id	sid	tid	token	lemma	pos
ajird-1325	1	1	american	american	PROPN
ajird-1325	1	2	journal	journal	PROPN
ajird-1325	1	3	of	of	ADP
ajird-1325	1	4	interdisciplinary	interdisciplinary	ADJ
ajird-1325	1	5	research	research	NOUN
ajird-1325	1	6	and	and	CCONJ
ajird-1325	1	7	development	development	NOUN
ajird-1325	1	8	issn	issn	PROPN
ajird-1325	1	9	online	online	NOUN
ajird-1325	1	10	:	:	PUNCT
ajird-1325	1	11	2771	2771	NUM
ajird-1325	1	12	-	-	SYM
ajird-1325	1	13	8948	8948	NUM
ajird-1325	1	14	website	website	NOUN
ajird-1325	1	15	:	:	PUNCT
ajird-1325	1	16	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-1325	1	17	volume	volume	NOUN
ajird-1325	1	18	33	33	NUM
ajird-1325	1	19	,	,	PUNCT
ajird-1325	1	20	october	october	PROPN
ajird-1325	1	21	2024	2024	NUM
ajird-1325	1	22	111	111	NUM
ajird-1325	1	23	|	|	ADV
ajird-1325	1	24	p	p	NOUN
ajird-1325	1	25	a	a	DET
ajird-1325	1	26	g	g	NOUN
ajird-1325	1	27	e	e	NOUN
ajird-1325	1	28	on	on	ADP
ajird-1325	1	29	τ	τ	PROPN
ajird-1325	1	30	-	-	PUNCT
ajird-1325	1	31	bounded	bound	VERB
ajird-1325	1	32	spaces	space	NOUN
ajird-1325	1	33	karimov	karimov	PROPN
ajird-1325	1	34	sardor	sardor	PROPN
ajird-1325	1	35	yashinovich	yashinovich	PROPN
ajird-1325	1	36	assistant	assistant	NOUN
ajird-1325	1	37	at	at	ADP
ajird-1325	1	38	the	the	DET
ajird-1325	1	39	department	department	NOUN
ajird-1325	1	40	of	of	ADP
ajird-1325	1	41	mathematics	mathematic	NOUN
ajird-1325	1	42	and	and	CCONJ
ajird-1325	1	43	natural	natural	ADJ
ajird-1325	1	44	sciences	science	NOUN
ajird-1325	1	45	,	,	PUNCT
ajird-1325	1	46	almalyk	almalyk	NOUN
ajird-1325	1	47	branch	branch	NOUN
ajird-1325	1	48	,	,	PUNCT
ajird-1325	1	49	tashkent	tashkent	PROPN
ajird-1325	1	50	state	state	PROPN
ajird-1325	1	51	technical	technical	PROPN
ajird-1325	1	52	university	university	NOUN
ajird-1325	1	53	email	email	NOUN
ajird-1325	1	54	:	:	PUNCT
ajird-1325	1	55	mr_man89@mail.ru	mr_man89@mail.ru	ADV
ajird-1325	1	56	abstract	abstract	ADJ
ajird-1325	1	57	:	:	PUNCT
ajird-1325	1	58	in	in	ADP
ajird-1325	1	59	this	this	DET
ajird-1325	1	60	article	article	NOUN
ajird-1325	1	61	,	,	PUNCT
ajird-1325	1	62	it	it	PRON
ajird-1325	1	63	has	have	AUX
ajird-1325	1	64	been	be	AUX
ajird-1325	1	65	proven	prove	VERB
ajird-1325	1	66	that	that	SCONJ
ajird-1325	1	67	τ	τ	PROPN
ajird-1325	1	68	-	-	PUNCT
ajird-1325	1	69	bounded	bound	VERB
ajird-1325	1	70	spaces	space	NOUN
ajird-1325	1	71	,	,	PUNCT
ajird-1325	1	72	if	if	SCONJ
ajird-1325	1	73	they	they	PRON
ajird-1325	1	74	satisfy	satisfy	VERB
ajird-1325	1	75	the	the	DET
ajird-1325	1	76	t2	t2	PROPN
ajird-1325	1	77	(	(	PUNCT
ajird-1325	1	78	hausdorff	hausdorff	NOUN
ajird-1325	1	79	space	space	NOUN
ajird-1325	1	80	)	)	PUNCT
ajird-1325	1	81	condition	condition	NOUN
ajird-1325	1	82	,	,	PUNCT
ajird-1325	1	83	are	be	AUX
ajird-1325	1	84	also	also	ADV
ajird-1325	1	85	t3	t3	NOUN
ajird-1325	1	86	(	(	PUNCT
ajird-1325	1	87	regular	regular	ADJ
ajird-1325	1	88	space	space	NOUN
ajird-1325	1	89	)	)	PUNCT
ajird-1325	1	90	but	but	CCONJ
ajird-1325	1	91	not	not	PART
ajird-1325	1	92	necessarily	necessarily	ADV
ajird-1325	1	93	t4	t4	PROPN
ajird-1325	1	94	(	(	PUNCT
ajird-1325	1	95	normal	normal	ADJ
ajird-1325	1	96	space	space	NOUN
ajird-1325	1	97	)	)	PUNCT
ajird-1325	1	98	,	,	PUNCT
ajird-1325	1	99	with	with	ADP
ajird-1325	1	100	a	a	DET
ajird-1325	1	101	counterexample	counterexample	NOUN
ajird-1325	1	102	provided	provide	VERB
ajird-1325	1	103	.	.	PUNCT
ajird-1325	2	1	additionally	additionally	ADV
ajird-1325	2	2	,	,	PUNCT
ajird-1325	2	3	the	the	DET
ajird-1325	2	4	relationship	relationship	NOUN
ajird-1325	2	5	between	between	ADP
ajird-1325	2	6	local	local	ADJ
ajird-1325	2	7	weak	weak	ADJ
ajird-1325	2	8	density	density	NOUN
ajird-1325	2	9	and	and	CCONJ
ajird-1325	2	10	local	local	ADJ
ajird-1325	2	11	density	density	NOUN
ajird-1325	2	12	has	have	AUX
ajird-1325	2	13	been	be	AUX
ajird-1325	2	14	examined	examine	VERB
ajird-1325	2	15	.	.	PUNCT
ajird-1325	3	1	keywords	keyword	NOUN
ajird-1325	3	2	:	:	PUNCT
ajird-1325	3	3	τ	τ	PROPN
ajird-1325	3	4	-	-	PUNCT
ajird-1325	3	5	bounded	bounded	ADJ
ajird-1325	3	6	space	space	NOUN
ajird-1325	3	7	,	,	PUNCT
ajird-1325	3	8	t2	t2	PROPN
ajird-1325	3	9	(	(	PUNCT
ajird-1325	3	10	hausdorff	hausdorff	NOUN
ajird-1325	3	11	space	space	NOUN
ajird-1325	3	12	)	)	PUNCT
ajird-1325	3	13	,	,	PUNCT
ajird-1325	3	14	t3	t3	PROPN
ajird-1325	3	15	(	(	PUNCT
ajird-1325	3	16	regular	regular	ADJ
ajird-1325	3	17	space	space	NOUN
ajird-1325	3	18	)	)	PUNCT
ajird-1325	3	19	,	,	PUNCT
ajird-1325	3	20	t4	t4	PROPN
ajird-1325	3	21	(	(	PUNCT
ajird-1325	3	22	normal	normal	ADJ
ajird-1325	3	23	space	space	NOUN
ajird-1325	3	24	)	)	PUNCT
ajird-1325	3	25	,	,	PUNCT
ajird-1325	3	26	local	local	ADJ
ajird-1325	3	27	weak	weak	ADJ
ajird-1325	3	28	density	density	NOUN
ajird-1325	3	29	,	,	PUNCT
ajird-1325	3	30	local	local	ADJ
ajird-1325	3	31	density	density	NOUN
ajird-1325	3	32	.	.	PUNCT
ajird-1325	4	1	introduction	introduction	NOUN
ajird-1325	4	2	here	here	ADV
ajird-1325	4	3	τ	τ	PROPN
ajird-1325	4	4	is	be	AUX
ajird-1325	4	5	cardinal	cardinal	ADJ
ajird-1325	4	6	number	number	NOUN
ajird-1325	4	7	.	.	PUNCT
ajird-1325	5	1	definition	definition	NOUN
ajird-1325	5	2	1	1	NUM
ajird-1325	5	3	.	.	PUNCT
ajird-1325	6	1	a	a	DET
ajird-1325	6	2	topological	topological	ADJ
ajird-1325	6	3	space	space	NOUN
ajird-1325	6	4	(	(	PUNCT
ajird-1325	6	5	𝑋	𝑋	PROPN
ajird-1325	6	6	,	,	PUNCT
ajird-1325	6	7	𝜏	𝜏	NOUN
ajird-1325	6	8	)	)	PUNCT
ajird-1325	6	9	is	be	AUX
ajird-1325	6	10	called	call	VERB
ajird-1325	6	11	𝜏-bounded	𝜏-bounde	VERB
ajird-1325	6	12	if	if	SCONJ
ajird-1325	6	13	the	the	DET
ajird-1325	6	14	closure	closure	NOUN
ajird-1325	6	15	of	of	ADP
ajird-1325	6	16	any	any	DET
ajird-1325	6	17	subset	subset	NOUN
ajird-1325	6	18	of	of	ADP
ajird-1325	6	19	the	the	DET
ajird-1325	6	20	space	space	NOUN
ajird-1325	6	21	,	,	PUNCT
ajird-1325	6	22	whose	whose	DET
ajird-1325	6	23	cardinality	cardinality	NOUN
ajird-1325	6	24	does	do	AUX
ajird-1325	6	25	not	not	PART
ajird-1325	6	26	exceed	exceed	VERB
ajird-1325	6	27	𝜏	𝜏	NOUN
ajird-1325	6	28	,	,	PUNCT
ajird-1325	6	29	is	be	AUX
ajird-1325	6	30	compact	compact	ADJ
ajird-1325	6	31	.	.	PUNCT
ajird-1325	7	1	definition	definition	NOUN
ajird-1325	7	2	2	2	NUM
ajird-1325	7	3	.	.	PUNCT
ajird-1325	8	1	in	in	ADP
ajird-1325	8	2	topology	topology	NOUN
ajird-1325	8	3	,	,	PUNCT
ajird-1325	8	4	compactness	compactness	NOUN
ajird-1325	8	5	means	mean	VERB
ajird-1325	8	6	that	that	SCONJ
ajird-1325	8	7	from	from	ADP
ajird-1325	8	8	any	any	DET
ajird-1325	8	9	open	open	ADJ
ajird-1325	8	10	cover	cover	NOUN
ajird-1325	8	11	of	of	ADP
ajird-1325	8	12	a	a	DET
ajird-1325	8	13	set	set	NOUN
ajird-1325	8	14	,	,	PUNCT
ajird-1325	8	15	a	a	DET
ajird-1325	8	16	finite	finite	NOUN
ajird-1325	8	17	subcover	subcover	PROPN
ajird-1325	8	18	can	can	AUX
ajird-1325	8	19	be	be	AUX
ajird-1325	8	20	chosen	choose	VERB
ajird-1325	8	21	.	.	PUNCT
ajird-1325	9	1	definition	definition	NOUN
ajird-1325	9	2	3	3	NUM
ajird-1325	9	3	.	.	PUNCT
ajird-1325	10	1	a	a	DET
ajird-1325	10	2	topological	topological	ADJ
ajird-1325	10	3	space	space	NOUN
ajird-1325	10	4	is	be	AUX
ajird-1325	10	5	called	call	VERB
ajird-1325	10	6	t2	t2	PROPN
ajird-1325	10	7	(	(	PUNCT
ajird-1325	10	8	hausdorff	hausdorff	NOUN
ajird-1325	10	9	space	space	NOUN
ajird-1325	10	10	)	)	PUNCT
ajird-1325	10	11	if	if	SCONJ
ajird-1325	10	12	,	,	PUNCT
ajird-1325	10	13	for	for	ADP
ajird-1325	10	14	any	any	DET
ajird-1325	10	15	two	two	NUM
ajird-1325	10	16	distinct	distinct	ADJ
ajird-1325	10	17	points	point	NOUN
ajird-1325	10	18	,	,	PUNCT
ajird-1325	10	19	there	there	PRON
ajird-1325	10	20	exist	exist	VERB
ajird-1325	10	21	non	non	ADJ
ajird-1325	10	22	-	-	ADJ
ajird-1325	10	23	intersecting	intersecting	ADJ
ajird-1325	10	24	open	open	ADJ
ajird-1325	10	25	sets	set	NOUN
ajird-1325	10	26	separating	separate	VERB
ajird-1325	10	27	them	they	PRON
ajird-1325	10	28	.	.	PUNCT
ajird-1325	11	1	that	that	PRON
ajird-1325	11	2	is	be	AUX
ajird-1325	11	3	,	,	PUNCT
ajird-1325	11	4	for	for	ADP
ajird-1325	11	5	each	each	DET
ajird-1325	11	6	𝑥	𝑥	DET
ajird-1325	11	7	≠	≠	PROPN
ajird-1325	11	8	𝑦	𝑦	NOUN
ajird-1325	11	9	,	,	PUNCT
ajird-1325	11	10	there	there	PRON
ajird-1325	11	11	exist	exist	VERB
ajird-1325	11	12	open	open	ADJ
ajird-1325	11	13	sets	set	NOUN
ajird-1325	11	14	𝑈	𝑈	PROPN
ajird-1325	11	15	and	and	CCONJ
ajird-1325	11	16	𝑉	𝑉	PROPN
ajird-1325	11	17	such	such	ADJ
ajird-1325	11	18	that	that	SCONJ
ajird-1325	11	19	𝑥	𝑥	PROPN
ajird-1325	11	20	∈	∈	PROPN
ajird-1325	11	21	𝑈	𝑈	PROPN
ajird-1325	11	22	,	,	PUNCT
ajird-1325	11	23	𝑦	𝑦	NOUN
ajird-1325	11	24	∈	∈	PROPN
ajird-1325	11	25	𝑉	𝑉	PROPN
ajird-1325	11	26	,	,	PUNCT
ajird-1325	11	27	and	and	CCONJ
ajird-1325	11	28	𝑈	𝑈	PROPN
ajird-1325	11	29	∩	∩	NOUN
ajird-1325	11	30	𝑉	𝑉	NOUN
ajird-1325	11	31	=	=	PUNCT
ajird-1325	11	32	∅.	∅.	NOUN
ajird-1325	11	33	definition	definition	NOUN
ajird-1325	11	34	4	4	NUM
ajird-1325	11	35	.	.	PUNCT
ajird-1325	12	1	a	a	DET
ajird-1325	12	2	topological	topological	ADJ
ajird-1325	12	3	space	space	NOUN
ajird-1325	12	4	is	be	AUX
ajird-1325	12	5	called	call	VERB
ajird-1325	12	6	t3	t3	PROPN
ajird-1325	12	7	(	(	PUNCT
ajird-1325	12	8	regular	regular	ADJ
ajird-1325	12	9	space	space	NOUN
ajird-1325	12	10	)	)	PUNCT
ajird-1325	12	11	if	if	SCONJ
ajird-1325	12	12	,	,	PUNCT
ajird-1325	12	13	for	for	ADP
ajird-1325	12	14	any	any	DET
ajird-1325	12	15	closed	closed	ADJ
ajird-1325	12	16	set	set	NOUN
ajird-1325	12	17	and	and	CCONJ
ajird-1325	12	18	a	a	DET
ajird-1325	12	19	point	point	NOUN
ajird-1325	12	20	not	not	PART
ajird-1325	12	21	belonging	belong	VERB
ajird-1325	12	22	to	to	ADP
ajird-1325	12	23	it	it	PRON
ajird-1325	12	24	,	,	PUNCT
ajird-1325	12	25	there	there	PRON
ajird-1325	12	26	exist	exist	VERB
ajird-1325	12	27	disjoint	disjoint	ADJ
ajird-1325	12	28	open	open	ADJ
ajird-1325	12	29	sets	set	NOUN
ajird-1325	12	30	separating	separate	VERB
ajird-1325	12	31	them	they	PRON
ajird-1325	12	32	.	.	PUNCT
ajird-1325	13	1	that	that	PRON
ajird-1325	13	2	is	be	AUX
ajird-1325	13	3	,	,	PUNCT
ajird-1325	13	4	for	for	ADP
ajird-1325	13	5	each	each	DET
ajird-1325	13	6	𝑥	𝑥	PRON
ajird-1325	13	7	∈	∈	PROPN
ajird-1325	13	8	𝑈	𝑈	PROPN
ajird-1325	13	9	and	and	CCONJ
ajird-1325	13	10	𝐴	𝐴	PROPN
ajird-1325	13	11	⊆	⊆	NUM
ajird-1325	13	12	𝑉	𝑉	PROPN
ajird-1325	13	13	,	,	PUNCT
ajird-1325	13	14	and	and	CCONJ
ajird-1325	13	15	𝑈	𝑈	PROPN
ajird-1325	13	16	∩	∩	ADJ
ajird-1325	13	17	𝑉	𝑉	NOUN
ajird-1325	13	18	=	=	NOUN
ajird-1325	13	19	∅	∅	NOUN
ajird-1325	13	20	and	and	CCONJ
ajird-1325	13	21	𝑈	𝑈	PROPN
ajird-1325	13	22	and	and	CCONJ
ajird-1325	13	23	𝑉	𝑉	PROPN
ajird-1325	13	24	are	be	AUX
ajird-1325	13	25	open	open	ADJ
ajird-1325	13	26	sets	set	NOUN
ajird-1325	13	27	such	such	ADJ
ajird-1325	13	28	that	that	SCONJ
ajird-1325	13	29	𝑉	𝑉	PROPN
ajird-1325	13	30	covers	cover	VERB
ajird-1325	13	31	𝐴.	𝐴.	ADJ
ajird-1325	13	32	definition	definition	NOUN
ajird-1325	13	33	5	5	NUM
ajird-1325	13	34	.	.	PUNCT
ajird-1325	14	1	a	a	DET
ajird-1325	14	2	set	set	VERB
ajird-1325	14	3	𝐷	𝐷	NOUN
ajird-1325	14	4	⊆	⊆	NUM
ajird-1325	14	5	𝑋	𝑋	NOUN
ajird-1325	14	6	is	be	AUX
ajird-1325	14	7	said	say	VERB
ajird-1325	14	8	to	to	PART
ajird-1325	14	9	be	be	AUX
ajird-1325	14	10	dense	dense	ADJ
ajird-1325	14	11	around	around	ADP
ajird-1325	14	12	a	a	DET
ajird-1325	14	13	point	point	NOUN
ajird-1325	14	14	𝑥	𝑥	DET
ajird-1325	14	15	∈	∈	NOUN
ajird-1325	14	16	𝑋	𝑋	NOUN
ajird-1325	14	17	if	if	SCONJ
ajird-1325	14	18	,	,	PUNCT
ajird-1325	14	19	for	for	ADP
ajird-1325	14	20	every	every	DET
ajird-1325	14	21	open	open	ADJ
ajird-1325	14	22	neighborhood	neighborhood	NOUN
ajird-1325	14	23	𝑈	𝑈	PROPN
ajird-1325	14	24	of	of	ADP
ajird-1325	14	25	𝑥	𝑥	PROPN
ajird-1325	14	26	,	,	PUNCT
ajird-1325	14	27	we	we	PRON
ajird-1325	14	28	have	have	VERB
ajird-1325	14	29	𝑈	𝑈	PROPN
ajird-1325	14	30	∩	∩	ADJ
ajird-1325	14	31	𝐷	𝐷	NOUN
ajird-1325	14	32	≠	≠	PROPN
ajird-1325	14	33	∅.	∅.	VERB
ajird-1325	14	34	in	in	ADP
ajird-1325	14	35	other	other	ADJ
ajird-1325	14	36	words	word	NOUN
ajird-1325	14	37	,	,	PUNCT
ajird-1325	14	38	a	a	DET
ajird-1325	14	39	set	set	NOUN
ajird-1325	14	40	is	be	AUX
ajird-1325	14	41	locally	locally	ADV
ajird-1325	14	42	dense	dense	ADJ
ajird-1325	14	43	if	if	SCONJ
ajird-1325	14	44	every	every	DET
ajird-1325	14	45	open	open	ADJ
ajird-1325	14	46	set	set	NOUN
ajird-1325	14	47	contains	contain	VERB
ajird-1325	14	48	at	at	ADV
ajird-1325	14	49	least	least	ADV
ajird-1325	14	50	one	one	NUM
ajird-1325	14	51	point	point	NOUN
ajird-1325	14	52	from	from	ADP
ajird-1325	14	53	the	the	DET
ajird-1325	14	54	dense	dense	ADJ
ajird-1325	14	55	set	set	NOUN
ajird-1325	14	56	.	.	PUNCT
ajird-1325	15	1	definition	definition	NOUN
ajird-1325	15	2	6	6	NUM
ajird-1325	15	3	.	.	PUNCT
ajird-1325	16	1	a	a	DET
ajird-1325	16	2	set	set	NOUN
ajird-1325	16	3	𝐴	𝐴	PROPN
ajird-1325	16	4	⊆	⊆	NUM
ajird-1325	16	5	𝑋	𝑋	PROPN
ajird-1325	16	6	is	be	AUX
ajird-1325	16	7	said	say	VERB
ajird-1325	16	8	to	to	PART
ajird-1325	16	9	be	be	AUX
ajird-1325	16	10	locally	locally	ADV
ajird-1325	16	11	weakly	weakly	ADV
ajird-1325	16	12	dense	dense	ADJ
ajird-1325	16	13	around	around	ADP
ajird-1325	16	14	a	a	DET
ajird-1325	16	15	point	point	NOUN
ajird-1325	16	16	𝑥	𝑥	DET
ajird-1325	16	17	∈	∈	NOUN
ajird-1325	16	18	𝑋	𝑋	NOUN
ajird-1325	16	19	if	if	SCONJ
ajird-1325	16	20	,	,	PUNCT
ajird-1325	16	21	for	for	ADP
ajird-1325	16	22	every	every	DET
ajird-1325	16	23	open	open	ADJ
ajird-1325	16	24	neighborhood	neighborhood	NOUN
ajird-1325	16	25	of	of	ADP
ajird-1325	16	26	𝑥	𝑥	NOUN
ajird-1325	16	27	,	,	PUNCT
ajird-1325	16	28	there	there	PRON
ajird-1325	16	29	exists	exist	VERB
ajird-1325	16	30	a	a	DET
ajird-1325	16	31	smaller	small	ADJ
ajird-1325	16	32	dense	dense	ADJ
ajird-1325	16	33	subset	subset	NOUN
ajird-1325	16	34	.	.	PUNCT
ajird-1325	17	1	theorem	theorem	NOUN
ajird-1325	17	2	1	1	NUM
ajird-1325	17	3	:	:	PUNCT
ajird-1325	17	4	if	if	SCONJ
ajird-1325	17	5	(	(	PUNCT
ajird-1325	17	6	𝑋	𝑋	NOUN
ajird-1325	17	7	,	,	PUNCT
ajird-1325	17	8	𝜏	𝜏	NOUN
ajird-1325	17	9	)	)	PUNCT
ajird-1325	17	10	is	be	AUX
ajird-1325	17	11	a	a	DET
ajird-1325	17	12	𝜏-bounded	𝜏-bounde	VERB
ajird-1325	17	13	and	and	CCONJ
ajird-1325	17	14	t2	t2	NOUN
ajird-1325	17	15	(	(	PUNCT
ajird-1325	17	16	hausdorff	hausdorff	NOUN
ajird-1325	17	17	)	)	PUNCT
ajird-1325	17	18	space	space	NOUN
ajird-1325	17	19	,	,	PUNCT
ajird-1325	17	20	then	then	ADV
ajird-1325	17	21	it	it	PRON
ajird-1325	17	22	is	be	AUX
ajird-1325	17	23	also	also	ADV
ajird-1325	17	24	t3	t3	NOUN
ajird-1325	17	25	(	(	PUNCT
ajird-1325	17	26	regular	regular	ADJ
ajird-1325	17	27	)	)	PUNCT
ajird-1325	17	28	.	.	PUNCT
ajird-1325	18	1	that	that	PRON
ajird-1325	18	2	is	be	AUX
ajird-1325	18	3	,	,	PUNCT
ajird-1325	18	4	for	for	ADP
ajird-1325	18	5	any	any	DET
ajird-1325	18	6	closed	closed	ADJ
ajird-1325	18	7	set	set	NOUN
ajird-1325	18	8	and	and	CCONJ
ajird-1325	18	9	a	a	DET
ajird-1325	18	10	point	point	NOUN
ajird-1325	18	11	outside	outside	ADP
ajird-1325	18	12	it	it	PRON
ajird-1325	18	13	,	,	PUNCT
ajird-1325	18	14	there	there	PRON
ajird-1325	18	15	exist	exist	VERB
ajird-1325	18	16	disjoint	disjoint	ADJ
ajird-1325	18	17	open	open	ADJ
ajird-1325	18	18	sets	set	NOUN
ajird-1325	18	19	separating	separate	VERB
ajird-1325	18	20	them	they	PRON
ajird-1325	18	21	.	.	PUNCT
ajird-1325	19	1	𝜏-boundedness	𝜏-boundedness	NOUN
ajird-1325	19	2	requires	require	VERB
ajird-1325	19	3	that	that	SCONJ
ajird-1325	19	4	the	the	DET
ajird-1325	19	5	closure	closure	NOUN
ajird-1325	19	6	of	of	ADP
ajird-1325	19	7	any	any	DET
ajird-1325	19	8	subset	subset	NOUN
ajird-1325	19	9	,	,	PUNCT
ajird-1325	19	10	whose	whose	DET
ajird-1325	19	11	cardinality	cardinality	NOUN
ajird-1325	19	12	does	do	AUX
ajird-1325	19	13	not	not	PART
ajird-1325	19	14	exceed	exceed	VERB
ajird-1325	19	15	𝜏	𝜏	NOUN
ajird-1325	19	16	,	,	PUNCT
ajird-1325	19	17	is	be	AUX
ajird-1325	19	18	compact	compact	ADJ
ajird-1325	19	19	.	.	PUNCT
ajird-1325	20	1	this	this	PRON
ajird-1325	20	2	means	mean	VERB
ajird-1325	20	3	that	that	SCONJ
ajird-1325	20	4	if	if	SCONJ
ajird-1325	20	5	a	a	DET
ajird-1325	20	6	set	set	NOUN
ajird-1325	20	7	is	be	AUX
ajird-1325	20	8	limited	limit	VERB
ajird-1325	20	9	in	in	ADP
ajird-1325	20	10	size	size	NOUN
ajird-1325	20	11	,	,	PUNCT
ajird-1325	20	12	its	its	PRON
ajird-1325	20	13	closure	closure	NOUN
ajird-1325	20	14	will	will	AUX
ajird-1325	20	15	be	be	AUX
ajird-1325	20	16	compact	compact	ADJ
ajird-1325	20	17	.	.	PUNCT
ajird-1325	21	1	this	this	DET
ajird-1325	21	2	property	property	NOUN
ajird-1325	21	3	is	be	AUX
ajird-1325	21	4	crucial	crucial	ADJ
ajird-1325	21	5	because	because	SCONJ
ajird-1325	21	6	compact	compact	ADJ
ajird-1325	21	7	sets	set	NOUN
ajird-1325	21	8	and	and	CCONJ
ajird-1325	21	9	their	their	PRON
ajird-1325	21	10	characteristics	characteristic	NOUN
ajird-1325	21	11	allow	allow	VERB
ajird-1325	21	12	the	the	DET
ajird-1325	21	13	separation	separation	NOUN
ajird-1325	21	14	of	of	ADP
ajird-1325	21	15	closed	closed	ADJ
ajird-1325	21	16	sets	set	NOUN
ajird-1325	21	17	and	and	CCONJ
ajird-1325	21	18	points	point	NOUN
ajird-1325	21	19	in	in	ADP
ajird-1325	21	20	a	a	DET
ajird-1325	21	21	t3	t3	NOUN
ajird-1325	21	22	space	space	NOUN
ajird-1325	21	23	.	.	PUNCT
ajird-1325	22	1	american	american	ADJ
ajird-1325	22	2	journal	journal	PROPN
ajird-1325	22	3	of	of	ADP
ajird-1325	22	4	interdisciplinary	interdisciplinary	ADJ
ajird-1325	22	5	research	research	NOUN
ajird-1325	22	6	and	and	CCONJ
ajird-1325	22	7	development	development	NOUN
ajird-1325	22	8	issn	issn	PROPN
ajird-1325	22	9	online	online	NOUN
ajird-1325	22	10	:	:	PUNCT
ajird-1325	22	11	2771	2771	NUM
ajird-1325	22	12	-	-	SYM
ajird-1325	22	13	8948	8948	NUM
ajird-1325	22	14	website	website	NOUN
ajird-1325	22	15	:	:	PUNCT
ajird-1325	22	16	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-1325	22	17	volume	volume	NOUN
ajird-1325	22	18	33	33	NUM
ajird-1325	22	19	,	,	PUNCT
ajird-1325	22	20	october	october	PROPN
ajird-1325	22	21	2024	2024	NUM
ajird-1325	22	22	112	112	NUM
ajird-1325	22	23	|	|	ADV
ajird-1325	22	24	p	p	X
ajird-1325	22	25	a	a	DET
ajird-1325	22	26	g	g	NOUN
ajird-1325	22	27	e	e	NOUN
ajird-1325	22	28	we	we	PRON
ajird-1325	22	29	know	know	VERB
ajird-1325	22	30	that	that	SCONJ
ajird-1325	22	31	one	one	NUM
ajird-1325	22	32	of	of	ADP
ajird-1325	22	33	the	the	DET
ajird-1325	22	34	properties	property	NOUN
ajird-1325	22	35	of	of	ADP
ajird-1325	22	36	hausdorff	hausdorff	NOUN
ajird-1325	22	37	spaces	space	NOUN
ajird-1325	22	38	is	be	AUX
ajird-1325	22	39	that	that	SCONJ
ajird-1325	22	40	compact	compact	ADJ
ajird-1325	22	41	sets	set	NOUN
ajird-1325	22	42	are	be	AUX
ajird-1325	22	43	closed	close	VERB
ajird-1325	22	44	,	,	PUNCT
ajird-1325	22	45	and	and	CCONJ
ajird-1325	22	46	they	they	PRON
ajird-1325	22	47	can	can	AUX
ajird-1325	22	48	also	also	ADV
ajird-1325	22	49	be	be	AUX
ajird-1325	22	50	separated	separate	VERB
ajird-1325	22	51	from	from	ADP
ajird-1325	22	52	points	point	NOUN
ajird-1325	22	53	.	.	PUNCT
ajird-1325	23	1	therefore	therefore	ADV
ajird-1325	23	2	,	,	PUNCT
ajird-1325	23	3	any	any	DET
ajird-1325	23	4	compact	compact	ADJ
ajird-1325	23	5	set	set	NOUN
ajird-1325	23	6	and	and	CCONJ
ajird-1325	23	7	point	point	NOUN
ajird-1325	23	8	can	can	AUX
ajird-1325	23	9	be	be	AUX
ajird-1325	23	10	separated	separate	VERB
ajird-1325	23	11	.	.	PUNCT
ajird-1325	24	1	now	now	ADV
ajird-1325	24	2	,	,	PUNCT
ajird-1325	24	3	let	let	VERB
ajird-1325	24	4	is	be	AUX
ajird-1325	24	5	consider	consider	VERB
ajird-1325	24	6	the	the	DET
ajird-1325	24	7	space	space	NOUN
ajird-1325	24	8	being	be	AUX
ajird-1325	24	9	𝜏-bounded	𝜏-bounde	VERB
ajird-1325	24	10	.	.	PUNCT
ajird-1325	25	1	if	if	SCONJ
ajird-1325	25	2	we	we	PRON
ajird-1325	25	3	are	be	AUX
ajird-1325	25	4	given	give	VERB
ajird-1325	25	5	a	a	DET
ajird-1325	25	6	closed	closed	ADJ
ajird-1325	25	7	set	set	VERB
ajird-1325	25	8	𝐴	𝐴	PROPN
ajird-1325	25	9	and	and	CCONJ
ajird-1325	25	10	a	a	DET
ajird-1325	25	11	point	point	NOUN
ajird-1325	25	12	𝑥	𝑥	PROPN
ajird-1325	25	13	∉	∉	PROPN
ajird-1325	25	14	𝐴	𝐴	PROPN
ajird-1325	25	15	,	,	PUNCT
ajird-1325	25	16	we	we	PRON
ajird-1325	25	17	take	take	VERB
ajird-1325	25	18	the	the	DET
ajird-1325	25	19	subset	subset	NOUN
ajird-1325	25	20	of	of	ADP
ajird-1325	25	21	𝐴	𝐴	PROPN
ajird-1325	25	22	whose	whose	DET
ajird-1325	25	23	cardinality	cardinality	NOUN
ajird-1325	25	24	does	do	AUX
ajird-1325	25	25	not	not	PART
ajird-1325	25	26	exceed	exceed	VERB
ajird-1325	25	27	𝜏	𝜏	PROPN
ajird-1325	25	28	(	(	PUNCT
ajird-1325	25	29	or	or	CCONJ
ajird-1325	25	30	the	the	DET
ajird-1325	25	31	set	set	NOUN
ajird-1325	25	32	itself	itself	PRON
ajird-1325	25	33	,	,	PUNCT
ajird-1325	25	34	if	if	SCONJ
ajird-1325	25	35	it	it	PRON
ajird-1325	25	36	is	be	AUX
ajird-1325	25	37	small	small	ADJ
ajird-1325	25	38	)	)	PUNCT
ajird-1325	25	39	.	.	PUNCT
ajird-1325	26	1	since	since	SCONJ
ajird-1325	26	2	the	the	DET
ajird-1325	26	3	space	space	NOUN
ajird-1325	26	4	is	be	AUX
ajird-1325	26	5	𝜏-bounded	𝜏-bounde	VERB
ajird-1325	26	6	,	,	PUNCT
ajird-1325	26	7	the	the	DET
ajird-1325	26	8	closure	closure	NOUN
ajird-1325	26	9	of	of	ADP
ajird-1325	26	10	this	this	DET
ajird-1325	26	11	set	set	NOUN
ajird-1325	26	12	will	will	AUX
ajird-1325	26	13	be	be	AUX
ajird-1325	26	14	compact	compact	ADJ
ajird-1325	26	15	.	.	PUNCT
ajird-1325	27	1	since	since	SCONJ
ajird-1325	27	2	the	the	DET
ajird-1325	27	3	set	set	NOUN
ajird-1325	27	4	is	be	AUX
ajird-1325	27	5	compact	compact	ADJ
ajird-1325	27	6	,	,	PUNCT
ajird-1325	27	7	we	we	PRON
ajird-1325	27	8	can	can	AUX
ajird-1325	27	9	use	use	VERB
ajird-1325	27	10	the	the	DET
ajird-1325	27	11	hausdorff	hausdorff	NOUN
ajird-1325	27	12	property	property	NOUN
ajird-1325	27	13	to	to	PART
ajird-1325	27	14	find	find	VERB
ajird-1325	27	15	disjoint	disjoint	NOUN
ajird-1325	27	16	open	open	ADJ
ajird-1325	27	17	sets	set	NOUN
ajird-1325	27	18	separating	separate	VERB
ajird-1325	27	19	𝐴	𝐴	PROPN
ajird-1325	27	20	and	and	CCONJ
ajird-1325	27	21	the	the	DET
ajird-1325	27	22	point	point	NOUN
ajird-1325	27	23	𝑥.	𝑥.	VERB
ajird-1325	27	24	the	the	DET
ajird-1325	27	25	hausdorff	hausdorff	NOUN
ajird-1325	27	26	property	property	NOUN
ajird-1325	27	27	allows	allow	VERB
ajird-1325	27	28	us	we	PRON
ajird-1325	27	29	to	to	PART
ajird-1325	27	30	find	find	VERB
ajird-1325	27	31	open	open	ADJ
ajird-1325	27	32	sets	set	NOUN
ajird-1325	27	33	𝑈	𝑈	PROPN
ajird-1325	27	34	containing	contain	VERB
ajird-1325	27	35	𝑥	𝑥	X
ajird-1325	27	36	and	and	CCONJ
ajird-1325	27	37	𝑉	𝑉	PROPN
ajird-1325	27	38	containing	contain	VERB
ajird-1325	27	39	𝐴	𝐴	PROPN
ajird-1325	27	40	such	such	ADJ
ajird-1325	27	41	that	that	PRON
ajird-1325	27	42	:	:	PUNCT
ajird-1325	28	1	𝑥	𝑥	X
ajird-1325	28	2	∈	∈	PROPN
ajird-1325	28	3	𝑈𝑥	𝑈𝑥	PROPN
ajird-1325	28	4	,	,	PUNCT
ajird-1325	28	5	𝐴	𝐴	PROPN
ajird-1325	28	6	⊆	⊆	NUM
ajird-1325	28	7	𝑉𝐴	𝑉𝐴	PROPN
ajird-1325	28	8	,	,	PUNCT
ajird-1325	28	9	𝑈	𝑈	PROPN
ajird-1325	28	10	∩	∩	NOUN
ajird-1325	28	11	𝑉	𝑉	NOUN
ajird-1325	28	12	=	=	NOUN
ajird-1325	28	13	∅	∅	NOUN
ajird-1325	28	14	this	this	PRON
ajird-1325	28	15	satisfies	satisfy	VERB
ajird-1325	28	16	the	the	DET
ajird-1325	28	17	regularity	regularity	NOUN
ajird-1325	28	18	(	(	PUNCT
ajird-1325	28	19	t3	t3	NOUN
ajird-1325	28	20	)	)	PUNCT
ajird-1325	28	21	condition	condition	NOUN
ajird-1325	28	22	.	.	PUNCT
ajird-1325	29	1	thus	thus	ADV
ajird-1325	29	2	,	,	PUNCT
ajird-1325	29	3	if	if	SCONJ
ajird-1325	29	4	a	a	DET
ajird-1325	29	5	𝜏-bounded	𝜏-bounde	VERB
ajird-1325	29	6	space	space	NOUN
ajird-1325	29	7	satisfies	satisfy	VERB
ajird-1325	29	8	the	the	DET
ajird-1325	29	9	t2	t2	PROPN
ajird-1325	29	10	(	(	PUNCT
ajird-1325	29	11	hausdorff	hausdorff	NOUN
ajird-1325	29	12	)	)	PUNCT
ajird-1325	29	13	condition	condition	NOUN
ajird-1325	29	14	,	,	PUNCT
ajird-1325	29	15	it	it	PRON
ajird-1325	29	16	will	will	AUX
ajird-1325	29	17	also	also	ADV
ajird-1325	29	18	satisfy	satisfy	VERB
ajird-1325	29	19	the	the	DET
ajird-1325	29	20	t3	t3	PROPN
ajird-1325	29	21	(	(	PUNCT
ajird-1325	29	22	regular	regular	ADJ
ajird-1325	29	23	)	)	PUNCT
ajird-1325	29	24	condition	condition	NOUN
ajird-1325	29	25	,	,	PUNCT
ajird-1325	29	26	meaning	mean	VERB
ajird-1325	29	27	any	any	DET
ajird-1325	29	28	closed	closed	ADJ
ajird-1325	29	29	set	set	NOUN
ajird-1325	29	30	and	and	CCONJ
ajird-1325	29	31	point	point	NOUN
ajird-1325	29	32	outside	outside	ADV
ajird-1325	29	33	it	it	PRON
ajird-1325	29	34	can	can	AUX
ajird-1325	29	35	be	be	AUX
ajird-1325	29	36	separated	separate	VERB
ajird-1325	29	37	by	by	ADP
ajird-1325	29	38	disjoint	disjoint	ADJ
ajird-1325	29	39	open	open	ADJ
ajird-1325	29	40	sets	set	NOUN
ajird-1325	29	41	.	.	PUNCT
ajird-1325	30	1	next	next	ADV
ajird-1325	30	2	,	,	PUNCT
ajird-1325	30	3	we	we	PRON
ajird-1325	30	4	will	will	AUX
ajird-1325	30	5	consider	consider	VERB
ajird-1325	30	6	a	a	DET
ajird-1325	30	7	counterexample	counterexample	NOUN
ajird-1325	30	8	showing	show	VERB
ajird-1325	30	9	that	that	SCONJ
ajird-1325	30	10	a	a	DET
ajird-1325	30	11	𝜏-bounded	𝜏-bounde	VERB
ajird-1325	30	12	space	space	NOUN
ajird-1325	30	13	satisfying	satisfy	VERB
ajird-1325	30	14	the	the	DET
ajird-1325	30	15	t2	t2	NOUN
ajird-1325	30	16	(	(	PUNCT
ajird-1325	30	17	hausdorff	hausdorff	NOUN
ajird-1325	30	18	space	space	NOUN
ajird-1325	30	19	)	)	PUNCT
ajird-1325	30	20	condition	condition	NOUN
ajird-1325	30	21	does	do	AUX
ajird-1325	30	22	not	not	PART
ajird-1325	30	23	necessarily	necessarily	ADV
ajird-1325	30	24	satisfy	satisfy	VERB
ajird-1325	30	25	the	the	DET
ajird-1325	30	26	t4	t4	PROPN
ajird-1325	30	27	(	(	PUNCT
ajird-1325	30	28	normal	normal	ADJ
ajird-1325	30	29	space	space	NOUN
ajird-1325	30	30	)	)	PUNCT
ajird-1325	30	31	condition	condition	NOUN
ajird-1325	30	32	.	.	PUNCT
ajird-1325	31	1	let	let	VERB
ajird-1325	31	2	us	we	PRON
ajird-1325	31	3	examine	examine	VERB
ajird-1325	31	4	the	the	DET
ajird-1325	31	5	product	product	NOUN
ajird-1325	31	6	space	space	NOUN
ajird-1325	31	7	𝑊	𝑊	PROPN
ajird-1325	31	8	×	×	NOUN
ajird-1325	31	9	𝑊0	𝑊0	NOUN
ajird-1325	31	10	where	where	SCONJ
ajird-1325	31	11	𝑊0	𝑊0	PROPN
ajird-1325	31	12	is	be	AUX
ajird-1325	31	13	the	the	DET
ajird-1325	31	14	space	space	NOUN
ajird-1325	31	15	of	of	ADP
ajird-1325	31	16	all	all	DET
ajird-1325	31	17	countable	countable	ADJ
ajird-1325	31	18	ordinal	ordinal	ADJ
ajird-1325	31	19	numbers	number	NOUN
ajird-1325	31	20	,	,	PUNCT
ajird-1325	31	21	and	and	CCONJ
ajird-1325	31	22	𝑊	𝑊	PROPN
ajird-1325	31	23	is	be	AUX
ajird-1325	31	24	the	the	DET
ajird-1325	31	25	space	space	NOUN
ajird-1325	31	26	of	of	ADP
ajird-1325	31	27	all	all	DET
ajird-1325	31	28	ordinal	ordinal	ADJ
ajird-1325	31	29	numbers	number	NOUN
ajird-1325	31	30	less	less	ADJ
ajird-1325	31	31	than	than	ADP
ajird-1325	31	32	or	or	CCONJ
ajird-1325	31	33	equal	equal	ADJ
ajird-1325	31	34	to	to	PART
ajird-1325	31	35	𝑤1	𝑤1	VERB
ajird-1325	31	36	.	.	PUNCT
ajird-1325	32	1	it	it	PRON
ajird-1325	32	2	is	be	AUX
ajird-1325	32	3	known	know	VERB
ajird-1325	32	4	that	that	SCONJ
ajird-1325	32	5	𝑊	𝑊	PROPN
ajird-1325	32	6	and	and	CCONJ
ajird-1325	32	7	𝑊0	𝑊0	PROPN
ajird-1325	32	8	are	be	AUX
ajird-1325	32	9	τ	τ	PROPN
ajird-1325	32	10	-	-	PUNCT
ajird-1325	32	11	bounded	bound	VERB
ajird-1325	32	12	and	and	CCONJ
ajird-1325	32	13	satisfy	satisfy	VERB
ajird-1325	32	14	the	the	DET
ajird-1325	32	15	t2	t2	PROPN
ajird-1325	32	16	(	(	PUNCT
ajird-1325	32	17	hausdorff	hausdorff	NOUN
ajird-1325	32	18	)	)	PUNCT
ajird-1325	32	19	condition	condition	NOUN
ajird-1325	32	20	,	,	PUNCT
ajird-1325	32	21	but	but	CCONJ
ajird-1325	32	22	their	their	PRON
ajird-1325	32	23	product	product	NOUN
ajird-1325	32	24	𝑊	𝑊	VERB
ajird-1325	32	25	×	×	NOUN
ajird-1325	32	26	𝑊0	𝑊0	NOUN
ajird-1325	32	27	does	do	AUX
ajird-1325	32	28	not	not	PART
ajird-1325	32	29	satisfy	satisfy	VERB
ajird-1325	32	30	the	the	DET
ajird-1325	32	31	normal	normal	ADJ
ajird-1325	32	32	space	space	NOUN
ajird-1325	32	33	condition	condition	NOUN
ajird-1325	32	34	.	.	PUNCT
ajird-1325	33	1	thus	thus	ADV
ajird-1325	33	2	,	,	PUNCT
ajird-1325	33	3	a	a	DET
ajird-1325	33	4	space	space	NOUN
ajird-1325	33	5	satisfying	satisfy	VERB
ajird-1325	33	6	the	the	DET
ajird-1325	33	7	t2	t2	NOUN
ajird-1325	33	8	(	(	PUNCT
ajird-1325	33	9	hausdorff	hausdorff	NOUN
ajird-1325	33	10	)	)	PUNCT
ajird-1325	33	11	condition	condition	NOUN
ajird-1325	33	12	does	do	AUX
ajird-1325	33	13	not	not	PART
ajird-1325	33	14	necessarily	necessarily	ADV
ajird-1325	33	15	satisfy	satisfy	VERB
ajird-1325	33	16	the	the	DET
ajird-1325	33	17	t4	t4	PROPN
ajird-1325	33	18	(	(	PUNCT
ajird-1325	33	19	normal	normal	ADJ
ajird-1325	33	20	space	space	NOUN
ajird-1325	33	21	)	)	PUNCT
ajird-1325	33	22	condition	condition	NOUN
ajird-1325	33	23	.	.	PUNCT
ajird-1325	34	1	theorem	theorem	NOUN
ajird-1325	34	2	2	2	NUM
ajird-1325	34	3	.	.	PUNCT
ajird-1325	35	1	if	if	SCONJ
ajird-1325	35	2	(	(	PUNCT
ajird-1325	35	3	𝑋	𝑋	NOUN
ajird-1325	35	4	,	,	PUNCT
ajird-1325	35	5	𝜏	𝜏	NOUN
ajird-1325	35	6	)	)	PUNCT
ajird-1325	35	7	is	be	AUX
ajird-1325	35	8	𝜏-bounded	𝜏-bounde	VERB
ajird-1325	35	9	space	space	NOUN
ajird-1325	35	10	,	,	PUNCT
ajird-1325	35	11	then	then	ADV
ajird-1325	35	12	𝑙𝑑𝑋	𝑙𝑑𝑋	PROPN
ajird-1325	35	13	=	=	PUNCT
ajird-1325	35	14	𝑙𝑤𝑑𝑋.	𝑙𝑤𝑑𝑋.	PROPN
ajird-1325	35	15	proof	proof	NOUN
ajird-1325	35	16	:	:	PUNCT
ajird-1325	35	17	it	it	PRON
ajird-1325	35	18	is	be	AUX
ajird-1325	35	19	known	know	VERB
ajird-1325	35	20	that	that	SCONJ
ajird-1325	35	21	in	in	ADP
ajird-1325	35	22	a	a	DET
ajird-1325	35	23	𝜏-bounded	𝜏-bounde	VERB
ajird-1325	35	24	space	space	NOUN
ajird-1325	35	25	,	,	PUNCT
ajird-1325	35	26	the	the	DET
ajird-1325	35	27	closure	closure	NOUN
ajird-1325	35	28	of	of	ADP
ajird-1325	35	29	any	any	DET
ajird-1325	35	30	subset	subset	NOUN
ajird-1325	35	31	whose	whose	DET
ajird-1325	35	32	cardinality	cardinality	NOUN
ajird-1325	35	33	does	do	AUX
ajird-1325	35	34	not	not	PART
ajird-1325	35	35	exceed	exceed	VERB
ajird-1325	35	36	𝜏	𝜏	NOUN
ajird-1325	35	37	is	be	AUX
ajird-1325	35	38	compact	compact	ADJ
ajird-1325	35	39	.	.	PUNCT
ajird-1325	36	1	this	this	PRON
ajird-1325	36	2	means	mean	VERB
ajird-1325	36	3	that	that	SCONJ
ajird-1325	36	4	sets	set	VERB
ajird-1325	36	5	with	with	ADP
ajird-1325	36	6	limited	limited	ADJ
ajird-1325	36	7	cardinality	cardinality	NOUN
ajird-1325	36	8	in	in	ADP
ajird-1325	36	9	such	such	ADJ
ajird-1325	36	10	spaces	space	NOUN
ajird-1325	36	11	possess	possess	VERB
ajird-1325	36	12	compactness	compactness	NOUN
ajird-1325	36	13	properties	property	NOUN
ajird-1325	36	14	.	.	PUNCT
ajird-1325	37	1	local	local	ADJ
ajird-1325	37	2	weak	weak	ADJ
ajird-1325	37	3	density	density	NOUN
ajird-1325	37	4	means	mean	VERB
ajird-1325	37	5	that	that	SCONJ
ajird-1325	37	6	for	for	ADP
ajird-1325	37	7	any	any	DET
ajird-1325	37	8	point	point	NOUN
ajird-1325	37	9	𝑥	𝑥	DET
ajird-1325	37	10	∈	∈	PROPN
ajird-1325	37	11	𝑋	𝑋	PROPN
ajird-1325	37	12	,	,	PUNCT
ajird-1325	37	13	there	there	PRON
ajird-1325	37	14	exists	exist	VERB
ajird-1325	37	15	a	a	DET
ajird-1325	37	16	smaller	small	ADJ
ajird-1325	37	17	dense	dense	ADJ
ajird-1325	37	18	subset	subset	NOUN
ajird-1325	37	19	around	around	ADP
ajird-1325	37	20	the	the	DET
ajird-1325	37	21	point	point	NOUN
ajird-1325	37	22	.	.	PUNCT
ajird-1325	38	1	this	this	PRON
ajird-1325	38	2	implies	imply	VERB
ajird-1325	38	3	that	that	SCONJ
ajird-1325	38	4	for	for	ADP
ajird-1325	38	5	any	any	DET
ajird-1325	38	6	open	open	ADJ
ajird-1325	38	7	neighborhood	neighborhood	NOUN
ajird-1325	38	8	𝑈	𝑈	PROPN
ajird-1325	38	9	of	of	ADP
ajird-1325	38	10	𝑥	𝑥	PROPN
ajird-1325	38	11	,	,	PUNCT
ajird-1325	38	12	there	there	PRON
ajird-1325	38	13	exists	exist	VERB
ajird-1325	38	14	a	a	DET
ajird-1325	38	15	dense	dense	ADJ
ajird-1325	38	16	subset	subset	NOUN
ajird-1325	38	17	𝐷	𝐷	PROPN
ajird-1325	38	18	⊆	⊆	NUM
ajird-1325	38	19	𝑈.	𝑈.	PROPN
ajird-1325	38	20	thus	thus	ADV
ajird-1325	38	21	,	,	PUNCT
ajird-1325	38	22	if	if	SCONJ
ajird-1325	38	23	a	a	DET
ajird-1325	38	24	set	set	NOUN
ajird-1325	38	25	𝐴	𝐴	PROPN
ajird-1325	38	26	is	be	AUX
ajird-1325	38	27	locally	locally	ADV
ajird-1325	38	28	weakly	weakly	ADV
ajird-1325	38	29	dense	dense	ADJ
ajird-1325	38	30	,	,	PUNCT
ajird-1325	38	31	we	we	PRON
ajird-1325	38	32	can	can	AUX
ajird-1325	38	33	find	find	VERB
ajird-1325	38	34	smaller	small	ADJ
ajird-1325	38	35	dense	dense	ADJ
ajird-1325	38	36	subsets	subset	NOUN
ajird-1325	38	37	in	in	ADP
ajird-1325	38	38	every	every	DET
ajird-1325	38	39	open	open	ADJ
ajird-1325	38	40	neighborhood	neighborhood	NOUN
ajird-1325	38	41	of	of	ADP
ajird-1325	38	42	any	any	DET
ajird-1325	38	43	point	point	NOUN
ajird-1325	38	44	.	.	PUNCT
ajird-1325	39	1	now	now	ADV
ajird-1325	39	2	,	,	PUNCT
ajird-1325	39	3	we	we	PRON
ajird-1325	39	4	move	move	VERB
ajird-1325	39	5	on	on	ADP
ajird-1325	39	6	to	to	ADP
ajird-1325	39	7	proving	prove	VERB
ajird-1325	39	8	local	local	ADJ
ajird-1325	39	9	density	density	NOUN
ajird-1325	39	10	.	.	PUNCT
ajird-1325	40	1	the	the	DET
ajird-1325	40	2	meaning	meaning	NOUN
ajird-1325	40	3	of	of	ADP
ajird-1325	40	4	local	local	ADJ
ajird-1325	40	5	density	density	NOUN
ajird-1325	40	6	is	be	AUX
ajird-1325	40	7	that	that	SCONJ
ajird-1325	40	8	within	within	ADP
ajird-1325	40	9	any	any	DET
ajird-1325	40	10	open	open	ADJ
ajird-1325	40	11	neighborhood	neighborhood	NOUN
ajird-1325	40	12	,	,	PUNCT
ajird-1325	40	13	the	the	DET
ajird-1325	40	14	set	set	NOUN
ajird-1325	40	15	itself	itself	PRON
ajird-1325	40	16	is	be	AUX
ajird-1325	40	17	dense	dense	ADJ
ajird-1325	40	18	.	.	PUNCT
ajird-1325	41	1	if	if	SCONJ
ajird-1325	41	2	𝐴	𝐴	PROPN
ajird-1325	41	3	⊆	⊆	NUM
ajird-1325	41	4	𝑋	𝑋	PROPN
ajird-1325	41	5	is	be	AUX
ajird-1325	41	6	locally	locally	ADV
ajird-1325	41	7	weakly	weakly	ADV
ajird-1325	41	8	dense	dense	ADJ
ajird-1325	41	9	,	,	PUNCT
ajird-1325	41	10	then	then	ADV
ajird-1325	41	11	within	within	ADP
ajird-1325	41	12	every	every	DET
ajird-1325	41	13	open	open	ADJ
ajird-1325	41	14	neighborhood	neighborhood	NOUN
ajird-1325	41	15	,	,	PUNCT
ajird-1325	41	16	smaller	small	ADJ
ajird-1325	41	17	dense	dense	ADJ
ajird-1325	41	18	subsets	subset	NOUN
ajird-1325	41	19	can	can	AUX
ajird-1325	41	20	be	be	AUX
ajird-1325	41	21	found	find	VERB
ajird-1325	41	22	.	.	PUNCT
ajird-1325	42	1	𝜏-boundedness	𝜏-boundedness	NOUN
ajird-1325	42	2	ensures	ensure	VERB
ajird-1325	42	3	that	that	SCONJ
ajird-1325	42	4	the	the	DET
ajird-1325	42	5	closure	closure	NOUN
ajird-1325	42	6	of	of	ADP
ajird-1325	42	7	any	any	DET
ajird-1325	42	8	subset	subset	NOUN
ajird-1325	42	9	with	with	ADP
ajird-1325	42	10	cardinality	cardinality	NOUN
ajird-1325	42	11	not	not	PART
ajird-1325	42	12	exceeding	exceed	VERB
ajird-1325	42	13	𝜏	𝜏	NOUN
ajird-1325	42	14	is	be	AUX
ajird-1325	42	15	compact	compact	ADJ
ajird-1325	42	16	.	.	PUNCT
ajird-1325	43	1	thus	thus	ADV
ajird-1325	43	2	,	,	PUNCT
ajird-1325	43	3	if	if	SCONJ
ajird-1325	43	4	we	we	PRON
ajird-1325	43	5	find	find	VERB
ajird-1325	43	6	dense	dense	ADJ
ajird-1325	43	7	subsets	subset	NOUN
ajird-1325	43	8	within	within	ADP
ajird-1325	43	9	smaller	small	ADJ
ajird-1325	43	10	parts	part	NOUN
ajird-1325	43	11	of	of	ADP
ajird-1325	43	12	𝐴	𝐴	PROPN
ajird-1325	43	13	,	,	PUNCT
ajird-1325	43	14	the	the	DET
ajird-1325	43	15	closure	closure	NOUN
ajird-1325	43	16	of	of	ADP
ajird-1325	43	17	these	these	DET
ajird-1325	43	18	subsets	subset	NOUN
ajird-1325	43	19	will	will	AUX
ajird-1325	43	20	be	be	AUX
ajird-1325	43	21	compact	compact	ADJ
ajird-1325	43	22	.	.	PUNCT
ajird-1325	44	1	american	american	ADJ
ajird-1325	44	2	journal	journal	PROPN
ajird-1325	44	3	of	of	ADP
ajird-1325	44	4	interdisciplinary	interdisciplinary	ADJ
ajird-1325	44	5	research	research	NOUN
ajird-1325	44	6	and	and	CCONJ
ajird-1325	44	7	development	development	NOUN
ajird-1325	44	8	issn	issn	PROPN
ajird-1325	44	9	online	online	NOUN
ajird-1325	44	10	:	:	PUNCT
ajird-1325	44	11	2771	2771	NUM
ajird-1325	44	12	-	-	SYM
ajird-1325	44	13	8948	8948	NUM
ajird-1325	44	14	website	website	NOUN
ajird-1325	44	15	:	:	PUNCT
ajird-1325	44	16	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-1325	44	17	volume	volume	NOUN
ajird-1325	44	18	33	33	NUM
ajird-1325	44	19	,	,	PUNCT
ajird-1325	44	20	october	october	PROPN
ajird-1325	44	21	2024	2024	NUM
ajird-1325	44	22	113	113	NUM
ajird-1325	45	1	|	|	ADV
ajird-1325	45	2	p	p	NOUN
ajird-1325	45	3	a	a	DET
ajird-1325	45	4	g	g	NOUN
ajird-1325	45	5	e	e	NOUN
ajird-1325	45	6	by	by	ADP
ajird-1325	45	7	finding	find	VERB
ajird-1325	45	8	dense	dense	ADJ
ajird-1325	45	9	parts	part	NOUN
ajird-1325	45	10	within	within	ADP
ajird-1325	45	11	any	any	DET
ajird-1325	45	12	open	open	ADJ
ajird-1325	45	13	neighborhood	neighborhood	NOUN
ajird-1325	45	14	,	,	PUNCT
ajird-1325	45	15	we	we	PRON
ajird-1325	45	16	ensure	ensure	VERB
ajird-1325	45	17	that	that	SCONJ
ajird-1325	45	18	𝐴	𝐴	PROPN
ajird-1325	45	19	itself	itself	PRON
ajird-1325	45	20	is	be	AUX
ajird-1325	45	21	dense	dense	ADJ
ajird-1325	45	22	.	.	PUNCT
ajird-1325	46	1	this	this	PRON
ajird-1325	46	2	shows	show	VERB
ajird-1325	46	3	that	that	SCONJ
ajird-1325	46	4	local	local	ADJ
ajird-1325	46	5	weak	weak	ADJ
ajird-1325	46	6	density	density	NOUN
ajird-1325	46	7	actually	actually	ADV
ajird-1325	46	8	guarantees	guarantee	VERB
ajird-1325	46	9	local	local	ADJ
ajird-1325	46	10	density	density	NOUN
ajird-1325	46	11	because	because	SCONJ
ajird-1325	46	12	smaller	small	ADJ
ajird-1325	46	13	dense	dense	ADJ
ajird-1325	46	14	subsets	subset	NOUN
ajird-1325	46	15	can	can	AUX
ajird-1325	46	16	be	be	AUX
ajird-1325	46	17	found	find	VERB
ajird-1325	46	18	within	within	ADP
ajird-1325	46	19	any	any	DET
ajird-1325	46	20	open	open	ADJ
ajird-1325	46	21	neighborhood	neighborhood	NOUN
ajird-1325	46	22	,	,	PUNCT
ajird-1325	46	23	making	make	VERB
ajird-1325	46	24	the	the	DET
ajird-1325	46	25	whole	whole	ADJ
ajird-1325	46	26	set	set	VERB
ajird-1325	46	27	dense	dense	ADJ
ajird-1325	46	28	.	.	PUNCT
ajird-1325	47	1	thus	thus	ADV
ajird-1325	47	2	,	,	PUNCT
ajird-1325	47	3	𝑙𝑑𝑋	𝑙𝑑𝑋	PROPN
ajird-1325	47	4	=	=	PUNCT
ajird-1325	47	5	𝑙𝑤𝑑𝑋.	𝑙𝑤𝑑𝑋.	PROPN
ajird-1325	47	6	references	reference	NOUN
ajird-1325	47	7	1	1	NUM
ajird-1325	47	8	.	.	PUNCT
ajird-1325	47	9	o.	o.	PROPN
ajird-1325	47	10	okunev	okunev	PROPN
ajird-1325	47	11	,	,	PUNCT
ajird-1325	47	12	"	"	PUNCT
ajird-1325	47	13	the	the	DET
ajird-1325	47	14	minitightness	minitightness	NOUN
ajird-1325	47	15	of	of	ADP
ajird-1325	47	16	products	product	NOUN
ajird-1325	47	17	,	,	PUNCT
ajird-1325	47	18	"	"	PUNCT
ajird-1325	47	19	topology	topology	NOUN
ajird-1325	47	20	and	and	CCONJ
ajird-1325	47	21	its	its	PRON
ajird-1325	47	22	applications	application	NOUN
ajird-1325	47	23	,	,	PUNCT
ajird-1325	47	24	vol	vol	NOUN
ajird-1325	47	25	.	.	PROPN
ajird-1325	47	26	208	208	NUM
ajird-1325	47	27	,	,	PUNCT
ajird-1325	47	28	2016	2016	NUM
ajird-1325	47	29	,	,	PUNCT
ajird-1325	47	30	pp	pp	ADJ
ajird-1325	47	31	.	.	PUNCT
ajird-1325	48	1	10	10	NUM
ajird-1325	48	2	-	-	SYM
ajird-1325	48	3	16	16	NUM
ajird-1325	48	4	.	.	PUNCT
ajird-1325	49	1	2	2	NUM
ajird-1325	49	2	.	.	X
ajird-1325	49	3	r.	r.	PROPN
ajird-1325	49	4	engelking	engelking	NOUN
ajird-1325	49	5	,	,	PUNCT
ajird-1325	49	6	general	general	ADJ
ajird-1325	49	7	topology	topology	NOUN
ajird-1325	49	8	,	,	PUNCT
ajird-1325	49	9	heldermann	heldermann	PROPN
ajird-1325	49	10	verlag	verlag	PROPN
ajird-1325	49	11	,	,	PUNCT
ajird-1325	49	12	berlin	berlin	PROPN
ajird-1325	49	13	,	,	PUNCT
ajird-1325	49	14	1989	1989	NUM
ajird-1325	49	15	.	.	PUNCT
ajird-1325	50	1	3	3	X
ajird-1325	50	2	.	.	X
ajird-1325	50	3	nodirbek	nodirbek	PROPN
ajird-1325	50	4	mamadaliev	mamadaliev	PROPN
ajird-1325	50	5	,	,	PUNCT
ajird-1325	50	6	sardor	sardor	NOUN
ajird-1325	50	7	karimov	karimov	NOUN
ajird-1325	50	8	,	,	PUNCT
ajird-1325	50	9	"	"	PUNCT
ajird-1325	50	10	on	on	ADP
ajird-1325	50	11	τ	τ	PROPN
ajird-1325	50	12	-	-	PUNCT
ajird-1325	50	13	bounded	bound	VERB
ajird-1325	50	14	spaces	space	NOUN
ajird-1325	50	15	,	,	PUNCT
ajird-1325	50	16	"	"	PUNCT
ajird-1325	50	17	problems	problem	NOUN
ajird-1325	50	18	of	of	ADP
ajird-1325	50	19	modern	modern	ADJ
ajird-1325	50	20	mathematics	mathematic	NOUN
ajird-1325	50	21	,	,	PUNCT
ajird-1325	50	22	70th	70th	ADJ
ajird-1325	50	23	anniversary	anniversary	NOUN
ajird-1325	50	24	of	of	ADP
ajird-1325	50	25	a.a	a.a	PROPN
ajird-1325	50	26	.	.	PROPN
ajird-1325	50	27	borubaev	borubaev	PROPN
ajird-1325	50	28	,	,	PUNCT
ajird-1325	50	29	june	june	PROPN
ajird-1325	50	30	15	15	NUM
ajird-1325	50	31	-	-	SYM
ajird-1325	50	32	19	19	NUM
ajird-1325	50	33	,	,	PUNCT
ajird-1325	50	34	2021	2021	NUM
ajird-1325	50	35	.	.	PUNCT
ajird-1325	51	1	4	4	X
ajird-1325	51	2	.	.	X
ajird-1325	51	3	adilbek	adilbek	NOUN
ajird-1325	51	4	zaitov	zaitov	NOUN
ajird-1325	51	5	,	,	PUNCT
ajird-1325	51	6	sardor	sardor	NOUN
ajird-1325	51	7	karimov	karimov	NOUN
ajird-1325	51	8	,	,	PUNCT
ajird-1325	51	9	"	"	PUNCT
ajird-1325	51	10	on	on	ADP
ajird-1325	51	11	the	the	DET
ajird-1325	51	12	weak	weak	ADJ
ajird-1325	51	13	density	density	NOUN
ajird-1325	51	14	of	of	ADP
ajird-1325	51	15	τ	τ	PROPN
ajird-1325	51	16	-	-	PUNCT
ajird-1325	51	17	bounded	bound	VERB
ajird-1325	51	18	spaces	space	NOUN
ajird-1325	51	19	,	,	PUNCT
ajird-1325	51	20	"	"	PUNCT
ajird-1325	51	21	modern	modern	ADJ
ajird-1325	51	22	problems	problem	NOUN
ajird-1325	51	23	of	of	ADP
ajird-1325	51	24	analysis	analysis	NOUN
ajird-1325	51	25	,	,	PUNCT
ajird-1325	51	26	june	june	PROPN
ajird-1325	51	27	2	2	NUM
ajird-1325	51	28	-	-	SYM
ajird-1325	51	29	3	3	NUM
ajird-1325	51	30	,	,	PUNCT
ajird-1325	51	31	2023	2023	NUM
ajird-1325	51	32	,	,	PUNCT
ajird-1325	51	33	karshi	karshi	NOUN
ajird-1325	51	34	.	.	PUNCT