id	sid	tid	token	lemma	pos
ejpam-3856	1	1	european	european	PROPN
ejpam-3856	1	2	journal	journal	PROPN
ejpam-3856	1	3	of	of	ADP
ejpam-3856	1	4	pure	pure	ADJ
ejpam-3856	1	5	and	and	CCONJ
ejpam-3856	1	6	applied	apply	VERB
ejpam-3856	1	7	mathematics	mathematic	NOUN
ejpam-3856	1	8	vol	vol	NOUN
ejpam-3856	1	9	.	.	PROPN
ejpam-3856	2	1	13	13	NUM
ejpam-3856	2	2	,	,	PUNCT
ejpam-3856	2	3	no	no	INTJ
ejpam-3856	2	4	.	.	NOUN
ejpam-3856	2	5	4	4	NUM
ejpam-3856	2	6	,	,	PUNCT
ejpam-3856	2	7	2020	2020	NUM
ejpam-3856	2	8	,	,	PUNCT
ejpam-3856	2	9	758	758	NUM
ejpam-3856	2	10	-	-	SYM
ejpam-3856	2	11	765	765	NUM
ejpam-3856	2	12	issn	issn	PROPN
ejpam-3856	2	13	1307	1307	NUM
ejpam-3856	2	14	-	-	SYM
ejpam-3856	2	15	5543	5543	NUM
ejpam-3856	2	16	–	–	PUNCT
ejpam-3856	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3856	2	18	published	publish	VERB
ejpam-3856	2	19	by	by	ADP
ejpam-3856	2	20	new	new	PROPN
ejpam-3856	2	21	york	york	PROPN
ejpam-3856	2	22	business	business	PROPN
ejpam-3856	2	23	global	global	PROPN
ejpam-3856	2	24	on	on	ADP
ejpam-3856	2	25	β	β	ADJ
ejpam-3856	2	26	-	-	ADJ
ejpam-3856	2	27	local	local	ADJ
ejpam-3856	2	28	functions	function	NOUN
ejpam-3856	2	29	in	in	ADP
ejpam-3856	2	30	ideal	ideal	ADJ
ejpam-3856	2	31	topological	topological	ADJ
ejpam-3856	2	32	spaces	space	NOUN
ejpam-3856	2	33	p.	p.	PROPN
ejpam-3856	2	34	l.	l.	PROPN
ejpam-3856	2	35	powar1,∗	powar1,∗	PROPN
ejpam-3856	2	36	,	,	PUNCT
ejpam-3856	2	37	t.	t.	NOUN
ejpam-3856	2	38	noiri2	noiri2	PROPN
ejpam-3856	2	39	,	,	PUNCT
ejpam-3856	2	40	shikha	shikha	PROPN
ejpam-3856	2	41	bhadauria3	bhadauria3	PROPN
ejpam-3856	2	42	1	1	NUM
ejpam-3856	2	43	department	department	NOUN
ejpam-3856	2	44	of	of	ADP
ejpam-3856	2	45	mathematics	mathematic	NOUN
ejpam-3856	2	46	and	and	CCONJ
ejpam-3856	2	47	computer	computer	NOUN
ejpam-3856	2	48	science	science	NOUN
ejpam-3856	2	49	,	,	PUNCT
ejpam-3856	2	50	r.	r.	PROPN
ejpam-3856	2	51	d.	d.	PROPN
ejpam-3856	2	52	university	university	PROPN
ejpam-3856	2	53	,	,	PUNCT
ejpam-3856	2	54	jabalpur	jabalpur	PROPN
ejpam-3856	2	55	,	,	PUNCT
ejpam-3856	2	56	india	india	PROPN
ejpam-3856	2	57	2	2	NUM
ejpam-3856	2	58	2949	2949	NUM
ejpam-3856	2	59	-	-	SYM
ejpam-3856	2	60	1	1	NUM
ejpam-3856	2	61	shiokita	shiokita	NOUN
ejpam-3856	2	62	-	-	PUNCT
ejpam-3856	2	63	cho	cho	ADJ
ejpam-3856	2	64	,	,	PUNCT
ejpam-3856	2	65	hinagu	hinagu	ADJ
ejpam-3856	2	66	,	,	PUNCT
ejpam-3856	2	67	yatsushiro	yatsushiro	PROPN
ejpam-3856	2	68	-	-	PUNCT
ejpam-3856	2	69	shi	shi	PROPN
ejpam-3856	2	70	,	,	PUNCT
ejpam-3856	2	71	kumamoto	kumamoto	PROPN
ejpam-3856	2	72	-	-	PUNCT
ejpam-3856	2	73	ken	ken	PROPN
ejpam-3856	2	74	,	,	PUNCT
ejpam-3856	2	75	869	869	NUM
ejpam-3856	2	76	-	-	SYM
ejpam-3856	2	77	5142	5142	NUM
ejpam-3856	2	78	japan	japan	PROPN
ejpam-3856	2	79	3	3	NUM
ejpam-3856	2	80	department	department	NOUN
ejpam-3856	2	81	of	of	ADP
ejpam-3856	2	82	mathematics	mathematic	NOUN
ejpam-3856	2	83	and	and	CCONJ
ejpam-3856	2	84	computer	computer	NOUN
ejpam-3856	2	85	science	science	NOUN
ejpam-3856	2	86	,	,	PUNCT
ejpam-3856	2	87	r.	r.	PROPN
ejpam-3856	2	88	d.	d.	PROPN
ejpam-3856	2	89	university	university	PROPN
ejpam-3856	2	90	,	,	PUNCT
ejpam-3856	2	91	jabalpur	jabalpur	PROPN
ejpam-3856	2	92	,	,	PUNCT
ejpam-3856	2	93	india	india	PROPN
ejpam-3856	2	94	abstract	abstract	NOUN
ejpam-3856	2	95	.	.	PUNCT
ejpam-3856	3	1	in	in	ADP
ejpam-3856	3	2	this	this	DET
ejpam-3856	3	3	paper	paper	NOUN
ejpam-3856	3	4	,	,	PUNCT
ejpam-3856	3	5	by	by	ADP
ejpam-3856	3	6	using	use	VERB
ejpam-3856	3	7	β	β	ADJ
ejpam-3856	3	8	-	-	ADJ
ejpam-3856	3	9	open	open	ADJ
ejpam-3856	3	10	sets	set	NOUN
ejpam-3856	3	11	in	in	ADP
ejpam-3856	3	12	[	[	X
ejpam-3856	3	13	1	1	X
ejpam-3856	3	14	]	]	PUNCT
ejpam-3856	3	15	we	we	PRON
ejpam-3856	3	16	introduce	introduce	VERB
ejpam-3856	3	17	and	and	CCONJ
ejpam-3856	3	18	investigate	investigate	VERB
ejpam-3856	3	19	the	the	DET
ejpam-3856	3	20	concepts	concept	NOUN
ejpam-3856	3	21	of	of	ADP
ejpam-3856	3	22	the	the	DET
ejpam-3856	3	23	β	β	ADJ
ejpam-3856	3	24	-	-	ADJ
ejpam-3856	3	25	local	local	ADJ
ejpam-3856	3	26	function	function	NOUN
ejpam-3856	3	27	,	,	PUNCT
ejpam-3856	3	28	is∗g	is∗g	PROPN
ejpam-3856	3	29	-	-	PUNCT
ejpam-3856	3	30	β	β	NOUN
ejpam-3856	3	31	-	-	PUNCT
ejpam-3856	3	32	closed	closed	ADJ
ejpam-3856	3	33	sets	set	NOUN
ejpam-3856	3	34	and	and	CCONJ
ejpam-3856	3	35	ig	ig	PROPN
ejpam-3856	3	36	-	-	ADJ
ejpam-3856	3	37	β	β	NOUN
ejpam-3856	3	38	-	-	PUNCT
ejpam-3856	3	39	closed	closed	ADJ
ejpam-3856	3	40	sets	set	NOUN
ejpam-3856	3	41	in	in	ADP
ejpam-3856	3	42	an	an	DET
ejpam-3856	3	43	ideal	ideal	ADJ
ejpam-3856	3	44	topological	topological	ADJ
ejpam-3856	3	45	space	space	NOUN
ejpam-3856	3	46	.	.	PUNCT
ejpam-3856	4	1	in	in	ADP
ejpam-3856	4	2	addition	addition	NOUN
ejpam-3856	4	3	to	to	ADP
ejpam-3856	4	4	the	the	DET
ejpam-3856	4	5	properties	property	NOUN
ejpam-3856	4	6	,	,	PUNCT
ejpam-3856	4	7	an	an	DET
ejpam-3856	4	8	operation	operation	NOUN
ejpam-3856	4	9	cl∗β	cl∗β	PROPN
ejpam-3856	4	10	is	be	AUX
ejpam-3856	4	11	defined	define	VERB
ejpam-3856	4	12	and	and	CCONJ
ejpam-3856	4	13	the	the	DET
ejpam-3856	4	14	properties	property	NOUN
ejpam-3856	4	15	are	be	AUX
ejpam-3856	4	16	obtained	obtain	VERB
ejpam-3856	4	17	similarly	similarly	ADV
ejpam-3856	4	18	with	with	ADP
ejpam-3856	4	19	the	the	DET
ejpam-3856	4	20	local	local	ADJ
ejpam-3856	4	21	function	function	NOUN
ejpam-3856	4	22	in	in	ADP
ejpam-3856	4	23	[	[	X
ejpam-3856	4	24	8	8	NUM
ejpam-3856	4	25	]	]	PUNCT
ejpam-3856	4	26	.	.	PUNCT
ejpam-3856	5	1	2020	2020	NUM
ejpam-3856	5	2	mathematics	mathematic	NOUN
ejpam-3856	5	3	subject	subject	NOUN
ejpam-3856	5	4	classifications	classification	NOUN
ejpam-3856	5	5	:	:	PUNCT
ejpam-3856	5	6	54c10	54c10	NUM
ejpam-3856	5	7	,	,	PUNCT
ejpam-3856	5	8	54a05	54a05	NUM
ejpam-3856	5	9	,	,	PUNCT
ejpam-3856	5	10	54d15	54d15	NUM
ejpam-3856	5	11	,	,	PUNCT
ejpam-3856	5	12	54d30	54d30	ADJ
ejpam-3856	5	13	key	key	ADJ
ejpam-3856	5	14	words	word	NOUN
ejpam-3856	5	15	and	and	CCONJ
ejpam-3856	5	16	phrases	phrase	NOUN
ejpam-3856	5	17	:	:	PUNCT
ejpam-3856	5	18	β	β	X
ejpam-3856	5	19	-	-	ADJ
ejpam-3856	5	20	open	open	ADJ
ejpam-3856	5	21	set	set	NOUN
ejpam-3856	5	22	,	,	PUNCT
ejpam-3856	5	23	β	β	ADJ
ejpam-3856	5	24	-	-	ADJ
ejpam-3856	5	25	local	local	ADJ
ejpam-3856	5	26	function	function	NOUN
ejpam-3856	5	27	,	,	PUNCT
ejpam-3856	5	28	operation	operation	NOUN
ejpam-3856	5	29	cl∗β	cl∗β	PROPN
ejpam-3856	5	30	,	,	PUNCT
ejpam-3856	5	31	is∗g	is∗g	PROPN
ejpam-3856	5	32	-	-	PUNCT
ejpam-3856	5	33	β	β	NOUN
ejpam-3856	5	34	-	-	PUNCT
ejpam-3856	5	35	closed	closed	ADJ
ejpam-3856	5	36	set	set	NOUN
ejpam-3856	5	37	,	,	PUNCT
ejpam-3856	5	38	ig	ig	PROPN
ejpam-3856	5	39	-	-	PUNCT
ejpam-3856	5	40	βclosed	βclose	VERB
ejpam-3856	5	41	set	set	NOUN
ejpam-3856	5	42	.	.	PUNCT
ejpam-3856	6	1	1	1	X
ejpam-3856	6	2	.	.	X
ejpam-3856	6	3	introduction	introduction	NOUN
ejpam-3856	6	4	kuratowski	kuratowski	NOUN
ejpam-3856	6	5	[	[	X
ejpam-3856	6	6	11	11	NUM
ejpam-3856	6	7	]	]	PUNCT
ejpam-3856	6	8	has	have	AUX
ejpam-3856	6	9	introduced	introduce	VERB
ejpam-3856	6	10	the	the	DET
ejpam-3856	6	11	concept	concept	NOUN
ejpam-3856	6	12	of	of	ADP
ejpam-3856	6	13	an	an	DET
ejpam-3856	6	14	ideal	ideal	ADJ
ejpam-3856	6	15	topological	topological	ADJ
ejpam-3856	6	16	space	space	NOUN
ejpam-3856	6	17	in	in	ADP
ejpam-3856	6	18	1930	1930	NUM
ejpam-3856	6	19	.	.	PUNCT
ejpam-3856	7	1	further	far	ADV
ejpam-3856	7	2	,	,	PUNCT
ejpam-3856	7	3	jankovic	jankovic	PROPN
ejpam-3856	7	4	and	and	CCONJ
ejpam-3856	7	5	hamlet	hamlet	PROPN
ejpam-3856	8	1	[	[	X
ejpam-3856	8	2	8	8	NUM
ejpam-3856	8	3	]	]	PUNCT
ejpam-3856	8	4	have	have	AUX
ejpam-3856	8	5	studied	study	VERB
ejpam-3856	8	6	ideal	ideal	ADJ
ejpam-3856	8	7	topological	topological	ADJ
ejpam-3856	8	8	spaces	space	NOUN
ejpam-3856	8	9	and	and	CCONJ
ejpam-3856	8	10	obtained	obtain	VERB
ejpam-3856	8	11	their	their	PRON
ejpam-3856	8	12	significant	significant	ADJ
ejpam-3856	8	13	properties	property	NOUN
ejpam-3856	8	14	.	.	PUNCT
ejpam-3856	9	1	they	they	PRON
ejpam-3856	9	2	introduced	introduce	VERB
ejpam-3856	9	3	the	the	DET
ejpam-3856	9	4	concept	concept	NOUN
ejpam-3856	9	5	of	of	ADP
ejpam-3856	9	6	i	i	NOUN
ejpam-3856	9	7	-	-	PUNCT
ejpam-3856	9	8	open	open	ADJ
ejpam-3856	9	9	sets	set	NOUN
ejpam-3856	9	10	and	and	CCONJ
ejpam-3856	9	11	studied	study	VERB
ejpam-3856	9	12	topologies	topology	NOUN
ejpam-3856	9	13	via	via	ADP
ejpam-3856	9	14	ideals	ideal	NOUN
ejpam-3856	9	15	quite	quite	ADV
ejpam-3856	9	16	extensively	extensively	ADV
ejpam-3856	9	17	.	.	PUNCT
ejpam-3856	10	1	abd	abd	PROPN
ejpam-3856	10	2	-	-	PUNCT
ejpam-3856	10	3	el	el	PROPN
ejpam-3856	10	4	-	-	PUNCT
ejpam-3856	10	5	monsef	monsef	PROPN
ejpam-3856	10	6	et	et	PROPN
ejpam-3856	10	7	al.[2	al.[2	PROPN
ejpam-3856	10	8	]	]	PUNCT
ejpam-3856	10	9	further	far	ADV
ejpam-3856	10	10	explored	explore	VERB
ejpam-3856	10	11	the	the	DET
ejpam-3856	10	12	ideas	idea	NOUN
ejpam-3856	10	13	of	of	ADP
ejpam-3856	10	14	i	i	NOUN
ejpam-3856	10	15	-	-	PUNCT
ejpam-3856	10	16	open	open	ADJ
ejpam-3856	10	17	sets	set	NOUN
ejpam-3856	10	18	.	.	PUNCT
ejpam-3856	11	1	the	the	DET
ejpam-3856	11	2	concept	concept	NOUN
ejpam-3856	11	3	of	of	ADP
ejpam-3856	11	4	ig	ig	PROPN
ejpam-3856	11	5	-	-	PUNCT
ejpam-3856	11	6	closed	closed	ADJ
ejpam-3856	11	7	sets	set	NOUN
ejpam-3856	11	8	has	have	AUX
ejpam-3856	11	9	been	be	AUX
ejpam-3856	11	10	given	give	VERB
ejpam-3856	11	11	by	by	ADP
ejpam-3856	11	12	dontchev	dontchev	PROPN
ejpam-3856	11	13	et	et	PROPN
ejpam-3856	11	14	al	al	PROPN
ejpam-3856	11	15	.	.	PUNCT
ejpam-3856	12	1	[	[	X
ejpam-3856	12	2	6	6	NUM
ejpam-3856	12	3	]	]	PUNCT
ejpam-3856	12	4	in	in	ADP
ejpam-3856	12	5	1999	1999	NUM
ejpam-3856	12	6	and	and	CCONJ
ejpam-3856	12	7	the	the	DET
ejpam-3856	12	8	idea	idea	NOUN
ejpam-3856	12	9	of	of	ADP
ejpam-3856	12	10	is∗g	is∗g	PROPN
ejpam-3856	12	11	-	-	PUNCT
ejpam-3856	12	12	closed	close	VERB
ejpam-3856	12	13	sets	set	NOUN
ejpam-3856	12	14	was	be	AUX
ejpam-3856	12	15	first	first	ADV
ejpam-3856	12	16	introduced	introduce	VERB
ejpam-3856	12	17	by	by	ADP
ejpam-3856	12	18	khan	khan	PROPN
ejpam-3856	12	19	and	and	CCONJ
ejpam-3856	12	20	hamza	hamza	PROPN
ejpam-3856	13	1	[	[	X
ejpam-3856	13	2	9	9	NUM
ejpam-3856	13	3	]	]	PUNCT
ejpam-3856	13	4	.	.	PUNCT
ejpam-3856	14	1	the	the	DET
ejpam-3856	14	2	concepts	concept	NOUN
ejpam-3856	14	3	of	of	ADP
ejpam-3856	14	4	the	the	DET
ejpam-3856	14	5	s	s	ADJ
ejpam-3856	14	6	-	-	ADJ
ejpam-3856	14	7	local	local	ADJ
ejpam-3856	14	8	function	function	NOUN
ejpam-3856	14	9	was	be	AUX
ejpam-3856	14	10	first	first	ADV
ejpam-3856	14	11	introduced	introduce	VERB
ejpam-3856	14	12	by	by	ADP
ejpam-3856	14	13	abd	abd	PROPN
ejpam-3856	14	14	.	.	PUNCT
ejpam-3856	15	1	el	el	PROPN
ejpam-3856	15	2	-	-	PUNCT
ejpam-3856	15	3	monsef	monsef	PROPN
ejpam-3856	15	4	et	et	PROPN
ejpam-3856	15	5	al	al	PROPN
ejpam-3856	15	6	.	.	PUNCT
ejpam-3856	16	1	[	[	X
ejpam-3856	16	2	3	3	X
ejpam-3856	16	3	]	]	PUNCT
ejpam-3856	16	4	and	and	CCONJ
ejpam-3856	16	5	further	far	ADV
ejpam-3856	16	6	investigated	investigate	VERB
ejpam-3856	16	7	by	by	ADP
ejpam-3856	16	8	khan	khan	PROPN
ejpam-3856	16	9	and	and	CCONJ
ejpam-3856	16	10	noiri	noiri	ADV
ejpam-3856	17	1	[	[	X
ejpam-3856	17	2	10	10	NUM
ejpam-3856	17	3	]	]	PUNCT
ejpam-3856	17	4	.	.	PUNCT
ejpam-3856	18	1	recently	recently	ADV
ejpam-3856	18	2	,	,	PUNCT
ejpam-3856	18	3	al	al	PROPN
ejpam-3856	18	4	-	-	PUNCT
ejpam-3856	18	5	omari	omari	PROPN
ejpam-3856	18	6	and	and	CCONJ
ejpam-3856	18	7	noiri	noiri	ADV
ejpam-3856	18	8	[	[	X
ejpam-3856	18	9	5	5	X
ejpam-3856	18	10	]	]	PUNCT
ejpam-3856	18	11	have	have	AUX
ejpam-3856	18	12	introduced	introduce	VERB
ejpam-3856	18	13	and	and	CCONJ
ejpam-3856	18	14	investigated	investigate	VERB
ejpam-3856	18	15	the	the	DET
ejpam-3856	18	16	notion	notion	NOUN
ejpam-3856	18	17	of	of	ADP
ejpam-3856	18	18	local	local	ADJ
ejpam-3856	18	19	function	function	NOUN
ejpam-3856	18	20	γ∗	γ∗	NOUN
ejpam-3856	18	21	in	in	ADP
ejpam-3856	18	22	an	an	DET
ejpam-3856	18	23	ideal	ideal	ADJ
ejpam-3856	18	24	topological	topological	ADJ
ejpam-3856	18	25	space	space	NOUN
ejpam-3856	18	26	and	and	CCONJ
ejpam-3856	18	27	showed	show	VERB
ejpam-3856	18	28	that	that	SCONJ
ejpam-3856	18	29	γ∗	γ∗	NOUN
ejpam-3856	18	30	is	be	AUX
ejpam-3856	18	31	equivalent	equivalent	ADJ
ejpam-3856	18	32	to	to	ADP
ejpam-3856	18	33	the	the	DET
ejpam-3856	18	34	δ	δ	PROPN
ejpam-3856	18	35	-	-	ADJ
ejpam-3856	18	36	local	local	ADJ
ejpam-3856	18	37	function	function	NOUN
ejpam-3856	18	38	due	due	ADP
ejpam-3856	18	39	to	to	PART
ejpam-3856	18	40	hatir	hatir	PROPN
ejpam-3856	18	41	et	et	PROPN
ejpam-3856	18	42	al	al	PROPN
ejpam-3856	18	43	.	.	PUNCT
ejpam-3856	19	1	[	[	X
ejpam-3856	19	2	7	7	NUM
ejpam-3856	19	3	]	]	PUNCT
ejpam-3856	19	4	.	.	PUNCT
ejpam-3856	20	1	in	in	ADP
ejpam-3856	20	2	this	this	DET
ejpam-3856	20	3	paper	paper	NOUN
ejpam-3856	20	4	,	,	PUNCT
ejpam-3856	20	5	by	by	ADP
ejpam-3856	20	6	using	use	VERB
ejpam-3856	20	7	β	β	ADJ
ejpam-3856	20	8	-	-	ADJ
ejpam-3856	20	9	open	open	ADJ
ejpam-3856	20	10	sets	set	NOUN
ejpam-3856	20	11	in	in	ADP
ejpam-3856	20	12	[	[	X
ejpam-3856	20	13	1	1	X
ejpam-3856	20	14	]	]	PUNCT
ejpam-3856	20	15	we	we	PRON
ejpam-3856	20	16	introduce	introduce	VERB
ejpam-3856	20	17	and	and	CCONJ
ejpam-3856	20	18	investigate	investigate	VERB
ejpam-3856	20	19	the	the	DET
ejpam-3856	20	20	concepts	concept	NOUN
ejpam-3856	20	21	of	of	ADP
ejpam-3856	20	22	the	the	DET
ejpam-3856	20	23	β	β	ADJ
ejpam-3856	20	24	-	-	ADJ
ejpam-3856	20	25	local	local	ADJ
ejpam-3856	20	26	function	function	NOUN
ejpam-3856	20	27	,	,	PUNCT
ejpam-3856	20	28	is∗g	is∗g	PROPN
ejpam-3856	20	29	-	-	PUNCT
ejpam-3856	20	30	β	β	NOUN
ejpam-3856	20	31	-	-	PUNCT
ejpam-3856	20	32	closed	closed	ADJ
ejpam-3856	20	33	sets	set	NOUN
ejpam-3856	20	34	and	and	CCONJ
ejpam-3856	20	35	ig	ig	PROPN
ejpam-3856	20	36	-	-	ADJ
ejpam-3856	20	37	β	β	NOUN
ejpam-3856	20	38	-	-	PUNCT
ejpam-3856	20	39	closed	closed	ADJ
ejpam-3856	20	40	sets	set	NOUN
ejpam-3856	20	41	in	in	ADP
ejpam-3856	20	42	an	an	DET
ejpam-3856	20	43	ideal	ideal	ADJ
ejpam-3856	20	44	topological	topological	ADJ
ejpam-3856	20	45	space	space	NOUN
ejpam-3856	20	46	.	.	PUNCT
ejpam-3856	21	1	and	and	CCONJ
ejpam-3856	21	2	also	also	ADV
ejpam-3856	21	3	,	,	PUNCT
ejpam-3856	21	4	an	an	DET
ejpam-3856	21	5	operation	operation	NOUN
ejpam-3856	21	6	cl∗β	cl∗β	PROPN
ejpam-3856	21	7	is	be	AUX
ejpam-3856	21	8	defined	define	VERB
ejpam-3856	21	9	and	and	CCONJ
ejpam-3856	21	10	the	the	DET
ejpam-3856	21	11	properties	property	NOUN
ejpam-3856	21	12	are	be	AUX
ejpam-3856	21	13	obtained	obtain	VERB
ejpam-3856	21	14	similarly	similarly	ADV
ejpam-3856	21	15	with	with	ADP
ejpam-3856	21	16	the	the	DET
ejpam-3856	21	17	local	local	ADJ
ejpam-3856	21	18	function	function	NOUN
ejpam-3856	21	19	in	in	ADP
ejpam-3856	21	20	[	[	X
ejpam-3856	21	21	8	8	NUM
ejpam-3856	21	22	]	]	PUNCT
ejpam-3856	21	23	.	.	PUNCT
ejpam-3856	22	1	∗corresponding	∗corresponde	VERB
ejpam-3856	22	2	author	author	NOUN
ejpam-3856	22	3	.	.	PUNCT
ejpam-3856	23	1	doi	doi	NOUN
ejpam-3856	23	2	:	:	PUNCT
ejpam-3856	23	3	https://doi.org/10.29020/nybg.ejpam.v13i4.3856	https://doi.org/10.29020/nybg.ejpam.v13i4.3856	NOUN
ejpam-3856	23	4	email	email	NOUN
ejpam-3856	23	5	addresses	address	NOUN
ejpam-3856	23	6	:	:	PUNCT
ejpam-3856	23	7	pvjrdvv@rediffmail.com	pvjrdvv@rediffmail.com	PROPN
ejpam-3856	23	8	(	(	PUNCT
ejpam-3856	23	9	p.	p.	PROPN
ejpam-3856	23	10	l.	l.	PROPN
ejpam-3856	23	11	powar	powar	PROPN
ejpam-3856	23	12	)	)	PUNCT
ejpam-3856	23	13	,	,	PUNCT
ejpam-3856	23	14	t.noiri@nifty.com	t.noiri@nifty.com	X
ejpam-3856	23	15	(	(	PUNCT
ejpam-3856	23	16	t.	t.	PROPN
ejpam-3856	23	17	noiri	noiri	PROPN
ejpam-3856	23	18	)	)	PUNCT
ejpam-3856	23	19	,	,	PUNCT
ejpam-3856	23	20	shikhabhadauriamaths@gmail.com	shikhabhadauriamaths@gmail.com	X
ejpam-3856	23	21	(	(	PUNCT
ejpam-3856	23	22	shikha	shikha	PROPN
ejpam-3856	23	23	bhadauria	bhadauria	PROPN
ejpam-3856	23	24	)	)	PUNCT
ejpam-3856	23	25	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-3856	23	26	758	758	NUM
ejpam-3856	24	1	c	c	NOUN
ejpam-3856	24	2	©	©	PROPN
ejpam-3856	24	3	2020	2020	NUM
ejpam-3856	24	4	ejpam	ejpam	VERB
ejpam-3856	24	5	all	all	DET
ejpam-3856	24	6	rights	right	NOUN
ejpam-3856	24	7	reserved	reserve	VERB
ejpam-3856	24	8	.	.	PUNCT
ejpam-3856	25	1	p.	p.	NOUN
ejpam-3856	25	2	l.	l.	PROPN
ejpam-3856	25	3	powar	powar	PROPN
ejpam-3856	25	4	,	,	PUNCT
ejpam-3856	25	5	t.	t.	PROPN
ejpam-3856	25	6	noiri	noiri	PROPN
ejpam-3856	25	7	,	,	PUNCT
ejpam-3856	25	8	shikha	shikha	PROPN
ejpam-3856	25	9	bhadauria	bhadauria	PROPN
ejpam-3856	25	10	/	/	SYM
ejpam-3856	25	11	eur	eur	PROPN
ejpam-3856	25	12	.	.	PUNCT
ejpam-3856	26	1	j.	j.	PROPN
ejpam-3856	26	2	pure	pure	PROPN
ejpam-3856	26	3	appl	appl	PROPN
ejpam-3856	26	4	.	.	PROPN
ejpam-3856	26	5	math	math	PROPN
ejpam-3856	26	6	,	,	PUNCT
ejpam-3856	26	7	13	13	NUM
ejpam-3856	26	8	(	(	PUNCT
ejpam-3856	26	9	4	4	NUM
ejpam-3856	26	10	)	)	PUNCT
ejpam-3856	26	11	(	(	PUNCT
ejpam-3856	26	12	2020	2020	NUM
ejpam-3856	26	13	)	)	PUNCT
ejpam-3856	26	14	,	,	PUNCT
ejpam-3856	26	15	758	758	NUM
ejpam-3856	26	16	-	-	SYM
ejpam-3856	26	17	765	765	NUM
ejpam-3856	26	18	759	759	NUM
ejpam-3856	26	19	2	2	NUM
ejpam-3856	26	20	.	.	PUNCT
ejpam-3856	26	21	preliminaries	preliminary	NOUN
ejpam-3856	26	22	throughout	throughout	ADP
ejpam-3856	26	23	this	this	DET
ejpam-3856	26	24	paper	paper	NOUN
ejpam-3856	26	25	(	(	PUNCT
ejpam-3856	26	26	x	x	X
ejpam-3856	26	27	,	,	PUNCT
ejpam-3856	26	28	τ	τ	X
ejpam-3856	26	29	)	)	PUNCT
ejpam-3856	26	30	and	and	CCONJ
ejpam-3856	26	31	(	(	PUNCT
ejpam-3856	26	32	x	x	X
ejpam-3856	26	33	,	,	PUNCT
ejpam-3856	26	34	τ	τ	PROPN
ejpam-3856	26	35	,	,	PUNCT
ejpam-3856	26	36	i	i	NOUN
ejpam-3856	26	37	)	)	PUNCT
ejpam-3856	26	38	denote	denote	VERB
ejpam-3856	26	39	a	a	DET
ejpam-3856	26	40	topological	topological	ADJ
ejpam-3856	26	41	space	space	NOUN
ejpam-3856	26	42	and	and	CCONJ
ejpam-3856	26	43	an	an	DET
ejpam-3856	26	44	ideal	ideal	ADJ
ejpam-3856	26	45	topological	topological	ADJ
ejpam-3856	26	46	space	space	NOUN
ejpam-3856	26	47	,	,	PUNCT
ejpam-3856	26	48	respectively	respectively	ADV
ejpam-3856	26	49	.	.	PUNCT
ejpam-3856	27	1	the	the	DET
ejpam-3856	27	2	collection	collection	NOUN
ejpam-3856	27	3	of	of	ADP
ejpam-3856	27	4	closed	closed	ADJ
ejpam-3856	27	5	sets	set	NOUN
ejpam-3856	27	6	in	in	ADP
ejpam-3856	27	7	x	x	PUNCT
ejpam-3856	27	8	is	be	AUX
ejpam-3856	27	9	denoted	denote	VERB
ejpam-3856	27	10	by	by	ADP
ejpam-3856	27	11	τf	τf	PRON
ejpam-3856	27	12	.	.	PUNCT
ejpam-3856	28	1	for	for	ADP
ejpam-3856	28	2	any	any	DET
ejpam-3856	28	3	subset	subset	NOUN
ejpam-3856	28	4	a	a	PRON
ejpam-3856	28	5	of	of	ADP
ejpam-3856	28	6	x	x	X
ejpam-3856	28	7	the	the	DET
ejpam-3856	28	8	closure	closure	NOUN
ejpam-3856	28	9	and	and	CCONJ
ejpam-3856	28	10	the	the	DET
ejpam-3856	28	11	interior	interior	NOUN
ejpam-3856	28	12	of	of	ADP
ejpam-3856	28	13	a	a	PRON
ejpam-3856	28	14	are	be	AUX
ejpam-3856	28	15	denoted	denote	VERB
ejpam-3856	28	16	by	by	ADP
ejpam-3856	28	17	cl(a	cl(a	NOUN
ejpam-3856	28	18	)	)	PUNCT
ejpam-3856	28	19	and	and	CCONJ
ejpam-3856	28	20	int(a	int(a	PROPN
ejpam-3856	28	21	)	)	PUNCT
ejpam-3856	28	22	,	,	PUNCT
ejpam-3856	28	23	respectively	respectively	ADV
ejpam-3856	28	24	.	.	PUNCT
ejpam-3856	29	1	we	we	PRON
ejpam-3856	29	2	now	now	ADV
ejpam-3856	29	3	recall	recall	VERB
ejpam-3856	29	4	certain	certain	ADJ
ejpam-3856	29	5	definitions	definition	NOUN
ejpam-3856	29	6	,	,	PUNCT
ejpam-3856	29	7	which	which	PRON
ejpam-3856	29	8	would	would	AUX
ejpam-3856	29	9	be	be	AUX
ejpam-3856	29	10	required	require	VERB
ejpam-3856	29	11	for	for	ADP
ejpam-3856	29	12	our	our	PRON
ejpam-3856	29	13	study	study	NOUN
ejpam-3856	29	14	.	.	PUNCT
ejpam-3856	30	1	definition	definition	NOUN
ejpam-3856	30	2	1	1	NUM
ejpam-3856	30	3	.	.	PUNCT
ejpam-3856	31	1	[	[	X
ejpam-3856	31	2	8	8	X
ejpam-3856	31	3	]	]	X
ejpam-3856	31	4	an	an	DET
ejpam-3856	31	5	ideal	ideal	NOUN
ejpam-3856	31	6	i	i	PRON
ejpam-3856	31	7	on	on	ADP
ejpam-3856	31	8	a	a	DET
ejpam-3856	31	9	topological	topological	ADJ
ejpam-3856	31	10	spaces	space	NOUN
ejpam-3856	31	11	(	(	PUNCT
ejpam-3856	31	12	x	x	X
ejpam-3856	31	13	,	,	PUNCT
ejpam-3856	31	14	τ	τ	X
ejpam-3856	31	15	)	)	PUNCT
ejpam-3856	31	16	is	be	AUX
ejpam-3856	31	17	a	a	DET
ejpam-3856	31	18	nonempty	nonempty	ADJ
ejpam-3856	31	19	collection	collection	NOUN
ejpam-3856	31	20	of	of	ADP
ejpam-3856	31	21	subsets	subset	NOUN
ejpam-3856	31	22	of	of	ADP
ejpam-3856	31	23	x	x	PROPN
ejpam-3856	31	24	,	,	PUNCT
ejpam-3856	31	25	which	which	PRON
ejpam-3856	31	26	satisfies	satisfy	VERB
ejpam-3856	31	27	the	the	DET
ejpam-3856	31	28	following	follow	VERB
ejpam-3856	31	29	conditions	condition	NOUN
ejpam-3856	31	30	:	:	PUNCT
ejpam-3856	31	31	•	•	ADP
ejpam-3856	31	32	a	a	PRON
ejpam-3856	31	33	∈	∈	NOUN
ejpam-3856	32	1	i	i	PRON
ejpam-3856	32	2	and	and	CCONJ
ejpam-3856	32	3	b	b	X
ejpam-3856	32	4	∈	∈	PROPN
ejpam-3856	32	5	i	i	PRON
ejpam-3856	32	6	implies	imply	VERB
ejpam-3856	32	7	a	a	DET
ejpam-3856	32	8	∪b	∪b	PUNCT
ejpam-3856	32	9	∈	∈	PROPN
ejpam-3856	32	10	i	i	PRON
ejpam-3856	32	11	,	,	PUNCT
ejpam-3856	32	12	•	•	ADP
ejpam-3856	32	13	a	a	PRON
ejpam-3856	32	14	∈	∈	NOUN
ejpam-3856	32	15	i	i	PRON
ejpam-3856	32	16	and	and	CCONJ
ejpam-3856	32	17	b	b	PROPN
ejpam-3856	32	18	⊂	⊂	PROPN
ejpam-3856	32	19	a	a	PRON
ejpam-3856	32	20	implies	imply	VERB
ejpam-3856	32	21	b	b	X
ejpam-3856	32	22	∈	∈	PROPN
ejpam-3856	32	23	i.	i.	NOUN
ejpam-3856	32	24	then	then	ADV
ejpam-3856	32	25	the	the	DET
ejpam-3856	32	26	triplet	triplet	NOUN
ejpam-3856	32	27	(	(	PUNCT
ejpam-3856	32	28	x	x	NOUN
ejpam-3856	32	29	,	,	PUNCT
ejpam-3856	32	30	τ	τ	PROPN
ejpam-3856	32	31	,	,	PUNCT
ejpam-3856	32	32	i	i	PROPN
ejpam-3856	32	33	)	)	PUNCT
ejpam-3856	32	34	is	be	AUX
ejpam-3856	32	35	called	call	VERB
ejpam-3856	32	36	an	an	DET
ejpam-3856	32	37	ideal	ideal	ADJ
ejpam-3856	32	38	topological	topological	ADJ
ejpam-3856	32	39	space	space	NOUN
ejpam-3856	32	40	.	.	PUNCT
ejpam-3856	33	1	definition	definition	NOUN
ejpam-3856	33	2	2	2	NUM
ejpam-3856	33	3	.	.	PUNCT
ejpam-3856	34	1	[	[	X
ejpam-3856	34	2	8	8	NUM
ejpam-3856	34	3	]	]	X
ejpam-3856	34	4	let	let	AUX
ejpam-3856	34	5	(	(	PUNCT
ejpam-3856	34	6	x	x	NOUN
ejpam-3856	34	7	,	,	PUNCT
ejpam-3856	34	8	τ	τ	PROPN
ejpam-3856	34	9	,	,	PUNCT
ejpam-3856	34	10	i	i	PRON
ejpam-3856	34	11	)	)	PUNCT
ejpam-3856	34	12	be	be	VERB
ejpam-3856	34	13	an	an	DET
ejpam-3856	34	14	ideal	ideal	ADJ
ejpam-3856	34	15	topological	topological	ADJ
ejpam-3856	34	16	space	space	NOUN
ejpam-3856	34	17	.	.	PUNCT
ejpam-3856	35	1	for	for	ADP
ejpam-3856	35	2	a	a	DET
ejpam-3856	35	3	set	set	NOUN
ejpam-3856	35	4	a	a	PRON
ejpam-3856	35	5	⊂	⊂	PROPN
ejpam-3856	35	6	x	x	SYM
ejpam-3856	35	7	,	,	PUNCT
ejpam-3856	35	8	a∗(x	a∗(x	PROPN
ejpam-3856	35	9	,	,	PUNCT
ejpam-3856	35	10	τ	τ	NOUN
ejpam-3856	35	11	)	)	PUNCT
ejpam-3856	35	12	=	=	NOUN
ejpam-3856	35	13	{	{	PUNCT
ejpam-3856	35	14	x	x	SYM
ejpam-3856	35	15	∈	∈	PROPN
ejpam-3856	35	16	x	x	X
ejpam-3856	35	17	:	:	PUNCT
ejpam-3856	35	18	a	a	DET
ejpam-3856	35	19	∩	∩	ADJ
ejpam-3856	35	20	u	u	NOUN
ejpam-3856	35	21	/∈	/∈	PUNCT
ejpam-3856	35	22	i	i	PRON
ejpam-3856	35	23	for	for	ADP
ejpam-3856	35	24	every	every	DET
ejpam-3856	35	25	u	u	PROPN
ejpam-3856	35	26	∈	∈	PROPN
ejpam-3856	35	27	τ(x	τ(x	NOUN
ejpam-3856	35	28	)	)	PUNCT
ejpam-3856	35	29	}	}	PUNCT
ejpam-3856	35	30	,	,	PUNCT
ejpam-3856	35	31	where	where	SCONJ
ejpam-3856	35	32	τ(x	τ(x	NOUN
ejpam-3856	35	33	)	)	PUNCT
ejpam-3856	35	34	=	=	PRON
ejpam-3856	35	35	{	{	PUNCT
ejpam-3856	35	36	u	u	X
ejpam-3856	35	37	∈	∈	PROPN
ejpam-3856	35	38	τ	τ	X
ejpam-3856	35	39	:	:	PUNCT
ejpam-3856	35	40	x	x	SYM
ejpam-3856	35	41	∈	∈	PROPN
ejpam-3856	35	42	u	u	NOUN
ejpam-3856	35	43	}	}	PUNCT
ejpam-3856	35	44	,	,	PUNCT
ejpam-3856	35	45	is	be	AUX
ejpam-3856	35	46	called	call	VERB
ejpam-3856	35	47	the	the	DET
ejpam-3856	35	48	localfunction	localfunction	NOUN
ejpam-3856	35	49	of	of	ADP
ejpam-3856	35	50	a	a	PRON
ejpam-3856	35	51	with	with	ADP
ejpam-3856	35	52	respect	respect	NOUN
ejpam-3856	35	53	to	to	ADP
ejpam-3856	35	54	i	i	PRON
ejpam-3856	35	55	and	and	CCONJ
ejpam-3856	35	56	τ	τ	PROPN
ejpam-3856	35	57	.	.	PUNCT
ejpam-3856	36	1	a∗(x	a∗(x	PROPN
ejpam-3856	36	2	,	,	PUNCT
ejpam-3856	36	3	τ	τ	X
ejpam-3856	36	4	)	)	PUNCT
ejpam-3856	36	5	is	be	AUX
ejpam-3856	36	6	simply	simply	ADV
ejpam-3856	36	7	denoted	denote	VERB
ejpam-3856	36	8	by	by	ADP
ejpam-3856	36	9	a∗.	a∗.	NOUN
ejpam-3856	36	10	definition	definition	NOUN
ejpam-3856	36	11	3	3	NUM
ejpam-3856	36	12	.	.	PUNCT
ejpam-3856	37	1	[	[	X
ejpam-3856	37	2	5	5	NUM
ejpam-3856	37	3	]	]	X
ejpam-3856	37	4	let	let	VERB
ejpam-3856	37	5	(	(	PUNCT
ejpam-3856	37	6	x	x	NOUN
ejpam-3856	37	7	,	,	PUNCT
ejpam-3856	37	8	τ	τ	PROPN
ejpam-3856	37	9	,	,	PUNCT
ejpam-3856	37	10	i	i	PRON
ejpam-3856	37	11	)	)	PUNCT
ejpam-3856	37	12	be	be	VERB
ejpam-3856	37	13	an	an	DET
ejpam-3856	37	14	ideal	ideal	ADJ
ejpam-3856	37	15	topological	topological	ADJ
ejpam-3856	37	16	space	space	NOUN
ejpam-3856	37	17	.	.	PUNCT
ejpam-3856	38	1	for	for	ADP
ejpam-3856	38	2	a	a	DET
ejpam-3856	38	3	set	set	NOUN
ejpam-3856	38	4	a	a	DET
ejpam-3856	38	5	⊂	⊂	PROPN
ejpam-3856	38	6	x	x	NOUN
ejpam-3856	38	7	,	,	PUNCT
ejpam-3856	38	8	γ∗(a)(i	γ∗(a)(i	NUM
ejpam-3856	38	9	,	,	PUNCT
ejpam-3856	38	10	τ	τ	X
ejpam-3856	38	11	)	)	PUNCT
ejpam-3856	38	12	=	=	NOUN
ejpam-3856	38	13	{	{	PUNCT
ejpam-3856	38	14	x	x	SYM
ejpam-3856	38	15	∈	∈	PROPN
ejpam-3856	38	16	x	x	X
ejpam-3856	38	17	:	:	PUNCT
ejpam-3856	38	18	a	a	DET
ejpam-3856	38	19	∩u	∩u	NOUN
ejpam-3856	38	20	/∈	/∈	PUNCT
ejpam-3856	39	1	i	i	PRON
ejpam-3856	39	2	for	for	ADP
ejpam-3856	39	3	every	every	DET
ejpam-3856	39	4	regular	regular	ADJ
ejpam-3856	39	5	open	open	ADJ
ejpam-3856	39	6	set	set	NOUN
ejpam-3856	39	7	u	u	NOUN
ejpam-3856	39	8	containing	contain	VERB
ejpam-3856	39	9	x	x	PRON
ejpam-3856	39	10	}	}	PUNCT
ejpam-3856	39	11	is	be	AUX
ejpam-3856	39	12	called	call	VERB
ejpam-3856	39	13	the	the	DET
ejpam-3856	39	14	local	local	ADJ
ejpam-3856	39	15	function	function	NOUN
ejpam-3856	39	16	γ∗	γ∗	NOUN
ejpam-3856	39	17	of	of	ADP
ejpam-3856	39	18	a	a	PRON
ejpam-3856	39	19	with	with	ADP
ejpam-3856	39	20	respect	respect	NOUN
ejpam-3856	39	21	to	to	ADP
ejpam-3856	39	22	i	i	PRON
ejpam-3856	39	23	and	and	CCONJ
ejpam-3856	39	24	τ	τ	PROPN
ejpam-3856	39	25	.	.	PUNCT
ejpam-3856	40	1	definition	definition	NOUN
ejpam-3856	40	2	4	4	NUM
ejpam-3856	40	3	.	.	PUNCT
ejpam-3856	41	1	[	[	X
ejpam-3856	41	2	3],[10	3],[10	X
ejpam-3856	41	3	]	]	X
ejpam-3856	41	4	let	let	AUX
ejpam-3856	41	5	(	(	PUNCT
ejpam-3856	41	6	x	x	NOUN
ejpam-3856	41	7	,	,	PUNCT
ejpam-3856	41	8	τ	τ	PROPN
ejpam-3856	41	9	,	,	PUNCT
ejpam-3856	41	10	i	i	PRON
ejpam-3856	41	11	)	)	PUNCT
ejpam-3856	41	12	be	be	VERB
ejpam-3856	41	13	an	an	DET
ejpam-3856	41	14	ideal	ideal	ADJ
ejpam-3856	41	15	topological	topological	ADJ
ejpam-3856	41	16	space	space	NOUN
ejpam-3856	41	17	and	and	CCONJ
ejpam-3856	41	18	a	a	DET
ejpam-3856	41	19	be	be	AUX
ejpam-3856	41	20	a	a	DET
ejpam-3856	41	21	subset	subset	NOUN
ejpam-3856	41	22	of	of	ADP
ejpam-3856	41	23	x.	x.	NOUN
ejpam-3856	41	24	then	then	ADV
ejpam-3856	41	25	(	(	PUNCT
ejpam-3856	42	1	a)∗	a)∗	PROPN
ejpam-3856	42	2	s	s	X
ejpam-3856	42	3	(	(	PUNCT
ejpam-3856	42	4	i	i	PROPN
ejpam-3856	42	5	,	,	PUNCT
ejpam-3856	42	6	τ	τ	PROPN
ejpam-3856	42	7	)	)	PUNCT
ejpam-3856	42	8	=	=	NOUN
ejpam-3856	42	9	{	{	PUNCT
ejpam-3856	42	10	x	x	SYM
ejpam-3856	42	11	∈	∈	PROPN
ejpam-3856	42	12	x	x	X
ejpam-3856	42	13	:	:	PUNCT
ejpam-3856	42	14	a	a	DET
ejpam-3856	42	15	∩	∩	ADJ
ejpam-3856	42	16	u	u	NOUN
ejpam-3856	42	17	/∈	/∈	PUNCT
ejpam-3856	42	18	i	i	PRON
ejpam-3856	42	19	for	for	ADP
ejpam-3856	42	20	every	every	DET
ejpam-3856	42	21	u	u	PROPN
ejpam-3856	42	22	∈	∈	PROPN
ejpam-3856	42	23	so(x	so(x	NOUN
ejpam-3856	42	24	,	,	PUNCT
ejpam-3856	42	25	x	x	NOUN
ejpam-3856	42	26	)	)	PUNCT
ejpam-3856	42	27	}	}	PUNCT
ejpam-3856	42	28	is	be	AUX
ejpam-3856	42	29	called	call	VERB
ejpam-3856	42	30	the	the	DET
ejpam-3856	42	31	semi	semi	ADJ
ejpam-3856	42	32	-	-	ADJ
ejpam-3856	42	33	local	local	ADJ
ejpam-3856	42	34	function	function	NOUN
ejpam-3856	42	35	of	of	ADP
ejpam-3856	42	36	a	a	PRON
ejpam-3856	42	37	with	with	ADP
ejpam-3856	42	38	respect	respect	NOUN
ejpam-3856	42	39	to	to	ADP
ejpam-3856	42	40	i	i	PRON
ejpam-3856	42	41	and	and	CCONJ
ejpam-3856	42	42	τ	τ	PROPN
ejpam-3856	42	43	,	,	PUNCT
ejpam-3856	42	44	where	where	SCONJ
ejpam-3856	42	45	so(x	so(x	NOUN
ejpam-3856	42	46	,	,	PUNCT
ejpam-3856	42	47	x	x	X
ejpam-3856	42	48	)	)	PUNCT
ejpam-3856	43	1	=	=	SYM
ejpam-3856	43	2	{	{	PUNCT
ejpam-3856	43	3	u	u	NOUN
ejpam-3856	43	4	∈	∈	PROPN
ejpam-3856	43	5	so(x)|x	so(x)|x	NOUN
ejpam-3856	43	6	∈	∈	PROPN
ejpam-3856	43	7	u	u	NOUN
ejpam-3856	43	8	}	}	PUNCT
ejpam-3856	43	9	.	.	PUNCT
ejpam-3856	44	1	when	when	SCONJ
ejpam-3856	44	2	there	there	PRON
ejpam-3856	44	3	is	be	VERB
ejpam-3856	44	4	no	no	DET
ejpam-3856	44	5	ambiguity	ambiguity	NOUN
ejpam-3856	44	6	we	we	PRON
ejpam-3856	44	7	write	write	VERB
ejpam-3856	44	8	a∗s	a∗s	PROPN
ejpam-3856	44	9	for	for	ADP
ejpam-3856	44	10	(	(	PUNCT
ejpam-3856	44	11	a)∗	a)∗	PROPN
ejpam-3856	44	12	s	s	X
ejpam-3856	44	13	(	(	PUNCT
ejpam-3856	44	14	i	i	PROPN
ejpam-3856	44	15	,	,	PUNCT
ejpam-3856	44	16	τ	τ	PROPN
ejpam-3856	44	17	)	)	PUNCT
ejpam-3856	44	18	.	.	PUNCT
ejpam-3856	45	1	definition	definition	NOUN
ejpam-3856	45	2	5	5	NUM
ejpam-3856	45	3	.	.	PUNCT
ejpam-3856	46	1	let	let	VERB
ejpam-3856	46	2	(	(	PUNCT
ejpam-3856	46	3	x	x	NOUN
ejpam-3856	46	4	,	,	PUNCT
ejpam-3856	46	5	τ	τ	X
ejpam-3856	46	6	)	)	PUNCT
ejpam-3856	46	7	be	be	VERB
ejpam-3856	46	8	a	a	DET
ejpam-3856	46	9	topological	topological	ADJ
ejpam-3856	46	10	space	space	NOUN
ejpam-3856	46	11	.	.	PUNCT
ejpam-3856	47	1	a	a	DET
ejpam-3856	47	2	subset	subset	NOUN
ejpam-3856	47	3	a	a	PRON
ejpam-3856	47	4	of	of	ADP
ejpam-3856	47	5	x	x	SYM
ejpam-3856	47	6	is	be	AUX
ejpam-3856	47	7	said	say	VERB
ejpam-3856	47	8	to	to	PART
ejpam-3856	47	9	be	be	AUX
ejpam-3856	47	10	:	:	PUNCT
ejpam-3856	47	11	(	(	PUNCT
ejpam-3856	47	12	i	i	NOUN
ejpam-3856	47	13	)	)	PUNCT
ejpam-3856	47	14	β	β	X
ejpam-3856	47	15	-	-	PUNCT
ejpam-3856	47	16	open	open	ADJ
ejpam-3856	47	17	[	[	X
ejpam-3856	47	18	1	1	NUM
ejpam-3856	47	19	]	]	X
ejpam-3856	47	20	if	if	SCONJ
ejpam-3856	47	21	a	a	DET
ejpam-3856	47	22	⊂	⊂	PROPN
ejpam-3856	47	23	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-3856	47	24	)	)	PUNCT
ejpam-3856	47	25	)	)	PUNCT
ejpam-3856	47	26	)	)	PUNCT
ejpam-3856	47	27	,	,	PUNCT
ejpam-3856	47	28	(	(	PUNCT
ejpam-3856	47	29	ii	ii	NOUN
ejpam-3856	47	30	)	)	PUNCT
ejpam-3856	47	31	semi	semi	ADJ
ejpam-3856	47	32	-	-	ADJ
ejpam-3856	47	33	open	open	ADJ
ejpam-3856	47	34	[	[	X
ejpam-3856	47	35	12	12	NUM
ejpam-3856	47	36	]	]	X
ejpam-3856	47	37	if	if	SCONJ
ejpam-3856	47	38	a	a	DET
ejpam-3856	47	39	⊂	⊂	PROPN
ejpam-3856	47	40	cl(int(a	cl(int(a	PROPN
ejpam-3856	47	41	)	)	PUNCT
ejpam-3856	47	42	)	)	PUNCT
ejpam-3856	47	43	,	,	PUNCT
ejpam-3856	47	44	(	(	PUNCT
ejpam-3856	47	45	iii	iii	NOUN
ejpam-3856	47	46	)	)	PUNCT
ejpam-3856	47	47	regular	regular	ADJ
ejpam-3856	47	48	-	-	PUNCT
ejpam-3856	47	49	open	open	NOUN
ejpam-3856	47	50	[	[	X
ejpam-3856	47	51	13	13	NUM
ejpam-3856	47	52	]	]	PUNCT
ejpam-3856	47	53	if	if	SCONJ
ejpam-3856	47	54	a	a	PRON
ejpam-3856	47	55	=	=	X
ejpam-3856	47	56	int(cl(a	int(cl(a	PROPN
ejpam-3856	47	57	)	)	PUNCT
ejpam-3856	47	58	)	)	PUNCT
ejpam-3856	47	59	.	.	PUNCT
ejpam-3856	48	1	the	the	DET
ejpam-3856	48	2	family	family	NOUN
ejpam-3856	48	3	of	of	ADP
ejpam-3856	48	4	all	all	DET
ejpam-3856	48	5	β	β	NOUN
ejpam-3856	48	6	-	-	ADJ
ejpam-3856	48	7	open	open	ADJ
ejpam-3856	48	8	(	(	PUNCT
ejpam-3856	48	9	resp	resp	NOUN
ejpam-3856	48	10	.	.	PUNCT
ejpam-3856	49	1	semi	semi	ADJ
ejpam-3856	49	2	-	-	ADJ
ejpam-3856	49	3	open	open	ADJ
ejpam-3856	49	4	,	,	PUNCT
ejpam-3856	49	5	regular	regular	ADJ
ejpam-3856	49	6	open	open	ADJ
ejpam-3856	49	7	)	)	PUNCT
ejpam-3856	49	8	sets	set	NOUN
ejpam-3856	49	9	in	in	ADP
ejpam-3856	49	10	x	x	VERB
ejpam-3856	49	11	is	be	AUX
ejpam-3856	49	12	denoted	denote	VERB
ejpam-3856	49	13	by	by	ADP
ejpam-3856	49	14	βo(x	βo(x	PUNCT
ejpam-3856	49	15	)	)	PUNCT
ejpam-3856	49	16	(	(	PUNCT
ejpam-3856	49	17	resp	resp	NOUN
ejpam-3856	49	18	.	.	PUNCT
ejpam-3856	49	19	so(x	so(x	NUM
ejpam-3856	49	20	)	)	PUNCT
ejpam-3856	49	21	,	,	PUNCT
ejpam-3856	49	22	ro(x	ro(x	ADJ
ejpam-3856	49	23	)	)	PUNCT
ejpam-3856	49	24	)	)	PUNCT
ejpam-3856	49	25	.	.	PUNCT
ejpam-3856	50	1	definition	definition	NOUN
ejpam-3856	50	2	6	6	NUM
ejpam-3856	50	3	.	.	PUNCT
ejpam-3856	51	1	[	[	X
ejpam-3856	51	2	1	1	X
ejpam-3856	51	3	]	]	X
ejpam-3856	51	4	let	let	VERB
ejpam-3856	51	5	(	(	PUNCT
ejpam-3856	51	6	x	x	NOUN
ejpam-3856	51	7	,	,	PUNCT
ejpam-3856	51	8	τ	τ	X
ejpam-3856	51	9	)	)	PUNCT
ejpam-3856	51	10	be	be	VERB
ejpam-3856	51	11	a	a	DET
ejpam-3856	51	12	topological	topological	ADJ
ejpam-3856	51	13	space	space	NOUN
ejpam-3856	51	14	.	.	PUNCT
ejpam-3856	52	1	a	a	DET
ejpam-3856	52	2	subset	subset	NOUN
ejpam-3856	52	3	a	a	PRON
ejpam-3856	52	4	of	of	ADP
ejpam-3856	52	5	x	x	SYM
ejpam-3856	52	6	is	be	AUX
ejpam-3856	52	7	said	say	VERB
ejpam-3856	52	8	to	to	PART
ejpam-3856	52	9	be	be	AUX
ejpam-3856	52	10	β	β	NOUN
ejpam-3856	52	11	-	-	VERB
ejpam-3856	52	12	closed	closed	ADJ
ejpam-3856	52	13	if	if	SCONJ
ejpam-3856	52	14	its	its	PRON
ejpam-3856	52	15	complement	complement	NOUN
ejpam-3856	52	16	is	be	AUX
ejpam-3856	52	17	β	β	NOUN
ejpam-3856	52	18	-	-	ADJ
ejpam-3856	52	19	open	open	ADJ
ejpam-3856	52	20	.	.	PUNCT
ejpam-3856	53	1	definition	definition	NOUN
ejpam-3856	53	2	7	7	NUM
ejpam-3856	53	3	.	.	PUNCT
ejpam-3856	54	1	[	[	X
ejpam-3856	54	2	4	4	X
ejpam-3856	54	3	]	]	X
ejpam-3856	54	4	let	let	VERB
ejpam-3856	54	5	(	(	PUNCT
ejpam-3856	54	6	x	x	NOUN
ejpam-3856	54	7	,	,	PUNCT
ejpam-3856	54	8	τ	τ	X
ejpam-3856	54	9	)	)	PUNCT
ejpam-3856	54	10	be	be	VERB
ejpam-3856	54	11	a	a	DET
ejpam-3856	54	12	topological	topological	ADJ
ejpam-3856	54	13	space	space	NOUN
ejpam-3856	54	14	and	and	CCONJ
ejpam-3856	54	15	a	a	DET
ejpam-3856	54	16	be	be	AUX
ejpam-3856	54	17	a	a	DET
ejpam-3856	54	18	subset	subset	NOUN
ejpam-3856	54	19	of	of	ADP
ejpam-3856	54	20	x.	x.	NOUN
ejpam-3856	54	21	the	the	DET
ejpam-3856	54	22	β	β	NOUN
ejpam-3856	54	23	-	-	NOUN
ejpam-3856	54	24	closure	closure	NOUN
ejpam-3856	54	25	of	of	ADP
ejpam-3856	54	26	a	a	PRON
ejpam-3856	54	27	is	be	AUX
ejpam-3856	54	28	defined	define	VERB
ejpam-3856	54	29	by	by	ADP
ejpam-3856	54	30	the	the	DET
ejpam-3856	54	31	intersection	intersection	NOUN
ejpam-3856	54	32	of	of	ADP
ejpam-3856	54	33	all	all	DET
ejpam-3856	54	34	β	β	ADJ
ejpam-3856	54	35	-	-	ADJ
ejpam-3856	54	36	closed	closed	ADJ
ejpam-3856	54	37	sets	set	NOUN
ejpam-3856	54	38	containing	contain	VERB
ejpam-3856	54	39	the	the	DET
ejpam-3856	54	40	set	set	NOUN
ejpam-3856	54	41	a	a	PRON
ejpam-3856	55	1	and	and	CCONJ
ejpam-3856	55	2	it	it	PRON
ejpam-3856	55	3	is	be	AUX
ejpam-3856	55	4	denoted	denote	VERB
ejpam-3856	55	5	by	by	ADP
ejpam-3856	55	6	βcl(a	βcl(a	NOUN
ejpam-3856	55	7	)	)	PUNCT
ejpam-3856	55	8	.	.	PUNCT
ejpam-3856	56	1	p.	p.	NOUN
ejpam-3856	56	2	l.	l.	PROPN
ejpam-3856	56	3	powar	powar	PROPN
ejpam-3856	56	4	,	,	PUNCT
ejpam-3856	56	5	t.	t.	PROPN
ejpam-3856	56	6	noiri	noiri	PROPN
ejpam-3856	56	7	,	,	PUNCT
ejpam-3856	56	8	shikha	shikha	PROPN
ejpam-3856	56	9	bhadauria	bhadauria	PROPN
ejpam-3856	56	10	/	/	SYM
ejpam-3856	56	11	eur	eur	PROPN
ejpam-3856	56	12	.	.	PUNCT
ejpam-3856	57	1	j.	j.	PROPN
ejpam-3856	57	2	pure	pure	PROPN
ejpam-3856	57	3	appl	appl	PROPN
ejpam-3856	57	4	.	.	PROPN
ejpam-3856	57	5	math	math	PROPN
ejpam-3856	57	6	,	,	PUNCT
ejpam-3856	57	7	13	13	NUM
ejpam-3856	57	8	(	(	PUNCT
ejpam-3856	57	9	4	4	NUM
ejpam-3856	57	10	)	)	PUNCT
ejpam-3856	57	11	(	(	PUNCT
ejpam-3856	57	12	2020	2020	NUM
ejpam-3856	57	13	)	)	PUNCT
ejpam-3856	57	14	,	,	PUNCT
ejpam-3856	57	15	758	758	NUM
ejpam-3856	57	16	-	-	SYM
ejpam-3856	57	17	765	765	NUM
ejpam-3856	57	18	760	760	NUM
ejpam-3856	57	19	3	3	NUM
ejpam-3856	57	20	.	.	PUNCT
ejpam-3856	57	21	β	β	X
ejpam-3856	57	22	-	-	ADJ
ejpam-3856	57	23	local	local	ADJ
ejpam-3856	57	24	functions	function	NOUN
ejpam-3856	57	25	in	in	ADP
ejpam-3856	57	26	order	order	NOUN
ejpam-3856	57	27	to	to	PART
ejpam-3856	57	28	define	define	VERB
ejpam-3856	57	29	the	the	DET
ejpam-3856	57	30	generalized	generalized	ADJ
ejpam-3856	57	31	version	version	NOUN
ejpam-3856	57	32	of	of	ADP
ejpam-3856	57	33	the	the	DET
ejpam-3856	57	34	local	local	ADJ
ejpam-3856	57	35	function	function	NOUN
ejpam-3856	57	36	[	[	X
ejpam-3856	57	37	8	8	NUM
ejpam-3856	57	38	]	]	PUNCT
ejpam-3856	57	39	,	,	PUNCT
ejpam-3856	57	40	we	we	PRON
ejpam-3856	57	41	now	now	ADV
ejpam-3856	57	42	introduce	introduce	VERB
ejpam-3856	57	43	the	the	DET
ejpam-3856	57	44	concept	concept	NOUN
ejpam-3856	57	45	of	of	ADP
ejpam-3856	57	46	the	the	DET
ejpam-3856	57	47	β	β	ADJ
ejpam-3856	57	48	-	-	ADJ
ejpam-3856	57	49	local	local	ADJ
ejpam-3856	57	50	function	function	NOUN
ejpam-3856	57	51	.	.	PUNCT
ejpam-3856	58	1	definition	definition	NOUN
ejpam-3856	58	2	8	8	NUM
ejpam-3856	58	3	.	.	PUNCT
ejpam-3856	59	1	let	let	VERB
ejpam-3856	59	2	(	(	PUNCT
ejpam-3856	59	3	x	x	X
ejpam-3856	59	4	,	,	PUNCT
ejpam-3856	59	5	τ	τ	PROPN
ejpam-3856	59	6	,	,	PUNCT
ejpam-3856	59	7	i	i	PRON
ejpam-3856	59	8	)	)	PUNCT
ejpam-3856	59	9	be	be	VERB
ejpam-3856	59	10	an	an	DET
ejpam-3856	59	11	ideal	ideal	ADJ
ejpam-3856	59	12	topological	topological	ADJ
ejpam-3856	59	13	space	space	NOUN
ejpam-3856	59	14	.	.	PUNCT
ejpam-3856	60	1	for	for	ADP
ejpam-3856	60	2	a	a	DET
ejpam-3856	60	3	set	set	NOUN
ejpam-3856	60	4	a	a	DET
ejpam-3856	60	5	⊂	⊂	PROPN
ejpam-3856	60	6	x	x	NOUN
ejpam-3856	60	7	,	,	PUNCT
ejpam-3856	60	8	a∗	a∗	PROPN
ejpam-3856	60	9	β(i	β(i	PRON
ejpam-3856	60	10	,	,	PUNCT
ejpam-3856	60	11	βo(x	βo(x	PUNCT
ejpam-3856	60	12	)	)	PUNCT
ejpam-3856	60	13	)	)	PUNCT
ejpam-3856	61	1	=	=	PRON
ejpam-3856	61	2	{	{	PUNCT
ejpam-3856	61	3	x	x	SYM
ejpam-3856	61	4	∈	∈	PROPN
ejpam-3856	61	5	x	x	X
ejpam-3856	61	6	:	:	PUNCT
ejpam-3856	61	7	a	a	DET
ejpam-3856	61	8	∩	∩	ADJ
ejpam-3856	61	9	u	u	NOUN
ejpam-3856	61	10	/∈	/∈	PUNCT
ejpam-3856	61	11	i	i	PRON
ejpam-3856	61	12	for	for	ADP
ejpam-3856	61	13	every	every	DET
ejpam-3856	61	14	u	u	PROPN
ejpam-3856	61	15	∈	∈	PROPN
ejpam-3856	61	16	βo(x	βo(x	PUNCT
ejpam-3856	61	17	)	)	PUNCT
ejpam-3856	61	18	}	}	PUNCT
ejpam-3856	61	19	,	,	PUNCT
ejpam-3856	61	20	where	where	SCONJ
ejpam-3856	61	21	βo(x	βo(x	PUNCT
ejpam-3856	61	22	)	)	PUNCT
ejpam-3856	61	23	=	=	SYM
ejpam-3856	61	24	{	{	PUNCT
ejpam-3856	61	25	u	u	NOUN
ejpam-3856	61	26	∈	∈	PROPN
ejpam-3856	61	27	βo(x	βo(x	PUNCT
ejpam-3856	61	28	)	)	PUNCT
ejpam-3856	61	29	:	:	PUNCT
ejpam-3856	61	30	x	x	X
ejpam-3856	61	31	∈	∈	X
ejpam-3856	61	32	u	u	NOUN
ejpam-3856	61	33	}	}	PUNCT
ejpam-3856	61	34	,	,	PUNCT
ejpam-3856	61	35	is	be	AUX
ejpam-3856	61	36	called	call	VERB
ejpam-3856	61	37	the	the	DET
ejpam-3856	61	38	β	β	ADJ
ejpam-3856	61	39	-	-	ADJ
ejpam-3856	61	40	local	local	ADJ
ejpam-3856	61	41	function	function	NOUN
ejpam-3856	61	42	of	of	ADP
ejpam-3856	61	43	a	a	PRON
ejpam-3856	61	44	with	with	ADP
ejpam-3856	61	45	respect	respect	NOUN
ejpam-3856	61	46	to	to	ADP
ejpam-3856	61	47	i	i	PRON
ejpam-3856	61	48	and	and	CCONJ
ejpam-3856	61	49	βo(x	βo(x	NUM
ejpam-3856	61	50	)	)	PUNCT
ejpam-3856	61	51	.	.	PUNCT
ejpam-3856	62	1	a∗	a∗	PROPN
ejpam-3856	62	2	β(i	β(i	PRON
ejpam-3856	62	3	,	,	PUNCT
ejpam-3856	62	4	βo(x	βo(x	PUNCT
ejpam-3856	62	5	)	)	PUNCT
ejpam-3856	62	6	)	)	PUNCT
ejpam-3856	62	7	is	be	AUX
ejpam-3856	62	8	simply	simply	ADV
ejpam-3856	62	9	denoted	denote	VERB
ejpam-3856	62	10	by	by	ADP
ejpam-3856	62	11	a∗	a∗	PROPN
ejpam-3856	62	12	β	β	PROPN
ejpam-3856	62	13	.	.	PUNCT
ejpam-3856	62	14	example	example	NOUN
ejpam-3856	63	1	1	1	NUM
ejpam-3856	63	2	.	.	PUNCT
ejpam-3856	64	1	let	let	VERB
ejpam-3856	64	2	x	x	PUNCT
ejpam-3856	64	3	=	=	PRON
ejpam-3856	64	4	{	{	PUNCT
ejpam-3856	64	5	a	a	PRON
ejpam-3856	64	6	,	,	PUNCT
ejpam-3856	64	7	b	b	NOUN
ejpam-3856	64	8	,	,	PUNCT
ejpam-3856	64	9	c	c	NOUN
ejpam-3856	64	10	,	,	PUNCT
ejpam-3856	64	11	d	d	AUX
ejpam-3856	64	12	}	}	PUNCT
ejpam-3856	64	13	be	be	AUX
ejpam-3856	64	14	a	a	DET
ejpam-3856	64	15	nonempty	nonempty	NOUN
ejpam-3856	64	16	set	set	VERB
ejpam-3856	64	17	with	with	ADP
ejpam-3856	64	18	the	the	DET
ejpam-3856	64	19	topology	topology	NOUN
ejpam-3856	64	20	τ	τ	X
ejpam-3856	64	21	=	=	SYM
ejpam-3856	64	22	{	{	PUNCT
ejpam-3856	64	23	φ	φ	PROPN
ejpam-3856	64	24	,	,	PUNCT
ejpam-3856	64	25	x	x	X
ejpam-3856	64	26	,	,	PUNCT
ejpam-3856	64	27	{	{	PUNCT
ejpam-3856	64	28	a	a	X
ejpam-3856	64	29	}	}	PUNCT
ejpam-3856	64	30	,	,	PUNCT
ejpam-3856	64	31	{	{	PUNCT
ejpam-3856	64	32	a	a	PRON
ejpam-3856	64	33	,	,	PUNCT
ejpam-3856	64	34	b	b	NOUN
ejpam-3856	64	35	,	,	PUNCT
ejpam-3856	64	36	c	c	NOUN
ejpam-3856	64	37	}	}	PUNCT
ejpam-3856	64	38	}	}	PUNCT
ejpam-3856	64	39	.	.	PUNCT
ejpam-3856	65	1	then	then	ADV
ejpam-3856	65	2	the	the	DET
ejpam-3856	65	3	collection	collection	NOUN
ejpam-3856	65	4	of	of	ADP
ejpam-3856	65	5	closed	closed	ADJ
ejpam-3856	65	6	sets	set	NOUN
ejpam-3856	65	7	is	be	AUX
ejpam-3856	65	8	τf	τf	ADP
ejpam-3856	65	9	=	=	SYM
ejpam-3856	65	10	{	{	PUNCT
ejpam-3856	65	11	x	x	PROPN
ejpam-3856	65	12	,	,	PUNCT
ejpam-3856	65	13	φ	φ	NUM
ejpam-3856	65	14	,	,	PUNCT
ejpam-3856	65	15	{	{	PUNCT
ejpam-3856	65	16	b	b	NOUN
ejpam-3856	65	17	,	,	PUNCT
ejpam-3856	65	18	c	c	NOUN
ejpam-3856	65	19	,	,	PUNCT
ejpam-3856	65	20	d	d	NOUN
ejpam-3856	65	21	}	}	PUNCT
ejpam-3856	65	22	,	,	PUNCT
ejpam-3856	65	23	{	{	PUNCT
ejpam-3856	65	24	d	d	NOUN
ejpam-3856	65	25	}	}	PUNCT
ejpam-3856	65	26	}	}	PUNCT
ejpam-3856	65	27	.	.	PUNCT
ejpam-3856	66	1	applying	apply	VERB
ejpam-3856	66	2	definition	definition	NOUN
ejpam-3856	66	3	5	5	NUM
ejpam-3856	66	4	,	,	PUNCT
ejpam-3856	66	5	we	we	PRON
ejpam-3856	66	6	compute	compute	VERB
ejpam-3856	66	7	the	the	DET
ejpam-3856	66	8	collection	collection	NOUN
ejpam-3856	66	9	βo(x)=	βo(x)=	X
ejpam-3856	66	10	{	{	PUNCT
ejpam-3856	66	11	φ	φ	PROPN
ejpam-3856	66	12	,	,	PUNCT
ejpam-3856	66	13	x	x	X
ejpam-3856	66	14	,	,	PUNCT
ejpam-3856	66	15	{	{	PUNCT
ejpam-3856	66	16	a	a	X
ejpam-3856	66	17	}	}	PUNCT
ejpam-3856	66	18	,	,	PUNCT
ejpam-3856	66	19	{	{	PUNCT
ejpam-3856	66	20	a	a	DET
ejpam-3856	66	21	,	,	PUNCT
ejpam-3856	66	22	b	b	NOUN
ejpam-3856	66	23	}	}	PUNCT
ejpam-3856	66	24	,	,	PUNCT
ejpam-3856	66	25	{	{	PUNCT
ejpam-3856	66	26	a	a	PRON
ejpam-3856	66	27	,	,	PUNCT
ejpam-3856	66	28	d	d	NOUN
ejpam-3856	66	29	}	}	PUNCT
ejpam-3856	66	30	,	,	PUNCT
ejpam-3856	66	31	{	{	PUNCT
ejpam-3856	66	32	a	a	DET
ejpam-3856	66	33	,	,	PUNCT
ejpam-3856	66	34	b	b	NOUN
ejpam-3856	66	35	,	,	PUNCT
ejpam-3856	66	36	c	c	NOUN
ejpam-3856	66	37	}	}	PUNCT
ejpam-3856	66	38	,	,	PUNCT
ejpam-3856	66	39	{	{	PUNCT
ejpam-3856	66	40	c	c	X
ejpam-3856	66	41	,	,	PUNCT
ejpam-3856	66	42	d	d	NOUN
ejpam-3856	66	43	,	,	PUNCT
ejpam-3856	66	44	a	a	PRON
ejpam-3856	66	45	}	}	PUNCT
ejpam-3856	66	46	,	,	PUNCT
ejpam-3856	66	47	{	{	PUNCT
ejpam-3856	66	48	d	d	X
ejpam-3856	66	49	,	,	PUNCT
ejpam-3856	66	50	a	a	DET
ejpam-3856	66	51	,	,	PUNCT
ejpam-3856	66	52	b	b	NOUN
ejpam-3856	66	53	}	}	PUNCT
ejpam-3856	66	54	,	,	PUNCT
ejpam-3856	66	55	{	{	PUNCT
ejpam-3856	66	56	a	a	PRON
ejpam-3856	66	57	,	,	PUNCT
ejpam-3856	66	58	c	c	NOUN
ejpam-3856	66	59	}	}	PUNCT
ejpam-3856	66	60	}	}	PUNCT
ejpam-3856	66	61	.	.	PUNCT
ejpam-3856	67	1	next	next	ADV
ejpam-3856	67	2	,	,	PUNCT
ejpam-3856	67	3	we	we	PRON
ejpam-3856	67	4	consider	consider	VERB
ejpam-3856	67	5	i	i	PRON
ejpam-3856	67	6	=	=	PUNCT
ejpam-3856	67	7	{	{	PUNCT
ejpam-3856	67	8	φ	φ	PROPN
ejpam-3856	67	9	,	,	PUNCT
ejpam-3856	67	10	{	{	PUNCT
ejpam-3856	67	11	b	b	NOUN
ejpam-3856	67	12	}	}	PUNCT
ejpam-3856	67	13	}	}	PUNCT
ejpam-3856	67	14	.	.	PUNCT
ejpam-3856	68	1	if	if	SCONJ
ejpam-3856	68	2	a	a	PRON
ejpam-3856	68	3	=	=	X
ejpam-3856	68	4	{	{	PUNCT
ejpam-3856	68	5	c	c	NOUN
ejpam-3856	68	6	,	,	PUNCT
ejpam-3856	68	7	d	d	NOUN
ejpam-3856	68	8	}	}	PUNCT
ejpam-3856	68	9	⊂	⊂	PROPN
ejpam-3856	68	10	x	x	X
ejpam-3856	68	11	then	then	ADV
ejpam-3856	68	12	it	it	PRON
ejpam-3856	68	13	may	may	AUX
ejpam-3856	68	14	be	be	AUX
ejpam-3856	68	15	easily	easily	ADV
ejpam-3856	68	16	verified	verify	VERB
ejpam-3856	68	17	that	that	SCONJ
ejpam-3856	68	18	a∗	a∗	PROPN
ejpam-3856	68	19	β	β	X
ejpam-3856	68	20	=	=	PUNCT
ejpam-3856	68	21	{	{	PUNCT
ejpam-3856	68	22	c	c	X
ejpam-3856	68	23	,	,	PUNCT
ejpam-3856	68	24	d	d	NOUN
ejpam-3856	68	25	}	}	PUNCT
ejpam-3856	68	26	and	and	CCONJ
ejpam-3856	68	27	a∗	a∗	PROPN
ejpam-3856	68	28	=	=	SYM
ejpam-3856	68	29	{	{	PUNCT
ejpam-3856	68	30	b	b	PROPN
ejpam-3856	68	31	,	,	PUNCT
ejpam-3856	68	32	c	c	NOUN
ejpam-3856	68	33	,	,	PUNCT
ejpam-3856	68	34	d	d	NOUN
ejpam-3856	68	35	}	}	PUNCT
ejpam-3856	68	36	.	.	PUNCT
ejpam-3856	69	1	we	we	PRON
ejpam-3856	69	2	need	need	VERB
ejpam-3856	69	3	the	the	DET
ejpam-3856	69	4	following	follow	VERB
ejpam-3856	69	5	lemma	lemma	PROPN
ejpam-3856	69	6	for	for	ADP
ejpam-3856	69	7	our	our	PRON
ejpam-3856	69	8	analysis	analysis	NOUN
ejpam-3856	69	9	.	.	PUNCT
ejpam-3856	70	1	lemma	lemma	PROPN
ejpam-3856	70	2	1	1	NUM
ejpam-3856	70	3	.	.	PUNCT
ejpam-3856	71	1	[	[	X
ejpam-3856	71	2	4	4	X
ejpam-3856	71	3	]	]	PUNCT
ejpam-3856	71	4	let	let	VERB
ejpam-3856	71	5	a	a	PRON
ejpam-3856	71	6	be	be	AUX
ejpam-3856	71	7	a	a	DET
ejpam-3856	71	8	subset	subset	NOUN
ejpam-3856	71	9	of	of	ADP
ejpam-3856	71	10	a	a	DET
ejpam-3856	71	11	topological	topological	ADJ
ejpam-3856	71	12	space	space	NOUN
ejpam-3856	71	13	(	(	PUNCT
ejpam-3856	71	14	x	x	X
ejpam-3856	71	15	,	,	PUNCT
ejpam-3856	71	16	τ	τ	PROPN
ejpam-3856	71	17	)	)	PUNCT
ejpam-3856	71	18	.	.	PUNCT
ejpam-3856	72	1	then	then	ADV
ejpam-3856	72	2	x	x	SYM
ejpam-3856	72	3	∈	∈	PROPN
ejpam-3856	72	4	βcl(a	βcl(a	PROPN
ejpam-3856	72	5	)	)	PUNCT
ejpam-3856	72	6	if	if	SCONJ
ejpam-3856	72	7	and	and	CCONJ
ejpam-3856	72	8	only	only	ADV
ejpam-3856	72	9	if	if	SCONJ
ejpam-3856	72	10	a	a	DET
ejpam-3856	72	11	∩	∩	ADJ
ejpam-3856	72	12	u	u	NOUN
ejpam-3856	72	13	6=	6=	PROPN
ejpam-3856	72	14	φ	φ	PROPN
ejpam-3856	72	15	for	for	ADP
ejpam-3856	72	16	every	every	DET
ejpam-3856	72	17	u	u	NOUN
ejpam-3856	72	18	in	in	ADP
ejpam-3856	72	19	βo(x	βo(x	PUNCT
ejpam-3856	72	20	)	)	PUNCT
ejpam-3856	72	21	.	.	PUNCT
ejpam-3856	73	1	theorem	theorem	NOUN
ejpam-3856	73	2	1	1	X
ejpam-3856	73	3	.	.	PUNCT
ejpam-3856	74	1	let	let	VERB
ejpam-3856	74	2	(	(	PUNCT
ejpam-3856	74	3	x	x	X
ejpam-3856	74	4	,	,	PUNCT
ejpam-3856	74	5	τ	τ	PROPN
ejpam-3856	74	6	,	,	PUNCT
ejpam-3856	74	7	i	i	PRON
ejpam-3856	74	8	)	)	PUNCT
ejpam-3856	74	9	be	be	VERB
ejpam-3856	74	10	an	an	DET
ejpam-3856	74	11	ideal	ideal	ADJ
ejpam-3856	74	12	topological	topological	ADJ
ejpam-3856	74	13	space	space	NOUN
ejpam-3856	74	14	and	and	CCONJ
ejpam-3856	74	15	a	a	DET
ejpam-3856	74	16	,	,	PUNCT
ejpam-3856	74	17	b	b	PROPN
ejpam-3856	74	18	be	be	AUX
ejpam-3856	74	19	subsets	subset	NOUN
ejpam-3856	74	20	of	of	ADP
ejpam-3856	74	21	x.	x.	NOUN
ejpam-3856	74	22	then	then	ADV
ejpam-3856	74	23	the	the	DET
ejpam-3856	74	24	following	follow	VERB
ejpam-3856	74	25	properties	property	NOUN
ejpam-3856	74	26	hold	hold	VERB
ejpam-3856	74	27	:	:	PUNCT
ejpam-3856	74	28	(	(	PUNCT
ejpam-3856	74	29	1	1	NUM
ejpam-3856	74	30	)	)	PUNCT
ejpam-3856	74	31	.	.	PUNCT
ejpam-3856	75	1	if	if	SCONJ
ejpam-3856	75	2	a	a	DET
ejpam-3856	75	3	⊂	⊂	PROPN
ejpam-3856	75	4	b	b	PROPN
ejpam-3856	75	5	then	then	ADV
ejpam-3856	75	6	a∗	a∗	PROPN
ejpam-3856	75	7	β	β	X
ejpam-3856	75	8	⊂	⊂	PROPN
ejpam-3856	75	9	b∗	b∗	PROPN
ejpam-3856	75	10	β	β	X
ejpam-3856	75	11	,	,	PUNCT
ejpam-3856	75	12	(	(	PUNCT
ejpam-3856	75	13	2	2	NUM
ejpam-3856	75	14	)	)	PUNCT
ejpam-3856	75	15	.	.	PUNCT
ejpam-3856	76	1	(	(	PUNCT
ejpam-3856	76	2	a	a	DET
ejpam-3856	76	3	∪b)∗β	∪b)∗β	X
ejpam-3856	76	4	=	=	SYM
ejpam-3856	76	5	a∗	a∗	PROPN
ejpam-3856	76	6	β	β	PROPN
ejpam-3856	76	7	∪b∗	∪b∗	NUM
ejpam-3856	76	8	β	β	X
ejpam-3856	76	9	,	,	PUNCT
ejpam-3856	76	10	(	(	PUNCT
ejpam-3856	76	11	3	3	NUM
ejpam-3856	76	12	)	)	PUNCT
ejpam-3856	76	13	.	.	PUNCT
ejpam-3856	77	1	(	(	PUNCT
ejpam-3856	77	2	a	a	DET
ejpam-3856	77	3	∩b)∗β	∩b)∗β	NOUN
ejpam-3856	77	4	⊂	⊂	X
ejpam-3856	77	5	a∗	a∗	PROPN
ejpam-3856	77	6	β	β	PROPN
ejpam-3856	77	7	∩b∗	∩b∗	X
ejpam-3856	77	8	β	β	X
ejpam-3856	77	9	,	,	PUNCT
ejpam-3856	77	10	(	(	PUNCT
ejpam-3856	77	11	4	4	NUM
ejpam-3856	77	12	)	)	PUNCT
ejpam-3856	77	13	.	.	PUNCT
ejpam-3856	78	1	(	(	PUNCT
ejpam-3856	78	2	a∗	a∗	PROPN
ejpam-3856	78	3	β)∗β	β)∗β	NOUN
ejpam-3856	78	4	⊂	⊂	PROPN
ejpam-3856	78	5	a∗	a∗	PROPN
ejpam-3856	78	6	β	β	PROPN
ejpam-3856	78	7	,	,	PUNCT
ejpam-3856	78	8	(	(	PUNCT
ejpam-3856	78	9	5	5	NUM
ejpam-3856	78	10	)	)	PUNCT
ejpam-3856	78	11	.	.	PUNCT
ejpam-3856	79	1	a∗	a∗	PROPN
ejpam-3856	79	2	β	β	X
ejpam-3856	79	3	=	=	PUNCT
ejpam-3856	79	4	βcl(a∗	βcl(a∗	PUNCT
ejpam-3856	79	5	β	β	NOUN
ejpam-3856	79	6	)	)	PUNCT
ejpam-3856	79	7	⊂	⊂	PROPN
ejpam-3856	79	8	βcl(a	βcl(a	PROPN
ejpam-3856	79	9	)	)	PUNCT
ejpam-3856	79	10	.	.	PUNCT
ejpam-3856	80	1	proof	proof	NOUN
ejpam-3856	80	2	.	.	PUNCT
ejpam-3856	81	1	(	(	PUNCT
ejpam-3856	81	2	1	1	NUM
ejpam-3856	81	3	)	)	PUNCT
ejpam-3856	81	4	.	.	PUNCT
ejpam-3856	82	1	if	if	SCONJ
ejpam-3856	82	2	x	x	PROPN
ejpam-3856	82	3	/∈	/∈	PUNCT
ejpam-3856	83	1	b∗	b∗	ADJ
ejpam-3856	83	2	β	β	X
ejpam-3856	83	3	,	,	PUNCT
ejpam-3856	83	4	then	then	ADV
ejpam-3856	83	5	there	there	PRON
ejpam-3856	83	6	exists	exist	VERB
ejpam-3856	83	7	u	u	PROPN
ejpam-3856	83	8	∈	∈	PROPN
ejpam-3856	83	9	βo(x	βo(x	PUNCT
ejpam-3856	83	10	)	)	PUNCT
ejpam-3856	84	1	such	such	ADJ
ejpam-3856	84	2	that	that	SCONJ
ejpam-3856	84	3	u	u	PROPN
ejpam-3856	84	4	∩b	∩b	PROPN
ejpam-3856	84	5	∈	∈	PROPN
ejpam-3856	84	6	i.	i.	NOUN
ejpam-3856	84	7	since	since	SCONJ
ejpam-3856	84	8	a	a	DET
ejpam-3856	84	9	⊂	⊂	PROPN
ejpam-3856	84	10	b	b	PROPN
ejpam-3856	84	11	,	,	PUNCT
ejpam-3856	84	12	u	u	NOUN
ejpam-3856	84	13	∩a	∩a	PROPN
ejpam-3856	84	14	∈	∈	PROPN
ejpam-3856	84	15	i	i	PRON
ejpam-3856	84	16	and	and	CCONJ
ejpam-3856	84	17	hence	hence	ADV
ejpam-3856	84	18	x	x	PROPN
ejpam-3856	84	19	/∈	/∈	PUNCT
ejpam-3856	84	20	a∗	a∗	PROPN
ejpam-3856	84	21	β	β	X
ejpam-3856	84	22	.	.	PUNCT
ejpam-3856	85	1	this	this	PRON
ejpam-3856	85	2	shows	show	VERB
ejpam-3856	85	3	that	that	SCONJ
ejpam-3856	85	4	a∗	a∗	PROPN
ejpam-3856	85	5	β	β	X
ejpam-3856	85	6	⊂	⊂	PROPN
ejpam-3856	85	7	b∗	b∗	PROPN
ejpam-3856	85	8	β	β	X
ejpam-3856	85	9	.	.	PUNCT
ejpam-3856	86	1	(	(	PUNCT
ejpam-3856	86	2	2	2	NUM
ejpam-3856	86	3	)	)	PUNCT
ejpam-3856	86	4	.	.	PUNCT
ejpam-3856	87	1	let	let	VERB
ejpam-3856	87	2	x	x	X
ejpam-3856	87	3	∈	∈	PROPN
ejpam-3856	87	4	(	(	PUNCT
ejpam-3856	87	5	a∪b)∗β	a∪b)∗β	PROPN
ejpam-3856	87	6	,	,	PUNCT
ejpam-3856	87	7	then	then	ADV
ejpam-3856	87	8	using	use	VERB
ejpam-3856	87	9	definition	definition	NOUN
ejpam-3856	87	10	8	8	NUM
ejpam-3856	87	11	,	,	PUNCT
ejpam-3856	87	12	we	we	PRON
ejpam-3856	87	13	have	have	VERB
ejpam-3856	87	14	u∩(a∪b	u∩(a∪b	NOUN
ejpam-3856	87	15	)	)	PUNCT
ejpam-3856	87	16	/∈	/∈	PUNCT
ejpam-3856	88	1	i	i	PRON
ejpam-3856	88	2	for	for	ADP
ejpam-3856	88	3	every	every	DET
ejpam-3856	88	4	u	u	PROPN
ejpam-3856	88	5	∈	∈	PROPN
ejpam-3856	88	6	βo(x	βo(x	PUNCT
ejpam-3856	88	7	)	)	PUNCT
ejpam-3856	88	8	and	and	CCONJ
ejpam-3856	88	9	(	(	PUNCT
ejpam-3856	88	10	u	u	NOUN
ejpam-3856	88	11	∩a	∩a	PROPN
ejpam-3856	88	12	)	)	PUNCT
ejpam-3856	88	13	∪	∪	NOUN
ejpam-3856	88	14	(	(	PUNCT
ejpam-3856	88	15	u	u	NOUN
ejpam-3856	88	16	∩b	∩b	PROPN
ejpam-3856	88	17	)	)	PUNCT
ejpam-3856	88	18	/∈	/∈	PUNCT
ejpam-3856	89	1	i.	i.	NOUN
ejpam-3856	89	2	now	now	ADV
ejpam-3856	89	3	,	,	PUNCT
ejpam-3856	89	4	since	since	SCONJ
ejpam-3856	89	5	i	i	PRON
ejpam-3856	89	6	is	be	AUX
ejpam-3856	89	7	an	an	DET
ejpam-3856	89	8	ideal	ideal	ADJ
ejpam-3856	89	9	,	,	PUNCT
ejpam-3856	89	10	three	three	NUM
ejpam-3856	89	11	cases	case	NOUN
ejpam-3856	89	12	can	can	AUX
ejpam-3856	89	13	be	be	AUX
ejpam-3856	89	14	possible	possible	ADJ
ejpam-3856	89	15	:	:	PUNCT
ejpam-3856	89	16	case	case	NOUN
ejpam-3856	89	17	i	i	PRON
ejpam-3856	89	18	(	(	PUNCT
ejpam-3856	89	19	u	u	NOUN
ejpam-3856	89	20	∩a	∩a	PROPN
ejpam-3856	89	21	)	)	PUNCT
ejpam-3856	89	22	/∈	/∈	PUNCT
ejpam-3856	90	1	i	i	PRON
ejpam-3856	90	2	and	and	CCONJ
ejpam-3856	90	3	(	(	PUNCT
ejpam-3856	90	4	u	u	NOUN
ejpam-3856	90	5	∩b	∩b	PROPN
ejpam-3856	90	6	)	)	PUNCT
ejpam-3856	90	7	/∈	/∈	PUNCT
ejpam-3856	91	1	i	i	PRON
ejpam-3856	91	2	,	,	PUNCT
ejpam-3856	91	3	case	case	NOUN
ejpam-3856	91	4	ii	ii	X
ejpam-3856	91	5	(	(	PUNCT
ejpam-3856	91	6	u	u	NOUN
ejpam-3856	91	7	∩a	∩a	PROPN
ejpam-3856	91	8	)	)	PUNCT
ejpam-3856	91	9	∈	∈	PROPN
ejpam-3856	92	1	i	i	PRON
ejpam-3856	92	2	and	and	CCONJ
ejpam-3856	92	3	(	(	PUNCT
ejpam-3856	92	4	u	u	NOUN
ejpam-3856	92	5	∩b	∩b	PROPN
ejpam-3856	92	6	)	)	PUNCT
ejpam-3856	92	7	/∈	/∈	PUNCT
ejpam-3856	93	1	i	i	PRON
ejpam-3856	93	2	,	,	PUNCT
ejpam-3856	93	3	case	case	NOUN
ejpam-3856	93	4	iii	iii	X
ejpam-3856	93	5	(	(	PUNCT
ejpam-3856	93	6	u	u	NOUN
ejpam-3856	93	7	∩a	∩a	PROPN
ejpam-3856	93	8	)	)	PUNCT
ejpam-3856	93	9	/∈	/∈	PUNCT
ejpam-3856	94	1	i	i	PRON
ejpam-3856	94	2	and	and	CCONJ
ejpam-3856	94	3	(	(	PUNCT
ejpam-3856	94	4	u	u	NOUN
ejpam-3856	94	5	∩b	∩b	PROPN
ejpam-3856	94	6	)	)	PUNCT
ejpam-3856	94	7	in	in	ADP
ejpam-3856	94	8	i.	i.	PROPN
ejpam-3856	94	9	it	it	PRON
ejpam-3856	94	10	may	may	AUX
ejpam-3856	94	11	be	be	AUX
ejpam-3856	94	12	seen	see	VERB
ejpam-3856	94	13	easily	easily	ADV
ejpam-3856	94	14	that	that	SCONJ
ejpam-3856	94	15	for	for	ADP
ejpam-3856	94	16	all	all	DET
ejpam-3856	94	17	three	three	NUM
ejpam-3856	94	18	cases	case	NOUN
ejpam-3856	94	19	x	x	SYM
ejpam-3856	94	20	∈	∈	PROPN
ejpam-3856	94	21	a∗	a∗	PROPN
ejpam-3856	94	22	β	β	PROPN
ejpam-3856	94	23	∪b∗	∪b∗	NUM
ejpam-3856	94	24	β	β	X
ejpam-3856	94	25	.	.	PUNCT
ejpam-3856	95	1	therefore	therefore	ADV
ejpam-3856	95	2	,	,	PUNCT
ejpam-3856	95	3	we	we	PRON
ejpam-3856	95	4	have	have	VERB
ejpam-3856	95	5	(	(	PUNCT
ejpam-3856	95	6	a∪b)∗β	a∪b)∗β	NOUN
ejpam-3856	95	7	⊂	⊂	PROPN
ejpam-3856	95	8	a∗	a∗	PROPN
ejpam-3856	95	9	β	β	PROPN
ejpam-3856	95	10	∪b∗	∪b∗	NUM
ejpam-3856	95	11	β	β	X
ejpam-3856	95	12	.	.	PUNCT
ejpam-3856	96	1	by	by	ADP
ejpam-3856	96	2	(	(	PUNCT
ejpam-3856	96	3	1	1	NUM
ejpam-3856	96	4	)	)	PUNCT
ejpam-3856	96	5	,	,	PUNCT
ejpam-3856	96	6	a∗	a∗	PROPN
ejpam-3856	96	7	β	β	X
ejpam-3856	96	8	⊂	⊂	PROPN
ejpam-3856	96	9	(	(	PUNCT
ejpam-3856	96	10	a∪b)∗β	a∪b)∗β	ADV
ejpam-3856	96	11	and	and	CCONJ
ejpam-3856	96	12	b∗	b∗	ADJ
ejpam-3856	96	13	β	β	X
ejpam-3856	96	14	⊂	⊂	PROPN
ejpam-3856	96	15	(	(	PUNCT
ejpam-3856	96	16	a∪b)∗β	a∪b)∗β	PROPN
ejpam-3856	96	17	.	.	PUNCT
ejpam-3856	97	1	hence	hence	ADV
ejpam-3856	97	2	,	,	PUNCT
ejpam-3856	97	3	a∗	a∗	PROPN
ejpam-3856	97	4	β	β	PROPN
ejpam-3856	97	5	∪b∗	∪b∗	NUM
ejpam-3856	97	6	β	β	X
ejpam-3856	97	7	⊂	⊂	X
ejpam-3856	97	8	(	(	PUNCT
ejpam-3856	97	9	a∪b)∗β	a∪b)∗β	NOUN
ejpam-3856	97	10	and	and	CCONJ
ejpam-3856	97	11	we	we	PRON
ejpam-3856	97	12	obtain	obtain	VERB
ejpam-3856	97	13	a∗	a∗	PROPN
ejpam-3856	97	14	β	β	PROPN
ejpam-3856	97	15	∪b∗	∪b∗	NUM
ejpam-3856	97	16	β	β	X
ejpam-3856	97	17	=	=	SYM
ejpam-3856	97	18	(	(	PUNCT
ejpam-3856	97	19	a	a	DET
ejpam-3856	97	20	∪b)∗β	∪b)∗β	NUM
ejpam-3856	97	21	.	.	PUNCT
ejpam-3856	98	1	(	(	PUNCT
ejpam-3856	98	2	3	3	NUM
ejpam-3856	98	3	)	)	PUNCT
ejpam-3856	98	4	.	.	PUNCT
ejpam-3856	99	1	since	since	SCONJ
ejpam-3856	99	2	,	,	PUNCT
ejpam-3856	99	3	a∩b	a∩b	PROPN
ejpam-3856	99	4	⊂	⊂	PROPN
ejpam-3856	99	5	a	a	PROPN
ejpam-3856	99	6	and	and	CCONJ
ejpam-3856	99	7	a∩b	a∩b	PROPN
ejpam-3856	99	8	⊂	⊂	PROPN
ejpam-3856	99	9	b	b	PROPN
ejpam-3856	99	10	,	,	PUNCT
ejpam-3856	99	11	by	by	ADP
ejpam-3856	99	12	(	(	PUNCT
ejpam-3856	99	13	1	1	NUM
ejpam-3856	99	14	)	)	PUNCT
ejpam-3856	99	15	,	,	PUNCT
ejpam-3856	99	16	(	(	PUNCT
ejpam-3856	99	17	a∩b)∗β	a∩b)∗β	ADV
ejpam-3856	99	18	⊂	⊂	X
ejpam-3856	99	19	(	(	PUNCT
ejpam-3856	99	20	a)∗β	a)∗β	NOUN
ejpam-3856	99	21	and	and	CCONJ
ejpam-3856	99	22	(	(	PUNCT
ejpam-3856	99	23	a∩b)∗β	a∩b)∗β	ADV
ejpam-3856	99	24	⊂	⊂	X
ejpam-3856	99	25	(	(	PUNCT
ejpam-3856	99	26	b)∗β	b)∗β	ADJ
ejpam-3856	99	27	and	and	CCONJ
ejpam-3856	99	28	hence	hence	ADV
ejpam-3856	99	29	,	,	PUNCT
ejpam-3856	99	30	p.	p.	PROPN
ejpam-3856	99	31	l.	l.	PROPN
ejpam-3856	99	32	powar	powar	PROPN
ejpam-3856	99	33	,	,	PUNCT
ejpam-3856	99	34	t.	t.	PROPN
ejpam-3856	99	35	noiri	noiri	PROPN
ejpam-3856	99	36	,	,	PUNCT
ejpam-3856	99	37	shikha	shikha	PROPN
ejpam-3856	99	38	bhadauria	bhadauria	PROPN
ejpam-3856	99	39	/	/	SYM
ejpam-3856	99	40	eur	eur	PROPN
ejpam-3856	99	41	.	.	PUNCT
ejpam-3856	100	1	j.	j.	PROPN
ejpam-3856	100	2	pure	pure	PROPN
ejpam-3856	100	3	appl	appl	PROPN
ejpam-3856	100	4	.	.	PROPN
ejpam-3856	100	5	math	math	PROPN
ejpam-3856	100	6	,	,	PUNCT
ejpam-3856	100	7	13	13	NUM
ejpam-3856	100	8	(	(	PUNCT
ejpam-3856	100	9	4	4	NUM
ejpam-3856	100	10	)	)	PUNCT
ejpam-3856	100	11	(	(	PUNCT
ejpam-3856	100	12	2020	2020	NUM
ejpam-3856	100	13	)	)	PUNCT
ejpam-3856	100	14	,	,	PUNCT
ejpam-3856	100	15	758	758	NUM
ejpam-3856	100	16	-	-	SYM
ejpam-3856	100	17	765	765	NUM
ejpam-3856	100	18	761	761	NUM
ejpam-3856	100	19	(	(	PUNCT
ejpam-3856	100	20	a	a	DET
ejpam-3856	100	21	∩b)∗β	∩b)∗β	PUNCT
ejpam-3856	101	1	⊂	⊂	X
ejpam-3856	101	2	(	(	PUNCT
ejpam-3856	101	3	a)∗β	a)∗β	NOUN
ejpam-3856	101	4	∩	∩	NOUN
ejpam-3856	101	5	(	(	PUNCT
ejpam-3856	101	6	b)∗β	b)∗β	ADJ
ejpam-3856	101	7	.	.	PUNCT
ejpam-3856	102	1	(	(	PUNCT
ejpam-3856	102	2	4	4	NUM
ejpam-3856	102	3	)	)	PUNCT
ejpam-3856	102	4	.	.	PUNCT
ejpam-3856	103	1	let	let	VERB
ejpam-3856	103	2	x	x	X
ejpam-3856	103	3	∈	∈	PROPN
ejpam-3856	103	4	(	(	PUNCT
ejpam-3856	103	5	a∗	a∗	PROPN
ejpam-3856	103	6	β)∗β	β)∗β	NOUN
ejpam-3856	103	7	,	,	PUNCT
ejpam-3856	103	8	then	then	ADV
ejpam-3856	103	9	by	by	ADP
ejpam-3856	103	10	definition	definition	NOUN
ejpam-3856	103	11	8	8	NUM
ejpam-3856	103	12	,	,	PUNCT
ejpam-3856	103	13	we	we	PRON
ejpam-3856	103	14	have	have	VERB
ejpam-3856	103	15	,	,	PUNCT
ejpam-3856	103	16	a∗	a∗	PROPN
ejpam-3856	103	17	β	β	X
ejpam-3856	103	18	∩	∩	ADJ
ejpam-3856	103	19	u	u	NOUN
ejpam-3856	103	20	/∈	/∈	PUNCT
ejpam-3856	104	1	i	i	PRON
ejpam-3856	104	2	for	for	ADP
ejpam-3856	104	3	every	every	DET
ejpam-3856	104	4	u	u	PROPN
ejpam-3856	104	5	∈	∈	PROPN
ejpam-3856	104	6	βo(x	βo(x	PUNCT
ejpam-3856	104	7	)	)	PUNCT
ejpam-3856	104	8	and	and	CCONJ
ejpam-3856	104	9	a∗	a∗	PROPN
ejpam-3856	104	10	β	β	X
ejpam-3856	104	11	∩	∩	PROPN
ejpam-3856	104	12	u	u	PROPN
ejpam-3856	104	13	6=	6=	PROPN
ejpam-3856	104	14	φ	φ	PROPN
ejpam-3856	104	15	.	.	PUNCT
ejpam-3856	105	1	now	now	ADV
ejpam-3856	105	2	,	,	PUNCT
ejpam-3856	105	3	let	let	VERB
ejpam-3856	105	4	y	y	PROPN
ejpam-3856	105	5	∈	∈	PROPN
ejpam-3856	105	6	a∗	a∗	PROPN
ejpam-3856	105	7	β	β	X
ejpam-3856	105	8	∩	∩	ADJ
ejpam-3856	105	9	u	u	PROPN
ejpam-3856	105	10	,	,	PUNCT
ejpam-3856	105	11	then	then	ADV
ejpam-3856	105	12	y	y	PROPN
ejpam-3856	105	13	∈	∈	PROPN
ejpam-3856	105	14	u	u	NOUN
ejpam-3856	105	15	and	and	CCONJ
ejpam-3856	105	16	u	u	PROPN
ejpam-3856	105	17	∈	∈	PROPN
ejpam-3856	105	18	βo(y	βo(y	PUNCT
ejpam-3856	105	19	)	)	PUNCT
ejpam-3856	105	20	.	.	PUNCT
ejpam-3856	106	1	since	since	SCONJ
ejpam-3856	106	2	y	y	PROPN
ejpam-3856	106	3	∈	∈	PROPN
ejpam-3856	106	4	a∗	a∗	PROPN
ejpam-3856	106	5	β	β	PROPN
ejpam-3856	106	6	,	,	PUNCT
ejpam-3856	106	7	a	a	DET
ejpam-3856	106	8	∩	∩	ADJ
ejpam-3856	106	9	u	u	NOUN
ejpam-3856	106	10	/∈	/∈	PROPN
ejpam-3856	106	11	i.	i.	PROPN
ejpam-3856	106	12	hence	hence	ADV
ejpam-3856	106	13	,	,	PUNCT
ejpam-3856	106	14	x	x	PUNCT
ejpam-3856	106	15	∈	∈	PROPN
ejpam-3856	106	16	a∗	a∗	PROPN
ejpam-3856	106	17	β	β	NOUN
ejpam-3856	106	18	.	.	PUNCT
ejpam-3856	107	1	therefore	therefore	ADV
ejpam-3856	107	2	,	,	PUNCT
ejpam-3856	107	3	(	(	PUNCT
ejpam-3856	107	4	a∗	a∗	PROPN
ejpam-3856	107	5	β)∗β	β)∗β	NOUN
ejpam-3856	107	6	⊂	⊂	PROPN
ejpam-3856	107	7	a∗	a∗	PROPN
ejpam-3856	107	8	β	β	PROPN
ejpam-3856	107	9	.	.	PUNCT
ejpam-3856	108	1	(	(	PUNCT
ejpam-3856	108	2	5	5	NUM
ejpam-3856	108	3	)	)	PUNCT
ejpam-3856	108	4	.	.	PUNCT
ejpam-3856	109	1	we	we	PRON
ejpam-3856	109	2	know	know	VERB
ejpam-3856	109	3	that	that	SCONJ
ejpam-3856	109	4	a∗	a∗	PROPN
ejpam-3856	109	5	β	β	X
ejpam-3856	109	6	⊂	⊂	PROPN
ejpam-3856	109	7	βcl(a∗	βcl(a∗	PUNCT
ejpam-3856	109	8	β	β	NOUN
ejpam-3856	109	9	)	)	PUNCT
ejpam-3856	109	10	.	.	PUNCT
ejpam-3856	110	1	we	we	PRON
ejpam-3856	110	2	show	show	VERB
ejpam-3856	110	3	that	that	SCONJ
ejpam-3856	110	4	βcl(a∗	βcl(a∗	PUNCT
ejpam-3856	110	5	β	β	X
ejpam-3856	110	6	)	)	PUNCT
ejpam-3856	110	7	⊂	⊂	PROPN
ejpam-3856	110	8	a∗	a∗	PROPN
ejpam-3856	110	9	β	β	X
ejpam-3856	110	10	.	.	PUNCT
ejpam-3856	111	1	let	let	VERB
ejpam-3856	111	2	x	x	SYM
ejpam-3856	111	3	∈	∈	PROPN
ejpam-3856	111	4	βcl(a∗	βcl(a∗	PUNCT
ejpam-3856	111	5	β	β	NOUN
ejpam-3856	111	6	)	)	PUNCT
ejpam-3856	111	7	.	.	PUNCT
ejpam-3856	112	1	then	then	ADV
ejpam-3856	112	2	by	by	ADP
ejpam-3856	112	3	lemma	lemma	PROPN
ejpam-3856	112	4	1	1	NUM
ejpam-3856	112	5	,	,	PUNCT
ejpam-3856	112	6	we	we	PRON
ejpam-3856	112	7	have	have	VERB
ejpam-3856	112	8	,	,	PUNCT
ejpam-3856	112	9	a∗	a∗	PROPN
ejpam-3856	112	10	β	β	X
ejpam-3856	112	11	∩u	∩u	PROPN
ejpam-3856	113	1	6=	6=	PROPN
ejpam-3856	114	1	φ	φ	PROPN
ejpam-3856	114	2	for	for	ADP
ejpam-3856	114	3	every	every	DET
ejpam-3856	114	4	u	u	PROPN
ejpam-3856	114	5	∈	∈	PROPN
ejpam-3856	114	6	βo(x	βo(x	PUNCT
ejpam-3856	114	7	)	)	PUNCT
ejpam-3856	114	8	.	.	PUNCT
ejpam-3856	115	1	now	now	ADV
ejpam-3856	115	2	,	,	PUNCT
ejpam-3856	115	3	let	let	VERB
ejpam-3856	115	4	y	y	PROPN
ejpam-3856	115	5	∈	∈	PROPN
ejpam-3856	115	6	a∗	a∗	PROPN
ejpam-3856	115	7	β	β	X
ejpam-3856	115	8	∩	∩	ADJ
ejpam-3856	115	9	u	u	PROPN
ejpam-3856	115	10	,	,	PUNCT
ejpam-3856	115	11	then	then	ADV
ejpam-3856	115	12	we	we	PRON
ejpam-3856	115	13	have	have	VERB
ejpam-3856	115	14	y	y	PROPN
ejpam-3856	115	15	∈	∈	PROPN
ejpam-3856	115	16	a∗	a∗	PROPN
ejpam-3856	115	17	β	β	X
ejpam-3856	115	18	and	and	CCONJ
ejpam-3856	115	19	y	y	PROPN
ejpam-3856	115	20	∈	∈	PROPN
ejpam-3856	115	21	u	u	PROPN
ejpam-3856	115	22	∈	∈	PROPN
ejpam-3856	115	23	βo(x	βo(x	PUNCT
ejpam-3856	115	24	)	)	PUNCT
ejpam-3856	115	25	.	.	PUNCT
ejpam-3856	116	1	therefore	therefore	ADV
ejpam-3856	116	2	,	,	PUNCT
ejpam-3856	116	3	we	we	PRON
ejpam-3856	116	4	have	have	VERB
ejpam-3856	116	5	,	,	PUNCT
ejpam-3856	116	6	a	a	DET
ejpam-3856	116	7	∩	∩	ADJ
ejpam-3856	116	8	u	u	NOUN
ejpam-3856	116	9	/∈	/∈	PUNCT
ejpam-3856	117	1	i	i	PRON
ejpam-3856	117	2	and	and	CCONJ
ejpam-3856	117	3	hence	hence	ADV
ejpam-3856	117	4	,	,	PUNCT
ejpam-3856	117	5	x	x	SYM
ejpam-3856	117	6	∈	∈	PROPN
ejpam-3856	117	7	a∗	a∗	PROPN
ejpam-3856	117	8	β	β	PROPN
ejpam-3856	117	9	.	.	PUNCT
ejpam-3856	118	1	therefore	therefore	ADV
ejpam-3856	118	2	,	,	PUNCT
ejpam-3856	118	3	βcl(a∗	βcl(a∗	X
ejpam-3856	118	4	β	β	X
ejpam-3856	118	5	)	)	PUNCT
ejpam-3856	118	6	⊂	⊂	PROPN
ejpam-3856	118	7	a∗	a∗	PROPN
ejpam-3856	118	8	β	β	PROPN
ejpam-3856	118	9	and	and	CCONJ
ejpam-3856	118	10	hence	hence	ADV
ejpam-3856	118	11	,	,	PUNCT
ejpam-3856	118	12	a∗	a∗	PROPN
ejpam-3856	118	13	β	β	X
ejpam-3856	118	14	=	=	PUNCT
ejpam-3856	118	15	βcl(a∗	βcl(a∗	PUNCT
ejpam-3856	118	16	β	β	NOUN
ejpam-3856	118	17	)	)	PUNCT
ejpam-3856	118	18	.	.	PUNCT
ejpam-3856	119	1	this	this	PRON
ejpam-3856	119	2	implies	imply	VERB
ejpam-3856	119	3	a∗	a∗	PROPN
ejpam-3856	119	4	β	β	X
ejpam-3856	119	5	is	be	AUX
ejpam-3856	119	6	β	β	NOUN
ejpam-3856	119	7	-	-	VERB
ejpam-3856	119	8	closed	closed	ADJ
ejpam-3856	119	9	.	.	PUNCT
ejpam-3856	120	1	next	next	ADV
ejpam-3856	120	2	,	,	PUNCT
ejpam-3856	120	3	we	we	PRON
ejpam-3856	120	4	show	show	VERB
ejpam-3856	120	5	that	that	SCONJ
ejpam-3856	120	6	a∗	a∗	PROPN
ejpam-3856	120	7	β	β	X
ejpam-3856	120	8	⊂	⊂	PROPN
ejpam-3856	120	9	βcl(a	βcl(a	PROPN
ejpam-3856	120	10	)	)	PUNCT
ejpam-3856	120	11	.	.	PUNCT
ejpam-3856	121	1	if	if	SCONJ
ejpam-3856	121	2	x	x	X
ejpam-3856	121	3	/∈	/∈	PUNCT
ejpam-3856	121	4	βcl(a	βcl(a	NUM
ejpam-3856	121	5	)	)	PUNCT
ejpam-3856	121	6	,	,	PUNCT
ejpam-3856	121	7	then	then	ADV
ejpam-3856	121	8	there	there	PRON
ejpam-3856	121	9	exists	exist	VERB
ejpam-3856	121	10	u	u	PROPN
ejpam-3856	121	11	∈	∈	PROPN
ejpam-3856	121	12	βo(x	βo(x	PUNCT
ejpam-3856	121	13	)	)	PUNCT
ejpam-3856	121	14	such	such	ADJ
ejpam-3856	121	15	that	that	SCONJ
ejpam-3856	121	16	u	u	NOUN
ejpam-3856	122	1	∩a	∩a	NOUN
ejpam-3856	122	2	=	=	PUNCT
ejpam-3856	123	1	φ	φ	PROPN
ejpam-3856	123	2	∈	∈	PROPN
ejpam-3856	124	1	i	i	PRON
ejpam-3856	124	2	and	and	CCONJ
ejpam-3856	124	3	x	x	PROPN
ejpam-3856	124	4	/∈	/∈	PUNCT
ejpam-3856	124	5	a∗	a∗	PROPN
ejpam-3856	124	6	β	β	PROPN
ejpam-3856	124	7	.	.	PUNCT
ejpam-3856	125	1	therefore	therefore	ADV
ejpam-3856	125	2	,	,	PUNCT
ejpam-3856	125	3	a∗	a∗	PROPN
ejpam-3856	125	4	β	β	X
ejpam-3856	125	5	⊂	⊂	PROPN
ejpam-3856	125	6	βcl(a	βcl(a	PROPN
ejpam-3856	125	7	)	)	PUNCT
ejpam-3856	125	8	.	.	PUNCT
ejpam-3856	126	1	remark	remark	PROPN
ejpam-3856	126	2	1	1	NUM
ejpam-3856	126	3	.	.	PUNCT
ejpam-3856	127	1	let	let	VERB
ejpam-3856	127	2	(	(	PUNCT
ejpam-3856	127	3	x	x	X
ejpam-3856	127	4	,	,	PUNCT
ejpam-3856	127	5	τ	τ	PROPN
ejpam-3856	127	6	,	,	PUNCT
ejpam-3856	127	7	i	i	PRON
ejpam-3856	127	8	)	)	PUNCT
ejpam-3856	127	9	be	be	VERB
ejpam-3856	127	10	an	an	DET
ejpam-3856	127	11	ideal	ideal	ADJ
ejpam-3856	127	12	topological	topological	ADJ
ejpam-3856	127	13	space	space	NOUN
ejpam-3856	127	14	and	and	CCONJ
ejpam-3856	127	15	a	a	DET
ejpam-3856	127	16	⊂	⊂	PROPN
ejpam-3856	127	17	x.	x.	NOUN
ejpam-3856	128	1	then	then	ADV
ejpam-3856	128	2	the	the	DET
ejpam-3856	128	3	following	follow	VERB
ejpam-3856	128	4	holds	hold	VERB
ejpam-3856	128	5	:	:	PUNCT
ejpam-3856	128	6	(	(	PUNCT
ejpam-3856	128	7	1	1	NUM
ejpam-3856	128	8	)	)	PUNCT
ejpam-3856	128	9	.	.	PUNCT
ejpam-3856	129	1	if	if	SCONJ
ejpam-3856	129	2	i	i	PRON
ejpam-3856	129	3	=	=	X
ejpam-3856	129	4	{	{	PUNCT
ejpam-3856	129	5	φ	φ	NOUN
ejpam-3856	129	6	}	}	PUNCT
ejpam-3856	129	7	,	,	PUNCT
ejpam-3856	129	8	then	then	ADV
ejpam-3856	129	9	a∗	a∗	PROPN
ejpam-3856	129	10	β	β	X
ejpam-3856	129	11	=	=	PUNCT
ejpam-3856	129	12	βcl(a	βcl(a	PROPN
ejpam-3856	129	13	)	)	PUNCT
ejpam-3856	129	14	,	,	PUNCT
ejpam-3856	129	15	(	(	PUNCT
ejpam-3856	129	16	2	2	NUM
ejpam-3856	129	17	)	)	PUNCT
ejpam-3856	129	18	.	.	PUNCT
ejpam-3856	130	1	if	if	SCONJ
ejpam-3856	130	2	k	k	PROPN
ejpam-3856	130	3	∈	∈	PROPN
ejpam-3856	130	4	i	i	PRON
ejpam-3856	130	5	,	,	PUNCT
ejpam-3856	130	6	then	then	ADV
ejpam-3856	130	7	k∗	k∗	VERB
ejpam-3856	130	8	β	β	X
ejpam-3856	130	9	=	=	SYM
ejpam-3856	130	10	φ	φ	PROPN
ejpam-3856	130	11	and	and	CCONJ
ejpam-3856	130	12	hence	hence	ADV
ejpam-3856	130	13	,	,	PUNCT
ejpam-3856	130	14	{	{	PUNCT
ejpam-3856	130	15	φ}∗β	φ}∗β	NOUN
ejpam-3856	130	16	=	=	SYM
ejpam-3856	130	17	φ	φ	PROPN
ejpam-3856	130	18	,	,	PUNCT
ejpam-3856	130	19	(	(	PUNCT
ejpam-3856	130	20	3	3	NUM
ejpam-3856	130	21	)	)	PUNCT
ejpam-3856	130	22	.	.	PUNCT
ejpam-3856	131	1	it	it	PRON
ejpam-3856	131	2	is	be	AUX
ejpam-3856	131	3	not	not	PART
ejpam-3856	131	4	necessary	necessary	ADJ
ejpam-3856	131	5	that	that	SCONJ
ejpam-3856	131	6	3(a	3(a	NUM
ejpam-3856	131	7	)	)	PUNCT
ejpam-3856	131	8	.	.	PUNCT
ejpam-3856	132	1	a	a	DET
ejpam-3856	132	2	⊂	⊂	PROPN
ejpam-3856	132	3	a∗	a∗	PROPN
ejpam-3856	132	4	β	β	PROPN
ejpam-3856	132	5	or	or	CCONJ
ejpam-3856	132	6	3(b	3(b	NUM
ejpam-3856	132	7	)	)	PUNCT
ejpam-3856	132	8	.	.	PUNCT
ejpam-3856	133	1	a∗	a∗	PROPN
ejpam-3856	133	2	β	β	X
ejpam-3856	133	3	⊂	⊂	PROPN
ejpam-3856	133	4	a	a	X
ejpam-3856	133	5	,	,	PUNCT
ejpam-3856	133	6	(	(	PUNCT
ejpam-3856	133	7	4	4	NUM
ejpam-3856	133	8	)	)	PUNCT
ejpam-3856	133	9	.	.	PUNCT
ejpam-3856	134	1	a∗	a∗	PROPN
ejpam-3856	134	2	β(i	β(i	PRON
ejpam-3856	134	3	,	,	PUNCT
ejpam-3856	134	4	βo(x	βo(x	PUNCT
ejpam-3856	134	5	)	)	PUNCT
ejpam-3856	134	6	)	)	PUNCT
ejpam-3856	135	1	=	=	SYM
ejpam-3856	135	2	(	(	PUNCT
ejpam-3856	135	3	a)∗	a)∗	PROPN
ejpam-3856	135	4	s	s	X
ejpam-3856	135	5	(	(	PUNCT
ejpam-3856	135	6	i	i	PROPN
ejpam-3856	135	7	,	,	PUNCT
ejpam-3856	135	8	τ	τ	PROPN
ejpam-3856	135	9	)	)	PUNCT
ejpam-3856	135	10	if	if	SCONJ
ejpam-3856	135	11	so(x	so(x	NOUN
ejpam-3856	135	12	)	)	PUNCT
ejpam-3856	135	13	=	=	PUNCT
ejpam-3856	135	14	βo(x	βo(x	PUNCT
ejpam-3856	135	15	)	)	PUNCT
ejpam-3856	135	16	,	,	PUNCT
ejpam-3856	135	17	(	(	PUNCT
ejpam-3856	135	18	5	5	NUM
ejpam-3856	135	19	)	)	PUNCT
ejpam-3856	135	20	.	.	PUNCT
ejpam-3856	136	1	a∗	a∗	PROPN
ejpam-3856	136	2	β(i	β(i	PRON
ejpam-3856	136	3	,	,	PUNCT
ejpam-3856	136	4	βo(x	βo(x	PUNCT
ejpam-3856	136	5	)	)	PUNCT
ejpam-3856	136	6	)	)	PUNCT
ejpam-3856	136	7	=	=	SYM
ejpam-3856	136	8	a∗(i	a∗(i	PROPN
ejpam-3856	136	9	,	,	PUNCT
ejpam-3856	136	10	τ	τ	X
ejpam-3856	136	11	)	)	PUNCT
ejpam-3856	136	12	if	if	SCONJ
ejpam-3856	136	13	βo(x	βo(x	PUNCT
ejpam-3856	136	14	)	)	PUNCT
ejpam-3856	136	15	=	=	SYM
ejpam-3856	136	16	τ(x	τ(x	NOUN
ejpam-3856	136	17	)	)	PUNCT
ejpam-3856	136	18	.	.	PUNCT
ejpam-3856	137	1	in	in	ADP
ejpam-3856	137	2	order	order	NOUN
ejpam-3856	137	3	to	to	PART
ejpam-3856	137	4	verify	verify	VERB
ejpam-3856	137	5	3(a	3(a	NUM
ejpam-3856	137	6	)	)	PUNCT
ejpam-3856	137	7	.	.	PUNCT
ejpam-3856	138	1	and	and	CCONJ
ejpam-3856	138	2	3(b	3(b	NUM
ejpam-3856	138	3	)	)	PUNCT
ejpam-3856	138	4	.	.	PUNCT
ejpam-3856	139	1	of	of	ADP
ejpam-3856	139	2	remark	remark	NOUN
ejpam-3856	139	3	1	1	NUM
ejpam-3856	139	4	,	,	PUNCT
ejpam-3856	139	5	we	we	PRON
ejpam-3856	139	6	explore	explore	VERB
ejpam-3856	139	7	the	the	DET
ejpam-3856	139	8	following	follow	VERB
ejpam-3856	139	9	example	example	NOUN
ejpam-3856	139	10	:	:	PUNCT
ejpam-3856	139	11	example	example	NOUN
ejpam-3856	139	12	2	2	NUM
ejpam-3856	139	13	.	.	X
ejpam-3856	139	14	letx	letx	NOUN
ejpam-3856	139	15	=	=	SYM
ejpam-3856	139	16	{	{	PUNCT
ejpam-3856	139	17	a	a	PRON
ejpam-3856	139	18	,	,	PUNCT
ejpam-3856	139	19	b	b	NOUN
ejpam-3856	139	20	,	,	PUNCT
ejpam-3856	139	21	c	c	NOUN
ejpam-3856	139	22	,	,	PUNCT
ejpam-3856	139	23	d	d	AUX
ejpam-3856	139	24	}	}	PUNCT
ejpam-3856	139	25	be	be	AUX
ejpam-3856	139	26	a	a	DET
ejpam-3856	139	27	nonempty	nonempty	NOUN
ejpam-3856	139	28	set	set	VERB
ejpam-3856	139	29	with	with	ADP
ejpam-3856	139	30	the	the	DET
ejpam-3856	139	31	topology	topology	NOUN
ejpam-3856	139	32	τ	τ	X
ejpam-3856	139	33	=	=	SYM
ejpam-3856	139	34	{	{	PUNCT
ejpam-3856	139	35	φ	φ	PROPN
ejpam-3856	139	36	,	,	PUNCT
ejpam-3856	139	37	x	x	X
ejpam-3856	139	38	,	,	PUNCT
ejpam-3856	139	39	{	{	PUNCT
ejpam-3856	139	40	a	a	X
ejpam-3856	139	41	}	}	PUNCT
ejpam-3856	139	42	,	,	PUNCT
ejpam-3856	139	43	{	{	PUNCT
ejpam-3856	139	44	a	a	PRON
ejpam-3856	139	45	,	,	PUNCT
ejpam-3856	139	46	b	b	NOUN
ejpam-3856	139	47	,	,	PUNCT
ejpam-3856	139	48	c	c	NOUN
ejpam-3856	139	49	}	}	PUNCT
ejpam-3856	139	50	}	}	PUNCT
ejpam-3856	139	51	and	and	CCONJ
ejpam-3856	139	52	the	the	DET
ejpam-3856	139	53	collection	collection	NOUN
ejpam-3856	139	54	of	of	ADP
ejpam-3856	139	55	closed	closed	ADJ
ejpam-3856	139	56	sets	set	NOUN
ejpam-3856	139	57	is	be	AUX
ejpam-3856	140	1	τf	τf	ADP
ejpam-3856	140	2	=	=	SYM
ejpam-3856	140	3	{	{	PUNCT
ejpam-3856	140	4	φ	φ	NOUN
ejpam-3856	140	5	,	,	PUNCT
ejpam-3856	140	6	x	x	X
ejpam-3856	140	7	,	,	PUNCT
ejpam-3856	140	8	{	{	PUNCT
ejpam-3856	140	9	b	b	NOUN
ejpam-3856	140	10	,	,	PUNCT
ejpam-3856	140	11	c	c	NOUN
ejpam-3856	140	12	,	,	PUNCT
ejpam-3856	140	13	d	d	NOUN
ejpam-3856	140	14	}	}	PUNCT
ejpam-3856	140	15	,	,	PUNCT
ejpam-3856	140	16	{	{	PUNCT
ejpam-3856	140	17	d	d	NOUN
ejpam-3856	140	18	}	}	PUNCT
ejpam-3856	140	19	}	}	PUNCT
ejpam-3856	140	20	.	.	PUNCT
ejpam-3856	141	1	next	next	ADV
ejpam-3856	141	2	,	,	PUNCT
ejpam-3856	141	3	by	by	ADP
ejpam-3856	141	4	applying	apply	VERB
ejpam-3856	141	5	definition	definition	NOUN
ejpam-3856	141	6	5	5	NUM
ejpam-3856	141	7	,	,	PUNCT
ejpam-3856	141	8	we	we	PRON
ejpam-3856	141	9	compute	compute	VERB
ejpam-3856	141	10	the	the	DET
ejpam-3856	141	11	collection	collection	NOUN
ejpam-3856	141	12	βo(x)=	βo(x)=	X
ejpam-3856	141	13	{	{	PUNCT
ejpam-3856	141	14	φ	φ	PROPN
ejpam-3856	141	15	,	,	PUNCT
ejpam-3856	141	16	x	x	X
ejpam-3856	141	17	,	,	PUNCT
ejpam-3856	141	18	{	{	PUNCT
ejpam-3856	141	19	a	a	X
ejpam-3856	141	20	}	}	PUNCT
ejpam-3856	141	21	,	,	PUNCT
ejpam-3856	141	22	{	{	PUNCT
ejpam-3856	141	23	a	a	DET
ejpam-3856	141	24	,	,	PUNCT
ejpam-3856	141	25	b	b	NOUN
ejpam-3856	141	26	}	}	PUNCT
ejpam-3856	141	27	,	,	PUNCT
ejpam-3856	141	28	{	{	PUNCT
ejpam-3856	141	29	a	a	PRON
ejpam-3856	141	30	,	,	PUNCT
ejpam-3856	141	31	d	d	NOUN
ejpam-3856	141	32	}	}	PUNCT
ejpam-3856	141	33	,	,	PUNCT
ejpam-3856	141	34	{	{	PUNCT
ejpam-3856	141	35	a	a	X
ejpam-3856	141	36	,	,	PUNCT
ejpam-3856	141	37	c	c	NOUN
ejpam-3856	141	38	}	}	PUNCT
ejpam-3856	141	39	,	,	PUNCT
ejpam-3856	141	40	{	{	PUNCT
ejpam-3856	141	41	a	a	DET
ejpam-3856	141	42	,	,	PUNCT
ejpam-3856	141	43	b	b	NOUN
ejpam-3856	141	44	,	,	PUNCT
ejpam-3856	141	45	c	c	NOUN
ejpam-3856	141	46	}	}	PUNCT
ejpam-3856	141	47	,	,	PUNCT
ejpam-3856	141	48	{	{	PUNCT
ejpam-3856	141	49	c	c	X
ejpam-3856	141	50	,	,	PUNCT
ejpam-3856	141	51	d	d	NOUN
ejpam-3856	141	52	,	,	PUNCT
ejpam-3856	141	53	a	a	PRON
ejpam-3856	141	54	}	}	PUNCT
ejpam-3856	141	55	,	,	PUNCT
ejpam-3856	141	56	{	{	PUNCT
ejpam-3856	141	57	d	d	X
ejpam-3856	141	58	,	,	PUNCT
ejpam-3856	141	59	a	a	DET
ejpam-3856	141	60	,	,	PUNCT
ejpam-3856	141	61	b	b	NOUN
ejpam-3856	141	62	}	}	PUNCT
ejpam-3856	141	63	}	}	PUNCT
ejpam-3856	141	64	.	.	PUNCT
ejpam-3856	142	1	considering	consider	VERB
ejpam-3856	142	2	i	i	PRON
ejpam-3856	142	3	=	=	SYM
ejpam-3856	142	4	{	{	PUNCT
ejpam-3856	142	5	φ	φ	PROPN
ejpam-3856	142	6	,	,	PUNCT
ejpam-3856	142	7	{	{	PUNCT
ejpam-3856	142	8	b	b	NOUN
ejpam-3856	142	9	}	}	PUNCT
ejpam-3856	142	10	}	}	PUNCT
ejpam-3856	142	11	and	and	CCONJ
ejpam-3856	142	12	a	a	DET
ejpam-3856	142	13	,	,	PUNCT
ejpam-3856	142	14	b	b	X
ejpam-3856	142	15	⊂	⊂	PROPN
ejpam-3856	142	16	x	x	PUNCT
ejpam-3856	142	17	where	where	SCONJ
ejpam-3856	142	18	,	,	PUNCT
ejpam-3856	142	19	a	a	DET
ejpam-3856	142	20	=	=	X
ejpam-3856	142	21	{	{	PUNCT
ejpam-3856	142	22	b	b	PROPN
ejpam-3856	142	23	,	,	PUNCT
ejpam-3856	142	24	c	c	NOUN
ejpam-3856	142	25	,	,	PUNCT
ejpam-3856	142	26	d	d	NOUN
ejpam-3856	142	27	}	}	PUNCT
ejpam-3856	142	28	and	and	CCONJ
ejpam-3856	142	29	b	b	X
ejpam-3856	142	30	=	=	NOUN
ejpam-3856	142	31	{	{	PUNCT
ejpam-3856	142	32	a	a	PRON
ejpam-3856	142	33	,	,	PUNCT
ejpam-3856	142	34	b	b	NOUN
ejpam-3856	142	35	,	,	PUNCT
ejpam-3856	142	36	c	c	NOUN
ejpam-3856	142	37	}	}	PUNCT
ejpam-3856	142	38	then	then	ADV
ejpam-3856	142	39	by	by	ADP
ejpam-3856	142	40	applying	apply	VERB
ejpam-3856	142	41	definition	definition	NOUN
ejpam-3856	142	42	8	8	NUM
ejpam-3856	142	43	,	,	PUNCT
ejpam-3856	142	44	a∗	a∗	PROPN
ejpam-3856	142	45	β	β	X
ejpam-3856	142	46	=	=	PUNCT
ejpam-3856	142	47	{	{	PUNCT
ejpam-3856	142	48	c	c	X
ejpam-3856	142	49	,	,	PUNCT
ejpam-3856	142	50	d	d	NOUN
ejpam-3856	142	51	}	}	PUNCT
ejpam-3856	142	52	and	and	CCONJ
ejpam-3856	142	53	b∗	b∗	ADJ
ejpam-3856	142	54	β	β	X
ejpam-3856	142	55	=	=	PUNCT
ejpam-3856	142	56	x.	x.	NOUN
ejpam-3856	142	57	in	in	ADP
ejpam-3856	142	58	view	view	NOUN
ejpam-3856	142	59	of	of	ADP
ejpam-3856	142	60	above	above	ADJ
ejpam-3856	142	61	assertions	assertion	NOUN
ejpam-3856	142	62	3(a	3(a	NUM
ejpam-3856	142	63	)	)	PUNCT
ejpam-3856	142	64	and	and	CCONJ
ejpam-3856	142	65	3(b	3(b	NUM
ejpam-3856	142	66	)	)	PUNCT
ejpam-3856	142	67	have	have	AUX
ejpam-3856	142	68	been	be	AUX
ejpam-3856	142	69	verified	verify	VERB
ejpam-3856	142	70	.	.	PUNCT
ejpam-3856	143	1	lemma	lemma	PROPN
ejpam-3856	143	2	2	2	X
ejpam-3856	143	3	.	.	PUNCT
ejpam-3856	144	1	let	let	VERB
ejpam-3856	144	2	(	(	PUNCT
ejpam-3856	144	3	x	x	X
ejpam-3856	144	4	,	,	PUNCT
ejpam-3856	144	5	τ	τ	PROPN
ejpam-3856	144	6	,	,	PUNCT
ejpam-3856	144	7	i	i	PRON
ejpam-3856	144	8	)	)	PUNCT
ejpam-3856	144	9	be	be	VERB
ejpam-3856	144	10	an	an	DET
ejpam-3856	144	11	ideal	ideal	ADJ
ejpam-3856	144	12	topological	topological	ADJ
ejpam-3856	144	13	space	space	NOUN
ejpam-3856	144	14	.	.	PUNCT
ejpam-3856	145	1	then	then	ADV
ejpam-3856	145	2	the	the	DET
ejpam-3856	145	3	following	follow	VERB
ejpam-3856	145	4	properties	property	NOUN
ejpam-3856	145	5	hold	hold	VERB
ejpam-3856	145	6	:	:	PUNCT
ejpam-3856	145	7	(	(	PUNCT
ejpam-3856	145	8	1	1	NUM
ejpam-3856	145	9	)	)	PUNCT
ejpam-3856	145	10	.	.	PUNCT
ejpam-3856	146	1	ro(x	ro(x	PUNCT
ejpam-3856	146	2	)	)	PUNCT
ejpam-3856	147	1	⊂	⊂	PROPN
ejpam-3856	147	2	τ	τ	PROPN
ejpam-3856	147	3	⊂	⊂	PROPN
ejpam-3856	147	4	so(x	so(x	NUM
ejpam-3856	147	5	)	)	PUNCT
ejpam-3856	147	6	⊂	⊂	PROPN
ejpam-3856	147	7	βo(x	βo(x	PUNCT
ejpam-3856	147	8	)	)	PUNCT
ejpam-3856	147	9	,	,	PUNCT
ejpam-3856	147	10	(	(	PUNCT
ejpam-3856	147	11	2	2	NUM
ejpam-3856	147	12	)	)	PUNCT
ejpam-3856	147	13	.	.	PUNCT
ejpam-3856	148	1	a∗	a∗	PROPN
ejpam-3856	148	2	β	β	X
ejpam-3856	148	3	⊂	⊂	PROPN
ejpam-3856	148	4	a∗s	a∗s	PROPN
ejpam-3856	148	5	⊂	⊂	PROPN
ejpam-3856	148	6	a∗	a∗	PROPN
ejpam-3856	148	7	⊂	⊂	PROPN
ejpam-3856	148	8	γ∗(a	γ∗(a	PROPN
ejpam-3856	148	9	)	)	PUNCT
ejpam-3856	148	10	for	for	ADP
ejpam-3856	148	11	every	every	DET
ejpam-3856	148	12	subset	subset	NOUN
ejpam-3856	148	13	a	a	PRON
ejpam-3856	148	14	of	of	ADP
ejpam-3856	148	15	x.	x.	NOUN
ejpam-3856	148	16	proof	proof	NOUN
ejpam-3856	148	17	.	.	PUNCT
ejpam-3856	149	1	(	(	PUNCT
ejpam-3856	149	2	1	1	NUM
ejpam-3856	149	3	)	)	PUNCT
ejpam-3856	149	4	.	.	PUNCT
ejpam-3856	150	1	the	the	DET
ejpam-3856	150	2	proof	proof	NOUN
ejpam-3856	150	3	is	be	AUX
ejpam-3856	150	4	obvious	obvious	ADJ
ejpam-3856	150	5	by	by	ADP
ejpam-3856	150	6	the	the	DET
ejpam-3856	150	7	definition	definition	NOUN
ejpam-3856	150	8	5	5	NUM
ejpam-3856	150	9	.	.	PUNCT
ejpam-3856	151	1	(	(	PUNCT
ejpam-3856	151	2	2	2	NUM
ejpam-3856	151	3	)	)	PUNCT
ejpam-3856	151	4	.	.	PUNCT
ejpam-3856	152	1	first	first	ADV
ejpam-3856	152	2	,	,	PUNCT
ejpam-3856	152	3	we	we	PRON
ejpam-3856	152	4	show	show	VERB
ejpam-3856	152	5	that	that	SCONJ
ejpam-3856	152	6	a∗	a∗	PROPN
ejpam-3856	152	7	β	β	X
ejpam-3856	152	8	⊂	⊂	PROPN
ejpam-3856	152	9	a∗s	a∗s	PROPN
ejpam-3856	152	10	.	.	PUNCT
ejpam-3856	153	1	let	let	VERB
ejpam-3856	153	2	x	x	SYM
ejpam-3856	153	3	∈	∈	PROPN
ejpam-3856	153	4	a∗	a∗	PROPN
ejpam-3856	153	5	β	β	X
ejpam-3856	153	6	.	.	PUNCT
ejpam-3856	154	1	then	then	ADV
ejpam-3856	154	2	,	,	PUNCT
ejpam-3856	154	3	a	a	DET
ejpam-3856	154	4	∩	∩	ADJ
ejpam-3856	154	5	u	u	NOUN
ejpam-3856	154	6	/∈	/∈	PUNCT
ejpam-3856	154	7	i	i	PRON
ejpam-3856	154	8	for	for	ADP
ejpam-3856	154	9	every	every	DET
ejpam-3856	154	10	u	u	PROPN
ejpam-3856	154	11	∈	∈	PROPN
ejpam-3856	154	12	βo(x	βo(x	PUNCT
ejpam-3856	154	13	)	)	PUNCT
ejpam-3856	154	14	.	.	PUNCT
ejpam-3856	155	1	since	since	SCONJ
ejpam-3856	155	2	so(x	so(x	NUM
ejpam-3856	155	3	)	)	PUNCT
ejpam-3856	155	4	⊂	⊂	PROPN
ejpam-3856	155	5	βo(x	βo(x	PUNCT
ejpam-3856	155	6	)	)	PUNCT
ejpam-3856	155	7	,	,	PUNCT
ejpam-3856	155	8	a	a	DET
ejpam-3856	155	9	∩	∩	ADJ
ejpam-3856	155	10	u	u	NOUN
ejpam-3856	155	11	/∈	/∈	PUNCT
ejpam-3856	155	12	i	i	PRON
ejpam-3856	155	13	for	for	ADP
ejpam-3856	155	14	every	every	DET
ejpam-3856	155	15	u	u	PROPN
ejpam-3856	155	16	∈	∈	PROPN
ejpam-3856	155	17	so(x	so(x	NOUN
ejpam-3856	155	18	)	)	PUNCT
ejpam-3856	155	19	and	and	CCONJ
ejpam-3856	155	20	x	x	PUNCT
ejpam-3856	155	21	∈	∈	PROPN
ejpam-3856	155	22	a∗s	a∗s	PROPN
ejpam-3856	155	23	.	.	PUNCT
ejpam-3856	156	1	hence	hence	ADV
ejpam-3856	156	2	,	,	PUNCT
ejpam-3856	156	3	we	we	PRON
ejpam-3856	156	4	have	have	VERB
ejpam-3856	156	5	a∗	a∗	PROPN
ejpam-3856	156	6	β	β	X
ejpam-3856	156	7	⊂	⊂	PROPN
ejpam-3856	156	8	a∗s	a∗s	PROPN
ejpam-3856	156	9	.	.	PUNCT
ejpam-3856	157	1	similarly	similarly	ADV
ejpam-3856	157	2	,	,	PUNCT
ejpam-3856	157	3	by	by	ADP
ejpam-3856	157	4	using	use	VERB
ejpam-3856	157	5	the	the	DET
ejpam-3856	157	6	fact	fact	NOUN
ejpam-3856	157	7	that	that	SCONJ
ejpam-3856	157	8	ro(x	ro(x	PUNCT
ejpam-3856	157	9	)	)	PUNCT
ejpam-3856	157	10	⊂	⊂	PROPN
ejpam-3856	157	11	τ	τ	PROPN
ejpam-3856	157	12	⊂	⊂	PROPN
ejpam-3856	157	13	so(x	so(x	NUM
ejpam-3856	157	14	)	)	PUNCT
ejpam-3856	157	15	,	,	PUNCT
ejpam-3856	157	16	we	we	PRON
ejpam-3856	157	17	may	may	AUX
ejpam-3856	157	18	establish	establish	VERB
ejpam-3856	157	19	a∗s	a∗s	PROPN
ejpam-3856	157	20	⊂	⊂	PROPN
ejpam-3856	157	21	a∗	a∗	PROPN
ejpam-3856	157	22	and	and	CCONJ
ejpam-3856	157	23	a∗	a∗	PROPN
ejpam-3856	157	24	⊂	⊂	PROPN
ejpam-3856	157	25	γ∗(a	γ∗(a	PROPN
ejpam-3856	157	26	)	)	PUNCT
ejpam-3856	157	27	.	.	PUNCT
ejpam-3856	158	1	definition	definition	NOUN
ejpam-3856	158	2	9	9	NUM
ejpam-3856	158	3	.	.	PUNCT
ejpam-3856	159	1	let	let	VERB
ejpam-3856	159	2	(	(	PUNCT
ejpam-3856	159	3	x	x	X
ejpam-3856	159	4	,	,	PUNCT
ejpam-3856	159	5	τ	τ	PROPN
ejpam-3856	159	6	,	,	PUNCT
ejpam-3856	159	7	i	i	PRON
ejpam-3856	159	8	)	)	PUNCT
ejpam-3856	159	9	be	be	VERB
ejpam-3856	159	10	an	an	DET
ejpam-3856	159	11	ideal	ideal	ADJ
ejpam-3856	159	12	topological	topological	ADJ
ejpam-3856	159	13	space	space	NOUN
ejpam-3856	159	14	.	.	PUNCT
ejpam-3856	160	1	we	we	PRON
ejpam-3856	160	2	define	define	VERB
ejpam-3856	160	3	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	160	4	)	)	PUNCT
ejpam-3856	160	5	=	=	PUNCT
ejpam-3856	160	6	a	a	DET
ejpam-3856	160	7	∪	∪	ADJ
ejpam-3856	160	8	a∗	a∗	PROPN
ejpam-3856	160	9	β	β	PROPN
ejpam-3856	160	10	for	for	ADP
ejpam-3856	160	11	every	every	DET
ejpam-3856	160	12	subset	subset	NOUN
ejpam-3856	160	13	a	a	PRON
ejpam-3856	160	14	of	of	ADP
ejpam-3856	160	15	x.	x.	PROPN
ejpam-3856	160	16	p.	p.	PROPN
ejpam-3856	160	17	l.	l.	PROPN
ejpam-3856	160	18	powar	powar	PROPN
ejpam-3856	160	19	,	,	PUNCT
ejpam-3856	160	20	t.	t.	PROPN
ejpam-3856	160	21	noiri	noiri	PROPN
ejpam-3856	160	22	,	,	PUNCT
ejpam-3856	160	23	shikha	shikha	PROPN
ejpam-3856	160	24	bhadauria	bhadauria	PROPN
ejpam-3856	160	25	/	/	SYM
ejpam-3856	160	26	eur	eur	PROPN
ejpam-3856	160	27	.	.	PUNCT
ejpam-3856	161	1	j.	j.	PROPN
ejpam-3856	161	2	pure	pure	PROPN
ejpam-3856	161	3	appl	appl	PROPN
ejpam-3856	161	4	.	.	PROPN
ejpam-3856	161	5	math	math	PROPN
ejpam-3856	161	6	,	,	PUNCT
ejpam-3856	161	7	13	13	NUM
ejpam-3856	161	8	(	(	PUNCT
ejpam-3856	161	9	4	4	NUM
ejpam-3856	161	10	)	)	PUNCT
ejpam-3856	161	11	(	(	PUNCT
ejpam-3856	161	12	2020	2020	NUM
ejpam-3856	161	13	)	)	PUNCT
ejpam-3856	161	14	,	,	PUNCT
ejpam-3856	161	15	758	758	NUM
ejpam-3856	161	16	-	-	SYM
ejpam-3856	161	17	765	765	NUM
ejpam-3856	161	18	762	762	NUM
ejpam-3856	161	19	theorem	theorem	NOUN
ejpam-3856	161	20	2	2	NUM
ejpam-3856	161	21	.	.	PUNCT
ejpam-3856	162	1	let	let	VERB
ejpam-3856	162	2	(	(	PUNCT
ejpam-3856	162	3	x	x	X
ejpam-3856	162	4	,	,	PUNCT
ejpam-3856	162	5	τ	τ	PROPN
ejpam-3856	162	6	,	,	PUNCT
ejpam-3856	162	7	i	i	PRON
ejpam-3856	162	8	)	)	PUNCT
ejpam-3856	162	9	be	be	VERB
ejpam-3856	162	10	an	an	DET
ejpam-3856	162	11	ideal	ideal	ADJ
ejpam-3856	162	12	topological	topological	ADJ
ejpam-3856	162	13	space	space	NOUN
ejpam-3856	162	14	.	.	PUNCT
ejpam-3856	163	1	then	then	ADV
ejpam-3856	163	2	the	the	DET
ejpam-3856	163	3	following	follow	VERB
ejpam-3856	163	4	properties	property	NOUN
ejpam-3856	163	5	hold	hold	VERB
ejpam-3856	163	6	:	:	PUNCT
ejpam-3856	163	7	(	(	PUNCT
ejpam-3856	163	8	1	1	NUM
ejpam-3856	163	9	)	)	PUNCT
ejpam-3856	163	10	.	.	PUNCT
ejpam-3856	164	1	a	a	DET
ejpam-3856	164	2	⊂	⊂	X
ejpam-3856	164	3	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	164	4	)	)	PUNCT
ejpam-3856	164	5	,	,	PUNCT
ejpam-3856	164	6	(	(	PUNCT
ejpam-3856	164	7	2	2	NUM
ejpam-3856	164	8	)	)	PUNCT
ejpam-3856	164	9	.	.	PUNCT
ejpam-3856	165	1	cl∗β(φ	cl∗β(φ	PROPN
ejpam-3856	165	2	)	)	PUNCT
ejpam-3856	166	1	=	=	PUNCT
ejpam-3856	166	2	φ	φ	PROPN
ejpam-3856	166	3	and	and	CCONJ
ejpam-3856	166	4	cl∗β(x	cl∗β(x	NOUN
ejpam-3856	166	5	)	)	PUNCT
ejpam-3856	167	1	=	=	SYM
ejpam-3856	167	2	x	x	NOUN
ejpam-3856	167	3	,	,	PUNCT
ejpam-3856	167	4	(	(	PUNCT
ejpam-3856	167	5	3	3	NUM
ejpam-3856	167	6	)	)	PUNCT
ejpam-3856	167	7	.	.	PUNCT
ejpam-3856	168	1	a	a	DET
ejpam-3856	168	2	⊂	⊂	PROPN
ejpam-3856	168	3	b	b	PROPN
ejpam-3856	168	4	implies	imply	VERB
ejpam-3856	168	5	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	168	6	)	)	PUNCT
ejpam-3856	168	7	⊂	⊂	PROPN
ejpam-3856	168	8	cl∗β(b	cl∗β(b	NOUN
ejpam-3856	168	9	)	)	PUNCT
ejpam-3856	168	10	,	,	PUNCT
ejpam-3856	168	11	(	(	PUNCT
ejpam-3856	168	12	4	4	NUM
ejpam-3856	168	13	)	)	PUNCT
ejpam-3856	168	14	.	.	PUNCT
ejpam-3856	169	1	cl∗β(a	cl∗β(a	NOUN
ejpam-3856	169	2	)	)	PUNCT
ejpam-3856	169	3	∪	∪	ADP
ejpam-3856	169	4	cl∗β(b	cl∗β(b	NOUN
ejpam-3856	169	5	)	)	PUNCT
ejpam-3856	169	6	=	=	SYM
ejpam-3856	169	7	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	169	8	∪b	∪b	PRON
ejpam-3856	169	9	)	)	PUNCT
ejpam-3856	169	10	,	,	PUNCT
ejpam-3856	169	11	(	(	PUNCT
ejpam-3856	169	12	5	5	NUM
ejpam-3856	169	13	)	)	PUNCT
ejpam-3856	169	14	.	.	PUNCT
ejpam-3856	170	1	(	(	PUNCT
ejpam-3856	170	2	cl∗β(a))∗β	cl∗β(a))∗β	PROPN
ejpam-3856	170	3	⊂	⊂	PROPN
ejpam-3856	170	4	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	170	5	)	)	PUNCT
ejpam-3856	170	6	=	=	SYM
ejpam-3856	171	1	cl∗β(cl∗β(a	cl∗β(cl∗β(a	PROPN
ejpam-3856	171	2	)	)	PUNCT
ejpam-3856	171	3	)	)	PUNCT
ejpam-3856	171	4	.	.	PUNCT
ejpam-3856	172	1	proof	proof	NOUN
ejpam-3856	172	2	.	.	PUNCT
ejpam-3856	173	1	(	(	PUNCT
ejpam-3856	173	2	1	1	NUM
ejpam-3856	173	3	)	)	PUNCT
ejpam-3856	173	4	.	.	PUNCT
ejpam-3856	174	1	this	this	PRON
ejpam-3856	174	2	follows	follow	VERB
ejpam-3856	174	3	directly	directly	ADV
ejpam-3856	174	4	by	by	ADP
ejpam-3856	174	5	the	the	DET
ejpam-3856	174	6	definition	definition	NOUN
ejpam-3856	174	7	9	9	NUM
ejpam-3856	174	8	.	.	PUNCT
ejpam-3856	175	1	(	(	PUNCT
ejpam-3856	175	2	2	2	NUM
ejpam-3856	175	3	)	)	PUNCT
ejpam-3856	175	4	.	.	PUNCT
ejpam-3856	176	1	we	we	PRON
ejpam-3856	176	2	have	have	AUX
ejpam-3856	176	3	cl∗β(φ	cl∗β(φ	PROPN
ejpam-3856	176	4	)	)	PUNCT
ejpam-3856	177	1	=	=	PRON
ejpam-3856	177	2	{	{	PUNCT
ejpam-3856	177	3	φ	φ	NOUN
ejpam-3856	177	4	}	}	PUNCT
ejpam-3856	177	5	∪	∪	ADJ
ejpam-3856	177	6	{	{	PUNCT
ejpam-3856	177	7	φ}∗β	φ}∗β	NOUN
ejpam-3856	177	8	=	=	NUM
ejpam-3856	177	9	φ	φ	PROPN
ejpam-3856	177	10	.	.	PUNCT
ejpam-3856	178	1	similarly	similarly	ADV
ejpam-3856	178	2	,	,	PUNCT
ejpam-3856	178	3	it	it	PRON
ejpam-3856	178	4	may	may	AUX
ejpam-3856	178	5	be	be	AUX
ejpam-3856	178	6	verified	verify	VERB
ejpam-3856	178	7	that	that	SCONJ
ejpam-3856	178	8	cl∗β(x	cl∗β(x	NOUN
ejpam-3856	178	9	)	)	PUNCT
ejpam-3856	179	1	=	=	PUNCT
ejpam-3856	179	2	x.	x.	NOUN
ejpam-3856	179	3	(	(	PUNCT
ejpam-3856	179	4	3	3	NUM
ejpam-3856	179	5	)	)	PUNCT
ejpam-3856	179	6	.	.	PUNCT
ejpam-3856	180	1	given	give	VERB
ejpam-3856	180	2	,	,	PUNCT
ejpam-3856	180	3	a	a	DET
ejpam-3856	180	4	⊂	⊂	X
ejpam-3856	180	5	b.	b.	AUX
ejpam-3856	180	6	by	by	ADP
ejpam-3856	180	7	definition	definition	NOUN
ejpam-3856	180	8	9	9	NUM
ejpam-3856	180	9	,	,	PUNCT
ejpam-3856	180	10	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	180	11	)	)	PUNCT
ejpam-3856	180	12	=	=	PUNCT
ejpam-3856	180	13	a	a	DET
ejpam-3856	180	14	∪	∪	ADJ
ejpam-3856	180	15	a∗	a∗	PROPN
ejpam-3856	180	16	β	β	X
ejpam-3856	180	17	and	and	CCONJ
ejpam-3856	180	18	cl∗β(b	cl∗β(b	NOUN
ejpam-3856	180	19	)	)	PUNCT
ejpam-3856	180	20	=	=	SYM
ejpam-3856	180	21	b	b	X
ejpam-3856	180	22	∪	∪	VERB
ejpam-3856	180	23	b∗	b∗	ADJ
ejpam-3856	180	24	β	β	NOUN
ejpam-3856	180	25	.	.	PUNCT
ejpam-3856	181	1	next	next	ADV
ejpam-3856	181	2	,	,	PUNCT
ejpam-3856	181	3	by	by	ADP
ejpam-3856	181	4	theorem	theorem	NOUN
ejpam-3856	181	5	1	1	NUM
ejpam-3856	181	6	(	(	PUNCT
ejpam-3856	181	7	1	1	NUM
ejpam-3856	181	8	)	)	PUNCT
ejpam-3856	181	9	,	,	PUNCT
ejpam-3856	181	10	we	we	PRON
ejpam-3856	181	11	have	have	VERB
ejpam-3856	181	12	a∗	a∗	PROPN
ejpam-3856	181	13	β	β	X
ejpam-3856	181	14	⊂	⊂	PROPN
ejpam-3856	181	15	b∗	b∗	PROPN
ejpam-3856	181	16	β	β	X
ejpam-3856	181	17	.	.	PUNCT
ejpam-3856	182	1	therefore	therefore	ADV
ejpam-3856	182	2	,	,	PUNCT
ejpam-3856	182	3	we	we	PRON
ejpam-3856	182	4	obtain	obtain	VERB
ejpam-3856	182	5	a	a	DET
ejpam-3856	182	6	∪	∪	ADJ
ejpam-3856	182	7	a∗	a∗	PROPN
ejpam-3856	182	8	β	β	X
ejpam-3856	182	9	⊂	⊂	PROPN
ejpam-3856	182	10	b	b	X
ejpam-3856	182	11	∪	∪	ADJ
ejpam-3856	182	12	b∗	b∗	ADJ
ejpam-3856	182	13	β	β	X
ejpam-3856	182	14	and	and	CCONJ
ejpam-3856	182	15	hence	hence	ADV
ejpam-3856	182	16	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	182	17	)	)	PUNCT
ejpam-3856	182	18	⊂	⊂	PROPN
ejpam-3856	182	19	cl∗β(b	cl∗β(b	NOUN
ejpam-3856	182	20	)	)	PUNCT
ejpam-3856	182	21	.	.	PUNCT
ejpam-3856	183	1	(	(	PUNCT
ejpam-3856	183	2	4	4	NUM
ejpam-3856	183	3	)	)	PUNCT
ejpam-3856	183	4	.	.	PUNCT
ejpam-3856	184	1	by	by	ADP
ejpam-3856	184	2	theorem	theorem	NOUN
ejpam-3856	184	3	1	1	NUM
ejpam-3856	184	4	(	(	PUNCT
ejpam-3856	184	5	2	2	NUM
ejpam-3856	184	6	)	)	PUNCT
ejpam-3856	184	7	,	,	PUNCT
ejpam-3856	184	8	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	184	9	∪b	∪b	PRON
ejpam-3856	184	10	)	)	PUNCT
ejpam-3856	184	11	=	=	SYM
ejpam-3856	184	12	(	(	PUNCT
ejpam-3856	184	13	a	a	DET
ejpam-3856	184	14	∪b	∪b	NOUN
ejpam-3856	184	15	)	)	PUNCT
ejpam-3856	184	16	∪	∪	NOUN
ejpam-3856	184	17	(	(	PUNCT
ejpam-3856	184	18	a∗	a∗	PROPN
ejpam-3856	184	19	β	β	PROPN
ejpam-3856	184	20	∪b∗	∪b∗	NUM
ejpam-3856	184	21	β	β	X
ejpam-3856	184	22	)	)	PUNCT
ejpam-3856	184	23	=	=	SYM
ejpam-3856	184	24	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	184	25	)	)	PUNCT
ejpam-3856	184	26	∪	∪	ADP
ejpam-3856	184	27	cl∗β(b	cl∗β(b	NOUN
ejpam-3856	184	28	)	)	PUNCT
ejpam-3856	184	29	.	.	PUNCT
ejpam-3856	185	1	hence	hence	ADV
ejpam-3856	185	2	,	,	PUNCT
ejpam-3856	185	3	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	185	4	∪b	∪b	PRON
ejpam-3856	185	5	)	)	PUNCT
ejpam-3856	185	6	=	=	SYM
ejpam-3856	185	7	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	185	8	)	)	PUNCT
ejpam-3856	185	9	∪	∪	ADP
ejpam-3856	185	10	cl∗β(b	cl∗β(b	NOUN
ejpam-3856	185	11	)	)	PUNCT
ejpam-3856	185	12	.	.	PUNCT
ejpam-3856	186	1	(	(	PUNCT
ejpam-3856	186	2	5	5	NUM
ejpam-3856	186	3	)	)	PUNCT
ejpam-3856	186	4	.	.	PUNCT
ejpam-3856	187	1	first	first	ADV
ejpam-3856	187	2	,	,	PUNCT
ejpam-3856	187	3	we	we	PRON
ejpam-3856	187	4	show	show	VERB
ejpam-3856	187	5	that	that	SCONJ
ejpam-3856	187	6	(	(	PUNCT
ejpam-3856	187	7	cl∗β(a))∗β	cl∗β(a))∗β	PROPN
ejpam-3856	187	8	⊂	⊂	PROPN
ejpam-3856	187	9	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	187	10	)	)	PUNCT
ejpam-3856	187	11	.	.	PUNCT
ejpam-3856	188	1	let	let	VERB
ejpam-3856	188	2	if	if	SCONJ
ejpam-3856	188	3	possible	possible	ADJ
ejpam-3856	188	4	x	x	INTJ
ejpam-3856	188	5	/∈	/∈	PUNCT
ejpam-3856	188	6	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	188	7	)	)	PUNCT
ejpam-3856	188	8	.	.	PUNCT
ejpam-3856	189	1	this	this	PRON
ejpam-3856	189	2	implies	imply	VERB
ejpam-3856	189	3	x	x	X
ejpam-3856	189	4	/∈	/∈	PUNCT
ejpam-3856	189	5	a∗	a∗	PROPN
ejpam-3856	189	6	β	β	X
ejpam-3856	189	7	and	and	CCONJ
ejpam-3856	189	8	there	there	PRON
ejpam-3856	189	9	exists	exist	VERB
ejpam-3856	189	10	u	u	PROPN
ejpam-3856	189	11	∈	∈	PROPN
ejpam-3856	189	12	βo(x	βo(x	PUNCT
ejpam-3856	189	13	)	)	PUNCT
ejpam-3856	189	14	such	such	ADJ
ejpam-3856	189	15	that	that	SCONJ
ejpam-3856	189	16	u	u	NOUN
ejpam-3856	189	17	∩a	∩a	PROPN
ejpam-3856	189	18	∈	∈	PROPN
ejpam-3856	190	1	i	i	PRON
ejpam-3856	190	2	and	and	CCONJ
ejpam-3856	190	3	we	we	PRON
ejpam-3856	190	4	conclude	conclude	VERB
ejpam-3856	190	5	that	that	SCONJ
ejpam-3856	190	6	u	u	NOUN
ejpam-3856	190	7	∩a∗	∩a∗	PUNCT
ejpam-3856	190	8	β	β	X
ejpam-3856	190	9	=	=	SYM
ejpam-3856	190	10	φ	φ	PROPN
ejpam-3856	190	11	and	and	CCONJ
ejpam-3856	190	12	φ	φ	PROPN
ejpam-3856	190	13	∈	∈	PROPN
ejpam-3856	190	14	i.	i.	NOUN
ejpam-3856	190	15	for	for	ADP
ejpam-3856	190	16	if	if	SCONJ
ejpam-3856	190	17	a∗	a∗	PROPN
ejpam-3856	190	18	β∩u	β∩u	PROPN
ejpam-3856	190	19	6=	6=	ADP
ejpam-3856	190	20	φ	φ	PROPN
ejpam-3856	190	21	then	then	ADV
ejpam-3856	190	22	there	there	PRON
ejpam-3856	190	23	exists	exist	VERB
ejpam-3856	190	24	y	y	PROPN
ejpam-3856	190	25	∈	∈	PROPN
ejpam-3856	190	26	a∗	a∗	PROPN
ejpam-3856	190	27	β	β	X
ejpam-3856	190	28	∩	∩	ADJ
ejpam-3856	190	29	u	u	NOUN
ejpam-3856	190	30	and	and	CCONJ
ejpam-3856	190	31	u	u	PROPN
ejpam-3856	190	32	∈	∈	PROPN
ejpam-3856	190	33	βo(y	βo(y	PUNCT
ejpam-3856	190	34	)	)	PUNCT
ejpam-3856	190	35	.	.	PUNCT
ejpam-3856	191	1	then	then	ADV
ejpam-3856	191	2	y	y	PROPN
ejpam-3856	191	3	∈	∈	PROPN
ejpam-3856	191	4	a∗	a∗	PROPN
ejpam-3856	191	5	β	β	PROPN
ejpam-3856	191	6	implies	imply	VERB
ejpam-3856	191	7	u	u	PROPN
ejpam-3856	191	8	∩	∩	NOUN
ejpam-3856	191	9	a	a	X
ejpam-3856	191	10	/∈	/∈	PUNCT
ejpam-3856	192	1	i	i	PRON
ejpam-3856	192	2	,	,	PUNCT
ejpam-3856	192	3	which	which	PRON
ejpam-3856	192	4	is	be	AUX
ejpam-3856	192	5	a	a	DET
ejpam-3856	192	6	contradiction	contradiction	NOUN
ejpam-3856	192	7	as	as	ADP
ejpam-3856	192	8	u	u	NOUN
ejpam-3856	192	9	∩a	∩a	PROPN
ejpam-3856	192	10	∈	∈	PROPN
ejpam-3856	192	11	i.	i.	NOUN
ejpam-3856	192	12	hence	hence	ADV
ejpam-3856	192	13	,	,	PUNCT
ejpam-3856	192	14	u	u	PROPN
ejpam-3856	192	15	∩a∗	∩a∗	PUNCT
ejpam-3856	192	16	β	β	X
ejpam-3856	192	17	=	=	SYM
ejpam-3856	192	18	φ	φ	PROPN
ejpam-3856	192	19	.	.	PUNCT
ejpam-3856	193	1	now	now	ADV
ejpam-3856	193	2	,	,	PUNCT
ejpam-3856	193	3	we	we	PRON
ejpam-3856	193	4	obtain	obtain	VERB
ejpam-3856	193	5	(	(	PUNCT
ejpam-3856	193	6	a	a	DET
ejpam-3856	193	7	∪a∗	∪a∗	NUM
ejpam-3856	193	8	β	β	NOUN
ejpam-3856	193	9	)	)	PUNCT
ejpam-3856	193	10	∩	∩	NOUN
ejpam-3856	193	11	u	u	NOUN
ejpam-3856	193	12	=	=	X
ejpam-3856	193	13	(	(	PUNCT
ejpam-3856	193	14	a	a	DET
ejpam-3856	193	15	∩	∩	ADJ
ejpam-3856	193	16	u	u	NOUN
ejpam-3856	193	17	)	)	PUNCT
ejpam-3856	193	18	∪	∪	ADV
ejpam-3856	193	19	(	(	PUNCT
ejpam-3856	193	20	a∗	a∗	PROPN
ejpam-3856	193	21	β	β	X
ejpam-3856	193	22	∩	∩	ADJ
ejpam-3856	193	23	u	u	NOUN
ejpam-3856	193	24	)	)	PUNCT
ejpam-3856	193	25	∈	∈	PROPN
ejpam-3856	193	26	i.	i.	NOUN
ejpam-3856	193	27	this	this	PRON
ejpam-3856	193	28	implies	imply	VERB
ejpam-3856	193	29	that	that	SCONJ
ejpam-3856	193	30	(	(	PUNCT
ejpam-3856	193	31	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	193	32	)	)	PUNCT
ejpam-3856	193	33	)	)	PUNCT
ejpam-3856	193	34	∩	∩	NOUN
ejpam-3856	193	35	u	u	PROPN
ejpam-3856	193	36	∈	∈	PROPN
ejpam-3856	193	37	i.	i.	NOUN
ejpam-3856	193	38	by	by	ADP
ejpam-3856	193	39	definition	definition	NOUN
ejpam-3856	193	40	8	8	NUM
ejpam-3856	193	41	,	,	PUNCT
ejpam-3856	193	42	we	we	PRON
ejpam-3856	193	43	obtain	obtain	VERB
ejpam-3856	193	44	,	,	PUNCT
ejpam-3856	193	45	x	x	X
ejpam-3856	193	46	/∈	/∈	PUNCT
ejpam-3856	194	1	(	(	PUNCT
ejpam-3856	194	2	cl∗β(a))∗β	cl∗β(a))∗β	PROPN
ejpam-3856	194	3	.	.	PUNCT
ejpam-3856	195	1	hence	hence	ADV
ejpam-3856	195	2	,	,	PUNCT
ejpam-3856	195	3	we	we	PRON
ejpam-3856	195	4	obtain	obtain	VERB
ejpam-3856	195	5	(	(	PUNCT
ejpam-3856	195	6	cl∗β(a))∗β	cl∗β(a))∗β	PROPN
ejpam-3856	195	7	⊂	⊂	PROPN
ejpam-3856	195	8	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	195	9	)	)	PUNCT
ejpam-3856	195	10	.	.	PUNCT
ejpam-3856	196	1	next	next	ADV
ejpam-3856	196	2	,	,	PUNCT
ejpam-3856	196	3	we	we	PRON
ejpam-3856	196	4	show	show	VERB
ejpam-3856	196	5	that	that	SCONJ
ejpam-3856	196	6	cl∗β(cl∗β(a	cl∗β(cl∗β(a	PROPN
ejpam-3856	196	7	)	)	PUNCT
ejpam-3856	196	8	)	)	PUNCT
ejpam-3856	197	1	=	=	SYM
ejpam-3856	197	2	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	197	3	)	)	PUNCT
ejpam-3856	197	4	.	.	PUNCT
ejpam-3856	198	1	now	now	ADV
ejpam-3856	198	2	,	,	PUNCT
ejpam-3856	198	3	we	we	PRON
ejpam-3856	198	4	have	have	VERB
ejpam-3856	198	5	cl∗β(cl∗β(a	cl∗β(cl∗β(a	PROPN
ejpam-3856	198	6	)	)	PUNCT
ejpam-3856	198	7	)	)	PUNCT
ejpam-3856	199	1	=	=	SYM
ejpam-3856	199	2	cl∗β(a)∪	cl∗β(a)∪	PROPN
ejpam-3856	199	3	(	(	PUNCT
ejpam-3856	199	4	cl∗β(a))∗β	cl∗β(a))∗β	PROPN
ejpam-3856	199	5	.	.	PUNCT
ejpam-3856	200	1	since	since	SCONJ
ejpam-3856	200	2	(	(	PUNCT
ejpam-3856	200	3	cl∗β(a))∗β	cl∗β(a))∗β	PROPN
ejpam-3856	200	4	⊂	⊂	PROPN
ejpam-3856	200	5	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	200	6	)	)	PUNCT
ejpam-3856	200	7	,	,	PUNCT
ejpam-3856	200	8	we	we	PRON
ejpam-3856	200	9	obtain	obtain	VERB
ejpam-3856	200	10	cl∗β(cl∗β(a	cl∗β(cl∗β(a	PROPN
ejpam-3856	200	11	)	)	PUNCT
ejpam-3856	200	12	)	)	PUNCT
ejpam-3856	201	1	⊂	⊂	PROPN
ejpam-3856	201	2	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	201	3	)	)	PUNCT
ejpam-3856	201	4	.	.	PUNCT
ejpam-3856	202	1	it	it	PRON
ejpam-3856	202	2	is	be	AUX
ejpam-3856	202	3	obvious	obvious	ADJ
ejpam-3856	202	4	that	that	SCONJ
ejpam-3856	202	5	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	202	6	)	)	PUNCT
ejpam-3856	202	7	⊂	⊂	PROPN
ejpam-3856	202	8	cl∗β(cl∗β(a	cl∗β(cl∗β(a	PROPN
ejpam-3856	202	9	)	)	PUNCT
ejpam-3856	202	10	)	)	PUNCT
ejpam-3856	202	11	.	.	PUNCT
ejpam-3856	203	1	therefore	therefore	ADV
ejpam-3856	203	2	,	,	PUNCT
ejpam-3856	203	3	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	203	4	)	)	PUNCT
ejpam-3856	203	5	=	=	SYM
ejpam-3856	203	6	cl∗β(cl∗β(a	cl∗β(cl∗β(a	PROPN
ejpam-3856	203	7	)	)	PUNCT
ejpam-3856	203	8	)	)	PUNCT
ejpam-3856	203	9	.	.	PUNCT
ejpam-3856	204	1	theorem	theorem	NOUN
ejpam-3856	204	2	3	3	X
ejpam-3856	204	3	.	.	PUNCT
ejpam-3856	205	1	let	let	VERB
ejpam-3856	205	2	(	(	PUNCT
ejpam-3856	205	3	x	x	X
ejpam-3856	205	4	,	,	PUNCT
ejpam-3856	205	5	τ	τ	PROPN
ejpam-3856	205	6	,	,	PUNCT
ejpam-3856	205	7	i	i	PRON
ejpam-3856	205	8	)	)	PUNCT
ejpam-3856	205	9	be	be	VERB
ejpam-3856	205	10	an	an	DET
ejpam-3856	205	11	ideal	ideal	ADJ
ejpam-3856	205	12	topological	topological	ADJ
ejpam-3856	205	13	space	space	NOUN
ejpam-3856	205	14	.	.	PUNCT
ejpam-3856	206	1	let	let	VERB
ejpam-3856	206	2	τ∗β	τ∗β	PUNCT
ejpam-3856	206	3	=	=	PRON
ejpam-3856	206	4	{	{	PUNCT
ejpam-3856	206	5	u	u	X
ejpam-3856	206	6	⊂	⊂	PROPN
ejpam-3856	206	7	x	x	X
ejpam-3856	206	8	:	:	PUNCT
ejpam-3856	206	9	cl∗β(x	cl∗β(x	NOUN
ejpam-3856	206	10	\u	\u	X
ejpam-3856	206	11	)	)	PUNCT
ejpam-3856	206	12	=	=	PUNCT
ejpam-3856	206	13	x	x	SYM
ejpam-3856	206	14	\	\	PROPN
ejpam-3856	206	15	u	u	NOUN
ejpam-3856	206	16	}	}	PUNCT
ejpam-3856	206	17	.	.	PUNCT
ejpam-3856	207	1	then	then	ADV
ejpam-3856	207	2	τ∗β	τ∗β	PUNCT
ejpam-3856	207	3	is	be	AUX
ejpam-3856	207	4	a	a	DET
ejpam-3856	207	5	topology	topology	NOUN
ejpam-3856	207	6	for	for	ADP
ejpam-3856	207	7	x	x	SYM
ejpam-3856	207	8	such	such	ADJ
ejpam-3856	207	9	that	that	SCONJ
ejpam-3856	207	10	τ∗	τ∗	NOUN
ejpam-3856	207	11	⊂	⊂	X
ejpam-3856	207	12	τ∗β	τ∗β	PUNCT
ejpam-3856	207	13	and	and	CCONJ
ejpam-3856	207	14	βo(x	βo(x	NUM
ejpam-3856	207	15	)	)	PUNCT
ejpam-3856	208	1	⊂	⊂	PRON
ejpam-3856	208	2	τ∗β	τ∗β	PUNCT
ejpam-3856	208	3	.	.	PUNCT
ejpam-3856	209	1	proof	proof	NOUN
ejpam-3856	209	2	.	.	PUNCT
ejpam-3856	210	1	by	by	ADP
ejpam-3856	210	2	theorem	theorem	NOUN
ejpam-3856	210	3	2	2	NUM
ejpam-3856	210	4	,	,	PUNCT
ejpam-3856	210	5	we	we	PRON
ejpam-3856	210	6	obtain	obtain	VERB
ejpam-3856	210	7	that	that	SCONJ
ejpam-3856	210	8	cl∗β(a	cl∗β(a	NOUN
ejpam-3856	210	9	)	)	PUNCT
ejpam-3856	211	1	=	=	NOUN
ejpam-3856	211	2	a	a	DET
ejpam-3856	211	3	∪	∪	ADJ
ejpam-3856	211	4	a∗	a∗	PROPN
ejpam-3856	211	5	β	β	X
ejpam-3856	211	6	is	be	AUX
ejpam-3856	211	7	a	a	DET
ejpam-3856	211	8	kuratowski	kuratowski	ADJ
ejpam-3856	211	9	closure	closure	NOUN
ejpam-3856	211	10	operator	operator	NOUN
ejpam-3856	211	11	.	.	PUNCT
ejpam-3856	212	1	therefore	therefore	ADV
ejpam-3856	212	2	,	,	PUNCT
ejpam-3856	212	3	τ∗β	τ∗β	PUNCT
ejpam-3856	212	4	is	be	AUX
ejpam-3856	212	5	the	the	DET
ejpam-3856	212	6	topology	topology	NOUN
ejpam-3856	212	7	for	for	ADP
ejpam-3856	212	8	x	x	PUNCT
ejpam-3856	212	9	generated	generate	VERB
ejpam-3856	212	10	by	by	ADP
ejpam-3856	212	11	cl∗β	cl∗β	PROPN
ejpam-3856	212	12	.	.	PUNCT
ejpam-3856	213	1	first	first	ADV
ejpam-3856	213	2	,	,	PUNCT
ejpam-3856	213	3	we	we	PRON
ejpam-3856	213	4	show	show	VERB
ejpam-3856	213	5	that	that	SCONJ
ejpam-3856	213	6	τ∗	τ∗	NOUN
ejpam-3856	213	7	⊂	⊂	PRON
ejpam-3856	213	8	τ∗β	τ∗β	PUNCT
ejpam-3856	213	9	.	.	PUNCT
ejpam-3856	214	1	by	by	ADP
ejpam-3856	214	2	lemma	lemma	PROPN
ejpam-3856	214	3	2(2	2(2	NUM
ejpam-3856	214	4	)	)	PUNCT
ejpam-3856	214	5	,	,	PUNCT
ejpam-3856	214	6	for	for	ADP
ejpam-3856	214	7	every	every	DET
ejpam-3856	214	8	subset	subset	NOUN
ejpam-3856	214	9	a	a	PRON
ejpam-3856	214	10	of	of	ADP
ejpam-3856	214	11	x	x	NOUN
ejpam-3856	214	12	,	,	PUNCT
ejpam-3856	214	13	cl∗β(a	cl∗β(a	ADJ
ejpam-3856	214	14	)	)	PUNCT
ejpam-3856	214	15	=	=	PUNCT
ejpam-3856	215	1	a∪a∗	a∪a∗	PROPN
ejpam-3856	215	2	β	β	X
ejpam-3856	215	3	⊂	⊂	PROPN
ejpam-3856	215	4	a	a	DET
ejpam-3856	215	5	∪	∪	ADJ
ejpam-3856	215	6	a∗	a∗	NOUN
ejpam-3856	215	7	=	=	SYM
ejpam-3856	215	8	cl∗(a	cl∗(a	NOUN
ejpam-3856	215	9	)	)	PUNCT
ejpam-3856	215	10	.	.	PUNCT
ejpam-3856	216	1	let	let	VERB
ejpam-3856	216	2	a	a	PRON
ejpam-3856	216	3	be	be	AUX
ejpam-3856	216	4	a	a	DET
ejpam-3856	216	5	τ∗-closed	τ∗-close	VERB
ejpam-3856	216	6	set	set	NOUN
ejpam-3856	216	7	,	,	PUNCT
ejpam-3856	216	8	then	then	ADV
ejpam-3856	216	9	cl∗(a	cl∗(a	NOUN
ejpam-3856	216	10	)	)	PUNCT
ejpam-3856	216	11	=	=	SYM
ejpam-3856	216	12	a	a	PRON
ejpam-3856	216	13	and	and	CCONJ
ejpam-3856	216	14	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	216	15	)	)	PUNCT
ejpam-3856	217	1	⊂	⊂	PROPN
ejpam-3856	217	2	a.	a.	NOUN
ejpam-3856	217	3	hence	hence	ADV
ejpam-3856	217	4	cl∗β(a	cl∗β(a	ADJ
ejpam-3856	217	5	)	)	PUNCT
ejpam-3856	218	1	=	=	SYM
ejpam-3856	218	2	a	a	PRON
ejpam-3856	218	3	and	and	CCONJ
ejpam-3856	218	4	a	a	PRON
ejpam-3856	218	5	is	is	AUX
ejpam-3856	218	6	τ∗β	τ∗β	PUNCT
ejpam-3856	218	7	-closed	-closed	ADJ
ejpam-3856	218	8	.	.	PUNCT
ejpam-3856	219	1	secondly	secondly	ADV
ejpam-3856	219	2	,	,	PUNCT
ejpam-3856	219	3	we	we	PRON
ejpam-3856	219	4	show	show	VERB
ejpam-3856	219	5	that	that	SCONJ
ejpam-3856	219	6	βo(x	βo(x	PUNCT
ejpam-3856	219	7	)	)	PUNCT
ejpam-3856	220	1	⊂	⊂	NOUN
ejpam-3856	220	2	τ∗β	τ∗β	PUNCT
ejpam-3856	220	3	.	.	PUNCT
ejpam-3856	220	4	suppose	suppose	VERB
ejpam-3856	220	5	that	that	SCONJ
ejpam-3856	220	6	a	a	PRON
ejpam-3856	220	7	is	be	AUX
ejpam-3856	220	8	β	β	NOUN
ejpam-3856	220	9	-	-	VERB
ejpam-3856	220	10	closed	closed	ADJ
ejpam-3856	220	11	.	.	PUNCT
ejpam-3856	221	1	if	if	SCONJ
ejpam-3856	221	2	x	x	X
ejpam-3856	221	3	/∈	/∈	NOUN
ejpam-3856	222	1	a	a	INTJ
ejpam-3856	222	2	,	,	PUNCT
ejpam-3856	222	3	then	then	ADV
ejpam-3856	222	4	by	by	ADP
ejpam-3856	222	5	lemma	lemma	PROPN
ejpam-3856	222	6	1	1	NUM
ejpam-3856	222	7	,	,	PUNCT
ejpam-3856	222	8	there	there	PRON
ejpam-3856	222	9	exists	exist	VERB
ejpam-3856	222	10	u	u	NOUN
ejpam-3856	222	11	in	in	ADP
ejpam-3856	222	12	βo(x	βo(x	PUNCT
ejpam-3856	222	13	)	)	PUNCT
ejpam-3856	222	14	such	such	ADJ
ejpam-3856	222	15	that	that	SCONJ
ejpam-3856	222	16	u	u	PROPN
ejpam-3856	222	17	∩	∩	NOUN
ejpam-3856	222	18	a	a	DET
ejpam-3856	222	19	=	=	SYM
ejpam-3856	222	20	φ	φ	PROPN
ejpam-3856	222	21	∈	∈	PROPN
ejpam-3856	222	22	i.	i.	NOUN
ejpam-3856	222	23	hence	hence	ADV
ejpam-3856	222	24	x	x	PROPN
ejpam-3856	222	25	/∈	/∈	PUNCT
ejpam-3856	223	1	a∗	a∗	PROPN
ejpam-3856	223	2	β	β	X
ejpam-3856	223	3	.	.	PUNCT
ejpam-3856	224	1	this	this	PRON
ejpam-3856	224	2	shows	show	VERB
ejpam-3856	224	3	a∗	a∗	PROPN
ejpam-3856	224	4	β	β	PROPN
ejpam-3856	224	5	⊂	⊂	PROPN
ejpam-3856	224	6	a.	a.	NOUN
ejpam-3856	224	7	therefore	therefore	ADV
ejpam-3856	224	8	,	,	PUNCT
ejpam-3856	224	9	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	224	10	)	)	PUNCT
ejpam-3856	224	11	=	=	PUNCT
ejpam-3856	225	1	a∪a∗	a∪a∗	PROPN
ejpam-3856	225	2	β	β	X
ejpam-3856	225	3	=	=	PUNCT
ejpam-3856	225	4	a	a	PROPN
ejpam-3856	225	5	and	and	CCONJ
ejpam-3856	225	6	a	a	PRON
ejpam-3856	225	7	is	be	AUX
ejpam-3856	225	8	τ∗β	τ∗β	PUNCT
ejpam-3856	225	9	-closed	-closed	ADJ
ejpam-3856	225	10	.	.	PUNCT
ejpam-3856	226	1	we	we	PRON
ejpam-3856	226	2	obtain	obtain	VERB
ejpam-3856	226	3	that	that	PRON
ejpam-3856	226	4	βo(x	βo(x	PUNCT
ejpam-3856	226	5	)	)	PUNCT
ejpam-3856	227	1	⊂	⊂	PRON
ejpam-3856	227	2	τ∗β	τ∗β	PUNCT
ejpam-3856	227	3	.	.	PUNCT
ejpam-3856	228	1	definition	definition	NOUN
ejpam-3856	228	2	10	10	NUM
ejpam-3856	228	3	.	.	PUNCT
ejpam-3856	229	1	let	let	VERB
ejpam-3856	229	2	(	(	PUNCT
ejpam-3856	229	3	x	x	X
ejpam-3856	229	4	,	,	PUNCT
ejpam-3856	229	5	τ	τ	PROPN
ejpam-3856	229	6	,	,	PUNCT
ejpam-3856	229	7	i	i	PRON
ejpam-3856	229	8	)	)	PUNCT
ejpam-3856	229	9	be	be	VERB
ejpam-3856	229	10	an	an	DET
ejpam-3856	229	11	ideal	ideal	ADJ
ejpam-3856	229	12	topological	topological	ADJ
ejpam-3856	229	13	space	space	NOUN
ejpam-3856	229	14	.	.	PUNCT
ejpam-3856	230	1	a	a	DET
ejpam-3856	230	2	subset	subset	NOUN
ejpam-3856	230	3	a	a	PRON
ejpam-3856	230	4	of	of	ADP
ejpam-3856	230	5	x	x	SYM
ejpam-3856	230	6	is	be	AUX
ejpam-3856	230	7	said	say	VERB
ejpam-3856	230	8	to	to	PART
ejpam-3856	230	9	be	be	AUX
ejpam-3856	230	10	ig	ig	PROPN
ejpam-3856	230	11	-	-	ADJ
ejpam-3856	230	12	β	β	NOUN
ejpam-3856	230	13	-	-	ADJ
ejpam-3856	230	14	closed	closed	ADJ
ejpam-3856	230	15	if	if	SCONJ
ejpam-3856	230	16	a∗	a∗	PROPN
ejpam-3856	230	17	β	β	X
ejpam-3856	230	18	⊂	⊂	PROPN
ejpam-3856	230	19	u	u	PROPN
ejpam-3856	230	20	whenever	whenever	SCONJ
ejpam-3856	230	21	a	a	DET
ejpam-3856	230	22	⊂	⊂	PROPN
ejpam-3856	230	23	u	u	NOUN
ejpam-3856	230	24	and	and	CCONJ
ejpam-3856	230	25	u	u	NOUN
ejpam-3856	230	26	in	in	ADP
ejpam-3856	230	27	βo(x	βo(x	PUNCT
ejpam-3856	230	28	)	)	PUNCT
ejpam-3856	230	29	.	.	PUNCT
ejpam-3856	231	1	theorem	theorem	ADJ
ejpam-3856	231	2	4	4	NUM
ejpam-3856	231	3	.	.	X
ejpam-3856	231	4	for	for	ADP
ejpam-3856	231	5	a	a	DET
ejpam-3856	231	6	subset	subset	NOUN
ejpam-3856	231	7	a	a	PRON
ejpam-3856	231	8	of	of	ADP
ejpam-3856	231	9	an	an	DET
ejpam-3856	231	10	ideal	ideal	ADJ
ejpam-3856	231	11	topological	topological	ADJ
ejpam-3856	231	12	space	space	NOUN
ejpam-3856	231	13	(	(	PUNCT
ejpam-3856	231	14	x	x	X
ejpam-3856	231	15	,	,	PUNCT
ejpam-3856	231	16	τ	τ	PROPN
ejpam-3856	231	17	,	,	PUNCT
ejpam-3856	231	18	i	i	PROPN
ejpam-3856	231	19	)	)	PUNCT
ejpam-3856	231	20	,	,	PUNCT
ejpam-3856	231	21	the	the	DET
ejpam-3856	231	22	following	follow	VERB
ejpam-3856	231	23	properties	property	NOUN
ejpam-3856	231	24	are	be	AUX
ejpam-3856	231	25	equivalent	equivalent	ADJ
ejpam-3856	231	26	:	:	PUNCT
ejpam-3856	231	27	p.	p.	PROPN
ejpam-3856	231	28	l.	l.	PROPN
ejpam-3856	231	29	powar	powar	PROPN
ejpam-3856	231	30	,	,	PUNCT
ejpam-3856	231	31	t.	t.	PROPN
ejpam-3856	231	32	noiri	noiri	PROPN
ejpam-3856	231	33	,	,	PUNCT
ejpam-3856	231	34	shikha	shikha	PROPN
ejpam-3856	231	35	bhadauria	bhadauria	PROPN
ejpam-3856	231	36	/	/	SYM
ejpam-3856	231	37	eur	eur	PROPN
ejpam-3856	231	38	.	.	PUNCT
ejpam-3856	232	1	j.	j.	PROPN
ejpam-3856	232	2	pure	pure	PROPN
ejpam-3856	232	3	appl	appl	PROPN
ejpam-3856	232	4	.	.	PROPN
ejpam-3856	232	5	math	math	PROPN
ejpam-3856	232	6	,	,	PUNCT
ejpam-3856	232	7	13	13	NUM
ejpam-3856	232	8	(	(	PUNCT
ejpam-3856	232	9	4	4	NUM
ejpam-3856	232	10	)	)	PUNCT
ejpam-3856	232	11	(	(	PUNCT
ejpam-3856	232	12	2020	2020	NUM
ejpam-3856	232	13	)	)	PUNCT
ejpam-3856	232	14	,	,	PUNCT
ejpam-3856	232	15	758	758	NUM
ejpam-3856	232	16	-	-	SYM
ejpam-3856	232	17	765	765	NUM
ejpam-3856	232	18	763	763	NUM
ejpam-3856	232	19	(	(	PUNCT
ejpam-3856	232	20	1	1	NUM
ejpam-3856	232	21	)	)	PUNCT
ejpam-3856	232	22	.	.	PUNCT
ejpam-3856	233	1	a	a	PRON
ejpam-3856	233	2	is	be	AUX
ejpam-3856	233	3	ig	ig	PROPN
ejpam-3856	233	4	-	-	ADJ
ejpam-3856	233	5	β	β	NOUN
ejpam-3856	233	6	-	-	VERB
ejpam-3856	233	7	closed	closed	ADJ
ejpam-3856	233	8	;	;	PUNCT
ejpam-3856	233	9	(	(	PUNCT
ejpam-3856	233	10	2	2	NUM
ejpam-3856	233	11	)	)	PUNCT
ejpam-3856	233	12	.	.	PUNCT
ejpam-3856	234	1	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	234	2	)	)	PUNCT
ejpam-3856	235	1	⊂	⊂	PROPN
ejpam-3856	235	2	u	u	NOUN
ejpam-3856	235	3	whenever	whenever	SCONJ
ejpam-3856	235	4	a	a	DET
ejpam-3856	235	5	⊂	⊂	PROPN
ejpam-3856	235	6	u	u	NOUN
ejpam-3856	235	7	and	and	CCONJ
ejpam-3856	235	8	u	u	NOUN
ejpam-3856	235	9	is	be	AUX
ejpam-3856	235	10	β	β	X
ejpam-3856	235	11	-	-	ADJ
ejpam-3856	235	12	open	open	ADJ
ejpam-3856	235	13	;	;	PUNCT
ejpam-3856	235	14	(	(	PUNCT
ejpam-3856	235	15	3	3	NUM
ejpam-3856	235	16	)	)	PUNCT
ejpam-3856	235	17	.	.	PUNCT
ejpam-3856	236	1	for	for	ADP
ejpam-3856	236	2	every	every	DET
ejpam-3856	236	3	x	x	PROPN
ejpam-3856	236	4	∈	∈	PROPN
ejpam-3856	236	5	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	236	6	)	)	PUNCT
ejpam-3856	236	7	,	,	PUNCT
ejpam-3856	236	8	βcl({x	βcl({x	NOUN
ejpam-3856	236	9	}	}	PUNCT
ejpam-3856	236	10	)	)	PUNCT
ejpam-3856	236	11	∩a	∩a	PROPN
ejpam-3856	236	12	6=	6=	PROPN
ejpam-3856	237	1	φ	φ	PROPN
ejpam-3856	237	2	;	;	PUNCT
ejpam-3856	237	3	(	(	PUNCT
ejpam-3856	237	4	4	4	NUM
ejpam-3856	237	5	)	)	PUNCT
ejpam-3856	237	6	.	.	PUNCT
ejpam-3856	238	1	cl∗β(a)−a	cl∗β(a)−a	PROPN
ejpam-3856	238	2	contains	contain	VERB
ejpam-3856	238	3	no	no	DET
ejpam-3856	238	4	nonempty	nonempty	ADV
ejpam-3856	238	5	β	β	ADJ
ejpam-3856	238	6	-	-	ADJ
ejpam-3856	238	7	closed	closed	ADJ
ejpam-3856	238	8	set	set	NOUN
ejpam-3856	238	9	;	;	PUNCT
ejpam-3856	238	10	(	(	PUNCT
ejpam-3856	238	11	5	5	NUM
ejpam-3856	238	12	)	)	PUNCT
ejpam-3856	238	13	.	.	PUNCT
ejpam-3856	239	1	a∗	a∗	PROPN
ejpam-3856	239	2	β	β	PROPN
ejpam-3856	239	3	−a	−a	NOUN
ejpam-3856	239	4	contains	contain	VERB
ejpam-3856	239	5	no	no	DET
ejpam-3856	239	6	nonempty	nonempty	ADV
ejpam-3856	239	7	β	β	ADJ
ejpam-3856	239	8	-	-	ADJ
ejpam-3856	239	9	closed	closed	ADJ
ejpam-3856	239	10	set	set	NOUN
ejpam-3856	239	11	.	.	PUNCT
ejpam-3856	240	1	proof	proof	NOUN
ejpam-3856	240	2	.	.	PUNCT
ejpam-3856	241	1	(	(	PUNCT
ejpam-3856	241	2	1	1	NUM
ejpam-3856	241	3	)	)	PUNCT
ejpam-3856	241	4	.	.	PUNCT
ejpam-3856	242	1	⇒	⇒	NOUN
ejpam-3856	242	2	(	(	PUNCT
ejpam-3856	242	3	2	2	NUM
ejpam-3856	242	4	)	)	PUNCT
ejpam-3856	242	5	.	.	PUNCT
ejpam-3856	243	1	by	by	ADP
ejpam-3856	243	2	hypothesis	hypothesis	NOUN
ejpam-3856	243	3	,	,	PUNCT
ejpam-3856	243	4	a	a	PRON
ejpam-3856	243	5	is	be	AUX
ejpam-3856	243	6	ig	ig	PROPN
ejpam-3856	243	7	-	-	ADJ
ejpam-3856	243	8	β	β	NOUN
ejpam-3856	243	9	-	-	ADJ
ejpam-3856	243	10	closed	closed	ADJ
ejpam-3856	243	11	.	.	PUNCT
ejpam-3856	244	1	therefore	therefore	ADV
ejpam-3856	244	2	,	,	PUNCT
ejpam-3856	244	3	a∗	a∗	PROPN
ejpam-3856	244	4	β	β	X
ejpam-3856	244	5	⊂	⊂	PROPN
ejpam-3856	244	6	u	u	PROPN
ejpam-3856	244	7	whenever	whenever	SCONJ
ejpam-3856	244	8	a	a	DET
ejpam-3856	244	9	⊂	⊂	PROPN
ejpam-3856	244	10	u	u	NOUN
ejpam-3856	244	11	and	and	CCONJ
ejpam-3856	244	12	u	u	NOUN
ejpam-3856	244	13	in	in	ADP
ejpam-3856	244	14	βo(x	βo(x	PUNCT
ejpam-3856	244	15	)	)	PUNCT
ejpam-3856	244	16	.	.	PUNCT
ejpam-3856	245	1	this	this	PRON
ejpam-3856	245	2	implies	imply	VERB
ejpam-3856	245	3	a∗	a∗	PROPN
ejpam-3856	245	4	β	β	SYM
ejpam-3856	245	5	∪	∪	ADP
ejpam-3856	245	6	a	a	DET
ejpam-3856	245	7	⊂	⊂	PROPN
ejpam-3856	245	8	u	u	NOUN
ejpam-3856	245	9	and	and	CCONJ
ejpam-3856	245	10	hence	hence	ADV
ejpam-3856	245	11	,	,	PUNCT
ejpam-3856	245	12	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	245	13	)	)	PUNCT
ejpam-3856	245	14	⊂	⊂	PROPN
ejpam-3856	245	15	u	u	NOUN
ejpam-3856	245	16	whenever	whenever	SCONJ
ejpam-3856	245	17	a	a	DET
ejpam-3856	245	18	⊂	⊂	PROPN
ejpam-3856	245	19	u	u	NOUN
ejpam-3856	245	20	and	and	CCONJ
ejpam-3856	245	21	u	u	NOUN
ejpam-3856	245	22	∈	∈	PROPN
ejpam-3856	245	23	βo(x	βo(x	PUNCT
ejpam-3856	245	24	)	)	PUNCT
ejpam-3856	245	25	.	.	PUNCT
ejpam-3856	246	1	(	(	PUNCT
ejpam-3856	246	2	2	2	NUM
ejpam-3856	246	3	)	)	PUNCT
ejpam-3856	246	4	.	.	PUNCT
ejpam-3856	247	1	⇒	⇒	NOUN
ejpam-3856	247	2	(	(	PUNCT
ejpam-3856	247	3	3	3	NUM
ejpam-3856	247	4	)	)	PUNCT
ejpam-3856	247	5	.	.	PUNCT
ejpam-3856	248	1	suppose	suppose	VERB
ejpam-3856	248	2	x	x	X
ejpam-3856	248	3	∈	∈	PROPN
ejpam-3856	248	4	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	248	5	)	)	PUNCT
ejpam-3856	248	6	.	.	PUNCT
ejpam-3856	249	1	if	if	SCONJ
ejpam-3856	249	2	βcl({x	βcl({x	NOUN
ejpam-3856	249	3	}	}	PUNCT
ejpam-3856	249	4	)	)	PUNCT
ejpam-3856	249	5	∩	∩	NOUN
ejpam-3856	249	6	a	a	DET
ejpam-3856	249	7	=	=	SYM
ejpam-3856	249	8	φ	φ	PROPN
ejpam-3856	249	9	,	,	PUNCT
ejpam-3856	249	10	then	then	ADV
ejpam-3856	249	11	a	a	PRON
ejpam-3856	249	12	⊂	⊂	PROPN
ejpam-3856	249	13	(	(	PUNCT
ejpam-3856	249	14	x	x	X
ejpam-3856	249	15	−	−	ADP
ejpam-3856	249	16	βcl({x	βcl({x	NOUN
ejpam-3856	249	17	}	}	PUNCT
ejpam-3856	249	18	)	)	PUNCT
ejpam-3856	249	19	)	)	PUNCT
ejpam-3856	249	20	,	,	PUNCT
ejpam-3856	249	21	where	where	SCONJ
ejpam-3856	249	22	(	(	PUNCT
ejpam-3856	249	23	x	x	NOUN
ejpam-3856	249	24	−	−	NOUN
ejpam-3856	249	25	βcl({x	βcl({x	NOUN
ejpam-3856	249	26	}	}	PUNCT
ejpam-3856	249	27	)	)	PUNCT
ejpam-3856	249	28	)	)	PUNCT
ejpam-3856	249	29	is	be	AUX
ejpam-3856	249	30	β	β	X
ejpam-3856	249	31	-	-	ADJ
ejpam-3856	249	32	open	open	ADJ
ejpam-3856	249	33	and	and	CCONJ
ejpam-3856	249	34	by	by	ADP
ejpam-3856	249	35	hypothesis	hypothesis	NOUN
ejpam-3856	249	36	,	,	PUNCT
ejpam-3856	249	37	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	249	38	)	)	PUNCT
ejpam-3856	249	39	⊂	⊂	PROPN
ejpam-3856	249	40	(	(	PUNCT
ejpam-3856	250	1	x	x	X
ejpam-3856	250	2	−	−	ADP
ejpam-3856	250	3	βcl({x	βcl({x	NOUN
ejpam-3856	250	4	}	}	PUNCT
ejpam-3856	250	5	)	)	PUNCT
ejpam-3856	250	6	)	)	PUNCT
ejpam-3856	250	7	.	.	PUNCT
ejpam-3856	251	1	therefore	therefore	ADV
ejpam-3856	251	2	,	,	PUNCT
ejpam-3856	251	3	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	251	4	)	)	PUNCT
ejpam-3856	251	5	∩	∩	NOUN
ejpam-3856	251	6	βcl({x	βcl({x	NOUN
ejpam-3856	251	7	}	}	PUNCT
ejpam-3856	251	8	)	)	PUNCT
ejpam-3856	251	9	=	=	SYM
ejpam-3856	251	10	φ	φ	PROPN
ejpam-3856	251	11	,	,	PUNCT
ejpam-3856	251	12	which	which	PRON
ejpam-3856	251	13	is	be	AUX
ejpam-3856	251	14	a	a	DET
ejpam-3856	251	15	contradiction	contradiction	NOUN
ejpam-3856	251	16	,	,	PUNCT
ejpam-3856	251	17	since	since	SCONJ
ejpam-3856	251	18	x	x	PROPN
ejpam-3856	251	19	∈	∈	PROPN
ejpam-3856	251	20	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	251	21	)	)	PUNCT
ejpam-3856	251	22	.	.	PUNCT
ejpam-3856	252	1	hence	hence	ADV
ejpam-3856	252	2	,	,	PUNCT
ejpam-3856	252	3	for	for	ADP
ejpam-3856	252	4	every	every	DET
ejpam-3856	252	5	x	x	PROPN
ejpam-3856	252	6	∈	∈	PROPN
ejpam-3856	252	7	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	252	8	)	)	PUNCT
ejpam-3856	252	9	,	,	PUNCT
ejpam-3856	252	10	βcl({x	βcl({x	NOUN
ejpam-3856	252	11	}	}	PUNCT
ejpam-3856	252	12	)	)	PUNCT
ejpam-3856	253	1	∩a	∩a	PROPN
ejpam-3856	253	2	6=	6=	PROPN
ejpam-3856	254	1	φ	φ	PROPN
ejpam-3856	254	2	(	(	PUNCT
ejpam-3856	254	3	3	3	NUM
ejpam-3856	254	4	)	)	PUNCT
ejpam-3856	254	5	.	.	PUNCT
ejpam-3856	255	1	⇒	⇒	NOUN
ejpam-3856	255	2	(	(	PUNCT
ejpam-3856	255	3	4	4	NUM
ejpam-3856	255	4	)	)	PUNCT
ejpam-3856	255	5	.	.	PUNCT
ejpam-3856	256	1	let	let	VERB
ejpam-3856	256	2	if	if	SCONJ
ejpam-3856	256	3	possible	possible	ADJ
ejpam-3856	256	4	,	,	PUNCT
ejpam-3856	256	5	f	f	PROPN
ejpam-3856	256	6	⊂	⊂	PROPN
ejpam-3856	256	7	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	256	8	)	)	PUNCT
ejpam-3856	257	1	−	−	PROPN
ejpam-3856	258	1	a	a	INTJ
ejpam-3856	258	2	,	,	PUNCT
ejpam-3856	258	3	where	where	SCONJ
ejpam-3856	258	4	f	f	PROPN
ejpam-3856	258	5	is	be	AUX
ejpam-3856	258	6	a	a	DET
ejpam-3856	258	7	nonempty	nonempty	ADJ
ejpam-3856	258	8	β	β	NOUN
ejpam-3856	258	9	-	-	ADJ
ejpam-3856	258	10	closed	closed	ADJ
ejpam-3856	258	11	set	set	NOUN
ejpam-3856	258	12	and	and	CCONJ
ejpam-3856	258	13	x	x	SYM
ejpam-3856	258	14	∈	∈	PROPN
ejpam-3856	258	15	f	f	X
ejpam-3856	258	16	.	.	PUNCT
ejpam-3856	259	1	this	this	PRON
ejpam-3856	259	2	implies	imply	VERB
ejpam-3856	259	3	f	f	PROPN
ejpam-3856	259	4	⊂	⊂	PROPN
ejpam-3856	259	5	x	x	PUNCT
ejpam-3856	259	6	−	−	NOUN
ejpam-3856	259	7	a	a	DET
ejpam-3856	259	8	and	and	CCONJ
ejpam-3856	259	9	f	f	PROPN
ejpam-3856	259	10	∩	∩	NOUN
ejpam-3856	259	11	a	a	DET
ejpam-3856	259	12	=	=	SYM
ejpam-3856	259	13	φ	φ	PROPN
ejpam-3856	259	14	.	.	PUNCT
ejpam-3856	260	1	therefore	therefore	ADV
ejpam-3856	260	2	,	,	PUNCT
ejpam-3856	260	3	βcl({x	βcl({x	NOUN
ejpam-3856	260	4	}	}	PUNCT
ejpam-3856	260	5	)	)	PUNCT
ejpam-3856	260	6	∩	∩	NOUN
ejpam-3856	260	7	a	a	DET
ejpam-3856	260	8	=	=	SYM
ejpam-3856	260	9	φ	φ	PROPN
ejpam-3856	260	10	,	,	PUNCT
ejpam-3856	260	11	which	which	PRON
ejpam-3856	260	12	is	be	AUX
ejpam-3856	260	13	a	a	DET
ejpam-3856	260	14	contradiction	contradiction	NOUN
ejpam-3856	260	15	to	to	ADP
ejpam-3856	260	16	our	our	PRON
ejpam-3856	260	17	hypothesis	hypothesis	NOUN
ejpam-3856	260	18	as	as	ADP
ejpam-3856	260	19	βcl({x	βcl({x	NOUN
ejpam-3856	260	20	}	}	PUNCT
ejpam-3856	260	21	)	)	PUNCT
ejpam-3856	260	22	∩	∩	NOUN
ejpam-3856	260	23	a	a	DET
ejpam-3856	260	24	6=	6=	NUM
ejpam-3856	260	25	φ	φ	PROPN
ejpam-3856	260	26	.	.	PUNCT
ejpam-3856	261	1	hence	hence	ADV
ejpam-3856	261	2	,	,	PUNCT
ejpam-3856	261	3	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	261	4	)	)	PUNCT
ejpam-3856	262	1	−	−	PROPN
ejpam-3856	262	2	a	a	PRON
ejpam-3856	262	3	contains	contain	VERB
ejpam-3856	262	4	no	no	DET
ejpam-3856	262	5	nonempty	nonempty	ADV
ejpam-3856	262	6	β	β	ADJ
ejpam-3856	262	7	-	-	ADJ
ejpam-3856	262	8	closed	closed	ADJ
ejpam-3856	262	9	set	set	NOUN
ejpam-3856	262	10	.	.	PUNCT
ejpam-3856	263	1	(	(	PUNCT
ejpam-3856	263	2	4	4	NUM
ejpam-3856	263	3	)	)	PUNCT
ejpam-3856	263	4	.	.	PUNCT
ejpam-3856	264	1	⇒	⇒	NOUN
ejpam-3856	264	2	(	(	PUNCT
ejpam-3856	264	3	5	5	NUM
ejpam-3856	264	4	)	)	PUNCT
ejpam-3856	264	5	.	.	PUNCT
ejpam-3856	265	1	the	the	DET
ejpam-3856	265	2	proof	proof	NOUN
ejpam-3856	265	3	is	be	AUX
ejpam-3856	265	4	obvious	obvious	ADJ
ejpam-3856	265	5	,	,	PUNCT
ejpam-3856	265	6	since	since	SCONJ
ejpam-3856	265	7	a∗	a∗	PROPN
ejpam-3856	265	8	β	β	X
ejpam-3856	265	9	⊂	⊂	X
ejpam-3856	265	10	cl∗β(a	cl∗β(a	PROPN
ejpam-3856	265	11	)	)	PUNCT
ejpam-3856	265	12	.	.	PUNCT
ejpam-3856	266	1	(	(	PUNCT
ejpam-3856	266	2	5	5	NUM
ejpam-3856	266	3	)	)	PUNCT
ejpam-3856	266	4	.	.	PUNCT
ejpam-3856	267	1	⇒	⇒	NOUN
ejpam-3856	267	2	(	(	PUNCT
ejpam-3856	267	3	1	1	NUM
ejpam-3856	267	4	)	)	PUNCT
ejpam-3856	267	5	.	.	PUNCT
ejpam-3856	268	1	let	let	VERB
ejpam-3856	268	2	a	a	DET
ejpam-3856	268	3	⊂	⊂	PROPN
ejpam-3856	268	4	u	u	NOUN
ejpam-3856	268	5	and	and	CCONJ
ejpam-3856	268	6	u	u	NOUN
ejpam-3856	268	7	is	be	AUX
ejpam-3856	268	8	any	any	DET
ejpam-3856	268	9	β	β	NOUN
ejpam-3856	268	10	-	-	ADJ
ejpam-3856	268	11	open	open	ADJ
ejpam-3856	268	12	set	set	NOUN
ejpam-3856	268	13	of	of	ADP
ejpam-3856	268	14	x.	x.	NOUN
ejpam-3856	268	15	by	by	ADP
ejpam-3856	268	16	theorem	theorem	ADJ
ejpam-3856	268	17	1(5	1(5	NUM
ejpam-3856	268	18	)	)	PUNCT
ejpam-3856	268	19	,	,	PUNCT
ejpam-3856	268	20	a∗	a∗	PROPN
ejpam-3856	268	21	β	β	X
ejpam-3856	268	22	is	be	AUX
ejpam-3856	268	23	β	β	NOUN
ejpam-3856	268	24	-	-	VERB
ejpam-3856	268	25	closed	closed	ADJ
ejpam-3856	268	26	and	and	CCONJ
ejpam-3856	268	27	a∗	a∗	PROPN
ejpam-3856	268	28	β	β	X
ejpam-3856	268	29	∩	∩	X
ejpam-3856	268	30	(	(	PUNCT
ejpam-3856	268	31	x	x	SYM
ejpam-3856	268	32	−u	−u	PROPN
ejpam-3856	268	33	)	)	PUNCT
ejpam-3856	269	1	⊂	⊂	PROPN
ejpam-3856	269	2	a∗	a∗	PROPN
ejpam-3856	269	3	β	β	PROPN
ejpam-3856	269	4	−a	−a	NOUN
ejpam-3856	269	5	,	,	PUNCT
ejpam-3856	269	6	where	where	SCONJ
ejpam-3856	269	7	,	,	PUNCT
ejpam-3856	269	8	a∗	a∗	PROPN
ejpam-3856	269	9	β	β	X
ejpam-3856	269	10	∩	∩	X
ejpam-3856	269	11	(	(	PUNCT
ejpam-3856	269	12	x	x	SYM
ejpam-3856	269	13	−u	−u	PROPN
ejpam-3856	269	14	)	)	PUNCT
ejpam-3856	269	15	is	be	AUX
ejpam-3856	269	16	β	β	NOUN
ejpam-3856	269	17	-	-	VERB
ejpam-3856	269	18	closed	closed	ADJ
ejpam-3856	269	19	.	.	PUNCT
ejpam-3856	270	1	by	by	ADP
ejpam-3856	270	2	(	(	PUNCT
ejpam-3856	270	3	5	5	NUM
ejpam-3856	270	4	)	)	PUNCT
ejpam-3856	270	5	,	,	PUNCT
ejpam-3856	270	6	a∗	a∗	PROPN
ejpam-3856	270	7	β	β	X
ejpam-3856	270	8	∩	∩	X
ejpam-3856	270	9	(	(	PUNCT
ejpam-3856	270	10	x	x	SYM
ejpam-3856	270	11	−u	−u	PROPN
ejpam-3856	270	12	)	)	PUNCT
ejpam-3856	270	13	=	=	PUNCT
ejpam-3856	271	1	φ	φ	PROPN
ejpam-3856	271	2	.	.	PUNCT
ejpam-3856	272	1	therefore	therefore	ADV
ejpam-3856	272	2	,	,	PUNCT
ejpam-3856	272	3	a∗	a∗	PROPN
ejpam-3856	272	4	β	β	X
ejpam-3856	272	5	⊂	⊂	PROPN
ejpam-3856	272	6	u	u	NOUN
ejpam-3856	272	7	and	and	CCONJ
ejpam-3856	272	8	hence	hence	ADV
ejpam-3856	272	9	a	a	PRON
ejpam-3856	272	10	is	be	AUX
ejpam-3856	272	11	ig	ig	PROPN
ejpam-3856	272	12	-	-	ADJ
ejpam-3856	272	13	β	β	NOUN
ejpam-3856	272	14	-	-	ADJ
ejpam-3856	272	15	closed	closed	ADJ
ejpam-3856	272	16	.	.	PUNCT
ejpam-3856	273	1	4	4	X
ejpam-3856	273	2	.	.	X
ejpam-3856	273	3	is∗g	is∗g	NUM
ejpam-3856	273	4	-	-	PUNCT
ejpam-3856	273	5	β	β	NOUN
ejpam-3856	273	6	-	-	PUNCT
ejpam-3856	273	7	closed	closed	ADJ
ejpam-3856	273	8	sets	set	NOUN
ejpam-3856	273	9	in	in	ADP
ejpam-3856	273	10	this	this	DET
ejpam-3856	273	11	section	section	NOUN
ejpam-3856	273	12	,	,	PUNCT
ejpam-3856	273	13	the	the	DET
ejpam-3856	273	14	notion	notion	NOUN
ejpam-3856	273	15	of	of	ADP
ejpam-3856	273	16	is∗g	is∗g	PROPN
ejpam-3856	273	17	-	-	PUNCT
ejpam-3856	273	18	β	β	NOUN
ejpam-3856	273	19	-	-	PUNCT
ejpam-3856	273	20	closed	closed	ADJ
ejpam-3856	273	21	sets	set	NOUN
ejpam-3856	273	22	is	be	AUX
ejpam-3856	273	23	defined	define	VERB
ejpam-3856	273	24	with	with	ADP
ejpam-3856	273	25	an	an	DET
ejpam-3856	273	26	illustrative	illustrative	ADJ
ejpam-3856	273	27	example	example	NOUN
ejpam-3856	273	28	.	.	PUNCT
ejpam-3856	274	1	moreover	moreover	ADV
ejpam-3856	274	2	,	,	PUNCT
ejpam-3856	274	3	some	some	DET
ejpam-3856	274	4	properties	property	NOUN
ejpam-3856	274	5	of	of	ADP
ejpam-3856	274	6	these	these	DET
ejpam-3856	274	7	closed	close	VERB
ejpam-3856	274	8	sets	set	NOUN
ejpam-3856	274	9	has	have	AUX
ejpam-3856	274	10	been	be	AUX
ejpam-3856	274	11	also	also	ADV
ejpam-3856	274	12	explored	explore	VERB
ejpam-3856	274	13	.	.	PUNCT
ejpam-3856	275	1	definition	definition	NOUN
ejpam-3856	275	2	11	11	NUM
ejpam-3856	275	3	.	.	PUNCT
ejpam-3856	276	1	let	let	VERB
ejpam-3856	276	2	(	(	PUNCT
ejpam-3856	276	3	x	x	X
ejpam-3856	276	4	,	,	PUNCT
ejpam-3856	276	5	τ	τ	PROPN
ejpam-3856	276	6	,	,	PUNCT
ejpam-3856	276	7	i	i	PRON
ejpam-3856	276	8	)	)	PUNCT
ejpam-3856	276	9	be	be	VERB
ejpam-3856	276	10	an	an	DET
ejpam-3856	276	11	ideal	ideal	ADJ
ejpam-3856	276	12	topological	topological	ADJ
ejpam-3856	276	13	space	space	NOUN
ejpam-3856	276	14	.	.	PUNCT
ejpam-3856	277	1	a	a	DET
ejpam-3856	277	2	subset	subset	NOUN
ejpam-3856	277	3	a	a	PRON
ejpam-3856	277	4	of	of	ADP
ejpam-3856	277	5	x	x	SYM
ejpam-3856	277	6	is	be	AUX
ejpam-3856	277	7	said	say	VERB
ejpam-3856	277	8	to	to	PART
ejpam-3856	277	9	be	be	AUX
ejpam-3856	277	10	is∗g	is∗g	PROPN
ejpam-3856	277	11	-	-	PUNCT
ejpam-3856	277	12	β	β	NOUN
ejpam-3856	277	13	-	-	ADJ
ejpam-3856	277	14	closed	closed	ADJ
ejpam-3856	277	15	(	(	PUNCT
ejpam-3856	277	16	resp	resp	NOUN
ejpam-3856	277	17	.	.	PUNCT
ejpam-3856	278	1	is∗g	is∗g	VERB
ejpam-3856	278	2	-	-	PUNCT
ejpam-3856	278	3	closed	close	VERB
ejpam-3856	278	4	[	[	X
ejpam-3856	278	5	9	9	NUM
ejpam-3856	278	6	]	]	SYM
ejpam-3856	278	7	)	)	PUNCT
ejpam-3856	278	8	if	if	SCONJ
ejpam-3856	278	9	a∗	a∗	PROPN
ejpam-3856	278	10	β	β	X
ejpam-3856	278	11	⊂	⊂	PROPN
ejpam-3856	278	12	u	u	PROPN
ejpam-3856	278	13	(	(	PUNCT
ejpam-3856	278	14	resp	resp	NOUN
ejpam-3856	278	15	.	.	PUNCT
ejpam-3856	279	1	a∗	a∗	PROPN
ejpam-3856	279	2	⊂	⊂	PROPN
ejpam-3856	279	3	u	u	PROPN
ejpam-3856	279	4	)	)	PUNCT
ejpam-3856	279	5	whenever	whenever	SCONJ
ejpam-3856	279	6	a	a	DET
ejpam-3856	279	7	⊂	⊂	PROPN
ejpam-3856	279	8	u	u	NOUN
ejpam-3856	279	9	and	and	CCONJ
ejpam-3856	279	10	u	u	NOUN
ejpam-3856	279	11	is	be	AUX
ejpam-3856	279	12	semi	semi	ADJ
ejpam-3856	279	13	-	-	ADJ
ejpam-3856	279	14	open	open	ADJ
ejpam-3856	279	15	.	.	PUNCT
ejpam-3856	280	1	the	the	DET
ejpam-3856	280	2	complement	complement	NOUN
ejpam-3856	280	3	of	of	ADP
ejpam-3856	280	4	an	an	DET
ejpam-3856	280	5	is∗g	is∗g	PROPN
ejpam-3856	280	6	-	-	PUNCT
ejpam-3856	280	7	β	β	NOUN
ejpam-3856	280	8	-	-	ADJ
ejpam-3856	280	9	closed	closed	ADJ
ejpam-3856	280	10	set	set	NOUN
ejpam-3856	280	11	is	be	AUX
ejpam-3856	280	12	said	say	VERB
ejpam-3856	280	13	to	to	PART
ejpam-3856	280	14	be	be	AUX
ejpam-3856	280	15	is∗g	is∗g	PROPN
ejpam-3856	280	16	-	-	PUNCT
ejpam-3856	280	17	β	β	NOUN
ejpam-3856	280	18	-	-	ADJ
ejpam-3856	280	19	open	open	ADJ
ejpam-3856	280	20	.	.	PUNCT
ejpam-3856	281	1	the	the	DET
ejpam-3856	281	2	family	family	NOUN
ejpam-3856	281	3	of	of	ADP
ejpam-3856	281	4	is∗g	is∗g	PROPN
ejpam-3856	281	5	-	-	PUNCT
ejpam-3856	281	6	β	β	NOUN
ejpam-3856	281	7	-	-	ADJ
ejpam-3856	281	8	closed	closed	ADJ
ejpam-3856	281	9	(	(	PUNCT
ejpam-3856	281	10	resp	resp	NOUN
ejpam-3856	281	11	.	.	PUNCT
ejpam-3856	282	1	is∗g	is∗g	VERB
ejpam-3856	282	2	-	-	PUNCT
ejpam-3856	282	3	closed	closed	ADJ
ejpam-3856	282	4	)	)	PUNCT
ejpam-3856	282	5	sets	set	NOUN
ejpam-3856	282	6	is	be	AUX
ejpam-3856	282	7	denoted	denote	VERB
ejpam-3856	282	8	by	by	ADP
ejpam-3856	282	9	is∗g	is∗g	PROPN
ejpam-3856	282	10	βc(x	βc(x	NUM
ejpam-3856	282	11	)	)	PUNCT
ejpam-3856	282	12	(	(	PUNCT
ejpam-3856	282	13	resp	resp	NOUN
ejpam-3856	282	14	.	.	PUNCT
ejpam-3856	283	1	is∗gc(x	is∗gc(x	NOUN
ejpam-3856	283	2	)	)	PUNCT
ejpam-3856	283	3	)	)	PUNCT
ejpam-3856	283	4	.	.	PUNCT
ejpam-3856	284	1	theorem	theorem	NOUN
ejpam-3856	284	2	5	5	NUM
ejpam-3856	284	3	.	.	PUNCT
ejpam-3856	285	1	let	let	VERB
ejpam-3856	285	2	(	(	PUNCT
ejpam-3856	285	3	x	x	X
ejpam-3856	285	4	,	,	PUNCT
ejpam-3856	285	5	τ	τ	PROPN
ejpam-3856	285	6	,	,	PUNCT
ejpam-3856	285	7	i	i	PRON
ejpam-3856	285	8	)	)	PUNCT
ejpam-3856	285	9	be	be	VERB
ejpam-3856	285	10	an	an	DET
ejpam-3856	285	11	ideal	ideal	ADJ
ejpam-3856	285	12	topological	topological	ADJ
ejpam-3856	285	13	space	space	NOUN
ejpam-3856	285	14	and	and	CCONJ
ejpam-3856	285	15	a	a	DET
ejpam-3856	285	16	a	a	DET
ejpam-3856	285	17	subset	subset	NOUN
ejpam-3856	285	18	of	of	ADP
ejpam-3856	285	19	x.	x.	NOUN
ejpam-3856	285	20	if	if	SCONJ
ejpam-3856	285	21	a	a	PRON
ejpam-3856	285	22	is	be	AUX
ejpam-3856	285	23	is∗g	is∗g	NOUN
ejpam-3856	285	24	-	-	PUNCT
ejpam-3856	285	25	closed	closed	ADJ
ejpam-3856	285	26	,	,	PUNCT
ejpam-3856	285	27	then	then	ADV
ejpam-3856	285	28	it	it	PRON
ejpam-3856	285	29	is	be	AUX
ejpam-3856	285	30	is∗g	is∗g	NUM
ejpam-3856	285	31	-	-	PUNCT
ejpam-3856	285	32	β	β	NOUN
ejpam-3856	285	33	-	-	VERB
ejpam-3856	285	34	closed	closed	ADJ
ejpam-3856	285	35	.	.	PUNCT
ejpam-3856	286	1	but	but	CCONJ
ejpam-3856	286	2	the	the	DET
ejpam-3856	286	3	converse	converse	NOUN
ejpam-3856	286	4	is	be	AUX
ejpam-3856	286	5	not	not	PART
ejpam-3856	286	6	always	always	ADV
ejpam-3856	286	7	true	true	ADJ
ejpam-3856	286	8	.	.	PUNCT
ejpam-3856	287	1	proof	proof	NOUN
ejpam-3856	287	2	.	.	PUNCT
ejpam-3856	288	1	suppose	suppose	VERB
ejpam-3856	288	2	that	that	SCONJ
ejpam-3856	288	3	a	a	PRON
ejpam-3856	288	4	is	be	AUX
ejpam-3856	288	5	is∗g	is∗g	NOUN
ejpam-3856	288	6	-	-	PUNCT
ejpam-3856	288	7	closed	closed	ADJ
ejpam-3856	288	8	.	.	PUNCT
ejpam-3856	289	1	for	for	ADP
ejpam-3856	289	2	every	every	DET
ejpam-3856	289	3	u	u	PROPN
ejpam-3856	289	4	∈	∈	PROPN
ejpam-3856	289	5	so(x	so(x	NOUN
ejpam-3856	289	6	)	)	PUNCT
ejpam-3856	289	7	containing	contain	VERB
ejpam-3856	289	8	a	a	PRON
ejpam-3856	289	9	,	,	PUNCT
ejpam-3856	289	10	we	we	PRON
ejpam-3856	289	11	have	have	VERB
ejpam-3856	289	12	a∗	a∗	PROPN
ejpam-3856	289	13	⊂	⊂	PROPN
ejpam-3856	289	14	u	u	PROPN
ejpam-3856	289	15	and	and	CCONJ
ejpam-3856	289	16	by	by	ADP
ejpam-3856	289	17	lemma	lemma	PROPN
ejpam-3856	289	18	2(2	2(2	NUM
ejpam-3856	289	19	)	)	PUNCT
ejpam-3856	289	20	,	,	PUNCT
ejpam-3856	289	21	a∗	a∗	PROPN
ejpam-3856	289	22	β	β	X
ejpam-3856	289	23	⊂	⊂	PROPN
ejpam-3856	289	24	a∗	a∗	PROPN
ejpam-3856	289	25	⊂	⊂	PROPN
ejpam-3856	289	26	u	u	PROPN
ejpam-3856	289	27	.	.	PUNCT
ejpam-3856	290	1	this	this	PRON
ejpam-3856	290	2	shows	show	VERB
ejpam-3856	290	3	that	that	SCONJ
ejpam-3856	290	4	a	a	PRON
ejpam-3856	290	5	is	be	AUX
ejpam-3856	290	6	is∗g	is∗g	NUM
ejpam-3856	290	7	-	-	PUNCT
ejpam-3856	290	8	β	β	NOUN
ejpam-3856	290	9	-	-	VERB
ejpam-3856	290	10	closed	closed	ADJ
ejpam-3856	290	11	.	.	PUNCT
ejpam-3856	291	1	example	example	NOUN
ejpam-3856	292	1	3	3	X
ejpam-3856	292	2	.	.	PUNCT
ejpam-3856	292	3	let	let	VERB
ejpam-3856	292	4	x	x	PUNCT
ejpam-3856	292	5	=	=	PRON
ejpam-3856	292	6	{	{	PUNCT
ejpam-3856	292	7	a	a	PRON
ejpam-3856	292	8	,	,	PUNCT
ejpam-3856	292	9	b	b	NOUN
ejpam-3856	292	10	,	,	PUNCT
ejpam-3856	292	11	c	c	NOUN
ejpam-3856	292	12	,	,	PUNCT
ejpam-3856	292	13	d	d	AUX
ejpam-3856	292	14	}	}	PUNCT
ejpam-3856	292	15	be	be	AUX
ejpam-3856	292	16	a	a	DET
ejpam-3856	292	17	nonempty	nonempty	NOUN
ejpam-3856	292	18	set	set	VERB
ejpam-3856	292	19	with	with	ADP
ejpam-3856	292	20	the	the	DET
ejpam-3856	292	21	topology	topology	NOUN
ejpam-3856	292	22	τ	τ	X
ejpam-3856	292	23	=	=	SYM
ejpam-3856	292	24	{	{	PUNCT
ejpam-3856	292	25	φ	φ	PROPN
ejpam-3856	292	26	,	,	PUNCT
ejpam-3856	292	27	x	x	X
ejpam-3856	292	28	,	,	PUNCT
ejpam-3856	292	29	{	{	PUNCT
ejpam-3856	292	30	b	b	NOUN
ejpam-3856	292	31	,	,	PUNCT
ejpam-3856	292	32	c	c	NOUN
ejpam-3856	292	33	}	}	PUNCT
ejpam-3856	292	34	,	,	PUNCT
ejpam-3856	292	35	{	{	PUNCT
ejpam-3856	292	36	a	a	DET
ejpam-3856	292	37	,	,	PUNCT
ejpam-3856	292	38	b	b	NOUN
ejpam-3856	292	39	,	,	PUNCT
ejpam-3856	292	40	c	c	NOUN
ejpam-3856	292	41	}	}	PUNCT
ejpam-3856	292	42	,	,	PUNCT
ejpam-3856	292	43	{	{	PUNCT
ejpam-3856	292	44	b	b	X
ejpam-3856	292	45	}	}	PUNCT
ejpam-3856	292	46	,	,	PUNCT
ejpam-3856	292	47	{	{	PUNCT
ejpam-3856	292	48	a	a	PRON
ejpam-3856	292	49	,	,	PUNCT
ejpam-3856	292	50	b	b	NOUN
ejpam-3856	292	51	}	}	PUNCT
ejpam-3856	292	52	}	}	PUNCT
ejpam-3856	292	53	and	and	CCONJ
ejpam-3856	292	54	the	the	DET
ejpam-3856	292	55	collection	collection	NOUN
ejpam-3856	292	56	of	of	ADP
ejpam-3856	292	57	closed	closed	ADJ
ejpam-3856	292	58	sets	set	NOUN
ejpam-3856	292	59	is	be	AUX
ejpam-3856	292	60	τf	τf	ADP
ejpam-3856	292	61	=	=	SYM
ejpam-3856	292	62	{	{	PUNCT
ejpam-3856	292	63	x	x	PROPN
ejpam-3856	292	64	,	,	PUNCT
ejpam-3856	292	65	φ	φ	PROPN
ejpam-3856	292	66	,	,	PUNCT
ejpam-3856	292	67	{	{	PUNCT
ejpam-3856	292	68	a	a	PRON
ejpam-3856	292	69	,	,	PUNCT
ejpam-3856	292	70	c	c	NOUN
ejpam-3856	292	71	,	,	PUNCT
ejpam-3856	292	72	d	d	NOUN
ejpam-3856	292	73	}	}	PUNCT
ejpam-3856	292	74	,	,	PUNCT
ejpam-3856	292	75	{	{	PUNCT
ejpam-3856	292	76	c	c	X
ejpam-3856	292	77	,	,	PUNCT
ejpam-3856	292	78	d	d	NOUN
ejpam-3856	292	79	}	}	PUNCT
ejpam-3856	292	80	,	,	PUNCT
ejpam-3856	292	81	{	{	PUNCT
ejpam-3856	292	82	a	a	PRON
ejpam-3856	292	83	,	,	PUNCT
ejpam-3856	292	84	d	d	NOUN
ejpam-3856	292	85	}	}	PUNCT
ejpam-3856	292	86	,	,	PUNCT
ejpam-3856	292	87	{	{	PUNCT
ejpam-3856	292	88	d	d	NOUN
ejpam-3856	292	89	}	}	PUNCT
ejpam-3856	292	90	}	}	PUNCT
ejpam-3856	292	91	.	.	PUNCT
ejpam-3856	293	1	applying	apply	VERB
ejpam-3856	293	2	the	the	DET
ejpam-3856	293	3	definition	definition	NOUN
ejpam-3856	293	4	5	5	NUM
ejpam-3856	293	5	,	,	PUNCT
ejpam-3856	293	6	we	we	PRON
ejpam-3856	293	7	compute	compute	VERB
ejpam-3856	293	8	the	the	DET
ejpam-3856	293	9	collection	collection	NOUN
ejpam-3856	293	10	βo(x)=	βo(x)=	X
ejpam-3856	293	11	{	{	PUNCT
ejpam-3856	293	12	φ	φ	PROPN
ejpam-3856	293	13	,	,	PUNCT
ejpam-3856	293	14	x	x	X
ejpam-3856	293	15	,	,	PUNCT
ejpam-3856	293	16	{	{	PUNCT
ejpam-3856	293	17	a	a	DET
ejpam-3856	293	18	,	,	PUNCT
ejpam-3856	293	19	b	b	NOUN
ejpam-3856	293	20	}	}	PUNCT
ejpam-3856	293	21	,	,	PUNCT
ejpam-3856	293	22	{	{	PUNCT
ejpam-3856	293	23	b	b	X
ejpam-3856	293	24	}	}	PUNCT
ejpam-3856	293	25	,	,	PUNCT
ejpam-3856	293	26	{	{	PUNCT
ejpam-3856	293	27	b	b	X
ejpam-3856	293	28	,	,	PUNCT
ejpam-3856	293	29	c	c	NOUN
ejpam-3856	293	30	}	}	PUNCT
ejpam-3856	293	31	,	,	PUNCT
ejpam-3856	293	32	{	{	PUNCT
ejpam-3856	293	33	b	b	X
ejpam-3856	293	34	,	,	PUNCT
ejpam-3856	293	35	d	d	NOUN
ejpam-3856	293	36	}	}	PUNCT
ejpam-3856	293	37	,	,	PUNCT
ejpam-3856	293	38	{	{	PUNCT
ejpam-3856	293	39	a	a	DET
ejpam-3856	293	40	,	,	PUNCT
ejpam-3856	293	41	b	b	NOUN
ejpam-3856	293	42	,	,	PUNCT
ejpam-3856	293	43	c	c	NOUN
ejpam-3856	293	44	}	}	PUNCT
ejpam-3856	293	45	,	,	PUNCT
ejpam-3856	293	46	{	{	PUNCT
ejpam-3856	293	47	b	b	X
ejpam-3856	293	48	,	,	PUNCT
ejpam-3856	293	49	c	c	NOUN
ejpam-3856	293	50	,	,	PUNCT
ejpam-3856	293	51	d	d	NOUN
ejpam-3856	293	52	}	}	PUNCT
ejpam-3856	293	53	,	,	PUNCT
ejpam-3856	293	54	{	{	PUNCT
ejpam-3856	293	55	d	d	X
ejpam-3856	293	56	,	,	PUNCT
ejpam-3856	293	57	a	a	DET
ejpam-3856	293	58	,	,	PUNCT
ejpam-3856	293	59	b	b	NOUN
ejpam-3856	293	60	}	}	PUNCT
ejpam-3856	293	61	}	}	PUNCT
ejpam-3856	293	62	.	.	PUNCT
ejpam-3856	294	1	considering	consider	VERB
ejpam-3856	294	2	i	i	PRON
ejpam-3856	294	3	=	=	SYM
ejpam-3856	294	4	{	{	PUNCT
ejpam-3856	294	5	φ	φ	PROPN
ejpam-3856	294	6	,	,	PUNCT
ejpam-3856	294	7	{	{	PUNCT
ejpam-3856	294	8	a	a	X
ejpam-3856	294	9	}	}	PUNCT
ejpam-3856	294	10	}	}	PUNCT
ejpam-3856	294	11	and	and	CCONJ
ejpam-3856	294	12	applying	apply	VERB
ejpam-3856	294	13	definition	definition	NOUN
ejpam-3856	294	14	11	11	NUM
ejpam-3856	294	15	,	,	PUNCT
ejpam-3856	294	16	we	we	PRON
ejpam-3856	294	17	compute	compute	VERB
ejpam-3856	294	18	the	the	DET
ejpam-3856	294	19	collection	collection	NOUN
ejpam-3856	294	20	of	of	ADP
ejpam-3856	294	21	is∗gβc(x)=	is∗gβc(x)=	PROPN
ejpam-3856	294	22	{	{	PUNCT
ejpam-3856	294	23	φ	φ	PROPN
ejpam-3856	294	24	,	,	PUNCT
ejpam-3856	294	25	x	x	X
ejpam-3856	294	26	,	,	PUNCT
ejpam-3856	294	27	{	{	PUNCT
ejpam-3856	294	28	a	a	PRON
ejpam-3856	294	29	,	,	PUNCT
ejpam-3856	294	30	c	c	NOUN
ejpam-3856	294	31	,	,	PUNCT
ejpam-3856	294	32	d	d	NOUN
ejpam-3856	294	33	}	}	PUNCT
ejpam-3856	294	34	,	,	PUNCT
ejpam-3856	294	35	{	{	PUNCT
ejpam-3856	294	36	c	c	X
ejpam-3856	294	37	,	,	PUNCT
ejpam-3856	294	38	d	d	NOUN
ejpam-3856	294	39	}	}	PUNCT
ejpam-3856	294	40	,	,	PUNCT
ejpam-3856	294	41	{	{	PUNCT
ejpam-3856	294	42	a	a	PRON
ejpam-3856	294	43	,	,	PUNCT
ejpam-3856	294	44	d	d	NOUN
ejpam-3856	294	45	}	}	PUNCT
ejpam-3856	294	46	,	,	PUNCT
ejpam-3856	294	47	{	{	PUNCT
ejpam-3856	294	48	a	a	X
ejpam-3856	294	49	,	,	PUNCT
ejpam-3856	294	50	c	c	NOUN
ejpam-3856	294	51	}	}	PUNCT
ejpam-3856	294	52	,	,	PUNCT
ejpam-3856	294	53	{	{	PUNCT
ejpam-3856	294	54	d	d	X
ejpam-3856	294	55	}	}	PUNCT
ejpam-3856	294	56	,	,	PUNCT
ejpam-3856	294	57	{	{	PUNCT
ejpam-3856	294	58	a	a	X
ejpam-3856	294	59	}	}	PUNCT
ejpam-3856	294	60	,	,	PUNCT
ejpam-3856	294	61	{	{	PUNCT
ejpam-3856	294	62	c	c	X
ejpam-3856	294	63	}	}	PUNCT
ejpam-3856	294	64	}	}	PUNCT
ejpam-3856	294	65	and	and	CCONJ
ejpam-3856	294	66	is∗gc(x	is∗gc(x	PROPN
ejpam-3856	294	67	)	)	PUNCT
ejpam-3856	295	1	=	=	PRON
ejpam-3856	295	2	{	{	PUNCT
ejpam-3856	295	3	φ	φ	PROPN
ejpam-3856	295	4	,	,	PUNCT
ejpam-3856	295	5	x	x	PRON
ejpam-3856	295	6	,	,	PUNCT
ejpam-3856	295	7	{	{	PUNCT
ejpam-3856	295	8	a	a	PRON
ejpam-3856	295	9	,	,	PUNCT
ejpam-3856	295	10	c	c	NOUN
ejpam-3856	295	11	,	,	PUNCT
ejpam-3856	295	12	d	d	NOUN
ejpam-3856	295	13	}	}	PUNCT
ejpam-3856	295	14	,	,	PUNCT
ejpam-3856	295	15	{	{	PUNCT
ejpam-3856	295	16	c	c	X
ejpam-3856	295	17	,	,	PUNCT
ejpam-3856	295	18	d	d	NOUN
ejpam-3856	295	19	}	}	PUNCT
ejpam-3856	295	20	,	,	PUNCT
ejpam-3856	295	21	{	{	PUNCT
ejpam-3856	295	22	a	a	PRON
ejpam-3856	295	23	,	,	PUNCT
ejpam-3856	295	24	d	d	NOUN
ejpam-3856	295	25	}	}	PUNCT
ejpam-3856	295	26	,	,	PUNCT
ejpam-3856	295	27	{	{	PUNCT
ejpam-3856	295	28	d	d	X
ejpam-3856	295	29	}	}	PUNCT
ejpam-3856	295	30	,	,	PUNCT
ejpam-3856	295	31	{	{	PUNCT
ejpam-3856	295	32	a	a	X
ejpam-3856	295	33	}	}	PUNCT
ejpam-3856	295	34	}	}	PUNCT
ejpam-3856	295	35	.	.	PUNCT
ejpam-3856	296	1	it	it	PRON
ejpam-3856	296	2	can	can	AUX
ejpam-3856	296	3	be	be	AUX
ejpam-3856	296	4	verified	verify	VERB
ejpam-3856	296	5	that	that	SCONJ
ejpam-3856	296	6	the	the	DET
ejpam-3856	296	7	subsets	subset	NOUN
ejpam-3856	296	8	{	{	PUNCT
ejpam-3856	296	9	a	a	PRON
ejpam-3856	296	10	,	,	PUNCT
ejpam-3856	296	11	c	c	NOUN
ejpam-3856	296	12	}	}	PUNCT
ejpam-3856	296	13	and	and	CCONJ
ejpam-3856	296	14	{	{	PUNCT
ejpam-3856	296	15	c	c	NOUN
ejpam-3856	296	16	}	}	PUNCT
ejpam-3856	296	17	of	of	ADP
ejpam-3856	296	18	x	x	NOUN
ejpam-3856	296	19	are	be	AUX
ejpam-3856	296	20	is∗gβ	is∗gβ	NOUN
ejpam-3856	296	21	-	-	PUNCT
ejpam-3856	296	22	closed	closed	ADJ
ejpam-3856	296	23	but	but	CCONJ
ejpam-3856	296	24	not	not	PART
ejpam-3856	296	25	is∗g	is∗g	PROPN
ejpam-3856	296	26	-	-	PUNCT
ejpam-3856	296	27	closed	close	VERB
ejpam-3856	296	28	.	.	PUNCT
ejpam-3856	297	1	references	reference	NOUN
ejpam-3856	297	2	764	764	NUM
ejpam-3856	297	3	theorem	theorem	VERB
ejpam-3856	297	4	6	6	NUM
ejpam-3856	297	5	.	.	PUNCT
ejpam-3856	298	1	let	let	VERB
ejpam-3856	298	2	(	(	PUNCT
ejpam-3856	298	3	x	x	X
ejpam-3856	298	4	,	,	PUNCT
ejpam-3856	298	5	τ	τ	PROPN
ejpam-3856	298	6	,	,	PUNCT
ejpam-3856	298	7	i	i	PRON
ejpam-3856	298	8	)	)	PUNCT
ejpam-3856	298	9	be	be	VERB
ejpam-3856	298	10	an	an	DET
ejpam-3856	298	11	ideal	ideal	ADJ
ejpam-3856	298	12	topological	topological	ADJ
ejpam-3856	298	13	space	space	NOUN
ejpam-3856	298	14	and	and	CCONJ
ejpam-3856	298	15	a	a	DET
ejpam-3856	298	16	,	,	PUNCT
ejpam-3856	298	17	b	b	PROPN
ejpam-3856	298	18	be	be	AUX
ejpam-3856	298	19	subsets	subset	NOUN
ejpam-3856	298	20	of	of	ADP
ejpam-3856	298	21	x.	x.	NOUN
ejpam-3856	298	22	(	(	PUNCT
ejpam-3856	298	23	1	1	NUM
ejpam-3856	298	24	)	)	PUNCT
ejpam-3856	298	25	.	.	PUNCT
ejpam-3856	299	1	if	if	SCONJ
ejpam-3856	299	2	a	a	PRON
ejpam-3856	299	3	and	and	CCONJ
ejpam-3856	299	4	b	b	NOUN
ejpam-3856	299	5	are	be	AUX
ejpam-3856	299	6	is∗g	is∗g	NUM
ejpam-3856	299	7	-	-	PUNCT
ejpam-3856	299	8	β	β	NOUN
ejpam-3856	299	9	-	-	VERB
ejpam-3856	299	10	closed	closed	ADJ
ejpam-3856	299	11	,	,	PUNCT
ejpam-3856	299	12	then	then	ADV
ejpam-3856	299	13	a	a	DET
ejpam-3856	299	14	∪b	∪b	PRON
ejpam-3856	299	15	is	be	AUX
ejpam-3856	299	16	is∗g	is∗g	PROPN
ejpam-3856	299	17	-	-	PUNCT
ejpam-3856	299	18	β	β	NOUN
ejpam-3856	299	19	-	-	VERB
ejpam-3856	299	20	closed	closed	ADJ
ejpam-3856	299	21	.	.	PUNCT
ejpam-3856	300	1	(	(	PUNCT
ejpam-3856	300	2	2	2	NUM
ejpam-3856	300	3	)	)	PUNCT
ejpam-3856	300	4	.	.	PUNCT
ejpam-3856	301	1	if	if	SCONJ
ejpam-3856	301	2	a	a	PRON
ejpam-3856	301	3	is	be	AUX
ejpam-3856	301	4	closed	close	VERB
ejpam-3856	301	5	in	in	ADP
ejpam-3856	301	6	x	x	NOUN
ejpam-3856	301	7	,	,	PUNCT
ejpam-3856	301	8	then	then	ADV
ejpam-3856	301	9	a	a	PRON
ejpam-3856	301	10	is	be	AUX
ejpam-3856	301	11	is∗g	is∗g	NUM
ejpam-3856	301	12	-	-	PUNCT
ejpam-3856	301	13	β	β	NOUN
ejpam-3856	301	14	-	-	VERB
ejpam-3856	301	15	closed	closed	ADJ
ejpam-3856	301	16	.	.	PUNCT
ejpam-3856	302	1	(	(	PUNCT
ejpam-3856	302	2	3	3	NUM
ejpam-3856	302	3	)	)	PUNCT
ejpam-3856	302	4	.	.	PUNCT
ejpam-3856	303	1	if	if	SCONJ
ejpam-3856	303	2	u	u	NOUN
ejpam-3856	303	3	is	be	AUX
ejpam-3856	303	4	open	open	ADJ
ejpam-3856	303	5	in	in	ADP
ejpam-3856	303	6	x	x	PUNCT
ejpam-3856	303	7	and	and	CCONJ
ejpam-3856	303	8	a	a	PRON
ejpam-3856	303	9	is	be	AUX
ejpam-3856	303	10	is∗g	is∗g	NUM
ejpam-3856	303	11	-	-	PUNCT
ejpam-3856	303	12	β	β	NOUN
ejpam-3856	303	13	-	-	ADJ
ejpam-3856	303	14	open	open	ADJ
ejpam-3856	303	15	,	,	PUNCT
ejpam-3856	303	16	then	then	ADV
ejpam-3856	303	17	u	u	NOUN
ejpam-3856	303	18	∩a	∩a	PROPN
ejpam-3856	303	19	is	be	AUX
ejpam-3856	303	20	is∗g	is∗g	PROPN
ejpam-3856	303	21	-	-	PUNCT
ejpam-3856	303	22	β	β	NOUN
ejpam-3856	303	23	-	-	ADJ
ejpam-3856	303	24	open	open	ADJ
ejpam-3856	303	25	.	.	PUNCT
ejpam-3856	304	1	proof	proof	NOUN
ejpam-3856	304	2	.	.	PUNCT
ejpam-3856	305	1	(	(	PUNCT
ejpam-3856	305	2	1	1	NUM
ejpam-3856	305	3	)	)	PUNCT
ejpam-3856	305	4	.	.	PUNCT
ejpam-3856	306	1	let	let	VERB
ejpam-3856	306	2	a	a	DET
ejpam-3856	306	3	∪b	∪b	X
ejpam-3856	306	4	⊂	⊂	PROPN
ejpam-3856	306	5	u	u	NOUN
ejpam-3856	306	6	and	and	CCONJ
ejpam-3856	306	7	u	u	PROPN
ejpam-3856	306	8	∈	∈	PROPN
ejpam-3856	306	9	so(x	so(x	NOUN
ejpam-3856	306	10	)	)	PUNCT
ejpam-3856	306	11	.	.	PUNCT
ejpam-3856	307	1	then	then	ADV
ejpam-3856	307	2	,	,	PUNCT
ejpam-3856	307	3	we	we	PRON
ejpam-3856	307	4	know	know	VERB
ejpam-3856	307	5	that	that	SCONJ
ejpam-3856	307	6	a	a	DET
ejpam-3856	307	7	⊂	⊂	PROPN
ejpam-3856	307	8	u	u	NOUN
ejpam-3856	307	9	and	and	CCONJ
ejpam-3856	307	10	b	b	PROPN
ejpam-3856	307	11	⊂	⊂	PROPN
ejpam-3856	307	12	u.	u.	PROPN
ejpam-3856	307	13	since	since	SCONJ
ejpam-3856	307	14	a	a	PRON
ejpam-3856	307	15	and	and	CCONJ
ejpam-3856	307	16	b	b	NOUN
ejpam-3856	307	17	both	both	PRON
ejpam-3856	307	18	are	be	AUX
ejpam-3856	307	19	is∗g	is∗g	PROPN
ejpam-3856	307	20	-	-	PUNCT
ejpam-3856	307	21	β	β	NOUN
ejpam-3856	307	22	-	-	VERB
ejpam-3856	307	23	closed	closed	ADJ
ejpam-3856	307	24	,	,	PUNCT
ejpam-3856	307	25	we	we	PRON
ejpam-3856	307	26	have	have	VERB
ejpam-3856	307	27	a∗	a∗	PROPN
ejpam-3856	307	28	β	β	X
ejpam-3856	307	29	⊂	⊂	PROPN
ejpam-3856	307	30	u	u	PROPN
ejpam-3856	307	31	and	and	CCONJ
ejpam-3856	307	32	b∗	b∗	ADJ
ejpam-3856	308	1	β	β	X
ejpam-3856	308	2	⊂	⊂	X
ejpam-3856	308	3	u.	u.	PROPN
ejpam-3856	308	4	hence	hence	ADV
ejpam-3856	308	5	,	,	PUNCT
ejpam-3856	308	6	a∗	a∗	PROPN
ejpam-3856	308	7	β	β	PROPN
ejpam-3856	308	8	∪b∗	∪b∗	NUM
ejpam-3856	308	9	β	β	X
ejpam-3856	308	10	⊂	⊂	PROPN
ejpam-3856	308	11	u	u	PROPN
ejpam-3856	308	12	.	.	PUNCT
ejpam-3856	309	1	now	now	ADV
ejpam-3856	309	2	by	by	ADP
ejpam-3856	309	3	theorem	theorem	NOUN
ejpam-3856	309	4	1(2	1(2	NUM
ejpam-3856	309	5	)	)	PUNCT
ejpam-3856	309	6	,	,	PUNCT
ejpam-3856	309	7	(	(	PUNCT
ejpam-3856	309	8	a∪b)∗β	a∪b)∗β	NOUN
ejpam-3856	309	9	=	=	SYM
ejpam-3856	309	10	a∗	a∗	PROPN
ejpam-3856	309	11	β	β	PROPN
ejpam-3856	309	12	∪b∗	∪b∗	ADP
ejpam-3856	309	13	β	β	X
ejpam-3856	309	14	⊂	⊂	PROPN
ejpam-3856	309	15	u.	u.	PROPN
ejpam-3856	309	16	hence	hence	ADV
ejpam-3856	309	17	,	,	PUNCT
ejpam-3856	309	18	we	we	PRON
ejpam-3856	309	19	obtain	obtain	VERB
ejpam-3856	309	20	a∪b	a∪b	NOUN
ejpam-3856	309	21	is	be	AUX
ejpam-3856	309	22	is∗g	is∗g	NUM
ejpam-3856	309	23	-	-	PUNCT
ejpam-3856	309	24	β	β	NOUN
ejpam-3856	309	25	-	-	VERB
ejpam-3856	309	26	closed	closed	ADJ
ejpam-3856	309	27	.	.	PUNCT
ejpam-3856	310	1	(	(	PUNCT
ejpam-3856	310	2	2	2	NUM
ejpam-3856	310	3	)	)	PUNCT
ejpam-3856	310	4	.	.	PUNCT
ejpam-3856	311	1	let	let	VERB
ejpam-3856	311	2	a	a	DET
ejpam-3856	311	3	⊂	⊂	X
ejpam-3856	311	4	u	u	NOUN
ejpam-3856	311	5	and	and	CCONJ
ejpam-3856	311	6	u	u	PROPN
ejpam-3856	311	7	∈	∈	PROPN
ejpam-3856	311	8	so(x	so(x	NOUN
ejpam-3856	311	9	)	)	PUNCT
ejpam-3856	311	10	.	.	PUNCT
ejpam-3856	312	1	by	by	ADP
ejpam-3856	312	2	lemma	lemma	PROPN
ejpam-3856	312	3	2	2	NUM
ejpam-3856	312	4	,	,	PUNCT
ejpam-3856	312	5	a∗	a∗	PROPN
ejpam-3856	312	6	β	β	X
ejpam-3856	312	7	⊂	⊂	PROPN
ejpam-3856	312	8	a∗	a∗	PROPN
ejpam-3856	312	9	⊂	⊂	PROPN
ejpam-3856	312	10	cl(a	cl(a	X
ejpam-3856	312	11	)	)	PUNCT
ejpam-3856	312	12	=	=	PUNCT
ejpam-3856	312	13	a	a	DET
ejpam-3856	312	14	⊂	⊂	PROPN
ejpam-3856	312	15	u	u	PROPN
ejpam-3856	312	16	.	.	PUNCT
ejpam-3856	313	1	this	this	PRON
ejpam-3856	313	2	shows	show	VERB
ejpam-3856	313	3	that	that	SCONJ
ejpam-3856	313	4	a	a	PRON
ejpam-3856	313	5	is	be	AUX
ejpam-3856	313	6	is∗g	is∗g	NUM
ejpam-3856	313	7	-	-	PUNCT
ejpam-3856	313	8	β	β	NOUN
ejpam-3856	313	9	-	-	VERB
ejpam-3856	313	10	closed	closed	ADJ
ejpam-3856	313	11	.	.	PUNCT
ejpam-3856	314	1	(	(	PUNCT
ejpam-3856	314	2	3	3	NUM
ejpam-3856	314	3	)	)	PUNCT
ejpam-3856	314	4	.	.	PUNCT
ejpam-3856	315	1	the	the	DET
ejpam-3856	315	2	proof	proof	NOUN
ejpam-3856	315	3	is	be	AUX
ejpam-3856	315	4	a	a	DET
ejpam-3856	315	5	direct	direct	ADJ
ejpam-3856	315	6	consequence	consequence	NOUN
ejpam-3856	315	7	of	of	ADP
ejpam-3856	315	8	(	(	PUNCT
ejpam-3856	315	9	1	1	NUM
ejpam-3856	315	10	)	)	PUNCT
ejpam-3856	315	11	and	and	CCONJ
ejpam-3856	315	12	(	(	PUNCT
ejpam-3856	315	13	2	2	NUM
ejpam-3856	315	14	)	)	PUNCT
ejpam-3856	315	15	.	.	PUNCT
ejpam-3856	316	1	5	5	X
ejpam-3856	316	2	.	.	X
ejpam-3856	316	3	conclusion	conclusion	NOUN
ejpam-3856	316	4	the	the	DET
ejpam-3856	316	5	concept	concept	NOUN
ejpam-3856	316	6	of	of	ADP
ejpam-3856	316	7	the	the	DET
ejpam-3856	316	8	β	β	ADJ
ejpam-3856	316	9	-	-	ADJ
ejpam-3856	316	10	local	local	ADJ
ejpam-3856	316	11	function	function	NOUN
ejpam-3856	316	12	,	,	PUNCT
ejpam-3856	316	13	the	the	DET
ejpam-3856	316	14	operation	operation	NOUN
ejpam-3856	316	15	cl∗β	cl∗β	PROPN
ejpam-3856	316	16	and	and	CCONJ
ejpam-3856	316	17	ig	ig	PROPN
ejpam-3856	316	18	-	-	ADJ
ejpam-3856	316	19	β	β	NOUN
ejpam-3856	316	20	-	-	PUNCT
ejpam-3856	316	21	closed	closed	ADJ
ejpam-3856	316	22	sets	set	NOUN
ejpam-3856	316	23	have	have	AUX
ejpam-3856	316	24	been	be	AUX
ejpam-3856	316	25	introduced	introduce	VERB
ejpam-3856	316	26	with	with	ADP
ejpam-3856	316	27	illustrative	illustrative	ADJ
ejpam-3856	316	28	examples	example	NOUN
ejpam-3856	316	29	.	.	PUNCT
ejpam-3856	317	1	moreover	moreover	ADV
ejpam-3856	317	2	,	,	PUNCT
ejpam-3856	317	3	certain	certain	ADJ
ejpam-3856	317	4	properties	property	NOUN
ejpam-3856	317	5	have	have	AUX
ejpam-3856	317	6	been	be	AUX
ejpam-3856	317	7	also	also	ADV
ejpam-3856	317	8	studied	study	VERB
ejpam-3856	317	9	and	and	CCONJ
ejpam-3856	317	10	explored	explore	VERB
ejpam-3856	317	11	.	.	PUNCT
ejpam-3856	318	1	it	it	PRON
ejpam-3856	318	2	may	may	AUX
ejpam-3856	318	3	be	be	AUX
ejpam-3856	318	4	concluded	conclude	VERB
ejpam-3856	318	5	that	that	SCONJ
ejpam-3856	318	6	the	the	DET
ejpam-3856	318	7	concept	concept	NOUN
ejpam-3856	318	8	of	of	ADP
ejpam-3856	318	9	the	the	DET
ejpam-3856	318	10	topology	topology	NOUN
ejpam-3856	318	11	τ∗β	τ∗β	PUNCT
ejpam-3856	318	12	is	be	AUX
ejpam-3856	318	13	more	more	ADV
ejpam-3856	318	14	generalized	generalized	ADJ
ejpam-3856	318	15	version	version	NOUN
ejpam-3856	318	16	of	of	ADP
ejpam-3856	318	17	τ∗	τ∗	NOUN
ejpam-3856	318	18	and	and	CCONJ
ejpam-3856	318	19	β	β	NOUN
ejpam-3856	318	20	-	-	ADJ
ejpam-3856	318	21	open	open	ADJ
ejpam-3856	318	22	sets	set	NOUN
ejpam-3856	318	23	,	,	PUNCT
ejpam-3856	318	24	which	which	PRON
ejpam-3856	318	25	may	may	AUX
ejpam-3856	318	26	be	be	AUX
ejpam-3856	318	27	further	far	ADV
ejpam-3856	318	28	useful	useful	ADJ
ejpam-3856	318	29	to	to	PART
ejpam-3856	318	30	enrich	enrich	VERB
ejpam-3856	318	31	the	the	DET
ejpam-3856	318	32	class	class	NOUN
ejpam-3856	318	33	of	of	ADP
ejpam-3856	318	34	continuous	continuous	ADJ
ejpam-3856	318	35	functions	function	NOUN
ejpam-3856	318	36	.	.	PUNCT
ejpam-3856	319	1	references	reference	NOUN
ejpam-3856	319	2	[	[	X
ejpam-3856	319	3	1	1	NUM
ejpam-3856	319	4	]	]	PUNCT
ejpam-3856	319	5	m.	m.	NOUN
ejpam-3856	319	6	e.	e.	PROPN
ejpam-3856	319	7	abd	abd	PROPN
ejpam-3856	319	8	el	el	PROPN
ejpam-3856	319	9	-	-	PROPN
ejpam-3856	319	10	monsef	monsef	PROPN
ejpam-3856	319	11	,	,	PUNCT
ejpam-3856	319	12	s.	s.	PROPN
ejpam-3856	319	13	n.	n.	PROPN
ejpam-3856	319	14	el	el	PROPN
ejpam-3856	319	15	-	-	PUNCT
ejpam-3856	319	16	deeb	deeb	PROPN
ejpam-3856	319	17	and	and	CCONJ
ejpam-3856	319	18	r.a	r.a	PROPN
ejpam-3856	319	19	.	.	PROPN
ejpam-3856	319	20	mahmoud	mahmoud	PROPN
ejpam-3856	319	21	,	,	PUNCT
ejpam-3856	319	22	β	β	NOUN
ejpam-3856	319	23	-	-	PUNCT
ejpam-3856	319	24	open	open	ADJ
ejpam-3856	319	25	and	and	CCONJ
ejpam-3856	319	26	β	β	ADJ
ejpam-3856	319	27	-	-	ADJ
ejpam-3856	319	28	continuous	continuous	ADJ
ejpam-3856	319	29	mappings	mapping	NOUN
ejpam-3856	319	30	,	,	PUNCT
ejpam-3856	319	31	bull	bull	NOUN
ejpam-3856	319	32	.	.	PUNCT
ejpam-3856	320	1	fac	fac	PROPN
ejpam-3856	320	2	.	.	PUNCT
ejpam-3856	321	1	sci	sci	PROPN
ejpam-3856	321	2	.	.	PUNCT
ejpam-3856	321	3	assiut	assiut	PROPN
ejpam-3856	321	4	univ	univ	PROPN
ejpam-3856	321	5	.	.	PROPN
ejpam-3856	321	6	,	,	PUNCT
ejpam-3856	321	7	12(1983	12(1983	NUM
ejpam-3856	321	8	)	)	PUNCT
ejpam-3856	321	9	,	,	PUNCT
ejpam-3856	321	10	77−	77−	PROPN
ejpam-3856	321	11	90	90	NUM
ejpam-3856	321	12	.	.	PUNCT
ejpam-3856	322	1	[	[	X
ejpam-3856	322	2	2	2	NUM
ejpam-3856	322	3	]	]	PUNCT
ejpam-3856	322	4	m.	m.	NOUN
ejpam-3856	322	5	e.	e.	PROPN
ejpam-3856	322	6	abd	abd	PROPN
ejpam-3856	322	7	el	el	PROPN
ejpam-3856	322	8	-	-	PROPN
ejpam-3856	322	9	monsef	monsef	PROPN
ejpam-3856	322	10	,	,	PUNCT
ejpam-3856	322	11	e.	e.	PROPN
ejpam-3856	322	12	f.	f.	PROPN
ejpam-3856	322	13	lashien	lashien	PROPN
ejpam-3856	322	14	and	and	CCONJ
ejpam-3856	322	15	a.	a.	NOUN
ejpam-3856	322	16	a.	a.	NOUN
ejpam-3856	322	17	nasef	nasef	PROPN
ejpam-3856	322	18	,	,	PUNCT
ejpam-3856	322	19	on	on	ADP
ejpam-3856	322	20	i	i	NOUN
ejpam-3856	322	21	-	-	PUNCT
ejpam-3856	322	22	open	open	ADJ
ejpam-3856	322	23	sets	set	NOUN
ejpam-3856	322	24	and	and	CCONJ
ejpam-3856	322	25	i	i	NOUN
ejpam-3856	322	26	-	-	PUNCT
ejpam-3856	322	27	continuous	continuous	ADJ
ejpam-3856	322	28	functions	function	NOUN
ejpam-3856	322	29	,	,	PUNCT
ejpam-3856	322	30	kyungpook	kyungpook	NOUN
ejpam-3856	322	31	math	math	NOUN
ejpam-3856	322	32	.	.	PUNCT
ejpam-3856	323	1	j.	j.	PROPN
ejpam-3856	323	2	,	,	PUNCT
ejpam-3856	323	3	32(1)(1992	32(1)(1992	NUM
ejpam-3856	323	4	)	)	PUNCT
ejpam-3856	323	5	,	,	PUNCT
ejpam-3856	323	6	21−	21−	NOUN
ejpam-3856	323	7	30	30	NUM
ejpam-3856	323	8	.	.	PUNCT
ejpam-3856	324	1	[	[	X
ejpam-3856	324	2	3	3	X
ejpam-3856	324	3	]	]	PUNCT
ejpam-3856	324	4	m.	m.	NOUN
ejpam-3856	324	5	e.	e.	PROPN
ejpam-3856	324	6	abd	abd	PROPN
ejpam-3856	324	7	el	el	PROPN
ejpam-3856	324	8	-	-	PROPN
ejpam-3856	324	9	monsef	monsef	PROPN
ejpam-3856	324	10	,	,	PUNCT
ejpam-3856	324	11	e.	e.	PROPN
ejpam-3856	324	12	f.	f.	PROPN
ejpam-3856	324	13	lashien	lashien	PROPN
ejpam-3856	324	14	and	and	CCONJ
ejpam-3856	324	15	a.	a.	NOUN
ejpam-3856	324	16	a.	a.	PROPN
ejpam-3856	324	17	nasef	nasef	PROPN
ejpam-3856	324	18	,	,	PUNCT
ejpam-3856	324	19	some	some	DET
ejpam-3856	324	20	topological	topological	ADJ
ejpam-3856	324	21	operators	operator	NOUN
ejpam-3856	324	22	via	via	ADP
ejpam-3856	324	23	ideals	ideal	NOUN
ejpam-3856	324	24	,	,	PUNCT
ejpam-3856	324	25	kyungpook	kyungpook	PROPN
ejpam-3856	324	26	j.	j.	PROPN
ejpam-3856	324	27	math	math	PROPN
ejpam-3856	324	28	.	.	PUNCT
ejpam-3856	324	29	,	,	PUNCT
ejpam-3856	324	30	32(2)(1992	32(2)(1992	NUM
ejpam-3856	324	31	)	)	PUNCT
ejpam-3856	324	32	,	,	PUNCT
ejpam-3856	324	33	273−	273−	NUM
ejpam-3856	324	34	284	284	NUM
ejpam-3856	324	35	.	.	PUNCT
ejpam-3856	325	1	[	[	X
ejpam-3856	325	2	4	4	X
ejpam-3856	325	3	]	]	PUNCT
ejpam-3856	325	4	m.	m.	PROPN
ejpam-3856	325	5	e.	e.	PROPN
ejpam-3856	325	6	abd	abd	PROPN
ejpam-3856	325	7	-	-	PUNCT
ejpam-3856	325	8	e	e	PROPN
ejpam-3856	325	9	-	-	NOUN
ejpam-3856	325	10	monsef	monsef	ADJ
ejpam-3856	325	11	,	,	PUNCT
ejpam-3856	325	12	r.	r.	PROPN
ejpam-3856	325	13	a.	a.	PROPN
ejpam-3856	325	14	mohmoud	mohmoud	PROPN
ejpam-3856	325	15	and	and	CCONJ
ejpam-3856	325	16	e.	e.	PROPN
ejpam-3856	325	17	r.	r.	PROPN
ejpam-3856	325	18	lashin	lashin	PROPN
ejpam-3856	325	19	,	,	PUNCT
ejpam-3856	325	20	β	β	NOUN
ejpam-3856	325	21	-	-	PUNCT
ejpam-3856	325	22	closure	closure	NOUN
ejpam-3856	325	23	and	and	CCONJ
ejpam-3856	325	24	β	β	NOUN
ejpam-3856	325	25	-	-	NOUN
ejpam-3856	325	26	interior	interior	ADJ
ejpam-3856	325	27	,	,	PUNCT
ejpam-3856	325	28	j.	j.	PROPN
ejpam-3856	325	29	fac	fac	PROPN
ejpam-3856	325	30	.	.	PUNCT
ejpam-3856	326	1	ed	ed	PROPN
ejpam-3856	326	2	.	.	PUNCT
ejpam-3856	326	3	ain	ain	PROPN
ejpam-3856	326	4	shans	shans	PROPN
ejpam-3856	326	5	univ	univ	PROPN
ejpam-3856	326	6	.	.	PROPN
ejpam-3856	326	7	,	,	PUNCT
ejpam-3856	326	8	10(1986	10(1986	NUM
ejpam-3856	326	9	)	)	PUNCT
ejpam-3856	326	10	,	,	PUNCT
ejpam-3856	326	11	235−	235−	NUM
ejpam-3856	326	12	245	245	NUM
ejpam-3856	326	13	.	.	PUNCT
ejpam-3856	327	1	[	[	X
ejpam-3856	327	2	5	5	NUM
ejpam-3856	327	3	]	]	PUNCT
ejpam-3856	327	4	a.	a.	PROPN
ejpam-3856	327	5	al	al	PROPN
ejpam-3856	327	6	-	-	PUNCT
ejpam-3856	327	7	omari	omari	PROPN
ejpam-3856	327	8	and	and	CCONJ
ejpam-3856	327	9	t.	t.	PROPN
ejpam-3856	327	10	noiri	noiri	PROPN
ejpam-3856	327	11	,	,	PUNCT
ejpam-3856	327	12	local	local	ADJ
ejpam-3856	327	13	function	function	NOUN
ejpam-3856	327	14	γ∗	γ∗	NOUN
ejpam-3856	327	15	in	in	ADP
ejpam-3856	327	16	ideal	ideal	ADJ
ejpam-3856	327	17	topological	topological	ADJ
ejpam-3856	327	18	spaces	space	NOUN
ejpam-3856	327	19	,	,	PUNCT
ejpam-3856	327	20	sci	sci	PROPN
ejpam-3856	327	21	.	.	PROPN
ejpam-3856	327	22	stud	stud	PROPN
ejpam-3856	327	23	.	.	PUNCT
ejpam-3856	328	1	res	re	NOUN
ejpam-3856	328	2	.	.	PUNCT
ejpam-3856	328	3	ser	ser	PROPN
ejpam-3856	328	4	.	.	PROPN
ejpam-3856	328	5	math	math	PROPN
ejpam-3856	328	6	.	.	PUNCT
ejpam-3856	329	1	inform	inform	NOUN
ejpam-3856	329	2	.	.	PUNCT
ejpam-3856	329	3	,	,	PUNCT
ejpam-3856	329	4	26(1)(2016	26(1)(2016	NUM
ejpam-3856	329	5	)	)	PUNCT
ejpam-3856	329	6	,	,	PUNCT
ejpam-3856	329	7	5−	5−	NUM
ejpam-3856	329	8	16	16	NUM
ejpam-3856	329	9	.	.	PUNCT
ejpam-3856	330	1	[	[	X
ejpam-3856	330	2	6	6	NUM
ejpam-3856	330	3	]	]	PUNCT
ejpam-3856	330	4	j.	j.	PROPN
ejpam-3856	330	5	dontchev	dontchev	PROPN
ejpam-3856	330	6	,	,	PUNCT
ejpam-3856	330	7	m.	m.	NOUN
ejpam-3856	330	8	ganster	ganster	NOUN
ejpam-3856	330	9	and	and	CCONJ
ejpam-3856	330	10	t.	t.	PROPN
ejpam-3856	330	11	noiri	noiri	PROPN
ejpam-3856	330	12	,	,	PUNCT
ejpam-3856	330	13	unified	unified	ADJ
ejpam-3856	330	14	operation	operation	NOUN
ejpam-3856	330	15	approach	approach	NOUN
ejpam-3856	330	16	of	of	ADP
ejpam-3856	330	17	generalized	generalized	ADJ
ejpam-3856	330	18	closed	close	VERB
ejpam-3856	330	19	sets	set	NOUN
ejpam-3856	330	20	via	via	ADP
ejpam-3856	330	21	topological	topological	ADJ
ejpam-3856	330	22	ideals	ideal	NOUN
ejpam-3856	330	23	,	,	PUNCT
ejpam-3856	330	24	math	math	NOUN
ejpam-3856	330	25	.	.	PUNCT
ejpam-3856	331	1	japon	japon	PROPN
ejpam-3856	331	2	.	.	PROPN
ejpam-3856	331	3	,	,	PUNCT
ejpam-3856	331	4	49(1999	49(1999	PROPN
ejpam-3856	331	5	)	)	PUNCT
ejpam-3856	331	6	,	,	PUNCT
ejpam-3856	331	7	395−	395−	PROPN
ejpam-3856	331	8	402	402	NUM
ejpam-3856	331	9	.	.	PUNCT
ejpam-3856	332	1	[	[	X
ejpam-3856	332	2	7	7	X
ejpam-3856	332	3	]	]	X
ejpam-3856	332	4	e.	e.	PROPN
ejpam-3856	332	5	hatir	hatir	PROPN
ejpam-3856	332	6	,	,	PUNCT
ejpam-3856	332	7	a.	a.	PROPN
ejpam-3856	332	8	al	al	PROPN
ejpam-3856	332	9	-	-	PUNCT
ejpam-3856	332	10	omari	omari	PROPN
ejpam-3856	332	11	and	and	CCONJ
ejpam-3856	332	12	s.	s.	PROPN
ejpam-3856	332	13	jafari	jafari	PROPN
ejpam-3856	332	14	,	,	PUNCT
ejpam-3856	332	15	δ	δ	PROPN
ejpam-3856	332	16	-	-	ADJ
ejpam-3856	332	17	local	local	ADJ
ejpam-3856	332	18	functions	function	NOUN
ejpam-3856	332	19	and	and	CCONJ
ejpam-3856	332	20	its	its	PRON
ejpam-3856	332	21	properties	property	NOUN
ejpam-3856	332	22	in	in	ADP
ejpam-3856	332	23	ideal	ideal	ADJ
ejpam-3856	332	24	topological	topological	ADJ
ejpam-3856	332	25	spaces	space	NOUN
ejpam-3856	332	26	,	,	PUNCT
ejpam-3856	332	27	fasciculi	fasciculi	PROPN
ejpam-3856	332	28	math	math	NOUN
ejpam-3856	332	29	.	.	PUNCT
ejpam-3856	332	30	,	,	PUNCT
ejpam-3856	332	31	53(2014	53(2014	NUM
ejpam-3856	332	32	)	)	PUNCT
ejpam-3856	332	33	,	,	PUNCT
ejpam-3856	332	34	53−	53−	NOUN
ejpam-3856	332	35	64	64	NUM
ejpam-3856	332	36	.	.	PUNCT
ejpam-3856	333	1	[	[	X
ejpam-3856	333	2	8	8	NUM
ejpam-3856	333	3	]	]	X
ejpam-3856	333	4	d.	d.	PROPN
ejpam-3856	333	5	jankovic	jankovic	PROPN
ejpam-3856	333	6	and	and	CCONJ
ejpam-3856	333	7	t.	t.	PROPN
ejpam-3856	333	8	r.	r.	PROPN
ejpam-3856	333	9	hamlett	hamlett	PROPN
ejpam-3856	333	10	,	,	PUNCT
ejpam-3856	333	11	new	new	ADJ
ejpam-3856	333	12	topologies	topology	NOUN
ejpam-3856	333	13	from	from	ADP
ejpam-3856	333	14	old	old	ADJ
ejpam-3856	333	15	via	via	ADP
ejpam-3856	333	16	ideals	ideal	NOUN
ejpam-3856	333	17	,	,	PUNCT
ejpam-3856	333	18	amer	amer	PROPN
ejpam-3856	333	19	.	.	PROPN
ejpam-3856	333	20	math	math	PROPN
ejpam-3856	333	21	.	.	PUNCT
ejpam-3856	334	1	monthly	monthly	ADJ
ejpam-3856	334	2	,	,	PUNCT
ejpam-3856	334	3	97(4)(1990	97(4)(1990	NUM
ejpam-3856	334	4	)	)	PUNCT
ejpam-3856	334	5	,	,	PUNCT
ejpam-3856	334	6	295−	295−	PROPN
ejpam-3856	334	7	310	310	NUM
ejpam-3856	334	8	.	.	PUNCT
ejpam-3856	335	1	references	reference	NOUN
ejpam-3856	335	2	765	765	NUM
ejpam-3856	336	1	[	[	X
ejpam-3856	336	2	9	9	NUM
ejpam-3856	336	3	]	]	PUNCT
ejpam-3856	336	4	m.	m.	NOUN
ejpam-3856	336	5	khan	khan	PROPN
ejpam-3856	336	6	and	and	CCONJ
ejpam-3856	336	7	m.	m.	PROPN
ejpam-3856	336	8	hamza	hamza	PROPN
ejpam-3856	336	9	,	,	PUNCT
ejpam-3856	336	10	is∗g	is∗g	NOUN
ejpam-3856	336	11	-	-	PUNCT
ejpam-3856	336	12	closed	close	VERB
ejpam-3856	336	13	sets	set	NOUN
ejpam-3856	336	14	in	in	ADP
ejpam-3856	336	15	ideal	ideal	ADJ
ejpam-3856	336	16	topological	topological	ADJ
ejpam-3856	336	17	spaces	space	NOUN
ejpam-3856	336	18	,	,	PUNCT
ejpam-3856	336	19	glob	glob	PROPN
ejpam-3856	336	20	.	.	PUNCT
ejpam-3856	337	1	j.	j.	PROPN
ejpam-3856	337	2	pure	pure	PROPN
ejpam-3856	337	3	appli	appli	PROPN
ejpam-3856	337	4	.	.	PUNCT
ejpam-3856	337	5	math	math	PROPN
ejpam-3856	337	6	.	.	PUNCT
ejpam-3856	337	7	,	,	PUNCT
ejpam-3856	338	1	7(1)(2011	7(1)(2011	NUM
ejpam-3856	338	2	)	)	PUNCT
ejpam-3856	339	1	,	,	PUNCT
ejpam-3856	339	2	89−	89−	NOUN
ejpam-3856	339	3	99	99	NUM
ejpam-3856	339	4	.	.	PUNCT
ejpam-3856	340	1	[	[	X
ejpam-3856	340	2	10	10	NUM
ejpam-3856	340	3	]	]	X
ejpam-3856	340	4	m.	m.	NOUN
ejpam-3856	340	5	khan	khan	PROPN
ejpam-3856	340	6	and	and	CCONJ
ejpam-3856	340	7	t.	t.	PROPN
ejpam-3856	340	8	noiri	noiri	PROPN
ejpam-3856	340	9	,	,	PUNCT
ejpam-3856	340	10	semi	semi	ADJ
ejpam-3856	340	11	-	-	ADJ
ejpam-3856	340	12	local	local	ADJ
ejpam-3856	340	13	functions	function	NOUN
ejpam-3856	340	14	in	in	ADP
ejpam-3856	340	15	ideal	ideal	ADJ
ejpam-3856	340	16	topological	topological	ADJ
ejpam-3856	340	17	spaces	space	NOUN
ejpam-3856	340	18	,	,	PUNCT
ejpam-3856	340	19	j.	j.	PROPN
ejpam-3856	340	20	adv	adv	PROPN
ejpam-3856	340	21	.	.	PUNCT
ejpam-3856	341	1	res	re	NOUN
ejpam-3856	341	2	.	.	PUNCT
ejpam-3856	342	1	pure	pure	ADJ
ejpam-3856	342	2	math	math	NOUN
ejpam-3856	342	3	.	.	PUNCT
ejpam-3856	342	4	,	,	PUNCT
ejpam-3856	342	5	2(1)(2010	2(1)(2010	NUM
ejpam-3856	342	6	)	)	PUNCT
ejpam-3856	342	7	,	,	PUNCT
ejpam-3856	342	8	36−	36−	VERB
ejpam-3856	342	9	42	42	NUM
ejpam-3856	342	10	.	.	PUNCT
ejpam-3856	343	1	[	[	X
ejpam-3856	343	2	11	11	NUM
ejpam-3856	343	3	]	]	PUNCT
ejpam-3856	343	4	k.	k.	PROPN
ejpam-3856	343	5	kuratowski	kuratowski	PROPN
ejpam-3856	343	6	,	,	PUNCT
ejpam-3856	343	7	topology	topology	PROPN
ejpam-3856	343	8	i	i	PRON
ejpam-3856	343	9	,	,	PUNCT
ejpam-3856	343	10	warszawa	warszawa	PROPN
ejpam-3856	343	11	(	(	PUNCT
ejpam-3856	343	12	1933	1933	NUM
ejpam-3856	343	13	)	)	PUNCT
ejpam-3856	343	14	.	.	PUNCT
ejpam-3856	344	1	[	[	X
ejpam-3856	344	2	12	12	NUM
ejpam-3856	344	3	]	]	X
ejpam-3856	344	4	n.	n.	PROPN
ejpam-3856	344	5	levine	levine	PROPN
ejpam-3856	344	6	,	,	PUNCT
ejpam-3856	344	7	semi	semi	ADJ
ejpam-3856	344	8	-	-	ADJ
ejpam-3856	344	9	open	open	ADJ
ejpam-3856	344	10	sets	set	NOUN
ejpam-3856	344	11	and	and	CCONJ
ejpam-3856	344	12	semi	semi	ADJ
ejpam-3856	344	13	-	-	NOUN
ejpam-3856	344	14	continuity	continuity	NOUN
ejpam-3856	344	15	in	in	ADP
ejpam-3856	344	16	topological	topological	ADJ
ejpam-3856	344	17	spaces	space	NOUN
ejpam-3856	344	18	,	,	PUNCT
ejpam-3856	344	19	amer	amer	PROPN
ejpam-3856	344	20	.	.	PROPN
ejpam-3856	344	21	math	math	PROPN
ejpam-3856	344	22	.	.	PUNCT
ejpam-3856	345	1	monthly	monthly	ADJ
ejpam-3856	345	2	,	,	PUNCT
ejpam-3856	345	3	70(1963	70(1963	NUM
ejpam-3856	345	4	)	)	PUNCT
ejpam-3856	345	5	,	,	PUNCT
ejpam-3856	345	6	36−	36−	VERB
ejpam-3856	345	7	41	41	NUM
ejpam-3856	345	8	.	.	PUNCT
ejpam-3856	346	1	[	[	X
ejpam-3856	346	2	13	13	NUM
ejpam-3856	346	3	]	]	PUNCT
ejpam-3856	346	4	p.	p.	PROPN
ejpam-3856	346	5	l.	l.	PROPN
ejpam-3856	346	6	powar	powar	PROPN
ejpam-3856	346	7	and	and	CCONJ
ejpam-3856	346	8	k.	k.	PROPN
ejpam-3856	346	9	rajak	rajak	PROPN
ejpam-3856	346	10	,	,	PUNCT
ejpam-3856	346	11	some	some	DET
ejpam-3856	346	12	new	new	ADJ
ejpam-3856	346	13	concepts	concept	NOUN
ejpam-3856	346	14	of	of	ADP
ejpam-3856	346	15	continuity	continuity	NOUN
ejpam-3856	346	16	in	in	ADP
ejpam-3856	346	17	generalized	generalized	ADJ
ejpam-3856	346	18	topological	topological	ADJ
ejpam-3856	346	19	space	space	NOUN
ejpam-3856	346	20	,	,	PUNCT
ejpam-3856	346	21	int	int	NOUN
ejpam-3856	346	22	.	.	PUNCT
ejpam-3856	347	1	j.	j.	PROPN
ejpam-3856	347	2	com	com	PROPN
ejpam-3856	347	3	.	.	PUNCT
ejpam-3856	347	4	appl	appl	PROPN
ejpam-3856	347	5	.	.	PROPN
ejpam-3856	347	6	,	,	PUNCT
ejpam-3856	347	7	38(5)(2012	38(5)(2012	NUM
ejpam-3856	347	8	)	)	PUNCT
ejpam-3856	347	9	,	,	PUNCT
ejpam-3856	347	10	12−	12−	NUM
ejpam-3856	347	11	17	17	NUM
ejpam-3856	347	12	.	.	PUNCT