id	sid	tid	token	lemma	pos
ejpam-2826	1	1	european	european	PROPN
ejpam-2826	1	2	journal	journal	PROPN
ejpam-2826	1	3	of	of	ADP
ejpam-2826	1	4	pure	pure	ADJ
ejpam-2826	1	5	and	and	CCONJ
ejpam-2826	1	6	applied	apply	VERB
ejpam-2826	1	7	mathematics	mathematic	NOUN
ejpam-2826	1	8	vol	vol	NOUN
ejpam-2826	1	9	.	.	PROPN
ejpam-2826	2	1	10	10	NUM
ejpam-2826	2	2	,	,	PUNCT
ejpam-2826	2	3	no	no	INTJ
ejpam-2826	2	4	.	.	NOUN
ejpam-2826	2	5	2	2	NUM
ejpam-2826	2	6	,	,	PUNCT
ejpam-2826	2	7	2017	2017	NUM
ejpam-2826	2	8	,	,	PUNCT
ejpam-2826	2	9	312	312	NUM
ejpam-2826	2	10	-	-	SYM
ejpam-2826	2	11	322	322	NUM
ejpam-2826	2	12	issn	issn	PROPN
ejpam-2826	2	13	1307	1307	NUM
ejpam-2826	2	14	-	-	SYM
ejpam-2826	2	15	5543	5543	NUM
ejpam-2826	2	16	–	–	PUNCT
ejpam-2826	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2826	2	18	published	publish	VERB
ejpam-2826	2	19	by	by	ADP
ejpam-2826	2	20	new	new	PROPN
ejpam-2826	2	21	york	york	PROPN
ejpam-2826	2	22	business	business	PROPN
ejpam-2826	2	23	global	global	PROPN
ejpam-2826	2	24	contra	contra	PROPN
ejpam-2826	2	25	δgb	δgb	PROPN
ejpam-2826	2	26	-	-	PUNCT
ejpam-2826	2	27	continuous	continuous	ADJ
ejpam-2826	2	28	functions	function	NOUN
ejpam-2826	2	29	in	in	ADP
ejpam-2826	2	30	topological	topological	ADJ
ejpam-2826	2	31	spaces	space	NOUN
ejpam-2826	3	1	s.s.benchalli1	s.s.benchalli1	VERB
ejpam-2826	3	2	,	,	PUNCT
ejpam-2826	3	3	p.g.patil	p.g.patil	PROPN
ejpam-2826	3	4	2,∗	2,∗	NUM
ejpam-2826	3	5	,	,	PUNCT
ejpam-2826	3	6	j.b.toranagatti3	j.b.toranagatti3	PROPN
ejpam-2826	3	7	,	,	PUNCT
ejpam-2826	3	8	s.r.vighneshi4	s.r.vighneshi4	NOUN
ejpam-2826	3	9	1,2	1,2	NUM
ejpam-2826	3	10	department	department	NOUN
ejpam-2826	3	11	of	of	ADP
ejpam-2826	3	12	mathematics	mathematics	PROPN
ejpam-2826	3	13	,	,	PUNCT
ejpam-2826	3	14	karnatak	karnatak	PROPN
ejpam-2826	3	15	university	university	PROPN
ejpam-2826	3	16	,	,	PUNCT
ejpam-2826	3	17	dharwad	dharwad	PROPN
ejpam-2826	3	18	,	,	PUNCT
ejpam-2826	3	19	india	india	PROPN
ejpam-2826	3	20	3	3	NUM
ejpam-2826	3	21	department	department	NOUN
ejpam-2826	3	22	of	of	ADP
ejpam-2826	3	23	mathematics	mathematics	PROPN
ejpam-2826	3	24	,	,	PUNCT
ejpam-2826	3	25	karnatak	karnatak	PROPN
ejpam-2826	3	26	university	university	PROPN
ejpam-2826	3	27	’s	’s	PART
ejpam-2826	3	28	karnatak	karnatak	PROPN
ejpam-2826	3	29	college	college	PROPN
ejpam-2826	3	30	,	,	PUNCT
ejpam-2826	3	31	dharwad	dharwad	PROPN
ejpam-2826	3	32	,	,	PUNCT
ejpam-2826	3	33	india	india	PROPN
ejpam-2826	3	34	4	4	NUM
ejpam-2826	3	35	department	department	NOUN
ejpam-2826	3	36	of	of	ADP
ejpam-2826	3	37	mathematics	mathematic	NOUN
ejpam-2826	3	38	,	,	PUNCT
ejpam-2826	3	39	r.l.s	r.l.s	ADJ
ejpam-2826	3	40	college	college	NOUN
ejpam-2826	3	41	,	,	PUNCT
ejpam-2826	3	42	dharwad	dharwad	PROPN
ejpam-2826	3	43	,	,	PUNCT
ejpam-2826	3	44	india	india	PROPN
ejpam-2826	3	45	abstract	abstract	NOUN
ejpam-2826	3	46	.	.	PUNCT
ejpam-2826	4	1	in	in	ADP
ejpam-2826	4	2	this	this	DET
ejpam-2826	4	3	paper	paper	NOUN
ejpam-2826	4	4	,	,	PUNCT
ejpam-2826	4	5	the	the	DET
ejpam-2826	4	6	notion	notion	NOUN
ejpam-2826	4	7	of	of	ADP
ejpam-2826	4	8	δgb	δgb	ADJ
ejpam-2826	4	9	-	-	PUNCT
ejpam-2826	4	10	open	open	ADJ
ejpam-2826	4	11	sets	set	NOUN
ejpam-2826	4	12	in	in	ADP
ejpam-2826	4	13	topological	topological	ADJ
ejpam-2826	4	14	spaces	space	NOUN
ejpam-2826	4	15	is	be	AUX
ejpam-2826	4	16	applied	apply	VERB
ejpam-2826	4	17	to	to	PART
ejpam-2826	4	18	study	study	VERB
ejpam-2826	4	19	a	a	DET
ejpam-2826	4	20	new	new	ADJ
ejpam-2826	4	21	class	class	NOUN
ejpam-2826	4	22	of	of	ADP
ejpam-2826	4	23	functions	function	NOUN
ejpam-2826	4	24	called	call	VERB
ejpam-2826	4	25	contra	contra	PROPN
ejpam-2826	4	26	δgb	δgb	PROPN
ejpam-2826	4	27	continuous	continuous	ADJ
ejpam-2826	4	28	functions	function	NOUN
ejpam-2826	4	29	as	as	ADP
ejpam-2826	4	30	a	a	DET
ejpam-2826	4	31	new	new	ADJ
ejpam-2826	4	32	generalization	generalization	NOUN
ejpam-2826	4	33	of	of	ADP
ejpam-2826	4	34	contra	contra	PROPN
ejpam-2826	4	35	continuity	continuity	NOUN
ejpam-2826	4	36	and	and	CCONJ
ejpam-2826	4	37	obtain	obtain	VERB
ejpam-2826	4	38	their	their	PRON
ejpam-2826	4	39	characterizations	characterization	NOUN
ejpam-2826	4	40	and	and	CCONJ
ejpam-2826	4	41	properties	property	NOUN
ejpam-2826	4	42	.	.	PUNCT
ejpam-2826	5	1	2010	2010	NUM
ejpam-2826	5	2	mathematics	mathematic	NOUN
ejpam-2826	5	3	subject	subject	NOUN
ejpam-2826	5	4	classifications	classification	NOUN
ejpam-2826	5	5	:	:	PUNCT
ejpam-2826	5	6	54c08	54c08	NUM
ejpam-2826	5	7	,	,	PUNCT
ejpam-2826	5	8	54c10	54c10	NUM
ejpam-2826	5	9	key	key	ADJ
ejpam-2826	5	10	words	word	NOUN
ejpam-2826	5	11	and	and	CCONJ
ejpam-2826	5	12	phrases	phrase	NOUN
ejpam-2826	5	13	:	:	PUNCT
ejpam-2826	5	14	δgb	δgb	ADJ
ejpam-2826	5	15	-	-	PUNCT
ejpam-2826	5	16	open	open	ADJ
ejpam-2826	5	17	,	,	PUNCT
ejpam-2826	5	18	δgb	δgb	ADV
ejpam-2826	5	19	-	-	PUNCT
ejpam-2826	5	20	closed	closed	ADJ
ejpam-2826	5	21	,	,	PUNCT
ejpam-2826	5	22	δgb	δgb	ADV
ejpam-2826	5	23	-	-	PUNCT
ejpam-2826	5	24	connected	connect	VERB
ejpam-2826	5	25	,	,	PUNCT
ejpam-2826	5	26	contra	contra	PROPN
ejpam-2826	5	27	δgb	δgb	PROPN
ejpam-2826	5	28	-	-	PUNCT
ejpam-2826	5	29	continuous	continuous	ADJ
ejpam-2826	5	30	,	,	PUNCT
ejpam-2826	5	31	δgbcontinuous	δgbcontinuous	ADJ
ejpam-2826	5	32	.	.	PUNCT
ejpam-2826	6	1	1	1	X
ejpam-2826	6	2	.	.	X
ejpam-2826	6	3	introduction	introduction	NOUN
ejpam-2826	6	4	and	and	CCONJ
ejpam-2826	6	5	preliminaries	preliminary	NOUN
ejpam-2826	6	6	in	in	ADP
ejpam-2826	6	7	1996	1996	NUM
ejpam-2826	6	8	,	,	PUNCT
ejpam-2826	6	9	dontchev[6	dontchev[6	PROPN
ejpam-2826	6	10	]	]	PUNCT
ejpam-2826	6	11	introduced	introduce	VERB
ejpam-2826	6	12	contra	contra	PROPN
ejpam-2826	6	13	continuous	continuous	ADJ
ejpam-2826	6	14	functions	function	NOUN
ejpam-2826	6	15	.	.	PUNCT
ejpam-2826	7	1	nasef	nasef	PROPN
ejpam-2826	8	1	[	[	X
ejpam-2826	8	2	10	10	NUM
ejpam-2826	8	3	]	]	PUNCT
ejpam-2826	8	4	introduced	introduce	VERB
ejpam-2826	8	5	and	and	CCONJ
ejpam-2826	8	6	studied	study	VERB
ejpam-2826	8	7	contra	contra	PROPN
ejpam-2826	8	8	b	b	PROPN
ejpam-2826	8	9	-	-	PUNCT
ejpam-2826	8	10	continuous	continuous	ADJ
ejpam-2826	8	11	functions.al	functions.al	PROPN
ejpam-2826	8	12	-	-	ADJ
ejpam-2826	8	13	omari	omari	PROPN
ejpam-2826	8	14	and	and	CCONJ
ejpam-2826	8	15	noorani[1	noorani[1	PROPN
ejpam-2826	8	16	]	]	PUNCT
ejpam-2826	8	17	introduced	introduce	VERB
ejpam-2826	8	18	the	the	DET
ejpam-2826	8	19	concept	concept	NOUN
ejpam-2826	8	20	of	of	ADP
ejpam-2826	8	21	contra	contra	PROPN
ejpam-2826	8	22	gb	gb	ADV
ejpam-2826	8	23	-	-	PUNCT
ejpam-2826	8	24	continuous	continuous	ADJ
ejpam-2826	8	25	functions	function	NOUN
ejpam-2826	8	26	.	.	PUNCT
ejpam-2826	9	1	recently	recently	ADV
ejpam-2826	9	2	benchalli	benchalli	PROPN
ejpam-2826	9	3	et.al.[5	et.al.[5	PROPN
ejpam-2826	9	4	]	]	PUNCT
ejpam-2826	9	5	introduced	introduce	VERB
ejpam-2826	9	6	and	and	CCONJ
ejpam-2826	9	7	studied	study	VERB
ejpam-2826	9	8	δgbcontinuous	δgbcontinuous	ADJ
ejpam-2826	9	9	functions.these	functions.these	ADJ
ejpam-2826	9	10	concepts	concept	NOUN
ejpam-2826	9	11	motivated	motivate	VERB
ejpam-2826	9	12	us	we	PRON
ejpam-2826	9	13	to	to	PART
ejpam-2826	9	14	define	define	VERB
ejpam-2826	9	15	a	a	DET
ejpam-2826	9	16	new	new	ADJ
ejpam-2826	9	17	class	class	NOUN
ejpam-2826	9	18	of	of	ADP
ejpam-2826	9	19	functions	function	NOUN
ejpam-2826	9	20	called	call	VERB
ejpam-2826	9	21	contra	contra	PROPN
ejpam-2826	9	22	δgb	δgb	PROPN
ejpam-2826	9	23	-	-	PUNCT
ejpam-2826	9	24	continuous	continuous	ADJ
ejpam-2826	9	25	functions	function	NOUN
ejpam-2826	9	26	.	.	PUNCT
ejpam-2826	10	1	throughout	throughout	ADP
ejpam-2826	10	2	this	this	DET
ejpam-2826	10	3	paper,(x	paper,(x	NOUN
ejpam-2826	10	4	,	,	PUNCT
ejpam-2826	10	5	τ),(y	τ),(y	PROPN
ejpam-2826	10	6	,	,	PUNCT
ejpam-2826	10	7	σ	σ	PROPN
ejpam-2826	10	8	)	)	PUNCT
ejpam-2826	10	9	and	and	CCONJ
ejpam-2826	10	10	(	(	PUNCT
ejpam-2826	10	11	z	z	NOUN
ejpam-2826	10	12	,	,	PUNCT
ejpam-2826	10	13	η)(or	η)(or	VERB
ejpam-2826	10	14	simply	simply	ADV
ejpam-2826	10	15	x	x	X
ejpam-2826	10	16	,	,	PUNCT
ejpam-2826	10	17	y	y	PROPN
ejpam-2826	10	18	and	and	CCONJ
ejpam-2826	10	19	z	z	PROPN
ejpam-2826	10	20	)	)	PUNCT
ejpam-2826	10	21	represent	represent	VERB
ejpam-2826	10	22	topological	topological	ADJ
ejpam-2826	10	23	spaces	space	NOUN
ejpam-2826	10	24	on	on	ADP
ejpam-2826	10	25	which	which	PRON
ejpam-2826	10	26	no	no	DET
ejpam-2826	10	27	separation	separation	NOUN
ejpam-2826	10	28	axioms	axiom	NOUN
ejpam-2826	10	29	are	be	AUX
ejpam-2826	10	30	assumed	assume	VERB
ejpam-2826	10	31	unless	unless	SCONJ
ejpam-2826	10	32	explicitly	explicitly	ADV
ejpam-2826	10	33	stated	state	VERB
ejpam-2826	10	34	.	.	PUNCT
ejpam-2826	11	1	for	for	ADP
ejpam-2826	11	2	a	a	DET
ejpam-2826	11	3	subset	subset	NOUN
ejpam-2826	11	4	a	a	PRON
ejpam-2826	11	5	of	of	ADP
ejpam-2826	11	6	a	a	DET
ejpam-2826	11	7	space	space	NOUN
ejpam-2826	11	8	x	x	NOUN
ejpam-2826	11	9	,	,	PUNCT
ejpam-2826	11	10	the	the	DET
ejpam-2826	11	11	closure	closure	NOUN
ejpam-2826	11	12	of	of	ADP
ejpam-2826	11	13	a	a	DET
ejpam-2826	11	14	,	,	PUNCT
ejpam-2826	11	15	interior	interior	NOUN
ejpam-2826	11	16	of	of	ADP
ejpam-2826	11	17	a	a	PRON
ejpam-2826	11	18	and	and	CCONJ
ejpam-2826	11	19	complement	complement	NOUN
ejpam-2826	11	20	of	of	ADP
ejpam-2826	11	21	a	a	PRON
ejpam-2826	11	22	are	be	AUX
ejpam-2826	11	23	denoted	denote	VERB
ejpam-2826	11	24	by	by	ADP
ejpam-2826	11	25	cl(a	cl(a	NOUN
ejpam-2826	11	26	)	)	PUNCT
ejpam-2826	11	27	,	,	PUNCT
ejpam-2826	11	28	int(a	int(a	PROPN
ejpam-2826	11	29	)	)	PUNCT
ejpam-2826	11	30	and	and	CCONJ
ejpam-2826	11	31	ac	ac	PROPN
ejpam-2826	11	32	respectively	respectively	ADV
ejpam-2826	11	33	.	.	PUNCT
ejpam-2826	12	1	definition	definition	NOUN
ejpam-2826	12	2	1	1	NUM
ejpam-2826	12	3	.	.	PUNCT
ejpam-2826	13	1	a	a	DET
ejpam-2826	13	2	subset	subset	NOUN
ejpam-2826	13	3	a	a	PRON
ejpam-2826	13	4	of	of	ADP
ejpam-2826	13	5	a	a	DET
ejpam-2826	13	6	topological	topological	ADJ
ejpam-2826	13	7	space	space	NOUN
ejpam-2826	13	8	x	x	PUNCT
ejpam-2826	13	9	is	be	AUX
ejpam-2826	13	10	called	call	VERB
ejpam-2826	13	11	a	a	DET
ejpam-2826	13	12	(	(	PUNCT
ejpam-2826	13	13	i	i	NOUN
ejpam-2826	13	14	)	)	PUNCT
ejpam-2826	13	15	pre	pre	ADJ
ejpam-2826	13	16	-	-	ADJ
ejpam-2826	13	17	closed	closed	ADJ
ejpam-2826	13	18	[	[	X
ejpam-2826	13	19	9	9	NUM
ejpam-2826	13	20	]	]	X
ejpam-2826	13	21	if	if	SCONJ
ejpam-2826	13	22	cl(int(a))⊆a	cl(int(a))⊆a	PROPN
ejpam-2826	13	23	(	(	PUNCT
ejpam-2826	13	24	ii	ii	NOUN
ejpam-2826	13	25	)	)	PUNCT
ejpam-2826	13	26	b	b	NOUN
ejpam-2826	13	27	-	-	PUNCT
ejpam-2826	13	28	closed	closed	ADJ
ejpam-2826	13	29	[	[	X
ejpam-2826	13	30	2	2	NUM
ejpam-2826	13	31	]	]	PUNCT
ejpam-2826	13	32	if	if	SCONJ
ejpam-2826	13	33	cl(int(a))∩int(cl(a))⊆a	cl(int(a))∩int(cl(a))⊆a	PROPN
ejpam-2826	13	34	(	(	PUNCT
ejpam-2826	13	35	iii	iii	NOUN
ejpam-2826	13	36	)	)	PUNCT
ejpam-2826	13	37	regular	regular	ADJ
ejpam-2826	13	38	-	-	PUNCT
ejpam-2826	13	39	closed	closed	ADJ
ejpam-2826	13	40	[	[	X
ejpam-2826	13	41	14	14	NUM
ejpam-2826	13	42	]	]	PUNCT
ejpam-2826	13	43	if	if	SCONJ
ejpam-2826	13	44	a	a	DET
ejpam-2826	13	45	=	=	NOUN
ejpam-2826	13	46	cl(int(a	cl(int(a	NOUN
ejpam-2826	13	47	)	)	PUNCT
ejpam-2826	13	48	)	)	PUNCT
ejpam-2826	14	1	(	(	PUNCT
ejpam-2826	14	2	iv	iv	X
ejpam-2826	14	3	)	)	PUNCT
ejpam-2826	14	4	δ	δ	PROPN
ejpam-2826	14	5	-	-	PUNCT
ejpam-2826	14	6	closed	close	VERB
ejpam-2826	14	7	[	[	X
ejpam-2826	14	8	17	17	NUM
ejpam-2826	14	9	]	]	PUNCT
ejpam-2826	14	10	if	if	SCONJ
ejpam-2826	14	11	a	a	DET
ejpam-2826	14	12	=	=	NOUN
ejpam-2826	14	13	clδ(a)where	clδ(a)where	NOUN
ejpam-2826	14	14	clδ(a)={x∈x	clδ(a)={x∈x	PROPN
ejpam-2826	14	15	:	:	PUNCT
ejpam-2826	14	16	int(cl(u))∩a	int(cl(u))∩a	VERB
ejpam-2826	14	17	6	6	NUM
ejpam-2826	14	18	=	=	SYM
ejpam-2826	14	19	φ	φ	NUM
ejpam-2826	14	20	,	,	PUNCT
ejpam-2826	14	21	u∈τ	u∈τ	ADJ
ejpam-2826	14	22	and	and	CCONJ
ejpam-2826	14	23	x∈u	x∈u	NOUN
ejpam-2826	14	24	}	}	PUNCT
ejpam-2826	14	25	∗corresponding	∗corresponde	VERB
ejpam-2826	14	26	author	author	NOUN
ejpam-2826	14	27	.	.	PUNCT
ejpam-2826	15	1	email	email	NOUN
ejpam-2826	15	2	addresses	address	NOUN
ejpam-2826	15	3	:	:	PUNCT
ejpam-2826	15	4	benchalliss@gmail.com(s.s.benchalli),pgpatil01@gmail.com(p.g.patil	benchalliss@gmail.com(s.s.benchalli),pgpatil01@gmail.com(p.g.patil	PROPN
ejpam-2826	15	5	)	)	PUNCT
ejpam-2826	15	6	,	,	PUNCT
ejpam-2826	15	7	jagadeeshbt2000@gmail.com(j.b.toranagatti),vighneshisr@gmail.com	jagadeeshbt2000@gmail.com(j.b.toranagatti),vighneshisr@gmail.com	PROPN
ejpam-2826	15	8	(	(	PUNCT
ejpam-2826	15	9	s.r.vighneshi	s.r.vighneshi	PROPN
ejpam-2826	15	10	)	)	PUNCT
ejpam-2826	15	11	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2826	16	1	312	312	NUM
ejpam-2826	16	2	c	c	X
ejpam-2826	16	3	©	©	PROPN
ejpam-2826	16	4	2017	2017	NUM
ejpam-2826	16	5	ejpam	ejpam	VERB
ejpam-2826	16	6	all	all	DET
ejpam-2826	16	7	rights	right	NOUN
ejpam-2826	16	8	reserved	reserve	VERB
ejpam-2826	16	9	.	.	PUNCT
ejpam-2826	17	1	s.s.benchalli	s.s.benchalli	PROPN
ejpam-2826	17	2	,	,	PUNCT
ejpam-2826	17	3	p.g.patil	p.g.patil	PROPN
ejpam-2826	17	4	,	,	PUNCT
ejpam-2826	17	5	j.b.toranagatti	j.b.toranagatti	PROPN
ejpam-2826	17	6	,	,	PUNCT
ejpam-2826	17	7	s.r.vighneshi	s.r.vighneshi	PROPN
ejpam-2826	17	8	/	/	SYM
ejpam-2826	17	9	eur	eur	PROPN
ejpam-2826	17	10	.	.	PUNCT
ejpam-2826	18	1	j.	j.	PROPN
ejpam-2826	18	2	pure	pure	PROPN
ejpam-2826	18	3	appl	appl	PROPN
ejpam-2826	18	4	.	.	PROPN
ejpam-2826	18	5	math	math	PROPN
ejpam-2826	18	6	,	,	PUNCT
ejpam-2826	18	7	10	10	NUM
ejpam-2826	18	8	(	(	PUNCT
ejpam-2826	18	9	2	2	NUM
ejpam-2826	18	10	)	)	PUNCT
ejpam-2826	18	11	(	(	PUNCT
ejpam-2826	18	12	2017	2017	NUM
ejpam-2826	18	13	)	)	PUNCT
ejpam-2826	18	14	,	,	PUNCT
ejpam-2826	18	15	312	312	NUM
ejpam-2826	18	16	-	-	SYM
ejpam-2826	18	17	322	322	NUM
ejpam-2826	18	18	313	313	NUM
ejpam-2826	18	19	(	(	PUNCT
ejpam-2826	18	20	v	v	NOUN
ejpam-2826	18	21	)	)	PUNCT
ejpam-2826	18	22	delta	delta	NOUN
ejpam-2826	18	23	generalized	generalize	VERB
ejpam-2826	18	24	b	b	X
ejpam-2826	18	25	-	-	PUNCT
ejpam-2826	18	26	closed	closed	ADJ
ejpam-2826	18	27	(	(	PUNCT
ejpam-2826	18	28	briefly	briefly	ADV
ejpam-2826	18	29	,	,	PUNCT
ejpam-2826	18	30	δgb	δgb	ADV
ejpam-2826	18	31	-	-	PUNCT
ejpam-2826	18	32	closed	closed	ADJ
ejpam-2826	18	33	)	)	PUNCT
ejpam-2826	19	1	[	[	X
ejpam-2826	19	2	4	4	X
ejpam-2826	19	3	]	]	X
ejpam-2826	19	4	if	if	SCONJ
ejpam-2826	19	5	bcl(a)⊆g	bcl(a)⊆g	ADP
ejpam-2826	19	6	whenever	whenever	SCONJ
ejpam-2826	19	7	a⊆g	a⊆g	PROPN
ejpam-2826	19	8	and	and	CCONJ
ejpam-2826	19	9	g	g	PROPN
ejpam-2826	19	10	is	be	AUX
ejpam-2826	19	11	δ	δ	NOUN
ejpam-2826	19	12	-	-	ADJ
ejpam-2826	19	13	open	open	ADJ
ejpam-2826	19	14	in	in	ADP
ejpam-2826	19	15	x.	x.	NOUN
ejpam-2826	19	16	the	the	DET
ejpam-2826	19	17	complements	complement	NOUN
ejpam-2826	19	18	of	of	ADP
ejpam-2826	19	19	the	the	DET
ejpam-2826	19	20	above	above	ADJ
ejpam-2826	19	21	mentioned	mention	VERB
ejpam-2826	19	22	closed	closed	ADJ
ejpam-2826	19	23	sets	set	NOUN
ejpam-2826	19	24	are	be	AUX
ejpam-2826	19	25	their	their	PRON
ejpam-2826	19	26	respective	respective	ADJ
ejpam-2826	19	27	open	open	ADJ
ejpam-2826	19	28	sets	set	NOUN
ejpam-2826	19	29	.	.	PUNCT
ejpam-2826	20	1	the	the	DET
ejpam-2826	20	2	b	b	NOUN
ejpam-2826	20	3	-	-	PUNCT
ejpam-2826	20	4	closure	closure	NOUN
ejpam-2826	20	5	of	of	ADP
ejpam-2826	20	6	a	a	DET
ejpam-2826	20	7	subset	subset	NOUN
ejpam-2826	20	8	a	a	PRON
ejpam-2826	20	9	of	of	ADP
ejpam-2826	20	10	x	x	NOUN
ejpam-2826	20	11	is	be	AUX
ejpam-2826	20	12	the	the	DET
ejpam-2826	20	13	intersection	intersection	NOUN
ejpam-2826	20	14	of	of	ADP
ejpam-2826	20	15	all	all	DET
ejpam-2826	20	16	b	b	NOUN
ejpam-2826	20	17	-	-	PUNCT
ejpam-2826	20	18	closed	closed	ADJ
ejpam-2826	20	19	sets	set	NOUN
ejpam-2826	20	20	containing	contain	VERB
ejpam-2826	20	21	a	a	PRON
ejpam-2826	20	22	and	and	CCONJ
ejpam-2826	20	23	is	be	AUX
ejpam-2826	20	24	denoted	denote	VERB
ejpam-2826	20	25	by	by	ADP
ejpam-2826	20	26	bcl(a	bcl(a	PROPN
ejpam-2826	20	27	)	)	PUNCT
ejpam-2826	20	28	.	.	PUNCT
ejpam-2826	21	1	definition	definition	NOUN
ejpam-2826	21	2	2	2	NUM
ejpam-2826	21	3	.	.	PUNCT
ejpam-2826	22	1	a	a	DET
ejpam-2826	22	2	function	function	NOUN
ejpam-2826	22	3	f	f	X
ejpam-2826	22	4	:	:	PUNCT
ejpam-2826	22	5	x→y	x→y	NUM
ejpam-2826	22	6	from	from	ADP
ejpam-2826	22	7	a	a	DET
ejpam-2826	22	8	topological	topological	ADJ
ejpam-2826	22	9	space	space	NOUN
ejpam-2826	22	10	x	x	PUNCT
ejpam-2826	22	11	into	into	ADP
ejpam-2826	22	12	a	a	DET
ejpam-2826	22	13	topological	topological	ADJ
ejpam-2826	22	14	space	space	NOUN
ejpam-2826	22	15	y	y	PROPN
ejpam-2826	22	16	is	be	AUX
ejpam-2826	22	17	called	call	VERB
ejpam-2826	22	18	a	a	DET
ejpam-2826	22	19	,	,	PUNCT
ejpam-2826	22	20	(	(	PUNCT
ejpam-2826	22	21	i	i	NOUN
ejpam-2826	22	22	)	)	PUNCT
ejpam-2826	22	23	contra	contra	PROPN
ejpam-2826	22	24	continuous	continuous	ADJ
ejpam-2826	22	25	[	[	X
ejpam-2826	22	26	6	6	NUM
ejpam-2826	22	27	]	]	PUNCT
ejpam-2826	22	28	if	if	SCONJ
ejpam-2826	22	29	f−1(g	f−1(g	PROPN
ejpam-2826	22	30	)	)	PUNCT
ejpam-2826	22	31	is	be	AUX
ejpam-2826	22	32	closed	close	VERB
ejpam-2826	22	33	in	in	ADP
ejpam-2826	22	34	x	x	PUNCT
ejpam-2826	22	35	for	for	ADP
ejpam-2826	22	36	every	every	DET
ejpam-2826	22	37	open	open	ADJ
ejpam-2826	22	38	set	set	NOUN
ejpam-2826	22	39	g	g	NOUN
ejpam-2826	22	40	of	of	ADP
ejpam-2826	22	41	y.	y.	PROPN
ejpam-2826	22	42	(	(	PUNCT
ejpam-2826	22	43	ii	ii	PROPN
ejpam-2826	22	44	)	)	PUNCT
ejpam-2826	22	45	contra	contra	PROPN
ejpam-2826	23	1	b	b	X
ejpam-2826	23	2	-	-	PUNCT
ejpam-2826	23	3	continuous	continuous	ADJ
ejpam-2826	23	4	[	[	X
ejpam-2826	23	5	10	10	NUM
ejpam-2826	23	6	]	]	X
ejpam-2826	23	7	if	if	SCONJ
ejpam-2826	23	8	f−1(g	f−1(g	PROPN
ejpam-2826	23	9	)	)	PUNCT
ejpam-2826	23	10	is	be	AUX
ejpam-2826	23	11	b	b	NOUN
ejpam-2826	23	12	-	-	PUNCT
ejpam-2826	23	13	closed	closed	ADJ
ejpam-2826	23	14	in	in	ADP
ejpam-2826	23	15	x	x	PUNCT
ejpam-2826	23	16	for	for	ADP
ejpam-2826	23	17	every	every	DET
ejpam-2826	23	18	open	open	ADJ
ejpam-2826	23	19	set	set	NOUN
ejpam-2826	23	20	g	g	NOUN
ejpam-2826	23	21	of	of	ADP
ejpam-2826	23	22	y.	y.	PROPN
ejpam-2826	23	23	(	(	PUNCT
ejpam-2826	23	24	iii	iii	PROPN
ejpam-2826	23	25	)	)	PUNCT
ejpam-2826	23	26	contra	contra	PROPN
ejpam-2826	23	27	rgb	rgb	PROPN
ejpam-2826	23	28	-	-	PROPN
ejpam-2826	23	29	continuous	continuous	ADJ
ejpam-2826	23	30	[	[	X
ejpam-2826	23	31	13	13	NUM
ejpam-2826	23	32	]	]	X
ejpam-2826	23	33	if	if	SCONJ
ejpam-2826	23	34	f−1(g	f−1(g	PROPN
ejpam-2826	23	35	)	)	PUNCT
ejpam-2826	23	36	is	be	AUX
ejpam-2826	23	37	rgb	rgb	PROPN
ejpam-2826	23	38	-	-	PUNCT
ejpam-2826	23	39	closed	closed	ADJ
ejpam-2826	23	40	in	in	ADP
ejpam-2826	23	41	x	x	PUNCT
ejpam-2826	23	42	for	for	SCONJ
ejpam-2826	23	43	every	every	DET
ejpam-2826	23	44	open	open	ADJ
ejpam-2826	23	45	set	set	NOUN
ejpam-2826	23	46	g	g	NOUN
ejpam-2826	23	47	of	of	ADP
ejpam-2826	23	48	y.	y.	PROPN
ejpam-2826	23	49	(	(	PUNCT
ejpam-2826	23	50	iv	iv	X
ejpam-2826	23	51	)	)	PUNCT
ejpam-2826	23	52	δgb	δgb	ADJ
ejpam-2826	23	53	-	-	PUNCT
ejpam-2826	23	54	continuous	continuous	ADJ
ejpam-2826	23	55	[	[	X
ejpam-2826	23	56	5	5	NUM
ejpam-2826	23	57	]	]	PUNCT
ejpam-2826	23	58	if	if	SCONJ
ejpam-2826	23	59	f−1(g	f−1(g	PROPN
ejpam-2826	23	60	)	)	PUNCT
ejpam-2826	23	61	is	be	AUX
ejpam-2826	23	62	δgb	δgb	ADV
ejpam-2826	23	63	-	-	PUNCT
ejpam-2826	23	64	open	open	ADJ
ejpam-2826	23	65	in	in	ADP
ejpam-2826	23	66	x	x	PUNCT
ejpam-2826	23	67	for	for	ADP
ejpam-2826	23	68	every	every	DET
ejpam-2826	23	69	open	open	ADJ
ejpam-2826	23	70	set	set	NOUN
ejpam-2826	23	71	g	g	NOUN
ejpam-2826	23	72	of	of	ADP
ejpam-2826	23	73	y.	y.	PROPN
ejpam-2826	23	74	(	(	PUNCT
ejpam-2826	23	75	v	v	NOUN
ejpam-2826	23	76	)	)	PUNCT
ejpam-2826	23	77	completely	completely	ADV
ejpam-2826	23	78	-	-	PUNCT
ejpam-2826	23	79	continuous	continuous	ADJ
ejpam-2826	23	80	[	[	X
ejpam-2826	23	81	3	3	NUM
ejpam-2826	23	82	]	]	X
ejpam-2826	23	83	if	if	SCONJ
ejpam-2826	23	84	f−1(g	f−1(g	PROPN
ejpam-2826	23	85	)	)	PUNCT
ejpam-2826	23	86	is	be	AUX
ejpam-2826	23	87	regular	regular	ADV
ejpam-2826	23	88	-	-	PUNCT
ejpam-2826	23	89	open	open	ADJ
ejpam-2826	23	90	in	in	ADP
ejpam-2826	23	91	x	x	PUNCT
ejpam-2826	23	92	for	for	ADP
ejpam-2826	23	93	every	every	DET
ejpam-2826	23	94	open	open	ADJ
ejpam-2826	23	95	set	set	NOUN
ejpam-2826	23	96	g	g	NOUN
ejpam-2826	23	97	of	of	ADP
ejpam-2826	23	98	y.	y.	PROPN
ejpam-2826	23	99	(	(	PUNCT
ejpam-2826	23	100	vi	vi	NOUN
ejpam-2826	23	101	)	)	PUNCT
ejpam-2826	23	102	perfectly	perfectly	ADV
ejpam-2826	23	103	-	-	PUNCT
ejpam-2826	23	104	continuous	continuous	ADJ
ejpam-2826	23	105	[	[	X
ejpam-2826	23	106	12	12	NUM
ejpam-2826	23	107	]	]	X
ejpam-2826	23	108	if	if	SCONJ
ejpam-2826	23	109	f−1(g	f−1(g	PROPN
ejpam-2826	23	110	)	)	PUNCT
ejpam-2826	23	111	is	be	AUX
ejpam-2826	23	112	clopen	clopen	ADJ
ejpam-2826	23	113	in	in	ADP
ejpam-2826	23	114	x	x	PUNCT
ejpam-2826	23	115	for	for	ADP
ejpam-2826	23	116	every	every	DET
ejpam-2826	23	117	open	open	ADJ
ejpam-2826	23	118	set	set	NOUN
ejpam-2826	23	119	g	g	NOUN
ejpam-2826	23	120	of	of	ADP
ejpam-2826	23	121	y.	y.	PROPN
ejpam-2826	23	122	(	(	PUNCT
ejpam-2826	23	123	vii	vii	PROPN
ejpam-2826	23	124	)	)	PUNCT
ejpam-2826	23	125	δ∗-continuous	δ∗-continuous	PROPN
ejpam-2826	23	126	if	if	SCONJ
ejpam-2826	23	127	f−1(g	f−1(g	PROPN
ejpam-2826	23	128	)	)	PUNCT
ejpam-2826	23	129	is	be	AUX
ejpam-2826	23	130	δ	δ	NOUN
ejpam-2826	23	131	-	-	ADJ
ejpam-2826	23	132	open	open	ADJ
ejpam-2826	23	133	in	in	ADP
ejpam-2826	23	134	x	x	PUNCT
ejpam-2826	23	135	for	for	ADP
ejpam-2826	23	136	every	every	DET
ejpam-2826	23	137	open	open	ADJ
ejpam-2826	23	138	set	set	NOUN
ejpam-2826	23	139	g	g	NOUN
ejpam-2826	23	140	of	of	ADP
ejpam-2826	23	141	y.	y.	PROPN
ejpam-2826	23	142	(	(	PUNCT
ejpam-2826	23	143	viii	viii	PROPN
ejpam-2826	23	144	)	)	PUNCT
ejpam-2826	23	145	contra	contra	PROPN
ejpam-2826	23	146	gb	gb	ADV
ejpam-2826	23	147	-	-	PUNCT
ejpam-2826	23	148	continuous	continuous	ADJ
ejpam-2826	23	149	[	[	X
ejpam-2826	23	150	1	1	NUM
ejpam-2826	23	151	]	]	X
ejpam-2826	23	152	if	if	SCONJ
ejpam-2826	23	153	f−1(g	f−1(g	PROPN
ejpam-2826	23	154	)	)	PUNCT
ejpam-2826	23	155	is	be	AUX
ejpam-2826	23	156	gb	gb	ADV
ejpam-2826	23	157	-	-	PUNCT
ejpam-2826	23	158	closed	closed	ADJ
ejpam-2826	23	159	in	in	ADP
ejpam-2826	23	160	x	x	PUNCT
ejpam-2826	23	161	for	for	ADP
ejpam-2826	23	162	every	every	DET
ejpam-2826	23	163	open	open	ADJ
ejpam-2826	23	164	set	set	NOUN
ejpam-2826	23	165	g	g	NOUN
ejpam-2826	23	166	of	of	ADP
ejpam-2826	23	167	y.	y.	PROPN
ejpam-2826	23	168	(	(	PUNCT
ejpam-2826	23	169	ix	ix	PROPN
ejpam-2826	23	170	)	)	PUNCT
ejpam-2826	23	171	pre	pre	ADJ
ejpam-2826	23	172	-	-	ADJ
ejpam-2826	23	173	closed	closed	ADJ
ejpam-2826	23	174	[	[	X
ejpam-2826	23	175	7]if	7]if	NUM
ejpam-2826	23	176	for	for	ADP
ejpam-2826	23	177	every	every	DET
ejpam-2826	23	178	closed	closed	NOUN
ejpam-2826	23	179	subset	subset	VERB
ejpam-2826	23	180	a	a	PRON
ejpam-2826	23	181	of	of	ADP
ejpam-2826	23	182	x	x	SYM
ejpam-2826	23	183	f(a	f(a	NOUN
ejpam-2826	23	184	)	)	PUNCT
ejpam-2826	23	185	is	be	AUX
ejpam-2826	23	186	pre	pre	ADJ
ejpam-2826	23	187	-	-	VERB
ejpam-2826	23	188	closed	closed	ADJ
ejpam-2826	23	189	in	in	ADP
ejpam-2826	23	190	y.	y.	PROPN
ejpam-2826	23	191	definition	definition	NOUN
ejpam-2826	24	1	3	3	NUM
ejpam-2826	24	2	.	.	PUNCT
ejpam-2826	25	1	[	[	X
ejpam-2826	25	2	5	5	NUM
ejpam-2826	25	3	]	]	PUNCT
ejpam-2826	25	4	a	a	DET
ejpam-2826	25	5	topological	topological	ADJ
ejpam-2826	25	6	space	space	NOUN
ejpam-2826	25	7	x	x	PRON
ejpam-2826	25	8	is	be	AUX
ejpam-2826	25	9	said	say	VERB
ejpam-2826	25	10	to	to	PART
ejpam-2826	25	11	be	be	AUX
ejpam-2826	25	12	a	a	DET
ejpam-2826	25	13	,	,	PUNCT
ejpam-2826	25	14	(	(	PUNCT
ejpam-2826	25	15	i	i	NOUN
ejpam-2826	25	16	)	)	PUNCT
ejpam-2826	25	17	tδgb	tδgb	ADJ
ejpam-2826	25	18	-	-	PUNCT
ejpam-2826	25	19	space	space	NOUN
ejpam-2826	25	20	if	if	SCONJ
ejpam-2826	25	21	every	every	DET
ejpam-2826	25	22	δgb	δgb	NOUN
ejpam-2826	25	23	-	-	PUNCT
ejpam-2826	25	24	closed	closed	ADJ
ejpam-2826	25	25	subset	subset	NOUN
ejpam-2826	25	26	of	of	ADP
ejpam-2826	25	27	x	x	PUNCT
ejpam-2826	25	28	is	be	AUX
ejpam-2826	25	29	closed	closed	ADJ
ejpam-2826	25	30	.	.	PUNCT
ejpam-2826	26	1	(	(	PUNCT
ejpam-2826	26	2	ii	ii	NOUN
ejpam-2826	26	3	)	)	PUNCT
ejpam-2826	26	4	δgbt	δgbt	NOUN
ejpam-2826	26	5	1	1	NUM
ejpam-2826	26	6	2	2	NUM
ejpam-2826	26	7	-space	-space	NOUN
ejpam-2826	26	8	if	if	SCONJ
ejpam-2826	26	9	every	every	DET
ejpam-2826	26	10	δgb	δgb	NOUN
ejpam-2826	26	11	-	-	PUNCT
ejpam-2826	26	12	closed	closed	ADJ
ejpam-2826	26	13	subset	subset	NOUN
ejpam-2826	26	14	of	of	ADP
ejpam-2826	26	15	x	x	PUNCT
ejpam-2826	26	16	is	be	AUX
ejpam-2826	26	17	b	b	NOUN
ejpam-2826	26	18	-	-	PUNCT
ejpam-2826	26	19	closed	closed	ADJ
ejpam-2826	26	20	.	.	PUNCT
ejpam-2826	27	1	2	2	X
ejpam-2826	27	2	.	.	X
ejpam-2826	27	3	contra	contra	PROPN
ejpam-2826	27	4	δgb	δgb	PROPN
ejpam-2826	27	5	-	-	PUNCT
ejpam-2826	27	6	continuous	continuous	ADJ
ejpam-2826	27	7	functions	function	NOUN
ejpam-2826	27	8	.	.	PUNCT
ejpam-2826	28	1	definition	definition	NOUN
ejpam-2826	28	2	4	4	NUM
ejpam-2826	28	3	.	.	PUNCT
ejpam-2826	29	1	a	a	DET
ejpam-2826	29	2	function	function	NOUN
ejpam-2826	29	3	f	f	NOUN
ejpam-2826	29	4	:	:	PUNCT
ejpam-2826	29	5	x→y	x→y	NUM
ejpam-2826	29	6	is	be	AUX
ejpam-2826	29	7	called	call	VERB
ejpam-2826	29	8	contra	contra	PROPN
ejpam-2826	29	9	δgb	δgb	PROPN
ejpam-2826	29	10	-	-	PUNCT
ejpam-2826	29	11	continuous	continuous	ADJ
ejpam-2826	29	12	if	if	SCONJ
ejpam-2826	29	13	f−1(v	f−1(v	PROPN
ejpam-2826	29	14	)	)	PUNCT
ejpam-2826	29	15	is	be	AUX
ejpam-2826	29	16	δgb	δgb	ADV
ejpam-2826	29	17	-	-	PUNCT
ejpam-2826	29	18	closed	closed	ADJ
ejpam-2826	29	19	in	in	ADP
ejpam-2826	29	20	x	x	PUNCT
ejpam-2826	29	21	for	for	SCONJ
ejpam-2826	29	22	each	each	DET
ejpam-2826	29	23	open	open	ADJ
ejpam-2826	29	24	set	set	VERB
ejpam-2826	29	25	v	v	NOUN
ejpam-2826	29	26	of	of	ADP
ejpam-2826	29	27	y.	y.	PROPN
ejpam-2826	29	28	clearly	clearly	ADV
ejpam-2826	29	29	,	,	PUNCT
ejpam-2826	29	30	f	f	X
ejpam-2826	29	31	:	:	PUNCT
ejpam-2826	29	32	x→y	x→y	NUM
ejpam-2826	29	33	is	be	AUX
ejpam-2826	29	34	contra	contra	PROPN
ejpam-2826	29	35	δgb	δgb	PROPN
ejpam-2826	29	36	-	-	PUNCT
ejpam-2826	29	37	continuous	continuous	ADJ
ejpam-2826	29	38	if	if	SCONJ
ejpam-2826	29	39	and	and	CCONJ
ejpam-2826	29	40	only	only	ADV
ejpam-2826	29	41	if	if	SCONJ
ejpam-2826	29	42	f−1(g	f−1(g	PROPN
ejpam-2826	29	43	)	)	PUNCT
ejpam-2826	29	44	is	be	AUX
ejpam-2826	29	45	δgb	δgb	ADV
ejpam-2826	29	46	-	-	PUNCT
ejpam-2826	29	47	open	open	ADJ
ejpam-2826	29	48	in	in	ADP
ejpam-2826	29	49	x	x	PUNCT
ejpam-2826	29	50	for	for	ADP
ejpam-2826	29	51	every	every	DET
ejpam-2826	29	52	closed	close	VERB
ejpam-2826	29	53	set	set	VERB
ejpam-2826	29	54	g	g	NOUN
ejpam-2826	29	55	in	in	ADP
ejpam-2826	29	56	y.	y.	PROPN
ejpam-2826	29	57	theorem	theorem	PROPN
ejpam-2826	29	58	1	1	NUM
ejpam-2826	29	59	.	.	PUNCT
ejpam-2826	30	1	if	if	SCONJ
ejpam-2826	30	2	f	f	X
ejpam-2826	30	3	:	:	PUNCT
ejpam-2826	30	4	x→y	x→y	NUM
ejpam-2826	30	5	is	be	AUX
ejpam-2826	30	6	contra	contra	PROPN
ejpam-2826	30	7	gb	gb	ADV
ejpam-2826	30	8	-	-	PUNCT
ejpam-2826	30	9	continuous	continuous	ADJ
ejpam-2826	30	10	then	then	ADV
ejpam-2826	30	11	it	it	PRON
ejpam-2826	30	12	is	be	AUX
ejpam-2826	30	13	contra	contra	PROPN
ejpam-2826	30	14	δgb	δgb	ADV
ejpam-2826	30	15	-	-	PUNCT
ejpam-2826	30	16	continuous	continuous	ADJ
ejpam-2826	30	17	.	.	PUNCT
ejpam-2826	31	1	proof	proof	NOUN
ejpam-2826	31	2	:	:	PUNCT
ejpam-2826	31	3	follows	follow	VERB
ejpam-2826	31	4	from	from	ADP
ejpam-2826	31	5	the	the	DET
ejpam-2826	31	6	fact	fact	NOUN
ejpam-2826	31	7	that	that	SCONJ
ejpam-2826	31	8	every	every	DET
ejpam-2826	31	9	gb	gb	ADV
ejpam-2826	31	10	-	-	PUNCT
ejpam-2826	31	11	closed	closed	ADJ
ejpam-2826	31	12	set	set	NOUN
ejpam-2826	31	13	is	be	AUX
ejpam-2826	31	14	δgb	δgb	ADV
ejpam-2826	31	15	-	-	PUNCT
ejpam-2826	31	16	closed	closed	ADJ
ejpam-2826	31	17	.	.	PUNCT
ejpam-2826	32	1	theorem	theorem	NOUN
ejpam-2826	32	2	2	2	NUM
ejpam-2826	32	3	.	.	PUNCT
ejpam-2826	33	1	if	if	SCONJ
ejpam-2826	33	2	f	f	X
ejpam-2826	33	3	:	:	PUNCT
ejpam-2826	33	4	x→y	x→y	NUM
ejpam-2826	33	5	is	be	AUX
ejpam-2826	33	6	contra	contra	PROPN
ejpam-2826	33	7	b	b	PROPN
ejpam-2826	33	8	-	-	PUNCT
ejpam-2826	33	9	continuous	continuous	ADJ
ejpam-2826	33	10	then	then	ADV
ejpam-2826	33	11	it	it	PRON
ejpam-2826	33	12	is	be	AUX
ejpam-2826	33	13	contra	contra	PROPN
ejpam-2826	33	14	δgb	δgb	ADV
ejpam-2826	33	15	-	-	PUNCT
ejpam-2826	33	16	continuous	continuous	ADJ
ejpam-2826	33	17	.	.	PUNCT
ejpam-2826	34	1	proof	proof	NOUN
ejpam-2826	34	2	:	:	PUNCT
ejpam-2826	34	3	follows	follow	VERB
ejpam-2826	34	4	from	from	ADP
ejpam-2826	34	5	the	the	DET
ejpam-2826	34	6	fact	fact	NOUN
ejpam-2826	34	7	that	that	SCONJ
ejpam-2826	34	8	every	every	DET
ejpam-2826	34	9	contra	contra	PROPN
ejpam-2826	34	10	b	b	X
ejpam-2826	34	11	-	-	PUNCT
ejpam-2826	34	12	continuous	continuous	ADJ
ejpam-2826	34	13	function	function	NOUN
ejpam-2826	34	14	is	be	AUX
ejpam-2826	34	15	contra	contra	PROPN
ejpam-2826	34	16	gb	gb	ADV
ejpam-2826	34	17	-	-	PUNCT
ejpam-2826	34	18	continuous	continuous	ADJ
ejpam-2826	34	19	and	and	CCONJ
ejpam-2826	34	20	theorem	theorem	ADJ
ejpam-2826	34	21	1	1	NUM
ejpam-2826	34	22	.	.	NOUN
ejpam-2826	34	23	remark	remark	NOUN
ejpam-2826	34	24	1	1	NUM
ejpam-2826	34	25	.	.	PUNCT
ejpam-2826	35	1	the	the	DET
ejpam-2826	35	2	converse	converse	NOUN
ejpam-2826	35	3	of	of	ADP
ejpam-2826	35	4	theorem	theorem	ADJ
ejpam-2826	35	5	1	1	NUM
ejpam-2826	35	6	and	and	CCONJ
ejpam-2826	35	7	theorem	theorem	VERB
ejpam-2826	35	8	2	2	NUM
ejpam-2826	35	9	need	need	AUX
ejpam-2826	35	10	not	not	PART
ejpam-2826	35	11	be	be	AUX
ejpam-2826	35	12	true	true	ADJ
ejpam-2826	35	13	as	as	SCONJ
ejpam-2826	35	14	seen	see	VERB
ejpam-2826	35	15	from	from	ADP
ejpam-2826	35	16	the	the	DET
ejpam-2826	35	17	following	follow	VERB
ejpam-2826	35	18	example	example	NOUN
ejpam-2826	35	19	.	.	PUNCT
ejpam-2826	36	1	s.s.benchalli	s.s.benchalli	PROPN
ejpam-2826	36	2	,	,	PUNCT
ejpam-2826	36	3	p.g.patil	p.g.patil	PROPN
ejpam-2826	36	4	,	,	PUNCT
ejpam-2826	36	5	j.b.toranagatti	j.b.toranagatti	PROPN
ejpam-2826	36	6	,	,	PUNCT
ejpam-2826	36	7	s.r.vighneshi	s.r.vighneshi	PROPN
ejpam-2826	36	8	/	/	SYM
ejpam-2826	36	9	eur	eur	PROPN
ejpam-2826	36	10	.	.	PUNCT
ejpam-2826	37	1	j.	j.	PROPN
ejpam-2826	37	2	pure	pure	PROPN
ejpam-2826	37	3	appl	appl	PROPN
ejpam-2826	37	4	.	.	PROPN
ejpam-2826	37	5	math	math	PROPN
ejpam-2826	37	6	,	,	PUNCT
ejpam-2826	37	7	10	10	NUM
ejpam-2826	37	8	(	(	PUNCT
ejpam-2826	37	9	2	2	NUM
ejpam-2826	37	10	)	)	PUNCT
ejpam-2826	37	11	(	(	PUNCT
ejpam-2826	37	12	2017	2017	NUM
ejpam-2826	37	13	)	)	PUNCT
ejpam-2826	37	14	,	,	PUNCT
ejpam-2826	37	15	312	312	NUM
ejpam-2826	37	16	-	-	SYM
ejpam-2826	37	17	322	322	NUM
ejpam-2826	37	18	314	314	NUM
ejpam-2826	37	19	example	example	NOUN
ejpam-2826	37	20	1	1	NUM
ejpam-2826	37	21	.	.	PUNCT
ejpam-2826	38	1	let	let	VERB
ejpam-2826	38	2	x	x	X
ejpam-2826	38	3	=	=	VERB
ejpam-2826	38	4	y={a	y={a	PROPN
ejpam-2826	38	5	,	,	PUNCT
ejpam-2826	38	6	b	b	NOUN
ejpam-2826	38	7	,	,	PUNCT
ejpam-2826	38	8	c	c	NOUN
ejpam-2826	38	9	}	}	PUNCT
ejpam-2826	38	10	.	.	PUNCT
ejpam-2826	39	1	let	let	VERB
ejpam-2826	39	2	τ={x	τ={x	NOUN
ejpam-2826	39	3	,	,	PUNCT
ejpam-2826	39	4	φ,{a	φ,{a	PROPN
ejpam-2826	39	5	}	}	PUNCT
ejpam-2826	39	6	}	}	PUNCT
ejpam-2826	39	7	and	and	CCONJ
ejpam-2826	39	8	σ={x	σ={x	ADJ
ejpam-2826	39	9	,	,	PUNCT
ejpam-2826	39	10	φ,{a},{b},{a	φ,{a},{b},{a	ADV
ejpam-2826	39	11	,	,	PUNCT
ejpam-2826	39	12	b	b	NOUN
ejpam-2826	39	13	}	}	PUNCT
ejpam-2826	39	14	}	}	PUNCT
ejpam-2826	39	15	be	be	AUX
ejpam-2826	39	16	topologies	topology	NOUN
ejpam-2826	39	17	on	on	ADP
ejpam-2826	39	18	x	x	PUNCT
ejpam-2826	39	19	and	and	CCONJ
ejpam-2826	39	20	y	y	PROPN
ejpam-2826	39	21	respectively	respectively	ADV
ejpam-2826	39	22	.	.	PUNCT
ejpam-2826	40	1	then	then	ADV
ejpam-2826	40	2	the	the	DET
ejpam-2826	40	3	identity	identity	NOUN
ejpam-2826	40	4	function	function	NOUN
ejpam-2826	40	5	f	f	X
ejpam-2826	40	6	:	:	PUNCT
ejpam-2826	40	7	x→y	x→y	NUM
ejpam-2826	40	8	is	be	AUX
ejpam-2826	40	9	contra	contra	PROPN
ejpam-2826	40	10	δgb	δgb	ADV
ejpam-2826	40	11	-	-	PUNCT
ejpam-2826	40	12	continuous	continuous	ADJ
ejpam-2826	40	13	but	but	CCONJ
ejpam-2826	40	14	neither	neither	CCONJ
ejpam-2826	40	15	contra	contra	PROPN
ejpam-2826	40	16	b	b	X
ejpam-2826	40	17	-	-	PUNCT
ejpam-2826	40	18	continuous	continuous	ADJ
ejpam-2826	40	19	and	and	CCONJ
ejpam-2826	40	20	nor	nor	CCONJ
ejpam-2826	40	21	contra	contra	PROPN
ejpam-2826	40	22	gb	gb	ADV
ejpam-2826	40	23	-	-	PUNCT
ejpam-2826	40	24	continuous	continuous	ADJ
ejpam-2826	40	25	,	,	PUNCT
ejpam-2826	40	26	since	since	SCONJ
ejpam-2826	40	27	{	{	PUNCT
ejpam-2826	40	28	a	a	PRON
ejpam-2826	40	29	}	}	PUNCT
ejpam-2826	40	30	is	be	AUX
ejpam-2826	40	31	open	open	ADJ
ejpam-2826	40	32	in	in	ADP
ejpam-2826	40	33	y	y	PROPN
ejpam-2826	40	34	but	but	CCONJ
ejpam-2826	40	35	f−1({a})={a	f−1({a})={a	ADJ
ejpam-2826	40	36	}	}	PUNCT
ejpam-2826	40	37	is	be	AUX
ejpam-2826	40	38	not	not	PART
ejpam-2826	40	39	gb	gb	ADV
ejpam-2826	40	40	-	-	PUNCT
ejpam-2826	40	41	closed	closed	ADJ
ejpam-2826	40	42	in	in	ADP
ejpam-2826	40	43	x	x	PUNCT
ejpam-2826	40	44	and	and	CCONJ
ejpam-2826	40	45	hence	hence	ADV
ejpam-2826	40	46	not	not	PART
ejpam-2826	40	47	b	b	NOUN
ejpam-2826	40	48	-	-	PUNCT
ejpam-2826	40	49	closed	closed	ADJ
ejpam-2826	40	50	in	in	ADP
ejpam-2826	40	51	x.	x.	NOUN
ejpam-2826	40	52	theorem	theorem	VERB
ejpam-2826	40	53	3	3	X
ejpam-2826	40	54	.	.	PUNCT
ejpam-2826	41	1	if	if	SCONJ
ejpam-2826	41	2	f	f	X
ejpam-2826	41	3	:	:	PUNCT
ejpam-2826	41	4	x→y	x→y	NUM
ejpam-2826	41	5	is	be	AUX
ejpam-2826	41	6	contra	contra	PROPN
ejpam-2826	41	7	δgb	δgb	PROPN
ejpam-2826	41	8	-	-	PUNCT
ejpam-2826	41	9	continuous	continuous	ADJ
ejpam-2826	41	10	then	then	ADV
ejpam-2826	41	11	it	it	PRON
ejpam-2826	41	12	is	be	AUX
ejpam-2826	41	13	contra	contra	PROPN
ejpam-2826	41	14	rgb	rgb	PROPN
ejpam-2826	41	15	-	-	ADJ
ejpam-2826	41	16	continuous	continuous	ADJ
ejpam-2826	41	17	.	.	PUNCT
ejpam-2826	42	1	proof	proof	NOUN
ejpam-2826	42	2	:	:	PUNCT
ejpam-2826	42	3	follows	follow	VERB
ejpam-2826	42	4	from	from	ADP
ejpam-2826	42	5	the	the	DET
ejpam-2826	42	6	fact	fact	NOUN
ejpam-2826	42	7	that	that	SCONJ
ejpam-2826	42	8	every	every	DET
ejpam-2826	42	9	δgb	δgb	ADV
ejpam-2826	42	10	-	-	PUNCT
ejpam-2826	42	11	closed	close	VERB
ejpam-2826	42	12	set	set	NOUN
ejpam-2826	42	13	is	be	AUX
ejpam-2826	42	14	rgb	rgb	PROPN
ejpam-2826	42	15	-	-	PUNCT
ejpam-2826	42	16	closed	closed	ADJ
ejpam-2826	42	17	.	.	PUNCT
ejpam-2826	43	1	remark	remark	NOUN
ejpam-2826	43	2	2	2	NUM
ejpam-2826	43	3	.	.	PUNCT
ejpam-2826	44	1	the	the	DET
ejpam-2826	44	2	converse	converse	NOUN
ejpam-2826	44	3	of	of	ADP
ejpam-2826	44	4	theorem	theorem	NOUN
ejpam-2826	44	5	3	3	NUM
ejpam-2826	44	6	need	need	AUX
ejpam-2826	44	7	not	not	PART
ejpam-2826	44	8	be	be	AUX
ejpam-2826	44	9	true	true	ADJ
ejpam-2826	44	10	as	as	SCONJ
ejpam-2826	44	11	seen	see	VERB
ejpam-2826	44	12	from	from	ADP
ejpam-2826	44	13	the	the	DET
ejpam-2826	44	14	following	follow	VERB
ejpam-2826	44	15	example	example	NOUN
ejpam-2826	44	16	.	.	PUNCT
ejpam-2826	45	1	example	example	NOUN
ejpam-2826	46	1	2	2	NUM
ejpam-2826	46	2	.	.	PUNCT
ejpam-2826	46	3	let	let	VERB
ejpam-2826	46	4	x	x	X
ejpam-2826	46	5	=	=	VERB
ejpam-2826	46	6	y={a	y={a	PROPN
ejpam-2826	46	7	,	,	PUNCT
ejpam-2826	46	8	b	b	NOUN
ejpam-2826	46	9	,	,	PUNCT
ejpam-2826	46	10	c	c	NOUN
ejpam-2826	46	11	}	}	PUNCT
ejpam-2826	46	12	.	.	PUNCT
ejpam-2826	47	1	let	let	VERB
ejpam-2826	47	2	τ={x	τ={x	NOUN
ejpam-2826	47	3	,	,	PUNCT
ejpam-2826	47	4	φ,{a},{b},{a	φ,{a},{b},{a	ADV
ejpam-2826	47	5	,	,	PUNCT
ejpam-2826	47	6	b	b	NOUN
ejpam-2826	47	7	}	}	PUNCT
ejpam-2826	47	8	}	}	PUNCT
ejpam-2826	47	9	and	and	CCONJ
ejpam-2826	47	10	σ={x	σ={x	ADJ
ejpam-2826	47	11	,	,	PUNCT
ejpam-2826	47	12	φ,{a	φ,{a	PROPN
ejpam-2826	47	13	}	}	PUNCT
ejpam-2826	47	14	}	}	PUNCT
ejpam-2826	47	15	be	be	AUX
ejpam-2826	47	16	topologies	topology	NOUN
ejpam-2826	47	17	on	on	ADP
ejpam-2826	47	18	x	x	PUNCT
ejpam-2826	47	19	and	and	CCONJ
ejpam-2826	48	1	y	y	PROPN
ejpam-2826	48	2	respectively.let	respectively.let	NOUN
ejpam-2826	48	3	f	f	X
ejpam-2826	48	4	:	:	PUNCT
ejpam-2826	48	5	x→y	x→y	NUM
ejpam-2826	48	6	be	be	AUX
ejpam-2826	48	7	a	a	DET
ejpam-2826	48	8	function	function	NOUN
ejpam-2826	48	9	defined	define	VERB
ejpam-2826	48	10	by	by	ADP
ejpam-2826	48	11	f(a)=a	f(a)=a	NOUN
ejpam-2826	48	12	=	=	SYM
ejpam-2826	48	13	f(b	f(b	X
ejpam-2826	48	14	)	)	PUNCT
ejpam-2826	48	15	and	and	CCONJ
ejpam-2826	48	16	f(c)=c	f(c)=c	PROPN
ejpam-2826	48	17	.	.	PUNCT
ejpam-2826	49	1	then	then	ADV
ejpam-2826	49	2	f	f	PROPN
ejpam-2826	49	3	is	be	AUX
ejpam-2826	49	4	contra	contra	PROPN
ejpam-2826	49	5	rgb	rgb	PROPN
ejpam-2826	49	6	-	-	ADJ
ejpam-2826	49	7	continuous	continuous	ADJ
ejpam-2826	49	8	but	but	CCONJ
ejpam-2826	49	9	not	not	PART
ejpam-2826	49	10	contra	contra	PROPN
ejpam-2826	49	11	δgb	δgb	ADV
ejpam-2826	49	12	-	-	PUNCT
ejpam-2826	49	13	continuous	continuous	ADJ
ejpam-2826	49	14	,	,	PUNCT
ejpam-2826	49	15	since	since	SCONJ
ejpam-2826	49	16	{	{	PUNCT
ejpam-2826	49	17	a	a	PRON
ejpam-2826	49	18	}	}	PUNCT
ejpam-2826	49	19	is	be	AUX
ejpam-2826	49	20	open	open	ADJ
ejpam-2826	49	21	in	in	ADP
ejpam-2826	49	22	y	y	PROPN
ejpam-2826	49	23	but	but	CCONJ
ejpam-2826	49	24	f−1({a})={a	f−1({a})={a	ADJ
ejpam-2826	49	25	,	,	PUNCT
ejpam-2826	49	26	b	b	NOUN
ejpam-2826	49	27	}	}	PUNCT
ejpam-2826	49	28	is	be	AUX
ejpam-2826	49	29	not	not	PART
ejpam-2826	49	30	δgb	δgb	ADV
ejpam-2826	49	31	-	-	PUNCT
ejpam-2826	49	32	closed	closed	ADJ
ejpam-2826	49	33	in	in	ADP
ejpam-2826	49	34	x	x	X
ejpam-2826	49	35	.	.	PUNCT
ejpam-2826	50	1	theorem	theorem	ADJ
ejpam-2826	50	2	4	4	NUM
ejpam-2826	50	3	.	.	PUNCT
ejpam-2826	51	1	let	let	VERB
ejpam-2826	51	2	f	f	X
ejpam-2826	51	3	:	:	PUNCT
ejpam-2826	51	4	x→y	x→y	NUM
ejpam-2826	51	5	be	be	AUX
ejpam-2826	51	6	a	a	DET
ejpam-2826	51	7	function	function	NOUN
ejpam-2826	51	8	.	.	PUNCT
ejpam-2826	52	1	(	(	PUNCT
ejpam-2826	52	2	i	i	NOUN
ejpam-2826	52	3	)	)	PUNCT
ejpam-2826	52	4	if	if	SCONJ
ejpam-2826	52	5	x	x	PRON
ejpam-2826	52	6	is	be	AUX
ejpam-2826	52	7	tδgb	tδgb	ADJ
ejpam-2826	52	8	-	-	PUNCT
ejpam-2826	52	9	space	space	NOUN
ejpam-2826	52	10	then	then	ADV
ejpam-2826	52	11	f	f	PROPN
ejpam-2826	52	12	is	be	AUX
ejpam-2826	52	13	contra	contra	PROPN
ejpam-2826	52	14	δgb	δgb	PROPN
ejpam-2826	52	15	-	-	PUNCT
ejpam-2826	52	16	continuous	continuous	ADJ
ejpam-2826	52	17	if	if	SCONJ
ejpam-2826	52	18	and	and	CCONJ
ejpam-2826	52	19	only	only	ADV
ejpam-2826	52	20	if	if	SCONJ
ejpam-2826	52	21	it	it	PRON
ejpam-2826	52	22	is	be	AUX
ejpam-2826	52	23	contra	contra	PROPN
ejpam-2826	52	24	continuous	continuous	ADJ
ejpam-2826	52	25	.	.	PUNCT
ejpam-2826	53	1	(	(	PUNCT
ejpam-2826	53	2	ii	ii	NOUN
ejpam-2826	53	3	)	)	PUNCT
ejpam-2826	53	4	if	if	SCONJ
ejpam-2826	53	5	x	x	PRON
ejpam-2826	53	6	is	be	AUX
ejpam-2826	53	7	δgbt	δgbt	NOUN
ejpam-2826	53	8	1	1	NUM
ejpam-2826	53	9	2	2	NUM
ejpam-2826	53	10	-space	-space	NOUN
ejpam-2826	53	11	then	then	ADV
ejpam-2826	53	12	f	f	PROPN
ejpam-2826	53	13	is	be	AUX
ejpam-2826	53	14	contra	contra	PROPN
ejpam-2826	53	15	δgb	δgb	PROPN
ejpam-2826	53	16	-	-	PUNCT
ejpam-2826	53	17	continuous	continuous	ADJ
ejpam-2826	53	18	if	if	SCONJ
ejpam-2826	53	19	and	and	CCONJ
ejpam-2826	53	20	only	only	ADV
ejpam-2826	53	21	if	if	SCONJ
ejpam-2826	53	22	it	it	PRON
ejpam-2826	53	23	is	be	AUX
ejpam-2826	53	24	contra	contra	PROPN
ejpam-2826	53	25	bcontinuous	bcontinuous	NOUN
ejpam-2826	53	26	.	.	PUNCT
ejpam-2826	54	1	proof:(i	proof:(i	PROPN
ejpam-2826	54	2	)	)	PUNCT
ejpam-2826	54	3	suppose	suppose	VERB
ejpam-2826	54	4	x	x	PRON
ejpam-2826	54	5	is	be	AUX
ejpam-2826	54	6	tδgb	tδgb	ADJ
ejpam-2826	54	7	-	-	PUNCT
ejpam-2826	54	8	space	space	NOUN
ejpam-2826	54	9	and	and	CCONJ
ejpam-2826	54	10	f	f	PROPN
ejpam-2826	54	11	is	be	AUX
ejpam-2826	54	12	contra	contra	PROPN
ejpam-2826	54	13	δgb	δgb	PROPN
ejpam-2826	54	14	-	-	PUNCT
ejpam-2826	54	15	continuous	continuous	ADJ
ejpam-2826	54	16	.	.	PUNCT
ejpam-2826	55	1	let	let	VERB
ejpam-2826	55	2	g	g	PRON
ejpam-2826	55	3	be	be	AUX
ejpam-2826	55	4	an	an	DET
ejpam-2826	55	5	open	open	ADJ
ejpam-2826	55	6	set	set	NOUN
ejpam-2826	55	7	in	in	ADP
ejpam-2826	55	8	y.	y.	PROPN
ejpam-2826	55	9	then	then	ADV
ejpam-2826	55	10	by	by	ADP
ejpam-2826	55	11	hypothesis	hypothesis	NOUN
ejpam-2826	55	12	f−1(g	f−1(g	PROPN
ejpam-2826	55	13	)	)	PUNCT
ejpam-2826	55	14	is	be	AUX
ejpam-2826	55	15	δgb	δgb	ADV
ejpam-2826	55	16	-	-	PUNCT
ejpam-2826	55	17	closed	closed	ADJ
ejpam-2826	55	18	in	in	ADP
ejpam-2826	55	19	x	x	X
ejpam-2826	55	20	and	and	CCONJ
ejpam-2826	55	21	hence	hence	ADV
ejpam-2826	55	22	f−1(g	f−1(g	PROPN
ejpam-2826	55	23	)	)	PUNCT
ejpam-2826	55	24	is	be	AUX
ejpam-2826	55	25	closed	close	VERB
ejpam-2826	55	26	in	in	ADP
ejpam-2826	55	27	x.therefore	x.therefore	PROPN
ejpam-2826	55	28	f	f	PROPN
ejpam-2826	55	29	is	be	AUX
ejpam-2826	55	30	contra	contra	PROPN
ejpam-2826	55	31	continuous	continuous	ADJ
ejpam-2826	55	32	.	.	PUNCT
ejpam-2826	56	1	converse	converse	NOUN
ejpam-2826	56	2	is	be	AUX
ejpam-2826	56	3	obvious	obvious	ADJ
ejpam-2826	56	4	.	.	PUNCT
ejpam-2826	57	1	(	(	PUNCT
ejpam-2826	57	2	ii)suppose	ii)suppose	NOUN
ejpam-2826	57	3	x	x	VERB
ejpam-2826	57	4	is	be	AUX
ejpam-2826	57	5	δgbt	δgbt	NOUN
ejpam-2826	57	6	1	1	NUM
ejpam-2826	57	7	2	2	NUM
ejpam-2826	57	8	-space	-space	NOUN
ejpam-2826	57	9	and	and	CCONJ
ejpam-2826	57	10	f	f	PROPN
ejpam-2826	57	11	is	be	AUX
ejpam-2826	57	12	contra	contra	PROPN
ejpam-2826	57	13	δgb	δgb	PROPN
ejpam-2826	57	14	-	-	PUNCT
ejpam-2826	57	15	continuous	continuous	ADJ
ejpam-2826	57	16	.	.	PUNCT
ejpam-2826	58	1	let	let	VERB
ejpam-2826	58	2	g	g	PRON
ejpam-2826	58	3	be	be	AUX
ejpam-2826	58	4	an	an	DET
ejpam-2826	58	5	open	open	ADJ
ejpam-2826	58	6	set	set	NOUN
ejpam-2826	58	7	in	in	ADP
ejpam-2826	58	8	y	y	PROPN
ejpam-2826	58	9	then	then	ADV
ejpam-2826	58	10	f−1(g	f−1(g	PROPN
ejpam-2826	58	11	)	)	PUNCT
ejpam-2826	58	12	is	be	AUX
ejpam-2826	58	13	δgb	δgb	ADV
ejpam-2826	58	14	-	-	PUNCT
ejpam-2826	58	15	closed	closed	ADJ
ejpam-2826	58	16	in	in	ADP
ejpam-2826	58	17	x	x	X
ejpam-2826	58	18	and	and	CCONJ
ejpam-2826	58	19	hence	hence	ADV
ejpam-2826	58	20	f−1(g	f−1(g	PROPN
ejpam-2826	58	21	)	)	PUNCT
ejpam-2826	58	22	is	be	AUX
ejpam-2826	58	23	b	b	NOUN
ejpam-2826	58	24	-	-	PUNCT
ejpam-2826	58	25	closed	closed	ADJ
ejpam-2826	58	26	in	in	ADP
ejpam-2826	58	27	x.therefore	x.therefore	PROPN
ejpam-2826	59	1	f	f	PROPN
ejpam-2826	59	2	is	be	AUX
ejpam-2826	59	3	contra	contra	PROPN
ejpam-2826	59	4	b	b	PROPN
ejpam-2826	59	5	-	-	PUNCT
ejpam-2826	59	6	continuous	continuous	ADJ
ejpam-2826	59	7	.	.	PUNCT
ejpam-2826	60	1	converse	converse	NOUN
ejpam-2826	60	2	is	be	AUX
ejpam-2826	60	3	follows	follow	VERB
ejpam-2826	60	4	from	from	ADP
ejpam-2826	60	5	the	the	DET
ejpam-2826	60	6	theorem	theorem	ADJ
ejpam-2826	60	7	2	2	NUM
ejpam-2826	60	8	.	.	PUNCT
ejpam-2826	60	9	theorem	theorem	NOUN
ejpam-2826	60	10	5	5	NUM
ejpam-2826	60	11	.	.	PUNCT
ejpam-2826	61	1	[	[	X
ejpam-2826	61	2	5	5	NUM
ejpam-2826	61	3	]	]	PUNCT
ejpam-2826	61	4	let	let	VERB
ejpam-2826	61	5	a⊆x.then	a⊆x.then	ADV
ejpam-2826	61	6	x	x	SYM
ejpam-2826	61	7	∈	∈	PROPN
ejpam-2826	61	8	δgbcl(a	δgbcl(a	PROPN
ejpam-2826	61	9	)	)	PUNCT
ejpam-2826	62	1	if	if	SCONJ
ejpam-2826	62	2	and	and	CCONJ
ejpam-2826	62	3	only	only	ADV
ejpam-2826	62	4	if	if	SCONJ
ejpam-2826	62	5	u∩a	u∩a	PROPN
ejpam-2826	62	6	6=	6=	NUM
ejpam-2826	62	7	φ	φ	PROPN
ejpam-2826	62	8	,	,	PUNCT
ejpam-2826	62	9	for	for	ADP
ejpam-2826	62	10	every	every	DET
ejpam-2826	62	11	δgb	δgb	ADJ
ejpam-2826	62	12	-	-	PUNCT
ejpam-2826	62	13	open	open	ADJ
ejpam-2826	62	14	set	set	NOUN
ejpam-2826	62	15	u	u	NOUN
ejpam-2826	62	16	containing	contain	VERB
ejpam-2826	62	17	x.	x.	NOUN
ejpam-2826	62	18	lemma	lemma	PROPN
ejpam-2826	62	19	1	1	NUM
ejpam-2826	62	20	.	.	PUNCT
ejpam-2826	63	1	[	[	X
ejpam-2826	63	2	8	8	X
ejpam-2826	63	3	]	]	X
ejpam-2826	63	4	the	the	DET
ejpam-2826	63	5	following	follow	VERB
ejpam-2826	63	6	properties	property	NOUN
ejpam-2826	63	7	are	be	AUX
ejpam-2826	63	8	hold	hold	ADJ
ejpam-2826	63	9	for	for	ADP
ejpam-2826	63	10	subsets	subset	NOUN
ejpam-2826	63	11	a	a	PRON
ejpam-2826	63	12	and	and	CCONJ
ejpam-2826	63	13	b	b	NOUN
ejpam-2826	63	14	of	of	ADP
ejpam-2826	63	15	a	a	DET
ejpam-2826	63	16	space	space	NOUN
ejpam-2826	63	17	x	x	X
ejpam-2826	63	18	:	:	PUNCT
ejpam-2826	63	19	(	(	PUNCT
ejpam-2826	63	20	i	i	NOUN
ejpam-2826	63	21	)	)	PUNCT
ejpam-2826	63	22	x∈ker(a	x∈ker(a	PROPN
ejpam-2826	63	23	)	)	PUNCT
ejpam-2826	64	1	if	if	SCONJ
ejpam-2826	64	2	and	and	CCONJ
ejpam-2826	64	3	only	only	ADV
ejpam-2826	64	4	if	if	SCONJ
ejpam-2826	64	5	a∩f	a∩f	PROPN
ejpam-2826	64	6	=	=	PROPN
ejpam-2826	64	7	φ	φ	NOUN
ejpam-2826	64	8	for	for	ADP
ejpam-2826	64	9	any	any	DET
ejpam-2826	64	10	closed	closed	ADJ
ejpam-2826	64	11	set	set	NOUN
ejpam-2826	64	12	f	f	PROPN
ejpam-2826	64	13	of	of	ADP
ejpam-2826	64	14	x	x	SYM
ejpam-2826	64	15	containing	contain	VERB
ejpam-2826	64	16	x.	x.	NOUN
ejpam-2826	64	17	(	(	PUNCT
ejpam-2826	64	18	ii	ii	NOUN
ejpam-2826	64	19	)	)	PUNCT
ejpam-2826	64	20	a⊆ker(a	a⊆ker(a	PUNCT
ejpam-2826	64	21	)	)	PUNCT
ejpam-2826	64	22	and	and	CCONJ
ejpam-2826	64	23	a	a	DET
ejpam-2826	64	24	=	=	X
ejpam-2826	64	25	ker(a	ker(a	PROPN
ejpam-2826	64	26	)	)	PUNCT
ejpam-2826	64	27	if	if	SCONJ
ejpam-2826	64	28	a	a	PRON
ejpam-2826	64	29	is	be	AUX
ejpam-2826	64	30	open	open	ADJ
ejpam-2826	64	31	in	in	ADP
ejpam-2826	64	32	x.	x.	PROPN
ejpam-2826	64	33	(	(	PUNCT
ejpam-2826	64	34	iii	iii	NOUN
ejpam-2826	64	35	)	)	PUNCT
ejpam-2826	64	36	if	if	SCONJ
ejpam-2826	64	37	a⊆b	a⊆b	PROPN
ejpam-2826	64	38	then	then	ADV
ejpam-2826	64	39	ker(a)⊆ker(b	ker(a)⊆ker(b	PROPN
ejpam-2826	64	40	)	)	PUNCT
ejpam-2826	64	41	.	.	PUNCT
ejpam-2826	65	1	theorem	theorem	NOUN
ejpam-2826	65	2	6	6	NUM
ejpam-2826	65	3	.	.	PUNCT
ejpam-2826	65	4	suppose	suppose	VERB
ejpam-2826	65	5	that	that	SCONJ
ejpam-2826	65	6	δgbc(x	δgbc(x	NOUN
ejpam-2826	65	7	)	)	PUNCT
ejpam-2826	65	8	is	be	AUX
ejpam-2826	65	9	closed	close	VERB
ejpam-2826	65	10	under	under	ADP
ejpam-2826	65	11	arbitrary	arbitrary	ADJ
ejpam-2826	65	12	intersections.then	intersections.then	ADP
ejpam-2826	65	13	the	the	DET
ejpam-2826	65	14	following	follow	VERB
ejpam-2826	65	15	are	be	AUX
ejpam-2826	65	16	equivalent	equivalent	ADJ
ejpam-2826	65	17	for	for	ADP
ejpam-2826	65	18	a	a	DET
ejpam-2826	65	19	function	function	NOUN
ejpam-2826	65	20	f	f	NOUN
ejpam-2826	65	21	:	:	PUNCT
ejpam-2826	65	22	x→y	x→y	NUM
ejpam-2826	65	23	:	:	PUNCT
ejpam-2826	65	24	(	(	PUNCT
ejpam-2826	65	25	i	i	NOUN
ejpam-2826	65	26	)	)	PUNCT
ejpam-2826	65	27	f	f	PROPN
ejpam-2826	65	28	is	be	AUX
ejpam-2826	65	29	contra	contra	PROPN
ejpam-2826	65	30	δgb	δgb	PROPN
ejpam-2826	65	31	-	-	PUNCT
ejpam-2826	65	32	continuous	continuous	ADJ
ejpam-2826	65	33	s.s.benchalli	s.s.benchalli	PROPN
ejpam-2826	65	34	,	,	PUNCT
ejpam-2826	65	35	p.g.patil	p.g.patil	PROPN
ejpam-2826	65	36	,	,	PUNCT
ejpam-2826	65	37	j.b.toranagatti	j.b.toranagatti	PROPN
ejpam-2826	65	38	,	,	PUNCT
ejpam-2826	65	39	s.r.vighneshi	s.r.vighneshi	PROPN
ejpam-2826	65	40	/	/	SYM
ejpam-2826	65	41	eur	eur	PROPN
ejpam-2826	65	42	.	.	PUNCT
ejpam-2826	66	1	j.	j.	PROPN
ejpam-2826	66	2	pure	pure	PROPN
ejpam-2826	66	3	appl	appl	PROPN
ejpam-2826	66	4	.	.	PROPN
ejpam-2826	66	5	math	math	PROPN
ejpam-2826	66	6	,	,	PUNCT
ejpam-2826	66	7	10	10	NUM
ejpam-2826	66	8	(	(	PUNCT
ejpam-2826	66	9	2	2	NUM
ejpam-2826	66	10	)	)	PUNCT
ejpam-2826	66	11	(	(	PUNCT
ejpam-2826	66	12	2017	2017	NUM
ejpam-2826	66	13	)	)	PUNCT
ejpam-2826	66	14	,	,	PUNCT
ejpam-2826	66	15	312	312	NUM
ejpam-2826	66	16	-	-	SYM
ejpam-2826	66	17	322	322	NUM
ejpam-2826	66	18	315	315	NUM
ejpam-2826	66	19	(	(	PUNCT
ejpam-2826	66	20	ii	ii	NOUN
ejpam-2826	66	21	)	)	PUNCT
ejpam-2826	66	22	for	for	ADP
ejpam-2826	66	23	each	each	DET
ejpam-2826	66	24	x	x	SYM
ejpam-2826	66	25	∈	∈	PROPN
ejpam-2826	66	26	x	x	X
ejpam-2826	66	27	and	and	CCONJ
ejpam-2826	66	28	each	each	PRON
ejpam-2826	66	29	closed	close	VERB
ejpam-2826	67	1	set	set	ADJ
ejpam-2826	67	2	b	b	PROPN
ejpam-2826	67	3	of	of	ADP
ejpam-2826	67	4	y	y	PROPN
ejpam-2826	67	5	containing	contain	VERB
ejpam-2826	67	6	f(x	f(x	PROPN
ejpam-2826	67	7	)	)	PUNCT
ejpam-2826	67	8	there	there	PRON
ejpam-2826	67	9	exists	exist	VERB
ejpam-2826	67	10	an	an	DET
ejpam-2826	67	11	δgb	δgb	ADV
ejpam-2826	67	12	-	-	PUNCT
ejpam-2826	67	13	open	open	NOUN
ejpam-2826	67	14	set	set	NOUN
ejpam-2826	67	15	a	a	PRON
ejpam-2826	67	16	of	of	ADP
ejpam-2826	67	17	x	x	PUNCT
ejpam-2826	67	18	containing	contain	VERB
ejpam-2826	67	19	x	x	PUNCT
ejpam-2826	68	1	such	such	ADJ
ejpam-2826	68	2	that	that	SCONJ
ejpam-2826	68	3	f(a)⊆b	f(a)⊆b	PROPN
ejpam-2826	68	4	(	(	PUNCT
ejpam-2826	68	5	iii	iii	NOUN
ejpam-2826	68	6	)	)	PUNCT
ejpam-2826	68	7	for	for	ADP
ejpam-2826	68	8	each	each	DET
ejpam-2826	68	9	x∈x	x∈x	NOUN
ejpam-2826	68	10	and	and	CCONJ
ejpam-2826	68	11	each	each	DET
ejpam-2826	68	12	open	open	ADJ
ejpam-2826	68	13	set	set	VERB
ejpam-2826	68	14	g	g	NOUN
ejpam-2826	68	15	of	of	ADP
ejpam-2826	68	16	y	y	PROPN
ejpam-2826	68	17	not	not	PART
ejpam-2826	68	18	containing	contain	VERB
ejpam-2826	68	19	f(x	f(x	PROPN
ejpam-2826	68	20	)	)	PUNCT
ejpam-2826	68	21	there	there	PRON
ejpam-2826	68	22	exists	exist	VERB
ejpam-2826	68	23	an	an	DET
ejpam-2826	68	24	δgb	δgb	ADV
ejpam-2826	68	25	-	-	PUNCT
ejpam-2826	68	26	closed	close	VERB
ejpam-2826	68	27	set	set	ADJ
ejpam-2826	68	28	h	h	NOUN
ejpam-2826	68	29	in	in	ADP
ejpam-2826	68	30	x	x	PUNCT
ejpam-2826	68	31	not	not	PART
ejpam-2826	68	32	containing	contain	VERB
ejpam-2826	68	33	x	x	PUNCT
ejpam-2826	68	34	such	such	ADJ
ejpam-2826	68	35	that	that	SCONJ
ejpam-2826	68	36	f−1(g)⊆h	f−1(g)⊆h	PROPN
ejpam-2826	68	37	(	(	PUNCT
ejpam-2826	68	38	iv	iv	X
ejpam-2826	68	39	)	)	PUNCT
ejpam-2826	68	40	f(δgbcl(a))⊆ker(f(a	f(δgbcl(a))⊆ker(f(a	NOUN
ejpam-2826	68	41	)	)	PUNCT
ejpam-2826	68	42	)	)	PUNCT
ejpam-2826	68	43	for	for	ADP
ejpam-2826	68	44	every	every	DET
ejpam-2826	68	45	subset	subset	NOUN
ejpam-2826	68	46	a	a	PRON
ejpam-2826	68	47	of	of	ADP
ejpam-2826	68	48	x	x	SYM
ejpam-2826	68	49	(	(	PUNCT
ejpam-2826	68	50	v	v	NOUN
ejpam-2826	68	51	)	)	PUNCT
ejpam-2826	68	52	δgbcl(f−1(b))⊆f−1(ker(b	δgbcl(f−1(b))⊆f−1(ker(b	NOUN
ejpam-2826	68	53	)	)	PUNCT
ejpam-2826	68	54	)	)	PUNCT
ejpam-2826	68	55	for	for	ADP
ejpam-2826	68	56	every	every	DET
ejpam-2826	68	57	subset	subset	NOUN
ejpam-2826	68	58	b	b	PROPN
ejpam-2826	68	59	of	of	ADP
ejpam-2826	68	60	y.	y.	PROPN
ejpam-2826	68	61	proof:(i)→(ii	proof:(i)→(ii	PROPN
ejpam-2826	68	62	)	)	PUNCT
ejpam-2826	68	63	let	let	VERB
ejpam-2826	68	64	b	b	X
ejpam-2826	68	65	be	be	AUX
ejpam-2826	68	66	a	a	DET
ejpam-2826	68	67	closed	closed	ADJ
ejpam-2826	68	68	set	set	NOUN
ejpam-2826	68	69	in	in	ADP
ejpam-2826	68	70	y	y	NOUN
ejpam-2826	68	71	containing	contain	VERB
ejpam-2826	68	72	f(x	f(x	PROPN
ejpam-2826	68	73	)	)	PUNCT
ejpam-2826	68	74	then	then	ADV
ejpam-2826	68	75	x	x	PUNCT
ejpam-2826	68	76	∈f−1(b	∈f−1(b	PROPN
ejpam-2826	68	77	)	)	PUNCT
ejpam-2826	68	78	.	.	PUNCT
ejpam-2826	69	1	by	by	ADP
ejpam-2826	69	2	(	(	PUNCT
ejpam-2826	69	3	i),f−1(b	i),f−1(b	PROPN
ejpam-2826	69	4	)	)	PUNCT
ejpam-2826	69	5	is	be	AUX
ejpam-2826	69	6	δgb	δgb	ADV
ejpam-2826	69	7	-	-	PUNCT
ejpam-2826	69	8	open	open	ADJ
ejpam-2826	69	9	set	set	NOUN
ejpam-2826	69	10	in	in	ADP
ejpam-2826	69	11	x	x	PUNCT
ejpam-2826	69	12	containing	contain	VERB
ejpam-2826	69	13	x.	x.	NOUN
ejpam-2826	69	14	let	let	VERB
ejpam-2826	69	15	a	a	DET
ejpam-2826	69	16	=	=	NOUN
ejpam-2826	69	17	f−1(f	f−1(f	NOUN
ejpam-2826	69	18	)	)	PUNCT
ejpam-2826	70	1	then	then	ADV
ejpam-2826	70	2	f(a)=f(f−1(b))⊆b	f(a)=f(f−1(b))⊆b	PROPN
ejpam-2826	70	3	.	.	PUNCT
ejpam-2826	71	1	(	(	PUNCT
ejpam-2826	71	2	ii)→(i	ii)→(i	NOUN
ejpam-2826	71	3	)	)	PUNCT
ejpam-2826	71	4	let	let	VERB
ejpam-2826	71	5	f	f	PRON
ejpam-2826	71	6	be	be	AUX
ejpam-2826	71	7	a	a	DET
ejpam-2826	71	8	closed	closed	ADJ
ejpam-2826	71	9	set	set	NOUN
ejpam-2826	71	10	in	in	ADP
ejpam-2826	71	11	y	y	NOUN
ejpam-2826	71	12	containing	contain	VERB
ejpam-2826	71	13	f(x	f(x	PROPN
ejpam-2826	71	14	)	)	PUNCT
ejpam-2826	71	15	then	then	ADV
ejpam-2826	71	16	x	x	X
ejpam-2826	71	17	∈f−1(f	∈f−1(f	ADJ
ejpam-2826	71	18	)	)	PUNCT
ejpam-2826	71	19	.	.	PUNCT
ejpam-2826	72	1	from	from	ADP
ejpam-2826	72	2	(	(	PUNCT
ejpam-2826	72	3	ii	ii	NOUN
ejpam-2826	72	4	)	)	PUNCT
ejpam-2826	72	5	,	,	PUNCT
ejpam-2826	72	6	there	there	PRON
ejpam-2826	72	7	exists	exist	VERB
ejpam-2826	72	8	δgb	δgb	ADJ
ejpam-2826	72	9	-	-	PUNCT
ejpam-2826	72	10	open	open	ADJ
ejpam-2826	72	11	set	set	VERB
ejpam-2826	72	12	gx	gx	PROPN
ejpam-2826	72	13	in	in	ADP
ejpam-2826	72	14	x	x	PUNCT
ejpam-2826	72	15	containing	contain	VERB
ejpam-2826	72	16	x	x	PUNCT
ejpam-2826	72	17	such	such	ADJ
ejpam-2826	72	18	that	that	SCONJ
ejpam-2826	72	19	f(gx)⊂f	f(gx)⊂f	NOUN
ejpam-2826	72	20	which	which	PRON
ejpam-2826	72	21	implies	imply	VERB
ejpam-2826	72	22	gx⊆f−1(f	gx⊆f−1(f	PROPN
ejpam-2826	72	23	)	)	PUNCT
ejpam-2826	72	24	.thus	.thus	ADV
ejpam-2826	73	1	f−1(f)=∪{ux	f−1(f)=∪{ux	NOUN
ejpam-2826	73	2	:	:	PUNCT
ejpam-2826	73	3	x	x	X
ejpam-2826	73	4	∈f−1(f	∈f−1(f	NOUN
ejpam-2826	73	5	)	)	PUNCT
ejpam-2826	73	6	}	}	PUNCT
ejpam-2826	73	7	which	which	PRON
ejpam-2826	73	8	is	be	AUX
ejpam-2826	73	9	δgb	δgb	ADV
ejpam-2826	73	10	-	-	PUNCT
ejpam-2826	73	11	open	open	ADJ
ejpam-2826	73	12	.	.	PUNCT
ejpam-2826	74	1	hence	hence	ADV
ejpam-2826	74	2	f−1(f	f−1(f	PROPN
ejpam-2826	74	3	)	)	PUNCT
ejpam-2826	74	4	is	be	AUX
ejpam-2826	74	5	δgb	δgb	ADV
ejpam-2826	74	6	-	-	PUNCT
ejpam-2826	74	7	open	open	ADJ
ejpam-2826	74	8	set	set	NOUN
ejpam-2826	74	9	in	in	ADP
ejpam-2826	74	10	x.	x.	PROPN
ejpam-2826	74	11	(	(	PUNCT
ejpam-2826	74	12	ii)→(iii	ii)→(iii	PROPN
ejpam-2826	74	13	)	)	PUNCT
ejpam-2826	74	14	let	let	VERB
ejpam-2826	74	15	g	g	NOUN
ejpam-2826	74	16	be	be	AUX
ejpam-2826	74	17	an	an	DET
ejpam-2826	74	18	open	open	ADJ
ejpam-2826	74	19	set	set	NOUN
ejpam-2826	74	20	in	in	ADP
ejpam-2826	74	21	y	y	PROPN
ejpam-2826	74	22	not	not	PART
ejpam-2826	74	23	containing	contain	VERB
ejpam-2826	74	24	f(x	f(x	PROPN
ejpam-2826	74	25	)	)	PUNCT
ejpam-2826	74	26	.	.	PUNCT
ejpam-2826	75	1	then	then	ADV
ejpam-2826	75	2	y	y	PROPN
ejpam-2826	75	3	-	-	PUNCT
ejpam-2826	75	4	g	g	PROPN
ejpam-2826	75	5	is	be	AUX
ejpam-2826	75	6	closed	close	VERB
ejpam-2826	75	7	set	set	VERB
ejpam-2826	75	8	in	in	ADP
ejpam-2826	75	9	y	y	NOUN
ejpam-2826	75	10	containing	contain	VERB
ejpam-2826	75	11	f(x	f(x	PROPN
ejpam-2826	75	12	)	)	PUNCT
ejpam-2826	75	13	.	.	PUNCT
ejpam-2826	76	1	from	from	ADP
ejpam-2826	76	2	(	(	PUNCT
ejpam-2826	76	3	ii	ii	NOUN
ejpam-2826	76	4	)	)	PUNCT
ejpam-2826	76	5	,	,	PUNCT
ejpam-2826	76	6	there	there	PRON
ejpam-2826	76	7	exists	exist	VERB
ejpam-2826	76	8	a	a	DET
ejpam-2826	76	9	δgb	δgb	ADV
ejpam-2826	76	10	-	-	PUNCT
ejpam-2826	76	11	open	open	ADJ
ejpam-2826	76	12	set	set	NOUN
ejpam-2826	76	13	f	f	PROPN
ejpam-2826	76	14	in	in	ADP
ejpam-2826	76	15	x	x	PUNCT
ejpam-2826	76	16	containing	contain	VERB
ejpam-2826	76	17	x	x	PUNCT
ejpam-2826	76	18	such	such	ADJ
ejpam-2826	76	19	that	that	SCONJ
ejpam-2826	76	20	f(f)⊆y	f(f)⊆y	NOUN
ejpam-2826	76	21	-	-	PUNCT
ejpam-2826	76	22	g	g	PROPN
ejpam-2826	76	23	.	.	PUNCT
ejpam-2826	77	1	this	this	PRON
ejpam-2826	77	2	implies	imply	VERB
ejpam-2826	77	3	f⊆f−1(y	f⊆f−1(y	PROPN
ejpam-2826	77	4	-	-	PUNCT
ejpam-2826	77	5	g)=x	g)=x	NOUN
ejpam-2826	77	6	-	-	PUNCT
ejpam-2826	77	7	f−1(g	f−1(g	NOUN
ejpam-2826	77	8	)	)	PUNCT
ejpam-2826	77	9	.	.	PUNCT
ejpam-2826	78	1	hence	hence	ADV
ejpam-2826	78	2	f−1(g)⊆x	f−1(g)⊆x	PROPN
ejpam-2826	78	3	-	-	PUNCT
ejpam-2826	78	4	f.	f.	PROPN
ejpam-2826	78	5	set	set	VERB
ejpam-2826	78	6	h	h	NOUN
ejpam-2826	78	7	=	=	NOUN
ejpam-2826	78	8	x	x	X
ejpam-2826	78	9	-	-	PUNCT
ejpam-2826	78	10	f	f	X
ejpam-2826	78	11	,	,	PUNCT
ejpam-2826	78	12	then	then	ADV
ejpam-2826	78	13	h	h	PROPN
ejpam-2826	78	14	is	be	AUX
ejpam-2826	78	15	δgb	δgb	ADV
ejpam-2826	78	16	-	-	PUNCT
ejpam-2826	78	17	closed	close	VERB
ejpam-2826	78	18	set	set	NOUN
ejpam-2826	78	19	not	not	PART
ejpam-2826	78	20	containing	contain	VERB
ejpam-2826	78	21	x	x	PUNCT
ejpam-2826	78	22	in	in	ADP
ejpam-2826	78	23	x	x	X
ejpam-2826	78	24	such	such	ADJ
ejpam-2826	78	25	that	that	DET
ejpam-2826	78	26	f−1(g	f−1(g	PROPN
ejpam-2826	78	27	)	)	PUNCT
ejpam-2826	78	28	⊆h	⊆h	PROPN
ejpam-2826	78	29	.	.	PUNCT
ejpam-2826	79	1	(	(	PUNCT
ejpam-2826	79	2	iii)→(ii	iii)→(ii	PROPN
ejpam-2826	79	3	)	)	PUNCT
ejpam-2826	79	4	let	let	VERB
ejpam-2826	79	5	f	f	PRON
ejpam-2826	79	6	be	be	AUX
ejpam-2826	79	7	a	a	DET
ejpam-2826	79	8	closed	closed	ADJ
ejpam-2826	79	9	set	set	NOUN
ejpam-2826	79	10	in	in	ADP
ejpam-2826	79	11	y	y	NOUN
ejpam-2826	79	12	containing	contain	VERB
ejpam-2826	79	13	f(x	f(x	PROPN
ejpam-2826	79	14	)	)	PUNCT
ejpam-2826	79	15	.	.	PUNCT
ejpam-2826	80	1	then	then	ADV
ejpam-2826	80	2	y	y	PROPN
ejpam-2826	80	3	-	-	PUNCT
ejpam-2826	80	4	f	f	PROPN
ejpam-2826	80	5	is	be	AUX
ejpam-2826	80	6	an	an	DET
ejpam-2826	80	7	open	open	ADJ
ejpam-2826	80	8	set	set	NOUN
ejpam-2826	80	9	in	in	ADP
ejpam-2826	80	10	y	y	PROPN
ejpam-2826	80	11	not	not	PART
ejpam-2826	80	12	containing	contain	VERB
ejpam-2826	80	13	f(x	f(x	PROPN
ejpam-2826	80	14	)	)	PUNCT
ejpam-2826	80	15	.	.	PUNCT
ejpam-2826	81	1	from	from	ADP
ejpam-2826	81	2	(	(	PUNCT
ejpam-2826	81	3	iii	iii	NOUN
ejpam-2826	81	4	)	)	PUNCT
ejpam-2826	81	5	,	,	PUNCT
ejpam-2826	81	6	there	there	PRON
ejpam-2826	81	7	exists	exist	VERB
ejpam-2826	81	8	δgb	δgb	ADV
ejpam-2826	81	9	-	-	PUNCT
ejpam-2826	81	10	closed	close	VERB
ejpam-2826	81	11	set	set	NOUN
ejpam-2826	81	12	k	k	PROPN
ejpam-2826	81	13	in	in	ADP
ejpam-2826	81	14	x	x	PUNCT
ejpam-2826	81	15	not	not	PART
ejpam-2826	81	16	containing	contain	VERB
ejpam-2826	81	17	x	x	PUNCT
ejpam-2826	81	18	such	such	ADJ
ejpam-2826	81	19	that	that	SCONJ
ejpam-2826	81	20	f−1(y	f−1(y	PROPN
ejpam-2826	81	21	-	-	PUNCT
ejpam-2826	81	22	f)⊆k.this	f)⊆k.this	PROPN
ejpam-2826	81	23	implies	imply	VERB
ejpam-2826	81	24	x	x	PROPN
ejpam-2826	81	25	-	-	PROPN
ejpam-2826	81	26	k	k	PROPN
ejpam-2826	81	27	⊆f−1(f	⊆f−1(f	PROPN
ejpam-2826	81	28	)	)	PUNCT
ejpam-2826	81	29	that	that	PRON
ejpam-2826	81	30	is	be	AUX
ejpam-2826	81	31	f(x	f(x	PROPN
ejpam-2826	81	32	-	-	PUNCT
ejpam-2826	81	33	k)⊆f.set	k)⊆f.set	VERB
ejpam-2826	81	34	u	u	NOUN
ejpam-2826	81	35	=	=	NOUN
ejpam-2826	81	36	x	x	PROPN
ejpam-2826	81	37	-	-	PUNCT
ejpam-2826	81	38	k	k	X
ejpam-2826	81	39	then	then	ADV
ejpam-2826	81	40	u	u	NOUN
ejpam-2826	81	41	is	be	AUX
ejpam-2826	81	42	δgb	δgb	ADV
ejpam-2826	81	43	-	-	PUNCT
ejpam-2826	81	44	open	open	ADJ
ejpam-2826	81	45	set	set	NOUN
ejpam-2826	81	46	containing	contain	VERB
ejpam-2826	81	47	x	x	PUNCT
ejpam-2826	81	48	in	in	ADP
ejpam-2826	81	49	x	x	X
ejpam-2826	81	50	such	such	ADJ
ejpam-2826	81	51	that	that	DET
ejpam-2826	81	52	f(u)⊆f	f(u)⊆f	NOUN
ejpam-2826	81	53	.	.	PUNCT
ejpam-2826	82	1	(	(	PUNCT
ejpam-2826	82	2	i)→(iv	i)→(iv	X
ejpam-2826	82	3	)	)	PUNCT
ejpam-2826	82	4	let	let	VERB
ejpam-2826	82	5	a	a	DET
ejpam-2826	82	6	be	be	AUX
ejpam-2826	82	7	any	any	DET
ejpam-2826	82	8	subset	subset	NOUN
ejpam-2826	82	9	of	of	ADP
ejpam-2826	82	10	x.	x.	PROPN
ejpam-2826	82	11	suppose	suppose	VERB
ejpam-2826	82	12	y	y	PROPN
ejpam-2826	82	13	/∈	/∈	PUNCT
ejpam-2826	82	14	ker(f(a	ker(f(a	PROPN
ejpam-2826	82	15	)	)	PUNCT
ejpam-2826	82	16	)	)	PUNCT
ejpam-2826	82	17	.	.	PUNCT
ejpam-2826	83	1	then	then	ADV
ejpam-2826	83	2	by	by	ADP
ejpam-2826	83	3	lemma	lemma	PROPN
ejpam-2826	83	4	1	1	NUM
ejpam-2826	83	5	,	,	PUNCT
ejpam-2826	83	6	there	there	PRON
ejpam-2826	83	7	exists	exist	VERB
ejpam-2826	83	8	a	a	DET
ejpam-2826	83	9	closed	closed	ADJ
ejpam-2826	83	10	set	set	VERB
ejpam-2826	83	11	f	f	PROPN
ejpam-2826	83	12	in	in	ADP
ejpam-2826	83	13	y	y	PROPN
ejpam-2826	83	14	containing	contain	VERB
ejpam-2826	83	15	y	y	PRON
ejpam-2826	83	16	such	such	ADJ
ejpam-2826	83	17	that	that	DET
ejpam-2826	83	18	f(a)∩f	f(a)∩f	NOUN
ejpam-2826	83	19	=	=	SYM
ejpam-2826	83	20	φ	φ	NOUN
ejpam-2826	83	21	.	.	PUNCT
ejpam-2826	84	1	hence	hence	ADV
ejpam-2826	84	2	we	we	PRON
ejpam-2826	84	3	have	have	VERB
ejpam-2826	84	4	a∩f−1(f)=φ	a∩f−1(f)=φ	NOUN
ejpam-2826	84	5	and	and	CCONJ
ejpam-2826	84	6	δgb	δgb	NOUN
ejpam-2826	84	7	-	-	PUNCT
ejpam-2826	84	8	cl(a)∩f−1(f)=φ	cl(a)∩f−1(f)=φ	PROPN
ejpam-2826	84	9	which	which	PRON
ejpam-2826	84	10	implies	imply	VERB
ejpam-2826	84	11	f(δgbcl(a))∩f	f(δgbcl(a))∩f	PROPN
ejpam-2826	84	12	=	=	NOUN
ejpam-2826	84	13	φ	φ	NOUN
ejpam-2826	84	14	and	and	CCONJ
ejpam-2826	84	15	hence	hence	ADV
ejpam-2826	84	16	y	y	PROPN
ejpam-2826	84	17	/∈	/∈	PUNCT
ejpam-2826	84	18	δgbcl(a	δgbcl(a	PROPN
ejpam-2826	84	19	)	)	PUNCT
ejpam-2826	84	20	.	.	PUNCT
ejpam-2826	85	1	therefore	therefore	ADV
ejpam-2826	85	2	f(δgbcl(a))⊂ker(f(a	f(δgbcl(a))⊂ker(f(a	NOUN
ejpam-2826	85	3	)	)	PUNCT
ejpam-2826	85	4	)	)	PUNCT
ejpam-2826	86	1	(	(	PUNCT
ejpam-2826	86	2	iv)→(v	iv)→(v	NOUN
ejpam-2826	86	3	)	)	PUNCT
ejpam-2826	86	4	let	let	VERB
ejpam-2826	86	5	b⊆y	b⊆y	PROPN
ejpam-2826	86	6	then	then	ADV
ejpam-2826	86	7	f−1(b)⊆x	f−1(b)⊆x	PROPN
ejpam-2826	86	8	.	.	PROPN
ejpam-2826	87	1	by	by	ADP
ejpam-2826	87	2	(	(	PUNCT
ejpam-2826	87	3	iv	iv	X
ejpam-2826	87	4	)	)	PUNCT
ejpam-2826	87	5	,	,	PUNCT
ejpam-2826	87	6	f	f	PROPN
ejpam-2826	87	7	(	(	PUNCT
ejpam-2826	87	8	δgbcl(f−1(b)))⊆	δgbcl(f−1(b)))⊆	PROPN
ejpam-2826	87	9	ker(f	ker(f	PROPN
ejpam-2826	87	10	(	(	PUNCT
ejpam-2826	87	11	f−1(b)))⊆ker(b	f−1(b)))⊆ker(b	PROPN
ejpam-2826	87	12	)	)	PUNCT
ejpam-2826	87	13	.	.	PUNCT
ejpam-2826	88	1	thus	thus	ADV
ejpam-2826	88	2	δgbcl(f−1(b))⊆f−1(ker(b	δgbcl(f−1(b))⊆f−1(ker(b	X
ejpam-2826	88	3	)	)	PUNCT
ejpam-2826	88	4	)	)	PUNCT
ejpam-2826	88	5	.	.	PUNCT
ejpam-2826	89	1	(	(	PUNCT
ejpam-2826	89	2	v)→(i	v)→(i	PROPN
ejpam-2826	89	3	)	)	PUNCT
ejpam-2826	89	4	let	let	VERB
ejpam-2826	89	5	v	v	PART
ejpam-2826	89	6	be	be	AUX
ejpam-2826	89	7	any	any	DET
ejpam-2826	89	8	open	open	ADJ
ejpam-2826	89	9	subset	subset	NOUN
ejpam-2826	89	10	of	of	ADP
ejpam-2826	89	11	y.	y.	PROPN
ejpam-2826	89	12	then	then	ADV
ejpam-2826	89	13	by	by	ADP
ejpam-2826	89	14	(	(	PUNCT
ejpam-2826	89	15	v	v	NOUN
ejpam-2826	89	16	)	)	PUNCT
ejpam-2826	89	17	and	and	CCONJ
ejpam-2826	89	18	lemma	lemma	PROPN
ejpam-2826	89	19	1	1	NUM
ejpam-2826	89	20	,	,	PUNCT
ejpam-2826	89	21	δgbcl(f−1(v)⊆f−1(ker(v))=f−1(v	δgbcl(f−1(v)⊆f−1(ker(v))=f−1(v	NUM
ejpam-2826	89	22	)	)	PUNCT
ejpam-2826	89	23	and	and	CCONJ
ejpam-2826	89	24	δgbcl(f−1(v	δgbcl(f−1(v	NOUN
ejpam-2826	89	25	)	)	PUNCT
ejpam-2826	89	26	)	)	PUNCT
ejpam-2826	90	1	=	=	SYM
ejpam-2826	90	2	f−1(v	f−1(v	NOUN
ejpam-2826	90	3	)	)	PUNCT
ejpam-2826	90	4	.	.	PUNCT
ejpam-2826	91	1	therefore	therefore	ADV
ejpam-2826	91	2	f−1(v	f−1(v	PROPN
ejpam-2826	91	3	)	)	PUNCT
ejpam-2826	91	4	is	be	AUX
ejpam-2826	91	5	δgb	δgb	ADV
ejpam-2826	91	6	-	-	PUNCT
ejpam-2826	91	7	closed	close	VERB
ejpam-2826	91	8	set	set	NOUN
ejpam-2826	91	9	in	in	ADP
ejpam-2826	91	10	x	x	PROPN
ejpam-2826	91	11	lemma	lemma	PROPN
ejpam-2826	91	12	2	2	NUM
ejpam-2826	91	13	.	.	PUNCT
ejpam-2826	92	1	[	[	X
ejpam-2826	92	2	16	16	NUM
ejpam-2826	92	3	]	]	PUNCT
ejpam-2826	92	4	for	for	ADP
ejpam-2826	92	5	a	a	DET
ejpam-2826	92	6	subset	subset	NOUN
ejpam-2826	92	7	a	a	PRON
ejpam-2826	92	8	of	of	ADP
ejpam-2826	92	9	a	a	DET
ejpam-2826	92	10	space	space	NOUN
ejpam-2826	92	11	x	x	NOUN
ejpam-2826	92	12	,	,	PUNCT
ejpam-2826	92	13	the	the	DET
ejpam-2826	92	14	following	follow	VERB
ejpam-2826	92	15	are	be	AUX
ejpam-2826	92	16	equivalent	equivalent	ADJ
ejpam-2826	92	17	:	:	PUNCT
ejpam-2826	92	18	(	(	PUNCT
ejpam-2826	92	19	i	i	NOUN
ejpam-2826	92	20	)	)	PUNCT
ejpam-2826	92	21	a	a	PRON
ejpam-2826	92	22	is	be	AUX
ejpam-2826	92	23	open	open	ADJ
ejpam-2826	92	24	and	and	CCONJ
ejpam-2826	92	25	gb	gb	ADV
ejpam-2826	92	26	-	-	PUNCT
ejpam-2826	92	27	closed	close	VERB
ejpam-2826	92	28	(	(	PUNCT
ejpam-2826	92	29	ii	ii	NOUN
ejpam-2826	92	30	)	)	PUNCT
ejpam-2826	92	31	a	a	PRON
ejpam-2826	92	32	is	be	AUX
ejpam-2826	92	33	regular	regular	ADJ
ejpam-2826	92	34	open	open	ADJ
ejpam-2826	92	35	.	.	PUNCT
ejpam-2826	93	1	theorem	theorem	VERB
ejpam-2826	93	2	7	7	NUM
ejpam-2826	93	3	.	.	PUNCT
ejpam-2826	94	1	[	[	X
ejpam-2826	94	2	4]if	4]if	NUM
ejpam-2826	94	3	a⊆x	a⊆x	NOUN
ejpam-2826	94	4	is	be	AUX
ejpam-2826	94	5	both	both	PRON
ejpam-2826	94	6	δ	δ	NOUN
ejpam-2826	94	7	-	-	ADJ
ejpam-2826	94	8	open	open	ADJ
ejpam-2826	94	9	and	and	CCONJ
ejpam-2826	94	10	δgb	δgb	ADV
ejpam-2826	94	11	-	-	PUNCT
ejpam-2826	94	12	closed	closed	ADJ
ejpam-2826	94	13	then	then	ADV
ejpam-2826	94	14	it	it	PRON
ejpam-2826	94	15	is	be	AUX
ejpam-2826	94	16	b	b	NOUN
ejpam-2826	94	17	-	-	PUNCT
ejpam-2826	94	18	closed	closed	ADJ
ejpam-2826	94	19	.	.	PUNCT
ejpam-2826	95	1	theorem	theorem	VERB
ejpam-2826	95	2	8	8	NUM
ejpam-2826	95	3	.	.	PUNCT
ejpam-2826	96	1	if	if	SCONJ
ejpam-2826	96	2	a⊆x	a⊆x	NOUN
ejpam-2826	96	3	is	be	AUX
ejpam-2826	96	4	regular	regular	ADJ
ejpam-2826	96	5	open	open	ADJ
ejpam-2826	96	6	then	then	ADV
ejpam-2826	96	7	it	it	PRON
ejpam-2826	96	8	is	be	AUX
ejpam-2826	96	9	b	b	NOUN
ejpam-2826	96	10	-	-	PUNCT
ejpam-2826	96	11	closed	closed	ADJ
ejpam-2826	96	12	.	.	PUNCT
ejpam-2826	97	1	lemma	lemma	PROPN
ejpam-2826	97	2	3	3	NUM
ejpam-2826	97	3	.	.	X
ejpam-2826	98	1	for	for	ADP
ejpam-2826	98	2	a	a	DET
ejpam-2826	98	3	subset	subset	NOUN
ejpam-2826	98	4	a	a	PRON
ejpam-2826	98	5	of	of	ADP
ejpam-2826	98	6	a	a	DET
ejpam-2826	98	7	space	space	NOUN
ejpam-2826	98	8	x	x	PUNCT
ejpam-2826	98	9	the	the	DET
ejpam-2826	98	10	following	follow	VERB
ejpam-2826	98	11	are	be	AUX
ejpam-2826	98	12	equivalent	equivalent	ADJ
ejpam-2826	98	13	:	:	PUNCT
ejpam-2826	98	14	(	(	PUNCT
ejpam-2826	98	15	i	i	NOUN
ejpam-2826	98	16	)	)	PUNCT
ejpam-2826	98	17	a	a	PRON
ejpam-2826	98	18	is	be	AUX
ejpam-2826	98	19	δ	δ	NOUN
ejpam-2826	98	20	-	-	ADJ
ejpam-2826	98	21	open	open	ADJ
ejpam-2826	98	22	and	and	CCONJ
ejpam-2826	98	23	δgb	δgb	ADV
ejpam-2826	98	24	-	-	PUNCT
ejpam-2826	98	25	closed	close	VERB
ejpam-2826	98	26	(	(	PUNCT
ejpam-2826	98	27	ii	ii	NOUN
ejpam-2826	98	28	)	)	PUNCT
ejpam-2826	98	29	a	a	PRON
ejpam-2826	98	30	is	be	AUX
ejpam-2826	98	31	regular	regular	ADJ
ejpam-2826	98	32	open	open	ADJ
ejpam-2826	98	33	s.s.benchalli	s.s.benchalli	NOUN
ejpam-2826	98	34	,	,	PUNCT
ejpam-2826	98	35	p.g.patil	p.g.patil	PROPN
ejpam-2826	98	36	,	,	PUNCT
ejpam-2826	98	37	j.b.toranagatti	j.b.toranagatti	PROPN
ejpam-2826	98	38	,	,	PUNCT
ejpam-2826	98	39	s.r.vighneshi	s.r.vighneshi	PROPN
ejpam-2826	98	40	/	/	SYM
ejpam-2826	98	41	eur	eur	PROPN
ejpam-2826	98	42	.	.	PUNCT
ejpam-2826	99	1	j.	j.	PROPN
ejpam-2826	99	2	pure	pure	PROPN
ejpam-2826	99	3	appl	appl	PROPN
ejpam-2826	99	4	.	.	PROPN
ejpam-2826	99	5	math	math	PROPN
ejpam-2826	99	6	,	,	PUNCT
ejpam-2826	99	7	10	10	NUM
ejpam-2826	99	8	(	(	PUNCT
ejpam-2826	99	9	2	2	NUM
ejpam-2826	99	10	)	)	PUNCT
ejpam-2826	99	11	(	(	PUNCT
ejpam-2826	99	12	2017	2017	NUM
ejpam-2826	99	13	)	)	PUNCT
ejpam-2826	99	14	,	,	PUNCT
ejpam-2826	99	15	312	312	NUM
ejpam-2826	99	16	-	-	SYM
ejpam-2826	99	17	322	322	NUM
ejpam-2826	99	18	316	316	NUM
ejpam-2826	99	19	(	(	PUNCT
ejpam-2826	99	20	iii	iii	NOUN
ejpam-2826	99	21	)	)	PUNCT
ejpam-2826	99	22	a	a	PRON
ejpam-2826	99	23	is	be	AUX
ejpam-2826	99	24	open	open	ADJ
ejpam-2826	99	25	and	and	CCONJ
ejpam-2826	99	26	b	b	X
ejpam-2826	99	27	-	-	PUNCT
ejpam-2826	99	28	closed	closed	ADJ
ejpam-2826	99	29	.	.	PUNCT
ejpam-2826	100	1	proof:(i)→(ii):let	proof:(i)→(ii):let	NOUN
ejpam-2826	100	2	a	a	PRON
ejpam-2826	100	3	be	be	AUX
ejpam-2826	100	4	an	an	DET
ejpam-2826	100	5	δ	δ	NOUN
ejpam-2826	100	6	-	-	ADJ
ejpam-2826	100	7	open	open	ADJ
ejpam-2826	100	8	and	and	CCONJ
ejpam-2826	100	9	δgb	δgb	ADV
ejpam-2826	100	10	-	-	PUNCT
ejpam-2826	100	11	closed	closed	ADJ
ejpam-2826	100	12	set.then	set.then	X
ejpam-2826	100	13	by	by	ADP
ejpam-2826	100	14	theorem	theorem	NOUN
ejpam-2826	100	15	7	7	NUM
ejpam-2826	100	16	,	,	PUNCT
ejpam-2826	100	17	a	a	PRON
ejpam-2826	100	18	is	be	AUX
ejpam-2826	100	19	b	b	NOUN
ejpam-2826	100	20	-	-	PUNCT
ejpam-2826	100	21	closed	closed	ADJ
ejpam-2826	100	22	that	that	PRON
ejpam-2826	100	23	is	be	AUX
ejpam-2826	100	24	bcl(a)⊆a	bcl(a)⊆a	NOUN
ejpam-2826	100	25	and	and	CCONJ
ejpam-2826	100	26	so	so	ADV
ejpam-2826	100	27	int(cl(a))⊆a	int(cl(a))⊆a	NOUN
ejpam-2826	100	28	.	.	PUNCT
ejpam-2826	101	1	since	since	SCONJ
ejpam-2826	101	2	a	a	PRON
ejpam-2826	101	3	is	be	AUX
ejpam-2826	101	4	δ	δ	NOUN
ejpam-2826	101	5	-	-	ADJ
ejpam-2826	101	6	open	open	ADJ
ejpam-2826	101	7	then	then	ADV
ejpam-2826	101	8	a	a	PRON
ejpam-2826	101	9	is	be	AUX
ejpam-2826	101	10	pre	pre	ADJ
ejpam-2826	101	11	-	-	ADJ
ejpam-2826	101	12	open	open	ADJ
ejpam-2826	101	13	and	and	CCONJ
ejpam-2826	101	14	thus	thus	ADV
ejpam-2826	101	15	a⊆int(cl(a)).hence	a⊆int(cl(a)).hence	NOUN
ejpam-2826	101	16	a	a	PRON
ejpam-2826	101	17	is	be	AUX
ejpam-2826	101	18	regular	regular	ADJ
ejpam-2826	101	19	open	open	ADJ
ejpam-2826	101	20	.	.	PUNCT
ejpam-2826	102	1	(	(	PUNCT
ejpam-2826	102	2	ii)→(i	ii)→(i	NOUN
ejpam-2826	102	3	):	):	PUNCT
ejpam-2826	102	4	follows	follow	VERB
ejpam-2826	102	5	from	from	ADP
ejpam-2826	102	6	the	the	DET
ejpam-2826	102	7	fact	fact	NOUN
ejpam-2826	102	8	that	that	SCONJ
ejpam-2826	102	9	every	every	DET
ejpam-2826	102	10	regular	regular	ADJ
ejpam-2826	102	11	open	open	ADJ
ejpam-2826	102	12	set	set	NOUN
ejpam-2826	102	13	is	be	AUX
ejpam-2826	102	14	δ	δ	NOUN
ejpam-2826	102	15	-	-	ADJ
ejpam-2826	102	16	open	open	ADJ
ejpam-2826	102	17	and	and	CCONJ
ejpam-2826	102	18	by	by	ADP
ejpam-2826	102	19	theorem	theorem	NOUN
ejpam-2826	102	20	8	8	NUM
ejpam-2826	102	21	.	.	PUNCT
ejpam-2826	103	1	(	(	PUNCT
ejpam-2826	103	2	ii)→(iii	ii)→(iii	PROPN
ejpam-2826	103	3	):	):	PUNCT
ejpam-2826	103	4	follows	follow	VERB
ejpam-2826	103	5	from	from	ADP
ejpam-2826	103	6	the	the	DET
ejpam-2826	103	7	fact	fact	NOUN
ejpam-2826	103	8	that	that	SCONJ
ejpam-2826	103	9	every	every	DET
ejpam-2826	103	10	regular	regular	ADJ
ejpam-2826	103	11	open	open	ADJ
ejpam-2826	103	12	set	set	NOUN
ejpam-2826	103	13	is	be	AUX
ejpam-2826	103	14	open	open	ADJ
ejpam-2826	103	15	and	and	CCONJ
ejpam-2826	103	16	theorem	theorem	VERB
ejpam-2826	103	17	8	8	NUM
ejpam-2826	103	18	.	.	PUNCT
ejpam-2826	103	19	(	(	PUNCT
ejpam-2826	103	20	iii)→(ii	iii)→(ii	PROPN
ejpam-2826	103	21	):	):	PUNCT
ejpam-2826	103	22	let	let	VERB
ejpam-2826	103	23	a	a	PRON
ejpam-2826	103	24	be	be	AUX
ejpam-2826	103	25	an	an	DET
ejpam-2826	103	26	open	open	ADJ
ejpam-2826	103	27	and	and	CCONJ
ejpam-2826	103	28	b	b	NOUN
ejpam-2826	103	29	-	-	PUNCT
ejpam-2826	103	30	closed	closed	ADJ
ejpam-2826	103	31	set	set	NOUN
ejpam-2826	103	32	then	then	ADV
ejpam-2826	103	33	bcl(a)⊆a	bcl(a)⊆a	NOUN
ejpam-2826	103	34	and	and	CCONJ
ejpam-2826	103	35	so	so	ADV
ejpam-2826	103	36	int(cl(a))⊆a	int(cl(a))⊆a	NOUN
ejpam-2826	103	37	.	.	PUNCT
ejpam-2826	104	1	since	since	SCONJ
ejpam-2826	104	2	a	a	PRON
ejpam-2826	104	3	is	be	AUX
ejpam-2826	104	4	open	open	ADJ
ejpam-2826	104	5	,	,	PUNCT
ejpam-2826	104	6	then	then	ADV
ejpam-2826	104	7	a	a	PRON
ejpam-2826	104	8	is	be	AUX
ejpam-2826	104	9	pre	pre	ADJ
ejpam-2826	104	10	-	-	ADJ
ejpam-2826	104	11	open	open	ADJ
ejpam-2826	104	12	and	and	CCONJ
ejpam-2826	104	13	thus	thus	ADV
ejpam-2826	104	14	a⊆int(cl(a)),which	a⊆int(cl(a)),which	PRON
ejpam-2826	104	15	implies	imply	VERB
ejpam-2826	104	16	a	a	DET
ejpam-2826	104	17	=	=	NOUN
ejpam-2826	104	18	int(cl(a	int(cl(a	PROPN
ejpam-2826	104	19	)	)	PUNCT
ejpam-2826	104	20	)	)	PUNCT
ejpam-2826	104	21	.	.	PUNCT
ejpam-2826	105	1	as	as	ADP
ejpam-2826	105	2	a	a	DET
ejpam-2826	105	3	consequence	consequence	NOUN
ejpam-2826	105	4	of	of	ADP
ejpam-2826	105	5	the	the	DET
ejpam-2826	105	6	above	above	ADJ
ejpam-2826	105	7	lemma	lemma	PROPN
ejpam-2826	105	8	,	,	PUNCT
ejpam-2826	105	9	we	we	PRON
ejpam-2826	105	10	have	have	VERB
ejpam-2826	105	11	the	the	DET
ejpam-2826	105	12	following	follow	VERB
ejpam-2826	105	13	result	result	NOUN
ejpam-2826	105	14	:	:	PUNCT
ejpam-2826	105	15	theorem	theorem	VERB
ejpam-2826	105	16	9	9	NUM
ejpam-2826	105	17	.	.	PUNCT
ejpam-2826	106	1	the	the	DET
ejpam-2826	106	2	following	follow	VERB
ejpam-2826	106	3	statements	statement	NOUN
ejpam-2826	106	4	are	be	AUX
ejpam-2826	106	5	equivalent	equivalent	ADJ
ejpam-2826	106	6	for	for	ADP
ejpam-2826	106	7	a	a	DET
ejpam-2826	106	8	function	function	NOUN
ejpam-2826	106	9	f	f	NOUN
ejpam-2826	106	10	:	:	PUNCT
ejpam-2826	106	11	x→y	x→y	NUM
ejpam-2826	106	12	:	:	PUNCT
ejpam-2826	106	13	(	(	PUNCT
ejpam-2826	106	14	i	i	NOUN
ejpam-2826	106	15	)	)	PUNCT
ejpam-2826	106	16	f	f	PROPN
ejpam-2826	106	17	is	be	AUX
ejpam-2826	106	18	completely	completely	ADV
ejpam-2826	106	19	continuous	continuous	ADJ
ejpam-2826	106	20	(	(	PUNCT
ejpam-2826	106	21	ii	ii	NOUN
ejpam-2826	106	22	)	)	PUNCT
ejpam-2826	106	23	f	f	PROPN
ejpam-2826	106	24	is	be	AUX
ejpam-2826	106	25	contra	contra	PROPN
ejpam-2826	106	26	δgb	δgb	ADV
ejpam-2826	106	27	-	-	PUNCT
ejpam-2826	106	28	continuous	continuous	ADJ
ejpam-2826	106	29	and	and	CCONJ
ejpam-2826	106	30	δ∗-continuous	δ∗-continuous	ADJ
ejpam-2826	106	31	(	(	PUNCT
ejpam-2826	106	32	iii	iii	NOUN
ejpam-2826	106	33	)	)	PUNCT
ejpam-2826	106	34	f	f	PROPN
ejpam-2826	106	35	is	be	AUX
ejpam-2826	106	36	contra	contra	PROPN
ejpam-2826	106	37	b	b	PROPN
ejpam-2826	106	38	-	-	PUNCT
ejpam-2826	106	39	continuous	continuous	ADJ
ejpam-2826	106	40	and	and	CCONJ
ejpam-2826	106	41	continuous	continuous	ADJ
ejpam-2826	106	42	.	.	PUNCT
ejpam-2826	107	1	definition	definition	NOUN
ejpam-2826	107	2	5	5	NUM
ejpam-2826	107	3	.	.	PUNCT
ejpam-2826	108	1	[	[	X
ejpam-2826	108	2	16	16	NUM
ejpam-2826	108	3	]	]	PUNCT
ejpam-2826	108	4	a	a	DET
ejpam-2826	108	5	subset	subset	NOUN
ejpam-2826	108	6	a	a	PRON
ejpam-2826	108	7	of	of	ADP
ejpam-2826	108	8	x	x	SYM
ejpam-2826	108	9	is	be	AUX
ejpam-2826	108	10	said	say	VERB
ejpam-2826	108	11	to	to	PART
ejpam-2826	108	12	be	be	AUX
ejpam-2826	108	13	q	q	ADJ
ejpam-2826	108	14	-	-	PUNCT
ejpam-2826	108	15	set	set	VERB
ejpam-2826	108	16	if	if	SCONJ
ejpam-2826	108	17	int(cl(a))=cl(int(a	int(cl(a))=cl(int(a	NOUN
ejpam-2826	108	18	)	)	PUNCT
ejpam-2826	108	19	)	)	PUNCT
ejpam-2826	108	20	.	.	PUNCT
ejpam-2826	109	1	definition	definition	NOUN
ejpam-2826	109	2	6	6	NUM
ejpam-2826	109	3	.	.	PUNCT
ejpam-2826	110	1	[	[	X
ejpam-2826	110	2	16	16	NUM
ejpam-2826	110	3	]	]	X
ejpam-2826	110	4	a	a	DET
ejpam-2826	110	5	function	function	NOUN
ejpam-2826	110	6	f	f	X
ejpam-2826	110	7	:	:	PUNCT
ejpam-2826	110	8	x→y	x→y	NUM
ejpam-2826	110	9	is	be	AUX
ejpam-2826	110	10	q	q	ADJ
ejpam-2826	110	11	-	-	ADJ
ejpam-2826	110	12	continuous	continuous	ADJ
ejpam-2826	110	13	if	if	SCONJ
ejpam-2826	110	14	f−1(v	f−1(v	PROPN
ejpam-2826	110	15	)	)	PUNCT
ejpam-2826	110	16	is	be	AUX
ejpam-2826	110	17	q	q	NOUN
ejpam-2826	110	18	-	-	PUNCT
ejpam-2826	110	19	set	set	VERB
ejpam-2826	110	20	in	in	ADP
ejpam-2826	110	21	x	x	PUNCT
ejpam-2826	110	22	for	for	SCONJ
ejpam-2826	110	23	every	every	DET
ejpam-2826	110	24	open	open	ADJ
ejpam-2826	110	25	set	set	VERB
ejpam-2826	110	26	v	v	NOUN
ejpam-2826	110	27	of	of	ADP
ejpam-2826	110	28	y.	y.	PROPN
ejpam-2826	110	29	theorem	theorem	VERB
ejpam-2826	110	30	10	10	NUM
ejpam-2826	110	31	.	.	PUNCT
ejpam-2826	111	1	for	for	ADP
ejpam-2826	111	2	a	a	DET
ejpam-2826	111	3	subset	subset	NOUN
ejpam-2826	111	4	a	a	PRON
ejpam-2826	111	5	of	of	ADP
ejpam-2826	111	6	a	a	DET
ejpam-2826	111	7	space	space	NOUN
ejpam-2826	111	8	x	x	PUNCT
ejpam-2826	111	9	the	the	DET
ejpam-2826	111	10	following	follow	VERB
ejpam-2826	111	11	are	be	AUX
ejpam-2826	111	12	equivalent	equivalent	ADJ
ejpam-2826	111	13	:	:	PUNCT
ejpam-2826	111	14	(	(	PUNCT
ejpam-2826	111	15	i	i	NOUN
ejpam-2826	111	16	)	)	PUNCT
ejpam-2826	111	17	a	a	PRON
ejpam-2826	111	18	is	be	AUX
ejpam-2826	111	19	clopen	clopen	ADJ
ejpam-2826	111	20	(	(	PUNCT
ejpam-2826	111	21	ii	ii	NOUN
ejpam-2826	111	22	)	)	PUNCT
ejpam-2826	111	23	a	a	PRON
ejpam-2826	111	24	is	be	AUX
ejpam-2826	111	25	δ	δ	NOUN
ejpam-2826	111	26	-	-	ADJ
ejpam-2826	111	27	open	open	ADJ
ejpam-2826	111	28	and	and	CCONJ
ejpam-2826	111	29	δ	δ	NOUN
ejpam-2826	111	30	-	-	PUNCT
ejpam-2826	111	31	closed	close	VERB
ejpam-2826	111	32	(	(	PUNCT
ejpam-2826	111	33	iii	iii	NOUN
ejpam-2826	111	34	)	)	PUNCT
ejpam-2826	111	35	a	a	PRON
ejpam-2826	111	36	is	be	AUX
ejpam-2826	111	37	regular	regular	ADJ
ejpam-2826	111	38	-	-	PUNCT
ejpam-2826	111	39	open	open	ADJ
ejpam-2826	111	40	and	and	CCONJ
ejpam-2826	111	41	regular	regular	ADV
ejpam-2826	111	42	-	-	PUNCT
ejpam-2826	111	43	closed	closed	ADJ
ejpam-2826	111	44	.	.	PUNCT
ejpam-2826	112	1	theorem	theorem	VERB
ejpam-2826	112	2	11	11	NUM
ejpam-2826	112	3	.	.	PUNCT
ejpam-2826	113	1	for	for	ADP
ejpam-2826	113	2	a	a	DET
ejpam-2826	113	3	subset	subset	NOUN
ejpam-2826	113	4	a	a	PRON
ejpam-2826	113	5	of	of	ADP
ejpam-2826	113	6	a	a	DET
ejpam-2826	113	7	space	space	NOUN
ejpam-2826	113	8	x	x	PUNCT
ejpam-2826	113	9	the	the	DET
ejpam-2826	113	10	following	follow	VERB
ejpam-2826	113	11	are	be	AUX
ejpam-2826	113	12	equivalent	equivalent	ADJ
ejpam-2826	113	13	:	:	PUNCT
ejpam-2826	113	14	(	(	PUNCT
ejpam-2826	113	15	i	i	NOUN
ejpam-2826	113	16	)	)	PUNCT
ejpam-2826	113	17	a	a	PRON
ejpam-2826	113	18	is	be	AUX
ejpam-2826	113	19	clopen	clopen	ADJ
ejpam-2826	113	20	(	(	PUNCT
ejpam-2826	113	21	ii	ii	NOUN
ejpam-2826	113	22	)	)	PUNCT
ejpam-2826	113	23	a	a	PRON
ejpam-2826	113	24	is	be	AUX
ejpam-2826	113	25	δ	δ	NOUN
ejpam-2826	113	26	-	-	ADJ
ejpam-2826	113	27	open	open	ADJ
ejpam-2826	113	28	,	,	PUNCT
ejpam-2826	113	29	q	q	NOUN
ejpam-2826	113	30	-	-	PUNCT
ejpam-2826	113	31	set	set	VERB
ejpam-2826	113	32	and	and	CCONJ
ejpam-2826	113	33	δgb	δgb	ADV
ejpam-2826	113	34	-	-	PUNCT
ejpam-2826	113	35	closed	close	VERB
ejpam-2826	113	36	(	(	PUNCT
ejpam-2826	113	37	iii	iii	NOUN
ejpam-2826	113	38	)	)	PUNCT
ejpam-2826	113	39	a	a	PRON
ejpam-2826	113	40	is	be	AUX
ejpam-2826	113	41	open	open	ADJ
ejpam-2826	113	42	,	,	PUNCT
ejpam-2826	113	43	q	q	NOUN
ejpam-2826	113	44	-	-	PUNCT
ejpam-2826	113	45	set	set	VERB
ejpam-2826	113	46	and	and	CCONJ
ejpam-2826	113	47	b	b	NOUN
ejpam-2826	113	48	-	-	PUNCT
ejpam-2826	113	49	closed	closed	ADJ
ejpam-2826	113	50	.	.	PUNCT
ejpam-2826	114	1	proof:(i)→(ii):let	proof:(i)→(ii):let	NOUN
ejpam-2826	114	2	a	a	DET
ejpam-2826	114	3	be	be	AUX
ejpam-2826	114	4	clopen	clopen	ADJ
ejpam-2826	114	5	then	then	ADV
ejpam-2826	114	6	by	by	ADP
ejpam-2826	114	7	theorem	theorem	NOUN
ejpam-2826	114	8	10	10	NUM
ejpam-2826	114	9	we	we	PRON
ejpam-2826	114	10	have	have	VERB
ejpam-2826	114	11	a	a	DET
ejpam-2826	114	12	=	=	NOUN
ejpam-2826	114	13	int(cl(a))=cl(int(a	int(cl(a))=cl(int(a	NOUN
ejpam-2826	114	14	)	)	PUNCT
ejpam-2826	114	15	)	)	PUNCT
ejpam-2826	114	16	.	.	PUNCT
ejpam-2826	115	1	hence	hence	ADV
ejpam-2826	115	2	a	a	PRON
ejpam-2826	115	3	is	be	AUX
ejpam-2826	115	4	q-set.again	q-set.again	VERB
ejpam-2826	115	5	by	by	ADP
ejpam-2826	115	6	theorem	theorem	NOUN
ejpam-2826	115	7	10	10	NUM
ejpam-2826	115	8	,	,	PUNCT
ejpam-2826	115	9	a	a	PRON
ejpam-2826	115	10	is	be	AUX
ejpam-2826	115	11	δ	δ	NOUN
ejpam-2826	115	12	-	-	ADJ
ejpam-2826	115	13	open	open	ADJ
ejpam-2826	115	14	and	and	CCONJ
ejpam-2826	115	15	δ	δ	NOUN
ejpam-2826	115	16	-	-	PUNCT
ejpam-2826	115	17	closed	closed	ADJ
ejpam-2826	115	18	.	.	PUNCT
ejpam-2826	116	1	since	since	SCONJ
ejpam-2826	116	2	every	every	DET
ejpam-2826	116	3	δ	δ	PROPN
ejpam-2826	116	4	-	-	PUNCT
ejpam-2826	116	5	closed	close	VERB
ejpam-2826	116	6	set	set	NOUN
ejpam-2826	116	7	is	be	AUX
ejpam-2826	116	8	δgb	δgb	ADV
ejpam-2826	116	9	-	-	PUNCT
ejpam-2826	116	10	closed	closed	ADJ
ejpam-2826	116	11	.	.	PUNCT
ejpam-2826	117	1	therefore	therefore	ADV
ejpam-2826	117	2	(	(	PUNCT
ejpam-2826	117	3	ii	ii	NOUN
ejpam-2826	117	4	)	)	PUNCT
ejpam-2826	117	5	holds	hold	VERB
ejpam-2826	117	6	.	.	PUNCT
ejpam-2826	118	1	(	(	PUNCT
ejpam-2826	118	2	ii)→(iii	ii)→(iii	PROPN
ejpam-2826	118	3	):	):	PUNCT
ejpam-2826	118	4	follows	follow	VERB
ejpam-2826	118	5	from	from	ADP
ejpam-2826	118	6	the	the	DET
ejpam-2826	118	7	theorem	theorem	NOUN
ejpam-2826	118	8	7	7	NUM
ejpam-2826	118	9	.	.	PUNCT
ejpam-2826	119	1	(	(	PUNCT
ejpam-2826	119	2	iii)→(i):let	iii)→(i):let	ADV
ejpam-2826	119	3	a	a	DET
ejpam-2826	119	4	be	be	AUX
ejpam-2826	119	5	an	an	DET
ejpam-2826	119	6	open	open	ADJ
ejpam-2826	119	7	,	,	PUNCT
ejpam-2826	119	8	q	q	NOUN
ejpam-2826	119	9	-	-	PUNCT
ejpam-2826	119	10	set	set	VERB
ejpam-2826	119	11	and	and	CCONJ
ejpam-2826	119	12	b	b	NOUN
ejpam-2826	119	13	-	-	PUNCT
ejpam-2826	119	14	closed	closed	ADJ
ejpam-2826	119	15	set	set	NOUN
ejpam-2826	119	16	then	then	ADV
ejpam-2826	119	17	by	by	ADP
ejpam-2826	119	18	lemma	lemma	PROPN
ejpam-2826	119	19	3	3	NUM
ejpam-2826	119	20	,	,	PUNCT
ejpam-2826	119	21	a	a	PRON
ejpam-2826	119	22	is	be	AUX
ejpam-2826	119	23	regular	regular	ADJ
ejpam-2826	119	24	open	open	ADJ
ejpam-2826	119	25	.	.	PUNCT
ejpam-2826	120	1	since	since	SCONJ
ejpam-2826	120	2	a	a	PRON
ejpam-2826	120	3	is	be	AUX
ejpam-2826	120	4	q	q	NOUN
ejpam-2826	120	5	-	-	PUNCT
ejpam-2826	120	6	set	set	ADJ
ejpam-2826	120	7	,	,	PUNCT
ejpam-2826	120	8	then	then	ADV
ejpam-2826	120	9	a	a	DET
ejpam-2826	120	10	=	=	NOUN
ejpam-2826	120	11	int(cl(a))=cl(int(a	int(cl(a))=cl(int(a	PROPN
ejpam-2826	120	12	)	)	PUNCT
ejpam-2826	120	13	)	)	PUNCT
ejpam-2826	120	14	which	which	PRON
ejpam-2826	120	15	implies	imply	VERB
ejpam-2826	120	16	a	a	DET
ejpam-2826	120	17	is	be	AUX
ejpam-2826	120	18	regular	regular	ADJ
ejpam-2826	120	19	closed	closed	ADJ
ejpam-2826	120	20	.	.	PUNCT
ejpam-2826	121	1	hence	hence	ADV
ejpam-2826	121	2	by	by	ADP
ejpam-2826	121	3	theorem	theorem	NOUN
ejpam-2826	121	4	10	10	NUM
ejpam-2826	121	5	,	,	PUNCT
ejpam-2826	121	6	a	a	PRON
ejpam-2826	121	7	is	be	AUX
ejpam-2826	121	8	clopen	clopen	ADJ
ejpam-2826	121	9	.	.	PUNCT
ejpam-2826	122	1	s.s.benchalli	s.s.benchalli	PROPN
ejpam-2826	122	2	,	,	PUNCT
ejpam-2826	122	3	p.g.patil	p.g.patil	PROPN
ejpam-2826	122	4	,	,	PUNCT
ejpam-2826	122	5	j.b.toranagatti	j.b.toranagatti	PROPN
ejpam-2826	122	6	,	,	PUNCT
ejpam-2826	122	7	s.r.vighneshi	s.r.vighneshi	PROPN
ejpam-2826	122	8	/	/	SYM
ejpam-2826	122	9	eur	eur	PROPN
ejpam-2826	122	10	.	.	PUNCT
ejpam-2826	123	1	j.	j.	PROPN
ejpam-2826	123	2	pure	pure	PROPN
ejpam-2826	123	3	appl	appl	PROPN
ejpam-2826	123	4	.	.	PROPN
ejpam-2826	123	5	math	math	PROPN
ejpam-2826	123	6	,	,	PUNCT
ejpam-2826	123	7	10	10	NUM
ejpam-2826	123	8	(	(	PUNCT
ejpam-2826	123	9	2	2	NUM
ejpam-2826	123	10	)	)	PUNCT
ejpam-2826	123	11	(	(	PUNCT
ejpam-2826	123	12	2017	2017	NUM
ejpam-2826	123	13	)	)	PUNCT
ejpam-2826	123	14	,	,	PUNCT
ejpam-2826	123	15	312	312	NUM
ejpam-2826	123	16	-	-	SYM
ejpam-2826	123	17	322	322	NUM
ejpam-2826	123	18	317	317	NUM
ejpam-2826	123	19	theorem	theorem	NOUN
ejpam-2826	123	20	12	12	NUM
ejpam-2826	123	21	.	.	PUNCT
ejpam-2826	124	1	the	the	DET
ejpam-2826	124	2	following	follow	VERB
ejpam-2826	124	3	statements	statement	NOUN
ejpam-2826	124	4	are	be	AUX
ejpam-2826	124	5	equivalent	equivalent	ADJ
ejpam-2826	124	6	for	for	ADP
ejpam-2826	124	7	a	a	DET
ejpam-2826	124	8	function	function	NOUN
ejpam-2826	124	9	f	f	NOUN
ejpam-2826	124	10	:	:	PUNCT
ejpam-2826	124	11	x→y	x→y	NUM
ejpam-2826	124	12	:	:	PUNCT
ejpam-2826	124	13	(	(	PUNCT
ejpam-2826	124	14	i	i	NOUN
ejpam-2826	124	15	)	)	PUNCT
ejpam-2826	124	16	f	f	PROPN
ejpam-2826	124	17	is	be	AUX
ejpam-2826	124	18	perfectly	perfectly	ADV
ejpam-2826	124	19	continuous	continuous	ADJ
ejpam-2826	124	20	(	(	PUNCT
ejpam-2826	124	21	ii	ii	NOUN
ejpam-2826	124	22	)	)	PUNCT
ejpam-2826	124	23	f	f	PROPN
ejpam-2826	124	24	is	be	AUX
ejpam-2826	124	25	δ∗-continuous	δ∗-continuous	ADJ
ejpam-2826	124	26	,	,	PUNCT
ejpam-2826	124	27	q	q	ADJ
ejpam-2826	124	28	-	-	ADJ
ejpam-2826	124	29	continuous	continuous	ADJ
ejpam-2826	124	30	and	and	CCONJ
ejpam-2826	124	31	contra	contra	PROPN
ejpam-2826	124	32	δgb	δgb	PROPN
ejpam-2826	124	33	-	-	PUNCT
ejpam-2826	124	34	continuous	continuous	ADJ
ejpam-2826	124	35	(	(	PUNCT
ejpam-2826	124	36	iii	iii	NOUN
ejpam-2826	124	37	)	)	PUNCT
ejpam-2826	124	38	f	f	PROPN
ejpam-2826	124	39	is	be	AUX
ejpam-2826	124	40	continuous	continuous	ADJ
ejpam-2826	124	41	,	,	PUNCT
ejpam-2826	124	42	q	q	ADJ
ejpam-2826	124	43	-	-	ADJ
ejpam-2826	124	44	continuous	continuous	ADJ
ejpam-2826	124	45	and	and	CCONJ
ejpam-2826	124	46	contra	contra	PROPN
ejpam-2826	124	47	b	b	X
ejpam-2826	124	48	-	-	PUNCT
ejpam-2826	124	49	continuous	continuous	ADJ
ejpam-2826	124	50	.	.	PUNCT
ejpam-2826	125	1	definition	definition	NOUN
ejpam-2826	125	2	7	7	NUM
ejpam-2826	125	3	.	.	PUNCT
ejpam-2826	126	1	a	a	DET
ejpam-2826	126	2	space	space	NOUN
ejpam-2826	126	3	x	x	PUNCT
ejpam-2826	126	4	is	be	AUX
ejpam-2826	126	5	called	call	VERB
ejpam-2826	126	6	locally	locally	ADV
ejpam-2826	126	7	δgb	δgb	ADJ
ejpam-2826	126	8	-	-	PUNCT
ejpam-2826	126	9	indiscrete	indiscrete	ADJ
ejpam-2826	126	10	if	if	SCONJ
ejpam-2826	126	11	every	every	DET
ejpam-2826	126	12	δgb	δgb	ADJ
ejpam-2826	126	13	-	-	PUNCT
ejpam-2826	126	14	open	open	ADJ
ejpam-2826	126	15	set	set	NOUN
ejpam-2826	126	16	is	be	AUX
ejpam-2826	126	17	closed	close	VERB
ejpam-2826	126	18	in	in	ADP
ejpam-2826	126	19	x.	x.	NOUN
ejpam-2826	126	20	theorem	theorem	VERB
ejpam-2826	126	21	13	13	NUM
ejpam-2826	126	22	.	.	PUNCT
ejpam-2826	127	1	if	if	SCONJ
ejpam-2826	127	2	f	f	X
ejpam-2826	127	3	:	:	PUNCT
ejpam-2826	127	4	x→y	x→y	NUM
ejpam-2826	127	5	is	be	AUX
ejpam-2826	127	6	a	a	DET
ejpam-2826	127	7	contra	contra	PROPN
ejpam-2826	127	8	δgb	δgb	ADV
ejpam-2826	127	9	-	-	PUNCT
ejpam-2826	127	10	continuous	continuous	ADJ
ejpam-2826	127	11	and	and	CCONJ
ejpam-2826	127	12	x	x	PRON
ejpam-2826	127	13	is	be	AUX
ejpam-2826	127	14	locally	locally	ADV
ejpam-2826	127	15	δgb	δgb	ADJ
ejpam-2826	127	16	-	-	PUNCT
ejpam-2826	127	17	indiscrete	indiscrete	ADJ
ejpam-2826	127	18	space	space	NOUN
ejpam-2826	127	19	then	then	ADV
ejpam-2826	127	20	f	f	PROPN
ejpam-2826	127	21	is	be	AUX
ejpam-2826	127	22	continuous	continuous	ADJ
ejpam-2826	127	23	.	.	PUNCT
ejpam-2826	128	1	proof	proof	NOUN
ejpam-2826	128	2	:	:	PUNCT
ejpam-2826	128	3	let	let	VERB
ejpam-2826	128	4	g	g	PRON
ejpam-2826	128	5	be	be	AUX
ejpam-2826	128	6	a	a	DET
ejpam-2826	128	7	closed	closed	ADJ
ejpam-2826	128	8	set	set	NOUN
ejpam-2826	128	9	in	in	ADP
ejpam-2826	128	10	y.since	y.since	NOUN
ejpam-2826	128	11	f	f	PROPN
ejpam-2826	128	12	is	be	AUX
ejpam-2826	128	13	contra	contra	PROPN
ejpam-2826	128	14	δgb	δgb	ADV
ejpam-2826	128	15	-	-	PUNCT
ejpam-2826	128	16	continuous	continuous	ADJ
ejpam-2826	128	17	and	and	CCONJ
ejpam-2826	128	18	x	x	NOUN
ejpam-2826	128	19	is	be	AUX
ejpam-2826	128	20	locally	locally	ADV
ejpam-2826	128	21	δgbindiscrete	δgbindiscrete	ADJ
ejpam-2826	128	22	space	space	NOUN
ejpam-2826	128	23	then	then	ADV
ejpam-2826	128	24	f−1(g)is	f−1(g)is	VERB
ejpam-2826	128	25	a	a	DET
ejpam-2826	128	26	closed	closed	ADJ
ejpam-2826	128	27	set	set	NOUN
ejpam-2826	128	28	in	in	ADP
ejpam-2826	128	29	x.	x.	NOUN
ejpam-2826	128	30	hence	hence	ADV
ejpam-2826	128	31	f	f	PROPN
ejpam-2826	128	32	is	be	AUX
ejpam-2826	128	33	continuous	continuous	ADJ
ejpam-2826	128	34	definition	definition	NOUN
ejpam-2826	128	35	8	8	NUM
ejpam-2826	128	36	.	.	PUNCT
ejpam-2826	129	1	[	[	X
ejpam-2826	129	2	11	11	NUM
ejpam-2826	129	3	]	]	PUNCT
ejpam-2826	129	4	a	a	DET
ejpam-2826	129	5	space	space	NOUN
ejpam-2826	129	6	x	x	PUNCT
ejpam-2826	129	7	is	be	AUX
ejpam-2826	129	8	called	call	VERB
ejpam-2826	129	9	locally	locally	ADV
ejpam-2826	129	10	indiscrete	indiscrete	ADJ
ejpam-2826	129	11	if	if	SCONJ
ejpam-2826	129	12	every	every	DET
ejpam-2826	129	13	open	open	ADJ
ejpam-2826	129	14	set	set	NOUN
ejpam-2826	129	15	is	be	AUX
ejpam-2826	129	16	closed	close	VERB
ejpam-2826	129	17	in	in	ADP
ejpam-2826	129	18	x.	x.	NOUN
ejpam-2826	129	19	theorem	theorem	VERB
ejpam-2826	129	20	14	14	NUM
ejpam-2826	129	21	.	.	PUNCT
ejpam-2826	130	1	if	if	SCONJ
ejpam-2826	130	2	f	f	X
ejpam-2826	130	3	:	:	PUNCT
ejpam-2826	130	4	x→y	x→y	NUM
ejpam-2826	130	5	is	be	AUX
ejpam-2826	130	6	a	a	DET
ejpam-2826	130	7	contra	contra	PROPN
ejpam-2826	130	8	δgb	δgb	ADJ
ejpam-2826	130	9	-	-	PUNCT
ejpam-2826	130	10	continuous	continuous	ADJ
ejpam-2826	130	11	preclosed	preclose	VERB
ejpam-2826	130	12	surjection	surjection	NOUN
ejpam-2826	130	13	and	and	CCONJ
ejpam-2826	130	14	x	x	NOUN
ejpam-2826	130	15	is	be	AUX
ejpam-2826	130	16	tδgbspace	tδgbspace	NOUN
ejpam-2826	130	17	then	then	ADV
ejpam-2826	130	18	y	y	PROPN
ejpam-2826	130	19	is	be	AUX
ejpam-2826	130	20	locally	locally	ADV
ejpam-2826	130	21	indiscrete	indiscrete	ADJ
ejpam-2826	130	22	.	.	PUNCT
ejpam-2826	131	1	proof	proof	NOUN
ejpam-2826	131	2	:	:	PUNCT
ejpam-2826	131	3	let	let	VERB
ejpam-2826	131	4	v	v	PART
ejpam-2826	131	5	be	be	AUX
ejpam-2826	131	6	an	an	DET
ejpam-2826	131	7	open	open	ADJ
ejpam-2826	131	8	set	set	NOUN
ejpam-2826	131	9	in	in	ADP
ejpam-2826	131	10	y.	y.	NOUN
ejpam-2826	131	11	since	since	SCONJ
ejpam-2826	131	12	f	f	PROPN
ejpam-2826	131	13	is	be	AUX
ejpam-2826	131	14	contra	contra	PROPN
ejpam-2826	131	15	δgb	δgb	ADV
ejpam-2826	131	16	-	-	PUNCT
ejpam-2826	131	17	continuous	continuous	ADJ
ejpam-2826	131	18	and	and	CCONJ
ejpam-2826	131	19	x	x	PRON
ejpam-2826	131	20	is	be	AUX
ejpam-2826	131	21	tδgb	tδgb	ADJ
ejpam-2826	131	22	-	-	PUNCT
ejpam-2826	131	23	space	space	NOUN
ejpam-2826	131	24	then	then	ADV
ejpam-2826	131	25	f−1(g)is	f−1(g)is	PROPN
ejpam-2826	131	26	closed	close	VERB
ejpam-2826	131	27	in	in	ADP
ejpam-2826	131	28	x.	x.	NOUN
ejpam-2826	131	29	also	also	ADV
ejpam-2826	131	30	f	f	PROPN
ejpam-2826	131	31	is	be	AUX
ejpam-2826	131	32	preclosed	preclose	VERB
ejpam-2826	131	33	then	then	ADV
ejpam-2826	131	34	v	v	NOUN
ejpam-2826	131	35	is	be	AUX
ejpam-2826	131	36	preclosed	preclose	VERB
ejpam-2826	131	37	in	in	ADP
ejpam-2826	131	38	y.	y.	PROPN
ejpam-2826	131	39	now	now	ADV
ejpam-2826	131	40	we	we	PRON
ejpam-2826	131	41	have	have	VERB
ejpam-2826	131	42	cl(v)=cl(int(v))⊆v	cl(v)=cl(int(v))⊆v	NOUN
ejpam-2826	131	43	.	.	PUNCT
ejpam-2826	132	1	this	this	PRON
ejpam-2826	132	2	means	mean	VERB
ejpam-2826	132	3	v	v	NOUN
ejpam-2826	132	4	is	be	AUX
ejpam-2826	132	5	closed	close	VERB
ejpam-2826	132	6	in	in	ADP
ejpam-2826	132	7	y	y	PROPN
ejpam-2826	132	8	and	and	CCONJ
ejpam-2826	132	9	hence	hence	ADV
ejpam-2826	132	10	y	y	PROPN
ejpam-2826	132	11	is	be	AUX
ejpam-2826	132	12	indiscrete	indiscrete	ADJ
ejpam-2826	132	13	.	.	PUNCT
ejpam-2826	133	1	theorem	theorem	ADJ
ejpam-2826	133	2	15	15	NUM
ejpam-2826	133	3	.	.	PUNCT
ejpam-2826	134	1	suppose	suppose	VERB
ejpam-2826	134	2	that	that	SCONJ
ejpam-2826	134	3	δgbc(x	δgbc(x	NOUN
ejpam-2826	134	4	)	)	PUNCT
ejpam-2826	134	5	is	be	AUX
ejpam-2826	134	6	closed	close	VERB
ejpam-2826	134	7	under	under	ADP
ejpam-2826	134	8	arbitrary	arbitrary	ADJ
ejpam-2826	134	9	intersections	intersection	NOUN
ejpam-2826	134	10	.	.	PUNCT
ejpam-2826	135	1	if	if	SCONJ
ejpam-2826	135	2	f	f	X
ejpam-2826	135	3	:	:	PUNCT
ejpam-2826	135	4	x→y	x→y	NUM
ejpam-2826	135	5	is	be	AUX
ejpam-2826	135	6	contra	contra	PROPN
ejpam-2826	135	7	δgb	δgb	PROPN
ejpam-2826	135	8	-	-	PUNCT
ejpam-2826	135	9	continuous	continuous	ADJ
ejpam-2826	135	10	and	and	CCONJ
ejpam-2826	135	11	y	y	PROPN
ejpam-2826	135	12	is	be	AUX
ejpam-2826	135	13	regular	regular	ADJ
ejpam-2826	136	1	then	then	ADV
ejpam-2826	136	2	f	f	PROPN
ejpam-2826	136	3	is	be	AUX
ejpam-2826	136	4	δgb	δgb	ADV
ejpam-2826	136	5	-	-	PUNCT
ejpam-2826	136	6	continuous	continuous	ADJ
ejpam-2826	136	7	.	.	PUNCT
ejpam-2826	137	1	proof	proof	NOUN
ejpam-2826	137	2	:	:	PUNCT
ejpam-2826	137	3	let	let	VERB
ejpam-2826	137	4	x∈x	x∈x	NOUN
ejpam-2826	137	5	and	and	CCONJ
ejpam-2826	137	6	v	v	X
ejpam-2826	137	7	be	be	AUX
ejpam-2826	137	8	an	an	DET
ejpam-2826	137	9	open	open	ADJ
ejpam-2826	137	10	set	set	NOUN
ejpam-2826	137	11	of	of	ADP
ejpam-2826	137	12	y	y	PROPN
ejpam-2826	137	13	containing	contain	VERB
ejpam-2826	137	14	f(x	f(x	PROPN
ejpam-2826	137	15	)	)	PUNCT
ejpam-2826	137	16	.	.	PUNCT
ejpam-2826	138	1	since	since	SCONJ
ejpam-2826	138	2	y	y	PROPN
ejpam-2826	138	3	is	be	AUX
ejpam-2826	138	4	regular	regular	ADJ
ejpam-2826	138	5	,	,	PUNCT
ejpam-2826	138	6	there	there	PRON
ejpam-2826	138	7	exists	exist	VERB
ejpam-2826	138	8	an	an	DET
ejpam-2826	138	9	open	open	ADJ
ejpam-2826	138	10	set	set	NOUN
ejpam-2826	138	11	g	g	NOUN
ejpam-2826	138	12	in	in	ADP
ejpam-2826	138	13	y	y	NOUN
ejpam-2826	138	14	containing	contain	VERB
ejpam-2826	138	15	f(x	f(x	PROPN
ejpam-2826	138	16	)	)	PUNCT
ejpam-2826	138	17	such	such	ADJ
ejpam-2826	138	18	that	that	DET
ejpam-2826	138	19	cl(g)⊆v	cl(g)⊆v	NOUN
ejpam-2826	138	20	.	.	PUNCT
ejpam-2826	139	1	since	since	SCONJ
ejpam-2826	139	2	f	f	PROPN
ejpam-2826	139	3	is	be	AUX
ejpam-2826	139	4	contra	contra	PROPN
ejpam-2826	139	5	δgb	δgb	PROPN
ejpam-2826	139	6	-	-	PUNCT
ejpam-2826	139	7	continuous	continuous	ADJ
ejpam-2826	139	8	,	,	PUNCT
ejpam-2826	139	9	there	there	PRON
ejpam-2826	139	10	exists	exist	VERB
ejpam-2826	139	11	an	an	DET
ejpam-2826	139	12	δgb	δgb	ADV
ejpam-2826	139	13	-	-	PUNCT
ejpam-2826	139	14	open	open	ADJ
ejpam-2826	139	15	set	set	NOUN
ejpam-2826	139	16	u	u	NOUN
ejpam-2826	139	17	in	in	ADP
ejpam-2826	139	18	x	x	PUNCT
ejpam-2826	139	19	containing	contain	VERB
ejpam-2826	139	20	x	x	PUNCT
ejpam-2826	139	21	such	such	ADJ
ejpam-2826	139	22	that	that	SCONJ
ejpam-2826	139	23	f(u)⊆cl(g).then	f(u)⊆cl(g).then	ADP
ejpam-2826	139	24	f(u)⊆cl(g)⊆v	f(u)⊆cl(g)⊆v	NOUN
ejpam-2826	139	25	.	.	PUNCT
ejpam-2826	140	1	hence	hence	ADV
ejpam-2826	140	2	f	f	PROPN
ejpam-2826	140	3	is	be	AUX
ejpam-2826	140	4	δgb	δgb	ADV
ejpam-2826	140	5	-	-	PUNCT
ejpam-2826	140	6	continuous	continuous	ADJ
ejpam-2826	140	7	.	.	PUNCT
ejpam-2826	141	1	recall	recall	VERB
ejpam-2826	141	2	that	that	PRON
ejpam-2826	141	3	for	for	ADP
ejpam-2826	141	4	a	a	DET
ejpam-2826	141	5	function	function	NOUN
ejpam-2826	141	6	f	f	NOUN
ejpam-2826	141	7	:	:	PUNCT
ejpam-2826	141	8	x→y	x→y	NUM
ejpam-2826	142	1	the	the	DET
ejpam-2826	142	2	subset	subset	NOUN
ejpam-2826	142	3	{	{	PUNCT
ejpam-2826	142	4	(	(	PUNCT
ejpam-2826	142	5	x	x	X
ejpam-2826	142	6	,	,	PUNCT
ejpam-2826	142	7	f(x	f(x	PROPN
ejpam-2826	142	8	)	)	PUNCT
ejpam-2826	142	9	):	):	PUNCT
ejpam-2826	142	10	x∈x}⊆x×y	x∈x}⊆x×y	PROPN
ejpam-2826	142	11	is	be	AUX
ejpam-2826	142	12	called	call	VERB
ejpam-2826	142	13	the	the	DET
ejpam-2826	142	14	graph	graph	NOUN
ejpam-2826	142	15	of	of	ADP
ejpam-2826	142	16	f	f	PROPN
ejpam-2826	142	17	and	and	CCONJ
ejpam-2826	142	18	is	be	AUX
ejpam-2826	142	19	denoted	denote	VERB
ejpam-2826	142	20	by	by	ADP
ejpam-2826	142	21	g(f	g(f	PROPN
ejpam-2826	142	22	)	)	PUNCT
ejpam-2826	142	23	.	.	PUNCT
ejpam-2826	143	1	definition	definition	NOUN
ejpam-2826	143	2	9	9	NUM
ejpam-2826	143	3	.	.	PUNCT
ejpam-2826	144	1	the	the	DET
ejpam-2826	144	2	graph	graph	NOUN
ejpam-2826	144	3	g(f	g(f	PROPN
ejpam-2826	144	4	)	)	PUNCT
ejpam-2826	144	5	of	of	ADP
ejpam-2826	144	6	a	a	DET
ejpam-2826	144	7	function	function	NOUN
ejpam-2826	144	8	f	f	NOUN
ejpam-2826	144	9	:	:	PUNCT
ejpam-2826	144	10	x→y	x→y	NUM
ejpam-2826	144	11	is	be	AUX
ejpam-2826	144	12	said	say	VERB
ejpam-2826	144	13	to	to	PART
ejpam-2826	144	14	be	be	AUX
ejpam-2826	144	15	contra	contra	PROPN
ejpam-2826	144	16	δgb	δgb	PROPN
ejpam-2826	144	17	-	-	PUNCT
ejpam-2826	144	18	closed	closed	ADJ
ejpam-2826	144	19	if	if	SCONJ
ejpam-2826	144	20	for	for	ADP
ejpam-2826	144	21	each	each	DET
ejpam-2826	144	22	(	(	PUNCT
ejpam-2826	144	23	x	x	NOUN
ejpam-2826	144	24	,	,	PUNCT
ejpam-2826	144	25	y)∈(x×y)-g(f	y)∈(x×y)-g(f	PROPN
ejpam-2826	144	26	)	)	PUNCT
ejpam-2826	144	27	there	there	PRON
ejpam-2826	144	28	exists	exist	VERB
ejpam-2826	144	29	u∈δgbo(x	u∈δgbo(x	PROPN
ejpam-2826	144	30	,	,	PUNCT
ejpam-2826	144	31	x	x	NOUN
ejpam-2826	144	32	)	)	PUNCT
ejpam-2826	144	33	and	and	CCONJ
ejpam-2826	144	34	v∈c(y	v∈c(y	PROPN
ejpam-2826	144	35	,	,	PUNCT
ejpam-2826	144	36	y	y	PROPN
ejpam-2826	144	37	)	)	PUNCT
ejpam-2826	144	38	such	such	ADJ
ejpam-2826	144	39	that	that	PRON
ejpam-2826	144	40	(	(	PUNCT
ejpam-2826	144	41	u×v)∩g(f)=φ	u×v)∩g(f)=φ	PROPN
ejpam-2826	144	42	.	.	PUNCT
ejpam-2826	145	1	theorem	theorem	VERB
ejpam-2826	145	2	16	16	NUM
ejpam-2826	145	3	.	.	PUNCT
ejpam-2826	146	1	the	the	DET
ejpam-2826	146	2	graph	graph	NOUN
ejpam-2826	146	3	g(f	g(f	PROPN
ejpam-2826	146	4	)	)	PUNCT
ejpam-2826	146	5	of	of	ADP
ejpam-2826	146	6	a	a	DET
ejpam-2826	146	7	function	function	NOUN
ejpam-2826	146	8	f	f	NOUN
ejpam-2826	146	9	:	:	PUNCT
ejpam-2826	146	10	x→y	x→y	NUM
ejpam-2826	146	11	is	be	AUX
ejpam-2826	146	12	contra	contra	PROPN
ejpam-2826	146	13	δgb	δgb	PROPN
ejpam-2826	146	14	-	-	PUNCT
ejpam-2826	146	15	closed	closed	ADJ
ejpam-2826	146	16	in	in	ADP
ejpam-2826	146	17	x×y	x×y	PROPN
ejpam-2826	146	18	if	if	SCONJ
ejpam-2826	146	19	and	and	CCONJ
ejpam-2826	146	20	only	only	ADV
ejpam-2826	146	21	for	for	ADP
ejpam-2826	146	22	each	each	DET
ejpam-2826	146	23	(	(	PUNCT
ejpam-2826	146	24	x	x	NOUN
ejpam-2826	146	25	,	,	PUNCT
ejpam-2826	146	26	y)∈(x×y)-g(f	y)∈(x×y)-g(f	PROPN
ejpam-2826	146	27	)	)	PUNCT
ejpam-2826	146	28	there	there	PRON
ejpam-2826	146	29	exists	exist	VERB
ejpam-2826	146	30	u∈δgbo(x	u∈δgbo(x	PROPN
ejpam-2826	146	31	,	,	PUNCT
ejpam-2826	146	32	x	x	NOUN
ejpam-2826	146	33	)	)	PUNCT
ejpam-2826	146	34	and	and	CCONJ
ejpam-2826	146	35	v∈c(y	v∈c(y	PROPN
ejpam-2826	146	36	,	,	PUNCT
ejpam-2826	146	37	y	y	PROPN
ejpam-2826	146	38	)	)	PUNCT
ejpam-2826	146	39	such	such	ADJ
ejpam-2826	146	40	that	that	DET
ejpam-2826	146	41	f(u)∩v	f(u)∩v	NOUN
ejpam-2826	146	42	=	=	SYM
ejpam-2826	146	43	φ	φ	PROPN
ejpam-2826	146	44	.	.	PUNCT
ejpam-2826	147	1	theorem	theorem	VERB
ejpam-2826	147	2	17	17	NUM
ejpam-2826	147	3	.	.	PUNCT
ejpam-2826	148	1	if	if	SCONJ
ejpam-2826	148	2	f	f	X
ejpam-2826	148	3	:	:	PUNCT
ejpam-2826	148	4	x→y	x→y	NUM
ejpam-2826	148	5	is	be	AUX
ejpam-2826	148	6	contra	contra	PROPN
ejpam-2826	148	7	δgb	δgb	PROPN
ejpam-2826	148	8	-	-	PUNCT
ejpam-2826	148	9	continuous	continuous	ADJ
ejpam-2826	148	10	and	and	CCONJ
ejpam-2826	148	11	y	y	PROPN
ejpam-2826	148	12	is	be	AUX
ejpam-2826	148	13	urysohn	urysohn	PRON
ejpam-2826	148	14	then	then	ADV
ejpam-2826	148	15	g(f	g(f	PROPN
ejpam-2826	148	16	)	)	PUNCT
ejpam-2826	148	17	is	be	AUX
ejpam-2826	148	18	contra	contra	PROPN
ejpam-2826	148	19	δgb	δgb	PROPN
ejpam-2826	148	20	-	-	PUNCT
ejpam-2826	148	21	closed	closed	ADJ
ejpam-2826	148	22	in	in	ADP
ejpam-2826	148	23	the	the	DET
ejpam-2826	148	24	product	product	NOUN
ejpam-2826	148	25	space	space	NOUN
ejpam-2826	148	26	x×y	x×y	PROPN
ejpam-2826	148	27	.	.	PUNCT
ejpam-2826	149	1	proof	proof	NOUN
ejpam-2826	149	2	:	:	PUNCT
ejpam-2826	149	3	let	let	VERB
ejpam-2826	149	4	(	(	PUNCT
ejpam-2826	149	5	x	x	NOUN
ejpam-2826	149	6	,	,	PUNCT
ejpam-2826	149	7	y)∈(x×y)-g(f	y)∈(x×y)-g(f	PROPN
ejpam-2826	149	8	)	)	PUNCT
ejpam-2826	149	9	,	,	PUNCT
ejpam-2826	149	10	then	then	ADV
ejpam-2826	149	11	y	y	PROPN
ejpam-2826	149	12	6	6	NUM
ejpam-2826	149	13	=	=	SYM
ejpam-2826	149	14	f(x	f(x	PROPN
ejpam-2826	149	15	)	)	PUNCT
ejpam-2826	149	16	and	and	CCONJ
ejpam-2826	149	17	there	there	PRON
ejpam-2826	149	18	exist	exist	VERB
ejpam-2826	149	19	open	open	ADJ
ejpam-2826	149	20	sets	set	NOUN
ejpam-2826	149	21	a	a	PRON
ejpam-2826	149	22	and	and	CCONJ
ejpam-2826	149	23	b	b	NOUN
ejpam-2826	149	24	such	such	ADJ
ejpam-2826	149	25	that	that	DET
ejpam-2826	149	26	f(x)∈a	f(x)∈a	NUM
ejpam-2826	149	27	,	,	PUNCT
ejpam-2826	149	28	y∈b	y∈b	NOUN
ejpam-2826	149	29	and	and	CCONJ
ejpam-2826	149	30	cl(a)∩cl(b)=φ	cl(a)∩cl(b)=φ	NOUN
ejpam-2826	149	31	.	.	PUNCT
ejpam-2826	150	1	since	since	SCONJ
ejpam-2826	150	2	f	f	PROPN
ejpam-2826	150	3	is	be	AUX
ejpam-2826	150	4	contra	contra	PROPN
ejpam-2826	150	5	δgb	δgb	ADV
ejpam-2826	150	6	-	-	PUNCT
ejpam-2826	150	7	continuous	continuous	ADJ
ejpam-2826	150	8	then	then	ADV
ejpam-2826	150	9	there	there	PRON
ejpam-2826	150	10	exists	exist	VERB
ejpam-2826	150	11	u∈δgbo(x	u∈δgbo(x	PROPN
ejpam-2826	150	12	,	,	PUNCT
ejpam-2826	150	13	x	x	NOUN
ejpam-2826	150	14	)	)	PUNCT
ejpam-2826	150	15	such	such	ADJ
ejpam-2826	150	16	that	that	DET
ejpam-2826	150	17	f(u)⊆cl(a	f(u)⊆cl(a	NUM
ejpam-2826	150	18	)	)	PUNCT
ejpam-2826	150	19	.	.	PUNCT
ejpam-2826	151	1	therefore	therefore	ADV
ejpam-2826	151	2	we	we	PRON
ejpam-2826	151	3	obtain	obtain	VERB
ejpam-2826	151	4	f(u)∩cl(b)=φ	f(u)∩cl(b)=φ	NOUN
ejpam-2826	151	5	.	.	PUNCT
ejpam-2826	152	1	this	this	PRON
ejpam-2826	152	2	shows	show	VERB
ejpam-2826	152	3	that	that	SCONJ
ejpam-2826	152	4	g(f	g(f	NOUN
ejpam-2826	152	5	)	)	PUNCT
ejpam-2826	152	6	is	be	AUX
ejpam-2826	152	7	contra	contra	PROPN
ejpam-2826	152	8	δgb	δgb	PROPN
ejpam-2826	152	9	-	-	PUNCT
ejpam-2826	152	10	closed	closed	ADJ
ejpam-2826	152	11	.	.	PUNCT
ejpam-2826	153	1	s.s.benchalli	s.s.benchalli	PROPN
ejpam-2826	153	2	,	,	PUNCT
ejpam-2826	153	3	p.g.patil	p.g.patil	PROPN
ejpam-2826	153	4	,	,	PUNCT
ejpam-2826	153	5	j.b.toranagatti	j.b.toranagatti	PROPN
ejpam-2826	153	6	,	,	PUNCT
ejpam-2826	153	7	s.r.vighneshi	s.r.vighneshi	PROPN
ejpam-2826	153	8	/	/	SYM
ejpam-2826	153	9	eur	eur	PROPN
ejpam-2826	153	10	.	.	PUNCT
ejpam-2826	154	1	j.	j.	PROPN
ejpam-2826	154	2	pure	pure	PROPN
ejpam-2826	154	3	appl	appl	PROPN
ejpam-2826	154	4	.	.	PROPN
ejpam-2826	154	5	math	math	PROPN
ejpam-2826	154	6	,	,	PUNCT
ejpam-2826	154	7	10	10	NUM
ejpam-2826	154	8	(	(	PUNCT
ejpam-2826	154	9	2	2	NUM
ejpam-2826	154	10	)	)	PUNCT
ejpam-2826	154	11	(	(	PUNCT
ejpam-2826	154	12	2017	2017	NUM
ejpam-2826	154	13	)	)	PUNCT
ejpam-2826	154	14	,	,	PUNCT
ejpam-2826	154	15	312	312	NUM
ejpam-2826	154	16	-	-	SYM
ejpam-2826	154	17	322	322	NUM
ejpam-2826	154	18	318	318	NUM
ejpam-2826	154	19	theorem	theorem	NOUN
ejpam-2826	154	20	18	18	NUM
ejpam-2826	154	21	.	.	PUNCT
ejpam-2826	155	1	if	if	SCONJ
ejpam-2826	155	2	f	f	X
ejpam-2826	155	3	:	:	PUNCT
ejpam-2826	155	4	x→y	x→y	NUM
ejpam-2826	155	5	is	be	AUX
ejpam-2826	155	6	δgb	δgb	ADV
ejpam-2826	155	7	-	-	PUNCT
ejpam-2826	155	8	continuous	continuous	ADJ
ejpam-2826	155	9	and	and	CCONJ
ejpam-2826	155	10	y	y	PROPN
ejpam-2826	155	11	is	be	AUX
ejpam-2826	155	12	t1	t1	NOUN
ejpam-2826	155	13	then	then	ADV
ejpam-2826	155	14	g(f	g(f	PROPN
ejpam-2826	155	15	)	)	PUNCT
ejpam-2826	155	16	is	be	AUX
ejpam-2826	155	17	contra	contra	PROPN
ejpam-2826	155	18	δgb	δgb	PROPN
ejpam-2826	155	19	-	-	PUNCT
ejpam-2826	155	20	closed	close	VERB
ejpam-2826	155	21	in	in	ADP
ejpam-2826	155	22	x×y	x×y	PROPN
ejpam-2826	155	23	.	.	PUNCT
ejpam-2826	156	1	proof	proof	NOUN
ejpam-2826	156	2	:	:	PUNCT
ejpam-2826	156	3	let	let	VERB
ejpam-2826	156	4	(	(	PUNCT
ejpam-2826	156	5	x	x	NOUN
ejpam-2826	156	6	,	,	PUNCT
ejpam-2826	156	7	y)∈(x×y)-g(f	y)∈(x×y)-g(f	PROPN
ejpam-2826	156	8	)	)	PUNCT
ejpam-2826	156	9	then	then	ADV
ejpam-2826	156	10	y	y	PROPN
ejpam-2826	156	11	6	6	NUM
ejpam-2826	156	12	=	=	SYM
ejpam-2826	156	13	f(x	f(x	PROPN
ejpam-2826	156	14	)	)	PUNCT
ejpam-2826	156	15	and	and	CCONJ
ejpam-2826	156	16	there	there	PRON
ejpam-2826	156	17	exists	exist	VERB
ejpam-2826	156	18	open	open	ADJ
ejpam-2826	156	19	set	set	VERB
ejpam-2826	156	20	u	u	PRON
ejpam-2826	156	21	such	such	ADJ
ejpam-2826	156	22	that	that	DET
ejpam-2826	156	23	f(x)∈u	f(x)∈u	PROPN
ejpam-2826	156	24	and	and	CCONJ
ejpam-2826	156	25	y/∈u	y/∈u	PROPN
ejpam-2826	156	26	.	.	PUNCT
ejpam-2826	157	1	since	since	SCONJ
ejpam-2826	157	2	f	f	PROPN
ejpam-2826	157	3	is	be	AUX
ejpam-2826	157	4	δgb	δgb	ADV
ejpam-2826	157	5	-	-	PUNCT
ejpam-2826	157	6	continuous	continuous	ADJ
ejpam-2826	157	7	,	,	PUNCT
ejpam-2826	157	8	then	then	ADV
ejpam-2826	157	9	there	there	PRON
ejpam-2826	157	10	exists	exist	VERB
ejpam-2826	157	11	v∈δgbo(x	v∈δgbo(x	PROPN
ejpam-2826	157	12	,	,	PUNCT
ejpam-2826	157	13	x	x	NOUN
ejpam-2826	157	14	)	)	PUNCT
ejpam-2826	157	15	such	such	ADJ
ejpam-2826	157	16	that	that	SCONJ
ejpam-2826	157	17	f(v)⊆u.therefore	f(v)⊆u.therefore	NOUN
ejpam-2826	157	18	we	we	PRON
ejpam-2826	157	19	obtain	obtain	VERB
ejpam-2826	157	20	f(v)∩(y	f(v)∩(y	NOUN
ejpam-2826	157	21	-	-	PUNCT
ejpam-2826	157	22	u)=φ	u)=φ	ADJ
ejpam-2826	157	23	and	and	CCONJ
ejpam-2826	157	24	y	y	PROPN
ejpam-2826	157	25	-	-	PUNCT
ejpam-2826	157	26	u∈c(y	u∈c(y	PROPN
ejpam-2826	157	27	,	,	PUNCT
ejpam-2826	157	28	y).this	y).this	PROPN
ejpam-2826	157	29	shows	show	VERB
ejpam-2826	157	30	that	that	SCONJ
ejpam-2826	157	31	g(f	g(f	NOUN
ejpam-2826	157	32	)	)	PUNCT
ejpam-2826	157	33	is	be	AUX
ejpam-2826	157	34	contra	contra	PROPN
ejpam-2826	157	35	δgb	δgb	PROPN
ejpam-2826	157	36	-	-	PUNCT
ejpam-2826	157	37	closed	closed	ADJ
ejpam-2826	157	38	.	.	PUNCT
ejpam-2826	158	1	theorem	theorem	NOUN
ejpam-2826	158	2	19	19	NUM
ejpam-2826	158	3	.	.	PUNCT
ejpam-2826	159	1	let	let	VERB
ejpam-2826	159	2	f	f	X
ejpam-2826	159	3	:	:	PUNCT
ejpam-2826	159	4	x→y	x→y	NUM
ejpam-2826	159	5	be	be	AUX
ejpam-2826	159	6	a	a	DET
ejpam-2826	159	7	function	function	NOUN
ejpam-2826	159	8	and	and	CCONJ
ejpam-2826	159	9	g	g	NOUN
ejpam-2826	159	10	:	:	PUNCT
ejpam-2826	159	11	x→x×y	x→x×y	PROPN
ejpam-2826	159	12	be	be	AUX
ejpam-2826	159	13	the	the	DET
ejpam-2826	159	14	graph	graph	NOUN
ejpam-2826	159	15	function	function	NOUN
ejpam-2826	159	16	of	of	ADP
ejpam-2826	159	17	f	f	PROPN
ejpam-2826	159	18	defined	define	VERB
ejpam-2826	159	19	by	by	ADP
ejpam-2826	159	20	g(x)=(x	g(x)=(x	PROPN
ejpam-2826	159	21	,	,	PUNCT
ejpam-2826	159	22	f(x	f(x	PROPN
ejpam-2826	159	23	)	)	PUNCT
ejpam-2826	159	24	)	)	PUNCT
ejpam-2826	159	25	for	for	ADP
ejpam-2826	159	26	each	each	DET
ejpam-2826	159	27	x∈x	x∈x	NOUN
ejpam-2826	159	28	.	.	PUNCT
ejpam-2826	160	1	if	if	SCONJ
ejpam-2826	160	2	g	g	PROPN
ejpam-2826	160	3	is	be	AUX
ejpam-2826	160	4	contra	contra	PROPN
ejpam-2826	160	5	δgb	δgb	ADV
ejpam-2826	160	6	-	-	PUNCT
ejpam-2826	160	7	continuous	continuous	ADJ
ejpam-2826	160	8	then	then	ADV
ejpam-2826	160	9	f	f	PROPN
ejpam-2826	160	10	is	be	AUX
ejpam-2826	160	11	contra	contra	PROPN
ejpam-2826	160	12	δgb	δgb	ADV
ejpam-2826	160	13	-	-	PUNCT
ejpam-2826	160	14	continuous	continuous	ADJ
ejpam-2826	160	15	.	.	PUNCT
ejpam-2826	161	1	proof	proof	NOUN
ejpam-2826	161	2	:	:	PUNCT
ejpam-2826	161	3	let	let	VERB
ejpam-2826	161	4	u	u	PRON
ejpam-2826	161	5	be	be	AUX
ejpam-2826	161	6	an	an	DET
ejpam-2826	161	7	open	open	ADJ
ejpam-2826	161	8	set	set	NOUN
ejpam-2826	161	9	in	in	ADP
ejpam-2826	161	10	y	y	PROPN
ejpam-2826	161	11	then	then	ADV
ejpam-2826	161	12	x×u	x×u	PROPN
ejpam-2826	161	13	is	be	AUX
ejpam-2826	161	14	an	an	DET
ejpam-2826	161	15	open	open	ADJ
ejpam-2826	161	16	set	set	NOUN
ejpam-2826	161	17	in	in	ADP
ejpam-2826	161	18	x×y	x×y	PROPN
ejpam-2826	161	19	.	.	PUNCT
ejpam-2826	162	1	since	since	SCONJ
ejpam-2826	162	2	g	g	PROPN
ejpam-2826	162	3	is	be	AUX
ejpam-2826	162	4	contra	contra	PROPN
ejpam-2826	162	5	δgb	δgb	ADV
ejpam-2826	162	6	-	-	PUNCT
ejpam-2826	162	7	continuous	continuous	ADJ
ejpam-2826	162	8	.	.	PUNCT
ejpam-2826	163	1	it	it	PRON
ejpam-2826	163	2	follows	follow	VERB
ejpam-2826	163	3	that	that	SCONJ
ejpam-2826	163	4	f−1(u)=g−1(x×u	f−1(u)=g−1(x×u	NOUN
ejpam-2826	163	5	)	)	PUNCT
ejpam-2826	163	6	is	be	AUX
ejpam-2826	163	7	δgb	δgb	ADV
ejpam-2826	163	8	-	-	PUNCT
ejpam-2826	163	9	closed	closed	ADJ
ejpam-2826	163	10	in	in	ADP
ejpam-2826	163	11	x.	x.	NOUN
ejpam-2826	164	1	thus	thus	ADV
ejpam-2826	164	2	f	f	PROPN
ejpam-2826	164	3	is	be	AUX
ejpam-2826	164	4	contra	contra	PROPN
ejpam-2826	164	5	δgb	δgb	ADV
ejpam-2826	164	6	-	-	PUNCT
ejpam-2826	164	7	continuous	continuous	ADJ
ejpam-2826	164	8	.	.	PUNCT
ejpam-2826	165	1	theorem	theorem	NOUN
ejpam-2826	165	2	20	20	NUM
ejpam-2826	165	3	.	.	PUNCT
ejpam-2826	166	1	if	if	SCONJ
ejpam-2826	166	2	f	f	X
ejpam-2826	166	3	:	:	PUNCT
ejpam-2826	166	4	x→y	x→y	NUM
ejpam-2826	166	5	is	be	AUX
ejpam-2826	166	6	contra	contra	PROPN
ejpam-2826	166	7	δgb	δgb	PROPN
ejpam-2826	166	8	-	-	PUNCT
ejpam-2826	166	9	continuous	continuous	ADJ
ejpam-2826	166	10	then	then	ADV
ejpam-2826	166	11	for	for	ADP
ejpam-2826	166	12	each	each	DET
ejpam-2826	166	13	x∈x	x∈x	NOUN
ejpam-2826	166	14	and	and	CCONJ
ejpam-2826	166	15	for	for	ADP
ejpam-2826	166	16	each	each	DET
ejpam-2826	166	17	closed	close	VERB
ejpam-2826	166	18	set	set	VERB
ejpam-2826	166	19	v	v	NOUN
ejpam-2826	166	20	in	in	ADP
ejpam-2826	166	21	y	y	NOUN
ejpam-2826	166	22	with	with	ADP
ejpam-2826	166	23	f(x)∈v	f(x)∈v	PRON
ejpam-2826	166	24	there	there	PRON
ejpam-2826	166	25	exists	exist	VERB
ejpam-2826	166	26	a	a	DET
ejpam-2826	166	27	δgb	δgb	ADV
ejpam-2826	166	28	-	-	PUNCT
ejpam-2826	166	29	open	open	ADJ
ejpam-2826	166	30	set	set	NOUN
ejpam-2826	166	31	u	u	NOUN
ejpam-2826	166	32	in	in	ADP
ejpam-2826	166	33	x	x	PUNCT
ejpam-2826	166	34	containing	contain	VERB
ejpam-2826	166	35	x	x	PUNCT
ejpam-2826	166	36	such	such	ADJ
ejpam-2826	166	37	that	that	DET
ejpam-2826	166	38	f(u)⊆v	f(u)⊆v	NOUN
ejpam-2826	166	39	.	.	PUNCT
ejpam-2826	167	1	proof	proof	NOUN
ejpam-2826	167	2	:	:	PUNCT
ejpam-2826	167	3	let	let	VERB
ejpam-2826	167	4	x∈x	x∈x	NOUN
ejpam-2826	167	5	and	and	CCONJ
ejpam-2826	167	6	v	v	NOUN
ejpam-2826	167	7	is	be	AUX
ejpam-2826	167	8	a	a	DET
ejpam-2826	167	9	closed	closed	ADJ
ejpam-2826	167	10	set	set	NOUN
ejpam-2826	167	11	in	in	ADP
ejpam-2826	167	12	y	y	PROPN
ejpam-2826	167	13	with	with	ADP
ejpam-2826	167	14	f(x)∈v	f(x)∈v	PROPN
ejpam-2826	167	15	then	then	ADV
ejpam-2826	167	16	x∈f−1(v	x∈f−1(v	PROPN
ejpam-2826	167	17	)	)	PUNCT
ejpam-2826	167	18	.	.	PUNCT
ejpam-2826	168	1	since	since	SCONJ
ejpam-2826	168	2	f	f	PROPN
ejpam-2826	168	3	is	be	AUX
ejpam-2826	168	4	contra	contra	PROPN
ejpam-2826	168	5	δgbcontinuous	δgbcontinuous	ADJ
ejpam-2826	168	6	,	,	PUNCT
ejpam-2826	168	7	f−1(v	f−1(v	PROPN
ejpam-2826	168	8	)	)	PUNCT
ejpam-2826	168	9	is	be	AUX
ejpam-2826	168	10	δgb	δgb	ADV
ejpam-2826	168	11	-	-	PUNCT
ejpam-2826	168	12	open	open	ADJ
ejpam-2826	168	13	in	in	ADP
ejpam-2826	168	14	x.	x.	NOUN
ejpam-2826	168	15	put	put	VERB
ejpam-2826	168	16	u	u	NOUN
ejpam-2826	168	17	=	=	NOUN
ejpam-2826	168	18	f−1(v	f−1(v	NOUN
ejpam-2826	168	19	)	)	PUNCT
ejpam-2826	169	1	then	then	ADV
ejpam-2826	169	2	x∈u	x∈u	PROPN
ejpam-2826	169	3	and	and	CCONJ
ejpam-2826	169	4	f(u)=f(f−1(v))⊆v	f(u)=f(f−1(v))⊆v	PROPN
ejpam-2826	169	5	.	.	PUNCT
ejpam-2826	170	1	definition	definition	NOUN
ejpam-2826	170	2	10	10	NUM
ejpam-2826	170	3	.	.	PUNCT
ejpam-2826	171	1	[	[	X
ejpam-2826	171	2	16	16	NUM
ejpam-2826	171	3	]	]	PUNCT
ejpam-2826	171	4	a	a	DET
ejpam-2826	171	5	space	space	NOUN
ejpam-2826	171	6	x	x	PUNCT
ejpam-2826	171	7	is	be	AUX
ejpam-2826	171	8	submaximal	submaximal	ADJ
ejpam-2826	171	9	and	and	CCONJ
ejpam-2826	171	10	extremally	extremally	ADV
ejpam-2826	171	11	disconnected	disconnected	ADJ
ejpam-2826	171	12	if	if	SCONJ
ejpam-2826	171	13	every	every	DET
ejpam-2826	171	14	b	b	X
ejpam-2826	171	15	-	-	PUNCT
ejpam-2826	171	16	open	open	ADJ
ejpam-2826	171	17	set	set	NOUN
ejpam-2826	171	18	is	be	AUX
ejpam-2826	171	19	open	open	ADJ
ejpam-2826	171	20	.	.	PUNCT
ejpam-2826	172	1	theorem	theorem	NOUN
ejpam-2826	172	2	21	21	NUM
ejpam-2826	172	3	.	.	PUNCT
ejpam-2826	173	1	if	if	SCONJ
ejpam-2826	173	2	a	a	PRON
ejpam-2826	173	3	and	and	CCONJ
ejpam-2826	173	4	b	b	NOUN
ejpam-2826	173	5	are	be	AUX
ejpam-2826	173	6	δgb	δgb	ADV
ejpam-2826	173	7	-	-	PUNCT
ejpam-2826	173	8	closed	close	VERB
ejpam-2826	173	9	sets	set	NOUN
ejpam-2826	173	10	in	in	ADP
ejpam-2826	173	11	submaximal	submaximal	ADJ
ejpam-2826	173	12	and	and	CCONJ
ejpam-2826	173	13	extremally	extremally	ADV
ejpam-2826	173	14	disconnected	disconnected	ADJ
ejpam-2826	173	15	space	space	NOUN
ejpam-2826	173	16	x	x	PUNCT
ejpam-2826	173	17	then	then	ADV
ejpam-2826	173	18	a∪b	a∪b	ADJ
ejpam-2826	173	19	is	be	AUX
ejpam-2826	173	20	δgb	δgb	ADV
ejpam-2826	173	21	-	-	PUNCT
ejpam-2826	173	22	closed	closed	ADJ
ejpam-2826	173	23	in	in	ADP
ejpam-2826	173	24	x.	x.	NOUN
ejpam-2826	173	25	proof	proof	NOUN
ejpam-2826	173	26	:	:	PUNCT
ejpam-2826	173	27	let	let	VERB
ejpam-2826	173	28	a∪b⊆g	a∪b⊆g	NOUN
ejpam-2826	173	29	where	where	SCONJ
ejpam-2826	173	30	g	g	PROPN
ejpam-2826	173	31	is	be	AUX
ejpam-2826	173	32	δ	δ	NOUN
ejpam-2826	173	33	-	-	ADJ
ejpam-2826	173	34	open	open	ADJ
ejpam-2826	173	35	in	in	ADP
ejpam-2826	173	36	x.	x.	NOUN
ejpam-2826	173	37	since	since	SCONJ
ejpam-2826	173	38	a⊆g	a⊆g	PROPN
ejpam-2826	173	39	,	,	PUNCT
ejpam-2826	173	40	b⊆g	b⊆g	PROPN
ejpam-2826	173	41	,	,	PUNCT
ejpam-2826	173	42	a	a	PRON
ejpam-2826	173	43	and	and	CCONJ
ejpam-2826	173	44	b	b	NOUN
ejpam-2826	173	45	are	be	AUX
ejpam-2826	173	46	δgb	δgb	ADV
ejpam-2826	173	47	-	-	PUNCT
ejpam-2826	173	48	closed	close	VERB
ejpam-2826	173	49	sets	set	NOUN
ejpam-2826	173	50	then	then	ADV
ejpam-2826	173	51	bcl(a)⊆g	bcl(a)⊆g	PUNCT
ejpam-2826	173	52	and	and	CCONJ
ejpam-2826	173	53	bcl(b)⊆g	bcl(b)⊆g	NOUN
ejpam-2826	173	54	.	.	PUNCT
ejpam-2826	174	1	as	as	SCONJ
ejpam-2826	174	2	x	x	PRON
ejpam-2826	174	3	is	be	AUX
ejpam-2826	174	4	submaximal	submaximal	ADJ
ejpam-2826	174	5	and	and	CCONJ
ejpam-2826	174	6	extremally	extremally	ADV
ejpam-2826	174	7	disconnected	disconnected	ADJ
ejpam-2826	174	8	,	,	PUNCT
ejpam-2826	174	9	bcl(m)=cl(m	bcl(m)=cl(m	PROPN
ejpam-2826	174	10	)	)	PUNCT
ejpam-2826	174	11	for	for	ADP
ejpam-2826	174	12	any	any	DET
ejpam-2826	174	13	m⊆x.therefore	m⊆x.therefore	PROPN
ejpam-2826	174	14	bcl(a∪b)=bcl(a)∪bcl(b)⊆g	bcl(a∪b)=bcl(a)∪bcl(b)⊆g	NOUN
ejpam-2826	174	15	and	and	CCONJ
ejpam-2826	174	16	hence	hence	ADV
ejpam-2826	174	17	a∪b	a∪b	ADJ
ejpam-2826	174	18	is	be	AUX
ejpam-2826	174	19	δgb	δgb	ADV
ejpam-2826	174	20	-	-	PUNCT
ejpam-2826	174	21	closed	closed	ADJ
ejpam-2826	174	22	.	.	PUNCT
ejpam-2826	175	1	corollary	corollary	ADJ
ejpam-2826	175	2	1	1	NUM
ejpam-2826	175	3	.	.	PUNCT
ejpam-2826	176	1	if	if	SCONJ
ejpam-2826	176	2	a	a	PRON
ejpam-2826	176	3	and	and	CCONJ
ejpam-2826	176	4	b	b	NOUN
ejpam-2826	176	5	are	be	AUX
ejpam-2826	176	6	δgb	δgb	ADV
ejpam-2826	176	7	-	-	PUNCT
ejpam-2826	176	8	open	open	ADJ
ejpam-2826	176	9	sets	set	NOUN
ejpam-2826	176	10	in	in	ADP
ejpam-2826	176	11	submaximal	submaximal	ADJ
ejpam-2826	176	12	and	and	CCONJ
ejpam-2826	176	13	extremally	extremally	ADV
ejpam-2826	176	14	disconnected	disconnected	ADJ
ejpam-2826	176	15	space	space	NOUN
ejpam-2826	176	16	x	x	X
ejpam-2826	176	17	then	then	ADV
ejpam-2826	176	18	a∩b	a∩b	PROPN
ejpam-2826	176	19	is	be	AUX
ejpam-2826	176	20	δgb	δgb	ADV
ejpam-2826	176	21	-	-	PUNCT
ejpam-2826	176	22	open	open	ADJ
ejpam-2826	176	23	in	in	ADP
ejpam-2826	176	24	x.	x.	NOUN
ejpam-2826	176	25	theorem	theorem	VERB
ejpam-2826	176	26	22	22	NUM
ejpam-2826	176	27	.	.	PUNCT
ejpam-2826	177	1	:	:	PUNCT
ejpam-2826	177	2	suppose	suppose	VERB
ejpam-2826	177	3	that	that	SCONJ
ejpam-2826	177	4	δgbc(x	δgbc(x	NOUN
ejpam-2826	177	5	)	)	PUNCT
ejpam-2826	177	6	is	be	AUX
ejpam-2826	177	7	closed	close	VERB
ejpam-2826	177	8	under	under	ADP
ejpam-2826	177	9	arbitrary	arbitrary	ADJ
ejpam-2826	177	10	intersections	intersection	NOUN
ejpam-2826	177	11	then	then	ADV
ejpam-2826	177	12	a⊆x	a⊆x	NOUN
ejpam-2826	177	13	is	be	AUX
ejpam-2826	177	14	δgb	δgb	ADV
ejpam-2826	177	15	-	-	PUNCT
ejpam-2826	177	16	closed	closed	ADJ
ejpam-2826	177	17	if	if	SCONJ
ejpam-2826	177	18	and	and	CCONJ
ejpam-2826	177	19	only	only	ADV
ejpam-2826	177	20	if	if	SCONJ
ejpam-2826	177	21	a	a	DET
ejpam-2826	177	22	=	=	NOUN
ejpam-2826	177	23	δgbcl(a	δgbcl(a	NOUN
ejpam-2826	177	24	)	)	PUNCT
ejpam-2826	177	25	theorem	theorem	VERB
ejpam-2826	177	26	23	23	NUM
ejpam-2826	177	27	.	.	PUNCT
ejpam-2826	177	28	suppose	suppose	VERB
ejpam-2826	177	29	that	that	SCONJ
ejpam-2826	177	30	δgbc(x	δgbc(x	NOUN
ejpam-2826	177	31	)	)	PUNCT
ejpam-2826	177	32	is	be	AUX
ejpam-2826	177	33	closed	close	VERB
ejpam-2826	177	34	under	under	ADP
ejpam-2826	177	35	arbitrary	arbitrary	ADJ
ejpam-2826	177	36	intersections	intersection	NOUN
ejpam-2826	177	37	.	.	PUNCT
ejpam-2826	178	1	if	if	SCONJ
ejpam-2826	178	2	f	f	X
ejpam-2826	178	3	:	:	PUNCT
ejpam-2826	178	4	x→y	x→y	NUM
ejpam-2826	178	5	and	and	CCONJ
ejpam-2826	178	6	g	g	NOUN
ejpam-2826	178	7	:	:	PUNCT
ejpam-2826	178	8	x→y	x→y	NUM
ejpam-2826	178	9	are	be	AUX
ejpam-2826	178	10	contra	contra	PROPN
ejpam-2826	178	11	δgb	δgb	ADV
ejpam-2826	178	12	-	-	PUNCT
ejpam-2826	178	13	continuous	continuous	ADJ
ejpam-2826	178	14	,	,	PUNCT
ejpam-2826	178	15	y	y	PROPN
ejpam-2826	178	16	is	be	AUX
ejpam-2826	178	17	urysohn	urysohn	PROPN
ejpam-2826	178	18	and	and	CCONJ
ejpam-2826	178	19	x	x	NOUN
ejpam-2826	178	20	is	be	AUX
ejpam-2826	178	21	submaximal	submaximal	ADJ
ejpam-2826	178	22	and	and	CCONJ
ejpam-2826	178	23	extremally	extremally	ADV
ejpam-2826	178	24	disconnected	disconnect	VERB
ejpam-2826	178	25	,	,	PUNCT
ejpam-2826	178	26	then	then	ADV
ejpam-2826	178	27	k={x∈x	k={x∈x	PROPN
ejpam-2826	178	28	:	:	PUNCT
ejpam-2826	178	29	f(x)=g(x	f(x)=g(x	NUM
ejpam-2826	178	30	)	)	PUNCT
ejpam-2826	178	31	}	}	PUNCT
ejpam-2826	178	32	δ	δ	PROPN
ejpam-2826	178	33	is	be	AUX
ejpam-2826	178	34	gb	gb	ADV
ejpam-2826	178	35	-	-	PUNCT
ejpam-2826	178	36	closed	closed	ADJ
ejpam-2826	178	37	in	in	ADP
ejpam-2826	178	38	x.	x.	NOUN
ejpam-2826	178	39	proof	proof	NOUN
ejpam-2826	178	40	:	:	PUNCT
ejpam-2826	178	41	let	let	VERB
ejpam-2826	178	42	x∈x-k.then	x∈x-k.then	PROPN
ejpam-2826	178	43	f(x	f(x	PROPN
ejpam-2826	178	44	)	)	PUNCT
ejpam-2826	178	45	6	6	NUM
ejpam-2826	178	46	=	=	SYM
ejpam-2826	178	47	g(x).since	g(x).since	PROPN
ejpam-2826	178	48	y	y	PROPN
ejpam-2826	178	49	is	be	AUX
ejpam-2826	178	50	urysohn	urysohn	PRON
ejpam-2826	178	51	there	there	PRON
ejpam-2826	178	52	exist	exist	VERB
ejpam-2826	178	53	open	open	ADJ
ejpam-2826	178	54	sets	set	NOUN
ejpam-2826	178	55	u	u	NOUN
ejpam-2826	178	56	and	and	CCONJ
ejpam-2826	178	57	v	v	ADP
ejpam-2826	178	58	such	such	ADJ
ejpam-2826	178	59	that	that	DET
ejpam-2826	178	60	f(x)∈u	f(x)∈u	NOUN
ejpam-2826	178	61	,	,	PUNCT
ejpam-2826	178	62	g(x)∈v	g(x)∈v	NUM
ejpam-2826	178	63	and	and	CCONJ
ejpam-2826	178	64	cl(u)∩cl(v)=φ	cl(u)∩cl(v)=φ	ADJ
ejpam-2826	178	65	.	.	PUNCT
ejpam-2826	179	1	since	since	SCONJ
ejpam-2826	179	2	f	f	PROPN
ejpam-2826	179	3	and	and	CCONJ
ejpam-2826	179	4	g	g	PROPN
ejpam-2826	179	5	are	be	AUX
ejpam-2826	179	6	contra	contra	PROPN
ejpam-2826	179	7	δgb	δgb	ADV
ejpam-2826	179	8	-	-	PUNCT
ejpam-2826	179	9	continuous	continuous	ADJ
ejpam-2826	179	10	,	,	PUNCT
ejpam-2826	179	11	f−1(cl(u	f−1(cl(u	NUM
ejpam-2826	179	12	)	)	PUNCT
ejpam-2826	179	13	)	)	PUNCT
ejpam-2826	179	14	and	and	CCONJ
ejpam-2826	179	15	g−1(cl(v	g−1(cl(v	NOUN
ejpam-2826	179	16	)	)	PUNCT
ejpam-2826	179	17	)	)	PUNCT
ejpam-2826	179	18	are	be	AUX
ejpam-2826	179	19	δgb	δgb	ADV
ejpam-2826	179	20	-	-	PUNCT
ejpam-2826	179	21	open	open	ADJ
ejpam-2826	179	22	sets	set	NOUN
ejpam-2826	179	23	in	in	ADP
ejpam-2826	179	24	x.	x.	NOUN
ejpam-2826	179	25	let	let	VERB
ejpam-2826	179	26	a	a	DET
ejpam-2826	179	27	=	=	NOUN
ejpam-2826	179	28	f−1(cl(u	f−1(cl(u	NOUN
ejpam-2826	179	29	)	)	PUNCT
ejpam-2826	179	30	)	)	PUNCT
ejpam-2826	179	31	and	and	CCONJ
ejpam-2826	179	32	b	b	X
ejpam-2826	179	33	=	=	NOUN
ejpam-2826	179	34	g−1(cl(v	g−1(cl(v	NOUN
ejpam-2826	179	35	)	)	PUNCT
ejpam-2826	179	36	)	)	PUNCT
ejpam-2826	179	37	.	.	PUNCT
ejpam-2826	180	1	then	then	ADV
ejpam-2826	180	2	a	a	PRON
ejpam-2826	180	3	and	and	CCONJ
ejpam-2826	180	4	b	b	NOUN
ejpam-2826	180	5	are	be	AUX
ejpam-2826	180	6	δgb	δgb	ADV
ejpam-2826	180	7	-	-	PUNCT
ejpam-2826	180	8	open	open	ADJ
ejpam-2826	180	9	sets	set	NOUN
ejpam-2826	180	10	containing	contain	VERB
ejpam-2826	180	11	x.set	x.set	PROPN
ejpam-2826	181	1	c	c	X
ejpam-2826	181	2	=	=	SYM
ejpam-2826	181	3	a∩b	a∩b	PROPN
ejpam-2826	181	4	,	,	PUNCT
ejpam-2826	181	5	then	then	ADV
ejpam-2826	181	6	c	c	PROPN
ejpam-2826	181	7	is	be	AUX
ejpam-2826	181	8	δgb	δgb	ADV
ejpam-2826	181	9	-	-	PUNCT
ejpam-2826	181	10	open	open	ADJ
ejpam-2826	181	11	set	set	NOUN
ejpam-2826	181	12	in	in	ADP
ejpam-2826	181	13	x.	x.	NOUN
ejpam-2826	181	14	hence	hence	ADV
ejpam-2826	181	15	f(c)∩g(c)=f(a∩b)∩g(a∩b)⊆f(a)∩g(b)=cl(u)∩cl(v)=φ	f(c)∩g(c)=f(a∩b)∩g(a∩b)⊆f(a)∩g(b)=cl(u)∩cl(v)=φ	PUNCT
ejpam-2826	181	16	.	.	PUNCT
ejpam-2826	182	1	therefore	therefore	ADV
ejpam-2826	182	2	c∩k	c∩k	VERB
ejpam-2826	182	3	=	=	PROPN
ejpam-2826	182	4	φ	φ	PROPN
ejpam-2826	182	5	.	.	PUNCT
ejpam-2826	183	1	by	by	ADP
ejpam-2826	183	2	theorem	theorem	NOUN
ejpam-2826	183	3	5	5	NUM
ejpam-2826	183	4	,	,	PUNCT
ejpam-2826	183	5	x/∈δgbcl(k	x/∈δgbcl(k	PROPN
ejpam-2826	183	6	)	)	PUNCT
ejpam-2826	183	7	.	.	PUNCT
ejpam-2826	184	1	hence	hence	ADV
ejpam-2826	184	2	k	k	PROPN
ejpam-2826	184	3	is	be	AUX
ejpam-2826	184	4	δgb	δgb	ADV
ejpam-2826	184	5	-	-	PUNCT
ejpam-2826	184	6	closed	close	VERB
ejpam-2826	184	7	in	in	ADP
ejpam-2826	184	8	x.	x.	NOUN
ejpam-2826	184	9	definition	definition	NOUN
ejpam-2826	184	10	11	11	NUM
ejpam-2826	184	11	.	.	PUNCT
ejpam-2826	185	1	a	a	DET
ejpam-2826	185	2	space	space	NOUN
ejpam-2826	185	3	x	x	PUNCT
ejpam-2826	185	4	is	be	AUX
ejpam-2826	185	5	called	call	VERB
ejpam-2826	185	6	δgb	δgb	ADV
ejpam-2826	185	7	-	-	PUNCT
ejpam-2826	185	8	connected	connect	VERB
ejpam-2826	185	9	provided	provide	VERB
ejpam-2826	185	10	that	that	SCONJ
ejpam-2826	185	11	x	x	PRON
ejpam-2826	185	12	is	be	AUX
ejpam-2826	185	13	not	not	PART
ejpam-2826	185	14	the	the	DET
ejpam-2826	185	15	union	union	NOUN
ejpam-2826	185	16	of	of	ADP
ejpam-2826	185	17	two	two	NUM
ejpam-2826	185	18	disjoint	disjoint	NOUN
ejpam-2826	185	19	nonempty	nonempty	ADJ
ejpam-2826	185	20	δgb	δgb	ADJ
ejpam-2826	185	21	-	-	PUNCT
ejpam-2826	185	22	open	open	ADJ
ejpam-2826	185	23	sets	set	NOUN
ejpam-2826	185	24	.	.	PUNCT
ejpam-2826	186	1	theorem	theorem	VERB
ejpam-2826	186	2	24	24	NUM
ejpam-2826	186	3	.	.	PUNCT
ejpam-2826	187	1	if	if	SCONJ
ejpam-2826	187	2	f	f	X
ejpam-2826	187	3	:	:	PUNCT
ejpam-2826	187	4	x→y	x→y	NUM
ejpam-2826	187	5	is	be	AUX
ejpam-2826	187	6	a	a	DET
ejpam-2826	187	7	contra	contra	PROPN
ejpam-2826	187	8	δgb	δgb	ADJ
ejpam-2826	187	9	-	-	PUNCT
ejpam-2826	187	10	continuous	continuous	ADJ
ejpam-2826	187	11	function	function	NOUN
ejpam-2826	187	12	from	from	ADP
ejpam-2826	187	13	a	a	DET
ejpam-2826	187	14	δgbconnected	δgbconnecte	VERB
ejpam-2826	187	15	space	space	NOUN
ejpam-2826	187	16	x	x	X
ejpam-2826	187	17	onto	onto	ADP
ejpam-2826	187	18	any	any	DET
ejpam-2826	187	19	space	space	NOUN
ejpam-2826	187	20	y	y	PROPN
ejpam-2826	187	21	then	then	ADV
ejpam-2826	187	22	y	y	PROPN
ejpam-2826	187	23	is	be	AUX
ejpam-2826	187	24	not	not	PART
ejpam-2826	187	25	a	a	DET
ejpam-2826	187	26	discrete	discrete	ADJ
ejpam-2826	187	27	space	space	NOUN
ejpam-2826	187	28	.	.	PUNCT
ejpam-2826	188	1	s.s.benchalli	s.s.benchalli	PROPN
ejpam-2826	188	2	,	,	PUNCT
ejpam-2826	188	3	p.g.patil	p.g.patil	PROPN
ejpam-2826	188	4	,	,	PUNCT
ejpam-2826	188	5	j.b.toranagatti	j.b.toranagatti	PROPN
ejpam-2826	188	6	,	,	PUNCT
ejpam-2826	188	7	s.r.vighneshi	s.r.vighneshi	PROPN
ejpam-2826	188	8	/	/	SYM
ejpam-2826	188	9	eur	eur	PROPN
ejpam-2826	188	10	.	.	PUNCT
ejpam-2826	189	1	j.	j.	PROPN
ejpam-2826	189	2	pure	pure	PROPN
ejpam-2826	189	3	appl	appl	PROPN
ejpam-2826	189	4	.	.	PROPN
ejpam-2826	189	5	math	math	PROPN
ejpam-2826	189	6	,	,	PUNCT
ejpam-2826	189	7	10	10	NUM
ejpam-2826	189	8	(	(	PUNCT
ejpam-2826	189	9	2	2	NUM
ejpam-2826	189	10	)	)	PUNCT
ejpam-2826	189	11	(	(	PUNCT
ejpam-2826	189	12	2017	2017	NUM
ejpam-2826	189	13	)	)	PUNCT
ejpam-2826	189	14	,	,	PUNCT
ejpam-2826	189	15	312	312	NUM
ejpam-2826	189	16	-	-	SYM
ejpam-2826	189	17	322	322	NUM
ejpam-2826	189	18	319	319	NUM
ejpam-2826	189	19	proof	proof	NOUN
ejpam-2826	189	20	:	:	PUNCT
ejpam-2826	189	21	since	since	SCONJ
ejpam-2826	189	22	f	f	PROPN
ejpam-2826	189	23	is	be	AUX
ejpam-2826	189	24	contra	contra	PROPN
ejpam-2826	189	25	δgb	δgb	ADV
ejpam-2826	189	26	-	-	PUNCT
ejpam-2826	189	27	continuous	continuous	ADJ
ejpam-2826	189	28	and	and	CCONJ
ejpam-2826	189	29	x	x	NOUN
ejpam-2826	189	30	is	be	AUX
ejpam-2826	189	31	δgb	δgb	ADV
ejpam-2826	189	32	-	-	PUNCT
ejpam-2826	189	33	connected	connect	VERB
ejpam-2826	189	34	space	space	NOUN
ejpam-2826	189	35	.	.	PUNCT
ejpam-2826	190	1	suppose	suppose	VERB
ejpam-2826	190	2	y	y	PRON
ejpam-2826	190	3	is	be	AUX
ejpam-2826	190	4	a	a	DET
ejpam-2826	190	5	discrete	discrete	ADJ
ejpam-2826	190	6	space	space	NOUN
ejpam-2826	190	7	.	.	PUNCT
ejpam-2826	191	1	let	let	VERB
ejpam-2826	191	2	v	v	PART
ejpam-2826	191	3	be	be	AUX
ejpam-2826	191	4	a	a	DET
ejpam-2826	191	5	proper	proper	ADJ
ejpam-2826	191	6	non	non	ADJ
ejpam-2826	191	7	empty	empty	ADJ
ejpam-2826	191	8	open	open	ADJ
ejpam-2826	191	9	and	and	CCONJ
ejpam-2826	191	10	closed	closed	ADJ
ejpam-2826	191	11	subset	subset	NOUN
ejpam-2826	191	12	of	of	ADP
ejpam-2826	191	13	y.	y.	PROPN
ejpam-2826	191	14	then	then	ADV
ejpam-2826	191	15	f−1(v	f−1(v	PROPN
ejpam-2826	191	16	)	)	PUNCT
ejpam-2826	191	17	is	be	AUX
ejpam-2826	191	18	proper	proper	ADJ
ejpam-2826	191	19	nonempty	nonempty	ADJ
ejpam-2826	191	20	δgb	δgb	NOUN
ejpam-2826	191	21	-	-	PUNCT
ejpam-2826	191	22	open	open	ADJ
ejpam-2826	191	23	and	and	CCONJ
ejpam-2826	191	24	δgb	δgb	ADV
ejpam-2826	191	25	-	-	PUNCT
ejpam-2826	191	26	closed	close	VERB
ejpam-2826	191	27	subset	subset	NOUN
ejpam-2826	191	28	of	of	ADP
ejpam-2826	191	29	x	x	PRON
ejpam-2826	191	30	,	,	PUNCT
ejpam-2826	191	31	which	which	PRON
ejpam-2826	191	32	contradicts	contradict	VERB
ejpam-2826	191	33	the	the	DET
ejpam-2826	191	34	fact	fact	NOUN
ejpam-2826	191	35	that	that	SCONJ
ejpam-2826	191	36	x	x	PRON
ejpam-2826	191	37	is	be	AUX
ejpam-2826	191	38	δgb	δgb	ADV
ejpam-2826	191	39	-	-	PUNCT
ejpam-2826	191	40	connected	connect	VERB
ejpam-2826	191	41	space	space	NOUN
ejpam-2826	191	42	.	.	PUNCT
ejpam-2826	192	1	hence	hence	ADV
ejpam-2826	192	2	y	y	PROPN
ejpam-2826	192	3	is	be	AUX
ejpam-2826	192	4	not	not	PART
ejpam-2826	192	5	a	a	DET
ejpam-2826	192	6	discrete	discrete	ADJ
ejpam-2826	192	7	space	space	NOUN
ejpam-2826	192	8	.	.	PUNCT
ejpam-2826	193	1	theorem	theorem	VERB
ejpam-2826	193	2	25	25	NUM
ejpam-2826	193	3	.	.	PUNCT
ejpam-2826	194	1	if	if	SCONJ
ejpam-2826	194	2	f	f	X
ejpam-2826	194	3	:	:	PUNCT
ejpam-2826	194	4	x→y	x→y	NUM
ejpam-2826	194	5	is	be	AUX
ejpam-2826	194	6	a	a	DET
ejpam-2826	194	7	contra	contra	PROPN
ejpam-2826	194	8	δgb	δgb	ADJ
ejpam-2826	194	9	-	-	PUNCT
ejpam-2826	194	10	continuous	continuous	ADJ
ejpam-2826	194	11	surjection	surjection	NOUN
ejpam-2826	194	12	and	and	CCONJ
ejpam-2826	194	13	x	x	NOUN
ejpam-2826	194	14	is	be	AUX
ejpam-2826	194	15	δgb	δgb	ADV
ejpam-2826	194	16	-	-	PUNCT
ejpam-2826	194	17	connected	connect	VERB
ejpam-2826	194	18	space	space	NOUN
ejpam-2826	194	19	then	then	ADV
ejpam-2826	194	20	y	y	PROPN
ejpam-2826	194	21	is	be	AUX
ejpam-2826	194	22	connected	connect	VERB
ejpam-2826	194	23	.	.	PUNCT
ejpam-2826	195	1	proof	proof	NOUN
ejpam-2826	195	2	:	:	PUNCT
ejpam-2826	195	3	let	let	VERB
ejpam-2826	195	4	f	f	X
ejpam-2826	195	5	:	:	PUNCT
ejpam-2826	195	6	x→y	x→y	NUM
ejpam-2826	195	7	is	be	AUX
ejpam-2826	195	8	a	a	DET
ejpam-2826	195	9	contra	contra	PROPN
ejpam-2826	195	10	δgb	δgb	ADV
ejpam-2826	195	11	-	-	PUNCT
ejpam-2826	195	12	continuous	continuous	ADJ
ejpam-2826	195	13	and	and	CCONJ
ejpam-2826	195	14	x	x	NOUN
ejpam-2826	195	15	is	be	AUX
ejpam-2826	195	16	δgb	δgb	ADV
ejpam-2826	195	17	-	-	PUNCT
ejpam-2826	195	18	connected	connect	VERB
ejpam-2826	195	19	space	space	NOUN
ejpam-2826	195	20	.	.	PUNCT
ejpam-2826	196	1	suppose	suppose	VERB
ejpam-2826	196	2	y	y	PRON
ejpam-2826	196	3	is	be	AUX
ejpam-2826	196	4	not	not	PART
ejpam-2826	196	5	connected	connect	VERB
ejpam-2826	196	6	.	.	PUNCT
ejpam-2826	197	1	then	then	ADV
ejpam-2826	197	2	there	there	PRON
ejpam-2826	197	3	exist	exist	VERB
ejpam-2826	197	4	disjoint	disjoint	ADJ
ejpam-2826	197	5	open	open	ADJ
ejpam-2826	197	6	sets	set	NOUN
ejpam-2826	197	7	u	u	NOUN
ejpam-2826	197	8	and	and	CCONJ
ejpam-2826	197	9	v	v	NOUN
ejpam-2826	197	10	in	in	ADP
ejpam-2826	197	11	y	y	PRON
ejpam-2826	197	12	such	such	ADJ
ejpam-2826	197	13	that	that	SCONJ
ejpam-2826	197	14	y	y	PROPN
ejpam-2826	197	15	=	=	NOUN
ejpam-2826	197	16	u∪v	u∪v	NOUN
ejpam-2826	197	17	.	.	PUNCT
ejpam-2826	198	1	therefore	therefore	ADV
ejpam-2826	198	2	u	u	PROPN
ejpam-2826	198	3	and	and	CCONJ
ejpam-2826	198	4	v	v	NOUN
ejpam-2826	198	5	are	be	AUX
ejpam-2826	198	6	clopen	clopen	ADJ
ejpam-2826	198	7	in	in	ADP
ejpam-2826	198	8	y.	y.	PROPN
ejpam-2826	198	9	since	since	SCONJ
ejpam-2826	198	10	f	f	PROPN
ejpam-2826	198	11	is	be	AUX
ejpam-2826	198	12	contra	contra	PROPN
ejpam-2826	198	13	δgb	δgb	PROPN
ejpam-2826	198	14	-	-	PUNCT
ejpam-2826	198	15	continuous	continuous	ADJ
ejpam-2826	198	16	f−1(u	f−1(u	NOUN
ejpam-2826	198	17	)	)	PUNCT
ejpam-2826	198	18	and	and	CCONJ
ejpam-2826	198	19	f−1(v	f−1(v	PROPN
ejpam-2826	198	20	)	)	PUNCT
ejpam-2826	198	21	are	be	AUX
ejpam-2826	198	22	δgb	δgb	ADV
ejpam-2826	198	23	-	-	PUNCT
ejpam-2826	198	24	open	open	ADJ
ejpam-2826	198	25	sets	set	NOUN
ejpam-2826	198	26	in	in	ADP
ejpam-2826	198	27	x.	x.	NOUN
ejpam-2826	198	28	further	further	PROPN
ejpam-2826	198	29	f	f	PROPN
ejpam-2826	198	30	is	be	AUX
ejpam-2826	198	31	surjective	surjective	ADJ
ejpam-2826	198	32	implies	implie	NOUN
ejpam-2826	198	33	,	,	PUNCT
ejpam-2826	198	34	f−1(u	f−1(u	PROPN
ejpam-2826	198	35	)	)	PUNCT
ejpam-2826	198	36	and	and	CCONJ
ejpam-2826	198	37	f−1(v	f−1(v	PROPN
ejpam-2826	198	38	)	)	PUNCT
ejpam-2826	198	39	are	be	AUX
ejpam-2826	198	40	non	non	X
ejpam-2826	198	41	empty	empty	ADJ
ejpam-2826	198	42	disjoint	disjoint	NOUN
ejpam-2826	198	43	and	and	CCONJ
ejpam-2826	198	44	x	x	SYM
ejpam-2826	198	45	=	=	NOUN
ejpam-2826	198	46	f−1(u)∪f−1(v	f−1(u)∪f−1(v	PROPN
ejpam-2826	198	47	)	)	PUNCT
ejpam-2826	198	48	.	.	PUNCT
ejpam-2826	199	1	this	this	PRON
ejpam-2826	199	2	contradicts	contradict	VERB
ejpam-2826	199	3	the	the	DET
ejpam-2826	199	4	fact	fact	NOUN
ejpam-2826	199	5	that	that	SCONJ
ejpam-2826	199	6	x	x	PRON
ejpam-2826	199	7	is	be	AUX
ejpam-2826	199	8	δgb	δgb	ADV
ejpam-2826	199	9	-	-	PUNCT
ejpam-2826	199	10	connected	connect	VERB
ejpam-2826	199	11	space	space	NOUN
ejpam-2826	199	12	.	.	PUNCT
ejpam-2826	200	1	therefore	therefore	ADV
ejpam-2826	200	2	y	y	PROPN
ejpam-2826	200	3	is	be	AUX
ejpam-2826	200	4	connected	connect	VERB
ejpam-2826	200	5	.	.	PUNCT
ejpam-2826	201	1	theorem	theorem	NOUN
ejpam-2826	201	2	26	26	NUM
ejpam-2826	201	3	.	.	PUNCT
ejpam-2826	202	1	let	let	VERB
ejpam-2826	202	2	x	x	PRON
ejpam-2826	202	3	be	be	AUX
ejpam-2826	202	4	a	a	DET
ejpam-2826	202	5	δgb	δgb	ADV
ejpam-2826	202	6	-	-	PUNCT
ejpam-2826	202	7	connected	connect	VERB
ejpam-2826	202	8	and	and	CCONJ
ejpam-2826	202	9	y	y	PROPN
ejpam-2826	202	10	be	be	AUX
ejpam-2826	202	11	t1	t1	NOUN
ejpam-2826	202	12	-	-	NOUN
ejpam-2826	202	13	space	space	NOUN
ejpam-2826	202	14	.	.	PUNCT
ejpam-2826	203	1	if	if	SCONJ
ejpam-2826	203	2	f	f	X
ejpam-2826	203	3	:	:	PUNCT
ejpam-2826	203	4	x→y	x→y	NUM
ejpam-2826	203	5	is	be	AUX
ejpam-2826	203	6	contra	contra	PROPN
ejpam-2826	203	7	δgbcontinuous	δgbcontinuous	PROPN
ejpam-2826	203	8	then	then	ADV
ejpam-2826	203	9	f	f	PROPN
ejpam-2826	203	10	is	be	AUX
ejpam-2826	203	11	constant	constant	ADJ
ejpam-2826	203	12	.	.	PUNCT
ejpam-2826	204	1	proof	proof	NOUN
ejpam-2826	204	2	:	:	PUNCT
ejpam-2826	204	3	since	since	SCONJ
ejpam-2826	204	4	y	y	PROPN
ejpam-2826	204	5	is	be	AUX
ejpam-2826	204	6	t1	t1	NOUN
ejpam-2826	204	7	-	-	PUNCT
ejpam-2826	204	8	space	space	NOUN
ejpam-2826	204	9	,	,	PUNCT
ejpam-2826	204	10	u={f−1(y	u={f−1(y	PROPN
ejpam-2826	204	11	):	):	PUNCT
ejpam-2826	204	12	y∈y	y∈y	PROPN
ejpam-2826	204	13	}	}	PUNCT
ejpam-2826	204	14	is	be	AUX
ejpam-2826	204	15	a	a	DET
ejpam-2826	204	16	disjoint	disjoint	ADJ
ejpam-2826	204	17	δgb	δgb	ADV
ejpam-2826	204	18	-	-	PUNCT
ejpam-2826	204	19	open	open	ADJ
ejpam-2826	204	20	partition	partition	NOUN
ejpam-2826	204	21	of	of	ADP
ejpam-2826	204	22	x.	x.	NOUN
ejpam-2826	204	23	if	if	SCONJ
ejpam-2826	204	24	|u|≥2	|u|≥2	PROPN
ejpam-2826	204	25	then	then	ADV
ejpam-2826	204	26	x	x	PUNCT
ejpam-2826	204	27	is	be	AUX
ejpam-2826	204	28	the	the	DET
ejpam-2826	204	29	union	union	NOUN
ejpam-2826	204	30	of	of	ADP
ejpam-2826	204	31	two	two	NUM
ejpam-2826	204	32	nonempty	nonempty	ADJ
ejpam-2826	204	33	δgb	δgb	NOUN
ejpam-2826	204	34	-	-	PUNCT
ejpam-2826	204	35	open	open	NOUN
ejpam-2826	204	36	sets.this	sets.this	PROPN
ejpam-2826	204	37	contradicts	contradict	VERB
ejpam-2826	204	38	the	the	DET
ejpam-2826	204	39	fact	fact	NOUN
ejpam-2826	204	40	that	that	SCONJ
ejpam-2826	204	41	x	x	PRON
ejpam-2826	204	42	is	be	AUX
ejpam-2826	204	43	δgb	δgb	ADV
ejpam-2826	204	44	-	-	PUNCT
ejpam-2826	204	45	connected	connect	VERB
ejpam-2826	204	46	.	.	PUNCT
ejpam-2826	205	1	therefore	therefore	ADV
ejpam-2826	205	2	|u|=1	|u|=1	NOUN
ejpam-2826	206	1	and	and	CCONJ
ejpam-2826	206	2	hence	hence	ADV
ejpam-2826	206	3	f	f	PROPN
ejpam-2826	206	4	is	be	AUX
ejpam-2826	206	5	constant	constant	ADJ
ejpam-2826	206	6	.	.	PUNCT
ejpam-2826	207	1	definition	definition	NOUN
ejpam-2826	207	2	12	12	NUM
ejpam-2826	207	3	.	.	PUNCT
ejpam-2826	208	1	[	[	X
ejpam-2826	208	2	4	4	X
ejpam-2826	208	3	]	]	X
ejpam-2826	208	4	a	a	DET
ejpam-2826	208	5	topological	topological	ADJ
ejpam-2826	208	6	space	space	NOUN
ejpam-2826	208	7	x	x	PRON
ejpam-2826	208	8	is	be	AUX
ejpam-2826	208	9	said	say	VERB
ejpam-2826	208	10	to	to	PART
ejpam-2826	208	11	be	be	AUX
ejpam-2826	208	12	δgb	δgb	ADJ
ejpam-2826	208	13	-	-	PUNCT
ejpam-2826	208	14	t2	t2	NOUN
ejpam-2826	208	15	space	space	NOUN
ejpam-2826	208	16	if	if	SCONJ
ejpam-2826	208	17	for	for	ADP
ejpam-2826	208	18	any	any	DET
ejpam-2826	208	19	pair	pair	NOUN
ejpam-2826	208	20	of	of	ADP
ejpam-2826	208	21	distinct	distinct	ADJ
ejpam-2826	208	22	points	point	NOUN
ejpam-2826	208	23	x	x	PUNCT
ejpam-2826	208	24	and	and	CCONJ
ejpam-2826	208	25	y	y	NOUN
ejpam-2826	208	26	there	there	PRON
ejpam-2826	208	27	exist	exist	VERB
ejpam-2826	208	28	disjoint	disjoint	ADJ
ejpam-2826	208	29	δgb	δgb	ADJ
ejpam-2826	208	30	-	-	PUNCT
ejpam-2826	208	31	open	open	ADJ
ejpam-2826	208	32	sets	set	NOUN
ejpam-2826	208	33	g	g	NOUN
ejpam-2826	208	34	and	and	CCONJ
ejpam-2826	208	35	h	h	NOUN
ejpam-2826	208	36	such	such	ADJ
ejpam-2826	208	37	that	that	DET
ejpam-2826	208	38	x∈g	x∈g	NOUN
ejpam-2826	208	39	and	and	CCONJ
ejpam-2826	208	40	y∈h	y∈h	NOUN
ejpam-2826	208	41	.	.	PUNCT
ejpam-2826	209	1	theorem	theorem	VERB
ejpam-2826	209	2	27	27	NUM
ejpam-2826	209	3	.	.	PUNCT
ejpam-2826	210	1	let	let	VERB
ejpam-2826	210	2	x	x	PRON
ejpam-2826	210	3	and	and	CCONJ
ejpam-2826	210	4	y	y	PROPN
ejpam-2826	210	5	be	be	AUX
ejpam-2826	210	6	topological	topological	ADJ
ejpam-2826	210	7	spaces	space	NOUN
ejpam-2826	210	8	.	.	PUNCT
ejpam-2826	211	1	if	if	SCONJ
ejpam-2826	211	2	(	(	PUNCT
ejpam-2826	211	3	i	i	NOUN
ejpam-2826	211	4	)	)	PUNCT
ejpam-2826	211	5	for	for	ADP
ejpam-2826	211	6	each	each	DET
ejpam-2826	211	7	pair	pair	NOUN
ejpam-2826	211	8	of	of	ADP
ejpam-2826	211	9	distinct	distinct	ADJ
ejpam-2826	211	10	points	point	NOUN
ejpam-2826	211	11	x	x	PUNCT
ejpam-2826	211	12	and	and	CCONJ
ejpam-2826	211	13	y	y	PROPN
ejpam-2826	211	14	in	in	ADP
ejpam-2826	211	15	x	x	SYM
ejpam-2826	211	16	there	there	PRON
ejpam-2826	211	17	exists	exist	VERB
ejpam-2826	211	18	a	a	DET
ejpam-2826	211	19	function	function	NOUN
ejpam-2826	211	20	f	f	NOUN
ejpam-2826	211	21	:	:	PUNCT
ejpam-2826	211	22	x→y	x→y	NUM
ejpam-2826	211	23	such	such	ADJ
ejpam-2826	211	24	that	that	DET
ejpam-2826	211	25	f(x)6	f(x)6	PROPN
ejpam-2826	211	26	=	=	SYM
ejpam-2826	211	27	f(y	f(y	NOUN
ejpam-2826	211	28	)	)	PUNCT
ejpam-2826	211	29	,	,	PUNCT
ejpam-2826	211	30	(	(	PUNCT
ejpam-2826	211	31	ii	ii	NOUN
ejpam-2826	211	32	)	)	PUNCT
ejpam-2826	211	33	y	y	PROPN
ejpam-2826	211	34	is	be	AUX
ejpam-2826	211	35	urysohn	urysohn	PROPN
ejpam-2826	211	36	space	space	NOUN
ejpam-2826	211	37	and	and	CCONJ
ejpam-2826	211	38	(	(	PUNCT
ejpam-2826	211	39	iii	iii	X
ejpam-2826	211	40	)	)	PUNCT
ejpam-2826	211	41	f	f	PROPN
ejpam-2826	211	42	is	be	AUX
ejpam-2826	211	43	contra	contra	PROPN
ejpam-2826	211	44	δgb	δgb	PROPN
ejpam-2826	211	45	-	-	PUNCT
ejpam-2826	211	46	continuous	continuous	ADJ
ejpam-2826	211	47	at	at	ADP
ejpam-2826	211	48	x	x	X
ejpam-2826	211	49	and	and	CCONJ
ejpam-2826	211	50	y.	y.	NOUN
ejpam-2826	211	51	then	then	ADV
ejpam-2826	211	52	x	x	PUNCT
ejpam-2826	211	53	is	be	AUX
ejpam-2826	211	54	δgb	δgb	NOUN
ejpam-2826	211	55	-	-	PUNCT
ejpam-2826	211	56	t2	t2	NOUN
ejpam-2826	211	57	.	.	PUNCT
ejpam-2826	212	1	proof	proof	NOUN
ejpam-2826	212	2	:	:	PUNCT
ejpam-2826	212	3	let	let	VERB
ejpam-2826	212	4	x	x	PRON
ejpam-2826	212	5	and	and	CCONJ
ejpam-2826	212	6	y	y	PROPN
ejpam-2826	212	7	be	be	AUX
ejpam-2826	212	8	any	any	DET
ejpam-2826	212	9	distinct	distinct	ADJ
ejpam-2826	212	10	points	point	NOUN
ejpam-2826	212	11	in	in	ADP
ejpam-2826	212	12	x	x	PUNCT
ejpam-2826	212	13	and	and	CCONJ
ejpam-2826	212	14	f	f	PROPN
ejpam-2826	212	15	is	be	AUX
ejpam-2826	212	16	a	a	DET
ejpam-2826	212	17	function	function	NOUN
ejpam-2826	212	18	such	such	ADJ
ejpam-2826	212	19	that	that	SCONJ
ejpam-2826	212	20	f(x	f(x	PROPN
ejpam-2826	212	21	)	)	PUNCT
ejpam-2826	212	22	6	6	NUM
ejpam-2826	212	23	=	=	SYM
ejpam-2826	212	24	f(y	f(y	NOUN
ejpam-2826	212	25	)	)	PUNCT
ejpam-2826	212	26	.	.	PUNCT
ejpam-2826	213	1	let	let	VERB
ejpam-2826	213	2	a	a	DET
ejpam-2826	213	3	=	=	NOUN
ejpam-2826	213	4	f(x	f(x	PROPN
ejpam-2826	213	5	)	)	PUNCT
ejpam-2826	213	6	and	and	CCONJ
ejpam-2826	213	7	b	b	X
ejpam-2826	213	8	=	=	SYM
ejpam-2826	213	9	f(y	f(y	NOUN
ejpam-2826	213	10	)	)	PUNCT
ejpam-2826	213	11	then	then	ADV
ejpam-2826	213	12	a	a	DET
ejpam-2826	213	13	6	6	NUM
ejpam-2826	213	14	=	=	NOUN
ejpam-2826	213	15	b.	b.	NOUN
ejpam-2826	213	16	since	since	SCONJ
ejpam-2826	213	17	y	y	PROPN
ejpam-2826	213	18	is	be	AUX
ejpam-2826	213	19	an	an	DET
ejpam-2826	213	20	urysohn	urysohn	ADJ
ejpam-2826	213	21	space	space	NOUN
ejpam-2826	213	22	there	there	PRON
ejpam-2826	213	23	exist	exist	VERB
ejpam-2826	213	24	open	open	ADJ
ejpam-2826	213	25	sets	set	NOUN
ejpam-2826	213	26	v	v	NOUN
ejpam-2826	213	27	and	and	CCONJ
ejpam-2826	213	28	w	w	NOUN
ejpam-2826	213	29	in	in	ADP
ejpam-2826	213	30	y	y	NOUN
ejpam-2826	213	31	containing	contain	VERB
ejpam-2826	213	32	a	a	DET
ejpam-2826	213	33	and	and	CCONJ
ejpam-2826	213	34	b	b	NOUN
ejpam-2826	213	35	respectively	respectively	ADV
ejpam-2826	213	36	such	such	ADJ
ejpam-2826	213	37	that	that	DET
ejpam-2826	213	38	cl(v)∩cl(w)=φ	cl(v)∩cl(w)=φ	NOUN
ejpam-2826	213	39	.	.	PUNCT
ejpam-2826	214	1	since	since	SCONJ
ejpam-2826	214	2	f	f	PROPN
ejpam-2826	214	3	is	be	AUX
ejpam-2826	214	4	contra	contra	PROPN
ejpam-2826	214	5	δgb	δgb	PROPN
ejpam-2826	214	6	-	-	PUNCT
ejpam-2826	214	7	continuous	continuous	ADJ
ejpam-2826	214	8	at	at	ADP
ejpam-2826	214	9	x	x	PROPN
ejpam-2826	214	10	and	and	CCONJ
ejpam-2826	214	11	y	y	PROPN
ejpam-2826	214	12	then	then	ADV
ejpam-2826	214	13	there	there	PRON
ejpam-2826	214	14	exist	exist	VERB
ejpam-2826	214	15	δgb	δgb	ADJ
ejpam-2826	214	16	-	-	PUNCT
ejpam-2826	214	17	open	open	ADJ
ejpam-2826	214	18	sets	set	NOUN
ejpam-2826	214	19	a	a	PRON
ejpam-2826	214	20	and	and	CCONJ
ejpam-2826	214	21	b	b	NOUN
ejpam-2826	214	22	in	in	ADP
ejpam-2826	214	23	x	x	PUNCT
ejpam-2826	214	24	containing	contain	VERB
ejpam-2826	214	25	x	x	X
ejpam-2826	214	26	and	and	CCONJ
ejpam-2826	214	27	y	y	PROPN
ejpam-2826	214	28	respectively	respectively	ADV
ejpam-2826	214	29	such	such	ADJ
ejpam-2826	214	30	that	that	SCONJ
ejpam-2826	214	31	f(a)⊆cl(v	f(a)⊆cl(v	NOUN
ejpam-2826	214	32	)	)	PUNCT
ejpam-2826	214	33	and	and	CCONJ
ejpam-2826	214	34	f(b)⊆cl(w	f(b)⊆cl(w	NOUN
ejpam-2826	214	35	)	)	PUNCT
ejpam-2826	214	36	.	.	PUNCT
ejpam-2826	215	1	we	we	PRON
ejpam-2826	215	2	have	have	VERB
ejpam-2826	215	3	a∩b⊆f−1(cl(v))∩f−1(cl(w))=f−1(φ)=φ	a∩b⊆f−1(cl(v))∩f−1(cl(w))=f−1(φ)=φ	NOUN
ejpam-2826	215	4	.	.	PUNCT
ejpam-2826	216	1	hence	hence	ADV
ejpam-2826	216	2	x	x	VERB
ejpam-2826	216	3	is	be	AUX
ejpam-2826	216	4	δgb	δgb	NOUN
ejpam-2826	216	5	-	-	PUNCT
ejpam-2826	216	6	t2	t2	NOUN
ejpam-2826	216	7	.	.	PUNCT
ejpam-2826	217	1	corollary	corollary	ADJ
ejpam-2826	217	2	2	2	NUM
ejpam-2826	217	3	.	.	PUNCT
ejpam-2826	218	1	let	let	VERB
ejpam-2826	218	2	f	f	X
ejpam-2826	218	3	:	:	PUNCT
ejpam-2826	218	4	x→y	x→y	NUM
ejpam-2826	218	5	be	be	AUX
ejpam-2826	218	6	a	a	DET
ejpam-2826	218	7	contra	contra	PROPN
ejpam-2826	218	8	δgb	δgb	ADJ
ejpam-2826	218	9	-	-	PUNCT
ejpam-2826	218	10	continuous	continuous	ADJ
ejpam-2826	218	11	injective	injective	ADJ
ejpam-2826	218	12	function	function	NOUN
ejpam-2826	218	13	from	from	ADP
ejpam-2826	218	14	a	a	DET
ejpam-2826	218	15	space	space	NOUN
ejpam-2826	218	16	x	x	PUNCT
ejpam-2826	218	17	into	into	ADP
ejpam-2826	218	18	urysohn	urysohn	PROPN
ejpam-2826	218	19	space	space	NOUN
ejpam-2826	219	1	y	y	PROPN
ejpam-2826	219	2	then	then	ADV
ejpam-2826	219	3	x	x	PUNCT
ejpam-2826	219	4	is	be	AUX
ejpam-2826	219	5	δgb	δgb	NOUN
ejpam-2826	219	6	-	-	PUNCT
ejpam-2826	219	7	t2	t2	NOUN
ejpam-2826	219	8	.	.	PUNCT
ejpam-2826	220	1	definition	definition	NOUN
ejpam-2826	220	2	13	13	NUM
ejpam-2826	220	3	.	.	PUNCT
ejpam-2826	221	1	[	[	X
ejpam-2826	221	2	14	14	NUM
ejpam-2826	221	3	]	]	PUNCT
ejpam-2826	221	4	a	a	DET
ejpam-2826	221	5	topological	topological	ADJ
ejpam-2826	221	6	space	space	NOUN
ejpam-2826	221	7	x	x	PUNCT
ejpam-2826	221	8	is	be	AUX
ejpam-2826	221	9	called	call	VERB
ejpam-2826	221	10	ultra	ultra	ADJ
ejpam-2826	221	11	hausdorff	hausdorff	NOUN
ejpam-2826	221	12	space	space	NOUN
ejpam-2826	221	13	if	if	SCONJ
ejpam-2826	221	14	for	for	ADP
ejpam-2826	221	15	every	every	DET
ejpam-2826	221	16	pair	pair	NOUN
ejpam-2826	221	17	of	of	ADP
ejpam-2826	221	18	distinct	distinct	ADJ
ejpam-2826	221	19	points	point	NOUN
ejpam-2826	221	20	x	x	PUNCT
ejpam-2826	221	21	and	and	CCONJ
ejpam-2826	221	22	y	y	PROPN
ejpam-2826	221	23	in	in	ADP
ejpam-2826	221	24	x	x	SYM
ejpam-2826	221	25	there	there	PRON
ejpam-2826	221	26	exist	exist	VERB
ejpam-2826	221	27	disjoint	disjoint	ADJ
ejpam-2826	221	28	clopen	clopen	ADJ
ejpam-2826	221	29	sets	set	NOUN
ejpam-2826	221	30	u	u	NOUN
ejpam-2826	221	31	and	and	CCONJ
ejpam-2826	221	32	v	v	NOUN
ejpam-2826	221	33	in	in	ADP
ejpam-2826	221	34	x	x	PUNCT
ejpam-2826	221	35	containing	contain	VERB
ejpam-2826	221	36	x	x	PROPN
ejpam-2826	221	37	and	and	CCONJ
ejpam-2826	221	38	y	y	PROPN
ejpam-2826	221	39	respectively	respectively	ADV
ejpam-2826	221	40	.	.	PUNCT
ejpam-2826	222	1	s.s.benchalli	s.s.benchalli	PROPN
ejpam-2826	222	2	,	,	PUNCT
ejpam-2826	222	3	p.g.patil	p.g.patil	PROPN
ejpam-2826	222	4	,	,	PUNCT
ejpam-2826	222	5	j.b.toranagatti	j.b.toranagatti	PROPN
ejpam-2826	222	6	,	,	PUNCT
ejpam-2826	222	7	s.r.vighneshi	s.r.vighneshi	PROPN
ejpam-2826	222	8	/	/	SYM
ejpam-2826	222	9	eur	eur	PROPN
ejpam-2826	222	10	.	.	PUNCT
ejpam-2826	223	1	j.	j.	PROPN
ejpam-2826	223	2	pure	pure	PROPN
ejpam-2826	223	3	appl	appl	PROPN
ejpam-2826	223	4	.	.	PROPN
ejpam-2826	223	5	math	math	PROPN
ejpam-2826	223	6	,	,	PUNCT
ejpam-2826	223	7	10	10	NUM
ejpam-2826	223	8	(	(	PUNCT
ejpam-2826	223	9	2	2	NUM
ejpam-2826	223	10	)	)	PUNCT
ejpam-2826	223	11	(	(	PUNCT
ejpam-2826	223	12	2017	2017	NUM
ejpam-2826	223	13	)	)	PUNCT
ejpam-2826	223	14	,	,	PUNCT
ejpam-2826	223	15	312	312	NUM
ejpam-2826	223	16	-	-	SYM
ejpam-2826	223	17	322	322	NUM
ejpam-2826	223	18	320	320	NUM
ejpam-2826	223	19	theorem	theorem	NOUN
ejpam-2826	223	20	28	28	NUM
ejpam-2826	223	21	.	.	PUNCT
ejpam-2826	224	1	if	if	SCONJ
ejpam-2826	224	2	f	f	X
ejpam-2826	224	3	:	:	PUNCT
ejpam-2826	224	4	x→y	x→y	NUM
ejpam-2826	224	5	be	be	AUX
ejpam-2826	224	6	contra	contra	PROPN
ejpam-2826	224	7	δgb	δgb	PROPN
ejpam-2826	224	8	-	-	PUNCT
ejpam-2826	224	9	continuous	continuous	ADJ
ejpam-2826	224	10	injective	injective	ADJ
ejpam-2826	224	11	function	function	NOUN
ejpam-2826	224	12	from	from	ADP
ejpam-2826	224	13	space	space	NOUN
ejpam-2826	224	14	x	x	PUNCT
ejpam-2826	224	15	into	into	ADP
ejpam-2826	224	16	a	a	DET
ejpam-2826	224	17	ultra	ultra	ADJ
ejpam-2826	224	18	hausdorff	hausdorff	NOUN
ejpam-2826	224	19	space	space	NOUN
ejpam-2826	224	20	y	y	PROPN
ejpam-2826	224	21	then	then	ADV
ejpam-2826	224	22	x	x	PUNCT
ejpam-2826	224	23	is	be	AUX
ejpam-2826	224	24	δgb	δgb	NOUN
ejpam-2826	224	25	-	-	PUNCT
ejpam-2826	224	26	t2	t2	NOUN
ejpam-2826	224	27	.	.	PUNCT
ejpam-2826	225	1	proof	proof	NOUN
ejpam-2826	225	2	:	:	PUNCT
ejpam-2826	225	3	let	let	VERB
ejpam-2826	225	4	x	x	PRON
ejpam-2826	225	5	and	and	CCONJ
ejpam-2826	225	6	y	y	PROPN
ejpam-2826	225	7	be	be	AUX
ejpam-2826	225	8	any	any	DET
ejpam-2826	225	9	two	two	NUM
ejpam-2826	225	10	distinct	distinct	ADJ
ejpam-2826	225	11	points	point	NOUN
ejpam-2826	225	12	in	in	ADP
ejpam-2826	225	13	x	x	X
ejpam-2826	225	14	.	.	PUNCT
ejpam-2826	226	1	since	since	SCONJ
ejpam-2826	226	2	f	f	PROPN
ejpam-2826	226	3	is	be	AUX
ejpam-2826	226	4	injective	injective	ADJ
ejpam-2826	226	5	f(x	f(x	PROPN
ejpam-2826	226	6	)	)	PUNCT
ejpam-2826	226	7	6	6	NUM
ejpam-2826	226	8	=	=	SYM
ejpam-2826	226	9	f(y	f(y	NOUN
ejpam-2826	226	10	)	)	PUNCT
ejpam-2826	226	11	and	and	CCONJ
ejpam-2826	226	12	y	y	PROPN
ejpam-2826	226	13	is	be	AUX
ejpam-2826	226	14	ultra	ultra	ADJ
ejpam-2826	226	15	hausdorff	hausdorff	NOUN
ejpam-2826	226	16	space	space	NOUN
ejpam-2826	226	17	implies	imply	VERB
ejpam-2826	226	18	there	there	PRON
ejpam-2826	226	19	exist	exist	VERB
ejpam-2826	226	20	disjoint	disjoint	ADJ
ejpam-2826	226	21	clopen	clopen	ADJ
ejpam-2826	226	22	sets	set	NOUN
ejpam-2826	226	23	u	u	NOUN
ejpam-2826	226	24	and	and	CCONJ
ejpam-2826	226	25	v	v	NOUN
ejpam-2826	226	26	of	of	ADP
ejpam-2826	226	27	y	y	NOUN
ejpam-2826	226	28	containing	contain	VERB
ejpam-2826	226	29	f(x	f(x	PROPN
ejpam-2826	226	30	)	)	PUNCT
ejpam-2826	226	31	and	and	CCONJ
ejpam-2826	226	32	f(y	f(y	NOUN
ejpam-2826	226	33	)	)	PUNCT
ejpam-2826	226	34	respectively	respectively	ADV
ejpam-2826	226	35	.	.	PUNCT
ejpam-2826	227	1	then	then	ADV
ejpam-2826	227	2	x∈f−1(u	x∈f−1(u	PROPN
ejpam-2826	227	3	)	)	PUNCT
ejpam-2826	227	4	and	and	CCONJ
ejpam-2826	227	5	y∈f−1(v	y∈f−1(v	NOUN
ejpam-2826	227	6	)	)	PUNCT
ejpam-2826	227	7	where	where	SCONJ
ejpam-2826	227	8	f−1(u	f−1(u	NOUN
ejpam-2826	227	9	)	)	PUNCT
ejpam-2826	227	10	and	and	CCONJ
ejpam-2826	227	11	f−1(v	f−1(v	PROPN
ejpam-2826	227	12	)	)	PUNCT
ejpam-2826	227	13	are	be	AUX
ejpam-2826	227	14	disjoint	disjoint	ADJ
ejpam-2826	227	15	δgb	δgb	ADV
ejpam-2826	227	16	-	-	PUNCT
ejpam-2826	227	17	open	open	ADJ
ejpam-2826	227	18	sets	set	NOUN
ejpam-2826	227	19	in	in	ADP
ejpam-2826	227	20	x	x	X
ejpam-2826	227	21	.	.	PUNCT
ejpam-2826	228	1	therefore	therefore	ADV
ejpam-2826	228	2	x	x	X
ejpam-2826	228	3	is	be	AUX
ejpam-2826	228	4	δgb	δgb	NOUN
ejpam-2826	228	5	-	-	PUNCT
ejpam-2826	228	6	t2	t2	NOUN
ejpam-2826	228	7	.	.	PUNCT
ejpam-2826	229	1	definition	definition	NOUN
ejpam-2826	229	2	14	14	NUM
ejpam-2826	229	3	.	.	PUNCT
ejpam-2826	230	1	[	[	X
ejpam-2826	230	2	14	14	NUM
ejpam-2826	230	3	]	]	PUNCT
ejpam-2826	230	4	a	a	DET
ejpam-2826	230	5	space	space	NOUN
ejpam-2826	230	6	x	x	PUNCT
ejpam-2826	230	7	is	be	AUX
ejpam-2826	230	8	called	call	VERB
ejpam-2826	230	9	ultra	ultra	ADJ
ejpam-2826	230	10	normal	normal	ADJ
ejpam-2826	230	11	space	space	NOUN
ejpam-2826	230	12	if	if	SCONJ
ejpam-2826	230	13	each	each	DET
ejpam-2826	230	14	pair	pair	NOUN
ejpam-2826	230	15	of	of	ADP
ejpam-2826	230	16	disjoint	disjoint	NOUN
ejpam-2826	230	17	closed	close	VERB
ejpam-2826	230	18	sets	set	NOUN
ejpam-2826	230	19	can	can	AUX
ejpam-2826	230	20	be	be	AUX
ejpam-2826	230	21	separated	separate	VERB
ejpam-2826	230	22	by	by	ADP
ejpam-2826	230	23	disjoint	disjoint	NOUN
ejpam-2826	230	24	clopen	clopen	ADJ
ejpam-2826	230	25	sets	set	NOUN
ejpam-2826	230	26	.	.	PUNCT
ejpam-2826	231	1	definition	definition	NOUN
ejpam-2826	231	2	15	15	NUM
ejpam-2826	231	3	.	.	PUNCT
ejpam-2826	232	1	[	[	X
ejpam-2826	232	2	4	4	X
ejpam-2826	232	3	]	]	X
ejpam-2826	232	4	a	a	DET
ejpam-2826	232	5	topological	topological	ADJ
ejpam-2826	232	6	space	space	NOUN
ejpam-2826	232	7	x	x	PRON
ejpam-2826	232	8	is	be	AUX
ejpam-2826	232	9	said	say	VERB
ejpam-2826	232	10	to	to	PART
ejpam-2826	232	11	be	be	AUX
ejpam-2826	232	12	δgb	δgb	NOUN
ejpam-2826	232	13	-	-	PUNCT
ejpam-2826	232	14	normal	normal	ADJ
ejpam-2826	232	15	if	if	SCONJ
ejpam-2826	232	16	each	each	DET
ejpam-2826	232	17	pair	pair	NOUN
ejpam-2826	232	18	of	of	ADP
ejpam-2826	232	19	disjoint	disjoint	NOUN
ejpam-2826	232	20	closed	close	VERB
ejpam-2826	232	21	sets	set	NOUN
ejpam-2826	232	22	can	can	AUX
ejpam-2826	232	23	be	be	AUX
ejpam-2826	232	24	separated	separate	VERB
ejpam-2826	232	25	by	by	ADP
ejpam-2826	232	26	disjoint	disjoint	ADJ
ejpam-2826	232	27	δgb	δgb	PROPN
ejpam-2826	232	28	-	-	PUNCT
ejpam-2826	232	29	open	open	ADJ
ejpam-2826	232	30	sets	set	NOUN
ejpam-2826	232	31	.	.	PUNCT
ejpam-2826	233	1	theorem	theorem	NOUN
ejpam-2826	233	2	29	29	NUM
ejpam-2826	233	3	.	.	PUNCT
ejpam-2826	234	1	if	if	SCONJ
ejpam-2826	234	2	f	f	X
ejpam-2826	234	3	:	:	PUNCT
ejpam-2826	234	4	x→y	x→y	NUM
ejpam-2826	234	5	be	be	AUX
ejpam-2826	234	6	contra	contra	PROPN
ejpam-2826	234	7	δgb	δgb	ADJ
ejpam-2826	234	8	-	-	PUNCT
ejpam-2826	234	9	continuous	continuous	ADJ
ejpam-2826	234	10	closed	closed	ADJ
ejpam-2826	234	11	injection	injection	NOUN
ejpam-2826	234	12	and	and	CCONJ
ejpam-2826	234	13	y	y	PROPN
ejpam-2826	234	14	is	be	AUX
ejpam-2826	234	15	ultra	ultra	ADJ
ejpam-2826	234	16	normal	normal	ADJ
ejpam-2826	234	17	then	then	ADV
ejpam-2826	234	18	x	x	PUNCT
ejpam-2826	234	19	is	be	AUX
ejpam-2826	234	20	δgb	δgb	ADV
ejpam-2826	234	21	-	-	PUNCT
ejpam-2826	234	22	normal	normal	ADJ
ejpam-2826	234	23	.	.	PUNCT
ejpam-2826	235	1	proof	proof	NOUN
ejpam-2826	235	2	:	:	PUNCT
ejpam-2826	235	3	let	let	VERB
ejpam-2826	235	4	e	e	NOUN
ejpam-2826	235	5	and	and	CCONJ
ejpam-2826	235	6	f	f	PROPN
ejpam-2826	235	7	be	be	AUX
ejpam-2826	235	8	disjoint	disjoint	X
ejpam-2826	235	9	closed	closed	ADJ
ejpam-2826	235	10	subsets	subset	NOUN
ejpam-2826	235	11	of	of	ADP
ejpam-2826	235	12	x.	x.	NOUN
ejpam-2826	235	13	since	since	SCONJ
ejpam-2826	235	14	f	f	PROPN
ejpam-2826	235	15	is	be	AUX
ejpam-2826	235	16	closed	closed	ADJ
ejpam-2826	235	17	and	and	CCONJ
ejpam-2826	235	18	injective	injective	ADJ
ejpam-2826	235	19	f(e	f(e	NOUN
ejpam-2826	235	20	)	)	PUNCT
ejpam-2826	235	21	and	and	CCONJ
ejpam-2826	235	22	f(f	f(f	PROPN
ejpam-2826	235	23	)	)	PUNCT
ejpam-2826	235	24	are	be	AUX
ejpam-2826	235	25	disjoint	disjoint	NOUN
ejpam-2826	235	26	closed	closed	ADJ
ejpam-2826	235	27	sets	set	NOUN
ejpam-2826	235	28	in	in	ADP
ejpam-2826	235	29	y.	y.	NOUN
ejpam-2826	235	30	since	since	SCONJ
ejpam-2826	235	31	y	y	PROPN
ejpam-2826	235	32	is	be	AUX
ejpam-2826	235	33	ultra	ultra	ADJ
ejpam-2826	235	34	normal	normal	ADJ
ejpam-2826	235	35	there	there	PRON
ejpam-2826	235	36	exists	exist	VERB
ejpam-2826	235	37	disjoint	disjoint	NOUN
ejpam-2826	235	38	clopen	clopen	ADJ
ejpam-2826	235	39	sets	set	NOUN
ejpam-2826	235	40	u	u	NOUN
ejpam-2826	235	41	and	and	CCONJ
ejpam-2826	235	42	v	v	NOUN
ejpam-2826	235	43	in	in	ADP
ejpam-2826	235	44	y	y	PRON
ejpam-2826	235	45	such	such	ADJ
ejpam-2826	235	46	that	that	DET
ejpam-2826	235	47	f(e)⊆u	f(e)⊆u	NOUN
ejpam-2826	235	48	and	and	CCONJ
ejpam-2826	235	49	f(f)⊆v	f(f)⊆v	NOUN
ejpam-2826	235	50	.	.	PUNCT
ejpam-2826	236	1	this	this	PRON
ejpam-2826	236	2	implies	imply	VERB
ejpam-2826	236	3	e⊆f−1(u	e⊆f−1(u	NOUN
ejpam-2826	236	4	)	)	PUNCT
ejpam-2826	236	5	and	and	CCONJ
ejpam-2826	236	6	f⊆f−1(v	f⊆f−1(v	PROPN
ejpam-2826	236	7	)	)	PUNCT
ejpam-2826	236	8	.	.	PUNCT
ejpam-2826	237	1	since	since	SCONJ
ejpam-2826	237	2	f	f	PROPN
ejpam-2826	237	3	is	be	AUX
ejpam-2826	237	4	contra	contra	PROPN
ejpam-2826	237	5	δgb	δgb	ADJ
ejpam-2826	237	6	-	-	PUNCT
ejpam-2826	237	7	continuous	continuous	ADJ
ejpam-2826	237	8	injection	injection	NOUN
ejpam-2826	237	9	,	,	PUNCT
ejpam-2826	237	10	f−1(u	f−1(u	PROPN
ejpam-2826	237	11	)	)	PUNCT
ejpam-2826	237	12	and	and	CCONJ
ejpam-2826	237	13	f−1(v	f−1(v	PROPN
ejpam-2826	237	14	)	)	PUNCT
ejpam-2826	237	15	are	be	AUX
ejpam-2826	237	16	disjoint	disjoint	ADJ
ejpam-2826	237	17	δgb	δgb	ADV
ejpam-2826	237	18	-	-	PUNCT
ejpam-2826	237	19	open	open	ADJ
ejpam-2826	237	20	sets	set	NOUN
ejpam-2826	237	21	in	in	ADP
ejpam-2826	237	22	x	x	X
ejpam-2826	237	23	.	.	PUNCT
ejpam-2826	238	1	this	this	PRON
ejpam-2826	238	2	shows	show	VERB
ejpam-2826	238	3	x	x	PRON
ejpam-2826	238	4	is	be	AUX
ejpam-2826	238	5	δgb	δgb	ADV
ejpam-2826	238	6	-	-	PUNCT
ejpam-2826	238	7	normal	normal	ADJ
ejpam-2826	238	8	.	.	PUNCT
ejpam-2826	239	1	remark	remark	PROPN
ejpam-2826	239	2	3	3	NUM
ejpam-2826	239	3	.	.	PUNCT
ejpam-2826	240	1	the	the	DET
ejpam-2826	240	2	composition	composition	NOUN
ejpam-2826	240	3	of	of	ADP
ejpam-2826	240	4	two	two	NUM
ejpam-2826	240	5	contra	contra	ADJ
ejpam-2826	240	6	-	-	ADJ
ejpam-2826	240	7	δgb	δgb	ADJ
ejpam-2826	240	8	-	-	PUNCT
ejpam-2826	240	9	continuous	continuous	ADJ
ejpam-2826	240	10	functions	function	NOUN
ejpam-2826	240	11	need	need	AUX
ejpam-2826	240	12	not	not	PART
ejpam-2826	240	13	be	be	AUX
ejpam-2826	240	14	contraδgb	contraδgb	ADJ
ejpam-2826	240	15	-	-	ADJ
ejpam-2826	240	16	continuous	continuous	ADJ
ejpam-2826	240	17	as	as	ADP
ejpam-2826	240	18	seen	see	VERB
ejpam-2826	240	19	from	from	ADP
ejpam-2826	240	20	the	the	DET
ejpam-2826	240	21	following	follow	VERB
ejpam-2826	240	22	example	example	NOUN
ejpam-2826	240	23	.	.	PUNCT
ejpam-2826	241	1	example	example	NOUN
ejpam-2826	242	1	3	3	X
ejpam-2826	242	2	.	.	PUNCT
ejpam-2826	243	1	let	let	VERB
ejpam-2826	243	2	x	x	SYM
ejpam-2826	243	3	=	=	NOUN
ejpam-2826	243	4	y	y	NOUN
ejpam-2826	243	5	=	=	NOUN
ejpam-2826	243	6	z={a	z={a	PROPN
ejpam-2826	243	7	,	,	PUNCT
ejpam-2826	243	8	b	b	NOUN
ejpam-2826	243	9	,	,	PUNCT
ejpam-2826	243	10	c	c	NOUN
ejpam-2826	243	11	}	}	PUNCT
ejpam-2826	243	12	,	,	PUNCT
ejpam-2826	243	13	τ={x	τ={x	NOUN
ejpam-2826	243	14	,	,	PUNCT
ejpam-2826	243	15	φ,{a},{b},{a	φ,{a},{b},{a	ADV
ejpam-2826	243	16	,	,	PUNCT
ejpam-2826	243	17	b}},σ={y	b}},σ={y	NOUN
ejpam-2826	243	18	,	,	PUNCT
ejpam-2826	243	19	φ,{a	φ,{a	PROPN
ejpam-2826	243	20	}	}	PUNCT
ejpam-2826	243	21	}	}	PUNCT
ejpam-2826	243	22	and	and	CCONJ
ejpam-2826	243	23	η={z	η={z	PROPN
ejpam-2826	243	24	,	,	PUNCT
ejpam-2826	243	25	φ,{b	φ,{b	PROPN
ejpam-2826	243	26	,	,	PUNCT
ejpam-2826	243	27	c	c	NOUN
ejpam-2826	243	28	}	}	PUNCT
ejpam-2826	243	29	}	}	PUNCT
ejpam-2826	243	30	be	be	AUX
ejpam-2826	243	31	topologies	topology	NOUN
ejpam-2826	243	32	on	on	ADP
ejpam-2826	243	33	x	x	PROPN
ejpam-2826	243	34	,	,	PUNCT
ejpam-2826	243	35	y	y	PROPN
ejpam-2826	243	36	and	and	CCONJ
ejpam-2826	243	37	z	z	PROPN
ejpam-2826	243	38	respectively.then	respectively.then	ADP
ejpam-2826	243	39	the	the	DET
ejpam-2826	243	40	identity	identity	NOUN
ejpam-2826	243	41	function	function	NOUN
ejpam-2826	243	42	f	f	NOUN
ejpam-2826	243	43	:	:	PUNCT
ejpam-2826	243	44	x→y	x→y	NUM
ejpam-2826	243	45	and	and	CCONJ
ejpam-2826	243	46	a	a	DET
ejpam-2826	243	47	function	function	NOUN
ejpam-2826	243	48	g	g	NOUN
ejpam-2826	243	49	:	:	PUNCT
ejpam-2826	243	50	y→z	y→z	NUM
ejpam-2826	243	51	defined	define	VERB
ejpam-2826	243	52	by	by	ADP
ejpam-2826	243	53	g(a)=b	g(a)=b	PROPN
ejpam-2826	243	54	,	,	PUNCT
ejpam-2826	243	55	g(b)=c	g(b)=c	PROPN
ejpam-2826	243	56	and	and	CCONJ
ejpam-2826	243	57	g(c)=a	g(c)=a	NOUN
ejpam-2826	243	58	are	be	AUX
ejpam-2826	243	59	contra	contra	PROPN
ejpam-2826	243	60	δgb	δgb	ADV
ejpam-2826	243	61	-	-	PUNCT
ejpam-2826	243	62	continuous	continuous	ADJ
ejpam-2826	243	63	but	but	CCONJ
ejpam-2826	243	64	g	g	ADP
ejpam-2826	243	65	◦	◦	NOUN
ejpam-2826	243	66	f	f	NOUN
ejpam-2826	243	67	:	:	PUNCT
ejpam-2826	243	68	x→z	x→z	NUM
ejpam-2826	243	69	is	be	AUX
ejpam-2826	243	70	not	not	PART
ejpam-2826	243	71	contra	contra	PROPN
ejpam-2826	243	72	δgb	δgb	ADV
ejpam-2826	243	73	-	-	PUNCT
ejpam-2826	243	74	continuous	continuous	ADJ
ejpam-2826	243	75	,	,	PUNCT
ejpam-2826	243	76	since	since	SCONJ
ejpam-2826	243	77	there	there	PRON
ejpam-2826	243	78	exists	exist	VERB
ejpam-2826	243	79	a	a	DET
ejpam-2826	243	80	open	open	ADJ
ejpam-2826	243	81	set	set	NOUN
ejpam-2826	243	82	{	{	PUNCT
ejpam-2826	243	83	b	b	NOUN
ejpam-2826	243	84	,	,	PUNCT
ejpam-2826	243	85	c	c	NOUN
ejpam-2826	243	86	}	}	PUNCT
ejpam-2826	243	87	in	in	ADP
ejpam-2826	243	88	z	z	NOUN
ejpam-2826	243	89	such	such	ADJ
ejpam-2826	243	90	that	that	SCONJ
ejpam-2826	243	91	(	(	PUNCT
ejpam-2826	243	92	g	g	NOUN
ejpam-2826	243	93	◦	◦	NOUN
ejpam-2826	243	94	f)−1{b	f)−1{b	NOUN
ejpam-2826	243	95	,	,	PUNCT
ejpam-2826	243	96	c}={a	c}={a	PROPN
ejpam-2826	243	97	,	,	PUNCT
ejpam-2826	243	98	b	b	NOUN
ejpam-2826	243	99	}	}	PUNCT
ejpam-2826	243	100	is	be	AUX
ejpam-2826	243	101	not	not	PART
ejpam-2826	243	102	δgb	δgb	ADV
ejpam-2826	243	103	-	-	PUNCT
ejpam-2826	243	104	closed	close	VERB
ejpam-2826	243	105	in	in	ADP
ejpam-2826	243	106	x.	x.	NOUN
ejpam-2826	243	107	theorem	theorem	VERB
ejpam-2826	243	108	30	30	NUM
ejpam-2826	243	109	.	.	PUNCT
ejpam-2826	244	1	let	let	VERB
ejpam-2826	244	2	f	f	X
ejpam-2826	244	3	:	:	PUNCT
ejpam-2826	244	4	x→y	x→y	NUM
ejpam-2826	244	5	and	and	CCONJ
ejpam-2826	244	6	g	g	NOUN
ejpam-2826	244	7	:	:	PUNCT
ejpam-2826	244	8	y→z	y→z	NUM
ejpam-2826	244	9	be	be	AUX
ejpam-2826	244	10	any	any	DET
ejpam-2826	244	11	two	two	NUM
ejpam-2826	244	12	functions	function	NOUN
ejpam-2826	244	13	.	.	PUNCT
ejpam-2826	245	1	(	(	PUNCT
ejpam-2826	245	2	i	i	NOUN
ejpam-2826	245	3	)	)	PUNCT
ejpam-2826	245	4	if	if	SCONJ
ejpam-2826	245	5	f	f	PROPN
ejpam-2826	245	6	is	be	AUX
ejpam-2826	245	7	contra	contra	PROPN
ejpam-2826	245	8	δgb	δgb	ADV
ejpam-2826	245	9	-	-	PUNCT
ejpam-2826	245	10	continuous	continuous	ADJ
ejpam-2826	245	11	and	and	CCONJ
ejpam-2826	245	12	g	g	NOUN
ejpam-2826	245	13	is	be	AUX
ejpam-2826	245	14	continuous	continuous	ADJ
ejpam-2826	245	15	then	then	ADV
ejpam-2826	245	16	g	g	ADP
ejpam-2826	245	17	◦	◦	NOUN
ejpam-2826	245	18	f	f	PROPN
ejpam-2826	245	19	is	be	AUX
ejpam-2826	245	20	contra	contra	PROPN
ejpam-2826	245	21	δgb	δgb	ADV
ejpam-2826	245	22	-	-	PUNCT
ejpam-2826	245	23	continuous	continuous	ADJ
ejpam-2826	245	24	.	.	PUNCT
ejpam-2826	246	1	(	(	PUNCT
ejpam-2826	246	2	ii	ii	NOUN
ejpam-2826	246	3	)	)	PUNCT
ejpam-2826	246	4	if	if	SCONJ
ejpam-2826	246	5	f	f	PROPN
ejpam-2826	246	6	is	be	AUX
ejpam-2826	246	7	contra	contra	PROPN
ejpam-2826	246	8	δgb	δgb	ADV
ejpam-2826	246	9	-	-	PUNCT
ejpam-2826	246	10	continuous	continuous	ADJ
ejpam-2826	246	11	and	and	CCONJ
ejpam-2826	246	12	g	g	PROPN
ejpam-2826	246	13	is	be	AUX
ejpam-2826	246	14	contra	contra	PROPN
ejpam-2826	246	15	continuous	continuous	ADJ
ejpam-2826	246	16	then	then	ADV
ejpam-2826	246	17	g	g	PROPN
ejpam-2826	246	18	◦	◦	NOUN
ejpam-2826	246	19	f	f	PROPN
ejpam-2826	246	20	is	be	AUX
ejpam-2826	246	21	δgb	δgb	ADV
ejpam-2826	246	22	-	-	PUNCT
ejpam-2826	246	23	continuous	continuous	ADJ
ejpam-2826	246	24	.	.	PUNCT
ejpam-2826	247	1	(	(	PUNCT
ejpam-2826	247	2	iii	iii	X
ejpam-2826	247	3	)	)	PUNCT
ejpam-2826	247	4	if	if	SCONJ
ejpam-2826	247	5	f	f	PROPN
ejpam-2826	247	6	is	be	AUX
ejpam-2826	247	7	δgb	δgb	ADV
ejpam-2826	247	8	-	-	PUNCT
ejpam-2826	247	9	continuous	continuous	ADJ
ejpam-2826	247	10	and	and	CCONJ
ejpam-2826	247	11	g	g	PROPN
ejpam-2826	247	12	is	be	AUX
ejpam-2826	247	13	contra	contra	PROPN
ejpam-2826	247	14	continuous	continuous	ADJ
ejpam-2826	247	15	then	then	ADV
ejpam-2826	247	16	g	g	PROPN
ejpam-2826	247	17	◦	◦	NOUN
ejpam-2826	247	18	f	f	PROPN
ejpam-2826	247	19	is	be	AUX
ejpam-2826	247	20	contra	contra	PROPN
ejpam-2826	247	21	δgb	δgb	ADV
ejpam-2826	247	22	-	-	PUNCT
ejpam-2826	247	23	continuous	continuous	ADJ
ejpam-2826	247	24	.	.	PUNCT
ejpam-2826	248	1	(	(	PUNCT
ejpam-2826	248	2	iv	iv	X
ejpam-2826	248	3	)	)	PUNCT
ejpam-2826	248	4	if	if	SCONJ
ejpam-2826	248	5	f	f	PROPN
ejpam-2826	248	6	is	be	AUX
ejpam-2826	248	7	δgb	δgb	NOUN
ejpam-2826	248	8	-	-	PUNCT
ejpam-2826	248	9	irresolute	irresolute	ADJ
ejpam-2826	248	10	and	and	CCONJ
ejpam-2826	248	11	g	g	PROPN
ejpam-2826	248	12	is	be	AUX
ejpam-2826	248	13	contra	contra	PROPN
ejpam-2826	248	14	δgb	δgb	ADV
ejpam-2826	248	15	-	-	PUNCT
ejpam-2826	248	16	continuous	continuous	ADJ
ejpam-2826	248	17	then	then	ADV
ejpam-2826	248	18	g	g	NOUN
ejpam-2826	248	19	◦	◦	NOUN
ejpam-2826	248	20	f	f	PROPN
ejpam-2826	248	21	is	be	AUX
ejpam-2826	248	22	contra	contra	PROPN
ejpam-2826	248	23	δgb	δgb	ADV
ejpam-2826	248	24	-	-	PUNCT
ejpam-2826	248	25	continuous	continuous	ADJ
ejpam-2826	248	26	.	.	PUNCT
ejpam-2826	248	27	proof:(i	proof:(i	PROPN
ejpam-2826	248	28	)	)	PUNCT
ejpam-2826	248	29	let	let	VERB
ejpam-2826	248	30	h	h	NOUN
ejpam-2826	248	31	=	=	NOUN
ejpam-2826	248	32	g	g	NOUN
ejpam-2826	248	33	◦	◦	NOUN
ejpam-2826	248	34	f	f	NOUN
ejpam-2826	248	35	and	and	CCONJ
ejpam-2826	248	36	v	v	NOUN
ejpam-2826	248	37	be	be	AUX
ejpam-2826	248	38	an	an	DET
ejpam-2826	248	39	open	open	ADJ
ejpam-2826	248	40	set	set	NOUN
ejpam-2826	248	41	in	in	ADP
ejpam-2826	248	42	z.	z.	PROPN
ejpam-2826	248	43	since	since	SCONJ
ejpam-2826	248	44	g	g	PROPN
ejpam-2826	248	45	is	be	AUX
ejpam-2826	248	46	continuous	continuous	ADJ
ejpam-2826	248	47	,	,	PUNCT
ejpam-2826	248	48	g−1(v	g−1(v	PROPN
ejpam-2826	248	49	)	)	PUNCT
ejpam-2826	248	50	is	be	AUX
ejpam-2826	248	51	open	open	ADJ
ejpam-2826	248	52	in	in	ADP
ejpam-2826	248	53	y.	y.	PROPN
ejpam-2826	248	54	therefore	therefore	ADV
ejpam-2826	248	55	f−1[g−1(v)]=h−1(v	f−1[g−1(v)]=h−1(v	NOUN
ejpam-2826	248	56	)	)	PUNCT
ejpam-2826	248	57	is	be	AUX
ejpam-2826	248	58	δgb	δgb	ADV
ejpam-2826	248	59	-	-	PUNCT
ejpam-2826	248	60	closed	closed	ADJ
ejpam-2826	248	61	in	in	ADP
ejpam-2826	248	62	x	x	PUNCT
ejpam-2826	248	63	because	because	SCONJ
ejpam-2826	248	64	f	f	PROPN
ejpam-2826	248	65	is	be	AUX
ejpam-2826	248	66	contra	contra	PROPN
ejpam-2826	248	67	δgb	δgb	ADV
ejpam-2826	248	68	-	-	PUNCT
ejpam-2826	248	69	continuous	continuous	ADJ
ejpam-2826	248	70	.	.	PUNCT
ejpam-2826	249	1	hence	hence	ADV
ejpam-2826	249	2	g	g	ADP
ejpam-2826	249	3	◦	◦	NOUN
ejpam-2826	249	4	f	f	PROPN
ejpam-2826	249	5	is	be	AUX
ejpam-2826	249	6	contra	contra	PROPN
ejpam-2826	249	7	δgb	δgb	ADV
ejpam-2826	249	8	-	-	PUNCT
ejpam-2826	249	9	continuous	continuous	ADJ
ejpam-2826	249	10	.	.	PUNCT
ejpam-2826	250	1	the	the	DET
ejpam-2826	250	2	proofs	proof	NOUN
ejpam-2826	250	3	of	of	ADP
ejpam-2826	250	4	(	(	PUNCT
ejpam-2826	250	5	ii),(iii	ii),(iii	X
ejpam-2826	250	6	)	)	PUNCT
ejpam-2826	250	7	and	and	CCONJ
ejpam-2826	250	8	(	(	PUNCT
ejpam-2826	250	9	iv	iv	X
ejpam-2826	250	10	)	)	PUNCT
ejpam-2826	250	11	are	be	AUX
ejpam-2826	250	12	similar	similar	ADJ
ejpam-2826	250	13	to	to	ADP
ejpam-2826	250	14	(	(	PUNCT
ejpam-2826	250	15	i	i	NOUN
ejpam-2826	250	16	)	)	PUNCT
ejpam-2826	250	17	.	.	PUNCT
ejpam-2826	251	1	theorem	theorem	NOUN
ejpam-2826	251	2	31	31	NUM
ejpam-2826	251	3	.	.	PUNCT
ejpam-2826	252	1	let	let	VERB
ejpam-2826	252	2	f	f	X
ejpam-2826	252	3	:	:	PUNCT
ejpam-2826	252	4	x→y	x→y	NUM
ejpam-2826	252	5	be	be	AUX
ejpam-2826	252	6	contra	contra	PROPN
ejpam-2826	252	7	δgb	δgb	ADV
ejpam-2826	252	8	-	-	PUNCT
ejpam-2826	252	9	continuous	continuous	ADJ
ejpam-2826	252	10	and	and	CCONJ
ejpam-2826	252	11	g	g	NOUN
ejpam-2826	252	12	:	:	PUNCT
ejpam-2826	252	13	y→z	y→z	NUM
ejpam-2826	252	14	be	be	AUX
ejpam-2826	252	15	δgb	δgb	ADV
ejpam-2826	252	16	-	-	PUNCT
ejpam-2826	252	17	continuous	continuous	ADJ
ejpam-2826	252	18	.	.	PUNCT
ejpam-2826	253	1	if	if	SCONJ
ejpam-2826	253	2	y	y	PROPN
ejpam-2826	253	3	is	be	AUX
ejpam-2826	253	4	tδgb	tδgb	ADJ
ejpam-2826	253	5	-	-	PUNCT
ejpam-2826	253	6	space	space	NOUN
ejpam-2826	253	7	,	,	PUNCT
ejpam-2826	253	8	then	then	ADV
ejpam-2826	253	9	g	g	ADP
ejpam-2826	253	10	◦	◦	NOUN
ejpam-2826	253	11	f	f	X
ejpam-2826	253	12	:	:	PUNCT
ejpam-2826	253	13	x→z	x→z	NUM
ejpam-2826	253	14	is	be	AUX
ejpam-2826	253	15	contra	contra	PROPN
ejpam-2826	253	16	δgbcontinuous	δgbcontinuous	ADJ
ejpam-2826	253	17	.	.	PUNCT
ejpam-2826	254	1	proof	proof	NOUN
ejpam-2826	254	2	:	:	PUNCT
ejpam-2826	254	3	let	let	VERB
ejpam-2826	254	4	v	v	PART
ejpam-2826	254	5	be	be	AUX
ejpam-2826	254	6	any	any	DET
ejpam-2826	254	7	open	open	ADJ
ejpam-2826	254	8	set	set	NOUN
ejpam-2826	254	9	in	in	ADP
ejpam-2826	254	10	z	z	PROPN
ejpam-2826	254	11	.	.	PUNCT
ejpam-2826	255	1	since	since	SCONJ
ejpam-2826	255	2	g	g	PROPN
ejpam-2826	255	3	is	be	AUX
ejpam-2826	255	4	δgb	δgb	ADV
ejpam-2826	255	5	-	-	PUNCT
ejpam-2826	255	6	continuous	continuous	ADJ
ejpam-2826	255	7	g−1(v	g−1(v	NOUN
ejpam-2826	255	8	)	)	PUNCT
ejpam-2826	255	9	is	be	AUX
ejpam-2826	255	10	δgb	δgb	ADV
ejpam-2826	255	11	-	-	PUNCT
ejpam-2826	255	12	open	open	ADJ
ejpam-2826	255	13	in	in	ADP
ejpam-2826	255	14	y	y	PROPN
ejpam-2826	255	15	and	and	CCONJ
ejpam-2826	255	16	since	since	SCONJ
ejpam-2826	255	17	y	y	PROPN
ejpam-2826	255	18	is	be	AUX
ejpam-2826	255	19	tδgb	tδgb	ADJ
ejpam-2826	255	20	-	-	PUNCT
ejpam-2826	255	21	space	space	NOUN
ejpam-2826	255	22	,	,	PUNCT
ejpam-2826	255	23	g−1(v	g−1(v	NOUN
ejpam-2826	255	24	)	)	PUNCT
ejpam-2826	255	25	open	open	ADJ
ejpam-2826	255	26	in	in	ADP
ejpam-2826	255	27	y.	y.	PROPN
ejpam-2826	255	28	since	since	SCONJ
ejpam-2826	255	29	f	f	PROPN
ejpam-2826	255	30	is	be	AUX
ejpam-2826	255	31	contra	contra	PROPN
ejpam-2826	255	32	δgb	δgb	ADV
ejpam-2826	255	33	-	-	PUNCT
ejpam-2826	255	34	continuous	continuous	ADJ
ejpam-2826	255	35	,	,	PUNCT
ejpam-2826	255	36	then	then	ADV
ejpam-2826	255	37	f−1(g−1(v	f−1(g−1(v	PROPN
ejpam-2826	255	38	)	)	PUNCT
ejpam-2826	255	39	)	)	PUNCT
ejpam-2826	256	1	=	=	PRON
ejpam-2826	256	2	(	(	PUNCT
ejpam-2826	256	3	g	g	NOUN
ejpam-2826	256	4	◦	◦	NOUN
ejpam-2826	256	5	f)−1(v	f)−1(v	NOUN
ejpam-2826	256	6	)	)	PUNCT
ejpam-2826	256	7	is	be	AUX
ejpam-2826	256	8	δgb	δgb	ADV
ejpam-2826	256	9	-	-	PUNCT
ejpam-2826	256	10	closed	close	VERB
ejpam-2826	256	11	set	set	NOUN
ejpam-2826	256	12	in	in	ADP
ejpam-2826	256	13	x.	x.	NOUN
ejpam-2826	256	14	therefore	therefore	ADV
ejpam-2826	256	15	g	g	PROPN
ejpam-2826	256	16	◦	◦	NOUN
ejpam-2826	256	17	f	f	PROPN
ejpam-2826	256	18	is	be	AUX
ejpam-2826	256	19	contra	contra	PROPN
ejpam-2826	256	20	δgb	δgb	ADV
ejpam-2826	256	21	-	-	PUNCT
ejpam-2826	256	22	continuous	continuous	ADJ
ejpam-2826	256	23	.	.	PUNCT
ejpam-2826	257	1	references	reference	NOUN
ejpam-2826	257	2	321	321	NUM
ejpam-2826	257	3	acknowledgements	acknowledgement	NOUN
ejpam-2826	257	4	the	the	DET
ejpam-2826	257	5	authors	author	NOUN
ejpam-2826	257	6	are	be	AUX
ejpam-2826	257	7	grateful	grateful	ADJ
ejpam-2826	257	8	to	to	ADP
ejpam-2826	257	9	the	the	DET
ejpam-2826	257	10	university	university	NOUN
ejpam-2826	257	11	grants	grant	NOUN
ejpam-2826	257	12	commission	commission	PROPN
ejpam-2826	257	13	,	,	PUNCT
ejpam-2826	257	14	new	new	PROPN
ejpam-2826	257	15	delhi	delhi	PROPN
ejpam-2826	257	16	,	,	PUNCT
ejpam-2826	257	17	india	india	PROPN
ejpam-2826	257	18	for	for	ADP
ejpam-2826	257	19	financial	financial	ADJ
ejpam-2826	257	20	support	support	NOUN
ejpam-2826	257	21	under	under	ADP
ejpam-2826	257	22	ugc	ugc	PROPN
ejpam-2826	257	23	sap	sap	PROPN
ejpam-2826	257	24	drs	drs	PROPN
ejpam-2826	257	25	-	-	PUNCT
ejpam-2826	257	26	iii	iii	NOUN
ejpam-2826	257	27	:	:	PUNCT
ejpam-2826	257	28	f-510/3	f-510/3	VERB
ejpam-2826	257	29	/	/	SYM
ejpam-2826	257	30	drs	drs	PROPN
ejpam-2826	257	31	-	-	PUNCT
ejpam-2826	257	32	iii/2016(sap	iii/2016(sap	NOUN
ejpam-2826	257	33	-	-	PUNCT
ejpam-2826	257	34	i	i	NOUN
ejpam-2826	257	35	)	)	PUNCT
ejpam-2826	257	36	dated	date	VERB
ejpam-2826	257	37	29th	29th	ADJ
ejpam-2826	257	38	feb	feb	NOUN
ejpam-2826	257	39	2016	2016	NUM
ejpam-2826	257	40	to	to	ADP
ejpam-2826	257	41	the	the	DET
ejpam-2826	257	42	department	department	NOUN
ejpam-2826	257	43	of	of	ADP
ejpam-2826	257	44	mathematics	mathematics	PROPN
ejpam-2826	257	45	,	,	PUNCT
ejpam-2826	257	46	karnatak	karnatak	PROPN
ejpam-2826	257	47	university	university	PROPN
ejpam-2826	257	48	,	,	PUNCT
ejpam-2826	257	49	dharwad	dharwad	PROPN
ejpam-2826	257	50	,	,	PUNCT
ejpam-2826	257	51	india	india	PROPN
ejpam-2826	257	52	.	.	PUNCT
ejpam-2826	258	1	references	reference	NOUN
ejpam-2826	258	2	[	[	X
ejpam-2826	258	3	1	1	NUM
ejpam-2826	258	4	]	]	X
ejpam-2826	258	5	a.al	a.al	PROPN
ejpam-2826	258	6	-	-	ADJ
ejpam-2826	258	7	omari	omari	ADJ
ejpam-2826	258	8	and	and	CCONJ
ejpam-2826	258	9	m.s.noorani	m.s.noorani	NOUN
ejpam-2826	258	10	,	,	PUNCT
ejpam-2826	258	11	decomposition	decomposition	NOUN
ejpam-2826	258	12	of	of	ADP
ejpam-2826	258	13	continuity	continuity	NOUN
ejpam-2826	258	14	via	via	ADP
ejpam-2826	258	15	b	b	NOUN
ejpam-2826	258	16	-	-	PUNCT
ejpam-2826	258	17	open	open	ADJ
ejpam-2826	258	18	set	set	NOUN
ejpam-2826	258	19	,	,	PUNCT
ejpam-2826	258	20	bol.soc.paran.mat	bol.soc.paran.mat	PROPN
ejpam-2826	258	21	.	.	NOUN
ejpam-2826	258	22	,	,	PUNCT
ejpam-2826	258	23	26(2008),53	26(2008),53	PROPN
ejpam-2826	258	24	-	-	SYM
ejpam-2826	258	25	64	64	NUM
ejpam-2826	258	26	.	.	PUNCT
ejpam-2826	259	1	[	[	X
ejpam-2826	259	2	2	2	X
ejpam-2826	259	3	]	]	X
ejpam-2826	259	4	d.	d.	PROPN
ejpam-2826	259	5	andrijivic	andrijivic	PROPN
ejpam-2826	259	6	,	,	PUNCT
ejpam-2826	259	7	on	on	ADP
ejpam-2826	259	8	b	b	X
ejpam-2826	259	9	-	-	PUNCT
ejpam-2826	259	10	open	open	ADJ
ejpam-2826	259	11	sets	set	NOUN
ejpam-2826	259	12	,	,	PUNCT
ejpam-2826	259	13	mat.vesnic	mat.vesnic	NOUN
ejpam-2826	259	14	,	,	PUNCT
ejpam-2826	259	15	48(1996	48(1996	NOUN
ejpam-2826	259	16	)	)	PUNCT
ejpam-2826	259	17	,	,	PUNCT
ejpam-2826	259	18	59	59	NUM
ejpam-2826	259	19	-	-	SYM
ejpam-2826	259	20	64	64	NUM
ejpam-2826	259	21	.	.	PUNCT
ejpam-2826	260	1	[	[	X
ejpam-2826	260	2	3	3	X
ejpam-2826	260	3	]	]	X
ejpam-2826	260	4	s.p.arya	s.p.arya	NOUN
ejpam-2826	260	5	and	and	CCONJ
ejpam-2826	260	6	r.gupta	r.gupta	NOUN
ejpam-2826	260	7	,	,	PUNCT
ejpam-2826	260	8	on	on	ADP
ejpam-2826	260	9	strongly	strongly	ADV
ejpam-2826	260	10	continuous	continuous	ADJ
ejpam-2826	260	11	mappings	mapping	NOUN
ejpam-2826	260	12	,	,	PUNCT
ejpam-2826	260	13	kyungpook	kyungpook	NOUN
ejpam-2826	260	14	math	math	NOUN
ejpam-2826	260	15	.	.	PUNCT
ejpam-2826	260	16	,	,	PUNCT
ejpam-2826	260	17	14(1974),131	14(1974),131	NUM
ejpam-2826	260	18	-	-	SYM
ejpam-2826	260	19	143	143	NUM
ejpam-2826	260	20	.	.	PUNCT
ejpam-2826	261	1	[	[	X
ejpam-2826	261	2	4	4	NUM
ejpam-2826	261	3	]	]	X
ejpam-2826	261	4	s.s.benchalli	s.s.benchalli	PROPN
ejpam-2826	261	5	,	,	PUNCT
ejpam-2826	261	6	p.g.patil	p.g.patil	PROPN
ejpam-2826	261	7	,	,	PUNCT
ejpam-2826	261	8	j	j	PROPN
ejpam-2826	261	9	,	,	PUNCT
ejpam-2826	261	10	b.toranagatti	b.toranagatti	NOUN
ejpam-2826	261	11	and	and	CCONJ
ejpam-2826	261	12	s.r.vighneshi	s.r.vighneshi	NOUN
ejpam-2826	261	13	,	,	PUNCT
ejpam-2826	261	14	δgb	δgb	ADJ
ejpam-2826	261	15	-	-	PUNCT
ejpam-2826	261	16	separation	separation	NOUN
ejpam-2826	261	17	axioms	axiom	NOUN
ejpam-2826	261	18	in	in	ADP
ejpam-2826	261	19	topological	topological	ADJ
ejpam-2826	261	20	spaces	space	NOUN
ejpam-2826	261	21	,	,	PUNCT
ejpam-2826	261	22	international	international	PROPN
ejpam-2826	261	23	mathematical	mathematical	ADJ
ejpam-2826	261	24	forum	forum	PROPN
ejpam-2826	261	25	,	,	PUNCT
ejpam-2826	261	26	23(2016),1117	23(2016),1117	NUM
ejpam-2826	261	27	-	-	SYM
ejpam-2826	261	28	1131	1131	NUM
ejpam-2826	261	29	.	.	PUNCT
ejpam-2826	262	1	[	[	X
ejpam-2826	262	2	5	5	NUM
ejpam-2826	262	3	]	]	PUNCT
ejpam-2826	262	4	s.s.benchalli	s.s.benchalli	PROPN
ejpam-2826	262	5	,	,	PUNCT
ejpam-2826	262	6	p.g.patil	p.g.patil	PROPN
ejpam-2826	262	7	and	and	CCONJ
ejpam-2826	262	8	j.b.toranagatti	j.b.toranagatti	PROPN
ejpam-2826	262	9	,	,	PUNCT
ejpam-2826	262	10	delta	delta	NOUN
ejpam-2826	262	11	generalized	generalize	VERB
ejpam-2826	262	12	b	b	X
ejpam-2826	262	13	-	-	PUNCT
ejpam-2826	262	14	continuous	continuous	ADJ
ejpam-2826	262	15	functions	function	NOUN
ejpam-2826	262	16	in	in	ADP
ejpam-2826	262	17	topological	topological	ADJ
ejpam-2826	262	18	spaces	space	NOUN
ejpam-2826	262	19	,	,	PUNCT
ejpam-2826	262	20	international	international	ADJ
ejpam-2826	262	21	journal	journal	NOUN
ejpam-2826	262	22	of	of	ADP
ejpam-2826	262	23	scientific	scientific	ADJ
ejpam-2826	262	24	and	and	CCONJ
ejpam-2826	262	25	innovative	innovative	ADJ
ejpam-2826	262	26	mathematical	mathematical	ADJ
ejpam-2826	262	27	research	research	NOUN
ejpam-2826	262	28	,	,	PUNCT
ejpam-2826	262	29	3(2015	3(2015	NUM
ejpam-2826	262	30	)	)	PUNCT
ejpam-2826	262	31	,	,	PUNCT
ejpam-2826	262	32	440	440	NUM
ejpam-2826	262	33	-	-	SYM
ejpam-2826	262	34	446	446	NUM
ejpam-2826	262	35	.	.	PUNCT
ejpam-2826	263	1	[	[	X
ejpam-2826	263	2	6	6	NUM
ejpam-2826	263	3	]	]	PUNCT
ejpam-2826	263	4	j.dontchev	j.dontchev	PROPN
ejpam-2826	263	5	,	,	PUNCT
ejpam-2826	263	6	contra	contra	PROPN
ejpam-2826	263	7	continuous	continuous	ADJ
ejpam-2826	263	8	functions	function	NOUN
ejpam-2826	263	9	and	and	CCONJ
ejpam-2826	263	10	strongly	strongly	ADV
ejpam-2826	263	11	s	s	AUX
ejpam-2826	263	12	-	-	PUNCT
ejpam-2826	263	13	closed	closed	ADJ
ejpam-2826	263	14	mappings	mapping	NOUN
ejpam-2826	263	15	,	,	PUNCT
ejpam-2826	263	16	int.j.math.sci	int.j.math.sci	PROPN
ejpam-2826	263	17	.	.	PUNCT
ejpam-2826	263	18	,19(1996),303	,19(1996),303	PUNCT
ejpam-2826	263	19	-	-	PUNCT
ejpam-2826	263	20	310	310	NUM
ejpam-2826	263	21	.	.	PUNCT
ejpam-2826	264	1	[	[	X
ejpam-2826	264	2	7	7	X
ejpam-2826	264	3	]	]	X
ejpam-2826	264	4	n.el	n.el	PROPN
ejpam-2826	264	5	-	-	PUNCT
ejpam-2826	264	6	deeb	deeb	PROPN
ejpam-2826	264	7	,	,	PUNCT
ejpam-2826	264	8	i.a.hasanein	i.a.hasanein	ADJ
ejpam-2826	264	9	,	,	PUNCT
ejpam-2826	264	10	a.s.mashhour	a.s.mashhour	ADJ
ejpam-2826	264	11	and	and	CCONJ
ejpam-2826	264	12	t.noiri	t.noiri	ADV
ejpam-2826	264	13	,	,	PUNCT
ejpam-2826	264	14	on	on	ADP
ejpam-2826	264	15	p	p	ADJ
ejpam-2826	264	16	-	-	PUNCT
ejpam-2826	264	17	regular	regular	ADJ
ejpam-2826	264	18	spaces	space	NOUN
ejpam-2826	264	19	,	,	PUNCT
ejpam-2826	264	20	bull	bull	NOUN
ejpam-2826	264	21	.	.	PUNCT
ejpam-2826	264	22	math	math	NOUN
ejpam-2826	264	23	.	.	PUNCT
ejpam-2826	265	1	soc.sci.math.r.s.roumanie	soc.sci.math.r.s.roumanie	PROPN
ejpam-2826	265	2	,	,	PUNCT
ejpam-2826	265	3	27(1983),311	27(1983),311	NUM
ejpam-2826	265	4	-	-	SYM
ejpam-2826	265	5	315	315	NUM
ejpam-2826	265	6	.	.	PUNCT
ejpam-2826	266	1	[	[	X
ejpam-2826	266	2	8	8	X
ejpam-2826	266	3	]	]	X
ejpam-2826	266	4	s.jafari	s.jafari	NOUN
ejpam-2826	266	5	and	and	CCONJ
ejpam-2826	266	6	t.noiri	t.noiri	ADV
ejpam-2826	266	7	,	,	PUNCT
ejpam-2826	266	8	contra	contra	PROPN
ejpam-2826	266	9	-	-	ADJ
ejpam-2826	266	10	super	super	ADJ
ejpam-2826	266	11	-	-	ADJ
ejpam-2826	266	12	continuous	continuous	ADJ
ejpam-2826	266	13	functions	function	NOUN
ejpam-2826	266	14	.	.	PUNCT
ejpam-2826	267	1	ann.univ.sci	ann.univ.sci	NOUN
ejpam-2826	267	2	.	.	PUNCT
ejpam-2826	268	1	budapest	budapest	PROPN
ejpam-2826	268	2	.	.	PUNCT
ejpam-2826	269	1	eotvos	eotvos	PROPN
ejpam-2826	269	2	sect.math	sect.math	PROPN
ejpam-2826	269	3	.	.	PUNCT
ejpam-2826	270	1	,42(1999),27	,42(1999),27	PUNCT
ejpam-2826	270	2	-	-	PUNCT
ejpam-2826	270	3	34	34	NUM
ejpam-2826	270	4	.	.	PUNCT
ejpam-2826	271	1	[	[	X
ejpam-2826	271	2	9	9	NUM
ejpam-2826	271	3	]	]	PUNCT
ejpam-2826	271	4	a.s.mashhour	a.s.mashhour	ADJ
ejpam-2826	271	5	,	,	PUNCT
ejpam-2826	271	6	m.e.abd	m.e.abd	PROPN
ejpam-2826	271	7	el	el	PROPN
ejpam-2826	271	8	-	-	PROPN
ejpam-2826	271	9	monsef	monsef	PROPN
ejpam-2826	271	10	and	and	CCONJ
ejpam-2826	271	11	s.	s.	PROPN
ejpam-2826	271	12	n.	n.	PROPN
ejpam-2826	271	13	el	el	PROPN
ejpam-2826	271	14	-	-	PROPN
ejpam-2826	271	15	deeb	deeb	PROPN
ejpam-2826	271	16	,	,	PUNCT
ejpam-2826	271	17	on	on	ADP
ejpam-2826	271	18	pre	pre	ADJ
ejpam-2826	271	19	-	-	ADJ
ejpam-2826	271	20	continuous	continuous	ADJ
ejpam-2826	271	21	and	and	CCONJ
ejpam-2826	271	22	weak	weak	ADJ
ejpam-2826	271	23	pre	pre	ADJ
ejpam-2826	271	24	continuous	continuous	ADJ
ejpam-2826	271	25	mappings	mapping	NOUN
ejpam-2826	271	26	,	,	PUNCT
ejpam-2826	271	27	proc	proc	NOUN
ejpam-2826	271	28	.	.	PUNCT
ejpam-2826	271	29	math	math	NOUN
ejpam-2826	271	30	and	and	CCONJ
ejpam-2826	271	31	phys.soc	phys.soc	NOUN
ejpam-2826	271	32	.	.	PUNCT
ejpam-2826	271	33	egypt,53(1982),47	egypt,53(1982),47	PROPN
ejpam-2826	271	34	-	-	PUNCT
ejpam-2826	271	35	53	53	NUM
ejpam-2826	271	36	.	.	PUNCT
ejpam-2826	272	1	[	[	X
ejpam-2826	272	2	10	10	NUM
ejpam-2826	272	3	]	]	X
ejpam-2826	272	4	a.a.nasef	a.a.nasef	NOUN
ejpam-2826	272	5	,	,	PUNCT
ejpam-2826	272	6	some	some	DET
ejpam-2826	272	7	properties	property	NOUN
ejpam-2826	272	8	of	of	ADP
ejpam-2826	272	9	contra	contra	PROPN
ejpam-2826	272	10	-	-	PUNCT
ejpam-2826	272	11	γ	γ	ADJ
ejpam-2826	272	12	-	-	ADJ
ejpam-2826	272	13	continuous	continuous	ADJ
ejpam-2826	272	14	functions	function	NOUN
ejpam-2826	272	15	,	,	PUNCT
ejpam-2826	272	16	chaos	chaos	NOUN
ejpam-2826	272	17	solitons	soliton	NOUN
ejpam-2826	272	18	and	and	CCONJ
ejpam-2826	272	19	fractals	fractal	NOUN
ejpam-2826	272	20	,	,	PUNCT
ejpam-2826	272	21	24(2005),471	24(2005),471	NUM
ejpam-2826	272	22	-	-	SYM
ejpam-2826	272	23	477	477	NUM
ejpam-2826	272	24	.	.	PUNCT
ejpam-2826	273	1	[	[	X
ejpam-2826	273	2	11	11	NUM
ejpam-2826	273	3	]	]	SYM
ejpam-2826	273	4	t.nieminen	t.nieminen	NUM
ejpam-2826	273	5	,	,	PUNCT
ejpam-2826	273	6	on	on	ADP
ejpam-2826	273	7	ultrapseudocompact	ultrapseudocompact	ADJ
ejpam-2826	273	8	and	and	CCONJ
ejpam-2826	273	9	related	related	ADJ
ejpam-2826	273	10	spaces	space	NOUN
ejpam-2826	273	11	,	,	PUNCT
ejpam-2826	273	12	ann.acad.sci.fenn.ser.a.i.math	ann.acad.sci.fenn.ser.a.i.math	NOUN
ejpam-2826	273	13	.	.	PUNCT
ejpam-2826	274	1	,3(1977),185	,3(1977),185	PUNCT
ejpam-2826	274	2	-	-	PUNCT
ejpam-2826	274	3	205	205	NUM
ejpam-2826	274	4	.	.	PUNCT
ejpam-2826	275	1	[	[	X
ejpam-2826	275	2	12	12	NUM
ejpam-2826	275	3	]	]	PUNCT
ejpam-2826	275	4	t.noiri	t.noiri	ADV
ejpam-2826	275	5	,	,	PUNCT
ejpam-2826	275	6	super	super	NOUN
ejpam-2826	275	7	-	-	NOUN
ejpam-2826	275	8	continuity	continuity	NOUN
ejpam-2826	275	9	and	and	CCONJ
ejpam-2826	275	10	some	some	DET
ejpam-2826	275	11	strong	strong	ADJ
ejpam-2826	275	12	forms	form	NOUN
ejpam-2826	275	13	of	of	ADP
ejpam-2826	275	14	continuity	continuity	NOUN
ejpam-2826	275	15	,	,	PUNCT
ejpam-2826	275	16	indian	indian	ADJ
ejpam-2826	275	17	j.pure	j.pure	NOUN
ejpam-2826	275	18	appl.math	appl.math	PROPN
ejpam-2826	275	19	.	.	PUNCT
ejpam-2826	275	20	,15(1984),241	,15(1984),241	X
ejpam-2826	275	21	-	-	PUNCT
ejpam-2826	275	22	250	250	NUM
ejpam-2826	275	23	.	.	PUNCT
ejpam-2826	276	1	[	[	X
ejpam-2826	276	2	13	13	NUM
ejpam-2826	276	3	]	]	X
ejpam-2826	276	4	s.sekar	s.sekar	ADJ
ejpam-2826	276	5	and	and	CCONJ
ejpam-2826	276	6	k.mariappa	k.mariappa	ADJ
ejpam-2826	276	7	,	,	PUNCT
ejpam-2826	276	8	almost	almost	ADV
ejpam-2826	276	9	contra	contra	PROPN
ejpam-2826	276	10	regular	regular	ADJ
ejpam-2826	276	11	generalized	generalize	VERB
ejpam-2826	276	12	b	b	X
ejpam-2826	276	13	-	-	PUNCT
ejpam-2826	276	14	continuous	continuous	ADJ
ejpam-2826	276	15	functions	function	NOUN
ejpam-2826	276	16	,	,	PUNCT
ejpam-2826	276	17	international	international	ADJ
ejpam-2826	276	18	journal	journal	NOUN
ejpam-2826	276	19	of	of	ADP
ejpam-2826	276	20	pure	pure	ADJ
ejpam-2826	276	21	and	and	CCONJ
ejpam-2826	276	22	applied	applied	ADJ
ejpam-2826	276	23	mathematics	mathematic	NOUN
ejpam-2826	276	24	,	,	PUNCT
ejpam-2826	276	25	97(2014),161	97(2014),161	NUM
ejpam-2826	276	26	-	-	SYM
ejpam-2826	276	27	176	176	NUM
ejpam-2826	276	28	.	.	PUNCT
ejpam-2826	277	1	references	reference	NOUN
ejpam-2826	277	2	322	322	NUM
ejpam-2826	278	1	[	[	X
ejpam-2826	278	2	14	14	NUM
ejpam-2826	278	3	]	]	PUNCT
ejpam-2826	278	4	r.staum	r.staum	NOUN
ejpam-2826	278	5	,	,	PUNCT
ejpam-2826	278	6	the	the	DET
ejpam-2826	278	7	algebra	algebra	NOUN
ejpam-2826	278	8	of	of	ADP
ejpam-2826	278	9	bounded	bounded	ADJ
ejpam-2826	278	10	continuous	continuous	ADJ
ejpam-2826	278	11	functions	function	NOUN
ejpam-2826	278	12	into	into	ADP
ejpam-2826	278	13	a	a	DET
ejpam-2826	278	14	non	non	ADJ
ejpam-2826	278	15	-	-	ADJ
ejpam-2826	278	16	archimedean	archimedean	ADJ
ejpam-2826	278	17	field	field	NOUN
ejpam-2826	278	18	,	,	PUNCT
ejpam-2826	278	19	pacific	pacific	PROPN
ejpam-2826	278	20	j.math	j.math	PROPN
ejpam-2826	278	21	.	.	PUNCT
ejpam-2826	278	22	,50(1974),169	,50(1974),169	PUNCT
ejpam-2826	278	23	-	-	PUNCT
ejpam-2826	278	24	185	185	NUM
ejpam-2826	278	25	.	.	PUNCT
ejpam-2826	279	1	[	[	X
ejpam-2826	279	2	15	15	NUM
ejpam-2826	279	3	]	]	SYM
ejpam-2826	279	4	m.stone	m.stone	NUM
ejpam-2826	279	5	,	,	PUNCT
ejpam-2826	279	6	application	application	NOUN
ejpam-2826	279	7	of	of	ADP
ejpam-2826	279	8	the	the	DET
ejpam-2826	279	9	theory	theory	NOUN
ejpam-2826	279	10	of	of	ADP
ejpam-2826	279	11	boolean	boolean	ADJ
ejpam-2826	279	12	rings	ring	NOUN
ejpam-2826	279	13	to	to	ADP
ejpam-2826	279	14	general	general	ADJ
ejpam-2826	279	15	topology	topology	NOUN
ejpam-2826	279	16	,	,	PUNCT
ejpam-2826	279	17	trans.amer.math.soc	trans.amer.math.soc	PROPN
ejpam-2826	279	18	.	.	PROPN
ejpam-2826	279	19	,41(1937),371	,41(1937),371	PUNCT
ejpam-2826	279	20	-	-	PUNCT
ejpam-2826	279	21	381	381	NUM
ejpam-2826	279	22	.	.	PUNCT
ejpam-2826	280	1	[	[	X
ejpam-2826	280	2	16	16	NUM
ejpam-2826	280	3	]	]	PUNCT
ejpam-2826	280	4	k.v.tamilselvi	k.v.tamilselvi	NOUN
ejpam-2826	280	5	,	,	PUNCT
ejpam-2826	280	6	p.thangaraj	p.thangaraj	ADJ
ejpam-2826	280	7	and	and	CCONJ
ejpam-2826	280	8	o.ravi	o.ravi	ADJ
ejpam-2826	280	9	,	,	PUNCT
ejpam-2826	280	10	on	on	ADP
ejpam-2826	280	11	contra	contra	PROPN
ejpam-2826	280	12	gγ	gγ	PROPN
ejpam-2826	280	13	-	-	PUNCT
ejpam-2826	280	14	continuous	continuous	ADJ
ejpam-2826	280	15	functions	function	NOUN
ejpam-2826	280	16	,	,	PUNCT
ejpam-2826	280	17	recent	recent	ADJ
ejpam-2826	280	18	and	and	CCONJ
ejpam-2826	280	19	innovation	innovation	NOUN
ejpam-2826	280	20	trends	trend	NOUN
ejpam-2826	280	21	in	in	ADP
ejpam-2826	280	22	computing	computing	NOUN
ejpam-2826	280	23	and	and	CCONJ
ejpam-2826	280	24	communication,4(2016),260	communication,4(2016),260	PROPN
ejpam-2826	280	25	-	-	SYM
ejpam-2826	280	26	269	269	NUM
ejpam-2826	280	27	.	.	PUNCT
ejpam-2826	281	1	[	[	X
ejpam-2826	281	2	17	17	NUM
ejpam-2826	281	3	]	]	X
ejpam-2826	281	4	n.v.veliko	n.v.veliko	NOUN
ejpam-2826	281	5	,	,	PUNCT
ejpam-2826	281	6	h	h	NOUN
ejpam-2826	281	7	-	-	PUNCT
ejpam-2826	281	8	closed	closed	ADJ
ejpam-2826	281	9	topological	topological	ADJ
ejpam-2826	281	10	spaces	space	NOUN
ejpam-2826	281	11	,	,	PUNCT
ejpam-2826	281	12	amer.math.soc.transl	amer.math.soc.transl	PROPN
ejpam-2826	281	13	.	.	PUNCT
ejpam-2826	281	14	,78(1968),103	,78(1968),103	PUNCT
ejpam-2826	281	15	-	-	PUNCT
ejpam-2826	281	16	118	118	NUM
ejpam-2826	281	17	.	.	PUNCT