id	sid	tid	token	lemma	pos
ejpam-131	1	1	european	european	PROPN
ejpam-131	1	2	journal	journal	PROPN
ejpam-131	1	3	of	of	ADP
ejpam-131	1	4	pure	pure	ADJ
ejpam-131	1	5	and	and	CCONJ
ejpam-131	1	6	applied	apply	VERB
ejpam-131	1	7	mathematics	mathematic	NOUN
ejpam-131	1	8	vol	vol	NOUN
ejpam-131	1	9	.	.	PROPN
ejpam-131	2	1	1	1	NUM
ejpam-131	2	2	,	,	PUNCT
ejpam-131	2	3	no	no	INTJ
ejpam-131	2	4	.	.	NOUN
ejpam-131	2	5	3	3	NUM
ejpam-131	2	6	,	,	PUNCT
ejpam-131	2	7	2008	2008	NUM
ejpam-131	2	8	,	,	PUNCT
ejpam-131	2	9	(	(	PUNCT
ejpam-131	2	10	3	3	NUM
ejpam-131	2	11	-	-	SYM
ejpam-131	2	12	9	9	NUM
ejpam-131	2	13	)	)	PUNCT
ejpam-131	2	14	issn	issn	PROPN
ejpam-131	2	15	1307	1307	NUM
ejpam-131	2	16	-	-	SYM
ejpam-131	2	17	5543	5543	NUM
ejpam-131	2	18	–	–	PUNCT
ejpam-131	2	19	www.ejpam.com	www.ejpam.com	X
ejpam-131	2	20	on	on	ADP
ejpam-131	2	21	ωb	ωb	NOUN
ejpam-131	2	22	-	-	PUNCT
ejpam-131	2	23	open	open	ADJ
ejpam-131	2	24	sets	set	NOUN
ejpam-131	2	25	and	and	CCONJ
ejpam-131	2	26	b	b	X
ejpam-131	2	27	-	-	PUNCT
ejpam-131	2	28	lindelöf	lindelöf	NOUN
ejpam-131	2	29	spaces	space	NOUN
ejpam-131	2	30	takashi	takashi	PROPN
ejpam-131	2	31	noiri1,∗†	noiri1,∗†	PROPN
ejpam-131	2	32	,	,	PUNCT
ejpam-131	2	33	ahmad	ahmad	PROPN
ejpam-131	2	34	al	al	PROPN
ejpam-131	2	35	-	-	PUNCT
ejpam-131	2	36	omari2	omari2	PROPN
ejpam-131	2	37	,	,	PUNCT
ejpam-131	2	38	and	and	CCONJ
ejpam-131	2	39	mohd	mohd	PROPN
ejpam-131	2	40	.	.	PUNCT
ejpam-131	3	1	salmi	salmi	PROPN
ejpam-131	3	2	md	md	PROPN
ejpam-131	3	3	.	.	PUNCT
ejpam-131	4	1	noorani2	noorani2	PROPN
ejpam-131	4	2	1	1	NUM
ejpam-131	4	3	2949	2949	NUM
ejpam-131	4	4	-	-	SYM
ejpam-131	4	5	1	1	NUM
ejpam-131	4	6	shiokita	shiokita	NOUN
ejpam-131	4	7	-	-	PUNCT
ejpam-131	4	8	cho	cho	ADJ
ejpam-131	4	9	,	,	PUNCT
ejpam-131	4	10	hinagu	hinagu	ADJ
ejpam-131	4	11	,	,	PUNCT
ejpam-131	4	12	yatsushiro	yatsushiro	PROPN
ejpam-131	4	13	-	-	PUNCT
ejpam-131	4	14	shi	shi	PROPN
ejpam-131	4	15	,	,	PUNCT
ejpam-131	4	16	kumamoto	kumamoto	PROPN
ejpam-131	4	17	-	-	PUNCT
ejpam-131	4	18	ken	ken	PROPN
ejpam-131	4	19	,	,	PUNCT
ejpam-131	4	20	869	869	NUM
ejpam-131	4	21	-	-	SYM
ejpam-131	4	22	5142	5142	NUM
ejpam-131	4	23	japan	japan	PROPN
ejpam-131	4	24	2	2	NUM
ejpam-131	4	25	school	school	NOUN
ejpam-131	4	26	of	of	ADP
ejpam-131	4	27	mathematical	mathematical	ADJ
ejpam-131	4	28	sciences	science	NOUN
ejpam-131	4	29	,	,	PUNCT
ejpam-131	4	30	faculty	faculty	NOUN
ejpam-131	4	31	of	of	ADP
ejpam-131	4	32	science	science	NOUN
ejpam-131	4	33	and	and	CCONJ
ejpam-131	4	34	technology	technology	NOUN
ejpam-131	4	35	,	,	PUNCT
ejpam-131	4	36	universiti	universiti	PROPN
ejpam-131	4	37	kebangsaan	kebangsaan	PROPN
ejpam-131	4	38	malaysia	malaysia	PROPN
ejpam-131	4	39	,	,	PUNCT
ejpam-131	4	40	43600	43600	NUM
ejpam-131	4	41	ukm	ukm	PROPN
ejpam-131	4	42	bangi	bangi	PROPN
ejpam-131	4	43	,	,	PUNCT
ejpam-131	4	44	selangor	selangor	PROPN
ejpam-131	4	45	,	,	PUNCT
ejpam-131	4	46	malaysia	malaysia	PROPN
ejpam-131	4	47	abstract	abstract	NOUN
ejpam-131	4	48	.	.	PUNCT
ejpam-131	5	1	in	in	ADP
ejpam-131	5	2	this	this	DET
ejpam-131	5	3	paper	paper	NOUN
ejpam-131	5	4	,	,	PUNCT
ejpam-131	5	5	we	we	PRON
ejpam-131	5	6	introduce	introduce	VERB
ejpam-131	5	7	and	and	CCONJ
ejpam-131	5	8	investigate	investigate	VERB
ejpam-131	5	9	a	a	DET
ejpam-131	5	10	new	new	ADJ
ejpam-131	5	11	class	class	NOUN
ejpam-131	5	12	of	of	ADP
ejpam-131	5	13	sets	set	NOUN
ejpam-131	5	14	called	call	VERB
ejpam-131	5	15	ωb	ωb	NOUN
ejpam-131	5	16	-	-	PUNCT
ejpam-131	5	17	open	open	ADJ
ejpam-131	5	18	sets	set	NOUN
ejpam-131	5	19	which	which	PRON
ejpam-131	5	20	is	be	AUX
ejpam-131	5	21	weaker	weak	ADJ
ejpam-131	5	22	than	than	ADP
ejpam-131	5	23	both	both	DET
ejpam-131	5	24	ω	ω	ADJ
ejpam-131	5	25	-	-	ADJ
ejpam-131	5	26	open	open	ADJ
ejpam-131	5	27	sets	set	NOUN
ejpam-131	5	28	and	and	CCONJ
ejpam-131	5	29	b	b	X
ejpam-131	5	30	-	-	PUNCT
ejpam-131	5	31	open	open	ADJ
ejpam-131	5	32	sets	set	NOUN
ejpam-131	5	33	.	.	PUNCT
ejpam-131	6	1	moreover	moreover	ADV
ejpam-131	6	2	,	,	PUNCT
ejpam-131	6	3	we	we	PRON
ejpam-131	6	4	obtain	obtain	VERB
ejpam-131	6	5	a	a	DET
ejpam-131	6	6	characterization	characterization	NOUN
ejpam-131	6	7	and	and	CCONJ
ejpam-131	6	8	preserving	preserve	VERB
ejpam-131	6	9	theorems	theorem	NOUN
ejpam-131	6	10	of	of	ADP
ejpam-131	6	11	b	b	NOUN
ejpam-131	6	12	-	-	PUNCT
ejpam-131	6	13	lindelöf	lindelöf	NOUN
ejpam-131	6	14	spaces	space	NOUN
ejpam-131	6	15	.	.	PUNCT
ejpam-131	7	1	ams	am	NOUN
ejpam-131	7	2	subject	subject	ADJ
ejpam-131	7	3	classifications	classification	NOUN
ejpam-131	7	4	:	:	PUNCT
ejpam-131	7	5	54c05	54c05	NUM
ejpam-131	7	6	,	,	PUNCT
ejpam-131	7	7	54c08	54c08	NUM
ejpam-131	7	8	,	,	PUNCT
ejpam-131	7	9	54c10	54c10	NUM
ejpam-131	7	10	.	.	PUNCT
ejpam-131	8	1	key	key	ADJ
ejpam-131	8	2	words	word	NOUN
ejpam-131	8	3	:	:	PUNCT
ejpam-131	8	4	b	b	X
ejpam-131	8	5	-	-	PUNCT
ejpam-131	8	6	open	open	ADJ
ejpam-131	8	7	set	set	NOUN
ejpam-131	8	8	,	,	PUNCT
ejpam-131	8	9	ω	ω	ADJ
ejpam-131	8	10	-	-	ADJ
ejpam-131	8	11	open	open	ADJ
ejpam-131	8	12	set	set	NOUN
ejpam-131	8	13	,	,	PUNCT
ejpam-131	8	14	,	,	PUNCT
ejpam-131	8	15	b	b	X
ejpam-131	8	16	-	-	PUNCT
ejpam-131	8	17	lindelöf	lindelöf	NOUN
ejpam-131	8	18	space	space	NOUN
ejpam-131	8	19	1	1	NUM
ejpam-131	8	20	.	.	PUNCT
ejpam-131	8	21	introduction	introduction	NOUN
ejpam-131	8	22	throughout	throughout	ADP
ejpam-131	8	23	this	this	DET
ejpam-131	8	24	paper	paper	NOUN
ejpam-131	8	25	,	,	PUNCT
ejpam-131	8	26	(	(	PUNCT
ejpam-131	8	27	x	x	X
ejpam-131	8	28	,	,	PUNCT
ejpam-131	8	29	τ	τ	PROPN
ejpam-131	8	30	)	)	PUNCT
ejpam-131	8	31	and	and	CCONJ
ejpam-131	8	32	(	(	PUNCT
ejpam-131	8	33	y	y	PROPN
ejpam-131	8	34	,	,	PUNCT
ejpam-131	8	35	σ	σ	PROPN
ejpam-131	8	36	)	)	PUNCT
ejpam-131	8	37	stand	stand	NOUN
ejpam-131	8	38	for	for	ADP
ejpam-131	8	39	topological	topological	ADJ
ejpam-131	8	40	spaces	space	NOUN
ejpam-131	8	41	with	with	ADP
ejpam-131	8	42	no	no	DET
ejpam-131	8	43	separation	separation	NOUN
ejpam-131	8	44	axioms	axiom	NOUN
ejpam-131	8	45	assumed	assume	VERB
ejpam-131	8	46	,	,	PUNCT
ejpam-131	8	47	unless	unless	SCONJ
ejpam-131	8	48	otherwise	otherwise	ADV
ejpam-131	8	49	stated	state	VERB
ejpam-131	8	50	.	.	PUNCT
ejpam-131	9	1	for	for	ADP
ejpam-131	9	2	a	a	DET
ejpam-131	9	3	subset	subset	NOUN
ejpam-131	9	4	a	a	PRON
ejpam-131	9	5	of	of	ADP
ejpam-131	9	6	x	x	PRON
ejpam-131	9	7	,	,	PUNCT
ejpam-131	9	8	the	the	DET
ejpam-131	9	9	closure	closure	NOUN
ejpam-131	9	10	of	of	ADP
ejpam-131	9	11	a	a	PRON
ejpam-131	9	12	and	and	CCONJ
ejpam-131	9	13	the	the	DET
ejpam-131	9	14	interior	interior	NOUN
ejpam-131	9	15	of	of	ADP
ejpam-131	9	16	a	a	PRON
ejpam-131	9	17	will	will	AUX
ejpam-131	9	18	be	be	AUX
ejpam-131	9	19	a	a	DET
ejpam-131	9	20	denoted	denote	VERB
ejpam-131	9	21	by	by	ADP
ejpam-131	9	22	cl(a	cl(a	NOUN
ejpam-131	9	23	)	)	PUNCT
ejpam-131	9	24	and	and	CCONJ
ejpam-131	9	25	int(a	int(a	PROPN
ejpam-131	9	26	)	)	PUNCT
ejpam-131	9	27	,	,	PUNCT
ejpam-131	9	28	respectively	respectively	ADV
ejpam-131	9	29	.	.	PUNCT
ejpam-131	10	1	let	let	AUX
ejpam-131	10	2	(	(	PUNCT
ejpam-131	10	3	x	x	X
ejpam-131	10	4	,	,	PUNCT
ejpam-131	10	5	τ	τ	X
ejpam-131	10	6	)	)	PUNCT
ejpam-131	10	7	be	be	VERB
ejpam-131	10	8	a	a	DET
ejpam-131	10	9	space	space	NOUN
ejpam-131	10	10	and	and	CCONJ
ejpam-131	10	11	let	let	VERB
ejpam-131	10	12	a	a	PRON
ejpam-131	10	13	be	be	AUX
ejpam-131	10	14	a	a	DET
ejpam-131	10	15	subset	subset	NOUN
ejpam-131	10	16	of	of	ADP
ejpam-131	10	17	x	x	X
ejpam-131	10	18	.	.	PUNCT
ejpam-131	11	1	a	a	DET
ejpam-131	11	2	point	point	NOUN
ejpam-131	11	3	x	x	X
ejpam-131	11	4	∈	∈	NOUN
ejpam-131	11	5	x	x	PUNCT
ejpam-131	11	6	is	be	AUX
ejpam-131	11	7	called	call	VERB
ejpam-131	11	8	a	a	DET
ejpam-131	11	9	condensation	condensation	NOUN
ejpam-131	11	10	point	point	NOUN
ejpam-131	11	11	of	of	ADP
ejpam-131	11	12	a	a	DET
ejpam-131	11	13	if	if	NOUN
ejpam-131	11	14	for	for	ADP
ejpam-131	11	15	each	each	DET
ejpam-131	11	16	u	u	NOUN
ejpam-131	11	17	∈	∈	PROPN
ejpam-131	11	18	τ	τ	X
ejpam-131	11	19	with	with	ADP
ejpam-131	11	20	x	x	PROPN
ejpam-131	11	21	∈	∈	PROPN
ejpam-131	11	22	u	u	NOUN
ejpam-131	11	23	,	,	PUNCT
ejpam-131	11	24	the	the	DET
ejpam-131	11	25	set	set	ADJ
ejpam-131	11	26	u	u	NOUN
ejpam-131	11	27	∩	∩	NOUN
ejpam-131	11	28	a	a	PRON
ejpam-131	11	29	is	be	AUX
ejpam-131	11	30	uncountable	uncountable	ADJ
ejpam-131	11	31	.	.	PUNCT
ejpam-131	12	1	a	a	PRON
ejpam-131	12	2	is	be	AUX
ejpam-131	12	3	said	say	VERB
ejpam-131	12	4	to	to	PART
ejpam-131	12	5	be	be	AUX
ejpam-131	12	6	ω	ω	NOUN
ejpam-131	12	7	-	-	ADJ
ejpam-131	12	8	closed	closed	ADJ
ejpam-131	12	9	[	[	X
ejpam-131	12	10	8	8	NUM
ejpam-131	12	11	]	]	X
ejpam-131	12	12	if	if	SCONJ
ejpam-131	12	13	it	it	PRON
ejpam-131	12	14	contains	contain	VERB
ejpam-131	12	15	all	all	DET
ejpam-131	12	16	its	its	PRON
ejpam-131	12	17	condensation	condensation	NOUN
ejpam-131	12	18	points	point	NOUN
ejpam-131	12	19	.	.	PUNCT
ejpam-131	13	1	the	the	DET
ejpam-131	13	2	complement	complement	NOUN
ejpam-131	13	3	of	of	ADP
ejpam-131	13	4	an	an	DET
ejpam-131	13	5	ω	ω	ADV
ejpam-131	13	6	-	-	PUNCT
ejpam-131	13	7	closed	closed	ADJ
ejpam-131	13	8	set	set	NOUN
ejpam-131	13	9	is	be	AUX
ejpam-131	13	10	said	say	VERB
ejpam-131	13	11	to	to	PART
ejpam-131	13	12	be	be	AUX
ejpam-131	13	13	ω	ω	NOUN
ejpam-131	13	14	-	-	NOUN
ejpam-131	13	15	open	open	ADJ
ejpam-131	13	16	.	.	PUNCT
ejpam-131	14	1	it	it	PRON
ejpam-131	14	2	is	be	AUX
ejpam-131	14	3	well	well	ADV
ejpam-131	14	4	known	know	VERB
ejpam-131	14	5	that	that	SCONJ
ejpam-131	14	6	a	a	DET
ejpam-131	14	7	subset	subset	NOUN
ejpam-131	14	8	w	w	NOUN
ejpam-131	14	9	of	of	ADP
ejpam-131	14	10	a	a	DET
ejpam-131	14	11	space	space	NOUN
ejpam-131	14	12	(	(	PUNCT
ejpam-131	14	13	x	x	X
ejpam-131	14	14	,	,	PUNCT
ejpam-131	14	15	τ	τ	X
ejpam-131	14	16	)	)	PUNCT
ejpam-131	14	17	is	be	AUX
ejpam-131	14	18	ω	ω	NOUN
ejpam-131	14	19	-	-	NOUN
ejpam-131	14	20	open	open	ADJ
ejpam-131	14	21	if	if	SCONJ
ejpam-131	14	22	and	and	CCONJ
ejpam-131	14	23	only	only	ADV
ejpam-131	14	24	if	if	SCONJ
ejpam-131	14	25	for	for	ADP
ejpam-131	14	26	each	each	DET
ejpam-131	14	27	x	x	SYM
ejpam-131	14	28	∈	∈	PROPN
ejpam-131	14	29	w	w	NOUN
ejpam-131	14	30	,	,	PUNCT
ejpam-131	14	31	there	there	PRON
ejpam-131	14	32	exists	exist	VERB
ejpam-131	14	33	u	u	PROPN
ejpam-131	14	34	∈	∈	PROPN
ejpam-131	14	35	τ	τ	X
ejpam-131	14	36	such	such	ADJ
ejpam-131	14	37	that	that	SCONJ
ejpam-131	14	38	x	x	SYM
ejpam-131	14	39	∈	∈	PROPN
ejpam-131	14	40	u	u	NOUN
ejpam-131	14	41	and	and	CCONJ
ejpam-131	14	42	u	u	PRON
ejpam-131	14	43	−w	−w	ADV
ejpam-131	14	44	is	be	AUX
ejpam-131	14	45	countable	countable	ADJ
ejpam-131	14	46	.	.	PUNCT
ejpam-131	15	1	the	the	DET
ejpam-131	15	2	family	family	NOUN
ejpam-131	15	3	of	of	ADP
ejpam-131	15	4	all	all	DET
ejpam-131	15	5	ω	ω	ADJ
ejpam-131	15	6	-	-	ADJ
ejpam-131	15	7	open	open	ADJ
ejpam-131	15	8	subsets	subset	NOUN
ejpam-131	15	9	of	of	ADP
ejpam-131	15	10	a	a	DET
ejpam-131	15	11	space	space	NOUN
ejpam-131	15	12	(	(	PUNCT
ejpam-131	15	13	x	x	X
ejpam-131	15	14	,	,	PUNCT
ejpam-131	15	15	τ	τ	PROPN
ejpam-131	15	16	)	)	PUNCT
ejpam-131	15	17	,	,	PUNCT
ejpam-131	15	18	denoted	denote	VERB
ejpam-131	15	19	by	by	ADP
ejpam-131	15	20	τω	τω	PRON
ejpam-131	15	21	or	or	CCONJ
ejpam-131	15	22	ωo(x	ωo(x	NUM
ejpam-131	15	23	)	)	PUNCT
ejpam-131	15	24	,	,	PUNCT
ejpam-131	15	25	forms	form	VERB
ejpam-131	15	26	a	a	DET
ejpam-131	15	27	topology	topology	NOUN
ejpam-131	15	28	on	on	ADP
ejpam-131	15	29	x	x	SYM
ejpam-131	15	30	finer	fine	ADJ
ejpam-131	15	31	than	than	ADP
ejpam-131	15	32	τ	τ	PROPN
ejpam-131	15	33	.	.	PUNCT
ejpam-131	16	1	the	the	DET
ejpam-131	16	2	ω	ω	NOUN
ejpam-131	16	3	-	-	NOUN
ejpam-131	16	4	closure	closure	NOUN
ejpam-131	16	5	and	and	CCONJ
ejpam-131	16	6	ω	ω	NOUN
ejpam-131	16	7	-	-	NOUN
ejpam-131	16	8	interior	interior	NOUN
ejpam-131	16	9	,	,	PUNCT
ejpam-131	16	10	that	that	PRON
ejpam-131	16	11	can	can	AUX
ejpam-131	16	12	be	be	AUX
ejpam-131	16	13	defined	define	VERB
ejpam-131	16	14	in	in	ADP
ejpam-131	16	15	the	the	DET
ejpam-131	16	16	same	same	ADJ
ejpam-131	16	17	way	way	NOUN
ejpam-131	16	18	as	as	ADP
ejpam-131	16	19	cl(a	cl(a	NUM
ejpam-131	16	20	)	)	PUNCT
ejpam-131	16	21	and	and	CCONJ
ejpam-131	16	22	int(a	int(a	PROPN
ejpam-131	16	23	)	)	PUNCT
ejpam-131	16	24	,	,	PUNCT
ejpam-131	16	25	respectively	respectively	ADV
ejpam-131	16	26	,	,	PUNCT
ejpam-131	16	27	will	will	AUX
ejpam-131	16	28	be	be	AUX
ejpam-131	16	29	denoted	denote	VERB
ejpam-131	16	30	by	by	ADP
ejpam-131	16	31	clω(a	clω(a	PROPN
ejpam-131	16	32	)	)	PUNCT
ejpam-131	16	33	and	and	CCONJ
ejpam-131	16	34	intω(a	intω(a	PROPN
ejpam-131	16	35	)	)	PUNCT
ejpam-131	16	36	,	,	PUNCT
ejpam-131	16	37	respectively	respectively	ADV
ejpam-131	16	38	.	.	PUNCT
ejpam-131	17	1	several	several	ADJ
ejpam-131	17	2	characterizations	characterization	NOUN
ejpam-131	17	3	ofω	ofω	ADV
ejpam-131	17	4	-	-	PUNCT
ejpam-131	17	5	closed	closed	ADJ
ejpam-131	17	6	subsets	subset	NOUN
ejpam-131	17	7	were	be	AUX
ejpam-131	17	8	provided	provide	VERB
ejpam-131	17	9	in	in	ADP
ejpam-131	17	10	[	[	X
ejpam-131	17	11	1,8,9	1,8,9	NUM
ejpam-131	17	12	]	]	PUNCT
ejpam-131	17	13	.	.	PUNCT
ejpam-131	18	1	andrijević	andrijević	VERB
ejpam-131	18	2	[	[	X
ejpam-131	18	3	4	4	X
ejpam-131	18	4	]	]	PUNCT
ejpam-131	18	5	introduced	introduce	VERB
ejpam-131	18	6	a	a	DET
ejpam-131	18	7	new	new	ADJ
ejpam-131	18	8	class	class	NOUN
ejpam-131	18	9	of	of	ADP
ejpam-131	18	10	generalized	generalized	ADJ
ejpam-131	18	11	open	open	ADJ
ejpam-131	18	12	sets	set	NOUN
ejpam-131	18	13	in	in	ADP
ejpam-131	18	14	a	a	DET
ejpam-131	18	15	topological	topological	ADJ
ejpam-131	18	16	space	space	NOUN
ejpam-131	18	17	,	,	PUNCT
ejpam-131	18	18	the	the	DET
ejpam-131	18	19	so	so	ADV
ejpam-131	18	20	-	-	PUNCT
ejpam-131	18	21	called	call	VERB
ejpam-131	18	22	b	b	NOUN
ejpam-131	18	23	-	-	PUNCT
ejpam-131	18	24	open	open	ADJ
ejpam-131	18	25	sets	set	NOUN
ejpam-131	18	26	.	.	PUNCT
ejpam-131	19	1	this	this	DET
ejpam-131	19	2	type	type	NOUN
ejpam-131	19	3	of	of	ADP
ejpam-131	19	4	sets	set	NOUN
ejpam-131	19	5	was	be	AUX
ejpam-131	19	6	discussed	discuss	VERB
ejpam-131	19	7	by	by	ADP
ejpam-131	19	8	[	[	X
ejpam-131	19	9	7	7	X
ejpam-131	19	10	]	]	PUNCT
ejpam-131	19	11	under	under	ADP
ejpam-131	19	12	the	the	DET
ejpam-131	19	13	name	name	NOUN
ejpam-131	19	14	of	of	ADP
ejpam-131	19	15	γ	γ	X
ejpam-131	19	16	-	-	ADJ
ejpam-131	19	17	open	open	ADJ
ejpam-131	19	18	sets	set	NOUN
ejpam-131	19	19	.	.	PUNCT
ejpam-131	20	1	the	the	DET
ejpam-131	20	2	class	class	NOUN
ejpam-131	20	3	of	of	ADP
ejpam-131	20	4	b	b	NOUN
ejpam-131	20	5	-	-	PUNCT
ejpam-131	20	6	open	open	ADJ
ejpam-131	20	7	sets	set	NOUN
ejpam-131	20	8	is	be	AUX
ejpam-131	20	9	contained	contain	VERB
ejpam-131	20	10	in	in	ADP
ejpam-131	20	11	the	the	DET
ejpam-131	20	12	class	class	NOUN
ejpam-131	20	13	of	of	ADP
ejpam-131	20	14	semi	semi	ADJ
ejpam-131	20	15	-	-	ADJ
ejpam-131	20	16	preopen	preopen	ADJ
ejpam-131	20	17	sets	set	NOUN
ejpam-131	20	18	and	and	CCONJ
ejpam-131	20	19	contains	contain	VERB
ejpam-131	20	20	all	all	DET
ejpam-131	20	21	semiopen	semiopen	ADJ
ejpam-131	20	22	sets	set	NOUN
ejpam-131	20	23	and	and	CCONJ
ejpam-131	20	24	preopen	preopen	ADJ
ejpam-131	20	25	sets	set	NOUN
ejpam-131	20	26	.	.	PUNCT
ejpam-131	21	1	the	the	DET
ejpam-131	21	2	class	class	NOUN
ejpam-131	21	3	of	of	ADP
ejpam-131	21	4	b	b	NOUN
ejpam-131	21	5	-	-	PUNCT
ejpam-131	21	6	open	open	ADJ
ejpam-131	21	7	sets	set	NOUN
ejpam-131	21	8	generates	generate	VERB
ejpam-131	21	9	the	the	DET
ejpam-131	21	10	same	same	ADJ
ejpam-131	21	11	topology	topology	NOUN
ejpam-131	21	12	as	as	ADP
ejpam-131	21	13	the	the	DET
ejpam-131	21	14	class	class	NOUN
ejpam-131	21	15	of	of	ADP
ejpam-131	21	16	preopen	preopen	ADJ
ejpam-131	21	17	sets	set	NOUN
ejpam-131	21	18	.	.	PUNCT
ejpam-131	22	1	since	since	SCONJ
ejpam-131	22	2	the	the	DET
ejpam-131	22	3	advent	advent	NOUN
ejpam-131	22	4	of	of	ADP
ejpam-131	22	5	these	these	DET
ejpam-131	22	6	notions	notion	NOUN
ejpam-131	22	7	,	,	PUNCT
ejpam-131	22	8	several	several	ADJ
ejpam-131	22	9	research	research	NOUN
ejpam-131	22	10	paper	paper	NOUN
ejpam-131	22	11	with	with	ADP
ejpam-131	22	12	interesting	interesting	ADJ
ejpam-131	22	13	results	result	NOUN
ejpam-131	22	14	in	in	ADP
ejpam-131	22	15	different	different	ADJ
ejpam-131	22	16	respects	respect	NOUN
ejpam-131	22	17	came	come	VERB
ejpam-131	22	18	to	to	ADP
ejpam-131	22	19	existence	existence	NOUN
ejpam-131	22	20	see	see	VERB
ejpam-131	22	21	[	[	X
ejpam-131	22	22	2,3,5,10,11	2,3,5,10,11	NUM
ejpam-131	22	23	]	]	PUNCT
ejpam-131	22	24	.	.	PUNCT
ejpam-131	23	1	∗corresponding	∗corresponde	VERB
ejpam-131	23	2	author	author	NOUN
ejpam-131	23	3	.	.	PUNCT
ejpam-131	24	1	email	email	NOUN
ejpam-131	24	2	addresses	address	NOUN
ejpam-131	24	3	:	:	PUNCT
ejpam-131	24	4	t.noiri@nifty.com	t.noiri@nifty.com	X
ejpam-131	24	5	(	(	PUNCT
ejpam-131	24	6	t.	t.	PROPN
ejpam-131	24	7	noiri	noiri	PROPN
ejpam-131	24	8	)	)	PUNCT
ejpam-131	24	9	,	,	PUNCT
ejpam-131	24	10	omarimutah1@yahoo.com	omarimutah1@yahoo.com	PROPN
ejpam-131	24	11	(	(	PUNCT
ejpam-131	24	12	a.	a.	NOUN
ejpam-131	24	13	omari	omari	PROPN
ejpam-131	24	14	)	)	PUNCT
ejpam-131	24	15	,	,	PUNCT
ejpam-131	24	16	msn@pkrisc.cc.ukm.my	msn@pkrisc.cc.ukm.my	PROPN
ejpam-131	24	17	(	(	PUNCT
ejpam-131	24	18	m.s	m.s	PROPN
ejpam-131	24	19	.	.	PROPN
ejpam-131	24	20	noorani	noorani	PROPN
ejpam-131	24	21	)	)	PUNCT
ejpam-131	25	1	†this	†this	DET
ejpam-131	25	2	work	work	NOUN
ejpam-131	25	3	is	be	AUX
ejpam-131	25	4	financially	financially	ADV
ejpam-131	25	5	supported	support	VERB
ejpam-131	25	6	by	by	ADP
ejpam-131	25	7	the	the	DET
ejpam-131	25	8	ministry	ministry	PROPN
ejpam-131	25	9	of	of	ADP
ejpam-131	25	10	science	science	PROPN
ejpam-131	25	11	,	,	PUNCT
ejpam-131	25	12	technology	technology	NOUN
ejpam-131	25	13	and	and	CCONJ
ejpam-131	25	14	innovation	innovation	NOUN
ejpam-131	25	15	,	,	PUNCT
ejpam-131	25	16	malaysia	malaysia	PROPN
ejpam-131	25	17	under	under	ADP
ejpam-131	25	18	science	science	NOUN
ejpam-131	25	19	fund	fund	NOUN
ejpam-131	25	20	grant	grant	VERB
ejpam-131	25	21	no	no	DET
ejpam-131	25	22	:	:	PUNCT
ejpam-131	25	23	06	06	NUM
ejpam-131	25	24	-	-	SYM
ejpam-131	25	25	01	01	NUM
ejpam-131	25	26	-	-	PUNCT
ejpam-131	25	27	02	02	NUM
ejpam-131	25	28	-	-	PUNCT
ejpam-131	25	29	sf0177	sf0177	NOUN
ejpam-131	25	30	.	.	PUNCT
ejpam-131	26	1	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-131	27	1	3	3	NUM
ejpam-131	27	2	c	c	X
ejpam-131	27	3	©	©	PROPN
ejpam-131	27	4	2008	2008	NUM
ejpam-131	27	5	ejpam	ejpam	VERB
ejpam-131	27	6	all	all	DET
ejpam-131	27	7	rights	right	NOUN
ejpam-131	27	8	reserved	reserve	VERB
ejpam-131	27	9	.	.	PUNCT
ejpam-131	28	1	t.	t.	PROPN
ejpam-131	28	2	noiri	noiri	PROPN
ejpam-131	28	3	,	,	PUNCT
ejpam-131	28	4	a.	a.	PROPN
ejpam-131	28	5	al	al	PROPN
ejpam-131	28	6	-	-	PUNCT
ejpam-131	28	7	omari	omari	PROPN
ejpam-131	28	8	and	and	CCONJ
ejpam-131	28	9	m.s.m	m.s.m	PROPN
ejpam-131	28	10	.	.	PROPN
ejpam-131	28	11	noorani	noorani	PROPN
ejpam-131	28	12	/	/	SYM
ejpam-131	28	13	eur	eur	PROPN
ejpam-131	28	14	.	.	PUNCT
ejpam-131	29	1	j.	j.	PROPN
ejpam-131	29	2	pure	pure	PROPN
ejpam-131	29	3	appl	appl	PROPN
ejpam-131	29	4	.	.	PROPN
ejpam-131	29	5	math	math	PROPN
ejpam-131	29	6	,	,	PUNCT
ejpam-131	29	7	1	1	NUM
ejpam-131	29	8	(	(	PUNCT
ejpam-131	29	9	2008	2008	NUM
ejpam-131	29	10	)	)	PUNCT
ejpam-131	29	11	,	,	PUNCT
ejpam-131	29	12	(	(	PUNCT
ejpam-131	29	13	3	3	NUM
ejpam-131	29	14	-	-	SYM
ejpam-131	29	15	9	9	NUM
ejpam-131	29	16	)	)	PUNCT
ejpam-131	29	17	4	4	NUM
ejpam-131	29	18	definition	definition	NOUN
ejpam-131	29	19	1.1	1.1	NUM
ejpam-131	29	20	.	.	PUNCT
ejpam-131	30	1	a	a	DET
ejpam-131	30	2	subset	subset	NOUN
ejpam-131	30	3	a	a	PRON
ejpam-131	30	4	of	of	ADP
ejpam-131	30	5	a	a	DET
ejpam-131	30	6	space	space	NOUN
ejpam-131	30	7	x	x	PUNCT
ejpam-131	30	8	is	be	AUX
ejpam-131	30	9	said	say	VERB
ejpam-131	30	10	to	to	PART
ejpam-131	30	11	be	be	AUX
ejpam-131	30	12	b	b	NOUN
ejpam-131	30	13	-	-	PUNCT
ejpam-131	30	14	open	open	ADJ
ejpam-131	30	15	[	[	X
ejpam-131	30	16	4	4	NUM
ejpam-131	30	17	]	]	X
ejpam-131	30	18	if	if	SCONJ
ejpam-131	30	19	a⊆	a⊆	PROPN
ejpam-131	30	20	cl(int(a))∪	cl(int(a))∪	VERB
ejpam-131	30	21	int(cl(a	int(cl(a	PROPN
ejpam-131	30	22	)	)	PUNCT
ejpam-131	30	23	)	)	PUNCT
ejpam-131	30	24	.	.	PUNCT
ejpam-131	31	1	the	the	DET
ejpam-131	31	2	complement	complement	NOUN
ejpam-131	31	3	of	of	ADP
ejpam-131	31	4	a	a	DET
ejpam-131	31	5	b	b	NOUN
ejpam-131	31	6	-	-	PUNCT
ejpam-131	31	7	open	open	ADJ
ejpam-131	31	8	set	set	NOUN
ejpam-131	31	9	is	be	AUX
ejpam-131	31	10	said	say	VERB
ejpam-131	31	11	to	to	PART
ejpam-131	31	12	be	be	AUX
ejpam-131	31	13	b	b	NOUN
ejpam-131	31	14	-	-	PUNCT
ejpam-131	31	15	closed	closed	ADJ
ejpam-131	31	16	[	[	X
ejpam-131	31	17	4	4	NUM
ejpam-131	31	18	]	]	PUNCT
ejpam-131	31	19	.	.	PUNCT
ejpam-131	32	1	the	the	DET
ejpam-131	32	2	intersection	intersection	NOUN
ejpam-131	32	3	of	of	ADP
ejpam-131	32	4	all	all	DET
ejpam-131	32	5	b	b	NOUN
ejpam-131	32	6	-	-	PUNCT
ejpam-131	32	7	closed	closed	ADJ
ejpam-131	32	8	sets	set	NOUN
ejpam-131	32	9	of	of	ADP
ejpam-131	32	10	x	x	PUNCT
ejpam-131	32	11	containing	contain	VERB
ejpam-131	32	12	a	a	PRON
ejpam-131	32	13	is	be	AUX
ejpam-131	32	14	called	call	VERB
ejpam-131	32	15	the	the	DET
ejpam-131	32	16	b	b	NOUN
ejpam-131	32	17	-	-	PUNCT
ejpam-131	32	18	closure	closure	NOUN
ejpam-131	32	19	of	of	ADP
ejpam-131	32	20	a	a	PRON
ejpam-131	32	21	and	and	CCONJ
ejpam-131	32	22	is	be	AUX
ejpam-131	32	23	denoted	denote	VERB
ejpam-131	32	24	by	by	ADP
ejpam-131	32	25	bcl(a	bcl(a	PROPN
ejpam-131	32	26	)	)	PUNCT
ejpam-131	32	27	.	.	PUNCT
ejpam-131	33	1	the	the	DET
ejpam-131	33	2	union	union	NOUN
ejpam-131	33	3	of	of	ADP
ejpam-131	33	4	all	all	DET
ejpam-131	33	5	b	b	NOUN
ejpam-131	33	6	-	-	PUNCT
ejpam-131	33	7	open	open	ADJ
ejpam-131	33	8	sets	set	NOUN
ejpam-131	33	9	of	of	ADP
ejpam-131	33	10	x	x	PUNCT
ejpam-131	33	11	contained	contain	VERB
ejpam-131	33	12	in	in	ADP
ejpam-131	33	13	a	a	PRON
ejpam-131	33	14	is	be	AUX
ejpam-131	33	15	called	call	VERB
ejpam-131	33	16	the	the	DET
ejpam-131	33	17	b	b	NOUN
ejpam-131	33	18	-	-	NOUN
ejpam-131	33	19	interior	interior	NOUN
ejpam-131	33	20	of	of	ADP
ejpam-131	33	21	a	a	PRON
ejpam-131	33	22	and	and	CCONJ
ejpam-131	33	23	is	be	AUX
ejpam-131	33	24	denoted	denote	VERB
ejpam-131	33	25	by	by	ADP
ejpam-131	33	26	bint(a	bint(a	PROPN
ejpam-131	33	27	)	)	PUNCT
ejpam-131	33	28	.	.	PUNCT
ejpam-131	34	1	the	the	DET
ejpam-131	34	2	family	family	NOUN
ejpam-131	34	3	of	of	ADP
ejpam-131	34	4	all	all	DET
ejpam-131	34	5	b	b	NOUN
ejpam-131	34	6	-	-	PUNCT
ejpam-131	34	7	open	open	ADJ
ejpam-131	34	8	(	(	PUNCT
ejpam-131	34	9	resp	resp	NOUN
ejpam-131	34	10	.	.	PUNCT
ejpam-131	35	1	b	b	X
ejpam-131	35	2	-	-	PUNCT
ejpam-131	35	3	closed	closed	ADJ
ejpam-131	35	4	)	)	PUNCT
ejpam-131	35	5	subsets	subset	NOUN
ejpam-131	35	6	of	of	ADP
ejpam-131	35	7	a	a	DET
ejpam-131	35	8	space	space	NOUN
ejpam-131	35	9	x	x	PUNCT
ejpam-131	35	10	is	be	AUX
ejpam-131	35	11	denoted	denote	VERB
ejpam-131	35	12	by	by	ADP
ejpam-131	35	13	bo(x	bo(x	NUM
ejpam-131	35	14	)	)	PUNCT
ejpam-131	35	15	(	(	PUNCT
ejpam-131	35	16	resp	resp	NOUN
ejpam-131	35	17	.	.	PUNCT
ejpam-131	35	18	bc(x	bc(x	NUM
ejpam-131	35	19	)	)	PUNCT
ejpam-131	35	20	)	)	PUNCT
ejpam-131	36	1	and	and	CCONJ
ejpam-131	36	2	the	the	DET
ejpam-131	36	3	collection	collection	NOUN
ejpam-131	36	4	of	of	ADP
ejpam-131	36	5	all	all	DET
ejpam-131	36	6	b	b	NOUN
ejpam-131	36	7	-	-	PUNCT
ejpam-131	36	8	open	open	ADJ
ejpam-131	36	9	subsets	subset	NOUN
ejpam-131	36	10	of	of	ADP
ejpam-131	36	11	x	x	PUNCT
ejpam-131	36	12	containing	contain	VERB
ejpam-131	36	13	a	a	DET
ejpam-131	36	14	fixed	fix	VERB
ejpam-131	36	15	point	point	NOUN
ejpam-131	36	16	x	x	VERB
ejpam-131	36	17	is	be	AUX
ejpam-131	36	18	denoted	denote	VERB
ejpam-131	36	19	by	by	ADP
ejpam-131	36	20	bo(x	bo(x	NUM
ejpam-131	36	21	,	,	PUNCT
ejpam-131	36	22	x	x	NOUN
ejpam-131	36	23	)	)	PUNCT
ejpam-131	36	24	.	.	PUNCT
ejpam-131	37	1	in	in	ADP
ejpam-131	37	2	this	this	DET
ejpam-131	37	3	paper	paper	NOUN
ejpam-131	37	4	,	,	PUNCT
ejpam-131	37	5	we	we	PRON
ejpam-131	37	6	introduce	introduce	VERB
ejpam-131	37	7	a	a	DET
ejpam-131	37	8	new	new	ADJ
ejpam-131	37	9	generalization	generalization	NOUN
ejpam-131	37	10	of	of	ADP
ejpam-131	37	11	ω	ω	VERB
ejpam-131	37	12	-	-	ADJ
ejpam-131	37	13	open	open	ADJ
ejpam-131	37	14	set	set	NOUN
ejpam-131	37	15	and	and	CCONJ
ejpam-131	37	16	b	b	NOUN
ejpam-131	37	17	-	-	PUNCT
ejpam-131	37	18	open	open	ADJ
ejpam-131	37	19	set	set	NOUN
ejpam-131	37	20	and	and	CCONJ
ejpam-131	37	21	investigate	investigate	VERB
ejpam-131	37	22	some	some	DET
ejpam-131	37	23	properties	property	NOUN
ejpam-131	37	24	of	of	ADP
ejpam-131	37	25	this	this	DET
ejpam-131	37	26	set	set	NOUN
ejpam-131	37	27	.	.	PUNCT
ejpam-131	38	1	moreover	moreover	ADV
ejpam-131	38	2	,	,	PUNCT
ejpam-131	38	3	we	we	PRON
ejpam-131	38	4	obtain	obtain	VERB
ejpam-131	38	5	a	a	DET
ejpam-131	38	6	characterization	characterization	NOUN
ejpam-131	38	7	and	and	CCONJ
ejpam-131	38	8	preserving	preserve	VERB
ejpam-131	38	9	theorems	theorem	NOUN
ejpam-131	38	10	of	of	ADP
ejpam-131	38	11	b	b	NOUN
ejpam-131	38	12	-	-	PUNCT
ejpam-131	38	13	lindelöf	lindelöf	NOUN
ejpam-131	38	14	spaces	space	NOUN
ejpam-131	38	15	.	.	PUNCT
ejpam-131	39	1	2	2	X
ejpam-131	39	2	.	.	X
ejpam-131	39	3	ωb	ωb	NOUN
ejpam-131	39	4	-	-	PUNCT
ejpam-131	39	5	open	open	ADJ
ejpam-131	39	6	sets	set	NOUN
ejpam-131	39	7	in	in	ADP
ejpam-131	39	8	this	this	DET
ejpam-131	39	9	section	section	NOUN
ejpam-131	39	10	we	we	PRON
ejpam-131	39	11	introduce	introduce	VERB
ejpam-131	39	12	the	the	DET
ejpam-131	39	13	following	following	ADJ
ejpam-131	39	14	notion	notion	NOUN
ejpam-131	39	15	:	:	PUNCT
ejpam-131	39	16	definition	definition	NOUN
ejpam-131	39	17	2.1	2.1	NUM
ejpam-131	39	18	.	.	PUNCT
ejpam-131	40	1	a	a	DET
ejpam-131	40	2	subset	subset	NOUN
ejpam-131	40	3	a	a	PRON
ejpam-131	40	4	of	of	ADP
ejpam-131	40	5	a	a	DET
ejpam-131	40	6	space	space	NOUN
ejpam-131	40	7	x	x	PUNCT
ejpam-131	40	8	is	be	AUX
ejpam-131	40	9	said	say	VERB
ejpam-131	40	10	to	to	PART
ejpam-131	40	11	be	be	AUX
ejpam-131	40	12	ωb	ωb	NOUN
ejpam-131	40	13	-	-	PUNCT
ejpam-131	40	14	open	open	ADJ
ejpam-131	40	15	if	if	SCONJ
ejpam-131	40	16	for	for	ADP
ejpam-131	40	17	every	every	DET
ejpam-131	40	18	x	x	SYM
ejpam-131	40	19	∈	∈	PROPN
ejpam-131	40	20	a	a	PRON
ejpam-131	40	21	,	,	PUNCT
ejpam-131	40	22	there	there	PRON
ejpam-131	40	23	exists	exist	VERB
ejpam-131	40	24	a	a	DET
ejpam-131	40	25	b	b	NOUN
ejpam-131	40	26	-	-	PUNCT
ejpam-131	40	27	open	open	ADJ
ejpam-131	40	28	subset	subset	NOUN
ejpam-131	40	29	ux	ux	PROPN
ejpam-131	40	30	⊆	⊆	NUM
ejpam-131	40	31	x	x	SYM
ejpam-131	40	32	containing	contain	VERB
ejpam-131	40	33	x	x	PUNCT
ejpam-131	40	34	such	such	ADJ
ejpam-131	40	35	that	that	DET
ejpam-131	40	36	ux	ux	PROPN
ejpam-131	40	37	−	−	PROPN
ejpam-131	40	38	a	a	PRON
ejpam-131	40	39	is	be	AUX
ejpam-131	40	40	countable	countable	ADJ
ejpam-131	40	41	.	.	PUNCT
ejpam-131	41	1	the	the	DET
ejpam-131	41	2	complement	complement	NOUN
ejpam-131	41	3	of	of	ADP
ejpam-131	41	4	an	an	DET
ejpam-131	41	5	ωb	ωb	NOUN
ejpam-131	41	6	-	-	PUNCT
ejpam-131	41	7	open	open	NOUN
ejpam-131	41	8	subset	subset	NOUN
ejpam-131	41	9	is	be	AUX
ejpam-131	41	10	said	say	VERB
ejpam-131	41	11	to	to	PART
ejpam-131	41	12	be	be	AUX
ejpam-131	41	13	ωb	ωb	NOUN
ejpam-131	41	14	-	-	PUNCT
ejpam-131	41	15	closed	closed	ADJ
ejpam-131	41	16	.	.	PUNCT
ejpam-131	42	1	lemma	lemma	PROPN
ejpam-131	42	2	2.2	2.2	NUM
ejpam-131	42	3	.	.	PUNCT
ejpam-131	43	1	for	for	ADP
ejpam-131	43	2	a	a	DET
ejpam-131	43	3	subset	subset	NOUN
ejpam-131	43	4	of	of	ADP
ejpam-131	43	5	a	a	DET
ejpam-131	43	6	topological	topological	ADJ
ejpam-131	43	7	space	space	NOUN
ejpam-131	43	8	,	,	PUNCT
ejpam-131	43	9	both	both	DET
ejpam-131	43	10	ω	ω	NOUN
ejpam-131	43	11	-	-	NOUN
ejpam-131	43	12	openness	openness	NOUN
ejpam-131	43	13	and	and	CCONJ
ejpam-131	43	14	b	b	NOUN
ejpam-131	43	15	-	-	PUNCT
ejpam-131	43	16	openness	openness	NOUN
ejpam-131	43	17	imply	imply	VERB
ejpam-131	43	18	ωbopenness	ωbopenness	NOUN
ejpam-131	43	19	.	.	PUNCT
ejpam-131	44	1	proof	proof	NOUN
ejpam-131	44	2	.	.	PUNCT
ejpam-131	45	1	(	(	PUNCT
ejpam-131	45	2	1	1	X
ejpam-131	45	3	)	)	PUNCT
ejpam-131	45	4	assume	assume	VERB
ejpam-131	45	5	a	a	PRON
ejpam-131	45	6	is	be	AUX
ejpam-131	45	7	ω	ω	NOUN
ejpam-131	45	8	-	-	NOUN
ejpam-131	45	9	open	open	ADJ
ejpam-131	45	10	then	then	ADV
ejpam-131	45	11	,	,	PUNCT
ejpam-131	45	12	for	for	ADP
ejpam-131	45	13	each	each	DET
ejpam-131	45	14	x	x	SYM
ejpam-131	45	15	∈	∈	PROPN
ejpam-131	45	16	a	a	PRON
ejpam-131	45	17	,	,	PUNCT
ejpam-131	45	18	there	there	PRON
ejpam-131	45	19	is	be	VERB
ejpam-131	45	20	an	an	DET
ejpam-131	45	21	open	open	ADJ
ejpam-131	45	22	set	set	NOUN
ejpam-131	45	23	containing	contain	VERB
ejpam-131	45	24	x	x	PUNCT
ejpam-131	45	25	such	such	ADJ
ejpam-131	45	26	that	that	DET
ejpam-131	45	27	ux	ux	PROPN
ejpam-131	45	28	−	−	PROPN
ejpam-131	45	29	a	a	PRON
ejpam-131	45	30	is	be	AUX
ejpam-131	45	31	countable	countable	ADJ
ejpam-131	45	32	set	set	NOUN
ejpam-131	45	33	.	.	PUNCT
ejpam-131	46	1	since	since	SCONJ
ejpam-131	46	2	every	every	DET
ejpam-131	46	3	open	open	ADJ
ejpam-131	46	4	set	set	NOUN
ejpam-131	46	5	is	be	AUX
ejpam-131	46	6	b	b	NOUN
ejpam-131	46	7	-	-	ADJ
ejpam-131	46	8	open	open	ADJ
ejpam-131	46	9	,	,	PUNCT
ejpam-131	46	10	a	a	PRON
ejpam-131	46	11	is	be	AUX
ejpam-131	46	12	ωb	ωb	NOUN
ejpam-131	46	13	-	-	PUNCT
ejpam-131	46	14	open	open	ADJ
ejpam-131	46	15	.	.	PUNCT
ejpam-131	47	1	(	(	PUNCT
ejpam-131	47	2	2	2	X
ejpam-131	47	3	)	)	PUNCT
ejpam-131	47	4	let	let	VERB
ejpam-131	47	5	a	a	PRON
ejpam-131	47	6	be	be	AUX
ejpam-131	47	7	b	b	NOUN
ejpam-131	47	8	-	-	ADJ
ejpam-131	47	9	open	open	ADJ
ejpam-131	47	10	.	.	PUNCT
ejpam-131	48	1	for	for	ADP
ejpam-131	48	2	each	each	DET
ejpam-131	48	3	x	x	SYM
ejpam-131	48	4	∈	∈	PROPN
ejpam-131	48	5	a	a	PRON
ejpam-131	48	6	,	,	PUNCT
ejpam-131	48	7	there	there	PRON
ejpam-131	48	8	exists	exist	VERB
ejpam-131	48	9	a	a	DET
ejpam-131	48	10	b	b	NOUN
ejpam-131	48	11	-	-	PUNCT
ejpam-131	48	12	open	open	ADJ
ejpam-131	48	13	set	set	NOUN
ejpam-131	48	14	ux	ux	PROPN
ejpam-131	48	15	=	=	PUNCT
ejpam-131	48	16	a	a	DET
ejpam-131	48	17	such	such	ADJ
ejpam-131	48	18	that	that	SCONJ
ejpam-131	48	19	x	x	SYM
ejpam-131	48	20	∈	∈	NOUN
ejpam-131	48	21	ux	ux	NOUN
ejpam-131	48	22	and	and	CCONJ
ejpam-131	48	23	ux	ux	INTJ
ejpam-131	48	24	−	−	CCONJ
ejpam-131	49	1	a=	a=	PROPN
ejpam-131	49	2	φ	φ	PROPN
ejpam-131	49	3	.	.	PUNCT
ejpam-131	50	1	therefore	therefore	ADV
ejpam-131	50	2	,	,	PUNCT
ejpam-131	50	3	a	a	PRON
ejpam-131	50	4	is	be	AUX
ejpam-131	50	5	ωb	ωb	NOUN
ejpam-131	50	6	-	-	PUNCT
ejpam-131	50	7	open	open	ADJ
ejpam-131	50	8	.	.	PUNCT
ejpam-131	51	1	the	the	DET
ejpam-131	51	2	following	follow	VERB
ejpam-131	51	3	diagram	diagram	NOUN
ejpam-131	51	4	shows	show	VERB
ejpam-131	51	5	the	the	DET
ejpam-131	51	6	implications	implication	NOUN
ejpam-131	51	7	for	for	ADP
ejpam-131	51	8	properties	property	NOUN
ejpam-131	51	9	of	of	ADP
ejpam-131	51	10	subsets	subset	NOUN
ejpam-131	51	11	open	open	VERB
ejpam-131	51	12	set	set	VERB
ejpam-131	51	13	−→	−→	NOUN
ejpam-131	51	14	b	b	NOUN
ejpam-131	51	15	-	-	PUNCT
ejpam-131	51	16	open	open	ADJ
ejpam-131	51	17	set	set	VERB
ejpam-131	51	18	↓	↓	PROPN
ejpam-131	51	19	↓	↓	PROPN
ejpam-131	51	20	ω	ω	PROPN
ejpam-131	51	21	-	-	ADJ
ejpam-131	51	22	open	open	ADJ
ejpam-131	51	23	set	set	VERB
ejpam-131	51	24	−→	−→	NOUN
ejpam-131	51	25	ωb	ωb	NOUN
ejpam-131	51	26	-	-	PUNCT
ejpam-131	51	27	open	open	NOUN
ejpam-131	51	28	set	set	NOUN
ejpam-131	51	29	the	the	DET
ejpam-131	51	30	converses	converse	NOUN
ejpam-131	51	31	need	need	AUX
ejpam-131	51	32	not	not	PART
ejpam-131	51	33	be	be	AUX
ejpam-131	51	34	true	true	ADJ
ejpam-131	51	35	as	as	SCONJ
ejpam-131	51	36	shown	show	VERB
ejpam-131	51	37	by	by	ADP
ejpam-131	51	38	the	the	DET
ejpam-131	51	39	following	follow	VERB
ejpam-131	51	40	examples	example	NOUN
ejpam-131	51	41	.	.	PUNCT
ejpam-131	52	1	example	example	NOUN
ejpam-131	52	2	2.3	2.3	NUM
ejpam-131	52	3	.	.	PUNCT
ejpam-131	53	1	let	let	VERB
ejpam-131	53	2	x	x	PUNCT
ejpam-131	53	3	=	=	PRON
ejpam-131	53	4	{	{	PUNCT
ejpam-131	53	5	a	a	PRON
ejpam-131	53	6	,	,	PUNCT
ejpam-131	53	7	b	b	NOUN
ejpam-131	53	8	,	,	PUNCT
ejpam-131	53	9	c	c	NOUN
ejpam-131	53	10	}	}	PUNCT
ejpam-131	53	11	and	and	CCONJ
ejpam-131	53	12	τ=	τ=	PRON
ejpam-131	53	13	{	{	PUNCT
ejpam-131	53	14	x	x	PROPN
ejpam-131	53	15	,	,	PUNCT
ejpam-131	53	16	φ	φ	PROPN
ejpam-131	53	17	,	,	PUNCT
ejpam-131	53	18	{	{	PUNCT
ejpam-131	53	19	a	a	X
ejpam-131	53	20	}	}	PUNCT
ejpam-131	53	21	}	}	PUNCT
ejpam-131	53	22	,	,	PUNCT
ejpam-131	53	23	{	{	PUNCT
ejpam-131	53	24	b	b	X
ejpam-131	53	25	}	}	PUNCT
ejpam-131	53	26	,	,	PUNCT
ejpam-131	53	27	{	{	PUNCT
ejpam-131	53	28	a	a	PRON
ejpam-131	53	29	,	,	PUNCT
ejpam-131	53	30	b	b	NOUN
ejpam-131	53	31	}	}	PUNCT
ejpam-131	53	32	}	}	PUNCT
ejpam-131	53	33	,	,	PUNCT
ejpam-131	53	34	then	then	ADV
ejpam-131	53	35	bo(x	bo(x	PUNCT
ejpam-131	53	36	)	)	PUNCT
ejpam-131	53	37	=	=	SYM
ejpam-131	54	1	{	{	PUNCT
ejpam-131	54	2	x	x	PROPN
ejpam-131	54	3	,	,	PUNCT
ejpam-131	54	4	φ	φ	PROPN
ejpam-131	54	5	,	,	PUNCT
ejpam-131	54	6	{	{	PUNCT
ejpam-131	54	7	a	a	X
ejpam-131	54	8	}	}	PUNCT
ejpam-131	54	9	,	,	PUNCT
ejpam-131	54	10	{	{	PUNCT
ejpam-131	54	11	b	b	NOUN
ejpam-131	54	12	}	}	PUNCT
ejpam-131	54	13	,	,	PUNCT
ejpam-131	54	14	{	{	PUNCT
ejpam-131	54	15	a	a	DET
ejpam-131	54	16	,	,	PUNCT
ejpam-131	54	17	b	b	NOUN
ejpam-131	54	18	}	}	PUNCT
ejpam-131	54	19	,	,	PUNCT
ejpam-131	54	20	{	{	PUNCT
ejpam-131	54	21	b	b	X
ejpam-131	54	22	,	,	PUNCT
ejpam-131	54	23	c	c	NOUN
ejpam-131	54	24	}	}	PUNCT
ejpam-131	54	25	,	,	PUNCT
ejpam-131	54	26	{	{	PUNCT
ejpam-131	54	27	a	a	PRON
ejpam-131	54	28	,	,	PUNCT
ejpam-131	54	29	c	c	NOUN
ejpam-131	54	30	}	}	PUNCT
ejpam-131	54	31	}	}	PUNCT
ejpam-131	54	32	.	.	PUNCT
ejpam-131	55	1	then	then	ADV
ejpam-131	55	2	{	{	PUNCT
ejpam-131	55	3	c	c	X
ejpam-131	55	4	}	}	PUNCT
ejpam-131	55	5	is	be	AUX
ejpam-131	55	6	ω	ω	NOUN
ejpam-131	55	7	-	-	ADJ
ejpam-131	55	8	open	open	ADJ
ejpam-131	55	9	(	(	PUNCT
ejpam-131	55	10	since	since	SCONJ
ejpam-131	55	11	x	x	PRON
ejpam-131	55	12	is	be	AUX
ejpam-131	55	13	a	a	DET
ejpam-131	55	14	countable	countable	ADJ
ejpam-131	55	15	set	set	NOUN
ejpam-131	55	16	)	)	PUNCT
ejpam-131	55	17	and	and	CCONJ
ejpam-131	55	18	it	it	PRON
ejpam-131	55	19	is	be	AUX
ejpam-131	55	20	not	not	PART
ejpam-131	55	21	b	b	NOUN
ejpam-131	55	22	-	-	PUNCT
ejpam-131	55	23	open	open	ADJ
ejpam-131	55	24	.	.	PUNCT
ejpam-131	55	25	example	example	NOUN
ejpam-131	56	1	2.4	2.4	NUM
ejpam-131	56	2	.	.	PUNCT
ejpam-131	57	1	let	let	VERB
ejpam-131	57	2	x	x	PUNCT
ejpam-131	57	3	=	=	PUNCT
ejpam-131	57	4	r	r	NOUN
ejpam-131	57	5	with	with	ADP
ejpam-131	57	6	the	the	DET
ejpam-131	57	7	usual	usual	ADJ
ejpam-131	57	8	topology	topology	NOUN
ejpam-131	57	9	τ	τ	PROPN
ejpam-131	57	10	.	.	PUNCT
ejpam-131	58	1	let	let	VERB
ejpam-131	58	2	a	a	DET
ejpam-131	58	3	=	=	NOUN
ejpam-131	58	4	q	q	NOUN
ejpam-131	58	5	be	be	AUX
ejpam-131	58	6	the	the	DET
ejpam-131	58	7	set	set	NOUN
ejpam-131	58	8	of	of	ADP
ejpam-131	58	9	all	all	DET
ejpam-131	58	10	rational	rational	ADJ
ejpam-131	58	11	numbers	number	NOUN
ejpam-131	58	12	.	.	PUNCT
ejpam-131	59	1	then	then	ADV
ejpam-131	59	2	a	a	PRON
ejpam-131	59	3	is	be	AUX
ejpam-131	59	4	b	b	NOUN
ejpam-131	59	5	-	-	PUNCT
ejpam-131	59	6	open	open	ADJ
ejpam-131	59	7	but	but	CCONJ
ejpam-131	59	8	it	it	PRON
ejpam-131	59	9	is	be	AUX
ejpam-131	59	10	not	not	PART
ejpam-131	59	11	ω	ω	NOUN
ejpam-131	59	12	-	-	NOUN
ejpam-131	59	13	open	open	ADJ
ejpam-131	59	14	.	.	PUNCT
ejpam-131	60	1	lemma	lemma	PROPN
ejpam-131	60	2	2.5	2.5	NUM
ejpam-131	60	3	.	.	PUNCT
ejpam-131	61	1	a	a	DET
ejpam-131	61	2	subset	subset	NOUN
ejpam-131	61	3	a	a	PRON
ejpam-131	61	4	of	of	ADP
ejpam-131	61	5	a	a	DET
ejpam-131	61	6	space	space	NOUN
ejpam-131	61	7	x	x	PUNCT
ejpam-131	61	8	is	be	AUX
ejpam-131	61	9	ωb	ωb	NOUN
ejpam-131	61	10	-	-	PUNCT
ejpam-131	61	11	open	open	ADJ
ejpam-131	61	12	if	if	SCONJ
ejpam-131	61	13	and	and	CCONJ
ejpam-131	61	14	only	only	ADV
ejpam-131	61	15	if	if	SCONJ
ejpam-131	61	16	for	for	ADP
ejpam-131	61	17	every	every	DET
ejpam-131	61	18	x	x	SYM
ejpam-131	61	19	∈	∈	PROPN
ejpam-131	61	20	a	a	PRON
ejpam-131	61	21	,	,	PUNCT
ejpam-131	61	22	there	there	PRON
ejpam-131	61	23	exists	exist	VERB
ejpam-131	61	24	a	a	DET
ejpam-131	61	25	b	b	NOUN
ejpam-131	61	26	-	-	PUNCT
ejpam-131	61	27	open	open	ADJ
ejpam-131	61	28	subset	subset	ADJ
ejpam-131	61	29	u	u	NOUN
ejpam-131	61	30	containing	contain	VERB
ejpam-131	61	31	x	x	PUNCT
ejpam-131	61	32	and	and	CCONJ
ejpam-131	61	33	a	a	DET
ejpam-131	61	34	countable	countable	ADJ
ejpam-131	61	35	subset	subset	NOUN
ejpam-131	61	36	c	c	NOUN
ejpam-131	61	37	such	such	ADJ
ejpam-131	61	38	that	that	DET
ejpam-131	61	39	u	u	NOUN
ejpam-131	62	1	−	−	PROPN
ejpam-131	62	2	c	c	NOUN
ejpam-131	62	3	⊆	⊆	NUM
ejpam-131	62	4	a.	a.	NOUN
ejpam-131	62	5	t.	t.	PROPN
ejpam-131	62	6	noiri	noiri	PROPN
ejpam-131	62	7	,	,	PUNCT
ejpam-131	62	8	a.	a.	PROPN
ejpam-131	62	9	al	al	PROPN
ejpam-131	62	10	-	-	PUNCT
ejpam-131	62	11	omari	omari	PROPN
ejpam-131	62	12	and	and	CCONJ
ejpam-131	62	13	m.s.m	m.s.m	PROPN
ejpam-131	62	14	.	.	PROPN
ejpam-131	62	15	noorani	noorani	PROPN
ejpam-131	62	16	/	/	SYM
ejpam-131	62	17	eur	eur	PROPN
ejpam-131	62	18	.	.	PUNCT
ejpam-131	63	1	j.	j.	PROPN
ejpam-131	63	2	pure	pure	PROPN
ejpam-131	63	3	appl	appl	PROPN
ejpam-131	63	4	.	.	PROPN
ejpam-131	63	5	math	math	PROPN
ejpam-131	63	6	,	,	PUNCT
ejpam-131	63	7	1	1	NUM
ejpam-131	63	8	(	(	PUNCT
ejpam-131	63	9	2008	2008	NUM
ejpam-131	63	10	)	)	PUNCT
ejpam-131	63	11	,	,	PUNCT
ejpam-131	63	12	(	(	PUNCT
ejpam-131	63	13	3	3	NUM
ejpam-131	63	14	-	-	SYM
ejpam-131	63	15	9	9	NUM
ejpam-131	63	16	)	)	PUNCT
ejpam-131	63	17	5	5	NUM
ejpam-131	63	18	proof	proof	NOUN
ejpam-131	63	19	.	.	PUNCT
ejpam-131	64	1	let	let	VERB
ejpam-131	64	2	a	a	DET
ejpam-131	64	3	beωb	beωb	NOUN
ejpam-131	64	4	-	-	PUNCT
ejpam-131	64	5	open	open	ADJ
ejpam-131	64	6	and	and	CCONJ
ejpam-131	64	7	x	x	SYM
ejpam-131	64	8	∈	∈	PROPN
ejpam-131	64	9	a	a	PRON
ejpam-131	64	10	,	,	PUNCT
ejpam-131	64	11	then	then	ADV
ejpam-131	64	12	there	there	PRON
ejpam-131	64	13	exists	exist	VERB
ejpam-131	64	14	a	a	DET
ejpam-131	64	15	b	b	NOUN
ejpam-131	64	16	-	-	PUNCT
ejpam-131	64	17	open	open	ADJ
ejpam-131	64	18	subset	subset	NOUN
ejpam-131	64	19	ux	ux	NOUN
ejpam-131	64	20	containing	contain	VERB
ejpam-131	64	21	x	x	PUNCT
ejpam-131	64	22	such	such	ADJ
ejpam-131	64	23	that	that	SCONJ
ejpam-131	64	24	|ux	|ux	NUM
ejpam-131	64	25	−	−	PROPN
ejpam-131	64	26	a|	a|	PROPN
ejpam-131	64	27	is	be	AUX
ejpam-131	64	28	countable	countable	ADJ
ejpam-131	64	29	.	.	PUNCT
ejpam-131	65	1	let	let	VERB
ejpam-131	65	2	c	c	NOUN
ejpam-131	65	3	=	=	SYM
ejpam-131	65	4	ux	ux	PROPN
ejpam-131	65	5	−	−	CCONJ
ejpam-131	65	6	a=	a=	ADV
ejpam-131	65	7	ux	ux	X
ejpam-131	65	8	∩	∩	NOUN
ejpam-131	65	9	(	(	PUNCT
ejpam-131	65	10	x	x	X
ejpam-131	65	11	−	−	PROPN
ejpam-131	65	12	a	a	NOUN
ejpam-131	65	13	)	)	PUNCT
ejpam-131	65	14	.	.	PUNCT
ejpam-131	66	1	then	then	ADV
ejpam-131	66	2	ux	ux	INTJ
ejpam-131	66	3	−	−	PROPN
ejpam-131	67	1	c	c	NOUN
ejpam-131	68	1	⊆	⊆	NUM
ejpam-131	68	2	a.	a.	NOUN
ejpam-131	68	3	conversely	conversely	ADV
ejpam-131	68	4	,	,	PUNCT
ejpam-131	68	5	let	let	VERB
ejpam-131	68	6	x	x	PUNCT
ejpam-131	68	7	∈	∈	VERB
ejpam-131	68	8	a.	a.	NOUN
ejpam-131	68	9	then	then	ADV
ejpam-131	68	10	there	there	PRON
ejpam-131	68	11	exists	exist	VERB
ejpam-131	68	12	a	a	DET
ejpam-131	68	13	b	b	NOUN
ejpam-131	68	14	-	-	PUNCT
ejpam-131	68	15	open	open	ADJ
ejpam-131	68	16	subset	subset	NOUN
ejpam-131	68	17	ux	ux	NOUN
ejpam-131	68	18	containing	contain	VERB
ejpam-131	68	19	x	x	PROPN
ejpam-131	68	20	and	and	CCONJ
ejpam-131	68	21	a	a	DET
ejpam-131	68	22	countable	countable	ADJ
ejpam-131	68	23	subset	subset	NOUN
ejpam-131	68	24	c	c	NOUN
ejpam-131	68	25	such	such	ADJ
ejpam-131	68	26	that	that	PRON
ejpam-131	69	1	ux	ux	PROPN
ejpam-131	70	1	−	−	PROPN
ejpam-131	70	2	c	c	NOUN
ejpam-131	70	3	⊆	⊆	NUM
ejpam-131	70	4	a.	a.	NOUN
ejpam-131	70	5	thus	thus	ADV
ejpam-131	70	6	ux	ux	ADV
ejpam-131	70	7	−	−	PROPN
ejpam-131	70	8	a⊆	a⊆	PROPN
ejpam-131	70	9	c	c	NOUN
ejpam-131	70	10	and	and	CCONJ
ejpam-131	70	11	ux	ux	INTJ
ejpam-131	70	12	−	−	PROPN
ejpam-131	71	1	a	a	PRON
ejpam-131	71	2	is	be	AUX
ejpam-131	71	3	countable	countable	ADJ
ejpam-131	71	4	set	set	NOUN
ejpam-131	71	5	.	.	PUNCT
ejpam-131	72	1	theorem	theorem	VERB
ejpam-131	72	2	2.6	2.6	NUM
ejpam-131	72	3	.	.	PUNCT
ejpam-131	73	1	let	let	VERB
ejpam-131	73	2	x	x	PRON
ejpam-131	73	3	be	be	AUX
ejpam-131	73	4	a	a	DET
ejpam-131	73	5	space	space	NOUN
ejpam-131	73	6	and	and	CCONJ
ejpam-131	73	7	c	c	NOUN
ejpam-131	73	8	⊆	⊆	NUM
ejpam-131	73	9	x	x	SYM
ejpam-131	73	10	.	.	PUNCT
ejpam-131	74	1	if	if	SCONJ
ejpam-131	74	2	c	c	PROPN
ejpam-131	74	3	is	be	AUX
ejpam-131	74	4	ωb	ωb	NOUN
ejpam-131	74	5	-	-	PUNCT
ejpam-131	74	6	closed	closed	ADJ
ejpam-131	74	7	,	,	PUNCT
ejpam-131	74	8	then	then	ADV
ejpam-131	74	9	c	c	PROPN
ejpam-131	74	10	⊆	⊆	NUM
ejpam-131	74	11	k	k	PROPN
ejpam-131	74	12	∪	∪	PROPN
ejpam-131	74	13	b	b	NOUN
ejpam-131	74	14	for	for	ADP
ejpam-131	74	15	some	some	DET
ejpam-131	74	16	b	b	NOUN
ejpam-131	74	17	-	-	PUNCT
ejpam-131	74	18	closed	closed	ADJ
ejpam-131	74	19	subset	subset	NOUN
ejpam-131	74	20	k	k	PROPN
ejpam-131	74	21	and	and	CCONJ
ejpam-131	74	22	a	a	DET
ejpam-131	74	23	countable	countable	ADJ
ejpam-131	74	24	subset	subset	NOUN
ejpam-131	74	25	b.	b.	NOUN
ejpam-131	74	26	proof	proof	NOUN
ejpam-131	74	27	.	.	PUNCT
ejpam-131	75	1	if	if	SCONJ
ejpam-131	75	2	c	c	PROPN
ejpam-131	75	3	is	be	AUX
ejpam-131	75	4	ωb	ωb	NOUN
ejpam-131	75	5	-	-	PUNCT
ejpam-131	75	6	closed	closed	ADJ
ejpam-131	75	7	,	,	PUNCT
ejpam-131	75	8	then	then	ADV
ejpam-131	75	9	x	x	INTJ
ejpam-131	75	10	−	−	PROPN
ejpam-131	75	11	c	c	PROPN
ejpam-131	75	12	is	be	AUX
ejpam-131	75	13	ωb	ωb	NOUN
ejpam-131	75	14	-	-	PUNCT
ejpam-131	75	15	open	open	ADJ
ejpam-131	75	16	and	and	CCONJ
ejpam-131	75	17	hence	hence	ADV
ejpam-131	75	18	for	for	ADP
ejpam-131	75	19	every	every	DET
ejpam-131	75	20	x	x	SYM
ejpam-131	75	21	∈	∈	PROPN
ejpam-131	75	22	x	x	PUNCT
ejpam-131	75	23	−	−	PROPN
ejpam-131	75	24	c	c	NOUN
ejpam-131	75	25	,	,	PUNCT
ejpam-131	75	26	there	there	PRON
ejpam-131	75	27	exists	exist	VERB
ejpam-131	75	28	a	a	DET
ejpam-131	75	29	b	b	NOUN
ejpam-131	75	30	-	-	PUNCT
ejpam-131	75	31	open	open	ADJ
ejpam-131	75	32	set	set	NOUN
ejpam-131	75	33	u	u	NOUN
ejpam-131	75	34	containing	contain	VERB
ejpam-131	75	35	x	x	PUNCT
ejpam-131	75	36	and	and	CCONJ
ejpam-131	75	37	a	a	DET
ejpam-131	75	38	countable	countable	ADJ
ejpam-131	75	39	set	set	NOUN
ejpam-131	75	40	b	b	NOUN
ejpam-131	75	41	such	such	ADJ
ejpam-131	75	42	that	that	DET
ejpam-131	75	43	u	u	NOUN
ejpam-131	76	1	−	−	PROPN
ejpam-131	76	2	b	b	PROPN
ejpam-131	76	3	⊆	⊆	NUM
ejpam-131	76	4	x	x	SYM
ejpam-131	76	5	−	−	PROPN
ejpam-131	76	6	c	c	NOUN
ejpam-131	76	7	.	.	PUNCT
ejpam-131	77	1	thus	thus	ADV
ejpam-131	77	2	c	c	X
ejpam-131	77	3	⊆	⊆	NUM
ejpam-131	77	4	x	x	SYM
ejpam-131	77	5	−	−	PROPN
ejpam-131	77	6	(	(	PUNCT
ejpam-131	77	7	u	u	NOUN
ejpam-131	77	8	−	−	PROPN
ejpam-131	77	9	b	b	NOUN
ejpam-131	77	10	)	)	PUNCT
ejpam-131	77	11	=	=	PUNCT
ejpam-131	78	1	x	x	X
ejpam-131	78	2	−	−	PROPN
ejpam-131	78	3	(	(	PUNCT
ejpam-131	78	4	u	u	NOUN
ejpam-131	78	5	∩	∩	NOUN
ejpam-131	78	6	(	(	PUNCT
ejpam-131	78	7	x	x	SYM
ejpam-131	78	8	−	−	PROPN
ejpam-131	78	9	b	b	NOUN
ejpam-131	78	10	)	)	PUNCT
ejpam-131	78	11	)	)	PUNCT
ejpam-131	79	1	=	=	PUNCT
ejpam-131	80	1	(	(	PUNCT
ejpam-131	80	2	x	x	X
ejpam-131	80	3	−	−	PROPN
ejpam-131	80	4	u)∪	u)∪	PROPN
ejpam-131	80	5	b.	b.	PROPN
ejpam-131	80	6	let	let	VERB
ejpam-131	80	7	k	k	NOUN
ejpam-131	80	8	=	=	PUNCT
ejpam-131	80	9	x	x	PUNCT
ejpam-131	80	10	−	−	PROPN
ejpam-131	80	11	u	u	NOUN
ejpam-131	80	12	.	.	PUNCT
ejpam-131	81	1	then	then	ADV
ejpam-131	81	2	k	k	PROPN
ejpam-131	81	3	is	be	AUX
ejpam-131	81	4	b	b	NOUN
ejpam-131	81	5	-	-	PUNCT
ejpam-131	81	6	closed	close	VERB
ejpam-131	81	7	such	such	ADJ
ejpam-131	81	8	that	that	SCONJ
ejpam-131	81	9	c	c	PROPN
ejpam-131	81	10	⊆	⊆	NUM
ejpam-131	81	11	k	k	PROPN
ejpam-131	81	12	∪	∪	PROPN
ejpam-131	81	13	b.	b.	PROPN
ejpam-131	81	14	lemma	lemma	PROPN
ejpam-131	81	15	2.7	2.7	NUM
ejpam-131	81	16	.	.	PUNCT
ejpam-131	82	1	[	[	X
ejpam-131	82	2	4	4	X
ejpam-131	82	3	]	]	X
ejpam-131	82	4	let	let	VERB
ejpam-131	82	5	(	(	PUNCT
ejpam-131	82	6	x	x	X
ejpam-131	82	7	,	,	PUNCT
ejpam-131	82	8	τ	τ	X
ejpam-131	82	9	)	)	PUNCT
ejpam-131	82	10	be	be	VERB
ejpam-131	82	11	a	a	DET
ejpam-131	82	12	topological	topological	ADJ
ejpam-131	82	13	space	space	NOUN
ejpam-131	82	14	.	.	PUNCT
ejpam-131	83	1	1	1	X
ejpam-131	83	2	.	.	X
ejpam-131	83	3	the	the	DET
ejpam-131	83	4	intersection	intersection	NOUN
ejpam-131	83	5	of	of	ADP
ejpam-131	83	6	an	an	DET
ejpam-131	83	7	open	open	ADJ
ejpam-131	83	8	set	set	NOUN
ejpam-131	83	9	and	and	CCONJ
ejpam-131	83	10	a	a	DET
ejpam-131	83	11	b	b	NOUN
ejpam-131	83	12	-	-	PUNCT
ejpam-131	83	13	open	open	ADJ
ejpam-131	83	14	set	set	NOUN
ejpam-131	83	15	is	be	AUX
ejpam-131	83	16	a	a	DET
ejpam-131	83	17	b	b	NOUN
ejpam-131	83	18	-	-	PUNCT
ejpam-131	83	19	open	open	ADJ
ejpam-131	83	20	set	set	NOUN
ejpam-131	83	21	.	.	PUNCT
ejpam-131	84	1	2	2	X
ejpam-131	84	2	.	.	X
ejpam-131	84	3	the	the	DET
ejpam-131	84	4	union	union	NOUN
ejpam-131	84	5	of	of	ADP
ejpam-131	84	6	any	any	DET
ejpam-131	84	7	family	family	NOUN
ejpam-131	84	8	of	of	ADP
ejpam-131	84	9	b	b	NOUN
ejpam-131	84	10	-	-	PUNCT
ejpam-131	84	11	open	open	ADJ
ejpam-131	84	12	sets	set	NOUN
ejpam-131	84	13	is	be	AUX
ejpam-131	84	14	a	a	DET
ejpam-131	84	15	b	b	NOUN
ejpam-131	84	16	-	-	PUNCT
ejpam-131	84	17	open	open	ADJ
ejpam-131	84	18	set	set	NOUN
ejpam-131	84	19	.	.	PUNCT
ejpam-131	85	1	proposition	proposition	NOUN
ejpam-131	85	2	2.8	2.8	NUM
ejpam-131	85	3	.	.	PUNCT
ejpam-131	86	1	the	the	DET
ejpam-131	86	2	intersection	intersection	NOUN
ejpam-131	86	3	of	of	ADP
ejpam-131	86	4	an	an	DET
ejpam-131	86	5	ωb	ωb	NOUN
ejpam-131	86	6	-	-	PUNCT
ejpam-131	86	7	open	open	NOUN
ejpam-131	86	8	set	set	NOUN
ejpam-131	86	9	and	and	CCONJ
ejpam-131	86	10	an	an	DET
ejpam-131	86	11	ω	ω	ADV
ejpam-131	86	12	-	-	ADJ
ejpam-131	86	13	open	open	ADJ
ejpam-131	86	14	set	set	NOUN
ejpam-131	86	15	is	be	AUX
ejpam-131	86	16	ωb	ωb	NOUN
ejpam-131	86	17	-	-	PUNCT
ejpam-131	86	18	open	open	ADJ
ejpam-131	86	19	.	.	PUNCT
ejpam-131	87	1	proof	proof	NOUN
ejpam-131	87	2	.	.	PUNCT
ejpam-131	88	1	let	let	VERB
ejpam-131	88	2	a	a	PRON
ejpam-131	88	3	be	be	AUX
ejpam-131	88	4	an	an	DET
ejpam-131	88	5	ωb	ωb	NOUN
ejpam-131	88	6	-	-	PUNCT
ejpam-131	88	7	open	open	NOUN
ejpam-131	88	8	set	set	NOUN
ejpam-131	88	9	and	and	CCONJ
ejpam-131	88	10	b	b	DET
ejpam-131	88	11	an	an	DET
ejpam-131	88	12	ω	ω	NOUN
ejpam-131	88	13	-	-	ADJ
ejpam-131	88	14	open	open	ADJ
ejpam-131	88	15	set	set	NOUN
ejpam-131	88	16	in	in	ADP
ejpam-131	88	17	a	a	DET
ejpam-131	88	18	space	space	NOUN
ejpam-131	88	19	x	x	X
ejpam-131	88	20	.	.	PUNCT
ejpam-131	89	1	let	let	VERB
ejpam-131	89	2	x	x	PRON
ejpam-131	89	3	be	be	AUX
ejpam-131	89	4	any	any	DET
ejpam-131	89	5	point	point	NOUN
ejpam-131	89	6	of	of	ADP
ejpam-131	89	7	a∩	a∩	PROPN
ejpam-131	89	8	b.	b.	PROPN
ejpam-131	89	9	since	since	SCONJ
ejpam-131	89	10	a	a	PRON
ejpam-131	89	11	is	be	AUX
ejpam-131	89	12	ωb	ωb	NOUN
ejpam-131	89	13	-	-	PUNCT
ejpam-131	89	14	open	open	ADJ
ejpam-131	89	15	,	,	PUNCT
ejpam-131	89	16	there	there	PRON
ejpam-131	89	17	exists	exist	VERB
ejpam-131	89	18	a	a	DET
ejpam-131	89	19	b	b	NOUN
ejpam-131	89	20	-	-	PUNCT
ejpam-131	89	21	open	open	ADJ
ejpam-131	89	22	set	set	NOUN
ejpam-131	89	23	ua	ua	NOUN
ejpam-131	89	24	containing	contain	VERB
ejpam-131	89	25	x	x	PUNCT
ejpam-131	89	26	such	such	ADJ
ejpam-131	89	27	that	that	SCONJ
ejpam-131	89	28	|ua−	|ua−	NOUN
ejpam-131	89	29	a|	a|	PROPN
ejpam-131	89	30	is	be	AUX
ejpam-131	89	31	countable	countable	ADJ
ejpam-131	89	32	.	.	PUNCT
ejpam-131	90	1	since	since	SCONJ
ejpam-131	90	2	b	b	PROPN
ejpam-131	90	3	is	be	AUX
ejpam-131	90	4	ω	ω	NOUN
ejpam-131	90	5	-	-	ADJ
ejpam-131	90	6	open	open	ADJ
ejpam-131	90	7	,	,	PUNCT
ejpam-131	90	8	there	there	PRON
ejpam-131	90	9	exists	exist	VERB
ejpam-131	90	10	an	an	DET
ejpam-131	90	11	open	open	ADJ
ejpam-131	90	12	set	set	NOUN
ejpam-131	90	13	ub	ub	NOUN
ejpam-131	90	14	containing	contain	VERB
ejpam-131	90	15	x	x	PUNCT
ejpam-131	90	16	such	such	ADJ
ejpam-131	90	17	that	that	SCONJ
ejpam-131	90	18	|ub	|ub	NUM
ejpam-131	90	19	−	−	NOUN
ejpam-131	90	20	b|	b|	PROPN
ejpam-131	90	21	is	be	AUX
ejpam-131	90	22	countable	countable	ADJ
ejpam-131	90	23	.	.	PUNCT
ejpam-131	91	1	by	by	ADP
ejpam-131	91	2	lemma	lemma	PROPN
ejpam-131	91	3	2.7	2.7	NUM
ejpam-131	91	4	,	,	PUNCT
ejpam-131	91	5	ua∩	ua∩	PROPN
ejpam-131	91	6	ub	ub	VERB
ejpam-131	91	7	is	be	AUX
ejpam-131	91	8	a	a	DET
ejpam-131	91	9	b	b	NOUN
ejpam-131	91	10	-	-	PUNCT
ejpam-131	91	11	open	open	ADJ
ejpam-131	91	12	set	set	NOUN
ejpam-131	91	13	containing	contain	VERB
ejpam-131	91	14	x	x	PUNCT
ejpam-131	91	15	and	and	CCONJ
ejpam-131	91	16	(	(	PUNCT
ejpam-131	91	17	ua∩	ua∩	X
ejpam-131	91	18	ub)−	ub)−	X
ejpam-131	91	19	(	(	PUNCT
ejpam-131	91	20	a∩	a∩	PROPN
ejpam-131	91	21	b	b	X
ejpam-131	91	22	)	)	PUNCT
ejpam-131	91	23	=	=	SYM
ejpam-131	92	1	(	(	PUNCT
ejpam-131	92	2	ua∩	ua∩	X
ejpam-131	92	3	ub)∩	ub)∩	PROPN
ejpam-131	93	1	[	[	X
ejpam-131	93	2	(	(	PUNCT
ejpam-131	93	3	x	x	INTJ
ejpam-131	93	4	−	−	PROPN
ejpam-131	94	1	a)∪	a)∪	INTJ
ejpam-131	94	2	(	(	PUNCT
ejpam-131	94	3	x	x	X
ejpam-131	94	4	−	−	PROPN
ejpam-131	94	5	b	b	NOUN
ejpam-131	94	6	)	)	PUNCT
ejpam-131	94	7	]	]	PUNCT
ejpam-131	95	1	=	=	PUNCT
ejpam-131	96	1	[	[	X
ejpam-131	96	2	ua∩	ua∩	X
ejpam-131	96	3	ub	ub	ADJ
ejpam-131	96	4	∩	∩	NOUN
ejpam-131	96	5	(	(	PUNCT
ejpam-131	96	6	x	x	SYM
ejpam-131	96	7	−	−	X
ejpam-131	96	8	a)]∪	a)]∪	PRON
ejpam-131	97	1	[	[	X
ejpam-131	97	2	ua∩	ua∩	ADJ
ejpam-131	97	3	ub	ub	ADJ
ejpam-131	97	4	∩	∩	NOUN
ejpam-131	97	5	(	(	PUNCT
ejpam-131	97	6	x	x	SYM
ejpam-131	97	7	−	−	PROPN
ejpam-131	97	8	b	b	NOUN
ejpam-131	97	9	)	)	PUNCT
ejpam-131	97	10	]	]	PUNCT
ejpam-131	98	1	⊆	⊆	X
ejpam-131	98	2	(	(	PUNCT
ejpam-131	98	3	ua∩	ua∩	X
ejpam-131	98	4	(	(	PUNCT
ejpam-131	98	5	x	x	SYM
ejpam-131	98	6	−	−	PROPN
ejpam-131	98	7	a))∪	a))∪	PROPN
ejpam-131	98	8	(	(	PUNCT
ejpam-131	98	9	ub	ub	X
ejpam-131	98	10	∩	∩	NOUN
ejpam-131	98	11	(	(	PUNCT
ejpam-131	98	12	x	x	SYM
ejpam-131	98	13	−	−	PROPN
ejpam-131	98	14	b	b	NOUN
ejpam-131	98	15	)	)	PUNCT
ejpam-131	98	16	)	)	PUNCT
ejpam-131	98	17	.	.	PUNCT
ejpam-131	99	1	since	since	SCONJ
ejpam-131	99	2	(	(	PUNCT
ejpam-131	99	3	ua∩	ua∩	X
ejpam-131	99	4	(	(	PUNCT
ejpam-131	99	5	x	x	NOUN
ejpam-131	99	6	−a))∪	−a))∪	NOUN
ejpam-131	99	7	(	(	PUNCT
ejpam-131	99	8	ub	ub	X
ejpam-131	99	9	∩	∩	NOUN
ejpam-131	99	10	(	(	PUNCT
ejpam-131	99	11	x	x	NOUN
ejpam-131	99	12	−b	−b	NOUN
ejpam-131	99	13	)	)	PUNCT
ejpam-131	99	14	)	)	PUNCT
ejpam-131	99	15	is	be	AUX
ejpam-131	99	16	a	a	DET
ejpam-131	99	17	countable	countable	ADJ
ejpam-131	99	18	set	set	NOUN
ejpam-131	99	19	,	,	PUNCT
ejpam-131	99	20	|(ua∩ub)−	|(ua∩ub)−	PROPN
ejpam-131	99	21	(	(	PUNCT
ejpam-131	99	22	a∩b)|	a∩b)|	NOUN
ejpam-131	99	23	is	be	AUX
ejpam-131	99	24	countable	countable	ADJ
ejpam-131	99	25	.	.	PUNCT
ejpam-131	100	1	this	this	PRON
ejpam-131	100	2	shows	show	VERB
ejpam-131	100	3	that	that	SCONJ
ejpam-131	100	4	a∩	a∩	PROPN
ejpam-131	100	5	b	b	PROPN
ejpam-131	100	6	is	be	AUX
ejpam-131	100	7	ωb	ωb	NOUN
ejpam-131	100	8	-	-	PUNCT
ejpam-131	100	9	open	open	ADJ
ejpam-131	100	10	.	.	PUNCT
ejpam-131	101	1	corollary	corollary	ADJ
ejpam-131	101	2	2.9	2.9	NUM
ejpam-131	101	3	.	.	PUNCT
ejpam-131	102	1	the	the	DET
ejpam-131	102	2	intersection	intersection	NOUN
ejpam-131	102	3	of	of	ADP
ejpam-131	102	4	an	an	DET
ejpam-131	102	5	ωb	ωb	NOUN
ejpam-131	102	6	-	-	PUNCT
ejpam-131	102	7	open	open	NOUN
ejpam-131	102	8	set	set	NOUN
ejpam-131	102	9	with	with	ADP
ejpam-131	102	10	an	an	DET
ejpam-131	102	11	open	open	ADJ
ejpam-131	102	12	set	set	NOUN
ejpam-131	102	13	is	be	AUX
ejpam-131	102	14	ωb	ωb	NOUN
ejpam-131	102	15	-	-	PUNCT
ejpam-131	102	16	open	open	ADJ
ejpam-131	102	17	.	.	PUNCT
ejpam-131	103	1	the	the	DET
ejpam-131	103	2	intersection	intersection	NOUN
ejpam-131	103	3	of	of	ADP
ejpam-131	103	4	two	two	NUM
ejpam-131	103	5	ωb	ωb	NOUN
ejpam-131	103	6	-	-	PUNCT
ejpam-131	103	7	open	open	ADJ
ejpam-131	103	8	sets	set	NOUN
ejpam-131	103	9	is	be	AUX
ejpam-131	103	10	not	not	PART
ejpam-131	103	11	always	always	ADV
ejpam-131	103	12	ωb	ωb	NOUN
ejpam-131	103	13	-	-	PUNCT
ejpam-131	103	14	open	open	ADJ
ejpam-131	103	15	.	.	PUNCT
ejpam-131	104	1	example	example	NOUN
ejpam-131	104	2	2.10	2.10	NUM
ejpam-131	104	3	.	.	PUNCT
ejpam-131	105	1	let	let	VERB
ejpam-131	105	2	x	x	PUNCT
ejpam-131	105	3	=	=	PUNCT
ejpam-131	105	4	r	r	NOUN
ejpam-131	105	5	with	with	ADP
ejpam-131	105	6	the	the	DET
ejpam-131	105	7	usual	usual	ADJ
ejpam-131	105	8	topology	topology	NOUN
ejpam-131	105	9	τ	τ	PROPN
ejpam-131	105	10	.	.	PUNCT
ejpam-131	106	1	let	let	VERB
ejpam-131	106	2	a	a	PRON
ejpam-131	106	3	=	=	X
ejpam-131	106	4	q	q	AUX
ejpam-131	106	5	be	be	AUX
ejpam-131	106	6	the	the	DET
ejpam-131	106	7	set	set	NOUN
ejpam-131	106	8	of	of	ADP
ejpam-131	106	9	all	all	DET
ejpam-131	106	10	rational	rational	ADJ
ejpam-131	106	11	numbers	number	NOUN
ejpam-131	106	12	and	and	CCONJ
ejpam-131	106	13	b	b	NOUN
ejpam-131	106	14	=	=	SYM
ejpam-131	107	1	[	[	X
ejpam-131	107	2	0,1	0,1	NUM
ejpam-131	107	3	)	)	PUNCT
ejpam-131	107	4	.	.	PUNCT
ejpam-131	108	1	then	then	ADV
ejpam-131	108	2	a	a	PRON
ejpam-131	108	3	and	and	CCONJ
ejpam-131	108	4	b	b	NOUN
ejpam-131	108	5	areωb	areωb	ADV
ejpam-131	108	6	-	-	PUNCT
ejpam-131	108	7	open	open	ADJ
ejpam-131	108	8	,	,	PUNCT
ejpam-131	108	9	but	but	CCONJ
ejpam-131	108	10	a∩b	a∩b	PROPN
ejpam-131	108	11	is	be	AUX
ejpam-131	108	12	notωb	notωb	ADV
ejpam-131	108	13	-	-	PUNCT
ejpam-131	108	14	open	open	ADJ
ejpam-131	108	15	,	,	PUNCT
ejpam-131	108	16	since	since	SCONJ
ejpam-131	108	17	each	each	DET
ejpam-131	108	18	b	b	NOUN
ejpam-131	108	19	-	-	PUNCT
ejpam-131	108	20	open	open	ADJ
ejpam-131	108	21	containing	contain	VERB
ejpam-131	108	22	0	0	NUM
ejpam-131	108	23	is	be	AUX
ejpam-131	108	24	uncountable	uncountable	ADJ
ejpam-131	108	25	set	set	NOUN
ejpam-131	108	26	.	.	PUNCT
ejpam-131	109	1	proposition	proposition	NOUN
ejpam-131	109	2	2.11	2.11	NUM
ejpam-131	109	3	.	.	PUNCT
ejpam-131	110	1	the	the	DET
ejpam-131	110	2	union	union	NOUN
ejpam-131	110	3	of	of	ADP
ejpam-131	110	4	any	any	DET
ejpam-131	110	5	family	family	NOUN
ejpam-131	110	6	of	of	ADP
ejpam-131	110	7	ωb	ωb	NOUN
ejpam-131	110	8	-	-	PUNCT
ejpam-131	110	9	open	open	ADJ
ejpam-131	110	10	sets	set	NOUN
ejpam-131	110	11	is	be	AUX
ejpam-131	110	12	ωb	ωb	NOUN
ejpam-131	110	13	-	-	PUNCT
ejpam-131	110	14	open	open	ADJ
ejpam-131	110	15	.	.	PUNCT
ejpam-131	111	1	t.	t.	PROPN
ejpam-131	111	2	noiri	noiri	PROPN
ejpam-131	111	3	,	,	PUNCT
ejpam-131	111	4	a.	a.	PROPN
ejpam-131	111	5	al	al	PROPN
ejpam-131	111	6	-	-	PUNCT
ejpam-131	111	7	omari	omari	PROPN
ejpam-131	111	8	and	and	CCONJ
ejpam-131	111	9	m.s.m	m.s.m	PROPN
ejpam-131	111	10	.	.	PROPN
ejpam-131	111	11	noorani	noorani	PROPN
ejpam-131	111	12	/	/	SYM
ejpam-131	111	13	eur	eur	PROPN
ejpam-131	111	14	.	.	PUNCT
ejpam-131	112	1	j.	j.	PROPN
ejpam-131	112	2	pure	pure	PROPN
ejpam-131	112	3	appl	appl	PROPN
ejpam-131	112	4	.	.	PROPN
ejpam-131	112	5	math	math	PROPN
ejpam-131	112	6	,	,	PUNCT
ejpam-131	112	7	1	1	NUM
ejpam-131	112	8	(	(	PUNCT
ejpam-131	112	9	2008	2008	NUM
ejpam-131	112	10	)	)	PUNCT
ejpam-131	112	11	,	,	PUNCT
ejpam-131	112	12	(	(	PUNCT
ejpam-131	112	13	3	3	NUM
ejpam-131	112	14	-	-	SYM
ejpam-131	112	15	9	9	NUM
ejpam-131	112	16	)	)	PUNCT
ejpam-131	112	17	6	6	NUM
ejpam-131	112	18	proof	proof	NOUN
ejpam-131	112	19	.	.	PUNCT
ejpam-131	113	1	if	if	SCONJ
ejpam-131	113	2	{	{	PUNCT
ejpam-131	113	3	aα	aα	NOUN
ejpam-131	113	4	:	:	PUNCT
ejpam-131	113	5	α	α	PROPN
ejpam-131	113	6	∈	∈	PROPN
ejpam-131	113	7	λ	λ	PROPN
ejpam-131	113	8	}	}	PUNCT
ejpam-131	113	9	is	be	AUX
ejpam-131	113	10	a	a	DET
ejpam-131	113	11	collection	collection	NOUN
ejpam-131	113	12	of	of	ADP
ejpam-131	113	13	ωb	ωb	NOUN
ejpam-131	113	14	-	-	PUNCT
ejpam-131	113	15	open	open	ADJ
ejpam-131	113	16	subsets	subset	NOUN
ejpam-131	113	17	of	of	ADP
ejpam-131	113	18	x	x	PRON
ejpam-131	113	19	,	,	PUNCT
ejpam-131	113	20	then	then	ADV
ejpam-131	113	21	for	for	ADP
ejpam-131	113	22	every	every	DET
ejpam-131	113	23	x	x	PROPN
ejpam-131	113	24	∈	∈	PROPN
ejpam-131	113	25	∪α∈λaα	∪α∈λaα	PROPN
ejpam-131	113	26	,	,	PUNCT
ejpam-131	113	27	x	x	PUNCT
ejpam-131	113	28	∈	∈	NOUN
ejpam-131	113	29	aβ	aβ	VERB
ejpam-131	113	30	for	for	ADP
ejpam-131	113	31	some	some	DET
ejpam-131	113	32	β	β	NOUN
ejpam-131	113	33	∈	∈	PROPN
ejpam-131	113	34	λ	λ	PROPN
ejpam-131	113	35	.	.	PUNCT
ejpam-131	114	1	hence	hence	ADV
ejpam-131	114	2	there	there	PRON
ejpam-131	114	3	exists	exist	VERB
ejpam-131	114	4	a	a	DET
ejpam-131	114	5	b	b	NOUN
ejpam-131	114	6	-	-	PUNCT
ejpam-131	114	7	open	open	ADJ
ejpam-131	114	8	subset	subset	ADJ
ejpam-131	114	9	u	u	NOUN
ejpam-131	114	10	of	of	ADP
ejpam-131	114	11	x	x	PUNCT
ejpam-131	114	12	containing	contain	VERB
ejpam-131	114	13	x	x	PUNCT
ejpam-131	114	14	such	such	ADJ
ejpam-131	114	15	that	that	SCONJ
ejpam-131	114	16	u\aβ	u\aβ	PROPN
ejpam-131	114	17	is	be	AUX
ejpam-131	114	18	countable	countable	ADJ
ejpam-131	114	19	.	.	PUNCT
ejpam-131	115	1	now	now	ADV
ejpam-131	115	2	as	as	ADP
ejpam-131	115	3	u\(∪α∈λaα	u\(∪α∈λaα	NOUN
ejpam-131	115	4	)	)	PUNCT
ejpam-131	115	5	⊆	⊆	NUM
ejpam-131	115	6	u\aβ	u\aβ	PROPN
ejpam-131	115	7	and	and	CCONJ
ejpam-131	115	8	thus	thus	ADV
ejpam-131	115	9	u\(∪α∈λaα	u\(∪α∈λaα	NUM
ejpam-131	115	10	)	)	PUNCT
ejpam-131	115	11	is	be	AUX
ejpam-131	115	12	countable	countable	ADJ
ejpam-131	115	13	.	.	PUNCT
ejpam-131	116	1	therefore	therefore	ADV
ejpam-131	116	2	,	,	PUNCT
ejpam-131	116	3	∪α∈λaα	∪α∈λaα	PROPN
ejpam-131	116	4	is	be	AUX
ejpam-131	116	5	ωb	ωb	NOUN
ejpam-131	116	6	-	-	PUNCT
ejpam-131	116	7	open	open	ADJ
ejpam-131	116	8	.	.	PUNCT
ejpam-131	117	1	the	the	DET
ejpam-131	117	2	intersection	intersection	NOUN
ejpam-131	117	3	of	of	ADP
ejpam-131	117	4	all	all	DET
ejpam-131	117	5	ωb	ωb	NOUN
ejpam-131	117	6	-	-	PUNCT
ejpam-131	117	7	closed	close	VERB
ejpam-131	117	8	sets	set	NOUN
ejpam-131	117	9	of	of	ADP
ejpam-131	117	10	x	x	PUNCT
ejpam-131	117	11	containing	contain	VERB
ejpam-131	117	12	a	a	PRON
ejpam-131	117	13	is	be	AUX
ejpam-131	117	14	called	call	VERB
ejpam-131	117	15	the	the	DET
ejpam-131	117	16	ωb	ωb	NOUN
ejpam-131	117	17	-	-	PUNCT
ejpam-131	117	18	closure	closure	NOUN
ejpam-131	117	19	of	of	ADP
ejpam-131	117	20	a	a	PRON
ejpam-131	117	21	and	and	CCONJ
ejpam-131	117	22	is	be	AUX
ejpam-131	117	23	denoted	denote	VERB
ejpam-131	117	24	by	by	ADP
ejpam-131	117	25	ωbcl(a	ωbcl(a	PROPN
ejpam-131	117	26	)	)	PUNCT
ejpam-131	117	27	.	.	PUNCT
ejpam-131	118	1	and	and	CCONJ
ejpam-131	118	2	the	the	DET
ejpam-131	118	3	union	union	NOUN
ejpam-131	118	4	of	of	ADP
ejpam-131	118	5	all	all	DET
ejpam-131	118	6	ωb	ωb	NOUN
ejpam-131	118	7	-	-	PUNCT
ejpam-131	118	8	open	open	ADJ
ejpam-131	118	9	sets	set	NOUN
ejpam-131	118	10	of	of	ADP
ejpam-131	118	11	x	x	PUNCT
ejpam-131	118	12	contained	contain	VERB
ejpam-131	118	13	in	in	ADP
ejpam-131	118	14	a	a	PRON
ejpam-131	118	15	is	be	AUX
ejpam-131	118	16	called	call	VERB
ejpam-131	118	17	the	the	DET
ejpam-131	118	18	ωb	ωb	NOUN
ejpam-131	118	19	-	-	PUNCT
ejpam-131	118	20	interior	interior	NOUN
ejpam-131	118	21	and	and	CCONJ
ejpam-131	118	22	is	be	AUX
ejpam-131	118	23	denoted	denote	VERB
ejpam-131	118	24	by	by	ADP
ejpam-131	118	25	ωbint(a	ωbint(a	PROPN
ejpam-131	118	26	)	)	PUNCT
ejpam-131	118	27	.	.	PUNCT
ejpam-131	119	1	theorem	theorem	VERB
ejpam-131	119	2	2.12	2.12	NUM
ejpam-131	119	3	.	.	PUNCT
ejpam-131	120	1	if	if	SCONJ
ejpam-131	120	2	each	each	DET
ejpam-131	120	3	non	non	ADJ
ejpam-131	120	4	-	-	ADJ
ejpam-131	120	5	empty	empty	ADJ
ejpam-131	120	6	b	b	NOUN
ejpam-131	120	7	-	-	PUNCT
ejpam-131	120	8	open	open	ADJ
ejpam-131	120	9	set	set	NOUN
ejpam-131	120	10	of	of	ADP
ejpam-131	120	11	a	a	DET
ejpam-131	120	12	space	space	NOUN
ejpam-131	120	13	x	x	PUNCT
ejpam-131	120	14	is	be	AUX
ejpam-131	120	15	uncountable	uncountable	ADJ
ejpam-131	120	16	,	,	PUNCT
ejpam-131	120	17	then	then	ADV
ejpam-131	120	18	bcl(a	bcl(a	PROPN
ejpam-131	120	19	)	)	PUNCT
ejpam-131	120	20	=	=	SYM
ejpam-131	120	21	ωbcl(a	ωbcl(a	PROPN
ejpam-131	120	22	)	)	PUNCT
ejpam-131	120	23	for	for	ADP
ejpam-131	120	24	each	each	DET
ejpam-131	120	25	open	open	ADJ
ejpam-131	120	26	set	set	VERB
ejpam-131	120	27	a	a	PRON
ejpam-131	120	28	of	of	ADP
ejpam-131	120	29	x	x	SYM
ejpam-131	120	30	.	.	PUNCT
ejpam-131	121	1	proof	proof	NOUN
ejpam-131	121	2	.	.	PUNCT
ejpam-131	122	1	clearly	clearly	ADV
ejpam-131	122	2	ωbcl(a	ωbcl(a	NUM
ejpam-131	122	3	)	)	PUNCT
ejpam-131	122	4	⊆	⊆	NUM
ejpam-131	122	5	bcl(a	bcl(a	NUM
ejpam-131	122	6	)	)	PUNCT
ejpam-131	122	7	.	.	PUNCT
ejpam-131	123	1	on	on	ADP
ejpam-131	123	2	the	the	DET
ejpam-131	123	3	other	other	ADJ
ejpam-131	123	4	hand	hand	NOUN
ejpam-131	123	5	,	,	PUNCT
ejpam-131	123	6	let	let	VERB
ejpam-131	123	7	x	x	X
ejpam-131	123	8	∈	∈	PROPN
ejpam-131	123	9	bcl(a	bcl(a	PROPN
ejpam-131	123	10	)	)	PUNCT
ejpam-131	123	11	and	and	CCONJ
ejpam-131	123	12	b	b	X
ejpam-131	123	13	be	be	AUX
ejpam-131	123	14	an	an	DET
ejpam-131	123	15	ωbopen	ωbopen	NOUN
ejpam-131	123	16	subset	subset	NOUN
ejpam-131	123	17	containing	contain	VERB
ejpam-131	123	18	x	x	X
ejpam-131	123	19	.	.	PUNCT
ejpam-131	124	1	then	then	ADV
ejpam-131	124	2	by	by	ADP
ejpam-131	124	3	lemma	lemma	PROPN
ejpam-131	124	4	2.5	2.5	NUM
ejpam-131	124	5	,	,	PUNCT
ejpam-131	124	6	there	there	PRON
ejpam-131	124	7	exists	exist	VERB
ejpam-131	124	8	a	a	DET
ejpam-131	124	9	b	b	NOUN
ejpam-131	124	10	-	-	PUNCT
ejpam-131	124	11	open	open	ADJ
ejpam-131	124	12	set	set	VERB
ejpam-131	124	13	v	v	NOUN
ejpam-131	124	14	containing	contain	VERB
ejpam-131	124	15	x	x	PUNCT
ejpam-131	124	16	and	and	CCONJ
ejpam-131	124	17	a	a	DET
ejpam-131	124	18	countable	countable	ADJ
ejpam-131	124	19	set	set	NOUN
ejpam-131	124	20	c	c	NOUN
ejpam-131	125	1	such	such	ADJ
ejpam-131	125	2	that	that	DET
ejpam-131	125	3	v	v	NOUN
ejpam-131	125	4	−	−	PROPN
ejpam-131	125	5	c	c	PROPN
ejpam-131	125	6	⊆	⊆	NUM
ejpam-131	125	7	b.	b.	NOUN
ejpam-131	125	8	thus	thus	ADV
ejpam-131	125	9	(	(	PUNCT
ejpam-131	125	10	v	v	NOUN
ejpam-131	125	11	−	−	PROPN
ejpam-131	125	12	c)∩	c)∩	PROPN
ejpam-131	125	13	a⊆	a⊆	PROPN
ejpam-131	125	14	b	b	NOUN
ejpam-131	125	15	∩	∩	NOUN
ejpam-131	125	16	a	a	X
ejpam-131	125	17	and	and	CCONJ
ejpam-131	125	18	so	so	ADV
ejpam-131	125	19	(	(	PUNCT
ejpam-131	125	20	v	v	NOUN
ejpam-131	125	21	∩	∩	NOUN
ejpam-131	125	22	a)−	a)−	PROPN
ejpam-131	125	23	c	c	NOUN
ejpam-131	125	24	⊆	⊆	NUM
ejpam-131	125	25	b	b	NOUN
ejpam-131	125	26	∩	∩	ADJ
ejpam-131	125	27	a.	a.	NOUN
ejpam-131	125	28	since	since	SCONJ
ejpam-131	125	29	x	x	PROPN
ejpam-131	125	30	∈	∈	PROPN
ejpam-131	125	31	v	v	NOUN
ejpam-131	125	32	and	and	CCONJ
ejpam-131	125	33	x	x	ADP
ejpam-131	125	34	∈	∈	PROPN
ejpam-131	125	35	bcl(a	bcl(a	PROPN
ejpam-131	125	36	)	)	PUNCT
ejpam-131	125	37	,	,	PUNCT
ejpam-131	125	38	v	v	X
ejpam-131	125	39	∩a	∩a	PROPN
ejpam-131	125	40	6=	6=	PROPN
ejpam-131	125	41	φ	φ	PROPN
ejpam-131	125	42	and	and	CCONJ
ejpam-131	125	43	v	v	ADP
ejpam-131	125	44	∩a	∩a	PROPN
ejpam-131	125	45	is	be	AUX
ejpam-131	125	46	b	b	NOUN
ejpam-131	125	47	-	-	PUNCT
ejpam-131	125	48	open	open	ADJ
ejpam-131	125	49	since	since	SCONJ
ejpam-131	125	50	v	v	NOUN
ejpam-131	125	51	is	be	AUX
ejpam-131	125	52	b	b	NOUN
ejpam-131	125	53	-	-	PUNCT
ejpam-131	125	54	open	open	ADJ
ejpam-131	125	55	and	and	CCONJ
ejpam-131	125	56	a	a	PRON
ejpam-131	125	57	is	be	AUX
ejpam-131	125	58	open	open	ADJ
ejpam-131	125	59	.	.	PUNCT
ejpam-131	126	1	by	by	ADP
ejpam-131	126	2	the	the	DET
ejpam-131	126	3	hypothesis	hypothesis	NOUN
ejpam-131	126	4	each	each	DET
ejpam-131	126	5	non	non	ADJ
ejpam-131	126	6	-	-	ADJ
ejpam-131	126	7	empty	empty	ADJ
ejpam-131	126	8	b	b	NOUN
ejpam-131	126	9	-	-	PUNCT
ejpam-131	126	10	open	open	ADJ
ejpam-131	126	11	set	set	NOUN
ejpam-131	126	12	of	of	ADP
ejpam-131	126	13	a	a	DET
ejpam-131	126	14	space	space	NOUN
ejpam-131	126	15	x	x	PUNCT
ejpam-131	126	16	is	be	AUX
ejpam-131	126	17	uncountable	uncountable	ADJ
ejpam-131	126	18	and	and	CCONJ
ejpam-131	126	19	so	so	ADV
ejpam-131	126	20	is	be	AUX
ejpam-131	126	21	(	(	PUNCT
ejpam-131	126	22	v	v	NOUN
ejpam-131	126	23	∩	∩	NOUN
ejpam-131	126	24	a)−	a)−	PROPN
ejpam-131	126	25	c	c	NOUN
ejpam-131	126	26	.	.	PUNCT
ejpam-131	127	1	thus	thus	ADV
ejpam-131	127	2	b	b	X
ejpam-131	127	3	∩	∩	NOUN
ejpam-131	127	4	a	a	PRON
ejpam-131	127	5	is	be	AUX
ejpam-131	127	6	uncountable	uncountable	ADJ
ejpam-131	127	7	.	.	PUNCT
ejpam-131	128	1	therefore	therefore	ADV
ejpam-131	128	2	,	,	PUNCT
ejpam-131	128	3	b	b	PROPN
ejpam-131	128	4	∩	∩	PROPN
ejpam-131	128	5	a	a	PRON
ejpam-131	128	6	6=	6=	NUM
ejpam-131	128	7	φ	φ	NUM
ejpam-131	128	8	which	which	PRON
ejpam-131	128	9	means	mean	VERB
ejpam-131	128	10	that	that	SCONJ
ejpam-131	128	11	x	x	PUNCT
ejpam-131	128	12	∈ωbcl(a	∈ωbcl(a	ADJ
ejpam-131	128	13	)	)	PUNCT
ejpam-131	128	14	.	.	PUNCT
ejpam-131	129	1	corollary	corollary	ADJ
ejpam-131	129	2	2.13	2.13	NUM
ejpam-131	129	3	.	.	PUNCT
ejpam-131	130	1	if	if	SCONJ
ejpam-131	130	2	each	each	DET
ejpam-131	130	3	non	non	ADJ
ejpam-131	130	4	-	-	ADJ
ejpam-131	130	5	empty	empty	ADJ
ejpam-131	130	6	b	b	NOUN
ejpam-131	130	7	-	-	PUNCT
ejpam-131	130	8	open	open	ADJ
ejpam-131	130	9	set	set	NOUN
ejpam-131	130	10	of	of	ADP
ejpam-131	130	11	a	a	DET
ejpam-131	130	12	space	space	NOUN
ejpam-131	130	13	x	x	PUNCT
ejpam-131	130	14	is	be	AUX
ejpam-131	130	15	uncountable	uncountable	ADJ
ejpam-131	130	16	,	,	PUNCT
ejpam-131	130	17	then	then	ADV
ejpam-131	130	18	bint(a	bint(a	NOUN
ejpam-131	130	19	)	)	PUNCT
ejpam-131	131	1	=	=	SYM
ejpam-131	131	2	ωbint(a	ωbint(a	NOUN
ejpam-131	131	3	)	)	PUNCT
ejpam-131	131	4	for	for	ADP
ejpam-131	131	5	each	each	DET
ejpam-131	131	6	closed	close	VERB
ejpam-131	131	7	set	set	VERB
ejpam-131	131	8	a	a	PRON
ejpam-131	131	9	of	of	ADP
ejpam-131	131	10	x	x	PUNCT
ejpam-131	131	11	.	.	PUNCT
ejpam-131	132	1	definition	definition	NOUN
ejpam-131	132	2	2.14	2.14	NUM
ejpam-131	132	3	.	.	PUNCT
ejpam-131	133	1	a	a	DET
ejpam-131	133	2	function	function	NOUN
ejpam-131	133	3	f	f	NOUN
ejpam-131	133	4	:	:	PUNCT
ejpam-131	133	5	x	x	X
ejpam-131	133	6	→	→	SYM
ejpam-131	133	7	y	y	PROPN
ejpam-131	133	8	is	be	AUX
ejpam-131	133	9	said	say	VERB
ejpam-131	133	10	to	to	PART
ejpam-131	133	11	be	be	AUX
ejpam-131	133	12	quasi	quasi	ADJ
ejpam-131	133	13	b	b	X
ejpam-131	133	14	-	-	PUNCT
ejpam-131	133	15	open	open	ADJ
ejpam-131	133	16	if	if	SCONJ
ejpam-131	133	17	the	the	DET
ejpam-131	133	18	image	image	NOUN
ejpam-131	133	19	of	of	ADP
ejpam-131	133	20	each	each	DET
ejpam-131	133	21	b	b	NOUN
ejpam-131	133	22	-	-	PUNCT
ejpam-131	133	23	open	open	ADJ
ejpam-131	133	24	set	set	NOUN
ejpam-131	133	25	in	in	ADP
ejpam-131	133	26	x	x	PUNCT
ejpam-131	133	27	is	be	AUX
ejpam-131	133	28	open	open	ADJ
ejpam-131	133	29	in	in	ADP
ejpam-131	133	30	y	y	PROPN
ejpam-131	133	31	.	.	PUNCT
ejpam-131	134	1	proposition	proposition	NOUN
ejpam-131	134	2	2.15	2.15	NUM
ejpam-131	134	3	.	.	PUNCT
ejpam-131	135	1	if	if	SCONJ
ejpam-131	135	2	f	f	PROPN
ejpam-131	135	3	:	:	PUNCT
ejpam-131	135	4	x	x	X
ejpam-131	135	5	→	→	SYM
ejpam-131	135	6	y	y	PROPN
ejpam-131	135	7	is	be	AUX
ejpam-131	135	8	quasi	quasi	ADJ
ejpam-131	135	9	b	b	X
ejpam-131	135	10	-	-	PUNCT
ejpam-131	135	11	open	open	ADJ
ejpam-131	135	12	,	,	PUNCT
ejpam-131	135	13	then	then	ADV
ejpam-131	135	14	the	the	DET
ejpam-131	135	15	image	image	NOUN
ejpam-131	135	16	of	of	ADP
ejpam-131	135	17	an	an	DET
ejpam-131	135	18	ωb	ωb	NOUN
ejpam-131	135	19	-	-	PUNCT
ejpam-131	135	20	open	open	ADJ
ejpam-131	135	21	set	set	NOUN
ejpam-131	135	22	of	of	ADP
ejpam-131	135	23	x	x	PROPN
ejpam-131	135	24	is	be	AUX
ejpam-131	135	25	ω	ω	NOUN
ejpam-131	135	26	-	-	NOUN
ejpam-131	135	27	open	open	ADJ
ejpam-131	135	28	in	in	ADP
ejpam-131	135	29	y	y	PROPN
ejpam-131	135	30	.	.	PUNCT
ejpam-131	136	1	proof	proof	NOUN
ejpam-131	136	2	.	.	PUNCT
ejpam-131	137	1	let	let	VERB
ejpam-131	137	2	f	f	NOUN
ejpam-131	137	3	:	:	PUNCT
ejpam-131	137	4	x	x	X
ejpam-131	137	5	→	→	SYM
ejpam-131	137	6	y	y	PROPN
ejpam-131	137	7	be	be	AUX
ejpam-131	137	8	quasi	quasi	ADJ
ejpam-131	137	9	b	b	NOUN
ejpam-131	137	10	-	-	PUNCT
ejpam-131	137	11	open	open	ADJ
ejpam-131	137	12	and	and	CCONJ
ejpam-131	137	13	w	w	ADP
ejpam-131	137	14	an	an	DET
ejpam-131	137	15	ωb	ωb	NOUN
ejpam-131	137	16	-	-	PUNCT
ejpam-131	137	17	open	open	NOUN
ejpam-131	137	18	subset	subset	NOUN
ejpam-131	137	19	of	of	ADP
ejpam-131	137	20	x	x	X
ejpam-131	137	21	.	.	PUNCT
ejpam-131	138	1	let	let	VERB
ejpam-131	138	2	y	y	PROPN
ejpam-131	138	3	∈	∈	PROPN
ejpam-131	138	4	f	f	X
ejpam-131	138	5	(	(	PUNCT
ejpam-131	138	6	w	w	PROPN
ejpam-131	138	7	)	)	PUNCT
ejpam-131	138	8	,	,	PUNCT
ejpam-131	138	9	there	there	PRON
ejpam-131	138	10	exists	exist	VERB
ejpam-131	138	11	x	x	X
ejpam-131	138	12	∈w	∈w	VERB
ejpam-131	138	13	such	such	ADJ
ejpam-131	138	14	that	that	SCONJ
ejpam-131	138	15	f	f	PROPN
ejpam-131	138	16	(	(	PUNCT
ejpam-131	138	17	x	x	X
ejpam-131	138	18	)	)	PUNCT
ejpam-131	138	19	=	=	SYM
ejpam-131	138	20	y	y	PROPN
ejpam-131	138	21	.	.	PUNCT
ejpam-131	139	1	since	since	SCONJ
ejpam-131	139	2	w	w	PROPN
ejpam-131	139	3	is	be	AUX
ejpam-131	139	4	ωb	ωb	NOUN
ejpam-131	139	5	-	-	PUNCT
ejpam-131	139	6	open	open	ADJ
ejpam-131	139	7	,	,	PUNCT
ejpam-131	139	8	there	there	PRON
ejpam-131	139	9	exists	exist	VERB
ejpam-131	139	10	a	a	DET
ejpam-131	139	11	b	b	NOUN
ejpam-131	139	12	-	-	PUNCT
ejpam-131	139	13	open	open	ADJ
ejpam-131	139	14	set	set	NOUN
ejpam-131	139	15	u	u	PRON
ejpam-131	139	16	such	such	ADJ
ejpam-131	139	17	that	that	SCONJ
ejpam-131	139	18	x	x	SYM
ejpam-131	139	19	∈	∈	PROPN
ejpam-131	139	20	u	u	NOUN
ejpam-131	139	21	and	and	CCONJ
ejpam-131	139	22	u	u	NOUN
ejpam-131	139	23	−w	−w	ADV
ejpam-131	139	24	=	=	PUNCT
ejpam-131	139	25	c	c	PROPN
ejpam-131	139	26	is	be	AUX
ejpam-131	139	27	countable	countable	ADJ
ejpam-131	139	28	.	.	PUNCT
ejpam-131	140	1	since	since	SCONJ
ejpam-131	140	2	f	f	PROPN
ejpam-131	140	3	is	be	AUX
ejpam-131	140	4	quasi	quasi	ADJ
ejpam-131	140	5	b	b	X
ejpam-131	140	6	-	-	PUNCT
ejpam-131	140	7	open	open	ADJ
ejpam-131	140	8	,	,	PUNCT
ejpam-131	140	9	f	f	PROPN
ejpam-131	140	10	(	(	PUNCT
ejpam-131	140	11	u	u	NOUN
ejpam-131	140	12	)	)	PUNCT
ejpam-131	140	13	is	be	AUX
ejpam-131	140	14	open	open	ADJ
ejpam-131	140	15	in	in	ADP
ejpam-131	140	16	y	y	PROPN
ejpam-131	140	17	such	such	ADJ
ejpam-131	140	18	that	that	SCONJ
ejpam-131	141	1	y	y	PROPN
ejpam-131	141	2	=	=	SYM
ejpam-131	141	3	f	f	PROPN
ejpam-131	141	4	(	(	PUNCT
ejpam-131	141	5	x	x	X
ejpam-131	141	6	)	)	PUNCT
ejpam-131	141	7	∈	∈	PROPN
ejpam-131	141	8	f	f	X
ejpam-131	141	9	(	(	PUNCT
ejpam-131	141	10	u	u	NOUN
ejpam-131	141	11	)	)	PUNCT
ejpam-131	141	12	and	and	CCONJ
ejpam-131	141	13	f	f	PROPN
ejpam-131	141	14	(	(	PUNCT
ejpam-131	141	15	u)−	u)−	PROPN
ejpam-131	141	16	f	f	PROPN
ejpam-131	141	17	(	(	PUNCT
ejpam-131	141	18	w	w	PROPN
ejpam-131	141	19	)	)	PUNCT
ejpam-131	141	20	⊆	⊆	NUM
ejpam-131	141	21	f	f	X
ejpam-131	141	22	(	(	PUNCT
ejpam-131	141	23	u	u	NOUN
ejpam-131	141	24	−w	−w	ADV
ejpam-131	141	25	)	)	PUNCT
ejpam-131	142	1	=	=	SYM
ejpam-131	142	2	f	f	X
ejpam-131	142	3	(	(	PUNCT
ejpam-131	142	4	c	c	NOUN
ejpam-131	142	5	)	)	PUNCT
ejpam-131	142	6	is	be	AUX
ejpam-131	142	7	countable	countable	ADJ
ejpam-131	142	8	.	.	PUNCT
ejpam-131	143	1	therefore	therefore	ADV
ejpam-131	143	2	,	,	PUNCT
ejpam-131	143	3	f	f	PROPN
ejpam-131	143	4	(	(	PUNCT
ejpam-131	143	5	w	w	PROPN
ejpam-131	143	6	)	)	PUNCT
ejpam-131	143	7	is	be	AUX
ejpam-131	143	8	ω	ω	NOUN
ejpam-131	143	9	-	-	NOUN
ejpam-131	143	10	open	open	ADJ
ejpam-131	143	11	in	in	ADP
ejpam-131	143	12	y	y	PROPN
ejpam-131	143	13	.	.	PUNCT
ejpam-131	144	1	3	3	X
ejpam-131	144	2	.	.	X
ejpam-131	144	3	b	b	X
ejpam-131	144	4	-	-	PUNCT
ejpam-131	144	5	lindelöf	lindelöf	NOUN
ejpam-131	144	6	spaces	space	NOUN
ejpam-131	144	7	definition	definition	NOUN
ejpam-131	144	8	3.1	3.1	NUM
ejpam-131	144	9	.	.	PUNCT
ejpam-131	145	1	(	(	PUNCT
ejpam-131	145	2	1	1	X
ejpam-131	145	3	)	)	PUNCT
ejpam-131	145	4	[	[	X
ejpam-131	145	5	6	6	NUM
ejpam-131	145	6	]	]	PUNCT
ejpam-131	145	7	a	a	DET
ejpam-131	145	8	space	space	NOUN
ejpam-131	145	9	x	x	PUNCT
ejpam-131	145	10	is	be	AUX
ejpam-131	145	11	said	say	VERB
ejpam-131	145	12	to	to	PART
ejpam-131	145	13	be	be	AUX
ejpam-131	145	14	b	b	NOUN
ejpam-131	145	15	-	-	PUNCT
ejpam-131	145	16	lindelöf	lindelöf	NOUN
ejpam-131	145	17	if	if	SCONJ
ejpam-131	145	18	every	every	DET
ejpam-131	145	19	b	b	NOUN
ejpam-131	145	20	-	-	PUNCT
ejpam-131	145	21	open	open	ADJ
ejpam-131	145	22	cover	cover	NOUN
ejpam-131	145	23	of	of	ADP
ejpam-131	145	24	x	x	PUNCT
ejpam-131	145	25	has	have	VERB
ejpam-131	145	26	a	a	DET
ejpam-131	145	27	countable	countable	ADJ
ejpam-131	145	28	subcover	subcover	NOUN
ejpam-131	145	29	.	.	PUNCT
ejpam-131	146	1	(	(	PUNCT
ejpam-131	146	2	2	2	X
ejpam-131	146	3	)	)	PUNCT
ejpam-131	146	4	a	a	DET
ejpam-131	146	5	subset	subset	NOUN
ejpam-131	146	6	a	a	PRON
ejpam-131	146	7	of	of	ADP
ejpam-131	146	8	a	a	DET
ejpam-131	146	9	space	space	NOUN
ejpam-131	146	10	x	x	PUNCT
ejpam-131	146	11	is	be	AUX
ejpam-131	146	12	said	say	VERB
ejpam-131	146	13	to	to	PART
ejpam-131	146	14	be	be	AUX
ejpam-131	146	15	b	b	NOUN
ejpam-131	146	16	-	-	PUNCT
ejpam-131	146	17	lindelöf	lindelöf	NOUN
ejpam-131	146	18	relative	relative	NOUN
ejpam-131	146	19	to	to	ADP
ejpam-131	146	20	x	x	PRON
ejpam-131	146	21	if	if	SCONJ
ejpam-131	146	22	every	every	DET
ejpam-131	146	23	cover	cover	NOUN
ejpam-131	146	24	of	of	ADP
ejpam-131	146	25	a	a	PRON
ejpam-131	146	26	by	by	ADP
ejpam-131	146	27	b	b	NOUN
ejpam-131	146	28	-	-	PUNCT
ejpam-131	146	29	open	open	ADJ
ejpam-131	146	30	sets	set	NOUN
ejpam-131	146	31	of	of	ADP
ejpam-131	146	32	x	x	PUNCT
ejpam-131	146	33	has	have	VERB
ejpam-131	146	34	a	a	DET
ejpam-131	146	35	countable	countable	ADJ
ejpam-131	146	36	subcover	subcover	PROPN
ejpam-131	146	37	.	.	PUNCT
ejpam-131	147	1	theorem	theorem	VERB
ejpam-131	147	2	3.2	3.2	NUM
ejpam-131	147	3	.	.	PUNCT
ejpam-131	148	1	if	if	SCONJ
ejpam-131	148	2	x	x	PRON
ejpam-131	148	3	is	be	AUX
ejpam-131	148	4	a	a	DET
ejpam-131	148	5	space	space	NOUN
ejpam-131	148	6	such	such	ADJ
ejpam-131	148	7	that	that	SCONJ
ejpam-131	148	8	every	every	DET
ejpam-131	148	9	b	b	X
ejpam-131	148	10	-	-	PUNCT
ejpam-131	148	11	open	open	ADJ
ejpam-131	148	12	subset	subset	NOUN
ejpam-131	148	13	of	of	ADP
ejpam-131	148	14	x	x	PROPN
ejpam-131	148	15	is	be	AUX
ejpam-131	148	16	b	b	NOUN
ejpam-131	148	17	-	-	PUNCT
ejpam-131	148	18	lindelöf	lindelöf	NOUN
ejpam-131	148	19	relative	relative	NOUN
ejpam-131	148	20	to	to	ADP
ejpam-131	148	21	x	x	PRON
ejpam-131	148	22	,	,	PUNCT
ejpam-131	148	23	then	then	ADV
ejpam-131	148	24	every	every	DET
ejpam-131	148	25	subset	subset	NOUN
ejpam-131	148	26	is	be	AUX
ejpam-131	148	27	b	b	NOUN
ejpam-131	148	28	-	-	PUNCT
ejpam-131	148	29	lindelöf	lindelöf	NOUN
ejpam-131	148	30	relative	relative	NOUN
ejpam-131	148	31	to	to	ADP
ejpam-131	148	32	x	x	PROPN
ejpam-131	148	33	.	.	PUNCT
ejpam-131	149	1	t.	t.	PROPN
ejpam-131	149	2	noiri	noiri	PROPN
ejpam-131	149	3	,	,	PUNCT
ejpam-131	149	4	a.	a.	PROPN
ejpam-131	149	5	al	al	PROPN
ejpam-131	149	6	-	-	PUNCT
ejpam-131	149	7	omari	omari	PROPN
ejpam-131	149	8	and	and	CCONJ
ejpam-131	149	9	m.s.m	m.s.m	PROPN
ejpam-131	149	10	.	.	PROPN
ejpam-131	149	11	noorani	noorani	PROPN
ejpam-131	149	12	/	/	SYM
ejpam-131	149	13	eur	eur	PROPN
ejpam-131	149	14	.	.	PUNCT
ejpam-131	150	1	j.	j.	PROPN
ejpam-131	150	2	pure	pure	PROPN
ejpam-131	150	3	appl	appl	PROPN
ejpam-131	150	4	.	.	PROPN
ejpam-131	150	5	math	math	PROPN
ejpam-131	150	6	,	,	PUNCT
ejpam-131	150	7	1	1	NUM
ejpam-131	150	8	(	(	PUNCT
ejpam-131	150	9	2008	2008	NUM
ejpam-131	150	10	)	)	PUNCT
ejpam-131	150	11	,	,	PUNCT
ejpam-131	150	12	(	(	PUNCT
ejpam-131	150	13	3	3	NUM
ejpam-131	150	14	-	-	SYM
ejpam-131	150	15	9	9	NUM
ejpam-131	150	16	)	)	PUNCT
ejpam-131	150	17	7	7	NUM
ejpam-131	150	18	proof	proof	NOUN
ejpam-131	150	19	.	.	PUNCT
ejpam-131	151	1	let	let	VERB
ejpam-131	151	2	b	b	X
ejpam-131	151	3	be	be	AUX
ejpam-131	151	4	an	an	DET
ejpam-131	151	5	arbitrary	arbitrary	ADJ
ejpam-131	151	6	subset	subset	NOUN
ejpam-131	151	7	of	of	ADP
ejpam-131	151	8	x	x	PUNCT
ejpam-131	151	9	and	and	CCONJ
ejpam-131	151	10	let	let	VERB
ejpam-131	151	11	{	{	PUNCT
ejpam-131	151	12	ui	ui	NOUN
ejpam-131	151	13	:	:	PUNCT
ejpam-131	151	14	i	i	PRON
ejpam-131	151	15	∈	∈	VERB
ejpam-131	152	1	i	i	PRON
ejpam-131	152	2	}	}	PUNCT
ejpam-131	152	3	be	be	VERB
ejpam-131	152	4	a	a	DET
ejpam-131	152	5	cover	cover	NOUN
ejpam-131	152	6	of	of	ADP
ejpam-131	152	7	b	b	NOUN
ejpam-131	152	8	by	by	ADP
ejpam-131	152	9	b	b	NOUN
ejpam-131	152	10	-	-	PUNCT
ejpam-131	152	11	open	open	ADJ
ejpam-131	152	12	sets	set	NOUN
ejpam-131	152	13	of	of	ADP
ejpam-131	152	14	x	x	X
ejpam-131	152	15	.	.	PUNCT
ejpam-131	153	1	then	then	ADV
ejpam-131	153	2	the	the	DET
ejpam-131	153	3	family	family	NOUN
ejpam-131	153	4	{	{	PUNCT
ejpam-131	153	5	ui	ui	NOUN
ejpam-131	153	6	:	:	PUNCT
ejpam-131	153	7	i	i	PRON
ejpam-131	153	8	∈	∈	VERB
ejpam-131	154	1	i	i	PRON
ejpam-131	154	2	}	}	PUNCT
ejpam-131	154	3	is	be	AUX
ejpam-131	154	4	a	a	DET
ejpam-131	154	5	b	b	NOUN
ejpam-131	154	6	-	-	PUNCT
ejpam-131	154	7	open	open	ADJ
ejpam-131	154	8	cover	cover	NOUN
ejpam-131	154	9	of	of	ADP
ejpam-131	154	10	the	the	DET
ejpam-131	154	11	b	b	NOUN
ejpam-131	154	12	-	-	PUNCT
ejpam-131	154	13	open	open	ADJ
ejpam-131	154	14	set	set	NOUN
ejpam-131	154	15	∪{ui	∪{ui	PROPN
ejpam-131	154	16	:	:	PUNCT
ejpam-131	154	17	i	i	PRON
ejpam-131	154	18	∈	∈	VERB
ejpam-131	154	19	i	i	X
ejpam-131	154	20	}	}	PUNCT
ejpam-131	154	21	by	by	ADP
ejpam-131	154	22	lemma	lemma	PROPN
ejpam-131	154	23	2.7	2.7	NUM
ejpam-131	154	24	.	.	PUNCT
ejpam-131	155	1	hence	hence	ADV
ejpam-131	155	2	by	by	ADP
ejpam-131	155	3	hypothesis	hypothesis	NOUN
ejpam-131	155	4	there	there	PRON
ejpam-131	155	5	is	be	VERB
ejpam-131	155	6	a	a	DET
ejpam-131	155	7	countable	countable	ADJ
ejpam-131	155	8	subfamily	subfamily	ADV
ejpam-131	155	9	{	{	PUNCT
ejpam-131	155	10	ui	ui	PROPN
ejpam-131	155	11	j	j	PROPN
ejpam-131	155	12	:	:	PUNCT
ejpam-131	155	13	j	j	PROPN
ejpam-131	155	14	∈	∈	PROPN
ejpam-131	155	15	n	n	CCONJ
ejpam-131	155	16	}	}	PUNCT
ejpam-131	155	17	which	which	PRON
ejpam-131	155	18	covers	cover	VERB
ejpam-131	155	19	∪{ui	∪{ui	PROPN
ejpam-131	155	20	:	:	PUNCT
ejpam-131	155	21	i	i	PRON
ejpam-131	155	22	∈	∈	VERB
ejpam-131	155	23	i	i	PRON
ejpam-131	155	24	}	}	PUNCT
ejpam-131	155	25	.	.	PUNCT
ejpam-131	156	1	this	this	PRON
ejpam-131	156	2	subfamily	subfamily	ADV
ejpam-131	156	3	is	be	AUX
ejpam-131	156	4	also	also	ADV
ejpam-131	156	5	a	a	DET
ejpam-131	156	6	cover	cover	NOUN
ejpam-131	156	7	of	of	ADP
ejpam-131	156	8	the	the	DET
ejpam-131	156	9	set	set	PROPN
ejpam-131	156	10	b.	b.	PROPN
ejpam-131	156	11	theorem	theorem	VERB
ejpam-131	156	12	3.3	3.3	NUM
ejpam-131	156	13	.	.	PUNCT
ejpam-131	157	1	for	for	ADP
ejpam-131	157	2	any	any	DET
ejpam-131	157	3	space	space	NOUN
ejpam-131	157	4	x	x	SYM
ejpam-131	157	5	,	,	PUNCT
ejpam-131	157	6	the	the	DET
ejpam-131	157	7	following	follow	VERB
ejpam-131	157	8	properties	property	NOUN
ejpam-131	157	9	are	be	AUX
ejpam-131	157	10	equivalent	equivalent	ADJ
ejpam-131	157	11	:	:	PUNCT
ejpam-131	157	12	1	1	X
ejpam-131	157	13	.	.	X
ejpam-131	157	14	x	x	X
ejpam-131	157	15	is	be	AUX
ejpam-131	157	16	b	b	NOUN
ejpam-131	157	17	-	-	PUNCT
ejpam-131	157	18	lindelöf	lindelöf	NOUN
ejpam-131	157	19	;	;	PUNCT
ejpam-131	157	20	2	2	X
ejpam-131	157	21	.	.	X
ejpam-131	158	1	every	every	DET
ejpam-131	158	2	ωb	ωb	NOUN
ejpam-131	158	3	-	-	PUNCT
ejpam-131	158	4	open	open	ADJ
ejpam-131	158	5	cover	cover	NOUN
ejpam-131	158	6	of	of	ADP
ejpam-131	158	7	x	x	PUNCT
ejpam-131	158	8	has	have	VERB
ejpam-131	158	9	a	a	DET
ejpam-131	158	10	countable	countable	ADJ
ejpam-131	158	11	subcover	subcover	NOUN
ejpam-131	158	12	.	.	PUNCT
ejpam-131	159	1	proof	proof	NOUN
ejpam-131	159	2	.	.	PUNCT
ejpam-131	160	1	(	(	PUNCT
ejpam-131	160	2	1)⇒	1)⇒	NUM
ejpam-131	160	3	(	(	PUNCT
ejpam-131	160	4	2	2	NUM
ejpam-131	160	5	):	):	PUNCT
ejpam-131	160	6	let	let	VERB
ejpam-131	160	7	{	{	PUNCT
ejpam-131	160	8	uα	uα	NOUN
ejpam-131	160	9	:	:	PUNCT
ejpam-131	160	10	α	α	PROPN
ejpam-131	160	11	∈	∈	PROPN
ejpam-131	160	12	λ	λ	NOUN
ejpam-131	160	13	}	}	PUNCT
ejpam-131	160	14	be	be	VERB
ejpam-131	160	15	any	any	DET
ejpam-131	160	16	ωb	ωb	NOUN
ejpam-131	160	17	-	-	PUNCT
ejpam-131	160	18	open	open	ADJ
ejpam-131	160	19	cover	cover	NOUN
ejpam-131	160	20	of	of	ADP
ejpam-131	160	21	x	x	X
ejpam-131	160	22	.	.	PUNCT
ejpam-131	161	1	for	for	ADP
ejpam-131	161	2	each	each	DET
ejpam-131	161	3	x	x	SYM
ejpam-131	161	4	∈	∈	PROPN
ejpam-131	161	5	x	x	X
ejpam-131	161	6	,	,	PUNCT
ejpam-131	161	7	there	there	PRON
ejpam-131	161	8	exists	exist	VERB
ejpam-131	161	9	α(x	α(x	NOUN
ejpam-131	161	10	)	)	PUNCT
ejpam-131	161	11	∈	∈	PROPN
ejpam-131	161	12	λ	λ	NOUN
ejpam-131	161	13	such	such	ADJ
ejpam-131	161	14	that	that	SCONJ
ejpam-131	161	15	x	x	SYM
ejpam-131	161	16	∈	∈	NOUN
ejpam-131	161	17	uα(x	uα(x	NOUN
ejpam-131	161	18	)	)	PUNCT
ejpam-131	161	19	.	.	PUNCT
ejpam-131	162	1	since	since	SCONJ
ejpam-131	162	2	uα(x	uα(x	NOUN
ejpam-131	162	3	)	)	PUNCT
ejpam-131	162	4	is	be	AUX
ejpam-131	162	5	ωb	ωb	NOUN
ejpam-131	162	6	-	-	PUNCT
ejpam-131	162	7	open	open	ADJ
ejpam-131	162	8	,	,	PUNCT
ejpam-131	162	9	there	there	PRON
ejpam-131	162	10	exists	exist	VERB
ejpam-131	162	11	a	a	DET
ejpam-131	162	12	b	b	NOUN
ejpam-131	162	13	-	-	PUNCT
ejpam-131	162	14	open	open	ADJ
ejpam-131	162	15	set	set	VERB
ejpam-131	162	16	vα(x	vα(x	NOUN
ejpam-131	162	17	)	)	PUNCT
ejpam-131	162	18	such	such	ADJ
ejpam-131	162	19	that	that	SCONJ
ejpam-131	162	20	x	x	SYM
ejpam-131	162	21	∈	∈	PROPN
ejpam-131	162	22	vα(x	vα(x	NOUN
ejpam-131	162	23	)	)	PUNCT
ejpam-131	162	24	and	and	CCONJ
ejpam-131	162	25	vα(x)\uα(x	vα(x)\uα(x	NOUN
ejpam-131	162	26	)	)	PUNCT
ejpam-131	162	27	is	be	AUX
ejpam-131	162	28	countable	countable	ADJ
ejpam-131	162	29	.	.	PUNCT
ejpam-131	163	1	the	the	DET
ejpam-131	163	2	family	family	NOUN
ejpam-131	163	3	{	{	PUNCT
ejpam-131	163	4	vα(x)|x	vα(x)|x	NOUN
ejpam-131	163	5	∈	∈	NOUN
ejpam-131	163	6	x	x	X
ejpam-131	163	7	}	}	PUNCT
ejpam-131	163	8	is	be	AUX
ejpam-131	163	9	a	a	DET
ejpam-131	163	10	b	b	NOUN
ejpam-131	163	11	-	-	PUNCT
ejpam-131	163	12	open	open	ADJ
ejpam-131	163	13	cover	cover	NOUN
ejpam-131	163	14	of	of	ADP
ejpam-131	163	15	x	x	PUNCT
ejpam-131	163	16	and	and	CCONJ
ejpam-131	163	17	x	x	ADJ
ejpam-131	163	18	is	be	AUX
ejpam-131	163	19	b	b	NOUN
ejpam-131	163	20	-	-	NOUN
ejpam-131	163	21	lindelöf	lindelöf	NOUN
ejpam-131	163	22	.	.	PUNCT
ejpam-131	164	1	there	there	PRON
ejpam-131	164	2	exists	exist	VERB
ejpam-131	164	3	a	a	DET
ejpam-131	164	4	countable	countable	ADJ
ejpam-131	164	5	subset	subset	NOUN
ejpam-131	164	6	,	,	PUNCT
ejpam-131	164	7	says	say	VERB
ejpam-131	164	8	α(x1),α(x2	α(x1),α(x2	NOUN
ejpam-131	164	9	)	)	PUNCT
ejpam-131	164	10	,	,	PUNCT
ejpam-131	164	11	·	·	PUNCT
ejpam-131	164	12	·	·	PUNCT
ejpam-131	164	13	·	·	PUNCT
ejpam-131	164	14	α(xn	α(xn	NUM
ejpam-131	164	15	)	)	PUNCT
ejpam-131	164	16	,	,	PUNCT
ejpam-131	164	17	·	·	PUNCT
ejpam-131	164	18	·	·	PUNCT
ejpam-131	164	19	·	·	PUNCT
ejpam-131	164	20	such	such	ADJ
ejpam-131	164	21	that	that	SCONJ
ejpam-131	164	22	x	x	NOUN
ejpam-131	164	23	=	=	PRON
ejpam-131	164	24	∪{vα(x	∪{vα(x	NOUN
ejpam-131	164	25	i)|i	i)|i	NOUN
ejpam-131	164	26	∈	∈	PROPN
ejpam-131	164	27	n	n	CCONJ
ejpam-131	164	28	}	}	PUNCT
ejpam-131	164	29	.	.	PUNCT
ejpam-131	165	1	now	now	ADV
ejpam-131	165	2	,	,	PUNCT
ejpam-131	165	3	we	we	PRON
ejpam-131	165	4	have	have	VERB
ejpam-131	165	5	x	x	NOUN
ejpam-131	165	6	=	=	VERB
ejpam-131	165	7	∪i∈n	∪i∈n	X
ejpam-131	165	8	{	{	PUNCT
ejpam-131	165	9	(	(	PUNCT
ejpam-131	165	10	vα(x	vα(x	NOUN
ejpam-131	165	11	i)\uα(x	i)\uα(x	PROPN
ejpam-131	165	12	i))∪	i))∪	NOUN
ejpam-131	165	13	uα(x	uα(x	PUNCT
ejpam-131	165	14	i	i	NOUN
ejpam-131	165	15	)	)	PUNCT
ejpam-131	165	16	}	}	PUNCT
ejpam-131	166	1	=[	=[	NOUN
ejpam-131	166	2	∪i∈n(vα(x	∪i∈n(vα(x	PROPN
ejpam-131	166	3	i)\uα(x	i)\uα(x	NOUN
ejpam-131	167	1	i))]∪	i))]∪	PROPN
ejpam-131	168	1	[	[	X
ejpam-131	168	2	∪i∈nuα(x	∪i∈nuα(x	PROPN
ejpam-131	168	3	i	i	PROPN
ejpam-131	168	4	)	)	PUNCT
ejpam-131	168	5	]	]	PUNCT
ejpam-131	168	6	.	.	PUNCT
ejpam-131	169	1	for	for	ADP
ejpam-131	169	2	each	each	DET
ejpam-131	169	3	α(x	α(x	PROPN
ejpam-131	169	4	i	i	PROPN
ejpam-131	169	5	)	)	PUNCT
ejpam-131	169	6	,	,	PUNCT
ejpam-131	169	7	vα(x	vα(x	X
ejpam-131	169	8	i)\uα(x	i)\uα(x	PROPN
ejpam-131	169	9	i	i	PROPN
ejpam-131	169	10	)	)	PUNCT
ejpam-131	169	11	is	be	AUX
ejpam-131	169	12	a	a	DET
ejpam-131	169	13	countable	countable	ADJ
ejpam-131	169	14	set	set	NOUN
ejpam-131	169	15	and	and	CCONJ
ejpam-131	169	16	there	there	PRON
ejpam-131	169	17	exists	exist	VERB
ejpam-131	169	18	a	a	DET
ejpam-131	169	19	countable	countable	ADJ
ejpam-131	169	20	subset	subset	NOUN
ejpam-131	169	21	λα(x	λα(x	PUNCT
ejpam-131	169	22	i	i	NOUN
ejpam-131	169	23	)	)	PUNCT
ejpam-131	169	24	of	of	ADP
ejpam-131	169	25	λ	λ	INTJ
ejpam-131	169	26	such	such	ADJ
ejpam-131	169	27	that	that	SCONJ
ejpam-131	169	28	vα(x	vα(x	NOUN
ejpam-131	169	29	i)\uα(x	i)\uα(x	PROPN
ejpam-131	169	30	i	i	PROPN
ejpam-131	169	31	)	)	PUNCT
ejpam-131	169	32	⊆	⊆	NUM
ejpam-131	169	33	∪{uα|α	∪{uα|α	PROPN
ejpam-131	169	34	∈	∈	NOUN
ejpam-131	169	35	λα(x	λα(x	X
ejpam-131	169	36	i	i	NOUN
ejpam-131	169	37	)	)	PUNCT
ejpam-131	169	38	}	}	PUNCT
ejpam-131	169	39	.	.	PUNCT
ejpam-131	170	1	therefore	therefore	ADV
ejpam-131	170	2	,	,	PUNCT
ejpam-131	170	3	we	we	PRON
ejpam-131	170	4	have	have	VERB
ejpam-131	170	5	x	x	NUM
ejpam-131	170	6	⊆	⊆	NUM
ejpam-131	170	7	[	[	X
ejpam-131	170	8	∪i∈n(∪{uα|α	∪i∈n(∪{uα|α	PROPN
ejpam-131	170	9	∈	∈	PROPN
ejpam-131	170	10	λα(x	λα(x	X
ejpam-131	170	11	i)})]∪	i)})]∪	PROPN
ejpam-131	171	1	[	[	X
ejpam-131	171	2	∪i∈nuα(x	∪i∈nuα(x	PROPN
ejpam-131	171	3	i	i	PROPN
ejpam-131	171	4	)	)	PUNCT
ejpam-131	171	5	]	]	PUNCT
ejpam-131	171	6	.	.	PUNCT
ejpam-131	172	1	(	(	PUNCT
ejpam-131	172	2	2)⇒	2)⇒	NUM
ejpam-131	172	3	(	(	PUNCT
ejpam-131	172	4	1	1	NUM
ejpam-131	172	5	):	):	PUNCT
ejpam-131	172	6	since	since	SCONJ
ejpam-131	172	7	every	every	DET
ejpam-131	172	8	b	b	X
ejpam-131	172	9	-	-	PUNCT
ejpam-131	172	10	open	open	ADJ
ejpam-131	172	11	is	be	AUX
ejpam-131	172	12	ωb	ωb	NOUN
ejpam-131	172	13	-	-	PUNCT
ejpam-131	172	14	open	open	ADJ
ejpam-131	172	15	,	,	PUNCT
ejpam-131	172	16	the	the	DET
ejpam-131	172	17	proof	proof	NOUN
ejpam-131	172	18	is	be	AUX
ejpam-131	172	19	obvious	obvious	ADJ
ejpam-131	172	20	.	.	PUNCT
ejpam-131	173	1	definition	definition	NOUN
ejpam-131	173	2	3.4	3.4	NUM
ejpam-131	173	3	.	.	PUNCT
ejpam-131	174	1	a	a	DET
ejpam-131	174	2	function	function	NOUN
ejpam-131	174	3	f	f	NOUN
ejpam-131	174	4	:	:	PUNCT
ejpam-131	174	5	x	x	X
ejpam-131	174	6	→	→	SYM
ejpam-131	174	7	y	y	PROPN
ejpam-131	174	8	is	be	AUX
ejpam-131	174	9	said	say	VERB
ejpam-131	174	10	to	to	PART
ejpam-131	174	11	be	be	AUX
ejpam-131	174	12	ωb	ωb	NOUN
ejpam-131	174	13	-	-	PUNCT
ejpam-131	174	14	continuous	continuous	ADJ
ejpam-131	174	15	if	if	SCONJ
ejpam-131	174	16	f	f	PROPN
ejpam-131	174	17	−1(v	−1(v	PROPN
ejpam-131	174	18	)	)	PUNCT
ejpam-131	174	19	is	be	AUX
ejpam-131	174	20	ωb	ωb	NOUN
ejpam-131	174	21	-	-	PUNCT
ejpam-131	174	22	open	open	ADJ
ejpam-131	174	23	in	in	ADP
ejpam-131	174	24	x	x	PUNCT
ejpam-131	174	25	for	for	SCONJ
ejpam-131	174	26	each	each	DET
ejpam-131	174	27	open	open	ADJ
ejpam-131	174	28	set	set	VERB
ejpam-131	174	29	v	v	NOUN
ejpam-131	174	30	in	in	ADP
ejpam-131	174	31	y	y	PROPN
ejpam-131	174	32	.	.	PUNCT
ejpam-131	175	1	theorem	theorem	VERB
ejpam-131	175	2	3.5	3.5	NUM
ejpam-131	175	3	.	.	PUNCT
ejpam-131	176	1	let	let	VERB
ejpam-131	176	2	f	f	PRON
ejpam-131	176	3	be	be	AUX
ejpam-131	176	4	an	an	DET
ejpam-131	176	5	ωb	ωb	NOUN
ejpam-131	176	6	-	-	PUNCT
ejpam-131	176	7	continuous	continuous	ADJ
ejpam-131	176	8	function	function	NOUN
ejpam-131	176	9	from	from	ADP
ejpam-131	176	10	a	a	DET
ejpam-131	176	11	space	space	NOUN
ejpam-131	176	12	x	x	X
ejpam-131	176	13	onto	onto	ADP
ejpam-131	176	14	a	a	DET
ejpam-131	176	15	space	space	NOUN
ejpam-131	176	16	y	y	NOUN
ejpam-131	176	17	.	.	PUNCT
ejpam-131	177	1	if	if	SCONJ
ejpam-131	177	2	x	x	PRON
ejpam-131	177	3	is	be	AUX
ejpam-131	177	4	b	b	NOUN
ejpam-131	177	5	-	-	PUNCT
ejpam-131	177	6	lindelöf	lindelöf	NOUN
ejpam-131	177	7	,	,	PUNCT
ejpam-131	177	8	then	then	ADV
ejpam-131	177	9	y	y	PROPN
ejpam-131	177	10	is	be	AUX
ejpam-131	177	11	lindelöf	lindelöf	NOUN
ejpam-131	177	12	.	.	PUNCT
ejpam-131	178	1	proof	proof	NOUN
ejpam-131	178	2	.	.	PUNCT
ejpam-131	179	1	let	let	VERB
ejpam-131	179	2	{	{	PUNCT
ejpam-131	179	3	vα	vα	X
ejpam-131	179	4	:	:	PUNCT
ejpam-131	179	5	α	α	PROPN
ejpam-131	179	6	∈	∈	PROPN
ejpam-131	179	7	λ	λ	NOUN
ejpam-131	179	8	}	}	PUNCT
ejpam-131	179	9	be	be	VERB
ejpam-131	179	10	an	an	DET
ejpam-131	179	11	open	open	ADJ
ejpam-131	179	12	cover	cover	NOUN
ejpam-131	179	13	of	of	ADP
ejpam-131	179	14	y	y	PROPN
ejpam-131	179	15	.	.	PUNCT
ejpam-131	180	1	then	then	ADV
ejpam-131	180	2	{	{	PUNCT
ejpam-131	180	3	f	f	PROPN
ejpam-131	180	4	−1(vα	−1(vα	PROPN
ejpam-131	180	5	)	)	PUNCT
ejpam-131	180	6	:	:	PUNCT
ejpam-131	180	7	α	α	PROPN
ejpam-131	180	8	∈	∈	PROPN
ejpam-131	180	9	λ	λ	NOUN
ejpam-131	180	10	}	}	PUNCT
ejpam-131	180	11	is	be	AUX
ejpam-131	180	12	anωb	anωb	ADJ
ejpam-131	180	13	-	-	PUNCT
ejpam-131	180	14	open	open	ADJ
ejpam-131	180	15	cover	cover	NOUN
ejpam-131	180	16	of	of	ADP
ejpam-131	180	17	x	x	X
ejpam-131	180	18	.	.	PUNCT
ejpam-131	181	1	since	since	SCONJ
ejpam-131	181	2	x	x	PROPN
ejpam-131	181	3	is	be	AUX
ejpam-131	181	4	b	b	NOUN
ejpam-131	181	5	-	-	PUNCT
ejpam-131	181	6	lindelöf	lindelöf	NOUN
ejpam-131	181	7	,	,	PUNCT
ejpam-131	181	8	by	by	ADP
ejpam-131	181	9	theorem	theorem	NOUN
ejpam-131	181	10	3.3	3.3	NUM
ejpam-131	181	11	,	,	PUNCT
ejpam-131	181	12	x	x	PUNCT
ejpam-131	181	13	has	have	VERB
ejpam-131	181	14	a	a	DET
ejpam-131	181	15	countable	countable	ADJ
ejpam-131	181	16	subcover	subcover	NOUN
ejpam-131	181	17	,	,	PUNCT
ejpam-131	181	18	say	say	VERB
ejpam-131	181	19	{	{	PUNCT
ejpam-131	181	20	f	f	X
ejpam-131	181	21	−1(vαi)}∞i=1	−1(vαi)}∞i=1	PROPN
ejpam-131	181	22	and	and	CCONJ
ejpam-131	181	23	vαi	vαi	PROPN
ejpam-131	181	24	∈	∈	PROPN
ejpam-131	181	25	{	{	PUNCT
ejpam-131	181	26	vα	vα	X
ejpam-131	181	27	:	:	PUNCT
ejpam-131	181	28	α	α	PROPN
ejpam-131	181	29	∈	∈	PROPN
ejpam-131	181	30	λ	λ	NOUN
ejpam-131	181	31	}	}	PUNCT
ejpam-131	181	32	.	.	PUNCT
ejpam-131	182	1	hence	hence	ADV
ejpam-131	182	2	{	{	PUNCT
ejpam-131	182	3	vαi}∞i=1	vαi}∞i=1	PROPN
ejpam-131	182	4	is	be	AUX
ejpam-131	182	5	a	a	DET
ejpam-131	182	6	countable	countable	ADJ
ejpam-131	182	7	subcover	subcover	NOUN
ejpam-131	182	8	of	of	ADP
ejpam-131	182	9	y	y	PROPN
ejpam-131	182	10	.	.	PUNCT
ejpam-131	183	1	hence	hence	ADV
ejpam-131	183	2	y	y	PROPN
ejpam-131	183	3	is	be	AUX
ejpam-131	183	4	lindelöf	lindelöf	NOUN
ejpam-131	183	5	.	.	PUNCT
ejpam-131	184	1	definition	definition	NOUN
ejpam-131	184	2	3.6	3.6	NUM
ejpam-131	184	3	.	.	PUNCT
ejpam-131	185	1	a	a	DET
ejpam-131	185	2	function	function	NOUN
ejpam-131	185	3	f	f	NOUN
ejpam-131	185	4	:	:	PUNCT
ejpam-131	185	5	x	x	X
ejpam-131	185	6	→	→	SYM
ejpam-131	185	7	y	y	PROPN
ejpam-131	185	8	is	be	AUX
ejpam-131	185	9	said	say	VERB
ejpam-131	185	10	to	to	PART
ejpam-131	185	11	be	be	AUX
ejpam-131	185	12	γ	γ	X
ejpam-131	185	13	-	-	ADJ
ejpam-131	185	14	continuous	continuous	ADJ
ejpam-131	185	15	[	[	X
ejpam-131	185	16	7	7	NUM
ejpam-131	185	17	]	]	X
ejpam-131	185	18	(	(	PUNCT
ejpam-131	185	19	resp	resp	NOUN
ejpam-131	185	20	.	.	PUNCT
ejpam-131	186	1	ω	ω	X
ejpam-131	186	2	-	-	ADJ
ejpam-131	186	3	continuous	continuous	ADJ
ejpam-131	186	4	[	[	X
ejpam-131	186	5	9	9	NUM
ejpam-131	186	6	]	]	SYM
ejpam-131	186	7	)	)	PUNCT
ejpam-131	186	8	if	if	SCONJ
ejpam-131	186	9	f	f	PROPN
ejpam-131	186	10	−1(v	−1(v	PROPN
ejpam-131	186	11	)	)	PUNCT
ejpam-131	186	12	is	be	AUX
ejpam-131	186	13	b	b	NOUN
ejpam-131	186	14	-	-	PUNCT
ejpam-131	186	15	open	open	ADJ
ejpam-131	186	16	(	(	PUNCT
ejpam-131	186	17	resp	resp	NOUN
ejpam-131	186	18	.	.	PUNCT
ejpam-131	187	1	ω	ω	X
ejpam-131	187	2	-	-	NOUN
ejpam-131	187	3	open	open	ADJ
ejpam-131	187	4	)	)	PUNCT
ejpam-131	187	5	for	for	ADP
ejpam-131	187	6	each	each	DET
ejpam-131	187	7	open	open	ADJ
ejpam-131	187	8	set	set	VERB
ejpam-131	187	9	v	v	NOUN
ejpam-131	187	10	in	in	ADP
ejpam-131	187	11	y	y	PROPN
ejpam-131	187	12	.	.	PUNCT
ejpam-131	188	1	since	since	SCONJ
ejpam-131	188	2	the	the	DET
ejpam-131	188	3	notion	notion	NOUN
ejpam-131	188	4	of	of	ADP
ejpam-131	188	5	b	b	NOUN
ejpam-131	188	6	-	-	PUNCT
ejpam-131	188	7	open	open	ADJ
ejpam-131	188	8	sets	set	NOUN
ejpam-131	188	9	and	and	CCONJ
ejpam-131	188	10	the	the	DET
ejpam-131	188	11	notion	notion	NOUN
ejpam-131	188	12	of	of	ADP
ejpam-131	188	13	γ	γ	ADJ
ejpam-131	188	14	-	-	ADJ
ejpam-131	188	15	open	open	ADJ
ejpam-131	188	16	sets	set	NOUN
ejpam-131	188	17	are	be	AUX
ejpam-131	188	18	same	same	ADJ
ejpam-131	188	19	,	,	PUNCT
ejpam-131	188	20	we	we	PRON
ejpam-131	188	21	will	will	AUX
ejpam-131	188	22	use	use	VERB
ejpam-131	188	23	the	the	DET
ejpam-131	188	24	term	term	NOUN
ejpam-131	188	25	b	b	NOUN
ejpam-131	188	26	-	-	PUNCT
ejpam-131	188	27	continuous	continuous	ADJ
ejpam-131	188	28	functions	function	NOUN
ejpam-131	188	29	instead	instead	ADV
ejpam-131	188	30	of	of	ADP
ejpam-131	188	31	γ	γ	NOUN
ejpam-131	188	32	-	-	ADJ
ejpam-131	188	33	continuous	continuous	ADJ
ejpam-131	188	34	functions	function	NOUN
ejpam-131	188	35	.	.	PUNCT
ejpam-131	189	1	corollary	corollary	ADJ
ejpam-131	189	2	3.7	3.7	NUM
ejpam-131	189	3	.	.	PUNCT
ejpam-131	190	1	let	let	VERB
ejpam-131	190	2	f	f	PRON
ejpam-131	190	3	be	be	AUX
ejpam-131	190	4	a	a	DET
ejpam-131	190	5	b	b	NOUN
ejpam-131	190	6	-	-	ADJ
ejpam-131	190	7	continuous	continuous	ADJ
ejpam-131	190	8	(	(	PUNCT
ejpam-131	190	9	or	or	CCONJ
ejpam-131	190	10	ω	ω	VERB
ejpam-131	190	11	-	-	ADJ
ejpam-131	190	12	continuous	continuous	ADJ
ejpam-131	190	13	)	)	PUNCT
ejpam-131	190	14	function	function	NOUN
ejpam-131	190	15	from	from	ADP
ejpam-131	190	16	a	a	DET
ejpam-131	190	17	space	space	NOUN
ejpam-131	190	18	x	x	X
ejpam-131	190	19	onto	onto	ADP
ejpam-131	190	20	a	a	DET
ejpam-131	190	21	space	space	NOUN
ejpam-131	190	22	y	y	NOUN
ejpam-131	190	23	.	.	PUNCT
ejpam-131	191	1	if	if	SCONJ
ejpam-131	191	2	x	x	PRON
ejpam-131	191	3	is	be	AUX
ejpam-131	191	4	b	b	NOUN
ejpam-131	191	5	-	-	PUNCT
ejpam-131	191	6	lindelöf	lindelöf	NOUN
ejpam-131	191	7	,	,	PUNCT
ejpam-131	191	8	then	then	ADV
ejpam-131	191	9	y	y	PROPN
ejpam-131	191	10	is	be	AUX
ejpam-131	191	11	lindelöf	lindelöf	NOUN
ejpam-131	191	12	.	.	PUNCT
ejpam-131	192	1	definition	definition	NOUN
ejpam-131	192	2	3.8	3.8	NUM
ejpam-131	192	3	.	.	PUNCT
ejpam-131	193	1	a	a	DET
ejpam-131	193	2	function	function	NOUN
ejpam-131	193	3	f	f	NOUN
ejpam-131	193	4	:	:	PUNCT
ejpam-131	193	5	x	x	X
ejpam-131	193	6	→	→	SYM
ejpam-131	193	7	y	y	PROPN
ejpam-131	193	8	is	be	AUX
ejpam-131	193	9	said	say	VERB
ejpam-131	193	10	to	to	PART
ejpam-131	193	11	be	be	AUX
ejpam-131	193	12	ωb∗-continuous	ωb∗-continuous	ADJ
ejpam-131	193	13	if	if	SCONJ
ejpam-131	193	14	f	f	PROPN
ejpam-131	193	15	−1(v	−1(v	PROPN
ejpam-131	193	16	)	)	PUNCT
ejpam-131	193	17	is	be	AUX
ejpam-131	193	18	ωb	ωb	NOUN
ejpam-131	193	19	-	-	PUNCT
ejpam-131	193	20	open	open	ADJ
ejpam-131	193	21	in	in	ADP
ejpam-131	193	22	x	x	PUNCT
ejpam-131	193	23	for	for	ADP
ejpam-131	193	24	each	each	DET
ejpam-131	193	25	b	b	NOUN
ejpam-131	193	26	-	-	PUNCT
ejpam-131	193	27	open	open	ADJ
ejpam-131	193	28	set	set	VERB
ejpam-131	193	29	v	v	NOUN
ejpam-131	193	30	in	in	ADP
ejpam-131	193	31	y	y	PROPN
ejpam-131	193	32	.	.	PUNCT
ejpam-131	194	1	t.	t.	PROPN
ejpam-131	194	2	noiri	noiri	PROPN
ejpam-131	194	3	,	,	PUNCT
ejpam-131	194	4	a.	a.	PROPN
ejpam-131	194	5	al	al	PROPN
ejpam-131	194	6	-	-	PUNCT
ejpam-131	194	7	omari	omari	PROPN
ejpam-131	194	8	and	and	CCONJ
ejpam-131	194	9	m.s.m	m.s.m	PROPN
ejpam-131	194	10	.	.	PROPN
ejpam-131	194	11	noorani	noorani	PROPN
ejpam-131	194	12	/	/	SYM
ejpam-131	194	13	eur	eur	PROPN
ejpam-131	194	14	.	.	PUNCT
ejpam-131	195	1	j.	j.	PROPN
ejpam-131	195	2	pure	pure	PROPN
ejpam-131	195	3	appl	appl	PROPN
ejpam-131	195	4	.	.	PROPN
ejpam-131	195	5	math	math	PROPN
ejpam-131	195	6	,	,	PUNCT
ejpam-131	195	7	1	1	NUM
ejpam-131	195	8	(	(	PUNCT
ejpam-131	195	9	2008	2008	NUM
ejpam-131	195	10	)	)	PUNCT
ejpam-131	195	11	,	,	PUNCT
ejpam-131	195	12	(	(	PUNCT
ejpam-131	195	13	3	3	NUM
ejpam-131	195	14	-	-	SYM
ejpam-131	195	15	9	9	NUM
ejpam-131	195	16	)	)	PUNCT
ejpam-131	195	17	8	8	NUM
ejpam-131	195	18	now	now	ADV
ejpam-131	195	19	we	we	PRON
ejpam-131	195	20	state	state	VERB
ejpam-131	195	21	the	the	DET
ejpam-131	195	22	following	following	NOUN
ejpam-131	195	23	theorem	theorem	NOUN
ejpam-131	195	24	whose	whose	DET
ejpam-131	195	25	proof	proof	NOUN
ejpam-131	195	26	is	be	AUX
ejpam-131	195	27	similar	similar	ADJ
ejpam-131	195	28	to	to	AUX
ejpam-131	195	29	theorem	theorem	VERB
ejpam-131	195	30	3.5	3.5	NUM
ejpam-131	195	31	.	.	PUNCT
ejpam-131	196	1	theorem	theorem	VERB
ejpam-131	196	2	3.9	3.9	NUM
ejpam-131	196	3	.	.	PUNCT
ejpam-131	197	1	let	let	VERB
ejpam-131	197	2	f	f	PRON
ejpam-131	197	3	be	be	AUX
ejpam-131	197	4	an	an	DET
ejpam-131	197	5	ωb∗-continuous	ωb∗-continuous	ADJ
ejpam-131	197	6	function	function	NOUN
ejpam-131	197	7	from	from	ADP
ejpam-131	197	8	a	a	DET
ejpam-131	197	9	space	space	NOUN
ejpam-131	197	10	x	x	X
ejpam-131	197	11	onto	onto	ADP
ejpam-131	197	12	a	a	DET
ejpam-131	197	13	space	space	NOUN
ejpam-131	197	14	y	y	NOUN
ejpam-131	197	15	.	.	PUNCT
ejpam-131	198	1	if	if	SCONJ
ejpam-131	198	2	x	x	PRON
ejpam-131	198	3	is	be	AUX
ejpam-131	198	4	b	b	NOUN
ejpam-131	198	5	-	-	PUNCT
ejpam-131	198	6	lindelöf	lindelöf	NOUN
ejpam-131	198	7	,	,	PUNCT
ejpam-131	198	8	then	then	ADV
ejpam-131	198	9	y	y	PROPN
ejpam-131	198	10	is	be	AUX
ejpam-131	198	11	b	b	NOUN
ejpam-131	198	12	-	-	PUNCT
ejpam-131	198	13	lindelöf	lindelöf	NOUN
ejpam-131	198	14	.	.	PUNCT
ejpam-131	199	1	proposition	proposition	NOUN
ejpam-131	199	2	3.10	3.10	NUM
ejpam-131	199	3	.	.	PUNCT
ejpam-131	200	1	an	an	DET
ejpam-131	200	2	ωb	ωb	NOUN
ejpam-131	200	3	-	-	PUNCT
ejpam-131	200	4	closed	close	VERB
ejpam-131	200	5	subset	subset	NOUN
ejpam-131	200	6	of	of	ADP
ejpam-131	200	7	a	a	DET
ejpam-131	200	8	b	b	NOUN
ejpam-131	200	9	-	-	PUNCT
ejpam-131	200	10	lindelöf	lindelöf	NOUN
ejpam-131	200	11	space	space	NOUN
ejpam-131	200	12	x	x	PUNCT
ejpam-131	200	13	is	be	AUX
ejpam-131	200	14	b	b	NOUN
ejpam-131	200	15	-	-	PUNCT
ejpam-131	200	16	lindelöf	lindelöf	NOUN
ejpam-131	200	17	relative	relative	NOUN
ejpam-131	200	18	to	to	ADP
ejpam-131	200	19	x	x	X
ejpam-131	200	20	.	.	PUNCT
ejpam-131	201	1	proof	proof	NOUN
ejpam-131	201	2	.	.	PUNCT
ejpam-131	202	1	let	let	VERB
ejpam-131	202	2	a	a	DET
ejpam-131	202	3	be	be	AUX
ejpam-131	202	4	an	an	DET
ejpam-131	202	5	ωb	ωb	NOUN
ejpam-131	202	6	-	-	PUNCT
ejpam-131	202	7	closed	close	VERB
ejpam-131	202	8	subset	subset	NOUN
ejpam-131	202	9	of	of	ADP
ejpam-131	202	10	x	x	X
ejpam-131	202	11	.	.	PUNCT
ejpam-131	203	1	let	let	VERB
ejpam-131	203	2	{	{	PUNCT
ejpam-131	203	3	uα	uα	NOUN
ejpam-131	203	4	:	:	PUNCT
ejpam-131	203	5	α	α	PROPN
ejpam-131	203	6	∈	∈	PROPN
ejpam-131	203	7	λ	λ	NOUN
ejpam-131	203	8	}	}	PUNCT
ejpam-131	203	9	be	be	VERB
ejpam-131	203	10	a	a	DET
ejpam-131	203	11	cover	cover	NOUN
ejpam-131	203	12	of	of	ADP
ejpam-131	203	13	a	a	PRON
ejpam-131	203	14	by	by	ADP
ejpam-131	203	15	b	b	NOUN
ejpam-131	203	16	-	-	PUNCT
ejpam-131	203	17	open	open	ADJ
ejpam-131	203	18	sets	set	NOUN
ejpam-131	203	19	of	of	ADP
ejpam-131	203	20	x	x	X
ejpam-131	203	21	.	.	PUNCT
ejpam-131	204	1	now	now	ADV
ejpam-131	204	2	for	for	ADP
ejpam-131	204	3	each	each	DET
ejpam-131	204	4	x	x	SYM
ejpam-131	204	5	∈	∈	PROPN
ejpam-131	204	6	x	x	X
ejpam-131	204	7	−	−	NOUN
ejpam-131	205	1	a	a	X
ejpam-131	205	2	,	,	PUNCT
ejpam-131	205	3	there	there	PRON
ejpam-131	205	4	is	be	VERB
ejpam-131	205	5	a	a	DET
ejpam-131	205	6	b	b	NOUN
ejpam-131	205	7	-	-	PUNCT
ejpam-131	205	8	open	open	ADJ
ejpam-131	205	9	set	set	NOUN
ejpam-131	205	10	vx	vx	ADP
ejpam-131	205	11	such	such	ADJ
ejpam-131	205	12	that	that	SCONJ
ejpam-131	205	13	vx	vx	PROPN
ejpam-131	205	14	∩	∩	PROPN
ejpam-131	205	15	a	a	PRON
ejpam-131	205	16	is	be	AUX
ejpam-131	205	17	countable	countable	ADJ
ejpam-131	205	18	.	.	PUNCT
ejpam-131	206	1	since	since	SCONJ
ejpam-131	206	2	{	{	PUNCT
ejpam-131	206	3	uα	uα	X
ejpam-131	206	4	:	:	PUNCT
ejpam-131	206	5	α	α	PROPN
ejpam-131	206	6	∈	∈	PROPN
ejpam-131	206	7	λ	λ	PROPN
ejpam-131	206	8	}	}	PUNCT
ejpam-131	206	9	∪	∪	NOUN
ejpam-131	206	10	{	{	PUNCT
ejpam-131	206	11	vx	vx	NOUN
ejpam-131	206	12	:	:	PUNCT
ejpam-131	206	13	x	x	SYM
ejpam-131	206	14	∈	∈	NOUN
ejpam-131	206	15	x	x	PUNCT
ejpam-131	206	16	−	−	NOUN
ejpam-131	206	17	a	a	PRON
ejpam-131	206	18	}	}	PUNCT
ejpam-131	206	19	is	be	AUX
ejpam-131	206	20	a	a	DET
ejpam-131	206	21	b	b	NOUN
ejpam-131	206	22	-	-	PUNCT
ejpam-131	206	23	open	open	ADJ
ejpam-131	206	24	cover	cover	NOUN
ejpam-131	206	25	of	of	ADP
ejpam-131	206	26	x	x	PUNCT
ejpam-131	206	27	and	and	CCONJ
ejpam-131	206	28	x	x	ADJ
ejpam-131	206	29	is	be	AUX
ejpam-131	206	30	b	b	NOUN
ejpam-131	206	31	-	-	PUNCT
ejpam-131	206	32	lindelöf	lindelöf	NOUN
ejpam-131	206	33	,	,	PUNCT
ejpam-131	206	34	there	there	PRON
ejpam-131	206	35	exists	exist	VERB
ejpam-131	206	36	a	a	DET
ejpam-131	206	37	countable	countable	ADJ
ejpam-131	206	38	subcover	subcover	NOUN
ejpam-131	206	39	{	{	PUNCT
ejpam-131	206	40	uαi	uαi	PROPN
ejpam-131	206	41	:	:	PUNCT
ejpam-131	206	42	i	i	PROPN
ejpam-131	206	43	∈	∈	PROPN
ejpam-131	206	44	n	n	CCONJ
ejpam-131	206	45	}	}	PUNCT
ejpam-131	206	46	∪	∪	NOUN
ejpam-131	206	47	{	{	PUNCT
ejpam-131	206	48	vx	vx	NOUN
ejpam-131	206	49	i	i	PRON
ejpam-131	206	50	:	:	PUNCT
ejpam-131	206	51	i	i	PROPN
ejpam-131	206	52	∈	∈	PROPN
ejpam-131	206	53	n	n	CCONJ
ejpam-131	206	54	}	}	PUNCT
ejpam-131	206	55	.	.	PUNCT
ejpam-131	207	1	since	since	SCONJ
ejpam-131	207	2	∪i∈n	∪i∈n	ADV
ejpam-131	207	3	(	(	PUNCT
ejpam-131	207	4	vx	vx	PROPN
ejpam-131	207	5	i	i	PROPN
ejpam-131	207	6	∩	∩	PROPN
ejpam-131	207	7	a	a	X
ejpam-131	207	8	)	)	PUNCT
ejpam-131	207	9	is	be	AUX
ejpam-131	207	10	countable	countable	ADJ
ejpam-131	207	11	,	,	PUNCT
ejpam-131	207	12	so	so	ADV
ejpam-131	207	13	for	for	SCONJ
ejpam-131	207	14	each	each	DET
ejpam-131	207	15	x	x	SYM
ejpam-131	207	16	j	j	PROPN
ejpam-131	207	17	∈	∈	PROPN
ejpam-131	207	18	∪(vx	∪(vx	PROPN
ejpam-131	207	19	i	i	PROPN
ejpam-131	207	20	∩	∩	NOUN
ejpam-131	207	21	a	a	X
ejpam-131	207	22	)	)	PUNCT
ejpam-131	207	23	,	,	PUNCT
ejpam-131	207	24	there	there	PRON
ejpam-131	207	25	is	be	VERB
ejpam-131	207	26	uα(x	uα(x	PROPN
ejpam-131	207	27	j	j	PROPN
ejpam-131	207	28	)	)	PUNCT
ejpam-131	207	29	∈	∈	PROPN
ejpam-131	207	30	{	{	PUNCT
ejpam-131	207	31	uα	uα	NOUN
ejpam-131	207	32	:	:	PUNCT
ejpam-131	207	33	α	α	PROPN
ejpam-131	207	34	∈	∈	PROPN
ejpam-131	207	35	λ	λ	NOUN
ejpam-131	207	36	}	}	PUNCT
ejpam-131	207	37	such	such	ADJ
ejpam-131	207	38	that	that	SCONJ
ejpam-131	207	39	x	x	PRON
ejpam-131	207	40	j	j	PROPN
ejpam-131	207	41	∈	∈	PROPN
ejpam-131	207	42	uα(x	uα(x	PROPN
ejpam-131	207	43	j	j	PROPN
ejpam-131	207	44	)	)	PUNCT
ejpam-131	207	45	and	and	CCONJ
ejpam-131	207	46	j	j	PROPN
ejpam-131	207	47	∈	∈	PROPN
ejpam-131	207	48	n.	n.	NOUN
ejpam-131	207	49	hence	hence	ADV
ejpam-131	207	50	{	{	PUNCT
ejpam-131	207	51	uαi	uαi	ADV
ejpam-131	207	52	:	:	PUNCT
ejpam-131	207	53	i	i	PROPN
ejpam-131	207	54	∈	∈	PROPN
ejpam-131	207	55	n	n	CCONJ
ejpam-131	207	56	}	}	PUNCT
ejpam-131	207	57	∪	∪	ADJ
ejpam-131	207	58	{	{	PUNCT
ejpam-131	207	59	uα(x	uα(x	NUM
ejpam-131	207	60	j	j	NOUN
ejpam-131	207	61	)	)	PUNCT
ejpam-131	207	62	:	:	PUNCT
ejpam-131	208	1	j	j	PROPN
ejpam-131	208	2	∈	∈	PROPN
ejpam-131	208	3	n	n	CCONJ
ejpam-131	208	4	}	}	PUNCT
ejpam-131	208	5	is	be	AUX
ejpam-131	208	6	a	a	DET
ejpam-131	208	7	countable	countable	ADJ
ejpam-131	208	8	subcover	subcover	NOUN
ejpam-131	208	9	of	of	ADP
ejpam-131	208	10	{	{	PUNCT
ejpam-131	208	11	uα	uα	X
ejpam-131	208	12	:	:	PUNCT
ejpam-131	208	13	α	α	PROPN
ejpam-131	208	14	∈	∈	PROPN
ejpam-131	208	15	λ	λ	X
ejpam-131	208	16	}	}	PUNCT
ejpam-131	208	17	and	and	CCONJ
ejpam-131	208	18	it	it	PRON
ejpam-131	208	19	covers	cover	VERB
ejpam-131	208	20	a.	a.	NOUN
ejpam-131	208	21	therefore	therefore	ADV
ejpam-131	208	22	,	,	PUNCT
ejpam-131	208	23	a	a	PRON
ejpam-131	208	24	is	be	AUX
ejpam-131	208	25	b	b	NOUN
ejpam-131	208	26	-	-	PUNCT
ejpam-131	208	27	lindelöf	lindelöf	NOUN
ejpam-131	208	28	relative	relative	NOUN
ejpam-131	208	29	to	to	ADP
ejpam-131	208	30	x	x	PROPN
ejpam-131	208	31	.	.	PUNCT
ejpam-131	209	1	corollary	corollary	ADJ
ejpam-131	209	2	3.11	3.11	NUM
ejpam-131	209	3	.	.	PUNCT
ejpam-131	210	1	if	if	SCONJ
ejpam-131	210	2	a	a	DET
ejpam-131	210	3	space	space	NOUN
ejpam-131	210	4	x	x	PUNCT
ejpam-131	210	5	is	be	AUX
ejpam-131	210	6	b	b	NOUN
ejpam-131	210	7	-	-	PUNCT
ejpam-131	210	8	lindelöf	lindelöf	NOUN
ejpam-131	210	9	and	and	CCONJ
ejpam-131	210	10	a	a	PRON
ejpam-131	210	11	is	be	AUX
ejpam-131	210	12	ω	ω	NOUN
ejpam-131	210	13	-	-	ADJ
ejpam-131	210	14	closed	closed	ADJ
ejpam-131	210	15	(	(	PUNCT
ejpam-131	210	16	or	or	CCONJ
ejpam-131	210	17	b	b	X
ejpam-131	210	18	-	-	PUNCT
ejpam-131	210	19	closed	closed	ADJ
ejpam-131	210	20	)	)	PUNCT
ejpam-131	210	21	,	,	PUNCT
ejpam-131	210	22	then	then	ADV
ejpam-131	210	23	a	a	PRON
ejpam-131	210	24	is	be	AUX
ejpam-131	210	25	b	b	NOUN
ejpam-131	210	26	-	-	PUNCT
ejpam-131	210	27	lindelöf	lindelöf	NOUN
ejpam-131	210	28	relative	relative	NOUN
ejpam-131	210	29	to	to	ADP
ejpam-131	210	30	x	x	PROPN
ejpam-131	210	31	.	.	PUNCT
ejpam-131	211	1	definition	definition	NOUN
ejpam-131	211	2	3.12	3.12	NUM
ejpam-131	211	3	.	.	PUNCT
ejpam-131	212	1	a	a	DET
ejpam-131	212	2	function	function	NOUN
ejpam-131	212	3	f	f	NOUN
ejpam-131	212	4	:	:	PUNCT
ejpam-131	212	5	x	x	X
ejpam-131	212	6	→	→	SYM
ejpam-131	212	7	y	y	PROPN
ejpam-131	212	8	is	be	AUX
ejpam-131	212	9	said	say	VERB
ejpam-131	212	10	to	to	PART
ejpam-131	212	11	be	be	AUX
ejpam-131	212	12	ωb	ωb	NOUN
ejpam-131	212	13	-	-	PUNCT
ejpam-131	212	14	closed	closed	ADJ
ejpam-131	212	15	if	if	SCONJ
ejpam-131	212	16	f	f	PROPN
ejpam-131	212	17	(	(	PUNCT
ejpam-131	212	18	a	a	PRON
ejpam-131	212	19	)	)	PUNCT
ejpam-131	212	20	ωb	ωb	NOUN
ejpam-131	212	21	-	-	PUNCT
ejpam-131	212	22	closed	close	VERB
ejpam-131	212	23	in	in	ADP
ejpam-131	212	24	y	y	PROPN
ejpam-131	212	25	for	for	ADP
ejpam-131	212	26	each	each	DET
ejpam-131	212	27	b	b	NOUN
ejpam-131	212	28	-	-	PUNCT
ejpam-131	212	29	closed	closed	ADJ
ejpam-131	212	30	set	set	NOUN
ejpam-131	212	31	a	a	PRON
ejpam-131	212	32	of	of	ADP
ejpam-131	212	33	x	x	SYM
ejpam-131	212	34	.	.	PUNCT
ejpam-131	213	1	theorem	theorem	NOUN
ejpam-131	213	2	3.13	3.13	NUM
ejpam-131	213	3	.	.	PUNCT
ejpam-131	214	1	if	if	SCONJ
ejpam-131	214	2	f	f	PROPN
ejpam-131	214	3	:	:	PUNCT
ejpam-131	214	4	x	x	X
ejpam-131	214	5	→	→	SYM
ejpam-131	214	6	y	y	PROPN
ejpam-131	214	7	is	be	AUX
ejpam-131	214	8	an	an	DET
ejpam-131	214	9	ωb	ωb	NOUN
ejpam-131	214	10	-	-	PUNCT
ejpam-131	214	11	closed	close	VERB
ejpam-131	214	12	surjection	surjection	NOUN
ejpam-131	214	13	such	such	ADJ
ejpam-131	214	14	that	that	SCONJ
ejpam-131	214	15	f	f	PROPN
ejpam-131	214	16	−1(y	−1(y	PROPN
ejpam-131	214	17	)	)	PUNCT
ejpam-131	214	18	is	be	AUX
ejpam-131	214	19	b	b	NOUN
ejpam-131	214	20	-	-	PUNCT
ejpam-131	214	21	lindelöf	lindelöf	NOUN
ejpam-131	214	22	relative	relative	NOUN
ejpam-131	214	23	to	to	ADP
ejpam-131	214	24	x	x	PRON
ejpam-131	214	25	and	and	CCONJ
ejpam-131	214	26	y	y	PROPN
ejpam-131	214	27	is	be	AUX
ejpam-131	214	28	b	b	NOUN
ejpam-131	214	29	-	-	PUNCT
ejpam-131	214	30	lindelöf	lindelöf	NOUN
ejpam-131	214	31	,	,	PUNCT
ejpam-131	214	32	then	then	ADV
ejpam-131	214	33	x	x	PUNCT
ejpam-131	214	34	is	be	AUX
ejpam-131	214	35	b	b	NOUN
ejpam-131	214	36	-	-	PUNCT
ejpam-131	214	37	lindelöf	lindelöf	NOUN
ejpam-131	214	38	.	.	PUNCT
ejpam-131	215	1	proof	proof	NOUN
ejpam-131	215	2	.	.	PUNCT
ejpam-131	216	1	let	let	VERB
ejpam-131	216	2	{	{	PUNCT
ejpam-131	216	3	uα	uα	NOUN
ejpam-131	216	4	:	:	PUNCT
ejpam-131	216	5	α	α	PROPN
ejpam-131	216	6	∈	∈	PROPN
ejpam-131	216	7	λ	λ	NOUN
ejpam-131	216	8	}	}	PUNCT
ejpam-131	216	9	be	be	VERB
ejpam-131	216	10	any	any	DET
ejpam-131	216	11	b	b	NOUN
ejpam-131	216	12	-	-	PUNCT
ejpam-131	216	13	open	open	ADJ
ejpam-131	216	14	cover	cover	NOUN
ejpam-131	216	15	of	of	ADP
ejpam-131	216	16	x	x	X
ejpam-131	216	17	.	.	PUNCT
ejpam-131	217	1	for	for	ADP
ejpam-131	217	2	each	each	DET
ejpam-131	217	3	y	y	PROPN
ejpam-131	217	4	∈	∈	PROPN
ejpam-131	217	5	y	y	PROPN
ejpam-131	217	6	,	,	PUNCT
ejpam-131	217	7	f	f	PROPN
ejpam-131	217	8	−1(y	−1(y	PROPN
ejpam-131	217	9	)	)	PUNCT
ejpam-131	217	10	is	be	AUX
ejpam-131	217	11	b	b	NOUN
ejpam-131	217	12	-	-	PUNCT
ejpam-131	217	13	lindelöf	lindelöf	NOUN
ejpam-131	217	14	relative	relative	NOUN
ejpam-131	217	15	to	to	ADP
ejpam-131	217	16	x	x	PUNCT
ejpam-131	217	17	and	and	CCONJ
ejpam-131	217	18	there	there	PRON
ejpam-131	217	19	exists	exist	VERB
ejpam-131	217	20	a	a	DET
ejpam-131	217	21	countable	countable	ADJ
ejpam-131	217	22	subset	subset	NOUN
ejpam-131	217	23	λ1(y	λ1(y	X
ejpam-131	217	24	)	)	PUNCT
ejpam-131	217	25	of	of	ADP
ejpam-131	217	26	λ	λ	PROPN
ejpam-131	217	27	such	such	ADJ
ejpam-131	217	28	that	that	SCONJ
ejpam-131	217	29	f	f	PROPN
ejpam-131	217	30	−1(y	−1(y	PROPN
ejpam-131	217	31	)	)	PUNCT
ejpam-131	218	1	⊂	⊂	PROPN
ejpam-131	218	2	∪{uα	∪{uα	NUM
ejpam-131	218	3	:	:	PUNCT
ejpam-131	218	4	α	α	PROPN
ejpam-131	218	5	∈	∈	PROPN
ejpam-131	218	6	λ1(y	λ1(y	PROPN
ejpam-131	218	7	)	)	PUNCT
ejpam-131	218	8	}	}	PUNCT
ejpam-131	218	9	.	.	PUNCT
ejpam-131	219	1	now	now	ADV
ejpam-131	219	2	we	we	PRON
ejpam-131	219	3	put	put	VERB
ejpam-131	219	4	u(y	u(y	NOUN
ejpam-131	219	5	)	)	PUNCT
ejpam-131	219	6	=	=	SYM
ejpam-131	220	1	∪{uα	∪{uα	NOUN
ejpam-131	220	2	:	:	PUNCT
ejpam-131	220	3	α	α	PROPN
ejpam-131	220	4	∈	∈	PROPN
ejpam-131	220	5	λ1(y	λ1(y	PROPN
ejpam-131	220	6	)	)	PUNCT
ejpam-131	220	7	}	}	PUNCT
ejpam-131	220	8	and	and	CCONJ
ejpam-131	220	9	v	v	X
ejpam-131	220	10	(	(	PUNCT
ejpam-131	220	11	y	y	NOUN
ejpam-131	220	12	)	)	PUNCT
ejpam-131	220	13	=	=	SYM
ejpam-131	221	1	y	y	PROPN
ejpam-131	221	2	−	−	PROPN
ejpam-131	221	3	f	f	PROPN
ejpam-131	221	4	(	(	PUNCT
ejpam-131	221	5	x	x	PROPN
ejpam-131	221	6	−	−	PROPN
ejpam-131	221	7	u(y	u(y	NOUN
ejpam-131	221	8	)	)	PUNCT
ejpam-131	221	9	)	)	PUNCT
ejpam-131	221	10	.	.	PUNCT
ejpam-131	222	1	then	then	ADV
ejpam-131	222	2	,	,	PUNCT
ejpam-131	222	3	since	since	SCONJ
ejpam-131	222	4	f	f	PROPN
ejpam-131	222	5	is	be	AUX
ejpam-131	222	6	ωb	ωb	NOUN
ejpam-131	222	7	-	-	PUNCT
ejpam-131	222	8	closed	closed	ADJ
ejpam-131	222	9	,	,	PUNCT
ejpam-131	222	10	v	v	NOUN
ejpam-131	222	11	(	(	PUNCT
ejpam-131	222	12	y	y	NOUN
ejpam-131	222	13	)	)	PUNCT
ejpam-131	222	14	is	be	AUX
ejpam-131	222	15	an	an	DET
ejpam-131	222	16	ωb	ωb	NOUN
ejpam-131	222	17	-	-	PUNCT
ejpam-131	222	18	open	open	NOUN
ejpam-131	222	19	set	set	NOUN
ejpam-131	222	20	in	in	ADP
ejpam-131	222	21	y	y	NOUN
ejpam-131	222	22	containing	contain	VERB
ejpam-131	222	23	y	y	PRON
ejpam-131	222	24	such	such	ADJ
ejpam-131	222	25	that	that	SCONJ
ejpam-131	222	26	f	f	PROPN
ejpam-131	222	27	−1(v	−1(v	X
ejpam-131	222	28	(	(	PUNCT
ejpam-131	222	29	y	y	NOUN
ejpam-131	222	30	)	)	PUNCT
ejpam-131	222	31	)	)	PUNCT
ejpam-131	223	1	⊂	⊂	PROPN
ejpam-131	223	2	u(y	u(y	PROPN
ejpam-131	223	3	)	)	PUNCT
ejpam-131	223	4	.	.	PUNCT
ejpam-131	224	1	since	since	SCONJ
ejpam-131	224	2	v	v	NOUN
ejpam-131	224	3	(	(	PUNCT
ejpam-131	224	4	y	y	NOUN
ejpam-131	224	5	)	)	PUNCT
ejpam-131	224	6	is	be	AUX
ejpam-131	224	7	ωb	ωb	NOUN
ejpam-131	224	8	-	-	PUNCT
ejpam-131	224	9	open	open	ADJ
ejpam-131	224	10	,	,	PUNCT
ejpam-131	224	11	there	there	PRON
ejpam-131	224	12	exists	exist	VERB
ejpam-131	224	13	a	a	DET
ejpam-131	224	14	b	b	NOUN
ejpam-131	224	15	-	-	PUNCT
ejpam-131	224	16	open	open	ADJ
ejpam-131	224	17	set	set	NOUN
ejpam-131	224	18	w	w	PROPN
ejpam-131	224	19	(	(	PUNCT
ejpam-131	224	20	y	y	NOUN
ejpam-131	224	21	)	)	PUNCT
ejpam-131	224	22	containing	contain	VERB
ejpam-131	224	23	y	y	PRON
ejpam-131	224	24	such	such	ADJ
ejpam-131	224	25	that	that	PRON
ejpam-131	224	26	w	w	NOUN
ejpam-131	224	27	(	(	PUNCT
ejpam-131	224	28	y)−	y)−	PROPN
ejpam-131	224	29	v	v	NOUN
ejpam-131	224	30	(	(	PUNCT
ejpam-131	224	31	y	y	NOUN
ejpam-131	224	32	)	)	PUNCT
ejpam-131	224	33	is	be	AUX
ejpam-131	224	34	a	a	DET
ejpam-131	224	35	countable	countable	ADJ
ejpam-131	224	36	set	set	NOUN
ejpam-131	224	37	.	.	PUNCT
ejpam-131	225	1	for	for	ADP
ejpam-131	225	2	each	each	DET
ejpam-131	225	3	y	y	PROPN
ejpam-131	225	4	∈	∈	PROPN
ejpam-131	225	5	y	y	PROPN
ejpam-131	225	6	,	,	PUNCT
ejpam-131	225	7	we	we	PRON
ejpam-131	225	8	have	have	VERB
ejpam-131	225	9	w	w	PROPN
ejpam-131	225	10	(	(	PUNCT
ejpam-131	225	11	y)⊂	y)⊂	PROPN
ejpam-131	225	12	(	(	PUNCT
ejpam-131	225	13	w	w	NOUN
ejpam-131	225	14	(	(	PUNCT
ejpam-131	225	15	y)−	y)−	PROPN
ejpam-131	225	16	v	v	NOUN
ejpam-131	225	17	(	(	PUNCT
ejpam-131	225	18	y))∪	y))∪	NOUN
ejpam-131	225	19	v	v	X
ejpam-131	225	20	(	(	PUNCT
ejpam-131	225	21	y	y	NOUN
ejpam-131	225	22	)	)	PUNCT
ejpam-131	225	23	and	and	CCONJ
ejpam-131	225	24	hence	hence	ADV
ejpam-131	225	25	f	f	PROPN
ejpam-131	226	1	−1(w	−1(w	X
ejpam-131	226	2	(	(	PUNCT
ejpam-131	226	3	y))⊂	y))⊂	NOUN
ejpam-131	226	4	f	f	PROPN
ejpam-131	226	5	−1(w	−1(w	X
ejpam-131	226	6	(	(	PUNCT
ejpam-131	226	7	y)−	y)−	PROPN
ejpam-131	226	8	v	v	NOUN
ejpam-131	226	9	(	(	PUNCT
ejpam-131	226	10	y))∪	y))∪	ADP
ejpam-131	226	11	f	f	PROPN
ejpam-131	226	12	−1(v	−1(v	PROPN
ejpam-131	226	13	(	(	PUNCT
ejpam-131	226	14	y	y	NOUN
ejpam-131	226	15	)	)	PUNCT
ejpam-131	226	16	)	)	PUNCT
ejpam-131	227	1	⊂	⊂	PROPN
ejpam-131	227	2	f	f	X
ejpam-131	228	1	−1(w	−1(w	ADV
ejpam-131	228	2	(	(	PUNCT
ejpam-131	228	3	y)−	y)−	PROPN
ejpam-131	228	4	v	v	NOUN
ejpam-131	228	5	(	(	PUNCT
ejpam-131	228	6	y))∪	y))∪	PROPN
ejpam-131	228	7	u(y	u(y	PROPN
ejpam-131	228	8	)	)	PUNCT
ejpam-131	228	9	.	.	PUNCT
ejpam-131	229	1	since	since	SCONJ
ejpam-131	229	2	w	w	PROPN
ejpam-131	229	3	(	(	PUNCT
ejpam-131	229	4	y)−	y)−	PROPN
ejpam-131	229	5	v	v	NOUN
ejpam-131	229	6	(	(	PUNCT
ejpam-131	229	7	y	y	NOUN
ejpam-131	229	8	)	)	PUNCT
ejpam-131	229	9	is	be	AUX
ejpam-131	229	10	a	a	DET
ejpam-131	229	11	countable	countable	ADJ
ejpam-131	229	12	set	set	NOUN
ejpam-131	229	13	and	and	CCONJ
ejpam-131	229	14	f	f	PROPN
ejpam-131	229	15	−1(y	−1(y	PROPN
ejpam-131	229	16	)	)	PUNCT
ejpam-131	229	17	is	be	AUX
ejpam-131	229	18	b	b	NOUN
ejpam-131	229	19	-	-	PUNCT
ejpam-131	229	20	lindelöf	lindelöf	NOUN
ejpam-131	229	21	relative	relative	NOUN
ejpam-131	229	22	to	to	ADP
ejpam-131	229	23	x	x	PRON
ejpam-131	229	24	,	,	PUNCT
ejpam-131	229	25	there	there	PRON
ejpam-131	229	26	exists	exist	VERB
ejpam-131	229	27	a	a	DET
ejpam-131	229	28	countable	countable	ADJ
ejpam-131	229	29	set	set	NOUN
ejpam-131	229	30	λ2(y	λ2(y	NOUN
ejpam-131	229	31	)	)	PUNCT
ejpam-131	229	32	of	of	ADP
ejpam-131	229	33	λ	λ	PROPN
ejpam-131	229	34	such	such	ADJ
ejpam-131	229	35	that	that	SCONJ
ejpam-131	229	36	f	f	PROPN
ejpam-131	229	37	−1(w	−1(w	X
ejpam-131	229	38	(	(	PUNCT
ejpam-131	229	39	y)−	y)−	PROPN
ejpam-131	229	40	v	v	NOUN
ejpam-131	229	41	(	(	PUNCT
ejpam-131	229	42	y))⊂	y))⊂	NOUN
ejpam-131	229	43	∪{uα	∪{uα	PROPN
ejpam-131	229	44	:	:	PUNCT
ejpam-131	229	45	α	α	PROPN
ejpam-131	229	46	∈	∈	PROPN
ejpam-131	230	1	λ2(y	λ2(y	VERB
ejpam-131	230	2	)	)	PUNCT
ejpam-131	230	3	}	}	PUNCT
ejpam-131	230	4	and	and	CCONJ
ejpam-131	230	5	hence	hence	ADV
ejpam-131	230	6	f	f	PROPN
ejpam-131	230	7	−1(w	−1(w	INTJ
ejpam-131	230	8	(	(	PUNCT
ejpam-131	230	9	y))⊂	y))⊂	NOUN
ejpam-131	231	1	[	[	X
ejpam-131	231	2	∪{uα	∪{uα	NUM
ejpam-131	231	3	:	:	PUNCT
ejpam-131	231	4	α	α	NUM
ejpam-131	231	5	∈	∈	NOUN
ejpam-131	231	6	λ2(y)}]∪	λ2(y)}]∪	X
ejpam-131	232	1	[	[	X
ejpam-131	232	2	u(y	u(y	NOUN
ejpam-131	232	3	)	)	PUNCT
ejpam-131	232	4	]	]	PUNCT
ejpam-131	232	5	.	.	PUNCT
ejpam-131	233	1	since	since	SCONJ
ejpam-131	233	2	{	{	PUNCT
ejpam-131	233	3	w	w	PROPN
ejpam-131	233	4	(	(	PUNCT
ejpam-131	233	5	y	y	NOUN
ejpam-131	233	6	)	)	PUNCT
ejpam-131	233	7	:	:	PUNCT
ejpam-131	233	8	y	y	PROPN
ejpam-131	233	9	∈	∈	PROPN
ejpam-131	233	10	y	y	PROPN
ejpam-131	233	11	}	}	PUNCT
ejpam-131	233	12	is	be	AUX
ejpam-131	233	13	a	a	DET
ejpam-131	233	14	b	b	NOUN
ejpam-131	233	15	-	-	PUNCT
ejpam-131	233	16	open	open	ADJ
ejpam-131	233	17	cover	cover	NOUN
ejpam-131	233	18	of	of	ADP
ejpam-131	233	19	the	the	DET
ejpam-131	233	20	b	b	NOUN
ejpam-131	233	21	-	-	PUNCT
ejpam-131	233	22	lindelöf	lindelöf	NOUN
ejpam-131	233	23	space	space	NOUN
ejpam-131	233	24	y	y	PROPN
ejpam-131	233	25	,	,	PUNCT
ejpam-131	233	26	there	there	PRON
ejpam-131	233	27	exist	exist	VERB
ejpam-131	233	28	countable	countable	ADJ
ejpam-131	233	29	points	point	NOUN
ejpam-131	233	30	of	of	ADP
ejpam-131	233	31	y	y	PROPN
ejpam-131	233	32	,	,	PUNCT
ejpam-131	233	33	say	say	INTJ
ejpam-131	233	34	,	,	PUNCT
ejpam-131	233	35	y1	y1	PROPN
ejpam-131	233	36	,	,	PUNCT
ejpam-131	233	37	y2	y2	PROPN
ejpam-131	233	38	,	,	PUNCT
ejpam-131	233	39	...	...	PUNCT
ejpam-131	233	40	,	,	PUNCT
ejpam-131	233	41	yn	yn	PROPN
ejpam-131	233	42	,	,	PUNCT
ejpam-131	233	43	...	...	PUNCT
ejpam-131	233	44	such	such	ADJ
ejpam-131	233	45	that	that	SCONJ
ejpam-131	233	46	y	y	PROPN
ejpam-131	233	47	=	=	SYM
ejpam-131	233	48	∪{w	∪{w	PROPN
ejpam-131	233	49	(	(	PUNCT
ejpam-131	233	50	yi	yi	PROPN
ejpam-131	233	51	)	)	PUNCT
ejpam-131	233	52	:	:	PUNCT
ejpam-131	233	53	i	i	PRON
ejpam-131	233	54	∈	∈	PROPN
ejpam-131	233	55	n	n	CCONJ
ejpam-131	233	56	}	}	PUNCT
ejpam-131	233	57	.	.	PUNCT
ejpam-131	234	1	therefore	therefore	ADV
ejpam-131	234	2	,	,	PUNCT
ejpam-131	234	3	we	we	PRON
ejpam-131	234	4	obtain	obtain	VERB
ejpam-131	234	5	x	x	X
ejpam-131	234	6	=	=	PUNCT
ejpam-131	234	7	∪i∈n	∪i∈n	X
ejpam-131	234	8	f	f	PROPN
ejpam-131	234	9	−1(w	−1(w	X
ejpam-131	234	10	(	(	PUNCT
ejpam-131	234	11	yi	yi	NOUN
ejpam-131	234	12	)	)	PUNCT
ejpam-131	234	13	)	)	PUNCT
ejpam-131	235	1	=	=	SYM
ejpam-131	235	2	∪i∈n[∪α∈λ2(yi)uα)∪	∪i∈n[∪α∈λ2(yi)uα)∪	PROPN
ejpam-131	235	3	(	(	PUNCT
ejpam-131	235	4	∪α∈λ1(yi)uα	∪α∈λ1(yi)uα	NOUN
ejpam-131	235	5	)	)	PUNCT
ejpam-131	235	6	]	]	PUNCT
ejpam-131	236	1	=	=	PUNCT
ejpam-131	236	2	∪{uα	∪{uα	NOUN
ejpam-131	236	3	:	:	PUNCT
ejpam-131	236	4	α	α	X
ejpam-131	236	5	∈	∈	PROPN
ejpam-131	236	6	λ1(yi)∪λ2(yi	λ1(yi)∪λ2(yi	PROPN
ejpam-131	236	7	)	)	PUNCT
ejpam-131	236	8	,	,	PUNCT
ejpam-131	236	9	i	i	PRON
ejpam-131	236	10	∈	∈	PROPN
ejpam-131	236	11	n	n	CCONJ
ejpam-131	236	12	}	}	PUNCT
ejpam-131	236	13	.	.	PUNCT
ejpam-131	237	1	this	this	PRON
ejpam-131	237	2	shows	show	VERB
ejpam-131	237	3	that	that	SCONJ
ejpam-131	237	4	x	x	PRON
ejpam-131	237	5	is	be	AUX
ejpam-131	237	6	b	b	NOUN
ejpam-131	237	7	-	-	PUNCT
ejpam-131	237	8	lindelöf	lindelöf	NOUN
ejpam-131	237	9	.	.	PUNCT
ejpam-131	238	1	references	reference	NOUN
ejpam-131	238	2	9	9	NUM
ejpam-131	238	3	references	reference	NOUN
ejpam-131	238	4	[	[	X
ejpam-131	238	5	1	1	NUM
ejpam-131	238	6	]	]	X
ejpam-131	238	7	a.al	a.al	PROPN
ejpam-131	238	8	-	-	NOUN
ejpam-131	238	9	omari	omari	ADJ
ejpam-131	238	10	and	and	CCONJ
ejpam-131	238	11	m.s.m.noorani	m.s.m.noorani	PROPN
ejpam-131	238	12	,	,	PUNCT
ejpam-131	238	13	"	"	PUNCT
ejpam-131	238	14	regular	regular	ADJ
ejpam-131	238	15	generalized	generalize	VERB
ejpam-131	238	16	ω	ω	VERB
ejpam-131	238	17	-	-	PUNCT
ejpam-131	238	18	closed	closed	ADJ
ejpam-131	238	19	sets	set	NOUN
ejpam-131	238	20	"	"	PUNCT
ejpam-131	238	21	,	,	PUNCT
ejpam-131	238	22	internat	internat	PROPN
ejpam-131	238	23	.	.	PUNCT
ejpam-131	239	1	j.	j.	PROPN
ejpam-131	239	2	math	math	PROPN
ejpam-131	239	3	.	.	PUNCT
ejpam-131	240	1	math	math	NOUN
ejpam-131	240	2	.	.	PUNCT
ejpam-131	241	1	sci	sci	PROPN
ejpam-131	241	2	.	.	PUNCT
ejpam-131	241	3	vol	vol	NOUN
ejpam-131	241	4	.	.	PROPN
ejpam-131	242	1	2007	2007	NUM
ejpam-131	242	2	,	,	PUNCT
ejpam-131	242	3	article	article	NOUN
ejpam-131	242	4	i	i	PROPN
ejpam-131	242	5	d	d	PROPN
ejpam-131	242	6	16292	16292	NUM
ejpam-131	242	7	,	,	PUNCT
ejpam-131	242	8	11	11	NUM
ejpam-131	242	9	page	page	NOUN
ejpam-131	242	10	.	.	PUNCT
ejpam-131	243	1	[	[	X
ejpam-131	243	2	2	2	NUM
ejpam-131	243	3	]	]	X
ejpam-131	243	4	a.al	a.al	PROPN
ejpam-131	243	5	-	-	NOUN
ejpam-131	243	6	omari	omari	ADJ
ejpam-131	243	7	and	and	CCONJ
ejpam-131	243	8	m.s.m.noorani	m.s.m.noorani	PROPN
ejpam-131	243	9	,	,	PUNCT
ejpam-131	243	10	"	"	PUNCT
ejpam-131	243	11	on	on	ADP
ejpam-131	243	12	generalzed	generalzed	ADJ
ejpam-131	243	13	b	b	X
ejpam-131	243	14	-	-	PUNCT
ejpam-131	243	15	closed	closed	ADJ
ejpam-131	243	16	sets	set	NOUN
ejpam-131	243	17	"	"	PUNCT
ejpam-131	243	18	,	,	PUNCT
ejpam-131	243	19	bull	bull	NOUN
ejpam-131	243	20	.	.	PUNCT
ejpam-131	244	1	malaysian	malaysian	ADJ
ejpam-131	244	2	math	math	PROPN
ejpam-131	244	3	.	.	PUNCT
ejpam-131	245	1	sc	sc	PROPN
ejpam-131	245	2	.	.	PROPN
ejpam-131	245	3	soc	soc	PROPN
ejpam-131	245	4	.	.	PUNCT
ejpam-131	246	1	appear	appear	VERB
ejpam-131	246	2	in	in	ADP
ejpam-131	246	3	32(1)(2009	32(1)(2009	NUM
ejpam-131	246	4	)	)	PUNCT
ejpam-131	246	5	.	.	PUNCT
ejpam-131	247	1	[	[	X
ejpam-131	247	2	3	3	NUM
ejpam-131	247	3	]	]	SYM
ejpam-131	247	4	d.andrijević	d.andrijević	PROPN
ejpam-131	247	5	,	,	PUNCT
ejpam-131	247	6	"	"	PUNCT
ejpam-131	247	7	semi	semi	ADJ
ejpam-131	247	8	-	-	ADJ
ejpam-131	247	9	preopen	preopen	ADJ
ejpam-131	247	10	sets	set	NOUN
ejpam-131	247	11	"	"	PUNCT
ejpam-131	247	12	,	,	PUNCT
ejpam-131	247	13	mat	mat	PROPN
ejpam-131	247	14	.	.	PROPN
ejpam-131	247	15	vesnik	vesnik	PROPN
ejpam-131	247	16	38(1)(1986	38(1)(1986	NUM
ejpam-131	247	17	)	)	PUNCT
ejpam-131	247	18	,	,	PUNCT
ejpam-131	247	19	24	24	NUM
ejpam-131	247	20	-	-	SYM
ejpam-131	247	21	32	32	NUM
ejpam-131	247	22	.	.	PUNCT
ejpam-131	248	1	[	[	X
ejpam-131	248	2	4	4	NUM
ejpam-131	248	3	]	]	SYM
ejpam-131	248	4	d.andrijević	d.andrijević	PROPN
ejpam-131	248	5	,	,	PUNCT
ejpam-131	248	6	"	"	PUNCT
ejpam-131	248	7	on	on	ADP
ejpam-131	248	8	b	b	X
ejpam-131	248	9	-	-	PUNCT
ejpam-131	248	10	open	open	ADJ
ejpam-131	248	11	sets	set	NOUN
ejpam-131	248	12	"	"	PUNCT
ejpam-131	248	13	,	,	PUNCT
ejpam-131	248	14	mat	mat	PROPN
ejpam-131	248	15	.	.	PROPN
ejpam-131	248	16	vesnik	vesnik	PROPN
ejpam-131	248	17	48(1996	48(1996	PROPN
ejpam-131	248	18	)	)	PUNCT
ejpam-131	248	19	,	,	PUNCT
ejpam-131	248	20	59	59	NUM
ejpam-131	248	21	-	-	SYM
ejpam-131	248	22	64	64	NUM
ejpam-131	248	23	.	.	PUNCT
ejpam-131	249	1	[	[	X
ejpam-131	249	2	5	5	NUM
ejpam-131	249	3	]	]	PUNCT
ejpam-131	249	4	m.caldas	m.caldas	PROPN
ejpam-131	249	5	and	and	CCONJ
ejpam-131	249	6	s.	s.	PROPN
ejpam-131	249	7	jafari	jafari	PROPN
ejpam-131	249	8	,	,	PUNCT
ejpam-131	249	9	"	"	PUNCT
ejpam-131	249	10	on	on	ADP
ejpam-131	249	11	some	some	DET
ejpam-131	249	12	applications	application	NOUN
ejpam-131	249	13	of	of	ADP
ejpam-131	249	14	b	b	NOUN
ejpam-131	249	15	-	-	PUNCT
ejpam-131	249	16	open	open	ADJ
ejpam-131	249	17	sets	set	NOUN
ejpam-131	249	18	in	in	ADP
ejpam-131	249	19	topological	topological	ADJ
ejpam-131	249	20	spaces	space	NOUN
ejpam-131	249	21	"	"	PUNCT
ejpam-131	249	22	,	,	PUNCT
ejpam-131	249	23	kochi	kochi	PROPN
ejpam-131	249	24	j.	j.	PROPN
ejpam-131	249	25	math	math	PROPN
ejpam-131	249	26	.	.	PUNCT
ejpam-131	250	1	2(2007	2(2007	X
ejpam-131	250	2	)	)	PUNCT
ejpam-131	250	3	,	,	PUNCT
ejpam-131	250	4	11	11	NUM
ejpam-131	250	5	-	-	SYM
ejpam-131	250	6	19	19	NUM
ejpam-131	250	7	.	.	PUNCT
ejpam-131	251	1	[	[	X
ejpam-131	251	2	6	6	NUM
ejpam-131	251	3	]	]	PUNCT
ejpam-131	251	4	e.ekici	e.ekici	NOUN
ejpam-131	251	5	and	and	CCONJ
ejpam-131	251	6	m.caldas	m.calda	NOUN
ejpam-131	251	7	,	,	PUNCT
ejpam-131	251	8	"	"	PUNCT
ejpam-131	251	9	slightly	slightly	ADV
ejpam-131	251	10	γ	γ	ADJ
ejpam-131	251	11	-	-	ADJ
ejpam-131	251	12	continuous	continuous	ADJ
ejpam-131	251	13	functions	function	NOUN
ejpam-131	251	14	"	"	PUNCT
ejpam-131	251	15	,	,	PUNCT
ejpam-131	251	16	bol	bol	NOUN
ejpam-131	251	17	.	.	PUNCT
ejpam-131	251	18	soc	soc	PROPN
ejpam-131	251	19	.	.	PUNCT
ejpam-131	252	1	paran	paran	PROPN
ejpam-131	252	2	.	.	PUNCT
ejpam-131	253	1	mat	mat	PROPN
ejpam-131	253	2	.	.	NOUN
ejpam-131	253	3	22(2)(2004	22(2)(2004	NUM
ejpam-131	253	4	)	)	PUNCT
ejpam-131	253	5	,	,	PUNCT
ejpam-131	253	6	63	63	NUM
ejpam-131	253	7	-	-	SYM
ejpam-131	253	8	74	74	NUM
ejpam-131	253	9	.	.	PUNCT
ejpam-131	254	1	[	[	X
ejpam-131	254	2	7	7	NUM
ejpam-131	254	3	]	]	X
ejpam-131	254	4	a.a	a.a	PROPN
ejpam-131	254	5	.	.	PROPN
ejpam-131	254	6	el	el	PROPN
ejpam-131	254	7	-	-	PUNCT
ejpam-131	254	8	atik	atik	PROPN
ejpam-131	254	9	,	,	PUNCT
ejpam-131	254	10	a	a	DET
ejpam-131	254	11	study	study	NOUN
ejpam-131	254	12	of	of	ADP
ejpam-131	254	13	some	some	DET
ejpam-131	254	14	types	type	NOUN
ejpam-131	254	15	of	of	ADP
ejpam-131	254	16	mappings	mapping	NOUN
ejpam-131	254	17	on	on	ADP
ejpam-131	254	18	topological	topological	ADJ
ejpam-131	254	19	spaces	space	NOUN
ejpam-131	254	20	,	,	PUNCT
ejpam-131	254	21	m.	m.	PROPN
ejpam-131	254	22	sc	sc	PROPN
ejpam-131	254	23	.	.	PUNCT
ejpam-131	255	1	thesis	thesis	PROPN
ejpam-131	255	2	,	,	PUNCT
ejpam-131	255	3	tanta	tanta	PROPN
ejpam-131	255	4	univ	univ	PROPN
ejpam-131	255	5	.	.	PROPN
ejpam-131	255	6	,	,	PUNCT
ejpam-131	255	7	egypt	egypt	PROPN
ejpam-131	255	8	,	,	PUNCT
ejpam-131	255	9	1997	1997	NUM
ejpam-131	255	10	.	.	PUNCT
ejpam-131	256	1	[	[	X
ejpam-131	256	2	8	8	NUM
ejpam-131	256	3	]	]	X
ejpam-131	256	4	h.z	h.z	PROPN
ejpam-131	256	5	.	.	PROPN
ejpam-131	256	6	hdeib	hdeib	PROPN
ejpam-131	256	7	,	,	PUNCT
ejpam-131	256	8	"	"	PUNCT
ejpam-131	256	9	ω	ω	VERB
ejpam-131	256	10	-	-	PUNCT
ejpam-131	256	11	closed	closed	ADJ
ejpam-131	256	12	mapping	mapping	NOUN
ejpam-131	256	13	"	"	PUNCT
ejpam-131	256	14	,	,	PUNCT
ejpam-131	256	15	rev	rev	PROPN
ejpam-131	256	16	.	.	PROPN
ejpam-131	256	17	colomb	colomb	PROPN
ejpam-131	256	18	.	.	PUNCT
ejpam-131	257	1	mat	mat	PROPN
ejpam-131	257	2	.	.	PROPN
ejpam-131	258	1	16	16	NUM
ejpam-131	258	2	(	(	PUNCT
ejpam-131	258	3	1	1	NUM
ejpam-131	258	4	-	-	SYM
ejpam-131	258	5	2	2	NUM
ejpam-131	258	6	)	)	PUNCT
ejpam-131	258	7	(	(	PUNCT
ejpam-131	258	8	1982	1982	NUM
ejpam-131	258	9	)	)	PUNCT
ejpam-131	258	10	,	,	PUNCT
ejpam-131	258	11	65	65	NUM
ejpam-131	258	12	-	-	SYM
ejpam-131	258	13	78	78	NUM
ejpam-131	258	14	.	.	PUNCT
ejpam-131	259	1	[	[	X
ejpam-131	259	2	9	9	NUM
ejpam-131	259	3	]	]	X
ejpam-131	259	4	h.z	h.z	PROPN
ejpam-131	259	5	.	.	PROPN
ejpam-131	259	6	hdeib	hdeib	PROPN
ejpam-131	259	7	,	,	PUNCT
ejpam-131	259	8	"	"	PUNCT
ejpam-131	259	9	ω	ω	ADJ
ejpam-131	259	10	-	-	ADJ
ejpam-131	259	11	continuous	continuous	ADJ
ejpam-131	259	12	functions	function	NOUN
ejpam-131	259	13	"	"	PUNCT
ejpam-131	259	14	,	,	PUNCT
ejpam-131	259	15	dirasat	dirasat	PROPN
ejpam-131	259	16	journal	journal	PROPN
ejpam-131	259	17	16	16	NUM
ejpam-131	259	18	(	(	PUNCT
ejpam-131	259	19	2	2	NUM
ejpam-131	259	20	)	)	PUNCT
ejpam-131	259	21	(	(	PUNCT
ejpam-131	259	22	1989	1989	NUM
ejpam-131	259	23	)	)	PUNCT
ejpam-131	259	24	,	,	PUNCT
ejpam-131	259	25	136	136	NUM
ejpam-131	259	26	-	-	SYM
ejpam-131	259	27	142	142	NUM
ejpam-131	259	28	.	.	PUNCT
ejpam-131	260	1	[	[	X
ejpam-131	260	2	10	10	NUM
ejpam-131	260	3	]	]	X
ejpam-131	260	4	a.a.nasef	a.a.nasef	NOUN
ejpam-131	260	5	,	,	PUNCT
ejpam-131	260	6	"	"	PUNCT
ejpam-131	260	7	on	on	ADP
ejpam-131	260	8	b	b	X
ejpam-131	260	9	-	-	PUNCT
ejpam-131	260	10	locally	locally	ADV
ejpam-131	260	11	closed	close	VERB
ejpam-131	260	12	sets	set	NOUN
ejpam-131	260	13	and	and	CCONJ
ejpam-131	260	14	related	related	ADJ
ejpam-131	260	15	topic	topic	NOUN
ejpam-131	260	16	"	"	PUNCT
ejpam-131	260	17	,	,	PUNCT
ejpam-131	260	18	chaos	chaos	NOUN
ejpam-131	260	19	solitions	solition	NOUN
ejpam-131	260	20	fractals	fractal	NOUN
ejpam-131	260	21	12	12	NUM
ejpam-131	260	22	(	(	PUNCT
ejpam-131	260	23	2001),19091915	2001),19091915	NUM
ejpam-131	260	24	.	.	PUNCT
ejpam-131	261	1	[	[	X
ejpam-131	261	2	11	11	NUM
ejpam-131	261	3	]	]	SYM
ejpam-131	261	4	j.h.park	j.h.park	NOUN
ejpam-131	261	5	,	,	PUNCT
ejpam-131	261	6	"	"	PUNCT
ejpam-131	261	7	strongly	strongly	ADV
ejpam-131	261	8	θ	θ	X
ejpam-131	261	9	-b	-b	PUNCT
ejpam-131	261	10	continuous	continuous	ADJ
ejpam-131	261	11	functions	function	NOUN
ejpam-131	261	12	"	"	PUNCT
ejpam-131	261	13	,	,	PUNCT
ejpam-131	261	14	acta	acta	PROPN
ejpam-131	261	15	math	math	PROPN
ejpam-131	261	16	.	.	PUNCT
ejpam-131	262	1	hungar	hungar	NOUN
ejpam-131	262	2	.	.	PUNCT
ejpam-131	263	1	110(4)(2006	110(4)(2006	X
ejpam-131	263	2	)	)	PUNCT
ejpam-131	263	3	,	,	PUNCT
ejpam-131	263	4	347	347	NUM
ejpam-131	263	5	-	-	SYM
ejpam-131	263	6	359	359	NUM
ejpam-131	263	7	.	.	PUNCT