id	sid	tid	token	lemma	pos
ejpam-3438	1	1	european	european	PROPN
ejpam-3438	1	2	journal	journal	PROPN
ejpam-3438	1	3	of	of	ADP
ejpam-3438	1	4	pure	pure	ADJ
ejpam-3438	1	5	and	and	CCONJ
ejpam-3438	1	6	applied	apply	VERB
ejpam-3438	1	7	mathematics	mathematic	NOUN
ejpam-3438	1	8	vol	vol	NOUN
ejpam-3438	1	9	.	.	PROPN
ejpam-3438	2	1	12	12	NUM
ejpam-3438	2	2	,	,	PUNCT
ejpam-3438	2	3	no	no	INTJ
ejpam-3438	2	4	.	.	NOUN
ejpam-3438	2	5	3	3	NUM
ejpam-3438	2	6	,	,	PUNCT
ejpam-3438	2	7	2019	2019	NUM
ejpam-3438	2	8	,	,	PUNCT
ejpam-3438	2	9	893	893	NUM
ejpam-3438	2	10	-	-	SYM
ejpam-3438	2	11	905	905	NUM
ejpam-3438	2	12	issn	issn	PROPN
ejpam-3438	2	13	1307	1307	NUM
ejpam-3438	2	14	-	-	SYM
ejpam-3438	2	15	5543	5543	NUM
ejpam-3438	2	16	–	–	PUNCT
ejpam-3438	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3438	2	18	published	publish	VERB
ejpam-3438	2	19	by	by	ADP
ejpam-3438	2	20	new	new	PROPN
ejpam-3438	2	21	york	york	PROPN
ejpam-3438	2	22	business	business	PROPN
ejpam-3438	2	23	global	global	PROPN
ejpam-3438	2	24	on	on	ADP
ejpam-3438	2	25	β	β	ADJ
ejpam-3438	2	26	-	-	ADJ
ejpam-3438	2	27	open	open	ADJ
ejpam-3438	2	28	sets	set	NOUN
ejpam-3438	2	29	and	and	CCONJ
ejpam-3438	2	30	ideals	ideal	NOUN
ejpam-3438	2	31	in	in	ADP
ejpam-3438	2	32	topological	topological	ADJ
ejpam-3438	2	33	spaces	space	NOUN
ejpam-3438	2	34	glaisa	glaisa	VERB
ejpam-3438	2	35	t.	t.	PROPN
ejpam-3438	2	36	catalan1,∗	catalan1,∗	NOUN
ejpam-3438	2	37	,	,	PUNCT
ejpam-3438	2	38	roberto	roberto	PROPN
ejpam-3438	2	39	n.	n.	PROPN
ejpam-3438	2	40	padua2	padua2	PROPN
ejpam-3438	2	41	,	,	PUNCT
ejpam-3438	2	42	michael	michael	PROPN
ejpam-3438	2	43	p.	p.	PROPN
ejpam-3438	2	44	baldado	baldado	NOUN
ejpam-3438	2	45	jr.2	jr.2	PROPN
ejpam-3438	2	46	1	1	NUM
ejpam-3438	2	47	negros	negros	PROPN
ejpam-3438	2	48	oriental	oriental	ADJ
ejpam-3438	2	49	state	state	PROPN
ejpam-3438	2	50	university	university	PROPN
ejpam-3438	2	51	siaton	siaton	PROPN
ejpam-3438	2	52	campus	campus	PROPN
ejpam-3438	2	53	,	,	PUNCT
ejpam-3438	2	54	siaton	siaton	NOUN
ejpam-3438	2	55	negros	negros	PROPN
ejpam-3438	2	56	oriental	oriental	ADJ
ejpam-3438	2	57	,	,	PUNCT
ejpam-3438	2	58	philippines	philippine	NOUN
ejpam-3438	2	59	2	2	NUM
ejpam-3438	2	60	mathematics	mathematics	NOUN
ejpam-3438	2	61	department	department	NOUN
ejpam-3438	2	62	,	,	PUNCT
ejpam-3438	2	63	negros	negros	PROPN
ejpam-3438	2	64	oriental	oriental	ADJ
ejpam-3438	2	65	state	state	PROPN
ejpam-3438	2	66	university	university	PROPN
ejpam-3438	2	67	main	main	ADJ
ejpam-3438	2	68	campus	campus	NOUN
ejpam-3438	2	69	,	,	PUNCT
ejpam-3438	2	70	dumaguete	dumaguete	PROPN
ejpam-3438	2	71	city	city	PROPN
ejpam-3438	2	72	,	,	PUNCT
ejpam-3438	2	73	philippines	philippine	NOUN
ejpam-3438	2	74	abstract	abstract	ADJ
ejpam-3438	2	75	.	.	PUNCT
ejpam-3438	3	1	let	let	VERB
ejpam-3438	3	2	x	x	PRON
ejpam-3438	3	3	be	be	AUX
ejpam-3438	3	4	a	a	DET
ejpam-3438	3	5	topological	topological	ADJ
ejpam-3438	3	6	space	space	NOUN
ejpam-3438	4	1	and	and	CCONJ
ejpam-3438	4	2	i	i	PRON
ejpam-3438	4	3	be	be	VERB
ejpam-3438	4	4	an	an	DET
ejpam-3438	4	5	ideal	ideal	NOUN
ejpam-3438	4	6	in	in	ADP
ejpam-3438	4	7	x.	x.	PROPN
ejpam-3438	4	8	a	a	DET
ejpam-3438	4	9	subset	subset	NOUN
ejpam-3438	4	10	a	a	PRON
ejpam-3438	4	11	of	of	ADP
ejpam-3438	4	12	a	a	DET
ejpam-3438	4	13	topological	topological	ADJ
ejpam-3438	4	14	space	space	NOUN
ejpam-3438	4	15	x	x	PUNCT
ejpam-3438	4	16	is	be	AUX
ejpam-3438	4	17	called	call	VERB
ejpam-3438	4	18	a	a	DET
ejpam-3438	4	19	β	β	NOUN
ejpam-3438	4	20	-	-	ADJ
ejpam-3438	4	21	open	open	ADJ
ejpam-3438	4	22	set	set	NOUN
ejpam-3438	4	23	if	if	SCONJ
ejpam-3438	4	24	a	a	DET
ejpam-3438	4	25	⊆	⊆	NUM
ejpam-3438	4	26	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-3438	4	27	)	)	PUNCT
ejpam-3438	4	28	)	)	PUNCT
ejpam-3438	4	29	)	)	PUNCT
ejpam-3438	4	30	.	.	PUNCT
ejpam-3438	5	1	a	a	DET
ejpam-3438	5	2	subset	subset	NOUN
ejpam-3438	5	3	a	a	PRON
ejpam-3438	5	4	of	of	ADP
ejpam-3438	5	5	x	x	PRON
ejpam-3438	5	6	is	be	AUX
ejpam-3438	5	7	called	call	VERB
ejpam-3438	5	8	β	β	VERB
ejpam-3438	5	9	-	-	VERB
ejpam-3438	5	10	open	open	ADJ
ejpam-3438	5	11	with	with	ADP
ejpam-3438	5	12	respect	respect	NOUN
ejpam-3438	5	13	to	to	ADP
ejpam-3438	5	14	the	the	DET
ejpam-3438	5	15	ideal	ideal	NOUN
ejpam-3438	5	16	i	i	PRON
ejpam-3438	5	17	,	,	PUNCT
ejpam-3438	5	18	or	or	CCONJ
ejpam-3438	5	19	βi	βi	PRON
ejpam-3438	5	20	-open	-open	ADJ
ejpam-3438	5	21	,	,	PUNCT
ejpam-3438	5	22	if	if	SCONJ
ejpam-3438	5	23	there	there	PRON
ejpam-3438	5	24	exists	exist	VERB
ejpam-3438	5	25	an	an	DET
ejpam-3438	5	26	open	open	ADJ
ejpam-3438	5	27	set	set	NOUN
ejpam-3438	5	28	u	u	PRON
ejpam-3438	5	29	such	such	ADJ
ejpam-3438	5	30	that	that	SCONJ
ejpam-3438	5	31	(	(	PUNCT
ejpam-3438	5	32	1	1	X
ejpam-3438	5	33	)	)	PUNCT
ejpam-3438	5	34	u	u	NOUN
ejpam-3438	5	35	−	−	PROPN
ejpam-3438	5	36	a	a	DET
ejpam-3438	5	37	∈	∈	PROPN
ejpam-3438	6	1	i	i	PRON
ejpam-3438	6	2	,	,	PUNCT
ejpam-3438	6	3	and	and	CCONJ
ejpam-3438	6	4	(	(	PUNCT
ejpam-3438	6	5	2	2	X
ejpam-3438	6	6	)	)	PUNCT
ejpam-3438	6	7	a	a	DET
ejpam-3438	6	8	−	−	PROPN
ejpam-3438	6	9	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	6	10	)	)	PUNCT
ejpam-3438	6	11	)	)	PUNCT
ejpam-3438	6	12	)	)	PUNCT
ejpam-3438	7	1	∈	∈	PROPN
ejpam-3438	7	2	i.	i.	NOUN
ejpam-3438	7	3	a	a	DET
ejpam-3438	7	4	space	space	NOUN
ejpam-3438	7	5	x	x	PUNCT
ejpam-3438	7	6	is	be	AUX
ejpam-3438	7	7	said	say	VERB
ejpam-3438	7	8	to	to	PART
ejpam-3438	7	9	be	be	AUX
ejpam-3438	7	10	a	a	DET
ejpam-3438	7	11	βi	βi	NOUN
ejpam-3438	7	12	-compact	-compact	NOUN
ejpam-3438	7	13	space	space	NOUN
ejpam-3438	7	14	if	if	SCONJ
ejpam-3438	7	15	it	it	PRON
ejpam-3438	7	16	is	be	AUX
ejpam-3438	7	17	βi	βi	PRON
ejpam-3438	7	18	-compact	-compact	NOUN
ejpam-3438	7	19	as	as	ADP
ejpam-3438	7	20	a	a	DET
ejpam-3438	7	21	subset	subset	NOUN
ejpam-3438	7	22	.	.	PUNCT
ejpam-3438	8	1	an	an	DET
ejpam-3438	8	2	ideal	ideal	ADJ
ejpam-3438	8	3	topological	topological	ADJ
ejpam-3438	8	4	space	space	NOUN
ejpam-3438	8	5	(	(	PUNCT
ejpam-3438	8	6	x	x	X
ejpam-3438	8	7	,	,	PUNCT
ejpam-3438	8	8	τ	τ	PROPN
ejpam-3438	8	9	,	,	PUNCT
ejpam-3438	8	10	i	i	PROPN
ejpam-3438	8	11	)	)	PUNCT
ejpam-3438	8	12	is	be	AUX
ejpam-3438	8	13	said	say	VERB
ejpam-3438	8	14	to	to	PART
ejpam-3438	8	15	be	be	AUX
ejpam-3438	8	16	a	a	DET
ejpam-3438	8	17	cβi	cβi	NOUN
ejpam-3438	8	18	-compact	-compact	NOUN
ejpam-3438	8	19	space	space	NOUN
ejpam-3438	8	20	if	if	SCONJ
ejpam-3438	8	21	it	it	PRON
ejpam-3438	8	22	is	be	AUX
ejpam-3438	8	23	cβi	cβi	NOUN
ejpam-3438	8	24	-compact	-compact	NOUN
ejpam-3438	8	25	as	as	ADP
ejpam-3438	8	26	a	a	DET
ejpam-3438	8	27	subset	subset	NOUN
ejpam-3438	8	28	.	.	PUNCT
ejpam-3438	9	1	an	an	DET
ejpam-3438	9	2	ideal	ideal	ADJ
ejpam-3438	9	3	topological	topological	ADJ
ejpam-3438	9	4	space	space	NOUN
ejpam-3438	9	5	(	(	PUNCT
ejpam-3438	9	6	x	x	X
ejpam-3438	9	7	,	,	PUNCT
ejpam-3438	9	8	τ	τ	PROPN
ejpam-3438	9	9	,	,	PUNCT
ejpam-3438	9	10	i	i	PROPN
ejpam-3438	9	11	)	)	PUNCT
ejpam-3438	9	12	is	be	AUX
ejpam-3438	9	13	said	say	VERB
ejpam-3438	9	14	to	to	PART
ejpam-3438	9	15	be	be	AUX
ejpam-3438	9	16	a	a	DET
ejpam-3438	9	17	countably	countably	ADJ
ejpam-3438	9	18	βi	βi	PRON
ejpam-3438	9	19	-compact	-compact	NOUN
ejpam-3438	9	20	space	space	NOUN
ejpam-3438	9	21	if	if	SCONJ
ejpam-3438	9	22	x	x	PRON
ejpam-3438	9	23	is	be	AUX
ejpam-3438	9	24	countably	countably	ADV
ejpam-3438	9	25	βi	βi	NOUN
ejpam-3438	9	26	-compact	-compact	NOUN
ejpam-3438	9	27	as	as	ADP
ejpam-3438	9	28	a	a	DET
ejpam-3438	9	29	subset	subset	NOUN
ejpam-3438	9	30	.	.	PUNCT
ejpam-3438	10	1	two	two	NUM
ejpam-3438	10	2	sets	set	VERB
ejpam-3438	10	3	a	a	PRON
ejpam-3438	10	4	and	and	CCONJ
ejpam-3438	10	5	b	b	NOUN
ejpam-3438	10	6	in	in	ADP
ejpam-3438	10	7	an	an	DET
ejpam-3438	10	8	ideal	ideal	ADJ
ejpam-3438	10	9	topological	topological	ADJ
ejpam-3438	10	10	space	space	NOUN
ejpam-3438	10	11	(	(	PUNCT
ejpam-3438	10	12	x	x	X
ejpam-3438	10	13	,	,	PUNCT
ejpam-3438	10	14	τ	τ	PROPN
ejpam-3438	10	15	,	,	PUNCT
ejpam-3438	10	16	i	i	PROPN
ejpam-3438	10	17	)	)	PUNCT
ejpam-3438	10	18	is	be	AUX
ejpam-3438	10	19	said	say	VERB
ejpam-3438	10	20	to	to	PART
ejpam-3438	10	21	be	be	AUX
ejpam-3438	10	22	βi	βi	PRON
ejpam-3438	10	23	-separated	-separate	VERB
ejpam-3438	10	24	if	if	SCONJ
ejpam-3438	10	25	clβi	clβi	NOUN
ejpam-3438	10	26	(	(	PUNCT
ejpam-3438	10	27	a)∩b	a)∩b	NOUN
ejpam-3438	10	28	=	=	SYM
ejpam-3438	10	29	∅	∅	NOUN
ejpam-3438	10	30	=	=	PUNCT
ejpam-3438	10	31	a∩	a∩	PROPN
ejpam-3438	10	32	clβ(b	clβ(b	PROPN
ejpam-3438	10	33	)	)	PUNCT
ejpam-3438	10	34	.	.	PUNCT
ejpam-3438	11	1	a	a	DET
ejpam-3438	11	2	subset	subset	NOUN
ejpam-3438	11	3	a	a	PRON
ejpam-3438	11	4	of	of	ADP
ejpam-3438	11	5	an	an	DET
ejpam-3438	11	6	ideal	ideal	ADJ
ejpam-3438	11	7	topological	topological	ADJ
ejpam-3438	11	8	space	space	NOUN
ejpam-3438	11	9	(	(	PUNCT
ejpam-3438	11	10	x	x	X
ejpam-3438	11	11	,	,	PUNCT
ejpam-3438	11	12	τ	τ	PROPN
ejpam-3438	11	13	,	,	PUNCT
ejpam-3438	11	14	i	i	PROPN
ejpam-3438	11	15	)	)	PUNCT
ejpam-3438	11	16	is	be	AUX
ejpam-3438	11	17	said	say	VERB
ejpam-3438	11	18	to	to	PART
ejpam-3438	11	19	be	be	AUX
ejpam-3438	11	20	βi	βi	PRON
ejpam-3438	11	21	-connected	-connected	ADJ
ejpam-3438	11	22	if	if	SCONJ
ejpam-3438	11	23	it	it	PRON
ejpam-3438	11	24	can	can	AUX
ejpam-3438	11	25	not	not	PART
ejpam-3438	11	26	be	be	AUX
ejpam-3438	11	27	expressed	express	VERB
ejpam-3438	11	28	as	as	ADP
ejpam-3438	11	29	a	a	DET
ejpam-3438	11	30	union	union	NOUN
ejpam-3438	11	31	of	of	ADP
ejpam-3438	11	32	two	two	NUM
ejpam-3438	11	33	βi	βi	ADV
ejpam-3438	11	34	-separated	-separate	VERB
ejpam-3438	11	35	sets	set	NOUN
ejpam-3438	11	36	.	.	PUNCT
ejpam-3438	12	1	an	an	DET
ejpam-3438	12	2	ideal	ideal	ADJ
ejpam-3438	12	3	topological	topological	ADJ
ejpam-3438	12	4	space	space	NOUN
ejpam-3438	12	5	(	(	PUNCT
ejpam-3438	12	6	x	x	X
ejpam-3438	12	7	,	,	PUNCT
ejpam-3438	12	8	τ	τ	PROPN
ejpam-3438	12	9	,	,	PUNCT
ejpam-3438	12	10	i	i	PROPN
ejpam-3438	12	11	)	)	PUNCT
ejpam-3438	12	12	is	be	AUX
ejpam-3438	12	13	said	say	VERB
ejpam-3438	12	14	to	to	PART
ejpam-3438	12	15	be	be	AUX
ejpam-3438	12	16	βi	βi	PRON
ejpam-3438	12	17	-connected	-connected	ADJ
ejpam-3438	12	18	if	if	SCONJ
ejpam-3438	12	19	x	x	PRON
ejpam-3438	12	20	βi	βi	PRON
ejpam-3438	12	21	-connected	-connecte	VERB
ejpam-3438	12	22	as	as	ADP
ejpam-3438	12	23	a	a	DET
ejpam-3438	12	24	subset	subset	NOUN
ejpam-3438	12	25	.	.	PUNCT
ejpam-3438	13	1	in	in	ADP
ejpam-3438	13	2	this	this	DET
ejpam-3438	13	3	study	study	NOUN
ejpam-3438	13	4	,	,	PUNCT
ejpam-3438	13	5	we	we	PRON
ejpam-3438	13	6	introduced	introduce	VERB
ejpam-3438	13	7	the	the	DET
ejpam-3438	13	8	notions	notion	NOUN
ejpam-3438	13	9	βi	βi	PRON
ejpam-3438	13	10	-open	-open	NOUN
ejpam-3438	13	11	set	set	NOUN
ejpam-3438	13	12	,	,	PUNCT
ejpam-3438	13	13	βi	βi	PRON
ejpam-3438	13	14	-compact	-compact	NOUN
ejpam-3438	13	15	,	,	PUNCT
ejpam-3438	13	16	cβi	cβi	NOUN
ejpam-3438	13	17	-compact	-compact	NOUN
ejpam-3438	13	18	,	,	PUNCT
ejpam-3438	13	19	βi	βi	PRON
ejpam-3438	13	20	-hyperconnected	-hyperconnecte	VERB
ejpam-3438	13	21	,	,	PUNCT
ejpam-3438	13	22	cβi	cβi	NOUN
ejpam-3438	13	23	-hyperconnected	-hyperconnecte	VERB
ejpam-3438	13	24	,	,	PUNCT
ejpam-3438	13	25	βi	βi	ADP
ejpam-3438	13	26	-connected	-connected	ADJ
ejpam-3438	13	27	and	and	CCONJ
ejpam-3438	13	28	βi	βi	PRON
ejpam-3438	13	29	-separated	-separate	VERB
ejpam-3438	13	30	.	.	PUNCT
ejpam-3438	14	1	moreover	moreover	ADV
ejpam-3438	14	2	,	,	PUNCT
ejpam-3438	14	3	we	we	PRON
ejpam-3438	14	4	investigated	investigate	VERB
ejpam-3438	14	5	the	the	DET
ejpam-3438	14	6	concept	concept	NOUN
ejpam-3438	14	7	β	β	X
ejpam-3438	14	8	-	-	ADJ
ejpam-3438	14	9	open	open	ADJ
ejpam-3438	14	10	set	set	VERB
ejpam-3438	14	11	by	by	ADP
ejpam-3438	14	12	determining	determine	VERB
ejpam-3438	14	13	some	some	PRON
ejpam-3438	14	14	of	of	ADP
ejpam-3438	14	15	its	its	PRON
ejpam-3438	14	16	properties	property	NOUN
ejpam-3438	14	17	relative	relative	ADJ
ejpam-3438	14	18	to	to	ADP
ejpam-3438	14	19	the	the	DET
ejpam-3438	14	20	above	above	ADV
ejpam-3438	14	21	-	-	PUNCT
ejpam-3438	14	22	mentioned	mention	VERB
ejpam-3438	14	23	notions	notion	NOUN
ejpam-3438	14	24	.	.	PUNCT
ejpam-3438	15	1	2010	2010	NUM
ejpam-3438	15	2	mathematics	mathematic	NOUN
ejpam-3438	15	3	subject	subject	NOUN
ejpam-3438	15	4	classifications	classification	NOUN
ejpam-3438	15	5	:	:	PUNCT
ejpam-3438	15	6	54	54	NUM
ejpam-3438	15	7	-	-	PUNCT
ejpam-3438	15	8	xx	xx	NUM
ejpam-3438	15	9	key	key	ADJ
ejpam-3438	15	10	words	word	NOUN
ejpam-3438	15	11	and	and	CCONJ
ejpam-3438	15	12	phrases	phrase	NOUN
ejpam-3438	15	13	:	:	PUNCT
ejpam-3438	15	14	β	β	X
ejpam-3438	15	15	-	-	ADJ
ejpam-3438	15	16	open	open	ADJ
ejpam-3438	15	17	sets	set	NOUN
ejpam-3438	15	18	,	,	PUNCT
ejpam-3438	15	19	βi	βi	X
ejpam-3438	15	20	-open	-open	NOUN
ejpam-3438	15	21	sets	set	NOUN
ejpam-3438	15	22	,	,	PUNCT
ejpam-3438	15	23	βi	βi	PRON
ejpam-3438	15	24	-compactness	-compactness	NOUN
ejpam-3438	15	25	,	,	PUNCT
ejpam-3438	15	26	cβi	cβi	NOUN
ejpam-3438	15	27	-compactness	-compactness	NOUN
ejpam-3438	15	28	,	,	PUNCT
ejpam-3438	15	29	βi	βi	PRON
ejpam-3438	15	30	hyperconnectedness	hyperconnectedness	NOUN
ejpam-3438	15	31	and	and	CCONJ
ejpam-3438	15	32	cβi	cβi	VERB
ejpam-3438	15	33	-hyperconnectednes	-hyperconnectedne	NOUN
ejpam-3438	15	34	1	1	NUM
ejpam-3438	15	35	.	.	PUNCT
ejpam-3438	16	1	introduction	introduction	NOUN
ejpam-3438	16	2	topology	topology	NOUN
ejpam-3438	16	3	is	be	AUX
ejpam-3438	16	4	an	an	DET
ejpam-3438	16	5	interesting	interesting	ADJ
ejpam-3438	16	6	area	area	NOUN
ejpam-3438	16	7	of	of	ADP
ejpam-3438	16	8	mathematics	mathematic	NOUN
ejpam-3438	16	9	.	.	PUNCT
ejpam-3438	17	1	it	it	PRON
ejpam-3438	17	2	is	be	AUX
ejpam-3438	17	3	new	new	ADJ
ejpam-3438	17	4	,	,	PUNCT
ejpam-3438	17	5	being	be	AUX
ejpam-3438	17	6	conceive	conceive	ADJ
ejpam-3438	17	7	in	in	ADP
ejpam-3438	17	8	the	the	DET
ejpam-3438	17	9	19th	19th	ADJ
ejpam-3438	17	10	century	century	NOUN
ejpam-3438	17	11	.	.	PUNCT
ejpam-3438	18	1	but	but	CCONJ
ejpam-3438	18	2	according	accord	VERB
ejpam-3438	18	3	to	to	ADP
ejpam-3438	18	4	morris	morris	PROPN
ejpam-3438	18	5	[	[	X
ejpam-3438	18	6	15	15	NUM
ejpam-3438	18	7	]	]	PUNCT
ejpam-3438	18	8	,	,	PUNCT
ejpam-3438	18	9	the	the	DET
ejpam-3438	18	10	influence	influence	NOUN
ejpam-3438	18	11	of	of	ADP
ejpam-3438	18	12	topology	topology	NOUN
ejpam-3438	18	13	is	be	AUX
ejpam-3438	18	14	so	so	ADV
ejpam-3438	18	15	vast	vast	ADJ
ejpam-3438	18	16	,	,	PUNCT
ejpam-3438	18	17	so	so	SCONJ
ejpam-3438	18	18	that	that	SCONJ
ejpam-3438	18	19	it	it	PRON
ejpam-3438	18	20	is	be	AUX
ejpam-3438	18	21	identifiable	identifiable	ADJ
ejpam-3438	18	22	in	in	ADP
ejpam-3438	18	23	various	various	ADJ
ejpam-3438	18	24	branches	branch	NOUN
ejpam-3438	18	25	of	of	ADP
ejpam-3438	18	26	mathematics	mathematic	NOUN
ejpam-3438	18	27	.	.	PUNCT
ejpam-3438	19	1	topological	topological	ADJ
ejpam-3438	19	2	ideas	idea	NOUN
ejpam-3438	19	3	are	be	AUX
ejpam-3438	19	4	present	present	ADJ
ejpam-3438	19	5	not	not	PART
ejpam-3438	19	6	only	only	ADV
ejpam-3438	19	7	in	in	ADP
ejpam-3438	19	8	mathematics	mathematic	NOUN
ejpam-3438	19	9	but	but	CCONJ
ejpam-3438	19	10	also	also	ADV
ejpam-3438	19	11	in	in	ADP
ejpam-3438	19	12	other	other	ADJ
ejpam-3438	19	13	areas	area	NOUN
ejpam-3438	19	14	,	,	PUNCT
ejpam-3438	19	15	for	for	ADP
ejpam-3438	19	16	example	example	NOUN
ejpam-3438	19	17	biochemistry	biochemistry	NOUN
ejpam-3438	20	1	[	[	X
ejpam-3438	20	2	16	16	NUM
ejpam-3438	20	3	]	]	PUNCT
ejpam-3438	20	4	and	and	CCONJ
ejpam-3438	20	5	information	information	NOUN
ejpam-3438	20	6	systems	system	NOUN
ejpam-3438	20	7	[	[	X
ejpam-3438	20	8	17	17	NUM
ejpam-3438	20	9	]	]	PUNCT
ejpam-3438	20	10	.	.	PUNCT
ejpam-3438	21	1	topology	topology	NOUN
ejpam-3438	21	2	as	as	ADP
ejpam-3438	21	3	a	a	DET
ejpam-3438	21	4	subject	subject	NOUN
ejpam-3438	21	5	has	have	VERB
ejpam-3438	21	6	several	several	ADJ
ejpam-3438	21	7	different	different	ADJ
ejpam-3438	21	8	branches	branch	NOUN
ejpam-3438	21	9	such	such	ADJ
ejpam-3438	21	10	as	as	ADP
ejpam-3438	21	11	point	point	NOUN
ejpam-3438	21	12	set	set	VERB
ejpam-3438	21	13	topology	topology	NOUN
ejpam-3438	21	14	,	,	PUNCT
ejpam-3438	21	15	algebraic	algebraic	ADJ
ejpam-3438	21	16	topology	topology	NOUN
ejpam-3438	21	17	,	,	PUNCT
ejpam-3438	21	18	differential	differential	NOUN
ejpam-3438	21	19	topology	topology	NOUN
ejpam-3438	21	20	,	,	PUNCT
ejpam-3438	21	21	etc	etc	X
ejpam-3438	21	22	.	.	X
ejpam-3438	22	1	the	the	DET
ejpam-3438	22	2	basic	basic	ADJ
ejpam-3438	22	3	component	component	NOUN
ejpam-3438	22	4	of	of	ADP
ejpam-3438	22	5	a	a	DET
ejpam-3438	22	6	topology	topology	NOUN
ejpam-3438	22	7	space	space	NOUN
ejpam-3438	22	8	are	be	AUX
ejpam-3438	22	9	open	open	ADJ
ejpam-3438	22	10	sets	set	NOUN
ejpam-3438	22	11	,	,	PUNCT
ejpam-3438	22	12	and	and	CCONJ
ejpam-3438	22	13	overtime	overtime	NOUN
ejpam-3438	22	14	there	there	PRON
ejpam-3438	22	15	have	have	AUX
ejpam-3438	22	16	been	be	AUX
ejpam-3438	22	17	so	so	ADV
ejpam-3438	22	18	many	many	ADJ
ejpam-3438	22	19	generalizations	generalization	NOUN
ejpam-3438	22	20	of	of	ADP
ejpam-3438	22	21	it	it	PRON
ejpam-3438	22	22	.	.	PUNCT
ejpam-3438	23	1	among	among	ADP
ejpam-3438	23	2	them	they	PRON
ejpam-3438	23	3	are	be	AUX
ejpam-3438	23	4	the	the	DET
ejpam-3438	23	5	following	following	NOUN
ejpam-3438	23	6	.	.	PUNCT
ejpam-3438	24	1	stone	stone	NOUN
ejpam-3438	25	1	[	[	X
ejpam-3438	25	2	18	18	NUM
ejpam-3438	25	3	]	]	PUNCT
ejpam-3438	25	4	introduced	introduce	VERB
ejpam-3438	25	5	∗corresponding	∗corresponde	VERB
ejpam-3438	25	6	author	author	NOUN
ejpam-3438	25	7	.	.	PUNCT
ejpam-3438	26	1	doi	doi	NOUN
ejpam-3438	26	2	:	:	PUNCT
ejpam-3438	26	3	https://doi.org/10.29020/nybg.ejpam.v12i3.3438	https://doi.org/10.29020/nybg.ejpam.v12i3.3438	PRON
ejpam-3438	26	4	email	email	NOUN
ejpam-3438	26	5	addresses	address	NOUN
ejpam-3438	26	6	:	:	PUNCT
ejpam-3438	26	7	never	never	ADV
ejpam-3438	26	8	glaisa@yahoo.com	glaisa@yahoo.com	PROPN
ejpam-3438	26	9	(	(	PUNCT
ejpam-3438	26	10	g.	g.	PROPN
ejpam-3438	26	11	catalan	catalan	PROPN
ejpam-3438	26	12	)	)	PUNCT
ejpam-3438	26	13	,	,	PUNCT
ejpam-3438	26	14	michaelpbaldadojr@yahoo.com	michaelpbaldadojr@yahoo.com	X
ejpam-3438	26	15	(	(	PUNCT
ejpam-3438	26	16	m.	m.	PROPN
ejpam-3438	26	17	baldado	baldado	PROPN
ejpam-3438	26	18	jr	jr	PROPN
ejpam-3438	26	19	.	.	PUNCT
ejpam-3438	26	20	)	)	PUNCT
ejpam-3438	26	21	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3438	27	1	893	893	NUM
ejpam-3438	27	2	c	c	X
ejpam-3438	27	3	©	©	PROPN
ejpam-3438	27	4	2019	2019	NUM
ejpam-3438	27	5	ejpam	ejpam	NOUN
ejpam-3438	27	6	all	all	DET
ejpam-3438	27	7	rights	right	NOUN
ejpam-3438	27	8	reserved	reserve	VERB
ejpam-3438	27	9	.	.	PUNCT
ejpam-3438	28	1	g.	g.	PROPN
ejpam-3438	28	2	catalan	catalan	PROPN
ejpam-3438	28	3	,	,	PUNCT
ejpam-3438	28	4	r.	r.	PROPN
ejpam-3438	28	5	padua	padua	PROPN
ejpam-3438	28	6	,	,	PUNCT
ejpam-3438	28	7	m.	m.	PROPN
ejpam-3438	28	8	baldado	baldado	PROPN
ejpam-3438	28	9	jr	jr	PROPN
ejpam-3438	28	10	.	.	PROPN
ejpam-3438	28	11	/	/	SYM
ejpam-3438	28	12	eur	eur	PROPN
ejpam-3438	28	13	.	.	PUNCT
ejpam-3438	29	1	j.	j.	PROPN
ejpam-3438	29	2	pure	pure	PROPN
ejpam-3438	29	3	appl	appl	PROPN
ejpam-3438	29	4	.	.	PROPN
ejpam-3438	29	5	math	math	PROPN
ejpam-3438	29	6	,	,	PUNCT
ejpam-3438	29	7	12	12	NUM
ejpam-3438	29	8	(	(	PUNCT
ejpam-3438	29	9	3	3	NUM
ejpam-3438	29	10	)	)	PUNCT
ejpam-3438	29	11	(	(	PUNCT
ejpam-3438	29	12	2019	2019	NUM
ejpam-3438	29	13	)	)	PUNCT
ejpam-3438	29	14	,	,	PUNCT
ejpam-3438	29	15	893	893	NUM
ejpam-3438	29	16	-	-	SYM
ejpam-3438	29	17	905	905	NUM
ejpam-3438	29	18	894	894	NUM
ejpam-3438	29	19	the	the	DET
ejpam-3438	29	20	concept	concept	NOUN
ejpam-3438	29	21	of	of	ADP
ejpam-3438	29	22	regular	regular	ADJ
ejpam-3438	29	23	open	open	ADJ
ejpam-3438	29	24	sets	set	NOUN
ejpam-3438	29	25	.	.	PUNCT
ejpam-3438	30	1	levine	levine	PROPN
ejpam-3438	30	2	[	[	X
ejpam-3438	30	3	19	19	NUM
ejpam-3438	30	4	]	]	PUNCT
ejpam-3438	30	5	introduced	introduce	VERB
ejpam-3438	30	6	the	the	DET
ejpam-3438	30	7	concept	concept	NOUN
ejpam-3438	30	8	of	of	ADP
ejpam-3438	30	9	semi	semi	ADJ
ejpam-3438	30	10	open	open	ADJ
ejpam-3438	30	11	sets	set	NOUN
ejpam-3438	30	12	.	.	PUNCT
ejpam-3438	31	1	najasted	najaste	VERB
ejpam-3438	32	1	[	[	X
ejpam-3438	32	2	20	20	NUM
ejpam-3438	32	3	]	]	PUNCT
ejpam-3438	32	4	introduced	introduce	VERB
ejpam-3438	32	5	the	the	DET
ejpam-3438	32	6	concept	concept	NOUN
ejpam-3438	32	7	of	of	ADP
ejpam-3438	32	8	α	α	NOUN
ejpam-3438	32	9	-	-	ADJ
ejpam-3438	32	10	open	open	ADJ
ejpam-3438	32	11	sets	set	NOUN
ejpam-3438	32	12	.	.	PUNCT
ejpam-3438	33	1	mashhour	mashhour	INTJ
ejpam-3438	33	2	et	et	PROPN
ejpam-3438	33	3	al	al	PROPN
ejpam-3438	33	4	.	.	PUNCT
ejpam-3438	34	1	[	[	X
ejpam-3438	34	2	31	31	NUM
ejpam-3438	34	3	]	]	PUNCT
ejpam-3438	34	4	introduced	introduce	VERB
ejpam-3438	34	5	the	the	DET
ejpam-3438	34	6	concept	concept	NOUN
ejpam-3438	34	7	of	of	ADP
ejpam-3438	34	8	pre	pre	ADJ
ejpam-3438	34	9	-	-	ADJ
ejpam-3438	34	10	open	open	ADJ
ejpam-3438	34	11	sets	set	NOUN
ejpam-3438	34	12	.	.	PUNCT
ejpam-3438	35	1	abd	abd	PROPN
ejpam-3438	35	2	el	el	PROPN
ejpam-3438	35	3	-	-	PROPN
ejpam-3438	35	4	monsef	monsef	PROPN
ejpam-3438	35	5	et	et	PROPN
ejpam-3438	35	6	al	al	PROPN
ejpam-3438	35	7	.	.	PUNCT
ejpam-3438	36	1	[	[	X
ejpam-3438	36	2	1	1	X
ejpam-3438	36	3	]	]	PUNCT
ejpam-3438	36	4	introduced	introduce	VERB
ejpam-3438	36	5	the	the	DET
ejpam-3438	36	6	concept	concept	NOUN
ejpam-3438	36	7	of	of	ADP
ejpam-3438	36	8	β	β	ADJ
ejpam-3438	36	9	-	-	ADJ
ejpam-3438	36	10	open	open	ADJ
ejpam-3438	36	11	sets	set	NOUN
ejpam-3438	36	12	.	.	PUNCT
ejpam-3438	37	1	apart	apart	ADV
ejpam-3438	37	2	from	from	ADP
ejpam-3438	37	3	introducing	introduce	VERB
ejpam-3438	37	4	β	β	ADJ
ejpam-3438	37	5	-	-	ADJ
ejpam-3438	37	6	open	open	ADJ
ejpam-3438	37	7	sets	set	NOUN
ejpam-3438	37	8	,	,	PUNCT
ejpam-3438	37	9	abd	abd	PROPN
ejpam-3438	37	10	el	el	PROPN
ejpam-3438	37	11	-	-	PROPN
ejpam-3438	37	12	monsef	monsef	PROPN
ejpam-3438	37	13	et	et	PROPN
ejpam-3438	37	14	al	al	PROPN
ejpam-3438	37	15	.	.	PUNCT
ejpam-3438	38	1	[	[	X
ejpam-3438	38	2	1	1	X
ejpam-3438	38	3	]	]	PUNCT
ejpam-3438	38	4	also	also	ADV
ejpam-3438	38	5	introduced	introduce	VERB
ejpam-3438	38	6	βcontinuous	βcontinuous	ADJ
ejpam-3438	38	7	mappings	mapping	NOUN
ejpam-3438	38	8	and	and	CCONJ
ejpam-3438	38	9	β	β	NOUN
ejpam-3438	38	10	-	-	ADJ
ejpam-3438	38	11	open	open	ADJ
ejpam-3438	38	12	mappings	mapping	NOUN
ejpam-3438	38	13	.	.	PUNCT
ejpam-3438	39	1	they	they	PRON
ejpam-3438	39	2	studied	study	VERB
ejpam-3438	39	3	their	their	PRON
ejpam-3438	39	4	properties	property	NOUN
ejpam-3438	39	5	and	and	CCONJ
ejpam-3438	39	6	discussed	discuss	VERB
ejpam-3438	39	7	the	the	DET
ejpam-3438	39	8	connections	connection	NOUN
ejpam-3438	39	9	of	of	ADP
ejpam-3438	39	10	these	these	DET
ejpam-3438	39	11	notions	notion	NOUN
ejpam-3438	39	12	with	with	ADP
ejpam-3438	39	13	the	the	DET
ejpam-3438	39	14	existing	exist	VERB
ejpam-3438	39	15	ones	one	NOUN
ejpam-3438	39	16	.	.	PUNCT
ejpam-3438	40	1	since	since	SCONJ
ejpam-3438	40	2	then	then	ADV
ejpam-3438	40	3	,	,	PUNCT
ejpam-3438	40	4	the	the	DET
ejpam-3438	40	5	concept	concept	NOUN
ejpam-3438	40	6	β	β	X
ejpam-3438	40	7	-	-	ADJ
ejpam-3438	40	8	open	open	ADJ
ejpam-3438	40	9	sets	set	NOUN
ejpam-3438	40	10	has	have	AUX
ejpam-3438	40	11	been	be	AUX
ejpam-3438	40	12	a	a	DET
ejpam-3438	40	13	subject	subject	NOUN
ejpam-3438	40	14	of	of	ADP
ejpam-3438	40	15	a	a	DET
ejpam-3438	40	16	couple	couple	NOUN
ejpam-3438	40	17	of	of	ADP
ejpam-3438	40	18	investigations	investigation	NOUN
ejpam-3438	40	19	.	.	PUNCT
ejpam-3438	41	1	among	among	ADP
ejpam-3438	41	2	them	they	PRON
ejpam-3438	41	3	were	be	AUX
ejpam-3438	41	4	the	the	DET
ejpam-3438	41	5	following	following	NOUN
ejpam-3438	41	6	.	.	PUNCT
ejpam-3438	42	1	abid	abid	PROPN
ejpam-3438	43	1	[	[	X
ejpam-3438	43	2	22	22	NUM
ejpam-3438	43	3	]	]	PUNCT
ejpam-3438	43	4	used	use	VERB
ejpam-3438	43	5	the	the	DET
ejpam-3438	43	6	concept	concept	NOUN
ejpam-3438	43	7	β	β	X
ejpam-3438	43	8	-	-	ADJ
ejpam-3438	43	9	open	open	ADJ
ejpam-3438	43	10	set	set	NOUN
ejpam-3438	43	11	to	to	PART
ejpam-3438	43	12	obtain	obtain	VERB
ejpam-3438	43	13	the	the	DET
ejpam-3438	43	14	properties	property	NOUN
ejpam-3438	43	15	of	of	ADP
ejpam-3438	43	16	the	the	DET
ejpam-3438	43	17	concept	concept	NOUN
ejpam-3438	43	18	non	non	ADJ
ejpam-3438	43	19	-	-	ADJ
ejpam-3438	43	20	semipredense	semipredense	ADJ
ejpam-3438	43	21	set	set	NOUN
ejpam-3438	43	22	.	.	PUNCT
ejpam-3438	44	1	tahiliani	tahiliani	PROPN
ejpam-3438	45	1	[	[	X
ejpam-3438	45	2	23	23	NUM
ejpam-3438	45	3	]	]	PUNCT
ejpam-3438	45	4	introduced	introduce	VERB
ejpam-3438	45	5	an	an	DET
ejpam-3438	45	6	operation	operation	NOUN
ejpam-3438	45	7	on	on	ADP
ejpam-3438	45	8	a	a	DET
ejpam-3438	45	9	family	family	NOUN
ejpam-3438	45	10	of	of	ADP
ejpam-3438	45	11	β	β	ADJ
ejpam-3438	45	12	-	-	ADJ
ejpam-3438	45	13	open	open	ADJ
ejpam-3438	45	14	sets	set	NOUN
ejpam-3438	45	15	;	;	PUNCT
ejpam-3438	45	16	and	and	CCONJ
ejpam-3438	45	17	using	use	VERB
ejpam-3438	45	18	the	the	DET
ejpam-3438	45	19	operation	operation	NOUN
ejpam-3438	45	20	,	,	PUNCT
ejpam-3438	45	21	the	the	DET
ejpam-3438	45	22	concept	concept	NOUN
ejpam-3438	45	23	β	β	NOUN
ejpam-3438	45	24	-	-	PUNCT
ejpam-3438	45	25	γ	γ	ADJ
ejpam-3438	45	26	-	-	ADJ
ejpam-3438	45	27	open	open	ADJ
ejpam-3438	45	28	sets	set	NOUN
ejpam-3438	45	29	was	be	AUX
ejpam-3438	45	30	defined	define	VERB
ejpam-3438	45	31	and	and	CCONJ
ejpam-3438	45	32	investigated	investigate	VERB
ejpam-3438	45	33	.	.	PUNCT
ejpam-3438	46	1	kannan	kannan	PROPN
ejpam-3438	46	2	and	and	CCONJ
ejpam-3438	46	3	nagaveni	nagaveni	ADJ
ejpam-3438	46	4	[	[	X
ejpam-3438	46	5	5	5	NUM
ejpam-3438	46	6	]	]	PUNCT
ejpam-3438	46	7	introduced	introduce	VERB
ejpam-3438	46	8	another	another	DET
ejpam-3438	46	9	generalization	generalization	NOUN
ejpam-3438	46	10	of	of	ADP
ejpam-3438	46	11	the	the	DET
ejpam-3438	46	12	concept	concept	NOUN
ejpam-3438	46	13	β	β	NOUN
ejpam-3438	46	14	-	-	ADJ
ejpam-3438	46	15	open	open	ADJ
ejpam-3438	46	16	sets	set	NOUN
ejpam-3438	46	17	,	,	PUNCT
ejpam-3438	46	18	called	call	VERB
ejpam-3438	46	19	β̂generalized	β̂generalized	ADJ
ejpam-3438	46	20	closed	closed	ADJ
ejpam-3438	46	21	sets	set	NOUN
ejpam-3438	46	22	.	.	PUNCT
ejpam-3438	47	1	mubarki	mubarki	INTJ
ejpam-3438	47	2	et	et	PROPN
ejpam-3438	47	3	al	al	PROPN
ejpam-3438	47	4	.	.	PUNCT
ejpam-3438	48	1	[	[	X
ejpam-3438	48	2	25	25	NUM
ejpam-3438	48	3	]	]	PUNCT
ejpam-3438	48	4	introduced	introduce	VERB
ejpam-3438	48	5	and	and	CCONJ
ejpam-3438	48	6	investigated	investigate	VERB
ejpam-3438	48	7	β∗-open	β∗-open	ADJ
ejpam-3438	48	8	sets	set	NOUN
ejpam-3438	48	9	,	,	PUNCT
ejpam-3438	48	10	which	which	PRON
ejpam-3438	48	11	is	be	AUX
ejpam-3438	48	12	also	also	ADV
ejpam-3438	48	13	a	a	DET
ejpam-3438	48	14	generalization	generalization	NOUN
ejpam-3438	48	15	of	of	ADP
ejpam-3438	48	16	the	the	DET
ejpam-3438	48	17	concept	concept	NOUN
ejpam-3438	48	18	β	β	NOUN
ejpam-3438	48	19	-	-	ADJ
ejpam-3438	48	20	open	open	ADJ
ejpam-3438	48	21	sets	set	NOUN
ejpam-3438	48	22	.	.	PUNCT
ejpam-3438	49	1	el	el	NOUN
ejpam-3438	49	2	-	-	PUNCT
ejpam-3438	49	3	mabhouh	mabhouh	NOUN
ejpam-3438	49	4	and	and	CCONJ
ejpam-3438	49	5	mizyed	mizye	VERB
ejpam-3438	49	6	[	[	X
ejpam-3438	49	7	26	26	NUM
ejpam-3438	49	8	]	]	PUNCT
ejpam-3438	49	9	introduced	introduce	VERB
ejpam-3438	49	10	the	the	DET
ejpam-3438	49	11	concept	concept	NOUN
ejpam-3438	49	12	βc	βc	X
ejpam-3438	49	13	-	-	PUNCT
ejpam-3438	49	14	open	open	ADJ
ejpam-3438	49	15	set	set	NOUN
ejpam-3438	49	16	which	which	PRON
ejpam-3438	49	17	is	be	AUX
ejpam-3438	49	18	a	a	DET
ejpam-3438	49	19	particular	particular	ADJ
ejpam-3438	49	20	class	class	NOUN
ejpam-3438	49	21	of	of	ADP
ejpam-3438	49	22	β	β	ADJ
ejpam-3438	49	23	-	-	ADJ
ejpam-3438	49	24	open	open	ADJ
ejpam-3438	49	25	sets	set	NOUN
ejpam-3438	49	26	.	.	PUNCT
ejpam-3438	50	1	they	they	PRON
ejpam-3438	50	2	also	also	ADV
ejpam-3438	50	3	showed	show	VERB
ejpam-3438	50	4	that	that	SCONJ
ejpam-3438	50	5	βc	βc	ADJ
ejpam-3438	50	6	-	-	PUNCT
ejpam-3438	50	7	open	open	ADJ
ejpam-3438	50	8	sets	set	NOUN
ejpam-3438	50	9	generates	generate	VERB
ejpam-3438	50	10	the	the	DET
ejpam-3438	50	11	same	same	ADJ
ejpam-3438	50	12	topology	topology	NOUN
ejpam-3438	50	13	as	as	ADP
ejpam-3438	50	14	the	the	DET
ejpam-3438	50	15	class	class	NOUN
ejpam-3438	50	16	of	of	ADP
ejpam-3438	50	17	θ	θ	ADJ
ejpam-3438	50	18	-	-	ADJ
ejpam-3438	50	19	open	open	ADJ
ejpam-3438	50	20	sets	set	NOUN
ejpam-3438	50	21	in	in	ADP
ejpam-3438	50	22	alexandroff	alexandroff	ADJ
ejpam-3438	50	23	space	space	NOUN
ejpam-3438	50	24	.	.	PUNCT
ejpam-3438	51	1	akdag	akdag	PROPN
ejpam-3438	51	2	and	and	CCONJ
ejpam-3438	51	3	ozkan	ozkan	X
ejpam-3438	52	1	[	[	X
ejpam-3438	52	2	27	27	NUM
ejpam-3438	52	3	]	]	PUNCT
ejpam-3438	52	4	adapted	adapt	VERB
ejpam-3438	52	5	the	the	DET
ejpam-3438	52	6	concept	concept	NOUN
ejpam-3438	52	7	β	β	X
ejpam-3438	52	8	-	-	ADJ
ejpam-3438	52	9	open	open	ADJ
ejpam-3438	52	10	set	set	NOUN
ejpam-3438	52	11	in	in	ADP
ejpam-3438	52	12	soft	soft	ADJ
ejpam-3438	52	13	topological	topological	ADJ
ejpam-3438	52	14	spaces	space	NOUN
ejpam-3438	52	15	,	,	PUNCT
ejpam-3438	52	16	and	and	CCONJ
ejpam-3438	52	17	defined	define	VERB
ejpam-3438	52	18	the	the	DET
ejpam-3438	52	19	concepts	concept	NOUN
ejpam-3438	52	20	soft	soft	ADJ
ejpam-3438	52	21	β	β	NOUN
ejpam-3438	52	22	-	-	ADJ
ejpam-3438	52	23	interior	interior	ADJ
ejpam-3438	52	24	and	and	CCONJ
ejpam-3438	52	25	soft	soft	ADJ
ejpam-3438	52	26	β	β	NOUN
ejpam-3438	52	27	-	-	NOUN
ejpam-3438	52	28	closure	closure	NOUN
ejpam-3438	52	29	,	,	PUNCT
ejpam-3438	52	30	and	and	CCONJ
ejpam-3438	52	31	gave	give	VERB
ejpam-3438	52	32	their	their	PRON
ejpam-3438	52	33	properties	property	NOUN
ejpam-3438	52	34	.	.	PUNCT
ejpam-3438	53	1	arockiarani	arockiarani	PROPN
ejpam-3438	53	2	and	and	CCONJ
ejpam-3438	53	3	arokia	arokia	PROPN
ejpam-3438	53	4	lancy	lancy	PROPN
ejpam-3438	54	1	[	[	X
ejpam-3438	54	2	28	28	NUM
ejpam-3438	54	3	]	]	PUNCT
ejpam-3438	54	4	presented	present	VERB
ejpam-3438	54	5	gβ	gβ	NOUN
ejpam-3438	54	6	-	-	PUNCT
ejpam-3438	54	7	closed	closed	ADJ
ejpam-3438	54	8	sets	set	NOUN
ejpam-3438	54	9	and	and	CCONJ
ejpam-3438	54	10	gsβclosed	gsβclose	VERB
ejpam-3438	54	11	sets	set	NOUN
ejpam-3438	54	12	(	(	PUNCT
ejpam-3438	54	13	which	which	PRON
ejpam-3438	54	14	were	be	AUX
ejpam-3438	54	15	defined	define	VERB
ejpam-3438	54	16	indirectly	indirectly	ADV
ejpam-3438	54	17	in	in	ADP
ejpam-3438	54	18	terms	term	NOUN
ejpam-3438	54	19	of	of	ADP
ejpam-3438	54	20	the	the	DET
ejpam-3438	54	21	notion	notion	NOUN
ejpam-3438	54	22	of	of	ADP
ejpam-3438	54	23	β	β	ADJ
ejpam-3438	54	24	-	-	ADJ
ejpam-3438	54	25	open	open	ADJ
ejpam-3438	54	26	sets	set	NOUN
ejpam-3438	54	27	)	)	PUNCT
ejpam-3438	54	28	and	and	CCONJ
ejpam-3438	54	29	introduced	introduce	VERB
ejpam-3438	54	30	parallel	parallel	ADJ
ejpam-3438	54	31	concepts	concept	NOUN
ejpam-3438	54	32	in	in	ADP
ejpam-3438	54	33	soft	soft	ADJ
ejpam-3438	54	34	topological	topological	ADJ
ejpam-3438	54	35	spaces	space	NOUN
ejpam-3438	54	36	.	.	PUNCT
ejpam-3438	55	1	let	let	VERB
ejpam-3438	55	2	x	x	PRON
ejpam-3438	55	3	be	be	AUX
ejpam-3438	55	4	a	a	DET
ejpam-3438	55	5	topological	topological	ADJ
ejpam-3438	55	6	space	space	NOUN
ejpam-3438	56	1	and	and	CCONJ
ejpam-3438	56	2	i	i	PRON
ejpam-3438	56	3	be	be	VERB
ejpam-3438	56	4	an	an	DET
ejpam-3438	56	5	ideal	ideal	NOUN
ejpam-3438	56	6	in	in	ADP
ejpam-3438	56	7	x.	x.	PROPN
ejpam-3438	56	8	a	a	DET
ejpam-3438	56	9	subset	subset	NOUN
ejpam-3438	56	10	a	a	PRON
ejpam-3438	56	11	of	of	ADP
ejpam-3438	56	12	a	a	DET
ejpam-3438	56	13	topological	topological	ADJ
ejpam-3438	56	14	space	space	NOUN
ejpam-3438	56	15	x	x	PUNCT
ejpam-3438	56	16	is	be	AUX
ejpam-3438	56	17	called	call	VERB
ejpam-3438	56	18	a	a	DET
ejpam-3438	56	19	β	β	NOUN
ejpam-3438	56	20	-	-	ADJ
ejpam-3438	56	21	open	open	ADJ
ejpam-3438	56	22	set	set	NOUN
ejpam-3438	56	23	if	if	SCONJ
ejpam-3438	56	24	a	a	DET
ejpam-3438	56	25	⊆	⊆	NUM
ejpam-3438	56	26	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-3438	56	27	)	)	PUNCT
ejpam-3438	56	28	)	)	PUNCT
ejpam-3438	56	29	)	)	PUNCT
ejpam-3438	56	30	.	.	PUNCT
ejpam-3438	57	1	a	a	DET
ejpam-3438	57	2	subset	subset	NOUN
ejpam-3438	57	3	a	a	PRON
ejpam-3438	57	4	of	of	ADP
ejpam-3438	57	5	x	x	PRON
ejpam-3438	57	6	is	be	AUX
ejpam-3438	57	7	called	call	VERB
ejpam-3438	57	8	β	β	VERB
ejpam-3438	57	9	-	-	VERB
ejpam-3438	57	10	open	open	ADJ
ejpam-3438	57	11	with	with	ADP
ejpam-3438	57	12	respect	respect	NOUN
ejpam-3438	57	13	to	to	ADP
ejpam-3438	57	14	the	the	DET
ejpam-3438	57	15	ideal	ideal	NOUN
ejpam-3438	57	16	i	i	PRON
ejpam-3438	57	17	,	,	PUNCT
ejpam-3438	57	18	or	or	CCONJ
ejpam-3438	57	19	βi	βi	PRON
ejpam-3438	57	20	-open	-open	ADJ
ejpam-3438	57	21	,	,	PUNCT
ejpam-3438	57	22	if	if	SCONJ
ejpam-3438	57	23	there	there	PRON
ejpam-3438	57	24	exists	exist	VERB
ejpam-3438	57	25	an	an	DET
ejpam-3438	57	26	open	open	ADJ
ejpam-3438	57	27	set	set	NOUN
ejpam-3438	57	28	u	u	PRON
ejpam-3438	57	29	such	such	ADJ
ejpam-3438	57	30	that	that	SCONJ
ejpam-3438	57	31	(	(	PUNCT
ejpam-3438	57	32	1	1	X
ejpam-3438	57	33	)	)	PUNCT
ejpam-3438	57	34	u	u	NOUN
ejpam-3438	57	35	−	−	PROPN
ejpam-3438	57	36	a	a	DET
ejpam-3438	57	37	∈	∈	PROPN
ejpam-3438	58	1	i	i	PRON
ejpam-3438	58	2	,	,	PUNCT
ejpam-3438	58	3	and	and	CCONJ
ejpam-3438	58	4	(	(	PUNCT
ejpam-3438	58	5	2	2	X
ejpam-3438	58	6	)	)	PUNCT
ejpam-3438	58	7	a	a	DET
ejpam-3438	58	8	−	−	PROPN
ejpam-3438	58	9	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	58	10	)	)	PUNCT
ejpam-3438	58	11	)	)	PUNCT
ejpam-3438	58	12	)	)	PUNCT
ejpam-3438	59	1	∈	∈	PROPN
ejpam-3438	59	2	i.	i.	NOUN
ejpam-3438	59	3	a	a	PRON
ejpam-3438	59	4	subset	subset	VERB
ejpam-3438	59	5	a	a	PRON
ejpam-3438	59	6	of	of	ADP
ejpam-3438	59	7	an	an	DET
ejpam-3438	59	8	ideal	ideal	ADJ
ejpam-3438	59	9	topological	topological	ADJ
ejpam-3438	59	10	space	space	NOUN
ejpam-3438	59	11	(	(	PUNCT
ejpam-3438	59	12	x	x	X
ejpam-3438	59	13	,	,	PUNCT
ejpam-3438	59	14	τ	τ	PROPN
ejpam-3438	59	15	,	,	PUNCT
ejpam-3438	59	16	i	i	PROPN
ejpam-3438	59	17	)	)	PUNCT
ejpam-3438	59	18	is	be	AUX
ejpam-3438	59	19	said	say	VERB
ejpam-3438	59	20	to	to	PART
ejpam-3438	59	21	be	be	AUX
ejpam-3438	59	22	βi	βi	PRON
ejpam-3438	59	23	-compact	-compact	NOUN
ejpam-3438	59	24	if	if	SCONJ
ejpam-3438	59	25	every	every	DET
ejpam-3438	59	26	cover	cover	NOUN
ejpam-3438	59	27	of	of	ADP
ejpam-3438	59	28	a	a	DET
ejpam-3438	59	29	by	by	ADP
ejpam-3438	59	30	βi	βi	PRON
ejpam-3438	59	31	-open	-open	NOUN
ejpam-3438	59	32	set	set	NOUN
ejpam-3438	59	33	has	have	VERB
ejpam-3438	59	34	a	a	DET
ejpam-3438	59	35	finite	finite	ADJ
ejpam-3438	59	36	sub	sub	NOUN
ejpam-3438	59	37	-	-	NOUN
ejpam-3438	59	38	cover	cover	NOUN
ejpam-3438	59	39	.	.	PUNCT
ejpam-3438	60	1	a	a	DET
ejpam-3438	60	2	space	space	NOUN
ejpam-3438	60	3	x	x	PUNCT
ejpam-3438	60	4	is	be	AUX
ejpam-3438	60	5	said	say	VERB
ejpam-3438	60	6	to	to	PART
ejpam-3438	60	7	be	be	AUX
ejpam-3438	60	8	a	a	DET
ejpam-3438	60	9	βi	βi	NOUN
ejpam-3438	60	10	-compact	-compact	NOUN
ejpam-3438	60	11	space	space	NOUN
ejpam-3438	60	12	if	if	SCONJ
ejpam-3438	60	13	it	it	PRON
ejpam-3438	60	14	is	be	AUX
ejpam-3438	60	15	βi	βi	PRON
ejpam-3438	60	16	-compact	-compact	NOUN
ejpam-3438	60	17	as	as	ADP
ejpam-3438	60	18	a	a	DET
ejpam-3438	60	19	subset	subset	NOUN
ejpam-3438	60	20	.	.	PUNCT
ejpam-3438	61	1	a	a	DET
ejpam-3438	61	2	subset	subset	NOUN
ejpam-3438	61	3	a	a	PRON
ejpam-3438	61	4	of	of	ADP
ejpam-3438	61	5	an	an	DET
ejpam-3438	61	6	ideal	ideal	ADJ
ejpam-3438	61	7	topological	topological	ADJ
ejpam-3438	61	8	space	space	NOUN
ejpam-3438	61	9	(	(	PUNCT
ejpam-3438	61	10	x	x	X
ejpam-3438	61	11	,	,	PUNCT
ejpam-3438	61	12	τ	τ	PROPN
ejpam-3438	61	13	,	,	PUNCT
ejpam-3438	61	14	i	i	PROPN
ejpam-3438	61	15	)	)	PUNCT
ejpam-3438	61	16	is	be	AUX
ejpam-3438	61	17	said	say	VERB
ejpam-3438	61	18	to	to	PART
ejpam-3438	61	19	be	be	AUX
ejpam-3438	61	20	a	a	DET
ejpam-3438	61	21	compatible	compatible	ADJ
ejpam-3438	61	22	βi	βi	PRON
ejpam-3438	61	23	-compact	-compact	NOUN
ejpam-3438	61	24	,	,	PUNCT
ejpam-3438	61	25	or	or	CCONJ
ejpam-3438	61	26	simply	simply	ADV
ejpam-3438	61	27	cβi	cβi	VERB
ejpam-3438	61	28	-compact	-compact	NOUN
ejpam-3438	61	29	,	,	PUNCT
ejpam-3438	61	30	if	if	SCONJ
ejpam-3438	61	31	every	every	DET
ejpam-3438	61	32	cover	cover	NOUN
ejpam-3438	61	33	{	{	PUNCT
ejpam-3438	61	34	uλ	uλ	NOUN
ejpam-3438	61	35	:	:	PUNCT
ejpam-3438	61	36	λ	λ	X
ejpam-3438	61	37	∈	∈	PROPN
ejpam-3438	61	38	λ	λ	PROPN
ejpam-3438	61	39	}	}	PUNCT
ejpam-3438	61	40	of	of	ADP
ejpam-3438	61	41	a	a	PRON
ejpam-3438	61	42	by	by	ADP
ejpam-3438	61	43	β	β	NOUN
ejpam-3438	61	44	-	-	ADJ
ejpam-3438	61	45	open	open	ADJ
ejpam-3438	61	46	set	set	NOUN
ejpam-3438	61	47	has	have	VERB
ejpam-3438	61	48	a	a	DET
ejpam-3438	61	49	finite	finite	ADJ
ejpam-3438	61	50	subset	subset	NOUN
ejpam-3438	61	51	λ0	λ0	NOUN
ejpam-3438	61	52	of	of	ADP
ejpam-3438	61	53	λ	λ	NOUN
ejpam-3438	61	54	such	such	ADJ
ejpam-3438	61	55	that	that	DET
ejpam-3438	61	56	a−	a−	PROPN
ejpam-3438	61	57	⋃	⋃	PROPN
ejpam-3438	61	58	{	{	PUNCT
ejpam-3438	61	59	uλ	uλ	NOUN
ejpam-3438	61	60	:	:	PUNCT
ejpam-3438	61	61	λ	λ	PROPN
ejpam-3438	61	62	∈	∈	NOUN
ejpam-3438	61	63	λ0	λ0	NOUN
ejpam-3438	61	64	}	}	PUNCT
ejpam-3438	61	65	∈	∈	PROPN
ejpam-3438	61	66	i.	i.	NOUN
ejpam-3438	61	67	an	an	DET
ejpam-3438	61	68	ideal	ideal	ADJ
ejpam-3438	61	69	topological	topological	ADJ
ejpam-3438	61	70	space	space	NOUN
ejpam-3438	61	71	(	(	PUNCT
ejpam-3438	61	72	x	x	X
ejpam-3438	61	73	,	,	PUNCT
ejpam-3438	61	74	τ	τ	PROPN
ejpam-3438	61	75	,	,	PUNCT
ejpam-3438	61	76	i	i	PROPN
ejpam-3438	61	77	)	)	PUNCT
ejpam-3438	61	78	is	be	AUX
ejpam-3438	61	79	said	say	VERB
ejpam-3438	61	80	to	to	PART
ejpam-3438	61	81	be	be	AUX
ejpam-3438	61	82	a	a	DET
ejpam-3438	61	83	cβi	cβi	NOUN
ejpam-3438	61	84	-compact	-compact	NOUN
ejpam-3438	61	85	space	space	NOUN
ejpam-3438	61	86	if	if	SCONJ
ejpam-3438	61	87	it	it	PRON
ejpam-3438	61	88	is	be	AUX
ejpam-3438	61	89	cβi	cβi	NOUN
ejpam-3438	61	90	-compact	-compact	NOUN
ejpam-3438	61	91	as	as	ADP
ejpam-3438	61	92	a	a	DET
ejpam-3438	61	93	subset	subset	NOUN
ejpam-3438	61	94	.	.	PUNCT
ejpam-3438	62	1	a	a	DET
ejpam-3438	62	2	subset	subset	NOUN
ejpam-3438	62	3	a	a	PRON
ejpam-3438	62	4	of	of	ADP
ejpam-3438	62	5	an	an	DET
ejpam-3438	62	6	ideal	ideal	ADJ
ejpam-3438	62	7	topological	topological	ADJ
ejpam-3438	62	8	space	space	NOUN
ejpam-3438	62	9	(	(	PUNCT
ejpam-3438	62	10	x	x	X
ejpam-3438	62	11	,	,	PUNCT
ejpam-3438	62	12	τ	τ	PROPN
ejpam-3438	62	13	,	,	PUNCT
ejpam-3438	62	14	i	i	PROPN
ejpam-3438	62	15	)	)	PUNCT
ejpam-3438	62	16	is	be	AUX
ejpam-3438	62	17	said	say	VERB
ejpam-3438	62	18	to	to	PART
ejpam-3438	62	19	be	be	AUX
ejpam-3438	62	20	countably	countably	ADV
ejpam-3438	62	21	βi	βi	PRON
ejpam-3438	62	22	-compact	-compact	NOUN
ejpam-3438	62	23	if	if	SCONJ
ejpam-3438	62	24	every	every	DET
ejpam-3438	62	25	countable	countable	ADJ
ejpam-3438	62	26	cover	cover	NOUN
ejpam-3438	62	27	{	{	PUNCT
ejpam-3438	62	28	un	un	PROPN
ejpam-3438	62	29	:	:	PUNCT
ejpam-3438	62	30	n	n	CCONJ
ejpam-3438	62	31	∈	∈	PROPN
ejpam-3438	62	32	n	n	CCONJ
ejpam-3438	62	33	}	}	PUNCT
ejpam-3438	62	34	of	of	ADP
ejpam-3438	62	35	a	a	DET
ejpam-3438	62	36	by	by	ADP
ejpam-3438	62	37	βi	βi	PRON
ejpam-3438	62	38	-open	-open	NOUN
ejpam-3438	62	39	set	set	NOUN
ejpam-3438	62	40	,	,	PUNCT
ejpam-3438	62	41	there	there	PRON
ejpam-3438	62	42	exists	exist	VERB
ejpam-3438	62	43	a	a	DET
ejpam-3438	62	44	finite	finite	NOUN
ejpam-3438	62	45	subset	subset	NOUN
ejpam-3438	62	46	{	{	PUNCT
ejpam-3438	62	47	i1	i1	PROPN
ejpam-3438	62	48	,	,	PUNCT
ejpam-3438	62	49	i2	i2	PROPN
ejpam-3438	62	50	,	,	PUNCT
ejpam-3438	62	51	.	.	PUNCT
ejpam-3438	62	52	.	.	PUNCT
ejpam-3438	63	1	.	.	PUNCT
ejpam-3438	64	1	,	,	PUNCT
ejpam-3438	64	2	ik	ik	PROPN
ejpam-3438	64	3	}	}	PUNCT
ejpam-3438	64	4	of	of	ADP
ejpam-3438	64	5	n	n	PRON
ejpam-3438	64	6	such	such	ADJ
ejpam-3438	64	7	that	that	DET
ejpam-3438	64	8	a−	a−	PROPN
ejpam-3438	64	9	⋃	⋃	PROPN
ejpam-3438	64	10	{	{	PUNCT
ejpam-3438	64	11	uij	uij	X
ejpam-3438	64	12	:	:	PUNCT
ejpam-3438	64	13	j	j	PROPN
ejpam-3438	64	14	=	=	SYM
ejpam-3438	64	15	1	1	NUM
ejpam-3438	64	16	,	,	PUNCT
ejpam-3438	64	17	2	2	NUM
ejpam-3438	64	18	,	,	PUNCT
ejpam-3438	64	19	.	.	PUNCT
ejpam-3438	64	20	.	.	PUNCT
ejpam-3438	64	21	.	.	PUNCT
ejpam-3438	65	1	,	,	PUNCT
ejpam-3438	65	2	k	k	X
ejpam-3438	65	3	}	}	PUNCT
ejpam-3438	65	4	∈	∈	PROPN
ejpam-3438	65	5	i.	i.	NOUN
ejpam-3438	65	6	an	an	DET
ejpam-3438	65	7	ideal	ideal	ADJ
ejpam-3438	65	8	topological	topological	ADJ
ejpam-3438	65	9	space	space	NOUN
ejpam-3438	65	10	(	(	PUNCT
ejpam-3438	65	11	x	x	X
ejpam-3438	65	12	,	,	PUNCT
ejpam-3438	65	13	τ	τ	PROPN
ejpam-3438	65	14	,	,	PUNCT
ejpam-3438	65	15	i	i	PROPN
ejpam-3438	65	16	)	)	PUNCT
ejpam-3438	65	17	is	be	AUX
ejpam-3438	65	18	said	say	VERB
ejpam-3438	65	19	to	to	PART
ejpam-3438	65	20	be	be	AUX
ejpam-3438	65	21	a	a	DET
ejpam-3438	65	22	countably	countably	ADJ
ejpam-3438	65	23	βi	βi	PRON
ejpam-3438	65	24	-compact	-compact	NOUN
ejpam-3438	65	25	space	space	NOUN
ejpam-3438	65	26	if	if	SCONJ
ejpam-3438	65	27	x	x	PRON
ejpam-3438	65	28	is	be	AUX
ejpam-3438	65	29	countably	countably	ADV
ejpam-3438	65	30	βi	βi	NOUN
ejpam-3438	65	31	-compact	-compact	NOUN
ejpam-3438	65	32	as	as	ADP
ejpam-3438	65	33	a	a	DET
ejpam-3438	65	34	subset	subset	NOUN
ejpam-3438	65	35	.	.	PUNCT
ejpam-3438	66	1	the	the	DET
ejpam-3438	66	2	concept	concept	NOUN
ejpam-3438	66	3	∗-hyperconnectedness	∗-hyperconnectedness	PRON
ejpam-3438	66	4	was	be	AUX
ejpam-3438	66	5	introduced	introduce	VERB
ejpam-3438	66	6	by	by	ADP
ejpam-3438	66	7	ekici	ekici	PROPN
ejpam-3438	66	8	et	et	PROPN
ejpam-3438	66	9	al	al	PROPN
ejpam-3438	66	10	.	.	PUNCT
ejpam-3438	67	1	[	[	X
ejpam-3438	67	2	2	2	NUM
ejpam-3438	67	3	]	]	PUNCT
ejpam-3438	67	4	,	,	PUNCT
ejpam-3438	67	5	and	and	CCONJ
ejpam-3438	67	6	the	the	DET
ejpam-3438	67	7	concept	concept	NOUN
ejpam-3438	67	8	i∗-hyperconnectedness	i∗-hyperconnectedness	ADV
ejpam-3438	67	9	was	be	AUX
ejpam-3438	67	10	introduced	introduce	VERB
ejpam-3438	67	11	by	by	ADP
ejpam-3438	67	12	abd	abd	PROPN
ejpam-3438	67	13	el	el	PROPN
ejpam-3438	67	14	-	-	PROPN
ejpam-3438	67	15	monsef	monsef	PROPN
ejpam-3438	67	16	et	et	PROPN
ejpam-3438	67	17	al	al	PROPN
ejpam-3438	67	18	.	.	PUNCT
ejpam-3438	68	1	[	[	X
ejpam-3438	68	2	7	7	NUM
ejpam-3438	68	3	]	]	PUNCT
ejpam-3438	68	4	.	.	PUNCT
ejpam-3438	69	1	as	as	SCONJ
ejpam-3438	69	2	defined	define	VERB
ejpam-3438	69	3	in	in	ADP
ejpam-3438	69	4	[	[	X
ejpam-3438	69	5	4	4	X
ejpam-3438	69	6	]	]	X
ejpam-3438	69	7	an	an	DET
ejpam-3438	69	8	ideal	ideal	ADJ
ejpam-3438	69	9	topological	topological	ADJ
ejpam-3438	69	10	space	space	NOUN
ejpam-3438	69	11	(	(	PUNCT
ejpam-3438	69	12	x	x	X
ejpam-3438	69	13	,	,	PUNCT
ejpam-3438	69	14	τ	τ	PROPN
ejpam-3438	69	15	,	,	PUNCT
ejpam-3438	69	16	i	i	PROPN
ejpam-3438	69	17	)	)	PUNCT
ejpam-3438	69	18	is	be	AUX
ejpam-3438	69	19	said	say	VERB
ejpam-3438	69	20	to	to	PART
ejpam-3438	69	21	be	be	AUX
ejpam-3438	69	22	∗-hyperconnected	∗-hyperconnecte	VERB
ejpam-3438	69	23	if	if	SCONJ
ejpam-3438	69	24	cl∗(a	cl∗(a	NOUN
ejpam-3438	69	25	)	)	PUNCT
ejpam-3438	69	26	=	=	PUNCT
ejpam-3438	70	1	x	x	PUNCT
ejpam-3438	70	2	for	for	SCONJ
ejpam-3438	70	3	every	every	DET
ejpam-3438	70	4	non	non	ADJ
ejpam-3438	70	5	-	-	ADJ
ejpam-3438	70	6	empty	empty	ADJ
ejpam-3438	70	7	open	open	NOUN
ejpam-3438	70	8	subset	subset	VERB
ejpam-3438	70	9	a	a	PRON
ejpam-3438	70	10	of	of	ADP
ejpam-3438	70	11	x	x	NOUN
ejpam-3438	70	12	,	,	PUNCT
ejpam-3438	70	13	and	and	CCONJ
ejpam-3438	70	14	as	as	SCONJ
ejpam-3438	70	15	defined	define	VERB
ejpam-3438	70	16	in	in	ADP
ejpam-3438	70	17	[	[	X
ejpam-3438	70	18	3	3	NUM
ejpam-3438	70	19	]	]	PUNCT
ejpam-3438	70	20	,	,	PUNCT
ejpam-3438	70	21	an	an	DET
ejpam-3438	70	22	ideal	ideal	ADJ
ejpam-3438	70	23	topological	topological	ADJ
ejpam-3438	70	24	space	space	NOUN
ejpam-3438	70	25	(	(	PUNCT
ejpam-3438	70	26	x	x	X
ejpam-3438	70	27	,	,	PUNCT
ejpam-3438	70	28	τ	τ	PROPN
ejpam-3438	70	29	,	,	PUNCT
ejpam-3438	70	30	i	i	PROPN
ejpam-3438	70	31	)	)	PUNCT
ejpam-3438	70	32	is	be	AUX
ejpam-3438	70	33	said	say	VERB
ejpam-3438	70	34	to	to	PART
ejpam-3438	70	35	be	be	AUX
ejpam-3438	70	36	i∗-hyperconnected	i∗-hyperconnecte	VERB
ejpam-3438	70	37	if	if	SCONJ
ejpam-3438	70	38	x	x	PUNCT
ejpam-3438	70	39	−	−	NOUN
ejpam-3438	70	40	cl∗(a	cl∗(a	NOUN
ejpam-3438	70	41	)	)	PUNCT
ejpam-3438	70	42	∈	∈	PROPN
ejpam-3438	70	43	i	i	PRON
ejpam-3438	70	44	for	for	ADP
ejpam-3438	70	45	every	every	DET
ejpam-3438	70	46	non	non	ADJ
ejpam-3438	70	47	-	-	ADJ
ejpam-3438	70	48	empty	empty	ADJ
ejpam-3438	70	49	open	open	NOUN
ejpam-3438	70	50	subset	subset	VERB
ejpam-3438	70	51	a	a	PRON
ejpam-3438	70	52	of	of	ADP
ejpam-3438	70	53	x.	x.	NOUN
ejpam-3438	70	54	given	give	VERB
ejpam-3438	70	55	these	these	DET
ejpam-3438	70	56	insights	insight	NOUN
ejpam-3438	70	57	,	,	PUNCT
ejpam-3438	70	58	we	we	PRON
ejpam-3438	70	59	introduce	introduce	VERB
ejpam-3438	70	60	the	the	DET
ejpam-3438	70	61	following	follow	VERB
ejpam-3438	70	62	parallel	parallel	ADJ
ejpam-3438	70	63	concept	concept	NOUN
ejpam-3438	70	64	.	.	PUNCT
ejpam-3438	71	1	an	an	DET
ejpam-3438	71	2	ideal	ideal	ADJ
ejpam-3438	71	3	topological	topological	ADJ
ejpam-3438	71	4	space	space	NOUN
ejpam-3438	71	5	(	(	PUNCT
ejpam-3438	71	6	x	x	X
ejpam-3438	71	7	,	,	PUNCT
ejpam-3438	71	8	τ	τ	PROPN
ejpam-3438	71	9	,	,	PUNCT
ejpam-3438	71	10	i	i	PROPN
ejpam-3438	71	11	)	)	PUNCT
ejpam-3438	71	12	is	be	AUX
ejpam-3438	71	13	said	say	VERB
ejpam-3438	71	14	to	to	PART
ejpam-3438	71	15	be	be	AUX
ejpam-3438	71	16	β∗i	β∗i	PUNCT
ejpam-3438	71	17	-hyperconnected	-hyperconnecte	VERB
ejpam-3438	71	18	if	if	SCONJ
ejpam-3438	71	19	x	x	NUM
ejpam-3438	71	20	−	−	NOUN
ejpam-3438	71	21	cl∗(a	cl∗(a	NOUN
ejpam-3438	71	22	)	)	PUNCT
ejpam-3438	71	23	∈	∈	PROPN
ejpam-3438	72	1	i	i	PRON
ejpam-3438	72	2	for	for	ADP
ejpam-3438	72	3	every	every	DET
ejpam-3438	72	4	non	non	ADJ
ejpam-3438	72	5	-	-	ADJ
ejpam-3438	72	6	empty	empty	ADJ
ejpam-3438	72	7	βi	βi	X
ejpam-3438	72	8	-open	-open	NOUN
ejpam-3438	72	9	subset	subset	VERB
ejpam-3438	72	10	a	a	PRON
ejpam-3438	72	11	of	of	ADP
ejpam-3438	72	12	x.	x.	PROPN
ejpam-3438	72	13	g.	g.	PROPN
ejpam-3438	72	14	catalan	catalan	PROPN
ejpam-3438	72	15	,	,	PUNCT
ejpam-3438	72	16	r.	r.	PROPN
ejpam-3438	72	17	padua	padua	PROPN
ejpam-3438	72	18	,	,	PUNCT
ejpam-3438	72	19	m.	m.	PROPN
ejpam-3438	72	20	baldado	baldado	PROPN
ejpam-3438	72	21	jr	jr	PROPN
ejpam-3438	72	22	.	.	PROPN
ejpam-3438	72	23	/	/	SYM
ejpam-3438	72	24	eur	eur	PROPN
ejpam-3438	72	25	.	.	PUNCT
ejpam-3438	73	1	j.	j.	PROPN
ejpam-3438	73	2	pure	pure	PROPN
ejpam-3438	73	3	appl	appl	PROPN
ejpam-3438	73	4	.	.	PROPN
ejpam-3438	73	5	math	math	PROPN
ejpam-3438	73	6	,	,	PUNCT
ejpam-3438	73	7	12	12	NUM
ejpam-3438	73	8	(	(	PUNCT
ejpam-3438	73	9	3	3	NUM
ejpam-3438	73	10	)	)	PUNCT
ejpam-3438	73	11	(	(	PUNCT
ejpam-3438	73	12	2019	2019	NUM
ejpam-3438	73	13	)	)	PUNCT
ejpam-3438	73	14	,	,	PUNCT
ejpam-3438	73	15	893	893	NUM
ejpam-3438	73	16	-	-	SYM
ejpam-3438	73	17	905	905	NUM
ejpam-3438	73	18	895	895	NUM
ejpam-3438	73	19	for	for	ADP
ejpam-3438	73	20	the	the	DET
ejpam-3438	73	21	concepts	concept	NOUN
ejpam-3438	73	22	that	that	PRON
ejpam-3438	73	23	were	be	AUX
ejpam-3438	73	24	not	not	PART
ejpam-3438	73	25	discussed	discuss	VERB
ejpam-3438	73	26	here	here	ADV
ejpam-3438	73	27	please	please	INTJ
ejpam-3438	73	28	refer	refer	VERB
ejpam-3438	73	29	to	to	ADP
ejpam-3438	73	30	[	[	X
ejpam-3438	73	31	6	6	NUM
ejpam-3438	73	32	,	,	PUNCT
ejpam-3438	73	33	14	14	NUM
ejpam-3438	73	34	,	,	PUNCT
ejpam-3438	73	35	15	15	NUM
ejpam-3438	73	36	]	]	PUNCT
ejpam-3438	73	37	.	.	PUNCT
ejpam-3438	74	1	2	2	X
ejpam-3438	74	2	.	.	X
ejpam-3438	74	3	β	β	X
ejpam-3438	74	4	-	-	ADJ
ejpam-3438	74	5	open	open	ADJ
ejpam-3438	74	6	sets	set	NOUN
ejpam-3438	74	7	with	with	ADP
ejpam-3438	74	8	respect	respect	NOUN
ejpam-3438	74	9	to	to	ADP
ejpam-3438	74	10	an	an	DET
ejpam-3438	74	11	ideal	ideal	NOUN
ejpam-3438	74	12	in	in	ADP
ejpam-3438	74	13	this	this	DET
ejpam-3438	74	14	section	section	NOUN
ejpam-3438	74	15	,	,	PUNCT
ejpam-3438	74	16	we	we	PRON
ejpam-3438	74	17	investigated	investigate	VERB
ejpam-3438	74	18	the	the	DET
ejpam-3438	74	19	concept	concept	NOUN
ejpam-3438	74	20	β	β	NOUN
ejpam-3438	74	21	-	-	VERB
ejpam-3438	74	22	open	open	ADJ
ejpam-3438	74	23	in	in	ADP
ejpam-3438	74	24	a	a	DET
ejpam-3438	74	25	direction	direction	NOUN
ejpam-3438	74	26	parallel	parallel	NOUN
ejpam-3438	74	27	to	to	ADP
ejpam-3438	74	28	the	the	DET
ejpam-3438	74	29	investigation	investigation	NOUN
ejpam-3438	74	30	of	of	ADP
ejpam-3438	74	31	semi	semi	ADJ
ejpam-3438	74	32	-	-	ADJ
ejpam-3438	74	33	open	open	ADJ
ejpam-3438	74	34	sets	set	NOUN
ejpam-3438	74	35	in	in	ADP
ejpam-3438	74	36	[	[	X
ejpam-3438	74	37	32	32	NUM
ejpam-3438	74	38	]	]	PUNCT
ejpam-3438	74	39	.	.	PUNCT
ejpam-3438	75	1	lemma	lemma	PROPN
ejpam-3438	75	2	1	1	X
ejpam-3438	75	3	.	.	PUNCT
ejpam-3438	76	1	let	let	AUX
ejpam-3438	76	2	(	(	PUNCT
ejpam-3438	76	3	x	x	NOUN
ejpam-3438	76	4	,	,	PUNCT
ejpam-3438	76	5	τ	τ	X
ejpam-3438	76	6	)	)	PUNCT
ejpam-3438	76	7	be	be	VERB
ejpam-3438	76	8	a	a	DET
ejpam-3438	76	9	topological	topological	ADJ
ejpam-3438	76	10	space	space	NOUN
ejpam-3438	76	11	and	and	CCONJ
ejpam-3438	76	12	a	a	DET
ejpam-3438	76	13	be	be	AUX
ejpam-3438	76	14	a	a	DET
ejpam-3438	76	15	subset	subset	NOUN
ejpam-3438	76	16	of	of	ADP
ejpam-3438	76	17	x.	x.	NOUN
ejpam-3438	76	18	then	then	ADV
ejpam-3438	76	19	int(a	int(a	PROPN
ejpam-3438	76	20	)	)	PUNCT
ejpam-3438	76	21	=	=	SYM
ejpam-3438	76	22	int(cl(a	int(cl(a	PROPN
ejpam-3438	76	23	)	)	PUNCT
ejpam-3438	76	24	)	)	PUNCT
ejpam-3438	77	1	=	=	PUNCT
ejpam-3438	77	2	int(cl(int(a	int(cl(int(a	PROPN
ejpam-3438	77	3	)	)	PUNCT
ejpam-3438	77	4	)	)	PUNCT
ejpam-3438	77	5	)	)	PUNCT
ejpam-3438	77	6	.	.	PUNCT
ejpam-3438	78	1	proof	proof	NOUN
ejpam-3438	78	2	.	.	PUNCT
ejpam-3438	79	1	let	let	VERB
ejpam-3438	79	2	(	(	PUNCT
ejpam-3438	79	3	x	x	NOUN
ejpam-3438	79	4	,	,	PUNCT
ejpam-3438	79	5	τ	τ	X
ejpam-3438	79	6	)	)	PUNCT
ejpam-3438	79	7	be	be	VERB
ejpam-3438	79	8	a	a	DET
ejpam-3438	79	9	topological	topological	ADJ
ejpam-3438	79	10	space	space	NOUN
ejpam-3438	79	11	and	and	CCONJ
ejpam-3438	79	12	a	a	DET
ejpam-3438	79	13	be	be	AUX
ejpam-3438	79	14	a	a	DET
ejpam-3438	79	15	subset	subset	NOUN
ejpam-3438	79	16	of	of	ADP
ejpam-3438	79	17	x.	x.	NOUN
ejpam-3438	79	18	then	then	ADV
ejpam-3438	79	19	we	we	PRON
ejpam-3438	79	20	have	have	VERB
ejpam-3438	79	21	,	,	PUNCT
ejpam-3438	79	22	int(cl(a	int(cl(a	PROPN
ejpam-3438	79	23	)	)	PUNCT
ejpam-3438	79	24	)	)	PUNCT
ejpam-3438	80	1	=	=	SYM
ejpam-3438	80	2	int(fr(a)∪	int(fr(a)∪	PROPN
ejpam-3438	81	1	int(a	int(a	NOUN
ejpam-3438	81	2	)	)	PUNCT
ejpam-3438	81	3	)	)	PUNCT
ejpam-3438	82	1	=	=	PUNCT
ejpam-3438	82	2	int(a	int(a	PROPN
ejpam-3438	82	3	)	)	PUNCT
ejpam-3438	82	4	=	=	NOUN
ejpam-3438	82	5	int(fr(int(a))∪	int(fr(int(a))∪	NOUN
ejpam-3438	82	6	int(a	int(a	PROPN
ejpam-3438	82	7	)	)	PUNCT
ejpam-3438	82	8	)	)	PUNCT
ejpam-3438	83	1	=	=	PUNCT
ejpam-3438	83	2	int(cl(int(a	int(cl(int(a	PROPN
ejpam-3438	83	3	)	)	PUNCT
ejpam-3438	83	4	)	)	PUNCT
ejpam-3438	83	5	)	)	PUNCT
ejpam-3438	83	6	.	.	PUNCT
ejpam-3438	84	1	lemma	lemma	PROPN
ejpam-3438	84	2	2	2	NUM
ejpam-3438	84	3	characterizes	characterize	VERB
ejpam-3438	84	4	β	β	X
ejpam-3438	84	5	-	-	ADJ
ejpam-3438	84	6	open	open	ADJ
ejpam-3438	84	7	sets	set	NOUN
ejpam-3438	84	8	.	.	PUNCT
ejpam-3438	85	1	lemma	lemma	PROPN
ejpam-3438	85	2	2	2	X
ejpam-3438	85	3	.	.	PUNCT
ejpam-3438	86	1	let	let	VERB
ejpam-3438	86	2	(	(	PUNCT
ejpam-3438	86	3	x	x	X
ejpam-3438	86	4	,	,	PUNCT
ejpam-3438	86	5	τ	τ	PROPN
ejpam-3438	86	6	,	,	PUNCT
ejpam-3438	86	7	i	i	PRON
ejpam-3438	86	8	)	)	PUNCT
ejpam-3438	86	9	be	be	VERB
ejpam-3438	86	10	an	an	DET
ejpam-3438	86	11	ideal	ideal	ADJ
ejpam-3438	86	12	topological	topological	ADJ
ejpam-3438	86	13	space	space	NOUN
ejpam-3438	86	14	.	.	PUNCT
ejpam-3438	87	1	a	a	DET
ejpam-3438	87	2	subset	subset	NOUN
ejpam-3438	87	3	a	a	PRON
ejpam-3438	87	4	of	of	ADP
ejpam-3438	87	5	x	x	NOUN
ejpam-3438	87	6	is	be	AUX
ejpam-3438	87	7	β	β	X
ejpam-3438	87	8	-	-	ADJ
ejpam-3438	87	9	open	open	ADJ
ejpam-3438	87	10	if	if	SCONJ
ejpam-3438	87	11	and	and	CCONJ
ejpam-3438	87	12	only	only	ADV
ejpam-3438	87	13	if	if	SCONJ
ejpam-3438	87	14	there	there	PRON
ejpam-3438	87	15	exists	exist	VERB
ejpam-3438	87	16	an	an	DET
ejpam-3438	87	17	open	open	ADJ
ejpam-3438	87	18	set	set	NOUN
ejpam-3438	87	19	u	u	PRON
ejpam-3438	87	20	such	such	ADJ
ejpam-3438	87	21	that	that	SCONJ
ejpam-3438	87	22	u	u	PROPN
ejpam-3438	87	23	⊆	⊆	NUM
ejpam-3438	87	24	a	a	DET
ejpam-3438	87	25	⊆	⊆	NUM
ejpam-3438	87	26	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	87	27	)	)	PUNCT
ejpam-3438	87	28	)	)	PUNCT
ejpam-3438	87	29	)	)	PUNCT
ejpam-3438	87	30	.	.	PUNCT
ejpam-3438	88	1	proof	proof	NOUN
ejpam-3438	88	2	.	.	PUNCT
ejpam-3438	89	1	assume	assume	VERB
ejpam-3438	89	2	that	that	SCONJ
ejpam-3438	89	3	a	a	PRON
ejpam-3438	89	4	is	be	AUX
ejpam-3438	89	5	β	β	NOUN
ejpam-3438	89	6	-	-	ADJ
ejpam-3438	89	7	open	open	ADJ
ejpam-3438	89	8	.	.	PUNCT
ejpam-3438	90	1	then	then	ADV
ejpam-3438	90	2	a	a	DET
ejpam-3438	90	3	⊆	⊆	NUM
ejpam-3438	90	4	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-3438	90	5	)	)	PUNCT
ejpam-3438	90	6	)	)	PUNCT
ejpam-3438	90	7	)	)	PUNCT
ejpam-3438	90	8	.	.	PUNCT
ejpam-3438	91	1	let	let	VERB
ejpam-3438	91	2	u	u	NOUN
ejpam-3438	91	3	=	=	VERB
ejpam-3438	91	4	int(a	int(a	PROPN
ejpam-3438	91	5	)	)	PUNCT
ejpam-3438	91	6	.	.	PUNCT
ejpam-3438	92	1	then	then	ADV
ejpam-3438	92	2	u	u	PRON
ejpam-3438	92	3	is	be	AUX
ejpam-3438	92	4	open	open	ADJ
ejpam-3438	92	5	and	and	CCONJ
ejpam-3438	92	6	,	,	PUNCT
ejpam-3438	92	7	by	by	ADP
ejpam-3438	92	8	lemma	lemma	PROPN
ejpam-3438	92	9	1	1	NUM
ejpam-3438	92	10	u	u	NOUN
ejpam-3438	92	11	⊆	⊆	NUM
ejpam-3438	92	12	a	a	DET
ejpam-3438	92	13	⊆	⊆	NUM
ejpam-3438	92	14	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-3438	92	15	)	)	PUNCT
ejpam-3438	92	16	)	)	PUNCT
ejpam-3438	92	17	)	)	PUNCT
ejpam-3438	93	1	=	=	PUNCT
ejpam-3438	93	2	cl(int(cl(int(a	cl(int(cl(int(a	NOUN
ejpam-3438	93	3	)	)	PUNCT
ejpam-3438	93	4	)	)	PUNCT
ejpam-3438	93	5	)	)	PUNCT
ejpam-3438	93	6	)	)	PUNCT
ejpam-3438	94	1	=	=	SYM
ejpam-3438	94	2	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	94	3	)	)	PUNCT
ejpam-3438	94	4	)	)	PUNCT
ejpam-3438	94	5	)	)	PUNCT
ejpam-3438	94	6	.	.	PUNCT
ejpam-3438	95	1	conversely	conversely	ADV
ejpam-3438	95	2	,	,	PUNCT
ejpam-3438	95	3	assume	assume	VERB
ejpam-3438	95	4	that	that	SCONJ
ejpam-3438	95	5	there	there	PRON
ejpam-3438	95	6	exists	exist	VERB
ejpam-3438	95	7	an	an	DET
ejpam-3438	95	8	open	open	ADJ
ejpam-3438	95	9	set	set	NOUN
ejpam-3438	95	10	u	u	PRON
ejpam-3438	95	11	such	such	ADJ
ejpam-3438	95	12	that	that	SCONJ
ejpam-3438	95	13	u	u	PROPN
ejpam-3438	95	14	⊆	⊆	NUM
ejpam-3438	95	15	a	a	DET
ejpam-3438	95	16	⊆	⊆	NUM
ejpam-3438	95	17	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	95	18	)	)	PUNCT
ejpam-3438	95	19	)	)	PUNCT
ejpam-3438	95	20	)	)	PUNCT
ejpam-3438	95	21	.	.	PUNCT
ejpam-3438	96	1	since	since	SCONJ
ejpam-3438	96	2	u	u	PRON
ejpam-3438	96	3	⊆	⊆	NUM
ejpam-3438	96	4	a	a	PRON
ejpam-3438	96	5	,	,	PUNCT
ejpam-3438	96	6	cl(u	cl(u	NOUN
ejpam-3438	96	7	)	)	PUNCT
ejpam-3438	96	8	⊆	⊆	NUM
ejpam-3438	96	9	cl(a	cl(a	NUM
ejpam-3438	96	10	)	)	PUNCT
ejpam-3438	96	11	.	.	PUNCT
ejpam-3438	97	1	hence	hence	ADV
ejpam-3438	97	2	,	,	PUNCT
ejpam-3438	97	3	int(cl(u	int(cl(u	PROPN
ejpam-3438	97	4	)	)	PUNCT
ejpam-3438	97	5	)	)	PUNCT
ejpam-3438	98	1	⊆	⊆	NUM
ejpam-3438	98	2	int(cl(a	int(cl(a	PROPN
ejpam-3438	98	3	)	)	PUNCT
ejpam-3438	98	4	)	)	PUNCT
ejpam-3438	98	5	.	.	PUNCT
ejpam-3438	99	1	and	and	CCONJ
ejpam-3438	99	2	so	so	ADV
ejpam-3438	99	3	,	,	PUNCT
ejpam-3438	99	4	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	99	5	)	)	PUNCT
ejpam-3438	99	6	)	)	PUNCT
ejpam-3438	99	7	)	)	PUNCT
ejpam-3438	100	1	⊆	⊆	NUM
ejpam-3438	100	2	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-3438	100	3	)	)	PUNCT
ejpam-3438	100	4	)	)	PUNCT
ejpam-3438	100	5	)	)	PUNCT
ejpam-3438	101	1	thus	thus	ADV
ejpam-3438	101	2	,	,	PUNCT
ejpam-3438	101	3	a	a	DET
ejpam-3438	101	4	⊆	⊆	NUM
ejpam-3438	101	5	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-3438	101	6	)	)	PUNCT
ejpam-3438	101	7	)	)	PUNCT
ejpam-3438	101	8	)	)	PUNCT
ejpam-3438	101	9	.	.	PUNCT
ejpam-3438	102	1	lemma	lemma	PROPN
ejpam-3438	102	2	3	3	NUM
ejpam-3438	102	3	says	say	VERB
ejpam-3438	102	4	that	that	SCONJ
ejpam-3438	102	5	every	every	DET
ejpam-3438	102	6	open	open	ADJ
ejpam-3438	102	7	set	set	NOUN
ejpam-3438	102	8	is	be	AUX
ejpam-3438	102	9	a	a	DET
ejpam-3438	102	10	βi	βi	X
ejpam-3438	102	11	-open	-open	NOUN
ejpam-3438	102	12	set	set	NOUN
ejpam-3438	102	13	,	,	PUNCT
ejpam-3438	102	14	every	every	DET
ejpam-3438	102	15	element	element	NOUN
ejpam-3438	102	16	of	of	ADP
ejpam-3438	102	17	the	the	DET
ejpam-3438	102	18	ideal	ideal	NOUN
ejpam-3438	102	19	is	be	AUX
ejpam-3438	102	20	a	a	DET
ejpam-3438	102	21	βi	βi	X
ejpam-3438	102	22	-open	-open	NOUN
ejpam-3438	102	23	set	set	NOUN
ejpam-3438	102	24	,	,	PUNCT
ejpam-3438	102	25	and	and	CCONJ
ejpam-3438	102	26	every	every	DET
ejpam-3438	102	27	β	β	X
ejpam-3438	102	28	-	-	ADJ
ejpam-3438	102	29	open	open	ADJ
ejpam-3438	102	30	set	set	NOUN
ejpam-3438	102	31	is	be	AUX
ejpam-3438	102	32	a	a	DET
ejpam-3438	102	33	βi	βi	X
ejpam-3438	102	34	-open	-open	NOUN
ejpam-3438	102	35	set	set	NOUN
ejpam-3438	102	36	.	.	PUNCT
ejpam-3438	103	1	lemma	lemma	PROPN
ejpam-3438	103	2	3	3	X
ejpam-3438	103	3	.	.	PUNCT
ejpam-3438	104	1	let	let	VERB
ejpam-3438	104	2	(	(	PUNCT
ejpam-3438	104	3	x	x	X
ejpam-3438	104	4	,	,	PUNCT
ejpam-3438	104	5	τ	τ	PROPN
ejpam-3438	104	6	,	,	PUNCT
ejpam-3438	104	7	i	i	PRON
ejpam-3438	104	8	)	)	PUNCT
ejpam-3438	104	9	be	be	VERB
ejpam-3438	104	10	an	an	DET
ejpam-3438	104	11	ideal	ideal	ADJ
ejpam-3438	104	12	topological	topological	ADJ
ejpam-3438	104	13	space	space	NOUN
ejpam-3438	104	14	.	.	PUNCT
ejpam-3438	105	1	(	(	PUNCT
ejpam-3438	105	2	i	i	NOUN
ejpam-3438	105	3	)	)	PUNCT
ejpam-3438	105	4	if	if	SCONJ
ejpam-3438	105	5	a	a	PRON
ejpam-3438	105	6	is	be	AUX
ejpam-3438	105	7	an	an	DET
ejpam-3438	105	8	open	open	ADJ
ejpam-3438	105	9	set	set	NOUN
ejpam-3438	105	10	,	,	PUNCT
ejpam-3438	105	11	then	then	ADV
ejpam-3438	105	12	a	a	PRON
ejpam-3438	105	13	is	be	AUX
ejpam-3438	105	14	an	an	DET
ejpam-3438	105	15	βi	βi	ADV
ejpam-3438	105	16	-	-	PUNCT
ejpam-3438	105	17	open	open	ADJ
ejpam-3438	105	18	set	set	NOUN
ejpam-3438	105	19	.	.	PUNCT
ejpam-3438	106	1	(	(	PUNCT
ejpam-3438	106	2	ii	ii	NOUN
ejpam-3438	106	3	)	)	PUNCT
ejpam-3438	106	4	if	if	SCONJ
ejpam-3438	106	5	a	a	DET
ejpam-3438	106	6	∈	∈	X
ejpam-3438	107	1	i	i	PRON
ejpam-3438	107	2	,	,	PUNCT
ejpam-3438	107	3	then	then	ADV
ejpam-3438	107	4	a	a	PRON
ejpam-3438	107	5	is	be	AUX
ejpam-3438	107	6	an	an	DET
ejpam-3438	107	7	βi	βi	ADV
ejpam-3438	107	8	-	-	PUNCT
ejpam-3438	107	9	open	open	ADJ
ejpam-3438	107	10	set	set	NOUN
ejpam-3438	107	11	.	.	PUNCT
ejpam-3438	108	1	(	(	PUNCT
ejpam-3438	108	2	iii	iii	X
ejpam-3438	108	3	)	)	PUNCT
ejpam-3438	108	4	if	if	SCONJ
ejpam-3438	108	5	a	a	PRON
ejpam-3438	108	6	is	be	AUX
ejpam-3438	108	7	a	a	DET
ejpam-3438	108	8	β	β	NOUN
ejpam-3438	108	9	-	-	ADJ
ejpam-3438	108	10	open	open	ADJ
ejpam-3438	108	11	set	set	NOUN
ejpam-3438	108	12	,	,	PUNCT
ejpam-3438	108	13	then	then	ADV
ejpam-3438	108	14	a	a	PRON
ejpam-3438	108	15	is	be	AUX
ejpam-3438	108	16	an	an	DET
ejpam-3438	108	17	βi	βi	ADV
ejpam-3438	108	18	-	-	PUNCT
ejpam-3438	108	19	open	open	ADJ
ejpam-3438	108	20	set	set	NOUN
ejpam-3438	108	21	.	.	PUNCT
ejpam-3438	109	1	proof	proof	NOUN
ejpam-3438	109	2	.	.	PUNCT
ejpam-3438	110	1	(	(	PUNCT
ejpam-3438	110	2	1	1	X
ejpam-3438	110	3	)	)	PUNCT
ejpam-3438	110	4	if	if	SCONJ
ejpam-3438	110	5	a	a	PRON
ejpam-3438	110	6	is	be	AUX
ejpam-3438	110	7	open	open	ADJ
ejpam-3438	110	8	,	,	PUNCT
ejpam-3438	110	9	then	then	ADV
ejpam-3438	110	10	we	we	PRON
ejpam-3438	110	11	let	let	VERB
ejpam-3438	110	12	u	u	PRON
ejpam-3438	110	13	=	=	NOUN
ejpam-3438	110	14	a.	a.	NOUN
ejpam-3438	110	15	observe	observe	VERB
ejpam-3438	110	16	that	that	SCONJ
ejpam-3438	110	17	u	u	NOUN
ejpam-3438	110	18	−	−	PROPN
ejpam-3438	110	19	a	a	DET
ejpam-3438	110	20	=	=	NOUN
ejpam-3438	110	21	∅	∅	NOUN
ejpam-3438	110	22	∈	∈	PROPN
ejpam-3438	110	23	i	i	PRON
ejpam-3438	110	24	,	,	PUNCT
ejpam-3438	110	25	and	and	CCONJ
ejpam-3438	110	26	a	a	DET
ejpam-3438	110	27	−	−	NOUN
ejpam-3438	110	28	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	110	29	)	)	PUNCT
ejpam-3438	110	30	)	)	PUNCT
ejpam-3438	110	31	)	)	PUNCT
ejpam-3438	111	1	=	=	PUNCT
ejpam-3438	111	2	a	a	DET
ejpam-3438	111	3	−	−	NOUN
ejpam-3438	111	4	cl(u	cl(u	NOUN
ejpam-3438	111	5	)	)	PUNCT
ejpam-3438	111	6	=	=	PUNCT
ejpam-3438	111	7	a	a	DET
ejpam-3438	111	8	−	−	NOUN
ejpam-3438	111	9	cl(a	cl(a	NUM
ejpam-3438	111	10	)	)	PUNCT
ejpam-3438	111	11	=	=	SYM
ejpam-3438	111	12	∅	∅	NOUN
ejpam-3438	111	13	∈	∈	PROPN
ejpam-3438	111	14	i.	i.	NOUN
ejpam-3438	111	15	this	this	PRON
ejpam-3438	111	16	shows	show	VERB
ejpam-3438	111	17	that	that	SCONJ
ejpam-3438	111	18	a	a	PRON
ejpam-3438	111	19	is	be	AUX
ejpam-3438	111	20	an	an	DET
ejpam-3438	111	21	βi	βi	PRON
ejpam-3438	111	22	open	open	ADJ
ejpam-3438	111	23	set	set	NOUN
ejpam-3438	111	24	.	.	PUNCT
ejpam-3438	112	1	(	(	PUNCT
ejpam-3438	112	2	2	2	X
ejpam-3438	112	3	)	)	PUNCT
ejpam-3438	112	4	if	if	SCONJ
ejpam-3438	112	5	a	a	PRON
ejpam-3438	112	6	∈	∈	X
ejpam-3438	112	7	i	i	PRON
ejpam-3438	112	8	,	,	PUNCT
ejpam-3438	112	9	then	then	ADV
ejpam-3438	112	10	we	we	PRON
ejpam-3438	112	11	let	let	VERB
ejpam-3438	112	12	u	u	PRON
ejpam-3438	112	13	=	=	PUNCT
ejpam-3438	112	14	∅.	∅.	VERB
ejpam-3438	112	15	observe	observe	VERB
ejpam-3438	112	16	that	that	SCONJ
ejpam-3438	112	17	u	u	NOUN
ejpam-3438	112	18	−	−	PROPN
ejpam-3438	112	19	a	a	DET
ejpam-3438	112	20	=	=	NOUN
ejpam-3438	112	21	∅	∅	NOUN
ejpam-3438	112	22	∈	∈	PROPN
ejpam-3438	112	23	i	i	PRON
ejpam-3438	112	24	,	,	PUNCT
ejpam-3438	112	25	and	and	CCONJ
ejpam-3438	112	26	a	a	DET
ejpam-3438	112	27	−	−	NOUN
ejpam-3438	112	28	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	112	29	)	)	PUNCT
ejpam-3438	112	30	)	)	PUNCT
ejpam-3438	112	31	)	)	PUNCT
ejpam-3438	113	1	=	=	PUNCT
ejpam-3438	113	2	a	a	DET
ejpam-3438	113	3	−	−	NOUN
ejpam-3438	113	4	∅	∅	NOUN
ejpam-3438	113	5	=	=	PUNCT
ejpam-3438	113	6	a	a	DET
ejpam-3438	113	7	∈	∈	PROPN
ejpam-3438	113	8	i.	i.	NOUN
ejpam-3438	113	9	this	this	PRON
ejpam-3438	113	10	shows	show	VERB
ejpam-3438	113	11	that	that	SCONJ
ejpam-3438	113	12	a	a	PRON
ejpam-3438	113	13	is	be	AUX
ejpam-3438	113	14	an	an	DET
ejpam-3438	113	15	βi	βi	X
ejpam-3438	113	16	-open	-open	NOUN
ejpam-3438	113	17	set	set	NOUN
ejpam-3438	113	18	.	.	PUNCT
ejpam-3438	114	1	(	(	PUNCT
ejpam-3438	114	2	3	3	X
ejpam-3438	114	3	)	)	PUNCT
ejpam-3438	114	4	if	if	SCONJ
ejpam-3438	114	5	a	a	PRON
ejpam-3438	114	6	is	be	AUX
ejpam-3438	114	7	β	β	NOUN
ejpam-3438	114	8	-	-	ADJ
ejpam-3438	114	9	open	open	ADJ
ejpam-3438	114	10	,	,	PUNCT
ejpam-3438	114	11	then	then	ADV
ejpam-3438	114	12	by	by	ADP
ejpam-3438	114	13	lemma	lemma	PROPN
ejpam-3438	114	14	2	2	NUM
ejpam-3438	114	15	there	there	PRON
ejpam-3438	114	16	exists	exist	VERB
ejpam-3438	114	17	an	an	DET
ejpam-3438	114	18	open	open	ADJ
ejpam-3438	114	19	set	set	NOUN
ejpam-3438	114	20	u	u	PRON
ejpam-3438	114	21	such	such	ADJ
ejpam-3438	114	22	that	that	SCONJ
ejpam-3438	114	23	u	u	PROPN
ejpam-3438	114	24	⊆	⊆	NUM
ejpam-3438	114	25	a	a	DET
ejpam-3438	114	26	⊆	⊆	NUM
ejpam-3438	114	27	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	114	28	)	)	PUNCT
ejpam-3438	114	29	)	)	PUNCT
ejpam-3438	114	30	)	)	PUNCT
ejpam-3438	114	31	.	.	PUNCT
ejpam-3438	115	1	observe	observe	VERB
ejpam-3438	115	2	that	that	SCONJ
ejpam-3438	115	3	u	u	NOUN
ejpam-3438	115	4	−	−	PROPN
ejpam-3438	115	5	a	a	DET
ejpam-3438	115	6	=	=	NOUN
ejpam-3438	115	7	∅	∅	NOUN
ejpam-3438	115	8	∈	∈	PROPN
ejpam-3438	115	9	i	i	PRON
ejpam-3438	115	10	,	,	PUNCT
ejpam-3438	115	11	and	and	CCONJ
ejpam-3438	115	12	a	a	DET
ejpam-3438	115	13	−	−	NOUN
ejpam-3438	115	14	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	115	15	)	)	PUNCT
ejpam-3438	115	16	)	)	PUNCT
ejpam-3438	115	17	)	)	PUNCT
ejpam-3438	116	1	=	=	PRON
ejpam-3438	116	2	∅	∅	NOUN
ejpam-3438	116	3	∈	∈	PROPN
ejpam-3438	116	4	i.	i.	NOUN
ejpam-3438	116	5	this	this	PRON
ejpam-3438	116	6	shows	show	VERB
ejpam-3438	116	7	that	that	SCONJ
ejpam-3438	116	8	a	a	PRON
ejpam-3438	116	9	is	be	AUX
ejpam-3438	116	10	an	an	DET
ejpam-3438	116	11	βi	βi	X
ejpam-3438	116	12	-open	-open	NOUN
ejpam-3438	116	13	set	set	NOUN
ejpam-3438	116	14	.	.	PUNCT
ejpam-3438	117	1	lemma	lemma	PROPN
ejpam-3438	117	2	4	4	NUM
ejpam-3438	117	3	says	say	VERB
ejpam-3438	117	4	that	that	SCONJ
ejpam-3438	117	5	if	if	SCONJ
ejpam-3438	117	6	i	i	PRON
ejpam-3438	117	7	is	be	AUX
ejpam-3438	117	8	the	the	DET
ejpam-3438	117	9	minimal	minimal	ADJ
ejpam-3438	117	10	ideal	ideal	NOUN
ejpam-3438	117	11	,	,	PUNCT
ejpam-3438	117	12	then	then	ADV
ejpam-3438	117	13	the	the	DET
ejpam-3438	117	14	βi	βi	PROPN
ejpam-3438	117	15	-open	-open	PROPN
ejpam-3438	117	16	sets	set	NOUN
ejpam-3438	117	17	are	be	AUX
ejpam-3438	117	18	precisely	precisely	ADV
ejpam-3438	117	19	the	the	DET
ejpam-3438	117	20	β	β	ADJ
ejpam-3438	117	21	-	-	ADJ
ejpam-3438	117	22	open	open	ADJ
ejpam-3438	117	23	sets	set	NOUN
ejpam-3438	117	24	.	.	PUNCT
ejpam-3438	118	1	g.	g.	PROPN
ejpam-3438	118	2	catalan	catalan	PROPN
ejpam-3438	118	3	,	,	PUNCT
ejpam-3438	118	4	r.	r.	PROPN
ejpam-3438	118	5	padua	padua	PROPN
ejpam-3438	118	6	,	,	PUNCT
ejpam-3438	118	7	m.	m.	PROPN
ejpam-3438	118	8	baldado	baldado	PROPN
ejpam-3438	118	9	jr	jr	PROPN
ejpam-3438	118	10	.	.	PROPN
ejpam-3438	118	11	/	/	SYM
ejpam-3438	118	12	eur	eur	PROPN
ejpam-3438	118	13	.	.	PUNCT
ejpam-3438	119	1	j.	j.	PROPN
ejpam-3438	119	2	pure	pure	PROPN
ejpam-3438	119	3	appl	appl	PROPN
ejpam-3438	119	4	.	.	PROPN
ejpam-3438	119	5	math	math	PROPN
ejpam-3438	119	6	,	,	PUNCT
ejpam-3438	119	7	12	12	NUM
ejpam-3438	119	8	(	(	PUNCT
ejpam-3438	119	9	3	3	NUM
ejpam-3438	119	10	)	)	PUNCT
ejpam-3438	119	11	(	(	PUNCT
ejpam-3438	119	12	2019	2019	NUM
ejpam-3438	119	13	)	)	PUNCT
ejpam-3438	119	14	,	,	PUNCT
ejpam-3438	119	15	893	893	NUM
ejpam-3438	119	16	-	-	SYM
ejpam-3438	119	17	905	905	NUM
ejpam-3438	119	18	896	896	NUM
ejpam-3438	119	19	lemma	lemma	PROPN
ejpam-3438	119	20	4	4	X
ejpam-3438	119	21	.	.	PUNCT
ejpam-3438	120	1	let	let	VERB
ejpam-3438	120	2	(	(	PUNCT
ejpam-3438	120	3	x	x	X
ejpam-3438	120	4	,	,	PUNCT
ejpam-3438	120	5	τ	τ	PROPN
ejpam-3438	120	6	,	,	PUNCT
ejpam-3438	120	7	i	i	PRON
ejpam-3438	120	8	)	)	PUNCT
ejpam-3438	120	9	be	be	VERB
ejpam-3438	120	10	an	an	DET
ejpam-3438	120	11	ideal	ideal	ADJ
ejpam-3438	120	12	topological	topological	ADJ
ejpam-3438	120	13	space	space	NOUN
ejpam-3438	120	14	.	.	PUNCT
ejpam-3438	121	1	if	if	SCONJ
ejpam-3438	121	2	i	i	PRON
ejpam-3438	121	3	is	be	AUX
ejpam-3438	121	4	not	not	PART
ejpam-3438	121	5	countably	countably	ADV
ejpam-3438	121	6	additive	additive	ADJ
ejpam-3438	121	7	,	,	PUNCT
ejpam-3438	121	8	then	then	ADV
ejpam-3438	121	9	the	the	DET
ejpam-3438	121	10	following	following	ADJ
ejpam-3438	121	11	statements	statement	NOUN
ejpam-3438	121	12	are	be	AUX
ejpam-3438	121	13	equivalent	equivalent	ADJ
ejpam-3438	121	14	.	.	PUNCT
ejpam-3438	122	1	(	(	PUNCT
ejpam-3438	122	2	i	i	NOUN
ejpam-3438	122	3	)	)	PUNCT
ejpam-3438	122	4	if	if	SCONJ
ejpam-3438	122	5	i	i	PRON
ejpam-3438	122	6	=	=	SYM
ejpam-3438	122	7	{	{	PUNCT
ejpam-3438	122	8	∅	∅	NOUN
ejpam-3438	122	9	}	}	PUNCT
ejpam-3438	122	10	.	.	PUNCT
ejpam-3438	123	1	(	(	PUNCT
ejpam-3438	123	2	ii	ii	NOUN
ejpam-3438	123	3	)	)	PUNCT
ejpam-3438	123	4	a	a	PRON
ejpam-3438	123	5	is	be	AUX
ejpam-3438	123	6	a	a	DET
ejpam-3438	123	7	β	β	NOUN
ejpam-3438	123	8	-	-	ADJ
ejpam-3438	123	9	open	open	ADJ
ejpam-3438	123	10	set	set	NOUN
ejpam-3438	123	11	if	if	SCONJ
ejpam-3438	123	12	and	and	CCONJ
ejpam-3438	123	13	only	only	ADV
ejpam-3438	123	14	if	if	SCONJ
ejpam-3438	123	15	a	a	PRON
ejpam-3438	123	16	is	be	AUX
ejpam-3438	123	17	a	a	DET
ejpam-3438	123	18	βi	βi	ADV
ejpam-3438	123	19	-	-	PUNCT
ejpam-3438	123	20	open	open	ADJ
ejpam-3438	123	21	set	set	NOUN
ejpam-3438	123	22	.	.	PUNCT
ejpam-3438	124	1	proof	proof	NOUN
ejpam-3438	124	2	.	.	PUNCT
ejpam-3438	125	1	assume	assume	VERB
ejpam-3438	125	2	that	that	SCONJ
ejpam-3438	125	3	i	i	PRON
ejpam-3438	125	4	=	=	PUNCT
ejpam-3438	125	5	{	{	PUNCT
ejpam-3438	125	6	∅	∅	NOUN
ejpam-3438	125	7	}	}	PUNCT
ejpam-3438	125	8	,	,	PUNCT
ejpam-3438	125	9	and	and	CCONJ
ejpam-3438	125	10	a	a	DET
ejpam-3438	125	11	be	be	AUX
ejpam-3438	125	12	a	a	DET
ejpam-3438	125	13	β	β	NOUN
ejpam-3438	125	14	-	-	ADJ
ejpam-3438	125	15	open	open	ADJ
ejpam-3438	125	16	set	set	NOUN
ejpam-3438	125	17	.	.	PUNCT
ejpam-3438	126	1	then	then	ADV
ejpam-3438	126	2	by	by	ADP
ejpam-3438	126	3	lemma	lemma	PROPN
ejpam-3438	126	4	3	3	NUM
ejpam-3438	126	5	,	,	PUNCT
ejpam-3438	126	6	a	a	PRON
ejpam-3438	126	7	is	be	AUX
ejpam-3438	126	8	a	a	DET
ejpam-3438	126	9	βi	βi	X
ejpam-3438	126	10	-open	-open	NOUN
ejpam-3438	126	11	set	set	NOUN
ejpam-3438	126	12	.	.	PUNCT
ejpam-3438	127	1	conversely	conversely	ADV
ejpam-3438	127	2	,	,	PUNCT
ejpam-3438	127	3	let	let	VERB
ejpam-3438	127	4	a	a	PRON
ejpam-3438	127	5	be	be	AUX
ejpam-3438	127	6	a	a	DET
ejpam-3438	127	7	βi	βi	NOUN
ejpam-3438	127	8	-open	-open	NOUN
ejpam-3438	127	9	set	set	NOUN
ejpam-3438	127	10	.	.	PUNCT
ejpam-3438	128	1	then	then	ADV
ejpam-3438	128	2	there	there	PRON
ejpam-3438	128	3	exists	exist	VERB
ejpam-3438	128	4	an	an	DET
ejpam-3438	128	5	open	open	ADJ
ejpam-3438	128	6	set	set	NOUN
ejpam-3438	128	7	u	u	PRON
ejpam-3438	128	8	such	such	ADJ
ejpam-3438	128	9	that	that	SCONJ
ejpam-3438	128	10	u−a	u−a	ADJ
ejpam-3438	128	11	∈	∈	NOUN
ejpam-3438	128	12	i	i	PRON
ejpam-3438	128	13	and	and	CCONJ
ejpam-3438	128	14	a−cl(int(cl(u	a−cl(int(cl(u	NOUN
ejpam-3438	128	15	)	)	PUNCT
ejpam-3438	128	16	)	)	PUNCT
ejpam-3438	128	17	)	)	PUNCT
ejpam-3438	129	1	∈	∈	PROPN
ejpam-3438	129	2	i.	i.	NOUN
ejpam-3438	129	3	since	since	SCONJ
ejpam-3438	129	4	∅	∅	NOUN
ejpam-3438	129	5	,	,	PUNCT
ejpam-3438	129	6	u−a	u−a	ADJ
ejpam-3438	129	7	∈	∈	NOUN
ejpam-3438	129	8	∅	∅	NOUN
ejpam-3438	129	9	and	and	CCONJ
ejpam-3438	129	10	a−cl(int(cl(u	a−cl(int(cl(u	NOUN
ejpam-3438	129	11	)	)	PUNCT
ejpam-3438	129	12	)	)	PUNCT
ejpam-3438	129	13	)	)	PUNCT
ejpam-3438	130	1	∈	∈	NOUN
ejpam-3438	130	2	∅	∅	NOUN
ejpam-3438	130	3	,	,	PUNCT
ejpam-3438	130	4	that	that	ADV
ejpam-3438	130	5	is	is	ADV
ejpam-3438	130	6	,	,	PUNCT
ejpam-3438	130	7	u	u	NOUN
ejpam-3438	130	8	⊆	⊆	NUM
ejpam-3438	130	9	a	a	PRON
ejpam-3438	130	10	and	and	CCONJ
ejpam-3438	130	11	a	a	DET
ejpam-3438	130	12	⊆	⊆	NUM
ejpam-3438	130	13	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	130	14	)	)	PUNCT
ejpam-3438	130	15	)	)	PUNCT
ejpam-3438	130	16	)	)	PUNCT
ejpam-3438	130	17	.	.	PUNCT
ejpam-3438	131	1	thus	thus	ADV
ejpam-3438	131	2	,	,	PUNCT
ejpam-3438	131	3	u	u	NOUN
ejpam-3438	131	4	⊆	⊆	NUM
ejpam-3438	131	5	a	a	DET
ejpam-3438	131	6	⊆	⊆	NUM
ejpam-3438	131	7	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	131	8	)	)	PUNCT
ejpam-3438	131	9	)	)	PUNCT
ejpam-3438	131	10	)	)	PUNCT
ejpam-3438	131	11	.	.	PUNCT
ejpam-3438	132	1	by	by	ADP
ejpam-3438	132	2	lemma	lemma	PROPN
ejpam-3438	132	3	2	2	NUM
ejpam-3438	132	4	,	,	PUNCT
ejpam-3438	132	5	a	a	PRON
ejpam-3438	132	6	is	be	AUX
ejpam-3438	132	7	a	a	DET
ejpam-3438	132	8	β	β	NOUN
ejpam-3438	132	9	-	-	ADJ
ejpam-3438	132	10	open	open	ADJ
ejpam-3438	132	11	set	set	NOUN
ejpam-3438	132	12	.	.	PUNCT
ejpam-3438	133	1	next	next	ADV
ejpam-3438	133	2	,	,	PUNCT
ejpam-3438	133	3	assume	assume	VERB
ejpam-3438	133	4	that	that	SCONJ
ejpam-3438	133	5	a	a	PRON
ejpam-3438	133	6	is	be	AUX
ejpam-3438	133	7	a	a	DET
ejpam-3438	133	8	β	β	NOUN
ejpam-3438	133	9	-	-	ADJ
ejpam-3438	133	10	open	open	ADJ
ejpam-3438	133	11	set	set	NOUN
ejpam-3438	133	12	if	if	SCONJ
ejpam-3438	133	13	and	and	CCONJ
ejpam-3438	133	14	only	only	ADV
ejpam-3438	133	15	if	if	SCONJ
ejpam-3438	133	16	a	a	PRON
ejpam-3438	133	17	is	be	AUX
ejpam-3438	133	18	a	a	DET
ejpam-3438	133	19	βi	βi	X
ejpam-3438	133	20	-open	-open	NOUN
ejpam-3438	133	21	set	set	NOUN
ejpam-3438	133	22	,	,	PUNCT
ejpam-3438	133	23	and	and	CCONJ
ejpam-3438	133	24	suppose	suppose	VERB
ejpam-3438	133	25	that	that	SCONJ
ejpam-3438	133	26	i	i	PRON
ejpam-3438	133	27	6=	6=	PRON
ejpam-3438	133	28	{	{	PUNCT
ejpam-3438	133	29	∅	∅	NOUN
ejpam-3438	133	30	}	}	PUNCT
ejpam-3438	133	31	.	.	PUNCT
ejpam-3438	134	1	let	let	VERB
ejpam-3438	134	2	b	b	X
ejpam-3438	134	3	∈	∈	PROPN
ejpam-3438	134	4	i	i	PRON
ejpam-3438	134	5	with	with	ADP
ejpam-3438	134	6	b	b	PROPN
ejpam-3438	134	7	6=	6=	NUM
ejpam-3438	134	8	∅.	∅.	NOUN
ejpam-3438	134	9	then	then	ADV
ejpam-3438	134	10	by	by	ADP
ejpam-3438	134	11	lemma	lemma	PROPN
ejpam-3438	134	12	3	3	NUM
ejpam-3438	134	13	,	,	PUNCT
ejpam-3438	134	14	b	b	PROPN
ejpam-3438	134	15	is	be	AUX
ejpam-3438	134	16	a	a	DET
ejpam-3438	134	17	βi	βi	X
ejpam-3438	134	18	-open	-open	NOUN
ejpam-3438	134	19	set	set	NOUN
ejpam-3438	134	20	.	.	PUNCT
ejpam-3438	135	1	by	by	ADP
ejpam-3438	135	2	assumption	assumption	NOUN
ejpam-3438	135	3	b	b	PROPN
ejpam-3438	135	4	is	be	AUX
ejpam-3438	135	5	a	a	DET
ejpam-3438	135	6	β	β	NOUN
ejpam-3438	135	7	-	-	ADJ
ejpam-3438	135	8	open	open	ADJ
ejpam-3438	135	9	set	set	NOUN
ejpam-3438	135	10	.	.	PUNCT
ejpam-3438	136	1	by	by	ADP
ejpam-3438	136	2	lemma	lemma	PROPN
ejpam-3438	136	3	2	2	NUM
ejpam-3438	136	4	,	,	PUNCT
ejpam-3438	136	5	there	there	PRON
ejpam-3438	136	6	exists	exist	VERB
ejpam-3438	136	7	an	an	DET
ejpam-3438	136	8	open	open	ADJ
ejpam-3438	136	9	set	set	VERB
ejpam-3438	136	10	u1	u1	NOUN
ejpam-3438	136	11	such	such	ADJ
ejpam-3438	136	12	that	that	DET
ejpam-3438	136	13	u1	u1	NOUN
ejpam-3438	136	14	⊆	⊆	NUM
ejpam-3438	136	15	b	b	PROPN
ejpam-3438	136	16	⊆	⊆	NUM
ejpam-3438	136	17	cl(int(cl(u1	cl(int(cl(u1	NUM
ejpam-3438	136	18	)	)	PUNCT
ejpam-3438	136	19	)	)	PUNCT
ejpam-3438	136	20	)	)	PUNCT
ejpam-3438	136	21	.	.	PUNCT
ejpam-3438	137	1	since	since	SCONJ
ejpam-3438	137	2	b	b	PROPN
ejpam-3438	137	3	∈	∈	PROPN
ejpam-3438	137	4	i	i	PRON
ejpam-3438	137	5	and	and	CCONJ
ejpam-3438	137	6	u1	u1	VERB
ejpam-3438	137	7	⊆	⊆	NUM
ejpam-3438	137	8	b	b	NOUN
ejpam-3438	137	9	,	,	PUNCT
ejpam-3438	137	10	u1	u1	PROPN
ejpam-3438	137	11	∈	∈	PROPN
ejpam-3438	137	12	i.	i.	NOUN
ejpam-3438	137	13	hence	hence	ADV
ejpam-3438	137	14	,	,	PUNCT
ejpam-3438	137	15	u1	u1	PROPN
ejpam-3438	137	16	∪	∪	NOUN
ejpam-3438	137	17	b	b	PROPN
ejpam-3438	137	18	∈	∈	PROPN
ejpam-3438	137	19	i.	i.	NOUN
ejpam-3438	137	20	by	by	ADP
ejpam-3438	137	21	lemma	lemma	PROPN
ejpam-3438	137	22	2	2	NUM
ejpam-3438	137	23	,	,	PUNCT
ejpam-3438	137	24	u1	u1	NOUN
ejpam-3438	137	25	∪	∪	NOUN
ejpam-3438	137	26	b	b	PROPN
ejpam-3438	137	27	is	be	AUX
ejpam-3438	137	28	a	a	DET
ejpam-3438	137	29	βi	βi	X
ejpam-3438	137	30	-open	-open	NOUN
ejpam-3438	137	31	set	set	NOUN
ejpam-3438	137	32	.	.	PUNCT
ejpam-3438	138	1	by	by	ADP
ejpam-3438	138	2	assumption	assumption	NOUN
ejpam-3438	138	3	u1	u1	NOUN
ejpam-3438	138	4	∪	∪	X
ejpam-3438	138	5	b	b	PROPN
ejpam-3438	138	6	is	be	AUX
ejpam-3438	138	7	a	a	DET
ejpam-3438	138	8	β	β	NOUN
ejpam-3438	138	9	-	-	ADJ
ejpam-3438	138	10	open	open	ADJ
ejpam-3438	138	11	set	set	NOUN
ejpam-3438	138	12	.	.	PUNCT
ejpam-3438	139	1	thus	thus	ADV
ejpam-3438	139	2	,	,	PUNCT
ejpam-3438	139	3	there	there	PRON
ejpam-3438	139	4	exists	exist	VERB
ejpam-3438	139	5	an	an	DET
ejpam-3438	139	6	open	open	ADJ
ejpam-3438	139	7	set	set	NOUN
ejpam-3438	139	8	u2	u2	NOUN
ejpam-3438	139	9	such	such	ADJ
ejpam-3438	139	10	that	that	DET
ejpam-3438	139	11	u2	u2	PROPN
ejpam-3438	139	12	⊆	⊆	NUM
ejpam-3438	139	13	(	(	PUNCT
ejpam-3438	139	14	u1	u1	PROPN
ejpam-3438	139	15	∪	∪	ADP
ejpam-3438	139	16	b	b	NOUN
ejpam-3438	139	17	)	)	PUNCT
ejpam-3438	139	18	⊆	⊆	NUM
ejpam-3438	139	19	cl(int(cl(u2	cl(int(cl(u2	NOUN
ejpam-3438	139	20	)	)	PUNCT
ejpam-3438	139	21	)	)	PUNCT
ejpam-3438	139	22	)	)	PUNCT
ejpam-3438	139	23	.	.	PUNCT
ejpam-3438	140	1	since	since	SCONJ
ejpam-3438	140	2	u1	u1	PROPN
ejpam-3438	140	3	∪	∪	VERB
ejpam-3438	140	4	b	b	NOUN
ejpam-3438	140	5	∈	∈	NOUN
ejpam-3438	141	1	i	i	PRON
ejpam-3438	141	2	and	and	CCONJ
ejpam-3438	141	3	u2	u2	PROPN
ejpam-3438	141	4	⊆	⊆	NUM
ejpam-3438	141	5	u1	u1	NOUN
ejpam-3438	141	6	∪	∪	ADP
ejpam-3438	141	7	b	b	NOUN
ejpam-3438	141	8	,	,	PUNCT
ejpam-3438	141	9	u2	u2	PROPN
ejpam-3438	141	10	∈	∈	PROPN
ejpam-3438	141	11	i.	i.	NOUN
ejpam-3438	141	12	hence	hence	ADV
ejpam-3438	141	13	,	,	PUNCT
ejpam-3438	141	14	u1	u1	NOUN
ejpam-3438	141	15	∪	∪	NOUN
ejpam-3438	141	16	u2	u2	PROPN
ejpam-3438	141	17	∪	∪	PROPN
ejpam-3438	141	18	b	b	PROPN
ejpam-3438	141	19	∈	∈	PROPN
ejpam-3438	141	20	i.	i.	NOUN
ejpam-3438	141	21	by	by	ADP
ejpam-3438	141	22	lemma	lemma	PROPN
ejpam-3438	141	23	2	2	NUM
ejpam-3438	141	24	,	,	PUNCT
ejpam-3438	141	25	u1	u1	NOUN
ejpam-3438	141	26	∪	∪	NOUN
ejpam-3438	141	27	u2	u2	PROPN
ejpam-3438	141	28	∪	∪	SYM
ejpam-3438	141	29	b	b	PROPN
ejpam-3438	141	30	is	be	AUX
ejpam-3438	141	31	a	a	DET
ejpam-3438	141	32	βi	βi	X
ejpam-3438	141	33	-open	-open	NOUN
ejpam-3438	141	34	set	set	NOUN
ejpam-3438	141	35	.	.	PUNCT
ejpam-3438	142	1	by	by	ADP
ejpam-3438	142	2	assumption	assumption	NOUN
ejpam-3438	142	3	u1	u1	NOUN
ejpam-3438	142	4	∪	∪	ADP
ejpam-3438	142	5	u2	u2	PROPN
ejpam-3438	142	6	∪	∪	SYM
ejpam-3438	142	7	b	b	PROPN
ejpam-3438	142	8	is	be	AUX
ejpam-3438	142	9	a	a	DET
ejpam-3438	142	10	β	β	NOUN
ejpam-3438	142	11	-	-	ADJ
ejpam-3438	142	12	open	open	ADJ
ejpam-3438	142	13	set	set	NOUN
ejpam-3438	142	14	.	.	PUNCT
ejpam-3438	143	1	thus	thus	ADV
ejpam-3438	143	2	,	,	PUNCT
ejpam-3438	143	3	there	there	PRON
ejpam-3438	143	4	exists	exist	VERB
ejpam-3438	143	5	an	an	DET
ejpam-3438	143	6	open	open	ADJ
ejpam-3438	143	7	set	set	VERB
ejpam-3438	143	8	u3	u3	NOUN
ejpam-3438	143	9	such	such	ADJ
ejpam-3438	143	10	that	that	SCONJ
ejpam-3438	143	11	u3	u3	NOUN
ejpam-3438	143	12	⊆	⊆	NUM
ejpam-3438	143	13	(	(	PUNCT
ejpam-3438	143	14	u1	u1	PROPN
ejpam-3438	143	15	∪	∪	ADP
ejpam-3438	143	16	u2	u2	PROPN
ejpam-3438	143	17	∪	∪	ADJ
ejpam-3438	143	18	b	b	NOUN
ejpam-3438	143	19	)	)	PUNCT
ejpam-3438	143	20	⊆	⊆	NUM
ejpam-3438	143	21	cl(int(cl(u3	cl(int(cl(u3	NOUN
ejpam-3438	143	22	)	)	PUNCT
ejpam-3438	143	23	)	)	PUNCT
ejpam-3438	143	24	)	)	PUNCT
ejpam-3438	143	25	.	.	PUNCT
ejpam-3438	144	1	since	since	SCONJ
ejpam-3438	144	2	u1	u1	PROPN
ejpam-3438	144	3	∪	∪	VERB
ejpam-3438	144	4	u2	u2	PROPN
ejpam-3438	144	5	∪	∪	NOUN
ejpam-3438	144	6	b	b	NOUN
ejpam-3438	144	7	∈	∈	PROPN
ejpam-3438	144	8	i	i	PRON
ejpam-3438	144	9	and	and	CCONJ
ejpam-3438	144	10	u3	u3	NOUN
ejpam-3438	144	11	⊆	⊆	NUM
ejpam-3438	144	12	u1	u1	NOUN
ejpam-3438	144	13	∪	∪	NOUN
ejpam-3438	144	14	u2	u2	PROPN
ejpam-3438	144	15	∪	∪	PROPN
ejpam-3438	144	16	b	b	NOUN
ejpam-3438	144	17	,	,	PUNCT
ejpam-3438	144	18	u3	u3	PROPN
ejpam-3438	144	19	∈	∈	PROPN
ejpam-3438	144	20	i.	i.	NOUN
ejpam-3438	144	21	hence	hence	ADV
ejpam-3438	144	22	,	,	PUNCT
ejpam-3438	144	23	u1	u1	NOUN
ejpam-3438	144	24	∪	∪	NOUN
ejpam-3438	144	25	u2	u2	PROPN
ejpam-3438	144	26	∪	∪	NOUN
ejpam-3438	144	27	u3	u3	NOUN
ejpam-3438	144	28	∪	∪	ADP
ejpam-3438	144	29	b	b	PROPN
ejpam-3438	144	30	∈	∈	PROPN
ejpam-3438	144	31	i.	i.	NOUN
ejpam-3438	144	32	continuing	continue	VERB
ejpam-3438	144	33	in	in	ADP
ejpam-3438	144	34	this	this	DET
ejpam-3438	144	35	manner	manner	NOUN
ejpam-3438	144	36	we	we	PRON
ejpam-3438	144	37	obtain	obtain	VERB
ejpam-3438	144	38	a	a	DET
ejpam-3438	144	39	sequence	sequence	NOUN
ejpam-3438	144	40	〈	〈	NOUN
ejpam-3438	144	41	u1	u1	NOUN
ejpam-3438	144	42	,	,	PUNCT
ejpam-3438	144	43	u2	u2	NOUN
ejpam-3438	144	44	,	,	PUNCT
ejpam-3438	144	45	u3	u3	NOUN
ejpam-3438	144	46	,	,	PUNCT
ejpam-3438	144	47	.	.	PUNCT
ejpam-3438	144	48	.	.	PUNCT
ejpam-3438	145	1	.	.	PUNCT
ejpam-3438	146	1	〉	〉	NOUN
ejpam-3438	146	2	of	of	ADP
ejpam-3438	146	3	set	set	NOUN
ejpam-3438	146	4	in	in	ADP
ejpam-3438	146	5	i	i	PRON
ejpam-3438	146	6	such	such	ADJ
ejpam-3438	146	7	that	that	DET
ejpam-3438	146	8	u1	u1	PROPN
ejpam-3438	146	9	∪	∪	ADP
ejpam-3438	146	10	u2	u2	PROPN
ejpam-3438	146	11	∪	∪	NOUN
ejpam-3438	146	12	u3	u3	NOUN
ejpam-3438	146	13	∪	∪	X
ejpam-3438	146	14	·	·	PUNCT
ejpam-3438	146	15	·	·	PUNCT
ejpam-3438	146	16	·	·	PUNCT
ejpam-3438	147	1	∈	∈	PROPN
ejpam-3438	147	2	i.	i.	NOUN
ejpam-3438	147	3	this	this	PRON
ejpam-3438	147	4	is	be	AUX
ejpam-3438	147	5	a	a	DET
ejpam-3438	147	6	contradiction	contradiction	NOUN
ejpam-3438	147	7	since	since	SCONJ
ejpam-3438	147	8	i	i	PRON
ejpam-3438	147	9	is	be	AUX
ejpam-3438	147	10	not	not	PART
ejpam-3438	147	11	countably	countably	ADV
ejpam-3438	147	12	additive	additive	ADJ
ejpam-3438	147	13	.	.	PUNCT
ejpam-3438	148	1	therefore	therefore	ADV
ejpam-3438	148	2	,	,	PUNCT
ejpam-3438	148	3	i	i	PRON
ejpam-3438	148	4	=	=	NOUN
ejpam-3438	148	5	{	{	PUNCT
ejpam-3438	148	6	∅	∅	NOUN
ejpam-3438	148	7	}	}	PUNCT
ejpam-3438	148	8	.	.	PUNCT
ejpam-3438	149	1	theorem	theorem	NOUN
ejpam-3438	149	2	1	1	NUM
ejpam-3438	149	3	says	say	VERB
ejpam-3438	149	4	that	that	SCONJ
ejpam-3438	149	5	if	if	SCONJ
ejpam-3438	149	6	i	i	PRON
ejpam-3438	149	7	is	be	AUX
ejpam-3438	149	8	the	the	DET
ejpam-3438	149	9	minimal	minimal	ADJ
ejpam-3438	149	10	ideal	ideal	NOUN
ejpam-3438	149	11	,	,	PUNCT
ejpam-3438	149	12	then	then	ADV
ejpam-3438	149	13	the	the	DET
ejpam-3438	149	14	notions	notion	NOUN
ejpam-3438	149	15	β	β	NOUN
ejpam-3438	149	16	-	-	ADJ
ejpam-3438	149	17	compact	compact	ADJ
ejpam-3438	149	18	,	,	PUNCT
ejpam-3438	149	19	βi	βi	PRON
ejpam-3438	149	20	-compact	-compact	VERB
ejpam-3438	149	21	and	and	CCONJ
ejpam-3438	149	22	cβi	cβi	VERB
ejpam-3438	149	23	-compact	-compact	NOUN
ejpam-3438	149	24	coincides	coincide	NOUN
ejpam-3438	149	25	.	.	PUNCT
ejpam-3438	150	1	theorem	theorem	NOUN
ejpam-3438	150	2	1	1	NUM
ejpam-3438	150	3	.	.	X
ejpam-3438	151	1	for	for	ADP
ejpam-3438	151	2	an	an	DET
ejpam-3438	151	3	ideal	ideal	ADJ
ejpam-3438	151	4	topological	topological	ADJ
ejpam-3438	151	5	space	space	NOUN
ejpam-3438	151	6	(	(	PUNCT
ejpam-3438	151	7	x	x	X
ejpam-3438	151	8	,	,	PUNCT
ejpam-3438	151	9	τ	τ	PROPN
ejpam-3438	151	10	,	,	PUNCT
ejpam-3438	151	11	i	i	PROPN
ejpam-3438	151	12	)	)	PUNCT
ejpam-3438	151	13	,	,	PUNCT
ejpam-3438	151	14	the	the	DET
ejpam-3438	151	15	following	follow	VERB
ejpam-3438	151	16	statements	statement	NOUN
ejpam-3438	151	17	are	be	AUX
ejpam-3438	151	18	equivalent	equivalent	ADJ
ejpam-3438	151	19	.	.	PUNCT
ejpam-3438	152	1	(	(	PUNCT
ejpam-3438	152	2	i	i	NOUN
ejpam-3438	152	3	)	)	PUNCT
ejpam-3438	152	4	(	(	PUNCT
ejpam-3438	152	5	x	x	X
ejpam-3438	152	6	,	,	PUNCT
ejpam-3438	152	7	τ	τ	X
ejpam-3438	152	8	)	)	PUNCT
ejpam-3438	152	9	is	be	AUX
ejpam-3438	152	10	a	a	DET
ejpam-3438	152	11	β	β	ADJ
ejpam-3438	152	12	-	-	ADJ
ejpam-3438	152	13	compact	compact	ADJ
ejpam-3438	152	14	space	space	NOUN
ejpam-3438	152	15	.	.	PUNCT
ejpam-3438	153	1	(	(	PUNCT
ejpam-3438	153	2	ii	ii	NOUN
ejpam-3438	153	3	)	)	PUNCT
ejpam-3438	153	4	(	(	PUNCT
ejpam-3438	153	5	x	x	X
ejpam-3438	153	6	,	,	PUNCT
ejpam-3438	153	7	τ	τ	X
ejpam-3438	153	8	,	,	PUNCT
ejpam-3438	153	9	{	{	PUNCT
ejpam-3438	153	10	∅	∅	NOUN
ejpam-3438	153	11	}	}	PUNCT
ejpam-3438	153	12	)	)	PUNCT
ejpam-3438	153	13	is	be	AUX
ejpam-3438	153	14	a	a	DET
ejpam-3438	153	15	βi	βi	ADJ
ejpam-3438	153	16	-	-	ADJ
ejpam-3438	153	17	compact	compact	ADJ
ejpam-3438	153	18	space	space	NOUN
ejpam-3438	153	19	.	.	PUNCT
ejpam-3438	154	1	(	(	PUNCT
ejpam-3438	154	2	iii	iii	X
ejpam-3438	154	3	)	)	PUNCT
ejpam-3438	154	4	(	(	PUNCT
ejpam-3438	154	5	x	x	X
ejpam-3438	154	6	,	,	PUNCT
ejpam-3438	154	7	τ	τ	X
ejpam-3438	154	8	,	,	PUNCT
ejpam-3438	154	9	{	{	PUNCT
ejpam-3438	154	10	∅	∅	NOUN
ejpam-3438	154	11	}	}	PUNCT
ejpam-3438	154	12	)	)	PUNCT
ejpam-3438	154	13	is	be	AUX
ejpam-3438	154	14	a	a	DET
ejpam-3438	154	15	cβi	cβi	NOUN
ejpam-3438	154	16	-	-	PUNCT
ejpam-3438	154	17	compact	compact	ADJ
ejpam-3438	154	18	space	space	NOUN
ejpam-3438	154	19	.	.	PUNCT
ejpam-3438	155	1	proof	proof	NOUN
ejpam-3438	155	2	.	.	PUNCT
ejpam-3438	156	1	(	(	PUNCT
ejpam-3438	156	2	1	1	X
ejpam-3438	156	3	)	)	PUNCT
ejpam-3438	156	4	⇒	⇒	NOUN
ejpam-3438	156	5	(	(	PUNCT
ejpam-3438	156	6	2	2	X
ejpam-3438	156	7	)	)	PUNCT
ejpam-3438	156	8	assume	assume	VERB
ejpam-3438	156	9	that	that	SCONJ
ejpam-3438	156	10	(	(	PUNCT
ejpam-3438	156	11	1	1	X
ejpam-3438	156	12	)	)	PUNCT
ejpam-3438	156	13	holds	hold	NOUN
ejpam-3438	156	14	,	,	PUNCT
ejpam-3438	156	15	and	and	CCONJ
ejpam-3438	156	16	let	let	VERB
ejpam-3438	156	17	{	{	PUNCT
ejpam-3438	156	18	uλ	uλ	X
ejpam-3438	156	19	:	:	PUNCT
ejpam-3438	156	20	λ	λ	X
ejpam-3438	156	21	∈	∈	PROPN
ejpam-3438	156	22	λ	λ	PROPN
ejpam-3438	156	23	}	}	PUNCT
ejpam-3438	156	24	be	be	VERB
ejpam-3438	156	25	a	a	DET
ejpam-3438	156	26	cover	cover	NOUN
ejpam-3438	156	27	of	of	ADP
ejpam-3438	156	28	x	x	PUNCT
ejpam-3438	156	29	by	by	ADP
ejpam-3438	156	30	β	β	ADJ
ejpam-3438	156	31	-	-	ADJ
ejpam-3438	156	32	open	open	ADJ
ejpam-3438	156	33	sets	set	NOUN
ejpam-3438	156	34	.	.	PUNCT
ejpam-3438	157	1	by	by	ADP
ejpam-3438	157	2	lemma	lemma	PROPN
ejpam-3438	157	3	3	3	NUM
ejpam-3438	157	4	uλ	uλ	NOUN
ejpam-3438	157	5	is	be	AUX
ejpam-3438	157	6	a	a	DET
ejpam-3438	157	7	βi	βi	NOUN
ejpam-3438	157	8	-open	-open	NOUN
ejpam-3438	157	9	set	set	VERB
ejpam-3438	157	10	for	for	ADP
ejpam-3438	157	11	all	all	DET
ejpam-3438	157	12	λ	λ	PROPN
ejpam-3438	157	13	∈	∈	PROPN
ejpam-3438	157	14	λ	λ	PROPN
ejpam-3438	157	15	.	.	PUNCT
ejpam-3438	158	1	since	since	SCONJ
ejpam-3438	158	2	x	x	PROPN
ejpam-3438	158	3	is	be	AUX
ejpam-3438	158	4	β	β	NOUN
ejpam-3438	158	5	-	-	ADJ
ejpam-3438	158	6	compact	compact	ADJ
ejpam-3438	158	7	,	,	PUNCT
ejpam-3438	158	8	there	there	PRON
ejpam-3438	158	9	exists	exist	VERB
ejpam-3438	158	10	a	a	DET
ejpam-3438	158	11	finite	finite	NOUN
ejpam-3438	158	12	subset	subset	NOUN
ejpam-3438	158	13	λ0	λ0	NOUN
ejpam-3438	158	14	of	of	ADP
ejpam-3438	158	15	λ	λ	PROPN
ejpam-3438	158	16	such	such	ADJ
ejpam-3438	158	17	that	that	SCONJ
ejpam-3438	158	18	{	{	PUNCT
ejpam-3438	158	19	uλ	uλ	X
ejpam-3438	158	20	:	:	PUNCT
ejpam-3438	158	21	λ	λ	PROPN
ejpam-3438	158	22	∈	∈	NOUN
ejpam-3438	158	23	λ0	λ0	NOUN
ejpam-3438	158	24	}	}	PUNCT
ejpam-3438	158	25	is	be	AUX
ejpam-3438	158	26	still	still	ADV
ejpam-3438	158	27	a	a	DET
ejpam-3438	158	28	cover	cover	NOUN
ejpam-3438	158	29	of	of	ADP
ejpam-3438	158	30	x.	x.	NOUN
ejpam-3438	158	31	by	by	ADP
ejpam-3438	158	32	lemma	lemma	PROPN
ejpam-3438	158	33	4	4	NUM
ejpam-3438	158	34	,	,	PUNCT
ejpam-3438	158	35	uλ	uλ	ADV
ejpam-3438	158	36	is	be	AUX
ejpam-3438	158	37	a	a	DET
ejpam-3438	158	38	βi	βi	NOUN
ejpam-3438	158	39	-open	-open	NOUN
ejpam-3438	158	40	set	set	VERB
ejpam-3438	158	41	for	for	ADP
ejpam-3438	158	42	all	all	DET
ejpam-3438	158	43	λ	λ	PROPN
ejpam-3438	158	44	∈	∈	NOUN
ejpam-3438	158	45	λ0	λ0	NOUN
ejpam-3438	158	46	.	.	PUNCT
ejpam-3438	159	1	hence	hence	ADV
ejpam-3438	159	2	,	,	PUNCT
ejpam-3438	159	3	there	there	PRON
ejpam-3438	159	4	exists	exist	VERB
ejpam-3438	159	5	a	a	DET
ejpam-3438	159	6	finite	finite	ADJ
ejpam-3438	159	7	subcover	subcover	NOUN
ejpam-3438	159	8	of	of	ADP
ejpam-3438	159	9	x	x	PUNCT
ejpam-3438	159	10	by	by	ADP
ejpam-3438	159	11	βi	βi	NOUN
ejpam-3438	159	12	-open	-open	NOUN
ejpam-3438	159	13	sets	set	NOUN
ejpam-3438	159	14	.	.	PUNCT
ejpam-3438	160	1	this	this	PRON
ejpam-3438	160	2	shows	show	VERB
ejpam-3438	160	3	that	that	SCONJ
ejpam-3438	160	4	x	x	PRON
ejpam-3438	160	5	is	be	AUX
ejpam-3438	160	6	a	a	DET
ejpam-3438	160	7	βi	βi	X
ejpam-3438	160	8	compact	compact	ADJ
ejpam-3438	160	9	space	space	NOUN
ejpam-3438	160	10	.	.	PUNCT
ejpam-3438	161	1	(	(	PUNCT
ejpam-3438	161	2	2	2	X
ejpam-3438	161	3	)	)	PUNCT
ejpam-3438	161	4	⇒	⇒	NOUN
ejpam-3438	161	5	(	(	PUNCT
ejpam-3438	161	6	3	3	X
ejpam-3438	161	7	)	)	PUNCT
ejpam-3438	161	8	assume	assume	VERB
ejpam-3438	161	9	that	that	SCONJ
ejpam-3438	161	10	(	(	PUNCT
ejpam-3438	161	11	2	2	X
ejpam-3438	161	12	)	)	PUNCT
ejpam-3438	161	13	holds	hold	NOUN
ejpam-3438	161	14	,	,	PUNCT
ejpam-3438	161	15	and	and	CCONJ
ejpam-3438	161	16	let	let	VERB
ejpam-3438	161	17	{	{	PUNCT
ejpam-3438	161	18	uλ	uλ	X
ejpam-3438	161	19	:	:	PUNCT
ejpam-3438	161	20	λ	λ	X
ejpam-3438	161	21	∈	∈	PROPN
ejpam-3438	161	22	λ	λ	PROPN
ejpam-3438	161	23	}	}	PUNCT
ejpam-3438	161	24	be	be	VERB
ejpam-3438	161	25	a	a	DET
ejpam-3438	161	26	cover	cover	NOUN
ejpam-3438	161	27	of	of	ADP
ejpam-3438	161	28	x	x	PUNCT
ejpam-3438	161	29	by	by	ADP
ejpam-3438	161	30	β	β	ADJ
ejpam-3438	161	31	-	-	ADJ
ejpam-3438	161	32	open	open	ADJ
ejpam-3438	161	33	sets	set	NOUN
ejpam-3438	161	34	.	.	PUNCT
ejpam-3438	162	1	by	by	ADP
ejpam-3438	162	2	lemma	lemma	PROPN
ejpam-3438	162	3	3	3	NUM
ejpam-3438	162	4	uλ	uλ	NOUN
ejpam-3438	162	5	is	be	AUX
ejpam-3438	162	6	a	a	DET
ejpam-3438	162	7	βi	βi	NOUN
ejpam-3438	162	8	-open	-open	NOUN
ejpam-3438	162	9	set	set	VERB
ejpam-3438	162	10	for	for	ADP
ejpam-3438	162	11	all	all	DET
ejpam-3438	162	12	λ	λ	PROPN
ejpam-3438	162	13	∈	∈	PROPN
ejpam-3438	162	14	λ	λ	PROPN
ejpam-3438	162	15	.	.	PUNCT
ejpam-3438	162	16	by	by	ADP
ejpam-3438	162	17	assumption	assumption	NOUN
ejpam-3438	162	18	,	,	PUNCT
ejpam-3438	162	19	there	there	PRON
ejpam-3438	162	20	exists	exist	VERB
ejpam-3438	162	21	a	a	DET
ejpam-3438	162	22	finite	finite	PROPN
ejpam-3438	162	23	subset	subset	PROPN
ejpam-3438	162	24	g.	g.	PROPN
ejpam-3438	162	25	catalan	catalan	PROPN
ejpam-3438	162	26	,	,	PUNCT
ejpam-3438	162	27	r.	r.	PROPN
ejpam-3438	162	28	padua	padua	PROPN
ejpam-3438	162	29	,	,	PUNCT
ejpam-3438	162	30	m.	m.	PROPN
ejpam-3438	162	31	baldado	baldado	PROPN
ejpam-3438	162	32	jr	jr	PROPN
ejpam-3438	162	33	.	.	PROPN
ejpam-3438	162	34	/	/	SYM
ejpam-3438	162	35	eur	eur	PROPN
ejpam-3438	162	36	.	.	PUNCT
ejpam-3438	163	1	j.	j.	PROPN
ejpam-3438	163	2	pure	pure	PROPN
ejpam-3438	163	3	appl	appl	PROPN
ejpam-3438	163	4	.	.	PROPN
ejpam-3438	163	5	math	math	PROPN
ejpam-3438	163	6	,	,	PUNCT
ejpam-3438	163	7	12	12	NUM
ejpam-3438	163	8	(	(	PUNCT
ejpam-3438	163	9	3	3	NUM
ejpam-3438	163	10	)	)	PUNCT
ejpam-3438	163	11	(	(	PUNCT
ejpam-3438	163	12	2019	2019	NUM
ejpam-3438	163	13	)	)	PUNCT
ejpam-3438	163	14	,	,	PUNCT
ejpam-3438	163	15	893	893	NUM
ejpam-3438	163	16	-	-	SYM
ejpam-3438	163	17	905	905	NUM
ejpam-3438	163	18	897	897	NUM
ejpam-3438	163	19	λ0	λ0	NOUN
ejpam-3438	163	20	of	of	ADP
ejpam-3438	163	21	λ	λ	NOUN
ejpam-3438	163	22	such	such	ADJ
ejpam-3438	163	23	that	that	SCONJ
ejpam-3438	163	24	{	{	PUNCT
ejpam-3438	163	25	uλ	uλ	X
ejpam-3438	163	26	:	:	PUNCT
ejpam-3438	163	27	λ	λ	PROPN
ejpam-3438	163	28	∈	∈	NOUN
ejpam-3438	163	29	λ0	λ0	NOUN
ejpam-3438	163	30	}	}	PUNCT
ejpam-3438	163	31	is	be	AUX
ejpam-3438	163	32	still	still	ADV
ejpam-3438	163	33	a	a	DET
ejpam-3438	163	34	cover	cover	NOUN
ejpam-3438	163	35	of	of	ADP
ejpam-3438	163	36	x	x	PRON
ejpam-3438	163	37	,	,	PUNCT
ejpam-3438	163	38	that	that	PRON
ejpam-3438	163	39	is	is	ADV
ejpam-3438	163	40	x	x	X
ejpam-3438	163	41	−	−	PROPN
ejpam-3438	163	42	⋃	⋃	PUNCT
ejpam-3438	163	43	λ	λ	X
ejpam-3438	163	44	∈	∈	NOUN
ejpam-3438	163	45	λ	λ	X
ejpam-3438	163	46	=	=	SYM
ejpam-3438	163	47	∅	∅	NOUN
ejpam-3438	163	48	∈	∈	PROPN
ejpam-3438	163	49	i.	i.	NOUN
ejpam-3438	163	50	this	this	PRON
ejpam-3438	163	51	show	show	VERB
ejpam-3438	163	52	that	that	SCONJ
ejpam-3438	163	53	(	(	PUNCT
ejpam-3438	163	54	3	3	X
ejpam-3438	163	55	)	)	PUNCT
ejpam-3438	163	56	holds	hold	VERB
ejpam-3438	163	57	.	.	PUNCT
ejpam-3438	164	1	(	(	PUNCT
ejpam-3438	164	2	3)⇒	3)⇒	NUM
ejpam-3438	164	3	(	(	PUNCT
ejpam-3438	164	4	1	1	NUM
ejpam-3438	164	5	)	)	PUNCT
ejpam-3438	164	6	assume	assume	VERB
ejpam-3438	164	7	that	that	SCONJ
ejpam-3438	164	8	(	(	PUNCT
ejpam-3438	164	9	3	3	X
ejpam-3438	164	10	)	)	PUNCT
ejpam-3438	164	11	holds	hold	NOUN
ejpam-3438	164	12	,	,	PUNCT
ejpam-3438	164	13	and	and	CCONJ
ejpam-3438	164	14	let	let	VERB
ejpam-3438	164	15	{	{	PUNCT
ejpam-3438	164	16	uλ	uλ	X
ejpam-3438	164	17	:	:	PUNCT
ejpam-3438	164	18	λ	λ	X
ejpam-3438	164	19	∈	∈	PROPN
ejpam-3438	164	20	λ	λ	PROPN
ejpam-3438	164	21	}	}	PUNCT
ejpam-3438	164	22	be	be	VERB
ejpam-3438	164	23	a	a	DET
ejpam-3438	164	24	cover	cover	NOUN
ejpam-3438	164	25	of	of	ADP
ejpam-3438	164	26	x	x	PUNCT
ejpam-3438	164	27	by	by	ADP
ejpam-3438	164	28	β	β	ADJ
ejpam-3438	164	29	-	-	ADJ
ejpam-3438	164	30	open	open	ADJ
ejpam-3438	164	31	sets	set	NOUN
ejpam-3438	164	32	.	.	PUNCT
ejpam-3438	165	1	by	by	ADP
ejpam-3438	165	2	lemma	lemma	PROPN
ejpam-3438	165	3	4	4	NUM
ejpam-3438	165	4	,	,	PUNCT
ejpam-3438	165	5	if	if	SCONJ
ejpam-3438	165	6	i	i	PRON
ejpam-3438	165	7	=	=	SYM
ejpam-3438	165	8	{	{	PUNCT
ejpam-3438	165	9	∅	∅	NOUN
ejpam-3438	165	10	}	}	PUNCT
ejpam-3438	165	11	,	,	PUNCT
ejpam-3438	165	12	then	then	ADV
ejpam-3438	165	13	a	a	DET
ejpam-3438	165	14	β	β	X
ejpam-3438	165	15	-	-	ADJ
ejpam-3438	165	16	open	open	ADJ
ejpam-3438	165	17	sets	set	NOUN
ejpam-3438	165	18	is	be	AUX
ejpam-3438	165	19	precisely	precisely	ADV
ejpam-3438	165	20	a	a	DET
ejpam-3438	165	21	βi	βi	NOUN
ejpam-3438	165	22	-open	-open	NOUN
ejpam-3438	165	23	set	set	NOUN
ejpam-3438	165	24	.	.	PUNCT
ejpam-3438	166	1	thus	thus	ADV
ejpam-3438	166	2	,	,	PUNCT
ejpam-3438	166	3	uλ	uλ	ADV
ejpam-3438	166	4	is	be	AUX
ejpam-3438	166	5	at	at	ADP
ejpam-3438	166	6	the	the	DET
ejpam-3438	166	7	same	same	ADJ
ejpam-3438	166	8	time	time	NOUN
ejpam-3438	166	9	a	a	DET
ejpam-3438	166	10	βi	βi	NOUN
ejpam-3438	166	11	-open	-open	NOUN
ejpam-3438	166	12	set	set	NOUN
ejpam-3438	166	13	for	for	ADP
ejpam-3438	166	14	all	all	DET
ejpam-3438	166	15	λ	λ	PROPN
ejpam-3438	166	16	∈	∈	PROPN
ejpam-3438	166	17	λ	λ	PROPN
ejpam-3438	166	18	.	.	PUNCT
ejpam-3438	166	19	by	by	ADP
ejpam-3438	166	20	assumption	assumption	NOUN
ejpam-3438	166	21	,	,	PUNCT
ejpam-3438	166	22	there	there	PRON
ejpam-3438	166	23	exists	exist	VERB
ejpam-3438	166	24	a	a	DET
ejpam-3438	166	25	finite	finite	NOUN
ejpam-3438	166	26	subset	subset	NOUN
ejpam-3438	166	27	λ0	λ0	NOUN
ejpam-3438	166	28	of	of	ADP
ejpam-3438	166	29	λ	λ	NOUN
ejpam-3438	166	30	such	such	ADJ
ejpam-3438	166	31	that	that	SCONJ
ejpam-3438	166	32	x	x	SYM
ejpam-3438	167	1	−	−	NOUN
ejpam-3438	167	2	⋃	⋃	NOUN
ejpam-3438	167	3	{	{	PUNCT
ejpam-3438	167	4	uλ	uλ	NOUN
ejpam-3438	167	5	:	:	PUNCT
ejpam-3438	167	6	λ	λ	PROPN
ejpam-3438	167	7	∈	∈	NOUN
ejpam-3438	167	8	λ0	λ0	NOUN
ejpam-3438	167	9	}	}	PUNCT
ejpam-3438	167	10	∈	∈	PROPN
ejpam-3438	167	11	i	i	PRON
ejpam-3438	167	12	,	,	PUNCT
ejpam-3438	167	13	that	that	ADV
ejpam-3438	167	14	is	is	ADV
ejpam-3438	167	15	x	x	X
ejpam-3438	167	16	−	−	PROPN
ejpam-3438	167	17	⋃	⋃	NOUN
ejpam-3438	167	18	{	{	PUNCT
ejpam-3438	167	19	uλ	uλ	NOUN
ejpam-3438	167	20	:	:	PUNCT
ejpam-3438	167	21	λ	λ	PROPN
ejpam-3438	167	22	∈	∈	NOUN
ejpam-3438	167	23	λ0	λ0	NOUN
ejpam-3438	167	24	}	}	PUNCT
ejpam-3438	167	25	=	=	SYM
ejpam-3438	167	26	∅	∅	NOUN
ejpam-3438	167	27	(	(	PUNCT
ejpam-3438	167	28	x	x	X
ejpam-3438	167	29	⊆	⊆	NUM
ejpam-3438	167	30	⋃	⋃	NOUN
ejpam-3438	167	31	{	{	PUNCT
ejpam-3438	167	32	uλ	uλ	NOUN
ejpam-3438	167	33	:	:	PUNCT
ejpam-3438	167	34	λ	λ	X
ejpam-3438	167	35	∈	∈	NOUN
ejpam-3438	167	36	λ0	λ0	NOUN
ejpam-3438	167	37	}	}	PUNCT
ejpam-3438	167	38	)	)	PUNCT
ejpam-3438	167	39	.	.	PUNCT
ejpam-3438	168	1	hence	hence	ADV
ejpam-3438	168	2	,	,	PUNCT
ejpam-3438	168	3	there	there	PRON
ejpam-3438	168	4	exists	exist	VERB
ejpam-3438	168	5	a	a	DET
ejpam-3438	168	6	finite	finite	NOUN
ejpam-3438	168	7	subset	subset	NOUN
ejpam-3438	168	8	λ0	λ0	NOUN
ejpam-3438	168	9	of	of	ADP
ejpam-3438	168	10	λ	λ	NOUN
ejpam-3438	168	11	such	such	ADJ
ejpam-3438	168	12	that	that	SCONJ
ejpam-3438	168	13	x	x	SYM
ejpam-3438	168	14	⊆	⊆	NUM
ejpam-3438	168	15	⋃	⋃	NOUN
ejpam-3438	168	16	{	{	PUNCT
ejpam-3438	168	17	uλ	uλ	NOUN
ejpam-3438	168	18	:	:	PUNCT
ejpam-3438	168	19	λ	λ	X
ejpam-3438	168	20	∈	∈	PROPN
ejpam-3438	168	21	λ0}.this	λ0}.this	DET
ejpam-3438	168	22	show	show	NOUN
ejpam-3438	168	23	that	that	SCONJ
ejpam-3438	168	24	(	(	PUNCT
ejpam-3438	168	25	1	1	X
ejpam-3438	168	26	)	)	PUNCT
ejpam-3438	168	27	holds	hold	NOUN
ejpam-3438	168	28	.	.	PUNCT
ejpam-3438	169	1	theorem	theorem	NOUN
ejpam-3438	169	2	2	2	NUM
ejpam-3438	169	3	characterizes	characterize	VERB
ejpam-3438	169	4	βi	βi	PRON
ejpam-3438	169	5	-compact	-compact	ADJ
ejpam-3438	169	6	space	space	NOUN
ejpam-3438	169	7	.	.	PUNCT
ejpam-3438	170	1	theorem	theorem	NOUN
ejpam-3438	170	2	2	2	NUM
ejpam-3438	170	3	.	.	X
ejpam-3438	170	4	for	for	ADP
ejpam-3438	170	5	an	an	DET
ejpam-3438	170	6	ideal	ideal	ADJ
ejpam-3438	170	7	topological	topological	ADJ
ejpam-3438	170	8	space	space	NOUN
ejpam-3438	170	9	(	(	PUNCT
ejpam-3438	170	10	x	x	X
ejpam-3438	170	11	,	,	PUNCT
ejpam-3438	170	12	τ	τ	PROPN
ejpam-3438	170	13	,	,	PUNCT
ejpam-3438	170	14	i	i	PROPN
ejpam-3438	170	15	)	)	PUNCT
ejpam-3438	170	16	,	,	PUNCT
ejpam-3438	170	17	the	the	DET
ejpam-3438	170	18	following	follow	VERB
ejpam-3438	170	19	statements	statement	NOUN
ejpam-3438	170	20	are	be	AUX
ejpam-3438	170	21	equivalent	equivalent	ADJ
ejpam-3438	170	22	.	.	PUNCT
ejpam-3438	171	1	(	(	PUNCT
ejpam-3438	171	2	i	i	NOUN
ejpam-3438	171	3	)	)	PUNCT
ejpam-3438	171	4	(	(	PUNCT
ejpam-3438	171	5	x	x	X
ejpam-3438	171	6	,	,	PUNCT
ejpam-3438	171	7	τ	τ	PROPN
ejpam-3438	171	8	,	,	PUNCT
ejpam-3438	171	9	i	i	PROPN
ejpam-3438	171	10	)	)	PUNCT
ejpam-3438	171	11	is	be	AUX
ejpam-3438	171	12	a	a	DET
ejpam-3438	171	13	βi	βi	ADJ
ejpam-3438	171	14	-	-	ADJ
ejpam-3438	171	15	compact	compact	ADJ
ejpam-3438	171	16	space	space	NOUN
ejpam-3438	171	17	.	.	PUNCT
ejpam-3438	172	1	(	(	PUNCT
ejpam-3438	172	2	ii	ii	NOUN
ejpam-3438	172	3	)	)	PUNCT
ejpam-3438	172	4	for	for	ADP
ejpam-3438	172	5	every	every	DET
ejpam-3438	172	6	family	family	NOUN
ejpam-3438	172	7	{	{	PUNCT
ejpam-3438	172	8	fλ	fλ	INTJ
ejpam-3438	172	9	:	:	PUNCT
ejpam-3438	172	10	λ	λ	X
ejpam-3438	172	11	∈	∈	PROPN
ejpam-3438	172	12	λ	λ	PROPN
ejpam-3438	172	13	}	}	PUNCT
ejpam-3438	172	14	of	of	ADP
ejpam-3438	172	15	βi	βi	ADJ
ejpam-3438	172	16	-	-	PUNCT
ejpam-3438	172	17	closed	close	VERB
ejpam-3438	172	18	sets	set	NOUN
ejpam-3438	172	19	such	such	ADJ
ejpam-3438	172	20	that	that	SCONJ
ejpam-3438	172	21	⋂	⋂	PROPN
ejpam-3438	172	22	{	{	PUNCT
ejpam-3438	172	23	fλ	fλ	INTJ
ejpam-3438	172	24	:	:	PUNCT
ejpam-3438	172	25	λ	λ	X
ejpam-3438	172	26	∈	∈	PROPN
ejpam-3438	172	27	λ	λ	NOUN
ejpam-3438	172	28	}	}	PUNCT
ejpam-3438	172	29	=	=	SYM
ejpam-3438	172	30	∅	∅	NOUN
ejpam-3438	172	31	,	,	PUNCT
ejpam-3438	172	32	there	there	PRON
ejpam-3438	172	33	exist	exist	VERB
ejpam-3438	172	34	a	a	DET
ejpam-3438	172	35	finite	finite	NOUN
ejpam-3438	172	36	subset	subset	NOUN
ejpam-3438	172	37	λ0	λ0	NOUN
ejpam-3438	172	38	of	of	ADP
ejpam-3438	172	39	λ	λ	PROPN
ejpam-3438	172	40	such	such	ADJ
ejpam-3438	172	41	that	that	SCONJ
ejpam-3438	172	42	⋂	⋂	PROPN
ejpam-3438	172	43	{	{	PUNCT
ejpam-3438	172	44	fλ	fλ	INTJ
ejpam-3438	172	45	:	:	PUNCT
ejpam-3438	172	46	λ	λ	PROPN
ejpam-3438	172	47	∈	∈	NOUN
ejpam-3438	172	48	λ0	λ0	NOUN
ejpam-3438	172	49	}	}	PUNCT
ejpam-3438	172	50	=	=	PUNCT
ejpam-3438	172	51	∅.	∅.	NOUN
ejpam-3438	172	52	proof	proof	NOUN
ejpam-3438	172	53	.	.	PUNCT
ejpam-3438	173	1	(	(	PUNCT
ejpam-3438	173	2	1)⇒	1)⇒	NUM
ejpam-3438	173	3	(	(	PUNCT
ejpam-3438	173	4	2	2	NUM
ejpam-3438	173	5	)	)	PUNCT
ejpam-3438	173	6	assume	assume	VERB
ejpam-3438	173	7	that	that	SCONJ
ejpam-3438	173	8	(	(	PUNCT
ejpam-3438	173	9	1	1	X
ejpam-3438	173	10	)	)	PUNCT
ejpam-3438	173	11	holds	hold	NOUN
ejpam-3438	173	12	,	,	PUNCT
ejpam-3438	173	13	and	and	CCONJ
ejpam-3438	173	14	let	let	VERB
ejpam-3438	173	15	{	{	PUNCT
ejpam-3438	173	16	fλ	fλ	X
ejpam-3438	173	17	:	:	PUNCT
ejpam-3438	173	18	λ	λ	X
ejpam-3438	173	19	∈	∈	PROPN
ejpam-3438	173	20	λ	λ	PROPN
ejpam-3438	173	21	}	}	PUNCT
ejpam-3438	173	22	be	be	VERB
ejpam-3438	173	23	a	a	DET
ejpam-3438	173	24	family	family	NOUN
ejpam-3438	173	25	of	of	ADP
ejpam-3438	173	26	βi	βi	PROPN
ejpam-3438	173	27	-closed	-close	VERB
ejpam-3438	173	28	sets	set	VERB
ejpam-3438	173	29	such	such	ADJ
ejpam-3438	173	30	that	that	SCONJ
ejpam-3438	173	31	⋂	⋂	PROPN
ejpam-3438	173	32	{	{	PUNCT
ejpam-3438	173	33	fλ	fλ	INTJ
ejpam-3438	173	34	:	:	PUNCT
ejpam-3438	173	35	λ	λ	X
ejpam-3438	173	36	∈	∈	PROPN
ejpam-3438	173	37	λ	λ	X
ejpam-3438	173	38	}	}	PUNCT
ejpam-3438	173	39	=	=	PUNCT
ejpam-3438	173	40	∅.	∅.	NOUN
ejpam-3438	173	41	if	if	SCONJ
ejpam-3438	173	42	⋂	⋂	PROPN
ejpam-3438	173	43	{	{	PUNCT
ejpam-3438	173	44	fλ	fλ	INTJ
ejpam-3438	173	45	:	:	PUNCT
ejpam-3438	173	46	λ	λ	X
ejpam-3438	173	47	∈	∈	PROPN
ejpam-3438	173	48	λ	λ	NOUN
ejpam-3438	173	49	}	}	PUNCT
ejpam-3438	173	50	=	=	SYM
ejpam-3438	173	51	∅	∅	NOUN
ejpam-3438	173	52	,	,	PUNCT
ejpam-3438	173	53	then	then	ADV
ejpam-3438	173	54	⋃	⋃	ADV
ejpam-3438	173	55	{	{	PUNCT
ejpam-3438	173	56	fcλ	fcλ	NOUN
ejpam-3438	173	57	:	:	PUNCT
ejpam-3438	173	58	λ	λ	X
ejpam-3438	173	59	∈	∈	PROPN
ejpam-3438	173	60	λ	λ	NOUN
ejpam-3438	173	61	}	}	PUNCT
ejpam-3438	173	62	=	=	SYM
ejpam-3438	173	63	(	(	PUNCT
ejpam-3438	173	64	⋂	⋂	PROPN
ejpam-3438	173	65	{	{	PUNCT
ejpam-3438	173	66	fλ	fλ	INTJ
ejpam-3438	173	67	:	:	PUNCT
ejpam-3438	173	68	λ	λ	X
ejpam-3438	173	69	∈	∈	PROPN
ejpam-3438	173	70	λ})c	λ})c	NOUN
ejpam-3438	173	71	=	=	PUNCT
ejpam-3438	173	72	x.	x.	NOUN
ejpam-3438	173	73	hence	hence	ADV
ejpam-3438	173	74	,	,	PUNCT
ejpam-3438	173	75	⋃	⋃	PROPN
ejpam-3438	173	76	{	{	PUNCT
ejpam-3438	173	77	fcλ	fcλ	NOUN
ejpam-3438	173	78	:	:	PUNCT
ejpam-3438	173	79	λ	λ	X
ejpam-3438	173	80	∈	∈	PROPN
ejpam-3438	173	81	λ	λ	PROPN
ejpam-3438	173	82	}	}	PUNCT
ejpam-3438	173	83	is	be	AUX
ejpam-3438	173	84	a	a	DET
ejpam-3438	173	85	covering	covering	NOUN
ejpam-3438	173	86	of	of	ADP
ejpam-3438	173	87	x	x	PUNCT
ejpam-3438	173	88	by	by	ADP
ejpam-3438	173	89	βi	βi	NOUN
ejpam-3438	173	90	-open	-open	NOUN
ejpam-3438	173	91	sets	set	NOUN
ejpam-3438	173	92	.	.	PUNCT
ejpam-3438	174	1	by	by	ADP
ejpam-3438	174	2	assumption	assumption	NOUN
ejpam-3438	174	3	there	there	PRON
ejpam-3438	174	4	exists	exist	VERB
ejpam-3438	174	5	a	a	DET
ejpam-3438	174	6	finite	finite	NOUN
ejpam-3438	174	7	subset	subset	NOUN
ejpam-3438	174	8	λ0	λ0	NOUN
ejpam-3438	174	9	of	of	ADP
ejpam-3438	174	10	λ	λ	NOUN
ejpam-3438	174	11	such	such	ADJ
ejpam-3438	174	12	that	that	SCONJ
ejpam-3438	174	13	⋃	⋃	PROPN
ejpam-3438	174	14	{	{	PUNCT
ejpam-3438	174	15	fcλ	fcλ	NOUN
ejpam-3438	174	16	:	:	PUNCT
ejpam-3438	174	17	λ	λ	PROPN
ejpam-3438	174	18	∈	∈	NOUN
ejpam-3438	174	19	λ0	λ0	NOUN
ejpam-3438	174	20	}	}	PUNCT
ejpam-3438	174	21	=	=	SYM
ejpam-3438	174	22	x	x	NOUN
ejpam-3438	174	23	,	,	PUNCT
ejpam-3438	174	24	that	that	PRON
ejpam-3438	174	25	is⋂	is⋂	ADJ
ejpam-3438	174	26	{	{	PUNCT
ejpam-3438	174	27	fλ	fλ	INTJ
ejpam-3438	174	28	:	:	PUNCT
ejpam-3438	174	29	λ	λ	PROPN
ejpam-3438	174	30	∈	∈	NOUN
ejpam-3438	174	31	λ0	λ0	NOUN
ejpam-3438	174	32	}	}	PUNCT
ejpam-3438	174	33	=	=	SYM
ejpam-3438	174	34	∅.	∅.	X
ejpam-3438	174	35	(	(	PUNCT
ejpam-3438	174	36	2	2	NUM
ejpam-3438	174	37	)	)	PUNCT
ejpam-3438	174	38	⇒	⇒	NOUN
ejpam-3438	174	39	(	(	PUNCT
ejpam-3438	174	40	1	1	X
ejpam-3438	174	41	)	)	PUNCT
ejpam-3438	174	42	assume	assume	VERB
ejpam-3438	174	43	that	that	SCONJ
ejpam-3438	174	44	(	(	PUNCT
ejpam-3438	174	45	2	2	X
ejpam-3438	174	46	)	)	PUNCT
ejpam-3438	174	47	holds	hold	NOUN
ejpam-3438	174	48	,	,	PUNCT
ejpam-3438	174	49	and	and	CCONJ
ejpam-3438	174	50	let	let	VERB
ejpam-3438	174	51	{	{	PUNCT
ejpam-3438	174	52	uλ	uλ	X
ejpam-3438	174	53	:	:	PUNCT
ejpam-3438	174	54	λ	λ	X
ejpam-3438	174	55	∈	∈	PROPN
ejpam-3438	174	56	λ	λ	PROPN
ejpam-3438	174	57	}	}	PUNCT
ejpam-3438	174	58	be	be	VERB
ejpam-3438	174	59	a	a	DET
ejpam-3438	174	60	cover	cover	NOUN
ejpam-3438	174	61	of	of	ADP
ejpam-3438	174	62	x	x	PUNCT
ejpam-3438	174	63	by	by	ADP
ejpam-3438	174	64	βi	βi	NOUN
ejpam-3438	174	65	-open	-open	NOUN
ejpam-3438	174	66	sets	set	NOUN
ejpam-3438	174	67	.	.	PUNCT
ejpam-3438	175	1	if	if	SCONJ
ejpam-3438	175	2	{	{	PUNCT
ejpam-3438	175	3	uλ	uλ	INTJ
ejpam-3438	175	4	:	:	PUNCT
ejpam-3438	175	5	λ	λ	X
ejpam-3438	175	6	∈	∈	PROPN
ejpam-3438	175	7	λ	λ	PROPN
ejpam-3438	175	8	}	}	PUNCT
ejpam-3438	175	9	is	be	AUX
ejpam-3438	175	10	a	a	DET
ejpam-3438	175	11	cover	cover	NOUN
ejpam-3438	175	12	of	of	ADP
ejpam-3438	175	13	x	x	PUNCT
ejpam-3438	175	14	by	by	ADP
ejpam-3438	175	15	βi	βi	NOUN
ejpam-3438	175	16	-open	-open	NOUN
ejpam-3438	175	17	sets	set	NOUN
ejpam-3438	175	18	,	,	PUNCT
ejpam-3438	175	19	that	that	ADV
ejpam-3438	175	20	is	is	ADV
ejpam-3438	175	21	⋃	⋃	PROPN
ejpam-3438	175	22	{	{	PUNCT
ejpam-3438	175	23	uλ	uλ	NOUN
ejpam-3438	175	24	:	:	PUNCT
ejpam-3438	176	1	λ	λ	X
ejpam-3438	176	2	∈	∈	PROPN
ejpam-3438	176	3	λ	λ	NOUN
ejpam-3438	176	4	}	}	PUNCT
ejpam-3438	176	5	=	=	SYM
ejpam-3438	176	6	x	x	NOUN
ejpam-3438	176	7	,	,	PUNCT
ejpam-3438	176	8	then⋂	then⋂	PROPN
ejpam-3438	176	9	{	{	PUNCT
ejpam-3438	176	10	ucλ	ucλ	INTJ
ejpam-3438	176	11	:	:	PUNCT
ejpam-3438	177	1	λ	λ	X
ejpam-3438	177	2	∈	∈	PROPN
ejpam-3438	177	3	λ	λ	NOUN
ejpam-3438	177	4	}	}	PUNCT
ejpam-3438	177	5	=	=	SYM
ejpam-3438	177	6	(	(	PUNCT
ejpam-3438	177	7	⋃	⋃	X
ejpam-3438	177	8	{	{	PUNCT
ejpam-3438	177	9	uλ	uλ	NOUN
ejpam-3438	177	10	:	:	PUNCT
ejpam-3438	177	11	λ	λ	X
ejpam-3438	177	12	∈	∈	PROPN
ejpam-3438	177	13	λ})c	λ})c	NOUN
ejpam-3438	177	14	=	=	PUNCT
ejpam-3438	177	15	∅.	∅.	NOUN
ejpam-3438	177	16	by	by	ADP
ejpam-3438	177	17	assumption	assumption	NOUN
ejpam-3438	177	18	,	,	PUNCT
ejpam-3438	177	19	there	there	PRON
ejpam-3438	177	20	exists	exist	VERB
ejpam-3438	177	21	a	a	DET
ejpam-3438	177	22	finite	finite	NOUN
ejpam-3438	177	23	subset	subset	NOUN
ejpam-3438	177	24	λ0	λ0	NOUN
ejpam-3438	177	25	of	of	ADP
ejpam-3438	177	26	λ	λ	PROPN
ejpam-3438	177	27	such	such	ADJ
ejpam-3438	177	28	that	that	SCONJ
ejpam-3438	177	29	⋂	⋂	PROPN
ejpam-3438	177	30	{	{	PUNCT
ejpam-3438	177	31	ucλ	ucλ	X
ejpam-3438	177	32	:	:	PUNCT
ejpam-3438	177	33	λ	λ	PROPN
ejpam-3438	177	34	∈	∈	NOUN
ejpam-3438	177	35	λ0	λ0	NOUN
ejpam-3438	177	36	}	}	PUNCT
ejpam-3438	177	37	=	=	SYM
ejpam-3438	177	38	∅	∅	NOUN
ejpam-3438	177	39	,	,	PUNCT
ejpam-3438	177	40	that	that	ADV
ejpam-3438	177	41	is	is	ADV
ejpam-3438	177	42	⋃	⋃	PROPN
ejpam-3438	177	43	{	{	PUNCT
ejpam-3438	177	44	uλ	uλ	NOUN
ejpam-3438	177	45	:	:	PUNCT
ejpam-3438	177	46	λ	λ	PROPN
ejpam-3438	177	47	∈	∈	NOUN
ejpam-3438	177	48	λ0	λ0	NOUN
ejpam-3438	177	49	}	}	PUNCT
ejpam-3438	177	50	=	=	PUNCT
ejpam-3438	177	51	x.	x.	NOUN
ejpam-3438	177	52	this	this	PRON
ejpam-3438	177	53	show	show	VERB
ejpam-3438	177	54	that	that	SCONJ
ejpam-3438	177	55	(	(	PUNCT
ejpam-3438	177	56	1	1	X
ejpam-3438	177	57	)	)	PUNCT
ejpam-3438	177	58	holds	hold	NOUN
ejpam-3438	177	59	.	.	PUNCT
ejpam-3438	178	1	theorem	theorem	NOUN
ejpam-3438	178	2	3	3	NUM
ejpam-3438	178	3	characterizes	characterize	VERB
ejpam-3438	178	4	βi	βi	PRON
ejpam-3438	178	5	-compact	-compact	ADJ
ejpam-3438	178	6	space	space	NOUN
ejpam-3438	178	7	.	.	PUNCT
ejpam-3438	179	1	theorem	theorem	NOUN
ejpam-3438	179	2	3	3	NUM
ejpam-3438	179	3	.	.	PUNCT
ejpam-3438	180	1	in	in	ADP
ejpam-3438	180	2	an	an	DET
ejpam-3438	180	3	ideal	ideal	ADJ
ejpam-3438	180	4	topological	topological	ADJ
ejpam-3438	180	5	space	space	NOUN
ejpam-3438	180	6	(	(	PUNCT
ejpam-3438	180	7	x	x	X
ejpam-3438	180	8	,	,	PUNCT
ejpam-3438	180	9	τ	τ	PROPN
ejpam-3438	180	10	,	,	PUNCT
ejpam-3438	180	11	i	i	PROPN
ejpam-3438	180	12	)	)	PUNCT
ejpam-3438	180	13	,	,	PUNCT
ejpam-3438	180	14	the	the	DET
ejpam-3438	180	15	following	follow	VERB
ejpam-3438	180	16	statements	statement	NOUN
ejpam-3438	180	17	are	be	AUX
ejpam-3438	180	18	equivalent	equivalent	ADJ
ejpam-3438	180	19	.	.	PUNCT
ejpam-3438	181	1	(	(	PUNCT
ejpam-3438	181	2	i	i	NOUN
ejpam-3438	181	3	)	)	PUNCT
ejpam-3438	181	4	(	(	PUNCT
ejpam-3438	181	5	x	x	X
ejpam-3438	181	6	,	,	PUNCT
ejpam-3438	181	7	τ	τ	PROPN
ejpam-3438	181	8	,	,	PUNCT
ejpam-3438	181	9	i	i	PROPN
ejpam-3438	181	10	)	)	PUNCT
ejpam-3438	181	11	is	be	AUX
ejpam-3438	181	12	a	a	DET
ejpam-3438	181	13	cβi	cβi	NOUN
ejpam-3438	181	14	-	-	PUNCT
ejpam-3438	181	15	compact	compact	ADJ
ejpam-3438	181	16	space	space	NOUN
ejpam-3438	181	17	.	.	PUNCT
ejpam-3438	182	1	(	(	PUNCT
ejpam-3438	182	2	ii	ii	NOUN
ejpam-3438	182	3	)	)	PUNCT
ejpam-3438	182	4	for	for	ADP
ejpam-3438	182	5	every	every	DET
ejpam-3438	182	6	family	family	NOUN
ejpam-3438	182	7	{	{	PUNCT
ejpam-3438	182	8	fλ	fλ	INTJ
ejpam-3438	182	9	:	:	PUNCT
ejpam-3438	182	10	λ	λ	X
ejpam-3438	182	11	∈	∈	PROPN
ejpam-3438	182	12	λ	λ	PROPN
ejpam-3438	182	13	}	}	PUNCT
ejpam-3438	182	14	of	of	ADP
ejpam-3438	182	15	βi	βi	ADJ
ejpam-3438	182	16	-	-	PUNCT
ejpam-3438	182	17	closed	close	VERB
ejpam-3438	182	18	sets	set	NOUN
ejpam-3438	182	19	such	such	ADJ
ejpam-3438	182	20	that	that	SCONJ
ejpam-3438	182	21	⋂	⋂	PROPN
ejpam-3438	182	22	{	{	PUNCT
ejpam-3438	182	23	fλ	fλ	INTJ
ejpam-3438	182	24	:	:	PUNCT
ejpam-3438	182	25	λ	λ	X
ejpam-3438	182	26	∈	∈	PROPN
ejpam-3438	182	27	λ	λ	NOUN
ejpam-3438	182	28	}	}	PUNCT
ejpam-3438	182	29	=	=	SYM
ejpam-3438	182	30	∅	∅	NOUN
ejpam-3438	182	31	,	,	PUNCT
ejpam-3438	182	32	there	there	PRON
ejpam-3438	182	33	exist	exist	VERB
ejpam-3438	182	34	a	a	DET
ejpam-3438	182	35	finite	finite	NOUN
ejpam-3438	182	36	subset	subset	NOUN
ejpam-3438	182	37	λ0	λ0	NOUN
ejpam-3438	182	38	of	of	ADP
ejpam-3438	182	39	λ	λ	PROPN
ejpam-3438	182	40	such	such	ADJ
ejpam-3438	182	41	that	that	SCONJ
ejpam-3438	182	42	⋂	⋂	PROPN
ejpam-3438	182	43	{	{	PUNCT
ejpam-3438	182	44	fλ	fλ	INTJ
ejpam-3438	182	45	:	:	PUNCT
ejpam-3438	182	46	λ	λ	PROPN
ejpam-3438	182	47	∈	∈	NOUN
ejpam-3438	182	48	λ0	λ0	NOUN
ejpam-3438	182	49	}	}	PUNCT
ejpam-3438	182	50	∈	∈	PROPN
ejpam-3438	182	51	i.	i.	NOUN
ejpam-3438	182	52	proof	proof	NOUN
ejpam-3438	182	53	.	.	PUNCT
ejpam-3438	183	1	(	(	PUNCT
ejpam-3438	183	2	1)⇒	1)⇒	NUM
ejpam-3438	183	3	(	(	PUNCT
ejpam-3438	183	4	2	2	NUM
ejpam-3438	183	5	)	)	PUNCT
ejpam-3438	183	6	assume	assume	VERB
ejpam-3438	183	7	that	that	SCONJ
ejpam-3438	183	8	(	(	PUNCT
ejpam-3438	183	9	1	1	X
ejpam-3438	183	10	)	)	PUNCT
ejpam-3438	183	11	holds	hold	NOUN
ejpam-3438	183	12	,	,	PUNCT
ejpam-3438	183	13	and	and	CCONJ
ejpam-3438	183	14	let	let	VERB
ejpam-3438	183	15	{	{	PUNCT
ejpam-3438	183	16	fλ	fλ	X
ejpam-3438	183	17	:	:	PUNCT
ejpam-3438	183	18	λ	λ	X
ejpam-3438	183	19	∈	∈	PROPN
ejpam-3438	183	20	λ	λ	PROPN
ejpam-3438	183	21	}	}	PUNCT
ejpam-3438	183	22	be	be	VERB
ejpam-3438	183	23	a	a	DET
ejpam-3438	183	24	family	family	NOUN
ejpam-3438	183	25	of	of	ADP
ejpam-3438	183	26	βi	βi	PROPN
ejpam-3438	183	27	-closed	-close	VERB
ejpam-3438	183	28	sets	set	VERB
ejpam-3438	183	29	such	such	ADJ
ejpam-3438	183	30	that	that	SCONJ
ejpam-3438	183	31	⋂	⋂	PROPN
ejpam-3438	183	32	{	{	PUNCT
ejpam-3438	183	33	fλ	fλ	INTJ
ejpam-3438	183	34	:	:	PUNCT
ejpam-3438	183	35	λ	λ	X
ejpam-3438	183	36	∈	∈	PROPN
ejpam-3438	183	37	λ	λ	X
ejpam-3438	183	38	}	}	PUNCT
ejpam-3438	183	39	=	=	PUNCT
ejpam-3438	183	40	∅.	∅.	NOUN
ejpam-3438	183	41	if	if	SCONJ
ejpam-3438	183	42	⋂	⋂	PROPN
ejpam-3438	183	43	{	{	PUNCT
ejpam-3438	183	44	fλ	fλ	INTJ
ejpam-3438	183	45	:	:	PUNCT
ejpam-3438	183	46	λ	λ	X
ejpam-3438	183	47	∈	∈	PROPN
ejpam-3438	183	48	λ	λ	NOUN
ejpam-3438	183	49	}	}	PUNCT
ejpam-3438	183	50	=	=	SYM
ejpam-3438	183	51	∅	∅	NOUN
ejpam-3438	183	52	,	,	PUNCT
ejpam-3438	183	53	then	then	ADV
ejpam-3438	183	54	⋃	⋃	ADV
ejpam-3438	183	55	{	{	PUNCT
ejpam-3438	183	56	fcλ	fcλ	NOUN
ejpam-3438	183	57	:	:	PUNCT
ejpam-3438	183	58	λ	λ	X
ejpam-3438	183	59	∈	∈	PROPN
ejpam-3438	183	60	λ	λ	NOUN
ejpam-3438	183	61	}	}	PUNCT
ejpam-3438	183	62	=	=	SYM
ejpam-3438	183	63	(	(	PUNCT
ejpam-3438	183	64	⋂	⋂	PROPN
ejpam-3438	183	65	{	{	PUNCT
ejpam-3438	183	66	fλ	fλ	INTJ
ejpam-3438	183	67	:	:	PUNCT
ejpam-3438	183	68	λ	λ	X
ejpam-3438	183	69	∈	∈	PROPN
ejpam-3438	183	70	λ})c	λ})c	NOUN
ejpam-3438	183	71	=	=	PUNCT
ejpam-3438	183	72	x.	x.	NOUN
ejpam-3438	183	73	hence	hence	ADV
ejpam-3438	183	74	,	,	PUNCT
ejpam-3438	183	75	⋃	⋃	PROPN
ejpam-3438	183	76	{	{	PUNCT
ejpam-3438	183	77	fcλ	fcλ	NOUN
ejpam-3438	183	78	:	:	PUNCT
ejpam-3438	183	79	λ	λ	X
ejpam-3438	183	80	∈	∈	PROPN
ejpam-3438	183	81	λ	λ	PROPN
ejpam-3438	183	82	}	}	PUNCT
ejpam-3438	183	83	is	be	AUX
ejpam-3438	183	84	a	a	DET
ejpam-3438	183	85	covering	covering	NOUN
ejpam-3438	183	86	of	of	ADP
ejpam-3438	183	87	x	x	PUNCT
ejpam-3438	183	88	by	by	ADP
ejpam-3438	183	89	βi	βi	NOUN
ejpam-3438	183	90	-open	-open	NOUN
ejpam-3438	183	91	sets	set	NOUN
ejpam-3438	183	92	.	.	PUNCT
ejpam-3438	184	1	by	by	ADP
ejpam-3438	184	2	assumption	assumption	NOUN
ejpam-3438	184	3	there	there	PRON
ejpam-3438	184	4	exists	exist	VERB
ejpam-3438	184	5	a	a	DET
ejpam-3438	184	6	finite	finite	NOUN
ejpam-3438	184	7	subset	subset	NOUN
ejpam-3438	184	8	λ0	λ0	NOUN
ejpam-3438	184	9	of	of	ADP
ejpam-3438	184	10	λ	λ	NOUN
ejpam-3438	184	11	such	such	ADJ
ejpam-3438	184	12	that	that	SCONJ
ejpam-3438	184	13	x	x	SYM
ejpam-3438	184	14	−	−	NOUN
ejpam-3438	184	15	⋃	⋃	NOUN
ejpam-3438	184	16	{	{	PUNCT
ejpam-3438	184	17	fcλ	fcλ	NOUN
ejpam-3438	184	18	:	:	PUNCT
ejpam-3438	184	19	λ	λ	PROPN
ejpam-3438	184	20	∈	∈	NOUN
ejpam-3438	184	21	λ0	λ0	NOUN
ejpam-3438	184	22	}	}	PUNCT
ejpam-3438	184	23	∈	∈	PROPN
ejpam-3438	184	24	i	i	PRON
ejpam-3438	184	25	,	,	PUNCT
ejpam-3438	184	26	that	that	PRON
ejpam-3438	184	27	is	be	AUX
ejpam-3438	184	28	⋂	⋂	PROPN
ejpam-3438	184	29	{	{	PUNCT
ejpam-3438	184	30	fλ	fλ	INTJ
ejpam-3438	184	31	:	:	PUNCT
ejpam-3438	184	32	λ	λ	PROPN
ejpam-3438	184	33	∈	∈	NOUN
ejpam-3438	184	34	λ0	λ0	NOUN
ejpam-3438	184	35	}	}	PUNCT
ejpam-3438	184	36	∈	∈	PROPN
ejpam-3438	184	37	i.	i.	NOUN
ejpam-3438	184	38	(	(	PUNCT
ejpam-3438	184	39	2	2	NUM
ejpam-3438	184	40	)	)	PUNCT
ejpam-3438	184	41	⇒	⇒	NOUN
ejpam-3438	184	42	(	(	PUNCT
ejpam-3438	184	43	1	1	X
ejpam-3438	184	44	)	)	PUNCT
ejpam-3438	184	45	assume	assume	VERB
ejpam-3438	184	46	that	that	SCONJ
ejpam-3438	184	47	(	(	PUNCT
ejpam-3438	184	48	2	2	X
ejpam-3438	184	49	)	)	PUNCT
ejpam-3438	184	50	holds	hold	NOUN
ejpam-3438	184	51	,	,	PUNCT
ejpam-3438	184	52	and	and	CCONJ
ejpam-3438	184	53	let	let	VERB
ejpam-3438	184	54	{	{	PUNCT
ejpam-3438	184	55	uλ	uλ	X
ejpam-3438	184	56	:	:	PUNCT
ejpam-3438	184	57	λ	λ	X
ejpam-3438	184	58	∈	∈	PROPN
ejpam-3438	184	59	λ	λ	PROPN
ejpam-3438	184	60	}	}	PUNCT
ejpam-3438	184	61	be	be	VERB
ejpam-3438	184	62	a	a	DET
ejpam-3438	184	63	cover	cover	NOUN
ejpam-3438	184	64	of	of	ADP
ejpam-3438	184	65	x	x	PUNCT
ejpam-3438	184	66	by	by	ADP
ejpam-3438	184	67	βi	βi	NOUN
ejpam-3438	184	68	-open	-open	NOUN
ejpam-3438	184	69	sets	set	NOUN
ejpam-3438	184	70	.	.	PUNCT
ejpam-3438	185	1	if	if	SCONJ
ejpam-3438	185	2	{	{	PUNCT
ejpam-3438	185	3	uλ	uλ	INTJ
ejpam-3438	185	4	:	:	PUNCT
ejpam-3438	185	5	λ	λ	X
ejpam-3438	185	6	∈	∈	PROPN
ejpam-3438	185	7	λ	λ	PROPN
ejpam-3438	185	8	}	}	PUNCT
ejpam-3438	185	9	is	be	AUX
ejpam-3438	185	10	a	a	DET
ejpam-3438	185	11	cover	cover	NOUN
ejpam-3438	185	12	of	of	ADP
ejpam-3438	185	13	x	x	PUNCT
ejpam-3438	185	14	by	by	ADP
ejpam-3438	185	15	βi	βi	NOUN
ejpam-3438	185	16	-open	-open	NOUN
ejpam-3438	185	17	sets	set	NOUN
ejpam-3438	185	18	,	,	PUNCT
ejpam-3438	185	19	that	that	ADV
ejpam-3438	185	20	is	is	ADV
ejpam-3438	185	21	⋃	⋃	PROPN
ejpam-3438	185	22	{	{	PUNCT
ejpam-3438	185	23	uλ	uλ	NOUN
ejpam-3438	185	24	:	:	PUNCT
ejpam-3438	186	1	λ	λ	X
ejpam-3438	186	2	∈	∈	PROPN
ejpam-3438	186	3	λ	λ	NOUN
ejpam-3438	186	4	}	}	PUNCT
ejpam-3438	186	5	=	=	SYM
ejpam-3438	186	6	x	x	NOUN
ejpam-3438	186	7	,	,	PUNCT
ejpam-3438	186	8	then	then	ADV
ejpam-3438	186	9	g.	g.	PROPN
ejpam-3438	186	10	catalan	catalan	PROPN
ejpam-3438	186	11	,	,	PUNCT
ejpam-3438	186	12	r.	r.	PROPN
ejpam-3438	186	13	padua	padua	PROPN
ejpam-3438	186	14	,	,	PUNCT
ejpam-3438	186	15	m.	m.	PROPN
ejpam-3438	186	16	baldado	baldado	PROPN
ejpam-3438	186	17	jr	jr	PROPN
ejpam-3438	186	18	.	.	PROPN
ejpam-3438	186	19	/	/	SYM
ejpam-3438	186	20	eur	eur	PROPN
ejpam-3438	186	21	.	.	PUNCT
ejpam-3438	187	1	j.	j.	PROPN
ejpam-3438	187	2	pure	pure	PROPN
ejpam-3438	187	3	appl	appl	PROPN
ejpam-3438	187	4	.	.	PROPN
ejpam-3438	187	5	math	math	PROPN
ejpam-3438	187	6	,	,	PUNCT
ejpam-3438	187	7	12	12	NUM
ejpam-3438	187	8	(	(	PUNCT
ejpam-3438	187	9	3	3	NUM
ejpam-3438	187	10	)	)	PUNCT
ejpam-3438	187	11	(	(	PUNCT
ejpam-3438	187	12	2019	2019	NUM
ejpam-3438	187	13	)	)	PUNCT
ejpam-3438	187	14	,	,	PUNCT
ejpam-3438	187	15	893	893	NUM
ejpam-3438	187	16	-	-	SYM
ejpam-3438	187	17	905	905	NUM
ejpam-3438	187	18	898⋂	898⋂	NOUN
ejpam-3438	187	19	{	{	PUNCT
ejpam-3438	187	20	ucλ	ucλ	INTJ
ejpam-3438	187	21	:	:	PUNCT
ejpam-3438	187	22	λ	λ	X
ejpam-3438	187	23	∈	∈	PROPN
ejpam-3438	187	24	λ	λ	NOUN
ejpam-3438	187	25	}	}	PUNCT
ejpam-3438	187	26	=	=	SYM
ejpam-3438	187	27	(	(	PUNCT
ejpam-3438	187	28	⋃	⋃	X
ejpam-3438	187	29	{	{	PUNCT
ejpam-3438	187	30	uλ	uλ	NOUN
ejpam-3438	187	31	:	:	PUNCT
ejpam-3438	187	32	λ	λ	X
ejpam-3438	187	33	∈	∈	PROPN
ejpam-3438	187	34	λ})c	λ})c	NOUN
ejpam-3438	188	1	=	=	PUNCT
ejpam-3438	188	2	∅.	∅.	NOUN
ejpam-3438	188	3	by	by	ADP
ejpam-3438	188	4	assumption	assumption	NOUN
ejpam-3438	188	5	,	,	PUNCT
ejpam-3438	188	6	there	there	PRON
ejpam-3438	188	7	exists	exist	VERB
ejpam-3438	188	8	a	a	DET
ejpam-3438	188	9	finite	finite	NOUN
ejpam-3438	188	10	subset	subset	NOUN
ejpam-3438	188	11	λ0	λ0	NOUN
ejpam-3438	188	12	of	of	ADP
ejpam-3438	188	13	λ	λ	PROPN
ejpam-3438	188	14	such	such	ADJ
ejpam-3438	188	15	that	that	SCONJ
ejpam-3438	188	16	⋂	⋂	PROPN
ejpam-3438	188	17	{	{	PUNCT
ejpam-3438	188	18	ucλ	ucλ	X
ejpam-3438	188	19	:	:	PUNCT
ejpam-3438	188	20	λ	λ	PROPN
ejpam-3438	188	21	∈	∈	NOUN
ejpam-3438	188	22	λ0	λ0	NOUN
ejpam-3438	188	23	}	}	PUNCT
ejpam-3438	188	24	∈	∈	PROPN
ejpam-3438	188	25	i	i	PRON
ejpam-3438	188	26	,	,	PUNCT
ejpam-3438	188	27	that	that	ADV
ejpam-3438	188	28	is	is	ADV
ejpam-3438	188	29	x	x	X
ejpam-3438	188	30	−	−	PROPN
ejpam-3438	188	31	⋃	⋃	NOUN
ejpam-3438	188	32	{	{	PUNCT
ejpam-3438	188	33	uλ	uλ	NOUN
ejpam-3438	188	34	:	:	PUNCT
ejpam-3438	188	35	λ	λ	PROPN
ejpam-3438	188	36	∈	∈	NOUN
ejpam-3438	188	37	λ0	λ0	NOUN
ejpam-3438	188	38	}	}	PUNCT
ejpam-3438	188	39	∈	∈	PROPN
ejpam-3438	188	40	i.	i.	NOUN
ejpam-3438	188	41	this	this	PRON
ejpam-3438	188	42	show	show	VERB
ejpam-3438	188	43	that	that	SCONJ
ejpam-3438	188	44	(	(	PUNCT
ejpam-3438	188	45	1	1	X
ejpam-3438	188	46	)	)	PUNCT
ejpam-3438	188	47	holds	hold	VERB
ejpam-3438	188	48	.	.	PUNCT
ejpam-3438	189	1	remark	remark	PROPN
ejpam-3438	189	2	1	1	NUM
ejpam-3438	189	3	.	.	PUNCT
ejpam-3438	190	1	[	[	X
ejpam-3438	190	2	30	30	NUM
ejpam-3438	190	3	]	]	X
ejpam-3438	190	4	let	let	VERB
ejpam-3438	190	5	(	(	PUNCT
ejpam-3438	190	6	x	x	NOUN
ejpam-3438	190	7	,	,	PUNCT
ejpam-3438	190	8	τ	τ	PROPN
ejpam-3438	190	9	,	,	PUNCT
ejpam-3438	190	10	i	i	PROPN
ejpam-3438	190	11	)	)	PUNCT
ejpam-3438	190	12	and	and	CCONJ
ejpam-3438	190	13	(	(	PUNCT
ejpam-3438	190	14	y	y	PROPN
ejpam-3438	190	15	,	,	PUNCT
ejpam-3438	190	16	σ	σ	PROPN
ejpam-3438	190	17	,	,	PUNCT
ejpam-3438	190	18	j	j	PROPN
ejpam-3438	190	19	)	)	PUNCT
ejpam-3438	190	20	be	be	AUX
ejpam-3438	190	21	ideal	ideal	ADJ
ejpam-3438	190	22	topological	topological	ADJ
ejpam-3438	190	23	spaces	space	NOUN
ejpam-3438	190	24	.	.	PUNCT
ejpam-3438	191	1	(	(	PUNCT
ejpam-3438	191	2	i	i	NOUN
ejpam-3438	191	3	)	)	PUNCT
ejpam-3438	191	4	if	if	SCONJ
ejpam-3438	191	5	f	f	PROPN
ejpam-3438	191	6	:	:	PUNCT
ejpam-3438	191	7	(	(	PUNCT
ejpam-3438	191	8	x	x	X
ejpam-3438	191	9	,	,	PUNCT
ejpam-3438	191	10	τ	τ	PROPN
ejpam-3438	191	11	,	,	PUNCT
ejpam-3438	191	12	i)→	i)→	PROPN
ejpam-3438	191	13	(	(	PUNCT
ejpam-3438	191	14	y	y	PROPN
ejpam-3438	191	15	,	,	PUNCT
ejpam-3438	191	16	σ	σ	PROPN
ejpam-3438	191	17	)	)	PUNCT
ejpam-3438	191	18	is	be	AUX
ejpam-3438	191	19	a	a	DET
ejpam-3438	191	20	function	function	NOUN
ejpam-3438	191	21	,	,	PUNCT
ejpam-3438	191	22	then	then	ADV
ejpam-3438	191	23	f(i	f(i	NUM
ejpam-3438	191	24	)	)	PUNCT
ejpam-3438	192	1	=	=	PRON
ejpam-3438	192	2	{	{	PUNCT
ejpam-3438	192	3	f(a	f(a	NOUN
ejpam-3438	192	4	)	)	PUNCT
ejpam-3438	192	5	:	:	PUNCT
ejpam-3438	192	6	a	a	DET
ejpam-3438	192	7	∈	∈	PROPN
ejpam-3438	192	8	i	i	X
ejpam-3438	192	9	}	}	PUNCT
ejpam-3438	192	10	is	be	AUX
ejpam-3438	192	11	an	an	DET
ejpam-3438	192	12	ideal	ideal	NOUN
ejpam-3438	192	13	in	in	ADP
ejpam-3438	192	14	y	y	PROPN
ejpam-3438	192	15	.	.	PUNCT
ejpam-3438	193	1	(	(	PUNCT
ejpam-3438	193	2	ii	ii	NOUN
ejpam-3438	193	3	)	)	PUNCT
ejpam-3438	193	4	if	if	SCONJ
ejpam-3438	193	5	f	f	PROPN
ejpam-3438	193	6	:	:	PUNCT
ejpam-3438	193	7	(	(	PUNCT
ejpam-3438	193	8	x	x	X
ejpam-3438	193	9	,	,	PUNCT
ejpam-3438	193	10	τ)→	τ)→	PROPN
ejpam-3438	193	11	(	(	PUNCT
ejpam-3438	193	12	y	y	PROPN
ejpam-3438	193	13	,	,	PUNCT
ejpam-3438	193	14	σ	σ	PROPN
ejpam-3438	193	15	,	,	PUNCT
ejpam-3438	193	16	j	j	PROPN
ejpam-3438	193	17	)	)	PUNCT
ejpam-3438	193	18	is	be	AUX
ejpam-3438	193	19	an	an	DET
ejpam-3438	193	20	injective	injective	ADJ
ejpam-3438	193	21	function	function	NOUN
ejpam-3438	193	22	,	,	PUNCT
ejpam-3438	193	23	then	then	ADV
ejpam-3438	193	24	f−1(j	f−1(j	PROPN
ejpam-3438	193	25	)	)	PUNCT
ejpam-3438	193	26	=	=	PRON
ejpam-3438	193	27	{	{	PUNCT
ejpam-3438	193	28	f−1(b	f−1(b	PROPN
ejpam-3438	193	29	)	)	PUNCT
ejpam-3438	193	30	:	:	PUNCT
ejpam-3438	194	1	b	b	X
ejpam-3438	194	2	∈	∈	PROPN
ejpam-3438	194	3	j	j	PROPN
ejpam-3438	194	4	}	}	PUNCT
ejpam-3438	194	5	is	be	AUX
ejpam-3438	194	6	an	an	DET
ejpam-3438	194	7	ideal	ideal	NOUN
ejpam-3438	194	8	in	in	ADP
ejpam-3438	194	9	x.	x.	NOUN
ejpam-3438	194	10	we	we	PRON
ejpam-3438	194	11	note	note	VERB
ejpam-3438	194	12	that	that	SCONJ
ejpam-3438	194	13	a	a	DET
ejpam-3438	194	14	function	function	NOUN
ejpam-3438	194	15	f	f	NOUN
ejpam-3438	194	16	:	:	PUNCT
ejpam-3438	194	17	(	(	PUNCT
ejpam-3438	194	18	x	x	X
ejpam-3438	194	19	,	,	PUNCT
ejpam-3438	194	20	τ	τ	PROPN
ejpam-3438	194	21	,	,	PUNCT
ejpam-3438	194	22	i)→	i)→	PROPN
ejpam-3438	194	23	(	(	PUNCT
ejpam-3438	194	24	y	y	PROPN
ejpam-3438	194	25	,	,	PUNCT
ejpam-3438	194	26	σ	σ	PROPN
ejpam-3438	194	27	,	,	PUNCT
ejpam-3438	194	28	j	j	PROPN
ejpam-3438	194	29	)	)	PUNCT
ejpam-3438	194	30	is	be	AUX
ejpam-3438	194	31	called	call	VERB
ejpam-3438	194	32	(	(	PUNCT
ejpam-3438	194	33	i	i	NOUN
ejpam-3438	194	34	)	)	PUNCT
ejpam-3438	194	35	βi	βi	AUX
ejpam-3438	194	36	-open	-open	VERB
ejpam-3438	194	37	function	function	NOUN
ejpam-3438	194	38	if	if	SCONJ
ejpam-3438	194	39	f(a	f(a	PROPN
ejpam-3438	194	40	)	)	PUNCT
ejpam-3438	194	41	is	be	AUX
ejpam-3438	194	42	βj	βj	PRON
ejpam-3438	194	43	-open	-open	ADJ
ejpam-3438	194	44	in	in	ADP
ejpam-3438	194	45	y	y	PROPN
ejpam-3438	194	46	for	for	SCONJ
ejpam-3438	194	47	each	each	DET
ejpam-3438	194	48	βi	βi	PRON
ejpam-3438	194	49	-open	-open	PROPN
ejpam-3438	194	50	set	set	VERB
ejpam-3438	194	51	a	a	PRON
ejpam-3438	194	52	in	in	ADP
ejpam-3438	194	53	x.	x.	PROPN
ejpam-3438	194	54	(	(	PUNCT
ejpam-3438	194	55	ii	ii	PROPN
ejpam-3438	194	56	)	)	PUNCT
ejpam-3438	194	57	βi	βi	PROPN
ejpam-3438	194	58	-irresolute	-irresolute	ADJ
ejpam-3438	194	59	function	function	NOUN
ejpam-3438	194	60	if	if	SCONJ
ejpam-3438	194	61	f−1(b	f−1(b	PROPN
ejpam-3438	194	62	)	)	PUNCT
ejpam-3438	194	63	is	be	AUX
ejpam-3438	194	64	βi	βi	PRON
ejpam-3438	194	65	-open	-open	ADJ
ejpam-3438	194	66	in	in	ADP
ejpam-3438	194	67	x	x	PUNCT
ejpam-3438	194	68	for	for	ADP
ejpam-3438	194	69	each	each	DET
ejpam-3438	194	70	βj	βj	PUNCT
ejpam-3438	194	71	-open	-open	ADJ
ejpam-3438	194	72	set	set	ADJ
ejpam-3438	194	73	b	b	NOUN
ejpam-3438	194	74	in	in	ADP
ejpam-3438	194	75	y	y	PROPN
ejpam-3438	194	76	.	.	PUNCT
ejpam-3438	195	1	(	(	PUNCT
ejpam-3438	195	2	iii	iii	X
ejpam-3438	195	3	)	)	PUNCT
ejpam-3438	195	4	βi	βi	ADP
ejpam-3438	195	5	-continuous	-continuous	ADJ
ejpam-3438	195	6	function	function	NOUN
ejpam-3438	195	7	if	if	SCONJ
ejpam-3438	195	8	f−1(b	f−1(b	PROPN
ejpam-3438	195	9	)	)	PUNCT
ejpam-3438	195	10	is	be	AUX
ejpam-3438	195	11	βi	βi	PRON
ejpam-3438	195	12	-open	-open	ADJ
ejpam-3438	195	13	in	in	ADP
ejpam-3438	195	14	x	x	PUNCT
ejpam-3438	195	15	for	for	ADP
ejpam-3438	195	16	each	each	DET
ejpam-3438	195	17	open	open	ADJ
ejpam-3438	195	18	set	set	VERB
ejpam-3438	195	19	b	b	PROPN
ejpam-3438	195	20	in	in	ADP
ejpam-3438	195	21	y	y	PROPN
ejpam-3438	195	22	.	.	PUNCT
ejpam-3438	196	1	the	the	DET
ejpam-3438	196	2	following	follow	VERB
ejpam-3438	196	3	theorems	theorem	NOUN
ejpam-3438	196	4	are	be	AUX
ejpam-3438	196	5	worth	worth	ADJ
ejpam-3438	196	6	-	-	PUNCT
ejpam-3438	196	7	noting	noting	NOUN
ejpam-3438	196	8	.	.	PUNCT
ejpam-3438	197	1	theorem	theorem	ADJ
ejpam-3438	197	2	4	4	NUM
ejpam-3438	197	3	.	.	PUNCT
ejpam-3438	198	1	if	if	SCONJ
ejpam-3438	198	2	f	f	PROPN
ejpam-3438	198	3	:	:	PUNCT
ejpam-3438	198	4	(	(	PUNCT
ejpam-3438	198	5	x	x	X
ejpam-3438	198	6	,	,	PUNCT
ejpam-3438	198	7	τ	τ	PROPN
ejpam-3438	198	8	,	,	PUNCT
ejpam-3438	198	9	i)→	i)→	PROPN
ejpam-3438	198	10	(	(	PUNCT
ejpam-3438	198	11	y	y	PROPN
ejpam-3438	198	12	,	,	PUNCT
ejpam-3438	198	13	σ	σ	PROPN
ejpam-3438	198	14	,	,	PUNCT
ejpam-3438	198	15	j	j	PROPN
ejpam-3438	198	16	)	)	PUNCT
ejpam-3438	198	17	is	be	AUX
ejpam-3438	198	18	a	a	DET
ejpam-3438	198	19	βi	βi	NOUN
ejpam-3438	198	20	-	-	PUNCT
ejpam-3438	198	21	irresolute	irresolute	ADJ
ejpam-3438	198	22	surjective	surjective	ADJ
ejpam-3438	198	23	function	function	NOUN
ejpam-3438	198	24	and	and	CCONJ
ejpam-3438	198	25	(	(	PUNCT
ejpam-3438	198	26	x	x	X
ejpam-3438	198	27	,	,	PUNCT
ejpam-3438	198	28	τ	τ	PROPN
ejpam-3438	198	29	,	,	PUNCT
ejpam-3438	198	30	i	i	PROPN
ejpam-3438	198	31	)	)	PUNCT
ejpam-3438	198	32	is	be	AUX
ejpam-3438	198	33	a	a	DET
ejpam-3438	198	34	cβi	cβi	NOUN
ejpam-3438	198	35	-	-	PUNCT
ejpam-3438	198	36	compact	compact	ADJ
ejpam-3438	198	37	space	space	NOUN
ejpam-3438	198	38	,	,	PUNCT
ejpam-3438	198	39	then	then	ADV
ejpam-3438	198	40	(	(	PUNCT
ejpam-3438	198	41	y	y	PROPN
ejpam-3438	198	42	,	,	PUNCT
ejpam-3438	198	43	σ	σ	PROPN
ejpam-3438	198	44	,	,	PUNCT
ejpam-3438	198	45	j	j	PROPN
ejpam-3438	198	46	)	)	PUNCT
ejpam-3438	198	47	is	be	AUX
ejpam-3438	198	48	also	also	ADV
ejpam-3438	198	49	a	a	DET
ejpam-3438	198	50	cβi	cβi	NOUN
ejpam-3438	198	51	-	-	PUNCT
ejpam-3438	198	52	compact	compact	ADJ
ejpam-3438	198	53	space	space	NOUN
ejpam-3438	198	54	.	.	PUNCT
ejpam-3438	199	1	proof	proof	NOUN
ejpam-3438	199	2	.	.	PUNCT
ejpam-3438	200	1	let	let	VERB
ejpam-3438	200	2	{	{	PUNCT
ejpam-3438	200	3	uλ	uλ	X
ejpam-3438	200	4	:	:	PUNCT
ejpam-3438	200	5	λ	λ	X
ejpam-3438	200	6	∈	∈	PROPN
ejpam-3438	200	7	λ	λ	PROPN
ejpam-3438	200	8	}	}	PUNCT
ejpam-3438	200	9	be	be	VERB
ejpam-3438	200	10	a	a	DET
ejpam-3438	200	11	cover	cover	NOUN
ejpam-3438	200	12	of	of	ADP
ejpam-3438	200	13	y	y	PROPN
ejpam-3438	200	14	by	by	ADP
ejpam-3438	200	15	βi	βi	NOUN
ejpam-3438	200	16	-open	-open	NOUN
ejpam-3438	200	17	sets	set	NOUN
ejpam-3438	200	18	.	.	PUNCT
ejpam-3438	201	1	since	since	SCONJ
ejpam-3438	201	2	f	f	PROPN
ejpam-3438	201	3	is	be	AUX
ejpam-3438	201	4	a	a	DET
ejpam-3438	201	5	βi	βi	ADV
ejpam-3438	201	6	-irresolute	-irresolute	ADJ
ejpam-3438	201	7	surjective	surjective	ADJ
ejpam-3438	201	8	function	function	NOUN
ejpam-3438	201	9	,	,	PUNCT
ejpam-3438	201	10	{	{	PUNCT
ejpam-3438	201	11	f−1(uλ	f−1(uλ	PROPN
ejpam-3438	201	12	)	)	PUNCT
ejpam-3438	201	13	:	:	PUNCT
ejpam-3438	202	1	λ	λ	X
ejpam-3438	202	2	∈	∈	PROPN
ejpam-3438	202	3	λ	λ	PROPN
ejpam-3438	202	4	}	}	PUNCT
ejpam-3438	202	5	is	be	AUX
ejpam-3438	202	6	a	a	DET
ejpam-3438	202	7	cover	cover	NOUN
ejpam-3438	202	8	of	of	ADP
ejpam-3438	202	9	x	x	PUNCT
ejpam-3438	202	10	by	by	ADP
ejpam-3438	202	11	βi	βi	NOUN
ejpam-3438	202	12	-open	-open	NOUN
ejpam-3438	202	13	sets	set	NOUN
ejpam-3438	202	14	.	.	PUNCT
ejpam-3438	203	1	since	since	SCONJ
ejpam-3438	203	2	x	x	PRON
ejpam-3438	203	3	is	be	AUX
ejpam-3438	203	4	cβi	cβi	NOUN
ejpam-3438	203	5	compact	compact	ADJ
ejpam-3438	203	6	,	,	PUNCT
ejpam-3438	203	7	there	there	PRON
ejpam-3438	203	8	exists	exist	VERB
ejpam-3438	203	9	a	a	DET
ejpam-3438	203	10	finite	finite	NOUN
ejpam-3438	203	11	subset	subset	NOUN
ejpam-3438	203	12	λ0	λ0	NOUN
ejpam-3438	203	13	of	of	ADP
ejpam-3438	203	14	λ	λ	NOUN
ejpam-3438	203	15	such	such	ADJ
ejpam-3438	203	16	that	that	SCONJ
ejpam-3438	203	17	x	x	SYM
ejpam-3438	204	1	−	−	X
ejpam-3438	204	2	⋃	⋃	PROPN
ejpam-3438	204	3	{	{	PUNCT
ejpam-3438	204	4	f−1(uλ	f−1(uλ	PROPN
ejpam-3438	204	5	)	)	PUNCT
ejpam-3438	204	6	:	:	PUNCT
ejpam-3438	205	1	λ	λ	X
ejpam-3438	205	2	∈	∈	NOUN
ejpam-3438	205	3	λ0	λ0	NOUN
ejpam-3438	205	4	}	}	PUNCT
ejpam-3438	205	5	∈	∈	PROPN
ejpam-3438	205	6	i.	i.	NOUN
ejpam-3438	205	7	by	by	ADP
ejpam-3438	205	8	remark	remark	NOUN
ejpam-3438	205	9	1	1	NUM
ejpam-3438	205	10	,	,	PUNCT
ejpam-3438	205	11	y	y	PROPN
ejpam-3438	205	12	−	−	PROPN
ejpam-3438	205	13	∪{f(uλ	∪{f(uλ	NUM
ejpam-3438	205	14	)	)	PUNCT
ejpam-3438	205	15	:	:	PUNCT
ejpam-3438	206	1	λ	λ	X
ejpam-3438	206	2	∈	∈	NOUN
ejpam-3438	206	3	λ0	λ0	NOUN
ejpam-3438	206	4	}	}	PUNCT
ejpam-3438	206	5	=	=	SYM
ejpam-3438	206	6	f(x	f(x	PROPN
ejpam-3438	206	7	−	−	PROPN
ejpam-3438	206	8	⋃	⋃	PROPN
ejpam-3438	206	9	{	{	PUNCT
ejpam-3438	206	10	f−1(uλ	f−1(uλ	PROPN
ejpam-3438	206	11	)	)	PUNCT
ejpam-3438	206	12	:	:	PUNCT
ejpam-3438	207	1	λ	λ	X
ejpam-3438	207	2	∈	∈	NOUN
ejpam-3438	207	3	λ0	λ0	NOUN
ejpam-3438	207	4	}	}	PUNCT
ejpam-3438	207	5	)	)	PUNCT
ejpam-3438	208	1	∈	∈	PROPN
ejpam-3438	208	2	j	j	PROPN
ejpam-3438	208	3	.	.	PUNCT
ejpam-3438	209	1	theorem	theorem	VERB
ejpam-3438	209	2	5	5	NUM
ejpam-3438	209	3	.	.	PUNCT
ejpam-3438	210	1	if	if	SCONJ
ejpam-3438	210	2	f	f	PROPN
ejpam-3438	210	3	:	:	PUNCT
ejpam-3438	210	4	(	(	PUNCT
ejpam-3438	210	5	x	x	X
ejpam-3438	210	6	,	,	PUNCT
ejpam-3438	210	7	τ	τ	PROPN
ejpam-3438	210	8	,	,	PUNCT
ejpam-3438	210	9	i	i	NOUN
ejpam-3438	210	10	)	)	PUNCT
ejpam-3438	210	11	→	→	SYM
ejpam-3438	210	12	(	(	PUNCT
ejpam-3438	210	13	y	y	PROPN
ejpam-3438	210	14	,	,	PUNCT
ejpam-3438	210	15	σ	σ	PROPN
ejpam-3438	210	16	,	,	PUNCT
ejpam-3438	210	17	j	j	PROPN
ejpam-3438	210	18	)	)	PUNCT
ejpam-3438	210	19	is	be	AUX
ejpam-3438	210	20	a	a	DET
ejpam-3438	210	21	βf−1(j)-open	βf−1(j)-open	ADJ
ejpam-3438	210	22	bijective	bijective	ADJ
ejpam-3438	210	23	function	function	NOUN
ejpam-3438	210	24	and	and	CCONJ
ejpam-3438	210	25	(	(	PUNCT
ejpam-3438	210	26	y	y	PROPN
ejpam-3438	210	27	,	,	PUNCT
ejpam-3438	210	28	σ	σ	PROPN
ejpam-3438	210	29	,	,	PUNCT
ejpam-3438	210	30	j	j	PROPN
ejpam-3438	210	31	)	)	PUNCT
ejpam-3438	210	32	is	be	AUX
ejpam-3438	210	33	a	a	DET
ejpam-3438	210	34	cβj	cβj	ADJ
ejpam-3438	210	35	-compact	-compact	NOUN
ejpam-3438	210	36	space	space	NOUN
ejpam-3438	210	37	,	,	PUNCT
ejpam-3438	210	38	then	then	ADV
ejpam-3438	210	39	(	(	PUNCT
ejpam-3438	210	40	x	x	X
ejpam-3438	210	41	,	,	PUNCT
ejpam-3438	210	42	τ	τ	PROPN
ejpam-3438	210	43	,	,	PUNCT
ejpam-3438	210	44	f−1(j	f−1(j	PROPN
ejpam-3438	210	45	)	)	PUNCT
ejpam-3438	210	46	)	)	PUNCT
ejpam-3438	211	1	is	be	AUX
ejpam-3438	211	2	also	also	ADV
ejpam-3438	211	3	a	a	DET
ejpam-3438	211	4	cβf−1(j)-compact	cβf−1(j)-compact	ADJ
ejpam-3438	211	5	space	space	NOUN
ejpam-3438	211	6	.	.	PUNCT
ejpam-3438	212	1	proof	proof	NOUN
ejpam-3438	212	2	.	.	PUNCT
ejpam-3438	213	1	let	let	VERB
ejpam-3438	213	2	{	{	PUNCT
ejpam-3438	213	3	uλ	uλ	X
ejpam-3438	213	4	:	:	PUNCT
ejpam-3438	213	5	λ	λ	X
ejpam-3438	213	6	∈	∈	PROPN
ejpam-3438	213	7	λ	λ	PROPN
ejpam-3438	213	8	}	}	PUNCT
ejpam-3438	213	9	be	be	VERB
ejpam-3438	213	10	a	a	DET
ejpam-3438	213	11	cover	cover	NOUN
ejpam-3438	213	12	of	of	ADP
ejpam-3438	213	13	x	x	PUNCT
ejpam-3438	213	14	by	by	ADP
ejpam-3438	213	15	βf−1(j)-open	βf−1(j)-open	ADJ
ejpam-3438	213	16	sets	set	NOUN
ejpam-3438	213	17	.	.	PUNCT
ejpam-3438	214	1	since	since	SCONJ
ejpam-3438	214	2	f	f	PROPN
ejpam-3438	214	3	is	be	AUX
ejpam-3438	214	4	an	an	DET
ejpam-3438	214	5	open	open	ADJ
ejpam-3438	214	6	bijective	bijective	ADJ
ejpam-3438	214	7	function	function	NOUN
ejpam-3438	214	8	,	,	PUNCT
ejpam-3438	214	9	{	{	PUNCT
ejpam-3438	214	10	f(uλ	f(uλ	PROPN
ejpam-3438	214	11	)	)	PUNCT
ejpam-3438	214	12	:	:	PUNCT
ejpam-3438	215	1	λ	λ	X
ejpam-3438	215	2	∈	∈	PROPN
ejpam-3438	215	3	λ	λ	PROPN
ejpam-3438	215	4	}	}	PUNCT
ejpam-3438	215	5	is	be	AUX
ejpam-3438	215	6	a	a	DET
ejpam-3438	215	7	cover	cover	NOUN
ejpam-3438	215	8	of	of	ADP
ejpam-3438	215	9	y	y	NOUN
ejpam-3438	215	10	by	by	ADP
ejpam-3438	215	11	βj	βj	NOUN
ejpam-3438	215	12	-open	-open	ADJ
ejpam-3438	215	13	sets	set	NOUN
ejpam-3438	215	14	.	.	PUNCT
ejpam-3438	216	1	since	since	SCONJ
ejpam-3438	216	2	y	y	PROPN
ejpam-3438	216	3	is	be	AUX
ejpam-3438	216	4	a	a	DET
ejpam-3438	216	5	cβj	cβj	ADJ
ejpam-3438	216	6	compact	compact	ADJ
ejpam-3438	216	7	space	space	NOUN
ejpam-3438	216	8	,	,	PUNCT
ejpam-3438	216	9	there	there	PRON
ejpam-3438	216	10	exists	exist	VERB
ejpam-3438	216	11	a	a	DET
ejpam-3438	216	12	finite	finite	NOUN
ejpam-3438	216	13	subset	subset	NOUN
ejpam-3438	216	14	λ0	λ0	NOUN
ejpam-3438	216	15	of	of	ADP
ejpam-3438	216	16	λ	λ	PROPN
ejpam-3438	216	17	such	such	ADJ
ejpam-3438	216	18	that	that	SCONJ
ejpam-3438	216	19	y	y	PROPN
ejpam-3438	217	1	−	−	PROPN
ejpam-3438	217	2	⋃	⋃	PROPN
ejpam-3438	217	3	{	{	PUNCT
ejpam-3438	217	4	f(uλ	f(uλ	PROPN
ejpam-3438	217	5	)	)	PUNCT
ejpam-3438	217	6	:	:	PUNCT
ejpam-3438	218	1	λ	λ	PROPN
ejpam-3438	218	2	∈	∈	NOUN
ejpam-3438	218	3	λ0	λ0	NOUN
ejpam-3438	218	4	}	}	PUNCT
ejpam-3438	218	5	∈	∈	PROPN
ejpam-3438	218	6	j	j	PROPN
ejpam-3438	218	7	,	,	PUNCT
ejpam-3438	218	8	that	that	ADV
ejpam-3438	218	9	is	is	ADV
ejpam-3438	218	10	x	x	PUNCT
ejpam-3438	218	11	−	−	PROPN
ejpam-3438	218	12	∪{uλ	∪{uλ	PROPN
ejpam-3438	218	13	:	:	PUNCT
ejpam-3438	218	14	λ	λ	PROPN
ejpam-3438	218	15	∈	∈	NOUN
ejpam-3438	218	16	λ0	λ0	NOUN
ejpam-3438	218	17	}	}	PUNCT
ejpam-3438	218	18	=	=	SYM
ejpam-3438	218	19	f−1(y	f−1(y	PROPN
ejpam-3438	218	20	−	−	PROPN
ejpam-3438	218	21	⋃	⋃	PROPN
ejpam-3438	218	22	{	{	PUNCT
ejpam-3438	218	23	f(uλ	f(uλ	PROPN
ejpam-3438	218	24	)	)	PUNCT
ejpam-3438	218	25	:	:	PUNCT
ejpam-3438	218	26	λ	λ	X
ejpam-3438	218	27	∈	∈	NOUN
ejpam-3438	218	28	λ0	λ0	NOUN
ejpam-3438	218	29	}	}	PUNCT
ejpam-3438	218	30	)	)	PUNCT
ejpam-3438	218	31	∈	∈	PROPN
ejpam-3438	219	1	j	j	PROPN
ejpam-3438	219	2	.	.	PUNCT
ejpam-3438	220	1	this	this	PRON
ejpam-3438	220	2	shows	show	VERB
ejpam-3438	220	3	that	that	SCONJ
ejpam-3438	220	4	(	(	PUNCT
ejpam-3438	220	5	x	x	X
ejpam-3438	220	6	,	,	PUNCT
ejpam-3438	220	7	τ	τ	PROPN
ejpam-3438	220	8	,	,	PUNCT
ejpam-3438	220	9	f−1(j	f−1(j	PROPN
ejpam-3438	220	10	)	)	PUNCT
ejpam-3438	220	11	)	)	PUNCT
ejpam-3438	220	12	is	be	AUX
ejpam-3438	220	13	a	a	DET
ejpam-3438	220	14	cβf−1(j)-compact	cβf−1(j)-compact	ADJ
ejpam-3438	220	15	space	space	NOUN
ejpam-3438	220	16	.	.	PUNCT
ejpam-3438	221	1	theorem	theorem	VERB
ejpam-3438	221	2	6	6	NUM
ejpam-3438	221	3	.	.	PUNCT
ejpam-3438	222	1	every	every	DET
ejpam-3438	222	2	cβi	cβi	NOUN
ejpam-3438	222	3	-	-	PUNCT
ejpam-3438	222	4	compact	compact	ADJ
ejpam-3438	222	5	space	space	NOUN
ejpam-3438	222	6	is	be	AUX
ejpam-3438	222	7	also	also	ADV
ejpam-3438	222	8	a	a	DET
ejpam-3438	222	9	countably	countably	ADV
ejpam-3438	222	10	βi	βi	ADJ
ejpam-3438	222	11	-	-	ADJ
ejpam-3438	222	12	compact	compact	ADJ
ejpam-3438	222	13	space	space	NOUN
ejpam-3438	222	14	.	.	PUNCT
ejpam-3438	223	1	proof	proof	NOUN
ejpam-3438	223	2	.	.	PUNCT
ejpam-3438	224	1	let	let	VERB
ejpam-3438	224	2	(	(	PUNCT
ejpam-3438	224	3	x	x	X
ejpam-3438	224	4	,	,	PUNCT
ejpam-3438	224	5	τ	τ	PROPN
ejpam-3438	224	6	,	,	PUNCT
ejpam-3438	224	7	i	i	PRON
ejpam-3438	224	8	)	)	PUNCT
ejpam-3438	224	9	be	be	AUX
ejpam-3438	224	10	cβi	cβi	NOUN
ejpam-3438	224	11	-compact	-compact	ADJ
ejpam-3438	224	12	space	space	NOUN
ejpam-3438	224	13	.	.	PUNCT
ejpam-3438	225	1	let	let	VERB
ejpam-3438	225	2	{	{	PUNCT
ejpam-3438	225	3	un	un	PROPN
ejpam-3438	225	4	:	:	PUNCT
ejpam-3438	225	5	n	n	CCONJ
ejpam-3438	225	6	∈	∈	PROPN
ejpam-3438	225	7	n	n	CCONJ
ejpam-3438	225	8	}	}	PUNCT
ejpam-3438	225	9	be	be	AUX
ejpam-3438	225	10	a	a	DET
ejpam-3438	225	11	countable	countable	ADJ
ejpam-3438	225	12	cover	cover	NOUN
ejpam-3438	225	13	of	of	ADP
ejpam-3438	225	14	x	x	PUNCT
ejpam-3438	225	15	by	by	ADP
ejpam-3438	225	16	βi	βi	NOUN
ejpam-3438	225	17	-open	-open	NOUN
ejpam-3438	225	18	sets	set	NOUN
ejpam-3438	225	19	.	.	PUNCT
ejpam-3438	226	1	since	since	SCONJ
ejpam-3438	226	2	x	x	PRON
ejpam-3438	226	3	is	be	AUX
ejpam-3438	226	4	a	a	DET
ejpam-3438	226	5	cβi	cβi	NOUN
ejpam-3438	226	6	-compact	-compact	NOUN
ejpam-3438	226	7	space	space	NOUN
ejpam-3438	226	8	,	,	PUNCT
ejpam-3438	226	9	there	there	PRON
ejpam-3438	226	10	exists	exist	VERB
ejpam-3438	226	11	a	a	DET
ejpam-3438	226	12	finite	finite	NOUN
ejpam-3438	226	13	subset	subset	NOUN
ejpam-3438	226	14	{	{	PUNCT
ejpam-3438	226	15	ij	ij	INTJ
ejpam-3438	226	16	:	:	PUNCT
ejpam-3438	226	17	j	j	PROPN
ejpam-3438	226	18	=	=	SYM
ejpam-3438	226	19	1	1	NUM
ejpam-3438	226	20	,	,	PUNCT
ejpam-3438	226	21	2	2	NUM
ejpam-3438	226	22	,	,	PUNCT
ejpam-3438	226	23	.	.	PUNCT
ejpam-3438	226	24	.	.	PUNCT
ejpam-3438	227	1	.	.	PUNCT
ejpam-3438	228	1	,	,	PUNCT
ejpam-3438	228	2	k	k	X
ejpam-3438	228	3	}	}	PUNCT
ejpam-3438	228	4	of	of	ADP
ejpam-3438	228	5	n	n	PRON
ejpam-3438	228	6	such	such	ADJ
ejpam-3438	228	7	that	that	SCONJ
ejpam-3438	228	8	x	x	PART
ejpam-3438	228	9	−	−	PROPN
ejpam-3438	228	10	⋃	⋃	NOUN
ejpam-3438	228	11	{	{	PUNCT
ejpam-3438	228	12	uij	uij	X
ejpam-3438	228	13	:	:	PUNCT
ejpam-3438	228	14	j	j	PROPN
ejpam-3438	228	15	=	=	SYM
ejpam-3438	228	16	1	1	NUM
ejpam-3438	228	17	,	,	PUNCT
ejpam-3438	228	18	2	2	NUM
ejpam-3438	228	19	,	,	PUNCT
ejpam-3438	228	20	.	.	PUNCT
ejpam-3438	228	21	.	.	PUNCT
ejpam-3438	228	22	.	.	PUNCT
ejpam-3438	229	1	,	,	PUNCT
ejpam-3438	229	2	k	k	X
ejpam-3438	229	3	}	}	PUNCT
ejpam-3438	229	4	∈	∈	PROPN
ejpam-3438	229	5	i.	i.	NOUN
ejpam-3438	229	6	this	this	PRON
ejpam-3438	229	7	shows	show	VERB
ejpam-3438	229	8	that	that	SCONJ
ejpam-3438	229	9	(	(	PUNCT
ejpam-3438	229	10	x	x	X
ejpam-3438	229	11	,	,	PUNCT
ejpam-3438	229	12	τ	τ	PROPN
ejpam-3438	229	13	,	,	PUNCT
ejpam-3438	229	14	i	i	PROPN
ejpam-3438	229	15	)	)	PUNCT
ejpam-3438	229	16	is	be	AUX
ejpam-3438	229	17	a	a	DET
ejpam-3438	229	18	countably	countably	ADV
ejpam-3438	229	19	βi	βi	PRON
ejpam-3438	229	20	-compact	-compact	NOUN
ejpam-3438	229	21	space	space	NOUN
ejpam-3438	229	22	.	.	PUNCT
ejpam-3438	230	1	g.	g.	PROPN
ejpam-3438	230	2	catalan	catalan	PROPN
ejpam-3438	230	3	,	,	PUNCT
ejpam-3438	230	4	r.	r.	PROPN
ejpam-3438	230	5	padua	padua	PROPN
ejpam-3438	230	6	,	,	PUNCT
ejpam-3438	230	7	m.	m.	PROPN
ejpam-3438	230	8	baldado	baldado	PROPN
ejpam-3438	230	9	jr	jr	PROPN
ejpam-3438	230	10	.	.	PROPN
ejpam-3438	230	11	/	/	SYM
ejpam-3438	230	12	eur	eur	PROPN
ejpam-3438	230	13	.	.	PUNCT
ejpam-3438	231	1	j.	j.	PROPN
ejpam-3438	231	2	pure	pure	PROPN
ejpam-3438	231	3	appl	appl	PROPN
ejpam-3438	231	4	.	.	PROPN
ejpam-3438	231	5	math	math	PROPN
ejpam-3438	231	6	,	,	PUNCT
ejpam-3438	231	7	12	12	NUM
ejpam-3438	231	8	(	(	PUNCT
ejpam-3438	231	9	3	3	NUM
ejpam-3438	231	10	)	)	PUNCT
ejpam-3438	231	11	(	(	PUNCT
ejpam-3438	231	12	2019	2019	NUM
ejpam-3438	231	13	)	)	PUNCT
ejpam-3438	231	14	,	,	PUNCT
ejpam-3438	231	15	893	893	NUM
ejpam-3438	231	16	-	-	SYM
ejpam-3438	231	17	905	905	NUM
ejpam-3438	231	18	899	899	NUM
ejpam-3438	231	19	3	3	NUM
ejpam-3438	231	20	.	.	PUNCT
ejpam-3438	231	21	hyperconnectedness	hyperconnectedness	NOUN
ejpam-3438	231	22	with	with	ADP
ejpam-3438	231	23	respect	respect	NOUN
ejpam-3438	231	24	to	to	AUX
ejpam-3438	231	25	ideals	ideal	NOUN
ejpam-3438	231	26	the	the	DET
ejpam-3438	231	27	concept	concept	NOUN
ejpam-3438	231	28	∗-hyperconnectedness	∗-hyperconnectedness	ADJ
ejpam-3438	231	29	was	be	AUX
ejpam-3438	231	30	introduced	introduce	VERB
ejpam-3438	231	31	by	by	ADP
ejpam-3438	231	32	ekici	ekici	PROPN
ejpam-3438	231	33	et	et	PROPN
ejpam-3438	231	34	al	al	PROPN
ejpam-3438	231	35	.	.	PUNCT
ejpam-3438	232	1	[	[	X
ejpam-3438	232	2	2	2	NUM
ejpam-3438	232	3	]	]	PUNCT
ejpam-3438	232	4	,	,	PUNCT
ejpam-3438	232	5	and	and	CCONJ
ejpam-3438	232	6	the	the	DET
ejpam-3438	232	7	concept	concept	NOUN
ejpam-3438	232	8	i∗-hyperconnectedness	i∗-hyperconnectedness	ADV
ejpam-3438	232	9	was	be	AUX
ejpam-3438	232	10	introduced	introduce	VERB
ejpam-3438	232	11	by	by	ADP
ejpam-3438	232	12	abd	abd	PROPN
ejpam-3438	232	13	el	el	PROPN
ejpam-3438	232	14	-	-	PROPN
ejpam-3438	232	15	monsef	monsef	PROPN
ejpam-3438	232	16	et	et	PROPN
ejpam-3438	232	17	al	al	PROPN
ejpam-3438	232	18	.	.	PUNCT
ejpam-3438	233	1	[	[	X
ejpam-3438	233	2	7	7	NUM
ejpam-3438	233	3	]	]	PUNCT
ejpam-3438	233	4	.	.	PUNCT
ejpam-3438	234	1	these	these	DET
ejpam-3438	234	2	insights	insight	NOUN
ejpam-3438	234	3	propelled	propel	VERB
ejpam-3438	234	4	us	we	PRON
ejpam-3438	234	5	to	to	PART
ejpam-3438	234	6	create	create	VERB
ejpam-3438	234	7	a	a	DET
ejpam-3438	234	8	parallel	parallel	ADJ
ejpam-3438	234	9	concept	concept	NOUN
ejpam-3438	234	10	called	call	VERB
ejpam-3438	234	11	βi∗-hyperconnectedness	βi∗-hyperconnectedness	ADV
ejpam-3438	234	12	,	,	PUNCT
ejpam-3438	234	13	and	and	CCONJ
ejpam-3438	234	14	the	the	DET
ejpam-3438	234	15	investigation	investigation	NOUN
ejpam-3438	234	16	in	in	ADP
ejpam-3438	234	17	this	this	DET
ejpam-3438	234	18	section	section	NOUN
ejpam-3438	234	19	is	be	AUX
ejpam-3438	234	20	parallel	parallel	ADJ
ejpam-3438	234	21	to	to	ADP
ejpam-3438	234	22	the	the	DET
ejpam-3438	234	23	investigation	investigation	NOUN
ejpam-3438	234	24	of	of	ADP
ejpam-3438	234	25	α	α	NOUN
ejpam-3438	234	26	-	-	ADJ
ejpam-3438	234	27	open	open	ADJ
ejpam-3438	234	28	sets	set	NOUN
ejpam-3438	234	29	in	in	ADP
ejpam-3438	234	30	[	[	X
ejpam-3438	234	31	24	24	NUM
ejpam-3438	234	32	]	]	PUNCT
ejpam-3438	234	33	.	.	PUNCT
ejpam-3438	235	1	theorem	theorem	VERB
ejpam-3438	235	2	7	7	NUM
ejpam-3438	235	3	.	.	PUNCT
ejpam-3438	236	1	every	every	DET
ejpam-3438	236	2	βi∗-hyperconnected	βi∗-hyperconnecte	VERB
ejpam-3438	236	3	space	space	NOUN
ejpam-3438	236	4	is	be	AUX
ejpam-3438	236	5	also	also	ADV
ejpam-3438	236	6	an	an	DET
ejpam-3438	236	7	i∗-hyperconnected	i∗-hyperconnecte	VERB
ejpam-3438	236	8	space	space	NOUN
ejpam-3438	236	9	.	.	PUNCT
ejpam-3438	237	1	proof	proof	NOUN
ejpam-3438	237	2	.	.	PUNCT
ejpam-3438	238	1	let	let	VERB
ejpam-3438	238	2	(	(	PUNCT
ejpam-3438	238	3	x	x	X
ejpam-3438	238	4	,	,	PUNCT
ejpam-3438	238	5	τ	τ	PROPN
ejpam-3438	238	6	,	,	PUNCT
ejpam-3438	238	7	i	i	PRON
ejpam-3438	238	8	)	)	PUNCT
ejpam-3438	238	9	be	be	VERB
ejpam-3438	238	10	a	a	DET
ejpam-3438	238	11	β∗i	β∗i	ADV
ejpam-3438	238	12	-hyperconnected	-hyperconnecte	VERB
ejpam-3438	238	13	space	space	NOUN
ejpam-3438	238	14	and	and	CCONJ
ejpam-3438	238	15	a	a	DET
ejpam-3438	238	16	be	be	AUX
ejpam-3438	238	17	an	an	DET
ejpam-3438	238	18	open	open	ADJ
ejpam-3438	238	19	set	set	NOUN
ejpam-3438	238	20	.	.	PUNCT
ejpam-3438	239	1	since	since	SCONJ
ejpam-3438	239	2	x	x	PRON
ejpam-3438	239	3	is	be	AUX
ejpam-3438	239	4	a	a	DET
ejpam-3438	239	5	β∗i	β∗i	ADV
ejpam-3438	239	6	-hyperconnected	-hyperconnecte	VERB
ejpam-3438	239	7	space	space	NOUN
ejpam-3438	239	8	and	and	CCONJ
ejpam-3438	239	9	every	every	DET
ejpam-3438	239	10	open	open	ADJ
ejpam-3438	239	11	set	set	NOUN
ejpam-3438	239	12	is	be	AUX
ejpam-3438	239	13	a	a	DET
ejpam-3438	239	14	βi	βi	X
ejpam-3438	239	15	-open	-open	NOUN
ejpam-3438	239	16	set	set	NOUN
ejpam-3438	239	17	,	,	PUNCT
ejpam-3438	239	18	x	x	PUNCT
ejpam-3438	239	19	−	−	NOUN
ejpam-3438	239	20	cl∗(a	cl∗(a	NOUN
ejpam-3438	239	21	)	)	PUNCT
ejpam-3438	239	22	∈	∈	PROPN
ejpam-3438	239	23	i.	i.	NOUN
ejpam-3438	239	24	thus	thus	ADV
ejpam-3438	239	25	,	,	PUNCT
ejpam-3438	239	26	(	(	PUNCT
ejpam-3438	239	27	x	x	X
ejpam-3438	239	28	,	,	PUNCT
ejpam-3438	239	29	τ	τ	PROPN
ejpam-3438	239	30	,	,	PUNCT
ejpam-3438	239	31	i	i	PROPN
ejpam-3438	239	32	)	)	PUNCT
ejpam-3438	239	33	is	be	AUX
ejpam-3438	239	34	also	also	ADV
ejpam-3438	239	35	an	an	DET
ejpam-3438	239	36	i∗-hyperconnected	i∗-hyperconnecte	VERB
ejpam-3438	239	37	space	space	NOUN
ejpam-3438	239	38	.	.	PUNCT
ejpam-3438	240	1	lemma	lemma	PROPN
ejpam-3438	240	2	5	5	NUM
ejpam-3438	240	3	.	.	PUNCT
ejpam-3438	241	1	the	the	DET
ejpam-3438	241	2	intersection	intersection	NOUN
ejpam-3438	241	3	of	of	ADP
ejpam-3438	241	4	any	any	DET
ejpam-3438	241	5	family	family	NOUN
ejpam-3438	241	6	of	of	ADP
ejpam-3438	241	7	ideals	ideal	NOUN
ejpam-3438	241	8	on	on	ADP
ejpam-3438	241	9	x	x	X
ejpam-3438	241	10	is	be	AUX
ejpam-3438	241	11	an	an	DET
ejpam-3438	241	12	ideal	ideal	NOUN
ejpam-3438	241	13	on	on	ADP
ejpam-3438	241	14	x.	x.	NOUN
ejpam-3438	241	15	theorem	theorem	VERB
ejpam-3438	241	16	8	8	NUM
ejpam-3438	241	17	say	say	VERB
ejpam-3438	241	18	that	that	SCONJ
ejpam-3438	241	19	if	if	SCONJ
ejpam-3438	241	20	i	i	PRON
ejpam-3438	241	21	is	be	AUX
ejpam-3438	241	22	the	the	DET
ejpam-3438	241	23	minimal	minimal	ADJ
ejpam-3438	241	24	ideal	ideal	NOUN
ejpam-3438	241	25	,	,	PUNCT
ejpam-3438	241	26	then	then	ADV
ejpam-3438	241	27	the	the	DET
ejpam-3438	241	28	notions	notion	NOUN
ejpam-3438	241	29	∗-hyperconnectedness	∗-hyperconnectedness	PART
ejpam-3438	241	30	and	and	CCONJ
ejpam-3438	241	31	i∗-hyperconnectedness	i∗-hyperconnectedness	ADV
ejpam-3438	241	32	are	be	AUX
ejpam-3438	241	33	the	the	DET
ejpam-3438	241	34	same	same	ADJ
ejpam-3438	241	35	.	.	PUNCT
ejpam-3438	242	1	moreover	moreover	ADV
ejpam-3438	242	2	,	,	PUNCT
ejpam-3438	242	3	if	if	SCONJ
ejpam-3438	242	4	the	the	DET
ejpam-3438	242	5	topological	topological	ADJ
ejpam-3438	242	6	space	space	NOUN
ejpam-3438	242	7	is	be	AUX
ejpam-3438	242	8	clopen	clopen	ADJ
ejpam-3438	242	9	,	,	PUNCT
ejpam-3438	242	10	then	then	ADV
ejpam-3438	242	11	the	the	DET
ejpam-3438	242	12	notions	notion	NOUN
ejpam-3438	242	13	∗-hyperconnectedness	∗-hyperconnectedness	NUM
ejpam-3438	242	14	,	,	PUNCT
ejpam-3438	242	15	i∗-hyperconnectedness	i∗-hyperconnectedness	NOUN
ejpam-3438	242	16	and	and	CCONJ
ejpam-3438	242	17	β∗i	β∗i	PUNCT
ejpam-3438	242	18	-hyperconnectedness	-hyperconnectedness	NOUN
ejpam-3438	242	19	are	be	AUX
ejpam-3438	242	20	the	the	DET
ejpam-3438	242	21	same	same	ADJ
ejpam-3438	242	22	.	.	PUNCT
ejpam-3438	243	1	theorem	theorem	ADJ
ejpam-3438	243	2	8	8	NUM
ejpam-3438	243	3	.	.	PUNCT
ejpam-3438	244	1	let	let	AUX
ejpam-3438	244	2	(	(	PUNCT
ejpam-3438	244	3	x	x	X
ejpam-3438	244	4	,	,	PUNCT
ejpam-3438	244	5	τ	τ	X
ejpam-3438	244	6	,	,	PUNCT
ejpam-3438	244	7	{	{	PUNCT
ejpam-3438	244	8	∅	∅	NOUN
ejpam-3438	244	9	}	}	PUNCT
ejpam-3438	244	10	)	)	PUNCT
ejpam-3438	244	11	be	be	AUX
ejpam-3438	244	12	an	an	DET
ejpam-3438	244	13	ideal	ideal	ADJ
ejpam-3438	244	14	topological	topological	ADJ
ejpam-3438	244	15	space	space	NOUN
ejpam-3438	244	16	.	.	PUNCT
ejpam-3438	245	1	(	(	PUNCT
ejpam-3438	245	2	i	i	NOUN
ejpam-3438	245	3	)	)	PUNCT
ejpam-3438	245	4	if	if	SCONJ
ejpam-3438	245	5	i	i	PRON
ejpam-3438	245	6	=	=	SYM
ejpam-3438	245	7	{	{	PUNCT
ejpam-3438	245	8	∅	∅	NOUN
ejpam-3438	245	9	}	}	PUNCT
ejpam-3438	245	10	,	,	PUNCT
ejpam-3438	245	11	then	then	ADV
ejpam-3438	245	12	the	the	DET
ejpam-3438	245	13	concepts	concept	NOUN
ejpam-3438	245	14	∗-hyperconnectedness	∗-hyperconnectedness	ADJ
ejpam-3438	245	15	and	and	CCONJ
ejpam-3438	245	16	i∗-hyperconnectedness	i∗-hyperconnectedness	ADV
ejpam-3438	245	17	are	be	AUX
ejpam-3438	245	18	equivalent	equivalent	ADJ
ejpam-3438	245	19	.	.	PUNCT
ejpam-3438	246	1	[	[	X
ejpam-3438	246	2	3	3	NUM
ejpam-3438	246	3	]	]	X
ejpam-3438	246	4	(	(	PUNCT
ejpam-3438	246	5	ii	ii	NOUN
ejpam-3438	246	6	)	)	PUNCT
ejpam-3438	246	7	if	if	SCONJ
ejpam-3438	246	8	i	i	PRON
ejpam-3438	246	9	=	=	SYM
ejpam-3438	246	10	{	{	PUNCT
ejpam-3438	246	11	∅	∅	NOUN
ejpam-3438	246	12	}	}	PUNCT
ejpam-3438	246	13	and	and	CCONJ
ejpam-3438	246	14	every	every	DET
ejpam-3438	246	15	open	open	ADJ
ejpam-3438	246	16	set	set	NOUN
ejpam-3438	246	17	is	be	AUX
ejpam-3438	246	18	closed	closed	ADJ
ejpam-3438	246	19	,	,	PUNCT
ejpam-3438	246	20	then	then	ADV
ejpam-3438	246	21	the	the	DET
ejpam-3438	246	22	concepts	concept	NOUN
ejpam-3438	246	23	∗-hyperconnectedness	∗-hyperconnectedness	NUM
ejpam-3438	246	24	,	,	PUNCT
ejpam-3438	246	25	i∗-hyperconnectedness	i∗-hyperconnectedness	NOUN
ejpam-3438	246	26	and	and	CCONJ
ejpam-3438	246	27	β∗i	β∗i	PUNCT
ejpam-3438	246	28	-hyperconnectedness	-hyperconnectedness	NOUN
ejpam-3438	246	29	are	be	AUX
ejpam-3438	246	30	equivalent	equivalent	ADJ
ejpam-3438	246	31	.	.	PUNCT
ejpam-3438	247	1	proof	proof	NOUN
ejpam-3438	247	2	.	.	PUNCT
ejpam-3438	248	1	(	(	PUNCT
ejpam-3438	248	2	1	1	X
ejpam-3438	248	3	)	)	PUNCT
ejpam-3438	248	4	if	if	SCONJ
ejpam-3438	248	5	(	(	PUNCT
ejpam-3438	248	6	x	x	NOUN
ejpam-3438	248	7	,	,	PUNCT
ejpam-3438	248	8	τ	τ	PROPN
ejpam-3438	248	9	,	,	PUNCT
ejpam-3438	248	10	i	i	PROPN
ejpam-3438	248	11	)	)	PUNCT
ejpam-3438	248	12	is	be	AUX
ejpam-3438	248	13	a	a	DET
ejpam-3438	248	14	β∗i	β∗i	ADV
ejpam-3438	248	15	-hyperconnected	-hyperconnecte	VERB
ejpam-3438	248	16	space	space	NOUN
ejpam-3438	248	17	and	and	CCONJ
ejpam-3438	248	18	a	a	PRON
ejpam-3438	248	19	is	be	AUX
ejpam-3438	248	20	a	a	DET
ejpam-3438	248	21	non	non	ADJ
ejpam-3438	248	22	-	-	ADJ
ejpam-3438	248	23	empty	empty	ADJ
ejpam-3438	248	24	open	open	ADJ
ejpam-3438	248	25	set	set	NOUN
ejpam-3438	248	26	,	,	PUNCT
ejpam-3438	248	27	then	then	ADV
ejpam-3438	248	28	cl∗(a	cl∗(a	NOUN
ejpam-3438	248	29	)	)	PUNCT
ejpam-3438	249	1	=	=	PUNCT
ejpam-3438	249	2	x.	x.	NOUN
ejpam-3438	249	3	hence	hence	ADV
ejpam-3438	249	4	,	,	PUNCT
ejpam-3438	249	5	x	x	PUNCT
ejpam-3438	249	6	−	−	NOUN
ejpam-3438	249	7	cl∗(a	cl∗(a	NOUN
ejpam-3438	249	8	)	)	PUNCT
ejpam-3438	249	9	=	=	SYM
ejpam-3438	249	10	∅	∅	NOUN
ejpam-3438	249	11	∈	∈	PROPN
ejpam-3438	249	12	i.	i.	NOUN
ejpam-3438	249	13	thus	thus	ADV
ejpam-3438	249	14	,	,	PUNCT
ejpam-3438	249	15	(	(	PUNCT
ejpam-3438	249	16	x	x	X
ejpam-3438	249	17	,	,	PUNCT
ejpam-3438	249	18	τ	τ	PROPN
ejpam-3438	249	19	,	,	PUNCT
ejpam-3438	249	20	i	i	PROPN
ejpam-3438	249	21	)	)	PUNCT
ejpam-3438	249	22	is	be	AUX
ejpam-3438	249	23	an	an	DET
ejpam-3438	249	24	i∗-hyperconnected	i∗-hyperconnecte	VERB
ejpam-3438	249	25	space	space	NOUN
ejpam-3438	249	26	.	.	PUNCT
ejpam-3438	250	1	conversely	conversely	ADV
ejpam-3438	250	2	,	,	PUNCT
ejpam-3438	250	3	if	if	SCONJ
ejpam-3438	250	4	(	(	PUNCT
ejpam-3438	250	5	x	x	NOUN
ejpam-3438	250	6	,	,	PUNCT
ejpam-3438	250	7	τ	τ	PROPN
ejpam-3438	250	8	,	,	PUNCT
ejpam-3438	250	9	i	i	PROPN
ejpam-3438	250	10	)	)	PUNCT
ejpam-3438	250	11	is	be	AUX
ejpam-3438	250	12	an	an	DET
ejpam-3438	250	13	i∗-hyperconnected	i∗-hyperconnecte	VERB
ejpam-3438	250	14	space	space	NOUN
ejpam-3438	250	15	and	and	CCONJ
ejpam-3438	250	16	a	a	PRON
ejpam-3438	250	17	is	be	AUX
ejpam-3438	250	18	a	a	DET
ejpam-3438	250	19	non	non	ADJ
ejpam-3438	250	20	-	-	ADJ
ejpam-3438	250	21	empty	empty	ADJ
ejpam-3438	250	22	open	open	ADJ
ejpam-3438	250	23	set	set	NOUN
ejpam-3438	250	24	,	,	PUNCT
ejpam-3438	250	25	then	then	ADV
ejpam-3438	250	26	x	x	ADP
ejpam-3438	250	27	−	−	PROPN
ejpam-3438	250	28	cl∗(a	cl∗(a	NOUN
ejpam-3438	250	29	)	)	PUNCT
ejpam-3438	250	30	∈	∈	PROPN
ejpam-3438	250	31	i.	i.	NOUN
ejpam-3438	250	32	since	since	SCONJ
ejpam-3438	250	33	i	i	PRON
ejpam-3438	250	34	=	=	PUNCT
ejpam-3438	250	35	{	{	PUNCT
ejpam-3438	250	36	∅	∅	NOUN
ejpam-3438	250	37	}	}	PUNCT
ejpam-3438	250	38	,	,	PUNCT
ejpam-3438	250	39	x	x	PUNCT
ejpam-3438	250	40	−	−	NOUN
ejpam-3438	250	41	cl∗(a	cl∗(a	NOUN
ejpam-3438	250	42	)	)	PUNCT
ejpam-3438	250	43	=	=	SYM
ejpam-3438	250	44	∅	∅	NOUN
ejpam-3438	250	45	,	,	PUNCT
ejpam-3438	250	46	that	that	PRON
ejpam-3438	250	47	is	be	AUX
ejpam-3438	250	48	cl∗(a	cl∗(a	ADJ
ejpam-3438	250	49	)	)	PUNCT
ejpam-3438	250	50	=	=	PUNCT
ejpam-3438	250	51	x.	x.	NOUN
ejpam-3438	251	1	thus	thus	ADV
ejpam-3438	251	2	,	,	PUNCT
ejpam-3438	251	3	(	(	PUNCT
ejpam-3438	251	4	x	x	X
ejpam-3438	251	5	,	,	PUNCT
ejpam-3438	251	6	τ	τ	PROPN
ejpam-3438	251	7	,	,	PUNCT
ejpam-3438	251	8	i	i	PROPN
ejpam-3438	251	9	)	)	PUNCT
ejpam-3438	251	10	is	be	AUX
ejpam-3438	251	11	an	an	DET
ejpam-3438	251	12	∗-hyperconnected	∗-hyperconnected	ADJ
ejpam-3438	251	13	space	space	NOUN
ejpam-3438	251	14	.	.	PUNCT
ejpam-3438	252	1	(	(	PUNCT
ejpam-3438	252	2	2	2	X
ejpam-3438	252	3	)	)	PUNCT
ejpam-3438	252	4	assume	assume	VERB
ejpam-3438	252	5	that	that	SCONJ
ejpam-3438	252	6	i	i	PRON
ejpam-3438	252	7	=	=	PUNCT
ejpam-3438	252	8	{	{	PUNCT
ejpam-3438	252	9	∅	∅	NOUN
ejpam-3438	252	10	}	}	PUNCT
ejpam-3438	252	11	and	and	CCONJ
ejpam-3438	252	12	every	every	DET
ejpam-3438	252	13	open	open	ADJ
ejpam-3438	252	14	set	set	NOUN
ejpam-3438	252	15	is	be	AUX
ejpam-3438	252	16	closed	close	VERB
ejpam-3438	252	17	.	.	PUNCT
ejpam-3438	253	1	claim	claim	NOUN
ejpam-3438	253	2	1	1	NUM
ejpam-3438	253	3	.	.	PUNCT
ejpam-3438	254	1	a	a	PRON
ejpam-3438	254	2	is	be	AUX
ejpam-3438	254	3	an	an	DET
ejpam-3438	254	4	open	open	ADJ
ejpam-3438	254	5	set	set	NOUN
ejpam-3438	254	6	if	if	SCONJ
ejpam-3438	254	7	and	and	CCONJ
ejpam-3438	254	8	only	only	ADV
ejpam-3438	254	9	if	if	SCONJ
ejpam-3438	254	10	a	a	PRON
ejpam-3438	254	11	is	be	AUX
ejpam-3438	254	12	a	a	DET
ejpam-3438	254	13	β	β	NOUN
ejpam-3438	254	14	-	-	ADJ
ejpam-3438	254	15	open	open	ADJ
ejpam-3438	254	16	set	set	NOUN
ejpam-3438	254	17	.	.	PUNCT
ejpam-3438	255	1	if	if	SCONJ
ejpam-3438	255	2	a	a	PRON
ejpam-3438	255	3	is	be	AUX
ejpam-3438	255	4	an	an	DET
ejpam-3438	255	5	open	open	ADJ
ejpam-3438	255	6	set	set	NOUN
ejpam-3438	255	7	,	,	PUNCT
ejpam-3438	255	8	then	then	ADV
ejpam-3438	255	9	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-3438	255	10	)	)	PUNCT
ejpam-3438	255	11	)	)	PUNCT
ejpam-3438	255	12	)	)	PUNCT
ejpam-3438	256	1	=	=	SYM
ejpam-3438	256	2	a.	a.	NOUN
ejpam-3438	256	3	hence	hence	ADV
ejpam-3438	256	4	,	,	PUNCT
ejpam-3438	256	5	a	a	PRON
ejpam-3438	256	6	is	be	AUX
ejpam-3438	256	7	a	a	DET
ejpam-3438	256	8	β	β	NOUN
ejpam-3438	256	9	-	-	ADJ
ejpam-3438	256	10	open	open	ADJ
ejpam-3438	256	11	set	set	NOUN
ejpam-3438	256	12	.	.	PUNCT
ejpam-3438	257	1	conversely	conversely	ADV
ejpam-3438	257	2	,	,	PUNCT
ejpam-3438	257	3	if	if	SCONJ
ejpam-3438	257	4	a	a	PRON
ejpam-3438	257	5	is	be	AUX
ejpam-3438	257	6	a	a	DET
ejpam-3438	257	7	β	β	NOUN
ejpam-3438	257	8	-	-	ADJ
ejpam-3438	257	9	open	open	ADJ
ejpam-3438	257	10	set	set	NOUN
ejpam-3438	257	11	,	,	PUNCT
ejpam-3438	257	12	then	then	ADV
ejpam-3438	257	13	by	by	ADP
ejpam-3438	257	14	lemma	lemma	PROPN
ejpam-3438	257	15	2	2	NUM
ejpam-3438	257	16	there	there	PRON
ejpam-3438	257	17	exists	exist	VERB
ejpam-3438	257	18	an	an	DET
ejpam-3438	257	19	open	open	ADJ
ejpam-3438	257	20	set	set	NOUN
ejpam-3438	257	21	u	u	PRON
ejpam-3438	257	22	such	such	ADJ
ejpam-3438	257	23	that	that	SCONJ
ejpam-3438	257	24	u	u	PROPN
ejpam-3438	257	25	⊆	⊆	NUM
ejpam-3438	257	26	a	a	DET
ejpam-3438	257	27	⊆	⊆	NUM
ejpam-3438	257	28	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	257	29	)	)	PUNCT
ejpam-3438	257	30	)	)	PUNCT
ejpam-3438	257	31	)	)	PUNCT
ejpam-3438	258	1	=	=	SYM
ejpam-3438	258	2	u	u	NOUN
ejpam-3438	258	3	,	,	PUNCT
ejpam-3438	258	4	that	that	PRON
ejpam-3438	258	5	is	be	AUX
ejpam-3438	258	6	a	a	DET
ejpam-3438	258	7	=	=	X
ejpam-3438	258	8	u	u	NOUN
ejpam-3438	258	9	.	.	PUNCT
ejpam-3438	259	1	hence	hence	ADV
ejpam-3438	259	2	,	,	PUNCT
ejpam-3438	259	3	a	a	PRON
ejpam-3438	259	4	is	be	AUX
ejpam-3438	259	5	an	an	DET
ejpam-3438	259	6	open	open	ADJ
ejpam-3438	259	7	set	set	NOUN
ejpam-3438	259	8	.	.	PUNCT
ejpam-3438	260	1	this	this	PRON
ejpam-3438	260	2	shows	show	VERB
ejpam-3438	260	3	the	the	DET
ejpam-3438	260	4	claim	claim	NOUN
ejpam-3438	260	5	.	.	PUNCT
ejpam-3438	261	1	claim	claim	NOUN
ejpam-3438	261	2	2	2	NUM
ejpam-3438	261	3	.	.	PUNCT
ejpam-3438	262	1	the	the	DET
ejpam-3438	262	2	concepts	concept	NOUN
ejpam-3438	262	3	i∗-hyperconnectedness	i∗-hyperconnectedness	PROPN
ejpam-3438	262	4	and	and	CCONJ
ejpam-3438	262	5	β∗i	β∗i	PUNCT
ejpam-3438	262	6	-hyperconnectedness	-hyperconnectedness	NOUN
ejpam-3438	262	7	are	be	AUX
ejpam-3438	262	8	equivalent	equivalent	ADJ
ejpam-3438	262	9	.	.	PUNCT
ejpam-3438	263	1	if	if	SCONJ
ejpam-3438	263	2	(	(	PUNCT
ejpam-3438	263	3	x	x	X
ejpam-3438	263	4	,	,	PUNCT
ejpam-3438	263	5	τ	τ	PROPN
ejpam-3438	263	6	,	,	PUNCT
ejpam-3438	263	7	i	i	PROPN
ejpam-3438	263	8	)	)	PUNCT
ejpam-3438	263	9	is	be	AUX
ejpam-3438	263	10	an	an	DET
ejpam-3438	263	11	i∗-hyperconnected	i∗-hyperconnecte	VERB
ejpam-3438	263	12	space	space	NOUN
ejpam-3438	263	13	and	and	CCONJ
ejpam-3438	263	14	a	a	PRON
ejpam-3438	263	15	is	be	AUX
ejpam-3438	263	16	a	a	DET
ejpam-3438	263	17	non	non	ADJ
ejpam-3438	263	18	-	-	ADJ
ejpam-3438	263	19	empty	empty	ADJ
ejpam-3438	263	20	open	open	ADJ
ejpam-3438	263	21	set	set	NOUN
ejpam-3438	263	22	,	,	PUNCT
ejpam-3438	263	23	then	then	ADV
ejpam-3438	263	24	x	x	ADP
ejpam-3438	263	25	−	−	PROPN
ejpam-3438	263	26	cl∗(a	cl∗(a	NOUN
ejpam-3438	263	27	)	)	PUNCT
ejpam-3438	263	28	∈	∈	PROPN
ejpam-3438	263	29	i.	i.	NOUN
ejpam-3438	263	30	by	by	ADP
ejpam-3438	263	31	claim	claim	NOUN
ejpam-3438	263	32	1	1	NUM
ejpam-3438	263	33	and	and	CCONJ
ejpam-3438	263	34	lemma	lemma	PROPN
ejpam-3438	263	35	5	5	NUM
ejpam-3438	263	36	,	,	PUNCT
ejpam-3438	263	37	any	any	DET
ejpam-3438	263	38	open	open	ADJ
ejpam-3438	263	39	set	set	NOUN
ejpam-3438	263	40	is	be	AUX
ejpam-3438	263	41	precisely	precisely	ADV
ejpam-3438	263	42	a	a	DET
ejpam-3438	263	43	βi	βi	NOUN
ejpam-3438	263	44	-open	-open	NOUN
ejpam-3438	263	45	set	set	NOUN
ejpam-3438	263	46	.	.	PUNCT
ejpam-3438	264	1	hence	hence	ADV
ejpam-3438	264	2	,	,	PUNCT
ejpam-3438	264	3	x	x	PUNCT
ejpam-3438	264	4	−	−	NOUN
ejpam-3438	264	5	cl∗(a	cl∗(a	NOUN
ejpam-3438	264	6	)	)	PUNCT
ejpam-3438	264	7	∈	∈	PROPN
ejpam-3438	264	8	i	i	PRON
ejpam-3438	264	9	for	for	ADP
ejpam-3438	264	10	every	every	DET
ejpam-3438	264	11	non	non	ADJ
ejpam-3438	264	12	-	-	ADJ
ejpam-3438	264	13	empty	empty	ADJ
ejpam-3438	264	14	βi	βi	X
ejpam-3438	264	15	-open	-open	NOUN
ejpam-3438	264	16	set	set	NOUN
ejpam-3438	264	17	.	.	PUNCT
ejpam-3438	265	1	thus	thus	ADV
ejpam-3438	265	2	,	,	PUNCT
ejpam-3438	265	3	(	(	PUNCT
ejpam-3438	265	4	x	x	X
ejpam-3438	265	5	,	,	PUNCT
ejpam-3438	265	6	τ	τ	PROPN
ejpam-3438	265	7	,	,	PUNCT
ejpam-3438	265	8	i	i	PROPN
ejpam-3438	265	9	)	)	PUNCT
ejpam-3438	265	10	is	be	AUX
ejpam-3438	265	11	a	a	DET
ejpam-3438	265	12	β∗i	β∗i	ADV
ejpam-3438	265	13	-hyperconnected	-hyperconnecte	VERB
ejpam-3438	265	14	space	space	NOUN
ejpam-3438	265	15	.	.	PUNCT
ejpam-3438	266	1	conversely	conversely	ADV
ejpam-3438	266	2	,	,	PUNCT
ejpam-3438	266	3	if	if	SCONJ
ejpam-3438	266	4	(	(	PUNCT
ejpam-3438	266	5	x	x	NOUN
ejpam-3438	266	6	,	,	PUNCT
ejpam-3438	266	7	τ	τ	PROPN
ejpam-3438	266	8	,	,	PUNCT
ejpam-3438	266	9	i	i	PROPN
ejpam-3438	266	10	)	)	PUNCT
ejpam-3438	266	11	is	be	AUX
ejpam-3438	266	12	a	a	DET
ejpam-3438	266	13	β∗i	β∗i	ADV
ejpam-3438	266	14	-hyperconnected	-hyperconnecte	VERB
ejpam-3438	266	15	space	space	NOUN
ejpam-3438	266	16	and	and	CCONJ
ejpam-3438	266	17	a	a	PRON
ejpam-3438	266	18	is	be	AUX
ejpam-3438	266	19	a	a	DET
ejpam-3438	266	20	non	non	ADJ
ejpam-3438	266	21	-	-	ADJ
ejpam-3438	266	22	empty	empty	ADJ
ejpam-3438	266	23	βi	βi	PRON
ejpam-3438	266	24	open	open	ADJ
ejpam-3438	266	25	set	set	NOUN
ejpam-3438	266	26	,	,	PUNCT
ejpam-3438	266	27	then	then	ADV
ejpam-3438	266	28	x	x	ADP
ejpam-3438	266	29	−	−	PROPN
ejpam-3438	266	30	cl∗(a	cl∗(a	NOUN
ejpam-3438	266	31	)	)	PUNCT
ejpam-3438	266	32	∈	∈	PROPN
ejpam-3438	266	33	i.	i.	NOUN
ejpam-3438	266	34	by	by	ADP
ejpam-3438	266	35	claim	claim	NOUN
ejpam-3438	266	36	1	1	NUM
ejpam-3438	266	37	and	and	CCONJ
ejpam-3438	266	38	lemma	lemma	PROPN
ejpam-3438	266	39	5	5	NUM
ejpam-3438	266	40	,	,	PUNCT
ejpam-3438	266	41	any	any	DET
ejpam-3438	266	42	open	open	ADJ
ejpam-3438	266	43	set	set	NOUN
ejpam-3438	266	44	is	be	AUX
ejpam-3438	266	45	precisely	precisely	ADV
ejpam-3438	266	46	a	a	DET
ejpam-3438	266	47	βi	βi	NOUN
ejpam-3438	266	48	-open	-open	NOUN
ejpam-3438	266	49	set	set	NOUN
ejpam-3438	266	50	.	.	PUNCT
ejpam-3438	267	1	hence	hence	ADV
ejpam-3438	267	2	,	,	PUNCT
ejpam-3438	267	3	x	x	PUNCT
ejpam-3438	267	4	−	−	NOUN
ejpam-3438	267	5	cl∗(a	cl∗(a	NOUN
ejpam-3438	267	6	)	)	PUNCT
ejpam-3438	267	7	∈	∈	PROPN
ejpam-3438	267	8	i	i	PRON
ejpam-3438	267	9	for	for	ADP
ejpam-3438	267	10	every	every	DET
ejpam-3438	267	11	non	non	ADJ
ejpam-3438	267	12	-	-	ADJ
ejpam-3438	267	13	empty	empty	ADJ
ejpam-3438	267	14	open	open	ADJ
ejpam-3438	267	15	set	set	NOUN
ejpam-3438	267	16	.	.	PUNCT
ejpam-3438	268	1	thus	thus	ADV
ejpam-3438	268	2	,	,	PUNCT
ejpam-3438	268	3	(	(	PUNCT
ejpam-3438	268	4	x	x	X
ejpam-3438	268	5	,	,	PUNCT
ejpam-3438	268	6	τ	τ	PROPN
ejpam-3438	268	7	,	,	PUNCT
ejpam-3438	268	8	i	i	PROPN
ejpam-3438	268	9	)	)	PUNCT
ejpam-3438	268	10	is	be	AUX
ejpam-3438	268	11	an	an	DET
ejpam-3438	268	12	i∗-hyperconnected	i∗-hyperconnecte	VERB
ejpam-3438	268	13	space	space	NOUN
ejpam-3438	268	14	.	.	PUNCT
ejpam-3438	269	1	accordingly	accordingly	ADV
ejpam-3438	269	2	,	,	PUNCT
ejpam-3438	269	3	by	by	ADP
ejpam-3438	269	4	statement	statement	NOUN
ejpam-3438	269	5	(	(	PUNCT
ejpam-3438	269	6	1	1	NUM
ejpam-3438	269	7	)	)	PUNCT
ejpam-3438	269	8	and	and	CCONJ
ejpam-3438	269	9	claim	claim	NOUN
ejpam-3438	269	10	2	2	NUM
ejpam-3438	269	11	,	,	PUNCT
ejpam-3438	269	12	statement	statement	NOUN
ejpam-3438	269	13	(	(	PUNCT
ejpam-3438	269	14	2	2	X
ejpam-3438	269	15	)	)	PUNCT
ejpam-3438	269	16	follows	follow	VERB
ejpam-3438	269	17	.	.	PUNCT
ejpam-3438	270	1	g.	g.	PROPN
ejpam-3438	270	2	catalan	catalan	PROPN
ejpam-3438	270	3	,	,	PUNCT
ejpam-3438	270	4	r.	r.	PROPN
ejpam-3438	270	5	padua	padua	PROPN
ejpam-3438	270	6	,	,	PUNCT
ejpam-3438	270	7	m.	m.	PROPN
ejpam-3438	270	8	baldado	baldado	PROPN
ejpam-3438	270	9	jr	jr	PROPN
ejpam-3438	270	10	.	.	PROPN
ejpam-3438	270	11	/	/	SYM
ejpam-3438	270	12	eur	eur	PROPN
ejpam-3438	270	13	.	.	PUNCT
ejpam-3438	271	1	j.	j.	PROPN
ejpam-3438	271	2	pure	pure	PROPN
ejpam-3438	271	3	appl	appl	PROPN
ejpam-3438	271	4	.	.	PROPN
ejpam-3438	271	5	math	math	PROPN
ejpam-3438	271	6	,	,	PUNCT
ejpam-3438	271	7	12	12	NUM
ejpam-3438	271	8	(	(	PUNCT
ejpam-3438	271	9	3	3	NUM
ejpam-3438	271	10	)	)	PUNCT
ejpam-3438	271	11	(	(	PUNCT
ejpam-3438	271	12	2019	2019	NUM
ejpam-3438	271	13	)	)	PUNCT
ejpam-3438	271	14	,	,	PUNCT
ejpam-3438	271	15	893	893	NUM
ejpam-3438	271	16	-	-	SYM
ejpam-3438	271	17	905	905	NUM
ejpam-3438	271	18	900	900	NUM
ejpam-3438	271	19	theorem	theorem	NOUN
ejpam-3438	271	20	9	9	NUM
ejpam-3438	271	21	.	.	PUNCT
ejpam-3438	272	1	if	if	SCONJ
ejpam-3438	272	2	an	an	DET
ejpam-3438	272	3	ideal	ideal	ADJ
ejpam-3438	272	4	topological	topological	ADJ
ejpam-3438	272	5	space	space	NOUN
ejpam-3438	272	6	(	(	PUNCT
ejpam-3438	272	7	x	x	X
ejpam-3438	272	8	,	,	PUNCT
ejpam-3438	272	9	τ	τ	X
ejpam-3438	272	10	,	,	PUNCT
ejpam-3438	272	11	{	{	PUNCT
ejpam-3438	272	12	∅	∅	NOUN
ejpam-3438	272	13	}	}	PUNCT
ejpam-3438	272	14	)	)	PUNCT
ejpam-3438	272	15	is	be	AUX
ejpam-3438	272	16	a	a	DET
ejpam-3438	272	17	βi∗-hyperconnected	βi∗-hyperconnected	ADJ
ejpam-3438	272	18	space	space	NOUN
ejpam-3438	272	19	,	,	PUNCT
ejpam-3438	272	20	then	then	ADV
ejpam-3438	272	21	x	x	PUNCT
ejpam-3438	272	22	−	−	PROPN
ejpam-3438	272	23	cl∗(a	cl∗(a	NOUN
ejpam-3438	272	24	)	)	PUNCT
ejpam-3438	272	25	∈	∈	PROPN
ejpam-3438	272	26	i	i	PRON
ejpam-3438	272	27	for	for	ADP
ejpam-3438	272	28	every	every	DET
ejpam-3438	272	29	non	non	ADJ
ejpam-3438	272	30	-	-	ADJ
ejpam-3438	272	31	empty	empty	ADJ
ejpam-3438	272	32	β	β	ADJ
ejpam-3438	272	33	-	-	ADJ
ejpam-3438	272	34	open	open	ADJ
ejpam-3438	272	35	subset	subset	VERB
ejpam-3438	272	36	a	a	PRON
ejpam-3438	272	37	of	of	ADP
ejpam-3438	272	38	x.	x.	NOUN
ejpam-3438	272	39	proof	proof	NOUN
ejpam-3438	272	40	.	.	PUNCT
ejpam-3438	273	1	let	let	VERB
ejpam-3438	273	2	(	(	PUNCT
ejpam-3438	273	3	x	x	X
ejpam-3438	273	4	,	,	PUNCT
ejpam-3438	273	5	τ	τ	PROPN
ejpam-3438	273	6	,	,	PUNCT
ejpam-3438	273	7	i	i	PRON
ejpam-3438	273	8	)	)	PUNCT
ejpam-3438	273	9	be	be	VERB
ejpam-3438	273	10	a	a	DET
ejpam-3438	273	11	β∗i	β∗i	ADV
ejpam-3438	273	12	-hyperconnected	-hyperconnecte	VERB
ejpam-3438	273	13	space	space	NOUN
ejpam-3438	273	14	and	and	CCONJ
ejpam-3438	273	15	a	a	PRON
ejpam-3438	273	16	is	be	AUX
ejpam-3438	273	17	a	a	DET
ejpam-3438	273	18	non	non	ADJ
ejpam-3438	273	19	-	-	ADJ
ejpam-3438	273	20	empty	empty	ADJ
ejpam-3438	273	21	β	β	ADJ
ejpam-3438	273	22	-	-	ADJ
ejpam-3438	273	23	open	open	ADJ
ejpam-3438	273	24	set	set	NOUN
ejpam-3438	273	25	.	.	PUNCT
ejpam-3438	274	1	since	since	SCONJ
ejpam-3438	274	2	by	by	ADP
ejpam-3438	274	3	lemma	lemma	PROPN
ejpam-3438	274	4	4	4	NUM
ejpam-3438	274	5	every	every	PRON
ejpam-3438	274	6	β	β	X
ejpam-3438	274	7	-	-	ADJ
ejpam-3438	274	8	open	open	ADJ
ejpam-3438	274	9	set	set	NOUN
ejpam-3438	274	10	is	be	AUX
ejpam-3438	274	11	a	a	DET
ejpam-3438	274	12	βi	βi	X
ejpam-3438	274	13	-open	-open	NOUN
ejpam-3438	274	14	set	set	NOUN
ejpam-3438	274	15	,	,	PUNCT
ejpam-3438	274	16	a	a	PRON
ejpam-3438	274	17	is	be	AUX
ejpam-3438	274	18	a	a	DET
ejpam-3438	274	19	non	non	ADJ
ejpam-3438	274	20	-	-	ADJ
ejpam-3438	274	21	empty	empty	ADJ
ejpam-3438	274	22	βi	βi	PRON
ejpam-3438	274	23	-open	-open	NOUN
ejpam-3438	274	24	set	set	VERB
ejpam-3438	274	25	also	also	ADV
ejpam-3438	274	26	.	.	PUNCT
ejpam-3438	275	1	if	if	SCONJ
ejpam-3438	275	2	(	(	PUNCT
ejpam-3438	275	3	x	x	X
ejpam-3438	275	4	,	,	PUNCT
ejpam-3438	275	5	τ	τ	PROPN
ejpam-3438	275	6	,	,	PUNCT
ejpam-3438	275	7	i	i	PROPN
ejpam-3438	275	8	)	)	PUNCT
ejpam-3438	275	9	is	be	AUX
ejpam-3438	275	10	a	a	DET
ejpam-3438	275	11	β∗i	β∗i	ADV
ejpam-3438	275	12	-hyperconnected	-hyperconnecte	VERB
ejpam-3438	275	13	space	space	NOUN
ejpam-3438	275	14	,	,	PUNCT
ejpam-3438	275	15	then	then	ADV
ejpam-3438	275	16	x	x	PUNCT
ejpam-3438	275	17	−	−	PROPN
ejpam-3438	275	18	cl∗(a	cl∗(a	NOUN
ejpam-3438	275	19	)	)	PUNCT
ejpam-3438	275	20	∈	∈	PROPN
ejpam-3438	275	21	i.	i.	NOUN
ejpam-3438	275	22	theorem	theorem	VERB
ejpam-3438	275	23	10	10	NUM
ejpam-3438	275	24	characterizes	characterize	VERB
ejpam-3438	275	25	a	a	DET
ejpam-3438	275	26	β∗i	β∗i	ADV
ejpam-3438	275	27	-hyperconnected	-hyperconnecte	VERB
ejpam-3438	275	28	space	space	NOUN
ejpam-3438	275	29	.	.	PUNCT
ejpam-3438	276	1	theorem	theorem	ADJ
ejpam-3438	276	2	10	10	NUM
ejpam-3438	276	3	.	.	PUNCT
ejpam-3438	277	1	for	for	ADP
ejpam-3438	277	2	an	an	DET
ejpam-3438	277	3	ideal	ideal	ADJ
ejpam-3438	277	4	topological	topological	ADJ
ejpam-3438	277	5	space	space	NOUN
ejpam-3438	277	6	(	(	PUNCT
ejpam-3438	277	7	x	x	X
ejpam-3438	277	8	,	,	PUNCT
ejpam-3438	277	9	τ	τ	PROPN
ejpam-3438	277	10	,	,	PUNCT
ejpam-3438	277	11	i	i	PROPN
ejpam-3438	277	12	)	)	PUNCT
ejpam-3438	277	13	,	,	PUNCT
ejpam-3438	277	14	the	the	DET
ejpam-3438	277	15	following	follow	VERB
ejpam-3438	277	16	statements	statement	NOUN
ejpam-3438	277	17	are	be	AUX
ejpam-3438	277	18	equivalent	equivalent	ADJ
ejpam-3438	277	19	.	.	PUNCT
ejpam-3438	278	1	(	(	PUNCT
ejpam-3438	278	2	i	i	NOUN
ejpam-3438	278	3	)	)	PUNCT
ejpam-3438	278	4	x	x	X
ejpam-3438	278	5	is	be	AUX
ejpam-3438	278	6	a	a	DET
ejpam-3438	278	7	β∗i	β∗i	ADV
ejpam-3438	278	8	-hyperconnected	-hyperconnecte	VERB
ejpam-3438	278	9	space	space	NOUN
ejpam-3438	278	10	.	.	PUNCT
ejpam-3438	279	1	(	(	PUNCT
ejpam-3438	279	2	ii	ii	NOUN
ejpam-3438	279	3	)	)	PUNCT
ejpam-3438	279	4	int∗(a	int∗(a	NOUN
ejpam-3438	279	5	)	)	PUNCT
ejpam-3438	279	6	∈	∈	NOUN
ejpam-3438	279	7	i	i	PRON
ejpam-3438	279	8	for	for	ADP
ejpam-3438	279	9	every	every	DET
ejpam-3438	279	10	proper	proper	ADJ
ejpam-3438	279	11	βi	βi	ADV
ejpam-3438	279	12	-	-	PUNCT
ejpam-3438	279	13	closed	closed	ADJ
ejpam-3438	279	14	subset	subset	VERB
ejpam-3438	279	15	a	a	PRON
ejpam-3438	279	16	of	of	ADP
ejpam-3438	279	17	x.	x.	NOUN
ejpam-3438	279	18	proof	proof	NOUN
ejpam-3438	279	19	.	.	PUNCT
ejpam-3438	280	1	(	(	PUNCT
ejpam-3438	280	2	1	1	X
ejpam-3438	280	3	)	)	PUNCT
ejpam-3438	280	4	⇒	⇒	NOUN
ejpam-3438	280	5	(	(	PUNCT
ejpam-3438	280	6	2	2	X
ejpam-3438	280	7	)	)	PUNCT
ejpam-3438	280	8	assume	assume	VERB
ejpam-3438	280	9	that	that	SCONJ
ejpam-3438	280	10	(	(	PUNCT
ejpam-3438	280	11	1	1	X
ejpam-3438	280	12	)	)	PUNCT
ejpam-3438	280	13	holds	hold	NOUN
ejpam-3438	280	14	,	,	PUNCT
ejpam-3438	280	15	and	and	CCONJ
ejpam-3438	280	16	let	let	VERB
ejpam-3438	280	17	b	b	X
ejpam-3438	280	18	be	be	AUX
ejpam-3438	280	19	a	a	DET
ejpam-3438	280	20	βi	βi	ADV
ejpam-3438	280	21	-closed	-close	VERB
ejpam-3438	280	22	set	set	NOUN
ejpam-3438	280	23	.	.	PUNCT
ejpam-3438	281	1	if	if	SCONJ
ejpam-3438	281	2	b	b	PROPN
ejpam-3438	281	3	is	be	AUX
ejpam-3438	281	4	a	a	DET
ejpam-3438	281	5	βi	βi	NOUN
ejpam-3438	281	6	closed	close	VERB
ejpam-3438	281	7	set	set	NOUN
ejpam-3438	281	8	,	,	PUNCT
ejpam-3438	281	9	then	then	ADV
ejpam-3438	281	10	x	x	SYM
ejpam-3438	281	11	−b	−b	NOUN
ejpam-3438	281	12	is	be	AUX
ejpam-3438	281	13	a	a	DET
ejpam-3438	281	14	βi	βi	X
ejpam-3438	281	15	-open	-open	NOUN
ejpam-3438	281	16	set	set	NOUN
ejpam-3438	281	17	.	.	PUNCT
ejpam-3438	282	1	moreover	moreover	ADV
ejpam-3438	282	2	,	,	PUNCT
ejpam-3438	282	3	if	if	SCONJ
ejpam-3438	282	4	b	b	PROPN
ejpam-3438	282	5	is	be	AUX
ejpam-3438	282	6	a	a	DET
ejpam-3438	282	7	proper	proper	ADJ
ejpam-3438	282	8	subset	subset	NOUN
ejpam-3438	282	9	,	,	PUNCT
ejpam-3438	282	10	then	then	ADV
ejpam-3438	282	11	bc	bc	PROPN
ejpam-3438	282	12	6=	6=	ADP
ejpam-3438	282	13	∅.	∅.	ADP
ejpam-3438	282	14	by	by	ADP
ejpam-3438	282	15	assumption	assumption	NOUN
ejpam-3438	282	16	,	,	PUNCT
ejpam-3438	282	17	int∗(b	int∗(b	X
ejpam-3438	282	18	)	)	PUNCT
ejpam-3438	282	19	=	=	SYM
ejpam-3438	283	1	x	x	PUNCT
ejpam-3438	283	2	−	−	NOUN
ejpam-3438	283	3	cl∗(x	cl∗(x	ADJ
ejpam-3438	283	4	−b	−b	NOUN
ejpam-3438	283	5	)	)	PUNCT
ejpam-3438	283	6	∈	∈	PROPN
ejpam-3438	283	7	i.	i.	NOUN
ejpam-3438	283	8	(	(	PUNCT
ejpam-3438	283	9	2	2	NUM
ejpam-3438	283	10	)	)	PUNCT
ejpam-3438	283	11	⇒	⇒	NOUN
ejpam-3438	283	12	(	(	PUNCT
ejpam-3438	283	13	1	1	X
ejpam-3438	283	14	)	)	PUNCT
ejpam-3438	283	15	assume	assume	VERB
ejpam-3438	283	16	that	that	SCONJ
ejpam-3438	283	17	(	(	PUNCT
ejpam-3438	283	18	2	2	X
ejpam-3438	283	19	)	)	PUNCT
ejpam-3438	283	20	holds	hold	NOUN
ejpam-3438	283	21	,	,	PUNCT
ejpam-3438	283	22	and	and	CCONJ
ejpam-3438	283	23	let	let	VERB
ejpam-3438	283	24	a	a	PRON
ejpam-3438	283	25	be	be	AUX
ejpam-3438	283	26	a	a	DET
ejpam-3438	283	27	non	non	ADJ
ejpam-3438	283	28	-	-	ADJ
ejpam-3438	283	29	empty	empty	ADJ
ejpam-3438	283	30	βi	βi	X
ejpam-3438	283	31	-open	-open	NOUN
ejpam-3438	283	32	set	set	NOUN
ejpam-3438	283	33	.	.	PUNCT
ejpam-3438	284	1	if	if	SCONJ
ejpam-3438	284	2	a	a	PRON
ejpam-3438	284	3	is	be	AUX
ejpam-3438	284	4	a	a	DET
ejpam-3438	284	5	non	non	ADJ
ejpam-3438	284	6	-	-	ADJ
ejpam-3438	284	7	empty	empty	ADJ
ejpam-3438	284	8	βi	βi	X
ejpam-3438	284	9	-open	-open	NOUN
ejpam-3438	284	10	set	set	NOUN
ejpam-3438	284	11	,	,	PUNCT
ejpam-3438	284	12	then	then	ADV
ejpam-3438	284	13	x	x	PUNCT
ejpam-3438	284	14	−	−	NOUN
ejpam-3438	284	15	a	a	PRON
ejpam-3438	284	16	is	be	AUX
ejpam-3438	284	17	a	a	DET
ejpam-3438	284	18	proper	proper	ADJ
ejpam-3438	284	19	βi	βi	NOUN
ejpam-3438	284	20	-open	-open	NOUN
ejpam-3438	284	21	subset	subset	NOUN
ejpam-3438	284	22	of	of	ADP
ejpam-3438	284	23	x.	x.	NOUN
ejpam-3438	284	24	by	by	ADP
ejpam-3438	284	25	assumption	assumption	NOUN
ejpam-3438	284	26	,	,	PUNCT
ejpam-3438	284	27	x	x	PUNCT
ejpam-3438	284	28	−	−	NOUN
ejpam-3438	284	29	cl∗(a	cl∗(a	NOUN
ejpam-3438	284	30	)	)	PUNCT
ejpam-3438	284	31	=	=	PUNCT
ejpam-3438	285	1	x	x	PUNCT
ejpam-3438	285	2	−	−	NOUN
ejpam-3438	285	3	cl∗(x	cl∗(x	NOUN
ejpam-3438	285	4	−	−	PROPN
ejpam-3438	285	5	(	(	PUNCT
ejpam-3438	285	6	x	x	X
ejpam-3438	285	7	−	−	PROPN
ejpam-3438	285	8	a	a	NOUN
ejpam-3438	285	9	)	)	PUNCT
ejpam-3438	285	10	)	)	PUNCT
ejpam-3438	286	1	=	=	NOUN
ejpam-3438	286	2	int∗(x	int∗(x	VERB
ejpam-3438	286	3	−	−	NOUN
ejpam-3438	286	4	a	a	X
ejpam-3438	286	5	)	)	PUNCT
ejpam-3438	286	6	∈	∈	PROPN
ejpam-3438	286	7	i.	i.	NOUN
ejpam-3438	286	8	this	this	PRON
ejpam-3438	286	9	shows	show	VERB
ejpam-3438	286	10	that	that	SCONJ
ejpam-3438	286	11	x	x	PRON
ejpam-3438	286	12	is	be	AUX
ejpam-3438	286	13	a	a	DET
ejpam-3438	286	14	β∗i	β∗i	PUNCT
ejpam-3438	286	15	hyperconnected	hyperconnecte	VERB
ejpam-3438	286	16	space	space	NOUN
ejpam-3438	286	17	.	.	PUNCT
ejpam-3438	287	1	4	4	X
ejpam-3438	287	2	.	.	X
ejpam-3438	287	3	separation	separation	NOUN
ejpam-3438	287	4	notions	notion	NOUN
ejpam-3438	287	5	with	with	ADP
ejpam-3438	287	6	respect	respect	NOUN
ejpam-3438	287	7	to	to	ADP
ejpam-3438	287	8	ideals	ideal	NOUN
ejpam-3438	287	9	let	let	VERB
ejpam-3438	287	10	(	(	PUNCT
ejpam-3438	287	11	x	x	X
ejpam-3438	287	12	,	,	PUNCT
ejpam-3438	287	13	τ	τ	PROPN
ejpam-3438	287	14	,	,	PUNCT
ejpam-3438	287	15	i	i	PRON
ejpam-3438	287	16	)	)	PUNCT
ejpam-3438	287	17	be	be	VERB
ejpam-3438	287	18	an	an	DET
ejpam-3438	287	19	ideal	ideal	ADJ
ejpam-3438	287	20	topological	topological	ADJ
ejpam-3438	287	21	space	space	NOUN
ejpam-3438	287	22	and	and	CCONJ
ejpam-3438	287	23	a	a	DET
ejpam-3438	287	24	be	be	AUX
ejpam-3438	287	25	a	a	DET
ejpam-3438	287	26	subset	subset	NOUN
ejpam-3438	287	27	of	of	ADP
ejpam-3438	287	28	x.	x.	NOUN
ejpam-3438	287	29	the	the	DET
ejpam-3438	287	30	βi	βi	PROPN
ejpam-3438	287	31	-closure	-closure	NOUN
ejpam-3438	287	32	of	of	ADP
ejpam-3438	287	33	a	a	PRON
ejpam-3438	287	34	is	be	AUX
ejpam-3438	287	35	the	the	DET
ejpam-3438	287	36	smallest	small	ADJ
ejpam-3438	287	37	βi	βi	SYM
ejpam-3438	287	38	-closed	-close	VERB
ejpam-3438	287	39	set	set	NOUN
ejpam-3438	287	40	containing	contain	VERB
ejpam-3438	287	41	a	a	PRON
ejpam-3438	287	42	,	,	PUNCT
ejpam-3438	287	43	denoted	denote	VERB
ejpam-3438	287	44	by	by	ADP
ejpam-3438	287	45	clβi	clβi	NOUN
ejpam-3438	287	46	(	(	PUNCT
ejpam-3438	287	47	a	a	NOUN
ejpam-3438	287	48	)	)	PUNCT
ejpam-3438	287	49	.	.	PUNCT
ejpam-3438	288	1	recall	recall	VERB
ejpam-3438	288	2	that	that	SCONJ
ejpam-3438	288	3	two	two	NUM
ejpam-3438	288	4	sets	set	VERB
ejpam-3438	288	5	a	a	PRON
ejpam-3438	288	6	and	and	CCONJ
ejpam-3438	288	7	b	b	NOUN
ejpam-3438	288	8	in	in	ADP
ejpam-3438	288	9	an	an	DET
ejpam-3438	288	10	ideal	ideal	ADJ
ejpam-3438	288	11	topological	topological	ADJ
ejpam-3438	288	12	space	space	NOUN
ejpam-3438	288	13	(	(	PUNCT
ejpam-3438	288	14	x	x	X
ejpam-3438	288	15	,	,	PUNCT
ejpam-3438	288	16	τ	τ	PROPN
ejpam-3438	288	17	,	,	PUNCT
ejpam-3438	288	18	i	i	PROPN
ejpam-3438	288	19	)	)	PUNCT
ejpam-3438	288	20	is	be	AUX
ejpam-3438	288	21	said	say	VERB
ejpam-3438	288	22	to	to	PART
ejpam-3438	288	23	be	be	AUX
ejpam-3438	289	1	βi	βi	PRON
ejpam-3438	289	2	separated	separate	VERB
ejpam-3438	289	3	if	if	SCONJ
ejpam-3438	289	4	clβi	clβi	NOUN
ejpam-3438	289	5	(	(	PUNCT
ejpam-3438	289	6	a	a	NOUN
ejpam-3438	289	7	)	)	PUNCT
ejpam-3438	289	8	∩	∩	ADJ
ejpam-3438	289	9	b	b	NOUN
ejpam-3438	289	10	=	=	NOUN
ejpam-3438	289	11	∅	∅	NOUN
ejpam-3438	289	12	=	=	PUNCT
ejpam-3438	289	13	a	a	DET
ejpam-3438	289	14	∩	∩	ADJ
ejpam-3438	289	15	clβ(b	clβ(b	NOUN
ejpam-3438	289	16	)	)	PUNCT
ejpam-3438	289	17	,	,	PUNCT
ejpam-3438	289	18	and	and	CCONJ
ejpam-3438	289	19	a	a	DET
ejpam-3438	289	20	subset	subset	NOUN
ejpam-3438	289	21	a	a	PRON
ejpam-3438	289	22	of	of	ADP
ejpam-3438	289	23	an	an	DET
ejpam-3438	289	24	ideal	ideal	ADJ
ejpam-3438	289	25	topological	topological	ADJ
ejpam-3438	289	26	space	space	NOUN
ejpam-3438	289	27	(	(	PUNCT
ejpam-3438	289	28	x	x	X
ejpam-3438	289	29	,	,	PUNCT
ejpam-3438	289	30	τ	τ	PROPN
ejpam-3438	289	31	,	,	PUNCT
ejpam-3438	289	32	i	i	PROPN
ejpam-3438	289	33	)	)	PUNCT
ejpam-3438	289	34	is	be	AUX
ejpam-3438	289	35	said	say	VERB
ejpam-3438	289	36	to	to	PART
ejpam-3438	289	37	be	be	AUX
ejpam-3438	289	38	βi	βi	PRON
ejpam-3438	289	39	-connected	-connected	ADJ
ejpam-3438	289	40	if	if	SCONJ
ejpam-3438	289	41	it	it	PRON
ejpam-3438	289	42	can	can	AUX
ejpam-3438	289	43	not	not	PART
ejpam-3438	289	44	be	be	AUX
ejpam-3438	289	45	expressed	express	VERB
ejpam-3438	289	46	as	as	ADP
ejpam-3438	289	47	a	a	DET
ejpam-3438	289	48	union	union	NOUN
ejpam-3438	289	49	of	of	ADP
ejpam-3438	289	50	two	two	NUM
ejpam-3438	289	51	βi	βi	ADV
ejpam-3438	289	52	-separated	-separate	VERB
ejpam-3438	289	53	sets	set	NOUN
ejpam-3438	289	54	.	.	PUNCT
ejpam-3438	290	1	an	an	DET
ejpam-3438	290	2	ideal	ideal	ADJ
ejpam-3438	290	3	topological	topological	ADJ
ejpam-3438	290	4	space	space	NOUN
ejpam-3438	290	5	(	(	PUNCT
ejpam-3438	290	6	x	x	X
ejpam-3438	290	7	,	,	PUNCT
ejpam-3438	290	8	τ	τ	PROPN
ejpam-3438	290	9	,	,	PUNCT
ejpam-3438	290	10	i	i	PROPN
ejpam-3438	290	11	)	)	PUNCT
ejpam-3438	290	12	is	be	AUX
ejpam-3438	290	13	said	say	VERB
ejpam-3438	290	14	to	to	PART
ejpam-3438	290	15	be	be	AUX
ejpam-3438	290	16	βi	βi	PRON
ejpam-3438	290	17	-connected	-connected	ADJ
ejpam-3438	290	18	if	if	SCONJ
ejpam-3438	290	19	x	x	PRON
ejpam-3438	290	20	βi	βi	PRON
ejpam-3438	290	21	-connected	-connecte	VERB
ejpam-3438	290	22	as	as	ADP
ejpam-3438	290	23	a	a	DET
ejpam-3438	290	24	subset	subset	NOUN
ejpam-3438	290	25	.	.	PUNCT
ejpam-3438	291	1	a	a	DET
ejpam-3438	291	2	subset	subset	NOUN
ejpam-3438	291	3	a	a	PRON
ejpam-3438	291	4	of	of	ADP
ejpam-3438	291	5	an	an	DET
ejpam-3438	291	6	ideal	ideal	ADJ
ejpam-3438	291	7	topological	topological	ADJ
ejpam-3438	291	8	space	space	NOUN
ejpam-3438	291	9	(	(	PUNCT
ejpam-3438	291	10	x	x	X
ejpam-3438	291	11	,	,	PUNCT
ejpam-3438	291	12	τ	τ	PROPN
ejpam-3438	291	13	,	,	PUNCT
ejpam-3438	291	14	i	i	PROPN
ejpam-3438	291	15	)	)	PUNCT
ejpam-3438	291	16	is	be	AUX
ejpam-3438	291	17	said	say	VERB
ejpam-3438	291	18	to	to	PART
ejpam-3438	291	19	be	be	AUX
ejpam-3438	291	20	βi	βi	PRON
ejpam-3438	291	21	-connected	-connected	ADJ
ejpam-3438	291	22	if	if	SCONJ
ejpam-3438	291	23	it	it	PRON
ejpam-3438	291	24	can	can	AUX
ejpam-3438	291	25	not	not	PART
ejpam-3438	291	26	be	be	AUX
ejpam-3438	291	27	expressed	express	VERB
ejpam-3438	291	28	as	as	ADP
ejpam-3438	291	29	a	a	DET
ejpam-3438	291	30	union	union	NOUN
ejpam-3438	291	31	of	of	ADP
ejpam-3438	291	32	two	two	NUM
ejpam-3438	291	33	βi	βi	ADV
ejpam-3438	291	34	-separated	-separate	VERB
ejpam-3438	291	35	sets	set	NOUN
ejpam-3438	291	36	.	.	PUNCT
ejpam-3438	292	1	an	an	DET
ejpam-3438	292	2	ideal	ideal	ADJ
ejpam-3438	292	3	topological	topological	ADJ
ejpam-3438	292	4	space	space	NOUN
ejpam-3438	292	5	(	(	PUNCT
ejpam-3438	292	6	x	x	X
ejpam-3438	292	7	,	,	PUNCT
ejpam-3438	292	8	τ	τ	PROPN
ejpam-3438	292	9	,	,	PUNCT
ejpam-3438	292	10	i	i	PROPN
ejpam-3438	292	11	)	)	PUNCT
ejpam-3438	292	12	is	be	AUX
ejpam-3438	292	13	said	say	VERB
ejpam-3438	292	14	to	to	PART
ejpam-3438	292	15	be	be	AUX
ejpam-3438	292	16	βi	βi	PRON
ejpam-3438	292	17	-connected	-connected	ADJ
ejpam-3438	292	18	if	if	SCONJ
ejpam-3438	292	19	x	x	PRON
ejpam-3438	292	20	βi	βi	PRON
ejpam-3438	292	21	-connected	-connecte	VERB
ejpam-3438	292	22	as	as	ADP
ejpam-3438	292	23	a	a	DET
ejpam-3438	292	24	subset	subset	NOUN
ejpam-3438	292	25	.	.	PUNCT
ejpam-3438	293	1	lemma	lemma	PROPN
ejpam-3438	293	2	6	6	NUM
ejpam-3438	293	3	.	.	PUNCT
ejpam-3438	294	1	let	let	VERB
ejpam-3438	294	2	(	(	PUNCT
ejpam-3438	294	3	x	x	X
ejpam-3438	294	4	,	,	PUNCT
ejpam-3438	294	5	τ	τ	PROPN
ejpam-3438	294	6	,	,	PUNCT
ejpam-3438	294	7	i	i	PRON
ejpam-3438	294	8	)	)	PUNCT
ejpam-3438	294	9	be	be	VERB
ejpam-3438	294	10	an	an	DET
ejpam-3438	294	11	ideal	ideal	ADJ
ejpam-3438	294	12	topological	topological	ADJ
ejpam-3438	294	13	space	space	NOUN
ejpam-3438	294	14	.	.	PUNCT
ejpam-3438	295	1	if	if	SCONJ
ejpam-3438	295	2	a	a	PRON
ejpam-3438	295	3	and	and	CCONJ
ejpam-3438	295	4	b	b	NOUN
ejpam-3438	295	5	are	be	AUX
ejpam-3438	295	6	non	non	ADJ
ejpam-3438	295	7	-	-	ADJ
ejpam-3438	295	8	empty	empty	ADJ
ejpam-3438	295	9	disjoint	disjoint	ADJ
ejpam-3438	295	10	subsets	subset	NOUN
ejpam-3438	295	11	of	of	ADP
ejpam-3438	295	12	x	x	SYM
ejpam-3438	295	13	such	such	ADJ
ejpam-3438	295	14	that	that	SCONJ
ejpam-3438	295	15	a	a	PRON
ejpam-3438	295	16	is	be	AUX
ejpam-3438	295	17	β	β	NOUN
ejpam-3438	295	18	-	-	ADJ
ejpam-3438	295	19	open	open	ADJ
ejpam-3438	295	20	and	and	CCONJ
ejpam-3438	295	21	b	b	NOUN
ejpam-3438	295	22	is	be	AUX
ejpam-3438	295	23	βi	βi	NOUN
ejpam-3438	295	24	-	-	ADJ
ejpam-3438	295	25	open	open	ADJ
ejpam-3438	295	26	,	,	PUNCT
ejpam-3438	295	27	then	then	ADV
ejpam-3438	295	28	a	a	PRON
ejpam-3438	295	29	and	and	CCONJ
ejpam-3438	295	30	b	b	NOUN
ejpam-3438	295	31	are	be	AUX
ejpam-3438	295	32	βi	βi	PRON
ejpam-3438	295	33	-	-	PUNCT
ejpam-3438	295	34	separated	separate	VERB
ejpam-3438	295	35	sets	set	NOUN
ejpam-3438	295	36	.	.	PUNCT
ejpam-3438	296	1	proof	proof	NOUN
ejpam-3438	296	2	.	.	PUNCT
ejpam-3438	297	1	suppose	suppose	VERB
ejpam-3438	297	2	that	that	SCONJ
ejpam-3438	297	3	a	a	PRON
ejpam-3438	297	4	and	and	CCONJ
ejpam-3438	297	5	b	b	NOUN
ejpam-3438	297	6	are	be	AUX
ejpam-3438	297	7	βi	βi	PRON
ejpam-3438	297	8	-separated	-separate	VERB
ejpam-3438	297	9	sets	set	NOUN
ejpam-3438	297	10	,	,	PUNCT
ejpam-3438	297	11	that	that	PRON
ejpam-3438	297	12	is	be	AUX
ejpam-3438	297	13	clβi	clβi	NOUN
ejpam-3438	297	14	(	(	PUNCT
ejpam-3438	297	15	a	a	NOUN
ejpam-3438	297	16	)	)	PUNCT
ejpam-3438	297	17	∩	∩	ADJ
ejpam-3438	297	18	b	b	PROPN
ejpam-3438	297	19	6=	6=	ADP
ejpam-3438	297	20	∅	∅	NOUN
ejpam-3438	297	21	or	or	CCONJ
ejpam-3438	297	22	a	a	DET
ejpam-3438	297	23	∩	∩	ADJ
ejpam-3438	297	24	clβ(b	clβ(b	NOUN
ejpam-3438	297	25	)	)	PUNCT
ejpam-3438	297	26	6=	6=	ADP
ejpam-3438	297	27	∅.	∅.	ADP
ejpam-3438	297	28	since	since	SCONJ
ejpam-3438	297	29	a	a	PRON
ejpam-3438	297	30	and	and	CCONJ
ejpam-3438	297	31	b	b	NOUN
ejpam-3438	297	32	are	be	AUX
ejpam-3438	297	33	non	non	ADJ
ejpam-3438	297	34	-	-	ADJ
ejpam-3438	297	35	empty	empty	ADJ
ejpam-3438	297	36	disjoint	disjoint	ADJ
ejpam-3438	297	37	subsets	subset	NOUN
ejpam-3438	297	38	of	of	ADP
ejpam-3438	297	39	x	x	PRON
ejpam-3438	297	40	,	,	PUNCT
ejpam-3438	297	41	a	a	DET
ejpam-3438	297	42	⊆	⊆	NUM
ejpam-3438	297	43	bc	bc	PROPN
ejpam-3438	297	44	and	and	CCONJ
ejpam-3438	297	45	b	b	PROPN
ejpam-3438	297	46	⊆	⊆	NUM
ejpam-3438	297	47	ac	ac	PROPN
ejpam-3438	297	48	.	.	PUNCT
ejpam-3438	298	1	if	if	SCONJ
ejpam-3438	298	2	a	a	PRON
ejpam-3438	298	3	is	be	AUX
ejpam-3438	298	4	β	β	NOUN
ejpam-3438	298	5	-	-	ADJ
ejpam-3438	298	6	open	open	ADJ
ejpam-3438	298	7	and	and	CCONJ
ejpam-3438	298	8	b	b	NOUN
ejpam-3438	298	9	is	be	AUX
ejpam-3438	298	10	βi	βi	PRON
ejpam-3438	298	11	-open	-open	ADJ
ejpam-3438	298	12	,	,	PUNCT
ejpam-3438	298	13	then	then	ADV
ejpam-3438	298	14	ac	ac	PROPN
ejpam-3438	298	15	is	be	AUX
ejpam-3438	298	16	β	β	NOUN
ejpam-3438	298	17	-	-	VERB
ejpam-3438	298	18	closed	closed	ADJ
ejpam-3438	298	19	and	and	CCONJ
ejpam-3438	298	20	bc	bc	PROPN
ejpam-3438	298	21	is	be	AUX
ejpam-3438	298	22	βi	βi	PRON
ejpam-3438	298	23	-closed	-close	VERB
ejpam-3438	298	24	.	.	PUNCT
ejpam-3438	299	1	hence	hence	ADV
ejpam-3438	299	2	,	,	PUNCT
ejpam-3438	299	3	bc	bc	PROPN
ejpam-3438	299	4	∩b	∩b	PROPN
ejpam-3438	299	5	⊇	⊇	PROPN
ejpam-3438	299	6	clβi	clβi	NOUN
ejpam-3438	299	7	(	(	PUNCT
ejpam-3438	299	8	a	a	X
ejpam-3438	299	9	)	)	PUNCT
ejpam-3438	299	10	∩b	∩b	NOUN
ejpam-3438	299	11	6=	6=	NOUN
ejpam-3438	299	12	∅	∅	NOUN
ejpam-3438	299	13	or	or	CCONJ
ejpam-3438	299	14	a	a	DET
ejpam-3438	299	15	∩ac	∩ac	PROPN
ejpam-3438	299	16	⊇	⊇	NOUN
ejpam-3438	299	17	a	a	DET
ejpam-3438	299	18	∩	∩	ADJ
ejpam-3438	299	19	clβ(b	clβ(b	NOUN
ejpam-3438	299	20	)	)	PUNCT
ejpam-3438	299	21	6=	6=	ADP
ejpam-3438	299	22	∅.	∅.	VERB
ejpam-3438	299	23	this	this	PRON
ejpam-3438	299	24	is	be	AUX
ejpam-3438	299	25	a	a	DET
ejpam-3438	299	26	contradiction	contradiction	NOUN
ejpam-3438	299	27	.	.	PUNCT
ejpam-3438	300	1	the	the	DET
ejpam-3438	300	2	next	next	ADJ
ejpam-3438	300	3	statement	statement	NOUN
ejpam-3438	300	4	,	,	PUNCT
ejpam-3438	300	5	lemma	lemma	PROPN
ejpam-3438	300	6	7	7	NUM
ejpam-3438	300	7	,	,	PUNCT
ejpam-3438	300	8	stressed	stress	VERB
ejpam-3438	300	9	that	that	SCONJ
ejpam-3438	300	10	every	every	DET
ejpam-3438	300	11	βi	βi	ADV
ejpam-3438	300	12	-connected	-connected	ADJ
ejpam-3438	300	13	space	space	NOUN
ejpam-3438	300	14	is	be	AUX
ejpam-3438	300	15	connected	connect	VERB
ejpam-3438	300	16	.	.	PUNCT
ejpam-3438	301	1	g.	g.	PROPN
ejpam-3438	301	2	catalan	catalan	PROPN
ejpam-3438	301	3	,	,	PUNCT
ejpam-3438	301	4	r.	r.	PROPN
ejpam-3438	301	5	padua	padua	PROPN
ejpam-3438	301	6	,	,	PUNCT
ejpam-3438	301	7	m.	m.	PROPN
ejpam-3438	301	8	baldado	baldado	PROPN
ejpam-3438	301	9	jr	jr	PROPN
ejpam-3438	301	10	.	.	PROPN
ejpam-3438	301	11	/	/	SYM
ejpam-3438	301	12	eur	eur	PROPN
ejpam-3438	301	13	.	.	PUNCT
ejpam-3438	302	1	j.	j.	PROPN
ejpam-3438	302	2	pure	pure	PROPN
ejpam-3438	302	3	appl	appl	PROPN
ejpam-3438	302	4	.	.	PROPN
ejpam-3438	302	5	math	math	PROPN
ejpam-3438	302	6	,	,	PUNCT
ejpam-3438	302	7	12	12	NUM
ejpam-3438	302	8	(	(	PUNCT
ejpam-3438	302	9	3	3	NUM
ejpam-3438	302	10	)	)	PUNCT
ejpam-3438	302	11	(	(	PUNCT
ejpam-3438	302	12	2019	2019	NUM
ejpam-3438	302	13	)	)	PUNCT
ejpam-3438	302	14	,	,	PUNCT
ejpam-3438	302	15	893	893	NUM
ejpam-3438	302	16	-	-	SYM
ejpam-3438	302	17	905	905	NUM
ejpam-3438	302	18	901	901	NUM
ejpam-3438	302	19	lemma	lemma	PROPN
ejpam-3438	302	20	7	7	NUM
ejpam-3438	302	21	.	.	PUNCT
ejpam-3438	303	1	if	if	SCONJ
ejpam-3438	303	2	an	an	DET
ejpam-3438	303	3	ideal	ideal	ADJ
ejpam-3438	303	4	topological	topological	ADJ
ejpam-3438	303	5	space	space	NOUN
ejpam-3438	303	6	(	(	PUNCT
ejpam-3438	303	7	x	x	X
ejpam-3438	303	8	,	,	PUNCT
ejpam-3438	303	9	τ	τ	PROPN
ejpam-3438	303	10	,	,	PUNCT
ejpam-3438	303	11	i	i	PROPN
ejpam-3438	303	12	)	)	PUNCT
ejpam-3438	303	13	is	be	AUX
ejpam-3438	303	14	βi	βi	NOUN
ejpam-3438	303	15	-	-	PUNCT
ejpam-3438	303	16	connected	connect	VERB
ejpam-3438	303	17	,	,	PUNCT
ejpam-3438	303	18	then	then	ADV
ejpam-3438	303	19	(	(	PUNCT
ejpam-3438	303	20	x	x	X
ejpam-3438	303	21	,	,	PUNCT
ejpam-3438	303	22	τ	τ	X
ejpam-3438	303	23	)	)	PUNCT
ejpam-3438	303	24	is	be	AUX
ejpam-3438	303	25	connected	connect	VERB
ejpam-3438	303	26	.	.	PUNCT
ejpam-3438	304	1	proof	proof	NOUN
ejpam-3438	304	2	.	.	PUNCT
ejpam-3438	305	1	suppose	suppose	VERB
ejpam-3438	305	2	that	that	SCONJ
ejpam-3438	305	3	x	x	PRON
ejpam-3438	305	4	is	be	AUX
ejpam-3438	305	5	not	not	PART
ejpam-3438	305	6	connected	connect	VERB
ejpam-3438	305	7	.	.	PUNCT
ejpam-3438	306	1	let	let	VERB
ejpam-3438	306	2	a	a	PRON
ejpam-3438	306	3	and	and	CCONJ
ejpam-3438	306	4	b	b	NOUN
ejpam-3438	306	5	be	be	AUX
ejpam-3438	306	6	non	non	ADJ
ejpam-3438	306	7	-	-	ADJ
ejpam-3438	306	8	empty	empty	ADJ
ejpam-3438	306	9	disjoint	disjoint	NOUN
ejpam-3438	306	10	open	open	ADJ
ejpam-3438	306	11	sets	set	NOUN
ejpam-3438	306	12	such	such	ADJ
ejpam-3438	306	13	that	that	SCONJ
ejpam-3438	306	14	x	x	SYM
ejpam-3438	306	15	=	=	SYM
ejpam-3438	306	16	a∪b	a∪b	PROPN
ejpam-3438	306	17	.	.	PUNCT
ejpam-3438	307	1	since	since	SCONJ
ejpam-3438	307	2	every	every	DET
ejpam-3438	307	3	open	open	ADJ
ejpam-3438	307	4	set	set	NOUN
ejpam-3438	307	5	is	be	AUX
ejpam-3438	307	6	both	both	PRON
ejpam-3438	307	7	β	β	NOUN
ejpam-3438	307	8	-	-	ADJ
ejpam-3438	307	9	open	open	ADJ
ejpam-3438	307	10	and	and	CCONJ
ejpam-3438	307	11	βi	βi	PRON
ejpam-3438	307	12	-open	-open	PROPN
ejpam-3438	307	13	,	,	PUNCT
ejpam-3438	307	14	a	a	PRON
ejpam-3438	307	15	and	and	CCONJ
ejpam-3438	307	16	b	b	NOUN
ejpam-3438	307	17	are	be	AUX
ejpam-3438	307	18	both	both	PRON
ejpam-3438	307	19	β	β	NOUN
ejpam-3438	307	20	-	-	ADJ
ejpam-3438	307	21	open	open	ADJ
ejpam-3438	307	22	and	and	CCONJ
ejpam-3438	307	23	βi	βi	PRON
ejpam-3438	307	24	-open	-open	NOUN
ejpam-3438	307	25	.	.	PUNCT
ejpam-3438	308	1	since	since	SCONJ
ejpam-3438	308	2	a	a	DET
ejpam-3438	308	3	=	=	SYM
ejpam-3438	308	4	bc	bc	PROPN
ejpam-3438	308	5	and	and	CCONJ
ejpam-3438	308	6	b	b	X
ejpam-3438	308	7	=	=	SYM
ejpam-3438	308	8	ac	ac	PROPN
ejpam-3438	308	9	,	,	PUNCT
ejpam-3438	308	10	a	a	PRON
ejpam-3438	308	11	and	and	CCONJ
ejpam-3438	308	12	b	b	NOUN
ejpam-3438	308	13	are	be	AUX
ejpam-3438	308	14	also	also	ADV
ejpam-3438	308	15	both	both	PRON
ejpam-3438	308	16	β	β	NOUN
ejpam-3438	308	17	-	-	VERB
ejpam-3438	308	18	closed	closed	ADJ
ejpam-3438	308	19	and	and	CCONJ
ejpam-3438	308	20	βi	βi	PRON
ejpam-3438	308	21	-closed	-close	VERB
ejpam-3438	308	22	.	.	PUNCT
ejpam-3438	309	1	thus	thus	ADV
ejpam-3438	309	2	a	a	DET
ejpam-3438	309	3	=	=	NOUN
ejpam-3438	309	4	clβi	clβi	NOUN
ejpam-3438	309	5	(	(	PUNCT
ejpam-3438	309	6	a	a	NOUN
ejpam-3438	309	7	)	)	PUNCT
ejpam-3438	309	8	and	and	CCONJ
ejpam-3438	309	9	b	b	X
ejpam-3438	309	10	=	=	SYM
ejpam-3438	309	11	clβ(b	clβ(b	PROPN
ejpam-3438	309	12	)	)	PUNCT
ejpam-3438	309	13	.	.	PUNCT
ejpam-3438	310	1	hence	hence	ADV
ejpam-3438	310	2	,	,	PUNCT
ejpam-3438	310	3	clβi	clβi	NOUN
ejpam-3438	310	4	(	(	PUNCT
ejpam-3438	310	5	a	a	NOUN
ejpam-3438	310	6	)	)	PUNCT
ejpam-3438	310	7	∩	∩	NOUN
ejpam-3438	310	8	b	b	X
ejpam-3438	310	9	=	=	PUNCT
ejpam-3438	310	10	a	a	DET
ejpam-3438	310	11	∩	∩	ADJ
ejpam-3438	310	12	b	b	NOUN
ejpam-3438	310	13	=	=	SYM
ejpam-3438	310	14	∅	∅	NOUN
ejpam-3438	310	15	and	and	CCONJ
ejpam-3438	310	16	a	a	DET
ejpam-3438	310	17	∩	∩	ADJ
ejpam-3438	310	18	clβ(b	clβ(b	NOUN
ejpam-3438	310	19	)	)	PUNCT
ejpam-3438	310	20	=	=	SYM
ejpam-3438	310	21	a	a	DET
ejpam-3438	310	22	∩b	∩b	NOUN
ejpam-3438	310	23	=	=	X
ejpam-3438	310	24	∅.	∅.	NOUN
ejpam-3438	310	25	therefore	therefore	ADV
ejpam-3438	310	26	,	,	PUNCT
ejpam-3438	310	27	(	(	PUNCT
ejpam-3438	310	28	x	x	X
ejpam-3438	310	29	,	,	PUNCT
ejpam-3438	310	30	τ	τ	PROPN
ejpam-3438	310	31	,	,	PUNCT
ejpam-3438	310	32	i	i	PROPN
ejpam-3438	310	33	)	)	PUNCT
ejpam-3438	310	34	is	be	AUX
ejpam-3438	310	35	not	not	PART
ejpam-3438	310	36	βi	βi	PRON
ejpam-3438	310	37	-connected	-connected	ADJ
ejpam-3438	310	38	.	.	PUNCT
ejpam-3438	311	1	theorem	theorem	VERB
ejpam-3438	311	2	11	11	NUM
ejpam-3438	311	3	.	.	PUNCT
ejpam-3438	312	1	let	let	VERB
ejpam-3438	312	2	(	(	PUNCT
ejpam-3438	312	3	x	x	X
ejpam-3438	312	4	,	,	PUNCT
ejpam-3438	312	5	τ	τ	PROPN
ejpam-3438	312	6	,	,	PUNCT
ejpam-3438	312	7	i	i	PRON
ejpam-3438	312	8	)	)	PUNCT
ejpam-3438	312	9	be	be	VERB
ejpam-3438	312	10	an	an	DET
ejpam-3438	312	11	ideal	ideal	ADJ
ejpam-3438	312	12	topological	topological	ADJ
ejpam-3438	312	13	space	space	NOUN
ejpam-3438	312	14	and	and	CCONJ
ejpam-3438	312	15	y	y	PROPN
ejpam-3438	312	16	be	be	AUX
ejpam-3438	312	17	an	an	DET
ejpam-3438	312	18	open	open	ADJ
ejpam-3438	312	19	set	set	NOUN
ejpam-3438	312	20	.	.	PUNCT
ejpam-3438	313	1	if	if	SCONJ
ejpam-3438	313	2	a	a	PRON
ejpam-3438	313	3	is	be	AUX
ejpam-3438	313	4	a	a	DET
ejpam-3438	313	5	βi	βi	NOUN
ejpam-3438	313	6	subset	subset	NOUN
ejpam-3438	313	7	of	of	ADP
ejpam-3438	313	8	x	x	PRON
ejpam-3438	313	9	,	,	PUNCT
ejpam-3438	313	10	then	then	ADV
ejpam-3438	313	11	a	a	DET
ejpam-3438	313	12	∩	∩	ADJ
ejpam-3438	313	13	y	y	NOUN
ejpam-3438	313	14	is	be	AUX
ejpam-3438	313	15	βiy	βiy	PROPN
ejpam-3438	313	16	-open	-open	PROPN
ejpam-3438	313	17	subset	subset	NOUN
ejpam-3438	313	18	of	of	ADP
ejpam-3438	313	19	y	y	PROPN
ejpam-3438	313	20	.	.	PUNCT
ejpam-3438	314	1	proof	proof	NOUN
ejpam-3438	314	2	.	.	PUNCT
ejpam-3438	315	1	if	if	SCONJ
ejpam-3438	315	2	a	a	PRON
ejpam-3438	315	3	is	be	AUX
ejpam-3438	315	4	a	a	DET
ejpam-3438	315	5	βi	βi	NOUN
ejpam-3438	315	6	subset	subset	NOUN
ejpam-3438	315	7	of	of	ADP
ejpam-3438	315	8	x	x	PRON
ejpam-3438	315	9	,	,	PUNCT
ejpam-3438	315	10	then	then	ADV
ejpam-3438	315	11	there	there	PRON
ejpam-3438	315	12	exists	exist	VERB
ejpam-3438	315	13	an	an	DET
ejpam-3438	315	14	open	open	ADJ
ejpam-3438	315	15	set	set	NOUN
ejpam-3438	315	16	u	u	NOUN
ejpam-3438	315	17	′	′	ADP
ejpam-3438	315	18	such	such	ADJ
ejpam-3438	316	1	that	that	SCONJ
ejpam-3438	316	2	u	u	PROPN
ejpam-3438	316	3	′	′	NOUN
ejpam-3438	316	4	−a	−a	NOUN
ejpam-3438	317	1	∈	∈	PROPN
ejpam-3438	318	1	i	i	PRON
ejpam-3438	318	2	and	and	CCONJ
ejpam-3438	318	3	a−	a−	PROPN
ejpam-3438	318	4	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	318	5	)	)	PUNCT
ejpam-3438	318	6	)	)	PUNCT
ejpam-3438	318	7	)	)	PUNCT
ejpam-3438	319	1	∈	∈	PROPN
ejpam-3438	319	2	i.	i.	NOUN
ejpam-3438	319	3	let	let	VERB
ejpam-3438	319	4	u	u	PRON
ejpam-3438	319	5	=	=	PUNCT
ejpam-3438	319	6	u	u	NOUN
ejpam-3438	319	7	′	′	NOUN
ejpam-3438	319	8	∩	∩	PROPN
ejpam-3438	319	9	y	y	PROPN
ejpam-3438	319	10	.	.	PUNCT
ejpam-3438	320	1	then	then	ADV
ejpam-3438	320	2	u	u	NOUN
ejpam-3438	320	3	−	−	PROPN
ejpam-3438	320	4	(	(	PUNCT
ejpam-3438	320	5	a	a	DET
ejpam-3438	320	6	∩	∩	ADJ
ejpam-3438	320	7	y	y	NOUN
ejpam-3438	320	8	)	)	PUNCT
ejpam-3438	320	9	=	=	SYM
ejpam-3438	320	10	u	u	PROPN
ejpam-3438	320	11	∩	∩	NOUN
ejpam-3438	320	12	(	(	PUNCT
ejpam-3438	320	13	a	a	DET
ejpam-3438	320	14	∩	∩	ADJ
ejpam-3438	320	15	y	y	NOUN
ejpam-3438	320	16	)	)	PUNCT
ejpam-3438	320	17	c	c	NOUN
ejpam-3438	321	1	=	=	SYM
ejpam-3438	321	2	(	(	PUNCT
ejpam-3438	321	3	u	u	NOUN
ejpam-3438	321	4	′	′	NUM
ejpam-3438	321	5	∩	∩	ADJ
ejpam-3438	321	6	y	y	PROPN
ejpam-3438	321	7	)	)	PUNCT
ejpam-3438	321	8	∩	∩	NOUN
ejpam-3438	321	9	(	(	PUNCT
ejpam-3438	321	10	ac	ac	PROPN
ejpam-3438	321	11	∪	∪	PROPN
ejpam-3438	321	12	y	y	PROPN
ejpam-3438	321	13	c	c	NOUN
ejpam-3438	321	14	)	)	PUNCT
ejpam-3438	321	15	=	=	SYM
ejpam-3438	322	1	(	(	PUNCT
ejpam-3438	322	2	u	u	NOUN
ejpam-3438	322	3	′	′	NOUN
ejpam-3438	322	4	∩	∩	NOUN
ejpam-3438	322	5	y	y	PROPN
ejpam-3438	322	6	∩ac	∩ac	PROPN
ejpam-3438	322	7	)	)	PUNCT
ejpam-3438	322	8	∪	∪	NOUN
ejpam-3438	322	9	(	(	PUNCT
ejpam-3438	322	10	u	u	NOUN
ejpam-3438	322	11	′	′	NOUN
ejpam-3438	322	12	∩	∩	NOUN
ejpam-3438	322	13	y	y	PROPN
ejpam-3438	322	14	∩	∩	PROPN
ejpam-3438	322	15	y	y	PROPN
ejpam-3438	322	16	c	c	NOUN
ejpam-3438	322	17	)	)	PUNCT
ejpam-3438	323	1	=	=	SYM
ejpam-3438	323	2	u	u	NOUN
ejpam-3438	323	3	′	′	NOUN
ejpam-3438	323	4	∩	∩	X
ejpam-3438	323	5	y	y	PROPN
ejpam-3438	323	6	∩ac	∩ac	PROPN
ejpam-3438	323	7	=	=	SYM
ejpam-3438	323	8	(	(	PUNCT
ejpam-3438	323	9	u	u	NOUN
ejpam-3438	323	10	′	′	NOUN
ejpam-3438	323	11	−a	−a	NOUN
ejpam-3438	323	12	)	)	PUNCT
ejpam-3438	323	13	∩	∩	PROPN
ejpam-3438	323	14	y	y	PROPN
ejpam-3438	323	15	∈	∈	PROPN
ejpam-3438	323	16	iy	iy	PROPN
ejpam-3438	323	17	.	.	PUNCT
ejpam-3438	324	1	moreover	moreover	ADV
ejpam-3438	324	2	,	,	PUNCT
ejpam-3438	324	3	(	(	PUNCT
ejpam-3438	324	4	a	a	DET
ejpam-3438	324	5	∩	∩	ADJ
ejpam-3438	324	6	y	y	NOUN
ejpam-3438	324	7	)	)	PUNCT
ejpam-3438	324	8	−	−	PROPN
ejpam-3438	324	9	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	324	10	)	)	PUNCT
ejpam-3438	324	11	)	)	PUNCT
ejpam-3438	324	12	)	)	PUNCT
ejpam-3438	325	1	=	=	PRON
ejpam-3438	325	2	(	(	PUNCT
ejpam-3438	325	3	a	a	DET
ejpam-3438	325	4	∩	∩	ADJ
ejpam-3438	325	5	y	y	NOUN
ejpam-3438	325	6	)	)	PUNCT
ejpam-3438	325	7	−	−	PROPN
ejpam-3438	325	8	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	325	9	′	′	NUM
ejpam-3438	325	10	∩	∩	PROPN
ejpam-3438	325	11	y	y	PROPN
ejpam-3438	325	12	)	)	PUNCT
ejpam-3438	325	13	)	)	PUNCT
ejpam-3438	325	14	)	)	PUNCT
ejpam-3438	326	1	=	=	PRON
ejpam-3438	326	2	(	(	PUNCT
ejpam-3438	326	3	a	a	DET
ejpam-3438	326	4	∩	∩	ADJ
ejpam-3438	326	5	y	y	NOUN
ejpam-3438	326	6	)	)	PUNCT
ejpam-3438	326	7	−	−	PROPN
ejpam-3438	326	8	cl(int(cl(u	cl(int(cl(u	PROPN
ejpam-3438	326	9	′	′	NUM
ejpam-3438	326	10	)	)	PUNCT
ejpam-3438	326	11	)	)	PUNCT
ejpam-3438	326	12	)	)	PUNCT
ejpam-3438	327	1	∩	∩	NOUN
ejpam-3438	327	2	y	y	NOUN
ejpam-3438	328	1	=	=	PUNCT
ejpam-3438	329	1	[	[	X
ejpam-3438	329	2	a−	a−	PROPN
ejpam-3438	329	3	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	329	4	′	′	NUM
ejpam-3438	329	5	)	)	PUNCT
ejpam-3438	329	6	)	)	PUNCT
ejpam-3438	329	7	)	)	PUNCT
ejpam-3438	329	8	]	]	PUNCT
ejpam-3438	329	9	∩	∩	PROPN
ejpam-3438	329	10	y	y	PROPN
ejpam-3438	329	11	∈	∈	PROPN
ejpam-3438	329	12	iy	iy	PROPN
ejpam-3438	329	13	.	.	PUNCT
ejpam-3438	330	1	therefore	therefore	ADV
ejpam-3438	330	2	,	,	PUNCT
ejpam-3438	330	3	a	a	DET
ejpam-3438	330	4	∩	∩	ADJ
ejpam-3438	330	5	y	y	NOUN
ejpam-3438	330	6	is	be	AUX
ejpam-3438	330	7	βiy	βiy	NOUN
ejpam-3438	330	8	-connected	-connected	ADJ
ejpam-3438	330	9	.	.	PUNCT
ejpam-3438	331	1	remark	remark	PROPN
ejpam-3438	331	2	2	2	NUM
ejpam-3438	331	3	.	.	PUNCT
ejpam-3438	332	1	let	let	VERB
ejpam-3438	332	2	(	(	PUNCT
ejpam-3438	332	3	x	x	X
ejpam-3438	332	4	,	,	PUNCT
ejpam-3438	332	5	τ	τ	PROPN
ejpam-3438	332	6	,	,	PUNCT
ejpam-3438	332	7	i	i	PRON
ejpam-3438	332	8	)	)	PUNCT
ejpam-3438	332	9	be	be	VERB
ejpam-3438	332	10	an	an	DET
ejpam-3438	332	11	ideal	ideal	ADJ
ejpam-3438	332	12	topological	topological	ADJ
ejpam-3438	332	13	space	space	NOUN
ejpam-3438	332	14	.	.	PUNCT
ejpam-3438	333	1	if	if	SCONJ
ejpam-3438	333	2	y	y	PROPN
ejpam-3438	333	3	⊆	⊆	NUM
ejpam-3438	333	4	x	x	NOUN
ejpam-3438	333	5	,	,	PUNCT
ejpam-3438	333	6	then	then	ADV
ejpam-3438	333	7	iy	iy	PROPN
ejpam-3438	333	8	=	=	PUNCT
ejpam-3438	333	9	{	{	PUNCT
ejpam-3438	333	10	a∩	a∩	PROPN
ejpam-3438	333	11	y	y	PROPN
ejpam-3438	333	12	:	:	PUNCT
ejpam-3438	333	13	a	a	DET
ejpam-3438	333	14	∈	∈	PROPN
ejpam-3438	333	15	i	i	X
ejpam-3438	333	16	}	}	PUNCT
ejpam-3438	333	17	is	be	AUX
ejpam-3438	333	18	a	a	DET
ejpam-3438	333	19	subset	subset	NOUN
ejpam-3438	333	20	of	of	ADP
ejpam-3438	333	21	i.	i.	NOUN
ejpam-3438	333	22	the	the	DET
ejpam-3438	333	23	succeeding	succeed	VERB
ejpam-3438	333	24	theorems	theorem	NOUN
ejpam-3438	333	25	are	be	AUX
ejpam-3438	333	26	worth	worth	ADJ
ejpam-3438	333	27	-	-	PUNCT
ejpam-3438	333	28	noting	noting	NOUN
ejpam-3438	333	29	.	.	PUNCT
ejpam-3438	334	1	theorem	theorem	NOUN
ejpam-3438	334	2	12	12	NUM
ejpam-3438	334	3	.	.	PUNCT
ejpam-3438	335	1	let	let	VERB
ejpam-3438	335	2	(	(	PUNCT
ejpam-3438	335	3	x	x	X
ejpam-3438	335	4	,	,	PUNCT
ejpam-3438	335	5	τ	τ	PROPN
ejpam-3438	335	6	,	,	PUNCT
ejpam-3438	335	7	i	i	PRON
ejpam-3438	335	8	)	)	PUNCT
ejpam-3438	335	9	be	be	VERB
ejpam-3438	335	10	an	an	DET
ejpam-3438	335	11	ideal	ideal	ADJ
ejpam-3438	335	12	topological	topological	ADJ
ejpam-3438	335	13	space	space	NOUN
ejpam-3438	335	14	,	,	PUNCT
ejpam-3438	335	15	y	y	PROPN
ejpam-3438	335	16	be	be	VERB
ejpam-3438	335	17	an	an	DET
ejpam-3438	335	18	open	open	ADJ
ejpam-3438	335	19	set	set	NOUN
ejpam-3438	335	20	,	,	PUNCT
ejpam-3438	335	21	and	and	CCONJ
ejpam-3438	335	22	a	a	DET
ejpam-3438	335	23	⊆	⊆	NUM
ejpam-3438	335	24	y	y	NOUN
ejpam-3438	335	25	.	.	PUNCT
ejpam-3438	336	1	a	a	PRON
ejpam-3438	336	2	is	be	AUX
ejpam-3438	336	3	a	a	DET
ejpam-3438	336	4	βiy	βiy	NOUN
ejpam-3438	336	5	-open	-open	NOUN
ejpam-3438	336	6	subset	subset	NOUN
ejpam-3438	336	7	of	of	ADP
ejpam-3438	336	8	y	y	PRON
ejpam-3438	336	9	if	if	SCONJ
ejpam-3438	337	1	and	and	CCONJ
ejpam-3438	337	2	only	only	ADV
ejpam-3438	337	3	if	if	SCONJ
ejpam-3438	337	4	it	it	PRON
ejpam-3438	337	5	is	be	AUX
ejpam-3438	337	6	a	a	DET
ejpam-3438	337	7	βi	βi	NOUN
ejpam-3438	337	8	-	-	ADJ
ejpam-3438	337	9	open	open	ADJ
ejpam-3438	337	10	subset	subset	NOUN
ejpam-3438	337	11	of	of	ADP
ejpam-3438	337	12	x.	x.	NOUN
ejpam-3438	337	13	proof	proof	PROPN
ejpam-3438	337	14	.	.	PUNCT
ejpam-3438	338	1	assume	assume	VERB
ejpam-3438	338	2	that	that	SCONJ
ejpam-3438	338	3	a	a	PRON
ejpam-3438	338	4	is	be	AUX
ejpam-3438	338	5	a	a	DET
ejpam-3438	338	6	βiy	βiy	NOUN
ejpam-3438	338	7	-open	-open	NOUN
ejpam-3438	338	8	set	set	NOUN
ejpam-3438	338	9	.	.	PUNCT
ejpam-3438	339	1	then	then	ADV
ejpam-3438	339	2	there	there	PRON
ejpam-3438	339	3	exists	exist	VERB
ejpam-3438	339	4	an	an	DET
ejpam-3438	339	5	open	open	ADJ
ejpam-3438	339	6	set	set	NOUN
ejpam-3438	339	7	u	u	PRON
ejpam-3438	339	8	such	such	ADJ
ejpam-3438	339	9	that	that	DET
ejpam-3438	339	10	u	u	NOUN
ejpam-3438	339	11	−	−	PROPN
ejpam-3438	339	12	a	a	DET
ejpam-3438	339	13	∈	∈	PROPN
ejpam-3438	339	14	iy	iy	PROPN
ejpam-3438	339	15	and	and	CCONJ
ejpam-3438	339	16	a	a	DET
ejpam-3438	339	17	−	−	NOUN
ejpam-3438	339	18	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	339	19	)	)	PUNCT
ejpam-3438	339	20	)	)	PUNCT
ejpam-3438	339	21	)	)	PUNCT
ejpam-3438	340	1	∈	∈	PROPN
ejpam-3438	340	2	iy	iy	INTJ
ejpam-3438	340	3	.	.	PUNCT
ejpam-3438	341	1	let	let	VERB
ejpam-3438	341	2	u	u	PRON
ejpam-3438	341	3	′	′	NOUN
ejpam-3438	341	4	=	=	SYM
ejpam-3438	341	5	u	u	NOUN
ejpam-3438	341	6	∩	∩	NOUN
ejpam-3438	341	7	y	y	PROPN
ejpam-3438	341	8	.	.	PUNCT
ejpam-3438	342	1	since	since	SCONJ
ejpam-3438	342	2	y	y	PROPN
ejpam-3438	342	3	is	be	AUX
ejpam-3438	342	4	open	open	ADJ
ejpam-3438	342	5	,	,	PUNCT
ejpam-3438	342	6	u	u	NOUN
ejpam-3438	342	7	′	′	NOUN
ejpam-3438	342	8	is	be	AUX
ejpam-3438	342	9	open	open	ADJ
ejpam-3438	342	10	.	.	PUNCT
ejpam-3438	343	1	thus	thus	ADV
ejpam-3438	343	2	,	,	PUNCT
ejpam-3438	343	3	by	by	ADP
ejpam-3438	343	4	remark	remark	NOUN
ejpam-3438	343	5	2	2	NUM
ejpam-3438	343	6	u	u	NOUN
ejpam-3438	343	7	′−a	′−a	NOUN
ejpam-3438	343	8	∈	∈	NOUN
ejpam-3438	344	1	i	i	PRON
ejpam-3438	344	2	and	and	CCONJ
ejpam-3438	344	3	a−	a−	PROPN
ejpam-3438	344	4	cl(int(cl(u	cl(int(cl(u	NOUN
ejpam-3438	344	5	′	′	NUM
ejpam-3438	344	6	)	)	PUNCT
ejpam-3438	344	7	)	)	PUNCT
ejpam-3438	344	8	)	)	PUNCT
ejpam-3438	345	1	∈	∈	PROPN
ejpam-3438	345	2	i.	i.	NOUN
ejpam-3438	345	3	this	this	PRON
ejpam-3438	345	4	shows	show	VERB
ejpam-3438	345	5	that	that	SCONJ
ejpam-3438	345	6	a	a	PRON
ejpam-3438	345	7	is	be	AUX
ejpam-3438	345	8	βi	βi	PRON
ejpam-3438	345	9	-open	-open	NOUN
ejpam-3438	345	10	.	.	PUNCT
ejpam-3438	346	1	conversely	conversely	ADV
ejpam-3438	346	2	,	,	PUNCT
ejpam-3438	346	3	assume	assume	VERB
ejpam-3438	346	4	that	that	SCONJ
ejpam-3438	346	5	a	a	PRON
ejpam-3438	346	6	is	be	AUX
ejpam-3438	346	7	a	a	DET
ejpam-3438	346	8	βi	βi	X
ejpam-3438	346	9	-open	-open	NOUN
ejpam-3438	346	10	set	set	NOUN
ejpam-3438	346	11	.	.	PUNCT
ejpam-3438	347	1	then	then	ADV
ejpam-3438	347	2	there	there	PRON
ejpam-3438	347	3	exists	exist	VERB
ejpam-3438	347	4	an	an	DET
ejpam-3438	347	5	open	open	ADJ
ejpam-3438	347	6	set	set	NOUN
ejpam-3438	347	7	u	u	PRON
ejpam-3438	347	8	such	such	ADJ
ejpam-3438	347	9	that	that	SCONJ
ejpam-3438	347	10	u−a	u−a	ADJ
ejpam-3438	347	11	∈	∈	NOUN
ejpam-3438	347	12	i	i	PRON
ejpam-3438	347	13	and	and	CCONJ
ejpam-3438	347	14	a−cl(int(cl(u	a−cl(int(cl(u	NOUN
ejpam-3438	347	15	)	)	PUNCT
ejpam-3438	347	16	)	)	PUNCT
ejpam-3438	347	17	)	)	PUNCT
ejpam-3438	347	18	∈	∈	PROPN
ejpam-3438	347	19	i.	i.	NOUN
ejpam-3438	347	20	note	note	VERB
ejpam-3438	347	21	that	that	SCONJ
ejpam-3438	347	22	y	y	PROPN
ejpam-3438	347	23	∩	∩	NOUN
ejpam-3438	347	24	(	(	PUNCT
ejpam-3438	347	25	u−a	u−a	ADJ
ejpam-3438	347	26	)	)	PUNCT
ejpam-3438	347	27	∈	∈	PROPN
ejpam-3438	347	28	iy	iy	PROPN
ejpam-3438	347	29	and	and	CCONJ
ejpam-3438	347	30	a−cl(int(cl(u	a−cl(int(cl(u	NOUN
ejpam-3438	347	31	)	)	PUNCT
ejpam-3438	347	32	)	)	PUNCT
ejpam-3438	347	33	)	)	PUNCT
ejpam-3438	348	1	∈	∈	PROPN
ejpam-3438	348	2	iy	iy	INTJ
ejpam-3438	348	3	,	,	PUNCT
ejpam-3438	348	4	that	that	ADV
ejpam-3438	348	5	is	is	ADV
ejpam-3438	348	6	(	(	PUNCT
ejpam-3438	348	7	y	y	PROPN
ejpam-3438	348	8	∩	∩	ADJ
ejpam-3438	348	9	u	u	NOUN
ejpam-3438	348	10	)	)	PUNCT
ejpam-3438	348	11	−	−	PROPN
ejpam-3438	348	12	a	a	DET
ejpam-3438	348	13	∈	∈	PROPN
ejpam-3438	348	14	iy	iy	PROPN
ejpam-3438	348	15	and	and	CCONJ
ejpam-3438	348	16	a	a	DET
ejpam-3438	348	17	−	−	PROPN
ejpam-3438	348	18	cl(int(cl(y	cl(int(cl(y	NOUN
ejpam-3438	348	19	∩	∩	ADJ
ejpam-3438	348	20	u	u	NOUN
ejpam-3438	348	21	)	)	PUNCT
ejpam-3438	348	22	)	)	PUNCT
ejpam-3438	348	23	)	)	PUNCT
ejpam-3438	349	1	∈	∈	PROPN
ejpam-3438	349	2	iy	iy	INTJ
ejpam-3438	349	3	.	.	PUNCT
ejpam-3438	350	1	this	this	PRON
ejpam-3438	350	2	shows	show	VERB
ejpam-3438	350	3	that	that	SCONJ
ejpam-3438	350	4	a	a	PRON
ejpam-3438	350	5	is	be	AUX
ejpam-3438	350	6	βiy	βiy	NOUN
ejpam-3438	350	7	-open	-open	NOUN
ejpam-3438	350	8	.	.	PUNCT
ejpam-3438	351	1	theorem	theorem	NOUN
ejpam-3438	351	2	13	13	NUM
ejpam-3438	351	3	.	.	PUNCT
ejpam-3438	352	1	let	let	VERB
ejpam-3438	352	2	(	(	PUNCT
ejpam-3438	352	3	x	x	X
ejpam-3438	352	4	,	,	PUNCT
ejpam-3438	352	5	τ	τ	PROPN
ejpam-3438	352	6	,	,	PUNCT
ejpam-3438	352	7	i	i	PRON
ejpam-3438	352	8	)	)	PUNCT
ejpam-3438	352	9	be	be	VERB
ejpam-3438	352	10	an	an	DET
ejpam-3438	352	11	ideal	ideal	ADJ
ejpam-3438	352	12	topological	topological	ADJ
ejpam-3438	352	13	space	space	NOUN
ejpam-3438	352	14	,	,	PUNCT
ejpam-3438	352	15	y	y	PROPN
ejpam-3438	352	16	be	be	VERB
ejpam-3438	352	17	an	an	DET
ejpam-3438	352	18	open	open	ADJ
ejpam-3438	352	19	set	set	NOUN
ejpam-3438	352	20	,	,	PUNCT
ejpam-3438	352	21	and	and	CCONJ
ejpam-3438	352	22	a	a	DET
ejpam-3438	352	23	⊆	⊆	NUM
ejpam-3438	352	24	y	y	NOUN
ejpam-3438	352	25	.	.	PUNCT
ejpam-3438	353	1	then	then	ADV
ejpam-3438	353	2	clβiy	clβiy	PROPN
ejpam-3438	353	3	(	(	PUNCT
ejpam-3438	353	4	a	a	NOUN
ejpam-3438	353	5	)	)	PUNCT
ejpam-3438	353	6	=	=	NOUN
ejpam-3438	353	7	clβi	clβi	NOUN
ejpam-3438	353	8	(	(	PUNCT
ejpam-3438	353	9	a	a	NOUN
ejpam-3438	353	10	)	)	PUNCT
ejpam-3438	353	11	∩	∩	PROPN
ejpam-3438	353	12	y	y	PROPN
ejpam-3438	353	13	.	.	PUNCT
ejpam-3438	354	1	g.	g.	PROPN
ejpam-3438	354	2	catalan	catalan	PROPN
ejpam-3438	354	3	,	,	PUNCT
ejpam-3438	354	4	r.	r.	PROPN
ejpam-3438	354	5	padua	padua	PROPN
ejpam-3438	354	6	,	,	PUNCT
ejpam-3438	354	7	m.	m.	PROPN
ejpam-3438	354	8	baldado	baldado	PROPN
ejpam-3438	354	9	jr	jr	PROPN
ejpam-3438	354	10	.	.	PROPN
ejpam-3438	354	11	/	/	SYM
ejpam-3438	354	12	eur	eur	PROPN
ejpam-3438	354	13	.	.	PUNCT
ejpam-3438	355	1	j.	j.	PROPN
ejpam-3438	355	2	pure	pure	PROPN
ejpam-3438	355	3	appl	appl	PROPN
ejpam-3438	355	4	.	.	PROPN
ejpam-3438	355	5	math	math	PROPN
ejpam-3438	355	6	,	,	PUNCT
ejpam-3438	355	7	12	12	NUM
ejpam-3438	355	8	(	(	PUNCT
ejpam-3438	355	9	3	3	NUM
ejpam-3438	355	10	)	)	PUNCT
ejpam-3438	355	11	(	(	PUNCT
ejpam-3438	355	12	2019	2019	NUM
ejpam-3438	355	13	)	)	PUNCT
ejpam-3438	355	14	,	,	PUNCT
ejpam-3438	355	15	893	893	NUM
ejpam-3438	355	16	-	-	SYM
ejpam-3438	355	17	905	905	NUM
ejpam-3438	355	18	902	902	NUM
ejpam-3438	355	19	proof	proof	NOUN
ejpam-3438	355	20	.	.	PUNCT
ejpam-3438	356	1	let	let	VERB
ejpam-3438	356	2	w	w	NOUN
ejpam-3438	356	3	/∈	/∈	PUNCT
ejpam-3438	356	4	clβi	clβi	NOUN
ejpam-3438	356	5	(	(	PUNCT
ejpam-3438	356	6	a	a	NOUN
ejpam-3438	356	7	)	)	PUNCT
ejpam-3438	356	8	∩	∩	PROPN
ejpam-3438	356	9	y	y	PROPN
ejpam-3438	356	10	.	.	PUNCT
ejpam-3438	357	1	then	then	ADV
ejpam-3438	357	2	w	w	PROPN
ejpam-3438	357	3	∈	∈	PROPN
ejpam-3438	357	4	x	x	PUNCT
ejpam-3438	357	5	−	−	NOUN
ejpam-3438	357	6	clβi	clβi	NOUN
ejpam-3438	357	7	(	(	PUNCT
ejpam-3438	357	8	a	a	NOUN
ejpam-3438	357	9	)	)	PUNCT
ejpam-3438	357	10	.	.	PUNCT
ejpam-3438	358	1	by	by	ADP
ejpam-3438	358	2	theorem	theorem	NOUN
ejpam-3438	358	3	12	12	NUM
ejpam-3438	358	4	,	,	PUNCT
ejpam-3438	358	5	(	(	PUNCT
ejpam-3438	358	6	x	x	NOUN
ejpam-3438	358	7	−	−	NOUN
ejpam-3438	358	8	clβi	clβi	NOUN
ejpam-3438	358	9	(	(	PUNCT
ejpam-3438	358	10	a	a	NOUN
ejpam-3438	358	11	)	)	PUNCT
ejpam-3438	358	12	)	)	PUNCT
ejpam-3438	358	13	∩	∩	NOUN
ejpam-3438	358	14	y	y	PROPN
ejpam-3438	358	15	is	be	AUX
ejpam-3438	358	16	a	a	DET
ejpam-3438	358	17	βiy	βiy	NOUN
ejpam-3438	358	18	-open	-open	NOUN
ejpam-3438	358	19	subset	subset	NOUN
ejpam-3438	358	20	of	of	ADP
ejpam-3438	358	21	y	y	PROPN
ejpam-3438	358	22	.	.	PUNCT
ejpam-3438	359	1	note	note	VERB
ejpam-3438	359	2	that	that	SCONJ
ejpam-3438	359	3	w	w	NOUN
ejpam-3438	359	4	must	must	AUX
ejpam-3438	359	5	be	be	AUX
ejpam-3438	359	6	in	in	ADP
ejpam-3438	359	7	(	(	PUNCT
ejpam-3438	359	8	x	x	NOUN
ejpam-3438	359	9	−	−	NOUN
ejpam-3438	359	10	clβi	clβi	NOUN
ejpam-3438	359	11	(	(	PUNCT
ejpam-3438	359	12	a	a	NOUN
ejpam-3438	359	13	)	)	PUNCT
ejpam-3438	359	14	)	)	PUNCT
ejpam-3438	359	15	.	.	PUNCT
ejpam-3438	360	1	hence	hence	ADV
ejpam-3438	360	2	,	,	PUNCT
ejpam-3438	360	3	y	y	PROPN
ejpam-3438	360	4	−	−	PROPN
ejpam-3438	361	1	[	[	X
ejpam-3438	361	2	(	(	PUNCT
ejpam-3438	361	3	x	x	SYM
ejpam-3438	361	4	−	−	NOUN
ejpam-3438	361	5	clβi	clβi	NOUN
ejpam-3438	361	6	(	(	PUNCT
ejpam-3438	361	7	a	a	NOUN
ejpam-3438	361	8	)	)	PUNCT
ejpam-3438	361	9	)	)	PUNCT
ejpam-3438	361	10	∩	∩	PROPN
ejpam-3438	361	11	y	y	PROPN
ejpam-3438	361	12	]	]	PUNCT
ejpam-3438	361	13	is	be	AUX
ejpam-3438	361	14	a	a	DET
ejpam-3438	361	15	βiy	βiy	NOUN
ejpam-3438	361	16	-closed	-close	VERB
ejpam-3438	361	17	set	set	NOUN
ejpam-3438	361	18	in	in	ADP
ejpam-3438	361	19	y	y	PROPN
ejpam-3438	361	20	,	,	PUNCT
ejpam-3438	361	21	which	which	PRON
ejpam-3438	361	22	does	do	AUX
ejpam-3438	361	23	not	not	PART
ejpam-3438	361	24	contain	contain	VERB
ejpam-3438	361	25	w.	w.	NOUN
ejpam-3438	361	26	thus	thus	ADV
ejpam-3438	361	27	,	,	PUNCT
ejpam-3438	361	28	x	x	X
ejpam-3438	361	29	/∈	/∈	PUNCT
ejpam-3438	361	30	clβi	clβi	NOUN
ejpam-3438	361	31	(	(	PUNCT
ejpam-3438	361	32	a	a	NOUN
ejpam-3438	361	33	)	)	PUNCT
ejpam-3438	361	34	.	.	PUNCT
ejpam-3438	362	1	therefore	therefore	ADV
ejpam-3438	362	2	,	,	PUNCT
ejpam-3438	362	3	clβiy	clβiy	NOUN
ejpam-3438	362	4	(	(	PUNCT
ejpam-3438	362	5	a	a	NOUN
ejpam-3438	362	6	)	)	PUNCT
ejpam-3438	362	7	⊆	⊆	NUM
ejpam-3438	362	8	clβi	clβi	NOUN
ejpam-3438	362	9	(	(	PUNCT
ejpam-3438	362	10	a	a	NOUN
ejpam-3438	362	11	)	)	PUNCT
ejpam-3438	362	12	∩	∩	PROPN
ejpam-3438	362	13	y	y	PROPN
ejpam-3438	362	14	.	.	PUNCT
ejpam-3438	363	1	next	next	ADV
ejpam-3438	363	2	,	,	PUNCT
ejpam-3438	363	3	let	let	VERB
ejpam-3438	363	4	z	z	NOUN
ejpam-3438	363	5	/∈	/∈	PUNCT
ejpam-3438	363	6	clβiy	clβiy	NOUN
ejpam-3438	363	7	(	(	PUNCT
ejpam-3438	363	8	a	a	NOUN
ejpam-3438	363	9	)	)	PUNCT
ejpam-3438	363	10	.	.	PUNCT
ejpam-3438	364	1	then	then	ADV
ejpam-3438	364	2	z	z	PROPN
ejpam-3438	364	3	∈	∈	PROPN
ejpam-3438	364	4	x	x	PUNCT
ejpam-3438	364	5	−	−	PROPN
ejpam-3438	364	6	clβiy	clβiy	NOUN
ejpam-3438	364	7	(	(	PUNCT
ejpam-3438	364	8	a	a	NOUN
ejpam-3438	364	9	)	)	PUNCT
ejpam-3438	364	10	.	.	PUNCT
ejpam-3438	365	1	by	by	ADP
ejpam-3438	365	2	theorem	theorem	NOUN
ejpam-3438	365	3	12	12	NUM
ejpam-3438	365	4	,	,	PUNCT
ejpam-3438	365	5	(	(	PUNCT
ejpam-3438	365	6	y	y	PROPN
ejpam-3438	365	7	−	−	PROPN
ejpam-3438	365	8	clβiy	clβiy	NOUN
ejpam-3438	365	9	(	(	PUNCT
ejpam-3438	365	10	a	a	NOUN
ejpam-3438	365	11	)	)	PUNCT
ejpam-3438	365	12	)	)	PUNCT
ejpam-3438	365	13	is	be	AUX
ejpam-3438	365	14	a	a	DET
ejpam-3438	365	15	βi	βi	X
ejpam-3438	365	16	-open	-open	NOUN
ejpam-3438	365	17	subset	subset	NOUN
ejpam-3438	365	18	of	of	ADP
ejpam-3438	365	19	x.	x.	PROPN
ejpam-3438	365	20	note	note	VERB
ejpam-3438	365	21	that	that	SCONJ
ejpam-3438	365	22	(	(	PUNCT
ejpam-3438	365	23	y	y	PROPN
ejpam-3438	365	24	−	−	PROPN
ejpam-3438	365	25	clβiy	clβiy	NOUN
ejpam-3438	365	26	(	(	PUNCT
ejpam-3438	365	27	a	a	NOUN
ejpam-3438	365	28	)	)	PUNCT
ejpam-3438	365	29	)	)	PUNCT
ejpam-3438	365	30	must	must	AUX
ejpam-3438	365	31	contain	contain	VERB
ejpam-3438	365	32	w.	w.	NOUN
ejpam-3438	365	33	thus	thus	ADV
ejpam-3438	365	34	,	,	PUNCT
ejpam-3438	365	35	x	x	PRON
ejpam-3438	365	36	−	−	PROPN
ejpam-3438	365	37	(	(	PUNCT
ejpam-3438	365	38	y	y	PROPN
ejpam-3438	365	39	−	−	PROPN
ejpam-3438	365	40	clβiy	clβiy	NOUN
ejpam-3438	365	41	(	(	PUNCT
ejpam-3438	365	42	a	a	NOUN
ejpam-3438	365	43	)	)	PUNCT
ejpam-3438	365	44	)	)	PUNCT
ejpam-3438	365	45	is	be	AUX
ejpam-3438	365	46	a	a	DET
ejpam-3438	365	47	βi	βi	ADV
ejpam-3438	365	48	-closed	-close	VERB
ejpam-3438	365	49	set	set	NOUN
ejpam-3438	365	50	in	in	ADP
ejpam-3438	365	51	x	x	NOUN
ejpam-3438	365	52	,	,	PUNCT
ejpam-3438	365	53	which	which	PRON
ejpam-3438	365	54	does	do	AUX
ejpam-3438	365	55	not	not	PART
ejpam-3438	365	56	contain	contain	VERB
ejpam-3438	365	57	z.	z.	PROPN
ejpam-3438	365	58	thus	thus	ADV
ejpam-3438	365	59	,	,	PUNCT
ejpam-3438	365	60	clβi	clβi	NOUN
ejpam-3438	365	61	(	(	PUNCT
ejpam-3438	365	62	a	a	X
ejpam-3438	365	63	)	)	PUNCT
ejpam-3438	365	64	=	=	SYM
ejpam-3438	365	65	⋂	⋂	PROPN
ejpam-3438	365	66	{	{	PUNCT
ejpam-3438	365	67	f	f	NOUN
ejpam-3438	365	68	:	:	PUNCT
ejpam-3438	365	69	f	f	PROPN
ejpam-3438	365	70	is	be	AUX
ejpam-3438	365	71	a	a	DET
ejpam-3438	365	72	βi	βi	ADV
ejpam-3438	365	73	-closed	-close	VERB
ejpam-3438	365	74	set	set	NOUN
ejpam-3438	365	75	and	and	CCONJ
ejpam-3438	365	76	a	a	DET
ejpam-3438	365	77	⊆	⊆	NUM
ejpam-3438	365	78	f	f	X
ejpam-3438	365	79	}	}	PUNCT
ejpam-3438	365	80	does	do	AUX
ejpam-3438	365	81	not	not	PART
ejpam-3438	365	82	contain	contain	VERB
ejpam-3438	365	83	z	z	NOUN
ejpam-3438	365	84	,	,	PUNCT
ejpam-3438	365	85	that	that	PRON
ejpam-3438	365	86	is	be	AUX
ejpam-3438	365	87	z	z	NOUN
ejpam-3438	365	88	/∈	/∈	PUNCT
ejpam-3438	365	89	clβi	clβi	NOUN
ejpam-3438	365	90	(	(	PUNCT
ejpam-3438	365	91	a	a	NOUN
ejpam-3438	365	92	)	)	PUNCT
ejpam-3438	365	93	.	.	PUNCT
ejpam-3438	366	1	therefore	therefore	ADV
ejpam-3438	366	2	,	,	PUNCT
ejpam-3438	366	3	clβiy	clβiy	NOUN
ejpam-3438	366	4	(	(	PUNCT
ejpam-3438	366	5	a	a	NOUN
ejpam-3438	366	6	)	)	PUNCT
ejpam-3438	366	7	⊇	⊇	NOUN
ejpam-3438	366	8	clβi	clβi	NOUN
ejpam-3438	366	9	(	(	PUNCT
ejpam-3438	366	10	a	a	NOUN
ejpam-3438	366	11	)	)	PUNCT
ejpam-3438	366	12	∩	∩	PROPN
ejpam-3438	366	13	y	y	PROPN
ejpam-3438	366	14	.	.	PUNCT
ejpam-3438	366	15	theorem	theorem	VERB
ejpam-3438	366	16	14	14	NUM
ejpam-3438	366	17	.	.	PUNCT
ejpam-3438	367	1	let	let	VERB
ejpam-3438	367	2	(	(	PUNCT
ejpam-3438	367	3	x	x	X
ejpam-3438	367	4	,	,	PUNCT
ejpam-3438	367	5	τ	τ	PROPN
ejpam-3438	367	6	,	,	PUNCT
ejpam-3438	367	7	i	i	PRON
ejpam-3438	367	8	)	)	PUNCT
ejpam-3438	367	9	be	be	VERB
ejpam-3438	367	10	an	an	DET
ejpam-3438	367	11	ideal	ideal	ADJ
ejpam-3438	367	12	topological	topological	ADJ
ejpam-3438	367	13	space	space	NOUN
ejpam-3438	367	14	,	,	PUNCT
ejpam-3438	367	15	y	y	PROPN
ejpam-3438	367	16	be	be	VERB
ejpam-3438	367	17	an	an	DET
ejpam-3438	367	18	open	open	ADJ
ejpam-3438	367	19	set	set	NOUN
ejpam-3438	367	20	,	,	PUNCT
ejpam-3438	367	21	and	and	CCONJ
ejpam-3438	367	22	a	a	PRON
ejpam-3438	367	23	and	and	CCONJ
ejpam-3438	367	24	b	b	NOUN
ejpam-3438	367	25	be	be	AUX
ejpam-3438	367	26	subsets	subset	NOUN
ejpam-3438	367	27	of	of	ADP
ejpam-3438	367	28	y	y	PROPN
ejpam-3438	367	29	.	.	PUNCT
ejpam-3438	368	1	then	then	ADV
ejpam-3438	368	2	the	the	DET
ejpam-3438	368	3	following	follow	VERB
ejpam-3438	368	4	statements	statement	NOUN
ejpam-3438	368	5	are	be	AUX
ejpam-3438	368	6	equivalent	equivalent	ADJ
ejpam-3438	368	7	.	.	PUNCT
ejpam-3438	369	1	(	(	PUNCT
ejpam-3438	369	2	i	i	NOUN
ejpam-3438	369	3	)	)	PUNCT
ejpam-3438	369	4	a	a	PRON
ejpam-3438	369	5	and	and	CCONJ
ejpam-3438	369	6	b	b	NOUN
ejpam-3438	369	7	are	be	AUX
ejpam-3438	369	8	βiy	βiy	NOUN
ejpam-3438	369	9	-separated	-separate	VERB
ejpam-3438	369	10	in	in	ADP
ejpam-3438	369	11	y	y	PROPN
ejpam-3438	369	12	.	.	PUNCT
ejpam-3438	370	1	(	(	PUNCT
ejpam-3438	370	2	ii	ii	NOUN
ejpam-3438	370	3	)	)	PUNCT
ejpam-3438	370	4	a	a	PROPN
ejpam-3438	370	5	and	and	CCONJ
ejpam-3438	370	6	b	b	NOUN
ejpam-3438	370	7	are	be	AUX
ejpam-3438	370	8	βi	βi	PRON
ejpam-3438	370	9	-	-	PUNCT
ejpam-3438	370	10	separated	separate	VERB
ejpam-3438	370	11	in	in	ADP
ejpam-3438	370	12	x.	x.	NOUN
ejpam-3438	370	13	proof	proof	NOUN
ejpam-3438	370	14	.	.	PUNCT
ejpam-3438	371	1	(	(	PUNCT
ejpam-3438	371	2	1)⇒	1)⇒	NUM
ejpam-3438	371	3	(	(	PUNCT
ejpam-3438	371	4	2	2	NUM
ejpam-3438	371	5	)	)	PUNCT
ejpam-3438	371	6	assume	assume	VERB
ejpam-3438	371	7	that	that	SCONJ
ejpam-3438	371	8	(	(	PUNCT
ejpam-3438	371	9	1	1	X
ejpam-3438	371	10	)	)	PUNCT
ejpam-3438	371	11	holds	hold	VERB
ejpam-3438	371	12	.	.	PUNCT
ejpam-3438	372	1	if	if	SCONJ
ejpam-3438	372	2	a	a	PRON
ejpam-3438	372	3	and	and	CCONJ
ejpam-3438	372	4	b	b	NOUN
ejpam-3438	372	5	are	be	AUX
ejpam-3438	372	6	βiy	βiy	NOUN
ejpam-3438	372	7	-separated	-separate	VERB
ejpam-3438	372	8	in	in	ADP
ejpam-3438	372	9	y	y	PROPN
ejpam-3438	372	10	,	,	PUNCT
ejpam-3438	372	11	then	then	ADV
ejpam-3438	372	12	by	by	ADP
ejpam-3438	372	13	the	the	DET
ejpam-3438	372	14	assumption	assumption	NOUN
ejpam-3438	372	15	and	and	CCONJ
ejpam-3438	372	16	by	by	ADP
ejpam-3438	372	17	theorem	theorem	ADJ
ejpam-3438	372	18	13	13	NUM
ejpam-3438	372	19	clβi	clβi	NOUN
ejpam-3438	372	20	(	(	PUNCT
ejpam-3438	372	21	a)∩b	a)∩b	NOUN
ejpam-3438	372	22	=	=	SYM
ejpam-3438	372	23	clβi	clβi	NOUN
ejpam-3438	372	24	(	(	PUNCT
ejpam-3438	372	25	a)∩	a)∩	X
ejpam-3438	372	26	(	(	PUNCT
ejpam-3438	372	27	b∩y	b∩y	PROPN
ejpam-3438	372	28	)	)	PUNCT
ejpam-3438	372	29	=	=	SYM
ejpam-3438	373	1	(	(	PUNCT
ejpam-3438	373	2	clβi	clβi	NOUN
ejpam-3438	373	3	(	(	PUNCT
ejpam-3438	373	4	a)∩y	a)∩y	PROPN
ejpam-3438	373	5	)	)	PUNCT
ejpam-3438	373	6	∩b	∩b	NOUN
ejpam-3438	373	7	=	=	SYM
ejpam-3438	373	8	clβiy	clβiy	NOUN
ejpam-3438	373	9	(	(	PUNCT
ejpam-3438	373	10	a)∩b	a)∩b	NOUN
ejpam-3438	373	11	=	=	SYM
ejpam-3438	373	12	∅	∅	NOUN
ejpam-3438	373	13	and	and	CCONJ
ejpam-3438	373	14	a∩clβ(b	a∩clβ(b	NOUN
ejpam-3438	373	15	)	)	PUNCT
ejpam-3438	373	16	=	=	SYM
ejpam-3438	373	17	(	(	PUNCT
ejpam-3438	373	18	a∩y	a∩y	PROPN
ejpam-3438	373	19	)	)	PUNCT
ejpam-3438	373	20	∩clβ(b	∩clβ(b	NOUN
ejpam-3438	373	21	)	)	PUNCT
ejpam-3438	373	22	=	=	SYM
ejpam-3438	373	23	a∩(clβ(b)∩y	a∩(clβ(b)∩y	X
ejpam-3438	373	24	)	)	PUNCT
ejpam-3438	373	25	=	=	SYM
ejpam-3438	374	1	a∩clβy	a∩clβy	X
ejpam-3438	374	2	(	(	PUNCT
ejpam-3438	374	3	b	b	NOUN
ejpam-3438	374	4	)	)	PUNCT
ejpam-3438	374	5	=	=	PUNCT
ejpam-3438	374	6	∅.	∅.	ADP
ejpam-3438	374	7	this	this	PRON
ejpam-3438	374	8	shows	show	VERB
ejpam-3438	374	9	that	that	SCONJ
ejpam-3438	374	10	a	a	PRON
ejpam-3438	374	11	and	and	CCONJ
ejpam-3438	374	12	b	b	NOUN
ejpam-3438	374	13	are	be	AUX
ejpam-3438	374	14	βi	βi	PRON
ejpam-3438	374	15	-separated	-separated	ADJ
ejpam-3438	374	16	.	.	PUNCT
ejpam-3438	375	1	(	(	PUNCT
ejpam-3438	375	2	2)⇒	2)⇒	NUM
ejpam-3438	375	3	(	(	PUNCT
ejpam-3438	375	4	1	1	X
ejpam-3438	375	5	)	)	PUNCT
ejpam-3438	375	6	assume	assume	VERB
ejpam-3438	375	7	that	that	SCONJ
ejpam-3438	375	8	(	(	PUNCT
ejpam-3438	375	9	2	2	X
ejpam-3438	375	10	)	)	PUNCT
ejpam-3438	375	11	holds	hold	VERB
ejpam-3438	375	12	.	.	PUNCT
ejpam-3438	376	1	ifa	ifa	PROPN
ejpam-3438	376	2	andb	andb	PROPN
ejpam-3438	376	3	are	be	AUX
ejpam-3438	376	4	βi	βi	PRON
ejpam-3438	376	5	-separated	-separated	ADJ
ejpam-3438	376	6	inx	inx	NOUN
ejpam-3438	376	7	,	,	PUNCT
ejpam-3438	376	8	then	then	ADV
ejpam-3438	376	9	by	by	ADP
ejpam-3438	376	10	the	the	DET
ejpam-3438	376	11	assumption	assumption	NOUN
ejpam-3438	376	12	and	and	CCONJ
ejpam-3438	376	13	by	by	ADP
ejpam-3438	376	14	theorem	theorem	ADJ
ejpam-3438	376	15	13	13	NUM
ejpam-3438	376	16	∅	∅	NOUN
ejpam-3438	376	17	=	=	NOUN
ejpam-3438	376	18	clβi	clβi	NOUN
ejpam-3438	376	19	(	(	PUNCT
ejpam-3438	376	20	a)∩b	a)∩b	NOUN
ejpam-3438	376	21	=	=	SYM
ejpam-3438	376	22	clβi	clβi	NOUN
ejpam-3438	376	23	(	(	PUNCT
ejpam-3438	376	24	a)∩(b∩y	a)∩(b∩y	NOUN
ejpam-3438	376	25	)	)	PUNCT
ejpam-3438	377	1	=	=	SYM
ejpam-3438	377	2	(	(	PUNCT
ejpam-3438	377	3	clβi	clβi	NOUN
ejpam-3438	377	4	(	(	PUNCT
ejpam-3438	377	5	a)∩y	a)∩y	PROPN
ejpam-3438	377	6	)	)	PUNCT
ejpam-3438	378	1	∩b	∩b	NOUN
ejpam-3438	378	2	=	=	SYM
ejpam-3438	378	3	clβiy	clβiy	NOUN
ejpam-3438	378	4	(	(	PUNCT
ejpam-3438	378	5	a)∩b	a)∩b	NOUN
ejpam-3438	378	6	and	and	CCONJ
ejpam-3438	378	7	∅	∅	NOUN
ejpam-3438	378	8	=	=	PUNCT
ejpam-3438	378	9	a	a	DET
ejpam-3438	378	10	∩	∩	ADJ
ejpam-3438	378	11	clβ(b	clβ(b	NOUN
ejpam-3438	378	12	)	)	PUNCT
ejpam-3438	378	13	=	=	SYM
ejpam-3438	378	14	(	(	PUNCT
ejpam-3438	378	15	a	a	DET
ejpam-3438	378	16	∩	∩	ADJ
ejpam-3438	378	17	y	y	NOUN
ejpam-3438	378	18	)	)	PUNCT
ejpam-3438	378	19	∩	∩	X
ejpam-3438	378	20	clβ(b	clβ(b	NOUN
ejpam-3438	378	21	)	)	PUNCT
ejpam-3438	378	22	=	=	SYM
ejpam-3438	378	23	a	a	DET
ejpam-3438	378	24	∩	∩	NOUN
ejpam-3438	378	25	(	(	PUNCT
ejpam-3438	378	26	clβ(b	clβ(b	NOUN
ejpam-3438	378	27	)	)	PUNCT
ejpam-3438	378	28	∩	∩	NOUN
ejpam-3438	378	29	y	y	NOUN
ejpam-3438	378	30	)	)	PUNCT
ejpam-3438	378	31	=	=	PUNCT
ejpam-3438	379	1	a	a	DET
ejpam-3438	379	2	∩	∩	ADJ
ejpam-3438	379	3	clβy	clβy	NOUN
ejpam-3438	379	4	(	(	PUNCT
ejpam-3438	379	5	b	b	NOUN
ejpam-3438	379	6	)	)	PUNCT
ejpam-3438	379	7	.	.	PUNCT
ejpam-3438	380	1	this	this	PRON
ejpam-3438	380	2	shows	show	VERB
ejpam-3438	380	3	that	that	SCONJ
ejpam-3438	380	4	a	a	PRON
ejpam-3438	380	5	and	and	CCONJ
ejpam-3438	380	6	b	b	NOUN
ejpam-3438	380	7	are	be	AUX
ejpam-3438	380	8	βiy	βiy	NOUN
ejpam-3438	380	9	-separated	-separate	VERB
ejpam-3438	380	10	.	.	PUNCT
ejpam-3438	381	1	theorem	theorem	VERB
ejpam-3438	381	2	15	15	NUM
ejpam-3438	381	3	.	.	PUNCT
ejpam-3438	382	1	an	an	DET
ejpam-3438	382	2	ideal	ideal	ADJ
ejpam-3438	382	3	topological	topological	ADJ
ejpam-3438	382	4	space	space	NOUN
ejpam-3438	382	5	(	(	PUNCT
ejpam-3438	382	6	x	x	X
ejpam-3438	382	7	,	,	PUNCT
ejpam-3438	382	8	τ	τ	PROPN
ejpam-3438	382	9	,	,	PUNCT
ejpam-3438	382	10	i	i	PROPN
ejpam-3438	382	11	)	)	PUNCT
ejpam-3438	382	12	is	be	AUX
ejpam-3438	382	13	a	a	DET
ejpam-3438	382	14	βi	βi	ADV
ejpam-3438	382	15	-	-	PUNCT
ejpam-3438	382	16	connected	connect	VERB
ejpam-3438	382	17	if	if	SCONJ
ejpam-3438	382	18	and	and	CCONJ
ejpam-3438	382	19	only	only	ADV
ejpam-3438	382	20	if	if	SCONJ
ejpam-3438	382	21	it	it	PRON
ejpam-3438	382	22	can	can	AUX
ejpam-3438	382	23	not	not	PART
ejpam-3438	382	24	be	be	AUX
ejpam-3438	382	25	written	write	VERB
ejpam-3438	382	26	as	as	ADP
ejpam-3438	382	27	a	a	DET
ejpam-3438	382	28	disjoint	disjoint	NOUN
ejpam-3438	382	29	union	union	NOUN
ejpam-3438	382	30	of	of	ADP
ejpam-3438	382	31	a	a	DET
ejpam-3438	382	32	non	non	ADJ
ejpam-3438	382	33	-	-	ADJ
ejpam-3438	382	34	empty	empty	ADJ
ejpam-3438	382	35	β	β	ADJ
ejpam-3438	382	36	-	-	ADJ
ejpam-3438	382	37	open	open	ADJ
ejpam-3438	382	38	set	set	NOUN
ejpam-3438	382	39	and	and	CCONJ
ejpam-3438	382	40	a	a	DET
ejpam-3438	382	41	βi	βi	NOUN
ejpam-3438	382	42	-	-	ADJ
ejpam-3438	382	43	open	open	ADJ
ejpam-3438	382	44	set	set	NOUN
ejpam-3438	382	45	.	.	PUNCT
ejpam-3438	383	1	proof	proof	NOUN
ejpam-3438	383	2	.	.	PUNCT
ejpam-3438	384	1	suppose	suppose	VERB
ejpam-3438	384	2	that	that	SCONJ
ejpam-3438	384	3	(	(	PUNCT
ejpam-3438	384	4	x	x	X
ejpam-3438	384	5	,	,	PUNCT
ejpam-3438	384	6	τ	τ	PROPN
ejpam-3438	384	7	,	,	PUNCT
ejpam-3438	384	8	i	i	PROPN
ejpam-3438	384	9	)	)	PUNCT
ejpam-3438	384	10	is	be	AUX
ejpam-3438	384	11	a	a	DET
ejpam-3438	384	12	βi	βi	ADV
ejpam-3438	384	13	-connected	-connected	ADJ
ejpam-3438	384	14	space	space	NOUN
ejpam-3438	384	15	and	and	CCONJ
ejpam-3438	384	16	x	x	PRON
ejpam-3438	384	17	can	can	AUX
ejpam-3438	384	18	be	be	AUX
ejpam-3438	384	19	written	write	VERB
ejpam-3438	384	20	as	as	ADP
ejpam-3438	384	21	a	a	DET
ejpam-3438	384	22	disjoint	disjoint	NOUN
ejpam-3438	384	23	union	union	NOUN
ejpam-3438	384	24	of	of	ADP
ejpam-3438	384	25	a	a	DET
ejpam-3438	384	26	non	non	ADJ
ejpam-3438	384	27	-	-	ADJ
ejpam-3438	384	28	empty	empty	ADJ
ejpam-3438	384	29	β	β	ADJ
ejpam-3438	384	30	-	-	ADJ
ejpam-3438	384	31	open	open	ADJ
ejpam-3438	384	32	set	set	NOUN
ejpam-3438	384	33	and	and	CCONJ
ejpam-3438	384	34	a	a	DET
ejpam-3438	384	35	βi	βi	NOUN
ejpam-3438	384	36	-open	-open	NOUN
ejpam-3438	384	37	set	set	NOUN
ejpam-3438	384	38	.	.	PUNCT
ejpam-3438	385	1	let	let	VERB
ejpam-3438	385	2	a	a	PRON
ejpam-3438	385	3	be	be	AUX
ejpam-3438	385	4	a	a	DET
ejpam-3438	385	5	non	non	ADJ
ejpam-3438	385	6	-	-	ADJ
ejpam-3438	385	7	empty	empty	ADJ
ejpam-3438	385	8	β	β	ADJ
ejpam-3438	385	9	-	-	ADJ
ejpam-3438	385	10	open	open	ADJ
ejpam-3438	385	11	set	set	NOUN
ejpam-3438	385	12	and	and	CCONJ
ejpam-3438	385	13	b	b	NOUN
ejpam-3438	385	14	be	be	AUX
ejpam-3438	385	15	a	a	DET
ejpam-3438	385	16	βi	βi	NOUN
ejpam-3438	385	17	-open	-open	NOUN
ejpam-3438	385	18	set	set	VERB
ejpam-3438	385	19	with	with	ADP
ejpam-3438	385	20	x	x	PROPN
ejpam-3438	385	21	=	=	PUNCT
ejpam-3438	385	22	a	a	PRON
ejpam-3438	385	23	∪b	∪b	X
ejpam-3438	385	24	and	and	CCONJ
ejpam-3438	385	25	a	a	DET
ejpam-3438	385	26	∩b	∩b	NOUN
ejpam-3438	385	27	6=	6=	PUNCT
ejpam-3438	385	28	∅.	∅.	VERB
ejpam-3438	385	29	if	if	SCONJ
ejpam-3438	385	30	x	x	PROPN
ejpam-3438	385	31	=	=	PUNCT
ejpam-3438	385	32	a	a	DET
ejpam-3438	385	33	∪b	∪b	X
ejpam-3438	385	34	and	and	CCONJ
ejpam-3438	385	35	a	a	DET
ejpam-3438	385	36	∩b	∩b	NOUN
ejpam-3438	385	37	6=	6=	NUM
ejpam-3438	385	38	∅	∅	NOUN
ejpam-3438	385	39	,	,	PUNCT
ejpam-3438	385	40	then	then	ADV
ejpam-3438	385	41	ac	ac	PROPN
ejpam-3438	385	42	=	=	PROPN
ejpam-3438	385	43	b	b	PROPN
ejpam-3438	385	44	and	and	CCONJ
ejpam-3438	385	45	bc	bc	PROPN
ejpam-3438	385	46	=	=	NOUN
ejpam-3438	385	47	a.	a.	NOUN
ejpam-3438	385	48	since	since	SCONJ
ejpam-3438	385	49	ac	ac	PROPN
ejpam-3438	385	50	=	=	PROPN
ejpam-3438	385	51	b	b	PROPN
ejpam-3438	385	52	and	and	CCONJ
ejpam-3438	385	53	bc	bc	PROPN
ejpam-3438	385	54	=	=	SYM
ejpam-3438	385	55	a	a	PROPN
ejpam-3438	385	56	,	,	PUNCT
ejpam-3438	385	57	a	a	PRON
ejpam-3438	385	58	is	be	AUX
ejpam-3438	385	59	a	a	DET
ejpam-3438	385	60	βi	βi	ADV
ejpam-3438	385	61	-closed	-close	VERB
ejpam-3438	385	62	set	set	NOUN
ejpam-3438	385	63	and	and	CCONJ
ejpam-3438	385	64	b	b	NOUN
ejpam-3438	385	65	be	be	AUX
ejpam-3438	385	66	a	a	DET
ejpam-3438	385	67	β	β	NOUN
ejpam-3438	385	68	-	-	ADJ
ejpam-3438	385	69	closed	closed	ADJ
ejpam-3438	385	70	set	set	NOUN
ejpam-3438	385	71	.	.	PUNCT
ejpam-3438	386	1	thus	thus	ADV
ejpam-3438	386	2	,	,	PUNCT
ejpam-3438	386	3	clβi	clβi	NOUN
ejpam-3438	386	4	(	(	PUNCT
ejpam-3438	386	5	a	a	X
ejpam-3438	386	6	)	)	PUNCT
ejpam-3438	386	7	∩b	∩b	NOUN
ejpam-3438	386	8	=	=	PUNCT
ejpam-3438	386	9	a	a	DET
ejpam-3438	386	10	∩b	∩b	NOUN
ejpam-3438	386	11	=	=	NOUN
ejpam-3438	386	12	∅	∅	NOUN
ejpam-3438	386	13	and	and	CCONJ
ejpam-3438	386	14	a	a	DET
ejpam-3438	386	15	∩	∩	ADJ
ejpam-3438	386	16	clβ(b	clβ(b	NOUN
ejpam-3438	386	17	)	)	PUNCT
ejpam-3438	386	18	=	=	SYM
ejpam-3438	386	19	a	a	DET
ejpam-3438	386	20	∩b	∩b	NOUN
ejpam-3438	386	21	=	=	X
ejpam-3438	386	22	∅.	∅.	NOUN
ejpam-3438	386	23	thus	thus	ADV
ejpam-3438	386	24	,	,	PUNCT
ejpam-3438	386	25	a	a	PRON
ejpam-3438	386	26	and	and	CCONJ
ejpam-3438	386	27	b	b	NOUN
ejpam-3438	386	28	are	be	AUX
ejpam-3438	386	29	βi	βi	ADP
ejpam-3438	386	30	-separated	-separate	VERB
ejpam-3438	386	31	sets	set	NOUN
ejpam-3438	386	32	.	.	PUNCT
ejpam-3438	387	1	this	this	PRON
ejpam-3438	387	2	is	be	AUX
ejpam-3438	387	3	a	a	DET
ejpam-3438	387	4	contradiction	contradiction	NOUN
ejpam-3438	387	5	since	since	SCONJ
ejpam-3438	387	6	x	x	PRON
ejpam-3438	387	7	is	be	AUX
ejpam-3438	387	8	a	a	DET
ejpam-3438	387	9	βi	βi	ADV
ejpam-3438	387	10	-connected	-connected	ADJ
ejpam-3438	387	11	space	space	NOUN
ejpam-3438	387	12	.	.	PUNCT
ejpam-3438	388	1	conversely	conversely	ADV
ejpam-3438	388	2	,	,	PUNCT
ejpam-3438	388	3	assume	assume	VERB
ejpam-3438	388	4	that	that	SCONJ
ejpam-3438	388	5	x	x	PRON
ejpam-3438	388	6	can	can	AUX
ejpam-3438	388	7	not	not	PART
ejpam-3438	388	8	be	be	AUX
ejpam-3438	388	9	written	write	VERB
ejpam-3438	388	10	as	as	ADP
ejpam-3438	388	11	a	a	DET
ejpam-3438	388	12	disjoint	disjoint	NOUN
ejpam-3438	388	13	union	union	NOUN
ejpam-3438	388	14	of	of	ADP
ejpam-3438	388	15	a	a	DET
ejpam-3438	388	16	non	non	ADJ
ejpam-3438	388	17	-	-	ADJ
ejpam-3438	388	18	empty	empty	ADJ
ejpam-3438	388	19	βopen	βopen	ADJ
ejpam-3438	388	20	set	set	NOUN
ejpam-3438	388	21	and	and	CCONJ
ejpam-3438	388	22	a	a	DET
ejpam-3438	388	23	βi	βi	NOUN
ejpam-3438	388	24	-open	-open	NOUN
ejpam-3438	388	25	set	set	NOUN
ejpam-3438	388	26	.	.	PUNCT
ejpam-3438	389	1	if	if	SCONJ
ejpam-3438	389	2	(	(	PUNCT
ejpam-3438	389	3	x	x	X
ejpam-3438	389	4	,	,	PUNCT
ejpam-3438	389	5	τ	τ	PROPN
ejpam-3438	389	6	,	,	PUNCT
ejpam-3438	389	7	i	i	PROPN
ejpam-3438	389	8	)	)	PUNCT
ejpam-3438	389	9	is	be	AUX
ejpam-3438	389	10	not	not	PART
ejpam-3438	389	11	βi	βi	PRON
ejpam-3438	389	12	-connected	-connected	ADJ
ejpam-3438	389	13	,	,	PUNCT
ejpam-3438	389	14	then	then	ADV
ejpam-3438	389	15	x	x	PRON
ejpam-3438	389	16	can	can	AUX
ejpam-3438	389	17	be	be	AUX
ejpam-3438	389	18	written	write	VERB
ejpam-3438	389	19	as	as	ADP
ejpam-3438	389	20	a	a	DET
ejpam-3438	389	21	union	union	NOUN
ejpam-3438	389	22	of	of	ADP
ejpam-3438	389	23	two	two	NUM
ejpam-3438	389	24	βi	βi	ADV
ejpam-3438	389	25	-separated	-separate	VERB
ejpam-3438	389	26	sets	set	NOUN
ejpam-3438	389	27	,	,	PUNCT
ejpam-3438	389	28	say	say	VERB
ejpam-3438	389	29	a	a	PRON
ejpam-3438	389	30	and	and	CCONJ
ejpam-3438	389	31	b	b	NOUN
ejpam-3438	389	32	,	,	PUNCT
ejpam-3438	389	33	with	with	ADP
ejpam-3438	389	34	x	x	X
ejpam-3438	389	35	=	=	PUNCT
ejpam-3438	389	36	a	a	DET
ejpam-3438	389	37	∪	∪	X
ejpam-3438	389	38	b.	b.	PROPN
ejpam-3438	389	39	thus	thus	ADV
ejpam-3438	389	40	,	,	PUNCT
ejpam-3438	389	41	clβi	clβi	NOUN
ejpam-3438	389	42	(	(	PUNCT
ejpam-3438	389	43	a	a	NOUN
ejpam-3438	389	44	)	)	PUNCT
ejpam-3438	389	45	∩	∩	ADJ
ejpam-3438	389	46	b	b	NOUN
ejpam-3438	389	47	=	=	SYM
ejpam-3438	389	48	∅	∅	NOUN
ejpam-3438	389	49	and	and	CCONJ
ejpam-3438	389	50	a	a	DET
ejpam-3438	389	51	∩	∩	ADJ
ejpam-3438	389	52	clβ(b	clβ(b	NOUN
ejpam-3438	389	53	)	)	PUNCT
ejpam-3438	389	54	=	=	NOUN
ejpam-3438	389	55	∅	∅	NOUN
ejpam-3438	389	56	,	,	PUNCT
ejpam-3438	389	57	that	that	PRON
ejpam-3438	389	58	is	be	AUX
ejpam-3438	389	59	clβi	clβi	NOUN
ejpam-3438	389	60	(	(	PUNCT
ejpam-3438	389	61	a	a	X
ejpam-3438	389	62	)	)	PUNCT
ejpam-3438	389	63	=	=	SYM
ejpam-3438	389	64	bc	bc	PROPN
ejpam-3438	389	65	and	and	CCONJ
ejpam-3438	389	66	clβ(b	clβ(b	PROPN
ejpam-3438	389	67	)	)	PUNCT
ejpam-3438	390	1	=	=	SYM
ejpam-3438	390	2	ac	ac	PROPN
ejpam-3438	390	3	.	.	PUNCT
ejpam-3438	391	1	this	this	PRON
ejpam-3438	391	2	implies	imply	VERB
ejpam-3438	391	3	that	that	SCONJ
ejpam-3438	391	4	a	a	PRON
ejpam-3438	391	5	is	be	AUX
ejpam-3438	391	6	a	a	DET
ejpam-3438	391	7	non	non	ADJ
ejpam-3438	391	8	-	-	ADJ
ejpam-3438	391	9	empty	empty	ADJ
ejpam-3438	391	10	β	β	ADJ
ejpam-3438	391	11	-	-	ADJ
ejpam-3438	391	12	open	open	ADJ
ejpam-3438	391	13	set	set	NOUN
ejpam-3438	391	14	and	and	CCONJ
ejpam-3438	391	15	b	b	NOUN
ejpam-3438	391	16	is	be	AUX
ejpam-3438	391	17	a	a	DET
ejpam-3438	391	18	βi	βi	X
ejpam-3438	391	19	-open	-open	NOUN
ejpam-3438	391	20	set	set	NOUN
ejpam-3438	391	21	.	.	PUNCT
ejpam-3438	392	1	this	this	PRON
ejpam-3438	392	2	is	be	AUX
ejpam-3438	392	3	a	a	DET
ejpam-3438	392	4	contradiction	contradiction	NOUN
ejpam-3438	392	5	since	since	SCONJ
ejpam-3438	392	6	x	x	PRON
ejpam-3438	392	7	can	can	AUX
ejpam-3438	392	8	not	not	PART
ejpam-3438	392	9	be	be	AUX
ejpam-3438	392	10	written	write	VERB
ejpam-3438	392	11	as	as	ADP
ejpam-3438	392	12	a	a	DET
ejpam-3438	392	13	disjoint	disjoint	NOUN
ejpam-3438	392	14	union	union	NOUN
ejpam-3438	392	15	of	of	ADP
ejpam-3438	392	16	a	a	DET
ejpam-3438	392	17	non	non	ADJ
ejpam-3438	392	18	-	-	ADJ
ejpam-3438	392	19	empty	empty	ADJ
ejpam-3438	392	20	β	β	ADJ
ejpam-3438	392	21	-	-	ADJ
ejpam-3438	392	22	open	open	ADJ
ejpam-3438	392	23	set	set	NOUN
ejpam-3438	392	24	and	and	CCONJ
ejpam-3438	392	25	a	a	DET
ejpam-3438	392	26	βi	βi	NOUN
ejpam-3438	392	27	-open	-open	NOUN
ejpam-3438	392	28	set	set	NOUN
ejpam-3438	392	29	.	.	PUNCT
ejpam-3438	393	1	theorem	theorem	VERB
ejpam-3438	393	2	16	16	NUM
ejpam-3438	393	3	.	.	PUNCT
ejpam-3438	394	1	let	let	VERB
ejpam-3438	394	2	(	(	PUNCT
ejpam-3438	394	3	x	x	X
ejpam-3438	394	4	,	,	PUNCT
ejpam-3438	394	5	τ	τ	PROPN
ejpam-3438	394	6	,	,	PUNCT
ejpam-3438	394	7	i	i	PRON
ejpam-3438	394	8	)	)	PUNCT
ejpam-3438	394	9	be	be	VERB
ejpam-3438	394	10	an	an	DET
ejpam-3438	394	11	ideal	ideal	ADJ
ejpam-3438	394	12	topological	topological	ADJ
ejpam-3438	394	13	space	space	NOUN
ejpam-3438	394	14	and	and	CCONJ
ejpam-3438	394	15	a	a	DET
ejpam-3438	394	16	be	be	AUX
ejpam-3438	394	17	an	an	DET
ejpam-3438	394	18	open	open	ADJ
ejpam-3438	394	19	set	set	NOUN
ejpam-3438	394	20	.	.	PUNCT
ejpam-3438	395	1	if	if	SCONJ
ejpam-3438	395	2	a	a	PRON
ejpam-3438	395	3	is	be	AUX
ejpam-3438	395	4	βi	βi	PRON
ejpam-3438	395	5	-	-	PUNCT
ejpam-3438	395	6	connected	connect	VERB
ejpam-3438	395	7	,	,	PUNCT
ejpam-3438	395	8	and	and	CCONJ
ejpam-3438	395	9	h	h	NOUN
ejpam-3438	395	10	and	and	CCONJ
ejpam-3438	395	11	g	g	PROPN
ejpam-3438	395	12	are	be	AUX
ejpam-3438	395	13	βi	βi	PRON
ejpam-3438	395	14	-	-	PUNCT
ejpam-3438	395	15	separated	separate	VERB
ejpam-3438	395	16	with	with	ADP
ejpam-3438	395	17	a	a	DET
ejpam-3438	395	18	⊆	⊆	NUM
ejpam-3438	395	19	h	h	NOUN
ejpam-3438	395	20	∪	∪	NOUN
ejpam-3438	395	21	g	g	NOUN
ejpam-3438	395	22	,	,	PUNCT
ejpam-3438	395	23	then	then	ADV
ejpam-3438	395	24	either	either	CCONJ
ejpam-3438	395	25	a	a	DET
ejpam-3438	395	26	⊆	⊆	NUM
ejpam-3438	395	27	h	h	NOUN
ejpam-3438	395	28	or	or	CCONJ
ejpam-3438	395	29	a	a	DET
ejpam-3438	395	30	⊆	⊆	NUM
ejpam-3438	395	31	g.	g.	NOUN
ejpam-3438	395	32	references	reference	NOUN
ejpam-3438	395	33	903	903	NUM
ejpam-3438	395	34	proof	proof	NOUN
ejpam-3438	395	35	.	.	PUNCT
ejpam-3438	396	1	suppose	suppose	VERB
ejpam-3438	396	2	that	that	SCONJ
ejpam-3438	396	3	a	a	DET
ejpam-3438	396	4	∩h	∩h	PROPN
ejpam-3438	396	5	6=	6=	ADP
ejpam-3438	396	6	∅	∅	NOUN
ejpam-3438	396	7	and	and	CCONJ
ejpam-3438	396	8	a	a	DET
ejpam-3438	396	9	∩	∩	ADJ
ejpam-3438	396	10	g	g	PROPN
ejpam-3438	396	11	6=	6=	X
ejpam-3438	396	12	∅.	∅.	NOUN
ejpam-3438	396	13	since	since	SCONJ
ejpam-3438	396	14	a	a	DET
ejpam-3438	396	15	⊆	⊆	NUM
ejpam-3438	396	16	h	h	NOUN
ejpam-3438	396	17	∪	∪	ADP
ejpam-3438	396	18	g	g	NOUN
ejpam-3438	396	19	,	,	PUNCT
ejpam-3438	396	20	a	a	PRON
ejpam-3438	396	21	=	=	X
ejpam-3438	396	22	(	(	PUNCT
ejpam-3438	396	23	a	a	DET
ejpam-3438	396	24	∩h	∩h	NOUN
ejpam-3438	396	25	)	)	PUNCT
ejpam-3438	396	26	∪	∪	NOUN
ejpam-3438	396	27	(	(	PUNCT
ejpam-3438	396	28	a	a	DET
ejpam-3438	396	29	∩	∩	ADJ
ejpam-3438	396	30	g	g	NOUN
ejpam-3438	396	31	)	)	PUNCT
ejpam-3438	396	32	.	.	PUNCT
ejpam-3438	397	1	since	since	SCONJ
ejpam-3438	397	2	h	h	PROPN
ejpam-3438	397	3	and	and	CCONJ
ejpam-3438	397	4	g	g	PROPN
ejpam-3438	397	5	are	be	AUX
ejpam-3438	397	6	βi	βi	PRON
ejpam-3438	397	7	-separated	-separated	ADJ
ejpam-3438	397	8	,	,	PUNCT
ejpam-3438	397	9	clβi	clβi	NOUN
ejpam-3438	397	10	(	(	PUNCT
ejpam-3438	397	11	a	a	DET
ejpam-3438	397	12	∩	∩	ADJ
ejpam-3438	397	13	h	h	NOUN
ejpam-3438	397	14	)	)	PUNCT
ejpam-3438	397	15	∩	∩	NOUN
ejpam-3438	397	16	(	(	PUNCT
ejpam-3438	397	17	a	a	DET
ejpam-3438	397	18	∩	∩	ADJ
ejpam-3438	397	19	g	g	NOUN
ejpam-3438	397	20	)	)	PUNCT
ejpam-3438	397	21	=	=	NOUN
ejpam-3438	397	22	clβi	clβi	NOUN
ejpam-3438	397	23	(	(	PUNCT
ejpam-3438	397	24	h	h	NOUN
ejpam-3438	397	25	)	)	PUNCT
ejpam-3438	397	26	∩	∩	NOUN
ejpam-3438	397	27	(	(	PUNCT
ejpam-3438	397	28	g	g	NOUN
ejpam-3438	397	29	)	)	PUNCT
ejpam-3438	397	30	=	=	NOUN
ejpam-3438	397	31	∅	∅	NOUN
ejpam-3438	397	32	and	and	CCONJ
ejpam-3438	397	33	(	(	PUNCT
ejpam-3438	397	34	a	a	DET
ejpam-3438	397	35	∩	∩	ADJ
ejpam-3438	397	36	h	h	NOUN
ejpam-3438	397	37	)	)	PUNCT
ejpam-3438	397	38	∩	∩	NOUN
ejpam-3438	397	39	clβ(a	clβ(a	PROPN
ejpam-3438	397	40	∩	∩	ADJ
ejpam-3438	397	41	g	g	NOUN
ejpam-3438	397	42	)	)	PUNCT
ejpam-3438	398	1	=	=	SYM
ejpam-3438	398	2	h	h	NOUN
ejpam-3438	398	3	∩	∩	NOUN
ejpam-3438	398	4	clβ(g	clβ(g	PROPN
ejpam-3438	398	5	)	)	PUNCT
ejpam-3438	398	6	=	=	PUNCT
ejpam-3438	398	7	∅.	∅.	ADP
ejpam-3438	398	8	thus	thus	ADV
ejpam-3438	398	9	,	,	PUNCT
ejpam-3438	398	10	[	[	X
ejpam-3438	398	11	clβi	clβi	NOUN
ejpam-3438	398	12	(	(	PUNCT
ejpam-3438	398	13	a	a	DET
ejpam-3438	398	14	∩h	∩h	NOUN
ejpam-3438	398	15	)	)	PUNCT
ejpam-3438	398	16	∩a	∩a	PROPN
ejpam-3438	398	17	]	]	PUNCT
ejpam-3438	398	18	∩	∩	NOUN
ejpam-3438	398	19	(	(	PUNCT
ejpam-3438	398	20	a	a	DET
ejpam-3438	398	21	∩	∩	ADJ
ejpam-3438	398	22	g	g	NOUN
ejpam-3438	398	23	)	)	PUNCT
ejpam-3438	398	24	=	=	NOUN
ejpam-3438	398	25	∅	∅	NOUN
ejpam-3438	398	26	and	and	CCONJ
ejpam-3438	398	27	(	(	PUNCT
ejpam-3438	398	28	a	a	DET
ejpam-3438	398	29	∩h	∩h	NOUN
ejpam-3438	398	30	)	)	PUNCT
ejpam-3438	398	31	∩	∩	NOUN
ejpam-3438	399	1	[	[	X
ejpam-3438	399	2	clβ(a	clβ(a	PROPN
ejpam-3438	399	3	∩g	∩g	PROPN
ejpam-3438	399	4	)	)	PUNCT
ejpam-3438	400	1	∩a	∩a	PROPN
ejpam-3438	400	2	]	]	X
ejpam-3438	401	1	=	=	PUNCT
ejpam-3438	401	2	∅.	∅.	X
ejpam-3438	401	3	by	by	ADP
ejpam-3438	401	4	theorem	theorem	ADJ
ejpam-3438	401	5	13	13	NUM
ejpam-3438	401	6	,	,	PUNCT
ejpam-3438	401	7	clβia	clβia	NOUN
ejpam-3438	401	8	(	(	PUNCT
ejpam-3438	401	9	a	a	DET
ejpam-3438	401	10	∩h	∩h	NOUN
ejpam-3438	401	11	)	)	PUNCT
ejpam-3438	401	12	∩	∩	NOUN
ejpam-3438	401	13	(	(	PUNCT
ejpam-3438	401	14	a	a	DET
ejpam-3438	401	15	∩	∩	ADJ
ejpam-3438	401	16	g	g	NOUN
ejpam-3438	401	17	)	)	PUNCT
ejpam-3438	401	18	=	=	NOUN
ejpam-3438	401	19	∅	∅	NOUN
ejpam-3438	401	20	and	and	CCONJ
ejpam-3438	401	21	(	(	PUNCT
ejpam-3438	401	22	a∩h)∩clβa(a∩g	a∩h)∩clβa(a∩g	NOUN
ejpam-3438	401	23	)	)	PUNCT
ejpam-3438	401	24	=	=	PUNCT
ejpam-3438	401	25	∅.	∅.	ADP
ejpam-3438	401	26	this	this	PRON
ejpam-3438	401	27	implies	imply	VERB
ejpam-3438	401	28	that	that	SCONJ
ejpam-3438	401	29	a	a	PRON
ejpam-3438	401	30	is	be	AUX
ejpam-3438	401	31	not	not	PART
ejpam-3438	401	32	βi	βi	PRON
ejpam-3438	401	33	-connected	-connected	ADJ
ejpam-3438	401	34	.	.	PUNCT
ejpam-3438	402	1	this	this	PRON
ejpam-3438	402	2	is	be	AUX
ejpam-3438	402	3	a	a	DET
ejpam-3438	402	4	contradiction	contradiction	NOUN
ejpam-3438	402	5	.	.	PUNCT
ejpam-3438	403	1	therefore	therefore	ADV
ejpam-3438	403	2	,	,	PUNCT
ejpam-3438	403	3	either	either	CCONJ
ejpam-3438	403	4	a	a	DET
ejpam-3438	403	5	∩h	∩h	NOUN
ejpam-3438	403	6	=	=	SYM
ejpam-3438	403	7	∅	∅	NOUN
ejpam-3438	403	8	or	or	CCONJ
ejpam-3438	403	9	a	a	DET
ejpam-3438	403	10	∩g	∩g	ADJ
ejpam-3438	403	11	=	=	SYM
ejpam-3438	403	12	∅	∅	NOUN
ejpam-3438	403	13	,	,	PUNCT
ejpam-3438	403	14	that	that	PRON
ejpam-3438	403	15	is	be	AUX
ejpam-3438	403	16	h	h	PRON
ejpam-3438	403	17	⊆	⊆	NUM
ejpam-3438	403	18	a	a	DET
ejpam-3438	403	19	or	or	CCONJ
ejpam-3438	403	20	g	g	NOUN
ejpam-3438	403	21	⊆	⊆	NUM
ejpam-3438	403	22	a.	a.	NOUN
ejpam-3438	403	23	theorem	theorem	NOUN
ejpam-3438	403	24	17	17	NUM
ejpam-3438	403	25	.	.	PUNCT
ejpam-3438	404	1	let	let	VERB
ejpam-3438	404	2	(	(	PUNCT
ejpam-3438	404	3	x	x	X
ejpam-3438	404	4	,	,	PUNCT
ejpam-3438	404	5	τ	τ	PROPN
ejpam-3438	404	6	,	,	PUNCT
ejpam-3438	404	7	i	i	PRON
ejpam-3438	404	8	)	)	PUNCT
ejpam-3438	404	9	be	be	VERB
ejpam-3438	404	10	an	an	DET
ejpam-3438	404	11	ideal	ideal	ADJ
ejpam-3438	404	12	topological	topological	ADJ
ejpam-3438	404	13	space	space	NOUN
ejpam-3438	404	14	and	and	CCONJ
ejpam-3438	404	15	,	,	PUNCT
ejpam-3438	404	16	a	a	PRON
ejpam-3438	404	17	and	and	CCONJ
ejpam-3438	404	18	b	b	NOUN
ejpam-3438	405	1	be	be	AUX
ejpam-3438	405	2	βi	βi	ADV
ejpam-3438	405	3	-	-	PUNCT
ejpam-3438	405	4	separated	separate	VERB
ejpam-3438	405	5	subsets	subset	NOUN
ejpam-3438	405	6	of	of	ADP
ejpam-3438	405	7	x.	x.	NOUN
ejpam-3438	405	8	if	if	SCONJ
ejpam-3438	405	9	c	c	PROPN
ejpam-3438	405	10	and	and	CCONJ
ejpam-3438	405	11	d	d	PROPN
ejpam-3438	405	12	are	be	AUX
ejpam-3438	405	13	two	two	NUM
ejpam-3438	405	14	non	non	ADJ
ejpam-3438	405	15	-	-	ADJ
ejpam-3438	405	16	empty	empty	ADJ
ejpam-3438	405	17	subsets	subset	NOUN
ejpam-3438	405	18	of	of	ADP
ejpam-3438	405	19	x	x	SYM
ejpam-3438	406	1	such	such	ADJ
ejpam-3438	406	2	that	that	SCONJ
ejpam-3438	406	3	c	c	PROPN
ejpam-3438	406	4	⊆	⊆	NUM
ejpam-3438	406	5	d	d	PROPN
ejpam-3438	406	6	and	and	CCONJ
ejpam-3438	406	7	d	d	PROPN
ejpam-3438	406	8	⊆	⊆	NUM
ejpam-3438	406	9	b	b	NOUN
ejpam-3438	406	10	,	,	PUNCT
ejpam-3438	406	11	then	then	ADV
ejpam-3438	406	12	c	c	PROPN
ejpam-3438	406	13	and	and	CCONJ
ejpam-3438	406	14	d	d	PROPN
ejpam-3438	406	15	are	be	AUX
ejpam-3438	406	16	also	also	ADV
ejpam-3438	406	17	βi	βi	ADV
ejpam-3438	406	18	-	-	PUNCT
ejpam-3438	406	19	separated	separate	VERB
ejpam-3438	406	20	.	.	PUNCT
ejpam-3438	407	1	proof	proof	NOUN
ejpam-3438	407	2	.	.	PUNCT
ejpam-3438	408	1	if	if	SCONJ
ejpam-3438	408	2	a	a	PRON
ejpam-3438	408	3	and	and	CCONJ
ejpam-3438	408	4	b	b	NOUN
ejpam-3438	408	5	are	be	AUX
ejpam-3438	408	6	βi	βi	PRON
ejpam-3438	408	7	-separated	-separated	ADJ
ejpam-3438	408	8	,	,	PUNCT
ejpam-3438	408	9	then	then	ADV
ejpam-3438	408	10	clβi	clβi	NOUN
ejpam-3438	408	11	(	(	PUNCT
ejpam-3438	408	12	a	a	NOUN
ejpam-3438	408	13	)	)	PUNCT
ejpam-3438	408	14	∩	∩	ADJ
ejpam-3438	408	15	b	b	NOUN
ejpam-3438	408	16	=	=	SYM
ejpam-3438	408	17	∅	∅	NOUN
ejpam-3438	408	18	and	and	CCONJ
ejpam-3438	408	19	a	a	DET
ejpam-3438	408	20	∩	∩	ADJ
ejpam-3438	408	21	clβ(b	clβ(b	NOUN
ejpam-3438	408	22	)	)	PUNCT
ejpam-3438	408	23	=	=	PUNCT
ejpam-3438	408	24	∅.	∅.	VERB
ejpam-3438	408	25	hence	hence	ADV
ejpam-3438	408	26	,	,	PUNCT
ejpam-3438	408	27	clβi	clβi	NOUN
ejpam-3438	408	28	(	(	PUNCT
ejpam-3438	408	29	c	c	NOUN
ejpam-3438	408	30	)	)	PUNCT
ejpam-3438	408	31	∩	∩	NOUN
ejpam-3438	408	32	d	d	ADP
ejpam-3438	408	33	⊆	⊆	NUM
ejpam-3438	408	34	clβi	clβi	NOUN
ejpam-3438	408	35	(	(	PUNCT
ejpam-3438	408	36	a	a	NOUN
ejpam-3438	408	37	)	)	PUNCT
ejpam-3438	408	38	∩	∩	ADJ
ejpam-3438	408	39	b	b	NOUN
ejpam-3438	408	40	=	=	SYM
ejpam-3438	408	41	∅	∅	NOUN
ejpam-3438	408	42	and	and	CCONJ
ejpam-3438	408	43	c	c	X
ejpam-3438	408	44	∩	∩	X
ejpam-3438	408	45	clβ(d	clβ(d	PROPN
ejpam-3438	408	46	)	)	PUNCT
ejpam-3438	408	47	=	=	PUNCT
ejpam-3438	408	48	a	a	DET
ejpam-3438	408	49	∩	∩	ADJ
ejpam-3438	408	50	clβ(b	clβ(b	NOUN
ejpam-3438	408	51	)	)	PUNCT
ejpam-3438	408	52	=	=	NOUN
ejpam-3438	408	53	∅	∅	NOUN
ejpam-3438	408	54	,	,	PUNCT
ejpam-3438	408	55	that	that	PRON
ejpam-3438	408	56	is	be	AUX
ejpam-3438	408	57	clβi	clβi	NOUN
ejpam-3438	408	58	(	(	PUNCT
ejpam-3438	408	59	c	c	NOUN
ejpam-3438	408	60	)	)	PUNCT
ejpam-3438	408	61	∩d	∩d	VERB
ejpam-3438	408	62	=	=	NOUN
ejpam-3438	409	1	∅	∅	NOUN
ejpam-3438	409	2	=	=	SYM
ejpam-3438	409	3	c	c	NOUN
ejpam-3438	409	4	∩	∩	X
ejpam-3438	409	5	clβ(d	clβ(d	PROPN
ejpam-3438	409	6	)	)	PUNCT
ejpam-3438	409	7	.	.	PUNCT
ejpam-3438	410	1	this	this	PRON
ejpam-3438	410	2	shows	show	VERB
ejpam-3438	410	3	that	that	SCONJ
ejpam-3438	410	4	c	c	PROPN
ejpam-3438	410	5	and	and	CCONJ
ejpam-3438	410	6	d	d	NOUN
ejpam-3438	410	7	are	be	AUX
ejpam-3438	410	8	βi	βi	PRON
ejpam-3438	410	9	-separated	-separate	VERB
ejpam-3438	410	10	.	.	PUNCT
ejpam-3438	411	1	theorem	theorem	NOUN
ejpam-3438	411	2	18	18	NUM
ejpam-3438	411	3	.	.	PUNCT
ejpam-3438	412	1	if	if	SCONJ
ejpam-3438	412	2	a	a	PRON
ejpam-3438	412	3	is	be	AUX
ejpam-3438	412	4	a	a	DET
ejpam-3438	412	5	βi	βi	ADV
ejpam-3438	412	6	-	-	PUNCT
ejpam-3438	412	7	connected	connect	VERB
ejpam-3438	412	8	subset	subset	NOUN
ejpam-3438	412	9	of	of	ADP
ejpam-3438	412	10	a	a	DET
ejpam-3438	412	11	βi	βi	ADV
ejpam-3438	412	12	-	-	PUNCT
ejpam-3438	412	13	connected	connect	VERB
ejpam-3438	412	14	ideal	ideal	ADJ
ejpam-3438	412	15	topological	topological	ADJ
ejpam-3438	412	16	space	space	NOUN
ejpam-3438	412	17	(	(	PUNCT
ejpam-3438	412	18	x	x	X
ejpam-3438	412	19	,	,	PUNCT
ejpam-3438	412	20	τ	τ	PROPN
ejpam-3438	412	21	,	,	PUNCT
ejpam-3438	412	22	i	i	NOUN
ejpam-3438	412	23	)	)	PUNCT
ejpam-3438	412	24	such	such	ADJ
ejpam-3438	412	25	that	that	SCONJ
ejpam-3438	412	26	x	x	SYM
ejpam-3438	412	27	−a	−a	NOUN
ejpam-3438	412	28	is	be	AUX
ejpam-3438	412	29	the	the	DET
ejpam-3438	412	30	union	union	NOUN
ejpam-3438	412	31	of	of	ADP
ejpam-3438	412	32	two	two	NUM
ejpam-3438	412	33	βi	βi	ADV
ejpam-3438	412	34	-	-	PUNCT
ejpam-3438	412	35	separated	separate	VERB
ejpam-3438	412	36	sets	set	NOUN
ejpam-3438	412	37	b	b	PROPN
ejpam-3438	412	38	and	and	CCONJ
ejpam-3438	412	39	c	c	NOUN
ejpam-3438	412	40	,	,	PUNCT
ejpam-3438	412	41	then	then	ADV
ejpam-3438	412	42	a	a	DET
ejpam-3438	412	43	∪b	∪b	X
ejpam-3438	412	44	and	and	CCONJ
ejpam-3438	412	45	a	a	DET
ejpam-3438	412	46	∪	∪	NOUN
ejpam-3438	412	47	c	c	NOUN
ejpam-3438	412	48	are	be	AUX
ejpam-3438	412	49	βi	βi	ADV
ejpam-3438	412	50	-	-	PUNCT
ejpam-3438	412	51	connected	connect	VERB
ejpam-3438	412	52	.	.	PUNCT
ejpam-3438	413	1	theorem	theorem	VERB
ejpam-3438	413	2	19	19	NUM
ejpam-3438	413	3	.	.	PUNCT
ejpam-3438	414	1	the	the	DET
ejpam-3438	414	2	continuous	continuous	ADJ
ejpam-3438	414	3	image	image	NOUN
ejpam-3438	414	4	a	a	DET
ejpam-3438	414	5	βi	βi	ADV
ejpam-3438	414	6	-	-	PUNCT
ejpam-3438	414	7	connected	connect	VERB
ejpam-3438	414	8	space	space	NOUN
ejpam-3438	414	9	is	be	AUX
ejpam-3438	414	10	connected	connect	VERB
ejpam-3438	414	11	.	.	PUNCT
ejpam-3438	415	1	theorem	theorem	ADJ
ejpam-3438	415	2	20	20	NUM
ejpam-3438	415	3	.	.	PUNCT
ejpam-3438	416	1	let	let	VERB
ejpam-3438	416	2	(	(	PUNCT
ejpam-3438	416	3	x	x	X
ejpam-3438	416	4	,	,	PUNCT
ejpam-3438	416	5	τ	τ	PROPN
ejpam-3438	416	6	,	,	PUNCT
ejpam-3438	416	7	i	i	PRON
ejpam-3438	416	8	)	)	PUNCT
ejpam-3438	416	9	be	be	VERB
ejpam-3438	416	10	an	an	DET
ejpam-3438	416	11	ideal	ideal	ADJ
ejpam-3438	416	12	topological	topological	ADJ
ejpam-3438	416	13	space	space	NOUN
ejpam-3438	416	14	.	.	PUNCT
ejpam-3438	417	1	if	if	SCONJ
ejpam-3438	417	2	the	the	DET
ejpam-3438	417	3	union	union	NOUN
ejpam-3438	417	4	of	of	ADP
ejpam-3438	417	5	two	two	NUM
ejpam-3438	417	6	βi	βi	ADV
ejpam-3438	417	7	-	-	PUNCT
ejpam-3438	417	8	separated	separate	VERB
ejpam-3438	417	9	sets	set	NOUN
ejpam-3438	417	10	is	be	AUX
ejpam-3438	417	11	a	a	DET
ejpam-3438	417	12	β	β	NOUN
ejpam-3438	417	13	-	-	ADJ
ejpam-3438	417	14	closed	closed	ADJ
ejpam-3438	417	15	set	set	NOUN
ejpam-3438	417	16	,	,	PUNCT
ejpam-3438	417	17	then	then	ADV
ejpam-3438	417	18	one	one	NUM
ejpam-3438	417	19	of	of	ADP
ejpam-3438	417	20	the	the	DET
ejpam-3438	417	21	sets	set	NOUN
ejpam-3438	417	22	is	be	AUX
ejpam-3438	417	23	β	β	NOUN
ejpam-3438	417	24	-	-	VERB
ejpam-3438	417	25	closed	closed	ADJ
ejpam-3438	417	26	and	and	CCONJ
ejpam-3438	417	27	the	the	DET
ejpam-3438	417	28	other	other	ADJ
ejpam-3438	417	29	is	be	AUX
ejpam-3438	417	30	βi	βi	NOUN
ejpam-3438	417	31	-	-	PUNCT
ejpam-3438	417	32	closed	closed	ADJ
ejpam-3438	417	33	.	.	PUNCT
ejpam-3438	418	1	proof	proof	NOUN
ejpam-3438	418	2	.	.	PUNCT
ejpam-3438	419	1	let	let	VERB
ejpam-3438	419	2	a	a	PRON
ejpam-3438	419	3	and	and	CCONJ
ejpam-3438	419	4	b	b	NOUN
ejpam-3438	419	5	be	be	VERB
ejpam-3438	419	6	βi	βi	PRON
ejpam-3438	419	7	-separated	-separate	VERB
ejpam-3438	419	8	such	such	ADJ
ejpam-3438	419	9	that	that	SCONJ
ejpam-3438	419	10	a	a	DET
ejpam-3438	419	11	∪	∪	NOUN
ejpam-3438	419	12	b	b	NOUN
ejpam-3438	419	13	is	be	AUX
ejpam-3438	419	14	β	β	NOUN
ejpam-3438	419	15	-	-	VERB
ejpam-3438	419	16	closed	closed	ADJ
ejpam-3438	419	17	.	.	PUNCT
ejpam-3438	420	1	if	if	SCONJ
ejpam-3438	420	2	a	a	PRON
ejpam-3438	420	3	and	and	CCONJ
ejpam-3438	420	4	b	b	NOUN
ejpam-3438	420	5	is	be	AUX
ejpam-3438	420	6	βi	βi	PRON
ejpam-3438	420	7	separated	separate	VERB
ejpam-3438	420	8	,	,	PUNCT
ejpam-3438	420	9	then	then	ADV
ejpam-3438	420	10	clβi	clβi	NOUN
ejpam-3438	420	11	(	(	PUNCT
ejpam-3438	420	12	a	a	X
ejpam-3438	420	13	)	)	PUNCT
ejpam-3438	420	14	∩b	∩b	NOUN
ejpam-3438	420	15	=	=	NOUN
ejpam-3438	420	16	∅	∅	NOUN
ejpam-3438	420	17	=	=	PUNCT
ejpam-3438	420	18	a	a	DET
ejpam-3438	420	19	∩	∩	ADJ
ejpam-3438	420	20	clβ(b	clβ(b	NOUN
ejpam-3438	420	21	)	)	PUNCT
ejpam-3438	421	1	=	=	PUNCT
ejpam-3438	421	2	∅.	∅.	VERB
ejpam-3438	421	3	moreover	moreover	ADV
ejpam-3438	421	4	,	,	PUNCT
ejpam-3438	421	5	if	if	SCONJ
ejpam-3438	421	6	a	a	DET
ejpam-3438	421	7	∪b	∪b	PRON
ejpam-3438	421	8	is	be	AUX
ejpam-3438	421	9	β	β	NOUN
ejpam-3438	421	10	-	-	VERB
ejpam-3438	421	11	closed	closed	ADJ
ejpam-3438	421	12	,	,	PUNCT
ejpam-3438	421	13	then	then	ADV
ejpam-3438	421	14	clβ(a∪b	clβ(a∪b	X
ejpam-3438	421	15	)	)	PUNCT
ejpam-3438	421	16	=	=	SYM
ejpam-3438	421	17	a∪b	a∪b	NOUN
ejpam-3438	421	18	.	.	PUNCT
ejpam-3438	422	1	thus	thus	ADV
ejpam-3438	422	2	,	,	PUNCT
ejpam-3438	422	3	a	a	DET
ejpam-3438	422	4	⊆	⊆	NUM
ejpam-3438	422	5	a∪b	a∪b	NOUN
ejpam-3438	422	6	implies	imply	VERB
ejpam-3438	422	7	clβi	clβi	NOUN
ejpam-3438	422	8	(	(	PUNCT
ejpam-3438	422	9	a	a	X
ejpam-3438	422	10	)	)	PUNCT
ejpam-3438	422	11	⊆	⊆	NUM
ejpam-3438	422	12	clβi	clβi	NOUN
ejpam-3438	422	13	(	(	PUNCT
ejpam-3438	422	14	a∪b	a∪b	ADJ
ejpam-3438	422	15	)	)	PUNCT
ejpam-3438	422	16	⊆	⊆	NUM
ejpam-3438	422	17	clβ(a∪b	clβ(a∪b	X
ejpam-3438	422	18	)	)	PUNCT
ejpam-3438	422	19	=	=	PUNCT
ejpam-3438	423	1	a∪b	a∪b	NOUN
ejpam-3438	423	2	.	.	PUNCT
ejpam-3438	424	1	hence	hence	ADV
ejpam-3438	424	2	,	,	PUNCT
ejpam-3438	424	3	clβi	clβi	NOUN
ejpam-3438	424	4	(	(	PUNCT
ejpam-3438	424	5	a	a	X
ejpam-3438	424	6	)	)	PUNCT
ejpam-3438	424	7	⊆	⊆	NUM
ejpam-3438	424	8	clβi	clβi	NOUN
ejpam-3438	424	9	(	(	PUNCT
ejpam-3438	424	10	a	a	NOUN
ejpam-3438	424	11	)	)	PUNCT
ejpam-3438	424	12	∩	∩	NOUN
ejpam-3438	424	13	(	(	PUNCT
ejpam-3438	424	14	a	a	DET
ejpam-3438	424	15	∪	∪	ADJ
ejpam-3438	424	16	b	b	NOUN
ejpam-3438	424	17	)	)	PUNCT
ejpam-3438	424	18	=	=	NOUN
ejpam-3438	424	19	clβi	clβi	NOUN
ejpam-3438	424	20	(	(	PUNCT
ejpam-3438	424	21	a	a	NOUN
ejpam-3438	424	22	)	)	PUNCT
ejpam-3438	424	23	∩	∩	NOUN
ejpam-3438	424	24	a	a	DET
ejpam-3438	424	25	∪	∪	ADJ
ejpam-3438	424	26	clβi	clβi	NOUN
ejpam-3438	424	27	(	(	PUNCT
ejpam-3438	424	28	a	a	NOUN
ejpam-3438	424	29	)	)	PUNCT
ejpam-3438	424	30	∩	∩	ADJ
ejpam-3438	424	31	b	b	NOUN
ejpam-3438	424	32	=	=	SYM
ejpam-3438	424	33	clβi	clβi	NOUN
ejpam-3438	424	34	(	(	PUNCT
ejpam-3438	424	35	a	a	NOUN
ejpam-3438	424	36	)	)	PUNCT
ejpam-3438	424	37	∩	∩	NOUN
ejpam-3438	424	38	a	a	PRON
ejpam-3438	424	39	=	=	SYM
ejpam-3438	424	40	a	a	NOUN
ejpam-3438	424	41	,	,	PUNCT
ejpam-3438	424	42	that	that	PRON
ejpam-3438	424	43	is	be	AUX
ejpam-3438	424	44	a	a	PRON
ejpam-3438	424	45	is	be	AUX
ejpam-3438	424	46	βi	βi	AUX
ejpam-3438	424	47	-closed	-close	VERB
ejpam-3438	424	48	.	.	PUNCT
ejpam-3438	425	1	similarly	similarly	ADV
ejpam-3438	425	2	,	,	PUNCT
ejpam-3438	425	3	b	b	PROPN
ejpam-3438	425	4	⊆	⊆	NUM
ejpam-3438	425	5	a	a	DET
ejpam-3438	425	6	∪	∪	ADJ
ejpam-3438	425	7	b	b	NOUN
ejpam-3438	425	8	implies	imply	VERB
ejpam-3438	425	9	clβ(b	clβ(b	X
ejpam-3438	425	10	)	)	PUNCT
ejpam-3438	425	11	⊆	⊆	NUM
ejpam-3438	425	12	clβ(a	clβ(a	PROPN
ejpam-3438	425	13	∪	∪	ADP
ejpam-3438	425	14	b	b	NOUN
ejpam-3438	425	15	)	)	PUNCT
ejpam-3438	425	16	=	=	NOUN
ejpam-3438	425	17	a	a	PRON
ejpam-3438	425	18	∪	∪	X
ejpam-3438	425	19	b.	b.	NOUN
ejpam-3438	425	20	hence	hence	ADV
ejpam-3438	425	21	,	,	PUNCT
ejpam-3438	425	22	clβ(b	clβ(b	PROPN
ejpam-3438	425	23	)	)	PUNCT
ejpam-3438	425	24	⊆	⊆	NUM
ejpam-3438	425	25	clβ(b	clβ(b	NOUN
ejpam-3438	425	26	)	)	PUNCT
ejpam-3438	425	27	∩	∩	NOUN
ejpam-3438	425	28	(	(	PUNCT
ejpam-3438	425	29	a	a	DET
ejpam-3438	425	30	∪	∪	X
ejpam-3438	425	31	b	b	NOUN
ejpam-3438	425	32	)	)	PUNCT
ejpam-3438	425	33	=	=	SYM
ejpam-3438	425	34	clβ(b	clβ(b	NOUN
ejpam-3438	425	35	)	)	PUNCT
ejpam-3438	425	36	∩	∩	NOUN
ejpam-3438	425	37	a	a	DET
ejpam-3438	425	38	∪	∪	ADJ
ejpam-3438	425	39	clβ(b	clβ(b	NOUN
ejpam-3438	425	40	)	)	PUNCT
ejpam-3438	425	41	∩	∩	ADJ
ejpam-3438	425	42	b	b	X
ejpam-3438	425	43	=	=	SYM
ejpam-3438	425	44	clβ(b	clβ(b	PROPN
ejpam-3438	425	45	)	)	PUNCT
ejpam-3438	425	46	∩	∩	NOUN
ejpam-3438	425	47	b	b	X
ejpam-3438	425	48	=	=	SYM
ejpam-3438	425	49	b	b	PROPN
ejpam-3438	425	50	,	,	PUNCT
ejpam-3438	425	51	that	that	PRON
ejpam-3438	425	52	is	be	AUX
ejpam-3438	425	53	b	b	NOUN
ejpam-3438	425	54	is	be	AUX
ejpam-3438	425	55	β	β	NOUN
ejpam-3438	425	56	-	-	VERB
ejpam-3438	425	57	closed	closed	ADJ
ejpam-3438	425	58	.	.	PUNCT
ejpam-3438	426	1	acknowledgements	acknowledgement	NOUN
ejpam-3438	426	2	the	the	DET
ejpam-3438	426	3	authors	author	NOUN
ejpam-3438	426	4	would	would	AUX
ejpam-3438	426	5	like	like	VERB
ejpam-3438	426	6	to	to	PART
ejpam-3438	426	7	thank	thank	VERB
ejpam-3438	426	8	rural	rural	ADJ
ejpam-3438	426	9	engineering	engineering	NOUN
ejpam-3438	426	10	and	and	CCONJ
ejpam-3438	426	11	technology	technology	NOUN
ejpam-3438	426	12	center	center	NOUN
ejpam-3438	426	13	of	of	ADP
ejpam-3438	426	14	negros	negros	PROPN
ejpam-3438	426	15	oriental	oriental	PROPN
ejpam-3438	426	16	state	state	PROPN
ejpam-3438	426	17	university	university	PROPN
ejpam-3438	426	18	for	for	ADP
ejpam-3438	426	19	supporting	support	VERB
ejpam-3438	426	20	this	this	DET
ejpam-3438	426	21	research	research	NOUN
ejpam-3438	426	22	.	.	PUNCT
ejpam-3438	427	1	references	reference	NOUN
ejpam-3438	427	2	[	[	X
ejpam-3438	427	3	1	1	NUM
ejpam-3438	427	4	]	]	X
ejpam-3438	427	5	m.e	m.e	PROPN
ejpam-3438	427	6	.	.	PROPN
ejpam-3438	427	7	abd	abd	PROPN
ejpam-3438	427	8	el	el	PROPN
ejpam-3438	427	9	-	-	PROPN
ejpam-3438	427	10	monsef	monsef	ADJ
ejpam-3438	427	11	,	,	PUNCT
ejpam-3438	427	12	s.n	s.n	PROPN
ejpam-3438	427	13	.	.	PROPN
ejpam-3438	427	14	el	el	PROPN
ejpam-3438	427	15	-	-	PUNCT
ejpam-3438	427	16	deep	deep	ADJ
ejpam-3438	427	17	,	,	PUNCT
ejpam-3438	427	18	r.a	r.a	PROPN
ejpam-3438	427	19	.	.	PROPN
ejpam-3438	427	20	mahmood	mahmood	PROPN
ejpam-3438	427	21	,	,	PUNCT
ejpam-3438	427	22	β	β	ADJ
ejpam-3438	427	23	-	-	ADJ
ejpam-3438	427	24	open	open	ADJ
ejpam-3438	427	25	sets	set	NOUN
ejpam-3438	427	26	and	and	CCONJ
ejpam-3438	427	27	β	β	ADJ
ejpam-3438	427	28	-	-	ADJ
ejpam-3438	427	29	continuous	continuous	ADJ
ejpam-3438	427	30	mappings	mapping	NOUN
ejpam-3438	427	31	,	,	PUNCT
ejpam-3438	427	32	bul	bul	PROPN
ejpam-3438	427	33	.	.	PUNCT
ejpam-3438	428	1	fac	fac	PROPN
ejpam-3438	428	2	.	.	PUNCT
ejpam-3438	429	1	sci	sci	PROPN
ejpam-3438	429	2	.	.	PUNCT
ejpam-3438	429	3	assiut	assiut	PROPN
ejpam-3438	429	4	univ	univ	PROPN
ejpam-3438	429	5	.	.	PROPN
ejpam-3438	430	1	12	12	NUM
ejpam-3438	430	2	(	(	PUNCT
ejpam-3438	430	3	1983	1983	NUM
ejpam-3438	430	4	)	)	PUNCT
ejpam-3438	430	5	77	77	NUM
ejpam-3438	430	6	-	-	SYM
ejpam-3438	430	7	90	90	NUM
ejpam-3438	430	8	.	.	PUNCT
ejpam-3438	431	1	[	[	X
ejpam-3438	431	2	2	2	X
ejpam-3438	431	3	]	]	PUNCT
ejpam-3438	431	4	e.	e.	PROPN
ejpam-3438	431	5	ekici	ekici	PROPN
ejpam-3438	431	6	,	,	PUNCT
ejpam-3438	431	7	t.	t.	PROPN
ejpam-3438	431	8	noiri	noiri	PROPN
ejpam-3438	431	9	,	,	PUNCT
ejpam-3438	431	10	∗-hyperconnected	∗-hyperconnected	ADJ
ejpam-3438	431	11	ideal	ideal	ADJ
ejpam-3438	431	12	topological	topological	ADJ
ejpam-3438	431	13	spaces	space	NOUN
ejpam-3438	431	14	,	,	PUNCT
ejpam-3438	431	15	analele	analele	ADP
ejpam-3438	431	16	stiintifice	stiintifice	PROPN
ejpam-3438	431	17	aie	aie	PROPN
ejpam-3438	431	18	universitatii	universitatii	PROPN
ejpam-3438	431	19	matematica	matematica	PROPN
ejpam-3438	431	20	(	(	PUNCT
ejpam-3438	431	21	2012	2012	NUM
ejpam-3438	431	22	)	)	PUNCT
ejpam-3438	431	23	121	121	NUM
ejpam-3438	431	24	-	-	SYM
ejpam-3438	431	25	129	129	NUM
ejpam-3438	431	26	.	.	PUNCT
ejpam-3438	432	1	[	[	X
ejpam-3438	432	2	3	3	X
ejpam-3438	432	3	]	]	X
ejpam-3438	432	4	m.e	m.e	PROPN
ejpam-3438	432	5	.	.	PROPN
ejpam-3438	432	6	abd	abd	PROPN
ejpam-3438	432	7	el	el	PROPN
ejpam-3438	432	8	-	-	PROPN
ejpam-3438	432	9	monsef	monsef	ADJ
ejpam-3438	432	10	,	,	PUNCT
ejpam-3438	432	11	a.a	a.a	PROPN
ejpam-3438	432	12	.	.	PROPN
ejpam-3438	432	13	nasef	nasef	PROPN
ejpam-3438	432	14	,	,	PUNCT
ejpam-3438	432	15	a.e	a.e	PROPN
ejpam-3438	432	16	.	.	PROPN
ejpam-3438	432	17	radwan	radwan	PROPN
ejpam-3438	432	18	,	,	PUNCT
ejpam-3438	432	19	f.a	f.a	PROPN
ejpam-3438	432	20	.	.	PROPN
ejpam-3438	432	21	ibrahem	ibrahem	PROPN
ejpam-3438	432	22	,	,	PUNCT
ejpam-3438	432	23	r.b	r.b	PROPN
ejpam-3438	432	24	.	.	PROPN
ejpam-3438	432	25	esmaeel	esmaeel	PROPN
ejpam-3438	432	26	,	,	PUNCT
ejpam-3438	432	27	some	some	DET
ejpam-3438	432	28	properties	property	NOUN
ejpam-3438	432	29	of	of	ADP
ejpam-3438	432	30	semi	semi	ADJ
ejpam-3438	432	31	-	-	ADJ
ejpam-3438	432	32	open	open	ADJ
ejpam-3438	432	33	sets	set	NOUN
ejpam-3438	432	34	with	with	ADP
ejpam-3438	432	35	respect	respect	NOUN
ejpam-3438	432	36	to	to	ADP
ejpam-3438	432	37	an	an	DET
ejpam-3438	432	38	ideal	ideal	NOUN
ejpam-3438	432	39	,	,	PUNCT
ejpam-3438	432	40	(	(	PUNCT
ejpam-3438	432	41	submitted	submit	VERB
ejpam-3438	432	42	)	)	PUNCT
ejpam-3438	432	43	.	.	PUNCT
ejpam-3438	433	1	references	reference	NOUN
ejpam-3438	433	2	904	904	NUM
ejpam-3438	434	1	[	[	X
ejpam-3438	434	2	4	4	NUM
ejpam-3438	434	3	]	]	PUNCT
ejpam-3438	434	4	e.	e.	PROPN
ejpam-3438	434	5	ekici	ekici	PROPN
ejpam-3438	434	6	and	and	CCONJ
ejpam-3438	434	7	t.	t.	PROPN
ejpam-3438	434	8	noiri	noiri	PROPN
ejpam-3438	434	9	,	,	PUNCT
ejpam-3438	434	10	∗-hyperconnected	∗-hyperconnected	ADJ
ejpam-3438	434	11	ideal	ideal	ADJ
ejpam-3438	434	12	topological	topological	ADJ
ejpam-3438	434	13	spaces	space	NOUN
ejpam-3438	434	14	,	,	PUNCT
ejpam-3438	434	15	analele	analele	ADP
ejpam-3438	434	16	stiintifice	stiintifice	PROPN
ejpam-3438	434	17	aie	aie	PROPN
ejpam-3438	434	18	universitatii	universitatii	PROPN
ejpam-3438	434	19	matematica	matematica	PROPN
ejpam-3438	434	20	(	(	PUNCT
ejpam-3438	434	21	2012	2012	NUM
ejpam-3438	434	22	)	)	PUNCT
ejpam-3438	434	23	,	,	PUNCT
ejpam-3438	434	24	121	121	NUM
ejpam-3438	434	25	-	-	SYM
ejpam-3438	434	26	129	129	NUM
ejpam-3438	434	27	.	.	PUNCT
ejpam-3438	435	1	[	[	X
ejpam-3438	435	2	5	5	X
ejpam-3438	435	3	]	]	PUNCT
ejpam-3438	435	4	k.	k.	PROPN
ejpam-3438	435	5	kannan	kannan	PROPN
ejpam-3438	435	6	,	,	PUNCT
ejpam-3438	435	7	n.	n.	PROPN
ejpam-3438	435	8	nagaveni	nagaveni	PROPN
ejpam-3438	435	9	,	,	PUNCT
ejpam-3438	435	10	on	on	ADP
ejpam-3438	435	11	β̂-generalized	β̂-generalized	ADJ
ejpam-3438	435	12	closed	closed	ADJ
ejpam-3438	435	13	sets	set	NOUN
ejpam-3438	435	14	in	in	ADP
ejpam-3438	435	15	topological	topological	ADJ
ejpam-3438	435	16	spaces	space	NOUN
ejpam-3438	435	17	,	,	PUNCT
ejpam-3438	435	18	international	international	ADJ
ejpam-3438	435	19	journal	journal	NOUN
ejpam-3438	435	20	of	of	ADP
ejpam-3438	435	21	math	math	NOUN
ejpam-3438	435	22	.	.	PUNCT
ejpam-3438	436	1	analysis	analysis	NOUN
ejpam-3438	436	2	,	,	PUNCT
ejpam-3438	436	3	6(57	6(57	NUM
ejpam-3438	436	4	)	)	PUNCT
ejpam-3438	436	5	(	(	PUNCT
ejpam-3438	436	6	2012	2012	NUM
ejpam-3438	436	7	)	)	PUNCT
ejpam-3438	436	8	,	,	PUNCT
ejpam-3438	436	9	2819	2819	NUM
ejpam-3438	436	10	-	-	SYM
ejpam-3438	436	11	2828	2828	NUM
ejpam-3438	436	12	.	.	PUNCT
ejpam-3438	437	1	[	[	X
ejpam-3438	437	2	6	6	NUM
ejpam-3438	437	3	]	]	PUNCT
ejpam-3438	437	4	k.	k.	PROPN
ejpam-3438	437	5	kuratowski	kuratowski	PROPN
ejpam-3438	437	6	,	,	PUNCT
ejpam-3438	437	7	topology	topology	NOUN
ejpam-3438	437	8	,	,	PUNCT
ejpam-3438	437	9	vol	vol	NOUN
ejpam-3438	437	10	i.	i.	PROPN
ejpam-3438	437	11	new	new	PROPN
ejpam-3438	437	12	york	york	PROPN
ejpam-3438	437	13	:	:	PUNCT
ejpam-3438	437	14	academic	academic	ADJ
ejpam-3438	437	15	press	press	NOUN
ejpam-3438	437	16	,	,	PUNCT
ejpam-3438	437	17	(	(	PUNCT
ejpam-3438	437	18	1933	1933	NUM
ejpam-3438	437	19	)	)	PUNCT
ejpam-3438	437	20	.	.	PUNCT
ejpam-3438	438	1	[	[	X
ejpam-3438	438	2	7	7	X
ejpam-3438	438	3	]	]	PUNCT
ejpam-3438	438	4	s.	s.	PROPN
ejpam-3438	438	5	n.	n.	PROPN
ejpam-3438	438	6	maheshwari	maheshwari	PROPN
ejpam-3438	438	7	and	and	CCONJ
ejpam-3438	438	8	s.	s.	PROPN
ejpam-3438	438	9	s.	s.	PROPN
ejpam-3438	438	10	thakur	thakur	PROPN
ejpam-3438	438	11	,	,	PUNCT
ejpam-3438	438	12	on	on	ADP
ejpam-3438	438	13	α	α	ADJ
ejpam-3438	438	14	-	-	ADJ
ejpam-3438	438	15	compact	compact	ADJ
ejpam-3438	438	16	spaces	space	NOUN
ejpam-3438	438	17	,	,	PUNCT
ejpam-3438	438	18	bulletin	bulletin	NOUN
ejpam-3438	438	19	of	of	ADP
ejpam-3438	438	20	the	the	DET
ejpam-3438	438	21	institute	institute	NOUN
ejpam-3438	438	22	of	of	ADP
ejpam-3438	438	23	mathematics	mathematics	PROPN
ejpam-3438	438	24	,	,	PUNCT
ejpam-3438	438	25	academia	academia	PROPN
ejpam-3438	438	26	sinice	sinice	PROPN
ejpam-3438	438	27	,	,	PUNCT
ejpam-3438	438	28	vol	vol	NOUN
ejpam-3438	438	29	.	.	PROPN
ejpam-3438	438	30	13	13	NUM
ejpam-3438	438	31	,	,	PUNCT
ejpam-3438	438	32	no	no	INTJ
ejpam-3438	438	33	.	.	NOUN
ejpam-3438	438	34	4	4	NUM
ejpam-3438	438	35	,	,	PUNCT
ejpam-3438	438	36	dece	dece	PROPN
ejpam-3438	438	37	.	.	PUNCT
ejpam-3438	439	1	(	(	PUNCT
ejpam-3438	439	2	1985	1985	NUM
ejpam-3438	439	3	)	)	PUNCT
ejpam-3438	439	4	,	,	PUNCT
ejpam-3438	439	5	341347	341347	NUM
ejpam-3438	439	6	.	.	PUNCT
ejpam-3438	440	1	[	[	X
ejpam-3438	440	2	8	8	NUM
ejpam-3438	440	3	]	]	PUNCT
ejpam-3438	440	4	a.m.	a.m.	NOUN
ejpam-3438	441	1	mubarki	mubarki	PROPN
ejpam-3438	441	2	,	,	PUNCT
ejpam-3438	441	3	m.m	m.m	PROPN
ejpam-3438	441	4	.	.	PROPN
ejpam-3438	441	5	al	al	PROPN
ejpam-3438	441	6	-	-	PUNCT
ejpam-3438	441	7	rshudi	rshudi	PROPN
ejpam-3438	441	8	,	,	PUNCT
ejpam-3438	441	9	m.a	m.a	PROPN
ejpam-3438	441	10	.	.	PROPN
ejpam-3438	441	11	al	al	PROPN
ejpam-3438	441	12	-	-	PUNCT
ejpam-3438	441	13	juhani	juhani	PROPN
ejpam-3438	441	14	,	,	PUNCT
ejpam-3438	441	15	β∗-open	β∗-open	ADJ
ejpam-3438	441	16	sets	set	NOUN
ejpam-3438	441	17	and	and	CCONJ
ejpam-3438	441	18	β∗-continuity	β∗-continuity	NOUN
ejpam-3438	441	19	in	in	ADP
ejpam-3438	441	20	topological	topological	ADJ
ejpam-3438	441	21	spaces	space	NOUN
ejpam-3438	441	22	,	,	PUNCT
ejpam-3438	441	23	journal	journal	NOUN
ejpam-3438	441	24	of	of	ADP
ejpam-3438	441	25	taibah	taibah	PROPN
ejpam-3438	441	26	university	university	PROPN
ejpam-3438	441	27	of	of	ADP
ejpam-3438	441	28	science	science	NOUN
ejpam-3438	441	29	,	,	PUNCT
ejpam-3438	441	30	8	8	NUM
ejpam-3438	441	31	(	(	PUNCT
ejpam-3438	441	32	2014	2014	NUM
ejpam-3438	441	33	)	)	PUNCT
ejpam-3438	441	34	,	,	PUNCT
ejpam-3438	441	35	142	142	NUM
ejpam-3438	441	36	-	-	SYM
ejpam-3438	441	37	148	148	NUM
ejpam-3438	441	38	.	.	PUNCT
ejpam-3438	442	1	[	[	X
ejpam-3438	442	2	9	9	NUM
ejpam-3438	442	3	]	]	PUNCT
ejpam-3438	442	4	s.	s.	PROPN
ejpam-3438	442	5	mel	mel	PROPN
ejpam-3438	442	6	,	,	PUNCT
ejpam-3438	442	7	m.	m.	NOUN
ejpam-3438	442	8	fhal	fhal	PROPN
ejpam-3438	442	9	,	,	PUNCT
ejpam-3438	442	10	new	new	ADJ
ejpam-3438	442	11	near	near	ADP
ejpam-3438	442	12	open	open	ADJ
ejpam-3438	442	13	sets	set	NOUN
ejpam-3438	442	14	in	in	ADP
ejpam-3438	442	15	topological	topological	ADJ
ejpam-3438	442	16	space	space	NOUN
ejpam-3438	442	17	,	,	PUNCT
ejpam-3438	442	18	journal	journal	NOUN
ejpam-3438	442	19	of	of	ADP
ejpam-3438	442	20	physical	physical	ADJ
ejpam-3438	442	21	mathematics	mathematic	NOUN
ejpam-3438	442	22	,	,	PUNCT
ejpam-3438	442	23	7(4	7(4	NUM
ejpam-3438	442	24	)	)	PUNCT
ejpam-3438	442	25	(	(	PUNCT
ejpam-3438	442	26	2016	2016	NUM
ejpam-3438	442	27	)	)	PUNCT
ejpam-3438	442	28	,	,	PUNCT
ejpam-3438	442	29	1	1	NUM
ejpam-3438	442	30	-	-	SYM
ejpam-3438	442	31	8	8	NUM
ejpam-3438	442	32	.	.	PUNCT
ejpam-3438	443	1	[	[	X
ejpam-3438	443	2	10	10	NUM
ejpam-3438	443	3	]	]	X
ejpam-3438	443	4	m.p	m.p	PROPN
ejpam-3438	443	5	.	.	PROPN
ejpam-3438	443	6	chaudhary	chaudhary	PROPN
ejpam-3438	443	7	,	,	PUNCT
ejpam-3438	443	8	v.	v.	PROPN
ejpam-3438	443	9	kumar	kumar	PROPN
ejpam-3438	443	10	,	,	PUNCT
ejpam-3438	443	11	s.	s.	PROPN
ejpam-3438	443	12	chowhary	chowhary	PROPN
ejpam-3438	443	13	,	,	PUNCT
ejpam-3438	443	14	on	on	ADP
ejpam-3438	443	15	topological	topological	ADJ
ejpam-3438	443	16	sets	set	NOUN
ejpam-3438	443	17	and	and	CCONJ
ejpam-3438	443	18	spaces	space	NOUN
ejpam-3438	443	19	,	,	PUNCT
ejpam-3438	443	20	global	global	ADJ
ejpam-3438	443	21	journal	journal	NOUN
ejpam-3438	443	22	of	of	ADP
ejpam-3438	443	23	science	science	PROPN
ejpam-3438	443	24	frontier	frontier	NOUN
ejpam-3438	443	25	research	research	NOUN
ejpam-3438	443	26	,	,	PUNCT
ejpam-3438	443	27	volume	volume	NOUN
ejpam-3438	443	28	xi	xi	X
ejpam-3438	443	29	.	.	PUNCT
ejpam-3438	444	1	[	[	X
ejpam-3438	444	2	11	11	NUM
ejpam-3438	444	3	]	]	X
ejpam-3438	444	4	s.g	s.g	PROPN
ejpam-3438	444	5	.	.	PROPN
ejpam-3438	444	6	crossley	crossley	PROPN
ejpam-3438	444	7	,	,	PUNCT
ejpam-3438	444	8	s.k	s.k	PROPN
ejpam-3438	444	9	.	.	PROPN
ejpam-3438	444	10	hildebrand	hildebrand	PROPN
ejpam-3438	444	11	,	,	PUNCT
ejpam-3438	444	12	semi	semi	ADJ
ejpam-3438	444	13	-	-	ADJ
ejpam-3438	444	14	topological	topological	ADJ
ejpam-3438	444	15	properties	property	NOUN
ejpam-3438	444	16	,	,	PUNCT
ejpam-3438	444	17	fund	fund	NOUN
ejpam-3438	444	18	.	.	PUNCT
ejpam-3438	444	19	math	math	NOUN
ejpam-3438	444	20	.	.	PUNCT
ejpam-3438	444	21	,	,	PUNCT
ejpam-3438	444	22	74	74	NUM
ejpam-3438	444	23	,	,	PUNCT
ejpam-3438	444	24	233254	233254	NUM
ejpam-3438	444	25	.	.	PUNCT
ejpam-3438	445	1	[	[	X
ejpam-3438	445	2	12	12	NUM
ejpam-3438	445	3	]	]	X
ejpam-3438	445	4	c.	c.	PROPN
ejpam-3438	445	5	dorsett	dorsett	PROPN
ejpam-3438	445	6	,	,	PUNCT
ejpam-3438	445	7	semi	semi	ADJ
ejpam-3438	445	8	-	-	ADJ
ejpam-3438	445	9	regular	regular	ADJ
ejpam-3438	445	10	spaces	space	NOUN
ejpam-3438	445	11	,	,	PUNCT
ejpam-3438	445	12	soochow	soochow	PROPN
ejpam-3438	445	13	j.	j.	PROPN
ejpam-3438	445	14	math	math	PROPN
ejpam-3438	445	15	.	.	PUNCT
ejpam-3438	445	16	,	,	PUNCT
ejpam-3438	445	17	8	8	NUM
ejpam-3438	445	18	,	,	PUNCT
ejpam-3438	445	19	45	45	NUM
ejpam-3438	445	20	-	-	SYM
ejpam-3438	445	21	53	53	NUM
ejpam-3438	445	22	.	.	PUNCT
ejpam-3438	446	1	[	[	X
ejpam-3438	446	2	13	13	NUM
ejpam-3438	446	3	]	]	X
ejpam-3438	446	4	c.	c.	PROPN
ejpam-3438	446	5	dorsett	dorsett	PROPN
ejpam-3438	446	6	,	,	PUNCT
ejpam-3438	446	7	semi	semi	ADJ
ejpam-3438	446	8	-	-	ADJ
ejpam-3438	446	9	normal	normal	ADJ
ejpam-3438	446	10	spaces	space	NOUN
ejpam-3438	446	11	,	,	PUNCT
ejpam-3438	446	12	kyungpook	kyungpook	PROPN
ejpam-3438	446	13	math	math	NOUN
ejpam-3438	446	14	.	.	PUNCT
ejpam-3438	447	1	j.	j.	PROPN
ejpam-3438	447	2	,	,	PUNCT
ejpam-3438	447	3	25	25	NUM
ejpam-3438	447	4	,	,	PUNCT
ejpam-3438	447	5	173	173	NUM
ejpam-3438	447	6	-	-	SYM
ejpam-3438	447	7	180	180	NUM
ejpam-3438	447	8	.	.	PUNCT
ejpam-3438	448	1	[	[	X
ejpam-3438	448	2	14	14	NUM
ejpam-3438	448	3	]	]	X
ejpam-3438	448	4	j.	j.	PROPN
ejpam-3438	448	5	dugundji	dugundji	PROPN
ejpam-3438	448	6	,	,	PUNCT
ejpam-3438	448	7	topology	topology	PROPN
ejpam-3438	448	8	,	,	PUNCT
ejpam-3438	448	9	usa	usa	PROPN
ejpam-3438	448	10	:	:	PUNCT
ejpam-3438	448	11	allyn	allyn	PROPN
ejpam-3438	448	12	and	and	CCONJ
ejpam-3438	448	13	bacon	bacon	PROPN
ejpam-3438	448	14	,	,	PUNCT
ejpam-3438	448	15	inc	inc	PROPN
ejpam-3438	448	16	.	.	PUNCT
ejpam-3438	449	1	[	[	X
ejpam-3438	449	2	15	15	NUM
ejpam-3438	449	3	]	]	X
ejpam-3438	449	4	s.	s.	PROPN
ejpam-3438	449	5	morris	morris	PROPN
ejpam-3438	449	6	,	,	PUNCT
ejpam-3438	449	7	topology	topology	NOUN
ejpam-3438	449	8	without	without	ADP
ejpam-3438	449	9	tears	tear	NOUN
ejpam-3438	449	10	,	,	PUNCT
ejpam-3438	449	11	version	version	NOUN
ejpam-3438	449	12	of	of	ADP
ejpam-3438	449	13	february	february	PROPN
ejpam-3438	449	14	20	20	NUM
ejpam-3438	449	15	,	,	PUNCT
ejpam-3438	449	16	2012	2012	NUM
ejpam-3438	449	17	.	.	PUNCT
ejpam-3438	450	1	[	[	X
ejpam-3438	450	2	16	16	NUM
ejpam-3438	450	3	]	]	PUNCT
ejpam-3438	450	4	p.	p.	PROPN
ejpam-3438	450	5	bhattacharya	bhattacharya	PROPN
ejpam-3438	450	6	,	,	PUNCT
ejpam-3438	450	7	b.k	b.k	PROPN
ejpam-3438	450	8	.	.	PROPN
ejpam-3438	450	9	lahiri	lahiri	PROPN
ejpam-3438	450	10	,	,	PUNCT
ejpam-3438	450	11	semi	semi	ADV
ejpam-3438	450	12	generalized	generalized	ADJ
ejpam-3438	450	13	closed	closed	ADJ
ejpam-3438	450	14	sets	set	NOUN
ejpam-3438	450	15	in	in	ADP
ejpam-3438	450	16	topology	topology	NOUN
ejpam-3438	450	17	,	,	PUNCT
ejpam-3438	450	18	indian	indian	PROPN
ejpam-3438	450	19	j	j	PROPN
ejpam-3438	450	20	math	math	PROPN
ejpam-3438	450	21	,	,	PUNCT
ejpam-3438	450	22	29	29	NUM
ejpam-3438	450	23	,	,	PUNCT
ejpam-3438	450	24	373	373	NUM
ejpam-3438	450	25	-	-	SYM
ejpam-3438	450	26	382	382	NUM
ejpam-3438	450	27	.	.	PUNCT
ejpam-3438	451	1	[	[	X
ejpam-3438	451	2	17	17	NUM
ejpam-3438	451	3	]	]	PUNCT
ejpam-3438	451	4	a.	a.	NOUN
ejpam-3438	451	5	skowron	skowron	PROPN
ejpam-3438	451	6	,	,	PUNCT
ejpam-3438	451	7	on	on	ADP
ejpam-3438	451	8	topology	topology	NOUN
ejpam-3438	451	9	information	information	NOUN
ejpam-3438	451	10	systems	system	NOUN
ejpam-3438	451	11	,	,	PUNCT
ejpam-3438	451	12	indian	indian	ADJ
ejpam-3438	451	13	bull	bull	NOUN
ejpam-3438	451	14	polish	polish	NOUN
ejpam-3438	451	15	acta	acta	PROPN
ejpam-3438	451	16	sci	sci	PROPN
ejpam-3438	451	17	math	math	PROPN
ejpam-3438	451	18	,	,	PUNCT
ejpam-3438	451	19	36178	36178	NUM
ejpam-3438	451	20	,	,	PUNCT
ejpam-3438	451	21	87	87	NUM
ejpam-3438	451	22	-	-	SYM
ejpam-3438	451	23	90	90	NUM
ejpam-3438	451	24	.	.	PUNCT
ejpam-3438	452	1	[	[	X
ejpam-3438	452	2	18	18	NUM
ejpam-3438	452	3	]	]	X
ejpam-3438	452	4	m.h	m.h	PROPN
ejpam-3438	452	5	.	.	PROPN
ejpam-3438	452	6	stone	stone	PROPN
ejpam-3438	452	7	,	,	PUNCT
ejpam-3438	452	8	applications	application	NOUN
ejpam-3438	452	9	of	of	ADP
ejpam-3438	452	10	the	the	DET
ejpam-3438	452	11	theory	theory	NOUN
ejpam-3438	452	12	of	of	ADP
ejpam-3438	452	13	boolean	boolean	ADJ
ejpam-3438	452	14	rings	ring	NOUN
ejpam-3438	452	15	to	to	ADP
ejpam-3438	452	16	general	general	ADJ
ejpam-3438	452	17	topology	topology	NOUN
ejpam-3438	452	18	,	,	PUNCT
ejpam-3438	452	19	indian	indian	PROPN
ejpam-3438	452	20	j	j	PROPN
ejpam-3438	452	21	math	math	PROPN
ejpam-3438	452	22	29	29	NUM
ejpam-3438	452	23	,	,	PUNCT
ejpam-3438	452	24	373	373	NUM
ejpam-3438	452	25	-	-	SYM
ejpam-3438	452	26	382	382	NUM
ejpam-3438	452	27	.	.	PUNCT
ejpam-3438	453	1	[	[	X
ejpam-3438	453	2	19	19	NUM
ejpam-3438	453	3	]	]	X
ejpam-3438	453	4	n.	n.	PROPN
ejpam-3438	453	5	levine	levine	PROPN
ejpam-3438	453	6	,	,	PUNCT
ejpam-3438	453	7	semi	semi	ADV
ejpam-3438	453	8	open	open	ADJ
ejpam-3438	453	9	sets	set	NOUN
ejpam-3438	453	10	and	and	CCONJ
ejpam-3438	453	11	semi	semi	ADV
ejpam-3438	453	12	continuous	continuous	ADJ
ejpam-3438	453	13	mappings	mapping	NOUN
ejpam-3438	453	14	in	in	ADP
ejpam-3438	453	15	topological	topological	ADJ
ejpam-3438	453	16	spaces	space	NOUN
ejpam-3438	453	17	,	,	PUNCT
ejpam-3438	453	18	.	.	PUNCT
ejpam-3438	454	1	amr	amr	PROPN
ejpam-3438	454	2	math	math	PROPN
ejpam-3438	454	3	monthly	monthly	ADV
ejpam-3438	454	4	,	,	PUNCT
ejpam-3438	454	5	70	70	NUM
ejpam-3438	454	6	,	,	PUNCT
ejpam-3438	454	7	36	36	NUM
ejpam-3438	454	8	-	-	SYM
ejpam-3438	454	9	41	41	NUM
ejpam-3438	454	10	.	.	PUNCT
ejpam-3438	455	1	[	[	X
ejpam-3438	455	2	20	20	NUM
ejpam-3438	455	3	]	]	X
ejpam-3438	455	4	o.	o.	PROPN
ejpam-3438	455	5	najastad	najastad	PROPN
ejpam-3438	455	6	,	,	PUNCT
ejpam-3438	455	7	on	on	ADP
ejpam-3438	455	8	some	some	DET
ejpam-3438	455	9	classes	class	NOUN
ejpam-3438	455	10	of	of	ADP
ejpam-3438	455	11	nearly	nearly	ADV
ejpam-3438	455	12	open	open	ADJ
ejpam-3438	455	13	sets	set	NOUN
ejpam-3438	455	14	,	,	PUNCT
ejpam-3438	455	15	pacific	pacific	PROPN
ejpam-3438	455	16	j	j	PROPN
ejpam-3438	455	17	math	math	PROPN
ejpam-3438	455	18	,	,	PUNCT
ejpam-3438	455	19	15	15	NUM
ejpam-3438	455	20	,	,	PUNCT
ejpam-3438	455	21	961	961	NUM
ejpam-3438	455	22	-	-	SYM
ejpam-3438	455	23	970	970	NUM
ejpam-3438	455	24	.	.	PUNCT
ejpam-3438	456	1	[	[	X
ejpam-3438	456	2	21	21	NUM
ejpam-3438	456	3	]	]	X
ejpam-3438	456	4	d.	d.	PROPN
ejpam-3438	456	5	andrijević	andrijević	PROPN
ejpam-3438	456	6	,	,	PUNCT
ejpam-3438	456	7	semi	semi	ADJ
ejpam-3438	456	8	-	-	ADJ
ejpam-3438	456	9	preopen	preopen	ADJ
ejpam-3438	456	10	sets	set	NOUN
ejpam-3438	456	11	,	,	PUNCT
ejpam-3438	456	12	matematički	matematički	PROPN
ejpam-3438	456	13	vesnik	vesnik	X
ejpam-3438	456	14	38.93	38.93	NUM
ejpam-3438	456	15	(	(	PUNCT
ejpam-3438	456	16	1986	1986	NUM
ejpam-3438	456	17	):	):	PUNCT
ejpam-3438	456	18	24	24	NUM
ejpam-3438	456	19	-	-	SYM
ejpam-3438	456	20	32	32	NUM
ejpam-3438	456	21	.	.	PUNCT
ejpam-3438	457	1	[	[	X
ejpam-3438	457	2	22	22	NUM
ejpam-3438	457	3	]	]	X
ejpam-3438	457	4	m.y	m.y	PROPN
ejpam-3438	457	5	.	.	PROPN
ejpam-3438	457	6	abid	abid	PROPN
ejpam-3438	457	7	,	,	PUNCT
ejpam-3438	457	8	non	non	ADJ
ejpam-3438	457	9	semipre	semipre	NOUN
ejpam-3438	457	10	-	-	PUNCT
ejpam-3438	457	11	denseness	denseness	NOUN
ejpam-3438	457	12	in	in	ADP
ejpam-3438	457	13	topological	topological	ADJ
ejpam-3438	457	14	spaces	space	NOUN
ejpam-3438	457	15	,	,	PUNCT
ejpam-3438	457	16	journal	journal	NOUN
ejpam-3438	457	17	of	of	ADP
ejpam-3438	457	18	kerbala	kerbala	PROPN
ejpam-3438	457	19	university	university	PROPN
ejpam-3438	457	20	,	,	PUNCT
ejpam-3438	457	21	vol	vol	NOUN
ejpam-3438	457	22	.	.	PROPN
ejpam-3438	457	23	5	5	NUM
ejpam-3438	457	24	no	no	NOUN
ejpam-3438	457	25	.	.	NOUN
ejpam-3438	457	26	2	2	NUM
ejpam-3438	457	27	scientific	scientific	ADJ
ejpam-3438	457	28	,	,	PUNCT
ejpam-3438	457	29	june	june	PROPN
ejpam-3438	457	30	2007	2007	NUM
ejpam-3438	457	31	.	.	PUNCT
ejpam-3438	458	1	references	reference	NOUN
ejpam-3438	458	2	905	905	NUM
ejpam-3438	459	1	[	[	X
ejpam-3438	459	2	23	23	NUM
ejpam-3438	459	3	]	]	PUNCT
ejpam-3438	459	4	s.	s.	PROPN
ejpam-3438	459	5	tahiliani	tahiliani	PROPN
ejpam-3438	459	6	,	,	PUNCT
ejpam-3438	459	7	operation	operation	NOUN
ejpam-3438	459	8	approach	approach	NOUN
ejpam-3438	459	9	to	to	ADP
ejpam-3438	459	10	β	β	ADJ
ejpam-3438	459	11	-	-	ADJ
ejpam-3438	459	12	open	open	ADJ
ejpam-3438	459	13	sets	set	NOUN
ejpam-3438	459	14	and	and	CCONJ
ejpam-3438	459	15	applications	application	NOUN
ejpam-3438	459	16	,	,	PUNCT
ejpam-3438	459	17	math	math	NOUN
ejpam-3438	459	18	commun	commun	PROPN
ejpam-3438	459	19	16	16	NUM
ejpam-3438	459	20	(	(	PUNCT
ejpam-3438	459	21	2011	2011	NUM
ejpam-3438	459	22	)	)	PUNCT
ejpam-3438	459	23	,	,	PUNCT
ejpam-3438	459	24	577	577	NUM
ejpam-3438	459	25	-	-	SYM
ejpam-3438	459	26	591	591	NUM
ejpam-3438	459	27	.	.	PUNCT
ejpam-3438	460	1	[	[	X
ejpam-3438	460	2	24	24	NUM
ejpam-3438	460	3	]	]	SYM
ejpam-3438	460	4	a.a	a.a	PROPN
ejpam-3438	460	5	.	.	PROPN
ejpam-3438	460	6	nasef	nasef	PROPN
ejpam-3438	460	7	,	,	PUNCT
ejpam-3438	460	8	a.e	a.e	PROPN
ejpam-3438	460	9	.	.	PROPN
ejpam-3438	460	10	radwan	radwan	PROPN
ejpam-3438	460	11	,	,	PUNCT
ejpam-3438	460	12	r.b	r.b	PROPN
ejpam-3438	460	13	.	.	PROPN
ejpam-3438	460	14	esmaeel	esmaeel	PROPN
ejpam-3438	460	15	,	,	PUNCT
ejpam-3438	460	16	some	some	DET
ejpam-3438	460	17	properties	property	NOUN
ejpam-3438	460	18	of	of	ADP
ejpam-3438	460	19	α	α	NOUN
ejpam-3438	460	20	-	-	ADJ
ejpam-3438	460	21	open	open	ADJ
ejpam-3438	460	22	sets	set	NOUN
ejpam-3438	460	23	with	with	ADP
ejpam-3438	460	24	respect	respect	NOUN
ejpam-3438	460	25	to	to	ADP
ejpam-3438	460	26	an	an	DET
ejpam-3438	460	27	ideal	ideal	ADJ
ejpam-3438	460	28	,	,	PUNCT
ejpam-3438	460	29	int	int	NOUN
ejpam-3438	460	30	journal	journal	NOUN
ejpam-3438	460	31	of	of	ADP
ejpam-3438	460	32	pure	pure	ADJ
ejpam-3438	460	33	and	and	CCONJ
ejpam-3438	460	34	applied	applied	ADJ
ejpam-3438	460	35	mathematics	mathematic	NOUN
ejpam-3438	460	36	,	,	PUNCT
ejpam-3438	460	37	vol	vol	NOUN
ejpam-3438	460	38	.	.	PROPN
ejpam-3438	460	39	102	102	NUM
ejpam-3438	460	40	,	,	PUNCT
ejpam-3438	460	41	no	no	INTJ
ejpam-3438	460	42	.	.	NOUN
ejpam-3438	460	43	57	57	NUM
ejpam-3438	460	44	2015	2015	NUM
ejpam-3438	460	45	,	,	PUNCT
ejpam-3438	460	46	613630	613630	NUM
ejpam-3438	460	47	.	.	PUNCT
ejpam-3438	461	1	[	[	X
ejpam-3438	461	2	25	25	NUM
ejpam-3438	461	3	]	]	PUNCT
ejpam-3438	461	4	a.m.	a.m.	NOUN
ejpam-3438	462	1	mubarki	mubarki	PROPN
ejpam-3438	462	2	,	,	PUNCT
ejpam-3438	462	3	a.	a.	NOUN
ejpam-3438	462	4	mizyed	mizye	VERB
ejpam-3438	462	5	,	,	PUNCT
ejpam-3438	462	6	on	on	ADP
ejpam-3438	462	7	the	the	DET
ejpam-3438	462	8	topology	topology	NOUN
ejpam-3438	462	9	generated	generate	VERB
ejpam-3438	462	10	by	by	ADP
ejpam-3438	462	11	βc	βc	NOUN
ejpam-3438	462	12	-	-	PUNCT
ejpam-3438	462	13	open	open	ADJ
ejpam-3438	462	14	sets	set	NOUN
ejpam-3438	462	15	,	,	PUNCT
ejpam-3438	462	16	international	international	ADJ
ejpam-3438	462	17	journal	journal	NOUN
ejpam-3438	462	18	of	of	ADP
ejpam-3438	462	19	math	math	NOUN
ejpam-3438	462	20	.	.	PUNCT
ejpam-3438	463	1	sci	sci	PROPN
ejpam-3438	463	2	.	.	PROPN
ejpam-3438	463	3	and	and	CCONJ
ejpam-3438	463	4	engg	engg	PROPN
ejpam-3438	463	5	.	.	PUNCT
ejpam-3438	464	1	appls	appls	PROPN
ejpam-3438	464	2	.	.	PUNCT
ejpam-3438	465	1	(	(	PUNCT
ejpam-3438	465	2	ijmsea	ijmsea	NOUN
ejpam-3438	465	3	)	)	PUNCT
ejpam-3438	465	4	,	,	PUNCT
ejpam-3438	465	5	vol	vol	NOUN
ejpam-3438	465	6	.	.	PROPN
ejpam-3438	466	1	9	9	NUM
ejpam-3438	466	2	no	no	NOUN
ejpam-3438	466	3	.	.	NOUN
ejpam-3438	466	4	2	2	NUM
ejpam-3438	466	5	scientific	scientific	ADJ
ejpam-3438	466	6	,	,	PUNCT
ejpam-3438	466	7	june	june	PROPN
ejpam-3438	466	8	2007	2007	NUM
ejpam-3438	466	9	.	.	PUNCT
ejpam-3438	467	1	[	[	X
ejpam-3438	467	2	26	26	NUM
ejpam-3438	467	3	]	]	PUNCT
ejpam-3438	467	4	a.	a.	NOUN
ejpam-3438	467	5	el	el	PROPN
ejpam-3438	467	6	mabhouh	mabhouh	PROPN
ejpam-3438	467	7	,	,	PUNCT
ejpam-3438	467	8	m.m	m.m	PROPN
ejpam-3438	467	9	.	.	PROPN
ejpam-3438	467	10	al	al	PROPN
ejpam-3438	467	11	-	-	PUNCT
ejpam-3438	467	12	rshudi	rshudi	PROPN
ejpam-3438	467	13	,	,	PUNCT
ejpam-3438	467	14	m.a	m.a	PROPN
ejpam-3438	467	15	.	.	PROPN
ejpam-3438	467	16	al	al	PROPN
ejpam-3438	467	17	-	-	PUNCT
ejpam-3438	467	18	juhani	juhani	PROPN
ejpam-3438	467	19	,	,	PUNCT
ejpam-3438	467	20	β∗-open	β∗-open	ADJ
ejpam-3438	467	21	sets	set	NOUN
ejpam-3438	467	22	and	and	CCONJ
ejpam-3438	467	23	β∗-continuity	β∗-continuity	NOUN
ejpam-3438	467	24	in	in	ADP
ejpam-3438	467	25	topological	topological	ADJ
ejpam-3438	467	26	spaces	space	NOUN
ejpam-3438	467	27	,	,	PUNCT
ejpam-3438	467	28	journal	journal	NOUN
ejpam-3438	467	29	of	of	ADP
ejpam-3438	467	30	kerbala	kerbala	PROPN
ejpam-3438	467	31	university	university	PROPN
ejpam-3438	467	32	,	,	PUNCT
ejpam-3438	467	33	vol	vol	NOUN
ejpam-3438	467	34	.	.	PROPN
ejpam-3438	467	35	5	5	NUM
ejpam-3438	467	36	no	no	NOUN
ejpam-3438	467	37	.	.	NOUN
ejpam-3438	467	38	1	1	NUM
ejpam-3438	467	39	,	,	PUNCT
ejpam-3438	467	40	march	march	PROPN
ejpam-3438	467	41	2015	2015	NUM
ejpam-3438	467	42	,	,	PUNCT
ejpam-3438	467	43	223	223	NUM
ejpam-3438	467	44	-	-	SYM
ejpam-3438	467	45	232	232	NUM
ejpam-3438	467	46	.	.	PUNCT
ejpam-3438	468	1	[	[	X
ejpam-3438	468	2	27	27	NUM
ejpam-3438	468	3	]	]	X
ejpam-3438	468	4	m.	m.	NOUN
ejpam-3438	468	5	akdag	akdag	PROPN
ejpam-3438	468	6	,	,	PUNCT
ejpam-3438	468	7	a.	a.	NOUN
ejpam-3438	468	8	ozkan	ozkan	PROPN
ejpam-3438	468	9	,	,	PUNCT
ejpam-3438	468	10	on	on	ADP
ejpam-3438	468	11	soft	soft	ADJ
ejpam-3438	468	12	β	β	NOUN
ejpam-3438	468	13	-	-	ADJ
ejpam-3438	468	14	open	open	ADJ
ejpam-3438	468	15	sets	set	NOUN
ejpam-3438	468	16	and	and	CCONJ
ejpam-3438	468	17	soft	soft	ADJ
ejpam-3438	468	18	β	β	ADJ
ejpam-3438	468	19	-	-	ADJ
ejpam-3438	468	20	continuous	continuous	ADJ
ejpam-3438	468	21	functions	function	NOUN
ejpam-3438	468	22	,	,	PUNCT
ejpam-3438	468	23	the	the	DET
ejpam-3438	468	24	scientific	scientific	ADJ
ejpam-3438	468	25	world	world	NOUN
ejpam-3438	468	26	journal	journal	PROPN
ejpam-3438	468	27	,	,	PUNCT
ejpam-3438	468	28	june	june	PROPN
ejpam-3438	468	29	2014	2014	NUM
ejpam-3438	468	30	.	.	PUNCT
ejpam-3438	469	1	[	[	X
ejpam-3438	469	2	28	28	NUM
ejpam-3438	469	3	]	]	X
ejpam-3438	469	4	m.	m.	NOUN
ejpam-3438	469	5	arockiarani	arockiarani	PROPN
ejpam-3438	469	6	,	,	PUNCT
ejpam-3438	469	7	a.	a.	NOUN
ejpam-3438	469	8	arokia	arokia	PROPN
ejpam-3438	469	9	lancy	lancy	PROPN
ejpam-3438	469	10	,	,	PUNCT
ejpam-3438	469	11	generalized	generalize	VERB
ejpam-3438	469	12	soft	soft	ADJ
ejpam-3438	469	13	gβ	gβ	NOUN
ejpam-3438	469	14	-	-	PUNCT
ejpam-3438	469	15	closed	closed	ADJ
ejpam-3438	469	16	sets	set	NOUN
ejpam-3438	469	17	and	and	CCONJ
ejpam-3438	469	18	soft	soft	ADJ
ejpam-3438	469	19	gsβ	gsβ	ADV
ejpam-3438	469	20	-	-	PUNCT
ejpam-3438	469	21	closed	close	VERB
ejpam-3438	469	22	sets	set	NOUN
ejpam-3438	469	23	in	in	ADP
ejpam-3438	469	24	soft	soft	ADJ
ejpam-3438	469	25	topological	topological	ADJ
ejpam-3438	469	26	spaces	space	NOUN
ejpam-3438	469	27	,	,	PUNCT
ejpam-3438	469	28	international	international	ADJ
ejpam-3438	469	29	journal	journal	NOUN
ejpam-3438	469	30	of	of	ADP
ejpam-3438	469	31	mathematical	mathematical	ADJ
ejpam-3438	469	32	archive	archive	NOUN
ejpam-3438	469	33	,	,	PUNCT
ejpam-3438	469	34	4	4	NUM
ejpam-3438	469	35	(	(	PUNCT
ejpam-3438	469	36	2	2	NUM
ejpam-3438	469	37	)	)	PUNCT
ejpam-3438	469	38	,	,	PUNCT
ejpam-3438	469	39	2013	2013	NUM
ejpam-3438	469	40	,	,	PUNCT
ejpam-3438	469	41	17	17	NUM
ejpam-3438	469	42	-	-	SYM
ejpam-3438	469	43	23	23	NUM
ejpam-3438	469	44	[	[	X
ejpam-3438	469	45	29	29	NUM
ejpam-3438	469	46	]	]	PUNCT
ejpam-3438	469	47	a.	a.	NOUN
ejpam-3438	469	48	devika	devika	PROPN
ejpam-3438	469	49	,	,	PUNCT
ejpam-3438	469	50	r.	r.	PROPN
ejpam-3438	469	51	vani	vani	PROPN
ejpam-3438	469	52	,	,	PUNCT
ejpam-3438	469	53	on	on	ADP
ejpam-3438	469	54	πg∗β	πg∗β	PROPN
ejpam-3438	469	55	-	-	PUNCT
ejpam-3438	469	56	closed	close	VERB
ejpam-3438	469	57	in	in	ADP
ejpam-3438	469	58	topological	topological	ADJ
ejpam-3438	469	59	spaces	space	NOUN
ejpam-3438	469	60	,	,	PUNCT
ejpam-3438	469	61	journal	journal	NOUN
ejpam-3438	469	62	of	of	ADP
ejpam-3438	469	63	applied	applied	ADJ
ejpam-3438	469	64	and	and	CCONJ
ejpam-3438	469	65	computational	computational	ADJ
ejpam-3438	469	66	mathematics	mathematic	NOUN
ejpam-3438	469	67	,	,	PUNCT
ejpam-3438	469	68	7	7	NUM
ejpam-3438	469	69	(	(	PUNCT
ejpam-3438	469	70	3	3	NUM
ejpam-3438	469	71	)	)	PUNCT
ejpam-3438	469	72	,	,	PUNCT
ejpam-3438	469	73	413	413	NUM
ejpam-3438	469	74	,	,	PUNCT
ejpam-3438	469	75	2018	2018	NUM
ejpam-3438	469	76	.	.	PUNCT
ejpam-3438	470	1	[	[	X
ejpam-3438	470	2	30	30	NUM
ejpam-3438	470	3	]	]	X
ejpam-3438	470	4	r.l	r.l	PROPN
ejpam-3438	470	5	.	.	PROPN
ejpam-3438	470	6	newcomb	newcomb	PROPN
ejpam-3438	470	7	,	,	PUNCT
ejpam-3438	470	8	topologies	topology	NOUN
ejpam-3438	470	9	which	which	PRON
ejpam-3438	470	10	are	be	AUX
ejpam-3438	470	11	compact	compact	ADJ
ejpam-3438	470	12	modulo	modulo	NOUN
ejpam-3438	470	13	an	an	DET
ejpam-3438	470	14	ideal	ideal	NOUN
ejpam-3438	470	15	,	,	PUNCT
ejpam-3438	470	16	ph.d	ph.d	PROPN
ejpam-3438	470	17	.	.	PUNCT
ejpam-3438	470	18	dissertation	dissertation	PROPN
ejpam-3438	470	19	,	,	PUNCT
ejpam-3438	470	20	univ	univ	PROPN
ejpam-3438	470	21	.	.	PROPN
ejpam-3438	470	22	of	of	ADP
ejpam-3438	470	23	cal	cal	PROPN
ejpam-3438	470	24	.	.	PUNCT
ejpam-3438	471	1	at	at	ADP
ejpam-3438	471	2	santa	santa	PROPN
ejpam-3438	471	3	barbara	barbara	PROPN
ejpam-3438	471	4	,	,	PUNCT
ejpam-3438	471	5	(	(	PUNCT
ejpam-3438	471	6	1967	1967	NUM
ejpam-3438	471	7	)	)	PUNCT
ejpam-3438	471	8	.	.	PUNCT
ejpam-3438	472	1	[	[	X
ejpam-3438	472	2	31	31	NUM
ejpam-3438	472	3	]	]	X
ejpam-3438	472	4	a.s	a.s	PROPN
ejpam-3438	472	5	.	.	PROPN
ejpam-3438	472	6	mashhour	mashhour	PROPN
ejpam-3438	472	7	,	,	PUNCT
ejpam-3438	472	8	m.e	m.e	PROPN
ejpam-3438	472	9	.	.	PROPN
ejpam-3438	472	10	abd	abd	PROPN
ejpam-3438	472	11	el	el	PROPN
ejpam-3438	472	12	-	-	PROPN
ejpam-3438	472	13	monsef	monsef	ADJ
ejpam-3438	472	14	,	,	PUNCT
ejpam-3438	472	15	s.n	s.n	PROPN
ejpam-3438	472	16	.	.	PROPN
ejpam-3438	472	17	el	el	PROPN
ejpam-3438	472	18	-	-	PUNCT
ejpam-3438	472	19	deeb	deeb	PROPN
ejpam-3438	472	20	,	,	PUNCT
ejpam-3438	472	21	on	on	ADP
ejpam-3438	472	22	pre	pre	ADJ
ejpam-3438	472	23	-	-	ADJ
ejpam-3438	472	24	continuous	continuous	ADJ
ejpam-3438	472	25	and	and	CCONJ
ejpam-3438	472	26	3	3	NUM
ejpam-3438	472	27	weak	weak	ADJ
ejpam-3438	472	28	pre	pre	ADJ
ejpam-3438	472	29	-	-	ADJ
ejpam-3438	472	30	continuous	continuous	ADJ
ejpam-3438	472	31	mappings	mapping	NOUN
ejpam-3438	472	32	,	,	PUNCT
ejpam-3438	472	33	proc	proc	NOUN
ejpam-3438	472	34	math	math	NOUN
ejpam-3438	472	35	and	and	CCONJ
ejpam-3438	472	36	phys	phy	NOUN
ejpam-3438	472	37	soc	soc	PROPN
ejpam-3438	472	38	egypt	egypt	PROPN
ejpam-3438	472	39	,	,	PUNCT
ejpam-3438	472	40	53	53	NUM
ejpam-3438	472	41	,	,	PUNCT
ejpam-3438	472	42	47	47	NUM
ejpam-3438	472	43	-	-	SYM
ejpam-3438	472	44	53	53	NUM
ejpam-3438	472	45	.	.	PUNCT
ejpam-3438	473	1	[	[	X
ejpam-3438	473	2	32	32	NUM
ejpam-3438	473	3	]	]	SYM
ejpam-3438	473	4	f.i	f.i	PROPN
ejpam-3438	473	5	.	.	PROPN
ejpam-3438	473	6	michael	michael	PROPN
ejpam-3438	473	7	,	,	PUNCT
ejpam-3438	473	8	on	on	ADP
ejpam-3438	473	9	the	the	DET
ejpam-3438	473	10	semi	semi	ADJ
ejpam-3438	473	11	-	-	ADJ
ejpam-3438	473	12	open	open	ADJ
ejpam-3438	473	13	sets	set	NOUN
ejpam-3438	473	14	with	with	ADP
ejpam-3438	473	15	respect	respect	NOUN
ejpam-3438	473	16	to	to	ADP
ejpam-3438	473	17	an	an	DET
ejpam-3438	473	18	ideal	ideal	ADJ
ejpam-3438	473	19	,	,	PUNCT
ejpam-3438	473	20	european	european	ADJ
ejpam-3438	473	21	journal	journal	PROPN
ejpam-3438	473	22	of	of	ADP
ejpam-3438	473	23	pure	pure	ADJ
ejpam-3438	473	24	and	and	CCONJ
ejpam-3438	473	25	applied	applied	ADJ
ejpam-3438	473	26	mathematics	mathematic	NOUN
ejpam-3438	473	27	,	,	PUNCT
ejpam-3438	473	28	vol	vol	NOUN
ejpam-3438	473	29	.	.	PROPN
ejpam-3438	474	1	6	6	NUM
ejpam-3438	474	2	,	,	PUNCT
ejpam-3438	474	3	no	no	INTJ
ejpam-3438	474	4	.	.	NOUN
ejpam-3438	474	5	1	1	NUM
ejpam-3438	474	6	,	,	PUNCT
ejpam-3438	474	7	2013	2013	NUM
ejpam-3438	474	8	,	,	PUNCT
ejpam-3438	474	9	53	53	NUM
ejpam-3438	474	10	-	-	SYM
ejpam-3438	474	11	58	58	NUM
ejpam-3438	474	12	.	.	PUNCT