id	sid	tid	token	lemma	pos
easat-1685	1	1	edelweiss	edelweiss	PROPN
easat-1685	1	2	applied	apply	VERB
easat-1685	1	3	science	science	NOUN
easat-1685	1	4	and	and	CCONJ
easat-1685	1	5	technology	technology	NOUN
easat-1685	1	6	issn	issn	PROPN
easat-1685	1	7	:	:	PUNCT
easat-1685	1	8	2576	2576	NUM
easat-1685	1	9	-	-	SYM
easat-1685	1	10	8484	8484	NUM
easat-1685	1	11	vol	vol	NOUN
easat-1685	1	12	.	.	PROPN
easat-1685	1	13	8	8	NUM
easat-1685	1	14	,	,	PUNCT
easat-1685	1	15	no	no	INTJ
easat-1685	1	16	.	.	NOUN
easat-1685	1	17	5	5	NUM
easat-1685	1	18	,	,	PUNCT
easat-1685	1	19	271	271	NUM
easat-1685	1	20	-	-	SYM
easat-1685	1	21	277	277	NUM
easat-1685	1	22	2024	2024	NUM
easat-1685	1	23	publisher	publisher	NOUN
easat-1685	1	24	:	:	PUNCT
easat-1685	1	25	learning	learn	VERB
easat-1685	1	26	gate	gate	NOUN
easat-1685	1	27	doi	doi	PROPN
easat-1685	1	28	:	:	PUNCT
easat-1685	1	29	10.55214/25768484.v8i5.1685	10.55214/25768484.v8i5.1685	NUM
easat-1685	1	30	©	©	ADP
easat-1685	1	31	2024	2024	NUM
easat-1685	1	32	by	by	ADP
easat-1685	1	33	the	the	DET
easat-1685	1	34	authors	author	NOUN
easat-1685	1	35	;	;	PUNCT
easat-1685	1	36	licensee	licensee	PROPN
easat-1685	1	37	learning	learning	NOUN
easat-1685	1	38	gate	gate	NOUN
easat-1685	1	39	©	©	PROPN
easat-1685	1	40	2024	2024	NUM
easat-1685	1	41	by	by	ADP
easat-1685	1	42	the	the	DET
easat-1685	1	43	authors	author	NOUN
easat-1685	1	44	;	;	PUNCT
easat-1685	1	45	licensee	licensee	PROPN
easat-1685	1	46	learning	learn	VERB
easat-1685	1	47	gate	gate	NOUN
easat-1685	1	48	*	*	PUNCT
easat-1685	1	49	correspondence	correspondence	NOUN
easat-1685	1	50	:	:	PUNCT
easat-1685	1	51	tomader.sayed2013m@csw.uobaghdad.edu.iq	tomader.sayed2013m@csw.uobaghdad.edu.iq	NOUN
easat-1685	1	52	on	on	ADP
easat-1685	1	53	some	some	DET
easat-1685	1	54	covering	cover	VERB
easat-1685	1	55	properties	property	NOUN
easat-1685	1	56	via	via	ADP
easat-1685	1	57	aopen	aopen	ADJ
easat-1685	1	58	sets	set	NOUN
easat-1685	1	59	tamadher	tamadher	PROPN
easat-1685	1	60	waleed	waleed	PROPN
easat-1685	1	61	said	say	VERB
easat-1685	1	62	ghani1	ghani1	PROPN
easat-1685	1	63	*	*	PROPN
easat-1685	1	64	,	,	PUNCT
easat-1685	1	65	jalal	jalal	PROPN
easat-1685	1	66	hatem	hatem	PROPN
easat-1685	1	67	hussein2	hussein2	PROPN
easat-1685	2	1	1,2department	1,2department	NUM
easat-1685	2	2	of	of	ADP
easat-1685	2	3	mathematics	mathematic	NOUN
easat-1685	2	4	,	,	PUNCT
easat-1685	2	5	college	college	NOUN
easat-1685	2	6	of	of	ADP
easat-1685	2	7	science	science	NOUN
easat-1685	2	8	for	for	ADP
easat-1685	2	9	women	woman	NOUN
easat-1685	2	10	,	,	PUNCT
easat-1685	2	11	university	university	NOUN
easat-1685	2	12	of	of	ADP
easat-1685	2	13	baghdad	baghdad	PROPN
easat-1685	2	14	,	,	PUNCT
easat-1685	2	15	baghdad	baghdad	PROPN
easat-1685	2	16	-	-	PUNCT
easat-1685	2	17	iraq	iraq	PROPN
easat-1685	2	18	;	;	PUNCT
easat-1685	2	19	tomader.sayed2013m@csw.uobaghdad.edu.iq	tomader.sayed2013m@csw.uobaghdad.edu.iq	PROPN
easat-1685	2	20	(	(	PUNCT
easat-1685	2	21	t.w.s.g	t.w.s.g	PROPN
easat-1685	2	22	.	.	PUNCT
easat-1685	2	23	)	)	PUNCT
easat-1685	3	1	jalalintuch@yahoo.com	jalalintuch@yahoo.com	X
easat-1685	4	1	(	(	PUNCT
easat-1685	4	2	j.h.h	j.h.h	PROPN
easat-1685	4	3	.	.	PUNCT
easat-1685	4	4	)	)	PUNCT
easat-1685	4	5	.	.	PUNCT
easat-1685	5	1	abstract	abstract	ADV
easat-1685	5	2	:	:	PUNCT
easat-1685	5	3	in	in	ADP
easat-1685	5	4	this	this	DET
easat-1685	5	5	paper	paper	NOUN
easat-1685	5	6	the	the	DET
easat-1685	5	7	notion	notion	NOUN
easat-1685	5	8	of	of	ADP
easat-1685	5	9	a	a	DET
easat-1685	5	10	-	-	PUNCT
easat-1685	5	11	open	open	ADJ
easat-1685	5	12	set	set	NOUN
easat-1685	5	13	used	use	VERB
easat-1685	5	14	as	as	ADP
easat-1685	5	15	a	a	DET
easat-1685	5	16	tool	tool	NOUN
easat-1685	5	17	to	to	PART
easat-1685	5	18	introduce	introduce	VERB
easat-1685	5	19	certain	certain	ADJ
easat-1685	5	20	types	type	NOUN
easat-1685	5	21	of	of	ADP
easat-1685	5	22	covering	cover	VERB
easat-1685	5	23	properties	property	NOUN
easat-1685	5	24	which	which	PRON
easat-1685	5	25	is	be	AUX
easat-1685	5	26	similar	similar	ADJ
easat-1685	5	27	to	to	ADP
easat-1685	5	28	the	the	DET
easat-1685	5	29	familiar	familiar	ADJ
easat-1685	5	30	property	property	NOUN
easat-1685	5	31	of	of	ADP
easat-1685	5	32	hurewicz	hurewicz	NOUN
easat-1685	5	33	,	,	PUNCT
easat-1685	5	34	and	and	CCONJ
easat-1685	5	35	prove	prove	VERB
easat-1685	5	36	that	that	SCONJ
easat-1685	5	37	we	we	PRON
easat-1685	5	38	can	can	AUX
easat-1685	5	39	use	use	VERB
easat-1685	5	40	a	a	DET
easat-1685	5	41	-	-	PUNCT
easat-1685	5	42	open	open	ADJ
easat-1685	5	43	sets	set	NOUN
easat-1685	5	44	instead	instead	ADV
easat-1685	5	45	of	of	ADP
easat-1685	5	46	open	open	ADJ
easat-1685	5	47	sets	set	NOUN
easat-1685	5	48	in	in	ADP
easat-1685	5	49	the	the	DET
easat-1685	5	50	definition	definition	NOUN
easat-1685	5	51	of	of	ADP
easat-1685	5	52	a	a	DET
easat-1685	5	53	-	-	PUNCT
easat-1685	5	54	compact	compact	ADJ
easat-1685	5	55	and	and	CCONJ
easat-1685	5	56	a	a	DET
easat-1685	5	57	-	-	PUNCT
easat-1685	5	58	hurewicz	hurewicz	NOUN
easat-1685	5	59	space	space	NOUN
easat-1685	5	60	and	and	CCONJ
easat-1685	5	61	investigated	investigate	VERB
easat-1685	5	62	.	.	PUNCT
easat-1685	6	1	some	some	DET
easat-1685	6	2	properties	property	NOUN
easat-1685	6	3	and	and	CCONJ
easat-1685	6	4	counter	counter	ADJ
easat-1685	6	5	examples	example	NOUN
easat-1685	6	6	are	be	AUX
easat-1685	6	7	given	give	VERB
easat-1685	6	8	,	,	PUNCT
easat-1685	6	9	also	also	ADV
easat-1685	6	10	the	the	DET
easat-1685	6	11	relationship	relationship	NOUN
easat-1685	6	12	between	between	ADP
easat-1685	6	13	these	these	DET
easat-1685	6	14	spaces	space	NOUN
easat-1685	6	15	was	be	AUX
easat-1685	6	16	considered	consider	VERB
easat-1685	6	17	.	.	PUNCT
easat-1685	7	1	keywords	keyword	NOUN
easat-1685	7	2	:	:	PUNCT
easat-1685	7	3	a	a	DET
easat-1685	7	4	-	-	PUNCT
easat-1685	7	5	compact	compact	ADJ
easat-1685	7	6	space	space	NOUN
easat-1685	7	7	,	,	PUNCT
easat-1685	7	8	a	a	DET
easat-1685	7	9	-	-	PUNCT
easat-1685	7	10	hurewicz	hurewicz	NOUN
easat-1685	7	11	space	space	NOUN
easat-1685	7	12	,	,	PUNCT
easat-1685	7	13	a	a	DET
easat-1685	7	14	-	-	PUNCT
easat-1685	7	15	open	open	ADJ
easat-1685	7	16	set	set	NOUN
easat-1685	7	17	,	,	PUNCT
easat-1685	7	18	covering	cover	VERB
easat-1685	7	19	property	property	NOUN
easat-1685	7	20	.	.	PUNCT
easat-1685	8	1	1	1	X
easat-1685	8	2	.	.	X
easat-1685	8	3	introduction	introduction	NOUN
easat-1685	8	4	the	the	DET
easat-1685	8	5	classical	classical	ADJ
easat-1685	8	6	hurewicz	hurewicz	NOUN
easat-1685	8	7	property	property	NOUN
easat-1685	8	8	has	have	VERB
easat-1685	8	9	a	a	DET
easat-1685	8	10	long	long	ADJ
easat-1685	8	11	history	history	NOUN
easat-1685	8	12	from	from	ADP
easat-1685	8	13	the	the	DET
easat-1685	8	14	paper	paper	NOUN
easat-1685	9	1	[	[	X
easat-1685	9	2	1	1	NUM
easat-1685	9	3	]	]	PUNCT
easat-1685	9	4	.	.	PUNCT
easat-1685	10	1	a	a	DET
easat-1685	10	2	topological	topological	ADJ
easat-1685	10	3	space	space	NOUN
easat-1685	10	4	𝑋	𝑋	NOUN
easat-1685	10	5	has	have	VERB
easat-1685	10	6	hurewicz	hurewicz	VERB
easat-1685	10	7	property	property	NOUN
easat-1685	10	8	if	if	SCONJ
easat-1685	10	9	for	for	ADP
easat-1685	10	10	each	each	DET
easat-1685	10	11	sequence	sequence	NOUN
easat-1685	10	12	(	(	PUNCT
easat-1685	10	13	𝑈𝑛)𝑛∈ℕ	𝑈𝑛)𝑛∈ℕ	NOUN
easat-1685	10	14	of	of	ADP
easat-1685	10	15	open	open	ADJ
easat-1685	10	16	covers	cover	NOUN
easat-1685	10	17	of	of	ADP
easat-1685	10	18	𝑋	𝑋	NOUN
easat-1685	10	19	there	there	PRON
easat-1685	10	20	exists	exist	VERB
easat-1685	10	21	a	a	DET
easat-1685	10	22	sequence	sequence	NOUN
easat-1685	10	23	(	(	PUNCT
easat-1685	10	24	𝑉𝑛)𝑛∈ℕ	𝑉𝑛)𝑛∈ℕ	VERB
easat-1685	10	25	where	where	SCONJ
easat-1685	10	26	for	for	ADP
easat-1685	10	27	each	each	PRON
easat-1685	10	28	𝑛	𝑛	PRON
easat-1685	10	29	∈	∈	PROPN
easat-1685	10	30	ℕ	ℕ	PROPN
easat-1685	10	31	𝑘	𝑘	NOUN
easat-1685	10	32	,	,	PUNCT
easat-1685	10	33	𝑉𝑛	𝑉𝑛	PRON
easat-1685	10	34	is	be	AUX
easat-1685	10	35	a	a	DET
easat-1685	10	36	finite	finite	NOUN
easat-1685	10	37	subset	subset	NOUN
easat-1685	10	38	of	of	ADP
easat-1685	10	39	𝑈𝑛	𝑈𝑛	PROPN
easat-1685	10	40	such	such	ADJ
easat-1685	10	41	that	that	PRON
easat-1685	10	42	for	for	ADP
easat-1685	10	43	each	each	DET
easat-1685	10	44	𝑥	𝑥	PRON
easat-1685	10	45	∈	∈	PROPN
easat-1685	10	46	𝑋	𝑋	PROPN
easat-1685	10	47	,	,	PUNCT
easat-1685	10	48	𝑥	𝑥	PRON
easat-1685	10	49	∈	∈	NOUN
easat-1685	10	50	⋃	⋃	NOUN
easat-1685	10	51	𝑉𝑛	𝑉𝑛	NOUN
easat-1685	10	52	for	for	ADP
easat-1685	10	53	all	all	PRON
easat-1685	10	54	but	but	CCONJ
easat-1685	10	55	finitely	finitely	ADV
easat-1685	10	56	many	many	ADJ
easat-1685	10	57	𝑛.	𝑛.	NOUN
easat-1685	10	58	recently	recently	ADV
easat-1685	10	59	,	,	PUNCT
easat-1685	10	60	several	several	ADJ
easat-1685	10	61	weak	weak	ADJ
easat-1685	10	62	variants	variant	NOUN
easat-1685	10	63	of	of	ADP
easat-1685	10	64	hurewicz	hurewicz	NOUN
easat-1685	10	65	property	property	NOUN
easat-1685	10	66	have	have	AUX
easat-1685	10	67	been	be	AUX
easat-1685	10	68	studied	study	VERB
easat-1685	10	69	after	after	ADP
easat-1685	10	70	applying	apply	VERB
easat-1685	10	71	the	the	DET
easat-1685	10	72	interior	interior	NOUN
easat-1685	10	73	and	and	CCONJ
easat-1685	10	74	the	the	DET
easat-1685	10	75	closure	closure	NOUN
easat-1685	10	76	operators	operator	NOUN
easat-1685	10	77	in	in	ADP
easat-1685	10	78	the	the	DET
easat-1685	10	79	definition	definition	NOUN
easat-1685	10	80	of	of	ADP
easat-1685	10	81	a	a	DET
easat-1685	10	82	hurewicz	hurewicz	NOUN
easat-1685	10	83	property	property	NOUN
easat-1685	10	84	.	.	PUNCT
easat-1685	11	1	also	also	ADV
easat-1685	11	2	,	,	PUNCT
easat-1685	11	3	the	the	DET
easat-1685	11	4	other	other	ADJ
easat-1685	11	5	ways	way	NOUN
easat-1685	11	6	have	have	AUX
easat-1685	11	7	been	be	AUX
easat-1685	11	8	examined	examine	VERB
easat-1685	11	9	when	when	SCONJ
easat-1685	11	10	the	the	DET
easat-1685	11	11	sequence	sequence	NOUN
easat-1685	11	12	of	of	ADP
easat-1685	11	13	open	open	ADJ
easat-1685	11	14	covers	cover	NOUN
easat-1685	11	15	are	be	AUX
easat-1685	11	16	replaced	replace	VERB
easat-1685	11	17	with	with	ADP
easat-1685	11	18	generalized	generalized	ADJ
easat-1685	11	19	open	open	ADJ
easat-1685	11	20	sets	set	NOUN
easat-1685	11	21	.	.	PUNCT
easat-1685	12	1	for	for	ADP
easat-1685	12	2	the	the	DET
easat-1685	12	3	study	study	NOUN
easat-1685	12	4	of	of	ADP
easat-1685	12	5	the	the	DET
easat-1685	12	6	variants	variant	NOUN
easat-1685	12	7	of	of	ADP
easat-1685	12	8	hurewicz	hurewicz	NOUN
easat-1685	12	9	spaces	space	NOUN
easat-1685	12	10	,	,	PUNCT
easat-1685	12	11	the	the	DET
easat-1685	12	12	readers	reader	NOUN
easat-1685	12	13	can	can	AUX
easat-1685	12	14	see	see	VERB
easat-1685	12	15	[	[	X
easat-1685	12	16	2	2	NUM
easat-1685	12	17	,	,	PUNCT
easat-1685	12	18	3	3	NUM
easat-1685	12	19	,	,	PUNCT
easat-1685	12	20	4	4	NUM
easat-1685	12	21	,	,	PUNCT
easat-1685	12	22	5	5	NUM
easat-1685	12	23	]	]	PUNCT
easat-1685	12	24	.	.	PUNCT
easat-1685	13	1	some	some	DET
easat-1685	13	2	types	type	NOUN
easat-1685	13	3	of	of	ADP
easat-1685	13	4	sets	set	NOUN
easat-1685	13	5	play	play	VERB
easat-1685	13	6	an	an	DET
easat-1685	13	7	important	important	ADJ
easat-1685	13	8	role	role	NOUN
easat-1685	13	9	in	in	ADP
easat-1685	13	10	the	the	DET
easat-1685	13	11	study	study	NOUN
easat-1685	13	12	of	of	ADP
easat-1685	13	13	various	various	ADJ
easat-1685	13	14	properties	property	NOUN
easat-1685	13	15	in	in	ADP
easat-1685	13	16	topological	topological	ADJ
easat-1685	13	17	spaces	space	NOUN
easat-1685	13	18	.	.	PUNCT
easat-1685	14	1	many	many	ADJ
easat-1685	14	2	authors	author	NOUN
easat-1685	14	3	introduced	introduce	VERB
easat-1685	14	4	and	and	CCONJ
easat-1685	14	5	studied	study	VERB
easat-1685	14	6	various	various	ADJ
easat-1685	14	7	generalized	generalized	ADJ
easat-1685	14	8	properties	property	NOUN
easat-1685	14	9	and	and	CCONJ
easat-1685	14	10	conditions	condition	NOUN
easat-1685	14	11	containing	contain	VERB
easat-1685	14	12	some	some	DET
easat-1685	14	13	forms	form	NOUN
easat-1685	14	14	of	of	ADP
easat-1685	14	15	sets	set	NOUN
easat-1685	14	16	in	in	ADP
easat-1685	14	17	topological	topological	ADJ
easat-1685	14	18	spaces	space	NOUN
easat-1685	14	19	.	.	PUNCT
easat-1685	15	1	in	in	ADP
easat-1685	15	2	this	this	DET
easat-1685	15	3	paper	paper	NOUN
easat-1685	15	4	,	,	PUNCT
easat-1685	15	5	we	we	PRON
easat-1685	15	6	investigate	investigate	VERB
easat-1685	15	7	some	some	DET
easat-1685	15	8	properties	property	NOUN
easat-1685	15	9	of	of	ADP
easat-1685	15	10	𝒶-open	𝒶-open	NOUN
easat-1685	15	11	sets	set	NOUN
easat-1685	15	12	.	.	PUNCT
easat-1685	16	1	moreover	moreover	ADV
easat-1685	16	2	,	,	PUNCT
easat-1685	16	3	the	the	DET
easat-1685	16	4	relationships	relationship	NOUN
easat-1685	16	5	among	among	ADP
easat-1685	16	6	open	open	ADJ
easat-1685	16	7	sets	set	NOUN
easat-1685	16	8	,	,	PUNCT
easat-1685	16	9	𝑎-open	𝑎-open	PROPN
easat-1685	16	10	sets	set	NOUN
easat-1685	16	11	and	and	CCONJ
easat-1685	16	12	the	the	DET
easat-1685	16	13	related	related	ADJ
easat-1685	16	14	classes	class	NOUN
easat-1685	16	15	of	of	ADP
easat-1685	16	16	sets	set	NOUN
easat-1685	16	17	are	be	AUX
easat-1685	16	18	investigated	investigate	VERB
easat-1685	16	19	.	.	PUNCT
easat-1685	17	1	in	in	ADP
easat-1685	17	2	this	this	DET
easat-1685	17	3	paper	paper	NOUN
easat-1685	17	4	,	,	PUNCT
easat-1685	17	5	spaces	space	VERB
easat-1685	17	6	𝑋	𝑋	NOUN
easat-1685	17	7	and	and	CCONJ
easat-1685	17	8	𝑌	𝑌	PROPN
easat-1685	17	9	mean	mean	VERB
easat-1685	17	10	topological	topological	ADJ
easat-1685	17	11	spaces	space	NOUN
easat-1685	17	12	.	.	PUNCT
easat-1685	18	1	for	for	ADP
easat-1685	18	2	a	a	DET
easat-1685	18	3	subset	subset	ADJ
easat-1685	18	4	𝐴	𝐴	NOUN
easat-1685	18	5	of	of	ADP
easat-1685	18	6	a	a	DET
easat-1685	18	7	space	space	NOUN
easat-1685	18	8	𝑋	𝑋	NOUN
easat-1685	18	9	,	,	PUNCT
easat-1685	18	10	𝑐𝑙(𝐴	𝑐𝑙(𝐴	PROPN
easat-1685	18	11	)	)	PUNCT
easat-1685	18	12	and	and	CCONJ
easat-1685	18	13	𝑖𝑛𝑡(𝐴	𝑖𝑛𝑡(𝐴	NUM
easat-1685	18	14	)	)	PUNCT
easat-1685	18	15	represent	represent	VERB
easat-1685	18	16	the	the	DET
easat-1685	18	17	closure	closure	NOUN
easat-1685	18	18	of	of	ADP
easat-1685	18	19	𝐴	𝐴	PROPN
easat-1685	18	20	and	and	CCONJ
easat-1685	18	21	the	the	DET
easat-1685	18	22	interior	interior	NOUN
easat-1685	18	23	of	of	ADP
easat-1685	18	24	𝐴	𝐴	PROPN
easat-1685	18	25	,	,	PUNCT
easat-1685	18	26	respectively	respectively	ADV
easat-1685	18	27	.	.	PUNCT
easat-1685	19	1	in	in	ADP
easat-1685	19	2	this	this	DET
easat-1685	19	3	paper	paper	NOUN
easat-1685	19	4	,	,	PUNCT
easat-1685	19	5	we	we	PRON
easat-1685	19	6	examine	examine	VERB
easat-1685	19	7	the	the	DET
easat-1685	19	8	covering	covering	NOUN
easat-1685	19	9	properties	property	NOUN
easat-1685	19	10	namely	namely	ADV
easat-1685	19	11	:	:	PUNCT
easat-1685	19	12	𝒶-hurewicz	𝒶-hurewicz	NUM
easat-1685	19	13	,	,	PUNCT
easat-1685	19	14	which	which	PRON
easat-1685	19	15	is	be	AUX
easat-1685	19	16	a	a	DET
easat-1685	19	17	like	like	NOUN
easat-1685	19	18	to	to	ADP
easat-1685	19	19	the	the	DET
easat-1685	19	20	classical	classical	ADJ
easat-1685	19	21	hurewicz	hurewicz	NOUN
easat-1685	19	22	property	property	NOUN
easat-1685	19	23	by	by	ADP
easat-1685	19	24	using	use	VERB
easat-1685	19	25	𝒶-open	𝒶-open	NOUN
easat-1685	19	26	sets	set	NOUN
easat-1685	19	27	see	see	VERB
easat-1685	19	28	[	[	X
easat-1685	19	29	6	6	NUM
easat-1685	19	30	,	,	PUNCT
easat-1685	19	31	7	7	NUM
easat-1685	19	32	,	,	PUNCT
easat-1685	19	33	8	8	NUM
easat-1685	19	34	,	,	PUNCT
easat-1685	19	35	9	9	NUM
easat-1685	19	36	]	]	PUNCT
easat-1685	19	37	.	.	PUNCT
easat-1685	20	1	the	the	DET
easat-1685	20	2	following	follow	VERB
easat-1685	20	3	generalizations	generalization	NOUN
easat-1685	20	4	of	of	ADP
easat-1685	20	5	open	open	ADJ
easat-1685	20	6	sets	set	NOUN
easat-1685	20	7	will	will	AUX
easat-1685	20	8	be	be	AUX
easat-1685	20	9	used	use	VERB
easat-1685	20	10	for	for	ADP
easat-1685	20	11	definitions	definition	NOUN
easat-1685	20	12	of	of	ADP
easat-1685	20	13	variations	variation	NOUN
easat-1685	20	14	on	on	ADP
easat-1685	20	15	the	the	DET
easat-1685	20	16	hurewicz	hurewicz	NOUN
easat-1685	20	17	property	property	NOUN
easat-1685	20	18	:	:	PUNCT
easat-1685	20	19	the	the	DET
easat-1685	20	20	paper	paper	NOUN
easat-1685	20	21	is	be	AUX
easat-1685	20	22	organized	organize	VERB
easat-1685	20	23	in	in	ADP
easat-1685	20	24	such	such	DET
easat-1685	20	25	a	a	DET
easat-1685	20	26	way	way	NOUN
easat-1685	20	27	that	that	SCONJ
easat-1685	20	28	after	after	ADP
easat-1685	20	29	this	this	DET
easat-1685	20	30	introduction	introduction	NOUN
easat-1685	20	31	in	in	ADP
easat-1685	20	32	section	section	NOUN
easat-1685	20	33	two	two	NUM
easat-1685	20	34	we	we	PRON
easat-1685	20	35	give	give	VERB
easat-1685	20	36	information	information	NOUN
easat-1685	20	37	about	about	ADP
easat-1685	20	38	terminology	terminology	NOUN
easat-1685	20	39	and	and	CCONJ
easat-1685	20	40	notation	notation	NOUN
easat-1685	20	41	.	.	PUNCT
easat-1685	21	1	in	in	ADP
easat-1685	21	2	section	section	NOUN
easat-1685	21	3	3and	3and	NUM
easat-1685	21	4	4	4	NUM
easat-1685	21	5	we	we	PRON
easat-1685	21	6	show	show	VERB
easat-1685	21	7	that	that	SCONJ
easat-1685	21	8	we	we	PRON
easat-1685	21	9	can	can	AUX
easat-1685	21	10	replace	replace	VERB
easat-1685	21	11	open	open	ADJ
easat-1685	21	12	sets	set	NOUN
easat-1685	21	13	with	with	ADP
easat-1685	21	14	𝒶-open	𝒶-open	NOUN
easat-1685	21	15	sets	set	NOUN
easat-1685	21	16	in	in	ADP
easat-1685	21	17	the	the	DET
easat-1685	21	18	definition	definition	NOUN
easat-1685	21	19	of	of	ADP
easat-1685	21	20	𝒶-hurewicz	𝒶-hurewicz	PROPN
easat-1685	21	21	spaces	space	NOUN
easat-1685	21	22	.	.	PUNCT
easat-1685	22	1	also	also	ADV
easat-1685	22	2	,	,	PUNCT
easat-1685	22	3	we	we	PRON
easat-1685	22	4	investigate	investigate	VERB
easat-1685	22	5	the	the	DET
easat-1685	22	6	behavior	behavior	NOUN
easat-1685	22	7	of	of	ADP
easat-1685	22	8	𝒶-hurewicz	𝒶-hurewicz	NOUN
easat-1685	22	9	properties	property	NOUN
easat-1685	22	10	with	with	ADP
easat-1685	22	11	respect	respect	NOUN
easat-1685	22	12	to	to	ADP
easat-1685	22	13	subspaces	subspace	NOUN
easat-1685	22	14	,	,	PUNCT
easat-1685	22	15	products	product	NOUN
easat-1685	22	16	and	and	CCONJ
easat-1685	22	17	𝒶-continuous	𝒶-continuous	ADJ
easat-1685	22	18	image	image	NOUN
easat-1685	22	19	.	.	PUNCT
easat-1685	23	1	2	2	X
easat-1685	23	2	.	.	X
easat-1685	23	3	background	background	NOUN
easat-1685	23	4	material	material	NOUN
easat-1685	23	5	the	the	DET
easat-1685	23	6	interior	interior	ADJ
easat-1685	23	7	and	and	CCONJ
easat-1685	23	8	closure	closure	NOUN
easat-1685	23	9	operators	operator	NOUN
easat-1685	23	10	in	in	ADP
easat-1685	23	11	topological	topological	ADJ
easat-1685	23	12	spaces	space	NOUN
easat-1685	23	13	play	play	VERB
easat-1685	23	14	a	a	DET
easat-1685	23	15	vital	vital	ADJ
easat-1685	23	16	role	role	NOUN
easat-1685	23	17	in	in	ADP
easat-1685	23	18	the	the	DET
easat-1685	23	19	generalization	generalization	NOUN
easat-1685	23	20	of	of	ADP
easat-1685	23	21	open	open	ADJ
easat-1685	23	22	sets	set	NOUN
easat-1685	23	23	and	and	CCONJ
easat-1685	23	24	closed	closed	ADJ
easat-1685	23	25	sets	set	NOUN
easat-1685	23	26	.	.	PUNCT
easat-1685	24	1	the	the	DET
easat-1685	24	2	relations	relation	NOUN
easat-1685	24	3	on	on	ADP
easat-1685	24	4	the	the	DET
easat-1685	24	5	interior	interior	ADJ
easat-1685	24	6	and	and	CCONJ
easat-1685	24	7	closure	closure	NOUN
easat-1685	24	8	operators	operator	NOUN
easat-1685	24	9	motivate	motivate	VERB
easat-1685	24	10	the	the	DET
easat-1685	24	11	point	point	NOUN
easat-1685	24	12	set	set	VERB
easat-1685	24	13	topologists	topologist	NOUN
easat-1685	24	14	to	to	PART
easat-1685	24	15	introduce	introduce	VERB
easat-1685	24	16	several	several	ADJ
easat-1685	24	17	forms	form	NOUN
easat-1685	24	18	of	of	ADP
easat-1685	24	19	𝒶-open	𝒶-open	NOUN
easat-1685	24	20	sets	set	NOUN
easat-1685	24	21	and	and	CCONJ
easat-1685	24	22	𝒶-closed	𝒶-close	VERB
easat-1685	24	23	sets	set	NOUN
easat-1685	24	24	.	.	PUNCT
easat-1685	25	1	some	some	PRON
easat-1685	25	2	of	of	ADP
easat-1685	25	3	them	they	PRON
easat-1685	25	4	are	be	AUX
easat-1685	25	5	given	give	VERB
easat-1685	25	6	in	in	ADP
easat-1685	25	7	the	the	DET
easat-1685	25	8	next	next	ADJ
easat-1685	25	9	definition	definition	NOUN
easat-1685	25	10	.	.	PUNCT
easat-1685	26	1	272	272	NUM
easat-1685	26	2	edelweiss	edelweiss	PROPN
easat-1685	26	3	applied	apply	VERB
easat-1685	26	4	science	science	NOUN
easat-1685	26	5	and	and	CCONJ
easat-1685	26	6	technology	technology	NOUN
easat-1685	26	7	issn	issn	PROPN
easat-1685	26	8	:	:	PUNCT
easat-1685	26	9	2576	2576	NUM
easat-1685	26	10	-	-	SYM
easat-1685	26	11	8484	8484	NUM
easat-1685	26	12	vol	vol	NOUN
easat-1685	26	13	.	.	PROPN
easat-1685	26	14	8	8	NUM
easat-1685	26	15	,	,	PUNCT
easat-1685	26	16	no	no	INTJ
easat-1685	26	17	.	.	NOUN
easat-1685	26	18	5	5	NUM
easat-1685	26	19	:	:	PUNCT
easat-1685	26	20	271	271	NUM
easat-1685	26	21	-	-	SYM
easat-1685	26	22	277	277	NUM
easat-1685	26	23	,	,	PUNCT
easat-1685	26	24	2024	2024	NUM
easat-1685	26	25	doi	doi	NOUN
easat-1685	26	26	:	:	PUNCT
easat-1685	26	27	10.55214/25768484.v8i5.1685	10.55214/25768484.v8i5.1685	NUM
easat-1685	26	28	©	©	ADP
easat-1685	26	29	2024	2024	NUM
easat-1685	26	30	by	by	ADP
easat-1685	26	31	the	the	DET
easat-1685	26	32	authors	author	NOUN
easat-1685	26	33	;	;	PUNCT
easat-1685	26	34	licensee	licensee	PROPN
easat-1685	26	35	learning	learning	NOUN
easat-1685	26	36	gate	gate	PROPN
easat-1685	26	37	definition	definition	NOUN
easat-1685	26	38	2.1	2.1	NUM
easat-1685	26	39	:	:	PUNCT
easat-1685	27	1	[	[	X
easat-1685	27	2	6,7	6,7	NUM
easat-1685	27	3	,	,	PUNCT
easat-1685	27	4	8	8	NUM
easat-1685	27	5	]	]	PUNCT
easat-1685	27	6	a	a	DET
easat-1685	27	7	subset	subset	NOUN
easat-1685	27	8	𝑈	𝑈	PROPN
easat-1685	27	9	of	of	ADP
easat-1685	27	10	a	a	DET
easat-1685	27	11	𝑇.	𝑇.	PROPN
easat-1685	27	12	𝑠	𝑠	PROPN
easat-1685	27	13	(	(	PUNCT
easat-1685	27	14	𝑋	𝑋	PROPN
easat-1685	27	15	,	,	PUNCT
easat-1685	27	16	𝒯	𝒯	PROPN
easat-1685	27	17	)	)	PUNCT
easat-1685	27	18	is	be	AUX
easat-1685	27	19	called	call	VERB
easat-1685	27	20	:	:	PUNCT
easat-1685	27	21	i.	i.	PROPN
easat-1685	27	22	regular	regular	PROPN
easat-1685	27	23	open	open	ADJ
easat-1685	27	24	(	(	PUNCT
easat-1685	27	25	𝑟-open	𝑟-open	NUM
easat-1685	27	26	)	)	PUNCT
easat-1685	27	27	,	,	PUNCT
easat-1685	27	28	if	if	SCONJ
easat-1685	27	29	𝐴	𝐴	PROPN
easat-1685	27	30	=	=	SYM
easat-1685	27	31	𝑖𝑛𝑡(𝑐𝑙(𝐴	𝑖𝑛𝑡(𝑐𝑙(𝐴	NUM
easat-1685	27	32	)	)	PUNCT
easat-1685	27	33	)	)	PUNCT
easat-1685	27	34	.	.	PUNCT
easat-1685	28	1	ii	ii	PROPN
easat-1685	28	2	.	.	PUNCT
easat-1685	29	1	𝛿-interior	𝛿-interior	PROPN
easat-1685	29	2	of	of	ADP
easat-1685	29	3	a	a	DET
easat-1685	29	4	subset	subset	ADJ
easat-1685	29	5	𝐴	𝐴	PROPN
easat-1685	29	6	of	of	ADP
easat-1685	29	7	𝑋	𝑋	PROPN
easat-1685	29	8	is	be	AUX
easat-1685	29	9	the	the	DET
easat-1685	29	10	union	union	NOUN
easat-1685	29	11	of	of	ADP
easat-1685	29	12	all	all	DET
easat-1685	29	13	𝑟-open	𝑟-open	NOUN
easat-1685	29	14	set	set	NOUN
easat-1685	29	15	of	of	ADP
easat-1685	29	16	𝑋	𝑋	PROPN
easat-1685	29	17	contained	contain	VERB
easat-1685	29	18	in	in	ADP
easat-1685	29	19	𝐴	𝐴	PROPN
easat-1685	29	20	and	and	CCONJ
easat-1685	29	21	it	it	PRON
easat-1685	29	22	is	be	AUX
easat-1685	29	23	denoted	denote	VERB
easat-1685	29	24	by	by	ADP
easat-1685	29	25	𝛿𝑖𝑛𝑡(𝐴	𝛿𝑖𝑛𝑡(𝐴	NOUN
easat-1685	29	26	)	)	PUNCT
easat-1685	29	27	.	.	PUNCT
easat-1685	30	1	iii	iii	X
easat-1685	30	2	.	.	PUNCT
easat-1685	30	3	𝛿-open	𝛿-open	VERB
easat-1685	30	4	if	if	SCONJ
easat-1685	30	5	𝐴	𝐴	PROPN
easat-1685	30	6	=	=	SYM
easat-1685	30	7	𝛿-𝑖𝑛𝑡(𝐴	𝛿-𝑖𝑛𝑡(𝐴	PROPN
easat-1685	30	8	)	)	PUNCT
easat-1685	30	9	.	.	PUNCT
easat-1685	31	1	iv	iv	X
easat-1685	31	2	.	.	PUNCT
easat-1685	32	1	the	the	DET
easat-1685	32	2	𝛿-closure	𝛿-closure	NOUN
easat-1685	32	3	of	of	ADP
easat-1685	32	4	a	a	DET
easat-1685	32	5	set	set	ADJ
easat-1685	32	6	𝐴	𝐴	PROPN
easat-1685	32	7	in	in	ADP
easat-1685	32	8	𝑋	𝑋	PROPN
easat-1685	32	9	denoted	denote	VERB
easat-1685	32	10	𝛿-𝑐𝑙(𝐴	𝛿-𝑐𝑙(𝐴	NOUN
easat-1685	32	11	)	)	PUNCT
easat-1685	32	12	and	and	CCONJ
easat-1685	32	13	defined	define	VERB
easat-1685	32	14	by	by	ADP
easat-1685	32	15	:	:	PUNCT
easat-1685	32	16	{	{	PUNCT
easat-1685	32	17	𝑥	𝑥	PRON
easat-1685	32	18	∈	∈	NOUN
easat-1685	32	19	𝑋	𝑋	PROPN
easat-1685	32	20	∶	∶	NOUN
easat-1685	32	21	𝐴⋂𝑖𝑛𝑡(𝑐𝑙(𝐵	𝐴⋂𝑖𝑛𝑡(𝑐𝑙(𝐵	ADV
easat-1685	32	22	)	)	PUNCT
easat-1685	32	23	)	)	PUNCT
easat-1685	33	1	≠	≠	PROPN
easat-1685	33	2	∅	∅	NOUN
easat-1685	33	3	,	,	PUNCT
easat-1685	33	4	𝐵	𝐵	PROPN
easat-1685	33	5	∈	∈	PROPN
easat-1685	33	6	𝒯	𝒯	PROPN
easat-1685	33	7	and	and	CCONJ
easat-1685	33	8	𝑥	𝑥	DET
easat-1685	33	9	∈	∈	PROPN
easat-1685	33	10	𝐵	𝐵	PROPN
easat-1685	33	11	}	}	PUNCT
easat-1685	33	12	.	.	PUNCT
easat-1685	34	1	iv	iv	X
easat-1685	34	2	.	.	X
easat-1685	34	3	𝒶-open	𝒶-open	PROPN
easat-1685	34	4	,	,	PUNCT
easat-1685	34	5	if	if	SCONJ
easat-1685	34	6	𝐴	𝐴	PROPN
easat-1685	34	7	⊆	⊆	NUM
easat-1685	34	8	𝑖𝑛𝑡(𝑐𝑙(𝛿-𝑖𝑛𝑡(𝐴	𝑖𝑛𝑡(𝑐𝑙(𝛿-𝑖𝑛𝑡(𝐴	NUM
easat-1685	34	9	)	)	PUNCT
easat-1685	34	10	)	)	PUNCT
easat-1685	34	11	)	)	PUNCT
easat-1685	34	12	.	.	PUNCT
easat-1685	35	1	example	example	NOUN
easat-1685	35	2	2.2	2.2	NUM
easat-1685	35	3	:	:	PUNCT
easat-1685	35	4	i.	i.	NOUN
easat-1685	35	5	in	in	ADP
easat-1685	35	6	(	(	PUNCT
easat-1685	35	7	ℝ	ℝ	PROPN
easat-1685	35	8	,	,	PUNCT
easat-1685	35	9	𝒯𝑈	𝒯𝑈	PROPN
easat-1685	35	10	)	)	PUNCT
easat-1685	35	11	a	a	DET
easat-1685	35	12	subset	subset	NOUN
easat-1685	35	13	(	(	PUNCT
easat-1685	35	14	0	0	NUM
easat-1685	35	15	,	,	PUNCT
easat-1685	35	16	1	1	NUM
easat-1685	35	17	)	)	PUNCT
easat-1685	35	18	is	be	AUX
easat-1685	35	19	an	an	DET
easat-1685	35	20	𝒶-open	𝒶-open	NOUN
easat-1685	35	21	.	.	PUNCT
easat-1685	36	1	ii	ii	PROPN
easat-1685	36	2	.	.	PUNCT
easat-1685	37	1	let	let	VERB
easat-1685	37	2	𝑋	𝑋	PROPN
easat-1685	37	3	=	=	SYM
easat-1685	37	4	{	{	PUNCT
easat-1685	37	5	𝑎	𝑎	PROPN
easat-1685	37	6	,	,	PUNCT
easat-1685	37	7	𝑏	𝑏	NOUN
easat-1685	37	8	,	,	PUNCT
easat-1685	37	9	𝑐	𝑐	NOUN
easat-1685	37	10	}	}	PUNCT
easat-1685	37	11	with	with	ADP
easat-1685	37	12	𝒯	𝒯	PROPN
easat-1685	37	13	=	=	PROPN
easat-1685	37	14	{	{	PUNCT
easat-1685	37	15	∅	∅	NOUN
easat-1685	37	16	,	,	PUNCT
easat-1685	37	17	𝑋	𝑋	PROPN
easat-1685	37	18	,	,	PUNCT
easat-1685	37	19	{	{	PUNCT
easat-1685	37	20	𝑎	𝑎	NOUN
easat-1685	37	21	}	}	PUNCT
easat-1685	37	22	,	,	PUNCT
easat-1685	37	23	{	{	PUNCT
easat-1685	37	24	𝑏	𝑏	NOUN
easat-1685	37	25	}	}	PUNCT
easat-1685	37	26	,	,	PUNCT
easat-1685	37	27	{	{	PUNCT
easat-1685	37	28	𝑎	𝑎	X
easat-1685	37	29	,	,	PUNCT
easat-1685	37	30	𝑏	𝑏	NOUN
easat-1685	37	31	}	}	PUNCT
easat-1685	37	32	}	}	PUNCT
easat-1685	37	33	.	.	PUNCT
easat-1685	38	1	a	a	DET
easat-1685	38	2	subset	subset	NOUN
easat-1685	38	3	{	{	PUNCT
easat-1685	38	4	𝑏	𝑏	NOUN
easat-1685	38	5	}	}	PUNCT
easat-1685	38	6	is	be	AUX
easat-1685	38	7	not	not	PART
easat-1685	38	8	𝒶-open	𝒶-open	ADJ
easat-1685	38	9	.	.	PUNCT
easat-1685	39	1	theorem	theorem	VERB
easat-1685	39	2	2.3	2.3	NUM
easat-1685	39	3	:	:	PUNCT
easat-1685	40	1	[	[	X
easat-1685	40	2	6	6	NUM
easat-1685	40	3	]	]	PUNCT
easat-1685	40	4	a	a	DET
easat-1685	40	5	subset	subset	NOUN
easat-1685	40	6	𝑈	𝑈	PROPN
easat-1685	40	7	of	of	ADP
easat-1685	40	8	a	a	DET
easat-1685	40	9	𝑇.	𝑇.	PROPN
easat-1685	40	10	𝑠	𝑠	PROPN
easat-1685	40	11	(	(	PUNCT
easat-1685	40	12	𝑋	𝑋	PROPN
easat-1685	40	13	,	,	PUNCT
easat-1685	40	14	𝒯	𝒯	PROPN
easat-1685	40	15	)	)	PUNCT
easat-1685	40	16	is	be	AUX
easat-1685	40	17	an	an	DET
easat-1685	40	18	𝒶-open	𝒶-open	NOUN
easat-1685	40	19	,	,	PUNCT
easat-1685	40	20	if	if	SCONJ
easat-1685	40	21	and	and	CCONJ
easat-1685	40	22	only	only	ADV
easat-1685	40	23	if	if	SCONJ
easat-1685	40	24	for	for	ADP
easat-1685	40	25	each	each	PRON
easat-1685	40	26	𝑥	𝑥	PRON
easat-1685	40	27	∈	∈	PROPN
easat-1685	40	28	𝑈	𝑈	PROPN
easat-1685	40	29	there	there	PRON
easat-1685	40	30	exists	exist	VERB
easat-1685	40	31	𝛿open	𝛿open	ADJ
easat-1685	40	32	set	set	VERB
easat-1685	40	33	𝑃	𝑃	NOUN
easat-1685	40	34	of	of	ADP
easat-1685	40	35	𝑋	𝑋	NOUN
easat-1685	40	36	such	such	ADJ
easat-1685	40	37	that	that	SCONJ
easat-1685	40	38	𝑥	𝑥	DET
easat-1685	40	39	∈	∈	NOUN
easat-1685	40	40	𝑃	𝑃	VERB
easat-1685	40	41	⊆	⊆	NUM
easat-1685	40	42	𝑈.	𝑈.	PROPN
easat-1685	40	43	remark	remark	NOUN
easat-1685	40	44	2.4	2.4	NUM
easat-1685	40	45	:	:	PUNCT
easat-1685	41	1	[	[	X
easat-1685	41	2	6	6	NUM
easat-1685	41	3	]	]	PUNCT
easat-1685	41	4	i.	i.	NOUN
easat-1685	41	5	the	the	DET
easat-1685	41	6	family	family	NOUN
easat-1685	41	7	of	of	ADP
easat-1685	41	8	all	all	DET
easat-1685	41	9	𝒶-open	𝒶-open	NOUN
easat-1685	41	10	sets	set	NOUN
easat-1685	41	11	of	of	ADP
easat-1685	41	12	a	a	DET
easat-1685	41	13	𝑇.	𝑇.	PROPN
easat-1685	41	14	𝑠	𝑠	PROPN
easat-1685	41	15	(	(	PUNCT
easat-1685	41	16	𝑋	𝑋	PROPN
easat-1685	41	17	,	,	PUNCT
easat-1685	41	18	𝒯	𝒯	PROPN
easat-1685	41	19	)	)	PUNCT
easat-1685	41	20	forms	form	VERB
easat-1685	41	21	a	a	DET
easat-1685	41	22	topology	topology	NOUN
easat-1685	41	23	on	on	ADP
easat-1685	41	24	𝑋	𝑋	PROPN
easat-1685	41	25	,	,	PUNCT
easat-1685	41	26	denoted	denote	VERB
easat-1685	41	27	by	by	ADP
easat-1685	41	28	𝒯𝒶.	𝒯𝒶.	PROPN
easat-1685	41	29	ii	ii	PROPN
easat-1685	41	30	.	.	PUNCT
easat-1685	42	1	for	for	ADP
easat-1685	42	2	any	any	DET
easat-1685	42	3	subset	subset	NOUN
easat-1685	42	4	of	of	ADP
easat-1685	42	5	a	a	DET
easat-1685	42	6	𝑇.	𝑇.	PROPN
easat-1685	42	7	𝑠	𝑠	PROPN
easat-1685	42	8	(	(	PUNCT
easat-1685	42	9	𝑋	𝑋	PROPN
easat-1685	42	10	,	,	PUNCT
easat-1685	42	11	𝒯	𝒯	PROPN
easat-1685	42	12	)	)	PUNCT
easat-1685	42	13	,	,	PUNCT
easat-1685	42	14	we	we	PRON
easat-1685	42	15	conclude	conclude	VERB
easat-1685	42	16	the	the	DET
easat-1685	42	17	following	follow	VERB
easat-1685	42	18	diagram	diagram	NOUN
easat-1685	42	19	:	:	PUNCT
easat-1685	42	20	definition	definition	NOUN
easat-1685	42	21	2.5	2.5	NUM
easat-1685	42	22	:	:	PUNCT
easat-1685	42	23	for	for	ADP
easat-1685	42	24	any	any	DET
easat-1685	42	25	subset	subset	NOUN
easat-1685	42	26	𝐴	𝐴	PROPN
easat-1685	42	27	a	a	DET
easat-1685	42	28	𝑇.	𝑇.	PROPN
easat-1685	42	29	𝑠	𝑠	PROPN
easat-1685	42	30	(	(	PUNCT
easat-1685	42	31	𝑋	𝑋	PROPN
easat-1685	42	32	,	,	PUNCT
easat-1685	42	33	𝒯	𝒯	PROPN
easat-1685	42	34	)	)	PUNCT
easat-1685	42	35	,	,	PUNCT
easat-1685	42	36	the	the	DET
easat-1685	42	37	following	follow	VERB
easat-1685	42	38	symbols	symbol	NOUN
easat-1685	42	39	denote	denote	VERB
easat-1685	42	40	:	:	PUNCT
easat-1685	42	41	i.	i.	PROPN
easat-1685	42	42	𝑐𝑙𝒶(𝐴	𝑐𝑙𝒶(𝐴	PROPN
easat-1685	42	43	)	)	PUNCT
easat-1685	42	44	is	be	AUX
easat-1685	42	45	the	the	DET
easat-1685	42	46	intersection	intersection	NOUN
easat-1685	42	47	of	of	ADP
easat-1685	42	48	all	all	DET
easat-1685	42	49	𝒶-closed	𝒶-close	VERB
easat-1685	42	50	subsets	subset	NOUN
easat-1685	42	51	of	of	ADP
easat-1685	42	52	𝑋	𝑋	NOUN
easat-1685	42	53	containing	contain	VERB
easat-1685	42	54	𝐴.	𝐴.	PROPN
easat-1685	42	55	ii	ii	PROPN
easat-1685	42	56	.	.	PUNCT
easat-1685	42	57	𝑖𝑛𝑡𝒶(𝐴	𝑖𝑛𝑡𝒶(𝐴	PROPN
easat-1685	42	58	)	)	PUNCT
easat-1685	43	1	is	be	AUX
easat-1685	43	2	the	the	DET
easat-1685	43	3	union	union	NOUN
easat-1685	43	4	of	of	ADP
easat-1685	43	5	all	all	DET
easat-1685	43	6	𝒶-open	𝒶-open	VERB
easat-1685	43	7	subsets	subset	NOUN
easat-1685	43	8	of	of	ADP
easat-1685	43	9	𝑋	𝑋	PROPN
easat-1685	43	10	contained	contain	VERB
easat-1685	43	11	in	in	ADP
easat-1685	43	12	𝐴.	𝐴.	PROPN
easat-1685	43	13	iii	iii	NOUN
easat-1685	43	14	.	.	PUNCT
easat-1685	44	1	𝐴	𝐴	PROPN
easat-1685	44	2	is	be	AUX
easat-1685	44	3	said	say	VERB
easat-1685	44	4	to	to	PART
easat-1685	44	5	be	be	AUX
easat-1685	44	6	𝒶-dense	𝒶-dense	PROPN
easat-1685	44	7	,	,	PUNCT
easat-1685	44	8	if	if	SCONJ
easat-1685	44	9	𝑐𝑙𝒶(𝐴	𝑐𝑙𝒶(𝐴	NUM
easat-1685	44	10	)	)	PUNCT
easat-1685	44	11	=	=	SYM
easat-1685	44	12	𝑋	𝑋	PROPN
easat-1685	44	13	.	.	PUNCT
easat-1685	45	1	recall	recall	VERB
easat-1685	45	2	that	that	SCONJ
easat-1685	45	3	a	a	DET
easat-1685	45	4	mapping	mapping	NOUN
easat-1685	45	5	𝑓	𝑓	NOUN
easat-1685	45	6	:	:	PUNCT
easat-1685	45	7	(	(	PUNCT
easat-1685	45	8	𝑋	𝑋	PROPN
easat-1685	45	9	,	,	PUNCT
easat-1685	45	10	𝒯	𝒯	PROPN
easat-1685	45	11	)	)	PUNCT
easat-1685	45	12	→	→	SYM
easat-1685	45	13	(	(	PUNCT
easat-1685	45	14	𝑌	𝑌	PROPN
easat-1685	45	15	,	,	PUNCT
easat-1685	45	16	𝒯′	𝒯′	NUM
easat-1685	45	17	)	)	PUNCT
easat-1685	45	18	is	be	AUX
easat-1685	45	19	said	say	VERB
easat-1685	45	20	to	to	PART
easat-1685	45	21	be	be	AUX
easat-1685	45	22			NUM
easat-1685	45	23	-continuous	-continuous	ADJ
easat-1685	45	24	,	,	PUNCT
easat-1685	45	25	if	if	SCONJ
easat-1685	45	26	𝑓−1(𝑈	𝑓−1(𝑈	NUM
easat-1685	45	27	)	)	PUNCT
easat-1685	45	28	is	be	AUX
easat-1685	45	29	𝛿-open	𝛿-open	VERB
easat-1685	45	30	set	set	NOUN
easat-1685	45	31	of	of	ADP
easat-1685	45	32	𝑋	𝑋	PROPN
easat-1685	45	33	for	for	ADP
easat-1685	45	34	every	every	DET
easat-1685	45	35	open	open	ADJ
easat-1685	45	36	set	set	VERB
easat-1685	45	37	𝑈	𝑈	PROPN
easat-1685	45	38	of	of	ADP
easat-1685	45	39	𝑌	𝑌	PROPN
easat-1685	45	40	,	,	PUNCT
easat-1685	45	41	[	[	X
easat-1685	45	42	9	9	NUM
easat-1685	45	43	]	]	PUNCT
easat-1685	45	44	.	.	PUNCT
easat-1685	46	1	definition	definition	NOUN
easat-1685	46	2	1.6	1.6	NUM
easat-1685	46	3	:	:	PUNCT
easat-1685	46	4	let	let	VERB
easat-1685	46	5	(	(	PUNCT
easat-1685	46	6	𝑋	𝑋	NOUN
easat-1685	46	7	,	,	PUNCT
easat-1685	46	8	𝒯	𝒯	PROPN
easat-1685	46	9	)	)	PUNCT
easat-1685	46	10	and	and	CCONJ
easat-1685	46	11	(	(	PUNCT
easat-1685	46	12	𝑌	𝑌	PROPN
easat-1685	46	13	,	,	PUNCT
easat-1685	46	14	𝒯′	𝒯′	PROPN
easat-1685	46	15	)	)	PUNCT
easat-1685	46	16	be	be	VERB
easat-1685	46	17	two	two	NUM
easat-1685	46	18	𝑇.	𝑇.	PROPN
easat-1685	46	19	𝑠	𝑠	ADP
easat-1685	46	20	'	'	PUNCT
easat-1685	46	21	s.	s.	PROPN
easat-1685	46	22	then	then	ADV
easat-1685	46	23	a	a	DET
easat-1685	46	24	mapping	mapping	NOUN
easat-1685	46	25	𝑓	𝑓	NOUN
easat-1685	46	26	:	:	PUNCT
easat-1685	46	27	(	(	PUNCT
easat-1685	46	28	𝑋	𝑋	PROPN
easat-1685	46	29	,	,	PUNCT
easat-1685	46	30	𝒯	𝒯	PROPN
easat-1685	46	31	)	)	PUNCT
easat-1685	46	32	→	→	SYM
easat-1685	46	33	(	(	PUNCT
easat-1685	46	34	𝑌	𝑌	PROPN
easat-1685	46	35	,	,	PUNCT
easat-1685	46	36	𝒯′	𝒯′	NUM
easat-1685	46	37	)	)	PUNCT
easat-1685	46	38	is	be	AUX
easat-1685	46	39	said	say	VERB
easat-1685	46	40	to	to	PART
easat-1685	46	41	be	be	AUX
easat-1685	46	42	:	:	PUNCT
easat-1685	46	43	i.	i.	PROPN
easat-1685	46	44	𝒶-continuous	𝒶-continuous	PROPN
easat-1685	46	45	,	,	PUNCT
easat-1685	46	46	if	if	SCONJ
easat-1685	46	47	𝑓−1(𝑉	𝑓−1(𝑉	NOUN
easat-1685	46	48	)	)	PUNCT
easat-1685	46	49	is	be	AUX
easat-1685	46	50	𝒶-open	𝒶-open	NOUN
easat-1685	46	51	set	set	VERB
easat-1685	46	52	of	of	ADP
easat-1685	46	53	𝑋	𝑋	PROPN
easat-1685	46	54	for	for	ADP
easat-1685	46	55	every	every	DET
easat-1685	46	56	open	open	ADJ
easat-1685	46	57	set	set	ADJ
easat-1685	46	58	𝑉	𝑉	PROPN
easat-1685	46	59	of	of	ADP
easat-1685	46	60	𝑌.	𝑌.	PROPN
easat-1685	46	61	ii	ii	PROPN
easat-1685	46	62	.	.	PUNCT
easat-1685	47	1	𝒶-irresolute	𝒶-irresolute	PROPN
easat-1685	47	2	continuous	continuous	ADJ
easat-1685	47	3	,	,	PUNCT
easat-1685	47	4	if	if	SCONJ
easat-1685	47	5	𝑓−1(𝑉	𝑓−1(𝑉	NOUN
easat-1685	47	6	)	)	PUNCT
easat-1685	47	7	is	be	AUX
easat-1685	47	8	𝒶-open	𝒶-open	NOUN
easat-1685	47	9	set	set	VERB
easat-1685	47	10	of	of	ADP
easat-1685	47	11	𝑋	𝑋	PROPN
easat-1685	47	12	for	for	ADP
easat-1685	47	13	every	every	DET
easat-1685	47	14	𝑎-open	𝑎-open	PROPN
easat-1685	47	15	set	set	VERB
easat-1685	47	16	𝑉	𝑉	PROPN
easat-1685	47	17	of	of	ADP
easat-1685	47	18	𝑌.	𝑌.	PROPN
easat-1685	47	19	example	example	NOUN
easat-1685	48	1	.2.7	.2.7	PROPN
easat-1685	48	2	:	:	PUNCT
easat-1685	48	3	let	let	VERB
easat-1685	48	4	𝑓	𝑓	DET
easat-1685	48	5	∶	∶	NOUN
easat-1685	48	6	(	(	PUNCT
easat-1685	48	7	ℕ	ℕ	PROPN
easat-1685	48	8	,	,	PUNCT
easat-1685	48	9	𝒯𝑖𝑛𝑑	𝒯𝑖𝑛𝑑	PROPN
easat-1685	48	10	)	)	PUNCT
easat-1685	48	11	⟶	⟶	NOUN
easat-1685	48	12	(	(	PUNCT
easat-1685	48	13	ℕ	ℕ	PROPN
easat-1685	48	14	,	,	PUNCT
easat-1685	48	15	𝒯𝑐𝑜𝑓	𝒯𝑐𝑜𝑓	PROPN
easat-1685	48	16	)	)	PUNCT
easat-1685	48	17	be	be	VERB
easat-1685	48	18	a	a	DET
easat-1685	48	19	mapping	mapping	NOUN
easat-1685	48	20	which	which	PRON
easat-1685	48	21	is	be	AUX
easat-1685	48	22	defined	define	VERB
easat-1685	48	23	by	by	ADP
easat-1685	48	24	𝑓(𝑥	𝑓(𝑥	NOUN
easat-1685	48	25	)	)	PUNCT
easat-1685	48	26	=	=	SYM
easat-1685	49	1	𝑥	𝑥	PROPN
easat-1685	49	2	for	for	ADP
easat-1685	49	3	all	all	PRON
easat-1685	49	4	𝑥	𝑥	DET
easat-1685	49	5	∈	∈	PROPN
easat-1685	49	6	ℕ.	ℕ.	PROPN
easat-1685	49	7	then	then	ADV
easat-1685	49	8	𝑓	𝑓	PRON
easat-1685	49	9	is	be	AUX
easat-1685	49	10	𝒶-continuous	𝒶-continuous	ADJ
easat-1685	49	11	.	.	PROPN
easat-1685	50	1	3	3	X
easat-1685	50	2	.	.	X
easat-1685	50	3	𝓪-compact	𝓪-compact	NOUN
easat-1685	50	4	space	space	NOUN
easat-1685	50	5	in	in	ADP
easat-1685	50	6	this	this	DET
easat-1685	50	7	section	section	NOUN
easat-1685	50	8	,	,	PUNCT
easat-1685	50	9	we	we	PRON
easat-1685	50	10	introduce	introduce	VERB
easat-1685	50	11	the	the	DET
easat-1685	50	12	concept	concept	NOUN
easat-1685	50	13	of	of	ADP
easat-1685	50	14	𝒶-compact	𝒶-compact	NOUN
easat-1685	50	15	spaces	space	NOUN
easat-1685	50	16	along	along	ADP
easat-1685	50	17	with	with	ADP
easat-1685	50	18	some	some	DET
easat-1685	50	19	basic	basic	ADJ
easat-1685	50	20	properties	property	NOUN
easat-1685	50	21	of	of	ADP
easat-1685	50	22	it	it	PRON
easat-1685	50	23	.	.	PUNCT
easat-1685	51	1	begin	begin	VERB
easat-1685	51	2	this	this	DET
easat-1685	51	3	section	section	NOUN
easat-1685	51	4	by	by	ADP
easat-1685	51	5	giving	give	VERB
easat-1685	51	6	some	some	DET
easat-1685	51	7	properties	property	NOUN
easat-1685	51	8	of	of	ADP
easat-1685	51	9	𝒶-irresolute	𝒶-irresolute	NOUN
easat-1685	51	10	continuous	continuous	ADJ
easat-1685	51	11	and	and	CCONJ
easat-1685	51	12	𝒶-continuous	𝒶-continuous	ADJ
easat-1685	51	13	mappings	mapping	NOUN
easat-1685	51	14	.	.	PUNCT
easat-1685	52	1	the	the	DET
easat-1685	52	2	prove	prove	NOUN
easat-1685	52	3	of	of	ADP
easat-1685	52	4	the	the	DET
easat-1685	52	5	following	follow	VERB
easat-1685	52	6	propositions	proposition	NOUN
easat-1685	52	7	is	be	AUX
easat-1685	52	8	obvious	obvious	ADJ
easat-1685	52	9	and	and	CCONJ
easat-1685	52	10	so	so	ADV
easat-1685	52	11	omitted	omit	VERB
easat-1685	52	12	definition	definition	NOUN
easat-1685	52	13	3.1	3.1	NUM
easat-1685	52	14	:	:	PUNCT
easat-1685	52	15	let	let	VERB
easat-1685	52	16	𝑓	𝑓	X
easat-1685	52	17	:	:	PUNCT
easat-1685	52	18	(	(	PUNCT
easat-1685	52	19	𝑋	𝑋	PROPN
easat-1685	52	20	,	,	PUNCT
easat-1685	52	21	𝒯	𝒯	PROPN
easat-1685	52	22	)	)	PUNCT
easat-1685	52	23	→	→	SYM
easat-1685	52	24	(	(	PUNCT
easat-1685	52	25	𝑌	𝑌	PROPN
easat-1685	52	26	,	,	PUNCT
easat-1685	52	27	𝒯′	𝒯′	NUM
easat-1685	52	28	)	)	PUNCT
easat-1685	52	29	be	be	AUX
easat-1685	52	30	a	a	DET
easat-1685	52	31	mapping	mapping	NOUN
easat-1685	52	32	.	.	PUNCT
easat-1685	53	1	then	then	ADV
easat-1685	53	2	𝑓	𝑓	PRON
easat-1685	53	3	is	be	AUX
easat-1685	53	4	said	say	VERB
easat-1685	53	5	to	to	PART
easat-1685	53	6	be	be	AUX
easat-1685	53	7	𝒶-open	𝒶-open	VERB
easat-1685	53	8	(	(	PUNCT
easat-1685	53	9	𝒶-closed	𝒶-closed	ADJ
easat-1685	53	10	,	,	PUNCT
easat-1685	53	11	resp	resp	NOUN
easat-1685	53	12	.	.	PUNCT
easat-1685	53	13	)	)	PUNCT
easat-1685	54	1	mapping	mapping	NOUN
easat-1685	54	2	,	,	PUNCT
easat-1685	54	3	if	if	SCONJ
easat-1685	54	4	𝑓(𝑈	𝑓(𝑈	NUM
easat-1685	54	5	)	)	PUNCT
easat-1685	54	6	is	be	AUX
easat-1685	54	7	𝒶-open	𝒶-open	NOUN
easat-1685	54	8	(	(	PUNCT
easat-1685	54	9	𝒶-closed	𝒶-close	VERB
easat-1685	54	10	)	)	PUNCT
easat-1685	54	11	set	set	NOUN
easat-1685	54	12	of	of	ADP
easat-1685	54	13	𝑌	𝑌	PROPN
easat-1685	54	14	for	for	ADP
easat-1685	54	15	every	every	DET
easat-1685	54	16	open	open	ADJ
easat-1685	54	17	(	(	PUNCT
easat-1685	54	18	closed	closed	ADJ
easat-1685	54	19	,	,	PUNCT
easat-1685	54	20	resp	resp	NOUN
easat-1685	54	21	.	.	PUNCT
easat-1685	54	22	)	)	PUNCT
easat-1685	55	1	set	set	VERB
easat-1685	55	2	𝑈	𝑈	PROPN
easat-1685	55	3	of	of	ADP
easat-1685	55	4	𝑋.	𝑋.	PROPN
easat-1685	55	5	proposition	proposition	NOUN
easat-1685	55	6	3.2	3.2	NUM
easat-1685	55	7	:	:	PUNCT
easat-1685	55	8	let	let	VERB
easat-1685	55	9	𝑓	𝑓	X
easat-1685	55	10	:	:	PUNCT
easat-1685	55	11	(	(	PUNCT
easat-1685	55	12	𝑋	𝑋	PROPN
easat-1685	55	13	,	,	PUNCT
easat-1685	55	14	𝒯	𝒯	PROPN
easat-1685	55	15	)	)	PUNCT
easat-1685	55	16	→	→	SYM
easat-1685	55	17	(	(	PUNCT
easat-1685	55	18	𝑌	𝑌	PROPN
easat-1685	55	19	,	,	PUNCT
easat-1685	55	20	𝒯′	𝒯′	NUM
easat-1685	55	21	)	)	PUNCT
easat-1685	55	22	be	be	AUX
easat-1685	55	23	a	a	DET
easat-1685	55	24	mapping	mapping	NOUN
easat-1685	55	25	.	.	PUNCT
easat-1685	56	1	then	then	ADV
easat-1685	56	2	the	the	DET
easat-1685	56	3	following	follow	VERB
easat-1685	56	4	statements	statement	NOUN
easat-1685	56	5	are	be	AUX
easat-1685	56	6	equivalent	equivalent	ADJ
easat-1685	56	7	:	:	PUNCT
easat-1685	56	8	i.	i.	PROPN
easat-1685	56	9	𝑓	𝑓	PROPN
easat-1685	56	10	is	be	AUX
easat-1685	56	11	𝒶-irresolute	𝒶-irresolute	NOUN
easat-1685	56	12	continuous	continuous	ADJ
easat-1685	56	13	.	.	PUNCT
easat-1685	56	14	ii	ii	PROPN
easat-1685	56	15	.	.	PROPN
easat-1685	56	16	𝑓−1(𝐹	𝑓−1(𝐹	X
easat-1685	56	17	)	)	PUNCT
easat-1685	56	18	is	be	AUX
easat-1685	56	19	𝒶-closed	𝒶-close	VERB
easat-1685	56	20	set	set	VERB
easat-1685	56	21	of	of	ADP
easat-1685	56	22	𝑋	𝑋	PROPN
easat-1685	56	23	for	for	ADP
easat-1685	56	24	every	every	DET
easat-1685	56	25	𝒶-closed	𝒶-close	VERB
easat-1685	56	26	set	set	VERB
easat-1685	56	27	𝐹	𝐹	PROPN
easat-1685	56	28	of	of	ADP
easat-1685	56	29	𝑌.	𝑌.	PROPN
easat-1685	56	30	iii	iii	PROPN
easat-1685	56	31	.	.	PUNCT
easat-1685	57	1	𝑐𝑙𝒶(𝑓−1(𝐵	𝑐𝑙𝒶(𝑓−1(𝐵	NOUN
easat-1685	57	2	)	)	PUNCT
easat-1685	57	3	)	)	PUNCT
easat-1685	58	1	⊆	⊆	NUM
easat-1685	58	2	𝑓−1(𝑐𝑙𝒶(𝐵	𝑓−1(𝑐𝑙𝒶(𝐵	NOUN
easat-1685	58	3	)	)	PUNCT
easat-1685	58	4	)	)	PUNCT
easat-1685	58	5	for	for	ADP
easat-1685	58	6	all	all	DET
easat-1685	58	7	𝐵	𝐵	PROPN
easat-1685	58	8	⊆	⊆	PROPN
easat-1685	58	9	𝑌.	𝑌.	PROPN
easat-1685	58	10	iv	iv	NUM
easat-1685	58	11	.	.	PUNCT
easat-1685	58	12	𝑓(𝑐𝑙𝒶(𝐴	𝑓(𝑐𝑙𝒶(𝐴	PROPN
easat-1685	58	13	)	)	PUNCT
easat-1685	58	14	)	)	PUNCT
easat-1685	59	1	⊆	⊆	NUM
easat-1685	59	2	𝑐𝑙𝒶(𝑓(𝐴	𝑐𝑙𝒶(𝑓(𝐴	NOUN
easat-1685	59	3	)	)	PUNCT
easat-1685	59	4	)	)	PUNCT
easat-1685	59	5	for	for	ADP
easat-1685	59	6	all	all	DET
easat-1685	59	7	𝐴	𝐴	PROPN
easat-1685	59	8	⊆	⊆	NUM
easat-1685	59	9	𝑋.	𝑋.	PROPN
easat-1685	59	10	𝑟-open	𝑟-open	PROPN
easat-1685	59	11	⟹	⟹	PUNCT
easat-1685	59	12	𝛿-open	𝛿-open	VERB
easat-1685	59	13	⟹	⟹	PUNCT
easat-1685	59	14	𝒶-open	𝒶-open	VERB
easat-1685	59	15	⟹	⟹	PUNCT
easat-1685	59	16	open	open	VERB
easat-1685	59	17	273	273	NUM
easat-1685	59	18	edelweiss	edelweiss	PROPN
easat-1685	59	19	applied	apply	VERB
easat-1685	59	20	science	science	NOUN
easat-1685	59	21	and	and	CCONJ
easat-1685	59	22	technology	technology	NOUN
easat-1685	59	23	issn	issn	PROPN
easat-1685	59	24	:	:	PUNCT
easat-1685	59	25	2576	2576	NUM
easat-1685	59	26	-	-	SYM
easat-1685	59	27	8484	8484	NUM
easat-1685	59	28	vol	vol	NOUN
easat-1685	59	29	.	.	PROPN
easat-1685	60	1	8	8	NUM
easat-1685	60	2	,	,	PUNCT
easat-1685	60	3	no	no	INTJ
easat-1685	60	4	.	.	NOUN
easat-1685	60	5	5	5	NUM
easat-1685	60	6	:	:	PUNCT
easat-1685	60	7	271	271	NUM
easat-1685	60	8	-	-	SYM
easat-1685	60	9	277	277	NUM
easat-1685	60	10	,	,	PUNCT
easat-1685	60	11	2024	2024	NUM
easat-1685	60	12	doi	doi	NOUN
easat-1685	60	13	:	:	PUNCT
easat-1685	60	14	10.55214/25768484.v8i5.1685	10.55214/25768484.v8i5.1685	NUM
easat-1685	60	15	©	©	ADP
easat-1685	60	16	2024	2024	NUM
easat-1685	60	17	by	by	ADP
easat-1685	60	18	the	the	DET
easat-1685	60	19	authors	author	NOUN
easat-1685	60	20	;	;	PUNCT
easat-1685	60	21	licensee	licensee	PROPN
easat-1685	60	22	learning	learning	NOUN
easat-1685	60	23	gate	gate	NOUN
easat-1685	60	24	v.	v.	ADP
easat-1685	60	25	𝑓−1(𝑖𝑛𝑡𝒶(𝐵	𝑓−1(𝑖𝑛𝑡𝒶(𝐵	PROPN
easat-1685	60	26	)	)	PUNCT
easat-1685	60	27	)	)	PUNCT
easat-1685	61	1	⊆	⊆	NUM
easat-1685	61	2	𝑖𝑛𝑡𝒶(𝑓−1(𝐵	𝑖𝑛𝑡𝒶(𝑓−1(𝐵	NUM
easat-1685	61	3	)	)	PUNCT
easat-1685	61	4	)	)	PUNCT
easat-1685	62	1	for	for	ADP
easat-1685	62	2	every	every	DET
easat-1685	62	3	𝐵	𝐵	PROPN
easat-1685	62	4	⊆	⊆	PROPN
easat-1685	62	5	𝑌.	𝑌.	PROPN
easat-1685	62	6	proposition	proposition	NOUN
easat-1685	62	7	3.3	3.3	NUM
easat-1685	62	8	:	:	PUNCT
easat-1685	62	9	i.	i.	NOUN
easat-1685	62	10	every	every	DET
easat-1685	62	11	continuous	continuous	ADJ
easat-1685	62	12	mapping	mapping	NOUN
easat-1685	62	13	is	be	AUX
easat-1685	62	14	𝒶-continuous	𝒶-continuous	ADJ
easat-1685	62	15	.	.	PUNCT
easat-1685	63	1	ii	ii	PROPN
easat-1685	63	2	.	.	PUNCT
easat-1685	64	1	composition	composition	NOUN
easat-1685	64	2	of	of	ADP
easat-1685	64	3	two	two	NUM
easat-1685	64	4	𝒶-irresolute	𝒶-irresolute	NUM
easat-1685	64	5	continuous	continuous	ADJ
easat-1685	64	6	mappings	mapping	NOUN
easat-1685	64	7	is	be	AUX
easat-1685	64	8	𝒶-irresolute	𝒶-irresolute	NOUN
easat-1685	64	9	continuous	continuous	ADJ
easat-1685	64	10	.	.	PUNCT
easat-1685	64	11	iii	iii	X
easat-1685	64	12	.	.	PUNCT
easat-1685	64	13	composition	composition	NOUN
easat-1685	64	14	of	of	ADP
easat-1685	64	15	𝒶-irresolute	𝒶-irresolute	NOUN
easat-1685	64	16	continuous	continuous	ADJ
easat-1685	64	17	and	and	CCONJ
easat-1685	64	18	𝒶-continuous	𝒶-continuous	ADJ
easat-1685	64	19	mappings	mapping	NOUN
easat-1685	64	20	is	be	AUX
easat-1685	64	21	𝒶-continuous	𝒶-continuous	PROPN
easat-1685	64	22	.	.	PUNCT
easat-1685	65	1	definition	definition	NOUN
easat-1685	65	2	3.4	3.4	NUM
easat-1685	65	3	:	:	PUNCT
easat-1685	65	4	a	a	DET
easat-1685	65	5	space	space	NOUN
easat-1685	65	6	𝑋	𝑋	NOUN
easat-1685	65	7	is	be	AUX
easat-1685	65	8	said	say	VERB
easat-1685	65	9	to	to	PART
easat-1685	65	10	be	be	AUX
easat-1685	65	11	𝒶-compact(resp	𝒶-compact(resp	VERB
easat-1685	65	12	.	.	PUNCT
easat-1685	66	1	𝒶lindelof	𝒶lindelof	NOUN
easat-1685	66	2	)	)	PUNCT
easat-1685	66	3	,	,	PUNCT
easat-1685	66	4	if	if	SCONJ
easat-1685	66	5	every	every	DET
easat-1685	66	6	𝒶-open	𝒶-open	NOUN
easat-1685	66	7	cover	cover	NOUN
easat-1685	66	8	of	of	ADP
easat-1685	66	9	𝑋	𝑋	NOUN
easat-1685	66	10	by	by	ADP
easat-1685	66	11	𝒶open	𝒶open	NOUN
easat-1685	66	12	subset	subset	NOUN
easat-1685	66	13	of	of	ADP
easat-1685	66	14	𝑋	𝑋	PROPN
easat-1685	66	15	has	have	VERB
easat-1685	66	16	a	a	DET
easat-1685	66	17	finite	finite	NOUN
easat-1685	66	18	(	(	PUNCT
easat-1685	66	19	resp	resp	NOUN
easat-1685	66	20	.	.	PUNCT
easat-1685	67	1	countable	countable	ADJ
easat-1685	67	2	)	)	PUNCT
easat-1685	67	3	subcover	subcover	PROPN
easat-1685	67	4	.	.	PUNCT
easat-1685	68	1	proposition	proposition	NOUN
easat-1685	68	2	3.5	3.5	NUM
easat-1685	68	3	:	:	PUNCT
easat-1685	68	4	the	the	DET
easat-1685	68	5	intersection	intersection	NOUN
easat-1685	68	6	of	of	ADP
easat-1685	68	7	𝛿	𝛿	PRON
easat-1685	68	8	open	open	ADJ
easat-1685	68	9	and	and	CCONJ
easat-1685	68	10	an	an	DET
easat-1685	68	11	𝒶open	𝒶open	ADJ
easat-1685	68	12	sets	set	NOUN
easat-1685	68	13	is	be	AUX
easat-1685	68	14	an	an	DET
easat-1685	68	15	𝒶-open	𝒶-open	NOUN
easat-1685	68	16	.	.	PUNCT
easat-1685	69	1	proof	proof	NOUN
easat-1685	69	2	:	:	PUNCT
easat-1685	69	3	let	let	VERB
easat-1685	69	4	a	a	PRON
easat-1685	69	5	is	be	AUX
easat-1685	69	6	𝛿-open	𝛿-open	VERB
easat-1685	69	7	and	and	CCONJ
easat-1685	69	8	b	b	NOUN
easat-1685	69	9	is	be	AUX
easat-1685	69	10	an	an	DET
easat-1685	69	11	𝒶	𝒶	NOUN
easat-1685	69	12	open	open	ADJ
easat-1685	69	13	sets	set	NOUN
easat-1685	69	14	in	in	ADP
easat-1685	69	15	𝒯𝑥.	𝒯𝑥.	PROPN
easat-1685	69	16	to	to	PART
easat-1685	69	17	show	show	VERB
easat-1685	69	18	that	that	SCONJ
easat-1685	69	19	a	a	DET
easat-1685	69	20	⋂	⋂	PROPN
easat-1685	69	21	b	b	PROPN
easat-1685	69	22	is	be	AUX
easat-1685	69	23	an	an	DET
easat-1685	69	24	𝒶-open	𝒶-open	NOUN
easat-1685	69	25	in	in	ADP
easat-1685	69	26	𝒯𝑦.	𝒯𝑦.	PROPN
easat-1685	69	27	a	a	DET
easat-1685	69	28	⋂	⋂	PROPN
easat-1685	69	29	b	b	NOUN
easat-1685	69	30	⊆	⊆	NUM
easat-1685	69	31	𝛿𝑖𝑛𝑡	𝛿𝑖𝑛𝑡	ADJ
easat-1685	69	32	(	(	PUNCT
easat-1685	69	33	a	a	X
easat-1685	69	34	)	)	PUNCT
easat-1685	69	35	⋂	⋂	PROPN
easat-1685	69	36	𝑖𝑛𝑡	𝑖𝑛𝑡	PROPN
easat-1685	69	37	(	(	PUNCT
easat-1685	69	38	𝑐𝑙	𝑐𝑙	ADV
easat-1685	69	39	(	(	PUNCT
easat-1685	69	40	𝛿𝑖𝑛𝑡	𝛿𝑖𝑛𝑡	ADJ
easat-1685	69	41	(	(	PUNCT
easat-1685	69	42	b	b	NOUN
easat-1685	69	43	)	)	PUNCT
easat-1685	69	44	)	)	PUNCT
easat-1685	69	45	)	)	PUNCT
easat-1685	69	46	.	.	PUNCT
easat-1685	70	1	⊆	⊆	NUM
easat-1685	70	2	𝑖𝑛𝑡𝐴(𝛿𝑖𝑛𝑡(a	𝑖𝑛𝑡𝐴(𝛿𝑖𝑛𝑡(a	SYM
easat-1685	70	3	)	)	PUNCT
easat-1685	70	4	⋂	⋂	PROPN
easat-1685	70	5	𝑖𝑛𝑡	𝑖𝑛𝑡	PRON
easat-1685	70	6	(	(	PUNCT
easat-1685	70	7	𝑐𝑙	𝑐𝑙	ADV
easat-1685	70	8	(	(	PUNCT
easat-1685	70	9	𝛿𝑖𝑛𝑡	𝛿𝑖𝑛𝑡	ADJ
easat-1685	70	10	(	(	PUNCT
easat-1685	70	11	b	b	NOUN
easat-1685	70	12	)	)	PUNCT
easat-1685	70	13	)	)	PUNCT
easat-1685	70	14	)	)	PUNCT
easat-1685	70	15	.	.	PUNCT
easat-1685	71	1	⊆	⊆	NUM
easat-1685	71	2	𝑖𝑛𝑡𝐴	𝑖𝑛𝑡𝐴	ADJ
easat-1685	71	3	(	(	PUNCT
easat-1685	71	4	𝑐𝑙	𝑐𝑙	X
easat-1685	71	5	(	(	PUNCT
easat-1685	71	6	𝛿𝑖𝑛𝑡(a	𝛿𝑖𝑛𝑡(a	NOUN
easat-1685	71	7	)	)	PUNCT
easat-1685	71	8	⋂	⋂	PROPN
easat-1685	71	9	𝑖𝑛𝑡	𝑖𝑛𝑡	PROPN
easat-1685	71	10	(	(	PUNCT
easat-1685	71	11	𝑐𝑙	𝑐𝑙	ADV
easat-1685	71	12	(	(	PUNCT
easat-1685	71	13	𝛿𝑖𝑛𝑡	𝛿𝑖𝑛𝑡	ADJ
easat-1685	71	14	(	(	PUNCT
easat-1685	71	15	b	b	NOUN
easat-1685	71	16	)	)	PUNCT
easat-1685	71	17	)	)	PUNCT
easat-1685	71	18	)	)	PUNCT
easat-1685	71	19	.	.	PUNCT
easat-1685	72	1	⊆	⊆	NUM
easat-1685	72	2	𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿𝑖𝑛𝑡	𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿𝑖𝑛𝑡	PUNCT
easat-1685	72	3	(	(	PUNCT
easat-1685	72	4	a	a	DET
easat-1685	72	5	⋂	⋂	PROPN
easat-1685	72	6	b	b	PROPN
easat-1685	72	7	)	)	PUNCT
easat-1685	72	8	)	)	PUNCT
easat-1685	72	9	)	)	PUNCT
easat-1685	72	10	.	.	PUNCT
easat-1685	73	1	⊆	⊆	NUM
easat-1685	73	2	𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿𝑖𝑛𝑡𝐴(a	𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿𝑖𝑛𝑡𝐴(a	ADP
easat-1685	73	3	⋂	⋂	PROPN
easat-1685	73	4	b	b	NOUN
easat-1685	73	5	)	)	PUNCT
easat-1685	73	6	)	)	PUNCT
easat-1685	73	7	)	)	PUNCT
easat-1685	73	8	.	.	PUNCT
easat-1685	74	1	since	since	SCONJ
easat-1685	74	2	𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿𝑖𝑛𝑡𝐴(a	𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿𝑖𝑛𝑡𝐴(a	PROPN
easat-1685	74	3	⋂	⋂	PROPN
easat-1685	74	4	b	b	NOUN
easat-1685	74	5	)	)	PUNCT
easat-1685	74	6	)	)	PUNCT
easat-1685	74	7	)	)	PUNCT
easat-1685	74	8	is	be	AUX
easat-1685	74	9	𝒶-open	𝒶-open	NOUN
easat-1685	74	10	set	set	VERB
easat-1685	74	11	in	in	ADP
easat-1685	74	12	𝒯𝑦,so	𝒯𝑦,so	PROPN
easat-1685	74	13	𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿𝑖𝑛𝑡𝐴(a	𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿𝑖𝑛𝑡𝐴(a	PROPN
easat-1685	74	14	⋂	⋂	PROPN
easat-1685	74	15	b	b	NOUN
easat-1685	74	16	)	)	PUNCT
easat-1685	74	17	)	)	PUNCT
easat-1685	74	18	)	)	PUNCT
easat-1685	75	1	=	=	PUNCT
easat-1685	76	1	𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿𝑖𝑛𝑡𝐴(a	𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿𝑖𝑛𝑡𝐴(a	ADV
easat-1685	76	2	⋂	⋂	PROPN
easat-1685	76	3	b	b	X
easat-1685	76	4	)	)	PUNCT
easat-1685	76	5	⋂	⋂	PROPN
easat-1685	76	6	b	b	PROPN
easat-1685	76	7	)	)	PUNCT
easat-1685	76	8	)	)	PUNCT
easat-1685	76	9	)	)	PUNCT
easat-1685	76	10	.	.	PUNCT
easat-1685	77	1	thus	thus	ADV
easat-1685	77	2	a	a	DET
easat-1685	77	3	⋂	⋂	PROPN
easat-1685	77	4	b	b	PROPN
easat-1685	77	5	⊆	⊆	NUM
easat-1685	77	6	𝑖𝑛𝑡𝐴(𝑐𝑙𝐴	𝑖𝑛𝑡𝐴(𝑐𝑙𝐴	NOUN
easat-1685	77	7	(	(	PUNCT
easat-1685	77	8	𝛿𝑖𝑛𝑡𝐴(a	𝛿𝑖𝑛𝑡𝐴(a	PROPN
easat-1685	77	9	⋂	⋂	PROPN
easat-1685	77	10	b	b	PROPN
easat-1685	77	11	)	)	PUNCT
easat-1685	77	12	)	)	PUNCT
easat-1685	77	13	)	)	PUNCT
easat-1685	77	14	.	.	PUNCT
easat-1685	78	1	theorem	theorem	VERB
easat-1685	78	2	3.6	3.6	NUM
easat-1685	78	3	:	:	PUNCT
easat-1685	78	4	a	a	DET
easat-1685	78	5	𝛿	𝛿	ADJ
easat-1685	78	6	-open	-open	NOUN
easat-1685	78	7	subset	subset	NOUN
easat-1685	78	8	𝑌	𝑌	PROPN
easat-1685	78	9	of	of	ADP
easat-1685	78	10	a	a	DET
easat-1685	78	11	space	space	NOUN
easat-1685	78	12	x	x	PUNCT
easat-1685	78	13	is	be	AUX
easat-1685	78	14	𝒶-compact	𝒶-compact	NOUN
easat-1685	78	15	if	if	SCONJ
easat-1685	78	16	and	and	CCONJ
easat-1685	78	17	only	only	ADV
easat-1685	78	18	if	if	SCONJ
easat-1685	78	19	every	every	DET
easat-1685	78	20	𝒶-open	𝒶-open	NOUN
easat-1685	78	21	cover	cover	NOUN
easat-1685	78	22	of	of	ADP
easat-1685	78	23	𝑌	𝑌	PROPN
easat-1685	78	24	by	by	ADP
easat-1685	78	25	the	the	DET
easat-1685	78	26	𝒶-open	𝒶-open	NOUN
easat-1685	78	27	subset	subset	NOUN
easat-1685	78	28	of	of	ADP
easat-1685	78	29	𝑋	𝑋	PROPN
easat-1685	78	30	has	have	VERB
easat-1685	78	31	a	a	DET
easat-1685	78	32	finite	finite	ADJ
easat-1685	78	33	subcover	subcover	PROPN
easat-1685	78	34	.	.	PUNCT
easat-1685	79	1	proof	proof	NOUN
easat-1685	79	2	:	:	PUNCT
easat-1685	79	3	let	let	VERB
easat-1685	79	4	𝑌	𝑌	PROPN
easat-1685	79	5	be	be	AUX
easat-1685	79	6	𝒶-compact	𝒶-compact	NOUN
easat-1685	79	7	subset	subset	NOUN
easat-1685	79	8	of	of	ADP
easat-1685	79	9	𝑋.	𝑋.	PROPN
easat-1685	79	10	let	let	VERB
easat-1685	79	11	{	{	PUNCT
easat-1685	79	12	𝐺𝜆	𝐺𝜆	NOUN
easat-1685	79	13	∶	∶	VERB
easat-1685	79	14	𝜆	𝜆	PRON
easat-1685	79	15	∈	∈	PROPN
easat-1685	79	16	𝛬	𝛬	AUX
easat-1685	79	17	}	}	PUNCT
easat-1685	79	18	be	be	AUX
easat-1685	79	19	𝒶-open	𝒶-open	NOUN
easat-1685	79	20	cover	cover	NOUN
easat-1685	79	21	of	of	ADP
easat-1685	79	22	𝑌	𝑌	PROPN
easat-1685	79	23	,	,	PUNCT
easat-1685	79	24	where	where	SCONJ
easat-1685	79	25	each	each	DET
easat-1685	79	26	𝐺𝜆	𝐺𝜆	PROPN
easat-1685	79	27	is	be	AUX
easat-1685	79	28	𝒶open	𝒶open	ADJ
easat-1685	79	29	set	set	NOUN
easat-1685	79	30	in	in	ADP
easat-1685	79	31	𝑋	𝑋	NOUN
easat-1685	79	32	for	for	ADP
easat-1685	79	33	all	all	PRON
easat-1685	79	34	𝜆	𝜆	DET
easat-1685	79	35	∈	∈	PROPN
easat-1685	79	36	𝛬.	𝛬.	NOUN
easat-1685	79	37	then	then	ADV
easat-1685	79	38	,	,	PUNCT
easat-1685	79	39	𝑌	𝑌	PROPN
easat-1685	79	40	⊆	⊆	NUM
easat-1685	79	41	⋃	⋃	ADP
easat-1685	79	42	𝐺𝜆𝜆∈𝛬	𝐺𝜆𝜆∈𝛬	NOUN
easat-1685	79	43	that	that	PRON
easat-1685	79	44	is	be	AUX
easat-1685	79	45	𝑌	𝑌	PROPN
easat-1685	79	46	⊆	⊆	NUM
easat-1685	79	47	⋃	⋃	ADJ
easat-1685	79	48	𝐺𝛾𝜆∈𝛬	𝐺𝛾𝜆∈𝛬	NOUN
easat-1685	79	49	⋂𝑌	⋂𝑌	NUM
easat-1685	79	50	,	,	PUNCT
easat-1685	79	51	where	where	SCONJ
easat-1685	79	52	each	each	DET
easat-1685	79	53	𝐺𝜆⋂𝑌	𝐺𝜆⋂𝑌	PROPN
easat-1685	79	54	is	be	AUX
easat-1685	79	55	𝒶-open	𝒶-open	VERB
easat-1685	79	56	in	in	ADP
easat-1685	79	57	ty	ty	NUM
easat-1685	79	58	by	by	ADP
easat-1685	79	59	the	the	DET
easat-1685	79	60	theorem	theorem	NOUN
easat-1685	79	61	(	(	PUNCT
easat-1685	79	62	3.5	3.5	NUM
easat-1685	79	63	)	)	PUNCT
easat-1685	79	64	.	.	PUNCT
easat-1685	80	1	therefore	therefore	ADV
easat-1685	80	2	,	,	PUNCT
easat-1685	80	3	by	by	ADP
easat-1685	80	4	𝒶-compactness	𝒶-compactness	NOUN
easat-1685	80	5	of	of	ADP
easat-1685	80	6	𝑌	𝑌	PROPN
easat-1685	80	7	,	,	PUNCT
easat-1685	80	8	there	there	PRON
easat-1685	80	9	is	be	VERB
easat-1685	80	10	a	a	DET
easat-1685	80	11	finite	finite	ADJ
easat-1685	80	12	subcollection	subcollection	NOUN
easat-1685	80	13	𝛬0	𝛬0	PRON
easat-1685	80	14	of	of	ADP
easat-1685	80	15	𝛬	𝛬	PROPN
easat-1685	80	16	with	with	ADP
easat-1685	80	17	𝑌	𝑌	PROPN
easat-1685	80	18	⊆	⊆	NUM
easat-1685	80	19	⋃	⋃	NOUN
easat-1685	80	20	𝐺𝜆𝜆∈𝛬0	𝐺𝜆𝜆∈𝛬0	NOUN
easat-1685	80	21	⋂𝑌	⋂𝑌	ADJ
easat-1685	80	22	,	,	PUNCT
easat-1685	80	23	so	so	ADV
easat-1685	80	24	𝑌	𝑌	PROPN
easat-1685	80	25	⊆	⊆	NUM
easat-1685	80	26	⋃	⋃	NOUN
easat-1685	80	27	𝐺𝜆𝜆∈𝛬0	𝐺𝜆𝜆∈𝛬0	NOUN
easat-1685	80	28	.	.	PUNCT
easat-1685	81	1	thus	thus	ADV
easat-1685	81	2	,	,	PUNCT
easat-1685	81	3	if	if	SCONJ
easat-1685	81	4	𝑌	𝑌	PROPN
easat-1685	81	5	is	be	AUX
easat-1685	81	6	𝒶-compact	𝒶-compact	NOUN
easat-1685	81	7	then	then	ADV
easat-1685	81	8	every	every	DET
easat-1685	81	9	𝒶-open	𝒶-open	NOUN
easat-1685	81	10	cover	cover	NOUN
easat-1685	81	11	of	of	ADP
easat-1685	81	12	𝑌	𝑌	PROPN
easat-1685	81	13	by	by	ADP
easat-1685	81	14	the	the	DET
easat-1685	81	15	𝒶open	𝒶open	ADJ
easat-1685	81	16	set	set	NOUN
easat-1685	81	17	of	of	ADP
easat-1685	81	18	𝑋	𝑋	PROPN
easat-1685	81	19	has	have	VERB
easat-1685	81	20	a	a	DET
easat-1685	81	21	finite	finite	ADJ
easat-1685	81	22	subcover	subcover	PROPN
easat-1685	81	23	.	.	PUNCT
easat-1685	82	1	conversely	conversely	ADV
easat-1685	82	2	,	,	PUNCT
easat-1685	82	3	let	let	VERB
easat-1685	82	4	{	{	PUNCT
easat-1685	82	5	𝑌𝜆	𝑌𝜆	ADJ
easat-1685	82	6	:	:	PUNCT
easat-1685	82	7	𝜆	𝜆	DET
easat-1685	82	8	∈	∈	PROPN
easat-1685	82	9	𝛬	𝛬	NOUN
easat-1685	82	10	}	}	PUNCT
easat-1685	82	11	an	an	DET
easat-1685	82	12	𝒶-open	𝒶-open	NOUN
easat-1685	82	13	cover	cover	NOUN
easat-1685	82	14	of	of	ADP
easat-1685	82	15	𝑌	𝑌	PROPN
easat-1685	82	16	by	by	ADP
easat-1685	82	17	the	the	DET
easat-1685	82	18	𝒶-open	𝒶-open	NOUN
easat-1685	82	19	sets	set	NOUN
easat-1685	82	20	of	of	ADP
easat-1685	82	21	𝑌.thus	𝑌.thu	NOUN
easat-1685	82	22	,	,	PUNCT
easat-1685	82	23	𝑌	𝑌	PROPN
easat-1685	82	24	⊆	⊆	NUM
easat-1685	82	25	⋃	⋃	PROPN
easat-1685	82	26	𝑌𝜆𝜆∈𝛬	𝑌𝜆𝜆∈𝛬	NOUN
easat-1685	82	27	.	.	PUNCT
easat-1685	83	1	since	since	SCONJ
easat-1685	83	2	𝑌	𝑌	PROPN
easat-1685	83	3	is	be	AUX
easat-1685	83	4	open	open	ADJ
easat-1685	83	5	,	,	PUNCT
easat-1685	83	6	𝑌𝜆	𝑌𝜆	PROPN
easat-1685	83	7	is	be	AUX
easat-1685	83	8	𝒶-open	𝒶-open	NOUN
easat-1685	83	9	set	set	VERB
easat-1685	83	10	in	in	ADP
easat-1685	83	11	𝑋	𝑋	NOUN
easat-1685	83	12	for	for	ADP
easat-1685	83	13	all	all	DET
easat-1685	83	14	𝜆	𝜆	DET
easat-1685	83	15	∈	∈	PROPN
easat-1685	83	16	𝛬.	𝛬.	NOUN
easat-1685	84	1	so	so	ADV
easat-1685	84	2	,	,	PUNCT
easat-1685	84	3	{	{	PUNCT
easat-1685	84	4	𝑌𝜆	𝑌𝜆	PROPN
easat-1685	84	5	∶	∶	NOUN
easat-1685	84	6	𝜆	𝜆	DET
easat-1685	84	7	∈	∈	PROPN
easat-1685	84	8	𝛬	𝛬	NOUN
easat-1685	84	9	}	}	PUNCT
easat-1685	84	10	is	be	AUX
easat-1685	84	11	𝒶-open	𝒶-open	NOUN
easat-1685	84	12	cover	cover	NOUN
easat-1685	84	13	of	of	ADP
easat-1685	84	14	𝑌	𝑌	PROPN
easat-1685	84	15	by	by	ADP
easat-1685	84	16	the	the	DET
easat-1685	84	17	𝒶-open	𝒶-open	NOUN
easat-1685	84	18	sets	set	NOUN
easat-1685	84	19	of	of	ADP
easat-1685	84	20	𝑋.	𝑋.	PROPN
easat-1685	84	21	then	then	ADV
easat-1685	84	22	by	by	ADP
easat-1685	84	23	the	the	DET
easat-1685	84	24	given	give	VERB
easat-1685	84	25	condition	condition	NOUN
easat-1685	84	26	,	,	PUNCT
easat-1685	84	27	there	there	PRON
easat-1685	84	28	is	be	VERB
easat-1685	84	29	a	a	DET
easat-1685	84	30	finite	finite	NOUN
easat-1685	84	31	subcover	subcover	PROPN
easat-1685	84	32	𝛬0	𝛬0	PROPN
easat-1685	84	33	of	of	ADP
easat-1685	84	34	𝛬	𝛬	PRON
easat-1685	84	35	such	such	ADJ
easat-1685	84	36	that	that	SCONJ
easat-1685	84	37	𝑌	𝑌	PROPN
easat-1685	84	38	⊆	⊆	NUM
easat-1685	84	39	⋃	⋃	ADP
easat-1685	84	40	𝐺𝜆𝜆∈𝛬	𝐺𝜆𝜆∈𝛬	PROPN
easat-1685	84	41	.	.	PUNCT
easat-1685	85	1	so	so	ADV
easat-1685	85	2	by	by	ADP
easat-1685	85	3	the	the	DET
easat-1685	85	4	definition	definition	NOUN
easat-1685	85	5	of	of	ADP
easat-1685	85	6	𝒶-compact	𝒶-compact	NOUN
easat-1685	85	7	space	space	NOUN
easat-1685	85	8	,	,	PUNCT
easat-1685	85	9	𝑌	𝑌	PROPN
easat-1685	85	10	is	be	AUX
easat-1685	85	11	𝒶-compact	𝒶-compact	NOUN
easat-1685	85	12	.	.	PUNCT
easat-1685	86	1	hence	hence	ADV
easat-1685	86	2	,	,	PUNCT
easat-1685	86	3	this	this	PRON
easat-1685	86	4	completes	complete	VERB
easat-1685	86	5	the	the	DET
easat-1685	86	6	proof	proof	NOUN
easat-1685	86	7	.	.	PUNCT
easat-1685	87	1	theorem	theorem	PROPN
easat-1685	87	2	.3.7	.3.7	NOUN
easat-1685	87	3	:	:	PUNCT
easat-1685	87	4	an	an	DET
easat-1685	87	5	𝒶-closed	𝒶-closed	ADJ
easat-1685	87	6	subset	subset	NOUN
easat-1685	87	7	of	of	ADP
easat-1685	87	8	an	an	DET
easat-1685	87	9	𝒶-compact	𝒶-compact	NOUN
easat-1685	87	10	space	space	NOUN
easat-1685	87	11	is	be	AUX
easat-1685	87	12	𝒶-compact	𝒶-compact	NOUN
easat-1685	87	13	.	.	PUNCT
easat-1685	88	1	proof	proof	NOUN
easat-1685	88	2	:	:	PUNCT
easat-1685	88	3	let	let	VERB
easat-1685	88	4	(	(	PUNCT
easat-1685	88	5	𝑋	𝑋	NOUN
easat-1685	88	6	,	,	PUNCT
easat-1685	88	7	𝒯	𝒯	PROPN
easat-1685	88	8	)	)	PUNCT
easat-1685	88	9	be	be	VERB
easat-1685	88	10	a	a	DET
easat-1685	88	11	𝒶-compact	𝒶-compact	ADJ
easat-1685	88	12	topological	topological	ADJ
easat-1685	88	13	space	space	NOUN
easat-1685	88	14	and	and	CCONJ
easat-1685	88	15	let	let	VERB
easat-1685	88	16	𝑌	𝑌	PROPN
easat-1685	88	17	be	be	AUX
easat-1685	88	18	an	an	DET
easat-1685	88	19	𝒶-closed	𝒶-closed	ADJ
easat-1685	88	20	subset	subset	NOUN
easat-1685	88	21	of	of	ADP
easat-1685	88	22	𝑋.	𝑋.	PROPN
easat-1685	88	23	now	now	ADV
easat-1685	88	24	we	we	PRON
easat-1685	88	25	have	have	VERB
easat-1685	88	26	to	to	PART
easat-1685	88	27	show	show	VERB
easat-1685	88	28	that	that	SCONJ
easat-1685	88	29	,	,	PUNCT
easat-1685	88	30	𝑌	𝑌	PROPN
easat-1685	88	31	is	be	AUX
easat-1685	88	32	𝒶-compact	𝒶-compact	NOUN
easat-1685	88	33	.	.	PUNCT
easat-1685	89	1	let	let	VERB
easat-1685	89	2	{	{	PUNCT
easat-1685	89	3	𝐺𝜆	𝐺𝜆	VERB
easat-1685	89	4	∶	∶	VERB
easat-1685	89	5	𝜆	𝜆	DET
easat-1685	89	6	∈	∈	PROPN
easat-1685	89	7	𝛬	𝛬	NOUN
easat-1685	89	8	}	}	PUNCT
easat-1685	89	9	be	be	AUX
easat-1685	89	10	an	an	DET
easat-1685	89	11	𝒶-open	𝒶-open	NOUN
easat-1685	89	12	cover	cover	NOUN
easat-1685	89	13	of	of	ADP
easat-1685	89	14	𝑌	𝑌	PROPN
easat-1685	89	15	,	,	PUNCT
easat-1685	89	16	where	where	SCONJ
easat-1685	89	17	each	each	DET
easat-1685	89	18	𝐺𝜆	𝐺𝜆	PROPN
easat-1685	89	19	is	be	AUX
easat-1685	89	20	𝒶-open	𝒶-open	NOUN
easat-1685	89	21	set	set	VERB
easat-1685	89	22	in	in	ADP
easat-1685	89	23	(	(	PUNCT
easat-1685	89	24	𝑋	𝑋	PROPN
easat-1685	89	25	,	,	PUNCT
easat-1685	89	26	𝒯	𝒯	PROPN
easat-1685	89	27	)	)	PUNCT
easat-1685	89	28	for	for	ADP
easat-1685	89	29	all	all	PRON
easat-1685	89	30	𝜆	𝜆	DET
easat-1685	89	31	∈	∈	PROPN
easat-1685	89	32	𝛬.	𝛬.	NOUN
easat-1685	89	33	then	then	ADV
easat-1685	89	34	𝑌	𝑌	PROPN
easat-1685	89	35	⊆	⊆	NUM
easat-1685	89	36	⋃	⋃	ADP
easat-1685	89	37	𝐺𝜆𝜆∈𝛬	𝐺𝜆𝜆∈𝛬	PROPN
easat-1685	89	38	.	.	PUNCT
easat-1685	90	1	so	so	ADV
easat-1685	90	2	,	,	PUNCT
easat-1685	90	3	𝑋	𝑋	PROPN
easat-1685	90	4	⊆	⊆	NUM
easat-1685	90	5	(	(	PUNCT
easat-1685	90	6	𝑋	𝑋	NOUN
easat-1685	90	7	\	\	PROPN
easat-1685	90	8	𝑌	𝑌	PROPN
easat-1685	90	9	)	)	PUNCT
easat-1685	90	10	⋃(⋃	⋃(⋃	PROPN
easat-1685	90	11	𝐺𝜆𝜆∈𝛬	𝐺𝜆𝜆∈𝛬	PROPN
easat-1685	90	12	)	)	PUNCT
easat-1685	90	13	.	.	PUNCT
easat-1685	91	1	since	since	SCONJ
easat-1685	91	2	𝑋	𝑋	PROPN
easat-1685	91	3	is	be	AUX
easat-1685	91	4	𝒶-compact	𝒶-compact	PROPN
easat-1685	91	5	,	,	PUNCT
easat-1685	91	6	there	there	PRON
easat-1685	91	7	exists	exist	VERB
easat-1685	91	8	a	a	DET
easat-1685	91	9	finite	finite	ADJ
easat-1685	91	10	collection	collection	NOUN
easat-1685	91	11	𝛬0	𝛬0	PROPN
easat-1685	91	12	of	of	ADP
easat-1685	91	13	𝛬	𝛬	PRON
easat-1685	91	14	such	such	ADJ
easat-1685	91	15	that	that	SCONJ
easat-1685	91	16	𝑋	𝑋	PROPN
easat-1685	91	17	⊆	⊆	NUM
easat-1685	91	18	(	(	PUNCT
easat-1685	91	19	𝑋	𝑋	NOUN
easat-1685	91	20	\	\	PROPN
easat-1685	91	21	𝑌	𝑌	PROPN
easat-1685	91	22	)	)	PUNCT
easat-1685	91	23	⋃(⋃	⋃(⋃	PROPN
easat-1685	91	24	𝐺𝜆𝜆∈𝛬0	𝐺𝜆𝜆∈𝛬0	PROPN
easat-1685	91	25	)	)	PUNCT
easat-1685	91	26	and	and	CCONJ
easat-1685	91	27	so	so	ADV
easat-1685	91	28	𝑌	𝑌	PROPN
easat-1685	91	29	⊆	⊆	NUM
easat-1685	91	30	⋃	⋃	NOUN
easat-1685	91	31	𝐺𝜆𝜆∈𝛬0	𝐺𝜆𝜆∈𝛬0	NOUN
easat-1685	91	32	.	.	PUNCT
easat-1685	92	1	hence	hence	ADV
easat-1685	92	2	every	every	DET
easat-1685	92	3	𝒶-open	𝒶-open	NOUN
easat-1685	92	4	cover	cover	NOUN
easat-1685	92	5	{	{	PUNCT
easat-1685	92	6	𝐺𝜆	𝐺𝜆	PROPN
easat-1685	92	7	∶	∶	NOUN
easat-1685	92	8	𝜆	𝜆	PRON
easat-1685	92	9	∈	∈	PROPN
easat-1685	93	1	𝛬	𝛬	NOUN
easat-1685	93	2	}	}	PUNCT
easat-1685	93	3	of	of	ADP
easat-1685	93	4	𝑌	𝑌	PROPN
easat-1685	93	5	has	have	VERB
easat-1685	93	6	a	a	DET
easat-1685	93	7	finite	finite	ADJ
easat-1685	93	8	subcover	subcover	PROPN
easat-1685	93	9	.	.	PUNCT
easat-1685	94	1	then	then	ADV
easat-1685	94	2	𝑌	𝑌	PROPN
easat-1685	94	3	is	be	AUX
easat-1685	94	4	an	an	DET
easat-1685	94	5	𝒶-compact	𝒶-compact	NOUN
easat-1685	94	6	.	.	PUNCT
easat-1685	95	1	hence	hence	ADV
easat-1685	95	2	,	,	PUNCT
easat-1685	95	3	the	the	DET
easat-1685	95	4	theorem	theorem	NOUN
easat-1685	95	5	is	be	AUX
easat-1685	95	6	done	do	VERB
easat-1685	95	7	.	.	PUNCT
easat-1685	96	1	theorem	theorem	VERB
easat-1685	96	2	3.8	3.8	NUM
easat-1685	96	3	:	:	PUNCT
easat-1685	96	4	let	let	VERB
easat-1685	96	5	𝑓	𝑓	PRON
easat-1685	96	6	be	be	AUX
easat-1685	96	7	𝑎-continuous	𝑎-continuous	ADJ
easat-1685	96	8	mapping	mapping	NOUN
easat-1685	96	9	from	from	ADP
easat-1685	96	10	(	(	PUNCT
easat-1685	96	11	𝑋	𝑋	PROPN
easat-1685	96	12	,	,	PUNCT
easat-1685	96	13	𝒯	𝒯	PROPN
easat-1685	96	14	)	)	PUNCT
easat-1685	96	15	to	to	ADP
easat-1685	96	16	(	(	PUNCT
easat-1685	96	17	𝑌	𝑌	PROPN
easat-1685	96	18	,	,	PUNCT
easat-1685	96	19	𝒯′	𝒯′	NUM
easat-1685	96	20	)	)	PUNCT
easat-1685	96	21	and	and	CCONJ
easat-1685	96	22	𝑉	𝑉	PROPN
easat-1685	96	23	be	be	VERB
easat-1685	96	24	a	a	DET
easat-1685	96	25	𝑎-open	𝑎-open	NOUN
easat-1685	96	26	set	set	VERB
easat-1685	96	27	in	in	ADP
easat-1685	96	28	𝑌.	𝑌.	PROPN
easat-1685	96	29	then	then	ADV
easat-1685	96	30	𝑓−1(𝑉	𝑓−1(𝑉	NOUN
easat-1685	96	31	)	)	PUNCT
easat-1685	96	32	is	be	AUX
easat-1685	96	33	an	an	DET
easat-1685	96	34	𝒶-open	𝒶-open	NOUN
easat-1685	96	35	in	in	ADP
easat-1685	96	36	𝑋.	𝑋.	PROPN
easat-1685	96	37	proof	proof	NOUN
easat-1685	96	38	:	:	PUNCT
easat-1685	96	39	let	let	VERB
easat-1685	96	40	𝑓	𝑓	X
easat-1685	96	41	:	:	PUNCT
easat-1685	96	42	(	(	PUNCT
easat-1685	96	43	𝑋	𝑋	PROPN
easat-1685	96	44	,	,	PUNCT
easat-1685	96	45	𝒯	𝒯	PROPN
easat-1685	96	46	)	)	PUNCT
easat-1685	96	47	→	→	SYM
easat-1685	96	48	(	(	PUNCT
easat-1685	96	49	𝑌	𝑌	PROPN
easat-1685	96	50	,	,	PUNCT
easat-1685	96	51	𝒯′	𝒯′	NUM
easat-1685	96	52	)	)	PUNCT
easat-1685	96	53	be	be	AUX
easat-1685	96	54	𝑎-continuous	𝑎-continuous	ADJ
easat-1685	96	55	mapping	mapping	NOUN
easat-1685	96	56	.	.	PUNCT
easat-1685	97	1	we	we	PRON
easat-1685	97	2	show	show	VERB
easat-1685	97	3	that	that	SCONJ
easat-1685	97	4	,	,	PUNCT
easat-1685	97	5	𝑓−1(𝑉	𝑓−1(𝑉	NOUN
easat-1685	97	6	)	)	PUNCT
easat-1685	97	7	is	be	AUX
easat-1685	97	8	𝒶-open	𝒶-open	ADJ
easat-1685	97	9	in	in	ADP
easat-1685	97	10	𝑋.	𝑋.	PROPN
easat-1685	97	11	since	since	SCONJ
easat-1685	97	12	𝑉	𝑉	PROPN
easat-1685	97	13	be	be	VERB
easat-1685	97	14	a	a	DET
easat-1685	97	15	𝛿-open	𝛿-open	NOUN
easat-1685	97	16	set	set	VERB
easat-1685	97	17	in	in	ADP
easat-1685	97	18	𝑌	𝑌	PROPN
easat-1685	97	19	,	,	PUNCT
easat-1685	97	20	then	then	ADV
easat-1685	97	21	from	from	ADP
easat-1685	97	22	𝛿-continuity	𝛿-continuity	NOUN
easat-1685	97	23	of	of	ADP
easat-1685	97	24	𝑓	𝑓	PRON
easat-1685	97	25	,	,	PUNCT
easat-1685	97	26	we	we	PRON
easat-1685	97	27	must	must	AUX
easat-1685	97	28	have	have	VERB
easat-1685	97	29	𝑓−1(𝑉	𝑓−1(𝑉	NOUN
easat-1685	97	30	)	)	PUNCT
easat-1685	97	31	is	be	AUX
easat-1685	97	32	an	an	DET
easat-1685	97	33	open	open	ADJ
easat-1685	97	34	set	set	NOUN
easat-1685	97	35	of	of	ADP
easat-1685	97	36	𝑋.	𝑋.	PROPN
easat-1685	97	37	that	that	PRON
easat-1685	97	38	is	be	AUX
easat-1685	97	39	,	,	PUNCT
easat-1685	97	40	for	for	ADP
easat-1685	97	41	every	every	DET
easat-1685	97	42	𝑥	𝑥	DET
easat-1685	97	43	∈	∈	PROPN
easat-1685	97	44	𝑓−1(𝑉	𝑓−1(𝑉	NOUN
easat-1685	97	45	)	)	PUNCT
easat-1685	97	46	,	,	PUNCT
easat-1685	97	47	there	there	PRON
easat-1685	97	48	exists	exist	VERB
easat-1685	97	49	a	a	DET
easat-1685	97	50	a	a	ADV
easat-1685	97	51	-	-	PUNCT
easat-1685	97	52	open	open	ADJ
easat-1685	97	53	set	set	NOUN
easat-1685	97	54	𝑊	𝑊	PROPN
easat-1685	97	55	in	in	ADP
easat-1685	97	56	𝑋	𝑋	NOUN
easat-1685	97	57	with	with	ADP
easat-1685	97	58	𝑥	𝑥	DET
easat-1685	97	59	∈	∈	PROPN
easat-1685	97	60	𝑊	𝑊	PROPN
easat-1685	97	61	⊆	⊆	NUM
easat-1685	97	62	𝑓−1(𝑉	𝑓−1(𝑉	NOUN
easat-1685	97	63	)	)	PUNCT
easat-1685	97	64	.let	.let	PUNCT
easat-1685	98	1	𝑥	𝑥	PRON
easat-1685	98	2	∈	∈	NOUN
easat-1685	98	3	𝑓−1(𝑉	𝑓−1(𝑉	NOUN
easat-1685	98	4	)	)	PUNCT
easat-1685	98	5	.	.	PUNCT
easat-1685	99	1	then	then	ADV
easat-1685	99	2	𝑓(𝑥	𝑓(𝑥	NOUN
easat-1685	99	3	)	)	PUNCT
easat-1685	99	4	∈	∈	PROPN
easat-1685	99	5	𝑉	𝑉	PROPN
easat-1685	99	6	,	,	PUNCT
easat-1685	99	7	since	since	SCONJ
easat-1685	99	8	𝑉	𝑉	PROPN
easat-1685	99	9	is	be	AUX
easat-1685	99	10	a	a	DET
easat-1685	99	11	-	-	PUNCT
easat-1685	99	12	open	open	ADJ
easat-1685	99	13	,	,	PUNCT
easat-1685	99	14	there	there	PRON
easat-1685	99	15	exists	exist	VERB
easat-1685	99	16	a	a	DET
easat-1685	99	17	a	a	ADV
easat-1685	99	18	-	-	PUNCT
easat-1685	99	19	open	open	ADJ
easat-1685	99	20	set	set	NOUN
easat-1685	99	21	𝑈	𝑈	PROPN
easat-1685	99	22	in	in	ADP
easat-1685	99	23	y	y	PROPN
easat-1685	99	24	such	such	ADJ
easat-1685	99	25	that	that	SCONJ
easat-1685	99	26	𝑓(𝑥	𝑓(𝑥	NOUN
easat-1685	99	27	)	)	PUNCT
easat-1685	99	28	∈	∈	PROPN
easat-1685	99	29	𝑈	𝑈	PROPN
easat-1685	99	30	⊆	⊆	NUM
easat-1685	99	31	𝑉	𝑉	PROPN
easat-1685	99	32	.	.	PUNCT
easat-1685	100	1	so	so	ADV
easat-1685	100	2	,	,	PUNCT
easat-1685	100	3	𝑥	𝑥	PRON
easat-1685	100	4	∈	∈	PROPN
easat-1685	100	5	𝑓−1(𝑈	𝑓−1(𝑈	NUM
easat-1685	100	6	)	)	PUNCT
easat-1685	100	7	⊆	⊆	NUM
easat-1685	100	8	𝑓−1(𝑉	𝑓−1(𝑉	NOUN
easat-1685	100	9	)	)	PUNCT
easat-1685	100	10	.	.	PUNCT
easat-1685	101	1	now	now	ADV
easat-1685	101	2	since	since	SCONJ
easat-1685	101	3	𝑈	𝑈	PROPN
easat-1685	101	4	is	be	AUX
easat-1685	101	5	𝑎-open	𝑎-open	ADJ
easat-1685	101	6	in	in	ADP
easat-1685	101	7	𝑌	𝑌	PROPN
easat-1685	101	8	and	and	CCONJ
easat-1685	101	9	𝑓	𝑓	PRON
easat-1685	101	10	is	be	AUX
easat-1685	101	11	𝑎-open	𝑎-open	ADJ
easat-1685	101	12	,	,	PUNCT
easat-1685	101	13	𝑎-continuous	𝑎-continuous	ADJ
easat-1685	101	14	mapping	mapping	NOUN
easat-1685	101	15	,	,	PUNCT
easat-1685	101	16	then	then	ADV
easat-1685	101	17	,	,	PUNCT
easat-1685	101	18	𝑓−1(𝑈	𝑓−1(𝑈	PROPN
easat-1685	101	19	)	)	PUNCT
easat-1685	101	20	is	be	AUX
easat-1685	101	21	𝑎-open	𝑎-open	PROPN
easat-1685	101	22	set	set	VERB
easat-1685	101	23	in	in	ADP
easat-1685	101	24	𝑋.	𝑋.	PROPN
easat-1685	101	25	hence	hence	ADV
easat-1685	101	26	𝑓−1(𝑉	𝑓−1(𝑉	NUM
easat-1685	101	27	)	)	PUNCT
easat-1685	101	28	is	be	AUX
easat-1685	101	29	𝑎-open	𝑎-open	ADJ
easat-1685	101	30	in	in	ADP
easat-1685	101	31	𝑋	𝑋	PROPN
easat-1685	101	32	,	,	PUNCT
easat-1685	101	33	therefore	therefore	ADV
easat-1685	101	34	𝑓−1(𝑉	𝑓−1(𝑉	NOUN
easat-1685	101	35	)	)	PUNCT
easat-1685	101	36	is	be	AUX
easat-1685	101	37	an	an	DET
easat-1685	101	38	𝒶-open	𝒶-open	NOUN
easat-1685	101	39	in	in	ADP
easat-1685	101	40	𝑋.	𝑋.	PROPN
easat-1685	101	41	theorem	theorem	VERB
easat-1685	101	42	3.9	3.9	NUM
easat-1685	101	43	:	:	PUNCT
easat-1685	101	44	let	let	VERB
easat-1685	101	45	𝑓	𝑓	X
easat-1685	101	46	:	:	PUNCT
easat-1685	101	47	(	(	PUNCT
easat-1685	101	48	𝑋	𝑋	PROPN
easat-1685	101	49	,	,	PUNCT
easat-1685	101	50	𝒯	𝒯	PROPN
easat-1685	101	51	)	)	PUNCT
easat-1685	101	52	→	→	SYM
easat-1685	101	53	(	(	PUNCT
easat-1685	101	54	𝑌	𝑌	PROPN
easat-1685	101	55	,	,	PUNCT
easat-1685	101	56	𝒯′	𝒯′	NUM
easat-1685	101	57	)	)	PUNCT
easat-1685	101	58	be	be	AUX
easat-1685	101	59	an	an	DET
easat-1685	101	60	𝒶-open	𝒶-open	NOUN
easat-1685	101	61	,	,	PUNCT
easat-1685	101	62	𝒶-continuous	𝒶-continuous	PROPN
easat-1685	101	63	mapping	mapping	NOUN
easat-1685	101	64	,	,	PUNCT
easat-1685	101	65	𝑋	𝑋	PROPN
easat-1685	101	66	is	be	AUX
easat-1685	101	67	an	an	DET
easat-1685	101	68	𝒶-compact	𝒶-compact	NOUN
easat-1685	101	69	.	.	PUNCT
easat-1685	102	1	then	then	ADV
easat-1685	102	2	𝑓(𝑋	𝑓(𝑋	NUM
easat-1685	102	3	)	)	PUNCT
easat-1685	102	4	is	be	AUX
easat-1685	102	5	an	an	DET
easat-1685	102	6	𝒶-compact	𝒶-compact	NOUN
easat-1685	102	7	subset	subset	VERB
easat-1685	102	8	𝑌.	𝑌.	PROPN
easat-1685	102	9	274	274	NUM
easat-1685	102	10	edelweiss	edelweiss	PROPN
easat-1685	102	11	applied	apply	VERB
easat-1685	102	12	science	science	NOUN
easat-1685	102	13	and	and	CCONJ
easat-1685	102	14	technology	technology	NOUN
easat-1685	102	15	issn	issn	PROPN
easat-1685	102	16	:	:	PUNCT
easat-1685	102	17	2576	2576	NUM
easat-1685	102	18	-	-	SYM
easat-1685	102	19	8484	8484	NUM
easat-1685	102	20	vol	vol	NOUN
easat-1685	102	21	.	.	PROPN
easat-1685	102	22	8	8	NUM
easat-1685	102	23	,	,	PUNCT
easat-1685	102	24	no	no	INTJ
easat-1685	102	25	.	.	NOUN
easat-1685	102	26	5	5	NUM
easat-1685	102	27	:	:	PUNCT
easat-1685	102	28	271	271	NUM
easat-1685	102	29	-	-	SYM
easat-1685	102	30	277	277	NUM
easat-1685	102	31	,	,	PUNCT
easat-1685	102	32	2024	2024	NUM
easat-1685	102	33	doi	doi	NOUN
easat-1685	102	34	:	:	PUNCT
easat-1685	102	35	10.55214/25768484.v8i5.1685	10.55214/25768484.v8i5.1685	NUM
easat-1685	102	36	©	©	ADP
easat-1685	102	37	2024	2024	NUM
easat-1685	102	38	by	by	ADP
easat-1685	102	39	the	the	DET
easat-1685	102	40	authors	author	NOUN
easat-1685	102	41	;	;	PUNCT
easat-1685	102	42	licensee	licensee	PROPN
easat-1685	102	43	learning	learn	VERB
easat-1685	102	44	gate	gate	NOUN
easat-1685	102	45	proof	proof	NOUN
easat-1685	102	46	:	:	PUNCT
easat-1685	102	47	let	let	VERB
easat-1685	102	48	{	{	PUNCT
easat-1685	102	49	𝑉𝜆	𝑉𝜆	NOUN
easat-1685	102	50	∶	∶	VERB
easat-1685	102	51	𝜆	𝜆	DET
easat-1685	102	52	∈	∈	PROPN
easat-1685	102	53	𝛬	𝛬	NOUN
easat-1685	102	54	}	}	PUNCT
easat-1685	102	55	be	be	AUX
easat-1685	102	56	an	an	DET
easat-1685	102	57	𝒶-open	𝒶-open	NOUN
easat-1685	102	58	cover	cover	NOUN
easat-1685	102	59	of	of	ADP
easat-1685	102	60	𝑓(𝑋	𝑓(𝑋	NUM
easat-1685	102	61	)	)	PUNCT
easat-1685	102	62	in	in	ADP
easat-1685	102	63	𝑌	𝑌	PROPN
easat-1685	102	64	,	,	PUNCT
easat-1685	102	65	so	so	ADV
easat-1685	102	66	𝑓(𝑋	𝑓(𝑋	NOUN
easat-1685	102	67	)	)	PUNCT
easat-1685	102	68	⊆	⊆	NUM
easat-1685	102	69	⋃	⋃	NOUN
easat-1685	102	70	𝑉𝜆𝜆∈𝛬	𝑉𝜆𝜆∈𝛬	NOUN
easat-1685	102	71	and	and	CCONJ
easat-1685	102	72	hence	hence	ADV
easat-1685	102	73	equality	equality	NOUN
easat-1685	102	74	is	be	AUX
easat-1685	102	75	hold	hold	NOUN
easat-1685	102	76	,	,	PUNCT
easat-1685	102	77	𝑋	𝑋	PROPN
easat-1685	102	78	⊆	⊆	NUM
easat-1685	102	79	𝑓−1(⋃	𝑓−1(⋃	NOUN
easat-1685	102	80	𝑉𝜆𝜆∈𝛬	𝑉𝜆𝜆∈𝛬	NOUN
easat-1685	102	81	)	)	PUNCT
easat-1685	103	1	=	=	PUNCT
easat-1685	103	2	⋃	⋃	NOUN
easat-1685	103	3	𝑉𝜆𝜆∈𝛬	𝑉𝜆𝜆∈𝛬	NOUN
easat-1685	103	4	𝑓−1(𝑉𝜆	𝑓−1(𝑉𝜆	NOUN
easat-1685	103	5	)	)	PUNCT
easat-1685	103	6	.	.	PUNCT
easat-1685	104	1	since	since	SCONJ
easat-1685	104	2	𝑓	𝑓	PRON
easat-1685	104	3	is	be	AUX
easat-1685	104	4	𝒶-open	𝒶-open	NOUN
easat-1685	104	5	and	and	CCONJ
easat-1685	104	6	𝒶-continuous	𝒶-continuous	ADJ
easat-1685	104	7	by	by	ADP
easat-1685	104	8	the	the	DET
easat-1685	104	9	previous	previous	ADJ
easat-1685	104	10	theorem	theorem	NOUN
easat-1685	104	11	(	(	PUNCT
easat-1685	104	12	3.8	3.8	NUM
easat-1685	104	13	)	)	PUNCT
easat-1685	104	14	,	,	PUNCT
easat-1685	104	15	each	each	DET
easat-1685	104	16	𝑓−1(𝑉𝜆	𝑓−1(𝑉𝜆	NUM
easat-1685	104	17	)	)	PUNCT
easat-1685	104	18	is	be	AUX
easat-1685	104	19	𝒶-open	𝒶-open	NOUN
easat-1685	104	20	in	in	ADP
easat-1685	104	21	𝑋.	𝑋.	PROPN
easat-1685	104	22	thus	thus	ADV
easat-1685	104	23	{	{	PUNCT
easat-1685	104	24	𝑓−1(𝑉𝜆	𝑓−1(𝑉𝜆	X
easat-1685	104	25	)	)	PUNCT
easat-1685	105	1	∶	∶	NOUN
easat-1685	105	2	𝜆	𝜆	PRON
easat-1685	105	3	∈	∈	PROPN
easat-1685	106	1	𝛬	𝛬	NOUN
easat-1685	106	2	}	}	PUNCT
easat-1685	106	3	is	be	AUX
easat-1685	106	4	a	a	DET
easat-1685	106	5	𝒶-open	𝒶-open	NOUN
easat-1685	106	6	cover	cover	NOUN
easat-1685	106	7	of	of	ADP
easat-1685	106	8	𝑋.	𝑋.	PROPN
easat-1685	106	9	consequently	consequently	ADV
easat-1685	106	10	,	,	PUNCT
easat-1685	106	11	and	and	CCONJ
easat-1685	106	12	from	from	ADP
easat-1685	106	13	𝒶-compactness	𝒶-compactness	NOUN
easat-1685	106	14	of	of	ADP
easat-1685	106	15	𝑋	𝑋	PROPN
easat-1685	106	16	,	,	PUNCT
easat-1685	106	17	there	there	PRON
easat-1685	106	18	exists	exist	VERB
easat-1685	106	19	{	{	PUNCT
easat-1685	106	20	𝑓−1(𝑉𝜆𝑖	𝑓−1(𝑉𝜆𝑖	PUNCT
easat-1685	106	21	)	)	PUNCT
easat-1685	106	22	𝑖	𝑖	SYM
easat-1685	107	1	=	=	SYM
easat-1685	107	2	1	1	NUM
easat-1685	107	3	,	,	PUNCT
easat-1685	107	4	2	2	NUM
easat-1685	107	5	,	,	PUNCT
easat-1685	107	6	…	…	PUNCT
easat-1685	107	7	,	,	PUNCT
easat-1685	107	8	𝑚	𝑚	NOUN
easat-1685	107	9	}	}	PUNCT
easat-1685	107	10	of	of	ADP
easat-1685	107	11	{	{	PUNCT
easat-1685	107	12	𝑓−1(𝑉𝜆	𝑓−1(𝑉𝜆	X
easat-1685	107	13	)	)	PUNCT
easat-1685	107	14	∶	∶	NOUN
easat-1685	107	15	𝜆	𝜆	DET
easat-1685	107	16	∈	∈	PROPN
easat-1685	107	17	𝛬	𝛬	NOUN
easat-1685	107	18	which	which	PRON
easat-1685	107	19	also	also	ADV
easat-1685	107	20	covers	cover	VERB
easat-1685	107	21	𝑋.	𝑋.	PROPN
easat-1685	107	22	thus	thus	ADV
easat-1685	107	23	𝑋	𝑋	PROPN
easat-1685	107	24	⊆	⊆	SYM
easat-1685	107	25	⋃	⋃	NOUN
easat-1685	107	26	𝑓−1(𝑉𝜆𝑖	𝑓−1(𝑉𝜆𝑖	PUNCT
easat-1685	107	27	)	)	PUNCT
easat-1685	107	28	𝑚	𝑚	X
easat-1685	107	29	𝑖=1	𝑖=1	PROPN
easat-1685	107	30	.	.	PUNCT
easat-1685	108	1	so	so	ADV
easat-1685	108	2	,	,	PUNCT
easat-1685	108	3	𝑓(𝑋	𝑓(𝑋	NUM
easat-1685	108	4	)	)	PUNCT
easat-1685	108	5	⊆	⊆	NUM
easat-1685	108	6	⋃	⋃	NOUN
easat-1685	108	7	𝑉𝜆𝑖	𝑉𝜆𝑖	ADP
easat-1685	108	8	𝑚	𝑚	X
easat-1685	108	9	𝑖=1	𝑖=1	PROPN
easat-1685	108	10	.	.	PUNCT
easat-1685	109	1	therefore	therefore	ADV
easat-1685	109	2	,	,	PUNCT
easat-1685	109	3	{	{	PUNCT
easat-1685	109	4	𝑉𝜆𝑖	𝑉𝜆𝑖	ADP
easat-1685	109	5	∶	∶	NOUN
easat-1685	109	6	𝑖	𝑖	NOUN
easat-1685	109	7	=	=	SYM
easat-1685	109	8	1	1	NUM
easat-1685	109	9	,	,	PUNCT
easat-1685	109	10	2	2	NUM
easat-1685	109	11	,	,	PUNCT
easat-1685	109	12	.	.	PUNCT
easat-1685	109	13	.	.	PUNCT
easat-1685	109	14	.	.	PUNCT
easat-1685	110	1	,	,	PUNCT
easat-1685	110	2	𝑚	𝑚	X
easat-1685	110	3	}	}	PUNCT
easat-1685	110	4	is	be	AUX
easat-1685	110	5	a	a	DET
easat-1685	110	6	finite	finite	ADJ
easat-1685	110	7	subcollection	subcollection	NOUN
easat-1685	110	8	of	of	ADP
easat-1685	110	9	{	{	PUNCT
easat-1685	110	10	𝑉𝜆	𝑉𝜆	PROPN
easat-1685	110	11	∶	∶	PROPN
easat-1685	110	12	𝜆	𝜆	DET
easat-1685	110	13	∈	∈	PROPN
easat-1685	110	14	𝛬	𝛬	NOUN
easat-1685	110	15	}	}	PUNCT
easat-1685	110	16	which	which	PRON
easat-1685	110	17	covers	cover	VERB
easat-1685	110	18	𝑓(𝑋	𝑓(𝑋	NOUN
easat-1685	110	19	)	)	PUNCT
easat-1685	110	20	.	.	PUNCT
easat-1685	111	1	so	so	ADV
easat-1685	111	2	,	,	PUNCT
easat-1685	111	3	𝑓(𝑋	𝑓(𝑋	PROPN
easat-1685	111	4	)	)	PUNCT
easat-1685	111	5	is	be	AUX
easat-1685	111	6	𝒶-compact	𝒶-compact	NOUN
easat-1685	111	7	in	in	ADP
easat-1685	111	8	(	(	PUNCT
easat-1685	111	9	𝑌	𝑌	PROPN
easat-1685	111	10	,	,	PUNCT
easat-1685	111	11	𝒯′	𝒯′	NUM
easat-1685	111	12	)	)	PUNCT
easat-1685	111	13	.	.	PUNCT
easat-1685	112	1	definition	definition	NOUN
easat-1685	112	2	3.10	3.10	NUM
easat-1685	112	3	:	:	PUNCT
easat-1685	112	4	let	let	VERB
easat-1685	112	5	(	(	PUNCT
easat-1685	112	6	𝑋	𝑋	PROPN
easat-1685	112	7	,	,	PUNCT
easat-1685	112	8	𝒯	𝒯	PROPN
easat-1685	112	9	)	)	PUNCT
easat-1685	112	10	be	be	VERB
easat-1685	112	11	a	a	DET
easat-1685	112	12	t.	t.	NOUN
easat-1685	112	13	𝑠	𝑠	PROPN
easat-1685	112	14	and	and	CCONJ
easat-1685	112	15	𝒜	𝒜	NOUN
easat-1685	112	16	be	be	AUX
easat-1685	112	17	a	a	DET
easat-1685	112	18	family	family	NOUN
easat-1685	112	19	of	of	ADP
easat-1685	112	20	subsets	subset	NOUN
easat-1685	112	21	of	of	ADP
easat-1685	112	22	𝒯𝒶.	𝒯𝒶.	PROPN
easat-1685	113	1	then	then	ADV
easat-1685	113	2	𝒶-star	𝒶-star	NOUN
easat-1685	113	3	of	of	ADP
easat-1685	113	4	𝐷	𝐷	PROPN
easat-1685	113	5	⊆	⊆	NUM
easat-1685	113	6	𝑋	𝑋	NOUN
easat-1685	113	7	with	with	ADP
easat-1685	113	8	respect	respect	NOUN
easat-1685	113	9	to	to	ADP
easat-1685	113	10	𝒜	𝒜	NOUN
easat-1685	113	11	is	be	AUX
easat-1685	113	12	the	the	DET
easat-1685	113	13	set	set	NOUN
easat-1685	113	14	:	:	PUNCT
easat-1685	113	15	𝑆𝑡𝒶(𝒜	𝑆𝑡𝒶(𝒜	PRON
easat-1685	113	16	،	،	PROPN
easat-1685	113	17	𝐷	𝐷	PROPN
easat-1685	113	18	)	)	PUNCT
easat-1685	113	19	=	=	SYM
easat-1685	113	20	⋃{𝑄	⋃{𝑄	NOUN
easat-1685	113	21	∈	∈	PROPN
easat-1685	113	22	𝒜	𝒜	NOUN
easat-1685	113	23	∶	∶	NOUN
easat-1685	113	24	𝑄⋂𝐷	𝑄⋂𝐷	X
easat-1685	113	25	≠	≠	PROPN
easat-1685	113	26	∅	∅	NOUN
easat-1685	113	27	}	}	PUNCT
easat-1685	113	28	.	.	PUNCT
easat-1685	114	1	remark	remark	VERB
easat-1685	114	2	3.11	3.11	NUM
easat-1685	114	3	:	:	PUNCT
easat-1685	114	4	a	a	DET
easat-1685	114	5	𝒶-star	𝒶-star	NOUN
easat-1685	114	6	of	of	ADP
easat-1685	114	7	a	a	DET
easat-1685	114	8	singleton	singleton	NOUN
easat-1685	114	9	set	set	NOUN
easat-1685	114	10	{	{	PUNCT
easat-1685	114	11	𝑥	𝑥	NOUN
easat-1685	114	12	}	}	PUNCT
easat-1685	114	13	,	,	PUNCT
easat-1685	114	14	𝑥	𝑥	DET
easat-1685	114	15	∈	∈	PROPN
easat-1685	114	16	𝑋	𝑋	NOUN
easat-1685	114	17	with	with	ADP
easat-1685	114	18	respect	respect	NOUN
easat-1685	114	19	to	to	ADP
easat-1685	114	20	𝒜	𝒜	NOUN
easat-1685	114	21	is	be	AUX
easat-1685	114	22	said	say	VERB
easat-1685	114	23	to	to	PART
easat-1685	114	24	be	be	AUX
easat-1685	114	25	a	a	DET
easat-1685	114	26	𝒶-star	𝒶-star	NOUN
easat-1685	114	27	of	of	ADP
easat-1685	114	28	a	a	DET
easat-1685	114	29	point	point	NOUN
easat-1685	114	30	and	and	CCONJ
easat-1685	114	31	defined	define	VERB
easat-1685	114	32	as	as	ADP
easat-1685	114	33	:	:	PUNCT
easat-1685	114	34	𝑆𝑡𝒶(𝒜	𝑆𝑡𝒶(𝒜	NOUN
easat-1685	114	35	,	,	PUNCT
easat-1685	114	36	{	{	PUNCT
easat-1685	114	37	𝑥	𝑥	NOUN
easat-1685	114	38	}	}	PUNCT
easat-1685	114	39	)	)	PUNCT
easat-1685	115	1	=	=	SYM
easat-1685	115	2	⋃{𝑄	⋃{𝑄	NOUN
easat-1685	115	3	∈	∈	PROPN
easat-1685	115	4	𝒜	𝒜	NOUN
easat-1685	115	5	∶	∶	NOUN
easat-1685	115	6	𝑄⋂{𝑥	𝑄⋂{𝑥	NOUN
easat-1685	115	7	}	}	PUNCT
easat-1685	115	8	≠	≠	PROPN
easat-1685	115	9	∅	∅	NOUN
easat-1685	115	10	}	}	PUNCT
easat-1685	115	11	.	.	PUNCT
easat-1685	116	1	example	example	NOUN
easat-1685	116	2	3.12	3.12	NUM
easat-1685	116	3	:	:	PUNCT
easat-1685	116	4	for	for	ADP
easat-1685	116	5	any	any	DET
easat-1685	116	6	non	non	ADJ
easat-1685	116	7	-	-	ADJ
easat-1685	116	8	empty	empty	ADJ
easat-1685	116	9	set	set	NOUN
easat-1685	116	10	𝑋.	𝑋.	PROPN
easat-1685	116	11	then	then	ADV
easat-1685	116	12	:	:	PUNCT
easat-1685	116	13	i.	i.	PROPN
easat-1685	116	14	in	in	ADP
easat-1685	116	15	(	(	PUNCT
easat-1685	116	16	𝑋	𝑋	PROPN
easat-1685	116	17	,	,	PUNCT
easat-1685	116	18	𝒯𝑑𝑖𝑠	𝒯𝑑𝑖𝑠	PROPN
easat-1685	116	19	)	)	PUNCT
easat-1685	117	1	,	,	PUNCT
easat-1685	117	2	it	it	PRON
easat-1685	117	3	follows	follow	VERB
easat-1685	117	4	that	that	SCONJ
easat-1685	117	5	𝒯𝒶	𝒯𝒶	PROPN
easat-1685	117	6	=	=	SYM
easat-1685	117	7	𝒯𝑑𝑖𝑠	𝒯𝑑𝑖𝑠	PROPN
easat-1685	117	8	and	and	CCONJ
easat-1685	117	9	𝑆𝑡𝒶(𝒜	𝑆𝑡𝒶(𝒜	DET
easat-1685	117	10	،	،	NOUN
easat-1685	117	11	{	{	PUNCT
easat-1685	117	12	𝑥	𝑥	NOUN
easat-1685	117	13	}	}	PUNCT
easat-1685	117	14	)	)	PUNCT
easat-1685	118	1	=	=	SYM
easat-1685	118	2	𝑋	𝑋	PROPN
easat-1685	118	3	for	for	ADP
easat-1685	118	4	any	any	DET
easat-1685	118	5	𝑥	𝑥	PRON
easat-1685	118	6	∈	∈	PROPN
easat-1685	118	7	𝑋.	𝑋.	PROPN
easat-1685	118	8	ii	ii	PROPN
easat-1685	118	9	.	.	PUNCT
easat-1685	119	1	in	in	ADP
easat-1685	119	2	(	(	PUNCT
easat-1685	119	3	𝑋	𝑋	PROPN
easat-1685	119	4	,	,	PUNCT
easat-1685	119	5	𝒯𝑖𝑛𝑑	𝒯𝑖𝑛𝑑	PROPN
easat-1685	119	6	)	)	PUNCT
easat-1685	119	7	,	,	PUNCT
easat-1685	119	8	it	it	PRON
easat-1685	119	9	follows	follow	VERB
easat-1685	119	10	that	that	SCONJ
easat-1685	119	11	𝒯𝒶	𝒯𝒶	PROPN
easat-1685	119	12	=	=	SYM
easat-1685	119	13	𝒯𝑑𝑖𝑠	𝒯𝑑𝑖𝑠	PROPN
easat-1685	119	14	and	and	CCONJ
easat-1685	119	15	𝑆𝑡𝒶(𝒜	𝑆𝑡𝒶(𝒜	DET
easat-1685	119	16	،	،	NOUN
easat-1685	119	17	𝑋	𝑋	PROPN
easat-1685	119	18	)	)	PUNCT
easat-1685	119	19	=	=	SYM
easat-1685	119	20	𝑋.	𝑋.	PROPN
easat-1685	119	21	theorem	theorem	VERB
easat-1685	119	22	3.13	3.13	NUM
easat-1685	119	23	:	:	PUNCT
easat-1685	119	24	let	let	VERB
easat-1685	119	25	(	(	PUNCT
easat-1685	119	26	𝑋	𝑋	PROPN
easat-1685	119	27	,	,	PUNCT
easat-1685	119	28	𝒯	𝒯	PROPN
easat-1685	119	29	)	)	PUNCT
easat-1685	119	30	be	be	VERB
easat-1685	119	31	a	a	DET
easat-1685	119	32	t.	t.	NOUN
easat-1685	119	33	𝑠	𝑠	PROPN
easat-1685	119	34	and	and	CCONJ
easat-1685	119	35	𝑈	𝑈	PROPN
easat-1685	119	36	be	be	AUX
easat-1685	119	37	an	an	DET
easat-1685	119	38	𝒶-open	𝒶-open	NOUN
easat-1685	119	39	of	of	ADP
easat-1685	119	40	𝑋.	𝑋.	PROPN
easat-1685	119	41	for	for	ADP
easat-1685	119	42	every	every	DET
easat-1685	119	43	𝒶-dense	𝒶-dense	NOUN
easat-1685	119	44	subspace	subspace	NOUN
easat-1685	119	45	𝑌	𝑌	PROPN
easat-1685	119	46	⊆	⊆	NUM
easat-1685	119	47	𝑋	𝑋	PROPN
easat-1685	119	48	there	there	PRON
easat-1685	119	49	exists	exist	VERB
easat-1685	119	50	a	a	DET
easat-1685	119	51	subset	subset	NOUN
easat-1685	119	52	𝐷	𝐷	PROPN
easat-1685	119	53	⊆	⊆	NUM
easat-1685	119	54	𝑌	𝑌	PROPN
easat-1685	119	55	such	such	ADJ
easat-1685	119	56	that	that	SCONJ
easat-1685	119	57	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	119	58	(	(	PUNCT
easat-1685	119	59	𝐷	𝐷	PROPN
easat-1685	119	60	,	,	PUNCT
easat-1685	119	61	𝑈	𝑈	PROPN
easat-1685	119	62	)	)	PUNCT
easat-1685	120	1	=	=	SYM
easat-1685	120	2	𝑋.	𝑋.	PROPN
easat-1685	120	3	proof	proof	NOUN
easat-1685	120	4	:	:	PUNCT
easat-1685	120	5	from	from	ADP
easat-1685	120	6	𝒶-density	𝒶-density	NOUN
easat-1685	120	7	of	of	ADP
easat-1685	120	8	𝑌	𝑌	PROPN
easat-1685	120	9	,	,	PUNCT
easat-1685	120	10	we	we	PRON
easat-1685	120	11	have	have	VERB
easat-1685	120	12	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	120	13	(	(	PUNCT
easat-1685	120	14	𝐷	𝐷	PROPN
easat-1685	120	15	,	,	PUNCT
easat-1685	120	16	𝑈	𝑈	PROPN
easat-1685	120	17	)	)	PUNCT
easat-1685	120	18	=	=	SYM
easat-1685	120	19	⋃{𝒪	⋃{𝒪	PROPN
easat-1685	120	20	⊆	⊆	NUM
easat-1685	120	21	𝑌	𝑌	PROPN
easat-1685	120	22	∶	∶	NOUN
easat-1685	120	23	𝐷⋂𝒪	𝐷⋂𝒪	X
easat-1685	120	24	≠	≠	PROPN
easat-1685	120	25	∅	∅	NOUN
easat-1685	120	26	}	}	PUNCT
easat-1685	120	27	.	.	PUNCT
easat-1685	121	1	that	that	PRON
easat-1685	121	2	is	be	AUX
easat-1685	121	3	,	,	PUNCT
easat-1685	121	4	for	for	ADP
easat-1685	121	5	any	any	DET
easat-1685	121	6	𝑈	𝑈	PROPN
easat-1685	121	7	an	an	DET
easat-1685	121	8	𝒶-open	𝒶-open	NOUN
easat-1685	121	9	of	of	ADP
easat-1685	121	10	𝑋	𝑋	PROPN
easat-1685	121	11	,	,	PUNCT
easat-1685	121	12	we	we	PRON
easat-1685	121	13	have	have	VERB
easat-1685	121	14	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	121	15	(	(	PUNCT
easat-1685	121	16	𝐷	𝐷	PROPN
easat-1685	121	17	,	,	PUNCT
easat-1685	121	18	𝑈	𝑈	PROPN
easat-1685	121	19	)	)	PUNCT
easat-1685	121	20	=	=	SYM
easat-1685	121	21	𝑋.	𝑋.	PROPN
easat-1685	121	22	remark	remark	VERB
easat-1685	121	23	3.14	3.14	NUM
easat-1685	121	24	:	:	PUNCT
easat-1685	121	25	for	for	ADP
easat-1685	121	26	every	every	DET
easat-1685	121	27	t.	t.	PROPN
easat-1685	121	28	𝑠	𝑠	PROPN
easat-1685	121	29	(	(	PUNCT
easat-1685	121	30	𝑋	𝑋	PROPN
easat-1685	121	31	,	,	PUNCT
easat-1685	121	32	𝒯	𝒯	PROPN
easat-1685	121	33	)	)	PUNCT
easat-1685	121	34	and	and	CCONJ
easat-1685	121	35	𝐷	𝐷	PROPN
easat-1685	121	36	is	be	AUX
easat-1685	121	37	a	a	DET
easat-1685	121	38	𝒶-open	𝒶-open	NOUN
easat-1685	121	39	cover	cover	NOUN
easat-1685	121	40	of	of	ADP
easat-1685	121	41	𝑋.	𝑋.	PROPN
easat-1685	121	42	it	it	PRON
easat-1685	121	43	follows	follow	VERB
easat-1685	121	44	that	that	SCONJ
easat-1685	121	45	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	121	46	(	(	PUNCT
easat-1685	121	47	𝐷	𝐷	PROPN
easat-1685	121	48	,	,	PUNCT
easat-1685	121	49	𝑈	𝑈	PROPN
easat-1685	121	50	)	)	PUNCT
easat-1685	121	51	is	be	AUX
easat-1685	121	52	an	an	DET
easat-1685	121	53	𝒶open	𝒶open	ADJ
easat-1685	121	54	set	set	NOUN
easat-1685	121	55	of	of	ADP
easat-1685	121	56	𝑋.	𝑋.	PROPN
easat-1685	121	57	definition	definition	NOUN
easat-1685	122	1	3.15	3.15	NUM
easat-1685	122	2	:	:	PUNCT
easat-1685	122	3	a	a	DET
easat-1685	122	4	t.	t.	NOUN
easat-1685	122	5	𝑠	𝑠	PROPN
easat-1685	122	6	(	(	PUNCT
easat-1685	122	7	𝑋	𝑋	PROPN
easat-1685	122	8	,	,	PUNCT
easat-1685	122	9	𝒯	𝒯	PROPN
easat-1685	122	10	)	)	PUNCT
easat-1685	122	11	is	be	AUX
easat-1685	122	12	said	say	VERB
easat-1685	122	13	to	to	PART
easat-1685	122	14	be	be	AUX
easat-1685	122	15	:	:	PUNCT
easat-1685	122	16	⦁	⦁	NUM
easat-1685	122	17	𝒶-starcompact	𝒶-starcompact	NOUN
easat-1685	122	18	,	,	PUNCT
easat-1685	122	19	if	if	SCONJ
easat-1685	122	20	for	for	ADP
easat-1685	122	21	every	every	DET
easat-1685	122	22	𝒶-open	𝒶-open	NOUN
easat-1685	122	23	covering	cover	VERB
easat-1685	122	24	𝒰	𝒰	PROPN
easat-1685	122	25	of	of	ADP
easat-1685	122	26	𝑋	𝑋	PROPN
easat-1685	122	27	,	,	PUNCT
easat-1685	122	28	there	there	PRON
easat-1685	122	29	exists	exist	VERB
easat-1685	122	30	a	a	DET
easat-1685	122	31	finite	finite	NOUN
easat-1685	122	32	subset	subset	NOUN
easat-1685	122	33	𝐹	𝐹	PROPN
easat-1685	122	34	of	of	ADP
easat-1685	122	35	𝒰	𝒰	PROPN
easat-1685	122	36	such	such	ADJ
easat-1685	122	37	that	that	SCONJ
easat-1685	122	38	𝑆𝑡𝒶(⋃	𝑆𝑡𝒶(⋃	PROPN
easat-1685	122	39	𝐹	𝐹	PROPN
easat-1685	122	40	,	,	PUNCT
easat-1685	122	41	𝒰	𝒰	PROPN
easat-1685	122	42	)	)	PUNCT
easat-1685	122	43	=	=	PUNCT
easat-1685	122	44	𝑋.	𝑋.	PROPN
easat-1685	122	45	⦁	⦁	NOUN
easat-1685	122	46	strong	strong	ADJ
easat-1685	122	47	𝒶-starcompact	𝒶-starcompact	NOUN
easat-1685	122	48	,	,	PUNCT
easat-1685	122	49	if	if	SCONJ
easat-1685	122	50	for	for	ADP
easat-1685	122	51	every	every	DET
easat-1685	122	52	𝒶-open	𝒶-open	NOUN
easat-1685	122	53	covering	cover	VERB
easat-1685	122	54	𝒰	𝒰	PROPN
easat-1685	122	55	of	of	ADP
easat-1685	122	56	𝑋	𝑋	PROPN
easat-1685	122	57	,	,	PUNCT
easat-1685	122	58	there	there	PRON
easat-1685	122	59	exists	exist	VERB
easat-1685	122	60	a	a	DET
easat-1685	122	61	finite	finite	NOUN
easat-1685	122	62	subset	subset	NOUN
easat-1685	122	63	𝐹	𝐹	PROPN
easat-1685	122	64	of	of	ADP
easat-1685	122	65	𝑋	𝑋	PROPN
easat-1685	122	66	such	such	ADJ
easat-1685	122	67	that	that	SCONJ
easat-1685	122	68	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	122	69	(	(	PUNCT
easat-1685	122	70	𝐹	𝐹	PROPN
easat-1685	122	71	,	,	PUNCT
easat-1685	122	72	𝒰	𝒰	PROPN
easat-1685	122	73	)	)	PUNCT
easat-1685	122	74	=	=	SYM
easat-1685	122	75	𝑋.	𝑋.	PROPN
easat-1685	122	76	⦁	⦁	VERB
easat-1685	122	77	𝒶	𝒶	NOUN
easat-1685	122	78	starlindelof	starlindelof	NOUN
easat-1685	122	79	,	,	PUNCT
easat-1685	122	80	if	if	SCONJ
easat-1685	122	81	for	for	ADP
easat-1685	122	82	𝒶	𝒶	NOUN
easat-1685	122	83	–	–	PUNCT
easat-1685	122	84	open	open	ADJ
easat-1685	122	85	cover	cover	NOUN
easat-1685	122	86	𝒰	𝒰	PROPN
easat-1685	122	87	of	of	ADP
easat-1685	122	88	𝑋	𝑋	PROPN
easat-1685	122	89	,	,	PUNCT
easat-1685	122	90	there	there	PRON
easat-1685	122	91	exists	exist	VERB
easat-1685	122	92	countable	countable	ADJ
easat-1685	122	93	subset	subset	NOUN
easat-1685	122	94	𝐹	𝐹	PROPN
easat-1685	122	95	of	of	ADP
easat-1685	122	96	𝒰	𝒰	PROPN
easat-1685	123	1	such	such	ADJ
easat-1685	123	2	that	that	SCONJ
easat-1685	123	3	𝑆𝑡𝒶(⋃	𝑆𝑡𝒶(⋃	PROPN
easat-1685	123	4	𝐹	𝐹	PROPN
easat-1685	123	5	,	,	PUNCT
easat-1685	123	6	𝒰	𝒰	PROPN
easat-1685	123	7	)	)	PUNCT
easat-1685	123	8	=	=	SYM
easat-1685	123	9	𝑋.	𝑋.	PROPN
easat-1685	123	10	⦁	⦁	NOUN
easat-1685	123	11	strong	strong	ADJ
easat-1685	123	12	𝒶	𝒶	NOUN
easat-1685	123	13	starlindelof	starlindelof	NOUN
easat-1685	123	14	,	,	PUNCT
easat-1685	123	15	if	if	SCONJ
easat-1685	123	16	for	for	ADP
easat-1685	123	17	𝒶	𝒶	NOUN
easat-1685	123	18	–	–	PUNCT
easat-1685	123	19	open	open	ADJ
easat-1685	123	20	cover	cover	NOUN
easat-1685	123	21	𝒰	𝒰	PROPN
easat-1685	123	22	of	of	ADP
easat-1685	123	23	𝑋	𝑋	PROPN
easat-1685	123	24	,	,	PUNCT
easat-1685	123	25	there	there	PRON
easat-1685	123	26	exists	exist	VERB
easat-1685	123	27	countable	countable	ADJ
easat-1685	123	28	subset	subset	NOUN
easat-1685	123	29	𝐹	𝐹	PROPN
easat-1685	123	30	of	of	ADP
easat-1685	123	31	𝒰	𝒰	PROPN
easat-1685	123	32	such	such	ADJ
easat-1685	123	33	that	that	SCONJ
easat-1685	123	34	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	123	35	(	(	PUNCT
easat-1685	123	36	𝐹	𝐹	PROPN
easat-1685	123	37	,	,	PUNCT
easat-1685	123	38	𝒰	𝒰	PROPN
easat-1685	123	39	)	)	PUNCT
easat-1685	123	40	=	=	SYM
easat-1685	123	41	𝑋.	𝑋.	PROPN
easat-1685	123	42	theorem	theorem	VERB
easat-1685	123	43	3.16	3.16	NUM
easat-1685	123	44	:	:	PUNCT
easat-1685	123	45	let	let	VERB
easat-1685	123	46	(	(	PUNCT
easat-1685	123	47	𝑋	𝑋	PROPN
easat-1685	123	48	,	,	PUNCT
easat-1685	123	49	𝒯	𝒯	PROPN
easat-1685	123	50	)	)	PUNCT
easat-1685	123	51	be	be	VERB
easat-1685	123	52	a	a	DET
easat-1685	123	53	𝒶-compact	𝒶-compact	NOUN
easat-1685	123	54	space	space	NOUN
easat-1685	123	55	and	and	CCONJ
easat-1685	123	56	𝒪	𝒪	NOUN
easat-1685	123	57	any	any	DET
easat-1685	123	58	𝒶-open	𝒶-open	NOUN
easat-1685	123	59	covering	covering	NOUN
easat-1685	123	60	of	of	ADP
easat-1685	123	61	𝑋.	𝑋.	PROPN
easat-1685	123	62	then	then	ADV
easat-1685	123	63	there	there	PRON
easat-1685	123	64	exists	exist	VERB
easat-1685	123	65	a	a	DET
easat-1685	123	66	finite	finite	NOUN
easat-1685	123	67	subset	subset	NOUN
easat-1685	123	68	𝐹	𝐹	PROPN
easat-1685	123	69	of	of	ADP
easat-1685	123	70	𝑋	𝑋	PROPN
easat-1685	123	71	such	such	ADJ
easat-1685	123	72	that	that	SCONJ
easat-1685	123	73	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	123	74	(	(	PUNCT
easat-1685	123	75	𝐹	𝐹	PROPN
easat-1685	123	76	,	,	PUNCT
easat-1685	123	77	𝒰	𝒰	PROPN
easat-1685	123	78	)	)	PUNCT
easat-1685	123	79	=	=	SYM
easat-1685	123	80	𝑋	𝑋	NOUN
easat-1685	123	81	and	and	CCONJ
easat-1685	123	82	hence	hence	ADV
easat-1685	123	83	(	(	PUNCT
easat-1685	123	84	𝑋	𝑋	PROPN
easat-1685	123	85	,	,	PUNCT
easat-1685	123	86	𝒯	𝒯	PROPN
easat-1685	123	87	)	)	PUNCT
easat-1685	123	88	is	be	AUX
easat-1685	123	89	𝒶-star	𝒶-star	NOUN
easat-1685	123	90	compact	compact	ADJ
easat-1685	123	91	.	.	PUNCT
easat-1685	124	1	proof	proof	NOUN
easat-1685	124	2	:	:	PUNCT
easat-1685	124	3	suppose	suppose	VERB
easat-1685	124	4	that	that	SCONJ
easat-1685	124	5	for	for	SCONJ
easat-1685	124	6	each	each	DET
easat-1685	124	7	finite	finite	NOUN
easat-1685	124	8	set	set	VERB
easat-1685	124	9	𝐹	𝐹	PROPN
easat-1685	124	10	of	of	ADP
easat-1685	124	11	𝑋	𝑋	PROPN
easat-1685	124	12	such	such	ADJ
easat-1685	124	13	that	that	SCONJ
easat-1685	124	14	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	124	15	(	(	PUNCT
easat-1685	124	16	𝐹	𝐹	PROPN
easat-1685	124	17	,	,	PUNCT
easat-1685	124	18	𝒰	𝒰	PROPN
easat-1685	124	19	)	)	PUNCT
easat-1685	124	20	is	be	AUX
easat-1685	124	21	a	a	DET
easat-1685	124	22	proper	proper	ADJ
easat-1685	124	23	subset	subset	NOUN
easat-1685	124	24	of	of	ADP
easat-1685	124	25	𝑋	𝑋	PROPN
easat-1685	124	26	,	,	PUNCT
easat-1685	124	27	such	such	ADJ
easat-1685	124	28	that	that	SCONJ
easat-1685	124	29	𝐹	𝐹	PROPN
easat-1685	124	30	=	=	PRON
easat-1685	124	31	{	{	PUNCT
easat-1685	124	32	𝑥1	𝑥1	NOUN
easat-1685	124	33	,	,	PUNCT
easat-1685	124	34	𝑥2	𝑥2	NOUN
easat-1685	124	35	,	,	PUNCT
easat-1685	124	36	,	,	PUNCT
easat-1685	124	37	…	…	PUNCT
easat-1685	124	38	.	.	PUNCT
easat-1685	124	39	,	,	PUNCT
easat-1685	124	40	𝑥𝑛	𝑥𝑛	VERB
easat-1685	124	41	}	}	PUNCT
easat-1685	124	42	.	.	PUNCT
easat-1685	125	1	suppose	suppose	VERB
easat-1685	125	2	that	that	SCONJ
easat-1685	125	3	a	a	DET
easat-1685	125	4	set	set	NOUN
easat-1685	125	5	𝐵	𝐵	NOUN
easat-1685	125	6	=	=	SYM
easat-1685	125	7	{	{	PUNCT
easat-1685	125	8	𝑥1	𝑥1	NOUN
easat-1685	125	9	,	,	PUNCT
easat-1685	125	10	𝑥2	𝑥2	NOUN
easat-1685	125	11	,	,	PUNCT
easat-1685	125	12	,	,	PUNCT
easat-1685	125	13	…	…	PUNCT
easat-1685	125	14	.	.	PUNCT
easat-1685	125	15	,	,	PUNCT
easat-1685	125	16	𝑥𝑛	𝑥𝑛	PROPN
easat-1685	125	17	,	,	PUNCT
easat-1685	125	18	…	…	PUNCT
easat-1685	125	19	}	}	PUNCT
easat-1685	125	20	⊆	⊆	NUM
easat-1685	125	21	𝑋	𝑋	NOUN
easat-1685	125	22	and	and	CCONJ
easat-1685	125	23	for	for	ADP
easat-1685	125	24	each	each	DET
easat-1685	125	25	𝑛	𝑛	PRON
easat-1685	125	26	≥	≥	NOUN
easat-1685	125	27	1	1	NUM
easat-1685	125	28	,	,	PUNCT
easat-1685	125	29	it	it	PRON
easat-1685	125	30	follows	follow	VERB
easat-1685	125	31	that	that	SCONJ
easat-1685	125	32	𝑥𝑛+1	𝑥𝑛+1	PROPN
easat-1685	125	33	∉	∉	PROPN
easat-1685	125	34	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	125	35	(	(	PUNCT
easat-1685	125	36	𝐹	𝐹	PROPN
easat-1685	125	37	,	,	PUNCT
easat-1685	125	38	𝒰	𝒰	PROPN
easat-1685	125	39	)	)	PUNCT
easat-1685	125	40	.	.	PUNCT
easat-1685	126	1	let	let	VERB
easat-1685	126	2	𝑦	𝑦	NOUN
easat-1685	126	3	∈	∈	NOUN
easat-1685	126	4	𝑐𝑙𝒶(𝐵	𝑐𝑙𝒶(𝐵	NUM
easat-1685	126	5	)	)	PUNCT
easat-1685	126	6	.	.	PUNCT
easat-1685	127	1	then	then	ADV
easat-1685	127	2	𝐵⋂𝑈	𝐵⋂𝑈	NUM
easat-1685	127	3	≠	≠	PROPN
easat-1685	127	4	∅	∅	NOUN
easat-1685	127	5	for	for	ADP
easat-1685	127	6	some	some	DET
easat-1685	127	7	𝑈	𝑈	PROPN
easat-1685	127	8	∈	∈	PROPN
easat-1685	127	9	𝒰	𝒰	PROPN
easat-1685	127	10	,	,	PUNCT
easat-1685	127	11	where	where	SCONJ
easat-1685	127	12	𝑦	𝑦	NOUN
easat-1685	127	13	∈	∈	PROPN
easat-1685	127	14	𝑈.	𝑈.	PROPN
easat-1685	127	15	let	let	VERB
easat-1685	127	16	𝜂	𝜂	NOUN
easat-1685	127	17	be	be	AUX
easat-1685	127	18	with	with	ADP
easat-1685	127	19	𝑥𝑛	𝑥𝑛	PROPN
easat-1685	127	20	∈	∈	PROPN
easat-1685	127	21	𝑈	𝑈	PROPN
easat-1685	127	22	such	such	ADJ
easat-1685	127	23	that	that	SCONJ
easat-1685	127	24	𝑦	𝑦	PROPN
easat-1685	127	25	∈	∈	ADJ
easat-1685	127	26	𝑆𝑡𝒶({𝑥1	𝑆𝑡𝒶({𝑥1	SYM
easat-1685	127	27	,	,	PUNCT
easat-1685	127	28	𝑥2	𝑥2	NOUN
easat-1685	127	29	,	,	PUNCT
easat-1685	127	30	,	,	PUNCT
easat-1685	127	31	…	…	PUNCT
easat-1685	127	32	.	.	PUNCT
easat-1685	127	33	,	,	PUNCT
easat-1685	127	34	𝑥𝜂	𝑥𝜂	NOUN
easat-1685	127	35	}	}	PUNCT
easat-1685	127	36	,	,	PUNCT
easat-1685	127	37	𝒰	𝒰	PROPN
easat-1685	127	38	)	)	PUNCT
easat-1685	127	39	.	.	PUNCT
easat-1685	128	1	then	then	ADV
easat-1685	128	2	{	{	PUNCT
easat-1685	128	3	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	128	4	(	(	PUNCT
easat-1685	128	5	{	{	PUNCT
easat-1685	128	6	𝑥1	𝑥1	NOUN
easat-1685	128	7	,	,	PUNCT
easat-1685	128	8	𝑥2	𝑥2	NOUN
easat-1685	128	9	,	,	PUNCT
easat-1685	128	10	,	,	PUNCT
easat-1685	128	11	…	…	PUNCT
easat-1685	128	12	.	.	PUNCT
easat-1685	128	13	,	,	PUNCT
easat-1685	128	14	𝑥𝑛	𝑥𝑛	VERB
easat-1685	128	15	}	}	PUNCT
easat-1685	128	16	,	,	PUNCT
easat-1685	128	17	𝒰	𝒰	PROPN
easat-1685	128	18	)	)	PUNCT
easat-1685	128	19	∶	∶	NOUN
easat-1685	128	20	𝑛	𝑛	DET
easat-1685	128	21	≥	≥	NOUN
easat-1685	128	22	1	1	NUM
easat-1685	128	23	}	}	PUNCT
easat-1685	128	24	is	be	AUX
easat-1685	128	25	a	a	DET
easat-1685	128	26	𝒶-open	𝒶-open	NOUN
easat-1685	128	27	cover	cover	NOUN
easat-1685	128	28	of	of	ADP
easat-1685	128	29	𝑐𝑙𝒶(𝐵	𝑐𝑙𝒶(𝐵	NOUN
easat-1685	128	30	)	)	PUNCT
easat-1685	128	31	.	.	PUNCT
easat-1685	129	1	consequently	consequently	ADV
easat-1685	129	2	,	,	PUNCT
easat-1685	129	3	𝑐𝑙𝒶(𝐵	𝑐𝑙𝒶(𝐵	PUNCT
easat-1685	129	4	)	)	PUNCT
easat-1685	129	5	is	be	AUX
easat-1685	129	6	a	a	DET
easat-1685	129	7	𝒶-compact	𝒶-compact	NOUN
easat-1685	129	8	set	set	VERB
easat-1685	129	9	.	.	PUNCT
easat-1685	130	1	but	but	CCONJ
easat-1685	130	2	,	,	PUNCT
easat-1685	130	3	by	by	ADP
easat-1685	130	4	construction	construction	NOUN
easat-1685	130	5	of	of	ADP
easat-1685	130	6	a	a	DET
easat-1685	130	7	set	set	ADJ
easat-1685	130	8	𝐵	𝐵	NOUN
easat-1685	130	9	,	,	PUNCT
easat-1685	130	10	and	and	CCONJ
easat-1685	130	11	so	so	ADV
easat-1685	130	12	{	{	PUNCT
easat-1685	130	13	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	130	14	(	(	PUNCT
easat-1685	130	15	{	{	PUNCT
easat-1685	130	16	𝑥1	𝑥1	NOUN
easat-1685	130	17	,	,	PUNCT
easat-1685	130	18	𝑥2	𝑥2	NOUN
easat-1685	130	19	,	,	PUNCT
easat-1685	130	20	,	,	PUNCT
easat-1685	130	21	…	…	PUNCT
easat-1685	130	22	.	.	PUNCT
easat-1685	131	1	,	,	PUNCT
easat-1685	131	2	𝑥𝑛	𝑥𝑛	AUX
easat-1685	131	3	}	}	PUNCT
easat-1685	131	4	,	,	PUNCT
easat-1685	131	5	𝒰	𝒰	PROPN
easat-1685	131	6	)	)	PUNCT
easat-1685	131	7	∶	∶	NOUN
easat-1685	131	8	𝑛	𝑛	DET
easat-1685	131	9	≥	≥	NOUN
easat-1685	131	10	1	1	NUM
easat-1685	131	11	}	}	PUNCT
easat-1685	131	12	has	have	VERB
easat-1685	131	13	no	no	DET
easat-1685	131	14	finite	finite	NOUN
easat-1685	131	15	𝒶-cover	𝒶-cover	PROPN
easat-1685	131	16	.	.	PUNCT
easat-1685	132	1	this	this	DET
easat-1685	132	2	contradiction	contradiction	NOUN
easat-1685	132	3	establishes	establish	VERB
easat-1685	132	4	the	the	DET
easat-1685	132	5	theorem	theorem	PROPN
easat-1685	132	6	.	.	PUNCT
easat-1685	132	7	theorem	theorem	VERB
easat-1685	132	8	3.17	3.17	NUM
easat-1685	132	9	:	:	PUNCT
easat-1685	132	10	every	every	DET
easat-1685	132	11	𝒶-compact	𝒶-compact	ADJ
easat-1685	132	12	topological	topological	ADJ
easat-1685	132	13	space	space	NOUN
easat-1685	132	14	is	be	AUX
easat-1685	132	15	strong	strong	ADJ
easat-1685	132	16	𝒶-starcompact	𝒶-starcompact	NOUN
easat-1685	132	17	space	space	NOUN
easat-1685	132	18	.	.	PUNCT
easat-1685	133	1	proof	proof	NOUN
easat-1685	133	2	:	:	PUNCT
easat-1685	133	3	let	let	VERB
easat-1685	133	4	𝒲	𝒲	NOUN
easat-1685	133	5	be	be	AUX
easat-1685	133	6	𝒶-open	𝒶-open	NOUN
easat-1685	133	7	cover	cover	NOUN
easat-1685	133	8	of	of	ADP
easat-1685	133	9	𝒶compact	𝒶compact	NOUN
easat-1685	133	10	space	space	NOUN
easat-1685	133	11	𝑋	𝑋	NOUN
easat-1685	133	12	.	.	PUNCT
easat-1685	134	1	then	then	ADV
easat-1685	134	2	there	there	PRON
easat-1685	134	3	exists	exist	VERB
easat-1685	134	4	a	a	DET
easat-1685	134	5	finite	finite	NOUN
easat-1685	134	6	subset	subset	NOUN
easat-1685	134	7	𝒲′	𝒲′	PUNCT
easat-1685	135	1	=	=	PRON
easat-1685	135	2	{	{	PUNCT
easat-1685	135	3	𝘞1	𝘞1	NOUN
easat-1685	135	4	,	,	PUNCT
easat-1685	135	5	𝘞2	𝘞2	NOUN
easat-1685	135	6	,	,	PUNCT
easat-1685	135	7	,	,	PUNCT
easat-1685	135	8	…	…	PUNCT
easat-1685	135	9	.	.	PUNCT
easat-1685	136	1	,	,	PUNCT
easat-1685	136	2	𝘞𝑛	𝘞𝑛	PROPN
easat-1685	136	3	,	,	PUNCT
easat-1685	136	4	…	…	PUNCT
easat-1685	136	5	}	}	PUNCT
easat-1685	136	6	⊆	⊆	NUM
easat-1685	136	7	𝒲	𝒲	NOUN
easat-1685	136	8	such	such	ADJ
easat-1685	136	9	that	that	SCONJ
easat-1685	136	10	⋃	⋃	ADP
easat-1685	136	11	𝒲′=	𝒲′=	NOUN
easat-1685	137	1	⋃	⋃	PUNCT
easat-1685	137	2	𝘞𝑖	𝘞𝑖	PROPN
easat-1685	137	3	𝑘	𝑘	PRON
easat-1685	137	4	𝑖=1	𝑖=1	PROPN
easat-1685	137	5	=	=	SYM
easat-1685	137	6	𝑋	𝑋	PROPN
easat-1685	137	7	.	.	PUNCT
easat-1685	138	1	now	now	ADV
easat-1685	138	2	take	take	VERB
easat-1685	138	3	𝑥𝑖	𝑥𝑖	ADP
easat-1685	138	4	∈	∈	PROPN
easat-1685	138	5	𝘞𝑖	𝘞𝑖	PROPN
easat-1685	138	6	for	for	ADP
easat-1685	138	7	each	each	DET
easat-1685	138	8	i=1,2,	i=1,2,	NOUN
easat-1685	138	9	…	…	PUNCT
easat-1685	138	10	,k	,k	PUNCT
easat-1685	138	11	and	and	CCONJ
easat-1685	138	12	from	from	ADP
easat-1685	138	13	a	a	DET
easat-1685	138	14	finite	finite	NOUN
easat-1685	138	15	set	set	VERB
easat-1685	138	16	𝐹={𝑥1	𝐹={𝑥1	NOUN
easat-1685	138	17	,	,	PUNCT
easat-1685	138	18	𝑥2	𝑥2	NOUN
easat-1685	138	19	,	,	PUNCT
easat-1685	138	20	,	,	PUNCT
easat-1685	138	21	…	…	PUNCT
easat-1685	138	22	.	.	PUNCT
easat-1685	139	1	,	,	PUNCT
easat-1685	139	2	𝑥𝑘	𝑥𝑘	X
easat-1685	139	3	}	}	PUNCT
easat-1685	139	4	,	,	PUNCT
easat-1685	139	5	then	then	ADV
easat-1685	139	6	𝑋	𝑋	PROPN
easat-1685	139	7	=	=	SYM
easat-1685	139	8	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	139	9	(	(	PUNCT
easat-1685	139	10	𝐹	𝐹	PROPN
easat-1685	139	11	,	,	PUNCT
easat-1685	139	12	𝒲	𝒲	PROPN
easat-1685	139	13	)	)	PUNCT
easat-1685	140	1	⊆	⊆	X
easat-1685	140	2	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	140	3	(	(	PUNCT
easat-1685	140	4	𝐹	𝐹	PROPN
easat-1685	140	5	,	,	PUNCT
easat-1685	140	6	𝒲′	𝒲′	PUNCT
easat-1685	140	7	)	)	PUNCT
easat-1685	141	1	=	=	SYM
easat-1685	141	2	𝑋	𝑋	NOUN
easat-1685	141	3	,	,	PUNCT
easat-1685	141	4	there	there	ADV
easat-1685	141	5	for	for	ADP
easat-1685	141	6	𝑋	𝑋	PROPN
easat-1685	141	7	is	be	AUX
easat-1685	141	8	a	a	DET
easat-1685	141	9	strong	strong	ADJ
easat-1685	141	10	𝒶-starcompact	𝒶-starcompact	NOUN
easat-1685	141	11	space	space	NOUN
easat-1685	141	12	.	.	PUNCT
easat-1685	142	1	example	example	NOUN
easat-1685	142	2	3.18	3.18	NUM
easat-1685	142	3	:	:	PUNCT
easat-1685	142	4	converse	converse	NOUN
easat-1685	142	5	of	of	ADP
easat-1685	142	6	the	the	DET
easat-1685	142	7	above	above	ADJ
easat-1685	142	8	theorem	theorem	NOUN
easat-1685	142	9	may	may	AUX
easat-1685	142	10	not	not	PART
easat-1685	142	11	be	be	AUX
easat-1685	142	12	true	true	ADJ
easat-1685	142	13	.	.	PUNCT
easat-1685	143	1	275	275	NUM
easat-1685	143	2	edelweiss	edelweiss	PROPN
easat-1685	143	3	applied	apply	VERB
easat-1685	143	4	science	science	NOUN
easat-1685	143	5	and	and	CCONJ
easat-1685	143	6	technology	technology	NOUN
easat-1685	143	7	issn	issn	PROPN
easat-1685	143	8	:	:	PUNCT
easat-1685	143	9	2576	2576	NUM
easat-1685	143	10	-	-	SYM
easat-1685	143	11	8484	8484	NUM
easat-1685	143	12	vol	vol	NOUN
easat-1685	143	13	.	.	PROPN
easat-1685	143	14	8	8	NUM
easat-1685	143	15	,	,	PUNCT
easat-1685	143	16	no	no	INTJ
easat-1685	143	17	.	.	NOUN
easat-1685	143	18	5	5	NUM
easat-1685	143	19	:	:	PUNCT
easat-1685	143	20	271	271	NUM
easat-1685	143	21	-	-	SYM
easat-1685	143	22	277	277	NUM
easat-1685	143	23	,	,	PUNCT
easat-1685	143	24	2024	2024	NUM
easat-1685	143	25	doi	doi	NOUN
easat-1685	143	26	:	:	PUNCT
easat-1685	143	27	10.55214/25768484.v8i5.1685	10.55214/25768484.v8i5.1685	NUM
easat-1685	143	28	©	©	ADP
easat-1685	143	29	2024	2024	NUM
easat-1685	143	30	by	by	ADP
easat-1685	143	31	the	the	DET
easat-1685	143	32	authors	author	NOUN
easat-1685	143	33	;	;	PUNCT
easat-1685	143	34	licensee	licensee	PROPN
easat-1685	143	35	learning	learn	VERB
easat-1685	143	36	gate	gate	NOUN
easat-1685	143	37	consider	consider	VERB
easat-1685	143	38	𝑋	𝑋	PROPN
easat-1685	143	39	the	the	DET
easat-1685	143	40	set	set	NOUN
easat-1685	143	41	of	of	ADP
easat-1685	143	42	natural	natural	ADJ
easat-1685	143	43	numbers	number	NOUN
easat-1685	143	44	and	and	CCONJ
easat-1685	143	45	the	the	DET
easat-1685	143	46	topology	topology	NOUN
easat-1685	143	47	𝜏	𝜏	X
easat-1685	143	48	=	=	NOUN
easat-1685	143	49	{	{	PUNCT
easat-1685	143	50	1,2,3	1,2,3	NUM
easat-1685	143	51	,	,	PUNCT
easat-1685	143	52	…	…	PUNCT
easat-1685	143	53	,	,	PUNCT
easat-1685	143	54	𝑛	𝑛	ADP
easat-1685	143	55	}	}	PUNCT
easat-1685	143	56	:	:	PUNCT
easat-1685	143	57	∈	∈	NUM
easat-1685	143	58	𝑛	𝑛	PRON
easat-1685	143	59	∈	∈	PROPN
easat-1685	143	60	ℕ	ℕ	PROPN
easat-1685	143	61	}	}	PUNCT
easat-1685	143	62	⋃{𝑥	⋃{𝑥	NOUN
easat-1685	143	63	,	,	PUNCT
easat-1685	143	64	∅	∅	NOUN
easat-1685	143	65	}	}	PUNCT
easat-1685	143	66	on	on	ADP
easat-1685	143	67	𝑋	𝑋	PROPN
easat-1685	143	68	,	,	PUNCT
easat-1685	143	69	then	then	ADV
easat-1685	143	70	for	for	ADP
easat-1685	143	71	a	a	DET
easat-1685	143	72	finite	finite	NOUN
easat-1685	143	73	subset	subset	NOUN
easat-1685	143	74	f	f	PROPN
easat-1685	143	75	=	=	PUNCT
easat-1685	143	76	{	{	PUNCT
easat-1685	143	77	1	1	NUM
easat-1685	143	78	}	}	SYM
easat-1685	143	79	⊆	⊆	NUM
easat-1685	143	80	𝑋	𝑋	NOUN
easat-1685	143	81	and	and	CCONJ
easat-1685	143	82	𝒲	𝒲	PROPN
easat-1685	143	83	be	be	VERB
easat-1685	143	84	an	an	DET
easat-1685	143	85	arbitrary	arbitrary	ADJ
easat-1685	143	86	𝒶open	𝒶open	ADJ
easat-1685	143	87	cover	cover	NOUN
easat-1685	143	88	of	of	ADP
easat-1685	143	89	𝑋,we	𝑋,we	PROPN
easat-1685	143	90	have	have	VERB
easat-1685	143	91	𝑆𝑡𝒶	𝑆𝑡𝒶	PROPN
easat-1685	143	92	(	(	PUNCT
easat-1685	143	93	𝐹	𝐹	PROPN
easat-1685	143	94	,	,	PUNCT
easat-1685	143	95	𝒲	𝒲	PROPN
easat-1685	143	96	)	)	PUNCT
easat-1685	143	97	=	=	SYM
easat-1685	143	98	⋃	⋃	NOUN
easat-1685	143	99	𝒲	𝒲	NOUN
easat-1685	143	100	=	=	SYM
easat-1685	143	101	𝑋	𝑋	PROPN
easat-1685	143	102	.hence	.hence	ADP
easat-1685	143	103	𝑋	𝑋	PROPN
easat-1685	143	104	is	be	AUX
easat-1685	143	105	strong	strong	ADJ
easat-1685	143	106	𝒶-starcompact	𝒶-starcompact	NOUN
easat-1685	143	107	space	space	NOUN
easat-1685	143	108	.	.	PUNCT
easat-1685	144	1	on	on	ADP
easat-1685	144	2	the	the	DET
easat-1685	144	3	other	other	ADJ
easat-1685	144	4	hand	hand	NOUN
easat-1685	144	5	,	,	PUNCT
easat-1685	144	6	let	let	VERB
easat-1685	144	7	𝒲	𝒲	NOUN
easat-1685	144	8	=	=	NOUN
easat-1685	144	9	{	{	PUNCT
easat-1685	144	10	𝒲𝑛	𝒲𝑛	PROPN
easat-1685	144	11	=	=	PRON
easat-1685	144	12	{	{	PUNCT
easat-1685	144	13	1,2,3	1,2,3	NUM
easat-1685	144	14	,	,	PUNCT
easat-1685	144	15	…	…	PUNCT
easat-1685	144	16	,	,	PUNCT
easat-1685	144	17	𝑛	𝑛	ADP
easat-1685	144	18	}	}	PUNCT
easat-1685	144	19	:	:	PUNCT
easat-1685	144	20	𝑛	𝑛	PROPN
easat-1685	144	21	∈	∈	PROPN
easat-1685	144	22	ℕ	ℕ	PROPN
easat-1685	144	23	}	}	PUNCT
easat-1685	144	24	is	be	AUX
easat-1685	144	25	an	an	DET
easat-1685	144	26	𝒶	𝒶	NOUN
easat-1685	144	27	–	–	PUNCT
easat-1685	144	28	open	open	ADJ
easat-1685	144	29	cover	cover	NOUN
easat-1685	144	30	of	of	ADP
easat-1685	144	31	𝑋.	𝑋.	PROPN
easat-1685	144	32	suppose	suppose	VERB
easat-1685	144	33	𝒲′	𝒲′	AUX
easat-1685	144	34	is	be	AUX
easat-1685	144	35	a	a	DET
easat-1685	144	36	finite	finite	ADJ
easat-1685	144	37	subcover	subcover	NOUN
easat-1685	144	38	of	of	ADP
easat-1685	144	39	it	it	PRON
easat-1685	144	40	.	.	PUNCT
easat-1685	145	1	by	by	ADP
easat-1685	145	2	the	the	DET
easat-1685	145	3	construction	construction	NOUN
easat-1685	145	4	of	of	ADP
easat-1685	145	5	𝑋	𝑋	PROPN
easat-1685	145	6	,	,	PUNCT
easat-1685	145	7	we	we	PRON
easat-1685	145	8	can	can	AUX
easat-1685	145	9	find	find	VERB
easat-1685	145	10	a	a	DET
easat-1685	145	11	largest	large	ADJ
easat-1685	145	12	set	set	NOUN
easat-1685	145	13	𝒲𝜆	𝒲𝜆	PROPN
easat-1685	145	14	∈	∈	PROPN
easat-1685	145	15	𝒲′,where	𝒲′,where	NOUN
easat-1685	145	16	𝜆	𝜆	PROPN
easat-1685	145	17	∈	∈	PROPN
easat-1685	145	18	ℕ,so	ℕ,so	PROPN
easat-1685	145	19	⋃	⋃	PROPN
easat-1685	145	20	𝒲=𝒲𝜆={1,2,3	𝒲=𝒲𝜆={1,2,3	PROPN
easat-1685	145	21	,	,	PUNCT
easat-1685	145	22	…	…	PUNCT
easat-1685	145	23	,	,	PUNCT
easat-1685	145	24	𝜆	𝜆	X
easat-1685	145	25	}	}	PUNCT
easat-1685	145	26	.	.	PUNCT
easat-1685	146	1	then	then	ADV
easat-1685	146	2	{	{	PUNCT
easat-1685	146	3	𝜆	𝜆	PROPN
easat-1685	146	4	+1	+1	PROPN
easat-1685	146	5	,	,	PUNCT
easat-1685	146	6	𝜆	𝜆	PROPN
easat-1685	146	7	+	+	ADJ
easat-1685	146	8	2	2	NUM
easat-1685	146	9	,	,	PUNCT
easat-1685	146	10	𝜆+3,	𝜆+3,	NOUN
easat-1685	146	11	…	…	PUNCT
easat-1685	146	12	}remains	}remains	PUNCT
easat-1685	146	13	without	without	ADP
easat-1685	146	14	cover	cover	NOUN
easat-1685	146	15	.thus	.thus	ADP
easat-1685	146	16	,	,	PUNCT
easat-1685	146	17	there	there	PRON
easat-1685	146	18	is	be	VERB
easat-1685	146	19	no	no	DET
easat-1685	146	20	finite	finite	NOUN
easat-1685	146	21	subcover	subcover	NOUN
easat-1685	146	22	for	for	ADP
easat-1685	146	23	𝒲	𝒲	PROPN
easat-1685	146	24	,	,	PUNCT
easat-1685	146	25	so	so	ADV
easat-1685	146	26	𝑋	𝑋	PROPN
easat-1685	146	27	is	be	AUX
easat-1685	146	28	not	not	PART
easat-1685	146	29	𝒶compact	𝒶compact	NOUN
easat-1685	146	30	space	space	NOUN
easat-1685	146	31	.	.	PUNCT
easat-1685	147	1	4	4	X
easat-1685	147	2	.	.	X
easat-1685	147	3	𝓪-hurewicz	𝓪-hurewicz	NOUN
easat-1685	147	4	spaces	space	VERB
easat-1685	147	5	definition	definition	NOUN
easat-1685	147	6	4.1	4.1	NUM
easat-1685	147	7	:	:	PUNCT
easat-1685	147	8	let	let	AUX
easat-1685	147	9	(	(	PUNCT
easat-1685	147	10	𝑋	𝑋	PROPN
easat-1685	147	11	,	,	PUNCT
easat-1685	147	12	𝒯	𝒯	PROPN
easat-1685	147	13	)	)	PUNCT
easat-1685	147	14	be	be	VERB
easat-1685	147	15	a	a	DET
easat-1685	147	16	t.	t.	PROPN
easat-1685	147	17	s.	s.	PROPN
easat-1685	147	18	and	and	CCONJ
easat-1685	147	19	𝐴	𝐴	PROPN
easat-1685	148	1	⊆	⊆	PROPN
easat-1685	148	2	𝑋.	𝑋.	PROPN
easat-1685	148	3	then	then	ADV
easat-1685	148	4	𝐴	𝐴	PROPN
easat-1685	148	5	is	be	AUX
easat-1685	148	6	said	say	VERB
easat-1685	148	7	to	to	PART
easat-1685	148	8	have	have	VERB
easat-1685	148	9	the	the	DET
easat-1685	148	10	𝒶-hurewicz	𝒶-hurewicz	NOUN
easat-1685	148	11	property	property	NOUN
easat-1685	148	12	,	,	PUNCT
easat-1685	148	13	if	if	SCONJ
easat-1685	148	14	for	for	ADP
easat-1685	148	15	any	any	DET
easat-1685	148	16	sequence	sequence	NOUN
easat-1685	148	17	(	(	PUNCT
easat-1685	148	18	𝑈𝑛	𝑈𝑛	PROPN
easat-1685	148	19	)	)	PUNCT
easat-1685	148	20	𝑛∈ℕ	𝑛∈ℕ	X
easat-1685	148	21	of	of	ADP
easat-1685	148	22	𝒶-open	𝒶-open	PROPN
easat-1685	148	23	covers	cover	NOUN
easat-1685	148	24	of	of	ADP
easat-1685	148	25	𝐴	𝐴	PROPN
easat-1685	148	26	,	,	PUNCT
easat-1685	148	27	there	there	PRON
easat-1685	148	28	is	be	VERB
easat-1685	148	29	a	a	DET
easat-1685	148	30	sequence	sequence	NOUN
easat-1685	148	31	(	(	PUNCT
easat-1685	148	32	𝑉𝑛)𝑛∈ℕ	𝑉𝑛)𝑛∈ℕ	VERB
easat-1685	148	33	for	for	ADP
easat-1685	148	34	any	any	DET
easat-1685	148	35	𝑛	𝑛	PRON
easat-1685	148	36	∈	∈	PROPN
easat-1685	148	37	ℕ	ℕ	PROPN
easat-1685	148	38	,	,	PUNCT
easat-1685	148	39	𝑉𝑛	𝑉𝑛	PROPN
easat-1685	148	40	is	be	AUX
easat-1685	148	41	a	a	DET
easat-1685	148	42	finite	finite	NOUN
easat-1685	148	43	subset	subset	NOUN
easat-1685	148	44	of	of	ADP
easat-1685	148	45	𝑈𝑛	𝑈𝑛	PROPN
easat-1685	148	46	and	and	CCONJ
easat-1685	148	47	for	for	ADP
easat-1685	148	48	each	each	DET
easat-1685	148	49	𝑥	𝑥	PRON
easat-1685	148	50	∈	∈	PROPN
easat-1685	148	51	𝐴	𝐴	PROPN
easat-1685	148	52	for	for	ADP
easat-1685	148	53	all	all	PRON
easat-1685	148	54	but	but	ADV
easat-1685	148	55	finitely	finitely	ADV
easat-1685	148	56	many	many	ADJ
easat-1685	148	57	𝑛	𝑛	ADP
easat-1685	148	58	,	,	PUNCT
easat-1685	148	59	with	with	ADP
easat-1685	148	60	𝑥	𝑥	DET
easat-1685	148	61	∈	∈	PROPN
easat-1685	148	62	⋃	⋃	NOUN
easat-1685	148	63	𝑉𝑛.	𝑉𝑛.	NOUN
easat-1685	148	64	we	we	PRON
easat-1685	148	65	say	say	VERB
easat-1685	148	66	that	that	SCONJ
easat-1685	148	67	𝑋	𝑋	PROPN
easat-1685	148	68	is	be	AUX
easat-1685	148	69	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	148	70	space	space	NOUN
easat-1685	148	71	,	,	PUNCT
easat-1685	148	72	if	if	SCONJ
easat-1685	148	73	the	the	DET
easat-1685	148	74	set	set	NOUN
easat-1685	148	75	𝑋	𝑋	NOUN
easat-1685	148	76	is	be	AUX
easat-1685	148	77	𝒶-hurewicz	𝒶-hurewicz	PROPN
easat-1685	148	78	.	.	PUNCT
easat-1685	148	79	example	example	NOUN
easat-1685	148	80	4.2	4.2	NUM
easat-1685	148	81	:	:	PUNCT
easat-1685	148	82	every	every	DET
easat-1685	148	83	𝒶-compact	𝒶-compact	NOUN
easat-1685	148	84	space	space	NOUN
easat-1685	148	85	is	be	AUX
easat-1685	148	86	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	148	87	space	space	NOUN
easat-1685	148	88	.	.	PUNCT
easat-1685	149	1	the	the	DET
easat-1685	149	2	convers	conver	NOUN
easat-1685	149	3	is	be	AUX
easat-1685	149	4	not	not	PART
easat-1685	149	5	true	true	ADJ
easat-1685	149	6	.	.	PUNCT
easat-1685	150	1	let	let	VERB
easat-1685	150	2	x	x	PUNCT
easat-1685	150	3	=	=	PUNCT
easat-1685	150	4	r	r	NOUN
easat-1685	150	5	with	with	ADP
easat-1685	150	6	the	the	DET
easat-1685	150	7	topology	topology	NOUN
easat-1685	150	8	𝒯=	𝒯=	PROPN
easat-1685	150	9	{	{	PUNCT
easat-1685	150	10	𝑈	𝑈	PROPN
easat-1685	150	11	⊆	⊆	NUM
easat-1685	150	12	𝑋	𝑋	NOUN
easat-1685	150	13	:	:	PUNCT
easat-1685	150	14	𝑈	𝑈	NOUN
easat-1685	150	15	=	=	PUNCT
easat-1685	150	16	∅	∅	NOUN
easat-1685	150	17	or	or	CCONJ
easat-1685	150	18	𝑋\𝑈	𝑋\𝑈	PROPN
easat-1685	150	19	is	be	AUX
easat-1685	150	20	countable	countable	ADJ
easat-1685	150	21	}	}	PUNCT
easat-1685	150	22	is	be	AUX
easat-1685	150	23	a	a	DET
easat-1685	150	24	𝑇1	𝑇1	NOUN
easat-1685	150	25	𝒶-hurewicz	𝒶-hurewicz	NOUN
easat-1685	150	26	space	space	NOUN
easat-1685	150	27	which	which	PRON
easat-1685	150	28	is	be	AUX
easat-1685	150	29	not	not	PART
easat-1685	150	30	𝒶compact	𝒶compact	NOUN
easat-1685	150	31	.	.	PUNCT
easat-1685	151	1	in	in	ADP
easat-1685	151	2	the	the	DET
easat-1685	151	3	following	following	ADJ
easat-1685	151	4	theorem	theorem	NOUN
easat-1685	151	5	,	,	PUNCT
easat-1685	151	6	we	we	PRON
easat-1685	151	7	put	put	VERB
easat-1685	151	8	a	a	DET
easat-1685	151	9	condition	condition	NOUN
easat-1685	151	10	to	to	PART
easat-1685	151	11	show	show	VERB
easat-1685	151	12	that	that	SCONJ
easat-1685	151	13	a	a	DET
easat-1685	151	14	subspace	subspace	NOUN
easat-1685	151	15	of	of	ADP
easat-1685	151	16	the	the	DET
easat-1685	151	17	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	151	18	space	space	NOUN
easat-1685	151	19	is	be	AUX
easat-1685	151	20	also	also	ADV
easat-1685	151	21	satisfied	satisfied	ADJ
easat-1685	151	22	.	.	PUNCT
easat-1685	152	1	theorem	theorem	VERB
easat-1685	152	2	4.3	4.3	NUM
easat-1685	152	3	:	:	PUNCT
easat-1685	152	4	let	let	VERB
easat-1685	152	5	𝑋	𝑋	NOUN
easat-1685	152	6	be	be	AUX
easat-1685	152	7	the	the	DET
easat-1685	152	8	𝒶-hurewicz	𝒶-hurewicz	NOUN
easat-1685	152	9	space	space	NOUN
easat-1685	152	10	and	and	CCONJ
easat-1685	152	11	𝑌	𝑌	PROPN
easat-1685	152	12	is	be	AUX
easat-1685	152	13	𝒶-clopen	𝒶-clopen	ADJ
easat-1685	152	14	subspace	subspace	NOUN
easat-1685	152	15	of	of	ADP
easat-1685	152	16	𝑋	𝑋	PROPN
easat-1685	152	17	,	,	PUNCT
easat-1685	152	18	then	then	ADV
easat-1685	152	19	𝑌	𝑌	PROPN
easat-1685	152	20	is	be	AUX
easat-1685	152	21	the	the	DET
easat-1685	152	22	𝒶	𝒶	NOUN
easat-1685	152	23	hurewicz	hurewicz	NOUN
easat-1685	152	24	space	space	NOUN
easat-1685	152	25	.	.	PUNCT
easat-1685	153	1	proof	proof	NOUN
easat-1685	153	2	:	:	PUNCT
easat-1685	153	3	suppose	suppose	VERB
easat-1685	153	4	that	that	SCONJ
easat-1685	153	5	𝑌	𝑌	PROPN
easat-1685	153	6	is	be	AUX
easat-1685	153	7	𝒶-clopen	𝒶-clopen	ADJ
easat-1685	153	8	subspace	subspace	NOUN
easat-1685	153	9	of	of	ADP
easat-1685	153	10	the	the	DET
easat-1685	153	11	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	153	12	space	space	NOUN
easat-1685	153	13	and	and	CCONJ
easat-1685	153	14	let	let	VERB
easat-1685	153	15	(	(	PUNCT
easat-1685	153	16	𝒰𝑛)𝑛∈ℕ	𝒰𝑛)𝑛∈ℕ	X
easat-1685	153	17	be	be	AUX
easat-1685	153	18	a	a	DET
easat-1685	153	19	sequence	sequence	NOUN
easat-1685	153	20	of	of	ADP
easat-1685	153	21	𝒶-open	𝒶-open	NOUN
easat-1685	153	22	covers	cover	NOUN
easat-1685	153	23	of	of	ADP
easat-1685	153	24	𝑌.	𝑌.	PROPN
easat-1685	153	25	it	it	PRON
easat-1685	153	26	easy	easy	ADJ
easat-1685	153	27	to	to	PART
easat-1685	153	28	see	see	VERB
easat-1685	153	29	that	that	SCONJ
easat-1685	153	30	every	every	DET
easat-1685	153	31	𝒶-open	𝒶-open	NOUN
easat-1685	153	32	subset	subset	NOUN
easat-1685	153	33	of	of	ADP
easat-1685	153	34	𝑎	𝑎	PROPN
easat-1685	153	35	−	−	PROPN
easat-1685	153	36	𝑐𝑙𝑜𝑝𝑒𝑛	𝑐𝑙𝑜𝑝𝑒𝑛	NOUN
easat-1685	153	37	𝑌	𝑌	PROPN
easat-1685	153	38	is	be	AUX
easat-1685	153	39	the	the	DET
easat-1685	153	40	intersection	intersection	NOUN
easat-1685	153	41	of	of	ADP
easat-1685	153	42	𝒶-open	𝒶-open	NOUN
easat-1685	153	43	subset	subset	NOUN
easat-1685	153	44	of	of	ADP
easat-1685	153	45	𝑋	𝑋	PROPN
easat-1685	153	46	with	with	ADP
easat-1685	153	47	𝑌.	𝑌.	PROPN
easat-1685	153	48	then	then	ADV
easat-1685	153	49	,	,	PUNCT
easat-1685	153	50	for	for	ADP
easat-1685	153	51	each	each	DET
easat-1685	153	52	n	n	PRON
easat-1685	153	53	∈	∈	NOUN
easat-1685	153	54	𝑁	𝑁	PROPN
easat-1685	153	55	and	and	CCONJ
easat-1685	153	56	each	each	DET
easat-1685	153	57	𝒰	𝒰	NOUN
easat-1685	153	58	∈	∈	PROPN
easat-1685	153	59	𝒰𝑛	𝒰𝑛	PROPN
easat-1685	153	60	there	there	PRON
easat-1685	153	61	exists	exist	VERB
easat-1685	153	62	an	an	DET
easat-1685	153	63	𝒶-open	𝒶-open	NOUN
easat-1685	153	64	set	set	VERB
easat-1685	153	65	𝒢𝑢	𝒢𝑢	PROPN
easat-1685	153	66	in	in	ADP
easat-1685	153	67	𝑋	𝑋	NOUN
easat-1685	154	1	such	such	ADJ
easat-1685	154	2	that	that	DET
easat-1685	154	3	𝒰	𝒰	PROPN
easat-1685	154	4	=	=	SYM
easat-1685	154	5	𝑌	𝑌	PROPN
easat-1685	154	6	⋂	⋂	PROPN
easat-1685	154	7	𝒢𝑢	𝒢𝑢	PROPN
easat-1685	154	8	.	.	PUNCT
easat-1685	155	1	let	let	VERB
easat-1685	155	2	𝔈𝑛	𝔈𝑛	NOUN
easat-1685	155	3	=	=	PUNCT
easat-1685	155	4	{	{	PUNCT
easat-1685	155	5	𝒢𝑢	𝒢𝑢	PROPN
easat-1685	155	6	:	:	PUNCT
easat-1685	155	7	𝒰	𝒰	PROPN
easat-1685	155	8	∈	∈	NOUN
easat-1685	156	1	𝒰𝑛	𝒰𝑛	PROPN
easat-1685	156	2	}	}	PUNCT
easat-1685	156	3	⋃	⋃	ADP
easat-1685	156	4	{	{	PUNCT
easat-1685	156	5	𝑋	𝑋	PROPN
easat-1685	156	6	\	\	PROPN
easat-1685	156	7	𝑌	𝑌	PROPN
easat-1685	156	8	}	}	PUNCT
easat-1685	156	9	,	,	PUNCT
easat-1685	156	10	𝑛	𝑛	DET
easat-1685	156	11	∈	∈	PROPN
easat-1685	156	12	𝑁.	𝑁.	PROPN
easat-1685	156	13	then	then	ADV
easat-1685	156	14	(	(	PUNCT
easat-1685	156	15	𝔈𝑛	𝔈𝑛	PROPN
easat-1685	156	16	)	)	PUNCT
easat-1685	156	17	𝑛∈𝑁	𝑛∈𝑁	NOUN
easat-1685	156	18	is	be	AUX
easat-1685	156	19	a	a	DET
easat-1685	156	20	sequence	sequence	NOUN
easat-1685	156	21	of	of	ADP
easat-1685	156	22	𝒶-open	𝒶-open	NOUN
easat-1685	156	23	covers	cover	NOUN
easat-1685	156	24	of	of	ADP
easat-1685	156	25	𝑋	𝑋	PROPN
easat-1685	156	26	.	.	PUNCT
easat-1685	157	1	the	the	DET
easat-1685	157	2	𝒶hurewiczness	𝒶hurewiczness	ADJ
easat-1685	157	3	property	property	NOUN
easat-1685	157	4	of	of	ADP
easat-1685	157	5	𝑋	𝑋	PROPN
easat-1685	157	6	,	,	PUNCT
easat-1685	157	7	implies	imply	VERB
easat-1685	157	8	the	the	DET
easat-1685	157	9	existence	existence	NOUN
easat-1685	157	10	of	of	ADP
easat-1685	157	11	a	a	DET
easat-1685	157	12	sequence	sequence	NOUN
easat-1685	157	13	(	(	PUNCT
easat-1685	157	14	𝒲𝑛	𝒲𝑛	PROPN
easat-1685	157	15	)	)	PUNCT
easat-1685	157	16	𝑛∈𝑁	𝑛∈𝑁	NOUN
easat-1685	157	17	with	with	ADP
easat-1685	157	18	𝒲𝑛	𝒲𝑛	PROPN
easat-1685	157	19	is	be	AUX
easat-1685	157	20	a	a	DET
easat-1685	157	21	finite	finite	NOUN
easat-1685	157	22	subset	subset	NOUN
easat-1685	157	23	of	of	ADP
easat-1685	157	24	𝔈𝑛	𝔈𝑛	PROPN
easat-1685	157	25	for	for	ADP
easat-1685	157	26	each	each	DET
easat-1685	157	27	𝑛	𝑛	PRON
easat-1685	157	28	∈	∈	NOUN
easat-1685	157	29	𝑁	𝑁	PROPN
easat-1685	157	30	and	and	CCONJ
easat-1685	157	31	𝑋=⋃𝑛∈𝑁	𝑋=⋃𝑛∈𝑁	PROPN
easat-1685	157	32	⋃𝒲𝑛	⋃𝒲𝑛	PROPN
easat-1685	157	33	.	.	PUNCT
easat-1685	158	1	if	if	SCONJ
easat-1685	158	2	we	we	PRON
easat-1685	158	3	put	put	VERB
easat-1685	158	4	for	for	ADP
easat-1685	158	5	each	each	DET
easat-1685	158	6	n	n	NOUN
easat-1685	158	7	,	,	PUNCT
easat-1685	158	8	𝒱𝑛=	𝒱𝑛=	PROPN
easat-1685	158	9	{	{	PUNCT
easat-1685	158	10	𝒰	𝒰	PROPN
easat-1685	158	11	∶	∶	NOUN
easat-1685	158	12	𝒢𝑢	𝒢𝑢	PROPN
easat-1685	158	13	∈	∈	PROPN
easat-1685	158	14	𝒲𝑛	𝒲𝑛	PROPN
easat-1685	158	15	}	}	PUNCT
easat-1685	158	16	,	,	PUNCT
easat-1685	158	17	we	we	PRON
easat-1685	158	18	obtain	obtain	VERB
easat-1685	158	19	the	the	DET
easat-1685	158	20	sequence	sequence	NOUN
easat-1685	158	21	(	(	PUNCT
easat-1685	158	22	𝒱𝑛	𝒱𝑛	PROPN
easat-1685	158	23	)	)	PUNCT
easat-1685	158	24	𝑛∈𝑁	𝑛∈𝑁	NOUN
easat-1685	158	25	is	be	AUX
easat-1685	158	26	a	a	DET
easat-1685	158	27	finite	finite	NOUN
easat-1685	158	28	subset	subset	NOUN
easat-1685	158	29	of	of	ADP
easat-1685	158	30	𝒰𝑛and	𝒰𝑛and	PROPN
easat-1685	158	31	each	each	PRON
easat-1685	158	32	𝑥	𝑥	PRON
easat-1685	158	33	∈	∈	PROPN
easat-1685	158	34	𝑌	𝑌	PROPN
easat-1685	158	35	for	for	ADP
easat-1685	158	36	all	all	PRON
easat-1685	158	37	but	but	ADV
easat-1685	158	38	finitely	finitely	ADV
easat-1685	158	39	many	many	ADJ
easat-1685	158	40	𝑛	𝑛	VERB
easat-1685	158	41	,	,	PUNCT
easat-1685	158	42	with	with	ADP
easat-1685	158	43	𝑥	𝑥	DET
easat-1685	158	44	∈	∈	PROPN
easat-1685	158	45	⋃𝒱𝑛	⋃𝒱𝑛	NUM
easat-1685	158	46	,	,	PUNCT
easat-1685	158	47	that	that	PRON
easat-1685	158	48	is	be	AUX
easat-1685	158	49	𝑌	𝑌	PROPN
easat-1685	158	50	an	an	DET
easat-1685	158	51	𝒶	𝒶	NOUN
easat-1685	158	52	hurewicz	hurewicz	NOUN
easat-1685	158	53	space	space	NOUN
easat-1685	158	54	.	.	PUNCT
easat-1685	159	1	the	the	DET
easat-1685	159	2	a	a	ADV
easat-1685	159	3	-	-	PUNCT
easat-1685	159	4	hurewiczness	hurewiczness	ADJ
easat-1685	159	5	is	be	AUX
easat-1685	159	6	an	an	DET
easat-1685	159	7	a	a	PRON
easat-1685	159	8	-	-	PUNCT
easat-1685	159	9	topological	topological	ADJ
easat-1685	159	10	property	property	NOUN
easat-1685	159	11	,	,	PUNCT
easat-1685	159	12	as	as	SCONJ
easat-1685	159	13	evidenced	evidence	VERB
easat-1685	159	14	by	by	ADP
easat-1685	159	15	the	the	DET
easat-1685	159	16	following	follow	VERB
easat-1685	159	17	theorem	theorem	PROPN
easat-1685	159	18	.	.	PUNCT
easat-1685	160	1	theorem	theorem	VERB
easat-1685	160	2	4.4	4.4	NUM
easat-1685	160	3	:	:	PUNCT
easat-1685	160	4	an	an	DET
easat-1685	160	5	𝒶irresolute	𝒶irresolute	ADJ
easat-1685	160	6	image	image	NOUN
easat-1685	160	7	of	of	ADP
easat-1685	160	8	an	an	DET
easat-1685	160	9	𝒶-hurewicz	𝒶-hurewicz	NOUN
easat-1685	160	10	space	space	NOUN
easat-1685	160	11	is	be	AUX
easat-1685	160	12	a	a	DET
easat-1685	160	13	hurewicz	hurewicz	NOUN
easat-1685	160	14	space	space	NOUN
easat-1685	160	15	.	.	PUNCT
easat-1685	161	1	proof	proof	NOUN
easat-1685	161	2	:	:	PUNCT
easat-1685	161	3	let	let	VERB
easat-1685	161	4	𝑋	𝑋	NOUN
easat-1685	161	5	be	be	AUX
easat-1685	161	6	an	an	DET
easat-1685	161	7	𝒶-hurewicz	𝒶-hurewicz	NOUN
easat-1685	161	8	space	space	NOUN
easat-1685	161	9	and	and	CCONJ
easat-1685	162	1	𝑌	𝑌	PROPN
easat-1685	162	2	=	=	SYM
easat-1685	162	3	f	f	PROPN
easat-1685	162	4	(	(	PUNCT
easat-1685	162	5	𝑋	𝑋	PROPN
easat-1685	162	6	)	)	PUNCT
easat-1685	162	7	its	its	PRON
easat-1685	162	8	image	image	NOUN
easat-1685	162	9	under	under	ADP
easat-1685	162	10	𝒶continuous	𝒶continuous	ADJ
easat-1685	162	11	mapping	mapping	NOUN
easat-1685	162	12	𝑓	𝑓	DET
easat-1685	162	13	∶	∶	NOUN
easat-1685	162	14	𝑋	𝑋	NOUN
easat-1685	162	15	⟶	⟶	NOUN
easat-1685	162	16	𝑌.	𝑌.	PROPN
easat-1685	162	17	let	let	NOUN
easat-1685	162	18	(	(	PUNCT
easat-1685	162	19	𝒱𝑛	𝒱𝑛	PROPN
easat-1685	162	20	)	)	PUNCT
easat-1685	162	21	𝑛∈𝑁	𝑛∈𝑁	NOUN
easat-1685	162	22	be	be	AUX
easat-1685	162	23	a	a	DET
easat-1685	162	24	sequence	sequence	NOUN
easat-1685	162	25	of	of	ADP
easat-1685	162	26	𝒶open	𝒶open	ADJ
easat-1685	162	27	covers	cover	NOUN
easat-1685	162	28	of	of	ADP
easat-1685	162	29	𝑌	𝑌	PROPN
easat-1685	162	30	and	and	CCONJ
easat-1685	162	31	𝑥	𝑥	DET
easat-1685	162	32	∈	∈	PROPN
easat-1685	162	33	𝑋	𝑋	NOUN
easat-1685	162	34	.	.	PUNCT
easat-1685	163	1	since	since	SCONJ
easat-1685	163	2	𝑓	𝑓	PROPN
easat-1685	163	3	is	be	AUX
easat-1685	163	4	𝒶irresolute	𝒶irresolute	ADJ
easat-1685	163	5	,	,	PUNCT
easat-1685	163	6	setting	set	VERB
easat-1685	163	7	𝒰𝑛	𝒰𝑛	PROPN
easat-1685	163	8	=	=	SYM
easat-1685	163	9	𝑓−1	𝑓−1	NUM
easat-1685	163	10	(	(	PUNCT
easat-1685	163	11	𝒱𝑛	𝒱𝑛	PROPN
easat-1685	163	12	)	)	PUNCT
easat-1685	163	13	,	,	PUNCT
easat-1685	163	14	𝑛	𝑛	DET
easat-1685	163	15	∈	∈	NOUN
easat-1685	163	16	𝑁	𝑁	PROPN
easat-1685	163	17	,	,	PUNCT
easat-1685	163	18	we	we	PRON
easat-1685	163	19	get	get	VERB
easat-1685	163	20	the	the	DET
easat-1685	163	21	sequence	sequence	NOUN
easat-1685	163	22	(	(	PUNCT
easat-1685	163	23	𝒰𝑛	𝒰𝑛	PROPN
easat-1685	163	24	)	)	PUNCT
easat-1685	163	25	𝑛∈𝑁	𝑛∈𝑁	NOUN
easat-1685	163	26	of	of	ADP
easat-1685	163	27	𝒶open	𝒶open	ADJ
easat-1685	163	28	covers	cover	NOUN
easat-1685	163	29	of	of	ADP
easat-1685	163	30	𝑋	𝑋	PROPN
easat-1685	163	31	.use	.use	PUNCT
easat-1685	163	32	the	the	DET
easat-1685	163	33	fact	fact	NOUN
easat-1685	163	34	𝑋	𝑋	NOUN
easat-1685	163	35	is	be	AUX
easat-1685	163	36	𝒶open	𝒶open	ADJ
easat-1685	163	37	covers	cover	NOUN
easat-1685	163	38	and	and	CCONJ
easat-1685	163	39	for	for	ADP
easat-1685	163	40	each	each	DET
easat-1685	163	41	𝑛	𝑛	PROPN
easat-1685	163	42	,	,	PUNCT
easat-1685	163	43	find	find	VERB
easat-1685	163	44	a	a	DET
easat-1685	163	45	finite	finite	NOUN
easat-1685	163	46	subset	subset	NOUN
easat-1685	164	1	ℋ𝑛	ℋ𝑛	NOUN
easat-1685	164	2	of	of	ADP
easat-1685	164	3	𝒰𝑛	𝒰𝑛	PROPN
easat-1685	164	4	with	with	ADP
easat-1685	164	5	for	for	ADP
easat-1685	164	6	each	each	DET
easat-1685	164	7	𝑥	𝑥	PRON
easat-1685	164	8	∈	∈	PROPN
easat-1685	164	9	𝑋.	𝑋.	PROPN
easat-1685	164	10	for	for	ADP
easat-1685	164	11	all	all	PRON
easat-1685	164	12	but	but	ADV
easat-1685	164	13	finitely	finitely	ADV
easat-1685	164	14	many	many	ADJ
easat-1685	164	15	𝑛	𝑛	DET
easat-1685	164	16	∈	∈	PROPN
easat-1685	164	17	𝑁	𝑁	PROPN
easat-1685	164	18	,	,	PUNCT
easat-1685	165	1	such	such	ADJ
easat-1685	165	2	that	that	SCONJ
easat-1685	165	3	𝑋=⋃𝑛∈𝑁	𝑋=⋃𝑛∈𝑁	NOUN
easat-1685	165	4	⋃ℋ𝑛.let	⋃ℋ𝑛.let	NOUN
easat-1685	165	5	𝒲𝑛	𝒲𝑛	PROPN
easat-1685	165	6	=	=	PROPN
easat-1685	165	7	𝑓(ℋ𝑛	𝑓(ℋ𝑛	NUM
easat-1685	165	8	)	)	PUNCT
easat-1685	165	9	,	,	PUNCT
easat-1685	165	10	𝑛	𝑛	DET
easat-1685	165	11	∈	∈	PROPN
easat-1685	165	12	𝑁.	𝑁.	PROPN
easat-1685	165	13	then	then	ADV
easat-1685	165	14	the	the	DET
easat-1685	165	15	sequence	sequence	NOUN
easat-1685	165	16	(	(	PUNCT
easat-1685	165	17	𝒲𝑛	𝒲𝑛	PROPN
easat-1685	165	18	)	)	PUNCT
easat-1685	165	19	𝑛∈𝑁	𝑛∈𝑁	NOUN
easat-1685	165	20	verifies	verifie	NOUN
easat-1685	165	21	for	for	ADP
easat-1685	165	22	(	(	PUNCT
easat-1685	165	23	𝒱𝑛	𝒱𝑛	PROPN
easat-1685	165	24	)	)	PUNCT
easat-1685	165	25	𝑛∈𝑁	𝑛∈𝑁	NOUN
easat-1685	165	26	that	that	PRON
easat-1685	165	27	𝑌	𝑌	PROPN
easat-1685	165	28	is	be	AUX
easat-1685	165	29	𝒶-hurewicz	𝒶-hurewicz	NOUN
easat-1685	165	30	space	space	NOUN
easat-1685	165	31	.	.	PUNCT
easat-1685	166	1	a	a	X
easat-1685	166	2	-	-	PUNCT
easat-1685	166	3	topological	topological	ADJ
easat-1685	166	4	property	property	NOUN
easat-1685	166	5	is	be	AUX
easat-1685	166	6	a	a	DET
easat-1685	166	7	property	property	NOUN
easat-1685	166	8	maintained	maintain	VERB
easat-1685	166	9	by	by	ADP
easat-1685	166	10	a	a	DET
easat-1685	166	11	-	-	PUNCT
easat-1685	166	12	homeomorphisms	homeomorphism	NOUN
easat-1685	166	13	.	.	PUNCT
easat-1685	167	1	theorem	theorem	VERB
easat-1685	167	2	4.5	4.5	NUM
easat-1685	167	3	:	:	PUNCT
easat-1685	167	4	if	if	SCONJ
easat-1685	167	5	𝑋	𝑋	PROPN
easat-1685	167	6	is	be	AUX
easat-1685	167	7	an	an	DET
easat-1685	167	8	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	167	9	space	space	NOUN
easat-1685	167	10	and	and	CCONJ
easat-1685	167	11	y	y	PROPN
easat-1685	167	12	𝒶compact	𝒶compact	NOUN
easat-1685	167	13	space	space	NOUN
easat-1685	167	14	,	,	PUNCT
easat-1685	167	15	then	then	ADV
easat-1685	167	16	𝑋	𝑋	PROPN
easat-1685	167	17	×	×	PROPN
easat-1685	167	18	𝑌	𝑌	PROPN
easat-1685	167	19	is	be	AUX
easat-1685	167	20	𝒶-hurewicz	𝒶-hurewicz	NOUN
easat-1685	167	21	space	space	NOUN
easat-1685	167	22	.	.	PUNCT
easat-1685	168	1	proof	proof	NOUN
easat-1685	168	2	:	:	PUNCT
easat-1685	168	3	let	let	VERB
easat-1685	168	4	𝑋	𝑋	PROPN
easat-1685	168	5	=	=	SYM
easat-1685	168	6	⋃{𝑋𝑘	⋃{𝑋𝑘	PROPN
easat-1685	168	7	:	:	PUNCT
easat-1685	168	8	𝑘	𝑘	PROPN
easat-1685	168	9	∈	∈	NOUN
easat-1685	168	10	𝑁},where	𝑁},where	X
easat-1685	168	11	each	each	DET
easat-1685	168	12	𝑋𝑘	𝑋𝑘	PROPN
easat-1685	168	13	is	be	AUX
easat-1685	168	14	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	168	15	.	.	PUNCT
easat-1685	169	1	let	let	AUX
easat-1685	169	2	{	{	PUNCT
easat-1685	169	3	𝒰𝑛	𝒰𝑛	VERB
easat-1685	169	4	:	:	PUNCT
easat-1685	169	5	𝑛	𝑛	PRON
easat-1685	169	6	∈	∈	PROPN
easat-1685	169	7	𝑁	𝑁	PROPN
easat-1685	169	8	}	}	PUNCT
easat-1685	169	9	be	be	AUX
easat-1685	169	10	a	a	DET
easat-1685	169	11	sequence	sequence	NOUN
easat-1685	169	12	of	of	ADP
easat-1685	169	13	𝒶	𝒶	DET
easat-1685	169	14	open	open	ADJ
easat-1685	169	15	covers	cover	NOUN
easat-1685	169	16	of	of	ADP
easat-1685	169	17	𝑋.	𝑋.	PROPN
easat-1685	169	18	for	for	ADP
easat-1685	169	19	each	each	DET
easat-1685	169	20	𝑘	𝑘	PRON
easat-1685	169	21	∈	∈	PROPN
easat-1685	169	22	𝑁	𝑁	PROPN
easat-1685	169	23	,	,	PUNCT
easat-1685	169	24	take	take	VERB
easat-1685	169	25	the	the	DET
easat-1685	169	26	sequence	sequence	NOUN
easat-1685	169	27	{	{	PUNCT
easat-1685	169	28	𝒰𝑛	𝒰𝑛	PROPN
easat-1685	169	29	∶	∶	NOUN
easat-1685	169	30	𝑛	𝑛	DET
easat-1685	169	31	≥	≥	NOUN
easat-1685	169	32	𝑘	𝑘	PROPN
easat-1685	169	33	}	}	PUNCT
easat-1685	169	34	.for	.for	ADP
easat-1685	169	35	each	each	DET
easat-1685	169	36	𝑘	𝑘	PRON
easat-1685	169	37	∈	∈	PROPN
easat-1685	169	38	𝑁	𝑁	PROPN
easat-1685	169	39	,	,	PUNCT
easat-1685	169	40	since	since	SCONJ
easat-1685	169	41	𝑋𝑘	𝑋𝑘	PROPN
easat-1685	169	42	is	be	AUX
easat-1685	169	43	𝒶	𝒶	PRON
easat-1685	169	44	hurewicz	hurewicz	NOUN
easat-1685	169	45	,	,	PUNCT
easat-1685	169	46	there	there	PRON
easat-1685	169	47	are	be	VERB
easat-1685	169	48	a	a	DET
easat-1685	169	49	dense	dense	ADJ
easat-1685	169	50	subset	subset	NOUN
easat-1685	169	51	s𝑘	s𝑘	NOUN
easat-1685	169	52	of	of	ADP
easat-1685	169	53	𝑋𝑘	𝑋𝑘	PROPN
easat-1685	169	54	and	and	CCONJ
easat-1685	169	55	a	a	DET
easat-1685	169	56	sequence(𝒱𝑛,𝑘	sequence(𝒱𝑛,𝑘	NOUN
easat-1685	169	57	:	:	PUNCT
easat-1685	169	58	𝑛	𝑛	PRON
easat-1685	169	59	≥	≥	NUM
easat-1685	169	60	𝑘	𝑘	NOUN
easat-1685	169	61	)	)	PUNCT
easat-1685	169	62	such	such	ADJ
easat-1685	169	63	that	that	PRON
easat-1685	169	64	for	for	ADP
easat-1685	169	65	each	each	DET
easat-1685	169	66	𝑛	𝑛	PRON
easat-1685	169	67	≥	≥	NUM
easat-1685	169	68	𝑘	𝑘	NOUN
easat-1685	169	69	,	,	PUNCT
easat-1685	169	70	(	(	PUNCT
easat-1685	169	71	𝒱𝑛,𝑘	𝒱𝑛,𝑘	NOUN
easat-1685	169	72	is	be	AUX
easat-1685	169	73	a	a	DET
easat-1685	169	74	finite	finite	NOUN
easat-1685	169	75	subset	subset	NOUN
easat-1685	169	76	of	of	ADP
easat-1685	169	77	𝒰𝑛and	𝒰𝑛and	PROPN
easat-1685	169	78	for	for	ADP
easat-1685	169	79	each	each	DET
easat-1685	169	80	𝑥	𝑥	PRON
easat-1685	169	81	∈	∈	NOUN
easat-1685	169	82	s𝑘	s𝑘	NOUN
easat-1685	169	83	,	,	PUNCT
easat-1685	169	84	𝑥	𝑥	PROPN
easat-1685	169	85	∈	∈	PROPN
easat-1685	169	86	⋃𝒱𝑛,𝑘	⋃𝒱𝑛,𝑘	PROPN
easat-1685	169	87	for	for	ADP
easat-1685	169	88	all	all	PRON
easat-1685	169	89	but	but	ADV
easat-1685	169	90	finitely	finitely	ADV
easat-1685	169	91	many	many	ADJ
easat-1685	169	92	𝑛	𝑛	PRON
easat-1685	169	93	≥	≥	NOUN
easat-1685	169	94	𝑘.	𝑘.	NOUN
easat-1685	169	95	let	let	VERB
easat-1685	169	96	s	s	NOUN
easat-1685	169	97	=	=	SYM
easat-1685	169	98	276	276	NUM
easat-1685	169	99	edelweiss	edelweiss	PROPN
easat-1685	169	100	applied	apply	VERB
easat-1685	169	101	science	science	NOUN
easat-1685	169	102	and	and	CCONJ
easat-1685	169	103	technology	technology	NOUN
easat-1685	169	104	issn	issn	PROPN
easat-1685	169	105	:	:	PUNCT
easat-1685	169	106	2576	2576	NUM
easat-1685	169	107	-	-	SYM
easat-1685	169	108	8484	8484	NUM
easat-1685	169	109	vol	vol	NOUN
easat-1685	169	110	.	.	PROPN
easat-1685	169	111	8	8	NUM
easat-1685	169	112	,	,	PUNCT
easat-1685	169	113	no	no	INTJ
easat-1685	169	114	.	.	NOUN
easat-1685	169	115	5	5	NUM
easat-1685	169	116	:	:	PUNCT
easat-1685	169	117	271	271	NUM
easat-1685	169	118	-	-	SYM
easat-1685	169	119	277	277	NUM
easat-1685	169	120	,	,	PUNCT
easat-1685	169	121	2024	2024	NUM
easat-1685	169	122	doi	doi	NOUN
easat-1685	169	123	:	:	PUNCT
easat-1685	169	124	10.55214/25768484.v8i5.1685	10.55214/25768484.v8i5.1685	NUM
easat-1685	169	125	©	©	ADP
easat-1685	169	126	2024	2024	NUM
easat-1685	169	127	by	by	ADP
easat-1685	169	128	the	the	DET
easat-1685	169	129	authors	author	NOUN
easat-1685	169	130	;	;	PUNCT
easat-1685	169	131	licensee	licensee	PROPN
easat-1685	169	132	learning	learning	NOUN
easat-1685	169	133	gate	gate	NOUN
easat-1685	169	134	⋃𝑘∈𝑁s𝑘.then	⋃𝑘∈𝑁s𝑘.then	PROPN
easat-1685	169	135	s	s	VERB
easat-1685	169	136	is	be	AUX
easat-1685	169	137	a	a	DET
easat-1685	169	138	dense	dense	ADJ
easat-1685	169	139	subset	subset	NOUN
easat-1685	169	140	of	of	ADP
easat-1685	169	141	𝑋	𝑋	PROPN
easat-1685	169	142	.	.	PUNCT
easat-1685	170	1	for	for	ADP
easat-1685	170	2	each	each	DET
easat-1685	170	3	𝑛	𝑛	PRON
easat-1685	170	4	∈	∈	PROPN
easat-1685	170	5	𝑁	𝑁	PROPN
easat-1685	170	6	,	,	PUNCT
easat-1685	170	7	let	let	VERB
easat-1685	170	8	⋃{(𝒱𝑛,𝑗	⋃{(𝒱𝑛,𝑗	ADJ
easat-1685	170	9	:	:	PUNCT
easat-1685	170	10	𝑗	𝑗	INTJ
easat-1685	170	11	≤	≤	NOUN
easat-1685	170	12	𝑛}.then	𝑛}.then	ADP
easat-1685	170	13	each	each	DET
easat-1685	170	14	𝒱𝑛	𝒱𝑛	PROPN
easat-1685	170	15	is	be	AUX
easat-1685	170	16	finite	finite	NOUN
easat-1685	170	17	subset	subset	NOUN
easat-1685	170	18	of	of	ADP
easat-1685	170	19	𝒰𝑛.the	𝒰𝑛.the	DET
easat-1685	170	20	dense	dense	ADJ
easat-1685	170	21	subset	subset	NOUN
easat-1685	170	22	s	s	NOUN
easat-1685	170	23	of	of	ADP
easat-1685	170	24	𝑋	𝑋	PROPN
easat-1685	170	25	and	and	CCONJ
easat-1685	170	26	the	the	DET
easat-1685	170	27	sequence	sequence	NOUN
easat-1685	170	28	(	(	PUNCT
easat-1685	170	29	𝒱𝑛	𝒱𝑛	NOUN
easat-1685	170	30	:	:	PUNCT
easat-1685	170	31	𝑛	𝑛	PRON
easat-1685	170	32	∈	∈	PROPN
easat-1685	170	33	𝑁	𝑁	PROPN
easat-1685	170	34	)	)	PUNCT
easat-1685	170	35	witness	witness	NOUN
easat-1685	170	36	that	that	SCONJ
easat-1685	170	37	𝑋	𝑋	PROPN
easat-1685	170	38	is	be	AUX
easat-1685	170	39	an	an	DET
easat-1685	170	40	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	170	41	space	space	NOUN
easat-1685	170	42	;	;	PUNCT
easat-1685	170	43	if	if	SCONJ
easat-1685	170	44	for	for	ADP
easat-1685	170	45	each	each	DET
easat-1685	170	46	𝑥	𝑥	PRON
easat-1685	170	47	∈	∈	PROPN
easat-1685	170	48	s	s	NOUN
easat-1685	170	49	,	,	PUNCT
easat-1685	170	50	there	there	PRON
easat-1685	170	51	exists	exist	VERB
easat-1685	170	52	some	some	PRON
easat-1685	170	53	𝑘	𝑘	PRON
easat-1685	170	54	∈	∈	NOUN
easat-1685	171	1	𝑁	𝑁	ADP
easat-1685	171	2	such	such	ADJ
easat-1685	171	3	that	that	SCONJ
easat-1685	171	4	𝑥	𝑥	DET
easat-1685	171	5	∈	∈	NOUN
easat-1685	171	6	s𝑘,then	s𝑘,then	ADP
easat-1685	171	7	𝑥	𝑥	PRON
easat-1685	171	8	∈	∈	PROPN
easat-1685	171	9	⋃𝒱𝑛	⋃𝒱𝑛	NUM
easat-1685	171	10	for	for	ADP
easat-1685	171	11	all	all	DET
easat-1685	171	12	but	but	ADV
easat-1685	171	13	finitely	finitely	ADV
easat-1685	171	14	many	many	ADJ
easat-1685	171	15	𝑛	𝑛	DET
easat-1685	171	16	≥	≥	NOUN
easat-1685	171	17	𝑘.	𝑘.	NOUN
easat-1685	171	18	remark	remark	NOUN
easat-1685	171	19	4.6	4.6	NUM
easat-1685	171	20	:	:	PUNCT
easat-1685	171	21	the	the	DET
easat-1685	171	22	product	product	NOUN
easat-1685	171	23	of	of	ADP
easat-1685	171	24	𝑎hurewicz	𝑎hurewicz	ADJ
easat-1685	171	25	space	space	NOUN
easat-1685	171	26	and	and	CCONJ
easat-1685	171	27	𝑎compact	𝑎compact	NOUN
easat-1685	171	28	is	be	AUX
easat-1685	171	29	𝑎hurewicz	𝑎hurewicz	ADJ
easat-1685	171	30	space	space	NOUN
easat-1685	171	31	,	,	PUNCT
easat-1685	171	32	as	as	SCONJ
easat-1685	171	33	demonstrated	demonstrate	VERB
easat-1685	171	34	by	by	ADP
easat-1685	171	35	the	the	DET
easat-1685	171	36	previous	previous	ADJ
easat-1685	171	37	theorem	theorem	NOUN
easat-1685	171	38	.	.	PUNCT
easat-1685	172	1	proposition	proposition	NOUN
easat-1685	172	2	4.7	4.7	NUM
easat-1685	172	3	:	:	PUNCT
easat-1685	172	4	𝒶-int	𝒶-int	NOUN
easat-1685	172	5	(	(	PUNCT
easat-1685	172	6	a	a	NOUN
easat-1685	172	7	)	)	PUNCT
easat-1685	172	8	⊆	⊆	NUM
easat-1685	172	9	int	int	NOUN
easat-1685	172	10	(	(	PUNCT
easat-1685	172	11	a	a	NOUN
easat-1685	172	12	)	)	PUNCT
easat-1685	172	13	.	.	PUNCT
easat-1685	173	1	proof	proof	NOUN
easat-1685	173	2	:	:	PUNCT
easat-1685	173	3	let	let	VERB
easat-1685	173	4	x	x	X
easat-1685	173	5	∈	∈	PROPN
easat-1685	173	6	𝒶-int	𝒶-int	NOUN
easat-1685	173	7	(	(	PUNCT
easat-1685	173	8	a	a	NOUN
easat-1685	173	9	)	)	PUNCT
easat-1685	173	10	,	,	PUNCT
easat-1685	173	11	there	there	PRON
easat-1685	173	12	is	be	VERB
easat-1685	173	13	𝒶	𝒶	DET
easat-1685	173	14	open	open	ADJ
easat-1685	173	15	set	set	NOUN
easat-1685	173	16	b	b	X
easat-1685	173	17	such	such	ADJ
easat-1685	173	18	that	that	SCONJ
easat-1685	173	19	x	x	SYM
easat-1685	173	20	∈	∈	NOUN
easat-1685	173	21	b	b	X
easat-1685	173	22	⊆	⊆	NUM
easat-1685	173	23	a.	a.	NOUN
easat-1685	174	1	so	so	ADV
easat-1685	174	2	,	,	PUNCT
easat-1685	174	3	there	there	PRON
easat-1685	174	4	is	be	VERB
easat-1685	174	5	open	open	ADJ
easat-1685	174	6	set	set	ADJ
easat-1685	174	7	b	b	X
easat-1685	174	8	such	such	ADJ
easat-1685	174	9	that	that	SCONJ
easat-1685	174	10	x	x	SYM
easat-1685	174	11	∈	∈	NOUN
easat-1685	174	12	b	b	PROPN
easat-1685	174	13	⊆	⊆	NUM
easat-1685	174	14	a	a	PRON
easat-1685	174	15	.	.	PUNCT
easat-1685	175	1	then	then	ADV
easat-1685	175	2	x	x	SYM
easat-1685	175	3	⊆	⊆	NUM
easat-1685	175	4	int	int	NOUN
easat-1685	175	5	(	(	PUNCT
easat-1685	175	6	a	a	NOUN
easat-1685	175	7	)	)	PUNCT
easat-1685	175	8	.hence	.hence	NOUN
easat-1685	175	9	,	,	PUNCT
easat-1685	175	10	(	(	PUNCT
easat-1685	175	11	a	a	X
easat-1685	175	12	)	)	PUNCT
easat-1685	175	13	⊆	⊆	NUM
easat-1685	175	14	int	int	NOUN
easat-1685	175	15	(	(	PUNCT
easat-1685	175	16	a	a	NOUN
easat-1685	175	17	)	)	PUNCT
easat-1685	175	18	.	.	PUNCT
easat-1685	176	1	proposition	proposition	NOUN
easat-1685	176	2	4.8	4.8	NUM
easat-1685	176	3	:	:	PUNCT
easat-1685	176	4	every	every	DET
easat-1685	176	5	𝒶-open	𝒶-open	NOUN
easat-1685	176	6	set	set	VERB
easat-1685	176	7	is	be	AUX
easat-1685	176	8	open	open	ADJ
easat-1685	176	9	.	.	PUNCT
easat-1685	177	1	proof	proof	NOUN
easat-1685	177	2	:	:	PUNCT
easat-1685	177	3	let	let	VERB
easat-1685	177	4	a	a	PRON
easat-1685	177	5	is	be	AUX
easat-1685	177	6	𝒶-open	𝒶-open	NOUN
easat-1685	177	7	set	set	VERB
easat-1685	177	8	.	.	PUNCT
easat-1685	178	1	suppose	suppose	VERB
easat-1685	178	2	x	x	SYM
easat-1685	178	3	∈	∈	PROPN
easat-1685	178	4	a	a	PRON
easat-1685	178	5	.	.	PUNCT
easat-1685	179	1	then	then	ADV
easat-1685	179	2	,	,	PUNCT
easat-1685	179	3	there	there	PRON
easat-1685	179	4	exists	exist	VERB
easat-1685	179	5	𝛿	𝛿	ADJ
easat-1685	179	6	–	–	PUNCT
easat-1685	179	7	open	open	ADJ
easat-1685	179	8	set	set	NOUN
easat-1685	179	9	b	b	X
easat-1685	179	10	such	such	ADJ
easat-1685	179	11	that	that	SCONJ
easat-1685	179	12	x	x	SYM
easat-1685	179	13	∈	∈	NOUN
easat-1685	179	14	b	b	NOUN
easat-1685	179	15	⊆	⊆	NUM
easat-1685	179	16	a.	a.	NOUN
easat-1685	179	17	since	since	SCONJ
easat-1685	179	18	b	b	PROPN
easat-1685	179	19	is	be	AUX
easat-1685	179	20	open	open	ADJ
easat-1685	179	21	.	.	PUNCT
easat-1685	180	1	then	then	ADV
easat-1685	180	2	,	,	PUNCT
easat-1685	180	3	for	for	ADP
easat-1685	180	4	every	every	DET
easat-1685	180	5	x	x	SYM
easat-1685	180	6	∈	∈	PROPN
easat-1685	180	7	a	a	DET
easat-1685	180	8	there	there	PRON
easat-1685	180	9	exists	exist	VERB
easat-1685	180	10	an	an	DET
easat-1685	180	11	open	open	ADJ
easat-1685	180	12	set	set	NOUN
easat-1685	180	13	b	b	NOUN
easat-1685	180	14	such	such	ADJ
easat-1685	180	15	that	that	SCONJ
easat-1685	180	16	x	x	SYM
easat-1685	180	17	∈	∈	NOUN
easat-1685	180	18	b	b	X
easat-1685	180	19	⊆	⊆	NUM
easat-1685	180	20	a.	a.	NOUN
easat-1685	180	21	hence	hence	ADV
easat-1685	180	22	a	a	PRON
easat-1685	180	23	is	be	AUX
easat-1685	180	24	open	open	ADJ
easat-1685	180	25	set	set	VERB
easat-1685	180	26	.	.	PUNCT
easat-1685	181	1	proposition	proposition	NOUN
easat-1685	181	2	4.9	4.9	NUM
easat-1685	181	3	:	:	PUNCT
easat-1685	181	4	if	if	SCONJ
easat-1685	181	5	a	a	DET
easat-1685	181	6	=	=	NOUN
easat-1685	181	7	𝒶-int	𝒶-int	NOUN
easat-1685	181	8	(	(	PUNCT
easat-1685	181	9	a	a	NOUN
easat-1685	181	10	)	)	PUNCT
easat-1685	181	11	then	then	ADV
easat-1685	181	12	a	a	PRON
easat-1685	181	13	is	be	AUX
easat-1685	181	14	an	an	DET
easat-1685	181	15	𝒶-open	𝒶-open	NOUN
easat-1685	181	16	set	set	NOUN
easat-1685	181	17	.	.	PUNCT
easat-1685	182	1	proof	proof	NOUN
easat-1685	182	2	:	:	PUNCT
easat-1685	182	3	𝒶-int	𝒶-int	NOUN
easat-1685	182	4	(	(	PUNCT
easat-1685	182	5	a	a	X
easat-1685	182	6	)	)	PUNCT
easat-1685	182	7	=	=	NOUN
easat-1685	182	8	{	{	PUNCT
easat-1685	182	9	∪	∪	ADP
easat-1685	182	10	b	b	NOUN
easat-1685	182	11	:	:	PUNCT
easat-1685	182	12	b	b	PROPN
easat-1685	182	13	⊆	⊆	NUM
easat-1685	182	14	a	a	PRON
easat-1685	182	15	,	,	PUNCT
easat-1685	182	16	b	b	NOUN
easat-1685	182	17	is	be	AUX
easat-1685	182	18	an	an	DET
easat-1685	182	19	𝒶-open	𝒶-open	NOUN
easat-1685	182	20	set	set	NOUN
easat-1685	182	21	}	}	PUNCT
easat-1685	182	22	.	.	PUNCT
easat-1685	183	1	let	let	VERB
easat-1685	183	2	a	a	DET
easat-1685	183	3	=	=	NOUN
easat-1685	183	4	𝒶-int	𝒶-int	NOUN
easat-1685	183	5	(	(	PUNCT
easat-1685	183	6	a	a	NOUN
easat-1685	183	7	)	)	PUNCT
easat-1685	183	8	to	to	PART
easat-1685	183	9	show	show	VERB
easat-1685	183	10	a	a	DET
easat-1685	183	11	is	be	AUX
easat-1685	183	12	𝒶-open	𝒶-open	NOUN
easat-1685	183	13	set	set	VERB
easat-1685	183	14	.	.	PUNCT
easat-1685	184	1	it	it	PRON
easat-1685	184	2	is	be	AUX
easat-1685	184	3	clearly	clearly	ADV
easat-1685	184	4	that	that	SCONJ
easat-1685	184	5	𝒶-int	𝒶-int	NOUN
easat-1685	184	6	(	(	PUNCT
easat-1685	184	7	a	a	X
easat-1685	184	8	)	)	PUNCT
easat-1685	184	9	⊆	⊆	NUM
easat-1685	184	10	a.	a.	NOUN
easat-1685	184	11	conversely	conversely	ADV
easat-1685	184	12	,	,	PUNCT
easat-1685	184	13	suppose	suppose	VERB
easat-1685	184	14	a	a	DET
easat-1685	184	15	⊆	⊆	NUM
easat-1685	184	16	int	int	NOUN
easat-1685	184	17	(	(	PUNCT
easat-1685	184	18	cl(𝛿-int(a	cl(𝛿-int(a	NOUN
easat-1685	184	19	)	)	PUNCT
easat-1685	184	20	)	)	PUNCT
easat-1685	184	21	)	)	PUNCT
easat-1685	185	1	such	such	ADJ
easat-1685	185	2	that	that	SCONJ
easat-1685	185	3	b=	b=	NOUN
easat-1685	185	4	cl(𝛿-int(a	cl(𝛿-int(a	PROPN
easat-1685	185	5	)	)	PUNCT
easat-1685	185	6	)	)	PUNCT
easat-1685	185	7	.	.	PUNCT
easat-1685	186	1	then	then	ADV
easat-1685	186	2	a	a	DET
easat-1685	186	3	⊆	⊆	NUM
easat-1685	186	4	int	int	NOUN
easat-1685	186	5	(	(	PUNCT
easat-1685	186	6	b	b	NOUN
easat-1685	186	7	)	)	PUNCT
easat-1685	186	8	.	.	PUNCT
easat-1685	187	1	then	then	ADV
easat-1685	187	2	,	,	PUNCT
easat-1685	187	3	𝒶-int	𝒶-int	NOUN
easat-1685	187	4	(	(	PUNCT
easat-1685	187	5	a	a	NOUN
easat-1685	187	6	)	)	PUNCT
easat-1685	187	7	⊆	⊆	NUM
easat-1685	187	8	int	int	NOUN
easat-1685	187	9	(	(	PUNCT
easat-1685	187	10	b	b	NOUN
easat-1685	187	11	)	)	PUNCT
easat-1685	187	12	.	.	PUNCT
easat-1685	188	1	so	so	ADV
easat-1685	188	2	,	,	PUNCT
easat-1685	188	3	a	a	DET
easat-1685	188	4	⊆	⊆	NUM
easat-1685	188	5	int	int	NOUN
easat-1685	188	6	(	(	PUNCT
easat-1685	188	7	cl(𝛿-int(a	cl(𝛿-int(a	NOUN
easat-1685	188	8	)	)	PUNCT
easat-1685	188	9	)	)	PUNCT
easat-1685	188	10	)	)	PUNCT
easat-1685	188	11	.	.	PUNCT
easat-1685	189	1	hence	hence	ADV
easat-1685	189	2	a	a	PRON
easat-1685	189	3	is	be	AUX
easat-1685	189	4	𝒶	𝒶	NOUN
easat-1685	189	5	–	–	PUNCT
easat-1685	189	6	open	open	ADJ
easat-1685	189	7	set	set	NOUN
easat-1685	189	8	.	.	PUNCT
easat-1685	190	1	a	a	DET
easat-1685	190	2	mapping	mapping	NOUN
easat-1685	190	3	𝑓	𝑓	DET
easat-1685	190	4	∶	∶	NOUN
easat-1685	190	5	𝑋	𝑋	NOUN
easat-1685	190	6	⟶	⟶	NOUN
easat-1685	190	7	𝑌	𝑌	PROPN
easat-1685	190	8	is	be	AUX
easat-1685	190	9	called	call	VERB
easat-1685	190	10	contra	contra	PROPN
easat-1685	190	11	𝒶continuous	𝒶continuous	NOUN
easat-1685	190	12	if	if	SCONJ
easat-1685	190	13	the	the	DET
easat-1685	190	14	preimage	preimage	NOUN
easat-1685	190	15	of	of	ADP
easat-1685	190	16	each	each	DET
easat-1685	190	17	𝒶-open	𝒶-open	NOUN
easat-1685	190	18	set	set	VERB
easat-1685	190	19	in	in	ADP
easat-1685	190	20	𝑌	𝑌	PROPN
easat-1685	190	21	is	be	AUX
easat-1685	190	22	𝒶	𝒶	PROPN
easat-1685	190	23	closed	close	VERB
easat-1685	190	24	in	in	ADP
easat-1685	190	25	𝑋	𝑋	NOUN
easat-1685	190	26	.a	.a	ADJ
easat-1685	191	1	mapping	mapping	NOUN
easat-1685	191	2	𝑓	𝑓	NOUN
easat-1685	191	3	is	be	AUX
easat-1685	191	4	called	call	VERB
easat-1685	191	5	pre𝒶continuous	pre𝒶continuous	ADJ
easat-1685	191	6	if	if	SCONJ
easat-1685	191	7	𝑓−1(𝑈)⊂	𝑓−1(𝑈)⊂	NUM
easat-1685	191	8	𝒶int	𝒶int	NOUN
easat-1685	191	9	(	(	PUNCT
easat-1685	191	10	𝒶-cl(𝑓−1(𝑈	𝒶-cl(𝑓−1(𝑈	NOUN
easat-1685	191	11	)	)	PUNCT
easat-1685	191	12	)	)	PUNCT
easat-1685	191	13	whenever	whenever	SCONJ
easat-1685	191	14	𝑈	𝑈	PROPN
easat-1685	191	15	is	be	AUX
easat-1685	191	16	𝒶-open	𝒶-open	NOUN
easat-1685	191	17	in	in	ADP
easat-1685	191	18	𝑌.	𝑌.	PROPN
easat-1685	191	19	theorem	theorem	NOUN
easat-1685	191	20	.4.10	.4.10	NOUN
easat-1685	191	21	:	:	PUNCT
easat-1685	191	22	a	a	DET
easat-1685	191	23	contra	contra	PROPN
easat-1685	191	24	𝒶continuous	𝒶continuous	ADJ
easat-1685	191	25	,	,	PUNCT
easat-1685	191	26	pre𝒶continuous	pre𝒶continuous	ADJ
easat-1685	191	27	image	image	NOUN
easat-1685	191	28	of	of	ADP
easat-1685	191	29	𝑌	𝑌	PROPN
easat-1685	191	30	of	of	ADP
easat-1685	191	31	an	an	DET
easat-1685	191	32	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	191	33	space	space	NOUN
easat-1685	191	34	𝑋	𝑋	PROPN
easat-1685	191	35	is	be	AUX
easat-1685	191	36	an	an	DET
easat-1685	191	37	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	191	38	space	space	NOUN
easat-1685	191	39	.	.	PUNCT
easat-1685	192	1	proof	proof	NOUN
easat-1685	192	2	:	:	PUNCT
easat-1685	192	3	let	let	VERB
easat-1685	192	4	(	(	PUNCT
easat-1685	192	5	𝑈𝑛	𝑈𝑛	VERB
easat-1685	192	6	:	:	PUNCT
easat-1685	192	7	𝑛	𝑛	PRON
easat-1685	192	8	∈	∈	PROPN
easat-1685	192	9	𝑁	𝑁	PROPN
easat-1685	192	10	)	)	PUNCT
easat-1685	192	11	be	be	VERB
easat-1685	192	12	a	a	DET
easat-1685	192	13	sequence	sequence	NOUN
easat-1685	192	14	of	of	ADP
easat-1685	192	15	𝒶-open	𝒶-open	NOUN
easat-1685	192	16	covers	cover	NOUN
easat-1685	192	17	of	of	ADP
easat-1685	192	18	𝑌.	𝑌.	PROPN
easat-1685	192	19	since	since	SCONJ
easat-1685	192	20	f	f	PROPN
easat-1685	192	21	is	be	AUX
easat-1685	192	22	contra	contra	PROPN
easat-1685	192	23	𝒶continuous	𝒶continuous	NOUN
easat-1685	192	24	,	,	PUNCT
easat-1685	192	25	for	for	ADP
easat-1685	192	26	each	each	DET
easat-1685	192	27	𝑛	𝑛	DET
easat-1685	192	28	∈	∈	NOUN
easat-1685	192	29	𝑁	𝑁	PROPN
easat-1685	192	30	and	and	CCONJ
easat-1685	192	31	for	for	ADP
easat-1685	192	32	each	each	DET
easat-1685	192	33	𝑈	𝑈	PROPN
easat-1685	192	34	∈	∈	PROPN
easat-1685	193	1	𝑈𝑛	𝑈𝑛	ADP
easat-1685	193	2	the	the	DET
easat-1685	193	3	set	set	NOUN
easat-1685	193	4	𝑓−1(𝑈	𝑓−1(𝑈	NUM
easat-1685	193	5	)	)	PUNCT
easat-1685	193	6	is	be	AUX
easat-1685	193	7	𝒶-closed	𝒶-close	VERB
easat-1685	193	8	in	in	ADP
easat-1685	193	9	𝑋	𝑋	PROPN
easat-1685	193	10	.	.	PUNCT
easat-1685	194	1	since	since	SCONJ
easat-1685	194	2	𝑓	𝑓	PRON
easat-1685	194	3	is	be	AUX
easat-1685	194	4	pre𝒶continuous	pre𝒶continuous	ADJ
easat-1685	194	5	𝑓−1(𝑈)⊂	𝑓−1(𝑈)⊂	NUM
easat-1685	194	6	𝒶int	𝒶int	NOUN
easat-1685	194	7	(	(	PUNCT
easat-1685	194	8	𝒶-cl(𝑓−1(𝑈)),so	𝒶-cl(𝑓−1(𝑈)),so	SYM
easat-1685	194	9	that	that	DET
easat-1685	194	10	𝑓−1(𝑈)⊂	𝑓−1(𝑈)⊂	NUM
easat-1685	194	11	𝒶int(𝑓−1(𝑈	𝒶int(𝑓−1(𝑈	NOUN
easat-1685	194	12	)	)	PUNCT
easat-1685	194	13	)	)	PUNCT
easat-1685	194	14	.on	.on	PUNCT
easat-1685	195	1	the	the	DET
easat-1685	195	2	other	other	ADJ
easat-1685	195	3	hand	hand	NOUN
easat-1685	195	4	,	,	PUNCT
easat-1685	195	5	𝒶int(𝑓−1(𝑈	𝒶int(𝑓−1(𝑈	NOUN
easat-1685	195	6	)	)	PUNCT
easat-1685	195	7	)	)	PUNCT
easat-1685	196	1	⊂𝑓−1(𝑈),hence𝑓−1(𝑈)=	⊂𝑓−1(𝑈),hence𝑓−1(𝑈)=	PROPN
easat-1685	196	2	𝒶	𝒶	PROPN
easat-1685	196	3	int	int	NOUN
easat-1685	196	4	(	(	PUNCT
easat-1685	196	5	𝒶-cl(𝑓−1(𝑈	𝒶-cl(𝑓−1(𝑈	PROPN
easat-1685	196	6	)	)	PUNCT
easat-1685	196	7	)	)	PUNCT
easat-1685	196	8	.	.	PUNCT
easat-1685	197	1	therefore	therefore	ADV
easat-1685	197	2	,	,	PUNCT
easat-1685	197	3	for	for	ADP
easat-1685	197	4	each	each	DET
easat-1685	197	5	𝑛	𝑛	PROPN
easat-1685	197	6	,	,	PUNCT
easat-1685	197	7	the	the	DET
easat-1685	197	8	set	set	PROPN
easat-1685	197	9	𝑉𝑛={𝑓−1(𝑈	𝑉𝑛={𝑓−1(𝑈	PROPN
easat-1685	197	10	):	):	PUNCT
easat-1685	197	11	𝑈	𝑈	PROPN
easat-1685	197	12	∈	∈	PROPN
easat-1685	197	13	𝑈𝑛	𝑈𝑛	PROPN
easat-1685	197	14	}	}	PUNCT
easat-1685	197	15	is	be	AUX
easat-1685	197	16	a	a	DET
easat-1685	197	17	cover	cover	NOUN
easat-1685	197	18	of	of	ADP
easat-1685	197	19	𝑋	𝑋	NOUN
easat-1685	197	20	by	by	ADP
easat-1685	197	21	𝒶-open	𝒶-open	NOUN
easat-1685	197	22	sets	set	NOUN
easat-1685	197	23	.	.	PUNCT
easat-1685	198	1	since	since	SCONJ
easat-1685	198	2	𝑋	𝑋	PROPN
easat-1685	198	3	is	be	AUX
easat-1685	198	4	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	198	5	space	space	NOUN
easat-1685	198	6	there	there	PRON
easat-1685	198	7	is	be	VERB
easat-1685	198	8	a	a	DET
easat-1685	198	9	sequence	sequence	NOUN
easat-1685	198	10	(	(	PUNCT
easat-1685	198	11	𝑔𝑛	𝑔𝑛	PROPN
easat-1685	198	12	:	:	PUNCT
easat-1685	198	13	𝑛	𝑛	PRON
easat-1685	198	14	∈	∈	PROPN
easat-1685	198	15	𝑁	𝑁	PROPN
easat-1685	198	16	)	)	PUNCT
easat-1685	198	17	such	such	ADJ
easat-1685	198	18	that	that	PRON
easat-1685	198	19	for	for	ADP
easat-1685	198	20	each	each	DET
easat-1685	198	21	𝑛	𝑛	NOUN
easat-1685	198	22	,	,	PUNCT
easat-1685	198	23	𝑔𝑛	𝑔𝑛	PROPN
easat-1685	198	24	is	be	AUX
easat-1685	198	25	finite	finite	ADJ
easat-1685	198	26	subset	subset	NOUN
easat-1685	198	27	of	of	ADP
easat-1685	198	28	𝑉𝑛	𝑉𝑛	PROPN
easat-1685	198	29	and	and	CCONJ
easat-1685	198	30	each	each	DET
easat-1685	198	31	𝑥	𝑥	PRON
easat-1685	198	32	∈	∈	PROPN
easat-1685	198	33	𝑋	𝑋	PROPN
easat-1685	198	34	belongs	belong	VERB
easat-1685	198	35	to	to	ADP
easat-1685	198	36	⋃	⋃	PROPN
easat-1685	198	37	{	{	PUNCT
easat-1685	198	38	𝒶-cl(g	𝒶-cl(g	NUM
easat-1685	198	39	):	):	PUNCT
easat-1685	198	40	g	g	PROPN
easat-1685	198	41	∈	∈	PROPN
easat-1685	198	42	𝑔𝑛}.hence	𝑔𝑛}.hence	PROPN
easat-1685	198	43	𝒲𝑛=	𝒲𝑛=	PROPN
easat-1685	198	44	{	{	PUNCT
easat-1685	198	45	𝑓(g	𝑓(g	NOUN
easat-1685	198	46	):	):	PUNCT
easat-1685	198	47	g	g	PROPN
easat-1685	198	48	∈	∈	PROPN
easat-1685	198	49	𝑔𝑛	𝑔𝑛	PROPN
easat-1685	198	50	}	}	PUNCT
easat-1685	198	51	is	be	AUX
easat-1685	198	52	a	a	DET
easat-1685	198	53	finite	finite	NOUN
easat-1685	198	54	subset	subset	NOUN
easat-1685	198	55	of	of	ADP
easat-1685	198	56	𝑈𝑛	𝑈𝑛	PROPN
easat-1685	198	57	for	for	ADP
easat-1685	198	58	each	each	DET
easat-1685	198	59	𝑛	𝑛	PRON
easat-1685	198	60	∈	∈	NOUN
easat-1685	198	61	𝑁	𝑁	PROPN
easat-1685	198	62	and	and	CCONJ
easat-1685	198	63	each	each	DET
easat-1685	198	64	𝑧	𝑧	PRON
easat-1685	198	65	∈	∈	PROPN
easat-1685	198	66	𝑌	𝑌	PROPN
easat-1685	198	67	belongs	belong	VERB
easat-1685	198	68	to	to	ADP
easat-1685	198	69	𝒶-cl(⋃𝒲𝑛	𝒶-cl(⋃𝒲𝑛	PROPN
easat-1685	198	70	)	)	PUNCT
easat-1685	198	71	for	for	ADP
easat-1685	198	72	all	all	PRON
easat-1685	198	73	but	but	CCONJ
easat-1685	198	74	finitely	finitely	ADV
easat-1685	198	75	many	many	ADJ
easat-1685	198	76	𝑛	𝑛	PRON
easat-1685	198	77	.	.	PUNCT
easat-1685	199	1	this	this	PRON
easat-1685	199	2	just	just	ADV
easat-1685	199	3	means	mean	VERB
easat-1685	199	4	that	that	SCONJ
easat-1685	199	5	𝑌	𝑌	PROPN
easat-1685	199	6	is	be	AUX
easat-1685	199	7	an	an	DET
easat-1685	199	8	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	199	9	space	space	NOUN
easat-1685	199	10	.	.	PUNCT
easat-1685	200	1	definition	definition	NOUN
easat-1685	200	2	4.11	4.11	NUM
easat-1685	200	3	:	:	PUNCT
easat-1685	200	4	a	a	DET
easat-1685	200	5	topological	topological	ADJ
easat-1685	200	6	space	space	NOUN
easat-1685	200	7	𝑋	𝑋	NOUN
easat-1685	200	8	is	be	AUX
easat-1685	200	9	:	:	PUNCT
easat-1685	200	10	⦁	⦁	NOUN
easat-1685	200	11	star	star	NOUN
easat-1685	200	12	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	200	13	space	space	NOUN
easat-1685	200	14	if	if	SCONJ
easat-1685	200	15	it	it	PRON
easat-1685	200	16	satisfies	satisfy	VERB
easat-1685	200	17	:	:	PUNCT
easat-1685	200	18	for	for	ADP
easat-1685	200	19	each	each	DET
easat-1685	200	20	sequence	sequence	NOUN
easat-1685	200	21	of	of	ADP
easat-1685	200	22	elements	element	NOUN
easat-1685	200	23	of	of	ADP
easat-1685	200	24	𝒶-open	𝒶-open	NOUN
easat-1685	200	25	cover	cover	NOUN
easat-1685	200	26	(	(	PUNCT
easat-1685	200	27	𝑈𝑛	𝑈𝑛	NOUN
easat-1685	200	28	:	:	PUNCT
easat-1685	200	29	𝑛	𝑛	DET
easat-1685	200	30	∈	∈	PROPN
easat-1685	200	31	𝑁	𝑁	PROPN
easat-1685	200	32	)	)	PUNCT
easat-1685	200	33	there	there	PRON
easat-1685	200	34	is	be	VERB
easat-1685	200	35	sequence	sequence	NOUN
easat-1685	200	36	(	(	PUNCT
easat-1685	200	37	𝑉𝑛	𝑉𝑛	NOUN
easat-1685	200	38	:	:	PUNCT
easat-1685	200	39	𝑛	𝑛	DET
easat-1685	200	40	∈	∈	PROPN
easat-1685	200	41	𝑁	𝑁	PROPN
easat-1685	200	42	)	)	PUNCT
easat-1685	200	43	such	such	ADJ
easat-1685	200	44	that	that	PRON
easat-1685	200	45	for	for	ADP
easat-1685	200	46	each	each	DET
easat-1685	200	47	𝑛	𝑛	PRON
easat-1685	200	48	∈	∈	NOUN
easat-1685	200	49	𝑁	𝑁	PROPN
easat-1685	200	50	,	,	PUNCT
easat-1685	200	51	𝑉𝑛	𝑉𝑛	PROPN
easat-1685	200	52	is	be	AUX
easat-1685	200	53	finite	finite	NOUN
easat-1685	200	54	subset	subset	NOUN
easat-1685	200	55	of	of	ADP
easat-1685	200	56	𝑈𝑛	𝑈𝑛	PROPN
easat-1685	200	57	,	,	PUNCT
easat-1685	200	58	and	and	CCONJ
easat-1685	200	59	each	each	DET
easat-1685	200	60	𝑥	𝑥	PRON
easat-1685	200	61	∈	∈	PROPN
easat-1685	200	62	𝑋	𝑋	NOUN
easat-1685	200	63	belong	belong	VERB
easat-1685	200	64	to	to	ADP
easat-1685	200	65	𝑆𝑡𝒶(⋃	𝑆𝑡𝒶(⋃	PROPN
easat-1685	200	66	𝑉𝑛	𝑉𝑛	PROPN
easat-1685	200	67	,	,	PUNCT
easat-1685	200	68	𝑈𝑛	𝑈𝑛	PROPN
easat-1685	200	69	)	)	PUNCT
easat-1685	200	70	for	for	ADP
easat-1685	200	71	all	all	PRON
easat-1685	200	72	but	but	ADV
easat-1685	200	73	finitely	finitely	ADV
easat-1685	200	74	many	many	ADJ
easat-1685	200	75	in	in	ADP
easat-1685	200	76	.	.	PUNCT
easat-1685	201	1	⦁	⦁	NOUN
easat-1685	201	2	strong	strong	ADJ
easat-1685	201	3	star	star	NOUN
easat-1685	201	4	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	201	5	space	space	NOUN
easat-1685	202	1	if	if	SCONJ
easat-1685	202	2	it	it	PRON
easat-1685	202	3	satisfies	satisfy	VERB
easat-1685	202	4	:	:	PUNCT
easat-1685	202	5	for	for	ADP
easat-1685	202	6	each	each	DET
easat-1685	202	7	sequence	sequence	NOUN
easat-1685	202	8	of	of	ADP
easat-1685	202	9	elements	element	NOUN
easat-1685	202	10	of	of	ADP
easat-1685	202	11	𝒶-open	𝒶-open	NOUN
easat-1685	202	12	cover	cover	NOUN
easat-1685	202	13	(	(	PUNCT
easat-1685	202	14	𝑈𝑛	𝑈𝑛	NOUN
easat-1685	202	15	:	:	PUNCT
easat-1685	202	16	𝑛	𝑛	DET
easat-1685	202	17	∈	∈	PROPN
easat-1685	202	18	𝑁	𝑁	PROPN
easat-1685	202	19	)	)	PUNCT
easat-1685	202	20	there	there	PRON
easat-1685	202	21	is	be	VERB
easat-1685	202	22	sequence	sequence	NOUN
easat-1685	202	23	(	(	PUNCT
easat-1685	202	24	𝐴𝑛	𝐴𝑛	NOUN
easat-1685	202	25	:	:	PUNCT
easat-1685	202	26	𝑛	𝑛	DET
easat-1685	202	27	∈	∈	PROPN
easat-1685	202	28	𝑁	𝑁	PROPN
easat-1685	202	29	)	)	PUNCT
easat-1685	202	30	such	such	ADJ
easat-1685	202	31	that	that	PRON
easat-1685	202	32	for	for	ADP
easat-1685	202	33	each	each	DET
easat-1685	202	34	𝑛	𝑛	DET
easat-1685	202	35	∈	∈	PROPN
easat-1685	202	36	𝑁	𝑁	PROPN
easat-1685	202	37	,	,	PUNCT
easat-1685	202	38	𝐴𝑛	𝐴𝑛	PROPN
easat-1685	202	39	is	be	AUX
easat-1685	202	40	finite	finite	ADJ
easat-1685	202	41	subset	subset	NOUN
easat-1685	202	42	of	of	ADP
easat-1685	202	43	𝑋	𝑋	PROPN
easat-1685	202	44	,	,	PUNCT
easat-1685	202	45	and	and	CCONJ
easat-1685	202	46	each	each	DET
easat-1685	202	47	𝑥	𝑥	PRON
easat-1685	202	48	∈	∈	PROPN
easat-1685	202	49	𝑋	𝑋	NOUN
easat-1685	202	50	belong	belong	VERB
easat-1685	202	51	to	to	ADP
easat-1685	202	52	𝑆𝑡𝒶(𝐴𝑛	𝑆𝑡𝒶(𝐴𝑛	PROPN
easat-1685	202	53	,	,	PUNCT
easat-1685	202	54	𝑈𝑛	𝑈𝑛	PROPN
easat-1685	202	55	)	)	PUNCT
easat-1685	202	56	for	for	ADP
easat-1685	202	57	all	all	PRON
easat-1685	202	58	but	but	ADV
easat-1685	202	59	finitely	finitely	ADV
easat-1685	202	60	many	many	ADJ
easat-1685	202	61	in	in	ADP
easat-1685	202	62	.	.	PUNCT
easat-1685	203	1	now	now	ADV
easat-1685	203	2	,	,	PUNCT
easat-1685	203	3	we	we	PRON
easat-1685	203	4	can	can	AUX
easat-1685	203	5	form	form	VERB
easat-1685	203	6	the	the	DET
easat-1685	203	7	following	follow	VERB
easat-1685	203	8	diagram	diagram	NOUN
easat-1685	203	9	.	.	PUNCT
easat-1685	204	1	strong	strong	ADJ
easat-1685	204	2	star	star	PROPN
easat-1685	204	3	𝒶compact	𝒶compact	NOUN
easat-1685	204	4	⟹	⟹	X
easat-1685	204	5	strong	strong	ADJ
easat-1685	204	6	star	star	NOUN
easat-1685	204	7	𝒶-hurewicz	𝒶-hurewicz	VERB
easat-1685	204	8	⟹	⟹	NUM
easat-1685	204	9	strong	strong	ADJ
easat-1685	204	10	star	star	NOUN
easat-1685	204	11	𝒶lindelof	𝒶lindelof	NOUN
easat-1685	204	12	⇓	⇓	PROPN
easat-1685	204	13	⇓	⇓	PROPN
easat-1685	204	14	⇓	⇓	PROPN
easat-1685	204	15	star	star	NOUN
easat-1685	204	16	𝒶compact	𝒶compact	NOUN
easat-1685	204	17	⟹	⟹	NUM
easat-1685	204	18	star	star	PROPN
easat-1685	204	19	𝒶-hurewicz	𝒶-hurewicz	PROPN
easat-1685	204	20	⟹	⟹	NUM
easat-1685	204	21	star	star	PROPN
easat-1685	204	22	𝒶lindelof	𝒶lindelof	PROPN
easat-1685	204	23	a	a	DET
easat-1685	204	24	space	space	NOUN
easat-1685	204	25	x	x	PUNCT
easat-1685	204	26	is	be	AUX
easat-1685	204	27	said	say	VERB
easat-1685	204	28	to	to	PART
easat-1685	204	29	be	be	AUX
easat-1685	204	30	σ	σ	NOUN
easat-1685	204	31	-	-	PUNCT
easat-1685	204	32	strongly	strongly	ADV
easat-1685	204	33	star	star	ADJ
easat-1685	204	34	𝑎-compact	𝑎-compact	NOUN
easat-1685	204	35	if	if	SCONJ
easat-1685	204	36	it	it	PRON
easat-1685	204	37	can	can	AUX
easat-1685	204	38	be	be	AUX
easat-1685	204	39	expressed	express	VERB
easat-1685	204	40	as	as	ADP
easat-1685	204	41	the	the	DET
easat-1685	204	42	union	union	NOUN
easat-1685	204	43	of	of	ADP
easat-1685	204	44	countably	countably	ADV
easat-1685	204	45	many	many	ADJ
easat-1685	204	46	σ	σ	NOUN
easat-1685	204	47	-	-	PUNCT
easat-1685	204	48	strongly	strongly	ADV
easat-1685	204	49	𝑎-compact	𝑎-compact	NOUN
easat-1685	204	50	spaces	space	NOUN
easat-1685	204	51	.	.	PUNCT
easat-1685	205	1	theorem	theorem	VERB
easat-1685	205	2	4.12	4.12	NUM
easat-1685	205	3	:	:	PUNCT
easat-1685	205	4	every	every	DET
easat-1685	205	5	σ	σ	PROPN
easat-1685	205	6	-	-	PUNCT
easat-1685	205	7	strongly	strongly	ADV
easat-1685	205	8	star	star	NOUN
easat-1685	205	9	𝒶compact	𝒶compact	NOUN
easat-1685	205	10	space	space	NOUN
easat-1685	205	11	is	be	AUX
easat-1685	205	12	strong	strong	ADJ
easat-1685	205	13	star	star	NOUN
easat-1685	205	14	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	205	15	space	space	NOUN
easat-1685	205	16	.	.	PUNCT
easat-1685	206	1	poof	poof	NOUN
easat-1685	206	2	:	:	PUNCT
easat-1685	206	3	let	let	VERB
easat-1685	206	4	σ	σ	PRON
easat-1685	206	5	-	-	PUNCT
easat-1685	206	6	strongly	strongly	ADV
easat-1685	206	7	star	star	NOUN
easat-1685	206	8	𝒶compact	𝒶compact	NOUN
easat-1685	206	9	space	space	NOUN
easat-1685	206	10	.suppose	.suppose	PUNCT
easat-1685	206	11	that	that	SCONJ
easat-1685	206	12	𝑌=	𝑌=	PROPN
easat-1685	206	13	⋃𝑛∈𝑁	⋃𝑛∈𝑁	PROPN
easat-1685	206	14	𝑌𝑛	𝑌𝑛	PROPN
easat-1685	206	15	,	,	PUNCT
easat-1685	206	16	where	where	SCONJ
easat-1685	206	17	each	each	DET
easat-1685	206	18	𝑌𝑛	𝑌𝑛	PROPN
easat-1685	206	19	is	be	AUX
easat-1685	206	20	strongly	strongly	ADV
easat-1685	206	21	star	star	NOUN
easat-1685	206	22	𝒶compact	𝒶compact	NOUN
easat-1685	206	23	.	.	PUNCT
easat-1685	207	1	let	let	VERB
easat-1685	207	2	𝑌1	𝑌1	PROPN
easat-1685	207	3	⊃	⊃	PROPN
easat-1685	207	4	𝑌2	𝑌2	X
easat-1685	207	5	⊃	⊃	PROPN
easat-1685	207	6	.	.	PUNCT
easat-1685	207	7	.	.	PUNCT
easat-1685	207	8	.	.	PUNCT
easat-1685	208	1	⊃	⊃	PROPN
easat-1685	208	2	𝑌𝑛	𝑌𝑛	PROPN
easat-1685	208	3	⊃	⊃	PROPN
easat-1685	208	4	.	.	PUNCT
easat-1685	208	5	.	.	PUNCT
easat-1685	208	6	.	.	PUNCT
easat-1685	209	1	,	,	PUNCT
easat-1685	209	2	since	since	SCONJ
easat-1685	209	3	the	the	DET
easat-1685	209	4	union	union	NOUN
easat-1685	209	5	of	of	ADP
easat-1685	209	6	finitely	finitely	ADV
easat-1685	209	7	many	many	ADJ
easat-1685	209	8	strongly	strongly	ADV
easat-1685	209	9	star	star	ADJ
easat-1685	209	10	𝒶	𝒶	PROPN
easat-1685	209	11	277	277	NUM
easat-1685	209	12	edelweiss	edelweiss	PROPN
easat-1685	209	13	applied	apply	VERB
easat-1685	209	14	science	science	NOUN
easat-1685	209	15	and	and	CCONJ
easat-1685	209	16	technology	technology	NOUN
easat-1685	209	17	issn	issn	PROPN
easat-1685	209	18	:	:	PUNCT
easat-1685	209	19	2576	2576	NUM
easat-1685	209	20	-	-	SYM
easat-1685	209	21	8484	8484	NUM
easat-1685	209	22	vol	vol	NOUN
easat-1685	209	23	.	.	PROPN
easat-1685	209	24	8	8	NUM
easat-1685	209	25	,	,	PUNCT
easat-1685	209	26	no	no	INTJ
easat-1685	209	27	.	.	NOUN
easat-1685	209	28	5	5	NUM
easat-1685	209	29	:	:	PUNCT
easat-1685	209	30	271	271	NUM
easat-1685	209	31	-	-	SYM
easat-1685	209	32	277	277	NUM
easat-1685	209	33	,	,	PUNCT
easat-1685	209	34	2024	2024	NUM
easat-1685	209	35	doi	doi	NOUN
easat-1685	209	36	:	:	PUNCT
easat-1685	209	37	10.55214/25768484.v8i5.1685	10.55214/25768484.v8i5.1685	NUM
easat-1685	210	1	©	©	ADP
easat-1685	210	2	2024	2024	NUM
easat-1685	210	3	by	by	ADP
easat-1685	210	4	the	the	DET
easat-1685	210	5	authors	author	NOUN
easat-1685	210	6	;	;	PUNCT
easat-1685	210	7	licensee	licensee	PROPN
easat-1685	210	8	learning	learn	VERB
easat-1685	210	9	gate	gate	PROPN
easat-1685	210	10	compact	compact	ADJ
easat-1685	210	11	spaces	space	NOUN
easat-1685	210	12	remains	remain	VERB
easat-1685	210	13	strongly	strongly	ADV
easat-1685	210	14	star	star	NOUN
easat-1685	210	15	𝒶compact	𝒶compact	NOUN
easat-1685	210	16	.	.	PUNCT
easat-1685	211	1	let	let	VERB
easat-1685	211	2	(	(	PUNCT
easat-1685	211	3	𝑌𝑛	𝑌𝑛	NOUN
easat-1685	211	4	:	:	PUNCT
easat-1685	211	5	𝑛	𝑛	PRON
easat-1685	211	6	∈	∈	PROPN
easat-1685	211	7	𝑁	𝑁	PROPN
easat-1685	211	8	)	)	PUNCT
easat-1685	211	9	be	be	VERB
easat-1685	211	10	a	a	DET
easat-1685	211	11	sequence	sequence	NOUN
easat-1685	211	12	of	of	ADP
easat-1685	211	13	𝒶-open	𝒶-open	NOUN
easat-1685	211	14	cover	cover	NOUN
easat-1685	211	15	of	of	ADP
easat-1685	211	16	𝑌	𝑌	PROPN
easat-1685	211	17	.for	.for	ADP
easat-1685	211	18	each	each	PRON
easat-1685	211	19	𝑛	𝑛	PRON
easat-1685	211	20	∈	∈	NOUN
easat-1685	211	21	𝑁	𝑁	PROPN
easat-1685	211	22	let	let	VERB
easat-1685	211	23	𝐴𝑛	𝐴𝑛	NOUN
easat-1685	211	24	be	be	AUX
easat-1685	211	25	a	a	DET
easat-1685	211	26	finite	finite	NOUN
easat-1685	211	27	subset	subset	NOUN
easat-1685	211	28	of	of	ADP
easat-1685	211	29	𝑌𝑛	𝑌𝑛	PRON
easat-1685	211	30	such	such	ADJ
easat-1685	211	31	that	that	SCONJ
easat-1685	211	32	𝑆𝑡𝒶(𝐴𝑛	𝑆𝑡𝒶(𝐴𝑛	PROPN
easat-1685	211	33	,	,	PUNCT
easat-1685	211	34	𝑌𝑛	𝑌𝑛	PROPN
easat-1685	211	35	)	)	PUNCT
easat-1685	211	36	⊃	⊃	PROPN
easat-1685	211	37	𝑌𝑛.it	𝑌𝑛.it	PROPN
easat-1685	211	38	follows	follow	VERB
easat-1685	211	39	that	that	SCONJ
easat-1685	211	40	each	each	DET
easat-1685	211	41	point	point	NOUN
easat-1685	211	42	of	of	ADP
easat-1685	211	43	𝑌	𝑌	PROPN
easat-1685	211	44	belongs	belong	VERB
easat-1685	211	45	to	to	ADP
easat-1685	211	46	all	all	DET
easat-1685	211	47	but	but	ADV
easat-1685	211	48	finitely	finitely	ADV
easat-1685	211	49	many	many	ADJ
easat-1685	211	50	sets	set	NOUN
easat-1685	211	51	of	of	ADP
easat-1685	211	52	𝑆𝑡𝒶(𝐴𝑛	𝑆𝑡𝒶(𝐴𝑛	PROPN
easat-1685	211	53	,	,	PUNCT
easat-1685	211	54	𝑌𝑛	𝑌𝑛	PROPN
easat-1685	211	55	)	)	PUNCT
easat-1685	211	56	.by	.by	PROPN
easat-1685	212	1	the	the	DET
easat-1685	212	2	sequence	sequence	NOUN
easat-1685	212	3	(	(	PUNCT
easat-1685	212	4	𝐴𝑛	𝐴𝑛	NOUN
easat-1685	212	5	:	:	PUNCT
easat-1685	212	6	𝑛	𝑛	DET
easat-1685	212	7	∈	∈	PROPN
easat-1685	212	8	𝑁	𝑁	PROPN
easat-1685	212	9	)	)	PUNCT
easat-1685	212	10	we	we	PRON
easat-1685	212	11	have	have	VERB
easat-1685	212	12	𝑌	𝑌	PROPN
easat-1685	212	13	is	be	AUX
easat-1685	212	14	strong	strong	ADJ
easat-1685	212	15	star	star	NOUN
easat-1685	212	16	𝒶hurewicz	𝒶hurewicz	ADJ
easat-1685	212	17	space	space	NOUN
easat-1685	212	18	.	.	PUNCT
easat-1685	213	1	copyright	copyright	NOUN
easat-1685	213	2	:	:	PUNCT
easat-1685	213	3	©	©	PROPN
easat-1685	213	4	2024	2024	NUM
easat-1685	213	5	by	by	ADP
easat-1685	213	6	the	the	DET
easat-1685	213	7	authors	author	NOUN
easat-1685	213	8	.	.	PUNCT
easat-1685	214	1	this	this	DET
easat-1685	214	2	article	article	NOUN
easat-1685	214	3	is	be	AUX
easat-1685	214	4	an	an	DET
easat-1685	214	5	open	open	ADJ
easat-1685	214	6	access	access	NOUN
easat-1685	214	7	article	article	NOUN
easat-1685	214	8	distributed	distribute	VERB
easat-1685	214	9	under	under	ADP
easat-1685	214	10	the	the	DET
easat-1685	214	11	terms	term	NOUN
easat-1685	214	12	and	and	CCONJ
easat-1685	214	13	conditions	condition	NOUN
easat-1685	214	14	of	of	ADP
easat-1685	214	15	the	the	DET
easat-1685	214	16	creative	creative	ADJ
easat-1685	214	17	commons	common	NOUN
easat-1685	214	18	attribution	attribution	NOUN
easat-1685	214	19	(	(	PUNCT
easat-1685	214	20	cc	cc	NOUN
easat-1685	214	21	by	by	ADP
easat-1685	214	22	)	)	PUNCT
easat-1685	214	23	license	license	NOUN
easat-1685	214	24	(	(	PUNCT
easat-1685	214	25	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-1685	214	26	)	)	PUNCT
easat-1685	214	27	.	.	PUNCT
easat-1685	215	1	references	reference	NOUN
easat-1685	215	2	[	[	X
easat-1685	215	3	1	1	X
easat-1685	215	4	]	]	X
easat-1685	215	5	miller	miller	PROPN
easat-1685	215	6	a.	a.	PROPN
easat-1685	215	7	and	and	CCONJ
easat-1685	215	8	fremlin	fremlin	PROPN
easat-1685	215	9	d.	d.	PROPN
easat-1685	215	10	,	,	PUNCT
easat-1685	215	11	"	"	PUNCT
easat-1685	215	12	on	on	ADP
easat-1685	215	13	some	some	DET
easat-1685	215	14	properties	property	NOUN
easat-1685	215	15	of	of	ADP
easat-1685	215	16	hurewicz	hurewicz	NOUN
easat-1685	215	17	,	,	PUNCT
easat-1685	215	18	menger	menger	PROPN
easat-1685	215	19	,	,	PUNCT
easat-1685	215	20	and	and	CCONJ
easat-1685	215	21	rothboerger	rothboerger	NOUN
easat-1685	215	22	"	"	PUNCT
easat-1685	215	23	,	,	PUNCT
easat-1685	215	24	fundamental	fundamental	ADJ
easat-1685	215	25	math	math	NOUN
easat-1685	215	26	.	.	PUNCT
easat-1685	216	1	,	,	PUNCT
easat-1685	216	2	vol	vol	NOUN
easat-1685	216	3	.	.	PROPN
easat-1685	217	1	129	129	NUM
easat-1685	217	2	(	(	PUNCT
easat-1685	217	3	1	1	NUM
easat-1685	217	4	)	)	PUNCT
easat-1685	217	5	,	,	PUNCT
easat-1685	217	6	1988	1988	NUM
easat-1685	217	7	,	,	PUNCT
easat-1685	217	8	17	17	NUM
easat-1685	217	9	-	-	SYM
easat-1685	217	10	33	33	NUM
easat-1685	217	11	.	.	PUNCT
easat-1685	218	1	[	[	X
easat-1685	218	2	2	2	X
easat-1685	218	3	]	]	X
easat-1685	218	4	sumit	sumit	PROPN
easat-1685	218	5	s.	s.	PROPN
easat-1685	218	6	and	and	CCONJ
easat-1685	218	7	kocinac	kocinac	PROPN
easat-1685	218	8	l.	l.	PROPN
easat-1685	218	9	,	,	PUNCT
easat-1685	218	10	"	"	PUNCT
easat-1685	218	11	star	star	NOUN
easat-1685	218	12	versions	version	NOUN
easat-1685	218	13	hurewicz	hurewicz	PROPN
easat-1685	218	14	spaces	space	VERB
easat-1685	218	15	"	"	PUNCT
easat-1685	218	16	,	,	PUNCT
easat-1685	218	17	hacettepe	hacettepe	ADJ
easat-1685	218	18	journal	journal	NOUN
easat-1685	218	19	of	of	ADP
easat-1685	218	20	mathematics	mathematic	NOUN
easat-1685	218	21	and	and	CCONJ
easat-1685	218	22	statistics	statistic	NOUN
easat-1685	218	23	,	,	PUNCT
easat-1685	218	24	vol	vol	NOUN
easat-1685	218	25	.	.	PUNCT
easat-1685	218	26	50(5	50(5	NUM
easat-1685	218	27	)	)	PUNCT
easat-1685	218	28	,	,	PUNCT
easat-1685	218	29	2021	2021	NUM
easat-1685	218	30	,	,	PUNCT
easat-1685	218	31	1325	1325	NUM
easat-1685	218	32	-	-	SYM
easat-1685	218	33	1333	1333	NUM
easat-1685	218	34	.	.	PUNCT
easat-1685	219	1	[	[	X
easat-1685	219	2	3	3	NUM
easat-1685	219	3	]	]	X
easat-1685	219	4	memet	memet	PROPN
easat-1685	219	5	k.	k.	PROPN
easat-1685	219	6	,	,	PUNCT
easat-1685	219	7	"	"	PUNCT
easat-1685	219	8	𝛽-menger	𝛽-menger	NOUN
easat-1685	219	9	and	and	CCONJ
easat-1685	219	10	𝛽-hurewicz	𝛽-hurewicz	ADJ
easat-1685	219	11	spaces	space	NOUN
easat-1685	219	12	"	"	PUNCT
easat-1685	219	13	,	,	PUNCT
easat-1685	219	14	hacettepe	hacettepe	ADJ
easat-1685	219	15	journal	journal	NOUN
easat-1685	219	16	of	of	ADP
easat-1685	219	17	mathematics	mathematic	NOUN
easat-1685	219	18	and	and	CCONJ
easat-1685	219	19	statistics	statistic	NOUN
easat-1685	219	20	,	,	PUNCT
easat-1685	219	21	vol	vol	NOUN
easat-1685	219	22	.	.	PROPN
easat-1685	219	23	51	51	NUM
easat-1685	219	24	(	(	PUNCT
easat-1685	219	25	1	1	NUM
easat-1685	219	26	)	)	PUNCT
easat-1685	219	27	,	,	PUNCT
easat-1685	219	28	2022	2022	NUM
easat-1685	219	29	,	,	PUNCT
easat-1685	219	30	1	1	NUM
easat-1685	219	31	-	-	SYM
easat-1685	219	32	7	7	NUM
easat-1685	219	33	.	.	PUNCT
easat-1685	220	1	[	[	X
easat-1685	220	2	4	4	NUM
easat-1685	220	3	]	]	X
easat-1685	220	4	alexander	alexander	PROPN
easat-1685	220	5	v.	v.	PROPN
easat-1685	220	6	,	,	PUNCT
easat-1685	220	7	"	"	PUNCT
easat-1685	220	8	a	a	DET
easat-1685	220	9	functional	functional	ADJ
easat-1685	220	10	characterization	characterization	NOUN
easat-1685	220	11	of	of	ADP
easat-1685	220	12	the	the	DET
easat-1685	220	13	hurewicz	hurewicz	NOUN
easat-1685	220	14	property	property	NOUN
easat-1685	220	15	"	"	PUNCT
easat-1685	220	16	,	,	PUNCT
easat-1685	220	17	iranian	iranian	ADJ
easat-1685	220	18	journal	journal	PROPN
easat-1685	220	19	of	of	ADP
easat-1685	220	20	mathematical	mathematical	ADJ
easat-1685	220	21	sciences	sciences	PROPN
easat-1685	220	22	and	and	CCONJ
easat-1685	220	23	informatics	informatic	NOUN
easat-1685	220	24	,	,	PUNCT
easat-1685	220	25	vol	vol	NOUN
easat-1685	220	26	.	.	PUNCT
easat-1685	220	27	17(1	17(1	NUM
easat-1685	220	28	)	)	PUNCT
easat-1685	220	29	,	,	PUNCT
easat-1685	220	30	2022	2022	NUM
easat-1685	220	31	,	,	PUNCT
easat-1685	220	32	99	99	NUM
easat-1685	220	33	-109	-109	NUM
easat-1685	220	34	.	.	PUNCT
easat-1685	221	1	[	[	X
easat-1685	221	2	5	5	X
easat-1685	221	3	]	]	X
easat-1685	221	4	kumar	kumar	PROPN
easat-1685	221	5	g.	g.	PROPN
easat-1685	221	6	,	,	PUNCT
easat-1685	221	7	mittal	mittal	PROPN
easat-1685	221	8	s.	s.	PROPN
easat-1685	221	9	and	and	CCONJ
easat-1685	221	10	brijk	brijk	NOUN
easat-1685	221	11	t.	t.	PROPN
easat-1685	221	12	,	,	PUNCT
easat-1685	221	13	"	"	PUNCT
easat-1685	221	14	on	on	ADP
easat-1685	221	15	𝜃-hurewicz	𝜃-hurewicz	NOUN
easat-1685	221	16	and	and	CCONJ
easat-1685	221	17	𝛼-hurewicz	𝛼-hurewicz	VERB
easat-1685	221	18	topological	topological	ADJ
easat-1685	221	19	spaces	space	NOUN
easat-1685	221	20	"	"	PUNCT
easat-1685	221	21	,	,	PUNCT
easat-1685	221	22	arxiv:2307.00487	arxiv:2307.00487	X
easat-1685	221	23	[	[	X
easat-1685	221	24	math.gn	math.gn	X
easat-1685	221	25	]	]	X
easat-1685	221	26	,	,	PUNCT
easat-1685	221	27	2023	2023	NUM
easat-1685	221	28	,	,	PUNCT
easat-1685	221	29	1	1	NUM
easat-1685	221	30	-14	-14	NUM
easat-1685	221	31	.	.	PUNCT
easat-1685	222	1	[	[	X
easat-1685	222	2	6	6	NUM
easat-1685	222	3	]	]	PUNCT
easat-1685	222	4	njastad	njastad	NOUN
easat-1685	222	5	o.	o.	PROPN
easat-1685	222	6	,	,	PUNCT
easat-1685	222	7	"	"	PUNCT
easat-1685	222	8	a	a	DET
easat-1685	222	9	note	note	NOUN
easat-1685	222	10	on	on	ADP
easat-1685	222	11	𝒶-open	𝒶-open	NOUN
easat-1685	222	12	sets	set	NOUN
easat-1685	222	13	and	and	CCONJ
easat-1685	222	14	𝑒∗-open	𝑒∗-open	ADJ
easat-1685	222	15	sets	set	NOUN
easat-1685	222	16	"	"	PUNCT
easat-1685	222	17	,	,	PUNCT
easat-1685	222	18	faculty	faculty	NOUN
easat-1685	222	19	of	of	ADP
easat-1685	222	20	sciences	science	NOUN
easat-1685	222	21	and	and	CCONJ
easat-1685	222	22	mathematics	mathematic	NOUN
easat-1685	222	23	,	,	PUNCT
easat-1685	222	24	university	university	PROPN
easat-1685	222	25	of	of	ADP
easat-1685	222	26	nis	nis	PROPN
easat-1685	222	27	,	,	PUNCT
easat-1685	222	28	serbia	serbia	PROPN
easat-1685	222	29	,	,	PUNCT
easat-1685	222	30	vol	vol	NOUN
easat-1685	222	31	.	.	PUNCT
easat-1685	222	32	22(1	22(1	NUM
easat-1685	222	33	)	)	PUNCT
easat-1685	222	34	,	,	PUNCT
easat-1685	222	35	2008	2008	NUM
easat-1685	222	36	,	,	PUNCT
easat-1685	222	37	89	89	NUM
easat-1685	222	38	-	-	SYM
easat-1685	222	39	96	96	NUM
easat-1685	222	40	.	.	PUNCT
easat-1685	223	1	[	[	X
easat-1685	223	2	7	7	NUM
easat-1685	223	3	]	]	X
easat-1685	223	4	ekici	ekici	PROPN
easat-1685	223	5	e.	e.	PROPN
easat-1685	223	6	,	,	PUNCT
easat-1685	223	7	"	"	PUNCT
easat-1685	223	8	on	on	ADP
easat-1685	223	9	𝒶-open	𝒶-open	NOUN
easat-1685	223	10	sets	set	NOUN
easat-1685	223	11	"	"	PUNCT
easat-1685	223	12	,	,	PUNCT
easat-1685	223	13	𝒜∗-sets	𝒜∗-set	NOUN
easat-1685	223	14	and	and	CCONJ
easat-1685	223	15	decompositions	decomposition	NOUN
easat-1685	223	16	of	of	ADP
easat-1685	223	17	continuity	continuity	NOUN
easat-1685	223	18	and	and	CCONJ
easat-1685	223	19	super	super	NOUN
easat-1685	223	20	-	-	NOUN
easat-1685	223	21	continuity	continuity	NOUN
easat-1685	223	22	,	,	PUNCT
easat-1685	223	23	submitted	submit	VERB
easat-1685	223	24	,	,	PUNCT
easat-1685	223	25	2008	2008	NUM
easat-1685	223	26	.	.	PUNCT
easat-1685	224	1	[	[	X
easat-1685	224	2	8	8	NUM
easat-1685	224	3	]	]	X
easat-1685	224	4	ekici	ekici	PROPN
easat-1685	224	5	e.	e.	PROPN
easat-1685	224	6	,	,	PUNCT
easat-1685	224	7	"	"	PUNCT
easat-1685	224	8	on	on	ADP
easat-1685	224	9	𝑒∗-open	𝑒∗-open	ADJ
easat-1685	224	10	sets	set	NOUN
easat-1685	224	11	and	and	CCONJ
easat-1685	224	12	(	(	PUNCT
easat-1685	224	13	𝒟	𝒟	NOUN
easat-1685	224	14	,	,	PUNCT
easat-1685	224	15	𝒮)∗-sets	𝒮)∗-set	NOUN
easat-1685	224	16	"	"	PUNCT
easat-1685	224	17	math	math	NOUN
easat-1685	224	18	.	.	PUNCT
easat-1685	225	1	moravica	moravica	PROPN
easat-1685	225	2	,	,	PUNCT
easat-1685	225	3	vol	vol	NOUN
easat-1685	225	4	.	.	PUNCT
easat-1685	226	1	13(1	13(1	NUM
easat-1685	226	2	)	)	PUNCT
easat-1685	226	3	,	,	PUNCT
easat-1685	226	4	2009	2009	NUM
easat-1685	226	5	,	,	PUNCT
easat-1685	226	6	29	29	NUM
easat-1685	226	7	-	-	SYM
easat-1685	226	8	36	36	NUM
easat-1685	226	9	.	.	PUNCT
easat-1685	227	1	[	[	X
easat-1685	227	2	9	9	NUM
easat-1685	227	3	]	]	X
easat-1685	227	4	mohammad	mohammad	PROPN
easat-1685	227	5	r.	r.	PROPN
easat-1685	227	6	,	,	PUNCT
easat-1685	227	7	"	"	PUNCT
easat-1685	227	8	delta	delta	NOUN
easat-1685	227	9	-	-	PUNCT
easat-1685	227	10	open	open	ADJ
easat-1685	227	11	sets	set	NOUN
easat-1685	227	12	and	and	CCONJ
easat-1685	227	13	delta	delta	NOUN
easat-1685	227	14	-continuous	-continuous	ADJ
easat-1685	227	15	functions	function	NOUN
easat-1685	227	16	"	"	PUNCT
easat-1685	227	17	,	,	PUNCT
easat-1685	227	18	international	international	ADJ
easat-1685	227	19	journal	journal	NOUN
easat-1685	227	20	of	of	ADP
easat-1685	227	21	pure	pure	ADJ
easat-1685	227	22	and	and	CCONJ
easat-1685	227	23	mathematics	mathematic	NOUN
easat-1685	227	24	,	,	PUNCT
easat-1685	227	25	vol	vol	NOUN
easat-1685	227	26	.	.	PUNCT
easat-1685	227	27	8(1	8(1	NOUN
easat-1685	227	28	)	)	PUNCT
easat-1685	227	29	,	,	PUNCT
easat-1685	227	30	2021	2021	NUM
easat-1685	227	31	,	,	PUNCT
easat-1685	227	32	123	123	NUM
easat-1685	227	33	.	.	PUNCT
easat-1685	228	1	[	[	X
easat-1685	228	2	10	10	NUM
easat-1685	228	3	]	]	X
easat-1685	228	4	jacobson	jacobson	PROPN
easat-1685	228	5	n.	n.	PROPN
easat-1685	228	6	,	,	PUNCT
easat-1685	228	7	"	"	PUNCT
easat-1685	228	8	basic	basic	ADJ
easat-1685	228	9	algebra	algebra	NOUN
easat-1685	228	10	"	"	PUNCT
easat-1685	228	11	,	,	PUNCT
easat-1685	228	12	(	(	PUNCT
easat-1685	228	13	https://en.m.wikipedia.org/wiki/nathan_jacobson	https://en.m.wikipedia.org/wiki/nathan_jacobson	PROPN
easat-1685	228	14	)	)	PUNCT
easat-1685	228	15	,	,	PUNCT
easat-1685	228	16	vol	vol	NOUN
easat-1685	228	17	.	.	PROPN
easat-1685	228	18	2	2	NUM
easat-1685	228	19	(	(	PUNCT
easat-1685	228	20	2nd	2nd	ADJ
easat-1685	228	21	ed	ed	NOUN
easat-1685	228	22	.	.	PUNCT
easat-1685	228	23	)	)	PUNCT
easat-1685	228	24	,	,	PUNCT
easat-1685	228	25	dover	dover	PROPN
easat-1685	228	26	,	,	PUNCT
easat-1685	228	27	isbn	isbn	ADJ
easat-1685	228	28	(	(	PUNCT
easat-1685	228	29	https://en.m.wikipedia.org/wiki/isbn_(identifier	https://en.m.wikipedia.org/wiki/isbn_(identifier	NOUN
easat-1685	228	30	)	)	PUNCT
easat-1685	228	31	)	)	PUNCT
easat-1685	228	32	2009	2009	NUM
easat-1685	228	33	.	.	PUNCT
easat-1685	229	1	[	[	X
easat-1685	229	2	11	11	NUM
easat-1685	229	3	]	]	PUNCT
easat-1685	229	4	ameen	ameen	PROPN
easat-1685	229	5	za	za	PROPN
easat-1685	229	6	,	,	PUNCT
easat-1685	229	7	asaad	asaad	PROPN
easat-1685	229	8	ba	ba	PROPN
easat-1685	229	9	,	,	PUNCT
easat-1685	229	10	muhammed	muhamme	VERB
easat-1685	229	11	ra	ra	PROPN
easat-1685	229	12	.	.	PUNCT
easat-1685	230	1	on	on	ADP
easat-1685	230	2	superclasses	superclass	NOUN
easat-1685	230	3	of	of	ADP
easat-1685	230	4	$	$	SYM
easat-1685	230	5	\delta	\delta	PROPN
easat-1685	230	6	$	$	SYM
easat-1685	230	7	-open	-open	ADJ
easat-1685	230	8	sets	set	NOUN
easat-1685	230	9	in	in	ADP
easat-1685	230	10	topological	topological	ADJ
easat-1685	230	11	spaces	space	NOUN
easat-1685	230	12	.	.	PUNCT
easat-1685	231	1	international	international	ADJ
easat-1685	231	2	journal	journal	PROPN
easat-1685	231	3	of	of	ADP
easat-1685	231	4	applied	apply	VERB
easat-1685	231	5	mathematics	mathematic	NOUN
easat-1685	231	6	.	.	PUNCT
easat-1685	232	1	2019;32(2):259	2019;32(2):259	NOUN
easat-1685	232	2	.	.	PUNCT
easat-1685	233	1	[	[	X
easat-1685	233	2	12	12	NUM
easat-1685	233	3	]	]	PUNCT
easat-1685	233	4	kocinac	kocinac	PROPN
easat-1685	233	5	ld	ld	PROPN
easat-1685	233	6	,	,	PUNCT
easat-1685	233	7	sabah	sabah	PROPN
easat-1685	233	8	a	a	PROPN
easat-1685	233	9	,	,	PUNCT
easat-1685	233	10	khan	khan	PROPN
easat-1685	233	11	mu	mu	PROPN
easat-1685	233	12	,	,	PUNCT
easat-1685	233	13	seba	seba	PROPN
easat-1685	233	14	d.	d.	PROPN
easat-1685	233	15	semi	semi	PROPN
easat-1685	233	16	-	-	ADJ
easat-1685	233	17	hurewicz	hurewicz	ADJ
easat-1685	233	18	spaces	space	NOUN
easat-1685	233	19	.	.	PUNCT
easat-1685	234	1	hacettepe	hacettepe	PROPN
easat-1685	234	2	journal	journal	PROPN
easat-1685	234	3	of	of	ADP
easat-1685	234	4	mathematics	mathematic	NOUN
easat-1685	234	5	and	and	CCONJ
easat-1685	234	6	statistics	statistic	NOUN
easat-1685	234	7	.	.	PUNCT
easat-1685	235	1	2017	2017	NUM
easat-1685	235	2	jan	jan	NOUN
easat-1685	235	3	1;46(1):53	1;46(1):53	NUM
easat-1685	235	4	-	-	NUM
easat-1685	235	5	66	66	NUM
easat-1685	235	6	.	.	PUNCT
easat-1685	236	1	[	[	X
easat-1685	236	2	13	13	NUM
easat-1685	236	3	]	]	X
easat-1685	236	4	bal	bal	PROPN
easat-1685	236	5	p	p	X
easat-1685	236	6	,	,	PUNCT
easat-1685	236	7	de	de	PROPN
easat-1685	236	8	r.	r.	PROPN
easat-1685	236	9	on	on	ADP
easat-1685	236	10	strongly	strongly	ADV
easat-1685	236	11	star	star	ADJ
easat-1685	236	12	semi	semi	ADJ
easat-1685	236	13	-	-	NOUN
easat-1685	236	14	compactness	compactness	NOUN
easat-1685	236	15	of	of	ADP
easat-1685	236	16	topological	topological	ADJ
easat-1685	236	17	spaces	space	NOUN
easat-1685	236	18	.	.	PUNCT
easat-1685	237	1	khayyam	khayyam	PROPN
easat-1685	237	2	journal	journal	PROPN
easat-1685	237	3	of	of	ADP
easat-1685	237	4	mathematics	mathematic	NOUN
easat-1685	237	5	.	.	PUNCT
easat-1685	238	1	2023	2023	NUM
easat-1685	238	2	jan	jan	NOUN
easat-1685	238	3	1;9(1):54	1;9(1):54	NUM
easat-1685	238	4	-	-	SYM
easat-1685	238	5	60	60	NUM
easat-1685	238	6	.	.	PUNCT
easat-1685	239	1	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN