id	sid	tid	token	lemma	pos
ejpam-4594	1	1	european	european	PROPN
ejpam-4594	1	2	journal	journal	PROPN
ejpam-4594	1	3	of	of	ADP
ejpam-4594	1	4	pure	pure	ADJ
ejpam-4594	1	5	and	and	CCONJ
ejpam-4594	1	6	applied	apply	VERB
ejpam-4594	1	7	mathematics	mathematic	NOUN
ejpam-4594	1	8	vol	vol	NOUN
ejpam-4594	1	9	.	.	PROPN
ejpam-4594	2	1	15	15	NUM
ejpam-4594	2	2	,	,	PUNCT
ejpam-4594	2	3	no	no	INTJ
ejpam-4594	2	4	.	.	NOUN
ejpam-4594	2	5	4	4	NUM
ejpam-4594	2	6	,	,	PUNCT
ejpam-4594	2	7	2022	2022	NUM
ejpam-4594	2	8	,	,	PUNCT
ejpam-4594	2	9	1948	1948	NUM
ejpam-4594	2	10	-	-	SYM
ejpam-4594	2	11	1956	1956	NUM
ejpam-4594	2	12	issn	issn	PROPN
ejpam-4594	2	13	1307	1307	NUM
ejpam-4594	2	14	-	-	SYM
ejpam-4594	2	15	5543	5543	NUM
ejpam-4594	2	16	–	–	PUNCT
ejpam-4594	2	17	ejpam.com	ejpam.com	X
ejpam-4594	2	18	published	publish	VERB
ejpam-4594	2	19	by	by	ADP
ejpam-4594	2	20	new	new	PROPN
ejpam-4594	2	21	york	york	PROPN
ejpam-4594	2	22	business	business	PROPN
ejpam-4594	2	23	global	global	ADJ
ejpam-4594	2	24	on	on	ADP
ejpam-4594	2	25	dual	dual	ADJ
ejpam-4594	2	26	b	b	NOUN
ejpam-4594	2	27	-	-	PUNCT
ejpam-4594	2	28	topological	topological	ADJ
ejpam-4594	2	29	spaces	space	NOUN
ejpam-4594	2	30	determined	determine	VERB
ejpam-4594	2	31	by	by	ADP
ejpam-4594	2	32	filterbase	filterbase	NOUN
ejpam-4594	2	33	and	and	CCONJ
ejpam-4594	2	34	some	some	DET
ejpam-4594	2	35	sets	set	NOUN
ejpam-4594	2	36	in	in	ADP
ejpam-4594	2	37	a	a	DET
ejpam-4594	2	38	dual	dual	ADJ
ejpam-4594	2	39	b	b	NOUN
ejpam-4594	2	40	-	-	PUNCT
ejpam-4594	2	41	algebra	algebra	PROPN
ejpam-4594	2	42	katrina	katrina	PROPN
ejpam-4594	2	43	e.	e.	PROPN
ejpam-4594	2	44	belleza1,∗	belleza1,∗	PROPN
ejpam-4594	2	45	,	,	PUNCT
ejpam-4594	2	46	jimboy	jimboy	PROPN
ejpam-4594	2	47	r.	r.	PROPN
ejpam-4594	2	48	albaracin2	albaracin2	PROPN
ejpam-4594	2	49	1	1	NUM
ejpam-4594	2	50	department	department	NOUN
ejpam-4594	2	51	of	of	ADP
ejpam-4594	2	52	computer	computer	NOUN
ejpam-4594	2	53	,	,	PUNCT
ejpam-4594	2	54	information	information	NOUN
ejpam-4594	2	55	science	science	NOUN
ejpam-4594	2	56	,	,	PUNCT
ejpam-4594	2	57	and	and	CCONJ
ejpam-4594	2	58	mathematics	mathematic	NOUN
ejpam-4594	2	59	,	,	PUNCT
ejpam-4594	2	60	school	school	NOUN
ejpam-4594	2	61	of	of	ADP
ejpam-4594	2	62	arts	art	NOUN
ejpam-4594	2	63	and	and	CCONJ
ejpam-4594	2	64	sciences	science	NOUN
ejpam-4594	2	65	,	,	PUNCT
ejpam-4594	2	66	university	university	NOUN
ejpam-4594	2	67	of	of	ADP
ejpam-4594	2	68	san	san	PROPN
ejpam-4594	2	69	carlos	carlos	PROPN
ejpam-4594	2	70	,	,	PUNCT
ejpam-4594	2	71	talamban	talamban	PROPN
ejpam-4594	2	72	,	,	PUNCT
ejpam-4594	2	73	cebu	cebu	NOUN
ejpam-4594	2	74	city	city	NOUN
ejpam-4594	2	75	,	,	PUNCT
ejpam-4594	2	76	philippines	philippine	NOUN
ejpam-4594	2	77	2	2	NUM
ejpam-4594	2	78	mathematics	mathematic	NOUN
ejpam-4594	2	79	and	and	CCONJ
ejpam-4594	2	80	statistics	statistic	NOUN
ejpam-4594	2	81	programs	program	NOUN
ejpam-4594	2	82	,	,	PUNCT
ejpam-4594	2	83	college	college	NOUN
ejpam-4594	2	84	of	of	ADP
ejpam-4594	2	85	science	science	NOUN
ejpam-4594	2	86	,	,	PUNCT
ejpam-4594	2	87	university	university	NOUN
ejpam-4594	2	88	of	of	ADP
ejpam-4594	2	89	the	the	DET
ejpam-4594	2	90	philippines	philippine	NOUN
ejpam-4594	2	91	cebu	cebu	NOUN
ejpam-4594	2	92	,	,	PUNCT
ejpam-4594	2	93	cebu	cebu	NOUN
ejpam-4594	2	94	city	city	NOUN
ejpam-4594	2	95	,	,	PUNCT
ejpam-4594	2	96	philippines	philippine	NOUN
ejpam-4594	2	97	abstract	abstract	ADJ
ejpam-4594	2	98	.	.	PUNCT
ejpam-4594	3	1	this	this	DET
ejpam-4594	3	2	paper	paper	NOUN
ejpam-4594	3	3	presents	present	VERB
ejpam-4594	3	4	dual	dual	ADJ
ejpam-4594	3	5	b	b	NOUN
ejpam-4594	3	6	-	-	PUNCT
ejpam-4594	3	7	topologies	topology	NOUN
ejpam-4594	3	8	that	that	PRON
ejpam-4594	3	9	are	be	AUX
ejpam-4594	3	10	determined	determine	VERB
ejpam-4594	3	11	by	by	ADP
ejpam-4594	3	12	filterbase	filterbase	NOUN
ejpam-4594	3	13	and	and	CCONJ
ejpam-4594	3	14	some	some	DET
ejpam-4594	3	15	sets	set	NOUN
ejpam-4594	3	16	in	in	ADP
ejpam-4594	3	17	a	a	DET
ejpam-4594	3	18	dual	dual	ADJ
ejpam-4594	3	19	b	b	NOUN
ejpam-4594	3	20	-	-	PUNCT
ejpam-4594	3	21	algebra	algebra	NOUN
ejpam-4594	3	22	.	.	PUNCT
ejpam-4594	4	1	also	also	ADV
ejpam-4594	4	2	,	,	PUNCT
ejpam-4594	4	3	some	some	DET
ejpam-4594	4	4	properties	property	NOUN
ejpam-4594	4	5	of	of	ADP
ejpam-4594	4	6	a	a	DET
ejpam-4594	4	7	filterbase	filterbase	NOUN
ejpam-4594	4	8	in	in	ADP
ejpam-4594	4	9	a	a	DET
ejpam-4594	4	10	dual	dual	ADJ
ejpam-4594	4	11	b	b	NOUN
ejpam-4594	4	12	-	-	PUNCT
ejpam-4594	4	13	topological	topological	ADJ
ejpam-4594	4	14	space	space	NOUN
ejpam-4594	4	15	are	be	AUX
ejpam-4594	4	16	provided	provide	VERB
ejpam-4594	4	17	.	.	PUNCT
ejpam-4594	5	1	in	in	ADP
ejpam-4594	5	2	particular	particular	ADJ
ejpam-4594	5	3	,	,	PUNCT
ejpam-4594	5	4	a	a	DET
ejpam-4594	5	5	commutative	commutative	ADJ
ejpam-4594	5	6	dual	dual	ADJ
ejpam-4594	5	7	b	b	NOUN
ejpam-4594	5	8	-	-	PUNCT
ejpam-4594	5	9	topological	topological	ADJ
ejpam-4594	5	10	space	space	NOUN
ejpam-4594	5	11	and	and	CCONJ
ejpam-4594	5	12	a	a	DET
ejpam-4594	5	13	symmetric	symmetric	ADJ
ejpam-4594	5	14	b	b	X
ejpam-4594	5	15	-	-	PUNCT
ejpam-4594	5	16	topological	topological	ADJ
ejpam-4594	5	17	space	space	NOUN
ejpam-4594	5	18	are	be	AUX
ejpam-4594	5	19	topological	topological	ADJ
ejpam-4594	5	20	dual	dual	ADJ
ejpam-4594	5	21	b	b	NOUN
ejpam-4594	5	22	-	-	PUNCT
ejpam-4594	5	23	algebras	algebras	X
ejpam-4594	5	24	.	.	PUNCT
ejpam-4594	6	1	2020	2020	NUM
ejpam-4594	6	2	mathematics	mathematics	PROPN
ejpam-4594	6	3	subject	subject	NOUN
ejpam-4594	6	4	classifications	classification	NOUN
ejpam-4594	6	5	:	:	PUNCT
ejpam-4594	6	6	46h10	46h10	NUM
ejpam-4594	6	7	,	,	PUNCT
ejpam-4594	6	8	54a05	54a05	NUM
ejpam-4594	6	9	,	,	PUNCT
ejpam-4594	6	10	54f65	54f65	NUM
ejpam-4594	6	11	,	,	PUNCT
ejpam-4594	6	12	54h99	54h99	NUM
ejpam-4594	6	13	,	,	PUNCT
ejpam-4594	6	14	55m99	55m99	NUM
ejpam-4594	6	15	key	key	ADJ
ejpam-4594	6	16	words	word	NOUN
ejpam-4594	6	17	and	and	CCONJ
ejpam-4594	6	18	phrases	phrase	NOUN
ejpam-4594	6	19	:	:	PUNCT
ejpam-4594	6	20	topological	topological	ADJ
ejpam-4594	6	21	algebra	algebra	NOUN
ejpam-4594	6	22	,	,	PUNCT
ejpam-4594	6	23	dual	dual	ADJ
ejpam-4594	6	24	b	b	NOUN
ejpam-4594	6	25	-	-	PUNCT
ejpam-4594	6	26	algebra	algebra	NOUN
ejpam-4594	6	27	,	,	PUNCT
ejpam-4594	6	28	topological	topological	ADJ
ejpam-4594	6	29	dual	dual	ADJ
ejpam-4594	6	30	b	b	NOUN
ejpam-4594	6	31	-	-	PUNCT
ejpam-4594	6	32	algebra	algebra	NOUN
ejpam-4594	6	33	,	,	PUNCT
ejpam-4594	6	34	dual	dual	ADJ
ejpam-4594	6	35	b	b	NOUN
ejpam-4594	6	36	-	-	PUNCT
ejpam-4594	6	37	topological	topological	ADJ
ejpam-4594	6	38	space	space	NOUN
ejpam-4594	6	39	,	,	PUNCT
ejpam-4594	6	40	tdb	tdb	PROPN
ejpam-4594	6	41	-	-	NOUN
ejpam-4594	6	42	algebra	algebra	PROPN
ejpam-4594	6	43	1	1	NUM
ejpam-4594	6	44	.	.	PUNCT
ejpam-4594	6	45	introduction	introduction	NOUN
ejpam-4594	6	46	in	in	ADP
ejpam-4594	6	47	1998	1998	NUM
ejpam-4594	6	48	,	,	PUNCT
ejpam-4594	6	49	d.s	d.s	PROPN
ejpam-4594	6	50	.	.	PROPN
ejpam-4594	6	51	lee	lee	PROPN
ejpam-4594	6	52	and	and	CCONJ
ejpam-4594	6	53	d.n	d.n	PROPN
ejpam-4594	6	54	.	.	PROPN
ejpam-4594	6	55	ryu	ryu	PROPN
ejpam-4594	7	1	[	[	X
ejpam-4594	7	2	5	5	NUM
ejpam-4594	7	3	]	]	PUNCT
ejpam-4594	7	4	introduced	introduce	VERB
ejpam-4594	7	5	the	the	DET
ejpam-4594	7	6	notion	notion	NOUN
ejpam-4594	7	7	of	of	ADP
ejpam-4594	7	8	a	a	DET
ejpam-4594	7	9	topological	topological	ADJ
ejpam-4594	7	10	bckalgebra	bckalgebra	NOUN
ejpam-4594	7	11	.	.	PUNCT
ejpam-4594	8	1	moreover	moreover	ADV
ejpam-4594	8	2	,	,	PUNCT
ejpam-4594	8	3	they	they	PRON
ejpam-4594	8	4	derived	derive	VERB
ejpam-4594	8	5	a	a	DET
ejpam-4594	8	6	filter	filter	NOUN
ejpam-4594	8	7	base	base	NOUN
ejpam-4594	8	8	generating	generate	VERB
ejpam-4594	8	9	a	a	DET
ejpam-4594	8	10	bck	bck	NOUN
ejpam-4594	8	11	-	-	PUNCT
ejpam-4594	8	12	algebra	algebra	NOUN
ejpam-4594	8	13	topology	topology	NOUN
ejpam-4594	8	14	.	.	PUNCT
ejpam-4594	9	1	on	on	ADP
ejpam-4594	9	2	the	the	DET
ejpam-4594	9	3	following	following	ADJ
ejpam-4594	9	4	year	year	NOUN
ejpam-4594	9	5	,	,	PUNCT
ejpam-4594	9	6	y.b	y.b	PROPN
ejpam-4594	9	7	.	.	PROPN
ejpam-4594	9	8	jun	jun	PROPN
ejpam-4594	9	9	et	et	PROPN
ejpam-4594	9	10	al	al	PROPN
ejpam-4594	9	11	.	.	PUNCT
ejpam-4594	10	1	[	[	X
ejpam-4594	10	2	4	4	X
ejpam-4594	10	3	]	]	PUNCT
ejpam-4594	10	4	gave	give	VERB
ejpam-4594	10	5	a	a	DET
ejpam-4594	10	6	filterbase	filterbase	NOUN
ejpam-4594	10	7	generating	generate	VERB
ejpam-4594	10	8	a	a	DET
ejpam-4594	10	9	bci	bci	NOUN
ejpam-4594	10	10	-	-	NOUN
ejpam-4594	10	11	topology	topology	NOUN
ejpam-4594	10	12	and	and	CCONJ
ejpam-4594	10	13	making	make	VERB
ejpam-4594	10	14	a	a	DET
ejpam-4594	10	15	bci	bci	NOUN
ejpam-4594	10	16	-	-	NOUN
ejpam-4594	10	17	algebra	algebra	NOUN
ejpam-4594	10	18	into	into	ADP
ejpam-4594	10	19	a	a	DET
ejpam-4594	10	20	topological	topological	ADJ
ejpam-4594	10	21	bci	bci	NOUN
ejpam-4594	10	22	-	-	NOUN
ejpam-4594	10	23	algebra	algebra	NOUN
ejpam-4594	10	24	for	for	ADP
ejpam-4594	10	25	which	which	PRON
ejpam-4594	10	26	the	the	DET
ejpam-4594	10	27	filterbase	filterbase	NOUN
ejpam-4594	10	28	is	be	AUX
ejpam-4594	10	29	a	a	DET
ejpam-4594	10	30	fundamental	fundamental	ADJ
ejpam-4594	10	31	system	system	NOUN
ejpam-4594	10	32	of	of	ADP
ejpam-4594	10	33	neighborhoods	neighborhood	NOUN
ejpam-4594	10	34	.	.	PUNCT
ejpam-4594	11	1	in	in	ADP
ejpam-4594	11	2	2019	2019	NUM
ejpam-4594	11	3	,	,	PUNCT
ejpam-4594	11	4	k.e	k.e	PROPN
ejpam-4594	11	5	.	.	PROPN
ejpam-4594	11	6	belleza	belleza	PROPN
ejpam-4594	11	7	and	and	CCONJ
ejpam-4594	11	8	j.p	j.p	PROPN
ejpam-4594	11	9	.	.	PROPN
ejpam-4594	11	10	vilela	vilela	NOUN
ejpam-4594	11	11	introduces	introduce	NOUN
ejpam-4594	11	12	and	and	CCONJ
ejpam-4594	11	13	characterized	characterize	VERB
ejpam-4594	11	14	the	the	DET
ejpam-4594	11	15	notion	notion	NOUN
ejpam-4594	11	16	of	of	ADP
ejpam-4594	11	17	a	a	DET
ejpam-4594	11	18	dual	dual	ADJ
ejpam-4594	11	19	b	b	NOUN
ejpam-4594	11	20	-	-	PUNCT
ejpam-4594	11	21	algebra	algebra	NOUN
ejpam-4594	11	22	[	[	X
ejpam-4594	11	23	2	2	NUM
ejpam-4594	11	24	]	]	PUNCT
ejpam-4594	11	25	.	.	PUNCT
ejpam-4594	12	1	moreover	moreover	ADV
ejpam-4594	12	2	on	on	ADP
ejpam-4594	12	3	the	the	DET
ejpam-4594	12	4	following	following	ADJ
ejpam-4594	12	5	year	year	NOUN
ejpam-4594	12	6	,	,	PUNCT
ejpam-4594	12	7	k.e	k.e	PROPN
ejpam-4594	12	8	.	.	PROPN
ejpam-4594	12	9	belleza	belleza	PROPN
ejpam-4594	12	10	introduces	introduce	VERB
ejpam-4594	12	11	the	the	DET
ejpam-4594	12	12	dual	dual	ADJ
ejpam-4594	12	13	b	b	NOUN
ejpam-4594	12	14	-	-	PUNCT
ejpam-4594	12	15	topological	topological	ADJ
ejpam-4594	12	16	space	space	NOUN
ejpam-4594	12	17	and	and	CCONJ
ejpam-4594	12	18	a	a	DET
ejpam-4594	12	19	tdb	tdb	NOUN
ejpam-4594	12	20	-	-	NOUN
ejpam-4594	12	21	algebra	algebra	NOUN
ejpam-4594	12	22	involving	involve	VERB
ejpam-4594	12	23	dual	dual	ADJ
ejpam-4594	12	24	b	b	NOUN
ejpam-4594	12	25	-	-	PUNCT
ejpam-4594	12	26	ideals	ideal	NOUN
ejpam-4594	12	27	and	and	CCONJ
ejpam-4594	12	28	dual	dual	ADJ
ejpam-4594	12	29	b	b	NOUN
ejpam-4594	12	30	-	-	PUNCT
ejpam-4594	12	31	subalgebras	subalgebras	X
ejpam-4594	12	32	.	.	PUNCT
ejpam-4594	13	1	2	2	X
ejpam-4594	13	2	.	.	X
ejpam-4594	13	3	preliminaries	preliminary	NOUN
ejpam-4594	13	4	definition	definition	NOUN
ejpam-4594	13	5	1	1	NUM
ejpam-4594	13	6	.	.	PUNCT
ejpam-4594	14	1	[	[	X
ejpam-4594	14	2	2	2	X
ejpam-4594	14	3	]	]	PUNCT
ejpam-4594	14	4	a	a	DET
ejpam-4594	14	5	dual	dual	ADJ
ejpam-4594	14	6	b	b	NOUN
ejpam-4594	14	7	-	-	PUNCT
ejpam-4594	14	8	algebra	algebra	NOUN
ejpam-4594	14	9	xd	xd	INTJ
ejpam-4594	14	10	is	be	AUX
ejpam-4594	14	11	a	a	DET
ejpam-4594	14	12	triple	triple	ADJ
ejpam-4594	14	13	(	(	PUNCT
ejpam-4594	14	14	xd	xd	ADV
ejpam-4594	14	15	,	,	PUNCT
ejpam-4594	14	16	◦	◦	NOUN
ejpam-4594	14	17	,	,	PUNCT
ejpam-4594	14	18	1	1	NUM
ejpam-4594	14	19	)	)	PUNCT
ejpam-4594	14	20	where	where	SCONJ
ejpam-4594	14	21	xd	xd	PRON
ejpam-4594	14	22	is	be	VERB
ejpam-4594	14	23	a	a	DET
ejpam-4594	14	24	non	non	ADJ
ejpam-4594	14	25	-	-	ADJ
ejpam-4594	14	26	empty	empty	ADJ
ejpam-4594	14	27	set	set	NOUN
ejpam-4594	14	28	with	with	ADP
ejpam-4594	14	29	a	a	DET
ejpam-4594	14	30	binary	binary	ADJ
ejpam-4594	14	31	operation	operation	NOUN
ejpam-4594	14	32	“	"	PUNCT
ejpam-4594	14	33	◦	◦	NOUN
ejpam-4594	14	34	”	"	PUNCT
ejpam-4594	14	35	and	and	CCONJ
ejpam-4594	14	36	a	a	DET
ejpam-4594	14	37	constant	constant	ADJ
ejpam-4594	14	38	1	1	NUM
ejpam-4594	14	39	satisfying	satisfy	VERB
ejpam-4594	14	40	the	the	DET
ejpam-4594	14	41	following	follow	VERB
ejpam-4594	14	42	axioms	axiom	NOUN
ejpam-4594	14	43	for	for	ADP
ejpam-4594	14	44	all	all	DET
ejpam-4594	14	45	x	x	NOUN
ejpam-4594	14	46	,	,	PUNCT
ejpam-4594	14	47	y	y	PROPN
ejpam-4594	14	48	,	,	PUNCT
ejpam-4594	14	49	z	z	VERB
ejpam-4594	14	50	in	in	ADP
ejpam-4594	14	51	xd	xd	ADP
ejpam-4594	14	52	:	:	PUNCT
ejpam-4594	14	53	∗corresponding	∗corresponde	VERB
ejpam-4594	14	54	author	author	NOUN
ejpam-4594	14	55	.	.	PUNCT
ejpam-4594	15	1	doi	doi	NOUN
ejpam-4594	15	2	:	:	PUNCT
ejpam-4594	15	3	https://doi.org/10.29020/nybg.ejpam.v15i4.4594	https://doi.org/10.29020/nybg.ejpam.v15i4.4594	NUM
ejpam-4594	15	4	email	email	NOUN
ejpam-4594	15	5	addresses	address	NOUN
ejpam-4594	15	6	:	:	PUNCT
ejpam-4594	15	7	kebelleza@usc.edu.ph	kebelleza@usc.edu.ph	PROPN
ejpam-4594	15	8	(	(	PUNCT
ejpam-4594	15	9	k.	k.	PROPN
ejpam-4594	15	10	belleza	belleza	PROPN
ejpam-4594	15	11	)	)	PUNCT
ejpam-4594	15	12	,	,	PUNCT
ejpam-4594	15	13	jralbaracin@up.edu.ph	jralbaracin@up.edu.ph	PROPN
ejpam-4594	15	14	(	(	PUNCT
ejpam-4594	15	15	j.	j.	PROPN
ejpam-4594	15	16	albaracin	albaracin	PROPN
ejpam-4594	15	17	)	)	PUNCT
ejpam-4594	15	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4594	15	19	1948	1948	NUM
ejpam-4594	16	1	©	©	ADP
ejpam-4594	16	2	2022	2022	NUM
ejpam-4594	16	3	ejpam	ejpam	VERB
ejpam-4594	16	4	all	all	DET
ejpam-4594	16	5	rights	right	NOUN
ejpam-4594	16	6	reserved	reserve	VERB
ejpam-4594	16	7	.	.	PUNCT
ejpam-4594	17	1	k.	k.	PROPN
ejpam-4594	17	2	belleza	belleza	PROPN
ejpam-4594	17	3	,	,	PUNCT
ejpam-4594	17	4	j.	j.	PROPN
ejpam-4594	17	5	albaracin	albaracin	PROPN
ejpam-4594	17	6	/	/	PUNCT
ejpam-4594	17	7	eur	eur	PROPN
ejpam-4594	17	8	.	.	PUNCT
ejpam-4594	18	1	j.	j.	PROPN
ejpam-4594	18	2	pure	pure	PROPN
ejpam-4594	18	3	appl	appl	PROPN
ejpam-4594	18	4	.	.	PROPN
ejpam-4594	18	5	math	math	PROPN
ejpam-4594	18	6	,	,	PUNCT
ejpam-4594	18	7	15	15	NUM
ejpam-4594	18	8	(	(	PUNCT
ejpam-4594	18	9	4	4	NUM
ejpam-4594	18	10	)	)	PUNCT
ejpam-4594	18	11	(	(	PUNCT
ejpam-4594	18	12	2022	2022	NUM
ejpam-4594	18	13	)	)	PUNCT
ejpam-4594	18	14	,	,	PUNCT
ejpam-4594	18	15	1948	1948	NUM
ejpam-4594	18	16	-	-	SYM
ejpam-4594	18	17	1956	1956	NUM
ejpam-4594	18	18	1949	1949	NUM
ejpam-4594	18	19	(	(	PUNCT
ejpam-4594	18	20	db1	db1	NOUN
ejpam-4594	18	21	)	)	PUNCT
ejpam-4594	18	22	x	x	SYM
ejpam-4594	19	1	◦	◦	NOUN
ejpam-4594	19	2	x	x	SYM
ejpam-4594	19	3	=	=	NOUN
ejpam-4594	19	4	1	1	NUM
ejpam-4594	19	5	;	;	PUNCT
ejpam-4594	19	6	(	(	PUNCT
ejpam-4594	19	7	db2	db2	NOUN
ejpam-4594	19	8	)	)	PUNCT
ejpam-4594	19	9	1	1	NUM
ejpam-4594	19	10	◦	◦	NOUN
ejpam-4594	19	11	x	x	SYM
ejpam-4594	19	12	=	=	SYM
ejpam-4594	19	13	x	x	X
ejpam-4594	19	14	;	;	PUNCT
ejpam-4594	19	15	(	(	PUNCT
ejpam-4594	19	16	db3	db3	PROPN
ejpam-4594	19	17	)	)	PUNCT
ejpam-4594	19	18	x	x	SYM
ejpam-4594	20	1	◦	◦	NOUN
ejpam-4594	20	2	(	(	PUNCT
ejpam-4594	20	3	y	y	PROPN
ejpam-4594	20	4	◦	◦	PROPN
ejpam-4594	20	5	z	z	PROPN
ejpam-4594	20	6	)	)	PUNCT
ejpam-4594	20	7	=	=	SYM
ejpam-4594	20	8	(	(	PUNCT
ejpam-4594	20	9	(	(	PUNCT
ejpam-4594	20	10	y	y	NOUN
ejpam-4594	20	11	◦	◦	NOUN
ejpam-4594	20	12	1	1	NUM
ejpam-4594	20	13	)	)	PUNCT
ejpam-4594	20	14	◦	◦	NOUN
ejpam-4594	20	15	x	x	SYM
ejpam-4594	20	16	)	)	PUNCT
ejpam-4594	20	17	◦	◦	NOUN
ejpam-4594	20	18	z.	z.	PROPN
ejpam-4594	20	19	lemma	lemma	PROPN
ejpam-4594	21	1	1	1	X
ejpam-4594	21	2	.	.	PUNCT
ejpam-4594	22	1	[	[	X
ejpam-4594	22	2	2	2	X
ejpam-4594	22	3	]	]	PUNCT
ejpam-4594	22	4	let	let	VERB
ejpam-4594	22	5	xd	xd	INTJ
ejpam-4594	22	6	be	be	AUX
ejpam-4594	22	7	a	a	DET
ejpam-4594	22	8	dual	dual	ADJ
ejpam-4594	22	9	b	b	NOUN
ejpam-4594	22	10	-	-	PUNCT
ejpam-4594	22	11	algebra	algebra	NOUN
ejpam-4594	22	12	.	.	PUNCT
ejpam-4594	23	1	for	for	ADP
ejpam-4594	23	2	any	any	DET
ejpam-4594	23	3	x	x	NOUN
ejpam-4594	23	4	,	,	PUNCT
ejpam-4594	23	5	y	y	PROPN
ejpam-4594	23	6	in	in	ADP
ejpam-4594	23	7	xd	xd	ADP
ejpam-4594	23	8	,	,	PUNCT
ejpam-4594	23	9	x	x	VERB
ejpam-4594	23	10	◦	◦	VERB
ejpam-4594	23	11	y	y	NOUN
ejpam-4594	23	12	=	=	SYM
ejpam-4594	23	13	1	1	NUM
ejpam-4594	23	14	implies	imply	VERB
ejpam-4594	23	15	x	x	X
ejpam-4594	23	16	=	=	SYM
ejpam-4594	23	17	y.	y.	NOUN
ejpam-4594	23	18	theorem	theorem	VERB
ejpam-4594	23	19	1	1	NUM
ejpam-4594	23	20	.	.	PUNCT
ejpam-4594	24	1	[	[	X
ejpam-4594	24	2	2	2	X
ejpam-4594	24	3	]	]	PUNCT
ejpam-4594	24	4	let	let	VERB
ejpam-4594	24	5	x	x	PUNCT
ejpam-4594	24	6	=	=	PUNCT
ejpam-4594	24	7	(	(	PUNCT
ejpam-4594	24	8	x	x	NOUN
ejpam-4594	24	9	,	,	PUNCT
ejpam-4594	24	10	◦	◦	NOUN
ejpam-4594	24	11	,	,	PUNCT
ejpam-4594	24	12	1	1	NUM
ejpam-4594	24	13	)	)	PUNCT
ejpam-4594	24	14	be	be	AUX
ejpam-4594	24	15	any	any	DET
ejpam-4594	24	16	algebra	algebra	NOUN
ejpam-4594	24	17	of	of	ADP
ejpam-4594	24	18	type	type	NOUN
ejpam-4594	24	19	(	(	PUNCT
ejpam-4594	24	20	2	2	NUM
ejpam-4594	24	21	,	,	PUNCT
ejpam-4594	24	22	0	0	NUM
ejpam-4594	24	23	)	)	PUNCT
ejpam-4594	24	24	.	.	PUNCT
ejpam-4594	25	1	then	then	ADV
ejpam-4594	25	2	x	x	PRON
ejpam-4594	25	3	is	be	AUX
ejpam-4594	25	4	a	a	DET
ejpam-4594	25	5	dual	dual	ADJ
ejpam-4594	25	6	balgebra	balgebra	NOUN
ejpam-4594	25	7	if	if	SCONJ
ejpam-4594	25	8	and	and	CCONJ
ejpam-4594	25	9	only	only	ADV
ejpam-4594	25	10	if	if	SCONJ
ejpam-4594	25	11	for	for	ADP
ejpam-4594	25	12	any	any	DET
ejpam-4594	25	13	x	x	NOUN
ejpam-4594	25	14	,	,	PUNCT
ejpam-4594	25	15	y	y	PROPN
ejpam-4594	25	16	,	,	PUNCT
ejpam-4594	25	17	z	z	PROPN
ejpam-4594	25	18	∈	∈	PROPN
ejpam-4594	25	19	x	x	X
ejpam-4594	25	20	,	,	PUNCT
ejpam-4594	25	21	(	(	PUNCT
ejpam-4594	25	22	i	i	NOUN
ejpam-4594	25	23	)	)	PUNCT
ejpam-4594	25	24	x	x	PUNCT
ejpam-4594	26	1	◦	◦	NOUN
ejpam-4594	26	2	x	x	SYM
ejpam-4594	26	3	=	=	NOUN
ejpam-4594	26	4	1	1	NUM
ejpam-4594	26	5	;	;	PUNCT
ejpam-4594	26	6	(	(	PUNCT
ejpam-4594	26	7	ii	ii	NOUN
ejpam-4594	26	8	)	)	PUNCT
ejpam-4594	26	9	x	x	X
ejpam-4594	27	1	=	=	PRON
ejpam-4594	27	2	(	(	PUNCT
ejpam-4594	27	3	x	x	SYM
ejpam-4594	27	4	◦	◦	NOUN
ejpam-4594	27	5	1	1	NUM
ejpam-4594	27	6	)	)	PUNCT
ejpam-4594	27	7	◦	◦	NOUN
ejpam-4594	27	8	1	1	NUM
ejpam-4594	27	9	;	;	PUNCT
ejpam-4594	27	10	(	(	PUNCT
ejpam-4594	27	11	iii	iii	X
ejpam-4594	27	12	)	)	PUNCT
ejpam-4594	27	13	(	(	PUNCT
ejpam-4594	27	14	x	x	X
ejpam-4594	27	15	◦	◦	VERB
ejpam-4594	27	16	y	y	NOUN
ejpam-4594	27	17	)	)	PUNCT
ejpam-4594	27	18	◦	◦	NOUN
ejpam-4594	27	19	(	(	PUNCT
ejpam-4594	27	20	x	x	PART
ejpam-4594	27	21	◦	◦	NOUN
ejpam-4594	27	22	z	z	NOUN
ejpam-4594	27	23	)	)	PUNCT
ejpam-4594	28	1	=	=	SYM
ejpam-4594	28	2	y	y	PROPN
ejpam-4594	28	3	◦	◦	NOUN
ejpam-4594	28	4	z.	z.	PROPN
ejpam-4594	28	5	definition	definition	NOUN
ejpam-4594	28	6	2	2	NUM
ejpam-4594	28	7	.	.	PUNCT
ejpam-4594	29	1	[	[	X
ejpam-4594	29	2	2	2	X
ejpam-4594	29	3	]	]	PUNCT
ejpam-4594	29	4	let	let	VERB
ejpam-4594	29	5	xd	xd	INTJ
ejpam-4594	29	6	be	be	AUX
ejpam-4594	29	7	a	a	DET
ejpam-4594	29	8	dual	dual	ADJ
ejpam-4594	29	9	b	b	NOUN
ejpam-4594	29	10	-	-	PUNCT
ejpam-4594	29	11	algebra	algebra	NOUN
ejpam-4594	29	12	.	.	PUNCT
ejpam-4594	30	1	define	define	VERB
ejpam-4594	30	2	a	a	DET
ejpam-4594	30	3	binary	binary	ADJ
ejpam-4594	30	4	operation	operation	NOUN
ejpam-4594	30	5	“	"	PUNCT
ejpam-4594	30	6	+	+	CCONJ
ejpam-4594	30	7	”	"	PUNCT
ejpam-4594	30	8	on	on	ADP
ejpam-4594	30	9	x	x	PUNCT
ejpam-4594	30	10	as	as	SCONJ
ejpam-4594	30	11	follows	follow	VERB
ejpam-4594	30	12	:	:	PUNCT
ejpam-4594	30	13	x+	x+	PROPN
ejpam-4594	30	14	y	y	NOUN
ejpam-4594	30	15	=	=	SYM
ejpam-4594	30	16	(	(	PUNCT
ejpam-4594	30	17	x	x	SYM
ejpam-4594	30	18	◦	◦	NOUN
ejpam-4594	30	19	1	1	NUM
ejpam-4594	30	20	)	)	PUNCT
ejpam-4594	30	21	◦	◦	NOUN
ejpam-4594	30	22	y	y	NOUN
ejpam-4594	30	23	for	for	ADP
ejpam-4594	30	24	all	all	DET
ejpam-4594	30	25	x	x	NOUN
ejpam-4594	30	26	,	,	PUNCT
ejpam-4594	30	27	y	y	PROPN
ejpam-4594	30	28	in	in	ADP
ejpam-4594	30	29	xd	xd	ADP
ejpam-4594	30	30	.	.	PUNCT
ejpam-4594	31	1	a	a	DET
ejpam-4594	31	2	dual	dual	ADJ
ejpam-4594	31	3	b	b	NOUN
ejpam-4594	31	4	-	-	PUNCT
ejpam-4594	31	5	algebra	algebra	NOUN
ejpam-4594	31	6	is	be	AUX
ejpam-4594	31	7	said	say	VERB
ejpam-4594	31	8	to	to	PART
ejpam-4594	31	9	be	be	AUX
ejpam-4594	31	10	commutative	commutative	ADJ
ejpam-4594	31	11	if	if	SCONJ
ejpam-4594	31	12	x+	x+	ADJ
ejpam-4594	32	1	y	y	PROPN
ejpam-4594	32	2	=	=	SYM
ejpam-4594	32	3	y	y	PROPN
ejpam-4594	32	4	+	+	CCONJ
ejpam-4594	32	5	x	x	X
ejpam-4594	32	6	,	,	PUNCT
ejpam-4594	32	7	that	that	ADV
ejpam-4594	32	8	is	is	ADV
ejpam-4594	32	9	,	,	PUNCT
ejpam-4594	32	10	(	(	PUNCT
ejpam-4594	32	11	x	x	X
ejpam-4594	32	12	◦	◦	NOUN
ejpam-4594	32	13	1	1	NUM
ejpam-4594	32	14	)	)	PUNCT
ejpam-4594	32	15	◦	◦	NOUN
ejpam-4594	32	16	y	y	NOUN
ejpam-4594	32	17	=	=	SYM
ejpam-4594	32	18	(	(	PUNCT
ejpam-4594	32	19	y	y	PROPN
ejpam-4594	32	20	◦	◦	NOUN
ejpam-4594	32	21	1	1	NUM
ejpam-4594	32	22	)	)	PUNCT
ejpam-4594	32	23	◦	◦	NOUN
ejpam-4594	32	24	x	x	SYM
ejpam-4594	32	25	for	for	ADP
ejpam-4594	32	26	all	all	DET
ejpam-4594	32	27	x	x	NOUN
ejpam-4594	32	28	,	,	PUNCT
ejpam-4594	32	29	y	y	PROPN
ejpam-4594	32	30	in	in	ADP
ejpam-4594	32	31	xd	xd	ADP
ejpam-4594	32	32	.	.	PUNCT
ejpam-4594	32	33	proposition	proposition	NOUN
ejpam-4594	32	34	1	1	NUM
ejpam-4594	32	35	.	.	PUNCT
ejpam-4594	33	1	[	[	X
ejpam-4594	33	2	2	2	X
ejpam-4594	33	3	]	]	PUNCT
ejpam-4594	33	4	suppose	suppose	VERB
ejpam-4594	33	5	xd	xd	INTJ
ejpam-4594	33	6	is	be	VERB
ejpam-4594	33	7	a	a	DET
ejpam-4594	33	8	commutative	commutative	ADJ
ejpam-4594	33	9	b	b	NOUN
ejpam-4594	33	10	-	-	PUNCT
ejpam-4594	33	11	algebra	algebra	NOUN
ejpam-4594	33	12	.	.	PUNCT
ejpam-4594	34	1	then	then	ADV
ejpam-4594	34	2	for	for	ADP
ejpam-4594	34	3	all	all	DET
ejpam-4594	34	4	x	x	NOUN
ejpam-4594	34	5	,	,	PUNCT
ejpam-4594	34	6	y	y	PROPN
ejpam-4594	34	7	in	in	ADP
ejpam-4594	34	8	xd	xd	ADP
ejpam-4594	34	9	,	,	PUNCT
ejpam-4594	34	10	x	x	PART
ejpam-4594	34	11	◦	◦	NOUN
ejpam-4594	34	12	(	(	PUNCT
ejpam-4594	34	13	y	y	PROPN
ejpam-4594	34	14	◦	◦	PROPN
ejpam-4594	34	15	z	z	PROPN
ejpam-4594	34	16	)	)	PUNCT
ejpam-4594	35	1	=	=	SYM
ejpam-4594	35	2	y	y	PROPN
ejpam-4594	35	3	◦	◦	NOUN
ejpam-4594	35	4	(	(	PUNCT
ejpam-4594	35	5	x	x	PART
ejpam-4594	35	6	◦	◦	NOUN
ejpam-4594	35	7	z	z	NOUN
ejpam-4594	35	8	)	)	PUNCT
ejpam-4594	35	9	.	.	PUNCT
ejpam-4594	36	1	let	let	VERB
ejpam-4594	36	2	xd	xd	INTJ
ejpam-4594	36	3	be	be	AUX
ejpam-4594	36	4	a	a	DET
ejpam-4594	36	5	dual	dual	ADJ
ejpam-4594	36	6	b	b	NOUN
ejpam-4594	36	7	-	-	PUNCT
ejpam-4594	36	8	algebra	algebra	NOUN
ejpam-4594	36	9	such	such	ADJ
ejpam-4594	36	10	that	that	SCONJ
ejpam-4594	36	11	x	x	X
ejpam-4594	36	12	◦	◦	NOUN
ejpam-4594	36	13	y	y	NOUN
ejpam-4594	36	14	=	=	PUNCT
ejpam-4594	36	15	y	y	PROPN
ejpam-4594	36	16	◦	◦	NOUN
ejpam-4594	36	17	x	x	PUNCT
ejpam-4594	36	18	for	for	ADP
ejpam-4594	36	19	all	all	DET
ejpam-4594	36	20	x	x	NOUN
ejpam-4594	36	21	,	,	PUNCT
ejpam-4594	36	22	y	y	PROPN
ejpam-4594	36	23	∈	∈	PROPN
ejpam-4594	37	1	xd	xd	INTJ
ejpam-4594	37	2	.	.	PUNCT
ejpam-4594	38	1	then	then	ADV
ejpam-4594	38	2	we	we	PRON
ejpam-4594	38	3	say	say	VERB
ejpam-4594	38	4	that	that	SCONJ
ejpam-4594	38	5	xd	xd	INTJ
ejpam-4594	38	6	satisfies	satisfy	VERB
ejpam-4594	38	7	a	a	DET
ejpam-4594	38	8	symmetric	symmetric	ADJ
ejpam-4594	38	9	condition	condition	NOUN
ejpam-4594	38	10	[	[	X
ejpam-4594	38	11	2	2	NUM
ejpam-4594	38	12	]	]	PUNCT
ejpam-4594	38	13	.	.	PUNCT
ejpam-4594	39	1	lemma	lemma	PROPN
ejpam-4594	39	2	2	2	NUM
ejpam-4594	39	3	.	.	PUNCT
ejpam-4594	40	1	[	[	X
ejpam-4594	40	2	2	2	X
ejpam-4594	40	3	]	]	PUNCT
ejpam-4594	40	4	let	let	VERB
ejpam-4594	40	5	xd	xd	INTJ
ejpam-4594	40	6	be	be	AUX
ejpam-4594	40	7	a	a	DET
ejpam-4594	40	8	dual	dual	ADJ
ejpam-4594	40	9	b	b	NOUN
ejpam-4594	40	10	-	-	PUNCT
ejpam-4594	40	11	algebra	algebra	NOUN
ejpam-4594	40	12	satisfying	satisfy	VERB
ejpam-4594	40	13	a	a	DET
ejpam-4594	40	14	symmetric	symmetric	ADJ
ejpam-4594	40	15	condition	condition	NOUN
ejpam-4594	40	16	.	.	PUNCT
ejpam-4594	41	1	then	then	ADV
ejpam-4594	41	2	for	for	ADP
ejpam-4594	41	3	all	all	DET
ejpam-4594	41	4	x	x	NOUN
ejpam-4594	41	5	,	,	PUNCT
ejpam-4594	41	6	y	y	PROPN
ejpam-4594	41	7	,	,	PUNCT
ejpam-4594	41	8	z	z	PROPN
ejpam-4594	41	9	∈	∈	PROPN
ejpam-4594	41	10	xd	xd	ADP
ejpam-4594	41	11	,	,	PUNCT
ejpam-4594	41	12	(	(	PUNCT
ejpam-4594	41	13	x	x	SYM
ejpam-4594	41	14	◦	◦	VERB
ejpam-4594	41	15	y	y	NOUN
ejpam-4594	41	16	)	)	PUNCT
ejpam-4594	41	17	◦	◦	NOUN
ejpam-4594	41	18	(	(	PUNCT
ejpam-4594	41	19	z	z	AUX
ejpam-4594	41	20	◦	◦	NOUN
ejpam-4594	41	21	y	y	NOUN
ejpam-4594	41	22	)	)	PUNCT
ejpam-4594	41	23	=	=	PUNCT
ejpam-4594	42	1	x	x	PUNCT
ejpam-4594	42	2	◦	◦	NOUN
ejpam-4594	42	3	z.	z.	PROPN
ejpam-4594	42	4	definition	definition	NOUN
ejpam-4594	42	5	3	3	NUM
ejpam-4594	42	6	.	.	PUNCT
ejpam-4594	43	1	[	[	X
ejpam-4594	43	2	1	1	X
ejpam-4594	43	3	]	]	PUNCT
ejpam-4594	43	4	let	let	VERB
ejpam-4594	43	5	xd	xd	INTJ
ejpam-4594	43	6	be	be	AUX
ejpam-4594	43	7	a	a	DET
ejpam-4594	43	8	dual	dual	ADJ
ejpam-4594	43	9	b	b	NOUN
ejpam-4594	43	10	-	-	PUNCT
ejpam-4594	43	11	algebra	algebra	NOUN
ejpam-4594	43	12	and	and	CCONJ
ejpam-4594	43	13	s	s	VERB
ejpam-4594	43	14	a	a	DET
ejpam-4594	43	15	nonempty	nonempty	ADJ
ejpam-4594	43	16	subset	subset	NOUN
ejpam-4594	43	17	of	of	ADP
ejpam-4594	43	18	xd	xd	PROPN
ejpam-4594	43	19	.	.	PUNCT
ejpam-4594	44	1	then	then	ADV
ejpam-4594	44	2	s	s	VERB
ejpam-4594	44	3	is	be	AUX
ejpam-4594	44	4	called	call	VERB
ejpam-4594	44	5	a	a	DET
ejpam-4594	44	6	dual	dual	ADJ
ejpam-4594	44	7	b	b	NOUN
ejpam-4594	44	8	-	-	PUNCT
ejpam-4594	44	9	subalgebra	subalgebra	NOUN
ejpam-4594	44	10	of	of	ADP
ejpam-4594	44	11	xd	xd	INTJ
ejpam-4594	44	12	if	if	SCONJ
ejpam-4594	44	13	s	s	PRON
ejpam-4594	44	14	itself	itself	PRON
ejpam-4594	44	15	is	be	AUX
ejpam-4594	44	16	a	a	DET
ejpam-4594	44	17	dual	dual	ADJ
ejpam-4594	44	18	b	b	NOUN
ejpam-4594	44	19	-	-	PUNCT
ejpam-4594	44	20	algebra	algebra	NOUN
ejpam-4594	44	21	with	with	ADP
ejpam-4594	44	22	binary	binary	ADJ
ejpam-4594	44	23	operation	operation	NOUN
ejpam-4594	44	24	of	of	ADP
ejpam-4594	44	25	xd	xd	INTJ
ejpam-4594	44	26	on	on	ADP
ejpam-4594	44	27	s.	s.	PROPN
ejpam-4594	44	28	definition	definition	NOUN
ejpam-4594	44	29	4	4	NUM
ejpam-4594	44	30	.	.	PUNCT
ejpam-4594	45	1	[	[	X
ejpam-4594	45	2	1	1	X
ejpam-4594	45	3	]	]	PUNCT
ejpam-4594	45	4	let	let	VERB
ejpam-4594	45	5	xd	xd	INTJ
ejpam-4594	45	6	be	be	AUX
ejpam-4594	45	7	a	a	DET
ejpam-4594	45	8	dual	dual	ADJ
ejpam-4594	45	9	b	b	NOUN
ejpam-4594	45	10	-	-	PUNCT
ejpam-4594	45	11	algebra	algebra	NOUN
ejpam-4594	45	12	.	.	PUNCT
ejpam-4594	46	1	a	a	DET
ejpam-4594	46	2	subset	subset	NOUN
ejpam-4594	46	3	f	f	NOUN
ejpam-4594	46	4	of	of	ADP
ejpam-4594	46	5	xd	xd	INTJ
ejpam-4594	46	6	is	be	AUX
ejpam-4594	46	7	called	call	VERB
ejpam-4594	46	8	a	a	DET
ejpam-4594	46	9	dual	dual	ADJ
ejpam-4594	46	10	b	b	NOUN
ejpam-4594	46	11	-	-	NOUN
ejpam-4594	46	12	filter	filter	NOUN
ejpam-4594	46	13	if	if	SCONJ
ejpam-4594	46	14	it	it	PRON
ejpam-4594	46	15	satisfies	satisfy	VERB
ejpam-4594	46	16	the	the	DET
ejpam-4594	46	17	following	follow	VERB
ejpam-4594	46	18	axioms	axiom	NOUN
ejpam-4594	46	19	:	:	PUNCT
ejpam-4594	46	20	for	for	ADP
ejpam-4594	46	21	all	all	DET
ejpam-4594	46	22	x	x	NOUN
ejpam-4594	46	23	,	,	PUNCT
ejpam-4594	46	24	y	y	PROPN
ejpam-4594	46	25	in	in	ADP
ejpam-4594	46	26	xd	xd	ADP
ejpam-4594	46	27	,	,	PUNCT
ejpam-4594	46	28	(	(	PUNCT
ejpam-4594	46	29	df1	df1	NOUN
ejpam-4594	46	30	)	)	PUNCT
ejpam-4594	46	31	1	1	NUM
ejpam-4594	46	32	∈	∈	PROPN
ejpam-4594	46	33	f	f	NOUN
ejpam-4594	46	34	;	;	PUNCT
ejpam-4594	46	35	(	(	PUNCT
ejpam-4594	46	36	df2	df2	NOUN
ejpam-4594	46	37	)	)	PUNCT
ejpam-4594	47	1	x	x	VERB
ejpam-4594	47	2	◦	◦	VERB
ejpam-4594	47	3	y	y	PROPN
ejpam-4594	47	4	∈	∈	PROPN
ejpam-4594	47	5	f	f	PROPN
ejpam-4594	47	6	and	and	CCONJ
ejpam-4594	47	7	x	x	PROPN
ejpam-4594	47	8	∈	∈	NOUN
ejpam-4594	48	1	f	f	X
ejpam-4594	48	2	imply	imply	VERB
ejpam-4594	48	3	y	y	PROPN
ejpam-4594	48	4	∈	∈	PROPN
ejpam-4594	48	5	f.	f.	PROPN
ejpam-4594	48	6	definition	definition	NOUN
ejpam-4594	48	7	5	5	NUM
ejpam-4594	48	8	.	.	PUNCT
ejpam-4594	49	1	[	[	X
ejpam-4594	49	2	3	3	X
ejpam-4594	49	3	]	]	X
ejpam-4594	49	4	let	let	VERB
ejpam-4594	49	5	x	x	PRON
ejpam-4594	49	6	be	be	AUX
ejpam-4594	49	7	a	a	DET
ejpam-4594	49	8	set	set	NOUN
ejpam-4594	49	9	.	.	PUNCT
ejpam-4594	50	1	a	a	DET
ejpam-4594	50	2	topology	topology	NOUN
ejpam-4594	50	3	(	(	PUNCT
ejpam-4594	50	4	or	or	CCONJ
ejpam-4594	50	5	topological	topological	ADJ
ejpam-4594	50	6	structure	structure	NOUN
ejpam-4594	50	7	)	)	PUNCT
ejpam-4594	50	8	in	in	ADP
ejpam-4594	50	9	x	x	PRON
ejpam-4594	50	10	is	be	AUX
ejpam-4594	50	11	a	a	DET
ejpam-4594	50	12	family	family	NOUN
ejpam-4594	50	13	τ	τ	PROPN
ejpam-4594	50	14	of	of	ADP
ejpam-4594	50	15	subsets	subset	NOUN
ejpam-4594	50	16	of	of	ADP
ejpam-4594	50	17	x	x	PRON
ejpam-4594	50	18	that	that	PRON
ejpam-4594	50	19	satisfies	satisfy	VERB
ejpam-4594	50	20	the	the	DET
ejpam-4594	50	21	following	following	NOUN
ejpam-4594	50	22	:	:	PUNCT
ejpam-4594	50	23	(	(	PUNCT
ejpam-4594	50	24	i	i	NOUN
ejpam-4594	50	25	)	)	PUNCT
ejpam-4594	50	26	each	each	DET
ejpam-4594	50	27	union	union	NOUN
ejpam-4594	50	28	of	of	ADP
ejpam-4594	50	29	members	member	NOUN
ejpam-4594	50	30	of	of	ADP
ejpam-4594	50	31	τ	τ	PROPN
ejpam-4594	50	32	is	be	AUX
ejpam-4594	50	33	also	also	ADV
ejpam-4594	50	34	a	a	DET
ejpam-4594	50	35	member	member	NOUN
ejpam-4594	50	36	of	of	ADP
ejpam-4594	50	37	τ	τ	PROPN
ejpam-4594	50	38	;	;	PUNCT
ejpam-4594	50	39	(	(	PUNCT
ejpam-4594	50	40	ii	ii	NOUN
ejpam-4594	50	41	)	)	PUNCT
ejpam-4594	50	42	each	each	DET
ejpam-4594	50	43	finite	finite	ADJ
ejpam-4594	50	44	intersection	intersection	NOUN
ejpam-4594	50	45	of	of	ADP
ejpam-4594	50	46	members	member	NOUN
ejpam-4594	50	47	of	of	ADP
ejpam-4594	50	48	τ	τ	PROPN
ejpam-4594	50	49	is	be	AUX
ejpam-4594	50	50	also	also	ADV
ejpam-4594	50	51	a	a	DET
ejpam-4594	50	52	member	member	NOUN
ejpam-4594	50	53	of	of	ADP
ejpam-4594	50	54	τ	τ	PROPN
ejpam-4594	50	55	;	;	PUNCT
ejpam-4594	50	56	and	and	CCONJ
ejpam-4594	50	57	(	(	PUNCT
ejpam-4594	50	58	iii	iii	X
ejpam-4594	50	59	)	)	PUNCT
ejpam-4594	50	60	∅	∅	NOUN
ejpam-4594	50	61	and	and	CCONJ
ejpam-4594	50	62	x	x	X
ejpam-4594	50	63	are	be	AUX
ejpam-4594	50	64	members	member	NOUN
ejpam-4594	50	65	of	of	ADP
ejpam-4594	50	66	τ	τ	PROPN
ejpam-4594	50	67	.	.	PUNCT
ejpam-4594	51	1	a	a	DET
ejpam-4594	51	2	couple	couple	NOUN
ejpam-4594	51	3	(	(	PUNCT
ejpam-4594	51	4	x	x	NOUN
ejpam-4594	51	5	,	,	PUNCT
ejpam-4594	51	6	τ	τ	X
ejpam-4594	51	7	)	)	PUNCT
ejpam-4594	51	8	consisting	consist	VERB
ejpam-4594	51	9	of	of	ADP
ejpam-4594	51	10	a	a	DET
ejpam-4594	51	11	set	set	NOUN
ejpam-4594	51	12	x	x	PUNCT
ejpam-4594	51	13	and	and	CCONJ
ejpam-4594	51	14	a	a	DET
ejpam-4594	51	15	topology	topology	NOUN
ejpam-4594	51	16	τ	τ	PROPN
ejpam-4594	51	17	in	in	ADP
ejpam-4594	51	18	x	x	PROPN
ejpam-4594	51	19	is	be	AUX
ejpam-4594	51	20	called	call	VERB
ejpam-4594	51	21	a	a	DET
ejpam-4594	51	22	topological	topological	ADJ
ejpam-4594	51	23	space	space	NOUN
ejpam-4594	51	24	.	.	PUNCT
ejpam-4594	52	1	we	we	PRON
ejpam-4594	52	2	also	also	ADV
ejpam-4594	52	3	say	say	VERB
ejpam-4594	52	4	“	"	PUNCT
ejpam-4594	52	5	τ	τ	X
ejpam-4594	52	6	is	be	AUX
ejpam-4594	52	7	the	the	DET
ejpam-4594	52	8	topology	topology	NOUN
ejpam-4594	52	9	of	of	ADP
ejpam-4594	52	10	the	the	DET
ejpam-4594	52	11	space	space	NOUN
ejpam-4594	52	12	x	x	NOUN
ejpam-4594	52	13	”	"	PUNCT
ejpam-4594	52	14	.	.	PUNCT
ejpam-4594	53	1	the	the	DET
ejpam-4594	53	2	members	member	NOUN
ejpam-4594	53	3	of	of	ADP
ejpam-4594	53	4	τ	τ	PROPN
ejpam-4594	53	5	are	be	AUX
ejpam-4594	53	6	called	call	VERB
ejpam-4594	53	7	open	open	ADJ
ejpam-4594	53	8	sets	set	NOUN
ejpam-4594	53	9	of	of	ADP
ejpam-4594	53	10	(	(	PUNCT
ejpam-4594	53	11	x	x	NOUN
ejpam-4594	53	12	,	,	PUNCT
ejpam-4594	53	13	τ	τ	PROPN
ejpam-4594	53	14	)	)	PUNCT
ejpam-4594	53	15	.	.	PUNCT
ejpam-4594	54	1	a	a	DET
ejpam-4594	54	2	family	family	NOUN
ejpam-4594	54	3	b	b	PROPN
ejpam-4594	54	4	⊂	⊂	PROPN
ejpam-4594	54	5	τ	τ	PROPN
ejpam-4594	54	6	is	be	AUX
ejpam-4594	54	7	called	call	VERB
ejpam-4594	54	8	a	a	DET
ejpam-4594	54	9	basis	basis	NOUN
ejpam-4594	54	10	for	for	ADP
ejpam-4594	54	11	τ	τ	PROPN
ejpam-4594	54	12	if	if	SCONJ
ejpam-4594	54	13	each	each	DET
ejpam-4594	54	14	open	open	ADJ
ejpam-4594	54	15	set	set	NOUN
ejpam-4594	54	16	is	be	AUX
ejpam-4594	54	17	the	the	DET
ejpam-4594	54	18	union	union	NOUN
ejpam-4594	54	19	of	of	ADP
ejpam-4594	54	20	members	member	NOUN
ejpam-4594	54	21	of	of	ADP
ejpam-4594	54	22	b.	b.	PROPN
ejpam-4594	54	23	let	let	AUX
ejpam-4594	54	24	(	(	PUNCT
ejpam-4594	54	25	x	x	NOUN
ejpam-4594	54	26	,	,	PUNCT
ejpam-4594	54	27	τx	τx	ADJ
ejpam-4594	54	28	)	)	PUNCT
ejpam-4594	54	29	and	and	CCONJ
ejpam-4594	54	30	(	(	PUNCT
ejpam-4594	54	31	y	y	PROPN
ejpam-4594	54	32	,	,	PUNCT
ejpam-4594	54	33	τy	τy	PART
ejpam-4594	54	34	)	)	PUNCT
ejpam-4594	54	35	be	be	AUX
ejpam-4594	54	36	topological	topological	ADJ
ejpam-4594	54	37	spaces	space	NOUN
ejpam-4594	54	38	.	.	PUNCT
ejpam-4594	55	1	a	a	DET
ejpam-4594	55	2	map	map	NOUN
ejpam-4594	55	3	f	f	X
ejpam-4594	55	4	:	:	PUNCT
ejpam-4594	55	5	x	x	X
ejpam-4594	55	6	→	→	SYM
ejpam-4594	55	7	y	y	PROPN
ejpam-4594	55	8	is	be	AUX
ejpam-4594	55	9	called	call	VERB
ejpam-4594	55	10	continuous	continuous	ADJ
ejpam-4594	55	11	if	if	SCONJ
ejpam-4594	55	12	the	the	DET
ejpam-4594	55	13	inverse	inverse	ADJ
ejpam-4594	55	14	image	image	NOUN
ejpam-4594	55	15	of	of	ADP
ejpam-4594	55	16	each	each	DET
ejpam-4594	55	17	open	open	ADJ
ejpam-4594	55	18	set	set	NOUN
ejpam-4594	55	19	in	in	ADP
ejpam-4594	55	20	y	y	PROPN
ejpam-4594	55	21	is	be	AUX
ejpam-4594	55	22	open	open	ADJ
ejpam-4594	55	23	in	in	ADP
ejpam-4594	55	24	x	x	PROPN
ejpam-4594	55	25	(	(	PUNCT
ejpam-4594	55	26	that	that	PRON
ejpam-4594	55	27	is	is	ADV
ejpam-4594	55	28	,	,	PUNCT
ejpam-4594	55	29	if	if	SCONJ
ejpam-4594	55	30	f−1	f−1	PROPN
ejpam-4594	55	31	maps	map	VERB
ejpam-4594	55	32	τy	τy	NOUN
ejpam-4594	55	33	into	into	ADP
ejpam-4594	55	34	τx	τx	NOUN
ejpam-4594	55	35	)	)	PUNCT
ejpam-4594	55	36	.	.	PUNCT
ejpam-4594	56	1	[	[	X
ejpam-4594	56	2	3	3	NUM
ejpam-4594	56	3	]	]	X
ejpam-4594	56	4	theorem	theorem	NOUN
ejpam-4594	56	5	2	2	NUM
ejpam-4594	56	6	.	.	PUNCT
ejpam-4594	57	1	[	[	X
ejpam-4594	57	2	3	3	X
ejpam-4594	57	3	]	]	X
ejpam-4594	57	4	let	let	VERB
ejpam-4594	57	5	b	b	PROPN
ejpam-4594	57	6	⊂	⊂	PROPN
ejpam-4594	57	7	τ	τ	PROPN
ejpam-4594	57	8	.	.	PUNCT
ejpam-4594	58	1	the	the	DET
ejpam-4594	58	2	following	follow	VERB
ejpam-4594	58	3	two	two	NUM
ejpam-4594	58	4	properties	property	NOUN
ejpam-4594	58	5	of	of	ADP
ejpam-4594	58	6	b	b	NOUN
ejpam-4594	58	7	are	be	AUX
ejpam-4594	58	8	equivalent	equivalent	ADJ
ejpam-4594	58	9	:	:	PUNCT
ejpam-4594	58	10	(	(	PUNCT
ejpam-4594	58	11	i	i	NOUN
ejpam-4594	58	12	)	)	PUNCT
ejpam-4594	58	13	b	b	PROPN
ejpam-4594	58	14	is	be	AUX
ejpam-4594	58	15	a	a	DET
ejpam-4594	58	16	basis	basis	NOUN
ejpam-4594	58	17	for	for	ADP
ejpam-4594	58	18	τ	τ	PROPN
ejpam-4594	58	19	;	;	PUNCT
ejpam-4594	58	20	(	(	PUNCT
ejpam-4594	58	21	ii	ii	NOUN
ejpam-4594	58	22	)	)	PUNCT
ejpam-4594	58	23	for	for	ADP
ejpam-4594	58	24	each	each	DET
ejpam-4594	58	25	g	g	PROPN
ejpam-4594	58	26	∈	∈	PROPN
ejpam-4594	58	27	τ	τ	X
ejpam-4594	58	28	and	and	CCONJ
ejpam-4594	58	29	each	each	DET
ejpam-4594	58	30	x	x	SYM
ejpam-4594	58	31	∈	∈	PROPN
ejpam-4594	58	32	g	g	NOUN
ejpam-4594	58	33	,	,	PUNCT
ejpam-4594	58	34	there	there	PRON
ejpam-4594	58	35	is	be	VERB
ejpam-4594	58	36	a	a	DET
ejpam-4594	58	37	u	u	NOUN
ejpam-4594	58	38	∈	∈	PROPN
ejpam-4594	58	39	b	b	NOUN
ejpam-4594	58	40	with	with	ADP
ejpam-4594	58	41	x	x	PROPN
ejpam-4594	58	42	∈	∈	PROPN
ejpam-4594	58	43	u	u	NOUN
ejpam-4594	58	44	⊂	⊂	PROPN
ejpam-4594	58	45	g.	g.	PROPN
ejpam-4594	58	46	k.	k.	PROPN
ejpam-4594	58	47	belleza	belleza	PROPN
ejpam-4594	58	48	,	,	PUNCT
ejpam-4594	58	49	j.	j.	PROPN
ejpam-4594	58	50	albaracin	albaracin	PROPN
ejpam-4594	58	51	/	/	PUNCT
ejpam-4594	58	52	eur	eur	PROPN
ejpam-4594	58	53	.	.	PUNCT
ejpam-4594	59	1	j.	j.	PROPN
ejpam-4594	59	2	pure	pure	PROPN
ejpam-4594	59	3	appl	appl	PROPN
ejpam-4594	59	4	.	.	PROPN
ejpam-4594	59	5	math	math	PROPN
ejpam-4594	59	6	,	,	PUNCT
ejpam-4594	59	7	15	15	NUM
ejpam-4594	59	8	(	(	PUNCT
ejpam-4594	59	9	4	4	NUM
ejpam-4594	59	10	)	)	PUNCT
ejpam-4594	59	11	(	(	PUNCT
ejpam-4594	59	12	2022	2022	NUM
ejpam-4594	59	13	)	)	PUNCT
ejpam-4594	59	14	,	,	PUNCT
ejpam-4594	59	15	1948	1948	NUM
ejpam-4594	59	16	-	-	SYM
ejpam-4594	59	17	1956	1956	NUM
ejpam-4594	59	18	1950	1950	NUM
ejpam-4594	59	19	definition	definition	NOUN
ejpam-4594	59	20	6	6	NUM
ejpam-4594	59	21	.	.	PUNCT
ejpam-4594	60	1	[	[	X
ejpam-4594	60	2	3	3	X
ejpam-4594	60	3	]	]	X
ejpam-4594	60	4	let	let	VERB
ejpam-4594	60	5	(	(	PUNCT
ejpam-4594	60	6	x	x	NOUN
ejpam-4594	60	7	,	,	PUNCT
ejpam-4594	60	8	τ	τ	X
ejpam-4594	60	9	)	)	PUNCT
ejpam-4594	60	10	be	be	VERB
ejpam-4594	60	11	a	a	DET
ejpam-4594	60	12	topological	topological	ADJ
ejpam-4594	60	13	space	space	NOUN
ejpam-4594	60	14	.	.	PUNCT
ejpam-4594	61	1	by	by	ADP
ejpam-4594	61	2	a	a	DET
ejpam-4594	61	3	neighborhood	neighborhood	NOUN
ejpam-4594	61	4	of	of	ADP
ejpam-4594	61	5	an	an	DET
ejpam-4594	61	6	element	element	NOUN
ejpam-4594	61	7	x	x	PUNCT
ejpam-4594	61	8	in	in	ADP
ejpam-4594	61	9	x	x	PROPN
ejpam-4594	61	10	(	(	PUNCT
ejpam-4594	61	11	denoted	denote	VERB
ejpam-4594	61	12	as	as	ADP
ejpam-4594	61	13	u(x	u(x	NOUN
ejpam-4594	61	14	)	)	PUNCT
ejpam-4594	61	15	)	)	PUNCT
ejpam-4594	61	16	is	be	AUX
ejpam-4594	61	17	meant	mean	VERB
ejpam-4594	61	18	any	any	DET
ejpam-4594	61	19	open	open	ADJ
ejpam-4594	61	20	set	set	NOUN
ejpam-4594	61	21	(	(	PUNCT
ejpam-4594	61	22	that	that	PRON
ejpam-4594	61	23	is	is	ADV
ejpam-4594	61	24	,	,	PUNCT
ejpam-4594	61	25	member	member	NOUN
ejpam-4594	61	26	of	of	ADP
ejpam-4594	61	27	τ	τ	PROPN
ejpam-4594	61	28	)	)	PUNCT
ejpam-4594	61	29	containing	contain	VERB
ejpam-4594	61	30	x.	x.	NOUN
ejpam-4594	61	31	definition	definition	NOUN
ejpam-4594	61	32	7	7	NUM
ejpam-4594	61	33	.	.	PUNCT
ejpam-4594	62	1	[	[	X
ejpam-4594	62	2	3	3	X
ejpam-4594	62	3	]	]	X
ejpam-4594	62	4	let	let	VERB
ejpam-4594	62	5	{	{	PUNCT
ejpam-4594	62	6	yα	yα	VERB
ejpam-4594	62	7	|	|	ADV
ejpam-4594	62	8	α	α	PROPN
ejpam-4594	62	9	∈	∈	PROPN
ejpam-4594	62	10	a	a	DET
ejpam-4594	62	11	}	}	PUNCT
ejpam-4594	62	12	be	be	AUX
ejpam-4594	62	13	any	any	DET
ejpam-4594	62	14	family	family	NOUN
ejpam-4594	62	15	of	of	ADP
ejpam-4594	62	16	topological	topological	ADJ
ejpam-4594	62	17	spaces	space	NOUN
ejpam-4594	62	18	.	.	PUNCT
ejpam-4594	63	1	for	for	ADP
ejpam-4594	63	2	each	each	DET
ejpam-4594	63	3	α	α	NOUN
ejpam-4594	63	4	∈	∈	PROPN
ejpam-4594	63	5	a	a	PRON
ejpam-4594	63	6	,	,	PUNCT
ejpam-4594	63	7	let	let	VERB
ejpam-4594	63	8	τα	τα	PRON
ejpam-4594	63	9	be	be	AUX
ejpam-4594	63	10	the	the	DET
ejpam-4594	63	11	topology	topology	NOUN
ejpam-4594	63	12	for	for	ADP
ejpam-4594	63	13	yα	yα	NOUN
ejpam-4594	63	14	.	.	PUNCT
ejpam-4594	64	1	the	the	DET
ejpam-4594	64	2	cartesian	cartesian	ADJ
ejpam-4594	64	3	product	product	NOUN
ejpam-4594	64	4	topology	topology	NOUN
ejpam-4594	64	5	in	in	ADP
ejpam-4594	64	6	∏	∏	PROPN
ejpam-4594	64	7	α	α	PROPN
ejpam-4594	64	8	yα	yα	NOUN
ejpam-4594	64	9	is	be	AUX
ejpam-4594	64	10	that	that	SCONJ
ejpam-4594	64	11	having	have	VERB
ejpam-4594	64	12	for	for	ADP
ejpam-4594	64	13	subbasis	subbasis	NOUN
ejpam-4594	64	14	all	all	DET
ejpam-4594	64	15	sets	set	NOUN
ejpam-4594	64	16	⟨uβ⟩	⟨uβ⟩	NOUN
ejpam-4594	64	17	=	=	SYM
ejpam-4594	65	1	ρ−1	ρ−1	PROPN
ejpam-4594	65	2	β	β	X
ejpam-4594	65	3	(	(	PUNCT
ejpam-4594	65	4	uβ	uβ	PROPN
ejpam-4594	65	5	)	)	PUNCT
ejpam-4594	65	6	,	,	PUNCT
ejpam-4594	65	7	where	where	SCONJ
ejpam-4594	65	8	ρ	ρ	NOUN
ejpam-4594	65	9	:	:	PUNCT
ejpam-4594	65	10	∏	∏	PROPN
ejpam-4594	65	11	α	α	NOUN
ejpam-4594	65	12	yα	yα	NOUN
ejpam-4594	65	13	→	→	SYM
ejpam-4594	65	14	yα	yα	NOUN
ejpam-4594	65	15	,	,	PUNCT
ejpam-4594	65	16	uβ	uβ	PROPN
ejpam-4594	65	17	ranges	range	VERB
ejpam-4594	65	18	over	over	ADP
ejpam-4594	65	19	all	all	DET
ejpam-4594	65	20	members	member	NOUN
ejpam-4594	65	21	of	of	ADP
ejpam-4594	65	22	τβ	τβ	NOUN
ejpam-4594	65	23	and	and	CCONJ
ejpam-4594	65	24	β	β	X
ejpam-4594	65	25	over	over	ADP
ejpam-4594	65	26	all	all	DET
ejpam-4594	65	27	elements	element	NOUN
ejpam-4594	65	28	of	of	ADP
ejpam-4594	65	29	a.	a.	NOUN
ejpam-4594	65	30	definition	definition	NOUN
ejpam-4594	65	31	8	8	NUM
ejpam-4594	65	32	.	.	PUNCT
ejpam-4594	66	1	[	[	X
ejpam-4594	66	2	1	1	X
ejpam-4594	66	3	]	]	PUNCT
ejpam-4594	66	4	let	let	VERB
ejpam-4594	66	5	xd	xd	INTJ
ejpam-4594	66	6	be	be	AUX
ejpam-4594	66	7	a	a	DET
ejpam-4594	66	8	dual	dual	ADJ
ejpam-4594	66	9	b	b	NOUN
ejpam-4594	66	10	-	-	PUNCT
ejpam-4594	66	11	algebra	algebra	NOUN
ejpam-4594	66	12	.	.	PUNCT
ejpam-4594	67	1	a	a	DET
ejpam-4594	67	2	topology	topology	NOUN
ejpam-4594	67	3	τ	τ	PROPN
ejpam-4594	67	4	on	on	ADP
ejpam-4594	67	5	xd	xd	INTJ
ejpam-4594	67	6	is	be	AUX
ejpam-4594	67	7	called	call	VERB
ejpam-4594	67	8	a	a	DET
ejpam-4594	67	9	dual	dual	ADJ
ejpam-4594	67	10	b	b	NOUN
ejpam-4594	67	11	-	-	NOUN
ejpam-4594	67	12	topology	topology	NOUN
ejpam-4594	67	13	and	and	CCONJ
ejpam-4594	67	14	the	the	DET
ejpam-4594	67	15	couple	couple	NOUN
ejpam-4594	67	16	(	(	PUNCT
ejpam-4594	67	17	xd	xd	ADV
ejpam-4594	67	18	,	,	PUNCT
ejpam-4594	67	19	τ	τ	PROPN
ejpam-4594	67	20	)	)	PUNCT
ejpam-4594	67	21	is	be	AUX
ejpam-4594	67	22	called	call	VERB
ejpam-4594	67	23	a	a	DET
ejpam-4594	67	24	dual	dual	ADJ
ejpam-4594	67	25	b	b	NOUN
ejpam-4594	67	26	-	-	PUNCT
ejpam-4594	67	27	topological	topological	ADJ
ejpam-4594	67	28	space	space	NOUN
ejpam-4594	67	29	.	.	PUNCT
ejpam-4594	68	1	remark	remark	PROPN
ejpam-4594	68	2	1	1	NUM
ejpam-4594	68	3	.	.	PUNCT
ejpam-4594	69	1	let	let	VERB
ejpam-4594	69	2	xd	xd	INTJ
ejpam-4594	69	3	be	be	AUX
ejpam-4594	69	4	a	a	DET
ejpam-4594	69	5	dual	dual	ADJ
ejpam-4594	69	6	b	b	NOUN
ejpam-4594	69	7	-	-	PUNCT
ejpam-4594	69	8	algebra	algebra	NOUN
ejpam-4594	69	9	and	and	CCONJ
ejpam-4594	69	10	nonempty	nonempty	X
ejpam-4594	69	11	a	a	DET
ejpam-4594	69	12	,	,	PUNCT
ejpam-4594	69	13	b	b	PROPN
ejpam-4594	69	14	∈	∈	PROPN
ejpam-4594	69	15	xd	xd	PROPN
ejpam-4594	69	16	.	.	PUNCT
ejpam-4594	70	1	then	then	ADV
ejpam-4594	70	2	a	a	DET
ejpam-4594	70	3	◦	◦	NOUN
ejpam-4594	70	4	b	b	NOUN
ejpam-4594	70	5	=	=	PUNCT
ejpam-4594	70	6	{	{	PUNCT
ejpam-4594	70	7	a	a	DET
ejpam-4594	70	8	◦	◦	NOUN
ejpam-4594	70	9	b	b	NOUN
ejpam-4594	70	10	|	|	ADV
ejpam-4594	70	11	a	a	DET
ejpam-4594	70	12	∈	∈	PROPN
ejpam-4594	70	13	a	a	PRON
ejpam-4594	70	14	,	,	PUNCT
ejpam-4594	70	15	b	b	PROPN
ejpam-4594	70	16	∈	∈	PROPN
ejpam-4594	70	17	b	b	NOUN
ejpam-4594	70	18	}	}	PUNCT
ejpam-4594	70	19	.	.	PUNCT
ejpam-4594	71	1	definition	definition	NOUN
ejpam-4594	71	2	9	9	NUM
ejpam-4594	71	3	.	.	PUNCT
ejpam-4594	72	1	[	[	X
ejpam-4594	72	2	1	1	X
ejpam-4594	72	3	]	]	PUNCT
ejpam-4594	72	4	the	the	DET
ejpam-4594	72	5	triple	triple	ADJ
ejpam-4594	72	6	(	(	PUNCT
ejpam-4594	72	7	xd	xd	ADV
ejpam-4594	72	8	,	,	PUNCT
ejpam-4594	72	9	◦	◦	NOUN
ejpam-4594	72	10	,	,	PUNCT
ejpam-4594	72	11	τ	τ	X
ejpam-4594	72	12	)	)	PUNCT
ejpam-4594	72	13	is	be	AUX
ejpam-4594	72	14	called	call	VERB
ejpam-4594	72	15	a	a	DET
ejpam-4594	72	16	topological	topological	ADJ
ejpam-4594	72	17	dual	dual	ADJ
ejpam-4594	72	18	b	b	NOUN
ejpam-4594	72	19	-	-	PUNCT
ejpam-4594	72	20	algebra	algebra	NOUN
ejpam-4594	72	21	(	(	PUNCT
ejpam-4594	72	22	or	or	CCONJ
ejpam-4594	72	23	tdbalgebra	tdbalgebra	NOUN
ejpam-4594	72	24	)	)	PUNCT
ejpam-4594	72	25	if	if	SCONJ
ejpam-4594	72	26	τ	τ	PROPN
ejpam-4594	72	27	is	be	AUX
ejpam-4594	72	28	a	a	DET
ejpam-4594	72	29	dual	dual	ADJ
ejpam-4594	72	30	b	b	NOUN
ejpam-4594	72	31	-	-	NOUN
ejpam-4594	72	32	topology	topology	NOUN
ejpam-4594	72	33	and	and	CCONJ
ejpam-4594	72	34	the	the	DET
ejpam-4594	72	35	binary	binary	PROPN
ejpam-4594	72	36	operation	operation	NOUN
ejpam-4594	72	37	◦	◦	NOUN
ejpam-4594	72	38	:	:	PUNCT
ejpam-4594	72	39	xd	xd	INTJ
ejpam-4594	72	40	×	×	INTJ
ejpam-4594	72	41	xd	xd	INTJ
ejpam-4594	72	42	→	→	SYM
ejpam-4594	72	43	xd	xd	INTJ
ejpam-4594	72	44	is	be	AUX
ejpam-4594	72	45	continuous	continuous	ADJ
ejpam-4594	72	46	where	where	SCONJ
ejpam-4594	72	47	the	the	DET
ejpam-4594	72	48	topology	topology	NOUN
ejpam-4594	72	49	on	on	ADP
ejpam-4594	72	50	xd	xd	INTJ
ejpam-4594	72	51	×xd	×xd	PROPN
ejpam-4594	72	52	is	be	AUX
ejpam-4594	72	53	the	the	DET
ejpam-4594	72	54	cartesian	cartesian	ADJ
ejpam-4594	72	55	product	product	NOUN
ejpam-4594	72	56	topology	topology	NOUN
ejpam-4594	72	57	.	.	PUNCT
ejpam-4594	73	1	theorem	theorem	VERB
ejpam-4594	73	2	3	3	NUM
ejpam-4594	73	3	.	.	PUNCT
ejpam-4594	74	1	[	[	X
ejpam-4594	74	2	1	1	X
ejpam-4594	74	3	]	]	PUNCT
ejpam-4594	74	4	let	let	VERB
ejpam-4594	74	5	xd	xd	INTJ
ejpam-4594	74	6	be	be	AUX
ejpam-4594	74	7	a	a	DET
ejpam-4594	74	8	dual	dual	ADJ
ejpam-4594	74	9	b	b	NOUN
ejpam-4594	74	10	-	-	PUNCT
ejpam-4594	74	11	algebra	algebra	NOUN
ejpam-4594	74	12	and	and	CCONJ
ejpam-4594	74	13	τ	τ	PROPN
ejpam-4594	74	14	a	a	DET
ejpam-4594	74	15	dual	dual	ADJ
ejpam-4594	74	16	b	b	NOUN
ejpam-4594	74	17	-	-	NOUN
ejpam-4594	74	18	topology	topology	NOUN
ejpam-4594	74	19	.	.	PUNCT
ejpam-4594	75	1	then	then	ADV
ejpam-4594	75	2	(	(	PUNCT
ejpam-4594	75	3	xd	xd	INTJ
ejpam-4594	75	4	,	,	PUNCT
ejpam-4594	75	5	◦	◦	NOUN
ejpam-4594	75	6	,	,	PUNCT
ejpam-4594	75	7	τ	τ	X
ejpam-4594	75	8	)	)	PUNCT
ejpam-4594	75	9	is	be	AUX
ejpam-4594	75	10	a	a	DET
ejpam-4594	75	11	tdb	tdb	NOUN
ejpam-4594	75	12	-	-	NOUN
ejpam-4594	75	13	algebra	algebra	NOUN
ejpam-4594	75	14	if	if	SCONJ
ejpam-4594	75	15	and	and	CCONJ
ejpam-4594	75	16	only	only	ADV
ejpam-4594	75	17	if	if	SCONJ
ejpam-4594	75	18	for	for	ADP
ejpam-4594	75	19	all	all	DET
ejpam-4594	75	20	x	x	NOUN
ejpam-4594	75	21	,	,	PUNCT
ejpam-4594	75	22	y	y	PROPN
ejpam-4594	75	23	∈	∈	PROPN
ejpam-4594	75	24	xd	xd	INTJ
ejpam-4594	75	25	and	and	CCONJ
ejpam-4594	75	26	u(x	u(x	VERB
ejpam-4594	75	27	◦	◦	NOUN
ejpam-4594	75	28	y	y	PROPN
ejpam-4594	75	29	)	)	PUNCT
ejpam-4594	75	30	,	,	PUNCT
ejpam-4594	75	31	there	there	PRON
ejpam-4594	75	32	exists	exist	VERB
ejpam-4594	75	33	u(x	u(x	NOUN
ejpam-4594	75	34	)	)	PUNCT
ejpam-4594	75	35	and	and	CCONJ
ejpam-4594	75	36	u(y	u(y	NOUN
ejpam-4594	75	37	)	)	PUNCT
ejpam-4594	75	38	such	such	ADJ
ejpam-4594	75	39	that	that	SCONJ
ejpam-4594	75	40	u(x	u(x	NOUN
ejpam-4594	75	41	)	)	PUNCT
ejpam-4594	75	42	◦	◦	NOUN
ejpam-4594	75	43	u(y	u(y	NOUN
ejpam-4594	75	44	)	)	PUNCT
ejpam-4594	75	45	⊆	⊆	NUM
ejpam-4594	75	46	u(x	u(x	PROPN
ejpam-4594	75	47	◦	◦	NOUN
ejpam-4594	75	48	y	y	NOUN
ejpam-4594	75	49	)	)	PUNCT
ejpam-4594	75	50	.	.	PUNCT
ejpam-4594	76	1	3	3	X
ejpam-4594	76	2	.	.	X
ejpam-4594	76	3	dual	dual	ADJ
ejpam-4594	76	4	b	b	X
ejpam-4594	76	5	-	-	PUNCT
ejpam-4594	76	6	topological	topological	ADJ
ejpam-4594	76	7	space	space	NOUN
ejpam-4594	76	8	determined	determine	VERB
ejpam-4594	76	9	by	by	ADP
ejpam-4594	76	10	some	some	DET
ejpam-4594	76	11	sets	set	NOUN
ejpam-4594	76	12	suppose	suppose	VERB
ejpam-4594	76	13	xd	xd	INTJ
ejpam-4594	76	14	is	be	VERB
ejpam-4594	76	15	a	a	DET
ejpam-4594	76	16	dual	dual	ADJ
ejpam-4594	76	17	b	b	NOUN
ejpam-4594	76	18	-	-	PUNCT
ejpam-4594	76	19	algebra	algebra	NOUN
ejpam-4594	76	20	.	.	PUNCT
ejpam-4594	77	1	for	for	ADP
ejpam-4594	77	2	each	each	PRON
ejpam-4594	77	3	v	v	ADP
ejpam-4594	77	4	⊆	⊆	NUM
ejpam-4594	77	5	xd	xd	ADV
ejpam-4594	77	6	and	and	CCONJ
ejpam-4594	77	7	x	x	PUNCT
ejpam-4594	77	8	∈	∈	PROPN
ejpam-4594	77	9	xd	xd	ADP
ejpam-4594	77	10	,	,	PUNCT
ejpam-4594	77	11	let	let	VERB
ejpam-4594	77	12	us	we	PRON
ejpam-4594	77	13	denote	denote	VERB
ejpam-4594	77	14	the	the	DET
ejpam-4594	77	15	following	following	ADJ
ejpam-4594	77	16	notations	notation	NOUN
ejpam-4594	77	17	:	:	PUNCT
ejpam-4594	77	18	(	(	PUNCT
ejpam-4594	77	19	i	i	NOUN
ejpam-4594	77	20	)	)	PUNCT
ejpam-4594	77	21	v	v	PART
ejpam-4594	78	1	[	[	X
ejpam-4594	79	1	x	x	X
ejpam-4594	79	2	]	]	X
ejpam-4594	79	3	=	=	X
ejpam-4594	79	4	{	{	PUNCT
ejpam-4594	79	5	y	y	PROPN
ejpam-4594	79	6	∈	∈	PROPN
ejpam-4594	79	7	xd	xd	INTJ
ejpam-4594	79	8	|	|	ADV
ejpam-4594	79	9	x	x	PART
ejpam-4594	79	10	◦	◦	VERB
ejpam-4594	79	11	y	y	PROPN
ejpam-4594	79	12	∈	∈	PROPN
ejpam-4594	79	13	v	v	ADP
ejpam-4594	79	14	}	}	PUNCT
ejpam-4594	79	15	;	;	PUNCT
ejpam-4594	79	16	(	(	PUNCT
ejpam-4594	79	17	ii	ii	NOUN
ejpam-4594	79	18	)	)	PUNCT
ejpam-4594	79	19	v	v	ADP
ejpam-4594	79	20	′[x	′[x	NOUN
ejpam-4594	79	21	]	]	PUNCT
ejpam-4594	80	1	=	=	PRON
ejpam-4594	81	1	{	{	PUNCT
ejpam-4594	81	2	y	y	PROPN
ejpam-4594	81	3	∈	∈	PROPN
ejpam-4594	82	1	xd	xd	INTJ
ejpam-4594	83	1	|	|	ADV
ejpam-4594	83	2	y	y	PROPN
ejpam-4594	83	3	◦	◦	NOUN
ejpam-4594	83	4	x	x	SYM
ejpam-4594	83	5	,	,	PUNCT
ejpam-4594	83	6	x	x	VERB
ejpam-4594	83	7	◦	◦	VERB
ejpam-4594	83	8	y	y	PROPN
ejpam-4594	83	9	∈	∈	PROPN
ejpam-4594	83	10	v	v	ADP
ejpam-4594	83	11	}	}	PUNCT
ejpam-4594	83	12	.	.	PUNCT
ejpam-4594	84	1	remark	remark	NOUN
ejpam-4594	84	2	2	2	NUM
ejpam-4594	84	3	.	.	PUNCT
ejpam-4594	85	1	let	let	VERB
ejpam-4594	85	2	xd	xd	INTJ
ejpam-4594	85	3	be	be	AUX
ejpam-4594	85	4	a	a	DET
ejpam-4594	85	5	dual	dual	ADJ
ejpam-4594	85	6	b	b	NOUN
ejpam-4594	85	7	-	-	PUNCT
ejpam-4594	85	8	algebra	algebra	NOUN
ejpam-4594	85	9	and	and	CCONJ
ejpam-4594	85	10	v	v	ADP
ejpam-4594	85	11	⊆	⊆	NUM
ejpam-4594	85	12	xd	xd	ADP
ejpam-4594	85	13	.	.	PUNCT
ejpam-4594	86	1	then	then	ADV
ejpam-4594	86	2	v	v	ADP
ejpam-4594	86	3	′[x	′[x	NOUN
ejpam-4594	86	4	]	]	PUNCT
ejpam-4594	87	1	⊆	⊆	NUM
ejpam-4594	87	2	v	v	PART
ejpam-4594	87	3	[	[	X
ejpam-4594	87	4	x	x	X
ejpam-4594	87	5	]	]	X
ejpam-4594	87	6	for	for	ADP
ejpam-4594	87	7	any	any	DET
ejpam-4594	87	8	x	x	SYM
ejpam-4594	87	9	∈	∈	PROPN
ejpam-4594	87	10	xd	xd	PROPN
ejpam-4594	87	11	.	.	PROPN
ejpam-4594	87	12	example	example	NOUN
ejpam-4594	87	13	1	1	NUM
ejpam-4594	87	14	.	.	X
ejpam-4594	87	15	consider	consider	VERB
ejpam-4594	87	16	the	the	DET
ejpam-4594	87	17	set	set	NOUN
ejpam-4594	87	18	xd	xd	INTJ
ejpam-4594	87	19	=	=	SYM
ejpam-4594	87	20	{	{	PUNCT
ejpam-4594	87	21	1	1	NUM
ejpam-4594	87	22	,	,	PUNCT
ejpam-4594	87	23	a	a	DET
ejpam-4594	87	24	,	,	PUNCT
ejpam-4594	87	25	b	b	NOUN
ejpam-4594	87	26	,	,	PUNCT
ejpam-4594	87	27	c	c	NOUN
ejpam-4594	87	28	,	,	PUNCT
ejpam-4594	87	29	d	d	NOUN
ejpam-4594	87	30	,	,	PUNCT
ejpam-4594	87	31	e	e	NOUN
ejpam-4594	87	32	}	}	PUNCT
ejpam-4594	87	33	and	and	CCONJ
ejpam-4594	87	34	binary	binary	ADJ
ejpam-4594	87	35	operation	operation	NOUN
ejpam-4594	87	36	◦	◦	NOUN
ejpam-4594	87	37	as	as	SCONJ
ejpam-4594	87	38	defined	define	VERB
ejpam-4594	87	39	in	in	ADP
ejpam-4594	87	40	the	the	DET
ejpam-4594	87	41	table	table	NOUN
ejpam-4594	87	42	below	below	ADV
ejpam-4594	87	43	.	.	PUNCT
ejpam-4594	88	1	◦	◦	NOUN
ejpam-4594	88	2	1	1	NUM
ejpam-4594	88	3	a	a	DET
ejpam-4594	88	4	b	b	NOUN
ejpam-4594	88	5	c	c	NOUN
ejpam-4594	88	6	d	d	X
ejpam-4594	88	7	e	e	PROPN
ejpam-4594	88	8	1	1	NUM
ejpam-4594	88	9	1	1	NUM
ejpam-4594	88	10	a	a	DET
ejpam-4594	88	11	b	b	NOUN
ejpam-4594	88	12	c	c	NOUN
ejpam-4594	88	13	d	d	PROPN
ejpam-4594	88	14	e	e	PROPN
ejpam-4594	88	15	a	a	PRON
ejpam-4594	88	16	b	b	PROPN
ejpam-4594	88	17	1	1	NUM
ejpam-4594	88	18	a	a	PRON
ejpam-4594	88	19	d	d	X
ejpam-4594	88	20	e	e	NOUN
ejpam-4594	88	21	c	c	NOUN
ejpam-4594	88	22	b	b	PROPN
ejpam-4594	88	23	a	a	DET
ejpam-4594	88	24	b	b	NOUN
ejpam-4594	88	25	1	1	NUM
ejpam-4594	88	26	e	e	NOUN
ejpam-4594	88	27	c	c	NOUN
ejpam-4594	88	28	d	d	NOUN
ejpam-4594	88	29	c	c	NOUN
ejpam-4594	88	30	c	c	NOUN
ejpam-4594	88	31	d	d	X
ejpam-4594	88	32	e	e	PROPN
ejpam-4594	88	33	1	1	NUM
ejpam-4594	88	34	a	a	DET
ejpam-4594	88	35	b	b	NOUN
ejpam-4594	88	36	d	d	X
ejpam-4594	88	37	d	d	PROPN
ejpam-4594	88	38	e	e	PROPN
ejpam-4594	88	39	c	c	NOUN
ejpam-4594	88	40	b	b	PROPN
ejpam-4594	88	41	1	1	NUM
ejpam-4594	88	42	a	a	DET
ejpam-4594	88	43	e	e	NOUN
ejpam-4594	88	44	e	e	NOUN
ejpam-4594	88	45	c	c	NOUN
ejpam-4594	88	46	d	d	X
ejpam-4594	88	47	a	a	DET
ejpam-4594	88	48	b	b	NOUN
ejpam-4594	88	49	1	1	NUM
ejpam-4594	88	50	then	then	ADV
ejpam-4594	88	51	xd	xd	INTJ
ejpam-4594	88	52	is	be	VERB
ejpam-4594	88	53	a	a	DET
ejpam-4594	88	54	dual	dual	ADJ
ejpam-4594	88	55	b	b	NOUN
ejpam-4594	88	56	-	-	PUNCT
ejpam-4594	88	57	algebra	algebra	NOUN
ejpam-4594	89	1	[	[	X
ejpam-4594	89	2	1	1	NUM
ejpam-4594	89	3	]	]	PUNCT
ejpam-4594	89	4	.	.	PUNCT
ejpam-4594	90	1	let	let	VERB
ejpam-4594	90	2	v	v	X
ejpam-4594	90	3	=	=	PUNCT
ejpam-4594	90	4	{	{	PUNCT
ejpam-4594	90	5	a	a	X
ejpam-4594	90	6	,	,	PUNCT
ejpam-4594	90	7	d	d	NOUN
ejpam-4594	90	8	,	,	PUNCT
ejpam-4594	90	9	e	e	NOUN
ejpam-4594	90	10	}	}	PUNCT
ejpam-4594	90	11	.	.	PUNCT
ejpam-4594	91	1	then	then	ADV
ejpam-4594	91	2	v	v	ADP
ejpam-4594	91	3	[	[	X
ejpam-4594	91	4	b	b	X
ejpam-4594	91	5	]	]	X
ejpam-4594	91	6	=	=	PUNCT
ejpam-4594	91	7	{	{	PUNCT
ejpam-4594	91	8	1	1	NUM
ejpam-4594	91	9	,	,	PUNCT
ejpam-4594	91	10	c	c	NOUN
ejpam-4594	91	11	,	,	PUNCT
ejpam-4594	91	12	e	e	NOUN
ejpam-4594	91	13	}	}	PUNCT
ejpam-4594	91	14	and	and	CCONJ
ejpam-4594	91	15	v	v	ADP
ejpam-4594	91	16	′[b	′[b	NOUN
ejpam-4594	91	17	]	]	X
ejpam-4594	91	18	=	=	SYM
ejpam-4594	91	19	{	{	PUNCT
ejpam-4594	91	20	c	c	NOUN
ejpam-4594	91	21	,	,	PUNCT
ejpam-4594	91	22	e	e	NOUN
ejpam-4594	91	23	}	}	PUNCT
ejpam-4594	91	24	.	.	PUNCT
ejpam-4594	92	1	k.	k.	PROPN
ejpam-4594	92	2	belleza	belleza	PROPN
ejpam-4594	92	3	,	,	PUNCT
ejpam-4594	92	4	j.	j.	PROPN
ejpam-4594	92	5	albaracin	albaracin	PROPN
ejpam-4594	92	6	/	/	PUNCT
ejpam-4594	92	7	eur	eur	PROPN
ejpam-4594	92	8	.	.	PUNCT
ejpam-4594	93	1	j.	j.	PROPN
ejpam-4594	93	2	pure	pure	PROPN
ejpam-4594	93	3	appl	appl	PROPN
ejpam-4594	93	4	.	.	PROPN
ejpam-4594	93	5	math	math	PROPN
ejpam-4594	93	6	,	,	PUNCT
ejpam-4594	93	7	15	15	NUM
ejpam-4594	93	8	(	(	PUNCT
ejpam-4594	93	9	4	4	NUM
ejpam-4594	93	10	)	)	PUNCT
ejpam-4594	93	11	(	(	PUNCT
ejpam-4594	93	12	2022	2022	NUM
ejpam-4594	93	13	)	)	PUNCT
ejpam-4594	93	14	,	,	PUNCT
ejpam-4594	93	15	1948	1948	NUM
ejpam-4594	93	16	-	-	SYM
ejpam-4594	93	17	1956	1956	NUM
ejpam-4594	93	18	1951	1951	NUM
ejpam-4594	93	19	proposition	proposition	NOUN
ejpam-4594	93	20	2	2	NUM
ejpam-4594	93	21	.	.	PUNCT
ejpam-4594	93	22	suppose	suppose	VERB
ejpam-4594	93	23	xd	xd	INTJ
ejpam-4594	93	24	is	be	AUX
ejpam-4594	93	25	a	a	DET
ejpam-4594	93	26	dual	dual	ADJ
ejpam-4594	93	27	b	b	NOUN
ejpam-4594	93	28	-	-	PUNCT
ejpam-4594	93	29	algebra	algebra	NOUN
ejpam-4594	93	30	and	and	CCONJ
ejpam-4594	93	31	v	v	ADP
ejpam-4594	93	32	⊆	⊆	NUM
ejpam-4594	93	33	xd	xd	ADP
ejpam-4594	93	34	such	such	ADJ
ejpam-4594	93	35	that	that	SCONJ
ejpam-4594	93	36	1	1	NUM
ejpam-4594	93	37	∈	∈	NOUN
ejpam-4594	93	38	v	v	NOUN
ejpam-4594	93	39	.	.	PUNCT
ejpam-4594	94	1	then	then	ADV
ejpam-4594	94	2	x	x	SYM
ejpam-4594	94	3	∈	∈	PROPN
ejpam-4594	94	4	v	v	ADP
ejpam-4594	94	5	′[x	′[x	NOUN
ejpam-4594	94	6	]	]	PUNCT
ejpam-4594	94	7	.	.	PUNCT
ejpam-4594	95	1	in	in	ADP
ejpam-4594	95	2	particular	particular	ADJ
ejpam-4594	95	3	,	,	PUNCT
ejpam-4594	95	4	v	v	NOUN
ejpam-4594	95	5	=	=	SYM
ejpam-4594	95	6	{	{	PUNCT
ejpam-4594	95	7	1	1	NUM
ejpam-4594	95	8	}	}	PUNCT
ejpam-4594	95	9	if	if	SCONJ
ejpam-4594	95	10	and	and	CCONJ
ejpam-4594	95	11	only	only	ADV
ejpam-4594	95	12	if	if	SCONJ
ejpam-4594	95	13	v	v	X
ejpam-4594	95	14	[	[	X
ejpam-4594	95	15	x	x	X
ejpam-4594	95	16	]	]	X
ejpam-4594	95	17	=	=	PUNCT
ejpam-4594	95	18	{	{	PUNCT
ejpam-4594	95	19	x	x	NOUN
ejpam-4594	95	20	}	}	PUNCT
ejpam-4594	95	21	=	=	PROPN
ejpam-4594	95	22	v	v	ADP
ejpam-4594	95	23	′[x	′[x	NOUN
ejpam-4594	95	24	]	]	PUNCT
ejpam-4594	95	25	for	for	ADP
ejpam-4594	95	26	any	any	DET
ejpam-4594	95	27	x	x	SYM
ejpam-4594	95	28	∈	∈	PROPN
ejpam-4594	95	29	x.	x.	NOUN
ejpam-4594	95	30	proof	proof	NOUN
ejpam-4594	95	31	.	.	PUNCT
ejpam-4594	96	1	suppose	suppose	VERB
ejpam-4594	96	2	xd	xd	INTJ
ejpam-4594	96	3	is	be	AUX
ejpam-4594	96	4	a	a	DET
ejpam-4594	96	5	dual	dual	ADJ
ejpam-4594	96	6	b	b	NOUN
ejpam-4594	96	7	-	-	PUNCT
ejpam-4594	96	8	algebra	algebra	NOUN
ejpam-4594	96	9	and	and	CCONJ
ejpam-4594	96	10	v	v	ADP
ejpam-4594	96	11	⊆	⊆	NUM
ejpam-4594	96	12	xd	xd	ADP
ejpam-4594	96	13	such	such	ADJ
ejpam-4594	96	14	that	that	SCONJ
ejpam-4594	96	15	1	1	NUM
ejpam-4594	96	16	∈	∈	NOUN
ejpam-4594	96	17	v	v	NOUN
ejpam-4594	96	18	.	.	PUNCT
ejpam-4594	97	1	by	by	ADP
ejpam-4594	97	2	(	(	PUNCT
ejpam-4594	97	3	db1	db1	NOUN
ejpam-4594	97	4	)	)	PUNCT
ejpam-4594	97	5	,	,	PUNCT
ejpam-4594	97	6	x	x	PUNCT
ejpam-4594	97	7	◦	◦	NOUN
ejpam-4594	97	8	x	x	SYM
ejpam-4594	97	9	=	=	SYM
ejpam-4594	97	10	1	1	NUM
ejpam-4594	97	11	∈	∈	NOUN
ejpam-4594	97	12	v	v	NOUN
ejpam-4594	97	13	for	for	ADP
ejpam-4594	97	14	all	all	DET
ejpam-4594	97	15	x	x	SYM
ejpam-4594	97	16	∈	∈	NOUN
ejpam-4594	97	17	xd	xd	PROPN
ejpam-4594	97	18	.	.	PUNCT
ejpam-4594	98	1	this	this	PRON
ejpam-4594	98	2	implies	imply	VERB
ejpam-4594	98	3	that	that	SCONJ
ejpam-4594	98	4	x	x	PUNCT
ejpam-4594	98	5	∈	∈	PROPN
ejpam-4594	98	6	v	v	ADP
ejpam-4594	98	7	′[x	′[x	NOUN
ejpam-4594	98	8	]	]	PUNCT
ejpam-4594	98	9	.	.	PUNCT
ejpam-4594	99	1	suppose	suppose	VERB
ejpam-4594	99	2	v	v	X
ejpam-4594	99	3	=	=	SYM
ejpam-4594	99	4	{	{	PUNCT
ejpam-4594	99	5	1	1	NUM
ejpam-4594	99	6	}	}	PUNCT
ejpam-4594	99	7	.	.	PUNCT
ejpam-4594	100	1	then	then	ADV
ejpam-4594	100	2	v	v	X
ejpam-4594	100	3	[	[	X
ejpam-4594	100	4	x	x	X
ejpam-4594	100	5	]	]	X
ejpam-4594	100	6	=	=	PUNCT
ejpam-4594	100	7	{	{	PUNCT
ejpam-4594	100	8	x	x	NOUN
ejpam-4594	100	9	}	}	PUNCT
ejpam-4594	100	10	=	=	PROPN
ejpam-4594	100	11	v	v	ADP
ejpam-4594	100	12	′[x	′[x	NOUN
ejpam-4594	100	13	]	]	PUNCT
ejpam-4594	100	14	for	for	ADP
ejpam-4594	100	15	any	any	DET
ejpam-4594	100	16	x	x	SYM
ejpam-4594	100	17	∈	∈	PROPN
ejpam-4594	100	18	x.	x.	NOUN
ejpam-4594	100	19	let	let	VERB
ejpam-4594	100	20	v	v	ADP
ejpam-4594	100	21	[	[	X
ejpam-4594	100	22	x	x	X
ejpam-4594	100	23	]	]	X
ejpam-4594	100	24	=	=	PUNCT
ejpam-4594	100	25	{	{	PUNCT
ejpam-4594	100	26	x	x	NOUN
ejpam-4594	100	27	}	}	PUNCT
ejpam-4594	100	28	=	=	PROPN
ejpam-4594	100	29	v	v	ADP
ejpam-4594	100	30	′[x	′[x	NOUN
ejpam-4594	100	31	]	]	PUNCT
ejpam-4594	100	32	.	.	PUNCT
ejpam-4594	101	1	then	then	ADV
ejpam-4594	101	2	x	x	PART
ejpam-4594	101	3	◦	◦	NOUN
ejpam-4594	101	4	x	x	SYM
ejpam-4594	101	5	∈	∈	NOUN
ejpam-4594	101	6	v	v	NOUN
ejpam-4594	101	7	.	.	PUNCT
ejpam-4594	102	1	thus	thus	ADV
ejpam-4594	102	2	,	,	PUNCT
ejpam-4594	102	3	1	1	NUM
ejpam-4594	102	4	∈	∈	NOUN
ejpam-4594	102	5	v	v	NOUN
ejpam-4594	102	6	.	.	PUNCT
ejpam-4594	102	7	suppose	suppose	VERB
ejpam-4594	102	8	a	a	DET
ejpam-4594	102	9	∈	∈	NOUN
ejpam-4594	102	10	v	v	ADP
ejpam-4594	102	11	such	such	DET
ejpam-4594	102	12	that	that	SCONJ
ejpam-4594	102	13	a	a	DET
ejpam-4594	102	14	̸=	̸=	PROPN
ejpam-4594	102	15	1	1	NUM
ejpam-4594	102	16	.	.	PUNCT
ejpam-4594	103	1	then	then	ADV
ejpam-4594	103	2	there	there	PRON
ejpam-4594	103	3	exists	exist	VERB
ejpam-4594	103	4	y	y	PROPN
ejpam-4594	103	5	∈	∈	PROPN
ejpam-4594	103	6	x	x	PUNCT
ejpam-4594	103	7	such	such	ADJ
ejpam-4594	103	8	that	that	SCONJ
ejpam-4594	103	9	x	x	X
ejpam-4594	103	10	◦	◦	NOUN
ejpam-4594	103	11	y	y	NOUN
ejpam-4594	103	12	=	=	PUNCT
ejpam-4594	103	13	a	a	PRON
ejpam-4594	103	14	with	with	ADP
ejpam-4594	103	15	x	x	PUNCT
ejpam-4594	103	16	̸=	̸=	PROPN
ejpam-4594	103	17	y.	y.	NOUN
ejpam-4594	103	18	hence	hence	ADV
ejpam-4594	103	19	,	,	PUNCT
ejpam-4594	103	20	y	y	PROPN
ejpam-4594	103	21	∈	∈	PROPN
ejpam-4594	103	22	v	v	ADP
ejpam-4594	103	23	[	[	X
ejpam-4594	103	24	x	x	X
ejpam-4594	103	25	]	]	X
ejpam-4594	103	26	,	,	PUNCT
ejpam-4594	103	27	a	a	DET
ejpam-4594	103	28	contradiction	contradiction	NOUN
ejpam-4594	103	29	.	.	PUNCT
ejpam-4594	104	1	therefore	therefore	ADV
ejpam-4594	104	2	,	,	PUNCT
ejpam-4594	104	3	v	v	X
ejpam-4594	104	4	=	=	SYM
ejpam-4594	104	5	{	{	PUNCT
ejpam-4594	104	6	1	1	NUM
ejpam-4594	104	7	}	}	PUNCT
ejpam-4594	104	8	.	.	PUNCT
ejpam-4594	105	1	proposition	proposition	NOUN
ejpam-4594	105	2	3	3	X
ejpam-4594	105	3	.	.	PUNCT
ejpam-4594	106	1	let	let	VERB
ejpam-4594	106	2	xd	xd	INTJ
ejpam-4594	106	3	be	be	AUX
ejpam-4594	106	4	a	a	DET
ejpam-4594	106	5	dual	dual	ADJ
ejpam-4594	106	6	b	b	NOUN
ejpam-4594	106	7	-	-	PUNCT
ejpam-4594	106	8	algebra	algebra	NOUN
ejpam-4594	106	9	and	and	CCONJ
ejpam-4594	106	10	u	u	NOUN
ejpam-4594	106	11	,	,	PUNCT
ejpam-4594	106	12	v	v	PROPN
ejpam-4594	106	13	⊂	⊂	PROPN
ejpam-4594	106	14	xd	xd	ADP
ejpam-4594	106	15	.	.	PUNCT
ejpam-4594	107	1	if	if	SCONJ
ejpam-4594	107	2	u	u	PROPN
ejpam-4594	107	3	⊆	⊆	NUM
ejpam-4594	107	4	v	v	NOUN
ejpam-4594	107	5	,	,	PUNCT
ejpam-4594	107	6	then	then	ADV
ejpam-4594	107	7	u	u	X
ejpam-4594	107	8	[	[	X
ejpam-4594	107	9	x	x	X
ejpam-4594	107	10	]	]	X
ejpam-4594	107	11	⊆	⊆	NUM
ejpam-4594	107	12	v	v	PART
ejpam-4594	107	13	[	[	X
ejpam-4594	107	14	x	x	X
ejpam-4594	107	15	]	]	X
ejpam-4594	107	16	and	and	CCONJ
ejpam-4594	107	17	u	u	NOUN
ejpam-4594	107	18	′[x	′[x	NOUN
ejpam-4594	107	19	]	]	PUNCT
ejpam-4594	107	20	⊆	⊆	NUM
ejpam-4594	107	21	v	v	ADP
ejpam-4594	107	22	′[x	′[x	NOUN
ejpam-4594	107	23	]	]	PUNCT
ejpam-4594	107	24	for	for	ADP
ejpam-4594	107	25	any	any	DET
ejpam-4594	107	26	x	x	SYM
ejpam-4594	107	27	∈	∈	PROPN
ejpam-4594	107	28	xd	xd	PROPN
ejpam-4594	107	29	.	.	PUNCT
ejpam-4594	107	30	proof	proof	NOUN
ejpam-4594	107	31	.	.	PUNCT
ejpam-4594	108	1	suppose	suppose	VERB
ejpam-4594	108	2	xd	xd	INTJ
ejpam-4594	108	3	is	be	AUX
ejpam-4594	108	4	a	a	DET
ejpam-4594	108	5	dual	dual	ADJ
ejpam-4594	108	6	b	b	NOUN
ejpam-4594	108	7	-	-	PUNCT
ejpam-4594	108	8	algebra	algebra	NOUN
ejpam-4594	108	9	and	and	CCONJ
ejpam-4594	108	10	u	u	NOUN
ejpam-4594	108	11	,	,	PUNCT
ejpam-4594	108	12	v	v	PROPN
ejpam-4594	108	13	⊂	⊂	PROPN
ejpam-4594	109	1	xd	xd	ADP
ejpam-4594	109	2	.	.	PUNCT
ejpam-4594	110	1	let	let	VERB
ejpam-4594	110	2	y	y	PROPN
ejpam-4594	110	3	∈	∈	PROPN
ejpam-4594	110	4	u	u	NOUN
ejpam-4594	110	5	[	[	X
ejpam-4594	110	6	x	x	X
ejpam-4594	110	7	]	]	X
ejpam-4594	110	8	.	.	PUNCT
ejpam-4594	111	1	then	then	ADV
ejpam-4594	111	2	x	x	PUNCT
ejpam-4594	111	3	◦	◦	NOUN
ejpam-4594	111	4	y	y	PROPN
ejpam-4594	111	5	∈	∈	PROPN
ejpam-4594	111	6	u	u	NOUN
ejpam-4594	111	7	⊆	⊆	NUM
ejpam-4594	111	8	v	v	NOUN
ejpam-4594	111	9	.	.	PUNCT
ejpam-4594	112	1	hence	hence	ADV
ejpam-4594	112	2	,	,	PUNCT
ejpam-4594	112	3	y	y	PROPN
ejpam-4594	112	4	∈	∈	PROPN
ejpam-4594	112	5	v	v	ADP
ejpam-4594	112	6	[	[	X
ejpam-4594	112	7	x	x	X
ejpam-4594	112	8	]	]	X
ejpam-4594	112	9	which	which	PRON
ejpam-4594	112	10	implies	imply	VERB
ejpam-4594	112	11	that	that	SCONJ
ejpam-4594	112	12	u	u	PRON
ejpam-4594	113	1	[	[	X
ejpam-4594	113	2	x	x	X
ejpam-4594	113	3	]	]	X
ejpam-4594	113	4	⊆	⊆	NUM
ejpam-4594	113	5	v	v	PART
ejpam-4594	113	6	[	[	X
ejpam-4594	113	7	x	x	X
ejpam-4594	113	8	]	]	X
ejpam-4594	113	9	.	.	PUNCT
ejpam-4594	114	1	similarly	similarly	ADV
ejpam-4594	114	2	,	,	PUNCT
ejpam-4594	114	3	u	u	NOUN
ejpam-4594	114	4	′[x	′[x	ADV
ejpam-4594	114	5	]	]	X
ejpam-4594	114	6	⊆	⊆	NUM
ejpam-4594	114	7	v	v	ADP
ejpam-4594	114	8	′[x	′[x	NOUN
ejpam-4594	114	9	]	]	PUNCT
ejpam-4594	114	10	.	.	PUNCT
ejpam-4594	115	1	proposition	proposition	NOUN
ejpam-4594	115	2	4	4	NUM
ejpam-4594	115	3	.	.	PUNCT
ejpam-4594	116	1	let	let	VERB
ejpam-4594	116	2	xd	xd	INTJ
ejpam-4594	116	3	be	be	AUX
ejpam-4594	116	4	a	a	DET
ejpam-4594	116	5	dual	dual	ADJ
ejpam-4594	116	6	b	b	NOUN
ejpam-4594	116	7	-	-	PUNCT
ejpam-4594	116	8	algebra	algebra	NOUN
ejpam-4594	116	9	satisfying	satisfy	VERB
ejpam-4594	116	10	the	the	DET
ejpam-4594	116	11	symmetric	symmetric	ADJ
ejpam-4594	116	12	condition	condition	NOUN
ejpam-4594	116	13	and	and	CCONJ
ejpam-4594	116	14	v	v	ADP
ejpam-4594	116	15	⊆	⊆	NUM
ejpam-4594	116	16	xd	xd	ADP
ejpam-4594	116	17	such	such	ADJ
ejpam-4594	116	18	that	that	PRON
ejpam-4594	116	19	for	for	ADP
ejpam-4594	116	20	all	all	DET
ejpam-4594	116	21	p	p	NOUN
ejpam-4594	116	22	,	,	PUNCT
ejpam-4594	116	23	q	q	PROPN
ejpam-4594	116	24	∈	∈	PROPN
ejpam-4594	116	25	v	v	NOUN
ejpam-4594	116	26	and	and	CCONJ
ejpam-4594	116	27	x	x	ADP
ejpam-4594	116	28	∈	∈	PROPN
ejpam-4594	116	29	xd	xd	ADP
ejpam-4594	116	30	,	,	PUNCT
ejpam-4594	116	31	p	p	NOUN
ejpam-4594	116	32	◦	◦	NOUN
ejpam-4594	116	33	(	(	PUNCT
ejpam-4594	116	34	x	x	PART
ejpam-4594	116	35	◦	◦	NOUN
ejpam-4594	116	36	q	q	NOUN
ejpam-4594	116	37	)	)	PUNCT
ejpam-4594	116	38	=	=	SYM
ejpam-4594	116	39	1	1	NUM
ejpam-4594	116	40	implies	imply	VERB
ejpam-4594	116	41	x	x	X
ejpam-4594	116	42	∈	∈	NOUN
ejpam-4594	116	43	v	v	NOUN
ejpam-4594	116	44	.	.	PUNCT
ejpam-4594	117	1	then	then	ADV
ejpam-4594	117	2	v	v	ADP
ejpam-4594	117	3	′[x	′[x	NOUN
ejpam-4594	117	4	]	]	PUNCT
ejpam-4594	117	5	◦	◦	NOUN
ejpam-4594	117	6	v	v	ADP
ejpam-4594	117	7	′[y	′[y	PROPN
ejpam-4594	117	8	]	]	PUNCT
ejpam-4594	117	9	⊆	⊆	NUM
ejpam-4594	117	10	v	v	ADP
ejpam-4594	117	11	′[x	′[x	NOUN
ejpam-4594	117	12	◦	◦	NOUN
ejpam-4594	117	13	y	y	NOUN
ejpam-4594	117	14	]	]	PUNCT
ejpam-4594	117	15	.	.	PUNCT
ejpam-4594	118	1	proof	proof	NOUN
ejpam-4594	118	2	.	.	PUNCT
ejpam-4594	119	1	let	let	VERB
ejpam-4594	119	2	p	p	PRON
ejpam-4594	119	3	◦	◦	VERB
ejpam-4594	119	4	q	q	NOUN
ejpam-4594	119	5	∈	∈	NOUN
ejpam-4594	119	6	v	v	ADP
ejpam-4594	119	7	′[x]	′[x]	PROPN
ejpam-4594	119	8	◦	◦	NOUN
ejpam-4594	119	9	v	v	NOUN
ejpam-4594	119	10	′[y	′[y	NOUN
ejpam-4594	119	11	]	]	PUNCT
ejpam-4594	119	12	where	where	SCONJ
ejpam-4594	119	13	p	p	PROPN
ejpam-4594	119	14	∈	∈	PROPN
ejpam-4594	119	15	v	v	ADP
ejpam-4594	119	16	′[x	′[x	NOUN
ejpam-4594	119	17	]	]	PUNCT
ejpam-4594	119	18	and	and	CCONJ
ejpam-4594	119	19	q	q	PROPN
ejpam-4594	119	20	∈	∈	PROPN
ejpam-4594	119	21	v	v	ADP
ejpam-4594	119	22	′[y	′[y	PROPN
ejpam-4594	119	23	]	]	PUNCT
ejpam-4594	119	24	.	.	PUNCT
ejpam-4594	120	1	then	then	ADV
ejpam-4594	120	2	p	p	X
ejpam-4594	120	3	◦	◦	NOUN
ejpam-4594	120	4	x	x	NOUN
ejpam-4594	120	5	,	,	PUNCT
ejpam-4594	120	6	x	x	X
ejpam-4594	120	7	◦	◦	NOUN
ejpam-4594	120	8	p	p	NOUN
ejpam-4594	120	9	,	,	PUNCT
ejpam-4594	120	10	q	q	PROPN
ejpam-4594	120	11	◦	◦	NOUN
ejpam-4594	120	12	y	y	PROPN
ejpam-4594	120	13	,	,	PUNCT
ejpam-4594	120	14	y	y	PROPN
ejpam-4594	120	15	◦	◦	NOUN
ejpam-4594	120	16	q	q	NOUN
ejpam-4594	120	17	∈	∈	PROPN
ejpam-4594	120	18	v	v	NOUN
ejpam-4594	120	19	.	.	PUNCT
ejpam-4594	121	1	by	by	ADP
ejpam-4594	121	2	(	(	PUNCT
ejpam-4594	121	3	db1	db1	NOUN
ejpam-4594	121	4	)	)	PUNCT
ejpam-4594	121	5	,	,	PUNCT
ejpam-4594	121	6	lemma	lemma	PROPN
ejpam-4594	121	7	2	2	NUM
ejpam-4594	121	8	,	,	PUNCT
ejpam-4594	121	9	(	(	PUNCT
ejpam-4594	121	10	db3	db3	PROPN
ejpam-4594	121	11	)	)	PUNCT
ejpam-4594	121	12	,	,	PUNCT
ejpam-4594	121	13	symmetric	symmetric	ADJ
ejpam-4594	121	14	condition	condition	NOUN
ejpam-4594	121	15	,	,	PUNCT
ejpam-4594	121	16	and	and	CCONJ
ejpam-4594	121	17	(	(	PUNCT
ejpam-4594	121	18	db2	db2	PROPN
ejpam-4594	121	19	)	)	PUNCT
ejpam-4594	121	20	,	,	PUNCT
ejpam-4594	121	21	1	1	X
ejpam-4594	121	22	=	=	SYM
ejpam-4594	121	23	(	(	PUNCT
ejpam-4594	121	24	p	p	NOUN
ejpam-4594	121	25	◦	◦	NOUN
ejpam-4594	121	26	x	x	NOUN
ejpam-4594	121	27	)	)	PUNCT
ejpam-4594	121	28	◦	◦	NOUN
ejpam-4594	121	29	(	(	PUNCT
ejpam-4594	121	30	p	p	NOUN
ejpam-4594	121	31	◦	◦	NOUN
ejpam-4594	121	32	x	x	X
ejpam-4594	121	33	)	)	PUNCT
ejpam-4594	121	34	=	=	SYM
ejpam-4594	121	35	(	(	PUNCT
ejpam-4594	121	36	p	p	X
ejpam-4594	121	37	◦	◦	NOUN
ejpam-4594	121	38	x	x	NOUN
ejpam-4594	121	39	)	)	PUNCT
ejpam-4594	121	40	◦	◦	NOUN
ejpam-4594	122	1	[	[	X
ejpam-4594	122	2	(	(	PUNCT
ejpam-4594	122	3	p	p	NOUN
ejpam-4594	122	4	◦	◦	NOUN
ejpam-4594	122	5	q	q	NOUN
ejpam-4594	122	6	)	)	PUNCT
ejpam-4594	122	7	◦	◦	NOUN
ejpam-4594	122	8	(	(	PUNCT
ejpam-4594	122	9	x	x	PART
ejpam-4594	122	10	◦	◦	VERB
ejpam-4594	122	11	q	q	NOUN
ejpam-4594	122	12	)	)	PUNCT
ejpam-4594	122	13	]	]	PUNCT
ejpam-4594	123	1	=	=	PUNCT
ejpam-4594	123	2	(	(	PUNCT
ejpam-4594	123	3	p	p	X
ejpam-4594	123	4	◦	◦	NOUN
ejpam-4594	123	5	x	x	NOUN
ejpam-4594	123	6	)	)	PUNCT
ejpam-4594	123	7	◦	◦	NOUN
ejpam-4594	123	8	[	[	PUNCT
ejpam-4594	123	9	(	(	PUNCT
ejpam-4594	123	10	p	p	PROPN
ejpam-4594	123	11	◦	◦	NOUN
ejpam-4594	123	12	q	q	NOUN
ejpam-4594	123	13	)	)	PUNCT
ejpam-4594	123	14	◦	◦	NOUN
ejpam-4594	124	1	[	[	X
ejpam-4594	124	2	(	(	PUNCT
ejpam-4594	124	3	x	x	SYM
ejpam-4594	124	4	◦	◦	VERB
ejpam-4594	124	5	y	y	NOUN
ejpam-4594	124	6	)	)	PUNCT
ejpam-4594	124	7	◦	◦	NOUN
ejpam-4594	124	8	(	(	PUNCT
ejpam-4594	124	9	q	q	PART
ejpam-4594	124	10	◦	◦	VERB
ejpam-4594	124	11	y	y	PROPN
ejpam-4594	124	12	)	)	PUNCT
ejpam-4594	124	13	]	]	PUNCT
ejpam-4594	124	14	]	]	PUNCT
ejpam-4594	125	1	=	=	PUNCT
ejpam-4594	125	2	(	(	PUNCT
ejpam-4594	125	3	p	p	X
ejpam-4594	125	4	◦	◦	NOUN
ejpam-4594	125	5	x	x	NOUN
ejpam-4594	125	6	)	)	PUNCT
ejpam-4594	125	7	◦	◦	NOUN
ejpam-4594	125	8	(	(	PUNCT
ejpam-4594	125	9	[	[	PUNCT
ejpam-4594	125	10	[	[	X
ejpam-4594	125	11	(	(	PUNCT
ejpam-4594	125	12	x	x	SYM
ejpam-4594	125	13	◦	◦	NOUN
ejpam-4594	125	14	y	y	NOUN
ejpam-4594	125	15	)	)	PUNCT
ejpam-4594	125	16	◦	◦	NOUN
ejpam-4594	125	17	1	1	NUM
ejpam-4594	125	18	]	]	X
ejpam-4594	125	19	◦	◦	NOUN
ejpam-4594	125	20	(	(	PUNCT
ejpam-4594	125	21	p	p	NOUN
ejpam-4594	125	22	◦	◦	NOUN
ejpam-4594	125	23	q	q	X
ejpam-4594	125	24	)	)	PUNCT
ejpam-4594	125	25	]	]	PUNCT
ejpam-4594	125	26	◦	◦	NOUN
ejpam-4594	125	27	(	(	PUNCT
ejpam-4594	125	28	q	q	PART
ejpam-4594	125	29	◦	◦	VERB
ejpam-4594	125	30	y	y	NOUN
ejpam-4594	125	31	)	)	PUNCT
ejpam-4594	125	32	)	)	PUNCT
ejpam-4594	126	1	=	=	PUNCT
ejpam-4594	126	2	(	(	PUNCT
ejpam-4594	126	3	p	p	X
ejpam-4594	126	4	◦	◦	NOUN
ejpam-4594	126	5	x	x	NOUN
ejpam-4594	126	6	)	)	PUNCT
ejpam-4594	126	7	◦	◦	NOUN
ejpam-4594	126	8	[	[	PUNCT
ejpam-4594	126	9	[	[	X
ejpam-4594	126	10	(	(	PUNCT
ejpam-4594	126	11	x	x	SYM
ejpam-4594	126	12	◦	◦	NOUN
ejpam-4594	126	13	y	y	NOUN
ejpam-4594	126	14	)	)	PUNCT
ejpam-4594	126	15	◦	◦	NOUN
ejpam-4594	126	16	(	(	PUNCT
ejpam-4594	126	17	p	p	NOUN
ejpam-4594	126	18	◦	◦	NOUN
ejpam-4594	126	19	q	q	X
ejpam-4594	126	20	)	)	PUNCT
ejpam-4594	126	21	]	]	PUNCT
ejpam-4594	127	1	◦	◦	NOUN
ejpam-4594	127	2	(	(	PUNCT
ejpam-4594	127	3	q	q	PART
ejpam-4594	127	4	◦	◦	VERB
ejpam-4594	127	5	y	y	PROPN
ejpam-4594	127	6	)	)	PUNCT
ejpam-4594	127	7	]	]	PUNCT
ejpam-4594	127	8	.	.	PUNCT
ejpam-4594	128	1	since	since	SCONJ
ejpam-4594	128	2	(	(	PUNCT
ejpam-4594	128	3	p	p	NOUN
ejpam-4594	128	4	◦	◦	NOUN
ejpam-4594	128	5	x	x	X
ejpam-4594	128	6	)	)	PUNCT
ejpam-4594	128	7	,	,	PUNCT
ejpam-4594	128	8	(	(	PUNCT
ejpam-4594	128	9	q	q	PUNCT
ejpam-4594	128	10	◦	◦	VERB
ejpam-4594	128	11	y	y	NOUN
ejpam-4594	128	12	)	)	PUNCT
ejpam-4594	128	13	∈	∈	PROPN
ejpam-4594	128	14	v	v	NOUN
ejpam-4594	128	15	and	and	CCONJ
ejpam-4594	128	16	(	(	PUNCT
ejpam-4594	128	17	p	p	PROPN
ejpam-4594	128	18	◦	◦	NOUN
ejpam-4594	128	19	x	x	NOUN
ejpam-4594	128	20	)	)	PUNCT
ejpam-4594	128	21	◦	◦	NOUN
ejpam-4594	128	22	[	[	PUNCT
ejpam-4594	128	23	[	[	X
ejpam-4594	128	24	(	(	PUNCT
ejpam-4594	128	25	x	x	NOUN
ejpam-4594	128	26	◦	◦	NOUN
ejpam-4594	128	27	y)	y)	NOUN
ejpam-4594	128	28	◦	◦	NOUN
ejpam-4594	128	29	(p	(p	NOUN
ejpam-4594	128	30	◦	◦	NOUN
ejpam-4594	128	31	q)]	q)]	ADJ
ejpam-4594	128	32	◦	◦	NOUN
ejpam-4594	128	33	(q	(q	PUNCT
ejpam-4594	128	34	◦	◦	NOUN
ejpam-4594	128	35	y	y	NOUN
ejpam-4594	128	36	)	)	PUNCT
ejpam-4594	128	37	]	]	PUNCT
ejpam-4594	129	1	=	=	PUNCT
ejpam-4594	129	2	1	1	NUM
ejpam-4594	129	3	,	,	PUNCT
ejpam-4594	129	4	this	this	PRON
ejpam-4594	129	5	implies	imply	VERB
ejpam-4594	129	6	that	that	SCONJ
ejpam-4594	129	7	(	(	PUNCT
ejpam-4594	129	8	x	x	X
ejpam-4594	129	9	◦	◦	NOUN
ejpam-4594	129	10	y)	y)	NOUN
ejpam-4594	129	11	◦	◦	NOUN
ejpam-4594	129	12	(p	(p	NOUN
ejpam-4594	129	13	◦	◦	NOUN
ejpam-4594	129	14	q	q	NOUN
ejpam-4594	129	15	)	)	PUNCT
ejpam-4594	129	16	∈	∈	NOUN
ejpam-4594	129	17	v	v	NOUN
ejpam-4594	129	18	by	by	ADP
ejpam-4594	129	19	the	the	DET
ejpam-4594	129	20	hypothesis	hypothesis	NOUN
ejpam-4594	129	21	.	.	PUNCT
ejpam-4594	130	1	similarly	similarly	ADV
ejpam-4594	130	2	,	,	PUNCT
ejpam-4594	130	3	(	(	PUNCT
ejpam-4594	130	4	p	p	NOUN
ejpam-4594	130	5	◦	◦	NOUN
ejpam-4594	130	6	q	q	NOUN
ejpam-4594	130	7	)	)	PUNCT
ejpam-4594	130	8	◦	◦	NOUN
ejpam-4594	130	9	(	(	PUNCT
ejpam-4594	130	10	x	x	PART
ejpam-4594	130	11	◦	◦	VERB
ejpam-4594	130	12	y	y	NOUN
ejpam-4594	130	13	)	)	PUNCT
ejpam-4594	130	14	∈	∈	PROPN
ejpam-4594	130	15	v	v	NOUN
ejpam-4594	130	16	.	.	PUNCT
ejpam-4594	131	1	hence	hence	ADV
ejpam-4594	131	2	,	,	PUNCT
ejpam-4594	131	3	p	p	PROPN
ejpam-4594	131	4	◦	◦	NOUN
ejpam-4594	131	5	q	q	NOUN
ejpam-4594	131	6	∈	∈	PROPN
ejpam-4594	131	7	v	v	ADP
ejpam-4594	131	8	′[x	′[x	NOUN
ejpam-4594	131	9	◦	◦	NOUN
ejpam-4594	131	10	y	y	X
ejpam-4594	131	11	]	]	X
ejpam-4594	131	12	.	.	PUNCT
ejpam-4594	132	1	therefore	therefore	ADV
ejpam-4594	132	2	,	,	PUNCT
ejpam-4594	132	3	v	v	ADP
ejpam-4594	132	4	′[x	′[x	NOUN
ejpam-4594	132	5	]	]	PUNCT
ejpam-4594	132	6	◦	◦	NOUN
ejpam-4594	132	7	v	v	NOUN
ejpam-4594	132	8	′[y	′[y	PROPN
ejpam-4594	132	9	]	]	PUNCT
ejpam-4594	132	10	⊆	⊆	NUM
ejpam-4594	132	11	v	v	ADP
ejpam-4594	132	12	′[x	′[x	NOUN
ejpam-4594	132	13	◦	◦	NOUN
ejpam-4594	132	14	y	y	X
ejpam-4594	132	15	]	]	PUNCT
ejpam-4594	132	16	.	.	PUNCT
ejpam-4594	133	1	theorem	theorem	ADJ
ejpam-4594	133	2	4	4	NUM
ejpam-4594	133	3	.	.	PUNCT
ejpam-4594	134	1	let	let	VERB
ejpam-4594	134	2	ω	ω	NUM
ejpam-4594	134	3	be	be	AUX
ejpam-4594	134	4	a	a	DET
ejpam-4594	134	5	family	family	NOUN
ejpam-4594	134	6	of	of	ADP
ejpam-4594	134	7	nonempty	nonempty	ADJ
ejpam-4594	134	8	subsets	subset	NOUN
ejpam-4594	134	9	in	in	ADP
ejpam-4594	134	10	a	a	DET
ejpam-4594	134	11	dual	dual	ADJ
ejpam-4594	134	12	b	b	NOUN
ejpam-4594	134	13	-	-	PUNCT
ejpam-4594	134	14	algebra	algebra	NOUN
ejpam-4594	134	15	xd	xd	INTJ
ejpam-4594	134	16	that	that	PRON
ejpam-4594	134	17	is	be	AUX
ejpam-4594	134	18	closed	close	VERB
ejpam-4594	134	19	under	under	ADP
ejpam-4594	134	20	finite	finite	ADJ
ejpam-4594	134	21	intersections	intersection	NOUN
ejpam-4594	134	22	.	.	PUNCT
ejpam-4594	135	1	then	then	ADV
ejpam-4594	135	2	the	the	DET
ejpam-4594	135	3	set	set	NOUN
ejpam-4594	135	4	τ	τ	X
ejpam-4594	135	5	=	=	PUNCT
ejpam-4594	135	6	{	{	PUNCT
ejpam-4594	135	7	u	u	NOUN
ejpam-4594	135	8	⊆	⊆	NUM
ejpam-4594	135	9	xd	xd	INTJ
ejpam-4594	135	10	|	|	ADV
ejpam-4594	135	11	∀x	∀x	X
ejpam-4594	135	12	∈	∈	PROPN
ejpam-4594	135	13	u,∃v	u,∃v	ADJ
ejpam-4594	135	14	∈	∈	PROPN
ejpam-4594	135	15	ω	ω	NUM
ejpam-4594	135	16	such	such	ADJ
ejpam-4594	135	17	that	that	PRON
ejpam-4594	135	18	v	v	PART
ejpam-4594	135	19	[	[	X
ejpam-4594	135	20	x	x	X
ejpam-4594	135	21	]	]	X
ejpam-4594	135	22	⊆	⊆	NUM
ejpam-4594	135	23	u	u	NOUN
ejpam-4594	135	24	}	}	PUNCT
ejpam-4594	135	25	is	be	AUX
ejpam-4594	135	26	a	a	DET
ejpam-4594	135	27	dual	dual	ADJ
ejpam-4594	135	28	b	b	NOUN
ejpam-4594	135	29	-	-	NOUN
ejpam-4594	135	30	topology	topology	NOUN
ejpam-4594	135	31	on	on	ADP
ejpam-4594	135	32	xd	xd	ADP
ejpam-4594	135	33	.	.	PUNCT
ejpam-4594	135	34	proof	proof	NOUN
ejpam-4594	135	35	.	.	PUNCT
ejpam-4594	136	1	let	let	VERB
ejpam-4594	136	2	xd	xd	INTJ
ejpam-4594	136	3	be	be	AUX
ejpam-4594	136	4	a	a	DET
ejpam-4594	136	5	dual	dual	ADJ
ejpam-4594	136	6	b	b	NOUN
ejpam-4594	136	7	-	-	PUNCT
ejpam-4594	136	8	algebra	algebra	NOUN
ejpam-4594	136	9	and	and	CCONJ
ejpam-4594	136	10	x	x	PART
ejpam-4594	136	11	∈	∈	PROPN
ejpam-4594	136	12	xd	xd	PROPN
ejpam-4594	136	13	.	.	PUNCT
ejpam-4594	136	14	note	note	VERB
ejpam-4594	136	15	that	that	SCONJ
ejpam-4594	136	16	v	v	X
ejpam-4594	136	17	[	[	X
ejpam-4594	136	18	x	x	X
ejpam-4594	136	19	]	]	X
ejpam-4594	136	20	⊆	⊆	NUM
ejpam-4594	136	21	xd	xd	INTJ
ejpam-4594	136	22	for	for	ADP
ejpam-4594	136	23	any	any	DET
ejpam-4594	136	24	v	v	PROPN
ejpam-4594	136	25	∈	∈	PROPN
ejpam-4594	136	26	ω	ω	NOUN
ejpam-4594	136	27	.	.	PUNCT
ejpam-4594	137	1	this	this	PRON
ejpam-4594	137	2	implies	imply	VERB
ejpam-4594	137	3	that	that	SCONJ
ejpam-4594	137	4	xd	xd	INTJ
ejpam-4594	137	5	∈	∈	PROPN
ejpam-4594	137	6	τ	τ	X
ejpam-4594	137	7	.	.	PUNCT
ejpam-4594	138	1	since	since	SCONJ
ejpam-4594	138	2	∅	∅	NOUN
ejpam-4594	138	3	does	do	AUX
ejpam-4594	138	4	not	not	PART
ejpam-4594	138	5	contain	contain	VERB
ejpam-4594	138	6	any	any	DET
ejpam-4594	138	7	element	element	NOUN
ejpam-4594	138	8	,	,	PUNCT
ejpam-4594	138	9	then	then	ADV
ejpam-4594	138	10	it	it	PRON
ejpam-4594	138	11	is	be	AUX
ejpam-4594	138	12	vacuously	vacuously	ADV
ejpam-4594	138	13	true	true	ADJ
ejpam-4594	138	14	that	that	SCONJ
ejpam-4594	138	15	∅	∅	NOUN
ejpam-4594	138	16	∈	∈	PROPN
ejpam-4594	138	17	τ	τ	X
ejpam-4594	138	18	.	.	PUNCT
ejpam-4594	139	1	suppose	suppose	VERB
ejpam-4594	139	2	u1	u1	NOUN
ejpam-4594	139	3	,	,	PUNCT
ejpam-4594	139	4	u2	u2	PROPN
ejpam-4594	139	5	∈	∈	PROPN
ejpam-4594	139	6	τ	τ	X
ejpam-4594	139	7	and	and	CCONJ
ejpam-4594	139	8	x	x	PROPN
ejpam-4594	139	9	∈	∈	PROPN
ejpam-4594	139	10	u1	u1	NOUN
ejpam-4594	139	11	∩	∩	NOUN
ejpam-4594	139	12	u2	u2	PROPN
ejpam-4594	139	13	.	.	PUNCT
ejpam-4594	140	1	then	then	ADV
ejpam-4594	140	2	there	there	PRON
ejpam-4594	140	3	exist	exist	VERB
ejpam-4594	140	4	v1	v1	NOUN
ejpam-4594	140	5	,	,	PUNCT
ejpam-4594	140	6	v2	v2	PROPN
ejpam-4594	140	7	∈	∈	PROPN
ejpam-4594	140	8	ω	ω	NOUN
ejpam-4594	140	9	such	such	ADJ
ejpam-4594	140	10	that	that	DET
ejpam-4594	140	11	v1[x	v1[x	PROPN
ejpam-4594	140	12	]	]	PUNCT
ejpam-4594	140	13	⊆	⊆	NUM
ejpam-4594	140	14	u1	u1	NOUN
ejpam-4594	140	15	and	and	CCONJ
ejpam-4594	140	16	v2[x	v2[x	NOUN
ejpam-4594	140	17	]	]	X
ejpam-4594	140	18	⊆	⊆	NUM
ejpam-4594	140	19	u2	u2	NOUN
ejpam-4594	140	20	.	.	PUNCT
ejpam-4594	141	1	since	since	SCONJ
ejpam-4594	141	2	v1∩v2	v1∩v2	PROPN
ejpam-4594	141	3	⊆	⊆	NUM
ejpam-4594	141	4	v1	v1	NOUN
ejpam-4594	141	5	,	,	PUNCT
ejpam-4594	141	6	v2	v2	PROPN
ejpam-4594	141	7	,	,	PUNCT
ejpam-4594	141	8	it	it	PRON
ejpam-4594	141	9	follows	follow	VERB
ejpam-4594	141	10	that	that	SCONJ
ejpam-4594	141	11	(	(	PUNCT
ejpam-4594	141	12	v1∩v2)[x	v1∩v2)[x	NOUN
ejpam-4594	141	13	]	]	X
ejpam-4594	141	14	⊆	⊆	NUM
ejpam-4594	141	15	v1[x	v1[x	ADJ
ejpam-4594	141	16	]	]	PUNCT
ejpam-4594	141	17	⊆	⊆	NUM
ejpam-4594	141	18	u1	u1	NOUN
ejpam-4594	141	19	and	and	CCONJ
ejpam-4594	141	20	(	(	PUNCT
ejpam-4594	141	21	v1	v1	NOUN
ejpam-4594	141	22	∩	∩	ADJ
ejpam-4594	141	23	v2)[x	v2)[x	NOUN
ejpam-4594	141	24	]	]	PUNCT
ejpam-4594	141	25	⊆	⊆	NUM
ejpam-4594	141	26	v2[x	v2[x	NOUN
ejpam-4594	141	27	]	]	PUNCT
ejpam-4594	141	28	⊆	⊆	NUM
ejpam-4594	141	29	u2	u2	NOUN
ejpam-4594	141	30	by	by	ADP
ejpam-4594	141	31	proposition	proposition	NOUN
ejpam-4594	141	32	3	3	NUM
ejpam-4594	141	33	.	.	PUNCT
ejpam-4594	142	1	moreover	moreover	ADV
ejpam-4594	142	2	by	by	ADP
ejpam-4594	142	3	the	the	DET
ejpam-4594	142	4	hypothesis	hypothesis	NOUN
ejpam-4594	142	5	,	,	PUNCT
ejpam-4594	142	6	v1	v1	NOUN
ejpam-4594	142	7	∩	∩	ADJ
ejpam-4594	142	8	v2	v2	PROPN
ejpam-4594	142	9	∈	∈	PROPN
ejpam-4594	142	10	ω	ω	NOUN
ejpam-4594	142	11	.	.	PUNCT
ejpam-4594	143	1	this	this	PRON
ejpam-4594	143	2	implies	imply	VERB
ejpam-4594	143	3	that	that	SCONJ
ejpam-4594	143	4	(	(	PUNCT
ejpam-4594	143	5	v1∩v2)[x	v1∩v2)[x	NOUN
ejpam-4594	143	6	]	]	X
ejpam-4594	143	7	⊆	⊆	NUM
ejpam-4594	143	8	u1	u1	NOUN
ejpam-4594	143	9	and	and	CCONJ
ejpam-4594	143	10	(	(	PUNCT
ejpam-4594	143	11	v1∩v2)[x	v1∩v2)[x	PROPN
ejpam-4594	143	12	]	]	X
ejpam-4594	143	13	⊆	⊆	NUM
ejpam-4594	143	14	u2	u2	NOUN
ejpam-4594	143	15	or	or	CCONJ
ejpam-4594	143	16	(	(	PUNCT
ejpam-4594	143	17	v1∩v2)[x	v1∩v2)[x	PROPN
ejpam-4594	143	18	]	]	X
ejpam-4594	143	19	⊆	⊆	NUM
ejpam-4594	143	20	u1∩u2	u1∩u2	PROPN
ejpam-4594	143	21	.	.	PUNCT
ejpam-4594	144	1	hence	hence	ADV
ejpam-4594	144	2	,	,	PUNCT
ejpam-4594	144	3	u1	u1	NOUN
ejpam-4594	144	4	∩	∩	NOUN
ejpam-4594	144	5	u2	u2	PROPN
ejpam-4594	144	6	∈	∈	PROPN
ejpam-4594	144	7	τ	τ	PROPN
ejpam-4594	144	8	.	.	PUNCT
ejpam-4594	144	9	suppose	suppose	VERB
ejpam-4594	144	10	x	x	SYM
ejpam-4594	144	11	∈	∈	PROPN
ejpam-4594	144	12	⋃	⋃	NOUN
ejpam-4594	144	13	i∈a	i∈a	ADJ
ejpam-4594	144	14	ui	ui	NOUN
ejpam-4594	144	15	where	where	SCONJ
ejpam-4594	144	16	ui	ui	PROPN
ejpam-4594	144	17	∈	∈	PROPN
ejpam-4594	144	18	τ	τ	PROPN
ejpam-4594	144	19	for	for	ADP
ejpam-4594	144	20	all	all	DET
ejpam-4594	144	21	i	i	PRON
ejpam-4594	144	22	∈	∈	VERB
ejpam-4594	144	23	a.	a.	NOUN
ejpam-4594	144	24	then	then	ADV
ejpam-4594	144	25	there	there	PRON
ejpam-4594	144	26	exists	exist	VERB
ejpam-4594	144	27	j	j	PROPN
ejpam-4594	144	28	∈	∈	PROPN
ejpam-4594	144	29	a	a	DET
ejpam-4594	144	30	such	such	ADJ
ejpam-4594	144	31	that	that	SCONJ
ejpam-4594	144	32	x	x	SYM
ejpam-4594	144	33	∈	∈	PROPN
ejpam-4594	144	34	uj	uj	PROPN
ejpam-4594	144	35	.	.	PUNCT
ejpam-4594	145	1	this	this	PRON
ejpam-4594	145	2	implies	imply	VERB
ejpam-4594	145	3	that	that	SCONJ
ejpam-4594	145	4	vj	vj	PROPN
ejpam-4594	145	5	[	[	X
ejpam-4594	145	6	x	x	X
ejpam-4594	145	7	]	]	X
ejpam-4594	145	8	⊆	⊆	NUM
ejpam-4594	145	9	uj	uj	NOUN
ejpam-4594	145	10	for	for	ADP
ejpam-4594	145	11	some	some	DET
ejpam-4594	145	12	vj	vj	PRON
ejpam-4594	145	13	∈	∈	PROPN
ejpam-4594	145	14	ω	ω	PROPN
ejpam-4594	145	15	.	.	PUNCT
ejpam-4594	146	1	hence	hence	ADV
ejpam-4594	146	2	,	,	PUNCT
ejpam-4594	146	3	vj	vj	X
ejpam-4594	147	1	[	[	X
ejpam-4594	147	2	x	x	X
ejpam-4594	147	3	]	]	X
ejpam-4594	147	4	⊆	⊆	NUM
ejpam-4594	147	5	⋃	⋃	ADV
ejpam-4594	147	6	i∈a	i∈a	ADJ
ejpam-4594	147	7	ui	ui	NOUN
ejpam-4594	147	8	.	.	PUNCT
ejpam-4594	148	1	it	it	PRON
ejpam-4594	148	2	follows	follow	VERB
ejpam-4594	148	3	that	that	SCONJ
ejpam-4594	148	4	⋃	⋃	ADV
ejpam-4594	148	5	i∈a	i∈a	ADJ
ejpam-4594	148	6	ui	ui	PROPN
ejpam-4594	148	7	∈	∈	PROPN
ejpam-4594	148	8	τ	τ	PROPN
ejpam-4594	148	9	.	.	PUNCT
ejpam-4594	149	1	therefore	therefore	ADV
ejpam-4594	149	2	,	,	PUNCT
ejpam-4594	149	3	τ	τ	PROPN
ejpam-4594	149	4	is	be	AUX
ejpam-4594	149	5	a	a	DET
ejpam-4594	149	6	dual	dual	ADJ
ejpam-4594	149	7	b	b	NOUN
ejpam-4594	149	8	-	-	NOUN
ejpam-4594	149	9	topology	topology	NOUN
ejpam-4594	149	10	on	on	ADP
ejpam-4594	149	11	xd	xd	ADP
ejpam-4594	149	12	.	.	PUNCT
ejpam-4594	150	1	henceforth	henceforth	ADV
ejpam-4594	150	2	,	,	PUNCT
ejpam-4594	150	3	the	the	DET
ejpam-4594	150	4	dual	dual	ADJ
ejpam-4594	150	5	b	b	NOUN
ejpam-4594	150	6	-	-	PUNCT
ejpam-4594	150	7	topology	topology	NOUN
ejpam-4594	150	8	τ	τ	NOUN
ejpam-4594	150	9	in	in	ADP
ejpam-4594	150	10	the	the	DET
ejpam-4594	150	11	following	follow	VERB
ejpam-4594	150	12	results	result	NOUN
ejpam-4594	150	13	is	be	AUX
ejpam-4594	150	14	the	the	DET
ejpam-4594	150	15	dual	dual	ADJ
ejpam-4594	150	16	b	b	NOUN
ejpam-4594	150	17	-	-	PUNCT
ejpam-4594	150	18	topology	topology	NOUN
ejpam-4594	150	19	in	in	ADP
ejpam-4594	150	20	theorem	theorem	ADJ
ejpam-4594	150	21	4	4	NUM
ejpam-4594	150	22	.	.	PUNCT
ejpam-4594	150	23	k.	k.	PROPN
ejpam-4594	150	24	belleza	belleza	PROPN
ejpam-4594	150	25	,	,	PUNCT
ejpam-4594	150	26	j.	j.	PROPN
ejpam-4594	150	27	albaracin	albaracin	PROPN
ejpam-4594	150	28	/	/	PUNCT
ejpam-4594	150	29	eur	eur	PROPN
ejpam-4594	150	30	.	.	PUNCT
ejpam-4594	151	1	j.	j.	PROPN
ejpam-4594	151	2	pure	pure	PROPN
ejpam-4594	151	3	appl	appl	PROPN
ejpam-4594	151	4	.	.	PROPN
ejpam-4594	151	5	math	math	PROPN
ejpam-4594	151	6	,	,	PUNCT
ejpam-4594	151	7	15	15	NUM
ejpam-4594	151	8	(	(	PUNCT
ejpam-4594	151	9	4	4	NUM
ejpam-4594	151	10	)	)	PUNCT
ejpam-4594	151	11	(	(	PUNCT
ejpam-4594	151	12	2022	2022	NUM
ejpam-4594	151	13	)	)	PUNCT
ejpam-4594	151	14	,	,	PUNCT
ejpam-4594	151	15	1948	1948	NUM
ejpam-4594	151	16	-	-	SYM
ejpam-4594	151	17	1956	1956	NUM
ejpam-4594	151	18	1952	1952	NUM
ejpam-4594	151	19	theorem	theorem	NOUN
ejpam-4594	151	20	5	5	NUM
ejpam-4594	151	21	.	.	PUNCT
ejpam-4594	151	22	suppose	suppose	VERB
ejpam-4594	151	23	xd	xd	INTJ
ejpam-4594	151	24	is	be	AUX
ejpam-4594	151	25	a	a	DET
ejpam-4594	151	26	dual	dual	ADJ
ejpam-4594	151	27	b	b	NOUN
ejpam-4594	151	28	-	-	PUNCT
ejpam-4594	151	29	topological	topological	ADJ
ejpam-4594	151	30	space	space	NOUN
ejpam-4594	151	31	and	and	CCONJ
ejpam-4594	151	32	ω	ω	PROPN
ejpam-4594	151	33	is	be	AUX
ejpam-4594	151	34	a	a	DET
ejpam-4594	151	35	family	family	NOUN
ejpam-4594	151	36	of	of	ADP
ejpam-4594	151	37	subsets	subset	NOUN
ejpam-4594	151	38	in	in	ADP
ejpam-4594	151	39	a	a	DET
ejpam-4594	151	40	dual	dual	ADJ
ejpam-4594	151	41	b	b	NOUN
ejpam-4594	151	42	-	-	PUNCT
ejpam-4594	151	43	algebra	algebra	NOUN
ejpam-4594	151	44	xd	xd	INTJ
ejpam-4594	151	45	that	that	PRON
ejpam-4594	151	46	is	be	AUX
ejpam-4594	151	47	closed	close	VERB
ejpam-4594	151	48	under	under	ADP
ejpam-4594	151	49	finite	finite	ADJ
ejpam-4594	151	50	intersections	intersection	NOUN
ejpam-4594	151	51	.	.	PUNCT
ejpam-4594	152	1	if	if	SCONJ
ejpam-4594	152	2	∅	∅	NOUN
ejpam-4594	152	3	∈	∈	PROPN
ejpam-4594	152	4	ω	ω	PROPN
ejpam-4594	152	5	,	,	PUNCT
ejpam-4594	152	6	then	then	ADV
ejpam-4594	152	7	xd	xd	INTJ
ejpam-4594	152	8	is	be	VERB
ejpam-4594	152	9	a	a	DET
ejpam-4594	152	10	tdb	tdb	NOUN
ejpam-4594	152	11	-	-	NOUN
ejpam-4594	152	12	algebra	algebra	NOUN
ejpam-4594	152	13	.	.	PUNCT
ejpam-4594	153	1	proof	proof	NOUN
ejpam-4594	153	2	.	.	PUNCT
ejpam-4594	154	1	suppose	suppose	VERB
ejpam-4594	154	2	xd	xd	INTJ
ejpam-4594	154	3	is	be	AUX
ejpam-4594	154	4	a	a	DET
ejpam-4594	154	5	dual	dual	ADJ
ejpam-4594	154	6	b	b	NOUN
ejpam-4594	154	7	-	-	PUNCT
ejpam-4594	154	8	topological	topological	ADJ
ejpam-4594	154	9	space	space	NOUN
ejpam-4594	154	10	and	and	CCONJ
ejpam-4594	154	11	ω	ω	PROPN
ejpam-4594	154	12	is	be	AUX
ejpam-4594	154	13	a	a	DET
ejpam-4594	154	14	family	family	NOUN
ejpam-4594	154	15	of	of	ADP
ejpam-4594	154	16	subsets	subset	NOUN
ejpam-4594	154	17	in	in	ADP
ejpam-4594	154	18	a	a	DET
ejpam-4594	154	19	dual	dual	ADJ
ejpam-4594	154	20	b	b	NOUN
ejpam-4594	154	21	-	-	PUNCT
ejpam-4594	154	22	algebra	algebra	NOUN
ejpam-4594	154	23	xd	xd	INTJ
ejpam-4594	154	24	that	that	PRON
ejpam-4594	154	25	is	be	AUX
ejpam-4594	154	26	closed	close	VERB
ejpam-4594	154	27	under	under	ADP
ejpam-4594	154	28	finite	finite	ADJ
ejpam-4594	154	29	intersections	intersection	NOUN
ejpam-4594	154	30	.	.	PUNCT
ejpam-4594	155	1	let	let	VERB
ejpam-4594	155	2	v	v	ADP
ejpam-4594	155	3	⊆	⊆	NUM
ejpam-4594	155	4	xd	xd	ADP
ejpam-4594	155	5	.	.	PUNCT
ejpam-4594	156	1	then	then	ADV
ejpam-4594	156	2	for	for	ADP
ejpam-4594	156	3	all	all	DET
ejpam-4594	156	4	x	x	SYM
ejpam-4594	156	5	∈	∈	NOUN
ejpam-4594	156	6	v	v	NOUN
ejpam-4594	156	7	,	,	PUNCT
ejpam-4594	156	8	there	there	PRON
ejpam-4594	156	9	exists	exist	VERB
ejpam-4594	156	10	∅	∅	NOUN
ejpam-4594	156	11	∈	∈	PROPN
ejpam-4594	156	12	ω	ω	NOUN
ejpam-4594	156	13	such	such	ADJ
ejpam-4594	156	14	that	that	SCONJ
ejpam-4594	156	15	∅[v	∅[v	PROPN
ejpam-4594	156	16	]	]	X
ejpam-4594	156	17	=	=	SYM
ejpam-4594	156	18	∅	∅	NOUN
ejpam-4594	156	19	⊆	⊆	NUM
ejpam-4594	156	20	v	v	NOUN
ejpam-4594	156	21	.	.	PUNCT
ejpam-4594	157	1	this	this	PRON
ejpam-4594	157	2	implies	imply	VERB
ejpam-4594	157	3	that	that	SCONJ
ejpam-4594	157	4	v	v	X
ejpam-4594	157	5	∈	∈	PROPN
ejpam-4594	157	6	τ	τ	X
ejpam-4594	157	7	.	.	PUNCT
ejpam-4594	158	1	hence	hence	ADV
ejpam-4594	158	2	p(xd	p(xd	NOUN
ejpam-4594	158	3	)	)	PUNCT
ejpam-4594	159	1	⊆	⊆	NUM
ejpam-4594	159	2	τ	τ	X
ejpam-4594	159	3	where	where	SCONJ
ejpam-4594	159	4	p(xd	p(xd	NOUN
ejpam-4594	159	5	)	)	PUNCT
ejpam-4594	159	6	is	be	AUX
ejpam-4594	159	7	the	the	DET
ejpam-4594	159	8	power	power	NOUN
ejpam-4594	159	9	set	set	NOUN
ejpam-4594	159	10	of	of	ADP
ejpam-4594	159	11	xd	xd	ADP
ejpam-4594	159	12	.	.	PUNCT
ejpam-4594	160	1	since	since	SCONJ
ejpam-4594	160	2	τ	τ	PROPN
ejpam-4594	160	3	⊆	⊆	NUM
ejpam-4594	160	4	p(xd	p(xd	NOUN
ejpam-4594	160	5	)	)	PUNCT
ejpam-4594	160	6	,	,	PUNCT
ejpam-4594	160	7	it	it	PRON
ejpam-4594	160	8	follows	follow	VERB
ejpam-4594	160	9	that	that	SCONJ
ejpam-4594	160	10	τ	τ	PROPN
ejpam-4594	160	11	=	=	PUNCT
ejpam-4594	160	12	p(xd	p(xd	PROPN
ejpam-4594	160	13	)	)	PUNCT
ejpam-4594	160	14	.	.	PUNCT
ejpam-4594	161	1	let	let	VERB
ejpam-4594	161	2	x	x	PRON
ejpam-4594	161	3	,	,	PUNCT
ejpam-4594	161	4	y	y	PROPN
ejpam-4594	161	5	∈	∈	PROPN
ejpam-4594	161	6	xd	xd	INTJ
ejpam-4594	161	7	and	and	CCONJ
ejpam-4594	161	8	u(x	u(x	VERB
ejpam-4594	161	9	◦	◦	NOUN
ejpam-4594	161	10	y	y	NOUN
ejpam-4594	161	11	)	)	PUNCT
ejpam-4594	161	12	∈	∈	PROPN
ejpam-4594	161	13	τ	τ	X
ejpam-4594	161	14	.	.	PUNCT
ejpam-4594	162	1	then	then	ADV
ejpam-4594	162	2	there	there	PRON
ejpam-4594	162	3	exist	exist	VERB
ejpam-4594	162	4	{	{	PUNCT
ejpam-4594	162	5	x	x	NOUN
ejpam-4594	162	6	}	}	PUNCT
ejpam-4594	162	7	,	,	PUNCT
ejpam-4594	162	8	{	{	PUNCT
ejpam-4594	162	9	y	y	NOUN
ejpam-4594	162	10	}	}	PUNCT
ejpam-4594	162	11	∈	∈	PROPN
ejpam-4594	162	12	τ	τ	X
ejpam-4594	162	13	such	such	ADJ
ejpam-4594	162	14	that	that	SCONJ
ejpam-4594	162	15	{	{	PUNCT
ejpam-4594	162	16	x	x	NOUN
ejpam-4594	162	17	}	}	PUNCT
ejpam-4594	162	18	◦	◦	NOUN
ejpam-4594	162	19	{	{	PUNCT
ejpam-4594	162	20	y	y	NOUN
ejpam-4594	162	21	}	}	PUNCT
ejpam-4594	162	22	=	=	SYM
ejpam-4594	162	23	{	{	PUNCT
ejpam-4594	162	24	x	x	PUNCT
ejpam-4594	162	25	◦	◦	VERB
ejpam-4594	162	26	y	y	NOUN
ejpam-4594	162	27	}	}	PUNCT
ejpam-4594	162	28	⊆	⊆	NUM
ejpam-4594	162	29	u(x	u(x	PROPN
ejpam-4594	162	30	◦	◦	NOUN
ejpam-4594	162	31	y	y	NOUN
ejpam-4594	162	32	)	)	PUNCT
ejpam-4594	162	33	.	.	PUNCT
ejpam-4594	163	1	then	then	ADV
ejpam-4594	163	2	xd	xd	INTJ
ejpam-4594	163	3	is	be	VERB
ejpam-4594	163	4	a	a	DET
ejpam-4594	163	5	tdb	tdb	NOUN
ejpam-4594	163	6	-	-	NOUN
ejpam-4594	163	7	algebra	algebra	NOUN
ejpam-4594	163	8	.	.	PUNCT
ejpam-4594	164	1	theorem	theorem	NOUN
ejpam-4594	164	2	6	6	NUM
ejpam-4594	164	3	.	.	PUNCT
ejpam-4594	165	1	let	let	VERB
ejpam-4594	165	2	ω	ω	NUM
ejpam-4594	165	3	be	be	AUX
ejpam-4594	165	4	a	a	DET
ejpam-4594	165	5	family	family	NOUN
ejpam-4594	165	6	of	of	ADP
ejpam-4594	165	7	subsets	subset	NOUN
ejpam-4594	165	8	in	in	ADP
ejpam-4594	165	9	the	the	DET
ejpam-4594	165	10	dual	dual	ADJ
ejpam-4594	165	11	b	b	NOUN
ejpam-4594	165	12	-	-	PUNCT
ejpam-4594	165	13	algebra	algebra	NOUN
ejpam-4594	165	14	xd	xd	INTJ
ejpam-4594	165	15	that	that	PRON
ejpam-4594	165	16	is	be	AUX
ejpam-4594	165	17	closed	close	VERB
ejpam-4594	165	18	under	under	ADP
ejpam-4594	165	19	finite	finite	ADJ
ejpam-4594	165	20	intersections	intersection	NOUN
ejpam-4594	165	21	.	.	PUNCT
ejpam-4594	166	1	suppose	suppose	VERB
ejpam-4594	166	2	that	that	SCONJ
ejpam-4594	166	3	for	for	ADP
ejpam-4594	166	4	each	each	DET
ejpam-4594	166	5	u	u	PROPN
ejpam-4594	166	6	∈	∈	PROPN
ejpam-4594	166	7	ω	ω	PROPN
ejpam-4594	166	8	,	,	PUNCT
ejpam-4594	166	9	1	1	NUM
ejpam-4594	166	10	∈	∈	NOUN
ejpam-4594	166	11	u	u	NOUN
ejpam-4594	166	12	and	and	CCONJ
ejpam-4594	166	13	for	for	ADP
ejpam-4594	166	14	each	each	DET
ejpam-4594	166	15	x	x	SYM
ejpam-4594	166	16	∈	∈	PROPN
ejpam-4594	166	17	u	u	NOUN
ejpam-4594	166	18	,	,	PUNCT
ejpam-4594	166	19	there	there	PRON
ejpam-4594	166	20	exists	exist	VERB
ejpam-4594	166	21	v	v	ADP
ejpam-4594	166	22	∈	∈	PROPN
ejpam-4594	166	23	ω	ω	NOUN
ejpam-4594	166	24	such	such	ADJ
ejpam-4594	166	25	that	that	PRON
ejpam-4594	166	26	v	v	PART
ejpam-4594	166	27	[	[	X
ejpam-4594	166	28	x	x	X
ejpam-4594	166	29	]	]	X
ejpam-4594	166	30	⊆	⊆	NUM
ejpam-4594	166	31	u	u	NOUN
ejpam-4594	166	32	.	.	PUNCT
ejpam-4594	167	1	then	then	ADV
ejpam-4594	167	2	the	the	DET
ejpam-4594	167	3	set	set	NOUN
ejpam-4594	167	4	b	b	NOUN
ejpam-4594	167	5	=	=	SYM
ejpam-4594	167	6	{	{	PUNCT
ejpam-4594	167	7	u	u	NOUN
ejpam-4594	167	8	[	[	X
ejpam-4594	167	9	a	a	X
ejpam-4594	167	10	]	]	X
ejpam-4594	167	11	|	|	NOUN
ejpam-4594	167	12	u	u	PROPN
ejpam-4594	167	13	∈	∈	PROPN
ejpam-4594	167	14	ω	ω	PROPN
ejpam-4594	167	15	,	,	PUNCT
ejpam-4594	167	16	a	a	DET
ejpam-4594	167	17	∈	∈	PROPN
ejpam-4594	167	18	xd	xd	ADP
ejpam-4594	167	19	}	}	PUNCT
ejpam-4594	167	20	is	be	AUX
ejpam-4594	167	21	a	a	DET
ejpam-4594	167	22	basis	basis	NOUN
ejpam-4594	167	23	for	for	ADP
ejpam-4594	167	24	the	the	DET
ejpam-4594	167	25	dual	dual	ADJ
ejpam-4594	167	26	b	b	NOUN
ejpam-4594	167	27	-	-	NOUN
ejpam-4594	167	28	topology	topology	NOUN
ejpam-4594	167	29	.	.	PUNCT
ejpam-4594	168	1	proof	proof	NOUN
ejpam-4594	168	2	.	.	PUNCT
ejpam-4594	169	1	first	first	ADV
ejpam-4594	169	2	,	,	PUNCT
ejpam-4594	169	3	we	we	PRON
ejpam-4594	169	4	will	will	AUX
ejpam-4594	169	5	show	show	VERB
ejpam-4594	169	6	that	that	SCONJ
ejpam-4594	169	7	b	b	NOUN
ejpam-4594	169	8	⊆	⊆	NUM
ejpam-4594	169	9	τ	τ	X
ejpam-4594	169	10	.	.	PUNCT
ejpam-4594	169	11	suppose	suppose	VERB
ejpam-4594	169	12	x	x	SYM
ejpam-4594	169	13	∈	∈	PRON
ejpam-4594	169	14	u	u	NOUN
ejpam-4594	169	15	[	[	X
ejpam-4594	169	16	a	a	X
ejpam-4594	169	17	]	]	X
ejpam-4594	169	18	∈	∈	PROPN
ejpam-4594	169	19	b	b	NOUN
ejpam-4594	169	20	for	for	ADP
ejpam-4594	169	21	any	any	DET
ejpam-4594	169	22	a	a	DET
ejpam-4594	169	23	∈	∈	PROPN
ejpam-4594	169	24	xd	xd	NOUN
ejpam-4594	169	25	and	and	CCONJ
ejpam-4594	170	1	u	u	PROPN
ejpam-4594	170	2	∈	∈	PROPN
ejpam-4594	170	3	ω	ω	PROPN
ejpam-4594	170	4	.	.	PUNCT
ejpam-4594	171	1	then	then	ADV
ejpam-4594	171	2	a	a	DET
ejpam-4594	171	3	◦	◦	NOUN
ejpam-4594	171	4	x	x	SYM
ejpam-4594	171	5	∈	∈	PROPN
ejpam-4594	171	6	u	u	NOUN
ejpam-4594	171	7	.	.	PUNCT
ejpam-4594	172	1	moreover	moreover	ADV
ejpam-4594	172	2	,	,	PUNCT
ejpam-4594	172	3	there	there	PRON
ejpam-4594	172	4	exists	exist	VERB
ejpam-4594	172	5	v	v	ADP
ejpam-4594	172	6	∈	∈	PROPN
ejpam-4594	172	7	ω	ω	NOUN
ejpam-4594	173	1	such	such	ADJ
ejpam-4594	173	2	that	that	PRON
ejpam-4594	173	3	v	v	ADP
ejpam-4594	173	4	[	[	X
ejpam-4594	173	5	a	a	DET
ejpam-4594	173	6	◦	◦	NOUN
ejpam-4594	173	7	x	x	NOUN
ejpam-4594	173	8	]	]	X
ejpam-4594	173	9	⊆	⊆	NUM
ejpam-4594	173	10	u	u	NOUN
ejpam-4594	173	11	.	.	PUNCT
ejpam-4594	174	1	let	let	VERB
ejpam-4594	174	2	y	y	PROPN
ejpam-4594	174	3	∈	∈	PROPN
ejpam-4594	174	4	v	v	ADP
ejpam-4594	174	5	[	[	X
ejpam-4594	174	6	x	x	X
ejpam-4594	174	7	]	]	X
ejpam-4594	174	8	.	.	PUNCT
ejpam-4594	175	1	then	then	ADV
ejpam-4594	175	2	x	x	X
ejpam-4594	175	3	◦	◦	VERB
ejpam-4594	175	4	y	y	PROPN
ejpam-4594	175	5	∈	∈	PROPN
ejpam-4594	175	6	v	v	NOUN
ejpam-4594	175	7	.	.	PUNCT
ejpam-4594	176	1	by	by	ADP
ejpam-4594	176	2	theorem	theorem	NOUN
ejpam-4594	176	3	1(iii	1(iii	NUM
ejpam-4594	176	4	)	)	PUNCT
ejpam-4594	176	5	,	,	PUNCT
ejpam-4594	176	6	(	(	PUNCT
ejpam-4594	176	7	a	a	DET
ejpam-4594	176	8	◦	◦	NOUN
ejpam-4594	176	9	x)	x)	NOUN
ejpam-4594	176	10	◦	◦	NOUN
ejpam-4594	176	11	(a	(a	NOUN
ejpam-4594	176	12	◦	◦	NOUN
ejpam-4594	176	13	y	y	NOUN
ejpam-4594	176	14	)	)	PUNCT
ejpam-4594	176	15	=	=	SYM
ejpam-4594	177	1	x	x	X
ejpam-4594	177	2	◦	◦	VERB
ejpam-4594	177	3	y	y	PROPN
ejpam-4594	177	4	∈	∈	PROPN
ejpam-4594	177	5	v	v	NOUN
ejpam-4594	177	6	.	.	PUNCT
ejpam-4594	178	1	hence	hence	ADV
ejpam-4594	178	2	,	,	PUNCT
ejpam-4594	178	3	a	a	DET
ejpam-4594	178	4	◦	◦	NOUN
ejpam-4594	178	5	y	y	NOUN
ejpam-4594	178	6	∈	∈	PROPN
ejpam-4594	178	7	v	v	ADP
ejpam-4594	178	8	[	[	X
ejpam-4594	178	9	a	a	DET
ejpam-4594	178	10	◦	◦	NOUN
ejpam-4594	178	11	x	x	NOUN
ejpam-4594	178	12	]	]	X
ejpam-4594	178	13	⊆	⊆	NUM
ejpam-4594	178	14	u	u	NOUN
ejpam-4594	178	15	.	.	PUNCT
ejpam-4594	179	1	this	this	PRON
ejpam-4594	179	2	implies	imply	VERB
ejpam-4594	179	3	that	that	SCONJ
ejpam-4594	179	4	y	y	PROPN
ejpam-4594	179	5	∈	∈	PROPN
ejpam-4594	179	6	u	u	NOUN
ejpam-4594	180	1	[	[	X
ejpam-4594	180	2	a	a	X
ejpam-4594	180	3	]	]	X
ejpam-4594	180	4	.	.	PUNCT
ejpam-4594	181	1	moreover	moreover	ADV
ejpam-4594	181	2	,	,	PUNCT
ejpam-4594	181	3	u	u	X
ejpam-4594	182	1	[	[	X
ejpam-4594	182	2	a	a	X
ejpam-4594	182	3	]	]	X
ejpam-4594	182	4	∈	∈	PROPN
ejpam-4594	182	5	τ	τ	X
ejpam-4594	182	6	.	.	PUNCT
ejpam-4594	183	1	that	that	PRON
ejpam-4594	183	2	is	be	AUX
ejpam-4594	183	3	,	,	PUNCT
ejpam-4594	183	4	v	v	X
ejpam-4594	183	5	[	[	X
ejpam-4594	183	6	x	x	X
ejpam-4594	183	7	]	]	X
ejpam-4594	183	8	⊆	⊆	NUM
ejpam-4594	183	9	u	u	NOUN
ejpam-4594	183	10	[	[	X
ejpam-4594	183	11	a	a	X
ejpam-4594	183	12	]	]	X
ejpam-4594	183	13	.	.	PUNCT
ejpam-4594	184	1	consequently	consequently	ADV
ejpam-4594	184	2	,	,	PUNCT
ejpam-4594	184	3	b	b	PROPN
ejpam-4594	184	4	⊆	⊆	NUM
ejpam-4594	184	5	τ	τ	X
ejpam-4594	184	6	.	.	PUNCT
ejpam-4594	184	7	now	now	ADV
ejpam-4594	184	8	let	let	VERB
ejpam-4594	184	9	u	u	PRON
ejpam-4594	184	10	∈	∈	PROPN
ejpam-4594	184	11	τ	τ	X
ejpam-4594	184	12	and	and	CCONJ
ejpam-4594	184	13	x	x	PROPN
ejpam-4594	184	14	∈	∈	PROPN
ejpam-4594	184	15	u	u	NOUN
ejpam-4594	184	16	.	.	PUNCT
ejpam-4594	185	1	then	then	ADV
ejpam-4594	185	2	there	there	PRON
ejpam-4594	185	3	exists	exist	VERB
ejpam-4594	185	4	v	v	ADP
ejpam-4594	185	5	∈	∈	PROPN
ejpam-4594	185	6	ω	ω	NOUN
ejpam-4594	185	7	such	such	ADJ
ejpam-4594	185	8	that	that	PRON
ejpam-4594	185	9	v	v	PART
ejpam-4594	185	10	[	[	X
ejpam-4594	185	11	x	x	X
ejpam-4594	185	12	]	]	X
ejpam-4594	185	13	⊆	⊆	NUM
ejpam-4594	185	14	u	u	NOUN
ejpam-4594	185	15	.	.	PUNCT
ejpam-4594	186	1	by	by	ADP
ejpam-4594	186	2	proposition	proposition	NOUN
ejpam-4594	186	3	2	2	NUM
ejpam-4594	186	4	and	and	CCONJ
ejpam-4594	186	5	remark	remark	NOUN
ejpam-4594	186	6	2	2	NUM
ejpam-4594	186	7	,	,	PUNCT
ejpam-4594	186	8	x	x	SYM
ejpam-4594	186	9	∈	∈	NOUN
ejpam-4594	186	10	v	v	PART
ejpam-4594	186	11	[	[	X
ejpam-4594	186	12	x	x	X
ejpam-4594	186	13	]	]	X
ejpam-4594	186	14	.	.	PUNCT
ejpam-4594	187	1	therefore	therefore	ADV
ejpam-4594	187	2	,	,	PUNCT
ejpam-4594	187	3	there	there	PRON
ejpam-4594	187	4	exists	exist	VERB
ejpam-4594	187	5	v	v	ADP
ejpam-4594	187	6	[	[	X
ejpam-4594	187	7	x	x	X
ejpam-4594	187	8	]	]	X
ejpam-4594	187	9	∈	∈	PROPN
ejpam-4594	187	10	b	b	NOUN
ejpam-4594	188	1	such	such	ADJ
ejpam-4594	188	2	that	that	SCONJ
ejpam-4594	188	3	x	x	SYM
ejpam-4594	188	4	∈	∈	NOUN
ejpam-4594	188	5	v	v	PART
ejpam-4594	188	6	[	[	X
ejpam-4594	188	7	x	x	X
ejpam-4594	188	8	]	]	X
ejpam-4594	188	9	⊆	⊆	NUM
ejpam-4594	188	10	u.	u.	NOUN
ejpam-4594	188	11	by	by	ADP
ejpam-4594	188	12	theorem	theorem	NOUN
ejpam-4594	188	13	2	2	NUM
ejpam-4594	188	14	,	,	PUNCT
ejpam-4594	188	15	b	b	NOUN
ejpam-4594	188	16	is	be	AUX
ejpam-4594	188	17	a	a	DET
ejpam-4594	188	18	basis	basis	NOUN
ejpam-4594	188	19	for	for	ADP
ejpam-4594	188	20	the	the	DET
ejpam-4594	188	21	dual	dual	ADJ
ejpam-4594	188	22	b	b	NOUN
ejpam-4594	188	23	-	-	PUNCT
ejpam-4594	188	24	topology	topology	NOUN
ejpam-4594	188	25	τ	τ	PROPN
ejpam-4594	188	26	.	.	PUNCT
ejpam-4594	189	1	theorem	theorem	VERB
ejpam-4594	189	2	7	7	NUM
ejpam-4594	189	3	.	.	PUNCT
ejpam-4594	190	1	let	let	VERB
ejpam-4594	190	2	ω	ω	NUM
ejpam-4594	190	3	be	be	AUX
ejpam-4594	190	4	a	a	DET
ejpam-4594	190	5	family	family	NOUN
ejpam-4594	190	6	of	of	ADP
ejpam-4594	190	7	subsets	subset	NOUN
ejpam-4594	190	8	in	in	ADP
ejpam-4594	190	9	a	a	DET
ejpam-4594	190	10	commutative	commutative	ADJ
ejpam-4594	190	11	dual	dual	ADJ
ejpam-4594	190	12	b	b	NOUN
ejpam-4594	190	13	-	-	PUNCT
ejpam-4594	190	14	algebra	algebra	NOUN
ejpam-4594	190	15	xd	xd	INTJ
ejpam-4594	190	16	that	that	PRON
ejpam-4594	190	17	is	be	AUX
ejpam-4594	190	18	closed	close	VERB
ejpam-4594	190	19	under	under	ADP
ejpam-4594	190	20	finite	finite	ADJ
ejpam-4594	190	21	intersections	intersection	NOUN
ejpam-4594	190	22	.	.	PUNCT
ejpam-4594	191	1	suppose	suppose	VERB
ejpam-4594	191	2	that	that	SCONJ
ejpam-4594	191	3	for	for	ADP
ejpam-4594	191	4	each	each	DET
ejpam-4594	191	5	v	v	ADP
ejpam-4594	191	6	∈	∈	PROPN
ejpam-4594	191	7	ω	ω	NOUN
ejpam-4594	191	8	,	,	PUNCT
ejpam-4594	191	9	1	1	NUM
ejpam-4594	191	10	∈	∈	NOUN
ejpam-4594	191	11	v	v	NOUN
ejpam-4594	191	12	and	and	CCONJ
ejpam-4594	191	13	for	for	ADP
ejpam-4594	191	14	each	each	DET
ejpam-4594	191	15	x	x	SYM
ejpam-4594	191	16	∈	∈	PROPN
ejpam-4594	191	17	v	v	ADP
ejpam-4594	191	18	∈	∈	PROPN
ejpam-4594	191	19	ω	ω	NOUN
ejpam-4594	191	20	,	,	PUNCT
ejpam-4594	191	21	there	there	PRON
ejpam-4594	191	22	exists	exist	VERB
ejpam-4594	191	23	u	u	PROPN
ejpam-4594	191	24	∈	∈	PROPN
ejpam-4594	191	25	ω	ω	NUM
ejpam-4594	191	26	such	such	ADJ
ejpam-4594	191	27	that	that	SCONJ
ejpam-4594	191	28	u	u	PRON
ejpam-4594	192	1	[	[	X
ejpam-4594	192	2	x	x	X
ejpam-4594	192	3	]	]	X
ejpam-4594	192	4	⊆	⊆	NUM
ejpam-4594	192	5	v	v	NOUN
ejpam-4594	192	6	.	.	PUNCT
ejpam-4594	193	1	then	then	ADV
ejpam-4594	193	2	xd	xd	INTJ
ejpam-4594	193	3	is	be	VERB
ejpam-4594	193	4	a	a	DET
ejpam-4594	193	5	tdb	tdb	NOUN
ejpam-4594	193	6	-	-	NOUN
ejpam-4594	193	7	algebra	algebra	NOUN
ejpam-4594	193	8	.	.	PUNCT
ejpam-4594	194	1	proof	proof	NOUN
ejpam-4594	194	2	.	.	PUNCT
ejpam-4594	195	1	let	let	VERB
ejpam-4594	195	2	x	x	PRON
ejpam-4594	195	3	,	,	PUNCT
ejpam-4594	195	4	y	y	PROPN
ejpam-4594	195	5	∈	∈	PROPN
ejpam-4594	196	1	xd	xd	INTJ
ejpam-4594	196	2	and	and	CCONJ
ejpam-4594	196	3	u	u	PROPN
ejpam-4594	196	4	∈	∈	PROPN
ejpam-4594	196	5	τ	τ	X
ejpam-4594	196	6	such	such	ADJ
ejpam-4594	196	7	that	that	SCONJ
ejpam-4594	196	8	x	x	X
ejpam-4594	196	9	◦	◦	VERB
ejpam-4594	196	10	y	y	PROPN
ejpam-4594	196	11	∈	∈	PROPN
ejpam-4594	196	12	u	u	PROPN
ejpam-4594	196	13	.	.	PUNCT
ejpam-4594	197	1	then	then	ADV
ejpam-4594	197	2	there	there	PRON
ejpam-4594	197	3	exists	exist	VERB
ejpam-4594	197	4	v	v	ADP
ejpam-4594	197	5	∈	∈	PROPN
ejpam-4594	197	6	ω	ω	NOUN
ejpam-4594	197	7	such	such	ADJ
ejpam-4594	197	8	that	that	PRON
ejpam-4594	197	9	v	v	PART
ejpam-4594	198	1	[	[	X
ejpam-4594	198	2	x	x	X
ejpam-4594	198	3	◦	◦	VERB
ejpam-4594	198	4	y	y	X
ejpam-4594	198	5	]	]	X
ejpam-4594	198	6	⊆	⊆	NUM
ejpam-4594	198	7	u	u	NOUN
ejpam-4594	198	8	.	.	PUNCT
ejpam-4594	199	1	by	by	ADP
ejpam-4594	199	2	remark	remark	NOUN
ejpam-4594	199	3	2	2	NUM
ejpam-4594	199	4	and	and	CCONJ
ejpam-4594	199	5	proposition	proposition	NOUN
ejpam-4594	199	6	2	2	NUM
ejpam-4594	199	7	respectively	respectively	ADV
ejpam-4594	199	8	,	,	PUNCT
ejpam-4594	199	9	v	v	ADP
ejpam-4594	199	10	′[x	′[x	NOUN
ejpam-4594	199	11	◦	◦	NOUN
ejpam-4594	199	12	y	y	NOUN
ejpam-4594	199	13	]	]	X
ejpam-4594	199	14	⊆	⊆	NUM
ejpam-4594	199	15	v	v	PART
ejpam-4594	199	16	[	[	X
ejpam-4594	199	17	x	x	X
ejpam-4594	199	18	◦	◦	NOUN
ejpam-4594	199	19	y	y	X
ejpam-4594	199	20	]	]	PUNCT
ejpam-4594	199	21	with	with	ADP
ejpam-4594	199	22	x	x	PROPN
ejpam-4594	199	23	∈	∈	PROPN
ejpam-4594	199	24	v	v	ADP
ejpam-4594	199	25	′[x	′[x	NOUN
ejpam-4594	199	26	]	]	PUNCT
ejpam-4594	199	27	and	and	CCONJ
ejpam-4594	199	28	y	y	PROPN
ejpam-4594	199	29	∈	∈	PROPN
ejpam-4594	199	30	v	v	ADP
ejpam-4594	199	31	′[y	′[y	PROPN
ejpam-4594	199	32	]	]	PUNCT
ejpam-4594	199	33	.	.	PUNCT
ejpam-4594	200	1	we	we	PRON
ejpam-4594	200	2	will	will	AUX
ejpam-4594	200	3	show	show	VERB
ejpam-4594	200	4	that	that	SCONJ
ejpam-4594	200	5	v	v	ADP
ejpam-4594	200	6	′[x	′[x	NOUN
ejpam-4594	200	7	]	]	PUNCT
ejpam-4594	200	8	◦	◦	NOUN
ejpam-4594	200	9	v	v	ADP
ejpam-4594	200	10	′[y	′[y	PROPN
ejpam-4594	200	11	]	]	PUNCT
ejpam-4594	200	12	⊆	⊆	NUM
ejpam-4594	200	13	v	v	ADP
ejpam-4594	200	14	′[x	′[x	NOUN
ejpam-4594	200	15	◦	◦	NOUN
ejpam-4594	200	16	y	y	NOUN
ejpam-4594	200	17	]	]	PUNCT
ejpam-4594	200	18	.	.	PUNCT
ejpam-4594	201	1	suppose	suppose	VERB
ejpam-4594	201	2	a	a	DET
ejpam-4594	201	3	∈	∈	NOUN
ejpam-4594	201	4	v	v	ADP
ejpam-4594	201	5	′[x	′[x	NOUN
ejpam-4594	201	6	]	]	PUNCT
ejpam-4594	201	7	.	.	PUNCT
ejpam-4594	202	1	then	then	ADV
ejpam-4594	202	2	a	a	DET
ejpam-4594	202	3	◦	◦	NOUN
ejpam-4594	202	4	x	x	SYM
ejpam-4594	202	5	,	,	PUNCT
ejpam-4594	202	6	x	x	PUNCT
ejpam-4594	202	7	◦	◦	VERB
ejpam-4594	202	8	a	a	DET
ejpam-4594	202	9	∈	∈	NOUN
ejpam-4594	202	10	v	v	NOUN
ejpam-4594	202	11	.	.	PUNCT
ejpam-4594	203	1	by	by	ADP
ejpam-4594	203	2	(	(	PUNCT
ejpam-4594	203	3	db1	db1	NOUN
ejpam-4594	203	4	)	)	PUNCT
ejpam-4594	203	5	,	,	PUNCT
ejpam-4594	203	6	theorem	theorem	VERB
ejpam-4594	203	7	1(iii	1(iii	NUM
ejpam-4594	203	8	)	)	PUNCT
ejpam-4594	203	9	,	,	PUNCT
ejpam-4594	203	10	and	and	CCONJ
ejpam-4594	203	11	proposition	proposition	NOUN
ejpam-4594	203	12	1	1	NUM
ejpam-4594	203	13	,	,	PUNCT
ejpam-4594	203	14	1	1	NUM
ejpam-4594	203	15	=	=	SYM
ejpam-4594	203	16	(	(	PUNCT
ejpam-4594	203	17	a	a	DET
ejpam-4594	203	18	◦	◦	NOUN
ejpam-4594	203	19	y	y	NOUN
ejpam-4594	203	20	)	)	PUNCT
ejpam-4594	203	21	◦	◦	NOUN
ejpam-4594	203	22	(	(	PUNCT
ejpam-4594	203	23	a	a	DET
ejpam-4594	203	24	◦	◦	NOUN
ejpam-4594	203	25	y	y	NOUN
ejpam-4594	203	26	)	)	PUNCT
ejpam-4594	203	27	=	=	SYM
ejpam-4594	204	1	(	(	PUNCT
ejpam-4594	204	2	a	a	DET
ejpam-4594	204	3	◦	◦	NOUN
ejpam-4594	204	4	y	y	NOUN
ejpam-4594	204	5	)	)	PUNCT
ejpam-4594	204	6	◦	◦	NOUN
ejpam-4594	205	1	[	[	X
ejpam-4594	205	2	(	(	PUNCT
ejpam-4594	205	3	x	x	SYM
ejpam-4594	205	4	◦	◦	VERB
ejpam-4594	205	5	a	a	PRON
ejpam-4594	205	6	)	)	PUNCT
ejpam-4594	205	7	◦	◦	NOUN
ejpam-4594	205	8	(	(	PUNCT
ejpam-4594	205	9	x	x	PART
ejpam-4594	205	10	◦	◦	VERB
ejpam-4594	205	11	y	y	PROPN
ejpam-4594	205	12	)	)	PUNCT
ejpam-4594	205	13	]	]	PUNCT
ejpam-4594	206	1	=	=	PUNCT
ejpam-4594	206	2	(	(	PUNCT
ejpam-4594	206	3	x	x	PART
ejpam-4594	206	4	◦	◦	VERB
ejpam-4594	206	5	a	a	X
ejpam-4594	206	6	)	)	PUNCT
ejpam-4594	206	7	◦	◦	NOUN
ejpam-4594	206	8	[	[	X
ejpam-4594	207	1	(	(	PUNCT
ejpam-4594	207	2	a	a	DET
ejpam-4594	207	3	◦	◦	NOUN
ejpam-4594	207	4	y	y	NOUN
ejpam-4594	207	5	)	)	PUNCT
ejpam-4594	207	6	◦	◦	NOUN
ejpam-4594	207	7	(	(	PUNCT
ejpam-4594	207	8	x	x	PART
ejpam-4594	207	9	◦	◦	VERB
ejpam-4594	207	10	y	y	PROPN
ejpam-4594	207	11	)	)	PUNCT
ejpam-4594	207	12	]	]	PUNCT
ejpam-4594	207	13	and	and	CCONJ
ejpam-4594	207	14	1	1	X
ejpam-4594	207	15	=	=	SYM
ejpam-4594	207	16	(	(	PUNCT
ejpam-4594	207	17	x	x	SYM
ejpam-4594	207	18	◦	◦	VERB
ejpam-4594	207	19	y	y	NOUN
ejpam-4594	207	20	)	)	PUNCT
ejpam-4594	207	21	◦	◦	NOUN
ejpam-4594	207	22	(	(	PUNCT
ejpam-4594	207	23	x	x	PART
ejpam-4594	207	24	◦	◦	VERB
ejpam-4594	207	25	y	y	NOUN
ejpam-4594	207	26	)	)	PUNCT
ejpam-4594	207	27	=	=	PRON
ejpam-4594	208	1	(	(	PUNCT
ejpam-4594	208	2	x	x	SYM
ejpam-4594	208	3	◦	◦	VERB
ejpam-4594	208	4	y	y	NOUN
ejpam-4594	208	5	)	)	PUNCT
ejpam-4594	208	6	◦	◦	NOUN
ejpam-4594	209	1	[	[	X
ejpam-4594	209	2	(	(	PUNCT
ejpam-4594	209	3	a	a	DET
ejpam-4594	209	4	◦	◦	NOUN
ejpam-4594	209	5	x	x	NOUN
ejpam-4594	209	6	)	)	PUNCT
ejpam-4594	209	7	◦	◦	NOUN
ejpam-4594	209	8	(	(	PUNCT
ejpam-4594	209	9	a	a	DET
ejpam-4594	209	10	◦	◦	NOUN
ejpam-4594	209	11	y	y	NOUN
ejpam-4594	209	12	)	)	PUNCT
ejpam-4594	209	13	]	]	PUNCT
ejpam-4594	210	1	=	=	PUNCT
ejpam-4594	210	2	(	(	PUNCT
ejpam-4594	210	3	a	a	DET
ejpam-4594	210	4	◦	◦	NOUN
ejpam-4594	210	5	x	x	NOUN
ejpam-4594	210	6	)	)	PUNCT
ejpam-4594	210	7	◦	◦	NOUN
ejpam-4594	210	8	[	[	X
ejpam-4594	210	9	(	(	PUNCT
ejpam-4594	210	10	x	x	SYM
ejpam-4594	210	11	◦	◦	VERB
ejpam-4594	210	12	y	y	NOUN
ejpam-4594	210	13	)	)	PUNCT
ejpam-4594	210	14	◦	◦	NOUN
ejpam-4594	210	15	(	(	PUNCT
ejpam-4594	210	16	a	a	DET
ejpam-4594	210	17	◦	◦	NOUN
ejpam-4594	210	18	y	y	NOUN
ejpam-4594	210	19	)	)	PUNCT
ejpam-4594	210	20	]	]	PUNCT
ejpam-4594	210	21	.	.	PUNCT
ejpam-4594	211	1	by	by	ADP
ejpam-4594	211	2	lemma	lemma	PROPN
ejpam-4594	211	3	1	1	NUM
ejpam-4594	211	4	,	,	PUNCT
ejpam-4594	211	5	it	it	PRON
ejpam-4594	211	6	follows	follow	VERB
ejpam-4594	211	7	that	that	SCONJ
ejpam-4594	211	8	(	(	PUNCT
ejpam-4594	211	9	a	a	DET
ejpam-4594	211	10	◦	◦	NOUN
ejpam-4594	211	11	y	y	NOUN
ejpam-4594	211	12	)	)	PUNCT
ejpam-4594	211	13	◦	◦	NOUN
ejpam-4594	211	14	(	(	PUNCT
ejpam-4594	211	15	x	x	PART
ejpam-4594	211	16	◦	◦	VERB
ejpam-4594	211	17	y	y	NOUN
ejpam-4594	211	18	)	)	PUNCT
ejpam-4594	212	1	=	=	PUNCT
ejpam-4594	213	1	x	x	PUNCT
ejpam-4594	213	2	◦	◦	VERB
ejpam-4594	213	3	a	a	DET
ejpam-4594	213	4	∈	∈	PROPN
ejpam-4594	213	5	v	v	NOUN
ejpam-4594	213	6	and	and	CCONJ
ejpam-4594	213	7	(	(	PUNCT
ejpam-4594	213	8	x	x	PART
ejpam-4594	213	9	◦	◦	VERB
ejpam-4594	213	10	y	y	NOUN
ejpam-4594	213	11	)	)	PUNCT
ejpam-4594	213	12	◦	◦	NOUN
ejpam-4594	213	13	(	(	PUNCT
ejpam-4594	213	14	a	a	DET
ejpam-4594	213	15	◦	◦	NOUN
ejpam-4594	213	16	y	y	NOUN
ejpam-4594	213	17	)	)	PUNCT
ejpam-4594	213	18	=	=	SYM
ejpam-4594	213	19	a	a	DET
ejpam-4594	213	20	◦	◦	NOUN
ejpam-4594	213	21	x	x	SYM
ejpam-4594	213	22	∈	∈	NOUN
ejpam-4594	213	23	v	v	NOUN
ejpam-4594	213	24	.	.	PUNCT
ejpam-4594	214	1	this	this	PRON
ejpam-4594	214	2	implies	imply	VERB
ejpam-4594	214	3	that	that	SCONJ
ejpam-4594	214	4	a	a	DET
ejpam-4594	214	5	◦	◦	NOUN
ejpam-4594	214	6	y	y	PROPN
ejpam-4594	214	7	∈	∈	PROPN
ejpam-4594	214	8	v	v	ADP
ejpam-4594	214	9	′[x	′[x	NOUN
ejpam-4594	214	10	◦	◦	NOUN
ejpam-4594	214	11	y	y	X
ejpam-4594	214	12	]	]	PUNCT
ejpam-4594	214	13	.	.	PUNCT
ejpam-4594	215	1	hence	hence	ADV
ejpam-4594	215	2	,	,	PUNCT
ejpam-4594	215	3	v	v	ADP
ejpam-4594	215	4	′[x	′[x	NOUN
ejpam-4594	215	5	]	]	PUNCT
ejpam-4594	215	6	◦	◦	NOUN
ejpam-4594	215	7	y	y	PROPN
ejpam-4594	215	8	⊆	⊆	NUM
ejpam-4594	215	9	v	v	ADP
ejpam-4594	215	10	′[x	′[x	NOUN
ejpam-4594	215	11	◦	◦	NOUN
ejpam-4594	215	12	y	y	X
ejpam-4594	215	13	]	]	PUNCT
ejpam-4594	215	14	.	.	PUNCT
ejpam-4594	216	1	suppose	suppose	VERB
ejpam-4594	216	2	b	b	X
ejpam-4594	216	3	∈	∈	PROPN
ejpam-4594	216	4	v	v	ADP
ejpam-4594	216	5	′[y	′[y	PROPN
ejpam-4594	216	6	]	]	PUNCT
ejpam-4594	216	7	.	.	PUNCT
ejpam-4594	217	1	then	then	ADV
ejpam-4594	217	2	b	b	X
ejpam-4594	217	3	◦	◦	NOUN
ejpam-4594	217	4	y	y	PROPN
ejpam-4594	217	5	,	,	PUNCT
ejpam-4594	217	6	y	y	PROPN
ejpam-4594	217	7	◦	◦	NOUN
ejpam-4594	217	8	b	b	PROPN
ejpam-4594	217	9	∈	∈	ADJ
ejpam-4594	217	10	v	v	NOUN
ejpam-4594	217	11	.	.	PUNCT
ejpam-4594	218	1	by	by	ADP
ejpam-4594	218	2	theorem	theorem	NOUN
ejpam-4594	218	3	1(iii	1(iii	NUM
ejpam-4594	218	4	)	)	PUNCT
ejpam-4594	218	5	,	,	PUNCT
ejpam-4594	218	6	(	(	PUNCT
ejpam-4594	218	7	x	x	X
ejpam-4594	218	8	◦	◦	NOUN
ejpam-4594	218	9	b	b	NOUN
ejpam-4594	218	10	)	)	PUNCT
ejpam-4594	218	11	◦	◦	NOUN
ejpam-4594	218	12	(	(	PUNCT
ejpam-4594	218	13	x	x	PART
ejpam-4594	218	14	◦	◦	VERB
ejpam-4594	218	15	y	y	NOUN
ejpam-4594	218	16	)	)	PUNCT
ejpam-4594	219	1	=	=	SYM
ejpam-4594	219	2	b	b	X
ejpam-4594	219	3	◦	◦	NOUN
ejpam-4594	219	4	y	y	PROPN
ejpam-4594	219	5	∈	∈	PROPN
ejpam-4594	219	6	v	v	NOUN
ejpam-4594	219	7	and	and	CCONJ
ejpam-4594	219	8	(	(	PUNCT
ejpam-4594	219	9	x	x	PART
ejpam-4594	219	10	◦	◦	VERB
ejpam-4594	219	11	y	y	NOUN
ejpam-4594	219	12	)	)	PUNCT
ejpam-4594	219	13	◦	◦	NOUN
ejpam-4594	219	14	(	(	PUNCT
ejpam-4594	219	15	x	x	PART
ejpam-4594	219	16	◦	◦	NOUN
ejpam-4594	219	17	b	b	NUM
ejpam-4594	219	18	)	)	PUNCT
ejpam-4594	220	1	=	=	SYM
ejpam-4594	220	2	y	y	PROPN
ejpam-4594	220	3	◦	◦	NOUN
ejpam-4594	220	4	b	b	PROPN
ejpam-4594	220	5	∈	∈	ADJ
ejpam-4594	220	6	v	v	NOUN
ejpam-4594	220	7	.	.	PUNCT
ejpam-4594	221	1	this	this	PRON
ejpam-4594	221	2	implies	imply	VERB
ejpam-4594	221	3	that	that	SCONJ
ejpam-4594	221	4	x	x	PUNCT
ejpam-4594	222	1	◦	◦	NOUN
ejpam-4594	222	2	b	b	SYM
ejpam-4594	222	3	∈	∈	NOUN
ejpam-4594	222	4	v	v	ADP
ejpam-4594	222	5	′[x	′[x	NOUN
ejpam-4594	222	6	◦	◦	NOUN
ejpam-4594	222	7	y	y	X
ejpam-4594	222	8	]	]	PUNCT
ejpam-4594	222	9	.	.	PUNCT
ejpam-4594	223	1	hence	hence	ADV
ejpam-4594	223	2	,	,	PUNCT
ejpam-4594	223	3	x	x	PUNCT
ejpam-4594	223	4	◦	◦	NOUN
ejpam-4594	223	5	v	v	ADP
ejpam-4594	223	6	′[y	′[y	PROPN
ejpam-4594	223	7	]	]	PUNCT
ejpam-4594	223	8	⊆	⊆	NUM
ejpam-4594	223	9	v	v	ADP
ejpam-4594	223	10	′[x	′[x	NOUN
ejpam-4594	223	11	◦	◦	NOUN
ejpam-4594	223	12	y	y	X
ejpam-4594	223	13	]	]	PUNCT
ejpam-4594	223	14	.	.	PUNCT
ejpam-4594	224	1	assume	assume	VERB
ejpam-4594	224	2	on	on	ADP
ejpam-4594	224	3	the	the	DET
ejpam-4594	224	4	contrary	contrary	NOUN
ejpam-4594	224	5	that	that	PRON
ejpam-4594	224	6	v	v	ADP
ejpam-4594	224	7	′[x	′[x	NOUN
ejpam-4594	224	8	]	]	PUNCT
ejpam-4594	224	9	◦	◦	NOUN
ejpam-4594	224	10	v	v	ADP
ejpam-4594	224	11	′[y	′[y	PROPN
ejpam-4594	224	12	]	]	PUNCT
ejpam-4594	224	13	⊈	⊈	PROPN
ejpam-4594	225	1	v	v	PART
ejpam-4594	225	2	′[x	′[x	NOUN
ejpam-4594	225	3	◦	◦	NOUN
ejpam-4594	225	4	y	y	NOUN
ejpam-4594	225	5	]	]	PUNCT
ejpam-4594	225	6	.	.	PUNCT
ejpam-4594	226	1	by	by	ADP
ejpam-4594	226	2	proposition	proposition	NOUN
ejpam-4594	226	3	2	2	NUM
ejpam-4594	226	4	,	,	PUNCT
ejpam-4594	226	5	v	v	ADP
ejpam-4594	226	6	′[x	′[x	NOUN
ejpam-4594	226	7	]	]	PUNCT
ejpam-4594	226	8	◦	◦	NOUN
ejpam-4594	226	9	y	y	PROPN
ejpam-4594	226	10	∈	∈	PROPN
ejpam-4594	226	11	v	v	ADP
ejpam-4594	226	12	′[x	′[x	NOUN
ejpam-4594	226	13	]	]	PUNCT
ejpam-4594	226	14	◦	◦	NOUN
ejpam-4594	226	15	v	v	ADP
ejpam-4594	226	16	′[y	′[y	PROPN
ejpam-4594	226	17	]	]	PUNCT
ejpam-4594	226	18	⊈	⊈	PROPN
ejpam-4594	226	19	v	v	PART
ejpam-4594	226	20	′[x	′[x	NOUN
ejpam-4594	226	21	◦	◦	NOUN
ejpam-4594	226	22	y	y	NOUN
ejpam-4594	226	23	]	]	PUNCT
ejpam-4594	226	24	and	and	CCONJ
ejpam-4594	226	25	x	x	PUNCT
ejpam-4594	226	26	◦	◦	NOUN
ejpam-4594	226	27	v	v	NUM
ejpam-4594	226	28	′[y	′[y	PROPN
ejpam-4594	226	29	]	]	X
ejpam-4594	226	30	∈	∈	PROPN
ejpam-4594	226	31	v	v	ADP
ejpam-4594	226	32	′[x	′[x	NOUN
ejpam-4594	226	33	]	]	PUNCT
ejpam-4594	226	34	◦	◦	NOUN
ejpam-4594	226	35	v	v	ADP
ejpam-4594	226	36	′[y	′[y	PROPN
ejpam-4594	226	37	]	]	PUNCT
ejpam-4594	226	38	⊈	⊈	PROPN
ejpam-4594	226	39	v	v	PART
ejpam-4594	226	40	′[x	′[x	NOUN
ejpam-4594	226	41	◦	◦	NOUN
ejpam-4594	226	42	y	y	NOUN
ejpam-4594	226	43	]	]	X
ejpam-4594	226	44	.	.	PUNCT
ejpam-4594	227	1	these	these	PRON
ejpam-4594	227	2	are	be	AUX
ejpam-4594	227	3	contradictions	contradiction	NOUN
ejpam-4594	227	4	.	.	PUNCT
ejpam-4594	228	1	therefore	therefore	ADV
ejpam-4594	228	2	,	,	PUNCT
ejpam-4594	228	3	xd	xd	INTJ
ejpam-4594	228	4	is	be	AUX
ejpam-4594	228	5	a	a	DET
ejpam-4594	228	6	tdb	tdb	NOUN
ejpam-4594	228	7	-	-	NOUN
ejpam-4594	228	8	algebra	algebra	NOUN
ejpam-4594	228	9	.	.	PUNCT
ejpam-4594	229	1	lemma	lemma	PROPN
ejpam-4594	229	2	3	3	X
ejpam-4594	229	3	.	.	PUNCT
ejpam-4594	229	4	suppose	suppose	VERB
ejpam-4594	229	5	ω	ω	NOUN
ejpam-4594	229	6	is	be	AUX
ejpam-4594	229	7	an	an	DET
ejpam-4594	229	8	arbitrary	arbitrary	ADJ
ejpam-4594	229	9	family	family	NOUN
ejpam-4594	229	10	of	of	ADP
ejpam-4594	229	11	dual	dual	ADJ
ejpam-4594	229	12	b	b	NOUN
ejpam-4594	229	13	-	-	PUNCT
ejpam-4594	229	14	filters	filter	NOUN
ejpam-4594	229	15	in	in	ADP
ejpam-4594	229	16	a	a	DET
ejpam-4594	229	17	dual	dual	ADJ
ejpam-4594	229	18	b	b	NOUN
ejpam-4594	229	19	-	-	PUNCT
ejpam-4594	229	20	algebra	algebra	NOUN
ejpam-4594	229	21	xd	xd	INTJ
ejpam-4594	229	22	.	.	PUNCT
ejpam-4594	230	1	then	then	ADV
ejpam-4594	230	2	for	for	ADP
ejpam-4594	230	3	all	all	DET
ejpam-4594	230	4	a	a	DET
ejpam-4594	230	5	∈	∈	NOUN
ejpam-4594	230	6	v	v	ADP
ejpam-4594	230	7	∈	∈	PROPN
ejpam-4594	230	8	ω	ω	PROPN
ejpam-4594	230	9	,	,	PUNCT
ejpam-4594	230	10	v	v	X
ejpam-4594	230	11	[	[	X
ejpam-4594	230	12	a	a	X
ejpam-4594	230	13	]	]	X
ejpam-4594	230	14	=	=	SYM
ejpam-4594	230	15	v	v	NOUN
ejpam-4594	230	16	.	.	PUNCT
ejpam-4594	231	1	k.	k.	PROPN
ejpam-4594	231	2	belleza	belleza	PROPN
ejpam-4594	231	3	,	,	PUNCT
ejpam-4594	231	4	j.	j.	PROPN
ejpam-4594	231	5	albaracin	albaracin	PROPN
ejpam-4594	231	6	/	/	PUNCT
ejpam-4594	231	7	eur	eur	PROPN
ejpam-4594	231	8	.	.	PUNCT
ejpam-4594	232	1	j.	j.	PROPN
ejpam-4594	232	2	pure	pure	PROPN
ejpam-4594	232	3	appl	appl	PROPN
ejpam-4594	232	4	.	.	PROPN
ejpam-4594	232	5	math	math	PROPN
ejpam-4594	232	6	,	,	PUNCT
ejpam-4594	232	7	15	15	NUM
ejpam-4594	232	8	(	(	PUNCT
ejpam-4594	232	9	4	4	NUM
ejpam-4594	232	10	)	)	PUNCT
ejpam-4594	232	11	(	(	PUNCT
ejpam-4594	232	12	2022	2022	NUM
ejpam-4594	232	13	)	)	PUNCT
ejpam-4594	232	14	,	,	PUNCT
ejpam-4594	232	15	1948	1948	NUM
ejpam-4594	232	16	-	-	SYM
ejpam-4594	232	17	1956	1956	NUM
ejpam-4594	232	18	1953	1953	NUM
ejpam-4594	232	19	proof	proof	NOUN
ejpam-4594	232	20	.	.	PUNCT
ejpam-4594	233	1	suppose	suppose	VERB
ejpam-4594	233	2	ω	ω	NOUN
ejpam-4594	233	3	is	be	AUX
ejpam-4594	233	4	an	an	DET
ejpam-4594	233	5	arbitrary	arbitrary	ADJ
ejpam-4594	233	6	family	family	NOUN
ejpam-4594	233	7	of	of	ADP
ejpam-4594	233	8	dual	dual	ADJ
ejpam-4594	233	9	b	b	NOUN
ejpam-4594	233	10	-	-	PUNCT
ejpam-4594	233	11	filters	filter	NOUN
ejpam-4594	233	12	in	in	ADP
ejpam-4594	233	13	a	a	DET
ejpam-4594	233	14	dual	dual	ADJ
ejpam-4594	233	15	b	b	NOUN
ejpam-4594	233	16	-	-	PUNCT
ejpam-4594	233	17	algebra	algebra	NOUN
ejpam-4594	233	18	xd	xd	INTJ
ejpam-4594	233	19	and	and	CCONJ
ejpam-4594	233	20	let	let	VERB
ejpam-4594	233	21	a	a	DET
ejpam-4594	233	22	∈	∈	NOUN
ejpam-4594	233	23	v	v	ADP
ejpam-4594	233	24	∈	∈	PROPN
ejpam-4594	233	25	ω	ω	PROPN
ejpam-4594	233	26	.	.	PUNCT
ejpam-4594	233	27	suppose	suppose	VERB
ejpam-4594	234	1	x	x	SYM
ejpam-4594	234	2	∈	∈	NOUN
ejpam-4594	234	3	v	v	ADP
ejpam-4594	234	4	[	[	X
ejpam-4594	234	5	a	a	X
ejpam-4594	234	6	]	]	X
ejpam-4594	234	7	.	.	PUNCT
ejpam-4594	235	1	then	then	ADV
ejpam-4594	235	2	a	a	DET
ejpam-4594	235	3	◦	◦	NOUN
ejpam-4594	235	4	x	x	SYM
ejpam-4594	235	5	∈	∈	NOUN
ejpam-4594	235	6	v	v	NOUN
ejpam-4594	235	7	.	.	PUNCT
ejpam-4594	236	1	since	since	SCONJ
ejpam-4594	236	2	v	v	NOUN
ejpam-4594	236	3	is	be	AUX
ejpam-4594	236	4	a	a	DET
ejpam-4594	236	5	dual	dual	ADJ
ejpam-4594	236	6	b	b	NOUN
ejpam-4594	236	7	-	-	NOUN
ejpam-4594	236	8	filter	filter	NOUN
ejpam-4594	236	9	and	and	CCONJ
ejpam-4594	236	10	a	a	DET
ejpam-4594	236	11	∈	∈	NOUN
ejpam-4594	236	12	v	v	NOUN
ejpam-4594	236	13	,	,	PUNCT
ejpam-4594	236	14	it	it	PRON
ejpam-4594	236	15	follows	follow	VERB
ejpam-4594	236	16	that	that	SCONJ
ejpam-4594	236	17	x	x	PUNCT
ejpam-4594	236	18	∈	∈	NOUN
ejpam-4594	236	19	v	v	ADP
ejpam-4594	236	20	implying	imply	VERB
ejpam-4594	236	21	that	that	SCONJ
ejpam-4594	236	22	v	v	ADP
ejpam-4594	236	23	[	[	X
ejpam-4594	236	24	a	a	X
ejpam-4594	236	25	]	]	PUNCT
ejpam-4594	236	26	⊆	⊆	NUM
ejpam-4594	236	27	v	v	NOUN
ejpam-4594	236	28	.	.	PUNCT
ejpam-4594	237	1	conversely	conversely	ADV
ejpam-4594	237	2	,	,	PUNCT
ejpam-4594	237	3	suppose	suppose	VERB
ejpam-4594	237	4	x	x	X
ejpam-4594	237	5	∈	∈	PROPN
ejpam-4594	237	6	v	v	NOUN
ejpam-4594	237	7	.	.	PUNCT
ejpam-4594	238	1	since	since	SCONJ
ejpam-4594	238	2	v	v	NOUN
ejpam-4594	238	3	is	be	AUX
ejpam-4594	238	4	a	a	DET
ejpam-4594	238	5	dual	dual	ADJ
ejpam-4594	238	6	b	b	NOUN
ejpam-4594	238	7	-	-	NOUN
ejpam-4594	238	8	filter	filter	NOUN
ejpam-4594	238	9	,	,	PUNCT
ejpam-4594	238	10	v	v	NOUN
ejpam-4594	238	11	is	be	AUX
ejpam-4594	238	12	a	a	DET
ejpam-4594	238	13	dual	dual	ADJ
ejpam-4594	238	14	b	b	NOUN
ejpam-4594	238	15	-	-	PUNCT
ejpam-4594	238	16	subalgebra	subalgebra	NOUN
ejpam-4594	238	17	of	of	ADP
ejpam-4594	238	18	xd	xd	ADP
ejpam-4594	238	19	.	.	PUNCT
ejpam-4594	239	1	then	then	ADV
ejpam-4594	239	2	a	a	DET
ejpam-4594	239	3	◦	◦	NOUN
ejpam-4594	239	4	x	x	SYM
ejpam-4594	239	5	∈	∈	PROPN
ejpam-4594	239	6	v	v	NOUN
ejpam-4594	239	7	.	.	PUNCT
ejpam-4594	240	1	hence	hence	ADV
ejpam-4594	240	2	,	,	PUNCT
ejpam-4594	240	3	x	x	PUNCT
ejpam-4594	240	4	∈	∈	NOUN
ejpam-4594	240	5	v	v	ADP
ejpam-4594	241	1	[	[	X
ejpam-4594	241	2	a	a	X
ejpam-4594	241	3	]	]	X
ejpam-4594	241	4	.	.	PUNCT
ejpam-4594	242	1	therefore	therefore	ADV
ejpam-4594	242	2	,	,	PUNCT
ejpam-4594	242	3	v	v	X
ejpam-4594	242	4	[	[	X
ejpam-4594	242	5	a	a	X
ejpam-4594	242	6	]	]	X
ejpam-4594	242	7	=	=	SYM
ejpam-4594	242	8	v	v	NOUN
ejpam-4594	242	9	.	.	PUNCT
ejpam-4594	243	1	the	the	DET
ejpam-4594	243	2	next	next	ADJ
ejpam-4594	243	3	corollary	corollary	NOUN
ejpam-4594	243	4	follows	follow	VERB
ejpam-4594	243	5	from	from	ADP
ejpam-4594	243	6	lemma	lemma	PROPN
ejpam-4594	243	7	3	3	NUM
ejpam-4594	243	8	and	and	CCONJ
ejpam-4594	243	9	theorem	theorem	VERB
ejpam-4594	243	10	7	7	NUM
ejpam-4594	243	11	.	.	PUNCT
ejpam-4594	243	12	corollary	corollary	ADJ
ejpam-4594	243	13	1	1	NUM
ejpam-4594	243	14	.	.	PUNCT
ejpam-4594	244	1	let	let	VERB
ejpam-4594	244	2	ω	ω	NUM
ejpam-4594	244	3	be	be	AUX
ejpam-4594	244	4	a	a	DET
ejpam-4594	244	5	family	family	NOUN
ejpam-4594	244	6	of	of	ADP
ejpam-4594	244	7	dual	dual	ADJ
ejpam-4594	244	8	b	b	NOUN
ejpam-4594	244	9	-	-	PUNCT
ejpam-4594	244	10	filters	filter	NOUN
ejpam-4594	244	11	in	in	ADP
ejpam-4594	244	12	a	a	DET
ejpam-4594	244	13	commutative	commutative	ADJ
ejpam-4594	244	14	dual	dual	ADJ
ejpam-4594	244	15	b	b	NOUN
ejpam-4594	244	16	-	-	PUNCT
ejpam-4594	244	17	algebra	algebra	NOUN
ejpam-4594	244	18	xd	xd	INTJ
ejpam-4594	244	19	closed	close	VERB
ejpam-4594	244	20	under	under	ADP
ejpam-4594	244	21	finite	finite	ADJ
ejpam-4594	244	22	intersections	intersection	NOUN
ejpam-4594	244	23	such	such	ADJ
ejpam-4594	244	24	that	that	SCONJ
ejpam-4594	244	25	1	1	NUM
ejpam-4594	244	26	∈	∈	NOUN
ejpam-4594	244	27	v	v	NOUN
ejpam-4594	244	28	for	for	ADP
ejpam-4594	244	29	all	all	PRON
ejpam-4594	244	30	v	v	ADP
ejpam-4594	244	31	∈	∈	PROPN
ejpam-4594	244	32	ω	ω	NOUN
ejpam-4594	244	33	.	.	PUNCT
ejpam-4594	245	1	then	then	ADV
ejpam-4594	245	2	xd	xd	INTJ
ejpam-4594	245	3	is	be	VERB
ejpam-4594	245	4	a	a	DET
ejpam-4594	245	5	tdb	tdb	NOUN
ejpam-4594	245	6	-	-	NOUN
ejpam-4594	245	7	algebra	algebra	NOUN
ejpam-4594	245	8	.	.	PUNCT
ejpam-4594	246	1	4	4	X
ejpam-4594	246	2	.	.	X
ejpam-4594	246	3	filterbase	filterbase	NOUN
ejpam-4594	246	4	in	in	ADP
ejpam-4594	246	5	a	a	DET
ejpam-4594	246	6	dual	dual	ADJ
ejpam-4594	246	7	b	b	NOUN
ejpam-4594	246	8	-	-	PUNCT
ejpam-4594	246	9	algebra	algebra	NOUN
ejpam-4594	246	10	definition	definition	NOUN
ejpam-4594	246	11	10	10	NUM
ejpam-4594	246	12	.	.	PUNCT
ejpam-4594	247	1	let	let	VERB
ejpam-4594	247	2	xd	xd	INTJ
ejpam-4594	247	3	be	be	AUX
ejpam-4594	247	4	a	a	DET
ejpam-4594	247	5	dual	dual	ADJ
ejpam-4594	247	6	b	b	NOUN
ejpam-4594	247	7	-	-	PUNCT
ejpam-4594	247	8	topological	topological	ADJ
ejpam-4594	247	9	space	space	NOUN
ejpam-4594	247	10	.	.	PUNCT
ejpam-4594	248	1	a	a	DET
ejpam-4594	248	2	filterbase	filterbase	NOUN
ejpam-4594	248	3	u	u	NOUN
ejpam-4594	248	4	in	in	ADV
ejpam-4594	248	5	xd	xd	INTJ
ejpam-4594	248	6	is	be	AUX
ejpam-4594	248	7	a	a	DET
ejpam-4594	248	8	family	family	NOUN
ejpam-4594	248	9	u	u	NOUN
ejpam-4594	248	10	=	=	PUNCT
ejpam-4594	248	11	{	{	PUNCT
ejpam-4594	248	12	aα	aα	NOUN
ejpam-4594	248	13	|	|	ADV
ejpam-4594	248	14	α	α	NOUN
ejpam-4594	248	15	∈	∈	PROPN
ejpam-4594	249	1	a	a	PRON
ejpam-4594	249	2	}	}	PUNCT
ejpam-4594	249	3	of	of	ADP
ejpam-4594	249	4	subsets	subset	NOUN
ejpam-4594	249	5	of	of	ADP
ejpam-4594	249	6	xd	xd	INTJ
ejpam-4594	249	7	having	have	VERB
ejpam-4594	249	8	two	two	NUM
ejpam-4594	249	9	properties	property	NOUN
ejpam-4594	249	10	:	:	PUNCT
ejpam-4594	249	11	(	(	PUNCT
ejpam-4594	249	12	i	i	NOUN
ejpam-4594	249	13	)	)	PUNCT
ejpam-4594	249	14	aα	aα	NOUN
ejpam-4594	249	15	̸=	̸=	PROPN
ejpam-4594	249	16	∅	∅	NOUN
ejpam-4594	249	17	for	for	ADP
ejpam-4594	249	18	all	all	DET
ejpam-4594	249	19	α	α	NOUN
ejpam-4594	249	20	∈	∈	NOUN
ejpam-4594	249	21	a	a	PRON
ejpam-4594	249	22	;	;	PUNCT
ejpam-4594	249	23	(	(	PUNCT
ejpam-4594	249	24	ii	ii	NOUN
ejpam-4594	249	25	)	)	PUNCT
ejpam-4594	249	26	for	for	ADP
ejpam-4594	249	27	all	all	DET
ejpam-4594	249	28	α	α	NOUN
ejpam-4594	249	29	,	,	PUNCT
ejpam-4594	249	30	β	β	X
ejpam-4594	249	31	∈	∈	PROPN
ejpam-4594	249	32	a	a	PRON
ejpam-4594	249	33	,	,	PUNCT
ejpam-4594	249	34	there	there	PRON
ejpam-4594	249	35	exists	exist	VERB
ejpam-4594	249	36	γ	γ	PROPN
ejpam-4594	249	37	∈	∈	PROPN
ejpam-4594	249	38	a	a	DET
ejpam-4594	249	39	such	such	ADJ
ejpam-4594	249	40	that	that	SCONJ
ejpam-4594	249	41	aγ	aγ	PRON
ejpam-4594	249	42	⊆	⊆	NUM
ejpam-4594	249	43	aα	aα	NOUN
ejpam-4594	249	44	∩aβ	∩aβ	NOUN
ejpam-4594	249	45	.	.	PUNCT
ejpam-4594	250	1	remark	remark	NOUN
ejpam-4594	250	2	3	3	NUM
ejpam-4594	250	3	.	.	PUNCT
ejpam-4594	251	1	let	let	VERB
ejpam-4594	251	2	xd	xd	INTJ
ejpam-4594	251	3	be	be	AUX
ejpam-4594	251	4	a	a	DET
ejpam-4594	251	5	dual	dual	ADJ
ejpam-4594	251	6	b	b	NOUN
ejpam-4594	251	7	-	-	PUNCT
ejpam-4594	251	8	topological	topological	ADJ
ejpam-4594	251	9	space	space	NOUN
ejpam-4594	251	10	and	and	CCONJ
ejpam-4594	251	11	x1	x1	PROPN
ejpam-4594	251	12	∈	∈	PROPN
ejpam-4594	252	1	xd	xd	PROPN
ejpam-4594	252	2	.	.	PUNCT
ejpam-4594	253	1	the	the	DET
ejpam-4594	253	2	family	family	NOUN
ejpam-4594	253	3	{	{	PUNCT
ejpam-4594	253	4	u(x1	u(x1	ADJ
ejpam-4594	253	5	)	)	PUNCT
ejpam-4594	253	6	}	}	PUNCT
ejpam-4594	253	7	is	be	AUX
ejpam-4594	253	8	a	a	DET
ejpam-4594	253	9	filterbase	filterbase	NOUN
ejpam-4594	253	10	called	call	VERB
ejpam-4594	253	11	the	the	DET
ejpam-4594	253	12	neighborhood	neighborhood	NOUN
ejpam-4594	253	13	filterbase	filterbase	NOUN
ejpam-4594	253	14	of	of	ADP
ejpam-4594	253	15	x1	x1	PROPN
ejpam-4594	253	16	.	.	PUNCT
ejpam-4594	253	17	example	example	NOUN
ejpam-4594	254	1	2	2	NUM
ejpam-4594	254	2	.	.	PUNCT
ejpam-4594	254	3	suppose	suppose	VERB
ejpam-4594	254	4	xd	xd	INTJ
ejpam-4594	254	5	is	be	AUX
ejpam-4594	254	6	a	a	DET
ejpam-4594	254	7	dual	dual	ADJ
ejpam-4594	254	8	b	b	NOUN
ejpam-4594	254	9	-	-	PUNCT
ejpam-4594	254	10	topological	topological	ADJ
ejpam-4594	254	11	space	space	NOUN
ejpam-4594	254	12	.	.	PUNCT
ejpam-4594	255	1	any	any	DET
ejpam-4594	255	2	family	family	NOUN
ejpam-4594	255	3	w	w	NOUN
ejpam-4594	255	4	of	of	ADP
ejpam-4594	255	5	subsets	subset	NOUN
ejpam-4594	255	6	of	of	ADP
ejpam-4594	255	7	xd	xd	INTJ
ejpam-4594	255	8	containing	contain	VERB
ejpam-4594	255	9	∅	∅	NOUN
ejpam-4594	255	10	is	be	AUX
ejpam-4594	255	11	not	not	PART
ejpam-4594	255	12	a	a	DET
ejpam-4594	255	13	filterbase	filterbase	NOUN
ejpam-4594	255	14	.	.	PUNCT
ejpam-4594	256	1	in	in	ADP
ejpam-4594	256	2	particular	particular	ADJ
ejpam-4594	256	3	,	,	PUNCT
ejpam-4594	256	4	the	the	DET
ejpam-4594	256	5	dual	dual	ADJ
ejpam-4594	256	6	b	b	NOUN
ejpam-4594	256	7	-	-	PUNCT
ejpam-4594	256	8	topology	topology	NOUN
ejpam-4594	256	9	τ	τ	NOUN
ejpam-4594	256	10	on	on	ADP
ejpam-4594	256	11	xd	xd	INTJ
ejpam-4594	256	12	is	be	AUX
ejpam-4594	256	13	not	not	PART
ejpam-4594	256	14	a	a	DET
ejpam-4594	256	15	filterbase	filterbase	NOUN
ejpam-4594	256	16	in	in	ADP
ejpam-4594	256	17	xd	xd	PROPN
ejpam-4594	256	18	.	.	PROPN
ejpam-4594	256	19	remark	remark	PROPN
ejpam-4594	256	20	4	4	NUM
ejpam-4594	256	21	.	.	PUNCT
ejpam-4594	257	1	the	the	DET
ejpam-4594	257	2	family	family	NOUN
ejpam-4594	257	3	of	of	ADP
ejpam-4594	257	4	dual	dual	ADJ
ejpam-4594	257	5	b	b	NOUN
ejpam-4594	257	6	-	-	PUNCT
ejpam-4594	257	7	filters	filter	NOUN
ejpam-4594	257	8	is	be	AUX
ejpam-4594	257	9	not	not	PART
ejpam-4594	257	10	a	a	DET
ejpam-4594	257	11	subclass	subclass	NOUN
ejpam-4594	257	12	of	of	ADP
ejpam-4594	257	13	a	a	DET
ejpam-4594	257	14	filterbase	filterbase	NOUN
ejpam-4594	257	15	in	in	ADP
ejpam-4594	257	16	a	a	DET
ejpam-4594	257	17	dual	dual	ADJ
ejpam-4594	257	18	b	b	NOUN
ejpam-4594	257	19	-	-	PUNCT
ejpam-4594	257	20	algebra	algebra	NOUN
ejpam-4594	257	21	xd	xd	NOUN
ejpam-4594	257	22	.	.	PROPN
ejpam-4594	257	23	example	example	NOUN
ejpam-4594	257	24	3	3	X
ejpam-4594	257	25	.	.	X
ejpam-4594	257	26	consider	consider	VERB
ejpam-4594	257	27	the	the	DET
ejpam-4594	257	28	dual	dual	ADJ
ejpam-4594	257	29	b	b	NOUN
ejpam-4594	257	30	-	-	PUNCT
ejpam-4594	257	31	algebra	algebra	NOUN
ejpam-4594	257	32	x	x	X
ejpam-4594	257	33	=	=	SYM
ejpam-4594	257	34	{	{	PUNCT
ejpam-4594	257	35	1	1	NUM
ejpam-4594	257	36	,	,	PUNCT
ejpam-4594	257	37	a	a	DET
ejpam-4594	257	38	,	,	PUNCT
ejpam-4594	257	39	b	b	NOUN
ejpam-4594	257	40	,	,	PUNCT
ejpam-4594	257	41	c	c	NOUN
ejpam-4594	257	42	,	,	PUNCT
ejpam-4594	257	43	d	d	NOUN
ejpam-4594	257	44	,	,	PUNCT
ejpam-4594	257	45	e	e	NOUN
ejpam-4594	257	46	}	}	PUNCT
ejpam-4594	257	47	in	in	ADP
ejpam-4594	257	48	example	example	NOUN
ejpam-4594	257	49	1	1	X
ejpam-4594	257	50	.	.	PUNCT
ejpam-4594	258	1	let	let	VERB
ejpam-4594	258	2	f	f	PROPN
ejpam-4594	258	3	=	=	PRON
ejpam-4594	258	4	{	{	PUNCT
ejpam-4594	258	5	{	{	PUNCT
ejpam-4594	258	6	1	1	NUM
ejpam-4594	258	7	,	,	PUNCT
ejpam-4594	258	8	e	e	NOUN
ejpam-4594	258	9	}	}	PUNCT
ejpam-4594	258	10	,	,	PUNCT
ejpam-4594	258	11	{	{	PUNCT
ejpam-4594	258	12	1	1	NUM
ejpam-4594	258	13	,	,	PUNCT
ejpam-4594	258	14	a	a	DET
ejpam-4594	258	15	,	,	PUNCT
ejpam-4594	258	16	b	b	NOUN
ejpam-4594	258	17	}	}	PUNCT
ejpam-4594	258	18	,	,	PUNCT
ejpam-4594	258	19	{	{	PUNCT
ejpam-4594	258	20	1	1	NUM
ejpam-4594	258	21	,	,	PUNCT
ejpam-4594	258	22	c	c	NOUN
ejpam-4594	258	23	}	}	PUNCT
ejpam-4594	258	24	}	}	PUNCT
ejpam-4594	258	25	.	.	PUNCT
ejpam-4594	259	1	then	then	ADV
ejpam-4594	259	2	f	f	PROPN
ejpam-4594	259	3	is	be	AUX
ejpam-4594	259	4	a	a	DET
ejpam-4594	259	5	family	family	NOUN
ejpam-4594	259	6	of	of	ADP
ejpam-4594	259	7	dual	dual	ADJ
ejpam-4594	259	8	b	b	NOUN
ejpam-4594	259	9	-	-	PUNCT
ejpam-4594	259	10	filters	filter	NOUN
ejpam-4594	259	11	in	in	ADP
ejpam-4594	259	12	xd	xd	NOUN
ejpam-4594	260	1	[	[	X
ejpam-4594	260	2	1	1	NUM
ejpam-4594	260	3	]	]	PUNCT
ejpam-4594	260	4	.	.	PUNCT
ejpam-4594	261	1	note	note	VERB
ejpam-4594	261	2	that	that	SCONJ
ejpam-4594	261	3	{	{	PUNCT
ejpam-4594	261	4	1	1	NUM
ejpam-4594	261	5	,	,	PUNCT
ejpam-4594	261	6	e	e	NOUN
ejpam-4594	261	7	}	}	PUNCT
ejpam-4594	261	8	,	,	PUNCT
ejpam-4594	261	9	{	{	PUNCT
ejpam-4594	261	10	1	1	NUM
ejpam-4594	261	11	,	,	PUNCT
ejpam-4594	261	12	a	a	DET
ejpam-4594	261	13	,	,	PUNCT
ejpam-4594	261	14	b	b	NOUN
ejpam-4594	261	15	}	}	PUNCT
ejpam-4594	261	16	∈	∈	PROPN
ejpam-4594	261	17	f	f	NOUN
ejpam-4594	261	18	but	but	CCONJ
ejpam-4594	261	19	{	{	PUNCT
ejpam-4594	261	20	1	1	NUM
ejpam-4594	261	21	,	,	PUNCT
ejpam-4594	261	22	e	e	NOUN
ejpam-4594	261	23	}	}	PUNCT
ejpam-4594	261	24	∩	∩	NOUN
ejpam-4594	261	25	{	{	PUNCT
ejpam-4594	261	26	1	1	NUM
ejpam-4594	261	27	,	,	PUNCT
ejpam-4594	261	28	a	a	DET
ejpam-4594	261	29	,	,	PUNCT
ejpam-4594	261	30	b	b	NOUN
ejpam-4594	261	31	}	}	PUNCT
ejpam-4594	261	32	=	=	SYM
ejpam-4594	261	33	{	{	PUNCT
ejpam-4594	261	34	1	1	NUM
ejpam-4594	261	35	}	}	PUNCT
ejpam-4594	261	36	/∈	/∈	PUNCT
ejpam-4594	262	1	f	f	PROPN
ejpam-4594	262	2	.	.	PUNCT
ejpam-4594	263	1	this	this	PRON
ejpam-4594	263	2	implies	imply	VERB
ejpam-4594	263	3	that	that	SCONJ
ejpam-4594	263	4	f	f	PROPN
ejpam-4594	263	5	is	be	AUX
ejpam-4594	263	6	not	not	PART
ejpam-4594	263	7	a	a	DET
ejpam-4594	263	8	filterbase	filterbase	NOUN
ejpam-4594	263	9	.	.	PUNCT
ejpam-4594	264	1	remark	remark	PROPN
ejpam-4594	264	2	5	5	NUM
ejpam-4594	264	3	.	.	PUNCT
ejpam-4594	265	1	a	a	DET
ejpam-4594	265	2	filterbase	filterbase	NOUN
ejpam-4594	265	3	is	be	AUX
ejpam-4594	265	4	not	not	PART
ejpam-4594	265	5	a	a	DET
ejpam-4594	265	6	subclass	subclass	NOUN
ejpam-4594	265	7	of	of	ADP
ejpam-4594	265	8	a	a	DET
ejpam-4594	265	9	family	family	NOUN
ejpam-4594	265	10	of	of	ADP
ejpam-4594	265	11	dual	dual	ADJ
ejpam-4594	265	12	b	b	NOUN
ejpam-4594	265	13	-	-	PUNCT
ejpam-4594	265	14	filters	filter	NOUN
ejpam-4594	265	15	in	in	ADP
ejpam-4594	265	16	a	a	DET
ejpam-4594	265	17	dual	dual	ADJ
ejpam-4594	265	18	b	b	NOUN
ejpam-4594	265	19	-	-	PUNCT
ejpam-4594	265	20	algebra	algebra	NOUN
ejpam-4594	265	21	.	.	PUNCT
ejpam-4594	265	22	example	example	NOUN
ejpam-4594	266	1	4	4	NUM
ejpam-4594	266	2	.	.	PUNCT
ejpam-4594	266	3	consider	consider	VERB
ejpam-4594	266	4	the	the	DET
ejpam-4594	266	5	dual	dual	ADJ
ejpam-4594	266	6	b	b	NOUN
ejpam-4594	266	7	-	-	PUNCT
ejpam-4594	266	8	algebra	algebra	NOUN
ejpam-4594	266	9	xd	xd	ADP
ejpam-4594	266	10	=	=	SYM
ejpam-4594	266	11	{	{	PUNCT
ejpam-4594	266	12	1	1	NUM
ejpam-4594	266	13	,	,	PUNCT
ejpam-4594	266	14	a	a	DET
ejpam-4594	266	15	,	,	PUNCT
ejpam-4594	266	16	b	b	NOUN
ejpam-4594	266	17	,	,	PUNCT
ejpam-4594	266	18	c	c	NOUN
ejpam-4594	266	19	,	,	PUNCT
ejpam-4594	266	20	d	d	NOUN
ejpam-4594	266	21	,	,	PUNCT
ejpam-4594	266	22	e	e	NOUN
ejpam-4594	266	23	}	}	PUNCT
ejpam-4594	266	24	in	in	ADP
ejpam-4594	266	25	example	example	NOUN
ejpam-4594	266	26	1	1	NUM
ejpam-4594	266	27	and	and	CCONJ
ejpam-4594	266	28	ω	ω	NUM
ejpam-4594	266	29	=	=	SYM
ejpam-4594	266	30	{	{	PUNCT
ejpam-4594	266	31	{	{	PUNCT
ejpam-4594	266	32	1	1	NUM
ejpam-4594	266	33	,	,	PUNCT
ejpam-4594	266	34	a	a	DET
ejpam-4594	266	35	,	,	PUNCT
ejpam-4594	266	36	b	b	NOUN
ejpam-4594	266	37	}	}	PUNCT
ejpam-4594	266	38	,	,	PUNCT
ejpam-4594	266	39	{	{	PUNCT
ejpam-4594	266	40	a	a	X
ejpam-4594	266	41	}	}	PUNCT
ejpam-4594	266	42	}	}	PUNCT
ejpam-4594	266	43	.	.	PUNCT
ejpam-4594	267	1	then	then	ADV
ejpam-4594	267	2	ω	ω	PROPN
ejpam-4594	267	3	is	be	AUX
ejpam-4594	267	4	a	a	DET
ejpam-4594	267	5	filterbase	filterbase	NOUN
ejpam-4594	267	6	of	of	ADP
ejpam-4594	267	7	xd	xd	INTJ
ejpam-4594	267	8	but	but	CCONJ
ejpam-4594	267	9	{	{	PUNCT
ejpam-4594	267	10	a	a	DET
ejpam-4594	267	11	}	}	PUNCT
ejpam-4594	267	12	∈	∈	PROPN
ejpam-4594	267	13	ω	ω	NOUN
ejpam-4594	267	14	is	be	AUX
ejpam-4594	267	15	not	not	PART
ejpam-4594	267	16	a	a	DET
ejpam-4594	267	17	dual	dual	ADJ
ejpam-4594	267	18	b	b	NOUN
ejpam-4594	267	19	-	-	NOUN
ejpam-4594	267	20	filter	filter	NOUN
ejpam-4594	267	21	of	of	ADP
ejpam-4594	267	22	xd	xd	INTJ
ejpam-4594	267	23	since	since	SCONJ
ejpam-4594	267	24	1	1	NUM
ejpam-4594	267	25	/∈	/∈	PUNCT
ejpam-4594	267	26	{	{	PUNCT
ejpam-4594	267	27	a	a	NOUN
ejpam-4594	267	28	}	}	PUNCT
ejpam-4594	267	29	.	.	PUNCT
ejpam-4594	268	1	the	the	DET
ejpam-4594	268	2	next	next	ADJ
ejpam-4594	268	3	results	result	NOUN
ejpam-4594	268	4	describes	describe	VERB
ejpam-4594	268	5	the	the	DET
ejpam-4594	268	6	dual	dual	ADJ
ejpam-4594	268	7	b	b	NOUN
ejpam-4594	268	8	-	-	PUNCT
ejpam-4594	268	9	topology	topology	NOUN
ejpam-4594	268	10	τ	τ	PROPN
ejpam-4594	268	11	determined	determine	VERB
ejpam-4594	268	12	by	by	ADP
ejpam-4594	268	13	a	a	DET
ejpam-4594	268	14	filterbase	filterbase	PROPN
ejpam-4594	268	15	ω	ω	PROPN
ejpam-4594	268	16	followed	follow	VERB
ejpam-4594	268	17	by	by	ADP
ejpam-4594	268	18	the	the	DET
ejpam-4594	268	19	relationship	relationship	NOUN
ejpam-4594	268	20	of	of	ADP
ejpam-4594	268	21	ω	ω	PROPN
ejpam-4594	268	22	and	and	CCONJ
ejpam-4594	268	23	τ	τ	NUM
ejpam-4594	268	24	if	if	SCONJ
ejpam-4594	268	25	ω	ω	PROPN
ejpam-4594	268	26	is	be	AUX
ejpam-4594	268	27	a	a	DET
ejpam-4594	268	28	family	family	NOUN
ejpam-4594	268	29	of	of	ADP
ejpam-4594	268	30	dual	dual	ADJ
ejpam-4594	268	31	b	b	NOUN
ejpam-4594	268	32	-	-	PUNCT
ejpam-4594	268	33	filters	filter	NOUN
ejpam-4594	268	34	.	.	PUNCT
ejpam-4594	269	1	theorem	theorem	ADJ
ejpam-4594	269	2	8	8	NUM
ejpam-4594	269	3	.	.	PUNCT
ejpam-4594	270	1	let	let	VERB
ejpam-4594	270	2	ω	ω	NUM
ejpam-4594	270	3	be	be	AUX
ejpam-4594	270	4	a	a	DET
ejpam-4594	270	5	filterbase	filterbase	NOUN
ejpam-4594	270	6	in	in	ADP
ejpam-4594	270	7	a	a	DET
ejpam-4594	270	8	dual	dual	ADJ
ejpam-4594	270	9	b	b	NOUN
ejpam-4594	270	10	-	-	PUNCT
ejpam-4594	270	11	algebra	algebra	NOUN
ejpam-4594	270	12	xd	xd	INTJ
ejpam-4594	270	13	.	.	PUNCT
ejpam-4594	271	1	then	then	ADV
ejpam-4594	271	2	the	the	DET
ejpam-4594	271	3	family	family	NOUN
ejpam-4594	271	4	τ	τ	X
ejpam-4594	271	5	=	=	PUNCT
ejpam-4594	271	6	{	{	PUNCT
ejpam-4594	271	7	o	o	NOUN
ejpam-4594	271	8	⊆	⊆	NUM
ejpam-4594	271	9	xd	xd	INTJ
ejpam-4594	271	10	|	|	ADV
ejpam-4594	271	11	∀a	∀a	NOUN
ejpam-4594	271	12	∈	∈	NOUN
ejpam-4594	271	13	o,∃v	o,∃v	VERB
ejpam-4594	271	14	∈	∈	PROPN
ejpam-4594	271	15	ω	ω	NOUN
ejpam-4594	271	16	such	such	ADJ
ejpam-4594	271	17	that	that	SCONJ
ejpam-4594	271	18	v	v	ADP
ejpam-4594	271	19	′[a	′[a	ADJ
ejpam-4594	271	20	]	]	PUNCT
ejpam-4594	271	21	⊆	⊆	NUM
ejpam-4594	271	22	o	o	NOUN
ejpam-4594	271	23	}	}	PUNCT
ejpam-4594	271	24	is	be	AUX
ejpam-4594	271	25	a	a	DET
ejpam-4594	271	26	dual	dual	ADJ
ejpam-4594	271	27	b	b	NOUN
ejpam-4594	271	28	-	-	NOUN
ejpam-4594	271	29	topology	topology	NOUN
ejpam-4594	271	30	on	on	ADP
ejpam-4594	271	31	xd	xd	ADP
ejpam-4594	271	32	.	.	PUNCT
ejpam-4594	271	33	k.	k.	PROPN
ejpam-4594	271	34	belleza	belleza	PROPN
ejpam-4594	271	35	,	,	PUNCT
ejpam-4594	271	36	j.	j.	PROPN
ejpam-4594	271	37	albaracin	albaracin	PROPN
ejpam-4594	271	38	/	/	PUNCT
ejpam-4594	271	39	eur	eur	PROPN
ejpam-4594	271	40	.	.	PUNCT
ejpam-4594	272	1	j.	j.	PROPN
ejpam-4594	272	2	pure	pure	PROPN
ejpam-4594	272	3	appl	appl	PROPN
ejpam-4594	272	4	.	.	PROPN
ejpam-4594	272	5	math	math	PROPN
ejpam-4594	272	6	,	,	PUNCT
ejpam-4594	272	7	15	15	NUM
ejpam-4594	272	8	(	(	PUNCT
ejpam-4594	272	9	4	4	NUM
ejpam-4594	272	10	)	)	PUNCT
ejpam-4594	272	11	(	(	PUNCT
ejpam-4594	272	12	2022	2022	NUM
ejpam-4594	272	13	)	)	PUNCT
ejpam-4594	272	14	,	,	PUNCT
ejpam-4594	272	15	1948	1948	NUM
ejpam-4594	272	16	-	-	SYM
ejpam-4594	272	17	1956	1956	NUM
ejpam-4594	272	18	1954	1954	NUM
ejpam-4594	272	19	proof	proof	NOUN
ejpam-4594	272	20	.	.	PUNCT
ejpam-4594	273	1	let	let	VERB
ejpam-4594	273	2	ω	ω	PRON
ejpam-4594	273	3	be	be	AUX
ejpam-4594	273	4	a	a	DET
ejpam-4594	273	5	filterbase	filterbase	NOUN
ejpam-4594	273	6	in	in	ADP
ejpam-4594	273	7	a	a	DET
ejpam-4594	273	8	dual	dual	ADJ
ejpam-4594	273	9	b	b	NOUN
ejpam-4594	273	10	-	-	PUNCT
ejpam-4594	273	11	algebra	algebra	NOUN
ejpam-4594	273	12	xd	xd	INTJ
ejpam-4594	273	13	.	.	PUNCT
ejpam-4594	274	1	since	since	SCONJ
ejpam-4594	274	2	v	v	NUM
ejpam-4594	274	3	′[a	′[a	NOUN
ejpam-4594	274	4	]	]	PUNCT
ejpam-4594	274	5	⊆	⊆	NUM
ejpam-4594	274	6	xd	xd	INTJ
ejpam-4594	274	7	for	for	ADP
ejpam-4594	274	8	all	all	PRON
ejpam-4594	274	9	v	v	ADP
ejpam-4594	274	10	∈	∈	PROPN
ejpam-4594	274	11	ω	ω	NOUN
ejpam-4594	274	12	and	and	CCONJ
ejpam-4594	274	13	a	a	DET
ejpam-4594	274	14	∈	∈	NOUN
ejpam-4594	274	15	xd	xd	ADP
ejpam-4594	274	16	,	,	PUNCT
ejpam-4594	274	17	it	it	PRON
ejpam-4594	274	18	follows	follow	VERB
ejpam-4594	274	19	that	that	SCONJ
ejpam-4594	274	20	xd	xd	INTJ
ejpam-4594	274	21	∈	∈	PROPN
ejpam-4594	274	22	τ	τ	X
ejpam-4594	274	23	.	.	PUNCT
ejpam-4594	275	1	since	since	SCONJ
ejpam-4594	275	2	∅	∅	NOUN
ejpam-4594	275	3	do	do	AUX
ejpam-4594	275	4	not	not	PART
ejpam-4594	275	5	have	have	VERB
ejpam-4594	275	6	any	any	DET
ejpam-4594	275	7	element	element	NOUN
ejpam-4594	275	8	,	,	PUNCT
ejpam-4594	275	9	then	then	ADV
ejpam-4594	275	10	vacuously	vacuously	ADV
ejpam-4594	275	11	∅	∅	VERB
ejpam-4594	275	12	∈	∈	PROPN
ejpam-4594	275	13	τ	τ	X
ejpam-4594	275	14	.	.	PUNCT
ejpam-4594	276	1	suppose	suppose	VERB
ejpam-4594	276	2	that	that	SCONJ
ejpam-4594	276	3	oα	oα	PROPN
ejpam-4594	276	4	,	,	PUNCT
ejpam-4594	276	5	oβ	oβ	PROPN
ejpam-4594	276	6	∈	∈	PROPN
ejpam-4594	276	7	τ	τ	X
ejpam-4594	276	8	and	and	CCONJ
ejpam-4594	276	9	a	a	DET
ejpam-4594	276	10	∈	∈	PROPN
ejpam-4594	276	11	oα	oα	NOUN
ejpam-4594	276	12	∩	∩	NOUN
ejpam-4594	276	13	oβ	oβ	PROPN
ejpam-4594	276	14	.	.	PUNCT
ejpam-4594	277	1	then	then	ADV
ejpam-4594	277	2	there	there	PRON
ejpam-4594	277	3	exist	exist	VERB
ejpam-4594	277	4	vα	vα	ADP
ejpam-4594	277	5	,	,	PUNCT
ejpam-4594	277	6	vβ	vβ	PRON
ejpam-4594	277	7	∈	∈	PROPN
ejpam-4594	277	8	ω	ω	NUM
ejpam-4594	277	9	such	such	ADJ
ejpam-4594	277	10	that	that	DET
ejpam-4594	277	11	v	v	NOUN
ejpam-4594	277	12	′	′	NUM
ejpam-4594	277	13	α[a	α[a	ADJ
ejpam-4594	277	14	]	]	PUNCT
ejpam-4594	277	15	⊆	⊆	NUM
ejpam-4594	277	16	oα	oα	NOUN
ejpam-4594	277	17	and	and	CCONJ
ejpam-4594	277	18	v	v	NOUN
ejpam-4594	277	19	′	′	NUM
ejpam-4594	277	20	β[a	β[a	NOUN
ejpam-4594	277	21	]	]	X
ejpam-4594	277	22	⊆	⊆	NUM
ejpam-4594	277	23	oβ	oβ	PROPN
ejpam-4594	277	24	.	.	PUNCT
ejpam-4594	278	1	since	since	SCONJ
ejpam-4594	278	2	ω	ω	PROPN
ejpam-4594	278	3	is	be	AUX
ejpam-4594	278	4	a	a	DET
ejpam-4594	278	5	filterbase	filterbase	NOUN
ejpam-4594	278	6	,	,	PUNCT
ejpam-4594	278	7	there	there	PRON
ejpam-4594	278	8	exists	exist	VERB
ejpam-4594	278	9	v	v	ADP
ejpam-4594	278	10	∈	∈	PROPN
ejpam-4594	278	11	ω	ω	NOUN
ejpam-4594	278	12	such	such	ADJ
ejpam-4594	278	13	that	that	SCONJ
ejpam-4594	278	14	v	v	ADP
ejpam-4594	278	15	⊆	⊆	NUM
ejpam-4594	278	16	vα∩vβ	vα∩vβ	NOUN
ejpam-4594	278	17	.	.	PUNCT
ejpam-4594	279	1	by	by	ADP
ejpam-4594	279	2	proposition	proposition	NOUN
ejpam-4594	279	3	3	3	NUM
ejpam-4594	279	4	,	,	PUNCT
ejpam-4594	279	5	v	v	X
ejpam-4594	279	6	′[a	′[a	X
ejpam-4594	279	7	]	]	X
ejpam-4594	279	8	⊆	⊆	NUM
ejpam-4594	279	9	(	(	PUNCT
ejpam-4594	279	10	vα∩vβ	vα∩vβ	PROPN
ejpam-4594	279	11	)	)	PUNCT
ejpam-4594	279	12	′[a	′[a	NOUN
ejpam-4594	279	13	]	]	PUNCT
ejpam-4594	279	14	⊆	⊆	NUM
ejpam-4594	279	15	v	v	ADP
ejpam-4594	279	16	′	′	NUM
ejpam-4594	279	17	α[a	α[a	ADJ
ejpam-4594	279	18	]	]	PUNCT
ejpam-4594	279	19	⊆	⊆	NUM
ejpam-4594	279	20	oα	oα	NOUN
ejpam-4594	279	21	.	.	PUNCT
ejpam-4594	279	22	similarly	similarly	ADV
ejpam-4594	279	23	,	,	PUNCT
ejpam-4594	279	24	v	v	ADP
ejpam-4594	279	25	′[a	′[a	X
ejpam-4594	279	26	]	]	X
ejpam-4594	279	27	⊆	⊆	NUM
ejpam-4594	279	28	oβ	oβ	NOUN
ejpam-4594	279	29	.	.	PUNCT
ejpam-4594	280	1	hence	hence	ADV
ejpam-4594	280	2	,	,	PUNCT
ejpam-4594	280	3	v	v	ADP
ejpam-4594	280	4	′[a	′[a	X
ejpam-4594	280	5	]	]	PUNCT
ejpam-4594	280	6	⊆	⊆	NUM
ejpam-4594	280	7	oα∩oβ	oα∩oβ	NOUN
ejpam-4594	280	8	.	.	PUNCT
ejpam-4594	281	1	this	this	PRON
ejpam-4594	281	2	implies	imply	VERB
ejpam-4594	281	3	that	that	SCONJ
ejpam-4594	281	4	oα∩oβ	oα∩oβ	PROPN
ejpam-4594	281	5	∈	∈	PROPN
ejpam-4594	281	6	τ	τ	X
ejpam-4594	281	7	.	.	PUNCT
ejpam-4594	282	1	suppose	suppose	VERB
ejpam-4594	282	2	oα	oα	PRON
ejpam-4594	282	3	∈	∈	PROPN
ejpam-4594	282	4	τ	τ	PROPN
ejpam-4594	282	5	for	for	ADP
ejpam-4594	282	6	all	all	DET
ejpam-4594	282	7	α	α	PRON
ejpam-4594	282	8	∈	∈	PROPN
ejpam-4594	282	9	a	a	PRON
ejpam-4594	282	10	and	and	CCONJ
ejpam-4594	282	11	let	let	VERB
ejpam-4594	282	12	a	a	DET
ejpam-4594	282	13	∈	∈	NOUN
ejpam-4594	282	14	⋃	⋃	NOUN
ejpam-4594	282	15	α∈a	α∈a	NOUN
ejpam-4594	282	16	oα	oα	NOUN
ejpam-4594	282	17	.	.	PUNCT
ejpam-4594	283	1	then	then	ADV
ejpam-4594	283	2	a	a	DET
ejpam-4594	283	3	∈	∈	PROPN
ejpam-4594	283	4	oβ	oβ	ADP
ejpam-4594	283	5	∈	∈	PROPN
ejpam-4594	283	6	τ	τ	PROPN
ejpam-4594	283	7	for	for	ADP
ejpam-4594	283	8	some	some	DET
ejpam-4594	283	9	β	β	NOUN
ejpam-4594	283	10	∈	∈	PROPN
ejpam-4594	283	11	a.	a.	NOUN
ejpam-4594	284	1	this	this	PRON
ejpam-4594	284	2	implies	imply	VERB
ejpam-4594	284	3	that	that	SCONJ
ejpam-4594	284	4	there	there	PRON
ejpam-4594	284	5	exists	exist	VERB
ejpam-4594	284	6	vβ	vβ	DET
ejpam-4594	284	7	∈	∈	PROPN
ejpam-4594	284	8	ω	ω	NUM
ejpam-4594	285	1	such	such	ADJ
ejpam-4594	285	2	that	that	DET
ejpam-4594	285	3	v	v	NOUN
ejpam-4594	285	4	′	′	NUM
ejpam-4594	285	5	β[a	β[a	NOUN
ejpam-4594	285	6	]	]	X
ejpam-4594	285	7	⊆	⊆	NUM
ejpam-4594	285	8	oβ	oβ	NOUN
ejpam-4594	285	9	.	.	PUNCT
ejpam-4594	286	1	hence	hence	ADV
ejpam-4594	286	2	,	,	PUNCT
ejpam-4594	286	3	v	v	X
ejpam-4594	286	4	′	′	NUM
ejpam-4594	286	5	β[a	β[a	NOUN
ejpam-4594	286	6	]	]	X
ejpam-4594	286	7	⊆	⊆	NUM
ejpam-4594	286	8	⋃	⋃	NOUN
ejpam-4594	286	9	α∈a	α∈a	NOUN
ejpam-4594	286	10	oα	oα	NOUN
ejpam-4594	286	11	.	.	PUNCT
ejpam-4594	287	1	it	it	PRON
ejpam-4594	287	2	follows	follow	VERB
ejpam-4594	287	3	that	that	SCONJ
ejpam-4594	287	4	⋃	⋃	ADV
ejpam-4594	287	5	α∈a	α∈a	NOUN
ejpam-4594	287	6	oα	oα	PROPN
ejpam-4594	287	7	∈	∈	PROPN
ejpam-4594	287	8	τ	τ	PROPN
ejpam-4594	287	9	.	.	PUNCT
ejpam-4594	288	1	therefore	therefore	ADV
ejpam-4594	288	2	,	,	PUNCT
ejpam-4594	288	3	τ	τ	PROPN
ejpam-4594	288	4	is	be	AUX
ejpam-4594	288	5	a	a	DET
ejpam-4594	288	6	dual	dual	ADJ
ejpam-4594	288	7	b	b	NOUN
ejpam-4594	288	8	-	-	NOUN
ejpam-4594	288	9	topology	topology	NOUN
ejpam-4594	288	10	.	.	PUNCT
ejpam-4594	289	1	theorem	theorem	VERB
ejpam-4594	289	2	9	9	NUM
ejpam-4594	289	3	.	.	PUNCT
ejpam-4594	290	1	let	let	VERB
ejpam-4594	290	2	xd	xd	INTJ
ejpam-4594	290	3	be	be	AUX
ejpam-4594	290	4	a	a	DET
ejpam-4594	290	5	dual	dual	ADJ
ejpam-4594	290	6	b	b	NOUN
ejpam-4594	290	7	-	-	PUNCT
ejpam-4594	290	8	topological	topological	ADJ
ejpam-4594	290	9	space	space	NOUN
ejpam-4594	290	10	and	and	CCONJ
ejpam-4594	290	11	ω	ω	NUM
ejpam-4594	290	12	a	a	DET
ejpam-4594	290	13	filterbase	filterbase	NOUN
ejpam-4594	290	14	in	in	ADP
ejpam-4594	290	15	xd	xd	INTJ
ejpam-4594	290	16	such	such	ADJ
ejpam-4594	290	17	that	that	SCONJ
ejpam-4594	290	18	ω	ω	PROPN
ejpam-4594	290	19	is	be	AUX
ejpam-4594	290	20	a	a	DET
ejpam-4594	290	21	family	family	NOUN
ejpam-4594	290	22	of	of	ADP
ejpam-4594	290	23	dual	dual	ADJ
ejpam-4594	290	24	b	b	NOUN
ejpam-4594	290	25	-	-	PUNCT
ejpam-4594	290	26	filters	filter	NOUN
ejpam-4594	290	27	of	of	ADP
ejpam-4594	290	28	xd	xd	ADP
ejpam-4594	290	29	.	.	PUNCT
ejpam-4594	291	1	then	then	ADV
ejpam-4594	291	2	ω	ω	PROPN
ejpam-4594	291	3	is	be	AUX
ejpam-4594	291	4	a	a	DET
ejpam-4594	291	5	proper	proper	ADJ
ejpam-4594	291	6	subclass	subclass	NOUN
ejpam-4594	291	7	of	of	ADP
ejpam-4594	291	8	τ	τ	PROPN
ejpam-4594	291	9	.	.	PUNCT
ejpam-4594	292	1	proof	proof	NOUN
ejpam-4594	292	2	.	.	PUNCT
ejpam-4594	293	1	suppose	suppose	VERB
ejpam-4594	293	2	xd	xd	INTJ
ejpam-4594	293	3	is	be	AUX
ejpam-4594	293	4	a	a	DET
ejpam-4594	293	5	dual	dual	ADJ
ejpam-4594	293	6	b	b	NOUN
ejpam-4594	293	7	-	-	PUNCT
ejpam-4594	293	8	topological	topological	ADJ
ejpam-4594	293	9	space	space	NOUN
ejpam-4594	293	10	and	and	CCONJ
ejpam-4594	293	11	ω	ω	NUM
ejpam-4594	293	12	a	a	DET
ejpam-4594	293	13	filterbase	filterbase	NOUN
ejpam-4594	293	14	in	in	ADP
ejpam-4594	293	15	xd	xd	INTJ
ejpam-4594	293	16	such	such	ADJ
ejpam-4594	293	17	that	that	SCONJ
ejpam-4594	293	18	ω	ω	PROPN
ejpam-4594	293	19	is	be	AUX
ejpam-4594	293	20	a	a	DET
ejpam-4594	293	21	family	family	NOUN
ejpam-4594	293	22	of	of	ADP
ejpam-4594	293	23	dual	dual	ADJ
ejpam-4594	293	24	b	b	NOUN
ejpam-4594	293	25	-	-	PUNCT
ejpam-4594	293	26	filters	filter	NOUN
ejpam-4594	293	27	of	of	ADP
ejpam-4594	293	28	xd	xd	ADP
ejpam-4594	293	29	.	.	PUNCT
ejpam-4594	293	30	note	note	VERB
ejpam-4594	293	31	that	that	SCONJ
ejpam-4594	293	32	∅	∅	NOUN
ejpam-4594	293	33	/∈	/∈	PUNCT
ejpam-4594	294	1	ω	ω	NUM
ejpam-4594	294	2	by	by	ADP
ejpam-4594	294	3	definition	definition	NOUN
ejpam-4594	294	4	10(i	10(i	NUM
ejpam-4594	294	5	)	)	PUNCT
ejpam-4594	294	6	but	but	CCONJ
ejpam-4594	294	7	∅	∅	NOUN
ejpam-4594	294	8	∈	∈	PROPN
ejpam-4594	294	9	τ	τ	X
ejpam-4594	294	10	.	.	PUNCT
ejpam-4594	295	1	this	this	PRON
ejpam-4594	295	2	implies	imply	VERB
ejpam-4594	295	3	that	that	SCONJ
ejpam-4594	295	4	ω	ω	PROPN
ejpam-4594	295	5	̸=	̸=	PROPN
ejpam-4594	295	6	τ	τ	X
ejpam-4594	295	7	.	.	PUNCT
ejpam-4594	296	1	let	let	VERB
ejpam-4594	296	2	o	o	X
ejpam-4594	296	3	∈	∈	PROPN
ejpam-4594	296	4	ω	ω	PROPN
ejpam-4594	296	5	and	and	CCONJ
ejpam-4594	296	6	x	x	PUNCT
ejpam-4594	296	7	∈	∈	PROPN
ejpam-4594	297	1	o.	o.	NOUN
ejpam-4594	297	2	it	it	PRON
ejpam-4594	297	3	remains	remain	VERB
ejpam-4594	297	4	to	to	PART
ejpam-4594	297	5	show	show	VERB
ejpam-4594	297	6	that	that	SCONJ
ejpam-4594	297	7	o′[x	o′[x	NOUN
ejpam-4594	297	8	]	]	PUNCT
ejpam-4594	298	1	⊆	⊆	NUM
ejpam-4594	298	2	o.	o.	NOUN
ejpam-4594	298	3	suppose	suppose	VERB
ejpam-4594	298	4	a	a	DET
ejpam-4594	298	5	∈	∈	NOUN
ejpam-4594	298	6	o′[x	o′[x	NOUN
ejpam-4594	298	7	]	]	PUNCT
ejpam-4594	298	8	.	.	PUNCT
ejpam-4594	299	1	then	then	ADV
ejpam-4594	299	2	a	a	DET
ejpam-4594	299	3	◦	◦	NOUN
ejpam-4594	299	4	x	x	SYM
ejpam-4594	299	5	,	,	PUNCT
ejpam-4594	299	6	x	x	PUNCT
ejpam-4594	299	7	◦	◦	VERB
ejpam-4594	299	8	a	a	DET
ejpam-4594	299	9	∈	∈	ADJ
ejpam-4594	299	10	o.	o.	NOUN
ejpam-4594	299	11	since	since	SCONJ
ejpam-4594	299	12	o	o	PROPN
ejpam-4594	299	13	is	be	AUX
ejpam-4594	299	14	a	a	DET
ejpam-4594	299	15	dual	dual	ADJ
ejpam-4594	299	16	b	b	NOUN
ejpam-4594	299	17	-	-	NOUN
ejpam-4594	299	18	filter	filter	NOUN
ejpam-4594	299	19	and	and	CCONJ
ejpam-4594	299	20	x	x	PART
ejpam-4594	299	21	∈	∈	NOUN
ejpam-4594	299	22	o	o	NOUN
ejpam-4594	299	23	,	,	PUNCT
ejpam-4594	299	24	it	it	PRON
ejpam-4594	299	25	follows	follow	VERB
ejpam-4594	299	26	that	that	SCONJ
ejpam-4594	299	27	a	a	DET
ejpam-4594	299	28	∈	∈	PROPN
ejpam-4594	299	29	o.	o.	NOUN
ejpam-4594	299	30	hence	hence	ADV
ejpam-4594	299	31	,	,	PUNCT
ejpam-4594	299	32	o′[x	o′[x	ADV
ejpam-4594	299	33	]	]	PUNCT
ejpam-4594	299	34	⊆	⊆	NUM
ejpam-4594	299	35	o.	o.	NOUN
ejpam-4594	299	36	this	this	PRON
ejpam-4594	299	37	implies	imply	VERB
ejpam-4594	299	38	that	that	SCONJ
ejpam-4594	299	39	o	o	PROPN
ejpam-4594	299	40	∈	∈	PROPN
ejpam-4594	299	41	τ	τ	X
ejpam-4594	299	42	.	.	PUNCT
ejpam-4594	300	1	therefore	therefore	ADV
ejpam-4594	300	2	,	,	PUNCT
ejpam-4594	300	3	ω	ω	PROPN
ejpam-4594	300	4	is	be	AUX
ejpam-4594	300	5	a	a	DET
ejpam-4594	300	6	proper	proper	ADJ
ejpam-4594	300	7	subclass	subclass	NOUN
ejpam-4594	300	8	of	of	ADP
ejpam-4594	300	9	τ	τ	PROPN
ejpam-4594	300	10	.	.	PUNCT
ejpam-4594	301	1	theorem	theorem	ADJ
ejpam-4594	301	2	10	10	NUM
ejpam-4594	301	3	.	.	PUNCT
ejpam-4594	302	1	suppose	suppose	VERB
ejpam-4594	302	2	xd	xd	INTJ
ejpam-4594	302	3	is	be	VERB
ejpam-4594	302	4	a	a	DET
ejpam-4594	302	5	dual	dual	ADJ
ejpam-4594	302	6	b	b	NOUN
ejpam-4594	302	7	-	-	PUNCT
ejpam-4594	302	8	topological	topological	ADJ
ejpam-4594	302	9	space	space	NOUN
ejpam-4594	302	10	and	and	CCONJ
ejpam-4594	302	11	let	let	VERB
ejpam-4594	302	12	ω	ω	NOUN
ejpam-4594	302	13	be	be	AUX
ejpam-4594	302	14	a	a	DET
ejpam-4594	302	15	filterbase	filterbase	NOUN
ejpam-4594	302	16	in	in	ADP
ejpam-4594	302	17	xd	xd	INTJ
ejpam-4594	302	18	such	such	ADJ
ejpam-4594	302	19	that	that	PRON
ejpam-4594	302	20	for	for	ADP
ejpam-4594	302	21	all	all	PRON
ejpam-4594	302	22	v	v	ADP
ejpam-4594	302	23	∈	∈	PROPN
ejpam-4594	302	24	ω	ω	NOUN
ejpam-4594	302	25	and	and	CCONJ
ejpam-4594	302	26	for	for	ADP
ejpam-4594	302	27	all	all	DET
ejpam-4594	302	28	p	p	NOUN
ejpam-4594	302	29	,	,	PUNCT
ejpam-4594	302	30	q	q	PROPN
ejpam-4594	302	31	∈	∈	PROPN
ejpam-4594	302	32	v	v	NOUN
ejpam-4594	302	33	,	,	PUNCT
ejpam-4594	302	34	(	(	PUNCT
ejpam-4594	302	35	i	i	NOUN
ejpam-4594	302	36	)	)	PUNCT
ejpam-4594	302	37	p	p	NOUN
ejpam-4594	302	38	◦	◦	NOUN
ejpam-4594	302	39	1	1	NUM
ejpam-4594	302	40	∈	∈	NOUN
ejpam-4594	302	41	v	v	NOUN
ejpam-4594	302	42	;	;	PUNCT
ejpam-4594	302	43	and	and	CCONJ
ejpam-4594	302	44	(	(	PUNCT
ejpam-4594	302	45	ii	ii	NOUN
ejpam-4594	302	46	)	)	PUNCT
ejpam-4594	302	47	(	(	PUNCT
ejpam-4594	302	48	p	p	NOUN
ejpam-4594	302	49	◦	◦	NOUN
ejpam-4594	302	50	x	x	NOUN
ejpam-4594	302	51	)	)	PUNCT
ejpam-4594	302	52	◦	◦	NOUN
ejpam-4594	302	53	q	q	NOUN
ejpam-4594	302	54	=	=	SYM
ejpam-4594	302	55	1	1	NUM
ejpam-4594	302	56	implies	imply	VERB
ejpam-4594	302	57	x	x	X
ejpam-4594	302	58	∈	∈	NOUN
ejpam-4594	302	59	v	v	NOUN
ejpam-4594	302	60	.	.	PUNCT
ejpam-4594	303	1	then	then	ADV
ejpam-4594	303	2	ω	ω	PROPN
ejpam-4594	303	3	is	be	AUX
ejpam-4594	303	4	the	the	DET
ejpam-4594	303	5	neighborhood	neighborhood	NOUN
ejpam-4594	303	6	filterbase	filterbase	NOUN
ejpam-4594	303	7	of	of	ADP
ejpam-4594	303	8	1	1	NUM
ejpam-4594	303	9	∈	∈	NOUN
ejpam-4594	303	10	xd	xd	NOUN
ejpam-4594	303	11	.	.	PUNCT
ejpam-4594	304	1	that	that	PRON
ejpam-4594	304	2	is	is	ADV
ejpam-4594	304	3	,	,	PUNCT
ejpam-4594	304	4	ω	ω	PROPN
ejpam-4594	304	5	is	be	AUX
ejpam-4594	304	6	a	a	DET
ejpam-4594	304	7	family	family	NOUN
ejpam-4594	304	8	of	of	ADP
ejpam-4594	304	9	neighborhoods	neighborhood	NOUN
ejpam-4594	304	10	of	of	ADP
ejpam-4594	304	11	1	1	NUM
ejpam-4594	304	12	(	(	PUNCT
ejpam-4594	304	13	∀v	∀v	PROPN
ejpam-4594	304	14	∈	∈	PROPN
ejpam-4594	304	15	ω	ω	NOUN
ejpam-4594	304	16	,	,	PUNCT
ejpam-4594	304	17	1	1	NUM
ejpam-4594	304	18	∈	∈	NOUN
ejpam-4594	304	19	v	v	NOUN
ejpam-4594	304	20	and	and	CCONJ
ejpam-4594	304	21	v	v	ADP
ejpam-4594	304	22	∈	∈	PROPN
ejpam-4594	304	23	τ	τ	PROPN
ejpam-4594	304	24	)	)	PUNCT
ejpam-4594	304	25	.	.	PUNCT
ejpam-4594	305	1	proof	proof	NOUN
ejpam-4594	305	2	.	.	PUNCT
ejpam-4594	306	1	suppose	suppose	VERB
ejpam-4594	306	2	xd	xd	INTJ
ejpam-4594	306	3	is	be	AUX
ejpam-4594	306	4	a	a	DET
ejpam-4594	306	5	dual	dual	ADJ
ejpam-4594	306	6	b	b	NOUN
ejpam-4594	306	7	-	-	PUNCT
ejpam-4594	306	8	topological	topological	ADJ
ejpam-4594	306	9	space	space	NOUN
ejpam-4594	306	10	and	and	CCONJ
ejpam-4594	306	11	let	let	VERB
ejpam-4594	306	12	ω	ω	NOUN
ejpam-4594	306	13	be	be	AUX
ejpam-4594	306	14	a	a	DET
ejpam-4594	306	15	filterbase	filterbase	NOUN
ejpam-4594	306	16	in	in	ADP
ejpam-4594	306	17	xd	xd	INTJ
ejpam-4594	306	18	and	and	CCONJ
ejpam-4594	306	19	p	p	NOUN
ejpam-4594	306	20	∈	∈	PROPN
ejpam-4594	306	21	v	v	NOUN
ejpam-4594	306	22	.	.	PUNCT
ejpam-4594	307	1	by	by	ADP
ejpam-4594	307	2	(	(	PUNCT
ejpam-4594	307	3	i	i	NOUN
ejpam-4594	307	4	)	)	PUNCT
ejpam-4594	307	5	,	,	PUNCT
ejpam-4594	307	6	p	p	NOUN
ejpam-4594	307	7	◦	◦	NOUN
ejpam-4594	307	8	1	1	NUM
ejpam-4594	307	9	∈	∈	NOUN
ejpam-4594	307	10	v	v	NOUN
ejpam-4594	307	11	.	.	PUNCT
ejpam-4594	308	1	by	by	ADP
ejpam-4594	308	2	(	(	PUNCT
ejpam-4594	308	3	db1	db1	NOUN
ejpam-4594	308	4	)	)	PUNCT
ejpam-4594	308	5	and	and	CCONJ
ejpam-4594	308	6	(	(	PUNCT
ejpam-4594	308	7	ii	ii	NOUN
ejpam-4594	308	8	)	)	PUNCT
ejpam-4594	308	9	,	,	PUNCT
ejpam-4594	308	10	(	(	PUNCT
ejpam-4594	308	11	p	p	NOUN
ejpam-4594	308	12	◦	◦	NOUN
ejpam-4594	308	13	1	1	NUM
ejpam-4594	308	14	)	)	PUNCT
ejpam-4594	308	15	◦	◦	NOUN
ejpam-4594	308	16	(	(	PUNCT
ejpam-4594	308	17	p	p	NOUN
ejpam-4594	308	18	◦	◦	NOUN
ejpam-4594	308	19	1	1	NUM
ejpam-4594	308	20	)	)	PUNCT
ejpam-4594	308	21	=	=	SYM
ejpam-4594	308	22	1	1	NUM
ejpam-4594	308	23	implying	imply	VERB
ejpam-4594	308	24	that	that	SCONJ
ejpam-4594	308	25	1	1	NUM
ejpam-4594	308	26	∈	∈	PROPN
ejpam-4594	308	27	v	v	NOUN
ejpam-4594	308	28	.	.	PUNCT
ejpam-4594	309	1	claim	claim	NOUN
ejpam-4594	309	2	:	:	PUNCT
ejpam-4594	309	3	v	v	X
ejpam-4594	309	4	′[p	′[p	X
ejpam-4594	309	5	]	]	X
ejpam-4594	309	6	⊆	⊆	NUM
ejpam-4594	309	7	v	v	NOUN
ejpam-4594	309	8	.	.	PUNCT
ejpam-4594	310	1	let	let	VERB
ejpam-4594	310	2	x	x	PUNCT
ejpam-4594	310	3	∈	∈	PROPN
ejpam-4594	310	4	v	v	ADP
ejpam-4594	310	5	′[p	′[p	PROPN
ejpam-4594	310	6	]	]	PUNCT
ejpam-4594	310	7	.	.	PUNCT
ejpam-4594	311	1	then	then	ADV
ejpam-4594	311	2	x	x	PUNCT
ejpam-4594	311	3	◦	◦	NOUN
ejpam-4594	311	4	p	p	X
ejpam-4594	311	5	,	,	PUNCT
ejpam-4594	311	6	p	p	NOUN
ejpam-4594	311	7	◦	◦	NOUN
ejpam-4594	311	8	x	x	SYM
ejpam-4594	311	9	∈	∈	NOUN
ejpam-4594	311	10	v	v	NOUN
ejpam-4594	311	11	.	.	PUNCT
ejpam-4594	312	1	this	this	PRON
ejpam-4594	312	2	implies	imply	VERB
ejpam-4594	312	3	that	that	SCONJ
ejpam-4594	312	4	p	p	PROPN
ejpam-4594	312	5	◦	◦	NOUN
ejpam-4594	312	6	x	x	X
ejpam-4594	312	7	=	=	NOUN
ejpam-4594	312	8	v	v	NOUN
ejpam-4594	312	9	for	for	ADP
ejpam-4594	312	10	some	some	DET
ejpam-4594	312	11	v	v	NOUN
ejpam-4594	312	12	∈	∈	PROPN
ejpam-4594	312	13	v	v	NOUN
ejpam-4594	312	14	.	.	PUNCT
ejpam-4594	313	1	by	by	ADP
ejpam-4594	313	2	(	(	PUNCT
ejpam-4594	313	3	db1	db1	NOUN
ejpam-4594	313	4	)	)	PUNCT
ejpam-4594	313	5	and	and	CCONJ
ejpam-4594	313	6	(	(	PUNCT
ejpam-4594	313	7	ii	ii	NOUN
ejpam-4594	313	8	)	)	PUNCT
ejpam-4594	313	9	,	,	PUNCT
ejpam-4594	313	10	1	1	NUM
ejpam-4594	313	11	=	=	SYM
ejpam-4594	313	12	v	v	ADP
ejpam-4594	313	13	◦	◦	NOUN
ejpam-4594	313	14	v	v	NOUN
ejpam-4594	313	15	=	=	PUNCT
ejpam-4594	313	16	(	(	PUNCT
ejpam-4594	313	17	p	p	NOUN
ejpam-4594	313	18	◦	◦	NOUN
ejpam-4594	313	19	x	x	NOUN
ejpam-4594	313	20	)	)	PUNCT
ejpam-4594	313	21	◦	◦	NOUN
ejpam-4594	313	22	v	v	ADP
ejpam-4594	313	23	implying	imply	VERB
ejpam-4594	313	24	that	that	SCONJ
ejpam-4594	313	25	x	x	PUNCT
ejpam-4594	313	26	∈	∈	NOUN
ejpam-4594	313	27	v	v	NOUN
ejpam-4594	313	28	.	.	PUNCT
ejpam-4594	314	1	this	this	PRON
ejpam-4594	314	2	proves	prove	VERB
ejpam-4594	314	3	the	the	DET
ejpam-4594	314	4	claim	claim	NOUN
ejpam-4594	314	5	.	.	PUNCT
ejpam-4594	315	1	by	by	ADP
ejpam-4594	315	2	the	the	DET
ejpam-4594	315	3	claim	claim	NOUN
ejpam-4594	315	4	,	,	PUNCT
ejpam-4594	315	5	v	v	X
ejpam-4594	315	6	∈	∈	PROPN
ejpam-4594	315	7	τ	τ	X
ejpam-4594	315	8	.	.	PUNCT
ejpam-4594	316	1	therefore	therefore	ADV
ejpam-4594	316	2	,	,	PUNCT
ejpam-4594	316	3	ω	ω	PROPN
ejpam-4594	316	4	is	be	AUX
ejpam-4594	316	5	the	the	DET
ejpam-4594	316	6	neighborhood	neighborhood	NOUN
ejpam-4594	316	7	filterbase	filterbase	NOUN
ejpam-4594	316	8	of	of	ADP
ejpam-4594	316	9	1	1	NUM
ejpam-4594	316	10	∈	∈	NOUN
ejpam-4594	316	11	xd	xd	PROPN
ejpam-4594	316	12	.	.	PUNCT
ejpam-4594	317	1	lemma	lemma	PROPN
ejpam-4594	317	2	4	4	X
ejpam-4594	317	3	.	.	PUNCT
ejpam-4594	317	4	suppose	suppose	VERB
ejpam-4594	317	5	xd	xd	INTJ
ejpam-4594	317	6	is	be	VERB
ejpam-4594	317	7	a	a	DET
ejpam-4594	317	8	dual	dual	ADJ
ejpam-4594	317	9	b	b	NOUN
ejpam-4594	317	10	-	-	PUNCT
ejpam-4594	317	11	topological	topological	ADJ
ejpam-4594	317	12	space	space	NOUN
ejpam-4594	317	13	and	and	CCONJ
ejpam-4594	317	14	let	let	VERB
ejpam-4594	317	15	ω	ω	NOUN
ejpam-4594	317	16	be	be	AUX
ejpam-4594	317	17	a	a	DET
ejpam-4594	317	18	filterbase	filterbase	NOUN
ejpam-4594	317	19	in	in	ADP
ejpam-4594	317	20	xd	xd	INTJ
ejpam-4594	317	21	such	such	ADJ
ejpam-4594	317	22	that	that	PRON
ejpam-4594	317	23	for	for	ADP
ejpam-4594	317	24	all	all	PRON
ejpam-4594	317	25	v	v	ADP
ejpam-4594	317	26	∈	∈	PROPN
ejpam-4594	317	27	ω	ω	NOUN
ejpam-4594	317	28	and	and	CCONJ
ejpam-4594	317	29	for	for	ADP
ejpam-4594	317	30	all	all	DET
ejpam-4594	317	31	p	p	NOUN
ejpam-4594	317	32	,	,	PUNCT
ejpam-4594	317	33	q	q	PROPN
ejpam-4594	317	34	∈	∈	PROPN
ejpam-4594	317	35	v	v	NOUN
ejpam-4594	317	36	,	,	PUNCT
ejpam-4594	317	37	(	(	PUNCT
ejpam-4594	317	38	i	i	NOUN
ejpam-4594	317	39	)	)	PUNCT
ejpam-4594	318	1	p	p	X
ejpam-4594	318	2	◦	◦	NOUN
ejpam-4594	318	3	1	1	NUM
ejpam-4594	318	4	∈	∈	NOUN
ejpam-4594	318	5	v	v	NOUN
ejpam-4594	318	6	;	;	PUNCT
ejpam-4594	318	7	and	and	CCONJ
ejpam-4594	318	8	(	(	PUNCT
ejpam-4594	318	9	ii	ii	NOUN
ejpam-4594	318	10	)	)	PUNCT
ejpam-4594	318	11	(	(	PUNCT
ejpam-4594	318	12	p	p	X
ejpam-4594	318	13	◦	◦	NOUN
ejpam-4594	318	14	x	x	NOUN
ejpam-4594	318	15	)	)	PUNCT
ejpam-4594	318	16	◦	◦	NOUN
ejpam-4594	318	17	q	q	PUNCT
ejpam-4594	318	18	=	=	SYM
ejpam-4594	318	19	1	1	NUM
ejpam-4594	318	20	implies	imply	VERB
ejpam-4594	318	21	x	x	X
ejpam-4594	318	22	∈	∈	NOUN
ejpam-4594	318	23	v	v	NOUN
ejpam-4594	318	24	.	.	PUNCT
ejpam-4594	319	1	then	then	ADV
ejpam-4594	319	2	v	v	ADP
ejpam-4594	319	3	′[a	′[a	NOUN
ejpam-4594	319	4	]	]	PUNCT
ejpam-4594	319	5	is	be	AUX
ejpam-4594	319	6	open	open	ADJ
ejpam-4594	319	7	in	in	ADP
ejpam-4594	319	8	xd	xd	ADV
ejpam-4594	319	9	for	for	ADP
ejpam-4594	319	10	all	all	DET
ejpam-4594	319	11	a	a	DET
ejpam-4594	319	12	∈	∈	NOUN
ejpam-4594	319	13	xd	xd	NOUN
ejpam-4594	319	14	.	.	PUNCT
ejpam-4594	319	15	proof	proof	NOUN
ejpam-4594	319	16	.	.	PUNCT
ejpam-4594	320	1	suppose	suppose	VERB
ejpam-4594	320	2	xd	xd	INTJ
ejpam-4594	320	3	is	be	AUX
ejpam-4594	320	4	a	a	DET
ejpam-4594	320	5	dual	dual	ADJ
ejpam-4594	320	6	b	b	NOUN
ejpam-4594	320	7	-	-	PUNCT
ejpam-4594	320	8	topological	topological	ADJ
ejpam-4594	320	9	space	space	NOUN
ejpam-4594	320	10	and	and	CCONJ
ejpam-4594	320	11	let	let	VERB
ejpam-4594	320	12	ω	ω	NOUN
ejpam-4594	320	13	be	be	AUX
ejpam-4594	320	14	a	a	DET
ejpam-4594	320	15	filterbase	filterbase	NOUN
ejpam-4594	320	16	in	in	ADP
ejpam-4594	320	17	xd	xd	PROPN
ejpam-4594	320	18	.	.	PUNCT
ejpam-4594	320	19	suppose	suppose	VERB
ejpam-4594	320	20	x	x	SYM
ejpam-4594	320	21	∈	∈	PROPN
ejpam-4594	320	22	v	v	NOUN
ejpam-4594	320	23	′[a	′[a	X
ejpam-4594	320	24	]	]	PUNCT
ejpam-4594	320	25	for	for	ADP
ejpam-4594	320	26	any	any	DET
ejpam-4594	320	27	a	a	DET
ejpam-4594	320	28	∈	∈	PROPN
ejpam-4594	320	29	xd	xd	ADP
ejpam-4594	320	30	.	.	PUNCT
ejpam-4594	321	1	then	then	ADV
ejpam-4594	321	2	a	a	DET
ejpam-4594	321	3	◦	◦	NOUN
ejpam-4594	321	4	x	x	SYM
ejpam-4594	321	5	,	,	PUNCT
ejpam-4594	321	6	x	x	PUNCT
ejpam-4594	321	7	◦	◦	VERB
ejpam-4594	321	8	a	a	DET
ejpam-4594	321	9	∈	∈	PROPN
ejpam-4594	321	10	v	v	NOUN
ejpam-4594	321	11	.	.	PUNCT
ejpam-4594	322	1	note	note	VERB
ejpam-4594	322	2	that	that	SCONJ
ejpam-4594	322	3	by	by	ADP
ejpam-4594	322	4	theorem	theorem	NOUN
ejpam-4594	322	5	10	10	NUM
ejpam-4594	322	6	,	,	PUNCT
ejpam-4594	322	7	v	v	NOUN
ejpam-4594	322	8	∈	∈	PROPN
ejpam-4594	322	9	τ	τ	X
ejpam-4594	322	10	.	.	PUNCT
ejpam-4594	323	1	by	by	ADP
ejpam-4594	323	2	theorem	theorem	NOUN
ejpam-4594	323	3	8	8	NUM
ejpam-4594	323	4	,	,	PUNCT
ejpam-4594	323	5	there	there	PRON
ejpam-4594	323	6	exist	exist	VERB
ejpam-4594	323	7	uα	uα	PROPN
ejpam-4594	323	8	,	,	PUNCT
ejpam-4594	323	9	uβ	uβ	PROPN
ejpam-4594	323	10	∈	∈	PROPN
ejpam-4594	323	11	ω	ω	NUM
ejpam-4594	324	1	such	such	ADJ
ejpam-4594	324	2	that	that	SCONJ
ejpam-4594	324	3	u	u	PROPN
ejpam-4594	324	4	′	′	NOUN
ejpam-4594	324	5	α[a	α[a	ADV
ejpam-4594	325	1	◦	◦	NOUN
ejpam-4594	325	2	x	x	X
ejpam-4594	325	3	]	]	X
ejpam-4594	325	4	,	,	PUNCT
ejpam-4594	325	5	u	u	NOUN
ejpam-4594	325	6	′	′	NOUN
ejpam-4594	325	7	β[x	β[x	NOUN
ejpam-4594	325	8	◦	◦	NOUN
ejpam-4594	325	9	a	a	DET
ejpam-4594	325	10	]	]	X
ejpam-4594	325	11	⊆	⊆	NUM
ejpam-4594	325	12	v	v	NOUN
ejpam-4594	325	13	.	.	PUNCT
ejpam-4594	326	1	since	since	SCONJ
ejpam-4594	326	2	ω	ω	PROPN
ejpam-4594	326	3	is	be	AUX
ejpam-4594	326	4	a	a	DET
ejpam-4594	326	5	filterbase	filterbase	NOUN
ejpam-4594	326	6	in	in	ADP
ejpam-4594	326	7	xd	xd	ADP
ejpam-4594	326	8	,	,	PUNCT
ejpam-4594	326	9	there	there	PRON
ejpam-4594	326	10	exist	exist	VERB
ejpam-4594	326	11	w	w	PROPN
ejpam-4594	326	12	∈	∈	PROPN
ejpam-4594	326	13	ω	ω	NOUN
ejpam-4594	326	14	such	such	ADJ
ejpam-4594	326	15	that	that	SCONJ
ejpam-4594	326	16	w	w	ADP
ejpam-4594	326	17	⊆	⊆	NUM
ejpam-4594	326	18	(	(	PUNCT
ejpam-4594	326	19	uα	uα	PROPN
ejpam-4594	326	20	∩	∩	PROPN
ejpam-4594	326	21	uβ	uβ	PROPN
ejpam-4594	326	22	)	)	PUNCT
ejpam-4594	326	23	.	.	PUNCT
ejpam-4594	327	1	this	this	PRON
ejpam-4594	327	2	implies	imply	VERB
ejpam-4594	327	3	that	that	SCONJ
ejpam-4594	327	4	w	w	ADP
ejpam-4594	327	5	⊆	⊆	NUM
ejpam-4594	327	6	uα	uα	NOUN
ejpam-4594	327	7	and	and	CCONJ
ejpam-4594	327	8	w	w	PROPN
ejpam-4594	327	9	⊆	⊆	NUM
ejpam-4594	327	10	uβ	uβ	PROPN
ejpam-4594	327	11	.	.	PUNCT
ejpam-4594	328	1	by	by	ADP
ejpam-4594	328	2	proposition	proposition	NOUN
ejpam-4594	328	3	3	3	NUM
ejpam-4594	328	4	,	,	PUNCT
ejpam-4594	328	5	it	it	PRON
ejpam-4594	328	6	follows	follow	VERB
ejpam-4594	328	7	that	that	SCONJ
ejpam-4594	328	8	w	w	ADP
ejpam-4594	328	9	′[a	′[a	X
ejpam-4594	328	10	◦	◦	NOUN
ejpam-4594	328	11	x	x	X
ejpam-4594	328	12	]	]	X
ejpam-4594	328	13	⊆	⊆	NUM
ejpam-4594	328	14	u	u	NOUN
ejpam-4594	328	15	′	′	NOUN
ejpam-4594	328	16	α[a	α[a	ADV
ejpam-4594	328	17	◦	◦	NOUN
ejpam-4594	328	18	x	x	X
ejpam-4594	328	19	]	]	X
ejpam-4594	328	20	⊆	⊆	NUM
ejpam-4594	328	21	v	v	NOUN
ejpam-4594	328	22	and	and	CCONJ
ejpam-4594	328	23	w	w	NOUN
ejpam-4594	328	24	′[x	′[x	PROPN
ejpam-4594	328	25	◦	◦	NOUN
ejpam-4594	328	26	a	a	DET
ejpam-4594	328	27	]	]	X
ejpam-4594	328	28	⊆	⊆	NUM
ejpam-4594	328	29	u	u	NOUN
ejpam-4594	328	30	′	′	NUM
ejpam-4594	328	31	β[x	β[x	NOUN
ejpam-4594	328	32	◦	◦	NOUN
ejpam-4594	328	33	a	a	DET
ejpam-4594	328	34	]	]	X
ejpam-4594	328	35	⊆	⊆	NUM
ejpam-4594	328	36	v	v	NOUN
ejpam-4594	328	37	.	.	PUNCT
ejpam-4594	329	1	claim	claim	NOUN
ejpam-4594	329	2	:	:	PUNCT
ejpam-4594	329	3	w	w	X
ejpam-4594	329	4	′[x	′[x	NOUN
ejpam-4594	329	5	]	]	X
ejpam-4594	329	6	⊆	⊆	NUM
ejpam-4594	329	7	v	v	ADP
ejpam-4594	329	8	′[a	′[a	NOUN
ejpam-4594	329	9	]	]	PUNCT
ejpam-4594	329	10	.	.	PUNCT
ejpam-4594	330	1	suppose	suppose	VERB
ejpam-4594	330	2	y	y	PROPN
ejpam-4594	330	3	∈	∈	PROPN
ejpam-4594	330	4	w	w	PROPN
ejpam-4594	330	5	′[x	′[x	NOUN
ejpam-4594	330	6	]	]	PUNCT
ejpam-4594	330	7	.	.	PUNCT
ejpam-4594	331	1	then	then	ADV
ejpam-4594	331	2	x	x	PUNCT
ejpam-4594	331	3	◦	◦	NOUN
ejpam-4594	331	4	y	y	PROPN
ejpam-4594	331	5	,	,	PUNCT
ejpam-4594	331	6	y	y	PROPN
ejpam-4594	331	7	◦	◦	NOUN
ejpam-4594	331	8	x	x	PUNCT
ejpam-4594	331	9	∈	∈	PROPN
ejpam-4594	331	10	w	w	NOUN
ejpam-4594	331	11	.	.	PUNCT
ejpam-4594	332	1	by	by	ADP
ejpam-4594	332	2	(	(	PUNCT
ejpam-4594	332	3	db1	db1	NOUN
ejpam-4594	332	4	)	)	PUNCT
ejpam-4594	332	5	and	and	CCONJ
ejpam-4594	332	6	theorem	theorem	VERB
ejpam-4594	332	7	1	1	NUM
ejpam-4594	332	8	(	(	PUNCT
ejpam-4594	332	9	iii	iii	NOUN
ejpam-4594	332	10	)	)	PUNCT
ejpam-4594	332	11	,	,	PUNCT
ejpam-4594	332	12	1	1	X
ejpam-4594	332	13	=	=	SYM
ejpam-4594	332	14	(	(	PUNCT
ejpam-4594	332	15	x	x	SYM
ejpam-4594	332	16	◦	◦	VERB
ejpam-4594	332	17	y	y	NOUN
ejpam-4594	332	18	)	)	PUNCT
ejpam-4594	332	19	◦	◦	NOUN
ejpam-4594	332	20	k.	k.	PROPN
ejpam-4594	332	21	belleza	belleza	PROPN
ejpam-4594	332	22	,	,	PUNCT
ejpam-4594	332	23	j.	j.	PROPN
ejpam-4594	332	24	albaracin	albaracin	PROPN
ejpam-4594	332	25	/	/	PUNCT
ejpam-4594	332	26	eur	eur	PROPN
ejpam-4594	332	27	.	.	PUNCT
ejpam-4594	333	1	j.	j.	PROPN
ejpam-4594	333	2	pure	pure	PROPN
ejpam-4594	333	3	appl	appl	PROPN
ejpam-4594	333	4	.	.	PROPN
ejpam-4594	333	5	math	math	PROPN
ejpam-4594	333	6	,	,	PUNCT
ejpam-4594	333	7	15	15	NUM
ejpam-4594	333	8	(	(	PUNCT
ejpam-4594	333	9	4	4	NUM
ejpam-4594	333	10	)	)	PUNCT
ejpam-4594	333	11	(	(	PUNCT
ejpam-4594	333	12	2022	2022	NUM
ejpam-4594	333	13	)	)	PUNCT
ejpam-4594	333	14	,	,	PUNCT
ejpam-4594	333	15	1948	1948	NUM
ejpam-4594	333	16	-	-	SYM
ejpam-4594	333	17	1956	1956	NUM
ejpam-4594	333	18	1955	1955	NUM
ejpam-4594	333	19	(	(	PUNCT
ejpam-4594	333	20	x	x	X
ejpam-4594	333	21	◦	◦	NOUN
ejpam-4594	333	22	y	y	NOUN
ejpam-4594	333	23	)	)	PUNCT
ejpam-4594	333	24	=	=	PUNCT
ejpam-4594	334	1	[	[	X
ejpam-4594	334	2	(	(	PUNCT
ejpam-4594	334	3	a	a	DET
ejpam-4594	334	4	◦	◦	NOUN
ejpam-4594	334	5	x	x	NOUN
ejpam-4594	334	6	)	)	PUNCT
ejpam-4594	334	7	◦	◦	NOUN
ejpam-4594	334	8	(	(	PUNCT
ejpam-4594	334	9	a	a	DET
ejpam-4594	334	10	◦	◦	NOUN
ejpam-4594	334	11	y	y	NOUN
ejpam-4594	334	12	)	)	PUNCT
ejpam-4594	334	13	]	]	X
ejpam-4594	334	14	◦	◦	NOUN
ejpam-4594	334	15	(	(	PUNCT
ejpam-4594	334	16	x	x	X
ejpam-4594	334	17	◦	◦	NOUN
ejpam-4594	334	18	y	y	NUM
ejpam-4594	334	19	)	)	PUNCT
ejpam-4594	334	20	.	.	PUNCT
ejpam-4594	335	1	similarly	similarly	ADV
ejpam-4594	335	2	,	,	PUNCT
ejpam-4594	335	3	1	1	X
ejpam-4594	335	4	=	=	SYM
ejpam-4594	335	5	(	(	PUNCT
ejpam-4594	335	6	y	y	PROPN
ejpam-4594	335	7	◦	◦	PROPN
ejpam-4594	335	8	x	x	NOUN
ejpam-4594	335	9	)	)	PUNCT
ejpam-4594	335	10	◦	◦	NOUN
ejpam-4594	335	11	(	(	PUNCT
ejpam-4594	335	12	y	y	PROPN
ejpam-4594	335	13	◦	◦	NOUN
ejpam-4594	335	14	x	x	NOUN
ejpam-4594	335	15	)	)	PUNCT
ejpam-4594	335	16	=	=	SYM
ejpam-4594	336	1	[	[	X
ejpam-4594	336	2	(	(	PUNCT
ejpam-4594	336	3	a	a	DET
ejpam-4594	336	4	◦	◦	NOUN
ejpam-4594	336	5	y	y	NOUN
ejpam-4594	336	6	)	)	PUNCT
ejpam-4594	336	7	◦	◦	NOUN
ejpam-4594	336	8	(	(	PUNCT
ejpam-4594	336	9	a	a	DET
ejpam-4594	336	10	◦	◦	NOUN
ejpam-4594	336	11	x	x	NOUN
ejpam-4594	336	12	)	)	PUNCT
ejpam-4594	336	13	]	]	X
ejpam-4594	336	14	◦	◦	NOUN
ejpam-4594	336	15	(	(	PUNCT
ejpam-4594	336	16	y	y	PROPN
ejpam-4594	336	17	◦	◦	PROPN
ejpam-4594	336	18	x	x	NOUN
ejpam-4594	336	19	)	)	PUNCT
ejpam-4594	336	20	.	.	PUNCT
ejpam-4594	337	1	hence	hence	ADV
ejpam-4594	337	2	by	by	ADP
ejpam-4594	337	3	(	(	PUNCT
ejpam-4594	337	4	db2	db2	PROPN
ejpam-4594	337	5	)	)	PUNCT
ejpam-4594	337	6	,	,	PUNCT
ejpam-4594	337	7	(	(	PUNCT
ejpam-4594	337	8	1	1	NUM
ejpam-4594	337	9	◦	◦	NOUN
ejpam-4594	337	10	[	[	X
ejpam-4594	337	11	(	(	PUNCT
ejpam-4594	337	12	a	a	DET
ejpam-4594	337	13	◦	◦	NOUN
ejpam-4594	337	14	x	x	NOUN
ejpam-4594	337	15	)	)	PUNCT
ejpam-4594	337	16	◦	◦	NOUN
ejpam-4594	337	17	(	(	PUNCT
ejpam-4594	337	18	a	a	DET
ejpam-4594	337	19	◦	◦	NOUN
ejpam-4594	337	20	y	y	NOUN
ejpam-4594	337	21	)	)	PUNCT
ejpam-4594	337	22	]	]	PUNCT
ejpam-4594	337	23	)	)	PUNCT
ejpam-4594	338	1	◦	◦	NOUN
ejpam-4594	338	2	(	(	PUNCT
ejpam-4594	338	3	x	x	PART
ejpam-4594	338	4	◦	◦	VERB
ejpam-4594	338	5	y	y	NOUN
ejpam-4594	338	6	)	)	PUNCT
ejpam-4594	338	7	=	=	SYM
ejpam-4594	338	8	1	1	NUM
ejpam-4594	338	9	and	and	CCONJ
ejpam-4594	338	10	(	(	PUNCT
ejpam-4594	338	11	1	1	NUM
ejpam-4594	338	12	◦	◦	NOUN
ejpam-4594	338	13	[	[	X
ejpam-4594	338	14	(	(	PUNCT
ejpam-4594	338	15	a	a	DET
ejpam-4594	338	16	◦	◦	NOUN
ejpam-4594	338	17	y	y	NOUN
ejpam-4594	338	18	)	)	PUNCT
ejpam-4594	338	19	◦	◦	NOUN
ejpam-4594	338	20	(	(	PUNCT
ejpam-4594	338	21	a	a	DET
ejpam-4594	338	22	◦	◦	NOUN
ejpam-4594	338	23	x	x	NOUN
ejpam-4594	338	24	)	)	PUNCT
ejpam-4594	338	25	]	]	PUNCT
ejpam-4594	338	26	)	)	PUNCT
ejpam-4594	339	1	◦	◦	NOUN
ejpam-4594	339	2	(	(	PUNCT
ejpam-4594	339	3	y	y	PROPN
ejpam-4594	339	4	◦	◦	NOUN
ejpam-4594	339	5	x	x	NOUN
ejpam-4594	339	6	)	)	PUNCT
ejpam-4594	339	7	=	=	SYM
ejpam-4594	339	8	1	1	X
ejpam-4594	339	9	.	.	PUNCT
ejpam-4594	339	10	by	by	ADP
ejpam-4594	339	11	theorem	theorem	ADJ
ejpam-4594	339	12	10	10	NUM
ejpam-4594	339	13	and	and	CCONJ
ejpam-4594	339	14	hypothesis	hypothesis	NOUN
ejpam-4594	339	15	(	(	PUNCT
ejpam-4594	339	16	ii	ii	NOUN
ejpam-4594	339	17	)	)	PUNCT
ejpam-4594	339	18	,	,	PUNCT
ejpam-4594	339	19	(	(	PUNCT
ejpam-4594	339	20	a	a	DET
ejpam-4594	339	21	◦	◦	NOUN
ejpam-4594	339	22	x	x	NOUN
ejpam-4594	339	23	)	)	PUNCT
ejpam-4594	339	24	◦	◦	NOUN
ejpam-4594	339	25	(	(	PUNCT
ejpam-4594	339	26	a	a	DET
ejpam-4594	339	27	◦	◦	NOUN
ejpam-4594	339	28	y	y	NOUN
ejpam-4594	339	29	)	)	PUNCT
ejpam-4594	339	30	∈	∈	PROPN
ejpam-4594	339	31	w	w	PROPN
ejpam-4594	339	32	and	and	CCONJ
ejpam-4594	339	33	(	(	PUNCT
ejpam-4594	339	34	a	a	DET
ejpam-4594	339	35	◦	◦	NOUN
ejpam-4594	339	36	y	y	NOUN
ejpam-4594	339	37	)	)	PUNCT
ejpam-4594	339	38	◦	◦	NOUN
ejpam-4594	339	39	(	(	PUNCT
ejpam-4594	339	40	a	a	DET
ejpam-4594	339	41	◦	◦	NOUN
ejpam-4594	339	42	x	x	SYM
ejpam-4594	339	43	)	)	PUNCT
ejpam-4594	339	44	∈	∈	PROPN
ejpam-4594	339	45	w	w	NOUN
ejpam-4594	339	46	.	.	PUNCT
ejpam-4594	340	1	this	this	PRON
ejpam-4594	340	2	implies	imply	VERB
ejpam-4594	340	3	that	that	SCONJ
ejpam-4594	340	4	a	a	DET
ejpam-4594	340	5	◦	◦	NOUN
ejpam-4594	340	6	y	y	PROPN
ejpam-4594	340	7	∈	∈	PROPN
ejpam-4594	340	8	w	w	PROPN
ejpam-4594	340	9	′[a	′[a	X
ejpam-4594	340	10	◦	◦	NOUN
ejpam-4594	340	11	x	x	X
ejpam-4594	340	12	]	]	X
ejpam-4594	340	13	⊆	⊆	NUM
ejpam-4594	340	14	u	u	NOUN
ejpam-4594	340	15	′	′	NOUN
ejpam-4594	340	16	α[a	α[a	ADJ
ejpam-4594	340	17	◦	◦	NOUN
ejpam-4594	340	18	x	x	X
ejpam-4594	340	19	]	]	X
ejpam-4594	340	20	⊆	⊆	NUM
ejpam-4594	340	21	v	v	NOUN
ejpam-4594	340	22	.	.	PUNCT
ejpam-4594	341	1	similarly	similarly	ADV
ejpam-4594	341	2	,	,	PUNCT
ejpam-4594	341	3	y	y	PROPN
ejpam-4594	341	4	◦	◦	VERB
ejpam-4594	341	5	a	a	DET
ejpam-4594	341	6	∈	∈	PROPN
ejpam-4594	341	7	w	w	PROPN
ejpam-4594	341	8	′[x	′[x	NOUN
ejpam-4594	341	9	◦	◦	NOUN
ejpam-4594	341	10	a	a	NOUN
ejpam-4594	341	11	]	]	X
ejpam-4594	341	12	⊆	⊆	NUM
ejpam-4594	341	13	u	u	NOUN
ejpam-4594	341	14	′	′	NUM
ejpam-4594	341	15	β[x	β[x	NOUN
ejpam-4594	341	16	◦	◦	NOUN
ejpam-4594	341	17	a	a	X
ejpam-4594	341	18	]	]	X
ejpam-4594	341	19	⊆	⊆	NUM
ejpam-4594	341	20	v	v	NOUN
ejpam-4594	341	21	.	.	PUNCT
ejpam-4594	342	1	it	it	PRON
ejpam-4594	342	2	follows	follow	VERB
ejpam-4594	342	3	that	that	SCONJ
ejpam-4594	342	4	y	y	PROPN
ejpam-4594	342	5	∈	∈	PROPN
ejpam-4594	342	6	v	v	ADP
ejpam-4594	342	7	′[a	′[a	NOUN
ejpam-4594	342	8	]	]	PUNCT
ejpam-4594	342	9	.	.	PUNCT
ejpam-4594	343	1	this	this	PRON
ejpam-4594	343	2	proves	prove	VERB
ejpam-4594	343	3	the	the	DET
ejpam-4594	343	4	claim	claim	NOUN
ejpam-4594	343	5	.	.	PUNCT
ejpam-4594	344	1	therefore	therefore	ADV
ejpam-4594	344	2	,	,	PUNCT
ejpam-4594	344	3	v	v	ADP
ejpam-4594	344	4	′[a	′[a	X
ejpam-4594	344	5	]	]	X
ejpam-4594	344	6	∈	∈	PROPN
ejpam-4594	344	7	τ	τ	X
ejpam-4594	344	8	.	.	PUNCT
ejpam-4594	345	1	that	that	PRON
ejpam-4594	345	2	is	be	AUX
ejpam-4594	345	3	,	,	PUNCT
ejpam-4594	345	4	v	v	ADP
ejpam-4594	345	5	′[a	′[a	NOUN
ejpam-4594	345	6	]	]	PUNCT
ejpam-4594	345	7	is	be	AUX
ejpam-4594	345	8	open	open	ADJ
ejpam-4594	345	9	in	in	ADP
ejpam-4594	345	10	xd	xd	ADV
ejpam-4594	345	11	for	for	ADP
ejpam-4594	345	12	all	all	DET
ejpam-4594	345	13	a	a	DET
ejpam-4594	345	14	∈	∈	NOUN
ejpam-4594	345	15	xd	xd	ADP
ejpam-4594	345	16	.	.	PUNCT
ejpam-4594	346	1	the	the	DET
ejpam-4594	346	2	next	next	ADJ
ejpam-4594	346	3	theorem	theorem	NOUN
ejpam-4594	346	4	identifies	identify	VERB
ejpam-4594	346	5	a	a	DET
ejpam-4594	346	6	dual	dual	ADJ
ejpam-4594	346	7	b	b	NOUN
ejpam-4594	346	8	-	-	PUNCT
ejpam-4594	346	9	topological	topological	ADJ
ejpam-4594	346	10	space	space	NOUN
ejpam-4594	346	11	determined	determine	VERB
ejpam-4594	346	12	by	by	ADP
ejpam-4594	346	13	a	a	DET
ejpam-4594	346	14	filterbase	filterbase	NOUN
ejpam-4594	346	15	to	to	PART
ejpam-4594	346	16	be	be	AUX
ejpam-4594	346	17	a	a	DET
ejpam-4594	346	18	tdb	tdb	NOUN
ejpam-4594	346	19	-	-	NOUN
ejpam-4594	346	20	algebra	algebra	NOUN
ejpam-4594	346	21	provided	provide	VERB
ejpam-4594	346	22	some	some	DET
ejpam-4594	346	23	conditions	condition	NOUN
ejpam-4594	346	24	.	.	PUNCT
ejpam-4594	347	1	theorem	theorem	NOUN
ejpam-4594	347	2	11	11	NUM
ejpam-4594	347	3	.	.	PUNCT
ejpam-4594	348	1	suppose	suppose	VERB
ejpam-4594	348	2	xd	xd	INTJ
ejpam-4594	348	3	is	be	VERB
ejpam-4594	348	4	a	a	DET
ejpam-4594	348	5	dual	dual	ADJ
ejpam-4594	348	6	b	b	NOUN
ejpam-4594	348	7	-	-	PUNCT
ejpam-4594	348	8	topological	topological	ADJ
ejpam-4594	348	9	space	space	NOUN
ejpam-4594	348	10	satisfying	satisfy	VERB
ejpam-4594	348	11	the	the	DET
ejpam-4594	348	12	symmetric	symmetric	ADJ
ejpam-4594	348	13	condition	condition	NOUN
ejpam-4594	348	14	and	and	CCONJ
ejpam-4594	348	15	ω	ω	NUM
ejpam-4594	348	16	a	a	DET
ejpam-4594	348	17	filterbase	filterbase	NOUN
ejpam-4594	348	18	in	in	ADP
ejpam-4594	348	19	xd	xd	INTJ
ejpam-4594	348	20	such	such	ADJ
ejpam-4594	348	21	that	that	PRON
ejpam-4594	348	22	for	for	ADP
ejpam-4594	348	23	all	all	PRON
ejpam-4594	348	24	v	v	ADP
ejpam-4594	348	25	∈	∈	PROPN
ejpam-4594	348	26	ω	ω	NOUN
ejpam-4594	348	27	and	and	CCONJ
ejpam-4594	348	28	for	for	ADP
ejpam-4594	348	29	all	all	DET
ejpam-4594	348	30	p	p	NOUN
ejpam-4594	348	31	,	,	PUNCT
ejpam-4594	348	32	q	q	PROPN
ejpam-4594	348	33	∈	∈	PROPN
ejpam-4594	348	34	v	v	NOUN
ejpam-4594	348	35	,	,	PUNCT
ejpam-4594	348	36	(	(	PUNCT
ejpam-4594	348	37	i	i	NOUN
ejpam-4594	348	38	)	)	PUNCT
ejpam-4594	348	39	p	p	NOUN
ejpam-4594	348	40	◦	◦	NOUN
ejpam-4594	348	41	1	1	NUM
ejpam-4594	348	42	∈	∈	NOUN
ejpam-4594	348	43	v	v	NOUN
ejpam-4594	348	44	;	;	PUNCT
ejpam-4594	348	45	and	and	CCONJ
ejpam-4594	348	46	(	(	PUNCT
ejpam-4594	348	47	ii	ii	NOUN
ejpam-4594	348	48	)	)	PUNCT
ejpam-4594	348	49	(	(	PUNCT
ejpam-4594	348	50	p	p	NOUN
ejpam-4594	348	51	◦	◦	NOUN
ejpam-4594	348	52	x	x	NOUN
ejpam-4594	348	53	)	)	PUNCT
ejpam-4594	348	54	◦	◦	NOUN
ejpam-4594	348	55	q	q	NOUN
ejpam-4594	348	56	=	=	SYM
ejpam-4594	348	57	1	1	NUM
ejpam-4594	348	58	implies	imply	VERB
ejpam-4594	348	59	x	x	X
ejpam-4594	348	60	∈	∈	NOUN
ejpam-4594	348	61	v	v	NOUN
ejpam-4594	348	62	.	.	PUNCT
ejpam-4594	349	1	then	then	ADV
ejpam-4594	349	2	xd	xd	INTJ
ejpam-4594	349	3	is	be	VERB
ejpam-4594	349	4	a	a	DET
ejpam-4594	349	5	tdb	tdb	NOUN
ejpam-4594	349	6	-	-	NOUN
ejpam-4594	349	7	algebra	algebra	NOUN
ejpam-4594	349	8	.	.	PUNCT
ejpam-4594	350	1	proof	proof	NOUN
ejpam-4594	350	2	.	.	PUNCT
ejpam-4594	351	1	suppose	suppose	VERB
ejpam-4594	351	2	xd	xd	INTJ
ejpam-4594	351	3	is	be	AUX
ejpam-4594	351	4	a	a	DET
ejpam-4594	351	5	dual	dual	ADJ
ejpam-4594	351	6	b	b	NOUN
ejpam-4594	351	7	-	-	PUNCT
ejpam-4594	351	8	topological	topological	ADJ
ejpam-4594	351	9	space	space	NOUN
ejpam-4594	351	10	satisfying	satisfy	VERB
ejpam-4594	351	11	the	the	DET
ejpam-4594	351	12	symmetric	symmetric	ADJ
ejpam-4594	351	13	condition	condition	NOUN
ejpam-4594	351	14	and	and	CCONJ
ejpam-4594	351	15	ω	ω	NUM
ejpam-4594	351	16	a	a	DET
ejpam-4594	351	17	filterbase	filterbase	NOUN
ejpam-4594	351	18	in	in	ADP
ejpam-4594	351	19	xd	xd	ADP
ejpam-4594	351	20	.	.	PUNCT
ejpam-4594	352	1	let	let	VERB
ejpam-4594	352	2	x	x	PART
ejpam-4594	352	3	◦	◦	VERB
ejpam-4594	352	4	y	y	NOUN
ejpam-4594	352	5	∈	∈	NOUN
ejpam-4594	352	6	o	o	X
ejpam-4594	352	7	∈	∈	PROPN
ejpam-4594	352	8	τ	τ	X
ejpam-4594	352	9	for	for	ADP
ejpam-4594	352	10	any	any	DET
ejpam-4594	352	11	x	x	NOUN
ejpam-4594	352	12	,	,	PUNCT
ejpam-4594	352	13	y	y	PROPN
ejpam-4594	352	14	∈	∈	PROPN
ejpam-4594	352	15	xd	xd	INTJ
ejpam-4594	352	16	.	.	PUNCT
ejpam-4594	353	1	by	by	ADP
ejpam-4594	353	2	theorem	theorem	NOUN
ejpam-4594	353	3	8	8	NUM
ejpam-4594	353	4	,	,	PUNCT
ejpam-4594	353	5	there	there	PRON
ejpam-4594	353	6	exists	exist	VERB
ejpam-4594	353	7	v	v	ADP
ejpam-4594	353	8	∈	∈	PROPN
ejpam-4594	353	9	ω	ω	NOUN
ejpam-4594	353	10	such	such	ADJ
ejpam-4594	353	11	that	that	DET
ejpam-4594	353	12	v	v	NOUN
ejpam-4594	353	13	′[x	′[x	NOUN
ejpam-4594	353	14	◦	◦	NOUN
ejpam-4594	353	15	y	y	X
ejpam-4594	353	16	]	]	X
ejpam-4594	353	17	⊆	⊆	NUM
ejpam-4594	353	18	o.	o.	NOUN
ejpam-4594	353	19	note	note	VERB
ejpam-4594	353	20	that	that	SCONJ
ejpam-4594	353	21	by	by	ADP
ejpam-4594	353	22	lemma	lemma	PROPN
ejpam-4594	353	23	4	4	NUM
ejpam-4594	353	24	,	,	PUNCT
ejpam-4594	353	25	theorem	theorem	VERB
ejpam-4594	353	26	10	10	NUM
ejpam-4594	353	27	,	,	PUNCT
ejpam-4594	353	28	and	and	CCONJ
ejpam-4594	353	29	proposition	proposition	NOUN
ejpam-4594	353	30	2	2	NUM
ejpam-4594	353	31	,	,	PUNCT
ejpam-4594	353	32	v	v	ADP
ejpam-4594	353	33	′[x	′[x	NOUN
ejpam-4594	353	34	]	]	PUNCT
ejpam-4594	353	35	,	,	PUNCT
ejpam-4594	353	36	v	v	ADP
ejpam-4594	353	37	′[y	′[y	PROPN
ejpam-4594	353	38	]	]	X
ejpam-4594	353	39	∈	∈	PROPN
ejpam-4594	353	40	τ	τ	X
ejpam-4594	353	41	with	with	ADP
ejpam-4594	353	42	x	x	PROPN
ejpam-4594	353	43	∈	∈	PROPN
ejpam-4594	353	44	v	v	ADP
ejpam-4594	353	45	′[x	′[x	NOUN
ejpam-4594	353	46	]	]	PUNCT
ejpam-4594	353	47	and	and	CCONJ
ejpam-4594	353	48	y	y	PROPN
ejpam-4594	353	49	∈	∈	PROPN
ejpam-4594	353	50	v	v	ADP
ejpam-4594	353	51	′[y	′[y	PROPN
ejpam-4594	353	52	]	]	PUNCT
ejpam-4594	353	53	.	.	PUNCT
ejpam-4594	354	1	by	by	ADP
ejpam-4594	354	2	proposition	proposition	NOUN
ejpam-4594	354	3	4	4	NUM
ejpam-4594	354	4	,	,	PUNCT
ejpam-4594	354	5	v	v	ADP
ejpam-4594	354	6	′[x	′[x	NOUN
ejpam-4594	354	7	]	]	PUNCT
ejpam-4594	354	8	◦	◦	NOUN
ejpam-4594	354	9	v	v	ADP
ejpam-4594	354	10	′[y	′[y	PROPN
ejpam-4594	354	11	]	]	PUNCT
ejpam-4594	354	12	⊆	⊆	NUM
ejpam-4594	354	13	o.	o.	NOUN
ejpam-4594	354	14	therefore	therefore	ADV
ejpam-4594	354	15	by	by	ADP
ejpam-4594	354	16	theorem	theorem	NOUN
ejpam-4594	354	17	3	3	NUM
ejpam-4594	354	18	,	,	PUNCT
ejpam-4594	354	19	xd	xd	INTJ
ejpam-4594	354	20	is	be	AUX
ejpam-4594	354	21	a	a	DET
ejpam-4594	354	22	tdb	tdb	NOUN
ejpam-4594	354	23	-	-	NOUN
ejpam-4594	354	24	algebra	algebra	NOUN
ejpam-4594	354	25	.	.	PUNCT
ejpam-4594	355	1	the	the	DET
ejpam-4594	355	2	last	last	ADJ
ejpam-4594	355	3	corollary	corollary	NOUN
ejpam-4594	355	4	follows	follow	VERB
ejpam-4594	355	5	from	from	ADP
ejpam-4594	355	6	theorem	theorem	ADJ
ejpam-4594	355	7	11	11	NUM
ejpam-4594	355	8	and	and	CCONJ
ejpam-4594	355	9	definition	definition	NOUN
ejpam-4594	355	10	4	4	NUM
ejpam-4594	355	11	of	of	ADP
ejpam-4594	355	12	a	a	DET
ejpam-4594	355	13	dual	dual	ADJ
ejpam-4594	355	14	b	b	NOUN
ejpam-4594	355	15	-	-	NOUN
ejpam-4594	355	16	filter	filter	NOUN
ejpam-4594	355	17	.	.	PUNCT
ejpam-4594	356	1	corollary	corollary	ADJ
ejpam-4594	356	2	2	2	PROPN
ejpam-4594	356	3	.	.	PUNCT
ejpam-4594	356	4	suppose	suppose	VERB
ejpam-4594	356	5	xd	xd	INTJ
ejpam-4594	356	6	is	be	AUX
ejpam-4594	356	7	a	a	DET
ejpam-4594	356	8	dual	dual	ADJ
ejpam-4594	356	9	b	b	NOUN
ejpam-4594	356	10	-	-	PUNCT
ejpam-4594	356	11	topological	topological	ADJ
ejpam-4594	356	12	space	space	NOUN
ejpam-4594	356	13	satisfying	satisfy	VERB
ejpam-4594	356	14	the	the	DET
ejpam-4594	356	15	symmetric	symmetric	ADJ
ejpam-4594	356	16	condition	condition	NOUN
ejpam-4594	356	17	and	and	CCONJ
ejpam-4594	356	18	ω	ω	NUM
ejpam-4594	356	19	a	a	DET
ejpam-4594	356	20	filterbase	filterbase	NOUN
ejpam-4594	356	21	in	in	ADP
ejpam-4594	356	22	xd	xd	INTJ
ejpam-4594	356	23	such	such	ADJ
ejpam-4594	356	24	that	that	PRON
ejpam-4594	356	25	for	for	ADP
ejpam-4594	356	26	all	all	DET
ejpam-4594	356	27	v	v	ADP
ejpam-4594	356	28	∈	∈	PROPN
ejpam-4594	356	29	ω	ω	NOUN
ejpam-4594	356	30	,	,	PUNCT
ejpam-4594	356	31	v	v	NOUN
ejpam-4594	356	32	is	be	AUX
ejpam-4594	356	33	a	a	DET
ejpam-4594	356	34	dual	dual	ADJ
ejpam-4594	356	35	b	b	NOUN
ejpam-4594	356	36	-	-	NOUN
ejpam-4594	356	37	filter	filter	NOUN
ejpam-4594	356	38	.	.	PUNCT
ejpam-4594	357	1	then	then	ADV
ejpam-4594	357	2	xd	xd	INTJ
ejpam-4594	357	3	is	be	VERB
ejpam-4594	357	4	a	a	DET
ejpam-4594	357	5	tdb	tdb	NOUN
ejpam-4594	357	6	-	-	NOUN
ejpam-4594	357	7	algebra	algebra	NOUN
ejpam-4594	357	8	.	.	PUNCT
ejpam-4594	358	1	5	5	X
ejpam-4594	358	2	.	.	X
ejpam-4594	358	3	conclusion	conclusion	NOUN
ejpam-4594	358	4	given	give	VERB
ejpam-4594	358	5	a	a	DET
ejpam-4594	358	6	dual	dual	ADJ
ejpam-4594	358	7	b	b	NOUN
ejpam-4594	358	8	-	-	PUNCT
ejpam-4594	358	9	algebra	algebra	NOUN
ejpam-4594	358	10	xd	xd	INTJ
ejpam-4594	358	11	and	and	CCONJ
ejpam-4594	358	12	a	a	DET
ejpam-4594	358	13	family	family	NOUN
ejpam-4594	358	14	ω	ω	NOUN
ejpam-4594	358	15	of	of	ADP
ejpam-4594	358	16	nonempty	nonempty	ADJ
ejpam-4594	358	17	subsets	subset	NOUN
ejpam-4594	358	18	of	of	ADP
ejpam-4594	358	19	xd	xd	INTJ
ejpam-4594	358	20	that	that	PRON
ejpam-4594	358	21	is	be	AUX
ejpam-4594	358	22	closed	close	VERB
ejpam-4594	358	23	under	under	ADP
ejpam-4594	358	24	finite	finite	ADJ
ejpam-4594	358	25	intersection	intersection	NOUN
ejpam-4594	358	26	,	,	PUNCT
ejpam-4594	358	27	we	we	PRON
ejpam-4594	358	28	can	can	AUX
ejpam-4594	358	29	construct	construct	VERB
ejpam-4594	358	30	a	a	DET
ejpam-4594	358	31	dual	dual	ADJ
ejpam-4594	358	32	b	b	NOUN
ejpam-4594	358	33	-	-	NOUN
ejpam-4594	358	34	topology	topology	NOUN
ejpam-4594	358	35	on	on	ADP
ejpam-4594	358	36	xd	xd	INTJ
ejpam-4594	358	37	given	give	VERB
ejpam-4594	358	38	by	by	ADP
ejpam-4594	358	39	τ	τ	X
ejpam-4594	358	40	=	=	PUNCT
ejpam-4594	358	41	{	{	PUNCT
ejpam-4594	358	42	u	u	NOUN
ejpam-4594	358	43	⊆	⊆	NUM
ejpam-4594	358	44	xd	xd	INTJ
ejpam-4594	358	45	|	|	ADV
ejpam-4594	358	46	∀x	∀x	X
ejpam-4594	358	47	∈	∈	PROPN
ejpam-4594	358	48	u,∃v	u,∃v	ADJ
ejpam-4594	358	49	∈	∈	PROPN
ejpam-4594	358	50	ω	ω	NUM
ejpam-4594	358	51	such	such	ADJ
ejpam-4594	358	52	that	that	PRON
ejpam-4594	358	53	v	v	PART
ejpam-4594	358	54	[	[	X
ejpam-4594	358	55	x	x	X
ejpam-4594	358	56	]	]	X
ejpam-4594	358	57	⊆	⊆	NUM
ejpam-4594	358	58	u	u	NOUN
ejpam-4594	358	59	}	}	PUNCT
ejpam-4594	358	60	.	.	PUNCT
ejpam-4594	359	1	if	if	SCONJ
ejpam-4594	359	2	the	the	DET
ejpam-4594	359	3	empty	empty	ADJ
ejpam-4594	359	4	set	set	NOUN
ejpam-4594	359	5	is	be	AUX
ejpam-4594	359	6	a	a	DET
ejpam-4594	359	7	member	member	NOUN
ejpam-4594	359	8	of	of	ADP
ejpam-4594	359	9	ω	ω	PROPN
ejpam-4594	359	10	,	,	PUNCT
ejpam-4594	359	11	then	then	ADV
ejpam-4594	359	12	xd	xd	INTJ
ejpam-4594	359	13	is	be	VERB
ejpam-4594	359	14	a	a	DET
ejpam-4594	359	15	tdb	tdb	PROPN
ejpam-4594	359	16	-	-	NOUN
ejpam-4594	359	17	algbera	algbera	NOUN
ejpam-4594	359	18	.	.	PUNCT
ejpam-4594	360	1	furthermore	furthermore	ADV
ejpam-4594	360	2	,	,	PUNCT
ejpam-4594	360	3	if	if	SCONJ
ejpam-4594	360	4	ω	ω	PROPN
ejpam-4594	360	5	is	be	AUX
ejpam-4594	360	6	a	a	DET
ejpam-4594	360	7	filterbase	filterbase	NOUN
ejpam-4594	360	8	of	of	ADP
ejpam-4594	360	9	xd	xd	ADP
ejpam-4594	360	10	,	,	PUNCT
ejpam-4594	360	11	then	then	ADV
ejpam-4594	360	12	τ	τ	X
ejpam-4594	360	13	=	=	PUNCT
ejpam-4594	360	14	{	{	PUNCT
ejpam-4594	360	15	o	o	NOUN
ejpam-4594	360	16	⊆	⊆	NUM
ejpam-4594	360	17	xd	xd	INTJ
ejpam-4594	360	18	|	|	ADV
ejpam-4594	360	19	∀a	∀a	NOUN
ejpam-4594	360	20	∈	∈	NOUN
ejpam-4594	360	21	o,∃v	o,∃v	VERB
ejpam-4594	360	22	∈	∈	PROPN
ejpam-4594	360	23	ω	ω	NOUN
ejpam-4594	360	24	such	such	ADJ
ejpam-4594	360	25	that	that	SCONJ
ejpam-4594	360	26	v	v	ADP
ejpam-4594	360	27	′[a	′[a	ADJ
ejpam-4594	360	28	]	]	PUNCT
ejpam-4594	360	29	⊆	⊆	NUM
ejpam-4594	360	30	o	o	NOUN
ejpam-4594	360	31	}	}	PUNCT
ejpam-4594	360	32	is	be	AUX
ejpam-4594	360	33	also	also	ADV
ejpam-4594	360	34	a	a	DET
ejpam-4594	360	35	dual	dual	ADJ
ejpam-4594	360	36	b	b	NOUN
ejpam-4594	360	37	-	-	NOUN
ejpam-4594	360	38	topology	topology	NOUN
ejpam-4594	360	39	on	on	ADP
ejpam-4594	360	40	xd	xd	ADP
ejpam-4594	360	41	.	.	PUNCT
ejpam-4594	361	1	if	if	SCONJ
ejpam-4594	361	2	the	the	DET
ejpam-4594	361	3	condition	condition	NOUN
ejpam-4594	361	4	is	be	AUX
ejpam-4594	361	5	imposed	impose	VERB
ejpam-4594	361	6	to	to	ADP
ejpam-4594	361	7	ω	ω	NUM
ejpam-4594	361	8	such	such	ADJ
ejpam-4594	361	9	that	that	SCONJ
ejpam-4594	361	10	for	for	ADP
ejpam-4594	361	11	all	all	PRON
ejpam-4594	361	12	v	v	ADP
ejpam-4594	361	13	∈	∈	PROPN
ejpam-4594	361	14	ω	ω	NOUN
ejpam-4594	361	15	and	and	CCONJ
ejpam-4594	361	16	for	for	ADP
ejpam-4594	361	17	all	all	DET
ejpam-4594	361	18	p	p	NOUN
ejpam-4594	361	19	,	,	PUNCT
ejpam-4594	361	20	q	q	PROPN
ejpam-4594	361	21	∈	∈	PROPN
ejpam-4594	361	22	v	v	NOUN
ejpam-4594	361	23	,	,	PUNCT
ejpam-4594	361	24	(	(	PUNCT
ejpam-4594	361	25	i	i	NOUN
ejpam-4594	361	26	)	)	PUNCT
ejpam-4594	361	27	p	p	NOUN
ejpam-4594	361	28	◦	◦	NOUN
ejpam-4594	361	29	1	1	NUM
ejpam-4594	361	30	∈	∈	NOUN
ejpam-4594	361	31	v	v	NOUN
ejpam-4594	361	32	;	;	PUNCT
ejpam-4594	361	33	and	and	CCONJ
ejpam-4594	361	34	(	(	PUNCT
ejpam-4594	361	35	ii	ii	NOUN
ejpam-4594	361	36	)	)	PUNCT
ejpam-4594	361	37	(	(	PUNCT
ejpam-4594	361	38	p	p	NOUN
ejpam-4594	361	39	◦	◦	NOUN
ejpam-4594	361	40	x	x	NOUN
ejpam-4594	361	41	)	)	PUNCT
ejpam-4594	361	42	◦	◦	NOUN
ejpam-4594	362	1	q	q	NOUN
ejpam-4594	362	2	=	=	SYM
ejpam-4594	362	3	1	1	NUM
ejpam-4594	362	4	implies	imply	VERB
ejpam-4594	362	5	x	x	X
ejpam-4594	362	6	∈	∈	NOUN
ejpam-4594	362	7	v	v	NOUN
ejpam-4594	362	8	,	,	PUNCT
ejpam-4594	362	9	then	then	ADV
ejpam-4594	362	10	xd	xd	INTJ
ejpam-4594	362	11	is	be	VERB
ejpam-4594	362	12	a	a	DET
ejpam-4594	362	13	tdb	tdb	NOUN
ejpam-4594	362	14	-	-	NOUN
ejpam-4594	362	15	algebra	algebra	NOUN
ejpam-4594	362	16	.	.	PUNCT
ejpam-4594	363	1	generally	generally	ADV
ejpam-4594	363	2	,	,	PUNCT
ejpam-4594	363	3	in	in	ADP
ejpam-4594	363	4	this	this	DET
ejpam-4594	363	5	paper	paper	NOUN
ejpam-4594	363	6	we	we	PRON
ejpam-4594	363	7	constructed	construct	VERB
ejpam-4594	363	8	two	two	NUM
ejpam-4594	363	9	dual	dual	ADJ
ejpam-4594	363	10	b	b	NOUN
ejpam-4594	363	11	-	-	PUNCT
ejpam-4594	363	12	topologies	topology	NOUN
ejpam-4594	363	13	on	on	ADP
ejpam-4594	363	14	xd	xd	ADV
ejpam-4594	363	15	and	and	CCONJ
ejpam-4594	363	16	proved	prove	VERB
ejpam-4594	363	17	with	with	ADP
ejpam-4594	363	18	some	some	DET
ejpam-4594	363	19	conditions	condition	NOUN
ejpam-4594	363	20	that	that	PRON
ejpam-4594	363	21	xd	xd	INTJ
ejpam-4594	363	22	with	with	ADP
ejpam-4594	363	23	these	these	DET
ejpam-4594	363	24	topologies	topology	NOUN
ejpam-4594	363	25	is	be	AUX
ejpam-4594	363	26	a	a	DET
ejpam-4594	363	27	tdb	tdb	NOUN
ejpam-4594	363	28	-	-	NOUN
ejpam-4594	363	29	algebra	algebra	NOUN
ejpam-4594	363	30	.	.	PUNCT
ejpam-4594	364	1	acknowledgement	acknowledgement	NOUN
ejpam-4594	364	2	this	this	DET
ejpam-4594	364	3	research	research	NOUN
ejpam-4594	364	4	is	be	AUX
ejpam-4594	364	5	financially	financially	ADV
ejpam-4594	364	6	supported	support	VERB
ejpam-4594	364	7	through	through	ADP
ejpam-4594	364	8	an	an	DET
ejpam-4594	364	9	approved	approve	VERB
ejpam-4594	364	10	research	research	NOUN
ejpam-4594	364	11	load	load	NOUN
ejpam-4594	364	12	by	by	ADP
ejpam-4594	364	13	the	the	DET
ejpam-4594	364	14	research	research	NOUN
ejpam-4594	364	15	,	,	PUNCT
ejpam-4594	364	16	development	development	NOUN
ejpam-4594	364	17	,	,	PUNCT
ejpam-4594	364	18	extension	extension	NOUN
ejpam-4594	364	19	and	and	CCONJ
ejpam-4594	364	20	publications	publication	NOUN
ejpam-4594	364	21	office	office	NOUN
ejpam-4594	364	22	(	(	PUNCT
ejpam-4594	364	23	rdepo	rdepo	NOUN
ejpam-4594	364	24	)	)	PUNCT
ejpam-4594	364	25	of	of	ADP
ejpam-4594	364	26	the	the	DET
ejpam-4594	364	27	university	university	PROPN
ejpam-4594	364	28	of	of	ADP
ejpam-4594	364	29	san	san	PROPN
ejpam-4594	364	30	carlos	carlos	PROPN
ejpam-4594	364	31	during	during	ADP
ejpam-4594	364	32	the	the	DET
ejpam-4594	364	33	2nd	2nd	ADJ
ejpam-4594	364	34	term	term	NOUN
ejpam-4594	364	35	of	of	ADP
ejpam-4594	364	36	a.y	a.y	PROPN
ejpam-4594	364	37	.	.	PROPN
ejpam-4594	364	38	2020	2020	NUM
ejpam-4594	364	39	-	-	SYM
ejpam-4594	364	40	2021	2021	NUM
ejpam-4594	364	41	.	.	PUNCT
ejpam-4594	365	1	the	the	DET
ejpam-4594	365	2	author	author	NOUN
ejpam-4594	365	3	would	would	AUX
ejpam-4594	365	4	like	like	VERB
ejpam-4594	365	5	to	to	PART
ejpam-4594	365	6	extend	extend	VERB
ejpam-4594	365	7	their	their	PRON
ejpam-4594	365	8	sincerest	sincere	ADJ
ejpam-4594	365	9	gratitude	gratitude	NOUN
ejpam-4594	365	10	for	for	ADP
ejpam-4594	365	11	this	this	DET
ejpam-4594	365	12	support	support	NOUN
ejpam-4594	365	13	.	.	PUNCT
ejpam-4594	366	1	references	reference	NOUN
ejpam-4594	366	2	1956	1956	NUM
ejpam-4594	366	3	references	reference	NOUN
ejpam-4594	366	4	[	[	X
ejpam-4594	366	5	1	1	NUM
ejpam-4594	366	6	]	]	X
ejpam-4594	366	7	belleza	belleza	PROPN
ejpam-4594	366	8	,	,	PUNCT
ejpam-4594	366	9	k.	k.	PROPN
ejpam-4594	366	10	,	,	PUNCT
ejpam-4594	366	11	and	and	CCONJ
ejpam-4594	366	12	albaracin	albaracin	PROPN
ejpam-4594	366	13	,	,	PUNCT
ejpam-4594	366	14	j.	j.	PROPN
ejpam-4594	366	15	,	,	PUNCT
ejpam-4594	366	16	on	on	ADP
ejpam-4594	366	17	dual	dual	ADJ
ejpam-4594	366	18	b	b	NOUN
ejpam-4594	366	19	-	-	PUNCT
ejpam-4594	366	20	filters	filter	NOUN
ejpam-4594	366	21	and	and	CCONJ
ejpam-4594	366	22	dual	dual	ADJ
ejpam-4594	366	23	b	b	NOUN
ejpam-4594	366	24	-	-	PUNCT
ejpam-4594	366	25	subalgebras	subalgebras	PROPN
ejpam-4594	366	26	in	in	ADP
ejpam-4594	366	27	a	a	DET
ejpam-4594	366	28	topological	topological	ADJ
ejpam-4594	366	29	dual	dual	ADJ
ejpam-4594	366	30	b	b	NOUN
ejpam-4594	366	31	-	-	PUNCT
ejpam-4594	366	32	algebra	algebra	NOUN
ejpam-4594	366	33	,	,	PUNCT
ejpam-4594	366	34	journal	journal	NOUN
ejpam-4594	366	35	of	of	ADP
ejpam-4594	366	36	mathematics	mathematic	NOUN
ejpam-4594	366	37	and	and	CCONJ
ejpam-4594	366	38	computer	computer	NOUN
ejpam-4594	366	39	science	science	NOUN
ejpam-4594	366	40	,	,	PUNCT
ejpam-4594	366	41	28	28	NUM
ejpam-4594	366	42	no.1	no.1	NOUN
ejpam-4594	366	43	(	(	PUNCT
ejpam-4594	366	44	2023	2023	NUM
ejpam-4594	366	45	)	)	PUNCT
ejpam-4594	366	46	,	,	PUNCT
ejpam-4594	366	47	1	1	NUM
ejpam-4594	366	48	-	-	SYM
ejpam-4594	366	49	10	10	NUM
ejpam-4594	366	50	.	.	PUNCT
ejpam-4594	367	1	[	[	X
ejpam-4594	367	2	2	2	NUM
ejpam-4594	367	3	]	]	X
ejpam-4594	367	4	belleza	belleza	PROPN
ejpam-4594	367	5	,	,	PUNCT
ejpam-4594	367	6	k.	k.	PROPN
ejpam-4594	367	7	and	and	CCONJ
ejpam-4594	367	8	vilela	vilela	PROPN
ejpam-4594	367	9	,	,	PUNCT
ejpam-4594	367	10	j.	j.	PROPN
ejpam-4594	367	11	,	,	PUNCT
ejpam-4594	367	12	the	the	DET
ejpam-4594	367	13	dual	dual	ADJ
ejpam-4594	367	14	b	b	NOUN
ejpam-4594	367	15	-	-	PUNCT
ejpam-4594	367	16	algebra	algebra	ADJ
ejpam-4594	367	17	,	,	PUNCT
ejpam-4594	367	18	european	european	ADJ
ejpam-4594	367	19	journal	journal	NOUN
ejpam-4594	367	20	of	of	ADP
ejpam-4594	367	21	pure	pure	ADJ
ejpam-4594	367	22	and	and	CCONJ
ejpam-4594	367	23	applied	applied	ADJ
ejpam-4594	367	24	mathematics	mathematic	NOUN
ejpam-4594	367	25	,	,	PUNCT
ejpam-4594	367	26	12	12	NUM
ejpam-4594	367	27	no.4	no.4	PROPN
ejpam-4594	367	28	(	(	PUNCT
ejpam-4594	367	29	2019	2019	NUM
ejpam-4594	367	30	)	)	PUNCT
ejpam-4594	367	31	,	,	PUNCT
ejpam-4594	367	32	1497	1497	NUM
ejpam-4594	367	33	-	-	SYM
ejpam-4594	367	34	1507	1507	NUM
ejpam-4594	367	35	.	.	PUNCT
ejpam-4594	368	1	[	[	X
ejpam-4594	368	2	3	3	NUM
ejpam-4594	368	3	]	]	PUNCT
ejpam-4594	368	4	dugunji	dugunji	NOUN
ejpam-4594	368	5	,	,	PUNCT
ejpam-4594	368	6	j.	j.	PROPN
ejpam-4594	368	7	,	,	PUNCT
ejpam-4594	368	8	topology	topology	NOUN
ejpam-4594	368	9	,	,	PUNCT
ejpam-4594	368	10	allyn	allyn	PROPN
ejpam-4594	368	11	and	and	CCONJ
ejpam-4594	368	12	bacon	bacon	PROPN
ejpam-4594	368	13	inc	inc	PROPN
ejpam-4594	368	14	.	.	PROPN
ejpam-4594	368	15	,	,	PUNCT
ejpam-4594	368	16	atlantic	atlantic	PROPN
ejpam-4594	368	17	avenue	avenue	PROPN
ejpam-4594	368	18	,	,	PUNCT
ejpam-4594	368	19	boston	boston	PROPN
ejpam-4594	368	20	(	(	PUNCT
ejpam-4594	368	21	1966	1966	NUM
ejpam-4594	368	22	)	)	PUNCT
ejpam-4594	368	23	.	.	PUNCT
ejpam-4594	369	1	[	[	X
ejpam-4594	369	2	4	4	NUM
ejpam-4594	369	3	]	]	X
ejpam-4594	369	4	jun	jun	PROPN
ejpam-4594	369	5	,	,	PUNCT
ejpam-4594	369	6	y.b	y.b	PROPN
ejpam-4594	369	7	.	.	PROPN
ejpam-4594	369	8	et	et	PROPN
ejpam-4594	369	9	al	al	PROPN
ejpam-4594	369	10	.	.	PROPN
ejpam-4594	369	11	,	,	PUNCT
ejpam-4594	369	12	on	on	ADP
ejpam-4594	369	13	topological	topological	ADJ
ejpam-4594	369	14	bci	bci	NOUN
ejpam-4594	369	15	-	-	PUNCT
ejpam-4594	369	16	algebras	algebra	NOUN
ejpam-4594	369	17	,	,	PUNCT
ejpam-4594	369	18	information	information	NOUN
ejpam-4594	369	19	sciences	science	NOUN
ejpam-4594	369	20	,	,	PUNCT
ejpam-4594	369	21	116	116	NUM
ejpam-4594	369	22	(	(	PUNCT
ejpam-4594	369	23	1999	1999	NUM
ejpam-4594	369	24	)	)	PUNCT
ejpam-4594	369	25	,	,	PUNCT
ejpam-4594	369	26	253	253	NUM
ejpam-4594	369	27	-	-	SYM
ejpam-4594	369	28	261	261	NUM
ejpam-4594	369	29	.	.	PUNCT
ejpam-4594	370	1	[	[	X
ejpam-4594	370	2	5	5	NUM
ejpam-4594	370	3	]	]	X
ejpam-4594	370	4	lee	lee	PROPN
ejpam-4594	370	5	,	,	PUNCT
ejpam-4594	370	6	d.s	d.s	PROPN
ejpam-4594	370	7	.	.	PROPN
ejpam-4594	370	8	and	and	CCONJ
ejpam-4594	370	9	ryu	ryu	PROPN
ejpam-4594	370	10	,	,	PUNCT
ejpam-4594	370	11	d.n	d.n	PROPN
ejpam-4594	370	12	.	.	PROPN
ejpam-4594	370	13	,	,	PUNCT
ejpam-4594	370	14	notes	note	NOUN
ejpam-4594	370	15	on	on	ADP
ejpam-4594	370	16	topological	topological	ADJ
ejpam-4594	370	17	bck	bck	PROPN
ejpam-4594	370	18	-	-	PUNCT
ejpam-4594	370	19	algebras	algebras	PROPN
ejpam-4594	370	20	,	,	PUNCT
ejpam-4594	370	21	scientiae	scientiae	NOUN
ejpam-4594	370	22	mathematicae	mathematicae	PROPN
ejpam-4594	370	23	japonicae	japonicae	PROPN
ejpam-4594	370	24	,	,	PUNCT
ejpam-4594	370	25	1	1	NUM
ejpam-4594	370	26	no	no	NOUN
ejpam-4594	370	27	.	.	NOUN
ejpam-4594	370	28	2	2	NUM
ejpam-4594	370	29	(	(	PUNCT
ejpam-4594	370	30	1998	1998	NUM
ejpam-4594	370	31	)	)	PUNCT
ejpam-4594	370	32	,	,	PUNCT
ejpam-4594	370	33	231	231	NUM
ejpam-4594	370	34	-	-	SYM
ejpam-4594	370	35	235	235	NUM
ejpam-4594	370	36	.	.	PUNCT