ejpam-2488	363	24	y	y	PROPN
ejpam-2488	363	25	∈	∈	PROPN
ejpam-2488	363	26	x	x	X
ejpam-2488	363	27	,	,	PUNCT
ejpam-2488	363	28	ieker({x	ieker({x	PROPN
ejpam-2488	363	29	}	}	PUNCT
ejpam-2488	363	30	)	)	PUNCT
ejpam-2488	363	31	6=	6=	ADP
ejpam-2488	363	32	ieker({y	ieker({y	PROPN
ejpam-2488	363	33	}	}	PUNCT
ejpam-2488	363	34	)	)	PUNCT
ejpam-2488	363	35	,	,	PUNCT
ejpam-2488	363	36	there	there	PRON
ejpam-2488	363	37	exist	exist	VERB
ejpam-2488	363	38	disjoint	disjoint	NOUN
ejpam-2488	363	39	e	e	NOUN
ejpam-2488	363	40	-	-	ADJ
ejpam-2488	363	41	i	i	PRON
ejpam-2488	363	42	-open	-open	NOUN
ejpam-2488	363	43	sets	set	VERB
ejpam-2488	363	44	u	u	NOUN
ejpam-2488	363	45	and	and	CCONJ
ejpam-2488	363	46	v	v	ADP
ejpam-2488	363	47	such	such	ADJ
ejpam-2488	363	48	that	that	SCONJ
ejpam-2488	363	49	cl∗e	cl∗e	PROPN
ejpam-2488	363	50	(	(	PUNCT
ejpam-2488	363	51	{	{	PUNCT
ejpam-2488	363	52	x	x	NOUN
ejpam-2488	363	53	}	}	PUNCT
ejpam-2488	363	54	)	)	PUNCT
ejpam-2488	363	55	⊂	⊂	PROPN
ejpam-2488	363	56	u	u	PROPN
ejpam-2488	363	57	and	and	CCONJ
ejpam-2488	363	58	cl∗e	cl∗e	PROPN
ejpam-2488	363	59	(	(	PUNCT
ejpam-2488	363	60	{	{	PUNCT
ejpam-2488	363	61	y	y	NOUN
ejpam-2488	363	62	}	}	PUNCT
ejpam-2488	363	63	)	)	PUNCT
ejpam-2488	364	1	⊂	⊂	PROPN
ejpam-2488	364	2	v	v	X
ejpam-2488	364	3	.	.	PUNCT
ejpam-2488	365	1	proof	proof	NOUN
ejpam-2488	365	2	.	.	PUNCT
ejpam-2488	366	1	it	it	PRON
ejpam-2488	366	2	follows	follow	VERB
ejpam-2488	366	3	from	from	ADP
ejpam-2488	366	4	lemma	lemma	PROPN
ejpam-2488	366	5	3	3	NUM
ejpam-2488	366	6	.	.	NOUN
ejpam-2488	366	7	remark	remark	NOUN
ejpam-2488	366	8	3	3	NUM
ejpam-2488	366	9	.	.	PUNCT
ejpam-2488	367	1	in	in	ADP
ejpam-2488	367	2	the	the	DET
ejpam-2488	367	3	following	follow	VERB
ejpam-2488	367	4	diagram	diagram	NOUN
ejpam-2488	367	5	we	we	PRON
ejpam-2488	367	6	denote	denote	VERB
ejpam-2488	367	7	by	by	ADP
ejpam-2488	367	8	arrows	arrow	NOUN
ejpam-2488	367	9	the	the	DET
ejpam-2488	367	10	implications	implication	NOUN
ejpam-2488	367	11	between	between	ADP
ejpam-2488	367	12	the	the	DET
ejpam-2488	367	13	separation	separation	NOUN
ejpam-2488	367	14	axioms	axiom	NOUN
ejpam-2488	367	15	which	which	PRON
ejpam-2488	367	16	we	we	PRON
ejpam-2488	367	17	have	have	AUX
ejpam-2488	367	18	introduced	introduce	VERB
ejpam-2488	367	19	and	and	CCONJ
ejpam-2488	367	20	discussed	discuss	VERB
ejpam-2488	367	21	in	in	ADP
ejpam-2488	367	22	this	this	DET
ejpam-2488	367	23	paper	paper	NOUN
ejpam-2488	367	24	and	and	CCONJ
ejpam-2488	367	25	examples	example	NOUN
ejpam-2488	367	26	show	show	VERB
ejpam-2488	367	27	that	that	SCONJ
ejpam-2488	367	28	no	no	DET
ejpam-2488	367	29	other	other	ADJ
ejpam-2488	367	30	implications	implication	NOUN
ejpam-2488	367	31	hold	hold	VERB
ejpam-2488	367	32	between	between	ADP
ejpam-2488	367	33	them	they	PRON
ejpam-2488	367	34	:	:	PUNCT
ejpam-2488	367	35	r1	r1	PROPN
ejpam-2488	367	36	//	//	SYM
ejpam-2488	367	37	�	�	PROPN
ejpam-2488	367	38	�	�	PROPN
ejpam-2488	367	39	r0	r0	PROPN
ejpam-2488	367	40	�	�	PROPN
ejpam-2488	367	41	�	�	PROPN
ejpam-2488	367	42	e	e	PROPN
ejpam-2488	367	43	-	-	PROPN
ejpam-2488	367	44	i	i	PRON
ejpam-2488	367	45	-r1	-r1	PROPN
ejpam-2488	367	46	�	�	PROPN
ejpam-2488	367	47	�	�	PROPN
ejpam-2488	367	48	//	//	NUM
ejpam-2488	367	49	e	e	PROPN
ejpam-2488	367	50	-	-	PROPN
ejpam-2488	367	51	i	i	PRON
ejpam-2488	367	52	-r0	-r0	PROPN
ejpam-2488	367	53	�	�	PROPN
ejpam-2488	367	54	�	�	PROPN
ejpam-2488	367	55	e	e	PROPN
ejpam-2488	367	56	-	-	NOUN
ejpam-2488	367	57	r1	r1	ADJ
ejpam-2488	367	58	//	//	NUM
ejpam-2488	367	59	e	e	NOUN
ejpam-2488	367	60	-	-	NOUN
ejpam-2488	367	61	r0	r0	ADJ
ejpam-2488	367	62	example	example	NOUN
ejpam-2488	367	63	3	3	X
ejpam-2488	367	64	.	.	PUNCT
ejpam-2488	368	1	let	let	VERB
ejpam-2488	368	2	x	x	PUNCT
ejpam-2488	368	3	=	=	PRON
ejpam-2488	368	4	{	{	PUNCT
ejpam-2488	368	5	a	a	DET
ejpam-2488	368	6	,	,	PUNCT
ejpam-2488	368	7	b	b	NOUN
ejpam-2488	368	8	,	,	PUNCT
ejpam-2488	368	9	c	c	NOUN
ejpam-2488	368	10	}	}	PUNCT
ejpam-2488	368	11	,	,	PUNCT
ejpam-2488	368	12	τ=	τ=	X
ejpam-2488	368	13	{	{	PUNCT
ejpam-2488	368	14	φ	φ	PROPN
ejpam-2488	368	15	,	,	PUNCT
ejpam-2488	368	16	{	{	PUNCT
ejpam-2488	368	17	a	a	X
ejpam-2488	368	18	}	}	PUNCT
ejpam-2488	368	19	,	,	PUNCT
ejpam-2488	368	20	{	{	PUNCT
ejpam-2488	368	21	b	b	NOUN
ejpam-2488	368	22	}	}	PUNCT
ejpam-2488	368	23	,	,	PUNCT
ejpam-2488	368	24	{	{	PUNCT
ejpam-2488	368	25	a	a	DET
ejpam-2488	368	26	,	,	PUNCT
ejpam-2488	368	27	b	b	NOUN
ejpam-2488	368	28	}	}	PUNCT
ejpam-2488	368	29	,	,	PUNCT
ejpam-2488	368	30	{	{	PUNCT
ejpam-2488	368	31	b	b	X
ejpam-2488	368	32	,	,	PUNCT
ejpam-2488	368	33	c	c	NOUN
ejpam-2488	368	34	}	}	PUNCT
ejpam-2488	368	35	,	,	PUNCT
ejpam-2488	368	36	x	x	SYM
ejpam-2488	368	37	}	}	PUNCT
ejpam-2488	368	38	and	and	CCONJ
ejpam-2488	368	39	i	i	PRON
ejpam-2488	368	40	=	=	PUNCT
ejpam-2488	368	41	{	{	PUNCT
ejpam-2488	368	42	φ	φ	PROPN
ejpam-2488	368	43	,	,	PUNCT
ejpam-2488	368	44	{	{	PUNCT
ejpam-2488	368	45	b	b	NOUN
ejpam-2488	368	46	}	}	PUNCT
ejpam-2488	368	47	}	}	PUNCT
ejpam-2488	368	48	.	.	PUNCT
ejpam-2488	369	1	eio	eio	PROPN
ejpam-2488	369	2	=	=	PRON
ejpam-2488	369	3	{	{	PUNCT
ejpam-2488	369	4	φ	φ	PROPN
ejpam-2488	369	5	,	,	PUNCT
ejpam-2488	369	6	{	{	PUNCT
ejpam-2488	369	7	a	a	X
ejpam-2488	369	8	}	}	PUNCT
ejpam-2488	369	9	,	,	PUNCT
ejpam-2488	369	10	{	{	PUNCT
ejpam-2488	369	11	b	b	NOUN
ejpam-2488	369	12	}	}	PUNCT
ejpam-2488	369	13	,	,	PUNCT
ejpam-2488	369	14	{	{	PUNCT
ejpam-2488	369	15	a	a	DET
ejpam-2488	369	16	,	,	PUNCT
ejpam-2488	369	17	b	b	NOUN
ejpam-2488	369	18	}	}	PUNCT
ejpam-2488	369	19	,	,	PUNCT
ejpam-2488	369	20	{	{	PUNCT
ejpam-2488	369	21	b	b	X
ejpam-2488	369	22	,	,	PUNCT
ejpam-2488	369	23	c	c	NOUN
ejpam-2488	369	24	}	}	PUNCT
ejpam-2488	369	25	,	,	PUNCT
ejpam-2488	369	26	x	x	SYM
ejpam-2488	369	27	}	}	PUNCT
ejpam-2488	369	28	.	.	PUNCT
ejpam-2488	370	1	then	then	ADV
ejpam-2488	370	2	(	(	PUNCT
ejpam-2488	370	3	x	x	X
ejpam-2488	370	4	,	,	PUNCT
ejpam-2488	370	5	τ	τ	PROPN
ejpam-2488	370	6	,	,	PUNCT
ejpam-2488	370	7	i	i	PROPN
ejpam-2488	370	8	)	)	PUNCT
ejpam-2488	370	9	is	be	AUX
ejpam-2488	370	10	e	e	NOUN
ejpam-2488	370	11	-	-	PUNCT
ejpam-2488	370	12	i	i	PRON
ejpam-2488	370	13	-r0	-r0	NOUN
ejpam-2488	370	14	but	but	CCONJ
ejpam-2488	370	15	not	not	PART
ejpam-2488	370	16	r0	r0	VERB
ejpam-2488	370	17	and	and	CCONJ
ejpam-2488	370	18	e	e	NOUN
ejpam-2488	370	19	-	-	PROPN
ejpam-2488	370	20	i	i	PRON
ejpam-2488	370	21	-r1	-r1	NOUN
ejpam-2488	370	22	.	.	PUNCT
ejpam-2488	370	23	example	example	NOUN
ejpam-2488	371	1	4	4	NUM
ejpam-2488	371	2	.	.	PUNCT
ejpam-2488	371	3	let	let	VERB
ejpam-2488	371	4	x	x	PUNCT
ejpam-2488	371	5	=	=	PRON
ejpam-2488	371	6	{	{	PUNCT
ejpam-2488	371	7	a	a	PRON
ejpam-2488	371	8	,	,	PUNCT
ejpam-2488	371	9	b	b	NOUN
ejpam-2488	371	10	,	,	PUNCT
ejpam-2488	371	11	c	c	NOUN
ejpam-2488	371	12	}	}	PUNCT
ejpam-2488	371	13	with	with	ADP
ejpam-2488	371	14	a	a	DET
ejpam-2488	371	15	topology	topology	NOUN
ejpam-2488	371	16	τ	τ	X
ejpam-2488	371	17	=	=	SYM
ejpam-2488	371	18	{	{	PUNCT
ejpam-2488	371	19	φ	φ	PROPN
ejpam-2488	371	20	,	,	PUNCT
ejpam-2488	371	21	x	x	INTJ
ejpam-2488	371	22	,	,	PUNCT
ejpam-2488	371	23	{	{	PUNCT
ejpam-2488	371	24	a	a	NOUN
ejpam-2488	371	25	}	}	PUNCT
ejpam-2488	371	26	,	,	PUNCT
ejpam-2488	371	27	{	{	PUNCT
ejpam-2488	371	28	b	b	NOUN
ejpam-2488	371	29	}	}	PUNCT
ejpam-2488	371	30	,	,	PUNCT
ejpam-2488	371	31	{	{	PUNCT
ejpam-2488	371	32	a	a	PRON
ejpam-2488	371	33	,	,	PUNCT
ejpam-2488	371	34	b	b	NOUN
ejpam-2488	371	35	}	}	PUNCT
ejpam-2488	371	36	}	}	PUNCT
ejpam-2488	371	37	and	and	CCONJ
ejpam-2488	371	38	i	i	PRON
ejpam-2488	371	39	=	=	PUNCT
ejpam-2488	371	40	{	{	PUNCT
ejpam-2488	371	41	φ	φ	PROPN
ejpam-2488	371	42	,	,	PUNCT
ejpam-2488	371	43	{	{	PUNCT
ejpam-2488	371	44	a	a	X
ejpam-2488	371	45	}	}	PUNCT
ejpam-2488	371	46	}	}	PUNCT
ejpam-2488	371	47	.	.	PUNCT
ejpam-2488	372	1	since	since	SCONJ
ejpam-2488	372	2	eio	eio	PROPN
ejpam-2488	372	3	=	=	PROPN
ejpam-2488	372	4	{	{	PUNCT
ejpam-2488	372	5	φ	φ	PROPN
ejpam-2488	372	6	,	,	PUNCT
ejpam-2488	372	7	x	x	INTJ
ejpam-2488	372	8	,	,	PUNCT
ejpam-2488	372	9	{	{	PUNCT
ejpam-2488	372	10	a	a	NOUN
ejpam-2488	372	11	}	}	PUNCT
ejpam-2488	372	12	,	,	PUNCT
ejpam-2488	372	13	{	{	PUNCT
ejpam-2488	372	14	b	b	NOUN
ejpam-2488	372	15	}	}	PUNCT
ejpam-2488	372	16	,	,	PUNCT
ejpam-2488	372	17	{	{	PUNCT
ejpam-2488	372	18	a	a	DET
ejpam-2488	372	19	,	,	PUNCT
ejpam-2488	372	20	b	b	NOUN
ejpam-2488	372	21	}	}	PUNCT
ejpam-2488	372	22	,	,	PUNCT
ejpam-2488	372	23	{	{	PUNCT
ejpam-2488	372	24	a	a	X
ejpam-2488	372	25	,	,	PUNCT
ejpam-2488	372	26	c	c	NOUN
ejpam-2488	372	27	}	}	PUNCT
ejpam-2488	372	28	,	,	PUNCT
ejpam-2488	372	29	{	{	PUNCT
ejpam-2488	372	30	b	b	X
ejpam-2488	372	31	,	,	PUNCT
ejpam-2488	372	32	c	c	NOUN
ejpam-2488	372	33	}	}	PUNCT
ejpam-2488	372	34	}	}	PUNCT
ejpam-2488	372	35	.	.	PUNCT
ejpam-2488	373	1	then	then	ADV
ejpam-2488	373	2	(	(	PUNCT
ejpam-2488	373	3	x	x	X
ejpam-2488	373	4	,	,	PUNCT
ejpam-2488	373	5	τ	τ	PROPN
ejpam-2488	373	6	,	,	PUNCT
ejpam-2488	373	7	i	i	PROPN
ejpam-2488	373	8	)	)	PUNCT
ejpam-2488	373	9	is	be	AUX
ejpam-2488	373	10	e	e	NOUN
ejpam-2488	373	11	-	-	PUNCT
ejpam-2488	373	12	i	i	PRON
ejpam-2488	373	13	-r0	-r0	NOUN
ejpam-2488	373	14	but	but	CCONJ
ejpam-2488	373	15	not	not	PART
ejpam-2488	373	16	r0	r0	NOUN
ejpam-2488	373	17	.	.	PUNCT
ejpam-2488	373	18	example	example	NOUN
ejpam-2488	374	1	5	5	NUM
ejpam-2488	374	2	.	.	PUNCT
ejpam-2488	374	3	let	let	VERB
ejpam-2488	374	4	x	x	PUNCT
ejpam-2488	374	5	=	=	PRON
ejpam-2488	374	6	{	{	PUNCT
ejpam-2488	374	7	a	a	PRON
ejpam-2488	374	8	,	,	PUNCT
ejpam-2488	374	9	b	b	NOUN
ejpam-2488	374	10	,	,	PUNCT
ejpam-2488	374	11	c	c	NOUN
ejpam-2488	374	12	}	}	PUNCT
ejpam-2488	374	13	with	with	ADP
ejpam-2488	374	14	a	a	DET
ejpam-2488	374	15	topology	topology	NOUN
ejpam-2488	374	16	τ	τ	X
ejpam-2488	374	17	=	=	SYM
ejpam-2488	374	18	{	{	PUNCT
ejpam-2488	374	19	φ	φ	PROPN
ejpam-2488	374	20	,	,	PUNCT
ejpam-2488	374	21	x	x	INTJ
ejpam-2488	374	22	,	,	PUNCT
ejpam-2488	374	23	{	{	PUNCT
ejpam-2488	374	24	a	a	PRON
ejpam-2488	374	25	,	,	PUNCT
ejpam-2488	374	26	b	b	NOUN
ejpam-2488	374	27	}	}	PUNCT
ejpam-2488	374	28	}	}	PUNCT
ejpam-2488	374	29	and	and	CCONJ
ejpam-2488	374	30	i	i	PRON
ejpam-2488	374	31	=	=	PUNCT
ejpam-2488	374	32	{	{	PUNCT
ejpam-2488	374	33	φ	φ	PROPN
ejpam-2488	374	34	,	,	PUNCT
ejpam-2488	374	35	{	{	PUNCT
ejpam-2488	374	36	c	c	NOUN
ejpam-2488	374	37	}	}	PUNCT
ejpam-2488	374	38	}	}	PUNCT
ejpam-2488	374	39	.	.	PUNCT
ejpam-2488	375	1	since	since	SCONJ
ejpam-2488	375	2	eio	eio	PROPN
ejpam-2488	375	3	=	=	PROPN
ejpam-2488	375	4	{	{	PUNCT
ejpam-2488	375	5	φ	φ	PROPN
ejpam-2488	375	6	,	,	PUNCT
ejpam-2488	375	7	x	x	INTJ
ejpam-2488	375	8	,	,	PUNCT
ejpam-2488	375	9	{	{	PUNCT
ejpam-2488	375	10	a	a	NOUN
ejpam-2488	375	11	}	}	PUNCT
ejpam-2488	375	12	,	,	PUNCT
ejpam-2488	375	13	{	{	PUNCT
ejpam-2488	375	14	b	b	NOUN
ejpam-2488	375	15	}	}	PUNCT
ejpam-2488	375	16	,	,	PUNCT
ejpam-2488	375	17	{	{	PUNCT
ejpam-2488	375	18	c	c	X
ejpam-2488	375	19	}	}	PUNCT
ejpam-2488	375	20	,	,	PUNCT
ejpam-2488	375	21	{	{	PUNCT
ejpam-2488	375	22	a	a	DET
ejpam-2488	375	23	,	,	PUNCT
ejpam-2488	375	24	b	b	NOUN
ejpam-2488	375	25	}	}	PUNCT
ejpam-2488	375	26	,	,	PUNCT
ejpam-2488	375	27	{	{	PUNCT
ejpam-2488	375	28	a	a	X
ejpam-2488	375	29	,	,	PUNCT
ejpam-2488	375	30	c	c	NOUN
ejpam-2488	375	31	}	}	PUNCT
ejpam-2488	375	32	,	,	PUNCT
ejpam-2488	375	33	{	{	PUNCT
ejpam-2488	375	34	b	b	X
ejpam-2488	375	35	,	,	PUNCT
ejpam-2488	375	36	c	c	NOUN
ejpam-2488	375	37	}	}	PUNCT
ejpam-2488	375	38	}	}	PUNCT
ejpam-2488	375	39	.	.	PUNCT
ejpam-2488	376	1	then	then	ADV
ejpam-2488	376	2	(	(	PUNCT
ejpam-2488	376	3	x	x	X
ejpam-2488	376	4	,	,	PUNCT
ejpam-2488	376	5	τ	τ	PROPN
ejpam-2488	376	6	,	,	PUNCT
ejpam-2488	376	7	i	i	PROPN
ejpam-2488	376	8	)	)	PUNCT
ejpam-2488	376	9	is	be	AUX
ejpam-2488	376	10	e	e	NOUN
ejpam-2488	376	11	-	-	NOUN
ejpam-2488	376	12	i	i	PRON
ejpam-2488	376	13	-r1	-r1	NOUN
ejpam-2488	376	14	but	but	CCONJ
ejpam-2488	376	15	not	not	PART
ejpam-2488	376	16	r1	r1	PROPN
ejpam-2488	376	17	.	.	PUNCT
ejpam-2488	377	1	w.	w.	PROPN
ejpam-2488	377	2	al	al	PROPN
ejpam-2488	377	3	-	-	PUNCT
ejpam-2488	377	4	omeri	omeri	ADJ
ejpam-2488	377	5	,	,	PUNCT
ejpam-2488	377	6	m.	m.	NOUN
ejpam-2488	377	7	noorani	noorani	PROPN
ejpam-2488	377	8	,	,	PUNCT
ejpam-2488	377	9	a.	a.	PROPN
ejpam-2488	377	10	al	al	PROPN
ejpam-2488	377	11	-	-	PUNCT
ejpam-2488	377	12	omari	omari	PROPN
ejpam-2488	377	13	,	,	PUNCT
ejpam-2488	377	14	and	and	CCONJ
ejpam-2488	377	15	t.	t.	PROPN
ejpam-2488	377	16	noiri	noiri	PROPN
ejpam-2488	377	17	/	/	SYM
ejpam-2488	377	18	eur	eur	PROPN
ejpam-2488	377	19	.	.	PUNCT
ejpam-2488	378	1	j.	j.	PROPN
ejpam-2488	378	2	pure	pure	PROPN
ejpam-2488	378	3	appl	appl	PROPN
ejpam-2488	378	4	.	.	PROPN
ejpam-2488	378	5	math	math	PROPN
ejpam-2488	378	6	,	,	PUNCT
ejpam-2488	378	7	8	8	NUM
ejpam-2488	378	8	(	(	PUNCT
ejpam-2488	378	9	2015	2015	NUM
ejpam-2488	378	10	)	)	PUNCT
ejpam-2488	378	11	,	,	PUNCT
ejpam-2488	378	12	502	502	NUM
ejpam-2488	378	13	-	-	SYM
ejpam-2488	378	14	513	513	NUM
ejpam-2488	378	15	510	510	NUM
ejpam-2488	378	16	example	example	NOUN
ejpam-2488	378	17	6	6	NUM
ejpam-2488	378	18	.	.	PUNCT
ejpam-2488	379	1	let	let	VERB
ejpam-2488	379	2	x	x	PUNCT
ejpam-2488	379	3	=	=	PRON
ejpam-2488	379	4	{	{	PUNCT
ejpam-2488	379	5	a	a	PRON
ejpam-2488	379	6	,	,	PUNCT
ejpam-2488	379	7	b	b	NOUN
ejpam-2488	379	8	,	,	PUNCT
ejpam-2488	379	9	c	c	NOUN
ejpam-2488	379	10	}	}	PUNCT
ejpam-2488	379	11	with	with	ADP
ejpam-2488	379	12	a	a	DET
ejpam-2488	379	13	topology	topology	NOUN
ejpam-2488	379	14	τ=	τ=	PRON
ejpam-2488	379	15	{	{	PUNCT
ejpam-2488	379	16	φ	φ	PROPN
ejpam-2488	379	17	,	,	PUNCT
ejpam-2488	379	18	x	x	INTJ
ejpam-2488	379	19	,	,	PUNCT
ejpam-2488	379	20	{	{	PUNCT
ejpam-2488	379	21	a	a	NOUN
ejpam-2488	379	22	}	}	PUNCT
ejpam-2488	379	23	,	,	PUNCT
ejpam-2488	379	24	{	{	PUNCT
ejpam-2488	379	25	a	a	PRON
ejpam-2488	379	26	,	,	PUNCT
ejpam-2488	379	27	b	b	NOUN
ejpam-2488	379	28	}	}	PUNCT
ejpam-2488	379	29	}	}	PUNCT
ejpam-2488	379	30	and	and	CCONJ
ejpam-2488	379	31	i	i	PRON
ejpam-2488	379	32	=	=	PUNCT
ejpam-2488	379	33	{	{	PUNCT
ejpam-2488	379	34	φ	φ	PROPN
ejpam-2488	379	35	,	,	PUNCT
ejpam-2488	379	36	{	{	PUNCT
ejpam-2488	379	37	a	a	X
ejpam-2488	379	38	}	}	PUNCT
ejpam-2488	379	39	,	,	PUNCT
ejpam-2488	379	40	{	{	PUNCT
ejpam-2488	379	41	b	b	NOUN
ejpam-2488	379	42	}	}	PUNCT
ejpam-2488	379	43	,	,	PUNCT
ejpam-2488	379	44	{	{	PUNCT
ejpam-2488	379	45	a	a	PRON
ejpam-2488	379	46	,	,	PUNCT
ejpam-2488	379	47	b	b	NOUN
ejpam-2488	379	48	}	}	PUNCT
ejpam-2488	379	49	}	}	PUNCT
ejpam-2488	379	50	.	.	PUNCT
ejpam-2488	380	1	since	since	SCONJ
ejpam-2488	380	2	eio	eio	PROPN
ejpam-2488	380	3	=	=	PROPN
ejpam-2488	380	4	{	{	PUNCT
ejpam-2488	380	5	φ	φ	PROPN
ejpam-2488	380	6	,	,	PUNCT
ejpam-2488	380	7	x	x	INTJ
ejpam-2488	380	8	,	,	PUNCT
ejpam-2488	380	9	{	{	PUNCT
ejpam-2488	380	10	a}{a	a}{a	NOUN
ejpam-2488	380	11	,	,	PUNCT
ejpam-2488	380	12	b	b	NOUN
ejpam-2488	380	13	}	}	PUNCT
ejpam-2488	380	14	,	,	PUNCT
ejpam-2488	380	15	{	{	PUNCT
ejpam-2488	380	16	a	a	PRON
ejpam-2488	380	17	,	,	PUNCT
ejpam-2488	380	18	c	c	NOUN
ejpam-2488	380	19	}	}	PUNCT
ejpam-2488	380	20	}	}	PUNCT
ejpam-2488	380	21	and	and	CCONJ
ejpam-2488	380	22	e	e	VERB
ejpam-2488	380	23	-	-	ADJ
ejpam-2488	380	24	open	open	ADJ
ejpam-2488	380	25	sets	set	NOUN
ejpam-2488	380	26	is	be	AUX
ejpam-2488	380	27	{	{	PUNCT
ejpam-2488	380	28	φ	φ	PROPN
ejpam-2488	380	29	,	,	PUNCT
ejpam-2488	380	30	x	x	INTJ
ejpam-2488	380	31	,	,	PUNCT
ejpam-2488	380	32	{	{	PUNCT
ejpam-2488	380	33	a	a	NOUN
ejpam-2488	380	34	}	}	PUNCT
ejpam-2488	380	35	,	,	PUNCT
ejpam-2488	380	36	{	{	PUNCT
ejpam-2488	380	37	b	b	NOUN
ejpam-2488	380	38	}	}	PUNCT
ejpam-2488	380	39	,	,	PUNCT
ejpam-2488	380	40	{	{	PUNCT
ejpam-2488	380	41	c	c	X
ejpam-2488	380	42	}	}	PUNCT
ejpam-2488	380	43	,	,	PUNCT
ejpam-2488	380	44	{	{	PUNCT
ejpam-2488	380	45	a	a	DET
ejpam-2488	380	46	,	,	PUNCT
ejpam-2488	380	47	b	b	NOUN
ejpam-2488	380	48	}	}	PUNCT
ejpam-2488	380	49	,	,	PUNCT
ejpam-2488	380	50	{	{	PUNCT
ejpam-2488	380	51	a	a	X
ejpam-2488	380	52	,	,	PUNCT
ejpam-2488	380	53	c	c	NOUN
ejpam-2488	380	54	}	}	PUNCT
ejpam-2488	380	55	,	,	PUNCT
ejpam-2488	380	56	{	{	PUNCT
ejpam-2488	380	57	b	b	X
ejpam-2488	380	58	,	,	PUNCT
ejpam-2488	380	59	c	c	NOUN
ejpam-2488	380	60	}	}	PUNCT
ejpam-2488	380	61	}	}	PUNCT
ejpam-2488	380	62	.	.	PUNCT
ejpam-2488	381	1	then	then	ADV
ejpam-2488	381	2	(	(	PUNCT
ejpam-2488	381	3	x	x	X
ejpam-2488	381	4	,	,	PUNCT
ejpam-2488	381	5	τ	τ	PROPN
ejpam-2488	381	6	,	,	PUNCT
ejpam-2488	381	7	i	i	PROPN
ejpam-2488	381	8	)	)	PUNCT
ejpam-2488	381	9	is	be	AUX
ejpam-2488	381	10	e	e	NOUN
ejpam-2488	381	11	-	-	NOUN
ejpam-2488	381	12	r0	r0	NOUN
ejpam-2488	381	13	but	but	CCONJ
ejpam-2488	381	14	not	not	PART
ejpam-2488	381	15	e	e	VERB
ejpam-2488	381	16	-	-	PROPN
ejpam-2488	381	17	i	i	PRON
ejpam-2488	381	18	-ro	-ro	X
ejpam-2488	381	19	.	.	PUNCT
ejpam-2488	382	1	theorem	theorem	VERB
ejpam-2488	382	2	9	9	NUM
ejpam-2488	382	3	.	.	PUNCT
ejpam-2488	383	1	the	the	DET
ejpam-2488	383	2	following	follow	VERB
ejpam-2488	383	3	properties	property	NOUN
ejpam-2488	383	4	are	be	AUX
ejpam-2488	383	5	equivalent	equivalent	ADJ
ejpam-2488	383	6	:	:	PUNCT
ejpam-2488	383	7	(	(	PUNCT
ejpam-2488	383	8	i	i	NOUN
ejpam-2488	383	9	)	)	PUNCT
ejpam-2488	383	10	(	(	PUNCT
ejpam-2488	383	11	x	x	X
ejpam-2488	383	12	,	,	PUNCT
ejpam-2488	383	13	τ	τ	PROPN
ejpam-2488	383	14	,	,	PUNCT
ejpam-2488	383	15	i	i	PROPN
ejpam-2488	383	16	)	)	PUNCT
ejpam-2488	383	17	is	be	AUX
ejpam-2488	383	18	e	e	NOUN
ejpam-2488	383	19	-	-	NOUN
ejpam-2488	383	20	i	i	PRON
ejpam-2488	383	21	-r1	-r1	NOUN
ejpam-2488	383	22	,	,	PUNCT
ejpam-2488	383	23	(	(	PUNCT
ejpam-2488	383	24	ii	ii	NOUN
ejpam-2488	383	25	)	)	PUNCT
ejpam-2488	383	26	for	for	ADP
ejpam-2488	383	27	each	each	PRON
ejpam-2488	383	28	x	x	NOUN
ejpam-2488	383	29	,	,	PUNCT
ejpam-2488	383	30	y	y	PROPN
ejpam-2488	383	31	∈	∈	PROPN
ejpam-2488	383	32	x	x	PUNCT
ejpam-2488	383	33	one	one	NUM
ejpam-2488	383	34	of	of	ADP
ejpam-2488	383	35	the	the	DET
ejpam-2488	383	36	following	follow	VERB
ejpam-2488	383	37	holds	hold	VERB
ejpam-2488	383	38	:	:	PUNCT
ejpam-2488	383	39	•	•	ADP
ejpam-2488	383	40	if	if	SCONJ
ejpam-2488	383	41	u	u	NOUN
ejpam-2488	383	42	is	be	AUX
ejpam-2488	383	43	e	e	NOUN
ejpam-2488	383	44	-	-	ADJ
ejpam-2488	383	45	i	i	PRON
ejpam-2488	383	46	-open	-open	NOUN
ejpam-2488	383	47	,	,	PUNCT
ejpam-2488	383	48	then	then	ADV
ejpam-2488	383	49	x	x	PART
ejpam-2488	383	50	∈	∈	PROPN
ejpam-2488	383	51	u	u	NOUN
ejpam-2488	383	52	if	if	SCONJ
ejpam-2488	383	53	and	and	CCONJ
ejpam-2488	383	54	only	only	ADV
ejpam-2488	383	55	if	if	SCONJ
ejpam-2488	383	56	y	y	PROPN
ejpam-2488	383	57	∈	∈	PROPN
ejpam-2488	383	58	u	u	PROPN
ejpam-2488	383	59	,	,	PUNCT
ejpam-2488	383	60	•	•	ADP
ejpam-2488	383	61	there	there	PRON
ejpam-2488	383	62	exist	exist	VERB
ejpam-2488	383	63	disjoint	disjoint	NOUN
ejpam-2488	383	64	e	e	NOUN
ejpam-2488	383	65	-	-	ADJ
ejpam-2488	383	66	i	i	PRON
ejpam-2488	383	67	-open	-open	NOUN
ejpam-2488	383	68	sets	set	VERB
ejpam-2488	383	69	u	u	NOUN
ejpam-2488	383	70	and	and	CCONJ
ejpam-2488	383	71	v	v	ADP
ejpam-2488	383	72	such	such	ADJ
ejpam-2488	383	73	that	that	SCONJ
ejpam-2488	383	74	x	x	SYM
ejpam-2488	383	75	∈	∈	PROPN
ejpam-2488	383	76	u	u	NOUN
ejpam-2488	383	77	and	and	CCONJ
ejpam-2488	383	78	y	y	PROPN
ejpam-2488	383	79	∈	∈	PROPN
ejpam-2488	383	80	v	v	NOUN
ejpam-2488	383	81	.	.	PUNCT
ejpam-2488	384	1	(	(	PUNCT
ejpam-2488	384	2	iii	iii	X
ejpam-2488	384	3	)	)	PUNCT
ejpam-2488	384	4	if	if	SCONJ
ejpam-2488	384	5	x	x	PRON
ejpam-2488	384	6	,	,	PUNCT
ejpam-2488	384	7	y	y	PROPN
ejpam-2488	384	8	∈	∈	PROPN
ejpam-2488	384	9	x	x	PUNCT
ejpam-2488	385	1	such	such	ADJ
ejpam-2488	385	2	that	that	SCONJ
ejpam-2488	385	3	cl∗e	cl∗e	PROPN
ejpam-2488	385	4	(	(	PUNCT
ejpam-2488	385	5	{	{	PUNCT
ejpam-2488	385	6	x	x	NOUN
ejpam-2488	385	7	}	}	PUNCT
ejpam-2488	385	8	)	)	PUNCT
ejpam-2488	385	9	6=	6=	ADP
ejpam-2488	385	10	cl∗e	cl∗e	X
ejpam-2488	385	11	(	(	PUNCT
ejpam-2488	385	12	{	{	PUNCT
ejpam-2488	385	13	y	y	NOUN
ejpam-2488	385	14	}	}	PUNCT
ejpam-2488	385	15	)	)	PUNCT
ejpam-2488	385	16	,	,	PUNCT
ejpam-2488	385	17	then	then	ADV
ejpam-2488	385	18	there	there	PRON
ejpam-2488	385	19	exist	exist	VERB
ejpam-2488	385	20	e	e	NOUN
ejpam-2488	385	21	-	-	PUNCT
ejpam-2488	385	22	i	i	PRON
ejpam-2488	385	23	-closed	-close	VERB
ejpam-2488	385	24	sets	set	NOUN
ejpam-2488	385	25	f1	f1	NOUN
ejpam-2488	385	26	and	and	CCONJ
ejpam-2488	385	27	f2	f2	NOUN
ejpam-2488	385	28	such	such	ADJ
ejpam-2488	385	29	that	that	SCONJ
ejpam-2488	385	30	x	x	SYM
ejpam-2488	385	31	∈	∈	PROPN
ejpam-2488	385	32	f1	f1	NOUN
ejpam-2488	385	33	,	,	PUNCT
ejpam-2488	385	34	y	y	PROPN
ejpam-2488	385	35	/∈	/∈	PUNCT
ejpam-2488	385	36	f1	f1	PROPN
ejpam-2488	385	37	,	,	PUNCT
ejpam-2488	385	38	y	y	PROPN
ejpam-2488	385	39	∈	∈	PROPN
ejpam-2488	385	40	f2	f2	PROPN
ejpam-2488	385	41	,	,	PUNCT
ejpam-2488	385	42	x	x	PROPN
ejpam-2488	385	43	/∈	/∈	PUNCT
ejpam-2488	385	44	f2	f2	PROPN
ejpam-2488	385	45	,	,	PUNCT
ejpam-2488	385	46	and	and	CCONJ
ejpam-2488	385	47	x	x	X
ejpam-2488	385	48	=	=	PUNCT
ejpam-2488	385	49	f1	f1	NOUN
ejpam-2488	385	50	∪	∪	NOUN
ejpam-2488	385	51	f2	f2	PROPN
ejpam-2488	385	52	.	.	PUNCT
ejpam-2488	386	1	proof	proof	NOUN
ejpam-2488	386	2	.	.	PUNCT
ejpam-2488	387	1	(	(	PUNCT
ejpam-2488	387	2	i	i	NOUN
ejpam-2488	387	3	)	)	PUNCT
ejpam-2488	387	4	⇒	⇒	PROPN
ejpam-2488	387	5	(	(	PUNCT
ejpam-2488	387	6	ii	ii	PROPN
ejpam-2488	387	7	):	):	PUNCT
ejpam-2488	387	8	let	let	VERB
ejpam-2488	387	9	x	x	PRON
ejpam-2488	387	10	,	,	PUNCT
ejpam-2488	387	11	y	y	PROPN
ejpam-2488	387	12	∈	∈	PROPN
ejpam-2488	387	13	x	x	X
ejpam-2488	387	14	.	.	PUNCT
ejpam-2488	388	1	then	then	ADV
ejpam-2488	388	2	cl∗e	cl∗e	PROPN
ejpam-2488	388	3	(	(	PUNCT
ejpam-2488	388	4	{	{	PUNCT
ejpam-2488	388	5	x	x	NOUN
ejpam-2488	388	6	}	}	PUNCT
ejpam-2488	388	7	)	)	PUNCT
ejpam-2488	388	8	=	=	SYM
ejpam-2488	388	9	cl∗e	cl∗e	PROPN
ejpam-2488	388	10	(	(	PUNCT
ejpam-2488	388	11	{	{	PUNCT
ejpam-2488	388	12	y	y	NOUN
ejpam-2488	388	13	}	}	PUNCT
ejpam-2488	388	14	)	)	PUNCT
ejpam-2488	388	15	or	or	CCONJ
ejpam-2488	388	16	cl∗e	cl∗e	X
ejpam-2488	388	17	(	(	PUNCT
ejpam-2488	388	18	{	{	PUNCT
ejpam-2488	388	19	x	x	NOUN
ejpam-2488	388	20	}	}	PUNCT
ejpam-2488	388	21	)	)	PUNCT
ejpam-2488	388	22	6=	6=	ADP
ejpam-2488	388	23	cl∗e	cl∗e	X
ejpam-2488	388	24	(	(	PUNCT
ejpam-2488	388	25	{	{	PUNCT
ejpam-2488	388	26	y	y	NOUN
ejpam-2488	388	27	}	}	PUNCT
ejpam-2488	388	28	)	)	PUNCT
ejpam-2488	388	29	.	.	PUNCT
ejpam-2488	389	1	if	if	SCONJ
ejpam-2488	389	2	cl∗e	cl∗e	PROPN
ejpam-2488	389	3	(	(	PUNCT
ejpam-2488	389	4	{	{	PUNCT
ejpam-2488	389	5	x	x	NOUN
ejpam-2488	389	6	}	}	PUNCT
ejpam-2488	389	7	)	)	PUNCT
ejpam-2488	389	8	=	=	SYM
ejpam-2488	389	9	cl∗e	cl∗e	PROPN
ejpam-2488	389	10	(	(	PUNCT
ejpam-2488	389	11	{	{	PUNCT
ejpam-2488	389	12	y	y	NOUN
ejpam-2488	389	13	}	}	PUNCT
ejpam-2488	389	14	)	)	PUNCT
ejpam-2488	389	15	and	and	CCONJ
ejpam-2488	389	16	u	u	NOUN
ejpam-2488	389	17	is	be	AUX
ejpam-2488	389	18	e	e	NOUN
ejpam-2488	389	19	-	-	ADJ
ejpam-2488	389	20	i	i	PRON
ejpam-2488	389	21	-open	-open	NOUN
ejpam-2488	389	22	,	,	PUNCT
ejpam-2488	389	23	then	then	ADV
ejpam-2488	389	24	x	x	PART
ejpam-2488	389	25	∈	∈	PROPN
ejpam-2488	389	26	u	u	NOUN
ejpam-2488	389	27	implies	imply	VERB
ejpam-2488	389	28	y	y	PROPN
ejpam-2488	389	29	∈	∈	PROPN
ejpam-2488	389	30	cl∗e	cl∗e	PROPN
ejpam-2488	389	31	(	(	PUNCT
ejpam-2488	389	32	{	{	PUNCT
ejpam-2488	389	33	x	x	NOUN
ejpam-2488	389	34	}	}	PUNCT
ejpam-2488	389	35	)	)	PUNCT
ejpam-2488	390	1	⊂	⊂	PROPN
ejpam-2488	390	2	u	u	PROPN
ejpam-2488	390	3	and	and	CCONJ
ejpam-2488	390	4	y	y	PROPN
ejpam-2488	390	5	∈	∈	PROPN
ejpam-2488	390	6	u	u	PROPN
ejpam-2488	390	7	implies	imply	VERB
ejpam-2488	390	8	x	x	X
ejpam-2488	390	9	∈	∈	PROPN
ejpam-2488	390	10	cl∗e	cl∗e	X
ejpam-2488	390	11	(	(	PUNCT
ejpam-2488	390	12	{	{	PUNCT
ejpam-2488	390	13	y	y	NOUN
ejpam-2488	390	14	}	}	PUNCT
ejpam-2488	390	15	)	)	PUNCT
ejpam-2488	391	1	⊂	⊂	PROPN
ejpam-2488	391	2	u	u	PROPN
ejpam-2488	391	3	.	.	PUNCT
ejpam-2488	391	4	thus	thus	ADV
ejpam-2488	391	5	consider	consider	VERB
ejpam-2488	391	6	the	the	DET
ejpam-2488	391	7	case	case	NOUN
ejpam-2488	391	8	that	that	SCONJ
ejpam-2488	391	9	cl∗e	cl∗e	PRON
ejpam-2488	391	10	(	(	PUNCT
ejpam-2488	391	11	{	{	PUNCT
ejpam-2488	391	12	x	x	NOUN
ejpam-2488	391	13	}	}	PUNCT
ejpam-2488	391	14	)	)	PUNCT
ejpam-2488	391	15	6=	6=	ADP
ejpam-2488	391	16	cl∗e	cl∗e	X
ejpam-2488	391	17	(	(	PUNCT
ejpam-2488	391	18	{	{	PUNCT
ejpam-2488	391	19	y	y	NOUN
ejpam-2488	391	20	}	}	PUNCT
ejpam-2488	391	21	)	)	PUNCT
ejpam-2488	391	22	.	.	PUNCT
ejpam-2488	392	1	then	then	ADV
ejpam-2488	392	2	there	there	PRON
ejpam-2488	392	3	exist	exist	VERB
ejpam-2488	392	4	disjoint	disjoint	NOUN
ejpam-2488	392	5	e	e	NOUN
ejpam-2488	392	6	-	-	ADJ
ejpam-2488	392	7	i	i	PRON
ejpam-2488	392	8	-open	-open	NOUN
ejpam-2488	392	9	sets	set	VERB
ejpam-2488	392	10	u	u	NOUN
ejpam-2488	392	11	and	and	CCONJ
ejpam-2488	392	12	v	v	ADP
ejpam-2488	392	13	such	such	ADJ
ejpam-2488	392	14	that	that	SCONJ
ejpam-2488	392	15	x	x	SYM
ejpam-2488	392	16	∈	∈	PRON
ejpam-2488	392	17	cl∗e	cl∗e	X
ejpam-2488	392	18	(	(	PUNCT
ejpam-2488	392	19	{	{	PUNCT
ejpam-2488	392	20	x	x	NOUN
ejpam-2488	392	21	}	}	PUNCT
ejpam-2488	392	22	)	)	PUNCT
ejpam-2488	392	23	⊂	⊂	PROPN
ejpam-2488	392	24	u	u	PROPN
ejpam-2488	392	25	and	and	CCONJ
ejpam-2488	392	26	y	y	PROPN
ejpam-2488	392	27	∈	∈	PROPN
ejpam-2488	392	28	cl∗e	cl∗e	PROPN
ejpam-2488	392	29	(	(	PUNCT
ejpam-2488	392	30	{	{	PUNCT
ejpam-2488	392	31	y	y	NOUN
ejpam-2488	392	32	}	}	PUNCT
ejpam-2488	392	33	)	)	PUNCT
ejpam-2488	393	1	⊂	⊂	PROPN
ejpam-2488	393	2	v	v	X
ejpam-2488	393	3	.	.	PUNCT
ejpam-2488	394	1	(	(	PUNCT
ejpam-2488	394	2	ii)⇒	ii)⇒	X
ejpam-2488	394	3	(	(	PUNCT
ejpam-2488	394	4	iii	iii	NOUN
ejpam-2488	394	5	):	):	PUNCT
ejpam-2488	394	6	let	let	VERB
ejpam-2488	394	7	x	x	PRON
ejpam-2488	394	8	,	,	PUNCT
ejpam-2488	394	9	y	y	PROPN
ejpam-2488	394	10	∈	∈	PROPN
ejpam-2488	394	11	x	x	PUNCT
ejpam-2488	394	12	such	such	ADJ
ejpam-2488	394	13	that	that	SCONJ
ejpam-2488	394	14	cl∗e	cl∗e	PROPN
ejpam-2488	394	15	(	(	PUNCT
ejpam-2488	394	16	{	{	PUNCT
ejpam-2488	394	17	x	x	NOUN
ejpam-2488	394	18	}	}	PUNCT
ejpam-2488	394	19	)	)	PUNCT
ejpam-2488	394	20	6=	6=	ADP
ejpam-2488	395	1	cl∗e	cl∗e	X
ejpam-2488	395	2	(	(	PUNCT
ejpam-2488	395	3	{	{	PUNCT
ejpam-2488	395	4	y	y	NOUN
ejpam-2488	395	5	}	}	PUNCT
ejpam-2488	395	6	)	)	PUNCT
ejpam-2488	395	7	.	.	PUNCT
ejpam-2488	396	1	then	then	ADV
ejpam-2488	396	2	x	x	X
ejpam-2488	396	3	/∈	/∈	PUNCT
ejpam-2488	396	4	cl∗e	cl∗e	PROPN
ejpam-2488	396	5	(	(	PUNCT
ejpam-2488	396	6	{	{	PUNCT
ejpam-2488	396	7	y	y	NOUN
ejpam-2488	396	8	}	}	PUNCT
ejpam-2488	396	9	)	)	PUNCT
ejpam-2488	396	10	or	or	CCONJ
ejpam-2488	396	11	y	y	PROPN
ejpam-2488	396	12	/∈	/∈	PUNCT
ejpam-2488	396	13	cl∗e	cl∗e	PROPN
ejpam-2488	396	14	(	(	PUNCT
ejpam-2488	396	15	{	{	PUNCT
ejpam-2488	396	16	x	x	NOUN
ejpam-2488	396	17	}	}	PUNCT
ejpam-2488	396	18	)	)	PUNCT
ejpam-2488	396	19	,	,	PUNCT
ejpam-2488	396	20	say	say	VERB
ejpam-2488	396	21	x	x	X
ejpam-2488	396	22	/∈	/∈	PUNCT
ejpam-2488	396	23	cl∗e	cl∗e	PROPN
ejpam-2488	396	24	(	(	PUNCT
ejpam-2488	396	25	{	{	PUNCT
ejpam-2488	396	26	y	y	NOUN
ejpam-2488	396	27	}	}	PUNCT
ejpam-2488	396	28	)	)	PUNCT
ejpam-2488	396	29	.	.	PUNCT
ejpam-2488	397	1	then	then	ADV
ejpam-2488	397	2	there	there	PRON
ejpam-2488	397	3	exists	exist	VERB
ejpam-2488	397	4	an	an	DET
ejpam-2488	397	5	e	e	NOUN
ejpam-2488	397	6	-	-	NOUN
ejpam-2488	397	7	i	i	PRON
ejpam-2488	397	8	-open	-open	NOUN
ejpam-2488	397	9	set	set	VERB
ejpam-2488	397	10	a	a	DET
ejpam-2488	397	11	such	such	ADJ
ejpam-2488	397	12	that	that	SCONJ
ejpam-2488	397	13	x	x	SYM
ejpam-2488	397	14	∈	∈	PROPN
ejpam-2488	397	15	a	a	PRON
ejpam-2488	397	16	and	and	CCONJ
ejpam-2488	397	17	y	y	PROPN
ejpam-2488	397	18	/∈	/∈	PROPN
ejpam-2488	398	1	a	a	PRON
ejpam-2488	398	2	,	,	PUNCT
ejpam-2488	398	3	which	which	PRON
ejpam-2488	398	4	implies	imply	VERB
ejpam-2488	398	5	there	there	PRON
ejpam-2488	398	6	exist	exist	VERB
ejpam-2488	398	7	disjoint	disjoint	NOUN
ejpam-2488	398	8	e	e	NOUN
ejpam-2488	398	9	-	-	ADJ
ejpam-2488	398	10	i	i	PRON
ejpam-2488	398	11	-open	-open	NOUN
ejpam-2488	398	12	sets	set	VERB
ejpam-2488	398	13	u	u	NOUN
ejpam-2488	398	14	and	and	CCONJ
ejpam-2488	398	15	v	v	ADP
ejpam-2488	398	16	such	such	ADJ
ejpam-2488	398	17	that	that	SCONJ
ejpam-2488	398	18	x	x	SYM
ejpam-2488	398	19	∈	∈	PROPN
ejpam-2488	398	20	u	u	NOUN
ejpam-2488	398	21	and	and	CCONJ
ejpam-2488	398	22	y	y	PROPN
ejpam-2488	398	23	∈	∈	PROPN
ejpam-2488	398	24	v	v	NOUN
ejpam-2488	398	25	.	.	PUNCT
ejpam-2488	399	1	then	then	ADV
ejpam-2488	399	2	f1	f1	PROPN
ejpam-2488	399	3	=	=	SYM
ejpam-2488	399	4	x\v	x\v	PROPN
ejpam-2488	399	5	and	and	CCONJ
ejpam-2488	399	6	f2	f2	PROPN
ejpam-2488	399	7	=	=	SYM
ejpam-2488	399	8	x\u	x\u	NOUN
ejpam-2488	399	9	are	be	AUX
ejpam-2488	399	10	e	e	NOUN
ejpam-2488	399	11	-	-	ADJ
ejpam-2488	399	12	i	i	PRON
ejpam-2488	399	13	-closed	-close	VERB
ejpam-2488	399	14	sets	set	VERB
ejpam-2488	399	15	such	such	ADJ
ejpam-2488	399	16	that	that	SCONJ
ejpam-2488	399	17	x	x	SYM
ejpam-2488	399	18	∈	∈	PROPN
ejpam-2488	399	19	f1	f1	NOUN
ejpam-2488	399	20	,	,	PUNCT
ejpam-2488	399	21	y	y	PROPN
ejpam-2488	399	22	/∈	/∈	PUNCT
ejpam-2488	399	23	f1	f1	PROPN
ejpam-2488	399	24	,	,	PUNCT
ejpam-2488	399	25	y	y	PROPN
ejpam-2488	399	26	∈	∈	PROPN
ejpam-2488	399	27	f2	f2	PROPN
ejpam-2488	399	28	,	,	PUNCT
ejpam-2488	399	29	x	x	PROPN
ejpam-2488	399	30	/∈	/∈	PUNCT
ejpam-2488	399	31	f2	f2	PROPN
ejpam-2488	399	32	,	,	PUNCT
ejpam-2488	399	33	and	and	CCONJ
ejpam-2488	399	34	x	x	X
ejpam-2488	399	35	=	=	PUNCT
ejpam-2488	399	36	f1	f1	NOUN
ejpam-2488	399	37	∪	∪	NOUN
ejpam-2488	399	38	f2	f2	PROPN
ejpam-2488	399	39	.	.	PUNCT
ejpam-2488	400	1	(	(	PUNCT
ejpam-2488	400	2	iii)⇒	iii)⇒	PROPN
ejpam-2488	400	3	(	(	PUNCT
ejpam-2488	400	4	i	i	NOUN
ejpam-2488	400	5	):	):	PUNCT
ejpam-2488	400	6	first	first	ADV
ejpam-2488	400	7	,	,	PUNCT
ejpam-2488	400	8	we	we	PRON
ejpam-2488	400	9	show	show	VERB
ejpam-2488	400	10	that	that	SCONJ
ejpam-2488	400	11	(	(	PUNCT
ejpam-2488	400	12	x	x	X
ejpam-2488	400	13	,	,	PUNCT
ejpam-2488	400	14	τ	τ	PROPN
ejpam-2488	400	15	,	,	PUNCT
ejpam-2488	400	16	i	i	PROPN
ejpam-2488	400	17	)	)	PUNCT
ejpam-2488	400	18	is	be	AUX
ejpam-2488	400	19	e	e	NOUN
ejpam-2488	400	20	-	-	PUNCT
ejpam-2488	400	21	i	i	PRON
ejpam-2488	400	22	-r0	-r0	INTJ
ejpam-2488	400	23	.	.	PUNCT
ejpam-2488	401	1	let	let	VERB
ejpam-2488	401	2	u	u	PRON
ejpam-2488	401	3	be	be	AUX
ejpam-2488	401	4	e	e	NOUN
ejpam-2488	401	5	-	-	NOUN
ejpam-2488	401	6	i	i	PRON
ejpam-2488	401	7	-open	-open	ADJ
ejpam-2488	401	8	and	and	CCONJ
ejpam-2488	401	9	let	let	VERB
ejpam-2488	401	10	x	x	PUNCT
ejpam-2488	401	11	∈	∈	PROPN
ejpam-2488	401	12	u	u	PROPN
ejpam-2488	401	13	.	.	PUNCT
ejpam-2488	401	14	suppose	suppose	VERB
ejpam-2488	401	15	that	that	SCONJ
ejpam-2488	401	16	cl∗e	cl∗e	PROPN
ejpam-2488	401	17	(	(	PUNCT
ejpam-2488	401	18	{	{	PUNCT
ejpam-2488	401	19	x	x	NOUN
ejpam-2488	401	20	}	}	PUNCT
ejpam-2488	401	21	)	)	PUNCT
ejpam-2488	401	22	6⊂	6⊂	NUM
ejpam-2488	401	23	u	u	NOUN
ejpam-2488	401	24	.	.	PUNCT
ejpam-2488	402	1	let	let	VERB
ejpam-2488	402	2	y	y	PROPN
ejpam-2488	402	3	∈	∈	PROPN
ejpam-2488	402	4	cl∗e	cl∗e	PROPN
ejpam-2488	402	5	(	(	PUNCT
ejpam-2488	402	6	{	{	PUNCT
ejpam-2488	402	7	x	x	NOUN
ejpam-2488	402	8	}	}	PUNCT
ejpam-2488	402	9	)	)	PUNCT
ejpam-2488	402	10	∩	∩	NOUN
ejpam-2488	402	11	(	(	PUNCT
ejpam-2488	402	12	x\u	x\u	PROPN
ejpam-2488	402	13	)	)	PUNCT
ejpam-2488	402	14	.	.	PUNCT
ejpam-2488	403	1	then	then	ADV
ejpam-2488	403	2	cl∗e	cl∗e	PRON
ejpam-2488	403	3	(	(	PUNCT
ejpam-2488	403	4	{	{	PUNCT
ejpam-2488	403	5	x	x	NOUN
ejpam-2488	403	6	}	}	PUNCT
ejpam-2488	403	7	)	)	PUNCT
ejpam-2488	403	8	6=	6=	ADP
ejpam-2488	403	9	cl∗e	cl∗e	X
ejpam-2488	403	10	(	(	PUNCT
ejpam-2488	403	11	{	{	PUNCT
ejpam-2488	403	12	y	y	NOUN
ejpam-2488	403	13	}	}	PUNCT
ejpam-2488	403	14	)	)	PUNCT
ejpam-2488	403	15	and	and	CCONJ
ejpam-2488	403	16	there	there	PRON
ejpam-2488	403	17	exist	exist	VERB
ejpam-2488	403	18	f1	f1	NOUN
ejpam-2488	403	19	,	,	PUNCT
ejpam-2488	403	20	f2	f2	PROPN
ejpam-2488	403	21	∈	∈	PROPN
ejpam-2488	403	22	ei	ei	X
ejpam-2488	403	23	c(x	c(x	NOUN
ejpam-2488	403	24	)	)	PUNCT
ejpam-2488	403	25	such	such	ADJ
ejpam-2488	403	26	that	that	SCONJ
ejpam-2488	403	27	x	x	SYM
ejpam-2488	403	28	∈	∈	PROPN
ejpam-2488	403	29	f1	f1	NOUN
ejpam-2488	403	30	,	,	PUNCT
ejpam-2488	403	31	y	y	PROPN
ejpam-2488	403	32	∈	∈	PROPN
ejpam-2488	403	33	f2	f2	PROPN
ejpam-2488	403	34	,	,	PUNCT
ejpam-2488	403	35	y	y	PROPN
ejpam-2488	403	36	/∈	/∈	PUNCT
ejpam-2488	403	37	f1	f1	PROPN
ejpam-2488	403	38	,	,	PUNCT
ejpam-2488	403	39	x	x	PROPN
ejpam-2488	403	40	/∈	/∈	PUNCT
ejpam-2488	403	41	f2	f2	PROPN
ejpam-2488	403	42	,	,	PUNCT
ejpam-2488	403	43	and	and	CCONJ
ejpam-2488	403	44	x	x	X
ejpam-2488	403	45	=	=	PUNCT
ejpam-2488	403	46	f1	f1	NOUN
ejpam-2488	403	47	∪	∪	NOUN
ejpam-2488	403	48	f2	f2	PROPN
ejpam-2488	403	49	.	.	PUNCT
ejpam-2488	404	1	then	then	ADV
ejpam-2488	404	2	y	y	PROPN
ejpam-2488	404	3	∈	∈	PROPN
ejpam-2488	404	4	f2\f1	f2\f1	PROPN
ejpam-2488	404	5	=	=	SYM
ejpam-2488	404	6	x\f1	x\f1	PROPN
ejpam-2488	404	7	,	,	PUNCT
ejpam-2488	404	8	which	which	PRON
ejpam-2488	404	9	is	be	AUX
ejpam-2488	404	10	e	e	NOUN
ejpam-2488	404	11	-	-	ADJ
ejpam-2488	404	12	i	i	PRON
ejpam-2488	404	13	-open	-open	ADJ
ejpam-2488	404	14	,	,	PUNCT
ejpam-2488	404	15	and	and	CCONJ
ejpam-2488	404	16	x	x	X
ejpam-2488	404	17	/∈	/∈	PROPN
ejpam-2488	404	18	x\f1	x\f1	PROPN
ejpam-2488	404	19	,	,	PUNCT
ejpam-2488	404	20	which	which	PRON
ejpam-2488	404	21	is	be	AUX
ejpam-2488	404	22	a	a	DET
ejpam-2488	404	23	contradiction	contradiction	NOUN
ejpam-2488	404	24	.	.	PUNCT
ejpam-2488	405	1	hence	hence	ADV
ejpam-2488	405	2	,	,	PUNCT
ejpam-2488	405	3	(	(	PUNCT
ejpam-2488	405	4	x	x	X
ejpam-2488	405	5	,	,	PUNCT
ejpam-2488	405	6	τ	τ	PROPN
ejpam-2488	405	7	,	,	PUNCT
ejpam-2488	405	8	i	i	PROPN
ejpam-2488	405	9	)	)	PUNCT
ejpam-2488	405	10	is	be	AUX
ejpam-2488	405	11	e	e	NOUN
ejpam-2488	405	12	-	-	PUNCT
ejpam-2488	405	13	i	i	PRON
ejpam-2488	405	14	-r0	-r0	NOUN
ejpam-2488	405	15	.	.	PUNCT
ejpam-2488	406	1	to	to	PART
ejpam-2488	406	2	show	show	VERB
ejpam-2488	406	3	x	x	PUNCT
ejpam-2488	406	4	to	to	PART
ejpam-2488	406	5	be	be	AUX
ejpam-2488	406	6	e	e	NOUN
ejpam-2488	406	7	-	-	NOUN
ejpam-2488	406	8	i	i	PRON
ejpam-2488	406	9	-r1	-r1	NOUN
ejpam-2488	406	10	assume	assume	VERB
ejpam-2488	406	11	that	that	SCONJ
ejpam-2488	406	12	a	a	DET
ejpam-2488	406	13	,	,	PUNCT
ejpam-2488	406	14	b	b	X
ejpam-2488	406	15	∈	∈	PROPN
ejpam-2488	406	16	x	x	PUNCT
ejpam-2488	406	17	such	such	ADJ
ejpam-2488	406	18	that	that	SCONJ
ejpam-2488	406	19	cl∗e	cl∗e	PROPN
ejpam-2488	406	20	(	(	PUNCT
ejpam-2488	406	21	{	{	PUNCT
ejpam-2488	406	22	a	a	NOUN
ejpam-2488	406	23	}	}	PUNCT
ejpam-2488	406	24	)	)	PUNCT
ejpam-2488	406	25	6=	6=	ADP
ejpam-2488	407	1	cl∗e	cl∗e	X
ejpam-2488	407	2	(	(	PUNCT
ejpam-2488	407	3	{	{	PUNCT
ejpam-2488	407	4	b	b	NOUN
ejpam-2488	407	5	}	}	PUNCT
ejpam-2488	407	6	)	)	PUNCT
ejpam-2488	407	7	.	.	PUNCT
ejpam-2488	408	1	then	then	ADV
ejpam-2488	408	2	there	there	PRON
ejpam-2488	408	3	exist	exist	VERB
ejpam-2488	408	4	p1	p1	NOUN
ejpam-2488	408	5	,	,	PUNCT
ejpam-2488	408	6	p2	p2	PROPN
ejpam-2488	408	7	∈	∈	PROPN
ejpam-2488	408	8	ei	ei	X
ejpam-2488	408	9	c(x	c(x	NOUN
ejpam-2488	408	10	)	)	PUNCT
ejpam-2488	408	11	such	such	ADJ
ejpam-2488	408	12	that	that	SCONJ
ejpam-2488	408	13	a	a	DET
ejpam-2488	408	14	∈	∈	PROPN
ejpam-2488	408	15	p1	p1	NOUN
ejpam-2488	408	16	,	,	PUNCT
ejpam-2488	408	17	b	b	PROPN
ejpam-2488	408	18	/∈	/∈	SYM
ejpam-2488	408	19	p1	p1	PROPN
ejpam-2488	408	20	,	,	PUNCT
ejpam-2488	408	21	a	a	DET
ejpam-2488	408	22	/∈	/∈	NOUN
ejpam-2488	408	23	p2	p2	NOUN
ejpam-2488	408	24	,	,	PUNCT
ejpam-2488	408	25	b	b	X
ejpam-2488	408	26	∈	∈	NOUN
ejpam-2488	408	27	p2	p2	NOUN
ejpam-2488	408	28	and	and	CCONJ
ejpam-2488	408	29	x	x	X
ejpam-2488	408	30	=	=	SYM
ejpam-2488	408	31	p1	p1	PROPN
ejpam-2488	408	32	∪	∪	ADJ
ejpam-2488	408	33	p2	p2	NOUN
ejpam-2488	408	34	.	.	PUNCT
ejpam-2488	409	1	thus	thus	ADV
ejpam-2488	409	2	a	a	DET
ejpam-2488	409	3	∈	∈	PROPN
ejpam-2488	409	4	p1\p2	p1\p2	NOUN
ejpam-2488	409	5	and	and	CCONJ
ejpam-2488	409	6	b	b	NOUN
ejpam-2488	409	7	∈	∈	PROPN
ejpam-2488	409	8	p2\p1	p2\p1	NOUN
ejpam-2488	409	9	,	,	PUNCT
ejpam-2488	409	10	which	which	PRON
ejpam-2488	409	11	are	be	AUX
ejpam-2488	409	12	e	e	NOUN
ejpam-2488	409	13	-	-	ADJ
ejpam-2488	409	14	i	i	PRON
ejpam-2488	409	15	-open	-open	ADJ
ejpam-2488	409	16	.	.	PUNCT
ejpam-2488	410	1	this	this	PRON
ejpam-2488	410	2	implies	imply	VERB
ejpam-2488	410	3	cl∗e	cl∗e	PROPN
ejpam-2488	410	4	(	(	PUNCT
ejpam-2488	410	5	{	{	PUNCT
ejpam-2488	410	6	a	a	NOUN
ejpam-2488	410	7	}	}	PUNCT
ejpam-2488	410	8	)	)	PUNCT
ejpam-2488	410	9	⊂	⊂	PROPN
ejpam-2488	410	10	p1\p2	p1\p2	PUNCT
ejpam-2488	411	1	=	=	PUNCT
ejpam-2488	411	2	x	x	SYM
ejpam-2488	411	3	−	−	NOUN
ejpam-2488	411	4	p2	p2	PROPN
ejpam-2488	411	5	∈	∈	PROPN
ejpam-2488	411	6	eio(x	eio(x	X
ejpam-2488	411	7	)	)	PUNCT
ejpam-2488	411	8	and	and	CCONJ
ejpam-2488	411	9	cl∗e	cl∗e	PROPN
ejpam-2488	411	10	(	(	PUNCT
ejpam-2488	411	11	{	{	PUNCT
ejpam-2488	411	12	b	b	NOUN
ejpam-2488	411	13	}	}	PUNCT
ejpam-2488	411	14	)	)	PUNCT
ejpam-2488	412	1	⊂	⊂	PROPN
ejpam-2488	412	2	p2\p1	p2\p1	NOUN
ejpam-2488	412	3	.	.	PUNCT
ejpam-2488	413	1	thus	thus	ADV
ejpam-2488	413	2	,	,	PUNCT
ejpam-2488	413	3	(	(	PUNCT
ejpam-2488	413	4	x	x	X
ejpam-2488	413	5	,	,	PUNCT
ejpam-2488	413	6	τ	τ	PROPN
ejpam-2488	413	7	,	,	PUNCT
ejpam-2488	413	8	i	i	PROPN
ejpam-2488	413	9	)	)	PUNCT
ejpam-2488	413	10	is	be	AUX
ejpam-2488	413	11	e	e	NOUN
ejpam-2488	413	12	-	-	NOUN
ejpam-2488	413	13	i	i	PRON
ejpam-2488	413	14	-r1	-r1	NOUN
ejpam-2488	413	15	.	.	PUNCT
ejpam-2488	414	1	theorem	theorem	VERB
ejpam-2488	414	2	10	10	NUM
ejpam-2488	414	3	.	.	PUNCT
ejpam-2488	415	1	the	the	DET
ejpam-2488	415	2	following	follow	VERB
ejpam-2488	415	3	properties	property	NOUN
ejpam-2488	415	4	are	be	AUX
ejpam-2488	415	5	equivalent	equivalent	ADJ
ejpam-2488	415	6	:	:	PUNCT
ejpam-2488	415	7	(	(	PUNCT
ejpam-2488	415	8	i	i	NOUN
ejpam-2488	415	9	)	)	PUNCT
ejpam-2488	415	10	(	(	PUNCT
ejpam-2488	415	11	x	x	X
ejpam-2488	415	12	,	,	PUNCT
ejpam-2488	415	13	τ	τ	PROPN
ejpam-2488	415	14	,	,	PUNCT
ejpam-2488	415	15	i	i	PROPN
ejpam-2488	415	16	)	)	PUNCT
ejpam-2488	415	17	is	be	AUX
ejpam-2488	415	18	e	e	NOUN
ejpam-2488	415	19	-	-	PROPN
ejpam-2488	415	20	i	i	PRON
ejpam-2488	415	21	-t2	-t2	VERB
ejpam-2488	415	22	,	,	PUNCT
ejpam-2488	415	23	(	(	PUNCT
ejpam-2488	415	24	ii	ii	NOUN
ejpam-2488	415	25	)	)	PUNCT
ejpam-2488	415	26	(	(	PUNCT
ejpam-2488	415	27	x	x	X
ejpam-2488	415	28	,	,	PUNCT
ejpam-2488	415	29	τ	τ	PROPN
ejpam-2488	415	30	,	,	PUNCT
ejpam-2488	415	31	i	i	PROPN
ejpam-2488	415	32	)	)	PUNCT
ejpam-2488	415	33	is	be	AUX
ejpam-2488	415	34	e	e	NOUN
ejpam-2488	415	35	-	-	NOUN
ejpam-2488	415	36	i	i	PRON
ejpam-2488	415	37	-r1	-r1	NOUN
ejpam-2488	415	38	and	and	CCONJ
ejpam-2488	415	39	e	e	X
ejpam-2488	415	40	-	-	NOUN
ejpam-2488	415	41	i	i	PRON
ejpam-2488	415	42	-t1	-t1	VERB
ejpam-2488	415	43	,	,	PUNCT
ejpam-2488	415	44	(	(	PUNCT
ejpam-2488	415	45	iii	iii	X
ejpam-2488	415	46	)	)	PUNCT
ejpam-2488	415	47	(	(	PUNCT
ejpam-2488	415	48	x	x	X
ejpam-2488	415	49	,	,	PUNCT
ejpam-2488	415	50	τ	τ	PROPN
ejpam-2488	415	51	,	,	PUNCT
ejpam-2488	415	52	i	i	PROPN
ejpam-2488	415	53	)	)	PUNCT
ejpam-2488	415	54	is	be	AUX
ejpam-2488	415	55	e	e	NOUN
ejpam-2488	415	56	-	-	NOUN
ejpam-2488	415	57	i	i	PRON
ejpam-2488	415	58	-r1	-r1	NOUN
ejpam-2488	415	59	and	and	CCONJ
ejpam-2488	415	60	e	e	X
ejpam-2488	415	61	-	-	NOUN
ejpam-2488	415	62	i	i	PRON
ejpam-2488	415	63	-t0	-t0	VERB
ejpam-2488	415	64	.	.	PUNCT
ejpam-2488	415	65	proof	proof	NOUN
ejpam-2488	415	66	.	.	PUNCT
ejpam-2488	416	1	(	(	PUNCT
ejpam-2488	416	2	i	i	NOUN
ejpam-2488	416	3	)	)	PUNCT
ejpam-2488	416	4	⇒	⇒	PROPN
ejpam-2488	416	5	(	(	PUNCT
ejpam-2488	416	6	ii	ii	PROPN
ejpam-2488	416	7	):	):	PUNCT
ejpam-2488	416	8	since	since	SCONJ
ejpam-2488	416	9	(	(	PUNCT
ejpam-2488	416	10	x	x	X
ejpam-2488	416	11	,	,	PUNCT
ejpam-2488	416	12	τ	τ	PROPN
ejpam-2488	416	13	,	,	PUNCT
ejpam-2488	416	14	i	i	PROPN
ejpam-2488	416	15	)	)	PUNCT
ejpam-2488	416	16	is	be	AUX
ejpam-2488	416	17	e	e	NOUN
ejpam-2488	416	18	-	-	NOUN
ejpam-2488	416	19	i	i	PRON
ejpam-2488	416	20	-t1	-t1	VERB
ejpam-2488	416	21	,	,	PUNCT
ejpam-2488	416	22	then	then	ADV
ejpam-2488	416	23	it	it	PRON
ejpam-2488	416	24	is	be	AUX
ejpam-2488	416	25	e	e	NOUN
ejpam-2488	416	26	-	-	NOUN
ejpam-2488	416	27	i	i	PRON
ejpam-2488	416	28	-t1	-t1	VERB
ejpam-2488	416	29	.	.	PUNCT
ejpam-2488	417	1	if	if	SCONJ
ejpam-2488	417	2	x	x	PRON
ejpam-2488	417	3	,	,	PUNCT
ejpam-2488	417	4	y	y	PROPN
ejpam-2488	417	5	∈	∈	PROPN
ejpam-2488	417	6	x	x	PUNCT
ejpam-2488	417	7	such	such	ADJ
ejpam-2488	417	8	that	that	SCONJ
ejpam-2488	417	9	cl∗e	cl∗e	PROPN
ejpam-2488	417	10	(	(	PUNCT
ejpam-2488	417	11	{	{	PUNCT
ejpam-2488	417	12	x	x	NOUN
ejpam-2488	417	13	}	}	PUNCT
ejpam-2488	417	14	)	)	PUNCT
ejpam-2488	417	15	6=	6=	ADP
ejpam-2488	418	1	cl∗e	cl∗e	X
ejpam-2488	418	2	(	(	PUNCT
ejpam-2488	418	3	{	{	PUNCT
ejpam-2488	418	4	y	y	NOUN
ejpam-2488	418	5	}	}	PUNCT
ejpam-2488	418	6	)	)	PUNCT
ejpam-2488	418	7	,	,	PUNCT
ejpam-2488	418	8	then	then	ADV
ejpam-2488	418	9	x	x	X
ejpam-2488	418	10	6=	6=	PROPN
ejpam-2488	418	11	y	y	PROPN
ejpam-2488	418	12	and	and	CCONJ
ejpam-2488	418	13	there	there	PRON
ejpam-2488	418	14	exist	exist	VERB
ejpam-2488	418	15	disjoint	disjoint	NOUN
ejpam-2488	418	16	e	e	NOUN
ejpam-2488	418	17	-	-	ADJ
ejpam-2488	418	18	i	i	PRON
ejpam-2488	418	19	-open	-open	NOUN
ejpam-2488	418	20	sets	set	VERB
ejpam-2488	418	21	u	u	NOUN
ejpam-2488	418	22	and	and	CCONJ
ejpam-2488	418	23	v	v	ADP
ejpam-2488	418	24	such	such	ADJ
ejpam-2488	418	25	that	that	SCONJ
ejpam-2488	418	26	x	x	SYM
ejpam-2488	418	27	∈	∈	PROPN
ejpam-2488	418	28	u	u	NOUN
ejpam-2488	418	29	and	and	CCONJ
ejpam-2488	418	30	y	y	PROPN
ejpam-2488	418	31	∈	∈	PROPN
ejpam-2488	418	32	v	v	NOUN
ejpam-2488	418	33	.	.	PUNCT
ejpam-2488	419	1	therefore	therefore	ADV
ejpam-2488	419	2	,	,	PUNCT
ejpam-2488	419	3	cl∗e	cl∗e	PROPN
ejpam-2488	419	4	(	(	PUNCT
ejpam-2488	419	5	{	{	PUNCT
ejpam-2488	419	6	x	x	NOUN
ejpam-2488	419	7	}	}	PUNCT
ejpam-2488	419	8	)	)	PUNCT
ejpam-2488	419	9	=	=	PRON
ejpam-2488	419	10	{	{	PUNCT
ejpam-2488	419	11	x	x	NOUN
ejpam-2488	419	12	}	}	PUNCT
ejpam-2488	419	13	⊂	⊂	PROPN
ejpam-2488	419	14	u	u	NOUN
ejpam-2488	419	15	and	and	CCONJ
ejpam-2488	419	16	cl∗e	cl∗e	PROPN
ejpam-2488	419	17	(	(	PUNCT
ejpam-2488	419	18	{	{	PUNCT
ejpam-2488	419	19	y	y	NOUN
ejpam-2488	419	20	}	}	PUNCT
ejpam-2488	419	21	)	)	PUNCT
ejpam-2488	420	1	=	=	PRON
ejpam-2488	420	2	{	{	PUNCT
ejpam-2488	420	3	y	y	PROPN
ejpam-2488	420	4	}	}	PUNCT
ejpam-2488	420	5	⊂	⊂	PROPN
ejpam-2488	420	6	v	v	NOUN
ejpam-2488	420	7	.	.	PUNCT
ejpam-2488	421	1	hence	hence	ADV
ejpam-2488	421	2	(	(	PUNCT
ejpam-2488	421	3	x	x	X
ejpam-2488	421	4	,	,	PUNCT
ejpam-2488	421	5	τ	τ	PROPN
ejpam-2488	421	6	,	,	PUNCT
ejpam-2488	421	7	i	i	PROPN
ejpam-2488	421	8	)	)	PUNCT
ejpam-2488	421	9	is	be	AUX
ejpam-2488	421	10	e	e	NOUN
ejpam-2488	421	11	-	-	NOUN
ejpam-2488	421	12	i	i	PRON
ejpam-2488	421	13	-r1	-r1	NOUN
ejpam-2488	421	14	.	.	PUNCT
ejpam-2488	422	1	(	(	PUNCT
ejpam-2488	422	2	ii)⇒	ii)⇒	PROPN
ejpam-2488	422	3	(	(	PUNCT
ejpam-2488	422	4	iii	iii	NOUN
ejpam-2488	422	5	):	):	PUNCT
ejpam-2488	422	6	since	since	SCONJ
ejpam-2488	422	7	(	(	PUNCT
ejpam-2488	422	8	x	x	X
ejpam-2488	422	9	,	,	PUNCT
ejpam-2488	422	10	τ	τ	PROPN
ejpam-2488	422	11	,	,	PUNCT
ejpam-2488	422	12	i	i	PROPN
ejpam-2488	422	13	)	)	PUNCT
ejpam-2488	422	14	is	be	AUX
ejpam-2488	422	15	e	e	NOUN
ejpam-2488	422	16	-	-	NOUN
ejpam-2488	422	17	i	i	PRON
ejpam-2488	422	18	-t1	-t1	VERB
ejpam-2488	422	19	,	,	PUNCT
ejpam-2488	422	20	then	then	ADV
ejpam-2488	422	21	(	(	PUNCT
ejpam-2488	422	22	x	x	X
ejpam-2488	422	23	,	,	PUNCT
ejpam-2488	422	24	τ	τ	PROPN
ejpam-2488	422	25	,	,	PUNCT
ejpam-2488	422	26	i	i	PROPN
ejpam-2488	422	27	)	)	PUNCT
ejpam-2488	422	28	is	be	AUX
ejpam-2488	422	29	e	e	NOUN
ejpam-2488	422	30	-	-	NOUN
ejpam-2488	422	31	i	i	PRON
ejpam-2488	422	32	-t0	-t0	PROPN
ejpam-2488	422	33	.	.	PUNCT
ejpam-2488	423	1	w.	w.	PROPN
ejpam-2488	423	2	al	al	PROPN
ejpam-2488	423	3	-	-	PUNCT
ejpam-2488	423	4	omeri	omeri	ADJ
ejpam-2488	423	5	,	,	PUNCT
ejpam-2488	423	6	m.	m.	NOUN
ejpam-2488	423	7	noorani	noorani	PROPN
ejpam-2488	423	8	,	,	PUNCT
ejpam-2488	423	9	a.	a.	PROPN
ejpam-2488	423	10	al	al	PROPN
ejpam-2488	423	11	-	-	PUNCT
ejpam-2488	423	12	omari	omari	PROPN
ejpam-2488	423	13	,	,	PUNCT
ejpam-2488	423	14	and	and	CCONJ
ejpam-2488	423	15	t.	t.	PROPN
ejpam-2488	423	16	noiri	noiri	PROPN
ejpam-2488	423	17	/	/	SYM
ejpam-2488	423	18	eur	eur	PROPN
ejpam-2488	423	19	.	.	PUNCT
ejpam-2488	424	1	j.	j.	PROPN
ejpam-2488	424	2	pure	pure	PROPN
ejpam-2488	424	3	appl	appl	PROPN
ejpam-2488	424	4	.	.	PROPN
ejpam-2488	424	5	math	math	PROPN
ejpam-2488	424	6	,	,	PUNCT
ejpam-2488	424	7	8	8	NUM
ejpam-2488	424	8	(	(	PUNCT
ejpam-2488	424	9	2015	2015	NUM
ejpam-2488	424	10	)	)	PUNCT
ejpam-2488	424	11	,	,	PUNCT
ejpam-2488	424	12	502	502	NUM
ejpam-2488	424	13	-	-	SYM
ejpam-2488	424	14	513	513	NUM
ejpam-2488	424	15	511	511	NUM
ejpam-2488	424	16	(	(	PUNCT
ejpam-2488	424	17	iii	iii	NOUN
ejpam-2488	424	18	)	)	PUNCT
ejpam-2488	424	19	⇒	⇒	NOUN
ejpam-2488	424	20	(	(	PUNCT
ejpam-2488	424	21	i	i	NOUN
ejpam-2488	424	22	):	):	PUNCT
ejpam-2488	424	23	since	since	SCONJ
ejpam-2488	424	24	(	(	PUNCT
ejpam-2488	424	25	x	x	X
ejpam-2488	424	26	,	,	PUNCT
ejpam-2488	424	27	τ	τ	PROPN
ejpam-2488	424	28	,	,	PUNCT
ejpam-2488	424	29	i	i	PROPN
ejpam-2488	424	30	)	)	PUNCT
ejpam-2488	424	31	is	be	AUX
ejpam-2488	424	32	e	e	NOUN
ejpam-2488	424	33	-	-	NOUN
ejpam-2488	424	34	i	i	PRON
ejpam-2488	424	35	-r1	-r1	NOUN
ejpam-2488	424	36	,	,	PUNCT
ejpam-2488	424	37	then	then	ADV
ejpam-2488	424	38	(	(	PUNCT
ejpam-2488	424	39	x	x	X
ejpam-2488	424	40	,	,	PUNCT
ejpam-2488	424	41	τ	τ	PROPN
ejpam-2488	424	42	,	,	PUNCT
ejpam-2488	424	43	i	i	PROPN
ejpam-2488	424	44	)	)	PUNCT
ejpam-2488	424	45	is	be	AUX
ejpam-2488	424	46	e	e	NOUN
ejpam-2488	424	47	-	-	PUNCT
ejpam-2488	424	48	i	i	PRON
ejpam-2488	424	49	-r0	-r0	PROPN
ejpam-2488	424	50	and	and	CCONJ
ejpam-2488	424	51	e	e	X
ejpam-2488	424	52	-	-	NOUN
ejpam-2488	424	53	i	i	PRON
ejpam-2488	424	54	-t0	-t0	VERB
ejpam-2488	424	55	and	and	CCONJ
ejpam-2488	424	56	hence	hence	ADV
ejpam-2488	424	57	by	by	ADP
ejpam-2488	424	58	theorem	theorem	NOUN
ejpam-2488	424	59	1	1	NUM
ejpam-2488	424	60	(	(	PUNCT
ejpam-2488	424	61	x	x	INTJ
ejpam-2488	424	62	,	,	PUNCT
ejpam-2488	424	63	τ	τ	PROPN
ejpam-2488	424	64	,	,	PUNCT
ejpam-2488	424	65	i	i	PROPN
ejpam-2488	424	66	)	)	PUNCT
ejpam-2488	424	67	is	be	AUX
ejpam-2488	424	68	e	e	NOUN
ejpam-2488	424	69	-	-	NOUN
ejpam-2488	424	70	i	i	PRON
ejpam-2488	424	71	-t1	-t1	VERB
ejpam-2488	424	72	.	.	PUNCT
ejpam-2488	425	1	let	let	VERB
ejpam-2488	425	2	x	x	PRON
ejpam-2488	425	3	,	,	PUNCT
ejpam-2488	425	4	y	y	PROPN
ejpam-2488	425	5	∈	∈	PROPN
ejpam-2488	425	6	x	x	PUNCT
ejpam-2488	425	7	such	such	ADJ
ejpam-2488	425	8	that	that	SCONJ
ejpam-2488	425	9	x	x	PROPN
ejpam-2488	425	10	6=	6=	NUM
ejpam-2488	425	11	y	y	PROPN
ejpam-2488	425	12	.	.	PUNCT
ejpam-2488	426	1	since	since	SCONJ
ejpam-2488	426	2	cl∗e	cl∗e	PROPN
ejpam-2488	426	3	(	(	PUNCT
ejpam-2488	426	4	{	{	PUNCT
ejpam-2488	426	5	x	x	NOUN
ejpam-2488	426	6	}	}	PUNCT
ejpam-2488	426	7	)	)	PUNCT
ejpam-2488	426	8	=	=	SYM
ejpam-2488	426	9	{	{	PUNCT
ejpam-2488	426	10	x	x	NOUN
ejpam-2488	426	11	}	}	PUNCT
ejpam-2488	426	12	6=	6=	NUM
ejpam-2488	426	13	{	{	PUNCT
ejpam-2488	426	14	y	y	NOUN
ejpam-2488	426	15	}	}	PUNCT
ejpam-2488	426	16	=	=	SYM
ejpam-2488	426	17	cl∗e	cl∗e	X
ejpam-2488	426	18	(	(	PUNCT
ejpam-2488	426	19	{	{	PUNCT
ejpam-2488	426	20	y	y	NOUN
ejpam-2488	426	21	}	}	PUNCT
ejpam-2488	426	22	)	)	PUNCT
ejpam-2488	426	23	,	,	PUNCT
ejpam-2488	426	24	then	then	ADV
ejpam-2488	426	25	there	there	PRON
ejpam-2488	426	26	exist	exist	VERB
ejpam-2488	426	27	disjoint	disjoint	NOUN
ejpam-2488	426	28	e	e	NOUN
ejpam-2488	426	29	-	-	ADJ
ejpam-2488	426	30	i	i	PRON
ejpam-2488	426	31	-open	-open	NOUN
ejpam-2488	426	32	sets	set	VERB
ejpam-2488	426	33	u	u	NOUN
ejpam-2488	426	34	and	and	CCONJ
ejpam-2488	426	35	v	v	ADP
ejpam-2488	426	36	such	such	ADJ
ejpam-2488	426	37	that	that	SCONJ
ejpam-2488	426	38	x	x	SYM
ejpam-2488	426	39	∈	∈	PROPN
ejpam-2488	426	40	u	u	NOUN
ejpam-2488	426	41	and	and	CCONJ
ejpam-2488	426	42	y	y	PROPN
ejpam-2488	426	43	∈	∈	PROPN
ejpam-2488	426	44	v	v	NOUN
ejpam-2488	426	45	.	.	PUNCT
ejpam-2488	427	1	hence	hence	ADV
ejpam-2488	427	2	,	,	PUNCT
ejpam-2488	427	3	(	(	PUNCT
ejpam-2488	427	4	x	x	X
ejpam-2488	427	5	,	,	PUNCT
ejpam-2488	427	6	τ	τ	PROPN
ejpam-2488	427	7	,	,	PUNCT
ejpam-2488	427	8	i	i	PROPN
ejpam-2488	427	9	)	)	PUNCT
ejpam-2488	427	10	is	be	AUX
ejpam-2488	427	11	e	e	NOUN
ejpam-2488	427	12	-	-	PROPN
ejpam-2488	427	13	i	i	PRON
ejpam-2488	427	14	-t2	-t2	VERB
ejpam-2488	427	15	.	.	PUNCT
ejpam-2488	428	1	in	in	ADP
ejpam-2488	428	2	view	view	NOUN
ejpam-2488	428	3	of	of	ADP
ejpam-2488	428	4	definition	definition	NOUN
ejpam-2488	428	5	3	3	NUM
ejpam-2488	428	6	,	,	PUNCT
ejpam-2488	428	7	it	it	PRON
ejpam-2488	428	8	follows	follow	VERB
ejpam-2488	428	9	that	that	SCONJ
ejpam-2488	428	10	theorem	theorem	VERB
ejpam-2488	428	11	11	11	NUM
ejpam-2488	428	12	.	.	PUNCT
ejpam-2488	429	1	an	an	DET
ejpam-2488	429	2	ideal	ideal	ADJ
ejpam-2488	429	3	topological	topological	ADJ
ejpam-2488	429	4	space	space	NOUN
ejpam-2488	429	5	(	(	PUNCT
ejpam-2488	429	6	x	x	X
ejpam-2488	429	7	,	,	PUNCT
ejpam-2488	429	8	τ	τ	PROPN
ejpam-2488	429	9	,	,	PUNCT
ejpam-2488	429	10	i	i	PROPN
ejpam-2488	429	11	)	)	PUNCT
ejpam-2488	429	12	is	be	AUX
ejpam-2488	429	13	e	e	NOUN
ejpam-2488	429	14	-	-	NOUN
ejpam-2488	429	15	i	i	PRON
ejpam-2488	429	16	-t2	-t2	VERB
ejpam-2488	429	17	if	if	SCONJ
ejpam-2488	430	1	and	and	CCONJ
ejpam-2488	430	2	only	only	ADV
ejpam-2488	430	3	if	if	SCONJ
ejpam-2488	430	4	for	for	ADP
ejpam-2488	430	5	x	x	X
ejpam-2488	430	6	,	,	PUNCT
ejpam-2488	430	7	y	y	PROPN
ejpam-2488	430	8	∈	∈	PROPN
ejpam-2488	430	9	x	x	PUNCT
ejpam-2488	430	10	such	such	ADJ
ejpam-2488	430	11	that	that	SCONJ
ejpam-2488	430	12	x	x	PROPN
ejpam-2488	430	13	6=	6=	NUM
ejpam-2488	430	14	y	y	PROPN
ejpam-2488	430	15	,	,	PUNCT
ejpam-2488	430	16	there	there	PRON
ejpam-2488	430	17	exist	exist	VERB
ejpam-2488	430	18	e	e	NOUN
ejpam-2488	430	19	-	-	PUNCT
ejpam-2488	430	20	i	i	PRON
ejpam-2488	430	21	-closed	-close	VERB
ejpam-2488	430	22	sets	set	NOUN
ejpam-2488	430	23	f1	f1	NOUN
ejpam-2488	430	24	and	and	CCONJ
ejpam-2488	430	25	f2	f2	NOUN
ejpam-2488	430	26	such	such	ADJ
ejpam-2488	430	27	that	that	SCONJ
ejpam-2488	430	28	x	x	SYM
ejpam-2488	430	29	∈	∈	PROPN
ejpam-2488	430	30	f1	f1	NOUN
ejpam-2488	430	31	,	,	PUNCT
ejpam-2488	430	32	y	y	PROPN
ejpam-2488	430	33	/∈	/∈	PUNCT
ejpam-2488	430	34	f1	f1	PROPN
ejpam-2488	430	35	,	,	PUNCT
ejpam-2488	430	36	y	y	PROPN
ejpam-2488	430	37	∈	∈	PROPN
ejpam-2488	430	38	f2	f2	PROPN
ejpam-2488	430	39	,	,	PUNCT
ejpam-2488	430	40	x	x	PROPN
ejpam-2488	430	41	/∈	/∈	PUNCT
ejpam-2488	430	42	f2	f2	PROPN
ejpam-2488	430	43	,	,	PUNCT
ejpam-2488	430	44	and	and	CCONJ
ejpam-2488	430	45	x	x	X
ejpam-2488	430	46	=	=	PUNCT
ejpam-2488	430	47	f1	f1	PROPN
ejpam-2488	430	48	∪	∪	ADP
ejpam-2488	430	49	f2	f2	PROPN
ejpam-2488	430	50	.	.	PUNCT
ejpam-2488	431	1	remark	remark	PROPN
ejpam-2488	431	2	4	4	NUM
ejpam-2488	431	3	.	.	PUNCT
ejpam-2488	432	1	let	let	VERB
ejpam-2488	432	2	{	{	PUNCT
ejpam-2488	432	3	xλ}λ∈λ	xλ}λ∈λ	PART
ejpam-2488	432	4	be	be	AUX
ejpam-2488	432	5	a	a	DET
ejpam-2488	432	6	net	net	NOUN
ejpam-2488	432	7	in	in	ADP
ejpam-2488	432	8	(	(	PUNCT
ejpam-2488	432	9	x	x	INTJ
ejpam-2488	432	10	,	,	PUNCT
ejpam-2488	432	11	τ	τ	PROPN
ejpam-2488	432	12	,	,	PUNCT
ejpam-2488	432	13	i	i	PROPN
ejpam-2488	432	14	)	)	PUNCT
ejpam-2488	432	15	and	and	CCONJ
ejpam-2488	432	16	ei	ei	ADP
ejpam-2488	432	17	l	l	NOUN
ejpam-2488	432	18	im({xλ}λ∈λ	im({xλ}λ∈λ	PROPN
ejpam-2488	432	19	)	)	PUNCT
ejpam-2488	432	20	denote	denote	VERB
ejpam-2488	432	21	{	{	PUNCT
ejpam-2488	432	22	x	x	SYM
ejpam-2488	432	23	∈	∈	PROPN
ejpam-2488	432	24	x	x	X
ejpam-2488	432	25	:	:	PUNCT
ejpam-2488	432	26	i	i	PRON
ejpam-2488	432	27	−	−	VERB
ejpam-2488	432	28	converges	converge	VERB
ejpam-2488	432	29	to	to	ADP
ejpam-2488	432	30	x	x	X
ejpam-2488	432	31	}	}	PUNCT
ejpam-2488	432	32	.	.	PUNCT
ejpam-2488	433	1	theorem	theorem	NOUN
ejpam-2488	433	2	12	12	NUM
ejpam-2488	433	3	.	.	PUNCT
ejpam-2488	434	1	the	the	DET
ejpam-2488	434	2	following	follow	VERB
ejpam-2488	434	3	properties	property	NOUN
ejpam-2488	434	4	are	be	AUX
ejpam-2488	434	5	equivalent	equivalent	ADJ
ejpam-2488	434	6	:	:	PUNCT
ejpam-2488	434	7	(	(	PUNCT
ejpam-2488	434	8	i	i	NOUN
ejpam-2488	434	9	)	)	PUNCT
ejpam-2488	434	10	(	(	PUNCT
ejpam-2488	434	11	x	x	X
ejpam-2488	434	12	,	,	PUNCT
ejpam-2488	434	13	τ	τ	PROPN
ejpam-2488	434	14	,	,	PUNCT
ejpam-2488	434	15	i	i	PROPN
ejpam-2488	434	16	)	)	PUNCT
ejpam-2488	434	17	is	be	AUX
ejpam-2488	434	18	e	e	NOUN
ejpam-2488	434	19	-	-	NOUN
ejpam-2488	434	20	i	i	PRON
ejpam-2488	434	21	-r1	-r1	NOUN
ejpam-2488	434	22	,	,	PUNCT
ejpam-2488	434	23	(	(	PUNCT
ejpam-2488	434	24	ii	ii	NOUN
ejpam-2488	434	25	)	)	PUNCT
ejpam-2488	434	26	for	for	ADP
ejpam-2488	434	27	x	x	X
ejpam-2488	434	28	,	,	PUNCT
ejpam-2488	434	29	y	y	PROPN
ejpam-2488	434	30	∈	∈	PROPN
ejpam-2488	434	31	x	x	SYM
ejpam-2488	434	32	cl∗e	cl∗e	PROPN
ejpam-2488	434	33	(	(	PUNCT
ejpam-2488	434	34	{	{	PUNCT
ejpam-2488	434	35	x	x	NOUN
ejpam-2488	434	36	}	}	PUNCT
ejpam-2488	434	37	)	)	PUNCT
ejpam-2488	435	1	=	=	SYM
ejpam-2488	435	2	cl∗e	cl∗e	PROPN
ejpam-2488	435	3	(	(	PUNCT
ejpam-2488	435	4	{	{	PUNCT
ejpam-2488	435	5	y	y	NOUN
ejpam-2488	435	6	}	}	PUNCT
ejpam-2488	435	7	)	)	PUNCT
ejpam-2488	435	8	,	,	PUNCT
ejpam-2488	435	9	whenever	whenever	SCONJ
ejpam-2488	435	10	there	there	PRON
ejpam-2488	435	11	exists	exist	VERB
ejpam-2488	435	12	a	a	DET
ejpam-2488	435	13	net	net	NOUN
ejpam-2488	435	14	{	{	PUNCT
ejpam-2488	435	15	xλ}λ∈a	xλ}λ∈a	ADP
ejpam-2488	435	16	such	such	ADJ
ejpam-2488	435	17	that	that	SCONJ
ejpam-2488	435	18	x	x	X
ejpam-2488	435	19	,	,	PUNCT
ejpam-2488	435	20	y	y	PROPN
ejpam-2488	435	21	∈	∈	PROPN
ejpam-2488	435	22	ei	ei	ADP
ejpam-2488	435	23	l	l	PROPN
ejpam-2488	435	24	im({xλ}λ∈a	im({xλ}λ∈a	PROPN
ejpam-2488	435	25	)	)	PUNCT
ejpam-2488	435	26	,	,	PUNCT
ejpam-2488	435	27	(	(	PUNCT
ejpam-2488	435	28	iii	iii	X
ejpam-2488	435	29	)	)	PUNCT
ejpam-2488	435	30	(	(	PUNCT
ejpam-2488	435	31	x	x	X
ejpam-2488	435	32	,	,	PUNCT
ejpam-2488	435	33	τ	τ	PROPN
ejpam-2488	435	34	,	,	PUNCT
ejpam-2488	435	35	i	i	PROPN
ejpam-2488	435	36	)	)	PUNCT
ejpam-2488	435	37	is	be	AUX
ejpam-2488	435	38	e	e	NOUN
ejpam-2488	435	39	-	-	PUNCT
ejpam-2488	435	40	i	i	PRON
ejpam-2488	435	41	-r0	-r0	NOUN
ejpam-2488	435	42	,	,	PUNCT
ejpam-2488	435	43	and	and	CCONJ
ejpam-2488	435	44	for	for	ADP
ejpam-2488	435	45	every	every	DET
ejpam-2488	435	46	e	e	NOUN
ejpam-2488	435	47	-	-	ADJ
ejpam-2488	435	48	i	i	PRON
ejpam-2488	435	49	-convergent	-convergent	ADJ
ejpam-2488	435	50	net	net	NOUN
ejpam-2488	435	51	{	{	PUNCT
ejpam-2488	435	52	xλ}λ∈a	xλ}λ∈a	X
ejpam-2488	435	53	in	in	ADP
ejpam-2488	435	54	x	x	X
ejpam-2488	435	55	,	,	PUNCT
ejpam-2488	435	56	ei	ei	X
ejpam-2488	435	57	l	l	NOUN
ejpam-2488	435	58	im({xλ}λ∈a	im({xλ}λ∈a	PROPN
ejpam-2488	435	59	)	)	PUNCT
ejpam-2488	435	60	=	=	SYM
ejpam-2488	435	61	cl∗e	cl∗e	PROPN
ejpam-2488	435	62	(	(	PUNCT
ejpam-2488	435	63	{	{	PUNCT
ejpam-2488	435	64	x	x	NOUN
ejpam-2488	435	65	}	}	PUNCT
ejpam-2488	435	66	)	)	PUNCT
ejpam-2488	435	67	for	for	ADP
ejpam-2488	435	68	some	some	DET
ejpam-2488	435	69	x	x	SYM
ejpam-2488	435	70	∈	∈	PROPN
ejpam-2488	435	71	x	x	X
ejpam-2488	435	72	.	.	PUNCT
ejpam-2488	436	1	proof	proof	NOUN
ejpam-2488	436	2	.	.	PUNCT
ejpam-2488	437	1	(	(	PUNCT
ejpam-2488	437	2	i)⇒	i)⇒	PROPN
ejpam-2488	437	3	(	(	PUNCT
ejpam-2488	437	4	ii	ii	NOUN
ejpam-2488	437	5	):	):	PUNCT
ejpam-2488	437	6	let	let	VERB
ejpam-2488	437	7	x	x	PRON
ejpam-2488	437	8	,	,	PUNCT
ejpam-2488	437	9	y	y	PROPN
ejpam-2488	437	10	∈	∈	PROPN
ejpam-2488	437	11	x	x	PUNCT
ejpam-2488	437	12	such	such	ADJ
ejpam-2488	437	13	that	that	SCONJ
ejpam-2488	437	14	there	there	PRON
ejpam-2488	437	15	exists	exist	VERB
ejpam-2488	437	16	a	a	DET
ejpam-2488	437	17	net	net	NOUN
ejpam-2488	437	18	{	{	PUNCT
ejpam-2488	437	19	xλ}λ∈a	xλ}λ∈a	X
ejpam-2488	437	20	in	in	ADP
ejpam-2488	437	21	x	x	PUNCT
ejpam-2488	437	22	such	such	ADJ
ejpam-2488	437	23	that	that	SCONJ
ejpam-2488	437	24	x	x	X
ejpam-2488	437	25	,	,	PUNCT
ejpam-2488	437	26	y	y	PROPN
ejpam-2488	437	27	∈	∈	PROPN
ejpam-2488	437	28	ei	ei	ADP
ejpam-2488	437	29	l	l	NOUN
ejpam-2488	437	30	im({xλ}λ∈a	im({xλ}λ∈a	PROPN
ejpam-2488	437	31	)	)	PUNCT
ejpam-2488	437	32	.	.	PUNCT
ejpam-2488	438	1	then	then	ADV
ejpam-2488	438	2	,	,	PUNCT
ejpam-2488	438	3	by	by	ADP
ejpam-2488	438	4	theorem	theorem	NOUN
ejpam-2488	438	5	9	9	NUM
ejpam-2488	438	6	,	,	PUNCT
ejpam-2488	438	7	(	(	PUNCT
ejpam-2488	438	8	a	a	X
ejpam-2488	438	9	)	)	PUNCT
ejpam-2488	438	10	if	if	SCONJ
ejpam-2488	438	11	u	u	NOUN
ejpam-2488	438	12	is	be	AUX
ejpam-2488	438	13	e	e	NOUN
ejpam-2488	438	14	-	-	ADJ
ejpam-2488	438	15	i	i	PRON
ejpam-2488	438	16	-open	-open	NOUN
ejpam-2488	438	17	,	,	PUNCT
ejpam-2488	438	18	then	then	ADV
ejpam-2488	438	19	x	x	PART
ejpam-2488	438	20	∈	∈	PROPN
ejpam-2488	438	21	u	u	NOUN
ejpam-2488	438	22	if	if	SCONJ
ejpam-2488	438	23	and	and	CCONJ
ejpam-2488	438	24	only	only	ADV
ejpam-2488	438	25	if	if	SCONJ
ejpam-2488	438	26	y	y	PROPN
ejpam-2488	438	27	∈	∈	PROPN
ejpam-2488	438	28	u	u	NOUN
ejpam-2488	438	29	or	or	CCONJ
ejpam-2488	438	30	(	(	PUNCT
ejpam-2488	438	31	b	b	X
ejpam-2488	438	32	)	)	PUNCT
ejpam-2488	438	33	there	there	PRON
ejpam-2488	438	34	exist	exist	VERB
ejpam-2488	438	35	disjoint	disjoint	NOUN
ejpam-2488	438	36	e	e	NOUN
ejpam-2488	438	37	-	-	ADJ
ejpam-2488	438	38	i	i	PRON
ejpam-2488	438	39	-open	-open	NOUN
ejpam-2488	438	40	sets	set	VERB
ejpam-2488	438	41	u	u	NOUN
ejpam-2488	438	42	and	and	CCONJ
ejpam-2488	438	43	v	v	ADP
ejpam-2488	438	44	such	such	ADJ
ejpam-2488	438	45	that	that	SCONJ
ejpam-2488	438	46	x	x	SYM
ejpam-2488	438	47	∈	∈	PROPN
ejpam-2488	438	48	u	u	NOUN
ejpam-2488	438	49	and	and	CCONJ
ejpam-2488	438	50	y	y	PROPN
ejpam-2488	438	51	∈	∈	PROPN
ejpam-2488	438	52	v	v	NOUN
ejpam-2488	438	53	.	.	PUNCT
ejpam-2488	439	1	since	since	SCONJ
ejpam-2488	439	2	x	x	X
ejpam-2488	439	3	,	,	PUNCT
ejpam-2488	439	4	y	y	PROPN
ejpam-2488	439	5	∈	∈	PROPN
ejpam-2488	439	6	ei	ei	ADP
ejpam-2488	439	7	l	l	PROPN
ejpam-2488	439	8	im({xλ}λ∈a	im({xλ}λ∈a	PROPN
ejpam-2488	439	9	)	)	PUNCT
ejpam-2488	439	10	,	,	PUNCT
ejpam-2488	439	11	then	then	ADV
ejpam-2488	439	12	(	(	PUNCT
ejpam-2488	439	13	a	a	X
ejpam-2488	439	14	)	)	PUNCT
ejpam-2488	439	15	is	be	AUX
ejpam-2488	439	16	satisfied	satisfied	ADJ
ejpam-2488	439	17	,	,	PUNCT
ejpam-2488	439	18	and	and	CCONJ
ejpam-2488	439	19	we	we	PRON
ejpam-2488	439	20	obtain	obtain	VERB
ejpam-2488	439	21	cl∗e	cl∗e	PROPN
ejpam-2488	439	22	(	(	PUNCT
ejpam-2488	439	23	{	{	PUNCT
ejpam-2488	439	24	x	x	NOUN
ejpam-2488	439	25	}	}	PUNCT
ejpam-2488	439	26	)	)	PUNCT
ejpam-2488	440	1	=	=	SYM
ejpam-2488	440	2	cl∗e	cl∗e	PROPN
ejpam-2488	440	3	(	(	PUNCT
ejpam-2488	440	4	{	{	PUNCT
ejpam-2488	440	5	y	y	NOUN
ejpam-2488	440	6	}	}	PUNCT
ejpam-2488	440	7	)	)	PUNCT
ejpam-2488	440	8	.	.	PUNCT
ejpam-2488	441	1	(	(	PUNCT
ejpam-2488	441	2	ii)⇒	ii)⇒	X
ejpam-2488	441	3	(	(	PUNCT
ejpam-2488	441	4	iii	iii	NOUN
ejpam-2488	441	5	):	):	PUNCT
ejpam-2488	441	6	let	let	VERB
ejpam-2488	441	7	u	u	PRON
ejpam-2488	441	8	∈	∈	PROPN
ejpam-2488	441	9	eio(x	eio(x	X
ejpam-2488	441	10	,	,	PUNCT
ejpam-2488	441	11	x	x	NOUN
ejpam-2488	441	12	)	)	PUNCT
ejpam-2488	441	13	.	.	PUNCT
ejpam-2488	442	1	let	let	VERB
ejpam-2488	442	2	y	y	PRON
ejpam-2488	442	3	/∈	/∈	PUNCT
ejpam-2488	443	1	u	u	PROPN
ejpam-2488	443	2	.	.	PUNCT
ejpam-2488	444	1	for	for	ADP
ejpam-2488	444	2	each	each	DET
ejpam-2488	444	3	n	n	PRON
ejpam-2488	444	4	∈	∈	PROPN
ejpam-2488	444	5	n	n	AUX
ejpam-2488	444	6	let	let	VERB
ejpam-2488	444	7	xn	xn	PUNCT
ejpam-2488	445	1	=	=	PUNCT
ejpam-2488	445	2	x	x	X
ejpam-2488	445	3	.	.	PUNCT
ejpam-2488	446	1	then	then	ADV
ejpam-2488	446	2	{	{	PUNCT
ejpam-2488	446	3	xn}n∈n	xn}n∈n	PUNCT
ejpam-2488	446	4	e	e	AUX
ejpam-2488	446	5	-	-	PROPN
ejpam-2488	446	6	i	i	PRON
ejpam-2488	446	7	converges	converge	VERB
ejpam-2488	446	8	to	to	ADP
ejpam-2488	446	9	x	x	PUNCT
ejpam-2488	446	10	and	and	CCONJ
ejpam-2488	446	11	since	since	SCONJ
ejpam-2488	446	12	cl∗e	cl∗e	PROPN
ejpam-2488	446	13	(	(	PUNCT
ejpam-2488	446	14	{	{	PUNCT
ejpam-2488	446	15	x	x	NOUN
ejpam-2488	446	16	}	}	PUNCT
ejpam-2488	446	17	)	)	PUNCT
ejpam-2488	447	1	6=	6=	ADP
ejpam-2488	448	1	cl∗e	cl∗e	X
ejpam-2488	448	2	(	(	PUNCT
ejpam-2488	448	3	{	{	PUNCT
ejpam-2488	448	4	y	y	NOUN
ejpam-2488	448	5	}	}	PUNCT
ejpam-2488	448	6	)	)	PUNCT
ejpam-2488	448	7	,	,	PUNCT
ejpam-2488	448	8	by	by	ADP
ejpam-2488	448	9	(	(	PUNCT
ejpam-2488	448	10	ii	ii	NOUN
ejpam-2488	448	11	)	)	PUNCT
ejpam-2488	448	12	{	{	PUNCT
ejpam-2488	448	13	xn	xn	X
ejpam-2488	448	14	}	}	PUNCT
ejpam-2488	448	15	does	do	AUX
ejpam-2488	448	16	not	not	PART
ejpam-2488	448	17	e	e	VERB
ejpam-2488	448	18	-	-	NOUN
ejpam-2488	448	19	i	i	PRON
ejpam-2488	448	20	-converge	-converge	NOUN
ejpam-2488	448	21	to	to	ADP
ejpam-2488	448	22	y	y	PROPN
ejpam-2488	448	23	and	and	CCONJ
ejpam-2488	448	24	there	there	PRON
ejpam-2488	448	25	exists	exist	VERB
ejpam-2488	448	26	a	a	DET
ejpam-2488	448	27	∈	∈	PROPN
ejpam-2488	448	28	eio(x	eio(x	X
ejpam-2488	448	29	)	)	PUNCT
ejpam-2488	448	30	such	such	ADJ
ejpam-2488	448	31	that	that	SCONJ
ejpam-2488	448	32	y	y	PROPN
ejpam-2488	448	33	∈	∈	PROPN
ejpam-2488	448	34	a	a	PRON
ejpam-2488	448	35	and	and	CCONJ
ejpam-2488	448	36	x	x	ADJ
ejpam-2488	448	37	/∈	/∈	PUNCT
ejpam-2488	448	38	a.	a.	PROPN
ejpam-2488	449	1	thus	thus	ADV
ejpam-2488	449	2	,	,	PUNCT
ejpam-2488	449	3	y	y	PROPN
ejpam-2488	449	4	/∈	/∈	PUNCT
ejpam-2488	449	5	cl∗e	cl∗e	PROPN
ejpam-2488	449	6	(	(	PUNCT
ejpam-2488	449	7	{	{	PUNCT
ejpam-2488	449	8	x	x	NOUN
ejpam-2488	449	9	}	}	PUNCT
ejpam-2488	449	10	)	)	PUNCT
ejpam-2488	449	11	and	and	CCONJ
ejpam-2488	449	12	cl∗e	cl∗e	PROPN
ejpam-2488	449	13	(	(	PUNCT
ejpam-2488	449	14	{	{	PUNCT
ejpam-2488	449	15	x	x	NOUN
ejpam-2488	449	16	}	}	PUNCT
ejpam-2488	449	17	)	)	PUNCT
ejpam-2488	450	1	⊂	⊂	PROPN
ejpam-2488	450	2	u	u	PROPN
ejpam-2488	450	3	.	.	PUNCT
ejpam-2488	451	1	hence	hence	ADV
ejpam-2488	451	2	(	(	PUNCT
ejpam-2488	451	3	x	x	X
ejpam-2488	451	4	,	,	PUNCT
ejpam-2488	451	5	τ	τ	PROPN
ejpam-2488	451	6	,	,	PUNCT
ejpam-2488	451	7	i	i	PROPN
ejpam-2488	451	8	)	)	PUNCT
ejpam-2488	451	9	is	be	AUX
ejpam-2488	451	10	e	e	NOUN
ejpam-2488	451	11	-	-	PUNCT
ejpam-2488	451	12	i	i	PRON
ejpam-2488	451	13	-r0	-r0	INTJ
ejpam-2488	451	14	.	.	PUNCT
ejpam-2488	452	1	let	let	AUX
ejpam-2488	452	2	{	{	PUNCT
ejpam-2488	452	3	xλ}λ∈a	xλ}λ∈a	PART
ejpam-2488	452	4	be	be	AUX
ejpam-2488	452	5	an	an	DET
ejpam-2488	452	6	e	e	NOUN
ejpam-2488	452	7	-	-	ADJ
ejpam-2488	452	8	i	i	PRON
ejpam-2488	452	9	-convergent	-convergent	ADJ
ejpam-2488	452	10	net	net	NOUN
ejpam-2488	452	11	in	in	ADP
ejpam-2488	452	12	x	x	X
ejpam-2488	452	13	.	.	PUNCT
ejpam-2488	453	1	let	let	VERB
ejpam-2488	453	2	x	x	PUNCT
ejpam-2488	453	3	∈	∈	PROPN
ejpam-2488	453	4	x	x	X
ejpam-2488	453	5	such	such	ADJ
ejpam-2488	453	6	that	that	SCONJ
ejpam-2488	453	7	{	{	PUNCT
ejpam-2488	453	8	xλ}λ∈a	xλ}λ∈a	PROPN
ejpam-2488	453	9	e	e	X
ejpam-2488	453	10	-	-	NOUN
ejpam-2488	453	11	i	i	PRON
ejpam-2488	453	12	-converges	-converge	NOUN
ejpam-2488	453	13	to	to	ADP
ejpam-2488	453	14	x	x	X
ejpam-2488	453	15	.	.	PUNCT
ejpam-2488	454	1	if	if	SCONJ
ejpam-2488	454	2	y	y	PROPN
ejpam-2488	454	3	∈	∈	PROPN
ejpam-2488	454	4	cl∗e	cl∗e	PROPN
ejpam-2488	454	5	(	(	PUNCT
ejpam-2488	454	6	{	{	PUNCT
ejpam-2488	454	7	x	x	NOUN
ejpam-2488	454	8	}	}	PUNCT
ejpam-2488	454	9	)	)	PUNCT
ejpam-2488	454	10	,	,	PUNCT
ejpam-2488	454	11	then	then	ADV
ejpam-2488	454	12	{	{	PUNCT
ejpam-2488	454	13	xλ}λ∈a	xλ}λ∈a	PROPN
ejpam-2488	454	14	e	e	X
ejpam-2488	454	15	-	-	NOUN
ejpam-2488	454	16	i	i	PRON
ejpam-2488	454	17	-converges	-converge	NOUN
ejpam-2488	454	18	to	to	ADP
ejpam-2488	454	19	y	y	PROPN
ejpam-2488	454	20	,	,	PUNCT
ejpam-2488	454	21	which	which	PRON
ejpam-2488	454	22	implies	imply	VERB
ejpam-2488	454	23	cl∗e	cl∗e	PROPN
ejpam-2488	454	24	(	(	PUNCT
ejpam-2488	454	25	{	{	PUNCT
ejpam-2488	454	26	x	x	NOUN
ejpam-2488	454	27	}	}	PUNCT
ejpam-2488	454	28	)	)	PUNCT
ejpam-2488	455	1	⊂	⊂	PROPN
ejpam-2488	455	2	ei	ei	X
ejpam-2488	455	3	l	l	PROPN
ejpam-2488	455	4	im({xλ}λ∈a	im({xλ}λ∈a	PROPN
ejpam-2488	455	5	)	)	PUNCT
ejpam-2488	455	6	.	.	PUNCT
ejpam-2488	456	1	let	let	VERB
ejpam-2488	456	2	y	y	PRON
ejpam-2488	456	3	∈	∈	PROPN
ejpam-2488	456	4	ei	ei	ADP
ejpam-2488	456	5	l	l	PROPN
ejpam-2488	456	6	im({xλ}λ∈a	im({xλ}λ∈a	PROPN
ejpam-2488	456	7	)	)	PUNCT
ejpam-2488	456	8	,	,	PUNCT
ejpam-2488	456	9	then	then	ADV
ejpam-2488	456	10	x	x	X
ejpam-2488	456	11	,	,	PUNCT
ejpam-2488	456	12	y	y	PROPN
ejpam-2488	456	13	∈	∈	PROPN
ejpam-2488	456	14	ei	ei	ADP
ejpam-2488	456	15	l	l	PROPN
ejpam-2488	456	16	im({xλ}λ∈a	im({xλ}λ∈a	PROPN
ejpam-2488	456	17	)	)	PUNCT
ejpam-2488	456	18	,	,	PUNCT
ejpam-2488	456	19	which	which	PRON
ejpam-2488	456	20	implies	imply	VERB
ejpam-2488	456	21	y	y	PROPN
ejpam-2488	456	22	∈	∈	PROPN
ejpam-2488	456	23	cl∗e	cl∗e	PROPN
ejpam-2488	456	24	(	(	PUNCT
ejpam-2488	456	25	{	{	PUNCT
ejpam-2488	456	26	y	y	NOUN
ejpam-2488	456	27	}	}	PUNCT
ejpam-2488	456	28	)	)	PUNCT
ejpam-2488	457	1	=	=	SYM
ejpam-2488	457	2	cl∗e	cl∗e	PROPN
ejpam-2488	457	3	(	(	PUNCT
ejpam-2488	457	4	{	{	PUNCT
ejpam-2488	457	5	x	x	NOUN
ejpam-2488	457	6	}	}	PUNCT
ejpam-2488	457	7	)	)	PUNCT
ejpam-2488	457	8	.	.	PUNCT
ejpam-2488	458	1	hence	hence	ADV
ejpam-2488	458	2	ei	ei	ADP
ejpam-2488	458	3	l	l	NOUN
ejpam-2488	458	4	im({xλ}λ∈a	im({xλ}λ∈a	PROPN
ejpam-2488	458	5	)	)	PUNCT
ejpam-2488	458	6	=	=	SYM
ejpam-2488	458	7	cl∗e	cl∗e	PROPN
ejpam-2488	458	8	(	(	PUNCT
ejpam-2488	458	9	{	{	PUNCT
ejpam-2488	458	10	x	x	NOUN
ejpam-2488	458	11	}	}	PUNCT
ejpam-2488	458	12	)	)	PUNCT
ejpam-2488	458	13	.	.	PUNCT
ejpam-2488	459	1	(	(	PUNCT
ejpam-2488	459	2	iii)⇒	iii)⇒	PROPN
ejpam-2488	459	3	(	(	PUNCT
ejpam-2488	459	4	i	i	NOUN
ejpam-2488	459	5	):	):	PUNCT
ejpam-2488	459	6	assume	assume	VERB
ejpam-2488	459	7	that	that	SCONJ
ejpam-2488	459	8	(	(	PUNCT
ejpam-2488	459	9	x	x	X
ejpam-2488	459	10	,	,	PUNCT
ejpam-2488	459	11	τ	τ	PROPN
ejpam-2488	459	12	,	,	PUNCT
ejpam-2488	459	13	i	i	PROPN
ejpam-2488	459	14	)	)	PUNCT
ejpam-2488	459	15	is	be	AUX
ejpam-2488	459	16	not	not	PART
ejpam-2488	459	17	e	e	NOUN
ejpam-2488	459	18	-	-	NOUN
ejpam-2488	459	19	i	i	PRON
ejpam-2488	459	20	-r1	-r1	NOUN
ejpam-2488	459	21	.	.	PUNCT
ejpam-2488	460	1	then	then	ADV
ejpam-2488	460	2	there	there	PRON
ejpam-2488	460	3	exist	exist	VERB
ejpam-2488	460	4	x	x	SYM
ejpam-2488	460	5	,	,	PUNCT
ejpam-2488	460	6	y	y	PROPN
ejpam-2488	460	7	∈	∈	PROPN
ejpam-2488	460	8	x	x	PUNCT
ejpam-2488	460	9	such	such	ADJ
ejpam-2488	460	10	that	that	SCONJ
ejpam-2488	460	11	cl∗e	cl∗e	PROPN
ejpam-2488	460	12	(	(	PUNCT
ejpam-2488	460	13	{	{	PUNCT
ejpam-2488	460	14	x	x	NOUN
ejpam-2488	460	15	}	}	PUNCT
ejpam-2488	460	16	)	)	PUNCT
ejpam-2488	460	17	6=	6=	ADP
ejpam-2488	461	1	cl∗e	cl∗e	X
ejpam-2488	461	2	(	(	PUNCT
ejpam-2488	461	3	{	{	PUNCT
ejpam-2488	461	4	y	y	NOUN
ejpam-2488	461	5	}	}	PUNCT
ejpam-2488	461	6	)	)	PUNCT
ejpam-2488	461	7	and	and	CCONJ
ejpam-2488	461	8	every	every	DET
ejpam-2488	461	9	e	e	NOUN
ejpam-2488	461	10	-	-	ADJ
ejpam-2488	461	11	i	i	PRON
ejpam-2488	461	12	-open	-open	NOUN
ejpam-2488	461	13	set	set	VERB
ejpam-2488	461	14	containing	contain	VERB
ejpam-2488	461	15	cl∗e	cl∗e	PROPN
ejpam-2488	461	16	(	(	PUNCT
ejpam-2488	461	17	{	{	PUNCT
ejpam-2488	461	18	x	x	NOUN
ejpam-2488	461	19	}	}	PUNCT
ejpam-2488	461	20	)	)	PUNCT
ejpam-2488	461	21	intersects	intersect	NOUN
ejpam-2488	461	22	every	every	DET
ejpam-2488	461	23	e	e	NOUN
ejpam-2488	461	24	-	-	ADJ
ejpam-2488	461	25	i	i	PRON
ejpam-2488	461	26	-open	-open	NOUN
ejpam-2488	461	27	set	set	VERB
ejpam-2488	461	28	containing	contain	VERB
ejpam-2488	461	29	cl∗e	cl∗e	PROPN
ejpam-2488	461	30	(	(	PUNCT
ejpam-2488	461	31	{	{	PUNCT
ejpam-2488	461	32	y	y	NOUN
ejpam-2488	461	33	}	}	PUNCT
ejpam-2488	461	34	)	)	PUNCT
ejpam-2488	461	35	.	.	PUNCT
ejpam-2488	462	1	since	since	SCONJ
ejpam-2488	462	2	(	(	PUNCT
ejpam-2488	462	3	x	x	X
ejpam-2488	462	4	,	,	PUNCT
ejpam-2488	462	5	τ	τ	PROPN
ejpam-2488	462	6	,	,	PUNCT
ejpam-2488	462	7	i	i	PROPN
ejpam-2488	462	8	)	)	PUNCT
ejpam-2488	462	9	is	be	AUX
ejpam-2488	462	10	e	e	NOUN
ejpam-2488	462	11	-	-	PUNCT
ejpam-2488	462	12	i	i	PRON
ejpam-2488	462	13	-r0	-r0	NOUN
ejpam-2488	462	14	,	,	PUNCT
ejpam-2488	462	15	then	then	ADV
ejpam-2488	462	16	every	every	DET
ejpam-2488	462	17	e	e	NOUN
ejpam-2488	462	18	-	-	ADJ
ejpam-2488	462	19	i	i	PRON
ejpam-2488	462	20	-open	-open	NOUN
ejpam-2488	462	21	set	set	VERB
ejpam-2488	462	22	containing	contain	VERB
ejpam-2488	462	23	x	x	PUNCT
ejpam-2488	462	24	contains	contain	VERB
ejpam-2488	462	25	cl∗e	cl∗e	PROPN
ejpam-2488	462	26	(	(	PUNCT
ejpam-2488	462	27	{	{	PUNCT
ejpam-2488	462	28	x	x	NOUN
ejpam-2488	462	29	}	}	PUNCT
ejpam-2488	462	30	)	)	PUNCT
ejpam-2488	462	31	and	and	CCONJ
ejpam-2488	462	32	every	every	DET
ejpam-2488	462	33	e	e	NOUN
ejpam-2488	462	34	-	-	ADJ
ejpam-2488	462	35	i	i	PRON
ejpam-2488	462	36	-open	-open	NOUN
ejpam-2488	462	37	set	set	VERB
ejpam-2488	462	38	containing	contain	VERB
ejpam-2488	462	39	y	y	PROPN
ejpam-2488	462	40	contains	contain	VERB
ejpam-2488	462	41	cl∗e	cl∗e	PROPN
ejpam-2488	462	42	(	(	PUNCT
ejpam-2488	462	43	{	{	PUNCT
ejpam-2488	462	44	y	y	NOUN
ejpam-2488	462	45	}	}	PUNCT
ejpam-2488	462	46	)	)	PUNCT
ejpam-2488	462	47	,	,	PUNCT
ejpam-2488	462	48	which	which	PRON
ejpam-2488	462	49	implies	imply	VERB
ejpam-2488	462	50	that	that	SCONJ
ejpam-2488	462	51	every	every	DET
ejpam-2488	462	52	e	e	NOUN
ejpam-2488	462	53	-	-	ADJ
ejpam-2488	462	54	i	i	PRON
ejpam-2488	462	55	-open	-open	NOUN
ejpam-2488	462	56	set	set	VERB
ejpam-2488	462	57	containing	contain	VERB
ejpam-2488	462	58	x	x	PUNCT
ejpam-2488	462	59	intersects	intersect	NOUN
ejpam-2488	462	60	every	every	DET
ejpam-2488	462	61	e	e	NOUN
ejpam-2488	462	62	-	-	ADJ
ejpam-2488	462	63	i	i	PRON
ejpam-2488	462	64	-open	-open	NOUN
ejpam-2488	462	65	set	set	VERB
ejpam-2488	462	66	containing	contain	VERB
ejpam-2488	462	67	y	y	PROPN
ejpam-2488	462	68	.	.	PUNCT
ejpam-2488	463	1	let	let	VERB
ejpam-2488	463	2	dx	dx	PROPN
ejpam-2488	463	3	=	=	PRON
ejpam-2488	463	4	{	{	PUNCT
ejpam-2488	463	5	u	u	X
ejpam-2488	463	6	⊂	⊂	PROPN
ejpam-2488	463	7	x	x	PUNCT
ejpam-2488	463	8	|u	|u	PROPN
ejpam-2488	463	9	∈	∈	PROPN
ejpam-2488	463	10	eio(x	eio(x	X
ejpam-2488	463	11	,	,	PUNCT
ejpam-2488	463	12	x	x	NOUN
ejpam-2488	463	13	)	)	PUNCT
ejpam-2488	463	14	}	}	PUNCT
ejpam-2488	463	15	.	.	PUNCT
ejpam-2488	464	1	let≥x	let≥x	PROPN
ejpam-2488	464	2	be	be	AUX
ejpam-2488	464	3	the	the	DET
ejpam-2488	464	4	binary	binary	ADJ
ejpam-2488	464	5	relation	relation	NOUN
ejpam-2488	464	6	on	on	ADP
ejpam-2488	464	7	dx	dx	PROPN
ejpam-2488	464	8	defined	define	VERB
ejpam-2488	464	9	by	by	ADP
ejpam-2488	464	10	u1	u1	NOUN
ejpam-2488	464	11	≥x	≥x	NOUN
ejpam-2488	464	12	u2	u2	PROPN
ejpam-2488	464	13	if	if	SCONJ
ejpam-2488	464	14	and	and	CCONJ
ejpam-2488	464	15	only	only	ADV
ejpam-2488	464	16	if	if	SCONJ
ejpam-2488	464	17	u1	u1	PROPN
ejpam-2488	464	18	⊂	⊂	PROPN
ejpam-2488	464	19	u2	u2	PROPN
ejpam-2488	464	20	.	.	PUNCT
ejpam-2488	465	1	then	then	ADV
ejpam-2488	465	2	,	,	PUNCT
ejpam-2488	465	3	clearly	clearly	ADV
ejpam-2488	465	4	(	(	PUNCT
ejpam-2488	465	5	dx	dx	PROPN
ejpam-2488	465	6	,	,	PUNCT
ejpam-2488	465	7	≥x	≥x	NOUN
ejpam-2488	465	8	)	)	PUNCT
ejpam-2488	465	9	is	be	AUX
ejpam-2488	465	10	a	a	DET
ejpam-2488	465	11	directed	direct	VERB
ejpam-2488	465	12	set	set	NOUN
ejpam-2488	465	13	.	.	PUNCT
ejpam-2488	466	1	let	let	VERB
ejpam-2488	466	2	dy	dy	X
ejpam-2488	466	3	=	=	PUNCT
ejpam-2488	466	4	{	{	PUNCT
ejpam-2488	466	5	u	u	X
ejpam-2488	466	6	⊂	⊂	PROPN
ejpam-2488	466	7	x	x	PUNCT
ejpam-2488	466	8	|u	|u	PROPN
ejpam-2488	466	9	∈	∈	PROPN
ejpam-2488	466	10	eio(x	eio(x	X
ejpam-2488	466	11	,	,	PUNCT
ejpam-2488	466	12	y	y	NOUN
ejpam-2488	466	13	)	)	PUNCT
ejpam-2488	466	14	}	}	PUNCT
ejpam-2488	466	15	and	and	CCONJ
ejpam-2488	466	16	let	let	VERB
ejpam-2488	466	17	≥y	≥y	NOUN
ejpam-2488	466	18	be	be	AUX
ejpam-2488	466	19	the	the	DET
ejpam-2488	466	20	binary	binary	ADJ
ejpam-2488	466	21	relation	relation	NOUN
ejpam-2488	466	22	on	on	ADP
ejpam-2488	466	23	dy	dy	NOUN
ejpam-2488	466	24	defined	define	VERB
ejpam-2488	466	25	by	by	ADP
ejpam-2488	466	26	u1	u1	NOUN
ejpam-2488	466	27	≥y	≥y	NOUN
ejpam-2488	466	28	u2	u2	PROPN
ejpam-2488	466	29	if	if	SCONJ
ejpam-2488	466	30	and	and	CCONJ
ejpam-2488	466	31	only	only	ADV
ejpam-2488	466	32	if	if	SCONJ
ejpam-2488	466	33	u1	u1	PROPN
ejpam-2488	466	34	⊂	⊂	PROPN
ejpam-2488	466	35	u2	u2	PROPN
ejpam-2488	466	36	.	.	PUNCT
ejpam-2488	467	1	then	then	ADV
ejpam-2488	467	2	,	,	PUNCT
ejpam-2488	467	3	(	(	PUNCT
ejpam-2488	467	4	dx	dx	PROPN
ejpam-2488	467	5	,	,	PUNCT
ejpam-2488	467	6	≥y	≥y	X
ejpam-2488	467	7	)	)	PUNCT
ejpam-2488	467	8	is	be	AUX
ejpam-2488	467	9	also	also	ADV
ejpam-2488	467	10	a	a	DET
ejpam-2488	467	11	directed	direct	VERB
ejpam-2488	467	12	set	set	NOUN
ejpam-2488	467	13	.	.	PUNCT
ejpam-2488	468	1	let	let	VERB
ejpam-2488	468	2	d	d	NOUN
ejpam-2488	468	3	=	=	PRON
ejpam-2488	468	4	{	{	PUNCT
ejpam-2488	468	5	(	(	PUNCT
ejpam-2488	468	6	u1	u1	NOUN
ejpam-2488	468	7	,	,	PUNCT
ejpam-2488	468	8	u2)|u1	u2)|u1	PROPN
ejpam-2488	468	9	∈	∈	PROPN
ejpam-2488	468	10	dx	dx	PROPN
ejpam-2488	468	11	and	and	CCONJ
ejpam-2488	468	12	u2	u2	PROPN
ejpam-2488	468	13	∈	∈	PROPN
ejpam-2488	468	14	dy	dy	NOUN
ejpam-2488	468	15	}	}	PUNCT
ejpam-2488	468	16	and	and	CCONJ
ejpam-2488	468	17	let≥	let≥	ADJ
ejpam-2488	468	18	be	be	AUX
ejpam-2488	468	19	the	the	DET
ejpam-2488	468	20	binary	binary	ADJ
ejpam-2488	468	21	relation	relation	NOUN
ejpam-2488	468	22	on	on	ADP
ejpam-2488	468	23	d	d	PROPN
ejpam-2488	468	24	defined	define	VERB
ejpam-2488	468	25	by	by	ADP
ejpam-2488	468	26	(	(	PUNCT
ejpam-2488	468	27	u1	u1	PROPN
ejpam-2488	468	28	,	,	PUNCT
ejpam-2488	468	29	u2)≥	u2)≥	X
ejpam-2488	468	30	(	(	PUNCT
ejpam-2488	468	31	v1	v1	NOUN
ejpam-2488	468	32	,	,	PUNCT
ejpam-2488	468	33	v2	v2	PROPN
ejpam-2488	468	34	)	)	PUNCT
ejpam-2488	469	1	if	if	SCONJ
ejpam-2488	469	2	and	and	CCONJ
ejpam-2488	469	3	only	only	ADV
ejpam-2488	469	4	if	if	SCONJ
ejpam-2488	469	5	u1	u1	NOUN
ejpam-2488	469	6	≥x	≥x	NOUN
ejpam-2488	469	7	v1	v1	NOUN
ejpam-2488	469	8	and	and	CCONJ
ejpam-2488	469	9	u2	u2	PROPN
ejpam-2488	469	10	≥y	≥y	PUNCT
ejpam-2488	469	11	v2	v2	PROPN
ejpam-2488	469	12	.	.	PUNCT
ejpam-2488	470	1	then	then	ADV
ejpam-2488	470	2	,	,	PUNCT
ejpam-2488	470	3	(	(	PUNCT
ejpam-2488	470	4	d,≥	d,≥	X
ejpam-2488	470	5	)	)	PUNCT
ejpam-2488	470	6	is	be	AUX
ejpam-2488	470	7	a	a	DET
ejpam-2488	470	8	directed	direct	VERB
ejpam-2488	470	9	set	set	NOUN
ejpam-2488	470	10	.	.	PUNCT
ejpam-2488	471	1	for	for	ADP
ejpam-2488	471	2	each	each	PRON
ejpam-2488	471	3	(	(	PUNCT
ejpam-2488	471	4	u1	u1	NOUN
ejpam-2488	471	5	,	,	PUNCT
ejpam-2488	471	6	u2	u2	NOUN
ejpam-2488	471	7	)	)	PUNCT
ejpam-2488	471	8	∈	∈	PROPN
ejpam-2488	472	1	d	d	NOUN
ejpam-2488	472	2	,	,	PUNCT
ejpam-2488	472	3	let	let	VERB
ejpam-2488	472	4	x(u1,u2	x(u1,u2	NOUN
ejpam-2488	472	5	)	)	PUNCT
ejpam-2488	473	1	∈	∈	PROPN
ejpam-2488	473	2	(	(	PUNCT
ejpam-2488	473	3	u1	u1	NOUN
ejpam-2488	473	4	,	,	PUNCT
ejpam-2488	473	5	u2	u2	PROPN
ejpam-2488	473	6	)	)	PUNCT
ejpam-2488	473	7	.	.	PUNCT
ejpam-2488	474	1	references	reference	NOUN
ejpam-2488	474	2	512	512	NUM
ejpam-2488	474	3	then	then	ADV
ejpam-2488	474	4	{	{	PUNCT
ejpam-2488	474	5	x(u1,u2	x(u1,u2	NOUN
ejpam-2488	474	6	)	)	PUNCT
ejpam-2488	474	7	}	}	PUNCT
ejpam-2488	474	8	(	(	PUNCT
ejpam-2488	474	9	u1,u2	u1,u2	PROPN
ejpam-2488	474	10	)	)	PUNCT
ejpam-2488	474	11	∈	∈	PROPN
ejpam-2488	475	1	d	d	NOUN
ejpam-2488	475	2	is	be	AUX
ejpam-2488	475	3	a	a	DET
ejpam-2488	475	4	net	net	NOUN
ejpam-2488	475	5	in	in	ADP
ejpam-2488	475	6	x	x	DET
ejpam-2488	475	7	that	that	SCONJ
ejpam-2488	475	8	e	e	NOUN
ejpam-2488	475	9	-	-	NOUN
ejpam-2488	475	10	i	i	PRON
ejpam-2488	475	11	-converges	-converge	NOUN
ejpam-2488	475	12	to	to	ADP
ejpam-2488	475	13	both	both	CCONJ
ejpam-2488	475	14	x	x	PROPN
ejpam-2488	475	15	and	and	CCONJ
ejpam-2488	475	16	y	y	PROPN
ejpam-2488	475	17	.	.	PUNCT
ejpam-2488	476	1	thus	thus	ADV
ejpam-2488	476	2	,	,	PUNCT
ejpam-2488	476	3	there	there	PRON
ejpam-2488	476	4	exists	exist	VERB
ejpam-2488	476	5	z	z	NOUN
ejpam-2488	476	6	∈	∈	PROPN
ejpam-2488	476	7	x	x	PUNCT
ejpam-2488	476	8	such	such	ADJ
ejpam-2488	476	9	that	that	PRON
ejpam-2488	476	10	ei	ei	NOUN
ejpam-2488	476	11	l	l	NOUN
ejpam-2488	476	12	im({x(u1,u2	im({x(u1,u2	PROPN
ejpam-2488	476	13	)	)	PUNCT
ejpam-2488	476	14	}	}	PUNCT
ejpam-2488	476	15	(	(	PUNCT
ejpam-2488	476	16	u1,u2)∈d	u1,u2)∈d	X
ejpam-2488	476	17	)	)	PUNCT
ejpam-2488	476	18	=	=	SYM
ejpam-2488	476	19	cl∗e	cl∗e	PROPN
ejpam-2488	476	20	(	(	PUNCT
ejpam-2488	476	21	{	{	PUNCT
ejpam-2488	476	22	z	z	NOUN
ejpam-2488	476	23	}	}	PUNCT
ejpam-2488	476	24	)	)	PUNCT
ejpam-2488	476	25	,	,	PUNCT
ejpam-2488	476	26	which	which	PRON
ejpam-2488	476	27	implies	imply	VERB
ejpam-2488	476	28	x	x	X
ejpam-2488	476	29	,	,	PUNCT
ejpam-2488	476	30	y	y	PROPN
ejpam-2488	476	31	∈	∈	PROPN
ejpam-2488	476	32	cl∗e	cl∗e	PROPN
ejpam-2488	476	33	(	(	PUNCT
ejpam-2488	476	34	{	{	PUNCT
ejpam-2488	476	35	z	z	NOUN
ejpam-2488	476	36	}	}	PUNCT
ejpam-2488	476	37	)	)	PUNCT
ejpam-2488	476	38	.	.	PUNCT
ejpam-2488	477	1	since	since	SCONJ
ejpam-2488	477	2	{	{	PUNCT
ejpam-2488	477	3	cl∗e	cl∗e	X
ejpam-2488	477	4	(	(	PUNCT
ejpam-2488	477	5	{	{	PUNCT
ejpam-2488	477	6	w	w	NOUN
ejpam-2488	477	7	}	}	PUNCT
ejpam-2488	477	8	)	)	PUNCT
ejpam-2488	477	9	:	:	PUNCT
ejpam-2488	477	10	w	w	X
ejpam-2488	477	11	∈	∈	NOUN
ejpam-2488	477	12	x	x	PUNCT
ejpam-2488	477	13	}	}	PUNCT
ejpam-2488	477	14	is	be	AUX
ejpam-2488	477	15	a	a	DET
ejpam-2488	477	16	decomposition	decomposition	NOUN
ejpam-2488	477	17	of	of	ADP
ejpam-2488	477	18	x	x	X
ejpam-2488	477	19	,	,	PUNCT
ejpam-2488	477	20	then	then	ADV
ejpam-2488	477	21	cl∗e	cl∗e	PROPN
ejpam-2488	477	22	(	(	PUNCT
ejpam-2488	477	23	{	{	PUNCT
ejpam-2488	477	24	x	x	NOUN
ejpam-2488	477	25	}	}	PUNCT
ejpam-2488	477	26	)	)	PUNCT
ejpam-2488	477	27	=	=	SYM
ejpam-2488	477	28	cl∗e	cl∗e	PROPN
ejpam-2488	477	29	(	(	PUNCT
ejpam-2488	477	30	{	{	PUNCT
ejpam-2488	477	31	z	z	NOUN
ejpam-2488	477	32	}	}	PUNCT
ejpam-2488	477	33	)	)	PUNCT
ejpam-2488	477	34	=	=	SYM
ejpam-2488	477	35	cl∗e	cl∗e	PROPN
ejpam-2488	477	36	(	(	PUNCT
ejpam-2488	477	37	{	{	PUNCT
ejpam-2488	477	38	y	y	NOUN
ejpam-2488	477	39	}	}	PUNCT
ejpam-2488	477	40	)	)	PUNCT
ejpam-2488	477	41	,	,	PUNCT
ejpam-2488	477	42	which	which	PRON
ejpam-2488	477	43	is	be	AUX
ejpam-2488	477	44	a	a	DET
ejpam-2488	477	45	contradiction	contradiction	NOUN
ejpam-2488	477	46	.	.	PUNCT
ejpam-2488	478	1	hence	hence	ADV
ejpam-2488	478	2	(	(	PUNCT
ejpam-2488	478	3	x	x	X
ejpam-2488	478	4	,	,	PUNCT
ejpam-2488	478	5	τ	τ	PROPN
ejpam-2488	478	6	,	,	PUNCT
ejpam-2488	478	7	i	i	PROPN
ejpam-2488	478	8	)	)	PUNCT
ejpam-2488	478	9	is	be	AUX
ejpam-2488	478	10	e	e	NOUN
ejpam-2488	478	11	-	-	NOUN
ejpam-2488	478	12	i	i	PRON
ejpam-2488	478	13	-r1	-r1	NOUN
ejpam-2488	478	14	.	.	PUNCT
ejpam-2488	479	1	acknowledgements	acknowledgement	VERB
ejpam-2488	479	2	the	the	DET
ejpam-2488	479	3	authors	author	NOUN
ejpam-2488	479	4	would	would	AUX
ejpam-2488	479	5	like	like	VERB
ejpam-2488	479	6	to	to	PART
ejpam-2488	479	7	acknowledge	acknowledge	VERB
ejpam-2488	479	8	the	the	DET
ejpam-2488	479	9	grant	grant	NOUN
ejpam-2488	479	10	:	:	PUNCT
ejpam-2488	479	11	ukm	ukm	PROPN
ejpam-2488	479	12	grant	grant	PROPN
ejpam-2488	479	13	dip2014	dip2014	NOUN
ejpam-2488	479	14	-	-	PUNCT
ejpam-2488	479	15	034	034	PROPN
ejpam-2488	479	16	and	and	CCONJ
ejpam-2488	479	17	ministry	ministry	PROPN
ejpam-2488	479	18	of	of	ADP
ejpam-2488	479	19	education	education	PROPN
ejpam-2488	479	20	,	,	PUNCT
ejpam-2488	479	21	malaysia	malaysia	PROPN
ejpam-2488	479	22	grant	grant	PROPN
ejpam-2488	479	23	frgs/1/2014	frgs/1/2014	NOUN
ejpam-2488	479	24	/	/	SYM
ejpam-2488	479	25	st06	st06	PROPN
ejpam-2488	479	26	/	/	SYM
ejpam-2488	479	27	ukm/01/1	ukm/01/1	NOUN
ejpam-2488	479	28	for	for	ADP
ejpam-2488	479	29	financial	financial	ADJ
ejpam-2488	479	30	support	support	NOUN
ejpam-2488	479	31	.	.	PUNCT
ejpam-2488	480	1	references	reference	NOUN
ejpam-2488	480	2	[	[	X
ejpam-2488	480	3	1	1	NUM
ejpam-2488	480	4	]	]	PUNCT
ejpam-2488	480	5	w.	w.	PROPN
ejpam-2488	480	6	al	al	PROPN
ejpam-2488	480	7	-	-	PUNCT
ejpam-2488	480	8	omeri	omeri	ADJ
ejpam-2488	480	9	,	,	PUNCT
ejpam-2488	480	10	m.	m.	NOUN
ejpam-2488	480	11	noorani	noorani	PROPN
ejpam-2488	480	12	,	,	PUNCT
ejpam-2488	480	13	and	and	CCONJ
ejpam-2488	480	14	a.	a.	PROPN
ejpam-2488	480	15	al	al	PROPN
ejpam-2488	480	16	-	-	PUNCT
ejpam-2488	480	17	omari	omari	PROPN
ejpam-2488	480	18	.	.	PUNCT
ejpam-2488	481	1	new	new	ADJ
ejpam-2488	481	2	forms	form	NOUN
ejpam-2488	481	3	of	of	ADP
ejpam-2488	481	4	contra	contra	NOUN
ejpam-2488	481	5	-	-	NOUN
ejpam-2488	481	6	continuity	continuity	NOUN
ejpam-2488	481	7	in	in	ADP
ejpam-2488	481	8	ideal	ideal	ADJ
ejpam-2488	481	9	topology	topology	NOUN
ejpam-2488	481	10	spaces	space	NOUN
ejpam-2488	481	11	.	.	PUNCT
ejpam-2488	482	1	missouri	missouri	PROPN
ejpam-2488	482	2	journal	journal	PROPN
ejpam-2488	482	3	of	of	ADP
ejpam-2488	482	4	mathematical	mathematical	ADJ
ejpam-2488	482	5	sciences	science	NOUN
ejpam-2488	482	6	,	,	PUNCT
ejpam-2488	482	7	26(1):33–47	26(1):33–47	NUM
ejpam-2488	482	8	,	,	PUNCT
ejpam-2488	482	9	2014	2014	NUM
ejpam-2488	482	10	.	.	PUNCT
ejpam-2488	483	1	[	[	X
ejpam-2488	483	2	2	2	X
ejpam-2488	483	3	]	]	PUNCT
ejpam-2488	483	4	w.	w.	PROPN
ejpam-2488	483	5	al	al	PROPN
ejpam-2488	483	6	-	-	PUNCT
ejpam-2488	483	7	omeri	omeri	ADJ
ejpam-2488	483	8	,	,	PUNCT
ejpam-2488	483	9	m.	m.	NOUN
ejpam-2488	483	10	noorani	noorani	PROPN
ejpam-2488	483	11	,	,	PUNCT
ejpam-2488	483	12	and	and	CCONJ
ejpam-2488	483	13	a.	a.	PROPN
ejpam-2488	483	14	al	al	PROPN
ejpam-2488	483	15	-	-	PUNCT
ejpam-2488	483	16	omari	omari	PROPN
ejpam-2488	483	17	.	.	PUNCT
ejpam-2488	484	1	on	on	ADP
ejpam-2488	484	2	e	e	PROPN
ejpam-2488	484	3	-	-	ADJ
ejpam-2488	484	4	i	i	PRON
ejpam-2488	484	5	-open	-open	NOUN
ejpam-2488	484	6	sets	set	NOUN
ejpam-2488	484	7	,	,	PUNCT
ejpam-2488	484	8	e	e	X
ejpam-2488	484	9	-	-	NOUN
ejpam-2488	484	10	i	i	PRON
ejpam-2488	484	11	-continuoues	-continuoue	NOUN
ejpam-2488	484	12	functions	function	NOUN
ejpam-2488	484	13	and	and	CCONJ
ejpam-2488	484	14	decomposition	decomposition	NOUN
ejpam-2488	484	15	of	of	ADP
ejpam-2488	484	16	continuity	continuity	NOUN
ejpam-2488	484	17	.	.	PUNCT
ejpam-2488	485	1	journal	journal	NOUN
ejpam-2488	485	2	of	of	ADP
ejpam-2488	485	3	mathematics	mathematic	NOUN
ejpam-2488	485	4	and	and	CCONJ
ejpam-2488	485	5	applications	application	NOUN
ejpam-2488	485	6	,	,	PUNCT
ejpam-2488	485	7	(	(	PUNCT
ejpam-2488	485	8	38):15–31	38):15–31	PROPN
ejpam-2488	485	9	,	,	PUNCT
ejpam-2488	485	10	2014	2014	NUM
ejpam-2488	485	11	.	.	PUNCT
ejpam-2488	486	1	[	[	X
ejpam-2488	486	2	3	3	X
ejpam-2488	486	3	]	]	X
ejpam-2488	486	4	f.	f.	PROPN
ejpam-2488	486	5	g.	g.	PROPN
ejpam-2488	486	6	arenas	arenas	PROPN
ejpam-2488	486	7	,	,	PUNCT
ejpam-2488	486	8	j.	j.	PROPN
ejpam-2488	486	9	dontchev	dontchev	PROPN
ejpam-2488	486	10	,	,	PUNCT
ejpam-2488	486	11	and	and	CCONJ
ejpam-2488	486	12	m.	m.	PROPN
ejpam-2488	486	13	l.	l.	PROPN
ejpam-2488	486	14	puertas	puertas	PROPN
ejpam-2488	486	15	.	.	PUNCT
ejpam-2488	487	1	idealization	idealization	NOUN
ejpam-2488	487	2	of	of	ADP
ejpam-2488	487	3	some	some	DET
ejpam-2488	487	4	weak	weak	ADJ
ejpam-2488	487	5	separation	separation	NOUN
ejpam-2488	487	6	axioms	axiom	NOUN
ejpam-2488	487	7	.	.	PUNCT
ejpam-2488	488	1	acta	acta	PROPN
ejpam-2488	488	2	mathematica	mathematica	PROPN
ejpam-2488	488	3	hungarica	hungarica	PROPN
ejpam-2488	488	4	,	,	PUNCT
ejpam-2488	488	5	89(1):47–53	89(1):47–53	NUM
ejpam-2488	488	6	,	,	PUNCT
ejpam-2488	488	7	2000	2000	NUM
ejpam-2488	488	8	.	.	PUNCT
ejpam-2488	489	1	[	[	X
ejpam-2488	489	2	4	4	NUM
ejpam-2488	489	3	]	]	PUNCT
ejpam-2488	489	4	a.	a.	PROPN
ejpam-2488	489	5	s.	s.	PROPN
ejpam-2488	489	6	davis	davis	PROPN
ejpam-2488	489	7	.	.	PUNCT
ejpam-2488	490	1	indexed	index	VERB
ejpam-2488	490	2	systems	system	NOUN
ejpam-2488	490	3	of	of	ADP
ejpam-2488	490	4	neighborhoods	neighborhood	NOUN
ejpam-2488	490	5	for	for	ADP
ejpam-2488	490	6	general	general	ADJ
ejpam-2488	490	7	topological	topological	ADJ
ejpam-2488	490	8	spaces	space	NOUN
ejpam-2488	490	9	.	.	PUNCT
ejpam-2488	491	1	american	american	PROPN
ejpam-2488	491	2	mathematical	mathematical	PROPN
ejpam-2488	491	3	society	society	NOUN
ejpam-2488	491	4	,	,	PUNCT
ejpam-2488	491	5	68:886–893	68:886–893	PROPN
ejpam-2488	491	6	,	,	PUNCT
ejpam-2488	491	7	1961	1961	NUM
ejpam-2488	491	8	.	.	PUNCT
ejpam-2488	492	1	[	[	X
ejpam-2488	492	2	5	5	X
ejpam-2488	492	3	]	]	PUNCT
ejpam-2488	492	4	j.	j.	PROPN
ejpam-2488	492	5	dontchev	dontchev	PROPN
ejpam-2488	492	6	.	.	PUNCT
ejpam-2488	493	1	strong	strong	ADJ
ejpam-2488	493	2	b	b	NOUN
ejpam-2488	493	3	-	-	PUNCT
ejpam-2488	493	4	sets	set	NOUN
ejpam-2488	493	5	and	and	CCONJ
ejpam-2488	493	6	another	another	DET
ejpam-2488	493	7	decomposition	decomposition	NOUN
ejpam-2488	493	8	of	of	ADP
ejpam-2488	493	9	continuity	continuity	NOUN
ejpam-2488	493	10	.	.	PUNCT
ejpam-2488	494	1	acta	acta	PROPN
ejpam-2488	494	2	mathematica	mathematica	PROPN
ejpam-2488	494	3	hungarica	hungarica	PROPN
ejpam-2488	494	4	,	,	PUNCT
ejpam-2488	494	5	75:259–265	75:259–265	NUM
ejpam-2488	494	6	,	,	PUNCT
ejpam-2488	494	7	1997	1997	NUM
ejpam-2488	494	8	.	.	PUNCT
ejpam-2488	495	1	[	[	X
ejpam-2488	495	2	6	6	NUM
ejpam-2488	495	3	]	]	PUNCT
ejpam-2488	495	4	k.	k.	PROPN
ejpam-2488	495	5	k.	k.	PROPN
ejpam-2488	495	6	dube	dube	PROPN
ejpam-2488	495	7	.	.	PUNCT
ejpam-2488	496	1	a	a	DET
ejpam-2488	496	2	note	note	NOUN
ejpam-2488	496	3	on	on	ADP
ejpam-2488	496	4	r0	r0	PROPN
ejpam-2488	496	5	topological	topological	ADJ
ejpam-2488	496	6	spaces	space	NOUN
ejpam-2488	496	7	.	.	PUNCT
ejpam-2488	497	1	matematicki	matematicki	NOUN
ejpam-2488	497	2	vesnik	vesnik	PROPN
ejpam-2488	497	3	,	,	PUNCT
ejpam-2488	497	4	11:203–208	11:203–208	NUM
ejpam-2488	497	5	,	,	PUNCT
ejpam-2488	497	6	1974	1974	NUM
ejpam-2488	497	7	.	.	PUNCT
ejpam-2488	498	1	[	[	X
ejpam-2488	498	2	7	7	X
ejpam-2488	498	3	]	]	X
ejpam-2488	498	4	e.	e.	PROPN
ejpam-2488	498	5	ekici	ekici	PROPN
ejpam-2488	498	6	.	.	PUNCT
ejpam-2488	499	1	some	some	DET
ejpam-2488	499	2	generalizations	generalization	NOUN
ejpam-2488	499	3	of	of	ADP
ejpam-2488	499	4	almost	almost	ADV
ejpam-2488	499	5	contra	contra	ADJ
ejpam-2488	499	6	-	-	ADJ
ejpam-2488	499	7	super	super	NOUN
ejpam-2488	499	8	-	-	NOUN
ejpam-2488	499	9	continuity	continuity	NOUN
ejpam-2488	499	10	.	.	PUNCT
ejpam-2488	500	1	filomat	filomat	NOUN
ejpam-2488	500	2	,	,	PUNCT
ejpam-2488	500	3	21(2):31–44	21(2):31–44	NUM
ejpam-2488	500	4	,	,	PUNCT
ejpam-2488	500	5	2007	2007	NUM
ejpam-2488	500	6	.	.	PUNCT
ejpam-2488	501	1	[	[	X
ejpam-2488	501	2	8	8	NUM
ejpam-2488	501	3	]	]	X
ejpam-2488	501	4	e.	e.	PROPN
ejpam-2488	501	5	ekici	ekici	PROPN
ejpam-2488	501	6	.	.	PUNCT
ejpam-2488	502	1	new	new	ADJ
ejpam-2488	502	2	forms	form	NOUN
ejpam-2488	502	3	of	of	ADP
ejpam-2488	502	4	contra	contra	NOUN
ejpam-2488	502	5	-	-	NOUN
ejpam-2488	502	6	continuity	continuity	NOUN
ejpam-2488	502	7	.	.	PUNCT
ejpam-2488	503	1	carpathian	carpathian	ADJ
ejpam-2488	503	2	journal	journal	PROPN
ejpam-2488	503	3	mathmatics	mathmatics	PROPN
ejpam-2488	503	4	,	,	PUNCT
ejpam-2488	503	5	24(1):37–45	24(1):37–45	NUM
ejpam-2488	503	6	,	,	PUNCT
ejpam-2488	503	7	2008	2008	NUM
ejpam-2488	503	8	.	.	PUNCT
ejpam-2488	504	1	[	[	X
ejpam-2488	504	2	9	9	X
ejpam-2488	504	3	]	]	PUNCT
ejpam-2488	504	4	e.	e.	PROPN
ejpam-2488	504	5	ekici	ekici	PROPN
ejpam-2488	504	6	.	.	PUNCT
ejpam-2488	505	1	on	on	ADP
ejpam-2488	505	2	e	e	VERB
ejpam-2488	505	3	-	-	ADJ
ejpam-2488	505	4	open	open	ADJ
ejpam-2488	505	5	sets	set	NOUN
ejpam-2488	505	6	,	,	PUNCT
ejpam-2488	505	7	dp∗-sets	dp∗-set	NOUN
ejpam-2488	505	8	and	and	CCONJ
ejpam-2488	505	9	dpe∗	dpe∗	NOUN
ejpam-2488	505	10	and	and	CCONJ
ejpam-2488	505	11	decompositions	decomposition	NOUN
ejpam-2488	505	12	of	of	ADP
ejpam-2488	505	13	continuity	continuity	NOUN
ejpam-2488	505	14	.	.	PUNCT
ejpam-2488	506	1	arabian	arabian	ADJ
ejpam-2488	506	2	journal	journal	PROPN
ejpam-2488	506	3	for	for	ADP
ejpam-2488	506	4	science	science	NOUN
ejpam-2488	506	5	and	and	CCONJ
ejpam-2488	506	6	engineering	engineering	NOUN
ejpam-2488	506	7	,	,	PUNCT
ejpam-2488	506	8	33(2a):269–282	33(2a):269–282	PROPN
ejpam-2488	506	9	,	,	PUNCT
ejpam-2488	506	10	2008	2008	NUM
ejpam-2488	506	11	.	.	PUNCT
ejpam-2488	507	1	[	[	X
ejpam-2488	507	2	10	10	NUM
ejpam-2488	507	3	]	]	X
ejpam-2488	507	4	d.	d.	PROPN
ejpam-2488	507	5	w.	w.	PROPN
ejpam-2488	507	6	hall	hall	PROPN
ejpam-2488	507	7	,	,	PUNCT
ejpam-2488	507	8	s.	s.	PROPN
ejpam-2488	507	9	k.	k.	PROPN
ejpam-2488	507	10	murphy	murphy	PROPN
ejpam-2488	507	11	,	,	PUNCT
ejpam-2488	507	12	and	and	CCONJ
ejpam-2488	507	13	j.	j.	PROPN
ejpam-2488	507	14	rozycki	rozycki	PROPN
ejpam-2488	507	15	.	.	PUNCT
ejpam-2488	508	1	on	on	ADP
ejpam-2488	508	2	spaces	space	NOUN
ejpam-2488	508	3	which	which	PRON
ejpam-2488	508	4	are	be	AUX
ejpam-2488	508	5	essentially	essentially	ADV
ejpam-2488	508	6	t1	t1	NUM
ejpam-2488	508	7	.	.	PUNCT
ejpam-2488	509	1	journal	journal	NOUN
ejpam-2488	509	2	of	of	ADP
ejpam-2488	509	3	the	the	DET
ejpam-2488	509	4	australian	australian	ADJ
ejpam-2488	509	5	mathematical	mathematical	ADJ
ejpam-2488	509	6	society	society	NOUN
ejpam-2488	509	7	,	,	PUNCT
ejpam-2488	509	8	12:451–455	12:451–455	NUM
ejpam-2488	509	9	,	,	PUNCT
ejpam-2488	509	10	1971	1971	NUM
ejpam-2488	509	11	.	.	PUNCT
ejpam-2488	510	1	[	[	X
ejpam-2488	510	2	11	11	NUM
ejpam-2488	510	3	]	]	X
ejpam-2488	510	4	d.	d.	PROPN
ejpam-2488	510	5	jankovic	jankovic	PROPN
ejpam-2488	510	6	and	and	CCONJ
ejpam-2488	510	7	t.	t.	PROPN
ejpam-2488	510	8	r.	r.	PROPN
ejpam-2488	510	9	hamlett	hamlett	PROPN
ejpam-2488	510	10	.	.	PUNCT
ejpam-2488	511	1	new	new	ADJ
ejpam-2488	511	2	topologies	topology	NOUN
ejpam-2488	511	3	from	from	ADP
ejpam-2488	511	4	old	old	ADJ
ejpam-2488	511	5	via	via	ADP
ejpam-2488	511	6	ideals	ideal	NOUN
ejpam-2488	511	7	.	.	PUNCT
ejpam-2488	512	1	american	american	PROPN
ejpam-2488	512	2	mathematical	mathematical	PROPN
ejpam-2488	512	3	monthly	monthly	PROPN
ejpam-2488	512	4	,	,	PUNCT
ejpam-2488	512	5	97(4):295–310	97(4):295–310	PROPN
ejpam-2488	512	6	,	,	PUNCT
ejpam-2488	512	7	1990	1990	NUM
ejpam-2488	512	8	.	.	PUNCT
ejpam-2488	513	1	[	[	X
ejpam-2488	513	2	12	12	NUM
ejpam-2488	513	3	]	]	PUNCT
ejpam-2488	513	4	m.	m.	NOUN
ejpam-2488	513	5	n.	n.	PROPN
ejpam-2488	513	6	mukherjee	mukherjee	PROPN
ejpam-2488	513	7	,	,	PUNCT
ejpam-2488	513	8	r.	r.	PROPN
ejpam-2488	513	9	bishwambhar	bishwambhar	PROPN
ejpam-2488	513	10	,	,	PUNCT
ejpam-2488	513	11	and	and	CCONJ
ejpam-2488	513	12	r.	r.	PROPN
ejpam-2488	513	13	sen	sen	PROPN
ejpam-2488	513	14	.	.	PROPN
ejpam-2488	513	15	on	on	ADP
ejpam-2488	513	16	extension	extension	NOUN
ejpam-2488	513	17	of	of	ADP
ejpam-2488	513	18	topological	topological	ADJ
ejpam-2488	513	19	spaces	space	NOUN
ejpam-2488	513	20	in	in	ADP
ejpam-2488	513	21	terms	term	NOUN
ejpam-2488	513	22	of	of	ADP
ejpam-2488	513	23	ideals	ideal	NOUN
ejpam-2488	513	24	.	.	PUNCT
ejpam-2488	514	1	topology	topology	NOUN
ejpam-2488	514	2	and	and	CCONJ
ejpam-2488	514	3	its	its	PRON
ejpam-2488	514	4	applications	application	NOUN
ejpam-2488	514	5	,	,	PUNCT
ejpam-2488	514	6	154(18):3167–3172	154(18):3167–3172	NUM
ejpam-2488	514	7	,	,	PUNCT
ejpam-2488	514	8	2007	2007	NUM
ejpam-2488	514	9	.	.	PUNCT
ejpam-2488	515	1	references	reference	NOUN
ejpam-2488	515	2	513	513	NUM
ejpam-2488	516	1	[	[	X
ejpam-2488	516	2	13	13	NUM
ejpam-2488	516	3	]	]	PUNCT
ejpam-2488	516	4	s.	s.	PROPN
ejpam-2488	516	5	a.	a.	PROPN
ejpam-2488	516	6	naimpally	naimpally	ADV
ejpam-2488	516	7	.	.	PUNCT
ejpam-2488	517	1	on	on	ADP
ejpam-2488	517	2	r0	r0	NOUN
ejpam-2488	517	3	-	-	PUNCT
ejpam-2488	517	4	topological	topological	ADJ
ejpam-2488	517	5	spaces	space	NOUN
ejpam-2488	517	6	.	.	PUNCT
ejpam-2488	518	1	annales	annales	PROPN
ejpam-2488	518	2	universitatis	universitatis	PROPN
ejpam-2488	518	3	scientiarum	scientiarum	PROPN
ejpam-2488	518	4	budapestinensis	budapestinensis	PROPN
ejpam-2488	518	5	,	,	PUNCT
ejpam-2488	518	6	sectio	sectio	PROPN
ejpam-2488	518	7	mathematica	mathematica	PROPN
ejpam-2488	518	8	,	,	PUNCT
ejpam-2488	518	9	10:53–54	10:53–54	NUM
ejpam-2488	518	10	,	,	PUNCT
ejpam-2488	518	11	1967	1967	NUM
ejpam-2488	518	12	.	.	PUNCT
ejpam-2488	519	1	[	[	X
ejpam-2488	519	2	14	14	NUM
ejpam-2488	519	3	]	]	PUNCT
ejpam-2488	519	4	a.	a.	NOUN
ejpam-2488	519	5	a.	a.	NOUN
ejpam-2488	519	6	nasef	nasef	PROPN
ejpam-2488	519	7	and	and	CCONJ
ejpam-2488	519	8	r.	r.	PROPN
ejpam-2488	519	9	a.	a.	PROPN
ejpam-2488	519	10	mahmoud	mahmoud	PROPN
ejpam-2488	519	11	.	.	PUNCT
ejpam-2488	520	1	some	some	DET
ejpam-2488	520	2	applications	application	NOUN
ejpam-2488	520	3	via	via	ADP
ejpam-2488	520	4	fuzzy	fuzzy	ADJ
ejpam-2488	520	5	ideals	ideal	NOUN
ejpam-2488	520	6	.	.	PUNCT
ejpam-2488	521	1	chaos	chaos	NOUN
ejpam-2488	521	2	,	,	PUNCT
ejpam-2488	521	3	solitons	soliton	NOUN
ejpam-2488	521	4	and	and	CCONJ
ejpam-2488	521	5	fractals	fractal	NOUN
ejpam-2488	521	6	,	,	PUNCT
ejpam-2488	521	7	13(4):825–831	13(4):825–831	NUM
ejpam-2488	521	8	,	,	PUNCT
ejpam-2488	521	9	2002	2002	NUM
ejpam-2488	521	10	.	.	PUNCT
ejpam-2488	522	1	[	[	X
ejpam-2488	522	2	15	15	NUM
ejpam-2488	522	3	]	]	X
ejpam-2488	522	4	n.	n.	NOUN
ejpam-2488	522	5	a.	a.	NOUN
ejpam-2488	522	6	shanin	shanin	PROPN
ejpam-2488	522	7	.	.	PUNCT
ejpam-2488	523	1	on	on	ADP
ejpam-2488	523	2	separation	separation	NOUN
ejpam-2488	523	3	in	in	ADP
ejpam-2488	523	4	topological	topological	ADJ
ejpam-2488	523	5	spaces	space	NOUN
ejpam-2488	523	6	.	.	PUNCT
ejpam-2488	524	1	doklady	doklady	PROPN
ejpam-2488	524	2	akademii	akademii	NOUN
ejpam-2488	524	3	nauk	nauk	NOUN
ejpam-2488	524	4	sssr	sssr	NOUN
ejpam-2488	524	5	,	,	PUNCT
ejpam-2488	524	6	38:110	38:110	NUM
ejpam-2488	524	7	–	–	PUNCT
ejpam-2488	524	8	113	113	NUM
ejpam-2488	524	9	,	,	PUNCT
ejpam-2488	524	10	1943	1943	NUM
ejpam-2488	524	11	.	.	PUNCT
ejpam-2488	525	1	[	[	X
ejpam-2488	525	2	16	16	NUM
ejpam-2488	525	3	]	]	PUNCT
ejpam-2488	525	4	m.	m.	NOUN
ejpam-2488	525	5	h.	h.	PROPN
ejpam-2488	525	6	stone	stone	PROPN
ejpam-2488	525	7	.	.	PUNCT
ejpam-2488	526	1	application	application	NOUN
ejpam-2488	526	2	of	of	ADP
ejpam-2488	526	3	the	the	DET
ejpam-2488	526	4	theory	theory	NOUN
ejpam-2488	526	5	of	of	ADP
ejpam-2488	526	6	boolean	boolean	ADJ
ejpam-2488	526	7	rings	ring	NOUN
ejpam-2488	526	8	to	to	ADP
ejpam-2488	526	9	general	general	ADJ
ejpam-2488	526	10	topology	topology	NOUN
ejpam-2488	526	11	.	.	PUNCT
ejpam-2488	527	1	transactions	transaction	NOUN
ejpam-2488	527	2	of	of	ADP
ejpam-2488	527	3	the	the	DET
ejpam-2488	527	4	american	american	PROPN
ejpam-2488	527	5	mathematical	mathematical	PROPN
ejpam-2488	527	6	society	society	NOUN
ejpam-2488	527	7	,	,	PUNCT
ejpam-2488	527	8	41:375–481	41:375–481	PROPN
ejpam-2488	527	9	,	,	PUNCT
ejpam-2488	527	10	1937	1937	NUM
ejpam-2488	527	11	.	.	PUNCT
ejpam-2488	528	1	[	[	X
ejpam-2488	528	2	17	17	NUM
ejpam-2488	528	3	]	]	X
ejpam-2488	528	4	r.	r.	PROPN
ejpam-2488	528	5	vaidyanathaswamy	vaidyanathaswamy	PROPN
ejpam-2488	528	6	.	.	PUNCT
ejpam-2488	529	1	the	the	DET
ejpam-2488	529	2	localization	localization	NOUN
ejpam-2488	529	3	theory	theory	NOUN
ejpam-2488	529	4	in	in	ADP
ejpam-2488	529	5	set	set	NOUN
ejpam-2488	529	6	-	-	PUNCT
ejpam-2488	529	7	topology	topology	NOUN
ejpam-2488	529	8	.	.	PUNCT
ejpam-2488	530	1	proceedings	proceeding	NOUN
ejpam-2488	530	2	of	of	ADP
ejpam-2488	530	3	the	the	DET
ejpam-2488	530	4	indian	indian	PROPN
ejpam-2488	530	5	academy	academy	PROPN
ejpam-2488	530	6	of	of	ADP
ejpam-2488	530	7	science	science	PROPN
ejpam-2488	530	8	,	,	PUNCT
ejpam-2488	530	9	20:51–61	20:51–61	NUM
ejpam-2488	530	10	,	,	PUNCT
ejpam-2488	530	11	1945	1945	NUM
ejpam-2488	530	12	.	.	PUNCT
ejpam-2488	531	1	[	[	X
ejpam-2488	531	2	18	18	NUM
ejpam-2488	531	3	]	]	X
ejpam-2488	531	4	n.	n.	NOUN
ejpam-2488	531	5	v.	v.	ADP
ejpam-2488	531	6	veliĉko	veliĉko	PROPN
ejpam-2488	531	7	.	.	PUNCT
ejpam-2488	532	1	h	h	NOUN
ejpam-2488	532	2	-	-	PUNCT
ejpam-2488	532	3	closed	close	VERB
ejpam-2488	532	4	topological	topological	ADJ
ejpam-2488	532	5	spaces	space	NOUN
ejpam-2488	532	6	.	.	PUNCT
ejpam-2488	533	1	transactions	transaction	NOUN
ejpam-2488	533	2	of	of	ADP
ejpam-2488	533	3	the	the	DET
ejpam-2488	533	4	american	american	PROPN
ejpam-2488	533	5	mathematical	mathematical	PROPN
ejpam-2488	533	6	society	society	NOUN
ejpam-2488	533	7	,	,	PUNCT
ejpam-2488	533	8	78(2):103–118	78(2):103–118	PROPN
ejpam-2488	533	9	,	,	PUNCT
ejpam-2488	533	10	1968	1968	NUM
ejpam-2488	533	11	.	.	PUNCT