id	sid	tid	token	lemma	pos
ejpam-3317	1	1	european	european	PROPN
ejpam-3317	1	2	journal	journal	PROPN
ejpam-3317	1	3	of	of	ADP
ejpam-3317	1	4	pure	pure	ADJ
ejpam-3317	1	5	and	and	CCONJ
ejpam-3317	1	6	applied	apply	VERB
ejpam-3317	1	7	mathematics	mathematic	NOUN
ejpam-3317	1	8	vol	vol	NOUN
ejpam-3317	1	9	.	.	PROPN
ejpam-3317	2	1	12	12	NUM
ejpam-3317	2	2	,	,	PUNCT
ejpam-3317	2	3	no	no	INTJ
ejpam-3317	2	4	.	.	NOUN
ejpam-3317	2	5	3	3	NUM
ejpam-3317	2	6	,	,	PUNCT
ejpam-3317	2	7	2019	2019	NUM
ejpam-3317	2	8	,	,	PUNCT
ejpam-3317	2	9	1082	1082	NUM
ejpam-3317	2	10	-	-	SYM
ejpam-3317	2	11	1095	1095	NUM
ejpam-3317	2	12	issn	issn	PROPN
ejpam-3317	2	13	1307	1307	NUM
ejpam-3317	2	14	-	-	SYM
ejpam-3317	2	15	5543	5543	NUM
ejpam-3317	2	16	–	–	PUNCT
ejpam-3317	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3317	2	18	published	publish	VERB
ejpam-3317	2	19	by	by	ADP
ejpam-3317	2	20	new	new	PROPN
ejpam-3317	2	21	york	york	PROPN
ejpam-3317	2	22	business	business	PROPN
ejpam-3317	2	23	global	global	ADJ
ejpam-3317	2	24	pre	pre	ADJ
ejpam-3317	2	25	-	-	ADJ
ejpam-3317	2	26	irresolute	irresolute	ADJ
ejpam-3317	2	27	functions	function	NOUN
ejpam-3317	2	28	in	in	ADP
ejpam-3317	2	29	closure	closure	NOUN
ejpam-3317	2	30	spaces	space	NOUN
ejpam-3317	2	31	halgwrd	halgwrd	VERB
ejpam-3317	2	32	m.	m.	PROPN
ejpam-3317	2	33	darwesh1	darwesh1	PROPN
ejpam-3317	2	34	,	,	PUNCT
ejpam-3317	2	35	sarhad	sarhad	PROPN
ejpam-3317	2	36	f.	f.	PROPN
ejpam-3317	2	37	namiq2,∗	namiq2,∗	PROPN
ejpam-3317	2	38	1department	1department	NUM
ejpam-3317	2	39	of	of	ADP
ejpam-3317	2	40	mathematics	mathematic	NOUN
ejpam-3317	2	41	,	,	PUNCT
ejpam-3317	2	42	college	college	NOUN
ejpam-3317	2	43	of	of	ADP
ejpam-3317	2	44	science	science	NOUN
ejpam-3317	2	45	,	,	PUNCT
ejpam-3317	2	46	university	university	NOUN
ejpam-3317	2	47	of	of	ADP
ejpam-3317	2	48	sulaimani	sulaimani	PROPN
ejpam-3317	2	49	,	,	PUNCT
ejpam-3317	2	50	kurdistan	kurdistan	ADJ
ejpam-3317	2	51	-	-	PUNCT
ejpam-3317	2	52	region	region	NOUN
ejpam-3317	2	53	,	,	PUNCT
ejpam-3317	2	54	iraq	iraq	PROPN
ejpam-3317	2	55	2	2	NUM
ejpam-3317	2	56	department	department	NOUN
ejpam-3317	2	57	of	of	ADP
ejpam-3317	2	58	mathematics	mathematic	NOUN
ejpam-3317	2	59	,	,	PUNCT
ejpam-3317	2	60	college	college	NOUN
ejpam-3317	2	61	of	of	ADP
ejpam-3317	2	62	education	education	NOUN
ejpam-3317	2	63	,	,	PUNCT
ejpam-3317	2	64	university	university	NOUN
ejpam-3317	2	65	of	of	ADP
ejpam-3317	2	66	garmian	garmian	PROPN
ejpam-3317	2	67	,	,	PUNCT
ejpam-3317	2	68	kurdistan	kurdistan	ADJ
ejpam-3317	2	69	-	-	PUNCT
ejpam-3317	2	70	region	region	NOUN
ejpam-3317	2	71	,	,	PUNCT
ejpam-3317	2	72	iraq	iraq	PROPN
ejpam-3317	2	73	abstract	abstract	NOUN
ejpam-3317	2	74	.	.	PUNCT
ejpam-3317	3	1	the	the	DET
ejpam-3317	3	2	preopen	preopen	ADJ
ejpam-3317	3	3	sets	set	NOUN
ejpam-3317	3	4	are	be	AUX
ejpam-3317	3	5	used	use	VERB
ejpam-3317	3	6	to	to	PART
ejpam-3317	3	7	define	define	VERB
ejpam-3317	3	8	pre	pre	ADJ
ejpam-3317	3	9	-	-	ADJ
ejpam-3317	3	10	open	open	ADJ
ejpam-3317	3	11	functions	function	NOUN
ejpam-3317	3	12	,	,	PUNCT
ejpam-3317	3	13	pre	pre	ADJ
ejpam-3317	3	14	-	-	ADJ
ejpam-3317	3	15	closed	closed	ADJ
ejpam-3317	3	16	functions	function	NOUN
ejpam-3317	3	17	,	,	PUNCT
ejpam-3317	3	18	precontinuous	precontinuous	ADJ
ejpam-3317	3	19	functions	function	NOUN
ejpam-3317	3	20	,	,	PUNCT
ejpam-3317	3	21	contra	contra	PROPN
ejpam-3317	3	22	-	-	ADJ
ejpam-3317	3	23	pre	pre	ADJ
ejpam-3317	3	24	-	-	ADJ
ejpam-3317	3	25	continuous	continuous	ADJ
ejpam-3317	3	26	functions	function	NOUN
ejpam-3317	3	27	and	and	CCONJ
ejpam-3317	3	28	pre	pre	ADJ
ejpam-3317	3	29	-	-	ADJ
ejpam-3317	3	30	irresolute	irresolute	ADJ
ejpam-3317	3	31	functions	function	NOUN
ejpam-3317	3	32	which	which	PRON
ejpam-3317	3	33	are	be	AUX
ejpam-3317	3	34	investigated	investigate	VERB
ejpam-3317	3	35	.	.	PUNCT
ejpam-3317	4	1	they	they	PRON
ejpam-3317	4	2	are	be	AUX
ejpam-3317	4	3	also	also	ADV
ejpam-3317	4	4	used	use	VERB
ejpam-3317	4	5	to	to	PART
ejpam-3317	4	6	introduce	introduce	VERB
ejpam-3317	4	7	a	a	DET
ejpam-3317	4	8	new	new	ADJ
ejpam-3317	4	9	type	type	NOUN
ejpam-3317	4	10	of	of	ADP
ejpam-3317	4	11	connectedness	connectedness	NOUN
ejpam-3317	4	12	and	and	CCONJ
ejpam-3317	4	13	compactness	compactness	NOUN
ejpam-3317	4	14	in	in	ADP
ejpam-3317	4	15	closure	closure	NOUN
ejpam-3317	4	16	spaces	space	NOUN
ejpam-3317	4	17	,	,	PUNCT
ejpam-3317	4	18	they	they	PRON
ejpam-3317	4	19	called	call	VERB
ejpam-3317	4	20	p	p	NOUN
ejpam-3317	4	21	-	-	PUNCT
ejpam-3317	4	22	connectedness	connectedness	NOUN
ejpam-3317	4	23	and	and	CCONJ
ejpam-3317	4	24	p	p	NOUN
ejpam-3317	4	25	-	-	PUNCT
ejpam-3317	4	26	compactness	compactness	NOUN
ejpam-3317	4	27	respectively	respectively	ADV
ejpam-3317	4	28	2010	2010	NUM
ejpam-3317	4	29	mathematics	mathematic	NOUN
ejpam-3317	4	30	subject	subject	NOUN
ejpam-3317	4	31	classifications	classification	NOUN
ejpam-3317	4	32	:	:	PUNCT
ejpam-3317	4	33	54a05	54a05	NUM
ejpam-3317	4	34	key	key	ADJ
ejpam-3317	4	35	words	word	NOUN
ejpam-3317	4	36	and	and	CCONJ
ejpam-3317	4	37	phrases	phrase	NOUN
ejpam-3317	4	38	:	:	PUNCT
ejpam-3317	4	39	closure	closure	NOUN
ejpam-3317	4	40	operator	operator	NOUN
ejpam-3317	4	41	,	,	PUNCT
ejpam-3317	4	42	closure	closure	NOUN
ejpam-3317	4	43	space	space	NOUN
ejpam-3317	4	44	,	,	PUNCT
ejpam-3317	4	45	pre	pre	ADJ
ejpam-3317	4	46	-	-	ADJ
ejpam-3317	4	47	open	open	ADJ
ejpam-3317	4	48	functions	function	NOUN
ejpam-3317	4	49	,	,	PUNCT
ejpam-3317	4	50	pre	pre	ADJ
ejpam-3317	4	51	-	-	ADJ
ejpam-3317	4	52	closed	closed	ADJ
ejpam-3317	4	53	functions	function	NOUN
ejpam-3317	4	54	,	,	PUNCT
ejpam-3317	4	55	contra	contra	PROPN
ejpam-3317	4	56	-	-	ADJ
ejpam-3317	4	57	pre	pre	ADJ
ejpam-3317	4	58	-	-	ADJ
ejpam-3317	4	59	continuous	continuous	ADJ
ejpam-3317	4	60	functions	function	NOUN
ejpam-3317	4	61	,	,	PUNCT
ejpam-3317	4	62	pre	pre	ADJ
ejpam-3317	4	63	-	-	ADJ
ejpam-3317	4	64	irresolute	irresolute	ADJ
ejpam-3317	4	65	functions	function	NOUN
ejpam-3317	4	66	,	,	PUNCT
ejpam-3317	4	67	p	p	NOUN
ejpam-3317	4	68	-	-	PUNCT
ejpam-3317	4	69	connectedness	connectedness	NOUN
ejpam-3317	4	70	,	,	PUNCT
ejpam-3317	4	71	p	p	NOUN
ejpam-3317	4	72	-	-	PUNCT
ejpam-3317	4	73	compactness	compactness	NOUN
ejpam-3317	4	74	,	,	PUNCT
ejpam-3317	4	75	tp	tp	NOUN
ejpam-3317	4	76	-	-	PUNCT
ejpam-3317	4	77	spaces	space	NOUN
ejpam-3317	4	78	1	1	NUM
ejpam-3317	4	79	.	.	PUNCT
ejpam-3317	5	1	introduction	introduction	NOUN
ejpam-3317	5	2	kazimierz	kazimierz	PROPN
ejpam-3317	5	3	kuratowski	kuratowski	PROPN
ejpam-3317	5	4	was	be	AUX
ejpam-3317	5	5	a	a	DET
ejpam-3317	5	6	polish	polish	ADJ
ejpam-3317	5	7	mathematician	mathematician	NOUN
ejpam-3317	5	8	and	and	CCONJ
ejpam-3317	5	9	logician	logician	ADJ
ejpam-3317	5	10	,	,	PUNCT
ejpam-3317	5	11	he	he	PRON
ejpam-3317	5	12	defined	define	VERB
ejpam-3317	5	13	[	[	X
ejpam-3317	5	14	14	14	NUM
ejpam-3317	5	15	]	]	X
ejpam-3317	5	16	closure	closure	NOUN
ejpam-3317	5	17	operator	operator	NOUN
ejpam-3317	5	18	by	by	ADP
ejpam-3317	5	19	the	the	DET
ejpam-3317	5	20	following	following	NOUN
ejpam-3317	5	21	:	:	PUNCT
ejpam-3317	5	22	let	let	VERB
ejpam-3317	5	23	x	x	PRON
ejpam-3317	5	24	be	be	AUX
ejpam-3317	5	25	a	a	DET
ejpam-3317	5	26	set	set	NOUN
ejpam-3317	5	27	and	and	CCONJ
ejpam-3317	5	28	p	p	NOUN
ejpam-3317	5	29	(	(	PUNCT
ejpam-3317	5	30	x	x	X
ejpam-3317	5	31	)	)	PUNCT
ejpam-3317	5	32	its	its	PRON
ejpam-3317	5	33	power	power	NOUN
ejpam-3317	5	34	set	set	NOUN
ejpam-3317	5	35	.	.	PUNCT
ejpam-3317	6	1	a	a	DET
ejpam-3317	6	2	kuratowski	kuratowski	ADJ
ejpam-3317	6	3	closure	closure	NOUN
ejpam-3317	6	4	operator	operator	NOUN
ejpam-3317	6	5	is	be	AUX
ejpam-3317	6	6	a	a	DET
ejpam-3317	6	7	function	function	NOUN
ejpam-3317	6	8	cl	cl	NOUN
ejpam-3317	6	9	:	:	PUNCT
ejpam-3317	6	10	p	p	X
ejpam-3317	6	11	(	(	PUNCT
ejpam-3317	6	12	x)→	x)→	PROPN
ejpam-3317	6	13	p	p	X
ejpam-3317	6	14	(	(	PUNCT
ejpam-3317	6	15	x	x	X
ejpam-3317	6	16	)	)	PUNCT
ejpam-3317	6	17	with	with	ADP
ejpam-3317	6	18	the	the	DET
ejpam-3317	6	19	following	follow	VERB
ejpam-3317	6	20	properties	property	NOUN
ejpam-3317	6	21	:	:	PUNCT
ejpam-3317	6	22	(	(	PUNCT
ejpam-3317	6	23	i	i	NOUN
ejpam-3317	6	24	)	)	PUNCT
ejpam-3317	6	25	cl	cl	NOUN
ejpam-3317	6	26	(	(	PUNCT
ejpam-3317	6	27	φ	φ	NOUN
ejpam-3317	6	28	)	)	PUNCT
ejpam-3317	6	29	=	=	SYM
ejpam-3317	6	30	φ	φ	PROPN
ejpam-3317	6	31	(	(	PUNCT
ejpam-3317	6	32	preservation	preservation	NOUN
ejpam-3317	6	33	of	of	ADP
ejpam-3317	6	34	nullary	nullary	ADJ
ejpam-3317	6	35	union	union	NOUN
ejpam-3317	6	36	)	)	PUNCT
ejpam-3317	6	37	(	(	PUNCT
ejpam-3317	6	38	ii	ii	NOUN
ejpam-3317	6	39	)	)	PUNCT
ejpam-3317	6	40	a	a	DET
ejpam-3317	6	41	⊆	⊆	NUM
ejpam-3317	6	42	cl	cl	NOUN
ejpam-3317	6	43	(	(	PUNCT
ejpam-3317	6	44	a	a	NOUN
ejpam-3317	6	45	)	)	PUNCT
ejpam-3317	6	46	for	for	ADP
ejpam-3317	6	47	every	every	DET
ejpam-3317	6	48	subset	subset	NOUN
ejpam-3317	6	49	a	a	DET
ejpam-3317	6	50	⊆	⊆	NUM
ejpam-3317	6	51	x	x	SYM
ejpam-3317	6	52	(	(	PUNCT
ejpam-3317	6	53	extensivity	extensivity	NOUN
ejpam-3317	6	54	)	)	PUNCT
ejpam-3317	6	55	(	(	PUNCT
ejpam-3317	6	56	iii	iii	NOUN
ejpam-3317	6	57	)	)	PUNCT
ejpam-3317	6	58	cl	cl	NOUN
ejpam-3317	6	59	(	(	PUNCT
ejpam-3317	6	60	a	a	DET
ejpam-3317	6	61	∪b	∪b	NOUN
ejpam-3317	6	62	)	)	PUNCT
ejpam-3317	6	63	=	=	SYM
ejpam-3317	6	64	cl	cl	NOUN
ejpam-3317	6	65	(	(	PUNCT
ejpam-3317	6	66	a)∪cl	a)∪cl	PROPN
ejpam-3317	6	67	(	(	PUNCT
ejpam-3317	6	68	b	b	NOUN
ejpam-3317	6	69	)	)	PUNCT
ejpam-3317	6	70	for	for	ADP
ejpam-3317	6	71	any	any	DET
ejpam-3317	6	72	subsets	subset	NOUN
ejpam-3317	6	73	a	a	PRON
ejpam-3317	6	74	,	,	PUNCT
ejpam-3317	6	75	b	b	NOUN
ejpam-3317	6	76	⊆	⊆	NUM
ejpam-3317	6	77	x	x	SYM
ejpam-3317	6	78	(	(	PUNCT
ejpam-3317	6	79	preservation	preservation	NOUN
ejpam-3317	6	80	of	of	ADP
ejpam-3317	6	81	binary	binary	PROPN
ejpam-3317	6	82	union	union	PROPN
ejpam-3317	6	83	)	)	PUNCT
ejpam-3317	6	84	.	.	PUNCT
ejpam-3317	7	1	(	(	PUNCT
ejpam-3317	7	2	iv	iv	X
ejpam-3317	7	3	)	)	PUNCT
ejpam-3317	7	4	cl	cl	NOUN
ejpam-3317	7	5	(	(	PUNCT
ejpam-3317	7	6	cl	cl	NOUN
ejpam-3317	7	7	(	(	PUNCT
ejpam-3317	7	8	a	a	NOUN
ejpam-3317	7	9	)	)	PUNCT
ejpam-3317	7	10	)	)	PUNCT
ejpam-3317	8	1	=	=	PUNCT
ejpam-3317	8	2	cl	cl	INTJ
ejpam-3317	8	3	(	(	PUNCT
ejpam-3317	8	4	a	a	NOUN
ejpam-3317	8	5	)	)	PUNCT
ejpam-3317	8	6	for	for	ADP
ejpam-3317	8	7	every	every	DET
ejpam-3317	8	8	subset	subset	NOUN
ejpam-3317	8	9	a	a	DET
ejpam-3317	8	10	⊆	⊆	NUM
ejpam-3317	8	11	x	x	SYM
ejpam-3317	8	12	(	(	PUNCT
ejpam-3317	8	13	idempotence	idempotence	NOUN
ejpam-3317	8	14	)	)	PUNCT
ejpam-3317	8	15	if	if	SCONJ
ejpam-3317	8	16	the	the	DET
ejpam-3317	8	17	last	last	ADJ
ejpam-3317	8	18	axiom(iv	axiom(iv	PROPN
ejpam-3317	8	19	)	)	PUNCT
ejpam-3317	8	20	,	,	PUNCT
ejpam-3317	8	21	idempotence	idempotence	NOUN
ejpam-3317	8	22	,	,	PUNCT
ejpam-3317	8	23	is	be	AUX
ejpam-3317	8	24	omitted	omit	VERB
ejpam-3317	8	25	,	,	PUNCT
ejpam-3317	8	26	then	then	ADV
ejpam-3317	8	27	the	the	DET
ejpam-3317	8	28	axioms	axiom	NOUN
ejpam-3317	8	29	define	define	VERB
ejpam-3317	8	30	a	a	DET
ejpam-3317	8	31	preclosure	preclosure	ADJ
ejpam-3317	8	32	operator	operator	NOUN
ejpam-3317	8	33	.	.	PUNCT
ejpam-3317	9	1	a	a	DET
ejpam-3317	9	2	consequence	consequence	NOUN
ejpam-3317	9	3	of	of	ADP
ejpam-3317	9	4	the	the	DET
ejpam-3317	9	5	third	third	ADJ
ejpam-3317	9	6	axiom(iii	axiom(iii	NOUN
ejpam-3317	9	7	)	)	PUNCT
ejpam-3317	9	8	is	be	AUX
ejpam-3317	9	9	:	:	PUNCT
ejpam-3317	9	10	a	a	DET
ejpam-3317	9	11	⊆	⊆	NUM
ejpam-3317	9	12	b	b	NOUN
ejpam-3317	9	13	then	then	ADV
ejpam-3317	9	14	cl	cl	INTJ
ejpam-3317	9	15	(	(	PUNCT
ejpam-3317	9	16	a	a	NOUN
ejpam-3317	9	17	)	)	PUNCT
ejpam-3317	9	18	⊆	⊆	NUM
ejpam-3317	9	19	cl	cl	NOUN
ejpam-3317	9	20	(	(	PUNCT
ejpam-3317	9	21	b	b	NOUN
ejpam-3317	9	22	)	)	PUNCT
ejpam-3317	9	23	(	(	PUNCT
ejpam-3317	9	24	preservation	preservation	NOUN
ejpam-3317	9	25	of	of	ADP
ejpam-3317	9	26	inclusion	inclusion	NOUN
ejpam-3317	9	27	)	)	PUNCT
ejpam-3317	9	28	.	.	PUNCT
ejpam-3317	10	1	then	then	ADV
ejpam-3317	10	2	cl	cl	VERB
ejpam-3317	10	3	,	,	PUNCT
ejpam-3317	10	4	together	together	ADV
ejpam-3317	10	5	with	with	ADP
ejpam-3317	10	6	the	the	DET
ejpam-3317	10	7	underlying	underlie	VERB
ejpam-3317	10	8	set	set	NOUN
ejpam-3317	10	9	x	x	NOUN
ejpam-3317	10	10	,	,	PUNCT
ejpam-3317	10	11	is	be	AUX
ejpam-3317	10	12	called	call	VERB
ejpam-3317	10	13	closure	closure	NOUN
ejpam-3317	10	14	space	space	NOUN
ejpam-3317	10	15	and	and	CCONJ
ejpam-3317	10	16	is	be	AUX
ejpam-3317	10	17	∗corresponding	∗corresponde	VERB
ejpam-3317	10	18	author	author	NOUN
ejpam-3317	10	19	.	.	PUNCT
ejpam-3317	11	1	doi	doi	NOUN
ejpam-3317	11	2	:	:	PUNCT
ejpam-3317	11	3	https://doi.org/10.29020/nybg.ejpam.v12i3.3317	https://doi.org/10.29020/nybg.ejpam.v12i3.3317	ADJ
ejpam-3317	11	4	email	email	NOUN
ejpam-3317	11	5	addresses	address	NOUN
ejpam-3317	11	6	:	:	PUNCT
ejpam-3317	11	7	halgwrd.darwesh@univsul.edu.iq	halgwrd.darwesh@univsul.edu.iq	NOUN
ejpam-3317	11	8	(	(	PUNCT
ejpam-3317	11	9	h.m	h.m	PROPN
ejpam-3317	11	10	.	.	PROPN
ejpam-3317	11	11	darwesh	darwesh	PROPN
ejpam-3317	11	12	)	)	PUNCT
ejpam-3317	11	13	,	,	PUNCT
ejpam-3317	11	14	sarhad1983@gmail.com	sarhad1983@gmail.com	NOUN
ejpam-3317	12	1	(	(	PUNCT
ejpam-3317	12	2	s.f	s.f	PROPN
ejpam-3317	12	3	.	.	PROPN
ejpam-3317	12	4	namiq	namiq	PROPN
ejpam-3317	12	5	)	)	PUNCT
ejpam-3317	12	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3317	13	1	1082	1082	NUM
ejpam-3317	13	2	c	c	X
ejpam-3317	13	3	©	©	PROPN
ejpam-3317	13	4	2019	2019	NUM
ejpam-3317	13	5	ejpam	ejpam	NOUN
ejpam-3317	13	6	all	all	DET
ejpam-3317	13	7	rights	right	NOUN
ejpam-3317	13	8	reserved	reserve	VERB
ejpam-3317	13	9	.	.	PUNCT
ejpam-3317	14	1	h.m	h.m	PROPN
ejpam-3317	14	2	.	.	PROPN
ejpam-3317	14	3	darwesh	darwesh	PROPN
ejpam-3317	14	4	,	,	PUNCT
ejpam-3317	14	5	s.f	s.f	PROPN
ejpam-3317	14	6	.	.	PROPN
ejpam-3317	14	7	namiq	namiq	PROPN
ejpam-3317	14	8	/	/	SYM
ejpam-3317	14	9	eur	eur	PROPN
ejpam-3317	14	10	.	.	PUNCT
ejpam-3317	15	1	j.	j.	PROPN
ejpam-3317	15	2	pure	pure	PROPN
ejpam-3317	15	3	appl	appl	PROPN
ejpam-3317	15	4	.	.	PROPN
ejpam-3317	15	5	math	math	PROPN
ejpam-3317	15	6	,	,	PUNCT
ejpam-3317	15	7	12	12	NUM
ejpam-3317	15	8	(	(	PUNCT
ejpam-3317	15	9	3	3	NUM
ejpam-3317	15	10	)	)	PUNCT
ejpam-3317	15	11	(	(	PUNCT
ejpam-3317	15	12	2019	2019	NUM
ejpam-3317	15	13	)	)	PUNCT
ejpam-3317	15	14	,	,	PUNCT
ejpam-3317	15	15	1082	1082	NUM
ejpam-3317	15	16	-	-	SYM
ejpam-3317	15	17	1095	1095	NUM
ejpam-3317	15	18	1083	1083	NUM
ejpam-3317	15	19	denoted	denote	VERB
ejpam-3317	15	20	by	by	ADP
ejpam-3317	15	21	(	(	PUNCT
ejpam-3317	15	22	x	x	NOUN
ejpam-3317	15	23	,	,	PUNCT
ejpam-3317	15	24	cl	cl	NOUN
ejpam-3317	15	25	)	)	PUNCT
ejpam-3317	15	26	.	.	PUNCT
ejpam-3317	16	1	in	in	ADP
ejpam-3317	16	2	1966	1966	NUM
ejpam-3317	16	3	,	,	PUNCT
ejpam-3317	16	4	eduard	eduard	PROPN
ejpam-3317	16	5	cech	cech	PROPN
ejpam-3317	16	6	defined	define	VERB
ejpam-3317	16	7	closure	closure	NOUN
ejpam-3317	16	8	operator	operator	NOUN
ejpam-3317	16	9	by	by	ADP
ejpam-3317	16	10	the	the	DET
ejpam-3317	16	11	following	following	NOUN
ejpam-3317	16	12	:	:	PUNCT
ejpam-3317	16	13	let	let	VERB
ejpam-3317	16	14	x	x	PRON
ejpam-3317	16	15	be	be	AUX
ejpam-3317	16	16	a	a	DET
ejpam-3317	16	17	set	set	NOUN
ejpam-3317	16	18	and	and	CCONJ
ejpam-3317	16	19	p	p	NOUN
ejpam-3317	16	20	(	(	PUNCT
ejpam-3317	16	21	x	x	X
ejpam-3317	16	22	)	)	PUNCT
ejpam-3317	16	23	its	its	PRON
ejpam-3317	16	24	power	power	NOUN
ejpam-3317	16	25	set	set	NOUN
ejpam-3317	16	26	.	.	PUNCT
ejpam-3317	17	1	a	a	DET
ejpam-3317	17	2	function	function	NOUN
ejpam-3317	17	3	c	c	NOUN
ejpam-3317	17	4	:	:	PUNCT
ejpam-3317	17	5	p	p	X
ejpam-3317	17	6	(	(	PUNCT
ejpam-3317	17	7	x	x	NOUN
ejpam-3317	17	8	)	)	PUNCT
ejpam-3317	17	9	→	→	SYM
ejpam-3317	18	1	p	p	X
ejpam-3317	18	2	(	(	PUNCT
ejpam-3317	18	3	x	x	X
ejpam-3317	18	4	)	)	PUNCT
ejpam-3317	18	5	with	with	ADP
ejpam-3317	18	6	the	the	DET
ejpam-3317	18	7	following	follow	VERB
ejpam-3317	18	8	properties	property	NOUN
ejpam-3317	18	9	:	:	PUNCT
ejpam-3317	18	10	(	(	PUNCT
ejpam-3317	18	11	i	i	NOUN
ejpam-3317	18	12	)	)	PUNCT
ejpam-3317	18	13	c	c	PROPN
ejpam-3317	18	14	(	(	PUNCT
ejpam-3317	18	15	φ	φ	NOUN
ejpam-3317	18	16	)	)	PUNCT
ejpam-3317	18	17	=	=	SYM
ejpam-3317	19	1	φ	φ	PROPN
ejpam-3317	19	2	.	.	PUNCT
ejpam-3317	19	3	(	(	PUNCT
ejpam-3317	19	4	ii	ii	NOUN
ejpam-3317	19	5	)	)	PUNCT
ejpam-3317	19	6	a	a	DET
ejpam-3317	19	7	⊆	⊆	NUM
ejpam-3317	19	8	c	c	NOUN
ejpam-3317	19	9	(	(	PUNCT
ejpam-3317	19	10	a	a	NOUN
ejpam-3317	19	11	)	)	PUNCT
ejpam-3317	19	12	for	for	ADP
ejpam-3317	19	13	every	every	DET
ejpam-3317	19	14	subset	subset	NOUN
ejpam-3317	19	15	a	a	DET
ejpam-3317	19	16	⊆	⊆	NUM
ejpam-3317	19	17	x.	x.	NOUN
ejpam-3317	19	18	(	(	PUNCT
ejpam-3317	19	19	iii	iii	NOUN
ejpam-3317	19	20	)	)	PUNCT
ejpam-3317	19	21	c	c	NOUN
ejpam-3317	19	22	(	(	PUNCT
ejpam-3317	19	23	a	a	DET
ejpam-3317	19	24	∪b	∪b	NOUN
ejpam-3317	19	25	)	)	PUNCT
ejpam-3317	19	26	=	=	SYM
ejpam-3317	20	1	c	c	X
ejpam-3317	20	2	(	(	PUNCT
ejpam-3317	20	3	a	a	NOUN
ejpam-3317	20	4	)	)	PUNCT
ejpam-3317	20	5	∪	∪	PROPN
ejpam-3317	20	6	c	c	X
ejpam-3317	20	7	(	(	PUNCT
ejpam-3317	20	8	b	b	NOUN
ejpam-3317	20	9	)	)	PUNCT
ejpam-3317	20	10	for	for	ADP
ejpam-3317	20	11	any	any	DET
ejpam-3317	20	12	subsets	subset	NOUN
ejpam-3317	20	13	a	a	PRON
ejpam-3317	20	14	,	,	PUNCT
ejpam-3317	20	15	b	b	PROPN
ejpam-3317	20	16	⊆	⊆	NUM
ejpam-3317	20	17	x.	x.	NOUN
ejpam-3317	20	18	then	then	ADV
ejpam-3317	20	19	c	c	AUX
ejpam-3317	20	20	,	,	PUNCT
ejpam-3317	20	21	together	together	ADV
ejpam-3317	20	22	with	with	ADP
ejpam-3317	20	23	the	the	DET
ejpam-3317	20	24	underlying	underlie	VERB
ejpam-3317	20	25	set	set	NOUN
ejpam-3317	20	26	x	x	NOUN
ejpam-3317	20	27	,	,	PUNCT
ejpam-3317	20	28	is	be	AUX
ejpam-3317	20	29	called	call	VERB
ejpam-3317	20	30	a	a	DET
ejpam-3317	20	31	cech	cech	NOUN
ejpam-3317	20	32	closure	closure	NOUN
ejpam-3317	20	33	space	space	NOUN
ejpam-3317	20	34	and	and	CCONJ
ejpam-3317	20	35	is	be	AUX
ejpam-3317	20	36	denoted	denote	VERB
ejpam-3317	20	37	by	by	ADP
ejpam-3317	20	38	(	(	PUNCT
ejpam-3317	20	39	x	x	NOUN
ejpam-3317	20	40	,	,	PUNCT
ejpam-3317	20	41	c	c	NOUN
ejpam-3317	20	42	)	)	PUNCT
ejpam-3317	20	43	.	.	PUNCT
ejpam-3317	21	1	if	if	SCONJ
ejpam-3317	21	2	c	c	PROPN
ejpam-3317	21	3	also	also	ADV
ejpam-3317	21	4	satisfies	satisfy	VERB
ejpam-3317	21	5	:	:	PUNCT
ejpam-3317	21	6	c	c	X
ejpam-3317	21	7	(	(	PUNCT
ejpam-3317	21	8	c	c	X
ejpam-3317	21	9	(	(	PUNCT
ejpam-3317	21	10	a	a	NOUN
ejpam-3317	21	11	)	)	PUNCT
ejpam-3317	21	12	)	)	PUNCT
ejpam-3317	22	1	=	=	PUNCT
ejpam-3317	22	2	c	c	X
ejpam-3317	22	3	(	(	PUNCT
ejpam-3317	22	4	a	a	NOUN
ejpam-3317	22	5	)	)	PUNCT
ejpam-3317	22	6	for	for	ADP
ejpam-3317	22	7	every	every	DET
ejpam-3317	22	8	subset	subset	NOUN
ejpam-3317	22	9	a	a	DET
ejpam-3317	22	10	⊆	⊆	NUM
ejpam-3317	22	11	x	x	NOUN
ejpam-3317	22	12	,	,	PUNCT
ejpam-3317	22	13	then	then	ADV
ejpam-3317	22	14	(	(	PUNCT
ejpam-3317	22	15	x	x	NOUN
ejpam-3317	22	16	,	,	PUNCT
ejpam-3317	22	17	c	c	NOUN
ejpam-3317	22	18	)	)	PUNCT
ejpam-3317	22	19	is	be	AUX
ejpam-3317	22	20	a	a	DET
ejpam-3317	22	21	topological	topological	ADJ
ejpam-3317	22	22	space	space	NOUN
ejpam-3317	22	23	.	.	PUNCT
ejpam-3317	23	1	in	in	ADP
ejpam-3317	23	2	2009	2009	NUM
ejpam-3317	23	3	,	,	PUNCT
ejpam-3317	23	4	jeeranunt	jeeranunt	NOUN
ejpam-3317	23	5	khampakdee	khampakdee	NOUN
ejpam-3317	23	6	[	[	X
ejpam-3317	23	7	15	15	NUM
ejpam-3317	23	8	]	]	PUNCT
ejpam-3317	23	9	defined	define	VERB
ejpam-3317	23	10	closure	closure	NOUN
ejpam-3317	23	11	operator	operator	NOUN
ejpam-3317	23	12	by	by	ADP
ejpam-3317	23	13	the	the	DET
ejpam-3317	23	14	following	following	NOUN
ejpam-3317	23	15	:	:	PUNCT
ejpam-3317	23	16	a	a	DET
ejpam-3317	23	17	function	function	NOUN
ejpam-3317	23	18	c	c	NOUN
ejpam-3317	23	19	:	:	PUNCT
ejpam-3317	23	20	p	p	X
ejpam-3317	23	21	(	(	PUNCT
ejpam-3317	23	22	x	x	NOUN
ejpam-3317	23	23	)	)	PUNCT
ejpam-3317	23	24	→	→	SYM
ejpam-3317	23	25	p	p	X
ejpam-3317	23	26	(	(	PUNCT
ejpam-3317	23	27	x	x	NOUN
ejpam-3317	23	28	)	)	PUNCT
ejpam-3317	23	29	defined	define	VERB
ejpam-3317	23	30	on	on	ADP
ejpam-3317	23	31	the	the	DET
ejpam-3317	23	32	power	power	NOUN
ejpam-3317	23	33	set	set	NOUN
ejpam-3317	23	34	p	p	PROPN
ejpam-3317	23	35	(	(	PUNCT
ejpam-3317	23	36	x	x	NOUN
ejpam-3317	23	37	)	)	PUNCT
ejpam-3317	23	38	of	of	ADP
ejpam-3317	23	39	a	a	DET
ejpam-3317	23	40	set	set	NOUN
ejpam-3317	23	41	x	x	PUNCT
ejpam-3317	23	42	is	be	AUX
ejpam-3317	23	43	called	call	VERB
ejpam-3317	23	44	a	a	DET
ejpam-3317	23	45	closure	closure	NOUN
ejpam-3317	23	46	operator	operator	NOUN
ejpam-3317	23	47	on	on	ADP
ejpam-3317	23	48	x	x	PUNCT
ejpam-3317	23	49	and	and	CCONJ
ejpam-3317	23	50	the	the	DET
ejpam-3317	23	51	pair	pair	NOUN
ejpam-3317	23	52	(	(	PUNCT
ejpam-3317	23	53	x	x	NOUN
ejpam-3317	23	54	,	,	PUNCT
ejpam-3317	23	55	c	c	NOUN
ejpam-3317	23	56	)	)	PUNCT
ejpam-3317	23	57	is	be	AUX
ejpam-3317	23	58	called	call	VERB
ejpam-3317	23	59	a	a	DET
ejpam-3317	23	60	closure	closure	NOUN
ejpam-3317	23	61	space	space	NOUN
ejpam-3317	23	62	if	if	SCONJ
ejpam-3317	23	63	the	the	DET
ejpam-3317	23	64	following	follow	VERB
ejpam-3317	23	65	axioms	axiom	NOUN
ejpam-3317	23	66	are	be	AUX
ejpam-3317	23	67	satisfied	satisfied	ADJ
ejpam-3317	23	68	:	:	PUNCT
ejpam-3317	23	69	(	(	PUNCT
ejpam-3317	23	70	i	i	NOUN
ejpam-3317	23	71	)	)	PUNCT
ejpam-3317	23	72	c	c	PROPN
ejpam-3317	23	73	(	(	PUNCT
ejpam-3317	23	74	φ	φ	NOUN
ejpam-3317	23	75	)	)	PUNCT
ejpam-3317	24	1	=	=	SYM
ejpam-3317	24	2	φ	φ	PROPN
ejpam-3317	24	3	.	.	PUNCT
ejpam-3317	24	4	(	(	PUNCT
ejpam-3317	24	5	ii	ii	NOUN
ejpam-3317	24	6	)	)	PUNCT
ejpam-3317	24	7	a	a	DET
ejpam-3317	24	8	⊆	⊆	NUM
ejpam-3317	24	9	c(a	c(a	NOUN
ejpam-3317	24	10	)	)	PUNCT
ejpam-3317	24	11	for	for	ADP
ejpam-3317	24	12	every	every	DET
ejpam-3317	24	13	a	a	DET
ejpam-3317	24	14	⊆	⊆	NUM
ejpam-3317	24	15	x.	x.	NOUN
ejpam-3317	24	16	(	(	PUNCT
ejpam-3317	24	17	iii	iii	NOUN
ejpam-3317	24	18	)	)	PUNCT
ejpam-3317	24	19	a	a	DET
ejpam-3317	24	20	⊆	⊆	NUM
ejpam-3317	24	21	b	b	NOUN
ejpam-3317	24	22	⇒	⇒	NOUN
ejpam-3317	24	23	c(a	c(a	PROPN
ejpam-3317	24	24	)	)	PUNCT
ejpam-3317	24	25	⊆	⊆	NUM
ejpam-3317	24	26	c(b	c(b	NOUN
ejpam-3317	24	27	)	)	PUNCT
ejpam-3317	24	28	,	,	PUNCT
ejpam-3317	24	29	for	for	ADP
ejpam-3317	24	30	all	all	DET
ejpam-3317	24	31	a	a	PRON
ejpam-3317	24	32	,	,	PUNCT
ejpam-3317	24	33	b	b	PROPN
ejpam-3317	24	34	⊆	⊆	NUM
ejpam-3317	24	35	x.	x.	NOUN
ejpam-3317	24	36	the	the	DET
ejpam-3317	24	37	concept	concept	NOUN
ejpam-3317	24	38	of	of	ADP
ejpam-3317	24	39	closure	closure	NOUN
ejpam-3317	24	40	operator	operator	NOUN
ejpam-3317	24	41	and	and	CCONJ
ejpam-3317	24	42	closure	closure	NOUN
ejpam-3317	24	43	spaces	space	NOUN
ejpam-3317	24	44	are	be	AUX
ejpam-3317	24	45	very	very	ADV
ejpam-3317	24	46	usefull	usefull	ADJ
ejpam-3317	24	47	material	material	NOUN
ejpam-3317	24	48	in	in	ADP
ejpam-3317	24	49	several	several	ADJ
ejpam-3317	24	50	branches	branch	NOUN
ejpam-3317	24	51	of	of	ADP
ejpam-3317	24	52	science	science	NOUN
ejpam-3317	24	53	,	,	PUNCT
ejpam-3317	24	54	such	such	ADJ
ejpam-3317	24	55	as	as	ADP
ejpam-3317	24	56	topology	topology	NOUN
ejpam-3317	24	57	[	[	X
ejpam-3317	24	58	2],[3],[4],[5]computer	2],[3],[4],[5]computer	NOUN
ejpam-3317	24	59	science	science	NOUN
ejpam-3317	24	60	theory[18	theory[18	PROPN
ejpam-3317	24	61	]	]	PUNCT
ejpam-3317	24	62	,	,	PUNCT
ejpam-3317	24	63	biochemistry[6	biochemistry[6	PROPN
ejpam-3317	24	64	]	]	PUNCT
ejpam-3317	24	65	.	.	PUNCT
ejpam-3317	25	1	the	the	DET
ejpam-3317	25	2	purpose	purpose	NOUN
ejpam-3317	25	3	of	of	ADP
ejpam-3317	25	4	this	this	DET
ejpam-3317	25	5	paper	paper	NOUN
ejpam-3317	25	6	is	be	AUX
ejpam-3317	25	7	to	to	PART
ejpam-3317	25	8	study	study	VERB
ejpam-3317	25	9	the	the	DET
ejpam-3317	25	10	concept	concept	NOUN
ejpam-3317	25	11	of	of	ADP
ejpam-3317	25	12	preopen	preopen	ADJ
ejpam-3317	25	13	sets	set	NOUN
ejpam-3317	25	14	in	in	ADP
ejpam-3317	25	15	closure	closure	NOUN
ejpam-3317	25	16	spaces	space	NOUN
ejpam-3317	25	17	.	.	PUNCT
ejpam-3317	26	1	closure	closure	NOUN
ejpam-3317	26	2	spaces	space	NOUN
ejpam-3317	26	3	were	be	AUX
ejpam-3317	26	4	introduced	introduce	VERB
ejpam-3317	26	5	by	by	ADP
ejpam-3317	26	6	e.cech	e.cech	NOUN
ejpam-3317	26	7	[	[	X
ejpam-3317	26	8	2	2	X
ejpam-3317	26	9	]	]	PUNCT
ejpam-3317	26	10	in	in	ADP
ejpam-3317	26	11	1966	1966	NUM
ejpam-3317	26	12	and	and	CCONJ
ejpam-3317	26	13	then	then	ADV
ejpam-3317	26	14	studied	study	VERB
ejpam-3317	26	15	by	by	ADP
ejpam-3317	26	16	many	many	ADJ
ejpam-3317	26	17	mathematicians	mathematician	NOUN
ejpam-3317	26	18	,	,	PUNCT
ejpam-3317	26	19	see	see	VERB
ejpam-3317	27	1	e.g.	e.g.	ADV
ejpam-3317	27	2	[	[	X
ejpam-3317	27	3	2],[3],[4],[5],[7],[10],[8],[9	2],[3],[4],[5],[7],[10],[8],[9	NUM
ejpam-3317	27	4	]	]	X
ejpam-3317	27	5	and	and	CCONJ
ejpam-3317	27	6	[	[	X
ejpam-3317	27	7	13	13	NUM
ejpam-3317	27	8	]	]	PUNCT
ejpam-3317	27	9	.	.	PUNCT
ejpam-3317	28	1	closure	closure	NOUN
ejpam-3317	28	2	spaces	space	NOUN
ejpam-3317	28	3	are	be	AUX
ejpam-3317	28	4	sets	set	NOUN
ejpam-3317	28	5	endowed	endow	VERB
ejpam-3317	28	6	with	with	ADP
ejpam-3317	28	7	a	a	DET
ejpam-3317	28	8	grounded	ground	VERB
ejpam-3317	28	9	,	,	PUNCT
ejpam-3317	28	10	extensive	extensive	ADJ
ejpam-3317	28	11	and	and	CCONJ
ejpam-3317	28	12	monotone	monotone	ADJ
ejpam-3317	28	13	closure	closure	NOUN
ejpam-3317	28	14	operator	operator	NOUN
ejpam-3317	28	15	.	.	PUNCT
ejpam-3317	29	1	mashhure	mashhure	NOUN
ejpam-3317	29	2	et	et	PROPN
ejpam-3317	29	3	al[17	al[17	PROPN
ejpam-3317	29	4	]	]	PUNCT
ejpam-3317	29	5	introduced	introduce	VERB
ejpam-3317	29	6	the	the	DET
ejpam-3317	29	7	concept	concept	NOUN
ejpam-3317	29	8	of	of	ADP
ejpam-3317	29	9	preopen	preopen	ADJ
ejpam-3317	29	10	sets	set	NOUN
ejpam-3317	29	11	and	and	CCONJ
ejpam-3317	29	12	pre	pre	ADJ
ejpam-3317	29	13	-	-	ADJ
ejpam-3317	29	14	continuous	continuous	ADJ
ejpam-3317	29	15	functions	function	NOUN
ejpam-3317	29	16	.	.	PUNCT
ejpam-3317	30	1	the	the	DET
ejpam-3317	30	2	preopen	preopen	ADJ
ejpam-3317	30	3	sets	set	NOUN
ejpam-3317	30	4	and	and	CCONJ
ejpam-3317	30	5	local	local	ADJ
ejpam-3317	30	6	dense	dense	ADJ
ejpam-3317	30	7	sets	set	NOUN
ejpam-3317	30	8	are	be	AUX
ejpam-3317	30	9	same	same	ADJ
ejpam-3317	30	10	in	in	ADP
ejpam-3317	30	11	topological	topological	ADJ
ejpam-3317	30	12	space	space	NOUN
ejpam-3317	30	13	,	,	PUNCT
ejpam-3317	30	14	also	also	ADV
ejpam-3317	30	15	the	the	DET
ejpam-3317	30	16	pre	pre	NOUN
ejpam-3317	30	17	-	-	NOUN
ejpam-3317	30	18	continuity	continuity	NOUN
ejpam-3317	30	19	and	and	CCONJ
ejpam-3317	30	20	almost	almost	ADV
ejpam-3317	30	21	-	-	PUNCT
ejpam-3317	30	22	continuity	continuity	NOUN
ejpam-3317	30	23	(	(	PUNCT
ejpam-3317	30	24	in	in	ADP
ejpam-3317	30	25	the	the	DET
ejpam-3317	30	26	sense	sense	NOUN
ejpam-3317	30	27	hussain)[12	hussain)[12	PROPN
ejpam-3317	30	28	]	]	PUNCT
ejpam-3317	30	29	are	be	AUX
ejpam-3317	30	30	same	same	ADJ
ejpam-3317	30	31	in	in	ADP
ejpam-3317	30	32	topological	topological	ADJ
ejpam-3317	30	33	spaces	space	NOUN
ejpam-3317	30	34	.	.	PUNCT
ejpam-3317	31	1	halgwrd	halgwrd	VERB
ejpam-3317	31	2	m.darwesh	m.darwesh	NOUN
ejpam-3317	32	1	[	[	X
ejpam-3317	32	2	11	11	NUM
ejpam-3317	32	3	]	]	PUNCT
ejpam-3317	32	4	used	use	VERB
ejpam-3317	32	5	the	the	DET
ejpam-3317	32	6	teqnique	teqnique	NOUN
ejpam-3317	32	7	of	of	ADP
ejpam-3317	32	8	mashhoury[12	mashhoury[12	NOUN
ejpam-3317	32	9	]	]	PUNCT
ejpam-3317	32	10	to	to	PART
ejpam-3317	32	11	introduce	introduce	VERB
ejpam-3317	32	12	and	and	CCONJ
ejpam-3317	32	13	study	study	VERB
ejpam-3317	32	14	the	the	DET
ejpam-3317	32	15	concept	concept	NOUN
ejpam-3317	32	16	of	of	ADP
ejpam-3317	32	17	preopen	preopen	ADJ
ejpam-3317	32	18	sets	set	NOUN
ejpam-3317	32	19	in	in	ADP
ejpam-3317	32	20	closure	closure	NOUN
ejpam-3317	32	21	spaces	space	NOUN
ejpam-3317	32	22	,	,	PUNCT
ejpam-3317	32	23	and	and	CCONJ
ejpam-3317	32	24	then	then	ADV
ejpam-3317	32	25	he	he	PRON
ejpam-3317	32	26	showed	show	VERB
ejpam-3317	32	27	that	that	SCONJ
ejpam-3317	32	28	its	its	PRON
ejpam-3317	32	29	differ	differ	NOUN
ejpam-3317	32	30	to	to	ADP
ejpam-3317	32	31	local	local	ADJ
ejpam-3317	32	32	dense	dense	ADJ
ejpam-3317	32	33	sets	set	NOUN
ejpam-3317	32	34	.	.	PUNCT
ejpam-3317	33	1	however	however	ADV
ejpam-3317	33	2	he	he	PRON
ejpam-3317	33	3	defined	define	VERB
ejpam-3317	33	4	the	the	DET
ejpam-3317	33	5	concept	concept	NOUN
ejpam-3317	33	6	of	of	ADP
ejpam-3317	33	7	pre	pre	ADJ
ejpam-3317	33	8	-	-	ADJ
ejpam-3317	33	9	continuous	continuous	ADJ
ejpam-3317	33	10	functions	function	NOUN
ejpam-3317	33	11	in	in	ADP
ejpam-3317	33	12	closure	closure	NOUN
ejpam-3317	33	13	spaces	space	NOUN
ejpam-3317	33	14	and	and	CCONJ
ejpam-3317	33	15	then	then	ADV
ejpam-3317	33	16	he	he	PRON
ejpam-3317	33	17	showed	show	VERB
ejpam-3317	33	18	that	that	SCONJ
ejpam-3317	33	19	the	the	DET
ejpam-3317	33	20	concepts	concept	NOUN
ejpam-3317	33	21	of	of	ADP
ejpam-3317	33	22	pre	pre	ADJ
ejpam-3317	33	23	-	-	NOUN
ejpam-3317	33	24	continuity	continuity	NOUN
ejpam-3317	33	25	and	and	CCONJ
ejpam-3317	33	26	almost	almost	ADV
ejpam-3317	33	27	-	-	PUNCT
ejpam-3317	33	28	continuity	continuity	NOUN
ejpam-3317	33	29	(	(	PUNCT
ejpam-3317	33	30	in	in	ADP
ejpam-3317	33	31	the	the	DET
ejpam-3317	33	32	sense	sense	NOUN
ejpam-3317	33	33	hussain	hussain	NOUN
ejpam-3317	33	34	)	)	PUNCT
ejpam-3317	34	1	[	[	X
ejpam-3317	34	2	11	11	NUM
ejpam-3317	34	3	]	]	PUNCT
ejpam-3317	34	4	are	be	AUX
ejpam-3317	34	5	independent	independent	ADJ
ejpam-3317	34	6	concepts	concept	NOUN
ejpam-3317	34	7	.	.	PUNCT
ejpam-3317	35	1	in	in	ADP
ejpam-3317	35	2	this	this	DET
ejpam-3317	35	3	paper	paper	NOUN
ejpam-3317	35	4	,	,	PUNCT
ejpam-3317	35	5	in	in	ADP
ejpam-3317	35	6	section	section	NOUN
ejpam-3317	35	7	3	3	NUM
ejpam-3317	35	8	,	,	PUNCT
ejpam-3317	35	9	we	we	PRON
ejpam-3317	35	10	introduce	introduce	VERB
ejpam-3317	35	11	the	the	DET
ejpam-3317	35	12	notion	notion	NOUN
ejpam-3317	35	13	of	of	ADP
ejpam-3317	35	14	pre	pre	ADJ
ejpam-3317	35	15	-	-	ADJ
ejpam-3317	35	16	open(pre	open(pre	ADJ
ejpam-3317	35	17	-	-	PUNCT
ejpam-3317	35	18	closed	closed	ADJ
ejpam-3317	35	19	)	)	PUNCT
ejpam-3317	35	20	functions	function	NOUN
ejpam-3317	35	21	,	,	PUNCT
ejpam-3317	35	22	contra	contra	PROPN
ejpam-3317	35	23	-	-	ADJ
ejpam-3317	35	24	pre	pre	ADJ
ejpam-3317	35	25	-	-	ADJ
ejpam-3317	35	26	continuous	continuous	ADJ
ejpam-3317	35	27	and	and	CCONJ
ejpam-3317	35	28	study	study	VERB
ejpam-3317	35	29	some	some	PRON
ejpam-3317	35	30	of	of	ADP
ejpam-3317	35	31	their	their	PRON
ejpam-3317	35	32	properties	property	NOUN
ejpam-3317	35	33	.	.	PUNCT
ejpam-3317	36	1	in	in	ADP
ejpam-3317	36	2	section	section	NOUN
ejpam-3317	36	3	4	4	NUM
ejpam-3317	36	4	,	,	PUNCT
ejpam-3317	36	5	we	we	PRON
ejpam-3317	36	6	introduce	introduce	VERB
ejpam-3317	36	7	and	and	CCONJ
ejpam-3317	36	8	discuss	discuss	VERB
ejpam-3317	36	9	pre	pre	ADJ
ejpam-3317	36	10	-	-	ADJ
ejpam-3317	36	11	irresolute	irresolute	ADJ
ejpam-3317	36	12	functions	function	NOUN
ejpam-3317	36	13	in	in	ADP
ejpam-3317	36	14	closure	closure	NOUN
ejpam-3317	36	15	spaces	space	NOUN
ejpam-3317	36	16	.	.	PUNCT
ejpam-3317	37	1	we	we	PRON
ejpam-3317	37	2	establish	establish	VERB
ejpam-3317	37	3	some	some	DET
ejpam-3317	37	4	basic	basic	ADJ
ejpam-3317	37	5	properties	property	NOUN
ejpam-3317	37	6	of	of	ADP
ejpam-3317	37	7	pre	pre	ADJ
ejpam-3317	37	8	-	-	ADJ
ejpam-3317	37	9	irresolute	irresolute	ADJ
ejpam-3317	37	10	functions	function	NOUN
ejpam-3317	37	11	in	in	ADP
ejpam-3317	37	12	section	section	NOUN
ejpam-3317	37	13	5	5	NUM
ejpam-3317	37	14	,	,	PUNCT
ejpam-3317	37	15	we	we	PRON
ejpam-3317	37	16	introduce	introduce	VERB
ejpam-3317	37	17	the	the	DET
ejpam-3317	37	18	notion	notion	NOUN
ejpam-3317	37	19	of	of	ADP
ejpam-3317	37	20	p	p	NOUN
ejpam-3317	37	21	-	-	PUNCT
ejpam-3317	37	22	connectedness	connectedness	NOUN
ejpam-3317	37	23	and	and	CCONJ
ejpam-3317	37	24	study	study	VERB
ejpam-3317	37	25	some	some	PRON
ejpam-3317	37	26	of	of	ADP
ejpam-3317	37	27	their	their	PRON
ejpam-3317	37	28	properties	property	NOUN
ejpam-3317	37	29	.	.	PUNCT
ejpam-3317	38	1	in	in	ADP
ejpam-3317	38	2	section	section	NOUN
ejpam-3317	38	3	6	6	NUM
ejpam-3317	38	4	,	,	PUNCT
ejpam-3317	38	5	we	we	PRON
ejpam-3317	38	6	introduce	introduce	VERB
ejpam-3317	38	7	the	the	DET
ejpam-3317	38	8	notion	notion	NOUN
ejpam-3317	38	9	of	of	ADP
ejpam-3317	38	10	p	p	NOUN
ejpam-3317	38	11	-	-	PUNCT
ejpam-3317	38	12	compactness	compactness	NOUN
ejpam-3317	38	13	and	and	CCONJ
ejpam-3317	38	14	study	study	VERB
ejpam-3317	38	15	some	some	PRON
ejpam-3317	38	16	of	of	ADP
ejpam-3317	38	17	their	their	PRON
ejpam-3317	38	18	properties	property	NOUN
ejpam-3317	38	19	.	.	PUNCT
ejpam-3317	39	1	h.m	h.m	PROPN
ejpam-3317	39	2	.	.	PROPN
ejpam-3317	39	3	darwesh	darwesh	PROPN
ejpam-3317	39	4	,	,	PUNCT
ejpam-3317	39	5	s.f	s.f	PROPN
ejpam-3317	39	6	.	.	PROPN
ejpam-3317	39	7	namiq	namiq	PROPN
ejpam-3317	39	8	/	/	SYM
ejpam-3317	39	9	eur	eur	PROPN
ejpam-3317	39	10	.	.	PUNCT
ejpam-3317	40	1	j.	j.	PROPN
ejpam-3317	40	2	pure	pure	PROPN
ejpam-3317	40	3	appl	appl	PROPN
ejpam-3317	40	4	.	.	PROPN
ejpam-3317	40	5	math	math	PROPN
ejpam-3317	40	6	,	,	PUNCT
ejpam-3317	40	7	12	12	NUM
ejpam-3317	40	8	(	(	PUNCT
ejpam-3317	40	9	3	3	NUM
ejpam-3317	40	10	)	)	PUNCT
ejpam-3317	40	11	(	(	PUNCT
ejpam-3317	40	12	2019	2019	NUM
ejpam-3317	40	13	)	)	PUNCT
ejpam-3317	40	14	,	,	PUNCT
ejpam-3317	40	15	1082	1082	NUM
ejpam-3317	40	16	-	-	SYM
ejpam-3317	40	17	1095	1095	NUM
ejpam-3317	40	18	1084	1084	NUM
ejpam-3317	40	19	2	2	NUM
ejpam-3317	40	20	.	.	PUNCT
ejpam-3317	40	21	preliminaries	preliminary	NOUN
ejpam-3317	40	22	a	a	DET
ejpam-3317	40	23	function	function	NOUN
ejpam-3317	40	24	c	c	NOUN
ejpam-3317	40	25	:	:	PUNCT
ejpam-3317	40	26	p	p	X
ejpam-3317	40	27	(	(	PUNCT
ejpam-3317	40	28	x	x	NOUN
ejpam-3317	40	29	)	)	PUNCT
ejpam-3317	40	30	→	→	SYM
ejpam-3317	40	31	p	p	X
ejpam-3317	40	32	(	(	PUNCT
ejpam-3317	40	33	x	x	NOUN
ejpam-3317	40	34	)	)	PUNCT
ejpam-3317	40	35	defined	define	VERB
ejpam-3317	40	36	on	on	ADP
ejpam-3317	40	37	the	the	DET
ejpam-3317	40	38	power	power	NOUN
ejpam-3317	40	39	set	set	NOUN
ejpam-3317	40	40	p	p	PROPN
ejpam-3317	40	41	(	(	PUNCT
ejpam-3317	40	42	x	x	NOUN
ejpam-3317	40	43	)	)	PUNCT
ejpam-3317	40	44	of	of	ADP
ejpam-3317	40	45	a	a	DET
ejpam-3317	40	46	set	set	NOUN
ejpam-3317	40	47	x	x	PUNCT
ejpam-3317	40	48	is	be	AUX
ejpam-3317	40	49	called	call	VERB
ejpam-3317	40	50	a	a	DET
ejpam-3317	40	51	closure	closure	NOUN
ejpam-3317	40	52	operator	operator	NOUN
ejpam-3317	40	53	on	on	ADP
ejpam-3317	40	54	x	x	PUNCT
ejpam-3317	40	55	and	and	CCONJ
ejpam-3317	40	56	the	the	DET
ejpam-3317	40	57	pair	pair	NOUN
ejpam-3317	40	58	(	(	PUNCT
ejpam-3317	40	59	x	x	NOUN
ejpam-3317	40	60	,	,	PUNCT
ejpam-3317	40	61	c	c	NOUN
ejpam-3317	40	62	)	)	PUNCT
ejpam-3317	40	63	is	be	AUX
ejpam-3317	40	64	called	call	VERB
ejpam-3317	40	65	a	a	DET
ejpam-3317	40	66	closure	closure	NOUN
ejpam-3317	40	67	space[15	space[15	NOUN
ejpam-3317	40	68	]	]	PUNCT
ejpam-3317	40	69	if	if	SCONJ
ejpam-3317	40	70	the	the	DET
ejpam-3317	40	71	following	follow	VERB
ejpam-3317	40	72	axioms	axiom	NOUN
ejpam-3317	40	73	are	be	AUX
ejpam-3317	40	74	satisfied	satisfied	ADJ
ejpam-3317	40	75	:	:	PUNCT
ejpam-3317	40	76	(	(	PUNCT
ejpam-3317	40	77	i	i	NOUN
ejpam-3317	40	78	)	)	PUNCT
ejpam-3317	40	79	c	c	PROPN
ejpam-3317	40	80	(	(	PUNCT
ejpam-3317	40	81	φ	φ	NOUN
ejpam-3317	40	82	)	)	PUNCT
ejpam-3317	40	83	=	=	SYM
ejpam-3317	41	1	φ	φ	PROPN
ejpam-3317	41	2	.	.	PUNCT
ejpam-3317	41	3	(	(	PUNCT
ejpam-3317	41	4	ii	ii	NOUN
ejpam-3317	41	5	)	)	PUNCT
ejpam-3317	41	6	a	a	DET
ejpam-3317	41	7	⊆	⊆	NUM
ejpam-3317	41	8	c(a	c(a	NOUN
ejpam-3317	41	9	)	)	PUNCT
ejpam-3317	41	10	for	for	ADP
ejpam-3317	41	11	every	every	DET
ejpam-3317	41	12	a	a	DET
ejpam-3317	41	13	⊆	⊆	NUM
ejpam-3317	41	14	x.	x.	NOUN
ejpam-3317	41	15	(	(	PUNCT
ejpam-3317	41	16	iii	iii	NOUN
ejpam-3317	41	17	)	)	PUNCT
ejpam-3317	41	18	a	a	DET
ejpam-3317	41	19	⊆	⊆	NUM
ejpam-3317	41	20	b	b	NOUN
ejpam-3317	41	21	⇒	⇒	NOUN
ejpam-3317	41	22	c(a	c(a	PROPN
ejpam-3317	41	23	)	)	PUNCT
ejpam-3317	41	24	⊆	⊆	NUM
ejpam-3317	41	25	c(b	c(b	NOUN
ejpam-3317	41	26	)	)	PUNCT
ejpam-3317	41	27	,	,	PUNCT
ejpam-3317	41	28	for	for	ADP
ejpam-3317	41	29	all	all	DET
ejpam-3317	41	30	a	a	PRON
ejpam-3317	41	31	,	,	PUNCT
ejpam-3317	41	32	b	b	PROPN
ejpam-3317	41	33	⊆	⊆	NUM
ejpam-3317	41	34	x.	x.	NOUN
ejpam-3317	41	35	definition	definition	NOUN
ejpam-3317	41	36	1	1	NUM
ejpam-3317	41	37	.	.	PUNCT
ejpam-3317	42	1	[	[	X
ejpam-3317	42	2	15	15	NUM
ejpam-3317	42	3	]	]	X
ejpam-3317	42	4	a	a	DET
ejpam-3317	42	5	closure	closure	NOUN
ejpam-3317	42	6	operator	operator	NOUN
ejpam-3317	42	7	c	c	NOUN
ejpam-3317	42	8	on	on	ADP
ejpam-3317	42	9	a	a	DET
ejpam-3317	42	10	set	set	NOUN
ejpam-3317	42	11	x	x	PUNCT
ejpam-3317	42	12	is	be	AUX
ejpam-3317	42	13	called	call	VERB
ejpam-3317	42	14	additive	additive	ADJ
ejpam-3317	42	15	if	if	SCONJ
ejpam-3317	42	16	c(a	c(a	PROPN
ejpam-3317	42	17	∪	∪	X
ejpam-3317	42	18	b	b	NOUN
ejpam-3317	42	19	)	)	PUNCT
ejpam-3317	42	20	=	=	SYM
ejpam-3317	42	21	c(a	c(a	PROPN
ejpam-3317	42	22	)	)	PUNCT
ejpam-3317	42	23	∪	∪	ADP
ejpam-3317	42	24	c(b	c(b	PROPN
ejpam-3317	42	25	)	)	PUNCT
ejpam-3317	42	26	,	,	PUNCT
ejpam-3317	42	27	for	for	ADP
ejpam-3317	42	28	all	all	DET
ejpam-3317	42	29	a	a	PRON
ejpam-3317	42	30	,	,	PUNCT
ejpam-3317	42	31	b	b	NOUN
ejpam-3317	42	32	⊆	⊆	NUM
ejpam-3317	42	33	x.	x.	NOUN
ejpam-3317	42	34	definition	definition	NOUN
ejpam-3317	42	35	2	2	NUM
ejpam-3317	42	36	.	.	PUNCT
ejpam-3317	43	1	[	[	X
ejpam-3317	43	2	15	15	NUM
ejpam-3317	43	3	]	]	X
ejpam-3317	43	4	a	a	DET
ejpam-3317	43	5	closure	closure	NOUN
ejpam-3317	43	6	operator	operator	NOUN
ejpam-3317	43	7	c	c	NOUN
ejpam-3317	43	8	on	on	ADP
ejpam-3317	43	9	a	a	DET
ejpam-3317	43	10	set	set	NOUN
ejpam-3317	43	11	x	x	PUNCT
ejpam-3317	43	12	is	be	AUX
ejpam-3317	43	13	called	call	VERB
ejpam-3317	43	14	idempotent	idempotent	ADJ
ejpam-3317	43	15	if	if	SCONJ
ejpam-3317	43	16	cc(a	cc(a	NOUN
ejpam-3317	43	17	)	)	PUNCT
ejpam-3317	43	18	=	=	SYM
ejpam-3317	43	19	c(a	c(a	PROPN
ejpam-3317	43	20	)	)	PUNCT
ejpam-3317	43	21	,	,	PUNCT
ejpam-3317	43	22	for	for	ADP
ejpam-3317	43	23	all	all	DET
ejpam-3317	43	24	a	a	DET
ejpam-3317	43	25	⊆	⊆	NUM
ejpam-3317	43	26	x.	x.	NOUN
ejpam-3317	43	27	definition	definition	NOUN
ejpam-3317	43	28	3	3	NUM
ejpam-3317	43	29	.	.	PUNCT
ejpam-3317	44	1	[	[	X
ejpam-3317	44	2	15	15	NUM
ejpam-3317	44	3	]	]	X
ejpam-3317	44	4	a	a	DET
ejpam-3317	44	5	subset	subset	NOUN
ejpam-3317	44	6	a	a	DET
ejpam-3317	44	7	⊆	⊆	NUM
ejpam-3317	44	8	x	x	NOUN
ejpam-3317	44	9	is	be	AUX
ejpam-3317	44	10	closed	close	VERB
ejpam-3317	44	11	in	in	ADP
ejpam-3317	44	12	the	the	DET
ejpam-3317	44	13	closure	closure	NOUN
ejpam-3317	44	14	space	space	NOUN
ejpam-3317	44	15	(	(	PUNCT
ejpam-3317	44	16	x	x	X
ejpam-3317	44	17	,	,	PUNCT
ejpam-3317	44	18	c	c	NOUN
ejpam-3317	44	19	)	)	PUNCT
ejpam-3317	44	20	if	if	SCONJ
ejpam-3317	44	21	c(a	c(a	ADV
ejpam-3317	44	22	)	)	PUNCT
ejpam-3317	45	1	=	=	PUNCT
ejpam-3317	45	2	a.	a.	NOUN
ejpam-3317	46	1	it	it	PRON
ejpam-3317	46	2	is	be	AUX
ejpam-3317	46	3	called	call	VERB
ejpam-3317	46	4	open	open	ADJ
ejpam-3317	46	5	,	,	PUNCT
ejpam-3317	46	6	if	if	SCONJ
ejpam-3317	46	7	its	its	PRON
ejpam-3317	46	8	complement	complement	NOUN
ejpam-3317	46	9	in	in	ADP
ejpam-3317	46	10	x	x	PROPN
ejpam-3317	46	11	is	be	AUX
ejpam-3317	46	12	closed	close	VERB
ejpam-3317	46	13	.	.	PUNCT
ejpam-3317	47	1	the	the	DET
ejpam-3317	47	2	empty	empty	ADJ
ejpam-3317	47	3	set	set	NOUN
ejpam-3317	47	4	and	and	CCONJ
ejpam-3317	47	5	the	the	DET
ejpam-3317	47	6	whole	whole	ADJ
ejpam-3317	47	7	space	space	NOUN
ejpam-3317	47	8	are	be	AUX
ejpam-3317	47	9	both	both	CCONJ
ejpam-3317	47	10	open	open	ADJ
ejpam-3317	47	11	and	and	CCONJ
ejpam-3317	47	12	closed	closed	ADJ
ejpam-3317	47	13	.	.	PUNCT
ejpam-3317	48	1	definition	definition	NOUN
ejpam-3317	48	2	4	4	NUM
ejpam-3317	48	3	.	.	PUNCT
ejpam-3317	49	1	[	[	X
ejpam-3317	49	2	11	11	NUM
ejpam-3317	49	3	]	]	PUNCT
ejpam-3317	49	4	a	a	DET
ejpam-3317	49	5	subset	subset	NOUN
ejpam-3317	49	6	a	a	PRON
ejpam-3317	49	7	of	of	ADP
ejpam-3317	49	8	a	a	DET
ejpam-3317	49	9	space	space	NOUN
ejpam-3317	49	10	(	(	PUNCT
ejpam-3317	49	11	x	x	X
ejpam-3317	49	12	,	,	PUNCT
ejpam-3317	49	13	c	c	NOUN
ejpam-3317	49	14	)	)	PUNCT
ejpam-3317	49	15	is	be	AUX
ejpam-3317	49	16	said	say	VERB
ejpam-3317	49	17	to	to	PART
ejpam-3317	49	18	be	be	AUX
ejpam-3317	49	19	a	a	DET
ejpam-3317	49	20	preopen	preopen	ADJ
ejpam-3317	49	21	set	set	NOUN
ejpam-3317	49	22	,	,	PUNCT
ejpam-3317	49	23	if	if	SCONJ
ejpam-3317	49	24	there	there	PRON
ejpam-3317	49	25	exists	exist	VERB
ejpam-3317	49	26	an	an	DET
ejpam-3317	49	27	open	open	ADJ
ejpam-3317	49	28	set	set	NOUN
ejpam-3317	49	29	g	g	PROPN
ejpam-3317	49	30	such	such	DET
ejpam-3317	49	31	that	that	SCONJ
ejpam-3317	49	32	a	a	DET
ejpam-3317	49	33	⊆	⊆	NUM
ejpam-3317	49	34	g	g	NOUN
ejpam-3317	49	35	⊆	⊆	NUM
ejpam-3317	49	36	c(a	c(a	NOUN
ejpam-3317	49	37	)	)	PUNCT
ejpam-3317	49	38	.	.	PUNCT
ejpam-3317	50	1	the	the	DET
ejpam-3317	50	2	complement	complement	NOUN
ejpam-3317	50	3	of	of	ADP
ejpam-3317	50	4	a	a	DET
ejpam-3317	50	5	preopen	preopen	ADJ
ejpam-3317	50	6	set	set	NOUN
ejpam-3317	50	7	is	be	AUX
ejpam-3317	50	8	called	call	VERB
ejpam-3317	50	9	preclosed	preclose	VERB
ejpam-3317	50	10	.	.	PUNCT
ejpam-3317	51	1	theorem	theorem	NOUN
ejpam-3317	51	2	1	1	NUM
ejpam-3317	51	3	.	.	PUNCT
ejpam-3317	52	1	[	[	X
ejpam-3317	52	2	11	11	NUM
ejpam-3317	52	3	]	]	PUNCT
ejpam-3317	52	4	a	a	DET
ejpam-3317	52	5	subset	subset	NOUN
ejpam-3317	52	6	a	a	PRON
ejpam-3317	52	7	of	of	ADP
ejpam-3317	52	8	a	a	DET
ejpam-3317	52	9	space(x	space(x	NOUN
ejpam-3317	52	10	,	,	PUNCT
ejpam-3317	52	11	c	c	PROPN
ejpam-3317	52	12	)	)	PUNCT
ejpam-3317	52	13	is	be	AUX
ejpam-3317	52	14	preclosed	preclose	VERB
ejpam-3317	52	15	if	if	SCONJ
ejpam-3317	52	16	and	and	CCONJ
ejpam-3317	52	17	only	only	ADV
ejpam-3317	52	18	if	if	SCONJ
ejpam-3317	52	19	there	there	PRON
ejpam-3317	52	20	exists	exist	VERB
ejpam-3317	52	21	a	a	DET
ejpam-3317	52	22	closed	closed	ADJ
ejpam-3317	52	23	set	set	NOUN
ejpam-3317	52	24	f	f	PROPN
ejpam-3317	52	25	such	such	ADJ
ejpam-3317	52	26	that	that	SCONJ
ejpam-3317	52	27	x\c(x\a	x\c(x\a	PROPN
ejpam-3317	52	28	)	)	PUNCT
ejpam-3317	53	1	⊆	⊆	NUM
ejpam-3317	53	2	f	f	SYM
ejpam-3317	53	3	⊆	⊆	NUM
ejpam-3317	53	4	a.	a.	NOUN
ejpam-3317	53	5	proposition	proposition	NOUN
ejpam-3317	53	6	1	1	NUM
ejpam-3317	53	7	.	.	PUNCT
ejpam-3317	54	1	[	[	X
ejpam-3317	54	2	15	15	NUM
ejpam-3317	54	3	]	]	X
ejpam-3317	54	4	let	let	VERB
ejpam-3317	54	5	(	(	PUNCT
ejpam-3317	54	6	x	x	NOUN
ejpam-3317	54	7	,	,	PUNCT
ejpam-3317	54	8	c	c	X
ejpam-3317	54	9	)	)	PUNCT
ejpam-3317	54	10	be	be	AUX
ejpam-3317	54	11	a	a	DET
ejpam-3317	54	12	closure	closure	NOUN
ejpam-3317	54	13	space	space	NOUN
ejpam-3317	54	14	and	and	CCONJ
ejpam-3317	54	15	{	{	PUNCT
ejpam-3317	54	16	gα}αεj	gα}αεj	PUNCT
ejpam-3317	54	17	be	be	AUX
ejpam-3317	54	18	a	a	DET
ejpam-3317	54	19	collection	collection	NOUN
ejpam-3317	54	20	of	of	ADP
ejpam-3317	54	21	subsets	subset	NOUN
ejpam-3317	54	22	of	of	ADP
ejpam-3317	54	23	x.	x.	NOUN
ejpam-3317	54	24	then	then	ADV
ejpam-3317	55	1	⋃	⋃	PROPN
ejpam-3317	55	2	αεj	αεj	NOUN
ejpam-3317	55	3	c(gα	c(gα	PROPN
ejpam-3317	55	4	)	)	PUNCT
ejpam-3317	55	5	⊆	⊆	NUM
ejpam-3317	55	6	c	c	X
ejpam-3317	55	7	(	(	PUNCT
ejpam-3317	55	8	⋃	⋃	NOUN
ejpam-3317	55	9	αεj	αεj	NOUN
ejpam-3317	55	10	gα	gα	PROPN
ejpam-3317	55	11	)	)	PUNCT
ejpam-3317	55	12	.	.	PUNCT
ejpam-3317	56	1	proposition	proposition	NOUN
ejpam-3317	56	2	2	2	NUM
ejpam-3317	56	3	.	.	PUNCT
ejpam-3317	57	1	[	[	X
ejpam-3317	57	2	15	15	NUM
ejpam-3317	57	3	]	]	X
ejpam-3317	57	4	the	the	DET
ejpam-3317	57	5	union(intersection)of	union(intersection)of	PROPN
ejpam-3317	57	6	any	any	DET
ejpam-3317	57	7	family	family	NOUN
ejpam-3317	57	8	of	of	ADP
ejpam-3317	57	9	open(closed)sets	open(closed)set	NOUN
ejpam-3317	57	10	in	in	ADP
ejpam-3317	57	11	a	a	DET
ejpam-3317	57	12	closure	closure	NOUN
ejpam-3317	57	13	space	space	NOUN
ejpam-3317	57	14	(	(	PUNCT
ejpam-3317	57	15	x	x	X
ejpam-3317	57	16	,	,	PUNCT
ejpam-3317	57	17	c	c	NOUN
ejpam-3317	57	18	)	)	PUNCT
ejpam-3317	57	19	is	be	AUX
ejpam-3317	57	20	open(closed	open(close	VERB
ejpam-3317	57	21	)	)	PUNCT
ejpam-3317	57	22	.	.	PUNCT
ejpam-3317	58	1	proposition	proposition	NOUN
ejpam-3317	58	2	3	3	NUM
ejpam-3317	58	3	.	.	PUNCT
ejpam-3317	59	1	[	[	X
ejpam-3317	59	2	11	11	NUM
ejpam-3317	59	3	]	]	PUNCT
ejpam-3317	59	4	the	the	DET
ejpam-3317	59	5	union(intersection)of	union(intersection)of	PROPN
ejpam-3317	59	6	any	any	DET
ejpam-3317	59	7	family	family	NOUN
ejpam-3317	59	8	of	of	ADP
ejpam-3317	59	9	preopen(preclosed)sets	preopen(preclosed)set	NOUN
ejpam-3317	59	10	in	in	ADP
ejpam-3317	59	11	a	a	DET
ejpam-3317	59	12	closure	closure	NOUN
ejpam-3317	59	13	space	space	NOUN
ejpam-3317	59	14	(	(	PUNCT
ejpam-3317	59	15	x	x	X
ejpam-3317	59	16	,	,	PUNCT
ejpam-3317	59	17	c	c	NOUN
ejpam-3317	59	18	)	)	PUNCT
ejpam-3317	59	19	is	be	AUX
ejpam-3317	59	20	preopen(preclosed	preopen(preclose	VERB
ejpam-3317	59	21	)	)	PUNCT
ejpam-3317	59	22	.	.	PUNCT
ejpam-3317	60	1	definition	definition	NOUN
ejpam-3317	60	2	5	5	NUM
ejpam-3317	60	3	.	.	PUNCT
ejpam-3317	61	1	[	[	X
ejpam-3317	61	2	11	11	NUM
ejpam-3317	61	3	]	]	PUNCT
ejpam-3317	61	4	the	the	DET
ejpam-3317	61	5	interior	interior	ADJ
ejpam-3317	61	6	operator	operator	NOUN
ejpam-3317	61	7	i	i	PRON
ejpam-3317	61	8	:	:	PUNCT
ejpam-3317	61	9	p	p	X
ejpam-3317	61	10	(	(	PUNCT
ejpam-3317	61	11	x	x	NOUN
ejpam-3317	61	12	)	)	PUNCT
ejpam-3317	61	13	→	→	SYM
ejpam-3317	61	14	p	p	X
ejpam-3317	61	15	(	(	PUNCT
ejpam-3317	61	16	x	x	NOUN
ejpam-3317	61	17	)	)	PUNCT
ejpam-3317	61	18	corresponding	correspond	VERB
ejpam-3317	61	19	to	to	ADP
ejpam-3317	61	20	the	the	DET
ejpam-3317	61	21	closure	closure	NOUN
ejpam-3317	61	22	operator	operator	NOUN
ejpam-3317	61	23	c	c	NOUN
ejpam-3317	61	24	on	on	X
ejpam-3317	61	25	x	x	AUX
ejpam-3317	61	26	is	be	AUX
ejpam-3317	61	27	given	give	VERB
ejpam-3317	61	28	by	by	ADP
ejpam-3317	61	29	;	;	PUNCT
ejpam-3317	61	30	i	i	PRON
ejpam-3317	61	31	(	(	PUNCT
ejpam-3317	61	32	a	a	X
ejpam-3317	61	33	)	)	PUNCT
ejpam-3317	61	34	=	=	SYM
ejpam-3317	61	35	x\c	x\c	PROPN
ejpam-3317	61	36	(	(	PUNCT
ejpam-3317	61	37	x\a	x\a	PROPN
ejpam-3317	61	38	)	)	PUNCT
ejpam-3317	61	39	.	.	PUNCT
ejpam-3317	62	1	theorem	theorem	NOUN
ejpam-3317	62	2	2	2	NUM
ejpam-3317	62	3	.	.	PUNCT
ejpam-3317	63	1	[	[	X
ejpam-3317	63	2	11	11	NUM
ejpam-3317	63	3	]	]	PUNCT
ejpam-3317	63	4	let	let	VERB
ejpam-3317	63	5	a	a	PRON
ejpam-3317	63	6	be	be	AUX
ejpam-3317	63	7	a	a	DET
ejpam-3317	63	8	subset	subset	NOUN
ejpam-3317	63	9	of	of	ADP
ejpam-3317	63	10	a	a	DET
ejpam-3317	63	11	closure	closure	NOUN
ejpam-3317	63	12	(	(	PUNCT
ejpam-3317	63	13	x	x	X
ejpam-3317	63	14	,	,	PUNCT
ejpam-3317	63	15	c	c	NOUN
ejpam-3317	63	16	)	)	PUNCT
ejpam-3317	63	17	.	.	PUNCT
ejpam-3317	64	1	if	if	SCONJ
ejpam-3317	64	2	x	x	PROPN
ejpam-3317	64	3	∈	∈	PROPN
ejpam-3317	64	4	c(a	c(a	PROPN
ejpam-3317	64	5	)	)	PUNCT
ejpam-3317	64	6	,	,	PUNCT
ejpam-3317	64	7	then	then	ADV
ejpam-3317	64	8	g∩a	g∩a	PROPN
ejpam-3317	64	9	6=	6=	PROPN
ejpam-3317	64	10	φ	φ	PROPN
ejpam-3317	64	11	,	,	PUNCT
ejpam-3317	64	12	for	for	SCONJ
ejpam-3317	64	13	each	each	DET
ejpam-3317	64	14	open	open	ADJ
ejpam-3317	64	15	subset	subset	VERB
ejpam-3317	64	16	g	g	NOUN
ejpam-3317	64	17	of	of	ADP
ejpam-3317	64	18	x	x	PUNCT
ejpam-3317	64	19	containing	contain	VERB
ejpam-3317	64	20	x.	x.	NOUN
ejpam-3317	64	21	proposition	proposition	NOUN
ejpam-3317	64	22	4	4	NUM
ejpam-3317	64	23	.	.	PUNCT
ejpam-3317	65	1	[	[	X
ejpam-3317	65	2	11	11	NUM
ejpam-3317	65	3	]	]	PUNCT
ejpam-3317	65	4	let	let	VERB
ejpam-3317	65	5	a	a	PRON
ejpam-3317	65	6	be	be	AUX
ejpam-3317	65	7	a	a	DET
ejpam-3317	65	8	subset	subset	NOUN
ejpam-3317	65	9	of	of	ADP
ejpam-3317	65	10	a	a	DET
ejpam-3317	65	11	closure	closure	NOUN
ejpam-3317	65	12	(	(	PUNCT
ejpam-3317	65	13	x	x	X
ejpam-3317	65	14	,	,	PUNCT
ejpam-3317	65	15	c	c	NOUN
ejpam-3317	65	16	)	)	PUNCT
ejpam-3317	65	17	and	and	CCONJ
ejpam-3317	65	18	c	c	PROPN
ejpam-3317	65	19	is	be	AUX
ejpam-3317	65	20	idempotent	idempotent	ADJ
ejpam-3317	65	21	on	on	ADP
ejpam-3317	65	22	x	x	NOUN
ejpam-3317	65	23	,	,	PUNCT
ejpam-3317	65	24	then	then	ADV
ejpam-3317	65	25	x	x	SYM
ejpam-3317	65	26	∈	∈	PROPN
ejpam-3317	65	27	c(a	c(a	PROPN
ejpam-3317	65	28	)	)	PUNCT
ejpam-3317	65	29	if	if	SCONJ
ejpam-3317	66	1	and	and	CCONJ
ejpam-3317	66	2	only	only	ADV
ejpam-3317	66	3	if	if	SCONJ
ejpam-3317	66	4	g	g	PROPN
ejpam-3317	66	5	∩a	∩a	PROPN
ejpam-3317	66	6	6=	6=	PROPN
ejpam-3317	66	7	φ	φ	PROPN
ejpam-3317	66	8	,	,	PUNCT
ejpam-3317	66	9	for	for	ADP
ejpam-3317	66	10	each	each	DET
ejpam-3317	66	11	open	open	ADJ
ejpam-3317	66	12	subset	subset	VERB
ejpam-3317	66	13	g	g	NOUN
ejpam-3317	66	14	of	of	ADP
ejpam-3317	66	15	x	x	PUNCT
ejpam-3317	66	16	containing	contain	VERB
ejpam-3317	66	17	x.	x.	PROPN
ejpam-3317	66	18	h.m	h.m	PROPN
ejpam-3317	66	19	.	.	PROPN
ejpam-3317	66	20	darwesh	darwesh	PROPN
ejpam-3317	66	21	,	,	PUNCT
ejpam-3317	66	22	s.f	s.f	PROPN
ejpam-3317	66	23	.	.	PROPN
ejpam-3317	66	24	namiq	namiq	PROPN
ejpam-3317	66	25	/	/	SYM
ejpam-3317	66	26	eur	eur	PROPN
ejpam-3317	66	27	.	.	PUNCT
ejpam-3317	67	1	j.	j.	PROPN
ejpam-3317	67	2	pure	pure	PROPN
ejpam-3317	67	3	appl	appl	PROPN
ejpam-3317	67	4	.	.	PROPN
ejpam-3317	67	5	math	math	PROPN
ejpam-3317	67	6	,	,	PUNCT
ejpam-3317	67	7	12	12	NUM
ejpam-3317	67	8	(	(	PUNCT
ejpam-3317	67	9	3	3	NUM
ejpam-3317	67	10	)	)	PUNCT
ejpam-3317	67	11	(	(	PUNCT
ejpam-3317	67	12	2019	2019	NUM
ejpam-3317	67	13	)	)	PUNCT
ejpam-3317	67	14	,	,	PUNCT
ejpam-3317	67	15	1082	1082	NUM
ejpam-3317	67	16	-	-	SYM
ejpam-3317	67	17	1095	1095	NUM
ejpam-3317	67	18	1085	1085	NUM
ejpam-3317	67	19	proposition	proposition	NOUN
ejpam-3317	67	20	5	5	NUM
ejpam-3317	67	21	.	.	PUNCT
ejpam-3317	68	1	[	[	X
ejpam-3317	68	2	11	11	NUM
ejpam-3317	68	3	]	]	PUNCT
ejpam-3317	68	4	let	let	VERB
ejpam-3317	68	5	c	c	PRON
ejpam-3317	68	6	be	be	AUX
ejpam-3317	68	7	an	an	DET
ejpam-3317	68	8	idempotent	idempotent	ADJ
ejpam-3317	68	9	closure	closure	NOUN
ejpam-3317	68	10	operator	operator	NOUN
ejpam-3317	68	11	on	on	ADP
ejpam-3317	68	12	a	a	DET
ejpam-3317	68	13	set	set	NOUN
ejpam-3317	68	14	x.	x.	NOUN
ejpam-3317	68	15	if	if	SCONJ
ejpam-3317	68	16	a	a	PRON
ejpam-3317	68	17	is	be	AUX
ejpam-3317	68	18	preopen	preopen	ADJ
ejpam-3317	68	19	in	in	ADP
ejpam-3317	68	20	x	x	X
ejpam-3317	68	21	and	and	CCONJ
ejpam-3317	68	22	b	b	NOUN
ejpam-3317	68	23	⊆	⊆	NUM
ejpam-3317	68	24	a	a	DET
ejpam-3317	68	25	⊆	⊆	NUM
ejpam-3317	68	26	c(b	c(b	NOUN
ejpam-3317	68	27	)	)	PUNCT
ejpam-3317	68	28	,	,	PUNCT
ejpam-3317	68	29	then	then	ADV
ejpam-3317	68	30	b	b	PROPN
ejpam-3317	68	31	is	be	AUX
ejpam-3317	68	32	preopen	preopen	ADJ
ejpam-3317	68	33	.	.	PUNCT
ejpam-3317	69	1	definition	definition	NOUN
ejpam-3317	69	2	6	6	NUM
ejpam-3317	69	3	.	.	PUNCT
ejpam-3317	70	1	[	[	X
ejpam-3317	70	2	15	15	NUM
ejpam-3317	70	3	]	]	X
ejpam-3317	70	4	a	a	DET
ejpam-3317	70	5	closure	closure	NOUN
ejpam-3317	70	6	space	space	NOUN
ejpam-3317	70	7	(	(	PUNCT
ejpam-3317	70	8	y	y	NOUN
ejpam-3317	70	9	,	,	PUNCT
ejpam-3317	70	10	v	v	NOUN
ejpam-3317	70	11	)	)	PUNCT
ejpam-3317	70	12	,	,	PUNCT
ejpam-3317	70	13	is	be	AUX
ejpam-3317	70	14	said	say	VERB
ejpam-3317	70	15	to	to	PART
ejpam-3317	70	16	be	be	AUX
ejpam-3317	70	17	a	a	DET
ejpam-3317	70	18	subspace	subspace	NOUN
ejpam-3317	70	19	of	of	ADP
ejpam-3317	70	20	(	(	PUNCT
ejpam-3317	70	21	x	x	NOUN
ejpam-3317	70	22	,	,	PUNCT
ejpam-3317	70	23	c	c	NOUN
ejpam-3317	70	24	)	)	PUNCT
ejpam-3317	70	25	,	,	PUNCT
ejpam-3317	70	26	if	if	SCONJ
ejpam-3317	70	27	y	y	PROPN
ejpam-3317	70	28	⊆	⊆	NUM
ejpam-3317	70	29	x	x	PUNCT
ejpam-3317	70	30	and	and	CCONJ
ejpam-3317	70	31	v	v	X
ejpam-3317	70	32	(	(	PUNCT
ejpam-3317	70	33	a	a	NOUN
ejpam-3317	70	34	)	)	PUNCT
ejpam-3317	70	35	=	=	SYM
ejpam-3317	70	36	c	c	X
ejpam-3317	70	37	(	(	PUNCT
ejpam-3317	70	38	a	a	NOUN
ejpam-3317	70	39	)	)	PUNCT
ejpam-3317	70	40	∩	∩	ADJ
ejpam-3317	70	41	y	y	PROPN
ejpam-3317	70	42	,	,	PUNCT
ejpam-3317	70	43	for	for	SCONJ
ejpam-3317	70	44	each	each	DET
ejpam-3317	70	45	subset	subset	VERB
ejpam-3317	70	46	a	a	DET
ejpam-3317	70	47	⊆	⊆	NUM
ejpam-3317	70	48	y.	y.	NOUN
ejpam-3317	70	49	theorem	theorem	NOUN
ejpam-3317	70	50	3	3	NUM
ejpam-3317	70	51	.	.	PUNCT
ejpam-3317	71	1	[	[	X
ejpam-3317	71	2	11	11	NUM
ejpam-3317	71	3	]	]	PUNCT
ejpam-3317	71	4	let	let	VERB
ejpam-3317	71	5	a	a	DET
ejpam-3317	71	6	⊆	⊆	NUM
ejpam-3317	71	7	y	y	SYM
ejpam-3317	71	8	⊆	⊆	NUM
ejpam-3317	71	9	x	x	NOUN
ejpam-3317	71	10	,	,	PUNCT
ejpam-3317	71	11	where	where	SCONJ
ejpam-3317	71	12	(	(	PUNCT
ejpam-3317	71	13	y	y	NOUN
ejpam-3317	71	14	,	,	PUNCT
ejpam-3317	71	15	υ	υ	NOUN
ejpam-3317	71	16	)	)	PUNCT
ejpam-3317	71	17	is	be	AUX
ejpam-3317	71	18	a	a	DET
ejpam-3317	71	19	subspace	subspace	NOUN
ejpam-3317	71	20	of	of	ADP
ejpam-3317	71	21	(	(	PUNCT
ejpam-3317	71	22	x	x	NOUN
ejpam-3317	71	23	,	,	PUNCT
ejpam-3317	71	24	c	c	NOUN
ejpam-3317	71	25	)	)	PUNCT
ejpam-3317	71	26	.	.	PUNCT
ejpam-3317	72	1	if	if	SCONJ
ejpam-3317	72	2	a	a	PRON
ejpam-3317	72	3	is	be	AUX
ejpam-3317	72	4	preopen	preopen	ADJ
ejpam-3317	72	5	in	in	ADP
ejpam-3317	72	6	x	x	NOUN
ejpam-3317	72	7	,	,	PUNCT
ejpam-3317	72	8	then	then	ADV
ejpam-3317	72	9	a	a	PRON
ejpam-3317	72	10	is	be	AUX
ejpam-3317	72	11	preopen	preopen	ADJ
ejpam-3317	72	12	in	in	ADP
ejpam-3317	72	13	y.	y.	PROPN
ejpam-3317	72	14	proposition	proposition	NOUN
ejpam-3317	72	15	6	6	NUM
ejpam-3317	72	16	.	.	PUNCT
ejpam-3317	73	1	[	[	X
ejpam-3317	73	2	1	1	X
ejpam-3317	73	3	]	]	PUNCT
ejpam-3317	73	4	the	the	DET
ejpam-3317	73	5	product	product	NOUN
ejpam-3317	73	6	of	of	ADP
ejpam-3317	73	7	a	a	DET
ejpam-3317	73	8	family	family	NOUN
ejpam-3317	73	9	{	{	PUNCT
ejpam-3317	73	10	(	(	PUNCT
ejpam-3317	73	11	xα	xα	INTJ
ejpam-3317	73	12	,	,	PUNCT
ejpam-3317	73	13	cα	cα	NOUN
ejpam-3317	73	14	)	)	PUNCT
ejpam-3317	73	15	:	:	PUNCT
ejpam-3317	74	1	α	α	X
ejpam-3317	74	2	∈	∈	PROPN
ejpam-3317	75	1	i	i	X
ejpam-3317	75	2	}	}	PUNCT
ejpam-3317	75	3	of	of	ADP
ejpam-3317	75	4	closure	closure	NOUN
ejpam-3317	75	5	spaces	space	NOUN
ejpam-3317	75	6	,	,	PUNCT
ejpam-3317	75	7	denoted	denote	VERB
ejpam-3317	75	8	by	by	ADP
ejpam-3317	75	9	∏	∏	PROPN
ejpam-3317	75	10	α∈i	α∈i	NUM
ejpam-3317	75	11	(	(	PUNCT
ejpam-3317	75	12	xα	xα	INTJ
ejpam-3317	75	13	,	,	PUNCT
ejpam-3317	75	14	cα	cα	NOUN
ejpam-3317	75	15	)	)	PUNCT
ejpam-3317	75	16	,	,	PUNCT
ejpam-3317	75	17	is	be	AUX
ejpam-3317	75	18	the	the	DET
ejpam-3317	75	19	closure	closure	NOUN
ejpam-3317	75	20	space	space	NOUN
ejpam-3317	75	21	(	(	PUNCT
ejpam-3317	75	22	∏	∏	X
ejpam-3317	75	23	α∈i	α∈i	X
ejpam-3317	75	24	xα	xα	ADJ
ejpam-3317	75	25	,	,	PUNCT
ejpam-3317	75	26	c	c	NOUN
ejpam-3317	75	27	)	)	PUNCT
ejpam-3317	75	28	,	,	PUNCT
ejpam-3317	75	29	where	where	SCONJ
ejpam-3317	75	30	∏	∏	PROPN
ejpam-3317	75	31	α∈i	α∈i	PRON
ejpam-3317	75	32	xα	xα	ADP
ejpam-3317	75	33	denotes	denote	VERB
ejpam-3317	75	34	the	the	DET
ejpam-3317	75	35	cartesian	cartesian	ADJ
ejpam-3317	75	36	product	product	NOUN
ejpam-3317	75	37	of	of	ADP
ejpam-3317	75	38	sets	set	NOUN
ejpam-3317	75	39	xα	xα	ADP
ejpam-3317	75	40	,	,	PUNCT
ejpam-3317	75	41	α	α	PROPN
ejpam-3317	75	42	∈	∈	PROPN
ejpam-3317	76	1	i	i	PRON
ejpam-3317	76	2	and	and	CCONJ
ejpam-3317	76	3	c	c	PROPN
ejpam-3317	76	4	is	be	AUX
ejpam-3317	76	5	the	the	DET
ejpam-3317	76	6	closure	closure	NOUN
ejpam-3317	76	7	operator	operator	NOUN
ejpam-3317	76	8	generated	generate	VERB
ejpam-3317	76	9	by	by	ADP
ejpam-3317	76	10	the	the	DET
ejpam-3317	76	11	projections	projection	NOUN
ejpam-3317	76	12	πα	πα	VERB
ejpam-3317	76	13	:	:	PUNCT
ejpam-3317	76	14	∏	∏	NUM
ejpam-3317	76	15	α∈i	α∈i	X
ejpam-3317	76	16	(	(	PUNCT
ejpam-3317	76	17	xα	xα	INTJ
ejpam-3317	76	18	,	,	PUNCT
ejpam-3317	76	19	c	c	NOUN
ejpam-3317	76	20	)	)	PUNCT
ejpam-3317	76	21	→	→	SYM
ejpam-3317	76	22	(	(	PUNCT
ejpam-3317	76	23	xα	xα	INTJ
ejpam-3317	76	24	,	,	PUNCT
ejpam-3317	76	25	c	c	NOUN
ejpam-3317	76	26	)	)	PUNCT
ejpam-3317	76	27	,	,	PUNCT
ejpam-3317	76	28	α	α	PROPN
ejpam-3317	76	29	∈	∈	PROPN
ejpam-3317	76	30	i	i	PRON
ejpam-3317	76	31	,	,	PUNCT
ejpam-3317	76	32	i.e.	i.e.	X
ejpam-3317	76	33	,	,	PUNCT
ejpam-3317	76	34	is	be	AUX
ejpam-3317	76	35	defined	define	VERB
ejpam-3317	76	36	by	by	ADP
ejpam-3317	76	37	c	c	PROPN
ejpam-3317	76	38	(	(	PUNCT
ejpam-3317	76	39	a	a	X
ejpam-3317	76	40	)	)	PUNCT
ejpam-3317	76	41	=	=	SYM
ejpam-3317	76	42	∏	∏	NUM
ejpam-3317	76	43	α∈i	α∈i	NUM
ejpam-3317	76	44	cαπα(a	cαπα(a	NOUN
ejpam-3317	76	45	)	)	PUNCT
ejpam-3317	76	46	,	,	PUNCT
ejpam-3317	76	47	a	a	DET
ejpam-3317	76	48	⊆	⊆	NUM
ejpam-3317	76	49	∏	∏	NUM
ejpam-3317	76	50	α∈i	α∈i	NUM
ejpam-3317	76	51	xα	xα	PROPN
ejpam-3317	76	52	.	.	PUNCT
ejpam-3317	77	1	then	then	ADV
ejpam-3317	77	2	the	the	DET
ejpam-3317	77	3	projection	projection	NOUN
ejpam-3317	77	4	function	function	NOUN
ejpam-3317	77	5	πα	πα	VERB
ejpam-3317	77	6	is	be	AUX
ejpam-3317	77	7	continuous	continuous	ADJ
ejpam-3317	77	8	.	.	PUNCT
ejpam-3317	78	1	proposition	proposition	NOUN
ejpam-3317	78	2	7	7	NUM
ejpam-3317	78	3	.	.	PUNCT
ejpam-3317	79	1	[	[	X
ejpam-3317	79	2	16	16	NUM
ejpam-3317	79	3	]	]	X
ejpam-3317	79	4	let	let	VERB
ejpam-3317	79	5	{	{	PUNCT
ejpam-3317	79	6	(	(	PUNCT
ejpam-3317	79	7	xα	xα	INTJ
ejpam-3317	79	8	,	,	PUNCT
ejpam-3317	79	9	cα	cα	NOUN
ejpam-3317	79	10	)	)	PUNCT
ejpam-3317	79	11	:	:	PUNCT
ejpam-3317	79	12	a	a	DET
ejpam-3317	79	13	∈	∈	PROPN
ejpam-3317	79	14	j	j	NOUN
ejpam-3317	79	15	}	}	PUNCT
ejpam-3317	79	16	be	be	VERB
ejpam-3317	79	17	a	a	DET
ejpam-3317	79	18	family	family	NOUN
ejpam-3317	79	19	of	of	ADP
ejpam-3317	79	20	closure	closure	NOUN
ejpam-3317	79	21	spaces	space	NOUN
ejpam-3317	79	22	.	.	PUNCT
ejpam-3317	80	1	then	then	ADV
ejpam-3317	80	2	fais	fais	PROPN
ejpam-3317	80	3	closed	close	VERB
ejpam-3317	80	4	in	in	ADP
ejpam-3317	80	5	(	(	PUNCT
ejpam-3317	80	6	xα	xα	INTJ
ejpam-3317	80	7	,	,	PUNCT
ejpam-3317	80	8	cα	cα	NOUN
ejpam-3317	80	9	)	)	PUNCT
ejpam-3317	80	10	,	,	PUNCT
ejpam-3317	80	11	for	for	ADP
ejpam-3317	80	12	all	all	DET
ejpam-3317	80	13	a	a	DET
ejpam-3317	80	14	∈	∈	ADJ
ejpam-3317	80	15	j	j	NOUN
ejpam-3317	80	16	if	if	SCONJ
ejpam-3317	81	1	and	and	CCONJ
ejpam-3317	81	2	only	only	ADV
ejpam-3317	81	3	if	if	SCONJ
ejpam-3317	81	4	∏	∏	PROPN
ejpam-3317	81	5	a∈j	a∈j	ADJ
ejpam-3317	81	6	fαis	fαis	NOUN
ejpam-3317	81	7	closed	close	VERB
ejpam-3317	81	8	in	in	ADP
ejpam-3317	81	9	∏	∏	PROPN
ejpam-3317	81	10	a∈j	a∈j	NOUN
ejpam-3317	81	11	(	(	PUNCT
ejpam-3317	81	12	xα	xα	INTJ
ejpam-3317	81	13	,	,	PUNCT
ejpam-3317	81	14	ca	ca	NOUN
ejpam-3317	81	15	)	)	PUNCT
ejpam-3317	81	16	.	.	PUNCT
ejpam-3317	82	1	proposition	proposition	NOUN
ejpam-3317	82	2	8	8	NUM
ejpam-3317	82	3	.	.	PUNCT
ejpam-3317	83	1	[	[	X
ejpam-3317	83	2	16	16	NUM
ejpam-3317	83	3	]	]	X
ejpam-3317	83	4	let	let	VERB
ejpam-3317	83	5	{	{	PUNCT
ejpam-3317	83	6	(	(	PUNCT
ejpam-3317	83	7	xα	xα	INTJ
ejpam-3317	83	8	,	,	PUNCT
ejpam-3317	83	9	cα	cα	NOUN
ejpam-3317	83	10	)	)	PUNCT
ejpam-3317	83	11	:	:	PUNCT
ejpam-3317	83	12	a	a	DET
ejpam-3317	83	13	∈	∈	PROPN
ejpam-3317	83	14	j	j	NOUN
ejpam-3317	83	15	}	}	PUNCT
ejpam-3317	83	16	be	be	AUX
ejpam-3317	83	17	a	a	DET
ejpam-3317	83	18	collection	collection	NOUN
ejpam-3317	83	19	of	of	ADP
ejpam-3317	83	20	closure	closure	NOUN
ejpam-3317	83	21	spaces	space	NOUN
ejpam-3317	83	22	,	,	PUNCT
ejpam-3317	83	23	g	g	PROPN
ejpam-3317	83	24	⊆∏	⊆∏	PROPN
ejpam-3317	83	25	αεj	αεj	NOUN
ejpam-3317	83	26	xα	xα	INTJ
ejpam-3317	83	27	.	.	PUNCT
ejpam-3317	84	1	if	if	SCONJ
ejpam-3317	84	2	g	g	PROPN
ejpam-3317	84	3	is	be	AUX
ejpam-3317	84	4	a	a	DET
ejpam-3317	84	5	open	open	ADJ
ejpam-3317	84	6	in	in	ADP
ejpam-3317	84	7	∏	∏	PROPN
ejpam-3317	84	8	αεj	αεj	NOUN
ejpam-3317	84	9	(	(	PUNCT
ejpam-3317	84	10	xα	xα	INTJ
ejpam-3317	84	11	,	,	PUNCT
ejpam-3317	84	12	cα	cα	NOUN
ejpam-3317	84	13	)	)	PUNCT
ejpam-3317	84	14	and	and	CCONJ
ejpam-3317	84	15	πα	πα	PROPN
ejpam-3317	84	16	is	be	AUX
ejpam-3317	84	17	a	a	DET
ejpam-3317	84	18	project	project	NOUN
ejpam-3317	84	19	function	function	NOUN
ejpam-3317	84	20	,	,	PUNCT
ejpam-3317	84	21	then	then	ADV
ejpam-3317	84	22	πα(g	πα(g	NUM
ejpam-3317	84	23	)	)	PUNCT
ejpam-3317	84	24	is	be	AUX
ejpam-3317	84	25	a	a	DET
ejpam-3317	84	26	open	open	NOUN
ejpam-3317	84	27	in	in	ADP
ejpam-3317	84	28	(	(	PUNCT
ejpam-3317	84	29	xα	xα	INTJ
ejpam-3317	84	30	,	,	PUNCT
ejpam-3317	84	31	cα	cα	NOUN
ejpam-3317	84	32	)	)	PUNCT
ejpam-3317	84	33	.	.	PUNCT
ejpam-3317	85	1	definition	definition	NOUN
ejpam-3317	85	2	7	7	NUM
ejpam-3317	85	3	.	.	PUNCT
ejpam-3317	86	1	[	[	X
ejpam-3317	86	2	16	16	NUM
ejpam-3317	86	3	]	]	X
ejpam-3317	86	4	let	let	VERB
ejpam-3317	86	5	(	(	PUNCT
ejpam-3317	86	6	x	x	NOUN
ejpam-3317	86	7	,	,	PUNCT
ejpam-3317	86	8	c1)and	c1)and	PROPN
ejpam-3317	87	1	(	(	PUNCT
ejpam-3317	87	2	y	y	PROPN
ejpam-3317	87	3	,	,	PUNCT
ejpam-3317	87	4	c2	c2	PROPN
ejpam-3317	87	5	)	)	PUNCT
ejpam-3317	87	6	be	be	VERB
ejpam-3317	87	7	closure	closure	NOUN
ejpam-3317	87	8	spaces	space	NOUN
ejpam-3317	87	9	.	.	PUNCT
ejpam-3317	88	1	a	a	DET
ejpam-3317	88	2	function	function	NOUN
ejpam-3317	88	3	f	f	NOUN
ejpam-3317	88	4	:	:	PUNCT
ejpam-3317	88	5	(	(	PUNCT
ejpam-3317	88	6	x	x	X
ejpam-3317	88	7	,	,	PUNCT
ejpam-3317	88	8	c1	c1	PROPN
ejpam-3317	88	9	)	)	PUNCT
ejpam-3317	88	10	→	→	PUNCT
ejpam-3317	88	11	(	(	PUNCT
ejpam-3317	88	12	y	y	PROPN
ejpam-3317	88	13	,	,	PUNCT
ejpam-3317	88	14	c2	c2	PROPN
ejpam-3317	88	15	)	)	PUNCT
ejpam-3317	88	16	is	be	AUX
ejpam-3317	88	17	called	call	VERB
ejpam-3317	88	18	open	open	ADJ
ejpam-3317	88	19	(	(	PUNCT
ejpam-3317	88	20	respectively	respectively	ADV
ejpam-3317	88	21	,	,	PUNCT
ejpam-3317	88	22	closed	closed	ADJ
ejpam-3317	88	23	)	)	PUNCT
ejpam-3317	88	24	if	if	SCONJ
ejpam-3317	88	25	the	the	DET
ejpam-3317	88	26	image	image	NOUN
ejpam-3317	88	27	of	of	ADP
ejpam-3317	88	28	every	every	PRON
ejpam-3317	88	29	open	open	ADJ
ejpam-3317	88	30	(	(	PUNCT
ejpam-3317	88	31	respectively	respectively	ADV
ejpam-3317	88	32	,	,	PUNCT
ejpam-3317	88	33	closed	closed	ADJ
ejpam-3317	88	34	)	)	PUNCT
ejpam-3317	88	35	set	set	VERB
ejpam-3317	88	36	in	in	ADP
ejpam-3317	88	37	(	(	PUNCT
ejpam-3317	88	38	x	x	NOUN
ejpam-3317	88	39	,	,	PUNCT
ejpam-3317	88	40	c1	c1	PROPN
ejpam-3317	88	41	)	)	PUNCT
ejpam-3317	88	42	is	be	AUX
ejpam-3317	88	43	open	open	ADJ
ejpam-3317	88	44	(	(	PUNCT
ejpam-3317	88	45	respectively	respectively	ADV
ejpam-3317	88	46	,	,	PUNCT
ejpam-3317	88	47	closed	closed	ADJ
ejpam-3317	88	48	)	)	PUNCT
ejpam-3317	88	49	in	in	ADP
ejpam-3317	88	50	(	(	PUNCT
ejpam-3317	88	51	y	y	PROPN
ejpam-3317	88	52	,	,	PUNCT
ejpam-3317	88	53	c2	c2	PROPN
ejpam-3317	88	54	)	)	PUNCT
ejpam-3317	88	55	.	.	PUNCT
ejpam-3317	89	1	proposition	proposition	NOUN
ejpam-3317	89	2	9	9	NUM
ejpam-3317	89	3	.	.	PUNCT
ejpam-3317	90	1	[	[	X
ejpam-3317	90	2	16	16	NUM
ejpam-3317	90	3	]	]	PUNCT
ejpam-3317	90	4	a	a	DET
ejpam-3317	90	5	function	function	NOUN
ejpam-3317	90	6	f	f	NOUN
ejpam-3317	90	7	:	:	PUNCT
ejpam-3317	90	8	(	(	PUNCT
ejpam-3317	90	9	x	x	X
ejpam-3317	90	10	,	,	PUNCT
ejpam-3317	90	11	c1)→	c1)→	NOUN
ejpam-3317	90	12	(	(	PUNCT
ejpam-3317	90	13	y	y	PROPN
ejpam-3317	90	14	,	,	PUNCT
ejpam-3317	90	15	c2	c2	PROPN
ejpam-3317	90	16	)	)	PUNCT
ejpam-3317	90	17	is	be	AUX
ejpam-3317	90	18	said	say	VERB
ejpam-3317	90	19	to	to	PART
ejpam-3317	90	20	be	be	AUX
ejpam-3317	90	21	continuous	continuous	ADJ
ejpam-3317	90	22	if	if	SCONJ
ejpam-3317	90	23	f(c1(a	f(c1(a	PROPN
ejpam-3317	90	24	)	)	PUNCT
ejpam-3317	90	25	)	)	PUNCT
ejpam-3317	91	1	⊆	⊆	NUM
ejpam-3317	91	2	c2f(a	c2f(a	PROPN
ejpam-3317	91	3	)	)	PUNCT
ejpam-3317	91	4	for	for	ADP
ejpam-3317	91	5	every	every	DET
ejpam-3317	91	6	subset	subset	NOUN
ejpam-3317	91	7	a	a	PRON
ejpam-3317	91	8	of	of	ADP
ejpam-3317	91	9	x.	x.	NOUN
ejpam-3317	91	10	proposition	proposition	NOUN
ejpam-3317	91	11	10	10	NUM
ejpam-3317	91	12	.	.	PUNCT
ejpam-3317	92	1	[	[	X
ejpam-3317	92	2	16	16	NUM
ejpam-3317	92	3	]	]	X
ejpam-3317	92	4	let	let	AUX
ejpam-3317	92	5	(	(	PUNCT
ejpam-3317	92	6	x	x	NOUN
ejpam-3317	92	7	,	,	PUNCT
ejpam-3317	92	8	c1	c1	PROPN
ejpam-3317	92	9	)	)	PUNCT
ejpam-3317	92	10	and	and	CCONJ
ejpam-3317	92	11	(	(	PUNCT
ejpam-3317	92	12	y	y	PROPN
ejpam-3317	92	13	,	,	PUNCT
ejpam-3317	92	14	c2	c2	PROPN
ejpam-3317	92	15	)	)	PUNCT
ejpam-3317	92	16	be	be	VERB
ejpam-3317	92	17	closure	closure	NOUN
ejpam-3317	92	18	spaces	space	NOUN
ejpam-3317	92	19	.	.	PUNCT
ejpam-3317	93	1	if	if	SCONJ
ejpam-3317	93	2	f	f	PROPN
ejpam-3317	93	3	:	:	PUNCT
ejpam-3317	93	4	(	(	PUNCT
ejpam-3317	93	5	x	x	X
ejpam-3317	93	6	,	,	PUNCT
ejpam-3317	93	7	c1)→	c1)→	NOUN
ejpam-3317	93	8	(	(	PUNCT
ejpam-3317	93	9	y	y	PROPN
ejpam-3317	93	10	,	,	PUNCT
ejpam-3317	93	11	c2	c2	PROPN
ejpam-3317	93	12	)	)	PUNCT
ejpam-3317	93	13	is	be	AUX
ejpam-3317	93	14	a	a	DET
ejpam-3317	93	15	continuous	continuous	ADJ
ejpam-3317	93	16	function	function	NOUN
ejpam-3317	93	17	,	,	PUNCT
ejpam-3317	93	18	then	then	ADV
ejpam-3317	93	19	the	the	DET
ejpam-3317	93	20	inverse	inverse	ADJ
ejpam-3317	93	21	image	image	NOUN
ejpam-3317	93	22	under	under	ADP
ejpam-3317	93	23	f	f	PROPN
ejpam-3317	93	24	of	of	ADP
ejpam-3317	93	25	each	each	DET
ejpam-3317	93	26	open	open	ADJ
ejpam-3317	93	27	set	set	NOUN
ejpam-3317	93	28	in	in	ADP
ejpam-3317	93	29	(	(	PUNCT
ejpam-3317	93	30	y	y	PROPN
ejpam-3317	93	31	,	,	PUNCT
ejpam-3317	93	32	c2)is	c2)is	PROPN
ejpam-3317	93	33	open	open	ADJ
ejpam-3317	93	34	in	in	ADP
ejpam-3317	93	35	(	(	PUNCT
ejpam-3317	93	36	x	x	NOUN
ejpam-3317	93	37	,	,	PUNCT
ejpam-3317	93	38	c1	c1	PROPN
ejpam-3317	93	39	)	)	PUNCT
ejpam-3317	93	40	.	.	PUNCT
ejpam-3317	94	1	proposition	proposition	NOUN
ejpam-3317	94	2	11	11	NUM
ejpam-3317	94	3	.	.	PUNCT
ejpam-3317	95	1	[	[	X
ejpam-3317	95	2	16	16	NUM
ejpam-3317	95	3	]	]	X
ejpam-3317	95	4	let	let	VERB
ejpam-3317	95	5	(	(	PUNCT
ejpam-3317	95	6	x	x	NOUN
ejpam-3317	95	7	,	,	PUNCT
ejpam-3317	95	8	c1	c1	PROPN
ejpam-3317	95	9	)	)	PUNCT
ejpam-3317	95	10	,	,	PUNCT
ejpam-3317	95	11	(	(	PUNCT
ejpam-3317	95	12	y	y	PROPN
ejpam-3317	95	13	,	,	PUNCT
ejpam-3317	95	14	c2	c2	PROPN
ejpam-3317	95	15	)	)	PUNCT
ejpam-3317	95	16	and	and	CCONJ
ejpam-3317	95	17	(	(	PUNCT
ejpam-3317	95	18	z	z	NOUN
ejpam-3317	95	19	,	,	PUNCT
ejpam-3317	95	20	c3)be	c3)be	NOUN
ejpam-3317	95	21	closure	closure	NOUN
ejpam-3317	95	22	spaces	space	NOUN
ejpam-3317	95	23	,	,	PUNCT
ejpam-3317	95	24	let	let	VERB
ejpam-3317	95	25	f	f	PRON
ejpam-3317	95	26	:	:	PUNCT
ejpam-3317	95	27	(	(	PUNCT
ejpam-3317	95	28	x	x	X
ejpam-3317	95	29	,	,	PUNCT
ejpam-3317	95	30	c1	c1	PROPN
ejpam-3317	95	31	)	)	PUNCT
ejpam-3317	95	32	→	→	PUNCT
ejpam-3317	95	33	(	(	PUNCT
ejpam-3317	95	34	y	y	PROPN
ejpam-3317	95	35	,	,	PUNCT
ejpam-3317	95	36	c2	c2	PROPN
ejpam-3317	95	37	)	)	PUNCT
ejpam-3317	95	38	and	and	CCONJ
ejpam-3317	95	39	g	g	NOUN
ejpam-3317	95	40	:	:	PUNCT
ejpam-3317	95	41	(	(	PUNCT
ejpam-3317	95	42	y	y	NOUN
ejpam-3317	95	43	,	,	PUNCT
ejpam-3317	95	44	c2)→	c2)→	X
ejpam-3317	95	45	(	(	PUNCT
ejpam-3317	95	46	z	z	NOUN
ejpam-3317	95	47	,	,	PUNCT
ejpam-3317	95	48	c3	c3	PROPN
ejpam-3317	95	49	)	)	PUNCT
ejpam-3317	95	50	be	be	AUX
ejpam-3317	95	51	functions	function	NOUN
ejpam-3317	95	52	.	.	PUNCT
ejpam-3317	96	1	then	then	ADV
ejpam-3317	96	2	:	:	PUNCT
ejpam-3317	96	3	(	(	PUNCT
ejpam-3317	96	4	i	i	NOUN
ejpam-3317	96	5	)	)	PUNCT
ejpam-3317	96	6	if	if	SCONJ
ejpam-3317	96	7	f	f	PROPN
ejpam-3317	96	8	and	and	CCONJ
ejpam-3317	96	9	g	g	PROPN
ejpam-3317	96	10	are	be	AUX
ejpam-3317	96	11	open	open	ADJ
ejpam-3317	96	12	,	,	PUNCT
ejpam-3317	96	13	then	then	ADV
ejpam-3317	96	14	so	so	ADV
ejpam-3317	96	15	is	be	AUX
ejpam-3317	96	16	gof	gof	NOUN
ejpam-3317	96	17	.	.	PUNCT
ejpam-3317	97	1	(	(	PUNCT
ejpam-3317	97	2	ii	ii	NOUN
ejpam-3317	97	3	)	)	PUNCT
ejpam-3317	97	4	if	if	SCONJ
ejpam-3317	97	5	gof	gof	NOUN
ejpam-3317	97	6	is	be	AUX
ejpam-3317	97	7	open	open	ADJ
ejpam-3317	97	8	and	and	CCONJ
ejpam-3317	97	9	f	f	PROPN
ejpam-3317	97	10	is	be	AUX
ejpam-3317	97	11	a	a	DET
ejpam-3317	97	12	continuous	continuous	ADJ
ejpam-3317	97	13	surjection	surjection	NOUN
ejpam-3317	97	14	,	,	PUNCT
ejpam-3317	97	15	then	then	ADV
ejpam-3317	97	16	g	g	PROPN
ejpam-3317	97	17	is	be	AUX
ejpam-3317	97	18	open	open	ADJ
ejpam-3317	97	19	.	.	PUNCT
ejpam-3317	98	1	(	(	PUNCT
ejpam-3317	98	2	iii	iii	X
ejpam-3317	98	3	)	)	PUNCT
ejpam-3317	98	4	if	if	SCONJ
ejpam-3317	98	5	gof	gof	NOUN
ejpam-3317	98	6	is	be	AUX
ejpam-3317	98	7	open	open	ADJ
ejpam-3317	98	8	and	and	CCONJ
ejpam-3317	98	9	g	g	NOUN
ejpam-3317	98	10	is	be	AUX
ejpam-3317	98	11	a	a	DET
ejpam-3317	98	12	continuous	continuous	ADJ
ejpam-3317	98	13	injection	injection	NOUN
ejpam-3317	98	14	,	,	PUNCT
ejpam-3317	98	15	then	then	ADV
ejpam-3317	98	16	f	f	PROPN
ejpam-3317	98	17	is	be	AUX
ejpam-3317	98	18	open	open	ADJ
ejpam-3317	98	19	.	.	PUNCT
ejpam-3317	99	1	proposition	proposition	NOUN
ejpam-3317	99	2	12	12	NUM
ejpam-3317	99	3	.	.	PUNCT
ejpam-3317	100	1	[	[	X
ejpam-3317	100	2	16	16	NUM
ejpam-3317	100	3	]	]	X
ejpam-3317	100	4	let	let	AUX
ejpam-3317	100	5	(	(	PUNCT
ejpam-3317	100	6	x	x	NOUN
ejpam-3317	100	7	,	,	PUNCT
ejpam-3317	100	8	c1	c1	PROPN
ejpam-3317	100	9	)	)	PUNCT
ejpam-3317	100	10	and	and	CCONJ
ejpam-3317	100	11	(	(	PUNCT
ejpam-3317	100	12	y	y	PROPN
ejpam-3317	100	13	,	,	PUNCT
ejpam-3317	100	14	c2	c2	PROPN
ejpam-3317	100	15	)	)	PUNCT
ejpam-3317	100	16	be	be	VERB
ejpam-3317	100	17	closure	closure	NOUN
ejpam-3317	100	18	spaces	space	NOUN
ejpam-3317	100	19	and	and	CCONJ
ejpam-3317	100	20	let	let	VERB
ejpam-3317	100	21	f	f	X
ejpam-3317	100	22	:	:	PUNCT
ejpam-3317	100	23	(	(	PUNCT
ejpam-3317	100	24	x	x	X
ejpam-3317	100	25	,	,	PUNCT
ejpam-3317	100	26	c1)→	c1)→	NOUN
ejpam-3317	100	27	(	(	PUNCT
ejpam-3317	100	28	y	y	PROPN
ejpam-3317	100	29	,	,	PUNCT
ejpam-3317	100	30	c2	c2	PROPN
ejpam-3317	100	31	)	)	PUNCT
ejpam-3317	100	32	be	be	AUX
ejpam-3317	100	33	a	a	DET
ejpam-3317	100	34	function	function	NOUN
ejpam-3317	100	35	.	.	PUNCT
ejpam-3317	101	1	if	if	SCONJ
ejpam-3317	101	2	f	f	PROPN
ejpam-3317	101	3	is	be	AUX
ejpam-3317	101	4	open	open	ADJ
ejpam-3317	101	5	,	,	PUNCT
ejpam-3317	101	6	then	then	ADV
ejpam-3317	101	7	for	for	ADP
ejpam-3317	101	8	every	every	DET
ejpam-3317	101	9	y	y	PROPN
ejpam-3317	101	10	∈	∈	PROPN
ejpam-3317	101	11	y	y	PROPN
ejpam-3317	101	12	and	and	CCONJ
ejpam-3317	101	13	every	every	DET
ejpam-3317	101	14	closed	close	VERB
ejpam-3317	101	15	subset	subset	NOUN
ejpam-3317	101	16	f	f	PROPN
ejpam-3317	101	17	of	of	ADP
ejpam-3317	101	18	(	(	PUNCT
ejpam-3317	101	19	x	x	NOUN
ejpam-3317	101	20	,	,	PUNCT
ejpam-3317	101	21	c1	c1	PROPN
ejpam-3317	101	22	)	)	PUNCT
ejpam-3317	101	23	such	such	ADJ
ejpam-3317	101	24	that	that	SCONJ
ejpam-3317	101	25	f−1	f−1	PROPN
ejpam-3317	101	26	(	(	PUNCT
ejpam-3317	101	27	{	{	PUNCT
ejpam-3317	101	28	y	y	NOUN
ejpam-3317	101	29	}	}	PUNCT
ejpam-3317	101	30	)	)	PUNCT
ejpam-3317	101	31	⊆	⊆	NUM
ejpam-3317	101	32	f	f	NOUN
ejpam-3317	101	33	,	,	PUNCT
ejpam-3317	101	34	there	there	PRON
ejpam-3317	101	35	exists	exist	VERB
ejpam-3317	101	36	a	a	DET
ejpam-3317	101	37	closed	closed	ADJ
ejpam-3317	101	38	subset	subset	NOUN
ejpam-3317	101	39	k	k	PROPN
ejpam-3317	101	40	of	of	ADP
ejpam-3317	101	41	(	(	PUNCT
ejpam-3317	101	42	y	y	PROPN
ejpam-3317	101	43	,	,	PUNCT
ejpam-3317	101	44	c2	c2	PROPN
ejpam-3317	101	45	)	)	PUNCT
ejpam-3317	101	46	such	such	ADJ
ejpam-3317	101	47	that	that	SCONJ
ejpam-3317	101	48	y	y	PROPN
ejpam-3317	101	49	∈	∈	PROPN
ejpam-3317	101	50	k	k	PROPN
ejpam-3317	101	51	and	and	CCONJ
ejpam-3317	101	52	f−1	f−1	PROPN
ejpam-3317	101	53	(	(	PUNCT
ejpam-3317	101	54	k	k	NOUN
ejpam-3317	101	55	)	)	PUNCT
ejpam-3317	101	56	⊆	⊆	PROPN
ejpam-3317	101	57	f.	f.	PROPN
ejpam-3317	101	58	h.m	h.m	PROPN
ejpam-3317	101	59	.	.	PROPN
ejpam-3317	101	60	darwesh	darwesh	PROPN
ejpam-3317	101	61	,	,	PUNCT
ejpam-3317	101	62	s.f	s.f	PROPN
ejpam-3317	101	63	.	.	PROPN
ejpam-3317	101	64	namiq	namiq	PROPN
ejpam-3317	101	65	/	/	SYM
ejpam-3317	101	66	eur	eur	PROPN
ejpam-3317	101	67	.	.	PUNCT
ejpam-3317	102	1	j.	j.	PROPN
ejpam-3317	102	2	pure	pure	PROPN
ejpam-3317	102	3	appl	appl	PROPN
ejpam-3317	102	4	.	.	PROPN
ejpam-3317	102	5	math	math	PROPN
ejpam-3317	102	6	,	,	PUNCT
ejpam-3317	102	7	12	12	NUM
ejpam-3317	102	8	(	(	PUNCT
ejpam-3317	102	9	3	3	NUM
ejpam-3317	102	10	)	)	PUNCT
ejpam-3317	102	11	(	(	PUNCT
ejpam-3317	102	12	2019	2019	NUM
ejpam-3317	102	13	)	)	PUNCT
ejpam-3317	102	14	,	,	PUNCT
ejpam-3317	102	15	1082	1082	NUM
ejpam-3317	102	16	-	-	SYM
ejpam-3317	102	17	1095	1095	NUM
ejpam-3317	102	18	1086	1086	NUM
ejpam-3317	102	19	proposition	proposition	NOUN
ejpam-3317	102	20	13	13	NUM
ejpam-3317	102	21	.	.	PUNCT
ejpam-3317	103	1	[	[	X
ejpam-3317	103	2	11	11	NUM
ejpam-3317	103	3	]	]	X
ejpam-3317	103	4	let	let	AUX
ejpam-3317	103	5	(	(	PUNCT
ejpam-3317	103	6	x	x	NOUN
ejpam-3317	103	7	,	,	PUNCT
ejpam-3317	103	8	c1	c1	PROPN
ejpam-3317	103	9	)	)	PUNCT
ejpam-3317	103	10	and	and	CCONJ
ejpam-3317	103	11	(	(	PUNCT
ejpam-3317	103	12	y	y	PROPN
ejpam-3317	103	13	,	,	PUNCT
ejpam-3317	103	14	c2	c2	PROPN
ejpam-3317	103	15	)	)	PUNCT
ejpam-3317	103	16	be	be	VERB
ejpam-3317	103	17	closure	closure	NOUN
ejpam-3317	103	18	spaces	space	NOUN
ejpam-3317	103	19	.	.	PUNCT
ejpam-3317	104	1	a	a	DET
ejpam-3317	104	2	function	function	NOUN
ejpam-3317	104	3	f	f	NOUN
ejpam-3317	104	4	:	:	PUNCT
ejpam-3317	104	5	(	(	PUNCT
ejpam-3317	104	6	x	x	X
ejpam-3317	104	7	,	,	PUNCT
ejpam-3317	104	8	c1)→	c1)→	NOUN
ejpam-3317	104	9	(	(	PUNCT
ejpam-3317	104	10	y	y	PROPN
ejpam-3317	104	11	,	,	PUNCT
ejpam-3317	104	12	c2	c2	PROPN
ejpam-3317	104	13	)	)	PUNCT
ejpam-3317	104	14	is	be	AUX
ejpam-3317	104	15	called	call	VERB
ejpam-3317	104	16	pre	pre	ADJ
ejpam-3317	104	17	-	-	ADJ
ejpam-3317	104	18	continuous	continuous	ADJ
ejpam-3317	104	19	if	if	SCONJ
ejpam-3317	104	20	the	the	DET
ejpam-3317	104	21	inverse	inverse	ADJ
ejpam-3317	104	22	image	image	NOUN
ejpam-3317	104	23	of	of	ADP
ejpam-3317	104	24	every	every	DET
ejpam-3317	104	25	open	open	ADJ
ejpam-3317	104	26	set	set	NOUN
ejpam-3317	104	27	in	in	ADP
ejpam-3317	104	28	(	(	PUNCT
ejpam-3317	104	29	y	y	PROPN
ejpam-3317	104	30	,	,	PUNCT
ejpam-3317	104	31	c2	c2	PROPN
ejpam-3317	104	32	)	)	PUNCT
ejpam-3317	104	33	is	be	AUX
ejpam-3317	104	34	preopen	preopen	ADJ
ejpam-3317	104	35	in	in	ADP
ejpam-3317	104	36	(	(	PUNCT
ejpam-3317	104	37	x	x	NOUN
ejpam-3317	104	38	,	,	PUNCT
ejpam-3317	104	39	c1	c1	PROPN
ejpam-3317	104	40	)	)	PUNCT
ejpam-3317	104	41	.	.	PUNCT
ejpam-3317	105	1	proposition	proposition	NOUN
ejpam-3317	105	2	14	14	NUM
ejpam-3317	105	3	.	.	PUNCT
ejpam-3317	106	1	[	[	X
ejpam-3317	106	2	11	11	NUM
ejpam-3317	106	3	]	]	X
ejpam-3317	106	4	let	let	AUX
ejpam-3317	106	5	(	(	PUNCT
ejpam-3317	106	6	x	x	NOUN
ejpam-3317	106	7	,	,	PUNCT
ejpam-3317	106	8	c1	c1	PROPN
ejpam-3317	106	9	)	)	PUNCT
ejpam-3317	106	10	and	and	CCONJ
ejpam-3317	106	11	(	(	PUNCT
ejpam-3317	106	12	y	y	PROPN
ejpam-3317	106	13	,	,	PUNCT
ejpam-3317	106	14	c2	c2	PROPN
ejpam-3317	106	15	)	)	PUNCT
ejpam-3317	106	16	be	be	VERB
ejpam-3317	106	17	closure	closure	NOUN
ejpam-3317	106	18	spaces	space	NOUN
ejpam-3317	106	19	.	.	PUNCT
ejpam-3317	107	1	a	a	DET
ejpam-3317	107	2	function	function	NOUN
ejpam-3317	107	3	f	f	NOUN
ejpam-3317	107	4	:	:	PUNCT
ejpam-3317	107	5	(	(	PUNCT
ejpam-3317	107	6	x	x	X
ejpam-3317	107	7	,	,	PUNCT
ejpam-3317	107	8	c1)→	c1)→	NOUN
ejpam-3317	107	9	(	(	PUNCT
ejpam-3317	107	10	y	y	PROPN
ejpam-3317	107	11	,	,	PUNCT
ejpam-3317	107	12	c2	c2	PROPN
ejpam-3317	107	13	)	)	PUNCT
ejpam-3317	107	14	is	be	AUX
ejpam-3317	107	15	called	call	VERB
ejpam-3317	107	16	pre	pre	ADJ
ejpam-3317	107	17	-	-	ADJ
ejpam-3317	107	18	continuous	continuous	ADJ
ejpam-3317	107	19	if	if	SCONJ
ejpam-3317	108	1	and	and	CCONJ
ejpam-3317	108	2	only	only	ADV
ejpam-3317	108	3	if	if	SCONJ
ejpam-3317	108	4	the	the	DET
ejpam-3317	108	5	inverse	inverse	ADJ
ejpam-3317	108	6	image	image	NOUN
ejpam-3317	108	7	of	of	ADP
ejpam-3317	108	8	every	every	DET
ejpam-3317	108	9	closed	close	VERB
ejpam-3317	108	10	set	set	VERB
ejpam-3317	108	11	in	in	ADP
ejpam-3317	108	12	(	(	PUNCT
ejpam-3317	108	13	y	y	PROPN
ejpam-3317	108	14	,	,	PUNCT
ejpam-3317	108	15	c2	c2	PROPN
ejpam-3317	108	16	)	)	PUNCT
ejpam-3317	108	17	is	be	AUX
ejpam-3317	108	18	preclosed	preclose	VERB
ejpam-3317	108	19	in	in	ADP
ejpam-3317	108	20	(	(	PUNCT
ejpam-3317	108	21	x	x	NOUN
ejpam-3317	108	22	,	,	PUNCT
ejpam-3317	108	23	c1	c1	PROPN
ejpam-3317	108	24	)	)	PUNCT
ejpam-3317	108	25	.	.	PUNCT
ejpam-3317	109	1	proposition	proposition	NOUN
ejpam-3317	109	2	15	15	NUM
ejpam-3317	109	3	.	.	PUNCT
ejpam-3317	110	1	[	[	X
ejpam-3317	110	2	11	11	NUM
ejpam-3317	110	3	]	]	X
ejpam-3317	110	4	let	let	VERB
ejpam-3317	110	5	(	(	PUNCT
ejpam-3317	110	6	x	x	NOUN
ejpam-3317	110	7	,	,	PUNCT
ejpam-3317	110	8	c1	c1	PROPN
ejpam-3317	110	9	)	)	PUNCT
ejpam-3317	110	10	,	,	PUNCT
ejpam-3317	110	11	(	(	PUNCT
ejpam-3317	110	12	y	y	PROPN
ejpam-3317	110	13	,	,	PUNCT
ejpam-3317	110	14	c2	c2	PROPN
ejpam-3317	110	15	)	)	PUNCT
ejpam-3317	110	16	and	and	CCONJ
ejpam-3317	110	17	(	(	PUNCT
ejpam-3317	110	18	z	z	NOUN
ejpam-3317	110	19	,	,	PUNCT
ejpam-3317	110	20	c3)be	c3)be	NOUN
ejpam-3317	110	21	closure	closure	NOUN
ejpam-3317	110	22	spaces	space	VERB
ejpam-3317	110	23	.	.	PUNCT
ejpam-3317	111	1	if	if	SCONJ
ejpam-3317	111	2	f	f	PROPN
ejpam-3317	111	3	:	:	PUNCT
ejpam-3317	111	4	(	(	PUNCT
ejpam-3317	111	5	x	x	X
ejpam-3317	111	6	,	,	PUNCT
ejpam-3317	111	7	c1	c1	PROPN
ejpam-3317	111	8	)	)	PUNCT
ejpam-3317	111	9	→	→	PUNCT
ejpam-3317	111	10	(	(	PUNCT
ejpam-3317	111	11	y	y	PROPN
ejpam-3317	111	12	,	,	PUNCT
ejpam-3317	111	13	c2	c2	PROPN
ejpam-3317	111	14	)	)	PUNCT
ejpam-3317	111	15	and	and	CCONJ
ejpam-3317	111	16	g	g	NOUN
ejpam-3317	111	17	:	:	PUNCT
ejpam-3317	111	18	(	(	PUNCT
ejpam-3317	111	19	y	y	NOUN
ejpam-3317	111	20	,	,	PUNCT
ejpam-3317	111	21	c2)→	c2)→	X
ejpam-3317	111	22	(	(	PUNCT
ejpam-3317	111	23	z	z	NOUN
ejpam-3317	111	24	,	,	PUNCT
ejpam-3317	111	25	c3	c3	PROPN
ejpam-3317	111	26	)	)	PUNCT
ejpam-3317	111	27	are	be	AUX
ejpam-3317	111	28	pre	pre	ADJ
ejpam-3317	111	29	-	-	ADJ
ejpam-3317	111	30	continuous	continuous	ADJ
ejpam-3317	111	31	and	and	CCONJ
ejpam-3317	111	32	continuous	continuous	ADJ
ejpam-3317	111	33	respectively.then	respectively.then	ADP
ejpam-3317	111	34	gof	gof	NOUN
ejpam-3317	111	35	:	:	PUNCT
ejpam-3317	111	36	(	(	PUNCT
ejpam-3317	111	37	x	x	X
ejpam-3317	111	38	,	,	PUNCT
ejpam-3317	111	39	c1)→	c1)→	NOUN
ejpam-3317	111	40	(	(	PUNCT
ejpam-3317	111	41	z	z	NOUN
ejpam-3317	111	42	,	,	PUNCT
ejpam-3317	111	43	c3	c3	PROPN
ejpam-3317	111	44	)	)	PUNCT
ejpam-3317	111	45	is	be	AUX
ejpam-3317	111	46	pre	pre	ADJ
ejpam-3317	111	47	-	-	ADJ
ejpam-3317	111	48	continuous	continuous	ADJ
ejpam-3317	111	49	.	.	PUNCT
ejpam-3317	112	1	proposition	proposition	NOUN
ejpam-3317	112	2	16	16	NUM
ejpam-3317	112	3	.	.	PUNCT
ejpam-3317	113	1	[	[	X
ejpam-3317	113	2	11	11	NUM
ejpam-3317	113	3	]	]	X
ejpam-3317	113	4	let	let	VERB
ejpam-3317	113	5	(	(	PUNCT
ejpam-3317	113	6	x	x	NOUN
ejpam-3317	113	7	,	,	PUNCT
ejpam-3317	113	8	c1	c1	PROPN
ejpam-3317	113	9	)	)	PUNCT
ejpam-3317	113	10	,	,	PUNCT
ejpam-3317	113	11	(	(	PUNCT
ejpam-3317	113	12	y	y	PROPN
ejpam-3317	113	13	,	,	PUNCT
ejpam-3317	113	14	c2	c2	PROPN
ejpam-3317	113	15	)	)	PUNCT
ejpam-3317	113	16	and	and	CCONJ
ejpam-3317	113	17	(	(	PUNCT
ejpam-3317	113	18	z	z	NOUN
ejpam-3317	113	19	,	,	PUNCT
ejpam-3317	113	20	c3	c3	PROPN
ejpam-3317	113	21	)	)	PUNCT
ejpam-3317	113	22	be	be	VERB
ejpam-3317	113	23	closure	closure	NOUN
ejpam-3317	113	24	spaces	space	NOUN
ejpam-3317	113	25	.	.	PUNCT
ejpam-3317	114	1	let	let	VERB
ejpam-3317	114	2	f	f	NOUN
ejpam-3317	114	3	:	:	PUNCT
ejpam-3317	114	4	(	(	PUNCT
ejpam-3317	114	5	x	x	X
ejpam-3317	114	6	,	,	PUNCT
ejpam-3317	114	7	c1)→	c1)→	NOUN
ejpam-3317	114	8	(	(	PUNCT
ejpam-3317	114	9	y	y	PROPN
ejpam-3317	114	10	,	,	PUNCT
ejpam-3317	114	11	c2	c2	PROPN
ejpam-3317	114	12	)	)	PUNCT
ejpam-3317	114	13	be	be	VERB
ejpam-3317	114	14	a	a	DET
ejpam-3317	114	15	surjective	surjective	ADJ
ejpam-3317	114	16	open	open	ADJ
ejpam-3317	114	17	continuous	continuous	ADJ
ejpam-3317	114	18	function	function	NOUN
ejpam-3317	114	19	and	and	CCONJ
ejpam-3317	114	20	g	g	NOUN
ejpam-3317	114	21	:	:	PUNCT
ejpam-3317	114	22	(	(	PUNCT
ejpam-3317	114	23	y	y	NOUN
ejpam-3317	114	24	,	,	PUNCT
ejpam-3317	114	25	c2)→	c2)→	X
ejpam-3317	114	26	(	(	PUNCT
ejpam-3317	114	27	z	z	NOUN
ejpam-3317	114	28	,	,	PUNCT
ejpam-3317	114	29	c3	c3	PROPN
ejpam-3317	114	30	)	)	PUNCT
ejpam-3317	114	31	is	be	AUX
ejpam-3317	114	32	a	a	DET
ejpam-3317	114	33	function	function	NOUN
ejpam-3317	114	34	such	such	ADJ
ejpam-3317	114	35	that	that	SCONJ
ejpam-3317	114	36	gof	gof	NOUN
ejpam-3317	114	37	:	:	PUNCT
ejpam-3317	114	38	(	(	PUNCT
ejpam-3317	114	39	x	x	X
ejpam-3317	114	40	,	,	PUNCT
ejpam-3317	114	41	c1)→	c1)→	NOUN
ejpam-3317	114	42	(	(	PUNCT
ejpam-3317	114	43	z	z	NOUN
ejpam-3317	114	44	,	,	PUNCT
ejpam-3317	114	45	c3	c3	PROPN
ejpam-3317	114	46	)	)	PUNCT
ejpam-3317	114	47	is	be	AUX
ejpam-3317	114	48	pre	pre	ADJ
ejpam-3317	114	49	-	-	ADJ
ejpam-3317	114	50	continuous	continuous	ADJ
ejpam-3317	114	51	.	.	PUNCT
ejpam-3317	115	1	then	then	ADV
ejpam-3317	115	2	g	g	NOUN
ejpam-3317	115	3	:	:	PUNCT
ejpam-3317	115	4	(	(	PUNCT
ejpam-3317	115	5	y	y	NOUN
ejpam-3317	115	6	,	,	PUNCT
ejpam-3317	115	7	c2)→	c2)→	X
ejpam-3317	115	8	(	(	PUNCT
ejpam-3317	115	9	z	z	NOUN
ejpam-3317	115	10	,	,	PUNCT
ejpam-3317	115	11	c3	c3	PROPN
ejpam-3317	115	12	)	)	PUNCT
ejpam-3317	115	13	is	be	AUX
ejpam-3317	115	14	pre	pre	ADJ
ejpam-3317	115	15	-	-	ADJ
ejpam-3317	115	16	continuous	continuous	ADJ
ejpam-3317	115	17	.	.	PUNCT
ejpam-3317	116	1	definition	definition	NOUN
ejpam-3317	116	2	8	8	NUM
ejpam-3317	116	3	.	.	PUNCT
ejpam-3317	117	1	[	[	X
ejpam-3317	117	2	16	16	NUM
ejpam-3317	117	3	]	]	X
ejpam-3317	117	4	a	a	DET
ejpam-3317	117	5	closure	closure	NOUN
ejpam-3317	117	6	space	space	NOUN
ejpam-3317	117	7	(	(	PUNCT
ejpam-3317	117	8	x	x	X
ejpam-3317	117	9	,	,	PUNCT
ejpam-3317	117	10	c	c	NOUN
ejpam-3317	117	11	)	)	PUNCT
ejpam-3317	117	12	is	be	AUX
ejpam-3317	117	13	said	say	VERB
ejpam-3317	117	14	to	to	PART
ejpam-3317	117	15	be	be	AUX
ejpam-3317	117	16	connected	connect	VERB
ejpam-3317	117	17	if	if	SCONJ
ejpam-3317	117	18	φ	φ	PROPN
ejpam-3317	117	19	and	and	CCONJ
ejpam-3317	117	20	x	x	NOUN
ejpam-3317	117	21	are	be	AUX
ejpam-3317	117	22	the	the	DET
ejpam-3317	117	23	only	only	ADJ
ejpam-3317	117	24	subsets	subset	NOUN
ejpam-3317	117	25	of	of	ADP
ejpam-3317	117	26	x	x	PUNCT
ejpam-3317	117	27	which	which	PRON
ejpam-3317	117	28	are	be	AUX
ejpam-3317	117	29	both	both	PRON
ejpam-3317	117	30	closed	closed	ADJ
ejpam-3317	117	31	and	and	CCONJ
ejpam-3317	117	32	open	open	ADJ
ejpam-3317	117	33	.	.	PUNCT
ejpam-3317	118	1	definition	definition	NOUN
ejpam-3317	118	2	9	9	NUM
ejpam-3317	118	3	.	.	PUNCT
ejpam-3317	119	1	[	[	X
ejpam-3317	119	2	16	16	NUM
ejpam-3317	119	3	]	]	PUNCT
ejpam-3317	119	4	a	a	DET
ejpam-3317	119	5	collection	collection	NOUN
ejpam-3317	119	6	{	{	PUNCT
ejpam-3317	119	7	ga}a∈j	ga}a∈j	NOUN
ejpam-3317	119	8	of	of	ADP
ejpam-3317	119	9	sets	set	NOUN
ejpam-3317	119	10	in	in	ADP
ejpam-3317	119	11	a	a	DET
ejpam-3317	119	12	closure	closure	NOUN
ejpam-3317	119	13	space(x	space(x	NOUN
ejpam-3317	119	14	,	,	PUNCT
ejpam-3317	119	15	c	c	NOUN
ejpam-3317	119	16	)	)	PUNCT
ejpam-3317	119	17	is	be	AUX
ejpam-3317	119	18	called	call	VERB
ejpam-3317	119	19	a	a	DET
ejpam-3317	119	20	cover	cover	NOUN
ejpam-3317	119	21	of	of	ADP
ejpam-3317	119	22	a	a	DET
ejpam-3317	119	23	subset	subset	NOUN
ejpam-3317	119	24	bof	bof	X
ejpam-3317	119	25	x	x	NOUN
ejpam-3317	119	26	if	if	SCONJ
ejpam-3317	119	27	b	b	PROPN
ejpam-3317	119	28	⊆	⊆	NUM
ejpam-3317	119	29	⋃	⋃	ADP
ejpam-3317	119	30	α∈j	α∈j	ADP
ejpam-3317	119	31	ga	ga	NOUN
ejpam-3317	119	32	if	if	SCONJ
ejpam-3317	119	33	holds	hold	VERB
ejpam-3317	119	34	,	,	PUNCT
ejpam-3317	119	35	and	and	CCONJ
ejpam-3317	119	36	an	an	DET
ejpam-3317	119	37	open	open	ADJ
ejpam-3317	119	38	cover	cover	NOUN
ejpam-3317	119	39	if	if	SCONJ
ejpam-3317	119	40	ga	ga	PROPN
ejpam-3317	119	41	is	be	AUX
ejpam-3317	119	42	open	open	ADJ
ejpam-3317	119	43	for	for	ADP
ejpam-3317	119	44	each	each	DET
ejpam-3317	119	45	α	α	PROPN
ejpam-3317	119	46	∈	∈	PROPN
ejpam-3317	119	47	j.	j.	PROPN
ejpam-3317	119	48	furthermore	furthermore	PROPN
ejpam-3317	119	49	,	,	PUNCT
ejpam-3317	119	50	a	a	DET
ejpam-3317	119	51	cover	cover	NOUN
ejpam-3317	119	52	{	{	PUNCT
ejpam-3317	119	53	ga}a∈j	ga}a∈j	NOUN
ejpam-3317	119	54	of	of	ADP
ejpam-3317	119	55	a	a	DET
ejpam-3317	119	56	subset	subset	NOUN
ejpam-3317	119	57	b	b	NOUN
ejpam-3317	119	58	contains	contain	VERB
ejpam-3317	119	59	a	a	DET
ejpam-3317	119	60	finite	finite	ADJ
ejpam-3317	119	61	subcover	subcover	NOUN
ejpam-3317	119	62	,	,	PUNCT
ejpam-3317	119	63	if	if	SCONJ
ejpam-3317	119	64	there	there	PRON
ejpam-3317	119	65	exists	exist	VERB
ejpam-3317	119	66	a	a	DET
ejpam-3317	119	67	finite	finite	NOUN
ejpam-3317	119	68	subset	subset	VERB
ejpam-3317	119	69	j0	j0	PROPN
ejpam-3317	119	70	of	of	ADP
ejpam-3317	119	71	j	j	PROPN
ejpam-3317	119	72	such	such	ADJ
ejpam-3317	119	73	that	that	PRON
ejpam-3317	119	74	b	b	NOUN
ejpam-3317	119	75	⊆	⊆	NUM
ejpam-3317	119	76	⋃	⋃	PROPN
ejpam-3317	119	77	α∈j0	α∈j0	PROPN
ejpam-3317	119	78	ga	ga	PROPN
ejpam-3317	119	79	.	.	PROPN
ejpam-3317	119	80	definition	definition	NOUN
ejpam-3317	119	81	10	10	NUM
ejpam-3317	119	82	.	.	PUNCT
ejpam-3317	120	1	[	[	X
ejpam-3317	120	2	16	16	NUM
ejpam-3317	120	3	]	]	PUNCT
ejpam-3317	120	4	a	a	DET
ejpam-3317	120	5	subset	subset	NOUN
ejpam-3317	120	6	a	a	PRON
ejpam-3317	120	7	of	of	ADP
ejpam-3317	120	8	a	a	DET
ejpam-3317	120	9	closure	closure	NOUN
ejpam-3317	120	10	space	space	NOUN
ejpam-3317	120	11	(	(	PUNCT
ejpam-3317	120	12	x	x	X
ejpam-3317	120	13	,	,	PUNCT
ejpam-3317	120	14	c	c	NOUN
ejpam-3317	120	15	)	)	PUNCT
ejpam-3317	120	16	is	be	AUX
ejpam-3317	120	17	compact	compact	ADJ
ejpam-3317	120	18	if	if	SCONJ
ejpam-3317	120	19	every	every	DET
ejpam-3317	120	20	open	open	ADJ
ejpam-3317	120	21	cover	cover	NOUN
ejpam-3317	120	22	of	of	ADP
ejpam-3317	120	23	a	a	PRON
ejpam-3317	120	24	contains	contain	VERB
ejpam-3317	120	25	a	a	DET
ejpam-3317	120	26	finite	finite	ADJ
ejpam-3317	120	27	subcover	subcover	PROPN
ejpam-3317	120	28	.	.	PUNCT
ejpam-3317	121	1	3	3	X
ejpam-3317	121	2	.	.	X
ejpam-3317	121	3	pre	pre	ADJ
ejpam-3317	121	4	-	-	ADJ
ejpam-3317	121	5	open	open	ADJ
ejpam-3317	121	6	(	(	PUNCT
ejpam-3317	121	7	pre	pre	ADJ
ejpam-3317	121	8	-	-	ADJ
ejpam-3317	121	9	closed	closed	ADJ
ejpam-3317	121	10	)	)	PUNCT
ejpam-3317	121	11	functions	function	NOUN
ejpam-3317	121	12	and	and	CCONJ
ejpam-3317	121	13	contra	contra	PROPN
ejpam-3317	121	14	-	-	ADJ
ejpam-3317	121	15	pre	pre	ADJ
ejpam-3317	121	16	-	-	ADJ
ejpam-3317	121	17	continuous	continuous	ADJ
ejpam-3317	121	18	in	in	ADP
ejpam-3317	121	19	the	the	DET
ejpam-3317	121	20	present	present	ADJ
ejpam-3317	121	21	section	section	NOUN
ejpam-3317	121	22	,	,	PUNCT
ejpam-3317	121	23	we	we	PRON
ejpam-3317	121	24	define	define	VERB
ejpam-3317	121	25	and	and	CCONJ
ejpam-3317	121	26	study	study	VERB
ejpam-3317	121	27	some	some	DET
ejpam-3317	121	28	properties	property	NOUN
ejpam-3317	121	29	of	of	ADP
ejpam-3317	121	30	pre	pre	ADJ
ejpam-3317	121	31	-	-	ADJ
ejpam-3317	121	32	open	open	ADJ
ejpam-3317	121	33	(	(	PUNCT
ejpam-3317	121	34	preclosed)functions	preclosed)function	NOUN
ejpam-3317	121	35	and	and	CCONJ
ejpam-3317	121	36	contra	contra	PROPN
ejpam-3317	121	37	-	-	ADJ
ejpam-3317	121	38	pre	pre	ADJ
ejpam-3317	121	39	-	-	ADJ
ejpam-3317	121	40	continuous	continuous	ADJ
ejpam-3317	121	41	.	.	PUNCT
ejpam-3317	122	1	definition	definition	NOUN
ejpam-3317	122	2	11	11	NUM
ejpam-3317	122	3	.	.	PUNCT
ejpam-3317	123	1	let	let	VERB
ejpam-3317	123	2	(	(	PUNCT
ejpam-3317	123	3	x	x	NOUN
ejpam-3317	123	4	,	,	PUNCT
ejpam-3317	123	5	c1	c1	PROPN
ejpam-3317	123	6	)	)	PUNCT
ejpam-3317	123	7	and	and	CCONJ
ejpam-3317	123	8	(	(	PUNCT
ejpam-3317	123	9	y	y	PROPN
ejpam-3317	123	10	,	,	PUNCT
ejpam-3317	123	11	c2	c2	PROPN
ejpam-3317	123	12	)	)	PUNCT
ejpam-3317	123	13	be	be	VERB
ejpam-3317	123	14	two	two	NUM
ejpam-3317	123	15	closure	closure	NOUN
ejpam-3317	123	16	spaces	space	NOUN
ejpam-3317	123	17	.	.	PUNCT
ejpam-3317	124	1	let	let	VERB
ejpam-3317	124	2	f	f	NOUN
ejpam-3317	124	3	:	:	PUNCT
ejpam-3317	124	4	(	(	PUNCT
ejpam-3317	124	5	x	x	X
ejpam-3317	124	6	,	,	PUNCT
ejpam-3317	124	7	c1	c1	PROPN
ejpam-3317	124	8	)	)	PUNCT
ejpam-3317	124	9	→	→	PUNCT
ejpam-3317	124	10	(	(	PUNCT
ejpam-3317	124	11	x	x	X
ejpam-3317	124	12	,	,	PUNCT
ejpam-3317	124	13	c2	c2	PROPN
ejpam-3317	124	14	)	)	PUNCT
ejpam-3317	124	15	be	be	VERB
ejpam-3317	124	16	a	a	DET
ejpam-3317	124	17	function	function	NOUN
ejpam-3317	124	18	.	.	PUNCT
ejpam-3317	125	1	then	then	ADV
ejpam-3317	125	2	f	f	PROPN
ejpam-3317	125	3	is	be	AUX
ejpam-3317	125	4	pre	pre	ADJ
ejpam-3317	125	5	-	-	ADJ
ejpam-3317	125	6	open	open	ADJ
ejpam-3317	125	7	(	(	PUNCT
ejpam-3317	125	8	or	or	CCONJ
ejpam-3317	125	9	preopen	preopen	ADJ
ejpam-3317	125	10	)	)	PUNCT
ejpam-3317	125	11	if	if	SCONJ
ejpam-3317	125	12	f	f	PROPN
ejpam-3317	125	13	(	(	PUNCT
ejpam-3317	125	14	g	g	NOUN
ejpam-3317	125	15	)	)	PUNCT
ejpam-3317	125	16	is	be	AUX
ejpam-3317	125	17	preopen	preopen	ADJ
ejpam-3317	125	18	in	in	ADP
ejpam-3317	125	19	y	y	PROPN
ejpam-3317	125	20	,	,	PUNCT
ejpam-3317	125	21	for	for	ADP
ejpam-3317	125	22	every	every	DET
ejpam-3317	125	23	open	open	NOUN
ejpam-3317	125	24	subset	subset	NOUN
ejpam-3317	125	25	g	g	NOUN
ejpam-3317	125	26	of	of	ADP
ejpam-3317	125	27	x.	x.	NOUN
ejpam-3317	125	28	definition	definition	NOUN
ejpam-3317	125	29	12	12	NUM
ejpam-3317	125	30	.	.	PUNCT
ejpam-3317	126	1	let	let	VERB
ejpam-3317	126	2	(	(	PUNCT
ejpam-3317	126	3	x	x	NOUN
ejpam-3317	126	4	,	,	PUNCT
ejpam-3317	126	5	c1	c1	PROPN
ejpam-3317	126	6	)	)	PUNCT
ejpam-3317	126	7	and	and	CCONJ
ejpam-3317	126	8	(	(	PUNCT
ejpam-3317	126	9	y	y	PROPN
ejpam-3317	126	10	,	,	PUNCT
ejpam-3317	126	11	c2	c2	PROPN
ejpam-3317	126	12	)	)	PUNCT
ejpam-3317	126	13	be	be	VERB
ejpam-3317	126	14	two	two	NUM
ejpam-3317	126	15	closure	closure	NOUN
ejpam-3317	126	16	spaces	space	NOUN
ejpam-3317	126	17	.	.	PUNCT
ejpam-3317	127	1	let	let	VERB
ejpam-3317	127	2	f	f	NOUN
ejpam-3317	127	3	:	:	PUNCT
ejpam-3317	127	4	(	(	PUNCT
ejpam-3317	127	5	x	x	X
ejpam-3317	127	6	,	,	PUNCT
ejpam-3317	127	7	c1	c1	PROPN
ejpam-3317	127	8	)	)	PUNCT
ejpam-3317	127	9	→	→	PUNCT
ejpam-3317	127	10	(	(	PUNCT
ejpam-3317	127	11	x	x	X
ejpam-3317	127	12	,	,	PUNCT
ejpam-3317	127	13	c2	c2	PROPN
ejpam-3317	127	14	)	)	PUNCT
ejpam-3317	127	15	be	be	VERB
ejpam-3317	127	16	a	a	DET
ejpam-3317	127	17	function	function	NOUN
ejpam-3317	127	18	.	.	PUNCT
ejpam-3317	128	1	then	then	ADV
ejpam-3317	128	2	f	f	PROPN
ejpam-3317	128	3	is	be	AUX
ejpam-3317	128	4	pre	pre	ADJ
ejpam-3317	128	5	-	-	ADJ
ejpam-3317	128	6	closed	closed	ADJ
ejpam-3317	128	7	(	(	PUNCT
ejpam-3317	128	8	or	or	CCONJ
ejpam-3317	128	9	preclosed	preclose	VERB
ejpam-3317	128	10	)	)	PUNCT
ejpam-3317	128	11	if	if	SCONJ
ejpam-3317	128	12	f	f	PROPN
ejpam-3317	128	13	(	(	PUNCT
ejpam-3317	128	14	k	k	NOUN
ejpam-3317	128	15	)	)	PUNCT
ejpam-3317	128	16	is	be	AUX
ejpam-3317	128	17	preclosed	preclose	VERB
ejpam-3317	128	18	in	in	ADP
ejpam-3317	128	19	y	y	PROPN
ejpam-3317	128	20	,	,	PUNCT
ejpam-3317	128	21	for	for	ADP
ejpam-3317	128	22	every	every	DET
ejpam-3317	128	23	closed	close	VERB
ejpam-3317	128	24	subset	subset	NOUN
ejpam-3317	129	1	k	k	PROPN
ejpam-3317	129	2	of	of	ADP
ejpam-3317	129	3	x.	x.	PROPN
ejpam-3317	129	4	proposition	proposition	PROPN
ejpam-3317	129	5	17	17	NUM
ejpam-3317	129	6	.	.	PUNCT
ejpam-3317	130	1	let	let	AUX
ejpam-3317	130	2	(	(	PUNCT
ejpam-3317	130	3	x	x	X
ejpam-3317	130	4	,	,	PUNCT
ejpam-3317	130	5	c1	c1	PROPN
ejpam-3317	130	6	)	)	PUNCT
ejpam-3317	130	7	,	,	PUNCT
ejpam-3317	130	8	(	(	PUNCT
ejpam-3317	130	9	y	y	PROPN
ejpam-3317	130	10	,	,	PUNCT
ejpam-3317	130	11	c2	c2	PROPN
ejpam-3317	130	12	)	)	PUNCT
ejpam-3317	130	13	and	and	CCONJ
ejpam-3317	130	14	(	(	PUNCT
ejpam-3317	130	15	z	z	NOUN
ejpam-3317	130	16	,	,	PUNCT
ejpam-3317	130	17	c3	c3	PROPN
ejpam-3317	130	18	)	)	PUNCT
ejpam-3317	130	19	be	be	VERB
ejpam-3317	130	20	closure	closure	NOUN
ejpam-3317	130	21	spaces	space	NOUN
ejpam-3317	130	22	,	,	PUNCT
ejpam-3317	130	23	let	let	VERB
ejpam-3317	130	24	f	f	PRON
ejpam-3317	130	25	:	:	PUNCT
ejpam-3317	130	26	(	(	PUNCT
ejpam-3317	130	27	x	x	X
ejpam-3317	130	28	,	,	PUNCT
ejpam-3317	130	29	c1)→	c1)→	NOUN
ejpam-3317	130	30	(	(	PUNCT
ejpam-3317	130	31	y	y	PROPN
ejpam-3317	130	32	,	,	PUNCT
ejpam-3317	130	33	c2	c2	PROPN
ejpam-3317	130	34	)	)	PUNCT
ejpam-3317	130	35	and	and	CCONJ
ejpam-3317	130	36	g	g	NOUN
ejpam-3317	130	37	:	:	PUNCT
ejpam-3317	130	38	(	(	PUNCT
ejpam-3317	130	39	y	y	NOUN
ejpam-3317	130	40	,	,	PUNCT
ejpam-3317	130	41	c2)→	c2)→	X
ejpam-3317	130	42	(	(	PUNCT
ejpam-3317	130	43	z	z	NOUN
ejpam-3317	130	44	,	,	PUNCT
ejpam-3317	130	45	c3	c3	PROPN
ejpam-3317	130	46	)	)	PUNCT
ejpam-3317	130	47	be	be	AUX
ejpam-3317	130	48	functions	function	NOUN
ejpam-3317	130	49	.	.	PUNCT
ejpam-3317	131	1	then	then	ADV
ejpam-3317	131	2	:	:	PUNCT
ejpam-3317	131	3	(	(	PUNCT
ejpam-3317	131	4	i	i	NOUN
ejpam-3317	131	5	)	)	PUNCT
ejpam-3317	131	6	if	if	SCONJ
ejpam-3317	131	7	f	f	PROPN
ejpam-3317	131	8	is	be	AUX
ejpam-3317	131	9	open	open	ADJ
ejpam-3317	131	10	and	and	CCONJ
ejpam-3317	131	11	g	g	NOUN
ejpam-3317	131	12	is	be	AUX
ejpam-3317	131	13	preopen	preopen	ADJ
ejpam-3317	131	14	,	,	PUNCT
ejpam-3317	131	15	then	then	ADV
ejpam-3317	131	16	gof	gof	NOUN
ejpam-3317	131	17	is	be	AUX
ejpam-3317	131	18	preopen	preopen	ADJ
ejpam-3317	131	19	.	.	PUNCT
ejpam-3317	132	1	(	(	PUNCT
ejpam-3317	132	2	ii	ii	NOUN
ejpam-3317	132	3	)	)	PUNCT
ejpam-3317	132	4	if	if	SCONJ
ejpam-3317	132	5	gof	gof	NOUN
ejpam-3317	132	6	is	be	AUX
ejpam-3317	132	7	preopen	preopen	ADJ
ejpam-3317	132	8	and	and	CCONJ
ejpam-3317	132	9	f	f	PROPN
ejpam-3317	132	10	is	be	AUX
ejpam-3317	132	11	a	a	DET
ejpam-3317	132	12	continuous	continuous	ADJ
ejpam-3317	132	13	surjection	surjection	NOUN
ejpam-3317	132	14	,	,	PUNCT
ejpam-3317	132	15	then	then	ADV
ejpam-3317	132	16	g	g	PROPN
ejpam-3317	132	17	is	be	AUX
ejpam-3317	132	18	preopen	preopen	ADJ
ejpam-3317	132	19	.	.	PUNCT
ejpam-3317	133	1	h.m	h.m	PROPN
ejpam-3317	133	2	.	.	PROPN
ejpam-3317	133	3	darwesh	darwesh	PROPN
ejpam-3317	133	4	,	,	PUNCT
ejpam-3317	133	5	s.f	s.f	PROPN
ejpam-3317	133	6	.	.	PROPN
ejpam-3317	133	7	namiq	namiq	PROPN
ejpam-3317	133	8	/	/	SYM
ejpam-3317	133	9	eur	eur	PROPN
ejpam-3317	133	10	.	.	PUNCT
ejpam-3317	134	1	j.	j.	PROPN
ejpam-3317	134	2	pure	pure	PROPN
ejpam-3317	134	3	appl	appl	PROPN
ejpam-3317	134	4	.	.	PROPN
ejpam-3317	134	5	math	math	PROPN
ejpam-3317	134	6	,	,	PUNCT
ejpam-3317	134	7	12	12	NUM
ejpam-3317	134	8	(	(	PUNCT
ejpam-3317	134	9	3	3	NUM
ejpam-3317	134	10	)	)	PUNCT
ejpam-3317	134	11	(	(	PUNCT
ejpam-3317	134	12	2019	2019	NUM
ejpam-3317	134	13	)	)	PUNCT
ejpam-3317	134	14	,	,	PUNCT
ejpam-3317	134	15	1082	1082	NUM
ejpam-3317	134	16	-	-	SYM
ejpam-3317	134	17	1095	1095	NUM
ejpam-3317	134	18	1087	1087	NUM
ejpam-3317	134	19	proof	proof	NOUN
ejpam-3317	134	20	.	.	PUNCT
ejpam-3317	135	1	(	(	PUNCT
ejpam-3317	135	2	i	i	NOUN
ejpam-3317	135	3	)	)	PUNCT
ejpam-3317	135	4	let	let	VERB
ejpam-3317	135	5	g	g	NOUN
ejpam-3317	135	6	be	be	AUX
ejpam-3317	135	7	an	an	DET
ejpam-3317	135	8	open	open	ADJ
ejpam-3317	135	9	subset	subset	NOUN
ejpam-3317	135	10	of	of	ADP
ejpam-3317	135	11	(	(	PUNCT
ejpam-3317	135	12	x	x	NOUN
ejpam-3317	135	13	,	,	PUNCT
ejpam-3317	135	14	c1	c1	PROPN
ejpam-3317	135	15	)	)	PUNCT
ejpam-3317	135	16	.	.	PUNCT
ejpam-3317	136	1	since	since	SCONJ
ejpam-3317	136	2	f	f	PROPN
ejpam-3317	136	3	is	be	AUX
ejpam-3317	136	4	open	open	ADJ
ejpam-3317	136	5	,	,	PUNCT
ejpam-3317	136	6	f(g	f(g	NOUN
ejpam-3317	136	7	)	)	PUNCT
ejpam-3317	136	8	is	be	AUX
ejpam-3317	136	9	open	open	ADJ
ejpam-3317	136	10	in	in	ADP
ejpam-3317	136	11	(	(	PUNCT
ejpam-3317	136	12	y	y	PROPN
ejpam-3317	136	13	,	,	PUNCT
ejpam-3317	136	14	c2	c2	PROPN
ejpam-3317	136	15	)	)	PUNCT
ejpam-3317	136	16	.	.	PUNCT
ejpam-3317	137	1	hence	hence	ADV
ejpam-3317	137	2	g(f(g	g(f(g	NOUN
ejpam-3317	137	3	)	)	PUNCT
ejpam-3317	137	4	)	)	PUNCT
ejpam-3317	137	5	is	be	AUX
ejpam-3317	137	6	preopen	preopen	ADJ
ejpam-3317	137	7	in	in	ADP
ejpam-3317	137	8	(	(	PUNCT
ejpam-3317	137	9	z	z	PROPN
ejpam-3317	137	10	,	,	PUNCT
ejpam-3317	137	11	c3	c3	PROPN
ejpam-3317	137	12	)	)	PUNCT
ejpam-3317	137	13	.	.	PUNCT
ejpam-3317	138	1	thus	thus	ADV
ejpam-3317	138	2	,	,	PUNCT
ejpam-3317	138	3	gof	gof	NOUN
ejpam-3317	138	4	is	be	AUX
ejpam-3317	138	5	preopen	preopen	ADJ
ejpam-3317	138	6	.	.	PUNCT
ejpam-3317	139	1	(	(	PUNCT
ejpam-3317	139	2	ii	ii	NOUN
ejpam-3317	139	3	)	)	PUNCT
ejpam-3317	139	4	let	let	VERB
ejpam-3317	139	5	g	g	NOUN
ejpam-3317	139	6	be	be	AUX
ejpam-3317	139	7	an	an	DET
ejpam-3317	139	8	open	open	ADJ
ejpam-3317	139	9	subset	subset	NOUN
ejpam-3317	139	10	of	of	ADP
ejpam-3317	139	11	(	(	PUNCT
ejpam-3317	139	12	y	y	PROPN
ejpam-3317	139	13	,	,	PUNCT
ejpam-3317	139	14	c2	c2	PROPN
ejpam-3317	139	15	)	)	PUNCT
ejpam-3317	139	16	.	.	PUNCT
ejpam-3317	140	1	since	since	SCONJ
ejpam-3317	140	2	f	f	PROPN
ejpam-3317	140	3	is	be	AUX
ejpam-3317	140	4	a	a	DET
ejpam-3317	140	5	continuous	continuous	ADJ
ejpam-3317	140	6	function	function	NOUN
ejpam-3317	140	7	,	,	PUNCT
ejpam-3317	140	8	f−1(g	f−1(g	PROPN
ejpam-3317	140	9	)	)	PUNCT
ejpam-3317	140	10	is	be	AUX
ejpam-3317	140	11	open	open	ADJ
ejpam-3317	140	12	in	in	ADP
ejpam-3317	140	13	(	(	PUNCT
ejpam-3317	140	14	x	x	INTJ
ejpam-3317	140	15	,	,	PUNCT
ejpam-3317	140	16	c1	c1	PROPN
ejpam-3317	140	17	)	)	PUNCT
ejpam-3317	140	18	.	.	PUNCT
ejpam-3317	141	1	since	since	SCONJ
ejpam-3317	141	2	gof	gof	NOUN
ejpam-3317	141	3	is	be	AUX
ejpam-3317	141	4	preopen	preopen	ADJ
ejpam-3317	141	5	,	,	PUNCT
ejpam-3317	141	6	gof(f−1(g	gof(f−1(g	NOUN
ejpam-3317	141	7	)	)	PUNCT
ejpam-3317	141	8	)	)	PUNCT
ejpam-3317	142	1	=	=	SYM
ejpam-3317	142	2	g(f(f−1(g	g(f(f−1(g	NOUN
ejpam-3317	142	3	)	)	PUNCT
ejpam-3317	142	4	)	)	PUNCT
ejpam-3317	142	5	)	)	PUNCT
ejpam-3317	142	6	is	be	AUX
ejpam-3317	142	7	preopen	preopen	ADJ
ejpam-3317	142	8	in	in	ADP
ejpam-3317	142	9	(	(	PUNCT
ejpam-3317	142	10	z	z	PROPN
ejpam-3317	142	11	,	,	PUNCT
ejpam-3317	142	12	c3	c3	PROPN
ejpam-3317	142	13	)	)	PUNCT
ejpam-3317	142	14	.	.	PUNCT
ejpam-3317	143	1	but	but	CCONJ
ejpam-3317	143	2	f	f	PROPN
ejpam-3317	143	3	is	be	AUX
ejpam-3317	143	4	surjection	surjection	NOUN
ejpam-3317	143	5	,	,	PUNCT
ejpam-3317	143	6	so	so	SCONJ
ejpam-3317	143	7	that	that	DET
ejpam-3317	143	8	gof(f−1(g	gof(f−1(g	NOUN
ejpam-3317	143	9	)	)	PUNCT
ejpam-3317	143	10	)	)	PUNCT
ejpam-3317	144	1	=	=	SYM
ejpam-3317	144	2	g(g	g(g	PROPN
ejpam-3317	144	3	)	)	PUNCT
ejpam-3317	144	4	.	.	PUNCT
ejpam-3317	145	1	hence	hence	ADV
ejpam-3317	145	2	,	,	PUNCT
ejpam-3317	145	3	g(g	g(g	PROPN
ejpam-3317	145	4	)	)	PUNCT
ejpam-3317	145	5	is	be	AUX
ejpam-3317	145	6	preopen	preopen	ADJ
ejpam-3317	145	7	in	in	ADP
ejpam-3317	145	8	(	(	PUNCT
ejpam-3317	145	9	z	z	PROPN
ejpam-3317	145	10	,	,	PUNCT
ejpam-3317	145	11	c3	c3	PROPN
ejpam-3317	145	12	)	)	PUNCT
ejpam-3317	145	13	.	.	PUNCT
ejpam-3317	146	1	therefore	therefore	ADV
ejpam-3317	146	2	,	,	PUNCT
ejpam-3317	146	3	g	g	PROPN
ejpam-3317	146	4	is	be	AUX
ejpam-3317	146	5	preopen	preopen	ADJ
ejpam-3317	146	6	.	.	PUNCT
ejpam-3317	147	1	proposition	proposition	NOUN
ejpam-3317	147	2	18	18	NUM
ejpam-3317	147	3	.	.	PUNCT
ejpam-3317	148	1	let	let	VERB
ejpam-3317	148	2	(	(	PUNCT
ejpam-3317	148	3	x	x	NOUN
ejpam-3317	148	4	,	,	PUNCT
ejpam-3317	148	5	c1	c1	PROPN
ejpam-3317	148	6	)	)	PUNCT
ejpam-3317	148	7	,	,	PUNCT
ejpam-3317	148	8	(	(	PUNCT
ejpam-3317	148	9	y	y	PROPN
ejpam-3317	148	10	,	,	PUNCT
ejpam-3317	148	11	c2	c2	PROPN
ejpam-3317	148	12	)	)	PUNCT
ejpam-3317	148	13	and	and	CCONJ
ejpam-3317	148	14	(	(	PUNCT
ejpam-3317	148	15	z	z	NOUN
ejpam-3317	148	16	,	,	PUNCT
ejpam-3317	148	17	c3	c3	PROPN
ejpam-3317	148	18	)	)	PUNCT
ejpam-3317	148	19	be	be	VERB
ejpam-3317	148	20	closure	closure	NOUN
ejpam-3317	148	21	spaces	space	NOUN
ejpam-3317	148	22	,	,	PUNCT
ejpam-3317	148	23	let	let	VERB
ejpam-3317	148	24	f	f	PRON
ejpam-3317	148	25	:	:	PUNCT
ejpam-3317	148	26	(	(	PUNCT
ejpam-3317	148	27	x	x	X
ejpam-3317	148	28	,	,	PUNCT
ejpam-3317	148	29	c1)→	c1)→	NOUN
ejpam-3317	148	30	(	(	PUNCT
ejpam-3317	148	31	y	y	PROPN
ejpam-3317	148	32	,	,	PUNCT
ejpam-3317	148	33	c2	c2	PROPN
ejpam-3317	148	34	)	)	PUNCT
ejpam-3317	148	35	and	and	CCONJ
ejpam-3317	148	36	g	g	NOUN
ejpam-3317	148	37	:	:	PUNCT
ejpam-3317	148	38	(	(	PUNCT
ejpam-3317	148	39	y	y	PROPN
ejpam-3317	148	40	,	,	PUNCT
ejpam-3317	148	41	c2	c2	PROPN
ejpam-3317	148	42	)	)	PUNCT
ejpam-3317	148	43	→	→	PUNCT
ejpam-3317	148	44	(	(	PUNCT
ejpam-3317	148	45	z	z	NOUN
ejpam-3317	148	46	,	,	PUNCT
ejpam-3317	148	47	c3	c3	PROPN
ejpam-3317	148	48	)	)	PUNCT
ejpam-3317	148	49	be	be	AUX
ejpam-3317	148	50	functions	function	NOUN
ejpam-3317	148	51	.	.	PUNCT
ejpam-3317	149	1	if	if	SCONJ
ejpam-3317	149	2	gof	gof	NOUN
ejpam-3317	149	3	is	be	AUX
ejpam-3317	149	4	open	open	ADJ
ejpam-3317	149	5	and	and	CCONJ
ejpam-3317	149	6	g	g	NOUN
ejpam-3317	149	7	is	be	AUX
ejpam-3317	149	8	a	a	DET
ejpam-3317	149	9	pre	pre	ADJ
ejpam-3317	149	10	-	-	ADJ
ejpam-3317	149	11	continuous	continuous	ADJ
ejpam-3317	149	12	injection	injection	NOUN
ejpam-3317	149	13	,	,	PUNCT
ejpam-3317	149	14	then	then	ADV
ejpam-3317	149	15	f	f	PROPN
ejpam-3317	149	16	is	be	AUX
ejpam-3317	149	17	preopen	preopen	ADJ
ejpam-3317	149	18	.	.	PUNCT
ejpam-3317	150	1	proof	proof	NOUN
ejpam-3317	150	2	.	.	PUNCT
ejpam-3317	151	1	let	let	VERB
ejpam-3317	151	2	g	g	PRON
ejpam-3317	151	3	be	be	AUX
ejpam-3317	151	4	an	an	DET
ejpam-3317	151	5	open	open	ADJ
ejpam-3317	151	6	subset	subset	NOUN
ejpam-3317	151	7	of	of	ADP
ejpam-3317	151	8	(	(	PUNCT
ejpam-3317	151	9	x	x	NOUN
ejpam-3317	151	10	,	,	PUNCT
ejpam-3317	151	11	c1	c1	PROPN
ejpam-3317	151	12	)	)	PUNCT
ejpam-3317	151	13	.	.	PUNCT
ejpam-3317	152	1	since	since	SCONJ
ejpam-3317	152	2	gof	gof	NOUN
ejpam-3317	152	3	is	be	AUX
ejpam-3317	152	4	open	open	ADJ
ejpam-3317	152	5	,	,	PUNCT
ejpam-3317	152	6	g(f(g	g(f(g	NOUN
ejpam-3317	152	7	)	)	PUNCT
ejpam-3317	152	8	)	)	PUNCT
ejpam-3317	152	9	is	be	AUX
ejpam-3317	152	10	open	open	ADJ
ejpam-3317	152	11	in	in	ADP
ejpam-3317	152	12	(	(	PUNCT
ejpam-3317	152	13	z	z	NOUN
ejpam-3317	152	14	,	,	PUNCT
ejpam-3317	152	15	c3	c3	PROPN
ejpam-3317	152	16	)	)	PUNCT
ejpam-3317	152	17	.	.	PUNCT
ejpam-3317	153	1	as	as	SCONJ
ejpam-3317	153	2	g	g	PROPN
ejpam-3317	153	3	is	be	AUX
ejpam-3317	153	4	pre	pre	ADJ
ejpam-3317	153	5	-	-	ADJ
ejpam-3317	153	6	continuous	continuous	ADJ
ejpam-3317	153	7	,	,	PUNCT
ejpam-3317	153	8	g−1(g(f	g−1(g(f	NOUN
ejpam-3317	153	9	(	(	PUNCT
ejpam-3317	153	10	g	g	NOUN
ejpam-3317	153	11	)	)	PUNCT
ejpam-3317	153	12	)	)	PUNCT
ejpam-3317	153	13	)	)	PUNCT
ejpam-3317	153	14	is	be	AUX
ejpam-3317	153	15	preopen	preopen	ADJ
ejpam-3317	153	16	in	in	ADP
ejpam-3317	153	17	(	(	PUNCT
ejpam-3317	153	18	y	y	PROPN
ejpam-3317	153	19	,	,	PUNCT
ejpam-3317	153	20	c2	c2	PROPN
ejpam-3317	153	21	)	)	PUNCT
ejpam-3317	153	22	.	.	PUNCT
ejpam-3317	154	1	but	but	CCONJ
ejpam-3317	154	2	g	g	PROPN
ejpam-3317	154	3	is	be	AUX
ejpam-3317	154	4	injective	injective	ADJ
ejpam-3317	154	5	,	,	PUNCT
ejpam-3317	154	6	so	so	SCONJ
ejpam-3317	154	7	that	that	SCONJ
ejpam-3317	154	8	g−1(g	g−1(g	PROPN
ejpam-3317	154	9	(	(	PUNCT
ejpam-3317	154	10	f	f	PROPN
ejpam-3317	154	11	(	(	PUNCT
ejpam-3317	154	12	g	g	NOUN
ejpam-3317	154	13	)	)	PUNCT
ejpam-3317	154	14	)	)	PUNCT
ejpam-3317	154	15	)	)	PUNCT
ejpam-3317	155	1	=	=	SYM
ejpam-3317	155	2	f	f	X
ejpam-3317	155	3	(	(	PUNCT
ejpam-3317	155	4	g	g	NOUN
ejpam-3317	155	5	)	)	PUNCT
ejpam-3317	155	6	is	be	AUX
ejpam-3317	155	7	preopen	preopen	ADJ
ejpam-3317	155	8	in	in	ADP
ejpam-3317	155	9	(	(	PUNCT
ejpam-3317	155	10	y	y	PROPN
ejpam-3317	155	11	,	,	PUNCT
ejpam-3317	155	12	c2	c2	PROPN
ejpam-3317	155	13	)	)	PUNCT
ejpam-3317	155	14	.	.	PUNCT
ejpam-3317	156	1	therefore	therefore	ADV
ejpam-3317	156	2	,	,	PUNCT
ejpam-3317	156	3	f	f	PROPN
ejpam-3317	156	4	is	be	AUX
ejpam-3317	156	5	preopen	preopen	ADJ
ejpam-3317	156	6	.	.	PUNCT
ejpam-3317	157	1	proposition	proposition	NOUN
ejpam-3317	157	2	19	19	NUM
ejpam-3317	157	3	.	.	PUNCT
ejpam-3317	158	1	let	let	AUX
ejpam-3317	158	2	(	(	PUNCT
ejpam-3317	158	3	x	x	NOUN
ejpam-3317	158	4	,	,	PUNCT
ejpam-3317	158	5	c1	c1	PROPN
ejpam-3317	158	6	)	)	PUNCT
ejpam-3317	158	7	and	and	CCONJ
ejpam-3317	158	8	(	(	PUNCT
ejpam-3317	158	9	y	y	PROPN
ejpam-3317	158	10	,	,	PUNCT
ejpam-3317	158	11	c2	c2	PROPN
ejpam-3317	158	12	)	)	PUNCT
ejpam-3317	158	13	be	be	VERB
ejpam-3317	158	14	closure	closure	NOUN
ejpam-3317	158	15	spaces	space	NOUN
ejpam-3317	158	16	.	.	PUNCT
ejpam-3317	159	1	if	if	SCONJ
ejpam-3317	159	2	f	f	PROPN
ejpam-3317	159	3	:	:	PUNCT
ejpam-3317	159	4	(	(	PUNCT
ejpam-3317	159	5	x	x	X
ejpam-3317	159	6	,	,	PUNCT
ejpam-3317	159	7	c1	c1	PROPN
ejpam-3317	159	8	)	)	PUNCT
ejpam-3317	159	9	→	→	PUNCT
ejpam-3317	159	10	(	(	PUNCT
ejpam-3317	159	11	y	y	PROPN
ejpam-3317	159	12	,	,	PUNCT
ejpam-3317	159	13	c2	c2	PROPN
ejpam-3317	159	14	)	)	PUNCT
ejpam-3317	159	15	is	be	AUX
ejpam-3317	159	16	a	a	DET
ejpam-3317	159	17	bijection	bijection	NOUN
ejpam-3317	159	18	,	,	PUNCT
ejpam-3317	159	19	then	then	ADV
ejpam-3317	159	20	the	the	DET
ejpam-3317	159	21	following	following	ADJ
ejpam-3317	159	22	statements	statement	NOUN
ejpam-3317	159	23	are	be	AUX
ejpam-3317	159	24	equivalent	equivalent	ADJ
ejpam-3317	159	25	:	:	PUNCT
ejpam-3317	159	26	(	(	PUNCT
ejpam-3317	159	27	i	i	NOUN
ejpam-3317	159	28	)	)	PUNCT
ejpam-3317	159	29	the	the	DET
ejpam-3317	159	30	inverse	inverse	NOUN
ejpam-3317	159	31	function	function	NOUN
ejpam-3317	159	32	f−1	f−1	PROPN
ejpam-3317	159	33	:	:	PUNCT
ejpam-3317	159	34	(	(	PUNCT
ejpam-3317	159	35	y	y	NOUN
ejpam-3317	159	36	,	,	PUNCT
ejpam-3317	159	37	c2)→	c2)→	X
ejpam-3317	159	38	(	(	PUNCT
ejpam-3317	159	39	x	x	NOUN
ejpam-3317	159	40	,	,	PUNCT
ejpam-3317	159	41	c1	c1	PROPN
ejpam-3317	159	42	)	)	PUNCT
ejpam-3317	159	43	is	be	AUX
ejpam-3317	159	44	pre	pre	ADJ
ejpam-3317	159	45	-	-	ADJ
ejpam-3317	159	46	continuous	continuous	ADJ
ejpam-3317	159	47	.	.	PUNCT
ejpam-3317	160	1	(	(	PUNCT
ejpam-3317	160	2	ii	ii	NOUN
ejpam-3317	160	3	)	)	PUNCT
ejpam-3317	160	4	f	f	PROPN
ejpam-3317	160	5	is	be	AUX
ejpam-3317	160	6	a	a	DET
ejpam-3317	160	7	preopen	preopen	ADJ
ejpam-3317	160	8	function	function	NOUN
ejpam-3317	160	9	.	.	PUNCT
ejpam-3317	161	1	(	(	PUNCT
ejpam-3317	161	2	iii	iii	X
ejpam-3317	161	3	)	)	PUNCT
ejpam-3317	161	4	f	f	PROPN
ejpam-3317	161	5	is	be	AUX
ejpam-3317	161	6	a	a	DET
ejpam-3317	161	7	preclosed	preclose	VERB
ejpam-3317	161	8	function	function	NOUN
ejpam-3317	161	9	.	.	PUNCT
ejpam-3317	162	1	proof	proof	NOUN
ejpam-3317	162	2	.	.	PUNCT
ejpam-3317	163	1	obvious	obvious	ADJ
ejpam-3317	163	2	definition	definition	NOUN
ejpam-3317	163	3	13	13	NUM
ejpam-3317	163	4	.	.	PUNCT
ejpam-3317	164	1	a	a	DET
ejpam-3317	164	2	closure	closure	NOUN
ejpam-3317	164	3	space	space	NOUN
ejpam-3317	164	4	(	(	PUNCT
ejpam-3317	164	5	x	x	X
ejpam-3317	164	6	,	,	PUNCT
ejpam-3317	164	7	c	c	NOUN
ejpam-3317	164	8	)	)	PUNCT
ejpam-3317	164	9	is	be	AUX
ejpam-3317	164	10	said	say	VERB
ejpam-3317	164	11	to	to	PART
ejpam-3317	164	12	be	be	AUX
ejpam-3317	164	13	a	a	DET
ejpam-3317	164	14	tp	tp	NOUN
ejpam-3317	164	15	-	-	PUNCT
ejpam-3317	164	16	space	space	NOUN
ejpam-3317	164	17	if	if	SCONJ
ejpam-3317	164	18	every	every	DET
ejpam-3317	164	19	preopen	preopen	NOUN
ejpam-3317	164	20	set	set	VERB
ejpam-3317	164	21	in	in	ADP
ejpam-3317	164	22	(	(	PUNCT
ejpam-3317	164	23	x	x	NOUN
ejpam-3317	164	24	,	,	PUNCT
ejpam-3317	164	25	c	c	NOUN
ejpam-3317	164	26	)	)	PUNCT
ejpam-3317	164	27	is	be	AUX
ejpam-3317	164	28	open	open	ADJ
ejpam-3317	164	29	.	.	PUNCT
ejpam-3317	165	1	the	the	DET
ejpam-3317	165	2	closure	closure	NOUN
ejpam-3317	165	3	space	space	NOUN
ejpam-3317	165	4	in	in	ADP
ejpam-3317	165	5	the	the	DET
ejpam-3317	165	6	following	follow	VERB
ejpam-3317	165	7	example	example	NOUN
ejpam-3317	165	8	is	be	AUX
ejpam-3317	165	9	a	a	DET
ejpam-3317	165	10	tp	tp	NOUN
ejpam-3317	165	11	-	-	NOUN
ejpam-3317	165	12	space	space	NOUN
ejpam-3317	165	13	.	.	PUNCT
ejpam-3317	165	14	example	example	NOUN
ejpam-3317	166	1	1	1	NUM
ejpam-3317	166	2	.	.	PUNCT
ejpam-3317	166	3	let	let	VERB
ejpam-3317	166	4	x	x	PUNCT
ejpam-3317	166	5	=	=	PRON
ejpam-3317	166	6	{	{	PUNCT
ejpam-3317	166	7	1	1	NUM
ejpam-3317	166	8	,	,	PUNCT
ejpam-3317	166	9	2	2	NUM
ejpam-3317	166	10	,	,	PUNCT
ejpam-3317	166	11	3	3	NUM
ejpam-3317	166	12	}	}	PUNCT
ejpam-3317	166	13	and	and	CCONJ
ejpam-3317	166	14	defined	define	VERB
ejpam-3317	166	15	a	a	DET
ejpam-3317	166	16	closure	closure	NOUN
ejpam-3317	166	17	operator	operator	NOUN
ejpam-3317	166	18	c	c	NOUN
ejpam-3317	166	19	:	:	PUNCT
ejpam-3317	166	20	p	p	X
ejpam-3317	166	21	(	(	PUNCT
ejpam-3317	166	22	x)→	x)→	PROPN
ejpam-3317	166	23	p	p	X
ejpam-3317	166	24	(	(	PUNCT
ejpam-3317	166	25	x	x	NOUN
ejpam-3317	166	26	)	)	PUNCT
ejpam-3317	166	27	by	by	ADP
ejpam-3317	166	28	:	:	PUNCT
ejpam-3317	166	29	c	c	X
ejpam-3317	166	30	(	(	PUNCT
ejpam-3317	166	31	a	a	X
ejpam-3317	166	32	)	)	PUNCT
ejpam-3317	166	33	=	=	PRON
ejpam-3317	166	34	{	{	PUNCT
ejpam-3317	166	35	a	a	X
ejpam-3317	166	36	if	if	SCONJ
ejpam-3317	166	37	a	a	DET
ejpam-3317	166	38	∈	∈	PROPN
ejpam-3317	166	39	{	{	PUNCT
ejpam-3317	166	40	φ	φ	NOUN
ejpam-3317	166	41	,	,	PUNCT
ejpam-3317	166	42	{	{	PUNCT
ejpam-3317	166	43	1	1	NUM
ejpam-3317	166	44	}	}	PUNCT
ejpam-3317	166	45	,	,	PUNCT
ejpam-3317	166	46	{	{	PUNCT
ejpam-3317	166	47	2	2	NUM
ejpam-3317	166	48	}	}	PUNCT
ejpam-3317	166	49	,	,	PUNCT
ejpam-3317	166	50	{	{	PUNCT
ejpam-3317	166	51	3	3	NUM
ejpam-3317	166	52	}	}	PUNCT
ejpam-3317	166	53	}	}	PUNCT
ejpam-3317	166	54	x	x	SYM
ejpam-3317	166	55	otherwise	otherwise	ADV
ejpam-3317	166	56	clearly	clearly	ADV
ejpam-3317	166	57	(	(	PUNCT
ejpam-3317	166	58	x	x	NOUN
ejpam-3317	166	59	,	,	PUNCT
ejpam-3317	166	60	c	c	NOUN
ejpam-3317	166	61	)	)	PUNCT
ejpam-3317	166	62	is	be	AUX
ejpam-3317	166	63	a	a	DET
ejpam-3317	166	64	tp	tp	NOUN
ejpam-3317	166	65	-	-	NOUN
ejpam-3317	166	66	space	space	NOUN
ejpam-3317	166	67	,	,	PUNCT
ejpam-3317	166	68	since	since	SCONJ
ejpam-3317	166	69	every	every	DET
ejpam-3317	166	70	preopen	preopen	ADJ
ejpam-3317	166	71	set	set	NOUN
ejpam-3317	166	72	is	be	AUX
ejpam-3317	166	73	open	open	ADJ
ejpam-3317	166	74	set	set	VERB
ejpam-3317	166	75	proposition	proposition	NOUN
ejpam-3317	166	76	20	20	NUM
ejpam-3317	166	77	.	.	PUNCT
ejpam-3317	167	1	let	let	AUX
ejpam-3317	167	2	(	(	PUNCT
ejpam-3317	167	3	x	x	NOUN
ejpam-3317	167	4	,	,	PUNCT
ejpam-3317	167	5	c1	c1	PROPN
ejpam-3317	167	6	)	)	PUNCT
ejpam-3317	167	7	and	and	CCONJ
ejpam-3317	167	8	(	(	PUNCT
ejpam-3317	167	9	y	y	PROPN
ejpam-3317	167	10	,	,	PUNCT
ejpam-3317	167	11	c2	c2	PROPN
ejpam-3317	167	12	)	)	PUNCT
ejpam-3317	167	13	be	be	VERB
ejpam-3317	167	14	closure	closure	NOUN
ejpam-3317	167	15	spaces	space	NOUN
ejpam-3317	167	16	and	and	CCONJ
ejpam-3317	167	17	(	(	PUNCT
ejpam-3317	167	18	y	y	PROPN
ejpam-3317	167	19	,	,	PUNCT
ejpam-3317	167	20	c2)be	c2)be	PROPN
ejpam-3317	167	21	a	a	DET
ejpam-3317	167	22	tp	tp	NOUN
ejpam-3317	167	23	-	-	NOUN
ejpam-3317	167	24	space	space	NOUN
ejpam-3317	167	25	.	.	PUNCT
ejpam-3317	168	1	if	if	SCONJ
ejpam-3317	168	2	f	f	PROPN
ejpam-3317	168	3	:	:	PUNCT
ejpam-3317	168	4	(	(	PUNCT
ejpam-3317	168	5	x	x	X
ejpam-3317	168	6	,	,	PUNCT
ejpam-3317	168	7	c1)→	c1)→	NOUN
ejpam-3317	168	8	(	(	PUNCT
ejpam-3317	168	9	y	y	PROPN
ejpam-3317	168	10	,	,	PUNCT
ejpam-3317	168	11	c2	c2	PROPN
ejpam-3317	168	12	)	)	PUNCT
ejpam-3317	168	13	and	and	CCONJ
ejpam-3317	168	14	g	g	NOUN
ejpam-3317	168	15	:	:	PUNCT
ejpam-3317	168	16	(	(	PUNCT
ejpam-3317	168	17	y	y	NOUN
ejpam-3317	168	18	,	,	PUNCT
ejpam-3317	168	19	c2)→	c2)→	X
ejpam-3317	168	20	(	(	PUNCT
ejpam-3317	168	21	z	z	NOUN
ejpam-3317	168	22	,	,	PUNCT
ejpam-3317	168	23	c3	c3	PROPN
ejpam-3317	168	24	)	)	PUNCT
ejpam-3317	168	25	are	be	AUX
ejpam-3317	168	26	pre	pre	ADJ
ejpam-3317	168	27	-	-	ADJ
ejpam-3317	168	28	continuous	continuous	ADJ
ejpam-3317	168	29	,	,	PUNCT
ejpam-3317	168	30	then	then	ADV
ejpam-3317	168	31	gof	gof	NOUN
ejpam-3317	168	32	is	be	AUX
ejpam-3317	168	33	pre	pre	ADJ
ejpam-3317	168	34	-	-	ADJ
ejpam-3317	168	35	continuous	continuous	ADJ
ejpam-3317	168	36	.	.	PUNCT
ejpam-3317	169	1	proof	proof	NOUN
ejpam-3317	169	2	.	.	PUNCT
ejpam-3317	170	1	let	let	VERB
ejpam-3317	170	2	h	h	PRON
ejpam-3317	170	3	be	be	AUX
ejpam-3317	170	4	open	open	ADJ
ejpam-3317	170	5	in	in	ADP
ejpam-3317	170	6	(	(	PUNCT
ejpam-3317	170	7	z	z	NOUN
ejpam-3317	170	8	,	,	PUNCT
ejpam-3317	170	9	c3	c3	PROPN
ejpam-3317	170	10	)	)	PUNCT
ejpam-3317	170	11	.	.	PUNCT
ejpam-3317	171	1	since	since	SCONJ
ejpam-3317	171	2	g	g	PROPN
ejpam-3317	171	3	is	be	AUX
ejpam-3317	171	4	pre	pre	ADJ
ejpam-3317	171	5	-	-	ADJ
ejpam-3317	171	6	continuous	continuous	ADJ
ejpam-3317	171	7	,	,	PUNCT
ejpam-3317	171	8	g−1	g−1	PROPN
ejpam-3317	171	9	(	(	PUNCT
ejpam-3317	171	10	h	h	NOUN
ejpam-3317	171	11	)	)	PUNCT
ejpam-3317	171	12	is	be	AUX
ejpam-3317	171	13	preopen	preopen	ADJ
ejpam-3317	171	14	in	in	ADP
ejpam-3317	171	15	(	(	PUNCT
ejpam-3317	171	16	y	y	PROPN
ejpam-3317	171	17	,	,	PUNCT
ejpam-3317	171	18	c2	c2	PROPN
ejpam-3317	171	19	)	)	PUNCT
ejpam-3317	171	20	.	.	PUNCT
ejpam-3317	172	1	but	but	CCONJ
ejpam-3317	172	2	(	(	PUNCT
ejpam-3317	172	3	y	y	PROPN
ejpam-3317	172	4	,	,	PUNCT
ejpam-3317	172	5	c2	c2	PROPN
ejpam-3317	172	6	)	)	PUNCT
ejpam-3317	172	7	is	be	AUX
ejpam-3317	172	8	a	a	DET
ejpam-3317	172	9	tp	tp	NOUN
ejpam-3317	172	10	-	-	PUNCT
ejpam-3317	172	11	space	space	NOUN
ejpam-3317	172	12	,	,	PUNCT
ejpam-3317	172	13	hence	hence	ADV
ejpam-3317	172	14	g−1(h)is	g−1(h)is	ADJ
ejpam-3317	172	15	open	open	ADJ
ejpam-3317	172	16	in	in	ADP
ejpam-3317	172	17	(	(	PUNCT
ejpam-3317	172	18	y	y	PROPN
ejpam-3317	172	19	,	,	PUNCT
ejpam-3317	172	20	c2	c2	PROPN
ejpam-3317	172	21	)	)	PUNCT
ejpam-3317	172	22	.	.	PUNCT
ejpam-3317	173	1	thus	thus	ADV
ejpam-3317	173	2	f−1	f−1	PROPN
ejpam-3317	173	3	(	(	PUNCT
ejpam-3317	173	4	g−1	g−1	PROPN
ejpam-3317	173	5	(	(	PUNCT
ejpam-3317	173	6	h	h	NOUN
ejpam-3317	173	7	)	)	PUNCT
ejpam-3317	173	8	)	)	PUNCT
ejpam-3317	173	9	=	=	SYM
ejpam-3317	173	10	(	(	PUNCT
ejpam-3317	173	11	gof)−1(h	gof)−1(h	PROPN
ejpam-3317	173	12	)	)	PUNCT
ejpam-3317	173	13	is	be	AUX
ejpam-3317	173	14	preopen	preopen	ADJ
ejpam-3317	173	15	in	in	ADP
ejpam-3317	173	16	(	(	PUNCT
ejpam-3317	173	17	x	x	NOUN
ejpam-3317	173	18	,	,	PUNCT
ejpam-3317	173	19	c1	c1	PROPN
ejpam-3317	173	20	)	)	PUNCT
ejpam-3317	173	21	.	.	PUNCT
ejpam-3317	174	1	therefore	therefore	ADV
ejpam-3317	174	2	,	,	PUNCT
ejpam-3317	174	3	gof	gof	PROPN
ejpam-3317	174	4	is	be	AUX
ejpam-3317	174	5	pre	pre	ADJ
ejpam-3317	174	6	-	-	ADJ
ejpam-3317	174	7	continuous	continuous	ADJ
ejpam-3317	174	8	h.m	h.m	PROPN
ejpam-3317	174	9	.	.	PROPN
ejpam-3317	174	10	darwesh	darwesh	PROPN
ejpam-3317	174	11	,	,	PUNCT
ejpam-3317	174	12	s.f	s.f	PROPN
ejpam-3317	174	13	.	.	PROPN
ejpam-3317	174	14	namiq	namiq	PROPN
ejpam-3317	174	15	/	/	SYM
ejpam-3317	174	16	eur	eur	PROPN
ejpam-3317	174	17	.	.	PUNCT
ejpam-3317	175	1	j.	j.	PROPN
ejpam-3317	175	2	pure	pure	PROPN
ejpam-3317	175	3	appl	appl	PROPN
ejpam-3317	175	4	.	.	PROPN
ejpam-3317	175	5	math	math	PROPN
ejpam-3317	175	6	,	,	PUNCT
ejpam-3317	175	7	12	12	NUM
ejpam-3317	175	8	(	(	PUNCT
ejpam-3317	175	9	3	3	NUM
ejpam-3317	175	10	)	)	PUNCT
ejpam-3317	175	11	(	(	PUNCT
ejpam-3317	175	12	2019	2019	NUM
ejpam-3317	175	13	)	)	PUNCT
ejpam-3317	175	14	,	,	PUNCT
ejpam-3317	175	15	1082	1082	NUM
ejpam-3317	175	16	-	-	SYM
ejpam-3317	175	17	1095	1095	NUM
ejpam-3317	175	18	1088	1088	NUM
ejpam-3317	175	19	theorem	theorem	NOUN
ejpam-3317	175	20	4	4	NUM
ejpam-3317	175	21	.	.	PUNCT
ejpam-3317	176	1	let	let	AUX
ejpam-3317	176	2	(	(	PUNCT
ejpam-3317	176	3	x	x	NOUN
ejpam-3317	176	4	,	,	PUNCT
ejpam-3317	176	5	c	c	X
ejpam-3317	176	6	)	)	PUNCT
ejpam-3317	176	7	be	be	AUX
ejpam-3317	176	8	a	a	DET
ejpam-3317	176	9	closure	closure	NOUN
ejpam-3317	176	10	space	space	NOUN
ejpam-3317	176	11	,	,	PUNCT
ejpam-3317	176	12	{	{	PUNCT
ejpam-3317	176	13	(	(	PUNCT
ejpam-3317	176	14	yα	yα	NOUN
ejpam-3317	176	15	,	,	PUNCT
ejpam-3317	176	16	cα	cα	NOUN
ejpam-3317	176	17	)	)	PUNCT
ejpam-3317	176	18	:	:	PUNCT
ejpam-3317	177	1	α	α	PROPN
ejpam-3317	177	2	∈	∈	PROPN
ejpam-3317	177	3	j	j	AUX
ejpam-3317	177	4	}	}	PUNCT
ejpam-3317	177	5	be	be	VERB
ejpam-3317	177	6	a	a	DET
ejpam-3317	177	7	family	family	NOUN
ejpam-3317	177	8	of	of	ADP
ejpam-3317	177	9	closure	closure	NOUN
ejpam-3317	177	10	spaces	space	NOUN
ejpam-3317	177	11	and	and	CCONJ
ejpam-3317	177	12	f	f	X
ejpam-3317	177	13	:	:	PUNCT
ejpam-3317	177	14	(	(	PUNCT
ejpam-3317	177	15	x	x	X
ejpam-3317	177	16	,	,	PUNCT
ejpam-3317	177	17	c)→	c)→	PROPN
ejpam-3317	177	18	∏	∏	PROPN
ejpam-3317	177	19	α∈j	α∈j	NOUN
ejpam-3317	177	20	(	(	PUNCT
ejpam-3317	177	21	yα	yα	NOUN
ejpam-3317	177	22	,	,	PUNCT
ejpam-3317	177	23	cα	cα	NOUN
ejpam-3317	177	24	)	)	PUNCT
ejpam-3317	177	25	be	be	AUX
ejpam-3317	177	26	a	a	DET
ejpam-3317	177	27	function	function	NOUN
ejpam-3317	177	28	.	.	PUNCT
ejpam-3317	178	1	if	if	SCONJ
ejpam-3317	178	2	f	f	PROPN
ejpam-3317	178	3	is	be	AUX
ejpam-3317	178	4	pre	pre	ADJ
ejpam-3317	178	5	-	-	ADJ
ejpam-3317	178	6	continuous	continuous	ADJ
ejpam-3317	178	7	and	and	CCONJ
ejpam-3317	178	8	πα	πα	ADJ
ejpam-3317	178	9	is	be	AUX
ejpam-3317	178	10	a	a	DET
ejpam-3317	178	11	projection	projection	NOUN
ejpam-3317	178	12	function	function	NOUN
ejpam-3317	178	13	,	,	PUNCT
ejpam-3317	178	14	then	then	ADV
ejpam-3317	178	15	παof	παof	NOUN
ejpam-3317	178	16	is	be	AUX
ejpam-3317	178	17	pre	pre	ADJ
ejpam-3317	178	18	-	-	ADJ
ejpam-3317	178	19	continuous	continuous	ADJ
ejpam-3317	178	20	for	for	ADP
ejpam-3317	178	21	each	each	DET
ejpam-3317	178	22	α	α	PROPN
ejpam-3317	178	23	∈	∈	PROPN
ejpam-3317	178	24	j	j	PROPN
ejpam-3317	178	25	.	.	PUNCT
ejpam-3317	179	1	proof	proof	NOUN
ejpam-3317	179	2	.	.	PUNCT
ejpam-3317	180	1	assume	assume	VERB
ejpam-3317	180	2	that	that	SCONJ
ejpam-3317	180	3	f	f	X
ejpam-3317	180	4	:	:	PUNCT
ejpam-3317	180	5	(	(	PUNCT
ejpam-3317	180	6	x	x	NOUN
ejpam-3317	180	7	,	,	PUNCT
ejpam-3317	180	8	c	c	NOUN
ejpam-3317	180	9	)	)	PUNCT
ejpam-3317	180	10	→	→	SYM
ejpam-3317	180	11	∏	∏	NUM
ejpam-3317	180	12	α∈j	α∈j	NOUN
ejpam-3317	180	13	(	(	PUNCT
ejpam-3317	180	14	yα	yα	NOUN
ejpam-3317	180	15	,	,	PUNCT
ejpam-3317	180	16	cα	cα	NOUN
ejpam-3317	180	17	)	)	PUNCT
ejpam-3317	180	18	is	be	AUX
ejpam-3317	180	19	pre	pre	ADJ
ejpam-3317	180	20	-	-	ADJ
ejpam-3317	180	21	continuous	continuous	ADJ
ejpam-3317	180	22	for	for	ADP
ejpam-3317	180	23	all	all	DET
ejpam-3317	180	24	α	α	PRON
ejpam-3317	180	25	∈	∈	PROPN
ejpam-3317	180	26	j	j	PROPN
ejpam-3317	180	27	.	.	PUNCT
ejpam-3317	181	1	since	since	SCONJ
ejpam-3317	181	2	πα	πα	PROPN
ejpam-3317	181	3	is	be	AUX
ejpam-3317	181	4	continuous	continuous	ADJ
ejpam-3317	181	5	,	,	PUNCT
ejpam-3317	181	6	παof	παof	ADJ
ejpam-3317	181	7	is	be	AUX
ejpam-3317	181	8	pre	pre	ADJ
ejpam-3317	181	9	-	-	ADJ
ejpam-3317	181	10	continuous	continuous	ADJ
ejpam-3317	181	11	for	for	ADP
ejpam-3317	181	12	each	each	DET
ejpam-3317	181	13	α	α	NOUN
ejpam-3317	181	14	∈	∈	PROPN
ejpam-3317	181	15	j	j	PROPN
ejpam-3317	181	16	by	by	ADP
ejpam-3317	181	17	proposition	proposition	NOUN
ejpam-3317	181	18	15	15	NUM
ejpam-3317	181	19	.	.	PUNCT
ejpam-3317	182	1	definition	definition	NOUN
ejpam-3317	182	2	14	14	NUM
ejpam-3317	182	3	.	.	PUNCT
ejpam-3317	183	1	let	let	AUX
ejpam-3317	183	2	(	(	PUNCT
ejpam-3317	183	3	x	x	NOUN
ejpam-3317	183	4	,	,	PUNCT
ejpam-3317	183	5	c1	c1	PROPN
ejpam-3317	183	6	)	)	PUNCT
ejpam-3317	183	7	and	and	CCONJ
ejpam-3317	183	8	(	(	PUNCT
ejpam-3317	183	9	y	y	PROPN
ejpam-3317	183	10	,	,	PUNCT
ejpam-3317	183	11	c2	c2	PROPN
ejpam-3317	183	12	)	)	PUNCT
ejpam-3317	183	13	be	be	VERB
ejpam-3317	183	14	closure	closure	NOUN
ejpam-3317	183	15	spaces	space	NOUN
ejpam-3317	183	16	and	and	CCONJ
ejpam-3317	183	17	let	let	VERB
ejpam-3317	183	18	f	f	X
ejpam-3317	183	19	:	:	PUNCT
ejpam-3317	183	20	(	(	PUNCT
ejpam-3317	183	21	x	x	X
ejpam-3317	183	22	,	,	PUNCT
ejpam-3317	183	23	c1	c1	PROPN
ejpam-3317	183	24	)	)	PUNCT
ejpam-3317	183	25	→	→	PUNCT
ejpam-3317	183	26	(	(	PUNCT
ejpam-3317	183	27	y	y	PROPN
ejpam-3317	183	28	,	,	PUNCT
ejpam-3317	183	29	c2	c2	PROPN
ejpam-3317	183	30	)	)	PUNCT
ejpam-3317	183	31	be	be	AUX
ejpam-3317	183	32	a	a	DET
ejpam-3317	183	33	function	function	NOUN
ejpam-3317	183	34	.	.	PUNCT
ejpam-3317	184	1	then	then	ADV
ejpam-3317	184	2	f	f	PROPN
ejpam-3317	184	3	is	be	AUX
ejpam-3317	184	4	contra	contra	PROPN
ejpam-3317	184	5	-	-	ADJ
ejpam-3317	184	6	pre	pre	ADJ
ejpam-3317	184	7	-	-	ADJ
ejpam-3317	184	8	continuous	continuous	ADJ
ejpam-3317	184	9	if	if	SCONJ
ejpam-3317	184	10	the	the	DET
ejpam-3317	184	11	inverse	inverse	NOUN
ejpam-3317	184	12	image	image	NOUN
ejpam-3317	184	13	under	under	ADP
ejpam-3317	184	14	f	f	PROPN
ejpam-3317	184	15	of	of	ADP
ejpam-3317	184	16	every	every	DET
ejpam-3317	184	17	open	open	ADJ
ejpam-3317	184	18	subset	subset	NOUN
ejpam-3317	184	19	of	of	ADP
ejpam-3317	184	20	(	(	PUNCT
ejpam-3317	184	21	y	y	PROPN
ejpam-3317	184	22	,	,	PUNCT
ejpam-3317	184	23	c2	c2	PROPN
ejpam-3317	184	24	)	)	PUNCT
ejpam-3317	184	25	is	be	AUX
ejpam-3317	184	26	preclosed	preclose	VERB
ejpam-3317	184	27	in	in	ADP
ejpam-3317	184	28	(	(	PUNCT
ejpam-3317	184	29	x	x	NOUN
ejpam-3317	184	30	,	,	PUNCT
ejpam-3317	184	31	c1	c1	PROPN
ejpam-3317	184	32	)	)	PUNCT
ejpam-3317	184	33	.	.	PUNCT
ejpam-3317	185	1	proposition	proposition	NOUN
ejpam-3317	185	2	21	21	NUM
ejpam-3317	185	3	.	.	PUNCT
ejpam-3317	186	1	let	let	AUX
ejpam-3317	186	2	(	(	PUNCT
ejpam-3317	186	3	x	x	NOUN
ejpam-3317	186	4	,	,	PUNCT
ejpam-3317	186	5	c1	c1	PROPN
ejpam-3317	186	6	)	)	PUNCT
ejpam-3317	186	7	and	and	CCONJ
ejpam-3317	186	8	(	(	PUNCT
ejpam-3317	186	9	y	y	PROPN
ejpam-3317	186	10	,	,	PUNCT
ejpam-3317	186	11	c2	c2	PROPN
ejpam-3317	186	12	)	)	PUNCT
ejpam-3317	186	13	be	be	VERB
ejpam-3317	186	14	closure	closure	NOUN
ejpam-3317	186	15	spaces	space	NOUN
ejpam-3317	186	16	and	and	CCONJ
ejpam-3317	186	17	let	let	VERB
ejpam-3317	186	18	f	f	X
ejpam-3317	186	19	:	:	PUNCT
ejpam-3317	186	20	(	(	PUNCT
ejpam-3317	186	21	x	x	X
ejpam-3317	186	22	,	,	PUNCT
ejpam-3317	186	23	c1)→	c1)→	NOUN
ejpam-3317	186	24	(	(	PUNCT
ejpam-3317	186	25	y	y	PROPN
ejpam-3317	186	26	,	,	PUNCT
ejpam-3317	186	27	c2	c2	PROPN
ejpam-3317	186	28	)	)	PUNCT
ejpam-3317	186	29	be	be	AUX
ejpam-3317	186	30	a	a	DET
ejpam-3317	186	31	function	function	NOUN
ejpam-3317	186	32	.	.	PUNCT
ejpam-3317	187	1	then	then	ADV
ejpam-3317	187	2	f	f	PROPN
ejpam-3317	187	3	is	be	AUX
ejpam-3317	187	4	contra	contra	PROPN
ejpam-3317	187	5	-	-	ADJ
ejpam-3317	187	6	pre	pre	ADJ
ejpam-3317	187	7	-	-	ADJ
ejpam-3317	187	8	continuous	continuous	ADJ
ejpam-3317	187	9	if	if	SCONJ
ejpam-3317	187	10	and	and	CCONJ
ejpam-3317	187	11	only	only	ADV
ejpam-3317	187	12	if	if	SCONJ
ejpam-3317	187	13	the	the	DET
ejpam-3317	187	14	inverse	inverse	NOUN
ejpam-3317	187	15	image	image	NOUN
ejpam-3317	187	16	under	under	ADP
ejpam-3317	187	17	f	f	PROPN
ejpam-3317	187	18	of	of	ADP
ejpam-3317	187	19	every	every	DET
ejpam-3317	187	20	closed	closed	ADJ
ejpam-3317	187	21	subset	subset	NOUN
ejpam-3317	187	22	of	of	ADP
ejpam-3317	187	23	(	(	PUNCT
ejpam-3317	187	24	y	y	PROPN
ejpam-3317	187	25	,	,	PUNCT
ejpam-3317	187	26	c2	c2	PROPN
ejpam-3317	187	27	)	)	PUNCT
ejpam-3317	187	28	is	be	AUX
ejpam-3317	187	29	preopen	preopen	ADJ
ejpam-3317	187	30	in	in	ADP
ejpam-3317	187	31	(	(	PUNCT
ejpam-3317	187	32	x	x	NOUN
ejpam-3317	187	33	,	,	PUNCT
ejpam-3317	187	34	c1	c1	PROPN
ejpam-3317	187	35	)	)	PUNCT
ejpam-3317	187	36	.	.	PUNCT
ejpam-3317	188	1	proof	proof	NOUN
ejpam-3317	188	2	.	.	PUNCT
ejpam-3317	189	1	let	let	VERB
ejpam-3317	189	2	f	f	PRON
ejpam-3317	189	3	be	be	AUX
ejpam-3317	189	4	a	a	DET
ejpam-3317	189	5	closed	closed	ADJ
ejpam-3317	189	6	subset	subset	NOUN
ejpam-3317	189	7	in	in	ADP
ejpam-3317	189	8	(	(	PUNCT
ejpam-3317	189	9	y	y	PROPN
ejpam-3317	189	10	,	,	PUNCT
ejpam-3317	189	11	c2	c2	PROPN
ejpam-3317	189	12	)	)	PUNCT
ejpam-3317	189	13	.	.	PUNCT
ejpam-3317	190	1	then	then	ADV
ejpam-3317	190	2	y	y	PROPN
ejpam-3317	190	3	/	/	SYM
ejpam-3317	190	4	f	f	PROPN
ejpam-3317	190	5	is	be	AUX
ejpam-3317	190	6	open	open	ADJ
ejpam-3317	190	7	in(y	in(y	PRON
ejpam-3317	190	8	,	,	PUNCT
ejpam-3317	190	9	c2	c2	PROPN
ejpam-3317	190	10	)	)	PUNCT
ejpam-3317	190	11	.	.	PUNCT
ejpam-3317	191	1	since	since	SCONJ
ejpam-3317	191	2	f	f	PROPN
ejpam-3317	191	3	is	be	AUX
ejpam-3317	191	4	contra	contra	PROPN
ejpam-3317	191	5	-	-	ADJ
ejpam-3317	191	6	pre	pre	ADJ
ejpam-3317	191	7	-	-	ADJ
ejpam-3317	191	8	continuous	continuous	ADJ
ejpam-3317	191	9	,	,	PUNCT
ejpam-3317	191	10	f−1(y	f−1(y	PROPN
ejpam-3317	191	11	/	/	SYM
ejpam-3317	191	12	f	f	PROPN
ejpam-3317	191	13	)	)	PUNCT
ejpam-3317	191	14	is	be	AUX
ejpam-3317	191	15	preclosed	preclose	VERB
ejpam-3317	191	16	.	.	PUNCT
ejpam-3317	192	1	but	but	CCONJ
ejpam-3317	192	2	f−1(y	f−1(y	PROPN
ejpam-3317	192	3	/	/	SYM
ejpam-3317	192	4	f	f	PROPN
ejpam-3317	192	5	)	)	PUNCT
ejpam-3317	193	1	=	=	PUNCT
ejpam-3317	193	2	x	x	X
ejpam-3317	193	3	/	/	SYM
ejpam-3317	193	4	f−1(f	f−1(f	PROPN
ejpam-3317	193	5	)	)	PUNCT
ejpam-3317	193	6	,	,	PUNCT
ejpam-3317	193	7	thus	thus	ADV
ejpam-3317	193	8	f−1(f	f−1(f	NOUN
ejpam-3317	193	9	)	)	PUNCT
ejpam-3317	193	10	is	be	AUX
ejpam-3317	193	11	preopen	preopen	ADJ
ejpam-3317	193	12	in	in	ADP
ejpam-3317	193	13	(	(	PUNCT
ejpam-3317	193	14	x	x	NOUN
ejpam-3317	193	15	,	,	PUNCT
ejpam-3317	193	16	c1	c1	PROPN
ejpam-3317	193	17	)	)	PUNCT
ejpam-3317	193	18	.	.	PUNCT
ejpam-3317	194	1	conversely	conversely	ADV
ejpam-3317	194	2	,	,	PUNCT
ejpam-3317	194	3	let	let	VERB
ejpam-3317	194	4	g	g	PRON
ejpam-3317	194	5	be	be	AUX
ejpam-3317	194	6	an	an	DET
ejpam-3317	194	7	open	open	ADJ
ejpam-3317	194	8	subset	subset	NOUN
ejpam-3317	194	9	in	in	ADP
ejpam-3317	194	10	(	(	PUNCT
ejpam-3317	194	11	y	y	PROPN
ejpam-3317	194	12	,	,	PUNCT
ejpam-3317	194	13	c2	c2	PROPN
ejpam-3317	194	14	)	)	PUNCT
ejpam-3317	194	15	.	.	PUNCT
ejpam-3317	195	1	then	then	ADV
ejpam-3317	195	2	y	y	PROPN
ejpam-3317	195	3	/	/	SYM
ejpam-3317	195	4	g	g	PROPN
ejpam-3317	195	5	is	be	AUX
ejpam-3317	195	6	closed	close	VERB
ejpam-3317	195	7	in	in	ADP
ejpam-3317	195	8	(	(	PUNCT
ejpam-3317	195	9	y	y	PROPN
ejpam-3317	195	10	,	,	PUNCT
ejpam-3317	195	11	c2	c2	PROPN
ejpam-3317	195	12	)	)	PUNCT
ejpam-3317	195	13	.	.	PUNCT
ejpam-3317	196	1	since	since	SCONJ
ejpam-3317	196	2	the	the	DET
ejpam-3317	196	3	inverse	inverse	NOUN
ejpam-3317	196	4	image	image	NOUN
ejpam-3317	196	5	of	of	ADP
ejpam-3317	196	6	each	each	DET
ejpam-3317	196	7	closed	close	VERB
ejpam-3317	196	8	subset	subset	NOUN
ejpam-3317	196	9	in	in	ADP
ejpam-3317	196	10	(	(	PUNCT
ejpam-3317	196	11	y	y	PROPN
ejpam-3317	196	12	,	,	PUNCT
ejpam-3317	196	13	c2)is	c2)is	ADJ
ejpam-3317	196	14	preopen	preopen	NOUN
ejpam-3317	196	15	in	in	ADP
ejpam-3317	196	16	(	(	PUNCT
ejpam-3317	196	17	x	x	NOUN
ejpam-3317	196	18	,	,	PUNCT
ejpam-3317	196	19	c1	c1	PROPN
ejpam-3317	196	20	)	)	PUNCT
ejpam-3317	196	21	,	,	PUNCT
ejpam-3317	196	22	f	f	PROPN
ejpam-3317	196	23	−1(y	−1(y	PROPN
ejpam-3317	196	24	/	/	SYM
ejpam-3317	196	25	g	g	NOUN
ejpam-3317	196	26	)	)	PUNCT
ejpam-3317	196	27	is	be	AUX
ejpam-3317	196	28	preopen	preopen	ADJ
ejpam-3317	196	29	in	in	ADP
ejpam-3317	196	30	(	(	PUNCT
ejpam-3317	196	31	x	x	NOUN
ejpam-3317	196	32	,	,	PUNCT
ejpam-3317	196	33	c1	c1	PROPN
ejpam-3317	196	34	)	)	PUNCT
ejpam-3317	196	35	.	.	PUNCT
ejpam-3317	197	1	but	but	CCONJ
ejpam-3317	197	2	f−1(y	f−1(y	PROPN
ejpam-3317	197	3	/	/	SYM
ejpam-3317	197	4	g	g	PROPN
ejpam-3317	197	5	)	)	PUNCT
ejpam-3317	197	6	=	=	SYM
ejpam-3317	198	1	x	x	X
ejpam-3317	198	2	/	/	SYM
ejpam-3317	198	3	f−1(g	f−1(g	PROPN
ejpam-3317	198	4	)	)	PUNCT
ejpam-3317	198	5	,	,	PUNCT
ejpam-3317	198	6	thus	thus	ADV
ejpam-3317	198	7	f−1(g	f−1(g	PROPN
ejpam-3317	198	8	)	)	PUNCT
ejpam-3317	198	9	is	be	AUX
ejpam-3317	198	10	preclosed	preclose	VERB
ejpam-3317	198	11	.	.	PUNCT
ejpam-3317	199	1	therefore	therefore	ADV
ejpam-3317	199	2	,	,	PUNCT
ejpam-3317	199	3	f	f	PROPN
ejpam-3317	199	4	is	be	AUX
ejpam-3317	199	5	contra	contra	PROPN
ejpam-3317	199	6	-	-	ADJ
ejpam-3317	199	7	pre	pre	ADJ
ejpam-3317	199	8	-	-	ADJ
ejpam-3317	199	9	continuous	continuous	ADJ
ejpam-3317	199	10	.	.	PUNCT
ejpam-3317	200	1	proposition	proposition	NOUN
ejpam-3317	200	2	22	22	NUM
ejpam-3317	200	3	.	.	PUNCT
ejpam-3317	201	1	let	let	VERB
ejpam-3317	201	2	(	(	PUNCT
ejpam-3317	201	3	x	x	NOUN
ejpam-3317	201	4	,	,	PUNCT
ejpam-3317	201	5	c1	c1	PROPN
ejpam-3317	201	6	)	)	PUNCT
ejpam-3317	201	7	,	,	PUNCT
ejpam-3317	201	8	(	(	PUNCT
ejpam-3317	201	9	y	y	PROPN
ejpam-3317	201	10	,	,	PUNCT
ejpam-3317	201	11	c2	c2	PROPN
ejpam-3317	201	12	)	)	PUNCT
ejpam-3317	201	13	and	and	CCONJ
ejpam-3317	201	14	(	(	PUNCT
ejpam-3317	201	15	z	z	NOUN
ejpam-3317	201	16	,	,	PUNCT
ejpam-3317	201	17	c3	c3	PROPN
ejpam-3317	201	18	)	)	PUNCT
ejpam-3317	201	19	be	be	VERB
ejpam-3317	201	20	closure	closure	NOUN
ejpam-3317	201	21	spaces	space	NOUN
ejpam-3317	201	22	,	,	PUNCT
ejpam-3317	201	23	let	let	VERB
ejpam-3317	201	24	f	f	PRON
ejpam-3317	201	25	:	:	PUNCT
ejpam-3317	201	26	(	(	PUNCT
ejpam-3317	201	27	x	x	X
ejpam-3317	201	28	,	,	PUNCT
ejpam-3317	201	29	c1)→	c1)→	NOUN
ejpam-3317	201	30	(	(	PUNCT
ejpam-3317	201	31	y	y	PROPN
ejpam-3317	201	32	,	,	PUNCT
ejpam-3317	201	33	c2	c2	PROPN
ejpam-3317	201	34	)	)	PUNCT
ejpam-3317	201	35	and	and	CCONJ
ejpam-3317	201	36	g	g	NOUN
ejpam-3317	201	37	:	:	PUNCT
ejpam-3317	201	38	(	(	PUNCT
ejpam-3317	201	39	y	y	PROPN
ejpam-3317	201	40	,	,	PUNCT
ejpam-3317	201	41	c2	c2	PROPN
ejpam-3317	201	42	)	)	PUNCT
ejpam-3317	201	43	→	→	PUNCT
ejpam-3317	201	44	(	(	PUNCT
ejpam-3317	201	45	z	z	NOUN
ejpam-3317	201	46	,	,	PUNCT
ejpam-3317	201	47	c3	c3	PROPN
ejpam-3317	201	48	)	)	PUNCT
ejpam-3317	201	49	be	be	AUX
ejpam-3317	201	50	functions	function	NOUN
ejpam-3317	201	51	.	.	PUNCT
ejpam-3317	202	1	if	if	SCONJ
ejpam-3317	202	2	gof	gof	NOUN
ejpam-3317	202	3	is	be	AUX
ejpam-3317	202	4	contra	contra	PROPN
ejpam-3317	202	5	-	-	ADJ
ejpam-3317	202	6	pre	pre	ADJ
ejpam-3317	202	7	-	-	ADJ
ejpam-3317	202	8	continuous	continuous	ADJ
ejpam-3317	202	9	and	and	CCONJ
ejpam-3317	202	10	g	g	NOUN
ejpam-3317	202	11	is	be	AUX
ejpam-3317	202	12	a	a	DET
ejpam-3317	202	13	closed	closed	ADJ
ejpam-3317	202	14	injection	injection	NOUN
ejpam-3317	202	15	,	,	PUNCT
ejpam-3317	202	16	then	then	ADV
ejpam-3317	202	17	f	f	PROPN
ejpam-3317	202	18	is	be	AUX
ejpam-3317	202	19	contra	contra	PROPN
ejpam-3317	202	20	-	-	ADJ
ejpam-3317	202	21	pre	pre	ADJ
ejpam-3317	202	22	-	-	ADJ
ejpam-3317	202	23	continuous	continuous	ADJ
ejpam-3317	202	24	.	.	PUNCT
ejpam-3317	203	1	proof	proof	NOUN
ejpam-3317	203	2	.	.	PUNCT
ejpam-3317	204	1	let	let	VERB
ejpam-3317	204	2	h	h	PRON
ejpam-3317	204	3	be	be	AUX
ejpam-3317	204	4	a	a	DET
ejpam-3317	204	5	closed	closed	ADJ
ejpam-3317	204	6	subset	subset	NOUN
ejpam-3317	204	7	of	of	ADP
ejpam-3317	204	8	(	(	PUNCT
ejpam-3317	204	9	y	y	PROPN
ejpam-3317	204	10	,	,	PUNCT
ejpam-3317	204	11	c2	c2	PROPN
ejpam-3317	204	12	)	)	PUNCT
ejpam-3317	204	13	.	.	PUNCT
ejpam-3317	205	1	since	since	SCONJ
ejpam-3317	205	2	g	g	PROPN
ejpam-3317	205	3	is	be	AUX
ejpam-3317	205	4	closed	closed	ADJ
ejpam-3317	205	5	,	,	PUNCT
ejpam-3317	205	6	g(h	g(h	NUM
ejpam-3317	205	7	)	)	PUNCT
ejpam-3317	205	8	is	be	AUX
ejpam-3317	205	9	closed	close	VERB
ejpam-3317	205	10	in	in	ADP
ejpam-3317	205	11	(	(	PUNCT
ejpam-3317	205	12	z	z	NOUN
ejpam-3317	205	13	,	,	PUNCT
ejpam-3317	205	14	c3	c3	PROPN
ejpam-3317	205	15	)	)	PUNCT
ejpam-3317	205	16	.	.	PUNCT
ejpam-3317	206	1	as	as	SCONJ
ejpam-3317	206	2	gof	gof	PROPN
ejpam-3317	206	3	is	be	AUX
ejpam-3317	206	4	contra	contra	PROPN
ejpam-3317	206	5	-	-	ADJ
ejpam-3317	206	6	pre	pre	ADJ
ejpam-3317	206	7	-	-	ADJ
ejpam-3317	206	8	continuous	continuous	ADJ
ejpam-3317	206	9	,	,	PUNCT
ejpam-3317	206	10	(	(	PUNCT
ejpam-3317	206	11	gof)−1	gof)−1	NOUN
ejpam-3317	206	12	(	(	PUNCT
ejpam-3317	206	13	g	g	PROPN
ejpam-3317	206	14	(	(	PUNCT
ejpam-3317	206	15	h	h	NOUN
ejpam-3317	206	16	)	)	PUNCT
ejpam-3317	206	17	)	)	PUNCT
ejpam-3317	207	1	=	=	SYM
ejpam-3317	207	2	f−1(g−1	f−1(g−1	PROPN
ejpam-3317	207	3	(	(	PUNCT
ejpam-3317	207	4	g	g	PROPN
ejpam-3317	207	5	(	(	PUNCT
ejpam-3317	207	6	h	h	NOUN
ejpam-3317	207	7	)	)	PUNCT
ejpam-3317	207	8	)	)	PUNCT
ejpam-3317	207	9	)	)	PUNCT
ejpam-3317	207	10	is	be	AUX
ejpam-3317	207	11	preopen	preopen	ADJ
ejpam-3317	207	12	in	in	ADP
ejpam-3317	207	13	(	(	PUNCT
ejpam-3317	207	14	x	x	NOUN
ejpam-3317	207	15	,	,	PUNCT
ejpam-3317	207	16	c1	c1	PROPN
ejpam-3317	207	17	)	)	PUNCT
ejpam-3317	207	18	by	by	ADP
ejpam-3317	207	19	proposition	proposition	NOUN
ejpam-3317	207	20	21	21	NUM
ejpam-3317	207	21	.	.	PUNCT
ejpam-3317	208	1	but	but	CCONJ
ejpam-3317	208	2	g	g	PROPN
ejpam-3317	208	3	is	be	AUX
ejpam-3317	208	4	injective	injective	ADJ
ejpam-3317	208	5	,	,	PUNCT
ejpam-3317	208	6	hence	hence	ADV
ejpam-3317	208	7	f−1	f−1	PROPN
ejpam-3317	208	8	(	(	PUNCT
ejpam-3317	208	9	g−1	g−1	PROPN
ejpam-3317	208	10	(	(	PUNCT
ejpam-3317	208	11	g	g	PROPN
ejpam-3317	208	12	(	(	PUNCT
ejpam-3317	208	13	h	h	NOUN
ejpam-3317	208	14	)	)	PUNCT
ejpam-3317	208	15	)	)	PUNCT
ejpam-3317	208	16	)	)	PUNCT
ejpam-3317	209	1	=	=	SYM
ejpam-3317	209	2	f−1(h	f−1(h	PROPN
ejpam-3317	209	3	)	)	PUNCT
ejpam-3317	209	4	.	.	PUNCT
ejpam-3317	210	1	therefore	therefore	ADV
ejpam-3317	210	2	,	,	PUNCT
ejpam-3317	210	3	f	f	PROPN
ejpam-3317	210	4	is	be	AUX
ejpam-3317	210	5	contra	contra	PROPN
ejpam-3317	210	6	-	-	ADJ
ejpam-3317	210	7	pre	pre	ADJ
ejpam-3317	210	8	-	-	ADJ
ejpam-3317	210	9	continuous	continuous	ADJ
ejpam-3317	210	10	.	.	PUNCT
ejpam-3317	211	1	proposition	proposition	NOUN
ejpam-3317	211	2	23	23	NUM
ejpam-3317	211	3	.	.	PUNCT
ejpam-3317	212	1	let	let	VERB
ejpam-3317	212	2	(	(	PUNCT
ejpam-3317	212	3	x	x	NOUN
ejpam-3317	212	4	,	,	PUNCT
ejpam-3317	212	5	c1	c1	PROPN
ejpam-3317	212	6	)	)	PUNCT
ejpam-3317	212	7	and	and	CCONJ
ejpam-3317	212	8	(	(	PUNCT
ejpam-3317	212	9	z	z	NOUN
ejpam-3317	212	10	,	,	PUNCT
ejpam-3317	212	11	c3)be	c3)be	NOUN
ejpam-3317	212	12	closure	closure	NOUN
ejpam-3317	212	13	spaces	space	NOUN
ejpam-3317	212	14	and	and	CCONJ
ejpam-3317	212	15	(	(	PUNCT
ejpam-3317	212	16	y	y	PROPN
ejpam-3317	212	17	,	,	PUNCT
ejpam-3317	212	18	c2	c2	PROPN
ejpam-3317	212	19	)	)	PUNCT
ejpam-3317	212	20	be	be	VERB
ejpam-3317	212	21	a	a	DET
ejpam-3317	212	22	tp	tp	NOUN
ejpam-3317	212	23	-	-	NOUN
ejpam-3317	212	24	space	space	NOUN
ejpam-3317	212	25	.	.	PUNCT
ejpam-3317	213	1	if	if	SCONJ
ejpam-3317	213	2	f	f	PROPN
ejpam-3317	213	3	:	:	PUNCT
ejpam-3317	213	4	(	(	PUNCT
ejpam-3317	213	5	x	x	X
ejpam-3317	213	6	,	,	PUNCT
ejpam-3317	213	7	c1	c1	PROPN
ejpam-3317	213	8	)	)	PUNCT
ejpam-3317	213	9	→	→	PUNCT
ejpam-3317	213	10	(	(	PUNCT
ejpam-3317	213	11	y	y	PROPN
ejpam-3317	213	12	,	,	PUNCT
ejpam-3317	213	13	c2	c2	PROPN
ejpam-3317	213	14	)	)	PUNCT
ejpam-3317	213	15	and	and	CCONJ
ejpam-3317	213	16	g	g	NOUN
ejpam-3317	213	17	:	:	PUNCT
ejpam-3317	213	18	(	(	PUNCT
ejpam-3317	213	19	y	y	PROPN
ejpam-3317	213	20	,	,	PUNCT
ejpam-3317	213	21	c2	c2	PROPN
ejpam-3317	213	22	)	)	PUNCT
ejpam-3317	213	23	→	→	PUNCT
ejpam-3317	213	24	(	(	PUNCT
ejpam-3317	213	25	z	z	NOUN
ejpam-3317	213	26	,	,	PUNCT
ejpam-3317	213	27	c3	c3	PROPN
ejpam-3317	213	28	)	)	PUNCT
ejpam-3317	213	29	are	be	AUX
ejpam-3317	213	30	contra	contra	PROPN
ejpam-3317	213	31	-	-	ADJ
ejpam-3317	213	32	pre	pre	ADJ
ejpam-3317	213	33	-	-	ADJ
ejpam-3317	213	34	continuous	continuous	ADJ
ejpam-3317	213	35	functions	function	NOUN
ejpam-3317	213	36	,	,	PUNCT
ejpam-3317	213	37	then	then	ADV
ejpam-3317	213	38	gof	gof	NOUN
ejpam-3317	213	39	is	be	AUX
ejpam-3317	213	40	pre	pre	ADJ
ejpam-3317	213	41	-	-	ADJ
ejpam-3317	213	42	continuous	continuous	ADJ
ejpam-3317	213	43	.	.	PUNCT
ejpam-3317	214	1	proof	proof	NOUN
ejpam-3317	214	2	.	.	PUNCT
ejpam-3317	215	1	let	let	VERB
ejpam-3317	215	2	h	h	NOUN
ejpam-3317	215	3	be	be	AUX
ejpam-3317	215	4	closed	close	VERB
ejpam-3317	215	5	in	in	ADP
ejpam-3317	215	6	(	(	PUNCT
ejpam-3317	215	7	z	z	NOUN
ejpam-3317	215	8	,	,	PUNCT
ejpam-3317	215	9	c3	c3	PROPN
ejpam-3317	215	10	)	)	PUNCT
ejpam-3317	215	11	.	.	PUNCT
ejpam-3317	216	1	since	since	SCONJ
ejpam-3317	216	2	g	g	PROPN
ejpam-3317	216	3	is	be	AUX
ejpam-3317	216	4	contra	contra	ADJ
ejpam-3317	216	5	-	-	ADJ
ejpam-3317	216	6	pre	pre	ADJ
ejpam-3317	216	7	-	-	ADJ
ejpam-3317	216	8	continuous	continuous	ADJ
ejpam-3317	216	9	,	,	PUNCT
ejpam-3317	216	10	g−1(h)is	g−1(h)is	ADJ
ejpam-3317	216	11	preopen	preopen	NOUN
ejpam-3317	216	12	in	in	ADP
ejpam-3317	216	13	(	(	PUNCT
ejpam-3317	216	14	y	y	PROPN
ejpam-3317	216	15	,	,	PUNCT
ejpam-3317	216	16	c2	c2	PROPN
ejpam-3317	216	17	)	)	PUNCT
ejpam-3317	216	18	.	.	PUNCT
ejpam-3317	217	1	but	but	CCONJ
ejpam-3317	217	2	(	(	PUNCT
ejpam-3317	217	3	y	y	PROPN
ejpam-3317	217	4	,	,	PUNCT
ejpam-3317	217	5	c2)is	c2)is	VERB
ejpam-3317	217	6	a	a	DET
ejpam-3317	217	7	tp	tp	NOUN
ejpam-3317	217	8	-	-	PUNCT
ejpam-3317	217	9	space	space	NOUN
ejpam-3317	217	10	,	,	PUNCT
ejpam-3317	217	11	hence	hence	ADV
ejpam-3317	217	12	g−1(h)is	g−1(h)is	ADJ
ejpam-3317	217	13	open	open	ADJ
ejpam-3317	217	14	in	in	ADP
ejpam-3317	217	15	(	(	PUNCT
ejpam-3317	217	16	y	y	PROPN
ejpam-3317	217	17	,	,	PUNCT
ejpam-3317	217	18	c2	c2	PROPN
ejpam-3317	217	19	)	)	PUNCT
ejpam-3317	217	20	.	.	PUNCT
ejpam-3317	218	1	as	as	SCONJ
ejpam-3317	218	2	f	f	PROPN
ejpam-3317	218	3	is	be	AUX
ejpam-3317	218	4	contrapre	contrapre	NOUN
ejpam-3317	218	5	-	-	ADJ
ejpam-3317	218	6	continuous	continuous	ADJ
ejpam-3317	218	7	by	by	ADP
ejpam-3317	218	8	proposition	proposition	NOUN
ejpam-3317	218	9	21	21	NUM
ejpam-3317	218	10	,	,	PUNCT
ejpam-3317	218	11	f−1	f−1	PROPN
ejpam-3317	218	12	(	(	PUNCT
ejpam-3317	218	13	g−1	g−1	PROPN
ejpam-3317	218	14	(	(	PUNCT
ejpam-3317	218	15	h	h	NOUN
ejpam-3317	218	16	)	)	PUNCT
ejpam-3317	218	17	)	)	PUNCT
ejpam-3317	218	18	=	=	SYM
ejpam-3317	218	19	(	(	PUNCT
ejpam-3317	218	20	gof)−1(h	gof)−1(h	PROPN
ejpam-3317	218	21	)	)	PUNCT
ejpam-3317	218	22	is	be	AUX
ejpam-3317	218	23	preclosed	preclose	VERB
ejpam-3317	218	24	in	in	ADP
ejpam-3317	218	25	(	(	PUNCT
ejpam-3317	218	26	x	x	NOUN
ejpam-3317	218	27	,	,	PUNCT
ejpam-3317	218	28	c1	c1	PROPN
ejpam-3317	218	29	)	)	PUNCT
ejpam-3317	218	30	.	.	PUNCT
ejpam-3317	219	1	therefore	therefore	ADV
ejpam-3317	219	2	,	,	PUNCT
ejpam-3317	219	3	gof	gof	PROPN
ejpam-3317	219	4	is	be	AUX
ejpam-3317	219	5	pre	pre	ADJ
ejpam-3317	219	6	-	-	ADJ
ejpam-3317	219	7	continuous	continuous	ADJ
ejpam-3317	219	8	by	by	ADP
ejpam-3317	219	9	proposition	proposition	NOUN
ejpam-3317	219	10	15	15	NUM
ejpam-3317	219	11	.	.	PUNCT
ejpam-3317	220	1	the	the	DET
ejpam-3317	220	2	following	follow	VERB
ejpam-3317	220	3	statement	statement	NOUN
ejpam-3317	220	4	is	be	AUX
ejpam-3317	220	5	evident	evident	ADJ
ejpam-3317	220	6	:	:	PUNCT
ejpam-3317	220	7	proposition	proposition	NOUN
ejpam-3317	220	8	24	24	NUM
ejpam-3317	220	9	.	.	PUNCT
ejpam-3317	221	1	let	let	VERB
ejpam-3317	221	2	(	(	PUNCT
ejpam-3317	221	3	x	x	NOUN
ejpam-3317	221	4	,	,	PUNCT
ejpam-3317	221	5	c1	c1	PROPN
ejpam-3317	221	6	)	)	PUNCT
ejpam-3317	221	7	,	,	PUNCT
ejpam-3317	221	8	(	(	PUNCT
ejpam-3317	221	9	y	y	PROPN
ejpam-3317	221	10	,	,	PUNCT
ejpam-3317	221	11	c2	c2	PROPN
ejpam-3317	221	12	)	)	PUNCT
ejpam-3317	221	13	and	and	CCONJ
ejpam-3317	221	14	(	(	PUNCT
ejpam-3317	221	15	z	z	NOUN
ejpam-3317	221	16	,	,	PUNCT
ejpam-3317	221	17	c3)be	c3)be	NOUN
ejpam-3317	221	18	closure	closure	NOUN
ejpam-3317	221	19	spaces	space	NOUN
ejpam-3317	221	20	and	and	CCONJ
ejpam-3317	221	21	let	let	VERB
ejpam-3317	221	22	f	f	X
ejpam-3317	221	23	:	:	PUNCT
ejpam-3317	221	24	(	(	PUNCT
ejpam-3317	221	25	x	x	X
ejpam-3317	221	26	,	,	PUNCT
ejpam-3317	221	27	c1	c1	PROPN
ejpam-3317	221	28	)	)	PUNCT
ejpam-3317	221	29	→	→	PUNCT
ejpam-3317	221	30	(	(	PUNCT
ejpam-3317	221	31	y	y	PROPN
ejpam-3317	221	32	,	,	PUNCT
ejpam-3317	221	33	c2	c2	PROPN
ejpam-3317	221	34	)	)	PUNCT
ejpam-3317	221	35	and	and	CCONJ
ejpam-3317	221	36	g	g	NOUN
ejpam-3317	221	37	:	:	PUNCT
ejpam-3317	221	38	(	(	PUNCT
ejpam-3317	221	39	y	y	PROPN
ejpam-3317	221	40	,	,	PUNCT
ejpam-3317	221	41	c2	c2	PROPN
ejpam-3317	221	42	)	)	PUNCT
ejpam-3317	221	43	→	→	PUNCT
ejpam-3317	221	44	(	(	PUNCT
ejpam-3317	221	45	z	z	NOUN
ejpam-3317	221	46	,	,	PUNCT
ejpam-3317	221	47	c3	c3	PROPN
ejpam-3317	221	48	)	)	PUNCT
ejpam-3317	221	49	be	be	AUX
ejpam-3317	221	50	functions	function	NOUN
ejpam-3317	221	51	.	.	PUNCT
ejpam-3317	222	1	if	if	SCONJ
ejpam-3317	222	2	f	f	PROPN
ejpam-3317	222	3	is	be	AUX
ejpam-3317	222	4	contra	contra	PROPN
ejpam-3317	222	5	-	-	ADJ
ejpam-3317	222	6	pre	pre	ADJ
ejpam-3317	222	7	-	-	ADJ
ejpam-3317	222	8	continuous	continuous	ADJ
ejpam-3317	222	9	and	and	CCONJ
ejpam-3317	222	10	g	g	NOUN
ejpam-3317	222	11	is	be	AUX
ejpam-3317	222	12	continuous	continuous	ADJ
ejpam-3317	222	13	,	,	PUNCT
ejpam-3317	222	14	then	then	ADV
ejpam-3317	222	15	gof	gof	NOUN
ejpam-3317	222	16	is	be	AUX
ejpam-3317	222	17	contra	contra	PROPN
ejpam-3317	222	18	-	-	ADJ
ejpam-3317	222	19	pre	pre	ADJ
ejpam-3317	222	20	-	-	ADJ
ejpam-3317	222	21	continuous	continuous	ADJ
ejpam-3317	222	22	.	.	PUNCT
ejpam-3317	223	1	h.m	h.m	PROPN
ejpam-3317	223	2	.	.	PROPN
ejpam-3317	223	3	darwesh	darwesh	PROPN
ejpam-3317	223	4	,	,	PUNCT
ejpam-3317	223	5	s.f	s.f	PROPN
ejpam-3317	223	6	.	.	PROPN
ejpam-3317	223	7	namiq	namiq	PROPN
ejpam-3317	223	8	/	/	SYM
ejpam-3317	223	9	eur	eur	PROPN
ejpam-3317	223	10	.	.	PUNCT
ejpam-3317	224	1	j.	j.	PROPN
ejpam-3317	224	2	pure	pure	PROPN
ejpam-3317	224	3	appl	appl	PROPN
ejpam-3317	224	4	.	.	PROPN
ejpam-3317	224	5	math	math	PROPN
ejpam-3317	224	6	,	,	PUNCT
ejpam-3317	224	7	12	12	NUM
ejpam-3317	224	8	(	(	PUNCT
ejpam-3317	224	9	3	3	NUM
ejpam-3317	224	10	)	)	PUNCT
ejpam-3317	224	11	(	(	PUNCT
ejpam-3317	224	12	2019	2019	NUM
ejpam-3317	224	13	)	)	PUNCT
ejpam-3317	224	14	,	,	PUNCT
ejpam-3317	224	15	1082	1082	NUM
ejpam-3317	224	16	-	-	SYM
ejpam-3317	224	17	1095	1095	NUM
ejpam-3317	224	18	1089	1089	NUM
ejpam-3317	224	19	as	as	ADP
ejpam-3317	224	20	a	a	DET
ejpam-3317	224	21	direct	direct	ADJ
ejpam-3317	224	22	consequence	consequence	NOUN
ejpam-3317	224	23	of	of	ADP
ejpam-3317	224	24	proposition	proposition	NOUN
ejpam-3317	224	25	24	24	NUM
ejpam-3317	224	26	,	,	PUNCT
ejpam-3317	224	27	we	we	PRON
ejpam-3317	224	28	have	have	AUX
ejpam-3317	224	29	:	:	PUNCT
ejpam-3317	224	30	proposition	proposition	NOUN
ejpam-3317	224	31	25	25	NUM
ejpam-3317	224	32	.	.	PUNCT
ejpam-3317	225	1	let	let	VERB
ejpam-3317	225	2	(	(	PUNCT
ejpam-3317	225	3	x	x	X
ejpam-3317	225	4	,	,	PUNCT
ejpam-3317	225	5	c)be	c)be	ADJ
ejpam-3317	225	6	a	a	DET
ejpam-3317	225	7	closure	closure	NOUN
ejpam-3317	225	8	space	space	NOUN
ejpam-3317	225	9	,	,	PUNCT
ejpam-3317	225	10	{	{	PUNCT
ejpam-3317	225	11	(	(	PUNCT
ejpam-3317	225	12	yα	yα	NOUN
ejpam-3317	225	13	,	,	PUNCT
ejpam-3317	225	14	cα	cα	NOUN
ejpam-3317	225	15	)	)	PUNCT
ejpam-3317	225	16	:	:	PUNCT
ejpam-3317	225	17	α	α	PROPN
ejpam-3317	225	18	∈	∈	PROPN
ejpam-3317	225	19	j}be	j}be	PROPN
ejpam-3317	225	20	a	a	DET
ejpam-3317	225	21	family	family	NOUN
ejpam-3317	225	22	of	of	ADP
ejpam-3317	225	23	closure	closure	NOUN
ejpam-3317	225	24	spaces	space	NOUN
ejpam-3317	225	25	and	and	CCONJ
ejpam-3317	225	26	f	f	X
ejpam-3317	225	27	:	:	PUNCT
ejpam-3317	225	28	(	(	PUNCT
ejpam-3317	225	29	x	x	X
ejpam-3317	225	30	,	,	PUNCT
ejpam-3317	225	31	c)→	c)→	PROPN
ejpam-3317	225	32	∏	∏	PROPN
ejpam-3317	225	33	α∈j	α∈j	NOUN
ejpam-3317	225	34	(	(	PUNCT
ejpam-3317	225	35	yα	yα	NOUN
ejpam-3317	225	36	,	,	PUNCT
ejpam-3317	225	37	cα	cα	NOUN
ejpam-3317	225	38	)	)	PUNCT
ejpam-3317	225	39	be	be	AUX
ejpam-3317	225	40	a	a	DET
ejpam-3317	225	41	function	function	NOUN
ejpam-3317	225	42	.	.	PUNCT
ejpam-3317	226	1	if	if	SCONJ
ejpam-3317	226	2	f	f	PROPN
ejpam-3317	226	3	is	be	AUX
ejpam-3317	226	4	contra	contra	PROPN
ejpam-3317	226	5	-	-	ADJ
ejpam-3317	226	6	pre	pre	ADJ
ejpam-3317	226	7	-	-	ADJ
ejpam-3317	226	8	continuous	continuous	ADJ
ejpam-3317	226	9	and	and	CCONJ
ejpam-3317	226	10	πα	πα	ADJ
ejpam-3317	226	11	is	be	AUX
ejpam-3317	226	12	a	a	DET
ejpam-3317	226	13	projection	projection	NOUN
ejpam-3317	226	14	function	function	NOUN
ejpam-3317	226	15	,	,	PUNCT
ejpam-3317	226	16	then	then	ADV
ejpam-3317	226	17	παof	παof	NOUN
ejpam-3317	226	18	is	be	AUX
ejpam-3317	226	19	contra	contra	ADJ
ejpam-3317	226	20	-	-	ADJ
ejpam-3317	226	21	pre	pre	ADJ
ejpam-3317	226	22	-	-	ADJ
ejpam-3317	226	23	continuous	continuous	ADJ
ejpam-3317	226	24	for	for	ADP
ejpam-3317	226	25	each	each	DET
ejpam-3317	226	26	α	α	PROPN
ejpam-3317	226	27	∈	∈	PROPN
ejpam-3317	226	28	j	j	PROPN
ejpam-3317	226	29	.	.	PUNCT
ejpam-3317	227	1	4	4	X
ejpam-3317	227	2	.	.	X
ejpam-3317	227	3	pre	pre	ADJ
ejpam-3317	227	4	-	-	ADJ
ejpam-3317	227	5	irresolute	irresolute	ADJ
ejpam-3317	227	6	functions	function	NOUN
ejpam-3317	227	7	in	in	ADP
ejpam-3317	227	8	view	view	NOUN
ejpam-3317	227	9	of	of	ADP
ejpam-3317	227	10	the	the	DET
ejpam-3317	227	11	definition	definition	NOUN
ejpam-3317	227	12	of	of	ADP
ejpam-3317	227	13	pre	pre	ADJ
ejpam-3317	227	14	-	-	ADJ
ejpam-3317	227	15	irresolute	irresolute	ADJ
ejpam-3317	227	16	functions	function	NOUN
ejpam-3317	227	17	,	,	PUNCT
ejpam-3317	227	18	we	we	PRON
ejpam-3317	227	19	define	define	VERB
ejpam-3317	227	20	pre	pre	ADJ
ejpam-3317	227	21	-	-	ADJ
ejpam-3317	227	22	irresolute	irresolute	ADJ
ejpam-3317	227	23	functions	function	NOUN
ejpam-3317	227	24	as	as	ADP
ejpam-3317	227	25	:	:	PUNCT
ejpam-3317	227	26	definition	definition	NOUN
ejpam-3317	227	27	15	15	NUM
ejpam-3317	227	28	.	.	PUNCT
ejpam-3317	228	1	let	let	AUX
ejpam-3317	228	2	(	(	PUNCT
ejpam-3317	228	3	x	x	NOUN
ejpam-3317	228	4	,	,	PUNCT
ejpam-3317	228	5	c1	c1	PROPN
ejpam-3317	228	6	)	)	PUNCT
ejpam-3317	228	7	and	and	CCONJ
ejpam-3317	228	8	(	(	PUNCT
ejpam-3317	228	9	y	y	PROPN
ejpam-3317	228	10	,	,	PUNCT
ejpam-3317	228	11	c2	c2	PROPN
ejpam-3317	228	12	)	)	PUNCT
ejpam-3317	228	13	be	be	VERB
ejpam-3317	228	14	closure	closure	NOUN
ejpam-3317	228	15	spaces	space	NOUN
ejpam-3317	228	16	.	.	PUNCT
ejpam-3317	229	1	a	a	DET
ejpam-3317	229	2	function	function	NOUN
ejpam-3317	229	3	f	f	NOUN
ejpam-3317	229	4	:	:	PUNCT
ejpam-3317	229	5	(	(	PUNCT
ejpam-3317	229	6	x	x	X
ejpam-3317	229	7	,	,	PUNCT
ejpam-3317	229	8	c1)→	c1)→	NOUN
ejpam-3317	229	9	(	(	PUNCT
ejpam-3317	229	10	y	y	PROPN
ejpam-3317	229	11	,	,	PUNCT
ejpam-3317	229	12	c2	c2	PROPN
ejpam-3317	229	13	)	)	PUNCT
ejpam-3317	229	14	is	be	AUX
ejpam-3317	229	15	called	call	VERB
ejpam-3317	229	16	pre	pre	ADJ
ejpam-3317	229	17	-	-	ADJ
ejpam-3317	229	18	irresolute	irresolute	ADJ
ejpam-3317	229	19	if	if	SCONJ
ejpam-3317	229	20	f−1(g	f−1(g	PROPN
ejpam-3317	229	21	)	)	PUNCT
ejpam-3317	229	22	is	be	AUX
ejpam-3317	229	23	preopen	preopen	ADJ
ejpam-3317	229	24	in	in	ADP
ejpam-3317	229	25	(	(	PUNCT
ejpam-3317	229	26	x	x	NOUN
ejpam-3317	229	27	,	,	PUNCT
ejpam-3317	229	28	c1	c1	PROPN
ejpam-3317	229	29	)	)	PUNCT
ejpam-3317	229	30	for	for	ADP
ejpam-3317	229	31	every	every	DET
ejpam-3317	229	32	preopen	preopen	NOUN
ejpam-3317	229	33	set	set	VERB
ejpam-3317	229	34	g	g	NOUN
ejpam-3317	229	35	in	in	ADP
ejpam-3317	229	36	(	(	PUNCT
ejpam-3317	229	37	y	y	PROPN
ejpam-3317	229	38	,	,	PUNCT
ejpam-3317	229	39	c2	c2	PROPN
ejpam-3317	229	40	)	)	PUNCT
ejpam-3317	229	41	.	.	PUNCT
ejpam-3317	230	1	proposition	proposition	NOUN
ejpam-3317	230	2	26	26	NUM
ejpam-3317	230	3	.	.	PUNCT
ejpam-3317	231	1	let	let	VERB
ejpam-3317	231	2	(	(	PUNCT
ejpam-3317	231	3	x	x	NOUN
ejpam-3317	231	4	,	,	PUNCT
ejpam-3317	231	5	c1	c1	PROPN
ejpam-3317	231	6	)	)	PUNCT
ejpam-3317	231	7	and	and	CCONJ
ejpam-3317	231	8	(	(	PUNCT
ejpam-3317	231	9	y	y	NOUN
ejpam-3317	231	10	,	,	PUNCT
ejpam-3317	231	11	c2)be	c2)be	ADJ
ejpam-3317	231	12	closure	closure	NOUN
ejpam-3317	231	13	spaces	space	NOUN
ejpam-3317	231	14	and	and	CCONJ
ejpam-3317	231	15	f	f	X
ejpam-3317	231	16	:	:	PUNCT
ejpam-3317	231	17	(	(	PUNCT
ejpam-3317	231	18	x	x	X
ejpam-3317	231	19	,	,	PUNCT
ejpam-3317	231	20	c1	c1	PROPN
ejpam-3317	231	21	)	)	PUNCT
ejpam-3317	231	22	→	→	PUNCT
ejpam-3317	231	23	(	(	PUNCT
ejpam-3317	231	24	y	y	PROPN
ejpam-3317	231	25	,	,	PUNCT
ejpam-3317	231	26	c2	c2	PROPN
ejpam-3317	231	27	)	)	PUNCT
ejpam-3317	231	28	be	be	AUX
ejpam-3317	231	29	a	a	DET
ejpam-3317	231	30	function	function	NOUN
ejpam-3317	231	31	.	.	PUNCT
ejpam-3317	232	1	then	then	ADV
ejpam-3317	232	2	f	f	PROPN
ejpam-3317	232	3	is	be	AUX
ejpam-3317	232	4	pre	pre	ADJ
ejpam-3317	232	5	-	-	ADJ
ejpam-3317	232	6	irresolute	irresolute	ADJ
ejpam-3317	232	7	if	if	SCONJ
ejpam-3317	232	8	and	and	CCONJ
ejpam-3317	232	9	only	only	ADV
ejpam-3317	232	10	if	if	SCONJ
ejpam-3317	232	11	f−1(b	f−1(b	PROPN
ejpam-3317	232	12	)	)	PUNCT
ejpam-3317	232	13	is	be	AUX
ejpam-3317	232	14	preclosed	preclose	VERB
ejpam-3317	232	15	in	in	ADP
ejpam-3317	232	16	(	(	PUNCT
ejpam-3317	232	17	x	x	NOUN
ejpam-3317	232	18	,	,	PUNCT
ejpam-3317	232	19	c1	c1	PROPN
ejpam-3317	232	20	)	)	PUNCT
ejpam-3317	232	21	,	,	PUNCT
ejpam-3317	232	22	whenever	whenever	SCONJ
ejpam-3317	232	23	b	b	NOUN
ejpam-3317	232	24	is	be	AUX
ejpam-3317	232	25	preclosed	preclose	VERB
ejpam-3317	232	26	in	in	ADP
ejpam-3317	232	27	(	(	PUNCT
ejpam-3317	232	28	y	y	PROPN
ejpam-3317	232	29	,	,	PUNCT
ejpam-3317	232	30	c2	c2	PROPN
ejpam-3317	232	31	)	)	PUNCT
ejpam-3317	232	32	.	.	PUNCT
ejpam-3317	233	1	proof	proof	NOUN
ejpam-3317	233	2	.	.	PUNCT
ejpam-3317	234	1	let	let	VERB
ejpam-3317	234	2	b	b	X
ejpam-3317	234	3	be	be	AUX
ejpam-3317	234	4	a	a	DET
ejpam-3317	234	5	preclosed	preclose	VERB
ejpam-3317	234	6	subset	subset	NOUN
ejpam-3317	234	7	of	of	ADP
ejpam-3317	234	8	(	(	PUNCT
ejpam-3317	234	9	y	y	PROPN
ejpam-3317	234	10	,	,	PUNCT
ejpam-3317	234	11	c2	c2	PROPN
ejpam-3317	234	12	)	)	PUNCT
ejpam-3317	234	13	.	.	PUNCT
ejpam-3317	235	1	then	then	ADV
ejpam-3317	235	2	y	y	PROPN
ejpam-3317	235	3	/	/	SYM
ejpam-3317	235	4	b	b	PROPN
ejpam-3317	235	5	is	be	AUX
ejpam-3317	235	6	preopen	preopen	ADJ
ejpam-3317	235	7	in	in	ADP
ejpam-3317	235	8	(	(	PUNCT
ejpam-3317	235	9	y	y	PROPN
ejpam-3317	235	10	,	,	PUNCT
ejpam-3317	235	11	c2	c2	PROPN
ejpam-3317	235	12	)	)	PUNCT
ejpam-3317	235	13	.	.	PUNCT
ejpam-3317	236	1	since	since	SCONJ
ejpam-3317	236	2	f	f	PROPN
ejpam-3317	236	3	:	:	PUNCT
ejpam-3317	236	4	(	(	PUNCT
ejpam-3317	236	5	x	x	X
ejpam-3317	236	6	,	,	PUNCT
ejpam-3317	236	7	c1	c1	PROPN
ejpam-3317	236	8	)	)	PUNCT
ejpam-3317	236	9	→	→	PUNCT
ejpam-3317	236	10	(	(	PUNCT
ejpam-3317	236	11	y	y	PROPN
ejpam-3317	236	12	,	,	PUNCT
ejpam-3317	236	13	c2	c2	PROPN
ejpam-3317	236	14	)	)	PUNCT
ejpam-3317	236	15	is	be	AUX
ejpam-3317	236	16	pre	pre	ADJ
ejpam-3317	236	17	-	-	ADJ
ejpam-3317	236	18	irresolute	irresolute	ADJ
ejpam-3317	236	19	,	,	PUNCT
ejpam-3317	236	20	f−1(y	f−1(y	PROPN
ejpam-3317	236	21	/	/	SYM
ejpam-3317	236	22	b	b	PROPN
ejpam-3317	236	23	)	)	PUNCT
ejpam-3317	236	24	is	be	AUX
ejpam-3317	236	25	preopen	preopen	ADJ
ejpam-3317	236	26	in	in	ADP
ejpam-3317	236	27	(	(	PUNCT
ejpam-3317	236	28	x	x	NOUN
ejpam-3317	236	29	,	,	PUNCT
ejpam-3317	236	30	c1	c1	PROPN
ejpam-3317	236	31	)	)	PUNCT
ejpam-3317	236	32	.	.	PUNCT
ejpam-3317	237	1	but	but	CCONJ
ejpam-3317	237	2	f−1(y	f−1(y	PROPN
ejpam-3317	237	3	/	/	SYM
ejpam-3317	237	4	b	b	PROPN
ejpam-3317	237	5	)	)	PUNCT
ejpam-3317	237	6	=	=	SYM
ejpam-3317	238	1	x	x	PUNCT
ejpam-3317	238	2	/f−1(b	/f−1(b	ADJ
ejpam-3317	238	3	)	)	PUNCT
ejpam-3317	238	4	,	,	PUNCT
ejpam-3317	238	5	so	so	SCONJ
ejpam-3317	238	6	that	that	SCONJ
ejpam-3317	238	7	f−1(b)is	f−1(b)is	PUNCT
ejpam-3317	238	8	preclosed	preclose	VERB
ejpam-3317	238	9	in	in	ADP
ejpam-3317	238	10	(	(	PUNCT
ejpam-3317	238	11	x	x	NOUN
ejpam-3317	238	12	,	,	PUNCT
ejpam-3317	238	13	c1	c1	PROPN
ejpam-3317	238	14	)	)	PUNCT
ejpam-3317	238	15	.	.	PUNCT
ejpam-3317	239	1	conversely	conversely	ADV
ejpam-3317	239	2	,	,	PUNCT
ejpam-3317	239	3	let	let	VERB
ejpam-3317	239	4	a	a	PRON
ejpam-3317	239	5	be	be	AUX
ejpam-3317	239	6	a	a	DET
ejpam-3317	239	7	preopen	preopen	ADJ
ejpam-3317	239	8	subset	subset	NOUN
ejpam-3317	239	9	in	in	ADP
ejpam-3317	239	10	(	(	PUNCT
ejpam-3317	239	11	y	y	PROPN
ejpam-3317	239	12	,	,	PUNCT
ejpam-3317	239	13	c2	c2	PROPN
ejpam-3317	239	14	)	)	PUNCT
ejpam-3317	239	15	.	.	PUNCT
ejpam-3317	240	1	then	then	ADV
ejpam-3317	240	2	y	y	PROPN
ejpam-3317	240	3	/	/	SYM
ejpam-3317	240	4	a	a	PRON
ejpam-3317	240	5	is	be	AUX
ejpam-3317	240	6	preclosed	preclose	VERB
ejpam-3317	240	7	in	in	ADP
ejpam-3317	240	8	(	(	PUNCT
ejpam-3317	240	9	y	y	PROPN
ejpam-3317	240	10	,	,	PUNCT
ejpam-3317	240	11	c2	c2	PROPN
ejpam-3317	240	12	)	)	PUNCT
ejpam-3317	240	13	.	.	PUNCT
ejpam-3317	241	1	by	by	ADP
ejpam-3317	241	2	the	the	DET
ejpam-3317	241	3	assumption	assumption	NOUN
ejpam-3317	241	4	,	,	PUNCT
ejpam-3317	241	5	f−1(y	f−1(y	PROPN
ejpam-3317	241	6	/	/	SYM
ejpam-3317	241	7	a	a	PRON
ejpam-3317	241	8	)	)	PUNCT
ejpam-3317	241	9	is	be	AUX
ejpam-3317	241	10	preclosed	preclose	VERB
ejpam-3317	241	11	in	in	ADP
ejpam-3317	241	12	(	(	PUNCT
ejpam-3317	241	13	x	x	NOUN
ejpam-3317	241	14	,	,	PUNCT
ejpam-3317	241	15	c1	c1	PROPN
ejpam-3317	241	16	)	)	PUNCT
ejpam-3317	241	17	.	.	PUNCT
ejpam-3317	242	1	but	but	CCONJ
ejpam-3317	242	2	f−1(y	f−1(y	PROPN
ejpam-3317	242	3	/	/	SYM
ejpam-3317	242	4	a	a	NOUN
ejpam-3317	242	5	)	)	PUNCT
ejpam-3317	242	6	=	=	SYM
ejpam-3317	243	1	x	x	X
ejpam-3317	243	2	/	/	SYM
ejpam-3317	243	3	f−1(a	f−1(a	NOUN
ejpam-3317	243	4	)	)	PUNCT
ejpam-3317	243	5	.	.	PUNCT
ejpam-3317	244	1	thus	thus	ADV
ejpam-3317	244	2	f−1(a	f−1(a	PROPN
ejpam-3317	244	3	)	)	PUNCT
ejpam-3317	244	4	is	be	AUX
ejpam-3317	244	5	preopen	preopen	ADJ
ejpam-3317	244	6	in	in	ADP
ejpam-3317	244	7	(	(	PUNCT
ejpam-3317	244	8	x	x	NOUN
ejpam-3317	244	9	,	,	PUNCT
ejpam-3317	244	10	c1	c1	PROPN
ejpam-3317	244	11	)	)	PUNCT
ejpam-3317	244	12	.	.	PUNCT
ejpam-3317	245	1	therefore	therefore	ADV
ejpam-3317	245	2	,	,	PUNCT
ejpam-3317	245	3	f	f	PROPN
ejpam-3317	245	4	is	be	AUX
ejpam-3317	245	5	pre	pre	ADJ
ejpam-3317	245	6	-	-	ADJ
ejpam-3317	245	7	irresolute	irresolute	ADJ
ejpam-3317	245	8	.	.	PUNCT
ejpam-3317	246	1	clearly	clearly	ADV
ejpam-3317	246	2	,	,	PUNCT
ejpam-3317	246	3	every	every	DET
ejpam-3317	246	4	pre	pre	ADJ
ejpam-3317	246	5	-	-	ADJ
ejpam-3317	246	6	irresolute	irresolute	ADJ
ejpam-3317	246	7	function	function	NOUN
ejpam-3317	246	8	is	be	AUX
ejpam-3317	246	9	pre	pre	ADJ
ejpam-3317	246	10	-	-	ADJ
ejpam-3317	246	11	continuous	continuous	ADJ
ejpam-3317	246	12	.	.	PUNCT
ejpam-3317	247	1	the	the	DET
ejpam-3317	247	2	converse	converse	NOUN
ejpam-3317	247	3	need	need	AUX
ejpam-3317	247	4	not	not	PART
ejpam-3317	247	5	be	be	AUX
ejpam-3317	247	6	true	true	ADJ
ejpam-3317	247	7	as	as	SCONJ
ejpam-3317	247	8	can	can	AUX
ejpam-3317	247	9	be	be	AUX
ejpam-3317	247	10	seen	see	VERB
ejpam-3317	247	11	from	from	ADP
ejpam-3317	247	12	the	the	DET
ejpam-3317	247	13	following	follow	VERB
ejpam-3317	247	14	example	example	NOUN
ejpam-3317	247	15	.	.	PUNCT
ejpam-3317	248	1	example	example	NOUN
ejpam-3317	249	1	2	2	NUM
ejpam-3317	249	2	.	.	PUNCT
ejpam-3317	249	3	let	let	VERB
ejpam-3317	249	4	x	x	PUNCT
ejpam-3317	249	5	=	=	PRON
ejpam-3317	249	6	{	{	PUNCT
ejpam-3317	249	7	1	1	NUM
ejpam-3317	249	8	,	,	PUNCT
ejpam-3317	249	9	2	2	NUM
ejpam-3317	249	10	,	,	PUNCT
ejpam-3317	249	11	3	3	NUM
ejpam-3317	249	12	}	}	PUNCT
ejpam-3317	249	13	=	=	SYM
ejpam-3317	249	14	y	y	PROPN
ejpam-3317	249	15	and	and	CCONJ
ejpam-3317	249	16	define	define	VERB
ejpam-3317	249	17	a	a	DET
ejpam-3317	249	18	closure	closure	NOUN
ejpam-3317	249	19	operator	operator	NOUN
ejpam-3317	249	20	c1	c1	NOUN
ejpam-3317	249	21	on	on	ADP
ejpam-3317	249	22	x	x	PUNCT
ejpam-3317	249	23	by	by	ADP
ejpam-3317	249	24	:	:	PUNCT
ejpam-3317	249	25	c1	c1	PROPN
ejpam-3317	249	26	(	(	PUNCT
ejpam-3317	249	27	a	a	PROPN
ejpam-3317	249	28	)	)	PUNCT
ejpam-3317	249	29	=	=	PUNCT
ejpam-3317	249	30			PUNCT
ejpam-3317	249	31	a	a	PRON
ejpam-3317	249	32	if	if	SCONJ
ejpam-3317	249	33	a	a	DET
ejpam-3317	249	34	∈	∈	PROPN
ejpam-3317	249	35	{	{	PUNCT
ejpam-3317	249	36	φ	φ	NOUN
ejpam-3317	249	37	,	,	PUNCT
ejpam-3317	249	38	{	{	PUNCT
ejpam-3317	249	39	3	3	NUM
ejpam-3317	249	40	}	}	PUNCT
ejpam-3317	249	41	}	}	PUNCT
ejpam-3317	249	42	{	{	PUNCT
ejpam-3317	249	43	1	1	NUM
ejpam-3317	249	44	,	,	PUNCT
ejpam-3317	249	45	2	2	NUM
ejpam-3317	249	46	}	}	PUNCT
ejpam-3317	249	47	if	if	SCONJ
ejpam-3317	249	48	a	a	PRON
ejpam-3317	249	49	=	=	X
ejpam-3317	249	50	{	{	PUNCT
ejpam-3317	249	51	1	1	NUM
ejpam-3317	249	52	}	}	PUNCT
ejpam-3317	249	53	{	{	PUNCT
ejpam-3317	249	54	2	2	NUM
ejpam-3317	249	55	,	,	PUNCT
ejpam-3317	249	56	3	3	NUM
ejpam-3317	249	57	}	}	PUNCT
ejpam-3317	249	58	if	if	SCONJ
ejpam-3317	249	59	a	a	PRON
ejpam-3317	249	60	=	=	X
ejpam-3317	249	61	{	{	PUNCT
ejpam-3317	249	62	2	2	NUM
ejpam-3317	249	63	}	}	PUNCT
ejpam-3317	249	64	x	x	NOUN
ejpam-3317	249	65	otherwise	otherwise	ADV
ejpam-3317	249	66	and	and	CCONJ
ejpam-3317	249	67	also	also	ADV
ejpam-3317	249	68	define	define	VERB
ejpam-3317	249	69	a	a	DET
ejpam-3317	249	70	closure	closure	NOUN
ejpam-3317	249	71	operator	operator	NOUN
ejpam-3317	249	72	c2	c2	PROPN
ejpam-3317	249	73	on	on	ADP
ejpam-3317	249	74	y	y	PROPN
ejpam-3317	249	75	by	by	ADP
ejpam-3317	249	76	:	:	PUNCT
ejpam-3317	249	77	c2	c2	PROPN
ejpam-3317	249	78	(	(	PUNCT
ejpam-3317	249	79	a	a	NOUN
ejpam-3317	249	80	)	)	PUNCT
ejpam-3317	249	81	=	=	PUNCT
ejpam-3317	249	82			PUNCT
ejpam-3317	249	83	a	a	INTJ
ejpam-3317	249	84	if	if	SCONJ
ejpam-3317	249	85	a	a	DET
ejpam-3317	249	86	=	=	SYM
ejpam-3317	249	87	φ	φ	X
ejpam-3317	249	88	{	{	PUNCT
ejpam-3317	249	89	1	1	NUM
ejpam-3317	249	90	,	,	PUNCT
ejpam-3317	249	91	3	3	X
ejpam-3317	249	92	}	}	PUNCT
ejpam-3317	249	93	if	if	SCONJ
ejpam-3317	249	94	a	a	PRON
ejpam-3317	249	95	=	=	X
ejpam-3317	249	96	{	{	PUNCT
ejpam-3317	249	97	1	1	NUM
ejpam-3317	249	98	}	}	PUNCT
ejpam-3317	249	99	{	{	PUNCT
ejpam-3317	249	100	2	2	NUM
ejpam-3317	249	101	,	,	PUNCT
ejpam-3317	249	102	3	3	NUM
ejpam-3317	249	103	}	}	PUNCT
ejpam-3317	249	104	if	if	SCONJ
ejpam-3317	249	105	a	a	PRON
ejpam-3317	249	106	=	=	X
ejpam-3317	249	107	{	{	PUNCT
ejpam-3317	249	108	2	2	NUM
ejpam-3317	249	109	}	}	PUNCT
ejpam-3317	249	110	y	y	PRON
ejpam-3317	249	111	otherwise	otherwise	ADV
ejpam-3317	249	112	let	let	VERB
ejpam-3317	249	113	f	f	NOUN
ejpam-3317	249	114	:	:	PUNCT
ejpam-3317	249	115	(	(	PUNCT
ejpam-3317	249	116	x	x	X
ejpam-3317	249	117	,	,	PUNCT
ejpam-3317	249	118	c1)→	c1)→	NOUN
ejpam-3317	249	119	(	(	PUNCT
ejpam-3317	249	120	y	y	PROPN
ejpam-3317	249	121	,	,	PUNCT
ejpam-3317	249	122	c2	c2	PROPN
ejpam-3317	249	123	)	)	PUNCT
ejpam-3317	249	124	be	be	VERB
ejpam-3317	249	125	the	the	DET
ejpam-3317	249	126	function	function	NOUN
ejpam-3317	249	127	defined	define	VERB
ejpam-3317	249	128	by	by	ADP
ejpam-3317	249	129	:	:	PUNCT
ejpam-3317	249	130	f	f	PROPN
ejpam-3317	249	131	(	(	PUNCT
ejpam-3317	249	132	x	x	X
ejpam-3317	249	133	)	)	PUNCT
ejpam-3317	249	134	=	=	PUNCT
ejpam-3317	250	1			PUNCT
ejpam-3317	250	2	1	1	NUM
ejpam-3317	250	3	2	2	NUM
ejpam-3317	250	4	3	3	NUM
ejpam-3317	250	5	if	if	SCONJ
ejpam-3317	250	6	x	x	PROPN
ejpam-3317	250	7	=	=	SYM
ejpam-3317	250	8	1	1	NUM
ejpam-3317	250	9	if	if	SCONJ
ejpam-3317	250	10	x	x	SYM
ejpam-3317	250	11	=	=	SYM
ejpam-3317	250	12	2	2	NUM
ejpam-3317	250	13	if	if	SCONJ
ejpam-3317	250	14	x	x	PROPN
ejpam-3317	250	15	=	=	SYM
ejpam-3317	250	16	3	3	NUM
ejpam-3317	250	17	h.m	h.m	PROPN
ejpam-3317	250	18	.	.	PROPN
ejpam-3317	250	19	darwesh	darwesh	PROPN
ejpam-3317	250	20	,	,	PUNCT
ejpam-3317	250	21	s.f	s.f	PROPN
ejpam-3317	250	22	.	.	PROPN
ejpam-3317	250	23	namiq	namiq	PROPN
ejpam-3317	250	24	/	/	SYM
ejpam-3317	250	25	eur	eur	PROPN
ejpam-3317	250	26	.	.	PUNCT
ejpam-3317	251	1	j.	j.	PROPN
ejpam-3317	251	2	pure	pure	PROPN
ejpam-3317	251	3	appl	appl	PROPN
ejpam-3317	251	4	.	.	PROPN
ejpam-3317	251	5	math	math	PROPN
ejpam-3317	251	6	,	,	PUNCT
ejpam-3317	251	7	12	12	NUM
ejpam-3317	251	8	(	(	PUNCT
ejpam-3317	251	9	3	3	NUM
ejpam-3317	251	10	)	)	PUNCT
ejpam-3317	251	11	(	(	PUNCT
ejpam-3317	251	12	2019	2019	NUM
ejpam-3317	251	13	)	)	PUNCT
ejpam-3317	251	14	,	,	PUNCT
ejpam-3317	251	15	1082	1082	NUM
ejpam-3317	251	16	-	-	SYM
ejpam-3317	251	17	1095	1095	NUM
ejpam-3317	251	18	1090	1090	NUM
ejpam-3317	251	19	the	the	DET
ejpam-3317	251	20	family	family	NOUN
ejpam-3317	251	21	of	of	ADP
ejpam-3317	251	22	all	all	DET
ejpam-3317	251	23	open	open	ADJ
ejpam-3317	251	24	sets	set	NOUN
ejpam-3317	251	25	with	with	ADP
ejpam-3317	251	26	respect	respect	NOUN
ejpam-3317	251	27	to	to	ADP
ejpam-3317	251	28	c1={φ	c1={φ	PROPN
ejpam-3317	251	29	,	,	PUNCT
ejpam-3317	251	30	{	{	PUNCT
ejpam-3317	251	31	1	1	NUM
ejpam-3317	251	32	,	,	PUNCT
ejpam-3317	251	33	2	2	NUM
ejpam-3317	251	34	}	}	PUNCT
ejpam-3317	251	35	,	,	PUNCT
ejpam-3317	251	36	x	x	X
ejpam-3317	251	37	}	}	PUNCT
ejpam-3317	251	38	po	po	NOUN
ejpam-3317	251	39	(	(	PUNCT
ejpam-3317	251	40	x	x	PROPN
ejpam-3317	251	41	,	,	PUNCT
ejpam-3317	251	42	c1	c1	PROPN
ejpam-3317	251	43	)	)	PUNCT
ejpam-3317	251	44	=	=	PRON
ejpam-3317	251	45	{	{	PUNCT
ejpam-3317	251	46	φ	φ	PROPN
ejpam-3317	251	47	,	,	PUNCT
ejpam-3317	251	48	{	{	PUNCT
ejpam-3317	251	49	1	1	NUM
ejpam-3317	251	50	}	}	PUNCT
ejpam-3317	251	51	,	,	PUNCT
ejpam-3317	251	52	{	{	PUNCT
ejpam-3317	251	53	1	1	NUM
ejpam-3317	251	54	,	,	PUNCT
ejpam-3317	251	55	2	2	NUM
ejpam-3317	251	56	}	}	PUNCT
ejpam-3317	251	57	,	,	PUNCT
ejpam-3317	251	58	{	{	PUNCT
ejpam-3317	251	59	1	1	NUM
ejpam-3317	251	60	,	,	PUNCT
ejpam-3317	251	61	3	3	NUM
ejpam-3317	251	62	}	}	PUNCT
ejpam-3317	251	63	,	,	PUNCT
ejpam-3317	251	64	{	{	PUNCT
ejpam-3317	251	65	2	2	NUM
ejpam-3317	251	66	,	,	PUNCT
ejpam-3317	251	67	3	3	NUM
ejpam-3317	251	68	}	}	PUNCT
ejpam-3317	251	69	,	,	PUNCT
ejpam-3317	251	70	x	x	NOUN
ejpam-3317	251	71	}	}	PUNCT
ejpam-3317	251	72	.	.	PUNCT
ejpam-3317	252	1	the	the	DET
ejpam-3317	252	2	family	family	NOUN
ejpam-3317	252	3	of	of	ADP
ejpam-3317	252	4	all	all	DET
ejpam-3317	252	5	open	open	ADJ
ejpam-3317	252	6	sets	set	NOUN
ejpam-3317	252	7	with	with	ADP
ejpam-3317	252	8	respect	respect	NOUN
ejpam-3317	252	9	to	to	ADP
ejpam-3317	252	10	c2={φ	c2={φ	PROPN
ejpam-3317	252	11	,	,	PUNCT
ejpam-3317	252	12	x	x	NOUN
ejpam-3317	252	13	}	}	PUNCT
ejpam-3317	252	14	po	po	NOUN
ejpam-3317	252	15	(	(	PUNCT
ejpam-3317	252	16	x	x	PROPN
ejpam-3317	252	17	,	,	PUNCT
ejpam-3317	252	18	c2	c2	PROPN
ejpam-3317	252	19	)	)	PUNCT
ejpam-3317	252	20	=	=	PRON
ejpam-3317	252	21	{	{	PUNCT
ejpam-3317	252	22	φ	φ	PROPN
ejpam-3317	252	23	,	,	PUNCT
ejpam-3317	252	24	{	{	PUNCT
ejpam-3317	252	25	3	3	NUM
ejpam-3317	252	26	}	}	PUNCT
ejpam-3317	252	27	,	,	PUNCT
ejpam-3317	252	28	{	{	PUNCT
ejpam-3317	252	29	1	1	NUM
ejpam-3317	252	30	,	,	PUNCT
ejpam-3317	252	31	2	2	NUM
ejpam-3317	252	32	}	}	PUNCT
ejpam-3317	252	33	,	,	PUNCT
ejpam-3317	252	34	{	{	PUNCT
ejpam-3317	252	35	1	1	NUM
ejpam-3317	252	36	,	,	PUNCT
ejpam-3317	252	37	3	3	NUM
ejpam-3317	252	38	}	}	PUNCT
ejpam-3317	252	39	,	,	PUNCT
ejpam-3317	252	40	{	{	PUNCT
ejpam-3317	252	41	2	2	NUM
ejpam-3317	252	42	,	,	PUNCT
ejpam-3317	252	43	3	3	NUM
ejpam-3317	252	44	}	}	PUNCT
ejpam-3317	252	45	,	,	PUNCT
ejpam-3317	252	46	x	x	X
ejpam-3317	252	47	}	}	PUNCT
ejpam-3317	252	48	.	.	PUNCT
ejpam-3317	253	1	then	then	ADV
ejpam-3317	253	2	f	f	PROPN
ejpam-3317	253	3	is	be	AUX
ejpam-3317	253	4	pre	pre	ADJ
ejpam-3317	253	5	-	-	ADJ
ejpam-3317	253	6	continuous	continuous	ADJ
ejpam-3317	253	7	but	but	CCONJ
ejpam-3317	253	8	not	not	PART
ejpam-3317	253	9	pre	pre	ADJ
ejpam-3317	253	10	-	-	NOUN
ejpam-3317	253	11	irresolute	irresolute	ADJ
ejpam-3317	253	12	because	because	SCONJ
ejpam-3317	253	13	{	{	PUNCT
ejpam-3317	253	14	3	3	X
ejpam-3317	253	15	}	}	PUNCT
ejpam-3317	253	16	is	be	AUX
ejpam-3317	253	17	preopen	preopen	ADJ
ejpam-3317	253	18	in	in	ADP
ejpam-3317	253	19	(	(	PUNCT
ejpam-3317	253	20	y	y	PROPN
ejpam-3317	253	21	,	,	PUNCT
ejpam-3317	253	22	c2	c2	PROPN
ejpam-3317	253	23	)	)	PUNCT
ejpam-3317	253	24	but	but	CCONJ
ejpam-3317	253	25	f−1({3	f−1({3	NOUN
ejpam-3317	253	26	}	}	PUNCT
ejpam-3317	253	27	)	)	PUNCT
ejpam-3317	254	1	=	=	PUNCT
ejpam-3317	254	2	{	{	PUNCT
ejpam-3317	254	3	3	3	NUM
ejpam-3317	254	4	}	}	PUNCT
ejpam-3317	254	5	is	be	AUX
ejpam-3317	254	6	not	not	PART
ejpam-3317	254	7	preopen	preopen	ADJ
ejpam-3317	254	8	in	in	ADP
ejpam-3317	254	9	(	(	PUNCT
ejpam-3317	254	10	x	x	NOUN
ejpam-3317	254	11	,	,	PUNCT
ejpam-3317	254	12	c1	c1	PROPN
ejpam-3317	254	13	)	)	PUNCT
ejpam-3317	254	14	.	.	PUNCT
ejpam-3317	255	1	proposition	proposition	NOUN
ejpam-3317	255	2	27	27	NUM
ejpam-3317	255	3	.	.	PUNCT
ejpam-3317	256	1	let	let	VERB
ejpam-3317	256	2	(	(	PUNCT
ejpam-3317	256	3	x	x	NOUN
ejpam-3317	256	4	,	,	PUNCT
ejpam-3317	256	5	c1	c1	PROPN
ejpam-3317	256	6	)	)	PUNCT
ejpam-3317	256	7	,	,	PUNCT
ejpam-3317	256	8	(	(	PUNCT
ejpam-3317	256	9	y	y	PROPN
ejpam-3317	256	10	,	,	PUNCT
ejpam-3317	256	11	c2	c2	PROPN
ejpam-3317	256	12	)	)	PUNCT
ejpam-3317	256	13	and	and	CCONJ
ejpam-3317	256	14	(	(	PUNCT
ejpam-3317	256	15	z	z	NOUN
ejpam-3317	256	16	,	,	PUNCT
ejpam-3317	256	17	c3	c3	PROPN
ejpam-3317	256	18	)	)	PUNCT
ejpam-3317	256	19	be	be	VERB
ejpam-3317	256	20	closure	closure	NOUN
ejpam-3317	256	21	spaces	space	NOUN
ejpam-3317	256	22	.	.	PUNCT
ejpam-3317	257	1	if	if	SCONJ
ejpam-3317	257	2	f	f	PROPN
ejpam-3317	257	3	:	:	PUNCT
ejpam-3317	257	4	(	(	PUNCT
ejpam-3317	257	5	x	x	X
ejpam-3317	257	6	,	,	PUNCT
ejpam-3317	257	7	c1)→	c1)→	NOUN
ejpam-3317	257	8	(	(	PUNCT
ejpam-3317	257	9	y	y	PROPN
ejpam-3317	257	10	,	,	PUNCT
ejpam-3317	257	11	c2	c2	PROPN
ejpam-3317	257	12	)	)	PUNCT
ejpam-3317	257	13	is	be	AUX
ejpam-3317	257	14	a	a	DET
ejpam-3317	257	15	pre	pre	ADJ
ejpam-3317	257	16	-	-	ADJ
ejpam-3317	257	17	irresolute	irresolute	ADJ
ejpam-3317	257	18	function	function	NOUN
ejpam-3317	257	19	and	and	CCONJ
ejpam-3317	257	20	g	g	NOUN
ejpam-3317	257	21	:	:	PUNCT
ejpam-3317	257	22	(	(	PUNCT
ejpam-3317	257	23	y	y	NOUN
ejpam-3317	257	24	,	,	PUNCT
ejpam-3317	257	25	c2)→	c2)→	X
ejpam-3317	257	26	(	(	PUNCT
ejpam-3317	257	27	z	z	NOUN
ejpam-3317	257	28	,	,	PUNCT
ejpam-3317	257	29	c3	c3	PROPN
ejpam-3317	257	30	)	)	PUNCT
ejpam-3317	257	31	is	be	AUX
ejpam-3317	257	32	a	a	DET
ejpam-3317	257	33	pre	pre	ADJ
ejpam-3317	257	34	-	-	ADJ
ejpam-3317	257	35	continuous	continuous	ADJ
ejpam-3317	257	36	function	function	NOUN
ejpam-3317	257	37	,	,	PUNCT
ejpam-3317	257	38	then	then	ADV
ejpam-3317	257	39	the	the	DET
ejpam-3317	257	40	composition	composition	NOUN
ejpam-3317	257	41	gof	gof	NOUN
ejpam-3317	257	42	:	:	PUNCT
ejpam-3317	257	43	(	(	PUNCT
ejpam-3317	257	44	x	x	X
ejpam-3317	257	45	,	,	PUNCT
ejpam-3317	257	46	c1)→	c1)→	NOUN
ejpam-3317	257	47	(	(	PUNCT
ejpam-3317	257	48	z	z	NOUN
ejpam-3317	257	49	,	,	PUNCT
ejpam-3317	257	50	c3	c3	PROPN
ejpam-3317	257	51	)	)	PUNCT
ejpam-3317	257	52	is	be	AUX
ejpam-3317	257	53	pre	pre	ADJ
ejpam-3317	257	54	-	-	ADJ
ejpam-3317	257	55	continuous	continuous	ADJ
ejpam-3317	257	56	.	.	PUNCT
ejpam-3317	258	1	proof	proof	NOUN
ejpam-3317	258	2	.	.	PUNCT
ejpam-3317	259	1	let	let	VERB
ejpam-3317	259	2	g	g	PRON
ejpam-3317	259	3	be	be	AUX
ejpam-3317	259	4	an	an	DET
ejpam-3317	259	5	open	open	ADJ
ejpam-3317	259	6	subset	subset	NOUN
ejpam-3317	259	7	of	of	ADP
ejpam-3317	259	8	(	(	PUNCT
ejpam-3317	259	9	z	z	PROPN
ejpam-3317	259	10	,	,	PUNCT
ejpam-3317	259	11	c3	c3	PROPN
ejpam-3317	259	12	)	)	PUNCT
ejpam-3317	259	13	.	.	PUNCT
ejpam-3317	260	1	then	then	ADV
ejpam-3317	260	2	g−1(g	g−1(g	PROPN
ejpam-3317	260	3	)	)	PUNCT
ejpam-3317	260	4	is	be	AUX
ejpam-3317	260	5	a	a	DET
ejpam-3317	260	6	preopen	preopen	ADJ
ejpam-3317	260	7	subset	subset	NOUN
ejpam-3317	260	8	of	of	ADP
ejpam-3317	260	9	(	(	PUNCT
ejpam-3317	260	10	y	y	PROPN
ejpam-3317	260	11	,	,	PUNCT
ejpam-3317	260	12	c2	c2	PROPN
ejpam-3317	260	13	)	)	PUNCT
ejpam-3317	260	14	as	as	SCONJ
ejpam-3317	260	15	g	g	PROPN
ejpam-3317	260	16	is	be	AUX
ejpam-3317	260	17	pre	pre	ADJ
ejpam-3317	260	18	-	-	ADJ
ejpam-3317	260	19	continuous	continuous	ADJ
ejpam-3317	260	20	.	.	PUNCT
ejpam-3317	261	1	hence	hence	ADV
ejpam-3317	261	2	,	,	PUNCT
ejpam-3317	261	3	f−1(g−1(g	f−1(g−1(g	PROPN
ejpam-3317	261	4	)	)	PUNCT
ejpam-3317	261	5	)	)	PUNCT
ejpam-3317	262	1	is	be	AUX
ejpam-3317	262	2	preopen	preopen	ADJ
ejpam-3317	262	3	in	in	ADP
ejpam-3317	262	4	(	(	PUNCT
ejpam-3317	262	5	x	x	NOUN
ejpam-3317	262	6	,	,	PUNCT
ejpam-3317	262	7	c1	c1	PROPN
ejpam-3317	262	8	)	)	PUNCT
ejpam-3317	262	9	because	because	SCONJ
ejpam-3317	262	10	f	f	PROPN
ejpam-3317	262	11	is	be	AUX
ejpam-3317	262	12	pre	pre	ADJ
ejpam-3317	262	13	-	-	ADJ
ejpam-3317	262	14	irresolute	irresolute	ADJ
ejpam-3317	262	15	.	.	PUNCT
ejpam-3317	263	1	thus	thus	ADV
ejpam-3317	263	2	,	,	PUNCT
ejpam-3317	263	3	gof	gof	PROPN
ejpam-3317	263	4	is	be	AUX
ejpam-3317	263	5	pre	pre	ADJ
ejpam-3317	263	6	-	-	ADJ
ejpam-3317	263	7	continuous	continuous	ADJ
ejpam-3317	263	8	.	.	PUNCT
ejpam-3317	264	1	the	the	DET
ejpam-3317	264	2	following	follow	VERB
ejpam-3317	264	3	statements	statement	NOUN
ejpam-3317	264	4	are	be	AUX
ejpam-3317	264	5	evident	evident	ADJ
ejpam-3317	264	6	:	:	PUNCT
ejpam-3317	264	7	proposition	proposition	NOUN
ejpam-3317	264	8	28	28	NUM
ejpam-3317	264	9	.	.	PUNCT
ejpam-3317	265	1	let	let	VERB
ejpam-3317	265	2	(	(	PUNCT
ejpam-3317	265	3	x	x	NOUN
ejpam-3317	265	4	,	,	PUNCT
ejpam-3317	265	5	c1	c1	PROPN
ejpam-3317	265	6	)	)	PUNCT
ejpam-3317	265	7	,	,	PUNCT
ejpam-3317	265	8	(	(	PUNCT
ejpam-3317	265	9	y	y	PROPN
ejpam-3317	265	10	,	,	PUNCT
ejpam-3317	265	11	c2	c2	PROPN
ejpam-3317	265	12	)	)	PUNCT
ejpam-3317	265	13	and	and	CCONJ
ejpam-3317	265	14	(	(	PUNCT
ejpam-3317	265	15	z	z	NOUN
ejpam-3317	265	16	,	,	PUNCT
ejpam-3317	265	17	c3	c3	PROPN
ejpam-3317	265	18	)	)	PUNCT
ejpam-3317	265	19	be	be	VERB
ejpam-3317	265	20	closure	closure	NOUN
ejpam-3317	265	21	spaces	space	NOUN
ejpam-3317	265	22	.	.	PUNCT
ejpam-3317	266	1	if	if	SCONJ
ejpam-3317	266	2	f	f	PROPN
ejpam-3317	266	3	:	:	PUNCT
ejpam-3317	266	4	(	(	PUNCT
ejpam-3317	266	5	x	x	X
ejpam-3317	266	6	,	,	PUNCT
ejpam-3317	266	7	c1)→	c1)→	NOUN
ejpam-3317	266	8	(	(	PUNCT
ejpam-3317	266	9	y	y	PROPN
ejpam-3317	266	10	,	,	PUNCT
ejpam-3317	266	11	c2	c2	PROPN
ejpam-3317	266	12	)	)	PUNCT
ejpam-3317	266	13	and	and	CCONJ
ejpam-3317	266	14	g	g	NOUN
ejpam-3317	266	15	:	:	PUNCT
ejpam-3317	266	16	(	(	PUNCT
ejpam-3317	266	17	y	y	NOUN
ejpam-3317	266	18	,	,	PUNCT
ejpam-3317	266	19	c2)→	c2)→	X
ejpam-3317	266	20	(	(	PUNCT
ejpam-3317	266	21	z	z	NOUN
ejpam-3317	266	22	,	,	PUNCT
ejpam-3317	266	23	c3	c3	PROPN
ejpam-3317	266	24	)	)	PUNCT
ejpam-3317	266	25	are	be	AUX
ejpam-3317	266	26	pre	pre	ADJ
ejpam-3317	266	27	-	-	ADJ
ejpam-3317	266	28	irresolute	irresolute	ADJ
ejpam-3317	266	29	,	,	PUNCT
ejpam-3317	266	30	then	then	ADV
ejpam-3317	266	31	gof	gof	NOUN
ejpam-3317	266	32	:	:	PUNCT
ejpam-3317	266	33	(	(	PUNCT
ejpam-3317	266	34	x	x	X
ejpam-3317	266	35	,	,	PUNCT
ejpam-3317	266	36	c1)→	c1)→	NOUN
ejpam-3317	266	37	(	(	PUNCT
ejpam-3317	266	38	z	z	NOUN
ejpam-3317	266	39	,	,	PUNCT
ejpam-3317	266	40	c3)is	c3)is	ADJ
ejpam-3317	266	41	pre	pre	NOUN
ejpam-3317	266	42	-	-	ADJ
ejpam-3317	266	43	irresolute	irresolute	ADJ
ejpam-3317	266	44	.	.	PUNCT
ejpam-3317	267	1	proof	proof	NOUN
ejpam-3317	267	2	.	.	PUNCT
ejpam-3317	268	1	obvious	obvious	ADJ
ejpam-3317	268	2	.	.	PUNCT
ejpam-3317	269	1	proposition	proposition	NOUN
ejpam-3317	269	2	29	29	NUM
ejpam-3317	269	3	.	.	PUNCT
ejpam-3317	270	1	let	let	VERB
ejpam-3317	270	2	(	(	PUNCT
ejpam-3317	270	3	x	x	NOUN
ejpam-3317	270	4	,	,	PUNCT
ejpam-3317	270	5	c1	c1	PROPN
ejpam-3317	270	6	)	)	PUNCT
ejpam-3317	270	7	and	and	CCONJ
ejpam-3317	270	8	(	(	PUNCT
ejpam-3317	270	9	z	z	NOUN
ejpam-3317	270	10	,	,	PUNCT
ejpam-3317	270	11	c3	c3	PROPN
ejpam-3317	270	12	)	)	PUNCT
ejpam-3317	270	13	be	be	VERB
ejpam-3317	270	14	closure	closure	NOUN
ejpam-3317	270	15	spaces	space	NOUN
ejpam-3317	270	16	and	and	CCONJ
ejpam-3317	270	17	(	(	PUNCT
ejpam-3317	270	18	y	y	PROPN
ejpam-3317	270	19	,	,	PUNCT
ejpam-3317	270	20	c2	c2	PROPN
ejpam-3317	270	21	)	)	PUNCT
ejpam-3317	270	22	be	be	AUX
ejpam-3317	270	23	a	a	DET
ejpam-3317	270	24	tp	tp	NOUN
ejpam-3317	270	25	-space	-space	NOUN
ejpam-3317	270	26	.	.	PUNCT
ejpam-3317	271	1	if	if	SCONJ
ejpam-3317	271	2	f	f	PROPN
ejpam-3317	271	3	:	:	PUNCT
ejpam-3317	271	4	(	(	PUNCT
ejpam-3317	271	5	x	x	X
ejpam-3317	271	6	,	,	PUNCT
ejpam-3317	271	7	c1)→	c1)→	NOUN
ejpam-3317	271	8	(	(	PUNCT
ejpam-3317	271	9	y	y	PROPN
ejpam-3317	271	10	,	,	PUNCT
ejpam-3317	271	11	c2	c2	PROPN
ejpam-3317	271	12	)	)	PUNCT
ejpam-3317	271	13	is	be	AUX
ejpam-3317	271	14	a	a	DET
ejpam-3317	271	15	pre	pre	ADJ
ejpam-3317	271	16	-	-	ADJ
ejpam-3317	271	17	continuous	continuous	ADJ
ejpam-3317	271	18	function	function	NOUN
ejpam-3317	271	19	and	and	CCONJ
ejpam-3317	271	20	g	g	NOUN
ejpam-3317	271	21	:	:	PUNCT
ejpam-3317	271	22	(	(	PUNCT
ejpam-3317	271	23	y	y	NOUN
ejpam-3317	271	24	,	,	PUNCT
ejpam-3317	271	25	c2)→	c2)→	X
ejpam-3317	271	26	(	(	PUNCT
ejpam-3317	271	27	z	z	NOUN
ejpam-3317	271	28	,	,	PUNCT
ejpam-3317	271	29	c3	c3	PROPN
ejpam-3317	271	30	)	)	PUNCT
ejpam-3317	271	31	is	be	AUX
ejpam-3317	271	32	a	a	DET
ejpam-3317	271	33	pre	pre	ADJ
ejpam-3317	271	34	-	-	ADJ
ejpam-3317	271	35	irresolute	irresolute	ADJ
ejpam-3317	271	36	function	function	NOUN
ejpam-3317	271	37	,	,	PUNCT
ejpam-3317	271	38	then	then	ADV
ejpam-3317	271	39	the	the	DET
ejpam-3317	271	40	composition	composition	NOUN
ejpam-3317	271	41	gof	gof	NOUN
ejpam-3317	271	42	:	:	PUNCT
ejpam-3317	271	43	(	(	PUNCT
ejpam-3317	271	44	x	x	X
ejpam-3317	271	45	,	,	PUNCT
ejpam-3317	271	46	c1)→	c1)→	NOUN
ejpam-3317	271	47	(	(	PUNCT
ejpam-3317	271	48	z	z	NOUN
ejpam-3317	271	49	,	,	PUNCT
ejpam-3317	271	50	c3)is	c3)is	ADJ
ejpam-3317	271	51	pre	pre	NOUN
ejpam-3317	271	52	-	-	ADJ
ejpam-3317	271	53	irresolute	irresolute	ADJ
ejpam-3317	271	54	.	.	PUNCT
ejpam-3317	272	1	proof	proof	NOUN
ejpam-3317	272	2	.	.	PUNCT
ejpam-3317	273	1	obvious	obvious	ADJ
ejpam-3317	273	2	.	.	PUNCT
ejpam-3317	274	1	proposition	proposition	NOUN
ejpam-3317	274	2	30	30	NUM
ejpam-3317	274	3	.	.	PUNCT
ejpam-3317	275	1	let	let	AUX
ejpam-3317	275	2	(	(	PUNCT
ejpam-3317	275	3	x	x	NOUN
ejpam-3317	275	4	,	,	PUNCT
ejpam-3317	275	5	c1	c1	PROPN
ejpam-3317	275	6	)	)	PUNCT
ejpam-3317	275	7	and	and	CCONJ
ejpam-3317	275	8	(	(	PUNCT
ejpam-3317	275	9	y	y	PROPN
ejpam-3317	275	10	,	,	PUNCT
ejpam-3317	275	11	c2	c2	PROPN
ejpam-3317	275	12	)	)	PUNCT
ejpam-3317	275	13	be	be	VERB
ejpam-3317	275	14	closure	closure	NOUN
ejpam-3317	275	15	spaces	space	NOUN
ejpam-3317	275	16	and	and	CCONJ
ejpam-3317	275	17	f	f	X
ejpam-3317	275	18	:	:	PUNCT
ejpam-3317	275	19	(	(	PUNCT
ejpam-3317	275	20	x	x	X
ejpam-3317	275	21	,	,	PUNCT
ejpam-3317	275	22	c1	c1	PROPN
ejpam-3317	275	23	)	)	PUNCT
ejpam-3317	275	24	→	→	PUNCT
ejpam-3317	275	25	(	(	PUNCT
ejpam-3317	275	26	y	y	PROPN
ejpam-3317	275	27	,	,	PUNCT
ejpam-3317	275	28	c2	c2	PROPN
ejpam-3317	275	29	)	)	PUNCT
ejpam-3317	275	30	be	be	VERB
ejpam-3317	275	31	a	a	DET
ejpam-3317	275	32	bijective	bijective	ADJ
ejpam-3317	275	33	function	function	NOUN
ejpam-3317	275	34	.	.	PUNCT
ejpam-3317	276	1	if	if	SCONJ
ejpam-3317	276	2	f	f	PROPN
ejpam-3317	276	3	and	and	CCONJ
ejpam-3317	276	4	f−1are	f−1are	VERB
ejpam-3317	276	5	continuous	continuous	ADJ
ejpam-3317	276	6	,	,	PUNCT
ejpam-3317	276	7	then	then	ADV
ejpam-3317	276	8	f	f	PROPN
ejpam-3317	276	9	and	and	CCONJ
ejpam-3317	276	10	f−1	f−1	PROPN
ejpam-3317	276	11	are	be	AUX
ejpam-3317	276	12	pre	pre	ADJ
ejpam-3317	276	13	-	-	ADJ
ejpam-3317	276	14	irresolute	irresolute	ADJ
ejpam-3317	276	15	.	.	PUNCT
ejpam-3317	277	1	proof	proof	NOUN
ejpam-3317	277	2	.	.	PUNCT
ejpam-3317	278	1	let	let	VERB
ejpam-3317	278	2	b	b	X
ejpam-3317	278	3	be	be	AUX
ejpam-3317	278	4	a	a	DET
ejpam-3317	278	5	preopen	preopen	ADJ
ejpam-3317	278	6	subset	subset	NOUN
ejpam-3317	278	7	of	of	ADP
ejpam-3317	278	8	(	(	PUNCT
ejpam-3317	278	9	y	y	PROPN
ejpam-3317	278	10	,	,	PUNCT
ejpam-3317	278	11	c2	c2	PROPN
ejpam-3317	278	12	)	)	PUNCT
ejpam-3317	278	13	.	.	PUNCT
ejpam-3317	279	1	then	then	ADV
ejpam-3317	279	2	there	there	PRON
ejpam-3317	279	3	exists	exist	VERB
ejpam-3317	279	4	an	an	DET
ejpam-3317	279	5	open	open	ADJ
ejpam-3317	279	6	set	set	NOUN
ejpam-3317	279	7	h	h	NOUN
ejpam-3317	279	8	in	in	ADP
ejpam-3317	279	9	(	(	PUNCT
ejpam-3317	279	10	y	y	PROPN
ejpam-3317	279	11	,	,	PUNCT
ejpam-3317	279	12	c2	c2	PROPN
ejpam-3317	279	13	)	)	PUNCT
ejpam-3317	280	1	such	such	ADJ
ejpam-3317	280	2	that	that	SCONJ
ejpam-3317	280	3	b	b	NOUN
ejpam-3317	280	4	⊆	⊆	NUM
ejpam-3317	280	5	h	h	NOUN
ejpam-3317	280	6	⊆	⊆	NUM
ejpam-3317	280	7	c2(b	c2(b	PROPN
ejpam-3317	280	8	)	)	PUNCT
ejpam-3317	280	9	,	,	PUNCT
ejpam-3317	280	10	hence	hence	ADV
ejpam-3317	280	11	f−1(b	f−1(b	PROPN
ejpam-3317	280	12	)	)	PUNCT
ejpam-3317	280	13	⊆	⊆	NUM
ejpam-3317	280	14	f−1(h	f−1(h	PROPN
ejpam-3317	280	15	)	)	PUNCT
ejpam-3317	280	16	⊆	⊆	NUM
ejpam-3317	280	17	f−1(c2(b	f−1(c2(b	NOUN
ejpam-3317	280	18	)	)	PUNCT
ejpam-3317	280	19	)	)	PUNCT
ejpam-3317	280	20	.	.	PUNCT
ejpam-3317	281	1	since	since	SCONJ
ejpam-3317	281	2	f−1	f−1	PROPN
ejpam-3317	281	3	is	be	AUX
ejpam-3317	281	4	continuous	continuous	ADJ
ejpam-3317	281	5	,	,	PUNCT
ejpam-3317	281	6	f−1(c2(b	f−1(c2(b	ADJ
ejpam-3317	281	7	)	)	PUNCT
ejpam-3317	281	8	)	)	PUNCT
ejpam-3317	282	1	⊆	⊆	NUM
ejpam-3317	282	2	c1f	c1f	X
ejpam-3317	282	3	−1	−1	NOUN
ejpam-3317	282	4	(	(	PUNCT
ejpam-3317	282	5	h	h	NOUN
ejpam-3317	282	6	)	)	PUNCT
ejpam-3317	282	7	.	.	PUNCT
ejpam-3317	283	1	but	but	CCONJ
ejpam-3317	283	2	f	f	PROPN
ejpam-3317	283	3	is	be	AUX
ejpam-3317	283	4	continuous	continuous	ADJ
ejpam-3317	283	5	.	.	PUNCT
ejpam-3317	284	1	thus	thus	ADV
ejpam-3317	284	2	f−1(h	f−1(h	PROPN
ejpam-3317	284	3	)	)	PUNCT
ejpam-3317	284	4	)	)	PUNCT
ejpam-3317	284	5	is	be	AUX
ejpam-3317	284	6	open	open	ADJ
ejpam-3317	284	7	in	in	ADP
ejpam-3317	284	8	(	(	PUNCT
ejpam-3317	284	9	x	x	NOUN
ejpam-3317	284	10	,	,	PUNCT
ejpam-3317	284	11	c1	c1	PROPN
ejpam-3317	284	12	)	)	PUNCT
ejpam-3317	284	13	.	.	PUNCT
ejpam-3317	285	1	hence	hence	ADV
ejpam-3317	285	2	,	,	PUNCT
ejpam-3317	285	3	f−1(b	f−1(b	PROPN
ejpam-3317	285	4	)	)	PUNCT
ejpam-3317	285	5	is	be	AUX
ejpam-3317	285	6	preopen	preopen	ADJ
ejpam-3317	285	7	in	in	ADP
ejpam-3317	285	8	(	(	PUNCT
ejpam-3317	285	9	x	x	NOUN
ejpam-3317	285	10	,	,	PUNCT
ejpam-3317	285	11	c1	c1	PROPN
ejpam-3317	285	12	)	)	PUNCT
ejpam-3317	285	13	.	.	PUNCT
ejpam-3317	286	1	therefore	therefore	ADV
ejpam-3317	286	2	,	,	PUNCT
ejpam-3317	286	3	f	f	PROPN
ejpam-3317	286	4	is	be	AUX
ejpam-3317	286	5	pre	pre	ADJ
ejpam-3317	286	6	-	-	ADJ
ejpam-3317	286	7	irresolute	irresolute	ADJ
ejpam-3317	286	8	.	.	PUNCT
ejpam-3317	287	1	let	let	VERB
ejpam-3317	287	2	a	a	PRON
ejpam-3317	287	3	be	be	AUX
ejpam-3317	287	4	a	a	DET
ejpam-3317	287	5	preopen	preopen	ADJ
ejpam-3317	287	6	subset	subset	NOUN
ejpam-3317	287	7	of	of	ADP
ejpam-3317	287	8	(	(	PUNCT
ejpam-3317	287	9	x	x	NOUN
ejpam-3317	287	10	,	,	PUNCT
ejpam-3317	287	11	c1	c1	PROPN
ejpam-3317	287	12	)	)	PUNCT
ejpam-3317	287	13	.	.	PUNCT
ejpam-3317	288	1	then	then	ADV
ejpam-3317	288	2	there	there	PRON
ejpam-3317	288	3	exists	exist	VERB
ejpam-3317	288	4	an	an	DET
ejpam-3317	288	5	open	open	ADJ
ejpam-3317	288	6	set	set	NOUN
ejpam-3317	288	7	g	g	NOUN
ejpam-3317	288	8	in	in	ADP
ejpam-3317	288	9	(	(	PUNCT
ejpam-3317	288	10	x	x	NOUN
ejpam-3317	288	11	,	,	PUNCT
ejpam-3317	288	12	c1	c1	PROPN
ejpam-3317	288	13	)	)	PUNCT
ejpam-3317	288	14	such	such	ADJ
ejpam-3317	288	15	that	that	SCONJ
ejpam-3317	288	16	a	a	DET
ejpam-3317	288	17	⊆	⊆	NUM
ejpam-3317	288	18	g	g	NOUN
ejpam-3317	288	19	⊆	⊆	NUM
ejpam-3317	288	20	c1(a	c1(a	NOUN
ejpam-3317	288	21	)	)	PUNCT
ejpam-3317	288	22	.	.	PUNCT
ejpam-3317	289	1	hence	hence	ADV
ejpam-3317	289	2	,	,	PUNCT
ejpam-3317	289	3	f	f	PROPN
ejpam-3317	289	4	(	(	PUNCT
ejpam-3317	289	5	a	a	NOUN
ejpam-3317	289	6	)	)	PUNCT
ejpam-3317	289	7	⊆	⊆	NUM
ejpam-3317	289	8	f	f	X
ejpam-3317	289	9	(	(	PUNCT
ejpam-3317	289	10	g	g	NOUN
ejpam-3317	289	11	)	)	PUNCT
ejpam-3317	289	12	⊆	⊆	NUM
ejpam-3317	289	13	f	f	X
ejpam-3317	289	14	(	(	PUNCT
ejpam-3317	289	15	c1	c1	PROPN
ejpam-3317	289	16	(	(	PUNCT
ejpam-3317	289	17	a	a	NOUN
ejpam-3317	289	18	)	)	PUNCT
ejpam-3317	289	19	)	)	PUNCT
ejpam-3317	289	20	.	.	PUNCT
ejpam-3317	290	1	as	as	SCONJ
ejpam-3317	290	2	f	f	PROPN
ejpam-3317	290	3	is	be	AUX
ejpam-3317	290	4	continuous	continuous	ADJ
ejpam-3317	290	5	,	,	PUNCT
ejpam-3317	290	6	f(c1a	f(c1a	NOUN
ejpam-3317	290	7	)	)	PUNCT
ejpam-3317	290	8	⊆	⊆	NUM
ejpam-3317	290	9	c2f(a	c2f(a	PROPN
ejpam-3317	290	10	)	)	PUNCT
ejpam-3317	290	11	.	.	PUNCT
ejpam-3317	291	1	since	since	SCONJ
ejpam-3317	291	2	f−1is	f−1is	VERB
ejpam-3317	291	3	continuous	continuous	ADJ
ejpam-3317	291	4	and	and	CCONJ
ejpam-3317	291	5	f(g	f(g	NOUN
ejpam-3317	291	6	)	)	PUNCT
ejpam-3317	291	7	is	be	AUX
ejpam-3317	291	8	the	the	DET
ejpam-3317	291	9	inverse	inverse	ADJ
ejpam-3317	291	10	image	image	NOUN
ejpam-3317	291	11	of	of	ADP
ejpam-3317	291	12	g	g	PROPN
ejpam-3317	291	13	under	under	ADP
ejpam-3317	291	14	f−1	f−1	PROPN
ejpam-3317	291	15	,	,	PUNCT
ejpam-3317	291	16	f(g	f(g	NOUN
ejpam-3317	291	17	)	)	PUNCT
ejpam-3317	291	18	is	be	AUX
ejpam-3317	291	19	open	open	ADJ
ejpam-3317	291	20	in	in	ADP
ejpam-3317	291	21	(	(	PUNCT
ejpam-3317	291	22	y	y	PROPN
ejpam-3317	291	23	,	,	PUNCT
ejpam-3317	291	24	c2	c2	PROPN
ejpam-3317	291	25	)	)	PUNCT
ejpam-3317	291	26	.	.	PUNCT
ejpam-3317	292	1	thus	thus	ADV
ejpam-3317	292	2	,	,	PUNCT
ejpam-3317	292	3	f(a	f(a	PROPN
ejpam-3317	292	4	)	)	PUNCT
ejpam-3317	292	5	is	be	AUX
ejpam-3317	292	6	preopen	preopen	ADJ
ejpam-3317	292	7	in	in	ADP
ejpam-3317	292	8	(	(	PUNCT
ejpam-3317	292	9	y	y	PROPN
ejpam-3317	292	10	,	,	PUNCT
ejpam-3317	292	11	c2	c2	PROPN
ejpam-3317	292	12	)	)	PUNCT
ejpam-3317	292	13	.	.	PUNCT
ejpam-3317	293	1	but	but	CCONJ
ejpam-3317	293	2	f(a	f(a	PROPN
ejpam-3317	293	3	)	)	PUNCT
ejpam-3317	293	4	is	be	AUX
ejpam-3317	293	5	the	the	DET
ejpam-3317	293	6	inverse	inverse	ADJ
ejpam-3317	293	7	image	image	NOUN
ejpam-3317	293	8	of	of	ADP
ejpam-3317	293	9	a	a	DET
ejpam-3317	293	10	under	under	ADP
ejpam-3317	293	11	f−1	f−1	PROPN
ejpam-3317	293	12	,	,	PUNCT
ejpam-3317	293	13	therefore	therefore	ADV
ejpam-3317	293	14	f−1is	f−1is	VERB
ejpam-3317	293	15	pre	pre	ADJ
ejpam-3317	293	16	-	-	ADJ
ejpam-3317	293	17	irresolute	irresolute	ADJ
ejpam-3317	293	18	.	.	PUNCT
ejpam-3317	294	1	h.m	h.m	PROPN
ejpam-3317	294	2	.	.	PROPN
ejpam-3317	294	3	darwesh	darwesh	PROPN
ejpam-3317	294	4	,	,	PUNCT
ejpam-3317	294	5	s.f	s.f	PROPN
ejpam-3317	294	6	.	.	PROPN
ejpam-3317	294	7	namiq	namiq	PROPN
ejpam-3317	294	8	/	/	SYM
ejpam-3317	294	9	eur	eur	PROPN
ejpam-3317	294	10	.	.	PUNCT
ejpam-3317	295	1	j.	j.	PROPN
ejpam-3317	295	2	pure	pure	PROPN
ejpam-3317	295	3	appl	appl	PROPN
ejpam-3317	295	4	.	.	PROPN
ejpam-3317	295	5	math	math	PROPN
ejpam-3317	295	6	,	,	PUNCT
ejpam-3317	295	7	12	12	NUM
ejpam-3317	295	8	(	(	PUNCT
ejpam-3317	295	9	3	3	NUM
ejpam-3317	295	10	)	)	PUNCT
ejpam-3317	295	11	(	(	PUNCT
ejpam-3317	295	12	2019	2019	NUM
ejpam-3317	295	13	)	)	PUNCT
ejpam-3317	295	14	,	,	PUNCT
ejpam-3317	295	15	1082	1082	NUM
ejpam-3317	295	16	-	-	SYM
ejpam-3317	295	17	1095	1095	NUM
ejpam-3317	295	18	1091	1091	NUM
ejpam-3317	295	19	5	5	NUM
ejpam-3317	295	20	.	.	PUNCT
ejpam-3317	296	1	p	p	X
ejpam-3317	296	2	-	-	PUNCT
ejpam-3317	296	3	connectedness	connectedness	NOUN
ejpam-3317	296	4	as	as	ADP
ejpam-3317	296	5	another	another	DET
ejpam-3317	296	6	application	application	NOUN
ejpam-3317	296	7	of	of	ADP
ejpam-3317	296	8	preopen	preopen	ADJ
ejpam-3317	296	9	sets	set	NOUN
ejpam-3317	296	10	,	,	PUNCT
ejpam-3317	296	11	a	a	DET
ejpam-3317	296	12	new	new	ADJ
ejpam-3317	296	13	kind	kind	NOUN
ejpam-3317	296	14	of	of	ADP
ejpam-3317	296	15	connectedness	connectedness	NOUN
ejpam-3317	296	16	,	,	PUNCT
ejpam-3317	296	17	namely	namely	ADV
ejpam-3317	296	18	pconnectedness	pconnectedness	NOUN
ejpam-3317	296	19	,	,	PUNCT
ejpam-3317	296	20	is	be	AUX
ejpam-3317	296	21	introduced	introduce	VERB
ejpam-3317	296	22	.	.	PUNCT
ejpam-3317	297	1	definition	definition	NOUN
ejpam-3317	297	2	16	16	NUM
ejpam-3317	297	3	.	.	PUNCT
ejpam-3317	298	1	a	a	DET
ejpam-3317	298	2	closure	closure	NOUN
ejpam-3317	298	3	space	space	NOUN
ejpam-3317	298	4	(	(	PUNCT
ejpam-3317	298	5	x	x	X
ejpam-3317	298	6	,	,	PUNCT
ejpam-3317	298	7	c)is	c)is	PROPN
ejpam-3317	298	8	said	say	VERB
ejpam-3317	298	9	to	to	PART
ejpam-3317	298	10	be	be	AUX
ejpam-3317	298	11	p	p	NOUN
ejpam-3317	298	12	-	-	PUNCT
ejpam-3317	298	13	connected	connect	VERB
ejpam-3317	298	14	if	if	SCONJ
ejpam-3317	298	15	φ	φ	PROPN
ejpam-3317	298	16	and	and	CCONJ
ejpam-3317	298	17	x	x	NOUN
ejpam-3317	298	18	are	be	AUX
ejpam-3317	298	19	the	the	DET
ejpam-3317	298	20	only	only	ADJ
ejpam-3317	298	21	subsets	subset	NOUN
ejpam-3317	298	22	of	of	ADP
ejpam-3317	298	23	x	x	PUNCT
ejpam-3317	298	24	which	which	PRON
ejpam-3317	298	25	are	be	AUX
ejpam-3317	298	26	both	both	PRON
ejpam-3317	298	27	preopen	preopen	ADJ
ejpam-3317	298	28	and	and	CCONJ
ejpam-3317	298	29	preclosed	preclose	VERB
ejpam-3317	298	30	.	.	PUNCT
ejpam-3317	299	1	clearly	clearly	ADV
ejpam-3317	299	2	,	,	PUNCT
ejpam-3317	299	3	if	if	SCONJ
ejpam-3317	299	4	(	(	PUNCT
ejpam-3317	299	5	x	x	NOUN
ejpam-3317	299	6	,	,	PUNCT
ejpam-3317	299	7	c	c	NOUN
ejpam-3317	299	8	)	)	PUNCT
ejpam-3317	299	9	is	be	AUX
ejpam-3317	299	10	p	p	ADV
ejpam-3317	299	11	-	-	PUNCT
ejpam-3317	299	12	connected	connect	VERB
ejpam-3317	299	13	,	,	PUNCT
ejpam-3317	299	14	then	then	ADV
ejpam-3317	299	15	(	(	PUNCT
ejpam-3317	299	16	x	x	X
ejpam-3317	299	17	,	,	PUNCT
ejpam-3317	299	18	c	c	NOUN
ejpam-3317	299	19	)	)	PUNCT
ejpam-3317	299	20	is	be	AUX
ejpam-3317	299	21	connected	connect	VERB
ejpam-3317	299	22	.	.	PUNCT
ejpam-3317	300	1	the	the	DET
ejpam-3317	300	2	converse	converse	NOUN
ejpam-3317	300	3	is	be	AUX
ejpam-3317	300	4	not	not	PART
ejpam-3317	300	5	true	true	ADJ
ejpam-3317	300	6	as	as	SCONJ
ejpam-3317	300	7	can	can	AUX
ejpam-3317	300	8	be	be	AUX
ejpam-3317	300	9	seen	see	VERB
ejpam-3317	300	10	from	from	ADP
ejpam-3317	300	11	the	the	DET
ejpam-3317	300	12	following	follow	VERB
ejpam-3317	300	13	example	example	NOUN
ejpam-3317	300	14	.	.	PUNCT
ejpam-3317	301	1	example	example	NOUN
ejpam-3317	302	1	3	3	X
ejpam-3317	302	2	.	.	PUNCT
ejpam-3317	302	3	let	let	VERB
ejpam-3317	302	4	x={1	x={1	ADJ
ejpam-3317	302	5	,	,	PUNCT
ejpam-3317	302	6	2	2	NUM
ejpam-3317	302	7	,	,	PUNCT
ejpam-3317	302	8	3	3	NUM
ejpam-3317	302	9	}	}	PUNCT
ejpam-3317	302	10	and	and	CCONJ
ejpam-3317	302	11	define	define	VERB
ejpam-3317	302	12	a	a	DET
ejpam-3317	302	13	closure	closure	NOUN
ejpam-3317	302	14	operator	operator	NOUN
ejpam-3317	302	15	c	c	NOUN
ejpam-3317	302	16	on	on	ADP
ejpam-3317	302	17	x	x	PUNCT
ejpam-3317	302	18	by	by	ADP
ejpam-3317	302	19	:	:	PUNCT
ejpam-3317	302	20	c	c	X
ejpam-3317	302	21	(	(	PUNCT
ejpam-3317	302	22	a	a	X
ejpam-3317	302	23	)	)	PUNCT
ejpam-3317	302	24	=	=	PUNCT
ejpam-3317	302	25			PUNCT
ejpam-3317	302	26	a	a	INTJ
ejpam-3317	302	27	if	if	SCONJ
ejpam-3317	302	28	a	a	DET
ejpam-3317	302	29	=	=	SYM
ejpam-3317	302	30	φ	φ	X
ejpam-3317	302	31	{	{	PUNCT
ejpam-3317	302	32	1	1	NUM
ejpam-3317	302	33	,	,	PUNCT
ejpam-3317	302	34	3	3	X
ejpam-3317	302	35	}	}	PUNCT
ejpam-3317	302	36	if	if	SCONJ
ejpam-3317	302	37	a	a	PRON
ejpam-3317	302	38	=	=	X
ejpam-3317	302	39	{	{	PUNCT
ejpam-3317	302	40	1	1	NUM
ejpam-3317	302	41	}	}	PUNCT
ejpam-3317	302	42	{	{	PUNCT
ejpam-3317	302	43	2	2	NUM
ejpam-3317	302	44	,	,	PUNCT
ejpam-3317	302	45	3	3	NUM
ejpam-3317	302	46	}	}	PUNCT
ejpam-3317	302	47	if	if	SCONJ
ejpam-3317	302	48	a	a	PRON
ejpam-3317	302	49	=	=	X
ejpam-3317	302	50	{	{	PUNCT
ejpam-3317	302	51	2	2	NUM
ejpam-3317	302	52	}	}	PUNCT
ejpam-3317	302	53	x	x	SYM
ejpam-3317	302	54	otherwise	otherwise	ADV
ejpam-3317	302	55	the	the	DET
ejpam-3317	302	56	family	family	NOUN
ejpam-3317	302	57	of	of	ADP
ejpam-3317	302	58	all	all	DET
ejpam-3317	302	59	open	open	ADJ
ejpam-3317	302	60	sets	set	NOUN
ejpam-3317	302	61	=	=	PRON
ejpam-3317	302	62	{	{	PUNCT
ejpam-3317	302	63	φ	φ	NUM
ejpam-3317	302	64	,	,	PUNCT
ejpam-3317	302	65	x	x	NOUN
ejpam-3317	302	66	}	}	PUNCT
ejpam-3317	302	67	po	po	NOUN
ejpam-3317	302	68	(	(	PUNCT
ejpam-3317	302	69	x	x	X
ejpam-3317	302	70	,	,	PUNCT
ejpam-3317	302	71	c	c	NOUN
ejpam-3317	302	72	)	)	PUNCT
ejpam-3317	302	73	=	=	SYM
ejpam-3317	302	74	{	{	PUNCT
ejpam-3317	302	75	φ	φ	PROPN
ejpam-3317	302	76	,	,	PUNCT
ejpam-3317	302	77	{	{	PUNCT
ejpam-3317	302	78	3	3	NUM
ejpam-3317	302	79	}	}	PUNCT
ejpam-3317	302	80	,	,	PUNCT
ejpam-3317	302	81	{	{	PUNCT
ejpam-3317	302	82	1	1	NUM
ejpam-3317	302	83	,	,	PUNCT
ejpam-3317	302	84	2	2	NUM
ejpam-3317	302	85	}	}	PUNCT
ejpam-3317	302	86	,	,	PUNCT
ejpam-3317	302	87	{	{	PUNCT
ejpam-3317	302	88	1	1	NUM
ejpam-3317	302	89	,	,	PUNCT
ejpam-3317	302	90	3	3	NUM
ejpam-3317	302	91	}	}	PUNCT
ejpam-3317	302	92	,	,	PUNCT
ejpam-3317	302	93	{	{	PUNCT
ejpam-3317	302	94	2	2	NUM
ejpam-3317	302	95	,	,	PUNCT
ejpam-3317	302	96	3	3	NUM
ejpam-3317	302	97	}	}	PUNCT
ejpam-3317	302	98	,	,	PUNCT
ejpam-3317	302	99	x	x	X
ejpam-3317	302	100	}	}	PUNCT
ejpam-3317	302	101	.	.	PUNCT
ejpam-3317	303	1	we	we	PRON
ejpam-3317	303	2	have	have	VERB
ejpam-3317	303	3	(	(	PUNCT
ejpam-3317	303	4	x	x	NOUN
ejpam-3317	303	5	,	,	PUNCT
ejpam-3317	303	6	c	c	NOUN
ejpam-3317	303	7	)	)	PUNCT
ejpam-3317	303	8	is	be	AUX
ejpam-3317	303	9	connected	connect	VERB
ejpam-3317	303	10	,	,	PUNCT
ejpam-3317	303	11	but	but	CCONJ
ejpam-3317	303	12	it	it	PRON
ejpam-3317	303	13	is	be	AUX
ejpam-3317	303	14	not	not	PART
ejpam-3317	303	15	p	p	NOUN
ejpam-3317	303	16	-	-	PUNCT
ejpam-3317	303	17	connected	connect	VERB
ejpam-3317	303	18	,	,	PUNCT
ejpam-3317	303	19	because	because	SCONJ
ejpam-3317	303	20	{	{	PUNCT
ejpam-3317	303	21	1	1	NUM
ejpam-3317	303	22	,	,	PUNCT
ejpam-3317	303	23	2	2	NUM
ejpam-3317	303	24	}	}	PUNCT
ejpam-3317	303	25	is	be	AUX
ejpam-3317	303	26	preopen	preopen	ADJ
ejpam-3317	303	27	set	set	VERB
ejpam-3317	303	28	and	and	CCONJ
ejpam-3317	303	29	preclosed	preclose	VERB
ejpam-3317	303	30	set	set	NOUN
ejpam-3317	303	31	.	.	PUNCT
ejpam-3317	304	1	proposition	proposition	NOUN
ejpam-3317	304	2	31	31	NUM
ejpam-3317	304	3	.	.	PUNCT
ejpam-3317	305	1	let	let	VERB
ejpam-3317	305	2	(	(	PUNCT
ejpam-3317	305	3	x	x	NOUN
ejpam-3317	305	4	,	,	PUNCT
ejpam-3317	305	5	c	c	X
ejpam-3317	305	6	)	)	PUNCT
ejpam-3317	305	7	be	be	AUX
ejpam-3317	305	8	a	a	DET
ejpam-3317	305	9	closure	closure	NOUN
ejpam-3317	305	10	space	space	NOUN
ejpam-3317	305	11	.	.	PUNCT
ejpam-3317	306	1	then	then	ADV
ejpam-3317	306	2	the	the	DET
ejpam-3317	306	3	following	follow	VERB
ejpam-3317	306	4	statements	statement	NOUN
ejpam-3317	306	5	are	be	AUX
ejpam-3317	306	6	equivalent	equivalent	ADJ
ejpam-3317	306	7	:	:	PUNCT
ejpam-3317	306	8	(	(	PUNCT
ejpam-3317	306	9	i	i	NOUN
ejpam-3317	306	10	)	)	PUNCT
ejpam-3317	306	11	x	x	X
ejpam-3317	306	12	is	be	AUX
ejpam-3317	306	13	p	p	ADV
ejpam-3317	306	14	-	-	PUNCT
ejpam-3317	306	15	connected	connect	VERB
ejpam-3317	306	16	.	.	PUNCT
ejpam-3317	307	1	(	(	PUNCT
ejpam-3317	307	2	ii	ii	NOUN
ejpam-3317	307	3	)	)	PUNCT
ejpam-3317	307	4	x	x	PRON
ejpam-3317	307	5	can	can	AUX
ejpam-3317	307	6	not	not	PART
ejpam-3317	307	7	be	be	AUX
ejpam-3317	307	8	expressed	express	VERB
ejpam-3317	307	9	as	as	ADP
ejpam-3317	307	10	the	the	DET
ejpam-3317	307	11	union	union	NOUN
ejpam-3317	307	12	of	of	ADP
ejpam-3317	307	13	two	two	NUM
ejpam-3317	307	14	disjoint	disjoint	ADJ
ejpam-3317	307	15	,	,	PUNCT
ejpam-3317	307	16	non	non	ADJ
ejpam-3317	307	17	-	-	ADJ
ejpam-3317	307	18	empty	empty	ADJ
ejpam-3317	307	19	,	,	PUNCT
ejpam-3317	307	20	preclosed	preclose	VERB
ejpam-3317	307	21	subsets	subset	NOUN
ejpam-3317	307	22	.	.	PUNCT
ejpam-3317	308	1	(	(	PUNCT
ejpam-3317	308	2	iii	iii	X
ejpam-3317	308	3	)	)	PUNCT
ejpam-3317	308	4	x	x	PRON
ejpam-3317	308	5	can	can	AUX
ejpam-3317	308	6	not	not	PART
ejpam-3317	308	7	be	be	AUX
ejpam-3317	308	8	expressed	express	VERB
ejpam-3317	308	9	as	as	ADP
ejpam-3317	308	10	the	the	DET
ejpam-3317	308	11	union	union	NOUN
ejpam-3317	308	12	of	of	ADP
ejpam-3317	308	13	two	two	NUM
ejpam-3317	308	14	disjoint	disjoint	ADJ
ejpam-3317	308	15	,	,	PUNCT
ejpam-3317	308	16	non	non	ADJ
ejpam-3317	308	17	-	-	ADJ
ejpam-3317	308	18	empty	empty	ADJ
ejpam-3317	308	19	,	,	PUNCT
ejpam-3317	308	20	preopen	preopen	ADJ
ejpam-3317	308	21	subsets	subset	NOUN
ejpam-3317	308	22	.	.	PUNCT
ejpam-3317	309	1	proof	proof	NOUN
ejpam-3317	309	2	.	.	PUNCT
ejpam-3317	310	1	statement(i	statement(i	NOUN
ejpam-3317	310	2	)	)	PUNCT
ejpam-3317	310	3	implies	imply	VERB
ejpam-3317	310	4	statement(ii	statement(ii	NOUN
ejpam-3317	310	5	):	):	PUNCT
ejpam-3317	310	6	suppose	suppose	VERB
ejpam-3317	310	7	that	that	SCONJ
ejpam-3317	310	8	x	x	X
ejpam-3317	310	9	=	=	PUNCT
ejpam-3317	310	10	u	u	NOUN
ejpam-3317	310	11	∪	∪	VERB
ejpam-3317	310	12	v	v	NOUN
ejpam-3317	310	13	,	,	PUNCT
ejpam-3317	310	14	where	where	SCONJ
ejpam-3317	310	15	u	u	NOUN
ejpam-3317	310	16	and	and	CCONJ
ejpam-3317	310	17	v	v	NOUN
ejpam-3317	310	18	are	be	AUX
ejpam-3317	310	19	non	non	ADJ
ejpam-3317	310	20	-	-	ADJ
ejpam-3317	310	21	empty	empty	ADJ
ejpam-3317	310	22	,	,	PUNCT
ejpam-3317	310	23	disjoint	disjoint	ADJ
ejpam-3317	310	24	,	,	PUNCT
ejpam-3317	310	25	preclosed	preclose	VERB
ejpam-3317	310	26	subsets	subset	NOUN
ejpam-3317	310	27	of	of	ADP
ejpam-3317	310	28	(	(	PUNCT
ejpam-3317	310	29	x	x	NOUN
ejpam-3317	310	30	,	,	PUNCT
ejpam-3317	310	31	c	c	NOUN
ejpam-3317	310	32	)	)	PUNCT
ejpam-3317	310	33	.	.	PUNCT
ejpam-3317	311	1	then	then	ADV
ejpam-3317	311	2	u	u	X
ejpam-3317	311	3	=	=	PUNCT
ejpam-3317	311	4	x	x	PROPN
ejpam-3317	311	5	/	/	SYM
ejpam-3317	311	6	v	v	NOUN
ejpam-3317	311	7	and	and	CCONJ
ejpam-3317	311	8	u	u	NOUN
ejpam-3317	311	9	is	be	AUX
ejpam-3317	311	10	preopen	preopen	ADJ
ejpam-3317	311	11	.	.	PUNCT
ejpam-3317	312	1	thus	thus	ADV
ejpam-3317	312	2	,	,	PUNCT
ejpam-3317	312	3	u	u	NOUN
ejpam-3317	312	4	is	be	AUX
ejpam-3317	312	5	a	a	DET
ejpam-3317	312	6	subset	subset	NOUN
ejpam-3317	312	7	of	of	ADP
ejpam-3317	312	8	x	x	PRON
ejpam-3317	312	9	which	which	PRON
ejpam-3317	312	10	is	be	AUX
ejpam-3317	312	11	both	both	CCONJ
ejpam-3317	312	12	preopen	preopen	ADJ
ejpam-3317	312	13	and	and	CCONJ
ejpam-3317	312	14	preclosed	preclose	VERB
ejpam-3317	312	15	but	but	CCONJ
ejpam-3317	312	16	u	u	NOUN
ejpam-3317	312	17	is	be	AUX
ejpam-3317	312	18	neither	neither	CCONJ
ejpam-3317	312	19	x	x	PROPN
ejpam-3317	312	20	nor	nor	CCONJ
ejpam-3317	312	21	φ	φ	NUM
ejpam-3317	312	22	.	.	PUNCT
ejpam-3317	313	1	hence	hence	ADV
ejpam-3317	313	2	,	,	PUNCT
ejpam-3317	313	3	(	(	PUNCT
ejpam-3317	313	4	x	x	NOUN
ejpam-3317	313	5	,	,	PUNCT
ejpam-3317	313	6	c	c	NOUN
ejpam-3317	313	7	)	)	PUNCT
ejpam-3317	313	8	is	be	AUX
ejpam-3317	313	9	not	not	PART
ejpam-3317	313	10	p	p	NOUN
ejpam-3317	313	11	-	-	PUNCT
ejpam-3317	313	12	connected	connect	VERB
ejpam-3317	313	13	.	.	PUNCT
ejpam-3317	314	1	statement	statement	NOUN
ejpam-3317	314	2	(	(	PUNCT
ejpam-3317	314	3	ii	ii	NOUN
ejpam-3317	314	4	)	)	PUNCT
ejpam-3317	314	5	implies	imply	VERB
ejpam-3317	314	6	statement	statement	NOUN
ejpam-3317	314	7	(	(	PUNCT
ejpam-3317	314	8	iii	iii	NOUN
ejpam-3317	314	9	):	):	PUNCT
ejpam-3317	314	10	suppose	suppose	VERB
ejpam-3317	314	11	that	that	SCONJ
ejpam-3317	314	12	x	x	X
ejpam-3317	314	13	=	=	PUNCT
ejpam-3317	314	14	a	a	DET
ejpam-3317	314	15	∪	∪	X
ejpam-3317	314	16	b	b	NOUN
ejpam-3317	314	17	where	where	SCONJ
ejpam-3317	314	18	a	a	PRON
ejpam-3317	314	19	and	and	CCONJ
ejpam-3317	314	20	b	b	NOUN
ejpam-3317	314	21	are	be	AUX
ejpam-3317	314	22	disjoint	disjoint	ADJ
ejpam-3317	314	23	non	non	ADJ
ejpam-3317	314	24	-	-	ADJ
ejpam-3317	314	25	empty	empty	ADJ
ejpam-3317	314	26	preopen	preopen	ADJ
ejpam-3317	314	27	subsets	subset	NOUN
ejpam-3317	314	28	of	of	ADP
ejpam-3317	314	29	(	(	PUNCT
ejpam-3317	314	30	x	x	NOUN
ejpam-3317	314	31	,	,	PUNCT
ejpam-3317	314	32	c	c	NOUN
ejpam-3317	314	33	)	)	PUNCT
ejpam-3317	314	34	.	.	PUNCT
ejpam-3317	315	1	then	then	ADV
ejpam-3317	315	2	x	x	X
ejpam-3317	315	3	/	/	SYM
ejpam-3317	315	4	a	a	DET
ejpam-3317	315	5	=	=	SYM
ejpam-3317	315	6	b	b	PROPN
ejpam-3317	315	7	and	and	CCONJ
ejpam-3317	315	8	x	x	PROPN
ejpam-3317	315	9	/	/	SYM
ejpam-3317	315	10	b	b	X
ejpam-3317	315	11	=	=	NOUN
ejpam-3317	315	12	a	a	PRON
ejpam-3317	315	13	are	be	AUX
ejpam-3317	315	14	both	both	DET
ejpam-3317	315	15	complements	complement	NOUN
ejpam-3317	315	16	of	of	ADP
ejpam-3317	315	17	preopen	preopen	ADJ
ejpam-3317	315	18	sets	set	NOUN
ejpam-3317	315	19	and	and	CCONJ
ejpam-3317	315	20	hence	hence	ADV
ejpam-3317	315	21	are	be	AUX
ejpam-3317	315	22	preclosed	preclose	VERB
ejpam-3317	315	23	.	.	PUNCT
ejpam-3317	316	1	thus	thus	ADV
ejpam-3317	316	2	,	,	PUNCT
ejpam-3317	316	3	x	x	X
ejpam-3317	316	4	=	=	PUNCT
ejpam-3317	316	5	a	a	PRON
ejpam-3317	316	6	∪b	∪b	PRON
ejpam-3317	316	7	is	be	AUX
ejpam-3317	316	8	an	an	DET
ejpam-3317	316	9	expression	expression	NOUN
ejpam-3317	316	10	of	of	ADP
ejpam-3317	316	11	x	x	PUNCT
ejpam-3317	316	12	as	as	ADP
ejpam-3317	316	13	the	the	DET
ejpam-3317	316	14	union	union	NOUN
ejpam-3317	316	15	of	of	ADP
ejpam-3317	316	16	two	two	NUM
ejpam-3317	316	17	disjoint	disjoint	ADJ
ejpam-3317	316	18	,	,	PUNCT
ejpam-3317	316	19	non	non	ADJ
ejpam-3317	316	20	-	-	ADJ
ejpam-3317	316	21	empty	empty	ADJ
ejpam-3317	316	22	,	,	PUNCT
ejpam-3317	316	23	preclosed	preclose	VERB
ejpam-3317	316	24	subset	subset	NOUN
ejpam-3317	316	25	of	of	ADP
ejpam-3317	316	26	(	(	PUNCT
ejpam-3317	316	27	x	x	NOUN
ejpam-3317	316	28	,	,	PUNCT
ejpam-3317	316	29	c	c	NOUN
ejpam-3317	316	30	)	)	PUNCT
ejpam-3317	316	31	,	,	PUNCT
ejpam-3317	316	32	which	which	PRON
ejpam-3317	316	33	contradicts	contradict	VERB
ejpam-3317	316	34	(	(	PUNCT
ejpam-3317	316	35	ii	ii	NOUN
ejpam-3317	316	36	)	)	PUNCT
ejpam-3317	316	37	.	.	PUNCT
ejpam-3317	317	1	statement	statement	NOUN
ejpam-3317	317	2	(	(	PUNCT
ejpam-3317	317	3	iii	iii	NOUN
ejpam-3317	317	4	)	)	PUNCT
ejpam-3317	317	5	implies	imply	VERB
ejpam-3317	317	6	statement	statement	NOUN
ejpam-3317	317	7	(	(	PUNCT
ejpam-3317	317	8	i	i	NOUN
ejpam-3317	317	9	):	):	PUNCT
ejpam-3317	317	10	suppose	suppose	VERB
ejpam-3317	317	11	that	that	SCONJ
ejpam-3317	317	12	a	a	PRON
ejpam-3317	317	13	is	be	AUX
ejpam-3317	317	14	a	a	DET
ejpam-3317	317	15	subset	subset	NOUN
ejpam-3317	317	16	of	of	ADP
ejpam-3317	317	17	x	x	PRON
ejpam-3317	317	18	which	which	PRON
ejpam-3317	317	19	is	be	AUX
ejpam-3317	317	20	both	both	CCONJ
ejpam-3317	317	21	preopen	preopen	ADJ
ejpam-3317	317	22	and	and	CCONJ
ejpam-3317	317	23	preclosed	preclose	VERB
ejpam-3317	317	24	,	,	PUNCT
ejpam-3317	317	25	but	but	CCONJ
ejpam-3317	317	26	a	a	PRON
ejpam-3317	317	27	is	be	AUX
ejpam-3317	317	28	neither	neither	CCONJ
ejpam-3317	317	29	x	x	PROPN
ejpam-3317	317	30	nor	nor	CCONJ
ejpam-3317	317	31	φ	φ	NUM
ejpam-3317	317	32	.	.	PUNCT
ejpam-3317	318	1	then	then	ADV
ejpam-3317	318	2	x/	x/	VERB
ejpam-3317	318	3	a	a	PRON
ejpam-3317	318	4	is	be	AUX
ejpam-3317	318	5	also	also	ADV
ejpam-3317	318	6	preclosed	preclose	VERB
ejpam-3317	318	7	,	,	PUNCT
ejpam-3317	318	8	preopen	preopen	ADJ
ejpam-3317	318	9	and	and	CCONJ
ejpam-3317	318	10	non	non	ADJ
ejpam-3317	318	11	-	-	ADJ
ejpam-3317	318	12	empty	empty	ADJ
ejpam-3317	318	13	.	.	PUNCT
ejpam-3317	319	1	thus	thus	ADV
ejpam-3317	319	2	,	,	PUNCT
ejpam-3317	319	3	x=	x=	PUNCT
ejpam-3317	319	4	(	(	PUNCT
ejpam-3317	319	5	x	x	X
ejpam-3317	319	6	/	/	SYM
ejpam-3317	319	7	a)∪a	a)∪a	PROPN
ejpam-3317	319	8	is	be	AUX
ejpam-3317	319	9	the	the	DET
ejpam-3317	319	10	expression	expression	NOUN
ejpam-3317	319	11	of	of	ADP
ejpam-3317	319	12	x	x	PUNCT
ejpam-3317	319	13	as	as	ADP
ejpam-3317	319	14	the	the	DET
ejpam-3317	319	15	union	union	NOUN
ejpam-3317	319	16	of	of	ADP
ejpam-3317	319	17	two	two	NUM
ejpam-3317	319	18	disjoint	disjoint	ADJ
ejpam-3317	319	19	,	,	PUNCT
ejpam-3317	319	20	non	non	ADJ
ejpam-3317	319	21	-	-	ADJ
ejpam-3317	319	22	empty	empty	ADJ
ejpam-3317	319	23	preopen	preopen	ADJ
ejpam-3317	319	24	subsets	subset	NOUN
ejpam-3317	319	25	,	,	PUNCT
ejpam-3317	319	26	which	which	PRON
ejpam-3317	319	27	contradicts	contradict	VERB
ejpam-3317	319	28	(	(	PUNCT
ejpam-3317	319	29	iii	iii	NOUN
ejpam-3317	319	30	)	)	PUNCT
ejpam-3317	319	31	.	.	PUNCT
ejpam-3317	320	1	the	the	DET
ejpam-3317	320	2	following	follow	VERB
ejpam-3317	320	3	statement	statement	NOUN
ejpam-3317	320	4	is	be	AUX
ejpam-3317	320	5	evident	evident	ADJ
ejpam-3317	320	6	:	:	PUNCT
ejpam-3317	320	7	h.m	h.m	PROPN
ejpam-3317	320	8	.	.	PROPN
ejpam-3317	320	9	darwesh	darwesh	PROPN
ejpam-3317	320	10	,	,	PUNCT
ejpam-3317	320	11	s.f	s.f	PROPN
ejpam-3317	320	12	.	.	PROPN
ejpam-3317	320	13	namiq	namiq	PROPN
ejpam-3317	320	14	/	/	SYM
ejpam-3317	320	15	eur	eur	PROPN
ejpam-3317	320	16	.	.	PUNCT
ejpam-3317	321	1	j.	j.	PROPN
ejpam-3317	321	2	pure	pure	PROPN
ejpam-3317	321	3	appl	appl	PROPN
ejpam-3317	321	4	.	.	PROPN
ejpam-3317	321	5	math	math	PROPN
ejpam-3317	321	6	,	,	PUNCT
ejpam-3317	321	7	12	12	NUM
ejpam-3317	321	8	(	(	PUNCT
ejpam-3317	321	9	3	3	NUM
ejpam-3317	321	10	)	)	PUNCT
ejpam-3317	321	11	(	(	PUNCT
ejpam-3317	321	12	2019	2019	NUM
ejpam-3317	321	13	)	)	PUNCT
ejpam-3317	321	14	,	,	PUNCT
ejpam-3317	321	15	1082	1082	NUM
ejpam-3317	321	16	-	-	SYM
ejpam-3317	321	17	1095	1095	NUM
ejpam-3317	321	18	1092	1092	NUM
ejpam-3317	321	19	proposition	proposition	NOUN
ejpam-3317	321	20	32	32	NUM
ejpam-3317	321	21	.	.	PUNCT
ejpam-3317	322	1	let	let	VERB
ejpam-3317	322	2	(	(	PUNCT
ejpam-3317	322	3	x	x	NOUN
ejpam-3317	322	4	,	,	PUNCT
ejpam-3317	322	5	c	c	X
ejpam-3317	322	6	)	)	PUNCT
ejpam-3317	322	7	be	be	AUX
ejpam-3317	322	8	a	a	DET
ejpam-3317	322	9	tp	tp	NOUN
ejpam-3317	322	10	-	-	NOUN
ejpam-3317	322	11	space	space	NOUN
ejpam-3317	322	12	.	.	PUNCT
ejpam-3317	323	1	then	then	ADV
ejpam-3317	323	2	(	(	PUNCT
ejpam-3317	323	3	x	x	X
ejpam-3317	323	4	,	,	PUNCT
ejpam-3317	323	5	c	c	NOUN
ejpam-3317	323	6	)	)	PUNCT
ejpam-3317	323	7	is	be	AUX
ejpam-3317	323	8	connected	connect	VERB
ejpam-3317	323	9	if	if	SCONJ
ejpam-3317	323	10	and	and	CCONJ
ejpam-3317	323	11	only	only	ADV
ejpam-3317	323	12	if	if	SCONJ
ejpam-3317	323	13	(	(	PUNCT
ejpam-3317	323	14	x	x	NOUN
ejpam-3317	323	15	,	,	PUNCT
ejpam-3317	323	16	c	c	NOUN
ejpam-3317	323	17	)	)	PUNCT
ejpam-3317	323	18	is	be	AUX
ejpam-3317	323	19	p	p	ADV
ejpam-3317	323	20	-	-	PUNCT
ejpam-3317	323	21	connected	connect	VERB
ejpam-3317	323	22	.	.	PUNCT
ejpam-3317	324	1	proof	proof	NOUN
ejpam-3317	324	2	.	.	PUNCT
ejpam-3317	325	1	obvious	obvious	ADJ
ejpam-3317	325	2	.	.	PUNCT
ejpam-3317	326	1	proposition	proposition	NOUN
ejpam-3317	326	2	33	33	NUM
ejpam-3317	326	3	.	.	PUNCT
ejpam-3317	327	1	let	let	VERB
ejpam-3317	327	2	(	(	PUNCT
ejpam-3317	327	3	x	x	NOUN
ejpam-3317	327	4	,	,	PUNCT
ejpam-3317	327	5	c1	c1	PROPN
ejpam-3317	327	6	)	)	PUNCT
ejpam-3317	327	7	be	be	AUX
ejpam-3317	327	8	a	a	DET
ejpam-3317	327	9	closure	closure	NOUN
ejpam-3317	327	10	space	space	NOUN
ejpam-3317	327	11	and	and	CCONJ
ejpam-3317	327	12	let	let	VERB
ejpam-3317	327	13	y	y	PROPN
ejpam-3317	327	14	=	=	PUNCT
ejpam-3317	327	15	{	{	PUNCT
ejpam-3317	327	16	0	0	NUM
ejpam-3317	327	17	,	,	PUNCT
ejpam-3317	327	18	1	1	NUM
ejpam-3317	327	19	}	}	PUNCT
ejpam-3317	327	20	and	and	CCONJ
ejpam-3317	327	21	c2	c2	PROPN
ejpam-3317	327	22	be	be	VERB
ejpam-3317	327	23	a	a	DET
ejpam-3317	327	24	closure	closure	NOUN
ejpam-3317	327	25	operator	operator	NOUN
ejpam-3317	327	26	on	on	ADP
ejpam-3317	327	27	y	y	PROPN
ejpam-3317	327	28	defined	define	VERB
ejpam-3317	327	29	by	by	ADP
ejpam-3317	327	30	:	:	PUNCT
ejpam-3317	327	31	c2	c2	PROPN
ejpam-3317	327	32	(	(	PUNCT
ejpam-3317	327	33	a	a	PROPN
ejpam-3317	327	34	)	)	PUNCT
ejpam-3317	327	35	=	=	SYM
ejpam-3317	327	36	a	a	PRON
ejpam-3317	327	37	,	,	PUNCT
ejpam-3317	327	38	for	for	ADP
ejpam-3317	327	39	all	all	PRON
ejpam-3317	327	40	subset	subset	VERB
ejpam-3317	327	41	a	a	PRON
ejpam-3317	327	42	of	of	ADP
ejpam-3317	327	43	y.	y.	NOUN
ejpam-3317	327	44	then	then	ADV
ejpam-3317	327	45	the	the	DET
ejpam-3317	327	46	following	follow	VERB
ejpam-3317	327	47	statements	statement	NOUN
ejpam-3317	327	48	are	be	AUX
ejpam-3317	327	49	equivalent	equivalent	ADJ
ejpam-3317	327	50	:	:	PUNCT
ejpam-3317	327	51	(	(	PUNCT
ejpam-3317	327	52	i	i	NOUN
ejpam-3317	327	53	)	)	PUNCT
ejpam-3317	327	54	the	the	DET
ejpam-3317	327	55	only	only	ADJ
ejpam-3317	327	56	contra	contra	PROPN
ejpam-3317	327	57	-	-	ADJ
ejpam-3317	327	58	pre	pre	ADJ
ejpam-3317	327	59	-	-	ADJ
ejpam-3317	327	60	continuous	continuous	ADJ
ejpam-3317	327	61	functions	function	NOUN
ejpam-3317	327	62	f	f	NOUN
ejpam-3317	327	63	:	:	PUNCT
ejpam-3317	327	64	(	(	PUNCT
ejpam-3317	327	65	x	x	X
ejpam-3317	327	66	,	,	PUNCT
ejpam-3317	327	67	c1	c1	PROPN
ejpam-3317	327	68	)	)	PUNCT
ejpam-3317	327	69	→	→	PUNCT
ejpam-3317	327	70	(	(	PUNCT
ejpam-3317	327	71	y	y	PROPN
ejpam-3317	327	72	,	,	PUNCT
ejpam-3317	327	73	c2	c2	PROPN
ejpam-3317	327	74	)	)	PUNCT
ejpam-3317	327	75	are	be	AUX
ejpam-3317	327	76	the	the	DET
ejpam-3317	327	77	constant	constant	ADJ
ejpam-3317	327	78	functions	function	NOUN
ejpam-3317	327	79	.	.	PUNCT
ejpam-3317	328	1	(	(	PUNCT
ejpam-3317	328	2	ii	ii	NOUN
ejpam-3317	328	3	)	)	PUNCT
ejpam-3317	328	4	a	a	DET
ejpam-3317	328	5	closure	closure	NOUN
ejpam-3317	328	6	space	space	NOUN
ejpam-3317	328	7	(	(	PUNCT
ejpam-3317	328	8	x	x	NOUN
ejpam-3317	328	9	,	,	PUNCT
ejpam-3317	328	10	c1	c1	PROPN
ejpam-3317	328	11	)	)	PUNCT
ejpam-3317	328	12	is	be	AUX
ejpam-3317	328	13	p	p	ADV
ejpam-3317	328	14	-	-	PUNCT
ejpam-3317	328	15	connected	connect	VERB
ejpam-3317	328	16	.	.	PUNCT
ejpam-3317	329	1	proof	proof	NOUN
ejpam-3317	329	2	.	.	PUNCT
ejpam-3317	330	1	statement	statement	NOUN
ejpam-3317	330	2	(	(	PUNCT
ejpam-3317	330	3	i	i	NOUN
ejpam-3317	330	4	)	)	PUNCT
ejpam-3317	330	5	implies	imply	VERB
ejpam-3317	330	6	statement	statement	NOUN
ejpam-3317	330	7	(	(	PUNCT
ejpam-3317	330	8	ii	ii	NOUN
ejpam-3317	330	9	):	):	PUNCT
ejpam-3317	330	10	suppose	suppose	VERB
ejpam-3317	330	11	that	that	SCONJ
ejpam-3317	330	12	there	there	PRON
ejpam-3317	330	13	is	be	VERB
ejpam-3317	330	14	a	a	DET
ejpam-3317	330	15	non	non	ADJ
ejpam-3317	330	16	-	-	ADJ
ejpam-3317	330	17	empty	empty	ADJ
ejpam-3317	330	18	subset	subset	NOUN
ejpam-3317	330	19	a	a	PRON
ejpam-3317	330	20	of	of	ADP
ejpam-3317	330	21	(	(	PUNCT
ejpam-3317	330	22	x	x	NOUN
ejpam-3317	330	23	,	,	PUNCT
ejpam-3317	330	24	c1	c1	PROPN
ejpam-3317	330	25	)	)	PUNCT
ejpam-3317	330	26	such	such	ADJ
ejpam-3317	330	27	that	that	SCONJ
ejpam-3317	330	28	a	a	DET
ejpam-3317	330	29	6=	6=	NOUN
ejpam-3317	330	30	x	x	NOUN
ejpam-3317	330	31	and	and	CCONJ
ejpam-3317	330	32	a	a	PRON
ejpam-3317	330	33	is	be	AUX
ejpam-3317	330	34	both	both	CCONJ
ejpam-3317	330	35	preopen	preopen	ADJ
ejpam-3317	330	36	and	and	CCONJ
ejpam-3317	330	37	preclosed	preclose	VERB
ejpam-3317	330	38	.	.	PUNCT
ejpam-3317	331	1	then	then	ADV
ejpam-3317	331	2	x	x	X
ejpam-3317	331	3	/	/	SYM
ejpam-3317	331	4	a	a	PRON
ejpam-3317	331	5	is	be	AUX
ejpam-3317	331	6	both	both	CCONJ
ejpam-3317	331	7	preopen	preopen	ADJ
ejpam-3317	331	8	and	and	CCONJ
ejpam-3317	331	9	preclosed	preclose	VERB
ejpam-3317	331	10	in	in	ADP
ejpam-3317	331	11	(	(	PUNCT
ejpam-3317	331	12	x	x	NOUN
ejpam-3317	331	13	,	,	PUNCT
ejpam-3317	331	14	c1	c1	PROPN
ejpam-3317	331	15	)	)	PUNCT
ejpam-3317	331	16	.	.	PUNCT
ejpam-3317	332	1	define	define	VERB
ejpam-3317	332	2	a	a	DET
ejpam-3317	332	3	function	function	NOUN
ejpam-3317	332	4	f	f	NOUN
ejpam-3317	332	5	:	:	PUNCT
ejpam-3317	332	6	(	(	PUNCT
ejpam-3317	332	7	x	x	X
ejpam-3317	332	8	,	,	PUNCT
ejpam-3317	332	9	c1)→	c1)→	NOUN
ejpam-3317	332	10	(	(	PUNCT
ejpam-3317	332	11	y	y	PROPN
ejpam-3317	332	12	,	,	PUNCT
ejpam-3317	332	13	c2	c2	PROPN
ejpam-3317	332	14	)	)	PUNCT
ejpam-3317	332	15	by	by	ADP
ejpam-3317	332	16	:	:	PUNCT
ejpam-3317	332	17	f	f	PROPN
ejpam-3317	332	18	(	(	PUNCT
ejpam-3317	332	19	x	x	X
ejpam-3317	332	20	)	)	PUNCT
ejpam-3317	332	21	=	=	PRON
ejpam-3317	332	22	{	{	PUNCT
ejpam-3317	332	23	0	0	NUM
ejpam-3317	332	24	x	x	SYM
ejpam-3317	332	25	∈	∈	PROPN
ejpam-3317	332	26	a	a	DET
ejpam-3317	332	27	1	1	NUM
ejpam-3317	332	28	x	x	SYM
ejpam-3317	332	29	∈	∈	PROPN
ejpam-3317	332	30	x	x	NOUN
ejpam-3317	332	31	/	/	X
ejpam-3317	332	32	a	a	DET
ejpam-3317	332	33	consequently	consequently	ADV
ejpam-3317	332	34	,	,	PUNCT
ejpam-3317	332	35	f−1	f−1	PROPN
ejpam-3317	332	36	(	(	PUNCT
ejpam-3317	332	37	φ	φ	NOUN
ejpam-3317	332	38	)	)	PUNCT
ejpam-3317	332	39	=	=	SYM
ejpam-3317	332	40	φ	φ	PROPN
ejpam-3317	332	41	,	,	PUNCT
ejpam-3317	332	42	f−1	f−1	PROPN
ejpam-3317	332	43	(	(	PUNCT
ejpam-3317	332	44	{	{	PUNCT
ejpam-3317	332	45	0	0	NUM
ejpam-3317	332	46	}	}	PUNCT
ejpam-3317	332	47	)	)	PUNCT
ejpam-3317	332	48	=	=	SYM
ejpam-3317	332	49	a	a	DET
ejpam-3317	332	50	,	,	PUNCT
ejpam-3317	332	51	f−1({1	f−1({1	NOUN
ejpam-3317	332	52	}	}	PUNCT
ejpam-3317	332	53	)	)	PUNCT
ejpam-3317	333	1	=	=	PUNCT
ejpam-3317	333	2	x	x	X
ejpam-3317	333	3	/	/	SYM
ejpam-3317	333	4	a	a	DET
ejpam-3317	333	5	=	=	SYM
ejpam-3317	333	6	b	b	PROPN
ejpam-3317	333	7	and	and	CCONJ
ejpam-3317	333	8	f−1(y	f−1(y	NOUN
ejpam-3317	333	9	)	)	PUNCT
ejpam-3317	334	1	=	=	PUNCT
ejpam-3317	334	2	x.	x.	NOUN
ejpam-3317	334	3	since	since	SCONJ
ejpam-3317	334	4	there	there	PRON
ejpam-3317	334	5	are	be	VERB
ejpam-3317	334	6	only	only	ADV
ejpam-3317	334	7	four	four	NUM
ejpam-3317	334	8	closed	closed	ADJ
ejpam-3317	334	9	subsets	subset	NOUN
ejpam-3317	334	10	of	of	ADP
ejpam-3317	334	11	(	(	PUNCT
ejpam-3317	334	12	y	y	PROPN
ejpam-3317	334	13	,	,	PUNCT
ejpam-3317	334	14	c2	c2	PROPN
ejpam-3317	334	15	)	)	PUNCT
ejpam-3317	334	16	,	,	PUNCT
ejpam-3317	334	17	namely	namely	ADV
ejpam-3317	334	18	φ	φ	PROPN
ejpam-3317	334	19	,	,	PUNCT
ejpam-3317	334	20	{	{	PUNCT
ejpam-3317	334	21	0	0	NUM
ejpam-3317	334	22	}	}	PUNCT
ejpam-3317	334	23	,	,	PUNCT
ejpam-3317	334	24	{	{	PUNCT
ejpam-3317	334	25	1	1	NUM
ejpam-3317	334	26	}	}	PUNCT
ejpam-3317	334	27	and	and	CCONJ
ejpam-3317	334	28	y	y	PROPN
ejpam-3317	334	29	,	,	PUNCT
ejpam-3317	334	30	the	the	DET
ejpam-3317	334	31	inverse	inverse	NOUN
ejpam-3317	334	32	image	image	NOUN
ejpam-3317	334	33	under	under	ADP
ejpam-3317	334	34	fof	fof	NOUN
ejpam-3317	334	35	any	any	DET
ejpam-3317	334	36	closed	closed	ADJ
ejpam-3317	334	37	subset	subset	NOUN
ejpam-3317	334	38	in	in	ADP
ejpam-3317	334	39	(	(	PUNCT
ejpam-3317	334	40	y	y	PROPN
ejpam-3317	334	41	,	,	PUNCT
ejpam-3317	334	42	c2	c2	PROPN
ejpam-3317	334	43	)	)	PUNCT
ejpam-3317	334	44	is	be	AUX
ejpam-3317	334	45	preopen	preopen	ADJ
ejpam-3317	334	46	in	in	ADP
ejpam-3317	334	47	(	(	PUNCT
ejpam-3317	334	48	x	x	NOUN
ejpam-3317	334	49	,	,	PUNCT
ejpam-3317	334	50	c1	c1	PROPN
ejpam-3317	334	51	)	)	PUNCT
ejpam-3317	334	52	.	.	PUNCT
ejpam-3317	335	1	thus	thus	ADV
ejpam-3317	335	2	,	,	PUNCT
ejpam-3317	335	3	f	f	PROPN
ejpam-3317	335	4	is	be	AUX
ejpam-3317	335	5	contra	contra	ADJ
ejpam-3317	335	6	-	-	ADJ
ejpam-3317	335	7	precontinuous	precontinuous	ADJ
ejpam-3317	335	8	but	but	CCONJ
ejpam-3317	335	9	non	non	ADJ
ejpam-3317	335	10	-	-	ADJ
ejpam-3317	335	11	constant	constant	ADJ
ejpam-3317	335	12	,	,	PUNCT
ejpam-3317	335	13	which	which	PRON
ejpam-3317	335	14	a	a	DET
ejpam-3317	335	15	contradiction	contradiction	NOUN
ejpam-3317	335	16	.	.	PUNCT
ejpam-3317	336	1	therefore	therefore	ADV
ejpam-3317	336	2	,	,	PUNCT
ejpam-3317	336	3	(	(	PUNCT
ejpam-3317	336	4	x	x	X
ejpam-3317	336	5	,	,	PUNCT
ejpam-3317	336	6	c1	c1	PROPN
ejpam-3317	336	7	)	)	PUNCT
ejpam-3317	336	8	is	be	AUX
ejpam-3317	336	9	p	p	ADV
ejpam-3317	336	10	-	-	PUNCT
ejpam-3317	336	11	connected	connect	VERB
ejpam-3317	336	12	.	.	PUNCT
ejpam-3317	337	1	statement	statement	NOUN
ejpam-3317	337	2	(	(	PUNCT
ejpam-3317	337	3	ii	ii	NOUN
ejpam-3317	337	4	)	)	PUNCT
ejpam-3317	337	5	implies	imply	VERB
ejpam-3317	337	6	statement	statement	NOUN
ejpam-3317	337	7	(	(	PUNCT
ejpam-3317	337	8	i	i	NOUN
ejpam-3317	337	9	):	):	PUNCT
ejpam-3317	337	10	suppose	suppose	VERB
ejpam-3317	337	11	that	that	SCONJ
ejpam-3317	337	12	a	a	DET
ejpam-3317	337	13	contra	contra	PROPN
ejpam-3317	337	14	-	-	ADJ
ejpam-3317	337	15	pre	pre	ADJ
ejpam-3317	337	16	-	-	ADJ
ejpam-3317	337	17	continuous	continuous	ADJ
ejpam-3317	337	18	function	function	NOUN
ejpam-3317	337	19	f	f	NOUN
ejpam-3317	337	20	:	:	PUNCT
ejpam-3317	337	21	(	(	PUNCT
ejpam-3317	337	22	x	x	X
ejpam-3317	337	23	,	,	PUNCT
ejpam-3317	337	24	c1)→	c1)→	NOUN
ejpam-3317	337	25	(	(	PUNCT
ejpam-3317	337	26	y	y	PROPN
ejpam-3317	337	27	,	,	PUNCT
ejpam-3317	337	28	c2	c2	PROPN
ejpam-3317	337	29	)	)	PUNCT
ejpam-3317	337	30	is	be	AUX
ejpam-3317	337	31	non	non	ADJ
ejpam-3317	337	32	-	-	ADJ
ejpam-3317	337	33	constant	constant	ADJ
ejpam-3317	337	34	,	,	PUNCT
ejpam-3317	337	35	where	where	SCONJ
ejpam-3317	337	36	the	the	DET
ejpam-3317	337	37	closure	closure	NOUN
ejpam-3317	337	38	operator	operator	NOUN
ejpam-3317	337	39	c2	c2	PROPN
ejpam-3317	337	40	on	on	ADP
ejpam-3317	337	41	y	y	PROPN
ejpam-3317	337	42	is	be	AUX
ejpam-3317	337	43	defined	define	VERB
ejpam-3317	337	44	by	by	ADP
ejpam-3317	337	45	:	:	PUNCT
ejpam-3317	337	46	c	c	X
ejpam-3317	337	47	(	(	PUNCT
ejpam-3317	337	48	a	a	NOUN
ejpam-3317	337	49	)	)	PUNCT
ejpam-3317	337	50	=	=	SYM
ejpam-3317	337	51	a	a	PRON
ejpam-3317	337	52	,	,	PUNCT
ejpam-3317	337	53	for	for	ADP
ejpam-3317	337	54	all	all	PRON
ejpam-3317	337	55	subset	subset	VERB
ejpam-3317	337	56	a	a	PRON
ejpam-3317	337	57	of	of	ADP
ejpam-3317	337	58	y.	y.	PROPN
ejpam-3317	337	59	then	then	ADV
ejpam-3317	337	60	f−1({0	f−1({0	PRON
ejpam-3317	337	61	}	}	PUNCT
ejpam-3317	337	62	)	)	PUNCT
ejpam-3317	337	63	and	and	CCONJ
ejpam-3317	337	64	f−1({1	f−1({1	NUM
ejpam-3317	337	65	}	}	PUNCT
ejpam-3317	337	66	)	)	PUNCT
ejpam-3317	337	67	are	be	AUX
ejpam-3317	337	68	non	non	ADJ
ejpam-3317	337	69	-	-	ADJ
ejpam-3317	337	70	empty	empty	ADJ
ejpam-3317	337	71	.	.	PUNCT
ejpam-3317	338	1	further	far	ADV
ejpam-3317	338	2	,	,	PUNCT
ejpam-3317	338	3	neither	neither	CCONJ
ejpam-3317	338	4	f−1({0	f−1({0	PRON
ejpam-3317	338	5	}	}	PUNCT
ejpam-3317	338	6	)	)	PUNCT
ejpam-3317	338	7	nor	nor	CCONJ
ejpam-3317	338	8	f−1({1	f−1({1	NOUN
ejpam-3317	338	9	}	}	PUNCT
ejpam-3317	338	10	)	)	PUNCT
ejpam-3317	338	11	are	be	AUX
ejpam-3317	338	12	equal	equal	ADJ
ejpam-3317	338	13	to	to	PART
ejpam-3317	338	14	x.	x.	VERB
ejpam-3317	338	15	since	since	SCONJ
ejpam-3317	338	16	{	{	PUNCT
ejpam-3317	338	17	0	0	NUM
ejpam-3317	338	18	}	}	PUNCT
ejpam-3317	338	19	and	and	CCONJ
ejpam-3317	338	20	{	{	PUNCT
ejpam-3317	338	21	1	1	X
ejpam-3317	338	22	}	}	PUNCT
ejpam-3317	338	23	are	be	AUX
ejpam-3317	338	24	closed	close	VERB
ejpam-3317	338	25	subset	subset	NOUN
ejpam-3317	338	26	of	of	ADP
ejpam-3317	338	27	(	(	PUNCT
ejpam-3317	338	28	y	y	PROPN
ejpam-3317	338	29	,	,	PUNCT
ejpam-3317	338	30	c2	c2	PROPN
ejpam-3317	338	31	)	)	PUNCT
ejpam-3317	338	32	and	and	CCONJ
ejpam-3317	338	33	f	f	PROPN
ejpam-3317	338	34	is	be	AUX
ejpam-3317	338	35	contra	contra	PROPN
ejpam-3317	338	36	pre	pre	ADJ
ejpam-3317	338	37	-	-	ADJ
ejpam-3317	338	38	continuous	continuous	ADJ
ejpam-3317	338	39	,	,	PUNCT
ejpam-3317	338	40	f−1({0	f−1({0	NOUN
ejpam-3317	338	41	}	}	PUNCT
ejpam-3317	338	42	)	)	PUNCT
ejpam-3317	338	43	and	and	CCONJ
ejpam-3317	338	44	f−1({1	f−1({1	NUM
ejpam-3317	338	45	}	}	PUNCT
ejpam-3317	338	46	)	)	PUNCT
ejpam-3317	338	47	are	be	AUX
ejpam-3317	338	48	preopen	preopen	ADJ
ejpam-3317	338	49	subsets	subset	NOUN
ejpam-3317	338	50	of	of	ADP
ejpam-3317	338	51	(	(	PUNCT
ejpam-3317	338	52	x	x	NOUN
ejpam-3317	338	53	,	,	PUNCT
ejpam-3317	338	54	c1	c1	PROPN
ejpam-3317	338	55	)	)	PUNCT
ejpam-3317	338	56	.	.	PUNCT
ejpam-3317	339	1	but	but	CCONJ
ejpam-3317	339	2	f−1	f−1	PROPN
ejpam-3317	339	3	(	(	PUNCT
ejpam-3317	339	4	{	{	PUNCT
ejpam-3317	339	5	0	0	NUM
ejpam-3317	339	6	}	}	PUNCT
ejpam-3317	339	7	)	)	PUNCT
ejpam-3317	339	8	=	=	PUNCT
ejpam-3317	340	1	x	x	X
ejpam-3317	340	2	/	/	SYM
ejpam-3317	340	3	f−1({1	f−1({1	NOUN
ejpam-3317	340	4	}	}	PUNCT
ejpam-3317	340	5	)	)	PUNCT
ejpam-3317	340	6	.	.	PUNCT
ejpam-3317	341	1	hence	hence	ADV
ejpam-3317	341	2	f−1({0	f−1({0	PRON
ejpam-3317	341	3	}	}	PUNCT
ejpam-3317	341	4	)	)	PUNCT
ejpam-3317	341	5	is	be	AUX
ejpam-3317	341	6	both	both	PRON
ejpam-3317	341	7	preclosed	preclose	VERB
ejpam-3317	341	8	and	and	CCONJ
ejpam-3317	341	9	preopen	preopen	ADJ
ejpam-3317	341	10	.	.	PUNCT
ejpam-3317	342	1	consequently	consequently	ADV
ejpam-3317	342	2	,	,	PUNCT
ejpam-3317	342	3	x	x	PRON
ejpam-3317	342	4	is	be	AUX
ejpam-3317	342	5	not	not	PART
ejpam-3317	342	6	p	p	NOUN
ejpam-3317	342	7	-	-	PUNCT
ejpam-3317	342	8	connected	connect	VERB
ejpam-3317	342	9	,	,	PUNCT
ejpam-3317	342	10	which	which	PRON
ejpam-3317	342	11	a	a	DET
ejpam-3317	342	12	contradiction	contradiction	NOUN
ejpam-3317	342	13	.	.	PUNCT
ejpam-3317	343	1	proposition	proposition	NOUN
ejpam-3317	343	2	34	34	NUM
ejpam-3317	343	3	.	.	PUNCT
ejpam-3317	344	1	let	let	VERB
ejpam-3317	344	2	(	(	PUNCT
ejpam-3317	344	3	x	x	NOUN
ejpam-3317	344	4	,	,	PUNCT
ejpam-3317	344	5	c1)and	c1)and	PROPN
ejpam-3317	344	6	(	(	PUNCT
ejpam-3317	344	7	y	y	NOUN
ejpam-3317	344	8	,	,	PUNCT
ejpam-3317	344	9	c2)be	c2)be	ADJ
ejpam-3317	344	10	closure	closure	NOUN
ejpam-3317	344	11	spaces	space	NOUN
ejpam-3317	344	12	and	and	CCONJ
ejpam-3317	344	13	f	f	X
ejpam-3317	344	14	:	:	PUNCT
ejpam-3317	344	15	(	(	PUNCT
ejpam-3317	344	16	x	x	X
ejpam-3317	344	17	,	,	PUNCT
ejpam-3317	344	18	c1	c1	PROPN
ejpam-3317	344	19	)	)	PUNCT
ejpam-3317	344	20	→	→	PUNCT
ejpam-3317	344	21	(	(	PUNCT
ejpam-3317	344	22	y	y	PROPN
ejpam-3317	344	23	,	,	PUNCT
ejpam-3317	344	24	c2	c2	PROPN
ejpam-3317	344	25	)	)	PUNCT
ejpam-3317	344	26	be	be	AUX
ejpam-3317	344	27	a	a	DET
ejpam-3317	344	28	function	function	NOUN
ejpam-3317	344	29	.	.	PUNCT
ejpam-3317	345	1	(	(	PUNCT
ejpam-3317	345	2	i	i	NOUN
ejpam-3317	345	3	)	)	PUNCT
ejpam-3317	345	4	if	if	SCONJ
ejpam-3317	345	5	f	f	PROPN
ejpam-3317	345	6	is	be	AUX
ejpam-3317	345	7	a	a	DET
ejpam-3317	345	8	contra	contra	PROPN
ejpam-3317	345	9	-	-	ADJ
ejpam-3317	345	10	pre	pre	ADJ
ejpam-3317	345	11	-	-	ADJ
ejpam-3317	345	12	continuous	continuous	ADJ
ejpam-3317	345	13	function	function	NOUN
ejpam-3317	345	14	from	from	ADP
ejpam-3317	345	15	(	(	PUNCT
ejpam-3317	345	16	x	x	NOUN
ejpam-3317	345	17	,	,	PUNCT
ejpam-3317	345	18	c1	c1	PROPN
ejpam-3317	345	19	)	)	PUNCT
ejpam-3317	345	20	onto	onto	ADP
ejpam-3317	345	21	(	(	PUNCT
ejpam-3317	345	22	y	y	PROPN
ejpam-3317	345	23	,	,	PUNCT
ejpam-3317	345	24	c2	c2	PROPN
ejpam-3317	345	25	)	)	PUNCT
ejpam-3317	345	26	and	and	CCONJ
ejpam-3317	345	27	(	(	PUNCT
ejpam-3317	345	28	x	x	X
ejpam-3317	345	29	,	,	PUNCT
ejpam-3317	345	30	c1	c1	PROPN
ejpam-3317	345	31	)	)	PUNCT
ejpam-3317	345	32	is	be	AUX
ejpam-3317	345	33	pconnected	pconnecte	VERB
ejpam-3317	345	34	,	,	PUNCT
ejpam-3317	345	35	then	then	ADV
ejpam-3317	345	36	(	(	PUNCT
ejpam-3317	345	37	y	y	PROPN
ejpam-3317	345	38	,	,	PUNCT
ejpam-3317	345	39	c2	c2	PROPN
ejpam-3317	345	40	)	)	PUNCT
ejpam-3317	345	41	is	be	AUX
ejpam-3317	345	42	connected	connect	VERB
ejpam-3317	345	43	.	.	PUNCT
ejpam-3317	346	1	(	(	PUNCT
ejpam-3317	346	2	ii	ii	NOUN
ejpam-3317	346	3	)	)	PUNCT
ejpam-3317	346	4	if	if	SCONJ
ejpam-3317	346	5	f	f	PROPN
ejpam-3317	346	6	is	be	AUX
ejpam-3317	346	7	a	a	DET
ejpam-3317	346	8	pre	pre	ADJ
ejpam-3317	346	9	-	-	ADJ
ejpam-3317	346	10	irresolute	irresolute	ADJ
ejpam-3317	346	11	function	function	NOUN
ejpam-3317	346	12	from	from	ADP
ejpam-3317	346	13	(	(	PUNCT
ejpam-3317	346	14	x	x	NOUN
ejpam-3317	346	15	,	,	PUNCT
ejpam-3317	346	16	c1)onto	c1)onto	ADJ
ejpam-3317	346	17	(	(	PUNCT
ejpam-3317	346	18	y	y	PROPN
ejpam-3317	346	19	,	,	PUNCT
ejpam-3317	346	20	c2	c2	PROPN
ejpam-3317	346	21	)	)	PUNCT
ejpam-3317	346	22	and	and	CCONJ
ejpam-3317	346	23	(	(	PUNCT
ejpam-3317	346	24	x	x	X
ejpam-3317	346	25	,	,	PUNCT
ejpam-3317	346	26	c1)is	c1)is	ADJ
ejpam-3317	346	27	p	p	NOUN
ejpam-3317	346	28	-	-	PUNCT
ejpam-3317	346	29	connected	connect	VERB
ejpam-3317	346	30	,	,	PUNCT
ejpam-3317	346	31	then	then	ADV
ejpam-3317	346	32	(	(	PUNCT
ejpam-3317	346	33	y	y	PROPN
ejpam-3317	346	34	,	,	PUNCT
ejpam-3317	346	35	c2	c2	PROPN
ejpam-3317	346	36	)	)	PUNCT
ejpam-3317	346	37	is	be	AUX
ejpam-3317	346	38	p	p	ADV
ejpam-3317	346	39	-	-	PUNCT
ejpam-3317	346	40	connected	connect	VERB
ejpam-3317	346	41	.	.	PUNCT
ejpam-3317	347	1	h.m	h.m	PROPN
ejpam-3317	347	2	.	.	PROPN
ejpam-3317	347	3	darwesh	darwesh	PROPN
ejpam-3317	347	4	,	,	PUNCT
ejpam-3317	347	5	s.f	s.f	PROPN
ejpam-3317	347	6	.	.	PROPN
ejpam-3317	347	7	namiq	namiq	PROPN
ejpam-3317	347	8	/	/	SYM
ejpam-3317	347	9	eur	eur	PROPN
ejpam-3317	347	10	.	.	PUNCT
ejpam-3317	348	1	j.	j.	PROPN
ejpam-3317	348	2	pure	pure	PROPN
ejpam-3317	348	3	appl	appl	PROPN
ejpam-3317	348	4	.	.	PROPN
ejpam-3317	348	5	math	math	PROPN
ejpam-3317	348	6	,	,	PUNCT
ejpam-3317	348	7	12	12	NUM
ejpam-3317	348	8	(	(	PUNCT
ejpam-3317	348	9	3	3	NUM
ejpam-3317	348	10	)	)	PUNCT
ejpam-3317	348	11	(	(	PUNCT
ejpam-3317	348	12	2019	2019	NUM
ejpam-3317	348	13	)	)	PUNCT
ejpam-3317	348	14	,	,	PUNCT
ejpam-3317	348	15	1082	1082	NUM
ejpam-3317	348	16	-	-	SYM
ejpam-3317	348	17	1095	1095	NUM
ejpam-3317	348	18	1093	1093	NUM
ejpam-3317	348	19	proof	proof	NOUN
ejpam-3317	348	20	.	.	PUNCT
ejpam-3317	349	1	(	(	PUNCT
ejpam-3317	349	2	ii	ii	NOUN
ejpam-3317	349	3	)	)	PUNCT
ejpam-3317	349	4	.	.	PUNCT
ejpam-3317	350	1	suppose	suppose	VERB
ejpam-3317	350	2	that	that	SCONJ
ejpam-3317	350	3	(	(	PUNCT
ejpam-3317	350	4	y	y	PROPN
ejpam-3317	350	5	,	,	PUNCT
ejpam-3317	350	6	c2	c2	PROPN
ejpam-3317	350	7	)	)	PUNCT
ejpam-3317	350	8	is	be	AUX
ejpam-3317	350	9	not	not	PART
ejpam-3317	350	10	p	p	NOUN
ejpam-3317	350	11	-	-	PUNCT
ejpam-3317	350	12	connected	connect	VERB
ejpam-3317	350	13	.	.	PUNCT
ejpam-3317	351	1	then	then	ADV
ejpam-3317	351	2	there	there	PRON
ejpam-3317	351	3	is	be	VERB
ejpam-3317	351	4	a	a	DET
ejpam-3317	351	5	non	non	ADJ
ejpam-3317	351	6	-	-	ADJ
ejpam-3317	351	7	empty	empty	ADJ
ejpam-3317	351	8	subset	subset	NOUN
ejpam-3317	351	9	a	a	PRON
ejpam-3317	351	10	of	of	ADP
ejpam-3317	351	11	y	y	PROPN
ejpam-3317	351	12	,	,	PUNCT
ejpam-3317	351	13	a	a	DET
ejpam-3317	351	14	6=	6=	NUM
ejpam-3317	351	15	y	y	PRON
ejpam-3317	351	16	such	such	ADJ
ejpam-3317	351	17	that	that	SCONJ
ejpam-3317	351	18	a	a	PRON
ejpam-3317	351	19	is	be	AUX
ejpam-3317	351	20	both	both	CCONJ
ejpam-3317	351	21	preopen	preopen	ADJ
ejpam-3317	351	22	and	and	CCONJ
ejpam-3317	351	23	preclosed	preclose	VERB
ejpam-3317	351	24	.	.	PUNCT
ejpam-3317	352	1	since	since	SCONJ
ejpam-3317	352	2	f	f	PROPN
ejpam-3317	352	3	is	be	AUX
ejpam-3317	352	4	pre	pre	ADJ
ejpam-3317	352	5	-	-	ADJ
ejpam-3317	352	6	irresolute	irresolute	ADJ
ejpam-3317	352	7	,	,	PUNCT
ejpam-3317	352	8	the	the	DET
ejpam-3317	352	9	set	set	ADJ
ejpam-3317	352	10	f−1(a	f−1(a	NOUN
ejpam-3317	352	11	)	)	PUNCT
ejpam-3317	352	12	is	be	AUX
ejpam-3317	352	13	both	both	CCONJ
ejpam-3317	352	14	preopen	preopen	ADJ
ejpam-3317	352	15	and	and	CCONJ
ejpam-3317	352	16	preclosed	preclose	VERB
ejpam-3317	352	17	.	.	PUNCT
ejpam-3317	353	1	since	since	SCONJ
ejpam-3317	353	2	f	f	PROPN
ejpam-3317	353	3	is	be	AUX
ejpam-3317	353	4	an	an	DET
ejpam-3317	353	5	onto	onto	ADP
ejpam-3317	353	6	function	function	NOUN
ejpam-3317	353	7	and	and	CCONJ
ejpam-3317	353	8	a	a	PRON
ejpam-3317	353	9	is	be	AUX
ejpam-3317	353	10	a	a	DET
ejpam-3317	353	11	non	non	ADJ
ejpam-3317	353	12	-	-	ADJ
ejpam-3317	353	13	empty	empty	ADJ
ejpam-3317	353	14	subset	subset	NOUN
ejpam-3317	353	15	of	of	ADP
ejpam-3317	353	16	y	y	PROPN
ejpam-3317	353	17	with	with	ADP
ejpam-3317	353	18	a	a	DET
ejpam-3317	353	19	6=	6=	NUM
ejpam-3317	353	20	y	y	PROPN
ejpam-3317	353	21	,	,	PUNCT
ejpam-3317	353	22	it	it	PRON
ejpam-3317	353	23	follows	follow	VERB
ejpam-3317	353	24	that	that	DET
ejpam-3317	353	25	f−1(a	f−1(a	NOUN
ejpam-3317	353	26	)	)	PUNCT
ejpam-3317	353	27	is	be	AUX
ejpam-3317	353	28	a	a	DET
ejpam-3317	353	29	non	non	ADJ
ejpam-3317	353	30	-	-	ADJ
ejpam-3317	353	31	empty	empty	ADJ
ejpam-3317	353	32	subset	subset	NOUN
ejpam-3317	353	33	of	of	ADP
ejpam-3317	353	34	x	x	PUNCT
ejpam-3317	353	35	with	with	ADP
ejpam-3317	353	36	f−1(a	f−1(a	NOUN
ejpam-3317	353	37	)	)	PUNCT
ejpam-3317	353	38	6=	6=	PUNCT
ejpam-3317	354	1	x	x	X
ejpam-3317	354	2	.	.	PUNCT
ejpam-3317	355	1	hence	hence	ADV
ejpam-3317	355	2	,	,	PUNCT
ejpam-3317	355	3	(	(	PUNCT
ejpam-3317	355	4	x	x	X
ejpam-3317	355	5	,	,	PUNCT
ejpam-3317	355	6	c1	c1	PROPN
ejpam-3317	355	7	)	)	PUNCT
ejpam-3317	355	8	is	be	AUX
ejpam-3317	355	9	not	not	PART
ejpam-3317	355	10	p	p	NOUN
ejpam-3317	355	11	-	-	PUNCT
ejpam-3317	355	12	connected	connect	VERB
ejpam-3317	355	13	which	which	PRON
ejpam-3317	355	14	a	a	DET
ejpam-3317	355	15	contradiction	contradiction	NOUN
ejpam-3317	355	16	.	.	PUNCT
ejpam-3317	356	1	therefore	therefore	ADV
ejpam-3317	356	2	,	,	PUNCT
ejpam-3317	356	3	(	(	PUNCT
ejpam-3317	356	4	y	y	PROPN
ejpam-3317	356	5	,	,	PUNCT
ejpam-3317	356	6	c2	c2	PROPN
ejpam-3317	356	7	)	)	PUNCT
ejpam-3317	356	8	is	be	AUX
ejpam-3317	356	9	connected	connect	VERB
ejpam-3317	356	10	.	.	PUNCT
ejpam-3317	357	1	the	the	DET
ejpam-3317	357	2	proof	proof	NOUN
ejpam-3317	357	3	of	of	ADP
ejpam-3317	357	4	(	(	PUNCT
ejpam-3317	357	5	i)is	i)is	PROPN
ejpam-3317	357	6	similar	similar	ADJ
ejpam-3317	357	7	to	to	ADP
ejpam-3317	357	8	that	that	PRON
ejpam-3317	357	9	of	of	ADP
ejpam-3317	357	10	(	(	PUNCT
ejpam-3317	357	11	ii	ii	NOUN
ejpam-3317	357	12	)	)	PUNCT
ejpam-3317	357	13	.	.	PUNCT
ejpam-3317	358	1	6	6	X
ejpam-3317	358	2	.	.	X
ejpam-3317	359	1	p	p	ADJ
ejpam-3317	359	2	-compactness	-compactness	NOUN
ejpam-3317	359	3	as	as	ADP
ejpam-3317	359	4	another	another	DET
ejpam-3317	359	5	application	application	NOUN
ejpam-3317	359	6	of	of	ADP
ejpam-3317	359	7	preopen	preopen	ADJ
ejpam-3317	359	8	sets	set	NOUN
ejpam-3317	359	9	,	,	PUNCT
ejpam-3317	359	10	a	a	DET
ejpam-3317	359	11	new	new	ADJ
ejpam-3317	359	12	kind	kind	NOUN
ejpam-3317	359	13	of	of	ADP
ejpam-3317	359	14	compactness	compactness	NOUN
ejpam-3317	359	15	,	,	PUNCT
ejpam-3317	359	16	namely	namely	ADV
ejpam-3317	359	17	p	p	NOUN
ejpam-3317	359	18	-	-	PUNCT
ejpam-3317	359	19	compactness	compactness	NOUN
ejpam-3317	359	20	,	,	PUNCT
ejpam-3317	359	21	is	be	AUX
ejpam-3317	359	22	introduced	introduce	VERB
ejpam-3317	359	23	.	.	PUNCT
ejpam-3317	360	1	definition	definition	NOUN
ejpam-3317	360	2	17	17	NUM
ejpam-3317	360	3	.	.	PUNCT
ejpam-3317	361	1	a	a	DET
ejpam-3317	361	2	collection	collection	NOUN
ejpam-3317	361	3	{	{	PUNCT
ejpam-3317	361	4	gα}α∈j	gα}α∈j	PROPN
ejpam-3317	361	5	of	of	ADP
ejpam-3317	361	6	preopen	preopen	ADJ
ejpam-3317	361	7	sets	set	NOUN
ejpam-3317	361	8	in	in	ADP
ejpam-3317	361	9	a	a	DET
ejpam-3317	361	10	closure	closure	NOUN
ejpam-3317	361	11	space	space	NOUN
ejpam-3317	361	12	(	(	PUNCT
ejpam-3317	361	13	x	x	NOUN
ejpam-3317	361	14	,	,	PUNCT
ejpam-3317	361	15	c1	c1	PROPN
ejpam-3317	361	16	)	)	PUNCT
ejpam-3317	361	17	is	be	AUX
ejpam-3317	361	18	called	call	VERB
ejpam-3317	361	19	a	a	DET
ejpam-3317	361	20	preopen	preopen	ADJ
ejpam-3317	361	21	cover	cover	NOUN
ejpam-3317	361	22	of	of	ADP
ejpam-3317	361	23	a	a	DET
ejpam-3317	361	24	subset	subset	NOUN
ejpam-3317	361	25	b	b	NOUN
ejpam-3317	361	26	of	of	ADP
ejpam-3317	361	27	x	x	PRON
ejpam-3317	361	28	if	if	SCONJ
ejpam-3317	361	29	b	b	PROPN
ejpam-3317	361	30	⊆	⊆	NUM
ejpam-3317	361	31	⋃	⋃	ADP
ejpam-3317	361	32	α∈j	α∈j	NOUN
ejpam-3317	361	33	gα	gα	ADP
ejpam-3317	361	34	holds	hold	NOUN
ejpam-3317	361	35	.	.	PUNCT
ejpam-3317	362	1	definition	definition	NOUN
ejpam-3317	362	2	18	18	NUM
ejpam-3317	362	3	.	.	PUNCT
ejpam-3317	363	1	a	a	DET
ejpam-3317	363	2	subset	subset	NOUN
ejpam-3317	363	3	a	a	PRON
ejpam-3317	363	4	of	of	ADP
ejpam-3317	363	5	a	a	DET
ejpam-3317	363	6	closure	closure	NOUN
ejpam-3317	363	7	space	space	NOUN
ejpam-3317	363	8	(	(	PUNCT
ejpam-3317	363	9	x	x	NOUN
ejpam-3317	363	10	,	,	PUNCT
ejpam-3317	363	11	c)is	c)is	PROPN
ejpam-3317	363	12	p	p	PROPN
ejpam-3317	363	13	-	-	PUNCT
ejpam-3317	363	14	compact	compact	ADJ
ejpam-3317	363	15	if	if	SCONJ
ejpam-3317	363	16	every	every	DET
ejpam-3317	363	17	preopen	preopen	ADJ
ejpam-3317	363	18	cover	cover	NOUN
ejpam-3317	363	19	of	of	ADP
ejpam-3317	363	20	a	a	PRON
ejpam-3317	363	21	contains	contain	VERB
ejpam-3317	363	22	a	a	DET
ejpam-3317	363	23	finite	finite	ADJ
ejpam-3317	363	24	subcover	subcover	PROPN
ejpam-3317	363	25	.	.	PUNCT
ejpam-3317	364	1	the	the	DET
ejpam-3317	364	2	following	follow	VERB
ejpam-3317	364	3	statements	statement	NOUN
ejpam-3317	364	4	are	be	AUX
ejpam-3317	364	5	evident	evident	ADJ
ejpam-3317	364	6	:	:	PUNCT
ejpam-3317	364	7	proposition	proposition	NOUN
ejpam-3317	364	8	35	35	NUM
ejpam-3317	364	9	.	.	PUNCT
ejpam-3317	365	1	let	let	VERB
ejpam-3317	365	2	(	(	PUNCT
ejpam-3317	365	3	x	x	NOUN
ejpam-3317	365	4	,	,	PUNCT
ejpam-3317	365	5	c1	c1	PROPN
ejpam-3317	365	6	)	)	PUNCT
ejpam-3317	365	7	be	be	AUX
ejpam-3317	365	8	a	a	DET
ejpam-3317	365	9	closure	closure	NOUN
ejpam-3317	365	10	space	space	NOUN
ejpam-3317	365	11	.	.	PUNCT
ejpam-3317	366	1	if	if	SCONJ
ejpam-3317	366	2	x	x	PRON
ejpam-3317	366	3	is	be	AUX
ejpam-3317	366	4	p	p	ADJ
ejpam-3317	366	5	-	-	ADJ
ejpam-3317	366	6	compact	compact	ADJ
ejpam-3317	366	7	and	and	CCONJ
ejpam-3317	366	8	b	b	NOUN
ejpam-3317	366	9	is	be	AUX
ejpam-3317	366	10	a	a	DET
ejpam-3317	366	11	preclosed	preclose	VERB
ejpam-3317	366	12	subset	subset	NOUN
ejpam-3317	366	13	of	of	ADP
ejpam-3317	366	14	x	x	PRON
ejpam-3317	366	15	,	,	PUNCT
ejpam-3317	366	16	then	then	ADV
ejpam-3317	366	17	b	b	PROPN
ejpam-3317	366	18	is	be	AUX
ejpam-3317	366	19	p	p	ADJ
ejpam-3317	366	20	-	-	PUNCT
ejpam-3317	366	21	compact	compact	ADJ
ejpam-3317	366	22	proof	proof	NOUN
ejpam-3317	366	23	.	.	PUNCT
ejpam-3317	367	1	let	let	VERB
ejpam-3317	367	2	{	{	PUNCT
ejpam-3317	367	3	gα}α∈j	gα}α∈j	VERB
ejpam-3317	367	4	be	be	AUX
ejpam-3317	367	5	a	a	DET
ejpam-3317	367	6	collection	collection	NOUN
ejpam-3317	367	7	of	of	ADP
ejpam-3317	367	8	preopen	preopen	ADJ
ejpam-3317	367	9	subsets	subset	NOUN
ejpam-3317	367	10	of	of	ADP
ejpam-3317	367	11	x	x	SYM
ejpam-3317	367	12	such	such	ADJ
ejpam-3317	367	13	that	that	DET
ejpam-3317	367	14	b	b	NOUN
ejpam-3317	367	15	⊆	⊆	NUM
ejpam-3317	367	16	⋃	⋃	NOUN
ejpam-3317	367	17	α∈j	α∈j	NOUN
ejpam-3317	367	18	gα	gα	NOUN
ejpam-3317	367	19	.	.	PUNCT
ejpam-3317	368	1	it	it	PRON
ejpam-3317	368	2	follows	follow	VERB
ejpam-3317	368	3	that	that	PRON
ejpam-3317	368	4	x=	x=	PUNCT
ejpam-3317	369	1	⋃	⋃	VERB
ejpam-3317	369	2	α∈j	α∈j	NOUN
ejpam-3317	369	3	gα	gα	ADP
ejpam-3317	369	4	∪	∪	ADJ
ejpam-3317	369	5	(	(	PUNCT
ejpam-3317	369	6	x	x	NOUN
ejpam-3317	369	7	/	/	SYM
ejpam-3317	369	8	b	b	NOUN
ejpam-3317	369	9	)	)	PUNCT
ejpam-3317	369	10	.	.	PUNCT
ejpam-3317	370	1	since	since	SCONJ
ejpam-3317	370	2	b	b	PROPN
ejpam-3317	370	3	is	be	AUX
ejpam-3317	370	4	preclosed	preclose	VERB
ejpam-3317	370	5	,	,	PUNCT
ejpam-3317	370	6	x	x	X
ejpam-3317	370	7	/	/	SYM
ejpam-3317	370	8	b	b	PROPN
ejpam-3317	370	9	is	be	AUX
ejpam-3317	370	10	preopen	preopen	ADJ
ejpam-3317	370	11	.	.	PUNCT
ejpam-3317	371	1	consequently,⋃	consequently,⋃	NOUN
ejpam-3317	371	2	α∈j	α∈j	ADP
ejpam-3317	371	3	gα	gα	ADP
ejpam-3317	371	4	∪	∪	ADJ
ejpam-3317	371	5	(	(	PUNCT
ejpam-3317	371	6	x	x	NOUN
ejpam-3317	371	7	/	/	SYM
ejpam-3317	371	8	b	b	NOUN
ejpam-3317	371	9	)	)	PUNCT
ejpam-3317	371	10	is	be	AUX
ejpam-3317	371	11	a	a	DET
ejpam-3317	371	12	preopen	preopen	ADJ
ejpam-3317	371	13	cover	cover	NOUN
ejpam-3317	371	14	of	of	ADP
ejpam-3317	371	15	x.	x.	NOUN
ejpam-3317	371	16	but	but	CCONJ
ejpam-3317	371	17	x	x	X
ejpam-3317	371	18	is	be	AUX
ejpam-3317	371	19	p	p	ADJ
ejpam-3317	371	20	-	-	PUNCT
ejpam-3317	371	21	compact	compact	ADJ
ejpam-3317	371	22	,	,	PUNCT
ejpam-3317	371	23	so	so	ADV
ejpam-3317	371	24	⋃	⋃	PUNCT
ejpam-3317	371	25	α∈j	α∈j	ADP
ejpam-3317	371	26	gα∪(x	gα∪(x	NOUN
ejpam-3317	371	27	/	/	SYM
ejpam-3317	371	28	b	b	PROPN
ejpam-3317	371	29	)	)	PUNCT
ejpam-3317	371	30	contains	contain	VERB
ejpam-3317	371	31	a	a	DET
ejpam-3317	371	32	finite	finite	ADJ
ejpam-3317	371	33	subcover	subcover	PROPN
ejpam-3317	371	34	,	,	PUNCT
ejpam-3317	371	35	i.e.	i.e.	X
ejpam-3317	371	36	there	there	PRON
ejpam-3317	371	37	exits	exit	VERB
ejpam-3317	371	38	a	a	DET
ejpam-3317	371	39	finite	finite	NOUN
ejpam-3317	371	40	subset	subset	VERB
ejpam-3317	371	41	j0	j0	PROPN
ejpam-3317	371	42	of	of	ADP
ejpam-3317	371	43	j	j	PROPN
ejpam-3317	371	44	such	such	ADJ
ejpam-3317	371	45	that	that	SCONJ
ejpam-3317	371	46	x=	x=	PUNCT
ejpam-3317	372	1	⋃	⋃	NOUN
ejpam-3317	372	2	α∈j0	α∈j0	NOUN
ejpam-3317	372	3	gα∪	gα∪	PUNCT
ejpam-3317	372	4	(	(	PUNCT
ejpam-3317	372	5	x	x	X
ejpam-3317	372	6	/	/	SYM
ejpam-3317	372	7	b	b	NOUN
ejpam-3317	372	8	)	)	PUNCT
ejpam-3317	372	9	.	.	PUNCT
ejpam-3317	373	1	since	since	SCONJ
ejpam-3317	373	2	b	b	PROPN
ejpam-3317	373	3	and	and	CCONJ
ejpam-3317	373	4	x	x	SYM
ejpam-3317	373	5	/	/	SYM
ejpam-3317	373	6	b	b	PROPN
ejpam-3317	373	7	are	be	AUX
ejpam-3317	373	8	disjoint	disjoint	ADJ
ejpam-3317	373	9	,	,	PUNCT
ejpam-3317	373	10	b	b	NOUN
ejpam-3317	373	11	⊆	⊆	NUM
ejpam-3317	373	12	⋃	⋃	NOUN
ejpam-3317	373	13	α∈j0	α∈j0	NOUN
ejpam-3317	373	14	gα	gα	NOUN
ejpam-3317	373	15	.	.	PUNCT
ejpam-3317	374	1	thus	thus	ADV
ejpam-3317	374	2	,	,	PUNCT
ejpam-3317	374	3	any	any	DET
ejpam-3317	374	4	preopen	preopen	ADJ
ejpam-3317	374	5	cover	cover	NOUN
ejpam-3317	374	6	{	{	PUNCT
ejpam-3317	374	7	gα}α∈j	gα}α∈j	NOUN
ejpam-3317	374	8	of	of	ADP
ejpam-3317	374	9	b	b	PROPN
ejpam-3317	374	10	contains	contain	VERB
ejpam-3317	374	11	a	a	DET
ejpam-3317	374	12	finite	finite	ADJ
ejpam-3317	374	13	subcover	subcover	PROPN
ejpam-3317	374	14	.	.	PUNCT
ejpam-3317	375	1	therefore	therefore	ADV
ejpam-3317	375	2	,	,	PUNCT
ejpam-3317	375	3	b	b	PROPN
ejpam-3317	375	4	is	be	AUX
ejpam-3317	375	5	p	p	ADJ
ejpam-3317	375	6	-	-	PUNCT
ejpam-3317	375	7	compact	compact	ADJ
ejpam-3317	375	8	.	.	PUNCT
ejpam-3317	376	1	proposition	proposition	NOUN
ejpam-3317	376	2	36	36	NUM
ejpam-3317	376	3	.	.	PUNCT
ejpam-3317	377	1	let	let	AUX
ejpam-3317	377	2	(	(	PUNCT
ejpam-3317	377	3	x	x	NOUN
ejpam-3317	377	4	,	,	PUNCT
ejpam-3317	377	5	c1	c1	PROPN
ejpam-3317	377	6	)	)	PUNCT
ejpam-3317	377	7	and	and	CCONJ
ejpam-3317	377	8	(	(	PUNCT
ejpam-3317	377	9	y	y	PROPN
ejpam-3317	377	10	,	,	PUNCT
ejpam-3317	377	11	c2	c2	PROPN
ejpam-3317	377	12	)	)	PUNCT
ejpam-3317	377	13	be	be	VERB
ejpam-3317	377	14	closure	closure	NOUN
ejpam-3317	377	15	spaces	space	NOUN
ejpam-3317	377	16	and	and	CCONJ
ejpam-3317	377	17	f	f	X
ejpam-3317	377	18	:	:	PUNCT
ejpam-3317	377	19	(	(	PUNCT
ejpam-3317	377	20	x	x	X
ejpam-3317	377	21	,	,	PUNCT
ejpam-3317	377	22	c1	c1	PROPN
ejpam-3317	377	23	)	)	PUNCT
ejpam-3317	377	24	→	→	PUNCT
ejpam-3317	377	25	(	(	PUNCT
ejpam-3317	377	26	y	y	PROPN
ejpam-3317	377	27	,	,	PUNCT
ejpam-3317	377	28	c2	c2	PROPN
ejpam-3317	377	29	)	)	PUNCT
ejpam-3317	377	30	be	be	AUX
ejpam-3317	377	31	a	a	DET
ejpam-3317	377	32	function	function	NOUN
ejpam-3317	377	33	.	.	PUNCT
ejpam-3317	378	1	if	if	SCONJ
ejpam-3317	378	2	f	f	PROPN
ejpam-3317	378	3	is	be	AUX
ejpam-3317	378	4	pre	pre	ADJ
ejpam-3317	378	5	-	-	ADJ
ejpam-3317	378	6	irresolute	irresolute	ADJ
ejpam-3317	378	7	and	and	CCONJ
ejpam-3317	378	8	a	a	DET
ejpam-3317	378	9	subset	subset	NOUN
ejpam-3317	378	10	b	b	NOUN
ejpam-3317	378	11	of	of	ADP
ejpam-3317	378	12	x	x	SYM
ejpam-3317	378	13	is	be	AUX
ejpam-3317	378	14	p	p	ADJ
ejpam-3317	378	15	-	-	PUNCT
ejpam-3317	378	16	compact	compact	ADJ
ejpam-3317	378	17	,	,	PUNCT
ejpam-3317	378	18	then	then	ADV
ejpam-3317	378	19	the	the	DET
ejpam-3317	378	20	image	image	NOUN
ejpam-3317	378	21	f	f	PROPN
ejpam-3317	378	22	(	(	PUNCT
ejpam-3317	378	23	b	b	NOUN
ejpam-3317	378	24	)	)	PUNCT
ejpam-3317	378	25	⊆	⊆	NUM
ejpam-3317	378	26	y	y	PROPN
ejpam-3317	378	27	is	be	AUX
ejpam-3317	378	28	p	p	ADJ
ejpam-3317	378	29	-	-	PUNCT
ejpam-3317	378	30	compact	compact	ADJ
ejpam-3317	378	31	.	.	PUNCT
ejpam-3317	379	1	proof	proof	NOUN
ejpam-3317	379	2	.	.	PUNCT
ejpam-3317	380	1	let{gα}α∈j	let{gα}α∈j	NOUN
ejpam-3317	380	2	be	be	AUX
ejpam-3317	380	3	a	a	DET
ejpam-3317	380	4	collection	collection	NOUN
ejpam-3317	380	5	of	of	ADP
ejpam-3317	380	6	preopen	preopen	ADJ
ejpam-3317	380	7	subsets	subset	NOUN
ejpam-3317	380	8	of	of	ADP
ejpam-3317	380	9	y	y	PRON
ejpam-3317	380	10	such	such	ADJ
ejpam-3317	380	11	that	that	DET
ejpam-3317	380	12	f(b	f(b	PROPN
ejpam-3317	380	13	)	)	PUNCT
ejpam-3317	381	1	⊆	⊆	NUM
ejpam-3317	381	2	⋃	⋃	ADP
ejpam-3317	381	3	α∈j	α∈j	NOUN
ejpam-3317	381	4	gα	gα	ADP
ejpam-3317	381	5	it	it	PRON
ejpam-3317	381	6	follow	follow	VERB
ejpam-3317	381	7	that	that	PRON
ejpam-3317	381	8	b	b	PROPN
ejpam-3317	381	9	⊆	⊆	NUM
ejpam-3317	381	10	f−1	f−1	PROPN
ejpam-3317	381	11	(	(	PUNCT
ejpam-3317	381	12	f	f	PROPN
ejpam-3317	381	13	(	(	PUNCT
ejpam-3317	381	14	b	b	NOUN
ejpam-3317	381	15	)	)	PUNCT
ejpam-3317	381	16	)	)	PUNCT
ejpam-3317	382	1	⊆	⊆	X
ejpam-3317	382	2	f−1	f−1	PROPN
ejpam-3317	382	3	{	{	PUNCT
ejpam-3317	382	4	⋃	⋃	NOUN
ejpam-3317	382	5	α∈j	α∈j	NOUN
ejpam-3317	382	6	gα	gα	ADP
ejpam-3317	382	7	}	}	PUNCT
ejpam-3317	382	8	=	=	SYM
ejpam-3317	382	9	⋃	⋃	ADP
ejpam-3317	382	10	α∈j	α∈j	NOUN
ejpam-3317	382	11	f	f	PROPN
ejpam-3317	382	12	−1(gα	−1(gα	NOUN
ejpam-3317	382	13	)	)	PUNCT
ejpam-3317	382	14	,	,	PUNCT
ejpam-3317	382	15	but	but	CCONJ
ejpam-3317	382	16	f	f	PROPN
ejpam-3317	382	17	is	be	AUX
ejpam-3317	382	18	pre	pre	ADJ
ejpam-3317	382	19	-	-	ADJ
ejpam-3317	382	20	irresolute	irresolute	ADJ
ejpam-3317	382	21	,	,	PUNCT
ejpam-3317	382	22	so	so	CCONJ
ejpam-3317	382	23	{	{	PUNCT
ejpam-3317	382	24	f−1(gα)}α∈j	f−1(gα)}α∈j	NOUN
ejpam-3317	382	25	is	be	AUX
ejpam-3317	382	26	a	a	DET
ejpam-3317	382	27	preopen	preopen	ADJ
ejpam-3317	382	28	cover	cover	NOUN
ejpam-3317	382	29	of	of	ADP
ejpam-3317	382	30	b.	b.	PROPN
ejpam-3317	382	31	since	since	SCONJ
ejpam-3317	382	32	b	b	PROPN
ejpam-3317	382	33	is	be	AUX
ejpam-3317	382	34	p	p	ADJ
ejpam-3317	382	35	-	-	PUNCT
ejpam-3317	382	36	compact	compact	ADJ
ejpam-3317	382	37	,	,	PUNCT
ejpam-3317	382	38	there	there	PRON
ejpam-3317	382	39	exists	exist	VERB
ejpam-3317	382	40	a	a	DET
ejpam-3317	382	41	finite	finite	NOUN
ejpam-3317	382	42	subset	subset	NOUN
ejpam-3317	382	43	j0of	j0of	PUNCT
ejpam-3317	383	1	j	j	PROPN
ejpam-3317	383	2	such	such	ADJ
ejpam-3317	383	3	that	that	DET
ejpam-3317	383	4	b	b	NOUN
ejpam-3317	383	5	⊆	⊆	NUM
ejpam-3317	383	6	⋃	⋃	NOUN
ejpam-3317	383	7	α∈j0	α∈j0	NOUN
ejpam-3317	383	8	f	f	PROPN
ejpam-3317	383	9	−1(gα	−1(gα	NOUN
ejpam-3317	383	10	)	)	PUNCT
ejpam-3317	383	11	.	.	PUNCT
ejpam-3317	384	1	it	it	PRON
ejpam-3317	384	2	follow	follow	VERB
ejpam-3317	384	3	that	that	SCONJ
ejpam-3317	384	4	f(b	f(b	PROPN
ejpam-3317	384	5	)	)	PUNCT
ejpam-3317	384	6	⊆	⊆	NUM
ejpam-3317	384	7	⋃	⋃	NOUN
ejpam-3317	384	8	α∈j0	α∈j0	NOUN
ejpam-3317	384	9	gα	gα	ADV
ejpam-3317	384	10	.	.	PUNCT
ejpam-3317	385	1	thus	thus	ADV
ejpam-3317	385	2	,	,	PUNCT
ejpam-3317	385	3	any	any	DET
ejpam-3317	385	4	preopen	preopen	ADJ
ejpam-3317	385	5	cover	cover	NOUN
ejpam-3317	385	6	{	{	PUNCT
ejpam-3317	385	7	gα}α∈j	gα}α∈j	NOUN
ejpam-3317	385	8	of	of	ADP
ejpam-3317	385	9	f	f	PROPN
ejpam-3317	385	10	(	(	PUNCT
ejpam-3317	385	11	b	b	NOUN
ejpam-3317	385	12	)	)	PUNCT
ejpam-3317	385	13	contains	contain	VERB
ejpam-3317	385	14	a	a	DET
ejpam-3317	385	15	finite	finite	PROPN
ejpam-3317	385	16	subcover	subcover	PROPN
ejpam-3317	385	17	.	.	PUNCT
ejpam-3317	386	1	therefore	therefore	ADV
ejpam-3317	386	2	,	,	PUNCT
ejpam-3317	386	3	f	f	PROPN
ejpam-3317	386	4	(	(	PUNCT
ejpam-3317	386	5	b	b	NOUN
ejpam-3317	386	6	)	)	PUNCT
ejpam-3317	386	7	is	be	AUX
ejpam-3317	386	8	p	p	ADJ
ejpam-3317	386	9	-	-	PUNCT
ejpam-3317	386	10	compact	compact	ADJ
ejpam-3317	386	11	.	.	PUNCT
ejpam-3317	387	1	proposition	proposition	NOUN
ejpam-3317	387	2	37	37	NUM
ejpam-3317	387	3	.	.	PUNCT
ejpam-3317	388	1	let	let	AUX
ejpam-3317	388	2	(	(	PUNCT
ejpam-3317	388	3	x	x	NOUN
ejpam-3317	388	4	,	,	PUNCT
ejpam-3317	388	5	c1	c1	PROPN
ejpam-3317	388	6	)	)	PUNCT
ejpam-3317	388	7	and	and	CCONJ
ejpam-3317	388	8	(	(	PUNCT
ejpam-3317	388	9	y	y	PROPN
ejpam-3317	388	10	,	,	PUNCT
ejpam-3317	388	11	c2	c2	PROPN
ejpam-3317	388	12	)	)	PUNCT
ejpam-3317	388	13	be	be	VERB
ejpam-3317	388	14	closure	closure	NOUN
ejpam-3317	388	15	spaces	space	NOUN
ejpam-3317	388	16	and	and	CCONJ
ejpam-3317	388	17	f	f	X
ejpam-3317	388	18	:	:	PUNCT
ejpam-3317	388	19	(	(	PUNCT
ejpam-3317	388	20	x	x	X
ejpam-3317	388	21	,	,	PUNCT
ejpam-3317	388	22	c1	c1	PROPN
ejpam-3317	388	23	)	)	PUNCT
ejpam-3317	388	24	→	→	PUNCT
ejpam-3317	388	25	(	(	PUNCT
ejpam-3317	388	26	y	y	PROPN
ejpam-3317	388	27	,	,	PUNCT
ejpam-3317	388	28	c2	c2	PROPN
ejpam-3317	388	29	)	)	PUNCT
ejpam-3317	388	30	be	be	AUX
ejpam-3317	388	31	a	a	DET
ejpam-3317	388	32	function	function	NOUN
ejpam-3317	388	33	.	.	PUNCT
ejpam-3317	389	1	if	if	SCONJ
ejpam-3317	389	2	f	f	PROPN
ejpam-3317	389	3	is	be	AUX
ejpam-3317	389	4	a	a	DET
ejpam-3317	389	5	pre	pre	ADJ
ejpam-3317	389	6	-	-	ADJ
ejpam-3317	389	7	continuous	continuous	ADJ
ejpam-3317	389	8	surjection	surjection	NOUN
ejpam-3317	389	9	and	and	CCONJ
ejpam-3317	389	10	x	x	NOUN
ejpam-3317	389	11	is	be	AUX
ejpam-3317	389	12	p	p	ADJ
ejpam-3317	389	13	-	-	PUNCT
ejpam-3317	389	14	compact	compact	ADJ
ejpam-3317	389	15	,	,	PUNCT
ejpam-3317	389	16	then	then	ADV
ejpam-3317	389	17	y	y	PROPN
ejpam-3317	389	18	is	be	AUX
ejpam-3317	389	19	compact	compact	ADJ
ejpam-3317	389	20	.	.	PUNCT
ejpam-3317	390	1	references	reference	NOUN
ejpam-3317	390	2	1094	1094	NUM
ejpam-3317	390	3	proof	proof	NOUN
ejpam-3317	390	4	.	.	PUNCT
ejpam-3317	391	1	let	let	VERB
ejpam-3317	391	2	{	{	PUNCT
ejpam-3317	391	3	gα}α∈j	gα}α∈j	VERB
ejpam-3317	391	4	be	be	AUX
ejpam-3317	391	5	a	a	DET
ejpam-3317	391	6	collection	collection	NOUN
ejpam-3317	391	7	of	of	ADP
ejpam-3317	391	8	open	open	ADJ
ejpam-3317	391	9	subsets	subset	NOUN
ejpam-3317	391	10	of	of	ADP
ejpam-3317	391	11	y	y	PRON
ejpam-3317	391	12	such	such	ADJ
ejpam-3317	391	13	that	that	SCONJ
ejpam-3317	391	14	y	y	PROPN
ejpam-3317	391	15	⊆	⊆	NUM
ejpam-3317	391	16	⋃	⋃	ADP
ejpam-3317	391	17	α∈j	α∈j	NOUN
ejpam-3317	391	18	gα	gα	NOUN
ejpam-3317	391	19	.	.	PUNCT
ejpam-3317	392	1	it	it	PRON
ejpam-3317	392	2	follows	follow	VERB
ejpam-3317	392	3	that	that	SCONJ
ejpam-3317	392	4	x	x	PUNCT
ejpam-3317	392	5	=	=	X
ejpam-3317	392	6	f−1	f−1	PROPN
ejpam-3317	392	7	(	(	PUNCT
ejpam-3317	392	8	y	y	PROPN
ejpam-3317	392	9	)	)	PUNCT
ejpam-3317	392	10	⊆	⊆	NUM
ejpam-3317	392	11	f−1	f−1	PROPN
ejpam-3317	392	12	(	(	PUNCT
ejpam-3317	392	13	⋃	⋃	NOUN
ejpam-3317	392	14	α∈j	α∈j	NOUN
ejpam-3317	392	15	gα	gα	NOUN
ejpam-3317	392	16	)	)	PUNCT
ejpam-3317	393	1	=	=	PUNCT
ejpam-3317	393	2	⋃	⋃	ADP
ejpam-3317	393	3	α∈j	α∈j	NOUN
ejpam-3317	393	4	f	f	NOUN
ejpam-3317	393	5	−1(gα	−1(gα	NOUN
ejpam-3317	393	6	)	)	PUNCT
ejpam-3317	393	7	.	.	PUNCT
ejpam-3317	394	1	but	but	CCONJ
ejpam-3317	394	2	f	f	PROPN
ejpam-3317	394	3	is	be	AUX
ejpam-3317	394	4	pre	pre	ADJ
ejpam-3317	394	5	-	-	ADJ
ejpam-3317	394	6	continuous	continuous	ADJ
ejpam-3317	394	7	,	,	PUNCT
ejpam-3317	394	8	so	so	CCONJ
ejpam-3317	394	9	{	{	PUNCT
ejpam-3317	394	10	f−1(gα)}α∈j	f−1(gα)}α∈j	NOUN
ejpam-3317	394	11	is	be	AUX
ejpam-3317	394	12	a	a	DET
ejpam-3317	394	13	preopen	preopen	ADJ
ejpam-3317	394	14	cover	cover	NOUN
ejpam-3317	394	15	of	of	ADP
ejpam-3317	394	16	x.	x.	NOUN
ejpam-3317	394	17	since	since	SCONJ
ejpam-3317	394	18	x	x	PRON
ejpam-3317	394	19	is	be	AUX
ejpam-3317	394	20	p	p	ADJ
ejpam-3317	394	21	-	-	PUNCT
ejpam-3317	394	22	compact	compact	ADJ
ejpam-3317	394	23	,	,	PUNCT
ejpam-3317	394	24	there	there	PRON
ejpam-3317	394	25	exists	exist	VERB
ejpam-3317	394	26	a	a	DET
ejpam-3317	394	27	finite	finite	NOUN
ejpam-3317	394	28	subset	subset	VERB
ejpam-3317	394	29	j0	j0	PROPN
ejpam-3317	394	30	of	of	ADP
ejpam-3317	394	31	j	j	PROPN
ejpam-3317	394	32	such	such	ADJ
ejpam-3317	394	33	that	that	PRON
ejpam-3317	394	34	x=	x=	PUNCT
ejpam-3317	395	1	⋃	⋃	PROPN
ejpam-3317	395	2	α∈j0	α∈j0	NOUN
ejpam-3317	395	3	f	f	PROPN
ejpam-3317	395	4	−1(gα	−1(gα	NOUN
ejpam-3317	395	5	)	)	PUNCT
ejpam-3317	395	6	.	.	PUNCT
ejpam-3317	396	1	it	it	PRON
ejpam-3317	396	2	follow	follow	VERB
ejpam-3317	396	3	that	that	SCONJ
ejpam-3317	396	4	y	y	PROPN
ejpam-3317	397	1	=	=	ADJ
ejpam-3317	397	2	f	f	X
ejpam-3317	397	3	(	(	PUNCT
ejpam-3317	397	4	⋃	⋃	PROPN
ejpam-3317	397	5	α∈j0	α∈j0	NOUN
ejpam-3317	397	6	f	f	PROPN
ejpam-3317	397	7	−1(gα	−1(gα	NOUN
ejpam-3317	397	8	)	)	PUNCT
ejpam-3317	397	9	)	)	PUNCT
ejpam-3317	398	1	=	=	PUNCT
ejpam-3317	398	2	f	f	PROPN
ejpam-3317	398	3	(	(	PUNCT
ejpam-3317	398	4	f−1	f−1	PROPN
ejpam-3317	398	5	(	(	PUNCT
ejpam-3317	398	6	⋃	⋃	NOUN
ejpam-3317	398	7	α∈j0	α∈j0	NOUN
ejpam-3317	398	8	(	(	PUNCT
ejpam-3317	398	9	gα	gα	NOUN
ejpam-3317	398	10	)	)	PUNCT
ejpam-3317	398	11	)	)	PUNCT
ejpam-3317	398	12	)	)	PUNCT
ejpam-3317	398	13	.	.	PUNCT
ejpam-3317	399	1	since	since	SCONJ
ejpam-3317	399	2	f	f	PROPN
ejpam-3317	399	3	is	be	AUX
ejpam-3317	399	4	a	a	DET
ejpam-3317	399	5	surjection	surjection	NOUN
ejpam-3317	399	6	,	,	PUNCT
ejpam-3317	399	7	y=	y=	PRON
ejpam-3317	399	8	⋃	⋃	NOUN
ejpam-3317	399	9	α∈j0	α∈j0	NOUN
ejpam-3317	399	10	gα	gα	ADP
ejpam-3317	399	11	thus	thus	ADV
ejpam-3317	399	12	,	,	PUNCT
ejpam-3317	399	13	any	any	DET
ejpam-3317	399	14	open	open	ADJ
ejpam-3317	399	15	cover	cover	NOUN
ejpam-3317	399	16	{	{	PUNCT
ejpam-3317	399	17	gα}α∈jof	gα}α∈jof	NOUN
ejpam-3317	399	18	y	y	NOUN
ejpam-3317	399	19	contains	contain	VERB
ejpam-3317	399	20	a	a	DET
ejpam-3317	399	21	finite	finite	PROPN
ejpam-3317	399	22	subcover	subcover	PROPN
ejpam-3317	399	23	.	.	PUNCT
ejpam-3317	400	1	therefore	therefore	ADV
ejpam-3317	400	2	,	,	PUNCT
ejpam-3317	400	3	y	y	PROPN
ejpam-3317	400	4	is	be	AUX
ejpam-3317	400	5	compact	compact	ADJ
ejpam-3317	400	6	.	.	PUNCT
ejpam-3317	401	1	proposition	proposition	NOUN
ejpam-3317	401	2	38	38	NUM
ejpam-3317	401	3	.	.	PUNCT
ejpam-3317	402	1	let	let	AUX
ejpam-3317	402	2	(	(	PUNCT
ejpam-3317	402	3	x	x	NOUN
ejpam-3317	402	4	,	,	PUNCT
ejpam-3317	402	5	c1	c1	PROPN
ejpam-3317	402	6	)	)	PUNCT
ejpam-3317	402	7	and	and	CCONJ
ejpam-3317	402	8	(	(	PUNCT
ejpam-3317	402	9	y	y	PROPN
ejpam-3317	402	10	,	,	PUNCT
ejpam-3317	402	11	c2	c2	PROPN
ejpam-3317	402	12	)	)	PUNCT
ejpam-3317	402	13	be	be	VERB
ejpam-3317	402	14	closure	closure	NOUN
ejpam-3317	402	15	spaces	space	NOUN
ejpam-3317	402	16	and	and	CCONJ
ejpam-3317	402	17	f	f	X
ejpam-3317	402	18	:	:	PUNCT
ejpam-3317	402	19	(	(	PUNCT
ejpam-3317	402	20	x	x	X
ejpam-3317	402	21	,	,	PUNCT
ejpam-3317	402	22	c1	c1	PROPN
ejpam-3317	402	23	)	)	PUNCT
ejpam-3317	402	24	→	→	PUNCT
ejpam-3317	402	25	(	(	PUNCT
ejpam-3317	402	26	y	y	PROPN
ejpam-3317	402	27	,	,	PUNCT
ejpam-3317	402	28	c2	c2	PROPN
ejpam-3317	402	29	)	)	PUNCT
ejpam-3317	402	30	be	be	AUX
ejpam-3317	402	31	a	a	DET
ejpam-3317	402	32	function	function	NOUN
ejpam-3317	402	33	.	.	PUNCT
ejpam-3317	403	1	if	if	SCONJ
ejpam-3317	403	2	f	f	PROPN
ejpam-3317	403	3	is	be	AUX
ejpam-3317	403	4	a	a	DET
ejpam-3317	403	5	preirresolute	preirresolute	PROPN
ejpam-3317	403	6	surjection	surjection	NOUN
ejpam-3317	403	7	and	and	CCONJ
ejpam-3317	403	8	x	x	NOUN
ejpam-3317	403	9	is	be	AUX
ejpam-3317	403	10	p	p	ADJ
ejpam-3317	403	11	-	-	PUNCT
ejpam-3317	403	12	compact	compact	ADJ
ejpam-3317	403	13	,	,	PUNCT
ejpam-3317	403	14	then	then	ADV
ejpam-3317	403	15	y	y	PROPN
ejpam-3317	403	16	is	be	AUX
ejpam-3317	403	17	p	p	ADJ
ejpam-3317	403	18	-	-	PUNCT
ejpam-3317	403	19	compact	compact	ADJ
ejpam-3317	403	20	.	.	PUNCT
ejpam-3317	404	1	proof	proof	NOUN
ejpam-3317	404	2	.	.	PUNCT
ejpam-3317	405	1	let	let	VERB
ejpam-3317	405	2	{	{	PUNCT
ejpam-3317	405	3	aα}α∈j	aα}α∈j	VERB
ejpam-3317	405	4	be	be	AUX
ejpam-3317	405	5	a	a	DET
ejpam-3317	405	6	collection	collection	NOUN
ejpam-3317	405	7	of	of	ADP
ejpam-3317	405	8	preopen	preopen	ADJ
ejpam-3317	405	9	subsets	subset	NOUN
ejpam-3317	405	10	of	of	ADP
ejpam-3317	405	11	y	y	PRON
ejpam-3317	405	12	such	such	ADJ
ejpam-3317	405	13	that	that	SCONJ
ejpam-3317	405	14	y	y	PROPN
ejpam-3317	405	15	⊆	⊆	NUM
ejpam-3317	405	16	⋃	⋃	ADP
ejpam-3317	405	17	α∈j	α∈j	ADJ
ejpam-3317	405	18	aα	aα	NOUN
ejpam-3317	405	19	.	.	PUNCT
ejpam-3317	406	1	it	it	PRON
ejpam-3317	406	2	follow	follow	VERB
ejpam-3317	406	3	that	that	SCONJ
ejpam-3317	406	4	x	x	NOUN
ejpam-3317	406	5	=	=	X
ejpam-3317	406	6	f−1	f−1	PROPN
ejpam-3317	406	7	(	(	PUNCT
ejpam-3317	406	8	y	y	PROPN
ejpam-3317	406	9	)	)	PUNCT
ejpam-3317	406	10	⊆	⊆	NUM
ejpam-3317	406	11	f−1	f−1	PROPN
ejpam-3317	406	12	(	(	PUNCT
ejpam-3317	406	13	⋃	⋃	PROPN
ejpam-3317	406	14	α∈j	α∈j	NOUN
ejpam-3317	406	15	aα	aα	NOUN
ejpam-3317	406	16	)	)	PUNCT
ejpam-3317	407	1	=	=	PUNCT
ejpam-3317	407	2	⋃	⋃	ADP
ejpam-3317	407	3	α∈j	α∈j	NOUN
ejpam-3317	407	4	f	f	PROPN
ejpam-3317	407	5	−1(aα	−1(aα	PROPN
ejpam-3317	407	6	)	)	PUNCT
ejpam-3317	407	7	.	.	PUNCT
ejpam-3317	408	1	but	but	CCONJ
ejpam-3317	408	2	f	f	PROPN
ejpam-3317	408	3	is	be	AUX
ejpam-3317	408	4	pre	pre	ADJ
ejpam-3317	408	5	-	-	ADJ
ejpam-3317	408	6	irresolute	irresolute	ADJ
ejpam-3317	408	7	,	,	PUNCT
ejpam-3317	408	8	hence	hence	ADV
ejpam-3317	408	9	{	{	PUNCT
ejpam-3317	408	10	f−1(aα)}α∈j	f−1(aα)}α∈j	PROPN
ejpam-3317	408	11	is	be	AUX
ejpam-3317	408	12	a	a	DET
ejpam-3317	408	13	preopen	preopen	ADJ
ejpam-3317	408	14	cover	cover	NOUN
ejpam-3317	408	15	of	of	ADP
ejpam-3317	408	16	x.	x.	NOUN
ejpam-3317	408	17	since	since	SCONJ
ejpam-3317	408	18	x	x	PRON
ejpam-3317	408	19	is	be	AUX
ejpam-3317	408	20	p	p	ADJ
ejpam-3317	408	21	-	-	PUNCT
ejpam-3317	408	22	compact	compact	ADJ
ejpam-3317	408	23	,	,	PUNCT
ejpam-3317	408	24	there	there	PRON
ejpam-3317	408	25	exists	exist	VERB
ejpam-3317	408	26	a	a	DET
ejpam-3317	408	27	finite	finite	NOUN
ejpam-3317	408	28	subset	subset	VERB
ejpam-3317	408	29	j0	j0	PROPN
ejpam-3317	408	30	of	of	ADP
ejpam-3317	408	31	j	j	PROPN
ejpam-3317	408	32	such	such	ADJ
ejpam-3317	408	33	that	that	SCONJ
ejpam-3317	408	34	x	x	X
ejpam-3317	408	35	=	=	PUNCT
ejpam-3317	408	36	⋃	⋃	NOUN
ejpam-3317	408	37	α∈j0	α∈j0	NOUN
ejpam-3317	408	38	f	f	PROPN
ejpam-3317	408	39	−1(aα	−1(aα	PROPN
ejpam-3317	408	40	)	)	PUNCT
ejpam-3317	408	41	.	.	PUNCT
ejpam-3317	409	1	hence	hence	ADV
ejpam-3317	409	2	,	,	PUNCT
ejpam-3317	409	3	(	(	PUNCT
ejpam-3317	409	4	y	y	NOUN
ejpam-3317	409	5	)	)	PUNCT
ejpam-3317	410	1	=	=	NOUN
ejpam-3317	410	2	f	f	X
ejpam-3317	410	3	(	(	PUNCT
ejpam-3317	410	4	⋃	⋃	PROPN
ejpam-3317	410	5	α∈j	α∈j	NOUN
ejpam-3317	410	6	f	f	PROPN
ejpam-3317	410	7	−1(aα	−1(aα	PROPN
ejpam-3317	410	8	)	)	PUNCT
ejpam-3317	410	9	)	)	PUNCT
ejpam-3317	411	1	=	=	SYM
ejpam-3317	411	2	f(f−1	f(f−1	X
ejpam-3317	411	3	(	(	PUNCT
ejpam-3317	411	4	⋃	⋃	NOUN
ejpam-3317	411	5	α∈j0	α∈j0	NOUN
ejpam-3317	411	6	(	(	PUNCT
ejpam-3317	411	7	aα	aα	NOUN
ejpam-3317	411	8	)	)	PUNCT
ejpam-3317	411	9	)	)	PUNCT
ejpam-3317	411	10	.	.	PUNCT
ejpam-3317	412	1	since	since	SCONJ
ejpam-3317	412	2	f	f	PROPN
ejpam-3317	412	3	is	be	AUX
ejpam-3317	412	4	a	a	DET
ejpam-3317	412	5	surjection	surjection	NOUN
ejpam-3317	412	6	y=	y=	PRON
ejpam-3317	412	7	⋃	⋃	PROPN
ejpam-3317	412	8	α∈j0	α∈j0	ADP
ejpam-3317	412	9	aα	aα	NOUN
ejpam-3317	412	10	of	of	ADP
ejpam-3317	412	11	y	y	PROPN
ejpam-3317	412	12	contains	contain	VERB
ejpam-3317	412	13	a	a	DET
ejpam-3317	412	14	finite	finite	PROPN
ejpam-3317	412	15	subcover	subcover	PROPN
ejpam-3317	412	16	.	.	PUNCT
ejpam-3317	413	1	therefore	therefore	ADV
ejpam-3317	413	2	,	,	PUNCT
ejpam-3317	413	3	y	y	PROPN
ejpam-3317	413	4	is	be	AUX
ejpam-3317	413	5	p	p	ADJ
ejpam-3317	413	6	-	-	PUNCT
ejpam-3317	413	7	compact	compact	ADJ
ejpam-3317	413	8	.	.	PUNCT
ejpam-3317	414	1	acknowledgements	acknowledgement	NOUN
ejpam-3317	414	2	as	as	ADP
ejpam-3317	414	3	a	a	DET
ejpam-3317	414	4	corresponding	corresponding	ADJ
ejpam-3317	414	5	author	author	NOUN
ejpam-3317	414	6	,	,	PUNCT
ejpam-3317	414	7	i	i	PRON
ejpam-3317	414	8	would	would	AUX
ejpam-3317	414	9	be	be	AUX
ejpam-3317	414	10	very	very	ADV
ejpam-3317	414	11	grateful	grateful	ADJ
ejpam-3317	414	12	to	to	PART
ejpam-3317	414	13	deeply	deeply	ADV
ejpam-3317	414	14	acknowledge	acknowledge	VERB
ejpam-3317	414	15	my	my	PRON
ejpam-3317	414	16	phd	phd	NOUN
ejpam-3317	414	17	supervisor	supervisor	NOUN
ejpam-3317	414	18	(	(	PUNCT
ejpam-3317	414	19	assist	assist	NOUN
ejpam-3317	414	20	.	.	PUNCT
ejpam-3317	415	1	prof	prof	PROPN
ejpam-3317	415	2	.	.	PUNCT
ejpam-3317	416	1	dr	dr	PROPN
ejpam-3317	416	2	.	.	PROPN
ejpam-3317	416	3	halgwrd	halgwrd	PROPN
ejpam-3317	416	4	m.	m.	PROPN
ejpam-3317	416	5	darwesh	darwesh	PROPN
ejpam-3317	416	6	)	)	PUNCT
ejpam-3317	416	7	for	for	ADP
ejpam-3317	416	8	helping	help	VERB
ejpam-3317	416	9	me	i	PRON
ejpam-3317	416	10	to	to	PART
ejpam-3317	416	11	publish	publish	VERB
ejpam-3317	416	12	this	this	DET
ejpam-3317	416	13	article	article	NOUN
ejpam-3317	416	14	,	,	PUNCT
ejpam-3317	416	15	which	which	PRON
ejpam-3317	416	16	is	be	AUX
ejpam-3317	416	17	a	a	DET
ejpam-3317	416	18	part	part	NOUN
ejpam-3317	416	19	of	of	ADP
ejpam-3317	416	20	ministry	ministry	NOUN
ejpam-3317	416	21	of	of	ADP
ejpam-3317	416	22	higher	high	ADJ
ejpam-3317	416	23	education	education	NOUN
ejpam-3317	416	24	and	and	CCONJ
ejpam-3317	416	25	scientific	scientific	ADJ
ejpam-3317	416	26	research	research	NOUN
ejpam-3317	416	27	-	-	PUNCT
ejpam-3317	416	28	krgiraq	krgiraq	NOUN
ejpam-3317	416	29	’s	’s	PART
ejpam-3317	416	30	requirements	requirement	NOUN
ejpam-3317	416	31	to	to	PART
ejpam-3317	416	32	achieve	achieve	VERB
ejpam-3317	416	33	my	my	PRON
ejpam-3317	416	34	phd	phd	NOUN
ejpam-3317	416	35	degree	degree	NOUN
ejpam-3317	416	36	.	.	PUNCT
ejpam-3317	417	1	i	i	PRON
ejpam-3317	417	2	would	would	AUX
ejpam-3317	417	3	also	also	ADV
ejpam-3317	417	4	acknowledge	acknowledge	VERB
ejpam-3317	417	5	the	the	DET
ejpam-3317	417	6	academic	academic	ADJ
ejpam-3317	417	7	staff	staff	NOUN
ejpam-3317	417	8	of	of	ADP
ejpam-3317	417	9	department	department	NOUN
ejpam-3317	417	10	of	of	ADP
ejpam-3317	417	11	mathematics	mathematics	PROPN
ejpam-3317	417	12	,	,	PUNCT
ejpam-3317	417	13	college	college	NOUN
ejpam-3317	417	14	of	of	ADP
ejpam-3317	417	15	science	science	NOUN
ejpam-3317	417	16	,	,	PUNCT
ejpam-3317	417	17	university	university	NOUN
ejpam-3317	417	18	of	of	ADP
ejpam-3317	417	19	sulaimani	sulaimani	NOUN
ejpam-3317	417	20	for	for	ADP
ejpam-3317	417	21	their	their	PRON
ejpam-3317	417	22	assistance	assistance	NOUN
ejpam-3317	417	23	as	as	ADP
ejpam-3317	417	24	phd	phd	NOUN
ejpam-3317	417	25	student	student	NOUN
ejpam-3317	417	26	in	in	ADP
ejpam-3317	417	27	their	their	PRON
ejpam-3317	417	28	department	department	NOUN
ejpam-3317	417	29	.	.	PUNCT
ejpam-3317	418	1	references	reference	NOUN
ejpam-3317	418	2	[	[	X
ejpam-3317	418	3	1	1	NUM
ejpam-3317	418	4	]	]	X
ejpam-3317	418	5	boonpok	boonpok	PROPN
ejpam-3317	418	6	,	,	PUNCT
ejpam-3317	418	7	c.	c.	PROPN
ejpam-3317	418	8	,	,	PUNCT
ejpam-3317	418	9	on	on	ADP
ejpam-3317	418	10	continuous	continuous	ADJ
ejpam-3317	418	11	maps	map	NOUN
ejpam-3317	418	12	in	in	ADP
ejpam-3317	418	13	closure	closure	NOUN
ejpam-3317	418	14	spaces	space	NOUN
ejpam-3317	418	15	,	,	PUNCT
ejpam-3317	418	16	general	general	ADJ
ejpam-3317	418	17	mthematics	mthematic	NOUN
ejpam-3317	418	18	,	,	PUNCT
ejpam-3317	418	19	17(2	17(2	NUM
ejpam-3317	418	20	)	)	PUNCT
ejpam-3317	418	21	(	(	PUNCT
ejpam-3317	418	22	2009	2009	NUM
ejpam-3317	418	23	)	)	PUNCT
ejpam-3317	418	24	,	,	PUNCT
ejpam-3317	418	25	127	127	NUM
ejpam-3317	418	26	-	-	SYM
ejpam-3317	418	27	134	134	NUM
ejpam-3317	418	28	.	.	PUNCT
ejpam-3317	419	1	[	[	X
ejpam-3317	419	2	2	2	NUM
ejpam-3317	419	3	]	]	PUNCT
ejpam-3317	419	4	cech	cech	NOUN
ejpam-3317	419	5	,	,	PUNCT
ejpam-3317	419	6	e.	e.	PROPN
ejpam-3317	419	7	,	,	PUNCT
ejpam-3317	419	8	on	on	ADP
ejpam-3317	419	9	bicompact	bicompact	ADJ
ejpam-3317	419	10	spaces	space	NOUN
ejpam-3317	419	11	,	,	PUNCT
ejpam-3317	419	12	annals	annal	NOUN
ejpam-3317	419	13	of	of	ADP
ejpam-3317	419	14	mathematics	mathematic	NOUN
ejpam-3317	419	15	,	,	PUNCT
ejpam-3317	419	16	6	6	NUM
ejpam-3317	419	17	(	(	PUNCT
ejpam-3317	419	18	1937)823	1937)823	NUM
ejpam-3317	419	19	-	-	PUNCT
ejpam-3317	419	20	844	844	NUM
ejpam-3317	419	21	.	.	PUNCT
ejpam-3317	420	1	[	[	X
ejpam-3317	420	2	3	3	NUM
ejpam-3317	420	3	]	]	X
ejpam-3317	420	4	cech	cech	NOUN
ejpam-3317	420	5	,	,	PUNCT
ejpam-3317	420	6	e.	e.	PROPN
ejpam-3317	420	7	,	,	PUNCT
ejpam-3317	420	8	topological	topological	ADJ
ejpam-3317	420	9	spaces	space	NOUN
ejpam-3317	420	10	,	,	PUNCT
ejpam-3317	420	11	inter	inter	ADJ
ejpam-3317	420	12	science	science	NOUN
ejpam-3317	420	13	publishers	publisher	NOUN
ejpam-3317	420	14	,	,	PUNCT
ejpam-3317	420	15	john	john	PROPN
ejpam-3317	420	16	wiley	wiley	PROPN
ejpam-3317	420	17	and	and	CCONJ
ejpam-3317	420	18	sons	son	NOUN
ejpam-3317	420	19	,	,	PUNCT
ejpam-3317	420	20	new	new	PROPN
ejpam-3317	420	21	york	york	PROPN
ejpam-3317	420	22	,	,	PUNCT
ejpam-3317	420	23	1966	1966	NUM
ejpam-3317	420	24	.	.	PUNCT
ejpam-3317	421	1	[	[	X
ejpam-3317	421	2	4	4	NUM
ejpam-3317	421	3	]	]	PUNCT
ejpam-3317	421	4	cech	cech	NOUN
ejpam-3317	421	5	,	,	PUNCT
ejpam-3317	421	6	e.	e.	PROPN
ejpam-3317	421	7	,	,	PUNCT
ejpam-3317	421	8	:	:	PUNCT
ejpam-3317	421	9	topological	topological	ADJ
ejpam-3317	421	10	spaces	space	NOUN
ejpam-3317	421	11	.	.	PUNCT
ejpam-3317	422	1	(	(	PUNCT
ejpam-3317	422	2	revised	revise	VERB
ejpam-3317	422	3	by	by	ADP
ejpam-3317	422	4	z.	z.	PROPN
ejpam-3317	422	5	frolk	frolk	PROPN
ejpam-3317	422	6	,	,	PUNCT
ejpam-3317	422	7	m.	m.	NOUN
ejpam-3317	422	8	kattov	kattov	PROPN
ejpam-3317	422	9	)	)	PUNCT
ejpam-3317	422	10	,	,	PUNCT
ejpam-3317	422	11	academia	academia	NOUN
ejpam-3317	422	12	,	,	PUNCT
ejpam-3317	422	13	prague	prague	NOUN
ejpam-3317	422	14	(	(	PUNCT
ejpam-3317	422	15	1966	1966	NUM
ejpam-3317	422	16	)	)	PUNCT
ejpam-3317	422	17	.	.	PUNCT
ejpam-3317	423	1	[	[	X
ejpam-3317	423	2	5	5	NUM
ejpam-3317	423	3	]	]	PUNCT
ejpam-3317	423	4	cech	cech	NOUN
ejpam-3317	423	5	,	,	PUNCT
ejpam-3317	423	6	e.	e.	PROPN
ejpam-3317	423	7	,	,	PUNCT
ejpam-3317	423	8	topological	topological	ADJ
ejpam-3317	423	9	spaces	space	NOUN
ejpam-3317	423	10	.	.	PUNCT
ejpam-3317	424	1	topological	topological	ADJ
ejpam-3317	424	2	papers	paper	NOUN
ejpam-3317	424	3	of	of	ADP
ejpam-3317	424	4	eduard	eduard	PROPN
ejpam-3317	424	5	ech	ech	PROPN
ejpam-3317	424	6	,	,	PUNCT
ejpam-3317	424	7	academia	academia	NOUN
ejpam-3317	424	8	,	,	PUNCT
ejpam-3317	424	9	prague	prague	NOUN
ejpam-3317	424	10	(	(	PUNCT
ejpam-3317	424	11	1968	1968	NUM
ejpam-3317	424	12	)	)	PUNCT
ejpam-3317	424	13	,	,	PUNCT
ejpam-3317	424	14	436	436	NUM
ejpam-3317	424	15	-	-	SYM
ejpam-3317	424	16	472	472	NUM
ejpam-3317	424	17	.	.	PUNCT
ejpam-3317	425	1	[	[	X
ejpam-3317	425	2	6	6	NUM
ejpam-3317	425	3	]	]	X
ejpam-3317	425	4	chandrasekhara	chandrasekhara	PROPN
ejpam-3317	425	5	rao	rao	PROPN
ejpam-3317	425	6	,	,	PUNCT
ejpam-3317	425	7	k.	k.	PROPN
ejpam-3317	425	8	,	,	PUNCT
ejpam-3317	425	9	gowri	gowri	PROPN
ejpam-3317	425	10	,	,	PUNCT
ejpam-3317	425	11	r.	r.	PROPN
ejpam-3317	425	12	,	,	PUNCT
ejpam-3317	425	13	and	and	CCONJ
ejpam-3317	425	14	swaminathan	swaminathan	ADV
ejpam-3317	425	15	,	,	PUNCT
ejpam-3317	425	16	v.	v.	ADV
ejpam-3317	425	17	,	,	PUNCT
ejpam-3317	425	18	cech	cech	NOUN
ejpam-3317	425	19	closure	closure	NOUN
ejpam-3317	425	20	space	space	NOUN
ejpam-3317	425	21	in	in	ADP
ejpam-3317	425	22	structural	structural	ADJ
ejpam-3317	425	23	configuration	configuration	NOUN
ejpam-3317	425	24	of	of	ADP
ejpam-3317	425	25	proteins	protein	NOUN
ejpam-3317	425	26	,	,	PUNCT
ejpam-3317	425	27	advanced	advanced	ADJ
ejpam-3317	425	28	studies	study	NOUN
ejpam-3317	425	29	in	in	ADP
ejpam-3317	425	30	biology	biology	NOUN
ejpam-3317	425	31	,	,	PUNCT
ejpam-3317	425	32	vol	vol	NOUN
ejpam-3317	425	33	.	.	PROPN
ejpam-3317	425	34	1	1	NUM
ejpam-3317	425	35	,	,	PUNCT
ejpam-3317	425	36	2009	2009	NUM
ejpam-3317	425	37	,	,	PUNCT
ejpam-3317	425	38	no	no	INTJ
ejpam-3317	425	39	.	.	NOUN
ejpam-3317	425	40	2	2	NUM
ejpam-3317	425	41	,	,	PUNCT
ejpam-3317	425	42	95	95	NUM
ejpam-3317	425	43	-	-	SYM
ejpam-3317	425	44	104	104	NUM
ejpam-3317	425	45	.	.	PUNCT
ejpam-3317	426	1	references	reference	NOUN
ejpam-3317	426	2	1095	1095	NUM
ejpam-3317	427	1	[	[	X
ejpam-3317	427	2	7	7	NUM
ejpam-3317	427	3	]	]	PUNCT
ejpam-3317	427	4	chvalina	chvalina	NOUN
ejpam-3317	427	5	j.	j.	PROPN
ejpam-3317	427	6	,	,	PUNCT
ejpam-3317	427	7	on	on	ADP
ejpam-3317	427	8	homeomorphic	homeomorphic	ADJ
ejpam-3317	427	9	topologies	topology	NOUN
ejpam-3317	427	10	and	and	CCONJ
ejpam-3317	427	11	equivalent	equivalent	ADJ
ejpam-3317	427	12	set	set	NOUN
ejpam-3317	427	13	-	-	PUNCT
ejpam-3317	427	14	system.arch.math.2	system.arch.math.2	NOUN
ejpam-3317	427	15	,	,	PUNCT
ejpam-3317	427	16	scripta	scripta	PROPN
ejpam-3317	427	17	fac.sci.nat	fac.sci.nat	PROPN
ejpam-3317	427	18	.	.	PUNCT
ejpam-3317	428	1	ujep	ujep	PROPN
ejpam-3317	428	2	brunensis	brunensis	NOUN
ejpam-3317	428	3	,	,	PUNCT
ejpam-3317	428	4	xii(1976),107	xii(1976),107	PROPN
ejpam-3317	428	5	-	-	PUNCT
ejpam-3317	428	6	116	116	NUM
ejpam-3317	428	7	.	.	PUNCT
ejpam-3317	429	1	[	[	X
ejpam-3317	429	2	8	8	NUM
ejpam-3317	429	3	]	]	PUNCT
ejpam-3317	429	4	chvalina	chvalina	ADJ
ejpam-3317	429	5	j.	j.	PROPN
ejpam-3317	429	6	,	,	PUNCT
ejpam-3317	429	7	separation	separation	NOUN
ejpam-3317	429	8	properties	property	NOUN
ejpam-3317	429	9	of	of	ADP
ejpam-3317	429	10	topologies	topology	NOUN
ejpam-3317	429	11	associated	associate	VERB
ejpam-3317	429	12	with	with	ADP
ejpam-3317	429	13	digraphs	digraph	NOUN
ejpam-3317	429	14	.	.	PUNCT
ejpam-3317	429	15	scripta	scripta	PROPN
ejpam-3317	429	16	fac	fac	PROPN
ejpam-3317	429	17	.	.	PUNCT
ejpam-3317	430	1	sci	sci	PROPN
ejpam-3317	430	2	.	.	PUNCT
ejpam-3317	431	1	nat	nat	PROPN
ejpam-3317	431	2	.	.	PUNCT
ejpam-3317	432	1	univ	univ	PROPN
ejpam-3317	432	2	.	.	PUNCT
ejpam-3317	433	1	purk	purk	PROPN
ejpam-3317	433	2	.	.	PUNCT
ejpam-3317	434	1	brun	brun	PROPN
ejpam-3317	434	2	.	.	PUNCT
ejpam-3317	434	3	8	8	NUM
ejpam-3317	434	4	(	(	PUNCT
ejpam-3317	434	5	10	10	NUM
ejpam-3317	434	6	)	)	PUNCT
ejpam-3317	434	7	(	(	PUNCT
ejpam-3317	434	8	1980	1980	NUM
ejpam-3317	434	9	)	)	PUNCT
ejpam-3317	434	10	,	,	PUNCT
ejpam-3317	434	11	399	399	NUM
ejpam-3317	434	12	-	-	SYM
ejpam-3317	434	13	410	410	NUM
ejpam-3317	434	14	.	.	PUNCT
ejpam-3317	435	1	[	[	X
ejpam-3317	435	2	9	9	NUM
ejpam-3317	435	3	]	]	PUNCT
ejpam-3317	435	4	chvalina	chvalina	NOUN
ejpam-3317	435	5	j.	j.	PROPN
ejpam-3317	435	6	,	,	PUNCT
ejpam-3317	435	7	stackbases	stackbase	VERB
ejpam-3317	435	8	in	in	ADP
ejpam-3317	435	9	power	power	NOUN
ejpam-3317	435	10	sets	set	NOUN
ejpam-3317	435	11	of	of	ADP
ejpam-3317	435	12	neighbourhood	neighbourhood	NOUN
ejpam-3317	435	13	spaces	space	NOUN
ejpam-3317	435	14	preserving	preserve	VERB
ejpam-3317	435	15	the	the	DET
ejpam-3317	435	16	continuity	continuity	NOUN
ejpam-3317	435	17	of	of	ADP
ejpam-3317	435	18	mappings	mapping	NOUN
ejpam-3317	435	19	.	.	PUNCT
ejpam-3317	436	1	arch	arch	NOUN
ejpam-3317	436	2	.	.	PUNCT
ejpam-3317	437	1	math	math	NOUN
ejpam-3317	437	2	.	.	PUNCT
ejpam-3317	438	1	2	2	NUM
ejpam-3317	438	2	,	,	PUNCT
ejpam-3317	438	3	scripta	scripta	PROPN
ejpam-3317	438	4	fac	fac	PROPN
ejpam-3317	438	5	.	.	PUNCT
ejpam-3317	439	1	sci	sci	PROPN
ejpam-3317	439	2	.	.	PUNCT
ejpam-3317	440	1	nat	nat	PROPN
ejpam-3317	440	2	.	.	PUNCT
ejpam-3317	441	1	ujep	ujep	VERB
ejpam-3317	441	2	brunensis	brunensis	NOUN
ejpam-3317	441	3	xvii(1981	xvii(1981	NOUN
ejpam-3317	441	4	)	)	PUNCT
ejpam-3317	441	5	81	81	NUM
ejpam-3317	441	6	-	-	SYM
ejpam-3317	441	7	86	86	NUM
ejpam-3317	441	8	.	.	PUNCT
ejpam-3317	442	1	[	[	X
ejpam-3317	442	2	10	10	NUM
ejpam-3317	442	3	]	]	X
ejpam-3317	442	4	david	david	PROPN
ejpam-3317	442	5	n.	n.	PROPN
ejpam-3317	442	6	roth	roth	PROPN
ejpam-3317	442	7	,	,	PUNCT
ejpam-3317	442	8	cech	cech	NOUN
ejpam-3317	442	9	closure	closure	NOUN
ejpam-3317	442	10	spaces	space	VERB
ejpam-3317	442	11	,	,	PUNCT
ejpam-3317	442	12	a	a	DET
ejpam-3317	442	13	thesis	thesis	NOUN
ejpam-3317	442	14	presented	present	VERB
ejpam-3317	442	15	to	to	ADP
ejpam-3317	442	16	the	the	DET
ejpam-3317	442	17	department	department	PROPN
ejpam-3317	442	18	of	of	ADP
ejpam-3317	442	19	mathematics	mathematics	PROPN
ejpam-3317	442	20	emporia	emporia	PROPN
ejpam-3317	442	21	state	state	PROPN
ejpam-3317	442	22	university	university	PROPN
ejpam-3317	442	23	,	,	PUNCT
ejpam-3317	442	24	july	july	PROPN
ejpam-3317	442	25	1979	1979	NUM
ejpam-3317	442	26	.	.	PUNCT
ejpam-3317	443	1	[	[	X
ejpam-3317	443	2	11	11	NUM
ejpam-3317	443	3	]	]	X
ejpam-3317	443	4	halgwrd	halgwrd	PROPN
ejpam-3317	443	5	m.	m.	PROPN
ejpam-3317	443	6	darwesh	darwesh	PROPN
ejpam-3317	443	7	,	,	PUNCT
ejpam-3317	443	8	almost	almost	ADV
ejpam-3317	443	9	-	-	PUNCT
ejpam-3317	443	10	continuity	continuity	NOUN
ejpam-3317	443	11	,	,	PUNCT
ejpam-3317	443	12	pre	pre	ADJ
ejpam-3317	443	13	-	-	NOUN
ejpam-3317	443	14	continuity	continuity	NOUN
ejpam-3317	443	15	and	and	CCONJ
ejpam-3317	443	16	preopen	preopen	ADJ
ejpam-3317	443	17	sets	set	NOUN
ejpam-3317	443	18	in	in	ADP
ejpam-3317	443	19	closure	closure	NOUN
ejpam-3317	443	20	spaces.(submit	spaces.(submit	NOUN
ejpam-3317	443	21	)	)	PUNCT
ejpam-3317	444	1	[	[	X
ejpam-3317	444	2	12	12	NUM
ejpam-3317	444	3	]	]	X
ejpam-3317	444	4	hussain	hussain	PROPN
ejpam-3317	444	5	,	,	PUNCT
ejpam-3317	444	6	t.	t.	PROPN
ejpam-3317	444	7	,	,	PUNCT
ejpam-3317	444	8	almost	almost	ADV
ejpam-3317	444	9	continuous	continuous	ADJ
ejpam-3317	444	10	mappings	mapping	NOUN
ejpam-3317	444	11	,	,	PUNCT
ejpam-3317	444	12	prace	prace	NOUN
ejpam-3317	444	13	.	.	PUNCT
ejpam-3317	445	1	mat	mat	PROPN
ejpam-3317	445	2	10	10	NUM
ejpam-3317	445	3	(	(	PUNCT
ejpam-3317	445	4	1966	1966	NUM
ejpam-3317	445	5	)	)	PUNCT
ejpam-3317	445	6	1	1	NUM
ejpam-3317	445	7	-	-	SYM
ejpam-3317	445	8	7	7	NUM
ejpam-3317	445	9	mr	mr	PROPN
ejpam-3317	445	10	36	36	NUM
ejpam-3317	445	11	3332	3332	NUM
ejpam-3317	445	12	[	[	X
ejpam-3317	445	13	13	13	NUM
ejpam-3317	445	14	]	]	X
ejpam-3317	445	15	halgwrd	halgwrd	PROPN
ejpam-3317	445	16	m.	m.	NOUN
ejpam-3317	445	17	darwesh	darwesh	PROPN
ejpam-3317	445	18	and	and	CCONJ
ejpam-3317	445	19	sarhad	sarhad	ADJ
ejpam-3317	445	20	f.namiq	f.namiq	NOUN
ejpam-3317	445	21	,	,	PUNCT
ejpam-3317	445	22	some	some	DET
ejpam-3317	445	23	properties	property	NOUN
ejpam-3317	445	24	of	of	ADP
ejpam-3317	445	25	preopen	preopen	ADJ
ejpam-3317	445	26	set	set	VERB
ejpam-3317	445	27	in	in	ADP
ejpam-3317	445	28	closure	closure	NOUN
ejpam-3317	445	29	spaces	space	NOUN
ejpam-3317	445	30	,	,	PUNCT
ejpam-3317	445	31	journal	journal	NOUN
ejpam-3317	445	32	of	of	ADP
ejpam-3317	445	33	garmian	garmian	PROPN
ejpam-3317	445	34	university	university	NOUN
ejpam-3317	445	35	.	.	PUNCT
ejpam-3317	446	1	volume	volume	NOUN
ejpam-3317	446	2	5	5	NUM
ejpam-3317	446	3	,	,	PUNCT
ejpam-3317	446	4	issue	issue	NOUN
ejpam-3317	446	5	2	2	NUM
ejpam-3317	446	6	,	,	PUNCT
ejpam-3317	446	7	spring	spring	NOUN
ejpam-3317	446	8	2018	2018	NUM
ejpam-3317	446	9	,	,	PUNCT
ejpam-3317	446	10	page	page	NOUN
ejpam-3317	446	11	229	229	NUM
ejpam-3317	446	12	-	-	SYM
ejpam-3317	446	13	246	246	NUM
ejpam-3317	446	14	.	.	PUNCT
ejpam-3317	447	1	[	[	X
ejpam-3317	447	2	14	14	NUM
ejpam-3317	447	3	]	]	X
ejpam-3317	447	4	kuratowski	kuratowski	PROPN
ejpam-3317	447	5	,	,	PUNCT
ejpam-3317	447	6	k.	k.	PROPN
ejpam-3317	447	7	(	(	PUNCT
ejpam-3317	447	8	1966	1966	NUM
ejpam-3317	447	9	)	)	PUNCT
ejpam-3317	448	1	[	[	X
ejpam-3317	448	2	1958	1958	NUM
ejpam-3317	448	3	]	]	PUNCT
ejpam-3317	448	4	,	,	PUNCT
ejpam-3317	448	5	topology	topology	NOUN
ejpam-3317	448	6	volume	volume	NOUN
ejpam-3317	448	7	i	i	PROPN
ejpam-3317	448	8	,	,	PUNCT
ejpam-3317	448	9	academic	academic	ADJ
ejpam-3317	448	10	press	press	NOUN
ejpam-3317	448	11	,	,	PUNCT
ejpam-3317	448	12	isbn	isbn	ADJ
ejpam-3317	448	13	0	0	NUM
ejpam-3317	448	14	-	-	SYM
ejpam-3317	448	15	12	12	NUM
ejpam-3317	448	16	-	-	PUNCT
ejpam-3317	448	17	4292011	4292011	NUM
ejpam-3317	449	1	[	[	X
ejpam-3317	449	2	15	15	NUM
ejpam-3317	449	3	]	]	X
ejpam-3317	449	4	khampakdee	khampakdee	PROPN
ejpam-3317	449	5	j.	j.	PROPN
ejpam-3317	449	6	(	(	PUNCT
ejpam-3317	449	7	2009	2009	NUM
ejpam-3317	449	8	)	)	PUNCT
ejpam-3317	449	9	,	,	PUNCT
ejpam-3317	449	10	semi	semi	ADJ
ejpam-3317	449	11	-	-	ADJ
ejpam-3317	449	12	open	open	ADJ
ejpam-3317	449	13	sets	set	NOUN
ejpam-3317	449	14	in	in	ADP
ejpam-3317	449	15	closure	closure	NOUN
ejpam-3317	449	16	spaces	space	NOUN
ejpam-3317	449	17	,	,	PUNCT
ejpam-3317	449	18	ph	ph	PROPN
ejpam-3317	449	19	.	.	PROPN
ejpam-3317	449	20	d.	d.	PROPN
ejpam-3317	449	21	thesis	thesis	PROPN
ejpam-3317	449	22	,	,	PUNCT
ejpam-3317	449	23	brno	brno	PROPN
ejpam-3317	449	24	university	university	PROPN
ejpam-3317	449	25	of	of	ADP
ejpam-3317	449	26	technology	technology	NOUN
ejpam-3317	449	27	,	,	PUNCT
ejpam-3317	449	28	brno	brno	NOUN
ejpam-3317	449	29	.	.	PUNCT
ejpam-3317	450	1	[	[	X
ejpam-3317	450	2	16	16	NUM
ejpam-3317	450	3	]	]	X
ejpam-3317	450	4	khampakdee	khampakdee	PROPN
ejpam-3317	450	5	j.	j.	PROPN
ejpam-3317	450	6	:	:	PUNCT
ejpam-3317	450	7	semi	semi	ADJ
ejpam-3317	450	8	-	-	ADJ
ejpam-3317	450	9	open	open	ADJ
ejpam-3317	450	10	sets	set	NOUN
ejpam-3317	450	11	in	in	ADP
ejpam-3317	450	12	biclosure	biclosure	NOUN
ejpam-3317	450	13	spaces	space	NOUN
ejpam-3317	450	14	.	.	PUNCT
ejpam-3317	451	1	discussiones	discussione	NOUN
ejpam-3317	451	2	mathematicae	mathematicae	VERB
ejpam-3317	451	3	,	,	PUNCT
ejpam-3317	451	4	general	general	ADJ
ejpam-3317	451	5	algebra	algebra	NOUN
ejpam-3317	451	6	and	and	CCONJ
ejpam-3317	451	7	applications	application	NOUN
ejpam-3317	451	8	,	,	PUNCT
ejpam-3317	451	9	53	53	NUM
ejpam-3317	451	10	(	(	PUNCT
ejpam-3317	451	11	2009),181	2009),181	ADV
ejpam-3317	451	12	-	-	NUM
ejpam-3317	451	13	201	201	NUM
ejpam-3317	451	14	.	.	PUNCT
ejpam-3317	452	1	[	[	X
ejpam-3317	452	2	17	17	NUM
ejpam-3317	452	3	]	]	X
ejpam-3317	452	4	mashhour	mashhour	PROPN
ejpam-3317	452	5	,	,	PUNCT
ejpam-3317	452	6	a.	a.	PROPN
ejpam-3317	452	7	s.	s.	PROPN
ejpam-3317	452	8	,	,	PUNCT
ejpam-3317	452	9	abd	abd	PROPN
ejpam-3317	452	10	el	el	PROPN
ejpam-3317	452	11	-	-	PUNCT
ejpam-3317	452	12	monsef	monsef	ADJ
ejpam-3317	452	13	,	,	PUNCT
ejpam-3317	452	14	m.	m.	PROPN
ejpam-3317	452	15	e.	e.	PROPN
ejpam-3317	452	16	,	,	PUNCT
ejpam-3317	452	17	and	and	CCONJ
ejpam-3317	452	18	el	el	PROPN
ejpam-3317	452	19	-	-	PUNCT
ejpam-3317	452	20	deeb	deeb	PROPN
ejpam-3317	452	21	,	,	PUNCT
ejpam-3317	452	22	s.	s.	PROPN
ejpam-3317	452	23	n.	n.	PROPN
ejpam-3317	452	24	,	,	PUNCT
ejpam-3317	452	25	on	on	ADP
ejpam-3317	452	26	precontinuous	precontinuous	ADJ
ejpam-3317	452	27	and	and	CCONJ
ejpam-3317	452	28	week	week	NOUN
ejpam-3317	452	29	precontinuous	precontinuous	ADJ
ejpam-3317	452	30	mappings	mapping	NOUN
ejpam-3317	452	31	,	,	PUNCT
ejpam-3317	452	32	proc	proc	NOUN
ejpam-3317	452	33	.	.	PUNCT
ejpam-3317	452	34	math	math	NOUN
ejpam-3317	452	35	.	.	PUNCT
ejpam-3317	453	1	phys	phy	NOUN
ejpam-3317	453	2	.	.	PUNCT
ejpam-3317	454	1	so	so	ADV
ejpam-3317	454	2	.	.	PUNCT
ejpam-3317	455	1	egypt	egypt	PROPN
ejpam-3317	455	2	,	,	PUNCT
ejpam-3317	455	3	53	53	NUM
ejpam-3317	455	4	(	(	PUNCT
ejpam-3317	455	5	1982	1982	NUM
ejpam-3317	455	6	)	)	PUNCT
ejpam-3317	455	7	,	,	PUNCT
ejpam-3317	455	8	47	47	NUM
ejpam-3317	455	9	-	-	SYM
ejpam-3317	455	10	53	53	NUM
ejpam-3317	455	11	.	.	PUNCT
ejpam-3317	456	1	[	[	X
ejpam-3317	456	2	18	18	NUM
ejpam-3317	456	3	]	]	PUNCT
ejpam-3317	456	4	smyth	smyth	NOUN
ejpam-3317	456	5	,	,	PUNCT
ejpam-3317	456	6	m.	m.	PROPN
ejpam-3317	456	7	b.	b.	PROPN
ejpam-3317	456	8	,	,	PUNCT
ejpam-3317	456	9	semi	semi	ADJ
ejpam-3317	456	10	-	-	NOUN
ejpam-3317	456	11	metrics	metric	NOUN
ejpam-3317	456	12	,	,	PUNCT
ejpam-3317	456	13	closure	closure	NOUN
ejpam-3317	456	14	spaces	space	NOUN
ejpam-3317	456	15	and	and	CCONJ
ejpam-3317	456	16	digital	digital	ADJ
ejpam-3317	456	17	topology	topology	NOUN
ejpam-3317	456	18	theoretical	theoretical	ADJ
ejpam-3317	456	19	computer	computer	NOUN
ejpam-3317	456	20	science	science	NOUN
ejpam-3317	456	21	151	151	NUM
ejpam-3317	456	22	(	(	PUNCT
ejpam-3317	456	23	1995	1995	NUM
ejpam-3317	456	24	)	)	PUNCT
ejpam-3317	456	25	257	257	NUM
ejpam-3317	456	26	-	-	SYM
ejpam-3317	456	27	276	276	NUM
ejpam-3317	456	28	.	.	PUNCT