id	sid	tid	token	lemma	pos
ejpam-3941	1	1	european	european	PROPN
ejpam-3941	1	2	journal	journal	PROPN
ejpam-3941	1	3	of	of	ADP
ejpam-3941	1	4	pure	pure	ADJ
ejpam-3941	1	5	and	and	CCONJ
ejpam-3941	1	6	applied	apply	VERB
ejpam-3941	1	7	mathematics	mathematic	NOUN
ejpam-3941	1	8	vol	vol	NOUN
ejpam-3941	1	9	.	.	PUNCT
ejpam-3941	2	1	14	14	NUM
ejpam-3941	2	2	,	,	PUNCT
ejpam-3941	2	3	no	no	INTJ
ejpam-3941	2	4	.	.	NOUN
ejpam-3941	2	5	2	2	NUM
ejpam-3941	2	6	,	,	PUNCT
ejpam-3941	2	7	2021	2021	NUM
ejpam-3941	2	8	,	,	PUNCT
ejpam-3941	2	9	351	351	NUM
ejpam-3941	2	10	-	-	SYM
ejpam-3941	2	11	357	357	NUM
ejpam-3941	2	12	issn	issn	PROPN
ejpam-3941	2	13	1307	1307	NUM
ejpam-3941	2	14	-	-	SYM
ejpam-3941	2	15	5543	5543	NUM
ejpam-3941	2	16	–	–	PUNCT
ejpam-3941	2	17	ejpam.com	ejpam.com	X
ejpam-3941	2	18	published	publish	VERB
ejpam-3941	2	19	by	by	ADP
ejpam-3941	2	20	new	new	PROPN
ejpam-3941	2	21	york	york	PROPN
ejpam-3941	2	22	business	business	PROPN
ejpam-3941	2	23	global	global	ADJ
ejpam-3941	2	24	results	result	NOUN
ejpam-3941	2	25	on	on	ADP
ejpam-3941	2	26	c2	c2	PROPN
ejpam-3941	2	27	-	-	PUNCT
ejpam-3941	2	28	paracompactness	paracompactness	PROPN
ejpam-3941	2	29	hala	hala	PROPN
ejpam-3941	2	30	alzumi1,2	alzumi1,2	PROPN
ejpam-3941	2	31	,	,	PUNCT
ejpam-3941	2	32	lutfi	lutfi	PROPN
ejpam-3941	2	33	kalantan1	kalantan1	PROPN
ejpam-3941	2	34	,	,	PUNCT
ejpam-3941	2	35	maha	maha	PROPN
ejpam-3941	2	36	mohammed	mohammed	PROPN
ejpam-3941	2	37	saeed1,∗	saeed1,∗	NOUN
ejpam-3941	2	38	1	1	NUM
ejpam-3941	2	39	department	department	NOUN
ejpam-3941	2	40	of	of	ADP
ejpam-3941	2	41	mathematics	mathematic	NOUN
ejpam-3941	2	42	,	,	PUNCT
ejpam-3941	2	43	king	king	PROPN
ejpam-3941	2	44	abdulaziz	abdulaziz	PROPN
ejpam-3941	2	45	university	university	PROPN
ejpam-3941	2	46	,	,	PUNCT
ejpam-3941	2	47	p.o.box	p.o.box	PROPN
ejpam-3941	2	48	80203	80203	NUM
ejpam-3941	2	49	,	,	PUNCT
ejpam-3941	2	50	jeddah	jeddah	PROPN
ejpam-3941	2	51	21589	21589	NUM
ejpam-3941	2	52	,	,	PUNCT
ejpam-3941	2	53	saudi	saudi	PROPN
ejpam-3941	2	54	arabia	arabia	PROPN
ejpam-3941	2	55	2	2	NUM
ejpam-3941	2	56	department	department	NOUN
ejpam-3941	2	57	of	of	ADP
ejpam-3941	2	58	mathematics	mathematics	PROPN
ejpam-3941	2	59	,	,	PUNCT
ejpam-3941	2	60	jeddah	jeddah	PROPN
ejpam-3941	2	61	university	university	PROPN
ejpam-3941	2	62	,	,	PUNCT
ejpam-3941	2	63	jeddah	jeddah	PROPN
ejpam-3941	2	64	,	,	PUNCT
ejpam-3941	3	1	saudi	saudi	PROPN
ejpam-3941	3	2	arabia	arabia	PROPN
ejpam-3941	3	3	abstract	abstract	NOUN
ejpam-3941	3	4	.	.	PUNCT
ejpam-3941	4	1	a	a	DET
ejpam-3941	4	2	c	c	NOUN
ejpam-3941	4	3	-	-	PUNCT
ejpam-3941	4	4	paracompact	paracompact	NOUN
ejpam-3941	4	5	is	be	AUX
ejpam-3941	4	6	a	a	DET
ejpam-3941	4	7	topological	topological	ADJ
ejpam-3941	4	8	space	space	NOUN
ejpam-3941	4	9	x	x	PUNCT
ejpam-3941	4	10	associated	associate	VERB
ejpam-3941	4	11	with	with	ADP
ejpam-3941	4	12	a	a	DET
ejpam-3941	4	13	paracompact	paracompact	ADJ
ejpam-3941	4	14	space	space	NOUN
ejpam-3941	4	15	y	y	PROPN
ejpam-3941	4	16	and	and	CCONJ
ejpam-3941	4	17	a	a	DET
ejpam-3941	4	18	bijective	bijective	ADJ
ejpam-3941	4	19	function	function	NOUN
ejpam-3941	4	20	f	f	NOUN
ejpam-3941	5	1	:	:	PUNCT
ejpam-3941	5	2	x	x	PUNCT
ejpam-3941	5	3	−→	−→	NOUN
ejpam-3941	5	4	y	y	NOUN
ejpam-3941	5	5	satisfying	satisfy	VERB
ejpam-3941	5	6	that	that	SCONJ
ejpam-3941	5	7	f	f	PROPN
ejpam-3941	5	8	�	�	PROPN
ejpam-3941	5	9	a	a	NOUN
ejpam-3941	5	10	:	:	PUNCT
ejpam-3941	5	11	a	a	DET
ejpam-3941	5	12	−→	−→	NOUN
ejpam-3941	5	13	f(a	f(a	NOUN
ejpam-3941	5	14	)	)	PUNCT
ejpam-3941	5	15	is	be	AUX
ejpam-3941	5	16	a	a	DET
ejpam-3941	5	17	homeomorphism	homeomorphism	NOUN
ejpam-3941	5	18	for	for	ADP
ejpam-3941	5	19	each	each	DET
ejpam-3941	5	20	compact	compact	ADJ
ejpam-3941	5	21	subspace	subspace	NOUN
ejpam-3941	5	22	a	a	DET
ejpam-3941	5	23	⊆	⊆	NUM
ejpam-3941	5	24	x.	x.	NOUN
ejpam-3941	5	25	furthermore	furthermore	ADV
ejpam-3941	5	26	,	,	PUNCT
ejpam-3941	5	27	x	x	VERB
ejpam-3941	5	28	is	be	AUX
ejpam-3941	5	29	called	call	VERB
ejpam-3941	5	30	c2	c2	PROPN
ejpam-3941	5	31	-	-	PUNCT
ejpam-3941	5	32	paracompact	paracompact	NOUN
ejpam-3941	5	33	if	if	SCONJ
ejpam-3941	5	34	y	y	PROPN
ejpam-3941	5	35	is	be	AUX
ejpam-3941	5	36	t2	t2	NOUN
ejpam-3941	5	37	paracompact	paracompact	NOUN
ejpam-3941	5	38	.	.	PUNCT
ejpam-3941	6	1	in	in	ADP
ejpam-3941	6	2	this	this	DET
ejpam-3941	6	3	article	article	NOUN
ejpam-3941	6	4	,	,	PUNCT
ejpam-3941	6	5	we	we	PRON
ejpam-3941	6	6	discuss	discuss	VERB
ejpam-3941	6	7	the	the	DET
ejpam-3941	6	8	above	above	ADJ
ejpam-3941	6	9	concepts	concept	NOUN
ejpam-3941	6	10	and	and	CCONJ
ejpam-3941	6	11	answer	answer	VERB
ejpam-3941	6	12	the	the	DET
ejpam-3941	6	13	problem	problem	NOUN
ejpam-3941	6	14	of	of	ADP
ejpam-3941	6	15	arhangel’skĭi	arhangel’skĭi	PROPN
ejpam-3941	6	16	.	.	PUNCT
ejpam-3941	7	1	moreover	moreover	ADV
ejpam-3941	7	2	,	,	PUNCT
ejpam-3941	7	3	we	we	PRON
ejpam-3941	7	4	prove	prove	VERB
ejpam-3941	7	5	that	that	SCONJ
ejpam-3941	7	6	the	the	DET
ejpam-3941	7	7	sigma	sigma	PROPN
ejpam-3941	7	8	product	product	NOUN
ejpam-3941	7	9	σ(0	σ(0	PROPN
ejpam-3941	7	10	)	)	PUNCT
ejpam-3941	7	11	can	can	AUX
ejpam-3941	7	12	not	not	PART
ejpam-3941	7	13	be	be	AUX
ejpam-3941	7	14	condensed	condense	VERB
ejpam-3941	7	15	onto	onto	ADP
ejpam-3941	7	16	a	a	DET
ejpam-3941	7	17	t2	t2	NOUN
ejpam-3941	7	18	paracompact	paracompact	ADJ
ejpam-3941	7	19	space	space	NOUN
ejpam-3941	7	20	.	.	PUNCT
ejpam-3941	8	1	2020	2020	NUM
ejpam-3941	8	2	mathematics	mathematic	NOUN
ejpam-3941	8	3	subject	subject	NOUN
ejpam-3941	8	4	classifications	classification	NOUN
ejpam-3941	8	5	:	:	PUNCT
ejpam-3941	8	6	54c10	54c10	NUM
ejpam-3941	8	7	,	,	PUNCT
ejpam-3941	8	8	54d20	54d20	NUM
ejpam-3941	8	9	key	key	ADJ
ejpam-3941	8	10	words	word	NOUN
ejpam-3941	8	11	and	and	CCONJ
ejpam-3941	8	12	phrases	phrase	NOUN
ejpam-3941	8	13	:	:	PUNCT
ejpam-3941	8	14	normal	normal	ADJ
ejpam-3941	8	15	,	,	PUNCT
ejpam-3941	8	16	paracompact	paracompact	ADJ
ejpam-3941	8	17	,	,	PUNCT
ejpam-3941	8	18	sigma	sigma	ADJ
ejpam-3941	8	19	product	product	NOUN
ejpam-3941	8	20	,	,	PUNCT
ejpam-3941	8	21	c	c	NOUN
ejpam-3941	8	22	-	-	PUNCT
ejpam-3941	8	23	paracompact	paracompact	ADJ
ejpam-3941	8	24	,	,	PUNCT
ejpam-3941	8	25	c2	c2	NOUN
ejpam-3941	8	26	-	-	PUNCT
ejpam-3941	8	27	paracompact	paracompact	NOUN
ejpam-3941	8	28	,	,	PUNCT
ejpam-3941	8	29	c	c	NOUN
ejpam-3941	8	30	-	-	ADJ
ejpam-3941	8	31	normal	normal	ADJ
ejpam-3941	8	32	,	,	PUNCT
ejpam-3941	8	33	sigma	sigma	ADJ
ejpam-3941	8	34	product	product	NOUN
ejpam-3941	8	35	,	,	PUNCT
ejpam-3941	8	36	open	open	ADJ
ejpam-3941	8	37	invariant	invariant	ADJ
ejpam-3941	8	38	,	,	PUNCT
ejpam-3941	8	39	alexandroff	alexandroff	ADJ
ejpam-3941	8	40	duplicate	duplicate	NOUN
ejpam-3941	8	41	1	1	NUM
ejpam-3941	8	42	.	.	PUNCT
ejpam-3941	8	43	introduction	introduction	NOUN
ejpam-3941	8	44	and	and	CCONJ
ejpam-3941	8	45	preliminaries	preliminary	NOUN
ejpam-3941	8	46	in	in	ADP
ejpam-3941	8	47	the	the	DET
ejpam-3941	8	48	present	present	ADJ
ejpam-3941	8	49	work	work	NOUN
ejpam-3941	8	50	,	,	PUNCT
ejpam-3941	8	51	we	we	PRON
ejpam-3941	8	52	give	give	VERB
ejpam-3941	8	53	some	some	DET
ejpam-3941	8	54	new	new	ADJ
ejpam-3941	8	55	results	result	NOUN
ejpam-3941	8	56	about	about	ADP
ejpam-3941	8	57	c	c	NOUN
ejpam-3941	8	58	-	-	PUNCT
ejpam-3941	8	59	paracompactness	paracompactness	NOUN
ejpam-3941	8	60	and	and	CCONJ
ejpam-3941	8	61	c2paracompactness	c2paracompactness	NOUN
ejpam-3941	8	62	[	[	X
ejpam-3941	8	63	8	8	NUM
ejpam-3941	8	64	]	]	PUNCT
ejpam-3941	8	65	and	and	CCONJ
ejpam-3941	8	66	answer	answer	VERB
ejpam-3941	8	67	a	a	DET
ejpam-3941	8	68	problem	problem	NOUN
ejpam-3941	8	69	of	of	ADP
ejpam-3941	8	70	arhangel’skĭi	arhangel’skĭi	PROPN
ejpam-3941	8	71	.	.	PUNCT
ejpam-3941	9	1	also	also	ADV
ejpam-3941	9	2	,	,	PUNCT
ejpam-3941	9	3	we	we	PRON
ejpam-3941	9	4	prove	prove	VERB
ejpam-3941	9	5	that	that	SCONJ
ejpam-3941	9	6	the	the	DET
ejpam-3941	9	7	sigma	sigma	PROPN
ejpam-3941	9	8	product	product	NOUN
ejpam-3941	9	9	σ(0	σ(0	PROPN
ejpam-3941	9	10	)	)	PUNCT
ejpam-3941	9	11	can	can	AUX
ejpam-3941	9	12	not	not	PART
ejpam-3941	9	13	be	be	AUX
ejpam-3941	9	14	condensed	condense	VERB
ejpam-3941	9	15	onto	onto	ADP
ejpam-3941	9	16	a	a	DET
ejpam-3941	9	17	t2	t2	NOUN
ejpam-3941	9	18	paracompact	paracompact	ADJ
ejpam-3941	9	19	space	space	NOUN
ejpam-3941	9	20	.	.	PUNCT
ejpam-3941	10	1	throughout	throughout	ADP
ejpam-3941	10	2	this	this	DET
ejpam-3941	10	3	paper	paper	NOUN
ejpam-3941	10	4	,	,	PUNCT
ejpam-3941	10	5	〈	〈	PROPN
ejpam-3941	10	6	x	x	X
ejpam-3941	10	7	,	,	PUNCT
ejpam-3941	10	8	y	y	PROPN
ejpam-3941	10	9	〉	〉	PROPN
ejpam-3941	10	10	denotes	denote	VERB
ejpam-3941	10	11	an	an	DET
ejpam-3941	10	12	ordered	ordered	ADJ
ejpam-3941	10	13	pair	pair	NOUN
ejpam-3941	10	14	,	,	PUNCT
ejpam-3941	10	15	n	n	PRON
ejpam-3941	10	16	denotes	denote	VERB
ejpam-3941	10	17	the	the	DET
ejpam-3941	10	18	set	set	NOUN
ejpam-3941	10	19	of	of	ADP
ejpam-3941	10	20	positive	positive	ADJ
ejpam-3941	10	21	integers	integer	NOUN
ejpam-3941	10	22	,	,	PUNCT
ejpam-3941	10	23	q	q	PUNCT
ejpam-3941	10	24	denotes	denote	VERB
ejpam-3941	10	25	the	the	DET
ejpam-3941	10	26	rational	rational	ADJ
ejpam-3941	10	27	numbers	number	NOUN
ejpam-3941	10	28	,	,	PUNCT
ejpam-3941	10	29	p	p	NOUN
ejpam-3941	10	30	denotes	denote	VERB
ejpam-3941	10	31	the	the	DET
ejpam-3941	10	32	irrational	irrational	ADJ
ejpam-3941	10	33	numbers	number	NOUN
ejpam-3941	10	34	,	,	PUNCT
ejpam-3941	10	35	and	and	CCONJ
ejpam-3941	10	36	r	r	NOUN
ejpam-3941	10	37	denotes	denote	VERB
ejpam-3941	10	38	the	the	DET
ejpam-3941	10	39	set	set	NOUN
ejpam-3941	10	40	of	of	ADP
ejpam-3941	10	41	real	real	ADJ
ejpam-3941	10	42	numbers	number	NOUN
ejpam-3941	10	43	.	.	PUNCT
ejpam-3941	11	1	t2	t2	NOUN
ejpam-3941	11	2	denotes	denote	VERB
ejpam-3941	11	3	the	the	DET
ejpam-3941	11	4	hausdorff	hausdorff	NOUN
ejpam-3941	11	5	property	property	NOUN
ejpam-3941	11	6	.	.	PUNCT
ejpam-3941	12	1	a	a	DET
ejpam-3941	12	2	t4	t4	PROPN
ejpam-3941	12	3	space	space	NOUN
ejpam-3941	12	4	is	be	AUX
ejpam-3941	12	5	a	a	DET
ejpam-3941	12	6	t1	t1	NOUN
ejpam-3941	12	7	normal	normal	ADJ
ejpam-3941	12	8	space	space	NOUN
ejpam-3941	12	9	and	and	CCONJ
ejpam-3941	12	10	a	a	DET
ejpam-3941	12	11	tychonoff	tychonoff	NOUN
ejpam-3941	12	12	space	space	NOUN
ejpam-3941	12	13	(	(	PUNCT
ejpam-3941	12	14	t3	t3	NOUN
ejpam-3941	12	15	1	1	NUM
ejpam-3941	12	16	2	2	NUM
ejpam-3941	12	17	)	)	PUNCT
ejpam-3941	12	18	is	be	AUX
ejpam-3941	12	19	a	a	DET
ejpam-3941	12	20	t1	t1	NOUN
ejpam-3941	12	21	completely	completely	ADV
ejpam-3941	12	22	regular	regular	ADJ
ejpam-3941	12	23	space	space	NOUN
ejpam-3941	12	24	.	.	PUNCT
ejpam-3941	13	1	we	we	PRON
ejpam-3941	13	2	do	do	AUX
ejpam-3941	13	3	not	not	PART
ejpam-3941	13	4	assume	assume	VERB
ejpam-3941	13	5	hausdorffness	hausdorffness	NOUN
ejpam-3941	13	6	in	in	ADP
ejpam-3941	13	7	the	the	DET
ejpam-3941	13	8	definition	definition	NOUN
ejpam-3941	13	9	of	of	ADP
ejpam-3941	13	10	compactness	compactness	NOUN
ejpam-3941	13	11	,	,	PUNCT
ejpam-3941	13	12	countable	countable	ADJ
ejpam-3941	13	13	compactness	compactness	NOUN
ejpam-3941	13	14	,	,	PUNCT
ejpam-3941	13	15	local	local	ADJ
ejpam-3941	13	16	compactness	compactness	NOUN
ejpam-3941	13	17	,	,	PUNCT
ejpam-3941	13	18	and	and	CCONJ
ejpam-3941	13	19	paracompactness	paracompactness	NOUN
ejpam-3941	13	20	.	.	PUNCT
ejpam-3941	14	1	so	so	ADV
ejpam-3941	14	2	,	,	PUNCT
ejpam-3941	14	3	a	a	DET
ejpam-3941	14	4	space	space	NOUN
ejpam-3941	14	5	is	be	AUX
ejpam-3941	14	6	paracompact	paracompact	ADJ
ejpam-3941	14	7	if	if	SCONJ
ejpam-3941	14	8	any	any	DET
ejpam-3941	14	9	open	open	ADJ
ejpam-3941	14	10	cover	cover	NOUN
ejpam-3941	14	11	has	have	VERB
ejpam-3941	14	12	a	a	DET
ejpam-3941	14	13	locally	locally	ADV
ejpam-3941	14	14	finite	finite	ADJ
ejpam-3941	14	15	open	open	ADJ
ejpam-3941	14	16	refinement	refinement	NOUN
ejpam-3941	14	17	.	.	PUNCT
ejpam-3941	15	1	the	the	DET
ejpam-3941	15	2	regularity	regularity	NOUN
ejpam-3941	15	3	of	of	ADP
ejpam-3941	15	4	lindelöfness	lindelöfness	NOUN
ejpam-3941	15	5	’s	’s	PART
ejpam-3941	15	6	definition	definition	NOUN
ejpam-3941	15	7	is	be	AUX
ejpam-3941	15	8	not	not	PART
ejpam-3941	15	9	assumed	assume	VERB
ejpam-3941	15	10	.	.	PUNCT
ejpam-3941	16	1	the	the	DET
ejpam-3941	16	2	interior	interior	PROPN
ejpam-3941	16	3	and	and	CCONJ
ejpam-3941	16	4	the	the	DET
ejpam-3941	16	5	closure	closure	NOUN
ejpam-3941	16	6	of	of	ADP
ejpam-3941	16	7	a	a	DET
ejpam-3941	16	8	subset	subset	NOUN
ejpam-3941	16	9	a	a	PRON
ejpam-3941	16	10	of	of	ADP
ejpam-3941	16	11	a	a	DET
ejpam-3941	16	12	space	space	NOUN
ejpam-3941	16	13	x	x	NOUN
ejpam-3941	16	14	,	,	PUNCT
ejpam-3941	16	15	are	be	AUX
ejpam-3941	16	16	denoted	denote	VERB
ejpam-3941	16	17	by	by	ADP
ejpam-3941	16	18	inta	inta	PROPN
ejpam-3941	16	19	and	and	CCONJ
ejpam-3941	16	20	a	a	PRON
ejpam-3941	16	21	,	,	PUNCT
ejpam-3941	16	22	respectively	respectively	ADV
ejpam-3941	16	23	.	.	PUNCT
ejpam-3941	17	1	an	an	DET
ejpam-3941	17	2	ordinal	ordinal	ADJ
ejpam-3941	17	3	γ	γ	NOUN
ejpam-3941	17	4	consists	consist	VERB
ejpam-3941	17	5	of	of	ADP
ejpam-3941	17	6	all	all	DET
ejpam-3941	17	7	ordinal	ordinal	ADJ
ejpam-3941	17	8	α	α	NOUN
ejpam-3941	17	9	that	that	SCONJ
ejpam-3941	17	10	satisfying	satisfy	VERB
ejpam-3941	17	11	α	α	PRON
ejpam-3941	17	12	<	<	X
ejpam-3941	17	13	γ	γ	X
ejpam-3941	17	14	.	.	PUNCT
ejpam-3941	18	1	the	the	DET
ejpam-3941	18	2	first	first	ADJ
ejpam-3941	18	3	infinite	infinite	ADJ
ejpam-3941	18	4	ordinal	ordinal	ADJ
ejpam-3941	18	5	is	be	AUX
ejpam-3941	18	6	ω0	ω0	NOUN
ejpam-3941	18	7	,	,	PUNCT
ejpam-3941	18	8	the	the	DET
ejpam-3941	18	9	first	first	ADJ
ejpam-3941	18	10	uncountable	uncountable	ADJ
ejpam-3941	18	11	ordinal	ordinal	NOUN
ejpam-3941	18	12	is	be	AUX
ejpam-3941	18	13	ω1	ω1	PROPN
ejpam-3941	18	14	,	,	PUNCT
ejpam-3941	18	15	and	and	CCONJ
ejpam-3941	18	16	the	the	DET
ejpam-3941	18	17	successor	successor	NOUN
ejpam-3941	18	18	cardinal	cardinal	NOUN
ejpam-3941	18	19	of	of	ADP
ejpam-3941	18	20	ω1	ω1	PROPN
ejpam-3941	18	21	is	be	AUX
ejpam-3941	18	22	ω2	ω2	ADJ
ejpam-3941	18	23	.	.	PUNCT
ejpam-3941	19	1	we	we	PRON
ejpam-3941	19	2	begin	begin	VERB
ejpam-3941	19	3	by	by	ADP
ejpam-3941	19	4	recalling	recall	VERB
ejpam-3941	19	5	the	the	DET
ejpam-3941	19	6	following	follow	VERB
ejpam-3941	19	7	definition	definition	NOUN
ejpam-3941	19	8	,	,	PUNCT
ejpam-3941	19	9	see	see	VERB
ejpam-3941	19	10	[	[	X
ejpam-3941	19	11	8	8	NUM
ejpam-3941	19	12	]	]	PUNCT
ejpam-3941	19	13	.	.	PUNCT
ejpam-3941	20	1	∗corresponding	∗corresponde	VERB
ejpam-3941	20	2	author	author	NOUN
ejpam-3941	20	3	.	.	PUNCT
ejpam-3941	21	1	doi	doi	NOUN
ejpam-3941	21	2	:	:	PUNCT
ejpam-3941	21	3	https://doi.org/10.29020/nybg.ejpam.v14i2.3941	https://doi.org/10.29020/nybg.ejpam.v14i2.3941	ADJ
ejpam-3941	21	4	email	email	NOUN
ejpam-3941	21	5	addresses	address	NOUN
ejpam-3941	21	6	:	:	PUNCT
ejpam-3941	21	7	hala.100@hotmail.com	hala.100@hotmail.com	X
ejpam-3941	21	8	(	(	PUNCT
ejpam-3941	21	9	h.	h.	PROPN
ejpam-3941	21	10	alzumi	alzumi	PROPN
ejpam-3941	21	11	)	)	PUNCT
ejpam-3941	21	12	,	,	PUNCT
ejpam-3941	21	13	lkalantan@kau.edu.sa	lkalantan@kau.edu.sa	PROPN
ejpam-3941	21	14	(	(	PUNCT
ejpam-3941	21	15	l.	l.	PROPN
ejpam-3941	21	16	kalantan	kalantan	PROPN
ejpam-3941	21	17	)	)	PUNCT
ejpam-3941	21	18	,	,	PUNCT
ejpam-3941	21	19	mmmohammed@kau.edu.sa	mmmohammed@kau.edu.sa	PROPN
ejpam-3941	21	20	(	(	PUNCT
ejpam-3941	21	21	m.	m.	NOUN
ejpam-3941	21	22	m.	m.	PROPN
ejpam-3941	21	23	saeed	saeed	PROPN
ejpam-3941	21	24	)	)	PUNCT
ejpam-3941	21	25	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3941	22	1	351	351	NUM
ejpam-3941	22	2	c	c	X
ejpam-3941	22	3	©	©	PROPN
ejpam-3941	22	4	2021	2021	NUM
ejpam-3941	22	5	ejpam	ejpam	VERB
ejpam-3941	22	6	all	all	DET
ejpam-3941	22	7	rights	right	NOUN
ejpam-3941	22	8	reserved	reserve	VERB
ejpam-3941	22	9	.	.	PUNCT
ejpam-3941	23	1	h.	h.	PROPN
ejpam-3941	23	2	alzumi	alzumi	PROPN
ejpam-3941	23	3	,	,	PUNCT
ejpam-3941	23	4	l.	l.	PROPN
ejpam-3941	23	5	kalantan	kalantan	PROPN
ejpam-3941	23	6	and	and	CCONJ
ejpam-3941	23	7	m.	m.	PROPN
ejpam-3941	23	8	mohammed	mohammed	PROPN
ejpam-3941	23	9	saeed	saeed	PROPN
ejpam-3941	23	10	/	/	SYM
ejpam-3941	23	11	eur	eur	PROPN
ejpam-3941	23	12	.	.	PUNCT
ejpam-3941	24	1	j.	j.	PROPN
ejpam-3941	24	2	pure	pure	PROPN
ejpam-3941	24	3	appl	appl	PROPN
ejpam-3941	24	4	.	.	PROPN
ejpam-3941	24	5	math	math	PROPN
ejpam-3941	24	6	,	,	PUNCT
ejpam-3941	24	7	14	14	NUM
ejpam-3941	24	8	(	(	PUNCT
ejpam-3941	24	9	2	2	NUM
ejpam-3941	24	10	)	)	PUNCT
ejpam-3941	24	11	(	(	PUNCT
ejpam-3941	24	12	2021	2021	NUM
ejpam-3941	24	13	)	)	PUNCT
ejpam-3941	24	14	,	,	PUNCT
ejpam-3941	24	15	351	351	NUM
ejpam-3941	24	16	-	-	SYM
ejpam-3941	24	17	357	357	NUM
ejpam-3941	24	18	352	352	NUM
ejpam-3941	24	19	definition	definition	NOUN
ejpam-3941	24	20	1	1	NUM
ejpam-3941	24	21	.	.	PUNCT
ejpam-3941	25	1	(	(	PUNCT
ejpam-3941	25	2	arhangel’skĭi	arhangel’skĭi	PROPN
ejpam-3941	25	3	,	,	PUNCT
ejpam-3941	25	4	2016	2016	NUM
ejpam-3941	25	5	.	.	PUNCT
ejpam-3941	25	6	)	)	PUNCT
ejpam-3941	26	1	a	a	DET
ejpam-3941	26	2	topological	topological	ADJ
ejpam-3941	26	3	space	space	NOUN
ejpam-3941	26	4	x	x	PUNCT
ejpam-3941	26	5	is	be	AUX
ejpam-3941	26	6	called	call	VERB
ejpam-3941	26	7	c	c	NOUN
ejpam-3941	26	8	-	-	PUNCT
ejpam-3941	26	9	paracompact	paracompact	ADJ
ejpam-3941	26	10	if	if	SCONJ
ejpam-3941	26	11	there	there	PRON
ejpam-3941	26	12	exist	exist	VERB
ejpam-3941	26	13	a	a	DET
ejpam-3941	26	14	paracompact	paracompact	ADJ
ejpam-3941	26	15	space	space	NOUN
ejpam-3941	26	16	y	y	PROPN
ejpam-3941	26	17	and	and	CCONJ
ejpam-3941	26	18	a	a	DET
ejpam-3941	26	19	bijective	bijective	ADJ
ejpam-3941	26	20	function	function	NOUN
ejpam-3941	27	1	f	f	NOUN
ejpam-3941	27	2	:	:	PUNCT
ejpam-3941	27	3	x	x	PUNCT
ejpam-3941	27	4	−→	−→	NOUN
ejpam-3941	27	5	y	y	PROPN
ejpam-3941	27	6	such	such	ADJ
ejpam-3941	27	7	that	that	SCONJ
ejpam-3941	27	8	the	the	DET
ejpam-3941	27	9	restriction	restriction	NOUN
ejpam-3941	27	10	f	f	PROPN
ejpam-3941	27	11	�	�	PROPN
ejpam-3941	27	12	a	a	NOUN
ejpam-3941	27	13	:	:	PUNCT
ejpam-3941	27	14	a	a	DET
ejpam-3941	27	15	−→	−→	NOUN
ejpam-3941	27	16	f(a	f(a	NOUN
ejpam-3941	27	17	)	)	PUNCT
ejpam-3941	27	18	is	be	AUX
ejpam-3941	27	19	a	a	DET
ejpam-3941	27	20	homeomorphism	homeomorphism	NOUN
ejpam-3941	27	21	for	for	ADP
ejpam-3941	27	22	each	each	DET
ejpam-3941	27	23	compact	compact	ADJ
ejpam-3941	27	24	subspace	subspace	NOUN
ejpam-3941	27	25	a	a	DET
ejpam-3941	27	26	⊆	⊆	NUM
ejpam-3941	27	27	x.	x.	NOUN
ejpam-3941	27	28	a	a	DET
ejpam-3941	27	29	topological	topological	ADJ
ejpam-3941	27	30	space	space	NOUN
ejpam-3941	27	31	x	x	PUNCT
ejpam-3941	27	32	is	be	AUX
ejpam-3941	27	33	called	call	VERB
ejpam-3941	27	34	c2	c2	PROPN
ejpam-3941	27	35	-	-	PUNCT
ejpam-3941	27	36	paracompact	paracompact	NOUN
ejpam-3941	27	37	if	if	SCONJ
ejpam-3941	27	38	there	there	PRON
ejpam-3941	27	39	exist	exist	VERB
ejpam-3941	27	40	a	a	DET
ejpam-3941	27	41	hausdorff	hausdorff	NOUN
ejpam-3941	27	42	paracompact	paracompact	NOUN
ejpam-3941	27	43	space	space	NOUN
ejpam-3941	27	44	y	y	PROPN
ejpam-3941	27	45	and	and	CCONJ
ejpam-3941	27	46	a	a	DET
ejpam-3941	27	47	bijective	bijective	ADJ
ejpam-3941	27	48	function	function	NOUN
ejpam-3941	28	1	f	f	NOUN
ejpam-3941	28	2	:	:	PUNCT
ejpam-3941	28	3	x	x	PUNCT
ejpam-3941	28	4	−→	−→	NOUN
ejpam-3941	28	5	y	y	PROPN
ejpam-3941	28	6	such	such	ADJ
ejpam-3941	28	7	that	that	SCONJ
ejpam-3941	28	8	the	the	DET
ejpam-3941	28	9	restriction	restriction	NOUN
ejpam-3941	28	10	f	f	PROPN
ejpam-3941	28	11	�	�	PROPN
ejpam-3941	28	12	a	a	NOUN
ejpam-3941	28	13	:	:	PUNCT
ejpam-3941	28	14	a	a	DET
ejpam-3941	28	15	−→	−→	NOUN
ejpam-3941	28	16	f(a	f(a	NOUN
ejpam-3941	28	17	)	)	PUNCT
ejpam-3941	28	18	is	be	AUX
ejpam-3941	28	19	a	a	DET
ejpam-3941	28	20	homeomorphism	homeomorphism	NOUN
ejpam-3941	28	21	for	for	ADP
ejpam-3941	28	22	each	each	DET
ejpam-3941	28	23	compact	compact	ADJ
ejpam-3941	28	24	subspace	subspace	NOUN
ejpam-3941	28	25	a	a	DET
ejpam-3941	28	26	⊆	⊆	NUM
ejpam-3941	28	27	x.	x.	NOUN
ejpam-3941	28	28	in	in	ADP
ejpam-3941	28	29	[	[	X
ejpam-3941	28	30	8	8	NUM
ejpam-3941	28	31	,	,	PUNCT
ejpam-3941	28	32	theorem	theorem	VERB
ejpam-3941	28	33	2.2	2.2	NUM
ejpam-3941	28	34	]	]	PUNCT
ejpam-3941	28	35	,	,	PUNCT
ejpam-3941	28	36	the	the	DET
ejpam-3941	28	37	following	follow	VERB
ejpam-3941	28	38	theorem	theorem	NOUN
ejpam-3941	28	39	was	be	AUX
ejpam-3941	28	40	proved	prove	VERB
ejpam-3941	28	41	.	.	PUNCT
ejpam-3941	29	1	theorem	theorem	NOUN
ejpam-3941	29	2	1	1	NUM
ejpam-3941	29	3	.	.	PUNCT
ejpam-3941	30	1	if	if	SCONJ
ejpam-3941	30	2	x	x	PRON
ejpam-3941	30	3	is	be	AUX
ejpam-3941	30	4	fréchet	fréchet	ADJ
ejpam-3941	30	5	and	and	CCONJ
ejpam-3941	30	6	c	c	NOUN
ejpam-3941	30	7	-	-	PUNCT
ejpam-3941	30	8	paracompact	paracompact	ADJ
ejpam-3941	30	9	(	(	PUNCT
ejpam-3941	30	10	c2	c2	NOUN
ejpam-3941	30	11	-	-	PUNCT
ejpam-3941	30	12	paracompact	paracompact	NOUN
ejpam-3941	30	13	)	)	PUNCT
ejpam-3941	30	14	,	,	PUNCT
ejpam-3941	30	15	then	then	ADV
ejpam-3941	30	16	any	any	DET
ejpam-3941	30	17	function	function	NOUN
ejpam-3941	30	18	witnesses	witness	VERB
ejpam-3941	30	19	its	its	PRON
ejpam-3941	30	20	c	c	NOUN
ejpam-3941	30	21	-	-	PUNCT
ejpam-3941	30	22	paracompactness	paracompactness	NOUN
ejpam-3941	30	23	(	(	PUNCT
ejpam-3941	30	24	c2	c2	PROPN
ejpam-3941	30	25	-	-	PUNCT
ejpam-3941	30	26	paracompactness	paracompactness	NOUN
ejpam-3941	30	27	)	)	PUNCT
ejpam-3941	30	28	is	be	AUX
ejpam-3941	30	29	continuous	continuous	ADJ
ejpam-3941	30	30	.	.	PUNCT
ejpam-3941	31	1	2	2	X
ejpam-3941	31	2	.	.	NOUN
ejpam-3941	31	3	results	result	NOUN
ejpam-3941	31	4	and	and	CCONJ
ejpam-3941	31	5	examples	example	NOUN
ejpam-3941	31	6	since	since	SCONJ
ejpam-3941	31	7	the	the	DET
ejpam-3941	31	8	paracompactness	paracompactness	NOUN
ejpam-3941	31	9	is	be	AUX
ejpam-3941	31	10	not	not	PART
ejpam-3941	31	11	multiplicative	multiplicative	ADJ
ejpam-3941	31	12	,	,	PUNCT
ejpam-3941	31	13	it	it	PRON
ejpam-3941	31	14	seems	seem	VERB
ejpam-3941	31	15	that	that	SCONJ
ejpam-3941	31	16	both	both	DET
ejpam-3941	31	17	c	c	NOUN
ejpam-3941	31	18	-	-	PUNCT
ejpam-3941	31	19	paracompactness	paracompactness	NOUN
ejpam-3941	31	20	and	and	CCONJ
ejpam-3941	31	21	c2	c2	PROPN
ejpam-3941	31	22	-	-	PUNCT
ejpam-3941	31	23	paracompactness	paracompactness	PROPN
ejpam-3941	31	24	are	be	AUX
ejpam-3941	31	25	not	not	PART
ejpam-3941	31	26	multiplicative	multiplicative	ADJ
ejpam-3941	31	27	,	,	PUNCT
ejpam-3941	31	28	but	but	CCONJ
ejpam-3941	31	29	we	we	PRON
ejpam-3941	31	30	still	still	ADV
ejpam-3941	31	31	could	could	AUX
ejpam-3941	31	32	not	not	PART
ejpam-3941	31	33	find	find	VERB
ejpam-3941	31	34	a	a	DET
ejpam-3941	31	35	counterexample	counterexample	NOUN
ejpam-3941	31	36	.	.	PUNCT
ejpam-3941	32	1	we	we	PRON
ejpam-3941	32	2	introduce	introduce	VERB
ejpam-3941	32	3	here	here	ADV
ejpam-3941	32	4	a	a	DET
ejpam-3941	32	5	case	case	NOUN
ejpam-3941	32	6	where	where	SCONJ
ejpam-3941	32	7	c	c	NOUN
ejpam-3941	32	8	-	-	PUNCT
ejpam-3941	32	9	paracompactness	paracompactness	PROPN
ejpam-3941	32	10	and	and	CCONJ
ejpam-3941	32	11	c2	c2	PROPN
ejpam-3941	32	12	-	-	PUNCT
ejpam-3941	32	13	paracompactness	paracompactness	PROPN
ejpam-3941	32	14	are	be	AUX
ejpam-3941	32	15	multiplicative	multiplicative	ADJ
ejpam-3941	32	16	.	.	PUNCT
ejpam-3941	33	1	theorem	theorem	NOUN
ejpam-3941	33	2	2	2	NUM
ejpam-3941	33	3	.	.	PUNCT
ejpam-3941	34	1	if	if	SCONJ
ejpam-3941	34	2	x	x	PRON
ejpam-3941	34	3	is	be	AUX
ejpam-3941	34	4	c	c	NOUN
ejpam-3941	34	5	-	-	PUNCT
ejpam-3941	34	6	paracompact	paracompact	ADJ
ejpam-3941	34	7	(	(	PUNCT
ejpam-3941	34	8	c2	c2	NOUN
ejpam-3941	34	9	-	-	PUNCT
ejpam-3941	34	10	paracompact	paracompact	NOUN
ejpam-3941	34	11	)	)	PUNCT
ejpam-3941	34	12	and	and	CCONJ
ejpam-3941	34	13	z	z	NOUN
ejpam-3941	34	14	is	be	AUX
ejpam-3941	34	15	a	a	DET
ejpam-3941	34	16	compact	compact	ADJ
ejpam-3941	34	17	t2	t2	NOUN
ejpam-3941	34	18	space	space	NOUN
ejpam-3941	34	19	,	,	PUNCT
ejpam-3941	34	20	then	then	ADV
ejpam-3941	34	21	x	x	SYM
ejpam-3941	34	22	×	×	PROPN
ejpam-3941	34	23	z	z	NOUN
ejpam-3941	34	24	is	be	AUX
ejpam-3941	34	25	c	c	NOUN
ejpam-3941	34	26	-	-	PUNCT
ejpam-3941	34	27	paracompact	paracompact	ADJ
ejpam-3941	34	28	(	(	PUNCT
ejpam-3941	34	29	c2	c2	NOUN
ejpam-3941	34	30	-	-	PUNCT
ejpam-3941	34	31	paracompact	paracompact	NOUN
ejpam-3941	34	32	)	)	PUNCT
ejpam-3941	34	33	.	.	PUNCT
ejpam-3941	35	1	proof	proof	NOUN
ejpam-3941	35	2	.	.	PUNCT
ejpam-3941	36	1	let	let	VERB
ejpam-3941	36	2	y	y	PRON
ejpam-3941	36	3	be	be	AUX
ejpam-3941	36	4	a	a	DET
ejpam-3941	36	5	paracompact	paracompact	NOUN
ejpam-3941	36	6	(	(	PUNCT
ejpam-3941	36	7	t2	t2	NOUN
ejpam-3941	36	8	paracompact	paracompact	NOUN
ejpam-3941	36	9	)	)	PUNCT
ejpam-3941	36	10	space	space	NOUN
ejpam-3941	36	11	and	and	CCONJ
ejpam-3941	36	12	f	f	NOUN
ejpam-3941	36	13	:	:	PUNCT
ejpam-3941	37	1	x	x	PUNCT
ejpam-3941	37	2	−→	−→	NOUN
ejpam-3941	37	3	y	y	NOUN
ejpam-3941	37	4	be	be	AUX
ejpam-3941	37	5	a	a	DET
ejpam-3941	37	6	bijective	bijective	ADJ
ejpam-3941	37	7	function	function	NOUN
ejpam-3941	37	8	such	such	ADJ
ejpam-3941	37	9	that	that	SCONJ
ejpam-3941	37	10	the	the	DET
ejpam-3941	37	11	restriction	restriction	NOUN
ejpam-3941	37	12	f	f	PROPN
ejpam-3941	37	13	�	�	PROPN
ejpam-3941	37	14	a	a	NOUN
ejpam-3941	37	15	:	:	PUNCT
ejpam-3941	37	16	a	a	DET
ejpam-3941	37	17	−→	−→	NOUN
ejpam-3941	37	18	f(a	f(a	NOUN
ejpam-3941	37	19	)	)	PUNCT
ejpam-3941	37	20	is	be	AUX
ejpam-3941	37	21	a	a	DET
ejpam-3941	37	22	homeomorphism	homeomorphism	NOUN
ejpam-3941	37	23	for	for	ADP
ejpam-3941	37	24	each	each	DET
ejpam-3941	37	25	compact	compact	ADJ
ejpam-3941	37	26	subspace	subspace	NOUN
ejpam-3941	37	27	a	a	DET
ejpam-3941	37	28	⊆	⊆	NUM
ejpam-3941	37	29	x.	x.	NOUN
ejpam-3941	37	30	consider	consider	VERB
ejpam-3941	37	31	the	the	DET
ejpam-3941	37	32	product	product	NOUN
ejpam-3941	37	33	space	space	NOUN
ejpam-3941	37	34	y	y	PROPN
ejpam-3941	37	35	×	×	PROPN
ejpam-3941	37	36	z	z	NOUN
ejpam-3941	37	37	which	which	PRON
ejpam-3941	37	38	is	be	AUX
ejpam-3941	37	39	paracompact	paracompact	ADJ
ejpam-3941	37	40	(	(	PUNCT
ejpam-3941	37	41	t2	t2	NOUN
ejpam-3941	37	42	paracompact	paracompact	NOUN
ejpam-3941	37	43	)	)	PUNCT
ejpam-3941	37	44	,	,	PUNCT
ejpam-3941	37	45	because	because	SCONJ
ejpam-3941	37	46	the	the	DET
ejpam-3941	37	47	product	product	NOUN
ejpam-3941	37	48	of	of	ADP
ejpam-3941	37	49	any	any	DET
ejpam-3941	37	50	paracompact	paracompact	ADJ
ejpam-3941	37	51	space	space	NOUN
ejpam-3941	37	52	with	with	ADP
ejpam-3941	37	53	a	a	DET
ejpam-3941	37	54	compact	compact	ADJ
ejpam-3941	37	55	space	space	NOUN
ejpam-3941	37	56	is	be	AUX
ejpam-3941	37	57	paracompact	paracompact	ADJ
ejpam-3941	37	58	,	,	PUNCT
ejpam-3941	37	59	see	see	VERB
ejpam-3941	37	60	[	[	X
ejpam-3941	37	61	4	4	NUM
ejpam-3941	37	62	,	,	PUNCT
ejpam-3941	37	63	5.1.36	5.1.36	NUM
ejpam-3941	37	64	]	]	PUNCT
ejpam-3941	37	65	.	.	PUNCT
ejpam-3941	38	1	define	define	VERB
ejpam-3941	38	2	g	g	NOUN
ejpam-3941	38	3	:	:	PUNCT
ejpam-3941	38	4	x	x	SYM
ejpam-3941	38	5	×	×	NOUN
ejpam-3941	38	6	z	z	NOUN
ejpam-3941	38	7	−→	−→	NOUN
ejpam-3941	38	8	y	y	PROPN
ejpam-3941	38	9	×	×	PROPN
ejpam-3941	38	10	z	z	NOUN
ejpam-3941	38	11	by	by	ADP
ejpam-3941	38	12	g(〈x	g(〈x	NOUN
ejpam-3941	38	13	,	,	PUNCT
ejpam-3941	38	14	i	i	PROPN
ejpam-3941	38	15	〉	〉	PROPN
ejpam-3941	38	16	)	)	PUNCT
ejpam-3941	38	17	=	=	PUNCT
ejpam-3941	39	1	〈	〈	PROPN
ejpam-3941	39	2	f(x	f(x	PROPN
ejpam-3941	39	3	)	)	PUNCT
ejpam-3941	39	4	,	,	PUNCT
ejpam-3941	39	5	i	i	PROPN
ejpam-3941	39	6	〉	〉	PROPN
ejpam-3941	39	7	.	.	PUNCT
ejpam-3941	40	1	then	then	ADV
ejpam-3941	40	2	g	g	PROPN
ejpam-3941	40	3	is	be	AUX
ejpam-3941	40	4	a	a	DET
ejpam-3941	40	5	bijective	bijective	ADJ
ejpam-3941	40	6	function	function	NOUN
ejpam-3941	40	7	and	and	CCONJ
ejpam-3941	40	8	g	g	NOUN
ejpam-3941	40	9	=	=	SYM
ejpam-3941	40	10	f	f	PROPN
ejpam-3941	40	11	×	×	PROPN
ejpam-3941	40	12	idz	idz	PROPN
ejpam-3941	40	13	,	,	PUNCT
ejpam-3941	40	14	where	where	SCONJ
ejpam-3941	40	15	idz	idz	NOUN
ejpam-3941	40	16	is	be	AUX
ejpam-3941	40	17	the	the	DET
ejpam-3941	40	18	identity	identity	NOUN
ejpam-3941	40	19	function	function	NOUN
ejpam-3941	40	20	on	on	ADP
ejpam-3941	40	21	z.	z.	PROPN
ejpam-3941	40	22	let	let	VERB
ejpam-3941	40	23	c	c	PRON
ejpam-3941	40	24	be	be	AUX
ejpam-3941	40	25	any	any	DET
ejpam-3941	40	26	compact	compact	ADJ
ejpam-3941	40	27	subspace	subspace	NOUN
ejpam-3941	40	28	of	of	ADP
ejpam-3941	40	29	x	x	SYM
ejpam-3941	40	30	×	×	PROPN
ejpam-3941	40	31	z.	z.	PROPN
ejpam-3941	40	32	then	then	ADV
ejpam-3941	40	33	c	c	PROPN
ejpam-3941	40	34	⊆	⊆	NUM
ejpam-3941	40	35	p1(c	p1(c	SYM
ejpam-3941	40	36	)	)	PUNCT
ejpam-3941	40	37	×	×	NOUN
ejpam-3941	40	38	p2(c	p2(c	PROPN
ejpam-3941	40	39	)	)	PUNCT
ejpam-3941	40	40	,	,	PUNCT
ejpam-3941	40	41	where	where	SCONJ
ejpam-3941	40	42	p1	p1	NOUN
ejpam-3941	40	43	and	and	CCONJ
ejpam-3941	40	44	p2	p2	PROPN
ejpam-3941	40	45	are	be	AUX
ejpam-3941	40	46	the	the	DET
ejpam-3941	40	47	usual	usual	ADJ
ejpam-3941	40	48	projection	projection	NOUN
ejpam-3941	40	49	functions	function	NOUN
ejpam-3941	40	50	.	.	PUNCT
ejpam-3941	41	1	p1(c	p1(c	X
ejpam-3941	41	2	)	)	PUNCT
ejpam-3941	41	3	is	be	AUX
ejpam-3941	41	4	a	a	DET
ejpam-3941	41	5	compact	compact	ADJ
ejpam-3941	41	6	subspace	subspace	NOUN
ejpam-3941	41	7	of	of	ADP
ejpam-3941	41	8	x	x	X
ejpam-3941	41	9	and	and	CCONJ
ejpam-3941	41	10	p2(c	p2(c	NUM
ejpam-3941	41	11	)	)	PUNCT
ejpam-3941	41	12	is	be	AUX
ejpam-3941	41	13	a	a	DET
ejpam-3941	41	14	compact	compact	ADJ
ejpam-3941	41	15	subspace	subspace	NOUN
ejpam-3941	41	16	of	of	ADP
ejpam-3941	41	17	z	z	PROPN
ejpam-3941	41	18	,	,	PUNCT
ejpam-3941	41	19	thus	thus	ADV
ejpam-3941	41	20	p1(c	p1(c	X
ejpam-3941	41	21	)	)	PUNCT
ejpam-3941	41	22	×	×	NOUN
ejpam-3941	41	23	p2(c	p2(c	PROPN
ejpam-3941	41	24	)	)	PUNCT
ejpam-3941	41	25	is	be	AUX
ejpam-3941	41	26	a	a	DET
ejpam-3941	41	27	compact	compact	ADJ
ejpam-3941	41	28	subspace	subspace	NOUN
ejpam-3941	41	29	of	of	ADP
ejpam-3941	41	30	x	x	X
ejpam-3941	41	31	×	×	PROPN
ejpam-3941	41	32	z.	z.	PROPN
ejpam-3941	42	1	now	now	ADV
ejpam-3941	42	2	,	,	PUNCT
ejpam-3941	42	3	f	f	PROPN
ejpam-3941	42	4	�	�	PROPN
ejpam-3941	42	5	p1(c	p1(c	PROPN
ejpam-3941	42	6	):	):	PUNCT
ejpam-3941	42	7	p1(c	p1(c	X
ejpam-3941	42	8	)	)	PUNCT
ejpam-3941	42	9	−→	−→	NOUN
ejpam-3941	42	10	f(p1(c	f(p1(c	PROPN
ejpam-3941	42	11	)	)	PUNCT
ejpam-3941	42	12	)	)	PUNCT
ejpam-3941	42	13	is	be	AUX
ejpam-3941	42	14	a	a	DET
ejpam-3941	42	15	homeomorphism	homeomorphism	PROPN
ejpam-3941	42	16	and	and	CCONJ
ejpam-3941	42	17	idz	idz	PROPN
ejpam-3941	42	18	�	�	PROPN
ejpam-3941	42	19	p2(c	p2(c	NUM
ejpam-3941	42	20	):	):	PUNCT
ejpam-3941	42	21	p2(c	p2(c	X
ejpam-3941	42	22	)	)	PUNCT
ejpam-3941	42	23	−→	−→	NOUN
ejpam-3941	42	24	p2(c	p2(c	PROPN
ejpam-3941	42	25	)	)	PUNCT
ejpam-3941	42	26	is	be	AUX
ejpam-3941	42	27	a	a	DET
ejpam-3941	42	28	homeomorphism	homeomorphism	NOUN
ejpam-3941	42	29	.	.	PUNCT
ejpam-3941	43	1	thus	thus	ADV
ejpam-3941	43	2	(	(	PUNCT
ejpam-3941	43	3	f	f	PROPN
ejpam-3941	43	4	×	×	PROPN
ejpam-3941	43	5	idz	idz	PROPN
ejpam-3941	43	6	)	)	PUNCT
ejpam-3941	43	7	�	�	PROPN
ejpam-3941	43	8	(	(	PUNCT
ejpam-3941	43	9	p1(c)×p2(c	p1(c)×p2(c	NOUN
ejpam-3941	43	10	)	)	PUNCT
ejpam-3941	43	11	):	):	PUNCT
ejpam-3941	43	12	p1(c)×	p1(c)×	PROPN
ejpam-3941	43	13	p2(c	p2(c	X
ejpam-3941	43	14	)	)	PUNCT
ejpam-3941	43	15	−→	−→	NOUN
ejpam-3941	43	16	f(p1(c))×	f(p1(c))×	NOUN
ejpam-3941	43	17	p2(c	p2(c	X
ejpam-3941	43	18	)	)	PUNCT
ejpam-3941	43	19	is	be	AUX
ejpam-3941	43	20	a	a	DET
ejpam-3941	43	21	homeomorphism	homeomorphism	NOUN
ejpam-3941	43	22	.	.	PUNCT
ejpam-3941	44	1	we	we	PRON
ejpam-3941	44	2	conclude	conclude	VERB
ejpam-3941	44	3	that	that	SCONJ
ejpam-3941	44	4	g	g	PROPN
ejpam-3941	44	5	�	�	PROPN
ejpam-3941	44	6	c	c	NOUN
ejpam-3941	44	7	:	:	PUNCT
ejpam-3941	44	8	c	c	PROPN
ejpam-3941	44	9	−→	−→	NOUN
ejpam-3941	44	10	g(c	g(c	NOUN
ejpam-3941	44	11	)	)	PUNCT
ejpam-3941	44	12	is	be	AUX
ejpam-3941	44	13	a	a	DET
ejpam-3941	44	14	homeomorphism	homeomorphism	NOUN
ejpam-3941	44	15	because	because	SCONJ
ejpam-3941	44	16	g	g	PROPN
ejpam-3941	44	17	�	�	PROPN
ejpam-3941	44	18	c=	c=	NOUN
ejpam-3941	44	19	(	(	PUNCT
ejpam-3941	44	20	(	(	PUNCT
ejpam-3941	44	21	f	f	X
ejpam-3941	44	22	×	×	PROPN
ejpam-3941	44	23	idz	idz	PROPN
ejpam-3941	44	24	)	)	PUNCT
ejpam-3941	44	25	�	�	PROPN
ejpam-3941	44	26	(	(	PUNCT
ejpam-3941	44	27	p1(c)×p2(c	p1(c)×p2(c	NOUN
ejpam-3941	44	28	)	)	PUNCT
ejpam-3941	44	29	)	)	PUNCT
ejpam-3941	44	30	)	)	PUNCT
ejpam-3941	45	1	�	�	PROPN
ejpam-3941	45	2	c	c	PROPN
ejpam-3941	45	3	.	.	PUNCT
ejpam-3941	46	1	corollary	corollary	ADJ
ejpam-3941	46	2	1	1	NUM
ejpam-3941	46	3	.	.	PUNCT
ejpam-3941	47	1	if	if	SCONJ
ejpam-3941	47	2	x	x	PRON
ejpam-3941	47	3	is	be	AUX
ejpam-3941	47	4	c2	c2	NOUN
ejpam-3941	47	5	-	-	PUNCT
ejpam-3941	47	6	paracompact	paracompact	NOUN
ejpam-3941	47	7	(	(	PUNCT
ejpam-3941	47	8	c	c	NOUN
ejpam-3941	47	9	-	-	PUNCT
ejpam-3941	47	10	paracompact	paracompact	ADJ
ejpam-3941	47	11	)	)	PUNCT
ejpam-3941	47	12	,	,	PUNCT
ejpam-3941	47	13	then	then	ADV
ejpam-3941	47	14	so	so	ADV
ejpam-3941	47	15	is	be	AUX
ejpam-3941	47	16	x	x	X
ejpam-3941	47	17	×	×	PROPN
ejpam-3941	47	18	i	i	PRON
ejpam-3941	47	19	,	,	PUNCT
ejpam-3941	47	20	where	where	SCONJ
ejpam-3941	47	21	i	i	PRON
ejpam-3941	47	22	is	be	AUX
ejpam-3941	47	23	the	the	DET
ejpam-3941	47	24	closed	closed	ADJ
ejpam-3941	47	25	unit	unit	NOUN
ejpam-3941	47	26	interval	interval	NOUN
ejpam-3941	47	27	[	[	X
ejpam-3941	47	28	0	0	NUM
ejpam-3941	47	29	,	,	PUNCT
ejpam-3941	47	30	1	1	NUM
ejpam-3941	47	31	]	]	PUNCT
ejpam-3941	47	32	considered	consider	VERB
ejpam-3941	47	33	with	with	ADP
ejpam-3941	47	34	its	its	PRON
ejpam-3941	47	35	usual	usual	ADJ
ejpam-3941	47	36	euclidean	euclidean	ADJ
ejpam-3941	47	37	metric	metric	ADJ
ejpam-3941	47	38	topology	topology	NOUN
ejpam-3941	47	39	.	.	PUNCT
ejpam-3941	48	1	we	we	PRON
ejpam-3941	48	2	still	still	ADV
ejpam-3941	48	3	do	do	AUX
ejpam-3941	48	4	not	not	PART
ejpam-3941	48	5	know	know	VERB
ejpam-3941	48	6	an	an	DET
ejpam-3941	48	7	answer	answer	NOUN
ejpam-3941	48	8	of	of	ADP
ejpam-3941	48	9	the	the	DET
ejpam-3941	48	10	converse	converse	NOUN
ejpam-3941	48	11	of	of	ADP
ejpam-3941	48	12	the	the	DET
ejpam-3941	48	13	above	above	ADJ
ejpam-3941	48	14	theorem	theorem	NOUN
ejpam-3941	48	15	which	which	PRON
ejpam-3941	48	16	is	be	AUX
ejpam-3941	48	17	the	the	DET
ejpam-3941	48	18	following	follow	VERB
ejpam-3941	48	19	statement	statement	NOUN
ejpam-3941	48	20	:	:	PUNCT
ejpam-3941	48	21	if	if	SCONJ
ejpam-3941	48	22	x	x	PRON
ejpam-3941	48	23	×	×	NOUN
ejpam-3941	48	24	i	i	PRON
ejpam-3941	48	25	is	be	AUX
ejpam-3941	48	26	c2	c2	PROPN
ejpam-3941	48	27	-	-	PUNCT
ejpam-3941	48	28	paracompact	paracompact	NOUN
ejpam-3941	48	29	,	,	PUNCT
ejpam-3941	48	30	is	be	AUX
ejpam-3941	48	31	then	then	ADV
ejpam-3941	48	32	x	x	PROPN
ejpam-3941	48	33	c2	c2	PROPN
ejpam-3941	48	34	-	-	PUNCT
ejpam-3941	48	35	paracompact	paracompact	NOUN
ejpam-3941	48	36	?	?	PUNCT
ejpam-3941	49	1	observe	observe	VERB
ejpam-3941	49	2	that	that	SCONJ
ejpam-3941	49	3	if	if	SCONJ
ejpam-3941	49	4	x	x	PRON
ejpam-3941	49	5	is	be	AUX
ejpam-3941	49	6	c2	c2	NOUN
ejpam-3941	49	7	-	-	PUNCT
ejpam-3941	49	8	paracompact	paracompact	PROPN
ejpam-3941	49	9	and	and	CCONJ
ejpam-3941	49	10	y	y	PROPN
ejpam-3941	49	11	is	be	AUX
ejpam-3941	49	12	t4	t4	PROPN
ejpam-3941	49	13	,	,	PUNCT
ejpam-3941	49	14	then	then	ADV
ejpam-3941	49	15	the	the	DET
ejpam-3941	49	16	natural	natural	ADJ
ejpam-3941	49	17	projection	projection	NOUN
ejpam-3941	49	18	p	p	NOUN
ejpam-3941	49	19	:	:	PUNCT
ejpam-3941	49	20	x	x	SYM
ejpam-3941	49	21	×	×	NOUN
ejpam-3941	49	22	y	y	PROPN
ejpam-3941	49	23	−→	−→	NOUN
ejpam-3941	49	24	y	y	PROPN
ejpam-3941	49	25	h.	h.	PROPN
ejpam-3941	49	26	alzumi	alzumi	PROPN
ejpam-3941	49	27	,	,	PUNCT
ejpam-3941	49	28	l.	l.	PROPN
ejpam-3941	49	29	kalantan	kalantan	PROPN
ejpam-3941	49	30	and	and	CCONJ
ejpam-3941	49	31	m.	m.	PROPN
ejpam-3941	49	32	mohammed	mohammed	PROPN
ejpam-3941	49	33	saeed	saeed	PROPN
ejpam-3941	49	34	/	/	SYM
ejpam-3941	49	35	eur	eur	PROPN
ejpam-3941	49	36	.	.	PUNCT
ejpam-3941	50	1	j.	j.	PROPN
ejpam-3941	50	2	pure	pure	PROPN
ejpam-3941	50	3	appl	appl	PROPN
ejpam-3941	50	4	.	.	PROPN
ejpam-3941	50	5	math	math	PROPN
ejpam-3941	50	6	,	,	PUNCT
ejpam-3941	50	7	14	14	NUM
ejpam-3941	50	8	(	(	PUNCT
ejpam-3941	50	9	2	2	NUM
ejpam-3941	50	10	)	)	PUNCT
ejpam-3941	50	11	(	(	PUNCT
ejpam-3941	50	12	2021	2021	NUM
ejpam-3941	50	13	)	)	PUNCT
ejpam-3941	50	14	,	,	PUNCT
ejpam-3941	50	15	351	351	NUM
ejpam-3941	50	16	-	-	SYM
ejpam-3941	50	17	357	357	NUM
ejpam-3941	50	18	353	353	NUM
ejpam-3941	50	19	may	may	AUX
ejpam-3941	50	20	not	not	PART
ejpam-3941	50	21	be	be	AUX
ejpam-3941	50	22	closed	close	VERB
ejpam-3941	50	23	.	.	PUNCT
ejpam-3941	51	1	for	for	ADP
ejpam-3941	51	2	example	example	NOUN
ejpam-3941	51	3	,	,	PUNCT
ejpam-3941	51	4	ω1	ω1	PROPN
ejpam-3941	51	5	is	be	AUX
ejpam-3941	51	6	c2	c2	PROPN
ejpam-3941	51	7	-	-	PUNCT
ejpam-3941	51	8	paracompact	paracompact	NOUN
ejpam-3941	51	9	being	being	NOUN
ejpam-3941	51	10	t2	t2	NOUN
ejpam-3941	51	11	locally	locally	ADV
ejpam-3941	51	12	compact	compact	ADJ
ejpam-3941	51	13	[	[	X
ejpam-3941	51	14	8	8	NUM
ejpam-3941	51	15	]	]	PUNCT
ejpam-3941	51	16	and	and	CCONJ
ejpam-3941	51	17	ω1	ω1	PROPN
ejpam-3941	51	18	+	+	CCONJ
ejpam-3941	51	19	1	1	NUM
ejpam-3941	51	20	is	be	AUX
ejpam-3941	51	21	t2	t2	NOUN
ejpam-3941	51	22	compact	compact	ADJ
ejpam-3941	51	23	,	,	PUNCT
ejpam-3941	51	24	hence	hence	ADV
ejpam-3941	51	25	t4	t4	PROPN
ejpam-3941	51	26	,	,	PUNCT
ejpam-3941	51	27	but	but	CCONJ
ejpam-3941	51	28	p	p	NOUN
ejpam-3941	51	29	:	:	PUNCT
ejpam-3941	51	30	ω1	ω1	PROPN
ejpam-3941	51	31	×	×	PROPN
ejpam-3941	51	32	(	(	PUNCT
ejpam-3941	51	33	ω1	ω1	PROPN
ejpam-3941	51	34	+	+	CCONJ
ejpam-3941	51	35	1	1	NUM
ejpam-3941	51	36	)	)	PUNCT
ejpam-3941	51	37	−→	−→	NOUN
ejpam-3941	51	38	ω1	ω1	NOUN
ejpam-3941	51	39	+	+	CCONJ
ejpam-3941	51	40	1	1	NUM
ejpam-3941	51	41	is	be	AUX
ejpam-3941	51	42	not	not	PART
ejpam-3941	51	43	closed	closed	ADJ
ejpam-3941	51	44	,	,	PUNCT
ejpam-3941	51	45	see	see	VERB
ejpam-3941	51	46	[	[	X
ejpam-3941	51	47	4	4	NUM
ejpam-3941	51	48	,	,	PUNCT
ejpam-3941	51	49	3.10.16	3.10.16	PROPN
ejpam-3941	51	50	]	]	PUNCT
ejpam-3941	51	51	.	.	PUNCT
ejpam-3941	52	1	referring	refer	VERB
ejpam-3941	52	2	to	to	ADP
ejpam-3941	52	3	theorem	theorem	NOUN
ejpam-3941	52	4	1	1	NUM
ejpam-3941	52	5	,	,	PUNCT
ejpam-3941	52	6	we	we	PRON
ejpam-3941	52	7	introduce	introduce	VERB
ejpam-3941	52	8	here	here	ADV
ejpam-3941	52	9	another	another	DET
ejpam-3941	52	10	case	case	NOUN
ejpam-3941	52	11	when	when	SCONJ
ejpam-3941	52	12	a	a	DET
ejpam-3941	52	13	product	product	NOUN
ejpam-3941	52	14	of	of	ADP
ejpam-3941	52	15	two	two	NUM
ejpam-3941	52	16	c2paracompact	c2paracompact	ADJ
ejpam-3941	52	17	spaces	space	NOUN
ejpam-3941	52	18	will	will	AUX
ejpam-3941	52	19	be	be	AUX
ejpam-3941	52	20	c2	c2	NOUN
ejpam-3941	52	21	-	-	PUNCT
ejpam-3941	52	22	paracompact	paracompact	NOUN
ejpam-3941	52	23	.	.	PUNCT
ejpam-3941	53	1	theorem	theorem	NOUN
ejpam-3941	53	2	3	3	NUM
ejpam-3941	53	3	.	.	PUNCT
ejpam-3941	54	1	if	if	SCONJ
ejpam-3941	54	2	x	x	PROPN
ejpam-3941	54	3	and	and	CCONJ
ejpam-3941	54	4	z	z	NOUN
ejpam-3941	54	5	are	be	AUX
ejpam-3941	54	6	c2	c2	NOUN
ejpam-3941	54	7	-	-	PUNCT
ejpam-3941	54	8	paracompact	paracompact	NOUN
ejpam-3941	54	9	spaces	space	NOUN
ejpam-3941	54	10	such	such	ADJ
ejpam-3941	54	11	that	that	SCONJ
ejpam-3941	54	12	x	x	PRON
ejpam-3941	54	13	is	be	AUX
ejpam-3941	54	14	fréchet	fréchet	ADJ
ejpam-3941	54	15	and	and	CCONJ
ejpam-3941	54	16	countably	countably	ADV
ejpam-3941	54	17	compact	compact	ADJ
ejpam-3941	54	18	,	,	PUNCT
ejpam-3941	54	19	then	then	ADV
ejpam-3941	54	20	x	x	SYM
ejpam-3941	54	21	×	×	PROPN
ejpam-3941	54	22	z	z	PROPN
ejpam-3941	54	23	is	be	AUX
ejpam-3941	54	24	c2	c2	NOUN
ejpam-3941	54	25	-	-	PUNCT
ejpam-3941	54	26	paracompact	paracompact	NOUN
ejpam-3941	54	27	.	.	PUNCT
ejpam-3941	55	1	proof	proof	NOUN
ejpam-3941	55	2	.	.	PUNCT
ejpam-3941	56	1	let	let	VERB
ejpam-3941	56	2	y	y	PRON
ejpam-3941	56	3	and	and	CCONJ
ejpam-3941	56	4	y	y	PROPN
ejpam-3941	56	5	′	′	NOUN
ejpam-3941	56	6	be	be	AUX
ejpam-3941	56	7	t2	t2	NOUN
ejpam-3941	56	8	paracompact	paracompact	NOUN
ejpam-3941	56	9	spaces	space	NOUN
ejpam-3941	56	10	,	,	PUNCT
ejpam-3941	56	11	f	f	X
ejpam-3941	56	12	:	:	PUNCT
ejpam-3941	56	13	x	x	PUNCT
ejpam-3941	56	14	−→	−→	NOUN
ejpam-3941	56	15	y	y	PROPN
ejpam-3941	56	16	and	and	CCONJ
ejpam-3941	56	17	f	f	PROPN
ejpam-3941	57	1	′	′	NUM
ejpam-3941	57	2	:	:	PUNCT
ejpam-3941	58	1	z	z	X
ejpam-3941	58	2	−→	−→	NOUN
ejpam-3941	58	3	y	y	PROPN
ejpam-3941	58	4	′	′	NOUN
ejpam-3941	58	5	be	be	AUX
ejpam-3941	58	6	bijective	bijective	ADJ
ejpam-3941	58	7	function	function	NOUN
ejpam-3941	58	8	such	such	ADJ
ejpam-3941	58	9	that	that	SCONJ
ejpam-3941	58	10	the	the	DET
ejpam-3941	58	11	restriction	restriction	NOUN
ejpam-3941	58	12	of	of	ADP
ejpam-3941	58	13	each	each	PRON
ejpam-3941	58	14	of	of	ADP
ejpam-3941	58	15	them	they	PRON
ejpam-3941	58	16	on	on	ADP
ejpam-3941	58	17	any	any	DET
ejpam-3941	58	18	compact	compact	ADJ
ejpam-3941	58	19	subspace	subspace	NOUN
ejpam-3941	58	20	is	be	AUX
ejpam-3941	58	21	a	a	DET
ejpam-3941	58	22	homeomorphism	homeomorphism	NOUN
ejpam-3941	58	23	.	.	PUNCT
ejpam-3941	59	1	define	define	VERB
ejpam-3941	59	2	g	g	NOUN
ejpam-3941	59	3	:	:	PUNCT
ejpam-3941	59	4	x×z	x×z	PROPN
ejpam-3941	59	5	−→	−→	NOUN
ejpam-3941	59	6	y	y	PROPN
ejpam-3941	59	7	×y	×y	ADV
ejpam-3941	59	8	′	′	VERB
ejpam-3941	59	9	by	by	ADP
ejpam-3941	59	10	g(〈x	g(〈x	NOUN
ejpam-3941	59	11	,	,	PUNCT
ejpam-3941	59	12	z	z	PROPN
ejpam-3941	59	13	〉	〉	NOUN
ejpam-3941	59	14	)	)	PUNCT
ejpam-3941	59	15	=	=	PUNCT
ejpam-3941	60	1	〈	〈	PROPN
ejpam-3941	60	2	f(x	f(x	PROPN
ejpam-3941	60	3	)	)	PUNCT
ejpam-3941	60	4	,	,	PUNCT
ejpam-3941	60	5	f	f	PROPN
ejpam-3941	60	6	′(z	′(z	NOUN
ejpam-3941	60	7	)	)	PUNCT
ejpam-3941	60	8	〉	〉	PROPN
ejpam-3941	60	9	,	,	PUNCT
ejpam-3941	60	10	i.e.	i.e.	X
ejpam-3941	60	11	,	,	PUNCT
ejpam-3941	60	12	g	g	PROPN
ejpam-3941	60	13	=	=	SYM
ejpam-3941	60	14	f	f	PROPN
ejpam-3941	60	15	×f	×f	PROPN
ejpam-3941	60	16	′.	′.	NOUN
ejpam-3941	60	17	then	then	ADV
ejpam-3941	60	18	g	g	PROPN
ejpam-3941	60	19	is	be	AUX
ejpam-3941	60	20	bijective	bijective	ADJ
ejpam-3941	60	21	.	.	PUNCT
ejpam-3941	61	1	now	now	ADV
ejpam-3941	61	2	,	,	PUNCT
ejpam-3941	61	3	x	x	PRON
ejpam-3941	61	4	is	be	AUX
ejpam-3941	61	5	fréchet	fréchet	VERB
ejpam-3941	61	6	gives	give	VERB
ejpam-3941	61	7	that	that	SCONJ
ejpam-3941	61	8	f	f	PROPN
ejpam-3941	61	9	is	be	AUX
ejpam-3941	61	10	continuous	continuous	ADJ
ejpam-3941	61	11	,	,	PUNCT
ejpam-3941	61	12	see	see	ADJ
ejpam-3941	61	13	theorem	theorem	NOUN
ejpam-3941	61	14	1	1	NUM
ejpam-3941	61	15	.	.	PUNCT
ejpam-3941	62	1	since	since	SCONJ
ejpam-3941	62	2	x	x	PRON
ejpam-3941	62	3	is	be	AUX
ejpam-3941	62	4	countably	countably	ADV
ejpam-3941	62	5	compact	compact	ADJ
ejpam-3941	62	6	and	and	CCONJ
ejpam-3941	62	7	f	f	PROPN
ejpam-3941	62	8	continuous	continuous	ADJ
ejpam-3941	62	9	surjective	surjective	NOUN
ejpam-3941	62	10	,	,	PUNCT
ejpam-3941	62	11	then	then	ADV
ejpam-3941	62	12	y	y	PROPN
ejpam-3941	62	13	is	be	AUX
ejpam-3941	62	14	countably	countably	ADV
ejpam-3941	62	15	compact	compact	ADJ
ejpam-3941	62	16	.	.	PUNCT
ejpam-3941	63	1	hence	hence	ADV
ejpam-3941	63	2	y	y	PROPN
ejpam-3941	63	3	is	be	AUX
ejpam-3941	63	4	compact	compact	ADJ
ejpam-3941	63	5	because	because	SCONJ
ejpam-3941	63	6	any	any	DET
ejpam-3941	63	7	t2	t2	NOUN
ejpam-3941	63	8	countably	countably	ADV
ejpam-3941	63	9	compact	compact	ADJ
ejpam-3941	63	10	paracompact	paracompact	NOUN
ejpam-3941	63	11	is	be	AUX
ejpam-3941	63	12	compact	compact	ADJ
ejpam-3941	63	13	[	[	X
ejpam-3941	63	14	4	4	NUM
ejpam-3941	63	15	,	,	PUNCT
ejpam-3941	63	16	5.1.20	5.1.20	NUM
ejpam-3941	63	17	]	]	PUNCT
ejpam-3941	63	18	.	.	PUNCT
ejpam-3941	64	1	since	since	SCONJ
ejpam-3941	64	2	a	a	DET
ejpam-3941	64	3	product	product	NOUN
ejpam-3941	64	4	of	of	ADP
ejpam-3941	64	5	a	a	DET
ejpam-3941	64	6	t2	t2	NOUN
ejpam-3941	64	7	paracompact	paracompact	NOUN
ejpam-3941	64	8	space	space	NOUN
ejpam-3941	64	9	with	with	ADP
ejpam-3941	64	10	a	a	DET
ejpam-3941	64	11	t2	t2	ADJ
ejpam-3941	64	12	compact	compact	ADJ
ejpam-3941	64	13	space	space	NOUN
ejpam-3941	64	14	is	be	AUX
ejpam-3941	64	15	t2	t2	NOUN
ejpam-3941	64	16	paracompact	paracompact	NOUN
ejpam-3941	64	17	[	[	X
ejpam-3941	64	18	4	4	NUM
ejpam-3941	64	19	,	,	PUNCT
ejpam-3941	64	20	5.1.36	5.1.36	PROPN
ejpam-3941	64	21	]	]	PUNCT
ejpam-3941	64	22	,	,	PUNCT
ejpam-3941	64	23	then	then	ADV
ejpam-3941	64	24	y	y	PROPN
ejpam-3941	64	25	×y	×y	ADV
ejpam-3941	64	26	′	′	NUM
ejpam-3941	64	27	is	be	AUX
ejpam-3941	64	28	t2	t2	NOUN
ejpam-3941	64	29	paracompact	paracompact	NOUN
ejpam-3941	64	30	.	.	PUNCT
ejpam-3941	65	1	now	now	ADV
ejpam-3941	65	2	,	,	PUNCT
ejpam-3941	65	3	similar	similar	ADJ
ejpam-3941	65	4	argument	argument	NOUN
ejpam-3941	65	5	as	as	ADP
ejpam-3941	65	6	in	in	ADP
ejpam-3941	65	7	the	the	DET
ejpam-3941	65	8	proof	proof	NOUN
ejpam-3941	65	9	of	of	ADP
ejpam-3941	65	10	theorem	theorem	ADJ
ejpam-3941	65	11	2	2	NUM
ejpam-3941	65	12	shows	show	VERB
ejpam-3941	65	13	that	that	SCONJ
ejpam-3941	65	14	the	the	DET
ejpam-3941	65	15	restriction	restriction	NOUN
ejpam-3941	65	16	of	of	ADP
ejpam-3941	65	17	g	g	NOUN
ejpam-3941	65	18	on	on	ADP
ejpam-3941	65	19	any	any	DET
ejpam-3941	65	20	compact	compact	ADJ
ejpam-3941	65	21	subspace	subspace	NOUN
ejpam-3941	65	22	c	c	PROPN
ejpam-3941	65	23	of	of	ADP
ejpam-3941	65	24	x	x	SYM
ejpam-3941	65	25	×	×	PROPN
ejpam-3941	65	26	z	z	NOUN
ejpam-3941	65	27	will	will	AUX
ejpam-3941	65	28	be	be	AUX
ejpam-3941	65	29	a	a	DET
ejpam-3941	65	30	homeomorphism	homeomorphism	NOUN
ejpam-3941	65	31	.	.	PUNCT
ejpam-3941	66	1	recall	recall	VERB
ejpam-3941	66	2	that	that	SCONJ
ejpam-3941	66	3	a	a	DET
ejpam-3941	66	4	topological	topological	ADJ
ejpam-3941	66	5	space	space	NOUN
ejpam-3941	66	6	(	(	PUNCT
ejpam-3941	66	7	x	x	X
ejpam-3941	66	8	,	,	PUNCT
ejpam-3941	66	9	τ	τ	PROPN
ejpam-3941	66	10	)	)	PUNCT
ejpam-3941	66	11	is	be	AUX
ejpam-3941	66	12	called	call	VERB
ejpam-3941	66	13	lower	low	ADJ
ejpam-3941	66	14	compact	compact	ADJ
ejpam-3941	66	15	if	if	SCONJ
ejpam-3941	66	16	there	there	PRON
ejpam-3941	66	17	exists	exist	VERB
ejpam-3941	66	18	a	a	DET
ejpam-3941	66	19	coarser	coarse	ADJ
ejpam-3941	66	20	topology	topology	NOUN
ejpam-3941	66	21	τ	τ	NOUN
ejpam-3941	66	22	′	′	NOUN
ejpam-3941	66	23	on	on	ADP
ejpam-3941	66	24	x	x	INTJ
ejpam-3941	66	25	such	such	ADJ
ejpam-3941	66	26	that	that	SCONJ
ejpam-3941	66	27	(	(	PUNCT
ejpam-3941	66	28	x	x	X
ejpam-3941	66	29	,	,	PUNCT
ejpam-3941	66	30	τ	τ	PROPN
ejpam-3941	66	31	′	′	NUM
ejpam-3941	66	32	)	)	PUNCT
ejpam-3941	66	33	is	be	AUX
ejpam-3941	66	34	t2	t2	NOUN
ejpam-3941	66	35	-	-	PUNCT
ejpam-3941	66	36	compact	compact	ADJ
ejpam-3941	66	37	,	,	PUNCT
ejpam-3941	67	1	[	[	X
ejpam-3941	67	2	8	8	NUM
ejpam-3941	67	3	]	]	PUNCT
ejpam-3941	67	4	.	.	PUNCT
ejpam-3941	68	1	it	it	PRON
ejpam-3941	68	2	was	be	AUX
ejpam-3941	68	3	proved	prove	VERB
ejpam-3941	68	4	in	in	ADP
ejpam-3941	68	5	[	[	X
ejpam-3941	68	6	8	8	NUM
ejpam-3941	68	7	,	,	PUNCT
ejpam-3941	68	8	2.21	2.21	NUM
ejpam-3941	68	9	]	]	PUNCT
ejpam-3941	68	10	that	that	SCONJ
ejpam-3941	68	11	“	"	PUNCT
ejpam-3941	68	12	if	if	SCONJ
ejpam-3941	68	13	x	x	PRON
ejpam-3941	68	14	is	be	AUX
ejpam-3941	68	15	c2	c2	NOUN
ejpam-3941	68	16	-	-	PUNCT
ejpam-3941	68	17	paracompact	paracompact	NOUN
ejpam-3941	68	18	countably	countably	ADV
ejpam-3941	68	19	compact	compact	ADJ
ejpam-3941	68	20	fréchet	fréchet	NOUN
ejpam-3941	68	21	,	,	PUNCT
ejpam-3941	68	22	then	then	ADV
ejpam-3941	68	23	x	x	PUNCT
ejpam-3941	68	24	is	be	AUX
ejpam-3941	68	25	lower	low	ADJ
ejpam-3941	68	26	compact	compact	ADJ
ejpam-3941	68	27	”	"	PUNCT
ejpam-3941	68	28	.	.	PUNCT
ejpam-3941	69	1	it	it	PRON
ejpam-3941	69	2	turns	turn	VERB
ejpam-3941	69	3	out	out	ADP
ejpam-3941	69	4	that	that	SCONJ
ejpam-3941	69	5	lower	low	ADJ
ejpam-3941	69	6	compactness	compactness	NOUN
ejpam-3941	69	7	is	be	AUX
ejpam-3941	69	8	enough	enough	ADJ
ejpam-3941	69	9	in	in	ADP
ejpam-3941	69	10	theorem	theorem	ADJ
ejpam-3941	69	11	3	3	NUM
ejpam-3941	69	12	.	.	PUNCT
ejpam-3941	69	13	theorem	theorem	NOUN
ejpam-3941	69	14	4	4	NUM
ejpam-3941	69	15	.	.	PUNCT
ejpam-3941	70	1	if	if	SCONJ
ejpam-3941	70	2	x	x	PRON
ejpam-3941	70	3	is	be	AUX
ejpam-3941	70	4	lower	low	ADJ
ejpam-3941	70	5	compact	compact	ADJ
ejpam-3941	70	6	and	and	CCONJ
ejpam-3941	70	7	z	z	NOUN
ejpam-3941	70	8	is	be	AUX
ejpam-3941	70	9	c2	c2	PROPN
ejpam-3941	70	10	-	-	PUNCT
ejpam-3941	70	11	paracompact	paracompact	NOUN
ejpam-3941	70	12	,	,	PUNCT
ejpam-3941	70	13	then	then	ADV
ejpam-3941	70	14	x×z	x×z	PROPN
ejpam-3941	70	15	is	be	AUX
ejpam-3941	70	16	c2	c2	NOUN
ejpam-3941	70	17	-	-	PUNCT
ejpam-3941	70	18	paracompact	paracompact	NOUN
ejpam-3941	70	19	.	.	PUNCT
ejpam-3941	71	1	proof	proof	NOUN
ejpam-3941	71	2	.	.	PUNCT
ejpam-3941	72	1	let	let	VERB
ejpam-3941	72	2	τ	τ	PROPN
ejpam-3941	72	3	denote	denote	VERB
ejpam-3941	72	4	the	the	DET
ejpam-3941	72	5	topology	topology	NOUN
ejpam-3941	72	6	on	on	ADP
ejpam-3941	72	7	x.	x.	PROPN
ejpam-3941	72	8	let	let	VERB
ejpam-3941	72	9	τ	τ	PROPN
ejpam-3941	72	10	′	′	NUM
ejpam-3941	72	11	be	be	AUX
ejpam-3941	72	12	a	a	DET
ejpam-3941	72	13	t2	t2	NOUN
ejpam-3941	72	14	compact	compact	ADJ
ejpam-3941	72	15	topology	topology	NOUN
ejpam-3941	72	16	on	on	ADP
ejpam-3941	72	17	x	x	SYM
ejpam-3941	72	18	coarser	coarse	ADJ
ejpam-3941	72	19	than	than	ADP
ejpam-3941	72	20	τ	τ	PROPN
ejpam-3941	72	21	.	.	PUNCT
ejpam-3941	73	1	pick	pick	VERB
ejpam-3941	73	2	a	a	DET
ejpam-3941	73	3	t2	t2	NOUN
ejpam-3941	73	4	paracompact	paracompact	NOUN
ejpam-3941	73	5	space	space	NOUN
ejpam-3941	73	6	y	y	PROPN
ejpam-3941	73	7	′	′	NOUN
ejpam-3941	73	8	and	and	CCONJ
ejpam-3941	73	9	a	a	DET
ejpam-3941	73	10	bijective	bijective	ADJ
ejpam-3941	73	11	function	function	NOUN
ejpam-3941	73	12	f	f	NOUN
ejpam-3941	73	13	′	′	NUM
ejpam-3941	73	14	:	:	PUNCT
ejpam-3941	74	1	z	z	X
ejpam-3941	74	2	−→	−→	NOUN
ejpam-3941	74	3	y	y	PROPN
ejpam-3941	74	4	′	′	NUM
ejpam-3941	74	5	such	such	ADJ
ejpam-3941	74	6	that	that	SCONJ
ejpam-3941	74	7	the	the	DET
ejpam-3941	74	8	restriction	restriction	NOUN
ejpam-3941	74	9	of	of	ADP
ejpam-3941	74	10	f	f	PROPN
ejpam-3941	74	11	′	′	NUM
ejpam-3941	74	12	on	on	ADP
ejpam-3941	74	13	any	any	DET
ejpam-3941	74	14	compact	compact	ADJ
ejpam-3941	74	15	subspace	subspace	NOUN
ejpam-3941	74	16	is	be	AUX
ejpam-3941	74	17	a	a	DET
ejpam-3941	74	18	homeomorphism	homeomorphism	NOUN
ejpam-3941	74	19	.	.	PUNCT
ejpam-3941	75	1	define	define	VERB
ejpam-3941	75	2	g	g	NOUN
ejpam-3941	75	3	:	:	PUNCT
ejpam-3941	75	4	x×z	x×z	PROPN
ejpam-3941	75	5	−→	−→	NOUN
ejpam-3941	75	6	x×y	x×y	PUNCT
ejpam-3941	75	7	′	′	NUM
ejpam-3941	75	8	by	by	ADP
ejpam-3941	75	9	g(〈x	g(〈x	NOUN
ejpam-3941	75	10	,	,	PUNCT
ejpam-3941	75	11	z	z	PROPN
ejpam-3941	75	12	〉	〉	NOUN
ejpam-3941	75	13	)	)	PUNCT
ejpam-3941	75	14	=	=	PUNCT
ejpam-3941	76	1	〈	〈	PROPN
ejpam-3941	76	2	x	x	PRON
ejpam-3941	76	3	,	,	PUNCT
ejpam-3941	76	4	f	f	PROPN
ejpam-3941	76	5	′(z	′(z	NOUN
ejpam-3941	76	6	)	)	PUNCT
ejpam-3941	76	7	〉	〉	PROPN
ejpam-3941	76	8	,	,	PUNCT
ejpam-3941	76	9	i.e.	i.e.	X
ejpam-3941	76	10	,	,	PUNCT
ejpam-3941	76	11	g	g	PROPN
ejpam-3941	76	12	=	=	PUNCT
ejpam-3941	76	13	idx×f	idx×f	PROPN
ejpam-3941	76	14	′	′	NUM
ejpam-3941	76	15	,	,	PUNCT
ejpam-3941	76	16	where	where	SCONJ
ejpam-3941	76	17	x	x	X
ejpam-3941	76	18	in	in	ADP
ejpam-3941	76	19	the	the	DET
ejpam-3941	76	20	codomain	codomain	NOUN
ejpam-3941	76	21	is	be	AUX
ejpam-3941	76	22	considered	consider	VERB
ejpam-3941	76	23	with	with	ADP
ejpam-3941	76	24	the	the	DET
ejpam-3941	76	25	topology	topology	NOUN
ejpam-3941	76	26	τ	τ	PROPN
ejpam-3941	76	27	′.	′.	NOUN
ejpam-3941	76	28	observe	observe	VERB
ejpam-3941	76	29	that	that	SCONJ
ejpam-3941	76	30	the	the	DET
ejpam-3941	76	31	restriction	restriction	NOUN
ejpam-3941	76	32	of	of	ADP
ejpam-3941	76	33	the	the	DET
ejpam-3941	76	34	identity	identity	NOUN
ejpam-3941	76	35	function	function	NOUN
ejpam-3941	76	36	idx	idx	NOUN
ejpam-3941	76	37	:	:	PUNCT
ejpam-3941	76	38	(	(	PUNCT
ejpam-3941	76	39	x	x	X
ejpam-3941	76	40	,	,	PUNCT
ejpam-3941	76	41	τ	τ	PROPN
ejpam-3941	76	42	)	)	PUNCT
ejpam-3941	76	43	−→	−→	NOUN
ejpam-3941	76	44	(	(	PUNCT
ejpam-3941	76	45	x	x	INTJ
ejpam-3941	76	46	,	,	PUNCT
ejpam-3941	76	47	τ	τ	PROPN
ejpam-3941	76	48	′	′	NUM
ejpam-3941	76	49	)	)	PUNCT
ejpam-3941	76	50	on	on	ADP
ejpam-3941	76	51	any	any	DET
ejpam-3941	76	52	compact	compact	ADJ
ejpam-3941	76	53	subspace	subspace	NOUN
ejpam-3941	76	54	a	a	PRON
ejpam-3941	76	55	in	in	ADP
ejpam-3941	76	56	(	(	PUNCT
ejpam-3941	76	57	x	x	INTJ
ejpam-3941	76	58	,	,	PUNCT
ejpam-3941	76	59	τ	τ	PROPN
ejpam-3941	76	60	)	)	PUNCT
ejpam-3941	76	61	is	be	AUX
ejpam-3941	76	62	a	a	DET
ejpam-3941	76	63	homeomorphism	homeomorphism	NOUN
ejpam-3941	76	64	,	,	PUNCT
ejpam-3941	76	65	see	see	VERB
ejpam-3941	76	66	[	[	X
ejpam-3941	76	67	4	4	NUM
ejpam-3941	76	68	,	,	PUNCT
ejpam-3941	76	69	3.1.13	3.1.13	NUM
ejpam-3941	76	70	]	]	PUNCT
ejpam-3941	76	71	.	.	PUNCT
ejpam-3941	77	1	also	also	ADV
ejpam-3941	77	2	,	,	PUNCT
ejpam-3941	77	3	the	the	DET
ejpam-3941	77	4	codomain	codomain	NOUN
ejpam-3941	77	5	of	of	ADP
ejpam-3941	77	6	g	g	NOUN
ejpam-3941	77	7	,	,	PUNCT
ejpam-3941	77	8	x	x	SYM
ejpam-3941	77	9	×	×	NOUN
ejpam-3941	77	10	y	y	NOUN
ejpam-3941	77	11	′	′	NOUN
ejpam-3941	77	12	is	be	AUX
ejpam-3941	77	13	t2	t2	PROPN
ejpam-3941	77	14	paracompact	paracompact	NOUN
ejpam-3941	77	15	being	be	AUX
ejpam-3941	77	16	a	a	DET
ejpam-3941	77	17	product	product	NOUN
ejpam-3941	77	18	of	of	ADP
ejpam-3941	77	19	a	a	DET
ejpam-3941	77	20	t2	t2	NOUN
ejpam-3941	77	21	compact	compact	ADJ
ejpam-3941	77	22	space	space	NOUN
ejpam-3941	77	23	(	(	PUNCT
ejpam-3941	77	24	x	x	X
ejpam-3941	77	25	,	,	PUNCT
ejpam-3941	77	26	τ	τ	PROPN
ejpam-3941	77	27	′	′	NUM
ejpam-3941	77	28	)	)	PUNCT
ejpam-3941	77	29	,	,	PUNCT
ejpam-3941	77	30	with	with	ADP
ejpam-3941	77	31	a	a	DET
ejpam-3941	77	32	t2	t2	NOUN
ejpam-3941	77	33	paracompact	paracompact	NOUN
ejpam-3941	77	34	space	space	NOUN
ejpam-3941	77	35	y	y	PROPN
ejpam-3941	77	36	′	′	NOUN
ejpam-3941	77	37	,	,	PUNCT
ejpam-3941	77	38	[	[	X
ejpam-3941	77	39	4	4	NUM
ejpam-3941	77	40	,	,	PUNCT
ejpam-3941	77	41	5.1.36	5.1.36	PROPN
ejpam-3941	77	42	]	]	PUNCT
ejpam-3941	77	43	.	.	PUNCT
ejpam-3941	78	1	now	now	ADV
ejpam-3941	78	2	,	,	PUNCT
ejpam-3941	78	3	similar	similar	ADJ
ejpam-3941	78	4	argument	argument	NOUN
ejpam-3941	78	5	as	as	ADP
ejpam-3941	78	6	in	in	ADP
ejpam-3941	78	7	the	the	DET
ejpam-3941	78	8	proof	proof	NOUN
ejpam-3941	78	9	of	of	ADP
ejpam-3941	78	10	theorem	theorem	ADJ
ejpam-3941	78	11	2	2	NUM
ejpam-3941	78	12	shows	show	VERB
ejpam-3941	78	13	that	that	SCONJ
ejpam-3941	78	14	the	the	DET
ejpam-3941	78	15	restriction	restriction	NOUN
ejpam-3941	78	16	of	of	ADP
ejpam-3941	78	17	g	g	NOUN
ejpam-3941	78	18	on	on	ADP
ejpam-3941	78	19	any	any	DET
ejpam-3941	78	20	compact	compact	ADJ
ejpam-3941	78	21	subspace	subspace	NOUN
ejpam-3941	78	22	c	c	PROPN
ejpam-3941	78	23	of	of	ADP
ejpam-3941	78	24	x	x	SYM
ejpam-3941	78	25	×	×	PROPN
ejpam-3941	78	26	z	z	NOUN
ejpam-3941	78	27	will	will	AUX
ejpam-3941	78	28	be	be	AUX
ejpam-3941	78	29	a	a	DET
ejpam-3941	78	30	homeomorphism	homeomorphism	NOUN
ejpam-3941	78	31	.	.	PUNCT
ejpam-3941	79	1	here	here	ADV
ejpam-3941	79	2	is	be	AUX
ejpam-3941	79	3	an	an	DET
ejpam-3941	79	4	example	example	NOUN
ejpam-3941	79	5	showing	show	VERB
ejpam-3941	79	6	that	that	SCONJ
ejpam-3941	79	7	the	the	DET
ejpam-3941	79	8	fréchet	fréchet	NOUN
ejpam-3941	79	9	property	property	NOUN
ejpam-3941	79	10	is	be	AUX
ejpam-3941	79	11	essential	essential	ADJ
ejpam-3941	79	12	in	in	ADP
ejpam-3941	79	13	theorem	theorem	ADJ
ejpam-3941	79	14	1	1	NUM
ejpam-3941	79	15	.	.	NOUN
ejpam-3941	79	16	example	example	NOUN
ejpam-3941	80	1	1	1	NUM
ejpam-3941	80	2	.	.	X
ejpam-3941	80	3	consider	consider	VERB
ejpam-3941	80	4	ω2	ω2	ADJ
ejpam-3941	80	5	,	,	PUNCT
ejpam-3941	80	6	the	the	DET
ejpam-3941	80	7	successor	successor	NOUN
ejpam-3941	80	8	cardinal	cardinal	ADJ
ejpam-3941	80	9	number	number	NOUN
ejpam-3941	80	10	of	of	ADP
ejpam-3941	80	11	the	the	DET
ejpam-3941	80	12	cardinal	cardinal	ADJ
ejpam-3941	80	13	number	number	NOUN
ejpam-3941	80	14	ω1	ω1	PROPN
ejpam-3941	80	15	.	.	PUNCT
ejpam-3941	81	1	let	let	VERB
ejpam-3941	81	2	[	[	X
ejpam-3941	81	3	ω2	ω2	ADJ
ejpam-3941	81	4	]	]	X
ejpam-3941	81	5	≤ω1	≤ω1	X
ejpam-3941	81	6	=	=	SYM
ejpam-3941	81	7	{	{	PUNCT
ejpam-3941	81	8	e	e	X
ejpam-3941	81	9	⊂	⊂	PROPN
ejpam-3941	81	10	ω2	ω2	ADJ
ejpam-3941	81	11	:	:	PUNCT
ejpam-3941	81	12	|e|	|e|	DET
ejpam-3941	81	13	≤	≤	PROPN
ejpam-3941	81	14	ω1	ω1	PROPN
ejpam-3941	81	15	}	}	PUNCT
ejpam-3941	81	16	.	.	PUNCT
ejpam-3941	82	1	let	let	VERB
ejpam-3941	82	2	i	i	PRON
ejpam-3941	82	3	6∈	6∈	PRON
ejpam-3941	82	4	ω2	ω2	ADJ
ejpam-3941	82	5	and	and	CCONJ
ejpam-3941	82	6	put	put	VERB
ejpam-3941	82	7	x	x	X
ejpam-3941	82	8	=	=	SYM
ejpam-3941	82	9	{	{	PUNCT
ejpam-3941	82	10	i	i	NOUN
ejpam-3941	82	11	}	}	PUNCT
ejpam-3941	82	12	∪	∪	X
ejpam-3941	82	13	ω2	ω2	PROPN
ejpam-3941	82	14	.	.	PUNCT
ejpam-3941	83	1	for	for	ADP
ejpam-3941	83	2	each	each	DET
ejpam-3941	83	3	α	α	PROPN
ejpam-3941	83	4	∈	∈	PROPN
ejpam-3941	83	5	ω2	ω2	PROPN
ejpam-3941	83	6	,	,	PUNCT
ejpam-3941	83	7	let	let	VERB
ejpam-3941	83	8	{	{	PUNCT
ejpam-3941	83	9	α	α	NOUN
ejpam-3941	83	10	}	}	PUNCT
ejpam-3941	83	11	be	be	AUX
ejpam-3941	83	12	open	open	ADJ
ejpam-3941	83	13	and	and	CCONJ
ejpam-3941	83	14	an	an	DET
ejpam-3941	83	15	open	open	ADJ
ejpam-3941	83	16	neighborhood	neighborhood	NOUN
ejpam-3941	83	17	of	of	ADP
ejpam-3941	83	18	i	i	PRON
ejpam-3941	83	19	is	be	AUX
ejpam-3941	83	20	of	of	ADP
ejpam-3941	83	21	the	the	DET
ejpam-3941	83	22	form	form	NOUN
ejpam-3941	83	23	u	u	NOUN
ejpam-3941	83	24	=	=	X
ejpam-3941	83	25	{	{	PUNCT
ejpam-3941	83	26	i	i	NOUN
ejpam-3941	83	27	}	}	PUNCT
ejpam-3941	83	28	∪	∪	ADJ
ejpam-3941	83	29	(	(	PUNCT
ejpam-3941	83	30	ω2	ω2	ADJ
ejpam-3941	83	31	\	\	PROPN
ejpam-3941	83	32	e	e	X
ejpam-3941	83	33	)	)	PUNCT
ejpam-3941	83	34	where	where	SCONJ
ejpam-3941	83	35	h.	h.	PROPN
ejpam-3941	83	36	alzumi	alzumi	PROPN
ejpam-3941	83	37	,	,	PUNCT
ejpam-3941	83	38	l.	l.	PROPN
ejpam-3941	83	39	kalantan	kalantan	PROPN
ejpam-3941	83	40	and	and	CCONJ
ejpam-3941	83	41	m.	m.	PROPN
ejpam-3941	83	42	mohammed	mohammed	PROPN
ejpam-3941	83	43	saeed	saeed	PROPN
ejpam-3941	83	44	/	/	SYM
ejpam-3941	83	45	eur	eur	PROPN
ejpam-3941	83	46	.	.	PUNCT
ejpam-3941	84	1	j.	j.	PROPN
ejpam-3941	84	2	pure	pure	PROPN
ejpam-3941	84	3	appl	appl	PROPN
ejpam-3941	84	4	.	.	PROPN
ejpam-3941	84	5	math	math	PROPN
ejpam-3941	84	6	,	,	PUNCT
ejpam-3941	84	7	14	14	NUM
ejpam-3941	84	8	(	(	PUNCT
ejpam-3941	84	9	2	2	NUM
ejpam-3941	84	10	)	)	PUNCT
ejpam-3941	84	11	(	(	PUNCT
ejpam-3941	84	12	2021	2021	NUM
ejpam-3941	84	13	)	)	PUNCT
ejpam-3941	84	14	,	,	PUNCT
ejpam-3941	84	15	351	351	NUM
ejpam-3941	84	16	-	-	SYM
ejpam-3941	84	17	357	357	NUM
ejpam-3941	84	18	354	354	NUM
ejpam-3941	84	19	e	e	NOUN
ejpam-3941	84	20	∈	∈	PROPN
ejpam-3941	84	21	[	[	X
ejpam-3941	84	22	ω2	ω2	ADJ
ejpam-3941	84	23	]	]	X
ejpam-3941	84	24	≤ω1	≤ω1	PROPN
ejpam-3941	84	25	.	.	PUNCT
ejpam-3941	85	1	then	then	ADV
ejpam-3941	85	2	x	x	PRON
ejpam-3941	85	3	is	be	AUX
ejpam-3941	85	4	not	not	PART
ejpam-3941	85	5	fréchet	fréchet	ADJ
ejpam-3941	86	1	because	because	SCONJ
ejpam-3941	86	2	i	i	PROPN
ejpam-3941	86	3	∈	∈	PROPN
ejpam-3941	86	4	ω2	ω2	ADJ
ejpam-3941	86	5	and	and	CCONJ
ejpam-3941	86	6	the	the	DET
ejpam-3941	86	7	only	only	ADJ
ejpam-3941	86	8	convergent	convergent	NOUN
ejpam-3941	86	9	sequence	sequence	NOUN
ejpam-3941	86	10	is	be	AUX
ejpam-3941	86	11	the	the	DET
ejpam-3941	86	12	eventually	eventually	ADV
ejpam-3941	86	13	constant	constant	ADJ
ejpam-3941	86	14	sequence	sequence	NOUN
ejpam-3941	86	15	.	.	PUNCT
ejpam-3941	87	1	(	(	PUNCT
ejpam-3941	87	2	so	so	ADV
ejpam-3941	87	3	,	,	PUNCT
ejpam-3941	87	4	x	x	PRON
ejpam-3941	87	5	is	be	AUX
ejpam-3941	87	6	not	not	PART
ejpam-3941	87	7	even	even	ADV
ejpam-3941	87	8	of	of	ADP
ejpam-3941	87	9	countable	countable	ADJ
ejpam-3941	87	10	tightness	tightness	NOUN
ejpam-3941	87	11	.	.	PUNCT
ejpam-3941	87	12	)	)	PUNCT
ejpam-3941	88	1	observe	observe	VERB
ejpam-3941	88	2	that	that	SCONJ
ejpam-3941	88	3	x	x	PRON
ejpam-3941	88	4	is	be	AUX
ejpam-3941	88	5	t2	t2	NOUN
ejpam-3941	88	6	paracompact	paracompact	NOUN
ejpam-3941	88	7	.	.	PUNCT
ejpam-3941	89	1	but	but	CCONJ
ejpam-3941	89	2	we	we	PRON
ejpam-3941	89	3	will	will	AUX
ejpam-3941	89	4	not	not	PART
ejpam-3941	89	5	treat	treat	VERB
ejpam-3941	89	6	x	x	PUNCT
ejpam-3941	89	7	in	in	ADP
ejpam-3941	89	8	this	this	DET
ejpam-3941	89	9	way	way	NOUN
ejpam-3941	89	10	.	.	PUNCT
ejpam-3941	90	1	a	a	DET
ejpam-3941	90	2	subspace	subspace	NOUN
ejpam-3941	90	3	a	a	PRON
ejpam-3941	90	4	of	of	ADP
ejpam-3941	90	5	x	x	NOUN
ejpam-3941	90	6	is	be	AUX
ejpam-3941	90	7	compact	compact	ADJ
ejpam-3941	90	8	if	if	SCONJ
ejpam-3941	91	1	and	and	CCONJ
ejpam-3941	91	2	only	only	ADV
ejpam-3941	91	3	if	if	SCONJ
ejpam-3941	91	4	a	a	PRON
ejpam-3941	91	5	is	be	AUX
ejpam-3941	91	6	finite	finite	ADJ
ejpam-3941	91	7	.	.	PUNCT
ejpam-3941	92	1	so	so	ADV
ejpam-3941	92	2	,	,	PUNCT
ejpam-3941	92	3	by	by	ADP
ejpam-3941	92	4	[	[	X
ejpam-3941	92	5	8	8	NUM
ejpam-3941	92	6	,	,	PUNCT
ejpam-3941	92	7	theorem	theorem	VERB
ejpam-3941	92	8	2.7	2.7	NUM
ejpam-3941	92	9	]	]	PUNCT
ejpam-3941	92	10	,	,	PUNCT
ejpam-3941	92	11	x	x	X
ejpam-3941	92	12	is	be	AUX
ejpam-3941	92	13	c2	c2	NOUN
ejpam-3941	92	14	-	-	PUNCT
ejpam-3941	92	15	paracompact	paracompact	PROPN
ejpam-3941	92	16	and	and	CCONJ
ejpam-3941	92	17	x	x	SYM
ejpam-3941	92	18	=	=	SYM
ejpam-3941	92	19	y	y	PROPN
ejpam-3941	92	20	with	with	ADP
ejpam-3941	92	21	the	the	DET
ejpam-3941	92	22	discrete	discrete	ADJ
ejpam-3941	92	23	topology	topology	NOUN
ejpam-3941	92	24	and	and	CCONJ
ejpam-3941	92	25	the	the	DET
ejpam-3941	92	26	identity	identity	NOUN
ejpam-3941	92	27	function	function	NOUN
ejpam-3941	92	28	witness	witness	VERB
ejpam-3941	92	29	the	the	DET
ejpam-3941	92	30	c2	c2	PROPN
ejpam-3941	92	31	-	-	PUNCT
ejpam-3941	92	32	paracompactness	paracompactness	PROPN
ejpam-3941	92	33	of	of	ADP
ejpam-3941	92	34	x	x	PROPN
ejpam-3941	92	35	and	and	CCONJ
ejpam-3941	92	36	clearly	clearly	ADV
ejpam-3941	92	37	the	the	DET
ejpam-3941	92	38	identity	identity	NOUN
ejpam-3941	92	39	function	function	NOUN
ejpam-3941	92	40	can	can	AUX
ejpam-3941	92	41	not	not	PART
ejpam-3941	92	42	be	be	AUX
ejpam-3941	92	43	continuous	continuous	ADJ
ejpam-3941	92	44	because	because	SCONJ
ejpam-3941	92	45	x	x	PRON
ejpam-3941	92	46	is	be	AUX
ejpam-3941	92	47	not	not	PART
ejpam-3941	92	48	discrete	discrete	ADJ
ejpam-3941	92	49	.	.	PUNCT
ejpam-3941	93	1	observe	observe	VERB
ejpam-3941	93	2	that	that	SCONJ
ejpam-3941	93	3	x	x	X
ejpam-3941	93	4	in	in	ADP
ejpam-3941	93	5	example	example	NOUN
ejpam-3941	93	6	1	1	NUM
ejpam-3941	93	7	is	be	AUX
ejpam-3941	93	8	not	not	PART
ejpam-3941	93	9	of	of	ADP
ejpam-3941	93	10	countable	countable	ADJ
ejpam-3941	93	11	tightness	tightness	NOUN
ejpam-3941	93	12	as	as	ADP
ejpam-3941	93	13	i	i	PROPN
ejpam-3941	93	14	∈	∈	PROPN
ejpam-3941	94	1	ω2	ω2	ADJ
ejpam-3941	95	1	but	but	CCONJ
ejpam-3941	95	2	there	there	PRON
ejpam-3941	95	3	is	be	VERB
ejpam-3941	95	4	no	no	DET
ejpam-3941	95	5	countable	countable	ADJ
ejpam-3941	95	6	subset	subset	NOUN
ejpam-3941	95	7	a	a	DET
ejpam-3941	95	8	of	of	ADP
ejpam-3941	95	9	ω2	ω2	ADJ
ejpam-3941	95	10	satisfies	satisfie	NOUN
ejpam-3941	95	11	i	i	PRON
ejpam-3941	95	12	∈	∈	VERB
ejpam-3941	95	13	a.	a.	NOUN
ejpam-3941	95	14	here	here	ADV
ejpam-3941	95	15	is	be	AUX
ejpam-3941	95	16	an	an	DET
ejpam-3941	95	17	example	example	NOUN
ejpam-3941	95	18	showing	show	VERB
ejpam-3941	95	19	that	that	SCONJ
ejpam-3941	95	20	the	the	DET
ejpam-3941	95	21	countable	countable	ADJ
ejpam-3941	95	22	tightness	tightness	NOUN
ejpam-3941	95	23	property	property	NOUN
ejpam-3941	95	24	is	be	AUX
ejpam-3941	95	25	not	not	PART
ejpam-3941	95	26	enough	enough	ADJ
ejpam-3941	95	27	in	in	ADP
ejpam-3941	95	28	theorem	theorem	ADJ
ejpam-3941	95	29	1	1	NUM
ejpam-3941	95	30	.	.	NOUN
ejpam-3941	95	31	example	example	NOUN
ejpam-3941	96	1	2	2	NUM
ejpam-3941	96	2	.	.	X
ejpam-3941	96	3	for	for	ADP
ejpam-3941	96	4	each	each	DET
ejpam-3941	96	5	i	i	PROPN
ejpam-3941	96	6	∈	∈	PROPN
ejpam-3941	96	7	n	n	CCONJ
ejpam-3941	96	8	,	,	PUNCT
ejpam-3941	96	9	let	let	VERB
ejpam-3941	96	10	xi	xi	X
ejpam-3941	97	1	=	=	PUNCT
ejpam-3941	97	2	{	{	PUNCT
ejpam-3941	97	3	ai	ai	VERB
ejpam-3941	97	4	}	}	PUNCT
ejpam-3941	97	5	∪	∪	ADJ
ejpam-3941	97	6	{	{	PUNCT
ejpam-3941	97	7	ai	ai	NOUN
ejpam-3941	97	8	,	,	PUNCT
ejpam-3941	97	9	j	j	PROPN
ejpam-3941	97	10	:	:	PUNCT
ejpam-3941	97	11	j	j	PROPN
ejpam-3941	97	12	∈	∈	PROPN
ejpam-3941	97	13	n	n	AUX
ejpam-3941	97	14	}	}	PUNCT
ejpam-3941	97	15	be	be	AUX
ejpam-3941	97	16	such	such	ADJ
ejpam-3941	97	17	that	that	SCONJ
ejpam-3941	97	18	xn	xn	PROPN
ejpam-3941	97	19	∩	∩	PROPN
ejpam-3941	97	20	xm	xm	NOUN
ejpam-3941	97	21	=	=	NOUN
ejpam-3941	97	22	∅	∅	NOUN
ejpam-3941	97	23	for	for	ADP
ejpam-3941	97	24	each	each	DET
ejpam-3941	97	25	n	n	NOUN
ejpam-3941	97	26	,	,	PUNCT
ejpam-3941	97	27	m	m	VERB
ejpam-3941	97	28	∈	∈	NOUN
ejpam-3941	97	29	n	n	NOUN
ejpam-3941	97	30	with	with	ADP
ejpam-3941	97	31	n	n	PROPN
ejpam-3941	97	32	6=	6=	PRON
ejpam-3941	97	33	m.	m.	NOUN
ejpam-3941	97	34	let	let	VERB
ejpam-3941	97	35	a	a	DET
ejpam-3941	97	36	6∈	6∈	NOUN
ejpam-3941	97	37	∪i∈nxi	∪i∈nxi	NOUN
ejpam-3941	97	38	and	and	CCONJ
ejpam-3941	97	39	put	put	VERB
ejpam-3941	97	40	x	x	X
ejpam-3941	97	41	=	=	X
ejpam-3941	97	42	{	{	PUNCT
ejpam-3941	97	43	a	a	PRON
ejpam-3941	97	44	}	}	PUNCT
ejpam-3941	97	45	∪	∪	NOUN
ejpam-3941	97	46	(	(	PUNCT
ejpam-3941	97	47	∪i∈nxi	∪i∈nxi	NOUN
ejpam-3941	97	48	)	)	PUNCT
ejpam-3941	97	49	.	.	PUNCT
ejpam-3941	98	1	generate	generate	VERB
ejpam-3941	98	2	a	a	DET
ejpam-3941	98	3	topology	topology	NOUN
ejpam-3941	98	4	on	on	ADP
ejpam-3941	98	5	x	x	PUNCT
ejpam-3941	98	6	by	by	ADP
ejpam-3941	98	7	the	the	DET
ejpam-3941	98	8	following	follow	VERB
ejpam-3941	98	9	neighborhood	neighborhood	NOUN
ejpam-3941	98	10	system	system	NOUN
ejpam-3941	98	11	:	:	PUNCT
ejpam-3941	98	12	for	for	ADP
ejpam-3941	98	13	each	each	DET
ejpam-3941	98	14	i	i	PROPN
ejpam-3941	98	15	,	,	PUNCT
ejpam-3941	98	16	j	j	PROPN
ejpam-3941	98	17	∈	∈	PROPN
ejpam-3941	98	18	n	n	CCONJ
ejpam-3941	98	19	,	,	PUNCT
ejpam-3941	98	20	let	let	VERB
ejpam-3941	98	21	b(ai	b(ai	PROPN
ejpam-3941	98	22	,	,	PUNCT
ejpam-3941	98	23	j	j	NOUN
ejpam-3941	98	24	)	)	PUNCT
ejpam-3941	99	1	=	=	PRON
ejpam-3941	99	2	{	{	PUNCT
ejpam-3941	99	3	{	{	PUNCT
ejpam-3941	99	4	ai	ai	PROPN
ejpam-3941	99	5	,	,	PUNCT
ejpam-3941	99	6	j	j	NOUN
ejpam-3941	99	7	}	}	PUNCT
ejpam-3941	99	8	}	}	PUNCT
ejpam-3941	99	9	.	.	PUNCT
ejpam-3941	100	1	for	for	ADP
ejpam-3941	100	2	each	each	DET
ejpam-3941	100	3	i	i	PROPN
ejpam-3941	100	4	∈	∈	PROPN
ejpam-3941	100	5	n	n	CCONJ
ejpam-3941	100	6	,	,	PUNCT
ejpam-3941	100	7	let	let	VERB
ejpam-3941	100	8	b(ai	b(ai	NOUN
ejpam-3941	100	9	)	)	PUNCT
ejpam-3941	100	10	=	=	PRON
ejpam-3941	100	11	{	{	PUNCT
ejpam-3941	100	12	ai	ai	AUX
ejpam-3941	100	13	}	}	PUNCT
ejpam-3941	100	14	∪	∪	ADJ
ejpam-3941	100	15	{	{	PUNCT
ejpam-3941	100	16	ai	ai	NOUN
ejpam-3941	100	17	,	,	PUNCT
ejpam-3941	100	18	j	j	PROPN
ejpam-3941	100	19	:	:	PUNCT
ejpam-3941	100	20	j	j	PROPN
ejpam-3941	100	21	≥	≥	NUM
ejpam-3941	100	22	k	k	NOUN
ejpam-3941	100	23	,	,	PUNCT
ejpam-3941	100	24	where	where	SCONJ
ejpam-3941	100	25	k	k	PROPN
ejpam-3941	100	26	∈	∈	PROPN
ejpam-3941	100	27	n	n	X
ejpam-3941	100	28	}	}	PUNCT
ejpam-3941	100	29	.	.	PUNCT
ejpam-3941	101	1	for	for	ADP
ejpam-3941	101	2	members	member	NOUN
ejpam-3941	101	3	of	of	ADP
ejpam-3941	101	4	b(a	b(a	PROPN
ejpam-3941	101	5	)	)	PUNCT
ejpam-3941	101	6	we	we	PRON
ejpam-3941	101	7	take	take	VERB
ejpam-3941	101	8	all	all	DET
ejpam-3941	101	9	sets	set	NOUN
ejpam-3941	101	10	obtained	obtain	VERB
ejpam-3941	101	11	from	from	ADP
ejpam-3941	101	12	x	x	PUNCT
ejpam-3941	101	13	by	by	ADP
ejpam-3941	101	14	removing	remove	VERB
ejpam-3941	101	15	a	a	DET
ejpam-3941	101	16	finite	finite	ADJ
ejpam-3941	101	17	numbers	number	NOUN
ejpam-3941	101	18	of	of	ADP
ejpam-3941	101	19	xi	xi	PROPN
ejpam-3941	101	20	’s	’s	NOUN
ejpam-3941	101	21	and	and	CCONJ
ejpam-3941	101	22	a	a	DET
ejpam-3941	101	23	finite	finite	ADJ
ejpam-3941	101	24	number	number	NOUN
ejpam-3941	101	25	of	of	ADP
ejpam-3941	101	26	points	point	NOUN
ejpam-3941	101	27	of	of	ADP
ejpam-3941	101	28	ai	ai	NOUN
ejpam-3941	101	29	,	,	PUNCT
ejpam-3941	101	30	j	j	PROPN
ejpam-3941	101	31	in	in	ADP
ejpam-3941	101	32	all	all	DET
ejpam-3941	101	33	the	the	DET
ejpam-3941	101	34	remaining	remain	VERB
ejpam-3941	101	35	xi	xi	NOUN
ejpam-3941	101	36	’s	’s	NOUN
ejpam-3941	101	37	.	.	PUNCT
ejpam-3941	102	1	so	so	ADV
ejpam-3941	102	2	,	,	PUNCT
ejpam-3941	102	3	if	if	SCONJ
ejpam-3941	102	4	u	u	PROPN
ejpam-3941	102	5	∈	∈	PROPN
ejpam-3941	102	6	b(a	b(a	NOUN
ejpam-3941	102	7	)	)	PUNCT
ejpam-3941	102	8	,	,	PUNCT
ejpam-3941	102	9	then	then	ADV
ejpam-3941	102	10	u	u	NOUN
ejpam-3941	102	11	is	be	AUX
ejpam-3941	102	12	of	of	ADP
ejpam-3941	102	13	the	the	DET
ejpam-3941	102	14	form	form	NOUN
ejpam-3941	102	15	u	u	NOUN
ejpam-3941	102	16	=	=	PUNCT
ejpam-3941	102	17	{	{	PUNCT
ejpam-3941	102	18	a	a	PRON
ejpam-3941	102	19	}	}	PUNCT
ejpam-3941	102	20	∪	∪	X
ejpam-3941	102	21	(	(	PUNCT
ejpam-3941	102	22	∪i∈(n\e)x	∪i∈(n\e)x	PROPN
ejpam-3941	102	23	′	′	NUM
ejpam-3941	103	1	i	i	NOUN
ejpam-3941	103	2	)	)	PUNCT
ejpam-3941	103	3	where	where	SCONJ
ejpam-3941	103	4	e	e	NOUN
ejpam-3941	103	5	is	be	AUX
ejpam-3941	103	6	a	a	DET
ejpam-3941	103	7	finite	finite	NOUN
ejpam-3941	103	8	subset	subset	NOUN
ejpam-3941	103	9	of	of	ADP
ejpam-3941	103	10	n	n	PROPN
ejpam-3941	103	11	and	and	CCONJ
ejpam-3941	103	12	x	x	PUNCT
ejpam-3941	103	13	′i	′i	NOUN
ejpam-3941	103	14	=	=	SYM
ejpam-3941	103	15	xi	xi	X
ejpam-3941	103	16	\	\	PROPN
ejpam-3941	104	1	ei	ei	X
ejpam-3941	104	2	where	where	SCONJ
ejpam-3941	104	3	ei	ei	NOUN
ejpam-3941	104	4	is	be	AUX
ejpam-3941	104	5	a	a	DET
ejpam-3941	104	6	finite	finite	NOUN
ejpam-3941	104	7	subset	subset	NOUN
ejpam-3941	104	8	of	of	ADP
ejpam-3941	104	9	{	{	PUNCT
ejpam-3941	104	10	ai	ai	PROPN
ejpam-3941	104	11	,	,	PUNCT
ejpam-3941	104	12	j	j	PROPN
ejpam-3941	104	13	:	:	PUNCT
ejpam-3941	104	14	j	j	PROPN
ejpam-3941	104	15	∈	∈	PROPN
ejpam-3941	104	16	n	n	CCONJ
ejpam-3941	104	17	}	}	PUNCT
ejpam-3941	104	18	.	.	PUNCT
ejpam-3941	105	1	it	it	PRON
ejpam-3941	105	2	is	be	AUX
ejpam-3941	105	3	well	well	ADV
ejpam-3941	105	4	-	-	PUNCT
ejpam-3941	105	5	known	know	VERB
ejpam-3941	105	6	that	that	SCONJ
ejpam-3941	105	7	x	x	PRON
ejpam-3941	105	8	is	be	AUX
ejpam-3941	105	9	zero	zero	NUM
ejpam-3941	105	10	-	-	PUNCT
ejpam-3941	105	11	dimensional	dimensional	ADJ
ejpam-3941	105	12	normal	normal	ADJ
ejpam-3941	105	13	space	space	NOUN
ejpam-3941	105	14	which	which	PRON
ejpam-3941	105	15	is	be	AUX
ejpam-3941	105	16	not	not	PART
ejpam-3941	105	17	fréchet	fréchet	VERB
ejpam-3941	105	18	[	[	X
ejpam-3941	105	19	4	4	NUM
ejpam-3941	105	20	,	,	PUNCT
ejpam-3941	105	21	1.6.19	1.6.19	NUM
ejpam-3941	105	22	]	]	PUNCT
ejpam-3941	105	23	.	.	PUNCT
ejpam-3941	106	1	let	let	VERB
ejpam-3941	106	2	z	z	NOUN
ejpam-3941	106	3	=	=	PUNCT
ejpam-3941	106	4	x	x	SYM
ejpam-3941	106	5	\	\	PROPN
ejpam-3941	106	6	{	{	PUNCT
ejpam-3941	106	7	ai	ai	INTJ
ejpam-3941	106	8	:	:	PUNCT
ejpam-3941	106	9	i	i	PRON
ejpam-3941	106	10	∈	∈	PROPN
ejpam-3941	106	11	n	n	INTJ
ejpam-3941	106	12	}	}	PUNCT
ejpam-3941	106	13	.	.	PUNCT
ejpam-3941	107	1	then	then	ADV
ejpam-3941	107	2	z	z	X
ejpam-3941	107	3	as	as	ADP
ejpam-3941	107	4	a	a	DET
ejpam-3941	107	5	subspace	subspace	NOUN
ejpam-3941	107	6	of	of	ADP
ejpam-3941	107	7	x	x	PUNCT
ejpam-3941	107	8	is	be	AUX
ejpam-3941	107	9	not	not	PART
ejpam-3941	107	10	sequential	sequential	ADJ
ejpam-3941	107	11	[	[	PUNCT
ejpam-3941	107	12	4	4	NUM
ejpam-3941	107	13	,	,	PUNCT
ejpam-3941	107	14	1.6.20	1.6.20	NUM
ejpam-3941	107	15	]	]	PUNCT
ejpam-3941	107	16	.	.	PUNCT
ejpam-3941	108	1	but	but	CCONJ
ejpam-3941	108	2	since	since	SCONJ
ejpam-3941	108	3	z	z	PROPN
ejpam-3941	108	4	is	be	AUX
ejpam-3941	108	5	countable	countable	ADJ
ejpam-3941	108	6	,	,	PUNCT
ejpam-3941	108	7	then	then	ADV
ejpam-3941	108	8	it	it	PRON
ejpam-3941	108	9	is	be	AUX
ejpam-3941	108	10	of	of	ADP
ejpam-3941	108	11	countable	countable	ADJ
ejpam-3941	108	12	tightness	tightness	NOUN
ejpam-3941	108	13	.	.	PUNCT
ejpam-3941	109	1	now	now	ADV
ejpam-3941	109	2	,	,	PUNCT
ejpam-3941	109	3	a	a	DET
ejpam-3941	109	4	subspace	subspace	NOUN
ejpam-3941	109	5	c	c	PROPN
ejpam-3941	109	6	of	of	ADP
ejpam-3941	109	7	z	z	PROPN
ejpam-3941	109	8	is	be	AUX
ejpam-3941	109	9	compact	compact	ADJ
ejpam-3941	109	10	if	if	SCONJ
ejpam-3941	110	1	and	and	CCONJ
ejpam-3941	110	2	only	only	ADV
ejpam-3941	110	3	if	if	SCONJ
ejpam-3941	110	4	c	c	PROPN
ejpam-3941	110	5	is	be	AUX
ejpam-3941	110	6	finite	finite	ADJ
ejpam-3941	110	7	.	.	PUNCT
ejpam-3941	111	1	since	since	SCONJ
ejpam-3941	111	2	z	z	PROPN
ejpam-3941	111	3	is	be	AUX
ejpam-3941	111	4	also	also	ADV
ejpam-3941	111	5	t1	t1	NOUN
ejpam-3941	111	6	,	,	PUNCT
ejpam-3941	111	7	then	then	ADV
ejpam-3941	111	8	by	by	ADP
ejpam-3941	111	9	[	[	X
ejpam-3941	111	10	8	8	NUM
ejpam-3941	111	11	,	,	PUNCT
ejpam-3941	111	12	theorem	theorem	VERB
ejpam-3941	111	13	2.7	2.7	NUM
ejpam-3941	111	14	]	]	PUNCT
ejpam-3941	111	15	,	,	PUNCT
ejpam-3941	111	16	y	y	PROPN
ejpam-3941	111	17	=	=	SYM
ejpam-3941	111	18	z	z	PROPN
ejpam-3941	111	19	with	with	ADP
ejpam-3941	111	20	the	the	DET
ejpam-3941	111	21	discrete	discrete	ADJ
ejpam-3941	111	22	topology	topology	NOUN
ejpam-3941	111	23	and	and	CCONJ
ejpam-3941	111	24	the	the	DET
ejpam-3941	111	25	identity	identity	NOUN
ejpam-3941	111	26	function	function	NOUN
ejpam-3941	111	27	witness	witness	VERB
ejpam-3941	111	28	the	the	DET
ejpam-3941	111	29	c2	c2	PROPN
ejpam-3941	111	30	-	-	PUNCT
ejpam-3941	111	31	paracompactness	paracompactness	PROPN
ejpam-3941	111	32	of	of	ADP
ejpam-3941	111	33	z	z	PROPN
ejpam-3941	111	34	and	and	CCONJ
ejpam-3941	111	35	since	since	SCONJ
ejpam-3941	111	36	z	z	NOUN
ejpam-3941	111	37	is	be	AUX
ejpam-3941	111	38	not	not	PART
ejpam-3941	111	39	discrete	discrete	ADJ
ejpam-3941	111	40	,	,	PUNCT
ejpam-3941	111	41	then	then	ADV
ejpam-3941	111	42	the	the	DET
ejpam-3941	111	43	identity	identity	NOUN
ejpam-3941	111	44	function	function	NOUN
ejpam-3941	111	45	is	be	AUX
ejpam-3941	111	46	not	not	PART
ejpam-3941	111	47	continuous	continuous	ADJ
ejpam-3941	111	48	.	.	PUNCT
ejpam-3941	112	1	let	let	VERB
ejpam-3941	112	2	x	x	PRON
ejpam-3941	112	3	be	be	AUX
ejpam-3941	112	4	any	any	DET
ejpam-3941	112	5	set	set	NOUN
ejpam-3941	112	6	containing	contain	VERB
ejpam-3941	112	7	more	more	ADJ
ejpam-3941	112	8	than	than	ADP
ejpam-3941	112	9	one	one	NUM
ejpam-3941	112	10	element	element	NOUN
ejpam-3941	112	11	.	.	PUNCT
ejpam-3941	113	1	fix	fix	VERB
ejpam-3941	113	2	an	an	DET
ejpam-3941	113	3	element	element	NOUN
ejpam-3941	113	4	p	p	PROPN
ejpam-3941	113	5	∈	∈	PROPN
ejpam-3941	113	6	x.	x.	NOUN
ejpam-3941	114	1	the	the	DET
ejpam-3941	114	2	topology	topology	NOUN
ejpam-3941	114	3	τ=	τ=	INTJ
ejpam-3941	114	4	{	{	PUNCT
ejpam-3941	114	5	∅}∪	∅}∪	X
ejpam-3941	114	6	{	{	PUNCT
ejpam-3941	114	7	w	w	NOUN
ejpam-3941	114	8	⊆	⊆	NUM
ejpam-3941	114	9	x	x	SYM
ejpam-3941	114	10	:	:	PUNCT
ejpam-3941	114	11	p	p	NOUN
ejpam-3941	114	12	∈w	∈w	NOUN
ejpam-3941	114	13	}	}	PUNCT
ejpam-3941	114	14	is	be	AUX
ejpam-3941	114	15	called	call	VERB
ejpam-3941	114	16	the	the	DET
ejpam-3941	114	17	particular	particular	ADJ
ejpam-3941	114	18	point	point	NOUN
ejpam-3941	114	19	topology	topology	NOUN
ejpam-3941	114	20	on	on	ADP
ejpam-3941	114	21	x	x	X
ejpam-3941	114	22	,	,	PUNCT
ejpam-3941	114	23	see	see	VERB
ejpam-3941	114	24	[	[	X
ejpam-3941	114	25	9	9	NUM
ejpam-3941	114	26	]	]	PUNCT
ejpam-3941	114	27	.	.	PUNCT
ejpam-3941	115	1	theorem	theorem	NOUN
ejpam-3941	115	2	5	5	NUM
ejpam-3941	115	3	.	.	PUNCT
ejpam-3941	116	1	let	let	AUX
ejpam-3941	116	2	(	(	PUNCT
ejpam-3941	116	3	x	x	X
ejpam-3941	116	4	,	,	PUNCT
ejpam-3941	116	5	τ	τ	PROPN
ejpam-3941	116	6	)	)	PUNCT
ejpam-3941	116	7	be	be	AUX
ejpam-3941	116	8	a	a	DET
ejpam-3941	116	9	fréchet	fréchet	NOUN
ejpam-3941	116	10	σ	σ	NOUN
ejpam-3941	116	11	-	-	ADJ
ejpam-3941	116	12	compact	compact	ADJ
ejpam-3941	116	13	non	non	ADJ
ejpam-3941	116	14	-	-	ADJ
ejpam-3941	116	15	compact	compact	ADJ
ejpam-3941	116	16	space	space	NOUN
ejpam-3941	116	17	such	such	ADJ
ejpam-3941	116	18	that	that	SCONJ
ejpam-3941	116	19	τ	τ	PROPN
ejpam-3941	116	20	is	be	AUX
ejpam-3941	116	21	coarser	coarse	ADJ
ejpam-3941	116	22	than	than	ADP
ejpam-3941	116	23	a	a	DET
ejpam-3941	116	24	particular	particular	ADJ
ejpam-3941	116	25	point	point	NOUN
ejpam-3941	116	26	topology	topology	NOUN
ejpam-3941	116	27	τ	τ	X
ejpam-3941	116	28	p	p	X
ejpam-3941	116	29	on	on	ADP
ejpam-3941	116	30	x	x	SYM
ejpam-3941	116	31	,	,	PUNCT
ejpam-3941	116	32	where	where	SCONJ
ejpam-3941	116	33	p	p	PROPN
ejpam-3941	116	34	∈	∈	PROPN
ejpam-3941	116	35	x	x	NOUN
ejpam-3941	116	36	,	,	PUNCT
ejpam-3941	116	37	then	then	ADV
ejpam-3941	116	38	(	(	PUNCT
ejpam-3941	116	39	x	x	X
ejpam-3941	116	40	,	,	PUNCT
ejpam-3941	116	41	τ	τ	PROPN
ejpam-3941	116	42	)	)	PUNCT
ejpam-3941	116	43	can	can	AUX
ejpam-3941	116	44	not	not	PART
ejpam-3941	116	45	be	be	AUX
ejpam-3941	116	46	c	c	NOUN
ejpam-3941	116	47	-	-	PUNCT
ejpam-3941	116	48	paracompact	paracompact	ADJ
ejpam-3941	116	49	.	.	PUNCT
ejpam-3941	117	1	proof	proof	NOUN
ejpam-3941	117	2	.	.	PUNCT
ejpam-3941	118	1	suppose	suppose	VERB
ejpam-3941	118	2	that	that	SCONJ
ejpam-3941	118	3	(	(	PUNCT
ejpam-3941	118	4	x	x	X
ejpam-3941	118	5	,	,	PUNCT
ejpam-3941	118	6	τ	τ	PROPN
ejpam-3941	118	7	)	)	PUNCT
ejpam-3941	118	8	is	be	AUX
ejpam-3941	118	9	c	c	NOUN
ejpam-3941	118	10	-	-	PUNCT
ejpam-3941	118	11	paracompact	paracompact	ADJ
ejpam-3941	118	12	.	.	PUNCT
ejpam-3941	119	1	pick	pick	VERB
ejpam-3941	119	2	a	a	DET
ejpam-3941	119	3	paracompact	paracompact	ADJ
ejpam-3941	119	4	space	space	NOUN
ejpam-3941	119	5	y	y	PROPN
ejpam-3941	119	6	and	and	CCONJ
ejpam-3941	119	7	a	a	DET
ejpam-3941	119	8	bijective	bijective	ADJ
ejpam-3941	119	9	function	function	NOUN
ejpam-3941	120	1	f	f	NOUN
ejpam-3941	120	2	:	:	PUNCT
ejpam-3941	120	3	x	x	PUNCT
ejpam-3941	120	4	−→	−→	NOUN
ejpam-3941	120	5	y	y	PROPN
ejpam-3941	120	6	such	such	ADJ
ejpam-3941	120	7	that	that	SCONJ
ejpam-3941	120	8	the	the	DET
ejpam-3941	120	9	restriction	restriction	NOUN
ejpam-3941	120	10	f	f	PROPN
ejpam-3941	120	11	�	�	PROPN
ejpam-3941	120	12	a	a	NOUN
ejpam-3941	120	13	:	:	PUNCT
ejpam-3941	120	14	a	a	DET
ejpam-3941	120	15	−→	−→	NOUN
ejpam-3941	120	16	f(a	f(a	NOUN
ejpam-3941	120	17	)	)	PUNCT
ejpam-3941	120	18	is	be	AUX
ejpam-3941	120	19	a	a	DET
ejpam-3941	120	20	homeomorphism	homeomorphism	NOUN
ejpam-3941	120	21	for	for	ADP
ejpam-3941	120	22	each	each	DET
ejpam-3941	120	23	compact	compact	ADJ
ejpam-3941	120	24	subspace	subspace	NOUN
ejpam-3941	120	25	a	a	DET
ejpam-3941	120	26	⊆	⊆	NUM
ejpam-3941	120	27	x.	x.	NOUN
ejpam-3941	120	28	since	since	SCONJ
ejpam-3941	120	29	x	x	PROPN
ejpam-3941	120	30	is	be	AUX
ejpam-3941	120	31	fréchet	fréchet	ADJ
ejpam-3941	120	32	,	,	PUNCT
ejpam-3941	120	33	then	then	ADV
ejpam-3941	120	34	f	f	PROPN
ejpam-3941	120	35	is	be	AUX
ejpam-3941	120	36	continuous	continuous	ADJ
ejpam-3941	120	37	,	,	PUNCT
ejpam-3941	120	38	see	see	ADJ
ejpam-3941	120	39	theorem	theorem	NOUN
ejpam-3941	120	40	1	1	NUM
ejpam-3941	120	41	.	.	PUNCT
ejpam-3941	121	1	so	so	ADV
ejpam-3941	121	2	,	,	PUNCT
ejpam-3941	121	3	for	for	ADP
ejpam-3941	121	4	any	any	DET
ejpam-3941	121	5	non	non	ADJ
ejpam-3941	121	6	-	-	ADJ
ejpam-3941	121	7	empty	empty	ADJ
ejpam-3941	121	8	open	open	ADJ
ejpam-3941	121	9	subset	subset	NOUN
ejpam-3941	121	10	w	w	NOUN
ejpam-3941	121	11	of	of	ADP
ejpam-3941	121	12	y	y	PRON
ejpam-3941	121	13	we	we	PRON
ejpam-3941	121	14	have	have	VERB
ejpam-3941	121	15	that	that	PRON
ejpam-3941	121	16	f−1(w	f−1(w	PROPN
ejpam-3941	121	17	)	)	PUNCT
ejpam-3941	121	18	is	be	AUX
ejpam-3941	121	19	open	open	ADJ
ejpam-3941	121	20	in	in	ADP
ejpam-3941	121	21	x	x	NOUN
ejpam-3941	121	22	,	,	PUNCT
ejpam-3941	121	23	hence	hence	ADV
ejpam-3941	121	24	p	p	NOUN
ejpam-3941	121	25	∈	∈	PROPN
ejpam-3941	121	26	f−1(w	f−1(w	PROPN
ejpam-3941	121	27	)	)	PUNCT
ejpam-3941	121	28	which	which	PRON
ejpam-3941	121	29	gives	give	VERB
ejpam-3941	121	30	that	that	DET
ejpam-3941	121	31	f(p	f(p	NOUN
ejpam-3941	121	32	)	)	PUNCT
ejpam-3941	121	33	∈w	∈w	NOUN
ejpam-3941	121	34	.	.	PUNCT
ejpam-3941	122	1	now	now	ADV
ejpam-3941	122	2	,	,	PUNCT
ejpam-3941	122	3	write	write	VERB
ejpam-3941	122	4	x	x	PUNCT
ejpam-3941	122	5	=	=	PUNCT
ejpam-3941	122	6	⋃	⋃	ADP
ejpam-3941	122	7	n∈nxn	n∈nxn	NOUN
ejpam-3941	122	8	where	where	SCONJ
ejpam-3941	122	9	xn	xn	PROPN
ejpam-3941	122	10	is	be	AUX
ejpam-3941	122	11	compact	compact	ADJ
ejpam-3941	122	12	for	for	ADP
ejpam-3941	122	13	each	each	DET
ejpam-3941	122	14	n	n	PRON
ejpam-3941	122	15	∈	∈	PROPN
ejpam-3941	122	16	n	n	NOUN
ejpam-3941	122	17	and	and	CCONJ
ejpam-3941	122	18	xn	xn	PROPN
ejpam-3941	123	1	⊂	⊂	PROPN
ejpam-3941	123	2	xn+1	xn+1	PROPN
ejpam-3941	123	3	for	for	ADP
ejpam-3941	123	4	each	each	DET
ejpam-3941	123	5	n	n	PRON
ejpam-3941	123	6	∈	∈	PROPN
ejpam-3941	123	7	n	n	CCONJ
ejpam-3941	123	8	,	,	PUNCT
ejpam-3941	123	9	i.e.	i.e.	X
ejpam-3941	123	10	,	,	PUNCT
ejpam-3941	123	11	the	the	DET
ejpam-3941	123	12	xn	xn	PROPN
ejpam-3941	123	13	’s	’s	PART
ejpam-3941	123	14	are	be	AUX
ejpam-3941	123	15	increasing	increase	VERB
ejpam-3941	123	16	.	.	PUNCT
ejpam-3941	124	1	since	since	SCONJ
ejpam-3941	124	2	x	x	PRON
ejpam-3941	124	3	is	be	AUX
ejpam-3941	124	4	not	not	PART
ejpam-3941	124	5	compact	compact	ADJ
ejpam-3941	124	6	,	,	PUNCT
ejpam-3941	124	7	there	there	PRON
ejpam-3941	124	8	exists	exist	VERB
ejpam-3941	124	9	an	an	DET
ejpam-3941	124	10	open	open	ADJ
ejpam-3941	124	11	cover	cover	NOUN
ejpam-3941	124	12	u	u	NOUN
ejpam-3941	124	13	=	=	PUNCT
ejpam-3941	124	14	{	{	PUNCT
ejpam-3941	124	15	uα	uα	X
ejpam-3941	124	16	:	:	PUNCT
ejpam-3941	124	17	α	α	PROPN
ejpam-3941	124	18	∈	∈	PROPN
ejpam-3941	124	19	λ	λ	PROPN
ejpam-3941	124	20	}	}	PUNCT
ejpam-3941	124	21	such	such	ADJ
ejpam-3941	124	22	that	that	SCONJ
ejpam-3941	124	23	for	for	ADP
ejpam-3941	124	24	any	any	DET
ejpam-3941	124	25	finite	finite	NOUN
ejpam-3941	124	26	subset	subset	VERB
ejpam-3941	124	27	f	f	PROPN
ejpam-3941	124	28	of	of	ADP
ejpam-3941	124	29	λ	λ	PROPN
ejpam-3941	124	30	there	there	PRON
ejpam-3941	124	31	exists	exist	VERB
ejpam-3941	124	32	an	an	DET
ejpam-3941	124	33	element	element	NOUN
ejpam-3941	124	34	x	x	SYM
ejpam-3941	124	35	∈	∈	PROPN
ejpam-3941	124	36	x	x	X
ejpam-3941	124	37	such	such	ADJ
ejpam-3941	124	38	that	that	SCONJ
ejpam-3941	124	39	x	x	SYM
ejpam-3941	124	40	6∈	6∈	PROPN
ejpam-3941	124	41	⋃	⋃	NOUN
ejpam-3941	124	42	α∈f	α∈f	NOUN
ejpam-3941	124	43	uα	uα	PROPN
ejpam-3941	124	44	.	.	PUNCT
ejpam-3941	125	1	now	now	ADV
ejpam-3941	125	2	,	,	PUNCT
ejpam-3941	125	3	u	u	NOUN
ejpam-3941	125	4	is	be	AUX
ejpam-3941	125	5	an	an	DET
ejpam-3941	125	6	open	open	ADJ
ejpam-3941	125	7	cover	cover	NOUN
ejpam-3941	125	8	for	for	ADP
ejpam-3941	125	9	x1	x1	PROPN
ejpam-3941	125	10	and	and	CCONJ
ejpam-3941	125	11	x1	x1	PROPN
ejpam-3941	125	12	is	be	AUX
ejpam-3941	125	13	compact	compact	ADJ
ejpam-3941	125	14	.	.	PUNCT
ejpam-3941	126	1	let	let	VERB
ejpam-3941	126	2	f1	f1	PROPN
ejpam-3941	126	3	be	be	AUX
ejpam-3941	126	4	a	a	DET
ejpam-3941	126	5	finite	finite	NOUN
ejpam-3941	126	6	subset	subset	NOUN
ejpam-3941	126	7	of	of	ADP
ejpam-3941	126	8	λ	λ	PROPN
ejpam-3941	126	9	such	such	ADJ
ejpam-3941	126	10	that	that	SCONJ
ejpam-3941	126	11	x1	x1	PROPN
ejpam-3941	126	12	⊆	⊆	NUM
ejpam-3941	126	13	⋃	⋃	ADP
ejpam-3941	126	14	α∈f1	α∈f1	NOUN
ejpam-3941	126	15	uα	uα	PROPN
ejpam-3941	126	16	=	=	PUNCT
ejpam-3941	126	17	v1	v1	PROPN
ejpam-3941	126	18	.	.	PUNCT
ejpam-3941	126	19	pick	pick	VERB
ejpam-3941	126	20	a2	a2	PROPN
ejpam-3941	126	21	∈	∈	PROPN
ejpam-3941	126	22	x	x	SYM
ejpam-3941	126	23	\	\	NOUN
ejpam-3941	126	24	v1	v1	NOUN
ejpam-3941	126	25	and	and	CCONJ
ejpam-3941	126	26	let	let	VERB
ejpam-3941	126	27	i2	i2	PROPN
ejpam-3941	126	28	∈	∈	PROPN
ejpam-3941	126	29	n	n	PRON
ejpam-3941	126	30	be	be	AUX
ejpam-3941	126	31	the	the	DET
ejpam-3941	126	32	minimal	minimal	ADJ
ejpam-3941	126	33	so	so	SCONJ
ejpam-3941	126	34	that	that	SCONJ
ejpam-3941	126	35	a2	a2	PROPN
ejpam-3941	126	36	∈	∈	PROPN
ejpam-3941	126	37	xi2	xi2	PROPN
ejpam-3941	126	38	,	,	PUNCT
ejpam-3941	126	39	i.e.	i.e.	X
ejpam-3941	126	40	,	,	PUNCT
ejpam-3941	126	41	if	if	SCONJ
ejpam-3941	126	42	j	j	PROPN
ejpam-3941	126	43	<	<	X
ejpam-3941	126	44	i2	i2	PROPN
ejpam-3941	126	45	,	,	PUNCT
ejpam-3941	126	46	then	then	ADV
ejpam-3941	126	47	a2	a2	PROPN
ejpam-3941	126	48	6∈	6∈	PROPN
ejpam-3941	126	49	xj	xj	PROPN
ejpam-3941	126	50	.	.	PUNCT
ejpam-3941	127	1	now	now	ADV
ejpam-3941	127	2	,	,	PUNCT
ejpam-3941	127	3	u	u	NOUN
ejpam-3941	127	4	is	be	AUX
ejpam-3941	127	5	an	an	DET
ejpam-3941	127	6	open	open	ADJ
ejpam-3941	127	7	cover	cover	NOUN
ejpam-3941	127	8	for	for	ADP
ejpam-3941	127	9	h.	h.	PROPN
ejpam-3941	127	10	alzumi	alzumi	PROPN
ejpam-3941	127	11	,	,	PUNCT
ejpam-3941	127	12	l.	l.	PROPN
ejpam-3941	127	13	kalantan	kalantan	PROPN
ejpam-3941	127	14	and	and	CCONJ
ejpam-3941	127	15	m.	m.	PROPN
ejpam-3941	127	16	mohammed	mohammed	PROPN
ejpam-3941	127	17	saeed	saeed	PROPN
ejpam-3941	127	18	/	/	SYM
ejpam-3941	127	19	eur	eur	PROPN
ejpam-3941	127	20	.	.	PUNCT
ejpam-3941	128	1	j.	j.	PROPN
ejpam-3941	128	2	pure	pure	PROPN
ejpam-3941	128	3	appl	appl	PROPN
ejpam-3941	128	4	.	.	PROPN
ejpam-3941	128	5	math	math	PROPN
ejpam-3941	128	6	,	,	PUNCT
ejpam-3941	128	7	14	14	NUM
ejpam-3941	128	8	(	(	PUNCT
ejpam-3941	128	9	2	2	NUM
ejpam-3941	128	10	)	)	PUNCT
ejpam-3941	128	11	(	(	PUNCT
ejpam-3941	128	12	2021	2021	NUM
ejpam-3941	128	13	)	)	PUNCT
ejpam-3941	128	14	,	,	PUNCT
ejpam-3941	128	15	351	351	NUM
ejpam-3941	128	16	-	-	SYM
ejpam-3941	128	17	357	357	NUM
ejpam-3941	128	18	355	355	NUM
ejpam-3941	128	19	xi2	xi2	NOUN
ejpam-3941	128	20	and	and	CCONJ
ejpam-3941	128	21	xi2	xi2	PROPN
ejpam-3941	128	22	is	be	AUX
ejpam-3941	128	23	compact	compact	ADJ
ejpam-3941	128	24	.	.	PUNCT
ejpam-3941	129	1	let	let	VERB
ejpam-3941	129	2	f2	f2	PROPN
ejpam-3941	129	3	be	be	AUX
ejpam-3941	129	4	a	a	DET
ejpam-3941	129	5	finite	finite	NOUN
ejpam-3941	129	6	subset	subset	NOUN
ejpam-3941	129	7	of	of	ADP
ejpam-3941	129	8	λ	λ	PROPN
ejpam-3941	129	9	such	such	ADJ
ejpam-3941	129	10	that	that	SCONJ
ejpam-3941	129	11	xi2	xi2	PROPN
ejpam-3941	130	1	⊆	⊆	NUM
ejpam-3941	130	2	⋃	⋃	ADP
ejpam-3941	130	3	α∈f2	α∈f2	NOUN
ejpam-3941	130	4	uα	uα	ADP
ejpam-3941	130	5	=	=	SYM
ejpam-3941	130	6	v2	v2	PROPN
ejpam-3941	130	7	.	.	PUNCT
ejpam-3941	131	1	if	if	SCONJ
ejpam-3941	131	2	m	m	VERB
ejpam-3941	131	3	∈	∈	VERB
ejpam-3941	132	1	n	n	PRON
ejpam-3941	132	2	so	so	ADV
ejpam-3941	132	3	that	that	PRON
ejpam-3941	132	4	am	be	AUX
ejpam-3941	132	5	∈	∈	PROPN
ejpam-3941	132	6	x	x	NOUN
ejpam-3941	132	7	,	,	PUNCT
ejpam-3941	132	8	i	i	PRON
ejpam-3941	132	9	m	m	PROPN
ejpam-3941	132	10	∈	∈	PROPN
ejpam-3941	132	11	n	n	CCONJ
ejpam-3941	132	12	,	,	PUNCT
ejpam-3941	132	13	fm	fm	PROPN
ejpam-3941	132	14	finite	finite	PROPN
ejpam-3941	132	15	subset	subset	PROPN
ejpam-3941	132	16	of	of	ADP
ejpam-3941	132	17	λ	λ	PROPN
ejpam-3941	132	18	,	,	PUNCT
ejpam-3941	132	19	and	and	CCONJ
ejpam-3941	132	20	vm	vm	PROPN
ejpam-3941	132	21	are	be	AUX
ejpam-3941	132	22	chosen	choose	VERB
ejpam-3941	132	23	,	,	PUNCT
ejpam-3941	132	24	then	then	ADV
ejpam-3941	132	25	pick	pick	VERB
ejpam-3941	132	26	am+1	am+1	PROPN
ejpam-3941	132	27	∈	∈	PROPN
ejpam-3941	132	28	x	x	PUNCT
ejpam-3941	132	29	\	\	PROPN
ejpam-3941	132	30	vm	vm	PROPN
ejpam-3941	132	31	and	and	CCONJ
ejpam-3941	132	32	let	let	VERB
ejpam-3941	132	33	im+1	im+1	PRON
ejpam-3941	132	34	∈	∈	PROPN
ejpam-3941	132	35	n	n	AUX
ejpam-3941	132	36	be	be	AUX
ejpam-3941	132	37	the	the	DET
ejpam-3941	132	38	minimal	minimal	ADJ
ejpam-3941	132	39	so	so	SCONJ
ejpam-3941	132	40	that	that	SCONJ
ejpam-3941	132	41	am+1	am+1	PROPN
ejpam-3941	132	42	∈	∈	PROPN
ejpam-3941	132	43	xim+1	xim+1	X
ejpam-3941	132	44	.	.	PUNCT
ejpam-3941	133	1	continue	continue	VERB
ejpam-3941	133	2	,	,	PUNCT
ejpam-3941	133	3	u	u	NOUN
ejpam-3941	133	4	is	be	AUX
ejpam-3941	133	5	an	an	DET
ejpam-3941	133	6	open	open	ADJ
ejpam-3941	133	7	cover	cover	NOUN
ejpam-3941	133	8	for	for	ADP
ejpam-3941	133	9	xim+1	xim+1	NOUN
ejpam-3941	133	10	and	and	CCONJ
ejpam-3941	133	11	xim+1	xim+1	PROPN
ejpam-3941	133	12	is	be	AUX
ejpam-3941	133	13	compact	compact	ADJ
ejpam-3941	133	14	.	.	PUNCT
ejpam-3941	134	1	let	let	VERB
ejpam-3941	134	2	fm+1	fm+1	NOUN
ejpam-3941	134	3	be	be	AUX
ejpam-3941	134	4	a	a	DET
ejpam-3941	134	5	finite	finite	NOUN
ejpam-3941	134	6	subset	subset	NOUN
ejpam-3941	134	7	of	of	ADP
ejpam-3941	134	8	λ	λ	PROPN
ejpam-3941	134	9	such	such	ADJ
ejpam-3941	134	10	that	that	SCONJ
ejpam-3941	134	11	xim+1	xim+1	ADP
ejpam-3941	134	12	⊆	⊆	NUM
ejpam-3941	134	13	⋃	⋃	PROPN
ejpam-3941	134	14	α∈fm+1	α∈fm+1	NOUN
ejpam-3941	134	15	uα	uα	PROPN
ejpam-3941	134	16	=	=	SYM
ejpam-3941	134	17	vm+1	vm+1	PROPN
ejpam-3941	134	18	.	.	PUNCT
ejpam-3941	135	1	so	so	ADV
ejpam-3941	135	2	,	,	PUNCT
ejpam-3941	135	3	we	we	PRON
ejpam-3941	135	4	have	have	AUX
ejpam-3941	135	5	constructed	construct	VERB
ejpam-3941	135	6	two	two	NUM
ejpam-3941	135	7	countably	countably	ADV
ejpam-3941	135	8	infinite	infinite	ADJ
ejpam-3941	135	9	families	family	NOUN
ejpam-3941	135	10	of	of	ADP
ejpam-3941	135	11	xim	xim	PROPN
ejpam-3941	135	12	’s	’s	PROPN
ejpam-3941	135	13	,	,	PUNCT
ejpam-3941	135	14	vm	vm	PROPN
ejpam-3941	135	15	’s	’s	NOUN
ejpam-3941	135	16	such	such	ADJ
ejpam-3941	135	17	that	that	SCONJ
ejpam-3941	135	18	xim	xim	PROPN
ejpam-3941	135	19	⊂	⊂	PROPN
ejpam-3941	135	20	xim+1	xim+1	PROPN
ejpam-3941	135	21	and	and	CCONJ
ejpam-3941	135	22	xim+1	xim+1	PRON
ejpam-3941	135	23	\	\	PROPN
ejpam-3941	135	24	vm	vm	PROPN
ejpam-3941	135	25	6=	6=	PROPN
ejpam-3941	135	26	∅	∅	NOUN
ejpam-3941	135	27	for	for	ADP
ejpam-3941	135	28	each	each	DET
ejpam-3941	135	29	m	m	NOUN
ejpam-3941	135	30	∈	∈	PROPN
ejpam-3941	135	31	n	n	PRON
ejpam-3941	135	32	as	as	ADP
ejpam-3941	135	33	am+1	am+1	PROPN
ejpam-3941	135	34	∈	∈	PROPN
ejpam-3941	135	35	xim+1	xim+1	PRON
ejpam-3941	135	36	\	\	PROPN
ejpam-3941	135	37	vm	vm	PROPN
ejpam-3941	135	38	.	.	PROPN
ejpam-3941	135	39	now	now	ADV
ejpam-3941	135	40	,	,	PUNCT
ejpam-3941	135	41	for	for	ADP
ejpam-3941	135	42	each	each	DET
ejpam-3941	135	43	m	m	PROPN
ejpam-3941	135	44	∈	∈	PROPN
ejpam-3941	135	45	n	n	CCONJ
ejpam-3941	135	46	,	,	PUNCT
ejpam-3941	135	47	f	f	PROPN
ejpam-3941	135	48	�	�	PROPN
ejpam-3941	135	49	xim	xim	PROPN
ejpam-3941	135	50	:	:	PUNCT
ejpam-3941	135	51	xim	xim	PROPN
ejpam-3941	135	52	−→	−→	PROPN
ejpam-3941	135	53	f(xim	f(xim	PROPN
ejpam-3941	135	54	)	)	PUNCT
ejpam-3941	135	55	is	be	AUX
ejpam-3941	135	56	a	a	DET
ejpam-3941	135	57	homeomorphism	homeomorphism	NOUN
ejpam-3941	135	58	,	,	PUNCT
ejpam-3941	135	59	where	where	SCONJ
ejpam-3941	135	60	i1	i1	PROPN
ejpam-3941	135	61	=	=	PROPN
ejpam-3941	135	62	1	1	X
ejpam-3941	135	63	.	.	X
ejpam-3941	136	1	we	we	PRON
ejpam-3941	136	2	have	have	VERB
ejpam-3941	136	3	vm∩xim+1	vm∩xim+1	PROPN
ejpam-3941	136	4	is	be	AUX
ejpam-3941	136	5	open	open	ADJ
ejpam-3941	136	6	in	in	ADP
ejpam-3941	136	7	xim+1	xim+1	X
ejpam-3941	136	8	for	for	ADP
ejpam-3941	136	9	each	each	DET
ejpam-3941	136	10	m	m	PROPN
ejpam-3941	136	11	∈	∈	PROPN
ejpam-3941	136	12	n	n	CCONJ
ejpam-3941	136	13	,	,	PUNCT
ejpam-3941	136	14	thus	thus	ADV
ejpam-3941	136	15	f(vm∩xim+1	f(vm∩xim+1	PROPN
ejpam-3941	136	16	)	)	PUNCT
ejpam-3941	136	17	is	be	AUX
ejpam-3941	136	18	open	open	ADJ
ejpam-3941	136	19	in	in	ADP
ejpam-3941	136	20	f(xim+1	f(xim+1	PROPN
ejpam-3941	136	21	)	)	PUNCT
ejpam-3941	136	22	for	for	ADP
ejpam-3941	136	23	each	each	DET
ejpam-3941	136	24	m	m	PROPN
ejpam-3941	136	25	∈	∈	PROPN
ejpam-3941	136	26	n.	n.	NOUN
ejpam-3941	136	27	hence	hence	ADV
ejpam-3941	136	28	,	,	PUNCT
ejpam-3941	136	29	for	for	ADP
ejpam-3941	136	30	each	each	DET
ejpam-3941	136	31	m	m	PROPN
ejpam-3941	136	32	∈	∈	NOUN
ejpam-3941	136	33	n	n	CCONJ
ejpam-3941	136	34	there	there	ADV
ejpam-3941	136	35	exists	exist	VERB
ejpam-3941	136	36	an	an	DET
ejpam-3941	136	37	open	open	ADJ
ejpam-3941	136	38	subset	subset	NOUN
ejpam-3941	136	39	wm	wm	PROPN
ejpam-3941	136	40	of	of	ADP
ejpam-3941	136	41	y	y	PROPN
ejpam-3941	136	42	such	such	ADJ
ejpam-3941	136	43	that	that	DET
ejpam-3941	136	44	wm∩	wm∩	NOUN
ejpam-3941	136	45	f(xim+1	f(xim+1	X
ejpam-3941	136	46	)	)	PUNCT
ejpam-3941	136	47	=	=	SYM
ejpam-3941	136	48	f(vm∩xim+1	f(vm∩xim+1	PROPN
ejpam-3941	136	49	)	)	PUNCT
ejpam-3941	136	50	.	.	PUNCT
ejpam-3941	137	1	observe	observe	VERB
ejpam-3941	137	2	that	that	SCONJ
ejpam-3941	137	3	f(am+1	f(am+1	PROPN
ejpam-3941	137	4	)	)	PUNCT
ejpam-3941	137	5	6∈wm	6∈wm	NUM
ejpam-3941	137	6	for	for	ADP
ejpam-3941	137	7	each	each	DET
ejpam-3941	137	8	m	m	PROPN
ejpam-3941	137	9	∈	∈	PROPN
ejpam-3941	137	10	n.	n.	NOUN
ejpam-3941	137	11	since	since	SCONJ
ejpam-3941	137	12	the	the	DET
ejpam-3941	137	13	family	family	NOUN
ejpam-3941	137	14	{	{	PUNCT
ejpam-3941	137	15	vm	vm	PROPN
ejpam-3941	137	16	:	:	PUNCT
ejpam-3941	137	17	m	m	VERB
ejpam-3941	137	18	∈	∈	PROPN
ejpam-3941	137	19	n	n	CCONJ
ejpam-3941	137	20	}	}	PUNCT
ejpam-3941	137	21	is	be	AUX
ejpam-3941	137	22	an	an	DET
ejpam-3941	137	23	open	open	ADJ
ejpam-3941	137	24	cover	cover	NOUN
ejpam-3941	137	25	for	for	ADP
ejpam-3941	137	26	x	x	X
ejpam-3941	137	27	,	,	PUNCT
ejpam-3941	137	28	then	then	ADV
ejpam-3941	137	29	we	we	PRON
ejpam-3941	137	30	have	have	VERB
ejpam-3941	137	31	that	that	SCONJ
ejpam-3941	137	32	the	the	DET
ejpam-3941	137	33	family	family	NOUN
ejpam-3941	137	34	{	{	PUNCT
ejpam-3941	137	35	wm	wm	PROPN
ejpam-3941	137	36	:	:	PUNCT
ejpam-3941	137	37	m	m	VERB
ejpam-3941	137	38	∈	∈	PROPN
ejpam-3941	137	39	n	n	CCONJ
ejpam-3941	137	40	}	}	PUNCT
ejpam-3941	137	41	is	be	AUX
ejpam-3941	137	42	an	an	DET
ejpam-3941	137	43	open	open	ADJ
ejpam-3941	137	44	cover	cover	NOUN
ejpam-3941	137	45	for	for	ADP
ejpam-3941	137	46	y	y	PROPN
ejpam-3941	137	47	consisting	consist	VERB
ejpam-3941	137	48	of	of	ADP
ejpam-3941	137	49	distinct	distinct	ADJ
ejpam-3941	137	50	proper	proper	ADJ
ejpam-3941	137	51	subsets	subset	NOUN
ejpam-3941	137	52	of	of	ADP
ejpam-3941	137	53	y	y	PROPN
ejpam-3941	137	54	.	.	PUNCT
ejpam-3941	138	1	since	since	SCONJ
ejpam-3941	138	2	each	each	DET
ejpam-3941	138	3	non	non	ADJ
ejpam-3941	138	4	-	-	ADJ
ejpam-3941	138	5	empty	empty	ADJ
ejpam-3941	138	6	open	open	ADJ
ejpam-3941	138	7	subset	subset	NOUN
ejpam-3941	138	8	of	of	ADP
ejpam-3941	138	9	y	y	PROPN
ejpam-3941	138	10	must	must	AUX
ejpam-3941	138	11	contain	contain	VERB
ejpam-3941	138	12	the	the	DET
ejpam-3941	138	13	element	element	NOUN
ejpam-3941	138	14	f(p	f(p	PROPN
ejpam-3941	138	15	)	)	PUNCT
ejpam-3941	138	16	,	,	PUNCT
ejpam-3941	138	17	then	then	ADV
ejpam-3941	138	18	the	the	DET
ejpam-3941	138	19	open	open	ADJ
ejpam-3941	138	20	cover	cover	NOUN
ejpam-3941	138	21	{	{	PUNCT
ejpam-3941	138	22	wm	wm	NOUN
ejpam-3941	138	23	:	:	PUNCT
ejpam-3941	138	24	m	m	VERB
ejpam-3941	138	25	∈	∈	PROPN
ejpam-3941	138	26	n	n	CCONJ
ejpam-3941	138	27	}	}	PUNCT
ejpam-3941	138	28	of	of	ADP
ejpam-3941	138	29	y	y	PROPN
ejpam-3941	138	30	has	have	VERB
ejpam-3941	138	31	no	no	DET
ejpam-3941	138	32	locally	locally	ADV
ejpam-3941	138	33	finite	finite	ADJ
ejpam-3941	138	34	open	open	ADJ
ejpam-3941	138	35	refinement	refinement	NOUN
ejpam-3941	138	36	,	,	PUNCT
ejpam-3941	138	37	which	which	PRON
ejpam-3941	138	38	is	be	AUX
ejpam-3941	138	39	a	a	DET
ejpam-3941	138	40	contradiction	contradiction	NOUN
ejpam-3941	138	41	.	.	PUNCT
ejpam-3941	139	1	therefore	therefore	ADV
ejpam-3941	139	2	,	,	PUNCT
ejpam-3941	139	3	(	(	PUNCT
ejpam-3941	139	4	x	x	X
ejpam-3941	139	5	,	,	PUNCT
ejpam-3941	139	6	τ	τ	PROPN
ejpam-3941	139	7	)	)	PUNCT
ejpam-3941	139	8	is	be	AUX
ejpam-3941	139	9	not	not	PART
ejpam-3941	139	10	c	c	NOUN
ejpam-3941	139	11	-	-	PUNCT
ejpam-3941	139	12	paracompact	paracompact	ADJ
ejpam-3941	139	13	.	.	PUNCT
ejpam-3941	140	1	non	non	ADJ
ejpam-3941	140	2	-	-	ADJ
ejpam-3941	140	3	compactness	compactness	ADJ
ejpam-3941	140	4	assumption	assumption	NOUN
ejpam-3941	140	5	is	be	AUX
ejpam-3941	140	6	essential	essential	ADJ
ejpam-3941	140	7	in	in	ADP
ejpam-3941	140	8	theorem	theorem	NOUN
ejpam-3941	140	9	5	5	NUM
ejpam-3941	140	10	,	,	PUNCT
ejpam-3941	140	11	for	for	ADP
ejpam-3941	140	12	example	example	NOUN
ejpam-3941	140	13	,	,	PUNCT
ejpam-3941	140	14	consider	consider	VERB
ejpam-3941	140	15	on	on	ADP
ejpam-3941	140	16	r	r	NOUN
ejpam-3941	140	17	the	the	DET
ejpam-3941	140	18	topology	topology	NOUN
ejpam-3941	140	19	τ=	τ=	PRON
ejpam-3941	140	20	{	{	PUNCT
ejpam-3941	140	21	∅,r	∅,r	PROPN
ejpam-3941	140	22	,	,	PUNCT
ejpam-3941	140	23	{	{	PUNCT
ejpam-3941	140	24	p	p	X
ejpam-3941	140	25	}	}	PUNCT
ejpam-3941	140	26	}	}	PUNCT
ejpam-3941	140	27	,	,	PUNCT
ejpam-3941	140	28	where	where	SCONJ
ejpam-3941	140	29	p	p	PROPN
ejpam-3941	140	30	∈	∈	PROPN
ejpam-3941	140	31	r.	r.	NOUN
ejpam-3941	140	32	the	the	DET
ejpam-3941	140	33	following	follow	VERB
ejpam-3941	140	34	example	example	NOUN
ejpam-3941	140	35	answers	answer	VERB
ejpam-3941	140	36	three	three	NUM
ejpam-3941	140	37	kinds	kind	NOUN
ejpam-3941	140	38	of	of	ADP
ejpam-3941	140	39	invariants	invariant	NOUN
ejpam-3941	140	40	.	.	PUNCT
ejpam-3941	141	1	we	we	PRON
ejpam-3941	141	2	used	use	VERB
ejpam-3941	141	3	two	two	NUM
ejpam-3941	141	4	well	well	ADV
ejpam-3941	141	5	-	-	PUNCT
ejpam-3941	141	6	known	know	VERB
ejpam-3941	141	7	spaces	space	NOUN
ejpam-3941	141	8	,	,	PUNCT
ejpam-3941	141	9	the	the	DET
ejpam-3941	141	10	alexandroff	alexandroff	ADJ
ejpam-3941	141	11	duplicate	duplicate	NOUN
ejpam-3941	141	12	space	space	NOUN
ejpam-3941	141	13	and	and	CCONJ
ejpam-3941	141	14	the	the	DET
ejpam-3941	141	15	closed	closed	ADJ
ejpam-3941	141	16	extension	extension	NOUN
ejpam-3941	141	17	space	space	NOUN
ejpam-3941	141	18	.	.	PUNCT
ejpam-3941	142	1	recall	recall	VERB
ejpam-3941	142	2	that	that	PRON
ejpam-3941	142	3	for	for	ADP
ejpam-3941	142	4	any	any	DET
ejpam-3941	142	5	t1	t1	NOUN
ejpam-3941	142	6	space	space	NOUN
ejpam-3941	142	7	x	x	NOUN
ejpam-3941	142	8	,	,	PUNCT
ejpam-3941	142	9	let	let	VERB
ejpam-3941	142	10	x	x	X
ejpam-3941	142	11	′	′	NUM
ejpam-3941	142	12	=	=	PUNCT
ejpam-3941	142	13	x	x	SYM
ejpam-3941	142	14	×	×	NOUN
ejpam-3941	142	15	{	{	PUNCT
ejpam-3941	142	16	1	1	NUM
ejpam-3941	142	17	}	}	PUNCT
ejpam-3941	142	18	.	.	PUNCT
ejpam-3941	143	1	let	let	VERB
ejpam-3941	143	2	a(x	a(x	NOUN
ejpam-3941	143	3	)	)	PUNCT
ejpam-3941	143	4	=	=	PUNCT
ejpam-3941	143	5	x	x	SYM
ejpam-3941	143	6	∪x	∪x	X
ejpam-3941	143	7	′.	′.	NOUN
ejpam-3941	143	8	for	for	ADP
ejpam-3941	143	9	simplicity	simplicity	NOUN
ejpam-3941	143	10	,	,	PUNCT
ejpam-3941	143	11	for	for	ADP
ejpam-3941	143	12	an	an	DET
ejpam-3941	143	13	element	element	NOUN
ejpam-3941	143	14	x	x	SYM
ejpam-3941	143	15	∈	∈	PROPN
ejpam-3941	143	16	x	x	X
ejpam-3941	143	17	,	,	PUNCT
ejpam-3941	143	18	we	we	PRON
ejpam-3941	143	19	denote	denote	VERB
ejpam-3941	143	20	the	the	DET
ejpam-3941	143	21	element	element	NOUN
ejpam-3941	143	22	〈	〈	PROPN
ejpam-3941	143	23	x	x	X
ejpam-3941	143	24	,	,	PUNCT
ejpam-3941	143	25	1	1	NUM
ejpam-3941	143	26	〉	〉	NUM
ejpam-3941	143	27	in	in	ADP
ejpam-3941	143	28	x	x	X
ejpam-3941	143	29	′	′	NUM
ejpam-3941	143	30	by	by	ADP
ejpam-3941	143	31	x′	x′	PROPN
ejpam-3941	143	32	and	and	CCONJ
ejpam-3941	143	33	for	for	ADP
ejpam-3941	143	34	a	a	DET
ejpam-3941	143	35	subset	subset	NOUN
ejpam-3941	143	36	b	b	NOUN
ejpam-3941	143	37	⊆	⊆	NUM
ejpam-3941	143	38	x	x	PRON
ejpam-3941	143	39	let	let	VERB
ejpam-3941	143	40	b′	b′	NOUN
ejpam-3941	143	41	=	=	PUNCT
ejpam-3941	143	42	{	{	PUNCT
ejpam-3941	143	43	x′	x′	PROPN
ejpam-3941	143	44	:	:	PUNCT
ejpam-3941	144	1	x	x	SYM
ejpam-3941	144	2	∈	∈	PROPN
ejpam-3941	144	3	b	b	AUX
ejpam-3941	144	4	}	}	PUNCT
ejpam-3941	144	5	=	=	SYM
ejpam-3941	144	6	b	b	SYM
ejpam-3941	144	7	×	×	NOUN
ejpam-3941	144	8	{	{	PUNCT
ejpam-3941	144	9	1	1	NUM
ejpam-3941	144	10	}	}	SYM
ejpam-3941	144	11	⊆	⊆	NUM
ejpam-3941	144	12	x	x	SYM
ejpam-3941	144	13	′.	′.	NOUN
ejpam-3941	144	14	for	for	ADP
ejpam-3941	144	15	each	each	DET
ejpam-3941	144	16	x′	x′	PROPN
ejpam-3941	144	17	∈	∈	PROPN
ejpam-3941	144	18	x	x	SYM
ejpam-3941	144	19	′	′	NOUN
ejpam-3941	144	20	,	,	PUNCT
ejpam-3941	144	21	let	let	VERB
ejpam-3941	144	22	b(x′	b(x′	NUM
ejpam-3941	144	23	)	)	PUNCT
ejpam-3941	144	24	=	=	PRON
ejpam-3941	144	25	{	{	PUNCT
ejpam-3941	144	26	{	{	PUNCT
ejpam-3941	144	27	x′	x′	NUM
ejpam-3941	144	28	}	}	PUNCT
ejpam-3941	144	29	}	}	PUNCT
ejpam-3941	144	30	.	.	PUNCT
ejpam-3941	145	1	for	for	ADP
ejpam-3941	145	2	each	each	DET
ejpam-3941	145	3	x	x	SYM
ejpam-3941	145	4	∈	∈	PROPN
ejpam-3941	145	5	x	x	NOUN
ejpam-3941	145	6	,	,	PUNCT
ejpam-3941	145	7	let	let	VERB
ejpam-3941	145	8	b(x	b(x	NOUN
ejpam-3941	145	9	)	)	PUNCT
ejpam-3941	146	1	=	=	PRON
ejpam-3941	146	2	{	{	PUNCT
ejpam-3941	146	3	u	u	NOUN
ejpam-3941	146	4	∪	∪	X
ejpam-3941	146	5	(	(	PUNCT
ejpam-3941	146	6	u	u	NOUN
ejpam-3941	146	7	′	′	NOUN
ejpam-3941	146	8	\	\	NOUN
ejpam-3941	146	9	{	{	PUNCT
ejpam-3941	146	10	x′	x′	NUM
ejpam-3941	146	11	}	}	PUNCT
ejpam-3941	146	12	)	)	PUNCT
ejpam-3941	146	13	:	:	PUNCT
ejpam-3941	146	14	u	u	NOUN
ejpam-3941	146	15	is	be	AUX
ejpam-3941	146	16	open	open	ADJ
ejpam-3941	146	17	in	in	ADP
ejpam-3941	146	18	x	x	PUNCT
ejpam-3941	146	19	with	with	ADP
ejpam-3941	146	20	x	x	PROPN
ejpam-3941	146	21	∈	∈	PROPN
ejpam-3941	146	22	u	u	NOUN
ejpam-3941	146	23	}	}	PUNCT
ejpam-3941	146	24	.	.	PUNCT
ejpam-3941	147	1	let	let	VERB
ejpam-3941	147	2	τ	τ	PROPN
ejpam-3941	147	3	denote	denote	VERB
ejpam-3941	147	4	the	the	DET
ejpam-3941	147	5	unique	unique	ADJ
ejpam-3941	147	6	topology	topology	NOUN
ejpam-3941	147	7	on	on	ADP
ejpam-3941	147	8	a(x	a(x	NOUN
ejpam-3941	147	9	)	)	PUNCT
ejpam-3941	147	10	which	which	PRON
ejpam-3941	147	11	has	have	AUX
ejpam-3941	147	12	{	{	PUNCT
ejpam-3941	147	13	b(x	b(x	NOUN
ejpam-3941	147	14	)	)	PUNCT
ejpam-3941	147	15	:	:	PUNCT
ejpam-3941	148	1	x	x	X
ejpam-3941	148	2	∈	∈	NOUN
ejpam-3941	148	3	x	x	SYM
ejpam-3941	148	4	}	}	PUNCT
ejpam-3941	148	5	∪	∪	ADJ
ejpam-3941	148	6	{	{	PUNCT
ejpam-3941	148	7	b(x′	b(x′	NUM
ejpam-3941	148	8	)	)	PUNCT
ejpam-3941	148	9	:	:	PUNCT
ejpam-3941	148	10	x′	x′	X
ejpam-3941	148	11	∈	∈	NOUN
ejpam-3941	148	12	x	x	NOUN
ejpam-3941	148	13	′	′	NOUN
ejpam-3941	148	14	}	}	PUNCT
ejpam-3941	148	15	as	as	ADP
ejpam-3941	148	16	its	its	PRON
ejpam-3941	148	17	neighborhood	neighborhood	NOUN
ejpam-3941	148	18	system	system	NOUN
ejpam-3941	148	19	.	.	PUNCT
ejpam-3941	149	1	a(x	a(x	NOUN
ejpam-3941	149	2	)	)	PUNCT
ejpam-3941	149	3	with	with	ADP
ejpam-3941	149	4	this	this	DET
ejpam-3941	149	5	topology	topology	NOUN
ejpam-3941	149	6	is	be	AUX
ejpam-3941	149	7	called	call	VERB
ejpam-3941	149	8	the	the	DET
ejpam-3941	149	9	alexandroff	alexandroff	ADJ
ejpam-3941	149	10	duplicate	duplicate	NOUN
ejpam-3941	149	11	of	of	ADP
ejpam-3941	149	12	x	x	PUNCT
ejpam-3941	150	1	[	[	X
ejpam-3941	150	2	3	3	NUM
ejpam-3941	150	3	]	]	PUNCT
ejpam-3941	150	4	.	.	PUNCT
ejpam-3941	151	1	in	in	ADP
ejpam-3941	151	2	[	[	X
ejpam-3941	151	3	8	8	NUM
ejpam-3941	151	4	]	]	PUNCT
ejpam-3941	151	5	,	,	PUNCT
ejpam-3941	151	6	it	it	PRON
ejpam-3941	151	7	was	be	AUX
ejpam-3941	151	8	shown	show	VERB
ejpam-3941	151	9	that	that	SCONJ
ejpam-3941	151	10	“	"	PUNCT
ejpam-3941	151	11	if	if	SCONJ
ejpam-3941	151	12	x	x	PRON
ejpam-3941	151	13	is	be	AUX
ejpam-3941	151	14	c2	c2	NOUN
ejpam-3941	151	15	-	-	PUNCT
ejpam-3941	151	16	paracompact	paracompact	NOUN
ejpam-3941	151	17	,	,	PUNCT
ejpam-3941	151	18	then	then	ADV
ejpam-3941	151	19	so	so	ADV
ejpam-3941	151	20	is	be	AUX
ejpam-3941	151	21	its	its	PRON
ejpam-3941	151	22	alexandroff	alexandroff	NOUN
ejpam-3941	151	23	duplicate	duplicate	VERB
ejpam-3941	151	24	a(x	a(x	NOUN
ejpam-3941	151	25	)	)	PUNCT
ejpam-3941	151	26	.	.	PUNCT
ejpam-3941	151	27	”	"	PUNCT
ejpam-3941	151	28	.	.	PUNCT
ejpam-3941	152	1	example	example	NOUN
ejpam-3941	153	1	3	3	X
ejpam-3941	153	2	.	.	X
ejpam-3941	153	3	consider	consider	VERB
ejpam-3941	153	4	the	the	DET
ejpam-3941	153	5	alexandroff	alexandroff	NOUN
ejpam-3941	153	6	duplicate	duplicate	ADJ
ejpam-3941	153	7	space	space	NOUN
ejpam-3941	153	8	a(r	a(r	NOUN
ejpam-3941	153	9	)	)	PUNCT
ejpam-3941	153	10	of	of	ADP
ejpam-3941	153	11	r	r	NOUN
ejpam-3941	153	12	with	with	ADP
ejpam-3941	153	13	its	its	PRON
ejpam-3941	153	14	usual	usual	ADJ
ejpam-3941	153	15	metric	metric	ADJ
ejpam-3941	153	16	topology	topology	NOUN
ejpam-3941	153	17	.	.	PUNCT
ejpam-3941	154	1	it	it	PRON
ejpam-3941	154	2	is	be	AUX
ejpam-3941	154	3	c2	c2	PROPN
ejpam-3941	154	4	-	-	PUNCT
ejpam-3941	154	5	paracompact	paracompact	NOUN
ejpam-3941	154	6	,	,	PUNCT
ejpam-3941	154	7	see	see	VERB
ejpam-3941	154	8	[	[	X
ejpam-3941	154	9	8	8	NUM
ejpam-3941	154	10	,	,	PUNCT
ejpam-3941	154	11	theorem	theorem	VERB
ejpam-3941	154	12	28	28	NUM
ejpam-3941	154	13	]	]	PUNCT
ejpam-3941	154	14	.	.	PUNCT
ejpam-3941	155	1	now	now	ADV
ejpam-3941	155	2	,	,	PUNCT
ejpam-3941	155	3	let	let	VERB
ejpam-3941	155	4	i	i	PRON
ejpam-3941	155	5	=	=	PUNCT
ejpam-3941	155	6	√	√	NUM
ejpam-3941	155	7	−1	−1	NOUN
ejpam-3941	155	8	6∈	6∈	NOUN
ejpam-3941	155	9	r	r	NOUN
ejpam-3941	155	10	and	and	CCONJ
ejpam-3941	155	11	put	put	VERB
ejpam-3941	155	12	x	x	PUNCT
ejpam-3941	155	13	=	=	PUNCT
ejpam-3941	155	14	r∪{i	r∪{i	ADV
ejpam-3941	155	15	}	}	PUNCT
ejpam-3941	155	16	.	.	PUNCT
ejpam-3941	156	1	let	let	VERB
ejpam-3941	156	2	τ	τ	PRON
ejpam-3941	156	3	be	be	AUX
ejpam-3941	156	4	the	the	DET
ejpam-3941	156	5	closed	closed	ADJ
ejpam-3941	156	6	extension	extension	NOUN
ejpam-3941	156	7	topology	topology	NOUN
ejpam-3941	156	8	on	on	ADP
ejpam-3941	156	9	x	x	PUNCT
ejpam-3941	156	10	generated	generate	VERB
ejpam-3941	156	11	from	from	ADP
ejpam-3941	156	12	r	r	NOUN
ejpam-3941	156	13	with	with	ADP
ejpam-3941	156	14	its	its	PRON
ejpam-3941	156	15	usual	usual	ADJ
ejpam-3941	156	16	metric	metric	ADJ
ejpam-3941	156	17	topology	topology	NOUN
ejpam-3941	156	18	and	and	CCONJ
ejpam-3941	156	19	i.	i.	NOUN
ejpam-3941	157	1	so	so	ADV
ejpam-3941	157	2	,	,	PUNCT
ejpam-3941	157	3	τ=	τ=	X
ejpam-3941	157	4	{	{	PUNCT
ejpam-3941	157	5	∅	∅	NOUN
ejpam-3941	157	6	}	}	PUNCT
ejpam-3941	157	7	∪	∪	NOUN
ejpam-3941	157	8	{	{	PUNCT
ejpam-3941	157	9	w	w	NOUN
ejpam-3941	157	10	∪	∪	X
ejpam-3941	157	11	{	{	PUNCT
ejpam-3941	157	12	i	i	NOUN
ejpam-3941	157	13	}	}	PUNCT
ejpam-3941	157	14	:	:	PUNCT
ejpam-3941	157	15	w	w	ADP
ejpam-3941	157	16	⊆	⊆	NUM
ejpam-3941	157	17	r;w	r;w	NOUN
ejpam-3941	157	18	is	be	AUX
ejpam-3941	157	19	open	open	ADJ
ejpam-3941	157	20	in	in	ADP
ejpam-3941	157	21	the	the	DET
ejpam-3941	157	22	usual	usual	ADJ
ejpam-3941	157	23	metric	metric	ADJ
ejpam-3941	157	24	topology	topology	NOUN
ejpam-3941	157	25	}	}	PUNCT
ejpam-3941	157	26	.	.	PUNCT
ejpam-3941	158	1	(	(	PUNCT
ejpam-3941	158	2	x	x	X
ejpam-3941	158	3	,	,	PUNCT
ejpam-3941	158	4	τ	τ	PROPN
ejpam-3941	158	5	)	)	PUNCT
ejpam-3941	158	6	is	be	AUX
ejpam-3941	158	7	not	not	PART
ejpam-3941	158	8	c	c	NOUN
ejpam-3941	158	9	-	-	NOUN
ejpam-3941	158	10	paracompact	paracompact	ADJ
ejpam-3941	158	11	because	because	SCONJ
ejpam-3941	158	12	it	it	PRON
ejpam-3941	158	13	is	be	AUX
ejpam-3941	158	14	fréchet	fréchet	PROPN
ejpam-3941	158	15	,	,	PUNCT
ejpam-3941	158	16	being	be	AUX
ejpam-3941	158	17	first	first	ADV
ejpam-3941	158	18	countable	countable	ADJ
ejpam-3941	158	19	,	,	PUNCT
ejpam-3941	158	20	non	non	ADJ
ejpam-3941	158	21	-	-	ADJ
ejpam-3941	158	22	compact	compact	ADJ
ejpam-3941	158	23	,	,	PUNCT
ejpam-3941	158	24	and	and	CCONJ
ejpam-3941	158	25	coarser	coarse	ADJ
ejpam-3941	158	26	than	than	ADP
ejpam-3941	158	27	the	the	DET
ejpam-3941	158	28	particular	particular	ADJ
ejpam-3941	158	29	point	point	NOUN
ejpam-3941	158	30	topology	topology	NOUN
ejpam-3941	158	31	on	on	ADP
ejpam-3941	158	32	x	x	SYM
ejpam-3941	158	33	where	where	SCONJ
ejpam-3941	158	34	the	the	DET
ejpam-3941	158	35	particular	particular	ADJ
ejpam-3941	158	36	point	point	NOUN
ejpam-3941	158	37	is	be	AUX
ejpam-3941	158	38	i	i	PRON
ejpam-3941	158	39	,	,	PUNCT
ejpam-3941	158	40	see	see	VERB
ejpam-3941	158	41	theorem	theorem	NOUN
ejpam-3941	158	42	5	5	NUM
ejpam-3941	158	43	.	.	PUNCT
ejpam-3941	158	44	define	define	VERB
ejpam-3941	158	45	g	g	NOUN
ejpam-3941	158	46	:	:	PUNCT
ejpam-3941	158	47	a(r	a(r	NOUN
ejpam-3941	158	48	)	)	PUNCT
ejpam-3941	158	49	−→	−→	NOUN
ejpam-3941	158	50	x	x	SYM
ejpam-3941	158	51	by	by	ADP
ejpam-3941	158	52	g(x	g(x	NOUN
ejpam-3941	158	53	)	)	PUNCT
ejpam-3941	159	1	=	=	PRON
ejpam-3941	159	2	{	{	PUNCT
ejpam-3941	159	3	i	i	INTJ
ejpam-3941	159	4	;	;	PUNCT
ejpam-3941	159	5	if	if	SCONJ
ejpam-3941	159	6	x	x	PUNCT
ejpam-3941	159	7	∈	∈	PROPN
ejpam-3941	159	8	r′	r′	PROPN
ejpam-3941	159	9	x	x	X
ejpam-3941	159	10	;	;	PUNCT
ejpam-3941	159	11	if	if	SCONJ
ejpam-3941	159	12	x	x	SYM
ejpam-3941	159	13	∈	∈	NOUN
ejpam-3941	159	14	r	r	NOUN
ejpam-3941	159	15	g	g	NOUN
ejpam-3941	159	16	is	be	AUX
ejpam-3941	159	17	an	an	DET
ejpam-3941	159	18	open	open	ADJ
ejpam-3941	159	19	surjection	surjection	NOUN
ejpam-3941	159	20	function	function	NOUN
ejpam-3941	159	21	.	.	PUNCT
ejpam-3941	160	1	thus	thus	ADV
ejpam-3941	160	2	c	c	X
ejpam-3941	160	3	-	-	PUNCT
ejpam-3941	160	4	paracompactness	paracompactness	PROPN
ejpam-3941	160	5	and	and	CCONJ
ejpam-3941	160	6	c2	c2	PROPN
ejpam-3941	160	7	-	-	PUNCT
ejpam-3941	160	8	paracompactness	paracompactness	PROPN
ejpam-3941	160	9	are	be	AUX
ejpam-3941	160	10	neither	neither	CCONJ
ejpam-3941	160	11	invariant	invariant	ADJ
ejpam-3941	160	12	,	,	PUNCT
ejpam-3941	160	13	open	open	ADJ
ejpam-3941	160	14	invariant	invariant	ADJ
ejpam-3941	160	15	,	,	PUNCT
ejpam-3941	160	16	nor	nor	CCONJ
ejpam-3941	160	17	quotient	quotient	NOUN
ejpam-3941	160	18	invariant	invariant	ADJ
ejpam-3941	160	19	.	.	PUNCT
ejpam-3941	161	1	references	reference	NOUN
ejpam-3941	161	2	356	356	NUM
ejpam-3941	161	3	now	now	ADV
ejpam-3941	161	4	we	we	PRON
ejpam-3941	161	5	show	show	VERB
ejpam-3941	161	6	that	that	SCONJ
ejpam-3941	161	7	c	c	NOUN
ejpam-3941	161	8	-	-	PUNCT
ejpam-3941	161	9	paracompactness	paracompactness	PROPN
ejpam-3941	161	10	and	and	CCONJ
ejpam-3941	161	11	c2	c2	PROPN
ejpam-3941	161	12	-	-	PUNCT
ejpam-3941	161	13	paracompactness	paracompactness	PROPN
ejpam-3941	161	14	are	be	AUX
ejpam-3941	161	15	both	both	PRON
ejpam-3941	161	16	not	not	PART
ejpam-3941	161	17	hereditary	hereditary	ADJ
ejpam-3941	161	18	.	.	PUNCT
ejpam-3941	162	1	recall	recall	VERB
ejpam-3941	162	2	that	that	SCONJ
ejpam-3941	162	3	a	a	DET
ejpam-3941	162	4	space	space	NOUN
ejpam-3941	162	5	x	x	PUNCT
ejpam-3941	162	6	is	be	AUX
ejpam-3941	162	7	called	call	VERB
ejpam-3941	162	8	c	c	NOUN
ejpam-3941	162	9	-	-	NOUN
ejpam-3941	162	10	normal	normal	ADJ
ejpam-3941	162	11	if	if	SCONJ
ejpam-3941	162	12	there	there	PRON
ejpam-3941	162	13	exist	exist	VERB
ejpam-3941	162	14	a	a	DET
ejpam-3941	162	15	normal	normal	ADJ
ejpam-3941	162	16	space	space	NOUN
ejpam-3941	162	17	y	y	PROPN
ejpam-3941	162	18	and	and	CCONJ
ejpam-3941	162	19	a	a	DET
ejpam-3941	162	20	bijective	bijective	ADJ
ejpam-3941	162	21	function	function	NOUN
ejpam-3941	163	1	f	f	NOUN
ejpam-3941	163	2	:	:	PUNCT
ejpam-3941	163	3	x	x	PUNCT
ejpam-3941	163	4	−→	−→	NOUN
ejpam-3941	163	5	y	y	PROPN
ejpam-3941	163	6	such	such	ADJ
ejpam-3941	163	7	that	that	SCONJ
ejpam-3941	163	8	the	the	DET
ejpam-3941	163	9	restriction	restriction	NOUN
ejpam-3941	163	10	f	f	PROPN
ejpam-3941	163	11	�	�	PROPN
ejpam-3941	163	12	a	a	NOUN
ejpam-3941	163	13	:	:	PUNCT
ejpam-3941	163	14	a	a	DET
ejpam-3941	163	15	−→	−→	NOUN
ejpam-3941	163	16	f(a	f(a	NOUN
ejpam-3941	163	17	)	)	PUNCT
ejpam-3941	163	18	is	be	AUX
ejpam-3941	163	19	a	a	DET
ejpam-3941	163	20	homeomorphism	homeomorphism	NOUN
ejpam-3941	163	21	for	for	ADP
ejpam-3941	163	22	each	each	DET
ejpam-3941	163	23	compact	compact	ADJ
ejpam-3941	163	24	subspace	subspace	NOUN
ejpam-3941	163	25	a	a	PRON
ejpam-3941	163	26	⊆	⊆	NUM
ejpam-3941	163	27	x	x	SYM
ejpam-3941	164	1	[	[	X
ejpam-3941	164	2	1	1	NUM
ejpam-3941	164	3	]	]	PUNCT
ejpam-3941	164	4	.	.	PUNCT
ejpam-3941	165	1	it	it	PRON
ejpam-3941	165	2	is	be	AUX
ejpam-3941	165	3	clear	clear	ADJ
ejpam-3941	165	4	that	that	SCONJ
ejpam-3941	165	5	any	any	DET
ejpam-3941	165	6	c2	c2	PROPN
ejpam-3941	165	7	-	-	PUNCT
ejpam-3941	165	8	paracompact	paracompact	NOUN
ejpam-3941	165	9	space	space	NOUN
ejpam-3941	165	10	is	be	AUX
ejpam-3941	165	11	c	c	NOUN
ejpam-3941	165	12	-	-	ADJ
ejpam-3941	165	13	normal	normal	ADJ
ejpam-3941	165	14	[	[	X
ejpam-3941	165	15	8	8	NUM
ejpam-3941	165	16	]	]	PUNCT
ejpam-3941	165	17	.	.	PUNCT
ejpam-3941	165	18	example	example	NOUN
ejpam-3941	166	1	4	4	X
ejpam-3941	166	2	.	.	PUNCT
ejpam-3941	166	3	consider	consider	VERB
ejpam-3941	166	4	2ω1	2ω1	NUM
ejpam-3941	166	5	,	,	PUNCT
ejpam-3941	166	6	where	where	SCONJ
ejpam-3941	166	7	2	2	X
ejpam-3941	166	8	=	=	SYM
ejpam-3941	166	9	{	{	PUNCT
ejpam-3941	166	10	0	0	NUM
ejpam-3941	166	11	,	,	PUNCT
ejpam-3941	166	12	1	1	NUM
ejpam-3941	166	13	}	}	PUNCT
ejpam-3941	166	14	with	with	ADP
ejpam-3941	166	15	the	the	DET
ejpam-3941	166	16	discrete	discrete	ADJ
ejpam-3941	166	17	topology	topology	NOUN
ejpam-3941	166	18	.	.	PUNCT
ejpam-3941	167	1	consider	consider	VERB
ejpam-3941	167	2	the	the	DET
ejpam-3941	167	3	subspace	subspace	NOUN
ejpam-3941	167	4	of	of	ADP
ejpam-3941	167	5	2ω1	2ω1	NUM
ejpam-3941	167	6	consisting	consist	VERB
ejpam-3941	167	7	of	of	ADP
ejpam-3941	167	8	all	all	DET
ejpam-3941	167	9	points	point	NOUN
ejpam-3941	167	10	with	with	ADP
ejpam-3941	167	11	at	at	ADP
ejpam-3941	167	12	most	most	ADV
ejpam-3941	167	13	countably	countably	ADV
ejpam-3941	167	14	many	many	ADJ
ejpam-3941	167	15	non	non	ADJ
ejpam-3941	167	16	-	-	ADJ
ejpam-3941	167	17	zero	zero	NUM
ejpam-3941	167	18	coordinates	coordinate	NOUN
ejpam-3941	167	19	,	,	PUNCT
ejpam-3941	167	20	i.e.	i.e.	X
ejpam-3941	167	21	,	,	PUNCT
ejpam-3941	167	22	the	the	DET
ejpam-3941	167	23	sigma	sigma	NOUN
ejpam-3941	167	24	product	product	NOUN
ejpam-3941	167	25	σ(0	σ(0	PROPN
ejpam-3941	167	26	)	)	PUNCT
ejpam-3941	167	27	.	.	PUNCT
ejpam-3941	168	1	put	put	VERB
ejpam-3941	168	2	x	x	PUNCT
ejpam-3941	168	3	=	=	SYM
ejpam-3941	168	4	2ω1	2ω1	NUM
ejpam-3941	168	5	×	×	NOUN
ejpam-3941	168	6	σ(0	σ(0	PROPN
ejpam-3941	168	7	)	)	PUNCT
ejpam-3941	168	8	.	.	PUNCT
ejpam-3941	169	1	raushan	raushan	PROPN
ejpam-3941	169	2	buzyakova	buzyakova	PROPN
ejpam-3941	169	3	proved	prove	VERB
ejpam-3941	169	4	that	that	SCONJ
ejpam-3941	169	5	x	x	PRON
ejpam-3941	169	6	can	can	AUX
ejpam-3941	169	7	not	not	PART
ejpam-3941	169	8	be	be	AUX
ejpam-3941	169	9	mapped	map	VERB
ejpam-3941	169	10	onto	onto	ADP
ejpam-3941	169	11	a	a	DET
ejpam-3941	169	12	normal	normal	ADJ
ejpam-3941	169	13	space	space	NOUN
ejpam-3941	169	14	y	y	NOUN
ejpam-3941	169	15	by	by	ADP
ejpam-3941	169	16	a	a	DET
ejpam-3941	169	17	bijective	bijective	ADJ
ejpam-3941	169	18	continuous	continuous	ADJ
ejpam-3941	169	19	function	function	NOUN
ejpam-3941	169	20	[	[	X
ejpam-3941	169	21	2	2	NUM
ejpam-3941	169	22	]	]	PUNCT
ejpam-3941	169	23	.	.	PUNCT
ejpam-3941	170	1	in	in	ADP
ejpam-3941	170	2	[	[	X
ejpam-3941	170	3	7	7	NUM
ejpam-3941	170	4	]	]	PUNCT
ejpam-3941	170	5	,	,	PUNCT
ejpam-3941	170	6	m.	m.	PROPN
ejpam-3941	170	7	saeed	saeed	PROPN
ejpam-3941	170	8	proved	prove	VERB
ejpam-3941	170	9	that	that	SCONJ
ejpam-3941	170	10	x	x	PRON
ejpam-3941	170	11	is	be	AUX
ejpam-3941	170	12	not	not	PART
ejpam-3941	170	13	c	c	NOUN
ejpam-3941	170	14	-	-	ADJ
ejpam-3941	170	15	normal	normal	ADJ
ejpam-3941	170	16	,	,	PUNCT
ejpam-3941	170	17	hence	hence	ADV
ejpam-3941	170	18	x	x	PUNCT
ejpam-3941	170	19	is	be	AUX
ejpam-3941	170	20	not	not	PART
ejpam-3941	170	21	c2	c2	NOUN
ejpam-3941	170	22	-	-	PUNCT
ejpam-3941	170	23	paracompact	paracompact	NOUN
ejpam-3941	170	24	.	.	PUNCT
ejpam-3941	171	1	since	since	SCONJ
ejpam-3941	171	2	x	x	PRON
ejpam-3941	171	3	is	be	AUX
ejpam-3941	171	4	a	a	DET
ejpam-3941	171	5	tychonoff	tychonoff	NOUN
ejpam-3941	171	6	non	non	ADJ
ejpam-3941	171	7	-	-	ADJ
ejpam-3941	171	8	compact	compact	ADJ
ejpam-3941	171	9	space	space	NOUN
ejpam-3941	171	10	,	,	PUNCT
ejpam-3941	171	11	any	any	DET
ejpam-3941	171	12	compactification	compactification	NOUN
ejpam-3941	171	13	of	of	ADP
ejpam-3941	171	14	x	x	SYM
ejpam-3941	171	15	is	be	AUX
ejpam-3941	171	16	c2	c2	NOUN
ejpam-3941	171	17	-	-	PUNCT
ejpam-3941	171	18	paracompact	paracompact	NOUN
ejpam-3941	171	19	while	while	SCONJ
ejpam-3941	171	20	x	x	PRON
ejpam-3941	171	21	is	be	AUX
ejpam-3941	171	22	not	not	PART
ejpam-3941	171	23	.	.	PUNCT
ejpam-3941	172	1	we	we	PRON
ejpam-3941	172	2	still	still	ADV
ejpam-3941	172	3	do	do	AUX
ejpam-3941	172	4	not	not	PART
ejpam-3941	172	5	know	know	VERB
ejpam-3941	172	6	if	if	SCONJ
ejpam-3941	172	7	c	c	NOUN
ejpam-3941	172	8	-	-	PUNCT
ejpam-3941	172	9	paracompactness	paracompactness	NOUN
ejpam-3941	172	10	(	(	PUNCT
ejpam-3941	172	11	c2	c2	PROPN
ejpam-3941	172	12	-	-	PUNCT
ejpam-3941	172	13	paracompactness	paracompactness	NOUN
ejpam-3941	172	14	)	)	PUNCT
ejpam-3941	172	15	is	be	AUX
ejpam-3941	172	16	hereditary	hereditary	ADJ
ejpam-3941	172	17	with	with	ADP
ejpam-3941	172	18	respect	respect	NOUN
ejpam-3941	172	19	to	to	ADP
ejpam-3941	172	20	closed	closed	ADJ
ejpam-3941	172	21	subspaces	subspace	NOUN
ejpam-3941	172	22	or	or	CCONJ
ejpam-3941	172	23	not	not	PART
ejpam-3941	172	24	.	.	PUNCT
ejpam-3941	173	1	now	now	ADV
ejpam-3941	173	2	,	,	PUNCT
ejpam-3941	173	3	here	here	ADV
ejpam-3941	173	4	is	be	AUX
ejpam-3941	173	5	our	our	PRON
ejpam-3941	173	6	first	first	ADJ
ejpam-3941	173	7	main	main	ADJ
ejpam-3941	173	8	result	result	NOUN
ejpam-3941	173	9	.	.	PUNCT
ejpam-3941	174	1	arhangel’skĭi	arhangel’skĭi	PROPN
ejpam-3941	174	2	stated	state	VERB
ejpam-3941	174	3	the	the	DET
ejpam-3941	174	4	following	follow	VERB
ejpam-3941	174	5	problem	problem	NOUN
ejpam-3941	174	6	,	,	PUNCT
ejpam-3941	174	7	see	see	VERB
ejpam-3941	174	8	[	[	X
ejpam-3941	174	9	8	8	NUM
ejpam-3941	174	10	]	]	PUNCT
ejpam-3941	174	11	:	:	PUNCT
ejpam-3941	174	12	“	"	PUNCT
ejpam-3941	174	13	is	be	AUX
ejpam-3941	174	14	there	there	PRON
ejpam-3941	174	15	a	a	DET
ejpam-3941	174	16	t4	t4	PROPN
ejpam-3941	174	17	space	space	NOUN
ejpam-3941	174	18	which	which	PRON
ejpam-3941	174	19	is	be	AUX
ejpam-3941	174	20	not	not	PART
ejpam-3941	174	21	c2	c2	NOUN
ejpam-3941	174	22	-	-	PUNCT
ejpam-3941	174	23	paracompact	paracompact	NOUN
ejpam-3941	174	24	?	?	PUNCT
ejpam-3941	174	25	”	"	PUNCT
ejpam-3941	174	26	.	.	PUNCT
ejpam-3941	175	1	we	we	PRON
ejpam-3941	175	2	will	will	AUX
ejpam-3941	175	3	answer	answer	VERB
ejpam-3941	175	4	this	this	DET
ejpam-3941	175	5	problem	problem	NOUN
ejpam-3941	175	6	in	in	ADP
ejpam-3941	175	7	positive	positive	ADJ
ejpam-3941	175	8	.	.	PUNCT
ejpam-3941	175	9	example	example	NOUN
ejpam-3941	176	1	5	5	NUM
ejpam-3941	176	2	.	.	PUNCT
ejpam-3941	176	3	consider	consider	VERB
ejpam-3941	176	4	the	the	DET
ejpam-3941	176	5	sigma	sigma	PROPN
ejpam-3941	176	6	product	product	NOUN
ejpam-3941	176	7	σ(0	σ(0	PROPN
ejpam-3941	176	8	)	)	PUNCT
ejpam-3941	176	9	as	as	ADP
ejpam-3941	176	10	a	a	DET
ejpam-3941	176	11	subspace	subspace	NOUN
ejpam-3941	176	12	of	of	ADP
ejpam-3941	176	13	2ω1	2ω1	NUM
ejpam-3941	176	14	,	,	PUNCT
ejpam-3941	176	15	where	where	SCONJ
ejpam-3941	176	16	2	2	X
ejpam-3941	176	17	=	=	SYM
ejpam-3941	176	18	{	{	PUNCT
ejpam-3941	176	19	0	0	NUM
ejpam-3941	176	20	,	,	PUNCT
ejpam-3941	176	21	1	1	NUM
ejpam-3941	176	22	}	}	PUNCT
ejpam-3941	176	23	with	with	ADP
ejpam-3941	176	24	the	the	DET
ejpam-3941	176	25	discrete	discrete	ADJ
ejpam-3941	176	26	topology	topology	NOUN
ejpam-3941	176	27	,	,	PUNCT
ejpam-3941	176	28	see	see	VERB
ejpam-3941	176	29	example	example	NOUN
ejpam-3941	176	30	4	4	X
ejpam-3941	176	31	.	.	X
ejpam-3941	177	1	we	we	PRON
ejpam-3941	177	2	have	have	VERB
ejpam-3941	177	3	that	that	DET
ejpam-3941	177	4	σ(0	σ(0	PROPN
ejpam-3941	177	5	)	)	PUNCT
ejpam-3941	177	6	is	be	AUX
ejpam-3941	177	7	t4	t4	PROPN
ejpam-3941	178	1	[	[	X
ejpam-3941	178	2	5	5	NUM
ejpam-3941	178	3	,	,	PUNCT
ejpam-3941	178	4	theorem	theorem	VERB
ejpam-3941	178	5	7.4	7.4	NUM
ejpam-3941	178	6	]	]	PUNCT
ejpam-3941	178	7	,	,	PUNCT
ejpam-3941	178	8	countably	countably	ADV
ejpam-3941	178	9	compact	compact	ADJ
ejpam-3941	178	10	[	[	X
ejpam-3941	178	11	6	6	NUM
ejpam-3941	178	12	,	,	PUNCT
ejpam-3941	178	13	theorem	theorem	VERB
ejpam-3941	178	14	6.10	6.10	NUM
ejpam-3941	178	15	]	]	PUNCT
ejpam-3941	178	16	,	,	PUNCT
ejpam-3941	178	17	fréchet	fréchet	VERB
ejpam-3941	179	1	[	[	X
ejpam-3941	179	2	4	4	NUM
ejpam-3941	179	3	,	,	PUNCT
ejpam-3941	179	4	3.10.d	3.10.d	NUM
ejpam-3941	179	5	]	]	PUNCT
ejpam-3941	179	6	,	,	PUNCT
ejpam-3941	179	7	hence	hence	ADV
ejpam-3941	179	8	it	it	PRON
ejpam-3941	179	9	is	be	AUX
ejpam-3941	179	10	a	a	DET
ejpam-3941	179	11	k	k	NOUN
ejpam-3941	179	12	-	-	NOUN
ejpam-3941	179	13	space	space	NOUN
ejpam-3941	179	14	[	[	X
ejpam-3941	179	15	4	4	NUM
ejpam-3941	179	16	,	,	PUNCT
ejpam-3941	179	17	3.10.d	3.10.d	NUM
ejpam-3941	179	18	]	]	X
ejpam-3941	179	19	.	.	PUNCT
ejpam-3941	180	1	also	also	ADV
ejpam-3941	180	2	σ(0	σ(0	PROPN
ejpam-3941	180	3	)	)	PUNCT
ejpam-3941	180	4	is	be	AUX
ejpam-3941	180	5	not	not	PART
ejpam-3941	180	6	paracompact	paracompact	ADJ
ejpam-3941	180	7	because	because	SCONJ
ejpam-3941	180	8	it	it	PRON
ejpam-3941	180	9	is	be	AUX
ejpam-3941	180	10	contained	contain	VERB
ejpam-3941	180	11	a	a	DET
ejpam-3941	180	12	copy	copy	NOUN
ejpam-3941	180	13	of	of	ADP
ejpam-3941	180	14	ω1	ω1	PROPN
ejpam-3941	180	15	as	as	ADP
ejpam-3941	180	16	a	a	DET
ejpam-3941	180	17	closed	closed	ADJ
ejpam-3941	180	18	subspace	subspace	NOUN
ejpam-3941	180	19	[	[	X
ejpam-3941	180	20	5	5	NUM
ejpam-3941	180	21	,	,	PUNCT
ejpam-3941	180	22	theorem	theorem	VERB
ejpam-3941	180	23	7.2	7.2	NUM
ejpam-3941	180	24	]	]	PUNCT
ejpam-3941	180	25	.	.	PUNCT
ejpam-3941	181	1	suppose	suppose	VERB
ejpam-3941	181	2	that	that	SCONJ
ejpam-3941	181	3	σ(0	σ(0	PROPN
ejpam-3941	181	4	)	)	PUNCT
ejpam-3941	181	5	is	be	AUX
ejpam-3941	181	6	c2	c2	PROPN
ejpam-3941	181	7	-	-	PUNCT
ejpam-3941	181	8	paracompact	paracompact	NOUN
ejpam-3941	181	9	.	.	PUNCT
ejpam-3941	182	1	by	by	ADP
ejpam-3941	182	2	theorem	theorem	NOUN
ejpam-3941	182	3	2	2	NUM
ejpam-3941	182	4	,	,	PUNCT
ejpam-3941	182	5	x	x	SYM
ejpam-3941	182	6	=	=	SYM
ejpam-3941	182	7	2ω1	2ω1	NUM
ejpam-3941	182	8	×	×	NOUN
ejpam-3941	182	9	σ(0	σ(0	PROPN
ejpam-3941	182	10	)	)	PUNCT
ejpam-3941	182	11	is	be	AUX
ejpam-3941	182	12	c2	c2	PROPN
ejpam-3941	182	13	-	-	PUNCT
ejpam-3941	182	14	paracompact	paracompact	NOUN
ejpam-3941	182	15	.	.	PUNCT
ejpam-3941	183	1	this	this	PRON
ejpam-3941	183	2	contradicts	contradict	VERB
ejpam-3941	183	3	m.	m.	PROPN
ejpam-3941	183	4	saeed	saeed	PROPN
ejpam-3941	183	5	’s	’s	PART
ejpam-3941	183	6	result	result	NOUN
ejpam-3941	184	1	[	[	X
ejpam-3941	184	2	7	7	X
ejpam-3941	184	3	]	]	PUNCT
ejpam-3941	184	4	and	and	CCONJ
ejpam-3941	184	5	buzyakova	buzyakova	PROPN
ejpam-3941	184	6	’s	’s	PART
ejpam-3941	184	7	result	result	NOUN
ejpam-3941	185	1	[	[	X
ejpam-3941	185	2	2	2	X
ejpam-3941	185	3	]	]	PUNCT
ejpam-3941	185	4	because	because	SCONJ
ejpam-3941	185	5	any	any	DET
ejpam-3941	185	6	t2	t2	NOUN
ejpam-3941	185	7	paracompact	paracompact	NOUN
ejpam-3941	185	8	space	space	NOUN
ejpam-3941	185	9	is	be	AUX
ejpam-3941	185	10	normal	normal	ADJ
ejpam-3941	185	11	.	.	PUNCT
ejpam-3941	186	1	here	here	ADV
ejpam-3941	186	2	is	be	AUX
ejpam-3941	186	3	our	our	PRON
ejpam-3941	186	4	second	second	ADJ
ejpam-3941	186	5	main	main	ADJ
ejpam-3941	186	6	result	result	NOUN
ejpam-3941	186	7	.	.	PUNCT
ejpam-3941	187	1	recall	recall	VERB
ejpam-3941	187	2	that	that	SCONJ
ejpam-3941	187	3	a	a	DET
ejpam-3941	187	4	function	function	NOUN
ejpam-3941	187	5	f	f	NOUN
ejpam-3941	187	6	:	:	PUNCT
ejpam-3941	187	7	x	x	PUNCT
ejpam-3941	188	1	−→	−→	NOUN
ejpam-3941	188	2	y	y	PROPN
ejpam-3941	188	3	is	be	AUX
ejpam-3941	188	4	called	call	VERB
ejpam-3941	188	5	condensation	condensation	NOUN
ejpam-3941	188	6	if	if	SCONJ
ejpam-3941	188	7	it	it	PRON
ejpam-3941	188	8	is	be	AUX
ejpam-3941	188	9	bijective	bijective	ADJ
ejpam-3941	188	10	and	and	CCONJ
ejpam-3941	188	11	continuous	continuous	ADJ
ejpam-3941	188	12	.	.	PUNCT
ejpam-3941	189	1	the	the	DET
ejpam-3941	189	2	sigma	sigma	PROPN
ejpam-3941	189	3	product	product	NOUN
ejpam-3941	189	4	σ(0	σ(0	PROPN
ejpam-3941	189	5	)	)	PUNCT
ejpam-3941	189	6	is	be	AUX
ejpam-3941	189	7	a	a	DET
ejpam-3941	189	8	k	k	NOUN
ejpam-3941	189	9	-	-	NOUN
ejpam-3941	189	10	space	space	NOUN
ejpam-3941	189	11	[	[	X
ejpam-3941	189	12	4	4	NUM
ejpam-3941	189	13	,	,	PUNCT
ejpam-3941	189	14	3.10.d	3.10.d	NUM
ejpam-3941	189	15	]	]	PUNCT
ejpam-3941	189	16	.	.	PUNCT
ejpam-3941	190	1	considering	consider	VERB
ejpam-3941	190	2	the	the	DET
ejpam-3941	190	3	theorem	theorem	NOUN
ejpam-3941	190	4	“	"	PUNCT
ejpam-3941	190	5	a	a	DET
ejpam-3941	190	6	function	function	NOUN
ejpam-3941	190	7	f	f	NOUN
ejpam-3941	190	8	of	of	ADP
ejpam-3941	190	9	a	a	DET
ejpam-3941	190	10	k	k	NOUN
ejpam-3941	190	11	-	-	NOUN
ejpam-3941	190	12	space	space	NOUN
ejpam-3941	190	13	x	x	PUNCT
ejpam-3941	190	14	to	to	ADP
ejpam-3941	190	15	a	a	DET
ejpam-3941	190	16	topological	topological	ADJ
ejpam-3941	190	17	space	space	NOUN
ejpam-3941	190	18	y	y	PROPN
ejpam-3941	190	19	is	be	AUX
ejpam-3941	190	20	continuous	continuous	ADJ
ejpam-3941	190	21	if	if	SCONJ
ejpam-3941	190	22	and	and	CCONJ
ejpam-3941	190	23	only	only	ADV
ejpam-3941	190	24	if	if	SCONJ
ejpam-3941	190	25	for	for	ADP
ejpam-3941	190	26	every	every	DET
ejpam-3941	190	27	compact	compact	ADJ
ejpam-3941	190	28	space	space	NOUN
ejpam-3941	190	29	c	c	NOUN
ejpam-3941	190	30	⊆	⊆	NUM
ejpam-3941	190	31	x	x	NOUN
ejpam-3941	190	32	the	the	DET
ejpam-3941	190	33	restriction	restriction	NOUN
ejpam-3941	190	34	f	f	PROPN
ejpam-3941	190	35	�	�	PROPN
ejpam-3941	190	36	c	c	PROPN
ejpam-3941	190	37	:	:	PUNCT
ejpam-3941	190	38	c	c	AUX
ejpam-3941	190	39	−→	−→	NOUN
ejpam-3941	190	40	y	y	PROPN
ejpam-3941	190	41	is	be	AUX
ejpam-3941	190	42	continuous	continuous	ADJ
ejpam-3941	190	43	”	"	PUNCT
ejpam-3941	190	44	,	,	PUNCT
ejpam-3941	190	45	[	[	X
ejpam-3941	190	46	4	4	NUM
ejpam-3941	190	47	,	,	PUNCT
ejpam-3941	190	48	3.3.21	3.3.21	PROPN
ejpam-3941	190	49	]	]	PUNCT
ejpam-3941	190	50	,	,	PUNCT
ejpam-3941	190	51	we	we	PRON
ejpam-3941	190	52	conclude	conclude	VERB
ejpam-3941	190	53	the	the	DET
ejpam-3941	190	54	following	follow	VERB
ejpam-3941	190	55	:	:	PUNCT
ejpam-3941	190	56	corollary	corollary	ADJ
ejpam-3941	190	57	2	2	NUM
ejpam-3941	190	58	.	.	PUNCT
ejpam-3941	191	1	the	the	DET
ejpam-3941	191	2	sigma	sigma	PROPN
ejpam-3941	191	3	product	product	NOUN
ejpam-3941	191	4	σ(0	σ(0	PROPN
ejpam-3941	191	5	)	)	PUNCT
ejpam-3941	191	6	can	can	AUX
ejpam-3941	191	7	not	not	PART
ejpam-3941	191	8	be	be	AUX
ejpam-3941	191	9	condensed	condense	VERB
ejpam-3941	191	10	onto	onto	ADP
ejpam-3941	191	11	any	any	DET
ejpam-3941	191	12	t2	t2	NOUN
ejpam-3941	191	13	paracompact	paracompact	NOUN
ejpam-3941	191	14	space	space	NOUN
ejpam-3941	191	15	.	.	PUNCT
ejpam-3941	192	1	open	open	ADJ
ejpam-3941	192	2	problem	problem	NOUN
ejpam-3941	192	3	:	:	PUNCT
ejpam-3941	192	4	is	be	AUX
ejpam-3941	192	5	c2	c2	PROPN
ejpam-3941	192	6	-	-	PUNCT
ejpam-3941	192	7	paracompactness	paracompactness	PROPN
ejpam-3941	192	8	multiplicative	multiplicative	ADJ
ejpam-3941	192	9	?	?	PUNCT
ejpam-3941	193	1	references	reference	NOUN
ejpam-3941	193	2	[	[	X
ejpam-3941	193	3	1	1	NUM
ejpam-3941	193	4	]	]	PUNCT
ejpam-3941	193	5	samirah	samirah	PROPN
ejpam-3941	193	6	alzahrani	alzahrani	PROPN
ejpam-3941	193	7	and	and	CCONJ
ejpam-3941	193	8	lutfi	lutfi	PROPN
ejpam-3941	193	9	kalantan	kalantan	PROPN
ejpam-3941	193	10	.	.	PUNCT
ejpam-3941	194	1	c	c	X
ejpam-3941	194	2	-	-	PUNCT
ejpam-3941	194	3	normal	normal	ADJ
ejpam-3941	194	4	topological	topological	ADJ
ejpam-3941	194	5	property	property	NOUN
ejpam-3941	194	6	.	.	PUNCT
ejpam-3941	195	1	filomat	filomat	PROPN
ejpam-3941	195	2	,	,	PUNCT
ejpam-3941	195	3	31:407–411	31:407–411	PROPN
ejpam-3941	195	4	,	,	PUNCT
ejpam-3941	195	5	2	2	NUM
ejpam-3941	195	6	2017	2017	NUM
ejpam-3941	195	7	.	.	PUNCT
ejpam-3941	196	1	references	reference	NOUN
ejpam-3941	196	2	357	357	NUM
ejpam-3941	196	3	[	[	X
ejpam-3941	196	4	2	2	NUM
ejpam-3941	196	5	]	]	X
ejpam-3941	196	6	r	r	NOUN
ejpam-3941	196	7	z	z	NOUN
ejpam-3941	196	8	buzyakova	buzyakova	NOUN
ejpam-3941	196	9	.	.	PUNCT
ejpam-3941	197	1	an	an	DET
ejpam-3941	197	2	example	example	NOUN
ejpam-3941	197	3	of	of	ADP
ejpam-3941	197	4	a	a	DET
ejpam-3941	197	5	product	product	NOUN
ejpam-3941	197	6	of	of	ADP
ejpam-3941	197	7	two	two	NUM
ejpam-3941	197	8	normal	normal	ADJ
ejpam-3941	197	9	groups	group	NOUN
ejpam-3941	197	10	that	that	PRON
ejpam-3941	197	11	can	can	AUX
ejpam-3941	197	12	not	not	PART
ejpam-3941	197	13	be	be	AUX
ejpam-3941	197	14	condensed	condense	VERB
ejpam-3941	197	15	onto	onto	ADP
ejpam-3941	197	16	a	a	DET
ejpam-3941	197	17	normal	normal	ADJ
ejpam-3941	197	18	space	space	NOUN
ejpam-3941	197	19	.	.	PUNCT
ejpam-3941	198	1	univ	univ	PROPN
ejpam-3941	198	2	.	.	PUNCT
ejpam-3941	198	3	math	math	NOUN
ejpam-3941	198	4	.	.	PUNCT
ejpam-3941	199	1	bull	bull	NOUN
ejpam-3941	199	2	,	,	PUNCT
ejpam-3941	199	3	52:42–42	52:42–42	NUM
ejpam-3941	199	4	,	,	PUNCT
ejpam-3941	199	5	3	3	NUM
ejpam-3941	199	6	.	.	PUNCT
ejpam-3941	200	1	[	[	X
ejpam-3941	200	2	3	3	NUM
ejpam-3941	200	3	]	]	X
ejpam-3941	200	4	r	r	NOUN
ejpam-3941	200	5	engelking	engelking	NOUN
ejpam-3941	200	6	.	.	PUNCT
ejpam-3941	201	1	on	on	ADP
ejpam-3941	201	2	the	the	DET
ejpam-3941	201	3	double	double	ADJ
ejpam-3941	201	4	circumference	circumference	NOUN
ejpam-3941	201	5	of	of	ADP
ejpam-3941	201	6	alexandroff	alexandroff	NOUN
ejpam-3941	201	7	.	.	PUNCT
ejpam-3941	202	1	bull	bull	NOUN
ejpam-3941	202	2	acad	acad	PROPN
ejpam-3941	202	3	pol	pol	PROPN
ejpam-3941	202	4	sci	sci	PROPN
ejpam-3941	202	5	ser	ser	PROPN
ejpam-3941	202	6	astron	astron	PROPN
ejpam-3941	202	7	math	math	PROPN
ejpam-3941	202	8	phys	phys	PROPN
ejpam-3941	202	9	,	,	PUNCT
ejpam-3941	202	10	16:629–634	16:629–634	PROPN
ejpam-3941	202	11	,	,	PUNCT
ejpam-3941	202	12	8	8	NUM
ejpam-3941	202	13	.	.	PUNCT
ejpam-3941	203	1	[	[	X
ejpam-3941	203	2	4	4	NUM
ejpam-3941	203	3	]	]	X
ejpam-3941	203	4	r	r	NOUN
ejpam-3941	203	5	engelking	engelking	NOUN
ejpam-3941	203	6	.	.	PUNCT
ejpam-3941	204	1	general	general	ADJ
ejpam-3941	204	2	topology	topology	PROPN
ejpam-3941	204	3	.	.	PUNCT
ejpam-3941	205	1	pwn	pwn	PROPN
ejpam-3941	205	2	,	,	PUNCT
ejpam-3941	205	3	1977	1977	NUM
ejpam-3941	205	4	.	.	PUNCT
ejpam-3941	206	1	[	[	X
ejpam-3941	206	2	5	5	X
ejpam-3941	206	3	]	]	PUNCT
ejpam-3941	206	4	t	t	NOUN
ejpam-3941	206	5	c	c	PROPN
ejpam-3941	206	6	przymusiński	przymusiński	PROPN
ejpam-3941	206	7	kunen	kunen	PROPN
ejpam-3941	206	8	k.	k.	PROPN
ejpam-3941	206	9	handbook	handbook	PROPN
ejpam-3941	206	10	of	of	ADP
ejpam-3941	206	11	set	set	ADJ
ejpam-3941	206	12	theoretic	theoretic	ADJ
ejpam-3941	206	13	topology	topology	NOUN
ejpam-3941	206	14	.	.	PUNCT
ejpam-3941	207	1	amsterdam	amsterdam	PROPN
ejpam-3941	207	2	,	,	PUNCT
ejpam-3941	207	3	1984	1984	NUM
ejpam-3941	207	4	.	.	PUNCT
ejpam-3941	208	1	[	[	X
ejpam-3941	208	2	6	6	NUM
ejpam-3941	208	3	]	]	X
ejpam-3941	208	4	w	w	PROPN
ejpam-3941	208	5	w	w	PROPN
ejpam-3941	208	6	comfort	comfort	PROPN
ejpam-3941	208	7	kunen	kunen	PROPN
ejpam-3941	208	8	k.	k.	PROPN
ejpam-3941	208	9	handbook	handbook	PROPN
ejpam-3941	208	10	of	of	ADP
ejpam-3941	208	11	set	set	ADJ
ejpam-3941	208	12	theoretic	theoretic	ADJ
ejpam-3941	208	13	topology	topology	NOUN
ejpam-3941	208	14	.	.	PUNCT
ejpam-3941	209	1	amsterdam	amsterdam	PROPN
ejpam-3941	209	2	,	,	PUNCT
ejpam-3941	209	3	1984	1984	NUM
ejpam-3941	209	4	.	.	PUNCT
ejpam-3941	210	1	[	[	X
ejpam-3941	210	2	7	7	X
ejpam-3941	210	3	]	]	X
ejpam-3941	210	4	maha	maha	PROPN
ejpam-3941	210	5	saeed	saeed	PROPN
ejpam-3941	210	6	.	.	PUNCT
ejpam-3941	211	1	countable	countable	ADJ
ejpam-3941	211	2	normality	normality	NOUN
ejpam-3941	211	3	,	,	PUNCT
ejpam-3941	211	4	2018	2018	NUM
ejpam-3941	211	5	.	.	PUNCT
ejpam-3941	212	1	[	[	X
ejpam-3941	212	2	8	8	NUM
ejpam-3941	212	3	]	]	PUNCT
ejpam-3941	212	4	maha	maha	PROPN
ejpam-3941	212	5	mohammed	mohammed	PROPN
ejpam-3941	212	6	saeed	saeed	PROPN
ejpam-3941	212	7	,	,	PUNCT
ejpam-3941	212	8	lutfi	lutfi	PROPN
ejpam-3941	212	9	kalantan	kalantan	PROPN
ejpam-3941	212	10	,	,	PUNCT
ejpam-3941	212	11	and	and	CCONJ
ejpam-3941	212	12	hala	hala	PROPN
ejpam-3941	212	13	alzumi	alzumi	PROPN
ejpam-3941	212	14	.	.	PUNCT
ejpam-3941	213	1	c	c	X
ejpam-3941	213	2	-	-	PUNCT
ejpam-3941	213	3	paracompactness	paracompactness	PROPN
ejpam-3941	213	4	and	and	CCONJ
ejpam-3941	213	5	c	c	PROPN
ejpam-3941	213	6	2	2	NUM
ejpam-3941	213	7	-	-	PUNCT
ejpam-3941	213	8	paracompactness	paracompactness	NOUN
ejpam-3941	213	9	.	.	PUNCT
ejpam-3941	214	1	[	[	X
ejpam-3941	214	2	9	9	NUM
ejpam-3941	214	3	]	]	SYM
ejpam-3941	214	4	l	l	NOUN
ejpam-3941	214	5	steen	steen	PROPN
ejpam-3941	214	6	and	and	CCONJ
ejpam-3941	214	7	j	j	PROPN
ejpam-3941	214	8	a	a	DET
ejpam-3941	214	9	seebach	seebach	NOUN
ejpam-3941	214	10	.	.	PUNCT
ejpam-3941	215	1	counterexamples	counterexample	NOUN
ejpam-3941	215	2	in	in	ADP
ejpam-3941	215	3	topology	topology	NOUN
ejpam-3941	215	4	.	.	PUNCT
ejpam-3941	216	1	dover	dover	PROPN
ejpam-3941	216	2	publications	publication	NOUN
ejpam-3941	216	3	,	,	PUNCT
ejpam-3941	216	4	1995	1995	NUM
ejpam-3941	216	5	.	.	PUNCT