id	sid	tid	token	lemma	pos
ejpam-3807	1	1	european	european	PROPN
ejpam-3807	1	2	journal	journal	PROPN
ejpam-3807	1	3	of	of	ADP
ejpam-3807	1	4	pure	pure	ADJ
ejpam-3807	1	5	and	and	CCONJ
ejpam-3807	1	6	applied	apply	VERB
ejpam-3807	1	7	mathematics	mathematic	NOUN
ejpam-3807	1	8	vol	vol	NOUN
ejpam-3807	1	9	.	.	PROPN
ejpam-3807	2	1	13	13	NUM
ejpam-3807	2	2	,	,	PUNCT
ejpam-3807	2	3	no	no	INTJ
ejpam-3807	2	4	.	.	NOUN
ejpam-3807	2	5	4	4	NUM
ejpam-3807	2	6	,	,	PUNCT
ejpam-3807	2	7	2020	2020	NUM
ejpam-3807	2	8	,	,	PUNCT
ejpam-3807	2	9	830	830	NUM
ejpam-3807	2	10	-	-	SYM
ejpam-3807	2	11	839	839	NUM
ejpam-3807	2	12	issn	issn	PROPN
ejpam-3807	2	13	1307	1307	NUM
ejpam-3807	2	14	-	-	SYM
ejpam-3807	2	15	5543	5543	NUM
ejpam-3807	2	16	–	–	PUNCT
ejpam-3807	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3807	2	18	published	publish	VERB
ejpam-3807	2	19	by	by	ADP
ejpam-3807	2	20	new	new	PROPN
ejpam-3807	2	21	york	york	PROPN
ejpam-3807	2	22	business	business	PROPN
ejpam-3807	2	23	global	global	PROPN
ejpam-3807	2	24	on	on	ADP
ejpam-3807	2	25	b	b	NOUN
ejpam-3807	2	26	-	-	PUNCT
ejpam-3807	2	27	ideals	ideal	NOUN
ejpam-3807	2	28	in	in	ADP
ejpam-3807	2	29	a	a	DET
ejpam-3807	2	30	topological	topological	ADJ
ejpam-3807	2	31	b	b	NOUN
ejpam-3807	2	32	-	-	PUNCT
ejpam-3807	2	33	algebra	algebra	NOUN
ejpam-3807	2	34	and	and	CCONJ
ejpam-3807	2	35	the	the	DET
ejpam-3807	2	36	uniform	uniform	ADJ
ejpam-3807	2	37	b	b	X
ejpam-3807	2	38	-	-	PUNCT
ejpam-3807	2	39	topological	topological	ADJ
ejpam-3807	2	40	space	space	NOUN
ejpam-3807	2	41	katrina	katrina	PROPN
ejpam-3807	2	42	e.	e.	PROPN
ejpam-3807	2	43	belleza1,∗	belleza1,∗	PROPN
ejpam-3807	2	44	,	,	PUNCT
ejpam-3807	2	45	jocelyn	jocelyn	PROPN
ejpam-3807	2	46	p.	p.	PROPN
ejpam-3807	2	47	vilela2	vilela2	PROPN
ejpam-3807	2	48	1	1	NUM
ejpam-3807	2	49	department	department	NOUN
ejpam-3807	2	50	of	of	ADP
ejpam-3807	2	51	computer	computer	NOUN
ejpam-3807	2	52	,	,	PUNCT
ejpam-3807	2	53	information	information	NOUN
ejpam-3807	2	54	sciences	science	NOUN
ejpam-3807	2	55	and	and	CCONJ
ejpam-3807	2	56	mathematics	mathematic	NOUN
ejpam-3807	2	57	,	,	PUNCT
ejpam-3807	2	58	school	school	NOUN
ejpam-3807	2	59	of	of	ADP
ejpam-3807	2	60	arts	art	NOUN
ejpam-3807	2	61	and	and	CCONJ
ejpam-3807	2	62	sciences	science	NOUN
ejpam-3807	2	63	,	,	PUNCT
ejpam-3807	2	64	university	university	NOUN
ejpam-3807	2	65	of	of	ADP
ejpam-3807	2	66	san	san	PROPN
ejpam-3807	2	67	carlos	carlos	PROPN
ejpam-3807	2	68	,	,	PUNCT
ejpam-3807	2	69	6000	6000	NUM
ejpam-3807	2	70	cebu	cebu	NOUN
ejpam-3807	2	71	city	city	NOUN
ejpam-3807	2	72	,	,	PUNCT
ejpam-3807	2	73	philippines	philippines	PROPN
ejpam-3807	2	74	2	2	NUM
ejpam-3807	2	75	department	department	NOUN
ejpam-3807	2	76	of	of	ADP
ejpam-3807	2	77	mathematics	mathematic	NOUN
ejpam-3807	2	78	and	and	CCONJ
ejpam-3807	2	79	statistics	statistic	NOUN
ejpam-3807	2	80	,	,	PUNCT
ejpam-3807	2	81	college	college	NOUN
ejpam-3807	2	82	of	of	ADP
ejpam-3807	2	83	science	science	NOUN
ejpam-3807	2	84	and	and	CCONJ
ejpam-3807	2	85	mathematics	mathematic	NOUN
ejpam-3807	2	86	,	,	PUNCT
ejpam-3807	2	87	center	center	NOUN
ejpam-3807	2	88	of	of	ADP
ejpam-3807	2	89	graph	graph	NOUN
ejpam-3807	2	90	theory	theory	NOUN
ejpam-3807	2	91	,	,	PUNCT
ejpam-3807	2	92	algebra	algebra	NOUN
ejpam-3807	2	93	and	and	CCONJ
ejpam-3807	2	94	analysis	analysis	NOUN
ejpam-3807	2	95	,	,	PUNCT
ejpam-3807	2	96	premier	premier	PROPN
ejpam-3807	2	97	research	research	PROPN
ejpam-3807	2	98	institute	institute	PROPN
ejpam-3807	2	99	of	of	ADP
ejpam-3807	2	100	science	science	NOUN
ejpam-3807	2	101	and	and	CCONJ
ejpam-3807	2	102	mathematics	mathematic	NOUN
ejpam-3807	2	103	,	,	PUNCT
ejpam-3807	2	104	mindanao	mindanao	PROPN
ejpam-3807	2	105	state	state	PROPN
ejpam-3807	2	106	university	university	PROPN
ejpam-3807	2	107	-	-	PUNCT
ejpam-3807	2	108	iligan	iligan	PROPN
ejpam-3807	2	109	institute	institute	PROPN
ejpam-3807	2	110	of	of	ADP
ejpam-3807	2	111	technology	technology	PROPN
ejpam-3807	2	112	,	,	PUNCT
ejpam-3807	2	113	9200	9200	NUM
ejpam-3807	2	114	iligan	iligan	ADJ
ejpam-3807	2	115	city	city	NOUN
ejpam-3807	2	116	,	,	PUNCT
ejpam-3807	2	117	philippines	philippine	NOUN
ejpam-3807	2	118	abstract	abstract	ADJ
ejpam-3807	2	119	.	.	PUNCT
ejpam-3807	3	1	this	this	DET
ejpam-3807	3	2	paper	paper	NOUN
ejpam-3807	3	3	presents	present	VERB
ejpam-3807	3	4	the	the	DET
ejpam-3807	3	5	characterizations	characterization	NOUN
ejpam-3807	3	6	and	and	CCONJ
ejpam-3807	3	7	properties	property	NOUN
ejpam-3807	3	8	of	of	ADP
ejpam-3807	3	9	b	b	NOUN
ejpam-3807	3	10	-	-	PUNCT
ejpam-3807	3	11	ideals	ideal	NOUN
ejpam-3807	3	12	in	in	ADP
ejpam-3807	3	13	a	a	DET
ejpam-3807	3	14	topological	topological	ADJ
ejpam-3807	3	15	b	b	NOUN
ejpam-3807	3	16	-	-	PUNCT
ejpam-3807	3	17	algebra	algebra	NOUN
ejpam-3807	3	18	and	and	CCONJ
ejpam-3807	3	19	introduces	introduce	VERB
ejpam-3807	3	20	the	the	DET
ejpam-3807	3	21	uniform	uniform	ADJ
ejpam-3807	3	22	topology	topology	NOUN
ejpam-3807	3	23	on	on	ADP
ejpam-3807	3	24	a	a	DET
ejpam-3807	3	25	b	b	NOUN
ejpam-3807	3	26	-	-	PUNCT
ejpam-3807	3	27	algebra	algebra	NOUN
ejpam-3807	3	28	in	in	ADP
ejpam-3807	3	29	terms	term	NOUN
ejpam-3807	3	30	of	of	ADP
ejpam-3807	3	31	its	its	PRON
ejpam-3807	3	32	b	b	NOUN
ejpam-3807	3	33	-	-	PUNCT
ejpam-3807	3	34	ideals	ideal	NOUN
ejpam-3807	3	35	.	.	PUNCT
ejpam-3807	4	1	moreover	moreover	ADV
ejpam-3807	4	2	,	,	PUNCT
ejpam-3807	4	3	this	this	DET
ejpam-3807	4	4	paper	paper	NOUN
ejpam-3807	4	5	shows	show	VERB
ejpam-3807	4	6	that	that	SCONJ
ejpam-3807	4	7	a	a	DET
ejpam-3807	4	8	uniform	uniform	ADJ
ejpam-3807	4	9	b	b	X
ejpam-3807	4	10	-	-	PUNCT
ejpam-3807	4	11	topological	topological	ADJ
ejpam-3807	4	12	space	space	NOUN
ejpam-3807	4	13	is	be	AUX
ejpam-3807	4	14	a	a	DET
ejpam-3807	4	15	topological	topological	ADJ
ejpam-3807	4	16	b	b	NOUN
ejpam-3807	4	17	-	-	PUNCT
ejpam-3807	4	18	algebra	algebra	NOUN
ejpam-3807	4	19	.	.	PUNCT
ejpam-3807	5	1	2020	2020	NUM
ejpam-3807	5	2	mathematics	mathematic	NOUN
ejpam-3807	5	3	subject	subject	NOUN
ejpam-3807	5	4	classifications	classification	NOUN
ejpam-3807	5	5	:	:	PUNCT
ejpam-3807	5	6	54a05	54a05	NUM
ejpam-3807	5	7	,	,	PUNCT
ejpam-3807	5	8	54f65	54f65	NUM
ejpam-3807	5	9	,	,	PUNCT
ejpam-3807	5	10	54h99	54h99	NUM
ejpam-3807	5	11	,	,	PUNCT
ejpam-3807	5	12	55m99	55m99	NUM
ejpam-3807	5	13	key	key	ADJ
ejpam-3807	5	14	words	word	NOUN
ejpam-3807	5	15	and	and	CCONJ
ejpam-3807	5	16	phrases	phrase	NOUN
ejpam-3807	5	17	:	:	PUNCT
ejpam-3807	5	18	b	b	X
ejpam-3807	5	19	-	-	PUNCT
ejpam-3807	5	20	ideals	ideal	NOUN
ejpam-3807	5	21	,	,	PUNCT
ejpam-3807	5	22	topological	topological	ADJ
ejpam-3807	5	23	b	b	NOUN
ejpam-3807	5	24	-	-	PUNCT
ejpam-3807	5	25	algebra	algebra	NOUN
ejpam-3807	5	26	,	,	PUNCT
ejpam-3807	5	27	uniform	uniform	ADJ
ejpam-3807	5	28	b	b	NOUN
ejpam-3807	5	29	-	-	PUNCT
ejpam-3807	5	30	topology	topology	NOUN
ejpam-3807	5	31	1	1	NUM
ejpam-3807	5	32	.	.	PUNCT
ejpam-3807	6	1	introduction	introduction	NOUN
ejpam-3807	6	2	y.b	y.b	PROPN
ejpam-3807	6	3	.	.	PROPN
ejpam-3807	6	4	jun	jun	PROPN
ejpam-3807	6	5	et	et	PROPN
ejpam-3807	6	6	al	al	PROPN
ejpam-3807	6	7	.	.	PUNCT
ejpam-3807	7	1	[	[	X
ejpam-3807	7	2	4	4	X
ejpam-3807	7	3	]	]	PUNCT
ejpam-3807	7	4	in	in	ADP
ejpam-3807	7	5	1999	1999	NUM
ejpam-3807	7	6	introduced	introduce	VERB
ejpam-3807	7	7	topological	topological	ADJ
ejpam-3807	7	8	bci	bci	NOUN
ejpam-3807	7	9	-	-	PUNCT
ejpam-3807	7	10	algebras	algebra	NOUN
ejpam-3807	7	11	,	,	PUNCT
ejpam-3807	7	12	provided	provide	VERB
ejpam-3807	7	13	some	some	DET
ejpam-3807	7	14	properties	property	NOUN
ejpam-3807	7	15	on	on	ADP
ejpam-3807	7	16	this	this	DET
ejpam-3807	7	17	structure	structure	NOUN
ejpam-3807	7	18	,	,	PUNCT
ejpam-3807	7	19	and	and	CCONJ
ejpam-3807	7	20	characterized	characterize	VERB
ejpam-3807	7	21	a	a	DET
ejpam-3807	7	22	topological	topological	ADJ
ejpam-3807	7	23	bci	bci	NOUN
ejpam-3807	7	24	-	-	NOUN
ejpam-3807	7	25	algebra	algebra	NOUN
ejpam-3807	7	26	in	in	ADP
ejpam-3807	7	27	terms	term	NOUN
ejpam-3807	7	28	of	of	ADP
ejpam-3807	7	29	neighborhoods	neighborhood	NOUN
ejpam-3807	7	30	.	.	PUNCT
ejpam-3807	8	1	in	in	ADP
ejpam-3807	8	2	2002	2002	NUM
ejpam-3807	8	3	,	,	PUNCT
ejpam-3807	8	4	j.neggers	j.negger	NOUN
ejpam-3807	8	5	and	and	CCONJ
ejpam-3807	8	6	h.s	h.s	PROPN
ejpam-3807	8	7	.	.	PROPN
ejpam-3807	8	8	kim	kim	PROPN
ejpam-3807	9	1	[	[	X
ejpam-3807	9	2	9	9	NUM
ejpam-3807	9	3	]	]	PUNCT
ejpam-3807	9	4	introduced	introduce	VERB
ejpam-3807	9	5	and	and	CCONJ
ejpam-3807	9	6	investigated	investigate	VERB
ejpam-3807	9	7	b	b	NOUN
ejpam-3807	9	8	-	-	PUNCT
ejpam-3807	9	9	algebras	algebras	X
ejpam-3807	9	10	.	.	PUNCT
ejpam-3807	10	1	in	in	ADP
ejpam-3807	10	2	2017	2017	NUM
ejpam-3807	10	3	,	,	PUNCT
ejpam-3807	10	4	s.	s.	PROPN
ejpam-3807	10	5	mehrshad	mehrshad	VERB
ejpam-3807	10	6	and	and	CCONJ
ejpam-3807	10	7	j.	j.	PROPN
ejpam-3807	10	8	golzarpoor	golzarpoor	PROPN
ejpam-3807	11	1	[	[	X
ejpam-3807	11	2	7	7	NUM
ejpam-3807	11	3	]	]	PUNCT
ejpam-3807	11	4	provided	provide	VERB
ejpam-3807	11	5	some	some	DET
ejpam-3807	11	6	properties	property	NOUN
ejpam-3807	11	7	of	of	ADP
ejpam-3807	11	8	uniform	uniform	ADJ
ejpam-3807	11	9	topology	topology	NOUN
ejpam-3807	11	10	and	and	CCONJ
ejpam-3807	11	11	topological	topological	ADJ
ejpam-3807	11	12	be	be	NOUN
ejpam-3807	11	13	-	-	PUNCT
ejpam-3807	11	14	algebras	algebras	X
ejpam-3807	11	15	.	.	PUNCT
ejpam-3807	12	1	a	a	DET
ejpam-3807	12	2	recent	recent	ADJ
ejpam-3807	12	3	study	study	NOUN
ejpam-3807	12	4	on	on	ADP
ejpam-3807	12	5	topological	topological	PROPN
ejpam-3807	12	6	b	b	X
ejpam-3807	12	7	-	-	PUNCT
ejpam-3807	12	8	algebras	algebras	PROPN
ejpam-3807	12	9	was	be	AUX
ejpam-3807	12	10	conducted	conduct	VERB
ejpam-3807	12	11	by	by	ADP
ejpam-3807	12	12	n.c	n.c	PROPN
ejpam-3807	12	13	.	.	PROPN
ejpam-3807	12	14	gonzaga	gonzaga	PROPN
ejpam-3807	12	15	,	,	PUNCT
ejpam-3807	12	16	jr	jr	PROPN
ejpam-3807	12	17	.	.	PUNCT
ejpam-3807	13	1	[	[	X
ejpam-3807	13	2	6	6	NUM
ejpam-3807	13	3	]	]	PUNCT
ejpam-3807	13	4	in	in	ADP
ejpam-3807	13	5	2019	2019	NUM
ejpam-3807	13	6	,	,	PUNCT
ejpam-3807	13	7	which	which	PRON
ejpam-3807	13	8	characterized	characterize	VERB
ejpam-3807	13	9	a	a	DET
ejpam-3807	13	10	topological	topological	ADJ
ejpam-3807	13	11	b	b	NOUN
ejpam-3807	13	12	-	-	PUNCT
ejpam-3807	13	13	algebra	algebra	NOUN
ejpam-3807	13	14	with	with	ADP
ejpam-3807	13	15	respect	respect	NOUN
ejpam-3807	13	16	to	to	ADP
ejpam-3807	13	17	neighborhoods	neighborhood	NOUN
ejpam-3807	13	18	.	.	PUNCT
ejpam-3807	14	1	this	this	DET
ejpam-3807	14	2	paper	paper	NOUN
ejpam-3807	14	3	provides	provide	VERB
ejpam-3807	14	4	some	some	DET
ejpam-3807	14	5	properties	property	NOUN
ejpam-3807	14	6	of	of	ADP
ejpam-3807	14	7	topological	topological	ADJ
ejpam-3807	14	8	b	b	NOUN
ejpam-3807	14	9	-	-	PUNCT
ejpam-3807	14	10	algebra	algebra	NOUN
ejpam-3807	14	11	,	,	PUNCT
ejpam-3807	14	12	describes	describe	VERB
ejpam-3807	14	13	the	the	DET
ejpam-3807	14	14	b	b	NOUN
ejpam-3807	14	15	-	-	PUNCT
ejpam-3807	14	16	ideals	ideal	NOUN
ejpam-3807	14	17	in	in	ADP
ejpam-3807	14	18	a	a	DET
ejpam-3807	14	19	topological	topological	ADJ
ejpam-3807	14	20	b	b	NOUN
ejpam-3807	14	21	-	-	PUNCT
ejpam-3807	14	22	algebra	algebra	NOUN
ejpam-3807	14	23	,	,	PUNCT
ejpam-3807	14	24	and	and	CCONJ
ejpam-3807	14	25	characterizes	characterize	VERB
ejpam-3807	14	26	uniform	uniform	ADJ
ejpam-3807	14	27	b	b	NOUN
ejpam-3807	14	28	-	-	PUNCT
ejpam-3807	14	29	topology	topology	NOUN
ejpam-3807	14	30	in	in	ADP
ejpam-3807	14	31	a	a	DET
ejpam-3807	14	32	b	b	NOUN
ejpam-3807	14	33	-	-	PUNCT
ejpam-3807	14	34	algebra	algebra	NOUN
ejpam-3807	14	35	.	.	PUNCT
ejpam-3807	15	1	∗corresponding	∗corresponde	VERB
ejpam-3807	15	2	author	author	NOUN
ejpam-3807	15	3	.	.	PUNCT
ejpam-3807	16	1	doi	doi	NOUN
ejpam-3807	16	2	:	:	PUNCT
ejpam-3807	16	3	https://doi.org/10.29020/nybg.ejpam.v13i4.3807	https://doi.org/10.29020/nybg.ejpam.v13i4.3807	ADJ
ejpam-3807	16	4	email	email	NOUN
ejpam-3807	16	5	addresses	address	NOUN
ejpam-3807	16	6	:	:	PUNCT
ejpam-3807	16	7	kebelleza@usc.edu.ph	kebelleza@usc.edu.ph	PROPN
ejpam-3807	16	8	(	(	PUNCT
ejpam-3807	16	9	k.	k.	PROPN
ejpam-3807	16	10	belleza	belleza	PROPN
ejpam-3807	16	11	)	)	PUNCT
ejpam-3807	16	12	,	,	PUNCT
ejpam-3807	16	13	jocelyn.vilela@g.msuiit.edu.ph	jocelyn.vilela@g.msuiit.edu.ph	PROPN
ejpam-3807	16	14	(	(	PUNCT
ejpam-3807	16	15	j.	j.	PROPN
ejpam-3807	16	16	vilela	vilela	PROPN
ejpam-3807	16	17	)	)	PUNCT
ejpam-3807	16	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-3807	17	1	830	830	NUM
ejpam-3807	17	2	c	c	NOUN
ejpam-3807	17	3	©	©	NOUN
ejpam-3807	17	4	2020	2020	NUM
ejpam-3807	17	5	ejpam	ejpam	VERB
ejpam-3807	17	6	all	all	DET
ejpam-3807	17	7	rights	right	NOUN
ejpam-3807	17	8	reserved	reserve	VERB
ejpam-3807	17	9	.	.	PUNCT
ejpam-3807	18	1	k.	k.	PROPN
ejpam-3807	18	2	belleza	belleza	PROPN
ejpam-3807	18	3	,	,	PUNCT
ejpam-3807	18	4	j.	j.	PROPN
ejpam-3807	18	5	vilela	vilela	PROPN
ejpam-3807	18	6	/	/	SYM
ejpam-3807	18	7	eur	eur	PROPN
ejpam-3807	18	8	.	.	PUNCT
ejpam-3807	19	1	j.	j.	PROPN
ejpam-3807	19	2	pure	pure	PROPN
ejpam-3807	19	3	appl	appl	PROPN
ejpam-3807	19	4	.	.	PROPN
ejpam-3807	19	5	math	math	PROPN
ejpam-3807	19	6	,	,	PUNCT
ejpam-3807	19	7	13	13	NUM
ejpam-3807	19	8	(	(	PUNCT
ejpam-3807	19	9	4	4	NUM
ejpam-3807	19	10	)	)	PUNCT
ejpam-3807	19	11	(	(	PUNCT
ejpam-3807	19	12	2020	2020	NUM
ejpam-3807	19	13	)	)	PUNCT
ejpam-3807	19	14	,	,	PUNCT
ejpam-3807	19	15	830	830	NUM
ejpam-3807	19	16	-	-	SYM
ejpam-3807	19	17	839	839	NUM
ejpam-3807	19	18	831	831	NUM
ejpam-3807	19	19	2	2	NUM
ejpam-3807	19	20	.	.	PUNCT
ejpam-3807	19	21	preliminaries	preliminary	NOUN
ejpam-3807	19	22	an	an	DET
ejpam-3807	19	23	algebra	algebra	NOUN
ejpam-3807	19	24	of	of	ADP
ejpam-3807	19	25	type	type	NOUN
ejpam-3807	19	26	(	(	PUNCT
ejpam-3807	19	27	2,0	2,0	NOUN
ejpam-3807	19	28	)	)	PUNCT
ejpam-3807	19	29	is	be	AUX
ejpam-3807	19	30	an	an	DET
ejpam-3807	19	31	algebra	algebra	NOUN
ejpam-3807	19	32	with	with	ADP
ejpam-3807	19	33	a	a	DET
ejpam-3807	19	34	binary	binary	ADJ
ejpam-3807	19	35	operation	operation	NOUN
ejpam-3807	19	36	and	and	CCONJ
ejpam-3807	19	37	a	a	DET
ejpam-3807	19	38	constant	constant	ADJ
ejpam-3807	19	39	element	element	NOUN
ejpam-3807	19	40	.	.	PUNCT
ejpam-3807	20	1	definition	definition	NOUN
ejpam-3807	20	2	1	1	NUM
ejpam-3807	20	3	.	.	PUNCT
ejpam-3807	21	1	[	[	X
ejpam-3807	21	2	9	9	NUM
ejpam-3807	21	3	]	]	PUNCT
ejpam-3807	21	4	a	a	DET
ejpam-3807	21	5	b	b	X
ejpam-3807	21	6	-	-	PUNCT
ejpam-3807	21	7	algebra	algebra	NOUN
ejpam-3807	21	8	is	be	AUX
ejpam-3807	21	9	a	a	DET
ejpam-3807	21	10	non	non	ADJ
ejpam-3807	21	11	-	-	ADJ
ejpam-3807	21	12	empty	empty	ADJ
ejpam-3807	21	13	set	set	NOUN
ejpam-3807	21	14	x	x	PUNCT
ejpam-3807	21	15	with	with	ADP
ejpam-3807	21	16	a	a	DET
ejpam-3807	21	17	constant	constant	ADJ
ejpam-3807	21	18	0	0	NUM
ejpam-3807	21	19	and	and	CCONJ
ejpam-3807	21	20	a	a	DET
ejpam-3807	21	21	binary	binary	ADJ
ejpam-3807	21	22	operation	operation	NOUN
ejpam-3807	21	23	“	"	PUNCT
ejpam-3807	21	24	∗	∗	NOUN
ejpam-3807	21	25	”	"	PUNCT
ejpam-3807	21	26	satisfying	satisfy	VERB
ejpam-3807	21	27	the	the	DET
ejpam-3807	21	28	following	following	ADJ
ejpam-3807	21	29	axioms	axiom	NOUN
ejpam-3807	21	30	for	for	ADP
ejpam-3807	21	31	all	all	DET
ejpam-3807	21	32	x	x	NOUN
ejpam-3807	21	33	,	,	PUNCT
ejpam-3807	21	34	y	y	PROPN
ejpam-3807	21	35	,	,	PUNCT
ejpam-3807	21	36	z	z	VERB
ejpam-3807	21	37	in	in	ADP
ejpam-3807	21	38	x	x	NOUN
ejpam-3807	21	39	:	:	PUNCT
ejpam-3807	21	40	(	(	PUNCT
ejpam-3807	21	41	b1	b1	NOUN
ejpam-3807	21	42	)	)	PUNCT
ejpam-3807	21	43	x	x	SYM
ejpam-3807	21	44	∗	∗	NOUN
ejpam-3807	21	45	x	x	X
ejpam-3807	22	1	=	=	SYM
ejpam-3807	22	2	0	0	NUM
ejpam-3807	22	3	(	(	PUNCT
ejpam-3807	22	4	b2	b2	NOUN
ejpam-3807	22	5	)	)	PUNCT
ejpam-3807	22	6	x	x	SYM
ejpam-3807	22	7	∗	∗	NOUN
ejpam-3807	22	8	0	0	NUM
ejpam-3807	23	1	=	=	SYM
ejpam-3807	23	2	x	x	X
ejpam-3807	23	3	(	(	PUNCT
ejpam-3807	23	4	b3	b3	PROPN
ejpam-3807	23	5	)	)	PUNCT
ejpam-3807	23	6	(	(	PUNCT
ejpam-3807	23	7	x	x	SYM
ejpam-3807	23	8	∗	∗	PROPN
ejpam-3807	23	9	y	y	NOUN
ejpam-3807	23	10	)	)	PUNCT
ejpam-3807	23	11	∗	∗	NOUN
ejpam-3807	23	12	z	z	NOUN
ejpam-3807	23	13	=	=	PUNCT
ejpam-3807	23	14	x	x	X
ejpam-3807	23	15	∗	∗	NOUN
ejpam-3807	24	1	[	[	X
ejpam-3807	24	2	z	z	X
ejpam-3807	24	3	∗	∗	NOUN
ejpam-3807	24	4	(	(	PUNCT
ejpam-3807	24	5	0	0	NUM
ejpam-3807	24	6	∗	∗	NOUN
ejpam-3807	24	7	y	y	PROPN
ejpam-3807	24	8	)	)	PUNCT
ejpam-3807	24	9	]	]	PUNCT
ejpam-3807	24	10	example	example	NOUN
ejpam-3807	25	1	1	1	NUM
ejpam-3807	25	2	.	.	PUNCT
ejpam-3807	26	1	[	[	X
ejpam-3807	26	2	8	8	NUM
ejpam-3807	26	3	]	]	PUNCT
ejpam-3807	26	4	let	let	VERB
ejpam-3807	26	5	x	x	PUNCT
ejpam-3807	26	6	=	=	PUNCT
ejpam-3807	26	7	{	{	PUNCT
ejpam-3807	26	8	0	0	NUM
ejpam-3807	26	9	,	,	PUNCT
ejpam-3807	26	10	a	a	DET
ejpam-3807	26	11	,	,	PUNCT
ejpam-3807	26	12	b	b	NOUN
ejpam-3807	26	13	,	,	PUNCT
ejpam-3807	26	14	c	c	NOUN
ejpam-3807	26	15	,	,	PUNCT
ejpam-3807	26	16	d	d	NOUN
ejpam-3807	26	17	,	,	PUNCT
ejpam-3807	26	18	e	e	AUX
ejpam-3807	26	19	}	}	PUNCT
ejpam-3807	26	20	be	be	AUX
ejpam-3807	26	21	a	a	DET
ejpam-3807	26	22	set	set	NOUN
ejpam-3807	26	23	with	with	ADP
ejpam-3807	26	24	the	the	DET
ejpam-3807	26	25	following	follow	VERB
ejpam-3807	26	26	cayley	cayley	ADJ
ejpam-3807	26	27	table	table	NOUN
ejpam-3807	26	28	:	:	PUNCT
ejpam-3807	26	29	∗	∗	NOUN
ejpam-3807	26	30	0	0	PUNCT
ejpam-3807	27	1	a	a	DET
ejpam-3807	27	2	b	b	NOUN
ejpam-3807	27	3	c	c	NOUN
ejpam-3807	27	4	d	d	X
ejpam-3807	27	5	e	e	X
ejpam-3807	27	6	0	0	NUM
ejpam-3807	27	7	0	0	NUM
ejpam-3807	27	8	b	b	PROPN
ejpam-3807	27	9	a	a	DET
ejpam-3807	27	10	c	c	NOUN
ejpam-3807	27	11	d	d	X
ejpam-3807	27	12	e	e	X
ejpam-3807	27	13	a	a	DET
ejpam-3807	27	14	a	a	DET
ejpam-3807	27	15	0	0	NUM
ejpam-3807	27	16	b	b	NOUN
ejpam-3807	27	17	d	d	PROPN
ejpam-3807	27	18	e	e	PROPN
ejpam-3807	27	19	c	c	PROPN
ejpam-3807	27	20	b	b	PROPN
ejpam-3807	27	21	b	b	PROPN
ejpam-3807	27	22	a	a	DET
ejpam-3807	27	23	0	0	NUM
ejpam-3807	27	24	e	e	NOUN
ejpam-3807	27	25	c	c	NOUN
ejpam-3807	27	26	d	d	X
ejpam-3807	27	27	c	c	NOUN
ejpam-3807	27	28	c	c	NOUN
ejpam-3807	27	29	d	d	X
ejpam-3807	27	30	e	e	X
ejpam-3807	27	31	0	0	PROPN
ejpam-3807	27	32	b	b	X
ejpam-3807	27	33	a	a	X
ejpam-3807	27	34	d	d	X
ejpam-3807	27	35	d	d	X
ejpam-3807	27	36	e	e	X
ejpam-3807	27	37	c	c	PROPN
ejpam-3807	27	38	a	a	DET
ejpam-3807	27	39	0	0	NUM
ejpam-3807	27	40	b	b	NOUN
ejpam-3807	27	41	e	e	NOUN
ejpam-3807	27	42	e	e	X
ejpam-3807	27	43	c	c	PROPN
ejpam-3807	27	44	d	d	PROPN
ejpam-3807	27	45	b	b	PROPN
ejpam-3807	27	46	a	a	DET
ejpam-3807	27	47	0	0	NUM
ejpam-3807	27	48	then	then	ADV
ejpam-3807	27	49	(	(	PUNCT
ejpam-3807	27	50	x	x	X
ejpam-3807	27	51	,	,	PUNCT
ejpam-3807	27	52	∗	∗	NOUN
ejpam-3807	27	53	,	,	PUNCT
ejpam-3807	27	54	0	0	NUM
ejpam-3807	27	55	)	)	PUNCT
ejpam-3807	27	56	is	be	AUX
ejpam-3807	27	57	a	a	DET
ejpam-3807	27	58	b	b	NOUN
ejpam-3807	27	59	-	-	PUNCT
ejpam-3807	27	60	algebra	algebra	NOUN
ejpam-3807	27	61	.	.	PUNCT
ejpam-3807	28	1	definition	definition	NOUN
ejpam-3807	28	2	2	2	NUM
ejpam-3807	28	3	.	.	PUNCT
ejpam-3807	29	1	[	[	X
ejpam-3807	29	2	6	6	NUM
ejpam-3807	29	3	]	]	PUNCT
ejpam-3807	29	4	let	let	VERB
ejpam-3807	29	5	a	a	PRON
ejpam-3807	29	6	and	and	CCONJ
ejpam-3807	29	7	b	b	NOUN
ejpam-3807	29	8	be	be	AUX
ejpam-3807	29	9	nonempty	nonempty	X
ejpam-3807	29	10	subsets	subset	NOUN
ejpam-3807	29	11	of	of	ADP
ejpam-3807	29	12	a	a	DET
ejpam-3807	29	13	b	b	NOUN
ejpam-3807	29	14	-	-	PUNCT
ejpam-3807	29	15	algebra	algebra	NOUN
ejpam-3807	29	16	x.	x.	NOUN
ejpam-3807	30	1	the	the	DET
ejpam-3807	30	2	product	product	NOUN
ejpam-3807	30	3	of	of	ADP
ejpam-3807	30	4	a	a	PRON
ejpam-3807	30	5	and	and	CCONJ
ejpam-3807	30	6	b	b	NOUN
ejpam-3807	30	7	,	,	PUNCT
ejpam-3807	30	8	denoted	denote	VERB
ejpam-3807	30	9	by	by	ADP
ejpam-3807	30	10	a	a	DET
ejpam-3807	30	11	∗b	∗b	NOUN
ejpam-3807	30	12	,	,	PUNCT
ejpam-3807	30	13	is	be	AUX
ejpam-3807	30	14	given	give	VERB
ejpam-3807	30	15	by	by	ADP
ejpam-3807	30	16	a	a	DET
ejpam-3807	30	17	∗b	∗b	NOUN
ejpam-3807	30	18	=	=	SYM
ejpam-3807	30	19	{	{	PUNCT
ejpam-3807	30	20	a	a	DET
ejpam-3807	30	21	∗	∗	NOUN
ejpam-3807	30	22	b|a	b|a	PUNCT
ejpam-3807	31	1	∈	∈	PROPN
ejpam-3807	31	2	a	a	PRON
ejpam-3807	31	3	,	,	PUNCT
ejpam-3807	31	4	b	b	PROPN
ejpam-3807	31	5	∈	∈	PROPN
ejpam-3807	31	6	b	b	NOUN
ejpam-3807	31	7	}	}	PUNCT
ejpam-3807	31	8	.	.	PUNCT
ejpam-3807	32	1	lemma	lemma	PROPN
ejpam-3807	32	2	1	1	NUM
ejpam-3807	32	3	.	.	PUNCT
ejpam-3807	33	1	[	[	X
ejpam-3807	33	2	9	9	NUM
ejpam-3807	33	3	]	]	X
ejpam-3807	33	4	let	let	VERB
ejpam-3807	33	5	(	(	PUNCT
ejpam-3807	33	6	x	x	NOUN
ejpam-3807	33	7	,	,	PUNCT
ejpam-3807	33	8	∗	∗	NOUN
ejpam-3807	33	9	,	,	PUNCT
ejpam-3807	33	10	0	0	NUM
ejpam-3807	33	11	)	)	PUNCT
ejpam-3807	33	12	be	be	AUX
ejpam-3807	33	13	a	a	DET
ejpam-3807	33	14	b	b	NOUN
ejpam-3807	33	15	-	-	PUNCT
ejpam-3807	33	16	algebra	algebra	NOUN
ejpam-3807	33	17	.	.	PUNCT
ejpam-3807	34	1	then	then	ADV
ejpam-3807	34	2	for	for	ADP
ejpam-3807	34	3	any	any	DET
ejpam-3807	34	4	x	x	NOUN
ejpam-3807	34	5	,	,	PUNCT
ejpam-3807	34	6	y	y	PROPN
ejpam-3807	34	7	∈	∈	PROPN
ejpam-3807	34	8	x	x	X
ejpam-3807	34	9	,	,	PUNCT
ejpam-3807	34	10	(	(	PUNCT
ejpam-3807	34	11	i	i	NOUN
ejpam-3807	34	12	)	)	PUNCT
ejpam-3807	34	13	x	x	PROPN
ejpam-3807	35	1	∗	∗	NOUN
ejpam-3807	35	2	y	y	NOUN
ejpam-3807	35	3	=	=	SYM
ejpam-3807	35	4	0	0	NUM
ejpam-3807	35	5	implies	imply	VERB
ejpam-3807	35	6	x	x	PUNCT
ejpam-3807	35	7	=	=	SYM
ejpam-3807	35	8	y	y	PROPN
ejpam-3807	35	9	;	;	PUNCT
ejpam-3807	35	10	(	(	PUNCT
ejpam-3807	35	11	ii	ii	NOUN
ejpam-3807	35	12	)	)	PUNCT
ejpam-3807	35	13	0	0	NUM
ejpam-3807	35	14	∗	∗	NOUN
ejpam-3807	35	15	x	x	X
ejpam-3807	35	16	=	=	SYM
ejpam-3807	35	17	0	0	NUM
ejpam-3807	35	18	∗	∗	NOUN
ejpam-3807	35	19	y	y	PROPN
ejpam-3807	35	20	implies	imply	VERB
ejpam-3807	35	21	x	x	PUNCT
ejpam-3807	35	22	=	=	SYM
ejpam-3807	35	23	y	y	PROPN
ejpam-3807	35	24	;	;	PUNCT
ejpam-3807	35	25	(	(	PUNCT
ejpam-3807	35	26	iii	iii	NOUN
ejpam-3807	35	27	)	)	PUNCT
ejpam-3807	35	28	0	0	NUM
ejpam-3807	35	29	∗	∗	NOUN
ejpam-3807	35	30	(	(	PUNCT
ejpam-3807	35	31	0	0	NUM
ejpam-3807	35	32	∗	∗	NOUN
ejpam-3807	35	33	x	x	NOUN
ejpam-3807	35	34	)	)	PUNCT
ejpam-3807	35	35	=	=	PUNCT
ejpam-3807	35	36	x.	x.	NOUN
ejpam-3807	35	37	definition	definition	NOUN
ejpam-3807	35	38	3	3	NUM
ejpam-3807	35	39	.	.	PUNCT
ejpam-3807	36	1	[	[	X
ejpam-3807	36	2	10	10	NUM
ejpam-3807	36	3	]	]	X
ejpam-3807	36	4	let	let	VERB
ejpam-3807	36	5	(	(	PUNCT
ejpam-3807	36	6	x	x	NOUN
ejpam-3807	36	7	,	,	PUNCT
ejpam-3807	36	8	∗	∗	NOUN
ejpam-3807	36	9	,	,	PUNCT
ejpam-3807	36	10	0	0	NUM
ejpam-3807	36	11	)	)	PUNCT
ejpam-3807	36	12	be	be	AUX
ejpam-3807	36	13	a	a	DET
ejpam-3807	36	14	b	b	NOUN
ejpam-3807	36	15	-	-	PUNCT
ejpam-3807	36	16	algebra	algebra	NOUN
ejpam-3807	36	17	.	.	PUNCT
ejpam-3807	37	1	a	a	DET
ejpam-3807	37	2	nonempty	nonempty	NOUN
ejpam-3807	37	3	subset	subset	VERB
ejpam-3807	37	4	n	n	PROPN
ejpam-3807	37	5	of	of	ADP
ejpam-3807	37	6	x	x	PROPN
ejpam-3807	37	7	is	be	AUX
ejpam-3807	37	8	called	call	VERB
ejpam-3807	37	9	a	a	DET
ejpam-3807	37	10	subalgebra	subalgebra	NOUN
ejpam-3807	37	11	of	of	ADP
ejpam-3807	37	12	x	x	PRON
ejpam-3807	37	13	if	if	SCONJ
ejpam-3807	37	14	x	x	PROPN
ejpam-3807	37	15	∗	∗	VERB
ejpam-3807	37	16	y	y	PROPN
ejpam-3807	37	17	∈	∈	PROPN
ejpam-3807	37	18	n	n	PROPN
ejpam-3807	37	19	for	for	ADP
ejpam-3807	37	20	any	any	DET
ejpam-3807	37	21	x	x	NOUN
ejpam-3807	37	22	,	,	PUNCT
ejpam-3807	37	23	y	y	PROPN
ejpam-3807	37	24	∈	∈	PROPN
ejpam-3807	37	25	n	n	ADV
ejpam-3807	37	26	.	.	PUNCT
ejpam-3807	38	1	lemma	lemma	PROPN
ejpam-3807	38	2	2	2	NUM
ejpam-3807	38	3	.	.	PUNCT
ejpam-3807	39	1	[	[	X
ejpam-3807	39	2	3	3	X
ejpam-3807	39	3	]	]	X
ejpam-3807	39	4	let	let	VERB
ejpam-3807	39	5	x	x	PRON
ejpam-3807	39	6	be	be	AUX
ejpam-3807	39	7	a	a	DET
ejpam-3807	39	8	b	b	NOUN
ejpam-3807	39	9	-	-	PUNCT
ejpam-3807	39	10	algebra	algebra	NOUN
ejpam-3807	39	11	.	.	PUNCT
ejpam-3807	40	1	if	if	SCONJ
ejpam-3807	40	2	{	{	PUNCT
ejpam-3807	40	3	nα	nα	NOUN
ejpam-3807	40	4	:	:	PUNCT
ejpam-3807	40	5	α	α	PROPN
ejpam-3807	40	6	∈	∈	PROPN
ejpam-3807	40	7	a	a	PRON
ejpam-3807	40	8	}	}	PUNCT
ejpam-3807	40	9	is	be	AUX
ejpam-3807	40	10	a	a	DET
ejpam-3807	40	11	nonempty	nonempty	ADJ
ejpam-3807	40	12	collection	collection	NOUN
ejpam-3807	40	13	of	of	ADP
ejpam-3807	40	14	subalgebras	subalgebras	PROPN
ejpam-3807	40	15	of	of	ADP
ejpam-3807	40	16	x	x	PROPN
ejpam-3807	40	17	,	,	PUNCT
ejpam-3807	40	18	then	then	ADV
ejpam-3807	40	19	⋂	⋂	PROPN
ejpam-3807	40	20	α∈a	α∈a	VERB
ejpam-3807	40	21	nα	nα	VERB
ejpam-3807	40	22	is	be	AUX
ejpam-3807	40	23	a	a	DET
ejpam-3807	40	24	subalgebra	subalgebra	NOUN
ejpam-3807	40	25	of	of	ADP
ejpam-3807	40	26	x.	x.	NOUN
ejpam-3807	40	27	definition	definition	NOUN
ejpam-3807	40	28	4	4	NUM
ejpam-3807	40	29	.	.	PUNCT
ejpam-3807	41	1	[	[	X
ejpam-3807	41	2	8	8	NUM
ejpam-3807	41	3	]	]	X
ejpam-3807	41	4	let	let	VERB
ejpam-3807	41	5	(	(	PUNCT
ejpam-3807	41	6	x	x	NOUN
ejpam-3807	41	7	,	,	PUNCT
ejpam-3807	41	8	∗	∗	NOUN
ejpam-3807	41	9	,	,	PUNCT
ejpam-3807	41	10	0	0	NUM
ejpam-3807	41	11	)	)	PUNCT
ejpam-3807	41	12	be	be	AUX
ejpam-3807	41	13	a	a	DET
ejpam-3807	41	14	b	b	NOUN
ejpam-3807	41	15	-	-	PUNCT
ejpam-3807	41	16	algebra	algebra	NOUN
ejpam-3807	41	17	.	.	PUNCT
ejpam-3807	42	1	a	a	DET
ejpam-3807	42	2	nonempty	nonempty	ADJ
ejpam-3807	42	3	subset	subset	VERB
ejpam-3807	42	4	s	s	NOUN
ejpam-3807	42	5	of	of	ADP
ejpam-3807	42	6	x	x	VERB
ejpam-3807	42	7	is	be	AUX
ejpam-3807	42	8	said	say	VERB
ejpam-3807	42	9	to	to	PART
ejpam-3807	42	10	be	be	AUX
ejpam-3807	42	11	normal	normal	ADJ
ejpam-3807	42	12	in	in	ADP
ejpam-3807	42	13	x	x	PUNCT
ejpam-3807	42	14	if	if	SCONJ
ejpam-3807	42	15	for	for	ADP
ejpam-3807	42	16	any	any	DET
ejpam-3807	42	17	x	x	PROPN
ejpam-3807	42	18	∗	∗	PROPN
ejpam-3807	42	19	y	y	PROPN
ejpam-3807	42	20	,	,	PUNCT
ejpam-3807	42	21	a	a	DET
ejpam-3807	42	22	∗	∗	NOUN
ejpam-3807	42	23	b	b	NOUN
ejpam-3807	42	24	∈	∈	ADJ
ejpam-3807	42	25	s	s	NOUN
ejpam-3807	42	26	,	,	PUNCT
ejpam-3807	42	27	(	(	PUNCT
ejpam-3807	42	28	x	x	X
ejpam-3807	42	29	∗	∗	NOUN
ejpam-3807	42	30	a	a	NOUN
ejpam-3807	42	31	)	)	PUNCT
ejpam-3807	42	32	∗	∗	NOUN
ejpam-3807	42	33	(	(	PUNCT
ejpam-3807	42	34	y	y	PROPN
ejpam-3807	42	35	∗	∗	X
ejpam-3807	42	36	b	b	NOUN
ejpam-3807	42	37	)	)	PUNCT
ejpam-3807	42	38	∈	∈	PROPN
ejpam-3807	42	39	s.	s.	PROPN
ejpam-3807	42	40	theorem	theorem	VERB
ejpam-3807	42	41	1	1	NUM
ejpam-3807	42	42	.	.	PUNCT
ejpam-3807	43	1	[	[	X
ejpam-3807	43	2	10	10	NUM
ejpam-3807	43	3	]	]	PUNCT
ejpam-3807	43	4	let	let	VERB
ejpam-3807	43	5	n	n	PRON
ejpam-3807	43	6	be	be	AUX
ejpam-3807	43	7	a	a	DET
ejpam-3807	43	8	subalgebra	subalgebra	NOUN
ejpam-3807	43	9	of	of	ADP
ejpam-3807	43	10	a	a	DET
ejpam-3807	43	11	b	b	NOUN
ejpam-3807	43	12	-	-	PUNCT
ejpam-3807	43	13	algebra	algebra	NOUN
ejpam-3807	43	14	x.	x.	NOUN
ejpam-3807	43	15	then	then	ADV
ejpam-3807	43	16	the	the	DET
ejpam-3807	43	17	following	follow	VERB
ejpam-3807	43	18	statements	statement	NOUN
ejpam-3807	43	19	are	be	AUX
ejpam-3807	43	20	equivalent	equivalent	ADJ
ejpam-3807	43	21	:	:	PUNCT
ejpam-3807	43	22	(	(	PUNCT
ejpam-3807	43	23	i	i	NOUN
ejpam-3807	43	24	)	)	PUNCT
ejpam-3807	43	25	n	n	PRON
ejpam-3807	43	26	is	be	AUX
ejpam-3807	43	27	a	a	DET
ejpam-3807	43	28	normal	normal	ADJ
ejpam-3807	43	29	subalgebra	subalgebra	NOUN
ejpam-3807	43	30	;	;	PUNCT
ejpam-3807	43	31	(	(	PUNCT
ejpam-3807	43	32	ii	ii	NOUN
ejpam-3807	43	33	)	)	PUNCT
ejpam-3807	43	34	if	if	SCONJ
ejpam-3807	43	35	x	x	PUNCT
ejpam-3807	43	36	∈	∈	PROPN
ejpam-3807	43	37	x	x	X
ejpam-3807	43	38	and	and	CCONJ
ejpam-3807	43	39	y	y	PROPN
ejpam-3807	43	40	∈	∈	PROPN
ejpam-3807	43	41	n	n	CCONJ
ejpam-3807	43	42	,	,	PUNCT
ejpam-3807	43	43	then	then	ADV
ejpam-3807	43	44	x	x	X
ejpam-3807	43	45	∗	∗	NOUN
ejpam-3807	43	46	(	(	PUNCT
ejpam-3807	43	47	x	x	X
ejpam-3807	43	48	∗	∗	PROPN
ejpam-3807	43	49	y	y	NOUN
ejpam-3807	43	50	)	)	PUNCT
ejpam-3807	43	51	∈	∈	PROPN
ejpam-3807	43	52	n.	n.	NOUN
ejpam-3807	43	53	suppose	suppose	VERB
ejpam-3807	43	54	(	(	PUNCT
ejpam-3807	43	55	x	x	X
ejpam-3807	43	56	,	,	PUNCT
ejpam-3807	43	57	∗	∗	NOUN
ejpam-3807	43	58	,	,	PUNCT
ejpam-3807	43	59	0	0	NUM
ejpam-3807	43	60	)	)	PUNCT
ejpam-3807	43	61	is	be	AUX
ejpam-3807	43	62	a	a	DET
ejpam-3807	43	63	b	b	NOUN
ejpam-3807	43	64	-	-	PUNCT
ejpam-3807	43	65	algebra	algebra	NOUN
ejpam-3807	43	66	and	and	CCONJ
ejpam-3807	43	67	i	i	PRON
ejpam-3807	43	68	a	a	DET
ejpam-3807	43	69	normal	normal	ADJ
ejpam-3807	43	70	subalgebra	subalgebra	NOUN
ejpam-3807	43	71	of	of	ADP
ejpam-3807	43	72	x.	x.	NOUN
ejpam-3807	43	73	the	the	DET
ejpam-3807	43	74	relation	relation	NOUN
ejpam-3807	43	75	“	"	PUNCT
ejpam-3807	43	76	∼=i	∼=i	NOUN
ejpam-3807	43	77	”	"	PUNCT
ejpam-3807	43	78	defined	define	VERB
ejpam-3807	43	79	by	by	ADP
ejpam-3807	43	80	x	x	SYM
ejpam-3807	43	81	∼=i	∼=i	NOUN
ejpam-3807	43	82	y	y	PROPN
ejpam-3807	44	1	if	if	SCONJ
ejpam-3807	44	2	and	and	CCONJ
ejpam-3807	44	3	only	only	ADV
ejpam-3807	44	4	if	if	SCONJ
ejpam-3807	44	5	x	x	X
ejpam-3807	44	6	∗	∗	PROPN
ejpam-3807	44	7	y	y	PROPN
ejpam-3807	44	8	,	,	PUNCT
ejpam-3807	44	9	y	y	PROPN
ejpam-3807	44	10	∗	∗	NOUN
ejpam-3807	45	1	x	x	X
ejpam-3807	45	2	∈	∈	NOUN
ejpam-3807	45	3	i	i	PRON
ejpam-3807	45	4	is	be	AUX
ejpam-3807	45	5	a	a	DET
ejpam-3807	45	6	congruence	congruence	NOUN
ejpam-3807	45	7	relation	relation	NOUN
ejpam-3807	45	8	on	on	ADP
ejpam-3807	45	9	x	x	PUNCT
ejpam-3807	45	10	for	for	ADP
ejpam-3807	45	11	any	any	DET
ejpam-3807	45	12	x	x	NOUN
ejpam-3807	45	13	,	,	PUNCT
ejpam-3807	45	14	y	y	PROPN
ejpam-3807	45	15	∈	∈	PROPN
ejpam-3807	45	16	x.	x.	NOUN
ejpam-3807	45	17	that	that	PRON
ejpam-3807	45	18	is	is	ADV
ejpam-3807	45	19	,	,	PUNCT
ejpam-3807	45	20	∼=i	∼=i	NOUN
ejpam-3807	45	21	is	be	AUX
ejpam-3807	45	22	an	an	DET
ejpam-3807	45	23	equivalence	equivalence	NOUN
ejpam-3807	45	24	relation	relation	NOUN
ejpam-3807	45	25	and	and	CCONJ
ejpam-3807	45	26	for	for	ADP
ejpam-3807	45	27	each	each	DET
ejpam-3807	45	28	a	a	PRON
ejpam-3807	45	29	,	,	PUNCT
ejpam-3807	45	30	b	b	NOUN
ejpam-3807	45	31	,	,	PUNCT
ejpam-3807	45	32	x	x	X
ejpam-3807	45	33	,	,	PUNCT
ejpam-3807	45	34	y	y	PROPN
ejpam-3807	45	35	∈	∈	PROPN
ejpam-3807	46	1	x	x	PRON
ejpam-3807	46	2	,	,	PUNCT
ejpam-3807	46	3	if	if	SCONJ
ejpam-3807	46	4	x	x	PRON
ejpam-3807	46	5	∼=i	∼=i	NOUN
ejpam-3807	46	6	y	y	PROPN
ejpam-3807	46	7	and	and	CCONJ
ejpam-3807	46	8	a	a	DET
ejpam-3807	46	9	∼=i	∼=i	NOUN
ejpam-3807	46	10	b	b	NOUN
ejpam-3807	46	11	,	,	PUNCT
ejpam-3807	46	12	then	then	ADV
ejpam-3807	46	13	a	a	DET
ejpam-3807	46	14	∗	∗	NOUN
ejpam-3807	46	15	x	x	PUNCT
ejpam-3807	46	16	∼=i	∼=i	NOUN
ejpam-3807	46	17	b	b	PROPN
ejpam-3807	46	18	∗	∗	X
ejpam-3807	46	19	y.	y.	PROPN
ejpam-3807	46	20	let	let	VERB
ejpam-3807	46	21	ix	ix	ADV
ejpam-3807	46	22	=	=	PUNCT
ejpam-3807	46	23	{	{	PUNCT
ejpam-3807	46	24	y	y	NOUN
ejpam-3807	46	25	:	:	PUNCT
ejpam-3807	46	26	y	y	PROPN
ejpam-3807	46	27	∼=i	∼=i	NOUN
ejpam-3807	46	28	x	x	VERB
ejpam-3807	46	29	}	}	PUNCT
ejpam-3807	46	30	denote	denote	VERB
ejpam-3807	46	31	the	the	DET
ejpam-3807	46	32	equivalence	equivalence	NOUN
ejpam-3807	46	33	class	class	NOUN
ejpam-3807	46	34	of	of	ADP
ejpam-3807	46	35	x	x	X
ejpam-3807	46	36	and	and	CCONJ
ejpam-3807	46	37	x	x	X
ejpam-3807	46	38	/	/	SYM
ejpam-3807	46	39	i	i	PRON
ejpam-3807	46	40	=	=	PUNCT
ejpam-3807	46	41	{	{	PUNCT
ejpam-3807	46	42	ix	ix	X
ejpam-3807	46	43	:	:	PUNCT
ejpam-3807	46	44	x	x	SYM
ejpam-3807	46	45	∈	∈	NOUN
ejpam-3807	46	46	x	x	X
ejpam-3807	46	47	}	}	PUNCT
ejpam-3807	46	48	.	.	PUNCT
ejpam-3807	47	1	then	then	ADV
ejpam-3807	47	2	x	x	X
ejpam-3807	47	3	/	/	SYM
ejpam-3807	47	4	i	i	PRON
ejpam-3807	47	5	is	be	AUX
ejpam-3807	47	6	a	a	DET
ejpam-3807	47	7	b	b	X
ejpam-3807	47	8	-	-	PUNCT
ejpam-3807	47	9	algebra	algebra	NOUN
ejpam-3807	47	10	called	call	VERB
ejpam-3807	47	11	the	the	DET
ejpam-3807	47	12	quotient	quotient	NOUN
ejpam-3807	47	13	b	b	NOUN
ejpam-3807	47	14	-	-	PUNCT
ejpam-3807	47	15	algebra	algebra	NOUN
ejpam-3807	47	16	under	under	ADP
ejpam-3807	47	17	the	the	DET
ejpam-3807	47	18	binary	binary	ADJ
ejpam-3807	47	19	operation	operation	NOUN
ejpam-3807	47	20	given	give	VERB
ejpam-3807	47	21	by	by	ADP
ejpam-3807	47	22	ix	ix	PROPN
ejpam-3807	47	23	∗	∗	NOUN
ejpam-3807	47	24	iy	iy	PROPN
ejpam-3807	48	1	=	=	PUNCT
ejpam-3807	48	2	ix∗y	ix∗y	X
ejpam-3807	48	3	[	[	X
ejpam-3807	48	4	3	3	NUM
ejpam-3807	48	5	]	]	PUNCT
ejpam-3807	48	6	.	.	PUNCT
ejpam-3807	49	1	k.	k.	PROPN
ejpam-3807	49	2	belleza	belleza	PROPN
ejpam-3807	49	3	,	,	PUNCT
ejpam-3807	49	4	j.	j.	PROPN
ejpam-3807	49	5	vilela	vilela	PROPN
ejpam-3807	49	6	/	/	SYM
ejpam-3807	49	7	eur	eur	PROPN
ejpam-3807	49	8	.	.	PUNCT
ejpam-3807	50	1	j.	j.	PROPN
ejpam-3807	50	2	pure	pure	PROPN
ejpam-3807	50	3	appl	appl	PROPN
ejpam-3807	50	4	.	.	PROPN
ejpam-3807	50	5	math	math	PROPN
ejpam-3807	50	6	,	,	PUNCT
ejpam-3807	50	7	13	13	NUM
ejpam-3807	50	8	(	(	PUNCT
ejpam-3807	50	9	4	4	NUM
ejpam-3807	50	10	)	)	PUNCT
ejpam-3807	50	11	(	(	PUNCT
ejpam-3807	50	12	2020	2020	NUM
ejpam-3807	50	13	)	)	PUNCT
ejpam-3807	50	14	,	,	PUNCT
ejpam-3807	50	15	830	830	NUM
ejpam-3807	50	16	-	-	SYM
ejpam-3807	50	17	839	839	NUM
ejpam-3807	50	18	832	832	NUM
ejpam-3807	50	19	definition	definition	NOUN
ejpam-3807	50	20	5	5	NUM
ejpam-3807	50	21	.	.	PUNCT
ejpam-3807	51	1	[	[	X
ejpam-3807	51	2	1	1	X
ejpam-3807	51	3	]	]	X
ejpam-3807	51	4	let	let	VERB
ejpam-3807	51	5	(	(	PUNCT
ejpam-3807	51	6	x	x	NOUN
ejpam-3807	51	7	,	,	PUNCT
ejpam-3807	51	8	∗	∗	NOUN
ejpam-3807	51	9	,	,	PUNCT
ejpam-3807	51	10	0	0	NUM
ejpam-3807	51	11	)	)	PUNCT
ejpam-3807	51	12	be	be	AUX
ejpam-3807	51	13	a	a	DET
ejpam-3807	51	14	b	b	NOUN
ejpam-3807	51	15	-	-	PUNCT
ejpam-3807	51	16	algebra	algebra	NOUN
ejpam-3807	52	1	and	and	CCONJ
ejpam-3807	52	2	i	i	PRON
ejpam-3807	52	3	a	a	DET
ejpam-3807	52	4	nonempty	nonempty	NOUN
ejpam-3807	52	5	subset	subset	NOUN
ejpam-3807	52	6	of	of	ADP
ejpam-3807	52	7	x.	x.	NOUN
ejpam-3807	52	8	then	then	ADV
ejpam-3807	52	9	i	i	PRON
ejpam-3807	52	10	is	be	AUX
ejpam-3807	52	11	called	call	VERB
ejpam-3807	52	12	a	a	DET
ejpam-3807	52	13	b	b	NOUN
ejpam-3807	52	14	-	-	PUNCT
ejpam-3807	52	15	ideal	ideal	NOUN
ejpam-3807	52	16	of	of	ADP
ejpam-3807	52	17	x	x	PRON
ejpam-3807	52	18	if	if	SCONJ
ejpam-3807	52	19	it	it	PRON
ejpam-3807	52	20	satisfies	satisfy	VERB
ejpam-3807	52	21	the	the	DET
ejpam-3807	52	22	following	following	NOUN
ejpam-3807	52	23	:	:	PUNCT
ejpam-3807	52	24	for	for	ADP
ejpam-3807	52	25	any	any	DET
ejpam-3807	52	26	x	x	NOUN
ejpam-3807	52	27	,	,	PUNCT
ejpam-3807	52	28	y	y	PROPN
ejpam-3807	52	29	in	in	ADP
ejpam-3807	52	30	x	x	PRON
ejpam-3807	52	31	,	,	PUNCT
ejpam-3807	52	32	(	(	PUNCT
ejpam-3807	52	33	i	i	NOUN
ejpam-3807	52	34	)	)	PUNCT
ejpam-3807	52	35	0	0	PUNCT
ejpam-3807	53	1	∈	∈	PROPN
ejpam-3807	54	1	i	i	PRON
ejpam-3807	54	2	;	;	PUNCT
ejpam-3807	54	3	(	(	PUNCT
ejpam-3807	54	4	ii	ii	NOUN
ejpam-3807	54	5	)	)	PUNCT
ejpam-3807	54	6	if	if	SCONJ
ejpam-3807	54	7	x	x	PROPN
ejpam-3807	54	8	∗	∗	VERB
ejpam-3807	54	9	y	y	NOUN
ejpam-3807	54	10	∈	∈	PROPN
ejpam-3807	55	1	i	i	PRON
ejpam-3807	55	2	and	and	CCONJ
ejpam-3807	55	3	y	y	PROPN
ejpam-3807	55	4	∈	∈	PROPN
ejpam-3807	56	1	i	i	PRON
ejpam-3807	56	2	,	,	PUNCT
ejpam-3807	56	3	then	then	ADV
ejpam-3807	56	4	x	x	PART
ejpam-3807	56	5	∈	∈	PROPN
ejpam-3807	56	6	i.	i.	NOUN
ejpam-3807	56	7	remark	remark	NOUN
ejpam-3807	56	8	1	1	NUM
ejpam-3807	56	9	.	.	PUNCT
ejpam-3807	56	10	not	not	PART
ejpam-3807	56	11	every	every	DET
ejpam-3807	56	12	b	b	NOUN
ejpam-3807	56	13	-	-	PUNCT
ejpam-3807	56	14	ideal	ideal	NOUN
ejpam-3807	56	15	is	be	AUX
ejpam-3807	56	16	a	a	DET
ejpam-3807	56	17	normal	normal	ADJ
ejpam-3807	56	18	subset	subset	NOUN
ejpam-3807	56	19	of	of	ADP
ejpam-3807	56	20	a	a	DET
ejpam-3807	56	21	b	b	NOUN
ejpam-3807	56	22	-	-	PUNCT
ejpam-3807	56	23	algebra	algebra	NOUN
ejpam-3807	56	24	x.	x.	NOUN
ejpam-3807	56	25	consider	consider	VERB
ejpam-3807	56	26	example	example	NOUN
ejpam-3807	56	27	1	1	NUM
ejpam-3807	56	28	.	.	PUNCT
ejpam-3807	57	1	then	then	ADV
ejpam-3807	57	2	{	{	PUNCT
ejpam-3807	57	3	0	0	NUM
ejpam-3807	57	4	,	,	PUNCT
ejpam-3807	57	5	c	c	NOUN
ejpam-3807	57	6	}	}	PUNCT
ejpam-3807	57	7	is	be	AUX
ejpam-3807	57	8	a	a	DET
ejpam-3807	57	9	b	b	NOUN
ejpam-3807	57	10	-	-	PUNCT
ejpam-3807	57	11	ideal	ideal	NOUN
ejpam-3807	57	12	but	but	CCONJ
ejpam-3807	57	13	is	be	AUX
ejpam-3807	57	14	not	not	PART
ejpam-3807	57	15	normal	normal	ADJ
ejpam-3807	57	16	since	since	SCONJ
ejpam-3807	57	17	c	c	NOUN
ejpam-3807	57	18	∗	∗	VERB
ejpam-3807	57	19	a	a	DET
ejpam-3807	57	20	=	=	SYM
ejpam-3807	57	21	c	c	NOUN
ejpam-3807	57	22	,	,	PUNCT
ejpam-3807	57	23	b	b	NOUN
ejpam-3807	57	24	∗	∗	X
ejpam-3807	57	25	d	d	NOUN
ejpam-3807	57	26	=	=	SYM
ejpam-3807	57	27	c	c	X
ejpam-3807	57	28	∈	∈	PROPN
ejpam-3807	57	29	{	{	PUNCT
ejpam-3807	57	30	0	0	NUM
ejpam-3807	57	31	,	,	PUNCT
ejpam-3807	57	32	c	c	NOUN
ejpam-3807	57	33	}	}	PUNCT
ejpam-3807	57	34	with	with	ADP
ejpam-3807	57	35	(	(	PUNCT
ejpam-3807	57	36	c	c	NOUN
ejpam-3807	57	37	∗	∗	PRON
ejpam-3807	57	38	b	b	NOUN
ejpam-3807	57	39	)	)	PUNCT
ejpam-3807	57	40	∗	∗	NOUN
ejpam-3807	57	41	(	(	PUNCT
ejpam-3807	57	42	a	a	DET
ejpam-3807	57	43	∗	∗	NOUN
ejpam-3807	57	44	d	d	NOUN
ejpam-3807	57	45	)	)	PUNCT
ejpam-3807	58	1	=	=	SYM
ejpam-3807	58	2	b	b	X
ejpam-3807	58	3	/∈	/∈	PUNCT
ejpam-3807	58	4	{	{	PUNCT
ejpam-3807	58	5	0	0	NUM
ejpam-3807	58	6	,	,	PUNCT
ejpam-3807	58	7	c	c	NOUN
ejpam-3807	58	8	}	}	PUNCT
ejpam-3807	58	9	.	.	PUNCT
ejpam-3807	59	1	remark	remark	NOUN
ejpam-3807	59	2	2	2	NUM
ejpam-3807	59	3	.	.	PUNCT
ejpam-3807	60	1	any	any	DET
ejpam-3807	60	2	nonempty	nonempty	ADJ
ejpam-3807	60	3	normal	normal	ADJ
ejpam-3807	60	4	subset	subset	NOUN
ejpam-3807	60	5	of	of	ADP
ejpam-3807	60	6	x	x	PUNCT
ejpam-3807	60	7	is	be	AUX
ejpam-3807	60	8	a	a	DET
ejpam-3807	60	9	subalgebra	subalgebra	NOUN
ejpam-3807	60	10	.	.	PUNCT
ejpam-3807	61	1	hence	hence	ADV
ejpam-3807	61	2	,	,	PUNCT
ejpam-3807	61	3	if	if	SCONJ
ejpam-3807	61	4	i	i	PRON
ejpam-3807	61	5	is	be	AUX
ejpam-3807	61	6	a	a	DET
ejpam-3807	61	7	normal	normal	ADJ
ejpam-3807	61	8	b	b	NOUN
ejpam-3807	61	9	-	-	PUNCT
ejpam-3807	61	10	ideal	ideal	ADJ
ejpam-3807	61	11	,	,	PUNCT
ejpam-3807	61	12	then	then	ADV
ejpam-3807	61	13	x	x	X
ejpam-3807	61	14	/	/	SYM
ejpam-3807	61	15	i	i	PRON
ejpam-3807	61	16	is	be	AUX
ejpam-3807	61	17	a	a	DET
ejpam-3807	61	18	b	b	NOUN
ejpam-3807	61	19	-	-	PUNCT
ejpam-3807	61	20	algebra	algebra	NOUN
ejpam-3807	61	21	.	.	PUNCT
ejpam-3807	62	1	let	let	VERB
ejpam-3807	62	2	x	x	PRON
ejpam-3807	62	3	be	be	AUX
ejpam-3807	62	4	a	a	DET
ejpam-3807	62	5	set	set	NOUN
ejpam-3807	62	6	.	.	PUNCT
ejpam-3807	63	1	a	a	DET
ejpam-3807	63	2	topology	topology	NOUN
ejpam-3807	63	3	(	(	PUNCT
ejpam-3807	63	4	or	or	CCONJ
ejpam-3807	63	5	topological	topological	ADJ
ejpam-3807	63	6	structure	structure	NOUN
ejpam-3807	63	7	)	)	PUNCT
ejpam-3807	63	8	in	in	ADP
ejpam-3807	63	9	x	x	PRON
ejpam-3807	63	10	is	be	AUX
ejpam-3807	63	11	a	a	DET
ejpam-3807	63	12	family	family	NOUN
ejpam-3807	63	13	τ	τ	PROPN
ejpam-3807	63	14	of	of	ADP
ejpam-3807	63	15	subsets	subset	NOUN
ejpam-3807	63	16	of	of	ADP
ejpam-3807	63	17	x	x	PRON
ejpam-3807	63	18	that	that	PRON
ejpam-3807	63	19	satisfies	satisfy	VERB
ejpam-3807	63	20	the	the	DET
ejpam-3807	63	21	following	following	NOUN
ejpam-3807	63	22	:	:	PUNCT
ejpam-3807	63	23	(	(	PUNCT
ejpam-3807	63	24	i	i	NOUN
ejpam-3807	63	25	)	)	PUNCT
ejpam-3807	63	26	each	each	DET
ejpam-3807	63	27	union	union	NOUN
ejpam-3807	63	28	of	of	ADP
ejpam-3807	63	29	members	member	NOUN
ejpam-3807	63	30	of	of	ADP
ejpam-3807	63	31	τ	τ	PROPN
ejpam-3807	63	32	is	be	AUX
ejpam-3807	63	33	also	also	ADV
ejpam-3807	63	34	a	a	DET
ejpam-3807	63	35	member	member	NOUN
ejpam-3807	63	36	of	of	ADP
ejpam-3807	63	37	τ	τ	PROPN
ejpam-3807	63	38	;	;	PUNCT
ejpam-3807	63	39	(	(	PUNCT
ejpam-3807	63	40	ii	ii	NOUN
ejpam-3807	63	41	)	)	PUNCT
ejpam-3807	63	42	each	each	DET
ejpam-3807	63	43	finite	finite	ADJ
ejpam-3807	63	44	intersection	intersection	NOUN
ejpam-3807	63	45	of	of	ADP
ejpam-3807	63	46	members	member	NOUN
ejpam-3807	63	47	of	of	ADP
ejpam-3807	63	48	τ	τ	PROPN
ejpam-3807	63	49	is	be	AUX
ejpam-3807	63	50	also	also	ADV
ejpam-3807	63	51	a	a	DET
ejpam-3807	63	52	member	member	NOUN
ejpam-3807	63	53	of	of	ADP
ejpam-3807	63	54	τ	τ	PROPN
ejpam-3807	63	55	;	;	PUNCT
ejpam-3807	63	56	and	and	CCONJ
ejpam-3807	63	57	(	(	PUNCT
ejpam-3807	63	58	iii	iii	X
ejpam-3807	63	59	)	)	PUNCT
ejpam-3807	63	60	∅	∅	NOUN
ejpam-3807	63	61	and	and	CCONJ
ejpam-3807	63	62	x	x	X
ejpam-3807	63	63	are	be	AUX
ejpam-3807	63	64	members	member	NOUN
ejpam-3807	63	65	of	of	ADP
ejpam-3807	63	66	τ	τ	PROPN
ejpam-3807	63	67	.	.	PUNCT
ejpam-3807	64	1	a	a	DET
ejpam-3807	64	2	couple	couple	NOUN
ejpam-3807	64	3	(	(	PUNCT
ejpam-3807	64	4	x	x	NOUN
ejpam-3807	64	5	,	,	PUNCT
ejpam-3807	64	6	τ	τ	X
ejpam-3807	64	7	)	)	PUNCT
ejpam-3807	64	8	consisting	consist	VERB
ejpam-3807	64	9	of	of	ADP
ejpam-3807	64	10	a	a	DET
ejpam-3807	64	11	set	set	NOUN
ejpam-3807	64	12	x	x	PUNCT
ejpam-3807	64	13	and	and	CCONJ
ejpam-3807	64	14	a	a	DET
ejpam-3807	64	15	topology	topology	NOUN
ejpam-3807	64	16	τ	τ	PROPN
ejpam-3807	64	17	in	in	ADP
ejpam-3807	64	18	x	x	PROPN
ejpam-3807	64	19	is	be	AUX
ejpam-3807	64	20	called	call	VERB
ejpam-3807	64	21	a	a	DET
ejpam-3807	64	22	topological	topological	ADJ
ejpam-3807	64	23	space	space	NOUN
ejpam-3807	64	24	.	.	PUNCT
ejpam-3807	65	1	we	we	PRON
ejpam-3807	65	2	also	also	ADV
ejpam-3807	65	3	say	say	VERB
ejpam-3807	65	4	“	"	PUNCT
ejpam-3807	65	5	τ	τ	X
ejpam-3807	65	6	is	be	AUX
ejpam-3807	65	7	the	the	DET
ejpam-3807	65	8	topology	topology	NOUN
ejpam-3807	65	9	of	of	ADP
ejpam-3807	65	10	the	the	DET
ejpam-3807	65	11	space	space	NOUN
ejpam-3807	65	12	x	x	NOUN
ejpam-3807	65	13	”	"	PUNCT
ejpam-3807	65	14	.	.	PUNCT
ejpam-3807	66	1	the	the	DET
ejpam-3807	66	2	members	member	NOUN
ejpam-3807	66	3	of	of	ADP
ejpam-3807	66	4	τ	τ	PROPN
ejpam-3807	66	5	are	be	AUX
ejpam-3807	66	6	called	call	VERB
ejpam-3807	66	7	open	open	ADJ
ejpam-3807	66	8	sets	set	NOUN
ejpam-3807	66	9	of	of	ADP
ejpam-3807	66	10	(	(	PUNCT
ejpam-3807	66	11	x	x	NOUN
ejpam-3807	66	12	,	,	PUNCT
ejpam-3807	66	13	τ	τ	PROPN
ejpam-3807	66	14	)	)	PUNCT
ejpam-3807	66	15	.	.	PUNCT
ejpam-3807	67	1	let	let	AUX
ejpam-3807	67	2	(	(	PUNCT
ejpam-3807	67	3	x	x	NOUN
ejpam-3807	67	4	,	,	PUNCT
ejpam-3807	67	5	τx	τx	ADJ
ejpam-3807	67	6	)	)	PUNCT
ejpam-3807	67	7	and	and	CCONJ
ejpam-3807	67	8	(	(	PUNCT
ejpam-3807	67	9	y	y	PROPN
ejpam-3807	67	10	,	,	PUNCT
ejpam-3807	67	11	τy	τy	PART
ejpam-3807	67	12	)	)	PUNCT
ejpam-3807	67	13	be	be	AUX
ejpam-3807	67	14	topological	topological	ADJ
ejpam-3807	67	15	spaces	space	NOUN
ejpam-3807	67	16	.	.	PUNCT
ejpam-3807	68	1	a	a	DET
ejpam-3807	68	2	map	map	NOUN
ejpam-3807	68	3	f	f	X
ejpam-3807	68	4	:	:	PUNCT
ejpam-3807	68	5	x	x	X
ejpam-3807	68	6	→	→	SYM
ejpam-3807	68	7	y	y	PROPN
ejpam-3807	68	8	is	be	AUX
ejpam-3807	68	9	called	call	VERB
ejpam-3807	68	10	continuous	continuous	ADJ
ejpam-3807	68	11	if	if	SCONJ
ejpam-3807	68	12	the	the	DET
ejpam-3807	68	13	inverse	inverse	ADJ
ejpam-3807	68	14	image	image	NOUN
ejpam-3807	68	15	of	of	ADP
ejpam-3807	68	16	each	each	DET
ejpam-3807	68	17	open	open	ADJ
ejpam-3807	68	18	set	set	NOUN
ejpam-3807	68	19	in	in	ADP
ejpam-3807	68	20	y	y	PROPN
ejpam-3807	68	21	is	be	AUX
ejpam-3807	68	22	open	open	ADJ
ejpam-3807	68	23	in	in	ADP
ejpam-3807	68	24	x	x	PROPN
ejpam-3807	68	25	(	(	PUNCT
ejpam-3807	68	26	that	that	PRON
ejpam-3807	68	27	is	is	ADV
ejpam-3807	68	28	,	,	PUNCT
ejpam-3807	68	29	if	if	SCONJ
ejpam-3807	68	30	f−1	f−1	PROPN
ejpam-3807	68	31	maps	map	VERB
ejpam-3807	68	32	τy	τy	NOUN
ejpam-3807	68	33	into	into	ADP
ejpam-3807	68	34	τx	τx	ADP
ejpam-3807	68	35	)	)	PUNCT
ejpam-3807	69	1	[	[	X
ejpam-3807	69	2	2	2	NUM
ejpam-3807	69	3	]	]	PUNCT
ejpam-3807	69	4	.	.	PUNCT
ejpam-3807	70	1	definition	definition	NOUN
ejpam-3807	70	2	6	6	NUM
ejpam-3807	70	3	.	.	PUNCT
ejpam-3807	71	1	[	[	X
ejpam-3807	71	2	2	2	NUM
ejpam-3807	71	3	]	]	X
ejpam-3807	71	4	let	let	AUX
ejpam-3807	71	5	{	{	PUNCT
ejpam-3807	71	6	yα|α	yα|α	PROPN
ejpam-3807	71	7	∈	∈	PROPN
ejpam-3807	71	8	a	a	DET
ejpam-3807	71	9	}	}	PUNCT
ejpam-3807	71	10	be	be	AUX
ejpam-3807	71	11	any	any	DET
ejpam-3807	71	12	family	family	NOUN
ejpam-3807	71	13	of	of	ADP
ejpam-3807	71	14	topological	topological	ADJ
ejpam-3807	71	15	spaces	space	NOUN
ejpam-3807	71	16	.	.	PUNCT
ejpam-3807	72	1	for	for	ADP
ejpam-3807	72	2	each	each	DET
ejpam-3807	72	3	α	α	NOUN
ejpam-3807	72	4	∈	∈	PROPN
ejpam-3807	72	5	a	a	PRON
ejpam-3807	72	6	,	,	PUNCT
ejpam-3807	72	7	let	let	VERB
ejpam-3807	72	8	τα	τα	PRON
ejpam-3807	72	9	be	be	AUX
ejpam-3807	72	10	the	the	DET
ejpam-3807	72	11	topology	topology	NOUN
ejpam-3807	72	12	for	for	ADP
ejpam-3807	72	13	yα	yα	NOUN
ejpam-3807	72	14	.	.	PUNCT
ejpam-3807	73	1	the	the	DET
ejpam-3807	73	2	cartersian	cartersian	ADJ
ejpam-3807	73	3	product	product	NOUN
ejpam-3807	73	4	topology	topology	NOUN
ejpam-3807	73	5	in	in	ADP
ejpam-3807	73	6	∏	∏	PROPN
ejpam-3807	73	7	α	α	PROPN
ejpam-3807	73	8	yα	yα	NOUN
ejpam-3807	73	9	is	be	AUX
ejpam-3807	73	10	that	that	SCONJ
ejpam-3807	73	11	having	have	VERB
ejpam-3807	73	12	for	for	ADP
ejpam-3807	73	13	subbasis	subbasis	NOUN
ejpam-3807	73	14	all	all	DET
ejpam-3807	73	15	sets	set	VERB
ejpam-3807	73	16	<	<	X
ejpam-3807	73	17	uβ	uβ	X
ejpam-3807	73	18	>	>	X
ejpam-3807	73	19	=	=	PUNCT
ejpam-3807	73	20	ρ−1β	ρ−1β	X
ejpam-3807	73	21	(	(	PUNCT
ejpam-3807	73	22	uβ	uβ	NOUN
ejpam-3807	73	23	)	)	PUNCT
ejpam-3807	73	24	,	,	PUNCT
ejpam-3807	73	25	where	where	SCONJ
ejpam-3807	73	26	ρ	ρ	NOUN
ejpam-3807	73	27	:	:	PUNCT
ejpam-3807	73	28	∏	∏	PROPN
ejpam-3807	73	29	α	α	NOUN
ejpam-3807	73	30	yα	yα	NOUN
ejpam-3807	73	31	→	→	SYM
ejpam-3807	73	32	yα	yα	NOUN
ejpam-3807	73	33	,	,	PUNCT
ejpam-3807	73	34	uβ	uβ	PROPN
ejpam-3807	73	35	ranges	range	VERB
ejpam-3807	73	36	over	over	ADP
ejpam-3807	73	37	all	all	DET
ejpam-3807	73	38	members	member	NOUN
ejpam-3807	73	39	of	of	ADP
ejpam-3807	73	40	τβ	τβ	NOUN
ejpam-3807	73	41	and	and	CCONJ
ejpam-3807	73	42	β	β	X
ejpam-3807	73	43	over	over	ADP
ejpam-3807	73	44	all	all	DET
ejpam-3807	73	45	elements	element	NOUN
ejpam-3807	73	46	of	of	ADP
ejpam-3807	73	47	a.	a.	NOUN
ejpam-3807	73	48	definition	definition	NOUN
ejpam-3807	73	49	7	7	NUM
ejpam-3807	73	50	.	.	PUNCT
ejpam-3807	74	1	[	[	X
ejpam-3807	74	2	6	6	NUM
ejpam-3807	74	3	]	]	PUNCT
ejpam-3807	74	4	let	let	VERB
ejpam-3807	74	5	x	x	PRON
ejpam-3807	74	6	be	be	AUX
ejpam-3807	74	7	a	a	DET
ejpam-3807	74	8	b	b	NOUN
ejpam-3807	74	9	-	-	PUNCT
ejpam-3807	74	10	algebra	algebra	NOUN
ejpam-3807	74	11	.	.	PUNCT
ejpam-3807	75	1	a	a	DET
ejpam-3807	75	2	topology	topology	NOUN
ejpam-3807	75	3	τ	τ	PROPN
ejpam-3807	75	4	furnished	furnish	VERB
ejpam-3807	75	5	on	on	ADP
ejpam-3807	75	6	x	x	VERB
ejpam-3807	75	7	is	be	AUX
ejpam-3807	75	8	called	call	VERB
ejpam-3807	75	9	a	a	DET
ejpam-3807	75	10	btopology	btopology	NOUN
ejpam-3807	75	11	on	on	ADP
ejpam-3807	75	12	x.	x.	PROPN
ejpam-3807	75	13	a	a	DET
ejpam-3807	75	14	b	b	X
ejpam-3807	75	15	-	-	PUNCT
ejpam-3807	75	16	topological	topological	ADJ
ejpam-3807	75	17	space	space	NOUN
ejpam-3807	75	18	(	(	PUNCT
ejpam-3807	75	19	x	x	X
ejpam-3807	75	20	,	,	PUNCT
ejpam-3807	75	21	τ	τ	X
ejpam-3807	75	22	)	)	PUNCT
ejpam-3807	75	23	is	be	AUX
ejpam-3807	75	24	called	call	VERB
ejpam-3807	75	25	a	a	DET
ejpam-3807	75	26	topological	topological	ADJ
ejpam-3807	75	27	b	b	NOUN
ejpam-3807	75	28	-	-	PUNCT
ejpam-3807	75	29	algebra	algebra	NOUN
ejpam-3807	75	30	if	if	SCONJ
ejpam-3807	75	31	τ	τ	PROPN
ejpam-3807	75	32	is	be	AUX
ejpam-3807	75	33	a	a	DET
ejpam-3807	75	34	b	b	NOUN
ejpam-3807	75	35	-	-	PUNCT
ejpam-3807	75	36	topology	topology	NOUN
ejpam-3807	75	37	on	on	ADP
ejpam-3807	75	38	x	x	PUNCT
ejpam-3807	75	39	and	and	CCONJ
ejpam-3807	75	40	the	the	DET
ejpam-3807	75	41	binary	binary	PROPN
ejpam-3807	75	42	operation	operation	NOUN
ejpam-3807	75	43	∗	∗	NOUN
ejpam-3807	75	44	:	:	PUNCT
ejpam-3807	75	45	x	x	X
ejpam-3807	75	46	×x	×x	ADP
ejpam-3807	75	47	→	→	SYM
ejpam-3807	75	48	x	x	X
ejpam-3807	75	49	is	be	AUX
ejpam-3807	75	50	continuous	continuous	ADJ
ejpam-3807	75	51	,	,	PUNCT
ejpam-3807	75	52	where	where	SCONJ
ejpam-3807	75	53	x	x	PUNCT
ejpam-3807	75	54	×x	×x	PRON
ejpam-3807	75	55	is	be	AUX
ejpam-3807	75	56	furnished	furnish	VERB
ejpam-3807	75	57	by	by	ADP
ejpam-3807	75	58	the	the	DET
ejpam-3807	75	59	cartesian	cartesian	ADJ
ejpam-3807	75	60	product	product	NOUN
ejpam-3807	75	61	topology	topology	NOUN
ejpam-3807	75	62	.	.	PUNCT
ejpam-3807	76	1	let	let	VERB
ejpam-3807	76	2	(	(	PUNCT
ejpam-3807	76	3	x	x	NOUN
ejpam-3807	76	4	,	,	PUNCT
ejpam-3807	76	5	τ	τ	X
ejpam-3807	76	6	)	)	PUNCT
ejpam-3807	76	7	be	be	VERB
ejpam-3807	76	8	a	a	DET
ejpam-3807	76	9	topological	topological	ADJ
ejpam-3807	76	10	space	space	NOUN
ejpam-3807	76	11	and	and	CCONJ
ejpam-3807	76	12	a	a	DET
ejpam-3807	76	13	⊂	⊂	X
ejpam-3807	76	14	x.	x.	NOUN
ejpam-3807	76	15	by	by	ADP
ejpam-3807	76	16	a	a	DET
ejpam-3807	76	17	neighborhood	neighborhood	NOUN
ejpam-3807	76	18	of	of	ADP
ejpam-3807	76	19	an	an	DET
ejpam-3807	76	20	element	element	NOUN
ejpam-3807	76	21	x	x	PUNCT
ejpam-3807	76	22	in	in	ADP
ejpam-3807	76	23	x	x	PROPN
ejpam-3807	76	24	(	(	PUNCT
ejpam-3807	76	25	denoted	denote	VERB
ejpam-3807	76	26	as	as	ADP
ejpam-3807	76	27	u(x	u(x	NOUN
ejpam-3807	76	28	)	)	PUNCT
ejpam-3807	76	29	)	)	PUNCT
ejpam-3807	76	30	is	be	AUX
ejpam-3807	76	31	meant	mean	VERB
ejpam-3807	76	32	any	any	DET
ejpam-3807	76	33	open	open	ADJ
ejpam-3807	76	34	set	set	NOUN
ejpam-3807	76	35	(	(	PUNCT
ejpam-3807	76	36	that	that	PRON
ejpam-3807	76	37	is	is	ADV
ejpam-3807	76	38	,	,	PUNCT
ejpam-3807	76	39	member	member	NOUN
ejpam-3807	76	40	of	of	ADP
ejpam-3807	76	41	τ	τ	PROPN
ejpam-3807	76	42	)	)	PUNCT
ejpam-3807	76	43	containing	contain	VERB
ejpam-3807	76	44	x.	x.	NOUN
ejpam-3807	76	45	the	the	DET
ejpam-3807	76	46	interior	interior	PROPN
ejpam-3807	76	47	int(a	int(a	PROPN
ejpam-3807	76	48	)	)	PUNCT
ejpam-3807	76	49	of	of	ADP
ejpam-3807	76	50	a	a	PRON
ejpam-3807	76	51	is	be	AUX
ejpam-3807	76	52	the	the	DET
ejpam-3807	76	53	largest	large	ADJ
ejpam-3807	76	54	open	open	ADJ
ejpam-3807	76	55	set	set	NOUN
ejpam-3807	76	56	contained	contain	VERB
ejpam-3807	76	57	in	in	ADP
ejpam-3807	76	58	a	a	PRON
ejpam-3807	76	59	,	,	PUNCT
ejpam-3807	76	60	that	that	ADV
ejpam-3807	76	61	is	is	ADV
ejpam-3807	76	62	,	,	PUNCT
ejpam-3807	76	63	int(a)=	int(a)=	INTJ
ejpam-3807	76	64	⋃	⋃	PROPN
ejpam-3807	76	65	{	{	PUNCT
ejpam-3807	76	66	u	u	PROPN
ejpam-3807	76	67	|u	|u	PROPN
ejpam-3807	76	68	∈	∈	PROPN
ejpam-3807	76	69	τ	τ	X
ejpam-3807	76	70	,	,	PUNCT
ejpam-3807	76	71	u	u	X
ejpam-3807	76	72	⊂	⊂	PROPN
ejpam-3807	76	73	a	a	X
ejpam-3807	76	74	}	}	PUNCT
ejpam-3807	76	75	.	.	PUNCT
ejpam-3807	77	1	a	a	DET
ejpam-3807	77	2	point	point	NOUN
ejpam-3807	77	3	a	a	PRON
ejpam-3807	77	4	is	be	AUX
ejpam-3807	77	5	an	an	DET
ejpam-3807	77	6	interior	interior	ADJ
ejpam-3807	77	7	point	point	NOUN
ejpam-3807	77	8	of	of	ADP
ejpam-3807	77	9	a	a	DET
ejpam-3807	77	10	if	if	SCONJ
ejpam-3807	77	11	a	a	DET
ejpam-3807	77	12	∈	∈	PROPN
ejpam-3807	77	13	int(a	int(a	PROPN
ejpam-3807	77	14	)	)	PUNCT
ejpam-3807	77	15	,	,	PUNCT
ejpam-3807	77	16	that	that	ADV
ejpam-3807	77	17	is	is	ADV
ejpam-3807	77	18	,	,	PUNCT
ejpam-3807	77	19	there	there	PRON
ejpam-3807	77	20	exists	exist	VERB
ejpam-3807	77	21	u(a	u(a	PROPN
ejpam-3807	77	22	)	)	PUNCT
ejpam-3807	77	23	∈	∈	PROPN
ejpam-3807	77	24	τ	τ	X
ejpam-3807	78	1	such	such	ADJ
ejpam-3807	78	2	that	that	SCONJ
ejpam-3807	78	3	u(a	u(a	PROPN
ejpam-3807	78	4	)	)	PUNCT
ejpam-3807	78	5	⊂	⊂	PROPN
ejpam-3807	79	1	a.	a.	NOUN
ejpam-3807	80	1	a	a	PRON
ejpam-3807	80	2	is	be	AUX
ejpam-3807	80	3	open	open	ADJ
ejpam-3807	80	4	if	if	SCONJ
ejpam-3807	80	5	and	and	CCONJ
ejpam-3807	80	6	only	only	ADV
ejpam-3807	80	7	if	if	SCONJ
ejpam-3807	80	8	int	int	NOUN
ejpam-3807	80	9	(	(	PUNCT
ejpam-3807	80	10	a)=	a)=	PART
ejpam-3807	80	11	a.	a.	NOUN
ejpam-3807	80	12	a	a	DET
ejpam-3807	80	13	set	set	NOUN
ejpam-3807	80	14	y	y	PROPN
ejpam-3807	80	15	⊂	⊂	PROPN
ejpam-3807	80	16	x	x	X
ejpam-3807	80	17	is	be	AUX
ejpam-3807	80	18	a	a	DET
ejpam-3807	80	19	closed	closed	ADJ
ejpam-3807	80	20	set	set	NOUN
ejpam-3807	80	21	in	in	ADP
ejpam-3807	80	22	x	x	PUNCT
ejpam-3807	80	23	if	if	SCONJ
ejpam-3807	80	24	its	its	PRON
ejpam-3807	80	25	complement	complement	NOUN
ejpam-3807	80	26	is	be	AUX
ejpam-3807	80	27	open	open	ADJ
ejpam-3807	80	28	.	.	PUNCT
ejpam-3807	81	1	a	a	DET
ejpam-3807	81	2	point	point	NOUN
ejpam-3807	81	3	x	x	X
ejpam-3807	81	4	∈	∈	NOUN
ejpam-3807	81	5	x	x	X
ejpam-3807	81	6	is	be	AUX
ejpam-3807	81	7	adherent	adherent	ADJ
ejpam-3807	81	8	to	to	ADP
ejpam-3807	81	9	y	y	PRON
ejpam-3807	81	10	if	if	SCONJ
ejpam-3807	81	11	each	each	DET
ejpam-3807	81	12	neighborhood	neighborhood	NOUN
ejpam-3807	81	13	of	of	ADP
ejpam-3807	81	14	x	x	PUNCT
ejpam-3807	81	15	contains	contain	VERB
ejpam-3807	81	16	at	at	ADV
ejpam-3807	81	17	least	least	ADV
ejpam-3807	81	18	one	one	NUM
ejpam-3807	81	19	point	point	NOUN
ejpam-3807	81	20	of	of	ADP
ejpam-3807	81	21	y	y	PROPN
ejpam-3807	81	22	.	.	PUNCT
ejpam-3807	82	1	the	the	DET
ejpam-3807	82	2	set	set	NOUN
ejpam-3807	82	3	y	y	PROPN
ejpam-3807	82	4	=	=	PRON
ejpam-3807	82	5	{	{	PUNCT
ejpam-3807	82	6	x	x	PUNCT
ejpam-3807	82	7	∈	∈	NOUN
ejpam-3807	82	8	x|∀u(x	x|∀u(x	NOUN
ejpam-3807	82	9	)	)	PUNCT
ejpam-3807	82	10	,	,	PUNCT
ejpam-3807	82	11	u(x	u(x	PROPN
ejpam-3807	82	12	)	)	PUNCT
ejpam-3807	82	13	∩	∩	PROPN
ejpam-3807	82	14	y	y	PROPN
ejpam-3807	82	15	6=	6=	PROPN
ejpam-3807	82	16	∅	∅	NOUN
ejpam-3807	82	17	}	}	PUNCT
ejpam-3807	82	18	of	of	ADP
ejpam-3807	82	19	all	all	DET
ejpam-3807	82	20	points	point	NOUN
ejpam-3807	82	21	in	in	ADP
ejpam-3807	82	22	x	x	PUNCT
ejpam-3807	82	23	adherent	adherent	NOUN
ejpam-3807	82	24	to	to	ADP
ejpam-3807	82	25	y	y	PROPN
ejpam-3807	82	26	is	be	AUX
ejpam-3807	82	27	called	call	VERB
ejpam-3807	82	28	the	the	DET
ejpam-3807	82	29	closure	closure	NOUN
ejpam-3807	82	30	of	of	ADP
ejpam-3807	82	31	y	y	PROPN
ejpam-3807	83	1	[	[	X
ejpam-3807	83	2	2	2	NUM
ejpam-3807	83	3	]	]	PUNCT
ejpam-3807	83	4	.	.	PUNCT
ejpam-3807	84	1	theorem	theorem	NOUN
ejpam-3807	84	2	2	2	NUM
ejpam-3807	84	3	.	.	PUNCT
ejpam-3807	85	1	[	[	X
ejpam-3807	85	2	6	6	NUM
ejpam-3807	85	3	]	]	PUNCT
ejpam-3807	85	4	let	let	VERB
ejpam-3807	85	5	x	x	SYM
ejpam-3807	85	6	=	=	SYM
ejpam-3807	85	7	(	(	PUNCT
ejpam-3807	85	8	x	x	X
ejpam-3807	85	9	,	,	PUNCT
ejpam-3807	85	10	∗	∗	NOUN
ejpam-3807	85	11	,	,	PUNCT
ejpam-3807	85	12	0	0	NUM
ejpam-3807	85	13	)	)	PUNCT
ejpam-3807	85	14	be	be	AUX
ejpam-3807	85	15	a	a	DET
ejpam-3807	85	16	b	b	NOUN
ejpam-3807	85	17	-	-	PUNCT
ejpam-3807	85	18	algebra	algebra	NOUN
ejpam-3807	85	19	and	and	CCONJ
ejpam-3807	85	20	τ	τ	PROPN
ejpam-3807	85	21	a	a	DET
ejpam-3807	85	22	b	b	NOUN
ejpam-3807	85	23	-	-	PUNCT
ejpam-3807	85	24	topology	topology	NOUN
ejpam-3807	85	25	on	on	ADP
ejpam-3807	85	26	the	the	DET
ejpam-3807	85	27	set	set	NOUN
ejpam-3807	85	28	x.	x.	NOUN
ejpam-3807	85	29	then	then	ADV
ejpam-3807	85	30	(	(	PUNCT
ejpam-3807	85	31	x	x	X
ejpam-3807	85	32	,	,	PUNCT
ejpam-3807	85	33	τ	τ	X
ejpam-3807	85	34	)	)	PUNCT
ejpam-3807	85	35	is	be	AUX
ejpam-3807	85	36	a	a	DET
ejpam-3807	85	37	topological	topological	ADJ
ejpam-3807	85	38	b	b	NOUN
ejpam-3807	85	39	-	-	PUNCT
ejpam-3807	85	40	algebra	algebra	NOUN
ejpam-3807	85	41	if	if	SCONJ
ejpam-3807	85	42	and	and	CCONJ
ejpam-3807	85	43	only	only	ADV
ejpam-3807	85	44	if	if	SCONJ
ejpam-3807	85	45	for	for	ADP
ejpam-3807	85	46	all	all	DET
ejpam-3807	85	47	x	x	NOUN
ejpam-3807	85	48	,	,	PUNCT
ejpam-3807	85	49	y	y	PROPN
ejpam-3807	85	50	in	in	ADP
ejpam-3807	85	51	x	x	X
ejpam-3807	85	52	and	and	CCONJ
ejpam-3807	85	53	for	for	ADP
ejpam-3807	85	54	every	every	DET
ejpam-3807	85	55	neighborhood	neighborhood	NOUN
ejpam-3807	85	56	w	w	NOUN
ejpam-3807	85	57	of	of	ADP
ejpam-3807	85	58	x∗y	x∗y	NUM
ejpam-3807	85	59	,	,	PUNCT
ejpam-3807	85	60	there	there	PRON
ejpam-3807	85	61	are	be	VERB
ejpam-3807	85	62	neighborhoods	neighborhood	NOUN
ejpam-3807	85	63	u	u	NOUN
ejpam-3807	85	64	and	and	CCONJ
ejpam-3807	85	65	v	v	NOUN
ejpam-3807	85	66	of	of	ADP
ejpam-3807	85	67	x	x	PROPN
ejpam-3807	85	68	and	and	CCONJ
ejpam-3807	85	69	y	y	PROPN
ejpam-3807	85	70	,	,	PUNCT
ejpam-3807	85	71	respectively	respectively	ADV
ejpam-3807	85	72	,	,	PUNCT
ejpam-3807	85	73	such	such	ADJ
ejpam-3807	85	74	that	that	SCONJ
ejpam-3807	85	75	u	u	NOUN
ejpam-3807	85	76	∗v	∗v	NOUN
ejpam-3807	85	77	⊆w	⊆w	NOUN
ejpam-3807	85	78	.	.	PUNCT
ejpam-3807	86	1	throughout	throughout	ADP
ejpam-3807	86	2	this	this	DET
ejpam-3807	86	3	article	article	NOUN
ejpam-3807	86	4	we	we	PRON
ejpam-3807	86	5	will	will	AUX
ejpam-3807	86	6	denote	denote	VERB
ejpam-3807	86	7	a	a	DET
ejpam-3807	86	8	b	b	PROPN
ejpam-3807	86	9	-	-	PUNCT
ejpam-3807	86	10	topological	topological	ADJ
ejpam-3807	86	11	space	space	NOUN
ejpam-3807	86	12	(	(	PUNCT
ejpam-3807	86	13	x	x	X
ejpam-3807	86	14	,	,	PUNCT
ejpam-3807	86	15	τ	τ	PROPN
ejpam-3807	86	16	)	)	PUNCT
ejpam-3807	86	17	,	,	PUNCT
ejpam-3807	86	18	topological	topological	ADJ
ejpam-3807	86	19	balgebra	balgebra	NOUN
ejpam-3807	86	20	(	(	PUNCT
ejpam-3807	86	21	x	x	X
ejpam-3807	86	22	,	,	PUNCT
ejpam-3807	86	23	∗	∗	NOUN
ejpam-3807	86	24	,	,	PUNCT
ejpam-3807	86	25	τ	τ	PROPN
ejpam-3807	86	26	)	)	PUNCT
ejpam-3807	86	27	,	,	PUNCT
ejpam-3807	86	28	or	or	CCONJ
ejpam-3807	86	29	a	a	DET
ejpam-3807	86	30	b	b	NOUN
ejpam-3807	86	31	-	-	PUNCT
ejpam-3807	86	32	algebra	algebra	NOUN
ejpam-3807	86	33	(	(	PUNCT
ejpam-3807	86	34	x	x	X
ejpam-3807	86	35	,	,	PUNCT
ejpam-3807	86	36	∗	∗	NOUN
ejpam-3807	86	37	,	,	PUNCT
ejpam-3807	86	38	0	0	NUM
ejpam-3807	86	39	)	)	PUNCT
ejpam-3807	86	40	as	as	ADP
ejpam-3807	86	41	simply	simply	ADV
ejpam-3807	86	42	,	,	PUNCT
ejpam-3807	86	43	x.	x.	PROPN
ejpam-3807	86	44	k.	k.	PROPN
ejpam-3807	86	45	belleza	belleza	PROPN
ejpam-3807	86	46	,	,	PUNCT
ejpam-3807	86	47	j.	j.	PROPN
ejpam-3807	86	48	vilela	vilela	PROPN
ejpam-3807	86	49	/	/	SYM
ejpam-3807	86	50	eur	eur	PROPN
ejpam-3807	86	51	.	.	PUNCT
ejpam-3807	87	1	j.	j.	PROPN
ejpam-3807	87	2	pure	pure	PROPN
ejpam-3807	87	3	appl	appl	PROPN
ejpam-3807	87	4	.	.	PROPN
ejpam-3807	87	5	math	math	PROPN
ejpam-3807	87	6	,	,	PUNCT
ejpam-3807	87	7	13	13	NUM
ejpam-3807	87	8	(	(	PUNCT
ejpam-3807	87	9	4	4	NUM
ejpam-3807	87	10	)	)	PUNCT
ejpam-3807	87	11	(	(	PUNCT
ejpam-3807	87	12	2020	2020	NUM
ejpam-3807	87	13	)	)	PUNCT
ejpam-3807	87	14	,	,	PUNCT
ejpam-3807	87	15	830	830	NUM
ejpam-3807	87	16	-	-	SYM
ejpam-3807	87	17	839	839	NUM
ejpam-3807	87	18	833	833	NUM
ejpam-3807	87	19	3	3	NUM
ejpam-3807	87	20	.	.	X
ejpam-3807	88	1	b	b	X
ejpam-3807	88	2	-	-	PUNCT
ejpam-3807	88	3	ideals	ideal	NOUN
ejpam-3807	88	4	in	in	ADP
ejpam-3807	88	5	topological	topological	PROPN
ejpam-3807	88	6	b	b	PROPN
ejpam-3807	88	7	-	-	PUNCT
ejpam-3807	88	8	algebras	algebras	ADJ
ejpam-3807	88	9	example	example	NOUN
ejpam-3807	88	10	2	2	X
ejpam-3807	88	11	.	.	X
ejpam-3807	88	12	consider	consider	VERB
ejpam-3807	88	13	the	the	DET
ejpam-3807	88	14	b	b	NOUN
ejpam-3807	88	15	-	-	PUNCT
ejpam-3807	88	16	algebra	algebra	NOUN
ejpam-3807	88	17	x	x	X
ejpam-3807	88	18	=	=	SYM
ejpam-3807	88	19	{	{	PUNCT
ejpam-3807	88	20	0	0	NUM
ejpam-3807	88	21	,	,	PUNCT
ejpam-3807	88	22	a	a	DET
ejpam-3807	88	23	,	,	PUNCT
ejpam-3807	88	24	b	b	NOUN
ejpam-3807	88	25	,	,	PUNCT
ejpam-3807	88	26	c	c	NOUN
ejpam-3807	88	27	}	}	PUNCT
ejpam-3807	88	28	with	with	ADP
ejpam-3807	88	29	the	the	DET
ejpam-3807	88	30	binary	binary	PROPN
ejpam-3807	88	31	operation	operation	NOUN
ejpam-3807	88	32	“	"	PUNCT
ejpam-3807	88	33	∗	∗	NOUN
ejpam-3807	88	34	”	"	PUNCT
ejpam-3807	88	35	defined	define	VERB
ejpam-3807	88	36	on	on	ADP
ejpam-3807	88	37	the	the	DET
ejpam-3807	88	38	cayley	cayley	ADJ
ejpam-3807	88	39	table	table	NOUN
ejpam-3807	88	40	provided	provide	VERB
ejpam-3807	88	41	.	.	PUNCT
ejpam-3807	89	1	let	let	VERB
ejpam-3807	89	2	τ	τ	PROPN
ejpam-3807	89	3	=	=	PRON
ejpam-3807	89	4	{	{	PUNCT
ejpam-3807	89	5	x,∅	x,∅	ADV
ejpam-3807	89	6	,	,	PUNCT
ejpam-3807	89	7	{	{	PUNCT
ejpam-3807	89	8	0	0	NUM
ejpam-3807	89	9	,	,	PUNCT
ejpam-3807	89	10	b	b	NOUN
ejpam-3807	89	11	}	}	PUNCT
ejpam-3807	89	12	,	,	PUNCT
ejpam-3807	89	13	{	{	PUNCT
ejpam-3807	89	14	a	a	PRON
ejpam-3807	89	15	,	,	PUNCT
ejpam-3807	89	16	c	c	NOUN
ejpam-3807	89	17	}	}	PUNCT
ejpam-3807	89	18	}	}	PUNCT
ejpam-3807	89	19	.	.	PUNCT
ejpam-3807	90	1	then	then	ADV
ejpam-3807	90	2	τ	τ	PROPN
ejpam-3807	90	3	is	be	AUX
ejpam-3807	90	4	a	a	DET
ejpam-3807	90	5	b	b	NOUN
ejpam-3807	90	6	-	-	PUNCT
ejpam-3807	90	7	topology	topology	NOUN
ejpam-3807	90	8	on	on	ADP
ejpam-3807	90	9	x	x	PUNCT
ejpam-3807	90	10	and	and	CCONJ
ejpam-3807	90	11	(	(	PUNCT
ejpam-3807	90	12	x	x	NOUN
ejpam-3807	90	13	,	,	PUNCT
ejpam-3807	90	14	∗	∗	NOUN
ejpam-3807	90	15	,	,	PUNCT
ejpam-3807	90	16	τ	τ	X
ejpam-3807	90	17	)	)	PUNCT
ejpam-3807	90	18	is	be	AUX
ejpam-3807	90	19	a	a	DET
ejpam-3807	90	20	topological	topological	ADJ
ejpam-3807	90	21	b	b	NOUN
ejpam-3807	90	22	-	-	PUNCT
ejpam-3807	90	23	algebra	algebra	NOUN
ejpam-3807	90	24	.	.	PUNCT
ejpam-3807	91	1	∗	∗	NOUN
ejpam-3807	91	2	0	0	NUM
ejpam-3807	92	1	a	a	DET
ejpam-3807	92	2	b	b	NOUN
ejpam-3807	92	3	c	c	NOUN
ejpam-3807	92	4	0	0	NUM
ejpam-3807	92	5	0	0	NUM
ejpam-3807	92	6	a	a	DET
ejpam-3807	92	7	b	b	NOUN
ejpam-3807	92	8	c	c	NOUN
ejpam-3807	92	9	a	a	DET
ejpam-3807	92	10	a	a	DET
ejpam-3807	92	11	0	0	NUM
ejpam-3807	92	12	c	c	NOUN
ejpam-3807	92	13	b	b	PROPN
ejpam-3807	92	14	b	b	PROPN
ejpam-3807	92	15	b	b	PROPN
ejpam-3807	92	16	c	c	PROPN
ejpam-3807	92	17	0	0	NUM
ejpam-3807	93	1	a	a	DET
ejpam-3807	93	2	c	c	NOUN
ejpam-3807	93	3	c	c	NOUN
ejpam-3807	93	4	b	b	PROPN
ejpam-3807	93	5	a	a	DET
ejpam-3807	93	6	0	0	NUM
ejpam-3807	93	7	remark	remark	NOUN
ejpam-3807	93	8	3	3	NUM
ejpam-3807	93	9	.	.	PUNCT
ejpam-3807	93	10	not	not	PART
ejpam-3807	93	11	every	every	DET
ejpam-3807	93	12	b	b	NOUN
ejpam-3807	93	13	-	-	PUNCT
ejpam-3807	93	14	ideal	ideal	NOUN
ejpam-3807	93	15	of	of	ADP
ejpam-3807	93	16	a	a	DET
ejpam-3807	93	17	b	b	NOUN
ejpam-3807	93	18	-	-	PUNCT
ejpam-3807	93	19	algebra	algebra	NOUN
ejpam-3807	93	20	x	x	PUNCT
ejpam-3807	93	21	is	be	AUX
ejpam-3807	93	22	either	either	CCONJ
ejpam-3807	93	23	an	an	DET
ejpam-3807	93	24	open	open	ADJ
ejpam-3807	93	25	or	or	CCONJ
ejpam-3807	93	26	closed	close	VERB
ejpam-3807	93	27	set	set	VERB
ejpam-3807	93	28	in	in	ADP
ejpam-3807	93	29	a	a	DET
ejpam-3807	93	30	topological	topological	ADJ
ejpam-3807	93	31	b	b	NOUN
ejpam-3807	93	32	-	-	PUNCT
ejpam-3807	93	33	algebra	algebra	NOUN
ejpam-3807	93	34	.	.	PUNCT
ejpam-3807	94	1	this	this	DET
ejpam-3807	94	2	remark	remark	NOUN
ejpam-3807	94	3	is	be	AUX
ejpam-3807	94	4	illustrated	illustrate	VERB
ejpam-3807	94	5	in	in	ADP
ejpam-3807	94	6	the	the	DET
ejpam-3807	94	7	next	next	ADJ
ejpam-3807	94	8	example	example	NOUN
ejpam-3807	94	9	.	.	PUNCT
ejpam-3807	95	1	example	example	NOUN
ejpam-3807	96	1	3	3	X
ejpam-3807	96	2	.	.	X
ejpam-3807	96	3	consider	consider	VERB
ejpam-3807	96	4	the	the	DET
ejpam-3807	96	5	topological	topological	ADJ
ejpam-3807	96	6	b	b	NOUN
ejpam-3807	96	7	-	-	PUNCT
ejpam-3807	96	8	algebra	algebra	NOUN
ejpam-3807	96	9	in	in	ADP
ejpam-3807	96	10	example	example	NOUN
ejpam-3807	96	11	2	2	NUM
ejpam-3807	96	12	.	.	PUNCT
ejpam-3807	97	1	let	let	VERB
ejpam-3807	97	2	i	i	PRON
ejpam-3807	97	3	=	=	PUNCT
ejpam-3807	97	4	{	{	PUNCT
ejpam-3807	97	5	0	0	NUM
ejpam-3807	97	6	,	,	PUNCT
ejpam-3807	97	7	c	c	NOUN
ejpam-3807	97	8	}	}	PUNCT
ejpam-3807	97	9	.	.	PUNCT
ejpam-3807	98	1	then	then	ADV
ejpam-3807	98	2	i	i	PRON
ejpam-3807	98	3	is	be	AUX
ejpam-3807	98	4	a	a	DET
ejpam-3807	98	5	b	b	NOUN
ejpam-3807	98	6	-	-	PUNCT
ejpam-3807	98	7	ideal	ideal	NOUN
ejpam-3807	98	8	of	of	ADP
ejpam-3807	98	9	x.	x.	NOUN
ejpam-3807	98	10	observe	observe	VERB
ejpam-3807	98	11	that	that	SCONJ
ejpam-3807	99	1	i	i	PRON
ejpam-3807	99	2	/∈	/∈	PUNCT
ejpam-3807	100	1	τ	τ	X
ejpam-3807	100	2	implying	imply	VERB
ejpam-3807	100	3	that	that	SCONJ
ejpam-3807	100	4	i	i	PRON
ejpam-3807	100	5	is	be	AUX
ejpam-3807	100	6	not	not	PART
ejpam-3807	100	7	an	an	DET
ejpam-3807	100	8	open	open	ADJ
ejpam-3807	100	9	set	set	NOUN
ejpam-3807	100	10	in	in	ADP
ejpam-3807	100	11	x.	x.	NOUN
ejpam-3807	100	12	also	also	ADV
ejpam-3807	100	13	,	,	PUNCT
ejpam-3807	100	14	x\i	x\i	PROPN
ejpam-3807	100	15	=	=	SYM
ejpam-3807	100	16	{	{	PUNCT
ejpam-3807	100	17	a	a	PRON
ejpam-3807	100	18	,	,	PUNCT
ejpam-3807	100	19	b	b	NOUN
ejpam-3807	100	20	}	}	PUNCT
ejpam-3807	100	21	/∈	/∈	PUNCT
ejpam-3807	101	1	τ	τ	PROPN
ejpam-3807	101	2	implying	imply	VERB
ejpam-3807	101	3	that	that	SCONJ
ejpam-3807	101	4	i	i	PRON
ejpam-3807	101	5	is	be	AUX
ejpam-3807	101	6	not	not	PART
ejpam-3807	101	7	a	a	DET
ejpam-3807	101	8	closed	closed	ADJ
ejpam-3807	101	9	set	set	NOUN
ejpam-3807	101	10	in	in	ADP
ejpam-3807	101	11	x.	x.	NOUN
ejpam-3807	101	12	however	however	ADV
ejpam-3807	101	13	if	if	SCONJ
ejpam-3807	101	14	a	a	DET
ejpam-3807	101	15	b	b	NOUN
ejpam-3807	101	16	-	-	PUNCT
ejpam-3807	101	17	ideal	ideal	NOUN
ejpam-3807	101	18	a	a	PRON
ejpam-3807	101	19	in	in	ADP
ejpam-3807	101	20	a	a	DET
ejpam-3807	101	21	topological	topological	ADJ
ejpam-3807	101	22	b	b	NOUN
ejpam-3807	101	23	-	-	PUNCT
ejpam-3807	101	24	algebra	algebra	NOUN
ejpam-3807	101	25	x	x	VERB
ejpam-3807	101	26	is	be	AUX
ejpam-3807	101	27	open	open	ADJ
ejpam-3807	101	28	,	,	PUNCT
ejpam-3807	101	29	it	it	PRON
ejpam-3807	101	30	is	be	AUX
ejpam-3807	101	31	also	also	ADV
ejpam-3807	101	32	a	a	DET
ejpam-3807	101	33	closed	closed	ADJ
ejpam-3807	101	34	b	b	NOUN
ejpam-3807	101	35	-	-	PUNCT
ejpam-3807	101	36	ideal	ideal	NOUN
ejpam-3807	101	37	in	in	ADP
ejpam-3807	101	38	x.	x.	NOUN
ejpam-3807	101	39	this	this	PRON
ejpam-3807	101	40	is	be	AUX
ejpam-3807	101	41	formally	formally	ADV
ejpam-3807	101	42	stated	state	VERB
ejpam-3807	101	43	in	in	ADP
ejpam-3807	101	44	the	the	DET
ejpam-3807	101	45	next	next	ADJ
ejpam-3807	101	46	theorem	theorem	PROPN
ejpam-3807	101	47	.	.	PUNCT
ejpam-3807	101	48	theorem	theorem	NOUN
ejpam-3807	101	49	3	3	NUM
ejpam-3807	101	50	.	.	PUNCT
ejpam-3807	102	1	if	if	SCONJ
ejpam-3807	102	2	a	a	PRON
ejpam-3807	102	3	is	be	AUX
ejpam-3807	102	4	an	an	DET
ejpam-3807	102	5	open	open	ADJ
ejpam-3807	102	6	b	b	NOUN
ejpam-3807	102	7	-	-	PUNCT
ejpam-3807	102	8	ideal	ideal	NOUN
ejpam-3807	102	9	of	of	ADP
ejpam-3807	102	10	a	a	DET
ejpam-3807	102	11	topological	topological	ADJ
ejpam-3807	102	12	b	b	NOUN
ejpam-3807	102	13	-	-	PUNCT
ejpam-3807	102	14	algebra	algebra	NOUN
ejpam-3807	102	15	x	x	NOUN
ejpam-3807	102	16	,	,	PUNCT
ejpam-3807	102	17	then	then	ADV
ejpam-3807	102	18	a	a	PRON
ejpam-3807	102	19	is	be	AUX
ejpam-3807	102	20	also	also	ADV
ejpam-3807	102	21	closed	closed	ADJ
ejpam-3807	102	22	.	.	PUNCT
ejpam-3807	103	1	proof	proof	NOUN
ejpam-3807	103	2	.	.	PUNCT
ejpam-3807	104	1	suppose	suppose	VERB
ejpam-3807	104	2	a	a	PRON
ejpam-3807	104	3	is	be	AUX
ejpam-3807	104	4	an	an	DET
ejpam-3807	104	5	open	open	ADJ
ejpam-3807	104	6	b	b	NOUN
ejpam-3807	104	7	-	-	PUNCT
ejpam-3807	104	8	ideal	ideal	NOUN
ejpam-3807	104	9	of	of	ADP
ejpam-3807	104	10	a	a	DET
ejpam-3807	104	11	topological	topological	ADJ
ejpam-3807	104	12	b	b	NOUN
ejpam-3807	104	13	-	-	PUNCT
ejpam-3807	104	14	algebra	algebra	NOUN
ejpam-3807	104	15	x.	x.	NOUN
ejpam-3807	104	16	let	let	VERB
ejpam-3807	104	17	x	x	PUNCT
ejpam-3807	104	18	∈	∈	PROPN
ejpam-3807	104	19	x\a	x\a	PROPN
ejpam-3807	104	20	.	.	PUNCT
ejpam-3807	105	1	since	since	SCONJ
ejpam-3807	105	2	a	a	PRON
ejpam-3807	105	3	is	be	AUX
ejpam-3807	105	4	a	a	DET
ejpam-3807	105	5	b	b	NOUN
ejpam-3807	105	6	-	-	PUNCT
ejpam-3807	105	7	ideal	ideal	NOUN
ejpam-3807	105	8	of	of	ADP
ejpam-3807	105	9	x	x	X
ejpam-3807	105	10	,	,	PUNCT
ejpam-3807	105	11	x	x	X
ejpam-3807	105	12	∗	∗	NOUN
ejpam-3807	105	13	x	x	X
ejpam-3807	105	14	=	=	SYM
ejpam-3807	105	15	0	0	NUM
ejpam-3807	105	16	∈	∈	PROPN
ejpam-3807	105	17	a	a	DET
ejpam-3807	105	18	by	by	X
ejpam-3807	105	19	(	(	PUNCT
ejpam-3807	105	20	b1	b1	NOUN
ejpam-3807	105	21	)	)	PUNCT
ejpam-3807	105	22	.	.	PUNCT
ejpam-3807	106	1	by	by	ADP
ejpam-3807	106	2	theorem	theorem	NOUN
ejpam-3807	106	3	2	2	NUM
ejpam-3807	106	4	,	,	PUNCT
ejpam-3807	106	5	there	there	PRON
ejpam-3807	106	6	exists	exist	VERB
ejpam-3807	106	7	u(x	u(x	NOUN
ejpam-3807	106	8	)	)	PUNCT
ejpam-3807	106	9	such	such	ADJ
ejpam-3807	106	10	that	that	SCONJ
ejpam-3807	106	11	u(x	u(x	NOUN
ejpam-3807	106	12	)	)	PUNCT
ejpam-3807	106	13	∗	∗	NOUN
ejpam-3807	106	14	u(x	u(x	NOUN
ejpam-3807	106	15	)	)	PUNCT
ejpam-3807	106	16	⊆	⊆	NUM
ejpam-3807	106	17	a.	a.	NOUN
ejpam-3807	106	18	we	we	PRON
ejpam-3807	106	19	claim	claim	VERB
ejpam-3807	106	20	that	that	SCONJ
ejpam-3807	106	21	u(x	u(x	VERB
ejpam-3807	106	22	)	)	PUNCT
ejpam-3807	107	1	⊆	⊆	NUM
ejpam-3807	107	2	x\a	x\a	PROPN
ejpam-3807	107	3	.	.	PUNCT
ejpam-3807	108	1	assume	assume	VERB
ejpam-3807	108	2	on	on	ADP
ejpam-3807	108	3	the	the	DET
ejpam-3807	108	4	contrary	contrary	NOUN
ejpam-3807	108	5	that	that	SCONJ
ejpam-3807	108	6	u(x	u(x	VERB
ejpam-3807	108	7	)	)	PUNCT
ejpam-3807	108	8	*	*	PUNCT
ejpam-3807	109	1	x\a	x\a	PROPN
ejpam-3807	109	2	,	,	PUNCT
ejpam-3807	109	3	that	that	ADV
ejpam-3807	109	4	is	is	ADV
ejpam-3807	109	5	,	,	PUNCT
ejpam-3807	109	6	u(x	u(x	PROPN
ejpam-3807	109	7	)	)	PUNCT
ejpam-3807	109	8	∩	∩	NOUN
ejpam-3807	109	9	a	a	DET
ejpam-3807	109	10	6=	6=	NUM
ejpam-3807	109	11	∅.	∅.	NOUN
ejpam-3807	109	12	then	then	ADV
ejpam-3807	109	13	there	there	PRON
ejpam-3807	109	14	exists	exist	VERB
ejpam-3807	109	15	y	y	PROPN
ejpam-3807	109	16	∈	∈	PROPN
ejpam-3807	109	17	u(x	u(x	PROPN
ejpam-3807	109	18	)	)	PUNCT
ejpam-3807	109	19	∩	∩	ADJ
ejpam-3807	109	20	a.	a.	NOUN
ejpam-3807	109	21	note	note	VERB
ejpam-3807	109	22	that	that	SCONJ
ejpam-3807	109	23	for	for	ADP
ejpam-3807	109	24	all	all	DET
ejpam-3807	109	25	z	z	NOUN
ejpam-3807	109	26	∈	∈	PROPN
ejpam-3807	109	27	u(x	u(x	NOUN
ejpam-3807	109	28	)	)	PUNCT
ejpam-3807	109	29	,	,	PUNCT
ejpam-3807	109	30	z	z	NOUN
ejpam-3807	109	31	∗	∗	NOUN
ejpam-3807	109	32	y	y	PROPN
ejpam-3807	109	33	∈	∈	PROPN
ejpam-3807	109	34	u(x	u(x	PROPN
ejpam-3807	109	35	)	)	PUNCT
ejpam-3807	109	36	∗u(x	∗u(x	NOUN
ejpam-3807	109	37	)	)	PUNCT
ejpam-3807	110	1	⊂	⊂	PROPN
ejpam-3807	110	2	a.	a.	NOUN
ejpam-3807	110	3	since	since	SCONJ
ejpam-3807	110	4	y	y	PROPN
ejpam-3807	110	5	∈	∈	PROPN
ejpam-3807	110	6	a	a	PRON
ejpam-3807	110	7	and	and	CCONJ
ejpam-3807	110	8	a	a	PRON
ejpam-3807	110	9	is	be	AUX
ejpam-3807	110	10	a	a	DET
ejpam-3807	110	11	b	b	NOUN
ejpam-3807	110	12	-	-	PUNCT
ejpam-3807	110	13	ideal	ideal	ADJ
ejpam-3807	110	14	,	,	PUNCT
ejpam-3807	110	15	z	z	PROPN
ejpam-3807	110	16	∈	∈	NOUN
ejpam-3807	110	17	a.	a.	NOUN
ejpam-3807	111	1	so	so	ADV
ejpam-3807	111	2	,	,	PUNCT
ejpam-3807	111	3	u(x	u(x	PROPN
ejpam-3807	111	4	)	)	PUNCT
ejpam-3807	112	1	⊆	⊆	NUM
ejpam-3807	112	2	a	a	DET
ejpam-3807	112	3	which	which	PRON
ejpam-3807	112	4	implies	imply	VERB
ejpam-3807	112	5	that	that	SCONJ
ejpam-3807	112	6	x	x	PUNCT
ejpam-3807	112	7	∈	∈	PROPN
ejpam-3807	112	8	a	a	X
ejpam-3807	112	9	,	,	PUNCT
ejpam-3807	112	10	a	a	DET
ejpam-3807	112	11	contradiction	contradiction	NOUN
ejpam-3807	112	12	.	.	PUNCT
ejpam-3807	113	1	hence	hence	ADV
ejpam-3807	113	2	,	,	PUNCT
ejpam-3807	113	3	x\a	x\a	PROPN
ejpam-3807	113	4	is	be	AUX
ejpam-3807	113	5	open	open	ADJ
ejpam-3807	113	6	.	.	PUNCT
ejpam-3807	114	1	therefore	therefore	ADV
ejpam-3807	114	2	,	,	PUNCT
ejpam-3807	114	3	a	a	PRON
ejpam-3807	114	4	is	be	AUX
ejpam-3807	114	5	closed	close	VERB
ejpam-3807	114	6	in	in	ADP
ejpam-3807	114	7	x.	x.	NOUN
ejpam-3807	114	8	the	the	DET
ejpam-3807	114	9	next	next	ADJ
ejpam-3807	114	10	theorem	theorem	NOUN
ejpam-3807	114	11	is	be	AUX
ejpam-3807	114	12	a	a	DET
ejpam-3807	114	13	characterization	characterization	NOUN
ejpam-3807	114	14	of	of	ADP
ejpam-3807	114	15	an	an	DET
ejpam-3807	114	16	open	open	ADJ
ejpam-3807	114	17	set	set	NOUN
ejpam-3807	114	18	(	(	PUNCT
ejpam-3807	114	19	containing	contain	VERB
ejpam-3807	114	20	0	0	NUM
ejpam-3807	114	21	)	)	PUNCT
ejpam-3807	114	22	in	in	ADP
ejpam-3807	114	23	a	a	DET
ejpam-3807	114	24	topological	topological	ADJ
ejpam-3807	114	25	b	b	NOUN
ejpam-3807	114	26	-	-	PUNCT
ejpam-3807	114	27	algebra	algebra	NOUN
ejpam-3807	114	28	.	.	PUNCT
ejpam-3807	115	1	theorem	theorem	NOUN
ejpam-3807	115	2	4	4	NUM
ejpam-3807	115	3	.	.	PUNCT
ejpam-3807	116	1	let	let	VERB
ejpam-3807	116	2	x	x	PRON
ejpam-3807	116	3	be	be	AUX
ejpam-3807	116	4	a	a	DET
ejpam-3807	116	5	topological	topological	ADJ
ejpam-3807	116	6	b	b	NOUN
ejpam-3807	116	7	-	-	PUNCT
ejpam-3807	116	8	algebra	algebra	NOUN
ejpam-3807	116	9	and	and	CCONJ
ejpam-3807	116	10	a	a	DET
ejpam-3807	116	11	⊂	⊂	PROPN
ejpam-3807	116	12	x	x	PUNCT
ejpam-3807	116	13	such	such	ADJ
ejpam-3807	116	14	that	that	SCONJ
ejpam-3807	116	15	0	0	NUM
ejpam-3807	116	16	∈	∈	NOUN
ejpam-3807	116	17	a.	a.	NOUN
ejpam-3807	116	18	then	then	ADV
ejpam-3807	116	19	a	a	PRON
ejpam-3807	116	20	is	be	AUX
ejpam-3807	116	21	open	open	ADJ
ejpam-3807	116	22	if	if	SCONJ
ejpam-3807	116	23	and	and	CCONJ
ejpam-3807	116	24	only	only	ADV
ejpam-3807	116	25	if	if	SCONJ
ejpam-3807	116	26	0	0	NUM
ejpam-3807	116	27	is	be	AUX
ejpam-3807	116	28	an	an	DET
ejpam-3807	116	29	interior	interior	ADJ
ejpam-3807	116	30	point	point	NOUN
ejpam-3807	116	31	of	of	ADP
ejpam-3807	116	32	a.	a.	NOUN
ejpam-3807	116	33	proof	proof	NOUN
ejpam-3807	116	34	.	.	PUNCT
ejpam-3807	117	1	suppose	suppose	VERB
ejpam-3807	117	2	a	a	PRON
ejpam-3807	117	3	is	be	AUX
ejpam-3807	117	4	open	open	ADJ
ejpam-3807	117	5	.	.	PUNCT
ejpam-3807	118	1	since	since	SCONJ
ejpam-3807	118	2	0	0	NUM
ejpam-3807	118	3	∈	∈	PROPN
ejpam-3807	118	4	a	a	PRON
ejpam-3807	118	5	,	,	PUNCT
ejpam-3807	118	6	0	0	NUM
ejpam-3807	118	7	is	be	AUX
ejpam-3807	118	8	an	an	DET
ejpam-3807	118	9	interior	interior	ADJ
ejpam-3807	118	10	point	point	NOUN
ejpam-3807	118	11	of	of	ADP
ejpam-3807	118	12	a.	a.	NOUN
ejpam-3807	118	13	conversely	conversely	ADV
ejpam-3807	118	14	,	,	PUNCT
ejpam-3807	118	15	suppose	suppose	VERB
ejpam-3807	118	16	0	0	NUM
ejpam-3807	118	17	is	be	AUX
ejpam-3807	118	18	an	an	DET
ejpam-3807	118	19	interior	interior	ADJ
ejpam-3807	118	20	point	point	NOUN
ejpam-3807	118	21	of	of	ADP
ejpam-3807	118	22	a.	a.	NOUN
ejpam-3807	118	23	then	then	ADV
ejpam-3807	118	24	there	there	PRON
ejpam-3807	118	25	exists	exist	VERB
ejpam-3807	118	26	u(0	u(0	NOUN
ejpam-3807	118	27	)	)	PUNCT
ejpam-3807	118	28	such	such	ADJ
ejpam-3807	118	29	that	that	SCONJ
ejpam-3807	118	30	u(0	u(0	NOUN
ejpam-3807	118	31	)	)	PUNCT
ejpam-3807	118	32	⊆	⊆	NUM
ejpam-3807	118	33	a.	a.	NOUN
ejpam-3807	118	34	let	let	VERB
ejpam-3807	118	35	y	y	PROPN
ejpam-3807	118	36	∈	∈	PROPN
ejpam-3807	118	37	a.	a.	NOUN
ejpam-3807	118	38	by	by	ADP
ejpam-3807	118	39	(	(	PUNCT
ejpam-3807	118	40	b1	b1	NOUN
ejpam-3807	118	41	)	)	PUNCT
ejpam-3807	118	42	,	,	PUNCT
ejpam-3807	119	1	y	y	PROPN
ejpam-3807	119	2	∗	∗	NOUN
ejpam-3807	119	3	y	y	PROPN
ejpam-3807	119	4	=	=	SYM
ejpam-3807	119	5	0	0	NUM
ejpam-3807	119	6	∈	∈	PROPN
ejpam-3807	119	7	u(0	u(0	PROPN
ejpam-3807	119	8	)	)	PUNCT
ejpam-3807	119	9	.	.	PUNCT
ejpam-3807	120	1	since	since	SCONJ
ejpam-3807	120	2	x	x	PRON
ejpam-3807	120	3	is	be	AUX
ejpam-3807	120	4	a	a	DET
ejpam-3807	120	5	topological	topological	ADJ
ejpam-3807	120	6	b	b	NOUN
ejpam-3807	120	7	-	-	PUNCT
ejpam-3807	120	8	algebra	algebra	NOUN
ejpam-3807	120	9	,	,	PUNCT
ejpam-3807	120	10	by	by	ADP
ejpam-3807	120	11	theorem	theorem	NOUN
ejpam-3807	120	12	2	2	NUM
ejpam-3807	120	13	,	,	PUNCT
ejpam-3807	120	14	there	there	PRON
ejpam-3807	120	15	exist	exist	VERB
ejpam-3807	120	16	u(y	u(y	NOUN
ejpam-3807	120	17	)	)	PUNCT
ejpam-3807	120	18	such	such	ADJ
ejpam-3807	120	19	that	that	SCONJ
ejpam-3807	120	20	u(y	u(y	NOUN
ejpam-3807	120	21	)	)	PUNCT
ejpam-3807	120	22	∗	∗	NOUN
ejpam-3807	120	23	u(y	u(y	NOUN
ejpam-3807	120	24	)	)	PUNCT
ejpam-3807	120	25	⊆	⊆	NUM
ejpam-3807	120	26	u(0	u(0	NOUN
ejpam-3807	120	27	)	)	PUNCT
ejpam-3807	120	28	.	.	PUNCT
ejpam-3807	121	1	it	it	PRON
ejpam-3807	121	2	remains	remain	VERB
ejpam-3807	121	3	to	to	PART
ejpam-3807	121	4	show	show	VERB
ejpam-3807	121	5	that	that	SCONJ
ejpam-3807	121	6	u(y	u(y	NOUN
ejpam-3807	121	7	)	)	PUNCT
ejpam-3807	121	8	⊆	⊆	NUM
ejpam-3807	121	9	a.	a.	NOUN
ejpam-3807	121	10	let	let	VERB
ejpam-3807	121	11	x	x	X
ejpam-3807	121	12	∈	∈	PROPN
ejpam-3807	121	13	u(y	u(y	PROPN
ejpam-3807	121	14	)	)	PUNCT
ejpam-3807	121	15	.	.	PUNCT
ejpam-3807	122	1	by	by	ADP
ejpam-3807	122	2	(	(	PUNCT
ejpam-3807	122	3	b1	b1	NOUN
ejpam-3807	122	4	)	)	PUNCT
ejpam-3807	122	5	,	,	PUNCT
ejpam-3807	122	6	x∗x	x∗x	PUNCT
ejpam-3807	122	7	=	=	SYM
ejpam-3807	122	8	0	0	NUM
ejpam-3807	122	9	∈	∈	NOUN
ejpam-3807	122	10	a.	a.	NOUN
ejpam-3807	122	11	if	if	SCONJ
ejpam-3807	122	12	x	x	PROPN
ejpam-3807	122	13	∈	∈	PROPN
ejpam-3807	122	14	a	a	X
ejpam-3807	122	15	,	,	PUNCT
ejpam-3807	122	16	we	we	PRON
ejpam-3807	122	17	are	be	AUX
ejpam-3807	122	18	done	do	VERB
ejpam-3807	122	19	.	.	PUNCT
ejpam-3807	123	1	suppose	suppose	VERB
ejpam-3807	124	1	x	x	X
ejpam-3807	124	2	/∈	/∈	PUNCT
ejpam-3807	124	3	a.	a.	NOUN
ejpam-3807	124	4	then	then	ADV
ejpam-3807	124	5	x	x	PROPN
ejpam-3807	124	6	/∈	/∈	PUNCT
ejpam-3807	124	7	u(0	u(0	NOUN
ejpam-3807	124	8	)	)	PUNCT
ejpam-3807	124	9	.	.	PUNCT
ejpam-3807	125	1	this	this	PRON
ejpam-3807	125	2	implies	imply	VERB
ejpam-3807	125	3	that	that	SCONJ
ejpam-3807	125	4	x	x	SYM
ejpam-3807	125	5	/∈	/∈	PUNCT
ejpam-3807	125	6	u(y	u(y	NOUN
ejpam-3807	125	7	)	)	PUNCT
ejpam-3807	125	8	∗	∗	NOUN
ejpam-3807	125	9	u(y	u(y	NOUN
ejpam-3807	125	10	)	)	PUNCT
ejpam-3807	125	11	.	.	PUNCT
ejpam-3807	126	1	by	by	ADP
ejpam-3807	126	2	(	(	PUNCT
ejpam-3807	126	3	b2	b2	NOUN
ejpam-3807	126	4	)	)	PUNCT
ejpam-3807	126	5	,	,	PUNCT
ejpam-3807	126	6	x	x	X
ejpam-3807	126	7	∗	∗	NOUN
ejpam-3807	126	8	0	0	NUM
ejpam-3807	127	1	=	=	SYM
ejpam-3807	127	2	x	x	PROPN
ejpam-3807	127	3	/∈	/∈	PUNCT
ejpam-3807	127	4	u(y	u(y	NOUN
ejpam-3807	127	5	)	)	PUNCT
ejpam-3807	127	6	∗	∗	NOUN
ejpam-3807	127	7	u(y	u(y	NOUN
ejpam-3807	127	8	)	)	PUNCT
ejpam-3807	127	9	.	.	PUNCT
ejpam-3807	128	1	this	this	PRON
ejpam-3807	128	2	implies	imply	VERB
ejpam-3807	128	3	that	that	SCONJ
ejpam-3807	128	4	x	x	SYM
ejpam-3807	128	5	/∈	/∈	PUNCT
ejpam-3807	128	6	u(y	u(y	NOUN
ejpam-3807	128	7	)	)	PUNCT
ejpam-3807	128	8	which	which	PRON
ejpam-3807	128	9	is	be	AUX
ejpam-3807	128	10	a	a	DET
ejpam-3807	128	11	contradiction	contradiction	NOUN
ejpam-3807	128	12	.	.	PUNCT
ejpam-3807	129	1	therefore	therefore	ADV
ejpam-3807	129	2	,	,	PUNCT
ejpam-3807	129	3	u(y	u(y	PROPN
ejpam-3807	129	4	)	)	PUNCT
ejpam-3807	129	5	⊆	⊆	NUM
ejpam-3807	129	6	a	a	PRON
ejpam-3807	129	7	and	and	CCONJ
ejpam-3807	129	8	a	a	PRON
ejpam-3807	129	9	is	be	AUX
ejpam-3807	129	10	open	open	ADJ
ejpam-3807	129	11	.	.	PUNCT
ejpam-3807	130	1	the	the	DET
ejpam-3807	130	2	following	follow	VERB
ejpam-3807	130	3	corollary	corollary	NOUN
ejpam-3807	130	4	follows	follow	VERB
ejpam-3807	130	5	from	from	ADP
ejpam-3807	130	6	theorem	theorem	ADJ
ejpam-3807	130	7	4	4	NUM
ejpam-3807	130	8	.	.	PUNCT
ejpam-3807	130	9	k.	k.	PROPN
ejpam-3807	130	10	belleza	belleza	PROPN
ejpam-3807	130	11	,	,	PUNCT
ejpam-3807	130	12	j.	j.	PROPN
ejpam-3807	130	13	vilela	vilela	PROPN
ejpam-3807	130	14	/	/	SYM
ejpam-3807	130	15	eur	eur	PROPN
ejpam-3807	130	16	.	.	PUNCT
ejpam-3807	131	1	j.	j.	PROPN
ejpam-3807	131	2	pure	pure	PROPN
ejpam-3807	131	3	appl	appl	PROPN
ejpam-3807	131	4	.	.	PROPN
ejpam-3807	131	5	math	math	PROPN
ejpam-3807	131	6	,	,	PUNCT
ejpam-3807	131	7	13	13	NUM
ejpam-3807	131	8	(	(	PUNCT
ejpam-3807	131	9	4	4	NUM
ejpam-3807	131	10	)	)	PUNCT
ejpam-3807	131	11	(	(	PUNCT
ejpam-3807	131	12	2020	2020	NUM
ejpam-3807	131	13	)	)	PUNCT
ejpam-3807	131	14	,	,	PUNCT
ejpam-3807	131	15	830	830	NUM
ejpam-3807	131	16	-	-	SYM
ejpam-3807	131	17	839	839	NUM
ejpam-3807	131	18	834	834	NUM
ejpam-3807	131	19	corollary	corollary	NOUN
ejpam-3807	131	20	1	1	NUM
ejpam-3807	131	21	.	.	PUNCT
ejpam-3807	132	1	let	let	VERB
ejpam-3807	132	2	x	x	PRON
ejpam-3807	132	3	be	be	AUX
ejpam-3807	132	4	a	a	DET
ejpam-3807	132	5	topological	topological	ADJ
ejpam-3807	132	6	b	b	NOUN
ejpam-3807	132	7	-	-	PUNCT
ejpam-3807	132	8	algebra	algebra	NOUN
ejpam-3807	132	9	.	.	PUNCT
ejpam-3807	133	1	then	then	ADV
ejpam-3807	133	2	0	0	NUM
ejpam-3807	133	3	is	be	AUX
ejpam-3807	133	4	an	an	DET
ejpam-3807	133	5	interior	interior	ADJ
ejpam-3807	133	6	point	point	NOUN
ejpam-3807	133	7	of	of	ADP
ejpam-3807	133	8	a	a	DET
ejpam-3807	133	9	b	b	NOUN
ejpam-3807	133	10	-	-	PUNCT
ejpam-3807	133	11	ideal	ideal	NOUN
ejpam-3807	134	1	i	i	PRON
ejpam-3807	134	2	if	if	SCONJ
ejpam-3807	134	3	and	and	CCONJ
ejpam-3807	134	4	only	only	ADV
ejpam-3807	134	5	if	if	SCONJ
ejpam-3807	134	6	i	i	PRON
ejpam-3807	134	7	is	be	AUX
ejpam-3807	134	8	open	open	ADJ
ejpam-3807	134	9	.	.	PUNCT
ejpam-3807	135	1	the	the	DET
ejpam-3807	135	2	next	next	ADJ
ejpam-3807	135	3	example	example	NOUN
ejpam-3807	135	4	illustrates	illustrate	VERB
ejpam-3807	135	5	that	that	SCONJ
ejpam-3807	135	6	an	an	DET
ejpam-3807	135	7	open	open	ADJ
ejpam-3807	135	8	subset	subset	NOUN
ejpam-3807	135	9	of	of	ADP
ejpam-3807	135	10	a	a	DET
ejpam-3807	135	11	topological	topological	ADJ
ejpam-3807	135	12	b	b	X
ejpam-3807	135	13	-	-	PUNCT
ejpam-3807	135	14	algebra	algebra	NOUN
ejpam-3807	135	15	may	may	AUX
ejpam-3807	135	16	not	not	PART
ejpam-3807	135	17	be	be	AUX
ejpam-3807	135	18	a	a	DET
ejpam-3807	135	19	b	b	NOUN
ejpam-3807	135	20	-	-	PUNCT
ejpam-3807	135	21	ideal	ideal	NOUN
ejpam-3807	135	22	in	in	ADP
ejpam-3807	135	23	which	which	PRON
ejpam-3807	135	24	the	the	DET
ejpam-3807	135	25	observation	observation	NOUN
ejpam-3807	135	26	is	be	AUX
ejpam-3807	135	27	formally	formally	ADV
ejpam-3807	135	28	stated	state	VERB
ejpam-3807	135	29	as	as	ADP
ejpam-3807	135	30	a	a	DET
ejpam-3807	135	31	remark	remark	NOUN
ejpam-3807	135	32	.	.	PUNCT
ejpam-3807	136	1	example	example	NOUN
ejpam-3807	136	2	4	4	NUM
ejpam-3807	136	3	.	.	PUNCT
ejpam-3807	137	1	consider	consider	VERB
ejpam-3807	137	2	the	the	DET
ejpam-3807	137	3	topological	topological	ADJ
ejpam-3807	137	4	b	b	NOUN
ejpam-3807	137	5	-	-	PUNCT
ejpam-3807	137	6	algebra	algebra	NOUN
ejpam-3807	137	7	in	in	ADP
ejpam-3807	137	8	example	example	NOUN
ejpam-3807	137	9	2	2	NUM
ejpam-3807	137	10	.	.	X
ejpam-3807	137	11	examine	examine	VERB
ejpam-3807	137	12	the	the	DET
ejpam-3807	137	13	open	open	ADJ
ejpam-3807	137	14	set	set	NOUN
ejpam-3807	137	15	{	{	PUNCT
ejpam-3807	137	16	a	a	NOUN
ejpam-3807	137	17	,	,	PUNCT
ejpam-3807	137	18	c	c	NOUN
ejpam-3807	137	19	}	}	PUNCT
ejpam-3807	137	20	.	.	PUNCT
ejpam-3807	138	1	note	note	VERB
ejpam-3807	138	2	that	that	SCONJ
ejpam-3807	138	3	{	{	PUNCT
ejpam-3807	138	4	a	a	PRON
ejpam-3807	138	5	,	,	PUNCT
ejpam-3807	138	6	c	c	NOUN
ejpam-3807	138	7	}	}	PUNCT
ejpam-3807	138	8	is	be	AUX
ejpam-3807	138	9	not	not	PART
ejpam-3807	138	10	a	a	DET
ejpam-3807	138	11	b	b	NOUN
ejpam-3807	138	12	-	-	PUNCT
ejpam-3807	138	13	ideal	ideal	NOUN
ejpam-3807	138	14	since	since	SCONJ
ejpam-3807	138	15	0	0	NUM
ejpam-3807	138	16	/∈	/∈	INTJ
ejpam-3807	138	17	{	{	PUNCT
ejpam-3807	138	18	a	a	X
ejpam-3807	138	19	,	,	PUNCT
ejpam-3807	138	20	c	c	NOUN
ejpam-3807	138	21	}	}	PUNCT
ejpam-3807	138	22	and	and	CCONJ
ejpam-3807	138	23	b	b	NOUN
ejpam-3807	138	24	∗	∗	NOUN
ejpam-3807	138	25	c	c	NOUN
ejpam-3807	138	26	=	=	PUNCT
ejpam-3807	138	27	a	a	PROPN
ejpam-3807	138	28	and	and	CCONJ
ejpam-3807	138	29	c	c	NOUN
ejpam-3807	138	30	∈	∈	PROPN
ejpam-3807	138	31	{	{	PUNCT
ejpam-3807	138	32	a	a	NOUN
ejpam-3807	138	33	,	,	PUNCT
ejpam-3807	138	34	c	c	NOUN
ejpam-3807	138	35	}	}	PUNCT
ejpam-3807	138	36	but	but	CCONJ
ejpam-3807	138	37	b	b	X
ejpam-3807	138	38	/∈	/∈	PUNCT
ejpam-3807	138	39	{	{	PUNCT
ejpam-3807	138	40	a	a	X
ejpam-3807	138	41	,	,	PUNCT
ejpam-3807	138	42	c	c	NOUN
ejpam-3807	138	43	}	}	PUNCT
ejpam-3807	138	44	.	.	PUNCT
ejpam-3807	139	1	remark	remark	PROPN
ejpam-3807	139	2	4	4	NUM
ejpam-3807	139	3	.	.	PUNCT
ejpam-3807	139	4	not	not	PART
ejpam-3807	139	5	every	every	DET
ejpam-3807	139	6	open	open	ADJ
ejpam-3807	139	7	subset	subset	NOUN
ejpam-3807	139	8	of	of	ADP
ejpam-3807	139	9	a	a	DET
ejpam-3807	139	10	topological	topological	ADJ
ejpam-3807	139	11	b	b	NOUN
ejpam-3807	139	12	-	-	PUNCT
ejpam-3807	139	13	algebra	algebra	NOUN
ejpam-3807	139	14	x	x	PUNCT
ejpam-3807	139	15	is	be	AUX
ejpam-3807	139	16	a	a	DET
ejpam-3807	139	17	b	b	NOUN
ejpam-3807	139	18	-	-	PUNCT
ejpam-3807	139	19	ideal	ideal	NOUN
ejpam-3807	139	20	of	of	ADP
ejpam-3807	139	21	x.	x.	NOUN
ejpam-3807	139	22	however	however	ADV
ejpam-3807	139	23	,	,	PUNCT
ejpam-3807	139	24	if	if	SCONJ
ejpam-3807	139	25	all	all	DET
ejpam-3807	139	26	open	open	ADJ
ejpam-3807	139	27	sets	set	NOUN
ejpam-3807	139	28	are	be	AUX
ejpam-3807	139	29	neighborhoods	neighborhood	NOUN
ejpam-3807	139	30	of	of	ADP
ejpam-3807	139	31	0	0	NUM
ejpam-3807	139	32	,	,	PUNCT
ejpam-3807	139	33	every	every	DET
ejpam-3807	139	34	open	open	ADJ
ejpam-3807	139	35	subset	subset	NOUN
ejpam-3807	139	36	of	of	ADP
ejpam-3807	139	37	a	a	DET
ejpam-3807	139	38	topological	topological	ADJ
ejpam-3807	139	39	b	b	NOUN
ejpam-3807	139	40	-	-	PUNCT
ejpam-3807	139	41	algebra	algebra	NOUN
ejpam-3807	139	42	x	x	PUNCT
ejpam-3807	139	43	is	be	AUX
ejpam-3807	139	44	a	a	DET
ejpam-3807	139	45	b	b	NOUN
ejpam-3807	139	46	-	-	PUNCT
ejpam-3807	139	47	ideal	ideal	NOUN
ejpam-3807	139	48	of	of	ADP
ejpam-3807	139	49	x.	x.	NOUN
ejpam-3807	139	50	this	this	PRON
ejpam-3807	139	51	is	be	AUX
ejpam-3807	139	52	formally	formally	ADV
ejpam-3807	139	53	stated	state	VERB
ejpam-3807	139	54	in	in	ADP
ejpam-3807	139	55	the	the	DET
ejpam-3807	139	56	next	next	ADJ
ejpam-3807	139	57	theorem	theorem	NOUN
ejpam-3807	139	58	which	which	PRON
ejpam-3807	139	59	is	be	AUX
ejpam-3807	139	60	a	a	DET
ejpam-3807	139	61	characterization	characterization	NOUN
ejpam-3807	139	62	of	of	ADP
ejpam-3807	139	63	a	a	DET
ejpam-3807	139	64	b	b	NOUN
ejpam-3807	139	65	-	-	PUNCT
ejpam-3807	139	66	ideal	ideal	NOUN
ejpam-3807	139	67	in	in	ADP
ejpam-3807	139	68	a	a	DET
ejpam-3807	139	69	topological	topological	ADJ
ejpam-3807	139	70	b	b	NOUN
ejpam-3807	139	71	-	-	PUNCT
ejpam-3807	139	72	algebra	algebra	NOUN
ejpam-3807	139	73	.	.	PUNCT
ejpam-3807	140	1	theorem	theorem	NOUN
ejpam-3807	140	2	5	5	NUM
ejpam-3807	140	3	.	.	PUNCT
ejpam-3807	141	1	let	let	VERB
ejpam-3807	141	2	x	x	PRON
ejpam-3807	141	3	be	be	AUX
ejpam-3807	141	4	a	a	DET
ejpam-3807	141	5	topological	topological	ADJ
ejpam-3807	141	6	b	b	NOUN
ejpam-3807	141	7	-	-	PUNCT
ejpam-3807	141	8	algebra	algebra	NOUN
ejpam-3807	141	9	and	and	CCONJ
ejpam-3807	141	10	i	i	PRON
ejpam-3807	141	11	an	an	DET
ejpam-3807	141	12	open	open	ADJ
ejpam-3807	141	13	subset	subset	NOUN
ejpam-3807	141	14	of	of	ADP
ejpam-3807	141	15	x.	x.	NOUN
ejpam-3807	142	1	if	if	SCONJ
ejpam-3807	142	2	0	0	NUM
ejpam-3807	142	3	∈	∈	PROPN
ejpam-3807	142	4	⋂	⋂	PROPN
ejpam-3807	142	5	u∈τ	u∈τ	ADJ
ejpam-3807	142	6	u	u	NOUN
ejpam-3807	142	7	,	,	PUNCT
ejpam-3807	142	8	then	then	ADV
ejpam-3807	142	9	i	i	PRON
ejpam-3807	142	10	is	be	AUX
ejpam-3807	142	11	a	a	DET
ejpam-3807	142	12	b	b	NOUN
ejpam-3807	142	13	-	-	PUNCT
ejpam-3807	142	14	ideal	ideal	NOUN
ejpam-3807	142	15	of	of	ADP
ejpam-3807	142	16	x.	x.	NOUN
ejpam-3807	142	17	proof	proof	NOUN
ejpam-3807	142	18	.	.	PUNCT
ejpam-3807	143	1	let	let	VERB
ejpam-3807	143	2	x	x	PRON
ejpam-3807	143	3	∗	∗	VERB
ejpam-3807	143	4	y	y	NOUN
ejpam-3807	143	5	∈	∈	PROPN
ejpam-3807	143	6	i	i	PRON
ejpam-3807	143	7	where	where	SCONJ
ejpam-3807	143	8	y	y	PROPN
ejpam-3807	143	9	∈	∈	PROPN
ejpam-3807	143	10	i.	i.	NOUN
ejpam-3807	143	11	since	since	SCONJ
ejpam-3807	143	12	i	i	PRON
ejpam-3807	143	13	is	be	AUX
ejpam-3807	143	14	open	open	ADJ
ejpam-3807	143	15	,	,	PUNCT
ejpam-3807	143	16	by	by	ADP
ejpam-3807	143	17	theorem	theorem	NOUN
ejpam-3807	143	18	2	2	NUM
ejpam-3807	143	19	,	,	PUNCT
ejpam-3807	143	20	there	there	PRON
ejpam-3807	143	21	exist	exist	VERB
ejpam-3807	143	22	v	v	ADP
ejpam-3807	143	23	(	(	PUNCT
ejpam-3807	143	24	x	x	NOUN
ejpam-3807	143	25	)	)	PUNCT
ejpam-3807	143	26	and	and	CCONJ
ejpam-3807	143	27	v	v	NOUN
ejpam-3807	143	28	(	(	PUNCT
ejpam-3807	143	29	y	y	NOUN
ejpam-3807	143	30	)	)	PUNCT
ejpam-3807	143	31	such	such	ADJ
ejpam-3807	143	32	that	that	DET
ejpam-3807	143	33	v	v	NOUN
ejpam-3807	143	34	(	(	PUNCT
ejpam-3807	143	35	x)∗v	x)∗v	X
ejpam-3807	143	36	(	(	PUNCT
ejpam-3807	143	37	y	y	X
ejpam-3807	143	38	)	)	PUNCT
ejpam-3807	143	39	⊆	⊆	NUM
ejpam-3807	143	40	u(x∗y	u(x∗y	NOUN
ejpam-3807	143	41	)	)	PUNCT
ejpam-3807	143	42	⊆	⊆	NUM
ejpam-3807	143	43	i.	i.	NOUN
ejpam-3807	143	44	by	by	ADP
ejpam-3807	143	45	(	(	PUNCT
ejpam-3807	143	46	b2	b2	NOUN
ejpam-3807	143	47	)	)	PUNCT
ejpam-3807	143	48	,	,	PUNCT
ejpam-3807	143	49	x	x	PUNCT
ejpam-3807	143	50	=	=	PUNCT
ejpam-3807	143	51	x∗0	x∗0	PROPN
ejpam-3807	143	52	∈	∈	PROPN
ejpam-3807	143	53	v	v	PROPN
ejpam-3807	143	54	(	(	PUNCT
ejpam-3807	143	55	x)∗v	x)∗v	X
ejpam-3807	143	56	(	(	PUNCT
ejpam-3807	143	57	y	y	X
ejpam-3807	143	58	)	)	PUNCT
ejpam-3807	143	59	⊆	⊆	NUM
ejpam-3807	143	60	i.	i.	NOUN
ejpam-3807	143	61	therefore	therefore	ADV
ejpam-3807	143	62	,	,	PUNCT
ejpam-3807	143	63	i	i	PRON
ejpam-3807	143	64	is	be	AUX
ejpam-3807	143	65	a	a	DET
ejpam-3807	143	66	b	b	NOUN
ejpam-3807	143	67	-	-	PUNCT
ejpam-3807	143	68	ideal	ideal	NOUN
ejpam-3807	143	69	of	of	ADP
ejpam-3807	143	70	x.	x.	PROPN
ejpam-3807	143	71	lemma	lemma	PROPN
ejpam-3807	144	1	3	3	X
ejpam-3807	144	2	.	.	PUNCT
ejpam-3807	145	1	let	let	VERB
ejpam-3807	145	2	x	x	PRON
ejpam-3807	145	3	be	be	AUX
ejpam-3807	145	4	a	a	DET
ejpam-3807	145	5	topological	topological	ADJ
ejpam-3807	145	6	b	b	NOUN
ejpam-3807	145	7	-	-	PUNCT
ejpam-3807	145	8	algebra	algebra	NOUN
ejpam-3807	145	9	,	,	PUNCT
ejpam-3807	145	10	i0	i0	PROPN
ejpam-3807	145	11	⊆	⊆	NUM
ejpam-3807	145	12	x	x	PUNCT
ejpam-3807	145	13	where	where	SCONJ
ejpam-3807	145	14	i0	i0	PROPN
ejpam-3807	145	15	contains	contain	VERB
ejpam-3807	145	16	0	0	NUM
ejpam-3807	145	17	is	be	AUX
ejpam-3807	145	18	such	such	ADJ
ejpam-3807	146	1	that	that	SCONJ
ejpam-3807	146	2	if	if	SCONJ
ejpam-3807	146	3	0	0	NUM
ejpam-3807	146	4	∈	∈	PROPN
ejpam-3807	146	5	u	u	NOUN
ejpam-3807	146	6	,	,	PUNCT
ejpam-3807	146	7	then	then	ADV
ejpam-3807	146	8	i0	i0	PROPN
ejpam-3807	146	9	⊆	⊆	NUM
ejpam-3807	146	10	u	u	NOUN
ejpam-3807	146	11	for	for	ADP
ejpam-3807	146	12	all	all	DET
ejpam-3807	146	13	u	u	PROPN
ejpam-3807	146	14	∈	∈	PROPN
ejpam-3807	146	15	τ	τ	X
ejpam-3807	146	16	.	.	PUNCT
ejpam-3807	146	17	then	then	ADV
ejpam-3807	146	18	for	for	ADP
ejpam-3807	146	19	any	any	DET
ejpam-3807	146	20	x	x	SYM
ejpam-3807	146	21	∈	∈	PROPN
ejpam-3807	146	22	i0	i0	PROPN
ejpam-3807	146	23	and	and	CCONJ
ejpam-3807	146	24	u(x	u(x	PROPN
ejpam-3807	146	25	)	)	PUNCT
ejpam-3807	146	26	∈	∈	PROPN
ejpam-3807	146	27	τ	τ	PROPN
ejpam-3807	146	28	,	,	PUNCT
ejpam-3807	146	29	i0	i0	PROPN
ejpam-3807	146	30	⊆	⊆	NUM
ejpam-3807	146	31	u(x	u(x	NOUN
ejpam-3807	146	32	)	)	PUNCT
ejpam-3807	146	33	.	.	PUNCT
ejpam-3807	147	1	proof	proof	NOUN
ejpam-3807	147	2	.	.	PUNCT
ejpam-3807	148	1	suppose	suppose	VERB
ejpam-3807	148	2	x	x	X
ejpam-3807	148	3	∈	∈	PROPN
ejpam-3807	148	4	i0	i0	PROPN
ejpam-3807	148	5	.	.	PUNCT
ejpam-3807	149	1	by	by	ADP
ejpam-3807	149	2	(	(	PUNCT
ejpam-3807	149	3	b2	b2	NOUN
ejpam-3807	149	4	)	)	PUNCT
ejpam-3807	149	5	,	,	PUNCT
ejpam-3807	149	6	x	x	X
ejpam-3807	149	7	∗	∗	NOUN
ejpam-3807	149	8	0	0	NUM
ejpam-3807	150	1	=	=	SYM
ejpam-3807	150	2	x	x	SYM
ejpam-3807	150	3	∈	∈	NOUN
ejpam-3807	150	4	u(x	u(x	NOUN
ejpam-3807	150	5	)	)	PUNCT
ejpam-3807	150	6	.	.	PUNCT
ejpam-3807	151	1	by	by	ADP
ejpam-3807	151	2	theorem	theorem	NOUN
ejpam-3807	151	3	2	2	NUM
ejpam-3807	151	4	,	,	PUNCT
ejpam-3807	151	5	there	there	PRON
ejpam-3807	151	6	exist	exist	VERB
ejpam-3807	151	7	v	v	ADP
ejpam-3807	151	8	(	(	PUNCT
ejpam-3807	151	9	x	x	NOUN
ejpam-3807	151	10	)	)	PUNCT
ejpam-3807	151	11	and	and	CCONJ
ejpam-3807	151	12	v	v	X
ejpam-3807	151	13	(	(	PUNCT
ejpam-3807	151	14	0	0	NUM
ejpam-3807	151	15	)	)	PUNCT
ejpam-3807	151	16	such	such	ADJ
ejpam-3807	151	17	that	that	PRON
ejpam-3807	151	18	v	v	NOUN
ejpam-3807	151	19	(	(	PUNCT
ejpam-3807	151	20	x	x	NOUN
ejpam-3807	151	21	)	)	PUNCT
ejpam-3807	151	22	∗	∗	NOUN
ejpam-3807	151	23	v	v	NOUN
ejpam-3807	151	24	(	(	PUNCT
ejpam-3807	151	25	0	0	NUM
ejpam-3807	151	26	)	)	PUNCT
ejpam-3807	151	27	⊆	⊆	NUM
ejpam-3807	151	28	u(x	u(x	NOUN
ejpam-3807	151	29	)	)	PUNCT
ejpam-3807	151	30	.	.	PUNCT
ejpam-3807	152	1	by	by	ADP
ejpam-3807	152	2	(	(	PUNCT
ejpam-3807	152	3	b1	b1	NOUN
ejpam-3807	152	4	)	)	PUNCT
ejpam-3807	152	5	and	and	CCONJ
ejpam-3807	152	6	the	the	DET
ejpam-3807	152	7	hypothesis	hypothesis	NOUN
ejpam-3807	152	8	,	,	PUNCT
ejpam-3807	152	9	0	0	PUNCT
ejpam-3807	152	10	=	=	SYM
ejpam-3807	152	11	x	x	SYM
ejpam-3807	152	12	∗	∗	NOUN
ejpam-3807	152	13	x	x	SYM
ejpam-3807	152	14	∈	∈	NOUN
ejpam-3807	152	15	v	v	NOUN
ejpam-3807	152	16	(	(	PUNCT
ejpam-3807	152	17	x	x	NOUN
ejpam-3807	152	18	)	)	PUNCT
ejpam-3807	152	19	∗	∗	NOUN
ejpam-3807	152	20	i0	i0	PROPN
ejpam-3807	152	21	⊆	⊆	NUM
ejpam-3807	152	22	v	v	PROPN
ejpam-3807	152	23	(	(	PUNCT
ejpam-3807	152	24	x	x	NOUN
ejpam-3807	152	25	)	)	PUNCT
ejpam-3807	152	26	∗	∗	NOUN
ejpam-3807	152	27	v	v	NOUN
ejpam-3807	152	28	(	(	PUNCT
ejpam-3807	152	29	0	0	NUM
ejpam-3807	152	30	)	)	PUNCT
ejpam-3807	152	31	⊆	⊆	NUM
ejpam-3807	152	32	u(x	u(x	NOUN
ejpam-3807	152	33	)	)	PUNCT
ejpam-3807	152	34	.	.	PUNCT
ejpam-3807	153	1	this	this	PRON
ejpam-3807	153	2	implies	imply	VERB
ejpam-3807	153	3	that	that	SCONJ
ejpam-3807	153	4	u(x	u(x	NOUN
ejpam-3807	153	5	)	)	PUNCT
ejpam-3807	153	6	is	be	AUX
ejpam-3807	153	7	an	an	DET
ejpam-3807	153	8	open	open	ADJ
ejpam-3807	153	9	set	set	NOUN
ejpam-3807	153	10	containing	contain	VERB
ejpam-3807	153	11	0	0	NUM
ejpam-3807	153	12	.	.	PUNCT
ejpam-3807	154	1	therefore	therefore	ADV
ejpam-3807	154	2	,	,	PUNCT
ejpam-3807	154	3	i0	i0	PROPN
ejpam-3807	154	4	⊆	⊆	NUM
ejpam-3807	154	5	u(x	u(x	NOUN
ejpam-3807	154	6	)	)	PUNCT
ejpam-3807	154	7	.	.	PUNCT
ejpam-3807	155	1	the	the	DET
ejpam-3807	155	2	next	next	ADJ
ejpam-3807	155	3	theorem	theorem	NOUN
ejpam-3807	155	4	gives	give	VERB
ejpam-3807	155	5	another	another	DET
ejpam-3807	155	6	characterization	characterization	NOUN
ejpam-3807	155	7	of	of	ADP
ejpam-3807	155	8	a	a	DET
ejpam-3807	155	9	b	b	NOUN
ejpam-3807	155	10	-	-	PUNCT
ejpam-3807	155	11	ideal	ideal	NOUN
ejpam-3807	155	12	.	.	PUNCT
ejpam-3807	156	1	theorem	theorem	NOUN
ejpam-3807	156	2	6	6	NUM
ejpam-3807	156	3	.	.	PUNCT
ejpam-3807	157	1	let	let	VERB
ejpam-3807	157	2	x	x	PRON
ejpam-3807	157	3	be	be	AUX
ejpam-3807	157	4	a	a	DET
ejpam-3807	157	5	topological	topological	ADJ
ejpam-3807	157	6	b	b	NOUN
ejpam-3807	157	7	-	-	PUNCT
ejpam-3807	157	8	algebra	algebra	NOUN
ejpam-3807	157	9	and	and	CCONJ
ejpam-3807	157	10	i	i	PRON
ejpam-3807	157	11	an	an	DET
ejpam-3807	157	12	open	open	ADJ
ejpam-3807	157	13	subset	subset	NOUN
ejpam-3807	157	14	of	of	ADP
ejpam-3807	157	15	x	x	PUNCT
ejpam-3807	157	16	containing	contain	VERB
ejpam-3807	157	17	0	0	NUM
ejpam-3807	157	18	such	such	ADJ
ejpam-3807	157	19	that	that	SCONJ
ejpam-3807	157	20	if	if	SCONJ
ejpam-3807	157	21	0	0	NUM
ejpam-3807	157	22	∈	∈	PROPN
ejpam-3807	157	23	u	u	NOUN
ejpam-3807	157	24	,	,	PUNCT
ejpam-3807	157	25	then	then	ADV
ejpam-3807	157	26	i	i	PRON
ejpam-3807	157	27	⊆	⊆	NUM
ejpam-3807	157	28	u	u	NOUN
ejpam-3807	157	29	for	for	ADP
ejpam-3807	157	30	all	all	DET
ejpam-3807	157	31	u	u	PROPN
ejpam-3807	157	32	∈	∈	PROPN
ejpam-3807	157	33	τ	τ	X
ejpam-3807	157	34	.	.	PUNCT
ejpam-3807	158	1	then	then	ADV
ejpam-3807	158	2	i	i	PRON
ejpam-3807	158	3	is	be	AUX
ejpam-3807	158	4	a	a	DET
ejpam-3807	158	5	b	b	NOUN
ejpam-3807	158	6	-	-	PUNCT
ejpam-3807	158	7	ideal	ideal	NOUN
ejpam-3807	158	8	of	of	ADP
ejpam-3807	158	9	x.	x.	NOUN
ejpam-3807	158	10	proof	proof	NOUN
ejpam-3807	158	11	.	.	PUNCT
ejpam-3807	159	1	suppose	suppose	VERB
ejpam-3807	159	2	x∗y	x∗y	X
ejpam-3807	159	3	,	,	PUNCT
ejpam-3807	159	4	y	y	PROPN
ejpam-3807	159	5	∈	∈	PROPN
ejpam-3807	159	6	i	i	PRON
ejpam-3807	159	7	for	for	ADP
ejpam-3807	159	8	any	any	DET
ejpam-3807	159	9	x	x	NOUN
ejpam-3807	159	10	,	,	PUNCT
ejpam-3807	159	11	y	y	PROPN
ejpam-3807	159	12	∈	∈	PROPN
ejpam-3807	159	13	x.	x.	NOUN
ejpam-3807	159	14	by	by	ADP
ejpam-3807	159	15	theorem	theorem	NOUN
ejpam-3807	159	16	2	2	NUM
ejpam-3807	159	17	,	,	PUNCT
ejpam-3807	159	18	there	there	PRON
ejpam-3807	159	19	exist	exist	VERB
ejpam-3807	159	20	u(x	u(x	NOUN
ejpam-3807	159	21	)	)	PUNCT
ejpam-3807	159	22	and	and	CCONJ
ejpam-3807	159	23	u(y	u(y	NOUN
ejpam-3807	159	24	)	)	PUNCT
ejpam-3807	159	25	such	such	ADJ
ejpam-3807	159	26	that	that	SCONJ
ejpam-3807	159	27	u(x	u(x	NOUN
ejpam-3807	159	28	)	)	PUNCT
ejpam-3807	159	29	∗u(y	∗u(y	PROPN
ejpam-3807	159	30	)	)	PUNCT
ejpam-3807	159	31	⊂	⊂	PROPN
ejpam-3807	159	32	i.	i.	PROPN
ejpam-3807	159	33	by	by	ADP
ejpam-3807	159	34	(	(	PUNCT
ejpam-3807	159	35	b2	b2	NOUN
ejpam-3807	159	36	)	)	PUNCT
ejpam-3807	159	37	,	,	PUNCT
ejpam-3807	159	38	the	the	DET
ejpam-3807	159	39	hypothesis	hypothesis	NOUN
ejpam-3807	159	40	,	,	PUNCT
ejpam-3807	159	41	and	and	CCONJ
ejpam-3807	159	42	lemma	lemma	PROPN
ejpam-3807	159	43	3	3	NUM
ejpam-3807	159	44	,	,	PUNCT
ejpam-3807	160	1	x	x	PUNCT
ejpam-3807	160	2	=	=	PUNCT
ejpam-3807	160	3	x	x	SYM
ejpam-3807	160	4	∗	∗	NOUN
ejpam-3807	160	5	0	0	NUM
ejpam-3807	160	6	∈	∈	PROPN
ejpam-3807	160	7	u(x	u(x	NOUN
ejpam-3807	160	8	)	)	PUNCT
ejpam-3807	160	9	∗	∗	NOUN
ejpam-3807	160	10	i	i	PRON
ejpam-3807	160	11	⊂	⊂	X
ejpam-3807	160	12	u(x	u(x	PROPN
ejpam-3807	160	13	)	)	PUNCT
ejpam-3807	160	14	∗	∗	NOUN
ejpam-3807	160	15	u(y	u(y	PROPN
ejpam-3807	160	16	)	)	PUNCT
ejpam-3807	160	17	⊂	⊂	PROPN
ejpam-3807	160	18	i.	i.	PROPN
ejpam-3807	161	1	hence	hence	ADV
ejpam-3807	161	2	,	,	PUNCT
ejpam-3807	161	3	i	i	PRON
ejpam-3807	161	4	is	be	AUX
ejpam-3807	161	5	a	a	DET
ejpam-3807	161	6	b	b	NOUN
ejpam-3807	161	7	-	-	PUNCT
ejpam-3807	161	8	ideal	ideal	NOUN
ejpam-3807	161	9	of	of	ADP
ejpam-3807	161	10	x.	x.	NOUN
ejpam-3807	161	11	note	note	VERB
ejpam-3807	161	12	that	that	SCONJ
ejpam-3807	161	13	the	the	DET
ejpam-3807	161	14	converse	converse	NOUN
ejpam-3807	161	15	of	of	ADP
ejpam-3807	161	16	theorem	theorem	NOUN
ejpam-3807	161	17	6	6	NUM
ejpam-3807	161	18	is	be	AUX
ejpam-3807	161	19	not	not	PART
ejpam-3807	161	20	always	always	ADV
ejpam-3807	161	21	true	true	ADJ
ejpam-3807	161	22	as	as	SCONJ
ejpam-3807	161	23	shown	show	VERB
ejpam-3807	161	24	in	in	ADP
ejpam-3807	161	25	the	the	DET
ejpam-3807	161	26	next	next	ADJ
ejpam-3807	161	27	example	example	NOUN
ejpam-3807	161	28	.	.	PUNCT
ejpam-3807	162	1	example	example	NOUN
ejpam-3807	163	1	5	5	NUM
ejpam-3807	163	2	.	.	PUNCT
ejpam-3807	164	1	let	let	VERB
ejpam-3807	164	2	x	x	PRON
ejpam-3807	164	3	be	be	AUX
ejpam-3807	164	4	the	the	DET
ejpam-3807	164	5	topological	topological	ADJ
ejpam-3807	164	6	b	b	NOUN
ejpam-3807	164	7	-	-	PUNCT
ejpam-3807	164	8	algebra	algebra	NOUN
ejpam-3807	164	9	in	in	ADP
ejpam-3807	164	10	example	example	NOUN
ejpam-3807	165	1	2	2	NUM
ejpam-3807	165	2	.	.	PUNCT
ejpam-3807	165	3	then	then	ADV
ejpam-3807	165	4	the	the	DET
ejpam-3807	165	5	trivial	trivial	ADJ
ejpam-3807	165	6	b	b	NOUN
ejpam-3807	165	7	-	-	PUNCT
ejpam-3807	165	8	ideal	ideal	NOUN
ejpam-3807	165	9	x	x	NOUN
ejpam-3807	165	10	is	be	AUX
ejpam-3807	165	11	not	not	PART
ejpam-3807	165	12	the	the	DET
ejpam-3807	165	13	smallest	small	ADJ
ejpam-3807	165	14	open	open	ADJ
ejpam-3807	165	15	set	set	NOUN
ejpam-3807	165	16	containing	contain	VERB
ejpam-3807	165	17	0	0	NUM
ejpam-3807	165	18	.	.	PUNCT
ejpam-3807	166	1	however	however	ADV
ejpam-3807	166	2	,	,	PUNCT
ejpam-3807	166	3	if	if	SCONJ
ejpam-3807	166	4	the	the	DET
ejpam-3807	166	5	b	b	NOUN
ejpam-3807	166	6	-	-	PUNCT
ejpam-3807	166	7	ideal	ideal	NOUN
ejpam-3807	166	8	i	i	PRON
ejpam-3807	166	9	is	be	AUX
ejpam-3807	166	10	closed	closed	ADJ
ejpam-3807	166	11	,	,	PUNCT
ejpam-3807	166	12	i	i	PRON
ejpam-3807	166	13	is	be	AUX
ejpam-3807	166	14	also	also	ADV
ejpam-3807	166	15	open	open	ADJ
ejpam-3807	166	16	but	but	CCONJ
ejpam-3807	166	17	may	may	AUX
ejpam-3807	166	18	not	not	PART
ejpam-3807	166	19	be	be	AUX
ejpam-3807	166	20	the	the	DET
ejpam-3807	166	21	smallest	small	ADJ
ejpam-3807	166	22	open	open	ADJ
ejpam-3807	166	23	set	set	NOUN
ejpam-3807	166	24	containing	contain	VERB
ejpam-3807	166	25	0	0	NUM
ejpam-3807	166	26	.	.	PUNCT
ejpam-3807	167	1	this	this	PRON
ejpam-3807	167	2	is	be	AUX
ejpam-3807	167	3	stated	state	VERB
ejpam-3807	167	4	in	in	ADP
ejpam-3807	167	5	the	the	DET
ejpam-3807	167	6	next	next	ADJ
ejpam-3807	167	7	theorem	theorem	NOUN
ejpam-3807	167	8	which	which	PRON
ejpam-3807	167	9	is	be	AUX
ejpam-3807	167	10	the	the	DET
ejpam-3807	167	11	converse	converse	NOUN
ejpam-3807	167	12	of	of	ADP
ejpam-3807	167	13	theorem	theorem	PROPN
ejpam-3807	167	14	3	3	NUM
ejpam-3807	167	15	k.	k.	PROPN
ejpam-3807	167	16	belleza	belleza	PROPN
ejpam-3807	167	17	,	,	PUNCT
ejpam-3807	167	18	j.	j.	PROPN
ejpam-3807	167	19	vilela	vilela	PROPN
ejpam-3807	167	20	/	/	SYM
ejpam-3807	167	21	eur	eur	PROPN
ejpam-3807	167	22	.	.	PUNCT
ejpam-3807	168	1	j.	j.	PROPN
ejpam-3807	168	2	pure	pure	PROPN
ejpam-3807	168	3	appl	appl	PROPN
ejpam-3807	168	4	.	.	PROPN
ejpam-3807	168	5	math	math	PROPN
ejpam-3807	168	6	,	,	PUNCT
ejpam-3807	168	7	13	13	NUM
ejpam-3807	168	8	(	(	PUNCT
ejpam-3807	168	9	4	4	NUM
ejpam-3807	168	10	)	)	PUNCT
ejpam-3807	168	11	(	(	PUNCT
ejpam-3807	168	12	2020	2020	NUM
ejpam-3807	168	13	)	)	PUNCT
ejpam-3807	168	14	,	,	PUNCT
ejpam-3807	168	15	830	830	NUM
ejpam-3807	168	16	-	-	SYM
ejpam-3807	168	17	839	839	NUM
ejpam-3807	168	18	835	835	NUM
ejpam-3807	168	19	theorem	theorem	NOUN
ejpam-3807	168	20	7	7	NUM
ejpam-3807	168	21	.	.	PUNCT
ejpam-3807	169	1	let	let	VERB
ejpam-3807	169	2	x	x	PRON
ejpam-3807	169	3	be	be	AUX
ejpam-3807	169	4	a	a	DET
ejpam-3807	169	5	topological	topological	ADJ
ejpam-3807	169	6	b	b	NOUN
ejpam-3807	169	7	-	-	PUNCT
ejpam-3807	169	8	algebra	algebra	NOUN
ejpam-3807	169	9	and	and	CCONJ
ejpam-3807	169	10	i	i	PRON
ejpam-3807	169	11	a	a	DET
ejpam-3807	169	12	closed	closed	ADJ
ejpam-3807	169	13	b	b	NOUN
ejpam-3807	169	14	-	-	PUNCT
ejpam-3807	169	15	ideal	ideal	NOUN
ejpam-3807	169	16	of	of	ADP
ejpam-3807	169	17	x.	x.	NOUN
ejpam-3807	169	18	then	then	ADV
ejpam-3807	169	19	i	i	PRON
ejpam-3807	169	20	is	be	AUX
ejpam-3807	169	21	also	also	ADV
ejpam-3807	169	22	open	open	ADJ
ejpam-3807	169	23	.	.	PUNCT
ejpam-3807	170	1	proof	proof	NOUN
ejpam-3807	170	2	.	.	PUNCT
ejpam-3807	171	1	suppose	suppose	VERB
ejpam-3807	171	2	i	i	PRON
ejpam-3807	171	3	is	be	AUX
ejpam-3807	171	4	a	a	DET
ejpam-3807	171	5	closed	closed	ADJ
ejpam-3807	171	6	b	b	NOUN
ejpam-3807	171	7	-	-	PUNCT
ejpam-3807	171	8	ideal	ideal	NOUN
ejpam-3807	171	9	of	of	ADP
ejpam-3807	171	10	x.	x.	NOUN
ejpam-3807	171	11	assume	assume	VERB
ejpam-3807	171	12	on	on	ADP
ejpam-3807	171	13	the	the	DET
ejpam-3807	171	14	contrary	contrary	NOUN
ejpam-3807	171	15	that	that	SCONJ
ejpam-3807	171	16	i	i	PRON
ejpam-3807	171	17	is	be	AUX
ejpam-3807	171	18	not	not	PART
ejpam-3807	171	19	an	an	DET
ejpam-3807	171	20	open	open	ADJ
ejpam-3807	171	21	set	set	NOUN
ejpam-3807	171	22	in	in	ADP
ejpam-3807	171	23	x.	x.	NOUN
ejpam-3807	171	24	by	by	ADP
ejpam-3807	171	25	theorem	theorem	NOUN
ejpam-3807	171	26	4	4	NUM
ejpam-3807	171	27	,	,	PUNCT
ejpam-3807	171	28	0	0	NUM
ejpam-3807	171	29	is	be	AUX
ejpam-3807	171	30	not	not	PART
ejpam-3807	171	31	an	an	DET
ejpam-3807	171	32	interior	interior	ADJ
ejpam-3807	171	33	point	point	NOUN
ejpam-3807	171	34	of	of	ADP
ejpam-3807	171	35	i.	i.	NOUN
ejpam-3807	171	36	this	this	PRON
ejpam-3807	171	37	implies	imply	VERB
ejpam-3807	171	38	that	that	SCONJ
ejpam-3807	171	39	for	for	ADP
ejpam-3807	171	40	all	all	DET
ejpam-3807	171	41	u(0	u(0	NOUN
ejpam-3807	171	42	)	)	PUNCT
ejpam-3807	171	43	∈	∈	PROPN
ejpam-3807	171	44	τ	τ	PROPN
ejpam-3807	171	45	,	,	PUNCT
ejpam-3807	171	46	u(0	u(0	PROPN
ejpam-3807	171	47	)	)	PUNCT
ejpam-3807	171	48	*	*	PUNCT
ejpam-3807	172	1	i.	i.	PROPN
ejpam-3807	172	2	let	let	VERB
ejpam-3807	172	3	i0	i0	PROPN
ejpam-3807	172	4	be	be	AUX
ejpam-3807	172	5	open	open	ADJ
ejpam-3807	172	6	with	with	ADP
ejpam-3807	172	7	property	property	NOUN
ejpam-3807	172	8	defined	define	VERB
ejpam-3807	172	9	in	in	ADP
ejpam-3807	172	10	lemma	lemma	PROPN
ejpam-3807	172	11	3	3	NUM
ejpam-3807	172	12	.	.	PUNCT
ejpam-3807	173	1	then	then	ADV
ejpam-3807	173	2	i0	i0	PROPN
ejpam-3807	173	3	*	*	PUNCT
ejpam-3807	173	4	i.	i.	PROPN
ejpam-3807	173	5	hence	hence	ADV
ejpam-3807	173	6	,	,	PUNCT
ejpam-3807	173	7	(	(	PUNCT
ejpam-3807	173	8	x\i)∩i0	x\i)∩i0	NOUN
ejpam-3807	173	9	6=	6=	ADP
ejpam-3807	173	10	∅	∅	NOUN
ejpam-3807	173	11	and	and	CCONJ
ejpam-3807	173	12	so	so	ADV
ejpam-3807	173	13	there	there	PRON
ejpam-3807	173	14	exists	exist	VERB
ejpam-3807	173	15	z	z	PROPN
ejpam-3807	173	16	∈	∈	PROPN
ejpam-3807	173	17	(	(	PUNCT
ejpam-3807	173	18	x\i)∩i0	x\i)∩i0	NOUN
ejpam-3807	173	19	.	.	PUNCT
ejpam-3807	174	1	note	note	VERB
ejpam-3807	174	2	that	that	SCONJ
ejpam-3807	174	3	(	(	PUNCT
ejpam-3807	174	4	x\i)∩i0	x\i)∩i0	NOUN
ejpam-3807	174	5	is	be	AUX
ejpam-3807	174	6	an	an	DET
ejpam-3807	174	7	open	open	ADJ
ejpam-3807	174	8	set	set	NOUN
ejpam-3807	174	9	containing	contain	VERB
ejpam-3807	174	10	z.	z.	PROPN
ejpam-3807	174	11	by	by	ADP
ejpam-3807	174	12	lemma	lemma	PROPN
ejpam-3807	174	13	3	3	NUM
ejpam-3807	174	14	,	,	PUNCT
ejpam-3807	174	15	i0	i0	PROPN
ejpam-3807	174	16	⊂	⊂	PROPN
ejpam-3807	174	17	(	(	PUNCT
ejpam-3807	174	18	x\i	x\i	PROPN
ejpam-3807	174	19	)	)	PUNCT
ejpam-3807	174	20	.	.	PUNCT
ejpam-3807	175	1	this	this	PRON
ejpam-3807	175	2	implies	imply	VERB
ejpam-3807	175	3	that	that	SCONJ
ejpam-3807	175	4	0	0	NUM
ejpam-3807	175	5	∈	∈	PROPN
ejpam-3807	175	6	x\i	x\i	ADP
ejpam-3807	175	7	which	which	PRON
ejpam-3807	175	8	is	be	AUX
ejpam-3807	175	9	a	a	DET
ejpam-3807	175	10	contradiction	contradiction	NOUN
ejpam-3807	175	11	.	.	PUNCT
ejpam-3807	176	1	therefore	therefore	ADV
ejpam-3807	176	2	,	,	PUNCT
ejpam-3807	176	3	i	i	PRON
ejpam-3807	176	4	is	be	AUX
ejpam-3807	176	5	open	open	ADJ
ejpam-3807	176	6	.	.	PUNCT
ejpam-3807	177	1	the	the	DET
ejpam-3807	177	2	next	next	ADJ
ejpam-3807	177	3	corollary	corollary	NOUN
ejpam-3807	177	4	follows	follow	VERB
ejpam-3807	177	5	directly	directly	ADV
ejpam-3807	177	6	from	from	ADP
ejpam-3807	177	7	theorems	theorem	NOUN
ejpam-3807	177	8	3	3	NUM
ejpam-3807	177	9	and	and	CCONJ
ejpam-3807	177	10	7	7	NUM
ejpam-3807	177	11	.	.	PUNCT
ejpam-3807	177	12	corollary	corollary	ADJ
ejpam-3807	177	13	2	2	NUM
ejpam-3807	177	14	.	.	PUNCT
ejpam-3807	177	15	suppose	suppose	VERB
ejpam-3807	177	16	x	x	PRON
ejpam-3807	177	17	is	be	AUX
ejpam-3807	177	18	a	a	DET
ejpam-3807	177	19	topological	topological	ADJ
ejpam-3807	177	20	b	b	NOUN
ejpam-3807	177	21	-	-	PUNCT
ejpam-3807	177	22	algebra	algebra	NOUN
ejpam-3807	177	23	and	and	CCONJ
ejpam-3807	177	24	i	i	PRON
ejpam-3807	177	25	a	a	DET
ejpam-3807	177	26	b	b	NOUN
ejpam-3807	177	27	-	-	PUNCT
ejpam-3807	177	28	ideal	ideal	NOUN
ejpam-3807	177	29	of	of	ADP
ejpam-3807	177	30	x.	x.	NOUN
ejpam-3807	178	1	then	then	ADV
ejpam-3807	178	2	i	i	PRON
ejpam-3807	178	3	is	be	AUX
ejpam-3807	178	4	an	an	DET
ejpam-3807	178	5	open	open	ADJ
ejpam-3807	178	6	subset	subset	NOUN
ejpam-3807	178	7	of	of	ADP
ejpam-3807	178	8	x	x	PRON
ejpam-3807	178	9	if	if	SCONJ
ejpam-3807	178	10	and	and	CCONJ
ejpam-3807	178	11	only	only	ADV
ejpam-3807	178	12	if	if	SCONJ
ejpam-3807	178	13	i	i	PRON
ejpam-3807	178	14	is	be	AUX
ejpam-3807	178	15	a	a	DET
ejpam-3807	178	16	closed	closed	ADJ
ejpam-3807	178	17	subset	subset	NOUN
ejpam-3807	178	18	of	of	ADP
ejpam-3807	178	19	x.	x.	PROPN
ejpam-3807	178	20	theorem	theorem	VERB
ejpam-3807	178	21	8	8	NUM
ejpam-3807	178	22	.	.	PUNCT
ejpam-3807	179	1	let	let	VERB
ejpam-3807	179	2	i	i	PRON
ejpam-3807	179	3	be	be	AUX
ejpam-3807	179	4	a	a	DET
ejpam-3807	179	5	family	family	NOUN
ejpam-3807	179	6	of	of	ADP
ejpam-3807	179	7	normal	normal	ADJ
ejpam-3807	179	8	b	b	NOUN
ejpam-3807	179	9	-	-	PUNCT
ejpam-3807	179	10	ideals	ideal	NOUN
ejpam-3807	179	11	in	in	ADP
ejpam-3807	179	12	a	a	DET
ejpam-3807	179	13	b	b	NOUN
ejpam-3807	179	14	-	-	PUNCT
ejpam-3807	179	15	algebra	algebra	NOUN
ejpam-3807	179	16	x.	x.	NOUN
ejpam-3807	179	17	then	then	ADV
ejpam-3807	179	18	there	there	PRON
ejpam-3807	179	19	is	be	VERB
ejpam-3807	179	20	a	a	DET
ejpam-3807	179	21	topology	topology	NOUN
ejpam-3807	179	22	τ	τ	X
ejpam-3807	179	23	=	=	PUNCT
ejpam-3807	179	24	{	{	PUNCT
ejpam-3807	179	25	u	u	NOUN
ejpam-3807	179	26	⊆	⊆	NUM
ejpam-3807	179	27	x|∀x	x|∀x	PROPN
ejpam-3807	179	28	∈	∈	PROPN
ejpam-3807	179	29	u,∃i	u,∃i	NOUN
ejpam-3807	179	30	∈	∈	PROPN
ejpam-3807	179	31	i	i	PRON
ejpam-3807	180	1	such	such	ADJ
ejpam-3807	180	2	that	that	SCONJ
ejpam-3807	180	3	ix	ix	ADP
ejpam-3807	180	4	⊆	⊆	NUM
ejpam-3807	180	5	u	u	NOUN
ejpam-3807	180	6	}	}	PUNCT
ejpam-3807	180	7	such	such	ADJ
ejpam-3807	180	8	that	that	SCONJ
ejpam-3807	180	9	(	(	PUNCT
ejpam-3807	180	10	x	x	X
ejpam-3807	180	11	,	,	PUNCT
ejpam-3807	180	12	∗	∗	NOUN
ejpam-3807	180	13	,	,	PUNCT
ejpam-3807	180	14	τ	τ	X
ejpam-3807	180	15	)	)	PUNCT
ejpam-3807	180	16	is	be	AUX
ejpam-3807	180	17	a	a	DET
ejpam-3807	180	18	topological	topological	ADJ
ejpam-3807	180	19	b	b	NOUN
ejpam-3807	180	20	-	-	PUNCT
ejpam-3807	180	21	algebra	algebra	NOUN
ejpam-3807	180	22	.	.	PUNCT
ejpam-3807	181	1	proof	proof	NOUN
ejpam-3807	181	2	.	.	PUNCT
ejpam-3807	182	1	note	note	VERB
ejpam-3807	182	2	that	that	SCONJ
ejpam-3807	182	3	for	for	ADP
ejpam-3807	182	4	all	all	DET
ejpam-3807	182	5	x	x	SYM
ejpam-3807	182	6	∈	∈	NOUN
ejpam-3807	182	7	x	x	NOUN
ejpam-3807	182	8	,	,	PUNCT
ejpam-3807	182	9	there	there	PRON
ejpam-3807	182	10	exists	exist	VERB
ejpam-3807	182	11	i	i	PRON
ejpam-3807	182	12	∈	∈	VERB
ejpam-3807	183	1	i	i	PRON
ejpam-3807	183	2	such	such	ADJ
ejpam-3807	183	3	that	that	SCONJ
ejpam-3807	183	4	ix	ix	ADP
ejpam-3807	183	5	⊆	⊆	NUM
ejpam-3807	183	6	x.	x.	NOUN
ejpam-3807	183	7	this	this	PRON
ejpam-3807	183	8	implies	imply	VERB
ejpam-3807	183	9	that	that	SCONJ
ejpam-3807	183	10	x	x	SYM
ejpam-3807	183	11	∈	∈	PROPN
ejpam-3807	183	12	τ	τ	X
ejpam-3807	183	13	.	.	PUNCT
ejpam-3807	183	14	suppose	suppose	VERB
ejpam-3807	183	15	∅	∅	NOUN
ejpam-3807	183	16	/∈	/∈	PUNCT
ejpam-3807	184	1	τ	τ	PROPN
ejpam-3807	184	2	.	.	PUNCT
ejpam-3807	185	1	then	then	ADV
ejpam-3807	185	2	there	there	PRON
ejpam-3807	185	3	exists	exist	VERB
ejpam-3807	185	4	x	x	X
ejpam-3807	185	5	∈	∈	NOUN
ejpam-3807	185	6	∅	∅	NOUN
ejpam-3807	185	7	such	such	ADJ
ejpam-3807	185	8	that	that	PRON
ejpam-3807	185	9	for	for	ADP
ejpam-3807	185	10	all	all	PRON
ejpam-3807	185	11	i	i	PRON
ejpam-3807	185	12	∈	∈	PROPN
ejpam-3807	186	1	i	i	PRON
ejpam-3807	186	2	,	,	PUNCT
ejpam-3807	186	3	ix	ix	PROPN
ejpam-3807	186	4	*	*	PUNCT
ejpam-3807	186	5	u	u	NOUN
ejpam-3807	186	6	which	which	PRON
ejpam-3807	186	7	is	be	AUX
ejpam-3807	186	8	a	a	DET
ejpam-3807	186	9	contradiction	contradiction	NOUN
ejpam-3807	186	10	.	.	PUNCT
ejpam-3807	187	1	hence	hence	ADV
ejpam-3807	187	2	,	,	PUNCT
ejpam-3807	187	3	∅	∅	NOUN
ejpam-3807	187	4	∈	∈	PROPN
ejpam-3807	187	5	τ	τ	X
ejpam-3807	187	6	.	.	PUNCT
ejpam-3807	188	1	let	let	VERB
ejpam-3807	188	2	y	y	PRON
ejpam-3807	188	3	∈	∈	PROPN
ejpam-3807	188	4	u1	u1	NOUN
ejpam-3807	188	5	∩	∩	NOUN
ejpam-3807	188	6	u2	u2	NOUN
ejpam-3807	188	7	,	,	PUNCT
ejpam-3807	188	8	where	where	SCONJ
ejpam-3807	188	9	u1	u1	NOUN
ejpam-3807	188	10	,	,	PUNCT
ejpam-3807	188	11	u2	u2	PROPN
ejpam-3807	188	12	∈	∈	PROPN
ejpam-3807	188	13	τ	τ	X
ejpam-3807	188	14	.	.	PUNCT
ejpam-3807	189	1	then	then	ADV
ejpam-3807	189	2	y	y	PROPN
ejpam-3807	189	3	∈	∈	PROPN
ejpam-3807	189	4	u1	u1	NOUN
ejpam-3807	189	5	and	and	CCONJ
ejpam-3807	189	6	y	y	PROPN
ejpam-3807	189	7	∈	∈	PROPN
ejpam-3807	189	8	u2	u2	PROPN
ejpam-3807	189	9	which	which	PRON
ejpam-3807	189	10	imply	imply	VERB
ejpam-3807	189	11	that	that	SCONJ
ejpam-3807	189	12	there	there	PRON
ejpam-3807	189	13	exist	exist	VERB
ejpam-3807	189	14	i1	i1	PROPN
ejpam-3807	189	15	,	,	PUNCT
ejpam-3807	189	16	i2	i2	PROPN
ejpam-3807	189	17	∈	∈	PROPN
ejpam-3807	190	1	i	i	PRON
ejpam-3807	190	2	such	such	ADJ
ejpam-3807	190	3	that	that	SCONJ
ejpam-3807	190	4	i1y	i1y	ADJ
ejpam-3807	190	5	⊆	⊆	NUM
ejpam-3807	190	6	u1	u1	NOUN
ejpam-3807	190	7	and	and	CCONJ
ejpam-3807	190	8	i2y	i2y	PROPN
ejpam-3807	190	9	⊆	⊆	NUM
ejpam-3807	190	10	u2	u2	NOUN
ejpam-3807	190	11	.	.	PUNCT
ejpam-3807	191	1	let	let	VERB
ejpam-3807	191	2	i	i	PRON
ejpam-3807	191	3	=	=	PROPN
ejpam-3807	191	4	i1	i1	PROPN
ejpam-3807	191	5	∩	∩	PROPN
ejpam-3807	191	6	i2	i2	PROPN
ejpam-3807	191	7	∈	∈	PROPN
ejpam-3807	191	8	i.	i.	NOUN
ejpam-3807	191	9	claim	claim	VERB
ejpam-3807	191	10	1	1	NUM
ejpam-3807	191	11	:	:	PUNCT
ejpam-3807	191	12	iy	iy	PROPN
ejpam-3807	191	13	⊆	⊆	NUM
ejpam-3807	191	14	i1y	i1y	PROPN
ejpam-3807	191	15	,	,	PUNCT
ejpam-3807	191	16	iy	iy	PROPN
ejpam-3807	191	17	⊆	⊆	NUM
ejpam-3807	191	18	i2y	i2y	PROPN
ejpam-3807	191	19	.	.	PUNCT
ejpam-3807	191	20	suppose	suppose	VERB
ejpam-3807	192	1	x	x	SYM
ejpam-3807	192	2	∈	∈	PROPN
ejpam-3807	192	3	iy	iy	PROPN
ejpam-3807	192	4	.	.	PUNCT
ejpam-3807	193	1	then	then	ADV
ejpam-3807	193	2	y	y	PROPN
ejpam-3807	193	3	∼=i	∼=i	NOUN
ejpam-3807	193	4	x	x	PUNCT
ejpam-3807	193	5	which	which	PRON
ejpam-3807	193	6	implies	imply	VERB
ejpam-3807	193	7	that	that	SCONJ
ejpam-3807	193	8	y	y	PROPN
ejpam-3807	193	9	∗	∗	NOUN
ejpam-3807	193	10	x	x	PUNCT
ejpam-3807	193	11	∈	∈	PROPN
ejpam-3807	193	12	i	i	PROPN
ejpam-3807	193	13	⊆	⊆	NUM
ejpam-3807	193	14	i1	i1	NOUN
ejpam-3807	193	15	.	.	PUNCT
ejpam-3807	194	1	hence	hence	ADV
ejpam-3807	194	2	,	,	PUNCT
ejpam-3807	194	3	y	y	PROPN
ejpam-3807	194	4	∼=i1	∼=i1	PROPN
ejpam-3807	194	5	x	x	PUNCT
ejpam-3807	194	6	implying	imply	VERB
ejpam-3807	194	7	that	that	SCONJ
ejpam-3807	194	8	x	x	PUNCT
ejpam-3807	194	9	∈	∈	PROPN
ejpam-3807	194	10	i1y	i1y	PROPN
ejpam-3807	194	11	so	so	SCONJ
ejpam-3807	194	12	that	that	SCONJ
ejpam-3807	194	13	iy	iy	PROPN
ejpam-3807	194	14	⊆	⊆	NUM
ejpam-3807	194	15	i1y	i1y	PROPN
ejpam-3807	194	16	.	.	PUNCT
ejpam-3807	195	1	similarly	similarly	ADV
ejpam-3807	195	2	,	,	PUNCT
ejpam-3807	195	3	iy	iy	PROPN
ejpam-3807	195	4	⊆	⊆	NUM
ejpam-3807	195	5	i2y	i2y	PROPN
ejpam-3807	195	6	.	.	PUNCT
ejpam-3807	196	1	this	this	PRON
ejpam-3807	196	2	proves	prove	VERB
ejpam-3807	196	3	claim	claim	NOUN
ejpam-3807	196	4	1	1	X
ejpam-3807	196	5	.	.	PUNCT
ejpam-3807	196	6	since	since	SCONJ
ejpam-3807	196	7	i1y	i1y	ADJ
ejpam-3807	196	8	⊆	⊆	NUM
ejpam-3807	196	9	u1	u1	NOUN
ejpam-3807	196	10	and	and	CCONJ
ejpam-3807	196	11	i2y	i2y	PROPN
ejpam-3807	196	12	⊆	⊆	NUM
ejpam-3807	196	13	u2	u2	NOUN
ejpam-3807	196	14	,	,	PUNCT
ejpam-3807	196	15	it	it	PRON
ejpam-3807	196	16	follows	follow	VERB
ejpam-3807	196	17	that	that	SCONJ
ejpam-3807	196	18	iy	iy	PROPN
ejpam-3807	196	19	⊆	⊆	NUM
ejpam-3807	196	20	(	(	PUNCT
ejpam-3807	196	21	u1	u1	NOUN
ejpam-3807	196	22	∩	∩	NOUN
ejpam-3807	196	23	u2	u2	NOUN
ejpam-3807	196	24	)	)	PUNCT
ejpam-3807	196	25	.	.	PUNCT
ejpam-3807	197	1	hence	hence	ADV
ejpam-3807	197	2	,	,	PUNCT
ejpam-3807	197	3	u1	u1	NOUN
ejpam-3807	197	4	∩	∩	NOUN
ejpam-3807	197	5	u2	u2	PROPN
ejpam-3807	197	6	∈	∈	PROPN
ejpam-3807	197	7	τ	τ	X
ejpam-3807	197	8	.	.	PUNCT
ejpam-3807	198	1	let	let	VERB
ejpam-3807	198	2	y	y	PROPN
ejpam-3807	198	3	∈	∈	PROPN
ejpam-3807	198	4	⋃	⋃	PROPN
ejpam-3807	198	5	α∈a	α∈a	NOUN
ejpam-3807	198	6	uα	uα	INTJ
ejpam-3807	198	7	where	where	SCONJ
ejpam-3807	198	8	uα	uα	PROPN
ejpam-3807	198	9	∈	∈	PROPN
ejpam-3807	198	10	τ	τ	PROPN
ejpam-3807	198	11	for	for	ADP
ejpam-3807	198	12	all	all	DET
ejpam-3807	198	13	α	α	PRON
ejpam-3807	198	14	∈	∈	NOUN
ejpam-3807	198	15	a.	a.	NOUN
ejpam-3807	198	16	then	then	ADV
ejpam-3807	198	17	y	y	PROPN
ejpam-3807	198	18	∈	∈	PROPN
ejpam-3807	198	19	uβ	uβ	NOUN
ejpam-3807	198	20	for	for	ADP
ejpam-3807	198	21	some	some	DET
ejpam-3807	198	22	β	β	NOUN
ejpam-3807	198	23	∈	∈	PROPN
ejpam-3807	198	24	a.	a.	NOUN
ejpam-3807	198	25	this	this	PRON
ejpam-3807	198	26	implies	imply	VERB
ejpam-3807	198	27	that	that	SCONJ
ejpam-3807	198	28	there	there	PRON
ejpam-3807	198	29	exists	exist	VERB
ejpam-3807	198	30	iβ	iβ	ADP
ejpam-3807	198	31	∈	∈	PROPN
ejpam-3807	198	32	i	i	PRON
ejpam-3807	198	33	such	such	ADJ
ejpam-3807	198	34	that	that	DET
ejpam-3807	198	35	iβy	iβy	PROPN
ejpam-3807	199	1	⊆	⊆	NUM
ejpam-3807	199	2	uβ	uβ	NOUN
ejpam-3807	199	3	⊆	⊆	NUM
ejpam-3807	199	4	⋃	⋃	NOUN
ejpam-3807	199	5	α∈a	α∈a	NOUN
ejpam-3807	199	6	uα	uα	PROPN
ejpam-3807	199	7	.	.	PUNCT
ejpam-3807	200	1	hence	hence	ADV
ejpam-3807	200	2	,	,	PUNCT
ejpam-3807	200	3	uα	uα	PROPN
ejpam-3807	200	4	α∈a	α∈a	PROPN
ejpam-3807	200	5	∈	∈	PROPN
ejpam-3807	200	6	τ	τ	PROPN
ejpam-3807	200	7	.	.	PUNCT
ejpam-3807	201	1	this	this	PRON
ejpam-3807	201	2	implies	imply	VERB
ejpam-3807	201	3	that	that	SCONJ
ejpam-3807	201	4	τ	τ	PROPN
ejpam-3807	201	5	is	be	AUX
ejpam-3807	201	6	a	a	DET
ejpam-3807	201	7	b	b	NOUN
ejpam-3807	201	8	-	-	PUNCT
ejpam-3807	201	9	topology	topology	NOUN
ejpam-3807	201	10	.	.	PUNCT
ejpam-3807	202	1	claim	claim	VERB
ejpam-3807	202	2	2	2	NUM
ejpam-3807	202	3	:	:	PUNCT
ejpam-3807	202	4	for	for	ADP
ejpam-3807	202	5	any	any	DET
ejpam-3807	202	6	i	i	PRON
ejpam-3807	202	7	∈	∈	PROPN
ejpam-3807	203	1	i	i	PRON
ejpam-3807	203	2	and	and	CCONJ
ejpam-3807	203	3	x	x	ADP
ejpam-3807	203	4	∈	∈	PROPN
ejpam-3807	203	5	x	x	NOUN
ejpam-3807	203	6	,	,	PUNCT
ejpam-3807	203	7	ix	ix	ADP
ejpam-3807	203	8	∈	∈	PROPN
ejpam-3807	203	9	τ	τ	X
ejpam-3807	203	10	let	let	VERB
ejpam-3807	203	11	y	y	PROPN
ejpam-3807	203	12	∈	∈	PROPN
ejpam-3807	203	13	ix	ix	PROPN
ejpam-3807	203	14	.	.	PUNCT
ejpam-3807	204	1	then	then	ADV
ejpam-3807	204	2	y	y	PROPN
ejpam-3807	204	3	∼=i	∼=i	NOUN
ejpam-3807	204	4	x.	x.	NOUN
ejpam-3807	205	1	we	we	PRON
ejpam-3807	205	2	will	will	AUX
ejpam-3807	205	3	show	show	VERB
ejpam-3807	205	4	that	that	SCONJ
ejpam-3807	205	5	iy	iy	PROPN
ejpam-3807	205	6	⊆	⊆	NUM
ejpam-3807	205	7	ix	ix	PROPN
ejpam-3807	205	8	.	.	PUNCT
ejpam-3807	206	1	let	let	VERB
ejpam-3807	206	2	z	z	PROPN
ejpam-3807	206	3	∈	∈	PROPN
ejpam-3807	206	4	iy	iy	PROPN
ejpam-3807	206	5	.	.	PUNCT
ejpam-3807	207	1	then	then	ADV
ejpam-3807	207	2	z	z	X
ejpam-3807	207	3	∼=i	∼=i	PROPN
ejpam-3807	207	4	y.	y.	NOUN
ejpam-3807	207	5	by	by	ADP
ejpam-3807	207	6	transitivity	transitivity	NOUN
ejpam-3807	207	7	,	,	PUNCT
ejpam-3807	207	8	z	z	NOUN
ejpam-3807	207	9	∼=i	∼=i	NOUN
ejpam-3807	208	1	x.	x.	NOUN
ejpam-3807	208	2	hence	hence	ADV
ejpam-3807	208	3	,	,	PUNCT
ejpam-3807	208	4	z	z	NOUN
ejpam-3807	208	5	∈	∈	PROPN
ejpam-3807	208	6	ix	ix	ADP
ejpam-3807	208	7	so	so	SCONJ
ejpam-3807	208	8	that	that	SCONJ
ejpam-3807	208	9	iy	iy	PROPN
ejpam-3807	208	10	⊆	⊆	NUM
ejpam-3807	208	11	ix	ix	PROPN
ejpam-3807	208	12	.	.	PUNCT
ejpam-3807	209	1	this	this	PRON
ejpam-3807	209	2	proves	prove	VERB
ejpam-3807	209	3	claim	claim	NOUN
ejpam-3807	209	4	2	2	X
ejpam-3807	209	5	.	.	PUNCT
ejpam-3807	209	6	suppose	suppose	VERB
ejpam-3807	209	7	x	x	X
ejpam-3807	209	8	∗	∗	VERB
ejpam-3807	209	9	y	y	PROPN
ejpam-3807	209	10	∈	∈	PROPN
ejpam-3807	209	11	u	u	PROPN
ejpam-3807	209	12	∈	∈	PROPN
ejpam-3807	209	13	τ	τ	X
ejpam-3807	209	14	.	.	PUNCT
ejpam-3807	210	1	then	then	ADV
ejpam-3807	210	2	there	there	PRON
ejpam-3807	210	3	exists	exist	VERB
ejpam-3807	210	4	i	i	PRON
ejpam-3807	210	5	∈	∈	VERB
ejpam-3807	211	1	i	i	PRON
ejpam-3807	211	2	such	such	ADJ
ejpam-3807	211	3	that	that	SCONJ
ejpam-3807	211	4	ix∗y	ix∗y	PROPN
ejpam-3807	211	5	⊆	⊆	NUM
ejpam-3807	211	6	u	u	PROPN
ejpam-3807	211	7	.	.	PUNCT
ejpam-3807	212	1	note	note	VERB
ejpam-3807	212	2	that	that	SCONJ
ejpam-3807	212	3	ix	ix	PROPN
ejpam-3807	212	4	and	and	CCONJ
ejpam-3807	212	5	iy	iy	PROPN
ejpam-3807	212	6	are	be	AUX
ejpam-3807	212	7	open	open	ADJ
ejpam-3807	212	8	sets	set	NOUN
ejpam-3807	212	9	containing	contain	VERB
ejpam-3807	212	10	x	x	PROPN
ejpam-3807	212	11	and	and	CCONJ
ejpam-3807	212	12	y	y	PROPN
ejpam-3807	212	13	,	,	PUNCT
ejpam-3807	212	14	respectively	respectively	ADV
ejpam-3807	212	15	.	.	PUNCT
ejpam-3807	213	1	then	then	ADV
ejpam-3807	213	2	ix	ix	ADP
ejpam-3807	213	3	∗	∗	PROPN
ejpam-3807	213	4	iy	iy	PROPN
ejpam-3807	214	1	=	=	PUNCT
ejpam-3807	214	2	ix∗y	ix∗y	VERB
ejpam-3807	214	3	⊆	⊆	NUM
ejpam-3807	214	4	u	u	NOUN
ejpam-3807	214	5	.	.	PUNCT
ejpam-3807	215	1	this	this	PRON
ejpam-3807	215	2	implies	imply	VERB
ejpam-3807	215	3	that	that	SCONJ
ejpam-3807	215	4	∗	∗	NOUN
ejpam-3807	215	5	is	be	AUX
ejpam-3807	215	6	continuous	continuous	ADJ
ejpam-3807	215	7	.	.	PUNCT
ejpam-3807	216	1	therefore	therefore	ADV
ejpam-3807	216	2	,	,	PUNCT
ejpam-3807	216	3	(	(	PUNCT
ejpam-3807	216	4	x	x	X
ejpam-3807	216	5	,	,	PUNCT
ejpam-3807	216	6	∗	∗	NOUN
ejpam-3807	216	7	,	,	PUNCT
ejpam-3807	216	8	τ	τ	X
ejpam-3807	216	9	)	)	PUNCT
ejpam-3807	216	10	is	be	AUX
ejpam-3807	216	11	a	a	DET
ejpam-3807	216	12	topological	topological	ADJ
ejpam-3807	216	13	b	b	NOUN
ejpam-3807	216	14	-	-	PUNCT
ejpam-3807	216	15	algebra	algebra	NOUN
ejpam-3807	216	16	by	by	ADP
ejpam-3807	216	17	theorem	theorem	NOUN
ejpam-3807	216	18	2	2	NUM
ejpam-3807	216	19	.	.	NOUN
ejpam-3807	216	20	4	4	NUM
ejpam-3807	216	21	.	.	X
ejpam-3807	216	22	uniform	uniform	ADJ
ejpam-3807	216	23	topology	topology	NOUN
ejpam-3807	216	24	on	on	ADP
ejpam-3807	216	25	b	b	NOUN
ejpam-3807	216	26	-	-	PUNCT
ejpam-3807	216	27	algebras	algebras	NOUN
ejpam-3807	216	28	throughout	throughout	ADP
ejpam-3807	216	29	this	this	DET
ejpam-3807	216	30	section	section	NOUN
ejpam-3807	216	31	,	,	PUNCT
ejpam-3807	216	32	all	all	DET
ejpam-3807	216	33	b	b	NOUN
ejpam-3807	216	34	-	-	PUNCT
ejpam-3807	216	35	ideals	ideal	NOUN
ejpam-3807	216	36	of	of	ADP
ejpam-3807	216	37	a	a	DET
ejpam-3807	216	38	b	b	NOUN
ejpam-3807	216	39	-	-	PUNCT
ejpam-3807	216	40	algebra	algebra	NOUN
ejpam-3807	216	41	x	x	VERB
ejpam-3807	216	42	are	be	AUX
ejpam-3807	216	43	normal	normal	ADJ
ejpam-3807	216	44	b	b	NOUN
ejpam-3807	216	45	-	-	PUNCT
ejpam-3807	216	46	ideals	ideal	NOUN
ejpam-3807	216	47	of	of	ADP
ejpam-3807	216	48	x.	x.	NOUN
ejpam-3807	216	49	the	the	DET
ejpam-3807	216	50	following	follow	VERB
ejpam-3807	216	51	definitions	definition	NOUN
ejpam-3807	216	52	are	be	AUX
ejpam-3807	216	53	parallel	parallel	ADJ
ejpam-3807	216	54	to	to	ADP
ejpam-3807	216	55	that	that	PRON
ejpam-3807	216	56	of	of	ADP
ejpam-3807	216	57	[	[	X
ejpam-3807	216	58	5	5	NUM
ejpam-3807	216	59	]	]	PUNCT
ejpam-3807	216	60	,	,	PUNCT
ejpam-3807	216	61	page	page	NOUN
ejpam-3807	216	62	340	340	NUM
ejpam-3807	216	63	-	-	SYM
ejpam-3807	216	64	341	341	NUM
ejpam-3807	216	65	.	.	PUNCT
ejpam-3807	217	1	suppose	suppose	VERB
ejpam-3807	217	2	x	x	PRON
ejpam-3807	217	3	is	be	AUX
ejpam-3807	217	4	a	a	DET
ejpam-3807	217	5	b	b	NOUN
ejpam-3807	217	6	-	-	PUNCT
ejpam-3807	217	7	algebra	algebra	NOUN
ejpam-3807	217	8	and	and	CCONJ
ejpam-3807	217	9	u	u	NOUN
ejpam-3807	217	10	,	,	PUNCT
ejpam-3807	217	11	v	v	ADP
ejpam-3807	217	12	⊆	⊆	NUM
ejpam-3807	217	13	x	x	SYM
ejpam-3807	217	14	×x	×x	X
ejpam-3807	217	15	,	,	PUNCT
ejpam-3807	217	16	consider	consider	VERB
ejpam-3807	217	17	the	the	DET
ejpam-3807	217	18	following	follow	VERB
ejpam-3807	217	19	notations	notation	NOUN
ejpam-3807	217	20	:	:	PUNCT
ejpam-3807	217	21	k.	k.	PROPN
ejpam-3807	217	22	belleza	belleza	PROPN
ejpam-3807	217	23	,	,	PUNCT
ejpam-3807	217	24	j.	j.	PROPN
ejpam-3807	217	25	vilela	vilela	PROPN
ejpam-3807	217	26	/	/	SYM
ejpam-3807	217	27	eur	eur	PROPN
ejpam-3807	217	28	.	.	PUNCT
ejpam-3807	218	1	j.	j.	PROPN
ejpam-3807	218	2	pure	pure	PROPN
ejpam-3807	218	3	appl	appl	PROPN
ejpam-3807	218	4	.	.	PROPN
ejpam-3807	218	5	math	math	PROPN
ejpam-3807	218	6	,	,	PUNCT
ejpam-3807	218	7	13	13	NUM
ejpam-3807	218	8	(	(	PUNCT
ejpam-3807	218	9	4	4	NUM
ejpam-3807	218	10	)	)	PUNCT
ejpam-3807	218	11	(	(	PUNCT
ejpam-3807	218	12	2020	2020	NUM
ejpam-3807	218	13	)	)	PUNCT
ejpam-3807	218	14	,	,	PUNCT
ejpam-3807	218	15	830	830	NUM
ejpam-3807	218	16	-	-	SYM
ejpam-3807	218	17	839	839	NUM
ejpam-3807	218	18	836	836	NUM
ejpam-3807	218	19	(	(	PUNCT
ejpam-3807	218	20	i	i	NOUN
ejpam-3807	218	21	)	)	PUNCT
ejpam-3807	219	1	u−1	u−1	PROPN
ejpam-3807	219	2	=	=	PRON
ejpam-3807	219	3	{	{	PUNCT
ejpam-3807	219	4	(	(	PUNCT
ejpam-3807	219	5	y	y	NOUN
ejpam-3807	219	6	,	,	PUNCT
ejpam-3807	219	7	x)|(x	x)|(x	PROPN
ejpam-3807	219	8	,	,	PUNCT
ejpam-3807	219	9	y	y	NOUN
ejpam-3807	219	10	)	)	PUNCT
ejpam-3807	219	11	∈	∈	PROPN
ejpam-3807	219	12	u	u	NOUN
ejpam-3807	219	13	}	}	PUNCT
ejpam-3807	219	14	;	;	PUNCT
ejpam-3807	219	15	(	(	PUNCT
ejpam-3807	219	16	iii	iii	X
ejpam-3807	219	17	)	)	PUNCT
ejpam-3807	219	18	u	u	NOUN
ejpam-3807	220	1	[	[	X
ejpam-3807	220	2	[	[	X
ejpam-3807	220	3	x	x	X
ejpam-3807	220	4	]	]	X
ejpam-3807	220	5	]	]	X
ejpam-3807	220	6	=	=	X
ejpam-3807	220	7	{	{	PUNCT
ejpam-3807	220	8	y|(x	y|(x	PROPN
ejpam-3807	220	9	,	,	PUNCT
ejpam-3807	220	10	y	y	NOUN
ejpam-3807	220	11	)	)	PUNCT
ejpam-3807	220	12	∈	∈	PROPN
ejpam-3807	220	13	u	u	NOUN
ejpam-3807	220	14	}	}	PUNCT
ejpam-3807	220	15	;	;	PUNCT
ejpam-3807	220	16	(	(	PUNCT
ejpam-3807	220	17	ii	ii	NOUN
ejpam-3807	220	18	)	)	PUNCT
ejpam-3807	220	19	u	u	NOUN
ejpam-3807	220	20	◦	◦	NOUN
ejpam-3807	220	21	v	v	NOUN
ejpam-3807	220	22	=	=	SYM
ejpam-3807	220	23	{	{	PUNCT
ejpam-3807	220	24	(	(	PUNCT
ejpam-3807	220	25	x	x	NOUN
ejpam-3807	220	26	,	,	PUNCT
ejpam-3807	220	27	z)|∃y	z)|∃y	PUNCT
ejpam-3807	220	28	∈	∈	PROPN
ejpam-3807	220	29	x	x	X
ejpam-3807	220	30	,	,	PUNCT
ejpam-3807	220	31	(	(	PUNCT
ejpam-3807	220	32	x	x	NOUN
ejpam-3807	220	33	,	,	PUNCT
ejpam-3807	220	34	y	y	NOUN
ejpam-3807	220	35	)	)	PUNCT
ejpam-3807	220	36	∈	∈	PROPN
ejpam-3807	220	37	v	v	NOUN
ejpam-3807	220	38	,	,	PUNCT
ejpam-3807	220	39	(	(	PUNCT
ejpam-3807	220	40	y	y	PROPN
ejpam-3807	220	41	,	,	PUNCT
ejpam-3807	220	42	z	z	NOUN
ejpam-3807	220	43	)	)	PUNCT
ejpam-3807	220	44	∈	∈	PROPN
ejpam-3807	220	45	u	u	NOUN
ejpam-3807	220	46	}	}	PUNCT
ejpam-3807	220	47	;	;	PUNCT
ejpam-3807	220	48	(	(	PUNCT
ejpam-3807	220	49	iv	iv	X
ejpam-3807	220	50	)	)	PUNCT
ejpam-3807	220	51	∆	∆	PROPN
ejpam-3807	220	52	=	=	PRON
ejpam-3807	220	53	{	{	PUNCT
ejpam-3807	220	54	(	(	PUNCT
ejpam-3807	220	55	x	x	X
ejpam-3807	220	56	,	,	PUNCT
ejpam-3807	220	57	x)|x	x)|x	X
ejpam-3807	220	58	∈	∈	PROPN
ejpam-3807	220	59	x	x	PRON
ejpam-3807	220	60	}	}	PUNCT
ejpam-3807	220	61	.	.	PUNCT
ejpam-3807	221	1	suppose	suppose	VERB
ejpam-3807	221	2	ω	ω	NOUN
ejpam-3807	221	3	is	be	AUX
ejpam-3807	221	4	an	an	DET
ejpam-3807	221	5	arbitrary	arbitrary	ADJ
ejpam-3807	221	6	family	family	NOUN
ejpam-3807	221	7	of	of	ADP
ejpam-3807	221	8	b	b	NOUN
ejpam-3807	221	9	-	-	PUNCT
ejpam-3807	221	10	ideals	ideal	NOUN
ejpam-3807	221	11	in	in	ADP
ejpam-3807	221	12	a	a	DET
ejpam-3807	221	13	b	b	NOUN
ejpam-3807	221	14	-	-	PUNCT
ejpam-3807	221	15	algebra	algebra	NOUN
ejpam-3807	221	16	x	x	PUNCT
ejpam-3807	221	17	and	and	CCONJ
ejpam-3807	221	18	a	a	DET
ejpam-3807	221	19	⊆	⊆	NUM
ejpam-3807	221	20	x.	x.	NOUN
ejpam-3807	221	21	consider	consider	VERB
ejpam-3807	221	22	the	the	DET
ejpam-3807	221	23	following	follow	VERB
ejpam-3807	221	24	notations	notation	NOUN
ejpam-3807	221	25	:	:	PUNCT
ejpam-3807	221	26	(	(	PUNCT
ejpam-3807	221	27	i	i	NOUN
ejpam-3807	221	28	)	)	PUNCT
ejpam-3807	221	29	ui	ui	NOUN
ejpam-3807	222	1	=	=	PUNCT
ejpam-3807	222	2	{	{	PUNCT
ejpam-3807	222	3	(	(	PUNCT
ejpam-3807	222	4	x	x	NOUN
ejpam-3807	222	5	,	,	PUNCT
ejpam-3807	222	6	y	y	NOUN
ejpam-3807	222	7	)	)	PUNCT
ejpam-3807	222	8	∈	∈	PROPN
ejpam-3807	222	9	x	x	PUNCT
ejpam-3807	222	10	×x|x	×x|x	PROPN
ejpam-3807	222	11	∼=i	∼=i	NOUN
ejpam-3807	222	12	y	y	PROPN
ejpam-3807	222	13	}	}	PUNCT
ejpam-3807	222	14	;	;	PUNCT
ejpam-3807	222	15	(	(	PUNCT
ejpam-3807	222	16	iii	iii	X
ejpam-3807	222	17	)	)	PUNCT
ejpam-3807	222	18	k	k	NOUN
ejpam-3807	223	1	=	=	PUNCT
ejpam-3807	223	2	{	{	PUNCT
ejpam-3807	223	3	u	u	NOUN
ejpam-3807	223	4	⊆	⊆	NUM
ejpam-3807	223	5	x	x	SYM
ejpam-3807	223	6	×x|ui	×x|ui	VERB
ejpam-3807	223	7	⊆	⊆	NUM
ejpam-3807	223	8	u,∃ui	u,∃ui	SYM
ejpam-3807	223	9	∈	∈	PROPN
ejpam-3807	223	10	k	k	NOUN
ejpam-3807	223	11	?	?	PUNCT
ejpam-3807	223	12	}	}	PUNCT
ejpam-3807	223	13	;	;	PUNCT
ejpam-3807	223	14	(	(	PUNCT
ejpam-3807	223	15	ii	ii	NOUN
ejpam-3807	223	16	)	)	PUNCT
ejpam-3807	223	17	k	k	NOUN
ejpam-3807	223	18	?	?	PUNCT
ejpam-3807	224	1	=	=	PRON
ejpam-3807	224	2	{	{	PUNCT
ejpam-3807	224	3	ui	ui	NOUN
ejpam-3807	224	4	:	:	PUNCT
ejpam-3807	225	1	i	i	PROPN
ejpam-3807	225	2	∈	∈	PROPN
ejpam-3807	225	3	ω	ω	PROPN
ejpam-3807	225	4	}	}	PUNCT
ejpam-3807	225	5	;	;	PUNCT
ejpam-3807	225	6	(	(	PUNCT
ejpam-3807	225	7	iv	iv	X
ejpam-3807	225	8	)	)	PUNCT
ejpam-3807	225	9	ui	ui	NOUN
ejpam-3807	226	1	[	[	X
ejpam-3807	226	2	[	[	X
ejpam-3807	226	3	a	a	X
ejpam-3807	226	4	]	]	X
ejpam-3807	226	5	]	]	X
ejpam-3807	226	6	=	=	X
ejpam-3807	226	7	⋃	⋃	VERB
ejpam-3807	226	8	a∈a	a∈a	ADJ
ejpam-3807	226	9	ui	ui	NOUN
ejpam-3807	227	1	[	[	X
ejpam-3807	227	2	[	[	X
ejpam-3807	227	3	a	a	X
ejpam-3807	227	4	]	]	X
ejpam-3807	227	5	]	]	PUNCT
ejpam-3807	227	6	.	.	PUNCT
ejpam-3807	227	7	remark	remark	PROPN
ejpam-3807	227	8	5	5	NUM
ejpam-3807	227	9	.	.	PUNCT
ejpam-3807	228	1	k	k	X
ejpam-3807	228	2	?	?	PUNCT
ejpam-3807	229	1	⊆	⊆	NUM
ejpam-3807	229	2	k.	k.	NOUN
ejpam-3807	229	3	definition	definition	NOUN
ejpam-3807	229	4	8	8	NUM
ejpam-3807	229	5	.	.	PUNCT
ejpam-3807	230	1	by	by	ADP
ejpam-3807	230	2	a	a	DET
ejpam-3807	230	3	uniformity	uniformity	NOUN
ejpam-3807	230	4	on	on	ADP
ejpam-3807	230	5	a	a	DET
ejpam-3807	230	6	b	b	NOUN
ejpam-3807	230	7	-	-	PUNCT
ejpam-3807	230	8	algebra	algebra	NOUN
ejpam-3807	230	9	x	x	NOUN
ejpam-3807	230	10	,	,	PUNCT
ejpam-3807	230	11	we	we	PRON
ejpam-3807	230	12	shall	shall	AUX
ejpam-3807	230	13	mean	mean	VERB
ejpam-3807	230	14	a	a	DET
ejpam-3807	230	15	nonempty	nonempty	ADJ
ejpam-3807	230	16	collection	collection	NOUN
ejpam-3807	230	17	k	k	PROPN
ejpam-3807	230	18	of	of	ADP
ejpam-3807	230	19	subsets	subset	NOUN
ejpam-3807	230	20	of	of	ADP
ejpam-3807	230	21	x	x	X
ejpam-3807	230	22	×x	×x	NUM
ejpam-3807	230	23	which	which	PRON
ejpam-3807	230	24	satisfies	satisfy	VERB
ejpam-3807	230	25	the	the	DET
ejpam-3807	230	26	following	follow	VERB
ejpam-3807	230	27	conditions	condition	NOUN
ejpam-3807	230	28	for	for	ADP
ejpam-3807	230	29	any	any	DET
ejpam-3807	230	30	u	u	NOUN
ejpam-3807	230	31	,	,	PUNCT
ejpam-3807	230	32	v	v	ADP
ejpam-3807	230	33	∈	∈	PROPN
ejpam-3807	230	34	k	k	NOUN
ejpam-3807	230	35	:	:	PUNCT
ejpam-3807	230	36	(	(	PUNCT
ejpam-3807	230	37	i	i	NOUN
ejpam-3807	230	38	)	)	PUNCT
ejpam-3807	230	39	∆	∆	PROPN
ejpam-3807	231	1	⊆	⊆	NUM
ejpam-3807	231	2	u	u	NOUN
ejpam-3807	231	3	;	;	PUNCT
ejpam-3807	231	4	(	(	PUNCT
ejpam-3807	231	5	iv	iv	X
ejpam-3807	231	6	)	)	PUNCT
ejpam-3807	231	7	u	u	NOUN
ejpam-3807	231	8	∩	∩	X
ejpam-3807	231	9	v	v	ADP
ejpam-3807	231	10	∈	∈	PROPN
ejpam-3807	231	11	k	k	NOUN
ejpam-3807	231	12	;	;	PUNCT
ejpam-3807	231	13	and	and	CCONJ
ejpam-3807	231	14	(	(	PUNCT
ejpam-3807	231	15	ii	ii	NOUN
ejpam-3807	231	16	)	)	PUNCT
ejpam-3807	231	17	u−1	u−1	PROPN
ejpam-3807	231	18	∈	∈	PROPN
ejpam-3807	231	19	k	k	NOUN
ejpam-3807	231	20	;	;	PUNCT
ejpam-3807	231	21	(	(	PUNCT
ejpam-3807	231	22	v	v	NOUN
ejpam-3807	231	23	)	)	PUNCT
ejpam-3807	231	24	if	if	SCONJ
ejpam-3807	231	25	u	u	NOUN
ejpam-3807	231	26	⊆w	⊆w	NOUN
ejpam-3807	231	27	⊆	⊆	NUM
ejpam-3807	231	28	x	x	X
ejpam-3807	231	29	×x	×x	VERB
ejpam-3807	231	30	then	then	ADV
ejpam-3807	231	31	w	w	PROPN
ejpam-3807	231	32	∈	∈	PROPN
ejpam-3807	231	33	k.	k.	PROPN
ejpam-3807	231	34	(	(	PUNCT
ejpam-3807	231	35	iii	iii	PROPN
ejpam-3807	231	36	)	)	PUNCT
ejpam-3807	231	37	w	w	PROPN
ejpam-3807	231	38	◦	◦	PROPN
ejpam-3807	231	39	w	w	ADP
ejpam-3807	231	40	⊆	⊆	NUM
ejpam-3807	231	41	u	u	NOUN
ejpam-3807	231	42	,	,	PUNCT
ejpam-3807	231	43	for	for	ADP
ejpam-3807	231	44	some	some	DET
ejpam-3807	231	45	w	w	PROPN
ejpam-3807	231	46	∈	∈	PROPN
ejpam-3807	231	47	k	k	NOUN
ejpam-3807	231	48	;	;	PUNCT
ejpam-3807	231	49	the	the	DET
ejpam-3807	231	50	pair	pair	NOUN
ejpam-3807	231	51	(	(	PUNCT
ejpam-3807	231	52	x	x	NOUN
ejpam-3807	231	53	,	,	PUNCT
ejpam-3807	231	54	k	k	NOUN
ejpam-3807	231	55	)	)	PUNCT
ejpam-3807	231	56	is	be	AUX
ejpam-3807	231	57	called	call	VERB
ejpam-3807	231	58	a	a	DET
ejpam-3807	231	59	uniform	uniform	ADJ
ejpam-3807	231	60	b	b	NOUN
ejpam-3807	231	61	-	-	NOUN
ejpam-3807	231	62	structure	structure	NOUN
ejpam-3807	231	63	.	.	PUNCT
ejpam-3807	231	64	example	example	NOUN
ejpam-3807	232	1	6	6	NUM
ejpam-3807	232	2	.	.	PUNCT
ejpam-3807	232	3	consider	consider	VERB
ejpam-3807	232	4	the	the	DET
ejpam-3807	232	5	b	b	NOUN
ejpam-3807	232	6	-	-	PUNCT
ejpam-3807	232	7	algebra	algebra	NOUN
ejpam-3807	232	8	x	x	X
ejpam-3807	232	9	=	=	SYM
ejpam-3807	232	10	{	{	PUNCT
ejpam-3807	232	11	0	0	NUM
ejpam-3807	232	12	,	,	PUNCT
ejpam-3807	232	13	a	a	DET
ejpam-3807	232	14	,	,	PUNCT
ejpam-3807	232	15	b	b	NOUN
ejpam-3807	232	16	,	,	PUNCT
ejpam-3807	232	17	c	c	NOUN
ejpam-3807	232	18	,	,	PUNCT
ejpam-3807	232	19	d	d	NOUN
ejpam-3807	232	20	,	,	PUNCT
ejpam-3807	232	21	e	e	NOUN
ejpam-3807	232	22	}	}	PUNCT
ejpam-3807	232	23	in	in	ADP
ejpam-3807	232	24	example	example	NOUN
ejpam-3807	232	25	1	1	X
ejpam-3807	232	26	.	.	PUNCT
ejpam-3807	233	1	the	the	DET
ejpam-3807	233	2	normal	normal	ADJ
ejpam-3807	233	3	bideals	bideal	NOUN
ejpam-3807	233	4	of	of	ADP
ejpam-3807	233	5	x	x	SYM
ejpam-3807	233	6	are	be	AUX
ejpam-3807	233	7	{	{	PUNCT
ejpam-3807	233	8	x	x	NOUN
ejpam-3807	233	9	,	,	PUNCT
ejpam-3807	233	10	i	i	NOUN
ejpam-3807	233	11	}	}	PUNCT
ejpam-3807	233	12	where	where	SCONJ
ejpam-3807	233	13	i	i	PRON
ejpam-3807	233	14	=	=	PUNCT
ejpam-3807	233	15	{	{	PUNCT
ejpam-3807	233	16	0	0	NUM
ejpam-3807	233	17	,	,	PUNCT
ejpam-3807	233	18	a	a	PRON
ejpam-3807	233	19	,	,	PUNCT
ejpam-3807	233	20	b	b	NOUN
ejpam-3807	233	21	}	}	PUNCT
ejpam-3807	233	22	.	.	PUNCT
ejpam-3807	234	1	by	by	ADP
ejpam-3807	234	2	routine	routine	ADJ
ejpam-3807	234	3	calculations	calculation	NOUN
ejpam-3807	234	4	,	,	PUNCT
ejpam-3807	234	5	(	(	PUNCT
ejpam-3807	234	6	x	x	NOUN
ejpam-3807	234	7	,	,	PUNCT
ejpam-3807	234	8	k	k	NOUN
ejpam-3807	234	9	)	)	PUNCT
ejpam-3807	234	10	is	be	AUX
ejpam-3807	234	11	a	a	DET
ejpam-3807	234	12	uniform	uniform	ADJ
ejpam-3807	234	13	b	b	NOUN
ejpam-3807	234	14	-	-	PUNCT
ejpam-3807	234	15	structure	structure	NOUN
ejpam-3807	234	16	where	where	SCONJ
ejpam-3807	234	17	k	k	X
ejpam-3807	234	18	?	?	PUNCT
ejpam-3807	235	1	=	=	PRON
ejpam-3807	235	2	{	{	PUNCT
ejpam-3807	235	3	ux	ux	INTJ
ejpam-3807	235	4	,	,	PUNCT
ejpam-3807	235	5	ui	ui	PROPN
ejpam-3807	235	6	}	}	PUNCT
ejpam-3807	235	7	,	,	PUNCT
ejpam-3807	235	8	ux	ux	PROPN
ejpam-3807	236	1	[	[	X
ejpam-3807	236	2	[	[	X
ejpam-3807	236	3	0	0	X
ejpam-3807	236	4	]	]	X
ejpam-3807	236	5	]	]	X
ejpam-3807	236	6	=	=	PUNCT
ejpam-3807	236	7	ux	ux	PROPN
ejpam-3807	237	1	[	[	X
ejpam-3807	237	2	[	[	X
ejpam-3807	237	3	a	a	X
ejpam-3807	237	4	]	]	X
ejpam-3807	237	5	]	]	X
ejpam-3807	237	6	=	=	PUNCT
ejpam-3807	237	7	ux	ux	PROPN
ejpam-3807	238	1	[	[	X
ejpam-3807	238	2	[	[	X
ejpam-3807	238	3	b	b	X
ejpam-3807	238	4	]	]	X
ejpam-3807	238	5	]	]	X
ejpam-3807	238	6	=	=	PUNCT
ejpam-3807	238	7	ux	ux	PROPN
ejpam-3807	239	1	[	[	X
ejpam-3807	239	2	[	[	X
ejpam-3807	239	3	c	c	X
ejpam-3807	239	4	]	]	X
ejpam-3807	239	5	]	]	X
ejpam-3807	239	6	=	=	PUNCT
ejpam-3807	239	7	ux	ux	PROPN
ejpam-3807	240	1	[	[	X
ejpam-3807	240	2	[	[	X
ejpam-3807	240	3	d	d	X
ejpam-3807	240	4	]	]	X
ejpam-3807	240	5	]	]	X
ejpam-3807	240	6	=	=	PUNCT
ejpam-3807	240	7	ux	ux	PROPN
ejpam-3807	241	1	[	[	X
ejpam-3807	241	2	[	[	X
ejpam-3807	241	3	e	e	X
ejpam-3807	241	4	]	]	X
ejpam-3807	241	5	]	]	X
ejpam-3807	241	6	=	=	SYM
ejpam-3807	241	7	{	{	PUNCT
ejpam-3807	241	8	0	0	NUM
ejpam-3807	241	9	,	,	PUNCT
ejpam-3807	241	10	a	a	DET
ejpam-3807	241	11	,	,	PUNCT
ejpam-3807	241	12	b	b	NOUN
ejpam-3807	241	13	,	,	PUNCT
ejpam-3807	241	14	c	c	NOUN
ejpam-3807	241	15	,	,	PUNCT
ejpam-3807	241	16	d	d	NOUN
ejpam-3807	241	17	,	,	PUNCT
ejpam-3807	241	18	e	e	NOUN
ejpam-3807	241	19	}	}	PUNCT
ejpam-3807	241	20	=	=	SYM
ejpam-3807	241	21	x	x	NOUN
ejpam-3807	241	22	,	,	PUNCT
ejpam-3807	241	23	ui	ui	PROPN
ejpam-3807	242	1	[	[	X
ejpam-3807	242	2	[	[	X
ejpam-3807	242	3	0	0	X
ejpam-3807	242	4	]	]	X
ejpam-3807	242	5	]	]	PUNCT
ejpam-3807	243	1	=	=	SYM
ejpam-3807	243	2	ui	ui	PROPN
ejpam-3807	244	1	[	[	X
ejpam-3807	244	2	[	[	X
ejpam-3807	244	3	a	a	X
ejpam-3807	244	4	]	]	X
ejpam-3807	244	5	]	]	X
ejpam-3807	244	6	=	=	SYM
ejpam-3807	244	7	ui	ui	PROPN
ejpam-3807	245	1	[	[	X
ejpam-3807	245	2	[	[	X
ejpam-3807	245	3	b	b	X
ejpam-3807	245	4	]	]	X
ejpam-3807	245	5	]	]	X
ejpam-3807	245	6	=	=	X
ejpam-3807	245	7	{	{	PUNCT
ejpam-3807	245	8	0	0	NUM
ejpam-3807	245	9	,	,	PUNCT
ejpam-3807	245	10	a	a	DET
ejpam-3807	245	11	,	,	PUNCT
ejpam-3807	245	12	b	b	NOUN
ejpam-3807	245	13	}	}	PUNCT
ejpam-3807	245	14	,	,	PUNCT
ejpam-3807	245	15	ui	ui	PROPN
ejpam-3807	246	1	[	[	X
ejpam-3807	246	2	[	[	X
ejpam-3807	246	3	c	c	X
ejpam-3807	246	4	]	]	X
ejpam-3807	246	5	]	]	X
ejpam-3807	246	6	=	=	SYM
ejpam-3807	246	7	ui	ui	NOUN
ejpam-3807	247	1	[	[	X
ejpam-3807	247	2	[	[	X
ejpam-3807	247	3	d]],=	d]],=	X
ejpam-3807	247	4	ui	ui	NOUN
ejpam-3807	248	1	[	[	X
ejpam-3807	248	2	[	[	X
ejpam-3807	248	3	e	e	X
ejpam-3807	248	4	]	]	X
ejpam-3807	248	5	]	]	X
ejpam-3807	248	6	=	=	X
ejpam-3807	248	7	{	{	PUNCT
ejpam-3807	248	8	c	c	NOUN
ejpam-3807	248	9	,	,	PUNCT
ejpam-3807	248	10	d	d	NOUN
ejpam-3807	248	11	,	,	PUNCT
ejpam-3807	248	12	e	e	NOUN
ejpam-3807	248	13	}	}	PUNCT
ejpam-3807	248	14	.	.	PUNCT
ejpam-3807	249	1	remark	remark	NOUN
ejpam-3807	249	2	6	6	NUM
ejpam-3807	249	3	.	.	PUNCT
ejpam-3807	250	1	(	(	PUNCT
ejpam-3807	250	2	x	x	X
ejpam-3807	250	3	,	,	PUNCT
ejpam-3807	250	4	k∗	k∗	PROPN
ejpam-3807	250	5	)	)	PUNCT
ejpam-3807	250	6	is	be	AUX
ejpam-3807	250	7	not	not	PART
ejpam-3807	250	8	a	a	DET
ejpam-3807	250	9	uniform	uniform	ADJ
ejpam-3807	250	10	b	b	NOUN
ejpam-3807	250	11	-	-	PUNCT
ejpam-3807	250	12	structure	structure	NOUN
ejpam-3807	250	13	as	as	SCONJ
ejpam-3807	250	14	shown	show	VERB
ejpam-3807	250	15	in	in	ADP
ejpam-3807	250	16	the	the	DET
ejpam-3807	250	17	next	next	ADJ
ejpam-3807	250	18	example	example	NOUN
ejpam-3807	250	19	.	.	PUNCT
ejpam-3807	251	1	example	example	NOUN
ejpam-3807	252	1	7	7	NUM
ejpam-3807	252	2	.	.	X
ejpam-3807	252	3	consider	consider	VERB
ejpam-3807	252	4	the	the	DET
ejpam-3807	252	5	b	b	NOUN
ejpam-3807	252	6	-	-	PUNCT
ejpam-3807	252	7	algebra	algebra	NOUN
ejpam-3807	252	8	x	x	X
ejpam-3807	252	9	=	=	SYM
ejpam-3807	252	10	{	{	PUNCT
ejpam-3807	252	11	0	0	NUM
ejpam-3807	252	12	,	,	PUNCT
ejpam-3807	252	13	a	a	DET
ejpam-3807	252	14	,	,	PUNCT
ejpam-3807	252	15	b	b	NOUN
ejpam-3807	252	16	,	,	PUNCT
ejpam-3807	252	17	c	c	NOUN
ejpam-3807	252	18	,	,	PUNCT
ejpam-3807	252	19	d	d	NOUN
ejpam-3807	252	20	,	,	PUNCT
ejpam-3807	252	21	e	e	NOUN
ejpam-3807	252	22	}	}	PUNCT
ejpam-3807	252	23	in	in	ADP
ejpam-3807	252	24	example	example	NOUN
ejpam-3807	252	25	1	1	NUM
ejpam-3807	252	26	and	and	CCONJ
ejpam-3807	252	27	the	the	DET
ejpam-3807	252	28	b	b	NOUN
ejpam-3807	252	29	-	-	PUNCT
ejpam-3807	252	30	ideal	ideal	NOUN
ejpam-3807	252	31	i	i	NOUN
ejpam-3807	252	32	=	=	PUNCT
ejpam-3807	252	33	{	{	PUNCT
ejpam-3807	252	34	0	0	NUM
ejpam-3807	252	35	,	,	PUNCT
ejpam-3807	252	36	a	a	DET
ejpam-3807	252	37	,	,	PUNCT
ejpam-3807	252	38	b	b	NOUN
ejpam-3807	252	39	}	}	PUNCT
ejpam-3807	252	40	in	in	ADP
ejpam-3807	252	41	example	example	NOUN
ejpam-3807	252	42	6	6	NUM
ejpam-3807	252	43	.	.	PUNCT
ejpam-3807	252	44	by	by	ADP
ejpam-3807	252	45	theorem	theorem	NOUN
ejpam-3807	252	46	1	1	NUM
ejpam-3807	252	47	,	,	PUNCT
ejpam-3807	252	48	x	x	PUNCT
ejpam-3807	252	49	and	and	CCONJ
ejpam-3807	252	50	i	i	PRON
ejpam-3807	252	51	are	be	AUX
ejpam-3807	252	52	normal	normal	ADJ
ejpam-3807	252	53	b	b	NOUN
ejpam-3807	252	54	-	-	PUNCT
ejpam-3807	252	55	ideals	ideal	NOUN
ejpam-3807	252	56	of	of	ADP
ejpam-3807	252	57	x.	x.	NOUN
ejpam-3807	252	58	hence	hence	ADV
ejpam-3807	252	59	,	,	PUNCT
ejpam-3807	252	60	k	k	PROPN
ejpam-3807	252	61	?	?	PUNCT
ejpam-3807	253	1	=	=	PRON
ejpam-3807	253	2	{	{	PUNCT
ejpam-3807	253	3	ux	ux	INTJ
ejpam-3807	253	4	,	,	PUNCT
ejpam-3807	253	5	ui	ui	PROPN
ejpam-3807	253	6	}	}	PUNCT
ejpam-3807	253	7	where	where	SCONJ
ejpam-3807	253	8	ux	ux	ADV
ejpam-3807	253	9	=	=	PRON
ejpam-3807	253	10	{	{	PUNCT
ejpam-3807	253	11	(	(	PUNCT
ejpam-3807	253	12	x	x	NOUN
ejpam-3807	253	13	,	,	PUNCT
ejpam-3807	253	14	y	y	NOUN
ejpam-3807	253	15	)	)	PUNCT
ejpam-3807	253	16	∈	∈	PROPN
ejpam-3807	254	1	x	x	X
ejpam-3807	254	2	×	×	NOUN
ejpam-3807	254	3	x|x	x|x	PROPN
ejpam-3807	255	1	∗	∗	NOUN
ejpam-3807	255	2	y	y	PROPN
ejpam-3807	255	3	,	,	PUNCT
ejpam-3807	255	4	y	y	PROPN
ejpam-3807	255	5	∗	∗	NOUN
ejpam-3807	255	6	x	x	PUNCT
ejpam-3807	255	7	∈	∈	NOUN
ejpam-3807	255	8	x	x	NOUN
ejpam-3807	255	9	}	}	PUNCT
ejpam-3807	255	10	=	=	PUNCT
ejpam-3807	255	11	x	x	SYM
ejpam-3807	255	12	×	×	NOUN
ejpam-3807	255	13	x	x	X
ejpam-3807	255	14	and	and	CCONJ
ejpam-3807	255	15	ui	ui	NOUN
ejpam-3807	255	16	=	=	NOUN
ejpam-3807	255	17	{	{	PUNCT
ejpam-3807	255	18	(	(	PUNCT
ejpam-3807	255	19	0	0	NUM
ejpam-3807	255	20	,	,	PUNCT
ejpam-3807	255	21	0	0	NUM
ejpam-3807	255	22	)	)	PUNCT
ejpam-3807	255	23	,	,	PUNCT
ejpam-3807	255	24	(	(	PUNCT
ejpam-3807	255	25	a	a	X
ejpam-3807	255	26	,	,	PUNCT
ejpam-3807	255	27	a	a	NOUN
ejpam-3807	255	28	)	)	PUNCT
ejpam-3807	255	29	,	,	PUNCT
ejpam-3807	255	30	(	(	PUNCT
ejpam-3807	255	31	b	b	X
ejpam-3807	255	32	,	,	PUNCT
ejpam-3807	255	33	b	b	NOUN
ejpam-3807	255	34	)	)	PUNCT
ejpam-3807	255	35	,	,	PUNCT
ejpam-3807	255	36	(	(	PUNCT
ejpam-3807	255	37	c	c	X
ejpam-3807	255	38	,	,	PUNCT
ejpam-3807	255	39	c	c	NOUN
ejpam-3807	255	40	)	)	PUNCT
ejpam-3807	255	41	,	,	PUNCT
ejpam-3807	255	42	(	(	PUNCT
ejpam-3807	255	43	d	d	X
ejpam-3807	255	44	,	,	PUNCT
ejpam-3807	255	45	d	d	NOUN
ejpam-3807	255	46	)	)	PUNCT
ejpam-3807	255	47	,	,	PUNCT
ejpam-3807	255	48	(	(	PUNCT
ejpam-3807	255	49	e	e	NOUN
ejpam-3807	255	50	,	,	PUNCT
ejpam-3807	255	51	e	e	NOUN
ejpam-3807	255	52	)	)	PUNCT
ejpam-3807	255	53	,	,	PUNCT
ejpam-3807	255	54	(	(	PUNCT
ejpam-3807	255	55	0	0	NUM
ejpam-3807	255	56	,	,	PUNCT
ejpam-3807	255	57	b	b	NOUN
ejpam-3807	255	58	)	)	PUNCT
ejpam-3807	255	59	,	,	PUNCT
ejpam-3807	255	60	(	(	PUNCT
ejpam-3807	255	61	b	b	NOUN
ejpam-3807	255	62	,	,	PUNCT
ejpam-3807	255	63	0	0	NUM
ejpam-3807	255	64	)	)	PUNCT
ejpam-3807	255	65	,	,	PUNCT
ejpam-3807	255	66	(	(	PUNCT
ejpam-3807	255	67	a	a	PRON
ejpam-3807	255	68	,	,	PUNCT
ejpam-3807	255	69	0	0	NUM
ejpam-3807	255	70	)	)	PUNCT
ejpam-3807	255	71	,	,	PUNCT
ejpam-3807	255	72	(	(	PUNCT
ejpam-3807	255	73	0	0	NUM
ejpam-3807	255	74	,	,	PUNCT
ejpam-3807	255	75	a	a	PRON
ejpam-3807	255	76	)	)	PUNCT
ejpam-3807	255	77	,	,	PUNCT
ejpam-3807	255	78	(	(	PUNCT
ejpam-3807	255	79	b	b	X
ejpam-3807	255	80	,	,	PUNCT
ejpam-3807	255	81	a	a	PRON
ejpam-3807	255	82	)	)	PUNCT
ejpam-3807	255	83	,	,	PUNCT
ejpam-3807	255	84	(	(	PUNCT
ejpam-3807	255	85	a	a	DET
ejpam-3807	255	86	,	,	PUNCT
ejpam-3807	255	87	b	b	NOUN
ejpam-3807	255	88	)	)	PUNCT
ejpam-3807	255	89	,	,	PUNCT
ejpam-3807	255	90	(	(	PUNCT
ejpam-3807	255	91	c	c	X
ejpam-3807	255	92	,	,	PUNCT
ejpam-3807	255	93	e	e	NOUN
ejpam-3807	255	94	)	)	PUNCT
ejpam-3807	255	95	,	,	PUNCT
ejpam-3807	255	96	(	(	PUNCT
ejpam-3807	255	97	e	e	NOUN
ejpam-3807	255	98	,	,	PUNCT
ejpam-3807	255	99	c	c	NOUN
ejpam-3807	255	100	)	)	PUNCT
ejpam-3807	255	101	,	,	PUNCT
ejpam-3807	255	102	(	(	PUNCT
ejpam-3807	255	103	d	d	X
ejpam-3807	255	104	,	,	PUNCT
ejpam-3807	255	105	c	c	NOUN
ejpam-3807	255	106	)	)	PUNCT
ejpam-3807	255	107	,	,	PUNCT
ejpam-3807	255	108	(	(	PUNCT
ejpam-3807	255	109	c	c	X
ejpam-3807	255	110	,	,	PUNCT
ejpam-3807	255	111	d	d	NOUN
ejpam-3807	255	112	)	)	PUNCT
ejpam-3807	255	113	,	,	PUNCT
ejpam-3807	255	114	(	(	PUNCT
ejpam-3807	255	115	e	e	NOUN
ejpam-3807	255	116	,	,	PUNCT
ejpam-3807	255	117	d	d	NOUN
ejpam-3807	255	118	)	)	PUNCT
ejpam-3807	255	119	,	,	PUNCT
ejpam-3807	255	120	(	(	PUNCT
ejpam-3807	255	121	d	d	X
ejpam-3807	255	122	,	,	PUNCT
ejpam-3807	255	123	e	e	NOUN
ejpam-3807	255	124	)	)	PUNCT
ejpam-3807	255	125	}	}	PUNCT
ejpam-3807	255	126	.	.	PUNCT
ejpam-3807	256	1	let	let	VERB
ejpam-3807	256	2	m	m	VERB
ejpam-3807	256	3	=	=	VERB
ejpam-3807	256	4	i	i	PRON
ejpam-3807	256	5	∪	∪	VERB
ejpam-3807	256	6	{	{	PUNCT
ejpam-3807	256	7	0	0	NUM
ejpam-3807	256	8	,	,	PUNCT
ejpam-3807	256	9	e	e	NOUN
ejpam-3807	256	10	}	}	PUNCT
ejpam-3807	256	11	=	=	SYM
ejpam-3807	256	12	{	{	PUNCT
ejpam-3807	256	13	0	0	NUM
ejpam-3807	256	14	,	,	PUNCT
ejpam-3807	256	15	a	a	DET
ejpam-3807	256	16	,	,	PUNCT
ejpam-3807	256	17	b	b	NOUN
ejpam-3807	256	18	,	,	PUNCT
ejpam-3807	256	19	e	e	NOUN
ejpam-3807	256	20	}	}	PUNCT
ejpam-3807	256	21	.	.	PUNCT
ejpam-3807	257	1	then	then	ADV
ejpam-3807	257	2	um	um	INTJ
ejpam-3807	257	3	=	=	SYM
ejpam-3807	257	4	ui	ui	NOUN
ejpam-3807	257	5	∪	∪	X
ejpam-3807	257	6	{	{	PUNCT
ejpam-3807	257	7	(	(	PUNCT
ejpam-3807	257	8	0	0	NUM
ejpam-3807	257	9	,	,	PUNCT
ejpam-3807	257	10	e	e	NOUN
ejpam-3807	257	11	)	)	PUNCT
ejpam-3807	257	12	,	,	PUNCT
ejpam-3807	257	13	(	(	PUNCT
ejpam-3807	257	14	e	e	NOUN
ejpam-3807	257	15	,	,	PUNCT
ejpam-3807	257	16	0	0	NUM
ejpam-3807	257	17	)	)	PUNCT
ejpam-3807	257	18	,	,	PUNCT
ejpam-3807	257	19	(	(	PUNCT
ejpam-3807	257	20	a	a	DET
ejpam-3807	257	21	,	,	PUNCT
ejpam-3807	257	22	d	d	NOUN
ejpam-3807	257	23	)	)	PUNCT
ejpam-3807	257	24	,	,	PUNCT
ejpam-3807	257	25	(	(	PUNCT
ejpam-3807	257	26	d	d	X
ejpam-3807	257	27	,	,	PUNCT
ejpam-3807	257	28	a	a	NOUN
ejpam-3807	257	29	)	)	PUNCT
ejpam-3807	257	30	,	,	PUNCT
ejpam-3807	257	31	(	(	PUNCT
ejpam-3807	257	32	b	b	X
ejpam-3807	257	33	,	,	PUNCT
ejpam-3807	257	34	c	c	NOUN
ejpam-3807	257	35	)	)	PUNCT
ejpam-3807	257	36	,	,	PUNCT
ejpam-3807	257	37	(	(	PUNCT
ejpam-3807	257	38	c	c	X
ejpam-3807	257	39	,	,	PUNCT
ejpam-3807	257	40	b	b	NOUN
ejpam-3807	257	41	)	)	PUNCT
ejpam-3807	257	42	}	}	PUNCT
ejpam-3807	257	43	.	.	PUNCT
ejpam-3807	258	1	note	note	VERB
ejpam-3807	258	2	that	that	SCONJ
ejpam-3807	258	3	ui	ui	PROPN
ejpam-3807	258	4	⊆	⊆	NUM
ejpam-3807	258	5	um	um	INTJ
ejpam-3807	258	6	⊆	⊆	NUM
ejpam-3807	258	7	x×x	x×x	PROPN
ejpam-3807	258	8	.	.	PUNCT
ejpam-3807	259	1	moreover	moreover	ADV
ejpam-3807	259	2	,	,	PUNCT
ejpam-3807	259	3	m	m	VERB
ejpam-3807	259	4	/∈	/∈	PROPN
ejpam-3807	260	1	ω	ω	PROPN
ejpam-3807	260	2	since	since	SCONJ
ejpam-3807	260	3	d∗a	d∗a	NOUN
ejpam-3807	260	4	=	=	SYM
ejpam-3807	260	5	e	e	NOUN
ejpam-3807	260	6	,	,	PUNCT
ejpam-3807	260	7	a	a	DET
ejpam-3807	260	8	∈m	∈m	NOUN
ejpam-3807	260	9	but	but	CCONJ
ejpam-3807	260	10	d	d	NOUN
ejpam-3807	260	11	/∈m	/∈m	PUNCT
ejpam-3807	260	12	.	.	PUNCT
ejpam-3807	261	1	hence	hence	ADV
ejpam-3807	261	2	,	,	PUNCT
ejpam-3807	261	3	um	um	INTJ
ejpam-3807	261	4	/∈	/∈	PUNCT
ejpam-3807	262	1	k	k	NOUN
ejpam-3807	262	2	?	?	PUNCT
ejpam-3807	262	3	.	.	PUNCT
ejpam-3807	263	1	this	this	PRON
ejpam-3807	263	2	implies	imply	VERB
ejpam-3807	263	3	that	that	SCONJ
ejpam-3807	263	4	k	k	X
ejpam-3807	263	5	?	?	PUNCT
ejpam-3807	263	6	does	do	AUX
ejpam-3807	263	7	not	not	PART
ejpam-3807	263	8	satisfy	satisfy	VERB
ejpam-3807	263	9	condition	condition	NOUN
ejpam-3807	263	10	(	(	PUNCT
ejpam-3807	263	11	v	v	NOUN
ejpam-3807	263	12	)	)	PUNCT
ejpam-3807	263	13	of	of	ADP
ejpam-3807	263	14	definition	definition	NOUN
ejpam-3807	263	15	8	8	NUM
ejpam-3807	263	16	.	.	PUNCT
ejpam-3807	264	1	however	however	ADV
ejpam-3807	264	2	,	,	PUNCT
ejpam-3807	264	3	in	in	ADP
ejpam-3807	264	4	view	view	NOUN
ejpam-3807	264	5	of	of	ADP
ejpam-3807	264	6	remark	remark	NOUN
ejpam-3807	264	7	5	5	NUM
ejpam-3807	264	8	,	,	PUNCT
ejpam-3807	264	9	the	the	DET
ejpam-3807	264	10	next	next	ADJ
ejpam-3807	264	11	theorem	theorem	ADJ
ejpam-3807	264	12	states	state	NOUN
ejpam-3807	264	13	that	that	SCONJ
ejpam-3807	264	14	the	the	DET
ejpam-3807	264	15	pair	pair	NOUN
ejpam-3807	264	16	(	(	PUNCT
ejpam-3807	264	17	x	x	NOUN
ejpam-3807	264	18	,	,	PUNCT
ejpam-3807	264	19	k	k	NOUN
ejpam-3807	264	20	)	)	PUNCT
ejpam-3807	264	21	is	be	AUX
ejpam-3807	264	22	a	a	DET
ejpam-3807	264	23	uniform	uniform	ADJ
ejpam-3807	264	24	b	b	NOUN
ejpam-3807	264	25	-	-	NOUN
ejpam-3807	264	26	structure	structure	NOUN
ejpam-3807	264	27	.	.	PUNCT
ejpam-3807	265	1	theorem	theorem	NOUN
ejpam-3807	265	2	9	9	NUM
ejpam-3807	265	3	.	.	PUNCT
ejpam-3807	266	1	let	let	VERB
ejpam-3807	266	2	ω	ω	NUM
ejpam-3807	266	3	be	be	AUX
ejpam-3807	266	4	an	an	DET
ejpam-3807	266	5	arbitrary	arbitrary	ADJ
ejpam-3807	266	6	family	family	NOUN
ejpam-3807	266	7	of	of	ADP
ejpam-3807	266	8	b	b	NOUN
ejpam-3807	266	9	-	-	PUNCT
ejpam-3807	266	10	ideals	ideal	NOUN
ejpam-3807	266	11	in	in	ADP
ejpam-3807	266	12	a	a	DET
ejpam-3807	266	13	b	b	NOUN
ejpam-3807	266	14	-	-	PUNCT
ejpam-3807	266	15	algebra	algebra	NOUN
ejpam-3807	266	16	x.	x.	NOUN
ejpam-3807	266	17	then	then	ADV
ejpam-3807	266	18	(	(	PUNCT
ejpam-3807	266	19	x	x	X
ejpam-3807	266	20	,	,	PUNCT
ejpam-3807	266	21	k	k	NOUN
ejpam-3807	266	22	)	)	PUNCT
ejpam-3807	266	23	is	be	AUX
ejpam-3807	266	24	a	a	DET
ejpam-3807	266	25	uniform	uniform	ADJ
ejpam-3807	266	26	b	b	NOUN
ejpam-3807	266	27	-	-	NOUN
ejpam-3807	266	28	structure	structure	NOUN
ejpam-3807	266	29	.	.	PUNCT
ejpam-3807	267	1	k.	k.	PROPN
ejpam-3807	267	2	belleza	belleza	PROPN
ejpam-3807	267	3	,	,	PUNCT
ejpam-3807	267	4	j.	j.	PROPN
ejpam-3807	267	5	vilela	vilela	PROPN
ejpam-3807	267	6	/	/	SYM
ejpam-3807	267	7	eur	eur	PROPN
ejpam-3807	267	8	.	.	PUNCT
ejpam-3807	268	1	j.	j.	PROPN
ejpam-3807	268	2	pure	pure	PROPN
ejpam-3807	268	3	appl	appl	PROPN
ejpam-3807	268	4	.	.	PROPN
ejpam-3807	268	5	math	math	PROPN
ejpam-3807	268	6	,	,	PUNCT
ejpam-3807	268	7	13	13	NUM
ejpam-3807	268	8	(	(	PUNCT
ejpam-3807	268	9	4	4	NUM
ejpam-3807	268	10	)	)	PUNCT
ejpam-3807	268	11	(	(	PUNCT
ejpam-3807	268	12	2020	2020	NUM
ejpam-3807	268	13	)	)	PUNCT
ejpam-3807	268	14	,	,	PUNCT
ejpam-3807	268	15	830	830	NUM
ejpam-3807	268	16	-	-	SYM
ejpam-3807	268	17	839	839	NUM
ejpam-3807	268	18	837	837	NUM
ejpam-3807	268	19	proof	proof	NOUN
ejpam-3807	268	20	.	.	PUNCT
ejpam-3807	269	1	suppose	suppose	VERB
ejpam-3807	269	2	ω	ω	NOUN
ejpam-3807	269	3	is	be	AUX
ejpam-3807	269	4	an	an	DET
ejpam-3807	269	5	arbitrary	arbitrary	ADJ
ejpam-3807	269	6	family	family	NOUN
ejpam-3807	269	7	of	of	ADP
ejpam-3807	269	8	b	b	NOUN
ejpam-3807	269	9	-	-	PUNCT
ejpam-3807	269	10	ideals	ideal	NOUN
ejpam-3807	269	11	in	in	ADP
ejpam-3807	269	12	a	a	DET
ejpam-3807	269	13	b	b	NOUN
ejpam-3807	269	14	-	-	PUNCT
ejpam-3807	269	15	algebra	algebra	NOUN
ejpam-3807	269	16	x	x	X
ejpam-3807	269	17	and	and	CCONJ
ejpam-3807	269	18	u	u	NOUN
ejpam-3807	269	19	,	,	PUNCT
ejpam-3807	269	20	v	v	PROPN
ejpam-3807	269	21	∈	∈	PROPN
ejpam-3807	269	22	k.	k.	NOUN
ejpam-3807	270	1	then	then	ADV
ejpam-3807	270	2	there	there	PRON
ejpam-3807	270	3	exist	exist	VERB
ejpam-3807	270	4	ui	ui	NOUN
ejpam-3807	270	5	,	,	PUNCT
ejpam-3807	270	6	uj	uj	PROPN
ejpam-3807	270	7	∈	∈	PROPN
ejpam-3807	270	8	k	k	PROPN
ejpam-3807	270	9	?	?	PUNCT
ejpam-3807	271	1	such	such	ADJ
ejpam-3807	271	2	that	that	DET
ejpam-3807	271	3	ui	ui	PROPN
ejpam-3807	271	4	⊆	⊆	NUM
ejpam-3807	271	5	u	u	NOUN
ejpam-3807	271	6	and	and	CCONJ
ejpam-3807	271	7	uj	uj	VERB
ejpam-3807	271	8	⊆	⊆	NUM
ejpam-3807	271	9	v	v	NOUN
ejpam-3807	271	10	,	,	PUNCT
ejpam-3807	271	11	respectively	respectively	ADV
ejpam-3807	271	12	.	.	PUNCT
ejpam-3807	272	1	(	(	PUNCT
ejpam-3807	272	2	i	i	NOUN
ejpam-3807	272	3	)	)	PUNCT
ejpam-3807	272	4	let	let	VERB
ejpam-3807	272	5	(	(	PUNCT
ejpam-3807	272	6	x	x	NOUN
ejpam-3807	272	7	,	,	PUNCT
ejpam-3807	272	8	x	x	SYM
ejpam-3807	272	9	)	)	PUNCT
ejpam-3807	272	10	∈	∈	PROPN
ejpam-3807	273	1	∆.	∆.	NOUN
ejpam-3807	273	2	since	since	SCONJ
ejpam-3807	273	3	x	x	SYM
ejpam-3807	273	4	∼=i	∼=i	NOUN
ejpam-3807	273	5	x	x	X
ejpam-3807	273	6	,	,	PUNCT
ejpam-3807	273	7	it	it	PRON
ejpam-3807	273	8	follows	follow	VERB
ejpam-3807	273	9	that	that	SCONJ
ejpam-3807	273	10	(	(	PUNCT
ejpam-3807	273	11	x	x	X
ejpam-3807	273	12	,	,	PUNCT
ejpam-3807	273	13	x	x	SYM
ejpam-3807	273	14	)	)	PUNCT
ejpam-3807	273	15	∈	∈	PROPN
ejpam-3807	273	16	ui	ui	PROPN
ejpam-3807	273	17	.	.	PUNCT
ejpam-3807	274	1	hence	hence	ADV
ejpam-3807	274	2	,	,	PUNCT
ejpam-3807	274	3	(	(	PUNCT
ejpam-3807	274	4	x	x	NOUN
ejpam-3807	274	5	,	,	PUNCT
ejpam-3807	274	6	x	x	SYM
ejpam-3807	274	7	)	)	PUNCT
ejpam-3807	274	8	∈	∈	NOUN
ejpam-3807	274	9	u	u	NOUN
ejpam-3807	274	10	so	so	SCONJ
ejpam-3807	274	11	that	that	SCONJ
ejpam-3807	274	12	∆	∆	PROPN
ejpam-3807	274	13	⊆	⊆	NUM
ejpam-3807	274	14	u	u	NOUN
ejpam-3807	274	15	.	.	PUNCT
ejpam-3807	275	1	(	(	PUNCT
ejpam-3807	275	2	ii	ii	NOUN
ejpam-3807	275	3	)	)	PUNCT
ejpam-3807	275	4	let	let	VERB
ejpam-3807	275	5	(	(	PUNCT
ejpam-3807	275	6	x	x	NOUN
ejpam-3807	275	7	,	,	PUNCT
ejpam-3807	275	8	y	y	NOUN
ejpam-3807	275	9	)	)	PUNCT
ejpam-3807	275	10	∈	∈	PROPN
ejpam-3807	275	11	ui	ui	PROPN
ejpam-3807	275	12	.	.	PUNCT
ejpam-3807	276	1	then	then	ADV
ejpam-3807	276	2	x	x	SYM
ejpam-3807	276	3	∼=i	∼=i	NOUN
ejpam-3807	276	4	y	y	PROPN
ejpam-3807	276	5	and	and	CCONJ
ejpam-3807	276	6	y	y	PROPN
ejpam-3807	276	7	∼=i	∼=i	NOUN
ejpam-3807	276	8	x.	x.	NOUN
ejpam-3807	277	1	this	this	PRON
ejpam-3807	277	2	implies	imply	VERB
ejpam-3807	277	3	that	that	SCONJ
ejpam-3807	277	4	(	(	PUNCT
ejpam-3807	277	5	y	y	NOUN
ejpam-3807	277	6	,	,	PUNCT
ejpam-3807	277	7	x	x	X
ejpam-3807	277	8	)	)	PUNCT
ejpam-3807	277	9	∈	∈	PROPN
ejpam-3807	277	10	ui	ui	PROPN
ejpam-3807	277	11	.	.	PUNCT
ejpam-3807	278	1	hence	hence	ADV
ejpam-3807	278	2	,	,	PUNCT
ejpam-3807	278	3	(	(	PUNCT
ejpam-3807	278	4	y	y	NOUN
ejpam-3807	278	5	,	,	PUNCT
ejpam-3807	278	6	x	x	X
ejpam-3807	278	7	)	)	PUNCT
ejpam-3807	278	8	∈	∈	PROPN
ejpam-3807	278	9	u	u	NOUN
ejpam-3807	278	10	.	.	PUNCT
ejpam-3807	279	1	it	it	PRON
ejpam-3807	279	2	follows	follow	VERB
ejpam-3807	279	3	that	that	SCONJ
ejpam-3807	279	4	(	(	PUNCT
ejpam-3807	279	5	x	x	X
ejpam-3807	279	6	,	,	PUNCT
ejpam-3807	279	7	y	y	NOUN
ejpam-3807	279	8	)	)	PUNCT
ejpam-3807	279	9	∈	∈	PROPN
ejpam-3807	279	10	u−1	u−1	PROPN
ejpam-3807	279	11	with	with	ADP
ejpam-3807	279	12	ui	ui	NOUN
ejpam-3807	279	13	⊆	⊆	NUM
ejpam-3807	279	14	u−1	u−1	PROPN
ejpam-3807	279	15	so	so	SCONJ
ejpam-3807	279	16	that	that	SCONJ
ejpam-3807	279	17	u−1	u−1	PROPN
ejpam-3807	279	18	∈	∈	PROPN
ejpam-3807	279	19	k.	k.	PROPN
ejpam-3807	279	20	(	(	PUNCT
ejpam-3807	279	21	iii	iii	X
ejpam-3807	279	22	)	)	PUNCT
ejpam-3807	279	23	consider	consider	VERB
ejpam-3807	279	24	ui	ui	PROPN
ejpam-3807	279	25	∈	∈	PROPN
ejpam-3807	279	26	k	k	PROPN
ejpam-3807	279	27	and	and	CCONJ
ejpam-3807	279	28	(	(	PUNCT
ejpam-3807	279	29	x	x	NOUN
ejpam-3807	279	30	,	,	PUNCT
ejpam-3807	279	31	z	z	NOUN
ejpam-3807	279	32	)	)	PUNCT
ejpam-3807	279	33	∈	∈	PROPN
ejpam-3807	279	34	ui	ui	PROPN
ejpam-3807	279	35	◦	◦	NOUN
ejpam-3807	279	36	ui	ui	PROPN
ejpam-3807	279	37	.	.	PUNCT
ejpam-3807	280	1	then	then	ADV
ejpam-3807	280	2	there	there	PRON
ejpam-3807	280	3	exists	exist	VERB
ejpam-3807	280	4	y	y	PROPN
ejpam-3807	280	5	∈	∈	PROPN
ejpam-3807	280	6	x	x	PUNCT
ejpam-3807	280	7	such	such	ADJ
ejpam-3807	280	8	that	that	SCONJ
ejpam-3807	280	9	(	(	PUNCT
ejpam-3807	280	10	x	x	NOUN
ejpam-3807	280	11	,	,	PUNCT
ejpam-3807	280	12	y	y	PROPN
ejpam-3807	280	13	)	)	PUNCT
ejpam-3807	280	14	,	,	PUNCT
ejpam-3807	280	15	(	(	PUNCT
ejpam-3807	280	16	y	y	NOUN
ejpam-3807	280	17	,	,	PUNCT
ejpam-3807	280	18	z	z	NOUN
ejpam-3807	280	19	)	)	PUNCT
ejpam-3807	280	20	∈	∈	PROPN
ejpam-3807	280	21	ui	ui	PROPN
ejpam-3807	280	22	.	.	PUNCT
ejpam-3807	281	1	this	this	PRON
ejpam-3807	281	2	implies	imply	VERB
ejpam-3807	281	3	that	that	SCONJ
ejpam-3807	281	4	x	x	SYM
ejpam-3807	281	5	∼=i	∼=i	NOUN
ejpam-3807	281	6	y	y	PROPN
ejpam-3807	281	7	and	and	CCONJ
ejpam-3807	281	8	y	y	PROPN
ejpam-3807	281	9	∼=i	∼=i	NOUN
ejpam-3807	281	10	z.	z.	PROPN
ejpam-3807	281	11	hence	hence	ADV
ejpam-3807	281	12	,	,	PUNCT
ejpam-3807	281	13	x	x	X
ejpam-3807	281	14	∼=i	∼=i	NOUN
ejpam-3807	281	15	z.	z.	PROPN
ejpam-3807	282	1	it	it	PRON
ejpam-3807	282	2	follows	follow	VERB
ejpam-3807	282	3	that	that	SCONJ
ejpam-3807	282	4	(	(	PUNCT
ejpam-3807	282	5	x	x	X
ejpam-3807	282	6	,	,	PUNCT
ejpam-3807	282	7	z	z	NOUN
ejpam-3807	282	8	)	)	PUNCT
ejpam-3807	282	9	∈	∈	PROPN
ejpam-3807	282	10	ui	ui	NOUN
ejpam-3807	283	1	⊆	⊆	NUM
ejpam-3807	283	2	u	u	NOUN
ejpam-3807	283	3	so	so	SCONJ
ejpam-3807	283	4	that	that	SCONJ
ejpam-3807	283	5	ui	ui	PROPN
ejpam-3807	283	6	◦	◦	NOUN
ejpam-3807	283	7	ui	ui	NOUN
ejpam-3807	283	8	⊆	⊆	NUM
ejpam-3807	283	9	u	u	NOUN
ejpam-3807	283	10	.	.	PUNCT
ejpam-3807	284	1	(	(	PUNCT
ejpam-3807	284	2	iv	iv	X
ejpam-3807	284	3	)	)	PUNCT
ejpam-3807	284	4	let	let	VERB
ejpam-3807	284	5	ui	ui	NOUN
ejpam-3807	284	6	,	,	PUNCT
ejpam-3807	284	7	uj	uj	PROPN
ejpam-3807	284	8	∈	∈	PROPN
ejpam-3807	284	9	k	k	X
ejpam-3807	284	10	?	?	PUNCT
ejpam-3807	284	11	.	.	PUNCT
ejpam-3807	285	1	note	note	VERB
ejpam-3807	285	2	that	that	SCONJ
ejpam-3807	285	3	a	a	DET
ejpam-3807	285	4	b	b	X
ejpam-3807	285	5	-	-	PUNCT
ejpam-3807	285	6	ideal	ideal	NOUN
ejpam-3807	285	7	is	be	AUX
ejpam-3807	285	8	a	a	DET
ejpam-3807	285	9	subalgebra	subalgebra	NOUN
ejpam-3807	285	10	of	of	ADP
ejpam-3807	285	11	x.	x.	NOUN
ejpam-3807	285	12	by	by	ADP
ejpam-3807	285	13	lemma	lemma	PROPN
ejpam-3807	285	14	2	2	NUM
ejpam-3807	285	15	,	,	PUNCT
ejpam-3807	285	16	ω	ω	PROPN
ejpam-3807	285	17	is	be	AUX
ejpam-3807	285	18	closed	close	VERB
ejpam-3807	285	19	under	under	ADP
ejpam-3807	285	20	finite	finite	ADJ
ejpam-3807	285	21	intersections	intersection	NOUN
ejpam-3807	285	22	.	.	PUNCT
ejpam-3807	286	1	claim	claim	NOUN
ejpam-3807	286	2	:	:	PUNCT
ejpam-3807	286	3	ui	ui	PROPN
ejpam-3807	286	4	∩	∩	ADJ
ejpam-3807	286	5	uj	uj	PROPN
ejpam-3807	286	6	=	=	SYM
ejpam-3807	286	7	ui∩j	ui∩j	PROPN
ejpam-3807	286	8	∈	∈	PROPN
ejpam-3807	286	9	k	k	NOUN
ejpam-3807	286	10	?	?	PUNCT
ejpam-3807	286	11	.	.	PUNCT
ejpam-3807	287	1	let	let	VERB
ejpam-3807	287	2	(	(	PUNCT
ejpam-3807	287	3	x	x	NOUN
ejpam-3807	287	4	,	,	PUNCT
ejpam-3807	287	5	y	y	NOUN
ejpam-3807	287	6	)	)	PUNCT
ejpam-3807	287	7	∈	∈	PROPN
ejpam-3807	287	8	ui	ui	NOUN
ejpam-3807	287	9	∩uj	∩uj	NOUN
ejpam-3807	287	10	.	.	PUNCT
ejpam-3807	288	1	then	then	ADV
ejpam-3807	288	2	x	x	SYM
ejpam-3807	288	3	∼=i	∼=i	NOUN
ejpam-3807	288	4	y	y	PROPN
ejpam-3807	288	5	and	and	CCONJ
ejpam-3807	288	6	x	x	X
ejpam-3807	288	7	∼=j	∼=j	ADP
ejpam-3807	288	8	y.	y.	NOUN
ejpam-3807	288	9	these	these	PRON
ejpam-3807	288	10	imply	imply	VERB
ejpam-3807	288	11	that	that	SCONJ
ejpam-3807	288	12	x∗y	x∗y	PRON
ejpam-3807	288	13	,	,	PUNCT
ejpam-3807	288	14	y	y	PROPN
ejpam-3807	288	15	∗x	∗x	PROPN
ejpam-3807	288	16	∈	∈	PROPN
ejpam-3807	289	1	i	i	PRON
ejpam-3807	289	2	,	,	PUNCT
ejpam-3807	289	3	j	j	PROPN
ejpam-3807	289	4	.	.	PUNCT
ejpam-3807	290	1	hence	hence	ADV
ejpam-3807	290	2	,	,	PUNCT
ejpam-3807	290	3	x∗y	x∗y	NUM
ejpam-3807	290	4	,	,	PUNCT
ejpam-3807	290	5	y∗x	y∗x	PRON
ejpam-3807	290	6	∈	∈	PROPN
ejpam-3807	290	7	i∩j	i∩j	NOUN
ejpam-3807	290	8	implying	imply	VERB
ejpam-3807	290	9	that	that	SCONJ
ejpam-3807	290	10	x	x	PUNCT
ejpam-3807	290	11	∼=i∩j	∼=i∩j	PROPN
ejpam-3807	290	12	y	y	PROPN
ejpam-3807	290	13	and	and	CCONJ
ejpam-3807	290	14	so	so	ADV
ejpam-3807	290	15	(	(	PUNCT
ejpam-3807	290	16	x	x	NOUN
ejpam-3807	290	17	,	,	PUNCT
ejpam-3807	290	18	y	y	NOUN
ejpam-3807	290	19	)	)	PUNCT
ejpam-3807	290	20	∈	∈	PROPN
ejpam-3807	290	21	ui∩j	ui∩j	ADV
ejpam-3807	290	22	.	.	PUNCT
ejpam-3807	291	1	therefore	therefore	ADV
ejpam-3807	291	2	,	,	PUNCT
ejpam-3807	291	3	ui∩uj	ui∩uj	PROPN
ejpam-3807	291	4	⊆	⊆	NUM
ejpam-3807	291	5	ui∩j	ui∩j	ADV
ejpam-3807	291	6	.	.	PUNCT
ejpam-3807	292	1	the	the	DET
ejpam-3807	292	2	converse	converse	NOUN
ejpam-3807	292	3	is	be	AUX
ejpam-3807	292	4	similar	similar	ADJ
ejpam-3807	292	5	.	.	PUNCT
ejpam-3807	293	1	this	this	PRON
ejpam-3807	293	2	proves	prove	VERB
ejpam-3807	293	3	the	the	DET
ejpam-3807	293	4	claim	claim	NOUN
ejpam-3807	293	5	.	.	PUNCT
ejpam-3807	294	1	let	let	VERB
ejpam-3807	294	2	(	(	PUNCT
ejpam-3807	294	3	x	x	NOUN
ejpam-3807	294	4	,	,	PUNCT
ejpam-3807	294	5	y	y	NOUN
ejpam-3807	294	6	)	)	PUNCT
ejpam-3807	294	7	∈	∈	PROPN
ejpam-3807	294	8	ui∩j	ui∩j	ADV
ejpam-3807	294	9	=	=	SYM
ejpam-3807	294	10	ui	ui	PROPN
ejpam-3807	294	11	∩	∩	PROPN
ejpam-3807	294	12	uj	uj	PROPN
ejpam-3807	294	13	.	.	PUNCT
ejpam-3807	295	1	then	then	ADV
ejpam-3807	295	2	x	x	SYM
ejpam-3807	295	3	∼=i	∼=i	NOUN
ejpam-3807	295	4	y	y	PROPN
ejpam-3807	295	5	and	and	CCONJ
ejpam-3807	295	6	x	x	X
ejpam-3807	295	7	∼=j	∼=j	ADP
ejpam-3807	295	8	y.	y.	PROPN
ejpam-3807	295	9	thus	thus	ADV
ejpam-3807	295	10	,	,	PUNCT
ejpam-3807	295	11	(	(	PUNCT
ejpam-3807	295	12	x	x	X
ejpam-3807	295	13	,	,	PUNCT
ejpam-3807	295	14	y	y	NOUN
ejpam-3807	295	15	)	)	PUNCT
ejpam-3807	295	16	∈	∈	PROPN
ejpam-3807	295	17	ui	ui	NOUN
ejpam-3807	296	1	and	and	CCONJ
ejpam-3807	296	2	(	(	PUNCT
ejpam-3807	296	3	x	x	NOUN
ejpam-3807	296	4	,	,	PUNCT
ejpam-3807	296	5	y	y	NOUN
ejpam-3807	296	6	)	)	PUNCT
ejpam-3807	296	7	∈	∈	PROPN
ejpam-3807	296	8	uj	uj	PROPN
ejpam-3807	296	9	.	.	PUNCT
ejpam-3807	297	1	hence	hence	ADV
ejpam-3807	297	2	,	,	PUNCT
ejpam-3807	297	3	(	(	PUNCT
ejpam-3807	297	4	x	x	X
ejpam-3807	297	5	,	,	PUNCT
ejpam-3807	297	6	y	y	NOUN
ejpam-3807	297	7	)	)	PUNCT
ejpam-3807	297	8	∈	∈	PROPN
ejpam-3807	297	9	u	u	NOUN
ejpam-3807	297	10	and	and	CCONJ
ejpam-3807	297	11	(	(	PUNCT
ejpam-3807	297	12	x	x	NOUN
ejpam-3807	297	13	,	,	PUNCT
ejpam-3807	297	14	y	y	NOUN
ejpam-3807	297	15	)	)	PUNCT
ejpam-3807	297	16	∈	∈	PROPN
ejpam-3807	297	17	v	v	NOUN
ejpam-3807	297	18	.	.	PUNCT
ejpam-3807	298	1	therefore	therefore	ADV
ejpam-3807	298	2	,	,	PUNCT
ejpam-3807	298	3	(	(	PUNCT
ejpam-3807	298	4	x	x	X
ejpam-3807	298	5	,	,	PUNCT
ejpam-3807	298	6	y	y	NOUN
ejpam-3807	298	7	)	)	PUNCT
ejpam-3807	298	8	∈	∈	PROPN
ejpam-3807	298	9	u	u	NOUN
ejpam-3807	298	10	∩	∩	NOUN
ejpam-3807	298	11	v	v	NOUN
ejpam-3807	298	12	so	so	SCONJ
ejpam-3807	298	13	that	that	SCONJ
ejpam-3807	298	14	ui∩j	ui∩j	ADV
ejpam-3807	298	15	⊆	⊆	NUM
ejpam-3807	298	16	u	u	NOUN
ejpam-3807	298	17	∩	∩	NOUN
ejpam-3807	298	18	v	v	NOUN
ejpam-3807	298	19	.	.	PUNCT
ejpam-3807	299	1	consequently	consequently	ADV
ejpam-3807	299	2	,	,	PUNCT
ejpam-3807	299	3	u	u	PROPN
ejpam-3807	299	4	∩	∩	PROPN
ejpam-3807	299	5	v	v	ADP
ejpam-3807	299	6	∈	∈	PROPN
ejpam-3807	299	7	k.	k.	NOUN
ejpam-3807	300	1	it	it	PRON
ejpam-3807	300	2	remains	remain	VERB
ejpam-3807	300	3	to	to	PART
ejpam-3807	300	4	show	show	VERB
ejpam-3807	300	5	that	that	SCONJ
ejpam-3807	300	6	k	k	PROPN
ejpam-3807	300	7	satisfies	satisfy	VERB
ejpam-3807	300	8	condition	condition	NOUN
ejpam-3807	300	9	(	(	PUNCT
ejpam-3807	300	10	v	v	NOUN
ejpam-3807	300	11	)	)	PUNCT
ejpam-3807	300	12	.	.	PUNCT
ejpam-3807	301	1	let	let	VERB
ejpam-3807	301	2	u	u	PRON
ejpam-3807	301	3	∈	∈	PROPN
ejpam-3807	301	4	k	k	ADP
ejpam-3807	301	5	such	such	ADJ
ejpam-3807	301	6	that	that	SCONJ
ejpam-3807	301	7	u	u	PROPN
ejpam-3807	301	8	⊆	⊆	NUM
ejpam-3807	301	9	v	v	ADP
ejpam-3807	301	10	⊆	⊆	NUM
ejpam-3807	301	11	x	x	SYM
ejpam-3807	301	12	×	×	NOUN
ejpam-3807	301	13	x.	x.	NOUN
ejpam-3807	301	14	then	then	ADV
ejpam-3807	301	15	there	there	PRON
ejpam-3807	301	16	exists	exist	VERB
ejpam-3807	301	17	ui	ui	PROPN
ejpam-3807	301	18	∈	∈	PROPN
ejpam-3807	302	1	k	k	PROPN
ejpam-3807	302	2	?	?	PUNCT
ejpam-3807	302	3	such	such	ADJ
ejpam-3807	302	4	that	that	DET
ejpam-3807	302	5	ui	ui	PROPN
ejpam-3807	302	6	⊆	⊆	NUM
ejpam-3807	302	7	u	u	NOUN
ejpam-3807	302	8	⊆	⊆	NUM
ejpam-3807	302	9	v	v	NOUN
ejpam-3807	302	10	.	.	PUNCT
ejpam-3807	303	1	hence	hence	ADV
ejpam-3807	303	2	,	,	PUNCT
ejpam-3807	303	3	v	v	PROPN
ejpam-3807	303	4	∈	∈	PROPN
ejpam-3807	303	5	k.	k.	PROPN
ejpam-3807	303	6	therefore	therefore	ADV
ejpam-3807	303	7	,	,	PUNCT
ejpam-3807	303	8	k	k	PROPN
ejpam-3807	303	9	satisfies	satisfy	VERB
ejpam-3807	303	10	condition	condition	NOUN
ejpam-3807	303	11	(	(	PUNCT
ejpam-3807	303	12	v	v	NOUN
ejpam-3807	303	13	)	)	PUNCT
ejpam-3807	303	14	and	and	CCONJ
ejpam-3807	303	15	(	(	PUNCT
ejpam-3807	303	16	x	x	NOUN
ejpam-3807	303	17	,	,	PUNCT
ejpam-3807	303	18	k	k	NOUN
ejpam-3807	303	19	)	)	PUNCT
ejpam-3807	303	20	is	be	AUX
ejpam-3807	303	21	a	a	DET
ejpam-3807	303	22	uniform	uniform	ADJ
ejpam-3807	303	23	b	b	NOUN
ejpam-3807	303	24	-	-	NOUN
ejpam-3807	303	25	structure	structure	NOUN
ejpam-3807	303	26	.	.	PUNCT
ejpam-3807	304	1	definition	definition	NOUN
ejpam-3807	304	2	9	9	NUM
ejpam-3807	304	3	.	.	PUNCT
ejpam-3807	305	1	let	let	AUX
ejpam-3807	305	2	(	(	PUNCT
ejpam-3807	305	3	x	x	NOUN
ejpam-3807	305	4	,	,	PUNCT
ejpam-3807	305	5	k	k	NOUN
ejpam-3807	305	6	)	)	PUNCT
ejpam-3807	305	7	be	be	AUX
ejpam-3807	305	8	a	a	DET
ejpam-3807	305	9	uniform	uniform	ADJ
ejpam-3807	305	10	b	b	NOUN
ejpam-3807	305	11	-	-	NOUN
ejpam-3807	305	12	structure	structure	NOUN
ejpam-3807	305	13	.	.	PUNCT
ejpam-3807	306	1	if	if	SCONJ
ejpam-3807	306	2	τ	τ	PROPN
ejpam-3807	306	3	is	be	AUX
ejpam-3807	306	4	a	a	DET
ejpam-3807	306	5	topology	topology	NOUN
ejpam-3807	306	6	on	on	ADP
ejpam-3807	306	7	x	x	NOUN
ejpam-3807	306	8	,	,	PUNCT
ejpam-3807	306	9	then	then	ADV
ejpam-3807	306	10	τ	τ	PROPN
ejpam-3807	306	11	is	be	AUX
ejpam-3807	306	12	called	call	VERB
ejpam-3807	306	13	a	a	DET
ejpam-3807	306	14	uniform	uniform	ADJ
ejpam-3807	306	15	b	b	NOUN
ejpam-3807	306	16	-	-	PUNCT
ejpam-3807	306	17	topology	topology	NOUN
ejpam-3807	306	18	and	and	CCONJ
ejpam-3807	306	19	the	the	DET
ejpam-3807	306	20	pair	pair	NOUN
ejpam-3807	306	21	(	(	PUNCT
ejpam-3807	306	22	x	x	X
ejpam-3807	306	23	,	,	PUNCT
ejpam-3807	306	24	τ	τ	X
ejpam-3807	306	25	)	)	PUNCT
ejpam-3807	306	26	is	be	AUX
ejpam-3807	306	27	called	call	VERB
ejpam-3807	306	28	a	a	DET
ejpam-3807	306	29	uniform	uniform	ADJ
ejpam-3807	306	30	b	b	PROPN
ejpam-3807	306	31	-	-	PUNCT
ejpam-3807	306	32	topological	topological	ADJ
ejpam-3807	306	33	space	space	NOUN
ejpam-3807	306	34	.	.	PUNCT
ejpam-3807	307	1	example	example	NOUN
ejpam-3807	307	2	8	8	NUM
ejpam-3807	307	3	.	.	PUNCT
ejpam-3807	308	1	consider	consider	VERB
ejpam-3807	308	2	the	the	DET
ejpam-3807	308	3	uniform	uniform	ADJ
ejpam-3807	308	4	structure	structure	NOUN
ejpam-3807	308	5	(	(	PUNCT
ejpam-3807	308	6	x	x	NOUN
ejpam-3807	308	7	,	,	PUNCT
ejpam-3807	308	8	k	k	NOUN
ejpam-3807	308	9	)	)	PUNCT
ejpam-3807	308	10	in	in	ADP
ejpam-3807	308	11	example	example	NOUN
ejpam-3807	308	12	6	6	NUM
ejpam-3807	308	13	.	.	PUNCT
ejpam-3807	309	1	then	then	ADV
ejpam-3807	309	2	the	the	DET
ejpam-3807	309	3	family	family	NOUN
ejpam-3807	309	4	τ	τ	PROPN
ejpam-3807	309	5	=	=	SYM
ejpam-3807	309	6	{	{	PUNCT
ejpam-3807	309	7	x,∅	x,∅	PROPN
ejpam-3807	309	8	,	,	PUNCT
ejpam-3807	309	9	{	{	PUNCT
ejpam-3807	309	10	0	0	NUM
ejpam-3807	309	11	,	,	PUNCT
ejpam-3807	309	12	a	a	DET
ejpam-3807	309	13	,	,	PUNCT
ejpam-3807	309	14	b	b	NOUN
ejpam-3807	309	15	}	}	PUNCT
ejpam-3807	309	16	,	,	PUNCT
ejpam-3807	309	17	{	{	PUNCT
ejpam-3807	309	18	c	c	X
ejpam-3807	309	19	,	,	PUNCT
ejpam-3807	309	20	d	d	NOUN
ejpam-3807	309	21	,	,	PUNCT
ejpam-3807	309	22	e	e	NOUN
ejpam-3807	309	23	}	}	PUNCT
ejpam-3807	309	24	}	}	PUNCT
ejpam-3807	309	25	is	be	AUX
ejpam-3807	309	26	a	a	DET
ejpam-3807	309	27	uniform	uniform	ADJ
ejpam-3807	309	28	b	b	NOUN
ejpam-3807	309	29	-	-	PUNCT
ejpam-3807	309	30	topology	topology	NOUN
ejpam-3807	309	31	on	on	ADP
ejpam-3807	309	32	x.	x.	PROPN
ejpam-3807	309	33	thus	thus	ADV
ejpam-3807	309	34	,	,	PUNCT
ejpam-3807	309	35	(	(	PUNCT
ejpam-3807	309	36	x	x	X
ejpam-3807	309	37	,	,	PUNCT
ejpam-3807	309	38	τ	τ	X
ejpam-3807	309	39	)	)	PUNCT
ejpam-3807	309	40	is	be	AUX
ejpam-3807	309	41	a	a	DET
ejpam-3807	309	42	uniform	uniform	ADJ
ejpam-3807	309	43	b	b	X
ejpam-3807	309	44	-	-	PUNCT
ejpam-3807	309	45	topological	topological	ADJ
ejpam-3807	309	46	space	space	NOUN
ejpam-3807	309	47	.	.	PUNCT
ejpam-3807	310	1	theorem	theorem	ADJ
ejpam-3807	310	2	10	10	NUM
ejpam-3807	310	3	.	.	PUNCT
ejpam-3807	311	1	suppose	suppose	VERB
ejpam-3807	311	2	(	(	PUNCT
ejpam-3807	311	3	x	x	X
ejpam-3807	311	4	,	,	PUNCT
ejpam-3807	311	5	k	k	NOUN
ejpam-3807	311	6	)	)	PUNCT
ejpam-3807	311	7	is	be	AUX
ejpam-3807	311	8	a	a	DET
ejpam-3807	311	9	uniform	uniform	ADJ
ejpam-3807	311	10	b	b	NOUN
ejpam-3807	311	11	-	-	NOUN
ejpam-3807	311	12	structure	structure	NOUN
ejpam-3807	311	13	.	.	PUNCT
ejpam-3807	312	1	then	then	ADV
ejpam-3807	312	2	τ	τ	X
ejpam-3807	312	3	=	=	PUNCT
ejpam-3807	312	4	{	{	PUNCT
ejpam-3807	312	5	g	g	PROPN
ejpam-3807	312	6	⊆	⊆	NUM
ejpam-3807	312	7	x|∀x	x|∀x	PROPN
ejpam-3807	312	8	∈	∈	PROPN
ejpam-3807	312	9	g,∃u	g,∃u	NOUN
ejpam-3807	312	10	∈	∈	PROPN
ejpam-3807	312	11	k	k	PROPN
ejpam-3807	312	12	,	,	PUNCT
ejpam-3807	312	13	u	u	X
ejpam-3807	313	1	[	[	X
ejpam-3807	313	2	[	[	X
ejpam-3807	313	3	x	x	X
ejpam-3807	313	4	]	]	X
ejpam-3807	313	5	]	]	X
ejpam-3807	313	6	⊆	⊆	NUM
ejpam-3807	313	7	g	g	NOUN
ejpam-3807	313	8	}	}	PUNCT
ejpam-3807	313	9	is	be	AUX
ejpam-3807	313	10	a	a	DET
ejpam-3807	313	11	uniform	uniform	ADJ
ejpam-3807	313	12	b	b	NOUN
ejpam-3807	313	13	-	-	PUNCT
ejpam-3807	313	14	topology	topology	NOUN
ejpam-3807	313	15	on	on	ADP
ejpam-3807	313	16	x.	x.	NOUN
ejpam-3807	313	17	proof	proof	PROPN
ejpam-3807	313	18	.	.	PUNCT
ejpam-3807	314	1	suppose	suppose	VERB
ejpam-3807	314	2	(	(	PUNCT
ejpam-3807	314	3	x	x	X
ejpam-3807	314	4	,	,	PUNCT
ejpam-3807	314	5	k	k	NOUN
ejpam-3807	314	6	)	)	PUNCT
ejpam-3807	314	7	is	be	AUX
ejpam-3807	314	8	a	a	DET
ejpam-3807	314	9	uniform	uniform	ADJ
ejpam-3807	314	10	b	b	NOUN
ejpam-3807	314	11	-	-	NOUN
ejpam-3807	314	12	structure	structure	NOUN
ejpam-3807	314	13	.	.	PUNCT
ejpam-3807	315	1	note	note	VERB
ejpam-3807	315	2	that	that	SCONJ
ejpam-3807	315	3	for	for	ADP
ejpam-3807	315	4	all	all	DET
ejpam-3807	315	5	x	x	SYM
ejpam-3807	315	6	∈	∈	PROPN
ejpam-3807	315	7	x	x	X
ejpam-3807	315	8	and	and	CCONJ
ejpam-3807	315	9	u	u	PROPN
ejpam-3807	315	10	∈	∈	PROPN
ejpam-3807	315	11	k	k	PROPN
ejpam-3807	315	12	,	,	PUNCT
ejpam-3807	315	13	u	u	X
ejpam-3807	315	14	[	[	X
ejpam-3807	315	15	[	[	X
ejpam-3807	315	16	x	x	X
ejpam-3807	315	17	]	]	X
ejpam-3807	315	18	]	]	X
ejpam-3807	315	19	⊆	⊆	NUM
ejpam-3807	315	20	x.	x.	NOUN
ejpam-3807	315	21	hence	hence	ADV
ejpam-3807	315	22	,	,	PUNCT
ejpam-3807	315	23	x	x	PROPN
ejpam-3807	315	24	∈	∈	PROPN
ejpam-3807	315	25	τ	τ	X
ejpam-3807	315	26	.	.	PUNCT
ejpam-3807	316	1	also	also	ADV
ejpam-3807	316	2	,	,	PUNCT
ejpam-3807	316	3	∅	∅	NOUN
ejpam-3807	316	4	∈	∈	PROPN
ejpam-3807	316	5	τ	τ	X
ejpam-3807	316	6	by	by	ADP
ejpam-3807	316	7	definition	definition	NOUN
ejpam-3807	316	8	.	.	PUNCT
ejpam-3807	317	1	let	let	VERB
ejpam-3807	317	2	x	x	PUNCT
ejpam-3807	317	3	∈	∈	NOUN
ejpam-3807	317	4	⋃	⋃	NOUN
ejpam-3807	317	5	gi∈τ	gi∈τ	NOUN
ejpam-3807	317	6	,	,	PUNCT
ejpam-3807	317	7	i∈a	i∈a	ADJ
ejpam-3807	317	8	gi	gi	INTJ
ejpam-3807	317	9	.	.	PUNCT
ejpam-3807	318	1	then	then	ADV
ejpam-3807	318	2	there	there	PRON
ejpam-3807	318	3	exists	exist	VERB
ejpam-3807	318	4	j	j	PROPN
ejpam-3807	318	5	∈	∈	PROPN
ejpam-3807	318	6	a	a	DET
ejpam-3807	318	7	such	such	ADJ
ejpam-3807	318	8	that	that	SCONJ
ejpam-3807	318	9	x	x	SYM
ejpam-3807	318	10	∈	∈	PROPN
ejpam-3807	318	11	gj	gj	NOUN
ejpam-3807	318	12	.	.	PUNCT
ejpam-3807	319	1	since	since	SCONJ
ejpam-3807	319	2	gj	gj	PROPN
ejpam-3807	319	3	∈	∈	PROPN
ejpam-3807	319	4	τ	τ	X
ejpam-3807	319	5	,	,	PUNCT
ejpam-3807	319	6	there	there	PRON
ejpam-3807	319	7	exists	exist	VERB
ejpam-3807	319	8	uj	uj	PROPN
ejpam-3807	319	9	∈	∈	PROPN
ejpam-3807	319	10	k	k	PROPN
ejpam-3807	320	1	such	such	ADJ
ejpam-3807	320	2	that	that	SCONJ
ejpam-3807	320	3	uj	uj	PROPN
ejpam-3807	321	1	[	[	X
ejpam-3807	321	2	[	[	X
ejpam-3807	321	3	x	x	X
ejpam-3807	321	4	]	]	X
ejpam-3807	321	5	]	]	X
ejpam-3807	321	6	⊆	⊆	NUM
ejpam-3807	321	7	gj	gj	NOUN
ejpam-3807	321	8	.	.	PUNCT
ejpam-3807	322	1	this	this	PRON
ejpam-3807	322	2	implies	imply	VERB
ejpam-3807	322	3	that	that	SCONJ
ejpam-3807	322	4	uj	uj	PROPN
ejpam-3807	323	1	[	[	X
ejpam-3807	323	2	[	[	X
ejpam-3807	323	3	x	x	X
ejpam-3807	323	4	]	]	X
ejpam-3807	323	5	]	]	X
ejpam-3807	323	6	⊆	⊆	NUM
ejpam-3807	323	7	⋃	⋃	NOUN
ejpam-3807	323	8	gi∈τ	gi∈τ	NOUN
ejpam-3807	323	9	,	,	PUNCT
ejpam-3807	323	10	i∈a	i∈a	ADJ
ejpam-3807	323	11	gi	gi	NOUN
ejpam-3807	323	12	.	.	PUNCT
ejpam-3807	324	1	hence	hence	ADV
ejpam-3807	324	2	,	,	PUNCT
ejpam-3807	324	3	⋃	⋃	ADP
ejpam-3807	324	4	gi∈τ	gi∈τ	NOUN
ejpam-3807	324	5	,	,	PUNCT
ejpam-3807	324	6	i∈a	i∈a	ADJ
ejpam-3807	324	7	gi	gi	ADP
ejpam-3807	324	8	∈	∈	PROPN
ejpam-3807	324	9	τ	τ	PROPN
ejpam-3807	324	10	.	.	PUNCT
ejpam-3807	324	11	suppose	suppose	VERB
ejpam-3807	324	12	g	g	NOUN
ejpam-3807	324	13	,	,	PUNCT
ejpam-3807	324	14	h	h	NOUN
ejpam-3807	324	15	∈	∈	PROPN
ejpam-3807	325	1	τ	τ	X
ejpam-3807	325	2	such	such	ADJ
ejpam-3807	325	3	that	that	SCONJ
ejpam-3807	325	4	x	x	SYM
ejpam-3807	325	5	∈	∈	PROPN
ejpam-3807	325	6	g	g	PROPN
ejpam-3807	325	7	∩	∩	PROPN
ejpam-3807	325	8	h.	h.	PROPN
ejpam-3807	325	9	then	then	ADV
ejpam-3807	325	10	there	there	PRON
ejpam-3807	325	11	exist	exist	VERB
ejpam-3807	325	12	u	u	NOUN
ejpam-3807	325	13	,	,	PUNCT
ejpam-3807	325	14	v	v	ADP
ejpam-3807	325	15	∈	∈	NOUN
ejpam-3807	325	16	k	k	NOUN
ejpam-3807	326	1	such	such	ADJ
ejpam-3807	326	2	that	that	SCONJ
ejpam-3807	326	3	u	u	PRON
ejpam-3807	327	1	[	[	X
ejpam-3807	327	2	[	[	X
ejpam-3807	327	3	x	x	X
ejpam-3807	327	4	]	]	X
ejpam-3807	327	5	]	]	X
ejpam-3807	327	6	⊆	⊆	NUM
ejpam-3807	327	7	g	g	NOUN
ejpam-3807	327	8	and	and	CCONJ
ejpam-3807	327	9	v	v	ADP
ejpam-3807	328	1	[	[	X
ejpam-3807	328	2	[	[	X
ejpam-3807	328	3	x	x	X
ejpam-3807	328	4	]	]	X
ejpam-3807	328	5	]	]	X
ejpam-3807	328	6	⊆	⊆	NUM
ejpam-3807	328	7	h.	h.	NOUN
ejpam-3807	328	8	let	let	VERB
ejpam-3807	328	9	w	w	NOUN
ejpam-3807	328	10	=	=	PUNCT
ejpam-3807	328	11	u	u	NOUN
ejpam-3807	328	12	∩	∩	NOUN
ejpam-3807	328	13	v	v	NOUN
ejpam-3807	328	14	.	.	PUNCT
ejpam-3807	329	1	by	by	ADP
ejpam-3807	329	2	definition	definition	NOUN
ejpam-3807	329	3	8(iv	8(iv	NUM
ejpam-3807	329	4	)	)	PUNCT
ejpam-3807	329	5	,	,	PUNCT
ejpam-3807	329	6	w	w	PROPN
ejpam-3807	329	7	∈	∈	PROPN
ejpam-3807	329	8	k.	k.	NOUN
ejpam-3807	329	9	claim	claim	NOUN
ejpam-3807	329	10	:	:	PUNCT
ejpam-3807	330	1	w	w	X
ejpam-3807	331	1	[	[	X
ejpam-3807	331	2	[	[	X
ejpam-3807	331	3	x	x	X
ejpam-3807	331	4	]	]	X
ejpam-3807	331	5	]	]	X
ejpam-3807	331	6	⊆	⊆	NUM
ejpam-3807	331	7	u	u	NOUN
ejpam-3807	331	8	[	[	X
ejpam-3807	331	9	[	[	X
ejpam-3807	331	10	x	x	X
ejpam-3807	331	11	]	]	X
ejpam-3807	331	12	]	]	X
ejpam-3807	331	13	∩	∩	NOUN
ejpam-3807	331	14	v	v	ADP
ejpam-3807	331	15	[	[	X
ejpam-3807	331	16	[	[	X
ejpam-3807	331	17	x	x	X
ejpam-3807	331	18	]	]	X
ejpam-3807	331	19	]	]	PUNCT
ejpam-3807	331	20	.	.	PUNCT
ejpam-3807	332	1	let	let	VERB
ejpam-3807	332	2	y	y	PROPN
ejpam-3807	332	3	∈	∈	PROPN
ejpam-3807	332	4	w	w	PUNCT
ejpam-3807	333	1	[	[	X
ejpam-3807	333	2	[	[	X
ejpam-3807	333	3	x	x	X
ejpam-3807	333	4	]	]	X
ejpam-3807	333	5	]	]	PUNCT
ejpam-3807	333	6	.	.	PUNCT
ejpam-3807	334	1	then	then	ADV
ejpam-3807	334	2	(	(	PUNCT
ejpam-3807	334	3	x	x	X
ejpam-3807	334	4	,	,	PUNCT
ejpam-3807	334	5	y	y	NOUN
ejpam-3807	334	6	)	)	PUNCT
ejpam-3807	334	7	∈	∈	PROPN
ejpam-3807	334	8	u	u	NOUN
ejpam-3807	334	9	and	and	CCONJ
ejpam-3807	334	10	(	(	PUNCT
ejpam-3807	334	11	x	x	NOUN
ejpam-3807	334	12	,	,	PUNCT
ejpam-3807	334	13	y	y	NOUN
ejpam-3807	334	14	)	)	PUNCT
ejpam-3807	334	15	∈	∈	NOUN
ejpam-3807	334	16	v	v	NOUN
ejpam-3807	334	17	.	.	PUNCT
ejpam-3807	335	1	this	this	PRON
ejpam-3807	335	2	implies	imply	VERB
ejpam-3807	335	3	that	that	SCONJ
ejpam-3807	335	4	y	y	PROPN
ejpam-3807	335	5	∈	∈	PROPN
ejpam-3807	335	6	u	u	NOUN
ejpam-3807	336	1	[	[	X
ejpam-3807	336	2	[	[	X
ejpam-3807	336	3	x	x	X
ejpam-3807	336	4	]	]	X
ejpam-3807	336	5	]	]	X
ejpam-3807	336	6	and	and	CCONJ
ejpam-3807	336	7	y	y	PROPN
ejpam-3807	336	8	∈	∈	PROPN
ejpam-3807	336	9	v	v	ADP
ejpam-3807	337	1	[	[	X
ejpam-3807	337	2	[	[	X
ejpam-3807	337	3	x	x	X
ejpam-3807	337	4	]	]	X
ejpam-3807	337	5	]	]	PUNCT
ejpam-3807	337	6	.	.	PUNCT
ejpam-3807	338	1	hence	hence	ADV
ejpam-3807	338	2	,	,	PUNCT
ejpam-3807	338	3	w	w	X
ejpam-3807	339	1	[	[	X
ejpam-3807	339	2	[	[	X
ejpam-3807	339	3	x	x	X
ejpam-3807	339	4	]	]	X
ejpam-3807	339	5	]	]	X
ejpam-3807	339	6	⊆	⊆	NUM
ejpam-3807	339	7	u	u	NOUN
ejpam-3807	340	1	[	[	X
ejpam-3807	340	2	[	[	X
ejpam-3807	340	3	x	x	X
ejpam-3807	340	4	]	]	X
ejpam-3807	340	5	]	]	X
ejpam-3807	340	6	∩	∩	NOUN
ejpam-3807	340	7	v	v	ADP
ejpam-3807	340	8	[	[	X
ejpam-3807	340	9	[	[	X
ejpam-3807	340	10	x	x	X
ejpam-3807	340	11	]	]	X
ejpam-3807	340	12	]	]	PUNCT
ejpam-3807	340	13	.	.	PUNCT
ejpam-3807	341	1	this	this	PRON
ejpam-3807	341	2	proves	prove	VERB
ejpam-3807	341	3	the	the	DET
ejpam-3807	341	4	claim	claim	NOUN
ejpam-3807	341	5	.	.	PUNCT
ejpam-3807	342	1	by	by	ADP
ejpam-3807	342	2	the	the	DET
ejpam-3807	342	3	claim	claim	NOUN
ejpam-3807	342	4	,	,	PUNCT
ejpam-3807	342	5	w	w	PROPN
ejpam-3807	343	1	[	[	X
ejpam-3807	343	2	[	[	X
ejpam-3807	343	3	x	x	X
ejpam-3807	343	4	]	]	X
ejpam-3807	343	5	]	]	X
ejpam-3807	343	6	⊆	⊆	NUM
ejpam-3807	343	7	u	u	NOUN
ejpam-3807	344	1	[	[	X
ejpam-3807	344	2	[	[	X
ejpam-3807	344	3	x	x	X
ejpam-3807	344	4	]	]	X
ejpam-3807	344	5	]	]	X
ejpam-3807	344	6	⊆	⊆	NUM
ejpam-3807	344	7	g	g	NOUN
ejpam-3807	344	8	and	and	CCONJ
ejpam-3807	344	9	w	w	NOUN
ejpam-3807	345	1	[	[	X
ejpam-3807	345	2	[	[	X
ejpam-3807	345	3	x	x	X
ejpam-3807	345	4	]	]	X
ejpam-3807	345	5	]	]	X
ejpam-3807	345	6	⊆	⊆	NUM
ejpam-3807	345	7	v	v	ADP
ejpam-3807	345	8	[	[	X
ejpam-3807	345	9	[	[	X
ejpam-3807	345	10	x	x	X
ejpam-3807	345	11	]	]	X
ejpam-3807	345	12	]	]	X
ejpam-3807	345	13	⊆	⊆	NUM
ejpam-3807	345	14	h.	h.	NOUN
ejpam-3807	345	15	hence	hence	ADV
ejpam-3807	345	16	,	,	PUNCT
ejpam-3807	345	17	w	w	PROPN
ejpam-3807	346	1	[	[	X
ejpam-3807	346	2	[	[	X
ejpam-3807	346	3	x	x	X
ejpam-3807	346	4	]	]	X
ejpam-3807	346	5	]	]	X
ejpam-3807	346	6	⊆	⊆	NUM
ejpam-3807	346	7	g∩h	g∩h	NOUN
ejpam-3807	346	8	.	.	PUNCT
ejpam-3807	347	1	this	this	PRON
ejpam-3807	347	2	implies	imply	VERB
ejpam-3807	347	3	that	that	SCONJ
ejpam-3807	347	4	g	g	PROPN
ejpam-3807	347	5	∩h	∩h	PROPN
ejpam-3807	347	6	∈	∈	PROPN
ejpam-3807	347	7	τ	τ	X
ejpam-3807	347	8	.	.	PUNCT
ejpam-3807	348	1	therefore	therefore	ADV
ejpam-3807	348	2	,	,	PUNCT
ejpam-3807	348	3	τ	τ	PROPN
ejpam-3807	348	4	is	be	AUX
ejpam-3807	348	5	a	a	DET
ejpam-3807	348	6	b	b	NOUN
ejpam-3807	348	7	-	-	PUNCT
ejpam-3807	348	8	topology	topology	NOUN
ejpam-3807	348	9	on	on	ADP
ejpam-3807	348	10	x.	x.	NOUN
ejpam-3807	348	11	the	the	DET
ejpam-3807	348	12	next	next	ADJ
ejpam-3807	348	13	remark	remark	NOUN
ejpam-3807	348	14	follows	follow	VERB
ejpam-3807	348	15	from	from	ADP
ejpam-3807	348	16	definition	definition	NOUN
ejpam-3807	348	17	9	9	NUM
ejpam-3807	348	18	and	and	CCONJ
ejpam-3807	348	19	theorem	theorem	VERB
ejpam-3807	348	20	10	10	NUM
ejpam-3807	348	21	.	.	PUNCT
ejpam-3807	348	22	remark	remark	PROPN
ejpam-3807	348	23	7	7	NUM
ejpam-3807	348	24	.	.	PUNCT
ejpam-3807	348	25	suppose	suppose	VERB
ejpam-3807	348	26	x	x	PRON
ejpam-3807	348	27	is	be	AUX
ejpam-3807	348	28	a	a	DET
ejpam-3807	348	29	b	b	PROPN
ejpam-3807	348	30	-	-	PUNCT
ejpam-3807	348	31	topological	topological	ADJ
ejpam-3807	348	32	space	space	NOUN
ejpam-3807	348	33	.	.	PUNCT
ejpam-3807	349	1	references	reference	NOUN
ejpam-3807	349	2	838	838	NUM
ejpam-3807	349	3	(	(	PUNCT
ejpam-3807	349	4	i	i	NOUN
ejpam-3807	349	5	)	)	PUNCT
ejpam-3807	349	6	then	then	ADV
ejpam-3807	349	7	(	(	PUNCT
ejpam-3807	349	8	x	x	X
ejpam-3807	349	9	,	,	PUNCT
ejpam-3807	349	10	τ	τ	X
ejpam-3807	349	11	)	)	PUNCT
ejpam-3807	349	12	in	in	ADP
ejpam-3807	349	13	theorem	theorem	NOUN
ejpam-3807	349	14	10	10	NUM
ejpam-3807	349	15	is	be	AUX
ejpam-3807	349	16	a	a	DET
ejpam-3807	349	17	uniform	uniform	ADJ
ejpam-3807	349	18	b	b	X
ejpam-3807	349	19	-	-	PUNCT
ejpam-3807	349	20	topological	topological	ADJ
ejpam-3807	349	21	space	space	NOUN
ejpam-3807	349	22	.	.	PUNCT
ejpam-3807	350	1	(	(	PUNCT
ejpam-3807	350	2	ii	ii	NOUN
ejpam-3807	350	3	)	)	PUNCT
ejpam-3807	350	4	for	for	ADP
ejpam-3807	350	5	any	any	DET
ejpam-3807	350	6	ui	ui	PROPN
ejpam-3807	350	7	∈	∈	PROPN
ejpam-3807	350	8	k∗	k∗	NOUN
ejpam-3807	350	9	and	and	CCONJ
ejpam-3807	350	10	x	x	SYM
ejpam-3807	350	11	∈	∈	PROPN
ejpam-3807	350	12	x	x	NOUN
ejpam-3807	350	13	,	,	PUNCT
ejpam-3807	350	14	x	x	SYM
ejpam-3807	350	15	∈	∈	NOUN
ejpam-3807	350	16	ui	ui	NOUN
ejpam-3807	351	1	[	[	X
ejpam-3807	351	2	[	[	X
ejpam-3807	351	3	x	x	X
ejpam-3807	351	4	]	]	X
ejpam-3807	351	5	]	]	PUNCT
ejpam-3807	351	6	and	and	CCONJ
ejpam-3807	351	7	ui	ui	X
ejpam-3807	352	1	[	[	X
ejpam-3807	352	2	[	[	X
ejpam-3807	352	3	x	x	X
ejpam-3807	352	4	]	]	X
ejpam-3807	352	5	]	]	X
ejpam-3807	352	6	∈	∈	PROPN
ejpam-3807	352	7	τ	τ	X
ejpam-3807	352	8	,	,	PUNCT
ejpam-3807	352	9	that	that	ADV
ejpam-3807	352	10	is	is	ADV
ejpam-3807	352	11	,	,	PUNCT
ejpam-3807	352	12	ui	ui	PROPN
ejpam-3807	353	1	[	[	X
ejpam-3807	353	2	[	[	X
ejpam-3807	353	3	x	x	X
ejpam-3807	353	4	]	]	X
ejpam-3807	353	5	]	]	X
ejpam-3807	353	6	is	be	AUX
ejpam-3807	353	7	a	a	DET
ejpam-3807	353	8	neighborhood	neighborhood	NOUN
ejpam-3807	353	9	of	of	ADP
ejpam-3807	353	10	x.	x.	NOUN
ejpam-3807	353	11	lemma	lemma	PROPN
ejpam-3807	353	12	4	4	X
ejpam-3807	353	13	.	.	PUNCT
ejpam-3807	353	14	suppose	suppose	VERB
ejpam-3807	353	15	x	x	PRON
ejpam-3807	353	16	is	be	AUX
ejpam-3807	353	17	a	a	DET
ejpam-3807	353	18	b	b	NOUN
ejpam-3807	353	19	-	-	PUNCT
ejpam-3807	353	20	algebra	algebra	NOUN
ejpam-3807	353	21	such	such	ADJ
ejpam-3807	353	22	that	that	SCONJ
ejpam-3807	353	23	u	u	PROPN
ejpam-3807	353	24	⊆	⊆	NUM
ejpam-3807	353	25	v	v	NOUN
ejpam-3807	353	26	for	for	ADP
ejpam-3807	353	27	any	any	DET
ejpam-3807	353	28	u	u	NOUN
ejpam-3807	353	29	,	,	PUNCT
ejpam-3807	353	30	v	v	PROPN
ejpam-3807	353	31	∈	∈	PROPN
ejpam-3807	354	1	k.	k.	NOUN
ejpam-3807	354	2	then	then	ADV
ejpam-3807	354	3	u	u	VERB
ejpam-3807	355	1	[	[	X
ejpam-3807	355	2	[	[	X
ejpam-3807	355	3	x	x	X
ejpam-3807	355	4	]	]	X
ejpam-3807	355	5	]	]	X
ejpam-3807	355	6	⊆	⊆	NUM
ejpam-3807	355	7	v	v	ADP
ejpam-3807	355	8	[	[	X
ejpam-3807	355	9	[	[	X
ejpam-3807	355	10	x	x	X
ejpam-3807	355	11	]	]	X
ejpam-3807	355	12	]	]	X
ejpam-3807	355	13	for	for	ADP
ejpam-3807	355	14	all	all	DET
ejpam-3807	355	15	x	x	SYM
ejpam-3807	355	16	∈	∈	ADJ
ejpam-3807	355	17	x.	x.	NOUN
ejpam-3807	355	18	proof	proof	NOUN
ejpam-3807	355	19	.	.	PUNCT
ejpam-3807	356	1	let	let	VERB
ejpam-3807	356	2	u	u	PRON
ejpam-3807	356	3	⊆	⊆	NUM
ejpam-3807	356	4	v	v	NOUN
ejpam-3807	356	5	for	for	ADP
ejpam-3807	356	6	any	any	DET
ejpam-3807	356	7	u	u	NOUN
ejpam-3807	356	8	,	,	PUNCT
ejpam-3807	356	9	v	v	ADP
ejpam-3807	356	10	∈	∈	PROPN
ejpam-3807	356	11	k	k	NOUN
ejpam-3807	356	12	and	and	CCONJ
ejpam-3807	356	13	x	x	SYM
ejpam-3807	356	14	∈	∈	PROPN
ejpam-3807	356	15	x.	x.	NOUN
ejpam-3807	356	16	suppose	suppose	VERB
ejpam-3807	356	17	a	a	DET
ejpam-3807	356	18	∈	∈	PROPN
ejpam-3807	356	19	u	u	NOUN
ejpam-3807	357	1	[	[	X
ejpam-3807	357	2	[	[	X
ejpam-3807	357	3	x	x	X
ejpam-3807	357	4	]	]	X
ejpam-3807	357	5	]	]	PUNCT
ejpam-3807	357	6	.	.	PUNCT
ejpam-3807	358	1	then	then	ADV
ejpam-3807	358	2	(	(	PUNCT
ejpam-3807	358	3	x	x	X
ejpam-3807	358	4	,	,	PUNCT
ejpam-3807	358	5	a	a	PRON
ejpam-3807	358	6	)	)	PUNCT
ejpam-3807	358	7	∈	∈	NOUN
ejpam-3807	358	8	u	u	NOUN
ejpam-3807	358	9	⊆	⊆	NUM
ejpam-3807	358	10	v	v	NOUN
ejpam-3807	358	11	.	.	PUNCT
ejpam-3807	359	1	this	this	PRON
ejpam-3807	359	2	implies	imply	VERB
ejpam-3807	359	3	that	that	SCONJ
ejpam-3807	359	4	(	(	PUNCT
ejpam-3807	359	5	x	x	NOUN
ejpam-3807	359	6	,	,	PUNCT
ejpam-3807	359	7	a	a	DET
ejpam-3807	359	8	)	)	PUNCT
ejpam-3807	359	9	∈	∈	NOUN
ejpam-3807	359	10	v	v	NOUN
ejpam-3807	359	11	.	.	PUNCT
ejpam-3807	360	1	therefore	therefore	ADV
ejpam-3807	360	2	,	,	PUNCT
ejpam-3807	360	3	a	a	DET
ejpam-3807	360	4	∈	∈	NOUN
ejpam-3807	360	5	v	v	ADP
ejpam-3807	360	6	[	[	X
ejpam-3807	360	7	[	[	X
ejpam-3807	360	8	x	x	X
ejpam-3807	360	9	]	]	X
ejpam-3807	360	10	]	]	PUNCT
ejpam-3807	360	11	.	.	PUNCT
ejpam-3807	361	1	theorem	theorem	NOUN
ejpam-3807	361	2	11	11	NUM
ejpam-3807	361	3	.	.	PUNCT
ejpam-3807	362	1	suppose	suppose	VERB
ejpam-3807	362	2	x	x	PRON
ejpam-3807	362	3	is	be	AUX
ejpam-3807	362	4	a	a	DET
ejpam-3807	362	5	uniform	uniform	ADJ
ejpam-3807	362	6	b	b	X
ejpam-3807	362	7	-	-	PUNCT
ejpam-3807	362	8	topological	topological	ADJ
ejpam-3807	362	9	space	space	NOUN
ejpam-3807	362	10	.	.	PUNCT
ejpam-3807	363	1	then	then	ADV
ejpam-3807	363	2	x	x	X
ejpam-3807	363	3	is	be	AUX
ejpam-3807	363	4	a	a	DET
ejpam-3807	363	5	topological	topological	ADJ
ejpam-3807	363	6	b	b	NOUN
ejpam-3807	363	7	-	-	PUNCT
ejpam-3807	363	8	algebra	algebra	NOUN
ejpam-3807	363	9	.	.	PUNCT
ejpam-3807	364	1	proof	proof	NOUN
ejpam-3807	364	2	.	.	PUNCT
ejpam-3807	365	1	let	let	VERB
ejpam-3807	365	2	(	(	PUNCT
ejpam-3807	365	3	x	x	NOUN
ejpam-3807	365	4	,	,	PUNCT
ejpam-3807	365	5	k	k	NOUN
ejpam-3807	365	6	)	)	PUNCT
ejpam-3807	365	7	be	be	VERB
ejpam-3807	365	8	a	a	DET
ejpam-3807	365	9	uniform	uniform	ADJ
ejpam-3807	365	10	structure	structure	NOUN
ejpam-3807	365	11	.	.	PUNCT
ejpam-3807	366	1	by	by	ADP
ejpam-3807	366	2	theorem	theorem	ADJ
ejpam-3807	366	3	10	10	NUM
ejpam-3807	366	4	and	and	CCONJ
ejpam-3807	366	5	remark	remark	NOUN
ejpam-3807	366	6	7(i	7(i	NUM
ejpam-3807	366	7	)	)	PUNCT
ejpam-3807	366	8	,	,	PUNCT
ejpam-3807	366	9	there	there	PRON
ejpam-3807	366	10	is	be	VERB
ejpam-3807	366	11	a	a	DET
ejpam-3807	366	12	uniform	uniform	ADJ
ejpam-3807	366	13	b	b	NOUN
ejpam-3807	366	14	-	-	PUNCT
ejpam-3807	366	15	topology	topology	NOUN
ejpam-3807	366	16	τ	τ	NOUN
ejpam-3807	366	17	=	=	PUNCT
ejpam-3807	366	18	{	{	PUNCT
ejpam-3807	366	19	g	g	PROPN
ejpam-3807	366	20	⊆	⊆	NUM
ejpam-3807	366	21	x|∀x	x|∀x	PROPN
ejpam-3807	366	22	∈	∈	PROPN
ejpam-3807	366	23	g,∃u	g,∃u	NOUN
ejpam-3807	366	24	∈	∈	PROPN
ejpam-3807	366	25	k	k	PROPN
ejpam-3807	366	26	,	,	PUNCT
ejpam-3807	366	27	u	u	X
ejpam-3807	367	1	[	[	X
ejpam-3807	367	2	[	[	X
ejpam-3807	367	3	x	x	X
ejpam-3807	367	4	]	]	X
ejpam-3807	367	5	]	]	X
ejpam-3807	367	6	⊆	⊆	NUM
ejpam-3807	367	7	g	g	NOUN
ejpam-3807	367	8	}	}	PUNCT
ejpam-3807	367	9	.	.	PUNCT
ejpam-3807	368	1	suppose	suppose	VERB
ejpam-3807	368	2	x	x	X
ejpam-3807	368	3	∗	∗	VERB
ejpam-3807	368	4	y	y	PROPN
ejpam-3807	368	5	∈	∈	PROPN
ejpam-3807	368	6	u(x	u(x	PROPN
ejpam-3807	368	7	∗	∗	NOUN
ejpam-3807	368	8	y	y	NOUN
ejpam-3807	368	9	)	)	PUNCT
ejpam-3807	368	10	where	where	SCONJ
ejpam-3807	368	11	x	x	X
ejpam-3807	368	12	,	,	PUNCT
ejpam-3807	368	13	y	y	PROPN
ejpam-3807	368	14	∈	∈	PROPN
ejpam-3807	368	15	x.	x.	NOUN
ejpam-3807	368	16	by	by	ADP
ejpam-3807	368	17	theorem	theorem	NOUN
ejpam-3807	368	18	10	10	NUM
ejpam-3807	368	19	,	,	PUNCT
ejpam-3807	368	20	there	there	PRON
ejpam-3807	368	21	exists	exist	VERB
ejpam-3807	368	22	g	g	PROPN
ejpam-3807	368	23	∈	∈	PROPN
ejpam-3807	368	24	k	k	PROPN
ejpam-3807	368	25	such	such	ADJ
ejpam-3807	368	26	that	that	SCONJ
ejpam-3807	368	27	g[[x	g[[x	PROPN
ejpam-3807	368	28	∗	∗	VERB
ejpam-3807	368	29	y	y	NOUN
ejpam-3807	368	30	]	]	X
ejpam-3807	368	31	]	]	X
ejpam-3807	368	32	⊆	⊆	NUM
ejpam-3807	368	33	u(x	u(x	PROPN
ejpam-3807	368	34	∗	∗	NOUN
ejpam-3807	368	35	y	y	NOUN
ejpam-3807	368	36	)	)	PUNCT
ejpam-3807	368	37	.	.	PUNCT
ejpam-3807	369	1	then	then	ADV
ejpam-3807	369	2	there	there	PRON
ejpam-3807	369	3	exists	exist	VERB
ejpam-3807	369	4	gi	gi	VERB
ejpam-3807	369	5	∈	∈	PROPN
ejpam-3807	369	6	k	k	NOUN
ejpam-3807	369	7	?	?	PUNCT
ejpam-3807	370	1	such	such	ADJ
ejpam-3807	370	2	that	that	PRON
ejpam-3807	370	3	gi	gi	VERB
ejpam-3807	370	4	⊆	⊆	NUM
ejpam-3807	370	5	g	g	NOUN
ejpam-3807	370	6	for	for	ADP
ejpam-3807	370	7	some	some	DET
ejpam-3807	370	8	b	b	NOUN
ejpam-3807	370	9	-	-	PUNCT
ejpam-3807	370	10	ideal	ideal	NOUN
ejpam-3807	370	11	i	i	PRON
ejpam-3807	370	12	of	of	ADP
ejpam-3807	370	13	x.	x.	NOUN
ejpam-3807	370	14	by	by	ADP
ejpam-3807	370	15	lemma	lemma	PROPN
ejpam-3807	370	16	4	4	NUM
ejpam-3807	370	17	,	,	PUNCT
ejpam-3807	370	18	gi	gi	X
ejpam-3807	371	1	[	[	X
ejpam-3807	371	2	[	[	X
ejpam-3807	371	3	x	x	X
ejpam-3807	371	4	∗	∗	X
ejpam-3807	371	5	y	y	NOUN
ejpam-3807	371	6	]	]	X
ejpam-3807	371	7	]	]	PUNCT
ejpam-3807	371	8	⊆	⊆	NUM
ejpam-3807	371	9	g[[x	g[[x	PROPN
ejpam-3807	371	10	∗	∗	VERB
ejpam-3807	371	11	y	y	NOUN
ejpam-3807	371	12	]	]	X
ejpam-3807	371	13	]	]	PUNCT
ejpam-3807	371	14	.	.	PUNCT
ejpam-3807	372	1	note	note	VERB
ejpam-3807	372	2	that	that	SCONJ
ejpam-3807	372	3	gi	gi	VERB
ejpam-3807	373	1	[	[	X
ejpam-3807	373	2	[	[	X
ejpam-3807	373	3	x	x	X
ejpam-3807	373	4	]	]	X
ejpam-3807	373	5	]	]	PUNCT
ejpam-3807	373	6	and	and	CCONJ
ejpam-3807	373	7	gi	gi	X
ejpam-3807	374	1	[	[	X
ejpam-3807	374	2	[	[	X
ejpam-3807	374	3	y	y	X
ejpam-3807	374	4	]	]	X
ejpam-3807	374	5	]	]	X
ejpam-3807	374	6	are	be	AUX
ejpam-3807	374	7	open	open	ADJ
ejpam-3807	374	8	neighborhoods	neighborhood	NOUN
ejpam-3807	374	9	of	of	ADP
ejpam-3807	374	10	x	x	X
ejpam-3807	374	11	and	and	CCONJ
ejpam-3807	374	12	y	y	PROPN
ejpam-3807	374	13	,	,	PUNCT
ejpam-3807	374	14	respectively	respectively	ADV
ejpam-3807	374	15	by	by	ADP
ejpam-3807	374	16	remark	remark	NOUN
ejpam-3807	374	17	7(ii	7(ii	PROPN
ejpam-3807	374	18	)	)	PUNCT
ejpam-3807	374	19	.	.	PUNCT
ejpam-3807	375	1	claim	claim	NOUN
ejpam-3807	375	2	:	:	PUNCT
ejpam-3807	376	1	gi	gi	X
ejpam-3807	377	1	[	[	X
ejpam-3807	377	2	[	[	X
ejpam-3807	377	3	x	x	X
ejpam-3807	377	4	]	]	X
ejpam-3807	377	5	]	]	X
ejpam-3807	377	6	∗gi	∗gi	PUNCT
ejpam-3807	378	1	[	[	X
ejpam-3807	378	2	[	[	X
ejpam-3807	378	3	y	y	X
ejpam-3807	378	4	]	]	X
ejpam-3807	378	5	]	]	X
ejpam-3807	378	6	⊆	⊆	NUM
ejpam-3807	378	7	gi	gi	NOUN
ejpam-3807	379	1	[	[	X
ejpam-3807	379	2	[	[	X
ejpam-3807	379	3	x	x	X
ejpam-3807	379	4	∗	∗	X
ejpam-3807	379	5	y	y	NOUN
ejpam-3807	379	6	]	]	X
ejpam-3807	379	7	]	]	PUNCT
ejpam-3807	379	8	.	.	PUNCT
ejpam-3807	379	9	suppose	suppose	VERB
ejpam-3807	379	10	a	a	DET
ejpam-3807	379	11	∗	∗	NOUN
ejpam-3807	379	12	b	b	NOUN
ejpam-3807	379	13	∈	∈	NOUN
ejpam-3807	379	14	gi	gi	X
ejpam-3807	380	1	[	[	X
ejpam-3807	380	2	[	[	X
ejpam-3807	380	3	x	x	X
ejpam-3807	380	4	]	]	X
ejpam-3807	380	5	]	]	X
ejpam-3807	380	6	∗	∗	NOUN
ejpam-3807	380	7	gi	gi	X
ejpam-3807	381	1	[	[	X
ejpam-3807	381	2	[	[	X
ejpam-3807	381	3	y	y	X
ejpam-3807	381	4	]	]	X
ejpam-3807	381	5	]	]	PUNCT
ejpam-3807	381	6	.	.	PUNCT
ejpam-3807	382	1	then	then	ADV
ejpam-3807	382	2	(	(	PUNCT
ejpam-3807	382	3	x	x	X
ejpam-3807	382	4	,	,	PUNCT
ejpam-3807	382	5	a	a	PRON
ejpam-3807	382	6	)	)	PUNCT
ejpam-3807	382	7	,	,	PUNCT
ejpam-3807	382	8	(	(	PUNCT
ejpam-3807	382	9	y	y	PROPN
ejpam-3807	382	10	,	,	PUNCT
ejpam-3807	382	11	b	b	NOUN
ejpam-3807	382	12	)	)	PUNCT
ejpam-3807	382	13	∈	∈	NOUN
ejpam-3807	382	14	gi	gi	INTJ
ejpam-3807	382	15	.	.	PUNCT
ejpam-3807	383	1	this	this	PRON
ejpam-3807	383	2	implies	imply	VERB
ejpam-3807	383	3	that	that	SCONJ
ejpam-3807	383	4	x	x	SYM
ejpam-3807	383	5	∼=i	∼=i	NOUN
ejpam-3807	383	6	a	a	NOUN
ejpam-3807	383	7	and	and	CCONJ
ejpam-3807	383	8	y	y	PROPN
ejpam-3807	383	9	∼=i	∼=i	PROPN
ejpam-3807	383	10	b.	b.	PROPN
ejpam-3807	383	11	hence	hence	ADV
ejpam-3807	383	12	,	,	PUNCT
ejpam-3807	383	13	x	x	PROPN
ejpam-3807	383	14	∗	∗	NOUN
ejpam-3807	383	15	y	y	PROPN
ejpam-3807	383	16	∼=i	∼=i	PROPN
ejpam-3807	383	17	a	a	DET
ejpam-3807	383	18	∗	∗	X
ejpam-3807	383	19	b.	b.	NOUN
ejpam-3807	384	1	it	it	PRON
ejpam-3807	384	2	follows	follow	VERB
ejpam-3807	384	3	that	that	SCONJ
ejpam-3807	384	4	a	a	DET
ejpam-3807	384	5	∗	∗	NOUN
ejpam-3807	384	6	b	b	NOUN
ejpam-3807	384	7	∈	∈	NOUN
ejpam-3807	384	8	gi	gi	NOUN
ejpam-3807	385	1	[	[	X
ejpam-3807	385	2	[	[	X
ejpam-3807	385	3	x	x	X
ejpam-3807	385	4	∗	∗	X
ejpam-3807	385	5	y	y	NOUN
ejpam-3807	385	6	]	]	X
ejpam-3807	385	7	]	]	PUNCT
ejpam-3807	385	8	.	.	PUNCT
ejpam-3807	386	1	this	this	PRON
ejpam-3807	386	2	proves	prove	VERB
ejpam-3807	386	3	the	the	DET
ejpam-3807	386	4	claim	claim	NOUN
ejpam-3807	386	5	.	.	PUNCT
ejpam-3807	387	1	hence	hence	ADV
ejpam-3807	387	2	,	,	PUNCT
ejpam-3807	387	3	gi	gi	X
ejpam-3807	388	1	[	[	X
ejpam-3807	388	2	[	[	X
ejpam-3807	388	3	x	x	X
ejpam-3807	388	4	]	]	X
ejpam-3807	388	5	]	]	X
ejpam-3807	388	6	∗gi	∗gi	PUNCT
ejpam-3807	389	1	[	[	X
ejpam-3807	389	2	[	[	X
ejpam-3807	389	3	y	y	X
ejpam-3807	389	4	]	]	X
ejpam-3807	389	5	]	]	X
ejpam-3807	389	6	⊆	⊆	NUM
ejpam-3807	389	7	u(x	u(x	PROPN
ejpam-3807	389	8	∗	∗	NOUN
ejpam-3807	389	9	y	y	NOUN
ejpam-3807	389	10	)	)	PUNCT
ejpam-3807	389	11	.	.	PUNCT
ejpam-3807	390	1	by	by	ADP
ejpam-3807	390	2	theorem	theorem	NOUN
ejpam-3807	390	3	2	2	NUM
ejpam-3807	390	4	,	,	PUNCT
ejpam-3807	390	5	x	x	X
ejpam-3807	390	6	is	be	AUX
ejpam-3807	390	7	a	a	DET
ejpam-3807	390	8	topological	topological	ADJ
ejpam-3807	390	9	b	b	NOUN
ejpam-3807	390	10	-	-	PUNCT
ejpam-3807	390	11	algebra	algebra	NOUN
ejpam-3807	390	12	.	.	PUNCT
ejpam-3807	391	1	the	the	DET
ejpam-3807	391	2	converse	converse	NOUN
ejpam-3807	391	3	of	of	ADP
ejpam-3807	391	4	theorem	theorem	ADJ
ejpam-3807	391	5	11	11	NUM
ejpam-3807	391	6	follows	follow	VERB
ejpam-3807	391	7	directly	directly	ADV
ejpam-3807	391	8	from	from	ADP
ejpam-3807	391	9	definition	definition	NOUN
ejpam-3807	391	10	9	9	NUM
ejpam-3807	391	11	provided	provide	VERB
ejpam-3807	391	12	that	that	SCONJ
ejpam-3807	391	13	the	the	DET
ejpam-3807	391	14	btopology	btopology	NOUN
ejpam-3807	391	15	is	be	AUX
ejpam-3807	391	16	a	a	DET
ejpam-3807	391	17	uniform	uniform	ADJ
ejpam-3807	391	18	b	b	NOUN
ejpam-3807	391	19	-	-	NOUN
ejpam-3807	391	20	topology	topology	NOUN
ejpam-3807	391	21	.	.	PUNCT
ejpam-3807	392	1	this	this	PRON
ejpam-3807	392	2	is	be	AUX
ejpam-3807	392	3	formally	formally	ADV
ejpam-3807	392	4	stated	state	VERB
ejpam-3807	392	5	in	in	ADP
ejpam-3807	392	6	the	the	DET
ejpam-3807	392	7	next	next	ADJ
ejpam-3807	392	8	corollary	corollary	NOUN
ejpam-3807	392	9	.	.	PUNCT
ejpam-3807	393	1	corollary	corollary	ADJ
ejpam-3807	393	2	3	3	NUM
ejpam-3807	393	3	.	.	PUNCT
ejpam-3807	393	4	suppose	suppose	VERB
ejpam-3807	393	5	x	x	PRON
ejpam-3807	393	6	is	be	AUX
ejpam-3807	393	7	a	a	DET
ejpam-3807	393	8	topological	topological	ADJ
ejpam-3807	393	9	b	b	NOUN
ejpam-3807	393	10	-	-	PUNCT
ejpam-3807	393	11	algebra	algebra	NOUN
ejpam-3807	393	12	.	.	PUNCT
ejpam-3807	394	1	if	if	SCONJ
ejpam-3807	394	2	τ	τ	PROPN
ejpam-3807	394	3	is	be	AUX
ejpam-3807	394	4	a	a	DET
ejpam-3807	394	5	uniform	uniform	ADJ
ejpam-3807	394	6	b	b	NOUN
ejpam-3807	394	7	-	-	PUNCT
ejpam-3807	394	8	topology	topology	NOUN
ejpam-3807	394	9	,	,	PUNCT
ejpam-3807	394	10	then	then	ADV
ejpam-3807	394	11	x	x	PUNCT
ejpam-3807	394	12	is	be	AUX
ejpam-3807	394	13	a	a	DET
ejpam-3807	394	14	uniform	uniform	ADJ
ejpam-3807	394	15	b	b	X
ejpam-3807	394	16	-	-	PUNCT
ejpam-3807	394	17	topological	topological	ADJ
ejpam-3807	394	18	space	space	NOUN
ejpam-3807	394	19	.	.	PUNCT
ejpam-3807	395	1	acknowledgements	acknowledgement	NOUN
ejpam-3807	395	2	this	this	DET
ejpam-3807	395	3	research	research	NOUN
ejpam-3807	395	4	is	be	AUX
ejpam-3807	395	5	funded	fund	VERB
ejpam-3807	395	6	by	by	ADP
ejpam-3807	395	7	the	the	DET
ejpam-3807	395	8	commission	commission	NOUN
ejpam-3807	395	9	on	on	ADP
ejpam-3807	395	10	higher	high	ADJ
ejpam-3807	395	11	education	education	NOUN
ejpam-3807	395	12	(	(	PUNCT
ejpam-3807	395	13	ched	che	VERB
ejpam-3807	395	14	)	)	PUNCT
ejpam-3807	395	15	and	and	CCONJ
ejpam-3807	395	16	mindanao	mindanao	PROPN
ejpam-3807	395	17	state	state	PROPN
ejpam-3807	395	18	university	university	PROPN
ejpam-3807	395	19	-	-	PUNCT
ejpam-3807	395	20	iligan	iligan	PROPN
ejpam-3807	395	21	institute	institute	PROPN
ejpam-3807	395	22	of	of	ADP
ejpam-3807	395	23	technology	technology	PROPN
ejpam-3807	395	24	,	,	PUNCT
ejpam-3807	395	25	philippines	philippine	NOUN
ejpam-3807	395	26	.	.	PUNCT
ejpam-3807	396	1	references	reference	NOUN
ejpam-3807	396	2	[	[	X
ejpam-3807	396	3	1	1	X
ejpam-3807	396	4	]	]	PUNCT
ejpam-3807	396	5	d.	d.	PROPN
ejpam-3807	396	6	al	al	PROPN
ejpam-3807	396	7	-	-	PUNCT
ejpam-3807	396	8	kadi	kadi	PROPN
ejpam-3807	396	9	.	.	PUNCT
ejpam-3807	397	1	anti	anti	ADJ
ejpam-3807	397	2	fuzzy	fuzzy	ADJ
ejpam-3807	397	3	ideals	ideal	NOUN
ejpam-3807	397	4	of	of	ADP
ejpam-3807	397	5	b	b	NOUN
ejpam-3807	397	6	-	-	PUNCT
ejpam-3807	397	7	algebra	algebra	NOUN
ejpam-3807	397	8	.	.	PUNCT
ejpam-3807	398	1	international	international	ADJ
ejpam-3807	398	2	journal	journal	NOUN
ejpam-3807	398	3	of	of	ADP
ejpam-3807	398	4	pure	pure	ADJ
ejpam-3807	398	5	and	and	CCONJ
ejpam-3807	398	6	applied	applied	ADJ
ejpam-3807	398	7	mathematics	mathematic	NOUN
ejpam-3807	398	8	,	,	PUNCT
ejpam-3807	398	9	117(3):437–445	117(3):437–445	NUM
ejpam-3807	398	10	,	,	PUNCT
ejpam-3807	398	11	2017	2017	NUM
ejpam-3807	398	12	.	.	PUNCT
ejpam-3807	399	1	[	[	X
ejpam-3807	399	2	2	2	X
ejpam-3807	399	3	]	]	PUNCT
ejpam-3807	399	4	j.	j.	PROPN
ejpam-3807	399	5	dugunji	dugunji	PROPN
ejpam-3807	399	6	.	.	PUNCT
ejpam-3807	399	7	topology	topology	PROPN
ejpam-3807	399	8	.	.	PUNCT
ejpam-3807	400	1	allyn	allyn	PROPN
ejpam-3807	400	2	and	and	CCONJ
ejpam-3807	400	3	bacon	bacon	PROPN
ejpam-3807	400	4	,	,	PUNCT
ejpam-3807	400	5	inc	inc	PROPN
ejpam-3807	400	6	.	.	PROPN
ejpam-3807	400	7	,	,	PUNCT
ejpam-3807	400	8	boston	boston	PROPN
ejpam-3807	400	9	,	,	PUNCT
ejpam-3807	400	10	1996	1996	NUM
ejpam-3807	400	11	.	.	PUNCT
ejpam-3807	401	1	[	[	X
ejpam-3807	401	2	3	3	X
ejpam-3807	401	3	]	]	X
ejpam-3807	401	4	j.	j.	PROPN
ejpam-3807	401	5	endam	endam	PROPN
ejpam-3807	401	6	and	and	CCONJ
ejpam-3807	401	7	j.	j.	PROPN
ejpam-3807	401	8	vilela	vilela	PROPN
ejpam-3807	401	9	.	.	PUNCT
ejpam-3807	402	1	the	the	DET
ejpam-3807	402	2	second	second	ADJ
ejpam-3807	402	3	isomorphism	isomorphism	NOUN
ejpam-3807	402	4	theorem	theorem	NOUN
ejpam-3807	402	5	for	for	ADP
ejpam-3807	402	6	b	b	NOUN
ejpam-3807	402	7	-	-	PUNCT
ejpam-3807	402	8	algebras	algebras	PROPN
ejpam-3807	402	9	.	.	PUNCT
ejpam-3807	403	1	applied	apply	VERB
ejpam-3807	403	2	mathematical	mathematical	ADJ
ejpam-3807	403	3	sciences	science	NOUN
ejpam-3807	403	4	,	,	PUNCT
ejpam-3807	403	5	8(38):1865–1872	8(38):1865–1872	NUM
ejpam-3807	403	6	,	,	PUNCT
ejpam-3807	403	7	2014	2014	NUM
ejpam-3807	403	8	.	.	PUNCT
ejpam-3807	404	1	references	reference	NOUN
ejpam-3807	404	2	839	839	NUM
ejpam-3807	404	3	[	[	X
ejpam-3807	404	4	4	4	NUM
ejpam-3807	404	5	]	]	X
ejpam-3807	404	6	y.b	y.b	PROPN
ejpam-3807	404	7	.	.	PROPN
ejpam-3807	404	8	jun	jun	PROPN
ejpam-3807	404	9	et	et	PROPN
ejpam-3807	404	10	al	al	PROPN
ejpam-3807	404	11	.	.	PROPN
ejpam-3807	405	1	on	on	ADP
ejpam-3807	405	2	topological	topological	ADJ
ejpam-3807	405	3	bci	bci	NOUN
ejpam-3807	405	4	-	-	PUNCT
ejpam-3807	405	5	algebras	algebra	NOUN
ejpam-3807	405	6	.	.	PUNCT
ejpam-3807	406	1	information	information	NOUN
ejpam-3807	406	2	sciences	sciences	PROPN
ejpam-3807	406	3	,	,	PUNCT
ejpam-3807	406	4	116:253–261	116:253–261	NUM
ejpam-3807	406	5	,	,	PUNCT
ejpam-3807	406	6	1999	1999	NUM
ejpam-3807	406	7	.	.	PUNCT
ejpam-3807	407	1	[	[	X
ejpam-3807	407	2	5	5	NUM
ejpam-3807	407	3	]	]	X
ejpam-3807	407	4	k.d	k.d	PROPN
ejpam-3807	407	5	.	.	PUNCT
ejpam-3807	407	6	joshi	joshi	PROPN
ejpam-3807	407	7	.	.	PUNCT
ejpam-3807	408	1	introduction	introduction	NOUN
ejpam-3807	408	2	to	to	ADP
ejpam-3807	408	3	general	general	ADJ
ejpam-3807	408	4	topology	topology	NOUN
ejpam-3807	408	5	.	.	PUNCT
ejpam-3807	409	1	new	new	ADJ
ejpam-3807	409	2	age	age	NOUN
ejpam-3807	409	3	international	international	ADJ
ejpam-3807	409	4	limited	limited	ADJ
ejpam-3807	409	5	publishers	publisher	NOUN
ejpam-3807	409	6	,	,	PUNCT
ejpam-3807	409	7	india	india	PROPN
ejpam-3807	409	8	,	,	PUNCT
ejpam-3807	409	9	1997	1997	NUM
ejpam-3807	409	10	.	.	PUNCT
ejpam-3807	410	1	[	[	X
ejpam-3807	410	2	6	6	NUM
ejpam-3807	410	3	]	]	X
ejpam-3807	410	4	n.c	n.c	PROPN
ejpam-3807	410	5	.	.	PROPN
ejpam-3807	410	6	gonzaga	gonzaga	PROPN
ejpam-3807	410	7	jr	jr	PROPN
ejpam-3807	410	8	.	.	PROPN
ejpam-3807	410	9	analyzing	analyze	VERB
ejpam-3807	410	10	some	some	DET
ejpam-3807	410	11	structural	structural	ADJ
ejpam-3807	410	12	properties	property	NOUN
ejpam-3807	410	13	of	of	ADP
ejpam-3807	410	14	topological	topological	ADJ
ejpam-3807	410	15	b	b	NOUN
ejpam-3807	410	16	-	-	PUNCT
ejpam-3807	410	17	algebra	algebra	NOUN
ejpam-3807	410	18	.	.	PUNCT
ejpam-3807	411	1	international	international	ADJ
ejpam-3807	411	2	journal	journal	PROPN
ejpam-3807	411	3	of	of	ADP
ejpam-3807	411	4	mathematics	mathematics	PROPN
ejpam-3807	411	5	and	and	CCONJ
ejpam-3807	411	6	mathematical	mathematical	ADJ
ejpam-3807	411	7	sciences	science	NOUN
ejpam-3807	411	8	,	,	PUNCT
ejpam-3807	411	9	2019:1–7	2019:1–7	NUM
ejpam-3807	411	10	,	,	PUNCT
ejpam-3807	411	11	2019	2019	NUM
ejpam-3807	411	12	.	.	PUNCT
ejpam-3807	412	1	[	[	X
ejpam-3807	412	2	7	7	X
ejpam-3807	412	3	]	]	X
ejpam-3807	412	4	s.	s.	PROPN
ejpam-3807	412	5	mehrshad	mehrshad	VERB
ejpam-3807	412	6	and	and	CCONJ
ejpam-3807	412	7	j.	j.	PROPN
ejpam-3807	412	8	golzarpoor	golzarpoor	PROPN
ejpam-3807	412	9	.	.	PUNCT
ejpam-3807	413	1	on	on	ADP
ejpam-3807	413	2	topological	topological	ADJ
ejpam-3807	413	3	be	be	NOUN
ejpam-3807	413	4	-	-	PUNCT
ejpam-3807	413	5	algebras	algebra	NOUN
ejpam-3807	413	6	.	.	PUNCT
ejpam-3807	414	1	mathematica	mathematica	PROPN
ejpam-3807	414	2	moravica	moravica	PROPN
ejpam-3807	414	3	,	,	PUNCT
ejpam-3807	414	4	21(2):1–13	21(2):1–13	NUM
ejpam-3807	414	5	,	,	PUNCT
ejpam-3807	414	6	2017	2017	NUM
ejpam-3807	414	7	.	.	PUNCT
ejpam-3807	415	1	[	[	X
ejpam-3807	415	2	8	8	X
ejpam-3807	415	3	]	]	X
ejpam-3807	415	4	j.	j.	PROPN
ejpam-3807	415	5	neggers	neggers	PROPN
ejpam-3807	415	6	and	and	CCONJ
ejpam-3807	415	7	h.	h.	PROPN
ejpam-3807	415	8	kim	kim	PROPN
ejpam-3807	415	9	.	.	PUNCT
ejpam-3807	416	1	a	a	DET
ejpam-3807	416	2	fundamental	fundamental	ADJ
ejpam-3807	416	3	theorem	theorem	NOUN
ejpam-3807	416	4	of	of	ADP
ejpam-3807	416	5	b	b	NOUN
ejpam-3807	416	6	-	-	PUNCT
ejpam-3807	416	7	homomorphism	homomorphism	NOUN
ejpam-3807	416	8	for	for	ADP
ejpam-3807	416	9	b	b	NOUN
ejpam-3807	416	10	-	-	PUNCT
ejpam-3807	416	11	algebras	algebras	PROPN
ejpam-3807	416	12	.	.	PUNCT
ejpam-3807	417	1	inter.math.j	inter.math.j	PROPN
ejpam-3807	417	2	.	.	PROPN
ejpam-3807	417	3	,	,	PUNCT
ejpam-3807	417	4	2:207–214	2:207–214	NUM
ejpam-3807	417	5	,	,	PUNCT
ejpam-3807	417	6	2002	2002	NUM
ejpam-3807	417	7	.	.	PUNCT
ejpam-3807	418	1	[	[	X
ejpam-3807	418	2	9	9	NUM
ejpam-3807	418	3	]	]	X
ejpam-3807	418	4	j.	j.	PROPN
ejpam-3807	418	5	neggers	neggers	PROPN
ejpam-3807	418	6	and	and	CCONJ
ejpam-3807	418	7	h.	h.	PROPN
ejpam-3807	418	8	kim	kim	PROPN
ejpam-3807	418	9	.	.	PUNCT
ejpam-3807	419	1	on	on	ADP
ejpam-3807	419	2	b	b	NOUN
ejpam-3807	419	3	-	-	PUNCT
ejpam-3807	419	4	algebras	algebras	PROPN
ejpam-3807	419	5	.	.	PUNCT
ejpam-3807	419	6	mat	mat	PROPN
ejpam-3807	419	7	.	.	PROPN
ejpam-3807	419	8	vesnik	vesnik	PROPN
ejpam-3807	419	9	,	,	PUNCT
ejpam-3807	419	10	54:21–29	54:21–29	NUM
ejpam-3807	419	11	,	,	PUNCT
ejpam-3807	419	12	2002	2002	NUM
ejpam-3807	419	13	.	.	PUNCT
ejpam-3807	420	1	[	[	X
ejpam-3807	420	2	10	10	NUM
ejpam-3807	420	3	]	]	PUNCT
ejpam-3807	420	4	a.	a.	NOUN
ejpam-3807	420	5	walendziak	walendziak	PROPN
ejpam-3807	420	6	.	.	PUNCT
ejpam-3807	421	1	a	a	DET
ejpam-3807	421	2	note	note	NOUN
ejpam-3807	421	3	on	on	ADP
ejpam-3807	421	4	normal	normal	ADJ
ejpam-3807	421	5	subalgebras	subalgebra	NOUN
ejpam-3807	421	6	in	in	ADP
ejpam-3807	421	7	b	b	NOUN
ejpam-3807	421	8	-	-	PUNCT
ejpam-3807	421	9	algebras	algebras	PROPN
ejpam-3807	421	10	.	.	PUNCT
ejpam-3807	422	1	scientiae	scientiae	PROPN
ejpam-3807	422	2	mathematicae	mathematicae	VERB
ejpam-3807	422	3	japonicae	japonicae	PROPN
ejpam-3807	422	4	online	online	ADV
ejpam-3807	422	5	,	,	PUNCT
ejpam-3807	422	6	pages	page	NOUN
ejpam-3807	422	7	49–53	49–53	NUM
ejpam-3807	422	8	,	,	PUNCT
ejpam-3807	422	9	2005	2005	NUM
ejpam-3807	422	10	.	.	PUNCT