id	sid	tid	token	lemma	pos
ejpam-4387	1	1	european	european	PROPN
ejpam-4387	1	2	journal	journal	PROPN
ejpam-4387	1	3	of	of	ADP
ejpam-4387	1	4	pure	pure	ADJ
ejpam-4387	1	5	and	and	CCONJ
ejpam-4387	1	6	applied	apply	VERB
ejpam-4387	1	7	mathematics	mathematic	NOUN
ejpam-4387	1	8	vol	vol	NOUN
ejpam-4387	1	9	.	.	PROPN
ejpam-4387	2	1	15	15	NUM
ejpam-4387	2	2	,	,	PUNCT
ejpam-4387	2	3	no	no	INTJ
ejpam-4387	2	4	.	.	NOUN
ejpam-4387	2	5	2	2	NUM
ejpam-4387	2	6	,	,	PUNCT
ejpam-4387	2	7	2022	2022	NUM
ejpam-4387	2	8	,	,	PUNCT
ejpam-4387	2	9	774	774	NUM
ejpam-4387	2	10	-	-	SYM
ejpam-4387	2	11	783	783	NUM
ejpam-4387	2	12	issn	issn	PROPN
ejpam-4387	2	13	1307	1307	NUM
ejpam-4387	2	14	-	-	SYM
ejpam-4387	2	15	5543	5543	NUM
ejpam-4387	2	16	–	–	PUNCT
ejpam-4387	2	17	ejpam.com	ejpam.com	X
ejpam-4387	2	18	published	publish	VERB
ejpam-4387	2	19	by	by	ADP
ejpam-4387	2	20	new	new	PROPN
ejpam-4387	2	21	york	york	PROPN
ejpam-4387	2	22	business	business	PROPN
ejpam-4387	2	23	global	global	ADJ
ejpam-4387	2	24	results	result	NOUN
ejpam-4387	2	25	about	about	ADP
ejpam-4387	2	26	p	p	NOUN
ejpam-4387	2	27	-normality	-normality	PROPN
ejpam-4387	2	28	lutfi	lutfi	PROPN
ejpam-4387	2	29	kalantan1	kalantan1	PROPN
ejpam-4387	2	30	,	,	PUNCT
ejpam-4387	2	31	mai	mai	PROPN
ejpam-4387	2	32	mansouri1,∗	mansouri1,∗	PROPN
ejpam-4387	2	33	1	1	NUM
ejpam-4387	2	34	king	king	PROPN
ejpam-4387	2	35	abdulaziz	abdulaziz	PROPN
ejpam-4387	2	36	university	university	PROPN
ejpam-4387	2	37	,	,	PUNCT
ejpam-4387	2	38	department	department	NOUN
ejpam-4387	2	39	of	of	ADP
ejpam-4387	2	40	mathematics	mathematic	NOUN
ejpam-4387	2	41	,	,	PUNCT
ejpam-4387	2	42	p.o.box	p.o.box	PROPN
ejpam-4387	2	43	80203	80203	NUM
ejpam-4387	2	44	,	,	PUNCT
ejpam-4387	2	45	jeddah	jeddah	PROPN
ejpam-4387	2	46	21589	21589	NUM
ejpam-4387	2	47	,	,	PUNCT
ejpam-4387	2	48	saudi	saudi	PROPN
ejpam-4387	2	49	arabia	arabia	PROPN
ejpam-4387	2	50	abstract	abstract	NOUN
ejpam-4387	2	51	.	.	PUNCT
ejpam-4387	3	1	a	a	DET
ejpam-4387	3	2	topological	topological	ADJ
ejpam-4387	3	3	spacex	spacex	NOUN
ejpam-4387	3	4	is	be	AUX
ejpam-4387	3	5	called	call	VERB
ejpam-4387	3	6	p	p	NOUN
ejpam-4387	3	7	-normal	-normal	ADJ
ejpam-4387	3	8	if	if	SCONJ
ejpam-4387	3	9	there	there	PRON
ejpam-4387	3	10	exist	exist	VERB
ejpam-4387	3	11	a	a	DET
ejpam-4387	3	12	normal	normal	ADJ
ejpam-4387	3	13	space	space	NOUN
ejpam-4387	3	14	y	y	PROPN
ejpam-4387	3	15	and	and	CCONJ
ejpam-4387	3	16	a	a	DET
ejpam-4387	3	17	bijective	bijective	ADJ
ejpam-4387	3	18	function	function	NOUN
ejpam-4387	4	1	f	f	NOUN
ejpam-4387	4	2	:	:	PUNCT
ejpam-4387	4	3	x	x	PUNCT
ejpam-4387	4	4	−→	−→	NOUN
ejpam-4387	4	5	y	y	PROPN
ejpam-4387	4	6	such	such	ADJ
ejpam-4387	4	7	that	that	SCONJ
ejpam-4387	4	8	the	the	DET
ejpam-4387	4	9	restriction	restriction	NOUN
ejpam-4387	4	10	f|a	f|a	PUNCT
ejpam-4387	4	11	:	:	PUNCT
ejpam-4387	4	12	a	a	DET
ejpam-4387	4	13	−→	−→	NOUN
ejpam-4387	4	14	f(a	f(a	NOUN
ejpam-4387	4	15	)	)	PUNCT
ejpam-4387	4	16	is	be	AUX
ejpam-4387	4	17	a	a	DET
ejpam-4387	4	18	homeomorphism	homeomorphism	NOUN
ejpam-4387	4	19	for	for	ADP
ejpam-4387	4	20	each	each	DET
ejpam-4387	4	21	paracompact	paracompact	ADJ
ejpam-4387	4	22	subspace	subspace	NOUN
ejpam-4387	4	23	a	a	DET
ejpam-4387	4	24	⊆	⊆	NUM
ejpam-4387	4	25	x.	x.	NOUN
ejpam-4387	4	26	in	in	ADP
ejpam-4387	4	27	this	this	DET
ejpam-4387	4	28	paper	paper	NOUN
ejpam-4387	4	29	we	we	PRON
ejpam-4387	4	30	present	present	VERB
ejpam-4387	4	31	some	some	DET
ejpam-4387	4	32	new	new	ADJ
ejpam-4387	4	33	results	result	NOUN
ejpam-4387	4	34	on	on	ADP
ejpam-4387	4	35	p	p	PRON
ejpam-4387	4	36	-normality	-normality	NOUN
ejpam-4387	4	37	.	.	PUNCT
ejpam-4387	5	1	we	we	PRON
ejpam-4387	5	2	study	study	VERB
ejpam-4387	5	3	the	the	DET
ejpam-4387	5	4	invariance	invariance	NOUN
ejpam-4387	5	5	and	and	CCONJ
ejpam-4387	5	6	inverse	inverse	NOUN
ejpam-4387	5	7	invariance	invariance	NOUN
ejpam-4387	5	8	of	of	ADP
ejpam-4387	5	9	p	p	NOUN
ejpam-4387	5	10	-normality	-normality	NOUN
ejpam-4387	5	11	as	as	ADP
ejpam-4387	5	12	a	a	DET
ejpam-4387	5	13	topological	topological	ADJ
ejpam-4387	5	14	property	property	NOUN
ejpam-4387	5	15	.	.	PUNCT
ejpam-4387	6	1	we	we	PRON
ejpam-4387	6	2	also	also	ADV
ejpam-4387	6	3	investigate	investigate	VERB
ejpam-4387	6	4	the	the	DET
ejpam-4387	6	5	alexandroff	alexandroff	ADJ
ejpam-4387	6	6	duplicate	duplicate	NOUN
ejpam-4387	6	7	of	of	ADP
ejpam-4387	6	8	a	a	DET
ejpam-4387	6	9	p	p	NOUN
ejpam-4387	6	10	-normal	-normal	ADJ
ejpam-4387	6	11	space	space	NOUN
ejpam-4387	6	12	,	,	PUNCT
ejpam-4387	6	13	the	the	DET
ejpam-4387	6	14	closed	closed	ADJ
ejpam-4387	6	15	extension	extension	NOUN
ejpam-4387	6	16	of	of	ADP
ejpam-4387	6	17	a	a	DET
ejpam-4387	6	18	p	p	NOUN
ejpam-4387	6	19	-normal	-normal	ADJ
ejpam-4387	6	20	space	space	NOUN
ejpam-4387	6	21	,	,	PUNCT
ejpam-4387	6	22	the	the	DET
ejpam-4387	6	23	discrete	discrete	ADJ
ejpam-4387	6	24	extension	extension	NOUN
ejpam-4387	6	25	of	of	ADP
ejpam-4387	6	26	a	a	DET
ejpam-4387	6	27	p	p	NOUN
ejpam-4387	6	28	-normal	-normal	ADJ
ejpam-4387	6	29	space	space	NOUN
ejpam-4387	6	30	and	and	CCONJ
ejpam-4387	6	31	the	the	DET
ejpam-4387	6	32	dowker	dowker	NOUN
ejpam-4387	6	33	topological	topological	ADJ
ejpam-4387	6	34	space	space	NOUN
ejpam-4387	6	35	.	.	PUNCT
ejpam-4387	7	1	furthermore	furthermore	ADV
ejpam-4387	7	2	,	,	PUNCT
ejpam-4387	7	3	we	we	PRON
ejpam-4387	7	4	introduce	introduce	VERB
ejpam-4387	7	5	a	a	DET
ejpam-4387	7	6	new	new	ADJ
ejpam-4387	7	7	property	property	NOUN
ejpam-4387	7	8	related	relate	VERB
ejpam-4387	7	9	to	to	ADP
ejpam-4387	7	10	p	p	NOUN
ejpam-4387	7	11	-normality	-normality	NOUN
ejpam-4387	7	12	which	which	PRON
ejpam-4387	7	13	we	we	PRON
ejpam-4387	7	14	call	call	VERB
ejpam-4387	7	15	strong	strong	ADJ
ejpam-4387	7	16	p	p	NOUN
ejpam-4387	7	17	-normality	-normality	NOUN
ejpam-4387	7	18	.	.	PUNCT
ejpam-4387	8	1	2020	2020	NUM
ejpam-4387	8	2	mathematics	mathematic	NOUN
ejpam-4387	8	3	subject	subject	NOUN
ejpam-4387	8	4	classifications	classification	NOUN
ejpam-4387	8	5	:	:	PUNCT
ejpam-4387	8	6	54d15	54d15	NUM
ejpam-4387	8	7	,	,	PUNCT
ejpam-4387	8	8	54c10	54c10	NUM
ejpam-4387	8	9	key	key	ADJ
ejpam-4387	8	10	words	word	NOUN
ejpam-4387	8	11	and	and	CCONJ
ejpam-4387	8	12	phrases	phrase	NOUN
ejpam-4387	8	13	:	:	PUNCT
ejpam-4387	8	14	normal	normal	ADJ
ejpam-4387	8	15	,	,	PUNCT
ejpam-4387	8	16	p	p	NOUN
ejpam-4387	8	17	-normal	-normal	NOUN
ejpam-4387	8	18	,	,	PUNCT
ejpam-4387	8	19	l	l	NOUN
ejpam-4387	8	20	-	-	ADJ
ejpam-4387	8	21	normal	normal	ADJ
ejpam-4387	8	22	,	,	PUNCT
ejpam-4387	8	23	c	c	NOUN
ejpam-4387	8	24	-	-	ADJ
ejpam-4387	8	25	normal	normal	ADJ
ejpam-4387	8	26	,	,	PUNCT
ejpam-4387	8	27	strong	strong	ADJ
ejpam-4387	8	28	p	p	NOUN
ejpam-4387	8	29	-noramlity	-noramlity	NOUN
ejpam-4387	8	30	,	,	PUNCT
ejpam-4387	8	31	alexandroff	alexandroff	NOUN
ejpam-4387	8	32	duplicate	duplicate	NOUN
ejpam-4387	8	33	,	,	PUNCT
ejpam-4387	8	34	invariance	invariance	NOUN
ejpam-4387	8	35	,	,	PUNCT
ejpam-4387	8	36	closed	closed	ADJ
ejpam-4387	8	37	extension	extension	NOUN
ejpam-4387	8	38	,	,	PUNCT
ejpam-4387	8	39	discrete	discrete	ADJ
ejpam-4387	8	40	extension	extension	NOUN
ejpam-4387	8	41	,	,	PUNCT
ejpam-4387	8	42	paracompact	paracompact	NOUN
ejpam-4387	8	43	,	,	PUNCT
ejpam-4387	8	44	product	product	NOUN
ejpam-4387	8	45	1	1	NUM
ejpam-4387	8	46	.	.	PUNCT
ejpam-4387	9	1	introduction	introduction	NOUN
ejpam-4387	9	2	we	we	PRON
ejpam-4387	9	3	introduced	introduce	VERB
ejpam-4387	9	4	p	p	NOUN
ejpam-4387	9	5	-normality	-normality	NOUN
ejpam-4387	9	6	in	in	ADP
ejpam-4387	9	7	our	our	PRON
ejpam-4387	9	8	previous	previous	ADJ
ejpam-4387	9	9	paper	paper	NOUN
ejpam-4387	10	1	[	[	X
ejpam-4387	10	2	10	10	NUM
ejpam-4387	10	3	]	]	PUNCT
ejpam-4387	10	4	.	.	PUNCT
ejpam-4387	11	1	the	the	DET
ejpam-4387	11	2	purpose	purpose	NOUN
ejpam-4387	11	3	of	of	ADP
ejpam-4387	11	4	this	this	DET
ejpam-4387	11	5	paper	paper	NOUN
ejpam-4387	11	6	is	be	AUX
ejpam-4387	11	7	to	to	PART
ejpam-4387	11	8	study	study	VERB
ejpam-4387	11	9	some	some	DET
ejpam-4387	11	10	new	new	ADJ
ejpam-4387	11	11	results	result	NOUN
ejpam-4387	11	12	about	about	ADP
ejpam-4387	11	13	p	p	NOUN
ejpam-4387	11	14	-normality	-normality	NOUN
ejpam-4387	11	15	.	.	PUNCT
ejpam-4387	12	1	we	we	PRON
ejpam-4387	12	2	investigate	investigate	VERB
ejpam-4387	12	3	some	some	DET
ejpam-4387	12	4	types	type	NOUN
ejpam-4387	12	5	of	of	ADP
ejpam-4387	12	6	invariance	invariance	NOUN
ejpam-4387	12	7	.	.	PUNCT
ejpam-4387	13	1	we	we	PRON
ejpam-4387	13	2	also	also	ADV
ejpam-4387	13	3	discuss	discuss	VERB
ejpam-4387	13	4	the	the	DET
ejpam-4387	13	5	alexandroff	alexandroff	NOUN
ejpam-4387	13	6	duplicate	duplicate	NOUN
ejpam-4387	13	7	,	,	PUNCT
ejpam-4387	13	8	the	the	DET
ejpam-4387	13	9	closed	closed	ADJ
ejpam-4387	13	10	extension	extension	NOUN
ejpam-4387	13	11	space	space	NOUN
ejpam-4387	13	12	and	and	CCONJ
ejpam-4387	13	13	the	the	DET
ejpam-4387	13	14	discrete	discrete	ADJ
ejpam-4387	13	15	extension	extension	NOUN
ejpam-4387	13	16	space	space	NOUN
ejpam-4387	13	17	of	of	ADP
ejpam-4387	13	18	a	a	DET
ejpam-4387	13	19	p	p	NOUN
ejpam-4387	13	20	-normal	-normal	ADJ
ejpam-4387	13	21	space	space	NOUN
ejpam-4387	13	22	.	.	PUNCT
ejpam-4387	14	1	we	we	PRON
ejpam-4387	14	2	examine	examine	VERB
ejpam-4387	14	3	whether	whether	SCONJ
ejpam-4387	14	4	p	p	PROPN
ejpam-4387	14	5	-normality	-normality	NOUN
ejpam-4387	14	6	is	be	AUX
ejpam-4387	14	7	preserved	preserve	VERB
ejpam-4387	14	8	in	in	ADP
ejpam-4387	14	9	these	these	DET
ejpam-4387	14	10	spaces	space	NOUN
ejpam-4387	14	11	or	or	CCONJ
ejpam-4387	14	12	not	not	PART
ejpam-4387	14	13	.	.	PUNCT
ejpam-4387	15	1	finally	finally	ADV
ejpam-4387	15	2	,	,	PUNCT
ejpam-4387	15	3	we	we	PRON
ejpam-4387	15	4	define	define	VERB
ejpam-4387	15	5	a	a	DET
ejpam-4387	15	6	new	new	ADJ
ejpam-4387	15	7	topological	topological	ADJ
ejpam-4387	15	8	property	property	NOUN
ejpam-4387	15	9	called	call	VERB
ejpam-4387	15	10	strong	strong	ADJ
ejpam-4387	15	11	p	p	NOUN
ejpam-4387	15	12	normality	normality	NOUN
ejpam-4387	15	13	.	.	PUNCT
ejpam-4387	16	1	throughout	throughout	ADP
ejpam-4387	16	2	this	this	DET
ejpam-4387	16	3	paper	paper	NOUN
ejpam-4387	16	4	,	,	PUNCT
ejpam-4387	16	5	we	we	PRON
ejpam-4387	16	6	denote	denote	VERB
ejpam-4387	16	7	an	an	DET
ejpam-4387	16	8	ordered	order	VERB
ejpam-4387	16	9	pair	pair	NOUN
ejpam-4387	16	10	by	by	ADP
ejpam-4387	16	11	⟨x	⟨x	NUM
ejpam-4387	16	12	,	,	PUNCT
ejpam-4387	16	13	y⟩	y⟩	NOUN
ejpam-4387	16	14	,	,	PUNCT
ejpam-4387	16	15	the	the	DET
ejpam-4387	16	16	set	set	NOUN
ejpam-4387	16	17	of	of	ADP
ejpam-4387	16	18	positive	positive	ADJ
ejpam-4387	16	19	integers	integer	NOUN
ejpam-4387	16	20	by	by	ADP
ejpam-4387	16	21	n	n	NOUN
ejpam-4387	16	22	and	and	CCONJ
ejpam-4387	16	23	the	the	DET
ejpam-4387	16	24	set	set	NOUN
ejpam-4387	16	25	of	of	ADP
ejpam-4387	16	26	real	real	ADJ
ejpam-4387	16	27	numbers	number	NOUN
ejpam-4387	16	28	by	by	ADP
ejpam-4387	16	29	r.	r.	PROPN
ejpam-4387	16	30	a	a	DET
ejpam-4387	16	31	t4	t4	PROPN
ejpam-4387	16	32	space	space	NOUN
ejpam-4387	16	33	is	be	AUX
ejpam-4387	16	34	a	a	DET
ejpam-4387	16	35	t1	t1	NOUN
ejpam-4387	16	36	normal	normal	ADJ
ejpam-4387	16	37	space	space	NOUN
ejpam-4387	16	38	and	and	CCONJ
ejpam-4387	16	39	a	a	DET
ejpam-4387	16	40	tychonoff	tychonoff	NOUN
ejpam-4387	16	41	space	space	NOUN
ejpam-4387	16	42	is	be	AUX
ejpam-4387	16	43	a	a	DET
ejpam-4387	16	44	t1	t1	NOUN
ejpam-4387	16	45	completely	completely	ADV
ejpam-4387	16	46	regular	regular	ADJ
ejpam-4387	16	47	space	space	NOUN
ejpam-4387	16	48	.	.	PUNCT
ejpam-4387	17	1	we	we	PRON
ejpam-4387	17	2	do	do	AUX
ejpam-4387	17	3	not	not	PART
ejpam-4387	17	4	assume	assume	VERB
ejpam-4387	17	5	t2	t2	NOUN
ejpam-4387	17	6	in	in	ADP
ejpam-4387	17	7	the	the	DET
ejpam-4387	17	8	definition	definition	NOUN
ejpam-4387	17	9	of	of	ADP
ejpam-4387	17	10	compactness	compactness	NOUN
ejpam-4387	17	11	,	,	PUNCT
ejpam-4387	17	12	paracompactness	paracompactness	NOUN
ejpam-4387	17	13	and	and	CCONJ
ejpam-4387	17	14	countable	countable	ADJ
ejpam-4387	17	15	compactness	compactness	NOUN
ejpam-4387	17	16	.	.	PUNCT
ejpam-4387	18	1	we	we	PRON
ejpam-4387	18	2	do	do	AUX
ejpam-4387	18	3	not	not	PART
ejpam-4387	18	4	assume	assume	VERB
ejpam-4387	18	5	regularity	regularity	NOUN
ejpam-4387	18	6	in	in	ADP
ejpam-4387	18	7	the	the	DET
ejpam-4387	18	8	definition	definition	NOUN
ejpam-4387	18	9	of	of	ADP
ejpam-4387	18	10	lindelöfness	lindelöfness	PROPN
ejpam-4387	18	11	.	.	PUNCT
ejpam-4387	19	1	for	for	ADP
ejpam-4387	19	2	a	a	DET
ejpam-4387	19	3	subset	subset	NOUN
ejpam-4387	19	4	a	a	PRON
ejpam-4387	19	5	of	of	ADP
ejpam-4387	19	6	a	a	DET
ejpam-4387	19	7	space	space	NOUN
ejpam-4387	19	8	x	x	NOUN
ejpam-4387	19	9	,	,	PUNCT
ejpam-4387	19	10	inta	inta	PROPN
ejpam-4387	19	11	and	and	CCONJ
ejpam-4387	19	12	a	a	DET
ejpam-4387	19	13	denote	denote	NOUN
ejpam-4387	19	14	the	the	DET
ejpam-4387	19	15	interior	interior	NOUN
ejpam-4387	19	16	and	and	CCONJ
ejpam-4387	19	17	the	the	DET
ejpam-4387	19	18	closure	closure	NOUN
ejpam-4387	19	19	of	of	ADP
ejpam-4387	19	20	a	a	PRON
ejpam-4387	19	21	,	,	PUNCT
ejpam-4387	19	22	respectively	respectively	ADV
ejpam-4387	19	23	.	.	PUNCT
ejpam-4387	20	1	an	an	DET
ejpam-4387	20	2	ordinal	ordinal	ADJ
ejpam-4387	20	3	γ	γ	X
ejpam-4387	20	4	is	be	AUX
ejpam-4387	20	5	the	the	DET
ejpam-4387	20	6	set	set	NOUN
ejpam-4387	20	7	of	of	ADP
ejpam-4387	20	8	all	all	DET
ejpam-4387	20	9	ordinal	ordinal	ADJ
ejpam-4387	20	10	α	α	PRON
ejpam-4387	20	11	such	such	ADJ
ejpam-4387	20	12	that	that	SCONJ
ejpam-4387	20	13	α	α	PROPN
ejpam-4387	20	14	<	<	X
ejpam-4387	20	15	γ	γ	X
ejpam-4387	20	16	.	.	PUNCT
ejpam-4387	21	1	the	the	DET
ejpam-4387	21	2	first	first	ADJ
ejpam-4387	21	3	infinite	infinite	ADJ
ejpam-4387	21	4	ordinal	ordinal	ADJ
ejpam-4387	21	5	is	be	AUX
ejpam-4387	21	6	ω0	ω0	NOUN
ejpam-4387	21	7	,	,	PUNCT
ejpam-4387	21	8	the	the	DET
ejpam-4387	21	9	first	first	ADJ
ejpam-4387	21	10	uncountable	uncountable	ADJ
ejpam-4387	21	11	ordinal	ordinal	NOUN
ejpam-4387	21	12	is	be	AUX
ejpam-4387	21	13	ω1	ω1	PROPN
ejpam-4387	21	14	,	,	PUNCT
ejpam-4387	21	15	and	and	CCONJ
ejpam-4387	21	16	the	the	DET
ejpam-4387	21	17	successor	successor	NOUN
ejpam-4387	21	18	cardinal	cardinal	NOUN
ejpam-4387	21	19	of	of	ADP
ejpam-4387	21	20	ω1	ω1	PROPN
ejpam-4387	21	21	is	be	AUX
ejpam-4387	21	22	ω2	ω2	ADJ
ejpam-4387	21	23	.	.	PUNCT
ejpam-4387	22	1	we	we	PRON
ejpam-4387	22	2	begin	begin	VERB
ejpam-4387	22	3	by	by	ADP
ejpam-4387	22	4	recalling	recall	VERB
ejpam-4387	22	5	the	the	DET
ejpam-4387	22	6	following	follow	VERB
ejpam-4387	22	7	definitions	definition	NOUN
ejpam-4387	22	8	:	:	PUNCT
ejpam-4387	22	9	∗corresponding	∗corresponde	VERB
ejpam-4387	22	10	author	author	NOUN
ejpam-4387	22	11	.	.	PUNCT
ejpam-4387	23	1	doi	doi	NOUN
ejpam-4387	23	2	:	:	PUNCT
ejpam-4387	23	3	https://doi.org/10.29020/nybg.ejpam.v15i2.4387	https://doi.org/10.29020/nybg.ejpam.v15i2.4387	NOUN
ejpam-4387	23	4	email	email	NOUN
ejpam-4387	23	5	addresses	address	NOUN
ejpam-4387	23	6	:	:	PUNCT
ejpam-4387	23	7	lnkalantan@hotmail.com	lnkalantan@hotmail.com	X
ejpam-4387	23	8	(	(	PUNCT
ejpam-4387	23	9	l.	l.	PROPN
ejpam-4387	23	10	kalantan	kalantan	PROPN
ejpam-4387	23	11	)	)	PUNCT
ejpam-4387	23	12	,	,	PUNCT
ejpam-4387	23	13	mfmansouri1@kau.edu.sa	mfmansouri1@kau.edu.sa	PROPN
ejpam-4387	23	14	(	(	PUNCT
ejpam-4387	23	15	m.	m.	NOUN
ejpam-4387	23	16	mansouri	mansouri	PROPN
ejpam-4387	23	17	)	)	PUNCT
ejpam-4387	23	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4387	23	19	774	774	NUM
ejpam-4387	24	1	©	©	ADP
ejpam-4387	24	2	2022	2022	NUM
ejpam-4387	24	3	ejpam	ejpam	VERB
ejpam-4387	24	4	all	all	DET
ejpam-4387	24	5	rights	right	NOUN
ejpam-4387	24	6	reserved	reserve	VERB
ejpam-4387	24	7	.	.	PUNCT
ejpam-4387	25	1	l.	l.	PROPN
ejpam-4387	25	2	kalantan	kalantan	PROPN
ejpam-4387	25	3	,	,	PUNCT
ejpam-4387	25	4	m.	m.	NOUN
ejpam-4387	25	5	mansouri	mansouri	PROPN
ejpam-4387	25	6	/	/	SYM
ejpam-4387	25	7	eur	eur	PROPN
ejpam-4387	25	8	.	.	PUNCT
ejpam-4387	26	1	j.	j.	PROPN
ejpam-4387	26	2	pure	pure	PROPN
ejpam-4387	26	3	appl	appl	PROPN
ejpam-4387	26	4	.	.	PROPN
ejpam-4387	26	5	math	math	PROPN
ejpam-4387	26	6	,	,	PUNCT
ejpam-4387	26	7	15	15	NUM
ejpam-4387	26	8	(	(	PUNCT
ejpam-4387	26	9	2	2	NUM
ejpam-4387	26	10	)	)	PUNCT
ejpam-4387	26	11	(	(	PUNCT
ejpam-4387	26	12	2022	2022	NUM
ejpam-4387	26	13	)	)	PUNCT
ejpam-4387	26	14	,	,	PUNCT
ejpam-4387	26	15	774	774	NUM
ejpam-4387	26	16	-	-	SYM
ejpam-4387	26	17	783	783	NUM
ejpam-4387	26	18	775	775	NUM
ejpam-4387	26	19	recall	recall	NOUN
ejpam-4387	26	20	that	that	SCONJ
ejpam-4387	26	21	a	a	DET
ejpam-4387	26	22	topological	topological	ADJ
ejpam-4387	26	23	space	space	NOUN
ejpam-4387	26	24	(	(	PUNCT
ejpam-4387	26	25	x	x	X
ejpam-4387	26	26	,	,	PUNCT
ejpam-4387	26	27	τ	τ	PROPN
ejpam-4387	26	28	)	)	PUNCT
ejpam-4387	26	29	is	be	AUX
ejpam-4387	26	30	paracompact	paracompact	ADJ
ejpam-4387	26	31	if	if	SCONJ
ejpam-4387	26	32	any	any	DET
ejpam-4387	26	33	open	open	ADJ
ejpam-4387	26	34	cover	cover	NOUN
ejpam-4387	26	35	has	have	VERB
ejpam-4387	26	36	a	a	DET
ejpam-4387	26	37	locally	locally	ADV
ejpam-4387	26	38	finite	finite	ADJ
ejpam-4387	26	39	open	open	ADJ
ejpam-4387	26	40	refinement	refinement	NOUN
ejpam-4387	26	41	.	.	PUNCT
ejpam-4387	27	1	for	for	ADP
ejpam-4387	27	2	a	a	DET
ejpam-4387	27	3	subspace	subspace	NOUN
ejpam-4387	27	4	a	a	PRON
ejpam-4387	27	5	of	of	ADP
ejpam-4387	27	6	x	x	PRON
ejpam-4387	27	7	,	,	PUNCT
ejpam-4387	27	8	a	a	PRON
ejpam-4387	27	9	is	be	AUX
ejpam-4387	27	10	paracompact	paracompact	ADJ
ejpam-4387	27	11	if	if	SCONJ
ejpam-4387	27	12	(	(	PUNCT
ejpam-4387	27	13	a	a	DET
ejpam-4387	27	14	,	,	PUNCT
ejpam-4387	27	15	τa	τa	PROPN
ejpam-4387	27	16	)	)	PUNCT
ejpam-4387	27	17	is	be	AUX
ejpam-4387	27	18	paracompact	paracompact	ADJ
ejpam-4387	27	19	,	,	PUNCT
ejpam-4387	27	20	i.e.	i.e.	X
ejpam-4387	27	21	,	,	PUNCT
ejpam-4387	27	22	any	any	PRON
ejpam-4387	27	23	open	open	ADJ
ejpam-4387	27	24	(	(	PUNCT
ejpam-4387	27	25	open	open	ADJ
ejpam-4387	27	26	in	in	ADP
ejpam-4387	27	27	the	the	DET
ejpam-4387	27	28	subspace	subspace	NOUN
ejpam-4387	27	29	)	)	PUNCT
ejpam-4387	27	30	cover	cover	NOUN
ejpam-4387	27	31	of	of	ADP
ejpam-4387	27	32	a	a	PRON
ejpam-4387	27	33	has	have	AUX
ejpam-4387	27	34	a	a	DET
ejpam-4387	27	35	locally	locally	ADV
ejpam-4387	27	36	finite	finite	ADJ
ejpam-4387	27	37	open	open	ADJ
ejpam-4387	27	38	(	(	PUNCT
ejpam-4387	27	39	open	open	ADJ
ejpam-4387	27	40	in	in	ADP
ejpam-4387	27	41	the	the	DET
ejpam-4387	27	42	subspace	subspace	NOUN
ejpam-4387	27	43	)	)	PUNCT
ejpam-4387	27	44	refinement	refinement	NOUN
ejpam-4387	27	45	.	.	PUNCT
ejpam-4387	28	1	we	we	PRON
ejpam-4387	28	2	do	do	AUX
ejpam-4387	28	3	not	not	PART
ejpam-4387	28	4	assume	assume	VERB
ejpam-4387	28	5	t2	t2	NOUN
ejpam-4387	28	6	in	in	ADP
ejpam-4387	28	7	the	the	DET
ejpam-4387	28	8	definition	definition	NOUN
ejpam-4387	28	9	.	.	PUNCT
ejpam-4387	29	1	definition	definition	NOUN
ejpam-4387	29	2	1	1	NUM
ejpam-4387	29	3	.	.	PUNCT
ejpam-4387	30	1	a	a	DET
ejpam-4387	30	2	topological	topological	ADJ
ejpam-4387	30	3	space	space	NOUN
ejpam-4387	30	4	x	x	PUNCT
ejpam-4387	30	5	is	be	AUX
ejpam-4387	30	6	called	call	VERB
ejpam-4387	30	7	p	p	NOUN
ejpam-4387	30	8	-normal	-normal	ADJ
ejpam-4387	30	9	if	if	SCONJ
ejpam-4387	30	10	there	there	PRON
ejpam-4387	30	11	exist	exist	VERB
ejpam-4387	30	12	a	a	DET
ejpam-4387	30	13	normal	normal	ADJ
ejpam-4387	30	14	space	space	NOUN
ejpam-4387	30	15	y	y	PROPN
ejpam-4387	30	16	and	and	CCONJ
ejpam-4387	30	17	a	a	DET
ejpam-4387	30	18	bijective	bijective	ADJ
ejpam-4387	30	19	function	function	NOUN
ejpam-4387	31	1	f	f	NOUN
ejpam-4387	31	2	:	:	PUNCT
ejpam-4387	31	3	x	x	PUNCT
ejpam-4387	31	4	−→	−→	NOUN
ejpam-4387	31	5	y	y	PROPN
ejpam-4387	31	6	such	such	ADJ
ejpam-4387	31	7	that	that	SCONJ
ejpam-4387	31	8	the	the	DET
ejpam-4387	31	9	restriction	restriction	NOUN
ejpam-4387	31	10	f|a	f|a	PUNCT
ejpam-4387	31	11	:	:	PUNCT
ejpam-4387	31	12	a	a	DET
ejpam-4387	31	13	−→	−→	NOUN
ejpam-4387	31	14	f(a	f(a	NOUN
ejpam-4387	31	15	)	)	PUNCT
ejpam-4387	31	16	is	be	AUX
ejpam-4387	31	17	a	a	DET
ejpam-4387	31	18	homeomorphism	homeomorphism	NOUN
ejpam-4387	31	19	for	for	ADP
ejpam-4387	31	20	each	each	DET
ejpam-4387	31	21	paracompact	paracompact	ADJ
ejpam-4387	31	22	subspace	subspace	NOUN
ejpam-4387	31	23	a	a	PRON
ejpam-4387	31	24	⊆	⊆	NUM
ejpam-4387	31	25	x	x	SYM
ejpam-4387	32	1	[	[	X
ejpam-4387	32	2	10	10	NUM
ejpam-4387	32	3	]	]	PUNCT
ejpam-4387	32	4	.	.	PUNCT
ejpam-4387	33	1	2	2	X
ejpam-4387	33	2	.	.	X
ejpam-4387	33	3	new	new	ADJ
ejpam-4387	33	4	results	result	NOUN
ejpam-4387	33	5	on	on	ADP
ejpam-4387	33	6	p	p	NOUN
ejpam-4387	33	7	-normality	-normality	NOUN
ejpam-4387	33	8	in	in	ADP
ejpam-4387	33	9	[	[	X
ejpam-4387	33	10	10	10	NUM
ejpam-4387	33	11	]	]	PUNCT
ejpam-4387	33	12	,	,	PUNCT
ejpam-4387	33	13	we	we	PRON
ejpam-4387	33	14	proved	prove	VERB
ejpam-4387	33	15	p	p	NOUN
ejpam-4387	33	16	-normality	-normality	PROPN
ejpam-4387	33	17	is	be	AUX
ejpam-4387	33	18	a	a	DET
ejpam-4387	33	19	topological	topological	ADJ
ejpam-4387	33	20	property	property	NOUN
ejpam-4387	33	21	,	,	PUNCT
ejpam-4387	33	22	studied	study	VERB
ejpam-4387	33	23	its	its	PRON
ejpam-4387	33	24	independence	independence	NOUN
ejpam-4387	33	25	of	of	ADP
ejpam-4387	33	26	other	other	ADJ
ejpam-4387	33	27	topological	topological	ADJ
ejpam-4387	33	28	properties	property	NOUN
ejpam-4387	33	29	,	,	PUNCT
ejpam-4387	33	30	and	and	CCONJ
ejpam-4387	33	31	investigated	investigate	VERB
ejpam-4387	33	32	whether	whether	SCONJ
ejpam-4387	33	33	or	or	CCONJ
ejpam-4387	33	34	not	not	PART
ejpam-4387	33	35	p	p	NOUN
ejpam-4387	33	36	-normality	-normality	PROPN
ejpam-4387	33	37	was	be	AUX
ejpam-4387	33	38	an	an	DET
ejpam-4387	33	39	additive	additive	ADJ
ejpam-4387	33	40	property	property	NOUN
ejpam-4387	33	41	,	,	PUNCT
ejpam-4387	33	42	a	a	DET
ejpam-4387	33	43	multiplicative	multiplicative	ADJ
ejpam-4387	33	44	property	property	NOUN
ejpam-4387	33	45	and	and	CCONJ
ejpam-4387	33	46	a	a	DET
ejpam-4387	33	47	hereditary	hereditary	ADJ
ejpam-4387	33	48	property	property	NOUN
ejpam-4387	33	49	.	.	PUNCT
ejpam-4387	34	1	until	until	ADP
ejpam-4387	34	2	now	now	ADV
ejpam-4387	34	3	,	,	PUNCT
ejpam-4387	34	4	we	we	PRON
ejpam-4387	34	5	still	still	ADV
ejpam-4387	34	6	do	do	AUX
ejpam-4387	34	7	n’t	not	PART
ejpam-4387	34	8	know	know	VERB
ejpam-4387	34	9	if	if	SCONJ
ejpam-4387	34	10	p	p	PROPN
ejpam-4387	34	11	-normality	-normality	NOUN
ejpam-4387	34	12	is	be	AUX
ejpam-4387	34	13	hereditary	hereditary	ADJ
ejpam-4387	34	14	with	with	ADP
ejpam-4387	34	15	respect	respect	NOUN
ejpam-4387	34	16	to	to	ADP
ejpam-4387	34	17	closed	closed	ADJ
ejpam-4387	34	18	subspaces	subspace	NOUN
ejpam-4387	34	19	.	.	PUNCT
ejpam-4387	35	1	but	but	CCONJ
ejpam-4387	35	2	under	under	ADP
ejpam-4387	35	3	some	some	DET
ejpam-4387	35	4	conditions	condition	NOUN
ejpam-4387	35	5	it	it	PRON
ejpam-4387	35	6	is	be	AUX
ejpam-4387	35	7	hereditary	hereditary	ADJ
ejpam-4387	35	8	with	with	ADP
ejpam-4387	35	9	respect	respect	NOUN
ejpam-4387	35	10	to	to	ADP
ejpam-4387	35	11	compact	compact	ADJ
ejpam-4387	35	12	subspaces	subspace	NOUN
ejpam-4387	35	13	.	.	PUNCT
ejpam-4387	36	1	before	before	SCONJ
ejpam-4387	36	2	we	we	PRON
ejpam-4387	36	3	state	state	VERB
ejpam-4387	36	4	such	such	ADJ
ejpam-4387	36	5	conditions	condition	NOUN
ejpam-4387	36	6	we	we	PRON
ejpam-4387	36	7	introduce	introduce	VERB
ejpam-4387	36	8	some	some	DET
ejpam-4387	36	9	results	result	NOUN
ejpam-4387	36	10	:	:	PUNCT
ejpam-4387	36	11	proposition	proposition	NOUN
ejpam-4387	36	12	1	1	NUM
ejpam-4387	36	13	.	.	PUNCT
ejpam-4387	37	1	if	if	SCONJ
ejpam-4387	37	2	x	x	PRON
ejpam-4387	37	3	is	be	AUX
ejpam-4387	37	4	a	a	DET
ejpam-4387	37	5	t1	t1	NOUN
ejpam-4387	37	6	space	space	NOUN
ejpam-4387	37	7	,	,	PUNCT
ejpam-4387	37	8	f	f	X
ejpam-4387	37	9	:	:	PUNCT
ejpam-4387	37	10	x	x	PUNCT
ejpam-4387	37	11	−→	−→	NOUN
ejpam-4387	37	12	y	y	PROPN
ejpam-4387	37	13	is	be	AUX
ejpam-4387	37	14	a	a	DET
ejpam-4387	37	15	one	one	NUM
ejpam-4387	37	16	to	to	ADP
ejpam-4387	37	17	one	one	NUM
ejpam-4387	37	18	and	and	CCONJ
ejpam-4387	37	19	onto	onto	ADP
ejpam-4387	37	20	map	map	NOUN
ejpam-4387	37	21	,	,	PUNCT
ejpam-4387	37	22	and	and	CCONJ
ejpam-4387	37	23	the	the	DET
ejpam-4387	37	24	restriction	restriction	NOUN
ejpam-4387	37	25	of	of	ADP
ejpam-4387	37	26	f	f	PROPN
ejpam-4387	37	27	on	on	ADP
ejpam-4387	37	28	any	any	DET
ejpam-4387	37	29	finite	finite	NOUN
ejpam-4387	37	30	subset	subset	NOUN
ejpam-4387	37	31	of	of	ADP
ejpam-4387	37	32	x	x	PUNCT
ejpam-4387	37	33	is	be	AUX
ejpam-4387	37	34	a	a	DET
ejpam-4387	37	35	homeomorphism	homeomorphism	NOUN
ejpam-4387	37	36	.	.	PUNCT
ejpam-4387	38	1	then	then	ADV
ejpam-4387	38	2	y	y	PROPN
ejpam-4387	38	3	is	be	AUX
ejpam-4387	38	4	also	also	ADV
ejpam-4387	38	5	t1	t1	NOUN
ejpam-4387	38	6	.	.	PUNCT
ejpam-4387	39	1	proof	proof	NOUN
ejpam-4387	39	2	.	.	PUNCT
ejpam-4387	40	1	sincex	sincex	PROPN
ejpam-4387	40	2	is	be	AUX
ejpam-4387	40	3	t1	t1	NUM
ejpam-4387	40	4	and	and	CCONJ
ejpam-4387	40	5	f	f	PROPN
ejpam-4387	40	6	is	be	AUX
ejpam-4387	40	7	a	a	DET
ejpam-4387	40	8	bijection	bijection	NOUN
ejpam-4387	40	9	then	then	ADV
ejpam-4387	40	10	y	y	PROPN
ejpam-4387	40	11	has	have	VERB
ejpam-4387	40	12	more	more	ADJ
ejpam-4387	40	13	than	than	ADP
ejpam-4387	40	14	one	one	NUM
ejpam-4387	40	15	element	element	NOUN
ejpam-4387	40	16	.	.	PUNCT
ejpam-4387	41	1	let	let	VERB
ejpam-4387	41	2	a	a	PRON
ejpam-4387	41	3	,	,	PUNCT
ejpam-4387	41	4	b	b	X
ejpam-4387	41	5	∈	∈	PROPN
ejpam-4387	41	6	y	y	PROPN
ejpam-4387	41	7	be	be	AUX
ejpam-4387	41	8	arbitrary	arbitrary	ADJ
ejpam-4387	41	9	such	such	ADJ
ejpam-4387	41	10	that	that	SCONJ
ejpam-4387	41	11	a	a	DET
ejpam-4387	41	12	̸=	̸=	PROPN
ejpam-4387	41	13	b.	b.	NOUN
ejpam-4387	41	14	then	then	ADV
ejpam-4387	41	15	there	there	PRON
ejpam-4387	41	16	exist	exist	VERB
ejpam-4387	41	17	unique	unique	ADJ
ejpam-4387	41	18	c	c	NOUN
ejpam-4387	41	19	,	,	PUNCT
ejpam-4387	41	20	d	d	PROPN
ejpam-4387	41	21	∈	∈	PROPN
ejpam-4387	41	22	x	x	PUNCT
ejpam-4387	41	23	such	such	ADJ
ejpam-4387	41	24	that	that	DET
ejpam-4387	41	25	f(c	f(c	PROPN
ejpam-4387	41	26	)	)	PUNCT
ejpam-4387	41	27	=	=	SYM
ejpam-4387	41	28	a	a	PRON
ejpam-4387	41	29	and	and	CCONJ
ejpam-4387	41	30	f(d	f(d	PROPN
ejpam-4387	41	31	)	)	PUNCT
ejpam-4387	41	32	=	=	SYM
ejpam-4387	41	33	b	b	PROPN
ejpam-4387	41	34	and	and	CCONJ
ejpam-4387	41	35	c	c	PROPN
ejpam-4387	41	36	̸=	̸=	PROPN
ejpam-4387	41	37	d.	d.	PROPN
ejpam-4387	41	38	now	now	ADV
ejpam-4387	41	39	,	,	PUNCT
ejpam-4387	41	40	{	{	PUNCT
ejpam-4387	41	41	c	c	X
ejpam-4387	41	42	,	,	PUNCT
ejpam-4387	41	43	d	d	NOUN
ejpam-4387	41	44	}	}	PUNCT
ejpam-4387	41	45	⊆	⊆	NUM
ejpam-4387	41	46	x	x	PUNCT
ejpam-4387	41	47	is	be	AUX
ejpam-4387	41	48	a	a	DET
ejpam-4387	41	49	finite	finite	NOUN
ejpam-4387	41	50	subset	subset	NOUN
ejpam-4387	41	51	of	of	ADP
ejpam-4387	41	52	x.	x.	PROPN
ejpam-4387	42	1	so	so	PROPN
ejpam-4387	42	2	f	f	PROPN
ejpam-4387	42	3	|{c	|{c	ADV
ejpam-4387	42	4	,	,	PUNCT
ejpam-4387	42	5	d	d	NOUN
ejpam-4387	42	6	}	}	PUNCT
ejpam-4387	42	7	:	:	PUNCT
ejpam-4387	42	8	{	{	PUNCT
ejpam-4387	42	9	c	c	X
ejpam-4387	42	10	,	,	PUNCT
ejpam-4387	42	11	d	d	NOUN
ejpam-4387	42	12	}	}	PUNCT
ejpam-4387	42	13	−→	−→	ADJ
ejpam-4387	42	14	{	{	PUNCT
ejpam-4387	42	15	a	a	DET
ejpam-4387	42	16	,	,	PUNCT
ejpam-4387	42	17	b	b	NOUN
ejpam-4387	42	18	}	}	PUNCT
ejpam-4387	42	19	is	be	AUX
ejpam-4387	42	20	a	a	DET
ejpam-4387	42	21	homeomorphism	homeomorphism	NOUN
ejpam-4387	42	22	.	.	PUNCT
ejpam-4387	43	1	now	now	ADV
ejpam-4387	43	2	,	,	PUNCT
ejpam-4387	43	3	a	a	DET
ejpam-4387	43	4	=	=	X
ejpam-4387	43	5	f(c	f(c	PROPN
ejpam-4387	43	6	)	)	PUNCT
ejpam-4387	43	7	and	and	CCONJ
ejpam-4387	43	8	c	c	PROPN
ejpam-4387	43	9	is	be	AUX
ejpam-4387	43	10	isolated	isolate	VERB
ejpam-4387	43	11	in	in	ADP
ejpam-4387	43	12	{	{	PUNCT
ejpam-4387	43	13	c	c	NOUN
ejpam-4387	43	14	,	,	PUNCT
ejpam-4387	43	15	d	d	NOUN
ejpam-4387	43	16	}	}	PUNCT
ejpam-4387	43	17	because	because	SCONJ
ejpam-4387	43	18	{	{	PUNCT
ejpam-4387	43	19	c	c	X
ejpam-4387	43	20	,	,	PUNCT
ejpam-4387	43	21	d	d	NOUN
ejpam-4387	43	22	}	}	PUNCT
ejpam-4387	43	23	⊆	⊆	NUM
ejpam-4387	43	24	x	x	PUNCT
ejpam-4387	43	25	is	be	AUX
ejpam-4387	43	26	discrete	discrete	ADJ
ejpam-4387	43	27	.	.	PUNCT
ejpam-4387	44	1	also	also	ADV
ejpam-4387	44	2	,	,	PUNCT
ejpam-4387	44	3	b	b	X
ejpam-4387	44	4	=	=	SYM
ejpam-4387	44	5	f(d	f(d	PROPN
ejpam-4387	44	6	)	)	PUNCT
ejpam-4387	44	7	and	and	CCONJ
ejpam-4387	44	8	d	d	NOUN
ejpam-4387	44	9	is	be	AUX
ejpam-4387	44	10	isolated	isolate	VERB
ejpam-4387	44	11	in	in	ADP
ejpam-4387	44	12	{	{	PUNCT
ejpam-4387	44	13	c	c	NOUN
ejpam-4387	44	14	,	,	PUNCT
ejpam-4387	44	15	d	d	NOUN
ejpam-4387	44	16	}	}	PUNCT
ejpam-4387	44	17	.	.	PUNCT
ejpam-4387	45	1	so	so	ADV
ejpam-4387	45	2	there	there	PRON
ejpam-4387	45	3	exist	exist	VERB
ejpam-4387	45	4	y	y	PROPN
ejpam-4387	45	5	-open	-open	PROPN
ejpam-4387	45	6	subset	subset	NOUN
ejpam-4387	45	7	u	u	NOUN
ejpam-4387	45	8	containing	contain	VERB
ejpam-4387	45	9	a	a	DET
ejpam-4387	45	10	such	such	ADJ
ejpam-4387	45	11	that	that	SCONJ
ejpam-4387	45	12	u	u	PROPN
ejpam-4387	45	13	∩	∩	NOUN
ejpam-4387	45	14	{	{	PUNCT
ejpam-4387	45	15	a	a	DET
ejpam-4387	45	16	,	,	PUNCT
ejpam-4387	45	17	b	b	NOUN
ejpam-4387	45	18	}	}	PUNCT
ejpam-4387	45	19	=	=	SYM
ejpam-4387	45	20	{	{	PUNCT
ejpam-4387	45	21	a	a	NOUN
ejpam-4387	45	22	}	}	PUNCT
ejpam-4387	45	23	,	,	PUNCT
ejpam-4387	45	24	and	and	CCONJ
ejpam-4387	45	25	there	there	PRON
ejpam-4387	45	26	exists	exist	VERB
ejpam-4387	45	27	y	y	PROPN
ejpam-4387	45	28	-open	-open	PROPN
ejpam-4387	45	29	subset	subset	VERB
ejpam-4387	45	30	v	v	ADP
ejpam-4387	45	31	containing	contain	VERB
ejpam-4387	45	32	b	b	NOUN
ejpam-4387	45	33	such	such	ADJ
ejpam-4387	45	34	that	that	DET
ejpam-4387	45	35	v	v	NOUN
ejpam-4387	45	36	∩	∩	NOUN
ejpam-4387	45	37	{	{	PUNCT
ejpam-4387	45	38	a	a	DET
ejpam-4387	45	39	,	,	PUNCT
ejpam-4387	45	40	b	b	NOUN
ejpam-4387	45	41	}	}	PUNCT
ejpam-4387	45	42	=	=	SYM
ejpam-4387	45	43	{	{	PUNCT
ejpam-4387	45	44	b	b	NOUN
ejpam-4387	45	45	}	}	PUNCT
ejpam-4387	45	46	.	.	PUNCT
ejpam-4387	46	1	then	then	ADV
ejpam-4387	46	2	b	b	X
ejpam-4387	46	3	/∈	/∈	PUNCT
ejpam-4387	46	4	u	u	NOUN
ejpam-4387	46	5	∋	∋	NOUN
ejpam-4387	46	6	a	a	NOUN
ejpam-4387	46	7	and	and	CCONJ
ejpam-4387	46	8	a	a	DET
ejpam-4387	46	9	/∈	/∈	NOUN
ejpam-4387	46	10	v	v	NOUN
ejpam-4387	46	11	∋	∋	NOUN
ejpam-4387	46	12	b.	b.	PROPN
ejpam-4387	46	13	which	which	PRON
ejpam-4387	46	14	implies	imply	VERB
ejpam-4387	46	15	that	that	SCONJ
ejpam-4387	46	16	y	y	PROPN
ejpam-4387	46	17	is	be	AUX
ejpam-4387	46	18	t1	t1	PROPN
ejpam-4387	46	19	.	.	PUNCT
ejpam-4387	47	1	corollary	corollary	ADJ
ejpam-4387	47	2	1	1	NUM
ejpam-4387	47	3	.	.	PUNCT
ejpam-4387	48	1	if	if	SCONJ
ejpam-4387	48	2	x	x	PRON
ejpam-4387	48	3	is	be	AUX
ejpam-4387	48	4	a	a	DET
ejpam-4387	48	5	t1	t1	NOUN
ejpam-4387	48	6	,	,	PUNCT
ejpam-4387	48	7	p	p	NOUN
ejpam-4387	48	8	-normal	-normal	ADJ
ejpam-4387	48	9	space	space	NOUN
ejpam-4387	48	10	then	then	ADV
ejpam-4387	48	11	the	the	DET
ejpam-4387	48	12	witness	witness	NOUN
ejpam-4387	48	13	y	y	PROPN
ejpam-4387	48	14	is	be	AUX
ejpam-4387	48	15	t4	t4	PROPN
ejpam-4387	48	16	.	.	PUNCT
ejpam-4387	49	1	using	use	VERB
ejpam-4387	49	2	the	the	DET
ejpam-4387	49	3	previous	previous	ADJ
ejpam-4387	49	4	proposition	proposition	NOUN
ejpam-4387	49	5	we	we	PRON
ejpam-4387	49	6	can	can	AUX
ejpam-4387	49	7	state	state	VERB
ejpam-4387	49	8	the	the	DET
ejpam-4387	49	9	following	follow	VERB
ejpam-4387	49	10	theorems	theorem	NOUN
ejpam-4387	49	11	:	:	PUNCT
ejpam-4387	49	12	corollary	corollary	ADJ
ejpam-4387	49	13	2	2	NUM
ejpam-4387	49	14	.	.	PUNCT
ejpam-4387	50	1	if	if	SCONJ
ejpam-4387	50	2	x	x	PRON
ejpam-4387	50	3	is	be	AUX
ejpam-4387	50	4	t1	t1	NOUN
ejpam-4387	50	5	and	and	CCONJ
ejpam-4387	50	6	the	the	DET
ejpam-4387	50	7	only	only	ADJ
ejpam-4387	50	8	paracompact	paracompact	ADJ
ejpam-4387	50	9	subspaces	subspace	NOUN
ejpam-4387	50	10	of	of	ADP
ejpam-4387	50	11	x	x	SYM
ejpam-4387	50	12	are	be	AUX
ejpam-4387	50	13	the	the	DET
ejpam-4387	50	14	finite	finite	ADJ
ejpam-4387	50	15	subspaces	subspace	NOUN
ejpam-4387	50	16	,	,	PUNCT
ejpam-4387	50	17	then	then	ADV
ejpam-4387	50	18	x	x	PUNCT
ejpam-4387	50	19	is	be	AUX
ejpam-4387	50	20	p	p	NOUN
ejpam-4387	50	21	-normal	-normal	NOUN
ejpam-4387	50	22	.	.	PUNCT
ejpam-4387	51	1	theorem	theorem	NOUN
ejpam-4387	51	2	1	1	NUM
ejpam-4387	51	3	.	.	PUNCT
ejpam-4387	52	1	let	let	VERB
ejpam-4387	52	2	x	x	PRON
ejpam-4387	52	3	be	be	AUX
ejpam-4387	52	4	a	a	DET
ejpam-4387	52	5	t1	t1	NOUN
ejpam-4387	52	6	,	,	PUNCT
ejpam-4387	52	7	fréchet	fréchet	NOUN
ejpam-4387	52	8	p	p	ADJ
ejpam-4387	52	9	-normal	-normal	ADJ
ejpam-4387	52	10	space	space	NOUN
ejpam-4387	52	11	.	.	PUNCT
ejpam-4387	53	1	then	then	ADV
ejpam-4387	53	2	,	,	PUNCT
ejpam-4387	53	3	any	any	DET
ejpam-4387	53	4	compact	compact	ADJ
ejpam-4387	53	5	subspace	subspace	NOUN
ejpam-4387	53	6	of	of	ADP
ejpam-4387	53	7	x	x	SYM
ejpam-4387	53	8	is	be	AUX
ejpam-4387	53	9	p	p	NOUN
ejpam-4387	53	10	-normal	-normal	NOUN
ejpam-4387	53	11	.	.	PUNCT
ejpam-4387	54	1	proof	proof	NOUN
ejpam-4387	54	2	.	.	PUNCT
ejpam-4387	55	1	let	let	VERB
ejpam-4387	55	2	y	y	PRON
ejpam-4387	55	3	and	and	CCONJ
ejpam-4387	55	4	f	f	PROPN
ejpam-4387	55	5	be	be	AUX
ejpam-4387	55	6	a	a	DET
ejpam-4387	55	7	witness	witness	NOUN
ejpam-4387	55	8	space	space	NOUN
ejpam-4387	55	9	and	and	CCONJ
ejpam-4387	55	10	function	function	NOUN
ejpam-4387	55	11	respectively	respectively	ADV
ejpam-4387	55	12	of	of	ADP
ejpam-4387	55	13	the	the	DET
ejpam-4387	55	14	p	p	NOUN
ejpam-4387	55	15	-normality	-normality	NOUN
ejpam-4387	55	16	of	of	ADP
ejpam-4387	55	17	x.	x.	NOUN
ejpam-4387	55	18	since	since	SCONJ
ejpam-4387	55	19	x	x	PROPN
ejpam-4387	55	20	is	be	AUX
ejpam-4387	55	21	t1	t1	NOUN
ejpam-4387	55	22	,	,	PUNCT
ejpam-4387	55	23	then	then	ADV
ejpam-4387	55	24	y	y	PROPN
ejpam-4387	55	25	is	be	AUX
ejpam-4387	55	26	t4	t4	PROPN
ejpam-4387	55	27	by	by	ADP
ejpam-4387	55	28	the	the	DET
ejpam-4387	55	29	above	above	ADJ
ejpam-4387	55	30	corollary	corollary	NOUN
ejpam-4387	55	31	.	.	PUNCT
ejpam-4387	56	1	now	now	ADV
ejpam-4387	56	2	,	,	PUNCT
ejpam-4387	56	3	f	f	PROPN
ejpam-4387	56	4	is	be	AUX
ejpam-4387	56	5	continuous	continuous	ADJ
ejpam-4387	56	6	since	since	SCONJ
ejpam-4387	56	7	x	x	SYM
ejpam-4387	56	8	is	be	AUX
ejpam-4387	56	9	fréchet	fréchet	VERB
ejpam-4387	56	10	by	by	ADP
ejpam-4387	56	11	[	[	X
ejpam-4387	56	12	10	10	NUM
ejpam-4387	56	13	,	,	PUNCT
ejpam-4387	56	14	theorem	theorem	VERB
ejpam-4387	56	15	5	5	NUM
ejpam-4387	56	16	]	]	PUNCT
ejpam-4387	56	17	.	.	PUNCT
ejpam-4387	57	1	let	let	VERB
ejpam-4387	57	2	a	a	DET
ejpam-4387	57	3	⊆	⊆	NUM
ejpam-4387	57	4	x	x	SYM
ejpam-4387	57	5	be	be	AUX
ejpam-4387	57	6	any	any	DET
ejpam-4387	57	7	compact	compact	ADJ
ejpam-4387	57	8	subset	subset	NOUN
ejpam-4387	57	9	of	of	ADP
ejpam-4387	57	10	x.	x.	NOUN
ejpam-4387	57	11	the	the	DET
ejpam-4387	57	12	continuous	continuous	ADJ
ejpam-4387	57	13	image	image	NOUN
ejpam-4387	57	14	of	of	ADP
ejpam-4387	57	15	a	a	DET
ejpam-4387	57	16	compact	compact	ADJ
ejpam-4387	57	17	subset	subset	NOUN
ejpam-4387	57	18	is	be	AUX
ejpam-4387	57	19	compact	compact	ADJ
ejpam-4387	57	20	so	so	SCONJ
ejpam-4387	57	21	f(a	f(a	NOUN
ejpam-4387	57	22	)	)	PUNCT
ejpam-4387	58	1	⊆	⊆	NUM
ejpam-4387	58	2	y	y	PROPN
ejpam-4387	58	3	is	be	AUX
ejpam-4387	58	4	compact	compact	ADJ
ejpam-4387	58	5	in	in	ADP
ejpam-4387	58	6	y	y	PROPN
ejpam-4387	58	7	.	.	PUNCT
ejpam-4387	59	1	moreover	moreover	ADV
ejpam-4387	59	2	,	,	PUNCT
ejpam-4387	59	3	since	since	SCONJ
ejpam-4387	59	4	y	y	PROPN
ejpam-4387	59	5	is	be	AUX
ejpam-4387	59	6	t2	t2	PROPN
ejpam-4387	59	7	that	that	PRON
ejpam-4387	59	8	means	mean	VERB
ejpam-4387	59	9	f(a	f(a	PROPN
ejpam-4387	59	10	)	)	PUNCT
ejpam-4387	59	11	is	be	AUX
ejpam-4387	59	12	closed	close	VERB
ejpam-4387	59	13	in	in	ADP
ejpam-4387	59	14	y	y	PROPN
ejpam-4387	59	15	.	.	PUNCT
ejpam-4387	60	1	y	y	PROPN
ejpam-4387	60	2	is	be	AUX
ejpam-4387	60	3	normal	normal	ADJ
ejpam-4387	60	4	and	and	CCONJ
ejpam-4387	60	5	normality	normality	NOUN
ejpam-4387	60	6	is	be	AUX
ejpam-4387	60	7	hereditary	hereditary	ADJ
ejpam-4387	60	8	with	with	ADP
ejpam-4387	60	9	respect	respect	NOUN
ejpam-4387	60	10	l.	l.	PROPN
ejpam-4387	60	11	kalantan	kalantan	PROPN
ejpam-4387	60	12	,	,	PUNCT
ejpam-4387	60	13	m.	m.	NOUN
ejpam-4387	60	14	mansouri	mansouri	PROPN
ejpam-4387	60	15	/	/	SYM
ejpam-4387	60	16	eur	eur	PROPN
ejpam-4387	60	17	.	.	PUNCT
ejpam-4387	61	1	j.	j.	PROPN
ejpam-4387	61	2	pure	pure	PROPN
ejpam-4387	61	3	appl	appl	PROPN
ejpam-4387	61	4	.	.	PROPN
ejpam-4387	61	5	math	math	PROPN
ejpam-4387	61	6	,	,	PUNCT
ejpam-4387	61	7	15	15	NUM
ejpam-4387	61	8	(	(	PUNCT
ejpam-4387	61	9	2	2	NUM
ejpam-4387	61	10	)	)	PUNCT
ejpam-4387	61	11	(	(	PUNCT
ejpam-4387	61	12	2022	2022	NUM
ejpam-4387	61	13	)	)	PUNCT
ejpam-4387	61	14	,	,	PUNCT
ejpam-4387	61	15	774	774	NUM
ejpam-4387	61	16	-	-	SYM
ejpam-4387	61	17	783	783	NUM
ejpam-4387	61	18	776	776	NUM
ejpam-4387	61	19	to	to	ADP
ejpam-4387	61	20	closed	close	VERB
ejpam-4387	61	21	sets	set	NOUN
ejpam-4387	61	22	so	so	SCONJ
ejpam-4387	61	23	f(a	f(a	NOUN
ejpam-4387	61	24	)	)	PUNCT
ejpam-4387	61	25	is	be	AUX
ejpam-4387	61	26	normal	normal	ADJ
ejpam-4387	61	27	and	and	CCONJ
ejpam-4387	61	28	hence	hence	ADV
ejpam-4387	61	29	will	will	AUX
ejpam-4387	61	30	be	be	AUX
ejpam-4387	61	31	a	a	DET
ejpam-4387	61	32	witness	witness	NOUN
ejpam-4387	61	33	for	for	ADP
ejpam-4387	61	34	the	the	DET
ejpam-4387	61	35	p	p	NOUN
ejpam-4387	61	36	-normality	-normality	NOUN
ejpam-4387	61	37	of	of	ADP
ejpam-4387	61	38	a.	a.	NOUN
ejpam-4387	61	39	let	let	VERB
ejpam-4387	61	40	g	g	PROPN
ejpam-4387	61	41	=	=	SYM
ejpam-4387	61	42	f	f	PROPN
ejpam-4387	61	43	|a	|a	VERB
ejpam-4387	61	44	:	:	PUNCT
ejpam-4387	61	45	a	a	DET
ejpam-4387	61	46	−→	−→	NOUN
ejpam-4387	61	47	f(a	f(a	NOUN
ejpam-4387	61	48	)	)	PUNCT
ejpam-4387	61	49	.	.	PUNCT
ejpam-4387	62	1	let	let	VERB
ejpam-4387	62	2	c	c	NOUN
ejpam-4387	62	3	⊆	⊆	NUM
ejpam-4387	62	4	a	a	DET
ejpam-4387	62	5	be	be	AUX
ejpam-4387	62	6	any	any	DET
ejpam-4387	62	7	paracompact	paracompact	ADJ
ejpam-4387	62	8	subspace	subspace	NOUN
ejpam-4387	62	9	of	of	ADP
ejpam-4387	62	10	a.	a.	NOUN
ejpam-4387	62	11	since	since	SCONJ
ejpam-4387	62	12	a	a	DET
ejpam-4387	62	13	subspace	subspace	NOUN
ejpam-4387	62	14	of	of	ADP
ejpam-4387	62	15	a	a	DET
ejpam-4387	62	16	subspace	subspace	NOUN
ejpam-4387	62	17	is	be	AUX
ejpam-4387	62	18	a	a	DET
ejpam-4387	62	19	subspace	subspace	NOUN
ejpam-4387	62	20	,	,	PUNCT
ejpam-4387	62	21	then	then	ADV
ejpam-4387	62	22	c	c	PROPN
ejpam-4387	62	23	is	be	AUX
ejpam-4387	62	24	paracompact	paracompact	ADJ
ejpam-4387	62	25	in	in	ADP
ejpam-4387	62	26	x.	x.	NOUN
ejpam-4387	62	27	therefore	therefore	ADV
ejpam-4387	62	28	,	,	PUNCT
ejpam-4387	62	29	g|c	g|c	PUNCT
ejpam-4387	63	1	=	=	PUNCT
ejpam-4387	63	2	f	f	X
ejpam-4387	63	3	|a|c	|a|c	PROPN
ejpam-4387	63	4	=	=	SYM
ejpam-4387	63	5	f	f	PROPN
ejpam-4387	63	6	|c	|c	X
ejpam-4387	63	7	is	be	AUX
ejpam-4387	63	8	a	a	DET
ejpam-4387	63	9	homeomorphism	homeomorphism	NOUN
ejpam-4387	63	10	.	.	PUNCT
ejpam-4387	64	1	which	which	PRON
ejpam-4387	64	2	means	mean	VERB
ejpam-4387	64	3	a	a	DET
ejpam-4387	64	4	,	,	PUNCT
ejpam-4387	64	5	the	the	DET
ejpam-4387	64	6	arbitrary	arbitrary	ADJ
ejpam-4387	64	7	compact	compact	ADJ
ejpam-4387	64	8	subset	subset	NOUN
ejpam-4387	64	9	of	of	ADP
ejpam-4387	64	10	x	x	X
ejpam-4387	64	11	,	,	PUNCT
ejpam-4387	64	12	is	be	AUX
ejpam-4387	64	13	p	p	NOUN
ejpam-4387	64	14	-normal	-normal	ADJ
ejpam-4387	64	15	and	and	CCONJ
ejpam-4387	64	16	we	we	PRON
ejpam-4387	64	17	are	be	AUX
ejpam-4387	64	18	done	do	VERB
ejpam-4387	64	19	.	.	PUNCT
ejpam-4387	65	1	in	in	ADP
ejpam-4387	65	2	the	the	DET
ejpam-4387	65	3	same	same	ADJ
ejpam-4387	65	4	way	way	NOUN
ejpam-4387	65	5	we	we	PRON
ejpam-4387	65	6	proved	prove	VERB
ejpam-4387	65	7	the	the	DET
ejpam-4387	65	8	previous	previous	ADJ
ejpam-4387	65	9	theorem	theorem	NOUN
ejpam-4387	65	10	,	,	PUNCT
ejpam-4387	65	11	we	we	PRON
ejpam-4387	65	12	can	can	AUX
ejpam-4387	65	13	deduce	deduce	VERB
ejpam-4387	65	14	the	the	DET
ejpam-4387	65	15	following	follow	VERB
ejpam-4387	65	16	corollary	corollary	NOUN
ejpam-4387	65	17	about	about	ADP
ejpam-4387	65	18	p	p	X
ejpam-4387	65	19	-normality	-normality	PROPN
ejpam-4387	65	20	being	be	AUX
ejpam-4387	65	21	hereditary	hereditary	ADJ
ejpam-4387	65	22	with	with	ADP
ejpam-4387	65	23	respect	respect	NOUN
ejpam-4387	65	24	to	to	ADP
ejpam-4387	65	25	countably	countably	ADV
ejpam-4387	65	26	compact	compact	ADJ
ejpam-4387	65	27	subspaces	subspace	NOUN
ejpam-4387	65	28	with	with	ADP
ejpam-4387	65	29	additional	additional	ADJ
ejpam-4387	65	30	conditions	condition	NOUN
ejpam-4387	65	31	.	.	PUNCT
ejpam-4387	66	1	recall	recall	VERB
ejpam-4387	66	2	that	that	SCONJ
ejpam-4387	66	3	a	a	DET
ejpam-4387	66	4	c	c	NOUN
ejpam-4387	66	5	-	-	PUNCT
ejpam-4387	66	6	closed	close	VERB
ejpam-4387	66	7	space	space	NOUN
ejpam-4387	66	8	y	y	PROPN
ejpam-4387	66	9	is	be	AUX
ejpam-4387	66	10	a	a	DET
ejpam-4387	66	11	t2	t2	NOUN
ejpam-4387	66	12	space	space	NOUN
ejpam-4387	66	13	where	where	SCONJ
ejpam-4387	66	14	every	every	DET
ejpam-4387	66	15	countably	countably	ADV
ejpam-4387	66	16	compact	compact	ADJ
ejpam-4387	66	17	subset	subset	VERB
ejpam-4387	66	18	a	a	DET
ejpam-4387	66	19	⊆	⊆	NUM
ejpam-4387	66	20	y	y	NOUN
ejpam-4387	66	21	is	be	AUX
ejpam-4387	66	22	closed	close	VERB
ejpam-4387	66	23	[	[	PUNCT
ejpam-4387	66	24	9	9	NUM
ejpam-4387	66	25	]	]	PUNCT
ejpam-4387	66	26	.	.	PUNCT
ejpam-4387	67	1	corollary	corollary	ADJ
ejpam-4387	67	2	3	3	X
ejpam-4387	67	3	.	.	PUNCT
ejpam-4387	68	1	let	let	VERB
ejpam-4387	68	2	x	x	PRON
ejpam-4387	68	3	be	be	AUX
ejpam-4387	68	4	a	a	DET
ejpam-4387	68	5	t1	t1	NOUN
ejpam-4387	68	6	,	,	PUNCT
ejpam-4387	68	7	fréchet	fréchet	NOUN
ejpam-4387	68	8	p	p	ADJ
ejpam-4387	68	9	-normal	-normal	ADJ
ejpam-4387	68	10	space	space	NOUN
ejpam-4387	68	11	.	.	PUNCT
ejpam-4387	69	1	let	let	VERB
ejpam-4387	69	2	y	y	PRON
ejpam-4387	69	3	a	a	DET
ejpam-4387	69	4	witness	witness	NOUN
ejpam-4387	69	5	of	of	ADP
ejpam-4387	69	6	the	the	DET
ejpam-4387	69	7	p	p	NOUN
ejpam-4387	69	8	-normality	-normality	NOUN
ejpam-4387	69	9	of	of	ADP
ejpam-4387	69	10	x	x	PART
ejpam-4387	69	11	be	be	AUX
ejpam-4387	69	12	a	a	DET
ejpam-4387	69	13	c	c	NOUN
ejpam-4387	69	14	-	-	PUNCT
ejpam-4387	69	15	closed	closed	ADJ
ejpam-4387	69	16	space	space	NOUN
ejpam-4387	69	17	.	.	PUNCT
ejpam-4387	70	1	then	then	ADV
ejpam-4387	70	2	,	,	PUNCT
ejpam-4387	70	3	any	any	DET
ejpam-4387	70	4	countably	countably	ADV
ejpam-4387	70	5	compact	compact	ADJ
ejpam-4387	70	6	subspace	subspace	NOUN
ejpam-4387	70	7	of	of	ADP
ejpam-4387	70	8	x	x	SYM
ejpam-4387	70	9	is	be	AUX
ejpam-4387	70	10	p	p	NOUN
ejpam-4387	70	11	-normal	-normal	NOUN
ejpam-4387	70	12	.	.	PUNCT
ejpam-4387	71	1	recall	recall	VERB
ejpam-4387	71	2	that	that	SCONJ
ejpam-4387	71	3	a	a	DET
ejpam-4387	71	4	topological	topological	ADJ
ejpam-4387	71	5	space	space	NOUN
ejpam-4387	71	6	(	(	PUNCT
ejpam-4387	71	7	x	x	X
ejpam-4387	71	8	,	,	PUNCT
ejpam-4387	71	9	τ	τ	PROPN
ejpam-4387	71	10	)	)	PUNCT
ejpam-4387	71	11	is	be	AUX
ejpam-4387	71	12	called	call	VERB
ejpam-4387	71	13	epinormal	epinormal	NOUN
ejpam-4387	71	14	if	if	SCONJ
ejpam-4387	71	15	there	there	PRON
ejpam-4387	71	16	is	be	VERB
ejpam-4387	71	17	a	a	DET
ejpam-4387	71	18	coarser	coarse	ADJ
ejpam-4387	71	19	topology	topology	NOUN
ejpam-4387	71	20	τ	τ	NOUN
ejpam-4387	71	21	′	′	NOUN
ejpam-4387	71	22	on	on	ADP
ejpam-4387	71	23	x	x	INTJ
ejpam-4387	71	24	such	such	ADJ
ejpam-4387	71	25	that	that	SCONJ
ejpam-4387	71	26	(	(	PUNCT
ejpam-4387	71	27	x	x	X
ejpam-4387	71	28	,	,	PUNCT
ejpam-4387	71	29	τ	τ	PROPN
ejpam-4387	71	30	′	′	NUM
ejpam-4387	71	31	)	)	PUNCT
ejpam-4387	71	32	is	be	AUX
ejpam-4387	71	33	t4	t4	PROPN
ejpam-4387	71	34	[	[	X
ejpam-4387	71	35	3	3	NUM
ejpam-4387	71	36	]	]	PUNCT
ejpam-4387	71	37	.	.	PUNCT
ejpam-4387	72	1	theorem	theorem	NOUN
ejpam-4387	72	2	2	2	NUM
ejpam-4387	72	3	.	.	PUNCT
ejpam-4387	73	1	let	let	VERB
ejpam-4387	73	2	x	x	PRON
ejpam-4387	73	3	be	be	AUX
ejpam-4387	73	4	a	a	DET
ejpam-4387	73	5	t1	t1	NOUN
ejpam-4387	73	6	,	,	PUNCT
ejpam-4387	73	7	fréchet	fréchet	NOUN
ejpam-4387	73	8	p	p	ADJ
ejpam-4387	73	9	-normal	-normal	ADJ
ejpam-4387	73	10	space	space	NOUN
ejpam-4387	73	11	,	,	PUNCT
ejpam-4387	73	12	then	then	ADV
ejpam-4387	73	13	x	x	PUNCT
ejpam-4387	73	14	is	be	AUX
ejpam-4387	73	15	epinormal	epinormal	ADJ
ejpam-4387	73	16	.	.	PUNCT
ejpam-4387	74	1	by	by	ADP
ejpam-4387	74	2	the	the	DET
ejpam-4387	74	3	above	above	ADJ
ejpam-4387	74	4	discussion	discussion	NOUN
ejpam-4387	74	5	we	we	PRON
ejpam-4387	74	6	have	have	AUX
ejpam-4387	74	7	seen	see	VERB
ejpam-4387	74	8	that	that	SCONJ
ejpam-4387	74	9	if	if	SCONJ
ejpam-4387	74	10	x	x	PRON
ejpam-4387	74	11	is	be	AUX
ejpam-4387	74	12	t1	t1	NOUN
ejpam-4387	74	13	and	and	CCONJ
ejpam-4387	74	14	p	p	NOUN
ejpam-4387	74	15	-normal	-normal	NOUN
ejpam-4387	74	16	then	then	ADV
ejpam-4387	74	17	y	y	PROPN
ejpam-4387	74	18	,	,	PUNCT
ejpam-4387	74	19	the	the	DET
ejpam-4387	74	20	witness	witness	NOUN
ejpam-4387	74	21	of	of	ADP
ejpam-4387	74	22	p	p	PROPN
ejpam-4387	74	23	-normality	-normality	NOUN
ejpam-4387	74	24	,	,	PUNCT
ejpam-4387	74	25	is	be	AUX
ejpam-4387	74	26	t4	t4	PROPN
ejpam-4387	74	27	.	.	PUNCT
ejpam-4387	75	1	now	now	ADV
ejpam-4387	75	2	,	,	PUNCT
ejpam-4387	75	3	since	since	SCONJ
ejpam-4387	75	4	x	x	PRON
ejpam-4387	75	5	is	be	AUX
ejpam-4387	75	6	fréchet	fréchet	VERB
ejpam-4387	75	7	that	that	PRON
ejpam-4387	75	8	means	mean	VERB
ejpam-4387	75	9	the	the	DET
ejpam-4387	75	10	witness	witness	NOUN
ejpam-4387	75	11	function	function	NOUN
ejpam-4387	75	12	f	f	PROPN
ejpam-4387	75	13	is	be	AUX
ejpam-4387	75	14	continuous	continuous	ADJ
ejpam-4387	75	15	by	by	ADP
ejpam-4387	75	16	[	[	X
ejpam-4387	75	17	10	10	NUM
ejpam-4387	75	18	,	,	PUNCT
ejpam-4387	75	19	theorem	theorem	VERB
ejpam-4387	75	20	5	5	NUM
ejpam-4387	75	21	]	]	PUNCT
ejpam-4387	75	22	.	.	PUNCT
ejpam-4387	76	1	this	this	PRON
ejpam-4387	76	2	allows	allow	VERB
ejpam-4387	76	3	us	we	PRON
ejpam-4387	76	4	to	to	PART
ejpam-4387	76	5	consider	consider	VERB
ejpam-4387	76	6	y	y	PRON
ejpam-4387	76	7	as	as	ADP
ejpam-4387	76	8	a	a	DET
ejpam-4387	76	9	coarser	coarse	ADJ
ejpam-4387	76	10	space	space	NOUN
ejpam-4387	76	11	of	of	ADP
ejpam-4387	76	12	x	x	PUNCT
ejpam-4387	77	1	[	[	X
ejpam-4387	77	2	7	7	NUM
ejpam-4387	77	3	,	,	PUNCT
ejpam-4387	77	4	2.4	2.4	NUM
ejpam-4387	77	5	]	]	PUNCT
ejpam-4387	77	6	.	.	PUNCT
ejpam-4387	78	1	note	note	VERB
ejpam-4387	78	2	that	that	SCONJ
ejpam-4387	78	3	since	since	SCONJ
ejpam-4387	78	4	in	in	ADP
ejpam-4387	78	5	this	this	DET
ejpam-4387	78	6	case	case	NOUN
ejpam-4387	78	7	y	y	NOUN
ejpam-4387	78	8	is	be	AUX
ejpam-4387	78	9	coarser	coarse	ADJ
ejpam-4387	78	10	than	than	SCONJ
ejpam-4387	78	11	x	x	PRON
ejpam-4387	78	12	and	and	CCONJ
ejpam-4387	78	13	y	y	PROPN
ejpam-4387	78	14	is	be	AUX
ejpam-4387	78	15	t4	t4	PROPN
ejpam-4387	78	16	hence	hence	ADV
ejpam-4387	78	17	t2	t2	PROPN
ejpam-4387	78	18	then	then	ADV
ejpam-4387	78	19	x	x	PUNCT
ejpam-4387	78	20	has	have	VERB
ejpam-4387	78	21	to	to	PART
ejpam-4387	78	22	be	be	AUX
ejpam-4387	78	23	t2	t2	NOUN
ejpam-4387	78	24	.	.	PUNCT
ejpam-4387	79	1	which	which	PRON
ejpam-4387	79	2	means	mean	VERB
ejpam-4387	79	3	,	,	PUNCT
ejpam-4387	79	4	if	if	SCONJ
ejpam-4387	79	5	a	a	DET
ejpam-4387	79	6	space	space	NOUN
ejpam-4387	79	7	x	x	PUNCT
ejpam-4387	79	8	is	be	AUX
ejpam-4387	79	9	not	not	PART
ejpam-4387	79	10	t2	t2	NOUN
ejpam-4387	79	11	,	,	PUNCT
ejpam-4387	79	12	but	but	CCONJ
ejpam-4387	79	13	t1	t1	NOUN
ejpam-4387	79	14	and	and	CCONJ
ejpam-4387	79	15	fréchet	fréchet	NOUN
ejpam-4387	79	16	then	then	ADV
ejpam-4387	79	17	it	it	PRON
ejpam-4387	79	18	can	can	AUX
ejpam-4387	79	19	not	not	PART
ejpam-4387	79	20	be	be	AUX
ejpam-4387	79	21	p	p	NOUN
ejpam-4387	79	22	-normal	-normal	NOUN
ejpam-4387	79	23	.	.	PUNCT
ejpam-4387	80	1	3	3	X
ejpam-4387	80	2	.	.	X
ejpam-4387	80	3	invariance	invariance	NOUN
ejpam-4387	80	4	we	we	PRON
ejpam-4387	80	5	begin	begin	VERB
ejpam-4387	80	6	by	by	ADP
ejpam-4387	80	7	studying	study	VERB
ejpam-4387	80	8	the	the	DET
ejpam-4387	80	9	invariant	invariant	ADJ
ejpam-4387	80	10	properties	property	NOUN
ejpam-4387	80	11	of	of	ADP
ejpam-4387	80	12	p	p	NOUN
ejpam-4387	80	13	-normality	-normality	NOUN
ejpam-4387	80	14	.	.	PUNCT
ejpam-4387	81	1	p	p	PRON
ejpam-4387	81	2	-normality	-normality	PROPN
ejpam-4387	81	3	is	be	AUX
ejpam-4387	81	4	not	not	PART
ejpam-4387	81	5	invariant	invariant	ADJ
ejpam-4387	81	6	in	in	ADP
ejpam-4387	81	7	general	general	ADJ
ejpam-4387	81	8	.	.	PUNCT
ejpam-4387	81	9	example	example	NOUN
ejpam-4387	82	1	1	1	NUM
ejpam-4387	82	2	.	.	PUNCT
ejpam-4387	83	1	in	in	ADP
ejpam-4387	83	2	[	[	X
ejpam-4387	83	3	10	10	NUM
ejpam-4387	83	4	]	]	PUNCT
ejpam-4387	83	5	we	we	PRON
ejpam-4387	83	6	showed	show	VERB
ejpam-4387	83	7	that	that	SCONJ
ejpam-4387	83	8	the	the	DET
ejpam-4387	83	9	dieudonné	dieudonné	NOUN
ejpam-4387	83	10	plank	plank	NOUN
ejpam-4387	83	11	(	(	PUNCT
ejpam-4387	83	12	x	x	X
ejpam-4387	83	13	,	,	PUNCT
ejpam-4387	83	14	τ	τ	PROPN
ejpam-4387	83	15	)	)	PUNCT
ejpam-4387	83	16	is	be	AUX
ejpam-4387	83	17	not	not	PART
ejpam-4387	83	18	p	p	NOUN
ejpam-4387	83	19	-normal	-normal	NOUN
ejpam-4387	83	20	.	.	PUNCT
ejpam-4387	84	1	now	now	ADV
ejpam-4387	84	2	,	,	PUNCT
ejpam-4387	84	3	consider	consider	VERB
ejpam-4387	84	4	(	(	PUNCT
ejpam-4387	84	5	x	x	NOUN
ejpam-4387	84	6	,	,	PUNCT
ejpam-4387	84	7	τ	τ	PROPN
ejpam-4387	84	8	′	′	NUM
ejpam-4387	84	9	)	)	PUNCT
ejpam-4387	84	10	,	,	PUNCT
ejpam-4387	84	11	where	where	SCONJ
ejpam-4387	84	12	τ	τ	PROPN
ejpam-4387	84	13	′	′	NOUN
ejpam-4387	84	14	is	be	AUX
ejpam-4387	84	15	generated	generate	VERB
ejpam-4387	84	16	by	by	ADP
ejpam-4387	84	17	making	make	VERB
ejpam-4387	84	18	any	any	DET
ejpam-4387	84	19	element	element	NOUN
ejpam-4387	84	20	on	on	ADP
ejpam-4387	84	21	the	the	DET
ejpam-4387	84	22	right	right	ADJ
ejpam-4387	84	23	side	side	NOUN
ejpam-4387	84	24	of	of	ADP
ejpam-4387	84	25	the	the	DET
ejpam-4387	84	26	plank	plank	NOUN
ejpam-4387	84	27	a	a	DET
ejpam-4387	84	28	isolated	isolate	VERB
ejpam-4387	84	29	.	.	PUNCT
ejpam-4387	85	1	consider	consider	VERB
ejpam-4387	85	2	idx	idx	NOUN
ejpam-4387	85	3	:	:	PUNCT
ejpam-4387	85	4	(	(	PUNCT
ejpam-4387	85	5	x	x	X
ejpam-4387	85	6	,	,	PUNCT
ejpam-4387	85	7	τ	τ	PROPN
ejpam-4387	85	8	′	′	NOUN
ejpam-4387	85	9	)	)	PUNCT
ejpam-4387	85	10	−→	−→	NOUN
ejpam-4387	85	11	(	(	PUNCT
ejpam-4387	85	12	x	x	X
ejpam-4387	85	13	,	,	PUNCT
ejpam-4387	85	14	τ	τ	PROPN
ejpam-4387	85	15	)	)	PUNCT
ejpam-4387	85	16	.	.	PUNCT
ejpam-4387	86	1	since	since	SCONJ
ejpam-4387	86	2	τ	τ	PROPN
ejpam-4387	86	3	is	be	AUX
ejpam-4387	86	4	coarser	coarse	ADJ
ejpam-4387	86	5	than	than	SCONJ
ejpam-4387	86	6	τ	τ	PROPN
ejpam-4387	86	7	′	′	NUM
ejpam-4387	86	8	then	then	ADV
ejpam-4387	86	9	idx	idx	VERB
ejpam-4387	86	10	:	:	PUNCT
ejpam-4387	86	11	(	(	PUNCT
ejpam-4387	86	12	x	x	X
ejpam-4387	86	13	,	,	PUNCT
ejpam-4387	86	14	τ	τ	PROPN
ejpam-4387	86	15	′	′	NOUN
ejpam-4387	86	16	)	)	PUNCT
ejpam-4387	86	17	−→	−→	NOUN
ejpam-4387	86	18	(	(	PUNCT
ejpam-4387	86	19	x	x	X
ejpam-4387	86	20	,	,	PUNCT
ejpam-4387	86	21	τ	τ	PROPN
ejpam-4387	86	22	)	)	PUNCT
ejpam-4387	86	23	is	be	AUX
ejpam-4387	86	24	continious	continious	ADJ
ejpam-4387	86	25	,	,	PUNCT
ejpam-4387	86	26	one	one	NUM
ejpam-4387	86	27	to	to	ADP
ejpam-4387	86	28	one	one	NUM
ejpam-4387	86	29	and	and	CCONJ
ejpam-4387	86	30	onto	onto	ADP
ejpam-4387	86	31	.	.	PUNCT
ejpam-4387	87	1	(	(	PUNCT
ejpam-4387	87	2	x	x	X
ejpam-4387	87	3	,	,	PUNCT
ejpam-4387	87	4	τ	τ	PROPN
ejpam-4387	87	5	′	′	NUM
ejpam-4387	87	6	)	)	PUNCT
ejpam-4387	87	7	is	be	AUX
ejpam-4387	87	8	p	p	NOUN
ejpam-4387	87	9	-normal	-normal	ADJ
ejpam-4387	87	10	being	be	AUX
ejpam-4387	87	11	t2	t2	NOUN
ejpam-4387	87	12	-	-	PUNCT
ejpam-4387	87	13	paracompact	paracompact	NOUN
ejpam-4387	87	14	i.e	i.e	PRON
ejpam-4387	87	15	normal	normal	ADJ
ejpam-4387	87	16	but	but	CCONJ
ejpam-4387	87	17	the	the	DET
ejpam-4387	87	18	dieudonné	dieudonné	NOUN
ejpam-4387	87	19	plank	plank	NOUN
ejpam-4387	87	20	(	(	PUNCT
ejpam-4387	87	21	x	x	X
ejpam-4387	87	22	,	,	PUNCT
ejpam-4387	87	23	τ	τ	PROPN
ejpam-4387	87	24	)	)	PUNCT
ejpam-4387	87	25	is	be	AUX
ejpam-4387	87	26	not	not	PART
ejpam-4387	87	27	.	.	PUNCT
ejpam-4387	88	1	by	by	ADP
ejpam-4387	88	2	a	a	DET
ejpam-4387	88	3	theorem	theorem	NOUN
ejpam-4387	88	4	of	of	ADP
ejpam-4387	88	5	ponomarev	ponomarev	PROPN
ejpam-4387	88	6	[	[	X
ejpam-4387	88	7	7	7	NUM
ejpam-4387	88	8	,	,	PUNCT
ejpam-4387	88	9	4.2.d	4.2.d	PROPN
ejpam-4387	88	10	]	]	PUNCT
ejpam-4387	88	11	which	which	PRON
ejpam-4387	88	12	says	say	VERB
ejpam-4387	88	13	:	:	PUNCT
ejpam-4387	88	14	“	"	PUNCT
ejpam-4387	88	15	a	a	DET
ejpam-4387	88	16	t0	t0	PROPN
ejpam-4387	88	17	spacex	spacex	PROPN
ejpam-4387	88	18	is	be	AUX
ejpam-4387	88	19	first	first	ADV
ejpam-4387	88	20	countable	countable	ADJ
ejpam-4387	88	21	if	if	SCONJ
ejpam-4387	88	22	and	and	CCONJ
ejpam-4387	88	23	only	only	ADV
ejpam-4387	88	24	if	if	SCONJ
ejpam-4387	88	25	x	x	PRON
ejpam-4387	88	26	is	be	AUX
ejpam-4387	88	27	a	a	DET
ejpam-4387	88	28	continuous	continuous	ADJ
ejpam-4387	88	29	image	image	NOUN
ejpam-4387	88	30	of	of	ADP
ejpam-4387	88	31	a	a	DET
ejpam-4387	88	32	metrizable	metrizable	ADJ
ejpam-4387	88	33	space	space	NOUN
ejpam-4387	88	34	under	under	ADP
ejpam-4387	88	35	an	an	DET
ejpam-4387	88	36	open	open	ADJ
ejpam-4387	88	37	mapping	mapping	NOUN
ejpam-4387	88	38	”	"	PUNCT
ejpam-4387	88	39	.	.	PUNCT
ejpam-4387	89	1	consider	consider	VERB
ejpam-4387	89	2	(	(	PUNCT
ejpam-4387	89	3	r	r	NOUN
ejpam-4387	89	4	,	,	PUNCT
ejpam-4387	89	5	rs	rs	NOUN
ejpam-4387	89	6	)	)	PUNCT
ejpam-4387	89	7	,	,	PUNCT
ejpam-4387	89	8	which	which	PRON
ejpam-4387	89	9	we	we	PRON
ejpam-4387	89	10	have	have	AUX
ejpam-4387	89	11	shown	show	VERB
ejpam-4387	89	12	is	be	AUX
ejpam-4387	89	13	not	not	PART
ejpam-4387	89	14	p	p	NOUN
ejpam-4387	89	15	-normal	-normal	ADJ
ejpam-4387	89	16	in	in	ADP
ejpam-4387	89	17	[	[	X
ejpam-4387	89	18	10	10	NUM
ejpam-4387	89	19	]	]	PUNCT
ejpam-4387	89	20	,	,	PUNCT
ejpam-4387	89	21	but	but	CCONJ
ejpam-4387	89	22	it	it	PRON
ejpam-4387	89	23	is	be	AUX
ejpam-4387	89	24	tychonoff	tychonoff	NOUN
ejpam-4387	89	25	and	and	CCONJ
ejpam-4387	89	26	first	first	ADV
ejpam-4387	89	27	countable	countable	ADJ
ejpam-4387	89	28	.	.	PUNCT
ejpam-4387	90	1	so	so	ADV
ejpam-4387	90	2	there	there	PRON
ejpam-4387	90	3	exists	exist	VERB
ejpam-4387	90	4	a	a	DET
ejpam-4387	90	5	metrizable	metrizable	ADJ
ejpam-4387	90	6	space	space	NOUN
ejpam-4387	90	7	x	x	PUNCT
ejpam-4387	90	8	and	and	CCONJ
ejpam-4387	90	9	an	an	DET
ejpam-4387	90	10	open	open	ADJ
ejpam-4387	90	11	function	function	NOUN
ejpam-4387	90	12	g	g	NOUN
ejpam-4387	90	13	:	:	PUNCT
ejpam-4387	90	14	x	x	PUNCT
ejpam-4387	90	15	−→	−→	NOUN
ejpam-4387	90	16	(	(	PUNCT
ejpam-4387	90	17	r	r	NOUN
ejpam-4387	90	18	,	,	PUNCT
ejpam-4387	90	19	rs	rs	NOUN
ejpam-4387	90	20	)	)	PUNCT
ejpam-4387	90	21	.	.	PUNCT
ejpam-4387	91	1	since	since	SCONJ
ejpam-4387	91	2	any	any	DET
ejpam-4387	91	3	metrizable	metrizable	ADJ
ejpam-4387	91	4	space	space	NOUN
ejpam-4387	91	5	is	be	AUX
ejpam-4387	91	6	p	p	NOUN
ejpam-4387	91	7	-normal	-normal	NOUN
ejpam-4387	91	8	,	,	PUNCT
ejpam-4387	91	9	then	then	ADV
ejpam-4387	91	10	this	this	DET
ejpam-4387	91	11	example	example	NOUN
ejpam-4387	91	12	shows	show	VERB
ejpam-4387	91	13	that	that	SCONJ
ejpam-4387	91	14	p	p	PROPN
ejpam-4387	91	15	-normality	-normality	NOUN
ejpam-4387	91	16	is	be	AUX
ejpam-4387	91	17	not	not	PART
ejpam-4387	91	18	open	open	ADJ
ejpam-4387	91	19	invariant	invariant	ADJ
ejpam-4387	91	20	and	and	CCONJ
ejpam-4387	91	21	hence	hence	ADV
ejpam-4387	91	22	can	can	AUX
ejpam-4387	91	23	not	not	PART
ejpam-4387	91	24	be	be	AUX
ejpam-4387	91	25	quotient	quotient	NOUN
ejpam-4387	91	26	invariant	invariant	ADJ
ejpam-4387	91	27	either	either	ADV
ejpam-4387	91	28	.	.	PUNCT
ejpam-4387	92	1	l.	l.	PROPN
ejpam-4387	92	2	kalantan	kalantan	PROPN
ejpam-4387	92	3	,	,	PUNCT
ejpam-4387	92	4	m.	m.	NOUN
ejpam-4387	92	5	mansouri	mansouri	PROPN
ejpam-4387	92	6	/	/	SYM
ejpam-4387	92	7	eur	eur	PROPN
ejpam-4387	92	8	.	.	PUNCT
ejpam-4387	93	1	j.	j.	PROPN
ejpam-4387	93	2	pure	pure	PROPN
ejpam-4387	93	3	appl	appl	PROPN
ejpam-4387	93	4	.	.	PROPN
ejpam-4387	93	5	math	math	PROPN
ejpam-4387	93	6	,	,	PUNCT
ejpam-4387	93	7	15	15	NUM
ejpam-4387	93	8	(	(	PUNCT
ejpam-4387	93	9	2	2	NUM
ejpam-4387	93	10	)	)	PUNCT
ejpam-4387	93	11	(	(	PUNCT
ejpam-4387	93	12	2022	2022	NUM
ejpam-4387	93	13	)	)	PUNCT
ejpam-4387	93	14	,	,	PUNCT
ejpam-4387	93	15	774	774	NUM
ejpam-4387	93	16	-	-	SYM
ejpam-4387	93	17	783	783	NUM
ejpam-4387	93	18	777	777	NUM
ejpam-4387	93	19	example	example	NOUN
ejpam-4387	93	20	2	2	NUM
ejpam-4387	93	21	.	.	PUNCT
ejpam-4387	94	1	the	the	DET
ejpam-4387	94	2	function	function	NOUN
ejpam-4387	94	3	f	f	NOUN
ejpam-4387	94	4	:	:	PUNCT
ejpam-4387	94	5	(	(	PUNCT
ejpam-4387	94	6	r	r	NOUN
ejpam-4387	94	7	,	,	PUNCT
ejpam-4387	94	8	rs	rs	NOUN
ejpam-4387	94	9	)	)	PUNCT
ejpam-4387	94	10	−→	−→	NOUN
ejpam-4387	94	11	(	(	PUNCT
ejpam-4387	94	12	{	{	PUNCT
ejpam-4387	94	13	0	0	NUM
ejpam-4387	94	14	,	,	PUNCT
ejpam-4387	94	15	1	1	NUM
ejpam-4387	94	16	}	}	PUNCT
ejpam-4387	94	17	,	,	PUNCT
ejpam-4387	94	18	d	d	PROPN
ejpam-4387	94	19	)	)	PUNCT
ejpam-4387	94	20	defined	define	VERB
ejpam-4387	94	21	by	by	ADP
ejpam-4387	94	22	f(x	f(x	PROPN
ejpam-4387	94	23	)	)	PUNCT
ejpam-4387	94	24	=	=	PRON
ejpam-4387	94	25	{	{	PUNCT
ejpam-4387	94	26	0	0	NUM
ejpam-4387	94	27	;	;	PUNCT
ejpam-4387	94	28	if	if	SCONJ
ejpam-4387	94	29	x	x	X
ejpam-4387	94	30	<	<	X
ejpam-4387	94	31	0	0	NUM
ejpam-4387	94	32	1	1	NUM
ejpam-4387	94	33	;	;	PUNCT
ejpam-4387	94	34	if	if	SCONJ
ejpam-4387	94	35	x	x	X
ejpam-4387	94	36	≥	≥	X
ejpam-4387	94	37	0	0	NUM
ejpam-4387	94	38	is	be	AUX
ejpam-4387	94	39	both	both	PRON
ejpam-4387	94	40	closed	closed	ADJ
ejpam-4387	94	41	and	and	CCONJ
ejpam-4387	94	42	open	open	ADJ
ejpam-4387	94	43	.	.	PUNCT
ejpam-4387	95	1	where	where	SCONJ
ejpam-4387	95	2	d	d	NOUN
ejpam-4387	95	3	is	be	AUX
ejpam-4387	95	4	the	the	DET
ejpam-4387	95	5	discrete	discrete	ADJ
ejpam-4387	95	6	topology	topology	NOUN
ejpam-4387	95	7	.	.	PUNCT
ejpam-4387	96	1	now	now	ADV
ejpam-4387	96	2	,	,	PUNCT
ejpam-4387	96	3	as	as	SCONJ
ejpam-4387	96	4	we	we	PRON
ejpam-4387	96	5	previously	previously	ADV
ejpam-4387	96	6	mentioned	mention	VERB
ejpam-4387	96	7	(	(	PUNCT
ejpam-4387	96	8	r	r	NOUN
ejpam-4387	96	9	,	,	PUNCT
ejpam-4387	96	10	rs	rs	NOUN
ejpam-4387	96	11	)	)	PUNCT
ejpam-4387	96	12	is	be	AUX
ejpam-4387	96	13	not	not	PART
ejpam-4387	96	14	p	p	NOUN
ejpam-4387	96	15	-normal	-normal	ADJ
ejpam-4387	96	16	but	but	CCONJ
ejpam-4387	96	17	(	(	PUNCT
ejpam-4387	96	18	{	{	PUNCT
ejpam-4387	96	19	0	0	NUM
ejpam-4387	96	20	,	,	PUNCT
ejpam-4387	96	21	1	1	NUM
ejpam-4387	96	22	}	}	PUNCT
ejpam-4387	96	23	,	,	PUNCT
ejpam-4387	96	24	d	d	PROPN
ejpam-4387	96	25	)	)	PUNCT
ejpam-4387	96	26	is	be	AUX
ejpam-4387	96	27	p	p	NOUN
ejpam-4387	96	28	-normal	-normal	ADJ
ejpam-4387	96	29	since	since	SCONJ
ejpam-4387	96	30	it	it	PRON
ejpam-4387	96	31	is	be	AUX
ejpam-4387	96	32	normal	normal	ADJ
ejpam-4387	96	33	.	.	PUNCT
ejpam-4387	97	1	this	this	DET
ejpam-4387	97	2	example	example	NOUN
ejpam-4387	97	3	shows	show	VERB
ejpam-4387	97	4	that	that	SCONJ
ejpam-4387	97	5	p	p	PROPN
ejpam-4387	97	6	-normality	-normality	NOUN
ejpam-4387	97	7	is	be	AUX
ejpam-4387	97	8	not	not	PART
ejpam-4387	97	9	inverse	inverse	ADJ
ejpam-4387	97	10	invariant	invariant	NOUN
ejpam-4387	97	11	in	in	ADP
ejpam-4387	97	12	general	general	ADJ
ejpam-4387	97	13	.	.	PUNCT
ejpam-4387	98	1	moreover	moreover	ADV
ejpam-4387	98	2	,	,	PUNCT
ejpam-4387	98	3	it	it	PRON
ejpam-4387	98	4	is	be	AUX
ejpam-4387	98	5	not	not	PART
ejpam-4387	98	6	inverse	inverse	ADJ
ejpam-4387	98	7	open	open	ADJ
ejpam-4387	98	8	invariant	invariant	ADJ
ejpam-4387	98	9	nor	nor	CCONJ
ejpam-4387	98	10	inverse	inverse	NOUN
ejpam-4387	98	11	closed	close	VERB
ejpam-4387	98	12	invariant	invariant	ADJ
ejpam-4387	98	13	.	.	PUNCT
ejpam-4387	99	1	4	4	X
ejpam-4387	99	2	.	.	X
ejpam-4387	99	3	generating	generate	VERB
ejpam-4387	99	4	spaces	space	NOUN
ejpam-4387	99	5	and	and	CCONJ
ejpam-4387	99	6	p	p	NOUN
ejpam-4387	99	7	-normality	-normality	NOUN
ejpam-4387	99	8	in	in	ADP
ejpam-4387	99	9	this	this	DET
ejpam-4387	99	10	section	section	NOUN
ejpam-4387	99	11	,	,	PUNCT
ejpam-4387	99	12	we	we	PRON
ejpam-4387	99	13	start	start	VERB
ejpam-4387	99	14	with	with	ADP
ejpam-4387	99	15	the	the	DET
ejpam-4387	99	16	study	study	NOUN
ejpam-4387	99	17	of	of	ADP
ejpam-4387	99	18	the	the	DET
ejpam-4387	99	19	alexandroff	alexandroff	NOUN
ejpam-4387	99	20	duplicate	duplicate	NOUN
ejpam-4387	99	21	space	space	NOUN
ejpam-4387	99	22	of	of	ADP
ejpam-4387	99	23	a	a	DET
ejpam-4387	99	24	p	p	X
ejpam-4387	99	25	normal	normal	ADJ
ejpam-4387	99	26	space	space	NOUN
ejpam-4387	99	27	.	.	PUNCT
ejpam-4387	100	1	let	let	VERB
ejpam-4387	100	2	us	we	PRON
ejpam-4387	100	3	first	first	ADV
ejpam-4387	100	4	recall	recall	VERB
ejpam-4387	100	5	the	the	DET
ejpam-4387	100	6	definition	definition	NOUN
ejpam-4387	100	7	of	of	ADP
ejpam-4387	100	8	the	the	DET
ejpam-4387	100	9	alexandroff	alexandroff	NOUN
ejpam-4387	100	10	duplicate	duplicate	VERB
ejpam-4387	100	11	topological	topological	ADJ
ejpam-4387	100	12	space	space	NOUN
ejpam-4387	100	13	.	.	PUNCT
ejpam-4387	101	1	let	let	VERB
ejpam-4387	101	2	x	x	PRON
ejpam-4387	101	3	be	be	AUX
ejpam-4387	101	4	an	an	DET
ejpam-4387	101	5	infinite	infinite	ADJ
ejpam-4387	101	6	topological	topological	ADJ
ejpam-4387	101	7	space	space	NOUN
ejpam-4387	101	8	.	.	PUNCT
ejpam-4387	102	1	we	we	PRON
ejpam-4387	102	2	denote	denote	VERB
ejpam-4387	102	3	the	the	DET
ejpam-4387	102	4	family	family	NOUN
ejpam-4387	102	5	of	of	ADP
ejpam-4387	102	6	all	all	DET
ejpam-4387	102	7	finite	finite	ADJ
ejpam-4387	102	8	subsets	subset	NOUN
ejpam-4387	102	9	of	of	ADP
ejpam-4387	102	10	x	x	PUNCT
ejpam-4387	102	11	by	by	ADP
ejpam-4387	102	12	[	[	X
ejpam-4387	102	13	x	x	X
ejpam-4387	102	14	]	]	X
ejpam-4387	102	15	<	<	X
ejpam-4387	102	16	ω0	ω0	PROPN
ejpam-4387	102	17	,	,	PUNCT
ejpam-4387	102	18	i.e.	i.e.	X
ejpam-4387	102	19	,	,	PUNCT
ejpam-4387	102	20	[	[	X
ejpam-4387	102	21	x	x	X
ejpam-4387	102	22	]	]	X
ejpam-4387	102	23	<	<	X
ejpam-4387	102	24	ω0	ω0	X
ejpam-4387	102	25	=	=	SYM
ejpam-4387	102	26	{	{	PUNCT
ejpam-4387	102	27	e	e	X
ejpam-4387	102	28	⊂	⊂	PROPN
ejpam-4387	102	29	x	x	X
ejpam-4387	102	30	:	:	PUNCT
ejpam-4387	102	31	e	e	NOUN
ejpam-4387	102	32	is	be	AUX
ejpam-4387	102	33	finite	finite	ADJ
ejpam-4387	102	34	}	}	PUNCT
ejpam-4387	102	35	.	.	PUNCT
ejpam-4387	103	1	put	put	VERB
ejpam-4387	103	2	x	x	PUNCT
ejpam-4387	103	3	′	′	NUM
ejpam-4387	104	1	=	=	PUNCT
ejpam-4387	104	2	x	x	SYM
ejpam-4387	104	3	×	×	NOUN
ejpam-4387	104	4	{	{	PUNCT
ejpam-4387	104	5	1	1	NUM
ejpam-4387	104	6	}	}	PUNCT
ejpam-4387	104	7	.	.	PUNCT
ejpam-4387	105	1	so	so	ADV
ejpam-4387	105	2	,	,	PUNCT
ejpam-4387	105	3	x	x	PUNCT
ejpam-4387	105	4	′	′	NOUN
ejpam-4387	105	5	is	be	AUX
ejpam-4387	105	6	just	just	ADV
ejpam-4387	105	7	a	a	DET
ejpam-4387	105	8	copy	copy	NOUN
ejpam-4387	105	9	of	of	ADP
ejpam-4387	105	10	x	x	PUNCT
ejpam-4387	105	11	and	and	CCONJ
ejpam-4387	105	12	we	we	PRON
ejpam-4387	105	13	have	have	VERB
ejpam-4387	105	14	x	x	NOUN
ejpam-4387	105	15	∩	∩	NOUN
ejpam-4387	105	16	x	x	SYM
ejpam-4387	105	17	′	′	NOUN
ejpam-4387	105	18	=	=	PUNCT
ejpam-4387	105	19	∅.	∅.	PRON
ejpam-4387	105	20	the	the	DET
ejpam-4387	105	21	ground	ground	NOUN
ejpam-4387	105	22	set	set	NOUN
ejpam-4387	105	23	of	of	ADP
ejpam-4387	105	24	the	the	DET
ejpam-4387	105	25	alexandroff	alexandroff	NOUN
ejpam-4387	105	26	duplicate	duplicate	VERB
ejpam-4387	105	27	space	space	NOUN
ejpam-4387	105	28	a(x	a(x	NOUN
ejpam-4387	105	29	)	)	PUNCT
ejpam-4387	105	30	of	of	ADP
ejpam-4387	105	31	x	x	PUNCT
ejpam-4387	105	32	is	be	AUX
ejpam-4387	105	33	a(x	a(x	NOUN
ejpam-4387	105	34	)	)	PUNCT
ejpam-4387	105	35	=	=	PUNCT
ejpam-4387	105	36	x	x	SYM
ejpam-4387	105	37	∪	∪	X
ejpam-4387	105	38	x	x	VERB
ejpam-4387	105	39	′.	′.	NOUN
ejpam-4387	105	40	to	to	PART
ejpam-4387	105	41	simplify	simplify	VERB
ejpam-4387	105	42	the	the	DET
ejpam-4387	105	43	symbols	symbol	NOUN
ejpam-4387	105	44	,	,	PUNCT
ejpam-4387	105	45	we	we	PRON
ejpam-4387	105	46	do	do	VERB
ejpam-4387	105	47	the	the	DET
ejpam-4387	105	48	following	following	NOUN
ejpam-4387	105	49	:	:	PUNCT
ejpam-4387	105	50	for	for	ADP
ejpam-4387	105	51	an	an	DET
ejpam-4387	105	52	element	element	NOUN
ejpam-4387	105	53	x	x	SYM
ejpam-4387	105	54	∈	∈	PROPN
ejpam-4387	105	55	x	x	X
ejpam-4387	105	56	,	,	PUNCT
ejpam-4387	105	57	we	we	PRON
ejpam-4387	105	58	denote	denote	VERB
ejpam-4387	105	59	the	the	DET
ejpam-4387	105	60	element	element	NOUN
ejpam-4387	105	61	⟨x	⟨x	VERB
ejpam-4387	105	62	,	,	PUNCT
ejpam-4387	105	63	1⟩	1⟩	NUM
ejpam-4387	105	64	in	in	ADP
ejpam-4387	105	65	x	x	X
ejpam-4387	105	66	′	′	NUM
ejpam-4387	105	67	by	by	ADP
ejpam-4387	105	68	x′	x′	PROPN
ejpam-4387	105	69	and	and	CCONJ
ejpam-4387	105	70	for	for	ADP
ejpam-4387	105	71	any	any	DET
ejpam-4387	105	72	subset	subset	NOUN
ejpam-4387	105	73	b	b	PROPN
ejpam-4387	105	74	⊆	⊆	NUM
ejpam-4387	105	75	x	x	NUM
ejpam-4387	105	76	,	,	PUNCT
ejpam-4387	105	77	put	put	VERB
ejpam-4387	105	78	b′	b′	NOUN
ejpam-4387	105	79	=	=	PUNCT
ejpam-4387	105	80	{	{	PUNCT
ejpam-4387	105	81	x′	x′	PROPN
ejpam-4387	105	82	:	:	PUNCT
ejpam-4387	106	1	x	x	SYM
ejpam-4387	106	2	∈	∈	PROPN
ejpam-4387	106	3	b	b	AUX
ejpam-4387	106	4	}	}	PUNCT
ejpam-4387	106	5	=	=	SYM
ejpam-4387	106	6	b	b	SYM
ejpam-4387	106	7	×	×	NOUN
ejpam-4387	106	8	{	{	PUNCT
ejpam-4387	106	9	1	1	NUM
ejpam-4387	106	10	}	}	SYM
ejpam-4387	106	11	⊆	⊆	NUM
ejpam-4387	106	12	x	x	SYM
ejpam-4387	106	13	′.	′.	NOUN
ejpam-4387	106	14	for	for	ADP
ejpam-4387	106	15	each	each	DET
ejpam-4387	106	16	x′	x′	PROPN
ejpam-4387	106	17	∈	∈	PROPN
ejpam-4387	106	18	x	x	SYM
ejpam-4387	106	19	′	′	NOUN
ejpam-4387	106	20	,	,	PUNCT
ejpam-4387	106	21	put	put	VERB
ejpam-4387	106	22	b(x′	b(x′	NUM
ejpam-4387	106	23	)	)	PUNCT
ejpam-4387	106	24	=	=	PRON
ejpam-4387	106	25	{	{	PUNCT
ejpam-4387	106	26	{	{	PUNCT
ejpam-4387	106	27	x′	x′	NUM
ejpam-4387	106	28	}	}	PUNCT
ejpam-4387	106	29	}	}	PUNCT
ejpam-4387	106	30	,	,	PUNCT
ejpam-4387	106	31	so	so	CCONJ
ejpam-4387	106	32	any	any	DET
ejpam-4387	106	33	element	element	NOUN
ejpam-4387	106	34	in	in	ADP
ejpam-4387	106	35	x	x	X
ejpam-4387	106	36	′	′	NOUN
ejpam-4387	106	37	will	will	AUX
ejpam-4387	106	38	be	be	AUX
ejpam-4387	106	39	isolated	isolate	VERB
ejpam-4387	106	40	in	in	ADP
ejpam-4387	106	41	a(x	a(x	NOUN
ejpam-4387	106	42	)	)	PUNCT
ejpam-4387	106	43	.	.	PUNCT
ejpam-4387	107	1	for	for	SCONJ
ejpam-4387	107	2	each	each	DET
ejpam-4387	107	3	x	x	SYM
ejpam-4387	107	4	∈	∈	PROPN
ejpam-4387	107	5	x	x	NOUN
ejpam-4387	107	6	,	,	PUNCT
ejpam-4387	107	7	put	put	VERB
ejpam-4387	107	8	b(x	b(x	NOUN
ejpam-4387	107	9	)	)	PUNCT
ejpam-4387	108	1	=	=	PRON
ejpam-4387	108	2	{	{	PUNCT
ejpam-4387	108	3	u	u	NOUN
ejpam-4387	108	4	∪	∪	X
ejpam-4387	108	5	(	(	PUNCT
ejpam-4387	108	6	u	u	NOUN
ejpam-4387	108	7	′	′	NOUN
ejpam-4387	108	8	\	\	NOUN
ejpam-4387	108	9	e′	e′	X
ejpam-4387	108	10	)	)	PUNCT
ejpam-4387	108	11	:	:	PUNCT
ejpam-4387	108	12	u	u	NOUN
ejpam-4387	108	13	is	be	AUX
ejpam-4387	108	14	open	open	ADJ
ejpam-4387	108	15	in	in	ADP
ejpam-4387	108	16	x	x	PUNCT
ejpam-4387	108	17	with	with	ADP
ejpam-4387	108	18	x	x	PROPN
ejpam-4387	108	19	∈	∈	PROPN
ejpam-4387	108	20	u	u	NOUN
ejpam-4387	108	21	and	and	CCONJ
ejpam-4387	108	22	e	e	NOUN
ejpam-4387	108	23	∈	∈	PROPN
ejpam-4387	109	1	[	[	X
ejpam-4387	109	2	x	x	X
ejpam-4387	109	3	]	]	X
ejpam-4387	109	4	<	<	X
ejpam-4387	109	5	ω0	ω0	ADV
ejpam-4387	109	6	}	}	PUNCT
ejpam-4387	109	7	.	.	PUNCT
ejpam-4387	110	1	then	then	ADV
ejpam-4387	110	2	b	b	X
ejpam-4387	110	3	=	=	PRON
ejpam-4387	110	4	{	{	PUNCT
ejpam-4387	110	5	b(y	b(y	PROPN
ejpam-4387	110	6	)	)	PUNCT
ejpam-4387	110	7	:	:	PUNCT
ejpam-4387	110	8	y	y	PROPN
ejpam-4387	110	9	∈	∈	PROPN
ejpam-4387	110	10	a(x	a(x	PROPN
ejpam-4387	110	11	)	)	PUNCT
ejpam-4387	110	12	}	}	PUNCT
ejpam-4387	110	13	generates	generate	VERB
ejpam-4387	110	14	a	a	DET
ejpam-4387	110	15	unique	unique	ADJ
ejpam-4387	110	16	topology	topology	NOUN
ejpam-4387	110	17	on	on	ADP
ejpam-4387	110	18	a(x	a(x	NOUN
ejpam-4387	110	19	)	)	PUNCT
ejpam-4387	110	20	such	such	ADJ
ejpam-4387	110	21	that	that	SCONJ
ejpam-4387	110	22	b	b	PROPN
ejpam-4387	110	23	is	be	AUX
ejpam-4387	110	24	its	its	PRON
ejpam-4387	110	25	neighborhood	neighborhood	NOUN
ejpam-4387	110	26	system	system	NOUN
ejpam-4387	110	27	.	.	PUNCT
ejpam-4387	111	1	a(x	a(x	NOUN
ejpam-4387	111	2	)	)	PUNCT
ejpam-4387	111	3	with	with	ADP
ejpam-4387	111	4	this	this	DET
ejpam-4387	111	5	topology	topology	NOUN
ejpam-4387	111	6	is	be	AUX
ejpam-4387	111	7	called	call	VERB
ejpam-4387	111	8	the	the	DET
ejpam-4387	111	9	alexandroff	alexandroff	ADJ
ejpam-4387	111	10	duplicate	duplicate	NOUN
ejpam-4387	111	11	of	of	ADP
ejpam-4387	111	12	x	x	X
ejpam-4387	111	13	,	,	PUNCT
ejpam-4387	111	14	see	see	VERB
ejpam-4387	111	15	[	[	X
ejpam-4387	111	16	2	2	X
ejpam-4387	111	17	]	]	PUNCT
ejpam-4387	111	18	and	and	CCONJ
ejpam-4387	111	19	[	[	X
ejpam-4387	111	20	6	6	NUM
ejpam-4387	111	21	]	]	PUNCT
ejpam-4387	111	22	.	.	PUNCT
ejpam-4387	112	1	observe	observe	VERB
ejpam-4387	112	2	that	that	SCONJ
ejpam-4387	112	3	for	for	ADP
ejpam-4387	112	4	any	any	DET
ejpam-4387	112	5	open	open	ADJ
ejpam-4387	112	6	set	set	NOUN
ejpam-4387	112	7	u	u	NOUN
ejpam-4387	112	8	in	in	ADP
ejpam-4387	112	9	x	x	SYM
ejpam-4387	112	10	,	,	PUNCT
ejpam-4387	112	11	we	we	PRON
ejpam-4387	112	12	have	have	VERB
ejpam-4387	112	13	that	that	SCONJ
ejpam-4387	112	14	u	u	PROPN
ejpam-4387	112	15	∪	∪	ADP
ejpam-4387	112	16	u	u	NOUN
ejpam-4387	112	17	′	′	NOUN
ejpam-4387	112	18	is	be	AUX
ejpam-4387	112	19	open	open	ADJ
ejpam-4387	112	20	in	in	ADP
ejpam-4387	112	21	a(x	a(x	NOUN
ejpam-4387	112	22	)	)	PUNCT
ejpam-4387	112	23	and	and	CCONJ
ejpam-4387	112	24	for	for	ADP
ejpam-4387	112	25	any	any	DET
ejpam-4387	112	26	x	x	SYM
ejpam-4387	112	27	∈	∈	PROPN
ejpam-4387	112	28	u	u	NOUN
ejpam-4387	112	29	,	,	PUNCT
ejpam-4387	112	30	we	we	PRON
ejpam-4387	112	31	have	have	VERB
ejpam-4387	112	32	u	u	NOUN
ejpam-4387	112	33	∪	∪	ADJ
ejpam-4387	112	34	(	(	PUNCT
ejpam-4387	112	35	u	u	NOUN
ejpam-4387	112	36	′	′	NOUN
ejpam-4387	112	37	\	\	NOUN
ejpam-4387	112	38	{	{	PUNCT
ejpam-4387	112	39	x′	x′	PROPN
ejpam-4387	112	40	}	}	PUNCT
ejpam-4387	112	41	)	)	PUNCT
ejpam-4387	112	42	is	be	AUX
ejpam-4387	112	43	a	a	DET
ejpam-4387	112	44	basic	basic	ADJ
ejpam-4387	112	45	open	open	ADJ
ejpam-4387	112	46	neighborhood	neighborhood	NOUN
ejpam-4387	112	47	of	of	ADP
ejpam-4387	112	48	x.	x.	NOUN
ejpam-4387	112	49	theorem	theorem	NOUN
ejpam-4387	112	50	3	3	X
ejpam-4387	112	51	.	.	PUNCT
ejpam-4387	113	1	if	if	SCONJ
ejpam-4387	113	2	x	x	PRON
ejpam-4387	113	3	is	be	AUX
ejpam-4387	113	4	p	p	NOUN
ejpam-4387	113	5	-normal	-normal	NOUN
ejpam-4387	113	6	,	,	PUNCT
ejpam-4387	113	7	then	then	ADV
ejpam-4387	113	8	so	so	ADV
ejpam-4387	113	9	is	be	AUX
ejpam-4387	113	10	its	its	PRON
ejpam-4387	113	11	alexandroff	alexandroff	NOUN
ejpam-4387	113	12	duplicate	duplicate	VERB
ejpam-4387	113	13	a(x	a(x	NOUN
ejpam-4387	113	14	)	)	PUNCT
ejpam-4387	113	15	.	.	PUNCT
ejpam-4387	114	1	proof	proof	NOUN
ejpam-4387	114	2	.	.	PUNCT
ejpam-4387	115	1	let	let	VERB
ejpam-4387	115	2	x	x	PRON
ejpam-4387	115	3	be	be	AUX
ejpam-4387	115	4	any	any	PRON
ejpam-4387	115	5	p	p	ADJ
ejpam-4387	115	6	-normal	-normal	ADJ
ejpam-4387	115	7	space	space	NOUN
ejpam-4387	115	8	.	.	PUNCT
ejpam-4387	116	1	pick	pick	VERB
ejpam-4387	116	2	a	a	DET
ejpam-4387	116	3	normal	normal	ADJ
ejpam-4387	116	4	space	space	NOUN
ejpam-4387	116	5	y	y	PROPN
ejpam-4387	116	6	and	and	CCONJ
ejpam-4387	116	7	a	a	DET
ejpam-4387	116	8	bijective	bijective	ADJ
ejpam-4387	116	9	function	function	NOUN
ejpam-4387	117	1	f	f	NOUN
ejpam-4387	117	2	:	:	PUNCT
ejpam-4387	117	3	x	x	PUNCT
ejpam-4387	117	4	−→	−→	NOUN
ejpam-4387	117	5	y	y	PROPN
ejpam-4387	117	6	such	such	ADJ
ejpam-4387	117	7	that	that	DET
ejpam-4387	117	8	f|c	f|c	NOUN
ejpam-4387	117	9	:	:	PUNCT
ejpam-4387	118	1	c	c	AUX
ejpam-4387	118	2	−→	−→	NOUN
ejpam-4387	118	3	f(c	f(c	PROPN
ejpam-4387	118	4	)	)	PUNCT
ejpam-4387	118	5	is	be	AUX
ejpam-4387	118	6	a	a	DET
ejpam-4387	118	7	homeomorphism	homeomorphism	NOUN
ejpam-4387	118	8	for	for	ADP
ejpam-4387	118	9	each	each	DET
ejpam-4387	118	10	paracompact	paracompact	ADJ
ejpam-4387	118	11	subspace	subspace	NOUN
ejpam-4387	118	12	c	c	PROPN
ejpam-4387	118	13	⊆	⊆	NUM
ejpam-4387	118	14	x.	x.	NOUN
ejpam-4387	118	15	consider	consider	VERB
ejpam-4387	118	16	the	the	DET
ejpam-4387	118	17	alexandroff	alexandroff	NOUN
ejpam-4387	118	18	duplicate	duplicate	NOUN
ejpam-4387	118	19	spaces	space	NOUN
ejpam-4387	118	20	a(x	a(x	NOUN
ejpam-4387	118	21	)	)	PUNCT
ejpam-4387	118	22	and	and	CCONJ
ejpam-4387	118	23	a(y	a(y	PROPN
ejpam-4387	118	24	)	)	PUNCT
ejpam-4387	118	25	of	of	ADP
ejpam-4387	118	26	x	x	X
ejpam-4387	118	27	and	and	CCONJ
ejpam-4387	118	28	y	y	PROPN
ejpam-4387	118	29	respectively	respectively	ADV
ejpam-4387	118	30	.	.	PUNCT
ejpam-4387	119	1	y	y	PROPN
ejpam-4387	119	2	is	be	AUX
ejpam-4387	119	3	normal	normal	ADJ
ejpam-4387	119	4	then	then	ADV
ejpam-4387	119	5	so	so	ADV
ejpam-4387	119	6	is	be	AUX
ejpam-4387	119	7	a(y	a(y	PROPN
ejpam-4387	119	8	)	)	PUNCT
ejpam-4387	120	1	[	[	X
ejpam-4387	120	2	2	2	NUM
ejpam-4387	120	3	]	]	PUNCT
ejpam-4387	120	4	.	.	PUNCT
ejpam-4387	121	1	let	let	VERB
ejpam-4387	121	2	us	we	PRON
ejpam-4387	121	3	define	define	VERB
ejpam-4387	121	4	g	g	NOUN
ejpam-4387	121	5	:	:	PUNCT
ejpam-4387	121	6	a(x	a(x	PROPN
ejpam-4387	121	7	)	)	PUNCT
ejpam-4387	121	8	−→	−→	NOUN
ejpam-4387	121	9	a(y	a(y	PROPN
ejpam-4387	121	10	)	)	PUNCT
ejpam-4387	121	11	by	by	ADP
ejpam-4387	121	12	g(a	g(a	PROPN
ejpam-4387	121	13	)	)	PUNCT
ejpam-4387	122	1	=	=	SYM
ejpam-4387	122	2	f(a	f(a	PROPN
ejpam-4387	122	3	)	)	PUNCT
ejpam-4387	122	4	if	if	SCONJ
ejpam-4387	122	5	a	a	DET
ejpam-4387	122	6	∈	∈	PROPN
ejpam-4387	122	7	x.	x.	NOUN
ejpam-4387	123	1	if	if	SCONJ
ejpam-4387	123	2	a	a	DET
ejpam-4387	123	3	∈	∈	PROPN
ejpam-4387	123	4	x	x	SYM
ejpam-4387	123	5	′	′	NOUN
ejpam-4387	123	6	,	,	PUNCT
ejpam-4387	123	7	let	let	VERB
ejpam-4387	123	8	b	b	X
ejpam-4387	123	9	be	be	AUX
ejpam-4387	123	10	the	the	DET
ejpam-4387	123	11	unique	unique	ADJ
ejpam-4387	123	12	element	element	NOUN
ejpam-4387	123	13	in	in	ADP
ejpam-4387	123	14	x	x	INTJ
ejpam-4387	123	15	such	such	ADJ
ejpam-4387	123	16	that	that	DET
ejpam-4387	123	17	b′	b′	NOUN
ejpam-4387	123	18	=	=	SYM
ejpam-4387	123	19	a	a	PRON
ejpam-4387	123	20	,	,	PUNCT
ejpam-4387	123	21	then	then	ADV
ejpam-4387	123	22	define	define	VERB
ejpam-4387	123	23	g(a	g(a	PROPN
ejpam-4387	123	24	)	)	PUNCT
ejpam-4387	124	1	=	=	PUNCT
ejpam-4387	124	2	(	(	PUNCT
ejpam-4387	124	3	f(b))′.	f(b))′.	NOUN
ejpam-4387	124	4	then	then	ADV
ejpam-4387	124	5	g	g	PROPN
ejpam-4387	124	6	is	be	AUX
ejpam-4387	124	7	a	a	DET
ejpam-4387	124	8	bijective	bijective	ADJ
ejpam-4387	124	9	function	function	NOUN
ejpam-4387	124	10	.	.	PUNCT
ejpam-4387	125	1	now	now	ADV
ejpam-4387	125	2	,	,	PUNCT
ejpam-4387	125	3	a	a	DET
ejpam-4387	125	4	subspace	subspace	NOUN
ejpam-4387	125	5	c	c	NOUN
ejpam-4387	125	6	⊆	⊆	NUM
ejpam-4387	125	7	a(x	a(x	NOUN
ejpam-4387	125	8	)	)	PUNCT
ejpam-4387	125	9	is	be	AUX
ejpam-4387	125	10	paracompact	paracompact	ADJ
ejpam-4387	125	11	in	in	ADP
ejpam-4387	125	12	a(x	a(x	NOUN
ejpam-4387	125	13	)	)	PUNCT
ejpam-4387	125	14	if	if	SCONJ
ejpam-4387	126	1	and	and	CCONJ
ejpam-4387	126	2	only	only	ADV
ejpam-4387	126	3	if	if	SCONJ
ejpam-4387	126	4	c	c	PROPN
ejpam-4387	126	5	∩x	∩x	NOUN
ejpam-4387	126	6	is	be	AUX
ejpam-4387	126	7	paracompact	paracompact	ADJ
ejpam-4387	126	8	in	in	ADP
ejpam-4387	126	9	x.	x.	NOUN
ejpam-4387	126	10	to	to	PART
ejpam-4387	126	11	prove	prove	VERB
ejpam-4387	126	12	this	this	PRON
ejpam-4387	126	13	,	,	PUNCT
ejpam-4387	126	14	assume	assume	VERB
ejpam-4387	126	15	that	that	SCONJ
ejpam-4387	126	16	c	c	PROPN
ejpam-4387	126	17	is	be	AUX
ejpam-4387	126	18	paracompact	paracompact	ADJ
ejpam-4387	126	19	in	in	ADP
ejpam-4387	126	20	a(x	a(x	NOUN
ejpam-4387	126	21	)	)	PUNCT
ejpam-4387	126	22	.	.	PUNCT
ejpam-4387	127	1	let	let	VERB
ejpam-4387	127	2	u	u	PRON
ejpam-4387	127	3	=	=	PUNCT
ejpam-4387	127	4	{	{	PUNCT
ejpam-4387	127	5	uα	uα	PROPN
ejpam-4387	127	6	⊆	⊆	NUM
ejpam-4387	127	7	c∩x	c∩x	NOUN
ejpam-4387	127	8	:	:	PUNCT
ejpam-4387	127	9	uα	uα	PROPN
ejpam-4387	127	10	is	be	AUX
ejpam-4387	127	11	open	open	ADJ
ejpam-4387	127	12	in	in	ADP
ejpam-4387	127	13	c	c	NOUN
ejpam-4387	127	14	∩x	∩x	NOUN
ejpam-4387	127	15	for	for	SCONJ
ejpam-4387	127	16	each	each	PRON
ejpam-4387	127	17	α	α	PRON
ejpam-4387	127	18	∈	∈	PROPN
ejpam-4387	127	19	λ	λ	PROPN
ejpam-4387	127	20	}	}	PUNCT
ejpam-4387	127	21	be	be	VERB
ejpam-4387	127	22	any	any	DET
ejpam-4387	127	23	open	open	ADJ
ejpam-4387	127	24	cover	cover	NOUN
ejpam-4387	127	25	(	(	PUNCT
ejpam-4387	127	26	open	open	ADJ
ejpam-4387	127	27	in	in	ADP
ejpam-4387	127	28	c	c	NOUN
ejpam-4387	127	29	∩x	∩x	NOUN
ejpam-4387	127	30	)	)	PUNCT
ejpam-4387	127	31	for	for	ADP
ejpam-4387	127	32	c	c	PROPN
ejpam-4387	127	33	∩x	∩x	PROPN
ejpam-4387	127	34	.	.	PUNCT
ejpam-4387	128	1	that	that	PRON
ejpam-4387	128	2	means	mean	VERB
ejpam-4387	128	3	for	for	ADP
ejpam-4387	128	4	each	each	DET
ejpam-4387	128	5	α	α	NOUN
ejpam-4387	128	6	∈	∈	PROPN
ejpam-4387	128	7	λ	λ	NOUN
ejpam-4387	128	8	there	there	PRON
ejpam-4387	128	9	exists	exist	VERB
ejpam-4387	128	10	vα	vα	ADP
ejpam-4387	128	11	⊆	⊆	NUM
ejpam-4387	128	12	x	x	NOUN
ejpam-4387	128	13	open	open	ADJ
ejpam-4387	128	14	in	in	ADP
ejpam-4387	128	15	x	x	INTJ
ejpam-4387	128	16	such	such	ADJ
ejpam-4387	128	17	that	that	PRON
ejpam-4387	128	18	uα	uα	PROPN
ejpam-4387	128	19	=	=	SYM
ejpam-4387	128	20	vα∩(c∩x	vα∩(c∩x	ADJ
ejpam-4387	128	21	)	)	PUNCT
ejpam-4387	128	22	.	.	PUNCT
ejpam-4387	129	1	now	now	ADV
ejpam-4387	129	2	,	,	PUNCT
ejpam-4387	129	3	for	for	ADP
ejpam-4387	129	4	every	every	DET
ejpam-4387	129	5	α	α	PROPN
ejpam-4387	129	6	∈	∈	PROPN
ejpam-4387	129	7	λ	λ	PROPN
ejpam-4387	129	8	,	,	PUNCT
ejpam-4387	129	9	vα	vα	PROPN
ejpam-4387	129	10	is	be	AUX
ejpam-4387	129	11	open	open	ADJ
ejpam-4387	129	12	in	in	ADP
ejpam-4387	129	13	x	x	X
ejpam-4387	129	14	and	and	CCONJ
ejpam-4387	129	15	therefore	therefore	ADV
ejpam-4387	129	16	vα∪v	vα∪v	ADJ
ejpam-4387	130	1	′	′	NUM
ejpam-4387	130	2	α	α	NOUN
ejpam-4387	130	3	is	be	AUX
ejpam-4387	130	4	open	open	ADJ
ejpam-4387	130	5	in	in	ADP
ejpam-4387	130	6	a(x	a(x	NOUN
ejpam-4387	130	7	)	)	PUNCT
ejpam-4387	130	8	.	.	PUNCT
ejpam-4387	131	1	so	so	ADV
ejpam-4387	131	2	(	(	PUNCT
ejpam-4387	131	3	vα∪v	vα∪v	ADJ
ejpam-4387	131	4	′	′	NUM
ejpam-4387	131	5	α)∩c	α)∩c	NOUN
ejpam-4387	131	6	is	be	AUX
ejpam-4387	131	7	open	open	ADJ
ejpam-4387	131	8	in	in	ADP
ejpam-4387	131	9	c.	c.	PROPN
ejpam-4387	131	10	this	this	PRON
ejpam-4387	131	11	implies	imply	VERB
ejpam-4387	131	12	(	(	PUNCT
ejpam-4387	131	13	vα∪v	vα∪v	NOUN
ejpam-4387	131	14	′	′	NUM
ejpam-4387	131	15	α)∩c	α)∩c	NOUN
ejpam-4387	131	16	=	=	SYM
ejpam-4387	131	17	(	(	PUNCT
ejpam-4387	131	18	vα∩c	vα∩c	X
ejpam-4387	131	19	)	)	PUNCT
ejpam-4387	131	20	⋃	⋃	NOUN
ejpam-4387	131	21	(	(	PUNCT
ejpam-4387	131	22	v	v	NOUN
ejpam-4387	131	23	′	′	NUM
ejpam-4387	131	24	α∩c	α∩c	NOUN
ejpam-4387	131	25	)	)	PUNCT
ejpam-4387	131	26	is	be	AUX
ejpam-4387	131	27	open	open	ADJ
ejpam-4387	131	28	in	in	ADP
ejpam-4387	131	29	c.	c.	NOUN
ejpam-4387	131	30	take	take	VERB
ejpam-4387	131	31	the	the	DET
ejpam-4387	131	32	unions	union	NOUN
ejpam-4387	131	33	of	of	ADP
ejpam-4387	131	34	these	these	DET
ejpam-4387	131	35	sets	set	NOUN
ejpam-4387	131	36	l.	l.	PROPN
ejpam-4387	131	37	kalantan	kalantan	PROPN
ejpam-4387	131	38	,	,	PUNCT
ejpam-4387	131	39	m.	m.	NOUN
ejpam-4387	131	40	mansouri	mansouri	PROPN
ejpam-4387	131	41	/	/	SYM
ejpam-4387	131	42	eur	eur	PROPN
ejpam-4387	131	43	.	.	PUNCT
ejpam-4387	132	1	j.	j.	PROPN
ejpam-4387	132	2	pure	pure	PROPN
ejpam-4387	132	3	appl	appl	PROPN
ejpam-4387	132	4	.	.	PROPN
ejpam-4387	132	5	math	math	PROPN
ejpam-4387	132	6	,	,	PUNCT
ejpam-4387	132	7	15	15	NUM
ejpam-4387	132	8	(	(	PUNCT
ejpam-4387	132	9	2	2	NUM
ejpam-4387	132	10	)	)	PUNCT
ejpam-4387	132	11	(	(	PUNCT
ejpam-4387	132	12	2022	2022	NUM
ejpam-4387	132	13	)	)	PUNCT
ejpam-4387	132	14	,	,	PUNCT
ejpam-4387	132	15	774	774	NUM
ejpam-4387	132	16	-	-	SYM
ejpam-4387	132	17	783	783	NUM
ejpam-4387	132	18	778	778	NUM
ejpam-4387	132	19	:	:	PUNCT
ejpam-4387	132	20	(	(	PUNCT
ejpam-4387	132	21	∪α∈λ(vα∩c	∪α∈λ(vα∩c	NUM
ejpam-4387	132	22	)	)	PUNCT
ejpam-4387	132	23	)	)	PUNCT
ejpam-4387	133	1	⋃	⋃	PROPN
ejpam-4387	133	2	(	(	PUNCT
ejpam-4387	133	3	∪α∈λ(v	∪α∈λ(v	PROPN
ejpam-4387	133	4	′	′	NUM
ejpam-4387	133	5	α∩c	α∩c	NOUN
ejpam-4387	133	6	)	)	PUNCT
ejpam-4387	133	7	)	)	PUNCT
ejpam-4387	133	8	.	.	PUNCT
ejpam-4387	134	1	for	for	ADP
ejpam-4387	134	2	each	each	DET
ejpam-4387	134	3	x′	x′	PROPN
ejpam-4387	134	4	∈	∈	PROPN
ejpam-4387	134	5	c	c	NOUN
ejpam-4387	134	6	\(∪α∈λ(v	\(∪α∈λ(v	VERB
ejpam-4387	134	7	′	′	NUM
ejpam-4387	134	8	α∩c	α∩c	NOUN
ejpam-4387	134	9	)	)	PUNCT
ejpam-4387	134	10	)	)	PUNCT
ejpam-4387	134	11	consider	consider	VERB
ejpam-4387	134	12	the	the	DET
ejpam-4387	134	13	singleton	singleton	NOUN
ejpam-4387	134	14	{	{	PUNCT
ejpam-4387	134	15	x′	x′	NUM
ejpam-4387	134	16	}	}	PUNCT
ejpam-4387	134	17	.	.	PUNCT
ejpam-4387	135	1	consider	consider	VERB
ejpam-4387	135	2	w	w	NOUN
ejpam-4387	135	3	=	=	PRON
ejpam-4387	135	4	{	{	PUNCT
ejpam-4387	135	5	(	(	PUNCT
ejpam-4387	135	6	vα	vα	PROPN
ejpam-4387	135	7	∩	∩	NOUN
ejpam-4387	135	8	c	c	NOUN
ejpam-4387	135	9	)	)	PUNCT
ejpam-4387	135	10	⋃	⋃	NOUN
ejpam-4387	135	11	(	(	PUNCT
ejpam-4387	135	12	v	v	NOUN
ejpam-4387	135	13	′	′	NUM
ejpam-4387	135	14	α	α	PROPN
ejpam-4387	135	15	∩	∩	ADJ
ejpam-4387	135	16	c	c	NOUN
ejpam-4387	135	17	)	)	PUNCT
ejpam-4387	135	18	,	,	PUNCT
ejpam-4387	135	19	{	{	PUNCT
ejpam-4387	135	20	x′	x′	NUM
ejpam-4387	135	21	}	}	PUNCT
ejpam-4387	135	22	:	:	PUNCT
ejpam-4387	136	1	α	α	PROPN
ejpam-4387	136	2	∈	∈	PROPN
ejpam-4387	136	3	λ	λ	PROPN
ejpam-4387	136	4	,	,	PUNCT
ejpam-4387	136	5	and	and	CCONJ
ejpam-4387	136	6	x′	x′	PROPN
ejpam-4387	136	7	∈	∈	PROPN
ejpam-4387	136	8	c	c	NOUN
ejpam-4387	136	9	\	\	PUNCT
ejpam-4387	137	1	(	(	PUNCT
ejpam-4387	137	2	∪α∈λ(v	∪α∈λ(v	NOUN
ejpam-4387	137	3	′	′	NUM
ejpam-4387	137	4	α	α	PROPN
ejpam-4387	137	5	∩	∩	ADJ
ejpam-4387	137	6	c	c	NOUN
ejpam-4387	137	7	)	)	PUNCT
ejpam-4387	137	8	)	)	PUNCT
ejpam-4387	137	9	}	}	PUNCT
ejpam-4387	137	10	.	.	PUNCT
ejpam-4387	138	1	this	this	PRON
ejpam-4387	138	2	is	be	AUX
ejpam-4387	138	3	an	an	DET
ejpam-4387	138	4	open	open	ADJ
ejpam-4387	138	5	cover	cover	NOUN
ejpam-4387	138	6	of	of	ADP
ejpam-4387	138	7	c	c	PROPN
ejpam-4387	138	8	in	in	ADP
ejpam-4387	138	9	a(x	a(x	NOUN
ejpam-4387	138	10	)	)	PUNCT
ejpam-4387	138	11	(	(	PUNCT
ejpam-4387	138	12	open	open	VERB
ejpam-4387	138	13	in	in	ADP
ejpam-4387	138	14	c	c	NOUN
ejpam-4387	138	15	)	)	PUNCT
ejpam-4387	138	16	.	.	PUNCT
ejpam-4387	139	1	since	since	SCONJ
ejpam-4387	139	2	c	c	PROPN
ejpam-4387	139	3	is	be	AUX
ejpam-4387	139	4	assumed	assume	VERB
ejpam-4387	139	5	to	to	PART
ejpam-4387	139	6	be	be	AUX
ejpam-4387	139	7	paracompact	paracompact	ADJ
ejpam-4387	139	8	in	in	ADP
ejpam-4387	139	9	a(x	a(x	NOUN
ejpam-4387	139	10	)	)	PUNCT
ejpam-4387	139	11	then	then	ADV
ejpam-4387	139	12	this	this	DET
ejpam-4387	139	13	cover	cover	NOUN
ejpam-4387	139	14	w	w	NOUN
ejpam-4387	139	15	has	have	VERB
ejpam-4387	139	16	a	a	DET
ejpam-4387	139	17	locally	locally	ADV
ejpam-4387	139	18	finite	finite	ADJ
ejpam-4387	139	19	open	open	ADJ
ejpam-4387	139	20	refinement	refinement	NOUN
ejpam-4387	139	21	(	(	PUNCT
ejpam-4387	139	22	in	in	ADP
ejpam-4387	139	23	c	c	NOUN
ejpam-4387	139	24	)	)	PUNCT
ejpam-4387	139	25	.	.	PUNCT
ejpam-4387	140	1	that	that	PRON
ejpam-4387	140	2	is	be	AUX
ejpam-4387	140	3	,	,	PUNCT
ejpam-4387	140	4	there	there	PRON
ejpam-4387	140	5	exists	exist	VERB
ejpam-4387	140	6	{	{	PUNCT
ejpam-4387	140	7	gs	gs	X
ejpam-4387	140	8	:	:	PUNCT
ejpam-4387	140	9	s	s	VERB
ejpam-4387	140	10	∈	∈	PROPN
ejpam-4387	140	11	s	s	PART
ejpam-4387	140	12	}	}	PUNCT
ejpam-4387	140	13	such	such	ADJ
ejpam-4387	140	14	that	that	SCONJ
ejpam-4387	140	15	for	for	ADP
ejpam-4387	140	16	each	each	DET
ejpam-4387	140	17	s	s	X
ejpam-4387	140	18	∈	∈	PROPN
ejpam-4387	140	19	s	s	NOUN
ejpam-4387	140	20	,	,	PUNCT
ejpam-4387	140	21	gs	gs	PROPN
ejpam-4387	140	22	is	be	AUX
ejpam-4387	140	23	open	open	ADJ
ejpam-4387	140	24	in	in	ADP
ejpam-4387	140	25	c	c	PROPN
ejpam-4387	140	26	and	and	CCONJ
ejpam-4387	140	27	for	for	ADP
ejpam-4387	140	28	each	each	DET
ejpam-4387	140	29	s	s	X
ejpam-4387	140	30	∈	∈	PROPN
ejpam-4387	140	31	s	s	PART
ejpam-4387	140	32	there	there	PRON
ejpam-4387	140	33	exists	exist	VERB
ejpam-4387	140	34	α	α	PRON
ejpam-4387	140	35	∈	∈	PROPN
ejpam-4387	140	36	λ	λ	NOUN
ejpam-4387	140	37	such	such	ADJ
ejpam-4387	140	38	that	that	SCONJ
ejpam-4387	140	39	either	either	CCONJ
ejpam-4387	140	40	gs	gs	PROPN
ejpam-4387	140	41	⊆	⊆	NUM
ejpam-4387	140	42	(	(	PUNCT
ejpam-4387	140	43	vα∩c	vα∩c	NUM
ejpam-4387	140	44	)	)	PUNCT
ejpam-4387	140	45	⋃	⋃	NOUN
ejpam-4387	140	46	(	(	PUNCT
ejpam-4387	140	47	v	v	NOUN
ejpam-4387	140	48	′	′	NUM
ejpam-4387	140	49	α∩c	α∩c	NOUN
ejpam-4387	140	50	)	)	PUNCT
ejpam-4387	140	51	or	or	CCONJ
ejpam-4387	140	52	gs	gs	ADJ
ejpam-4387	140	53	=	=	PUNCT
ejpam-4387	140	54	{	{	PUNCT
ejpam-4387	140	55	x′	x′	NUM
ejpam-4387	140	56	}	}	PUNCT
ejpam-4387	140	57	for	for	ADP
ejpam-4387	140	58	some	some	PRON
ejpam-4387	140	59	x′	x′	PROPN
ejpam-4387	140	60	∈	∈	PROPN
ejpam-4387	140	61	c	c	NOUN
ejpam-4387	140	62	\	\	PUNCT
ejpam-4387	141	1	(	(	PUNCT
ejpam-4387	141	2	∪α∈λ(v	∪α∈λ(v	PROPN
ejpam-4387	141	3	′	′	NUM
ejpam-4387	141	4	α∩c	α∩c	NOUN
ejpam-4387	141	5	)	)	PUNCT
ejpam-4387	141	6	)	)	PUNCT
ejpam-4387	141	7	}	}	PUNCT
ejpam-4387	141	8	.	.	PUNCT
ejpam-4387	142	1	let	let	VERB
ejpam-4387	142	2	s	s	PRON
ejpam-4387	142	3	′	′	VERB
ejpam-4387	142	4	=	=	PUNCT
ejpam-4387	142	5	{	{	PUNCT
ejpam-4387	142	6	s	s	NOUN
ejpam-4387	142	7	∈	∈	NOUN
ejpam-4387	142	8	s	s	PART
ejpam-4387	142	9	:	:	PUNCT
ejpam-4387	142	10	gs	gs	PROPN
ejpam-4387	142	11	⊆	⊆	NUM
ejpam-4387	142	12	vα	vα	ADP
ejpam-4387	142	13	∩	∩	NOUN
ejpam-4387	142	14	c	c	NOUN
ejpam-4387	142	15	}	}	PUNCT
ejpam-4387	142	16	.	.	PUNCT
ejpam-4387	143	1	then	then	ADV
ejpam-4387	143	2	{	{	PUNCT
ejpam-4387	143	3	gs	gs	X
ejpam-4387	143	4	:	:	PUNCT
ejpam-4387	143	5	s	s	VERB
ejpam-4387	143	6	∈	∈	PROPN
ejpam-4387	143	7	s	s	PART
ejpam-4387	143	8	′	′	NOUN
ejpam-4387	143	9	}	}	PUNCT
ejpam-4387	143	10	is	be	AUX
ejpam-4387	143	11	a	a	DET
ejpam-4387	143	12	sub	sub	NOUN
ejpam-4387	143	13	-	-	NOUN
ejpam-4387	143	14	family	family	NOUN
ejpam-4387	143	15	of	of	ADP
ejpam-4387	143	16	{	{	PUNCT
ejpam-4387	143	17	gs	gs	X
ejpam-4387	143	18	:	:	PUNCT
ejpam-4387	143	19	s	s	VERB
ejpam-4387	143	20	∈	∈	PROPN
ejpam-4387	143	21	s	s	PART
ejpam-4387	143	22	}	}	PUNCT
ejpam-4387	143	23	which	which	PRON
ejpam-4387	143	24	is	be	AUX
ejpam-4387	143	25	locally	locally	ADV
ejpam-4387	143	26	finite	finite	ADJ
ejpam-4387	143	27	,	,	PUNCT
ejpam-4387	143	28	hence	hence	ADV
ejpam-4387	143	29	{	{	PUNCT
ejpam-4387	143	30	gs	gs	X
ejpam-4387	143	31	:	:	PUNCT
ejpam-4387	143	32	s	s	VERB
ejpam-4387	143	33	∈	∈	PROPN
ejpam-4387	143	34	s	s	PART
ejpam-4387	143	35	′	′	NOUN
ejpam-4387	143	36	}	}	PUNCT
ejpam-4387	143	37	is	be	AUX
ejpam-4387	143	38	locally	locally	ADV
ejpam-4387	143	39	finite	finite	ADJ
ejpam-4387	143	40	as	as	ADV
ejpam-4387	143	41	well	well	ADV
ejpam-4387	143	42	.	.	PUNCT
ejpam-4387	144	1	so	so	ADV
ejpam-4387	144	2	{	{	PUNCT
ejpam-4387	144	3	gs	gs	X
ejpam-4387	144	4	:	:	PUNCT
ejpam-4387	144	5	s	s	VERB
ejpam-4387	144	6	∈	∈	PROPN
ejpam-4387	144	7	s	s	PART
ejpam-4387	144	8	′	′	NOUN
ejpam-4387	144	9	}	}	PUNCT
ejpam-4387	144	10	is	be	AUX
ejpam-4387	144	11	a	a	DET
ejpam-4387	144	12	locally	locally	ADV
ejpam-4387	144	13	finite	finite	ADJ
ejpam-4387	144	14	open	open	ADJ
ejpam-4387	144	15	(	(	PUNCT
ejpam-4387	144	16	open	open	ADJ
ejpam-4387	144	17	in	in	ADP
ejpam-4387	144	18	c	c	NOUN
ejpam-4387	144	19	∩	∩	ADJ
ejpam-4387	144	20	x	x	X
ejpam-4387	144	21	)	)	PUNCT
ejpam-4387	144	22	refinement	refinement	NOUN
ejpam-4387	144	23	of	of	ADP
ejpam-4387	144	24	u	u	PROPN
ejpam-4387	144	25	.	.	PUNCT
ejpam-4387	145	1	on	on	ADP
ejpam-4387	145	2	the	the	DET
ejpam-4387	145	3	other	other	ADJ
ejpam-4387	145	4	hand	hand	NOUN
ejpam-4387	145	5	,	,	PUNCT
ejpam-4387	145	6	assume	assume	VERB
ejpam-4387	145	7	c	c	NOUN
ejpam-4387	145	8	∩	∩	PROPN
ejpam-4387	145	9	x	x	PRON
ejpam-4387	145	10	is	be	AUX
ejpam-4387	145	11	a	a	DET
ejpam-4387	145	12	paracompact	paracompact	NOUN
ejpam-4387	145	13	subset	subset	NOUN
ejpam-4387	145	14	in	in	ADP
ejpam-4387	145	15	x.	x.	NOUN
ejpam-4387	145	16	let	let	VERB
ejpam-4387	145	17	g	g	NOUN
ejpam-4387	145	18	=	=	PUNCT
ejpam-4387	145	19	{	{	PUNCT
ejpam-4387	145	20	gα	gα	ADP
ejpam-4387	145	21	∩	∩	ADJ
ejpam-4387	145	22	c	c	NOUN
ejpam-4387	145	23	:	:	PUNCT
ejpam-4387	145	24	α	α	PROPN
ejpam-4387	145	25	∈	∈	PROPN
ejpam-4387	145	26	λ;gα	λ;gα	VERB
ejpam-4387	145	27	⊆	⊆	NUM
ejpam-4387	145	28	a(x	a(x	NOUN
ejpam-4387	145	29	)	)	PUNCT
ejpam-4387	145	30	open	open	ADJ
ejpam-4387	145	31	in	in	ADP
ejpam-4387	145	32	a(x	a(x	NOUN
ejpam-4387	145	33	)	)	PUNCT
ejpam-4387	145	34	for	for	ADP
ejpam-4387	145	35	each	each	DET
ejpam-4387	145	36	α	α	PROPN
ejpam-4387	145	37	∈	∈	PROPN
ejpam-4387	145	38	λ	λ	PROPN
ejpam-4387	145	39	}	}	PUNCT
ejpam-4387	145	40	be	be	VERB
ejpam-4387	145	41	an	an	DET
ejpam-4387	145	42	arbitrary	arbitrary	ADJ
ejpam-4387	145	43	open	open	ADJ
ejpam-4387	145	44	(	(	PUNCT
ejpam-4387	145	45	open	open	ADJ
ejpam-4387	145	46	in	in	ADP
ejpam-4387	145	47	c	c	NOUN
ejpam-4387	145	48	)	)	PUNCT
ejpam-4387	145	49	cover	cover	NOUN
ejpam-4387	145	50	of	of	ADP
ejpam-4387	145	51	c	c	PROPN
ejpam-4387	145	52	in	in	ADP
ejpam-4387	145	53	a(x	a(x	NOUN
ejpam-4387	145	54	)	)	PUNCT
ejpam-4387	145	55	.	.	PUNCT
ejpam-4387	146	1	so	so	ADV
ejpam-4387	146	2	c	c	NOUN
ejpam-4387	146	3	⊆	⊆	NUM
ejpam-4387	146	4	⋃	⋃	NOUN
ejpam-4387	146	5	g.	g.	NOUN
ejpam-4387	146	6	consider	consider	VERB
ejpam-4387	146	7	gx	gx	PROPN
ejpam-4387	146	8	=	=	PRON
ejpam-4387	146	9	{	{	PUNCT
ejpam-4387	146	10	(	(	PUNCT
ejpam-4387	146	11	gα	gα	ADP
ejpam-4387	146	12	∩	∩	ADJ
ejpam-4387	146	13	c	c	NOUN
ejpam-4387	146	14	)	)	PUNCT
ejpam-4387	146	15	∩x	∩x	NOUN
ejpam-4387	146	16	:	:	PUNCT
ejpam-4387	147	1	α	α	PROPN
ejpam-4387	147	2	∈	∈	PROPN
ejpam-4387	147	3	λ	λ	X
ejpam-4387	147	4	}	}	PUNCT
ejpam-4387	147	5	=	=	SYM
ejpam-4387	147	6	{	{	PUNCT
ejpam-4387	147	7	gα	gα	ADP
ejpam-4387	147	8	∩	∩	NOUN
ejpam-4387	147	9	(	(	PUNCT
ejpam-4387	147	10	c	c	NOUN
ejpam-4387	147	11	∩x	∩x	NOUN
ejpam-4387	147	12	)	)	PUNCT
ejpam-4387	147	13	:	:	PUNCT
ejpam-4387	148	1	α	α	X
ejpam-4387	148	2	∈	∈	PROPN
ejpam-4387	149	1	λ}.then	λ}.then	X
ejpam-4387	149	2	c	c	NOUN
ejpam-4387	149	3	∩x	∩x	NOUN
ejpam-4387	149	4	⊆	⊆	NUM
ejpam-4387	149	5	⋃	⋃	PROPN
ejpam-4387	149	6	gx	gx	PROPN
ejpam-4387	149	7	,	,	PUNCT
ejpam-4387	149	8	i.e	i.e	PROPN
ejpam-4387	149	9	,	,	PUNCT
ejpam-4387	149	10	this	this	PRON
ejpam-4387	149	11	is	be	AUX
ejpam-4387	149	12	an	an	DET
ejpam-4387	149	13	open	open	ADJ
ejpam-4387	149	14	(	(	PUNCT
ejpam-4387	149	15	open	open	ADJ
ejpam-4387	149	16	in	in	ADP
ejpam-4387	149	17	c	c	NOUN
ejpam-4387	149	18	∩	∩	ADJ
ejpam-4387	149	19	x	x	X
ejpam-4387	149	20	)	)	PUNCT
ejpam-4387	149	21	cover	cover	VERB
ejpam-4387	149	22	for	for	ADP
ejpam-4387	149	23	c	c	NOUN
ejpam-4387	149	24	∩	∩	X
ejpam-4387	149	25	x.	x.	NOUN
ejpam-4387	149	26	by	by	ADP
ejpam-4387	149	27	assumption	assumption	NOUN
ejpam-4387	149	28	there	there	PRON
ejpam-4387	149	29	exists	exist	VERB
ejpam-4387	149	30	{	{	PUNCT
ejpam-4387	149	31	hs	hs	INTJ
ejpam-4387	149	32	:	:	PUNCT
ejpam-4387	149	33	s	s	PART
ejpam-4387	149	34	∈	∈	PROPN
ejpam-4387	149	35	s	s	PART
ejpam-4387	149	36	}	}	PUNCT
ejpam-4387	149	37	locally	locally	ADV
ejpam-4387	149	38	finite	finite	VERB
ejpam-4387	149	39	open	open	ADJ
ejpam-4387	149	40	(	(	PUNCT
ejpam-4387	149	41	open	open	ADJ
ejpam-4387	149	42	in	in	ADP
ejpam-4387	149	43	c	c	NOUN
ejpam-4387	149	44	∩x	∩x	NOUN
ejpam-4387	149	45	)	)	PUNCT
ejpam-4387	149	46	refinement	refinement	NOUN
ejpam-4387	149	47	of	of	ADP
ejpam-4387	149	48	gx	gx	PROPN
ejpam-4387	149	49	.	.	PUNCT
ejpam-4387	150	1	that	that	PRON
ejpam-4387	150	2	is	be	AUX
ejpam-4387	150	3	,	,	PUNCT
ejpam-4387	150	4	for	for	SCONJ
ejpam-4387	150	5	each	each	DET
ejpam-4387	150	6	s	s	X
ejpam-4387	150	7	∈	∈	PROPN
ejpam-4387	150	8	s	s	PART
ejpam-4387	150	9	there	there	PRON
ejpam-4387	150	10	exists	exist	VERB
ejpam-4387	150	11	αs	αs	ADP
ejpam-4387	150	12	∈	∈	PROPN
ejpam-4387	150	13	λ	λ	NOUN
ejpam-4387	150	14	such	such	ADJ
ejpam-4387	150	15	that	that	SCONJ
ejpam-4387	150	16	hs	hs	PROPN
ejpam-4387	150	17	⊆	⊆	NUM
ejpam-4387	150	18	gαs	gαs	PROPN
ejpam-4387	150	19	.	.	PUNCT
ejpam-4387	151	1	for	for	ADP
ejpam-4387	151	2	every	every	DET
ejpam-4387	151	3	s	s	X
ejpam-4387	151	4	∈	∈	NOUN
ejpam-4387	151	5	s	s	PART
ejpam-4387	151	6	there	there	PRON
ejpam-4387	151	7	exists	exist	VERB
ejpam-4387	151	8	ks	ks	PROPN
ejpam-4387	151	9	⊆	⊆	NUM
ejpam-4387	151	10	x	x	PUNCT
ejpam-4387	151	11	open	open	ADJ
ejpam-4387	151	12	in	in	ADP
ejpam-4387	151	13	x	x	X
ejpam-4387	151	14	such	such	ADJ
ejpam-4387	151	15	that	that	SCONJ
ejpam-4387	151	16	ks	ks	NOUN
ejpam-4387	151	17	∩	∩	NOUN
ejpam-4387	151	18	(	(	PUNCT
ejpam-4387	151	19	x	x	PROPN
ejpam-4387	151	20	∩	∩	ADJ
ejpam-4387	151	21	c	c	X
ejpam-4387	151	22	)	)	PUNCT
ejpam-4387	151	23	=	=	SYM
ejpam-4387	152	1	hs	hs	X
ejpam-4387	152	2	.	.	PROPN
ejpam-4387	152	3	consider	consider	VERB
ejpam-4387	152	4	{	{	PUNCT
ejpam-4387	152	5	(	(	PUNCT
ejpam-4387	152	6	ks	ks	X
ejpam-4387	152	7	∪k	∪k	PROPN
ejpam-4387	152	8	′	′	NUM
ejpam-4387	152	9	s	s	X
ejpam-4387	152	10	)	)	PUNCT
ejpam-4387	152	11	∪	∪	ADP
ejpam-4387	152	12	c	c	NOUN
ejpam-4387	152	13	:	:	PUNCT
ejpam-4387	152	14	s	s	X
ejpam-4387	152	15	∈	∈	PROPN
ejpam-4387	152	16	s}=	s}=	NOUN
ejpam-4387	152	17	{	{	PUNCT
ejpam-4387	152	18	(	(	PUNCT
ejpam-4387	152	19	ks	ks	PROPN
ejpam-4387	152	20	∩	∩	NOUN
ejpam-4387	152	21	c	c	NOUN
ejpam-4387	152	22	)	)	PUNCT
ejpam-4387	152	23	⋃	⋃	PROPN
ejpam-4387	152	24	(	(	PUNCT
ejpam-4387	152	25	k	k	NOUN
ejpam-4387	152	26	′	′	NOUN
ejpam-4387	152	27	s	s	VERB
ejpam-4387	152	28	∩	∩	NOUN
ejpam-4387	152	29	c	c	NOUN
ejpam-4387	152	30	)	)	PUNCT
ejpam-4387	152	31	:	:	PUNCT
ejpam-4387	152	32	s	s	VERB
ejpam-4387	152	33	∈	∈	PROPN
ejpam-4387	152	34	s	s	PART
ejpam-4387	152	35	}	}	PUNCT
ejpam-4387	152	36	.	.	PUNCT
ejpam-4387	153	1	since	since	SCONJ
ejpam-4387	153	2	{	{	PUNCT
ejpam-4387	153	3	ks	ks	PROPN
ejpam-4387	153	4	∩	∩	PROPN
ejpam-4387	153	5	c	c	NOUN
ejpam-4387	153	6	:	:	PUNCT
ejpam-4387	153	7	s	s	VERB
ejpam-4387	153	8	∈	∈	PROPN
ejpam-4387	153	9	s	s	PART
ejpam-4387	153	10	}	}	PUNCT
ejpam-4387	153	11	is	be	AUX
ejpam-4387	153	12	locally	locally	ADV
ejpam-4387	153	13	finite	finite	ADJ
ejpam-4387	153	14	then	then	ADV
ejpam-4387	153	15	so	so	ADV
ejpam-4387	153	16	is	be	AUX
ejpam-4387	153	17	{	{	PUNCT
ejpam-4387	153	18	(	(	PUNCT
ejpam-4387	153	19	ks	ks	PROPN
ejpam-4387	153	20	∩	∩	NOUN
ejpam-4387	153	21	c	c	NOUN
ejpam-4387	153	22	)	)	PUNCT
ejpam-4387	153	23	⋃	⋃	PROPN
ejpam-4387	154	1	(	(	PUNCT
ejpam-4387	154	2	k	k	NOUN
ejpam-4387	154	3	′	′	NOUN
ejpam-4387	154	4	s	s	VERB
ejpam-4387	154	5	∩	∩	NOUN
ejpam-4387	154	6	c	c	NOUN
ejpam-4387	154	7	)	)	PUNCT
ejpam-4387	154	8	:	:	PUNCT
ejpam-4387	154	9	s	s	VERB
ejpam-4387	154	10	∈	∈	PROPN
ejpam-4387	154	11	s	s	PART
ejpam-4387	154	12	}	}	PUNCT
ejpam-4387	154	13	.	.	PUNCT
ejpam-4387	155	1	for	for	ADP
ejpam-4387	155	2	every	every	DET
ejpam-4387	155	3	x′	x′	PROPN
ejpam-4387	155	4	∈	∈	PROPN
ejpam-4387	155	5	(	(	PUNCT
ejpam-4387	155	6	c	c	NOUN
ejpam-4387	155	7	∩x	∩x	NOUN
ejpam-4387	155	8	′	′	NUM
ejpam-4387	155	9	)	)	PUNCT
ejpam-4387	155	10	\	\	PUNCT
ejpam-4387	155	11	(	(	PUNCT
ejpam-4387	155	12	⋂	⋂	NUM
ejpam-4387	155	13	s∈s(k	s∈s(k	NOUN
ejpam-4387	155	14	′	′	NOUN
ejpam-4387	155	15	s	s	PART
ejpam-4387	155	16	∩	∩	NOUN
ejpam-4387	155	17	c	c	NOUN
ejpam-4387	155	18	)	)	PUNCT
ejpam-4387	155	19	)	)	PUNCT
ejpam-4387	156	1	there	there	PRON
ejpam-4387	156	2	exists	exist	VERB
ejpam-4387	156	3	αx′	αx′	PROPN
ejpam-4387	156	4	∈	∈	PROPN
ejpam-4387	156	5	λ	λ	NOUN
ejpam-4387	156	6	such	such	ADJ
ejpam-4387	156	7	that	that	SCONJ
ejpam-4387	156	8	x′	x′	PROPN
ejpam-4387	156	9	∈	∈	PROPN
ejpam-4387	156	10	gα′	gα′	PROPN
ejpam-4387	156	11	x	x	PUNCT
ejpam-4387	156	12	.	.	PUNCT
ejpam-4387	157	1	consider	consider	VERB
ejpam-4387	157	2	k	k	NOUN
ejpam-4387	157	3	=	=	PRON
ejpam-4387	157	4	{	{	PUNCT
ejpam-4387	157	5	(	(	PUNCT
ejpam-4387	157	6	ks	ks	PROPN
ejpam-4387	157	7	∩	∩	ADJ
ejpam-4387	157	8	c	c	X
ejpam-4387	157	9	)	)	PUNCT
ejpam-4387	157	10	∪	∪	NOUN
ejpam-4387	157	11	(	(	PUNCT
ejpam-4387	157	12	k	k	NOUN
ejpam-4387	157	13	′	′	NOUN
ejpam-4387	157	14	s	s	VERB
ejpam-4387	157	15	∩	∩	NOUN
ejpam-4387	157	16	c	c	NOUN
ejpam-4387	157	17	)	)	PUNCT
ejpam-4387	157	18	,	,	PUNCT
ejpam-4387	157	19	{	{	PUNCT
ejpam-4387	157	20	x′	x′	NUM
ejpam-4387	157	21	}	}	PUNCT
ejpam-4387	157	22	:	:	PUNCT
ejpam-4387	157	23	s	s	VERB
ejpam-4387	157	24	∈	∈	PROPN
ejpam-4387	157	25	s;x′	s;x′	NOUN
ejpam-4387	157	26	∈	∈	NOUN
ejpam-4387	157	27	(	(	PUNCT
ejpam-4387	157	28	c	c	NOUN
ejpam-4387	157	29	∩x	∩x	NOUN
ejpam-4387	157	30	′	′	NUM
ejpam-4387	157	31	)	)	PUNCT
ejpam-4387	157	32	\	\	PUNCT
ejpam-4387	158	1	(	(	PUNCT
ejpam-4387	158	2	⋂	⋂	NUM
ejpam-4387	158	3	s∈s(k	s∈s(k	NOUN
ejpam-4387	158	4	′	′	NOUN
ejpam-4387	158	5	s	s	PART
ejpam-4387	158	6	∩	∩	ADJ
ejpam-4387	158	7	c	c	NOUN
ejpam-4387	158	8	)	)	PUNCT
ejpam-4387	158	9	)	)	PUNCT
ejpam-4387	158	10	}	}	PUNCT
ejpam-4387	158	11	.	.	PUNCT
ejpam-4387	159	1	k	k	PROPN
ejpam-4387	159	2	is	be	AUX
ejpam-4387	159	3	a	a	DET
ejpam-4387	159	4	locally	locally	ADV
ejpam-4387	159	5	finite	finite	ADJ
ejpam-4387	159	6	open	open	ADJ
ejpam-4387	159	7	refinement	refinement	NOUN
ejpam-4387	159	8	(	(	PUNCT
ejpam-4387	159	9	open	open	VERB
ejpam-4387	159	10	in	in	ADP
ejpam-4387	159	11	c	c	NOUN
ejpam-4387	159	12	)	)	PUNCT
ejpam-4387	159	13	of	of	ADP
ejpam-4387	159	14	g.	g.	PROPN
ejpam-4387	159	15	now	now	ADV
ejpam-4387	159	16	,	,	PUNCT
ejpam-4387	159	17	let	let	VERB
ejpam-4387	159	18	c	c	NOUN
ejpam-4387	159	19	⊆	⊆	NUM
ejpam-4387	159	20	a(x	a(x	NOUN
ejpam-4387	159	21	)	)	PUNCT
ejpam-4387	159	22	be	be	VERB
ejpam-4387	159	23	any	any	DET
ejpam-4387	159	24	paracompact	paracompact	ADJ
ejpam-4387	159	25	subspace	subspace	NOUN
ejpam-4387	159	26	.	.	PUNCT
ejpam-4387	160	1	we	we	PRON
ejpam-4387	160	2	show	show	VERB
ejpam-4387	160	3	g|c	g|c	PROPN
ejpam-4387	160	4	:	:	PUNCT
ejpam-4387	160	5	c	c	X
ejpam-4387	160	6	−→	−→	NOUN
ejpam-4387	160	7	g(c	g(c	NOUN
ejpam-4387	160	8	)	)	PUNCT
ejpam-4387	160	9	is	be	AUX
ejpam-4387	160	10	a	a	DET
ejpam-4387	160	11	homeomorphism	homeomorphism	NOUN
ejpam-4387	160	12	.	.	PUNCT
ejpam-4387	161	1	let	let	VERB
ejpam-4387	161	2	a	a	DET
ejpam-4387	161	3	∈	∈	NOUN
ejpam-4387	161	4	c	c	AUX
ejpam-4387	161	5	be	be	AUX
ejpam-4387	161	6	arbitrary	arbitrary	ADJ
ejpam-4387	161	7	.	.	PUNCT
ejpam-4387	162	1	if	if	SCONJ
ejpam-4387	162	2	a	a	DET
ejpam-4387	162	3	∈	∈	NOUN
ejpam-4387	162	4	c	c	NOUN
ejpam-4387	162	5	∩x	∩x	NOUN
ejpam-4387	163	1	′	′	NOUN
ejpam-4387	163	2	,	,	PUNCT
ejpam-4387	163	3	let	let	VERB
ejpam-4387	163	4	b	b	X
ejpam-4387	163	5	∈	∈	PROPN
ejpam-4387	163	6	x	x	AUX
ejpam-4387	163	7	be	be	AUX
ejpam-4387	163	8	the	the	DET
ejpam-4387	163	9	unique	unique	ADJ
ejpam-4387	163	10	element	element	NOUN
ejpam-4387	163	11	such	such	ADJ
ejpam-4387	163	12	that	that	DET
ejpam-4387	163	13	b′	b′	NUM
ejpam-4387	163	14	=	=	NOUN
ejpam-4387	163	15	a.	a.	NOUN
ejpam-4387	163	16	for	for	ADP
ejpam-4387	163	17	the	the	DET
ejpam-4387	163	18	smallest	small	ADJ
ejpam-4387	163	19	basic	basic	ADJ
ejpam-4387	163	20	open	open	ADJ
ejpam-4387	163	21	neighborhood	neighborhood	NOUN
ejpam-4387	163	22	{	{	PUNCT
ejpam-4387	163	23	(	(	PUNCT
ejpam-4387	163	24	f(b))′	f(b))′	PROPN
ejpam-4387	163	25	}	}	PUNCT
ejpam-4387	163	26	of	of	ADP
ejpam-4387	163	27	the	the	DET
ejpam-4387	163	28	point	point	NOUN
ejpam-4387	163	29	g(a	g(a	PROPN
ejpam-4387	163	30	)	)	PUNCT
ejpam-4387	163	31	we	we	PRON
ejpam-4387	163	32	have	have	VERB
ejpam-4387	163	33	that	that	SCONJ
ejpam-4387	163	34	{	{	PUNCT
ejpam-4387	163	35	a	a	PRON
ejpam-4387	163	36	}	}	PUNCT
ejpam-4387	163	37	is	be	AUX
ejpam-4387	163	38	open	open	ADJ
ejpam-4387	163	39	in	in	ADP
ejpam-4387	163	40	c	c	NOUN
ejpam-4387	163	41	∩x	∩x	PUNCT
ejpam-4387	163	42	′	′	NUM
ejpam-4387	163	43	and	and	CCONJ
ejpam-4387	163	44	g({a	g({a	PROPN
ejpam-4387	163	45	}	}	PUNCT
ejpam-4387	163	46	)	)	PUNCT
ejpam-4387	164	1	⊆	⊆	X
ejpam-4387	164	2	{	{	PUNCT
ejpam-4387	164	3	(	(	PUNCT
ejpam-4387	164	4	f(b))′	f(b))′	X
ejpam-4387	164	5	}	}	PUNCT
ejpam-4387	164	6	.	.	PUNCT
ejpam-4387	165	1	if	if	SCONJ
ejpam-4387	165	2	a	a	DET
ejpam-4387	165	3	∈	∈	PROPN
ejpam-4387	165	4	c	c	NOUN
ejpam-4387	165	5	∩x	∩x	NOUN
ejpam-4387	165	6	.	.	PUNCT
ejpam-4387	166	1	let	let	VERB
ejpam-4387	166	2	w	w	NOUN
ejpam-4387	166	3	be	be	AUX
ejpam-4387	166	4	any	any	DET
ejpam-4387	166	5	open	open	ADJ
ejpam-4387	166	6	set	set	NOUN
ejpam-4387	166	7	in	in	ADP
ejpam-4387	166	8	y	y	PROPN
ejpam-4387	166	9	such	such	ADJ
ejpam-4387	166	10	that	that	PRON
ejpam-4387	166	11	g(a	g(a	PROPN
ejpam-4387	166	12	)	)	PUNCT
ejpam-4387	167	1	=	=	SYM
ejpam-4387	167	2	f(a	f(a	X
ejpam-4387	167	3	)	)	PUNCT
ejpam-4387	167	4	∈	∈	PROPN
ejpam-4387	167	5	w	w	PROPN
ejpam-4387	167	6	.	.	PUNCT
ejpam-4387	168	1	consider	consider	VERB
ejpam-4387	168	2	h	h	NOUN
ejpam-4387	168	3	=	=	PUNCT
ejpam-4387	169	1	(	(	PUNCT
ejpam-4387	169	2	w	w	NOUN
ejpam-4387	169	3	∪	∪	X
ejpam-4387	169	4	(	(	PUNCT
ejpam-4387	169	5	w	w	PROPN
ejpam-4387	169	6	′	′	NOUN
ejpam-4387	169	7	\	\	NOUN
ejpam-4387	169	8	{	{	PUNCT
ejpam-4387	169	9	f(a)′	f(a)′	PROPN
ejpam-4387	169	10	}	}	PUNCT
ejpam-4387	169	11	)	)	PUNCT
ejpam-4387	169	12	)	)	PUNCT
ejpam-4387	169	13	∩	∩	ADJ
ejpam-4387	169	14	g(c	g(c	NOUN
ejpam-4387	169	15	)	)	PUNCT
ejpam-4387	169	16	which	which	PRON
ejpam-4387	169	17	is	be	AUX
ejpam-4387	169	18	a	a	DET
ejpam-4387	169	19	basic	basic	ADJ
ejpam-4387	169	20	open	open	ADJ
ejpam-4387	169	21	neighborhood	neighborhood	NOUN
ejpam-4387	169	22	of	of	ADP
ejpam-4387	169	23	f(a	f(a	NOUN
ejpam-4387	169	24	)	)	PUNCT
ejpam-4387	169	25	in	in	ADP
ejpam-4387	169	26	g(c	g(c	NOUN
ejpam-4387	169	27	)	)	PUNCT
ejpam-4387	169	28	.	.	PUNCT
ejpam-4387	170	1	since	since	SCONJ
ejpam-4387	170	2	f|c∩x	f|c∩x	NUM
ejpam-4387	170	3	:	:	PUNCT
ejpam-4387	170	4	c	c	NOUN
ejpam-4387	170	5	∩	∩	X
ejpam-4387	170	6	x	x	SYM
ejpam-4387	170	7	−→	−→	ADJ
ejpam-4387	170	8	f(c	f(c	PROPN
ejpam-4387	170	9	∩	∩	NOUN
ejpam-4387	170	10	x	x	X
ejpam-4387	170	11	)	)	PUNCT
ejpam-4387	170	12	is	be	AUX
ejpam-4387	170	13	a	a	DET
ejpam-4387	170	14	homeomorphism	homeomorphism	NOUN
ejpam-4387	170	15	,	,	PUNCT
ejpam-4387	170	16	then	then	ADV
ejpam-4387	170	17	there	there	PRON
ejpam-4387	170	18	exists	exist	VERB
ejpam-4387	170	19	an	an	DET
ejpam-4387	170	20	open	open	ADJ
ejpam-4387	170	21	set	set	NOUN
ejpam-4387	170	22	u	u	NOUN
ejpam-4387	170	23	in	in	ADP
ejpam-4387	170	24	x	x	PUNCT
ejpam-4387	170	25	with	with	ADP
ejpam-4387	170	26	a	a	DET
ejpam-4387	170	27	∈	∈	PROPN
ejpam-4387	170	28	u	u	NOUN
ejpam-4387	170	29	and	and	CCONJ
ejpam-4387	170	30	f|c∩x	f|c∩x	NUM
ejpam-4387	170	31	(	(	PUNCT
ejpam-4387	170	32	u	u	NOUN
ejpam-4387	170	33	∩c	∩c	NOUN
ejpam-4387	170	34	)	)	PUNCT
ejpam-4387	170	35	⊆	⊆	NUM
ejpam-4387	170	36	w	w	NOUN
ejpam-4387	170	37	.	.	PUNCT
ejpam-4387	171	1	now	now	ADV
ejpam-4387	171	2	,	,	PUNCT
ejpam-4387	171	3	(	(	PUNCT
ejpam-4387	171	4	u∪(u	u∪(u	VERB
ejpam-4387	171	5	′\{a′}))∩c	′\{a′}))∩c	PROPN
ejpam-4387	171	6	=	=	PUNCT
ejpam-4387	171	7	g	g	NOUN
ejpam-4387	171	8	is	be	AUX
ejpam-4387	171	9	open	open	ADJ
ejpam-4387	171	10	in	in	ADP
ejpam-4387	171	11	c∩x	c∩x	PROPN
ejpam-4387	171	12	such	such	ADJ
ejpam-4387	171	13	that	that	SCONJ
ejpam-4387	171	14	a	a	DET
ejpam-4387	171	15	∈	∈	PROPN
ejpam-4387	171	16	g	g	NOUN
ejpam-4387	171	17	and	and	CCONJ
ejpam-4387	171	18	g|c	g|c	PROPN
ejpam-4387	171	19	(	(	PUNCT
ejpam-4387	171	20	g	g	NOUN
ejpam-4387	171	21	)	)	PUNCT
ejpam-4387	172	1	⊆	⊆	NUM
ejpam-4387	172	2	h.	h.	PROPN
ejpam-4387	172	3	therefore	therefore	ADV
ejpam-4387	172	4	,	,	PUNCT
ejpam-4387	172	5	g|c	g|c	PROPN
ejpam-4387	172	6	is	be	AUX
ejpam-4387	172	7	continuous	continuous	ADJ
ejpam-4387	172	8	.	.	PUNCT
ejpam-4387	173	1	now	now	ADV
ejpam-4387	173	2	,	,	PUNCT
ejpam-4387	173	3	we	we	PRON
ejpam-4387	173	4	show	show	VERB
ejpam-4387	173	5	that	that	SCONJ
ejpam-4387	173	6	g|c	g|c	PROPN
ejpam-4387	173	7	is	be	AUX
ejpam-4387	173	8	open	open	ADJ
ejpam-4387	173	9	.	.	PUNCT
ejpam-4387	174	1	let	let	VERB
ejpam-4387	174	2	k	k	PROPN
ejpam-4387	174	3	∪	∪	X
ejpam-4387	174	4	(	(	PUNCT
ejpam-4387	174	5	k	k	NOUN
ejpam-4387	174	6	′	′	NOUN
ejpam-4387	174	7	\	\	PROPN
ejpam-4387	174	8	{	{	PUNCT
ejpam-4387	174	9	k′	k′	PROPN
ejpam-4387	174	10	}	}	PUNCT
ejpam-4387	174	11	)	)	PUNCT
ejpam-4387	174	12	,	,	PUNCT
ejpam-4387	174	13	where	where	SCONJ
ejpam-4387	174	14	k	k	PROPN
ejpam-4387	174	15	∈	∈	PROPN
ejpam-4387	174	16	k	k	PROPN
ejpam-4387	174	17	and	and	CCONJ
ejpam-4387	174	18	k	k	PROPN
ejpam-4387	174	19	is	be	AUX
ejpam-4387	174	20	open	open	ADJ
ejpam-4387	174	21	in	in	ADP
ejpam-4387	174	22	x	x	X
ejpam-4387	174	23	,	,	PUNCT
ejpam-4387	174	24	be	be	AUX
ejpam-4387	174	25	any	any	DET
ejpam-4387	174	26	basic	basic	ADJ
ejpam-4387	174	27	open	open	ADJ
ejpam-4387	174	28	set	set	NOUN
ejpam-4387	174	29	in	in	ADP
ejpam-4387	174	30	a(x	a(x	NOUN
ejpam-4387	174	31	)	)	PUNCT
ejpam-4387	174	32	,	,	PUNCT
ejpam-4387	174	33	then	then	ADV
ejpam-4387	174	34	(	(	PUNCT
ejpam-4387	174	35	k	k	NOUN
ejpam-4387	174	36	∩c)∪	∩c)∪	PROPN
ejpam-4387	174	37	(	(	PUNCT
ejpam-4387	174	38	(	(	PUNCT
ejpam-4387	174	39	k	k	NOUN
ejpam-4387	174	40	′	′	NUM
ejpam-4387	174	41	∩c	∩c	NOUN
ejpam-4387	174	42	)	)	PUNCT
ejpam-4387	174	43	\	\	PROPN
ejpam-4387	174	44	{	{	PUNCT
ejpam-4387	174	45	k′	k′	PROPN
ejpam-4387	174	46	}	}	PUNCT
ejpam-4387	174	47	)	)	PUNCT
ejpam-4387	174	48	is	be	AUX
ejpam-4387	174	49	a	a	DET
ejpam-4387	174	50	basic	basic	ADJ
ejpam-4387	174	51	open	open	ADJ
ejpam-4387	174	52	set	set	NOUN
ejpam-4387	174	53	in	in	ADP
ejpam-4387	174	54	c.	c.	NOUN
ejpam-4387	174	55	since	since	SCONJ
ejpam-4387	174	56	x	x	PROPN
ejpam-4387	174	57	∩c	∩c	NOUN
ejpam-4387	174	58	is	be	AUX
ejpam-4387	174	59	compact	compact	ADJ
ejpam-4387	174	60	in	in	ADP
ejpam-4387	174	61	x	x	NOUN
ejpam-4387	174	62	,	,	PUNCT
ejpam-4387	174	63	then	then	ADV
ejpam-4387	174	64	g|c	g|c	PROPN
ejpam-4387	175	1	(	(	PUNCT
ejpam-4387	175	2	k	k	X
ejpam-4387	175	3	∩	∩	X
ejpam-4387	175	4	(	(	PUNCT
ejpam-4387	175	5	x	x	SYM
ejpam-4387	175	6	∩c	∩c	NOUN
ejpam-4387	175	7	)	)	PUNCT
ejpam-4387	175	8	)	)	PUNCT
ejpam-4387	176	1	=	=	X
ejpam-4387	176	2	f|x∩c	f|x∩c	PRON
ejpam-4387	177	1	(	(	PUNCT
ejpam-4387	177	2	k	k	X
ejpam-4387	177	3	∩	∩	X
ejpam-4387	177	4	(	(	PUNCT
ejpam-4387	177	5	x	x	SYM
ejpam-4387	177	6	∩c	∩c	NOUN
ejpam-4387	177	7	)	)	PUNCT
ejpam-4387	177	8	)	)	PUNCT
ejpam-4387	177	9	is	be	AUX
ejpam-4387	177	10	open	open	ADJ
ejpam-4387	177	11	in	in	ADP
ejpam-4387	177	12	y	y	PROPN
ejpam-4387	177	13	∩f(c∩x	∩f(c∩x	PROPN
ejpam-4387	177	14	)	)	PUNCT
ejpam-4387	177	15	as	as	SCONJ
ejpam-4387	177	16	f|x∩c	f|x∩c	NOUN
ejpam-4387	177	17	is	be	AUX
ejpam-4387	177	18	a	a	DET
ejpam-4387	177	19	homeomorphism	homeomorphism	NOUN
ejpam-4387	177	20	.	.	PUNCT
ejpam-4387	178	1	thus	thus	ADV
ejpam-4387	178	2	k∩c	k∩c	PROPN
ejpam-4387	178	3	is	be	AUX
ejpam-4387	178	4	open	open	ADJ
ejpam-4387	178	5	in	in	ADP
ejpam-4387	178	6	y	y	PROPN
ejpam-4387	178	7	∩f(x∩c	∩f(x∩c	PROPN
ejpam-4387	178	8	)	)	PUNCT
ejpam-4387	178	9	.	.	PUNCT
ejpam-4387	179	1	also	also	ADV
ejpam-4387	179	2	,	,	PUNCT
ejpam-4387	179	3	g((k	g((k	NOUN
ejpam-4387	179	4	′	′	NUM
ejpam-4387	179	5	∩c	∩c	NOUN
ejpam-4387	179	6	)	)	PUNCT
ejpam-4387	179	7	\	\	PROPN
ejpam-4387	179	8	{	{	PUNCT
ejpam-4387	179	9	k′	k′	PROPN
ejpam-4387	179	10	}	}	PUNCT
ejpam-4387	179	11	)	)	PUNCT
ejpam-4387	179	12	is	be	AUX
ejpam-4387	179	13	open	open	ADJ
ejpam-4387	179	14	in	in	ADP
ejpam-4387	179	15	y	y	PROPN
ejpam-4387	179	16	′	′	NUM
ejpam-4387	179	17	∩	∩	ADJ
ejpam-4387	179	18	g(c	g(c	NOUN
ejpam-4387	179	19	)	)	PUNCT
ejpam-4387	179	20	being	be	AUX
ejpam-4387	179	21	a	a	DET
ejpam-4387	179	22	set	set	NOUN
ejpam-4387	179	23	of	of	ADP
ejpam-4387	179	24	isolated	isolated	ADJ
ejpam-4387	179	25	points	point	NOUN
ejpam-4387	179	26	.	.	PUNCT
ejpam-4387	180	1	thus	thus	ADV
ejpam-4387	180	2	g|c	g|c	PROPN
ejpam-4387	180	3	is	be	AUX
ejpam-4387	180	4	an	an	DET
ejpam-4387	180	5	open	open	ADJ
ejpam-4387	180	6	function	function	NOUN
ejpam-4387	180	7	.	.	PUNCT
ejpam-4387	181	1	therefore	therefore	ADV
ejpam-4387	181	2	,	,	PUNCT
ejpam-4387	181	3	g|c	g|c	PROPN
ejpam-4387	181	4	is	be	AUX
ejpam-4387	181	5	a	a	DET
ejpam-4387	181	6	homeomorphism	homeomorphism	NOUN
ejpam-4387	181	7	.	.	PUNCT
ejpam-4387	182	1	next	next	ADV
ejpam-4387	182	2	,	,	PUNCT
ejpam-4387	182	3	we	we	PRON
ejpam-4387	182	4	present	present	VERB
ejpam-4387	182	5	a	a	DET
ejpam-4387	182	6	result	result	NOUN
ejpam-4387	182	7	about	about	ADP
ejpam-4387	182	8	dowker	dowker	NOUN
ejpam-4387	182	9	topological	topological	ADJ
ejpam-4387	182	10	spaces	space	NOUN
ejpam-4387	182	11	.	.	PUNCT
ejpam-4387	183	1	this	this	DET
ejpam-4387	183	2	result	result	NOUN
ejpam-4387	183	3	may	may	AUX
ejpam-4387	183	4	seem	seem	VERB
ejpam-4387	183	5	repeated	repeat	VERB
ejpam-4387	183	6	as	as	SCONJ
ejpam-4387	183	7	it	it	PRON
ejpam-4387	183	8	was	be	AUX
ejpam-4387	183	9	mentioned	mention	VERB
ejpam-4387	183	10	about	about	ADP
ejpam-4387	183	11	l	l	NOUN
ejpam-4387	183	12	-	-	NOUN
ejpam-4387	183	13	normality	normality	NOUN
ejpam-4387	183	14	in	in	ADP
ejpam-4387	183	15	[	[	X
ejpam-4387	183	16	11	11	NUM
ejpam-4387	183	17	]	]	PUNCT
ejpam-4387	183	18	but	but	CCONJ
ejpam-4387	183	19	it	it	PRON
ejpam-4387	183	20	is	be	AUX
ejpam-4387	183	21	so	so	ADV
ejpam-4387	183	22	interesting	interesting	ADJ
ejpam-4387	183	23	that	that	SCONJ
ejpam-4387	183	24	we	we	PRON
ejpam-4387	183	25	mention	mention	VERB
ejpam-4387	183	26	it	it	PRON
ejpam-4387	183	27	again	again	ADV
ejpam-4387	183	28	with	with	ADP
ejpam-4387	183	29	regards	regard	NOUN
ejpam-4387	183	30	to	to	ADP
ejpam-4387	183	31	p	p	PROPN
ejpam-4387	183	32	-normality	-normality	NOUN
ejpam-4387	183	33	.	.	PUNCT
ejpam-4387	184	1	recall	recall	VERB
ejpam-4387	184	2	that	that	SCONJ
ejpam-4387	184	3	a	a	DET
ejpam-4387	184	4	dowker	dowker	NOUN
ejpam-4387	184	5	space	space	NOUN
ejpam-4387	184	6	is	be	AUX
ejpam-4387	184	7	a	a	DET
ejpam-4387	184	8	t4	t4	PROPN
ejpam-4387	184	9	space	space	NOUN
ejpam-4387	184	10	whose	whose	DET
ejpam-4387	184	11	product	product	NOUN
ejpam-4387	184	12	with	with	ADP
ejpam-4387	184	13	i	i	PRON
ejpam-4387	184	14	,	,	PUNCT
ejpam-4387	184	15	i	i	PRON
ejpam-4387	184	16	=	=	PUNCT
ejpam-4387	185	1	[	[	X
ejpam-4387	185	2	0	0	NUM
ejpam-4387	185	3	,	,	PUNCT
ejpam-4387	185	4	1	1	NUM
ejpam-4387	185	5	]	]	PUNCT
ejpam-4387	185	6	with	with	ADP
ejpam-4387	185	7	its	its	PRON
ejpam-4387	185	8	usual	usual	ADJ
ejpam-4387	185	9	metric	metric	NOUN
ejpam-4387	185	10	,	,	PUNCT
ejpam-4387	185	11	is	be	AUX
ejpam-4387	185	12	not	not	PART
ejpam-4387	185	13	normal	normal	ADJ
ejpam-4387	185	14	.	.	PUNCT
ejpam-4387	186	1	m.	m.	PROPN
ejpam-4387	186	2	e.	e.	PROPN
ejpam-4387	186	3	rudin	rudin	PROPN
ejpam-4387	186	4	used	use	VERB
ejpam-4387	186	5	the	the	DET
ejpam-4387	186	6	existence	existence	NOUN
ejpam-4387	186	7	of	of	ADP
ejpam-4387	186	8	a	a	DET
ejpam-4387	186	9	suslin	suslin	ADJ
ejpam-4387	186	10	line	line	NOUN
ejpam-4387	186	11	to	to	PART
ejpam-4387	186	12	obtain	obtain	VERB
ejpam-4387	186	13	a	a	DET
ejpam-4387	186	14	dowker	dowker	NOUN
ejpam-4387	186	15	space	space	NOUN
ejpam-4387	186	16	which	which	PRON
ejpam-4387	186	17	is	be	AUX
ejpam-4387	186	18	hereditarily	hereditarily	ADV
ejpam-4387	186	19	separable	separable	ADJ
ejpam-4387	186	20	and	and	CCONJ
ejpam-4387	186	21	first	first	ADJ
ejpam-4387	186	22	countable	countable	ADJ
ejpam-4387	186	23	[	[	X
ejpam-4387	186	24	12	12	NUM
ejpam-4387	186	25	]	]	PUNCT
ejpam-4387	186	26	.	.	PUNCT
ejpam-4387	187	1	using	use	VERB
ejpam-4387	187	2	ch	ch	NOUN
ejpam-4387	187	3	,	,	PUNCT
ejpam-4387	187	4	i.	i.	PROPN
ejpam-4387	187	5	juhász	juhász	PROPN
ejpam-4387	187	6	,	,	PUNCT
ejpam-4387	187	7	k.	k.	PROPN
ejpam-4387	187	8	kunen	kunen	PROPN
ejpam-4387	187	9	,	,	PUNCT
ejpam-4387	187	10	and	and	CCONJ
ejpam-4387	187	11	m.	m.	PROPN
ejpam-4387	187	12	e.	e.	PROPN
ejpam-4387	187	13	rudin	rudin	PROPN
ejpam-4387	187	14	constructed	construct	VERB
ejpam-4387	187	15	a	a	DET
ejpam-4387	187	16	first	first	ADJ
ejpam-4387	187	17	countable	countable	ADJ
ejpam-4387	187	18	hereditarily	hereditarily	ADV
ejpam-4387	187	19	separable	separable	ADJ
ejpam-4387	187	20	real	real	ADJ
ejpam-4387	187	21	compact	compact	ADJ
ejpam-4387	187	22	dowker	dowker	NOUN
ejpam-4387	187	23	space	space	NOUN
ejpam-4387	187	24	[	[	X
ejpam-4387	187	25	8	8	NUM
ejpam-4387	187	26	]	]	PUNCT
ejpam-4387	187	27	.	.	PUNCT
ejpam-4387	188	1	weiss	weiss	PROPN
ejpam-4387	188	2	constructed	construct	VERB
ejpam-4387	188	3	a	a	DET
ejpam-4387	188	4	first	first	ADJ
ejpam-4387	188	5	countable	countable	ADJ
ejpam-4387	188	6	separable	separable	NOUN
ejpam-4387	188	7	locally	locally	ADV
ejpam-4387	188	8	compact	compact	ADJ
ejpam-4387	188	9	dowker	dowker	NOUN
ejpam-4387	188	10	space	space	NOUN
ejpam-4387	188	11	whose	whose	DET
ejpam-4387	188	12	existence	existence	NOUN
ejpam-4387	188	13	is	be	AUX
ejpam-4387	188	14	consistent	consistent	ADJ
ejpam-4387	188	15	with	with	ADP
ejpam-4387	188	16	ma	ma	PROPN
ejpam-4387	188	17	l.	l.	PROPN
ejpam-4387	188	18	kalantan	kalantan	PROPN
ejpam-4387	188	19	,	,	PUNCT
ejpam-4387	188	20	m.	m.	NOUN
ejpam-4387	188	21	mansouri	mansouri	PROPN
ejpam-4387	188	22	/	/	SYM
ejpam-4387	188	23	eur	eur	PROPN
ejpam-4387	188	24	.	.	PUNCT
ejpam-4387	189	1	j.	j.	PROPN
ejpam-4387	189	2	pure	pure	PROPN
ejpam-4387	189	3	appl	appl	PROPN
ejpam-4387	189	4	.	.	PROPN
ejpam-4387	189	5	math	math	PROPN
ejpam-4387	189	6	,	,	PUNCT
ejpam-4387	189	7	15	15	NUM
ejpam-4387	189	8	(	(	PUNCT
ejpam-4387	189	9	2	2	NUM
ejpam-4387	189	10	)	)	PUNCT
ejpam-4387	189	11	(	(	PUNCT
ejpam-4387	189	12	2022	2022	NUM
ejpam-4387	189	13	)	)	PUNCT
ejpam-4387	189	14	,	,	PUNCT
ejpam-4387	189	15	774	774	NUM
ejpam-4387	189	16	-	-	SYM
ejpam-4387	189	17	783	783	NUM
ejpam-4387	189	18	779	779	NUM
ejpam-4387	189	19	+	+	CCONJ
ejpam-4387	189	20	¬	¬	PROPN
ejpam-4387	189	21	ch	ch	NOUN
ejpam-4387	190	1	[	[	X
ejpam-4387	190	2	14	14	NUM
ejpam-4387	190	3	]	]	PUNCT
ejpam-4387	190	4	.	.	PUNCT
ejpam-4387	191	1	we	we	PRON
ejpam-4387	191	2	already	already	ADV
ejpam-4387	191	3	know	know	VERB
ejpam-4387	191	4	that	that	SCONJ
ejpam-4387	191	5	such	such	ADJ
ejpam-4387	191	6	spaces	space	NOUN
ejpam-4387	191	7	are	be	AUX
ejpam-4387	191	8	consistent	consistent	ADJ
ejpam-4387	191	9	examples	example	NOUN
ejpam-4387	191	10	of	of	ADP
ejpam-4387	191	11	dowker	dowker	NOUN
ejpam-4387	191	12	spaces	space	NOUN
ejpam-4387	191	13	whose	whose	DET
ejpam-4387	191	14	product	product	NOUN
ejpam-4387	191	15	with	with	ADP
ejpam-4387	191	16	i	i	PRON
ejpam-4387	191	17	are	be	AUX
ejpam-4387	191	18	not	not	PART
ejpam-4387	191	19	l	l	NOUN
ejpam-4387	191	20	-	-	ADJ
ejpam-4387	191	21	normal	normal	ADJ
ejpam-4387	191	22	[	[	X
ejpam-4387	191	23	11	11	NUM
ejpam-4387	191	24	]	]	PUNCT
ejpam-4387	191	25	.	.	PUNCT
ejpam-4387	192	1	this	this	PRON
ejpam-4387	192	2	means	mean	VERB
ejpam-4387	192	3	that	that	SCONJ
ejpam-4387	192	4	they	they	PRON
ejpam-4387	192	5	can	can	AUX
ejpam-4387	192	6	not	not	PART
ejpam-4387	192	7	be	be	AUX
ejpam-4387	192	8	p	p	NOUN
ejpam-4387	192	9	-normal	-normal	ADJ
ejpam-4387	192	10	either	either	ADV
ejpam-4387	192	11	;	;	PUNCT
ejpam-4387	192	12	since	since	SCONJ
ejpam-4387	192	13	any	any	DET
ejpam-4387	192	14	regular	regular	ADJ
ejpam-4387	192	15	p	p	NOUN
ejpam-4387	192	16	-normal	-normal	ADJ
ejpam-4387	192	17	space	space	NOUN
ejpam-4387	192	18	is	be	AUX
ejpam-4387	192	19	l	l	NOUN
ejpam-4387	192	20	-	-	ADJ
ejpam-4387	192	21	normal	normal	ADJ
ejpam-4387	192	22	[	[	X
ejpam-4387	192	23	10	10	NUM
ejpam-4387	192	24	]	]	PUNCT
ejpam-4387	192	25	.	.	PUNCT
ejpam-4387	193	1	we	we	PRON
ejpam-4387	193	2	move	move	VERB
ejpam-4387	193	3	on	on	ADP
ejpam-4387	193	4	to	to	ADP
ejpam-4387	193	5	studying	study	VERB
ejpam-4387	193	6	the	the	DET
ejpam-4387	193	7	p	p	ADJ
ejpam-4387	193	8	-normality	-normality	NOUN
ejpam-4387	193	9	of	of	ADP
ejpam-4387	193	10	the	the	DET
ejpam-4387	193	11	closed	closed	ADJ
ejpam-4387	193	12	extension	extension	NOUN
ejpam-4387	193	13	.	.	PUNCT
ejpam-4387	194	1	let	let	VERB
ejpam-4387	194	2	(	(	PUNCT
ejpam-4387	194	3	x	x	X
ejpam-4387	194	4	,	,	PUNCT
ejpam-4387	194	5	τ	τ	PROPN
ejpam-4387	194	6	)	)	PUNCT
ejpam-4387	194	7	be	be	AUX
ejpam-4387	194	8	a	a	DET
ejpam-4387	194	9	topological	topological	ADJ
ejpam-4387	194	10	space	space	NOUN
ejpam-4387	194	11	and	and	CCONJ
ejpam-4387	194	12	let	let	VERB
ejpam-4387	194	13	p	p	PRON
ejpam-4387	194	14	be	be	AUX
ejpam-4387	194	15	an	an	DET
ejpam-4387	194	16	object	object	NOUN
ejpam-4387	194	17	not	not	PART
ejpam-4387	194	18	in	in	ADP
ejpam-4387	194	19	x	x	NOUN
ejpam-4387	194	20	,	,	PUNCT
ejpam-4387	194	21	i.e.	i.e.	X
ejpam-4387	194	22	,	,	PUNCT
ejpam-4387	194	23	p	p	PROPN
ejpam-4387	194	24	̸∈	̸∈	PROPN
ejpam-4387	194	25	x.	x.	PROPN
ejpam-4387	194	26	put	put	VERB
ejpam-4387	194	27	xp	xp	NOUN
ejpam-4387	195	1	=	=	NOUN
ejpam-4387	195	2	x	x	SYM
ejpam-4387	195	3	∪	∪	ADP
ejpam-4387	195	4	{	{	PUNCT
ejpam-4387	195	5	p	p	NOUN
ejpam-4387	195	6	}	}	PUNCT
ejpam-4387	195	7	.	.	PUNCT
ejpam-4387	196	1	define	define	VERB
ejpam-4387	196	2	a	a	DET
ejpam-4387	196	3	topology	topology	NOUN
ejpam-4387	196	4	τ	τ	X
ejpam-4387	196	5	⋆	⋆	VERB
ejpam-4387	196	6	on	on	ADP
ejpam-4387	196	7	xp	xp	INTJ
ejpam-4387	196	8	by	by	ADP
ejpam-4387	196	9	τ	τ	PROPN
ejpam-4387	196	10	⋆	⋆	X
ejpam-4387	196	11	=	=	NOUN
ejpam-4387	196	12	{	{	PUNCT
ejpam-4387	196	13	∅	∅	NOUN
ejpam-4387	196	14	}	}	PUNCT
ejpam-4387	196	15	∪	∪	VERB
ejpam-4387	196	16	{	{	PUNCT
ejpam-4387	196	17	u	u	NOUN
ejpam-4387	196	18	∪	∪	NOUN
ejpam-4387	196	19	{	{	PUNCT
ejpam-4387	196	20	p	p	NOUN
ejpam-4387	196	21	}	}	PUNCT
ejpam-4387	196	22	:	:	PUNCT
ejpam-4387	196	23	u	u	PROPN
ejpam-4387	196	24	∈	∈	PROPN
ejpam-4387	196	25	τ	τ	X
ejpam-4387	196	26	}	}	PUNCT
ejpam-4387	196	27	.	.	PUNCT
ejpam-4387	197	1	the	the	DET
ejpam-4387	197	2	space	space	NOUN
ejpam-4387	197	3	(	(	PUNCT
ejpam-4387	197	4	xp	xp	INTJ
ejpam-4387	197	5	,	,	PUNCT
ejpam-4387	197	6	τ	τ	PROPN
ejpam-4387	197	7	⋆	⋆	VERB
ejpam-4387	197	8	)	)	PUNCT
ejpam-4387	197	9	is	be	AUX
ejpam-4387	197	10	called	call	VERB
ejpam-4387	197	11	the	the	DET
ejpam-4387	197	12	closed	closed	ADJ
ejpam-4387	197	13	extension	extension	NOUN
ejpam-4387	197	14	space	space	NOUN
ejpam-4387	197	15	of	of	ADP
ejpam-4387	197	16	(	(	PUNCT
ejpam-4387	197	17	x	x	INTJ
ejpam-4387	197	18	,	,	PUNCT
ejpam-4387	197	19	τ	τ	PROPN
ejpam-4387	197	20	)	)	PUNCT
ejpam-4387	197	21	,	,	PUNCT
ejpam-4387	197	22	[	[	X
ejpam-4387	197	23	13	13	NUM
ejpam-4387	197	24	,	,	PUNCT
ejpam-4387	197	25	example	example	NOUN
ejpam-4387	197	26	12	12	NUM
ejpam-4387	197	27	]	]	PUNCT
ejpam-4387	197	28	.	.	PUNCT
ejpam-4387	198	1	since	since	SCONJ
ejpam-4387	198	2	characterizing	characterize	VERB
ejpam-4387	198	3	all	all	DET
ejpam-4387	198	4	paracompact	paracompact	ADJ
ejpam-4387	198	5	subspaces	subspace	NOUN
ejpam-4387	198	6	[	[	X
ejpam-4387	198	7	10	10	NUM
ejpam-4387	198	8	]	]	PUNCT
ejpam-4387	198	9	is	be	AUX
ejpam-4387	198	10	a	a	DET
ejpam-4387	198	11	core	core	NOUN
ejpam-4387	198	12	subject	subject	NOUN
ejpam-4387	198	13	in	in	ADP
ejpam-4387	198	14	the	the	DET
ejpam-4387	198	15	notion	notion	NOUN
ejpam-4387	198	16	of	of	ADP
ejpam-4387	198	17	p	p	PROPN
ejpam-4387	198	18	-normality	-normality	NOUN
ejpam-4387	198	19	,	,	PUNCT
ejpam-4387	198	20	we	we	PRON
ejpam-4387	198	21	will	will	AUX
ejpam-4387	198	22	start	start	VERB
ejpam-4387	198	23	with	with	ADP
ejpam-4387	198	24	characterizing	characterize	VERB
ejpam-4387	198	25	all	all	DET
ejpam-4387	198	26	paracompact	paracompact	ADJ
ejpam-4387	198	27	subspaces	subspace	NOUN
ejpam-4387	198	28	of	of	ADP
ejpam-4387	198	29	the	the	DET
ejpam-4387	198	30	closed	closed	ADJ
ejpam-4387	198	31	extension	extension	NOUN
ejpam-4387	198	32	space	space	NOUN
ejpam-4387	198	33	(	(	PUNCT
ejpam-4387	198	34	xp	xp	INTJ
ejpam-4387	198	35	,	,	PUNCT
ejpam-4387	198	36	τ	τ	PROPN
ejpam-4387	198	37	⋆	⋆	NOUN
ejpam-4387	198	38	)	)	PUNCT
ejpam-4387	198	39	of	of	ADP
ejpam-4387	198	40	a	a	DET
ejpam-4387	198	41	given	give	VERB
ejpam-4387	198	42	space	space	NOUN
ejpam-4387	198	43	(	(	PUNCT
ejpam-4387	198	44	x	x	X
ejpam-4387	198	45	,	,	PUNCT
ejpam-4387	198	46	τ	τ	PROPN
ejpam-4387	198	47	)	)	PUNCT
ejpam-4387	198	48	.	.	PUNCT
ejpam-4387	199	1	proposition	proposition	NOUN
ejpam-4387	199	2	2	2	NUM
ejpam-4387	199	3	.	.	PUNCT
ejpam-4387	200	1	let	let	AUX
ejpam-4387	200	2	(	(	PUNCT
ejpam-4387	200	3	x	x	X
ejpam-4387	200	4	,	,	PUNCT
ejpam-4387	200	5	τ	τ	PROPN
ejpam-4387	200	6	)	)	PUNCT
ejpam-4387	200	7	be	be	AUX
ejpam-4387	200	8	a	a	DET
ejpam-4387	200	9	topological	topological	ADJ
ejpam-4387	200	10	space	space	NOUN
ejpam-4387	200	11	.	.	PUNCT
ejpam-4387	201	1	consider	consider	VERB
ejpam-4387	201	2	the	the	DET
ejpam-4387	201	3	closed	closed	ADJ
ejpam-4387	201	4	extension	extension	NOUN
ejpam-4387	201	5	space	space	NOUN
ejpam-4387	201	6	(	(	PUNCT
ejpam-4387	201	7	xp	xp	INTJ
ejpam-4387	201	8	,	,	PUNCT
ejpam-4387	201	9	τ	τ	PROPN
ejpam-4387	201	10	⋆	⋆	NOUN
ejpam-4387	201	11	)	)	PUNCT
ejpam-4387	201	12	of	of	ADP
ejpam-4387	201	13	(	(	PUNCT
ejpam-4387	201	14	x	x	INTJ
ejpam-4387	201	15	,	,	PUNCT
ejpam-4387	201	16	τ	τ	PROPN
ejpam-4387	201	17	)	)	PUNCT
ejpam-4387	201	18	.	.	PUNCT
ejpam-4387	202	1	let	let	VERB
ejpam-4387	202	2	a	a	DET
ejpam-4387	202	3	⊆	⊆	NUM
ejpam-4387	202	4	xp	xp	NOUN
ejpam-4387	202	5	.	.	PUNCT
ejpam-4387	203	1	if	if	SCONJ
ejpam-4387	203	2	p	p	PROPN
ejpam-4387	203	3	̸∈	̸∈	PROPN
ejpam-4387	203	4	a	a	PROPN
ejpam-4387	203	5	,	,	PUNCT
ejpam-4387	203	6	then	then	ADV
ejpam-4387	203	7	a	a	PRON
ejpam-4387	203	8	is	be	AUX
ejpam-4387	203	9	a	a	DET
ejpam-4387	203	10	paracompact	paracompact	NOUN
ejpam-4387	203	11	subset	subset	NOUN
ejpam-4387	203	12	in	in	ADP
ejpam-4387	203	13	(	(	PUNCT
ejpam-4387	203	14	xp	xp	INTJ
ejpam-4387	203	15	,	,	PUNCT
ejpam-4387	203	16	τ	τ	PROPN
ejpam-4387	203	17	⋆	⋆	NOUN
ejpam-4387	203	18	)	)	PUNCT
ejpam-4387	203	19	if	if	SCONJ
ejpam-4387	203	20	and	and	CCONJ
ejpam-4387	203	21	only	only	ADV
ejpam-4387	203	22	if	if	SCONJ
ejpam-4387	203	23	a	a	PRON
ejpam-4387	203	24	is	be	AUX
ejpam-4387	203	25	a	a	DET
ejpam-4387	203	26	paracompact	paracompact	NOUN
ejpam-4387	203	27	subset	subset	NOUN
ejpam-4387	203	28	in	in	ADP
ejpam-4387	203	29	(	(	PUNCT
ejpam-4387	203	30	x	x	INTJ
ejpam-4387	203	31	,	,	PUNCT
ejpam-4387	203	32	τ	τ	PROPN
ejpam-4387	203	33	)	)	PUNCT
ejpam-4387	203	34	.	.	PUNCT
ejpam-4387	204	1	if	if	SCONJ
ejpam-4387	204	2	p	p	PROPN
ejpam-4387	204	3	∈	∈	PROPN
ejpam-4387	204	4	a	a	PRON
ejpam-4387	204	5	,	,	PUNCT
ejpam-4387	204	6	then	then	ADV
ejpam-4387	204	7	a	a	PRON
ejpam-4387	204	8	is	be	AUX
ejpam-4387	204	9	a	a	DET
ejpam-4387	204	10	paracompact	paracompact	NOUN
ejpam-4387	204	11	subset	subset	NOUN
ejpam-4387	204	12	in	in	ADP
ejpam-4387	204	13	(	(	PUNCT
ejpam-4387	204	14	xp	xp	INTJ
ejpam-4387	204	15	,	,	PUNCT
ejpam-4387	204	16	τ	τ	PROPN
ejpam-4387	204	17	⋆	⋆	NOUN
ejpam-4387	204	18	)	)	PUNCT
ejpam-4387	204	19	if	if	SCONJ
ejpam-4387	204	20	and	and	CCONJ
ejpam-4387	204	21	only	only	ADV
ejpam-4387	204	22	if	if	SCONJ
ejpam-4387	204	23	a	a	PRON
ejpam-4387	204	24	\	\	NOUN
ejpam-4387	204	25	{	{	PUNCT
ejpam-4387	204	26	p	p	X
ejpam-4387	204	27	}	}	PUNCT
ejpam-4387	204	28	is	be	AUX
ejpam-4387	204	29	a	a	DET
ejpam-4387	204	30	compact	compact	ADJ
ejpam-4387	204	31	subset	subset	NOUN
ejpam-4387	204	32	in	in	ADP
ejpam-4387	204	33	(	(	PUNCT
ejpam-4387	204	34	x	x	INTJ
ejpam-4387	204	35	,	,	PUNCT
ejpam-4387	204	36	τ	τ	PROPN
ejpam-4387	204	37	)	)	PUNCT
ejpam-4387	204	38	.	.	PUNCT
ejpam-4387	205	1	the	the	DET
ejpam-4387	205	2	proof	proof	NOUN
ejpam-4387	205	3	of	of	ADP
ejpam-4387	205	4	this	this	DET
ejpam-4387	205	5	proposition	proposition	NOUN
ejpam-4387	205	6	can	can	AUX
ejpam-4387	205	7	be	be	AUX
ejpam-4387	205	8	found	find	VERB
ejpam-4387	205	9	in	in	ADP
ejpam-4387	205	10	[	[	X
ejpam-4387	205	11	5	5	NUM
ejpam-4387	205	12	]	]	PUNCT
ejpam-4387	205	13	.	.	PUNCT
ejpam-4387	206	1	a	a	DET
ejpam-4387	206	2	space	space	NOUN
ejpam-4387	206	3	is	be	AUX
ejpam-4387	206	4	called	call	VERB
ejpam-4387	206	5	ultra	ultra	ADJ
ejpam-4387	206	6	-	-	VERB
ejpam-4387	206	7	connected	connected	ADJ
ejpam-4387	206	8	if	if	SCONJ
ejpam-4387	206	9	any	any	DET
ejpam-4387	206	10	two	two	NUM
ejpam-4387	206	11	non	non	ADJ
ejpam-4387	206	12	-	-	ADJ
ejpam-4387	206	13	empty	empty	ADJ
ejpam-4387	206	14	closed	closed	ADJ
ejpam-4387	206	15	sets	set	NOUN
ejpam-4387	206	16	intersect	intersect	ADJ
ejpam-4387	206	17	[	[	X
ejpam-4387	206	18	13	13	NUM
ejpam-4387	206	19	]	]	PUNCT
ejpam-4387	206	20	.	.	PUNCT
ejpam-4387	207	1	since	since	SCONJ
ejpam-4387	207	2	any	any	DET
ejpam-4387	207	3	normal	normal	ADJ
ejpam-4387	207	4	space	space	NOUN
ejpam-4387	207	5	is	be	AUX
ejpam-4387	207	6	p	p	NOUN
ejpam-4387	207	7	-normal	-normal	ADJ
ejpam-4387	207	8	(	(	PUNCT
ejpam-4387	207	9	just	just	ADV
ejpam-4387	207	10	by	by	ADP
ejpam-4387	207	11	taking	take	VERB
ejpam-4387	207	12	in	in	ADP
ejpam-4387	207	13	definition	definition	NOUN
ejpam-4387	207	14	1	1	NUM
ejpam-4387	207	15	,	,	PUNCT
ejpam-4387	207	16	y	y	PROPN
ejpam-4387	207	17	=	=	PUNCT
ejpam-4387	207	18	x	x	PROPN
ejpam-4387	207	19	and	and	CCONJ
ejpam-4387	207	20	f	f	X
ejpam-4387	207	21	to	to	PART
ejpam-4387	207	22	be	be	AUX
ejpam-4387	207	23	the	the	DET
ejpam-4387	207	24	identity	identity	NOUN
ejpam-4387	207	25	function	function	NOUN
ejpam-4387	207	26	)	)	PUNCT
ejpam-4387	207	27	then	then	ADV
ejpam-4387	207	28	by	by	ADP
ejpam-4387	207	29	[	[	X
ejpam-4387	207	30	5	5	NUM
ejpam-4387	207	31	,	,	PUNCT
ejpam-4387	207	32	theorem	theorem	VERB
ejpam-4387	207	33	1.4	1.4	NUM
ejpam-4387	207	34	]	]	PUNCT
ejpam-4387	207	35	,	,	PUNCT
ejpam-4387	207	36	we	we	PRON
ejpam-4387	207	37	get	get	VERB
ejpam-4387	207	38	the	the	DET
ejpam-4387	207	39	following	following	NOUN
ejpam-4387	207	40	theorem	theorem	VERB
ejpam-4387	207	41	:	:	PUNCT
ejpam-4387	207	42	if	if	SCONJ
ejpam-4387	207	43	(	(	PUNCT
ejpam-4387	207	44	x	x	X
ejpam-4387	207	45	,	,	PUNCT
ejpam-4387	207	46	τ	τ	PROPN
ejpam-4387	207	47	)	)	PUNCT
ejpam-4387	207	48	is	be	AUX
ejpam-4387	207	49	ultra	ultra	ADJ
ejpam-4387	207	50	-	-	ADJ
ejpam-4387	207	51	connected	connected	ADJ
ejpam-4387	207	52	,	,	PUNCT
ejpam-4387	207	53	then	then	ADV
ejpam-4387	207	54	its	its	PRON
ejpam-4387	207	55	closed	closed	ADJ
ejpam-4387	207	56	extension	extension	NOUN
ejpam-4387	207	57	(	(	PUNCT
ejpam-4387	207	58	xp	xp	INTJ
ejpam-4387	207	59	,	,	PUNCT
ejpam-4387	207	60	τ	τ	PROPN
ejpam-4387	207	61	⋆	⋆	VERB
ejpam-4387	207	62	)	)	PUNCT
ejpam-4387	207	63	is	be	AUX
ejpam-4387	207	64	p	p	X
ejpam-4387	207	65	-normal	-normal	ADJ
ejpam-4387	207	66	[	[	PUNCT
ejpam-4387	207	67	5	5	NUM
ejpam-4387	207	68	]	]	PUNCT
ejpam-4387	207	69	.	.	PUNCT
ejpam-4387	208	1	recall	recall	VERB
ejpam-4387	208	2	that	that	SCONJ
ejpam-4387	208	3	a	a	DET
ejpam-4387	208	4	topological	topological	ADJ
ejpam-4387	208	5	space	space	NOUN
ejpam-4387	208	6	x	x	PUNCT
ejpam-4387	208	7	is	be	AUX
ejpam-4387	208	8	called	call	VERB
ejpam-4387	208	9	c	c	NOUN
ejpam-4387	208	10	-	-	NOUN
ejpam-4387	208	11	normal	normal	ADJ
ejpam-4387	208	12	if	if	SCONJ
ejpam-4387	208	13	there	there	PRON
ejpam-4387	208	14	exist	exist	VERB
ejpam-4387	208	15	a	a	DET
ejpam-4387	208	16	normal	normal	ADJ
ejpam-4387	208	17	space	space	NOUN
ejpam-4387	208	18	y	y	PROPN
ejpam-4387	208	19	and	and	CCONJ
ejpam-4387	208	20	a	a	DET
ejpam-4387	208	21	bijective	bijective	ADJ
ejpam-4387	208	22	function	function	NOUN
ejpam-4387	209	1	f	f	NOUN
ejpam-4387	209	2	:	:	PUNCT
ejpam-4387	209	3	x	x	PUNCT
ejpam-4387	209	4	−→	−→	NOUN
ejpam-4387	209	5	y	y	PROPN
ejpam-4387	209	6	such	such	ADJ
ejpam-4387	209	7	that	that	SCONJ
ejpam-4387	209	8	the	the	DET
ejpam-4387	209	9	restriction	restriction	NOUN
ejpam-4387	209	10	f|c	f|c	NOUN
ejpam-4387	209	11	:	:	PUNCT
ejpam-4387	209	12	c	c	AUX
ejpam-4387	209	13	−→	−→	NOUN
ejpam-4387	209	14	f(c	f(c	PROPN
ejpam-4387	209	15	)	)	PUNCT
ejpam-4387	209	16	is	be	AUX
ejpam-4387	209	17	a	a	DET
ejpam-4387	209	18	homeomorphism	homeomorphism	NOUN
ejpam-4387	209	19	for	for	ADP
ejpam-4387	209	20	each	each	DET
ejpam-4387	209	21	compact	compact	ADJ
ejpam-4387	209	22	subspace	subspace	NOUN
ejpam-4387	209	23	c	c	NOUN
ejpam-4387	209	24	⊆	⊆	NUM
ejpam-4387	209	25	x	x	PUNCT
ejpam-4387	210	1	[	[	X
ejpam-4387	210	2	4	4	NUM
ejpam-4387	210	3	]	]	PUNCT
ejpam-4387	210	4	.	.	PUNCT
ejpam-4387	211	1	now	now	ADV
ejpam-4387	211	2	,	,	PUNCT
ejpam-4387	211	3	in	in	ADP
ejpam-4387	211	4	[	[	X
ejpam-4387	211	5	5	5	X
ejpam-4387	211	6	]	]	PUNCT
ejpam-4387	211	7	it	it	PRON
ejpam-4387	211	8	was	be	AUX
ejpam-4387	211	9	proved	prove	VERB
ejpam-4387	211	10	that	that	SCONJ
ejpam-4387	211	11	the	the	DET
ejpam-4387	211	12	closed	closed	ADJ
ejpam-4387	211	13	extension	extension	NOUN
ejpam-4387	211	14	space	space	NOUN
ejpam-4387	211	15	(	(	PUNCT
ejpam-4387	211	16	xp	xp	INTJ
ejpam-4387	211	17	,	,	PUNCT
ejpam-4387	211	18	τ	τ	PROPN
ejpam-4387	211	19	⋆	⋆	VERB
ejpam-4387	211	20	)	)	PUNCT
ejpam-4387	211	21	is	be	AUX
ejpam-4387	211	22	not	not	PART
ejpam-4387	211	23	c	c	NOUN
ejpam-4387	211	24	-	-	NOUN
ejpam-4387	211	25	normal	normal	ADJ
ejpam-4387	211	26	if	if	SCONJ
ejpam-4387	211	27	(	(	PUNCT
ejpam-4387	211	28	x	x	X
ejpam-4387	211	29	,	,	PUNCT
ejpam-4387	211	30	τ	τ	PROPN
ejpam-4387	211	31	)	)	PUNCT
ejpam-4387	211	32	is	be	AUX
ejpam-4387	211	33	not	not	PART
ejpam-4387	211	34	ultra	ultra	ADJ
ejpam-4387	211	35	-	-	VERB
ejpam-4387	211	36	connected	connected	ADJ
ejpam-4387	211	37	.	.	PUNCT
ejpam-4387	212	1	in	in	ADP
ejpam-4387	212	2	[	[	X
ejpam-4387	212	3	10	10	NUM
ejpam-4387	212	4	]	]	PUNCT
ejpam-4387	212	5	,	,	PUNCT
ejpam-4387	212	6	we	we	PRON
ejpam-4387	212	7	showed	show	VERB
ejpam-4387	212	8	that	that	SCONJ
ejpam-4387	212	9	p	p	PROPN
ejpam-4387	212	10	-normality	-normality	PROPN
ejpam-4387	212	11	implies	imply	VERB
ejpam-4387	212	12	c	c	NOUN
ejpam-4387	212	13	-	-	NOUN
ejpam-4387	212	14	normality	normality	NOUN
ejpam-4387	212	15	.	.	PUNCT
ejpam-4387	213	1	combining	combine	VERB
ejpam-4387	213	2	all	all	DET
ejpam-4387	213	3	the	the	DET
ejpam-4387	213	4	information	information	NOUN
ejpam-4387	213	5	above	above	ADP
ejpam-4387	213	6	together	together	ADV
ejpam-4387	213	7	we	we	PRON
ejpam-4387	213	8	get	get	AUX
ejpam-4387	213	9	:	:	PUNCT
ejpam-4387	213	10	theorem	theorem	ADJ
ejpam-4387	213	11	4	4	NUM
ejpam-4387	213	12	.	.	PUNCT
ejpam-4387	214	1	if	if	SCONJ
ejpam-4387	214	2	(	(	PUNCT
ejpam-4387	214	3	x	x	X
ejpam-4387	214	4	,	,	PUNCT
ejpam-4387	214	5	τ	τ	PROPN
ejpam-4387	214	6	)	)	PUNCT
ejpam-4387	214	7	is	be	AUX
ejpam-4387	214	8	not	not	PART
ejpam-4387	214	9	ultra	ultra	ADJ
ejpam-4387	214	10	-	-	VERB
ejpam-4387	214	11	connected	connected	ADJ
ejpam-4387	214	12	,	,	PUNCT
ejpam-4387	214	13	then	then	ADV
ejpam-4387	214	14	the	the	DET
ejpam-4387	214	15	closed	closed	ADJ
ejpam-4387	214	16	extension	extension	NOUN
ejpam-4387	214	17	space	space	NOUN
ejpam-4387	214	18	(	(	PUNCT
ejpam-4387	214	19	xp	xp	INTJ
ejpam-4387	214	20	,	,	PUNCT
ejpam-4387	214	21	τ	τ	PROPN
ejpam-4387	214	22	⋆	⋆	VERB
ejpam-4387	214	23	)	)	PUNCT
ejpam-4387	214	24	is	be	AUX
ejpam-4387	214	25	not	not	PART
ejpam-4387	214	26	p	p	NOUN
ejpam-4387	214	27	-normal	-normal	ADJ
ejpam-4387	214	28	.	.	PUNCT
ejpam-4387	215	1	now	now	ADV
ejpam-4387	215	2	,	,	PUNCT
ejpam-4387	215	3	we	we	PRON
ejpam-4387	215	4	discuss	discuss	VERB
ejpam-4387	215	5	a	a	DET
ejpam-4387	215	6	new	new	ADJ
ejpam-4387	215	7	result	result	NOUN
ejpam-4387	215	8	about	about	ADP
ejpam-4387	215	9	p	p	NOUN
ejpam-4387	215	10	-normality	-normality	NOUN
ejpam-4387	215	11	and	and	CCONJ
ejpam-4387	215	12	whether	whether	SCONJ
ejpam-4387	215	13	it	it	PRON
ejpam-4387	215	14	’s	’s	AUX
ejpam-4387	215	15	preserved	preserve	VERB
ejpam-4387	215	16	or	or	CCONJ
ejpam-4387	215	17	not	not	PART
ejpam-4387	215	18	in	in	ADP
ejpam-4387	215	19	the	the	DET
ejpam-4387	215	20	discrete	discrete	ADJ
ejpam-4387	215	21	extension	extension	NOUN
ejpam-4387	215	22	space	space	NOUN
ejpam-4387	215	23	.	.	PUNCT
ejpam-4387	216	1	to	to	PART
ejpam-4387	216	2	do	do	VERB
ejpam-4387	216	3	this	this	PRON
ejpam-4387	216	4	let	let	VERB
ejpam-4387	216	5	us	we	PRON
ejpam-4387	216	6	recall	recall	VERB
ejpam-4387	216	7	the	the	DET
ejpam-4387	216	8	definition	definition	NOUN
ejpam-4387	216	9	of	of	ADP
ejpam-4387	216	10	the	the	DET
ejpam-4387	216	11	discrete	discrete	ADJ
ejpam-4387	216	12	extension	extension	NOUN
ejpam-4387	216	13	space	space	NOUN
ejpam-4387	216	14	:	:	PUNCT
ejpam-4387	216	15	let	let	VERB
ejpam-4387	216	16	m	m	PRON
ejpam-4387	216	17	be	be	AUX
ejpam-4387	216	18	a	a	DET
ejpam-4387	216	19	non	non	ADJ
ejpam-4387	216	20	-	-	ADJ
ejpam-4387	216	21	empty	empty	ADJ
ejpam-4387	216	22	proper	proper	ADJ
ejpam-4387	216	23	subset	subset	NOUN
ejpam-4387	216	24	of	of	ADP
ejpam-4387	216	25	a	a	DET
ejpam-4387	216	26	topological	topological	ADJ
ejpam-4387	216	27	space	space	NOUN
ejpam-4387	216	28	(	(	PUNCT
ejpam-4387	216	29	x	x	X
ejpam-4387	216	30	,	,	PUNCT
ejpam-4387	216	31	τ	τ	PROPN
ejpam-4387	216	32	)	)	PUNCT
ejpam-4387	216	33	.	.	PUNCT
ejpam-4387	217	1	define	define	VERB
ejpam-4387	217	2	a	a	DET
ejpam-4387	217	3	new	new	ADJ
ejpam-4387	217	4	topology	topology	NOUN
ejpam-4387	217	5	τ	τ	X
ejpam-4387	217	6	(	(	PUNCT
ejpam-4387	217	7	m	m	NOUN
ejpam-4387	217	8	)	)	PUNCT
ejpam-4387	217	9	on	on	ADP
ejpam-4387	217	10	x	x	PUNCT
ejpam-4387	217	11	as	as	SCONJ
ejpam-4387	217	12	follows	follow	VERB
ejpam-4387	217	13	:	:	PUNCT
ejpam-4387	217	14	τ	τ	PROPN
ejpam-4387	217	15	(	(	PUNCT
ejpam-4387	217	16	m	m	NOUN
ejpam-4387	217	17	)	)	PUNCT
ejpam-4387	217	18	=	=	PRON
ejpam-4387	217	19	{	{	PUNCT
ejpam-4387	217	20	u	u	NOUN
ejpam-4387	217	21	∪k	∪k	PROPN
ejpam-4387	217	22	:	:	PUNCT
ejpam-4387	217	23	u	u	PROPN
ejpam-4387	217	24	∈	∈	PROPN
ejpam-4387	217	25	τ	τ	X
ejpam-4387	217	26	and	and	CCONJ
ejpam-4387	217	27	k	k	PROPN
ejpam-4387	217	28	⊆	⊆	NUM
ejpam-4387	217	29	x	x	X
ejpam-4387	217	30	\m	\m	NOUN
ejpam-4387	217	31	}	}	PUNCT
ejpam-4387	217	32	.	.	PUNCT
ejpam-4387	218	1	(	(	PUNCT
ejpam-4387	218	2	x	x	X
ejpam-4387	218	3	,	,	PUNCT
ejpam-4387	218	4	τ	τ	PROPN
ejpam-4387	218	5	(	(	PUNCT
ejpam-4387	218	6	m	m	NOUN
ejpam-4387	218	7	)	)	PUNCT
ejpam-4387	218	8	)	)	PUNCT
ejpam-4387	218	9	is	be	AUX
ejpam-4387	218	10	called	call	VERB
ejpam-4387	218	11	a	a	DET
ejpam-4387	218	12	discrete	discrete	ADJ
ejpam-4387	218	13	extension	extension	NOUN
ejpam-4387	218	14	of	of	ADP
ejpam-4387	218	15	(	(	PUNCT
ejpam-4387	218	16	x	x	INTJ
ejpam-4387	218	17	,	,	PUNCT
ejpam-4387	218	18	τ	τ	PROPN
ejpam-4387	218	19	)	)	PUNCT
ejpam-4387	218	20	and	and	CCONJ
ejpam-4387	218	21	we	we	PRON
ejpam-4387	218	22	denote	denote	VERB
ejpam-4387	218	23	it	it	PRON
ejpam-4387	218	24	by	by	ADP
ejpam-4387	218	25	xm	xm	PROPN
ejpam-4387	219	1	[	[	X
ejpam-4387	219	2	13	13	NUM
ejpam-4387	219	3	]	]	PUNCT
ejpam-4387	219	4	,	,	PUNCT
ejpam-4387	219	5	see	see	VERB
ejpam-4387	219	6	also	also	ADV
ejpam-4387	219	7	[	[	X
ejpam-4387	219	8	7	7	NUM
ejpam-4387	219	9	,	,	PUNCT
ejpam-4387	219	10	5.1.22	5.1.22	NUM
ejpam-4387	219	11	]	]	PUNCT
ejpam-4387	219	12	.	.	PUNCT
ejpam-4387	220	1	we	we	PRON
ejpam-4387	220	2	will	will	AUX
ejpam-4387	220	3	now	now	ADV
ejpam-4387	220	4	show	show	VERB
ejpam-4387	220	5	that	that	SCONJ
ejpam-4387	220	6	p	p	PRON
ejpam-4387	220	7	-normality	-normality	NOUN
ejpam-4387	220	8	is	be	AUX
ejpam-4387	220	9	not	not	PART
ejpam-4387	220	10	preserved	preserve	VERB
ejpam-4387	220	11	by	by	ADP
ejpam-4387	220	12	a	a	DET
ejpam-4387	220	13	discrete	discrete	ADJ
ejpam-4387	220	14	extension	extension	NOUN
ejpam-4387	220	15	.	.	PUNCT
ejpam-4387	221	1	that	that	PRON
ejpam-4387	221	2	is	be	AUX
ejpam-4387	221	3	,	,	PUNCT
ejpam-4387	221	4	the	the	DET
ejpam-4387	221	5	discrete	discrete	ADJ
ejpam-4387	221	6	extension	extension	NOUN
ejpam-4387	221	7	of	of	ADP
ejpam-4387	221	8	a	a	DET
ejpam-4387	221	9	p	p	NOUN
ejpam-4387	221	10	-normal	-normal	ADJ
ejpam-4387	221	11	space	space	NOUN
ejpam-4387	221	12	need	need	AUX
ejpam-4387	221	13	not	not	PART
ejpam-4387	221	14	be	be	AUX
ejpam-4387	221	15	p	p	NOUN
ejpam-4387	221	16	-normal	-normal	NOUN
ejpam-4387	221	17	.	.	PUNCT
ejpam-4387	222	1	example	example	NOUN
ejpam-4387	223	1	3	3	X
ejpam-4387	223	2	.	.	X
ejpam-4387	224	1	we	we	PRON
ejpam-4387	224	2	know	know	VERB
ejpam-4387	224	3	that	that	SCONJ
ejpam-4387	224	4	(	(	PUNCT
ejpam-4387	224	5	r	r	NOUN
ejpam-4387	224	6	,	,	PUNCT
ejpam-4387	224	7	rs	rs	NOUN
ejpam-4387	224	8	)	)	PUNCT
ejpam-4387	224	9	where	where	SCONJ
ejpam-4387	224	10	rs	rs	NOUN
ejpam-4387	224	11	is	be	AUX
ejpam-4387	224	12	the	the	DET
ejpam-4387	224	13	rational	rational	ADJ
ejpam-4387	224	14	sequence	sequence	NOUN
ejpam-4387	224	15	topology	topology	NOUN
ejpam-4387	224	16	on	on	ADP
ejpam-4387	224	17	r	r	NOUN
ejpam-4387	224	18	,	,	PUNCT
ejpam-4387	224	19	is	be	AUX
ejpam-4387	224	20	a	a	DET
ejpam-4387	224	21	tychonoff	tychonoff	NOUN
ejpam-4387	224	22	locally	locally	ADV
ejpam-4387	224	23	compact	compact	ADJ
ejpam-4387	224	24	non	non	ADJ
ejpam-4387	224	25	compact	compact	ADJ
ejpam-4387	224	26	space	space	NOUN
ejpam-4387	224	27	[	[	X
ejpam-4387	224	28	13	13	NUM
ejpam-4387	224	29	,	,	PUNCT
ejpam-4387	224	30	example	example	NOUN
ejpam-4387	224	31	65	65	NUM
ejpam-4387	224	32	]	]	PUNCT
ejpam-4387	224	33	.	.	PUNCT
ejpam-4387	225	1	thus	thus	ADV
ejpam-4387	225	2	r	r	NOUN
ejpam-4387	225	3	with	with	ADP
ejpam-4387	225	4	the	the	DET
ejpam-4387	225	5	l.	l.	PROPN
ejpam-4387	225	6	kalantan	kalantan	PROPN
ejpam-4387	225	7	,	,	PUNCT
ejpam-4387	225	8	m.	m.	NOUN
ejpam-4387	225	9	mansouri	mansouri	PROPN
ejpam-4387	225	10	/	/	SYM
ejpam-4387	225	11	eur	eur	PROPN
ejpam-4387	225	12	.	.	PUNCT
ejpam-4387	226	1	j.	j.	PROPN
ejpam-4387	226	2	pure	pure	PROPN
ejpam-4387	226	3	appl	appl	PROPN
ejpam-4387	226	4	.	.	PROPN
ejpam-4387	226	5	math	math	PROPN
ejpam-4387	226	6	,	,	PUNCT
ejpam-4387	226	7	15	15	NUM
ejpam-4387	226	8	(	(	PUNCT
ejpam-4387	226	9	2	2	NUM
ejpam-4387	226	10	)	)	PUNCT
ejpam-4387	226	11	(	(	PUNCT
ejpam-4387	226	12	2022	2022	NUM
ejpam-4387	226	13	)	)	PUNCT
ejpam-4387	226	14	,	,	PUNCT
ejpam-4387	226	15	774	774	NUM
ejpam-4387	226	16	-	-	SYM
ejpam-4387	226	17	783	783	NUM
ejpam-4387	226	18	780	780	NUM
ejpam-4387	226	19	rational	rational	ADJ
ejpam-4387	226	20	sequence	sequence	NOUN
ejpam-4387	226	21	topology	topology	NOUN
ejpam-4387	226	22	has	have	VERB
ejpam-4387	226	23	a	a	DET
ejpam-4387	226	24	one	one	NUM
ejpam-4387	226	25	-	-	PUNCT
ejpam-4387	226	26	point	point	NOUN
ejpam-4387	226	27	compactification	compactification	NOUN
ejpam-4387	226	28	.	.	PUNCT
ejpam-4387	227	1	let	let	VERB
ejpam-4387	227	2	x	x	PUNCT
ejpam-4387	227	3	=	=	PUNCT
ejpam-4387	227	4	r	r	NOUN
ejpam-4387	227	5	∪	∪	X
ejpam-4387	227	6	{	{	PUNCT
ejpam-4387	227	7	p	p	NOUN
ejpam-4387	227	8	}	}	PUNCT
ejpam-4387	227	9	,	,	PUNCT
ejpam-4387	227	10	where	where	SCONJ
ejpam-4387	227	11	p	p	PROPN
ejpam-4387	227	12	̸∈	̸∈	PROPN
ejpam-4387	227	13	r	r	PROPN
ejpam-4387	227	14	,	,	PUNCT
ejpam-4387	227	15	be	be	AUX
ejpam-4387	227	16	a	a	DET
ejpam-4387	227	17	one	one	NUM
ejpam-4387	227	18	-	-	PUNCT
ejpam-4387	227	19	point	point	NOUN
ejpam-4387	227	20	compactification	compactification	NOUN
ejpam-4387	227	21	of	of	ADP
ejpam-4387	227	22	r.	r.	PROPN
ejpam-4387	227	23	since	since	SCONJ
ejpam-4387	227	24	x	x	PROPN
ejpam-4387	227	25	is	be	AUX
ejpam-4387	227	26	t2	t2	NOUN
ejpam-4387	227	27	-	-	PUNCT
ejpam-4387	227	28	compact	compact	ADJ
ejpam-4387	227	29	,	,	PUNCT
ejpam-4387	227	30	then	then	ADV
ejpam-4387	227	31	it	it	PRON
ejpam-4387	227	32	is	be	AUX
ejpam-4387	227	33	t4	t4	PROPN
ejpam-4387	227	34	,	,	PUNCT
ejpam-4387	227	35	hence	hence	ADV
ejpam-4387	227	36	p	p	X
ejpam-4387	227	37	-normal	-normal	ADJ
ejpam-4387	227	38	[	[	X
ejpam-4387	227	39	10	10	NUM
ejpam-4387	227	40	]	]	PUNCT
ejpam-4387	227	41	.	.	PUNCT
ejpam-4387	228	1	now	now	ADV
ejpam-4387	228	2	,	,	PUNCT
ejpam-4387	228	3	take	take	VERB
ejpam-4387	228	4	the	the	DET
ejpam-4387	228	5	discrete	discrete	ADJ
ejpam-4387	228	6	extension	extension	NOUN
ejpam-4387	228	7	of	of	ADP
ejpam-4387	228	8	x	x	PUNCT
ejpam-4387	228	9	denoted	denote	VERB
ejpam-4387	228	10	by	by	ADP
ejpam-4387	228	11	xr	xr	PROPN
ejpam-4387	228	12	.	.	PUNCT
ejpam-4387	228	13	observe	observe	VERB
ejpam-4387	228	14	that	that	SCONJ
ejpam-4387	228	15	in	in	ADP
ejpam-4387	228	16	xr	xr	PROPN
ejpam-4387	228	17	,	,	PUNCT
ejpam-4387	228	18	the	the	DET
ejpam-4387	228	19	singleton	singleton	NOUN
ejpam-4387	228	20	{	{	PUNCT
ejpam-4387	228	21	p	p	X
ejpam-4387	228	22	}	}	PUNCT
ejpam-4387	228	23	is	be	AUX
ejpam-4387	228	24	closed	closed	ADJ
ejpam-4387	228	25	-	-	PUNCT
ejpam-4387	228	26	and	and	CCONJ
ejpam-4387	228	27	-	-	PUNCT
ejpam-4387	228	28	open	open	ADJ
ejpam-4387	228	29	.	.	PUNCT
ejpam-4387	229	1	xr	xr	PROPN
ejpam-4387	229	2	is	be	AUX
ejpam-4387	229	3	first	first	ADV
ejpam-4387	229	4	countable	countable	ADJ
ejpam-4387	229	5	and	and	CCONJ
ejpam-4387	229	6	tychonoff	tychonoff	NOUN
ejpam-4387	229	7	because	because	SCONJ
ejpam-4387	229	8	r	r	NOUN
ejpam-4387	229	9	with	with	ADP
ejpam-4387	229	10	the	the	DET
ejpam-4387	229	11	rational	rational	ADJ
ejpam-4387	229	12	sequence	sequence	NOUN
ejpam-4387	229	13	topology	topology	NOUN
ejpam-4387	229	14	is	be	AUX
ejpam-4387	229	15	,	,	PUNCT
ejpam-4387	229	16	thus	thus	ADV
ejpam-4387	229	17	xr	xr	PROPN
ejpam-4387	229	18	is	be	AUX
ejpam-4387	229	19	of	of	ADP
ejpam-4387	229	20	countable	countable	ADJ
ejpam-4387	229	21	tightness	tightness	NOUN
ejpam-4387	229	22	.	.	PUNCT
ejpam-4387	230	1	xr	xr	PROPN
ejpam-4387	230	2	is	be	AUX
ejpam-4387	230	3	also	also	ADV
ejpam-4387	230	4	separable	separable	ADJ
ejpam-4387	230	5	because	because	SCONJ
ejpam-4387	230	6	(	(	PUNCT
ejpam-4387	230	7	r	r	NOUN
ejpam-4387	230	8	,	,	PUNCT
ejpam-4387	230	9	rs	rs	NOUN
ejpam-4387	230	10	)	)	PUNCT
ejpam-4387	230	11	is	be	AUX
ejpam-4387	230	12	separable	separable	ADJ
ejpam-4387	230	13	and	and	CCONJ
ejpam-4387	230	14	q	q	NOUN
ejpam-4387	230	15	∪	∪	X
ejpam-4387	230	16	{	{	PUNCT
ejpam-4387	230	17	p	p	NOUN
ejpam-4387	230	18	}	}	PUNCT
ejpam-4387	230	19	is	be	AUX
ejpam-4387	230	20	a	a	DET
ejpam-4387	230	21	countable	countable	ADJ
ejpam-4387	230	22	dense	dense	ADJ
ejpam-4387	230	23	subset	subset	NOUN
ejpam-4387	230	24	of	of	ADP
ejpam-4387	230	25	xr	xr	PROPN
ejpam-4387	231	1	[	[	X
ejpam-4387	231	2	1	1	NUM
ejpam-4387	231	3	]	]	PUNCT
ejpam-4387	231	4	.	.	PUNCT
ejpam-4387	232	1	now	now	ADV
ejpam-4387	232	2	,	,	PUNCT
ejpam-4387	232	3	r	r	NOUN
ejpam-4387	232	4	with	with	ADP
ejpam-4387	232	5	the	the	DET
ejpam-4387	232	6	rational	rational	ADJ
ejpam-4387	232	7	sequence	sequence	NOUN
ejpam-4387	232	8	topology	topology	NOUN
ejpam-4387	232	9	is	be	AUX
ejpam-4387	232	10	not	not	PART
ejpam-4387	232	11	normal	normal	ADJ
ejpam-4387	232	12	.	.	PUNCT
ejpam-4387	233	1	since	since	SCONJ
ejpam-4387	233	2	r	r	NOUN
ejpam-4387	233	3	is	be	AUX
ejpam-4387	233	4	closed	close	VERB
ejpam-4387	233	5	in	in	ADP
ejpam-4387	233	6	xr	xr	PROPN
ejpam-4387	233	7	,	,	PUNCT
ejpam-4387	233	8	we	we	PRON
ejpam-4387	233	9	conclude	conclude	VERB
ejpam-4387	233	10	that	that	SCONJ
ejpam-4387	233	11	xr	xr	PROPN
ejpam-4387	233	12	is	be	AUX
ejpam-4387	233	13	not	not	PART
ejpam-4387	233	14	normal	normal	ADJ
ejpam-4387	233	15	.	.	PUNCT
ejpam-4387	234	1	using	use	VERB
ejpam-4387	234	2	the	the	DET
ejpam-4387	234	3	theorem	theorem	NOUN
ejpam-4387	234	4	:	:	PUNCT
ejpam-4387	234	5	“	"	PUNCT
ejpam-4387	234	6	if	if	SCONJ
ejpam-4387	234	7	y	y	PROPN
ejpam-4387	234	8	is	be	AUX
ejpam-4387	234	9	t3	t3	PROPN
ejpam-4387	234	10	,	,	PUNCT
ejpam-4387	234	11	separable	separable	NOUN
ejpam-4387	234	12	,	,	PUNCT
ejpam-4387	234	13	p	p	NOUN
ejpam-4387	234	14	-normal	-normal	ADJ
ejpam-4387	234	15	and	and	CCONJ
ejpam-4387	234	16	of	of	ADP
ejpam-4387	234	17	countable	countable	ADJ
ejpam-4387	234	18	tightness	tightness	NOUN
ejpam-4387	234	19	,	,	PUNCT
ejpam-4387	234	20	then	then	ADV
ejpam-4387	234	21	y	y	PROPN
ejpam-4387	234	22	is	be	AUX
ejpam-4387	234	23	normal	normal	ADJ
ejpam-4387	234	24	.	.	PUNCT
ejpam-4387	234	25	”	"	PUNCT
ejpam-4387	235	1	[	[	X
ejpam-4387	235	2	10	10	NUM
ejpam-4387	235	3	]	]	PUNCT
ejpam-4387	235	4	,	,	PUNCT
ejpam-4387	235	5	we	we	PRON
ejpam-4387	235	6	conclude	conclude	VERB
ejpam-4387	235	7	that	that	SCONJ
ejpam-4387	235	8	xr	xr	PROPN
ejpam-4387	235	9	can	can	AUX
ejpam-4387	235	10	not	not	PART
ejpam-4387	235	11	be	be	AUX
ejpam-4387	235	12	p	p	NOUN
ejpam-4387	235	13	-normal	-normal	NOUN
ejpam-4387	235	14	.	.	PUNCT
ejpam-4387	236	1	this	this	DET
ejpam-4387	236	2	example	example	NOUN
ejpam-4387	236	3	shows	show	VERB
ejpam-4387	236	4	that	that	SCONJ
ejpam-4387	236	5	[	[	X
ejpam-4387	236	6	1	1	NUM
ejpam-4387	236	7	,	,	PUNCT
ejpam-4387	236	8	theorem	theorem	VERB
ejpam-4387	236	9	12	12	NUM
ejpam-4387	236	10	]	]	PUNCT
ejpam-4387	236	11	is	be	AUX
ejpam-4387	236	12	not	not	PART
ejpam-4387	236	13	true	true	ADJ
ejpam-4387	236	14	for	for	ADP
ejpam-4387	236	15	p	p	PROPN
ejpam-4387	236	16	-normality	-normality	NOUN
ejpam-4387	236	17	.	.	PUNCT
ejpam-4387	237	1	that	that	PRON
ejpam-4387	237	2	is	be	AUX
ejpam-4387	237	3	if	if	SCONJ
ejpam-4387	237	4	y	y	PROPN
ejpam-4387	237	5	is	be	AUX
ejpam-4387	237	6	a	a	DET
ejpam-4387	237	7	tychonoff	tychonoff	NOUN
ejpam-4387	237	8	space	space	NOUN
ejpam-4387	237	9	,	,	PUNCT
ejpam-4387	237	10	then	then	ADV
ejpam-4387	237	11	a	a	DET
ejpam-4387	237	12	discrete	discrete	ADJ
ejpam-4387	237	13	extension	extension	NOUN
ejpam-4387	237	14	xm	xm	PROPN
ejpam-4387	237	15	of	of	ADP
ejpam-4387	237	16	any	any	DET
ejpam-4387	237	17	compactification	compactification	NOUN
ejpam-4387	237	18	x	x	PUNCT
ejpam-4387	237	19	of	of	ADP
ejpam-4387	237	20	y	y	PRON
ejpam-4387	237	21	need	need	AUX
ejpam-4387	237	22	not	not	PART
ejpam-4387	237	23	be	be	AUX
ejpam-4387	237	24	p	p	NOUN
ejpam-4387	237	25	-normal	-normal	NOUN
ejpam-4387	237	26	.	.	PUNCT
ejpam-4387	238	1	5	5	NUM
ejpam-4387	238	2	.	.	X
ejpam-4387	238	3	strong	strong	ADJ
ejpam-4387	238	4	p	p	NOUN
ejpam-4387	238	5	-normality	-normality	PROPN
ejpam-4387	238	6	definition	definition	NOUN
ejpam-4387	238	7	2	2	NUM
ejpam-4387	238	8	.	.	PUNCT
ejpam-4387	239	1	a	a	DET
ejpam-4387	239	2	topological	topological	ADJ
ejpam-4387	239	3	space	space	NOUN
ejpam-4387	239	4	x	x	PUNCT
ejpam-4387	239	5	is	be	AUX
ejpam-4387	239	6	called	call	VERB
ejpam-4387	239	7	strongly	strongly	ADV
ejpam-4387	239	8	p	p	X
ejpam-4387	239	9	-normal	-normal	ADJ
ejpam-4387	239	10	if	if	SCONJ
ejpam-4387	239	11	there	there	PRON
ejpam-4387	239	12	exists	exist	VERB
ejpam-4387	239	13	a	a	DET
ejpam-4387	239	14	bijective	bijective	ADJ
ejpam-4387	239	15	function	function	NOUN
ejpam-4387	239	16	f	f	NOUN
ejpam-4387	239	17	:	:	PUNCT
ejpam-4387	239	18	x	x	PUNCT
ejpam-4387	239	19	−→	−→	NOUN
ejpam-4387	239	20	i	i	PRON
ejpam-4387	239	21	,	,	PUNCT
ejpam-4387	239	22	where	where	SCONJ
ejpam-4387	239	23	i	i	PRON
ejpam-4387	239	24	=	=	X
ejpam-4387	239	25	[	[	PUNCT
ejpam-4387	239	26	0	0	NUM
ejpam-4387	239	27	,	,	PUNCT
ejpam-4387	239	28	1	1	NUM
ejpam-4387	239	29	]	]	PUNCT
ejpam-4387	239	30	the	the	DET
ejpam-4387	239	31	closed	closed	ADJ
ejpam-4387	239	32	unit	unit	NOUN
ejpam-4387	239	33	interval	interval	NOUN
ejpam-4387	239	34	considered	consider	VERB
ejpam-4387	239	35	with	with	ADP
ejpam-4387	239	36	its	its	PRON
ejpam-4387	239	37	usual	usual	ADJ
ejpam-4387	239	38	metric	metric	ADJ
ejpam-4387	239	39	topology	topology	NOUN
ejpam-4387	239	40	,	,	PUNCT
ejpam-4387	239	41	such	such	ADJ
ejpam-4387	239	42	that	that	SCONJ
ejpam-4387	239	43	the	the	DET
ejpam-4387	239	44	restriction	restriction	NOUN
ejpam-4387	239	45	f|a	f|a	PUNCT
ejpam-4387	239	46	:	:	PUNCT
ejpam-4387	239	47	a	a	DET
ejpam-4387	239	48	−→	−→	NOUN
ejpam-4387	239	49	f(a	f(a	NOUN
ejpam-4387	239	50	)	)	PUNCT
ejpam-4387	239	51	is	be	AUX
ejpam-4387	239	52	a	a	DET
ejpam-4387	239	53	homeomorphism	homeomorphism	NOUN
ejpam-4387	239	54	for	for	ADP
ejpam-4387	239	55	each	each	DET
ejpam-4387	239	56	paracompact	paracompact	ADJ
ejpam-4387	239	57	subspace	subspace	NOUN
ejpam-4387	239	58	a	a	DET
ejpam-4387	239	59	⊆	⊆	NUM
ejpam-4387	239	60	x.	x.	NOUN
ejpam-4387	239	61	it	it	PRON
ejpam-4387	239	62	is	be	AUX
ejpam-4387	239	63	clear	clear	ADJ
ejpam-4387	239	64	from	from	ADP
ejpam-4387	239	65	the	the	DET
ejpam-4387	239	66	definition	definition	NOUN
ejpam-4387	239	67	that	that	SCONJ
ejpam-4387	239	68	any	any	DET
ejpam-4387	239	69	strongly	strongly	ADV
ejpam-4387	239	70	p	p	X
ejpam-4387	239	71	-normal	-normal	ADJ
ejpam-4387	239	72	space	space	NOUN
ejpam-4387	239	73	is	be	AUX
ejpam-4387	239	74	p	p	NOUN
ejpam-4387	239	75	-normal	-normal	NOUN
ejpam-4387	239	76	.	.	PUNCT
ejpam-4387	240	1	the	the	DET
ejpam-4387	240	2	converse	converse	NOUN
ejpam-4387	240	3	is	be	AUX
ejpam-4387	240	4	not	not	PART
ejpam-4387	240	5	always	always	ADV
ejpam-4387	240	6	true	true	ADJ
ejpam-4387	240	7	.	.	PUNCT
ejpam-4387	240	8	example	example	NOUN
ejpam-4387	241	1	4	4	NUM
ejpam-4387	241	2	.	.	X
ejpam-4387	242	1	ω2	ω2	ADJ
ejpam-4387	242	2	+	+	NOUN
ejpam-4387	242	3	1	1	NUM
ejpam-4387	242	4	with	with	ADP
ejpam-4387	242	5	its	its	PRON
ejpam-4387	242	6	usual	usual	ADJ
ejpam-4387	242	7	ordered	order	VERB
ejpam-4387	242	8	topology	topology	NOUN
ejpam-4387	242	9	is	be	AUX
ejpam-4387	242	10	p	p	NOUN
ejpam-4387	242	11	-normal	-normal	ADJ
ejpam-4387	242	12	because	because	SCONJ
ejpam-4387	242	13	it	it	PRON
ejpam-4387	242	14	is	be	AUX
ejpam-4387	242	15	normal	normal	ADJ
ejpam-4387	242	16	being	be	AUX
ejpam-4387	242	17	t2	t2	NOUN
ejpam-4387	242	18	compact	compact	ADJ
ejpam-4387	242	19	.	.	PUNCT
ejpam-4387	243	1	but	but	CCONJ
ejpam-4387	243	2	ω2	ω2	ADV
ejpam-4387	243	3	+	+	CCONJ
ejpam-4387	243	4	1	1	NUM
ejpam-4387	243	5	can	can	AUX
ejpam-4387	243	6	not	not	PART
ejpam-4387	243	7	be	be	AUX
ejpam-4387	243	8	strongly	strongly	ADV
ejpam-4387	243	9	p	p	X
ejpam-4387	243	10	-normal	-normal	ADJ
ejpam-4387	243	11	because	because	SCONJ
ejpam-4387	243	12	|	|	ADV
ejpam-4387	243	13	[	[	PUNCT
ejpam-4387	243	14	0	0	NUM
ejpam-4387	243	15	,	,	PUNCT
ejpam-4387	243	16	1	1	NUM
ejpam-4387	243	17	]	]	PUNCT
ejpam-4387	243	18	|	|	NOUN
ejpam-4387	243	19	=	=	SYM
ejpam-4387	243	20	|r|	|r|	X
ejpam-4387	243	21	=	=	PUNCT
ejpam-4387	243	22	c	c	X
ejpam-4387	243	23	<	<	X
ejpam-4387	243	24	ω2	ω2	NOUN
ejpam-4387	243	25	=	=	PUNCT
ejpam-4387	243	26	|ω2	|ω2	PROPN
ejpam-4387	243	27	+	+	PUNCT
ejpam-4387	243	28	1|	1|	NUM
ejpam-4387	243	29	.	.	PUNCT
ejpam-4387	243	30	example	example	NOUN
ejpam-4387	244	1	5	5	NUM
ejpam-4387	244	2	.	.	PUNCT
ejpam-4387	245	1	(	(	PUNCT
ejpam-4387	245	2	r	r	NOUN
ejpam-4387	245	3	,	,	PUNCT
ejpam-4387	245	4	u	u	NOUN
ejpam-4387	245	5	)	)	PUNCT
ejpam-4387	245	6	is	be	AUX
ejpam-4387	245	7	not	not	PART
ejpam-4387	245	8	strongly	strongly	ADV
ejpam-4387	245	9	p	p	ADJ
ejpam-4387	245	10	-normal	-normal	NOUN
ejpam-4387	245	11	.	.	PUNCT
ejpam-4387	246	1	(	(	PUNCT
ejpam-4387	246	2	r	r	NOUN
ejpam-4387	246	3	,	,	PUNCT
ejpam-4387	246	4	u	u	NOUN
ejpam-4387	246	5	)	)	PUNCT
ejpam-4387	246	6	is	be	AUX
ejpam-4387	246	7	homeomorphic	homeomorphic	ADJ
ejpam-4387	246	8	to	to	ADP
ejpam-4387	246	9	the	the	DET
ejpam-4387	246	10	open	open	ADJ
ejpam-4387	246	11	interval	interval	NOUN
ejpam-4387	246	12	(	(	PUNCT
ejpam-4387	246	13	0	0	NUM
ejpam-4387	246	14	,	,	PUNCT
ejpam-4387	246	15	1	1	NUM
ejpam-4387	246	16	)	)	PUNCT
ejpam-4387	246	17	with	with	ADP
ejpam-4387	246	18	the	the	DET
ejpam-4387	246	19	usual	usual	ADJ
ejpam-4387	246	20	topology	topology	NOUN
ejpam-4387	246	21	.	.	PUNCT
ejpam-4387	247	1	so	so	ADV
ejpam-4387	247	2	,	,	PUNCT
ejpam-4387	247	3	if	if	SCONJ
ejpam-4387	247	4	(	(	PUNCT
ejpam-4387	247	5	r	r	NOUN
ejpam-4387	247	6	,	,	PUNCT
ejpam-4387	247	7	u	u	NOUN
ejpam-4387	247	8	)	)	PUNCT
ejpam-4387	247	9	is	be	AUX
ejpam-4387	247	10	strongly	strongly	ADV
ejpam-4387	247	11	p	p	ADJ
ejpam-4387	247	12	-normal	-normal	NOUN
ejpam-4387	247	13	,	,	PUNCT
ejpam-4387	247	14	then	then	ADV
ejpam-4387	247	15	there	there	PRON
ejpam-4387	247	16	will	will	AUX
ejpam-4387	247	17	be	be	AUX
ejpam-4387	247	18	a	a	DET
ejpam-4387	247	19	bijection	bijection	ADJ
ejpam-4387	247	20	f	f	NOUN
ejpam-4387	247	21	:	:	PUNCT
ejpam-4387	247	22	(	(	PUNCT
ejpam-4387	247	23	0	0	NUM
ejpam-4387	247	24	,	,	PUNCT
ejpam-4387	247	25	1	1	X
ejpam-4387	247	26	)	)	PUNCT
ejpam-4387	248	1	−→	−→	NOUN
ejpam-4387	249	1	[	[	X
ejpam-4387	249	2	0	0	NUM
ejpam-4387	249	3	,	,	PUNCT
ejpam-4387	249	4	1	1	NUM
ejpam-4387	249	5	]	]	PUNCT
ejpam-4387	249	6	such	such	ADJ
ejpam-4387	249	7	that	that	SCONJ
ejpam-4387	249	8	f	f	PROPN
ejpam-4387	249	9	|a	|a	VERB
ejpam-4387	249	10	is	be	AUX
ejpam-4387	249	11	a	a	DET
ejpam-4387	249	12	homeomorphism	homeomorphism	NOUN
ejpam-4387	249	13	for	for	ADP
ejpam-4387	249	14	every	every	DET
ejpam-4387	249	15	paracompact	paracompact	NOUN
ejpam-4387	249	16	subset	subset	VERB
ejpam-4387	249	17	a	a	PRON
ejpam-4387	249	18	⊆	⊆	NUM
ejpam-4387	249	19	(	(	PUNCT
ejpam-4387	249	20	0	0	NUM
ejpam-4387	249	21	,	,	PUNCT
ejpam-4387	249	22	1	1	NUM
ejpam-4387	249	23	)	)	PUNCT
ejpam-4387	249	24	.	.	PUNCT
ejpam-4387	250	1	since	since	SCONJ
ejpam-4387	250	2	(	(	PUNCT
ejpam-4387	250	3	0	0	NUM
ejpam-4387	250	4	,	,	PUNCT
ejpam-4387	250	5	1	1	NUM
ejpam-4387	250	6	)	)	PUNCT
ejpam-4387	250	7	is	be	AUX
ejpam-4387	250	8	fréchet	fréchet	PROPN
ejpam-4387	250	9	,	,	PUNCT
ejpam-4387	250	10	f	f	PROPN
ejpam-4387	250	11	is	be	AUX
ejpam-4387	250	12	continuous	continuous	ADJ
ejpam-4387	250	13	by	by	ADP
ejpam-4387	250	14	[	[	X
ejpam-4387	250	15	10	10	NUM
ejpam-4387	250	16	,	,	PUNCT
ejpam-4387	250	17	theorem	theorem	VERB
ejpam-4387	250	18	5	5	NUM
ejpam-4387	250	19	]	]	PUNCT
ejpam-4387	250	20	.	.	PUNCT
ejpam-4387	251	1	since	since	SCONJ
ejpam-4387	251	2	f	f	PROPN
ejpam-4387	251	3	is	be	AUX
ejpam-4387	251	4	bijection	bijection	ADJ
ejpam-4387	251	5	,	,	PUNCT
ejpam-4387	251	6	there	there	PRON
ejpam-4387	251	7	is	be	VERB
ejpam-4387	251	8	unique	unique	ADJ
ejpam-4387	251	9	a	a	PRON
ejpam-4387	251	10	,	,	PUNCT
ejpam-4387	251	11	b	b	X
ejpam-4387	251	12	∈	∈	PROPN
ejpam-4387	251	13	(	(	PUNCT
ejpam-4387	251	14	0	0	NUM
ejpam-4387	251	15	,	,	PUNCT
ejpam-4387	251	16	1	1	NUM
ejpam-4387	251	17	)	)	PUNCT
ejpam-4387	251	18	such	such	ADJ
ejpam-4387	251	19	that	that	DET
ejpam-4387	251	20	f(a	f(a	NOUN
ejpam-4387	251	21	)	)	PUNCT
ejpam-4387	251	22	=	=	SYM
ejpam-4387	251	23	0	0	NUM
ejpam-4387	251	24	and	and	CCONJ
ejpam-4387	251	25	f(b	f(b	PROPN
ejpam-4387	251	26	)	)	PUNCT
ejpam-4387	251	27	=	=	SYM
ejpam-4387	251	28	1	1	X
ejpam-4387	251	29	.	.	X
ejpam-4387	251	30	assume	assume	VERB
ejpam-4387	251	31	without	without	ADP
ejpam-4387	251	32	loss	loss	NOUN
ejpam-4387	251	33	of	of	ADP
ejpam-4387	251	34	generality	generality	NOUN
ejpam-4387	251	35	that	that	PRON
ejpam-4387	251	36	a	a	DET
ejpam-4387	251	37	<	<	X
ejpam-4387	251	38	b.	b.	PROPN
ejpam-4387	251	39	then	then	ADV
ejpam-4387	251	40	,	,	PUNCT
ejpam-4387	251	41	(	(	PUNCT
ejpam-4387	251	42	0	0	NUM
ejpam-4387	251	43	,	,	PUNCT
ejpam-4387	251	44	1	1	NUM
ejpam-4387	251	45	)	)	PUNCT
ejpam-4387	251	46	\	\	PUNCT
ejpam-4387	252	1	[	[	X
ejpam-4387	252	2	a	a	DET
ejpam-4387	252	3	,	,	PUNCT
ejpam-4387	252	4	b	b	NOUN
ejpam-4387	252	5	]	]	X
ejpam-4387	252	6	̸=	̸=	PROPN
ejpam-4387	252	7	∅	∅	NOUN
ejpam-4387	252	8	and	and	CCONJ
ejpam-4387	252	9	clearly	clearly	ADV
ejpam-4387	252	10	f	f	PROPN
ejpam-4387	252	11	is	be	AUX
ejpam-4387	252	12	continuous	continuous	ADJ
ejpam-4387	252	13	on	on	ADP
ejpam-4387	252	14	[	[	X
ejpam-4387	252	15	a	a	X
ejpam-4387	252	16	,	,	PUNCT
ejpam-4387	252	17	b	b	NOUN
ejpam-4387	252	18	]	]	X
ejpam-4387	252	19	.	.	PUNCT
ejpam-4387	253	1	now	now	ADV
ejpam-4387	253	2	using	use	VERB
ejpam-4387	253	3	the	the	DET
ejpam-4387	253	4	intermediate	intermediate	ADJ
ejpam-4387	253	5	value	value	NOUN
ejpam-4387	253	6	,	,	PUNCT
ejpam-4387	253	7	for	for	ADP
ejpam-4387	253	8	every	every	DET
ejpam-4387	253	9	y	y	PROPN
ejpam-4387	253	10	∈	∈	PROPN
ejpam-4387	253	11	(	(	PUNCT
ejpam-4387	253	12	f(a	f(a	NOUN
ejpam-4387	253	13	)	)	PUNCT
ejpam-4387	253	14	,	,	PUNCT
ejpam-4387	253	15	f(b	f(b	PROPN
ejpam-4387	253	16	)	)	PUNCT
ejpam-4387	253	17	)	)	PUNCT
ejpam-4387	254	1	=	=	PUNCT
ejpam-4387	254	2	(	(	PUNCT
ejpam-4387	254	3	0	0	NUM
ejpam-4387	254	4	,	,	PUNCT
ejpam-4387	254	5	1	1	NUM
ejpam-4387	254	6	)	)	PUNCT
ejpam-4387	254	7	,	,	PUNCT
ejpam-4387	254	8	there	there	PRON
ejpam-4387	254	9	exists	exist	VERB
ejpam-4387	254	10	x	x	X
ejpam-4387	254	11	∈	∈	PROPN
ejpam-4387	254	12	(	(	PUNCT
ejpam-4387	254	13	a	a	DET
ejpam-4387	254	14	,	,	PUNCT
ejpam-4387	254	15	b	b	NOUN
ejpam-4387	254	16	)	)	PUNCT
ejpam-4387	254	17	such	such	ADJ
ejpam-4387	254	18	that	that	SCONJ
ejpam-4387	254	19	f(x	f(x	NOUN
ejpam-4387	254	20	)	)	PUNCT
ejpam-4387	254	21	=	=	PUNCT
ejpam-4387	255	1	y.	y.	PROPN
ejpam-4387	255	2	hence	hence	ADV
ejpam-4387	255	3	,	,	PUNCT
ejpam-4387	255	4	f−1([0	f−1([0	ADJ
ejpam-4387	255	5	,	,	PUNCT
ejpam-4387	255	6	1	1	NUM
ejpam-4387	255	7	]	]	PUNCT
ejpam-4387	255	8	)	)	PUNCT
ejpam-4387	256	1	⊆	⊆	NUM
ejpam-4387	256	2	[	[	X
ejpam-4387	256	3	a	a	X
ejpam-4387	256	4	,	,	PUNCT
ejpam-4387	256	5	b	b	NOUN
ejpam-4387	256	6	]	]	X
ejpam-4387	256	7	⊂	⊂	X
ejpam-4387	256	8	(	(	PUNCT
ejpam-4387	256	9	0	0	NUM
ejpam-4387	256	10	,	,	PUNCT
ejpam-4387	256	11	1	1	NUM
ejpam-4387	256	12	)	)	PUNCT
ejpam-4387	256	13	.	.	PUNCT
ejpam-4387	257	1	this	this	PRON
ejpam-4387	257	2	implies	imply	VERB
ejpam-4387	257	3	that	that	SCONJ
ejpam-4387	257	4	for	for	ADP
ejpam-4387	257	5	every	every	DET
ejpam-4387	257	6	x	x	SYM
ejpam-4387	257	7	∈	∈	PROPN
ejpam-4387	257	8	(	(	PUNCT
ejpam-4387	257	9	0	0	NUM
ejpam-4387	257	10	,	,	PUNCT
ejpam-4387	257	11	1	1	NUM
ejpam-4387	257	12	)	)	PUNCT
ejpam-4387	257	13	\	\	PUNCT
ejpam-4387	258	1	[	[	X
ejpam-4387	258	2	a	a	DET
ejpam-4387	258	3	,	,	PUNCT
ejpam-4387	258	4	b	b	NOUN
ejpam-4387	258	5	]	]	X
ejpam-4387	258	6	(	(	PUNCT
ejpam-4387	258	7	which	which	PRON
ejpam-4387	258	8	is	be	AUX
ejpam-4387	258	9	non	non	ADJ
ejpam-4387	258	10	-	-	ADJ
ejpam-4387	258	11	empty	empty	ADJ
ejpam-4387	258	12	)	)	PUNCT
ejpam-4387	258	13	,	,	PUNCT
ejpam-4387	258	14	x	x	PRON
ejpam-4387	258	15	has	have	VERB
ejpam-4387	258	16	no	no	DET
ejpam-4387	258	17	image	image	NOUN
ejpam-4387	258	18	in	in	ADP
ejpam-4387	258	19	[	[	X
ejpam-4387	258	20	0	0	NUM
ejpam-4387	258	21	,	,	PUNCT
ejpam-4387	258	22	1	1	NUM
ejpam-4387	258	23	]	]	PUNCT
ejpam-4387	258	24	which	which	PRON
ejpam-4387	258	25	contradicts	contradict	VERB
ejpam-4387	258	26	that	that	SCONJ
ejpam-4387	258	27	f	f	PROPN
ejpam-4387	258	28	is	be	AUX
ejpam-4387	258	29	a	a	DET
ejpam-4387	258	30	function	function	NOUN
ejpam-4387	258	31	.	.	PUNCT
ejpam-4387	259	1	hence	hence	ADV
ejpam-4387	259	2	there	there	PRON
ejpam-4387	259	3	is	be	VERB
ejpam-4387	259	4	no	no	DET
ejpam-4387	259	5	continuous	continuous	ADJ
ejpam-4387	259	6	bijection	bijection	NOUN
ejpam-4387	259	7	between	between	ADP
ejpam-4387	259	8	r	r	NOUN
ejpam-4387	259	9	and	and	CCONJ
ejpam-4387	259	10	i.	i.	NOUN
ejpam-4387	259	11	therefore	therefore	ADV
ejpam-4387	259	12	,	,	PUNCT
ejpam-4387	259	13	(	(	PUNCT
ejpam-4387	259	14	r	r	NOUN
ejpam-4387	259	15	,	,	PUNCT
ejpam-4387	259	16	u	u	NOUN
ejpam-4387	259	17	)	)	PUNCT
ejpam-4387	259	18	is	be	AUX
ejpam-4387	259	19	not	not	PART
ejpam-4387	259	20	strongly	strongly	ADV
ejpam-4387	259	21	p	p	X
ejpam-4387	259	22	-normal	-normal	NOUN
ejpam-4387	259	23	.	.	PUNCT
ejpam-4387	260	1	theorem	theorem	ADJ
ejpam-4387	260	2	5	5	NUM
ejpam-4387	260	3	.	.	PUNCT
ejpam-4387	261	1	strong	strong	ADJ
ejpam-4387	261	2	p	p	NOUN
ejpam-4387	261	3	-normality	-normality	PROPN
ejpam-4387	261	4	is	be	AUX
ejpam-4387	261	5	a	a	DET
ejpam-4387	261	6	topological	topological	ADJ
ejpam-4387	261	7	property	property	NOUN
ejpam-4387	261	8	.	.	PUNCT
ejpam-4387	262	1	l.	l.	PROPN
ejpam-4387	262	2	kalantan	kalantan	PROPN
ejpam-4387	262	3	,	,	PUNCT
ejpam-4387	262	4	m.	m.	NOUN
ejpam-4387	262	5	mansouri	mansouri	PROPN
ejpam-4387	262	6	/	/	SYM
ejpam-4387	262	7	eur	eur	PROPN
ejpam-4387	262	8	.	.	PUNCT
ejpam-4387	263	1	j.	j.	PROPN
ejpam-4387	263	2	pure	pure	PROPN
ejpam-4387	263	3	appl	appl	PROPN
ejpam-4387	263	4	.	.	PROPN
ejpam-4387	263	5	math	math	PROPN
ejpam-4387	263	6	,	,	PUNCT
ejpam-4387	263	7	15	15	NUM
ejpam-4387	263	8	(	(	PUNCT
ejpam-4387	263	9	2	2	NUM
ejpam-4387	263	10	)	)	PUNCT
ejpam-4387	263	11	(	(	PUNCT
ejpam-4387	263	12	2022	2022	NUM
ejpam-4387	263	13	)	)	PUNCT
ejpam-4387	263	14	,	,	PUNCT
ejpam-4387	263	15	774	774	NUM
ejpam-4387	263	16	-	-	SYM
ejpam-4387	263	17	783	783	NUM
ejpam-4387	263	18	781	781	NUM
ejpam-4387	263	19	proof	proof	NOUN
ejpam-4387	263	20	.	.	PUNCT
ejpam-4387	264	1	let	let	VERB
ejpam-4387	264	2	x	x	PRON
ejpam-4387	264	3	be	be	AUX
ejpam-4387	264	4	any	any	DET
ejpam-4387	264	5	strongly	strongly	ADV
ejpam-4387	264	6	p	p	X
ejpam-4387	264	7	-normal	-normal	ADJ
ejpam-4387	264	8	space	space	NOUN
ejpam-4387	264	9	.	.	PUNCT
ejpam-4387	265	1	assume	assume	VERB
ejpam-4387	265	2	that	that	SCONJ
ejpam-4387	265	3	x	x	PUNCT
ejpam-4387	265	4	∼=	∼=	NOUN
ejpam-4387	265	5	z	z	NOUN
ejpam-4387	265	6	,	,	PUNCT
ejpam-4387	265	7	so	so	SCONJ
ejpam-4387	265	8	there	there	PRON
ejpam-4387	265	9	exists	exist	VERB
ejpam-4387	265	10	a	a	DET
ejpam-4387	265	11	homeomorphism	homeomorphism	NOUN
ejpam-4387	266	1	k	k	NOUN
ejpam-4387	266	2	:	:	PUNCT
ejpam-4387	266	3	z	z	PUNCT
ejpam-4387	266	4	−→	−→	NOUN
ejpam-4387	266	5	x.	x.	NOUN
ejpam-4387	266	6	since	since	SCONJ
ejpam-4387	266	7	x	x	PRON
ejpam-4387	266	8	is	be	AUX
ejpam-4387	266	9	strongly	strongly	ADV
ejpam-4387	266	10	p	p	ADJ
ejpam-4387	266	11	-normal	-normal	ADJ
ejpam-4387	266	12	then	then	ADV
ejpam-4387	266	13	there	there	PRON
ejpam-4387	266	14	exists	exist	VERB
ejpam-4387	266	15	a	a	DET
ejpam-4387	266	16	witness	witness	NOUN
ejpam-4387	266	17	function	function	NOUN
ejpam-4387	266	18	h	h	NOUN
ejpam-4387	266	19	:	:	PUNCT
ejpam-4387	266	20	x	x	PUNCT
ejpam-4387	266	21	−→	−→	NOUN
ejpam-4387	266	22	i	i	PRON
ejpam-4387	266	23	which	which	PRON
ejpam-4387	266	24	is	be	AUX
ejpam-4387	266	25	a	a	DET
ejpam-4387	266	26	bijection	bijection	NOUN
ejpam-4387	266	27	with	with	ADP
ejpam-4387	266	28	the	the	DET
ejpam-4387	266	29	restriction	restriction	NOUN
ejpam-4387	266	30	h|c	h|c	PUNCT
ejpam-4387	266	31	:	:	PUNCT
ejpam-4387	266	32	c	c	AUX
ejpam-4387	266	33	−→	−→	NOUN
ejpam-4387	266	34	f(c	f(c	PROPN
ejpam-4387	266	35	)	)	PUNCT
ejpam-4387	266	36	is	be	AUX
ejpam-4387	266	37	a	a	DET
ejpam-4387	266	38	homeomorphism	homeomorphism	NOUN
ejpam-4387	266	39	for	for	ADP
ejpam-4387	266	40	any	any	DET
ejpam-4387	266	41	paracompact	paracompact	ADJ
ejpam-4387	266	42	subspace	subspace	NOUN
ejpam-4387	266	43	c	c	NOUN
ejpam-4387	266	44	of	of	ADP
ejpam-4387	266	45	x.	x.	PROPN
ejpam-4387	266	46	.	.	PUNCT
ejpam-4387	267	1	then	then	ADV
ejpam-4387	267	2	h	h	PROPN
ejpam-4387	267	3	◦	◦	NOUN
ejpam-4387	267	4	k	k	X
ejpam-4387	267	5	:	:	PUNCT
ejpam-4387	267	6	z	z	X
ejpam-4387	268	1	−→	−→	NOUN
ejpam-4387	268	2	i	i	PRON
ejpam-4387	268	3	satisfies	satisfy	VERB
ejpam-4387	268	4	the	the	DET
ejpam-4387	268	5	requirements	requirement	NOUN
ejpam-4387	268	6	.	.	PUNCT
ejpam-4387	269	1	theorem	theorem	VERB
ejpam-4387	269	2	6	6	NUM
ejpam-4387	269	3	.	.	PUNCT
ejpam-4387	270	1	if	if	SCONJ
ejpam-4387	270	2	x	x	PRON
ejpam-4387	270	3	is	be	AUX
ejpam-4387	270	4	fréchet	fréchet	ADJ
ejpam-4387	270	5	and	and	CCONJ
ejpam-4387	270	6	strongly	strongly	ADV
ejpam-4387	270	7	p	p	ADJ
ejpam-4387	270	8	-normal	-normal	NOUN
ejpam-4387	270	9	,	,	PUNCT
ejpam-4387	270	10	then	then	ADV
ejpam-4387	270	11	any	any	DET
ejpam-4387	270	12	function	function	NOUN
ejpam-4387	270	13	witnessing	witness	VERB
ejpam-4387	270	14	the	the	DET
ejpam-4387	270	15	strong	strong	ADJ
ejpam-4387	270	16	p	p	NOUN
ejpam-4387	270	17	-normality	-normality	NOUN
ejpam-4387	270	18	of	of	ADP
ejpam-4387	270	19	x	x	PRON
ejpam-4387	270	20	is	be	AUX
ejpam-4387	270	21	continuous	continuous	ADJ
ejpam-4387	270	22	.	.	PUNCT
ejpam-4387	271	1	proof	proof	NOUN
ejpam-4387	271	2	.	.	PUNCT
ejpam-4387	272	1	assume	assume	VERB
ejpam-4387	272	2	that	that	SCONJ
ejpam-4387	272	3	x	x	PRON
ejpam-4387	272	4	is	be	AUX
ejpam-4387	272	5	strongly	strongly	ADV
ejpam-4387	272	6	p	p	X
ejpam-4387	272	7	-normal	-normal	ADJ
ejpam-4387	272	8	and	and	CCONJ
ejpam-4387	272	9	fréchet	fréchet	NOUN
ejpam-4387	272	10	.	.	PUNCT
ejpam-4387	273	1	let	let	VERB
ejpam-4387	273	2	f	f	NOUN
ejpam-4387	273	3	:	:	PUNCT
ejpam-4387	273	4	x	x	PUNCT
ejpam-4387	274	1	−→	−→	NOUN
ejpam-4387	274	2	i	i	PRON
ejpam-4387	274	3	be	be	VERB
ejpam-4387	274	4	a	a	DET
ejpam-4387	274	5	witness	witness	NOUN
ejpam-4387	274	6	of	of	ADP
ejpam-4387	274	7	the	the	DET
ejpam-4387	274	8	strong	strong	ADJ
ejpam-4387	274	9	p	p	NOUN
ejpam-4387	274	10	-normality	-normality	NOUN
ejpam-4387	274	11	of	of	ADP
ejpam-4387	274	12	x.	x.	NOUN
ejpam-4387	274	13	let	let	VERB
ejpam-4387	274	14	a	a	DET
ejpam-4387	274	15	⊆	⊆	NUM
ejpam-4387	274	16	x	x	PUNCT
ejpam-4387	274	17	and	and	CCONJ
ejpam-4387	274	18	pick	pick	VERB
ejpam-4387	274	19	y	y	PROPN
ejpam-4387	274	20	∈	∈	PROPN
ejpam-4387	274	21	f(a	f(a	PROPN
ejpam-4387	274	22	)	)	PUNCT
ejpam-4387	274	23	.	.	PUNCT
ejpam-4387	275	1	pick	pick	VERB
ejpam-4387	275	2	the	the	DET
ejpam-4387	275	3	unique	unique	ADJ
ejpam-4387	275	4	x	x	SYM
ejpam-4387	275	5	∈	∈	NOUN
ejpam-4387	275	6	x	x	PUNCT
ejpam-4387	275	7	such	such	ADJ
ejpam-4387	275	8	that	that	SCONJ
ejpam-4387	275	9	f(x	f(x	NOUN
ejpam-4387	275	10	)	)	PUNCT
ejpam-4387	275	11	=	=	PUNCT
ejpam-4387	276	1	y.	y.	NOUN
ejpam-4387	276	2	thus	thus	ADV
ejpam-4387	276	3	x	x	X
ejpam-4387	276	4	∈	∈	NOUN
ejpam-4387	276	5	a.	a.	NOUN
ejpam-4387	276	6	since	since	SCONJ
ejpam-4387	276	7	x	x	PROPN
ejpam-4387	276	8	is	be	AUX
ejpam-4387	276	9	fréchet	fréchet	VERB
ejpam-4387	276	10	,	,	PUNCT
ejpam-4387	276	11	there	there	PRON
ejpam-4387	276	12	exist	exist	VERB
ejpam-4387	276	13	a	a	DET
ejpam-4387	276	14	sequence	sequence	NOUN
ejpam-4387	276	15	(	(	PUNCT
ejpam-4387	276	16	an	an	NOUN
ejpam-4387	276	17	)	)	PUNCT
ejpam-4387	276	18	⊆	⊆	PROPN
ejpam-4387	276	19	a	a	DET
ejpam-4387	276	20	such	such	ADJ
ejpam-4387	276	21	that	that	SCONJ
ejpam-4387	276	22	an	an	DET
ejpam-4387	276	23	−→	−→	NOUN
ejpam-4387	276	24	x.	x.	NOUN
ejpam-4387	276	25	the	the	DET
ejpam-4387	276	26	subspace	subspace	PROPN
ejpam-4387	276	27	b	b	PROPN
ejpam-4387	276	28	=	=	PRON
ejpam-4387	276	29	{	{	PUNCT
ejpam-4387	276	30	x	x	NOUN
ejpam-4387	276	31	,	,	PUNCT
ejpam-4387	276	32	an	an	PRON
ejpam-4387	276	33	:	:	PUNCT
ejpam-4387	276	34	n	n	CCONJ
ejpam-4387	276	35	∈	∈	PROPN
ejpam-4387	276	36	n	n	CCONJ
ejpam-4387	276	37	}	}	PUNCT
ejpam-4387	276	38	of	of	ADP
ejpam-4387	276	39	x	x	SYM
ejpam-4387	276	40	is	be	AUX
ejpam-4387	276	41	paracompact	paracompact	ADJ
ejpam-4387	276	42	being	be	AUX
ejpam-4387	276	43	compact	compact	ADJ
ejpam-4387	276	44	,	,	PUNCT
ejpam-4387	276	45	thus	thus	ADV
ejpam-4387	276	46	f|b	f|b	VERB
ejpam-4387	276	47	:	:	PUNCT
ejpam-4387	276	48	b	b	X
ejpam-4387	276	49	−→	−→	ADJ
ejpam-4387	276	50	f(b	f(b	PROPN
ejpam-4387	276	51	)	)	PUNCT
ejpam-4387	276	52	is	be	AUX
ejpam-4387	276	53	a	a	DET
ejpam-4387	276	54	homeomorphism	homeomorphism	NOUN
ejpam-4387	276	55	.	.	PUNCT
ejpam-4387	277	1	now	now	ADV
ejpam-4387	277	2	,	,	PUNCT
ejpam-4387	277	3	let	let	VERB
ejpam-4387	277	4	w	w	PRON
ejpam-4387	277	5	⊆	⊆	NUM
ejpam-4387	277	6	i	i	PRON
ejpam-4387	277	7	be	be	VERB
ejpam-4387	277	8	any	any	DET
ejpam-4387	277	9	open	open	ADJ
ejpam-4387	277	10	neighborhood	neighborhood	NOUN
ejpam-4387	277	11	of	of	ADP
ejpam-4387	277	12	y	y	PROPN
ejpam-4387	277	13	,	,	PUNCT
ejpam-4387	277	14	then	then	ADV
ejpam-4387	277	15	w	w	PROPN
ejpam-4387	277	16	∩	∩	ADJ
ejpam-4387	277	17	f(b	f(b	PROPN
ejpam-4387	277	18	)	)	PUNCT
ejpam-4387	277	19	is	be	AUX
ejpam-4387	277	20	open	open	ADJ
ejpam-4387	277	21	in	in	ADP
ejpam-4387	277	22	the	the	DET
ejpam-4387	277	23	subspace	subspace	NOUN
ejpam-4387	277	24	f(b	f(b	PROPN
ejpam-4387	277	25	)	)	PUNCT
ejpam-4387	277	26	containing	contain	VERB
ejpam-4387	277	27	y.	y.	NOUN
ejpam-4387	277	28	by	by	ADP
ejpam-4387	277	29	continuity	continuity	NOUN
ejpam-4387	277	30	of	of	ADP
ejpam-4387	277	31	the	the	DET
ejpam-4387	277	32	homeomorphism	homeomorphism	PROPN
ejpam-4387	277	33	f|b	f|b	PROPN
ejpam-4387	277	34	,	,	PUNCT
ejpam-4387	277	35	f	f	PROPN
ejpam-4387	277	36	−1(w	−1(w	ADV
ejpam-4387	277	37	∩	∩	ADJ
ejpam-4387	277	38	f(b	f(b	PROPN
ejpam-4387	277	39	)	)	PUNCT
ejpam-4387	277	40	)	)	PUNCT
ejpam-4387	278	1	=	=	SYM
ejpam-4387	278	2	f−1(w	f−1(w	ADJ
ejpam-4387	278	3	)	)	PUNCT
ejpam-4387	278	4	∩	∩	NOUN
ejpam-4387	278	5	b	b	X
ejpam-4387	278	6	is	be	AUX
ejpam-4387	278	7	an	an	DET
ejpam-4387	278	8	open	open	ADJ
ejpam-4387	278	9	neighborhood	neighborhood	NOUN
ejpam-4387	278	10	of	of	ADP
ejpam-4387	278	11	x	x	PUNCT
ejpam-4387	278	12	in	in	ADP
ejpam-4387	278	13	b.	b.	PROPN
ejpam-4387	278	14	then,(f−1(w	then,(f−1(w	X
ejpam-4387	278	15	)	)	PUNCT
ejpam-4387	278	16	∩	∩	ADJ
ejpam-4387	278	17	b	b	X
ejpam-4387	278	18	)	)	PUNCT
ejpam-4387	278	19	∩	∩	NOUN
ejpam-4387	278	20	{	{	PUNCT
ejpam-4387	278	21	an	an	PRON
ejpam-4387	278	22	:	:	PUNCT
ejpam-4387	278	23	n	n	CCONJ
ejpam-4387	278	24	∈	∈	PROPN
ejpam-4387	278	25	n	n	CCONJ
ejpam-4387	278	26	}	}	PUNCT
ejpam-4387	278	27	=	=	NOUN
ejpam-4387	278	28	̸	̸	ADV
ejpam-4387	278	29	∅.	∅.	ADV
ejpam-4387	278	30	so	so	ADV
ejpam-4387	278	31	(	(	PUNCT
ejpam-4387	278	32	f−1(w	f−1(w	PROPN
ejpam-4387	278	33	)	)	PUNCT
ejpam-4387	278	34	∩	∩	ADJ
ejpam-4387	278	35	b	b	X
ejpam-4387	278	36	)	)	PUNCT
ejpam-4387	278	37	∩	∩	NOUN
ejpam-4387	278	38	a	a	DET
ejpam-4387	278	39	̸=	̸=	PROPN
ejpam-4387	278	40	∅.	∅.	VERB
ejpam-4387	278	41	therefore	therefore	ADV
ejpam-4387	278	42	we	we	PRON
ejpam-4387	278	43	have	have	VERB
ejpam-4387	278	44	,	,	PUNCT
ejpam-4387	278	45	∅	∅	NOUN
ejpam-4387	278	46	̸=	̸=	PROPN
ejpam-4387	278	47	f((f−1(w	f((f−1(w	NOUN
ejpam-4387	278	48	)	)	PUNCT
ejpam-4387	278	49	∩	∩	ADJ
ejpam-4387	278	50	b	b	X
ejpam-4387	278	51	)	)	PUNCT
ejpam-4387	278	52	∩	∩	NOUN
ejpam-4387	278	53	a	a	X
ejpam-4387	278	54	)	)	PUNCT
ejpam-4387	278	55	⊆	⊆	NUM
ejpam-4387	278	56	f(f−1(w	f(f−1(w	PROPN
ejpam-4387	278	57	)	)	PUNCT
ejpam-4387	278	58	∩	∩	NOUN
ejpam-4387	278	59	a	a	X
ejpam-4387	278	60	)	)	PUNCT
ejpam-4387	278	61	=	=	SYM
ejpam-4387	278	62	w	w	PROPN
ejpam-4387	278	63	∩	∩	ADJ
ejpam-4387	278	64	f(a	f(a	PROPN
ejpam-4387	278	65	)	)	PUNCT
ejpam-4387	278	66	then	then	ADV
ejpam-4387	278	67	w	w	PROPN
ejpam-4387	278	68	∩	∩	ADJ
ejpam-4387	278	69	f(a	f(a	NOUN
ejpam-4387	278	70	)	)	PUNCT
ejpam-4387	278	71	̸=	̸=	PROPN
ejpam-4387	278	72	∅.	∅.	PRON
ejpam-4387	278	73	hence	hence	ADV
ejpam-4387	278	74	y	y	PROPN
ejpam-4387	278	75	∈	∈	PROPN
ejpam-4387	278	76	f(a	f(a	PROPN
ejpam-4387	278	77	)	)	PUNCT
ejpam-4387	278	78	,	,	PUNCT
ejpam-4387	278	79	thus	thus	ADV
ejpam-4387	278	80	f(a	f(a	X
ejpam-4387	278	81	)	)	PUNCT
ejpam-4387	278	82	⊆	⊆	NUM
ejpam-4387	278	83	f(a	f(a	NOUN
ejpam-4387	278	84	)	)	PUNCT
ejpam-4387	278	85	.	.	PUNCT
ejpam-4387	279	1	therefore	therefore	ADV
ejpam-4387	279	2	,	,	PUNCT
ejpam-4387	279	3	f	f	PROPN
ejpam-4387	279	4	is	be	AUX
ejpam-4387	279	5	continuous	continuous	ADJ
ejpam-4387	279	6	.	.	PUNCT
ejpam-4387	279	7	example	example	NOUN
ejpam-4387	280	1	6	6	NUM
ejpam-4387	280	2	.	.	PUNCT
ejpam-4387	281	1	it	it	PRON
ejpam-4387	281	2	is	be	AUX
ejpam-4387	281	3	clear	clear	ADJ
ejpam-4387	281	4	that	that	SCONJ
ejpam-4387	281	5	i	i	PRON
ejpam-4387	281	6	with	with	ADP
ejpam-4387	281	7	its	its	PRON
ejpam-4387	281	8	usual	usual	ADJ
ejpam-4387	281	9	metric	metric	ADJ
ejpam-4387	281	10	topology	topology	NOUN
ejpam-4387	281	11	is	be	AUX
ejpam-4387	281	12	strongly	strongly	ADV
ejpam-4387	281	13	p	p	ADJ
ejpam-4387	281	14	-normal	-normal	NOUN
ejpam-4387	281	15	.	.	PUNCT
ejpam-4387	282	1	we	we	PRON
ejpam-4387	282	2	show	show	VERB
ejpam-4387	282	3	that	that	SCONJ
ejpam-4387	282	4	the	the	DET
ejpam-4387	282	5	product	product	NOUN
ejpam-4387	282	6	i	i	PRON
ejpam-4387	282	7	×	×	VERB
ejpam-4387	282	8	i	i	PRON
ejpam-4387	282	9	is	be	AUX
ejpam-4387	282	10	not	not	PART
ejpam-4387	282	11	strongly	strongly	ADV
ejpam-4387	282	12	p	p	X
ejpam-4387	282	13	-normal	-normal	NOUN
ejpam-4387	282	14	.	.	PUNCT
ejpam-4387	283	1	proof	proof	NOUN
ejpam-4387	283	2	.	.	PUNCT
ejpam-4387	284	1	suppose	suppose	VERB
ejpam-4387	284	2	to	to	ADP
ejpam-4387	284	3	the	the	DET
ejpam-4387	284	4	contrary	contrary	NOUN
ejpam-4387	284	5	that	that	PRON
ejpam-4387	284	6	i	i	PRON
ejpam-4387	284	7	×	×	VERB
ejpam-4387	284	8	i	i	PRON
ejpam-4387	284	9	is	be	AUX
ejpam-4387	284	10	strongly	strongly	ADV
ejpam-4387	284	11	p	p	X
ejpam-4387	284	12	-normal	-normal	NOUN
ejpam-4387	284	13	.	.	PUNCT
ejpam-4387	285	1	pick	pick	VERB
ejpam-4387	285	2	a	a	DET
ejpam-4387	285	3	bijection	bijection	NOUN
ejpam-4387	285	4	f	f	NOUN
ejpam-4387	285	5	:	:	PUNCT
ejpam-4387	286	1	i	i	PRON
ejpam-4387	286	2	×	×	VERB
ejpam-4387	287	1	i	i	PRON
ejpam-4387	287	2	−→	−→	VERB
ejpam-4387	287	3	i	i	PRON
ejpam-4387	287	4	such	such	ADJ
ejpam-4387	287	5	that	that	SCONJ
ejpam-4387	287	6	f	f	PROPN
ejpam-4387	287	7	|a	|a	VERB
ejpam-4387	287	8	:	:	PUNCT
ejpam-4387	287	9	a	a	DET
ejpam-4387	287	10	−→	−→	NOUN
ejpam-4387	287	11	f(a	f(a	NOUN
ejpam-4387	287	12	)	)	PUNCT
ejpam-4387	287	13	is	be	AUX
ejpam-4387	287	14	a	a	DET
ejpam-4387	287	15	homeomorphism	homeomorphism	NOUN
ejpam-4387	287	16	for	for	ADP
ejpam-4387	287	17	each	each	DET
ejpam-4387	287	18	paracompact	paracompact	ADJ
ejpam-4387	287	19	subspace	subspace	NOUN
ejpam-4387	287	20	a	a	DET
ejpam-4387	287	21	⊆	⊆	NUM
ejpam-4387	287	22	i	i	PRON
ejpam-4387	287	23	×	×	PROPN
ejpam-4387	287	24	i.	i.	NOUN
ejpam-4387	288	1	now	now	ADV
ejpam-4387	289	1	i	i	PRON
ejpam-4387	289	2	×	×	VERB
ejpam-4387	289	3	i	i	PRON
ejpam-4387	289	4	is	be	AUX
ejpam-4387	289	5	first	first	ADV
ejpam-4387	289	6	countable	countable	ADJ
ejpam-4387	289	7	and	and	CCONJ
ejpam-4387	289	8	hence	hence	ADV
ejpam-4387	289	9	fréchet	fréchet	PROPN
ejpam-4387	289	10	.	.	PUNCT
ejpam-4387	290	1	this	this	PRON
ejpam-4387	290	2	implies	imply	VERB
ejpam-4387	290	3	that	that	SCONJ
ejpam-4387	290	4	f	f	PROPN
ejpam-4387	290	5	is	be	AUX
ejpam-4387	290	6	continuous	continuous	ADJ
ejpam-4387	290	7	,	,	PUNCT
ejpam-4387	290	8	see	see	ADJ
ejpam-4387	290	9	theorem	theorem	VERB
ejpam-4387	290	10	6	6	NUM
ejpam-4387	290	11	,	,	PUNCT
ejpam-4387	290	12	which	which	PRON
ejpam-4387	290	13	contradicts	contradict	VERB
ejpam-4387	290	14	the	the	DET
ejpam-4387	290	15	fact	fact	NOUN
ejpam-4387	290	16	that	that	SCONJ
ejpam-4387	290	17	there	there	PRON
ejpam-4387	290	18	exists	exist	VERB
ejpam-4387	290	19	no	no	DET
ejpam-4387	290	20	continuous	continuous	ADJ
ejpam-4387	290	21	bijection	bijection	NOUN
ejpam-4387	290	22	f	f	NOUN
ejpam-4387	290	23	:	:	PUNCT
ejpam-4387	291	1	i	i	PRON
ejpam-4387	291	2	×	×	VERB
ejpam-4387	292	1	i	i	PRON
ejpam-4387	292	2	−→	−→	ADJ
ejpam-4387	292	3	i.	i.	NOUN
ejpam-4387	292	4	because	because	SCONJ
ejpam-4387	292	5	if	if	SCONJ
ejpam-4387	292	6	there	there	PRON
ejpam-4387	292	7	were	be	VERB
ejpam-4387	292	8	,	,	PUNCT
ejpam-4387	292	9	then	then	ADV
ejpam-4387	292	10	f	f	PROPN
ejpam-4387	292	11	would	would	AUX
ejpam-4387	292	12	be	be	AUX
ejpam-4387	292	13	a	a	DET
ejpam-4387	292	14	homeomorphism	homeomorphism	NOUN
ejpam-4387	292	15	since	since	SCONJ
ejpam-4387	292	16	i	i	PRON
ejpam-4387	292	17	×	×	VERB
ejpam-4387	292	18	i	i	PRON
ejpam-4387	292	19	is	be	AUX
ejpam-4387	292	20	compact	compact	ADJ
ejpam-4387	293	1	and	and	CCONJ
ejpam-4387	293	2	i	i	PRON
ejpam-4387	293	3	is	be	AUX
ejpam-4387	293	4	t2	t2	NOUN
ejpam-4387	293	5	.	.	PUNCT
ejpam-4387	294	1	this	this	PRON
ejpam-4387	294	2	is	be	AUX
ejpam-4387	294	3	a	a	DET
ejpam-4387	294	4	contradiction	contradiction	NOUN
ejpam-4387	294	5	since	since	SCONJ
ejpam-4387	294	6	i	i	PRON
ejpam-4387	294	7	×	×	VERB
ejpam-4387	294	8	i	i	PRON
ejpam-4387	294	9	is	be	AUX
ejpam-4387	294	10	connected	connect	VERB
ejpam-4387	294	11	with	with	ADP
ejpam-4387	294	12	no	no	DET
ejpam-4387	294	13	cut	cut	NOUN
ejpam-4387	294	14	points	point	NOUN
ejpam-4387	294	15	(	(	PUNCT
ejpam-4387	294	16	i	i	PRON
ejpam-4387	294	17	×	×	VERB
ejpam-4387	294	18	i	i	NOUN
ejpam-4387	294	19	)	)	PUNCT
ejpam-4387	294	20	\	\	NOUN
ejpam-4387	294	21	{	{	PUNCT
ejpam-4387	294	22	⟨x	⟨x	NUM
ejpam-4387	294	23	,	,	PUNCT
ejpam-4387	294	24	y⟩	y⟩	NOUN
ejpam-4387	294	25	}	}	PUNCT
ejpam-4387	294	26	is	be	AUX
ejpam-4387	294	27	connected	connect	VERB
ejpam-4387	294	28	for	for	ADP
ejpam-4387	294	29	every	every	DET
ejpam-4387	294	30	point	point	NOUN
ejpam-4387	294	31	⟨x	⟨x	VERB
ejpam-4387	294	32	,	,	PUNCT
ejpam-4387	294	33	y⟩	y⟩	NOUN
ejpam-4387	294	34	∈	∈	PROPN
ejpam-4387	295	1	i	i	PRON
ejpam-4387	295	2	×	×	VERB
ejpam-4387	295	3	i	i	PRON
ejpam-4387	295	4	,	,	PUNCT
ejpam-4387	295	5	while	while	SCONJ
ejpam-4387	295	6	i	i	PRON
ejpam-4387	295	7	is	be	AUX
ejpam-4387	295	8	connected	connect	VERB
ejpam-4387	295	9	with	with	ADP
ejpam-4387	295	10	cut	cut	ADJ
ejpam-4387	295	11	points	point	NOUN
ejpam-4387	295	12	(	(	PUNCT
ejpam-4387	295	13	take	take	VERB
ejpam-4387	295	14	any	any	DET
ejpam-4387	295	15	x	x	SYM
ejpam-4387	295	16	∈	∈	PROPN
ejpam-4387	295	17	(	(	PUNCT
ejpam-4387	295	18	0	0	NUM
ejpam-4387	295	19	,	,	PUNCT
ejpam-4387	295	20	1	1	NUM
ejpam-4387	295	21	)	)	PUNCT
ejpam-4387	296	1	⊂	⊂	PROPN
ejpam-4387	297	1	i	i	PRON
ejpam-4387	297	2	,	,	PUNCT
ejpam-4387	297	3	then	then	ADV
ejpam-4387	297	4	i	i	PRON
ejpam-4387	297	5	\	\	PUNCT
ejpam-4387	297	6	{	{	PUNCT
ejpam-4387	297	7	x	x	X
ejpam-4387	297	8	}	}	PUNCT
ejpam-4387	297	9	=	=	PUNCT
ejpam-4387	298	1	[	[	X
ejpam-4387	298	2	0	0	NUM
ejpam-4387	298	3	,	,	PUNCT
ejpam-4387	298	4	x	x	NOUN
ejpam-4387	298	5	)	)	PUNCT
ejpam-4387	298	6	∪	∪	ADV
ejpam-4387	298	7	(	(	PUNCT
ejpam-4387	298	8	x	x	X
ejpam-4387	298	9	,	,	PUNCT
ejpam-4387	298	10	1	1	NUM
ejpam-4387	298	11	]	]	PUNCT
ejpam-4387	298	12	where	where	SCONJ
ejpam-4387	298	13	both	both	PRON
ejpam-4387	298	14	[	[	X
ejpam-4387	298	15	0	0	NUM
ejpam-4387	298	16	,	,	PUNCT
ejpam-4387	298	17	x	x	NOUN
ejpam-4387	298	18	)	)	PUNCT
ejpam-4387	298	19	and	and	CCONJ
ejpam-4387	298	20	(	(	PUNCT
ejpam-4387	298	21	x	x	X
ejpam-4387	298	22	,	,	PUNCT
ejpam-4387	298	23	1	1	NUM
ejpam-4387	298	24	]	]	PUNCT
ejpam-4387	298	25	are	be	AUX
ejpam-4387	298	26	non	non	ADJ
ejpam-4387	298	27	-	-	ADJ
ejpam-4387	298	28	empty	empty	ADJ
ejpam-4387	298	29	disjoint	disjoint	ADJ
ejpam-4387	298	30	open	open	ADJ
ejpam-4387	298	31	subsets	subset	NOUN
ejpam-4387	298	32	of	of	ADP
ejpam-4387	298	33	i	i	PROPN
ejpam-4387	298	34	)	)	PUNCT
ejpam-4387	298	35	.	.	PUNCT
ejpam-4387	299	1	therefore	therefore	ADV
ejpam-4387	299	2	,	,	PUNCT
ejpam-4387	299	3	i	i	PRON
ejpam-4387	299	4	×	×	VERB
ejpam-4387	299	5	i	i	PRON
ejpam-4387	299	6	is	be	AUX
ejpam-4387	299	7	not	not	PART
ejpam-4387	299	8	strongly	strongly	ADV
ejpam-4387	299	9	p	p	ADJ
ejpam-4387	299	10	-normal	-normal	NOUN
ejpam-4387	299	11	.	.	PUNCT
ejpam-4387	300	1	so	so	ADV
ejpam-4387	300	2	,	,	PUNCT
ejpam-4387	300	3	a	a	DET
ejpam-4387	300	4	product	product	NOUN
ejpam-4387	300	5	of	of	ADP
ejpam-4387	300	6	two	two	NUM
ejpam-4387	300	7	strongly	strongly	ADV
ejpam-4387	300	8	p	p	ADJ
ejpam-4387	300	9	-normal	-normal	ADJ
ejpam-4387	300	10	spaces	space	NOUN
ejpam-4387	300	11	may	may	AUX
ejpam-4387	300	12	not	not	PART
ejpam-4387	300	13	be	be	AUX
ejpam-4387	300	14	strongly	strongly	ADV
ejpam-4387	300	15	p	p	X
ejpam-4387	300	16	-normal	-normal	NOUN
ejpam-4387	300	17	.	.	PUNCT
ejpam-4387	301	1	but	but	CCONJ
ejpam-4387	301	2	,	,	PUNCT
ejpam-4387	301	3	a	a	DET
ejpam-4387	301	4	product	product	NOUN
ejpam-4387	301	5	of	of	ADP
ejpam-4387	301	6	two	two	NUM
ejpam-4387	301	7	strongly	strongly	ADV
ejpam-4387	301	8	p	p	ADJ
ejpam-4387	301	9	-normal	-normal	ADJ
ejpam-4387	301	10	spaces	space	NOUN
ejpam-4387	301	11	is	be	AUX
ejpam-4387	301	12	p	p	NOUN
ejpam-4387	301	13	-normal	-normal	NOUN
ejpam-4387	301	14	.	.	PUNCT
ejpam-4387	302	1	to	to	PART
ejpam-4387	302	2	show	show	VERB
ejpam-4387	302	3	this	this	PRON
ejpam-4387	302	4	,	,	PUNCT
ejpam-4387	302	5	we	we	PRON
ejpam-4387	302	6	start	start	VERB
ejpam-4387	302	7	with	with	ADP
ejpam-4387	302	8	a	a	DET
ejpam-4387	302	9	lemma	lemma	PROPN
ejpam-4387	302	10	.	.	PUNCT
ejpam-4387	303	1	lemma	lemma	PROPN
ejpam-4387	303	2	1	1	NUM
ejpam-4387	303	3	.	.	PUNCT
ejpam-4387	304	1	if	if	SCONJ
ejpam-4387	304	2	a	a	PRON
ejpam-4387	304	3	is	be	AUX
ejpam-4387	304	4	a	a	DET
ejpam-4387	304	5	paracompact	paracompact	ADJ
ejpam-4387	304	6	subset	subset	NOUN
ejpam-4387	304	7	of	of	ADP
ejpam-4387	304	8	the	the	DET
ejpam-4387	304	9	product	product	NOUN
ejpam-4387	304	10	x	x	PUNCT
ejpam-4387	304	11	×z	×z	NOUN
ejpam-4387	304	12	,	,	PUNCT
ejpam-4387	304	13	then	then	ADV
ejpam-4387	304	14	p1(a	p1(a	NOUN
ejpam-4387	304	15	)	)	PUNCT
ejpam-4387	304	16	and	and	CCONJ
ejpam-4387	304	17	p2(a	p2(a	NOUN
ejpam-4387	304	18	)	)	PUNCT
ejpam-4387	304	19	are	be	AUX
ejpam-4387	304	20	both	both	PRON
ejpam-4387	304	21	paracompact	paracompact	ADJ
ejpam-4387	304	22	in	in	ADP
ejpam-4387	304	23	x	x	X
ejpam-4387	304	24	and	and	CCONJ
ejpam-4387	304	25	z	z	NOUN
ejpam-4387	304	26	respectively	respectively	ADV
ejpam-4387	304	27	.	.	PUNCT
ejpam-4387	305	1	where	where	SCONJ
ejpam-4387	305	2	p1	p1	NOUN
ejpam-4387	305	3	and	and	CCONJ
ejpam-4387	305	4	p2	p2	PROPN
ejpam-4387	305	5	are	be	AUX
ejpam-4387	305	6	the	the	DET
ejpam-4387	305	7	natural	natural	ADJ
ejpam-4387	305	8	projection	projection	NOUN
ejpam-4387	305	9	functions	function	NOUN
ejpam-4387	305	10	.	.	PUNCT
ejpam-4387	306	1	proof	proof	NOUN
ejpam-4387	306	2	.	.	PUNCT
ejpam-4387	307	1	let	let	VERB
ejpam-4387	307	2	a	a	PRON
ejpam-4387	307	3	be	be	AUX
ejpam-4387	307	4	a	a	DET
ejpam-4387	307	5	paracompact	paracompact	ADJ
ejpam-4387	307	6	subset	subset	NOUN
ejpam-4387	307	7	of	of	ADP
ejpam-4387	307	8	the	the	DET
ejpam-4387	307	9	product	product	NOUN
ejpam-4387	307	10	x×z	x×z	PROPN
ejpam-4387	307	11	.	.	PUNCT
ejpam-4387	307	12	suppose	suppose	VERB
ejpam-4387	307	13	that	that	SCONJ
ejpam-4387	307	14	p1(a	p1(a	NOUN
ejpam-4387	307	15	)	)	PUNCT
ejpam-4387	307	16	is	be	AUX
ejpam-4387	307	17	not	not	PART
ejpam-4387	307	18	paracompact	paracompact	NOUN
ejpam-4387	307	19	subset	subset	NOUN
ejpam-4387	307	20	in	in	ADP
ejpam-4387	307	21	x	x	PRON
ejpam-4387	307	22	,	,	PUNCT
ejpam-4387	307	23	i.e.	i.e.	X
ejpam-4387	307	24	,	,	PUNCT
ejpam-4387	307	25	p1(a	p1(a	NOUN
ejpam-4387	307	26	)	)	PUNCT
ejpam-4387	307	27	as	as	ADP
ejpam-4387	307	28	a	a	DET
ejpam-4387	307	29	subspace	subspace	NOUN
ejpam-4387	307	30	of	of	ADP
ejpam-4387	307	31	x	x	PUNCT
ejpam-4387	307	32	is	be	AUX
ejpam-4387	307	33	not	not	PART
ejpam-4387	307	34	paracompact	paracompact	ADJ
ejpam-4387	307	35	.	.	PUNCT
ejpam-4387	308	1	then	then	ADV
ejpam-4387	308	2	there	there	PRON
ejpam-4387	308	3	references	reference	VERB
ejpam-4387	308	4	782	782	NUM
ejpam-4387	308	5	exist	exist	VERB
ejpam-4387	308	6	an	an	DET
ejpam-4387	308	7	open	open	ADJ
ejpam-4387	308	8	cover	cover	NOUN
ejpam-4387	308	9	u	u	NOUN
ejpam-4387	308	10	=	=	PUNCT
ejpam-4387	308	11	{	{	PUNCT
ejpam-4387	308	12	uα	uα	PROPN
ejpam-4387	308	13	⊆	⊆	NUM
ejpam-4387	308	14	p1(a	p1(a	NOUN
ejpam-4387	308	15	)	)	PUNCT
ejpam-4387	308	16	:	:	PUNCT
ejpam-4387	308	17	uα	uα	PROPN
ejpam-4387	308	18	is	be	AUX
ejpam-4387	308	19	open	open	ADJ
ejpam-4387	308	20	in	in	ADP
ejpam-4387	308	21	p1(a	p1(a	NOUN
ejpam-4387	308	22	)	)	PUNCT
ejpam-4387	308	23	for	for	ADP
ejpam-4387	308	24	each	each	DET
ejpam-4387	308	25	α	α	NOUN
ejpam-4387	308	26	∈	∈	PROPN
ejpam-4387	308	27	λ	λ	PROPN
ejpam-4387	308	28	}	}	PUNCT
ejpam-4387	308	29	for	for	ADP
ejpam-4387	308	30	p1(a	p1(a	NOUN
ejpam-4387	308	31	)	)	PUNCT
ejpam-4387	308	32	such	such	ADJ
ejpam-4387	308	33	that	that	SCONJ
ejpam-4387	308	34	any	any	DET
ejpam-4387	308	35	open	open	ADJ
ejpam-4387	308	36	(	(	PUNCT
ejpam-4387	308	37	open	open	ADJ
ejpam-4387	308	38	in	in	ADP
ejpam-4387	308	39	p1(a	p1(a	NOUN
ejpam-4387	308	40	)	)	PUNCT
ejpam-4387	308	41	)	)	PUNCT
ejpam-4387	308	42	refinement	refinement	NOUN
ejpam-4387	308	43	of	of	ADP
ejpam-4387	308	44	u	u	NOUN
ejpam-4387	308	45	is	be	AUX
ejpam-4387	308	46	not	not	PART
ejpam-4387	308	47	locally	locally	ADV
ejpam-4387	308	48	finite	finite	ADJ
ejpam-4387	308	49	.	.	PUNCT
ejpam-4387	309	1	now	now	ADV
ejpam-4387	309	2	,	,	PUNCT
ejpam-4387	309	3	let	let	VERB
ejpam-4387	309	4	x	x	X
ejpam-4387	309	5	∈	∈	PROPN
ejpam-4387	309	6	p1(a	p1(a	PROPN
ejpam-4387	309	7	)	)	PUNCT
ejpam-4387	309	8	and	and	CCONJ
ejpam-4387	309	9	fix	fix	VERB
ejpam-4387	309	10	an	an	DET
ejpam-4387	309	11	αx	αx	ADV
ejpam-4387	309	12	∈	∈	PROPN
ejpam-4387	309	13	λ	λ	NOUN
ejpam-4387	309	14	such	such	ADJ
ejpam-4387	309	15	that	that	SCONJ
ejpam-4387	309	16	x	x	SYM
ejpam-4387	309	17	∈	∈	PROPN
ejpam-4387	309	18	uαx	uαx	PROPN
ejpam-4387	309	19	.	.	PUNCT
ejpam-4387	310	1	for	for	ADP
ejpam-4387	310	2	each	each	DET
ejpam-4387	310	3	z	z	NOUN
ejpam-4387	310	4	∈	∈	PROPN
ejpam-4387	310	5	z	z	NOUN
ejpam-4387	311	1	such	such	ADJ
ejpam-4387	311	2	that	that	SCONJ
ejpam-4387	311	3	there	there	PRON
ejpam-4387	311	4	exists	exist	VERB
ejpam-4387	311	5	x	x	X
ejpam-4387	311	6	∈	∈	PROPN
ejpam-4387	311	7	p1(a	p1(a	NOUN
ejpam-4387	311	8	)	)	PUNCT
ejpam-4387	311	9	with	with	ADP
ejpam-4387	311	10	⟨x	⟨x	NUM
ejpam-4387	311	11	,	,	PUNCT
ejpam-4387	311	12	z⟩	z⟩	PROPN
ejpam-4387	311	13	∈	∈	PROPN
ejpam-4387	311	14	a	a	PRON
ejpam-4387	311	15	,	,	PUNCT
ejpam-4387	311	16	let	let	VERB
ejpam-4387	311	17	wz	wz	PART
ejpam-4387	311	18	be	be	AUX
ejpam-4387	311	19	an	an	DET
ejpam-4387	311	20	open	open	ADJ
ejpam-4387	311	21	neighborhood	neighborhood	NOUN
ejpam-4387	311	22	of	of	ADP
ejpam-4387	311	23	z	z	PROPN
ejpam-4387	311	24	in	in	ADP
ejpam-4387	311	25	z.	z.	PROPN
ejpam-4387	311	26	note	note	VERB
ejpam-4387	311	27	that	that	SCONJ
ejpam-4387	311	28	a	a	DET
ejpam-4387	311	29	⊆	⊆	NUM
ejpam-4387	311	30	p1(a)×	p1(a)×	PROPN
ejpam-4387	311	31	p2(a	p2(a	PROPN
ejpam-4387	311	32	)	)	PUNCT
ejpam-4387	311	33	.	.	PUNCT
ejpam-4387	312	1	consider	consider	VERB
ejpam-4387	312	2	the	the	DET
ejpam-4387	312	3	family	family	NOUN
ejpam-4387	312	4	k	k	PROPN
ejpam-4387	312	5	=	=	PRON
ejpam-4387	312	6	{	{	PUNCT
ejpam-4387	312	7	(	(	PUNCT
ejpam-4387	312	8	u)αx	u)αx	PROPN
ejpam-4387	312	9	×wz	×wz	PROPN
ejpam-4387	312	10	)	)	PUNCT
ejpam-4387	312	11	∩	∩	NOUN
ejpam-4387	312	12	a	a	PRON
ejpam-4387	312	13	:	:	PUNCT
ejpam-4387	312	14	⟨x	⟨x	NUM
ejpam-4387	312	15	,	,	PUNCT
ejpam-4387	312	16	z⟩	z⟩	PROPN
ejpam-4387	312	17	∈	∈	PROPN
ejpam-4387	312	18	a	a	PRON
ejpam-4387	312	19	}	}	PUNCT
ejpam-4387	312	20	which	which	PRON
ejpam-4387	312	21	is	be	AUX
ejpam-4387	312	22	an	an	DET
ejpam-4387	312	23	open	open	ADJ
ejpam-4387	312	24	(	(	PUNCT
ejpam-4387	312	25	open	open	ADJ
ejpam-4387	312	26	in	in	ADP
ejpam-4387	312	27	a	a	DET
ejpam-4387	312	28	)	)	PUNCT
ejpam-4387	312	29	cover	cover	NOUN
ejpam-4387	312	30	for	for	ADP
ejpam-4387	312	31	a.	a.	NOUN
ejpam-4387	312	32	claim	claim	NOUN
ejpam-4387	312	33	:	:	PUNCT
ejpam-4387	312	34	k	k	PROPN
ejpam-4387	312	35	has	have	VERB
ejpam-4387	312	36	no	no	DET
ejpam-4387	312	37	locally	locally	ADV
ejpam-4387	312	38	finite	finite	ADJ
ejpam-4387	312	39	open	open	ADJ
ejpam-4387	312	40	refinement	refinement	NOUN
ejpam-4387	312	41	.	.	PUNCT
ejpam-4387	313	1	proof	proof	NOUN
ejpam-4387	313	2	of	of	ADP
ejpam-4387	313	3	claim	claim	NOUN
ejpam-4387	313	4	:	:	PUNCT
ejpam-4387	313	5	suppose	suppose	VERB
ejpam-4387	313	6	that	that	SCONJ
ejpam-4387	313	7	k	k	PROPN
ejpam-4387	313	8	has	have	VERB
ejpam-4387	313	9	a	a	DET
ejpam-4387	313	10	locally	locally	ADV
ejpam-4387	313	11	finite	finite	ADJ
ejpam-4387	313	12	open	open	ADJ
ejpam-4387	313	13	refinement	refinement	NOUN
ejpam-4387	313	14	,	,	PUNCT
ejpam-4387	313	15	say	say	VERB
ejpam-4387	313	16	{	{	PUNCT
ejpam-4387	313	17	gs	gs	INTJ
ejpam-4387	313	18	×	×	NOUN
ejpam-4387	313	19	hs	hs	INTJ
ejpam-4387	313	20	:	:	PUNCT
ejpam-4387	313	21	s	s	VERB
ejpam-4387	313	22	∈	∈	PROPN
ejpam-4387	313	23	s	s	PART
ejpam-4387	313	24	}	}	PUNCT
ejpam-4387	313	25	(	(	PUNCT
ejpam-4387	313	26	we	we	PRON
ejpam-4387	313	27	can	can	AUX
ejpam-4387	313	28	assume	assume	VERB
ejpam-4387	313	29	that	that	SCONJ
ejpam-4387	313	30	this	this	DET
ejpam-4387	313	31	refinement	refinement	NOUN
ejpam-4387	313	32	is	be	AUX
ejpam-4387	313	33	of	of	ADP
ejpam-4387	313	34	the	the	DET
ejpam-4387	313	35	basic	basic	ADJ
ejpam-4387	313	36	open	open	ADJ
ejpam-4387	313	37	set	set	VERB
ejpam-4387	313	38	form	form	NOUN
ejpam-4387	313	39	in	in	ADP
ejpam-4387	313	40	the	the	DET
ejpam-4387	313	41	product	product	NOUN
ejpam-4387	313	42	x	x	X
ejpam-4387	313	43	×	×	PROPN
ejpam-4387	313	44	z	z	PROPN
ejpam-4387	313	45	)	)	PUNCT
ejpam-4387	313	46	,	,	PUNCT
ejpam-4387	313	47	then	then	ADV
ejpam-4387	313	48	the	the	DET
ejpam-4387	313	49	family	family	NOUN
ejpam-4387	313	50	{	{	PUNCT
ejpam-4387	313	51	gs	gs	PROPN
ejpam-4387	313	52	∩	∩	NOUN
ejpam-4387	313	53	pa(a	pa(a	NUM
ejpam-4387	313	54	)	)	PUNCT
ejpam-4387	313	55	:	:	PUNCT
ejpam-4387	313	56	s	s	VERB
ejpam-4387	313	57	∈	∈	PROPN
ejpam-4387	313	58	s	s	AUX
ejpam-4387	313	59	}	}	PUNCT
ejpam-4387	313	60	would	would	AUX
ejpam-4387	313	61	be	be	AUX
ejpam-4387	313	62	a	a	DET
ejpam-4387	313	63	locally	locally	ADV
ejpam-4387	313	64	finite	finite	ADJ
ejpam-4387	313	65	open	open	ADJ
ejpam-4387	313	66	refinement	refinement	NOUN
ejpam-4387	313	67	of	of	ADP
ejpam-4387	313	68	u	u	PRON
ejpam-4387	313	69	which	which	PRON
ejpam-4387	313	70	is	be	AUX
ejpam-4387	313	71	a	a	DET
ejpam-4387	313	72	contradiction	contradiction	NOUN
ejpam-4387	313	73	and	and	CCONJ
ejpam-4387	313	74	claim	claim	NOUN
ejpam-4387	313	75	is	be	AUX
ejpam-4387	313	76	proved	prove	VERB
ejpam-4387	313	77	.	.	PUNCT
ejpam-4387	314	1	so	so	ADV
ejpam-4387	314	2	,	,	PUNCT
ejpam-4387	314	3	k	k	PROPN
ejpam-4387	314	4	is	be	AUX
ejpam-4387	314	5	an	an	DET
ejpam-4387	314	6	open	open	ADJ
ejpam-4387	314	7	(	(	PUNCT
ejpam-4387	314	8	open	open	ADJ
ejpam-4387	314	9	in	in	ADP
ejpam-4387	314	10	a	a	DET
ejpam-4387	314	11	)	)	PUNCT
ejpam-4387	314	12	cover	cover	NOUN
ejpam-4387	314	13	for	for	ADP
ejpam-4387	314	14	a	a	PRON
ejpam-4387	314	15	which	which	PRON
ejpam-4387	314	16	has	have	VERB
ejpam-4387	314	17	no	no	DET
ejpam-4387	314	18	locally	locally	ADV
ejpam-4387	314	19	finite	finite	ADJ
ejpam-4387	314	20	open	open	ADJ
ejpam-4387	314	21	refinement	refinement	NOUN
ejpam-4387	314	22	and	and	CCONJ
ejpam-4387	314	23	this	this	PRON
ejpam-4387	314	24	contradicts	contradict	VERB
ejpam-4387	314	25	that	that	SCONJ
ejpam-4387	314	26	a	a	PRON
ejpam-4387	314	27	is	be	AUX
ejpam-4387	314	28	a	a	DET
ejpam-4387	314	29	paracompact	paracompact	NOUN
ejpam-4387	314	30	subset	subset	NOUN
ejpam-4387	314	31	in	in	ADP
ejpam-4387	314	32	x	x	X
ejpam-4387	314	33	×	×	PROPN
ejpam-4387	314	34	z.	z.	PROPN
ejpam-4387	314	35	therefore	therefore	ADV
ejpam-4387	314	36	,	,	PUNCT
ejpam-4387	314	37	p1(a	p1(a	NOUN
ejpam-4387	314	38	)	)	PUNCT
ejpam-4387	314	39	is	be	AUX
ejpam-4387	314	40	a	a	DET
ejpam-4387	314	41	paracompact	paracompact	NOUN
ejpam-4387	314	42	subset	subset	NOUN
ejpam-4387	314	43	in	in	ADP
ejpam-4387	314	44	x.	x.	NOUN
ejpam-4387	314	45	similarly	similarly	ADV
ejpam-4387	314	46	,	,	PUNCT
ejpam-4387	314	47	p2(a	p2(a	PROPN
ejpam-4387	314	48	)	)	PUNCT
ejpam-4387	314	49	is	be	AUX
ejpam-4387	314	50	a	a	DET
ejpam-4387	314	51	paracompact	paracompact	ADJ
ejpam-4387	314	52	subset	subset	NOUN
ejpam-4387	314	53	of	of	ADP
ejpam-4387	314	54	z.	z.	PROPN
ejpam-4387	314	55	theorem	theorem	VERB
ejpam-4387	314	56	7	7	NUM
ejpam-4387	314	57	.	.	PUNCT
ejpam-4387	315	1	if	if	SCONJ
ejpam-4387	315	2	x	x	PROPN
ejpam-4387	315	3	and	and	CCONJ
ejpam-4387	315	4	z	z	NOUN
ejpam-4387	315	5	are	be	AUX
ejpam-4387	315	6	both	both	PRON
ejpam-4387	315	7	strongly	strongly	ADV
ejpam-4387	315	8	p	p	ADJ
ejpam-4387	315	9	-normal	-normal	NOUN
ejpam-4387	315	10	,	,	PUNCT
ejpam-4387	315	11	then	then	ADV
ejpam-4387	315	12	x	x	SYM
ejpam-4387	315	13	×	×	PROPN
ejpam-4387	315	14	z	z	NOUN
ejpam-4387	315	15	is	be	AUX
ejpam-4387	315	16	p	p	NOUN
ejpam-4387	315	17	-normal	-normal	NOUN
ejpam-4387	315	18	.	.	PUNCT
ejpam-4387	316	1	proof	proof	NOUN
ejpam-4387	316	2	.	.	PUNCT
ejpam-4387	317	1	assume	assume	VERB
ejpam-4387	317	2	that	that	SCONJ
ejpam-4387	317	3	x	x	PRON
ejpam-4387	317	4	and	and	CCONJ
ejpam-4387	317	5	z	z	NOUN
ejpam-4387	317	6	are	be	AUX
ejpam-4387	317	7	both	both	PRON
ejpam-4387	317	8	strongly	strongly	ADV
ejpam-4387	317	9	p	p	ADJ
ejpam-4387	317	10	-normal	-normal	NOUN
ejpam-4387	317	11	.	.	PUNCT
ejpam-4387	318	1	pick	pick	VERB
ejpam-4387	318	2	two	two	NUM
ejpam-4387	318	3	bijection	bijection	NOUN
ejpam-4387	318	4	functions	function	NOUN
ejpam-4387	318	5	f	f	NOUN
ejpam-4387	318	6	:	:	PUNCT
ejpam-4387	318	7	x	x	PUNCT
ejpam-4387	319	1	−→	−→	NOUN
ejpam-4387	319	2	i	i	PRON
ejpam-4387	319	3	and	and	CCONJ
ejpam-4387	319	4	g	g	NOUN
ejpam-4387	319	5	:	:	PUNCT
ejpam-4387	319	6	z	z	X
ejpam-4387	320	1	−→	−→	NOUN
ejpam-4387	320	2	i	i	PRON
ejpam-4387	320	3	such	such	ADJ
ejpam-4387	320	4	f|a	f|a	NOUN
ejpam-4387	320	5	:	:	PUNCT
ejpam-4387	320	6	a	a	DET
ejpam-4387	320	7	−→	−→	NOUN
ejpam-4387	320	8	f(a	f(a	NOUN
ejpam-4387	320	9	)	)	PUNCT
ejpam-4387	320	10	is	be	AUX
ejpam-4387	320	11	a	a	DET
ejpam-4387	320	12	homeomorphism	homeomorphism	NOUN
ejpam-4387	320	13	for	for	ADP
ejpam-4387	320	14	each	each	DET
ejpam-4387	320	15	paracompact	paracompact	ADJ
ejpam-4387	320	16	subspace	subspace	NOUN
ejpam-4387	320	17	a	a	DET
ejpam-4387	320	18	⊆	⊆	NUM
ejpam-4387	320	19	x	x	NOUN
ejpam-4387	320	20	and	and	CCONJ
ejpam-4387	320	21	g|a	g|a	PROPN
ejpam-4387	320	22	:	:	PUNCT
ejpam-4387	320	23	a	a	DET
ejpam-4387	320	24	−→	−→	NOUN
ejpam-4387	320	25	f(a	f(a	NOUN
ejpam-4387	320	26	)	)	PUNCT
ejpam-4387	320	27	is	be	AUX
ejpam-4387	320	28	a	a	DET
ejpam-4387	320	29	homeomorphism	homeomorphism	NOUN
ejpam-4387	320	30	for	for	ADP
ejpam-4387	320	31	each	each	DET
ejpam-4387	320	32	paracompact	paracompact	ADJ
ejpam-4387	320	33	subspace	subspace	NOUN
ejpam-4387	320	34	a	a	PRON
ejpam-4387	320	35	⊆	⊆	NUM
ejpam-4387	320	36	z.	z.	NOUN
ejpam-4387	321	1	i	i	PRON
ejpam-4387	321	2	×	×	VERB
ejpam-4387	321	3	i	i	PRON
ejpam-4387	321	4	is	be	AUX
ejpam-4387	321	5	normal	normal	ADJ
ejpam-4387	321	6	being	be	AUX
ejpam-4387	321	7	t2	t2	NOUN
ejpam-4387	321	8	compact	compact	ADJ
ejpam-4387	321	9	.	.	PUNCT
ejpam-4387	322	1	put	put	VERB
ejpam-4387	322	2	h	h	NOUN
ejpam-4387	323	1	=	=	SYM
ejpam-4387	323	2	f	f	X
ejpam-4387	323	3	×	×	NOUN
ejpam-4387	323	4	g	g	NOUN
ejpam-4387	323	5	,	,	PUNCT
ejpam-4387	323	6	i.e.	i.e.	X
ejpam-4387	323	7	,	,	PUNCT
ejpam-4387	323	8	h	h	NOUN
ejpam-4387	323	9	:	:	PUNCT
ejpam-4387	323	10	x×z	x×z	PROPN
ejpam-4387	323	11	−→	−→	NOUN
ejpam-4387	323	12	i×i	i×i	PROPN
ejpam-4387	323	13	is	be	AUX
ejpam-4387	323	14	defined	define	VERB
ejpam-4387	323	15	by	by	ADP
ejpam-4387	323	16	h(⟨x	h(⟨x	NOUN
ejpam-4387	323	17	,	,	PUNCT
ejpam-4387	323	18	z⟩	z⟩	NOUN
ejpam-4387	323	19	)	)	PUNCT
ejpam-4387	323	20	=	=	SYM
ejpam-4387	323	21	⟨f(x	⟨f(x	NOUN
ejpam-4387	323	22	)	)	PUNCT
ejpam-4387	323	23	,	,	PUNCT
ejpam-4387	323	24	g(z)⟩	g(z)⟩	VERB
ejpam-4387	323	25	for	for	ADP
ejpam-4387	323	26	each	each	DET
ejpam-4387	323	27	⟨x	⟨x	VERB
ejpam-4387	323	28	,	,	PUNCT
ejpam-4387	323	29	z⟩	z⟩	PROPN
ejpam-4387	323	30	∈	∈	PROPN
ejpam-4387	323	31	x×z	x×z	PROPN
ejpam-4387	323	32	.	.	PUNCT
ejpam-4387	324	1	it	it	PRON
ejpam-4387	324	2	is	be	AUX
ejpam-4387	324	3	clear	clear	ADJ
ejpam-4387	324	4	that	that	SCONJ
ejpam-4387	324	5	h	h	NOUN
ejpam-4387	324	6	is	be	AUX
ejpam-4387	324	7	a	a	DET
ejpam-4387	324	8	bijection	bijection	ADJ
ejpam-4387	324	9	function	function	NOUN
ejpam-4387	324	10	.	.	PUNCT
ejpam-4387	325	1	let	let	VERB
ejpam-4387	325	2	a	a	DET
ejpam-4387	325	3	be	be	AUX
ejpam-4387	325	4	any	any	DET
ejpam-4387	325	5	paracompact	paracompact	NOUN
ejpam-4387	325	6	subset	subset	NOUN
ejpam-4387	325	7	of	of	ADP
ejpam-4387	325	8	x	x	SYM
ejpam-4387	325	9	×	×	PROPN
ejpam-4387	325	10	z.	z.	PROPN
ejpam-4387	325	11	by	by	ADP
ejpam-4387	325	12	lemma	lemma	PROPN
ejpam-4387	325	13	1	1	NUM
ejpam-4387	325	14	,	,	PUNCT
ejpam-4387	325	15	we	we	PRON
ejpam-4387	325	16	have	have	AUX
ejpam-4387	325	17	that	that	DET
ejpam-4387	325	18	p1(a	p1(a	NOUN
ejpam-4387	325	19	)	)	PUNCT
ejpam-4387	325	20	is	be	AUX
ejpam-4387	325	21	a	a	DET
ejpam-4387	325	22	paracompact	paracompact	ADJ
ejpam-4387	325	23	subset	subset	NOUN
ejpam-4387	325	24	of	of	ADP
ejpam-4387	325	25	x	x	X
ejpam-4387	325	26	and	and	CCONJ
ejpam-4387	325	27	p2(a	p2(a	PROPN
ejpam-4387	325	28	)	)	PUNCT
ejpam-4387	325	29	is	be	AUX
ejpam-4387	325	30	a	a	DET
ejpam-4387	325	31	paracompact	paracompact	ADJ
ejpam-4387	325	32	subset	subset	NOUN
ejpam-4387	325	33	of	of	ADP
ejpam-4387	325	34	z.	z.	PROPN
ejpam-4387	325	35	thus	thus	ADV
ejpam-4387	325	36	f|p1(a	f|p1(a	PROPN
ejpam-4387	325	37	)	)	PUNCT
ejpam-4387	325	38	:	:	PUNCT
ejpam-4387	326	1	p1(a	p1(a	NOUN
ejpam-4387	326	2	)	)	PUNCT
ejpam-4387	326	3	−→	−→	NOUN
ejpam-4387	326	4	f(p1(a	f(p1(a	PROPN
ejpam-4387	326	5	)	)	PUNCT
ejpam-4387	326	6	)	)	PUNCT
ejpam-4387	326	7	is	be	AUX
ejpam-4387	326	8	a	a	DET
ejpam-4387	326	9	homeomorphism	homeomorphism	NOUN
ejpam-4387	326	10	and	and	CCONJ
ejpam-4387	326	11	g|p2(a	g|p2(a	NOUN
ejpam-4387	326	12	)	)	PUNCT
ejpam-4387	326	13	:	:	PUNCT
ejpam-4387	326	14	p2(a	p2(a	X
ejpam-4387	326	15	)	)	PUNCT
ejpam-4387	326	16	−→	−→	ADJ
ejpam-4387	326	17	g(p2(a	g(p2(a	NOUN
ejpam-4387	326	18	)	)	PUNCT
ejpam-4387	326	19	)	)	PUNCT
ejpam-4387	327	1	is	be	AUX
ejpam-4387	327	2	a	a	DET
ejpam-4387	327	3	homeomorphism	homeomorphism	NOUN
ejpam-4387	327	4	.	.	PUNCT
ejpam-4387	328	1	since	since	SCONJ
ejpam-4387	328	2	a	a	DET
ejpam-4387	328	3	product	product	NOUN
ejpam-4387	328	4	of	of	ADP
ejpam-4387	328	5	two	two	NUM
ejpam-4387	328	6	homeomorphisms	homeomorphism	NOUN
ejpam-4387	328	7	is	be	AUX
ejpam-4387	328	8	a	a	DET
ejpam-4387	328	9	homeomorphism	homeomorphism	NOUN
ejpam-4387	328	10	[	[	X
ejpam-4387	328	11	7	7	NUM
ejpam-4387	328	12	]	]	PUNCT
ejpam-4387	328	13	,	,	PUNCT
ejpam-4387	328	14	we	we	PRON
ejpam-4387	328	15	get	get	VERB
ejpam-4387	328	16	that	that	DET
ejpam-4387	328	17	hp1(a)×p2(a	hp1(a)×p2(a	NOUN
ejpam-4387	328	18	)	)	PUNCT
ejpam-4387	329	1	=	=	SYM
ejpam-4387	329	2	(	(	PUNCT
ejpam-4387	329	3	f|p1(a	f|p1(a	NOUN
ejpam-4387	329	4	)	)	PUNCT
ejpam-4387	329	5	)	)	PUNCT
ejpam-4387	330	1	×	×	NOUN
ejpam-4387	330	2	(	(	PUNCT
ejpam-4387	330	3	g|p2(a	g|p2(a	NOUN
ejpam-4387	330	4	)	)	PUNCT
ejpam-4387	330	5	)	)	PUNCT
ejpam-4387	330	6	:	:	PUNCT
ejpam-4387	331	1	p1(a)×	p1(a)×	PROPN
ejpam-4387	331	2	p2(a	p2(a	NOUN
ejpam-4387	331	3	)	)	PUNCT
ejpam-4387	331	4	−→	−→	NOUN
ejpam-4387	331	5	(	(	PUNCT
ejpam-4387	331	6	f(p1(a)))×	f(p1(a)))×	PROPN
ejpam-4387	331	7	(	(	PUNCT
ejpam-4387	331	8	g(p2(a	g(p2(a	NOUN
ejpam-4387	331	9	)	)	PUNCT
ejpam-4387	331	10	)	)	PUNCT
ejpam-4387	331	11	)	)	PUNCT
ejpam-4387	332	1	=	=	PUNCT
ejpam-4387	332	2	h(p1(a	h(p1(a	PROPN
ejpam-4387	332	3	)	)	PUNCT
ejpam-4387	332	4	×	×	NOUN
ejpam-4387	332	5	p2(a	p2(a	NOUN
ejpam-4387	332	6	)	)	PUNCT
ejpam-4387	332	7	)	)	PUNCT
ejpam-4387	332	8	is	be	AUX
ejpam-4387	332	9	a	a	DET
ejpam-4387	332	10	homeomorphism	homeomorphism	NOUN
ejpam-4387	332	11	.	.	PUNCT
ejpam-4387	333	1	since	since	SCONJ
ejpam-4387	333	2	a	a	DET
ejpam-4387	333	3	⊆	⊆	NUM
ejpam-4387	333	4	p1(a	p1(a	NOUN
ejpam-4387	333	5	)	)	PUNCT
ejpam-4387	333	6	×	×	NOUN
ejpam-4387	333	7	p2(a	p2(a	PROPN
ejpam-4387	333	8	)	)	PUNCT
ejpam-4387	333	9	and	and	CCONJ
ejpam-4387	333	10	a	a	DET
ejpam-4387	333	11	restriction	restriction	NOUN
ejpam-4387	333	12	of	of	ADP
ejpam-4387	333	13	a	a	DET
ejpam-4387	333	14	homeomorphism	homeomorphism	NOUN
ejpam-4387	333	15	is	be	AUX
ejpam-4387	333	16	a	a	DET
ejpam-4387	333	17	homeomorphism	homeomorphism	NOUN
ejpam-4387	333	18	,	,	PUNCT
ejpam-4387	333	19	we	we	PRON
ejpam-4387	333	20	conclude	conclude	VERB
ejpam-4387	333	21	that	that	SCONJ
ejpam-4387	333	22	ha	ha	INTJ
ejpam-4387	333	23	:	:	PUNCT
ejpam-4387	333	24	a	a	DET
ejpam-4387	333	25	−→	−→	NOUN
ejpam-4387	333	26	h(a	h(a	PROPN
ejpam-4387	333	27	)	)	PUNCT
ejpam-4387	333	28	is	be	AUX
ejpam-4387	333	29	a	a	DET
ejpam-4387	333	30	homeomorphism	homeomorphism	NOUN
ejpam-4387	333	31	.	.	PUNCT
ejpam-4387	334	1	therefore	therefore	ADV
ejpam-4387	334	2	,	,	PUNCT
ejpam-4387	334	3	x	x	X
ejpam-4387	334	4	×	×	PROPN
ejpam-4387	334	5	z	z	NOUN
ejpam-4387	334	6	is	be	AUX
ejpam-4387	334	7	p	p	NOUN
ejpam-4387	334	8	-normal	-normal	NOUN
ejpam-4387	334	9	.	.	PUNCT
ejpam-4387	335	1	the	the	DET
ejpam-4387	335	2	following	follow	VERB
ejpam-4387	335	3	problems	problem	NOUN
ejpam-4387	335	4	are	be	AUX
ejpam-4387	335	5	still	still	ADV
ejpam-4387	335	6	open	open	ADJ
ejpam-4387	335	7	:	:	PUNCT
ejpam-4387	335	8	1	1	X
ejpam-4387	335	9	.	.	X
ejpam-4387	335	10	is	be	AUX
ejpam-4387	335	11	p	p	NOUN
ejpam-4387	335	12	-normality	-normality	ADJ
ejpam-4387	335	13	hereditary	hereditary	ADJ
ejpam-4387	335	14	with	with	ADP
ejpam-4387	335	15	respect	respect	NOUN
ejpam-4387	335	16	to	to	ADP
ejpam-4387	335	17	closed	closed	ADJ
ejpam-4387	335	18	sets	set	NOUN
ejpam-4387	335	19	?	?	PUNCT
ejpam-4387	336	1	2	2	X
ejpam-4387	336	2	.	.	X
ejpam-4387	336	3	is	be	AUX
ejpam-4387	336	4	there	there	PRON
ejpam-4387	336	5	a	a	DET
ejpam-4387	336	6	tychonoff	tychonoff	NOUN
ejpam-4387	336	7	p	p	NOUN
ejpam-4387	336	8	-normal	-normal	ADJ
ejpam-4387	336	9	space	space	NOUN
ejpam-4387	336	10	which	which	PRON
ejpam-4387	336	11	is	be	AUX
ejpam-4387	336	12	not	not	PART
ejpam-4387	336	13	normal	normal	ADJ
ejpam-4387	336	14	?	?	PUNCT
ejpam-4387	337	1	references	reference	NOUN
ejpam-4387	337	2	[	[	X
ejpam-4387	337	3	1	1	NUM
ejpam-4387	337	4	]	]	PUNCT
ejpam-4387	337	5	l	l	NOUN
ejpam-4387	337	6	kalantan	kalantan	PROPN
ejpam-4387	337	7	a	a	DET
ejpam-4387	337	8	alawadi	alawadi	NOUN
ejpam-4387	337	9	and	and	CCONJ
ejpam-4387	337	10	m	m	NOUN
ejpam-4387	337	11	saeed	saeed	PROPN
ejpam-4387	337	12	.	.	PUNCT
ejpam-4387	338	1	on	on	ADP
ejpam-4387	338	2	the	the	DET
ejpam-4387	338	3	discrete	discrete	ADJ
ejpam-4387	338	4	extension	extension	NOUN
ejpam-4387	338	5	spaces	space	NOUN
ejpam-4387	338	6	.	.	PUNCT
ejpam-4387	339	1	journal	journal	PROPN
ejpam-4387	339	2	of	of	ADP
ejpam-4387	339	3	mathematical	mathematical	ADJ
ejpam-4387	339	4	analysis	analysis	NOUN
ejpam-4387	339	5	,	,	PUNCT
ejpam-4387	339	6	9(2):150–157	9(2):150–157	NOUN
ejpam-4387	339	7	.	.	PROPN
ejpam-4387	339	8	,	,	PUNCT
ejpam-4387	339	9	2018	2018	NUM
ejpam-4387	339	10	.	.	PUNCT
ejpam-4387	340	1	references	reference	NOUN
ejpam-4387	340	2	783	783	NUM
ejpam-4387	341	1	[	[	X
ejpam-4387	341	2	2	2	NUM
ejpam-4387	341	3	]	]	X
ejpam-4387	341	4	p	p	X
ejpam-4387	341	5	s	s	NOUN
ejpam-4387	341	6	alexandroff	alexandroff	NOUN
ejpam-4387	342	1	and	and	CCONJ
ejpam-4387	342	2	p	p	X
ejpam-4387	342	3	s	s	VERB
ejpam-4387	342	4	urysohn	urysohn	NOUN
ejpam-4387	342	5	.	.	PUNCT
ejpam-4387	343	1	mémoire	mémoire	PROPN
ejpam-4387	343	2	sur	sur	PROPN
ejpam-4387	343	3	les	les	X
ejpam-4387	343	4	espaces	espace	NOUN
ejpam-4387	343	5	topologiques	topologique	NOUN
ejpam-4387	343	6	compacts	compact	NOUN
ejpam-4387	343	7	.	.	PUNCT
ejpam-4387	344	1	verh	verh	ADJ
ejpam-4387	344	2	.	.	PUNCT
ejpam-4387	345	1	konink	konink	PROPN
ejpam-4387	345	2	.	.	PUNCT
ejpam-4387	346	1	acad	acad	PROPN
ejpam-4387	346	2	.	.	PUNCT
ejpam-4387	347	1	wetensch	wetensch	PROPN
ejpam-4387	347	2	.	.	PUNCT
ejpam-4387	348	1	amsterdam	amsterdam	PROPN
ejpam-4387	348	2	,	,	PUNCT
ejpam-4387	348	3	14:1–96	14:1–96	PROPN
ejpam-4387	348	4	.	.	PROPN
ejpam-4387	348	5	,	,	PUNCT
ejpam-4387	348	6	1929	1929	NUM
ejpam-4387	348	7	.	.	PUNCT
ejpam-4387	349	1	[	[	X
ejpam-4387	349	2	3	3	NUM
ejpam-4387	349	3	]	]	X
ejpam-4387	349	4	s	s	PART
ejpam-4387	349	5	alzahrani	alzahrani	NOUN
ejpam-4387	349	6	and	and	CCONJ
ejpam-4387	349	7	l	l	PROPN
ejpam-4387	349	8	kalantan	kalantan	PROPN
ejpam-4387	349	9	.	.	PUNCT
ejpam-4387	350	1	epinormality	epinormality	NOUN
ejpam-4387	350	2	.	.	PUNCT
ejpam-4387	351	1	journal	journal	PROPN
ejpam-4387	351	2	of	of	ADP
ejpam-4387	351	3	nonlinear	nonlinear	PROPN
ejpam-4387	351	4	sciences	sciences	PROPN
ejpam-4387	351	5	&	&	CCONJ
ejpam-4387	351	6	applications	application	NOUN
ejpam-4387	351	7	,	,	PUNCT
ejpam-4387	351	8	9(9):5398–5402	9(9):5398–5402	PROPN
ejpam-4387	351	9	.	.	PROPN
ejpam-4387	351	10	,	,	PUNCT
ejpam-4387	351	11	2016	2016	NUM
ejpam-4387	351	12	.	.	PUNCT
ejpam-4387	352	1	[	[	X
ejpam-4387	352	2	4	4	NUM
ejpam-4387	352	3	]	]	SYM
ejpam-4387	352	4	s	s	PART
ejpam-4387	352	5	alzahrani	alzahrani	NOUN
ejpam-4387	352	6	and	and	CCONJ
ejpam-4387	352	7	l	l	PROPN
ejpam-4387	352	8	kalantan	kalantan	PROPN
ejpam-4387	352	9	.	.	PUNCT
ejpam-4387	353	1	c	c	X
ejpam-4387	353	2	-	-	PUNCT
ejpam-4387	353	3	normal	normal	ADJ
ejpam-4387	353	4	topological	topological	ADJ
ejpam-4387	353	5	property	property	NOUN
ejpam-4387	353	6	.	.	PUNCT
ejpam-4387	354	1	filomat	filomat	NOUN
ejpam-4387	354	2	,	,	PUNCT
ejpam-4387	354	3	31(2):407–411	31(2):407–411	PROPN
ejpam-4387	354	4	.	.	PROPN
ejpam-4387	354	5	,	,	PUNCT
ejpam-4387	354	6	2017	2017	NUM
ejpam-4387	354	7	.	.	PUNCT
ejpam-4387	355	1	[	[	X
ejpam-4387	355	2	5	5	NUM
ejpam-4387	355	3	]	]	X
ejpam-4387	355	4	s	s	PART
ejpam-4387	355	5	al	al	PROPN
ejpam-4387	355	6	-	-	PUNCT
ejpam-4387	355	7	qarhi	qarhi	PROPN
ejpam-4387	355	8	d	d	PROPN
ejpam-4387	355	9	abuzaid	abuzaid	ADJ
ejpam-4387	355	10	and	and	CCONJ
ejpam-4387	355	11	l	l	NOUN
ejpam-4387	355	12	kalantani	kalantani	NOUN
ejpam-4387	355	13	.	.	PUNCT
ejpam-4387	356	1	on	on	ADP
ejpam-4387	356	2	the	the	DET
ejpam-4387	356	3	closed	closed	ADJ
ejpam-4387	356	4	extension	extension	NOUN
ejpam-4387	356	5	spaces	space	NOUN
ejpam-4387	356	6	.	.	PUNCT
ejpam-4387	357	1	to	to	PART
ejpam-4387	357	2	appear	appear	VERB
ejpam-4387	357	3	.	.	PUNCT
ejpam-4387	358	1	[	[	X
ejpam-4387	358	2	6	6	NUM
ejpam-4387	358	3	]	]	X
ejpam-4387	358	4	r	r	NOUN
ejpam-4387	358	5	engelking	engelking	NOUN
ejpam-4387	358	6	.	.	PUNCT
ejpam-4387	359	1	on	on	ADP
ejpam-4387	359	2	the	the	DET
ejpam-4387	359	3	double	double	ADJ
ejpam-4387	359	4	circumference	circumference	NOUN
ejpam-4387	359	5	of	of	ADP
ejpam-4387	359	6	alexandroff	alexandroff	NOUN
ejpam-4387	359	7	.	.	PUNCT
ejpam-4387	360	1	bull	bull	NOUN
ejpam-4387	360	2	.	.	PUNCT
ejpam-4387	361	1	acad	acad	PROPN
ejpam-4387	361	2	.	.	PUNCT
ejpam-4387	362	1	pol	pol	PROPN
ejpam-4387	362	2	.	.	PUNCT
ejpam-4387	363	1	sci	sci	PROPN
ejpam-4387	363	2	.	.	PUNCT
ejpam-4387	363	3	ser	ser	PROPN
ejpam-4387	363	4	.	.	PUNCT
ejpam-4387	364	1	astron	astron	PROPN
ejpam-4387	364	2	.	.	PUNCT
ejpam-4387	364	3	math	math	NOUN
ejpam-4387	364	4	.	.	PUNCT
ejpam-4387	365	1	phys	phy	NOUN
ejpam-4387	365	2	.	.	PUNCT
ejpam-4387	365	3	,	,	PUNCT
ejpam-4387	365	4	16(8):629–634	16(8):629–634	PROPN
ejpam-4387	365	5	.	.	PROPN
ejpam-4387	365	6	,	,	PUNCT
ejpam-4387	365	7	1968	1968	NUM
ejpam-4387	365	8	.	.	PUNCT
ejpam-4387	366	1	[	[	X
ejpam-4387	366	2	7	7	NUM
ejpam-4387	366	3	]	]	X
ejpam-4387	366	4	r	r	NOUN
ejpam-4387	366	5	engelking	engelking	NOUN
ejpam-4387	366	6	.	.	PUNCT
ejpam-4387	367	1	general	general	ADJ
ejpam-4387	367	2	topology	topology	PROPN
ejpam-4387	367	3	.	.	PUNCT
ejpam-4387	368	1	pwn	pwn	PROPN
ejpam-4387	368	2	,	,	PUNCT
ejpam-4387	368	3	warszawa	warszawa	PROPN
ejpam-4387	368	4	,	,	PUNCT
ejpam-4387	368	5	1977	1977	NUM
ejpam-4387	368	6	.	.	PUNCT
ejpam-4387	369	1	[	[	X
ejpam-4387	369	2	8	8	NUM
ejpam-4387	369	3	]	]	X
ejpam-4387	369	4	k	k	PROPN
ejpam-4387	369	5	kunen	kunen	PROPN
ejpam-4387	370	1	i	i	PRON
ejpam-4387	370	2	juhász	juhász	ADV
ejpam-4387	370	3	and	and	CCONJ
ejpam-4387	370	4	m	m	PROPN
ejpam-4387	370	5	e	e	NOUN
ejpam-4387	370	6	rudin	rudin	PROPN
ejpam-4387	370	7	.	.	PUNCT
ejpam-4387	371	1	two	two	NUM
ejpam-4387	371	2	more	more	ADV
ejpam-4387	371	3	hereditarily	hereditarily	ADV
ejpam-4387	371	4	separable	separable	ADJ
ejpam-4387	371	5	non	non	PROPN
ejpam-4387	371	6	-	-	NOUN
ejpam-4387	371	7	lindelöf	lindelöf	ADJ
ejpam-4387	371	8	spaces	space	NOUN
ejpam-4387	371	9	.	.	PUNCT
ejpam-4387	372	1	canadian	canadian	ADJ
ejpam-4387	372	2	journal	journal	PROPN
ejpam-4387	372	3	of	of	ADP
ejpam-4387	372	4	mathematics	mathematic	NOUN
ejpam-4387	372	5	,	,	PUNCT
ejpam-4387	372	6	28(5):998–1005	28(5):998–1005	NUM
ejpam-4387	372	7	.	.	NOUN
ejpam-4387	372	8	,	,	PUNCT
ejpam-4387	372	9	1976	1976	NUM
ejpam-4387	372	10	.	.	PUNCT
ejpam-4387	373	1	[	[	X
ejpam-4387	373	2	9	9	NUM
ejpam-4387	373	3	]	]	X
ejpam-4387	373	4	m	m	VERB
ejpam-4387	373	5	ismail	ismail	NOUN
ejpam-4387	373	6	and	and	CCONJ
ejpam-4387	373	7	p	p	NOUN
ejpam-4387	373	8	nyikos	nyikos	ADV
ejpam-4387	373	9	.	.	PUNCT
ejpam-4387	374	1	on	on	ADP
ejpam-4387	374	2	spaces	space	NOUN
ejpam-4387	374	3	in	in	ADP
ejpam-4387	374	4	which	which	PRON
ejpam-4387	374	5	countably	countably	ADV
ejpam-4387	374	6	compact	compact	ADJ
ejpam-4387	374	7	sets	set	NOUN
ejpam-4387	374	8	are	be	AUX
ejpam-4387	374	9	closed	closed	ADJ
ejpam-4387	374	10	and	and	CCONJ
ejpam-4387	374	11	hereditary	hereditary	ADJ
ejpam-4387	374	12	properties	property	NOUN
ejpam-4387	374	13	.	.	PUNCT
ejpam-4387	375	1	topology	topology	NOUN
ejpam-4387	375	2	and	and	CCONJ
ejpam-4387	375	3	its	its	PRON
ejpam-4387	375	4	applications	application	NOUN
ejpam-4387	375	5	,	,	PUNCT
ejpam-4387	375	6	11(3):281–292	11(3):281–292	NUM
ejpam-4387	375	7	.	.	PUNCT
ejpam-4387	375	8	,	,	PUNCT
ejpam-4387	375	9	1980	1980	NUM
ejpam-4387	375	10	.	.	PUNCT
ejpam-4387	376	1	[	[	X
ejpam-4387	376	2	10	10	NUM
ejpam-4387	376	3	]	]	X
ejpam-4387	376	4	l	l	NOUN
ejpam-4387	376	5	kalantan	kalantan	PROPN
ejpam-4387	376	6	and	and	CCONJ
ejpam-4387	376	7	m	m	PROPN
ejpam-4387	376	8	mansouri	mansouri	ADJ
ejpam-4387	376	9	.	.	PUNCT
ejpam-4387	377	1	p	p	X
ejpam-4387	377	2	-normality	-normality	PROPN
ejpam-4387	377	3	.	.	PUNCT
ejpam-4387	378	1	journal	journal	PROPN
ejpam-4387	378	2	of	of	ADP
ejpam-4387	378	3	mathematical	mathematical	ADJ
ejpam-4387	378	4	analysis	analysis	NOUN
ejpam-4387	378	5	,	,	PUNCT
ejpam-4387	378	6	12(6):1–8	12(6):1–8	NUM
ejpam-4387	378	7	.	.	NOUN
ejpam-4387	378	8	,	,	PUNCT
ejpam-4387	378	9	2021	2021	NUM
ejpam-4387	378	10	.	.	PUNCT
ejpam-4387	379	1	[	[	X
ejpam-4387	379	2	11	11	NUM
ejpam-4387	379	3	]	]	PUNCT
ejpam-4387	379	4	l	l	NOUN
ejpam-4387	379	5	kalantan	kalantan	PROPN
ejpam-4387	379	6	and	and	CCONJ
ejpam-4387	379	7	m	m	PROPN
ejpam-4387	379	8	saeed	saeed	PROPN
ejpam-4387	379	9	.	.	PUNCT
ejpam-4387	380	1	l	l	NOUN
ejpam-4387	380	2	-	-	NOUN
ejpam-4387	380	3	normality	normality	NOUN
ejpam-4387	380	4	.	.	PUNCT
ejpam-4387	381	1	topology	topology	NOUN
ejpam-4387	381	2	proceedings	proceeding	NOUN
ejpam-4387	381	3	,	,	PUNCT
ejpam-4387	381	4	50:141–149	50:141–149	NUM
ejpam-4387	381	5	.	.	NOUN
ejpam-4387	381	6	,	,	PUNCT
ejpam-4387	381	7	2017	2017	NUM
ejpam-4387	381	8	.	.	PUNCT
ejpam-4387	382	1	[	[	X
ejpam-4387	382	2	12	12	NUM
ejpam-4387	382	3	]	]	X
ejpam-4387	382	4	m	m	VERB
ejpam-4387	382	5	e	e	NOUN
ejpam-4387	382	6	rudin	rudin	NOUN
ejpam-4387	382	7	.	.	PUNCT
ejpam-4387	383	1	a	a	DET
ejpam-4387	383	2	separable	separable	ADJ
ejpam-4387	383	3	dowker	dowker	NOUN
ejpam-4387	383	4	space	space	NOUN
ejpam-4387	383	5	.	.	PUNCT
ejpam-4387	384	1	symposia	symposia	PROPN
ejpam-4387	384	2	mathematica	mathematica	PROPN
ejpam-4387	384	3	,	,	PUNCT
ejpam-4387	384	4	16:125–132	16:125–132	PROPN
ejpam-4387	384	5	.	.	PUNCT
ejpam-4387	384	6	,	,	PUNCT
ejpam-4387	384	7	1973	1973	NUM
ejpam-4387	384	8	.	.	PUNCT
ejpam-4387	385	1	[	[	X
ejpam-4387	385	2	13	13	NUM
ejpam-4387	385	3	]	]	SYM
ejpam-4387	385	4	l	l	NOUN
ejpam-4387	385	5	steen	steen	PROPN
ejpam-4387	385	6	and	and	CCONJ
ejpam-4387	385	7	j	j	PROPN
ejpam-4387	385	8	a	a	DET
ejpam-4387	385	9	seebach	seebach	NOUN
ejpam-4387	385	10	.	.	PUNCT
ejpam-4387	386	1	counterexamples	counterexample	NOUN
ejpam-4387	386	2	in	in	ADP
ejpam-4387	386	3	topology	topology	NOUN
ejpam-4387	386	4	.	.	PUNCT
ejpam-4387	387	1	dover	dover	PROPN
ejpam-4387	387	2	publications	publications	PROPN
ejpam-4387	387	3	inc	inc	PROPN
ejpam-4387	387	4	,	,	PUNCT
ejpam-4387	387	5	usa	usa	PROPN
ejpam-4387	387	6	,	,	PUNCT
ejpam-4387	387	7	1995	1995	NUM
ejpam-4387	387	8	.	.	PUNCT
ejpam-4387	388	1	[	[	X
ejpam-4387	388	2	14	14	NUM
ejpam-4387	388	3	]	]	X
ejpam-4387	388	4	w	w	PROPN
ejpam-4387	388	5	weiss	weiss	PROPN
ejpam-4387	388	6	.	.	PUNCT
ejpam-4387	389	1	small	small	ADJ
ejpam-4387	389	2	dowker	dowker	NOUN
ejpam-4387	389	3	spaces	space	NOUN
ejpam-4387	389	4	.	.	PUNCT
ejpam-4387	390	1	pacific	pacific	PROPN
ejpam-4387	390	2	journal	journal	PROPN
ejpam-4387	390	3	of	of	ADP
ejpam-4387	390	4	mathematics	mathematic	NOUN
ejpam-4387	390	5	,	,	PUNCT
ejpam-4387	390	6	24(2):485–492	24(2):485–492	PROPN
ejpam-4387	390	7	.	.	PROPN
ejpam-4387	390	8	,	,	PUNCT
ejpam-4387	390	9	1981	1981	NUM
ejpam-4387	390	10	.	.	PUNCT